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\section{Introduction}
Machine Learning is used in a wide range of systems, such as health care \cite{de2018clinically,kourou2015machine}, education \cite{oneto2017dropout,oneto2019learning,papamitsiou2014learning}, financial lending \cite{byanjankar2015predicting,malekipirbazari2015risk}, and social services \cite{now2018litigating,eubanks2018automating}, to facilitate decision making and automate services which has a critical implications to individuals and communities. This extensive use of machine learning creates a growing concern, as algorithms might introduce far-reaching bias that treats individuals or groups unfairly, based on certain characteristics such as age, race, gender, or political affiliation. Thus, it is becoming very important to develop fairness aware algorithms.
In recent years many methods have been developed addressing both individual-based and group-based fairness. Most of the works tackles issues of discovering discrimination, and adding solutions to eliminate such discrimination to have fair and accurate decisions \cite{zhang2019fairness,zliobaite2015survey}. The bias mitigation approaches on either the training data or the learned model can be categorized into three main groups \cite{d2017conscientious}: $(1)$ Pre-processing approaches such as reweighting \cite{calders2009building}, massaging \cite{kamiran2009classifying}, aiming to eliminate discrimination at the data level; $(2)$ In-processing algorithms such as \cite{bhaskaruni2019improving,kamiran2010discrimination} addressing discrimination by modifying algorithms; $(3)$ post-processing methods such as \cite{hajian2015discrimination,kamiran2010discrimination} take the learned model and change the predicted labels appropriately to meet fairness requirements.
Algorithms proposed recently in bias mitigation has focused on neural networks. However, the efficiency and explainability of tree ensembles for many applications makes them preferable to be implemented in many areas. Even though there are few works focused on studying fairness for trees and tree ensembles, notably \cite{kamiran2010discrimination,raff2018fair,zhangfaht,grari2020achieving}, most of them are focused on single decision tree classifiers and in-processing approaches. Our interest mainly lies in developing fair random forest classifiers with post-processing approaches designed to relabel leaves with accuracy and discrimination constraints. We take advantage of implementing a post-process approach, in which we do not require to know the training process.
{\bf Contributions.}
We focus on decision tree ensembles for binary classification tasks susceptible to group discrimination
with respect to attributes sensitive classes such as age, gender, race, etc.
We propose a post-processing approach named EiFFFeL --Enforcing Fairness in Forests by Flipping Leaves-- that given a forest,
however trained, selects a subset of its leaves and changes their predictions so as to reduce the discrimination degree of the forest.
We summarize the main contributions of our work as follows.
\begin{enumerate}[noitemsep,topsep=4pt]
\item We propose an iterative leaf flipping post-processing algorithm to ensure group fairness .
\item We devise tree-based and leaf-based flipping methodologies on top of random forest classifier to enforce fairness.
\item We report experimental evaluations of group fairness on three different datasets, aiming to empirically show the effectiveness of our method.
\end{enumerate}
\section{Fairness in Machine Learning}
Without loss of generality, we consider a binary classifier $g:\mathcal{X \to Y}$ that maps an input feature vector $\bm{x} \in \mathcal{X}$ to a binary class label $y \in \mathcal{Y} = \{0,1\}$.
Among the attributes in the feature space $\mathcal{X}$,
a binary attribute called \textit{sensitive feature} ${S} \in \{0,1\}$ identifies the aspects of data which are socio-culturally precarious for the application of machine learning. Specifically, given $x \in \mathcal{X}$ and $x.S$ the value of the sensitive attribute $S$ for the given instance, if $x.S=0$ then we say that $x$ belongs to the \textit{unprivileged group} that could possibly be discriminated.
\subsection{Fairness and Discrimination Definitions}
To achieve non-discriminatory and fair machine learning model, it is essential to first define \textit{fairness}. In a broad context, fairness can be seen from an individual or a group point of view. \textit{Individual fairness} requires that similar individuals being treated similarly.
\textit{Group fairness} requires fairness of the classification model to apply on the two groups, defined through the binary sensitive feature $S$ \cite{dwork2012fairness}. Our work focuses on group fairness, in which a group of individuals identified by $S$ risks for experiencing discrimination.
We define the discrimination of a classifier measured by \textit{group fairness} as follows.
Recall that attribute $S=0$ identifies the unprivileged group, while $S=1$ corresponds to the privileged one, whose members are not discriminated but rather favoured by a learnt ML model.
Moreover, we assume that the values $1$ and $0$ of class label $Y$ represent \textit{favorable} and \textit{unfavorable} outcomes, respectively. For example, $Y=1$ might correspond to the decision of granting a loan, thus favouring a bank customer.
A classifier $g$ applied over $x \in \mathcal{X}$ is non-discriminatory if its prediction $g(\bm{x})$ is statistically independent of the sensitive attribute $S$. Hence, a classifier is fair if both groups have equal probability of being classified as belonging to the favorable class, which is the desirable outcome.
Using the problem formalization by \cite{kamiran2010discrimination}, the discrimination of a model $g$ with respect to a sensitive attribute $S$ and a dataset $\mathcal{D}=\{(\bm{x}_i, y_i)\}_{i=1}^N$ can be computed as follows:
\begin{multline*}
disc_{\mathcal{D},S,g} := \frac{|\{ (\bm{x},y) \in \mathcal{D} ~|~ \bm{x}.S=1 \wedge g(\bm{x})=1\}|}{|\{(\bm{x},y) \in \mathcal{D}~|~ \bm{x}.S=1\}|}\\ - \frac{|\{(\bm{x},y) \in \mathcal{D}~|~ \bm{x}.S=0 \wedge g(\bm{x})=1\}|}{|\{ (\bm{x},y) \in \mathcal{D} ~|~ \bm{x}.S=0 \}|} ,
\end{multline*}
\noindent where $\bm{x}.S$ refers to the sensitive attribute of the instance $\bm{x}$. When $S$ and $\mathcal{D}$ are clear from the context we simply use the notation $disc_g$.
To clarify the above definition, let's consider the case of a classifier $g$ used by the HR staff of a company. The classifier $g$ suggests hiring when $g(\bm{x})=1$ vs.\ not hiring when $g(\bm{x})=0$. We may wonder whether the classifier favours \textit{men} ($S=1$) over \textit{women} ($S=0$). The value of $disc_g$ is large if the ratio of men with a favorable hiring prediction is larger than the ratio of women with a favorable hiring prediction.
By minimizing $disc_g$ we can provide a fairer classifier w.r.t.\ the gender attribute.
\subsection{Related Works}
Notably, in recent years works identifying and solving bias in machine learning algorithms have progressed. \textit{Pre-processing}, \textit{in-processing}, and \textit{post-processing} approaches have been used to mitigate and quantify bias coming from training data, learning algorithms, or the interaction between the twos.
Algorithms which are identified in the \textit{Pre-processing} category deal with discrimination at the dataset level by altering its distribution to ensure there is no bias against a specific group or individual. This can be achieved by removing the sensitive attribute, re-sampling the data, or changing class labels. One of the well known pre-processing method is massaging \cite{kamiran2009classifying}, which changes the class labels of a subset of carefully selected instances. Another work in this category is re-weighting \cite{calders2009building}, which assigns different weights to different groups of the dataset to reduce bias. A re-sampling approach in \cite{calmon2017optimized} limits the sample size to control discrimination.
In the \textit{In-processing} bias mitigation algorithms, discrimination is accounted during the training phase of the learning algorithm. Strategies in this group take different approaches to discount discrimination by including fairness penalty into the loss function such as in \cite{zafar2017fairness}, which integrates decision boundary covariance constraint for logistic regression. In \cite{aghaei2019learning} regularization terms are added to penalize discrimination in mixed-integer optimization framework of decision tree. Another interesting work is \cite{calders2010three}, which proposes three approaches for fairness-aware Na\"ive Bayes classifiers. The approaches are: altering the decision distribution until there is no more discrimination, building a separate model for each sensitive group to remove the correlation between sensitive attribute and class label, and adding latent variable representing unbiased label.
Kamiran \textit{et al.} \cite{kamiran2010discrimination} included a discrimination factor into the information gain splitting criterion of a single decision tree classifier by considering the split of a node under the influence of a sensitive feature, i.e., before a node split happens not only the usual purity w.r.t.\ to the target label is calculated, but also the purity of the split w.r.t.\ the sensitive feature. Three alternative splitting criteria are given based on the way discrimination is accounted. The first option is subtracting discrimination gain from accuracy gain, which allows for a split if it is non-discriminatory, second option is an accuracy-discrimination trade-off split where the accuracy gain is divided by discrimination gain to have the final gain value. The third option is adding the accuracy and discrimination gain to decide the best feature to split a node. The authors claim the additive information gain criterion produces a lower discrimination. We also implement this method for the base trees of our forest and evaluate the impact of it to the overall forest discrimination value.
Finally, authors propose an additional relabeling of some leaves of the tree so as to further reduce its discrimination degree.
A recent work, called Distributed Fair Random Forest (DFRF) \cite{fantin2020distributed} exploits randomly generated decision trees and filters them by their fairness before adding them to the forest.
This is achieved through a hyper-parameter fairness constraint, which forces to accept only decision trees with statistical parity below the given threshold. The generation and fairness thresholding of each individual tree can be done in distributed framework that optimizes the trade-of between discrimination and accuracy of the tree before being added to the forest. Furthermore, this algorithm uses randomness constraint to train base trees in which one feature is randomly selected to split a node for building a randomized decision tree.
\textit{Post-processing} mitigation approaches focus on adjusting the final output of the trained model rather than the underline loss function or training data.
The algorithms discussed in \cite{hardt2016equality,pleiss2017fairness} aim at achieving same error rates between privileged and unprivileged groups,
\cite{hardt2016equality} uses equalized odd and equalized opportunity to promote features which are more dependent on the target label than the sensitive attribute. While
in \cite{pleiss2017fairness} the proposed algorithm aims to achieve both privileged and unprivileged groups to have the same false negative rate and false positive rate
by taking into account a calibrated probability estimates.
Another post-processing algorithm called Reject Option based Classification (ROC) \cite{kamiran2012decision} takes in to consideration the decision boundary of classifiers; in a region where uncertainty is high, it gives favorable outcomes to the unprivileged group and unfavorable outcomes to the privileged group to reduce discrimination.
Among the various works, the closest to our proposal is \cite{kamiran2010discrimination}, which relabels leaves of a single tree classifier with a small effect on the model accuracy.
We borrow from this approach and propose a novel algorithm for enforcing fairness in forests of decision trees.
\begin{table}[H]
\centering
\footnotesize
\caption{\label{tab:notation} Notation Summary}
\begin{tabular}{@{}c|p{.6\columnwidth}@{}}
\toprule
Symbol & Meaning \\
\midrule
$\mathcal{D}$ & Dataset \\
$S$ & Sensitive feature\\
$\lambda$ & leaf \\
$\Lambda$ & Set of Leaves to be flipped\\
$disc_{ \mathcal{F}}$ & Forest discrimination\\
$disc_{ \mathcal{T}}$ & Tree discrimination\\
$accu_{ \mathcal{F}}$ & Forest accuracy \\
$\Delta disc_{ \mathcal{\lambda}}$ & change in discrimination after flipping\\
$\Delta accu_{ \mathcal{\lambda}}$ & change on accuracy after flipping \\
$\delta$ & Ratio of change in accuracy and discrimination \\
\bottomrule
\end{tabular}
\end{table}
\section{The EiFFFel Algorithm}
We propose a novel post-processing algorithm named EiFFFeL that, given a forest of decision trees for a binary classification task, modifies the prediction of a carefully chosen set of leaves so as to reduce the forest's discrimination degree.
This process is named leaf relabeling, or, since we are focusing on a binary prediction task, \emph{leaf flipping}.
The rationale is to flip the prediction of the leaves that contribute the most to the model discrimination degree so as to make them fair.
Recall that the score $disc_{\mathcal{D},S,g}$ adopted to evaluate the model's discrimination depends on the number of privileged/unprivileged instances with a favorable prediction.
Therefore, by flipping a leaf label we can increase or decrease the number of instances that contribute to the discrimination score.
Note that, while leaf relabeling can be done judiciously so as to reduce discrimination, modifying the leaf predictions determined at training time may reduce the accuracy of the whole forest.
Therefore the goal of EiFFFeL is to find a sweet-spot in the accuracy vs.\ discrimination trade-off.
While leaf relabeling was introduced by \cite{kamiran2010discrimination} for a single tree, we improve such strategy and extend it to a forest of decision trees.
In this work we focus on Random Forests ensembles, which, for their high accuracy and limited bias, are an optimal candidate for building a fair classifier. The approach is however general and we leave to future work the application to other tree ensembles, such as those obtained by bagging and boosting approaches.
The proposed EiFFFel algorithm accepts a user-defined maximum discrimination constraint $\epsilon$ and a minimum relative accuracy drop constraint $\alpha$.
Given a forest $\mathcal{F}$, it iteratively modifies the prediction associated with a subset of the leaves of $\mathcal{F}$, until either the desired discrimination $\epsilon$ is achieved, or the maximum required accuracy drop $\alpha$ is hit.
Below we first illustrate the \emph{Leaf Scoring} strategy used to find the most discriminative leaves of a tree, and then we illustrate two variants of the EiFFFeL algorithm.
\begin{algorithm}[t]
\renewcommand{\algorithmicrequire}{\textbf{Input:}}
\renewcommand{\algorithmicensure}{\textbf{Output:}}
\caption{\sc Score\_Leaves}
\label{alg:getL}
\begin{algorithmic}[1]
\small
\REQUIRE Decision Tree $\mathcal{T}$ \\ Dataset $\mathcal{D}$ \\ Sensitive feature $S$
\ENSURE Candidate flipping leaves $\Lambda$
\STATE $\Lambda \gets \emptyset$
\FORALL{$\lambda \in \mathcal{T} ~|~ \neg \lambda.flipped$}
\STATE $\Delta accu_{ \lambda}\gets -abs\left( \frac{ | \mathcal{D}^\lambda_{y= 1} | - |\mathcal{D}^\lambda_{y= 0 }| }{|\mathcal{D}|} \right)$ \label{lst:line:line4}
\STATE $\Delta disc_{\lambda} \gets sign\left(| \mathcal{D}^\lambda_{y=1} | - |\mathcal{D}^\lambda_{y=0} |\right)\cdot\left( \frac{ |\mathcal{D}^\lambda_{S=1} |}{|\mathcal{D}_{S=1} |}
- \frac{|\mathcal{D}^\lambda_{S=0}|}{|\mathcal{D}_{S=0}|} \right) $ \label{lst:line:line3}
\STATE $\delta \gets \frac{\Delta disc_{\lambda}}{\Delta accu_{ \lambda}}$ \label{lst:line:line5}
\IF{$\delta \geq 0$}
\STATE $\lambda.score \gets \delta$
\STATE $\Lambda \gets \Lambda \cup \{\lambda\}$
\ENDIF
\ENDFOR
\RETURN $\Lambda $
\end{algorithmic}
\end{algorithm}
\subsection{Leaf Scoring}
EiFFFeL borrows from \cite{kamiran2010discrimination} a simple strategy for scoring leaves according to their impact $\Delta accu_{ \lambda}$ and $\Delta disc_{ \lambda}$ on accuracy and discrimination respectively. Then, the ratio $\delta$ between the two is used as a score to greedily select the best leaves to be \emph{flipped}.
We proceed as described in Alg.~\ref{alg:getL}.
We consider only leaves of the tree that were not flipped during previous iteration of the EiFFFeL algorithm (see subsection below). For those leaves we compute the accuracy and discrimination variation in the case of flipping the leaf prediction. We illustrate shortly the computations below, please refer to \cite{kamiran2010discrimination} for a more detailed description.
The change in accuracy $\Delta accu_{ \lambda}$ clearly depends on the number of instances of $\mathcal{D}$ that fall into the leaf $\lambda$ denoted with $\mathcal{D}^\lambda$. The training process sets the leaf prediction to the majority class among such instances. Therefore, when flipping the leaf prediction the accuracy may only decrease depending on the instances with label $1$ and $0$, denoted by $\mathcal{D}^\lambda_{y=1}$ and $\mathcal{D}^\lambda_{y=0}$ respectively. The difference between the size of these two sets results in the accuracy loss as computed in line~\ref{lst:line:line4}.
The change in discrimination $\Delta disc_{ \lambda}$ depends on the number of privileged and unpriviledged instances that fall in the leaf $\lambda$ respectively denoted by $\mathcal{D}^\lambda_{S=1}$ and $\mathcal{D}^\lambda_{S=0}$, and on their analogous on the whole dataset $\mathcal{D}_{S=1}$ and $\mathcal{D}_{S=0}$. If the leaf prediction equals $1$ (favourable class), then increasing $\mathcal{D}^\lambda_{S=1}$ would increase the discrimination, while increasing $\mathcal{D}^\lambda_{S=0}$ would decrease it. The opposite holds if the prediction of the leaf equals $0$ (unfavourable class). As the original leaf prediction depends on the majority of the instances between $\mathcal{D}^\lambda_{y=1}$ and $\mathcal{D}^\lambda_{y=0}$, the sign of their difference is used to correct the above contributions as computed in line~\ref{lst:line:line3}.
The ratio $\delta=\Delta disc_{ \lambda}/\Delta accu_{ \lambda}$ is positive if the flipping generates a discrimination drop, and it is large if the benefit to discrimination is larger than the harm to accuracy. If the value of $\delta$ is positive, then this is stored with the leaf $\lambda$, and $\lambda$ is recorded into the set of candidate leaves $\Lambda$. The set $\Lambda$ is eventually returned and exploited during the iterations of EiFFFeL.
\subsection{EiFFFeL Leaf Flipping Strategies}
By exploiting the scoring technique discussed before, we propose two strategies to choose which trees and which leaves in those trees to flip.
The first strategy, named \textit{Tree-based Flipping}, is illustrated in Alg.~\ref{alg:mlr}.
During each iteration of EiFFFeL, the tree $\mathcal{T}^\dagger$ with the largest discrimination degree is greedily selected: this is the best tree to be attacked in order to significantly reduce the discrimination of the full forest. Then, we use the previous scoring technique to find the set of leaves $\Lambda$ in $\mathcal{T}^\dagger$ that should be relabeled. If $\Lambda$ is not empty, the predictions $\lambda.pred$ of such leaves will be flipped. Then, the whole tree is marked as already flipped. The selection is repeated by considering only the remaining non-flipped trees. The algorithm ends when all trees have been flipped, or when the desired discrimination $\epsilon$ is achieved, or when tolerated accuracy drop $\alpha$ is met. Note that the accuracy drop is computed by comparing the accuracy of the original forest with the accuracy of the current forest after the flipping step.
\begin{algorithm}[t]
\small
\renewcommand{\algorithmicrequire}{\textbf{Input:}}
\renewcommand{\algorithmicensure}{\textbf{Output:}}
\caption{EiFFFeL-TF (Tree-based Flipping)}
\label{alg:mlr}
\begin{algorithmic}[1]
\REQUIRE Random Forest classifier $\mathcal{F}$ \\ Discrimination Constraint $\epsilon \in [0,1]$ \\ Accuracy Constraint $\alpha \in [0,1]$ \\ Training Dataset $\mathcal{D}$ \\ Sensitive feature $S$
\ENSURE Fair Random Forest $\mathcal{F}$
\FORALL{$\mathcal{T} \in \mathcal{F}$}
\STATE{$\mathcal{T}.flipped \gets false$}
\FORALL{$\lambda \in \mathcal{T}$}
\STATE{$\lambda.flipped \gets false$}
\ENDFOR
\ENDFOR
\STATE $accu_{ \mathcal{F}}^*\gets \frac{ | \mathcal{D}_{y= 1} \wedge \mathcal{F}(\bm{x})=1 |~+~ |\mathcal{D}_{y= 0 }\wedge \mathcal{F}(\bm{x})=0| }{|\mathcal{D}|} $
\STATE{$\Delta accu_{\mathcal{F}} \gets 0$}
\WHILE{$|\{\mathcal{T} \in \mathcal{F} ~|~ \neg\mathcal{T}.flipped\}|>0 \wedge\hfill \hspace*{\fill}\linebreak
\hspace*{3em}\ disc_{\mathcal{D},S,\mathcal{F}} > \epsilon\ \wedge\ \Delta accu_{\mathcal{F}} < \alpha$}
\STATE $\mathcal{T}^\dagger \gets \argmax_{\mathcal{T} \in \mathcal{F}}\ disc_{\mathcal{D},S,\mathcal{T}}$
\STATE $\Lambda \gets$ {\sc Score\_Leaves($\mathcal{T}^\dagger,\mathcal{D},S$)}
\IF{$\Lambda\neq\emptyset$}
\FORALL{$\lambda \in \Lambda$}
\STATE{$\lambda.pred = 1 - \lambda.pred$}
\ENDFOR
\ENDIF
\STATE{$\mathcal{T}^\dagger.flipped = true$}
\STATE{$accu_{ \mathcal{F}}\gets \frac{ | \mathcal{D}_{y= 1} \wedge \mathcal{F}(\bm{x})=1 |~+~ |\mathcal{D}_{y= 0 }\wedge \mathcal{F}(\bm{x})=0| }{|\mathcal{D}|} $}
\STATE{$\Delta accu_{\mathcal{F}} \gets accu_{ \mathcal{F}}^* - accu_{ \mathcal{F}}$}
\ENDWHILE
\RETURN $\mathcal{F}$
\end{algorithmic}
\end{algorithm}
Such tree-based strategy might be too aggressive, as it immediately flips all the candidate leaves of the selected tree. Indeed, only a few leaves may be sufficient to meet our discrimination and accuracy requirements.
Therefore we propose a second strategy, named \textit{Leaf-Based Flipping}, illustrated in Alg~\ref{alg:slr}. As in the former strategy, we first select the tree $\mathcal{T}^\dagger$ with the largest discrimination. Then we use the leaf scoring technique to find a set of candidate leaves from $\mathcal{T}^\dagger$. If such set is empty, e.g., because they were already flipped or they cannot improve the discrimination, the full tree is marked as flipped and the procedure is repeated on the remaining non-flipped trees. Otherwise, the leaf with the largest score $\lambda^\dagger$ is selected, marked as flipped, while its prediction is inverted. The process is repeated until all trees have been flipped, or the desired discrimination $\epsilon$ is achieved, or the tolerated accuracy drop $\alpha$ is met.
We argued that the Leaf-based approach exploits a more fine-grained tuning of the given forest, and therefore it can achieve the desired accuracy with a smaller set of alterations. Indeed, reducing the flips applied to the forest provides a larger accuracy.
\begin{algorithm}[t]
\small
\renewcommand{\algorithmicrequire}{\textbf{Input:}}
\renewcommand{\algorithmicensure}{\textbf{Output:}}
\caption{EiFFFeL-LF (Leaf-based Flipping)}
\algsetup{linenosize=\small}
\label{alg:slr}
\begin{algorithmic}[1]
\REQUIRE Random Forest classifier $\mathcal{F}$ \\ Discrimination Constraint $\epsilon \in [0,1]$ \\ Accuracy Constraint $\alpha \in [0,1]$ \\ Training Dataset $\mathcal{D}$ \\ Sensitive feature $S$
\ENSURE Fair Random Forest $\mathcal{F}$
\FORALL{$\mathcal{T} \in \mathcal{F}$}
\STATE{$\mathcal{T}.flipped \gets false$}
\FORALL{$\lambda \in \mathcal{T}$}
\STATE{$\lambda.flipped \gets false$}
\ENDFOR
\ENDFOR
\STATE $accu_{ \mathcal{F}}^*\gets \frac{ | \mathcal{D}_{y= 1} \wedge \mathcal{F}(\bm{x})=1 |~+~ |\mathcal{D}_{y= 0 }\wedge \mathcal{F}(\bm{x})=0| }{|\mathcal{D}|} $
\STATE{$\Delta accu_{\mathcal{F}} \gets 0$}
\WHILE{$|\{\mathcal{T} \in \mathcal{F} ~|~ \neg\mathcal{T}.flipped\}|>0 \wedge\hfill \hspace*{\fill}\linebreak
\hspace*{3em}\ disc_{\mathcal{D},S,\mathcal{F}} > \epsilon\ \wedge\ \Delta accu_{\mathcal{F}} < \alpha$}
\STATE $\mathcal{T}^\dagger \gets \argmax_{\mathcal{T} \in \mathcal{F}}\ disc_{\mathcal{D},S,\mathcal{T}}$
\STATE $\Lambda \gets$ {\sc Score\_Leaves($\mathcal{T}^\dagger,\mathcal{D},S$)}
\IF{$\Lambda=\emptyset$}
\STATE{$\mathcal{T}.flipped \gets true$}
\ELSE
\STATE $\lambda^\dagger \gets \argmax_{\lambda \in \Lambda}\ \lambda.score$
\STATE{$\lambda^\dagger.flipped = true$}
\STATE{$\lambda^\dagger.pred = 1 - \lambda.pred$}
\STATE{$accu_{ \mathcal{F}}\gets \frac{ | \mathcal{D}_{y= 1} \wedge \mathcal{F}(\bm{x})=1 |~+~ |\mathcal{D}_{y= 0 }\wedge \mathcal{F}(\bm{x})=0| }{|\mathcal{D}|} $}
\STATE{$\Delta accu_{\mathcal{F}} \gets accu_{ \mathcal{F}}^* - accu_{ \mathcal{F}}$}
\ENDIF
\ENDWHILE
\RETURN $\mathcal{F}$
\end{algorithmic}
\end{algorithm}
\begin{table*}[ht]
\caption{Comparison of accuracy reduction and discrimination decrease on Adult dataset with respect to baseline accuracy of 0.85 and discrimination 0.2.
Along with $\Delta$Accu and $\Delta$Disc, we also report (within parentheses) the final accuracy and discrimination values obtained.
\label{tab:adult}}
\center
\begin{tabular}{ccc|ccc|ccc|ccc|ccc|ccc|ccc|}
\cline{4-15}
& & & \multicolumn{2}{ c|| }{DFRF} & \multicolumn{2}{ c|| }{EOP} & \multicolumn{2}{ c|| }{EiFFFeL-TF} & \multicolumn{2}{ c|| }{EiFFFeL-LF}& \multicolumn{2}{ c||}{EiFFFeL-TF$^{\star}$} & \multicolumn{2}{ c|| }{EiFFFeL-LF$^{\star}$} \\
\cline{4-15}
& & & \multicolumn{1}{ c| }{$\Delta$Accu$\downarrow$} & \multicolumn{1}{ c|| }{$\Delta$Disc $\uparrow$} & \multicolumn{1}{ c| }{$\Delta$Accu$\downarrow$} & \multicolumn{1}{ c|| }{$\Delta$Disc$\uparrow$} &
\multicolumn{1}{ c|}{$\Delta$Accu$\downarrow$} & \multicolumn{1}{ c||}{$\Delta$Disc$\uparrow$} & \multicolumn{1}{ c| }{$\Delta$Accu$\downarrow$} & \multicolumn{1}{ c|| }{$\Delta$Disc$\uparrow$} & \multicolumn{1}{ c| }{$\Delta$Accu$\downarrow$} & \multicolumn{1}{ c|| }{$\Delta$Disc$\uparrow$} &
\multicolumn{1}{ c||}{$\Delta$Accu$\downarrow$} & \multicolumn{1}{ c|| }{$\Delta$Disc$\uparrow$} \\
\toprule
\multicolumn{1}{ |c}{\multirow{4}{*}{Adult} } &
\multicolumn{1}{ |c }{\multirow{4}{*}{\rotatebox[origin=c]{0}{$\epsilon$}}} &
\multicolumn{1}{|c|}{0.01} &
\multicolumn{1}{c|}{{7(0.78)}} & \multicolumn{1}{c||}{{18(0.02)}} &
\multicolumn{1}{ c| }{\multirow{4}{*}{\rotatebox[origin=c]{0}{2(0.83)}}} &
\multicolumn{1}{ c|| }{\multirow{4}{*}{\rotatebox[origin=c]{0}{7(0.13)}}} &
\multicolumn{1}{c|}{4(0.81)} & \multicolumn{1}{c||}{19(0.01)} &
\multicolumn{1}{c|}{{4(0.81)}} & \multicolumn{1}{c||}{\color{blue}{\textbf{20(0)}}}
& \multicolumn{1}{c|}{{6(0.79)}} & \multicolumn{1}{c||}{15(0.05)}
& \multicolumn{1}{c|}{{3(0.82)}} & \multicolumn{1}{c||}{17(0.03)}
\\
\cline{3-5}
\cline{8-15}
\multicolumn{1}{|c}{} & \multicolumn{1}{|c}{} &
\multicolumn{1}{|c|}{0.05} &
\multicolumn{1}{c|}{3(0.82)} & \multicolumn{1}{c||}{13(0.07)} &
\multicolumn{1}{ c| }{\multirow{4}{*}{\rotatebox[origin=c]{0}{}}} &
\multicolumn{1}{ c || }{\multirow{4}{*}{\rotatebox[origin=c]{0}{}}} &
\multicolumn{1}{c|}{3(0.82)} &
\multicolumn{1}{c||}{\color{blue}{\textbf{16(0.04)}}} & \multicolumn{1}{c|}{{2(0.83)}} &
\multicolumn{1}{c||}{{15(0.05)}}
& \multicolumn{1}{c|}{{6(0.79)}} & \multicolumn{1}{c||}{16(0.04)}
& \multicolumn{1}{c|}{{3(0.82)}} & \multicolumn{1}{c||}{\color{blue}{\textbf{16(0.04)}}}
\\
\cline{3-5}
\cline{8-15}
\multicolumn{1}{|c}{} & \multicolumn{1}{|c}{} &
\multicolumn{1}{|c|}{0.10} &
\multicolumn{1}{c|}{4(0.81)} & \multicolumn{1}{c||}{15(0.05)} &
\multicolumn{1}{ c| }{\multirow{4}{*}{\rotatebox[origin=c]{0}{}}} &
\multicolumn{1}{ c|| }{\multirow{4}{*}{\rotatebox[origin=c]{0}{}}} &
\multicolumn{1}{c|}{2(0.83)} & \multicolumn{1}{c||}{12(0.08)} &
\multicolumn{1}{c|}{{1(0.84)}} & \multicolumn{1}{c||}{{\color{blue}{\textbf{12(0.08)}}}}
& \multicolumn{1}{c|}{{1(0.84)}} & \multicolumn{1}{c||}{\color{blue}{\textbf{12(0.08)}}}
& \multicolumn{1}{c|}{{2(0.83)}} & \multicolumn{1}{c||}{10(0.1)}
\\
\cline{3-5}
\cline{8-15}
\multicolumn{1}{|c}{} & \multicolumn{1}{|c}{} &
\multicolumn{1}{|c|}{0.15} &
\multicolumn{1}{c|}{2(0.83)} & \multicolumn{1}{c||}{10(0.1)} &
\multicolumn{1}{ c| }{\multirow{4}{*}{\rotatebox[origin=c]{0}{}}} &
\multicolumn{1}{ c|| }{\multirow{4}{*}{\rotatebox[origin=c]{0}{}}} &
\multicolumn{1}{c|}{0(0.85)} & \multicolumn{1}{c||}{8(0.12)} & \multicolumn{1}{c|}{{0(0.85)}} & \multicolumn{1}{c||}{{ \color{blue}{\textbf{9(0.11)}}}}
& \multicolumn{1}{c|}{{0(0.85)}} & \multicolumn{1}{c||}{7(0.13)}
& \multicolumn{1}{c|}{{0(0.85)}} & \multicolumn{1}{c||}{7(0.13)}
\\
\bottomrule
\end{tabular}
\end{table*}
\begin{table*}[ht]
\caption{Comparison of accuracy reduction and discrimination decrease on Bank dataset with respect to baseline accuracy of 0.82 and discrimination 0.18. Along with $\Delta$Accu and $\Delta$Disc, we also report (within parentheses) the final accuracy and discrimination values obtained.
\label{tab:bank}}
\center
\footnotesize
\begin{tabular}{ccc|ccc|ccc|ccc|ccc|ccc|ccc|}
\cline{4-15}
& & & \multicolumn{2}{ c|| }{DFRF} & \multicolumn{2}{ c|| }{EOP} & \multicolumn{2}{ c|| }{EiFFFeL-TF} & \multicolumn{2}{ c|| }{EiFFFeL-LF}& \multicolumn{2}{ c||}{EiFFFeL-TF$^{\star}$} & \multicolumn{2}{ c|| }{EiFFFeL-LF$^{\star}$} \\
\cline{4-15}
& & & \multicolumn{1}{ c| }{$\Delta$Accu$\downarrow$} & \multicolumn{1}{ c|| }{$\Delta$Disc $\uparrow$} & \multicolumn{1}{ c| }{$\Delta$Accu$\downarrow$} & \multicolumn{1}{ c|| }{$\Delta$Disc$\uparrow$} &
\multicolumn{1}{ c| }{$\Delta$Accu$\downarrow$} & \multicolumn{1}{ c|| }{$\Delta$Disc$\uparrow$} & \multicolumn{1}{ c| }{$\Delta$Accu$\downarrow$} & \multicolumn{1}{ c|| }{$\Delta$Disc$\uparrow$} & \multicolumn{1}{ c| }{$\Delta$Accu$\downarrow$} & \multicolumn{1}{ c|| }{$\Delta$Disc$\uparrow$} &
\multicolumn{1}{ c||}{$\Delta$Accu$\downarrow$} & \multicolumn{1}{ c|| }{$\Delta$Disc$\uparrow$} \\
\toprule
\multicolumn{1}{ |c}{\multirow{4}{*}{Bank} } &
\multicolumn{1}{ |c }{\multirow{4}{*}{\rotatebox[origin=c]{0}{$\epsilon$}}} &
\multicolumn{1}{|c|}{0.01} &
\multicolumn{1}{c|}{{9(0.73)}} & \multicolumn{1}{c||}{{13(0.05)}} &
\multicolumn{1}{ c| }{\multirow{4}{*}{\rotatebox[origin=c]{0}{0(0.82)}}} &
\multicolumn{1}{ c|| }{\multirow{4}{*}{\rotatebox[origin=c]{0}{14(0.04)}}} &
\multicolumn{1}{c|}{7(0.75)} & \multicolumn{1}{c||}{17(0.01)} & \multicolumn{1}{c|}{{10(0.72)}} & \multicolumn{1}{c||}{15(0.03)}
& \multicolumn{1}{c|}{{8(0.74)}} & \multicolumn{1}{c||}{14(0.04)}
& \multicolumn{1}{c|}{{5(0.77)}} & \multicolumn{1}{c||}{10(0.08)}
\\
\cline{3-5}
\cline{8-15}
\multicolumn{1}{|c}{} & \multicolumn{1}{|c}{} &
\multicolumn{1}{|c|}{0.05} &
\multicolumn{1}{c|}{4(0.78)} & \multicolumn{1}{c||}{11(0.07)} &
\multicolumn{1}{ c| }{\multirow{4}{*}{\rotatebox[origin=c]{0}{}}} &
\multicolumn{1}{ c|| }{\multirow{4}{*}{\rotatebox[origin=c]{0}{}}} &
\multicolumn{1}{c|}{3(0.79)} & \multicolumn{1}{c||}{13(0.05)} & \multicolumn{1}{c|}{{8(0.74)}} & \multicolumn{1}{c||}{{14(0.04)}}
& \multicolumn{1}{c|}{{8(0.74)}} & \multicolumn{1}{c||}{13(0.05)}
& \multicolumn{1}{c|}{{5(0.77)}} & \multicolumn{1}{c||}{13(0.05)}
\\
\cline{3-5}
\cline{8-15}
\multicolumn{1}{|c}{} & \multicolumn{1}{|c}{} &
\multicolumn{1}{|c|}{0.10} &
\multicolumn{1}{c|}{4(0.78)} & \multicolumn{1}{c||}{6(0.12)} &
\multicolumn{1}{ c| }{\multirow{4}{*}{\rotatebox[origin=c]{0}{}}} &
\multicolumn{1}{ c|| }{\multirow{4}{*}{\rotatebox[origin=c]{0}{}}} &
\multicolumn{1}{c|}{2(0.80)} & \multicolumn{1}{c||}{10(0.08)} & \multicolumn{1}{c|}{{1(0.81)}} & \multicolumn{1}{c||}{{7(0.11)}}
& \multicolumn{1}{c|}{{7(0.75)}} & \multicolumn{1}{c||}{9(0.09)}
& \multicolumn{1}{c|}{{4(0.78)}} & \multicolumn{1}{c||}{8(0.10)}
\\
\cline{3-5}
\cline{8-15}
\multicolumn{1}{|c}{} & \multicolumn{1}{|c}{} &
\multicolumn{1}{|c|}{0.15} &
\multicolumn{1}{c|}{4(0.78)} & \multicolumn{1}{c||}{9(0.09)} &
\multicolumn{1}{ c| }{\multirow{4}{*}{\rotatebox[origin=c]{0}{}}} &
\multicolumn{1}{ c|| }{\multirow{4}{*}{\rotatebox[origin=c]{0}{}}} &
\multicolumn{1}{c|}{0(0.82)} & \multicolumn{1}{c||}{4(0.14)} & \multicolumn{1}{c|}{{0(0.82)}} & \multicolumn{1}{c||}{{ 4(0.14)}}
& \multicolumn{1}{c|}{{6(0.76)}} & \multicolumn{1}{c||}{5(0.13)}
& \multicolumn{1}{c|}{{2(0.80)}} & \multicolumn{1}{c||}{5(0.13)}
\\
\bottomrule
\end{tabular}
\end{table*}
\begin{table*}[ht]
\caption{Comparison of accuracy reduction and discrimination decrease on Compas dataset with respect to baseline accuracy of 0.69 and discrimination 0.3.
Along with $\Delta$Accu and $\Delta$Disc, we also report (within parentheses) the final accuracy and discrimination values obtained.
\label{tab:compas}}
\center
\footnotesize
\begin{tabular}{ccc|ccc|ccc|ccc|ccc|ccc|ccc|}
\cline{4-15}
& & & \multicolumn{2}{ c|| }{DFRF} & \multicolumn{2}{ c|| }{EOP} & \multicolumn{2}{ c|| }{EiFFFeL-TF} & \multicolumn{2}{ c|| }{EiFFFeL-LF}& \multicolumn{2}{ c||}{EiFFFeL-TF$^{\star}$} & \multicolumn{2}{ c|| }{EiFFFeL-LF$^{\star}$} \\
\cline{4-15}
& & & \multicolumn{1}{ c| }{$\Delta$Accu$\downarrow$} & \multicolumn{1}{ c|| }{$\Delta$Disc $\uparrow$} & \multicolumn{1}{ c| }{$\Delta$Accu$\downarrow$} & \multicolumn{1}{ c|| }{$\Delta$Disc$\uparrow$} &
\multicolumn{1}{ c| }{$\Delta$Accu$\downarrow$} & \multicolumn{1}{ c|| }{$\Delta$Disc$\uparrow$} & \multicolumn{1}{ c| }{$\Delta$Accu$\downarrow$} & \multicolumn{1}{ c|| }{$\Delta$Disc$\uparrow$} & \multicolumn{1}{ c| }{$\Delta$Accu$\downarrow$} & \multicolumn{1}{ c|| }{$\Delta$Disc$\uparrow$} &
\multicolumn{1}{ c||}{$\Delta$Accu$\downarrow$} & \multicolumn{1}{ c|| }{$\Delta$Disc$\uparrow$} \\
\toprule
\multicolumn{1}{ |c}{\multirow{4}{*}{COMPAS} } &
\multicolumn{1}{ |c }{\multirow{4}{*}{\rotatebox[origin=c]{0}{$\epsilon$}}} &
\multicolumn{1}{|c|}{0.01} &
\multicolumn{1}{c|}{{11(0.58)}} & \multicolumn{1}{c||}{{28(0.02)}} &
\multicolumn{1}{ c| }{\multirow{4}{*}{\rotatebox[origin=c]{0}{4(0.65)}}} &
\multicolumn{1}{ c|| }{\multirow{4}{*}{\rotatebox[origin=c]{0}{5(0.25)}}} &
\multicolumn{1}{c|}{25(0.44)} & \multicolumn{1}{c||}{29(0.01)} & \multicolumn{1}{c|}{{5(0.64)}} & \multicolumn{1}{c||}{26(0.04)}
& \multicolumn{1}{c|}{{9(0.60)}} & \multicolumn{1}{c||}{29(0.01)}
& \multicolumn{1}{c|}{{1(0.68)}} & \multicolumn{1}{c||}{7(0.23)}
\\
\cline{3-5}
\cline{8-15}
\multicolumn{1}{|c}{} & \multicolumn{1}{|c}{} &
\multicolumn{1}{|c|}{0.05} &
\multicolumn{1}{c|}{5(0.64)} & \multicolumn{1}{c||}{13(0.17)} &
\multicolumn{1}{ c| }{\multirow{4}{*}{\rotatebox[origin=c]{0}{}}} &
\multicolumn{1}{ c|| }{\multirow{4}{*}{\rotatebox[origin=c]{0}{}}} &
\multicolumn{1}{c|}{12(0.57)} & \multicolumn{1}{c||}{28(0.02)} & \multicolumn{1}{c|}{{5(0.64)}} & \multicolumn{1}{c||}{{22(0.08)}}
& \multicolumn{1}{c|}{{9(0.60)}} & \multicolumn{1}{c||}{28(0.02)}
& \multicolumn{1}{c|}{{1(0.68)}} & \multicolumn{1}{c||}{7(0.23)}
\\
\cline{3-5}
\cline{8-15}
\multicolumn{1}{|c}{} & \multicolumn{1}{|c}{} &
\multicolumn{1}{|c|}{0.10} &
\multicolumn{1}{c|}{4(0.65)} & \multicolumn{1}{c||}{7(0.23)} &
\multicolumn{1}{ c| }{\multirow{4}{*}{\rotatebox[origin=c]{0}{}}} &
\multicolumn{1}{ c|| }{\multirow{4}{*}{\rotatebox[origin=c]{0}{}}} &
\multicolumn{1}{c|}{7(0.62)} & \multicolumn{1}{c||}{21(0.09)} & \multicolumn{1}{c|}{{5(0.64)}} & \multicolumn{1}{c||}{{21(0.09)}}
& \multicolumn{1}{c|}{{1(0.68)}} & \multicolumn{1}{c||}{21(0.09)}
& \multicolumn{1}{c|}{{1(0.68)}} & \multicolumn{1}{c||}{7(0.23)}
\\
\cline{3-5}
\cline{8-15}
\multicolumn{1}{|c}{} & \multicolumn{1}{|c}{} &
\multicolumn{1}{|c|}{0.15} &
\multicolumn{1}{c|}{2(0.67)} & \multicolumn{1}{c||}{6(0.24)} &
\multicolumn{1}{ c| }{\multirow{4}{*}{\rotatebox[origin=c]{0}{}}} &
\multicolumn{1}{ c|| }{\multirow{4}{*}{\rotatebox[origin=c]{0}{}}} &
\multicolumn{1}{c|}{1(0.68)} & \multicolumn{1}{c||}{19(0.11)} & \multicolumn{1}{c|}{{2(0.67)}} & \multicolumn{1}{c||}{{ 15(0.15)}}
& \multicolumn{1}{c|}{{0(0.69)}} & \multicolumn{1}{c||}{16(0.14)}
& \multicolumn{1}{c|}{{1(0.68)}} & \multicolumn{1}{c||}{7(0.23)}
\\
\bottomrule
\end{tabular}
\end{table*}
\section{Experimental Evaluation}
\subsection{Datasets.}
We use datasets publicly available, widely used in fairness literature, concerning binary classification. We pre-process them using one-hot encoding for categorical features, binary encoding of sensitive feature, and removing of instances containing missing values. Moreover, we use an 80/20 training/test split.
\begin{itemize}
\item \textit{Adult}: The Adult UCI income dataset \cite{Dua:2019} contains 14 demographic attributes of more than 45,000 individuals, together with class labels which states whether their income is higher than \$50K or not. As sensitive attribute, we use the \textit{gender} encoded as a binary attribute 1/0 for male/female respectively.
\item \textit{COMPAS}: The COMPAS dataset \cite{angwin2016machine} contains data collected on the use of the COMPAS (Correctional Offender Management Profiling for Alternative Sanctions) risk assessment tool. It contains 13 attributes of more than 7,000 convicted criminals, with class labels that state whether or not the individual reoffend within 2 years of her/his most recent crime. We use \textit{race} as sensitive attribute encoded as a binary attribute 1/0 for Others/African-American, respectively.
\item \textit{Bank}: Bank marketing dataset \cite{moro2014data} contains 16 features about 45,211 clients of direct marketing campaigns of a Portuguese banking institution. The goal is to predict whether the client will subscribe or not to a term deposit. We consider the \textit{age} as sensitive attribute, encoded as a binary attribute 1/0, indicating whether the client’s age is $\geq$25 or $<$25, respectively.
\end{itemize}
\begin{figure*} [ht]
\centering
\includegraphics[width=0.7\textwidth]{Ac_Vs_dis.png}
\caption{ Accuracy vs. discrimination scores after relabeling for constraints $\epsilon=0.01,0.05,0.1,0.15$. }
\label{fig:plot1}
\end{figure*}
\begin{figure*}[ht]
\centering
\includegraphics[width=0.7\textwidth]{Ac_Vs_ep.png}
\caption{Accuracy of the model as a function the $\epsilon$ constraint.}
\label{fig:plot2}
\end{figure*}
\begin{figure*}[ht]
\centering
\includegraphics[width=0.7\textwidth]{Dis_Vs_ep.png}
\caption{Discrimination scores as a function of the $\epsilon$ constraint.}
\label{fig:plot3}
\end{figure*}
\subsection{Experimental Setup.}
We apply our proposed EiFFFeL algorithm over a Random Forest classifier with/without the fair splitting of nodes for individual base trees, and evaluate the performance of the algorithms in terms of model accuracy and discrimination over the three datasets mentioned above.
We compare our results against a DFRF classifier (\textit{Distributed fair random forest}) \cite{fantin2020distributed}, which only includes fair decision trees within the forest. The setting of hyper-parameters of DFRF are the same as the one described in the original work. We use fair split and sensitive feature as hyper-parameters, along with
tree number and maximum tree depth.
Additionally,
we also compare our results against EOP (\textit{Equalized Odds Post-processing}) \cite{hardt2016equality,pleiss2017fairness}, a random forest classifier with the same number of base estimators and maximum depth as ours. After training and achieving the desired equalized odd we score the discrimination in the same approach we used for our experiments.
In conclusion, the comparisons of accuracy and discrimination values are among the following methods:
\begin{itemize}
\item DFRF \cite{fantin2020distributed},\footnote{\url{https://github.com/pjlake98/Distributed-Fair-Random-Forest}} which adds base trees to the forest only if they are fair;
\item EOP \cite{hardt2016equality,pleiss2017fairness},\footnote{\url{https://github.com/Trusted-AI/AIF360/blob/master/aif360/algorithms/postprocessing/calibrated_eq_odds_postprocessing.py}} which adopts a post-processing method based on achieving equalized odds requiring the privileged and unprivileged groups to have the same false negative rate and same false positive rate;
\item our implementations of EiFFFeL-TF and EiFFFeL-LF algorithms, whose post-processing is applied to a plain Random Forest of trees;
\item the same post-processing techniques of EiFFFeL-TF and EiFFFeL-LF applied on top of a random forest with discrimination aware base trees \cite{kamiran2010discrimination}. These versions are denoted by EiFFFeL-TF$^{\star}$ and EiFFFeL-LF$^{\star}$.
\end{itemize}
Finally, the \textit{baseline accuracy} and \textit{discrimination} used to compare the various methods are the ones obtained by a plain Random Forest of trees, trained on the three datasets through the scikit-learn algorithm Random Forest Classifier\footnote{\url{https://scikit-learn.org/stable/modules/generated/sklearn.ensemble.RandomForestClassifier.html}}. The various EiFFFeL methods are applied to the same baseline Random Forest.
\subsection{Results.}
Tables \ref{tab:adult}), \ref{tab:bank}), and \ref{tab:compas}) compare the decreases in accuracy and discrimination, obtained by the different algorithms, on the three datasets with respect to the baseline results obtained by plain Random Forest models.
Recall that increasing $\epsilon$, we reduce the space for improving discrimination, and as a side effect, we preserves the baseline accuracy. Indeed, in these experiments the accuracy constraint $\alpha$ was set to 1, so that there are no limits in the possible accuracy reduction $\Delta$Accu. This allows us to compare our methods against DFRF and EOP, which do not have this $\alpha$ constraint. Indeed, EOP is completely parameter free, and does not support neither $\alpha$ nor $\epsilon$.
In more details, Tables \ref{tab:adult}), \ref{tab:bank}), and \ref{tab:compas}) report, for different values of $\epsilon$ in the set $\{0.01, 0.05, 0.10, 0.15\}$, the $\Delta$Accu and $\Delta$Disc values obtained by the different algorithms, where $\Delta$Accu and $\Delta$Disc indicate the \textit{absolute difference} in accuracy and discrimination w.r.t. the baselines.
Indeed, we express these $\Delta$ absolute differences in points/hundredths (each point corresponds to $1/100$).
Note that while greater values of $\Delta$Disc are better, greater values of $\Delta$Accu are worse, so a trade-off is needed. In addition, besides the absolute $\Delta$ values, we also report (within parentheses) the final values for accuracy and discrimination score obtained by the various techniques.
For example, for the \textit{Adult} dataset (Table \ref{tab:adult}) and $\epsilon=0.01$, EiFFFeL-TF can reach a very low discrimination score of $0.01$, by only losing $4$ points in accuracy (from $0.85$ of the baseline to $0.81$). In comparison, the best results we can obtain with DFRF in terms of discrimination is a score of $0.02$, by losing 7 points in accuracy (from $0.85$ of the baseline to $0.78$). Overall, our algorithms are capable of reducing discrimination better than DFRF while maintaining the same accuracy. Also EOP does not work well, as the best discrimination score is only $0.13$, by losing 2 points in accuracy. In addition, using $\epsilon=0.15$ for EiFFFeL-TF and EiFFFeL-LF (also EiFFFeL-TF$^{\star}$ and EiFFFeL-LF$^{\star}$), we can decrease the baseline discrimination of about $7-9$ points, by keeping the same accuracy of the baseline.
Results for the Bank dataset (Table \ref{tab:bank}) shows that EiFFFeL-TF can reach for $\epsilon=0.01$ the desired discrimination score, but losing 7 points in accuracy (from 0.82 to 0.75), whereas DFRF has worse discrimination score of 0.05 and a worse accuracy of 0.73. EOP does not lose any accuracy for lowering the discrimination score by 14 points to 0.04.
Finally, considering the results obtained for the COMPAS dataset (Table \ref{tab:compas}), we observe in some cases DFRF works pretty well, but always one of our algorithms gets better results. For example, for $\epsilon=0.01$, the best discrimination score of $0.01$ is obtained by EiFFFeL-TF$^\star$, by only losing 9 points in accuracy, against the 11 points lost by DFRF with a discrimination score of $0.02$.
\medskip
Figures~\ref{fig:plot1}, \ref{fig:plot2}, and \ref{fig:plot3} report the same data of the above tables, where we varied the discrimination constraint $\epsilon=\{0.01, 0.05, 0.1, 0.5\}$, with no constraints on accuracy. The results obtained by EOP are not plotted, as its results are always worse than the competitors and do not vary with $\epsilon$.
Specifically, Figure \ref{fig:plot1} reports results for the three datasets, and aims at showing the tradeoff
of accuracy vs. discrimination when we vary $\epsilon$. Recall that we are interested in achieving low discrimination and high accuracy, and thus the best tradeoff corresponds to points of curves falling in the top-left quadrant.
First, we highlight that DRFR performs poorly on most settings compared to the proposed EIFFFeL variants.
On the Adult dataset, EIFFFeL-LF dominates the other algorithms for all values of $\epsilon$ and achieves the desired or better discrimination with the largest accuracy. To appreciate the strict relationships between of the setting of $\epsilon$ and the discrimination/accuracy obtained, the reader can refer to the other two Figures \ref{fig:plot1} and \ref{fig:plot2}.
Returning to Figure \ref{fig:plot1}, the COMPAS EIFFFeL-LF provides the best performance together with EIFFFeL-TF$^{\star}$. This is the only dataset where EIFFFeL-TF$^{\star}$ provides interesting performance, and thus the discrimination aware splitting at training time provides some benefits. We also highlight that when using $\epsilon=0.15$ (see Figure \ref{fig:plot3}) the algorithm DFRF only gets a discrimination score of 0.25. Note that EIFFFeL-LF$^{\star}$ is not able to provide better performance when varying $\epsilon$, thus resulting in a constant curve.
Finally, on the Bank Dataset, EIFFFeL-TF and EIFFFeL-LF achieve the best results, with an advantage for EIFFFeL-TF for smaller values of $\epsilon$. Finally, the results show how we can obtain the desired discrimination degree with a limited drop in accuracy. Overall, the proposed EIFFFEL algorithm outperforms the competitor DFRF, and, on average, it is advisable to avoid the discrimination aware node splitting. We believe that working only at post-processing allows us to exploit a richer set of trees grown, by exploring a larger and unconstrained search space.
\begin{table*}
\caption{Accuracy and discrimination scores on the Adult dataset for $\epsilon=0.01$ and $\alpha=0.01,0.02,0.03,0.05$. The baseline accuracy and discrimination score are $0.85$ and $0.2$, respectively.
\label{tab:alpha}}
\center
\footnotesize
\begin{tabular}{cccc|ccc|ccc|ccc|}
\cline{5-12}
& & & & \multicolumn{2}{ c|| }{EiFFFeL-TF} & \multicolumn{2}{ c|| }{EiFFFeL-LF}& \multicolumn{2}{ c||}{EiFFFeL-TF$^{\star}$} & \multicolumn{2}{ c|| }{EiFFFeL-LF$^{\star}$} \\
\cline{5-12}
& & & & \multicolumn{1}{ c| }{Accu} & \multicolumn{1}{ c|| }{Disc} & \multicolumn{1}{ c| }{Accu} & \multicolumn{1}{ c|| }{Disc} & \multicolumn{1}{ c| }{Accu} & \multicolumn{1}{ c|| }{Disc} & \multicolumn{1}{ c||}{Accu} & \multicolumn{1}{ c|| }{Disc} \\
\toprule
\multicolumn{1}{ |c}{\multirow{4}{*}{Adult} } &
\multicolumn{1}{ |c }{\multirow{4}{*}{\rotatebox[origin=c]{0}{$\epsilon$=0.01}}} &
\multicolumn{1}{ |c }{\multirow{4}{*}{\rotatebox[origin=c]{0}{$\alpha$}}} &
\multicolumn{1}{|c|}{0.01} &
\multicolumn{1}{c|}{0.83} & \multicolumn{1}{c||}{0.09} &
\multicolumn{1}{c|}{{0.84}} & \multicolumn{1}{c||}{\bf0.08}
& \multicolumn{1}{c|}{{0.84}} & \multicolumn{1}{c||}{0.10}
& \multicolumn{1}{c|}{{0.84}} & \multicolumn{1}{c||}{0.11}
\\
\cline{4-12}
\multicolumn{1}{|c}{} &
\multicolumn{1}{|c}{} &
\multicolumn{1}{|c}{} &
\multicolumn{1}{|c|}{0.02} &
\multicolumn{1}{c|}{0.83} & \multicolumn{1}{c||}{0.09} &
\multicolumn{1}{c|}{{0.83}} & \multicolumn{1}{c||}{{\bf0.06}}
& \multicolumn{1}{c|}{{0.83}} & \multicolumn{1}{c||}{0.10}
& \multicolumn{1}{c|}{{0.83}} & \multicolumn{1}{c||}{0.07}
\\
\cline{4-12}
\multicolumn{1}{|c}{} &
\multicolumn{1}{|c}{} &
\multicolumn{1}{|c}{} &
\multicolumn{1}{|c|}{0.03} &
\multicolumn{1}{c|}{0.82} & \multicolumn{1}{c||}{\bf0.04} &
\multicolumn{1}{c|}{{0.82}} & \multicolumn{1}{c||}{{0.05}}
& \multicolumn{1}{c|}{{0.82}} & \multicolumn{1}{c||}{0.07}
& \multicolumn{1}{c|}{{0.82}} & \multicolumn{1}{c||}{\bf0.04}
\\
\cline{4-12}
\multicolumn{1}{|c}{} &
\multicolumn{1}{|c}{} &
\multicolumn{1}{|c}{} &
\multicolumn{1}{|c|}{0.05} &
\multicolumn{1}{c|}{0.81} & \multicolumn{1}{c||}{0.01} &
\multicolumn{1}{c|}{{0.81}} & \multicolumn{1}{c||}{{\bf 0.00}}
& \multicolumn{1}{c|}{{0.80}} & \multicolumn{1}{c||}{0.08}
& \multicolumn{1}{c|}{{0.82}} & \multicolumn{1}{c||}{0.03}
\\
\bottomrule
\end{tabular}
\end{table*}
The effect of varying the discrimination constraint $\epsilon$ without constraining accuracy can be observed in Figure \ref{fig:plot2}, where we discover that lower discrimination is achieved with large accuracy reduction. This is due to the fact that a small discrimination threshold allows our flipping strategies to force the change of many leaves, thus changing more the classification decision regions, with a final lower accuracy.
However our approach of selecting potential leaves to relabel seems better than training random forest with only fair trees.
In addition, training and then rejecting trees (because they are not fair) makes longer the training of the forest,
particularly when we fail often in finding fair trees.
Finally, Figure~\ref{fig:plot3} contrasts the discrimination measured on the test set against the desired discrimination constraint $\epsilon$. Clearly, the twos do not always match. In particular, DFRF has an unstable behaviour, meaning that filtering the tree to be added to the forest is not the best option. Conversely, EiFFFeL-TF and EiFFFeL-LF provide a much more stable behaviour.
\medskip
We also discuss the results of other experiments, aiming to evaluate the effects of different values for the $\alpha$ constraints. Note that only the EiFFFeL algorithms support the $\alpha$ parameter, so we cannot reports any results for the competitors DFRF and EOP.
Specifically, Table \ref{tab:alpha} reports results relative to the Adult dataset, where,
for a fixed $\epsilon=0.01$, we vary the $\alpha$ constraint over the expected accuracy, with values ranging in the set \{0.01,0.02,0.03,0.05\}. For each $\alpha$ value, we show in bold the best results in terms of discrimination score.
We observe that the accuracy constraint $\alpha$ has an indirect impact on the final discrimination score obtained. Using EiFFFeL-LF with $\alpha=0.01$, the loss in accuracy is 1 point as expected, while the baseline discrimination score decreases by more than half (from $0.2$ to $0.08$). Furthermore, as the $\alpha$ value increases, discrimination score decreases further. With $\alpha=0.05$, EiFFFeL-LF is able to reduce by 4 points the final accuracy, by also achieving a discrimination score of 0, thus showing the power of our method in achieving a very good trade-off between accuracy and discrimination.
\section{Conclusion}
In this work we deal with fairness in machine learning, and specifically in binary classifiers trained by a Random Forest algorithm. We are interested in group fairness, so as to mitigate the effect of bias against specific groups, which may comes from biased training datasets or algorithm design.
We develop EiFFFeL, a novel post-process approach, which maintains good predictive performance of the trained model with a low discrimination score. Our approaches flips the label of selected leaf (or leaves) of base trees in a random forest by using two algorithms: $(i)$ an aggressive tree-based approach, which flips all candidate leaves of a tree, and $(ii)$ a leaf-based strategy which only flips the label of the most discriminative leaf of a tree. Both strategies are implemented by considering accuracy and discrimination constraints. Indeed, the constraints are used to control the minimum accuracy decrease we can tolerate in order to achieve the desired discrimination value.
In addition, we have tested the impact of incorporating discrimination aware node split strategies for base trees of the forest, by adding discrimination gain value in their node splitting criterion \cite{kamiran2010discrimination}.
By using three publicly available datasets, our experimental results show that effective non-discriminative models can be obtained, while keeping a strict control over both accuracy and discrimination level. Compared to state-of-the-art methods, which adopt both in-process and post-process bias mitigation approaches, EiFFFeL resulted to produce the most accurate models that also exhibit the best levels of fairness.
As part of the future work, we plan to extend our methods by studying the effect of multiple sensitive features in relation to discrimination and accuracy, by also extending our work to other tree ensemble learning methods.
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2024-02-18T23:40:24.353Z
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2022-05-17T02:43:34.000Z
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{"paloma_paragraphs":[]}
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{"arxiv_id":"2112.14435","language":"en","timestamp":1652755414000,"url":"https:\/\/arxiv.org\/abs\/2112.14435","yymm":"2112"}
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proofpile-arXiv_000-10006
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{"provenance":"002.jsonl.gz:10007"}
| null | null |
\section{Introduction}
A digital model of quantum computing can be considered as a natural extension of the classical computing model, which operates with quantum analogs of information bits known as qubits.
During last decades, remarkable progress in the development of quantum information processing devices on various physical platforms has been performed~\cite{Martinis2019,Pan2020,Blatt2018-2,Wright2019,Browaeys2021,Lukin2021}.
A peculiar feature of most existing physical systems used for quantum computing is the ability to operate in the larger states spaces, which are formed, for example,
by accessible additional energy levels of ions or (artificial) atoms~\cite{Ruben2018,Zeilinger2018,Sanders2020}.
The use of such additional degrees of freedom is the basic idea of qudit-based quantum information processing.
In the case of quantum computing, the use of qudits may help to reduce resources required for implementing quantum algorithms.
This is possible, first, by decomposing qudits on a set of two-level systems (qubits).
This approach has been widely studied both theoretically
and experimentally~\cite{Farhi1998,Kessel1999,Kessel2000,Kessel2002,Muthukrishnan2000,Nielsen2002,Berry2002,Klimov2003,Bagan2003,Vlasov2003,Clark2004,Leary2006,Ralph2007,White2008,Ionicioiu2009,Ivanov2012,Li2013,Kiktenko2015,Kiktenko2015-2, Song2016,Frydryszak2017,Bocharov2017,Gokhale2019,Pan2019,Low2020,Jin2021,Martinis2009,White2009,Wallraff2012,Mischuck2012,Gustavsson2015,Martinis2014,Ustinov2015, Morandotti2017,Balestro2017,Low2020,Sawant2020,Pavlidis2021,Rambow2021}.
Despite its simplicity and efficiency, there is a number of limitations associated with finding optimal qudit-to-qubit mappings~\cite{Nikolaeva2021}.
The second approach is to employ higher qudit levels can be used for substituting ancilla qubits, which is of specific interest in the problem of decomposing multi-qubit gates~\cite{Barenco1995},
such as the Toffoli gate~\cite{Ralph2007,White2009,Ionicioiu2009,Wallraff2012,Kwek2020,Baker2020,Kiktenko2020,Kwek2021,Galda2021,Gu2021}.
We note that in the pioneering experimental work on the realization of the Toffoli gate, the third energy level of the superconducting transmon qubit has been used~\cite{Wallraff2012};
this has been further exploited in Ref.~\cite{Galda2021} in experiments with fixed-frequency transmons on the IBM cloud-based superconducting processor and generalized~\cite{Gu2021} for the $n$ qubits (qutrits) coupled within linear topology.
Systematic studies of this approach have demonstrated an ability to efficiently decompose multi-qubit gates: $2N-3$ qubit-qudit gates are required for decomposing the $N$-qubit generalized Toffoli gate~\cite{Kiktenko2020}.
However, this requires one to satisfy a general relation between the dimensionality of qudits and the topology of coupling map, specifically,
for a given qudit one should have $d\geq{k+1}$, where $d$ is its dimension and $k$ is the number of its connections to other qudits used within the decomposition~\cite{Kiktenko2020}.
The limited connectivity of existing noisy intermediate-scale quantum (NISQ) devices and sufficiently high level of errors in the operation with high energy levels poses a problem of relaxing conditions for the efficient implementation of multi-qubit gates.
Many works are then focused on revealing the potential of qutrits~\cite{Klimov2003,Bocharov2017,Gokhale2019,Galda2021},
which are possible to operate efficiently in experiments~\cite{White2008,Wallraff2012,Song2016,Jin2021,Galda2021,Wu2020,Blok2021,Hill2021,Ringbauer2021}.
We note that the use of qudits offers advantages also in quantum teleportation~\cite{Pan2019} and quantum communications~\cite{Gisin2002,Boyd2015},
as well as opens up opportunities for uncovering fundamental concepts of quantum mechanics~\cite{Li2013,Frydryszak2017,Zyczkowski2022}.
\begin{figure}
\center{\includegraphics[width=1\linewidth]{fig1.pdf}}
\caption{Decomposition of two-qutrit gate \ensuremath{U_{i\rightarrow j}}~ (a) and its inverse $U_{i\rightarrow j}^\dagger$ (b) using $\mathtt{iSWAP^{02}}$ gate, which is native for superconducting qutrit-based platforms, is shown.
In (c) the transformation of $\mathtt{iSWAP^{02}}$ into $\mathtt{iSWAP^{20}}$ using local operations is depicted.}
\label{fig::uab}
\end{figure}
\begin{figure*}[ht!]
\includegraphics[width=1\linewidth]{fig2.pdf}
\vskip-3mm
\caption{
Three types of operations with the tree that correspond to $\ensuremath{\mathtt{C}^{N-1}\mathtt{Z}}~$ gate decomposition \BLUE{and evolution of the qutrits' states during these operations.
Before the decomposition, qutrits are initialized with the state of zero population on the ancillary level, therefore corresponding truth tables are presented for these qubit states of qutrits only.}
In (a) the circuit structure for elementary folding operation is presented. It changes the tree structure by collapsing all siblings ${\bf s|1},\ldots, {\bf s}|n({\bf s})$ to their parent ${\bf s}$.
In (b) the circuit structure for the basic operation that applies a phase factor $-1$ to the input state if all qutrit of the level-one subtree are in the state $\ket{1}$ is shown. It does not change single-level tree structure.
In (c) the circuit structure for elementary unfolding operation that serves as uncomputation of elementary folding operation is illustrated.
}\label{fig::main}
\end{figure*}
In the present work, we improve recent results by developing a scheme for decomposing $N$-qubit generalized Toffoli gate using $2N-3$ two-qutrit gates by employing qutrits $(d=3)$ {\it only}; for summary, see Tab.~\ref{tab1}.
The fixed number of the required additional levels (the choice of qutrits is optimal) and the use of the \ensuremath{\mathtt{iSWAP}}~ gate as a native operation make our approach directly applicable for superconducting quantum processors,
specifically, for the Aspen-9 processor supporting quantum operations with qutrits~\cite{Hill2021}.
Although, we are focused here on the superconducting quantum computing setup, our approach in principle can be adopted for other physical platforms,
in particular, for trapped ions, for whose an ability to realize a multi-qudit processor has been demonstrated~\cite{Ringbauer2021}.
Our work is organized as follows.
In Sec.~\ref{sec:qutrit-proc}, we introduce main concepts behind the idea of a qutrit-based processor and describe the required quantum gates for its operation.
In Sec.~\ref{sec:Toffoli}, we describe the details of the proposed approach for decomposing the generalized Toffoli gate with qutrits.
In Sec.~\ref{sec:Aspen}, we discuss the application to existing architecture of superconducting qutirt-based processor.
We summarize our results in Sec.~\ref{sec:Conclusion}.
\section{Qutrit-based processor and gates}\label{sec:qutrit-proc}
We consider a system of $N$ qutrits, which are composed by three levels $\ket{0}$, $\ket{1}$, and $\ket{2}$, labeled by indices $i \in \{1,\dots,N\}$.
The first two levels $\ket{0}_i, \ket{1}_i$ of $i$th qutrit are considered as a qubit, whereas the third level $\ket{2}_i$ is as an ancillary level.
We focus on the problem of the $N$-qubit gate decomposition, and then assume that the initial and the final states of the $N$-qutrit system can be considered an $N$-qubit states, i.e., population of the ancillary states $\ket{2}_{i}$ should be zero.
As basic single-qutrit operations, we employ the following gates:
\begin{equation}
R_{x(y)}^{01}(\varphi)=e^{-\imath \lambda_{1(2)} \varphi}, \quad
R_{x(y)}^{12}(\varphi)=e^{-\imath \lambda_{6(7)} \varphi}.
\end{equation}
Here standard notations of Gell-Mann matrices are used:
\begin{equation}
\begin{aligned}
\lambda_{1}&={\begin{pmatrix}0&1&0\\1&0&0\\0&0&0\end{pmatrix}}, \quad
\lambda_{2}&={\begin{pmatrix}0&-\imath&0\\\imath&0&0\\0&0&0\end{pmatrix}},\\
\lambda_{6}&={\begin{pmatrix}0&0&0\\0&0&1\\0&1&0\end{pmatrix}}, \quad
\lambda_{7}&={\begin{pmatrix}0&0&0\\0&0&-\imath\\0&\imath&0\end{pmatrix}}.
\end{aligned}
\end{equation}
We note that these operations correspond to experimentally accessible transitions in superconducting qutrit-based processors~\cite{Blok2021}.
The $0\to2$ transition, \BLUE{which cannot be accessed directly at least in certain experimental setups~\cite{Blok2021}, can be implemented via the sequence of} intermediate operations:
\begin{equation}
R_{x(y)}^{02}(\varphi)=e^{-\imath(\pi/2)\lambda_6}e^{-\imath(\varphi/2)\lambda_{1(2)}}e^{\imath(\pi/2)\lambda_6}.
\end{equation}
\BLUE{If the $0\to2$ transition is directly available, this provides the further simplification of the proposed scheme.}
As native two-qutrit operations, we consider $\mathtt{iSWAP^{02}(\theta)}$ and $\mathtt{iSWAP^{20}(\theta)}$ gates representing a `standard' $\mathtt{iSWAP}$ operation in the $|11\rangle$-$|02\rangle$ and $|11\rangle$-$|20\rangle$ subspaces,
respectively, with an additional off-diagonal controllable phase \cite{Hill2021}:
\begin{equation}
\begin{aligned}
\ensuremath{\mathtt{iSWAP}}^{02}(\theta)\ket{11}&=-\imath e^{-\imath \theta}\ket{02},\\
\ensuremath{\mathtt{iSWAP}}^{02}(\theta)\ket{02}&=-\imath e^{-\imath \theta}\ket{11},\\
\ensuremath{\mathtt{iSWAP}}^{02}(\theta)\ket{xy}&=\ket{xy}, ~\text{for}~ xy \neq 11, 02,
\end{aligned}
\end{equation}
and
\begin{equation}
\begin{aligned}
\ensuremath{\mathtt{iSWAP}}^{20}(\theta)\ket{11}&=-\imath e^{-\imath \theta}\ket{20},\\
\ensuremath{\mathtt{iSWAP}}^{20}(\theta)\ket{20}&=-\imath e^{-\imath \theta}\ket{11},\\
\ensuremath{\mathtt{iSWAP}}^{20}(\theta)\ket{xy}&=\ket{xy}, ~\text{for}~ xy \neq 11, 20.
\end{aligned}
\end{equation}
Such quantum gates are native for superconducting quantum computing platforms~\cite{Blok2021,Oliver2019,Hill2021}.
\begin{table*}
\caption{
Comparison of qudit-based \ensuremath{\mathtt{C}^{N-1}\mathtt{Z}}~ (\ensuremath{\mathtt{C}^{N-1}\mathtt{X}}~) gate decompositions.
$^{*}$The logarithmic scale of the decomposition depth is achieved for the optimal structure of the tree $\widetilde{E}$, that can be chosen i.e. within all-to-all coupling map.
$^{**}$Here $d_i$ is a dimension of a particular qudit, and $k_i$ is a number of other qudits connected to this qudit within decomposition.
$^{***}$Due to the specificity of the decomposition, the exact number of two-qudit gates is given for the case $N=2^\alpha$, $\alpha\geq 2$.
$^{****}$The number is given for the case of no measurement-based feed-forward operations.
}
\begin{ruledtabular}
\begin{tabular}{ccccccc}
&This work & Ref.~\cite{Gu2021} &Ref.~\cite{Gokhale2019}& Ref.~\cite{Kiktenko2020} & Refs.~\cite{White2009,Ralph2007} &Ref.~\cite{Ionicioiu2019} \\ \hline
Depth & {$\bm{\mathcal{O}(\log N)}^*$ }& $\mathcal{O}(N)$ & $\bm{\mathcal{O}(\log N)}$ & $\bm{\mathcal{O}(\log N)}^*$& $\mathcal{O}(N)$ & $\mathcal{O}(N)$\\
\multirow{2}{*}{Qudit type} & \multirow{2}{*}{\bf qutrits} & \multirow{2}{*}{\bf qutrits} & \multirow{2}{*}{\bf qutrits} & qudits, & single qudit, & single qudit, \\
& & & & ${d_i\geq k_i+1}^{**}$ & $d=N+1$ & $d=N$ \\
\# of ancillas &{\bf0} &{\bf 0} &{\bf 0}&{\bf 0}&{\bf 0}& 1 \\
Coupling map & {\bf arbitrary} &linear &all-to-all & {\bf arbitrary} & star & star\\
\# of 2-body gates & $\bm{2N-3}$ & $\bm{2N-3}$ & ${6N-11}^{***}$ &$\bm{2N-3}$& $\bm{2N-3}$ &$\bm{2N-3}^{****}$\\
\end{tabular}
\end{ruledtabular}\label{tab1}
\end{table*}
To define a coupling map that determines the possibility of applying at least one of the described two-qutrit gates,
we introduce $E$ as a set of ordered pairs $(i, j)$, such that $i, j \in \{1,\dots, N\}, i < j$.
We then suppose that it is possible to execute $\mathtt{iSWAP^{02}(\theta)}$ or $\mathtt{iSWAP^{20}(\theta)}$ between qutrits $i$ and $j$ if and only if $(i,j)\in E$.
We also assume the graph corresponded to the coupling map is connected.
Combinations of native two-qutrit gates together with single-qutrit rotations give us more complex gates.
An illustrative example is an implementation of a standard controlled-phase gate $\ensuremath{\mathtt{CZ}}~$ that can be obtained as sequence of two \ensuremath{\mathtt{iSWAP}}$^{02}(0)$ (or \ensuremath{\mathtt{iSWAP}}$^{20}(0)$) gates:
the \ensuremath{\mathtt{CZ}}~ gate then can be transformed into the \ensuremath{\mathtt{CX}}~ gate by surrounding \ensuremath{\mathtt{CZ}}~ with $R_y^{01}(-\pi/2)$ and $R_y^{01}(\pi/2)$ rotations.
We also introduce an auxiliary gate $U_{i\rightarrow j}$ acting on $i$th and $j$th qudit obtained as a sequence of $R_x^{01}(\pi)$ and \ensuremath{\mathtt{iSWAP}}$^{02}(0)$ [see Fig.~\ref{fig::uab}(a)].
The idea behind of $U_{i\rightarrow j}$ is that it leaves $i$th qutrit in state $\ket{1}_{i}$ only if both qutrits $i$ and $j$ are in the state $\ket{1}$.
Otherwise, the state of $i$th qutrit becomes $\ket{0}_{i}$.
The action of $U_{i\rightarrow j}$ on qubit levels of input qutrits is as follows:
\begin{equation}
\begin{aligned}
U_{i\rightarrow j}\ket{00}_{ij}&=\ket{01}_{ij},\\
U_{i\rightarrow j}\ket{01}_{ij}&=\ket{00}_{ij},\\
U_{i\rightarrow j}\ket{10}_{ij}&=-\imath\ket{02}_{ij},\\
U_{i\rightarrow j}\ket{11}_{ij}&=\ket{10}_{ij}.
\end{aligned}
\end{equation}
We note that $U_{i\rightarrow j}$ can be also obtained from
\ensuremath{\mathtt{iSWAP}}$^{20}$ gate using \ensuremath{\mathtt{iSWAP}}$^{20}$ decomposition shown in Fig.~\ref{fig::uab}(c).
\section{Generalized Toffoli gate decomposition}\label{sec:Toffoli}
The generalized $N$-qubit Toffoli gate $\ensuremath{\mathtt{C}^{N-1}\mathtt{X}}~$ inverts the state of the target qubit if and only if all $N-1$ control qubits are in the state $\ket{1}$, and acts as identity otherwise.
Its implementation can be reduced to surrounding the generalized $N$-qubit controlled-phase gate \ensuremath{\mathtt{C}^{N-1}\mathtt{Z}}~ by $R_y(\pi/2)$ and $R_y(-\pi/2)$ on the target qubit.
We note that \ensuremath{\mathtt{C}^{N-1}\mathtt{Z}}~ has a symmetrical form and is given by
\begin{equation}\label{eq:global_CZ_gate}
\begin{aligned}
&\ensuremath{\mathtt{C}^{N-1}\mathtt{Z}}~\ket{1\ldots 1}_{1\ldots N} = -\ket{1\ldots 1}_{1\ldots N},\\
&\ensuremath{\mathtt{C}^{N-1}\mathtt{Z}}~\ket{x_1\ldots x_N}_{1\ldots N} = \ket{x_1\ldots x_N}_{1\ldots N},
\end{aligned}
\end{equation}
for $\prod_i x_i\neq 1$.
To construct the \ensuremath{\mathtt{C}^{N-1}\mathtt{Z}}~ gate decomposition, we first choose an acyclic connected graph (tree) $\widetilde{E}$ within the coupling graph $E$ by removing, if necessary, some edges from $E$.
All two-qutrit gates that we use in our decomposition are taken from $\widetilde{E}$ only.
We also choose the root of the tree.
As it is shown below, the optimal choice of $\widetilde{E}$ and the root has to provide the minimal height of tree, i.e. the minimal number of edges between the root an the furthest node.
We relabel qutrits according to the choice of $\widetilde{E}$ and the root.
Each node (qutrit) is identified by its `address' in the tree.
We denote the root node as ${\bf 1}$.
The siblings of the node $({\bf s})$ are denoted by ${\bf s|1}, {\bf s|2}, \ldots {\bf s}|n({\bf s})$, where $n({\bf s})$ is a number of siblings of $({\bf s})$.
Our method for the \ensuremath{\mathtt{C}^{N-1}\mathtt{Z}}~~gate decomposition is along the lines of the approach of Ref.~\cite{Kiktenko2020}.
Specifically, we introduce three groups of operations with a tree $\widetilde{E}$, namely: (i) folding operation, (ii) basic operation, and (iii) unfolding operation [see Fig.~\ref{fig::main}(b)].
Each group of operations corresponds to adding gates to the decomposition circuit that is initialized as empty one.
We first apply the folding operation that transforms the original tree $\widetilde{E}$ to a one-level form in which only the root $\bf 1$ and its siblings ${\bf 1|1},\ldots,{\bf 1}|n({\bf 1})$ remain.
To achieve this, we sequentially apply the sequence of so-called elementary folding operations [see Fig.~\ref{fig::main}(a)].
Each elementary folding operation collapses a set of leaves ${\bf s|1},\ldots,{\bf s}|n({\bf s})$ having the same parent ${\bf s}$.
At the same time, we add $U_{\bf s \rightarrow {\bf s}|\ell}$ gate for $\ell={\bf 1},\ldots, n({\bf s})$ to the decomposition circuit [see Fig.~\ref{fig::main}(a)].
Application of $U_{{\bf s} \rightarrow{\bf s}|\ell}$ to the qutrits ${\bf s}$ and ${\bf s}|\ell$ keeps ${\bf s}$ in the state $\ket{1}_{\bf s}$ if and only if ${\bf s}$ and ${\bf s}|\ell$ were initially in the state $\ket{11}_{{\bf s},{\bf s}|\ell}$.
Thus, after elementary folding operation on a subtree ${\bf s}$, ${\bf s|1},\ldots,{\bf s}|n({\bf s)}$,
qutrit ${\bf s}$ is left in the state $\ket{1}_{\bf s}$ if and only if all qutrits ${\bf s}$, ${\bf s|1},\ldots,{\bf s}|n({\bf s)}$ were in the state $\ket{1}$ before its start.
Otherwise, the ${\bf s}$ turns into the state 0 and some of leaves ${\bf s}|\ell$ turn into the ancillary state 2.
We note that after the whole folding procedure the root's siblings ${\bf 1|1,\ldots,1}|n({\bf 1})$ transform into the state $\ket{1\ldots1}_{{\bf 1|1,\ldots,1}|n({\bf 1})}$ if and only if all the qutrits except the root have been initialized in the state $\ket{1}$.
\BLUE{The transformation of initial qubit states of the qutrits after application of the elementary folding operation is presented in the truth table in Fig.~\ref{fig::main}(a)}.
Further, we apply the basic operation that involves only qutrits that correspond to the to the root and its leaves [see Fig.~\ref{fig::main}(b)].
In analogy with the elementary folding operation, the basic operation consists of sequential application of $U_{\bf 1 \rightarrow 1|\ell}$ for $\ell={\bf 1},\ldots n({\bf 1})-1$.
We then apply the \ensuremath{\mathtt{CZ}}~~gate to the root and the last leaf ${\bf 1}|n({\bf 1})$, and repeat the first $n({\bf 1})-1$ operations in reverse order replacing $U_{\bf 1 \rightarrow 1|\ell}$ by $U_{\bf 1 \rightarrow 1|\ell}^\dagger$.
The idea of the basic operation is to acquire a phase factor $-1$ if all qutrits of the level-one subtree are in the state $\ket{1}$.
At the last step, we employ the unfolding operation that is a `mirror reflection' of the folding.
It consists of a sequence of elementary unfolding operations that return the the graph to its original form, and serves for an uncomputation purposes [see Fig. \ref{fig::main}(c)].
As the result, \BLUE{after the application of all three groups of operations,} the state taken from computational basis and belonging to the qubit subspace acquires the phase factor of $-1$ if all qutrits were in the state $\ket{1}$,
and remains the same otherwise.
One can see that by the design of the scheme $2N-3$ two-qutrit gates are required: $2N-4$ $\ensuremath{\mathtt{iSWAP}}$ gates (or its reversals) and single \ensuremath{\mathtt{CZ}}~ gate.
We note that (un)folding operations applied to siblings ${\bf{p}|1},\ldots,{\bf{p}}|n({\bf{p}})$ and ${\bf{q}|1},\ldots,{\bf{q}}|n({\bf{q}})$ of different parent nodes ${\bf{p}}$ and $\bf{q}$ can be performed simultaneously.
Therefore, some blocks of the decomposition circuit can be performed in parallel and the decomposition circuit depth depends on the tree height and its structure, more specifically on the number of siblings of each node.
If the tree is a complete $\kappa$-ary tree, where $\kappa$ is fixed parameter, then circuit depth belongs to $\mathcal{O}(\log N)$.
\begin{figure}[ht!]
\center{\includegraphics[width=0.675\linewidth]{fig3.pdf}}
\caption{
Example of the $\mathtt{C}^{5}\mathtt{Z}$ gate decomposition for the existing qutrit-based superconducting quantum processor.
In (a) coupling map of the Aspen-9 architecture~\cite{Hill2021} is shown.
Each incoming arrow indicates a transmon that transitions to the second excited state $\ket{2}$ in the corresponding two-qutrit operation.
Subgraph $\widetilde{E}$ corresponds to the qutrits that are chosen for decomposition of $\mathtt{C}^{5}\mathtt{Z}$ gate.
In (b) the resulting decomposition circuit for the $\mathtt{C}^{5}\mathtt{Z}$ gate is illustrated.}
\label{fig::aspen9}
\end{figure}
We provide the detailed comparison of our approach with other qudit-based \ensuremath{\mathtt{C}^{N-1}\mathtt{Z}}~ (\ensuremath{\mathtt{C}^{N-1}\mathtt{X}}~) gate decompositions in Tab. \ref{tab1}.
One can see that our approach combines the best features of previously proposed schemes in terms of circuit depth, number of ancillas, qudit type (our approach uses qutrits only), coupling map, and the number of required two-qutrit gates.
Our approach allows decomposing the $N$-qubit \ensuremath{\mathtt{C}^{N-1}\mathtt{Z}}~ (\ensuremath{\mathtt{C}^{N-1}\mathtt{X}}~) gate with $2N-3$ two-qutrit gates with arbitrary coupling map.
We also note that in comparison with the method of Ref.~\cite{Kiktenko2020}, our decomposition requires less time for the execution.
This is because we use the \ensuremath{\mathtt{iSWAP}}~ gate as the main operation, whereas the approach introduced in Ref.~\cite{Kiktenko2020} uses the \ensuremath{\mathtt{CZ}}~ gate; the realization of the \ensuremath{\mathtt{CZ}}~ gates requires us to implement the \ensuremath{\mathtt{iSWAP}}~ gate twice.
\section{Application to existing architecture of superconducting qutirt-based processor}\label{sec:Aspen}
We use our method for the decomposition of the $\mathtt{C}^{5}\mathtt{Z}$ gate for the Aspen-9 processor containing 32 qutrits with a honeycomb-like coupling map [see Fig. \ref{fig::aspen9}a].
We note that calibration of this processor is organized in such a way that the for each pair of coupled qutrits, only one type of interaction ($\ensuremath{\mathtt{iSWAP}}^{02}$ or $\ensuremath{\mathtt{iSWAP}}^{20}$) is chosen.
Let us consider 6-qubit $\mathtt{C}^{5}\mathtt{Z}$ decomposition on qutrits with coupling map, which is illustrated in Fig. \ref{fig::aspen9}a.
On the subgraph $\tilde{E}$~ 6-qubit $\mathtt{C}^{5}\mathtt{Z}$ is the implemented with 9 two-qutrit gates, 8 \ensuremath{\mathtt{iSWAP}}~ and 1 \ensuremath{\mathtt{CZ}}~ gate [see Fig. \ref{fig::aspen9}b].
We note that types of \ensuremath{\mathtt{iSWAP}}~gates depends on the chosen nodes/qutrits in $\tilde{E}$.
If in $\tilde{E}$ a parent node ${\bf s}$ has an incoming arrow from its sibling ${\bf s}|\ell$, then $\ensuremath{U_{i\rightarrow j}}~$ gate must be implemented with $\ensuremath{\mathtt{iSWAP}}_{20}$ in its core.
This result is directly applicable for the decomposition of multi-qubit gates of the Aspen-9 processor.
\section{Conclusion}\label{sec:Conclusion}
We have developed a scheme for the generalized $N$-qubit Toffoli gate, which is efficient in terms of the required number of $2N-3$ two-qutrit gates, types of qudits (qutrits only), and coupling between information carriers.
The developed approach can combined with other techniques allowing one to reduce the complexity of the implementation of quantum algorithms by using qudits~\cite{Nikolaeva2021}.
We have used native gates of the superconducting quantum processors for our scheme, which makes it directly applicable for such category of NISQ devices.
We also expect that approach can be extended to the use for other platforms, for example, the trapped ion platform, in which qudits can be efficiently controlled~\cite{Ringbauer2021,Low2020}, and molecules~\cite{Sawant2020,Ruben2018}.
\section*{Acknowledgments}
The work was supported by the Russian Science Foundation Grant No. 19-71-10091.
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2024-02-18T23:40:24.356Z
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2022-04-05T02:32:51.000Z
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{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14535","language":"en","timestamp":1649125971000,"url":"https:\/\/arxiv.org\/abs\/2112.14535","yymm":"2112"}
|
proofpile-arXiv_000-10007
|
{"provenance":"002.jsonl.gz:10008"}
| null | null |
\section{Introduction}\label{S:intro}
Let $f$ be a holomorphic map defined on a neighbourhood of $0$ in the complex plane $\mathbb{C}$, and assume that $0$ is an
\textbf{irrationally indifferent fixed point} of $f$, that is, $f$ is of the form $e^{2\pi i \alpha} z+ O(z^2)$ near $0$ with $\alpha \in \D{R} \setminus \D{Q}$.
It is well-known that such systems exhibit highly non-trivial dynamical behaviour which depends on the
arithmetic properties of $\alpha$, see for instance \cite{Sie42,Brj71,Yoc95,McM98,GrSw03,PZ04,Zha11}.
When the system is unstable near $0$, the local dynamics remains mysterious, even for simple looking non-linear maps such as the quadratic polynomials
$e^{2\pi i \alpha}z+ z^2$.
A powerful tool for the study of irrationally indifferent fixed points is renormalisation.
A \textbf{renormalisation scheme} consists of a class of maps (dynamical systems), and an operator which preserves that class of maps.
The operator assigns a new dynamical system to a given dynamical system, using suitable iterates of the
original system.
A fundamental renormalisation scheme for the systems with an irrationally indifferent fixed point is the
sector renormalisation of Yoccoz, illustrated in \refF{F:renormalisation}.
In this renormalisation, the new system is obtained from considering the return map to a sector landing at $0$, and the sector is
identified with a neighbourhood of $0$ using a change of coordinate.
A remarkable semi-local version of the local Yoccoz renormalisation is built by Inuo and Shishikura in \cite{IS06},
which is defined on a sector uniformly large in size, and hence captures more dynamical information about the original system.
\begin{figure}[h]
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\rput(2.5,3){\small $f$}
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\rput(5.3,-.1){\small return map}
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\rput(8.2,0){\small renormalisation of $f$}
\end{pspicture}
\label{F:renormalisation}
\caption{In the left hand picture,
the (canonically defined) sector landing at $0$ is bounded by a curve landing at $0$, the image of that curve by $f$, and a line segment connecting the end points of those curves.
By glueing the sides of the sector according to $f$, one obtains a Riemann surface, isomorphic to a
punctured disk.
}
\end{figure}
A main technical issue with employing a renormalisation method for irrationally indifferent fixed points is the highly distorting nature of the
changes of coordinates that appear in successive applications of the renormalisation.
One is lead to analysing the local and global distorting behaviours of the successive changes of coordinates
in conjunction with the fine arithmetic features of $\alpha$.
In this paper with build a geometric model for the (semi-local version of the) renormalisation of irrationally
indifferent fixed points.
\smallskip
\begin{thm}[Renormalisation model]\label{T:toy-model-renormalisation}
There exists a class of maps
\[\mathbb{F}=
\big\{\mathbb{T}_\alpha: \mathbb{M}_\alpha \to \mathbb{M}_\alpha \big\}
_{\alpha \in \mathbb{R}\setminus \mathbb{Q}}\]
and a renormalisation operator $\mathcal{R}_m:\mathbb{F} \to \mathbb{F}$ satisfying the following properties:
\begin{itemize}
\item[(i)]for every $\alpha \in \mathbb{R} \setminus \mathbb{Q}$,
$\mathbb{M}_\alpha \subset \mathbb{C}$ is a compact star-like set with $\{0, +1\} \subset \mathbb{M}_\alpha$,
\item[(ii)]for every $\alpha \in \mathbb{R} \setminus \mathbb{Q}$,
$\mathbb{T}_\alpha: \mathbb{M}_\alpha \to \mathbb{M}_\alpha$ is a homeomorphism which
acts as rotation by $2\pi \alpha$ in the tangential direction,
\item[(iii)] for $\alpha \in (-1/2, 1/2) \setminus \mathbb{Q}$,
\[\mathcal{R}_m \left(\mathbb{T}_\alpha: \mathbb{M}_\alpha \to \mathbb{M}_\alpha\right)
= \left(\mathbb{T}_{-1/\alpha}: \mathbb{M}_{-1/\alpha} \to \mathbb{M}_{-1/\alpha}\right),\]
\item[(iv)] for every $\alpha \in \mathbb{R} \setminus \mathbb{Q}$ and every integer $k$ with
$0 \leq k \leq 1/|\alpha|$,
\[\frac{C^{-1}}{1+ \min \{k, |\alpha|^{-1} - k\}} \leq |\mathbb{T}_\alpha\co{k}(+1)| \leq \frac{C}{1+ \min \{k, |\alpha|^{-1}-k\}},\]
for some constant $C$ independent of $k$ and $\alpha$.
\item[(v)] for every $\alpha \in \mathbb{R} \setminus \mathbb{Q}$,
$\mathbb{M}_{\alpha+1}= \mathbb{M}_{\alpha}$, $\mathbb{T}_{\alpha+1}=\mathbb{T}_{\alpha}$,
$\mathbb{M}_{-\alpha}= s(\mathbb{M}_{\alpha})$, and $\mathbb{T}_{-\alpha}=s \circ \mathbb{T}_{\alpha} \circ s$,
where $s$ denotes the complex conjugation map,
\item[(vi)] $\mathbb{M}_\alpha$ depends continuously on $\alpha \in \mathbb{R} \setminus \mathbb{Q}$,
in the Hausdorff topology.
\end{itemize}
\end{thm}
The map $\mathbb{T}_\alpha$ is a topological model for the map $f(z)=e^{2\pi i \alpha}z+ O(z^2)$.
The set $\mathbb{M}_\alpha$ is a topological model for the maximal invariant set of $f$ at $0$ on which $f$ is injective.
The model for the renormalisation, $\mathcal{R}_m$, is defined using the return map of $\mathbb{T}_\alpha$ to a cone of angle $2\pi \alpha$ landing at $0$;
with a change of coordinate which preserves rays landing at 0, while exhibiting the non-linear behaviour of the actual change of coordinate for the sector
renormalisation of $f$.
As in the sector renormalisation, $\mathcal{R}_m$ induces the Gauss map $\alpha \mapsto -1/\alpha$ on the asymptotic rotation numbers at $0$.
The point $+1$ is the largest real number which belongs to $\mathbb{M}_\alpha$.
It plays the role of a certain critical point of $f$, which we will explain in a moment.
The geometry of the orbit of $+1$ under $\mathbb{T}_\alpha$, for one return of the dynamics, is explained in part (iv) of \refT{T:toy-model-renormalisation}.
This form of geometry is ubiquitous for maps with an irrationally indifferent fixed point; it requires a non-zero second derivative at $0$.
For instance, it holds for the quadratic polynomials $e^{2\pi i \alpha} z + z^2$.
By employing the renormalisation scheme $(\mathcal{R}_m, \mathbb{F})$, we build a topological model for the dynamics
near an irrationally indifferent fixed point, as we explain below.
By the classic work of Fatou \cite{Fat19} and Mane \cite{Man87}, when $f$ is a polynomial or a rational function of
the Riemann sphere, there is a recurrent critical point of $f$ which ``interacts'' with the fixed point at $0$.
The topological boundary of $\mathbb{M}_\alpha$,
\[\mathbb{A}_\alpha= \partial \mathbb{M}_\alpha,\]
is equal to the closure of the orbit of $+1$ for the iterates of $\mathbb{T}_\alpha$.
The set $\mathbb{A}_\alpha$ is a topological model for the closure of the orbit of that recurrent critical point of $f$,
which is the measure theoretic attractor of $f$ for the orbits that remain near $0$.
We explain the topology of the sets $\mathbb{A}_\alpha$ in terms of the arithmetic nature of $\alpha$.
\smallskip
\begin{thm}[Trichotomy of the maximal invariant set]\label{T:model-trichotomy-thm}
For every $\alpha \in \mathbb{R} \setminus \mathbb{Q}$ one of the following statements hold:
\begin{itemize}
\item[(i)] $\alpha$ is a Herman number and $\mathbb{A}_\alpha$ is a Jordan curve,
\item[(ii)] $\alpha$ is a Brjuno but not a Herman number, and $\mathbb{A}_\alpha$ is a one-sided hairy Jordan curve,
\item[(iii)] $\alpha$ is not a Brjuno number, and $\mathbb{A}_\alpha$ is a Cantor bouquet.
\end{itemize}
\end{thm}
\begin{figure}[t]
\begin{pspicture}(14,5)
\epsfxsize=4cm
\rput(2.5,2.5){\epsfbox{Figs/Jordan.eps}}
\rput(7,2.5){\epsfbox{Figs/HC.eps}}
\rput(11.5,2.5){\epsfbox{Figs/bouquet.eps}}
\end{pspicture}
\caption{Computer simulations for the three topologies in \refT{T:model-trichotomy-thm}; from left to right, a Jordan curve, a hairy Jordan curve, and a Cantor bouquet.}
\end{figure}
The Brjuno type was discovered by Siegel-Brjuno \cite{Sie42,Brj71} in their now classic work on the study of the linearisation problem for
irrationally indifferent fixed points.
It is the set of $\alpha$ such that the denominators $q_n$ of the best rational approximants of $\alpha$ satisfy
\[\textstyle{\sum}_{n=1}^\infty q_n^{-1} \log q_{n+1} < +\infty.\]
The arithmetic class of Herman was discovered by Herman-Yoccoz \cite{Her79,Yoc02}
in their landmark study of the analytic linearisation problem for analytic diffeomorphisms of the circle.
The set of Herman numbers is more complicated to characterise in terms of the arithmetic of $\alpha$,
see Section~\ref{S:arithmetic}. It forms an $F_{\sigma,\delta}$ subset of $\D{R}$.
However, any Herman number is a Brjuno number, the set of Herman numbers has full measure in $\D{R}$,
while the set of num-Brjuno numbers is topologically generic in $\D{R}$.
The arguments presented in this paper do not rely on the optimality properties of the Brjuno and Herman
numbers for the linearisation problems.
These arithmetic conditions naturally emerge in the study of the model $\mathbb{A}_\alpha$.
A Cantor bouquet is a compact subset of $\mathbb{C}$ which has empty interior, consists of a collection of Jordan arcs only meeting at $0$,
and every arc is accumulated from both sides by arcs.
A (one-sided) hairy Jordan curve is a similar object consisting of a collection of Jordan arcs all attached to (one side of) a Jordan arc.
See \refS{SS:CB-HJC} for the precise definitions of these objects.
While one-sided hairy Jordan curves and Cantor bouquets are universal topological objects,
our construction is geometric, featuring delicate metric properties.
That is, the models for dissimilar values of non-Brjuno numbers, and also for dissimilar values of Brjuno but non-Herman numbers,
have very distinct metric properties.
The maps $\mathbb{T}_\alpha: \mathbb{A}_\alpha \to \mathbb{A}_\alpha$ exhibit peculiar dynamical behaviour, especially
when $\alpha$ is not a Herman number.
Every point in $\mathbb{A}_\alpha$ is topologically recurrent.
There are points in $\mathbb{A}_\alpha$ with dense orbits.
There are uncountably many distinct closed fully invariant sets.
In contrast to the linearisable examples where an invariant set consists of disjoint unions of closed analytic curves, here there is only a
one-parameter family of closed invariant sets.
The set of accumulation points of the orbit of a point $z\in \mathbb{A}_\alpha$, denoted by $\omega(z)$, is a notable example of a closed invariant set.
It turns out that these are the only ones, as we state below.
Define $r_\alpha \in [0,1]$ according to
\[[r_\alpha, 1]=\{ z \in \mathbb{A}_\alpha \mid \Im z=0 , \Re z \geq 0\}.\]
\begin{thm}[Topological dynamics]\label{T:dynamics-on-model}
For every $\alpha \in \mathbb{R} \setminus \mathbb{Q}$ the map
$\mathbb{T}_\alpha: \mathbb{A}_\alpha \to \mathbb{A}_\alpha$ satisfies the following properties:
\begin{itemize}
\item[(i)] the map $\mathbb{T}_\alpha: \mathbb{A}_\alpha \to \mathbb{A}_\alpha$ is topologically recurrent.
\item[(ii)] the map
\[\omega: [r_\alpha, 1] \to \{X \subseteq \mathbb{A}_\alpha \mid X \text{ is non-empty, closed and invariant}\}\]
is a homeomorphism with respect to the Hausdorff metric on the range.
\item[(iii)] the map $\omega$ on $[r_\alpha, 1]$ is strictly increasing with respect to
the inclusion.
\item[(iv)]if $\alpha$ is a Brjuno number, for every $t \in (r_\alpha, 1]$, $\omega(t)$ is a hairy Jordan curve.
\item[(v)] if $\alpha$ is not a Brjuno number, for every $t \in (r_\alpha, 1]$, $\omega(t)$ is a Cantor bouquet.
\end{itemize}
\end{thm}
\begin{comment}
\begin{thm}[Measurable dynamics]
For every $\alpha \in \mathbb{R} \setminus \mathbb{Q}$,
$\mathbb{T}_\alpha : \mathbb{A}_\alpha \to \mathbb{A}_\alpha$ is uniquely ergodic.
Moreover,
\begin{itemize}
\item if $\alpha$ is a non-Brjuno number, the unique invariant probability measure on $\mathbb{A}_\alpha$ is the
delta-Dirac measure at $0$,
\item if $\alpha$ is a Brjuno number, the unique invariant probability measure on $\mathbb{A}_\alpha$ is the radial
projection of the Lebesgue measure on $\mathbb{A}_\alpha$.
\end{itemize}
\end{thm}
\end{comment}
Based on the above results for the model, we propose the following conjecture.
\begin{conj}[Trichotomy of the irrationally indifferent attractors]\label{C:trichotomy-main-conj}
Let $f$ be a rational function of the Riemann sphere $\hat{\mathbb{C}}$ of degree at least $2$, with
$f(0)=0$, $f'(0)=e^{2\pi i \alpha}$, and $\alpha \in \D{R} \setminus \D{Q}$.
There exists a critical point of $f$, such that the closure of its orbit, denoted by $\Lambda(f)$, satisfies the following:
\begin{itemize}
\item[(i)] if $\alpha$ is a Herman number, $\Lambda(f)$ is a Jordan curve,
\item[(ii)] if $\alpha$ is a Brjuno but not a Herman number, $\Lambda(f)$ is a one-sided hairy Jordan curve,
\item[(iii)] if $\alpha$ is not a Brjuno number, $\Lambda(f)$ is a Cantor bouquet.
\end{itemize}
Moreover, in cases (i) and (ii), $f$ preserves the connected component of $\hat{\D{C}} \setminus \Lambda(f)$
containing $0$, and in case (iii), $\Lambda(f)$ contains $0$.
\end{conj}
The above conjecture immediately implies a number of important conjectures in the study of irrationally indifferent
fixed points for rational functions.
Notably, it implies the optimality of the Brjuno condition for the linearisability of irrationally indifferent fixed
points (Douady-Yoccoz conjecture 1986), the optimality of the Herman condition for the presence of a critical
point on the boundary of the Siegel disk (Herman conjecture 1985),
the Siegel disk are Jordan domains (Douady-Sullivan conjecture 1987).
In a counterpart paper \cite{Che17}, we employ the topological model in \refT{T:toy-model-renormalisation},
and the near-parabolic renormalisation scheme of Inou and Shishikura \cite{IS06},
to prove the above conjecture for a class of maps and rotation numbers.
In order to explain this relation, we briefly outline the construction of the model.
Here we take an upside-down approach to the renormalisation.
In contrast to defining renormalisation for a given map defined on a domain, we start by building
the changes of coordinates for the renormalisation, one for each rotation number.
Repeatedly applying the Gauss map to a fixed $\alpha$, one obtains a sequence of rotation numbers,
and the corresponding changes of coordinates.
Then, there is a maximal set on which the infinite chain of the changes of coordinates is defined.
Finally we build the map $\mathbb{T}_\alpha$ on the maximal set so that its return map via the change of
coordinate becomes $\mathbb{T}_{G(\alpha)}$.
The advantage of this approach is that any given renormalisation scheme for irrationally indifferent fixed
points may be compared to the model $(\mathcal{R}_m, \mathbb{F})$ by comparing the corresponding
changes of coordinates.
This provides a streamlined approach to the study of the dynamics of irrationally indifferent fixed points.
It is also unified, in the sense that one does not need to reconsider the role of the arithmetic properties in
a given renormalisation scheme.
In the counterpart paper \cite{Che17}, we show that if a renormalisation scheme consists of a change of
coordinate which is sufficiently close to the change of coordinate for the renormalisation model,
then the corresponding maps are topologically conjugate on the corresponding maximal domain of
renormalisations.
See also \cite{ShY18} for partial progress towards the above conjecture.
An alternative construction for $\mathbb{M}_\alpha$ was suggested by Buff and Ch\'eritat in 2009 \cite{BC09}.
Our thoughts in this direction was influenced and motivated by their construction based on employing toy models for the changes of coordinates.
In contrast to the conformal changes of coordinates considered by those authors, the models for the changes of coordinates
presented here are not conformal, but preserve straight rays landing at $0$, while maintaining the correct nonlinear behaviour in the radial directions.
This flexibility allows us to incorporate some remarkable functional relations, which in turn allow us to avoid taking Hausdorff limits in the construction.
The explicit construction presented here allow us to promote the models for $\mathbb{M}_\alpha$ to build the maps $\mathbb{T}_\alpha$, and a renormalisation
scheme on those maps.
\subsection*{Acknowledgement}
I would like to thank Xavier Buff and Arnaud Cheritat for sharing with me the manuscript \cite{BC09},
and also for many useful discussions around the topic during the author's visit to Univeriste Paul Sabatier
in 2012.
I am also grateful to EPSRC (UK) for the five-year fellowship EP/M01746X/1, which allowed carrying
out this project.
\section{The changes of coordinates for the renormalisation \texorpdfstring{$\mathcal{R}_m$}{Rm}}\label{S:change-coordinates}
In this section we introduce a one-parameter family of real analytic diffeomorphisms.
When we set the parameter as a rotation number, the diffeomorphism becomes the change of coordinate for the
renormalisation of a map with that asymptotic rotation number at $0$.
These are the toy models for the change of coordinates in the toy model for the renormalisation scheme.
\subsection{Explicit formula for the changes of coordinates}\label{SS:coordinate-core}
Consider the set
\[\D{H}'= \{w\in \D{C} \mid \Im w > -1 \}.\]
For $r \in (0,1/2]$, we define the map $Y_r : \overline{\D{H}'} \to \D{C}$
as \footnote{$\overline{X}$ denotes the topological closure of a given set $X$.}
\[Y_r(w)= r\Re w +
\frac{ i }{2\pi} \log \Big |\frac{e^{-3\pi r}- e^{-\pi r i }e^{-2\pi r i w}}{e^{-3\pi r}- e^{\pi r i }}\Big|.\]
Evidently, $Y_r$ maps a vertical line in $\D{H}'$ to a vertical line.
Also, one can see that $Y_r(0)=0$ and $Y_r(1/(2r)-i)$ is uniformly close to $1/2+ i (\log 1/r)/(2\pi)$.
\refF{F:model-Y_r} shows the behaviour of $Y_r$ on horizontal and vertical lines.
\begin{figure}[ht]
\includegraphics[width=0.8\textwidth]{Figs/Y-images.ps}
\caption{The curves in black are the images of the horizontal lines $y=-1, 0, 1, \dots, 8$ under $Y_r$.
The vertical lines in blue, from left to right, are the images of the vertical lines $\Re w=-1$, $\Re w=0$,
and $\Re w=1/\alpha$, under $Y_r$. Here, $r= 1/(10+1/(1+1/(1+1/(1+\dots))))$.}
\label{F:model-Y_r}
\end{figure}
\begin{lem}\label{L:Y-domain}
For every $r \in (0, 1/2]$, $Y_r: \overline{\D{H}'} \to \D{H}'$ is well-defined, real-analytic in $\Re w$ and $\Im w$, and is
injective.
Moreover, for every $r \in (0, 1/2)$ and every $y \geq 0$, we have
\[H_r \left (\{w\in \D{C} \mid \Im w \geq y-1 \}\right ) \subset \{w\in \D{C} \mid \Im w \geq \Im H_r(i y)-1\}.\]
\end{lem}
\begin{proof}
The proof is based on some elementary calculations. Note that
\[10 \leq 1+ (12/5)+ (12/5)^2/2!+ \dots + (12/5)^5/5! \leq e^{12/5}.\]
Using $0 \leq r \leq 1/2$, $3\leq \pi \leq 4$, and the above inequality, we obtain
\[\log (4+ 3\pi r) + 2\pi r \leq \log(e^{12/5})+ \pi \leq \log e^{4\pi/5} +\pi \leq 9\pi/5.\]
This implies that for all $r\in [0, 1/2]$, $e^{9\pi/5} \geq (4+3\pi r)e^{2\pi r}$, and therefore,
\[\pi e^{\pi r} (e^{9\pi/5} - (4+3\pi r)e^{2\pi r})) \geq 0.\]
Fix $r\in [0,1/2]$. Integrating the above inequality from $0$ to $r$, we conclude that
\[e^{9\pi/5} (e^{\pi r} - 1) -(\pi r+1)e^{3\pi r} +1 \geq 0.\]
This implies that for all $r\in (0,1/2]$, we have
\begin{equation}\label{E:proof-Y_r-basic-1}
\frac{e^{\pi r} -1}{(\pi r+1)e^{3\pi r} -1} \geq e^{-9\pi/5}.
\end{equation}
On the other hand, for $w\in \overline{\D{H}'}$, by the triangle inequality,
\[|e^{-3\pi r} - e^{-\pi r i }e^{-2\pi r i w} | \geq |e^{-\pi r i }e^{-2\pi r i w} | - |e^{-3\pi r}|
\geq e^{-2\pi r} - e^{-3\pi r},\]
and
\[|e^{-3\pi r}- e^{\pi r i }| \leq |e^{-3\pi r}- 1| + |1-e^{\pi r i }| \leq (1- e^{-3\pi r}) + \pi r.\]
Combining the above two inequalities with \refE{E:proof-Y_r-basic-1} we obtain
\[\left| \frac{e^{-3\pi r}- e^{-\pi r i }e^{-2\pi r i w}}{e^{-3\pi r}- e^{\pi r i }} \right|
\geq \frac{e^{-2\pi r} - e^{-3\pi r}}{\pi r + 1- e^{-3\pi r}}= \frac{e^{\pi r} -1}{(\pi r+1)e^{3\pi r} -1}
\geq e^{-9\pi/5}.\]
The above inequality implies that for all $w\in \D{H}'$ we have
\[\Im Y_r(w) \geq \frac{1}{2\pi} \log e^{-9\pi/5} > -1.\]
In particular, $Y_r$ is well-defined, and maps $\overline{\D{H}'}$ into $\D{H}'$. This also implies that
$Y_r$ is real-analytic in $\Re w$ and $\Im w$, for $w$ in $\overline{\D{H}'}$.
To prove that $Y_r$ is injective, assume that $w_1$ and $w_2$ are two distinct points in $\overline{\D{H}'}$.
If $\Re w_1 \neq \Re w_2$, then $\Re Y_r(w_1) \neq \Re Y_r(w_2)$.
If $\Re w_1 = \Re w_2$ but $\Im w_1 \neq \Im w_2$, then
\[|e^{-3\pi r} - e^{-\pi r i } e^{-2\pi r i w_1}| \neq |e^{-3\pi r} - e^{-\pi r i } e^{-2\pi r i w_2}|.\]
This implies that $\Im Y_r(w_1) \neq \Im Y_r(w_2)$.
One may employ similar elementary calculations to derive the latter part of the lemma.
\end{proof}
\subsection{Uniform contraction of the changes of coordinates}
A key property of the maps $Y_r$ is stated in the following lemma.
\begin{lem}\label{L:uniform-contraction-Y_r}
For every $ r\in (0,1/2]$, and every $w_1, w_2$ in $\overline{\D{H}'}$, we have
\[|Y_r(w_1)- Y_r(w_2)| \leq 0.9 |w_1-w_2|.\]
\end{lem}
The precise contraction factor $0.9$ in the above lemma is not crucial, any constant strictly less than $1$ suffices.
\begin{proof}
Let $g(w)= (e^{-3\pi r}- e^{-\pi r i } e^{-2\pi r i w})(e^{-3\pi r}- e^{\pi r i } e^{2\pi r i \overline{w}})$.
Then, $g(w)$ is of the form $\zeta \overline{\zeta}$, for some $\zeta\in \D{C}$, and hence it produces positive real values for
$w\in \overline{\D{H}'}$.
We have
\[\partial g(w)/\partial w
= 2\pi r i e^{-\pi r i } e^{-2\pi r i w} (e^{-3\pi r}- e^{\pi r i } e^{2\pi r i \overline{w}}),\]
and
\[\partial g(w)/\partial \overline{w}
= -2 \pi r i e^{\pi r i } e^{2\pi r i \overline{w}} (e^{-3\pi r}- e^{-\pi r i } e^{-2\pi r i w}) .\]
Therefore, by the complex chain rule,
\[\frac{\partial }{\partial w}\big(\log g(w)\big) = \frac{1}{g(w)} \frac{\partial g}{\partial w}
= \frac{2\pi r i e^{-\pi r i } e^{-2\pi r i w}}{e^{-3\pi r}- e^{-\pi r i } e^{-2\pi r i w}}
=\frac{2\pi r i }{e^{-3\pi r} e^{\pi r i } e^{2\pi r i w} - 1},\]
and
\[\frac{\partial}{\partial \overline{w}}\big(\log g(w)\big) = \frac{1}{g(w)} \frac{\partial g}{\partial \overline{w}}
= \frac{-2 \pi r i e^{\pi r i } e^{2\pi r i \overline{w}}}{e^{-3\pi r}- e^{\pi r i } e^{2\pi r i \overline{w}}}
=\frac{-2 \pi r i }{e^{-3\pi r} e^{-\pi r i } e^{-2\pi r i \overline{w}}-1}.\]
We rewrite $Y_r$ in the following form
\[Y_r(w)
=r \cdot \frac{w+\overline{w}}{2} + \frac{ i }{2\pi} \cdot \frac{1}{2} \log g(w) -
\frac{ i }{2\pi} \log |e^{-3\pi r}-e^{\pi r i }|.\]
Then, by the above calculations,
\[\frac{\partial Y_r}{\partial w}(w)
= \frac{r}{2} + \frac{ i }{4\pi} \cdot \frac{2\pi r i }{e^{-3\pi r} e^{\pi r i } e^{2\pi r i w} - 1}
= \frac{r}{2} \Big (1 - \frac{1}{e^{-3\pi r} e^{\pi r i } e^{2\pi r i w} - 1}\Big ),\]
and
\[\frac{\partial Y_r}{\partial \overline{w}}(w)
=\frac{r}{2} + \frac{ i }{4\pi} \cdot \frac{-2 \pi r i }{e^{-3\pi r} e^{-\pi r i } e^{-2\pi r i \overline{w}}-1}
= \frac{r}{2} \Big (1+ \frac{1}{e^{-3\pi r} e^{-\pi r i } e^{-2\pi r i \overline{w}}-1}\Big).\]
Let $\xi=e^{-3\pi r} e^{\pi r i } e^{2\pi r i w}$. For $w\in \overline{\D{H}'}$, $|\xi| \leq e^{-\pi r}$.
For the maximum size of the directional derivative of $Y_r$ we have
\begin{align*}
\max_{\theta \in [0,2\pi]} \big | \operatorname{D} Y_r (w) \cdot e^{ i \theta}\big|
&= \Big |\frac{\partial Y_r}{\partial w}(w) \Big | + \Big | \frac{\partial Y_r}{\partial \overline{w}}(w) \Big | \\
&\leq \frac{r}{2} \cdot \Big |1-\frac{1}{\xi -1}\Big | + \frac{r}{2} \cdot \Big |1+ \frac{1}{\overline{\xi}-1}\Big | \\
&\leq \frac{r}{2} \cdot \frac{2+ e^{-\pi r}}{1-e^{-\pi r}} + \frac{r}{2} \cdot \frac{e^{-\pi r}}{1-e^{-\pi r}}
=r \cdot \frac{e^{\pi r}+1 }{e^{\pi r}-1}.
\end{align*}
For $r \geq 0$, $e^{\pi r}-1 \geq \pi r + \pi ^2 r^2/2$, (the first two terms of the Taylor series with positive terms).
This gives us
\[r \cdot \frac{e^{\pi r}+1 }{e^{\pi r}-1} = r\left (1+ \frac{2}{e^{\pi r}-1}\right )
\leq r\left (1+ \frac{2}{\pi r + \pi ^2 r^2/2}\right)
= \frac{2\pi r + \pi^2 r^2+4}{2\pi + \pi^2 r}.\]
The last function in the above equation is increasing on $(0, 1/2)$,
because it has a non-negative derivative $(4\pi r+\pi^2 r^2)/(2+\pi r)^2$.
Then, it is bounded by its value at $1/2$, which, using $\pi \geq 3$, gives us
\[\frac{2\pi r + \pi^2 r^2+4}{2\pi + \pi^2 r}
\leq \frac{\pi +\pi^2/4+4}{2\pi+ \pi^2/2}
= \frac{1}{2}+ \frac{4}{2\pi+\pi^2/2}
\leq \frac{1}{2} + \frac{4}{6+4} = \frac{9}{10}.
\qedhere\]
\end{proof}
\subsection{Remarkable functional relations for the changes of coordinates}
The maps $Y_r$ satisfy two crucial functional relations, one at a large scale, and one at a small scale,
both of which are illustrated in \refF{F:model-Y_r}. We present these properties in the following lemma.
\begin{lem}\label{L:Y_r-commutation}
For every $r\in (0,1/2]$, we have
\begin{itemize}
\item[(i)] for every $w\in \overline{\D{H}'}$,
\[Y_r(w+1/r) = Y_r(w)+1\]
\item[(ii)] for every $t \geq -1$,
\[Y_r( i t+ 1/r-1)= Y_r( i t)+ 1- r.\]
\end{itemize}
\end{lem}
\begin{proof}
Part (i) of the lemma readily follows from the formula defining $Y_r$.
To prove part (ii) of the lemma, first note that
\begin{align*}
\big |e^{-3\pi r}- e^{-\pi r i } e^{-2\pi r i(it+1/r -1)}\big|
& =\big |e^{-3\pi r}- e^{-\pi r i } e^{2\pi r t} e^{2\pi r i} \big| \\
&= \big| e^{-3\pi r}- e^{\pi r i } e^{2\pi r t} \big| =\big |e^{-3\pi r}- e^{-\pi r i } e^{2\pi r t}\big|.
\end{align*}
Above, the first and second ``$=$'' are simple multiplications of complex numbers, while for the third ``$=$'' we have
used that $|x-z|=|x-\overline{z}|$, for real numbers $x$ and complex numbers $z$. Thus,
\begin{equation*}
Y_r( i t+ 1/r-1)=
r (1/r-1)+ \frac{i}{2\pi}\log \Big |\frac{e^{-3\pi r}-e^{-\pi r i } e^{2\pi r t}}{e^{-3\pi r}- e^{\pi r i }}\Big|
=(1-r) + Y_{r}( i t). \qedhere
\end{equation*}
\end{proof}
\section{The sets \texorpdfstring{$\mathbb{M}_\alpha$}{M-ga}}\label{S:M-ga}
In this section we build the sets $\mathbb{M}_\alpha$ introduced in \refS{S:intro}.
These sets are defined by successively applying the changes of coordinates $Y_r$, for a sequence of parameters $r$.
\subsection{Successive rotation numbers \texorpdfstring{$\alpha_n$}{ga-n} and signs \texorpdfstring{$\varepsilon_n$}{epsilon-n}}\label{SS:modified-fractions-mini}
For $x\in \D{R}$, define $d(x, \D{Z})= \min_{k\in \D{Z}} |x-k|$. Let us fix an irrational number $\alpha \in \D{R}$.
Define the numbers $\alpha_n\in (0,1/2)$, for $n\geq 0$, according to
\begin{equation} \label{E:rotations-rest}
\alpha_0=d(\alpha, \D{Z}), \quad \alpha_{n+1}=d(1/\alpha_n, \D{Z}).
\end{equation}
Then, there are unique integers $a_n$, for $n\geq -1$, and $\varepsilon_n \in \{+1, -1\}$, for $n\geq 0$, such that
\begin{equation}\label{E:rotations-relations}
\alpha= a_{-1}+ \varepsilon_0 \alpha_0, \quad 1/\alpha_n= a_n + \varepsilon_{n+1} \alpha_{n+1}.
\end{equation}
Evidently, for all $n\geq 0$,
\begin{equation}\label{E:rotations-signs-epsilon-n}
1/\alpha_n \in (a_n-1/2, a_n+1/2), \quad a_n\geq 2,
\end{equation}
and
\begin{equation}\label{E:signs-introduced}
\varepsilon_{n+1}=
\begin{cases}
+1 & \text{if } 1/\alpha_n \in (a_n, a_n+1/2), \\
-1 & \text{if } 1/\alpha_n \in (a_n-1/2, a_n).
\end{cases}
\end{equation}
To unify the upcoming notations, we set $\alpha_{-1}=+1$.
\subsection{Successive changes of coordinates}
Recall that $s(w)=\overline{w}$ denote the complex conjugation.
For $n\geq0$ we define
\begin{equation}\label{E:Y_n}
Y_n(w) =
\begin{cases}
Y_{\alpha_n}(w) & \text{if } \varepsilon_n=-1\\
- s\circ Y_{\alpha_n}(w) & \text{if } \varepsilon_n=+1.
\end{cases}
\end{equation}
Each $Y_n$ is either orientation preserving or reversing, depending on the sing of $\varepsilon_n$.
For $n\geq 0$, we have \footnote{We define $ i X= \{ i x \mid x\in X \}$, for a given set $X \subseteq \D{C}$.}
\begin{equation} \label{E:invariant-imaginary-line}
Y_n( i [-1, +\infty)) \subset i (-1, +\infty), \quad \text{ and } \quad Y_n(0)=0.
\end{equation}
\refL{L:uniform-contraction-Y_r} implies that for all $n\geq 0$ and all $w_1, w_2$ in $\overline{\D{H}'}$, we have
\begin{equation}\label{E:uniform-contraction-Y}
|Y_n(w_1)- Y_n(w_2)| \leq 0.9 |w_1-w_2|.
\end{equation}
It follows from \refL{L:Y_r-commutation} that for all $n\geq 0$ and all $w\in \overline{\D{H}'}$,
\begin{equation} \label{E:Y_n-comm-1}
Y_n(w+1/\alpha_n) =
\begin{cases}
Y_n(w)+1 & \text{if } \varepsilon_n=-1,\\
Y_n(w)-1 & \text{if } \varepsilon_n=+1.
\end{cases}
\end{equation}
Also, by the same lemma, for all $n\geq 0$, and all $t\geq -1$,
\begin{equation}\label{E:Y_n-comm-2}
Y_n( i t+ 1/\alpha_n-1)=
\begin{cases}
Y_n( i t)+ (1-\alpha_n) & \text{if } \varepsilon_n=-1, \\
Y_n( i t) +(\alpha_n-1) & \text{if } \varepsilon_n=+1.
\end{cases}
\end{equation}
\subsection{Equivariant tiling of the tower}\label{SS:tilings-nest}
For $n\geq 0$ let
\begin{equation}\label{E:I_n-J_n-K_n}
\begin{gathered}
I_n^0 = \{w\in \overline{\D{H}'} \mid \Re w\in [0, 1/\alpha_n]\}, \\
J_n^0 = \{w\in I_n^0 \mid \Re w \in [1/\alpha_n-1, 1/\alpha_n]\}, \\
K_n^0 = \{w\in I_n^0 \mid \Re w \in [0, 1/\alpha_n-1] \}.
\end{gathered}
\end{equation}
We inductively defined the sets $I_n^j$, $J_n^j$, and $K_n^j$, for $j \geq 1$ and $n\geq 0$.
Assume that $I_n^j$, $J_n^j$, and $K_n^j$ are defined for some $j$ and all $n \geq 0$.
We define these sets for $j+1$ and all $n\geq 0$ as follows.
Fix an arbitrary $n \geq 0$.
If $\varepsilon_{n+1}=-1$, let
\begin{equation}\label{E:I_n^j--1}
I_n^{j+1}
= \bigcup_{l=0}^{a_n-2} \big( Y_{n+1} ( I_{n+1}^j)+ l \big) \bigcup \big( Y_{n+1}(K_{n+1}^j)+ a_n-1\big).
\end{equation}
If $\varepsilon_{n+1}=+1$, let
\begin{equation}\label{E:I_n^j-+1}
I_n^{j+1}
= \bigcup_{l=1}^{a_n} \big( Y_{n+1} ( I_{n+1}^j)+ l \big) \bigcup \big(Y_{n+1}(J_{n+1}^j)+ a_n+1\big ).
\end{equation}
Regardless of the sign of $\varepsilon_{n+1}$, define
\[J_n^{j+1} = \{w\in I_n^{j+1} \mid \Re w \in [1/\alpha_n-1, 1/\alpha_n]\},\]
\[K_n^{j+1} = \{w\in I_n^{j+1} \mid \Re w \in [0, 1/\alpha_n-1] \}.\]
\refF{F:topological-model} presents two generations of these domains.
We summarise the basic features of these sets in the following lemma.
\begin{comment}
\begin{figure}
\begin{center}
\includegraphics[width=0.8\textwidth]{Figs/partitions.ps}
\end{center}
\caption{The left hand picture is for $\varepsilon_n=-1$ and the right hand picture is for $\varepsilon_n=+1$.
The sets $K_n^0$ and $J_n^0$ are on the lower row, and the set $I_{n-1}^1$ is on the upper row.}
\label{F:topological-model}
\end{figure}
\end{comment}
\begin{figure}[ht]
\begin{pspicture}(14,11)
\epsfxsize=6cm
\rput(3,8){\epsfbox{Figs/neg-epsil.eps}}
\rput(11,8){\epsfbox{Figs/pos-epsil.eps}}
\pspolygon[linecolor=newcyan,fillstyle=solid,fillcolor=newcyan](0,0)(5.4545,0)(5.4545,3)(0,3)
\pspolygon[linecolor=neworange,fillstyle=solid,fillcolor=neworange](5.4545,0)(6,0)(6,3)(5.4545,3)
\rput(3,1.5){\small $K_n^0$}
\rput(5.7,1.5){\small $J_n^0$}
\pspolygon[origin={8,0},linecolor=newcyan,fillstyle=solid,fillcolor=newcyan](0,0)(5.4545,0)(5.4545,3)(0,3)
\pspolygon[origin={8,0},linecolor=neworange,fillstyle=solid,fillcolor=neworange](5.4545,0)(6,0)(6,3)(5.4545,3)
\rput(11,1.5){\small $K_n^0$}
\rput(13.7,1.5){\small $J_n^0$}
\psline[linewidth=.4pt]{->}(0,3.05)(0,5.8)
\psline[linewidth=.4pt]{->}(5.95,3.05)(1.25,5.85)
\psline[linewidth=.4pt]{->}(0.05,3.05)(4.9,5.8)
\psline[linewidth=.4pt]{->}(5.4545,3.05)(6,5.8)
\rput(1,5){\small $Y_n$}
\rput(4.8,5){\small $Y_n+a_{n-1}-1$}
\psline[origin={8,0},linewidth=.4pt]{->}(0,3.05)(1.15,5.95)
\psline[origin={8,0},linewidth=.4pt]{->}(5.95,3.05)(0,5.9)
\psline[origin={8,0},linewidth=.4pt]{->}(5.95,3.05)(5.9,5.9)
\psline[origin={8,0},linewidth=.4pt]{->}(5.4545,3.05)(6,5.9)
\rput(8.5,5){\small $Y_n$}
\rput(12.8,5.5){\small $Y_n+a_{n-1}+1$}
\end{pspicture}
\caption{The left hand picture is for $\varepsilon_n=-1$ and the right hand picture is for $\varepsilon_n=+1$.
The sets $K_n^0$ and $J_n^0$ are on the lower row, and the set $I_{n-1}^1$ is on the upper row.}
\label{F:topological-model}
\end{figure}
\begin{lem}\label{L:I_n^j-basic-features}
For all $n \geq 0$ and $j\geq 0$, the sets $I_n^j$, $J_n^j$, and $K_n^j$ are closed and connected subsets of
$\D{C}$ which are bounded by piece-wise analytic curves.
Moreover,
\begin{itemize}
\item[(i)] $\{\Re w \mid w\in I_n^j\}= [0, 1/\alpha_n]$;
\item[(ii)] if $\varepsilon_{n+1}=-1$, we have
\begin{equation}\label{E:lemma-I_n^j--1}
\begin{gathered}
\{w\in I_n^j \mid \Re w=0\} \subseteq Y_{n+1}( i [-1, \infty)), \\
\{w\in I_n^j \mid \Re w=1/\alpha_n\} \subseteq Y_{n+1}( i [-1, \infty)+ 1/\alpha_{n+1}-1) +a_n-1;
\end{gathered}
\end{equation}
\item[(iii)] if $\varepsilon_{n+1}=+1$, we have
\begin{equation}\label{E:lemma-I_n^j-+1}
\begin{gathered}
\{w\in I_n^j \mid \Re w=0\} \subseteq Y_{n+1}( i [-1, \infty)+1/\alpha_n)+1, \\
\{w\in I_n^j \mid \Re w=1/\alpha_n\} \subseteq Y_{n+1}( i [-1, \infty)+ 1/\alpha_{n+1}-1) +a_n+1.
\end{gathered}
\end{equation}
\end{itemize}
\end{lem}
\begin{proof}
The proof is elementary and is left to the reader. One only needs to follow the basic arithmetic relations among
$\alpha_n$, $a_n$ and $\varepsilon_{n+1}$ in \eqref{E:rotations-rest}-\eqref{E:rotations-signs-epsilon-n},
and use the functional relations in \eqref{E:Y_n-comm-1} and \eqref{E:Y_n-comm-2}.
\end{proof}
\begin{lem}\label{L:model-almost-periodic}
For every $n\geq 0$ and $j\geq 0$, the following hold:
\begin{itemize}
\item[(i)] for all $w$ with $\Re w\in [0, 1/\alpha_n-1]$, $w \in I_n^j$ if and only if $w+1 \in I_n^j$;
\item[(ii)] for real values of $t$, $it \in I_n^j$ if and only if $it +1/\alpha_n \in I_n^j$.
\end{itemize}
\end{lem}
\begin{proof}
We shall prove both parts at once by an inductive argument on $j$.
Clearly, both statements hold for $j=0$ and all $n\geq 0$.
Assume that both parts of the lemma hold for some $j \geq 0$ and all $n\geq 0$.
Part (ii) for $j$ and \refE{E:Y_n-comm-1} together imply that for real values of $t$, $it \in I_n^{j+1}$ if and only
if $it+1 \in I_n^{j+1}$. Then, by the definition of $I_n^{j+1}$ in \eqref{E:I_n^j--1} and \eqref{E:I_n^j-+1},
one concludes part (i) for $j+1$ and all $n\geq 0$.
To prove part (ii) of the lemma for $j+1$ and all $n\geq 0$ we need to consider two cases.
First assume that $\varepsilon_{n+1}=-1$. By \eqref{E:Y_n-comm-2} and \eqref{E:rotations-relations},
for $t$ and $t'$ in $[-1, +\infty)$ satisfying $Y_{n+1}(it')=it$, we have
\[Y_{n+1}(it'+ 1/\alpha_{n+1} - 1)+a_n-1=
Y_{n+1}(it')+ (1-\alpha_{n+ 1}) + a_n - 1=
it + 1/\alpha_n .\]
If $i t \in I_n^{j+1}$, then by \refE{E:I_n^j--1}, there is $it' \in I_{n+1}^j$ with $Y_{n+1}(it')=it$.
By the induction hypotheses (part (i) and (ii) for $j$ and all $n\geq 0$), $it'+ 1/\alpha_{n+1} - 1 \in I_{n+1}^j$.
Then, $it'+ 1/\alpha_{n+1} - 1 \in K_{n+1}^j$. Therefore, by the above equation and \eqref{E:I_n^j--1},
$it+1/\alpha_n \in I_n^{j+1}$.
On the other hand, if $it+1/\alpha_n \in I_n^{j+1}$, then by \eqref{E:I_n^j--1} and the induction hypotheses,
there is $it'\in I_{n+1}^j$ such that $it'+1/\alpha_{n+1} -1 \in K_{n+1}^j$ and
\[Y_{n+1}(it'+1/\alpha_{n+1} -1) + (a_n-1) = it+1/\alpha_n.\]
Again, by \eqref{E:Y_n-comm-2} and \eqref{E:rotations-relations}, this implies that
$it+1/\alpha_n=Y_{n+1}(it')+1/\alpha_n$.
Hence, $it= Y_{n+1}(it')$, which implies that $it\in I_n^{j+1}$.
The proof when $\varepsilon_{n+1}=+1$ is similar, using \eqref{E:I_n^j-+1} and \eqref{E:Y_n-comm-2}.
\end{proof}
Recall that $\alpha_{-1}=+1$. Let $I_{-1}^0=\{w\in \overline{\D{H}'} \mid \Re w \in [0, 1/\alpha_{-1}]\}$, and for $j\geq 1$,
consider the sets
\begin{equation}\label{E:I--1}
I_{-1}^j= Y_0(I_0^{j-1}) + (\varepsilon_0+1)/2.
\end{equation}
By \refL{L:Y-domain}, $I_n^1 \subset I_n^0$, for $n\geq -1$. By an inductive argument, this implies that for all
$n\geq -1$ and all $j\geq 0$,
\begin{equation}\label{E:I_n^j-forms-nest}
I_n^{j+1} \subset I_n^j.
\end{equation}
\subsection{The sets \texorpdfstring{$\mathbb{M}_\alpha$}{M-ga}}\label{SS:M-ga}
For $n\geq -1$, we define
\[I_{n}= \cap_{j\geq 1} I_{n}^j.\]
Each $I_n$ consists of closed half-infinite vertical lines tending to $+ i \infty$. However, $I_n$ may or may not
be connected.
We note that $\Re I_{-1} \subset [0,1]$.
Indeed, by \refL{L:model-almost-periodic}, for real $t$, $it \in I_{-1}$ if and only if $(it +1) \in I_{-1}$.
Thus, we may define
\begin{equation}\label{E:M_ga}
\mathbb{M}_\alpha= \{s(e^{2\pi i w}) \mid w \in I_{-1}\} \cup \{0\}.
\end{equation}
\begin{rem}
There may be alternative (simpler) approaches to build the set $\mathbb{M}_\alpha$ using the maps $Y_n$.
For the sake of applications, here we have selected an approach which closely mimics a construction
in the renormalisation scheme.
However, it is worth noting that one cannot simply define the domains $I_n^j$, and subsequently
$\mathbb{M}_\alpha$, by first extending the maps $Y_n$ $1/\alpha_n$-periodically onto $\D{H}'$,
and then iterating them on $\D{H}'$.
That is because, such a construction would lead to a set in the limit which is periodic under translations
by $+1$ and some irrational number.
It would follow that the interior of that set must be the region above a horizontal line, which cannot be the case for
arbitrary $\alpha$.
\end{rem}
\begin{propo}\label{P:M-ga-relations-1}
For every $\alpha \in \mathbb{R} \setminus \mathbb{Q}$ we have the following:
\begin{itemize}
\item[(i)] $\mathbb{M}_\alpha$ is a compact star-like set about $0$, $+1\in \mathbb{M}_\alpha$,
and $\mathbb{M}_\alpha \cap (1, \infty)=\emptyset$;
\item[(ii)] $\mathbb{M}_{\alpha+1}= \mathbb{M}_\alpha$ and $s(\mathbb{M}_\alpha)= \mathbb{M}_{-\alpha}$.
\end{itemize}
\end{propo}
\begin{proof}
Since every map $Y_n$ sends vertical lines to vertical lines, each of $I_n^j$ is the region above the
graph of a function.
This implies that for every $n\geq -1$, $I_n$ consists of some half-infinite vertical lines.
Thus, $\mathbb{M}_\alpha$ is star-like about $0$.
Also, $\mathbb{M}_\alpha$ is bounded and closed, hence a compact set.
On the other hand, for all $n\geq -1$, $0\in I_n$, which implies that $+1 \in \mathbb{M}_\alpha$.
Indeed, by the uniform contraction of the maps $Y_n$, \refE{E:uniform-contraction-Y}, $+1$ is an
end point of a ray in $\mathbb{M}_\alpha$, that is, for every $\delta>0$, $1+\delta \notin \mathbb{M}_\alpha$.
Recall that in order to define $\mathbb{M}_\alpha$, which only depends on $\alpha$,
we first define the sequences $(\alpha_n)_{n\geq 0}$, $(a_n)_{n\geq 0}$, and $(\varepsilon_n)_{n\geq 0}$.
These fully determine the sequence of maps $(Y_n)_{n\geq 0}$, and hence the set $\mathbb{M}_\alpha$.
The irrational numbers $\alpha$ and $\alpha+1$ produce the same sequences $(\alpha_n)_{n\geq 0}$, $(a_n)_{n\geq 0}$
and $(\varepsilon_n)_{n\geq 0}$.
Therefore, the sequence of the maps $Y_n$ are the same for both $\alpha$ and $\alpha+1$.
This implies that $\mathbb{M}_{\alpha+1}= \mathbb{M}_\alpha$.
To determine $\mathbb{M}_{-\alpha}$, we need to compare the corresponding sequences $(\alpha_n)_{n\geq 0}$,
$(a_n)_{n\geq 0}$ and $(\varepsilon_n)_{n\geq 0}$ for $\alpha$ and $-\alpha$.
Let us denote the corresponding objects for $-\alpha$ using the same notations as the ones for $\alpha$ but with a prime,
that is, $\alpha_n'$, $a_n'$, $\varepsilon'_n$, $Y'_n$, $I'_n$, $K'_n$, $J'_n$, etc.
Using $\alpha= a_{-1}+ \varepsilon_0 \alpha_0$, we note that $a'_{-1}=-a_{-1}$, $\alpha_0'=\alpha_0$, $\varepsilon'_{0}=-\varepsilon_0$.
These imply that $Y_0'=-s \circ Y_0$.
However, since $\alpha_0'=\alpha_0$, we conclude that for all $n \geq 1$ we have $a'_{n-1}=a_{n-1}$,
$\alpha'_n=\alpha_n$, $\varepsilon'_n=\varepsilon_n$.
Thus, for all $n\geq 1$, $Y_n'=Y_n$.
These imply that $I_0= I_0'$.
In particular, in the definition of $\mathbb{M}_{-\alpha}$, the only difference with the definition of
$\mathbb{M}_\alpha$ is that $Y_0$ changes to $-s \circ Y_0$.
Therefore, using \refE{E:I--1},
\begin{align*}
I'_{-1} = Y_0'(I'_0) + \frac{1+\epsilon_0'}{2}
& = -s \circ Y_0(I_0') + \frac{1-\epsilon_0}{2} \\
&= -s \left( Y_0(I_0) + \frac{1+\epsilon_0}{2} \right) + 1
= -s (I_{-1}) +1.
\end{align*}
Comparing to \refE{E:M_ga}, we have
\begin{align*}
\mathbb{M}_{-\alpha}
& = \{ s(e^{2\pi i w}) \mid w \in I'_{-1}\} \cup \{0\} \\
& = \{ s(e^{2\pi i w}) \mid w \in -s (I_{-1}) \} \cup \{0\} \\
&= \{ s \circ s(e^{2\pi i w}) \mid w \in I_{-1} \} \cup \{0\} \\
&= s \left (\{s(e^{2\pi i w}) \mid w \in I_{-1} \}\right) \cup \{0\}
= s (\mathbb{M}_\alpha). \qedhere
\end{align*}
\end{proof}
\begin{lem}\label{L:M-ga-relations-2}
For every $\alpha \in \mathbb{R} \setminus \mathbb{Q}$ we have the following:
\begin{itemize}
\item[(i)] if $\alpha \in (0, 1/2)$, $\mathbb{M}_{1/\alpha}= \{s(e^{2\pi i w}) \mid w \in I_0 \} \cup \{0\}$,
\item[(ii)] if $\alpha \in (-1/2, 0)$, $\mathbb{M}_{-1/\alpha}= \{s(e^{2\pi i w}) \mid w \in I_0 \} \cup \{0\}$.
\end{itemize}
\end{lem}
\begin{proof}
For $\alpha$, let $(\alpha_n)_{n\geq 0}$, $(a_n)_{n \geq 0}$ and $(\varepsilon_n)_{n\geq 0}$ denote the sequences defined in
\refS{SS:modified-fractions-mini}.
Define $\alpha'= \varepsilon_1 \alpha_1$.
Let us use the notations $(\alpha'_n)_{n\geq 0}$, $(a'_n)_{n \geq 0}$ and $(\varepsilon'_n)_{n\geq 0}$ for the sequences
$(\alpha_n)_{n \geq 0}$, $(a_n)_{n \geq 0}$ and $(\varepsilon_n)_{n\geq 0}$ corresponding to $\alpha'$.
We have $\alpha'_0=\alpha_1$, and then
\[\varepsilon_1 \alpha_1= \alpha'= a'_{-1}+\varepsilon'_0 \alpha'_0= a'_{-1}+\varepsilon'_0 \alpha_1.\]
As $\alpha' \in (-1/2, 1/2)$, we must have $a'_{-1}=0$, and hence $\varepsilon'_0=\varepsilon_1$.
To determine $Y_0'$, we consider two cases.
If $\varepsilon'_0=\varepsilon_1=-1$, we have $Y'_0= Y_{\alpha'_0}=Y_{\alpha_1}= Y_1$,
and if $\varepsilon'_0=\varepsilon_1=+1$, we have $Y'_0= -s \circ Y_{\alpha'_0}= -s \circ Y_{\alpha_1}= Y_1$.
The relation $\alpha'_0=\alpha_1$ implies that for all $n\geq 1$ we have $\alpha'_n= \alpha_{n+1}$, $a'_n=a_{n+1}$ and
$\varepsilon'_n = \varepsilon_{n+1}$.
Hence, for all $n\geq 1$, $Y'_{n}= Y_{n+1}$.
These imply that $I'_{-1}=I_0$.
Therefore, according to \refE{E:M_ga}, we must have
\begin{equation}\label{E:I_0-projection}
\mathbb{M}_{\varepsilon_1 \alpha_1}=\mathbb{M}_{\alpha'}= \{ s(e^{2\pi i w}) \mid w \in I'_{-1}\} \cup \{0\}
= \{ s(e^{2\pi i w}) \mid w \in I_0 \} \cup \{0\}.
\end{equation}
If $\alpha\in (0, 1/2)$, we have $1/\alpha=1/\alpha_0 = a_{-1}+ \varepsilon_1 \alpha_1$, which using \refL{P:M-ga-relations-1}-(ii),
we obtain
\[\mathbb{M}_{1/\alpha}=\mathbb{M}_{a_{-1}+\varepsilon_1 \alpha_1}= \mathbb{M}_{\varepsilon_1 \alpha_1}.\]
If $\alpha \in (-1/2, 0)$, we have $-1/\alpha= 1/\alpha_0= a_{-1}+ \varepsilon_1 \alpha_1$, and hence, using \refL{P:M-ga-relations-1}-(ii),
we get
\[\mathbb{M}_{-1/\alpha}= \mathbb{M}_{a_{-1}+\varepsilon_1 \alpha_1} = \mathbb{M}_{\varepsilon_1 \alpha_1}.\]
Combining the above equations, we obtain the desired properties in parts (i) and (ii).
\end{proof}
\section{The map \texorpdfstring{$\mathbb{T}_\alpha$}{T} on \texorpdfstring{$\mathbb{M}_\alpha$}{M-ga}}
\label{S:T-on-M}
In this section we define the map
\begin{equation}
\mathbb{T}_\alpha: \mathbb{M}_\alpha \to \mathbb{M}_\alpha,
\end{equation}
where $\mathbb{M}_\alpha$ is the topological model defined in \refS{S:M-ga}.
The topological description of $\mathbb{M}_\alpha$, which is presented in \refS{S:topology-A}, does not employ
the map $\mathbb{T}_\alpha$ in any ways.
However, the topological description can be used to identify the map $\mathbb{T}_\alpha$ as follows.
When $\mathbb{M}_\alpha$ is a Jordan curve, it is the graph of a function of the argument.
Thus, there is a unique homeomorphism of $\mathbb{M}_\alpha$ which acts as rotation by $2\pi \alpha$ in the
tangential direction.
Similarly, when, $\mathbb{M}_\alpha$ is a hairy Jordan curve, there is a unique homeomorphism of the base
Jordan curve which acts as rotation by $2\pi \alpha$.
This map can be extended onto the end points of the Jordan arcs attached to the Jordan curve by matching the
corresponding end points.
Since the set of end points of those arcs is dense in $\mathbb{M}_\alpha$, there may be a unique homeomorphism
of $\mathbb{M}_\alpha$ which acts as rotation by $2\pi \alpha$ on the base Jordan curve.
However, it is not clear if this map continuously extents to the whole hairy Cantor set.
Similarly, there may be a unique homeomorphism of a Cantor bouquet $\mathbb{M}_\alpha$ which acts as rotation
by $2\pi \alpha$ in the tangential direction.
Here we take a different approach to build $\mathbb{T}_\alpha$ on $\mathbb{M}_\alpha$.
We give a presentation which is aligned with the action of the map on the renormalisation tower; compare with
\cite{Che13,AC18}.
This helps us later when describing the dynamics of $\mathbb{T}_\alpha$ on $\mathbb{M}_\alpha$, and may also
be employed to link the toy renormalisation scheme we build here to an actual renormalisation scheme.
\subsection{Definition of the lift of \texorpdfstring{$\mathbb{T}_\alpha$}{T-ga}}\label{SS:T-defn}
Let us fix $\alpha \in \D{R}\setminus \D{Q}$, and let $I_n$, for $n\geq -1$, be the sets in \refS{SS:tilings-nest}.
Given $w_{-1} \in I_{-1}$, we inductively identify the integers $l_i$ and then the points $w_{i+1} \in I_{i+1}$ so that
\[0 \leq \Re (w_i -l_i) < 1, \ \text{if } \varepsilon_{i+1}=-1; \quad -1 < \Re (w_i -l_i) \leq 0 , \quad \text{if } \varepsilon_{i+1}=+1;\]
and
\[Y_{i+1}(w_{i+1})+l_{i}= w_i.\]
It follows that for all $n\geq 0$, we have
\begin{equation}\label{E:trajectory-condition-1}
w_{-1}=(Y_0+l_{-1}) \circ (Y_1 + l_0) \circ \dots \circ (Y_n+1_{n-1})(w_n).
\end{equation}
Also, by the definition of $I_i$ in \eqref{E:I_n^j--1} and \eqref{E:I_n^j-+1}, for all $i \geq 0$,
\begin{equation}\label{E:trajectory-real-parts}
0 \leq l_i \leq a_i + \varepsilon_{i+1}, \quad \text{ and } \quad 0 \leq \Re w_i < 1/\alpha_i.
\end{equation}
We refer to the sequence $(w_i ; l_i)_{i \geq -1}$ as the \textbf{trajectory} of $w_{-1}$, with respect to $\alpha$, or simply, as
the trajectory of $w_{-1}$, when it is clear from the context what irrational number is used.
We define the map
\begin{equation}
\tilde{T}_\alpha:I_{-1} \to I_{-1},
\end{equation}
as follows.
Let $w_{-1}$ be an arbitrary point in $I_{-1}$, and let $(w_i; l_i)_{i\geq -1}$ denote the trajectory of $w_{-1}$.
Then,
\begin{itemize}
\item[(i)] if there is $n \geq 0$ such that $w_n \in K_n$, and for all $0 \leq i \leq n-1$, $w_i \in I_i \setminus K_i$, then
\[\tilde{T}_\alpha(w_{-1})= \left(Y_0+\frac{\varepsilon_0+1}{2}\right ) \circ \left(Y_1+\frac{\varepsilon_1+1}{2}\right )
\circ \cdots \circ \left (Y_n +\frac{\varepsilon_{n}+1}{2} \right)(w_n+1);\]
\item[(ii)] if for all $n \geq 0$, $w_n \in I_n \setminus K_n$, then
\[\tilde{T}_\alpha(w_{-1})= \lim_{n \to +\infty} \left(Y_0+\frac{\varepsilon_0+1}{2}\right ) \circ \left(Y_1+\frac{\varepsilon_1+1}{2}\right )
\circ \cdots \circ \left (Y_n +\frac{\varepsilon_{n}+1}{2} \right) (w_n+1-1/\alpha_n).\]
\end{itemize}
It might not be clear that the limit in case (ii) exists. We look into this within the proof of \refL{L:model-map-lift}.
\subsection{The continuity}\label{SS:continuity-T-ga}
\begin{lem}\label{L:model-map-lift}
For every $\alpha \in \D{R} \setminus \D{Q}$, the map $\tilde{T}_\alpha : I_{-1} \to I_{-1}$ induces a well-defined,
continuous and injective map \footnote{Here, $\D{Z}$ acts on $I_{-1}$ by horizontal translations by integers.}
\[\tilde{T}_{\alpha}: I_{-1}/\D{Z} \to I_{-1}/ \D{Z}.\]
\end{lem}
The main idea of the proof for the above statement is to partition the set $I_{-1}$ into infinitely many pieces,
where the map is continuous on each piece. Then, we show that the maps on the pieces match at the boundary points.
Below we introduce the partition pieces.
For $w_{-1} \in I_{-1}$, let $(w_i; l_i)_{i \geq -1}$ denote the trajectory of $w_{-1}$.
For each $n\geq 0$, let
\[W^n= \{ w_{-1} \in I_{-1} \mid \text{ for all } 0 \leq i \leq n-1, w_i \in I_i \setminus K_i, \text{ and } w_n \in K_n\},\]
\[V^n =\{w_{-1} \in I_{-1} \mid \text{ for all } 0 \leq i \leq n, w_i \in I_i \setminus K_i\}.\]
We set $V^\infty= \cap_{n\geq 0} V^n$.
Evidently, we have
\begin{equation}\label{E:I_-1-decomposed}
I_{-1}=\cup_{n \geq 0} W^n \cup V^\infty.
\end{equation}
It is also convenient to use some simplified notations for the compositions of the maps which appear in the definition
of $\mathbb{T}_\alpha$.
That is, for $m \geq n-1$, let \footnote{\textrm{id} denotes the identity map.}
\begin{equation*}
X_n^{n-1}=\textrm{id}, \quad X_n^{n}= \left(Y_{n}+\frac{\varepsilon_{n}+1}{2}\right ), \quad
X_n^{n+1}= \left(Y_{n}+\frac{\varepsilon_{n}+1}{2}\right ) \circ \left(Y_{n+1}+\frac{\varepsilon_{n+1}+1}{2}\right )
\end{equation*}
and for all $m\geq n+2$,
\begin{equation*}
X_n^m= \left(Y_n+\frac{\varepsilon_n+1}{2}\right ) \circ \left(Y_{n+1}+\frac{\varepsilon_{n+1}+1}{2}\right )
\circ \cdots \circ \left (Y_{m} +\frac{\varepsilon_{m}+1}{2} \right).
\end{equation*}
We break the proof of \refL{L:model-map-lift} into several lemmas.
\begin{lem}\label{L:P:model-map-lift-1}
For all $n\geq 1$, the map $w_{-1} \mapsto w_n$ is continuous and injective on the sets $W^n$ and $V^n$.
In particular, for all $n \geq 1$, $\tilde{T}_\alpha: W^n \to I_{-1}$ is continuous and injective.
\end{lem}
\begin{proof}
For each $i\geq 0$, if $w_i \in I_i \setminus K_i$ and $w_{i+1} \in K_{i+1}$, then $l_i=a_i+ (\varepsilon_{i+1}-1)/2$.
Similarly, if $w_i \in I_i \setminus K_i$ and $w_{i+1} \in I_{i+1} \setminus K_{i+1}$, then $l_i=a_i+ (3\varepsilon_{i+1}-1)/2$.
These imply that for all $n\geq 1$ and all $w_{-1} \in W^n$, the entries $(l_i)_{i=-1}^{n-1}$ in the trajectory of $w_{-1}$
is independent of $w_{-1}$.
Similarly, for all $n\geq 1$ the entries $(l_i)_{i=-1}^{n-1}$ in the trajectory of $w_{-1} \in V^n$ is independent of $w_{-1}$.
In particular, the map $w_{-1} \mapsto w_n$ is continuous and injective on each of $W^n$ and $V^n$.
As each $Y_j$ is continuous and injective, we conclude that $\tilde{T}_\alpha$ is continuous and injective on $W^n$.
\end{proof}
\begin{lem} \label{L:P:model-map-lift-2}
The map $\tilde{T}_\alpha$ is well-defined, continuous and injective on $V^\infty$.
\end{lem}
\begin{proof}
We know from \refL{L:P:model-map-lift-1} that for every $n \geq 0$, the map $w_{-1} \mapsto w_n$ is continuous and
injective on $V^n$.
The image of this map covers $(1/\alpha_n-1, 1/\alpha_n) \cap I_n$, due to the choice we made in \refE{E:trajectory-real-parts}.
Since the inverse map $w_n \mapsto w_{-1}$ is also continuous and injective on $(1/\alpha_n-1, 1/\alpha_n) \cap I_n$, it follows that
$V^n$ is relatively open in $I_{-1}$. It is possible that for some values of $\alpha$, the nest $\cap_{n\geq 0} V^n$ is empty
(for instance when $\varepsilon_i=-1$, for all $i\geq 0$).
Below we assume that $V^\infty$ is not empty.
For each $n \geq 0$ and $0 \leq i \leq n$, we define the sets $V^n_i$ as the set of $w_i$, for $w_{-1} \in V^n$.
Then, define $V^\infty_i = \cap_{n \geq i} V^n_i$. It follows that $w_{-1} \mapsto w_i$ is continuous and injective
from $V^\infty$ to $V^\infty_i$, for all $i\geq 0$.
Moreover, by the uniform contraction of the map $Y_j$, each $V^\infty_i$ is a closed half-infinite vertical line.
For $n \geq 0$ and $m \geq n$, we define the map $E_n^m: V_n^m \to I_n$ as follows
\begin{equation}\label{E:L:model-map-lift-2-1}
E_n^m(w_n) = X_{n+1}^m (w_m+1-1/\alpha_m).
\end{equation}
By the above paragraphs, this is a continuous and injective map on $V_n^m$.
Note that for $w_n \in I_n \setminus K_n$ we have
\[|Y_{n+1}(w_{n+1}) + (\varepsilon_{n+1}+1)/2 - (w_n+1-1/\alpha_n)| \leq 1.\]
This is because, $\Im Y_{n+1}(w_{n+1})= \Im w_n$, $\Re w_n +1 -1/\alpha_n \in [0,1]$, and
$\Re Y_{n+1} (w_{n+1})+ (\varepsilon_{n+1}+1)/2$ belongs to $[0,+1]$.
By \refE{E:Y_n-comm-1}, $Y_{n+1}(w- 1/\alpha_{n+1})=Y_{n+1}(w) + \varepsilon_{n+1}$, and
by \refL{L:uniform-contraction-Y_r},
$|Y_{n+1}(w +1)-Y_{n+1}(w)| \leq 0.9$.
Combining with the above inequality, we conclude that for all $w_n \in V_n^{n+1}$,
\begin{align*}
|E_n^{n+1} (w_n)- E_n^n(w_n)|
&= \left | Y_{n+1} (w_{n+1}+1-1/\alpha_{n+1})+(\varepsilon_{n+1}+1)/2 - (w_n +1 -1/\alpha_n)\right | \\
&\leq \left | Y_{n+1} (w_{n+1}) + \varepsilon_{n+1}+ (\varepsilon_{n+1}+1)/2 - (w_n +1 -1/\alpha_n)\right |+ 0.9\\
&= |\varepsilon_{n+1}| + 1 + 0.9 \leq 3.
\end{align*}
Therefore, using the uniform contraction of $Y_j$ in \refL{L:uniform-contraction-Y_r}, we conclude that for all
$m \geq n+2 $, and all $w_n \in V^{m+1}_n$, we have
\begin{equation}\label{E:L:model-map-lift-2-2}
|E_n^{m+1}(w_n)- E_n^m(w_n)|
=|X_{n+1}^m \circ E_m^{m+1}(w_m) - X_{n+1}^m \circ E_m^m(w_m)|
\leq (0.9)^{m-n} \cdot 3.
\end{equation}
In particular, for $w_n \in V_n^\infty$, the above inequality holds for all $m \geq n$, and hence $E_n^m$ forms a
uniformly Cauchy sequence. Thus, the map
\begin{equation}\label{E:E_n-defn}
E_n = \lim_{m \to \infty} E_n^m: V^\infty_n \to I_n
\end{equation}
is well-define, and continuous.
In particular, since $\tilde{T}_\alpha(w_{-1})= (Y_0+(\varepsilon_0+1)/2) \circ E_0 (w_0)$, we conclude that $\tilde{T}_\alpha$ is
continuous on $V^\infty$.
In order to show that $\tilde{T}_\alpha$ is injective, it is enough to show that $E_0$ is injective.
To this end, we first note that for all $n\geq 0$ and all $w_n \in V_n^\infty$, we have
\begin{equation}\label{E:L:model-map-lift-2-3}
|E_n(w_n) - (w_n+1-1/\alpha_n) | \leq \sum_{j=n}^\infty |E_n^{j+1}(w_n) - E_n^j(w_n)| \leq
3 \sum_{j=n}^\infty 0.9^{j-n} \leq 30.
\end{equation}
Also, for $m \geq n+1$, we may rewrite \refE{E:L:model-map-lift-2-1} as
\begin{equation}\label{E:L:model-map-lift-2-5}
E_n^m(w_n)= (Y_{n+1}+ (\varepsilon_{n+1}+1)/2) \circ E_{n+1}^m(w_{n+1}),
\end{equation}
and then take limits as $m \to \infty$ to obtain
\begin{equation}\label{E:L:model-map-lift-2-4}
E_n(w_n)= (Y_{n+1}+ (\varepsilon_{n+1}+1)/2) \circ E_{n+1}(w_{n+1}).
\end{equation}
The above relation holds for all $n \geq 0$ and all $w_{n+1} \in V_{n+1}^\infty$.
Let $w_{-1}$ and $w'_{-1}$ be distinct elements in $V^\infty$. Let $(w_i ; l_i)_{i\geq -1}$ and $(w_i'; l_i)_{i\geq -1}$ denote the
trajectories of $w_{-1}$ and $w'_{-1}$, respectively.
Then, for each $n \geq 0$, both $w_n$ and $w'_n$ belong to $V_n^\infty$.
By the uniform contraction of $Y_j$, there is $n \geq 0$ such that $|w_n - w_n'| \geq 61$.
By virtue of the uniform bound in \refE{E:L:model-map-lift-2-3}, $E_n(w_n) \neq E_n(w_n')$.
Inductive using \eqref{E:L:model-map-lift-2-4}, and the injectivity of $Y_j$, we conclude that
$E_n(w_n)\neq E_n(w_n')$, for all $n\geq 0$.
In particular, $E_0$ is injective.
\end{proof}
\begin{lem}\label{L:P:model-map-lift-3}
For all $n \geq 1$, $\tilde{T}_\alpha: \overline{W^n} \to I_{-1}/\D{Z}$ is continuous.
\end{lem}
\begin{proof}
Fix an arbitrary $n \geq 1$.
By definition, $W^n \subset V^{n-1}$.
Recall from the proof of \refL{L:P:model-map-lift-2} that the map $w_{-1} \mapsto w_{n-1}$ is a homeomorphism from
$V^{n-1}$ onto $(1/\alpha_{n-1}-1, 1/\alpha_{n-1}) \cap I_{n-1}$. Thus, $V^{n-1}$ is relatively open in $I_{-1}$.
Indeed, for $w_{-1} \in V^{n-1}$, the map $\Re w_{-1} \to \Re w_{n-1}$ is a translation independent of $w_{-1}$.
We have
\begin{equation}\label{E:L:P:model-map-lift-3-1}
\begin{aligned}
\{w_{n-1} \mid w_{-1} \in W^n \}=\{ w \in I_{n-1} \mid \Re w \in [a_{n-1}-1, 1/\alpha_{n-1})\}, \quad \text{if } \varepsilon_n=-1; \\
\{w_{n-1} \mid w_{-1} \in W^n\}=\{ w \in I_{n-1} \mid \Re w \in (1/\alpha_{n-1}-1, a_{n-1}]\}, \quad \text{if } \varepsilon_n=+1.
\end{aligned}
\end{equation}
Combining with the above paragraph, we conclude that there are real numbers $x_n$ and $y_n$ such that
either $W^n=\{w\in I_{-1} \mid \Re w \in (x_n, y_n]\}$ or $W^n=\{w\in I_{-1} \mid \Re w \in [x_n, y_n)\}$.
Let $(w^i)_{i\geq 0}$ be a sequence in $W^n$ converging to some $w \in \overline{W^n}$.
If $w \in W^n$, the continuity at $w$ follows from \refL{L:P:model-map-lift-1}.
So, below we assume that $w\in \overline{W_n} \setminus W^n$.
Note that either $\Re w^i < \Re w$ or $\Re w^i > \Re w$ holds for all $i \geq 0$.
Let $(w_j ; l_j)_{j\geq -1}$ denote the trajectory of $w$, and for each $i \geq 0$, let $(w^i_j, l^i_j)_{j\geq -1}$ denote
the trajectory of $w^i$. By \eqref{E:L:P:model-map-lift-3-1}, either $\varepsilon_n=+1$ and
$\Re w^i_{n-1} \searrow 1/\alpha_{n-1}-1$, or $\varepsilon_n=-1$ and $\Re w^i_{n-1} \nearrow 1/\alpha_{n-1}$.\footnote{We use the
notation $x_i \searrow x$ to mean that the sequence $x_i $ converges to $x$ and $\Re x_i > x$. Similarly, $x_i \nearrow x$
means that the sequence $x_i$ converges to $x$ and $\Re x_i < x$.}
We consider two cases based on these scenarios.
(i) $\varepsilon_n=-1$ and $\Re w^i_{n-1} \nearrow 1/\alpha_{n-1}$, as $i\to \infty$.
For integers $j$ with $-1 \leq j \leq n-3$, $\lim_{i \to \infty} \Re w^i_j \notin \mathbb{Z}$.
That is because, if there is $j$ with $0 \leq j \leq n-3$ and $\lim_{i \to \infty} \Re w^i_j \in \mathbb{Z}$ then we must have
$\lim_{i \to \infty} \Re w^i_{j+1}= 1/\alpha_{j+1}$, then $\lim_{i \to \infty} \Re w^i_{j+2}= 1/\alpha_{j+2}-1$,
and then $\lim_{i \to \infty} \Re w^i_{n-1}= 1/\alpha_{n-1}-1$.
The last property is not possible in this case.
Similarly, if $\lim_{i \to \infty} \Re w^i_{-1} \in \mathbb{Z}$ then we must have
$\lim_{i \to \infty} \Re w^i_{0}= 1/\alpha_{0}$, then $\lim_{i \to \infty} \Re w^i_{1}= 1/\alpha_{1}-1$,
and then $\lim_{i \to \infty} \Re w^i_{n-1}= 1/\alpha_{n-1}-1$.
The same contradiction.
By the previous paragraph, $l^i_j=l_j$, for all $-1 \leq j \leq n-3$.
This implies that for $-1\leq j \leq n-2$, $\lim_{i \to \infty} w^i_j= w_j$.
The integer $n-1$ is the smallest integer with $w_{n-1} \in K_{n-1}$.
To see this, first note that either $\varepsilon_{n-1}=-1$ and $\Re w^i_{n-2} \nearrow a_{n-2}-1$, or $\varepsilon_{n-1}=+1$ and
$\Re w^i_{n-2} \searrow a_{n-2}$.
Thus, $\Re w_{n-2}= \lim_{i \to \infty} \Re w_{n-2}^i \in \mathbb{Z}$. Hence, $\Re w_{n-1}=0$.
On the other hand, for $0 \leq j \leq n-3$, $\Re w_j \neq 1/\alpha_j -1$, since otherwise $\Re w_{n-1}= 1/\alpha_{n-1}-1$.
With this paragraph, we conclude that
\[\tilde{T}_\alpha (w)=X_0^{n-1} (w_{n-1}+1).\]
Let $w_{n-1}'= \lim_{i\to \infty} w^i_{n-1}$ and $w_{n}'= \lim_{i \to \infty} w^i_{n}$.
We must have $\Re w'_{n-1}= 1/\alpha_{n-1}$ and $\Re w'_n= 1/\alpha_{n}-1$.
By \refE{E:Y_n-comm-1}, we have $w'_{n-1}= w_{n-1}+1/\alpha_{n-1}$, and then using \refE{E:Y_n-comm-2}, we get
$w'_{n}+1= w_{n}+1/\alpha_n$.
Therefore,
\begin{align*}
\lim_{i \to \infty} \tilde{T}_\alpha(w^i) = \lim_{i \to \infty} X_0^{n} (w^i_n+1) = X_0^n (w'_n+1)
&= X_0^{n-1} \circ Y_n (w_n'+1) \\
&= X_0^{n-1} \circ Y_n (w_n+1/\alpha_n) \\
&=X_0^{n-1} (Y_n (w_n)+1)=X_0^{n-1} (w_{n-1}+1).
\end{align*}
This completes the proof in this case.
(ii) $\varepsilon_n=+1$ and $\Re w^i_{n-1} \searrow 1/\alpha_{n-1}-1$, as $i\to \infty$.
Let us first assume that for all $1 \leq j \leq n-1$ we have $\varepsilon_j=-1$.
This implies that $\Re w^i_j \searrow 1/\alpha_j-1$, for all $0 \leq j \leq n-1$.
In particular, $\Re w_j=1/\alpha_j-1$, for all $j \geq 0$.
For $0 \leq j \leq n$, choose $t_j \geq -1$ such that $w_j=1/\alpha_j-1+it_j$.
Then, by \refE{E:Y_n-comm-1},
\begin{align*}
\tilde{T}_\alpha(w)= \left(Y_0+\frac{\varepsilon_0+1}{2}\right)(w_0+1) = Y_0 (1/\alpha_0 + i t_0) + \frac{\varepsilon_0+1}{2}
= Y_0(it_0) - \varepsilon_0 + \frac{\varepsilon_0+1}{2}.
\end{align*}
On the other hand, by \eqref{E:Y_n-comm-2},
\[\Im Y_j(i t_j)= \Im Y_j(i t_j +1/\alpha_j-1)= \Im Y_j(w_j)= \Im w_{j-1}= t_{j-1},\]
and hence $Y_j(it_j)= i t_{j-1}$.
Therefore,
\begin{align*}
\lim_{i \to \infty} \tilde{T}_\alpha (w^i) = \lim_{i \to \infty} X_0^n(w^i_n+1)
&= X_0^n(w_n+1) \\
&= Y_0 \circ \dots \circ Y_{n-1} \circ (Y_n + 1)(1/\alpha_n + i t_n) + \frac{\varepsilon_0+1}{2} \\
&= Y_0 \circ \dots \circ Y_{n-1} \circ (Y_n( i t_{n})) + \frac{\varepsilon_0+1}{2}\\
&= Y_0(it_0) + \frac{\varepsilon_0+1}{2}.
\end{align*}
By the above equations, $\lim_{i \to \infty} \tilde{T}_\alpha (w^i) /\mathbb{Z}= \tilde{T}_\alpha(w)/\mathbb{Z}$.
Now assume that there is $1 \leq m \leq n-1$ with $\varepsilon_m=+1$.
Assume that $m$ is the largest integer with $1 \leq m \leq n-1$ and $\varepsilon_m=+1$.
As in the above paragraph, we note that for all $j$ with $m \leq j \leq n-1$ we have $\Re w^i_j \searrow 1/\alpha_j-1$.
Since $\varepsilon_m=+1$, we must have $\Re w^i_{m-1} \nearrow 1/\alpha_{m-1}$.
As in case (i), this implies that for $0 \leq j\leq m-3$, $\lim_{i \to \infty} \Re w^i_{j} \notin \mathbb{Z}$.
Hence, for $-1 \leq j \leq m-2$, $w_j= \lim_{i\to \infty} w^i_j$.
Moreover, $\lim_{i \to \infty} \Re w^i_{m-2} \in \mathbb{Z}$.
By the above paragraph, $\Re w_{m-1}=0$, and $m-1$ is the smallest positive integer with $w_{m-1} \in K_{m-1}$.
Then,
\[\tilde{T}_\alpha (w)=X_0^{m-1} (w_{m-1}+1).\]
For $m-1 \leq j \leq n $, let $w'_j= \lim_{i \to \infty} w^i_j$.
We have $w_j=w_j'+1-1/\alpha_j$.
Then, as in the previous cases, we note that
\begin{align*}
\lim_{i \to \infty} \tilde{T}_\alpha(w^i) = \lim_{i \to \infty} X_0^{n} (w^i_n+1) &= X_0^n (w'_n+1) \\
&= X_0^{m-1} \circ (Y_m+1) \circ Y_{m+1} \dots \circ Y_{n-1} \circ (Y_n+1) (w_n'+1) \\
&= X_0^{m-1} \circ (Y_m+1) \circ Y_{m+1} \dots \circ Y_{n-1} \circ (Y_n+1) (w_n+1/\alpha_n) \\
&= X_0^{m-1} \circ (Y_m+1) \circ Y_{m+1} \dots \circ Y_{n-1} (w_{n-1}) \\
&= X_0^{m-1} (w_{m-1}+1).
\end{align*}
This completes the proof in this case.
\end{proof}
\begin{lem}\label{L:P:model-map-lift-4}
The set $W^0$ is closed, and $\tilde{T}: W^0/\D{Z} \to I_{-1}/\D{Z}$ is well-defined, continuous, and injective.
\end{lem}
\begin{proof}
This is straightforward and is left to the reader.
\end{proof}
\begin{proof}[Proof of \refL{L:model-map-lift}]
First we prove the injectivity.
For $n\geq 1$, each equivalence class in $W^n/\D{Z}$ consists of a single element, since the horizontal width of $W^n$
is at most $\alpha_0 \alpha_1 \dots \alpha_{n-1} \leq 1/2$.
Also, the width of $\tilde{T}_\alpha (W^n)$ is at most $1/2$, which implies that each equivalence class in $\tilde{T}_\alpha(W^n)/\D{Z}$
consists of a single element.
Then, by \refL{L:P:model-map-lift-1}, $\tilde{T}_\alpha: W^n/\D{Z} \to I_{-1}/\D{Z}$ is injective.
By \refL{L:P:model-map-lift-4}, $\tilde{T}_\alpha: W^0/\D{Z} \to I_{-1}/\D{Z}$ is injective.
Similarly, $V^\infty$ consists of a single half-infinite vertical line which is mapped to a single half-infinite vertical
line by $\tilde{T}_\alpha$.
Thus, by \refL{L:P:model-map-lift-2}, $\tilde{T}_\alpha : V^\infty/\D{Z} \to V^\infty/\D{Z}$ is injective.
on the other hand, for $0 \leq n < m$, the sets $\tilde{T}_\alpha (W^n)$, $\tilde{T}_\alpha (W^m)$ and
$\tilde{T}_\alpha (V^\infty)$, are pairwise disjoint. Therefore, $\tilde{T}_\alpha: I_{-1}/\D{Z} \to I_{-1}/\D{Z}$ is injective.
Now we prove the continuity.
Let $(w^i)_{i\geq 0}$ be a convergent sequence in $I_{-1}/\D{Z}$. Without loss of generality, we may assume that
$(w^i)_{i\geq 0}$ converges in $I_{-1}$.
By Lemmas \ref{L:P:model-map-lift-2} and \ref{L:P:model-map-lift-3},
if a subsequence of this sequence lies in some $W^n$ or in $V^\infty$, then $\tilde{T}_\alpha$ is continuous along that subsequence.
Therefore, it is enough to deal with the case of $w^i \in W^{j_i}$, with $j_i \to +\infty$ as $i \to +\infty$.
Without loss of generality (in order to simplify the notations) we may assume that $w^i \in W^{i+1}$, for $i\geq 0$.
Let $w=\lim_{i \to \infty} w^i$. Below, we consider three cases, based on the location of $w$.
{\em Case 1: $w \in V^\infty$.}
Let $(w_j ; l_j)_{j \geq -1}$ denote the trajectory of $w$, and for $i\geq 0$, let $(w^i_j ; l^i_j)_{j\geq -1}$
denote the trajectory of $w^i$.
Recall from the proof of \refL{L:P:model-map-lift-2} that
\[\tilde{T}_\alpha(w_{-1})=(Y_0+(\varepsilon_0+1)/2) \circ E_0(w_0),\]
and $E_0^m \to E_0$, uniformly on $V^\infty_0$.
Fix an arbitrary $\varepsilon>0$.
Let us choose an integer $m \geq 5$ such that for all $i \geq m$ we have
\[|E_0(w_0) - E_0^i(w_0)| \leq \varepsilon/2, \qquad 63 \cdot (0.9)^m \leq \varepsilon/2.\]
For $i\geq m$, $w^i \in W^{i+1} \subset V^i \subset V^m$, and $w \in V^\infty \subset V^m$.
Then, by \refL{L:P:model-map-lift-1}, $w^i_m \to w_m$, as $i \to \infty$.
In particular, there is $N \geq 0$ such that for all $i \geq N$ we have $|w_m - w^i_m| \leq 1$.
Below we assume that $i \geq \max\{N, m\}$, and we aim to show \refE{E:P:model-map-lift-3}.
Since both $w^i_m$ and $w_m$ belong to $V^i_m$, we may employ the estimate in \refE{E:L:model-map-lift-2-2}, to obtain
\begin{equation}\label{E:P:model-map-lift-2}
\begin{aligned}
|E_m^{i}(w_m) & - E_m^{i}(w_m^i) | \\
&\leq \sum_{l=m}^{i-1}|E_m^{l+1}(w_m) - E_m^{l}(w_m)|
+ |w_m-w_m^i| + \sum_{l=m}^{i-1}|E_m^{l+1}(w^i_m) - E_m^{l}(w^i_m)| \\
& \leq 2 \cdot 3 \sum_{l=m}^{\infty} (0.9)^{l-m} + |w_m-w^i_m| \leq 60 +1=61.
\end{aligned}
\end{equation}
Note that
\[\left | (w^i_{i}+1-1/\alpha_{i}) - \left (Y_{i+1}+\frac{\varepsilon_{i+1}+1}{2}\right ) (w^i_{i+1})\right | \leq 1.\]
Combining the above inequality with the uniform contraction in \refL{L:uniform-contraction-Y_r}, we obtain
\begin{align*}
\bigg | E_{i}^{i}(w^i_{i}) & - \left (Y_{i+1}+\frac{\varepsilon_{i+1}+1}{2}\right ) (w^i_{i+1}+1) \bigg | \\
& = \left | (w^i_{i}+1-1/\alpha_{i}) - \left (Y_{i+1}+\frac{\varepsilon_{i+1}+1}{2}\right ) (w^i_{i+1}) \right. \\
& \left. \quad \quad + \left (Y_{i+1}+\frac{\varepsilon_{i+1}+1}{2}\right ) (w^i_{i+1})
- \left (Y_{i+1}+\frac{\varepsilon_{i+1}+1}{2}\right ) (w^i_{i+1}+1) \right |
\leq 1 + 0.9 \cdot 1 \leq 2.
\end{align*}
Using the uniform contraction one more time, this give us
\begin{align*}
\left| E_m^{i}(w_m^i) - X_{m+1}^{i+1} (w^i_{i+1}+1) \right |
= \left| X_{m+1}^{i} \circ E_{i}^{i}(w^i_{i}) - X_{m+1}^{i} \circ \left (Y_{i+1} +\frac{\varepsilon_{i}+1}{2} \right) (w^i_{i+1}+1) \right |
\leq 2.
\end{align*}
Combining the above bound with the bound in \refE{E:P:model-map-lift-2}, we conclude that
\[\left| E_m^{i}(w_m) - X_{m+1}^{i+1} (w^i_{i+1}+1) \right | \leq 61+2=63.\]
Using the relation in \refE{E:L:model-map-lift-2-5} several times, we note that
$(Y_0+(\varepsilon_0+1)/2) \circ E_0^i(w_0) =X_0^{m-1} \circ E_m^i(w_m)$.
Thus, using the above equation with $m=0$, we obtain
\begin{equation}\label{E:P:model-map-lift-3}
\begin{aligned}
|\tilde{T}_\alpha(w) - \tilde{T}_\alpha(w^i)|
& = \left | (Y_0+(\varepsilon_0+1)/2) \circ E_0(w_0) - X_0^{m-1} \circ X_m^{i+1}(w^i_{i+1}+1) \right | \\
& \leq | (Y_0+(\varepsilon_0+1)/2) \circ E_0(w_0)- (Y_0+(\varepsilon_0+1)/2) \circ E_0^i(w_0)| \\
& \qquad \qquad \qquad + | X_0^{m-1} \circ E_m^i(w_m) - X_0^{m-1} \circ X_m^{i+1}(w^i_{i+1}+1)| \\
& \leq (0.9) \cdot \varepsilon/2 + (0.9)^m \cdot (63) \leq \varepsilon.
\end{aligned}
\end{equation}
This completes the proof in Case 1.
{\em Case 2: $w \in W^n$ for some $n\geq 1$.}
Recall from the proof of \refL{L:P:model-map-lift-2}, that there are real numbers $x_n$ and $y_n$ such that either
$W^n= \{w\in I_{-1} \mid \Re w \in [x_n, y_n)\}$ or $W^n= \{w\in I_{-1} \mid \Re w \in (y_n, x_n]\}$.
Since, $W^n$ and $W^{i+1}$ are pairwise disjoint, for $i+1 > n$, and $w_i \in W^{i+1}$, we must have $\Re w= x_n$.
Note that for integers $j$ with $0 \leq j \leq n-2$, $\Re w_j \notin \D{Z}$.
This implies that $w^i_{n-1} \to w_{n-1}$, as $i \to \infty$.
It follows from \refE{E:L:P:model-map-lift-3-1} that $\Re w_{n-1} \in \D{Z}$.
Hence, $\Re w_n=0$.
Below we consider two scenarios.
{\em Case 2-i: $\varepsilon_n=-1$.}
Since $w^i_n \notin K_n$, for large $i$, we must have $\Re w^i_{n-1} \nearrow \Re w_{n-1}$.
It follows that $\Re w^i_n \nearrow 1/\alpha_n + w_n$.
Let us define $w_n'= 1/\alpha_n + w_n$.
For $i \geq n$, let $(w'_i; l_i')_{i\geq n}$ denote the trajectory of $w_n'$, defined in the same fashion according to
\refE{E:trajectory-real-parts}.
We must have $\Re w'_{i}=1/\alpha_{i}-1$, for all $i \geq n+1$, and by \refE{E:Y_n-comm-2}, $w_i=w_i' + 1-1/\alpha_i$, for all $i\geq n$.
If $\varepsilon_{n+1}=-1$, then we must have $\Re w_{n+1}^i \nearrow 1/\alpha_{n+1}-1$.
This is a contradiction, since this implies that $w^i \in W^{n+1}$, for sufficiently large $i$.
Therefore, we have $\varepsilon_{n+1}=+1$ and $\Re w^i_{n+1} \searrow 1/\alpha_{n+1}-1$.
Indeed, for the same reason, we must have $\varepsilon_i=-1$, for all $i\geq n+2$.
Assume that $i \geq n$. We have
\[\left| (w_{i} - \left (Y_{i+1}+\frac{\varepsilon_{i+1}+1}{2}\right ) (w^i_{i+1}) \right| \leq 1.\]
Therefore, applying $X_{n+1}^i$, we obtain
\begin{align*}
\left | w_n - X_{n+1}^{i+1}(w^i_{i+1}+1) \right | \leq 1 \cdot (0.9) ^{i-n}.
\end{align*}
Then,
\begin{align*}
\left |\tilde{T}_\alpha (w) - \tilde{T}_\alpha (w^i) \right |
&=\left | X_0^{n} (w_n) - X_0^{i+1}(w^i_{i+1}+1) \right | \\
& \leq \left | X_0^{n} (w_n) - X_0^n \circ X_{n+1}^{i+1}(w^i_{i+1}+1) \right |
\leq 1 \cdot (0.9)^{i+1}.
\end{align*}
This completes the proof in this case.
{\em Case 2-ii: $\varepsilon_n=+1$.}
Here, we must have $\Re w^i_{n-1} \searrow \Re w_{n-1}$.
It follows that $w^i_n \nearrow 1/\alpha_n + w_n$. Let $w_n'= 1/\alpha_n + w_n$.
Let $(w'_i; l'_i)_{i\geq n}$ denote the trajectory of $w'_n$.
We must have $\varepsilon_{n+1}=+1$, otherwise, $\Re w^i_{n+1} \nearrow 1/\alpha_{n+1}-1$ and hence $w^i \in W^n$
for sufficiently large $i$. For the same reason, we must have $\varepsilon_i=-1$, for all $i \geq n+2$.
As in the previous case, we obtain $\tilde{T}_\alpha (w^i) \to \tilde{T}_\alpha (w)$.
{\em Case 3: $w \in W_0$.}
If $\varepsilon_0=-1$, either $\Re w^i \nearrow 1$ and hence $\Re w^i_0 \nearrow 1/\alpha_0$, or $\Re w^i \searrow 1-\alpha_0$
and hence $\Re w^i_0 \searrow 1/\alpha_0-1$.
Similarly, if $\varepsilon_0=+1$, either $\Re w^i \nearrow \alpha_0$ and hence $\Re w^i_0 \searrow 1/\alpha_0-1$,
or $\Re w^i \searrow 0$ and hence $\Re w^i_0 \nearrow 1/\alpha_0$.
All these scenarios may be dealt with as in Case 2.
\end{proof}
\subsection{The map \texorpdfstring{$\mathbb{T}_\alpha$}{T-ga}, and its properties}\label{SS:T-ga-properties}
\begin{propo}\label{P:T-ga-tangential}
For every $\alpha \in \D{R}\setminus \D{Q}$, the map $\tilde{T}_{\alpha}: I_{-1} \to I_{-1}$ induces
a homeomorphism
\[\mathbb{T}_\alpha:\mathbb{M}_\alpha \to \mathbb{M}_\alpha,\]
via the projection $w \mapsto s(e^{2\pi i w})$, that is,
$s(e^{2\pi i \tilde{T}_\alpha(w)})= \mathbb{T}_\alpha (s(e^{2\pi i w}))$ for all $w\in I_{-1}$.
Moreover, $\mathbb{T}_\alpha$ acts as rotation by $2\pi \alpha$ in the tangential direction,
that is, there is a function $g_\alpha$ such that
\[\mathbb{T}_\alpha(r e^{2\pi i \theta})= g_\alpha(r, \theta) e^{2\pi i (\theta+\alpha)}\]
for every $r e^{2\pi i\theta} \in \mathbb{M}_\alpha$.
\end{propo}
\begin{proof}
By \refL{L:model-map-lift}, $\tilde{T}_{\alpha}: I_{-1}/\D{Z} \to I_{-1}/ \D{Z}$ induces a continuous and injective map
$\mathbb{T}_\alpha$ of the set $\{s(e^{2\pi i w}) \mid w\in I_{-1}\}$.
From the construction, we note that as $\Im z \to \infty$ in $I_{-1}/\mathbb{Z}$,
$\Im \tilde{T}_\alpha(z) \to +\infty$.
This implies that we may continuously extend $\mathbb{T}_\alpha$ onto $0$ by setting $\mathbb{T}_\alpha(0)=0$.
Since $\mathbb{M}_\alpha$ is compact, and $\mathbb{T}_{\alpha}: \mathbb{M}_\alpha \to \mathbb{M}_\alpha$ is continuous
and injective, it must be a homeomorphism.
In order to show that $\mathbb{T}_\alpha$ acts as rotation by $2\pi \alpha$ in the tangential direction, it is enough to show
that $\tilde{T}_\alpha$ acts as translation by $-\alpha$ on $\cup_{n\geq 0}W^n \cup V^\infty$.
However, because $V^\infty$ has empty interior, by the continuity of $\tilde{T}_\alpha$, it is enough to prove this
on the sets $W^n$.
The latter property follows from the definition of $\tilde{T}_\alpha$ and the functional relation in \refE{E:Y_n-comm-1}.
We present the details below.
For $w_{-1} \in W^0$, we have
\[w_{-1}= (Y_0+(1+\epsilon_0)/2)(w_0), \qquad \tilde{T}_\alpha(w_1)= (Y_0+(1+\epsilon_0)/2)(w_0+1).\]
Thus, by the definition of $Y_0$,
\[\Re (Y_0+(1+\epsilon_0)/2)(w_0+1) = \Re w_{-1} -\epsilon_0 \alpha_0.\]
Therefore,
\[\arg \left(\mathbb{T}_{\alpha}(s(e^{2\pi i w_{-1}}))\right)
= \arg (s(e^{2\pi i w_{-1}})) + \epsilon_0\alpha_0
=\arg (s(e^{2\pi i w_{-1}})) + \alpha.\]
Now, fix an arbitrary $n\geq 1$, and let $w_{-1} \in W^n$ be an arbitrary point with trajectory $(w_i, l_i)_{i\geq -1}$.
For $j=n, n-1, n-2, \dots, -1$, let us define the points $\xi_j= X_{j+1}^n(w_n+1)$.
We note that in this case we must have $\varepsilon_n=-1$, and $\varepsilon_i=+1$ for all $i=n-1, n-2, \dots, 1$.
Then, using $1/\alpha_{n-1}= a_{n-1} -\alpha_n$ and \refE{E:I_n^j--1}, we obtain
\begin{align*}
\Re w_{n-1} - \Re \xi_{n-1}
&= \left( \Re (Y_n(w_n)) + a_{n-1}-1 \right) - \Re (Y_n(w_n+1)) \\
&= (a_{n-1} -1) - \alpha_n \\
&= \left(\frac{1}{\alpha_{n-1}} + \alpha_n -1 \right) - \alpha_n = \frac{1}{\alpha_{n-1}}-1.
\end{align*}
For the next step, we have $1/\alpha_{n-2}= a_{n-2} + \alpha_{n-1}$, and use \refE{E:I_n^j-+1}, to obtain
\[\Re w_{n-2} - \Re \xi_{n-2}= -\alpha_{n-1}\left(\frac{1}{\alpha_{n-1}}-1\right) + a_{n-2}= \frac{1}{\alpha_{n-2}} -1.\]
Repeating the above process for levels $n-3, n-4, \dots, 0$, we end up with
\[\Re w_{0} - \Re \xi_{0}= \frac{1}{\alpha_0}-1.\]
In the last stage we apply the map $Y_0+(1+\varepsilon_0)/2$, to obtain,
\[\Re w_{-1} - \Re \xi_{-1}= -\varepsilon_0 \alpha_0 \left( \frac{1}{\alpha_0}-1\right) =- \varepsilon_0 + \varepsilon_0\alpha_0.\]
Thus, $\tilde{T}_\alpha$ acts as translation by $-\varepsilon_0\alpha_0$ which is the same as $-\alpha$ modulo $\mathbb{Z}$.
This completes the argument for the set $W^n$.
\end{proof}
\begin{propo}\label{P:T-ga-relations}
For every $\alpha \in \mathbb{R} \setminus \mathbb{Q}$, we have $s \circ \mathbb{T}_\alpha \circ s = \mathbb{T}_{-\alpha}$
on $\mathbb{M}_{-\alpha}$.
\end{propo}
\begin{proof}
Recall that by \refP{P:M-ga-relations-1}, we have $s(\mathbb{M}_{\alpha})= \mathbb{M}_{-\alpha}$.
Thus both of the maps $\mathbb{T}_{-\alpha}$ and $s\circ \mathbb{T}_\alpha \circ s$ are defined on $\mathbb{M}_{-\alpha}$
and map into $\mathbb{M}_{-\alpha}$.
When $\alpha$ changes to $-\alpha$, $\varepsilon_0$ changes to $-\varepsilon_0$, but all the remaining numbers $\alpha_i$
and $\varepsilon_{i+1}$, for $i \geq 0$, remain the same.
Let $I_{-1}$ and $I'_{-1}$ denote the corresponding sets for $\alpha$ and $-\alpha$, respectively.
We have $-s(I'_{-1})= I_{-1}$; see proof of \refP{P:M-ga-relations-1}.
From the definition of the maps $\tilde{T}_\alpha$ and $\tilde{T}_{-\alpha}$, we can see that
$(-s) \circ \tilde{T}_{\alpha} \circ (- s) = \tilde{T}_{-\alpha}$ on $I'_{-1}$.
Projecting onto $\mathbb{M}_{-\alpha}$ via $w \mapsto s(e^{2\pi i w})$ we obtain the desired relation.
\end{proof}
\begin{propo} \label{P:orbit-size}
There is a constant $C$ such that for every $\alpha \in (-1/2, 1/2) \setminus \mathbb{Q}$ and every integer
$k$ satisfying $0 \leq k < 1/|\alpha|$ we have
\[\frac{C^{-1}}{1+ \min \{k, |\alpha|^{-1}-k\}} \leq |\mathbb{T}_\alpha\co{k}(+1)| \leq \frac{C}{1+ \min \{k,|\alpha|^{-1}-k\}}.\]
\end{propo}
\begin{proof}
By virtue of \refP{P:T-ga-relations}, it is enough to show this for $\alpha \in (0, 1/2) \setminus \mathbb{Q}$.
In that case, we have $\alpha_0=\alpha$, $\varepsilon_0=+1$, and $Y_0=-s \circ Y_{\alpha_0}$.
The point $+1$ may be lifted under the projection $w \mapsto s(e^{2\pi i w})$ to the point $+1$ in $I_{-1}$.
In the trajectory of $+1$, we have $l_{-1}=+1$ and $w_0=0$.
Therefore, by the definition of $\tilde{T}_{\alpha}$, for all integers $k$ satisfying $0 \leq k < 1/\alpha$, we have
$\tilde{T}_\alpha\co{k} (+1)= Y_0(k)+(1+\varepsilon_0)/2=Y_0(k)+1$.
This implies that
\[|\mathbb{T}_\alpha\co{k}(+1)| =\left | s\left (e^{2\pi i Y_0(k)} \right) \right |
= \Big |\frac{e^{-3\pi\alpha}- e^{\pi\alpha i}}{e^{-3\pi\alpha}- e^{-\pi\alpha i}e^{-2\pi\alpha i k }}\Big|.\]
We need to estimate the right hand side of the above equation.
The numerator of that formula is proportional to $\alpha$;
see Equations \eqref{E:P:Y_r-vs-h_r-0} and \eqref{E:P:Y_r-vs-h_r^-1} for a more precise estimate on this.
For the denominator we have,
\begin{align*}
\left |e^{-3\pi\alpha}- e^{-\pi\alpha i}e^{-2\pi\alpha i k }\right |
&\leq \left|e^{-3\pi \alpha}-1 \right| + \left|1 - e^{-\pi \alpha i}\right| + \left| e^{-\pi \alpha i} - e^{-\pi\alpha i}e^{-2\pi\alpha ik}\right| \\
& \leq 3 \pi \alpha + \pi \alpha + 2\pi \alpha \min \{k, \alpha^{-1} -k\} \\
& \leq 4 \pi \alpha \left ( 1+ \min \{k, \alpha^{-1} -k\}\right ).
\end{align*}
To give a lower bound on the left hand side of the above equation, let us consider two cases.
If $k \leq 5$, or $\alpha^{-1} -k \leq 5$, we have
\begin{align*}
\left |e^{-3\pi\alpha}- e^{-\pi\alpha i}e^{-2\pi\alpha i k }\right |
& \geq \left| e^{-3\pi \alpha} -1 \right| \\
& \geq \alpha \\
& \geq \frac{\alpha}{6} \left (1+ \min \{k, \alpha^{-1} -k\}\right ).
\end{align*}
If $k$ satisfies $5 \leq k \leq \alpha^{-1} -5$, by the triangle inequality, we have
\begin{align*}
\left |e^{-3\pi\alpha}- e^{-\pi\alpha i}e^{-2\pi\alpha i k }\right |
&\geq \left|e^{-\pi\alpha i}e^{-2\pi\alpha ik} - e^{-\pi \alpha i} \right| - \left|e^{-\pi \alpha i} - e^{-3\pi \alpha} \right| \\
& \geq 2\pi \alpha \min \{k, \alpha^{-1} -k\} - (\pi \alpha + 3\pi \alpha) \\
& \geq 2 \pi \alpha \left (\min \{k, \alpha^{-1} -k\} -2 \right) \\
& \geq \pi \alpha \left (\min \{k, \alpha^{-1} -k\} + 1 \right).
\end{align*}
Combining the above inequalities, we obtain the desired estimate in the proposition for a suitable constant $C$.
\end{proof}
The estimate of the sizes of the orbit presented in \refP{P:orbit-size} widely holds for holomorphic map of the
from $e^{2\pi i \alpha} z+ a_2 z^2+ \dots$, with $a_2 \neq 0$.
More precisely, for a fixed nonlinearity (higher order terms), and a choice of a point close enough to the fixed point
at $0$, there is a constant $C$ such that the upper and lower bounds hold for those integers.
The constant $C$ is independent of the rotation number $\alpha$ and the number of iterates $k$.
For large values of $\alpha$, this mostly follows from continuity of the map.
For smaller values of $\alpha$, one employs the existence of perturbed Fatou-coordinates, and elementary estimates on
their behaviour. See \cite{Sh00} for more details on this.
\section{The renormalisation operator \texorpdfstring{$\mathcal{R}_m$}{R-m}}\label{S:renormalisation-model}
In this section we define the renormalisation of $\mathbb{T}_\alpha: \mathbb{M}_\alpha \to \mathbb{M}_\alpha$
using a suitable return map to a fundamental set in $\mathbb{M}_\alpha$.
This construction is in the spirit of the sector renormalisation, qualitatively presented by Douady and
Ghys \cite{Do86}, and quantitatively employed by Yoccoz in the study of the small divisors problem in
complex dimension \cite{Yoc95}.
Our construction of renormalisation relies on the model for the changes of coordinates we defined
in \refS{S:change-coordinates}.
\subsection{Definition of \texorpdfstring{$\mathcal{R}_m$}{R-m}}\label{SS:definition-Renorm}
Fix an arbitrary $\alpha \in \mathbb{R} \setminus \mathbb{Q}$.
Since $\mathbb{M}_{\alpha+1}= \mathbb{M}_\alpha$ and $\mathbb{T}_{\alpha+1}= \mathbb{T}_\alpha$, we may assume that
$\alpha \in (-1/2,1/2)$.
Let us first assume that $\alpha \in (0, 1/2)$.
Consider the set
\[S_\alpha= \{ z \in \mathbb{M}_\alpha \setminus \{0\} \mid \arg z \in ([0, 2\pi \alpha)+2 \pi \mathbb{Z})\}.\]
This consists of all the non-zero points in $\mathbb{M}_\alpha$ which lie in a ``sector'' of angle $2\pi \alpha$ at $0$.
Because $\mathbb{T}_\alpha$ acts as rotation by $2\pi \alpha$ in the tangential direction, \refP{P:T-ga-tangential},
for every $z \in S_\alpha$, there is the smallest integer $k_z \geq 2$ such that $\mathbb{T}_\alpha^{\circ k_z}(z) \in S_\alpha$.
Indeed, for every $z \in S$, either $k_z$ is equal to the integer part of $1/\alpha$ or $k_z$ is equal to the
integer part of $1/\alpha$ plus $1$.
The map $\mathbb{T}_\alpha^{\circ k_z}$ is the first return map to $S_\alpha$ in the dynamical system
$\mathbb{T}_\alpha: \mathbb{M}_\alpha \to \mathbb{M}_\alpha$.
Recall that $Y_0(w+ 1/\alpha_0)= Y_0(w) + (1+\varepsilon_0)/2=Y_0+1$.
Let us consider the map
\[\psi_\alpha: \mathbb{H}' \to \mathbb{C} \setminus \{0\}\]
defined as
\[\psi_\alpha(w)= s \left( e^{2\pi i Y_0(w)} \right)
= s \left( e^{2\pi i (-s\circ Y_{\alpha_0}(w))} \right)
=\Big |\frac{e^{-3\pi\alpha}- e^{\pi\alpha i}}{e^{-3\pi\alpha}- e^{-\pi\alpha i}e^{-2\pi\alpha iw}}\Big| e^{2\pi i \alpha \Re w}.\]
We note that
\begin{equation}\label{E:renorm-def-1}
S_\alpha \subset \psi_\alpha \left( \{w \in \mathbb{H}' \mid \Re w \in [0, 1)\}\right).
\end{equation}
The map $\psi_\alpha$ is continuous and injective on $ \{w \in \mathbb{H}' \mid \Re w \in [0, 1)\}$, and
it sends half-infinite vertical lines in that set to straight rays landing at $0$.
There is a continuous inverse branch of $\psi_\alpha$ defined on $S_\alpha$, that is,
\begin{equation}\label{E:renorm-def-2}
\phi_\alpha: S_\alpha \to \{w \in \mathbb{H}' \mid \Re w\in [0, 1)\}.
\end{equation}
The above map is continuous and injective.
(This map is the analogue of the perturbed Fatou coordinate for $\mathbb{T}_\alpha$)
The return map $\mathbb{T}_\alpha\co{k_z}: S_\alpha \to S_\alpha$ induces a map
\[h_\alpha: \phi_\alpha(S_\alpha) \to \phi_\alpha(S_\alpha)\]
via $\psi_\alpha$ and $\phi_\alpha$, that is,
\[h_\alpha(w)= \phi_\alpha \circ \mathbb{T}_\alpha^{\circ k_{\psi_\alpha(w)}} \circ \psi_\alpha(w).\]
Using the projection $w \mapsto e^{2\pi i w}$, the map $h_\alpha$ projects to a map $E_\alpha$, defined on
$e^{2\pi i \phi_\alpha(S_\alpha)} \subset \mathbb{C}\setminus \{0\}$.
That is, $e^{2\pi i h_\alpha(w)}= E_\alpha (e^{2\pi i w})$ for all $w\in \phi_\alpha(S_\alpha)$.
We may extend $E_\alpha$ onto $0$ by letting it map $0$ to $0$.
We call the map $E_\alpha$, and its domain of definition, the renormalisation of
$\mathbb{T}_\alpha: \mathbb{M}_\alpha \to \mathbb{M}_\alpha$, and denote it by
$\mathcal{R}_m(\mathbb{T}_\alpha: \mathbb{M}_\alpha \to \mathbb{M}_\alpha)$.
\begin{rem}
In the above definition, one may replace the set $S_\alpha$ by any set of the form
\[S_\alpha^\eta
=\{z \in \mathbb{M}_\alpha \setminus \{0\} \mid \arg z \in \left([2\pi \eta, 2\pi\eta+2\pi\alpha) + 2\pi\mathbb{Z}\right)\}\]
for a fixed $\eta \in \mathbb{R}$, and use the return map of $\mathbb{T}_\alpha$ on $S_\alpha^\eta$ to
define the renormalisation of $\mathbb{T}_\alpha: \mathbb{M}_\alpha \to \mathbb{M}_\alpha$ in the same fashion.
The definition of renormalisation is independent of the choice of $\eta$.
That is, one obtains the same map defined on the same set of points.
We avoid considering this generality for the sake of simplicity, although, the independence from $\eta$ will
become evident from the proof of \refP{P:renormalisation-relations}.
Also, when projecting the map $h_\alpha$ to obtain $E_\alpha$ we have used the projection map $w \mapsto e^{2\pi i w}$
for the sake of consistency with how the sector renormalisation is defined.
This may appear in contrast to the projection map $w\mapsto s\left(e^{2\pi i w} \right)$ used in the definition of the
sets $\mathbb{M}_\alpha$.
Due to the properties in \refP{P:M-ga-relations-1} and \refL{L:M-ga-relations-2}, this does not cause any problems.
See the proof of \ref{P:renormalisation-relations} for more details.
\end{rem}
The above process defines the renormalisation of $\mathbb{T}_\alpha: \mathbb{M}_\alpha \to \mathbb{M}_\alpha$
when $\alpha \in (0, 1/2)$.
For $\alpha \in (-1/2, 0)$, we first use dynamical conjugation via the complex conjugation map so that the
rotation number of the map becomes $-\alpha \in (0, 1/2)$ and then repeat the above process.
By \refP{P:M-ga-relations-1}, we have $s(\mathbb{M}_\alpha)= \mathbb{M}_{-\alpha}$ and by \refP{P:T-ga-relations}
we have $s \circ \mathbb{T}_\alpha \circ s= \mathbb{T}_{-\alpha}$.
Therefore, for $\alpha \in (-1/2, 0)$, we define
\begin{equation}\label{E:def-renorm-neg-alpha}
\begin{aligned}
\mathcal{R}_m(\mathbb{T}_\alpha: \mathbb{M}_\alpha \to \mathbb{M}_\alpha)
&=s \Big ( \mathcal{R}_m \big(s \circ \mathbb{T}_\alpha \circ s: s\left(\mathbb{M}_\alpha\right) \to s\left(\mathbb{M}_\alpha\right)\big)\Big ) \\
&= s\big(\mathcal{R}_m(\mathbb{T}_{-\alpha}: \mathbb{M}_{-\alpha} \to \mathbb{M}_{-\alpha}) \big),
\end{aligned}
\end{equation}
where the complex conjugation $s$ on the right hand side of the above equation means that we consider the
dynamical conjugate of the map and its domain of definition using the complex conjugation map.
More precisely, in terms of our notations, that means
\[s(f: X \to Y)= \left( s\circ f \circ s: s(X) \to s(Y)\right).\]
\subsection{Invariance of the class of maps under the renormalisation operator}\label{SS:invariance-renormalisation}
\begin{propo}\label{P:renormalisation-relations}
For every $\alpha\in (-1/2, 1/2) \setminus \mathbb{Q}$ we have
\[\mathcal{R}_m(\mathbb{T}_\alpha: \mathbb{M}_\alpha \to \mathbb{M}_\alpha)
= (\mathbb{T}_{-1/\alpha}: \mathbb{M}_{-1/\alpha} \to \mathbb{M}_{-1/\alpha}).\]
\end{propo}
\begin{proof}
Let us first assume that $\alpha \in (0, 1/2) \setminus \mathbb{Q}$.
Recall the numbers $(\alpha_n)_{n\geq 0}$ and $(\varepsilon_n)_{n\geq 0}$ defined in \refS{SS:modified-fractions-mini}.
We shall also use the notations $I_n$, $K_n$ and $J_n$ introduced in \refS{SS:tilings-nest}.
Let us consider the set
\[\hat{S}_\alpha= \{w\in I_0 \mid \Re w \in [0, 1)\}.\]
By the definition of $\mathbb{M}_\alpha$ in \refS{S:M-ga}, in particular Equations \eqref{E:I--1} and \eqref{E:M_ga},
we have $\psi_\alpha(I_0) \cup\{0\}= \mathbb{M}_\alpha$, and
\[\psi_\alpha (\hat{S}_\alpha)=S_\alpha.\]
Indeed, $\psi_\alpha: \hat{S}_\alpha \to S_\alpha$ is a homeomorphism.
This implies that
\[\phi_\alpha: S_\alpha \to \hat{S}_\alpha\]
is a homeomorphism as well.
Therefore, we have
\[h_\alpha: \hat{S}_\alpha \to \hat{S}_\alpha.\]
Because $\alpha \in (0, 1/2)$ and $I_0$ is periodic of period $+1$, by \refP{P:M-ga-relations-1}
and \refL{L:M-ga-relations-2}, we have
\[\{e^{2\pi i w} \mid w \in \hat{S}_\alpha\} \cup \{0\} = s (\mathbb{M}_{1/\alpha})= \mathbb{M}_{-1/\alpha}.\]
This shows that the map $h_\alpha$ projects to the map $E_\alpha$ defined on $\mathbb{M}_{-1/\alpha}$, via the projection
$w \mapsto e^{2\pi i w}$.
In other words, the domain of definition of
$\mathcal{R}_m(\mathbb{T}_\alpha: \mathbb{M}_\alpha \to \mathbb{M}_\alpha)$ is $\mathbb{M}_{-1/\alpha}$.
Now, we need to show that $E_\alpha= \mathbb{T}_{-1/\alpha}$.
We continue to assume that $\alpha \in (0, 1/2)$.
Let us consider the set
\[\tilde{S}_\alpha= \{w \in I_{-1} \mid \Re w \in (1-\alpha, 1].\]
The map $w \mapsto s(e^{2\pi i w})$ is a homeomorphism from $\tilde{S}_\alpha$ to $S_\alpha$.
We also have the homeomorphism
\[Y_0+1: \hat{S}_\alpha \to \tilde{S}_\alpha.\]
Recall from \refS{S:T-on-M}, that $\mathbb{T}_\alpha$ on $\mathbb{M}_\alpha$
is induced from $\tilde{T}_\alpha$ on $I_{-1}$, via the projection $w \mapsto s(e^{2\pi i w})$.
Therefore, the return map of $\mathbb{T}_\alpha: \mathbb{M}_\alpha \to \mathbb{M}_\alpha$ on $S_\alpha$
corresponds to the return map of $\tilde{T}_\alpha: I_{-1} \to I_{-1}$ on $\tilde{S}_\alpha$.
The map $Y_0+ (1+\varepsilon_0)/2= Y_0+1: I_0 \to I_{-1}$ is a homeomorphism.
Let us consider the map
\[\hat{T}_\alpha= (Y_0+1)^{-1} \circ \tilde{T}_\alpha \circ (Y_0+1): I_0 \to I_0.\]
Then, through the conjugacy $Y_0+1$, the return map of $\tilde{T}_\alpha: I_{-1} \to I_{-1}$ on $\tilde{S}_\alpha$
corresponds to the return map of $\hat{T}_\alpha: I_0 \to I_0$ on $\hat{S}_\alpha$.
Because $\hat{S}_\alpha = \phi_\alpha(S_\alpha)$, by the definition of renormalisation, the return map of
$\hat{T}_\alpha: I_0 \to I_0$ on $\hat{S}_\alpha$ projects via $w \mapsto e^{2\pi i w}$ to the map $E_\alpha$.
Below, we investigate this return map in more details.
Let $w\in I_{-1}$ be an arbitrary point, and let $(w_i; l_i)_{i\geq -1}$ denote the trajectory of $w$ defined in
\refS{SS:T-defn}.
By the definition of $\tilde{T}_\alpha$, if $w_0 \in K_0$ we have
$\tilde{T}_\alpha(w_{-1})= (Y_0+1)((Y_0+1)^{-1}(w_{-1})+1)$.
This implies that for $w\in K_0$, $\hat{T}_\alpha (w)=w+1$.
On the other hand, if $w_0 \in I_0\setminus K_0$, $\hat{T}_\alpha(w_0)$ is defined as follows:
\begin{itemize}
\item[(i)] if there is $n \geq 1$ such that $w_n \in K_n$, and for all $1 \leq i \leq n-1$, $w_i \in I_i \setminus K_i$, then
\[\hat{T}_\alpha (w_0)= \left(Y_1+\frac{\varepsilon_1+1}{2}\right ) \circ \left(Y_2+\frac{\varepsilon_2+1}{2}\right )
\circ \cdots \circ \left (Y_n +\frac{\varepsilon_{n}+1}{2} \right)(w_n+1);\]
\item[(ii)] if for all $i \geq 1$, $w_i \in I_i \setminus K_i$, then
\[\tilde{T}_\alpha(w_0)= \lim_{n \to +\infty} \left(Y_1+\frac{\varepsilon_1+1}{2}\right ) \circ \left(Y_2+\frac{\varepsilon_2+1}{2}\right )
\circ \cdots \circ \left (Y_n +\frac{\varepsilon_{n}+1}{2} \right) (w_n+1-1/\alpha_n).\]
\end{itemize}
Combining the above together, we conclude that the return map of $\hat{T}_\alpha: I_0 \to I_0$ on the set
$\hat{S}_\alpha$ consists of a finite number of translations by $+1$ which take a point in
$\hat{S}_\alpha$ to a point in $I_0 \setminus K_0$, and then one iterate of either the map in
item (i) or the the map in item (ii), depending on which scenario takes place.
Since any point in $I_0$ and its integer translations are sent by $w \mapsto s(e^{2\pi i w})$ to the same point,
each translation by +1 on $K_0$ induce the identity map via the projection $w \mapsto s(e^{2\pi i w})$.
Thus, by the above paragraph, the rerun map of $\hat{T}_\alpha: I_0 \to I_0$ on the set $\hat{S}_\alpha$
and the map $\hat{T}_\alpha: I_0\setminus K_0 \to \hat{S}_\alpha$ induce the same map via the
projection $w \mapsto s(e^{2\pi i w})$.
On the other hand, by the definition in \refS{SS:T-defn}, the map specified in items (i) and (ii) is
$\tilde{T}_{\varepsilon_1\alpha_1}: I_0/\mathbb{Z} \to I_0/\mathbb{Z}$.
Therefore, the rerun map of $\hat{T}_\alpha: I_0 \to I_0$ on the set $\hat{S}_\alpha$, and the map
$\tilde{T}_{\varepsilon_1\alpha_1}: I_0/\mathbb{Z} \to I_0/\mathbb{Z}$ induce the same map via the projection
$w \mapsto s(e^{2\pi i w})$.
By the definition in \refS{SS:T-defn}, $\tilde{T}_{\varepsilon_1 \alpha_1}: I_0/\mathbb{Z} \to I_0/\mathbb{Z}$
induces $\mathbb{T}_{\varepsilon_1 \alpha_1}: \mathbb{M}_{\varepsilon_1\alpha_1} \to \mathbb{M}_{\varepsilon_1\alpha_1}$
via the projection $w \mapsto s(e^{2\pi i w})$.
If we project via $w \mapsto e^{2\pi i w}$, $\tilde{T}_{\varepsilon_1 \alpha_1}$ induces
$s \circ \mathbb{T}_{\varepsilon_1 \alpha_1} \circ s$ on $s(\mathbb{M}_{\varepsilon_1\alpha_1})$.
By \refP{P:M-ga-relations-1},
$s(\mathbb{M}_{\varepsilon_1 \alpha_1})= \mathbb{M}_{-\varepsilon_1\alpha_1} = \mathbb{M}_{-1/\alpha_0}= \mathbb{M}_{-1/\alpha}$
and by \refP{P:T-ga-relations}, $s \circ \mathbb{T}_{\varepsilon_1 \alpha_1} \circ s= \mathbb{T}_{-\varepsilon_1 \alpha_1}
= \mathbb{T}_{-1/\alpha}$.
Note that here we have used $\alpha \in (0, 1/2)$, which implies that $\alpha_0=\alpha$ and
$1/\alpha_0= -\varepsilon_1 \alpha_1, \mod \mathbb{Z}$.
Now assume that $\alpha \in (-1/2,0)$.
By the definition of renormalisation for $\alpha \in (-1/2, 0)$, we have
\begin{align*}
\mathcal{R}_m (\mathbb{T}_\alpha: \mathbb{M}_\alpha \to \mathbb{M}_\alpha)
& = s \left ( \mathcal{R}_m (\mathbb{T}_{-\alpha}: \mathbb{M}_{-\alpha} \to \mathbb{M}_{-\alpha})\right) \\
& = s \left ( \mathbb{T}_{1/\alpha}: \mathbb{M}_{1/\alpha} \to \mathbb{M}_{1/\alpha}\right) \\
& = s \circ \mathbb{T}_{1/\alpha} \circ : s (\mathbb{M}_{1/\alpha}) \to s(\mathbb{M}_{1/\alpha}) \\
& = \mathbb{T}_{-1/\alpha}: \mathbb{M}_{-1/\alpha} \to \mathbb{M}_{-1/\alpha}. \qedhere
\end{align*}
\end{proof}
\section{Arithmetic classes of Brjuno and Herman}\label{S:arithmetic}
In this section we define the arithmetic classes of Brjuno and Herman.
This requires the action of the modular group $\mathrm{PGL}(2, \D{Z})$ on the real line, which produces
continued fraction type representation of irrational numbers.
To study the action of this group, one may choose a fundamental interval for the action of $z \mapsto z+1$
and study the action of $z \mapsto 1/z$ on that interval.
When the interval $(0,1)$ is chosen, one obtains the standard representation (continued fraction).
In \cite{Yoc02}, Yoccoz uses this representation to identify the arithmetic class $\E{H}$,
stated in \refD{D:Herman-Yoccoz-criterion}.
Due to the nature of the renormalisation, we work with the fundamental interval $(-1/2,1/2)$ for the translation.
This is partly due to the symmetry of the renormalisation scheme $\{\mathbb{T}_\alpha, \mathbb{M}_\alpha$
with respect to the rotation stated in part (v) of \refT{T:toy-model-renormalisation}.
This choice of the fundamental interval leads to a modified representation (continued fraction) of irrationals.
The main aim of this section is to identify the equivalent form of the Herman condition in terms of the modified
continued fraction.
We shall only use the modified representation beyond this section.
For basic properties of continued fractions one may consult \cite{Khin64}.
\subsection{Standard continued fraction}\label{SS:standard-fraction}
For $x\in \D{R}$, let $\langle x \rangle \in (0,1)$ denote the fractional part of $x$, that is, $x\in \D{Z}+\langle x \rangle$.
For $\alpha \in \D{R}\setminus \D{Q}$, we may define the numbers $\tilde{\alpha}_n \in (0,1)$ as $\tilde{\alpha}_0=\langle \alpha \rangle$ and
$\tilde{\alpha}_{n+1}=\langle 1/\tilde{\alpha}_n \rangle$, for $n\geq 0$.
Then we identify the unique integers $\tilde{a}_n$, for $n\geq -1$, according to
\begin{equation}\label{E:recursive-factions-standard}
\alpha = \tilde{a}_{-1}+ \tilde{\alpha}_0, \quad 1/\tilde{\alpha}_{n}=\tilde{a}_{n} + \tilde{\alpha}_{n+1}.
\end{equation}
These may be combined to obtain
\begin{equation*}\label{E:modified-expansion-alpha-standard}
\alpha= \tilde{a}_{-1}+\cfrac{1}{\tilde{a}_0+\cfrac{1}{\ddots + \cfrac{1}{\tilde{a}_n + \tilde{\alpha}_{n+1}}}}, \; \text{for } n\geq -1.
\end{equation*}
The $n$-th convergent of $\alpha$ is define as
\[\frac{\tilde{p}_n}{\tilde{q}_n}=\tilde{a}_{-1}+\cfrac{1}{\tilde{a}_0+\cfrac{1}{\ddots + \cfrac{1}{\tilde{a}_n}}}, \; \text{for } n\geq -1.\]
\begin{comment}
The numbers $\tilde{p}_n$ and $\tilde{q}_n$ may be also generated using the recursive relations
\begin{equation}\label{E:p_n-q_n-recursive}
\begin{aligned}
p_{-2}&=1, & p_{-1}&= \tilde{a}_{-1}, & p_n&=\tilde{a}_n p_{n-1}+ p_{n-2}, \; \text{ for all } n\geq 0, \\
q_{-2}&=0, & q_{-1}&= 1, & q_n&=\tilde{a}_n q_{n-1}+ q_{n-2}, \; \text{ for all } n\geq 0.
\end{aligned}
\end{equation}
\end{comment}
Then,
\begin{equation*}\label{E:ga-ga_n-formula}
\alpha=(\tilde{p}_n+\tilde{p}_{n-1}\tilde{\alpha}_{n+1})/(\tilde{q}_n+ \tilde{q}_{n-1}\tilde{\alpha}_{n+1}), \; \text{for } n\geq -1,
\end{equation*}
which implies
\begin{equation}\label{E:ga_n-ga-formula}
\alpha_{n+1}=-(\alpha \tilde{q}_n- \tilde{p}_n)/(\alpha \tilde{q}_{n-1}-\tilde{p}_{n-1}), \; \text{for } n\geq -1.
\end{equation}
In \cite{Brj71}, Brjuno introduced the important series $\sum_{n=-1}^{+\infty} \tilde{q}_n^{-1} \log \tilde{q}_{n+1}$.
In \cite{Yoc95}, Yoccoz defines a closely related series which enjoys remarkable equivariant properties
with respect to the action of $\mathrm{PGL}(2, \D{Z})$.
To define that, we need to introduce the numbers
\[\tilde{\beta}_{-2}=\alpha, \quad \tilde{\beta}_{-1}=+1, \quad \tilde{\beta}_n= \prod_{i=0}^n \tilde{\alpha}_i, \; \text{for } n \geq 0.\]
In terms of the convergents, \refE{E:ga_n-ga-formula} gives us
\begin{equation}\label{E:beta_n-p_n-q_n}
\tilde{\beta}_n= (-1)^n (\alpha \tilde{q}_{n-1} - \tilde{p}_{n-1}), \; \text{for } n\geq -1.
\end{equation}
Define the (standard Brjuno) function $\tilde{\C{B}}: \D{R}\setminus \D{Q} \to (0, +\infty) \cup \{+\infty\}$ as
\begin{equation}\label{E:Brjuno-condition-standard}
\tilde{\C{B}}(\alpha)= \sum_{n=-1}^{+\infty} \tilde{\beta}_{n} \log \frac{1}{\tilde{\alpha}_{n+1}}
= \sum_{n=-1}^{+\infty} (-1)^n (\alpha \tilde{q}_{n-1} -\tilde{p}_{n-1}) \log \frac{\alpha \tilde{q}_{n-1}-\tilde{p}_{n-1}}{\tilde{p}_n-\alpha \tilde{q}_n}.
\end{equation}
This is a highly irregular function; $\tilde{\C{B}}(\alpha)=+\infty$ for generic choice of $\alpha \in \D{R}$.
One may refer to \cite{MMY97,MMY01,JM18}, and the extensive list of references therein,
for detailed analysis of the regularity properties of this function.
In this paper we are not concerned with the regularity, but only exploit the equivariant properties of the Brjuno
function with respect to the action of $PGL(2,\mathbb{Z})$.
\begin{def-numbered}\label{D:brjuno-numbers-beta-version}
An irrational number $\alpha$ is called a \textbf{Brjuno number} if $\tilde{\C{B}}(\alpha)<+\infty$.
\end{def-numbered}
The function $\tilde{\C{B}}$ satisfies the remarkable relations
\begin{equation}\label{E:Brjuno-functional-equations-standard}
\tilde{\C{B}}(\alpha)= \tilde{\C{B}}(\alpha+n), \;
\tilde{\C{B}}(\alpha)= \alpha \tilde{\C{B}}(1/\alpha) + \log (1/\alpha),
\end{equation}
for all $\alpha \in (0,1)$ and all $n\in \D{Z}$. These show that the set of Brjuno numbers is
$\mathrm{PGL}(2, \D{Z})$-invariant.
Let $\E{B}$ denote the set of Brjuno numbers.
The definition of the Brjuno numbers given in the introduction is consistent with the one given in
\refD{D:brjuno-numbers-beta-version}.
That is because, $| \sum_{n=-1}^{+\infty} \tilde{q}_n^{-1} \log \tilde{q}_{n+1} - \tilde{\C{B}}(\alpha) |$ is uniformly
bounded from above.
\begin{comment}
This follows from the next proposition.
\begin{propo}\label{P:Brjuno-Yoccoz-equivalent}
For all $\alpha \in \D{R}\setminus \D{Q}$ we have $| \sum_{n=-1}^{+\infty} q_n^{-1} \log q_{n+1} - \tilde{\C{B}}(\alpha) | \leq 32$.
\end{propo}
\begin{proof}
By an inductive argument, \eqref{E:p_n-q_n-recursive} and \eqref{E:beta_n-p_n-q_n} imply that
for all $n\geq -1$,
\begin{equation*}\label{E:q_n-beta_n-formula}
q_n p_{n-1}- p_n q_{n-1}=(-1)^{n+1}, \quad q_{n} \tilde{\beta}_n + q_{n-1} \tilde{\beta}_{n+1}= +1.
\end{equation*}
Using $\tilde{\beta}_{n+1}= \tilde{\beta}_n \tilde{\alpha}_{n+1}$, the latter formula in the above equation gives us
\begin{equation*}\label{E:q_n-beta_n-inequality}
1/(q_{n}+ q_{n-1}) < \tilde{\beta}_n= 1/(q_{n}+ q_{n-1} \tilde{\alpha}_{n+1}) < 1/q_{n}.
\end{equation*}
In particular, for all $ n \geq -1$,
\[1/q_n - \tilde{\beta}_{n} = q_{n-1} \tilde{\beta}_{n+1}/q_n, \quad 1/2 < q_{n}\tilde{\beta}_n < 1.\]
Note that $q_{n-1}> q_{n-2}$, for $n\geq 0$.
Therefore, for all $n\geq 0$,
\begin{align*}
\big| q_{n-1}^{-1} \log q_{n} - \tilde{\beta}_{n-1} \log (1/\tilde{\beta}_n) \big |
& \leq q_{n-1}^{-1} \big|\log q_{n} - \log 1/\tilde{\beta}_{n} \big|
+ \big|q_{n-1}^{-1}-\tilde{\beta}_{n-1}\big| \big|\log (1/\tilde{\beta}_n)\big|\\
& \leq q_{n-1}^{-1} \log 2 + \big|q_{n-2} \tilde{\beta}_{n}/ q_{n-1} \big| \big|\log (1/\tilde{\beta}_n)\big | \\
& \leq 2 \tilde{\beta}_{n-1} \log 2 + \big|\tilde{\beta}_n \log (\tilde{\beta}_n)\big|.
\end{align*}
Hence,
\begin{align*}
\big| q_{n-1}^{-1} \log q_{n} - \tilde{\beta}_{n-1} \log (1/\tilde{\alpha}_n) \big |
& \leq \big| q_{n-1}^{-1} \log q_{n} - \tilde{\beta}_{n-1} \log (1/\tilde{\beta}_n) \big |
+ \big| \tilde{\beta}_{n-1} \log (1/\tilde{\beta}_n) - \tilde{\beta}_{n-1} \log (1/\tilde{\alpha}_n) \big | \\
& \leq 2 \tilde{\beta}_{n-1} \log 2 + \big|\tilde{\beta}_n \log (\tilde{\beta}_n)\big|
+ \big| \tilde{\beta}_{n-1} \log \tilde{\beta}_{n-1}\big|.
\end{align*}
On the other hand, since $\tilde{\alpha}_n \tilde{\alpha}_{n+1} < 1/2$ for all $n\geq 0$,
\begin{equation}\label{E:beta-sum}
\textstyle{\sum_{n=0}^{+\infty} \tilde{\beta}_{n-1} \leq (\tilde{\beta}_{-1}+ \tilde{\beta}_0) \sum_{n=0}^{+\infty} 2^{-n} \leq 4,}
\end{equation}
and since $|x \log x| \leq 2 \sqrt{x}$ for $x\in (0,1)$,
\begin{equation}\label{E:P:Brjuno-Yoccoz-equivalent}
\begin{aligned}
\textstyle{ \sum_{n=0}^{+\infty} |\tilde{\beta}_{n}\log \tilde{\beta}_{n}| }
\textstyle{ \leq 2 \sum_{n=0}^{+\infty} (\tilde{\beta}_{n})^{1/2} }
& \textstyle{ \leq 2 \sum_{n=0}^{+\infty} (\tilde{\beta}_{2n})^{1/2} + 2 \sum_{n=0}^{+\infty} (\tilde{\beta}_{2n+1})^{1/2} } \\
& \textstyle{ \leq 2 (\tilde{\beta}_0)^{1/2} \sum_{n=0}^{+\infty} 2^{-n/2} + 2 \sum_{n=1}^{+\infty} 2^{-n/2} }
\leq 6+ 4 \cdot 2^{1/2}.
\end{aligned}
\end{equation}
Recall that $\tilde{\beta}_{-1} \log \tilde{\beta}_{-1}=0$. Combining the above inequalities we obtain
\begin{align*}
\textstyle{ \left | \sum_{n=0}^{k} q_{n-1}^{-1} \log q_{n} - \sum_{n=0}^k \tilde{\beta}_{n-1} \log (1/\tilde{\alpha}_n)\right | }
&\leq \textstyle{ \sum_{n=0}^{+\infty} \left (2 \tilde{\beta}_{n-1} \log 2 + 2 | \tilde{\beta}_{n-1} \log \tilde{{\beta}}_{n-1}| \right ) } \\
& \leq 8 \log 2 + 12 + 8 \sqrt{2}\leq 32. \qedhere
\end{align*}
\end{proof}
\end{comment}
The set of Herman numbers is defined in a different fashion.
To that end, we need to consider the functions $h_r: \D{R} \to (0, +\infty)$, for $r\in (0,1)$:
\[h_r(y)=
\begin{cases}
r^{-1} (y- \log r^{-1} +1) & \text{if } y \geq \log r^{-1}, \\
e^{y} & \text{if } y \leq \log r^{-1}.
\end{cases}
\]
The function $h_r$ is $C^1$ on $\D{R}$, and satisfies
\begin{equation}\label{E:h_r-properties}
\begin{gathered}
h_r (\log r^{-1}) = h_r'(\log r^{-1}) = r^{-1}, \\
e^y \geq h_r(y) \geq y+1, \forall y\in \D{R}, \\
h_r'(y)\geq 1, \forall y\geq 0.
\end{gathered}
\end{equation}
Following Yoccoz \cite{Yoc02}, we give the following definition.
\begin{def-numbered}\label{D:Herman-Yoccoz-criterion}
An irrational number $\alpha$ is of \textbf{Herman type}, if for all $n\geq 0$ there is $m\geq n$ such that
\[h_{\tilde{\alpha}_{m-1}} \circ \dots \circ h_{\tilde{\alpha}_n} (0)\geq \tilde{\C{B}}(\tilde{\alpha}_{m}).\]
\end{def-numbered}
In the above definition, the composition $h_{\tilde{\alpha}_{m-1}} \circ \dots \circ h_{\tilde{\alpha}_n}$ is understood as the
identity map when $m=n$, and as $h_{\tilde{\alpha}_n}$ when $m=n+1$.
The set of Herman numbers is denoted by $\E{H}$.
It follows from the definition that
\[\E{H} \subset \E{B}.\]
That is because, if $\alpha \notin \E{B}$, then $\tilde{\C{B}}(\tilde{\alpha})=\tilde{\C{B}}(\tilde{\alpha}_{0})= +\infty$.
Repeatedly using the functional equations in \refE{E:Brjuno-functional-equations-standard}, one concludes that
$\tilde{\C{B}}(\tilde{\alpha}_{m})=+\infty$ for all $m\geq 0$.
In particular, the inequality in the above definition never holds.
In \refD{D:Herman-Yoccoz-criterion}, one may only require that for large $n$ there is $m$ such that the inequality
holds. That is because, if $m'$ works for some $n'$, then the same $m'$ works for all $n \leq n'$.
This shows that $\tilde{\alpha}_0 \in \E{H}$ if and only if $\tilde{\alpha}_1 \in \E{H}$.
On the other hand, since $\alpha$ and $\alpha+1$ produce the same sequence of $\tilde{\alpha}_i$, we see that $\E{H}$ is
invariant under $z\mapsto z+1$. These show that $\E{H}$ is invariant under the action of
$\mathrm{PGL}(2,\D{Z})$.
Recall that $\alpha$ is a Diophantine number, if there are $\tau \geq 0$ and $c>0$ such that
for all $p/q \in \D{Q}$ with $q\geq 1$ we have $|\alpha-p/q|\geq c/q^{2+\tau}$.
Any Diophantine number is of Herman type. Since the set of Diophantine numbers has full Lebesgue measure in
$\D{R}$, the sets $\E{H}$ and $\E{B}$ have full Lebesgue measure in $\D{R}$.
\begin{comment}
\begin{lem}
Every Diophantine number is contain in $\E{H}$. In particular, $\E{H}$ and $\E{B}$ have full Lebesgue measure
in $\D{R}$.
\end{lem}
\begin{proof}
Let $\alpha$ be a Diophantine number with $\tau$ and $c$, as in the definition preceding the lemma.
Choosing the convergent $p_n/q_n$ as $p/q$ in the definition, and using \refE{E:beta_n-p_n-q_n} and
$q_n\tilde{\beta}_n<1$, we get $\tilde{\beta}_{n+1}= |\alpha q_n -p_n|\geq c/q_n^{1+\tau}\geq c \tilde{\beta}_n^{1+\tau}$.
Let $m\geq 0$. By the above inequality, for all $j \geq m$ we have
$1/ \tilde{\alpha}_{j+1} \leq 1/(c\tilde{\beta}_j^{\tau})$. Therefore,
\begin{align*}
\tilde{\C{B}}(\tilde{\alpha}_m)
&= \log (1/\tilde{\alpha}_m) + \sum_{j=m}^{+\infty} (\tilde{\beta}_{j}/\tilde{\beta}_{m-1}) \log (1/\tilde{\alpha}_{j+1}) \\
& \leq \log (1/\tilde{\alpha}_m)
+ \sum_{j=m}^{+\infty} (\tilde{\beta}_{j}/\tilde{\beta}_{m-1}) (\tau \log (1/\tilde{\beta}_{j}))
+ \sum_{j=m}^{+\infty} (\tilde{\beta}_{j}/\tilde{\beta}_{m-1}) \log (1/c)
\end{align*}
Using the inequalities $x \log x$
\end{proof}
\end{comment}
\begin{lem}
The set $\E{B} \setminus \E{H}$ is dense in $\D{R}$.
\end{lem}
\begin{proof}
Let $\alpha$ be an irrational number such that there is an integer $k \geq 0$ such that
$\tilde{a}_{k}=1$, and for all $i\geq k$ we have $e^{\tilde{a}_i} \leq \tilde{a}_{i+1} \leq e^{2 \tilde{a}_i}-1$.
Evidently, the set of such irrational numbers is dense in $\D{R}$. Below we show that any such $\alpha$ belongs to
$\E{B} \setminus \E{H}$.
For integers $i \geq k$,
\[\tilde{\beta}_i \log (1/\tilde{\alpha}_{i+1})
\leq \tilde{\beta}_i \log (\tilde{a}_{i+1}+1)
\leq \tilde{\beta}_i 2 \tilde{a}_{i}
\leq 2 \tilde{\beta}_{i-1}.\]
Then,
\begin{align*}
\tilde{\C{B}}(\alpha)
&= \textstyle{ \sum_{i=-1}^{k-1}\tilde{\beta}_i\log (1/\tilde{\alpha}_{i+1}) + \sum_{i=k}^{+\infty}\tilde{\beta}_i\log(1/\tilde{\alpha}_{i+1}) } \\
& \leq \textstyle{ \sum_{i=-1}^{k-1}\tilde{\beta}_i\log (1/\tilde{\alpha}_{i+1}) + 2 \sum_{i=k}^{+\infty} \tilde{\beta}_{i-1} }
\leq \textstyle{ \sum_{i=-1}^{k-1}\tilde{\beta}_i \log (1/\tilde{\alpha}_{i+1}) + 8.}
\end{align*}
This proves that $\alpha$ belongs to $\E{B}$.
On the other hand, for all $i \geq k+1$,
\[\tilde{a}_{i-1} \leq \log \tilde{a}_{i} \leq \log (1/\tilde{\alpha}_i) \leq \tilde{\C{B}}(\tilde{\alpha}_i).\]
Then, by an inductive argument, for all $i \geq k+1$, $h_{\tilde{\alpha}_{i-1}} \circ \dots \circ h_{\tilde{\alpha}_k}(0) \leq \tilde{a}_{i-1}$.
Therefore, for all integers $m\geq n \geq k+1$,
\[h_{\tilde{\alpha}_{m-1}} \circ \dots \circ h_{\tilde{\alpha}_n}(0)
\leq h_{\tilde{\alpha}_{m-1}} \circ \dots \circ h_{\tilde{\alpha}_n} (h_{\tilde{\alpha}_{n-1}} \circ \dots \circ h_{\tilde{\alpha}_k}(0))
\leq \tilde{a}_{m-1} \leq \tilde{\C{B}}(\tilde{\alpha}_m).\]
This shows that $\alpha \notin \E{H}$.
\begin{comment}
For integers $m\geq 1$ and $j\geq 0$, consider the sets
\[B_{m,j}= \big\{ \alpha \in \D{R} \setminus \D{Q} \mid e^{-m/\alpha_j} \leq \alpha_{j+1} \leq 2 e^{-(1/\alpha_j)^{1/m}}\}.\]
We claim that
\[\bigcup_{m\geq 1} \bigcap_{j\geq 1} B_{m,j} \subseteq \E{B} \setminus \E{H}. \]
First note that
\[\tilde{\C{B}} (\alpha)=\sum_{i=-1}^{+\infty} \tilde{\beta}_{i} \log (1/\tilde{\alpha}_{i+1})
\leq \log (1/\tilde{\alpha}_0) + \sum_{i=0}^{+\infty} \tilde{\beta}_{i} (1/\tilde{\alpha}_i)
\leq \log (1/\tilde{\alpha}_0) + \sum_{i=0}^{+\infty} \tilde{\beta}_{i-1}.\]
For all $i\geq 0$, $\tilde{\alpha}_i \tilde{\alpha}_{i+1}\leq 1/2$. This implies that $\sum_{i=0}^{+\infty} \tilde{\beta}_{i-1}$
is a convergent series, and hence $\alpha\in \E{B}$.
first fine $x_0>0$ such that for all $x\geq x_0$ we have $\log x \leq x^{1/m}$.
consider two cases:
(1) there is $n \geq 1$ such that $\log (1/\alpha_{n+1}) \geq x_0$ and
\[h_{\tilde{\alpha}_n} \circ h_{\tilde{\alpha}_{n-1}} \circ \dots \circ h_{\tilde{\alpha}_1}(0) \geq \log (1/\alpha_{n+1}).\]
Then by an induction argument, and $\alpha \in \cap_{j\geq 1} B_{m,j}$, this shows that for all $m\geq n$ we have
\[h_{\tilde{\alpha}_m} \circ h_{\tilde{\alpha}_{n-1}} \circ \dots \circ h_{\tilde{\alpha}_1}(0) \geq \log (1/\alpha_{m+1}).\]
This shows that the compositions $h_{\tilde{\alpha}_n} \circ h_{\tilde{\alpha}_{n-1}} \circ \dots \circ h_{\tilde{\alpha}_1}(0)$
are based on the linear part of each $h_{\tilde{\alpha}_j}$.
Starting with the inequality $\tilde{\C{B}}(\alpha_1)\geq 1$, and using an inductive argument, one can see that
for all $n\geq 1$ we have
\[h_{\tilde{\alpha}_n} \circ h_{\tilde{\alpha}_{n-1}} \circ \dots \circ h_{\tilde{\alpha}_1}(0) \leq \tilde{\C{B}}(\tilde{\alpha}_{n+1}).\]
This implies that for arbitrary $m \geq n \geq 1$, we have
\[h_{\tilde{\alpha}_{m-1}} \circ \dots \circ h_{\tilde{\alpha}_n} (0)
\leq h_{\tilde{\alpha}_{m-1}} \circ \dots \circ h_{\tilde{\alpha}_n} (h_{\tilde{\alpha}_{n-1}} \circ \dots \circ h_{\tilde{\alpha}_1}(0))
\leq \tilde{\C{B}}(\tilde{\alpha}_{m}).\]
Therefore, $\alpha$ does not belong to $\E{H}$.
(2) for all sufficiently large $n$ we have
\[h_{\tilde{\alpha}_n} \circ h_{\tilde{\alpha}_{n-1}} \circ \dots \circ h_{\tilde{\alpha}_1}(0) \leq \log (1/\alpha_{n+1}).\]
Then use $\log (1/\alpha_{n+1}) \leq \C{B}(\alpha_{n+1})$ to complete the argument.
\end{comment}
\end{proof}
By classical results of Siegel \cite{Sie42} and Brjuno \cite{Brj71}, if $\alpha \in \E{B}$, then every germ of
holomorphic map $f(z)=e^{2\pi i \alpha} z+ O(z^2)$ is locally conformally conjugate to the rotation by $2\pi\alpha$
near $0$. On the other hand, Yoccoz in \cite{Yoc1988,Yoc95} proved that this condition is optimal in the quadratic family
$e^{2\pi i \alpha} z+ z^2$, i.e.\ if $\alpha \notin \E{B}$ then $e^{2\pi i \alpha} z+ z^2$ is not linearisable near $0$.
His approach is geometric, and avoids formidable calculations one encounters in the study of
small-divisors.
The optimality of this condition has been (re)confirmed for several classes of maps
\cite{PM93,Ge01,BC04,Ok04,Ok2005,FMS2018,Che19}, but in its general form for rational functions remains a
significant challenge in the field of holomorphic dynamics.
In this paper we do not rely on the optimality of this condition in any class of maps.
In \cite{Her79}, Herman carried out a comprehensive study of the problem of linearisation of orientation-preserving
diffeomorphisms of the circle $\D{R}/\D{Z}$ with irrational rotation number.
In particular, he presented a rather technical arithmetic condition which guaranteed the linearisation of such analytic
diffeomorphisms.
Although the linearisation problem for analytic circle diffeomorphisms close to rotations was successfully studied
earlier by Arnold \cite{Ar61}, no progress had been made in between.
Enhancing the work of Herman, Yoccoz identified the optimal arithmetic condition $\E{H}$ for the analytic
linearisation of analytic diffeomorphisms of the circle, \cite{Yoc95-ICM,Yoc02}.
The name, Herman numbers, for the class $\E{H}$ was suggested by Yoccoz in honour of the work of Herman on this problem.
Similarly, in this paper we do not use this form of the optimality of the condition $\E{H}$.
\subsection{Modified continued fraction, and the equivalent form of Herman numbers}
\label{SS:modified-fractions}
Let us recall the modified continued fraction algorithm we mentioned in \ref{SS:modified-fractions-mini}.
For $x\in \D{R}$, define $d(x, \D{Z})= \min_{k\in \D{Z}} |x-k|$. Let us fix an irrational number $\alpha \in \D{R}$.
Define the numbers $\alpha_n\in (0,1/2)$, for $n\geq 0$, according to
\begin{equation}
\alpha_0=d(\alpha, \D{Z}), \quad \alpha_{n+1}=d(1/\alpha_n, \D{Z}),
\end{equation}
Then, there are unique integers $a_n$, for $n\geq -1$, and $\varepsilon_n \in \{+1, -1\}$, for $n\geq 0$, such that
\begin{equation}
\alpha= a_{-1}+ \varepsilon_0 \alpha_0, \quad 1/\alpha_n= a_n + \varepsilon_{n+1} \alpha_{n+1}.
\end{equation}
Evidently, for all $n\geq 0$,
\begin{equation}
1/\alpha_n \in (a_n-1/2, a_n+1/2), \quad a_n\geq 2,
\end{equation}
and
\begin{equation}
\varepsilon_{n+1}=
\begin{cases}
+1 & \text{if } 1/\alpha_n \in (a_n, a_n+1/2), \\
-1 & \text{if } 1/\alpha_n \in (a_n-1/2, a_n).
\end{cases}
\end{equation}
We also defined $\alpha_{-1}=+1$.
The sequences $\{a_n\}$ and $\{\varepsilon_n\}$ provide us with the infinite continued fraction
\begin{equation*}\label{E:modified-expansion-alpha}
\alpha=a_{-1}+\cfrac{\varepsilon_0}{a_0+\cfrac{\varepsilon_1}{a_1+\cfrac{\varepsilon_2}{a_2+\dots}}}.
\end{equation*}
Consider the numbers
\[\beta_{-2}=\alpha, \; \beta_{-1}=+1, \; \beta_n= \beta_n(\alpha)= \textstyle{ \prod_{i=0}^n \alpha_i, \; \text{for } n \geq 0}.\]
In \cite{Yoc95}, Yoccoz defines the arithmetic series
\begin{equation}\label{E:Brjuno-condition}
\textstyle{ \C{B}(\alpha)= \sum_{n=0}^\infty \beta_{n-1} \log \alpha_n^{-1},}
\end{equation}
and calls it the Brjuno function. This function is defined on the set of irrational numbers, and takes values in
$(0, +\infty]$. One may extend $\C{B}$ onto $\D{Q}$, by setting $\C{B}(p/q)=+\infty$, for all $p/q\in \D{Q}$.
The Brjuno function satisfies the remarkable relations
\begin{equation}\label{E:Brjuno-functional-equations}
\begin{gathered}
\C{B}(\alpha)= \C{B}(\alpha+1)= \C{B}(-\alpha), \; \text{for } \alpha \in \D{R}, \\
\C{B}(\alpha)= \alpha \C{B}(1/\alpha)+ \log 1/\alpha, \; \text{for } \alpha \in (0,1/2).
\end{gathered}
\end{equation}
These show that one may think of the Brjuno function as a $\mathrm{PGL}(2,\D{Z})$-cocycle.
This point of view drives some of the technical arguments we present later in the paper,
notably in \refS{SS:Herman-tower}.
One may formally define the arithmetic classes of $\E{B}$ and $\E{H}$ using the modified continued fraction and the
modified function $\C{B}$ in the same fashion. The following two propositions guarantee that through this
we identify the same classes of irrational numbers.
\begin{propo}\label{P:brjuno-standard-vs-modified}
For all $\alpha \in \D{R}\setminus \D{Q}$ we have $|\C{B}(\alpha)- \tilde{\C{B}}(\alpha)| \leq 29$.
In particular, $\alpha$ is a Brjuno number if and only if $\C{B}(\alpha)< +\infty$.
\end{propo}
\begin{propo}\label{P:Herman-Yoccoz-criterion}
An irrational number $\alpha$ is a Herman number if and only if for all $n\geq 0$ there is $m\geq n$ such that
\[h_{\alpha_{m-1}} \circ \dots \circ h_{\alpha_n} (0)\geq \C{B}(\alpha_{m}).\]
\end{propo}
Although the criterion in the above proposition appears identical to the one given in
\refD{D:Herman-Yoccoz-criterion}, the value of $m$ for a given $n$ may be different.
As before, here $h_{\alpha_{m-1}} \circ \dots \circ h_{\alpha_n}$ is understood as the identity map when $m=n$,
and as $h_{\alpha_n}$ when $m=n+1$.
The remaining of this section is devoted to the proof of the above two propositions.
The main reason here is that the sequences $\{\tilde{\alpha}_n\}$ and $\{\alpha_n\}$ are closely related, and it is possible to
identify one from the other using an algorithm.
Let us define the sequence
\[ \textstyle{ c(-1)=-1,\; c(n)=-1+ \sum_{i=0}^{n} (3-\varepsilon_i)/2,\; \text{for } n\geq 0}.\]
That is, $c(n)$ is obtained from $c(n-1)$ by adding $+2$ if $\varepsilon_n=-1$ or adding $+1$ if $\varepsilon_n=+1$.
Clearly, $n \mapsto c(n)$ is strictly monotone with $c(n) \to +\infty$ as $n \to +\infty$.
\begin{lem}\label{L:fractions-related}
For all $n\geq -1$, the following hold:
\begin{itemize}
\item[(i)] if $\varepsilon_{n+1}=-1$, $\alpha_{n+1}= 1- \tilde{\alpha}_{c(n)+1}$,
and if $\varepsilon_{n+1}=+1$, $\alpha_{n+1}=\tilde{\alpha}_{c(n)+1}$;
\item[(ii)] $\alpha_{n+1}= \prod_{c(n)+1}^{c(n+1)} \tilde{\alpha}_i$;
\item[(iii)] $\tilde{\beta}_{c(n)}=\beta_n$;
\item[(iv)] $\C{B}(\tilde{\alpha}_{c(n)+1})= \C{B}(\alpha_{n+1})$.
\end{itemize}
\end{lem}
\begin{proof}
We prove (i) by induction on $n$.
We start with $n=-1$. If $\varepsilon_0=+1$, $\alpha_0=\tilde{\alpha}_0= \tilde{\alpha}_{c(-1)+1}$.
If $\varepsilon_0=-1$, $\alpha_0 =1-\tilde{\alpha}_0=1-\tilde{\alpha}_{c(-1)+1}$.
Now assume that the assertion in (i) is true for $n-1$. To prove it for $n$, we consider two cases:
First assume that $\varepsilon_n=+1$. By the induction hypothesis for $n-1$, $\alpha_n= \tilde{\alpha}_{c(n-1)+1}=\tilde{\alpha}_{c(n)}$.
Hence, $1/\alpha_n=1/\tilde{\alpha}_{c(n)}$.
Now, if $\varepsilon_{n+1}=+1$, $1/\alpha_n=1/\tilde{\alpha}_{c(n)}$ leads to $\alpha_{n+1}=\tilde{\alpha}_{c(n)+1}$.
If $\varepsilon_{n+1}=-1$, $1/\alpha_n=1/\tilde{\alpha}_{c(n)}$ leads to $\alpha_{n+1}=1-\tilde{\alpha}_{c(n)+1}$.
Now assume that $\varepsilon_n=-1$. By the induction hypothesis, $\alpha_n=1-\tilde{\alpha}_{c(n-1)+1}=1-\tilde{\alpha}_{c(n)-1}$.
As $\alpha_n \in (0,1/2)$, $\tilde{\alpha}_{c(n)-1} \in (1/2,1)$.
Then, $\tilde{\alpha}_{c(n)}= 1/\tilde{\alpha}_{c(n)-1}-1= 1/(1-\alpha_n)-1= \alpha_n /(1-\alpha_n)$.
Hence, $1/\tilde{\alpha}_{c(n)}= 1/\alpha_n-1$.
Now, if $\varepsilon_{n+1}=+1$, $1/\tilde{\alpha}_{c(n)}= 1/\alpha_n-1$ leads to $\tilde{\alpha}_{c(n)+1}= \alpha_{n+1}$.
If $\varepsilon_{n+1}=-1$, $1/\tilde{\alpha}_{c(n)}= 1/\alpha_n-1$ leads to $1-\tilde{\alpha}_{c(n)+1}=\alpha_{n+1}$.
Part (ii): If $\varepsilon_{n+1}=+1$, by Part (i), $\alpha_{n+1}= \tilde{\alpha}_{c(n)+1}= \tilde{\alpha}_{c(n+1)}$.
If $\varepsilon_{n+1}=-1$, by Part (i), $\alpha_{n+1}=1-\tilde{\alpha}_{c(n)+1}$. As
$\alpha_{n+1}\in (0,1/2)$ we conclude that $\tilde{\alpha}_{c(n)+1}\in (1/2,1)$, which implies that
$\tilde{\alpha}_{c(n)+2}= 1/\tilde{\alpha}_{c(n)+1}-1$.
Therefore,
\[\tilde{\alpha}_{c(n+1)} \tilde{\alpha}_{c(n)+1}=\tilde{\alpha}_{c(n)+2} \tilde{\alpha}_{c(n)+1}=1-\tilde{\alpha}_{c(n)+1}= \alpha_{n+1}.\]
Part (iii): By the formula in Part (ii), and the definition of $c(n)$,
\[\textstyle{ \beta_n=\prod_{m=0}^n \alpha_m =\prod_{m=0}^n \Big (\prod_{i=c(m-1)+1}^{c(m)} \tilde{\alpha}_i\Big)
=\prod_{i=0}^{c(n)} \tilde{\alpha}_i= \tilde{\beta}_{c(n)}. }\]
Part (iv): If $\varepsilon_{n+1}=+1$, by Part (i), $\alpha_{n+1}= \tilde{\alpha}_{c(n)+1}$, and hence
$\C{B}(\tilde{\alpha}_{c(n)+1})= \C{B}(\alpha_{n+1})$.
If $\varepsilon_{n+1}=-1$, by Part (i), $\alpha_{n+1}= 1-\tilde{\alpha}_{c(n)+1}$. Using \refE{E:Brjuno-functional-equations},
\[\C{B}(\alpha_{n+1})=\C{B}(1-\tilde{\alpha}_{c(n)+1})=\C{B}(-\tilde{\alpha}_{c(n)+1})= \C{B}(\tilde{\alpha}_{c(n)+1}). \qedhere\]
\end{proof}
\begin{proof}[Proof of \refP{P:brjuno-standard-vs-modified}]
Fix $n \geq 0$.
If $\varepsilon_n=+1$ then $c(n)=c(n-1)+1$, and by \refL{L:fractions-related}-(iii),
\begin{equation}\label{E:P:brjuno-standard-vs-modified-1}
\beta_{n-1}\log \beta_n = \tilde{\beta}_{c(n-1)}\log \tilde{\beta}_{c(n-1)+1}= \tilde{\beta}_{c(n)-1}\log \tilde{\beta}_{c(n)}.
\end{equation}
If $\varepsilon_n=-1$ then $c(n)= c(n-1)+2$, and by \refL{L:fractions-related}-(iii),
\[\tilde{\beta}_{c(n)-1}
=\tilde{\beta}_{c(n-1)+1} = \tilde{\beta}_{c(n-1)} \tilde{\alpha}_{c(n-1)+1} = \beta_{n-1} (1-\alpha_n)= \beta_{n-1}- \beta_{n}.\]
and therefore
\begin{equation}\label{E:P:brjuno-standard-vs-modified-2}
\begin{aligned}
\big(\tilde{\beta}_{c(n-1)}\log\tilde{\beta}_{c(n-1)+1}&+\tilde{\beta}_{c(n)-1}\log\tilde{\beta}_{c(n)} \big) -\beta_{n-1}\log\beta_n\\
&=\big( \beta_{n-1}\log (\beta_{n-1}- \beta_n) + (\beta_{n-1} - \beta_n) \log \beta_{n}\big) - \beta_{n-1}\log\beta_n\\
&= \beta_{n-1}\log (\beta_{n-1}- \beta_n) - \beta_n \log \beta_{n}
\end{aligned}
\end{equation}
Combing \eqref{E:P:brjuno-standard-vs-modified-1} and \eqref{E:P:brjuno-standard-vs-modified-2},
and using $\beta_{n-1}- \beta_n = \beta_{n-1}(1-\alpha_n)$, we conclude that for all $m\geq 0$ we have
\begin{align*}
\textstyle{ \sum_{i=0}^{c(m)} \tilde{\beta}_{i-1} \log \tilde{\beta}_i} & - \textstyle{\sum_{n=0}^m \beta_{n-1} \log \beta_n} \\
&=\textstyle{ \sum_{n=0}^m \left (\sum_{i=c(n-1)+1}^{c(n)} (\tilde{\beta}_{i-1} \log \tilde{\beta}_i) - \beta_{n-1}\log \beta_n \right) }\\
&= \textstyle{ \sum_{n=0\, ;\, \varepsilon_n=-1}^m \left (\beta_{n-1}\log \beta_{n-1} + \beta_{n-1} \log (1-\alpha_n) - \beta_n \log \beta_n\right).}
\end{align*}
On the other hand,
\begin{equation*}
\textstyle{ \sum_{n=0}^m \beta_{n-1} \log (1/\alpha_n) + \sum_{n=0}^m \beta_{n-1} \log \beta_n
= \sum_{n=0}^m \beta_{n-1} \log \beta_{n-1}, }
\end{equation*}
and similarly,
\[\textstyle{ - \sum_{i=0}^{c(m)} \tilde{\beta}_{i-1} \log \tilde{\beta}_i - \sum_{i=0}^{c(m)} \tilde{\beta}_{i-1} \log (1/\tilde{\alpha}_i)
= - \sum_{i=0}^{c(m)} \tilde{\beta}_{i-1} \log \tilde{\beta}_{i-1}. }\]
Recall that $\tilde{\beta}_{-1}=\beta_{-1}=1$. Adding the above three equations, we conclude that
\begin{equation*}
\begin{aligned}
\textstyle{ \left |\sum_{n=0}^m \beta_{n-1} \log (1/\alpha_n) \right. } & \textstyle{ \left .- \sum_{i=0}^{c(m)} \tilde{\beta}_{i-1} \log (1/\tilde{\alpha}_i) \right |} \\
& \textstyle{ \leq 3 \sum_{n=0}^{+\infty} |\beta_{n} \log \beta_{n}| + \sum_{n=0}^{+\infty} |\beta_{n-1} \log (1-\alpha_n)| }
+ \textstyle{ \sum_{i=0}^{+\infty} |\tilde{\beta}_{i} \log \tilde{\beta}_{i}|}
\end{aligned}
\end{equation*}
Since $|x \log x| \leq 2 \sqrt{x}$ for $x\in (0,1)$,
\begin{equation}\label{E:P:Brjuno-Yoccoz-equivalent}
\begin{aligned}
\textstyle{ \sum_{n=0}^{+\infty} |\tilde{\beta}_{n}\log \tilde{\beta}_{n}| }
\textstyle{ \leq 2 \sum_{n=0}^{+\infty} (\tilde{\beta}_{n})^{1/2} }
& \textstyle{ \leq 2 \sum_{n=0}^{+\infty} (\tilde{\beta}_{2n})^{1/2} + 2 \sum_{n=0}^{+\infty} (\tilde{\beta}_{2n+1})^{1/2} } \\
& \textstyle{ \leq 2 (\tilde{\beta}_0)^{1/2} \sum_{n=0}^{+\infty} 2^{-n/2} + 2 \sum_{n=1}^{+\infty} 2^{-n/2} } \\
& \leq 6+ 4 \cdot 2^{1/2}.
\end{aligned}
\end{equation}
On the other hand, $\beta_n\leq 2^{-n-1}$, for all $n\geq 0$.
Using $|x \log x| \geq 2 \sqrt{x}$, for $x \in (0,1)$, $\alpha_j \in (0, 1/2)$, we obtain
\begin{equation*}
\textstyle{ \sum_{n=0}^k |\beta_{n} \log \beta_{n}| \leq 2 \sum_{n=0}^{+\infty} \sqrt{\beta_{n}} \leq
2 \sum_{n=0}^{+\infty} 1/2^{(n+1)/2} = 2+ 2 \sqrt{2}, }
\end{equation*}
and
\[ \textstyle{\sum_{n=0}^{+\infty} |\beta_{n-1} \log (1-\alpha_n)| \leq \log 2 \sum_{n=0}^{+\infty} \beta_{n-1} = 2 \log 2. }\]
This completes the proof of the proposition.
\end{proof}
\begin{lem}\label{L:h-vs-cocycle}
We have,
\begin{itemize}
\item[(i)] for all $r \in (0,1)$,
\[h_r (\C{B}(r)) \geq \C{B}(1/r)+1, \quad h_r (\tilde{\C{B}}(r)) \geq \tilde{\C{B}}(1/r)+1.\]
\item[(ii)] if there are $m\geq n \geq 0$ satisfying
$h_{\tilde{\alpha}_{m-1}} \circ \dots \circ h_{\tilde{\alpha}_n} (0)\geq \tilde{\C{B}}(\tilde{\alpha}_{m})$,
then
\[\lim_{m \to +\infty} h_{\tilde{\alpha}_{m-1}} \circ \dots \circ h_{\tilde{\alpha}_n} (0) - \tilde{\C{B}}(\tilde{\alpha}_{m})=+\infty.\]
\item[(iii)] if there are $m\geq n \geq 0$ satisfying
$h_{\alpha_{m-1}} \circ \dots \circ h_{\alpha_n} (0)\geq \C{B}(\alpha_{m})$, then
\[\lim_{m\to +\infty} h_{\alpha_{m-1}} \circ \dots \circ h_{\alpha_n} (0) - \C{B}(\alpha_{m})=+\infty.\]
\end{itemize}
\end{lem}
\begin{proof}
Note that for all $r\in (0,1)$ and all $y \in \D{R}$,
\[h_r(y) \geq r^{-1} y + r^{-1} \log r+ 1.\]
If $y\geq \log r^{-1}$, $h_r(y)=r^{-1}y + r^{-1} \log r + r^{-1} \geq r^{-1}y + r^{-1} \log r +1$.
Using the inequality $x \geq 1+ \log x$, for $x>0$, we note that
$r e^y \geq 1+ \log (re^y) = y + \log r +1 \geq y + \log r + r$.
This implies the above inequality for $y < \log r^{-1}$.
By the above inequality, as well as \eqref{E:Brjuno-functional-equations-standard} and \eqref{E:Brjuno-functional-equations}, we obtain
\[h_r (\C{B}(r)) \geq r^{-1} \C{B}(r)+ \log r +1 = \C{B}(1/r)+1, \quad
h_r (\tilde{\C{B}}(r)) \geq r^{-1} \tilde{\C{B}}(r)+ \log r +1 = \tilde{\C{B}}(1/r)+1.\]
If the inequalities in (ii) and (iii) hold, we may use the inequality in (i) and $h_r(y+1)\geq h_r(y)+1$ in
\refE{E:h_r-properties}, to obtain
\[h_{\tilde{\alpha}_{m+j-1}} \circ \dots \circ h_{\tilde{\alpha}_n} (0)\geq \tilde{\C{B}}(\tilde{\alpha}_{m+j})+j, \quad
h_{\alpha_{m+j-1}} \circ \dots \circ h_{\alpha_n} (0)\geq \C{B}(\alpha_{m+j})+j.\qedhere\]
\end{proof}
\begin{lem}\label{L:herman-blocks}
Let $r_1 \in (1/2,1)$, $r_2=1/r_1-1\in (0,1)$, and $r=r_1 r_2 \in (0, 1/2)$. Then, for all $y \geq e^2$ we have
\[\big| h^{-1}_r(y)- h^{-1}_{r_1} \circ h^{-1}_{r_2}(y)\big| \leq 1+ e^{-1}.\]
\end{lem}
\begin{proof}
The inverse map $h_r^{-1}: (0, +\infty) \to \D{R}$ is given by the formula
\[h_r^{-1}(y)=
\begin{cases}
r y + \log r^{-1} -1 & \text{if } y \geq 1/r, \\
\log y & \text{if } 0 < y \leq 1/r.
\end{cases}\]
Since $y\geq e^2$, one can see that $h_{r_2}^{-1}(y)\geq 1+ r_2 = 1/r_1$. Thus,
$h^{-1}_{r_1} \circ h^{-1}_{r_2}(y)= r_1 h_{r_2}^{-1}(y)+ \log r_1^{-1}-1$.
Using $r_2=1/r_1-1$ and the elementary inequality $|x \log x| \leq 1/e$, for $x\in (0,1)$, we have
\[(1-r_1) \log (1/r_2) = (r_2/(1+ r_2)) \log r^{-1}_2 \leq e^{-1}/(1+r_2) \leq e^{-1}.\]
We consider three cases:
\noindent (1) $y \leq 1/r_2$:
Since $1/r_2 \leq 1/r$, we get
\begin{align*}
\big| h^{-1}_r(y)- h^{-1}_{r_1} \circ h^{-1}_{r_2}(y)\big|
&= \big | \log y - (r_1 \log y + \log r_1^{-1} -1)\big| \\
& \leq (1-r_1) \log y + \big |1- \log r^{-1}_1 \big| \\
& \leq (1-r_1) \log r_2^{-1} + \big |1- \log r^{-1}_1 \big| \leq e^{-1}+ 1.
\end{align*}
\noindent (2) $1/r_2 \leq y \leq 1/r$:
Then,
\begin{align*}
h^{-1}_r(y)- h^{-1}_{r_1} \circ h^{-1}_{r_2}(y)
& = \log y - (r_1 (r_2 y+\log r_2^{-1}-1) + \log r_1^{-1} -1) \\
&= \log (yr_1) - r_1 \log r_2^{-1} + r_1 (1-r_2 y) + 1.
\end{align*}
On the other hand, we have
\[-1/2 \leq r_1 -1 \leq r_1 - ry = r_1 - r_1 r_2 y = r_1(1-r_2y) \leq 0,\]
and, using $y r_1 r_2= y r \leq 1$, we get
\[\log yr_1- r_1 \log r_2^{-1} \leq \log r^{-1}_2 - r_1 \log r_2^{-1} = (1-r_1) \log r_2^{-1} \leq e^{-1},\]
and, using $y\geq 1/r_2$,
\[\log yr_1- r_1 \log r_2^{-1} \geq \log (r_1/r_2) - r_1 \log r_2^{-1} = \log r_1 + (1-r_1) \log r_2 ^{-1}
\geq \log r_1 \geq - \log 2.\]
Combining the above inequalities we get
\[-1 - e^{-1} \leq 1-1/2 -\log 2 \leq h^{-1}_r(y)- h^{-1}_{r_1} \circ h^{-1}_{r_2}(y) \leq 1+ e^{-1}.\]
\noindent (3) $y \geq 1/r$:
Using $r_1 r_2=r$, we get
\begin{align*}
\big| h^{-1}_r(y)- h^{-1}_{r_1} \circ h^{-1}_{r_2}(y)\big|
& = \big | r y + \log r^{-1} -1 - (r_1 (r_2 y+\log r_2^{-1}-1) + \log r_1^{-1} -1)\big| \\
&= (1-r_1) \log r_2^{-1} + r_1 \leq e^{-1}+ 1.
\end{align*}
This completes the proof of the lemma.
\end{proof}
\begin{lem}\label{L:herman-chains}
Let $m > n \geq 0$ and $y \geq e^2$. Assume that at least one of the following holds:
\begin{itemize}
\item[(i)] $h^{-1}_{\alpha_n} \circ h^{-1}_{\alpha_{n+1}} \circ \dots \circ h^{-1}_{\alpha_m}(y)$ is defined and is at least $e^2$,
\item[(ii)]
$h^{-1}_{\tilde{\alpha}_{c(n-1)+1}} \circ h^{-1}_{\tilde{\alpha}_{c(n-1)+2}} \circ \dots \circ h^{-1}_{\tilde{\alpha}_{c(m)}}(y)$
is defined and is at least $e^2$.
\end{itemize}
Then,
\[\big |h^{-1}_{\alpha_n} \circ h^{-1}_{\alpha_{n+1}} \circ \dots \circ h^{-1}_{\alpha_m}(y)
- h^{-1}_{\tilde{\alpha}_{c(n-1)+1}} \circ h^{-1}_{\tilde{\alpha}_{c(n-1)+2}} \circ \dots \circ h^{-1}_{\tilde{\alpha}_{c(m)}}(y)
\big |\leq 2(1+e^{-1}).\]
\end{lem}
In the above lemma, it is part of the conclusion that if one of the compositions in items (i) and (ii) is defined and
is at least $e^2$, then the composition in the other item is defined as well.
\begin{proof}
First assume that item (i) holds.
By \refE{E:h_r-properties}, $h_r(t) \geq t$ for all $t \geq 0$ and $r\in (0,1)$.
This implies that for all $j$ with $n \leq j \leq m$,
$h^{-1}_{\alpha_j} \circ h^{-1}_{\alpha_{j+1}} \circ \dots \circ h^{-1}_{\alpha_m}(y)\geq e^2$.
Let
\[y_j=h^{-1}_{\alpha_j} \circ h^{-1}_{\alpha_{j+1}} \circ \dots \circ h^{-1}_{\alpha_m}(y), \; \text{for } n \leq j \leq m.\]
By an inductive argument, we show that for all $j$ with $n \leq j \leq m$,
\[\tilde{y}_j
=h^{-1}_{\tilde{\alpha}_{c(j-1)+1}} \circ h^{-1}_{\tilde{\alpha}_{c(n-1)+2}} \circ \dots \circ h^{-1}_{\tilde{\alpha}_{c(m)}}(y),\]
is defined, and satisfies
\[|y_j -\tilde{y}_j| \leq 2 (1+ e^{-1}).\]
We start with $j=m$.
If $\varepsilon_m=+1$, $c(m-1)+1=c(m)$, and since $y \geq e^2 > 0$, $\tilde{y}_m=h^{-1}_{\tilde{\alpha}_{c(m)}}(y)$ is
defined. Moreover, by \refL{L:fractions-related}, $\alpha_m=\tilde{\alpha}_{c(m-1)+1}= \tilde{\alpha}_{c(m)}$, and hence,
$y_m=\tilde{y}_m$.
If $\varepsilon_m=-1$, $c(m-1)+1=c(m)-1$. Since $y\geq e^2 >1$,
$h^{-1}_{\tilde{\alpha}_{c(m)}}(y) > h^{-1}_{\tilde{\alpha}_{c(m)}}(1)=0$, and hence
$\tilde{y}_m=h^{-1}_{\tilde{\alpha}_{c(m-1)+1}} \circ h^{-1}_{\tilde{\alpha}_{c(m)}}(y)$ is defined.
Moreover, by \refL{L:fractions-related}, we have $\alpha_m=1- \tilde{\alpha}_{c(m-1)+1}$ and
$\alpha_m=\tilde{\alpha}_{c(m-1)+1} \tilde{\alpha}_{c(m)}$.
As $\alpha_m \in (0,1/2)$, $\tilde{\alpha}_{c(m-1)+1}\in (1/2,1)$, and hence
$\tilde{\alpha}_{c(m)}=\tilde{\alpha}_{c(m-1)+2}=1/\tilde{\alpha}_{c(m-1)+1}-1$.
We use \refL{L:herman-blocks} with $r=\alpha_m$, $r_1=\tilde{\alpha}_{c(m-1)+1}$ and $r_2=\tilde{\alpha}_{c(m)}$
to conclude that
$|y_m-\tilde{y}_m|= |h_{\alpha_m}^{-1}(y)- h^{-1}_{\tilde{\alpha}_c(m-1)+1} \circ h^{-1}_{\tilde{\alpha}_{c(m)}}(y)|
\leq 1+e^{-1}$.
Now assume that the assertion holds for $j+1$, with $n < j+1 \leq m$. To prove it for $j$ we consider two cases.
If $\varepsilon_j=+1$, $c(j-1)+1=c(j)$. By the induction hypothesis for $j+1$, $|y_{j+1}-\tilde{y}_{j+1}| \leq 2 (1+e^{-1})$.
Combining with $y_{j+1}\geq e^2$, we obtain $\tilde{y}_{j+1}\geq e^2 - 2 (1+e^{-1})>4$. This implies that
$\tilde{y}_j=h^{-1}_{\tilde{\alpha}_{c(j)}}(\tilde{y}_{j+1})$ is defined.
On the other hand, by \refL{L:fractions-related}, $\alpha_j=\tilde{\alpha}_{c(j-1)+1}= \tilde{\alpha}_{c(j)}$.
Moreover, for all $t \geq 4$ and all $r \in (0,1)$, $(h_r^{-1})'(t)\leq 1/2$.
Therefore,
\begin{align*}
|y_j-\tilde{y}_j|= |h^{-1}_{\alpha_j}(y_{j+1}) - h^{-1}_{\tilde{\alpha}_{c(j)}}(\tilde{y}_{j+1})|
& = |h^{-1}_{\alpha_j}(y_{j+1}) - h^{-1}_{\alpha_j}(\tilde{y}_{j+1})| \\
& \leq (1/2) |y_{j+1}- \tilde{y}_{j+1}| \leq (1/2) 2 (1+e^{-1}) \leq 2 (1+e^{-1}).
\end{align*}
If $\varepsilon_j=-1$, $c(j-1)+1=c(j)-1$. As in the previous case, the induction hypothesis for $j+1$ gives us
$|y_{j+1}-\tilde{y}_{j+1}| \leq 2 (1+e^{-1})$.
Combining with $y_{j+1}\geq e^2$ we obtain $\tilde{y}_{j+1}\geq e^2 - 2 (1+e^{-1})>4$.
This implies that $\tilde{y}_j=h^{-1}_{\tilde{\alpha}_{c(j)-1}}\circ h^{-1}_{\tilde{\alpha}_{c(j)}}(\tilde{y}_{j+1})$ is defined.
On the other hand, by \refL{L:fractions-related}, $\alpha_j=1- \tilde{\alpha}_{c(j-1)+1}$, and
$\tilde{\alpha}_{c(j)}=\tilde{\alpha}_{c(j-1)+2}=1/\tilde{\alpha}_{c(j-1)+1}-1$.
Using \refL{L:herman-blocks} with $r=\alpha_j$, $r_1=\tilde{\alpha}_{c(j-1)+1}$ and $r_2=\tilde{\alpha}_{c(j)}$
we get
\[|h_{\alpha_j}(y_j) - h^{-1}_{\tilde{\alpha}_{c(j)-1}}\circ h^{-1}_{\tilde{\alpha}_{c(j)}}(y_j) |\leq 1+e^{-1}.\]
By elementary calculations one may see that for all $t\geq 4$,
$\big(h^{-1}_{\tilde{\alpha}_{c(j)-1}}\circ h^{-1}_{\tilde{\alpha}_{c(j)}}\big)'(t)\leq 1/2$.
Therefore,
\begin{align*}
|y_j-\tilde{y}_j|
&=|h^{-1}_{\alpha_j}(y_{j+1})-h^{-1}_{\tilde{\alpha}_{c(j)-1}}\circ h^{-1}_{\tilde{\alpha}_{c(j)}}(\tilde{y}_{j+1})|\\
& \leq |h^{-1}_{\alpha_j}(y_{j+1}) - h^{-1}_{\tilde{\alpha}_{c(j)-1}}\circ h^{-1}_{\tilde{\alpha}_{c(j)}}(y_{j+1})| \\
& \quad \quad + |h^{-1}_{\tilde{\alpha}_{c(j)-1}}\circ h^{-1}_{\tilde{\alpha}_{c(j)}}(y_{j+1})
- h^{-1}_{\tilde{\alpha}_{c(j)-1}}\circ h^{-1}_{\tilde{\alpha}_{c(j)}}(\tilde{y}_{j+1})| \\
& \leq (1+ e^{-1})+ (1/2) \cdot |y_{j+1}- \tilde{y}_{j+1}| \\
& \leq (1+ e^{-1})+ (1/2) \cdot 2 (1+e^{-1}) = 2 (1+e^{-1}).
\end{align*}
This completes the proof of the induction.
The proof of the lemma when item (ii) holds is similar, indeed easier, and is left to the reader.
\end{proof}
\begin{proof}[Proof of \refP{P:Herman-Yoccoz-criterion}]
Let $\exp^{\circ i}$ denote the $i$-fold composition of the exponential map $x\mapsto e^x$.
Assume that $\alpha$ satisfies the criterion in \refP{P:Herman-Yoccoz-criterion}.
Fix an arbitrary $n \geq 0$.
Let $n'$ be an integer with
\begin{equation}\label{E:P:Herman-Yoccoz-criterion-2}
n' \geq n + \exp^{\circ 2}(2) + 2 (1+e^{-1}).
\end{equation}
Applying the criterion in \refP{P:Herman-Yoccoz-criterion} with $n'$, there is $m' \geq n'$ such that
\[h_{\alpha_{m'-1}} \circ \dots \circ h_{\alpha_{n'}} (0) \geq \C{B}(\alpha_{m'}).\]
By \refL{L:h-vs-cocycle}-(iii), there is $m \geq n'$ such that
\[h_{\alpha_{m}} \circ h_{\alpha_{m-1}} \circ \dots \circ h_{\alpha_{n'}} (0)
\geq \C{B}(\alpha_{m+1})+29.\]
We are going to show that the pair $c(m)+1, n$ satisfies the inequality in \refD{D:Herman-Yoccoz-criterion}.
By \refP{P:brjuno-standard-vs-modified} and \refL{L:fractions-related}-(iv),
\[\C{B}(\alpha_{m+1})+ 29= \C{B}(\tilde{\alpha}_{c(m)+1})+ 29 \geq \tilde{\C{B}}(\tilde{\alpha}_{c(m)+1}).\]
Combining with the previous inequality, we obtain
\begin{equation*}
h_{\alpha_{m}} \circ h_{\alpha_{m-1}} \circ \dots \circ h_{\alpha_{n'}} (0) \geq \tilde{\C{B}}(\tilde{\alpha}_{c(m)+1}).
\end{equation*}
This implies that there is an integer $j$, with $n' \leq j \leq m$ such that
\begin{equation}\label{E:P:Herman-Yoccoz-criterion-1}
h^{-1}_{\alpha_j} \circ \dots \circ h^{-1}_{\alpha_{m}} (\tilde{\C{B}}(\tilde{\alpha}_{c(m)+1})) \leq 0.
\end{equation}
Now we consider two cases based on the size of $\tilde{\C{B}}(\tilde{\alpha}_{c(m)+1})$.
First assume that $\tilde{\C{B}}(\tilde{\alpha}_{c(m)+1}) < e^2$.
Using $h_r(y)\geq y+1$ several times, we obtain
\[e^2 \leq n'-n \leq m-n \leq c(m)-n \leq h_{\tilde{\alpha}_{c(m)}} \circ \dots \circ h_{\tilde{\alpha}_n} (0).\]
Thus, we have the desired inequality
$\tilde{\C{B}}(\tilde{\alpha}_{c(m)+1}) \leq h_{\tilde{\alpha}_{c(m)}} \circ \dots \circ h_{\tilde{\alpha}_n} (0)$.
Now assume that $\tilde{\C{B}}(\tilde{\alpha}_{c(m)+1}) \geq e^2$.
Combining with \refE{E:P:Herman-Yoccoz-criterion-1}, there is an integer $j'$, with $j \leq j' \leq m$, such that
\[h^{-1}_{\alpha_{j'}} \circ \dots \circ h^{-1}_{\alpha_{m}} (\tilde{\C{B}}(\tilde{\alpha}_{c(m)+1}))
\in [e^2,\exp^{\circ 1}(e^2)].\]
This is because any such orbit must pass through the interval $[e^2,\exp^{\circ 1}(e^2)]$.
We may now apply \refL{L:herman-chains}, to conclude that
$h^{-1}_{\tilde{\alpha}_{c(j'-1)+1}} \circ \dots \circ h^{-1}_{\tilde{\alpha}_{c(m)}}(\tilde{\C{B}}(\tilde{\alpha}_{c(m)+1}))$
is defined and
\begin{align*}
h^{-1}_{\tilde{\alpha}_{c(j'-1)+1}} \circ \dots \circ & h^{-1}_{\tilde{\alpha}_{c(m)}}(\tilde{\C{B}}(\tilde{\alpha}_{c(m)+1})) \\
& \leq
h^{-1}_{\tilde{\alpha}_{c(j'-1)+1}} \circ \dots \circ h^{-1}_{\tilde{\alpha}_{c(m)}}(\tilde{\C{B}}(\tilde{\alpha}_{c(m)+1}))
- h^{-1}_{\alpha_j'} \circ \dots \circ h^{-1}_{\alpha_{m}}(\tilde{\C{B}}(\tilde{\alpha}_{c(m)+1})) \\
& \qquad \qquad + h^{-1}_{\alpha_{j'}} \circ \dots \circ h^{-1}_{\alpha_{m}}(\tilde{\C{B}}(\tilde{\alpha}_{c(m)+1})) \\
& \leq 2(1+e^{-1}) + \exp^{\circ 1}(e^2) = 2 (1+ e^{-1}) + \exp^{\circ 2}(2).
\end{align*}
On the other hand, for all $j'\geq 0$, $j' \leq c(j')$.
Then, combining with \refE{E:P:Herman-Yoccoz-criterion-2}, we have
\[c(j'-1)-n+1 \geq j'-1-n+1 \geq j - n \geq n' -n \geq 2 (1+e^{-1})+\exp^{\circ 2}(2).\]
Now, using $h_r(y)\geq y+1$ several times, we obtain
\[2(1+e^{-1}) +\exp^{\circ 2}(2) \leq c(j'-1)-n+1 \leq h_{\tilde{\alpha}_{c(j'-1)}} \circ \dots \circ h_{\tilde{\alpha}_n}(0).\]
Combining the above inequalities we obtain
\[h^{-1}_{\tilde{\alpha}_{c(j'-1)+1}} \circ \dots \circ h^{-1}_{\tilde{\alpha}_{c(m)}}(\tilde{\C{B}}(\tilde{\alpha}_{c(m)+1}))
\leq h_{\tilde{\alpha}_{c(j'-1)}} \circ \dots \circ h_{\tilde{\alpha}_n}(0),\]
which gives the desired relation
\[\tilde{\C{B}}(\tilde{\alpha}_{c(m)+1}) \leq h_{\tilde{\alpha}_{c(m)}} \circ \dots \circ h_{\tilde{\alpha}_n}(0).\]
This completes the proof of one side of the proposition.
The other side of the proposition may be proved along the same lines, but there are some technical
differences.
Assume that $\alpha$ satisfies \refD{D:Herman-Yoccoz-criterion}.
Fix an arbitrary $n \geq 0$.
Let $n'$ be an integer with
\begin{equation}\label{E:P:Herman-Yoccoz-criterion-3}
n' \geq 2n + 2 \exp^{\circ 3}(2) + 4 (1+e^{-1}).
\end{equation}
Applying \refD{D:Herman-Yoccoz-criterion} with $n'$, there is $m' \geq n'$ such that
\[h_{\tilde{\alpha}_{m'-1}} \circ \dots \circ h_{\tilde{\alpha}_{n'}} (0) \geq \tilde{\C{B}}(\tilde{\alpha}_{m'}).\]
Recall that $c(m)\to +\infty$ as $m\to +\infty$. By \refL{L:h-vs-cocycle}-(ii), there is $m \geq n'$ such that
\[h_{\tilde{\alpha}_{c(m)}} \circ h_{\tilde{\alpha}_{c(m)-1}} \circ \dots \circ h_{\tilde{\alpha}_{n'}} (0)
\geq \tilde{\C{B}}(\tilde{\alpha}_{c(m)+1})+29.\]
Note that $c(m)\geq m \geq n'$.
We are going to show that the pair $m+1, n$ satisfies the inequality in \refP{P:Herman-Yoccoz-criterion}.
By \refP{P:brjuno-standard-vs-modified} and \refL{L:fractions-related}-(iv),
\[ \tilde{\C{B}}(\tilde{\alpha}_{c(m)+1})+29 \geq \C{B}(\tilde{\alpha}_{c(m)+1})= \C{B}(\alpha_{m+1}).\]
Combining with the previous inequality, we obtain
\begin{equation*}
h_{\tilde{\alpha}_{c(m)}} \circ h_{\tilde{\alpha}_{c(m)-1}} \circ \dots \circ h_{\tilde{\alpha}_{n'}} (0)
\geq \C{B}(\alpha_{m+1}).
\end{equation*}
This implies that there is an integer $j$, with $n' \leq j \leq c(m)$ such that
\begin{equation}\label{E:P:Herman-Yoccoz-criterion-4}
h^{-1}_{\tilde{\alpha}_j} \circ \dots \circ h^{-1}_{\tilde{\alpha}_{c(m)}} (\C{B}(\alpha_{m+1})) \leq 0.
\end{equation}
Now we consider two cases based on the size of $\C{B}(\alpha_{m+1})$.
First assume that $\C{B}(\alpha_{m+1}) < \exp^{\circ 3}(2)$.
Using $h_r(y)\geq y+1$ several times, we obtain
\[\exp^{\circ 3}(2) \leq n'-n \leq m-n \leq h_{\alpha_{m}} \circ \dots \circ h_{\alpha_n} (0).\]
Thus, we have the desired inequality $\C{B}(\alpha_{m+1}) \leq h_{\alpha_{m}} \circ \dots \circ h_{\alpha_n} (0)$.
Now assume that $\C{B}(\alpha_{m+1}) \geq \exp^{\circ 3}(2)= \exp^{\circ 2}(e^2)$.
Combining with \refE{E:P:Herman-Yoccoz-criterion-4}, there is an integer $j'$, with $j \leq c(j')-1 \leq c(m)$, such that
\[h^{-1}_{\tilde{\alpha}_{c(j')-1}} \circ\dots\circ h^{-1}_{\tilde{\alpha}_{c(m)}} (\C{B}(\alpha_{m+1}))
\in [e^2,\exp^{\circ 2}(e^2)].\]
This is because any such orbit has two consecutive elements in the interval $[e^2,\exp^{\circ 3}(2)]$,
and the image of the map $i \mapsto c(i)$ covers at least one element of any pair of consecutive integers.
We may now apply \refL{L:herman-chains}, to conclude that
$h^{-1}_{\alpha_{j'}} \circ \dots \circ h^{-1}_{\alpha_{m}}( \C{B}(\alpha_{m+1}))$ is defined and
\begin{align*}
h^{-1}_{\alpha_{j'}} \circ \dots \circ h^{-1}_{\alpha_{m}}( \C{B}(\alpha_{m+1}))
& \leq
h^{-1}_{\alpha_{j'}} \circ \dots \circ h^{-1}_{\alpha_{m}}( \C{B}(\alpha_{m+1}))
- h^{-1}_{\tilde{\alpha}_{c(j')-1}} \circ\dots\circ h^{-1}_{\tilde{\alpha}_{c(m)}} (\C{B}(\alpha_{m+1})) \\
& \qquad + h^{-1}_{\tilde{\alpha}_{c(j')-1}} \circ\dots\circ h^{-1}_{\tilde{\alpha}_{c(m)}} (\C{B}(\alpha_{m+1})) \\
& \leq 2(1+e^{-1}) +\exp^{\circ 3}(2).
\end{align*}
On the other hand, for all $j'\geq 0$, $1+ 2j' \geq c(j')$.
Then, combining with our choice of $n'$ in \refE{E:P:Herman-Yoccoz-criterion-3}, we have
\[j'-n \geq (c(j')-1)/2 -n \geq j/2 -n \geq n'/2 -n \geq \exp^{\circ 3}(2) + 2 (1+e^{-1}).\]
Now, using $h_r(y)\geq y+1$ several times, we obtain
\[2(1+e^{-1}) +\exp^{\circ 3}(2) \leq j'-n \leq h_{\alpha_{j'-1}} \circ \dots \circ h_{\alpha_n}(0).\]
Combining the above inequalities we obtain
\[h^{-1}_{\alpha_{j'}} \circ \dots \circ h^{-1}_{\alpha_{m}}( \C{B}(\alpha_{m+1}))\leq h_{\alpha_{j'-1}}\circ\dots\circ h_{\alpha_n}(0),\]
which gives the desired relation $\C{B}(\alpha_{m+1}) \leq h_{\alpha_{m}} \circ \dots \circ h_{\alpha_n}(0)$.
This completes the proof of the other side of the proposition.
\end{proof}
\section{Elementary properties of the change of coordinates}\label{S:elementary-properties-change-coord}
In this section we establish some basic properties of the changes of coordinates $Y_r$ introduced in
\refS{S:change-coordinates}.
The important properties are the relation between $Y_r$ and the function $h_r$ employed in the definition of the
Herman numbers in \refS{S:arithmetic}, and the relation between $Y_r$ and the functional relation
for the Brjuno function in \refE{E:Brjuno-functional-equations}.
In \refP{P:Y_r-vs-h_r^-1} we show that a suitable rescaling of the restriction of $Y_r$ to the
$y$-axis is uniformly close to $h_r$.
In \refL{L:Y_n-on-horizontals} we relate the behaviour of $Y_r$ to the main functional relation
for the Brjuno function.
In \refL{L:Y_r-distances} we list some geometric properties of the mapping $Y_r: \mathbb{H}' \to \mathbb{H}'$.
Recall from \refS{SS:standard-fraction} that $h_r$ is a diffeomorphism from $\D{R}$ onto $(0, +\infty)$.
\begin{propo}\label{P:Y_r-vs-h_r^-1}
For all $r \in (0, 1/2]$ and all $y \geq 1$,
\[|2 \pi \Im Y_r( i y/(2\pi)) - h_r^{-1}(y)| \leq \pi.\]
\end{propo}
\begin{proof}
The map $h_r^{-1}: (0, +\infty) \to \D{R}$ is given by the formula
\[h_r^{-1}(y)=
\begin{cases}
r y + \log r^{-1} -1 & \text{if } y \geq 1/r, \\
\log y & \text{if } 0 < y \leq 1/r.
\end{cases}\]
We first give some basic estimates needed for the proof.
For all $t\in [0,1]$ we have $t\leq e^t-1 \leq e t$, for all $t\in [0, 3\pi/2]$ we have $1-e^{-t}\leq t$, and for
all $t\in \D{R}$ we have $|1-e^{it}|\leq |t|$.
Then, for all $r \in (0, 1/2]$, we have
\begin{equation}\label{E:P:Y_r-vs-h_r-0}
|e^{-3\pi r}-e^{\pi r i}| \leq |e^{-3\pi r} -1| + |1-e^{\pi r i}| \leq 3\pi r+\pi r= 4\pi r,
\end{equation}
and
\begin{equation}\label{E:P:Y_r-vs-h_r^-1}
|e^{-3\pi r}-e^{\pi r i}| \geq |\Im (e^{-3\pi r}-e^{\pi r i})| =\sin (\pi r) \geq \pi r/2.
\end{equation}
Now we consider two cases. First assume that $1 \leq y \leq 1/r$. We note that
\begin{align*}
\big| e^{-3\pi r}-e^{-\pi r i}e^{r y} \big|
&\leq |e^{-3\pi r} -1| + |1- e^{-\pi r i}| + |e^{-\pi r i} - e^{-\pi r i}e^{r y}| \\
&\leq 3\pi r + \pi r + e r y= 4 \pi r+ e r y,
\end{align*}
and
\begin{equation*}
\big| e^{-3\pi r}-e^{-\pi r i}e^{r y} \big| \geq |e^{\pi r i} e^{r y}| - |e^{-3\pi r}| \geq e^{r y} -1 \geq r y.
\end{equation*}
These imply that
\begin{equation*}
\Big|\frac{e^{-3\pi r}-e^{-\pi r i}e^{r y}}{y(e^{-3\pi r}-e^{\pi r i}) }\Big|
\leq \frac{4\pi r+ e r y}{y \pi r/2}
\leq \frac{4\pi/y + e}{\pi/2}
\leq \frac{4\pi + e}{\pi/2} \leq e^{\pi},
\end{equation*}
and
\begin{equation*}
\Big|\frac{e^{-3\pi r}-e^{-\pi r i}e^{r y}}{y(e^{-3\pi r}-e^{\pi r i}) }\Big|
\geq \frac{r y}{y(4\pi r)} = \frac{1}{4\pi}\geq e^{-\pi}.
\end{equation*}
For $1\leq y \leq 1/r$, we have
\begin{equation*}
2\pi \Im Y_r ( i y/(2\pi)) - h_r^{-1}(y) =
\log \Big| \frac{e^{-3\pi r}-e^{-\pi r i}e^{r y}}{e^{-3\pi r}-e^{\pi r i}}\Big|
- \log y
= \log \Big| \frac{e^{-3\pi r}-e^{-\pi r i}e^{r y}}{y(e^{-3\pi r}-e^{\pi r i}) }\Big|.
\end{equation*}
Combining these, one obtains the inequality in the proposition when $1\leq y \leq 1/r$.
Now assume that $y \geq 1/r$.
First note that
\[|e^{-3\pi r}-e^{-\pi r i}e^{r y}| \leq |e^{-3\pi r}| + |e^{-\pi r i}e^{r y}| \leq 1+ e^{ry}
\leq 2 e^{ry},\]
and
\begin{equation*}
|e^{-3\pi r}-e^{-\pi r i}e^{r y}| \geq |e^{-\pi r i}e^{r y}| - |e^{-3\pi r}| \geq e^{ry} -1 \geq e^{ry}/2.
\end{equation*}
For $y \geq 1/r$, we have
\begin{align*}
2\pi \Im Y_r ( i y/(2\pi)) - h_r^{-1}(y) &=
\log \Big| \frac{e^{-3\pi r} - e^{-\pi r i}e^{r y}}{e^{-3\pi r}-e^{\pi r i}}\Big|
- r y - \log r^{-1} + 1\\
&=\log \Big|\frac{r(e^{-3\pi r} - e^{-\pi r i}e^{r y})}
{e^{r y}(e^{-3\pi r}-e^{\pi r i})}\Big| + 1 \\
&=\log \Big| \frac{r}{e^{-3\pi r} - e^{\pi r i}}\Big|
+ \log \Big| \frac{e^{-3\pi r}-e^{-\pi r i}e^{r y}}{e^{r y}}\Big|+ 1.
\end{align*}
Using $\log (2/\pi)+ \log 2 + 1 \leq \pi$ and $- \log (4\pi) -\log 2+1 \geq -\pi$, these imply the desired inequality in
the proposition when $y \geq 1/r$.
\end{proof}
Compare the item (iii) in the following lemma to the second functional equation in
\eqref{E:Brjuno-functional-equations}.
\begin{lem}\label{L:Y_n-on-horizontals}
Let $r\in (0, 1/2]$. We have
\begin{itemize}
\item[(i)] for all $y\geq 0$ and all $x_0 \in [0, 1/r]$,
\[\max_{x\in x_0+\D{Z}} \Im Y_r (x+iy) = \Im Y_r(x'+iy),\]
where $x' \in (x_0 + \D{Z}) \cap [1/(2r)-1, 1/(2r)]$;
\item[(ii)] for all $y\geq 0$, and all $x\in[1/(2r)-1, 1/(2r)]$ we have
\[2\pi ry+ \log (1/r) - 4 \leq 2\pi \Im Y_r (x+iy) \leq 2\pi ry + \log (1/r) + 2.\]
\end{itemize}
\end{lem}
\begin{proof}
(i): Note that
\begin{align*}
\max_{x\in x_0+\D{Z}} \Im Y_r (x+iy)
& =\frac{1}{2\pi}\max_{x \in x_0+\D{Z}}
\log\Big|\frac{e^{-3\pi r}-e^{-\pi ri}e^{-2\pi rix}e^{2\pi r y}}{e^{-3\pi r}-e^{\pi ri}}
\Big|.
\end{align*}
The maximum of $|e^{-3\pi r}-e^{-\pi ri}e^{-2\pi ri(x_0+k)}e^{2\pi r y}|$ occurs when
$e^{-\pi ri}e^{-2\pi ri(x_0+k)}$ is closest to the negative real axis.
This happens when $|\pi r i+ 2\pi ri (x_0+k) -\pi i| \leq \pi r$, which implies that $|(x_0+k)-(1/(2r) -1/2)|\leq 1/2$.
(ii): For $x \in \D{R}$,
\begin{align*}
2\pi \Im Y_r (x+iy) - 2\pi ry - \log \frac{1}{r}
& =\log\Big|\frac{e^{-3\pi r}-e^{-\pi ri}e^{-2\pi rix}e^{2\pi r y}}{e^{-3\pi r}-e^{\pi ri}}
\cdot r e^{-2\pi ry} \Big |.
\end{align*}
Assume that $2\pi ry\geq 1$. By \refE{E:P:Y_r-vs-h_r^-1},
\begin{align*}
\Big| \frac{e^{-3\pi r}-e^{-\pi ri}e^{-2\pi rix}e^{2\pi r y}}{e^{-3\pi r}-e^{\pi ri}}
\cdot r e^{-2\pi ry} \Big |
& \leq \frac{|e^{-3\pi r}|+ |e^{-\pi ri}e^{-2\pi rix}e^{2\pi r y}|}{|e^{-3\pi r}-e^{\pi ri}|}
\cdot r e^{-2\pi ry} \\
& \leq \frac{2 e^{2\pi ry}}{\pi r/2} \cdot r e^{-2\pi ry}
= \frac{4}{\pi} \leq e^2.
\end{align*}
On the other hand, by \refE{E:P:Y_r-vs-h_r-0},
\begin{align*}
\Big| \frac{e^{-3\pi r}-e^{-\pi ri}e^{-2\pi rix}e^{2\pi r y}}{e^{-3\pi r}-e^{\pi ri}}
\cdot r e^{-2\pi ry} \Big|
& \geq \frac{|e^{-\pi ri}e^{-2\pi rix}e^{2\pi r y}| - |e^{-3\pi r}|}{|e^{-3\pi r}-e^{\pi ri}|}
\cdot r e^{-2\pi ry} \\
& \geq \frac{e^{2\pi ry} -1}{|e^{-3\pi r}-e^{\pi ri}|} \cdot r e^{-2\pi ry} \\
& \geq \frac{e^{2\pi ry}/2}{4\pi r} \cdot r e^{-2\pi ry}
= \frac{1}{8\pi} \geq e^{-4}.
\end{align*}
Now assume that $0 \leq 2\pi ry \leq 1$.
For all $x\in \D{R}$, by \refE{E:P:Y_r-vs-h_r^-1},
\begin{align*}
\Big| \frac{e^{-3\pi r}-e^{-\pi ri}e^{-2\pi rix}e^{2\pi r y}}{e^{-3\pi r}-e^{\pi ri}}
\cdot r e^{-2\pi ry} \Big |
&\leq \Big| \frac{e^{-3\pi r}+ e^{2\pi r y}}{e^{-3\pi r}-e^{\pi ri}} \Big | \cdot r e^{-2\pi ry}\\
&\leq \frac{3\pi r+ e 2\pi ry}{\pi r/2} \cdot r e^{-2\pi ry} \\
&= (6r+ e 4ry) e^{-2\pi ry}.
\end{align*}
Hence, using $y\leq 1/(2\pi r)$ and $r\in (0,1/2]$, we obtain
\[ (6r+ e 4ry) e^{-2\pi ry}
\leq (3 + \frac{2e}{\pi}) e^{-2\pi ry}
\leq 3 + \frac{2e}{\pi} \leq e^2.\]
On the other hand, for $x\in [1/(2r)-1, 1/(2r)]$,
we have
\[|e^{-3\pi r}-e^{-\pi ri}e^{-2\pi rix}e^{2\pi r y}| \geq e^{2\pi ry}.\]
Hence, by \refE{E:P:Y_r-vs-h_r-0},
\[\Big| \frac{e^{-3\pi r}-e^{-\pi ri}e^{-2\pi rix}e^{2\pi r y}}{e^{-3\pi r}-e^{\pi ri}}
\cdot r e^{-2\pi ry} \Big |
\geq \frac{e^{2\pi ry}}{4\pi r} \cdot r e^{-2\pi ry} = \frac{1}{4\pi} \geq e^{-4}.
\]
These imply the desired inequality in part (iii).
\end{proof}
\begin{lem}\label{L:Y_r-distances}
For all $r\in (0, 1/2]$, we have
\begin{itemize}
\item[(i)] for all $x \in [0,1/r]$ and all $y\geq -1$,
\[\Im Y_r(x+ iy) \geq \Im Y_r(iy) -1/(2\pi);\]
\item[(ii)]for all $x\in [0, 1/r]$ and all $y_1 \geq y_2 \geq -1$,
\[\Im Y_r(x+iy_1) - \Im Y_r(x+ iy_2) \leq \Im Y_r(iy_1)- \Im Y_r(iy_2) + 1/(2\pi);\]
\item[(iii)]for all $y_1 \geq y_2 \geq -1$ and $y \geq 0$,
\[\Im Y_r(i y+iy_1)- \Im Y_r(i y+iy_2) \leq \Im Y_r(iy_1)- \Im Y_r(iy_2)+ 1/(4\pi);\]
\item[(iv)]for all $y_1 \geq y_2\geq -1$ and $y \in [0,5/\pi]$,
\[\Im Y_r(iy_1) - \Im Y_r(iy_2) \leq \Im Y_r(iy+ iy_1)- \Im Y_r(iy+ iy_2)+ 5/\pi.\]
\end{itemize}
\end{lem}
\begin{proof}
Part (i): Recall from the proof of \refL{L:uniform-contraction-Y_r} that
\begin{align*}
\frac{\partial Y_r}{\partial s}(w)=\frac{\partial Y_r}{\partial w}(w) + \frac{\partial Y_r}{\partial \overline{w}}(w)
= r+ \frac{i r \Im \xi}{|\xi-1|^2},
\end{align*}
where $w=s+iy$ and $\xi=e^{-3\pi r} e^{\pi r i } e^{2\pi r i w}$.
We note that when $s \in [0, 1/(2r)-1/2] \cup [1/r-1/2, 1/r]$, $\Im \partial Y_r (s+iy)/\partial s \geq 0$,
and when $s \in [1/(2r)-1/2, 1/r-1/2]$, $\Im \partial Y_r (s+iy)/\partial s \leq 0$.
Moreover, when $s \in [1/r-1, 1/r-1/2]$, we have
\begin{align*}
\Big| \Im \frac{\partial Y_r}{\partial s}(s+iy)\Big|
= \frac{r |\Im \xi|}{|\xi-1|^2} \leq \frac{r |e^{-3\pi r}e^{2\pi r} \sin (\pi r+ 2\pi rs)|}{(1-e^{-\pi r})^2}
\leq \frac{r e^{-\pi r} \sin {(\pi r)}}{1+ e^{-2\pi r}-2e^{-\pi r}}\leq \frac{1}{\pi}.
\end{align*}
To see the last inequality, it is enough to show that $g(u)=u \sin u-e^{u}-e^{-u}+2 \leq 0$ for all $u=\pi r\in[0,\pi/2]$.
By Taylor's remainder theorem, for all $u \in [0, \pi/2]$, there is $u_0 \in [0, u]$ such that
\[g(u)= g(0)+ u g'(0)+ u^2 g''(0)/2+u_0^3 g^{(3)}(u_0)/6
= u_0^3 (-3\sin u_0 -u_0 \cos u_0 - e^{u_0}+ e^{-u_0})/6 \leq 0.\]
For $x\in [0,1/r]$, we use the formula
\begin{align*}
\Im Y_r(x+iy) - \Im Y_r(iy) = \int_0^x \Im \frac{\partial Y_r(s+iy)}{\partial s} \, ds
\end{align*}
to obtain a lower bound. We consider four cases:
$\bullet$ When $x\in [0, 1/(2r)-1/2]$, the integrand is non-negative, and hence
\[\int_0^x \Im \frac{\partial Y_r(s+iy)}{\partial s} \, ds\geq 0.\]
$\bullet$ When $x\in [1/(2r)-1/2, 1/r-1]$, then $\frac{1}{r}-1-x \in [0, 1/(2r)-1/2]$ and by the previous case,
\begin{align*}
\int_0^x \Im \frac{\partial Y_r(s+iy)}{\partial s} \, ds
& =\int_0^{1/r-1-x} \Im \frac{\partial Y_r(s+iy)}{\partial s} \, ds
+ \int_{1/r-1-x}^{x} \Im \frac{\partial Y_r(s+iy)}{\partial s} \, ds \\
&\geq \int_{1/r-1-x}^{x} \Im \frac{\partial Y_r(s+iy)}{\partial s} \, ds \\
&= \int_{1/r-1-x}^{x} \frac{r \sin (\pi r+ 2\pi rs)}{1+e^{-6\pi r - 4\pi ry} -2 e^{-3\pi r -2\pi ry} \cos (\pi r+ 2\pi rs)} \,ds\\
&= \int_{2\pi -\pi r-2\pi rx}^{\pi r+ 2\pi rx} \frac{r \sin (\theta)}{1+e^{-6\pi r - 4\pi ry} -2 e^{-3\pi r -2\pi ry} \cos \theta} \, \frac{d\theta}{2\pi r}=0.
\end{align*}
$\bullet$ When $x\in [1/r-1, 1/r-1/2]$, by the previous case and the above estimate,
\begin{align*}
\int_0^x \Im \frac{\partial Y_r(s+iy)}{\partial s} \, ds
& =\int_0^{1/r-1} \Im \frac{\partial Y_r(s+iy)}{\partial s} \, ds
+ \int_{1/r-1}^{1/r-1/2} \Im \frac{\partial Y_r(s+iy)}{\partial s} \, ds \\
&\geq \int_{1/r-1}^{1/r-1/2} \Im \frac{\partial Y_r(s+iy)}{\partial s} \, ds
\geq \frac{-1}{\pi} \cdot \frac{1}{2}= \frac{-1}{2\pi}.
\end{align*}
$\bullet$ When $x \in [1/r-1/2, 1/r]$, by the previous case,
\begin{align*}
\int_0^x \Im \frac{\partial Y_r(s+iy)}{\partial s} \, ds
& =\int_0^{1/r-1/2} \Im \frac{\partial Y_r(s+iy)}{\partial s} \, ds
+ \int_{1/r-1/2}^{x} \Im \frac{\partial Y_r(s+iy)}{\partial s} \, ds \\
&\geq \frac{-1}{2\pi} + \int_{1/r-1/2}^{x} \Im \frac{\partial Y_r(s+iy)}{\partial s} \, ds \geq \frac{-1}{2\pi}.
\end{align*}
This completes the proof of Part (i).
Part (ii): Let
\[g(\theta, t_1, t_2)= \frac{1+ e^{2t_1} - 2e^{t_1}\cos \theta}{1+ e^{2t_2} - 2e^{t_2}\cos \theta},
\quad \theta \in \D{R}, t_1\geq t_2 > 0.\]
Since $1+ e^{2t_2} - 2e^{t_2}\cos \theta \geq 1+ e^{2t_2} - 2e^{t_2} = (e^{t_2}-1)^2>0$, $g$ is well-defined,
and is positive.
Moreover,
\[\partial_\theta g(\theta, t_1, t_2)= \frac{-2\sin \theta (e^{t_1}-e^{t_2}) (e^{t_1+t_2}-1)}{(1+ e^{2t_2} - 2e^{t_2}\cos \theta)^2}.\]
Thus, $g$ is a decreasing function of $\theta$ on $[0,\pi]+2\pi \D{Z}$, and increasing on $[\pi, 2\pi]+ 2\pi \D{Z}$.
This implies that $g(\theta, t_1, t_2)\leq g(0, t_1, t_2)$, for all $\theta\in \D{R}$.
Applying this inequality with $\theta=-\pi r-2\pi rx$, $t_1=3\pi r+2\pi r y_1$, and $t_2=3\pi r+2\pi r y_2$, one obtains
\[\Big|\frac{e^{-3\pi r}-e^{-\pi r i} e^{-2\pi r ix} e^{2\pi r y_1}}{e^{-3\pi r} - e^{-\pi r i} e^{-2\pi r ix}e^{2\pi r y_2}}\Big|^2
= g(\theta, t_1, t_2) \leq g(0, t_1, t_2)
= \Big|\frac{e^{-3\pi r} - e^{2\pi r y_1}}{e^{-3\pi r} - e^{2\pi r y_2}} \Big|^2.\]
Therefore, for all $y_1 \geq y_2 \geq -1$, we have
\begin{align*}
\Im Y_r(x+iy_1) - \Im Y_r(x+ iy_2)
& = \frac{1}{2\pi}\log\Big|\frac{e^{-3\pi r} - e^{-\pi r i} e^{-2\pi r ix} e^{2\pi r y_1}}
{e^{-3\pi r} - e^{-\pi r i} e^{-2\pi r ix} e^{2\pi r y_2}} \Big | \\
& \leq \frac{1}{2\pi}\log \Big|\frac{e^{-3\pi r} - e^{2\pi r y_1}}{e^{-3\pi r} - e^{2\pi r y_2}} \Big | \\
&= \Im Y_r(1/r-1/2+iy_1) - \Im Y_r(1/r-1/2+ iy_2).
\end{align*}
On the other hand,
\[\Im Y_r(1/r-1/2+iy_1)= \frac{1}{2\pi} \log \Big|\frac{e^{-3\pi r} - e^{2\pi r y_1}}{e^{-3\pi r} - e^{-\pi ri}} \Big |
\leq \frac{1}{2\pi} \log \Big|\frac{e^{-3\pi r} - e^{-\pi ri} e^{2\pi r y_1}}{e^{-3\pi r} - e^{-\pi ri}} \Big |
=\Im Y_r(iy_1),\]
and by the inequality in Part (i),
\[\Im Y_r(1/r-1/2+ iy_2)\geq \Im Y_r(iy_2)-1/(2\pi).\]
Combining the above inequalities together, we conclude Part (ii) of the lemma.
Part (iii):
If $y_1=y_2$, the inequality trivially holds. Below we assume that $y_1> y_2$.
By the definition of $Y_r$, it is enough to prove that for all $r\in (0, 1/2]$, $y_1\geq y_2\geq 0$, and $y\geq 0$
we have
\[\Big |\frac{e^{-3\pi r}- e^{-\pi r i }e^{2\pi r (y+y_1)}}{e^{-3\pi r}- e^{-\pi r i }e^{2\pi r (y+y_2)}}\Big|
\leq \sqrt{2} \Big |\frac{e^{-3\pi r}- e^{-\pi r i }e^{2\pi r y_1}}{e^{-3\pi r}- e^{-\pi r i }e^{2\pi r y_2}}\Big|.\]
Let us fix $y_1> y_2$, and consider the M\"obius transformations
\[M(z)=\frac{e^{-3\pi r}- e^{-\pi ri}e^{2\pi r y_1}z}{e^{-3\pi r}- e^{-\pi ri}e^{2\pi r y_2}z}, \quad
M_1(z)=\frac{e^{-3\pi r} + e^{2\pi r y_1}z}{e^{-3\pi r} + e^{2\pi r y_2}z}.\]
We have $M(0)=M_1(0)=1$, and $M(\infty)= M_1(\infty)=e^{2\pi r(y_1-y_2)}$.
Then, there is a M\"obius transformation $M_2$ with $M_2(1)=1$, $M_2(e^{2\pi r(y_1-y_2)})=e^{2\pi r(y_1-y_2)}$,
and $M=M_2 \circ M_1$. The precise form of $M_2$ is not relevant here.
An elementary algebraic calculation shows that
\[M_1(1)\geq (1+ e^{2\pi r(y_1-y_2)})/2.\]
Since $M_1$ maps the real interval $[0, +\infty]$ to $[1, e^{2\pi r (y_1-y_2)}]$, we must have
\[M_1(e^{2\pi ry}) \geq M_1(1) \geq (1+ e^{2\pi r(y_1-y_2)})/2.\]
That is, $M_1(1)$ and $M_1(e^{2\pi ry})$ belong to the right hand of the middle point of the interval
$[1, e^{2\pi r(y_1-y_2)}]$.
Note that $|M_2'(1)| = |M'(0)/M_1'(0)|=1$. Thus, $M_2$ preserves the line $\Re z= (1+ e^{2\pi r(y_1-y_2)})/2$,
and maps the real interval $[(1+ e^{2\pi r(y_1-y_2)})/2, e^{2\pi r (y_1-y_2)}]$ to an arc $\gamma$ of a circle
whose center $B$ lies on the vertical line $\Re z= (1+ e^{2\pi r(y_1-y_2)})/2$.
By the above paragraph, $M(e^{2\pi ry})$ and $M(1)$ belong to $\gamma$.
By looking at $\arg M_2'(1)$, one can see that $\gamma$ meets the line $\Im z=0$ at $e^{2\pi r(y_1-y_2)}$
with asymptotic
angle $+ \pi r \leq \pi/2$. That is, $\gamma$ and $B$ lie in the same component of $\D{C}\setminus \D{R}$.
See Figure~\ref{F:L:Y_r-distances} for an illustration of the following argument.
\begin{figure}[ht]
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\psline[linewidth=.5pt]{->}(0.5,2)(7.5,2)
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\psarc[linewidth=.8pt](5,2.8){1}{-53}{90}
\psdots[dotsize=2.5pt](5,3.8)(5,2.8)
\psdots[dotsize=2.5pt](4.4,2)(5.6,2)
\psdot[dotsize=2.5pt](5.98,3)
\psline[linewidth=.5pt](1,2)(5.98,3)
\psline[linewidth=.5pt](5,2.8)(5,3.8)
\psline[linewidth=.5pt](1,2)(5,3.8)
\rput(1.2,1.7){\small $O$}
\rput(4.4,1.7){\small $1$}
\rput(6.1,1.7){\small $e^{2\pi r(y_1-y_2)}$}
\rput(5,4){\small $A$}
\rput(5.1,2.6){\small $B$}
\rput(6.2,3){\small $C$}
\rput(5.6,3.9){\tiny $M(1)$}
\rput(6.4,3.6){\tiny $M(e^{2\pi ry})$}
\end{pspicture}
\caption{Illustration of the inequality in \refL{L:Y_r-distances}-(iii). The curve $\gamma$ is the arc on the dotted circle.}
\label{F:L:Y_r-distances}
\end{figure}
Let $C$ be the point on $\gamma$ where $|C|$ achieves its maximum, and let $O$ denote the origin.
Then, the line from $O$ to $C$ must pass through $B$.
Let $A$ denote the point where the arc $\gamma$ meets the line $\Re z= (1+ e^{2\pi r(y_1-y_2)})/2$.
In the triangle $OAB$, the angle $\theta$ at the vertex $B$ is at least $\pi/2$.
By the cosine formula,
\[|OA|^2=|OB|^2+ |AB|^2-2|OB||AB| \cos \theta \geq |OB|^2 + |AB|^2.\]
Using the inequality $|OB|^2 + |AB|^2 \geq 2 |OB| |AB|$, we conclude that
\[2|OA|^2 \geq |OB|^2 + |AB|^2 + 2 |OB||AB|= (|OB|+|AB|)^2.\]
By the above discussion, $M(1)$ and $M(e^{2\pi ry})$ belong to $\gamma$, with $M(1)$ lying between $A$ and
$M(e^{2\pi ry})$.
Now, we consider two cases. If $M(1)$ lies between $C$ and $e^{2\pi r (y_1-y_2)}$, then we must have
$M(e^{2\pi ry})/M(1) \leq 1$.
If $M(1)$ lies between $A$ and $C$, by the above inequality, we obtain
\[\frac{|M(e^{2\pi ry})|}{|M(1)|} \leq \frac{|OC|}{|OA|} = \frac{|OB|+ |BA|}{|OA|} \leq \sqrt{2}.\]
This completes the proof of the desired inequality.
Part (iv): Recall that $Y_r(0)=0$ and hence by \refL{L:uniform-contraction-Y_r}, $\Im Y_r(-i) \geq -9/10$.
Moreover, $\Im Y_r(iy+ iy_1) \geq \Im Y_r(iy_1)$.
By rearranging terms and using the inequality in Part (iii), we note that
\begin{align*}
\Im Y_r(iy_1) - \Im Y_r(iy_2) & - (\Im Y_r(iy+ iy_1) - \Im Y_r(iy+ iy_2)) \\
&=\big(\Im Y_r(iy+ iy_2) - \Im Y_r(iy_2)\big) + \big( \Im Y_r(iy_1)- \Im Y_r(iy+ iy_1)\big) \\
& \leq \Im Y_r(iy+ iy_2) - \Im Y_r(iy_2) \\
& \leq \Im Y_r(i(y-1)) - \Im Y_r(-i) + 1/(4\pi) \\
& \leq \Im Y_r(i (5-\pi)/\pi) + 9/10 + 1/(4\pi).
\end{align*}
On the other hand, by \refE{E:P:Y_r-vs-h_r^-1}, for all $r\in (0, 1/2]$, we have
\begin{align*}
Y_r(i (5-\pi)/\pi) &=\frac{1}{2\pi} \log \Big| \frac{e^{-3\pi r} - e^{-\pi r i} e^{2r(5-\pi)}}{e^{-3\pi r}- e^{\pi ri}}\Big| \\
&\leq \frac{1}{2\pi} \log \frac{|e^{-3\pi r}-1|+ |1-e^{-\pi r i}|+ |e^{-\pi ri} - e^{-\pi ri}e^{2r(5-\pi)}|}{\pi r/2} \\
&\leq \frac{1}{2\pi} \log \frac{3\pi r+ \pi r + (e^{2r(5-\pi)}-1)}{\pi r/2} \\
&\leq \frac{1}{2\pi} \log \frac{3\pi r+ \pi r + (e^3-8)\pi r/2}{\pi r/2} = \frac{3}{2\pi}.
\end{align*}
Using the inequality $9/10 \leq 3/\pi$, we obtain the desired estimate in Part (iv).
\end{proof}
\section{Topology of the sets \texorpdfstring{$\mathbb{A}_\alpha$}{A-ga}}\label{S:topology-A}
\subsection{Cantour bouquets and hairy Jordan curves}\label{SS:CB-HJC}
In this section we describe the topology of the sets $\mathbb{A}_\alpha$ in terms of the arithmetic properties of $\alpha$.
In particular, here we will prove \refT{T:model-trichotomy-thm}.
Let us start by presenting the definition of the two main topological objects which appear in the theorem.
A \textbf{Cantor bouquet} is any subset of the plane which is ambiently homeomorphic to a set of the form
\begin{equation*}\label{E:straight-cantor-bouquet}
\{re^{2\pi i \theta} \in \D{C} \mid 0 \leq \theta \leq 1, 0 \leq r \leq R(\theta) \}
\end{equation*}
where $R: \D{R}/\D{Z} \to [0, 1]$ satisfies the following:
\begin{itemize}
\item[(a)] $R=0$ on a dense subset of $\D{R}/\D{Z}$, and $R > 0$ on a dense subset of $\D{R}/\D{Z}$,
\item[(b)] for each $\theta_0\in \D{R}/\D{Z}$ we have
\[\limsup_{\theta \to \theta_0^+ } R(\theta) = R(\theta_0) = \limsup_{\theta \to \theta_0^-} R(\theta).\]
\end{itemize}
A \textbf{one-sided hairy Jordan curve} is any subset of the plane which is ambiently homeomorphic to a set
of the form
\begin{equation*}\label{E:straight-hairy-circle}
\{re^{2\pi i \theta} \in \D{C} \mid 0 \leq \theta \leq 1, 1\leq r \leq 1+ R(\theta) \}
\end{equation*}
where $R: \D{R}/\D{Z} \to [0,1]$ satisfies properties (a) and (b) in the above definition.
The Cantor bouquet and one-sided hairy Jordan curve enjoy similar topological feature as the standard Cantor set.
Under an additional mild condition (topological smoothness) they are uniquely characterised by some
topological axioms, see \cite{AaOv93}.
To study the topology of the sets $\mathbb{M}_\alpha$ (and $\mathbb{A}_\alpha$), it is convenient to work with the sets
$I_n^j$ and $I_n$ from \refS{SS:tilings-nest}. That is because each of the sets $I_n^j$ and $I_n$ is the region
above the graph of a function. Since the sets $I_n^j$, for $j\geq 0$, forms a nest, one is led to an increasing
collection of functions on a fixed domain.
Since the nest may shrink to an empty set along a fixed vertical line, we are led to considering functions which attain
$+\infty$ at some points.
It turns out that there are two main collections of functions required to capture the topological features of these sets.
In the next section we build these functions and study their properties.
\subsection{Height functions}\label{SS:height-functions}
\footnote{The letter $b$ stands for ``base'' and ``p'' for ``pinnacle''; the reason for these will become clear in a moment.}
Recall that the sets $I_n^j$ and $I_n$ consist of closed half-infinite vertical lines.
For $n\geq -1$, and $j\geq 0$, define $b_n^j:[0, 1/\alpha_n] \to [-1, +\infty)$ as
\begin{equation}\label{E:I_n^j-b_n^j}
b_n^j(x)= \min \{y \mid x+iy \in I_n^j\}.
\end{equation}
Since each $Y_n$ preserves vertical lines, it follows that
\[I_n^j= \{w\in \D{C} \mid 0 \leq \Re w \leq 1/\alpha_n, \Im w\geq b_n^j(\Re w)\}.\]
By the definition of the sets $I_n^j$ and the functional equations \eqref{E:Y_n-comm-1}--\eqref{E:Y_n-comm-2},
one can see that for all $n\geq -1$ and $j\geq 0$, $b_n^j:[0, 1/\alpha_n] \to [-1, +\infty)$ is continuous.
Moreover, by \refE{E:I_n^j-forms-nest}, we must have $b_n^{j+1} \geq b_n^j$ on $[0, 1/\alpha_n]$.
For $n\geq -1$, we define $b_n:[0, 1/\alpha_n] \to [-1,+\infty]$ as
\[b_n(x)= \lim_{j\to + \infty} b_n^j(x)= \sup_{j\geq 1} b_n^j(x).\]
Note that $b_n$ is allowed to take $+\infty$. The function $b_n$ describes the set $I_n$, that is,
\begin{equation}\label{E:I_n-b_n}
I_n= \{w\in \D{C} \mid 0 \leq \Re w \leq 1/\alpha_n, \Im w \geq b_n(\Re w)\}.
\end{equation}
By \refL{L:model-almost-periodic}, $b_n^j(0)=b_n^j(1/\alpha_n)$, and $b_n^j(x+1)= b_n^j(x)$ for all $x\in [0, 1/\alpha_n-1]$.
Taking limits as $j\to +\infty$, we obtain
\begin{equation}\label{E:b_n^j-cont-periodic}
b_n(0)=b_n(1/\alpha_n), \qquad b_n(x+1)=b_n(x), \text{ for all } x\in [0, 1/\alpha_n-1] \text{ and } n\geq -1.
\end{equation}
Only when $\alpha$ is a Brjuno number, for $n\geq -1$ and $j\geq 0$ we inductively define the functions
\[p_n^j: [0,1/\alpha_n] \to [-1, +\infty).\]
For all $n\geq -1$, we set $p_n^0\equiv (\C{B}(\alpha_{n+1})+5\pi)/(2\pi)$.
Assume that $p_n^j$ is defined for some $j\geq 0$ and all $n\geq -1$.
We define $p_n^{j+1}$ on $[0, 1/\alpha_n]$ as follows.
For $x_n \in [0,1/\alpha_n]$, we find $x_{n+1} \in [0, 1/\alpha_{n+1}]$ and $l_{n}\in \D{Z}$ such that
$-\varepsilon_{n+1} \alpha_{n+1} x_{n+1}= x_n - l_n$, and define
\[p_n^{j+1}(x_n)=\Im Y_{n+1}\big(x_{n+1} + i p_{n+1}^j (x_{n+1})\big).\]
In other words, the graph of $p_n^{j+1}$ is obtained from applying $Y_{n+1}$ to the graph of $p_{n+1}^j$,
and then applying suitable translations by integers.
Evidently, for all $n\geq -1$ and all $j\geq 0$, we have
$p_n^{j+1}(x+1)= p_n^{j+1}(x)$, for $x \in [0, 1/\alpha_n-1]$.
Moreover, it follows from \eqref{E:Y_n-comm-1} and \eqref{E:Y_n-comm-2} that each
$p_n^j: [0,1/\alpha_n]\to \D{R}$ is continuous, and $p_n^j(0)=p_n^j(1/\alpha_n)$.
Note that for every $n\geq -1$, by \refL{L:Y_n-on-horizontals}-(i)-(ii) and \refE{E:Brjuno-functional-equations},
we have
\begin{align*}
p_n^1& \leq \max_{x \in [0, 1/\alpha_{n+1}]} \Im Y_{n+1}\big (x+ i (\C{B}(\alpha_{n+2})+5\pi)/(2\pi)\big ) \\
& \leq \alpha_{n+1} \frac{\C{B}(\alpha_{n+2})+5\pi}{2\pi} + \frac{1}{2\pi} \log \frac{1}{\alpha_{n+1}}+\frac{1}{\pi} \\
&=\frac{1}{2\pi}\big(\alpha_{n+1}\C{B}(\alpha_{n+2})+\log\frac{1}{\alpha_{n+1}}\big)
+\frac{\alpha_{n+1}5\pi}{2\pi}+\frac{1}{\pi}\\
& \leq \frac{1}{2\pi} \C{B}(\alpha_{n+1}) + \frac{5}{4} + \frac{1}{\pi}
\leq p_n^0.
\end{align*}
Using an induction argument, starting with $p_n^ 1 \leq p_n^0$, for every $n\geq -1$, one may show that for all
$n\geq -1$ and $j\geq 0$ we have
\[p_n^{j+1}(x) \leq p_n^j(x), \quad \forall x\in [0, 1/\alpha_n].\]
Therefore, we may define the functions
\[p_n(x)= \lim_{j \to +\infty} p_n^j(x), \quad \forall x\in [0, 1/\alpha_n].\]
It follows that
\begin{equation}\label{E:p_n-peiodic}
p_n(0)=p_n(1/\alpha_n), \quad p_n(x)=p_{n}(x+1), \forall x\in [0, 1/\alpha_n-1].
\end{equation}
On the other hand, by definition, $p_n^0 \geq b_n^0$, for all $n\geq -1$. Since the graphs of $b_{n+1}^0$
and $p_{n+1}^0$ are mapped to the graphs of $b_n^1$ and $p_n^1$, respectively, by $Y_{n+1}$
and its integer translations, we must have $p_{n}^1 \geq b_n^1$, for all $n\geq -1$.
By induction, this implies that for all $n\geq -1$ and all $j\geq 0$,
\begin{equation}\label{E:p_n>=b_n}
p_n^j(x) \geq b_n^j(x), \quad \forall x\in [0, 1/\alpha_n].
\end{equation}
In particular, $p_n \geq b_n$ on $[0, 1/\alpha_n]$.
\subsection{Accumulation of the hairs}\label{SS:hairs-accumulate}
\begin{propo}\label{P:b_n-liminfs}
For all $n\geq -1$, we have \footnote{To be clear, the notation $s \to x^+$ means that $s \to x$ and $s>x$. Similarly,
$s\to x^-$ means that $s\to x$ and $s< x$.}
\begin{itemize}
\item[(i)] for all $x\in [0, 1/\alpha_n)$, $\liminf_{s\to x^+} b_n(s)= b_n(x)$;
\item[(ii)] for all $x\in (0, 1/\alpha_n]$, $\liminf_{s\to x^-} b_n(s)= b_n(x)$.
\end{itemize}
\end{propo}
\begin{proof}
By \refL{L:I_n^j-basic-features}, the sets $I_n^j$ are closed. Thus, their intersection $I_n$ is also a closed set.
Since $I_n$ is the set above the graph of $b_n$, this implies that
\begin{gather*}
\forall x\in [0, 1/\alpha_n), \liminf_{s\to x^+} b_n(s)\geq b_n(x), \quad \text{ and } \quad
\forall x\in (0, 1/\alpha_n], \liminf_{s\to x^-} b_n(s) \geq b_n(x).
\end{gather*}
So we need to show that the equality holds in both cases. If $b_n(x)=+\infty$, then we automatically have the
equalities. Indeed, $\liminf$ can be replaced by $\lim$ in that case.
Below we assume that $b_n(x)\neq \infty$.
Fix $x_n \in [0, 1/\alpha_n]$, and let $e_n \in \{+1, -1\}$ be arbitrary, except that $e_n=+1$ if $x_n=0$, and $e_n=-1$
if $x_n=1/\alpha_n$.
Define the sequence $\{e_m\}_{m\geq n}$ according to $e_{m+1}= - \varepsilon_{m+1} e_m$.
We use $e_n$ to deal with both statements at once.
That is, to prove the desired equalities, it is enough to show that for every $\delta>0$ there is $x_n'$ strictly
between $x_n$ and $x_n+ \delta e_n$ such that $b_n(x_n')\leq b_n(x_n)+\delta$.
The idea of the proof is rather elementary.
We map $x_n+i b_n(x_n) \in I_n$ to $z_m \in I_m$ using the maps $Y_j^{-1}$, for large $m\geq n$.
Then, using \refE{E:b_n^j-cont-periodic}, we find $z_m'=z_m+e_m \in I_m$, which may be mapped to
$z'_n \in I_n$ using $Y_j$.
It follows that $\sign (\Re z'_n - x_n)=e_n$, and by the uniform contraction of $Y_j$,
$z'_n$ is close to $x_n +i b_n(x_n)$.
However, there are some technical difficulties due to $\varepsilon_j=\pm 1$ and $z'_m=z_m+e_m \notin I_m$.
We present the details in several steps.
\smallskip
{\em Step 1.} There is an infinite sequence $\{(l_m, x_{m+1})\}_{m\geq n}$, with $l_m \in \D{Z}$ and
$x_{m+1} \in [0, 1/\alpha_{m+1}]$ such that $\Re Y_{m+1}(x_{m+1}) + l_m= x_m$ for all $m\geq n$.
Moreover, if $x_m=0$ for some $m\geq n$, then $e_m=+1$.
\smallskip
We define the sequence inductively. Assume that $x_m \in [0,1/\alpha_m]$ is defined for some $m\geq n$.
To define $l_m$ and $x_{m+1}$ we proceed as follows:
\begin{itemize}
\item[(a)]if $x_m \in [0, 1/\alpha_m] \cap \D{Z}$, we let $l_m= x_m+ (e_m+\varepsilon_{m+1})/2$ and define
\[x_{m+1}=\frac{1}{-\varepsilon_{m+1}\alpha_{m+1}} \cdot (x_m-l_m) =
\frac{1+ \varepsilon_{m+1}e_m}{2\alpha_{m+1}}.\]
\item[(b)] if $x_m \in [0, 1/\alpha_m] \setminus \D{Z}$ we choose $l_m \in \D{Z}$ such that
$x_m-l_m \in ((-1-\varepsilon_{m+1})/2,(1-\varepsilon_{m+1})/2)$, and define
\[x_{m+1} = \frac{x_m- l_m}{-\varepsilon_{m+1}\alpha_{m+1}}.\]
\end{itemize}
In part (a), $x_{m+1}\in \{0, 1/\alpha_{m+1}\}$, depending on the sign of $\varepsilon_{m+1} e_m$.
In part (b), if $\varepsilon_{m+1}=-1$, then $x_m-l_m \in (0,1)$ and therefore $x_{m+1} \in (0, 1/\alpha_{m+1})$.
If $\varepsilon_{m+1}=+1$, then $x_m - l_m \in (-1,0)$ and therefore $x_{m+1} \in (0, 1/\alpha_{m+1})$.
Thus, in both cases $x_{m+1}$ belongs to $[0, 1/\alpha_{m+1}]$.
By \refE{E:Y_n}, $\Re Y_{m+1}(x_{m+1})= -\varepsilon_{m+1} \alpha_{m+1}x_{m+1}$.
Thus, $\Re Y_{m+1}(x_{m+1})+l_m=x_m$.
To prove the latter part of Step 1, note that if $j=n$ then $e_{j}=+1$ by the definition of $e_n$ at the beginning of
the proof. If $j > n$, by the definition of the sequence $(x_m, l_m)$, $x_j$ must be generated in part (a), since part (b)
produces values in $(0, 1/\alpha_{m})$.
But, part (a) produces $x_{j}=0$ only if $\varepsilon_{j} e_{j-1}=-1$. Then, $e_{j}= - \varepsilon_{j} e_{j-1}=+1$.
For all $m\geq n$ we have
\begin{equation}\label{E:P:bounds-on-l_m}
(1+\varepsilon_{m+1})/2 \leq l_m \leq a_m + \varepsilon_{m+1}.
\end{equation}
That is because, if $l_m$ is produced in (b) and $\varepsilon_{m+1}=-1$ then the largest integer in
$[0, 1/\alpha_{m}]$ is $a_m-1$ and there is $0 \leq l_m \leq a_m-1$ with $x_m-l_m\in (0,1)$.
If $l_m$ is produced in (b) and $\varepsilon_{m+1}=+1$ then the largest integer in $[0,1/\alpha_m]$ is $a_m$ and there is
$1 \leq l_m \leq a_m+1$ with $x_m-l_m \in (-1,0)$.
If $l_m$ is produced in (a), then $x_m \leq a_m + (\varepsilon_{m+1}-1)/2$ and therefore
$l_m \leq a_m+\varepsilon_{m+1} +e_m/2-1/2 \leq a_m+\varepsilon_{m+1}$. On the other hand, in (a)
$l_m = x_m + (e_m + \varepsilon_{m+1})/2 \geq (1+\varepsilon_{m+1})/2$, since, by the above paragraph, if $x_m=0$ then
$e_m=+1$.
We are done with Step 1.
Let us say that some level $m \geq n$ is \textbf{eligible}, if one of the following four cases occurs:
\begin{itemize}
\item[$(E_1)$] $(e_m, \varepsilon_{m+1})= (+1, -1)$ and $x_m$ belongs to
\[[0, 1/\alpha_m-2] \cup [a_m-2-\alpha_{m+1}, a_m-2] \cup [a_m-1-\alpha_{m+1}, a_m-1];\]
\item[$(E_2)$] $(e_m, \varepsilon_{m+1})= (+1, +1)$ and $x_m$ belongs to
\[[0, 1/\alpha_m-2] \cup [a_m-1-\alpha_{m+1}, a_m-1] \cup [a_m-\alpha_{m+1}, a_m];\]
\item[$(E_3)$] $(e_m, \varepsilon_{m+1})= (-1, -1)$ and $x_m$ belongs to
\[[1, 1/\alpha_m-1] \cup [1-\alpha_{m+1},1] \cup [a_m-1, a_m-1+\alpha_{m+1}];\]
\item[$(E_4)$] $(e_m, \varepsilon_{m+1})= (-1, +1)$ and $x_m$ belongs to
\[[1, 1/\alpha_m-1] \cup [\alpha_{m+1}, 2\alpha_{m+1}] \cup [a_m, a_m+\alpha_{m+1}].\]
\end{itemize}
Recall the numbers $\beta_j$ introduced in \refS{SS:modified-fractions}.
\smallskip
{\em Step 2.} Let $k \geq j \geq n$ be integers. Either there is an eligible $m \in [j,k]$, or
\begin{equation}\label{E:P:tiny-end-intervals}
x_j \in
\begin{cases}
[0,\beta_k/\beta_j] \cup [1/\alpha_j-1, 1/\alpha_j-1+ \beta_k/\beta_j] & \text{if } e_j=-1 \\
[1/\alpha_j-1-\beta_k/\beta_j, 1/\alpha_j-1] \cup [1/\alpha_j-\beta_k/\beta_j, 1/\alpha_j] & \text{if } e_j=+1.
\end{cases}
\end{equation}
\smallskip
We prove this by induction on $k-j$.
Assume that $k-j=0$.
If $e_j=+1$ then either $x_j \in [0, 1/\alpha_j-2]$ and $j$ is eligible as in ($E_1$) and $(E_2)$,
or $x_j \in [1/\alpha_j-2, 1/\alpha_j-1] \cup [1/\alpha_j-1, 1/\alpha_j]$ and \eqref{E:P:tiny-end-intervals} holds.
If $e_j=-1$ then either $x_j \in [1, 1/\alpha_j-1]$ and $j$ is eligible as in ($E_3$) and $(E_4)$, or
$x_j \in [0,1] \cup [1/\alpha_j-1, 1/\alpha_j]$ and \eqref{E:P:tiny-end-intervals} holds.
Now assume that the statement holds for integers $k$ and $j$ with $k-j=t \geq 0$.
We aim to prove it for integers $k$ and $j$ with $k-j=t+1$.
By the induction hypotheses applied to the pair $j+1$ and $k$ we conclude that either there is an eligible $m\in [j+1, k]$,
or
\begin{equation}\label{E:P:tiny-end-intervals-2}
x_{j+1} \in
\begin{cases}
[0,\beta_k/\beta_{j+1}] \cup [1/\alpha_{j+1}-1, 1/\alpha_{j+1}-1+\beta_k/\beta_{j+1}] & \text{if } e_{j+1}=-1 \\
[1/\alpha_{j+1}-1-\beta_k/\beta_{j+1}, 1/\alpha_{j+1}-1] \cup [1/\alpha_{j+1}-\beta_k/\beta_{j+1}, 1/\alpha_{j+1}] & \text{if } e_{j+1}=+1.
\end{cases}
\end{equation}
If there is an eligible $m \in [j+1, k] \subset [j,k]$ then we are done. We show that if
\eqref{E:P:tiny-end-intervals-2} holds, either $j$ is eligible or \eqref{E:P:tiny-end-intervals} holds.
To prove this, we consider four cases based on the values of $e_j$ and $\varepsilon_{j+1}$.
\smallskip
I) If $(e_j, \varepsilon_{j+1})=(+1,-1)$, $e_{j+1}= -\varepsilon_{j+1} e_j=+1$, and by \refE{E:P:tiny-end-intervals-2},
\[x_{j+1}\in [1/\alpha_{j+1}-1-\beta_k/\beta_{j+1}, 1/\alpha_{j+1}-1] \cup [1/\alpha_{j+1}-\beta_k/\beta_{j+1}, 1/\alpha_{j+1}].\]
Since $x_m - l_m=-\varepsilon_{j+1}\alpha_{j+1}x_{j+1}= \alpha_{j+1}x_{j+1}$, this implies that
\[x_j \in \big([1-\alpha_{j+1} - \beta_k/\beta_j, 1-\alpha_{j+1}] \cup [1 - \beta_k/\beta_j, 1]\big) +\D{Z}.\]
If $x_j \in [0, 1/\alpha_j-2]$ then $j$ is eligible through ($E_1$).
If $x_j \in [1/\alpha_j-2, 1/\alpha_j]$, using $1/\alpha_j=a_j-\alpha_{j+1}$, $x_j$ must belong to one of the intervals
\begin{gather*}
[a_j-2-\beta_k/\beta_j, a_j-2], \; [a_j-1- \beta_k/\beta_j, a_j-1] \\
[1/\alpha_j-1-\beta_k/\beta_j, 1/\alpha_j-1], \; [1/\alpha_j - \beta_k/\beta_j, 1/\alpha_j].
\end{gather*}
If $x_j$ belongs to one of the bottom two intervals then \eqref{E:P:tiny-end-intervals} holds.
If $x_j$ belongs to one of the top two interval, then $j$ is eligible though ($E_1$), since $[a_j-2-\beta_k/\beta_j, a_j-2]$
is contained in $[a_j-2-\alpha_{j+1}, a_j-2]$ and $[a_j-1- \beta_k/\beta_j, a_j-1]$ is contained in $[a_j-1- \alpha_{j+1}, a_j-1]$.
Here we use $\beta_k/\beta_j\leq \alpha_{j+1}$, which is valid due to $k\geq j+1$.
\smallskip
II) If $(e_j, \varepsilon_{j+1})=(+1,+1)$, $e_{j+1}=-\varepsilon_{j+1} e_j= -1$, and by \refE{E:P:tiny-end-intervals-2},
\[x_{j+1}\in [0,\beta_k/\beta_{j+1}] \cup [1/\alpha_{j+1}-1, 1/\alpha_{j+1}-1+\beta_k/\beta_{j+1}] .\]
Since $x_j-l_j= -\varepsilon_{j+1}\alpha_{j+1} x_{j+1}=-\alpha_{j+1} x_{j+1}$, this implies that
\[x_j \in \big([-\beta_k/\beta_j,0] \cup [-1+\alpha_{j+1} - \beta_k/\beta_j, -1+\alpha_{j+1}]\big)+\D{Z}.\]
If $x_j \in [0, 1/\alpha_j-2]$ then $j$ is eligible though ($E_2$).
If $x_j \in [1/\alpha_j-2, 1/\alpha_j]$, using $1/\alpha_j= a_j + \alpha_{j+1}$, $x_j$ must belong to one of the intervals
\begin{gather*}
[a_j- \beta_k/\beta_j, a_j], \; [a_j-1 - \beta_k/\beta_j, a_j-1] \\
[1/\alpha_j-1 - \beta_k/\beta_j, 1/\alpha_j-1], \; [1/\alpha_j-\beta_k/\beta_j, 1/\alpha_j].
\end{gather*}
If $x_j$ belongs to one of the bottom two intervals, then \eqref{E:P:tiny-end-intervals} holds.
If $x_j$ belongs to one of the top two intervals then $j$ is eligible as in ($E_2$), since $[a_j- \beta_k/\beta_j, a_j]$
is contained in $[a_j- \alpha_{j+1}, a_j]$ and $[a_j-1 - \beta_k/\beta_j, a_j-1]$ is contained in $[a_j-1 - \alpha_{j+1}, a_j-1]$.
\smallskip
III) If $(e_j, \varepsilon_{j+1})=(-1,-1)$, $e_{j+1}=-\varepsilon_{j+1} e_j=-1$, and by \refE{E:P:tiny-end-intervals-2},
\[x_{j+1}\in [0,\beta_k/\beta_{j+1}] \cup [1/\alpha_{j+1}-1, 1/\alpha_{j+1}-1+\beta_k/\beta_{j+1}] .\]
Since $x_j-l_j= -\varepsilon_{j+1}\alpha_{j+1} x_{j+1}=\alpha_{j+1} x_{j+1}$, this implies that
\[x_j \in \big([0,\beta_k/\beta_j] \cup [1-\alpha_{j+1}, 1-\alpha_{j+1} +\beta_k/\beta_j]\big) +\D{Z}.\]
If $x_j \in [1, 1/\alpha_j-1]$ then $j$ is eligible through ($E_3$).
If $x_j \in [0,1] \cup [1/\alpha_j-1, 1/\alpha_j]$, using $1/\alpha_j=a_j-\alpha_{j+1}$, $x_j$ must belong to one of the intervals
\begin{gather*}
[0, \beta_k/\beta_j], [1/\alpha_j-1, 1/\alpha_j-1 + \beta_k/\beta_j] \\
[1-\alpha_{j+1},1-\alpha_{j+1} + \beta_k/\beta_j], [a_j-1, a_j-1+ \beta_k/\beta_j].
\end{gather*}
If $x_j$ belongs to one of the top two intervals, then \eqref{E:P:tiny-end-intervals} holds.
If $x_j$ belongs to one of the bottom two interval then $j$ is eligible through ($E_3$) since
$[1-\alpha_{j+1},1-\alpha_{j+1}+\beta_k/\beta_j]$ is contained in $[1-\alpha_{j+1},1]$
and $[a_j-1, a_j-1+ \beta_k/\beta_j]$ is contained in $[a_j-1, a_j-1+ \alpha_{j+1}]$.
\smallskip
IV) If $(e_j, \varepsilon_{j+1})=(-1,+1)$, $e_{j+1}= - \varepsilon_{j+1} e_j= +1$, and by \refE{E:P:tiny-end-intervals-2},
\[x_{j+1}\in [1/\alpha_{j+1}-1-\beta_k/\beta_{j+1}, 1/\alpha_{j+1}-1] \cup [1/\alpha_{j+1}-\beta_k/\beta_{j+1}, 1/\alpha_{j+1}].\]
Since $x_j-l_j= -\varepsilon_{j+1}\alpha_{j+1} x_{j+1}=-\alpha_{j+1} x_{j+1}$, this implies that
\[x_j \in \big([-1+\alpha_{j+1}, -1+\alpha_j + \beta_k/\beta_j] \cup [-1, -1 + \beta_k/\beta_j]\big) +\D{Z}.\]
If $x_j \in [1, 1/\alpha_j-1]$ then $j$ is eligible through ($E_4$).
If $x_j \in [0,1] \cup [1/\alpha_j-1, 1/\alpha_j]$, using $1/\alpha_j=a_j + \alpha_{j+1}$, $x_j$ must belong to one of the intervals
\begin{gather*}
[\alpha_{j+1}, \alpha_{j+1}+ \beta_k/\beta_j], \; [a_j,a_j+ \beta_k/\beta_j]\\
[0, \beta_k/\beta_j], \; [1/\alpha_j-1, 1/\alpha_j-1+ \beta_k/\beta_j]
\end{gather*}
If $x_j$ belongs to one of the bottom two intervals, then \eqref{E:P:tiny-end-intervals} holds.
If $x_j$ belongs to one of the top two intervals then $j$ is eligible through ($E_4$),
since $[\alpha_{j+1}, \alpha_{j+1}+ \beta_k/\beta_j]$ is contained in $[\alpha_{j+1}, 2\alpha_{j+1}]$ and $[a_j,a_j+ \beta_k/\beta_j]$
is contained in $[a_j, a_j+ \alpha_{j+1}]$.
\smallskip
{\em Step 3.} Either there is $j \geq n$ such that for all $m \geq j$ we have $x_m=1/\alpha_m-1$, or there are arbitrarily
large eligible $m \geq n$.
\smallskip
If there is $j \geq n$ such that $x_j \in \{1/\alpha_j-1, 1/\alpha_j\}$, then for all $m \geq j+1$ we have $x_m=1/\alpha_m-1$.
That is because, by the definition of the sequence $\{(x_m, l_m)\}$, if $x_j \in \{1/\alpha_j-1, 1/\alpha_j\}$ then
$x_{j+1}\in \{1/\alpha_{j+1}-1\}$.
Therefore, for all $m \geq j+1$ we have $x_m \in \{1/\alpha_m-1\}$.
Assume that there is $j\geq n$ such that $x_j=0$.
Recall from Step 1 that whenever $x_m=0$, $e_m=+1$.
By the definition of the sequence $(x_m, l_m)$, if some $x_m=0$ then either
$\varepsilon_{m+1}= -1$ and hence $x_{m+1}=0$, or $\varepsilon_{m+1}=+1$ and hence $x_{m+1}=1/\alpha_{m+1}$.
By the above paragraph, it follows that either eventually $x_m=0$, or eventually $x_m= 1/\alpha_m-1$.
When eventually $x_m=0$, all sufficiently large $m$ becomes eligible through ($E_1$) and ($E_2$).
By the above paragraphs, if there is $j \geq n$ with $x_j \in \{0, 1/\alpha_j-1, 1/\alpha_j\}$, we are done.
Below we assume that there are no such $j$.
Fix an arbitrary $j \geq n$. Since $\beta_k/\beta_j \to 0$ as $k\to \infty$, there is $k \geq j$ such that
$x_j \notin [0, \beta_k/\beta_j] \cup [1/\alpha_j-1- \beta_k/\beta_j, 1/\alpha_j-1 + \beta_k/\beta_j] \cup [1/\alpha_j - \beta_k/\beta_j]$.
It follows from Step 2 that there must be an eligible $j' \in [j, k]$.
This proves that there are arbitrarily large eligible $m\geq n$.
\smallskip
{\em Step 4.} For every $\delta>0$ there is $x_n'$ strictly between $x_n$ and $x_n+ \delta e_n$ such that
$b_n(x_n')\leq b_n(x_n)+\delta$.
\smallskip
For $m\geq n$, let $z_m=x_m+ i b_m(x_m) \in I_m$, which is the lowest point on $I_m$ with real part
equal to $x_m$.
By Step 1, for $m\geq n$, we have $\Re Y_{m+1}(x_{m+1}) + l_m= x_m$.
It follows from the definition of the sets $I_n^j$ and $I_n$ that $Y_{m+1}(z_{m+1})+l_m=z_m$.
For $m\geq 0$, let $K_m= \cap _{j\geq 1} K_m^j$ and $J_m= \cap_{j\geq 1} J_m^j$.
Assume that $m\geq n$, and either $x_m=1/\alpha_m-1$ or $m$ is eligible.
We claim that there is $z_m'\in I_m$ satisfying the following properties:
\begin{itemize}
\item[(i)] $\sign (\Re z'_m - \Re z_m)=e_m$,
\item[(ii)] $|z'_m-z_m|\leq 2$,
\item[(iii)] either both $z'_m$ and $z_m$ belong to $K_m$, or both $z'_m$ and $z_m$ belong to $J_m$.
\end{itemize}
If $x_m = 1/\alpha_m-1$ we simply let $z'_m=z_m+e_m$. By \refE{E:b_n^j-cont-periodic}, $z'_m \in I_m$.
Moreover, $\{z_m, z'_m\} \subset K_m$ if $e_m=-1$ and $\{z_m, z'_m\} \subset J_m$ if $e_m=+1$.
If $m$ is eligible, we show this by looking at cases $(E_1)$ through $(E_4)$ in the definition of eligibility.
The main tool for each of those cases is the periodic property of $I_m$ and $I_{m+1}$ in \refE{E:b_n^j-cont-periodic}.
$(E_1)$: If $x_m \in [0, 1/\alpha_m-2]$ we let $z'_m=z_m+1=z_m+e_m$. Here $\{z_m ,z'_m\} \subset K_m$.
If $x_m \in [a_m-2-\alpha_{m+1},a_m-2]= [1/\alpha_m-2, a_m-2]$, then $x_{m+1}\in [1/\alpha_{m+1}-1,1/\alpha_{m+1}]$.
Choose an integer $k_{m+1}$ with $x_{m+1}-k_{m+1} \in [0,1]$ and define
$z'_m= Y_{m+1}(z_{m+1}-k_{m+1})+ a_m-2$.
Note that $\Re z_m \leq a_m-2 \leq \Re z'_m\leq a_m-2+\alpha_{m+1} \leq 1/\alpha_m-1$.
This implies that $\sign (\Re z_m' -\Re z_m)=+1=e_m$, and $\{z_m, z'_m\} \subset K_m$.
Here, $a_m = 1/\alpha_m+\alpha_{m+1}\geq 2+ \alpha_{m+1}$, so $a_m\geq 3$.
Using \refE{E:Y_n-comm-1} with $e_m=+1$ and \refE{E:uniform-contraction-Y},
\begin{align*}
|z_m-z'_m| &= |(Y_{m+1}(z_{m+1})+a_m-3) - (Y_{m+1}(z_{m+1} -k_{m+1}) + a_m-2)| \\
&= |Y_{m+1}(z_{m+1}) -1 - Y_{m+1}(z_{m+1} -k_{m+1})| \\
&= |Y_{m+1}(z_{m+1}-1/\alpha_{m+1}) - Y_{m+1}(z_{m+1} -k_{m+1})| \\
&\leq 0.9 |(z_{m+1} -1/\alpha_{m+1}) - (z_{m+1}-k_{m+1})| \\
&= 0.9 |1/\alpha_{m+1} -k_{m+1}| \leq 0.9 \cdot 2 \leq 2.
\end{align*}
If $x_m \in [a_m-1-\alpha_{m+1},a_m-1]= [1/\alpha_m-1, a_m-1]$ we let $z'_m= Y_{m+1}(z_{m+1}-k_{m+1})+ a_m-1$.
In this case, $\Re z_m \leq a_m-1 \leq \Re z'_m \leq a_m-1 +\alpha_{m+1} \leq 1/\alpha_m$.
Hence, $\sign (\Re z_m' -\Re z_m)=+1=e_m$, and $\{z_m, z'_m\} \subset J_m$.
As in the previous case, $|z_m-z'_m|\leq 2$.
$(E_2)$: If $x_m \in [0, 1/\alpha_m-2]$ we let $z'_m=z_m+1=z_m+e_m$.
If $x_m$ belongs to $[a_m-1-\alpha_{m+1}, a_m-1]$, then $x_{m+1} \in [0,1]$.
Choose an integer $k_{m+1}$ with $x_{m+1}+k_{m+1} \in [1/\alpha_{m+1}-1, 1/\alpha_{m+1}]$ and define
$z'_m = Y_{m+1}(z_{m+1}+ k_{m+1})+ a_m$.
If $x_m$ belongs to $[a_m-\alpha_{m+1}, a_m]$, we let $z'_m = Y_{m+1}(z_{m+1}+ k_{m+1})+ a_m+1$.
As in the previous case, one can see that $z'_m$ enjoys the desired properties.
$(E_3)$: If $x_m$ belongs to $[1, 1/\alpha_m-1]$, we let $z'_m=z_m-1=z_m+e_m$.
If $x_m$ belongs to $[1-\alpha_{m+1},1]$, then $x_{m+1} \in [1/\alpha_{m+1}-1, 1/\alpha_{m+1}]$.
We define $z'_m= Y_{m+1}(z_{m+1}-1)$.
If $x_m$ belongs to $[a_m-1, a_m-1+\alpha_{m+1}]$ then $x_{m+1}\in [0,1]$.
We choose an integer $k_{m+1}$ with $x_{m+1}+k_{m+1}\in [1/\alpha_{m+1}-1, 1/\alpha_{m+1}]$ and define
$z'_m= Y_{m+1}(z_{m+1}+k_{m+1})+ a_m-2$.
One can see that $z'_m$ enjoys the desired properties.
$(E_4)$: If $x_m$ belongs to $[1, 1/\alpha_m-1]$, we let $z'_m=z_m-1=z_m+e_m$.
If $x_m$ belongs to $[\alpha_{m+1}, 2\alpha_{m+1}]$ then $x_{m+1}\in [1/\alpha_{m+1}-2,1/\alpha_{m+1}-1]$.
We define $z'_m=Y_{m+1}(z_{m+1}+1)+1$.
If $x_m$ belongs to $[a_m, a_m+\alpha_{m+1}]=[a_m, 1/\alpha_m]$, then $x_{m+1} \in [1/\alpha_{m+1}-1, 1/\alpha_{m+1}]$.
Choose an integer $k_{m+1}$ with $x_{m+1}-k_{m+1}\in [0,1]$ and define
$z'_m= Y_{m+1}(z_{m+1}-k_{m+1})+ a_m$.
One can see that $z'_m$ enjoys the desired properties.
This completes the proof of the existence of $z'_m$.
By the definition of the domains $I_n^j$, and \refE{E:P:bounds-on-l_m}, for all $m\geq n$
\begin{itemize}
\item either $Y_{m+1}(K_{m+1})+l_m \subset K_m$ or $Y_{m+1}(K_{m+1})+l_m \subset J_m$,
\item either $Y_{m+1}(J_{m+1})+l_m \subset K_m$ or $Y_{m+1}(J_{m+1})+l_m \subset J_m$.
\end{itemize}
It follows from the above properties that any composition of the form $(Y_{n+1}+l_n) \circ \dots \circ (Y_{m+1}+l_m)$
is defined and continuous on both $K_m$ and $J_m$.
Fix an arbitrary $\delta>0$ and choose $n' \geq n$ such that $2 \cdot 0.9^{(n'-n)}< \delta$.
By Step 3 there is $m \geq n'$ such that either $x_m=1/\alpha_m-1$ or $m$ is eligible.
Then, by the above argument, there is $z'_m$ satisfying the three items listed above.
Let us define $z_n^m= (Y_{n+1}+l_n) \circ \dots \circ (Y_{m+1}+l_m)(z'_m) \in I_n$.
By \refE{E:uniform-contraction-Y}, we have $|z_n^m -z_n|\leq 2 \cdot 0.9^{m-n}< \delta$.
Let $x'_n =x_n^m= \Re z_n^m$. We have $|x'_n - x_n| = |\Re z_n^m - \Re z_n| < \delta$ and
$b_n(x_n^m)- b_n(x_n) \leq \Im z_n^m-\Im z_n < \delta$.
Moreover, the relations $\sign (\Re z'_m-\Re z_m)=e_m$, $e_{m-1}= -\varepsilon_{m}e_{m}$, and
$\Re Y_{m}(x)= -\varepsilon_{m} \alpha_{m} x$ imply that $\sign (x_n^m-x_n)= \sign (\Re x_n^m - \Re z_n)=e_n$.
\end{proof}
\subsection{The Brjuno condition in the renormalisation tower}\label{SS:Brjuno-Tower}
\begin{propo}\label{P:b_n-sup-B}
For all $\alpha \in \E{B}$ and all $n\geq -1$ we have
\[\Big|2 \pi \sup_{x\in [0, 1/\alpha_n]} b_n(x) - \C{B}(\alpha_{n+1})\Big | \leq 5 \pi.\]
\end{propo}
\begin{proof}
For $n\geq -1$ and $j\geq 0$ we define
\[D_n^j= \max \{b_n^j(x) \mid x\in [0, 1/\alpha_n]\}.\]
We first show that the numbers $D_n^j$ nearly satisfy the recursive relation for the
Brjuno function, see \refE{E:Brjuno-functional-equations}. That is, for all $n\geq 0$ and all $j\geq 1$, we have
\begin{equation}\label{E:L:max-relations}
\big |2\pi D_{n-1}^{j} - 2\pi \alpha_n D_{n}^{j-1} - \log (1/\alpha_n) \big | \leq 4.
\end{equation}
Since $b_{n-1}^j$ and $b_n^{j-1}$ are periodic of period $+1$, we may choose
$x_{n-1} \in [1/(2\alpha_{n-1})-1,1/(2\alpha_{n-1})]$ and $x_n \in [1/(2\alpha_n)-1,1/(2\alpha_n)]$ such that
$b_{n-1}^j(x_{n-1})=D_{n-1}^j$ and $b_{n}^{j-1}(x_{n})=D_{n}^{j-1}$.
Choose $x_n' \in [0, 1/\alpha_n]$ such that $-\varepsilon_n \alpha_n x'_n \in x_{n-1}+\D{Z}$.
By \refL{L:Y_n-on-horizontals}-(i), we must have $x_n' \in [1/(2\alpha_n)-1,1/(2\alpha_n)]$.
We apply \ref{L:Y_n-on-horizontals}-(ii) with $y=b_n^{j-1}(x_n')$ and $x=x_n'$, to obtain
\begin{align*}
2\pi \alpha_n D_{n}^{j-1}+ \log 1/\alpha_n = 2\pi \alpha_n b_{n}^{j-1}(x_n) + \log 1/\alpha_n
& \geq 2\pi \alpha_n b_{n}^{j-1}(x'_n) + \log 1/\alpha_n \\
& \geq 2\pi \Im Y_{n}(x_n'+i b_n^{j-1}(x_n')) - 2\\
&= 2\pi b_{n-1}^j(x_{n-1}) - 2= 2\pi D_{n-1}^j -2.
\end{align*}
Similarly,
\begin{align*}
2\pi \alpha_n D_n^{j-1} + \log 1/\alpha_n = 2\pi \alpha_n b_{n}^{j-1}(x_n) + \log 1/\alpha_n
& \leq 2\pi \Im(x_n+ ib_n^{j-1}(x_n)) + 4 \\
& \leq 2\pi b_{n-1}^j(x_{n-1}) + 4 = 2\pi D_{n-1}^j+ 4.
\end{align*}
This completes the proof of inequality \eqref{E:L:max-relations}.
Fix an arbitrary $n\geq -1$ and $j\geq 1$. Let us define $\beta_{n+i}(\alpha_{n+1}) = \prod_{l=1}^{i} \alpha_{n+l}$ for $i\geq 1$,
and $\beta_{n}(\alpha_{n+1})=1$.
Then for integers $k\in [0, j]$ define the numbers
\[X_k= 2\pi \beta_{n+k}(\alpha_{n+1}) D_{n+k}^{j-k} + \sum_{i=1}^{k} \beta_{n+i-1}(\alpha_{n+1}) \log 1/\alpha_{n+i}.\]
We have $X_{0}= 2 \pi D_{n}^{j}$.
With this notation, we form a telescoping sum
\[2\pi D_n^j = \sum_{k=0}^{j-1} (X_{k} - X_{k+1}) + X_j.\]
By \refE{E:L:max-relations},
\[|X_k-X_{k+1}|
=\big |\beta_{n+k}(\alpha_{n+1})\big (2\pi D_{n+k}^{j-k}-2\pi \alpha_{n+k+1} D_{n+k+1}^{j-k-1}-\log (1/\alpha_{n+k+1})\big)\big|
\leq \beta_{n+k}(\alpha_{n+1}) 4. \]
On the other hand, since $D_{n+j}^0=-1$, we have
\[\big |X_j - \sum_{i=1}^{j} \beta_{n+i-1}(\alpha_{n+1}) \log 1/\alpha_{n+i} \big |
= |2\pi \beta_{n+j}(\alpha_{n+1})D_{n+j}^0| \leq 2\pi.\]
Combining the above inequalities, and using \refE{E:rotations-rest}, we conclude that
\begin{align*}
\Big| 2\pi D_n^j- \sum_{i=1}^{j} \beta_{n+i-1}(\alpha_{n+1}) \log 1/\alpha_{n+i} \Big |
& \leq \sum_{k=0}^{j-1} \beta_{n+k}(\alpha_{n+1})4 + 2\pi \\
& \leq \sum_{k=0}^{j-1} 2^{-k} 4 + 2\pi \leq 8 +2\pi.
\end{align*}
By \refE{E:I_n^j-forms-nest}, $b_n^j \geq b_n^{j-1}$, which implies that $D_{n}^{j}\geq D_{n}^{j-1}$.
Hence, for each $n\geq -1$, $D_{n}^{j}$ forms an increasing sequence. Combining with the above inequality,
\[\Big | 2\pi \lim_{j\to +\infty} D_n^j - \C{B}(\alpha_{n+1})\Big | \leq 8 +2\pi.\]
Note that since $b_n^j \leq b_n$ and $b_n^j \to b_n$ point-wise, we must have
$\sup_{x\in [0, 1/\alpha_n]} b_n=\lim_{j\to +\infty} D_n^j$.
\end{proof}
\subsection{The Herman condition in the renormalisation tower}\label{SS:Herman-tower}
In this section we establish an equivalent criterion for the arithmetic class $\E{H}$ in terms of the maps $Y_n$.
The key idea here is that the equivalent criterion in \refP{P:Herman-Yoccoz-criterion} is stable under uniform changes
to the maps $h_{\alpha_n}$. That is, if one replaces $h_{\alpha_n}$ by uniformly nearby maps, say $Y_n^{-1}$,
the corresponding set of rotation numbers stays the same.
See \refP{P:Y_r-vs-h_r^-1}.
Note that for arbitrary $m>n\geq 0$ and $y\geq 0$, the compositions
$h_{\alpha_n}^{-1} \circ \dots \circ h_{\alpha_m}^{-1}(y)$ may not be defined. This happens when an intermediate
iterate falls into $(-\infty, 0]$.
\begin{lem}\label{L:herman-Y-iterates-close}
Assume that for some integers $m > n\geq 0$, and $y \in (1, +\infty)$, the composition
$h_{\alpha_n}^{-1} \circ \dots \circ h_{\alpha_m}^{-1}(y)$ is defined and is positive. Then,
\[\big| 2\pi \Im Y_{n} \circ \dots \circ Y_m( i y/(2\pi)) - h_{\alpha_n}^{-1} \circ \dots \circ h_{\alpha_m}^{-1}(y)\big|
\leq 10 \pi.\]
\end{lem}
\begin{proof}
For integers $j$ with $n \leq j \leq m-1$, we may introduce
\begin{gather*}
G_{m,j-1}(y)=h_{\alpha_j}^{-1} \circ \dots \circ h_{\alpha_{m}}^{-1}(y), G_{m,m-1}(y) = h_{\alpha_m}^{-1}(y), G_{m, m}(y)= y.
\end{gather*}
By the assumptions in the lemma, all the above values are positive.
Also, for integers $j$ with $n+1 \leq j \leq m$, and $t\geq -1$, we may introduce the maps
\begin{gather*}
\Psi_{j,n-1}(t)= 2\pi \Im Y_{n} \circ \dots \circ Y_{j}(i t/(2\pi)),
\Psi_{n,n-1}(t)=2\pi \Im Y_n(i t/(2\pi)), \Psi_{n-1, n-1}(t)=t.
\end{gather*}
With the above notations, we may form a telescoping sum, to obtain
\begin{align*}
\big|h_{\alpha_n}^{-1} \circ \dots \circ h_{\alpha_m}^{-1}(y) & - 2\pi \Im Y_{n} \circ \dots \circ Y_m(i y/(2\pi))\big| \\
&= \big|G_{m,n-1}(y)- \Psi_{m,n-1}(y) \big | \\
&= \Big | \sum_{j=n}^{m} \Psi_{j-1,n-1}(G_{m,j-1}(y)) - \Psi_{j, n-1}(G_{m,j}(y)) \Big|
\end{align*}
By \refE{E:uniform-contraction-Y}, for all $s$ and $t$ in $(0, +\infty)$,
$|\Psi_{j-1,n-1}(s) - \Psi_{j-1, n-1}(t)\big| \leq 0.9^{(j-n)} \cdot |s-t|$.
Thus,
\begin{align*}
\big |\Psi_{j-1,n-1}(G_{m,j-1}(y))-&\Psi_{j, n-1}(G_{m,j}(y))\big| \\
&= \big |\Psi_{j-1,n-1}(G_{m,j-1}(y)) - \Psi_{j-1, n-1}(-2\pi i Y_j (i G_{m,j}(y)/(2\pi)))\big| \\
& \leq 0.9^{(j-n)} \big |G_{m,j-1}(y) + 2\pi i Y_j (i G_{m,j}(y)/(2\pi)) \big|
\end{align*}
On the other hand, by \refP{P:Y_r-vs-h_r^-1} and \refE{E:Y_n}, for all $j \geq 0$ and all $t \geq 1$, we have
\[|h_{\alpha_j}^{-1}(t) - 2 \pi \Im Y_j(i t/(2\pi))| \leq \pi.\]
Also, note that since $G_{m,n-1}(y)>0$ and $h_{\alpha_n}(0)=1$, $G_{m, j}(y)>1$, for $n\leq j \leq m$.
Therefore,
\begin{align*}
\big |G_{m,j-1}(y) + 2\pi i Y_j (i G_{m,j}(y)/(2\pi)) \big|
=\big |h_{\alpha_j}^{-1} (G_{m,j}(y)) - 2\pi\Im Y_j (i G_{m,j}(y)/(2\pi))\big| \leq \pi.
\end{align*}
Combining the above inequalities together, we obtain
\begin{align*}
\big|h_{\alpha_n}^{-1} \circ \dots \circ h_{\alpha_m}^{-1}(y) & - 2\pi \Im Y_{n} \circ \dots \circ Y_m(i y/(2\pi))\big|
\leq \sum_{j=n}^m 0.9^{(j-n)} \pi
\leq 10\pi.\qedhere
\end{align*}
\end{proof}
\begin{propo}\label{P:Herman-Y-iterates}
An irrational number $\alpha$ belongs to $\E{H}$, if and only if, for all $x>0$ there is $m \geq 1$ such that
\[\Im Y_0 \circ \dots \circ Y_{m-1}(i \C{B}(\alpha_{m})/(2\pi)) \leq x.\]
\end{propo}
\begin{proof}
Fix an arbitrary $\alpha \in \E{H}$ and $x >0$.
Choose $n\geq 0$ such that $0.9^n(10 \pi + 1)/(2\pi) \leq x$.
By the criterion in \refP{P:Herman-Yoccoz-criterion}, there is $m\geq n$ such that
\begin{equation}\label{E:P-Herman-Y-iterates}
h_{\alpha_{m-1}}\circ \cdots \circ h_{\alpha_n} (0) \geq \C{B}(\alpha_m).
\end{equation}
Now we consider two cases. First assume that $\C{B}(\alpha_m)\leq 1$.
Recall that $\C{B}(\alpha_m)>0$. By \eqref{E:invariant-imaginary-line} and \eqref{E:uniform-contraction-Y},
\[\Im Y_0 \circ \dots \circ Y_{m-1}(i \C{B}(\alpha_{m})/(2\pi)) \leq 0.9^{m} \cdot 1/ (2\pi) \leq 0.9^{n}/(2\pi) \leq x.\]
Thus, we have the desired inequality in the proposition in this case.
Now assume that $\C{B}(\alpha_m) > 1$. The composition of the maps in \refE{E:P-Herman-Y-iterates} is understood
as the identity map when $m=n$, and as the map $h_{\alpha_n}$ when $m=n+1$. Also recall that $h_{\alpha_n}(0)=1$.
Then, in this case, we must have $n< m-1$.
Let $k \in [n, m-1]$ be the smallest integer such that
\[h_{\alpha_k}^{-1} \circ \dots \circ h_{\alpha_{m-1}}^{-1}(\C{B}(\alpha_{m}))\]
is defined and is positive. By \refE{E:P-Herman-Y-iterates}, $k$ exists and $k>n$. Moreover, by the minimality of
$k$, we must have $h_{\alpha_k}^{-1} \circ \dots \circ h_{\alpha_{m-1}}^{-1}(\C{B}(\alpha_{m})) \leq +1$.
Now, we may use \refL{L:herman-Y-iterates-close} with $y=\C{B}(\alpha_m)$ and
$h_{\alpha_k}^{-1} \circ \dots \circ h_{\alpha_{m-1}}^{-1}(\C{B}(\alpha_{m}))$ to conclude that
\begin{align*}
|2\pi \Im & \, Y_k \circ \dots \circ Y_{m-1}(i \C{B}(\alpha_{m})/(2\pi))| \\
& \qquad \leq |2\pi \Im Y_k \circ \dots \circ Y_{m-1}(i \C{B}(\alpha_{m})/(2\pi))
- h_{\alpha_k}^{-1} \circ \dots \circ h_{\alpha_{m-1}}^{-1}(\C{B}(\alpha_{m}))| \\
& \qquad \qquad + |h_{\alpha_k}^{-1} \circ \dots \circ h_{\alpha_{m-1}}^{-1}(\C{B}(\alpha_{m}))| \\
& \qquad \leq 10\pi+1.
\end{align*}
Then, by \eqref{E:invariant-imaginary-line} and \eqref{E:uniform-contraction-Y}, we obtain
\begin{align*}
\Im Y_0 \circ \dots \circ Y_{k-1} \circ Y_k \circ \cdots \circ Y_{m-1}(i \C{B}(\alpha_{m})/(2\pi))
& \leq Y_0 \circ \dots \circ Y_{k-1} ((10\pi+1)/(2\pi)) \\
&\leq 0.9^k \frac{10 \pi+1}{2\pi} \leq 0.9^n \frac{10 \pi +1}{2\pi} \leq x.
\end{align*}
Thus, the desired inequality in the proposition also holds in this case.
To prove the other direction of the proposition, fix an arbitrary $n\geq 0$.
We shall prove that there is $m\geq n$ satisfying the inequality in \refP{P:Herman-Yoccoz-criterion}.
First note that for all $j\geq 0$ and $y\geq 0$, $h_{\alpha_j}^{-1}(y)\leq y-1$. This implies that there are $k > l \geq n$
such that
\begin{equation}\label{E:herm-equiv-k}
h_{\alpha_l}^{-1} \circ \dots \circ h_{\alpha_k}^{-1}(12 \pi)\leq 0.
\end{equation}
In particular, the composition in the above equation is defined. Note that in general one cannot choose $l=n$.
Now, choose $x>0$ such that
\begin{equation}\label{E:herm-equiv-1}
x < \Im Y_0 \circ \dots \circ Y_k(i),
\end{equation}
and
\begin{equation}\label{E:herm-equiv-2}
x < \min \big\{\Im Y_0 \circ \dots \circ Y_j(i \C{B}(\alpha_{j+1})/(2\pi))\mid \forall j\in [0, k-1] \cap \mathbb{Z} \big\},
\end{equation}
By the hypothesis in the proposition, there is $m\geq 1$ such that
\begin{equation}\label{E:P:herm-equiv-second-side}
\Im Y_0 \circ \dots \circ Y_{m-1}(i \C{B}(\alpha_{m})/(2\pi)) \leq x.
\end{equation}
Note that by \refE{E:herm-equiv-2} we must have $m\geq k+1$. In particular, $m\geq n$.
By \refE{E:herm-equiv-1}, we must have
\begin{equation}\label{E:P:herman-equiv-second-side-truncate}
\Im Y_{k+1} \circ \dots \circ Y_{m-1}(i \C{B}(\alpha_{m})/(2\pi)) \leq 1.
\end{equation}
Otherwise, by the injectivity of the maps $Y_j$ and \refE{E:invariant-imaginary-line},
\begin{align*}
\Im Y_0 \circ \dots \circ Y_{m-1}(i \C{B}(\alpha_{m})/(2\pi))
&=\Im Y_0 \circ \dots \circ Y_k (Y_{k+1} \circ \dots \circ Y_{m-1}(i \C{B}(\alpha_{m})/(2\pi)) \\
&> \Im Y_0 \circ \dots \circ Y_k (i) > x,
\end{align*}
which contradicts \refE{E:P:herm-equiv-second-side}.
Now we consider two cases.
First assume that $\C{B}(\alpha_m)> 1$ and $h_{\alpha_{k+1}}^{-1} \circ\dots\circ h_{\alpha_{m-1}}^{-1}(\C{B}(\alpha_{m}))$
is defined. Here, we may apply \refL{L:herman-Y-iterates-close} to this composition with $y=\C{B}(\alpha_m)$, and use
\refE{E:P:herman-equiv-second-side-truncate}, to get
\begin{align*}
h_{\alpha_{k+1}}^{-1} \circ \dots & \circ h_{\alpha_{m-1}}^{-1}(\C{B}(\alpha_{m})) \\
& \leq |h_{\alpha_{k+1}}^{-1} \circ \dots \circ h_{\alpha_{m-1}}^{-1}(\C{B}(\alpha_{m}))
- 2\pi \Im Y_{k+1} \circ \dots \circ Y_{m-1}(i \C{B}(\alpha_{m})/(2\pi)) | \\
& \qquad \qquad + |2\pi \Im Y_{k+1} \circ \dots \circ Y_{m-1}(i \C{B}(\alpha_{m})/(2\pi))| \\
&\leq 10 \pi + 2\pi.
\end{align*}
Combining this with \refE{E:herm-equiv-k}, and using the monotonicity of the maps $h_{\alpha_j}$, we conclude that
there is $l' \in [l, k]$ such that
\begin{align*}
h_{\alpha_{l'}}^{-1} \circ \dots \circ h_{\alpha_{m-1}}^{-1}(\C{B}(\alpha_{m}))
& = h_{\alpha_{l'}}^{-1} \circ \dots \circ h_{\alpha_k}^{-1}
(h_{\alpha_{k+1}}^{-1}\circ\cdots \circ h_{\alpha_{m-1}}^{-1}(\C{B}(\alpha_{m}))) \\
&\leq h_{\alpha_{l'}}^{-1} \circ \dots \circ h_{\alpha_k}^{-1} (12 \pi) < 0.
\end{align*}
Here we need to choose $l'\geq l$ so that the compositions in the above equation are defined.
The above inequality implies that $\C{B}(\alpha_m)< h_{\alpha_{m-1}} \circ \dots \circ h_{\alpha_{l'}}(0)$.
Note that $l'\geq l \geq n$.
On the other hand, since $h_{\alpha_j}(y)\geq y+1$ for all $j$ and $y > 0$, we must have
$h_{\alpha_{l'-1}} \circ \cdots \circ h_{\alpha_n}(0)>0$. Therefore,
\[\C{B}(\alpha_m)< h_{\alpha_{m-1}} \circ \dots \circ h_{\alpha_{l'}}(0)
< h_{\alpha_{m-1}} \circ \dots \circ h_{\alpha_n}(0).\]
This completes the argument in this case.
Now assume that either $\C{B}(\alpha_m)\leq 1$, or
$h_{\alpha_{k+1}}^{-1} \circ\dots\circ h_{\alpha_{m-1}}^{-1}(\C{B}(\alpha_{m}))$ is not defined.
These imply that there is $j$ in $[k+2, m-1]$ such that
$h_{\alpha_{j}}^{-1} \circ\dots\circ h_{\alpha_{m-1}}^{-1}(\C{B}(\alpha_{m}))\leq 0$, and hence
$\C{B}(\alpha_m) \leq h_{\alpha_{m-1}} \circ \dots \circ h_{\alpha_{j}}(0)$.
Note that $j \geq k+2 \geq n+2$.
As in the previous case, $h_{\alpha_{j-1}} \circ \cdots \circ h_{\alpha_n}(0)>0$. Therefore,
\[\C{B}(\alpha_m)< h_{\alpha_{m-1}} \circ \dots \circ h_{\alpha_j}(0)
< h_{\alpha_{m-1}} \circ \dots \circ h_{\alpha_n}(0).\]
This completes the argument in this case.
\end{proof}
\begin{propo}\label{P:optimal-herman-p_n-b_n}
Assume that $\alpha \in \E{B}$. Then, $\alpha\in \E{H}$ if and only if $p_n(0)=b_n(0)=0$ for all $n\geq -1$.
\end{propo}
\begin{proof}
Recall that $p_n(0)=p_n(1/\alpha_n)$ and $b_n(0)=b_n(1/\alpha_n)$, for all $n\geq -1$. Also, $Y_{n+1}$ maps the graphs of
$p_{n+1}$ and $b_{n+1}$ to the graphs of $p_n$ and $b_n$, respectively. These imply that $p_{n+1}(0)=b_{n+1}(0)$
if and only if $p_n(0)=b_n(0)$, for all $n\geq -1$.
Therefore, to prove the proposition, it is enough to show that $\alpha\in \E{H}$ if and only if $p_{-1}(0)=b_{-1}(0)=0$.
Assume that $\alpha\in \E{H}$. Fix an arbitrary $\varepsilon>0$. Choose $m_0 \geq 1$ satisfying $5 \cdot 0.9^{m_0}\leq \varepsilon$.
Let $\varepsilon' \leq \varepsilon/2$ be a positive constant. We may apply \refP{P:Herman-Y-iterates} with $x=\varepsilon'$ and obtain
$m\geq 1$ satisfying the inequality in that proposition.
By making $\varepsilon'$ small enough, we may make $m \geq m_0$.
Now, using the uniform contraction of the maps $Y_n$ in \refE{E:uniform-contraction-Y},
\begin{align*}
p_{-1}(0) \leq p_{-1}^m(0)
& = \Im Y_0 \circ \dots \circ Y_{m-1}(ip_{m-1}^0(0)) \\
&= \Im Y_0 \circ \dots \circ Y_{m-1}(i\C{B}(\alpha_m)/(2\pi)+5/2)
- \Im Y_0 \circ \dots \circ Y_{m-1}(i\C{B}(\alpha_m)/(2\pi)) \\
& \qquad \qquad + \Im Y_0 \circ \dots \circ Y_{m-1}(i\C{B}(\alpha_m)/(2\pi)) \\
&\leq 0.9^m 5/2 + \varepsilon' \leq \varepsilon.
\end{align*}
That is, $b_{-1}(0)=0\leq p_{-1}(0) \leq \varepsilon$, for all $\varepsilon > 0$. This implies that $b_{-1}(0)=p_{-1}(0)$.
Now assume that $b_{-1}(0)=p_{-1}(0)=0$. Fix $x >0$. Since $p_{-1}^m(0) \to p_{-1}(0)$, as $m\to +\infty$, there
is $m\geq 1$ such that $p_{-1}^m(0)< x$.
Then, by the monotonicity of the maps $t \mapsto \Im Y_l(it)$,
\[\Im Y_0 \circ \dots \circ Y_{m-1}(i \C{B}(\alpha_m)/(2\pi))
\leq \Im Y_0 \circ \dots \circ Y_{m-1}(i\C{B}(\alpha_m)/(2\pi)+5/2)
= p_{-1}^m(0) \leq x.
\]
By \refP{P:Herman-Y-iterates}, this implies that $\alpha \in \E{H}$.
\end{proof}
\subsection{Hairs, or no hairs}
\begin{propo}\label{P:b_n-p_n-dense-touches}
For every $n\geq -1$, we have
\begin{itemize}
\item[(i)] if $\alpha \in \E{B}\setminus \E{H}$, each of $b_n=p_n$ and $b_n< p_n$ hold on a dense subset of $[0, 1/\alpha_n]$;
\item[(ii)] if $\alpha \notin \E{B}$, each of $b_n=+\infty$ and $b_n < +\infty$ hold on a dense subset of $[0,1/\alpha_n]$.
\end{itemize}
\end{propo}
\begin{proof}
Each of the strict inequalities in items (i) and (ii) hold at least at one point. That is because when
$\alpha \in \E{B} \setminus \E{H}$, by \refP{P:optimal-herman-p_n-b_n}, we have $p_n(0) \neq b_n(0)$, and
when $\alpha \notin \E{B}$, $b_n(0) = 0 < +\infty$.
The main task is to show that each of the equalities in (i) and (ii) hold at least at one point.
To that end we show that there is $x_n \in [0,1/\alpha_n]$ such that if $\alpha \in \E{B}\setminus \E{H}$ we have
$p_n(x_n)=b_n(x_n)$ and if $\alpha \notin \E{B}$ we have $b_n(x_n)=+\infty$.
There is an algorithm to identify the points $x_n$, as we explain below.
There are $x_n\in [0, 1/\alpha_n]$ and $l_n \in \D{Z}$, for $n\geq -1$, such that
\[x_{n+1} \in [1/(2\alpha_{n+1})-1, 1/(2\alpha_{n+1})], \quad -\varepsilon_{n+1}\alpha_{n+1} x_{n+1} \in x_n - l_n.\]
By \refL{L:Y_n-on-horizontals}-(ii), for all $n\geq -1$ and all $j\geq 0$ we have
\[| 2\pi b_{n-1}^j(x_{n-1}) - 2 \pi \alpha_n b_n^{j-1}(x_n) - \log (1/\alpha_n)| \leq 4.\]
One may literally repeat the latter part of the proof of \refP{P:b_n-sup-B} with $D_n^j=b_n^j(x_n)$ to conclude that
for all $n\geq -1$,
\begin{equation}\label{E:P:b_n-p_n-dense-touches-1}
\big |2\pi b_{n}(x_n) - \C{B}(\alpha_{n+1})\big | \leq 8+2\pi.
\end{equation}
Now, if $\alpha \notin \E{B}$ we must have $b_n(x_n)=+\infty$.
Assume that $\alpha \in \E{B}$. We have
\begin{align*}
p_n^{j}(x_n) &= \Im (Y_n+l_n) \circ \dots \circ (Y_{n+j}(x_{n+j}+ ip_{n+j}^0(x_{n+j}))+l_{n+j-1}) \\
&= \Im (Y_n+l_n) \circ \dots \circ (Y_{n+j}(x_{n+j}+ i\C{B}(\alpha_{n+j+1})/(2\pi)+5/2)+l_{n+j-1}),
\end{align*}
and
\begin{align*}
b_n(x_n) = \Im (Y_n+l_n) \circ \dots \circ (Y_{n+j}(x_{n+j}+ i b_{n+j}(x_{n+j}))+l_{n+j-1}).
\end{align*}
Therefore, by \eqref{E:uniform-contraction-Y} and \eqref{E:P:b_n-p_n-dense-touches-1},
\begin{align*}
p_n^{j}(& x_n) - b_n(x_n) \leq 0.9^{j} \Big(\frac{5}{2}+ \frac{8+2\pi}{2\pi}\Big).
\end{align*}
Since $p_n \geq b_n$ on $[0,1/\alpha_n]$, we conclude that $p_n(x_n)=\lim_{j\to +\infty} p_n^j(x_n)=b_n(x_n)$.
To discuss items (i) and (ii) in the proposition at once, let us define $p_n \equiv +\infty$, for all $n\geq -1$,
when $\alpha \notin \E{B}$.
Recall that the graphs of $p_n$ and $b_n$ are obtained from the graphs of $p_{n+1}$ and $b_{n+1}$,
respectively, using the map $Y_{n+1}$ and its integer translations.
Thus, if $p_n=b_n$ at some $y \in [0, 1/\alpha_n]$, then $p_{n-1}=b_{n-1}$ at $-\varepsilon_n\alpha_n y + (1+ \varepsilon_n)/2$.
Similarly, if $p_n\neq b_n$ at some $y\in [0,1/\alpha_n]$, then $p_{n-1}\neq b_{n-1}$ at $-\varepsilon_n\alpha_n y+(1+\varepsilon_n)/2$.
On the other hand, $p_n$ and $b_n$ are periodic of period $+1$, for all $n\geq -1$.
One infers from these properties, and the first part of the proof, that $p_n=b_n$ on a dense subset of $[0, 1/\alpha_n]$,
and $p_n\neq b_n$ on a dense subset of $[0, 1/\alpha_n]$, for all $n\geq -1$.
\end{proof}
\begin{propo} \label{P:p_n-cont} \label{P:b_n-p_n-identical-or-not}
Assume that $\alpha \in \E{B}$. For all $n\geq -1$, $p_n: [0, 1/\alpha_n] \to [1, +\infty)$ is continuous.
Moreover, if $\alpha\in \E{H}$ then $p_n=b_n$ on $[0, 1/\alpha_n]$, for all $n\geq -1$.
\end{propo}
\begin{proof}
We aim to show that $p_n^j$ uniformly converges to $p_n$ on $[0, 1/\alpha_n]$. By the continuity of the maps $p_n^j$,
this implies the first part of the proposition. At the same time, we show that $p_n^j$ uniformly converges to $b_n$, when
$\alpha \in \E{H}$, which implies the latter part of the proposition. We present the details in several steps.
\smallskip
{\em Step 1.} For all $n\geq -1$, all $j\geq 0$, and all $x\in [0,1/\alpha_n]$,
\[p_n^j(x)\geq p_n^j(0)-5/\pi, \quad b_n^j(x) \geq b_n^j(0)-5/\pi.\]
\smallskip
We prove this by induction on $j$.
When $j=0$, $p_n^0\equiv (\C{B}(\alpha_{n+1})+5\pi)/(2\pi)$ and $b_n^0 \equiv -1$.
Therefore, $p_n^0 (x) \geq p_n^0(0)-5/\pi$ and $b_n^0(x)\geq b_n^0(0)-5/\pi$.
Now assume that both inequalities hold for some $j-1 \geq 0$ and all $n\geq -1$.
Below we prove them for $j$ and all $n\geq -1$.
Fix an arbitrary $n\geq 0$, and let $x_{n-1} \in [0, 1/\alpha_{n-1}]$ be arbitrary. Choose $x_{n}\in [0, 1/\alpha_{n}]$
and $l_{n-1}\in \D{Z}$ with $-\varepsilon_{n}\alpha_{n} x_{n}= x_{n-1}+l_{n-1}$.
If $p_n^{j-1}(x_n) \geq p_n^{j-1}(0)$, by the monotonicity of $y\mapsto \Im Y_n(x_n+iy)$, we have
\[\Im Y_{n}(x_{n}+ip_{n}^{j-1}(x_{n})) - \Im Y_{n}(x_{n}+ip_{n}^{j-1}(0)) \geq 0\geq -9/(2\pi).\]
If $p_n^{j-1}(x_n) \leq p_n^{j-1}(0)$, by the uniform contraction of $Y_{n}$ in \refE{E:uniform-contraction-Y}
and the induction hypothesis, we have
\begin{align*}
\Im Y_{n}(x_{n}+ip_{n}^{j-1}(x_{n})) - \Im Y_{n}(x_{n}+ip_{n}^{j-1}(0))
& \geq 0.9 (p_{n}^{j-1}(x_{n}) - p_{n}^{j-1}(0)) \\
& \geq 0.9(-5/\pi)= -9/(2\pi).
\end{align*}
On the other hand, by \refL{L:Y_r-distances}-(i),
\[\Im Y_{n}(x_{n}+ip_{n}^{j-1}(0)) \geq \Im Y_{n}(ip_{n}^{j-1}(0)) - 1/(2\pi) =p_{n-1}^{j}(0) -1/(2\pi).\]
Therefore, by the above inequalities and the definition of $p_{n-1}^{j}$, we have
\begin{align*}
p_{n-1}^{j}(x_{n-1})& =\Im Y_{n}(x_{n}+ ip_{n}^{j-1}(x_{n})) \\
&=\big(\Im Y_{n}(x_{n}+ip_{n}^{j-1}(x_{n}))-\Im Y_{n}(x_{n}+ip_{n}^{j-1}(0))\big)
+\Im Y_{n}(x_{n}+ip_{n}^{j-1}(0))\\
& \geq -9/(2\pi) + p_{n-1}^{j}(0) -1/(2\pi) = p_{n-1}^{j}(0) -5/\pi.
\end{align*}
The same argument applies to the map $b_{n-1}^j$.
As $j\to + \infty$, $p_n^j\to p_n$ and $b_n^j \to b_n$ point-wise on $[0, 1/\alpha_n]$.
These lead to
\begin{equation}\label{E:P:p_n-cont-1}
p_n(x)\geq p_n(0)-5/\pi, \quad b_n(x) \geq b_n(0)-5/\pi.
\end{equation}
\smallskip
{\em Step 2.} For all $n\geq -1$ and $j\geq 0$ we have
\[p_n^j(x) -p_n(x) \leq p_n^j(0)-p_n(0) +55/\pi, \quad \text{ and } \quad p_n^j(x)-b_n^j(x) \leq p_n^j(0)-b_n^j(0) + 55/\pi.\]
\smallskip
We shall prove these by induction on $j$. If $j=0$, by \refE{E:P:p_n-cont-1},
\[p_n^0(x)- p_n(x)= p_n^0(0)- p_n(x) \leq p_n^0(0) - p_n(0)+5/\pi, \]
and
\[p_n^0(x)- b_n^0(x)= p_n^0(0) - b_n^0(0) \leq p_n^0(0) - b_n^0(0)+5/\pi.\]
Now assume that both inequalities in Step 2 hold for some $j\geq 0$ and all $n\geq -1$.
We aim to prove it for $j+1$ and all $n\geq -1$.
Fix an arbitrary $x_{n-1} \in [0, 1/\alpha_{n-1}]$. Choose $x_{n}\in [0, 1/\alpha_{n}]$ and $l_{n-1} \in \D{Z}$ such that
$- \varepsilon_{n} \alpha_{n} x_{n}= x_{n-1} +l_{n-1}$.
Then, using \refL{L:Y_r-distances}-(ii),
\begin{align*}
p_{n-1}^{j+1}(x_{n-1}) - p_{n-1}(x_{n-1}) &= \Im Y_{n}(x_{n} + i p_{n}^j(x_n)) - \Im Y_{n}(x_{n} + i p_{n}(x_n)) \\
& \leq \Im Y_{n}(i p_{n}^j(x_n)) - \Im Y_{n}(i p_{n}(x_n)) + 1/(2\pi).
\end{align*}
We consider two cases.
(1) Assume that $p_n(x_n) \geq p_n(0)$. Using \refL{L:Y_r-distances}-(iii), the induction hypothesis, and
\refE{E:uniform-contraction-Y}, respectively, we obtain
\begin{align*}
\Im Y_{n}(i p_{n}^j(x_n))& - \Im Y_{n}(i p_{n}(x_n)) \\
& \leq \Im Y_{n}(i (p_{n}^j(x_n) - p_n(x_n) +p_n(0)) - \Im Y_{n}(i p_{n}(0)) + 1/(4\pi) \\
& \leq \Im Y_{n}(i(p_{n}^j(0) + 55/\pi) - p_{n-1}(0) + 1/(4\pi) \\
& \leq \Im Y_{n}(ip_{n}^j(0)) + (9/10) \cdot (55/\pi)- p_{n-1}(0) + 1/(4\pi) \\
& =p_{n-1}^{j+1}(0) - p_{n-1}(0) + 199/(4\pi).
\end{align*}
(2) Assume that $p_n(x_n) < p_n(0)$. By \refE{E:P:p_n-cont-1}, we must have $p_n(0)-p_n(x_n)\in [0, 5/\pi]$.
Then we may apply \refL{L:Y_r-distances}-(iv), and the induction hypothesis, to obtain
\begin{align*}
\Im Y_{n}(i p_{n}^j(x_n)) & - \Im Y_{n}(i p_{n}(x_n)) \\
& \leq \Im Y_{n}(i(p_{n}^j(x_n) - p_n(x_n)+ p_n(0)) - \Im Y_{n}(ip_{n}(0))+ 5/\pi \\
& \leq \Im Y_{n}(ip_{n}^j(0) + 55/\pi) - p_{n-1}(0) + 5/\pi \\
& \leq \Im Y_{n}(i p_{n}^j(0)) + 0.9 \cdot 55/\pi - p_{n-1}(0) + 5/\pi \\
& =p_{n-1}^{j+1}(0) - p_{n-1}(0) + 109/(2\pi).
\end{align*}
Combining the above inequalities, we obtain the first inequality in Step 2 for $j+1$.
The same argument applies to the difference $p_{n-1}^{j+1}(x_{n-1}) - b_{n-1}^{j+1}(x_{n-1})$.
\smallskip
{\em Step 3.} For every $n\geq -1$, $p_n^j$ uniformly converges to $p_n$ on $[0, 1/\alpha_n]$, as $j\to +\infty$.
\smallskip
Fix $n\geq -1$, and let $\varepsilon>0$ be arbitrary. Choose $m \geq n$ such that $0.9 ^{m-n} (1+55/\pi) < \varepsilon$.
Since $p_m^j(0) \to p_m(0)$, as $j\to + \infty$, there is $j_0 >0$ such that for all $j\geq j_0$ we have
$|p_m^j(0)-p_m(0)| < 1$. By Step 2, this implies that for all $x' \in [0, 1/\alpha_m]$ we have
$|p_m^j(x')-p_m(x')| \leq 1+55/\pi$.
By the uniform contraction of $Y_l$ in \refE{E:uniform-contraction-Y}, and since $Y_l$ maps the graph of $p_l$ to
$p_{l-1}$ and the graph of $p_l^k$ to the graph of $p_{l-1}^{k+1}$, we conclude that
$|p_n^{j+m-n}(x)- p_n(x)| \leq 0.9^{m-n} (1+55/\pi) < \varepsilon$.
\smallskip
{\em Step 4.} If $\alpha\in \E{H}$, for every $n\geq -1$, $p_n^j \to b_n$ on $[0, 1/\alpha_n]$.
\smallskip
By \refP{P:optimal-herman-p_n-b_n}, if $\alpha \in \E{H}$, we have $b_n(0)=p_n(0)$ for all $n\geq -1$.
Taking limits as $j\to +\infty$ in the second inequality in Step 2, we conclude that for all $n\geq -1$ we have
$|p_n(x)-b_n(x)| \leq 55/\pi$.
By the uniform contraction of $Y_l$, and equivariant property of the graphs of these functions,
we conclude that $p_n \equiv b_n$, for all $n\geq -1$.
\end{proof}
\subsection{Proof of Theorem \ref{T:model-trichotomy-thm}}\label{SS:proof-thm-thrichotomy}
In this section we combine the results from the previous sections to characterise the topology of the model
$\mathbb{A}_\alpha$.
Note that the topological boundary of $\mathbb{M}_\alpha$, $\mathbb{A}_\alpha$, satisfies the relations
\begin{equation}\label{E:A_ga}
\begin{gathered}
\mathbb{A}_\alpha = \left\{s(e^{2\pi i w}) \mid w \in I_{-1}, \Im w \leq p_{-1}(\Re w) \right \},\quad \text{if } \alpha \in \E{B}, \\
\mathbb{A}_\alpha= \left \{s(e^{2\pi i w}) \mid w \in I_{-1} \right\} \cup \left\{0 \right\}, \quad \text{if } \alpha \notin \E{B}.
\end{gathered}
\end{equation}
\begin{proof}[Proof of Theorem~\ref{T:model-trichotomy-thm}]
Recall that $I_{-1}$ is the set above the graph of $b_{-1}$, and $\mathbb{A}_\alpha$ is obtained from $I_{-1}$
through a projection, see \eqref{E:I_n-b_n} and \eqref{E:A_ga}.
By \refL{L:model-almost-periodic} and \refE{E:b_n^j-cont-periodic} we may extend $I_{-1}$ and $b_{-1}$
periodically with period $+1$. That is, $b_{-1}$ is defined on $\D{R}$ and is +1-periodic, and $I_{-1}$ becomes
$I_{-1}+\D{Z}$.
When $\alpha$ is a Brjuno number, we also have the function $p_{-1}$, which is $+1$-periodic by \refE{E:p_n-peiodic}.
We may extend this function $+1$-periodically onto $\D{R}$ as well.
{\em Part (i)}
By \refP{P:b_n-p_n-identical-or-not}, $b_{-1}\equiv p_{-1}$ on $[0,1]$, and hence $b_{-1}$ is a continuous function.
It follows that $\partial I_{-1}$ is equal to the graph of $b_{-1}$.
The map $e^{2\pi ix} \mapsto e^{-2\pi ix} e^{-2\pi b_{-1}(x)}$ from the unit circle to $\mathbb{A}_\alpha$ is continuous
and injective.
{\em Part (ii)}
By \refP{P:b_n-sup-B}, $b_{-1}(x)$ is finite for every $x\in \D{R}$, and by \refE{E:p_n>=b_n}, $b_{-1} \leq p_{-1}$.
Consider the function $R_\alpha:\D{R}/\D{Z} \to [1, +\infty)$, $R_\alpha(x)=e^{2\pi (p_{-1}(x)-b_{-1}(x))}$, and then the set
\[A'_\alpha=\{r e^{2\pi i x} \mid 1\leq r \leq R_\alpha(x)\}.\]
From \refP{P:b_n-p_n-dense-touches} we infer that each of $R_\alpha(x)=1$ and $R_\alpha(x) \neq 1$ hold on
a dense subset of $\D{R}/\D{Z}$.
Using the continuity of $p_{-1}$ in \refP{P:p_n-cont}, and \refP{P:b_n-liminfs}, we note that
for every $x\in \D{R}/\D{Z}$, we have
\[\limsup_{s\to x^+} R_\alpha(s)= \exp \big(2\pi p_{-1} (x) - 2\pi \liminf_{s \to x^+} b_{-1}(s)\big) = R_\alpha(x).\]
Similarly, $\limsup_{s\to x^-} R_\alpha(s)= R_\alpha(x)$.
Therefore, by the definition in the introduction, $A'_\alpha$ is a one-sided hairy Jordan curve.
Since $I_{-1}$ is a closed set, $\partial I_{-1} \subset I_{-1}$.
On the other hand, $p_{-1}$ is continuous, and $p_{-1}=b_{-1}$ on a dense subset of $\D{R}$.
It follows that
\[\partial I_{-1}= \{x+iy \mid x\in \D{R}, b_{-1}(x) \leq y\leq p_{-1}(x)\}.\]
Hence,
\[\mathbb{A}_\alpha=\{ e^{2\pi ix} e^{-2\pi y} \mid x\in \D{R}/\D{Z}, b_{-1}(x) \leq y\leq p_{-1}(x) \}.\]
The map $r e^{2\pi ix} \mapsto r e^{-2\pi i x} e^{-2\pi p_{-1}(x)}$ from $\mathbb{A}_\alpha'$ to $\mathbb{A}_\alpha$ is a homeomorphism.
{\em Part (iii)}
Consider the set
\[A'_\alpha = \{r e^{2\pi ix} \mid 0 \leq r \leq e^{-2\pi b_{-1}(x)}\}.\]
By virtue of propositions \ref{P:b_n-liminfs} and \ref{P:b_n-p_n-dense-touches}, $\mathbb{A}_\alpha'$ is a Cantor bouquet.
Here, $I_{-1}$ is the set above the graph of $b_{-1}$, and $b_{-1}$ takes $+\infty$ on a dense subset of $\D{R}$,
see \refP{P:b_n-p_n-dense-touches}.
It follows that
\[I_{-1}= \{x+iy \mid x \in \D{R}, b_{-1}(x) \leq y\},\]
and hence,
\[\mathbb{A}_\alpha=\{r e^{2\pi ix} \mid 0 \leq r \leq e^{-2\pi b_{-1}(x)} \}.\]
Here, $r e^{2\pi ix} \mapsto r e^{-2\pi i x}$ provides a homeomorphism from $\mathbb{A}_\alpha'$ to $\mathbb{A}_\alpha$.
Thus, $\mathbb{A}_\alpha$ is a Cantor bouquet.
\end{proof}
\begin{cor}
For every Brjuno number $\alpha$, we have the following properties:
\begin{itemize}
\item $\mathbb{M}_\alpha$ contains the ball of radius $e^{-\mathcal{B}(\alpha)-5\pi}$ about $0$,
\item $\mathbb{M}_\alpha$ does not contain any ball of radius more than $e^{-\mathcal{B}(\alpha)+5\pi}$ about $0$.
\end{itemize}
\end{cor}
\begin{proof}
Recall that any point $w$ with $\Re w \in [0,1]$ and $\Im w \geq p_{-1}(\Re w)$ belongs to
$I_{-1}$.
Recall that $P_{-1}^j$ is a decreasing sequence of functions converging to $p_{-1}$,
and $p_{-1}^0 \equiv (\mathcal{B}(\alpha)+5\pi)/(2\pi)$.
Through projection $w \mapsto s(e^{2\pi i w})$, and adding $0$, we obtain the desired ball.
On the other hand, by \refP{P:b_n-sup-B}, $ \sup_{x\in [0,1]} b_{-1}(x)$ is at least $(\mathcal{B}(\alpha)-5\pi)/(2\pi)$.
With the project, we obtain a point with modulus at most $e^{-\mathcal{B}(\alpha)+ 5\pi}$ outside $\mathbb{M}_\alpha$.
\end{proof}
\section{Dynamics of \texorpdfstring{$\mathbb{T}_\alpha$}{T-ga} on \texorpdfstring{$\mathbb{A}_\alpha$}{A-ga}}\label{SS:dynamics on the model}
Here, we study the dynamics of $\mathbb{T}_\alpha$ on $\mathbb{A}_\alpha$ and
classify the closed invariant subset of this model.
In particular, we prove all parts of \refT{T:dynamics-on-model} in this section.
\subsection{Topological recurrence}\label{SS:recurrence}
Recall that a map $f: X \to X$, of a topological space $X$, is called \textbf{topologically recurrent}, if for every $x\in X$ there
is a strictly increasing sequence of positive integers $(m_i)_{i\geq 0}$ such that $f\co{m_i}(x) \to x$ as $i \to +\infty$.
\begin{propo}\label{P:model-recurrent}
For every $\alpha \in \D{R} \setminus \D{Q}$, $\mathbb{T}_\alpha : \mathbb{A}_\alpha \to \mathbb{A}_\alpha$ is topologically recurrent.
\end{propo}
\begin{proof}
Evidently, it is enough to show that $\tilde{T}_\alpha: I_{-1}/\D{Z} \to I_{-1}/\D{Z}$ is topologically recurrent.
To that end, fix an arbitrary $w_{-1} \in I_{-1}$, and assume that $(w_i ; l_i)_{i\geq -1}$ denotes the trajectory of $w_{-1}$.
We consider two cases:
\begin{itemize}
\item[(i)] there are arbitrarily large integers $m$ with $w_m \in K_m$,
\item[(ii)] there is an integer $m \geq -1$ such that for all $i \geq m$ we have $w_i \in I_i \setminus K_i$.
\end{itemize}
Let us first assume that (i) holds. Fix an arbitrary $\varepsilon>0$. There is $m \geq 4$ such that $w_m \in K_m$ and
$(0.9)^m \sqrt{2} \leq \varepsilon$.
If $w_m+1 \in K_m$, then we note that
\[(Y_0+(\varepsilon_0+1)/2) \circ (Y_1 + l_0) \circ \dots \circ (Y_m + l_{m-1})(w_m+1)\]
is defined and belongs to $I_{-1}$.
It follows from an induction argument that there is an integer $n\geq 0$ such that the above point is equal to
$\tilde{T}_\alpha\co{n}(w_{-1})$.
Then, by \refE{E:trajectory-condition-1} and the uniform contraction of $Y_j$ in
\refL{L:uniform-contraction-Y_r},
we have $|\tilde{T}_\alpha\co{n}(w_{-1}) -w_{-1}| \leq (0.9)^{m+1} \cdot 1 \leq \varepsilon$.
If $w_m+1 \notin K_m$, we may not directly apply the above argument, since $Y_m+l_{m-1}(w_m+1)$
may not belong to $I_{m-1}$.
However, there is an integer $l' \geq (\varepsilon_m+1)/2$ such that
\[|\Re (Y_m+l')(w_m+1) - \Re w_{m-1}| \leq 1,\]
and either both $(Y_m+l')(w_m+1)$ and $w_{m-1}$ belong to $K_{m-1}$, or both $(Y_m+l')(w_m+1)$ and $w_{m-1}$ belong
to $I_{m-1} \setminus K_{m-1}$.
Note that, by \refL{L:uniform-contraction-Y_r}, we have
\begin{align*}
| \Im (Y_m + l')(w_m+1) - \Im w_{m-1}| & = |\Im (Y_m + l')(w_m+1) - \Im (Y_m + l_{m-1})(w_m) | \\
& \leq | Y_m (w_m+1) - Y_m(w_m) | \\
& \leq 0.9 \cdot 1 \leq 1.
\end{align*}
Combining with the upper bound on the difference of the real parts, we obtain
\[|(Y_m + l')(w_m+1) - w_{m-1}| \leq \sqrt{2}.\]
Now we consider the point
\[(Y_0+(\varepsilon_0+1)/2) \circ (Y_1 + l_0) \circ \dots \circ (Y_{m-1}+ l_{m-2}) \circ (Y_m + l')(w_m+1),\]
which is defined and belongs to $I_{-1}$.
In the same fashion, there is an integer $n\geq 0$ such that the above point is equal to $\tilde{T}_\alpha \co{n}(w_{-1})$.
Using the uniform contraction of $Y_j$ again, and the above bound on $|(Y_m + l')(w_m+1) - w_{m-1}|$, we conclude that
$|\tilde{T}_\alpha\co{n}(w_{-1}) - w_{-1}| \leq (0.9)^{m} \cdot \sqrt{2} \leq \varepsilon$.
Now assume that case (ii) holds.
Given $\varepsilon>0$, we may choose $m \geq 4$ such that for all $i \geq m$ we have $w_i \in I_i \setminus K_i$, and
$0.9 ^{m+1} \cdot 31 \leq \varepsilon$.
There is $w'_{-1}$ in $I_{-1}$ whose trajectory $(w'_i; l_i')_{i\geq -1}$ satisfies $w_i \in I_i \setminus K_i$,
for $0 \leq i \leq m$, and $w'_m = w_m$. It follows that there is an integer $n_1 \geq 0$ such that
$\tilde{T}_\alpha \co{n_1}(w_{-1})= w'_{-1}$.
The point $w'_{-1}$ belongs to $V^\infty$, and $\tilde{T}_\alpha$ at $w'_{-1}$ is defined using the second case in the definition of
$\tilde{T}_\alpha$. Recall the map $E_n$ defined in \refE{E:E_n-defn}.
Let us choose the integer $l \geq (\varepsilon_{m+1}+1)/2$ such that $E_m(w_m)+l \in I_m\setminus K_m$.
It follows that $|\Re E_m(w_m) + l - \Re w_m | \leq 1$.
By \refE{E:L:model-map-lift-2-3}, we have $|\Im E_m(w_m) + l - \Im w_m | \leq 30$.
Thus, $|E_m(w_m) +l - w_m | \leq 31$.
Consider the point
\[(Y_0+(\varepsilon_0+1)/2) \circ (Y_1 + l_0) \circ \dots \circ (Y_{m-1}+ l_{m-2}) \circ (Y_m + l)(E_m(w_m)),\]
which is defined and belongs to $I_{-1}$.
The above point is equal to $\tilde{T}_\alpha\co {n_2} (\tilde{T}_\alpha(w'_{-1}))$.
By the uniform contraction of $Y_j$, we conclude that
$|\tilde{T}_\alpha\co {n_2} (\tilde{T}_\alpha(w'_{-1})) - \tilde{T}_\alpha (w_{-1}')| \leq 0.9^{m+1} \cdot 31$.
By the above paragraph, this implies that
$|\tilde{T}_\alpha\co {n_2+1+n_1}(w_{-1}) - \tilde{T}_\alpha^{n_1+1} (w_{-1}) | \leq 0.9^{m+1} \cdot 31 \leq \varepsilon$.
As $n_1$ is independent of $\varepsilon$, this completes the proof in case (ii).
\end{proof}
Recall that a set $K \subset \mathbb{A}_\alpha$ is called \textbf{forward invariant} under $\mathbb{T}_\alpha$,
if $\mathbb{T}_\alpha (K) \subseteq K$.
The set $K$ is called \textbf{invariant}, or \textbf{fully invariant}, under $\mathbb{T}_\alpha$,
if $\mathbb{T}_\alpha ^{-1}(K)=K$.
\begin{propo}\label{P:invariant-fully-invariant}
Let $\alpha \in \D{R} \setminus \D{Q}$. If $K \subset \mathbb{A}_\alpha$ is closed and forward invariant under
$\mathbb{T}_\alpha$, then $K$ is fully invariant under $\mathbb{T}_\alpha$.
\end{propo}
\begin{proof}
Fix an arbitrary $z\in K$.
By \refP{P:model-recurrent}, there is an increasing sequence of positive integers $m_i$
such that $\mathbb{T}_\alpha\co {m_i}(z) \to z$ as $i \to \infty$.
Since, $K$ is closed, the sequence $\mathbb{T}_\alpha \co {m_i-1}(z)$ has a convergence
subsequence, which converges to some $z'$ in $K$. Evidently, $\mathbb{T}_\alpha(z')=z$, and hence
$z'= \mathbb{T}_\alpha ^{-1}(z) \in K$.
\end{proof}
\subsection{Closed invariant subsets}\label{SS:invariants-model}
In this section we build a family of closed invariant sets for $\mathbb{T}_\alpha: \mathbb{A}_\alpha \to \mathbb{A}_\alpha$, parametrised on a closed interval
in $\D{R}$.
The process is in analogy with how the set $\mathbb{A}_\alpha$ is built in \refS{SS:tilings-nest}.
Fix an arbitrary $y \geq 0$, and inductively define $y_n \geq 0$, for $n\geq -1$, according to
\begin{equation}\label{E:invariants-imaginary-traces}
y_{-1}= y, \qquad y_{n+1}= \Im Y_{n+1}^{-1}(i y_n).
\end{equation}
For $n\geq 0$, let
\begin{equation}\label{E:I_n-J_n-K_n-general}
\begin{gathered}
\prescript{y}{}{I}_n^0 = \{w \in \D{C} \mid \Re w\in [0, 1/\alpha_n], \Im w \geq y_n -1\}, \\
\prescript{y}{}{J}_n^0 = \{w \in \prescript{y}{}{I}_n^0 \mid \Re w \in [1/\alpha_n-1, 1/\alpha_n]\}, \quad
\prescript{y}{}{K}_n^0 = \{w \in \prescript{y}{}{I}_n^0 \mid \Re w \in [0, 1/\alpha_n-1] \}.
\end{gathered}
\end{equation}
As in \refS{SS:tilings-nest}, we inductively defined the sets $\prescript{y}{}{I}_n^j$, $\prescript{y}{}{J}_n^j$, and
$\prescript{y}{}{K}_n^j$, for $j \geq 1$ and $n\geq 0$.
Assume that $\prescript{y}{}{I}_n^j$, $\prescript{y}{}{J}_n^j$, and $\prescript{y}{}{K}_n^j$ are defined for some $j$ and
all $n \geq 0$.
When $\varepsilon_{n+1}=-1$, we let
\begin{equation}\label{E:I_n^j--1-general}
\prescript{y}{}{I}_n^{j+1}
= \bigcup_{l=0}^{a_n-2} \big( Y_{n+1} (\prescript{y}{}{I}_{n+1}^j)+ l \big)
\bigcup \big( Y_{n+1}(\prescript{y}{}{K}_{n+1}^j)+ a_n-1\big).
\end{equation}
When $\varepsilon_{n+1}=+1$, we let
\begin{equation}\label{E:I_n^j-+1-general}
\prescript{y}{}{I}_n^{j+1}
= \bigcup_{l=1}^{a_n} \big( Y_{n+1} (\prescript{y}{}{I}_{n+1}^j)+ l \big)
\bigcup \big(Y_{n+1}(\prescript{y}{}{J}_{n+1}^j)+ a_n+1\big ).
\end{equation}
Then, define
\[\prescript{y}{}{J}_n^{j+1} = \{w \in \prescript{y}{}{I}_n^{j+1} \mid \Re w \in [1/\alpha_n-1, 1/\alpha_n]\}, \;
\prescript{y}{}{K}_n^{j+1} = \{w\in \prescript{y}{}{I}_n^{j+1} \mid \Re w \in [0, 1/\alpha_n-1] \}.\]
Let
\[\prescript{y}{}{I}_{-1}^0=\{w \in \D{C} \mid \Re w \in [0, 1/\alpha_{-1}], \Im w \geq y_{-1}-1\},\]
and for $j\geq 1$, consider the sets
\[\prescript{y}{}{I}_{-1}^j= Y_0(\prescript{y}{}{I}_0^{j-1}) + (\varepsilon_0+1)/2.\]
By the latter part of \refL{L:Y-domain}, $\prescript{y}{}{I}_n^1 \subset \prescript{y}{}{I}_n^0$, for $n\geq -1$.
This implies that for all $n\geq -1$ and all $j\geq 0$,
\begin{equation}\label{E:I_n^j-forms-nest-general}
\prescript{y}{}{I}_n^{j+1} \subset \prescript{y}{}{I}_n^j.
\end{equation}
For $n\geq -1$, we define the closed sets
\[\prescript{y}{}{I}_{n}= \bigcap_{j\geq 0} \prescript{y}{}{I}_{n}^j.\]
Evidently, when $y=0$, we have $y_n=0$, for all $n\geq 0$, and hence, $\prescript{0}{}I_n=I_n$, for all $n\geq -1$.
Note that, $iy \in \prescript{y}{}I_{-1}$, for any $y\geq 0$, and
\begin{equation}\label{E:invariant-sets-lift-nest}
\prescript{x}{}I_{-1} \subsetneq \prescript{y}{}I_{-1}, \qquad \text{if } x> y \geq 0.
\end{equation}
Moreover, by the uniform contraction of $Y_n$ in \refL{L:uniform-contraction-Y_r},
\begin{equation}\label{E:invariant-set-end}
iy' \notin \prescript{y}{}{I}_{-1}, \quad \text{if } y' < y.
\end{equation}
Recall that for all $\alpha \in \D{R} \setminus \D{Q}$, $\max (\mathbb{A}_\alpha \cap \D{R})=+1$.
We define $r_\alpha \geq 0$ according to
\[[r_\alpha, 1]= \mathbb{A}_\alpha \cap [0, +\infty).\]
If $\alpha \notin \E{B}$, $r_\alpha=0$, and if $\alpha \in \E{B}$, $r_\alpha=e^{-2\pi p_{-1}(0)}$.
When $\alpha \notin \E{B}$ and $t \in (0,1]$, choose $y \geq 0$ so that $t=e^{-2\pi y}$ and define
\[\prescript{t}{}{\mathbb{A}}_\alpha=\left \{s(e^{2\pi i w}) \mid w\in\prescript{y}{}{I}_{-1} \right\} \cup \left\{0 \right\}.\]
We extend this notation by setting $\prescript{0}{}{\mathbb{A}_\alpha}= \{0\}$.
When $\alpha \in \E{B}$ and $t \in [r_\alpha,1]$, choose $y \geq 0$ so that $t=e^{-2\pi y}$, and define
\[\prescript{t}{}{\mathbb{A}}_\alpha= \left\{s(e^{2\pi i w}) \mid w\in\prescript{y}{}{I}_{-1}, \Im w\leq p_{-1}(\Re w) \right \}.\]
For all $\alpha \in \D{R} \setminus \D{Q}$, $\prescript{1}{}{\mathbb{A}}_\alpha= \mathbb{A}_\alpha$.
By \eqref{E:invariant-sets-lift-nest} and \eqref{E:invariant-set-end}, for every $r_\alpha \leq s < t \leq 1$,
we have
\begin{equation}\label{E:invariants-model-nested}
\prescript{s}{}{\mathbb{A}_\alpha} \subsetneq \prescript{t}{}{\mathbb{A}_\alpha}, \quad \text{ and } \quad t
\notin \prescript{s}{}{\mathbb{A}_\alpha}.
\end{equation}
\begin{propo}\label{P:^tA_ga-invariant}
For any $\alpha \in \D{R} \setminus \D{Q}$ and any $t \in [r_\alpha, 1]$, $\prescript{t}{}{\mathbb{A}}_\alpha$ is fully
invariant under $\mathbb{T}_\alpha: \mathbb{A}_\alpha \to \mathbb{A}_\alpha$.
\end{propo}
\begin{proof}
By \refP{P:invariant-fully-invariant}, it is enough to show that $\prescript{t}{}{\mathbb{A}}_\alpha$ is forward invariant.
By the definition of $\prescript{t}{}{\mathbb{A}}_\alpha$, it is enough to show that for all $y\geq 0$, $\prescript{y}{}{I}_{-1}$
is forward invariant under $\tilde{T}_\alpha$. Let us fix $y \geq 0$.
Recall the decomposition $I_{-1}=\cup_{n\geq 0} W^n \cup V^\infty$ in \refE{E:I_-1-decomposed}.
Let $w \in \prescript{y}{}I_{-1} \cap W^n$, for some $n\geq 0$.
Let $(w_j, l_j)_{j\geq -1}$ denote the trajectory of $w$. By the definition of $W^n$, $w_{n} \in K_n$, and by the definition of
$\prescript{y}{}I_{-1}$, $w_n \in \prescript{y}{}I_{n}$.
Since, $\prescript{y}{}I_{n}$ is translation invariant, $w_n +1 \in \prescript{y}{}I_{n}$.
Then, it follows from the definitions of $\tilde{T}_\alpha$ and $\prescript{y}{}I_{-1}$ that
$\tilde{T}_{\alpha}(w) \in \prescript{y}{}I_{-1}$.
By the above paragraph, $\tilde{T}_\alpha$ maps $\cup_{n\geq 0} W^n \cap \prescript{y}{}I_{-1}$ into $\prescript{y}{}I_{-1}$.
Since $\tilde{T}_\alpha: I_{-1}/\D{Z} \to I_{-1}/\D{Z}$ is continuous and $\prescript{y}{}I_{-1}/\D{Z}$ is closed,
$\tilde{T}_{\alpha}$ maps the closure of $(\cup_{n\geq 0} W^n \cap \prescript{y}{}I_{-1})/\D{Z}$ into
$\prescript{y}{}I_{-1}/\D{Z}$.
The closure of $(\cup_{n\geq 0} W^n \cap \prescript{y}{}I_{-1})/\D{Z}$ is equal to $\prescript{y}{}I_{-1}/\D{Z}$.
\end{proof}
\subsection{Closures of orbits}
\begin{lem}\label{L:dense-orbits-model}
For every $\alpha \in \D{R} \setminus \D{Q}$, the following hold:
\begin{itemize}
\item[(i)] if $\alpha \notin \E{B}$, then for all $y \geq 0$, the orbit of $iy$ under $\tilde{T}_\alpha$
is dense in $\prescript{y}{}{I}_{-1}$;
\item[(ii)] if $\alpha \in \E{B}$, then for all $y$ with $0 \leq y \leq p_{-1}(0)$, the orbit of $iy$
under $\tilde{T}_\alpha$ is dense in
\[\{ w\in \prescript{y}{}{I}_{-1}\mid \Im w\leq p_{-1}(\Re w)\}.\]
\end{itemize}
\end{lem}
\begin{proof}
To simplify the notations, let us first consider the orbit of $y=0$.
When $\alpha \in \E{H}$, the set in item (ii) becomes the graph of the function $b_{-1}= p_{-1}$.
Then, the statement follows from the continuity of $b_{-1}$ in \refP{P:p_n-cont}.
The non-trivial case is to prove the statement when $\alpha \notin \E{H}$.
Let $\langle x \rangle$ denote the fractional part of $x \in \D{R}$, that is, $\langle x \rangle \in [0,1)$ and
$ x \in \langle x \rangle + \D{Z}$.
By \refE{E:A_ga}, in order to prove the proposition, it is enough to show that the set
\[\Big \{\langle - m \varepsilon_0 \alpha_0 + (1+\varepsilon_0)/2 \rangle + i b_{-1}(\langle -m \varepsilon_0 \alpha_0 + (1+\varepsilon_0)/2 \rangle)
\Bigm \vert m \in \D{N}\Big\}\]
is dense in $\partial I_{-1}$. This is because $s(e^{2 \pi i (-m \varepsilon_0 \alpha_0)})= e^{2\pi i m\alpha}$.
It is possible to prove both items in the proposition at once.
Assume that $z$ is an arbitrary point, such that either $\alpha \notin \E{B}$ and $z \in I_{-1}$,
or $\alpha \in \E{B}$ and $z \in I_{-1}$ with $\Im z \leq p_{-1}(\Re z)$.
Also, fix an arbitrary $\delta>0$.
We aim to identify an element of the orbit of $y$ in the $\delta$ neighbourhood of $z$.
Recall that $I_{-1}= \cap_{j\geq 1} I_{-1}^j$.
Choose $j_0 \geq 1$ such that $(9/10)^{j_0} \sqrt{2} \leq \delta/2$.
There are $j \geq j_0$ and $z' \in \partial I_{-1}^{j+1}$ such that $|z'-z|\leq \delta/2$ and $\Re z'\in (0,1)$.
By the definition of $I_{-1}^{j+1}$, there must be $w_{j} \in \partial I_{j}^0 $ and integers $l_k \in \D{Z}$,
for $-1\leq k\leq j$, such that
\[z'=(Y_0+l_{-1})\circ (Y_1+l_0) \circ \dots \circ (Y_{j}+l_{j-1})(w_j).\]
There is an integer $l_j$ with $0 \leq l_j \leq 1/\alpha_j$ such that $|l_j - w_j| \leq \sqrt{2}$.
Let
\[z''=(Y_0+l_{-1}) \circ (Y_1+l_1) \circ \dots \circ (Y_j+l_{j-1})(l_j).\]
By the uniform contraction of the maps $Y_k$ in \refE{E:uniform-contraction-Y},
$|z'-z''| \leq (9/10)^{j+1}\sqrt{2}\leq \delta/2$.
In particular, $|z''-z|\leq \delta$.
On the other hand, since $\Re Y_k(x) = - \varepsilon_k \alpha_k \Re x$, one may verify that
\[\Re z''= \sum_{k=0}^{j} \Big(l_k \prod_{n=0}^{k} (-\varepsilon_n \alpha_n)\Big)+ l_{-1}.\]
Any value of the above form is equal to $- m\varepsilon_0\alpha_0 + (1+\varepsilon_0)/2$, for some $m\in \D{N}$, modulo $\D{Z}$.
This may be proved by induction on $j$.
From the definition of $I_{-1}$ we note that $l_{-1}= (1+\varepsilon_0)/2$. This implies the statement for $j=0$.
Now assume that the statement is true for all integers less than $j$. To prove it for $j$, one uses the relation
$\varepsilon_j \alpha_j= 1/\alpha_{j-1}-a_{j-1}$ to reduce the statement to $j-1$.
The proof for non-zero values of $y$ is identical to the above one; one only needs to employ the translation
invariance of the sets $\prescript{y}{}{I}_n^j$.
\end{proof}
\begin{propo}\label{P:invariants-ordered-model}
For any $\alpha \in \D{R} \setminus \D{Q}$, and any $t \in [r_\alpha,1]$, $\omega(t)= \prescript{t}{}{\mathbb{A}}_\alpha$.
In particular,
\begin{itemize}
\item if $s > t$, $s \notin \omega(t)$,
\item the obit of $+1$ is dense in $\mathbb{A}_\alpha$.
\end{itemize}
\end{propo}
\begin{proof}
This is an immediate corollary of \refL{L:dense-orbits-model} and the property in \refE{E:invariant-set-end}.
\end{proof}
\begin{propo}\label{P:invariants-classified-model}
Let $\alpha \in \D{R} \setminus \D{Q}$. For every non-empty closed invariant set $X$ of
$\mathbb{T}_\alpha: \mathbb{A}_\alpha \to \mathbb{A}_\alpha$, there is $t \in [r_\alpha,1]$ such that
$X=\prescript{t}{}{\mathbb{A}}_\alpha$.
\end{propo}
\begin{proof}
If $\alpha \notin \E{B}$ and $X=\{0\}$, we let $t=0$.
Otherwise, let $X_{-1} \subset I_{-1}$ be the lift of $X \setminus \{0\}$, which is a closed set in $\mathbb{C}$.
We inductively define the closed sets $X_n$, for $n\geq 0$, according to
\[X_{n+1}=\{w \in I_{n+1} \mid Y_{n+1}(w)+(\varepsilon_{n+1}+1)/2 \in X_n\}.\]
We consider two cases below.
{\em Case (I)} There is $N\geq -1$ such that for all $n\geq N$, $\min \Im X_n \geq 4$.
We define the functions $h_n^j:[0, 1/\alpha_n] \to [-1, +\infty)$, for $n\geq -1$ and $j\geq N+1$, as follows.
For $n\geq N$, let
\[h_n^0(x) = \min \Im X_n -5.\]
Then, define $h_n^{j+1}$ as the lift of $h_{n+1}^j$ by $Y_{n+1}$.
That is, for $x\in [0,1]$, let
\[h_n^{j+1}(x)= \Im Y_{n+1} (- \varepsilon_{n+1}x / \alpha_{n+1}+ i h_{n+1}^j(- \varepsilon_{n+1}x / \alpha_{n+1})),\]
and then extend $h_n^{j+1}$ over $[0, 1/\alpha_n]$ using $h_n^{j+1}(x)= h_n^{j+1}(x+1)$.
We claim that for all $n \geq -1$ and all $j\geq N+1$, $h_n^{j+1} \geq h_n^j$.
We show this by induction on $j$.
Let $n \geq N$.
Since $X_{-1}$ is invariant under $\tilde{T}_\alpha$, $X_{n+1}$ is $+1$-periodic.
This implies that every point on the graph of $h_{n+1}^0$ is at distance at most $(5+ 1/2)$ from some element of $X_{n+1}$.
Then, by the uniform contraction of $Y_{n+1}$ in \refL{L:uniform-contraction-Y_r}, any point on the graph of $h_n^1$
is at distance at most $(1/2+5) \cdot 0.9 \leq 5$ from some element of $X_n$.
This implies that $h_n^1 \geq \min \Im X_n -5 = h_n^0$, for all $n \geq N$.
Now repeatedly applying the maps $Y_l$, we obtain the desired property.
By the above paragraph, $h_n^j$ converges to some function $h_n: [0, 1/\alpha_n] \to [-1, +\infty]$, as $j \to +\infty$.
Moreover, by the relations in \eqref{E:Y_n-comm-1} and \eqref{E:Y_n-comm-2}, for any $n\geq -1$ we have
\[h_n(0)=h_n(1/\alpha_n), \qquad h_n(x)=h_n(x+1), \quad \text{for } n\geq -1, x\in [0, 1/\alpha_n-1].\]
One may repeat the argument in Step 1 of the proof of \refP{P:p_n-cont} (replace $p_n^j$ with $h_n^j$) to conclude that for
all $n \geq N$ and all $x\in [0, 1/\alpha_n]$, $h_n(x) \geq h_n(0) -5/\pi$.
Since $X_N$ is not empty, and lies above the graph of $h_N$, there must be $x \in [0,1/\alpha_N]$ such that $h_N(x) < +\infty$.
In particular, $h_N(0)\neq +\infty$, and hence $h_{-1}(0) \neq +\infty$.
Let us introduce $y=h_{-1}(0)$.
We claim that $iy \in X_{-1}$.
To see this, fix an arbitrary $n\geq N$.
There is a net of points on $X_{n}-5 i$, at distance one from one another, which lie on the graph of $h_n^0$.
Lifting these points using $Y_j+l_j$, we obtain a net of points on the graph of $h_{-1}^n$, which by the uniform contraction of
$Y_j$ in \refL{L:uniform-contraction-Y_r} must be at distances at most $1 \cdot 0.9^{n+1}$ from one another.
Moreover, those points on the graph of $h_{-1}^n$ lie at distances at most $5 \cdot 0.9^{n+1}$ from some elements of $X_{-1}$.
As $h_{-1}^n \to h_{-1}(0)$, one may conclude that there is a sequence of points in $X_{-1}$ which converges to $i h_{-1}(0)= iy$.
Since $X_{-1}$ is closed, $iy \in X_{-1}$.
Let us consider the set $\prescript{y}{}I_{-1}$ and the associated function $\prescript{y}{}b_n$.
We claim that $h_n \equiv \prescript{y}{}{b}_n$, on $[0, 1/\alpha_n]$, for all $n\geq -1$.
Recall the values $y_n \geq 0$ from \refE{E:invariants-imaginary-traces}, and note that $h_n(0)=y_n$, for all $n\geq -1$.
Also, recall the functions $\prescript{y}{}b_n^j: [0, 1/\alpha_n] \to [-1, +\infty]$.
Since $X_n$ lies above the graph of $h_n$, for $n\geq N$ we have
\[h_n^0(x)=\min \Im X_n -5 \geq \min_{x\in [0, 1/\alpha_n]} h_n(x) -5 \geq h_n(0)-5/\pi-5.\]
On the other hand, $h_n(0) \geq h_n^0(0)= h_n^0(x)$, for all $x \in [0, 1/\alpha_n]$.
Recall that $\prescript{y}{}b^0_n(x) \equiv y_n-1=h_n(0)-1$, for $x\in [0, 1/\alpha_n]$.
Therefore, for all $n \geq N$ and all $x\in [0, 1/\alpha_n]$, $|h_n^0(x)-\prescript{y}{}{b}_n^0(x)| \leq 6$.
By the uniform contraction of $Y_l$ in \refL{L:uniform-contraction-Y_r}, for all $n \geq -1$, $x \in [0, 1/\alpha_n]$ and
$j \geq N+1$, we must have $|h_n^j(x)- \prescript{y}{}b_n^j(x)| \leq 6 \cdot 0.9^j$.
Taking limits as $j \to +\infty$, we obtain $h_n(x) \equiv \prescript{y}{}{b}_n(x)$.
Since $X_{-1}$ lies above the graph of $h_{-1}$, and $h_{-1}= \prescript{y}{}b_{-1}$, we must have
\[X_{-1} \subseteq \prescript{y}{}{I}_{-1}, \; \text{if } \alpha \notin \E{B}, \quad \text{ and } \quad
X_{-1} \subseteq \{ w\in \prescript{y}{}{I}_{-1} \mid \Im w \leq p_{-1}(\Re w) \}, \; \text{if } \alpha \in \E{B}.\]
On the other hand, as $iy\in X_{-1}$, $X_{-1}$ is closed, and $X_{-1}$ is invariant under $\tilde{T}_\alpha$, the closure of the orbit
of $iy$ under $\tilde{T}_\alpha$ must be contained in $X_{-1}$.
By virtue of \refL{L:dense-orbits-model}, we conclude that the inclusions in the above equation are equalities.
{\em Case (II)} There are arbitrarily large $n$ with $\min \Im X_n \leq 4$.
Here, we may introduce $h_n^0(x)\equiv -1$, for all $n \geq -1$, and define the functions $h_n^j$ as in case (I).
We note that $h_n^j$ is an increasing sequence of functions, which converges to some $h_n: [0, 1/\alpha_n] \to [-1, +\infty]$.
Moreover, these functions also enjoy the functional relations $h_n(x+1)=h_n(x)$, for $x\in [0, 1/\alpha_n]$, and
$h_n(0)=h_n(1/\alpha_n)$. Then, as in the above paragraphs, we must have $h_n(0) \leq h_n(x) +5/\pi$, for all $n\geq 0$.
Since $X_n$ lies above the graph of $h_n$, combining with the hypothesis in this case, there must be arbitrarily large $n$
with $h_n(0) \leq 4+ 5/\pi$.
By the uniform contraction of $Y_l$, we conclude that $h_{-1}(0)=0$.
Thus, $0 \in X_{-1}$.
Since $X_{-1}$ is closed, and invariant under $\tilde{T}_\alpha$, the closure of the orbit of $0$ must be in $X_{-1}$.
Using \refL{L:dense-orbits-model}, we complete the proof in this case.
\end{proof}
\subsection{Topology of the closed invariant subsets}\label{SS:top-invariant-sets}
In order to explain the topological properties of the sets $\prescript{t}{}{\mathbb{A}}_\alpha$, we use height functions as in
\refS{SS:height-functions} to study the structure of the sets $\prescript{y}{}I_{-1}$.
Since each $Y_n$ preserves vertical lines, each of $I_n^j$ and $I_n$, for $n\geq -1$ and $j\geq 0$, consists of closed
half-infinite vertical lines.
For $n\geq -1$ and $j\geq 0$, define $\prescript{y}{}b_n^j:[0, 1/\alpha_n] \to [-1, +\infty)$ according to
\begin{equation}\label{E:I_n^j-b_n^j-general}
\prescript{y}{}I_n^j= \{w\in \D{C} \mid 0 \leq \Re w \leq 1/\alpha_n, \Im w \geq \prescript{y}{}b_n^j(\Re w)\}.
\end{equation}
By the relations in \eqref{E:Y_n-comm-1}--\eqref{E:Y_n-comm-2}, for all $n\geq -1$ and $j\geq 0$,
$\prescript{y}{}b_n^j:[0, 1/\alpha_n] \to [-1, +\infty)$ is continuous.
Moreover, by \eqref{E:I_n^j-forms-nest-general} and \eqref{E:I_n^j-b_n^j-general},
$\prescript{y}{}b_n^{j+1} \geq \prescript{y}{}b_n^j$ on $[0, 1/\alpha_n]$.
Thus, for $n\geq -1$, we may define $\prescript{y}{}b_n:[0, 1/\alpha_n] \to [-1,+\infty]$ as
\[\prescript{y}{}b_n(x)= \lim_{j\to + \infty} \prescript{y}{}b_n^j(x)= \sup_{j\geq 0} \prescript{y}{}b_n^j(x).\]
The function $\prescript{y}{}b_n$ describes the set $I_n$, that is,
\begin{equation}\label{E:I_n-b_n-general}
\prescript{y}{}I_n= \{w\in \D{C} \mid 0 \leq \Re w \leq 1/\alpha_n, \Im w \geq \prescript{y}{}b_n(\Re w)\}.
\end{equation}
By the definition of the sets $\prescript{y}{}I_n^j$, $\prescript{y}{}b_n^j(0)=\prescript{y}{}b_n^j(1/\alpha_n)$, and
$\prescript{y}{}b_n^j(x+1)= \prescript{y}{}b_n^j(x)$ for all $x\in [0, 1/\alpha_n-1]$.
Taking limits as $j\to +\infty$, we note that for all $n\geq -1$,
\begin{equation}\label{E:b_n^j-cont-periodic-general}
\prescript{y}{}b_n(0)=\prescript{y}{}b_n(1/\alpha_n), \quad
\prescript{y}{}b_n(x+1)=\prescript{y}{}b_n(x), \quad \forall x \in [0, 1/\alpha_n-1].
\end{equation}
These are the key functional relations required to explain the topology of $\prescript{y}{}I_{-1}$.
From \refS{SS:height-functions}, recall the height functions $p_n : [0, 1/\alpha_n] \to [-1, +\infty)$, for $n\geq -1$.
\begin{propo}\label{P:invariants-topology-model}
For every $\alpha \in \D{R} \setminus \D{Q}$ the following hold:
\begin{itemize}
\item[(i)] if $\alpha \notin \E{B}$, for every $t \in (r_\alpha, 1]$, $\prescript{t}{}{\mathbb{A}}_\alpha$ is a Cantor bouquet;
\item[(ii)] if $\alpha \in \E{B} \setminus \E{H}$, for every $t \in (r_\alpha, 1]$, $\prescript{t}{}{\mathbb{A}}_\alpha$ is a one-sided hairy Jordan curve.
\end{itemize}
\end{propo}
\begin{proof}
One may repeat the proof of \refP{P:b_n-liminfs}, replacing $b_n$ with $\prescript{y}{}b_n$, and using the
relations in \refE{E:b_n^j-cont-periodic-general} instead of the corresponding ones in \refE{E:b_n^j-cont-periodic}.
(Indeed, one only needs to verify the first and the last paragraphs in the proof of \refP{P:b_n-liminfs}.)
That leads to the properties
\begin{itemize}
\item[(a)] for all $x\in [0, 1/\alpha_n)$, $\liminf_{s\to x^+} \prescript{y}{}{b}_n(s)= \prescript{y}{}{b}_n(x)$;
\item[(b)] for all $x\in (0, 1/\alpha_n]$, $\liminf_{s\to x^-} \prescript{y}{}{b}_n(s)= \prescript{y}{}{b}_n(x)$;
\end{itemize}
for each $n\geq -1$ and $y\geq 0$.
In the same way, as in the proof of \refP{P:b_n-p_n-dense-touches}, we also note that for every $n\geq -1$,
\begin{itemize}
\item[(c)] if $\alpha \notin \E{B}$, each of $\prescript{y}{}b_n=+\infty$ and $\prescript{y}{}b_n < +\infty$
hold on a dense subset of
$[0,1/\alpha_n]$;
\item[(d)] if $\alpha \in \E{B}\setminus \E{H}$ and $y < p_{-1}(0)$,
each of $\prescript{y}{}b_n=p_n$ and $\prescript{y}{}b_n < p_n$ hold on a dense subset of $[0, 1/\alpha_n]$.
\end{itemize}
Then, one may repeat the proof of \refT{T:model-trichotomy-thm}, using the above properties (a)--(d)
to establish the desired result.
\end{proof}
\subsection{Dependence on the parameter}
\begin{propo}\label{P:invariants-cont-depen}
For every $\alpha \in \D{R} \setminus \D{Q}$, the map $t \mapsto \prescript{t}{}{\mathbb{A}}_\alpha$, $t \in [r_\alpha, 1]$,
is continuous with respect to the Hausdorff metric on compact subsets of $\D{C}$.
\end{propo}
\begin{proof}
By the definition of $\prescript{t}{}{\mathbb{A}}_\alpha$, it is enough to show that the map $y \mapsto \prescript{y}{}I_{-1}$,
for $y\geq 0$, is continuous with respect to the Hausdorff distance.
To that end, we need to show that for any $y \geq 0$ and any $\varepsilon >0$, there is $\delta>0$ such that
if $|x-y| \leq \delta$ and $x\geq 0$, then $\prescript{x}{}I_{-1} \subset B_\varepsilon (\prescript{y}{}I_{-1})$
and $\prescript{y}{}I_{-1} \subset B_\varepsilon (\prescript{x}{}I_{-1})$.
Here, $B_\varepsilon(\prescript{z}{}I_{-1})$ denotes the $\varepsilon$-neighbourhood of $\prescript{z}{}I_{-1}$.
Fix an arbitrary $y \geq 0$ and $\varepsilon >0$.
We claim that
\begin{itemize}
\item[(i)] $\prescript{y}{}I_{-1}= \overline{\cup_{x>y} \prescript{x}{}I_{-1}}$,
\item[(ii)] if $y> 0$, $\prescript{y}{}I_{-1} = \cap_{x<y} \prescript{x}{}I_{-1}$.
\end{itemize}
To prove these, it is enough to show that for all $t \in [0, 1]$, $\prescript{x}{}b_{-1}(t) \to \prescript{y}{}b_{-1}(t)$, as $x \to y$.
Let $(x_n)_{n \geq -1}$ and $(y_n)_{n\geq -1}$ denote the sequences defined according to \refE{E:invariants-imaginary-traces}
for the values $x$ and $y$ respectively.
If $|x-y|$ is small enough, $|x_n - y_n| \leq 1$.
The inequality $|\prescript{x}{}b_n^0 - \prescript{y}{}b_n^0| \leq 1$ and the uniform contraction in
\refL{L:uniform-contraction-Y_r} imply that $|\prescript{x}{}b_{-1}^{n+1} - \prescript{y}{}b_{-1}^{n+1}| \leq 0.9 ^{n+1}$.
Since $\prescript{x}{}b_{-1}^n(t) \to \prescript{x}{}b_{-1}(t)$ and $\prescript{y}{}b_{-1}^n(t) \to \prescript{y}{}b_{-1}(t)$,
as $n\to \infty$, one infers that $\prescript{x}{}b_{-1}(t) \to \prescript{y}{}b_{-1}(t)$, as $x \to y$.
By property (i) above, $\cup_{x>y} B_\varepsilon (\prescript{x}{}I_{-1})$ provides an open cover of $\prescript{y}{}I_{-1}$.
Note that for any $x>y$, $B_\varepsilon (\prescript{x}{}I_{-1})$ covers all points in $\prescript{0}{}I_{-1}$ above some imaginary
value.
It follows (choose a finite cover) that there is $x_0 > y$ such that if $y < x \leq x_0$,
$\prescript{y}{}I_{-1} \subset B_\varepsilon (\prescript{x}{}I_{-1})$.
On the other hand, if $y>0$, property (ii) implies that there is $x_1 < y$ such that for all $x_1 < x < y$,
we have $\prescript{x}{}I_{-1} \subset B_\varepsilon (\prescript{y}{}I_{-1})$.
Let $\delta = \min \{|x_0-y|, |x_1-y|\}$.
Below assume that $|x-y| \leq \delta$, for some $x \geq 0$.
If $x< y$, by \refE{E:invariant-sets-lift-nest}, $\prescript{y}{}I_{-1} \subset \prescript{x}{}I_{-1}$ and hence
$\prescript{y}{}I_{-1} \subset B_\varepsilon(\prescript{x}{}I_{-1})$.
If $x>y$, by the above paragraph, we have $\prescript{y}{}I_{-1} \subset B_\varepsilon(\prescript{x}{}I_{-1})$.
Similarly, if $x > y$, by \refE{E:invariant-sets-lift-nest}, $\prescript{x}{}I_{-1} \subset \prescript{y}{}I_{-1}$ and hence
$\prescript{x}{}I_{-1} \subset B_\varepsilon(\prescript{y}{}I_{-1})$.
If $x< y$, by the above paragraphs, $\prescript{x}{}I_{-1} \subset B_\varepsilon (\prescript{y}{}I_{-1})$.
\end{proof}
This completes the proof of \refT{T:dynamics-on-model}.
More precisely, part (i) is proved in \refP{P:model-recurrent}.
Part (ii) follows from Propositions \ref{P:invariants-ordered-model} and \ref{P:invariants-cont-depen}.
Part (iii) follows from \refP{P:invariants-classified-model} and \refE{E:invariants-model-nested}.
Parts (iv) and (v) follow from Propositions \ref{P:invariants-classified-model} and \ref{P:invariants-topology-model}.
Here, when $\alpha \in \E{H}$, we have $r_\alpha=1$ and there is nothing to prove.
|
2024-02-18T23:40:24.359Z
|
2022-04-07T02:18:14.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14557","language":"en","timestamp":1649297894000,"url":"https:\/\/arxiv.org\/abs\/2112.14557","yymm":"2112"}
|
proofpile-arXiv_000-10008
|
{"provenance":"002.jsonl.gz:10009"}
| null | null |
\section{Introduction}
Strongly correlated fermionic many-body systems play an essential role in the study of modern condensed matter physics. In particular, a simplified model describing electrons in crystals, proposed by Kanamori, Gutzwiller, and Hubbard, has been extensively examined theoretically \cite{Gutzwiller1963,Hubbard1963,Kanamori1963} and is currently known as the Hubbard model. Despite its simple mathematical definition, the Hubbard model has been confirmed numerically to describe various phenomena such as metal-insulator transition and magnetic orders.
On the other hand, rigorous analysis of the Hubbard model is quite challenging, and mathematical proofs of these numerical predictions, which imply a wide variety of phenomena, are often still missing.
In these circumstances, mathematical studies of the magnetic properties of the ground states of the Hubbard model are somewhat of an exception, and several mathematical theorems are known \cite{Lieb2004,Tasaki2020};
Nagaoka and Thouless investigated a many-electron system with very high strength of the Coulomb interaction and exactly one hole on the lattice, and proved that the ground state of such a system is ferromagnetic \cite{Nagaoka1965,Thouless_1965}. The Nagaoka--Thouless theorem is the first result concerning the rigorous study of ferromagnetic ground states in the Hubbard model, and has greatly influenced subsequent works on magnetism in the model. Aizenman and Lieb extended the Nagaoka and Thouless result to finite temperatures by applying a method known as the random loop representations to the partition function \cite{Aizenman1990}.
Furthermore, the author of this paper extends the Nagaoka--Thouless and Aizenman--Lieb theorems to systems with electron-phonon interaction and electron-quantized radiation field interaction \cite{Miyao2017,Miyao2020-2}.
\medskip
Recent progress in experimental technologies for ultracold atoms has allowed realizing an extension of the conventional spin-$1/2$ Hubbard model, the so-called $\mathrm{SU}(n)$Hubbard model with general $\mathrm{SU}(n)$ symmetry \cite{Cazalilla_2014,PhysRevX.6.021030,Pagano2014,Scazza2014,Zhang2014}. Theoretical analysis of the $\mathrm{SU}(n)$ model has been developed mainly through numerical calculations, and it has become clear that phenomena that do not appear in systems described by the usual $\mathrm{SU}(2)$ Hubbard model occur in systems of the $\mathrm{SU}(n)$ Hubbard model \cite{PhysRevB.77.144520, PhysRevLett.98.160405, Titvinidze2011,Zhao2007}.
On the other hand, as in the case of the $\mathrm{SU}(2)$ model, rigorous analysis is known to be very challenging. Therefore, the question of how the rigorous results of the $\mathrm{SU}(2)$ model can be extended to the $\mathrm{SU}(n)$ model is intriguing from both a physical and a mathematical point of view.
As for studies in this direction, we refer to \cite{Katsura2013,LIU20191490, PhysRevB.96.075149,Tamura2021,Yoshida2021}.
In particular, in \cite{Katsura2013}, Katsura and Tanaka extend the Nagaoka and Thouless result to the $\mathrm{SU}(n) $Hubbard model by carefully examining the connectivity condition, which is the key to the proof of the Nagaoka--Thouless theorem.
The purpose of this paper is to extend the Aizenman--Lieb theorem, which is a finite temperature version of the Nagaoka--Thouless theorem, to the $\mathrm{SU}(n)$ Hubbard model. The strategy of the proof is to extend the random-loop representation used in the description of the $\mathrm{SU}(2)$ model appropriately to the analysis of the partition function of the $\mathrm{SU}(n)$ Hubbard model.
Random-loop representations for the partition functions of quantum spin systems are well known as powerful analytical tools in the rigorous study of critical phenomena
\cite{cmp/1104270709}, and the random-loop representation constructed in this paper is expected to have further applications.
\medskip
The organization of this paper is as follows. In Section \ref{Sec2}, we define the $\mathrm{SU}(n)$ Hubbard model and explicitly state the main theorems of this paper. In addition, we make it clear that these main theorems are extensions of the Aizenman--Lieb theorem. Then, in Section \ref{Sec3}, we provide necessary preparations for proving the main theorems: we construct the Feynman--Kac--It{\^o} formulas for the heat semigroup generated by the $\mathrm{SU}(n)$ Hubbard model. In Section \ref{Sec4}, we first construct a random loop representation for the partition function. Then, using this representation, we give proofs of the main theorems stated in Section \ref{Sec2}.
\subsection*{Acknowledgements}
This work was supported by JSPS KAKENHI Grant Numbers 18K03315, 20KK0304.
\section{Main results}\label{Sec2}
For $d\ge 2$, let $\Lambda$ be a $d$-dimensional hypercube lattice with one side of length $2\ell$: $\Lambda=(\mathbb{Z}\cap[-\ell, \ell))^d$.
The ${\rm SU}(n)$ Hubbard Hamiltonian on $\Lambda$ is defined by
\begin{equation}
H_{\Lambda}^{\rm H}=\sum_{\sigma=1}^n\sum_{x, y\in \Lambda} t_{x, y} c_{x, \sigma}^*c_{y, \sigma}+\sum_{x\in \Lambda} \mu_x n_x
+\sum_{x, y\in \Lambda} U_{x, y} n_xn_y.
\end{equation}
Here, $c_{x, \sigma}^*$ and $c_{x, \sigma}$ represent the creation and annihilation operators of a fermion with the site $x$ and the flavour $\sigma$, satisfying the usual anti-commutation relations:
\begin{align}
\{c_{x, \sigma}, c_{y, \tau}\}=0=\{c_{x, \sigma}^*, c_{y, \tau}^*\},\ \ \{c_{x, \sigma}, c_{y, \tau}^*\}=\delta_{x, y} \delta_{\sigma, \tau}.
\end{align}
$n_{x}$ is the number operator of fermions at the site $x$:
\begin{equation}
n_{x}=\sum_{\sigma=1}^n c_{x, \sigma}^*c_{x, \sigma}.
\end{equation}
$T=\{t_{x, y}: x, y\in \Lambda\}$ is a hopping matrix of fermions. In this paper, we assume that $T$ describes the nearest neighbor hopping:\footnote{Let $x, y\in \Lambda$. We say that $x$ and $y$ are nearest neighbor if
$\|x-y\|_{\infty}=1$, where $\|x\|_{\infty}=\max_{j=1}^d |x_j|$.}
\begin{equation}
t_{x, y}
=\begin{cases}
t & \mbox{$x$ and $y$ are nearest neighbor}\\
0 & \mbox{otherwise.}
\end{cases}
\end{equation}
We suppose that the Hamiltonian acts on the $N=|\Lambda|-1$ particle space:
\begin{equation}
\bigwedge^N \ell^2(\Lambda) \otimes \mathbb{C}^n.
\end{equation}
For the sake of convenience, we assume that the on-site Coulomb interaction is uniform:
\begin{equation}
U=U_{x, x}\ \ \mbox{for all $x\in \Lambda$}.
\end{equation}
We introduce the operators that will play a fundamental role in this paper by
\begin{equation}
h_{\sigma}=N_{\sigma}-N_{\sigma+1},\ \ \ \sigma=1, \dots, n-1,
\end{equation}
where $N_{\sigma}$ is the number operator of fermions with the flavour $\sigma$:
\begin{equation}
N_{\sigma}=\sum_{x\in \Lambda} n_{x, \sigma},\ \ n_{x, \sigma}=c_{x, \sigma}^*c_{x, \sigma}.
\end{equation}
If we set
\begin{equation}
e_{\sigma, \sigma\rq{}}=\sum_{x\in \Lambda} c_{x, \sigma}^* c_{x, \sigma\rq{}},\ \ \sigma, \sigma\rq{}=1,\dots, n,\ \ \sigma\neq \sigma\rq{},
\end{equation}
the family $\{h_{\sigma}, e_{\sigma, \sigma\rq{}}\}$ of operators gives a representation of the $\mathfrak{su}(n)$ Lie algebra.
The family of operators $\{h_{\sigma}\}$ provides a representation of the Cartan subalgebra of $\mathfrak{su}(n)$ and is known to play an essential role in the representation theory of Lie algebras; see, e.g., \cite{Humphreys1972}.
In the case of $n=2$, $H_{\Lambda}$ is the conventional Hubbard Hamiltonian, and $h_1$ corresponds to the third component $S_{\rm tot}^{(3)}$ of the total spin operators.
Extending the wording of the $n=2$ case, we call the following Hamiltonian a model describing an interaction between an external magnetic field ${\boldsymbol b}=(b_1, \dots, b_{n-1})\in \mathbb{R}^{n-1}$ and fermions for convenience:
\begin{equation}
H^{\rm H}_{\Lambda}({\boldsymbol b})=\sum_{\sigma=1}^n\sum_{x, y\in \Lambda} t_{x, y} c_{x, \sigma}^*c_{y, \sigma}+\sum_{x\in \Lambda} \mu_x n_x
+\sum_{x, y\in \Lambda} U_{x, y} n_xn_y-\sum_{\sigma=1}^{n-1} b_{\sigma} h_{\sigma}.
\end{equation}
In this paper, we examine the system with $U=\infty$. In order to describe the effective Hamiltonian for such a system, we need some preparations.
First, define an orthogonal projection $Q_{\Lambda}$ as follows: let $E_{n_x}(\cdot)$ be the spectral measure of $n_x$, and let $Q_{\Lambda, x}=E_{n_x}(\{1\})$. Define
\begin{equation}
Q_{\Lambda}=\prod_{x\in \Lambda}Q_{\Lambda, x}. \label{DefQ}
\end{equation}
Then we define the subspace $\mathfrak{F}_N$ of the $N$-particle space by
\begin{equation}
\mathfrak{F}_{N}=Q_{\Lambda}\bigwedge^N \ell^2(\Lambda) \otimes \mathbb{C}^n. \label{DefF_N}
\end{equation}
$\mathfrak{F}_N$ is a Hilbert space of state vectors describing a system with exactly one hole in $\Lambda$ and a single fermion occupying each site except the hole site.
In the case of $U=\infty$, the energy required for two or more fermions to occupy a single site is infinite.
Therefore, the Hilbert space of states of such a system is $\mathfrak{F}_N$, and the effective Hamiltonian is given by
\begin{equation}
H_{\Lambda}({\boldsymbol b})=Q_{\Lambda} H_{\Lambda}^{{\rm H}, U=0}({\boldsymbol b})Q_{\Lambda},
\end{equation}
where $H_{\Lambda}^{{\rm H}, U=0}({\boldsymbol b})$ denotes the linear operator set to $U=0$ in the definition of $H_{\Lambda}^{{\rm H}}({\boldsymbol b})$.
The following proposition is a mathematical expression of the intuitive explanation given above:
\begin{Prop}\label{EffH}
In the limit of $U\to \infty$, $H_{\Lambda}^{\rm H}({\boldsymbol b})$ converges to $H_{\Lambda}({\boldsymbol b})$ in the following sense:
\begin{equation}
\lim_{U\to \infty} \Big\| (H_{\Lambda}^{\rm H}({\boldsymbol b})-z)^{-1}-(H_{\Lambda}({\boldsymbol b})-z)^{-1} Q_{\Lambda} \Big\|=0,\ \ z\in \mathbb{C}\setminus \mathbb{R},
\end{equation}
where $\|\cdot\|$ stands for the operator norm.
\end{Prop}
Proposition \ref{EffH} can be proved by similar arguments as in the proof of Theorem 2.5 in \cite{Miyao2017}.
In order to state the first main theorem, consider the partition function for $H_{\Lambda}({\boldsymbol b})$:
\begin{equation}
Z_{\Lambda}(\beta; {\boldsymbol b})=\mathrm{Tr}_{\mathfrak{F}_N}\Big[
e^{-\beta H_{\Lambda}({\boldsymbol b})}
\Big], \ \ \ \beta\ge 0.
\end{equation}
\begin{Thm}\label{Main1}
Let $P_N$ denote the entire partitions of $N.$\footnote{Thus, each ${\boldsymbol n}=\{n_j\}_{j=1}^k\in P_N$ satisfies $n_j\in \mathbb{N}$ and $\sum_{j=1}^k n_j=N$.}
For any ${\boldsymbol n}\in P_N$, there exists a positive number $D_{\beta}({\boldsymbol n})$ such that
\begin{equation}
Z_{\Lambda}(\beta; {\boldsymbol b})=\sum_{{\boldsymbol n}\in P_N}D_{\beta}({\boldsymbol n}) \mathcal{G}_{\beta}({\boldsymbol n}; {\boldsymbol b}),\label{ExZ}
\end{equation}
where, for each ${\boldsymbol n} =\{n_j\}_{j=1}^k\in P_N$, $\mathcal{G}_{\beta}({\boldsymbol n}; {\boldsymbol b})$ is defined by
\begin{align}
\mathcal{G}_{\beta}({\boldsymbol n}; {\boldsymbol b}) &= \prod_{j=1}^k G_{\beta}(n_j; {\boldsymbol b}),\label{DefG1}\\
G_{\beta}(m; {\boldsymbol b}) &= e^{\beta m b_1}+e^{-\beta m b_{n-1}}+\sum_{\sigma=2}^{n-1} e^{\beta m (-b_{\sigma-1}+b_{\sigma})}.
\end{align}
\end{Thm}
\begin{Rem}
\rm
For $n=2$, Theorem \ref{Main1} gives
\begin{equation}
Z_{\Lambda}(\beta; {\boldsymbol b}) =\sum_{{\boldsymbol n} \in P_N} D_{\beta}({\boldsymbol n}) \prod_{j=1}^k 2 \cosh(\beta b_1 n_j), \label{AL}
\end{equation}
reproducing the result of Aizenman--Lieb \cite{Aizenman1990} for the ordinary Hubbard model.
\end{Rem}
To state the next result, we introduce the following symbols:
\begin{equation}
B_{\sigma}=\begin{cases}
b_1 & \mbox{if $\sigma=1$}\\
b_{\sigma}-b_{\sigma-1} & \mbox{if $2\le \sigma \le n-2$}\\
-b_{n-1} & \mbox{if $\sigma=n$}.
\end{cases}
\end{equation}
\begin{Thm}\label{Main2}
Let $n\ge 2$.
Fix $\sigma\in \{1, 2, \dots, n-1\}$, arbitrarily.
Assume that $B_{\sigma}>B_{\tau}$ for all $\tau\neq\sigma$.
Then one obtains
\begin{equation}
\langle h_{\sigma}\rangle_{\beta}\ge \frac{f_{\beta, \sigma}(\boldsymbol b)}{1+g_{\beta, \sigma}({\boldsymbol b})} N, \label{MainInq}
\end{equation}
where $\langle h_{\sigma}\rangle_{\beta}$ stands for the thermal expectation\footnote{To be precise,
$\langle h_{\sigma}\rangle_{\beta}$ is defined by
$\langle h_{\sigma}\rangle_{\beta}=\mathrm{Tr}_{\mathfrak{F}_{N}}[h_{\sigma} e^{-\beta H_{\Lambda}({\boldsymbol b})}] \Big/ Z_{\Lambda}(\beta; {\boldsymbol b})$.
}
of $h_{\sigma}$, and
the functions $f_{\beta, \sigma}({\boldsymbol b})$ and $g_{\beta, \sigma} ({\boldsymbol b})$ are respectively given by
\begin{align}
f_{\beta, \sigma}({\boldsymbol b}) &= \frac{1-e^{-\beta (B_{\sigma}-B_{\sigma+1})}}{1+e^{-\beta (B_{\sigma}-B_{\sigma+1})}},\label{Deff}\\
g_{\beta, \sigma} ({\boldsymbol b}) &= \sum_{\tau \neq \sigma, \sigma+1} e^{\beta (B_{\tau}-B_{\sigma})}. \label{Defg}
\end{align}
For $n=2$, we understand that $g_{\beta, \sigma}({\boldsymbol b})=0$.
\end{Thm}
\begin{Rem}
\begin{itemize}
\item
By using \eqref{MainInq} and the fact $g_{\beta, \sigma}({\boldsymbol b}) \le n-2$, we get
\begin{equation}
\langle h_{\sigma}\rangle_{\beta} \ge \frac{f_{\beta, \sigma}(\boldsymbol b)}{n-1} N. \label{CoroMain}
\end{equation}
Note that this inequality holds even for $n=2$.
\item
For $n=2$, if we set $B_1=-B_2=b>0$, we can re-derive Aizenman--Lieb's result \cite{Aizenman1990} from \eqref{CoroMain}:
\begin{equation}
\langle h_1\rangle_{\beta}\ge \tanh(\beta b)N.
\end{equation}
\end{itemize}
\end{Rem}
\section{Functional integral representations for the semigroup generated by the Hamiltonian}\label{Sec3}
\subsection{Preliminaries}
\subsubsection{Case of a single fermion}
We need a functional integral representation for $e^{-\beta H_{\Lambda}({\boldsymbol b})}$ to prove the main theorems.
In this section, for the convenience of the readers,
we will outline how to construct the functional integral representation for the $\mathrm{SU}(N)$ Hubbard model.
States of a single fermion are represented by normalized vectors in the Hilbert space:
\begin{equation}
\ell^2(\Lambda) \otimes \mathbb{C}^n=\ell^2(\Omega),
\end{equation}
where $\Omega=\Lambda\times \{1, \dots, n\}$. The inner product in $\ell^2(\Omega)$ is given by
\begin{equation}
\langle f|g\rangle=\sum_{\sigma=1}^n \sum_{x\in \Lambda} f(x, \sigma)^* g(x, \sigma),\ \ f, g\in \ell^2(\Omega).
\end{equation}
Define the free Hamiltonian $h_0$ of a single fermion by
\begin{equation}
(h_0f)(x, \sigma)=\sum_{\sigma=1}^n \sum_{y\in \Lambda}t_{x, y}\Big(
f(x, \sigma)-f(y, \sigma)
\Big),\ \ f\in \ell^2(\Omega).
\end{equation}
To represent fundamental physical observables, we introduce the function $\delta_{(x, \sigma)}\ ((x, \sigma)\in \Omega)$ on $\Omega$ by
\begin{equation}
\delta_{(x, \sigma)}(y, \tau)=\delta_{x, y} \delta_{\sigma, \tau},\ \ (y, \tau)\in \Omega.
\end{equation}
Now, define the function $k_{\sigma}$ by
\begin{equation}
k_{\sigma}=\sum_{x\in \Lambda} \{ \delta_{(x, \sigma)}-\delta_{(x, \sigma+1)} \},\ \ \sigma=1, \dots, n-1.
\label{Defk}
\end{equation}
In this paper, for a given function $v$ on $\Omega$, the multiplication operator by $v$ will be denoted by the same symbol:
\begin{equation}
(vf)(x, \sigma)=v(x, \sigma) f(x, \sigma),\ \ f\in \ell^2(\Omega).
\end{equation}
Under this convention,
for
${\boldsymbol b}=(b_1, \dots, b_{n-1}) \in \mathbb{R}^{n-1}$ and a function $\mu : (x, \sigma) \ni\Omega\to \mu_x\in \mathbb{R}$, we define the self-adjoint operator $h_{\mu}({\boldsymbol b})$ on $\ell^2(\Omega)$ by
\begin{equation}
h_{\mu}({\boldsymbol b})=h_0+\mu-\sum_{\sigma=1}^{n-1} b_{\sigma} k_{\sigma}.
\end{equation}
Note that $\mu_x$ does not depend on $\sigma$.
The function $\mu$ represents the on-site potential and $\boldsymbol b$ represents the external field.
For notational simplicity, set $\mathbb{Z}_+=\mathbb{Z}\cap [0, \infty)$.
Let $(Y_n)_{n\in \mathbb{Z}_+}$ be a discrete time Markov chain with state-space $\Omega$,
which is characterized by
\begin{align}
P(Y_n=X|Y_{n-1}=Y)=
\delta_{\sigma, \tau} \frac{t_{x, y}}{d(y)}, \ \ \ d(x)=\sum_{y\in \Lambda} t_{x, y}
\end{align}
for $n\in \mathbb{N}, X=(x, \sigma)\in \Omega$ and $Y=(y, \tau)\in \Omega$.
In the rest of this paper, we work with the fixed probability space $(M, \mathcal{F}, P)$.
Let $(T_n)_{n\in \mathbb{Z}_+}$ be independent exponentially distributed random variables of parameter $1$, independent of
$(Y_n)_{n\in \mathbb{Z}_+}$.
Then we set \begin{align}
S_n=\frac{T_n}{d(Y_{n-1})},\ \ \ J_n=S_1+\cdots +S_n
\end{align}
and
\begin{align}
X_t=\sum_{n\in \mathbb{Z}_+} \mathbbm{ 1}_{\{ J_n\le t < J_{n+1}\}} Y_n,
\end{align}
where $\mathbbm{1}_S$ represents the indicator function of the set $S$.
Under this setting, we see that $(X_t)_{t\ge 0}$ is a right continuous process. Furthemore,
$J_0:=0, J_1, J_2, \dots$ are the jump times of $(X_t)_{t\ge 0}$ and
$S_1, S_2, \dots$ are the holding times of $(X_t)_{t\ge 0}$.
Now, set $P_X(\cdot)=P(\cdot | X_0=X)$ and let $(\mathcal{F}_t)_{t\ge 0}$ be the filtration defined by
$\mathcal{F}_t=\sigma(X_s| s\le t)$. Then we can check that
$
(M, \mathcal{F}, (\mathcal{F}_{t})_{t\ge 0}, (P_X)_{X\in \Omega})
$
is a strong Markov process, see, e.g., \cite[Theorems 2.8.1 and 6.5.4]{Norris1997}.
The following Feynman--Kac--It{\^o} formula
is well-known:
\begin{align}
\big(
e^{-t h_{\mu}({\boldsymbol b})} f
\big)(X)=\mathbb{E}_X\Big[
e^{-\int_0^tv(X_t) dt}f(X_t)
\Big], \ \ f\in \ell^2(\Omega),\ \ X\in \Omega, \label{FKI1}
\end{align}
where
$\mathbb{E}_X[f]$ is the expected value of $f$ with respect to $P_X$, and
the function $v$ on $\Omega$ is defined by
\begin{equation}
v(X)=\mu(X)-\sum_{\sigma=1}^{n-1}b_{\sigma} k_{\sigma}(X).
\end{equation}
See, e.g., \cite{Gneysu2015} for a detailed proof.
The representation \eqref{FKI1} is the first step to perform our analysis.
\subsubsection{Case of $N$ fermions}
Here, we examine an $N$- fermion system.
The non-interacting Hamiltonian is given by
\begin{align}
T=\underbrace{h_{\mu}({\boldsymbol b}) \otimes 1 \otimes \cdots \otimes 1}_N +1 \otimes h_{\mu}({\boldsymbol b})\otimes 1 \otimes \cdots \otimes 1 +\cdots+ 1 \otimes \cdots \otimes 1\otimes h_{\mu}({\boldsymbol b}).
\label{NELHAMI}
\end{align}
The operator $T $ is a self-adjoint operator acting in $\bigotimes^N\ell^2(\Omega)$, the $N$-fold tensor product of $\ell^2(\Omega)$.
In the remainder of this paper, we will freely use the following identification:
\begin{equation}
\bigotimes^N\ell^2(\Omega)= \ell^2(\Omega^N). \label{IdnTens}
\end{equation}
Note that this identification is implemented by the unitary operator $\iota: \bigotimes^N\ell^2(\Omega) \to \ell^2(\Omega^N)$ given by
\begin{equation}
\iota( f_1\otimes \cdots \otimes f_n) = (f_1(X_1) \cdots f_n(X_n))_{(X_1, \dots, X_n)\in \Omega^N}.
\end{equation}
Let $\delta_{(x, \sigma)}^{(j)}$ be the multiplication operator on $\bigotimes^N_{j=1}\ell^2(\Omega)$ given by
\begin{align}
\delta_{(x, \sigma)}^{(j)}=\underbrace{1\otimes \cdots \otimes 1 \otimes \overbrace{\delta_{(x, \sigma)} }^{j^{\mathrm{th}}}\otimes 1\otimes \cdots \otimes 1}_N,\ \ j=1, \dots, N. \label{DefVod}
\end{align}
The Coulomb interaction term is
described as the multiplication operator $V$ defined by
\begin{align}
V&=V_{\mathrm{o}}+V_{\mathrm{d}},
\end{align}
where
\begin{align}
V_{\mathrm{d}}=U\sum_{x\in \Lambda}\sum_{\sigma, \tau=1}^n\sum_{i, j=1}^N \delta_{(x, \sigma)}^{(i)} \delta_{(x, \tau)}^{(j)},\ \
V_{\mathrm{o}}=\sum_{\sigma, \tau=1}^n\sum_{x\neq y}\sum_{i, j=1}^N U_{x, y} \delta_{(x, \sigma)}^{(i)}\delta_{(y, \tau)}^{(j)}. \label{VasMulti}
\end{align}
Based on the above setups, we define the Hamiltonian describing the interacting $N$-fermion system by
\begin{equation}
L({\boldsymbol b}) =T+V.
\end{equation}
Since the particles are fermions, the Fermi-Dirac statistics have to be taken into account.
For this purpose, we introduce the {\it antisymmetrizer} $A_N$
on $\ell^2(\Omega^N)$ by
\begin{align}
(A_NF)({\boldsymbol X})=\sum_{\tau\in \mathfrak{S}_N} \frac{\mathrm{sgn}(\tau) }{N!} F(\tau^{-1}{\boldsymbol X}) \label{DefAN}
\end{align}
for $F\in \ell^2(\Omega^N)$ and ${\boldsymbol X}=(X^{(1)}, \dots, X^{(N)}) \in \Omega^N$,
where $\mathfrak{S}_N$ indicates the permutation group on the set $\{1, \dots, N\}$, and $\tau {\boldsymbol X}:=(X^{(\tau(1))}, \dots, X^{(\tau(N))}) $ for $\tau\in \mathfrak{S}_N$.
Note that $A_N$ is the orthogonal projection from $\ell^2(\Omega^N)$ onto $\ell^2_{\mathrm{as}}(\Omega^N)$, the space of all antisymmetric functions on $\Omega^N$.
Then the Hamiltonian we are interested in, $H^{\rm H}({\boldsymbol b})$, can be expressed as
\begin{equation}
H_{\Lambda}^{\rm H}({\boldsymbol b})=A_N L({\boldsymbol b})A_N.\label{RepHami}
\end{equation}
Using the representation \eqref{RepHami},
we wish to construct a Feynman--Kac--It\^o formula for the semigroup generated by $H^{\rm H}({\boldsymbol b})$. For this purpose, let
\begin{align}
\Omega_{\neq}^N=\Big\{{\boldsymbol X}\in \Omega^N\, \Big|\, X^{(i)} \neq X^{(j)} \ \mbox{for all $i, j\in \{1, \dots, N\}$ with $i\neq j$}\Big\}.
\end{align}
Using Eq.\eqref{IdnTens}, we have the following identification:
\begin{equation}
\bigwedge^N \ell^2(\Lambda)\otimes \mathbb{C}^n=\ell_{\rm as}^2(\Omega_{\neq}^N).
\end{equation}
Given ${\boldsymbol m}=(m_1. \dots, m_N)\in (M)^N$, we set
$X_s^{(j)}({\boldsymbol m})=X_s(m_j)$.
For each ${\boldsymbol m} \in (M)^N$, a right-continuous $\Omega^N$-valued function
\begin{align}
({\boldsymbol X}_t({\boldsymbol m}))_{t\ge 0}=\big(
X^{(1)}_t({\boldsymbol m}), \dots, X_t^{(N)}({\boldsymbol m})
\big)
\end{align}
is simply called a {\it path}.
If we write $X_t^{(j)}({\boldsymbol m})=(x_t^{(j)}({\boldsymbol m}), \sigma_t^{(j)}({\boldsymbol m}))$, then $\sigma^{(j)}_t({\boldsymbol m})$ is called the {\it flavor component} of $X_t^{(j)}({\boldsymbol m})$, and
$
x_t^{(j)}({\boldsymbol m})
$
is called the {\it spatial component} of $
X_t^{(j)}({\boldsymbol m})
$, respectively. The collection of the spatial components $(x_t^{(1)}({\boldsymbol m}), \dots, x_t^{(N)}({\boldsymbol m}))$
represents a trajectory of the $N$-fermions.
Define the event by
\begin{align}
D&=D_{{\rm O}}\cap D_{\mathrm{S}},
\end{align}
where
\begin{align}
D_{{\rm O}}&=
\Big\{{\boldsymbol m} \in (M)^N\, \Big|\,
\mbox{
${\boldsymbol X}_s({\boldsymbol m}) \in \Omega_{\neq}^N$ for all $s\in [0, \infty)$
}
\Big\} ,\\
D_{{\rm S }}&=\Big\{{\boldsymbol m} \in (M)^N\, \Big|\,
\mbox{
$\sigma^{(j)}_s({\boldsymbol m})=\sigma^{(j)}_0({\boldsymbol m})$ for all $j=\{1. \dots, N\}$ and $s\in [0, \infty)$
}
\Big\}.
\end{align}
Here, in the definition of $D_{\mathrm{S}}$, $\sigma_s^{(j)}(\boldsymbol m)$ denotes the flavor part of $X_s^{(j)}({\boldsymbol m})$.\footnote{
To be precise,
$X_s^{(j)}({\boldsymbol m})=(x_s^{(j)}({\boldsymbol m}), \sigma_s^{(j)}({\boldsymbol m}))$.}
Note that, for a given ${\boldsymbol m}\in D$,
the path $({\boldsymbol X}_t({\boldsymbol m}))_{t\ge 0}$ has characteristics such that
$
\sigma^{(j)}_t({\boldsymbol m})
$ are constant in time, and
fermions of the equal flavor never meet each other.
By using the Feynman--Kac--It\^o formula for a single fermion \eqref{FKI1} and Trotter's product formula, one obtains the following:
\begin{Prop}\label{NFermi}
For every $
{\boldsymbol X}=\big(X^{(1)},\dots, X^{(N)}\big)\in \Omega_{\neq}^N
$
and $F\in \ell^2_{\rm as}(\Omega^N)$, we have
\begin{align}
\Big(e^{-t H_{\Lambda}^{\rm H}({\boldsymbol b})} F
\Big)({\boldsymbol X})
=\mathbb{E}_{\boldsymbol X}\Bigg[ \mathbbm{1}_{D}
\exp\Bigg\{
-\int_0^{t} W({\boldsymbol X}_s)ds
\Bigg\}F({\boldsymbol X}_{t})
\Bigg], \label{LFK}
\end{align}
where $\mathbb{E}_{\boldsymbol X}[F]$ represents the expected value of $F$ associated with the probability measure $\otimes_{j=1}^N P_{X^{(j)}}$ on $(\Omega^N, M^N)$, and
\begin{equation}
W({\boldsymbol X})=V({\boldsymbol X})+\sum_{j=1}^N v(X^{(j)}),\ \ {\boldsymbol X}\in \Omega_{\neq}^N.
\end{equation}
\end{Prop}
\subsubsection{The system of $U=\infty$}
Here, we provide a Feynman--Kac--It{\^o} formula for the semigroup generated by the Hamiltonian $H_{\Lambda}({\boldsymbol b})$ describing the system of $U=\infty$.
Let
\begin{align}
\Omega_{\neq, \infty}^N=\Big\{
{\boldsymbol X}\in \Omega^N\, \Big|\, x^{(i)} \neq x^{(j)}\ \ \mbox{for all $i, j\in \{1, \dots, N\}$ with $i\neq j$}
\Big\}.
\end{align}
In the above definition, we used the following notations: ${\boldsymbol X}=(X^{(1)}, \dots, X^{(N)})
$ with $X^{(j)}=(x^{(j)}, \sigma^{(j)})$.
In the remainder of this paper, we freely use the following natural identification:
\begin{equation}
\mathfrak{F}_N=\ell_{\rm as}^2(\Omega_{\neq, \infty}^N),
\end{equation}
where $\mathfrak{F}_N$ is defined by \eqref{DefF_N}.
Given $\beta>0$, we set
$D_{\infty}(\beta)=D_{\mathrm{O}, \infty}(\beta) \cap D$ with
\begin{align}
D_{\mathrm{O}, \infty}(\beta)=\Big\{
{\boldsymbol m}\in (M)^N\, \Big|\, {\boldsymbol X}_s({\boldsymbol m})\in \Omega_{\neq, \infty}^N\, \mbox{ for all $s\in [0, \beta]$}
\Big\}.
\end{align}
Note that, for each ${\boldsymbol m} \in D_{\infty}(\beta)$, there are no fermion encounters in the corresponding path
$({\boldsymbol X}_t({\boldsymbol m}))_{t\in [0, \beta]}$.
Now we are ready to construct a Feynman--Kac--It{\^o} formula for $e^{-\beta H_{\Lambda}({\boldsymbol b})}$.
\begin{Thm}\label{FKIHubbrdUInf}
For every ${\boldsymbol X} \in \Omega_{\neq, \infty}^N$ and $F\in \ell^2_{\rm as}(\Omega_{\neq, \infty}^N)$, we have
\begin{align}
\Big(e^{-\beta H_{\Lambda}({\boldsymbol b})} F
\Big)({\boldsymbol X})
=\mathbb{E}_{\boldsymbol X}\Bigg[ \mathbbm{1}_{D_{\infty}(\beta)}
\exp\Bigg\{-\int_0^{\beta} W_{\infty}({\boldsymbol X}_s) ds
\Bigg\} F({\boldsymbol X}_t)
\Bigg], \label{InftyFKF}
\end{align}
where
\begin{equation}
W_{\infty}({\boldsymbol X})=V_{\rm o}({\boldsymbol X})+\sum_{j=1}^Nv(X^{(j)}),\ \ {\boldsymbol X}\in \Omega_{\neq, \infty}^N.
\end{equation}
Here, recall that $V_{\rm o}$ is given by \eqref{VasMulti}.
\end{Thm}
\begin{proof}
By Proposition \ref{EffH}, we have
\begin{align}
\lim_{U\to \infty} e^{-\beta H_{\Lambda}^{\rm H} ({\boldsymbol b})} F
= e^{-\beta H_{\Lambda}({\boldsymbol b})} F. \label{UInfty}
\end{align}
Denote by $\mathbbm{1}_D G_U({\boldsymbol X}_{\bullet})$ the integrand in the right hand side of \eqref{LFK}.
We split
$\mathbb{E}_{{\boldsymbol X}}[\mathbbm{1}_DG_U({\boldsymbol X}_{\bullet})]$
into two parts as follows:
\begin{align}
\mathbb{E}_{{\boldsymbol X}}[\mathbbm{1}_DG_U({\boldsymbol X}_{\bullet})]=\mathbb{E}_{{\boldsymbol X}}[\mathbbm{1}_{D_{\infty}(\beta) }G_{U=0}({\boldsymbol X}_{\bullet})]
+\mathbb{E}_{{\boldsymbol X}}[\mathbbm{1}_{D\setminus D_{\infty}(\beta)} G_{U=0}({\boldsymbol X}_{\bullet})].
\end{align}
Because $
\lim_{U\to \infty} G_U({\boldsymbol X}_{\bullet}({\boldsymbol m}))=0
$ for all ${\boldsymbol m} \in D\setminus D_{\infty}(\beta)$, we have
\begin{align}
\lim_{U\to \infty}\mathbb{E}_{{\boldsymbol X}}[\mathbbm{1}_DG_U({\boldsymbol X}_{\bullet})]=\mathbb{E}_{{\boldsymbol X}}[\mathbbm{1}_{D_{\infty}(\beta) }G_{U=0}({\boldsymbol X}_{\bullet})]\label{InftyUMain}
\end{align}
by the dominated convergence theorem.
Combining (\ref{UInfty}) and (\ref{InftyUMain}), we obtain the desired assertion in Theorem \ref{FKIHubbrdUInf}.
\end{proof}
\subsubsection{A Feynman--Kac--It{\^o} formula for the partition function}
Given $\beta>0$, let
\begin{align}
D_{\mathrm{P}, \infty}(\beta)=\big\{{\boldsymbol m}\in (M)^N\, \big|\, \mbox{$\exists\tau \in \mathfrak{S}_{N}({\boldsymbol X}_0({\boldsymbol m}))$ such that ${\boldsymbol X}_{\beta}({\boldsymbol m})=\tau {\boldsymbol X}_0({\boldsymbol m})$} \big\}, \label{DefDP}
\end{align}
where $\mathfrak{S}_N({\boldsymbol X})$ is a subset of $\mathfrak{S}_N$, and since its definition is a bit complicated, we will provide its precise definition in Definition \ref{DynP} below.
We then define the event by
\begin{align}
L_{\beta}=D_{\infty}(\beta) \cap D_{\mathrm{P}, \infty}(\beta).
\end{align}
The purpose here is to prove the following theorem:
\begin{Thm}\label{FKIUINFINITY}
For every $\beta >0$, there exists a measeure $\mu_{\beta}$ on $L_{\beta}$ such that
\begin{align}
Z_{\Lambda}(\beta; {\boldsymbol b})
=
\int_{L_{\beta}}d \mu_{\beta}
\prod_{j=1}^{N}\prod_{\sigma=1}^{n-1}\exp\Bigg\{\int_0^{\beta} b_{\sigma} k_{\sigma}(X^{(j)}_s)ds\Bigg\},
\label{FKITrUInf}
\end{align}
where $k_{\sigma}$ is given by \eqref{Defk}.
\end{Thm}
In order to prove Theorem \ref{FKIUINFINITY}, we need some preparations.
First, let us construct a complete orthonormal system (CONS) for $\mathfrak{F}_N=\ell^2_{\mathrm{as}}(\Omega_{\neq, \infty}^N)$.
Given ${\boldsymbol X} \in \Omega^N$, we set
\begin{equation}
\delta_{\boldsymbol X}=\otimes_{j=1}^N\delta_{X^{(j)}} \in \ell^2(\Omega^N)
\end{equation}
and $e_{{\boldsymbol X}} =A_N \delta_{{\boldsymbol X}}$, where $A_N$ is the antisymmetrizer defined by \eqref{DefAN}. Then we readily confirm that $\{\delta_{{\boldsymbol X}}\, |\, {\boldsymbol X} \in \Omega^N\}$ is a CONS for $\ell^2(\Omega^N)$.
To construct a CONS for $\ell_{\mathrm{as}}^2(\Omega_{\neq, \infty}^N)$, we do a little more preparation. Note that $e_{\tau {\boldsymbol X}}=\mathrm{sgn}(\tau) e_{{\boldsymbol X}}$ holds for all $\tau\in \mathfrak{S}_N$. With this in mind, we introduce an equivalence relation in $\Omega_{\neq, \infty}^N$ as follows:
Let ${\boldsymbol X}, {\boldsymbol Y} \in \Omega_{\neq, \infty}^N$. If there exists a $\tau\in \mathfrak{S}_N$ satisfying ${\boldsymbol Y}=\tau {\boldsymbol X}$, then we write ${\boldsymbol X} \equiv {\boldsymbol Y}$.
It is easy to see that this binary relation gives an equivalence relation.
Let $[{\boldsymbol X}]$ be the equivalence class to which ${\boldsymbol X}$
belongs. We will often abbreviate $[{\boldsymbol X}]$ to ${\boldsymbol X}$ if no confusion occurs.
We denote by $[\Omega_{\neq, \infty}^N]$ the quotient set $\Omega_{\neq, \infty}^N/\equiv $.
Then we readily confirm that $\{e_{\boldsymbol X}\, |\, {\boldsymbol X} \in [{\Omega_{\neq, \infty}^N}]\}$ is a CONS for $\ell^2_{\mathrm{as}}(\Omega_{\neq, \infty}^N)$. This CONS is useful in our analysis below.
Let $Q_{\Lambda}$ be the orthogonal projection from $\ell^2(\Omega^N)$ to $\mathfrak{F}_N$ given by \eqref{DefQ}.
We denote by ${\boldsymbol T}$ the Hamiltonian of the free fermions:
\begin{equation}
{\boldsymbol T}=Q_{\Lambda} T_{{\boldsymbol b} ={\boldsymbol 0}, \mu=0}Q_{\Lambda},
\end{equation}
where $T_{{\boldsymbol b}={\boldsymbol 0}, \mu=0}$ denotes the operator defined as ${\boldsymbol b}={\boldsymbol 0}$ and $\mu=0$ in the defining equation of $T$, i.e., \eqref{NELHAMI}.
We are now ready to state the precise definition of $\mathfrak{S}_N({\boldsymbol X})$ that appears in Eq. \eqref{DefDP}:
\begin{Def}\label{DynP}\rm
Let ${\boldsymbol X}\in \Omega^N_{\neq, \infty}$.
We say that a permutation $\tau\in \mathfrak{S}_N$ is {\it dynamically allowed}
associated with ${\boldsymbol X}$ if there exists an $n\in\mathbb{Z}_+$
such that
\begin{align}
\langle \delta_{{\boldsymbol X}}|{\boldsymbol T}^n \delta_{\tau{\boldsymbol X}}\rangle \neq 0. \label{DyAlll1}
\end{align}
We denote by $\mathfrak{S}_N({\boldsymbol X})$
the set of all dynamically allowed permutations associated with ${\boldsymbol X}$.
Note that if $\tau$ is dynamically allowed, then $\tau$ is always even, i.e., $\mathrm{sgn}(\tau)=1$ \cite{Aizenman1990}.
\end{Def}
In order to give a characterization of the dynamically allowed permutations, let us introduce some terms.
Given ${\boldsymbol X}=(X^{(j)})_{j=1}^N\in \Omega^N_{\neq, \infty}$ and $ {\boldsymbol Y}=(Y^{(j)})_{j=1}^N\in \Omega^N_{\neq, \infty}$, define the distance
between ${\boldsymbol X}$ and ${\boldsymbol Y}$ by
\begin{equation}
\|{\boldsymbol X}-{\boldsymbol Y}\|_{\infty}=\max_{j=1, \dots, N}\|x^{(j)}-y^{(j)}\|_{\infty},
\end{equation} where $x^{(j)}$ (resp. $y^{(j)}$)
is the spatial component of $X^{(j)}$ (resp. $Y^{(j)}$). We say that ${\boldsymbol X}$ and ${\boldsymbol Y}$ are {\it neighbors}
if $\|{\boldsymbol X}-{\boldsymbol Y}\|_{\infty}=1$ and the flavor components of $X^{(j)}$ and $Y^{(j)}$ are equal for all $j=1, \dots, n$. A pair $\{ {\boldsymbol X}, {\boldsymbol Y}\} \in \Omega^N_{\neq, \infty} \times \Omega^N_{\neq, \infty}$ is called an {\it edge} if ${\boldsymbol X}$ and ${\boldsymbol Y}$ are neighbors.
A sequence $({\boldsymbol X}_i)_{i=1}^m \subset \Omega_{\neq, \infty}^N$ is called a {\it path}, if
$\{{\boldsymbol X}_i, {\boldsymbol X_{i+1}}\}$ is an edge for all $i$.
For a given edge $\{{\boldsymbol X}, {\boldsymbol Y}\}$, define the linear operator acting in $\ell^2(\Omega_{\neq, \infty}^N)$
by
\begin{align}
Q({\boldsymbol X}, {\boldsymbol Y})= |\delta_{{\boldsymbol X}}\rangle \langle \delta_{{\boldsymbol Y}}|.
\end{align}
This operator is employed to describe the following lemma, which characterizes the dynamically allowed permutations.
\begin{Lemm}\label{EquivPath}
Let $\tau\in \mathfrak{S}_N$ and let ${\boldsymbol X}\in\Omega^N_{\neq, \infty}$.
The following {\rm (i)} and {\rm (ii)} are mutually equivalent:
\begin{itemize}
\item[{\rm (i)}] $\tau$ is dynamically allowed associated with ${\boldsymbol X}$;
\item[{\rm (ii)}] there exists a path $({\boldsymbol X}_i)_{i=1}^m$ satisfying the following:
\begin{itemize}
\item[] ${\boldsymbol X}_1={\boldsymbol X}$ and ${\boldsymbol X}_m=\tau {\boldsymbol X}$;
\item[] $
\langle \delta_{{\boldsymbol X}}|Q({\boldsymbol X}_1, {\boldsymbol X}_2) Q({\boldsymbol X}_2, {\boldsymbol X}_3) \cdots Q({\boldsymbol X}_{m-1}, {\boldsymbol X}_m) \delta_{\tau {\boldsymbol X}}\rangle >0.
$
\end{itemize}
\end{itemize}
\end{Lemm}
See \cite{Miyao2020-2} for a proof of this lemma.
We are now ready to prove Theorem \ref{FKIUINFINITY}.
\begin{proof}[Proof of Theorem \ref{FKIUINFINITY}]
We give only a brief outline of the proof. For details, see \cite{Miyao2020-2}.
We divide the proof into two parts.
{\bf Step 1.}
Let
$\ell^{\infty}_{\rm s}(\Omega^N)$
be the set of all symmetric functions on $\Omega^N$.
Let $F_0, F_1, \dots, F_{n-1}$ be elements in $ \ell_{\rm s}^{\infty}(\Omega^N)$ that are strictly positive.
We set \begin{equation}
K_n
=F_0 e^{-t_1H_{\Lambda}({\boldsymbol b})} F_1 e^{-(t_2-t_1)H_{\Lambda}({\boldsymbol b})}F_2 \cdots F_{n-1}e^{-(\beta-t_{n-1}) H_{\Lambda}({\boldsymbol b})} .
\end{equation}
Fix ${\boldsymbol X}\in \Omega_{\neq, \infty}^N$, arbitrarily.
We claim that if $\tau$ is {\it not} dynamically allowed associated with ${\boldsymbol X}$, then one obtains, for all $n\in \mathbb{N}$ and $
0<t_1<t_2<\cdots <t_{n-1}<\beta
$, that
\begin{align}
\langle \delta_{{\boldsymbol X}}|K_{n} \delta_{\tau{\boldsymbol X}}\rangle=0. \label{DAConc}
\end{align}
We shall prove this equation in a step-by-step manner.
First, let us consider the case where $W_{\infty}\equiv0$.
For simplicity, suppose that $n=2$.
Because $F_0$ and $F_1$ are multipication operators, it holds that
\begin{align}
\langle \delta_{{\boldsymbol X}}|F_0 (-{\boldsymbol T})^{n_1} F_1 (-{\boldsymbol T})^{n_2}\delta_{\tau{\boldsymbol X}}\rangle=0 \label{DyAll2}
\end{align}
for all $n_1, n_2\in \mathbb{Z}_+$.
To show this, note that we can express ${\boldsymbol T}$ as
\begin{align}
{\boldsymbol T}=\sum_{\{{\boldsymbol X}, {\boldsymbol Y}\}} C_{{\boldsymbol X}, {\boldsymbol Y}}Q({\boldsymbol X}, {\boldsymbol Y})
+\mathcal{D}, \label{OfDD}
\end{align}
where
$\sum_{\{{\boldsymbol X}, {\boldsymbol Y}\}}$ means sum over all edges, the coefficients $C_{{\boldsymbol X}, {\boldsymbol Y}}$ satisfy
$C_{{\boldsymbol X}, {\boldsymbol Y}}<0
$ for each edge $\{{\boldsymbol X}, {\boldsymbol Y}\}$ and $\mathcal{D}$ is some multiplication operator.
By using the formula \eqref{OfDD},
the equation (\ref{DyAll2}) follows from the following property:
\begin{align}
\langle \delta_{{\boldsymbol X}}|Q({\boldsymbol X}_1, {\boldsymbol X}_2) Q({\boldsymbol X}_2, {\boldsymbol X}_3) \cdots Q({\boldsymbol X}_{m-1}, {\boldsymbol X}_m) \delta_{\tau {\boldsymbol X}}\rangle =0
\end{align}
for any path $({\boldsymbol X}_i)_{i=1}^m$. But this is obvious from Lemma \ref{EquivPath}.
Using (\ref{DyAll2}), we can prove (\ref{DAConc}) as:
\begin{align}
\langle \delta_{{\boldsymbol X}}|K_{n} \delta_{\tau{\boldsymbol X}}\rangle
=\sum_{n_1=1}^{\infty}\sum_{n_2=1}^{\infty}\frac{t_1^{n_1} (t_2-t_1)^{n_2}}{n_1!n_2!}\langle \delta_{{\boldsymbol X}}|F_0 (-{\boldsymbol T})^{n_1} F_1 (-{\boldsymbol T})^{n_2}\delta_{\tau{\boldsymbol X}}\rangle=0.
\end{align}
Similarly, we can prove \eqref{DAConc} for general $n$, when $W_{\infty}\equiv 0$.
Next, let us consider the case where $W_{\infty} \neq 0$.
Again, we will consider the case $n=2$ for simplicity.
By using Trotter's formula, we have
\begin{align}
&\langle \delta_{{\boldsymbol X}}|K_{n} \delta_{\tau{\boldsymbol X}}\rangle\nonumber \\
=&\lim_{N_1\to \infty}\lim_{N_2\to \infty}
\big\langle \delta_{\boldsymbol X}| F_0\big(
e^{-t_1 {\boldsymbol T}/N_1} e^{-t_1W_{\infty}/N_1}
\big)^{N_1 } F_1\big (
e^{-(t_2-t_1) {\boldsymbol T}/N_2} e^{-(t_2-t_1)W_{\infty}/N_2}
\big)^{N_1 }\delta_{\tau {\boldsymbol X}}\big\rangle. \label{TKN2}
\end{align}
By applying the claim for the case where $W_{\infty} \equiv 0$, we see that the right hand side of (\ref{TKN2}) equals zero.
Similarly, we can prove the assertion for general $n$.
We are now done with the proof of Eq. \eqref{DAConc}.
{\bf Step 2.}
By using Theorem \ref{FKIHubbrdUInf} and (\ref{DAConc}), we get
\begin{align}
Z_{\Lambda}(\beta; {\boldsymbol b})
=&\sum_{{\boldsymbol X} \in [\Omega_{\neq, \infty}^N]} \langle e_{{\boldsymbol X}}\, |\, e^{-\beta H_{\Lambda}({\boldsymbol b})}e_{{\boldsymbol X}}\rangle\nonumber \\
=& \sum_{{\boldsymbol X} \in [\Omega_{\neq, \infty}^N]} \sum_{\tau \in \mathfrak{S}_N} \frac{\mathrm{sgn}(\tau)}{N!}
\langle \delta_{{\boldsymbol X}}\, |\, e^{-\beta H_{\Lambda}({\boldsymbol b})} \delta_{\tau {\boldsymbol X}}\rangle\nonumber \\
=& \sum_{{\boldsymbol X} \in [\Omega_{\neq, \infty}^N]} \sum_{\tau \in \mathfrak{S}_N({\boldsymbol X})} \frac{\mathrm{sgn}(\tau)}{N!}
\langle \delta_{{\boldsymbol X}}\, |\, e^{-\beta H_{\Lambda}({\boldsymbol b})} \delta_{\tau {\boldsymbol X}}\rangle\nonumber \\
=& \sum_{{\boldsymbol X} \in [\Omega_{\neq, \infty}^N]} \sum_{\tau \in \mathfrak{S}_N({\boldsymbol X})} \frac{\mathrm{sgn}(\tau)}{N!}
\mathbb{E}_{{\boldsymbol X}} \Big[
\mathbbm{1}_{\{{\boldsymbol X}_{\beta}=\tau{\boldsymbol X}\}\cap D_{\infty}(\beta)} e^{-\int_0^{\beta} W_{\infty}({\boldsymbol X}_s) ds}
\Big]. \label{TraF1}
\end{align}
Now let us define the measure on $L_{\beta}$ by
\begin{align}
\mu_{\beta}(B)=\sum_{{\boldsymbol X} \in [\Omega_{\neq, \infty}^N]} \sum_{\tau\in \mathfrak{S}_N({\boldsymbol X})} \frac{1}{N!}
\mathbb{E}_{\boldsymbol X}\Bigg[
\mathbbm{1}_B \mathbbm{1}_{\{{\boldsymbol X}_{\beta}=\tau{\boldsymbol X}\} \cap D_{\infty}(\beta)}
e^{-\int_0^{\beta} W_{\infty, {\boldsymbol b}={\boldsymbol 0}}({\boldsymbol X}_s) ds}
\Bigg], \label{Measurenu}
\end{align}
where $W_{\infty, {\boldsymbol b}={\boldsymbol 0}}({\boldsymbol X})$ is defined by setting ${\boldsymbol b}={\boldsymbol 0}$ in the defining equation of $W_{\infty}({\boldsymbol X})$.
Because $\mathrm{sgn}(\tau)=1$ for all $\tau\in \mathfrak{S}_N({\boldsymbol X})$, we finally obtain the desired assertion in Theorem \ref{FKIUINFINITY}.
\end{proof}
\section{Proofs of Theorems \ref{Main1} and \ref{Main2}} \label{Sec4}
\subsection{Random loop representations}
For a given ${\boldsymbol m}\in L_{\beta}$, we define the particle world lines of the path $({\boldsymbol X}_s({\boldsymbol m}))_{s\in [0, \beta]}$ by
\begin{equation}
{\boldsymbol x}_t({\boldsymbol m})=(x_t^{(1)}({\boldsymbol m}), \dots, x_t^{(N)}({\boldsymbol m})),
\end{equation}
where $x^{(i)}_t({\boldsymbol m})$ denotes the spatial component of $X_t^{(i)}({\boldsymbol m})$.
Since each $x_t^{(i)}$ takes values on $\Lambda$, is piecewise constant concerning time, and hops to nearest neighbor sites at random times, the particle world lines can be illustrated as a collection of polygonal lines in the space-time picture. Figure \ref{Fig21} depicts typical world lines in a two-dimensional system.
\begin{figure}[t]
\centering
\includegraphics[scale=0.5]{test4.pdf}
\caption{
Loops of the eight-particle world lines on the lattice $\Lambda=\{-1, 0, 1\}\times \{-1, 0, 1\}$. The colors of the world lines represent the flavor of fermions.
}\label{Fig21}
\end{figure}
Following \cite{Aizenman1990}, we can associate arbitrary particle world lines with a collection of loops in the space-time picture by the following procedure:
\begin{itemize}
\item we begin by plotting the loops from the position of each of the particles at time $t=0$;
\item we trace the time evolution of a particle's position in space-time; note that because the strength of the Coulomb interaction is infinite, the particles never encounter each other;
\item when
the trace line reaches the time at $t=\beta$, the trace line reappears in the same location at time $t=0$ by regarding time as periodic;
\item we continue the above procedures until the trace line is closed.
\end{itemize}
Each path is completely characterized by the following three conditions:
The initial configuration of the particles: ${\boldsymbol x}_0$, the particle world lines, and the flavor assigned along the world lines.
Therefore, we can specify each path by assigning a flavor to each loop.
Figure \ref{Fig21} depicts typical loops consisting of particle world lines in a two-dimensional system.
Now, suppose we are given the set of loops $\{\ell_1, \dots, \ell_k\}$ corresponding to a path $({\boldsymbol X}_s({\boldsymbol m}))_{s\in [0, \beta]}$.
In general, a pair of a loop $\ell$ and a flavor $\sigma$, $\gamma=(\ell, \sigma)$, is called a {\it flavored loop}.
Then the path $({\boldsymbol X}_s({\boldsymbol m}))_{s\in [0, \beta]}$ can be identified with the collection of the flavored loops $\varGamma({\boldsymbol m})=\{\gamma_1, \dots, \gamma_k\}$, where $\gamma_j=(\ell_j, \sigma_j)$.
The set $\varGamma({\boldsymbol m})$ is called the {\it flavored random loop}.
By using Theorem \ref{FKIUINFINITY}, we obtain the following:
\begin{Thm}
The partition function has the following random loop representation:
\begin{align}
Z_{\Lambda}(\beta; {\boldsymbol b})
=\int_{L_{\beta}} d\mu_{\beta}
\prod_{\gamma\in \varGamma}
\exp\{ \beta w_{\gamma} f(\sigma_{\gamma})\},
\end{align}
where, for each $\gamma=(\ell_{\gamma}, \sigma_{\gamma})\in \varGamma$,
$w_{\gamma}$ denotes the absolute value of
the winding number of the loop $\ell_{\gamma}$, and the function $f$ on $\{1, \dots, n\}$ is given by
\begin{align}
f(\sigma)
=\begin{cases}
b_1 & (\sigma=1)\\
-b_{\sigma-1}+b_{\sigma} & (2\le \sigma\le n-1)\\
- b_{n-1} & (\sigma=n).
\end{cases}
\end{align}
\end{Thm}
\begin{proof}
Given ${\boldsymbol m}\in L_{\beta}$, consider the flavored loops $\varGamma(\boldsymbol m)=\{\gamma_1, \dots, \gamma_k\}$ corresponding to the path $(\boldsymbol X_s({\boldsymbol m}))_{s}$.
When expressed as $\gamma_j=(\ell_j, \sigma_j)$, each loop $\ell_j$ can be represented in terms of the particle world lines: $\ell_j=\cup_{s\in [0, \beta]}\cup_{i\in I_j} x_s^{(i)}(\boldsymbol m)$,
where we denote by $I_j$ the set of particle labels that comprise each $\ell_j$.
Using these symbols, we get
\begin{align}
\prod_{j=1}^{N}\prod_{\sigma=1}^{n-1}\exp\Bigg\{\int_0^{\beta} b_{\sigma} k_{\sigma}(X^{(j)}_s({\boldsymbol m}))ds\Bigg\}
&=\prod_{j=1}^k \underbrace{\prod_{\sigma=1}^{n-1} \prod_{i\in I_j} \exp\Bigg\{\int_0^{\beta} b_{\sigma} k_{\sigma}(X^{(j)}_s({\boldsymbol m}))ds\Bigg\}}_{=: \mathscr{C}(\gamma_j)}\nonumber \\
&=\prod_{j=1}^k\mathscr{C}(\gamma_j).
\end{align}
Recalling \eqref{Defk}, we obtain
\begin{align}
\sum_{i\in I_j} \sum_{\sigma=1}^{n-1} b_{\sigma} k_{\sigma}(X_s^{(i)}({\boldsymbol m}))
&=\sum_{i\in I_j} \sum_{x\in \Lambda}\sum_{\sigma=1}^{n-1} b_{\sigma}\Big\{
\delta_{(x, \sigma)} (X_s^{(i)}({\boldsymbol m}))-\delta_{(x, \sigma+1)} (X_s^{(i)}({\boldsymbol m}))
\Big\}\nonumber \\
&=\sum_{i\in I_j} \sum_{x\in \Lambda}\delta_{x, x_s^{(i)}({\boldsymbol m})} f(\sigma_{j})=|I_j| f(\sigma_{j}).
\end{align}
Combining this with the fact $|I_j|=w(\gamma_j)$, we conclude that $\mathscr{C}(\gamma)=e^{\beta w(\gamma) f(\sigma_{\gamma})}$.
\end{proof}
\subsection{Proof of Theorem \ref{Main1}}
Note that for a given flavored random loops $\varGamma({\boldsymbol m})=\{\gamma_1, \dots, \gamma_k\}$, the collection of corresponding winding numbers ${\boldsymbol w}_{\varGamma({\boldsymbol m })}=(w_{\gamma_1}, \dots, w_{\gamma_k})$ is a partition of $N$:
\begin{equation}
\sum_{j=1}^k w_{\gamma_j}=N.
\end{equation}
Thus, as ${\boldsymbol m}$ runs in $L_{\beta}$, ${\boldsymbol w}_{\varGamma({\boldsymbol m})}$ runs through various partitions of $N$.
To state a technical lemma,
we prepare some symbols.
Let $L$ and $F$ be the collections of the loops and the flavors associated with $\varGamma(\boldsymbol m)$, respectively:
\begin{equation}
L(\varGamma(\boldsymbol m))=(\ell_{\gamma} : \gamma\in \varGamma(\boldsymbol m)),
\ \
F(\varGamma(\boldsymbol m))=(\sigma_{\gamma} : \gamma\in \varGamma(\boldsymbol m)).
\end{equation}
Note that, if $\# \varGamma({\boldsymbol m})=k$, then
$F(\varGamma({\boldsymbol m}))$ belongs to $\mathscr{F}_k:=\{1, \dots, n\}^k$.
Given $F\in \{1, \dots, n\}^k$, we set
\begin{equation}
S(k; F)=\{ {\boldsymbol m} \in L_{\beta} : \#\varGamma(\boldsymbol m)=k,\ \ F(\varGamma({\boldsymbol m}))=F\}.
\end{equation}
Next, we divide the set of partitions of $N$, $P_N$, as follows:
\begin{equation}
P_N=\bigsqcup_{k=1}^NP_N(k),
\end{equation}
where we set $P_N(k)=\{ {\boldsymbol n} \in P_N : \#{\boldsymbol n}=k\}$.
For each $F\in \mathscr{F}_k$ and ${\boldsymbol n}\in P_N(k)$, define
\begin{equation}
S(k; F; {\boldsymbol n})=\{{\boldsymbol m} \in S(k; F) : {\boldsymbol w}_{\varGamma}={\boldsymbol n}\},
\end{equation}
where ${\boldsymbol w}_{\varGamma}$ is the collection of the winding numbers of the loops associated with $\varGamma$.
It is essential in the proof of Theorem \ref{Main1} that $S(k; F)$ can be partitioned as follows:
\begin{equation}
S(k; F)=\bigsqcup_{{\boldsymbol n} \in P_N(k)} S(k; F; {\boldsymbol n}).
\end{equation}
\begin{Lemm}\label{Key}
Fix
$k\in \{1, \dots, n\}$, arbitrarily.
We also fix ${\boldsymbol n}\in P_N(k)$.
Then $\mu_{\beta}( S(k; F; {\boldsymbol n}))$ is independent of $F$ and constant on $\mathscr{F}_k$.
Setting
\begin{equation}
D_{\beta}(\boldsymbol n)=\mu_{\beta}( S(k; F; {\boldsymbol n})),
\end{equation}
we obtain the following identity:
\begin{equation}
\sum_{F\in \mathscr{F}_k}\int_{S(k; F; {\boldsymbol n})} d\mu_{\beta}
\prod_{\gamma\in \varGamma}
\exp\{ \beta w_{\gamma} f(\sigma_{\gamma})\}=
D_{\beta}({\boldsymbol n}) \mathcal{G}_{\beta}({\boldsymbol n}; {\boldsymbol b}),
\end{equation}
where $ \mathcal{G}_{\beta}({\boldsymbol n}; {\boldsymbol b})$ is given by \eqref{DefG1}.
\end{Lemm}
\begin{proof}
Let $\mathfrak{S}_n$ be the permutation group on the set $\{1, 2, \dots, n\}$.
For any ${\boldsymbol\varepsilon}=(\varepsilon_j)_{j=1}^N\in \mathfrak{S}_n^N$, we define the unitary operator $U_{\boldsymbol \varepsilon}$ on $\mathfrak{F}_N$ by
\begin{equation}
(U_{\varepsilon}F)({\boldsymbol X})=F({\boldsymbol X}_{\boldsymbol \varepsilon}),\ \ F\in \mathfrak{F}_N,
\end{equation}
where
${\boldsymbol X_{\varepsilon}}=(X^{(1)}_{\boldsymbol \varepsilon}, \dots, X_{\boldsymbol \varepsilon}^{(N)})$ is defined by
$X^{(j)}_{\boldsymbol \varepsilon}=(x^{(j)}, \varepsilon_j(\sigma^{(j)}))$.
Because
\begin{equation}
U_{\boldsymbol \varepsilon} H_{\Lambda}({\boldsymbol 0}) U_{\boldsymbol \varepsilon}^{-1} =H_{\Lambda}({\boldsymbol 0})
\end{equation}
holds for any ${\boldsymbol \varepsilon}\in \mathfrak{S}_n^N$,
we readily confirm that $\mu_{\beta}( S(k; F; {\boldsymbol n}))$ is independent of $F$ and constant on $\mathscr{F}_k$.
Since the function $\prod_{\gamma\in \varGamma}
\exp\{ \beta w_{\gamma} f(\sigma_{\gamma})\}$ is constant on
$S(k; F; {\boldsymbol n})$, we see that
\begin{equation}
\int_{S(k; F; {\boldsymbol n})} d\mu_{\beta} \prod_{\gamma\in \varGamma}
\exp\{ \beta w_{\gamma} f(\sigma_{\gamma})\}
=\mu_{\beta}(S(k; F; {\boldsymbol n}))\prod_{j=1}^k \exp\{ \beta n_j f(\sigma_j)\},
\end{equation}
where we set $F=(\sigma_1, \dots, \sigma_k)$ and ${\boldsymbol n}=(n_1, \dots, n_k)$.
Therefore, by combining the first half of the statement with this fact, we get the following:
\begin{align}
\sum_{F\in \mathscr{F}_k}\int_{S(k; F; {\boldsymbol n})} d\mu_{\beta}
\prod_{\gamma\in \varGamma}
\exp\{ \beta w_{\gamma} f(\sigma_{\gamma})\}&=
\sum_{F\in \mathscr{F}_k}D_{\beta}({\boldsymbol n})
\prod_{j=1}^k
\exp\{ \beta n_j f(\sigma_j)\}\nonumber \\
&=D_{\beta}({\boldsymbol n}) \prod_{j=1}^k \sum_{\sigma=1}^n\exp\{ \beta n_j f(\sigma)\}\nonumber \\
&=D_{\beta}({\boldsymbol n}) \mathcal{G}_{\beta}({\boldsymbol n}; {\boldsymbol b}).
\end{align}
We are now done with the proof of Lemma \ref{Key}.
\end{proof}
\begin{proof}[Completion of the proof of Theorem \ref{Main1}]
Using the fact that $L_{\beta}$ can be divided as
\begin{equation}
L_{\beta}
=\bigsqcup_{k=1}^N \bigsqcup_{F\in \mathscr{F}} S(k; F)=
\bigsqcup_{k=1}^N \bigsqcup_{F\in \mathscr{F}} \bigsqcup_{{\boldsymbol n} \in P_N(k)} S(k; F; {\boldsymbol n}),
\end{equation}
the partition function can be expressed as follows:
\begin{align}
Z_{\Lambda}(\beta; {\boldsymbol b}) =&\sum_{k=1}^N \sum_{{\boldsymbol n}\in P_N(k)}\sum_{F\in \mathscr{F}_k} \int_{S(k; F; {\boldsymbol n})} d\mu_{\beta}
\prod_{\gamma\in \varGamma}
\exp\{ \beta w_{\gamma} f(\sigma_{\gamma})\}\nonumber \\
=&\sum_{k=1}^N \sum_{{\boldsymbol n} \in P_N(k)} D_{\beta}({\boldsymbol n}) \mathcal{G}_{\beta}({\boldsymbol n}; {\boldsymbol b})\nonumber \\
=& \mbox{the RHS of \eqref{ExZ}}.
\end{align}
This completes the proof of Theorem \ref{Main1}.
\end{proof}
\subsection{Proof of Theorem \ref{Main2}}
First, note that the expected value $\langle h_{\sigma}\rangle$ can be expressed as follows
\begin{align}
\beta \langle h_{\sigma} \rangle_{\beta}&=\frac{\partial}{\partial b_{\sigma}} \log Z_{\Lambda}(\beta; \boldsymbol b)\nonumber \\
&=Z_{\Lambda}(\beta; {\boldsymbol b})^{-1} \sum_{{\boldsymbol n} \in P_N}\sum_{j=1}^k D_{\beta}(\boldsymbol n)\mathcal{G}_{\beta}({\boldsymbol n}; {\boldsymbol b})\frac{\frac{\partial }{\partial b_{\sigma}} G_{\beta}(n_j;{\boldsymbol b})}{ G_{\beta}(n_j; {\boldsymbol b})}. \label{ExpVG}
\end{align}
In the following, we estimate
$\frac{\partial}{\partial b_{\sigma}} G_{\beta}(m; {\boldsymbol b}) \Big/G_{\beta}(m; {\boldsymbol b})$
from below.
To this end, we set
\begin{align}
C_{\sigma}(m) &=e^{\beta m B_{\sigma}}+e^{\beta m B_{\sigma+1}} ,\\
S_{\sigma}(m) &=e^{\beta m B_{\sigma}}-e^{\beta m B_{\sigma+1}} .
\end{align}
Then $G_{\beta}(m; {\boldsymbol b})$ can be expressed as
\begin{equation}
G_{\beta}(m; {\boldsymbol b})=\sum_{\sigma=1}^{n-1} e^{\beta m B_{\sigma}}
=C_{\sigma}(m)+H_{\sigma}(m),
\end{equation}
where
\begin{equation}
H_{\sigma}=\sum_{\tau\neq \sigma, \sigma+1} e^{\beta m B_{\tau}}.
\end{equation}
From this representation, the equality
\begin{equation}
\frac{\partial}{\partial b_{\sigma}} G_{\beta}(m; {\boldsymbol b})=\beta m S_{\sigma}(m)
\end{equation}
follows immediately. Hence, we obtain
\begin{equation}
\frac{\partial}{\partial b_{\sigma}} G_{\beta}(m; {\boldsymbol b}) \Big/G_{\beta}(m; {\boldsymbol b})=\beta m\frac{S_{\sigma}/C_{\sigma}}{1+H_{\sigma}/C_{\sigma}}.
\end{equation}
If we define the function $f$ by
$f(x)=\frac{1-e^{-x}}{1+e^{-x}}$, then since $f$ is monotonically increasing, we have the following inequality:
\begin{equation}
\frac{S_{\sigma}(m)}{C_{\sigma}(m)} =f(\beta m (B_{\sigma}-B_{\sigma+1})) \ge f(\beta(B_{\sigma}-B_{\sigma+1}))=f_{\beta, \sigma}(\boldsymbol b),
\end{equation}
where we use the fact that $B_{\sigma}>B_{\sigma+1}$ and $f_{\beta, \sigma}({\boldsymbol b})$ is defined by \eqref{Deff}.
A quick examination, on the other hand, also reveals the following inequality:
\begin{align}
\frac{H_{\sigma}(m)}{C_{\sigma}(m)}&=\frac{\sum_{\tau\neq \sigma, \sigma+1} e^{\beta m (B_{\tau}-B_{\sigma})}}{1+e^{-\beta m (B_{\sigma}-B_{\sigma+1})}}\nonumber \\
&\le \sum_{\tau\neq \sigma, \sigma+1} e^{\beta m (B_{\tau}-B_{\sigma})}\le g_{\beta, \sigma}(\boldsymbol b),
\end{align}
where $g_{\beta, \sigma}(\boldsymbol b)$ is given by \eqref{Defg}. Here, we used the assumption that $B_{\sigma}>B_{\tau}\ (\tau\neq \sigma)$ in deriving the second inequality.
Putting the above inequalities together, we get
\begin{align}
\frac{\partial}{\partial b_{\sigma}} G_{\beta}(m; {\boldsymbol b}) \Big/G_{\beta}(m; {\boldsymbol b})\ge \beta m \frac{f_{\beta, \sigma}(\boldsymbol b)} {1+g_{\beta, \sigma}(\boldsymbol b)},
\end{align}
which implies that
\begin{align}
\mbox{the RHS of \eqref{ExpVG} }\ge \beta \frac{ f_{\beta, \sigma}(\boldsymbol b)}{1+g_{\beta, \sigma}(\boldsymbol b)} N,
\end{align}
where we use the fact $\sum_{j=1}^k n_j=N$.
We are now done with the proof of Theorem \ref{Main2}.
\qed
|
2024-02-18T23:40:24.378Z
|
2021-12-30T02:24:05.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14475","language":"en","timestamp":1640831045000,"url":"https:\/\/arxiv.org\/abs\/2112.14475","yymm":"2112"}
|
proofpile-arXiv_000-10009
|
{"provenance":"002.jsonl.gz:10010"}
| null | null |
\section{Introduction}
\label{sect:intro}
The origin of intercycle variations in the series of 11-year solar
activity cycles and the possibilities to predict them have been
the focus of intense research over the past decade
(\citen{Petrovay:LRSP2}, \citen{Charbonneau:LRSP2}, \citen{Nandy:review}).
In a physical system displaying periodic or quasiperiodic behaviour,
intercycle variations must be due to either inherent nonlinearities or
external forcing. A plausible example is the way in which the strictly sinusoidal,
periodic solutions of the simple linear oscillator equation turn into
less regular quasiperiodic variations when nonlinear or stochastic
forcing terms are added. Indeed, under some simplifying assumptions,
the equations describing a solar flux transport dynamo can be
truncated to a nonlinear oscillator equation (\citen{Lopes+:oscillrev}).
Here, "external'' forcing refers to the physical model
under consideration and not necessarily to the object of study, namely: in mean-field
models of the solar dynamo, stochastic variation triggered by
small-scale disturbances that are not explicitly modelled is the most plausible
source of such forcing. Possibilities for making solar cycle predictions are
determined by the nature and interplay of these nonlinear and stochastic
effects.
Based on its observed characteristics, the solar dynamo is likely to be an
$\alpha\omega$-dynamo, wherein strong toroidal magnetic fields are
generated by the windup of a weaker poloidal (north--south) field by
differential rotation. The strong toroidal field buried below the
surface then occasionally and locally emerges to the surface and
protrudes into the atmosphere in the form of east--west oriented
magnetic flux loops that are observed as solar active regions. Besides their
dominant azimuthal magnetic field component, these loops, presumably
under the action of Coriolis force, also develop a poloidal field
component which is typically oriented opposite to the preexisting
global poloidal field. The evolution of the poloidal magnetic field
on the solar surface (where it is effectively radial) can be followed
on solar full-disk magnetograms and found to be well described by
phenomenological surface flux transport (SFT) models
(\citen{Jiang+:SFTreview}). In such SFT models, the evolution of the
radial magnetic field at the surface is described by an
advective-diffusive transport equation, with an added source term
representing emerging active regions and an optional sink term to
describe the effects of radial diffusion. Advection is attributed to the
poleward meridional flow, while diffusion is due to supergranular
motions.
At solar minimum, the field is found to be strongly concentrated at the
Sun's poles, so the dominant contribution to the solar dipole moment, namely:\
\begin{equation}
D(t) = \frac32 \int_{-\pi/2}^{\pi/2}
{B}(\lambda,t)\sin\lambda\cos\lambda\, \mathrm{d}\lambda,
\label{eq:dipmom}
\end{equation}
(where $\lambda$: latitude, $t$: time, $B$: azimuthally averaged
field strength) comes from the polar caps. It is this polar field that
serves as seed for the windup into the toroidal field of the next solar
cycle, so for intercycle variations it is sufficient to consider the
azimuthally averaged SFT equation. This is expressed as:\
\begin{eqnarray}
\label{eq33}
\pdv Bt &=&
\frac{1}{R\cos{\lambda}}\pdv{}\lambda(B\,u\,\cos{\lambda})
\nonumber \\
&&+\frac{\eta}{R^2\cos{\lambda}}
\pdv{}\lambda\left(\cos{\lambda}\pdv B\lambda\right)
-\frac{B}{\tau} + s(\lambda,t),
\label{eq:transp}
\end{eqnarray}
where $R$ is the solar radius, $\eta$ is the supergranular
diffusivity, $U$ is the meridional flow speed, $\tau$ is the time
scale of decay due to radial diffusion, and $s$ is the source
representing flux emergence.
As Eq.~(\ref{eq:transp}) is linear, the amplitude of the dipole
moment, $D_{i}$, built up by the end of cycle $i-1$ is uniquely set by
the inhomogeneous term, $s$. The windup of the poloidal field, $D_i$,
in turn, is expected to be linearly related to the peak amplitude of
the toroidal field in cycle, $i$, and, hence, to the amplitude of the
next activity cycle --- this is indeed confirmed by both observations
and dynamo models (\citen{Munozjara+:polar.precursor},
\citen{Kumar2021}). The only step where nonlinearities and stochastic
noise can enter this process is thus via flux emergence, linking the
subsurface toroidal field to the poloidal source $s(\lambda,t)$ at the
surface. As $s(\lambda,t)$ is directly accessible to observations,
this implies that the nonlinearities involved may, in principle, be
empirically constrained.
Indeed, \cite{Dasi-Espuig+} reported that the average tilt of the axis of
bipolar active regions relative to the azimuthal direction
is anticorrelated with regard to the cycle amplitude. Starting with the pioneering work
of \cite{Cameron+:tiltprecursor}, this tilt quenching (TQ) effect
has become widely used in solar dynamo modeling, especially as the
product of the mean tilt with the cycle amplitude has been shown to be a
good predictor of the amplitude of the next cycle. The observational
evidence, however, is still inconclusive,
and the numerical form of the effect remains insufficiently constrained
{ (see, however, \citen{jiao2021} for more robust recent
evidence { and also \citen{Jha2020} for the effect's dependence on
field strength}). }
\cite{Jiang:nonlin} recently called attention to another nonlinear
modulation mechanism: latitude quenching (LQ). This is based on the
emprirical finding based on an analysis of a long sunspot record, which shows that
the mean latitude where active regions emerge in a given phase of the
solar cycle is correlated with the cycle amplitude. From higher latitudes, a
lower fraction of leading flux can manage become diffused across the
equator, leaving less trailing flux to contribute to the polar fields.
Therefore, the correlation found here represents a negative feedback
effect. Assuming a linear dependence for both the mean tilt and the mean
latitude on cycle amplitude, based on one particular SFT setup,
\cite{Jiang:nonlin} found that TQ and LQ yield comparable
contributions to the overall nonlinearity in the process of the
regeneration of the poloidal field from the poloidal source. The net
result showed that the net dipole moment change during a cycle tends to be
saturated for stronger cycles.
Our objective in the present work is to further explore the respective
roles played by TQ and LQ in the solar dynamo. In Section 2, we extend
the analysis to other SFT model setups and study how the respective
importance of TQ and LQ depend on model parameters. In Section 3, we
further consider a case where the form of TQ is explicitly assumed to
be nonlinear, namely: the 2x2D dynamo model. Our conclusions are summarized in
Section 4.
\section{Quadratic nonlinearities in a surface flux transport model}
\label{sect:SFT}
\subsection{Model}
\label{sect:model}
For the study of how different nonlinearities compare in SFT models
with different parameter combinations, our model setup is a
generalization of the approach of \cite{paper1} (hereafter
Paper~1).
In the prior study, our intention was to model ``typical'' or ``average''
solar cycles, so that our source function would not consist of individual
ARs but a smooth distribution (interpreted as an ensemble average)
representing the probability distribution of the emergence of
leading and trailing polarities on the solar surface.
By nature, this source is thus axially symmetric, so our whole SFT
model is reduced to one dimension, as described by
Eq.~(\ref{eq:transp}).
For the meridional flow, we consider a sinusoidal profile with a dead zone
around the poles,
\begin{equation}
\label{eq5}
u_{c} =
\left\{
\begin{array}{ll}
u_{0}\sin(\pi\lambda/\lambda_{0})
& \mbox{if } |\lambda| < \lambda_{0} \\
0 & \mbox{otherwise, }
\end{array}
\right.
\end{equation}
with $\lambda_0=75^\circ$. This profile has been used in many other
studies, including \cite{Jiang:nonlin}, who took the values
$u_0=11\,$m$/$s, $\eta=250\,$km$^2/$s, and $\tau=\infty$.
Our source term is a generalization of the source used in Paper~1. The
source is a smooth distribution representing an ensemble average or
probability distribution of the emergence of leading and trailing polarities
on the solar surface (\citen{Cameron2007}, \citen{Munozjara2010}),
represented as a pair of rings of opposite magnetic polarities:
\begin{eqnarray}
s(\lambda,t)&=&
kA_n f_{\mbox{\scriptsize\rm sph}} s_1(t)
s_2\left[\lambda;\lambda_0(t)-\Delta\lambda(t),\delta\lambda\right]
\nonumber \\
&&-kA_n s_1(t)
s_2\left[\lambda;\lambda_0(t)+\Delta\lambda(t),\delta\lambda\right]
\nonumber \\
&& +kA_n s_1(t)
s_2\left[\lambda;-\lambda_0(t)-\Delta\lambda(t),\delta\lambda\right]
\nonumber \\
&& -kA_n f_{\mbox{\scriptsize\rm sph}} s_1(t)
s_2\left[\lambda;-\lambda_0(t)+\Delta\lambda(t),\delta\lambda\right],
\label{eq:source}
\end{eqnarray}
where $k=\pm 1$ is a factor that depends on the sign of the toroidal
field and
$A_n$ is the amplitude for cycle $n$.
In the numerical implementation of this source profile, care was taken to
ensure zero net flux on the spherical surface by reducing the amplitude of the
equatorward member of each pair by an appropriate sphericity factor
$f_{\mbox{\scriptsize\rm sph}}$.
$s_1$ is the time profile of activity in a cycle while
$s_2$ characterizes the latitude dependence of activity at a given cycle phase.
In contrast to Paper~1 where a series of identical cycles was
considered, here we allow for intercycle variations, so $A_n$ will be
different for each cycle.
The time profile of solar activity in a typical cycle was determined
by \cite{hathaway1994shape} from the average of many cycles as:
\begin{equation}
s_1(t)= at^{3}_{c}/[\exp(t^{2}_{c}/b^{2})- c],
\label{eq:ampli}
\end{equation}
with $a = 0.00185$, $b = 48.7$, $c= 0.71$, where $t_c$ is the time since the
last cycle minimum.
The latitudinal profile
$s_2\left[\lambda;\lambda_0(t),\delta\lambda\right]$ is a Gaussian
migrating equatorward during the course of a cycle:
\begin{equation}
s_2(\lambda;\lambda_0,\delta\lambda) = \frac{\delta\lambda_0}{\delta\lambda}
\exp\left[-(\lambda-\lambda_0)^2/2\delta\lambda^2\right],
\end{equation}
with the constant fixed as $\delta\lambda_0=6{{}^{\circ}\hspace{-0.3em}.}26$.
Following the empirical results of \cite{Jiang+:1700a},
the standard deviation is given by:
\begin{equation}
\label{eq:fwhm}
\delta \lambda = [ 0.14+1.05(t/P)-0.78(t/P)^{2}]\lambda_{0}
,\end{equation}
where $P=11\,$year is the cycle period. We note that alternate fitting
formulae were determined by \citen{Lemerle1} on the basis of a
single cycle.
We obtain the latitudinal separation $2\Delta\lambda$ of the rings as a
consequence of Joy's law, while the mean latitude $\lambda_0$ was
again empirically determined by \cite{Jiang+:1700a}. These
parameters, in turn, are now dependent on cycle amplitude introducing
two nonlinearities into the problem.
{\it Tilt Quenching (TQ):} Joy's law varies depending on the amplitude
of the cycle as
\begin{equation}
\Delta\lambda=1{{}^{\circ}\hspace{-0.3em}.}5\,{\sin\lambda_0}
\left( 1-b_{\text{joy}}\frac{A_n-A_0}{A_0}\right) .
\label{eq:mod_joyslaw}
\end{equation}
Here, $A_0$ is an arbitrary reference value corresponding to a
``typical'' or average cycle amplitude, as set by the nonlinear
limitation of the dynamo. Equation~(\ref{eq:mod_joyslaw}), thus,
measures the departure of the feedback $\Delta\lambda=f(A_n)$ from
its value in this reference state; its assumed linear form should be
valid for a restricted range in $\Delta\lambda$ and is not
incompatible with the rather lax observational constraints,--- cf.
Fig.~1b in \cite{Jiang:nonlin}.
The nonlinearity parameter $b_{\text{joy}}$ is another free parameter to be
explored; our reference value for it will be 0.15.\footnote{We note that
alternate functional forms for Joy's law, admitted by the
observational constraints, such as
$\Delta\lambda\propto\lambda$ or
$\Delta\lambda\propto\sin^{1/2}\lambda$ have also been tested in the
context of the linear case (Paper 1) and found to have only a minor
impact on the results.}
{\it Latitude Quenching (LQ):} following \cite{Jiang+:1700a} for the
mean latitude $\lambda_0$ of activity at a given phase of cycle $i$ we have:
\begin{equation}
\label{eq:lambdan}
\lambda_0(t;i) = [ 26.4 - 34.2 (t/P) + 16.1(t/P)^2 ]
(\lambda_{i}/14{{}^{\circ}\hspace{-0.3em}.}6),
\end{equation}
with the mean latitude in cycle $i$ given by:
\begin{equation}
\lambda_i = 14{{}^{\circ}\hspace{-0.3em}.}6+b_{\text{lat}}\frac{A_n-A_0}{A_0}.\\
\label{eq:latit_quen}
\end{equation}
Here, the nonlinearity parameter is fairly tightly constrained by the
empirical results of \cite{Jiang+:1700a} as $b_{\text{lat}}\simeq 2.4$.
For the SFT model parameters, our model grid will be a subset of the
grid described in Paper~1. For $\tau,$ we only consider two values here:
a decay time scale of eight years, comparable to the cycle length and
supported by a number of studies (see Paper~1 and references therein),
and the reference case with effectively no decay ($\tau=${ $\infty$} years).
For $u_0=10\,$m/s, the full set of $\eta$ values in the grid is
considered; and similarly for $\eta=500\,$km$^2/$s, the full set of
$u_0$ values is studied (as listed in Table~\ref{table:dev}).
Furthermore, for the study of the effects of individual
nonlinearities, for each parameter combination in the SFT model four
cases will be considered, as listed in Table~\ref{table:cases}.
\begin{center}
\begin{table}
\caption{Notation of cases with different combinations
of the nonlinearity parameters}
\label{table:cases}
\begin{tabular}{lcc}
Case & $b_{\text{lat}}$ & $b_{\text{joy}}$ \\
\hline
noQ & 0 & 0 \\
TQ & 0 & 0.15 \\
LQ & 2.4 & 0 \\
LQTQ & 2.4 & 0.15
\end{tabular}
\end{table}
\end{center}
For each combination of the parameters $u_0$, $\eta$, $\tau$, $b_{\text{lat}}$,
and $b_{\text{joy}}$ we run the SFT code solving Eq. (\ref{eq:transp}) for
1000 solar cycles to produce a statistically meaningful sample of
cycles.
\begin{figure}[htb]
\centering
\includegraphics[width=\hsize]{case5_250_8}
\caption{%
Net contribution of a cycle to the solar dipolar moment vs. cycle amplitude for
the linear case and for the cases with tilt quenching and/or latitude
quenching, with quadratic fits.
Parameter values for this run were $u_0=5\,$m$/$s, $\eta=250\,$km$^2/$s
and $\tau = 8$ years.
}
\label{fig:quad_fit}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[width=\hsize]{case20_600_inf}
\caption
Net contribution of a cycle to the solar dipolar moment vs. cycle amplitude for
the linear case and for the cases with tilt quenching and/or latitude
quenching, with quadratic fits.
Parameter values for this run were $u_0=20\,$m$/$s, $\eta=600\,$km$^2/$s
and $\tau$ = $\infty$ years.}
\label{fig:dev_180}
\end{figure}
\subsection{Results}
\label{sect:SFTresults}
In order to study how the nonlinearities affect the dipole moment
built up during cycles of different amplitudes, here we consider the
purely stochastic case where individual cycle amplitudes are drawn
from a statistical distribution without any regard for previous solar
activity. The factor $k=\pm 1$ in Eq.~(\ref{eq:source}) is assumed
to alternate between even and odd cycles. The form of the statistical
distribution chosen is not important for our current purpose but, for
concreteness, fluctuations are represented as multiplicative noise
applied to the amplitude of the poloidal field source. This noise then
follows lognormal statistics:
$
A_n=A_{0} \times 10^{G}
$,
where $G(0,\sigma)$ is a gaussian random variable of zero mean and
standard deviation of $\sigma = 0.13$. The coefficient for an average
cycle is set to $A_0=0.001 \,e^{7/\tau}$ to ensure that the resulting
dipolar moments roughly agree with observed values (in Gauss).
{
We note that the latitude-integrated amplitude of our source term $s(\lambda,t)$
must be linearly related to the observed value of the
smoothed international sunspot number:
\[ S =K\int_{-\pi/2}^{\pi/2} s(\lambda,t)\,d\lambda . \]
The proportionality coefficient, $K,$ was arbitrarily tuned to
closely reproduce the observed mean value of $S$ for each value of
$\tau$.
}
Introducing the notation $r= e^{-11/\tau}$ (with $\tau$ in years), the net
contribution of cycle $n$ to the change in dipolar moment is given by
$\Delta D_n = D_{n+1}-r D_n$.
Figures~\ref{fig:quad_fit} and \ref{fig:dev_180} display $|\Delta D|$
against the cycle amplitude $S_n$ for some example runs.
The linear case ({ noQ}) is obviously well suited to a linear fit, while
all three nonlinear cases are seen to be well represented by quadratic
fits forced to intersect the linear fit at the mean value of $S_n$.
The respective importance of TQ and LQ is seen to differ for these
cases. For a quantitative characterization of the influence of
nonlinearities we use the deviation of $|\Delta D_n|$ from the linear
case at an arbitrary selected value of $S_n$ (roughly twice the mean),
as illustrated by the solid vertical lines in
Figs.~\ref{fig:quad_fit}--\ref{fig:dev_180}. To give an example, in
the TQ case, the deviation is $\mbox{dev}_{\scriptsize TQ}=7.2281-6.1647=1.0634$ for the
model in Fig.~\ref{fig:quad_fit}. The relative importance of latitude
quenching versus tilt quenching is then characterized by the ratio of
these deviations for the LQ and TQ cases, $\mbox{dev}_{\scriptsize LQ}/\mbox{dev}_{\scriptsize TQ}$. Comparing
these values to several parameter combinations reveals that the
relative importance of the two effects is primarily determined by the
ratio $u_0/\eta$. This is borne out in Fig.~\ref{fig:ratio_u0_eta_all}.
\begin{table}
\begin{center}
\caption{Parameters of the models and their dynamo effectivity range
}
\label{table:dev}
\begin{tabular}{crrrr}
\hline \\
case & $\tau$ [yr] &$u_0$ [m$/$s] & $\eta$ [km$^2/$s] & $\lambda_R$ [$^\circ$] \\
\\
\hline
a & { $\infty$} & 5 & 250 & 12.35 \\
b & { $\infty$} & 5 & 600 & 18.52 \\
c & { $\infty$} & 10 & 250 & 9.44 \\
d & { $\infty$} & 10 & 600 & 13.41 \\
e & { $\infty$} & 20 & 250 & 7.79 \\
f & { $\infty$} & 20 & 600 & 10.05 \\
g & 8 & 5 & 250 & 12.35 \\
h & 8 & 5 & 600 & 18.52 \\
i & 8 & 10 & 250 & 9.44 \\
j & 8 & 10 & 600 & 13.41 \\
k & 8 & 20 & 250 & 7.79 \\
l & 8 & 20 & 600 & 10.05 \\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{figure}[htb]
\centering
\includegraphics[width=\hsize]{dev_2}
\caption
Relative importance of LQ vs TQ, plotted against the
parameter $u_0/\eta$ for the parameter combinations listed in
Table~\ref{table:dev}. Separate quadratic fits for the case
$\tau$ ={ $\infty$} (orange) and $\tau=8$ years (blue) are shown.
}
\label{fig:ratio_u0_eta_all}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[width=\hsize]{lambdar_quad_sep.png}
\caption
Relative importance of LQ vs TQ, plotted against the dynamo
effectivity range, $\lambda_R,$ for the parameter combinations listed
in Table~\ref{table:dev}. Separate fits of the form
$C_1+C_2/\lambda_R^2$ are shown for the case
$\tau =$ { $\infty$} (orange) and $\tau=8$ years (blue).
Large coloured symbols mark the loci of SFT and dynamo models studied
in recent literature:
\cite{Bhowmik_Nandy_2018} (\textcolor{red}{$\bullet$}),
\cite{Jiang+Cao} (\textcolor{green}{X}),
\cite{Lemerle2} (\textcolor{yellow}{$\star$}),
\cite{Whitbread+:howmany} ($+$).
}
\label{fig:lambdar_quad_sep}
\end{figure}
\subsection{Interpretation}
Over a short interval of time $[t,t+dt],$ the smooth source distribution assumed,
Eq.~(\ref{eq:source}) is reduced to a bipole with a flux of
\begin{eqnarray}
\Phi(t)\,dt &=&
2\pi A_n s_1(t) \cos\lambda_0 \,R^2
\int_{-\pi/2}^{\pi/2} s_2\,d\lambda \,dt \nonumber \\
&\simeq & (2\pi)^{3/2} A_n s_1(t) \cos\lambda_0 \,R^2 \,dt
,\end{eqnarray}
in each polarity.
The ultimate contribution of such a bipole to the solar axial dipole moment
after several years has been analytically derived by
\cite{Petrovay+:algebraic1} as:
\begin{equation}
\delta D_U = f_\infty\, \delta D_1 \, e^{(t-t_{n+1})/\tau}
\label{eq:dDU}
,\end{equation}
where $\delta D_1$ is the initial contribution of the bipole to the global
dipole moment, $t_{n+1}$ is the time of the minimum of the next cycle, while the
asymptotic dipole contribution factor $f_\infty$ is given by:
\begin{equation}
f_\infty= \frac a{\lambda_R} \exp
\left(\frac{-\lambda_0^2}{2\lambda_R^2}\right).
\label{eq:finfty}
\end{equation}
Here, $a$ is a constant and defining $\Delta_u$ as the divergence of
the meridional flow at the equator and the dynamo effectivity range
$\lambda_R$ is a universal function of $\Delta_u/\eta$ for plausible
choices of the meridional flow profile.
For the source considered here:
\begin{equation}
\delta D_1(t)=\frac{3\Phi}{\pi R^2}\,\Delta\lambda\,\cos\lambda_0 =
6(2\pi)^{1/2} A_n\, \Delta\lambda\, s_1 \, {
\cos^2\lambda_0}
\label{eq:dD1}
\end{equation}
Combining the last three equations and integrating for the full duration of
the cycle yields the net change in the solar dipole moment:
\begin{equation}
\Delta D_n \equiv D_{n+1}-rD_n = \int_{t_n}^{t_{n+1}} \delta D_U(t) \,dt =
\frac{6a}{\lambda_R} (2\pi)^{1/2} A_n I_1
\label{eq:interpret}
,\end{equation}
with
\begin{equation}
I_1 = \int_{t_n}^{t_{n+1}} \Delta\lambda\, \cos^2\lambda_0\, s_1(t)\,
\exp\left( \frac{t-t_{n+1}}\tau - \frac{\lambda_0^2}{2\lambda_R^2} \right) \,dt
.\end{equation}
This shows that $|\Delta D_n|$ and, hence, the nonlinear effects in
it, only depend on model parameters through $\tau$ and the dynamo
effectivity range $\lambda_R$. For a fixed meridional flow profile, as
used in this work, $\Delta_u\sim u_0$, hence $\lambda_R$ is uniquely
related to $u_0/\eta$, explaining the dependence on the latter
variable, as shown in Fig.~\ref{fig:ratio_u0_eta_all}. On the other
hand, usinq Eqs. (26) and (27) of \cite{Petrovay+:algebraic1},
$\lambda_R$ can be derived for each of our models, allowing us to plot
the underlying relation of the nonlinearity ratios with $\lambda_R$.
This plot is displayed in Fig.~\ref{fig:lambdar_quad_sep}.
In order to understand the form of this relation we recall from
Eqs. (\ref{eq:mod_joyslaw})--(\ref{eq:latit_quen}) that
\begin{equation} \Delta\lambda \sim 1-b_{\text{joy}}\frac{A_n-A_0}{A_0} \qquad
\mbox{and} \qquad \lambda_0 \sim 1+b_{\text{lat}}\frac{A_n-A_0}{A_0}
.\end{equation}
Upon substituting the Eqs.{
(\ref{eq:mod_joyslaw})--(\ref{eq:latit_quen})} into
(\ref{eq:interpret}), the deviations from the linear case correspond to
the terms containing $b_{\text{joy}}$ and $b_{\text{lat}}$. We will consider these terms
in the limit $\tau\rightarrow\infty$ and, relying on the mean value
theorem for integrals, time integrals will be substituted by the
product of the cycle period and the value of the integrand for an
appropriately defined average. These weighted averages are not
expected to differ from the unweighted average $\overline{\lambda_0}$
by more than a few degrees, so they will be substituted here simply by
$\overline{\lambda_0}$ for an approximation.
In the TQ case, { using Eq. (\ref{eq:mod_joyslaw}),}
the term containing $b_{\text{joy}}$ is immediately found to scale as
\begin{equation}
\mbox{dev}_{TQ}\sim \sin\ovl \cos^2\ovl
\exp\left( -\frac{\ovl^2}{2\lambda_R^2}\right).
\label{eq:devtq}
\end{equation}
The LQ case is slightly more complex. In the limit
$\tau\rightarrow\infty$, the dependence on the latitude of activity
enters Eq. (\ref{eq:interpret}) through the combination
$h(\lambda_0)=\sin\lambda_0\cos^2\lambda_0\exp (-
{\lambda_0^2}/{2\lambda_R^2})$. A Taylor expansion yields:
\begin{eqnarray}
&& h(\lambda_0+\varepsilon\lambda_0)= h(\lambda_0)
+\varepsilon{\lambda_0} \cos\lambda_0
\exp\left(-\frac{\lambda_0^2}{2\lambda_R^2} \right) \times \nonumber \\
&& \qquad (\cos^2\lambda_0 -2\sin^2\lambda_0 +
\lambda_0\sin\lambda_0\cos\lambda_0 /\lambda_R^2).
\end{eqnarray}
Substituting here $\varepsilon=b_{\text{lat}}\frac{A_n-A_0}{A_0}$ and plugging the
expression back into Eq. (\ref{eq:interpret}), the term involving
$b_{\text{lat}}$ is found to scale as:
\begin{eqnarray}
&& \mbox{dev}_{LQ}\sim \ovl \cos\ovl
\exp\left( -\frac{\ovl^2}{2\lambda_R^2}\right) \times \nonumber \\
&& \qquad (\cos^2\ovl -2\sin^2\ovl+\ovl\sin\ovl\cos\ovl/\lambda_R^2).
\label{eq:devlq}
\end{eqnarray}
Dividing Eq. (\ref{eq:devlq}) by (\ref{eq:devtq}) yields
\begin{equation}
\mbox{dev}_{LQ}/\mbox{dev}_{TQ}\sim
C_1(\ovl) + C_2(\ovl)/\lambda_R^2 .
\label{eq:devlqtq}
\end{equation}
This expectation is indeed confirmed by the fits in
Fig.~\ref{fig:lambdar_quad_sep}.
\subsection{Discussion}
It is to be noted in Fig.~\ref{fig:lambdar_quad_sep} that for high
values of $\lambda_R$, the value of $\mbox{dev}_{\scriptsize LQ}/\mbox{dev}_{\scriptsize TQ}$ becomes slightly negative.
This is due to the circumstance that latitudinal quenching can affect
$\Delta D$ in two ways:\\
(i) by modulating the fraction of active regions above $\lambda_R$
that do not contribute to $\Delta D$, which results in a negative
feedback; (ii) by modulating the mean tilt angle of active regions in accordance
with Joy's law, which is a positive feedback effect.
For $\lambda_R\la\ovl$ the first effect dominates. However,
when $\lambda_R$ significantly exceeds $\ovl$ the fraction of ARs
above $\lambda_R$ becomes negligible, so only the second, positive
feedback effect remains. As a result, $\mbox{dev}_{\scriptsize LQ}$ changes sign.
A slight dependence on $\tau$ is also noticed. This can be attributed
to the fact that for shorter values of $\tau$ the dipole moment
contribution of ARs appearing early in the cycle will have more time
to decay until the next minimum, hence, these ARs will have a lower
relative contribution to $|\Delta D|$. But the ARs in the early part
of the cycle are the ones at the highest latitudes, that is, they are most affected
by LQ via effect (i) above. Hence, for shorter values of $\tau,$
the relative importance of LQ is expected to be slightly suppressed. This
suppression may be expected to vanish towards higher values of
$\lambda_R$, in agreement with Fig.~\ref{fig:lambdar_quad_sep}.
The larger, coloured symbols in Fig.~\ref{fig:lambdar_quad_sep} mark
the positions of various dynamo and SFT models discussed in recent
publications. It is apparent that the case where LQ results in a
positive feedback is not strictly of academic interest: at least one SFT
model discussed recently, namely, that of \cite{Whitbread+:howmany} is in
this parameter range. The parameter choice for this model was
motivated by an overall optimization of observed versus simulated
magnetic supersynoptic maps (\citen{Whitbread+:SFT}).
At the other end of the scale, the low diffusivity, medium flow speed
models of \cite{Bhowmik_Nandy_2018} and \cite{Jiang+Cao} are in a
range where LQ is manifest as a negative feedback giving the
dominating contribution to the nonlinearity, although the effect of TQ
remains non-negligible.
Conversely, the locus of the SFT component of the 2x2D model of
\cite{Lemerle2} in Fig.~\ref{fig:lambdar_quad_sep} indicates that if
the TQ and LQ effects were formulated as in our model, the
contributions of TQ should dominate but the negative feedback due to
LQ would also contribute quite significantly to the nonlinearity. We take a closer look
at this model in the following section.
{
As a caveat regarding the validity of these conclusions, we recall that
the results presented in this section are based on the assumption of
one particular form of the time profile of the source function,
Eq. (\ref{eq:ampli}), with constant parameters and a strictly
fixed periodicity of 11 years. As long as deviations from it are
stochastic, this source may still be considered representative as an
ensemble average. However, there have also been indications of systematic
deviations.
It is well known that the length of solar cycles is inversely related
to their amplitude: this effect implies an inverse correlation of the
parameter $b$ in Eq. (\ref{eq:ampli}) with $a$ (or, equivalently,
$S_{\rm max}$). It is, however, straightforward to see that a variation
in $b$ will not affect our results. If $c$ is kept constant, as
suggested by \cite{hathaway1994shape}, $b$ is the only time scale
appearing in $s_1$, so, with $S_{\rm max}$ fixed, its variation will
simply manifest as a self-similar temporal expansion or contraction of
the profile (\ref{eq:ampli}). As a result, cycle lengths will then
vary and so will the area below the curve, namely, the total amount of
emerging flux, and, hence, the amplitude of the dipole moment source. As
a result, the dipole moment built up by the end of the cycle will
linearly scale with $b/{\overline b}$. This scaling factor, however,
will be the same for the TQ, LQ, and subsequent cases, so the ratio between their
deviations, plotted in Figure~\ref{fig:lambdar_quad_sep}, remains unaffected.
Furthermore, \cite{jiang2018predictability} noted that actual cycle
profiles tend to show a systematic upward deviation from the profile
(\ref{eq:ampli}) in their late phases. Such an effect would indeed
had an impact on our results somewhat. As in these late phases the activity is
concentrated near the equator, at latitudes well below $\lambda_R$,
the extra contribution to the dipole moment due to this excess is not
expected to be subject to LQ, while it is still affected by TQ. Hence,
$\mbox{dev}_{\scriptsize LQ}/\mbox{dev}_{\scriptsize TQ}$, plotted in Figure~\ref{fig:lambdar_quad_sep}, is
expected to be somewhat lower with this correction. This relative
difference, in turn, may be expected to vanish towards higher values of
$\lambda_R$, where LQ becomes insignificant anyway.
}
\begin{figure}[htb]
\centering
\includegraphics[width=\hsize]{2x2D_SSN_DeltaD__rfaktor}
\caption
\ Contribution of all active regions in a cycle to the net change in
the solar dipole moment during dynamo cycles vs. the cycle amplitude in
the 2x2D dynamo model.
Solid line: Mean; dashed curves: $\pm 1\sigma$ range.
}
\label{fig:2x2D_logSSN_DeltaD}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[width=\hsize]{2x2D_SSN-D}
\caption
Solar dipole moment at cycle minimum vs. the amplitude of the subsequent cycle
in the 2x2D dynamo model.
}
\label{fig:2x2D_SSN-D}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[width=\hsize]{2x2D_mapping}
\caption
Cycle amplitude vs. the amplitude of the subsequent cycle
in the 2x2D dynamo model.
Solid line: Mean; dashed curves: $\pm 1\sigma$ range.
}
\label{fig:2x2D_mapping}
\end{figure}
\section{Nonlinear effects in the 2x2D dynamo model}
\label{sect:2x2D}
In this section, we consider the role of nonlinearities in a full
dynamo model, namely, the 2x2D dynamo model (\citen{Lemerle1} and
\citen{Lemerle2}). This model, also used for solar cycle forecast
(\citen{Labonville2019}), couples a 2D surface SFT model with an
axisymmetric model of the dynamo operating in the convective zone. The
azimuthal average of the poloidal magnetic field resulting from the
SFT component is used as upper boundary condition in the subsurface
component, while the source term in the SFT model is a set of bipolar
magnetic regions (BMRs) introduced instantaneously with a probability
related to the amount of toroidal flux in a layer near the base of the
convective zone, as a simplified representation of the flux emergence
process. The model has a large number of parameters optimized for best
reproduction of the observed characteristics of solar cycle 21.
\subsection{Nonlinearity vs. stochastic scatter in the 2x2D dynamo}
Figure~\ref{fig:2x2D_logSSN_DeltaD} presents the equivalent of
Figs.~\ref{fig:quad_fit} and \ref{fig:dev_180} for 5091 cycles
simulated in the 2x2D model. A marked nonlinearity is clearly seen:
the net contribution of a cycle to the dipole moment increases slower
than in the linear case with cycle amplitude, representing a negative feedback,
similarly to the case of the pure SFT model.
In the dynamo model, the dipole moment built up during a cycle feeds
back into the cycle as the amplitude of the toroidal field generated
by the windup of the poloidal fields (and, hence, the amplitude of the
upcoming cycle) increases with the amplitude of the poloidal field.
This is borne out in Fig.~\ref{fig:2x2D_SSN-D}. We note that the relation
is not exactly linear and there is a slight threshold effect at work, which is
presumably related to the requirement that the toroidal field must
exceed a minimal field strength for emergence.
The combined result of the two feedbacks presented in
Figs.~\ref{fig:2x2D_logSSN_DeltaD} and \ref{fig:2x2D_SSN-D} is a
nonlinear coupling between the amplitudes of subsequent cycles as
shown in Fig.~\ref{fig:2x2D_mapping}. The convex shape of the median
curve implies that on average, very weak cycles will be followed by
stronger cycles and very strong cycles will be followed by weak
cycles. However, there is a very large amount of scatter present in
the plot. A comparison with the scatter in
Figs.~\ref{fig:2x2D_logSSN_DeltaD} and \ref{fig:2x2D_SSN-D} shows that
the main contribution to the scatter comes from
Fig.~\ref{fig:2x2D_logSSN_DeltaD}. As $\Delta D$ is uniquely
determined by the source term in the SFT equation representing flux
emergence, the main cause of random scatter around the mean nonlinear
relation is identified as the stochastic nature of the flux emergence
process, namely, the scatter in the time, location, and properties of
individual active regions (BMRs in the 2x2D model) statistically
following the evolution of the underlying toroidal field.
\begin{figure}[htb]
\centering
\includegraphics[width=\hsize]{LQ_2x2D_total_hemisph}
\caption
Latitude quenching in the the 2x2D dynamo model. Mean latitude of BMRs in a
cycle vs. cycle amplitude.
}
\label{fig:LQ_2x2D_total_hemisph}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[width=\hsize]{meanLatitude_fluxweight_cycledep220_yy15a}
\includegraphics[width=\hsize]{meanLatitude_fluxweight_cycledep220_yy15d}
\caption
Butterfly tracks in the 2x2D dynamo model sorted by cycle amplitude:
(a) tracks of the active latitude vs.\ cycle phase;
(b) tracks of the active latitude vs.\ time;
Red, black, and blue curves represent flux-weighted mean latitudes for subsamples of
strong, average, weak cycles, respectively, as described in the text.
Shaded areas mark the standard deviations of the actual data around
the mean curves.
}
\label{fig:meanLatitude_fluxweight_cycledep220}
\end{figure}
\subsection{Latitude quenching in the 2x2D dynamo model}
The question arises as to whether the nonlinearity plotted in
Fig.~\ref{fig:2x2D_logSSN_DeltaD} is a consequence of TQ, LQ, or both.
Apart from a threshold effect which was not found to be strong or even
necessarily needed in the optimized model, the only nonlinearity
explicitly built into the 2x2D model is TQ. The tilt angle (i.e.,
initial dipole contribution $\delta D_1$) of a BMR is assumed to
scale with
\begin{equation}
\alpha_q = \frac{\alpha}{1 + (B_{\phi}/B_q)^2}
\label{eq:tiltquenching}
,\end{equation}
where $B_q$ is the quenching field amplitude.
As the latitudinal distribution of BMRs is fed into the SFT component
of the 2x2D model from the mean-field component, LQ cannot be imposed.
Nevertheless, this does not mean that LQ is not present. Indeed,
as demonstrated in Fig.~\ref{fig:LQ_2x2D_total_hemisph}, the 2x2D
model displays a very marked latitude quenching effect, with a tight
correlation between the mean emergence latitude of BMRs and cycle
amplitude. This LQ is even stronger than witnessed in the Sun, $\ovl$
varying by $\sim 10^\circ$ as $S_n/\overline{S_n}$ varies between
$0.1$ and $2$. In contrast, according to Eq.
(\ref{eq:latit_quen}), $\ovl$ only varies by about $5^\circ$ (from
$12^\circ$ to $17^\circ$) for the same range in $S_n$.
The origin of this LQ effect is elucidated in
Fig.~\ref{fig:meanLatitude_fluxweight_cycledep220}. Three subsamples
of all simulated cycles are considered: cycles stronger or weaker than
the mean amplitude by at least 1 standard deviation are displayed in
red or blue, respectively, while black marks cycles not farther from the
mean amplitude than 0.1 standard deviation. The plots clearly
demonstrate that tracks of latitudinal migration of these three
subsets are systematically shifted relative to each other. The
alternative possibility that all cycles follow the same track with LQ
resulting from the different temporal variation of the activity level
can thus be discarded.
\begin{figure}[htb]
\centering
\includegraphics[width=\hsize]{meanLatitude_fluxweight_cycledep220_yy15b}
\includegraphics[width=\hsize]{meanLatitude_fluxweight_cycledep220_yy15c}
\caption
Universality of the cycle decay phase in the 2x2D dynamo model: (a) Pseudo sunspot
number against flux-weighted mean latitude of BMRs. Curves show
flux-weighted mean values; (b) Width $2\delta\lambda$ of the
latitudinal distribution of BMRs vs. their flux-weighted mean
latitude. Colour coding is the same as in
Fig.~\ref{fig:meanLatitude_fluxweight_cycledep220} Shaded areas mark
the standard deviations of actual data around the mean curves. Solid
lines correspond to a $0.5^{\circ}$ binning, dotted lines do to a
$2^{\circ}$ binning.
}
\label{fig:universal}
\end{figure}
\cite{Cameron+Schussler:turbdiff} considered the time evolution of
mean latitude and width of the activity belts in observed solar
cycles, interpreting the decay phase of cycles as a result of the
cancellation of oppositely oriented toroidal flux bundles across the
equator by turbulent diffusion. They pointed out that this leads to a
certain universality of the characteristics of the decay phases in
different solar cycles. Our finding that the tracks of migrating
toroidal flux bundles systematically differ for strong and weak cycles
may at first sight seem to contradict this interpretation. However, a
closer look at the properties of decay phases of cycles simulated in
the 2x2D model confirms the universality posited by
\cite{Cameron+Schussler:turbdiff}. In Fig.~\ref{fig:universal}a, we
present the evolution of the (pseudo) sunspot number simulated in the
models against the latitude of activity belts, rather than against
time, while Fig.~\ref{fig:universal}b presents the evolution of the
width of the latitudinal distribution of BMRs in a similar way (see
Figs. 3 and 4 in \citen{Cameron+Schussler:turbdiff} for the
observational equivalents of these plots.) It is apparent that the
curves for the strong, average and weak subsamples are nicely aligned
towards their low-latitude (i.e., late-phase) ends, confirming the
universality proposition.
\section{Conclusion}
\label{sect:concl}
In this work, we explore the relative importance and the
determining factors behind the LQ and TQ effects. The degree of
nonlinearity induced by TQ, LQ, and their combination was
systematically probed in a grid of surface flux transport (SFT)
models. The relative importance of LQ versus TQ has been found to
correlate with the ratio $u_0/\eta$ in the SFT model grid, where $u_0$
is the meridional flow amplitude and $\eta$ is diffusivity. An
analytical interpretation of this result has been given, further
showing that the main underlying parameter is the dynamo effectivity
range $\lambda_R$, which, in turn, is determined by the ratio of
equatorial flow divergence to diffusivity. The relative importance of
LQ versus TQ was found to scale as $C_1+C_2/\lambda_R^2$.
As displayed in Fig.~\ref{fig:lambdar_quad_sep}, for various dynamo
and SFT models considered in the literature the contribution of LQ to
the amplitude modulation covers a broad range from being insignificant
to being the dominant form of feedback. On the other hand, the
contribution of a TQ effect (with the usually assumed amplitude) is
never negligible.
The role of TQ and LQ was also explored in the 2x2D dynamo model
optimized to reproduce the statistical behaviour of real solar cycles.
We demonstrated the presence of latitude quenching in the 2x2D dynamo.
The locus of the SFT component of the model in
Fig.~\ref{fig:lambdar_quad_sep} suggests that LQ contributes to the
nonlinear modulation by an amount comparable to TQ. This LQ effect is
present in the model despite the lack of any modulation in the
meridional flow. We note that the meridional inflow module of the 2x2D model
developed by \citen{Nagy+:inflow} was not used for these runs. This
agrees with the results of \cite{Karak2020} who showed the presence of
latitude quenching in the STABLE dynamo model with a steady meridional
flow. As LQ is rather well constrained on the observational side
(\citen{Jiang:nonlin}), its presence and character may potentially be used in the future as a test or merit function in the process of fine-tuning
of dynamo models to solar observations.
\begin{acknowledgements}
This research was supported by the Hungarian National Research,
Development and Innovation Fund (grants no. NKFI K-128384 and TKP2021-NKTA-64), by the
European Union's Horizon 2020 research and innovation programme under
grant agreement No.~955620, and by the Fonds de Recherche du Qu\'ebec
-- Nature et Technologie (Programme de recherche coll\'egiale). The
collaboration of the authors was facilitated by support from the
International Space Science Institute in ISSI Team 474.
\end{acknowledgements}
\bibliographystyle{aa}
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\subsubsection*{Appendix: \ Other Identities}
There are many other $SL\left( 2
\mathbb{C}
\right) $ \textquotedblleft roads\textquotedblright\ that lead to a closed
form expression for the oscillator propagator (i.e. the \textquotedblleft
Rome\textquotedblright\ of this problem).\ \ Here we list a few more of them
involving the operators $x$ and $p$ with $\left[ x,p\right] =i\hbar$. \ For
simplicity, we omit the $\lambda/x^{2}$\ potential, but the identities to
follow are valid even with that potential term upon substituting
$p^{2}\rightarrow p^{2}+\lambda/x^{2}$.
In addition to the identity in the main text, (\ref{TSvKIdentity}), there are
many relations paired by conjugation plus $t\rightarrow-t$. \ For example
\begin{gather}
\exp\left( -\frac{it}{\hbar}\left( \frac{1}{2m}p^{2}+\frac{1}{2}m\omega
^{2}x^{2}\right) \right) =\exp\left( -i\alpha x^{2}\right) \exp\left(
-i\gamma\left( xp+px\right) \right) \exp\left( -i\beta p^{2}\right)
\tag{A1a}\\
\text{with \ \ }e^{2\gamma\hbar}=\cos\left( \omega t\right) \ ,\ \ \ \alpha
=\frac{m\omega}{2\hbar}\tan\left( \omega t\right) \ ,\ \ \ \beta=\frac
{1}{2\hbar m\omega}\tan\left( \omega t\right) \nonumber
\end{gather
\begin{gather}
\exp\left( -\frac{it}{\hbar}\left( \frac{1}{2m}p^{2}+\frac{1}{2}m\omega
^{2}x^{2}\right) \right) =\exp\left( -i\beta p^{2}\right) \exp\left(
-i\gamma\left( xp+px\right) \right) \exp\left( -i\alpha x^{2}\right)
\tag{A1b}\\
\text{with \ \ }e^{-2\gamma\hbar}=\cos\left( \omega t\right) \ ,\ \ \ \alpha
=\frac{m\omega}{2\hbar}\tan\left( \omega t\right) \ ,\ \ \ \beta=\frac
{1}{2\hbar m\omega}\tan\left( \omega t\right) \nonumber
\end{gather
\begin{gather}
\exp\left( -\frac{it}{\hbar}\left( \frac{1}{2m}p^{2}+\frac{1}{2}m\omega
^{2}x^{2}\right) \right) =\exp\left( -i\gamma\left( xp+px\right) \right)
\exp\left( -i\alpha x^{2}\right) \exp\left( -i\beta p^{2}\right)
\tag{A2a}\\
\text{with \ \ }e^{2\gamma\hbar}=\cos\left( \omega t\right) \ ,\ \ \ \alpha
=\frac{m\omega}{2\hbar}\sin\left( \omega t\right) \cos\left( \omega
t\right) \ ,\ \ \ \beta=\frac{1}{2\hbar m\omega}\tan\left( \omega t\right)
\nonumber
\end{gather
\begin{gather}
\exp\left( -\frac{it}{\hbar}\left( \frac{1}{2m}p^{2}+\frac{1}{2}m\omega
^{2}x^{2}\right) \right) =\exp\left( -i\beta p^{2}\right) \exp\left(
-i\alpha x^{2}\right) \exp\left( -i\gamma\left( xp+px\right) \right)
\tag{A2b}\\
\text{with \ \ }e^{-2\gamma\hbar}=\cos\left( \omega t\right) \ ,\ \ \ \alpha
=\frac{m\omega}{2\hbar}\sin\left( \omega t\right) \cos\left( \omega
t\right) \ ,\ \ \ \beta=\frac{1}{2\hbar m\omega}\tan\left( \omega t\right)
\nonumber
\end{gather
\begin{gather}
\exp\left( -\frac{it}{\hbar}\left( \frac{1}{2m}p^{2}+\frac{1}{2}m\omega
^{2}x^{2}\right) \right) =\exp\left( -i\alpha x^{2}\right) \exp\left(
-i\beta p^{2}\right) \exp\left( -i\gamma\left( xp+px\right) \right)
\tag{A3a}\\
\text{with \ \ }e^{2\gamma\hbar}=\cos\left( \omega t\right) \ ,\ \ \ \alpha
=\frac{m\omega}{2\hbar}\tan\left( \omega t\right) \ ,\ \ \ \beta=\frac
{\sin\left( \omega t\right) \cos\left( \omega t\right) }{2\hbar m\omega
}\nonumber
\end{gather
\begin{gather}
\exp\left( -\frac{it}{\hbar}\left( \frac{1}{2m}p^{2}+\frac{1}{2}m\omega
^{2}x^{2}\right) \right) =\exp\left( -i\gamma\left( xp+px\right) \right)
\exp\left( -i\beta p^{2}\right) \exp\left( -i\alpha x^{2}\right)
\tag{A3b}\\
\text{with \ \ }e^{-2\gamma\hbar}=\cos\left( \omega t\right) \ ,\ \ \ \alpha
=\frac{m\omega}{2\hbar}\tan\left( \omega t\right) \ ,\ \ \ \beta=\frac
{\sin\left( \omega t\right) \cos\left( \omega t\right) }{2\hbar m\omega
}\nonumber
\end{gather}
These identities immediately lead to the propagator (\ref{SHO}), or
(\ref{HalflineSHO}) after restoration of the $\lambda/x^{2}$ term, upon taking
into account the effects of $\exp\left( -i\gamma\left( xp+px\right)
\right) $ to rescale position eigenstates. \ Namely
\begin{equation}
\exp\left( -i\gamma\left( xp+px\right) \right) ~\left\vert x_{2
\right\rangle =\exp\left( \hbar\gamma\right) ~\left\vert x_{2}\exp\left(
2\hbar\gamma\right) \right\rangle \tag{A4
\end{equation}
etc.
\newpage
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\section{Introduction}
\label{sec:intro}
\input{intro}
\begin{figure}[t]
\centering
\subfigure[Transformer Layer]{
\includegraphics[width=0.13\textwidth]{Figures/preliminary/transformer_layer.pdf}
\label{fig:transformer}
}
\subfigure[Transformer Layer with Mixture-of-Expert]{
\includegraphics[width=0.32\textwidth]{Figures/preliminary/moe_transformer_layer.pdf}
\label{fig:moe_transformer}
}
\caption{A brief architecture of Transformer Encoder Layer and Transformer with Mixture-of-Expert Layer. The Transformer encoder layer contains two main components: a Multi-Head Self-Attention Layer and a Position-wise Feed-Forward Layer. Based on Transformer layer, the transformer with MoE replaces the FFN with a series of FFNs and introduce a gate network.}
\label{fig:preliminary}
\end{figure}
\section{Preliminary}
\label{sec:preliminary}
\subsection{Transformer}
The model architecture of Transformer~\cite{vaswani2017attention} has demonstrated its superior performance in many sequence-to-sequence natural language processing (NLP) tasks, which contains several encoder layers and decoder layers. Each encoder layer is stacked by a multi-head self-attention network and a position-wise feed-forward network (FFN), which is illustrated in Figure~\ref{fig:transformer}.
It employs a residual connection on each of these two sub-layers, followed by a normalization layer~\cite{DBLP:journals/corr/layernorm}. Formally, each sub-layer, e.g., attention and FFN, produces its output as $\text{LayerNorm}(x + \text{Sublayer}(x))$.
The decoder is similarly constructed, except for an additional cross-attention mechanism between attention and FFN to introduce the output of the encoder. For a sequence of input tokens $(x_{1},...,x_{s})$ $\in$ $\mathbb{R}^{D}$, we formulate the function of each sub-layer in following:
\begin{table}[t]
\centering
\renewcommand{\multirowsetup}{\centering}
\caption{Notations}
\label{tab:notation}
\vspace{-2mm}
\begin{tabular}{cc}
\toprule
Symbols & Definitions \\
\midrule
$Q$ & Queries in self-attention module \\
\midrule
$K$ & Keys in self-attention module \\
\midrule
$V$ & Values in self-attention module\\
\midrule
$d_k$ & Feature dimension of each Query/Key \\
\midrule
$S$ & A set of input tokens \\
\midrule
$E$ & An series of experts\\
\midrule
$D$ & Feature dimension of each token \\
\midrule
$N$ & Number of experts in each MoE layer \\
\midrule
$e_{i}(x_s)$ & The output of $i$-$th$ Expert by taking input $x_s$ \\
\midrule
$g(x_s)_i$ & The routing score for $x_s$ on $i$-$th$ Expert\\
\midrule
$c$ & Threshold for expert selection \\
\midrule
$G_{S, E}$ & The routing score for $S$ on $E$ \\
\midrule
$Id_{S}$ & The set of selected expert id on $S$\\
\bottomrule
\end{tabular}
\end{table}
\textbf{Attention:} The attention module~\cite{DBLP:conf/nips/VaswaniSPUJGKP17} could capture the dependencies between tokens in the sequence, and is effective in sequence modeling. It performs as a ternary function, which maps the input queries (Q), keys (K) and values (V) to the output (O). Equation~\ref{equ:attention} represents the Scaled Dot-Product Attention~\cite{DBLP:conf/nips/VaswaniSPUJGKP17}, which performs dot products of each query with all keys, divides each by $\sqrt{d_k}$ and then adopts the softmax function to get the weight of each value. In addition, $d_k$ is the dimension of queries and keys.
\begin{equation}
\texttt{Attention}(Q, K, V) = \texttt{softmax}(\frac{QK^T}{\sqrt{d_k}})V
\label{equ:attention}
\end{equation}
\textbf{Multi-Head Attention:} ~\citet{DBLP:conf/nips/VaswaniSPUJGKP17} proposed the multi-head attention mechanism to jointly learn from different representation subspaces at different positions and thus improved the model performance. The multi-head attention linearly projected the queries, keys and values $h$ times with learned linear projections to $d_k$, $d_k$ and $d_v$, dimensions, respectively.
\begin{align}
\texttt{MultiHead}&(Q, K, V) = \texttt{Concat}(head_1, ..., head_h){W^O} \\
\text{where}&\ head_i = \texttt{Attention}(QW_{i}^Q, KW_{i}^K, VW_{i}^V) \nonumber
\label{equ:multi_head}
\end{align}
\noindent The projections are the trainable parameter matrices, where $W_{i}^Q \in \mathbb{R}^{d_{model} \times d_k}$, $W_{i}^K \in \mathbb{R}^{d_{model} \times d_k}$, $W_{i}^V \in \mathbb{R}^{d_{model} \times d_v}$. Meanwhile, $h$ is the number of heads, and $d_k = d_v = d_{model}/h$. Because the dimension of each head is reduced from $d_{model}$ to $d_{model}/h$, the time cost of multi-head attention is similar to that of the original attention. In addition, the decoder employs a masked self-attention, which only sees the tokens on the left of the sequence.
\textbf{Position-wise Feed-Forward Networks:} Each transformer layer also includes a fully connected feed-forward network (Equation~\ref{equ:ffn}), which consists of two fully connected networks and a ReLU activation function.
\begin{equation}
\texttt{FFN}(x_{s}) = W_{2} \cdot \texttt{ReLU}(W_{1} \cdot x_{s} + b_{1}) + b_{2}
\label{equ:ffn}
\end{equation}
\begin{figure}[t]
\centering
\includegraphics[width=0.415\textwidth]{Figures/preliminary/moe_layer.pdf}
\caption{Illustration on the workflow of Mixture-of-Expert (MoE). The input token $x_s$ is first processed by the gate network to calculate similarities between $x_s$ and each expert. Then, it performs Top-K operation on these similarities to determine the target activated experts. Finally, $y_s$ is produced by the linearly weighted combination of each expert's output on the token by the gate's output. }
\label{fig:moe_layer}
\end{figure}
\begin{figure*}[t]
\centering
\subfigure[Expert Loads Distribution .]{
\includegraphics[width=0.22\textwidth]{Figures/expert_loads_conformity.pdf}
\label{fig:duration_distribution}
}
\subfigure[Unstable Routing Pattern for Token ``the'']{
\includegraphics[width=0.68\textwidth]{Figures/expert_selection.pdf}
\label{fig:weights_distribution}
}
\caption{The observations of GPT-MoE with 16 experts and the Top-1 gate among 12 MoE-layers (totally 24 layer).
Figure~\ref{fig:duration_distribution} shows the expert load distribution (deeper color represents heavier workload, i.e., more tokens to be processed) and Figure~\ref{fig:weights_distribution} shows the expert selection for a specific token ``the''.}
\label{fig:dense_moe}
\end{figure*}
\subsection{Mixture of Experts}
Because larger pretrained models always achieve better model quality~\cite{DBLP:scaling_laws}, the size of state-of-the-art NLP models has been increasing $10\times$ per year, e.g., BERT~\cite{DBLP:conf/naacl/BERT}, GPT~\cite{gpt}, T5~\cite{DBLP:googleT5}, GPT-2~\cite{gpt2}, GPT-3~\cite{DBLP:conf/nips/gpt3}, which require increasing compute budgets. To improve the model capacity without increasing computation budgets, researchers sparsely scale transformers recently as Figure~\ref{fig:moe_transformer} by replacing the feed-forward network with the mixture of experts (MoE) architecture and activating only a subset of these experts for each input sample~\cite{DBLP:lstm_moe, DBLP:switch, DBLP:gshard}.
The main components of the MoE architecture include an expert network $E$ for scaling model capacity and a sparse gate network $G$ for introducing model sparsity.
\textbf{Expert Network:}
The expert network $E$ includes a series of experts $\{e_{1},...,e_{N}\}$ to increase the model capacity, where each expert $e_{i}$ represents a single neural network, e.g., \text{FFN}, and contains its own parameters. In Figure~\ref{fig:moe_layer}, the MoE layer consists of three FFN networks.
For each expert $e_{i}$ ($e_{i}: \mathbb{R}^{D} \rightarrow \mathbb{R}^{D}$), it takes the token $x_{s}$ as an input to produce its own output $e_{i}(x_{s})$. The final output of the expert network $y_{s}$ is the linearly weighted combination of each expert’s output on the token by the gate's output, formulated as Equation \ref{equ:moe_weighted_sum}.
\begin{equation}
y_{s} = \sum_{i=1}^{N}g(x_{s})_{i} \cdot e_{i}(x_{s})
\label{equ:moe_weighted_sum}
\end{equation}
\noindent In Figure~\ref{fig:moe_layer}, the expert network takes the input token $x_{s}: [-0.2, 0.4, 1.5]$ and produces the output of each individual expert on $x_s$, e.g., $e_{0}(x_s)$, $e_{1}(x_s)$ and $e_{2}(x_s)$. By combining the gate's output, i.e., $[0.35, 0.65, 0]$, the output of this MoE layer is $y_s = 0.35*e_{0}(x_s) + 0.65*e_{1}(x_s)$.
\textbf{Sparse Gate Network:}
The sparse gate network $G$ is the key component to introduce model sparsity, which takes a batch of tokens $\{x_{1},...,x_{s}\}$ as input and produces the probability of them with respective to all experts $\{e_{1},...,e_{N}\}$.
\citet{DBLP:lstm_moe} proposes the Top-K gating as Equation~\ref{equ:topk_gate}, which keeps only the top k values before the softmax function. In addition, $W_{g}$ is a trainable variable ($W_{g} \in \mathbb{R}^{D \times N}$) and determine the targeted experts for each token.
\begin{equation}
g(x_s) = softmax(TopK(x_{s} \cdot W_{g}) )
\label{equ:topk_gate}
\end{equation}
\noindent We illustrate the workflow of a MoE layer in Figure~\ref{fig:moe_layer}, where $k=2$ and $W_g$ is a 3 $\times$ 3 (i.e., feature dimension $\times$ number of experts) matrix to represents the parameter of gate network. We first perform a dot-product on $x_s$ and $W_g$ to calculate similarity between the input token and the experts. The result, $[2.01, 2.64, 1.8]$, indicates that the input prefers $e_1 > e_0 > e_2$ and we only activate $e_0$ and $e_1$ as $k = 2$. Finally, we conduct a softmax function to get the weight score of each expert and perform a weighted sum to get the final output $y_s$.
Previous work mainly focuses on how to improve the quality and efficiency of training such sparse gate network. ~\citet{DBLP:lstm_moe} proposed the noisy Top-K gating on Long Short-TerM memory (LSTM) kayers~\cite{DBLP:journals/neco/lstm} and ~\citet{DBLP:gshard} introduced MoE with Top-2 gate into Transformer.
~\citet{lewis2021base} adopted the numerous solution for balanced token-to-expert routing and \citet{roller2021hash} utilized the hash-based routing strategy.
\textbf{Distributed Training of MoE Models:}
Expert parallel training is a specific method of parallelism for MoE models, which is first proposed by GShard~\cite{DBLP:gshard}. Experts are placed on different workers and each worker takes a different batch of training samples. For non-MoE layers, expert parallelism behaves the same as data parallelism. In MoE layers, tokens in the sequence are sent to workers where their desired experts reside. Similar to model parallelism, the outputs of each MoE layer are exchanged again to be organized back into original sequences for the computation of the next layer. As MoE models often have numerous experts, expert parallelism can scale up with model size better than model parallelism.
\subsection{Observation and Motivation}
\label{sec:motivation}
In this section, we revisit the learning process of MoE models and introduce our two key findings in the following, which motivates us to design our {EvoMoE}\xspace framework.
\textbf{Conformity in Mixture of Experts:} One interesting finding is conformity. During the early training stage, existing join-training methods of sparse MoE make the routing decision to comply with most tokens.
Here we train a GPT model including 24 transformer layers, with every FFN layer replaced by 16-expert MoE layer using the Top-1 gate.
Figure~\ref{fig:duration_distribution} shows that most tokens keep concentrating on the 8-th expert at first, since it has been greedily reinforced.
After around hundreds of training steps (i.e., 1 duration equals 40 training steps), the other experts gradually catch up and the workload becomes balanced. Such phenomenon motivates us to focus on training a common expert and utilize the computational resources to accelerate the early stage.
\textbf{Instability in Mixture of Experts:} Another important finding is the instability. We take a single token ``the'' as an example and Figure~\ref{fig:weights_distribution} shows its expert selection results for a longer training process. As we can see, the selection is highly unstable later since both the gate network and the experts are not knowledgeable enough to obtain a stable routing pattern, especially at the early stage of the training process. This indicates that a pre-defined gates (e.g., Top-K) in existing works, which assumes a fixed number of activated experts, could limit the exploration of potential valuable experts. Aggressively increasing the number of activated experts could improve the model capacity but inherently violates the original design intention of sparse MoE. Such a dilemma motivates us to design an adaptive solution to balance the trade-off between the convergence performance and computation costs.
\begin{figure}[t]
\centering
\includegraphics[width=0.25\textwidth]{Figures/shared_to_diverse.pdf}
\caption{
To spawn multiple diverse experts from the shared-expert, \ourmethods{} adopt the random masking technique. Specifically,
part of the shared expert's weight are masked as 0.}
\label{fig:shared_to_diverse}
\end{figure}
\section{Methods}
\label{sec:methods}
The observations in Section~\ref{sec:motivation} motivates EvoMoE, a two-phase framework that gradually and adaptively training MoE-based models, which is different from existing methods that jointly train the gate network and the expert network over a pre-defined sparse (e.g., Top-1 or Top-2) gate and a series of randomly initialized experts.
As shown in Figure~\ref{fig:overview}, EvoMoE contains two phases: an expert-diversify phase and a gate-sparsify phase.
In the expert-diversify phase, \ourmethods{} shares the weights among experts in one MoE layer for several training steps and then makes experts diverse by randomly masking.
In the gate-sparsify phase, \ourmethods{} introduces the dense-to-sparse (i.e., DTS) gate, which begins routing as a dense gate that routes tokens to all experts and then adaptively learns the weights of routing to each expert and gradually anneals to standard Top-1 gating.
\subsection{Problem Formulation}
Given an input token $x_s$, a series of experts $\{e_1,..., e_N\}$ and a learn-able gate with parameter $W_g$, \texttt{Func} is adopted by the gate network to determine the targeted experts for it, i.e., the token-to-expert assignment, formulated in Equation~\ref{equ:gate_problem}. $g(x_s)$ is a 1$\times$N vector, which represents the scores of $x_s$ with respect to experts.
Meanwhile, each expert will process the input token separately as $e_i(x_s)$ and combine their output as Equation~\ref{equ:moe_weighted_sum_problem}.
\begin{equation}
g(x_s) = \texttt{Func}(x_{s} \cdot W_{g})
\label{equ:gate_problem}
\end{equation}
\begin{equation}
y_{s} = \sum_{i=1}^{N}g(x_{s})_{i} \cdot e_{i}(x_{s})
\label{equ:moe_weighted_sum_problem}
\end{equation}
Existing work adopts a pre-defined Top-K as $Func$, such as Top-2 for GShard ~\cite{DBLP:gshard} and Top-1 for Switch-Transformer~\cite{DBLP:switch}. However, due to the non-derivability of Top-K, only the selected experts would back-propagate their gradients to the gate network and update their corresponding columns in $W_g$.
For example, only 1 expert is selected and 1 column of the gate would be updated in Switch-Transformer. So it is hard for Top-K gate to optimize this \textbf{expert-selection} problem.
Moreover, as observed in Figure~\ref{fig:duration_distribution}, the loads of experts are extremely imbalanced at the early stage of training and thus most GPUs suffer from low utilization due to stragglers in expert parallelism.
\setlength{\textfloatsep}{0.1cm}
\begin{algorithm}[t]
\SetAlgoLined
\SetKwProg{Fn}{Function}{}{end}
\KwData{$x_{S}$: a group of tokens of size $S$, \\
\quad \quad $E$: expert network \\
\quad \quad $T_{S}$: number of iterations for shared-expert, \\
\quad \quad $T_{D}$: number of iterations for dense-gate, \\
\quad \quad $T$: number of training iterations.}
\SetInd{0.61em}{0.61em}
\For{$i \leftarrow 1\ to\ T_{S}$}{
$y_{S} \leftarrow e(x_{S})$ \;
}
// Diversify experts from the shared \;
\For{$e_{i} \in E$}{
$e_{i} \leftarrow diversify(e, i)$ \;
}
\For{$i \leftarrow T_{S}\ to\ T$}{
$\tau \leftarrow \text{\textit{temperature scheduler}}(i)$ \;
// Get selected expert ids and weights for each token \;
$G_{S,\ E}, Id_{S} \leftarrow DTS\_Gate(x_{S},\ \tau, T_{D})$\;
\For{$s \leftarrow 1\ to\ S$}{
$y_{s} \leftarrow 0$
\For{$id \in id_{s}$}{
$y_{s} \leftarrow y_{s} + G_{s, \ id} * e_{id}(x_{s})$ \;
}
}
}
\caption{Training MoE in the {EvoMoE}\xspace{} Framework}
\label{alg:std_algo}
\end{algorithm}
\subsection{Stage 1: Expert-Diversify}
As the gate network and expert network are both randomly initialized, it requires a vast amount of computation budget for trial and errors, which is inefficient for models' training. Based on the observation in Section~\ref{sec:motivation} that most tokens are processed by the same expert and other experts waste their computation budget, we train one shared-expert instead of $N$ individual experts in the early stage (illustrated as the left of Figure~\ref{fig:overview}). Because all experts within the same MoE layer share weights, the model is equal to its corresponding non-MoE model as a small dense model.
Algorithm~\ref{alg:std_algo} illustrates the MoE training process in our \ourmethods{} framework. First, input tokens are processed by the shared expert $e_{0}$ (line 1-2). Then \ourmethods{} switches the training into standard MoE models' training, by adding a gate network at each MoE layer and diversifying all experts from the shared expert (line 4-5). After this expert-diversify phase, \ourmethods{} steps into the gate-sparsify phase, where it schedules the gate temperature coefficients and then obtains the token-to-expert routing relation from DTS-gate (line 7-9). Tokens will be dispatched to corresponding experts and aggregated together by weighted sum operating (line 10-12).
Multiple $diversify$ techniques can be adopted to spawn multiple diverse experts from one expert, such as noise, NAS, random mask.
\ourmethods{} adopts the random mask, which masks part of the shared expert's weights as 0 (shown as Figure~\ref{fig:shared_to_diverse}). For example, expert-1 is initialized by masking the central value. The proposed expert-diversify stage avoids joint training from scratch and the well-trained diverse experts could be treated as a better initialization to benefit the following model convergence.
\subsection{Stage 2: Gate-Sparsify}
Although sparse gating has demonstrated its superior model efficiency in both training and inference, prior work tends to convergence to a sub-optimal model under the fixed computation budget or the dataset size due to the jointly training of the randomly initialized gate network and expert network.
In this paper, we propose a new mechanism for training the gate network, named \texttt{Dense-to-Sparse} gate (DTS-Gate, as illustrated in Algorithm~\ref{alg:dts_algo} ), which starts as a dense gate that routes tokens to most experts and then gradually becomes sparser.
DTS-Gate benefits from the sufficient training of experts in the early stage and then make the experts selection becomes sparser on the basis of specialized experts.
This dense-to-sparse process only occupies a small fraction compared with the total training time, which usually takes days to weeks.
\textbf{Gate with Temperature:}
In order to control the sparsity during training, we adopt the softmax temperature to adjust the weights distribution among experts. Formulated as Equation~\ref{equ:gumbel_softmax}, $W_{g}$ is the parameter of gate, $\zeta$ is the extra noise and sampled from \texttt{Gumbel}$(0, 1)$ distribution~\citep{DBLP:gumbel_softmax}, and $\tau$ is the softmax temperature which controls the distribution.
When the $\tau$ increases, the distribution of $g'(x_{s})$ becomes more uniform, which evolves more experts into the computation of each token.
As the $\tau$ approaching 0, the distribution becomes one-hot, which is more confident for the gate network.
\begin{equation}
g'(x_{s}) = \frac {e^{(x_{s} \cdot W_{g}+\zeta)/\tau }}{\sum _ {s'=1}^ {N}e^{(x_{s'} \cdot W_{g}+\zeta)/\tau }}
\label{equ:gumbel_softmax}
\end{equation}
\textbf{Content-based Sparsity:}
Different from existing static Top-K based gate~\cite{DBLP:switch}~\cite{DBLP:gshard}, \ourmethods{} adopts the \textit{content-based sparsity} method to determine the number of activated experts, which keeps the value beyond an threshold $c$. As formulated by Equation~\ref{equ:adaptive_sparsity}, we drop the experts whose weights fall below the threshold $c$ and no extra communication or computation will be wasted.
It's worth noting that the sum of selected experts' score can not be equal to 1 because we don't normalize them after dropping. It is useful to remain the original score, especially only one expert is selected, which was verified in Switch. To meet the demand of above two designs, we enable each expert with this content-based gate to make them well specialized. transformer~\cite{DBLP:switch}.
\begin{equation}
g(x_{s})_i = \begin{cases}
g'(x_{s})_i, &\text{if}\quad {g(x_{s})_i} > c \\
0, &\text{else}
\end{cases}
\label{equ:adaptive_sparsity}
\end{equation}
\textbf{Sparsity Scheduler:}
With temperature $\tau$ increasing, the distribution tends to be uniform and more experts will be selected. So the sparsity decreases and the training cost of the neural network would increases.
On the opposite, less experts are involved into computation and thus the sparsity increases.
By scheduling the temperature of Equation~\ref{equ:gumbel_softmax}, we can control the sparsity of the MoE layer over different training stages. There is a trade-off between model quality and training cost for the selection of temperature.
For example, when the distribution of experts is nearly one-hot, it would lead to large variance of gradients between experts and thus make the learning of MoE learning difficult. To optimize this problem, our DTS-Gate starts at a large temperature that routes tokens to most experts and then anneals to a small temperature that gradually sparsifies the MoE layer.
\textbf{Balance Loss:} Similar to Switch transformer~\cite{DBLP:switch}, we utilize the balance loss $\mathcal{L}_{balance}$ to avoid imbalanced assignments for different experts which would cause the straggler problem and thus lead to low training efficiency.
\begin{equation}
\mathcal{L}_{balance} = \alpha N \sum_{i=1}^N(\frac{\sum_{x_s \in \mathcal{B}}{\mathbb{I}\{g(x_s)_i} > 0\}}{|\mathcal{B}|^2} \sum_{x_s \in \mathcal{B}}{g'(x_s)_i})
\label{equ:balance_loss}
\end{equation}
\noindent As formulated in Equation~\ref{equ:balance_loss}, $\alpha$ is a hyper-parameter and $\mathcal{B}$ represents current batch of tokens. $\sum_{x_s \in \mathcal{B}}{\mathbb{I}\{g(x_s)_i} > 0\}$ represents the number of tokens dispatch to expert $i$ and $\sum_{x_s \in \mathcal{B}}{g'(x_s)_i}$ represents the gate probability allocated for expert $i$. Intuitively, the balance loss will reduce the amount of data for overloaded-experts and move towards to balanced loads at the batch data level.
\textbf{Training Objective:}
In the first stage, the experts of each MoE layer share same weights and thus the loads can be divided to them equally. The training objective is to optimize the model quality (i.e., $\mathcal{L}_{task}$). In the second stage, both the model quality and training efficiency (i.e., balanced workloads between experts) need to be considered.
\begin{equation}
\mathcal{L} = \begin{cases}
\mathcal{L}_{task}, &\text{if}\quad $stage = 1$ \\
\mathcal{L}_{task} + \mathcal{L}_{balance}, &\text{else}
\end{cases}
\label{equ:train_obj}
\end{equation}
\setlength{\textfloatsep}{0.1cm}
\begin{algorithm}[t]
\SetAlgoLined
\SetKwProg{Fn}{Function}{}{end}
\KwData{$x_{S}$: a group of tokens of size $S$, $\tau$: temperature,\\ \quad\quad $T_{D}$: number of iterations of dense-gate.}
\KwResult{$G_{S,\ E}$: group combine weights, $Id_{S}$: Index of selected experts}
\SetInd{0.61em}{0.61em}
\Fn{DTS\_Gate($x_{S},\ \tau$, $T_{D}$):}{
$g_{S,\ E} \leftarrow gumbel\_softmax(x_{S} \cdot W_{g},\ \tau)$ \;
\If{current\_iteration $<$ $T_{D}$}{
//select experts for token, N $\geq$ len($id_{s}$) $\geq$1 \;
$Id_{S} \leftarrow select\ g_{S,\ E} \textgreater threshold$ \;
}
\Else{
//select Top-1 expert for token, len($id_{s}$) = 1 \;
$Id_{S} \leftarrow select\ Top1(g_{S,\ E})$\;
}
\For{$s \leftarrow 1\ to\ S$}{
\For{$id \in ids$}{
$G_{s,\ id} \leftarrow g_{s,\ id}$ \;
}
}
Return $Id_{S}, \ G_{S,\ E}$ \;
}
\caption{Dense-to-Sparse Gate Mechanism}
\label{alg:dts_algo}
\end{algorithm}
\section{Implementation}
\label{sec:impl}
\input{implementation}
\section{Experiments}
\label{sec:experiment}
\input{experiment}
\section{Related Work}
\label{sec:related_work}
\input{related_work}
\section{Conclusion and Future Work}
\label{sec:conclusion}
MoE models suffer from the training efficiency challenge due to the difficulty of training many experts and the gate network jointly.
In this work, we presented an MoE training framework {EvoMoE}\xspace that decouples the training of experts and the gate network by first spawning multiple diverse experts from one single well-trained base expert and then learning a increasingly sparse gate from a dense gate.
Our evaluations show that {EvoMoE}\xspace can not only achieve better model quality in Transformers with given computation budget but also achieve better FLOPs-efficiency when comparing with previous works in MoE training.
On the other hand, {EvoMoE}\xspace opens challenges for system execution due to the computation in the early stage and the adaptive capacity of experts.
In the future, we would like to design and implement system-level optimizations to achieve efficient training in both model quality and system execution.
\balance
\subsection{Experimental Setup}
\subsubsection{Machine Environment}
We conduct experiments on DGX-A100, where each server is equipped 2 AMD CPUs and 8 NVIDIA Ampere A100 (40GB) GPUs, with Ubuntu 20.04, CUDA 11.3, CuDNN 8.2.0 and NCCL 2.12.7. GPUs inside a server are connected via NVLink 3.0
with and servers are connected with 8 InfiniBand NICs via 8*200 Gbps bandwidth totally. The RDMA is used by default and the PyTorch version is 1.11. \\ \\
\subsubsection{Baselines}
To verify the effectiveness of our method, we compare it with the several representative baselines, including Switch-Transformer~\cite{DBLP:switch}, BASELayer~\cite{DBLP:conf/icml/baselayer}, HashLayer~\cite{roller2021hash}, DSelectK~\cite{DBLP:conf/nips/dselectk}
and StableMoE~\cite{DBLP:conf/acl/stablemoe}.
Switch-Transformer~\cite{DBLP:switch} proposed to adopt Top-1 gate for the training of large-scale models.
BASELayer~\cite{lewis2021base} formulates the token-expert routing as a linear assignment problem and guarantees balanced compute loads by employing numerous algorithms.
HashLayer~\cite{roller2021hash} replaces the gating network with a hash-based routing strategy (e.g., random hash, clustered hash). DSelectK~\cite{DBLP:conf/nips/dselectk} involves sparse gates (Top-K) in the multi-gate MoE (i.e., MMoE) architecture for better parameter sharing among different tasks and trains gates from dense to sparse for smoothness. StableMoE~\cite{DBLP:conf/acl/stablemoe} also proposed two training stages, which learn the gate as well as distill it into a lightweight one in the first stage, and freezes the parameter of gate for stable routing in the second stage.
Our {EvoMoE}\xspace mainly contains two phases: an \texttt{expert-diversify}
phase to spawn multiple diverse experts from one single well-trained base expert,
and a \texttt{gate-sparsify} phase that gradually and adaptively learns a increasingly sparse gate from a dense gate.
\subsubsection{Benchmark and Datasets}
We evaluate {EvoMoE}\xspace{} on three popular tasks, including the machine translation (MT) task for domain-specific models, the Masked Language Modeling (MLM) task and the language modeling (LM) task for pre-trained models.
We adopt standard Transformer architecture~\cite{vaswani2017attention} (Encoder-Decoder)
for the MT task and train models on four popular translation datasets, WMT17 (English to German/German to English)\footnote{https://www.statmt.org/wmt17/}, and WMT14 (English to French/French to English)\footnote{https://www.statmt.org/wmt14/}. BLEU scores of the test sets are reported for comparison.
We adopt RoBERTa architecture architecture~\cite{liu2019roberta} (Encoder-only)
for the MLM task and train models on the combination of datasets, including Wikipedia\footnote{https://dumps.wikimedia.org/enwiki/latest/enwiki-latest-abstract.xml.gz}, BooksCorpus\footnote{https://battle.shawwn.com/sdb/books1/books1.tar.gz}, OpenWebText\footnote{https://zenodo.org/record/3834942/files/openwebtext.tar.xz} and CC-100\footnote{https://data.statmt.org/cc-100/}. Moreover, these datasets are tokenized by byte-pair encoding with a vocabulary size of 50257. Models are validated on the famous General Language Understanding Evaluation(GLUE) benchmark~\cite{DBLP:conf/emnlp/glue} for comparison.
We adopt GPT architecture architecture~\cite{gpt2} (Decoder-only)
for the LM task and train models on OpenWebText as ~\citet{gpt2}. We report train/valid/test perplexity (PPL) for comparison.
We also report the inference FLOPs of each model, which represents the speed of deploying this model at industry. All the training data are downloaded and pre-processed by following the example scripts from Fairseq\footnote{https://github.com/facebookresearch/fairseq/tree/main/examples}.
\subsubsection{Hyper-Parameter Detail}
We sparsely scale these models by replacing every other the \textit{feed-forward} layer (FFN) with MoE-FFN Layer, which contains a series of FFN experts.
All models use the GeLU activation functions~\cite{gelu_activation}, polynomial learning rate scheduler and Adam optimizer~\cite{DBLP:journals/corr/adam_optim}, where $\beta_1 = 0.9$ and $\beta_2 = 0.98$ .
We set clip norm as 0.0, weight decay as 0.1 and dropout rate as 0.1.
We use CrossEntropy as the criterion and utilize the label smoothed technique with coefficient of 0.1 for the MT task.
The coefficient of balance loss is set as 0.1 in Switch-Transformer~\cite{DBLP:switch}, StableMoE~\cite{DBLP:conf/acl/stablemoe} and our EvoMoE.
We set the threshold $c$ of our dense-to-sparse gate as 0.001 over training steps, which determines how large the expert's weight is important and is a trade-off between training cost and model quality from our point of view.
\begin{table*}[t]
\caption{Perplexity results of language modeling task.}
\begin{center}
\begin{tabular}{l|cccc|c}
\toprule
\textbf{Models} & \textbf{\#Shared Params.} & \textbf{\#Expert}& \textbf{\#Expert Params.} & \textbf{FLOPs} &
\textbf{Perplexity($\downarrow$)} \\
\midrule
Standard TRM& 345M& - & - & 207B& 15.14\\
Larger TRM (wider)& 360M & - & - & 220B& 14.92\\
\midrule
Switch TRM & 249M& 192& 1536M& 220B& 13.12\\
BASE Layer & 249M& 192& 1536M& 220B& 12.45\\
Hash Layer& 249M& 192& 1536M& 220B& 12.87\\
StableMoE & 249M& 192& 1536M&220B & 12.91\\
EvoMoE & 249M& 192& 1536M& 220B& \textbf{12.24}\\
\bottomrule
\end{tabular}
\end{center}
\label{tab:gpt_lm}
\end{table*}
\begin{table}[t]
\caption{BLEU score on each machine translation datasets}
\begin{center}
\begin{tabular}{c|cccc}
\toprule
\textbf{Models}
& \textbf{En-De} & \textbf{De-En} & \textbf{En-Fr} & \textbf{Fr-En}\\
\midrule
{TRM-Base}
& {28.1} &{34.8}
& {39.2} &{38.1}\\
{Switch-TRM}
& {28.4} &{34.6}
& {39.1} &{38.2} \\
\midrule
{{EvoMoE}\xspace{}}
& {\textbf{29.6}} &{\textbf{36.7}}
& {\textbf{40.3}} &{39.2} \\
{\quad\quad\quad\quad w/o \ stage\ 1}
& {\textbf{29.6}} &{36.5}
& {40.2} &{\textbf{39.3}} \\
{\quad\quad\quad\quad w/o \ stage\ 2}
& {28.7} &{35.2}
& {39.4} &{38.3} \\
\bottomrule
\end{tabular}
\end{center}
\label{tab:trans_tasks}
\end{table}
\subsection{GLUE Results}
\textbf{Model Architecture:} We pretrain the representative RoBERTa model for the masked language modeling task, where we set standard Transformer(TRM) with 24 encoder layers, hidden dimension as 1024 and number of attention heads as 16. We replace every other FFN layer in standard Transformer with the MoE layer (16 experts per layers) to construct the MoE models. The standard Transformer is a dense model and contains 355M parameters totally, whose inference FLOPs is 207B. Meanwhile, the sparse MoE model contains 1759M parameters totally, including 259M parameters for shared backbone and 1536M parameters for the expert network. In our setting that only 1 expert is active at a time, each input token will activate 335M parameters of the sparse MoE models, which is the same as standard Transformer model except for the gate network. To exactly match the inference speed (FLOPs) of MoE models, we slightly increase the FFN hidden size of standard TRM to construct the larger TRM.
\textbf{Model Performance:}
We pretrained each model for 100k steps totally, 5k of which was the warm-up phase. For our \ourmethods{}, we scheduled the first 5k steps as the expert-diversify stage and the following 5k steps for annealing temperature from 2.0 to 0.3. After the pre-training stage, we finetune the pre-trained models on each GLUE task and summarized the results in Table~\ref{tab:roberta_glue}. As for RTE, we finetune it starting from the MNLI model rather than the pretrained model as ~\citet{liu2019roberta}.
Compared with other baselines, \ourmethods{} achieves state-of-the-art results on 7 out of 8 tasks and the best averaged score. The MoE model is constructed by adding the gate network and replacing the original FFN layer of standard Transformer, which increases its model size and thus enlarges its capacity. Thus all MoE models outperform their backbone model (standard TRM), e.g., 89.750 for standard TRM and 90.313 (+ 0.563) for Switch TRM with respect to the avg score. Larger TRM slightly outperforms standard TRM because of its large model size. As verified by ~\citet{DBLP:scaling_laws}, larger models tend to be more sample-efficient, which represents better model quality with fixed training data/steps.
Compared with other MoE methods, \ourmethods{} benefits from the sufficient training of experts in the early stage and then make the experts selection becomes sparser on the basis of specialized experts.
Specifically, \ourmethods{} outperforms other MoE methods on GLUE benchmark up to 0.562 and 0.403 on average.
Switch TRM~\cite{DBLP:switch} jointly trains the randomly initialized experts and gates, which aims to learn better parameter as well as balanced routing. It is hard to optimize them simultaneously and thus performs bad among MoE models. To alleviate this problem, StableMoE~\cite{DBLP:conf/acl/stablemoe} freezes the parameter of gate network after the early training stage and improves over Switch-TRM. Hash Layer~\cite{roller2021hash} utilizes the fixed hash strategy to route tokens, which is based on the input embedding. Because both the hash strategy and input embedding is fixed, Hash Layers only need to learn the parameter of experts. However, it may lead to sub-optimal because the hash strategy is selected based on human knowledge and may be inappropriate. BASE Layer~\cite{DBLP:conf/icml/baselayer} enforces a balanced token-to-expert assignment through a linear assignment problem, which simplify the training in another way. All these work find the problem of jointly training and targeting at alleviate it.
\subsection{Language Modeling Results}
\textbf{Model Architecture:} We pretrain the representative GPT model for the language modeling task, where we set standard Transformer(TRM) with 24 decoder layers, hidden dimension as 1024 and number of attention heads as 16.
Every other FFN layer is replaced by in standard Transformer with the MoE layer (16 experts per layers) to construct the MoE models. There totally exists 12 MoE layers and thus 192 experts (i.e., $12 \times 16$).
Meanwhile, larger TRM is scaled by increasing its FFN hidden size.
\textbf{Model Performance:}
We pretrained each model on the OpenWebText dataset for 200k steps totally, 10k of which was the warm-up phase. For our \ourmethods{}, we scheduled the first 10k steps as the expert-diversify stage and the following 5k steps for annealing temperature from 2.0 to 0.3. We report the perplexity on the test set. Results are summarized in Table~\ref{tab:gpt_lm}.
Compared with other baselines, our \ourmethods{} achieves the best result among all baselines. Specifically, the perplexity of \ourmethods{} is 12.24, which achieves a 2.90 improvement compared with 15.14 of standard TRM. Meanwhile, all MoE models outperform their backbone model (standard TRM) because of their large model capacity. Larger TRM slightly outperforms standard TRM because of its large model size, which demonstrates the sample-efficient of large models.
Compared with other MoE methods, \ourmethods{} benefits from the sufficient training of experts in the early stage and then make the experts selection becomes sparser on the basis of specialized experts. Specifically, \ourmethods{} outperforms other MoE methods up to 0.88 ppl and 0.545 ppl on average. The analysis between different methods is same as that in GLUE results.
\begin{table}[t]
\caption{BLEU performance of MoE models with different expert number. }
\begin{center}
\begin{tabular}{c|ccc}
\toprule
{} & \multicolumn{3}{c}{\textbf{Number\ of\ Experts }} \\
&{\textbf{4}} & {\textbf{8}} &{\textbf{16}}\\
\midrule
{Switch}
&{28.4}
&{28.6} &{28.7}\\
{\ourmethods{}}
&\textbf{29.6} &\textbf{29.9}
&{30.0} \\
{\quad\quad\quad\quad w/o \ stage\ 1}
&\textbf{29.6} &\textbf{29.9}
&\textbf{30.1} \\
{\quad\quad\quad\quad w/o \ stage\ 2}
&{28.7} &{28.9}
&{28.9} \\
\bottomrule
\end{tabular}
\end{center}
\label{tab:expert_number}
\end{table}
\begin{table}[t]
\caption{Efficiency of MoE models with different expert number, and the results are normalized over Switch.}
\begin{center}
\begin{tabular}{c|ccc}
\toprule
{} & \multicolumn{3}{c}{\textbf{Number\ of\ Experts }} \\
&{\textbf{4}} & {\textbf{8}} &{\textbf{16}}\\
\midrule
{Switch}
&{1}
&{1} &{1}\\
{\ourmethods{}}
&\textbf{0.82}
&\textbf{0.78} &\textbf{0.75} \\
{\quad\quad\quad\quad w/o \ stage\ 1}
&{0.86}
&{0.82} &{0.81} \\
{\quad\quad\quad\quad w/o \ stage\ 2}
&{0.95}
&{0.93} &{0.92} \\
\bottomrule
\end{tabular}
\end{center}
\label{tab:efficiency}
\end{table}
\begin{figure*}[t]
\begin{center}
\subfigure[Validation PPL over steps]{
\includegraphics[width=0.35\textwidth]{Figures/e2e_iteration.pdf}
\label{fig:iters_ppl} }
\subfigure[Validation PPL over FLOPs]{
\includegraphics[width=0.35\textwidth]{Figures/e2e_flops.pdf}
\label{fig:flops_ppl}
}
\caption{End-to-end performance comparison between GPT-ori, GPT-Switch and GPT-DTS. Figure~\ref{fig:iters_ppl} and Figure~\ref{fig:flops_ppl} represent the curve of PPL over iterations and FLOPs, where GPT-DTS can obtain $2.0$x speed-up to reach the same validation perplexity, as well as higher FLOPs-efficiency of a $1.42$x speed-up.
}
\label{fig:end2end}
\end{center}
\end{figure*}
\subsection{Machine Translation Results}
\textbf{Model Architecture:} We pretrain the representative T5 model for the machine translation task, where we set standard Transformer(TRM) with 12 encoder-decoder layers, hidden dimension as 768 and number of attention heads as 12.
Every other FFN layer is replaced by in standard Transformer with the MoE layer (4 experts per layers) to construct the MoE models.
\textbf{Model Performance:}
We compare {EvoMoE}\xspace{} with Transformer and Switch-Transformer on four language-pair datasets, including English to German, German to English, English to French and French to English. Results are shown in Table~\ref{tab:trans_tasks},~\ref{tab:expert_number},~\ref{tab:efficiency}.
We remark that these models all have the same inference speed even if MoE models enlarge the parameter size.
We show the BLEU score on the test set of each dataset in Table~\ref{tab:trans_tasks}.
\ourmethods{} outperforms other methods by 1 BLEU score on average.
Although Switch-Transformer scale the model size, it still achieves a similar performance with Transformer-Base, which is parameter-efficient.
Table~\ref{tab:expert_number} shows the BLEU performance of different expert number on the English-German datasets. \ourmethods{} can still outperform the Switch-Transformer about 1.3 BLEU with the increasing number of experts. Because of the datasets' quality, the effect of increasing expert number is limited.
\textbf{Model-Efficency:} Table~\ref{tab:efficiency} show the model efficiency of \ourmethods{} and Switch Gate on the English-German datasets. \ourmethods{} is efficient at the speed of converge. For example, \ourmethods{} need only $75\%$ compute budget of Switch-Transformer to reach the same PPL. It is worth noting that the speedup over Switch-Transformer improves as the expert number increases.
\textbf{Ablation Study:}
We present a ablation study on \ourmethods{} to show the influence of two stages by removing the expert-diversify phase and the gate-sparsify phase respectively. Results are summarized in Table~\ref{tab:trans_tasks}~\ref{tab:expert_number}~\ref{tab:efficiency}.
As for the model performance metric, it will lead to performance degradation when \ourmethods{} removes the gate-sparsify stage, such as 38.3/39.2 in Fr-En of Table~\ref{tab:trans_tasks}.
Meanwhile, it is worth noting that influence is little as for w/ and w/o the expert-diversify stage, which encourages us to involve this stage for saving computation budget.
As for the FLOPs-efficiency metric, the gate-sparsify phase can improve the FLOPs-efficiency by $17\%$. By introducing the expert-diversify stage, \ourmethods{} can obtain an extra $4\%$ improvement.
In summary, the gate-sparsify phase can both improve the model performance and FLOPs-efficiency significantly and the expert-diversify phase can introduce extra FLOPs-efficiency without performance degradation. In the following sections, we will detail analyze the gate-sparsify phase and evaluate it at large scale.
\subsection{Breakdown on the Gate-Sparsify Phase}
\textbf{Model Architecture:} We pretrain the representative GPT model for the language modeling task, where we set GPT-ori with 24 decoder layers, hidden dimension as 1024 and number of attention heads as 16.
Every other FFN layer is replaced by in standard Transformer with the MoE layer (16 experts per layers) to construct the GPT-MoE model.
GPT-Switch represents training MoE models with Switch gate, which keeps Top-1 selection.
GPT-DTS represents training MoE models with the \texttt{dense-to-sparse} gate, which starts as a dense gate that routes tokens to most experts and then gradually becomes sparser.
We compare the required FLOPs to train models to show the FLOPs-efficiency of different methods.
The FLOPs-efficiency is defined as the best model performance (PPL) can be achieved given the fixed number of floating-point operations (computation budget).
Because the actual training time could be affected by the system engineering efforts on the implementation details, which are not our focus in this approach. Instead, in our experiments, we prefer to choose the computation complexity for fair comparisons.
\begin{figure*}[t]
\centering
\subfigure[More Experts]{
\includegraphics[width=0.33\textwidth]{Figures/expert_scalability_flops.pdf}
\label{fig:expert_scalability}
}
\subfigure[More MoE Layers]{
\includegraphics[width=0.33\textwidth]{Figures/moe_layer_scalability_flops.pdf}
\label{fig:moe_layer_scalability}
}
\caption{Scalability for DTS gate. It shows that more experts or more MoE-layers (larger models with constant FLOPs), will lead to better FLOPs-efficiency.
}
\label{fig:scalability}
\end{figure*}
\textbf{Model Performance:}
We pretrained each model on the OpenWebText dataset for 300k steps totally, 10k of which was the warm-up phase. For our \ourmethods{}, we scheduled the first 10k steps for annealing temperature from 2.0 to 0.3. We report the perplexity on the validation set. Results are shown in Figure~\ref{fig:end2end}.
To improve the computation efficiency, only part parameters are used for each token in sparse models with the cost of model performance.
DTS-Gate aims to shift the model training from dense to sparse, and keep the inference cost same as sparse models.
Experiments show that compared with the state-of-the-art Switch-Transformer in GPT-MoE model with OpenWebText dataset, GPT-DTS can obtain
2.0x speed-up to reach the same validation perplexity (Figure~\ref{fig:iters_ppl}), as well as higher FLOPs-efficiency of a 1.42x speed-up (Figure~\ref{fig:flops_ppl}).
Experiments also verify the ability of \texttt{dense-to-sparse} gate for scaling models with more experts or more MoE layers.
Comparison with Sparse Models.
MoE-Switch pre-defines its static Top-1 gating network and jointly training the gate and experts networks. Different from GPT-Switch, GPT-DTS utilizes temperature to adjust the distribution of the token-to-experts (one-hot or uniform) and threshold to remove computation of experts with low weights. \ourmethods{} performs better than GPT-Switch in sample-efficiency because of more experts involved in training and updates at the beginning, shown as Figure~\ref{fig:iters_ppl}. As for FLOPs-efficiency, DTS-Gate first involves more experts into warm-up training, which is poor FLOPs-efficency. But with the training going on, GPT-DTS can obtain greater than $25\%$ improvements in FLOPs-efficiency compared with the state-of-the-art Switch-Transformer in GPT-MoE model with OpenWebText dataset.
\subsection{Scalability}
In this subsection, we investigate different experiment settings to validate the scalability of our DTS-Gate.
\textbf{Model Architecture:} We choose the GPT-small as the backbone model for the language modeling task, where we set the model with 12 decoder layers, hidden dimension as 768 and number of attention heads as 12.
\textbf{Increase the Expert Number:} Based on GPT-Small model with 117M parameters, we replace the 7-th FFN layer by one MoE layer and vary its experts number within $\{1, 4, 8, 16\}$.
As shown by Figure~\ref{fig:expert_scalability}, with increasing expert numbers, \ourmethods{} keeps consistent improvements (i.e., lower PPL) during training.
\textbf{Increase the MoE layer number:} Similarly, we also vary the number of MoE layers to validate the performance of DTS gate.
We insert $k$ MoE layers in GPT-Small, where $k \in \{0, 1, 4, 6\}$ and each MoE layer contains 8 experts.
Figure~\ref{fig:moe_layer_scalability} shows that by increasing MoE layers, \ourmethods{} can achieve better model performance with same FLOPs.
\subsection{Effect of Sparsity Scheduler}
It is worth noting that several hyper-parameters are introduced in Dense-To-Sparse gate, such as max/min temperature and decay iterations.
In this section, we analyze the effect of different hyper-parameter setting by conducting experiments of various settings. The training model is GPT-MoE, 24-layer decoder with 12 MoE-layer (16 experts per layer) and the dataset is OpenWebText.
We decay temperature $\tau$ from $max\_value$ to $min\_value$ in the first 15000 iterations and switch to Top1 then. Experiments with different $max\_value$ to $min\_value$ are evaluated, and the results are shown in Figure~\ref{fig:scheduler}.
\begin{figure}[t]
\centering
\includegraphics[width=0.35\textwidth]{Figures/temperature_scheduler_flops.pdf}
\caption{Effect of different temperature scheduler.}
\label{fig:scheduler}
\end{figure}
\textbf{Max/Min Temperature} Under small temperatures, the weight distribution of experts is close to one-hot, which leads to the one-token-one-expert distribution and low training cost,
but the variance of gradients is large.
In contrast, large temperatures result in nearly uniform distribution gate weights,
which evolves more experts into training but the variance of gradients is small. As shown in Figure~\ref{fig:scheduler}, we find the these two hyper-parameters have low influence on the model quality under the same training budget, except for the extrame value, e.g., 1.0 for $max\_value$ and 0.1 for $min\_value$.
\begin{table}[t]
\caption{The most frequent tokens assigned to each expert in the validation set, which shows that some experts assignment decisions are made based on local contexts. For many other experts, the assignment decision depends on longer context, and is harder to visualize.}
\begin{center}
\begin{tabular}{c|ccccc|c}
\toprule
\textbf{Expert} & \multicolumn{5}{c|}{\textbf{Top5 Proceeding Tokens}} & \textbf{Descriptions} \\
\midrule
1 & is & was & be & are & have & auxiliary verbs \\
3 & .& \textbackslash n & /& (&; & punctuations \\
4 & in& , & of& and& from & prepositions \\
6 & and& I & it& that& they & possessive cases\\
12 & out& up & go& back& down & directional prepositions \\
\bottomrule
\end{tabular}
\end{center}
\label{tab:visual}
\end{table}
\subsection{Visualization of Expert Specialization}
We visualize the routing strategy of the pre-trained GPT model by \ourmethods{} in Table~\ref{tab:visual} through its corresponding input embedding, where each MoE layer contains 16 experts. For each expert, we present the Top5 proceeding tokens assigned and give descriptions for explanations from our points of view. For example, Expert 1 captures the auxiliary verbs and Expert 6 captures possessive cases. These experts can capture local contexts of each embedding well. For other experts, it is diff cult to visualize because of the long contexts' influence.
|
2024-02-18T23:40:24.385Z
|
2022-10-11T02:16:53.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14397","language":"en","timestamp":1665454613000,"url":"https:\/\/arxiv.org\/abs\/2112.14397","yymm":"2112"}
|
proofpile-arXiv_000-10012
|
{"provenance":"002.jsonl.gz:10013"}
| null | null |
\section{Introduction}
\label{sec:Introduction}
\input{intro}
\section{ML Model under DP}
\label{sec:formulation}
\input{formulation}
\section{DP Forward-Propagation (DP-FP)}
\label{sec:dpfp}
\input{dp-fp}
\section{Experiments}
\label{sec:Experiments}
\input{exps}
\section{Related Work}
\label{sec:RelatedWork}
\input{related}
\section{Discussion}
It is instructive to compare the types of information that DP-SGD and DP-FP protect under DP.
As illustrated in Eq.~(\ref{eqn:clipping-dpsgd}), in the DP-SGD, the label information is embedded in the gradient,
thus the data record under DP protection is each pair of training data samples and their labels.
While in the DP-FP, as illustrated in Eq.~(\ref{eqn:clipping-dpfp}), because the DP operation is performed in the forward stage, the data record under DP protection is simply the training data sample without labels.
But, it is interesting to note that for the majority of classification tasks, one only needs to protect the privacy of the data sample, as the labels themselves are finite and not constitute privacy information as long as they cannot be connected to the training data sample.
Consider the SST-2 sentence classification task as an illustration, which contains 67,500 sentences that require protection under DP.
Additionally, each sentence is labeled with ``positive'' or ``negative''.
Because DP-FP ensures that the adversary almost never recovers any sentence using the fine-tuned model, labels cannot be associated with the training sentences.
However, in DP-FP for the generation task, protecting the label information is required and difficult, and further model architecture design is required. We plan to investigate it as part of our future work.
\section{Conclusion}
In this paper, we have introduced the differentially private forward propagation (DP-FP) method for applying differential privacy to large pretrained models on classification tasks.
The key design of DP-FP exploits differential privacy's post-processing property, ensuring privacy by protecting the latent representation in the forward stage rather than the conventional wisdom of DP Stochastic Gradient Descent (DP-SGD), which protects the gradient in the backward stage.
DP-FP has the same memory cost as an off-the-shelf SGD-based optimizer, an unbiased gradient, and significantly lower noise power that scales only with the latent representation dimension, as opposed to DP-SGD, which has a large memory cost, a biased gradient, and total noise power that scales with the huge model size.
We have also created micro-batches that are unique to DP-FP in order to reduce the noise power for each coordinate.
As a result, on a large model like RoBERTa-large, DP-FP achieves an average accuracy of 91.34\% on four downstream tasks with $\epsilon$ less than 3, which is only within 0.9\% lower than the non-private baseline and 3.81\% better than the state-of-the-art DP-SGD method.
\subsection{Forward Propagation under DP}
Let $X$ denote the entire training dataset and $\text S(X)$ be the subsampling scheme used to build a batch for each step of model update,
such as shuffling and sampling with/without replacement, Poisson sampling.
The latent representation is then denoted in a composition form of $h\circ \text S(X)$
with $h(\cdot)$ being the ML model for latent representation computation, e.g., the hidden state or pooler output of \texttt{[CLS]}\footnote{In this paper, we simply use the pooler output of \texttt{[CLS]} as our choice of $h(\cdot)$, since it is straightforward to
integrate our DP at the outside of encoders for classification tasks, without changing any code inside encoders. Furthermore, it is much smaller in size than the transformer layers preceding the pooler.}.
Note that subsampling schemes provide DP amplifications~\cite{wang2019subsampled,dong2021gaussian}, which reduce the amount of noise required under the same privacy budget.
To achieve the forward propagation under DP, we first stabilize the latent representation.
Because training data is random, the output of $h\circ \text S(X)$ can vary significantly, implying that the model will vary significantly if different data records are used for training. As a result, data privacy is at risk of being compromised by a membership attack.
Thus, we constrain $h$'s output range by clipping the representation that corresponds to each data record, such as a sentence in a language model.
We clip the output of $h(\cdot)$, which shrinks the latent representation whenever its $\ell_2$ norm exceeds a certain threshold $C$, similarly to DP-SGD.
More formally, the clipped latent representation is given by
\begin{equation}
\label{eqn:clipping-dpfp}
\mathrm{Clip}\left(h(\cdot), C\right)
\triangleq
h\cdot\min\left(1, \frac{C}{\|h(\cdot)\|_2}\right).
\end{equation}
The clipping operation also implies that the greatest variation output for a pair of neighboring datasets in terms of the $\ell_2$-norm is given by
\begin{align*}
C = \underset{ X \sim X^{\prime} }{\max}||h\circ \text S(X)-h\circ\text S(X^{\prime})||_2.
\end{align*}
Following clipping, Gaussian noise is added to ensure DP, with details on noise power $\sigma^2$ calibration provided later in this subsection. Before the downstream classification layer, the latent representation is computed as follows:
\begin{equation}
\label{eqn:dp-fp}
\mathcal M(X)\triangleq \mathrm{Clip}\left(h\circ \text S(X), C\right) + \mathcal{N}\left(0, \sigma^2I_{k}\right).
\end{equation}
Since the input data is under DP according to Eq.~(\ref{eqn:dp-fp}), the model updated based on the result of Eq.~(\ref{eqn:dp-fp}) still follows the same DP assurance according to the post-processing property of DP~\cite[Proposition~2.1]{dwork2014algorithmic}.
As a result, after Eq. (\ref{eqn:dp-fp}), a conventional SGD-based optimizer, such as Adam, can be used and the privacy of $X$ is still guaranteed.
Until now, we identify the advantages of DP-FP over DP-SGD in the following propositions:
\noindent
\begin{proposition}
\label{prop:noise-dim}
\textbf{Small noise dimension in DP-FP}:
It is worth noting that $I_k$ represents a $k$-dimension identity matrix, and therefore the noise vector dimension $k$ in DP-FP is much smaller than $d$, which is the noise vector dimension in Eq.~(\ref{eq:dpsgd}) that equals to the model dimension in DP-SGD.
Take BERT for example, $d\approx$ 110M, whereas $k$ is only 768 if using the hidden state of \texttt{[CLS]} for downstream task. Thus, DP-FP saves significant total noise power than that for DP-SGD due to a significant noise dimension reduction.
\end{proposition}
\begin{proposition}
\label{prop:unbiased-g}
\textbf{ Unbiased gradient in DP-FP}: The results of Eq.~(\ref{eqn:dp-fp}) are then fed to the classifier,
which predicts label distribution, computes the loss further, and then performs standard backpropagation via an off-the-shelf optimizer such as Adam for model update.
As a result, the DP-FP backpropation inherits all of the advantages of the non-DP optimizer and produces an unbiased estimate of the true gradient.
\end{proposition}
\subsection{Micro-batch for DP Amplification\footnote{The micro-batch construction in our DP-FP is used to achieve DP amplification in order to reduce calibrated noise variance without sacrificing privacy, which is distinct from the micro-batch functionality used in Tensorflow Privacy to reduce the memory cost of the DP-SGD implementation.} in DP-FP}
Intuitively, privacy amplification by subsampling is caused by the fact that individual data record has complete privacy if it is not included in the subsample. Based on this intuition, DP-SGD benefits from batch subsampling for DP amplification \citep{li2021large, yu2021large}, which reduces the calibrated noise power for each coordinate significantly.
In contrast to DP-SGD, we show that additional DP amplification can be achieved by subsampling out $M$ micro-batches that comprise a batch, resulting in lower noise power for each coordinate of the $k$-dimension noise vector in Eq.~(\ref{eqn:dp-fp}). Because of the unique structure for DP operations in forward propagation, this DP amplification is unique to DP-FP and does not exist in DP-SGD, as discussed further below.
More concretely, an independent Bernoulli trial for all data records, i.e., sentences in a dataset, is performed to construct each micro-batch with subsampling probability $p$.
Clipping is applied to each latent representation corresponding to the input data record, i.e., clip the hidden state of \texttt{[CLS]} in the BERT model.
In the following, we evaluate the privacy cost using the Gaussian DP (GDP) framework~\citep{dong2021gaussian}, which measures the privacy profile $(\epsilon, \delta)$ in terms of $\mu = C/\sigma$ using Eq.~(\ref{eqn:compute_eps}) and (\ref{eqn:mu}).
To make our paper self-contained, we include a preliminary of GDP calculation in Appendix~\ref{app:GDP}, as well as a more detailed procedure for privacy accounting described below.
\begin{table*}
\centering
\scalebox{0.9}{
\begin{tabular}{c|c c c c }
\hline
& DP-SGD & DP-FP & RGP & SGD \\
\hline
Memory cost & $\mathcal O(md)$ & $\mathcal O(d)$ &
$\mathcal O(m r w)$ &
$\mathcal O(d)$ \\
Computational cost &
$\mathcal O(md)$ & $\mathcal O(md)$
& $
\mathcal{O}\left(m d+K r d+K r^{2}w\right)
$ & $\mathcal O(md)$ \\
Coordinates \# to add noise & $\mathcal O(d)$ & $\mathcal O(k)$ & $\mathcal O(w)$ & -- \\
Noise power at each coordinate & $\sigma^2$ & $\frac{1}{M}\sigma^2$ &$\sigma^2$ &-- \\
\hline
\end{tabular}}
\caption{For all methods, $m$ is the batch size, and $d$ is the model size. In our DP-FP, $k$ and $M$ are the latent representation dimension used for downstream tasks and micro-batch number, respectively. Note that $k\ll d$, e.g. in the experiment $k=768$ while $d=3$ Million for BERT model. Specifically for RGP in~\citet{yu2021large}, $w$ is the model width, $r$ is the reparametrization rank, and
$K$ is the number of power iterations.\protect\footnotemark{}}
\label{table:cmp}
\end{table*}
According to the subsampling DP amplification in the Gaussian DP framework~\citep{bu2020deep},
the privacy cost corresponding to each micro-batch is given by
\begin{equation}
p\cdot G_{\mu}+(1-p) \text { Id },
\end{equation}
where $
G_{\mu}$ is a function of $\mathcal{N}(0,1)$ and $\mathcal{N}(\mu, 1)$ with $\mu = C/\sigma$
and $\text { Id }(\alpha)=1-\alpha$.
The details of function $
G_{\mu}(\alpha)$ is given in the appendix.
In each step, the training stage executes $M$ micro-batches and updates $T$ steps based on the training dataset. Even if each micro-step is DP protected with a privacy cost of $(\epsilon, \delta)$, the question is whether all $T\times M$ micro-batches are private when combined, and if so, how privacy degrades as the number of steps increases, a process known as DP composition.
We have the total privacy cost
according to the central limit theorem for $T\cdot M$ rounds Gaussian DP composition given by
\begin{equation}
\label{eqn:gdp-clt-dpfp}
\mu_{\text{tot}}
= p\cdot \sqrt{T\cdot M
\left(\mathrm{e}^{(C/\sigma)^2}-1\right)},
\end{equation}
with details provided in the appendix. It is evident that the smaller $p$ is, the smaller the total privacy cost denoted by $\mu_{\text{tot}}$.
Similarly in DP-SGD~\cite{li2021large} with subsampling probability $\widetilde p$ to construct the mini-batch, the total privacy cost is given by
\begin{equation}
\label{eqn:gdp-clt-dpsgd}
\widetilde \mu_{\text{tot}}
= \widetilde p\cdot \sqrt{T
\left(\mathrm{e}^{\widetilde \mu^2}-1\right)},
\end{equation}
where $\widetilde \mu = C/\widetilde \sigma$.
To make a fair comparison, DP-FC is set to have the same batch size in expectation as DP-SGD by setting $\widetilde p=p \cdot M$.
In the strong DP regime, $\mu$ and $\widetilde \mu$ are very small positive values close to zero. Thus, by taking the Taylor series expansion of the exponential function in (\ref{eqn:gdp-clt-dpfp}) and (\ref{eqn:gdp-clt-dpsgd}), respectively and taking into account the fact that
$
\mu_{\text{tot}} = \widetilde \mu_{\text{tot}}
$ for the same privacy budget,
we obtain $$ \frac{\sigma^2}{\widetilde\sigma^{2}}= \frac{1}{{M}}.$$
As a result, we have the third advantage of DP-FP over DP-SGD:
\begin{proposition}
\label{prop:noise-power}\textbf{ As with DP-SGD, DP-FP requires less than $1/M$ noise power per coordinate.}
Note that the comparison is
under the same batch-size, privacy budget, and clipping value. However, as demonstrated in~\citet{li2021large}, larger batch sizes improve performance in their DP-SGD. As demonstrated later in the experiment, our DP-FP, on the other hand, prefers small batch sizes. Due to the fact that each batch is constructed by randomly sampling each data record, the batch size decreases as $p$ decreases. Thus, $\sigma^2$ is even smaller than $\frac{1}{M}\widetilde{\sigma}^2$ according to Eq.~(\ref{eqn:gdp-clt-dpfp}).
\end{proposition}
Finally, we summarize the DP-FP algorithm in Algorithm~\ref{alg:DP-FP}, including the micro-batch construction for DP amplification.
Since the noise power in each step is calibrated according to the
DP budget of $(\epsilon, \delta)$ and total steps $T$, $(\epsilon, \delta)$
is spent out after $T$ steps.
\begin{algorithm}[t]
\caption{DP-FP Training}
\label{alg:DP-FP}
\begin{algorithmic}[1]
\REQUIRE DP budget $(\epsilon,\delta)$, sampling rate $p$, clipping threshold $C$, and representation $h(\cdot)$.
\STATE Put $(\epsilon, \delta)$ into (\ref{eqn:compute_eps}) and compute $\mu$ as $\mu_{\text{tot}}$.
\STATE Calibrate noise power $\sigma^2$ by substituting $\mu_{\text{tot}}$, $T$, $M$, and $C$ into~(\ref{eqn:gdp-clt-dpfp}).
\FOR{$t=1, \ldots, T$}
\FOR[$\triangleright$ $M$ micro-batches in each step:]{$m=1, \ldots, M$ in parallel}
\STATE Sample micro-batch $x_m$ with Bernoulli trail for each data record
\STATE $\widetilde x_m \gets \text{Clip}\left(h(x_m);C\right) + \mathcal{N}(0, \sigma^2 I_k)$
\STATE $w\gets \text{optimizer}(\widetilde x_m, w )$ \COMMENT{$\triangleright$ SGD-based off-the-shelf optimizer for model update}
\ENDFOR
\ENDFOR
\end{algorithmic}
\end{algorithm}
\subsection{Comparison to existing methods}
In contrast to SGD, DP-SGD necessitates the computation and storage of individual gradients, resulting in batch- and model-size-dependent memory costs. Furthermore, the total noise power in DP-SGD scales linearly with model size. As a result, applying DP-SGD to large-scale models is difficult.
\citet{li2021large} propose ghost clipping, a memory-saving technique that allows clipping in DP-SGD to run without instantiating per-example gradients. By extending the method in \citet{lee2020scaling}, ghost clipping consumes nearly the same memory as non-private SGD training while doubling the computation throughput via numerical validation. As a result, we use the memory and computational cost of SGD as a reference for ghost clipping.
We also compare the costs of the most recent memory efficient method, reparameterized gradient perturbation (RGP)~\citep{yu2021large}. While RGP is faster per update, it requires more than three times as many epochs as the ghost clipping method as pointed out by~\citet{li2021large}.
\footnotetext{We use slightly different symbol notion for complexity analysis from that in~\cite{yu2021large} without assuming the weight matrix is square as that in ~\cite{yu2021large}.}
\subsection{Data and Settings}
Following \newcite{li2021large}, we mainly focus on fine-tuning large pretrained models on classification tasks,
including MNLI, QQP, QNLI, and SST-2 from the GLUE benchmark~\cite{wang2018glue} that each has more than 60k training instances.
We provide the data statistics for the four tasks in Appendix~\ref{app:statistics}.
Our non differential privacy (Non-DP) baselines are finetuned BERT-base, RoBERTa-base, and RoBERTa-large.
For classification task, following common settings, we add a linear layer over the output of the pooler layer for encoder,
the pooler layer simply takes the hidden state of \texttt{[CLS]} token as input, and applies another linear dense layer and a $tanh(\cdot)$ activation.
We train our baselines on four data sets for 5 epochs with a learning rate $2\times 10^{-5}$, a batch size $32$, and a max input length $128$.
We save and validate each model for each epoch, and report the best accuracy scores as our baseline systems.
During the fine-tuning stage, our DP-FP method adds noise and clipping operations (as in Algorithm~\ref{alg:DP-FP}) before the linear classification layer based on the following two facts.
First this approach treats pretrained model encoder classes as black boxes and does not require any code change inside encoder classes.
Second, typically for large-pretrained models, the noise dimension, i.e., 768 for BERT-base and RoBERTa-base, and 1024 for RoBERT-large, is fixed and much smaller than the model size, i.e., 110M for BERT-base.
Then we apply standard AdamW~\cite{adamw} optimizer in the back-propagation stage.
Please note that we do not add any noise and clippings in the inference time, as we only need to protect the fine-tuning training data.
In our following experiments, we use the following hyper-parameters as our default settings for our DP-FP method:
total fine-tuning epoch is three, $M = 32$, $C=1.0$, learning rate is $5 \times 10^{-6}$, max input length is $128$, the micro-batch subsampling rate $p=\frac{B}{M\cdot D}$ and the expected batch size $B=32$.
We consider a practical scenario in which the total amount of privacy budget $(\epsilon, \delta)$ is constrained. For both DP-FP and DP-SGD, this privacy budget constraint corresponds to a constraint on the number of data samples used in the tuning process.
As a result, we report accuracy scores on development sets once the privacy budget has been depleted.
\footnotetext{The results for SGD$^*$ and RGP$^*$ in Table~\ref{table:main_results} are from documented numbers in~\citet{yu2021large}. These results are under the DP guarantees of $\delta=10^{-5})$. These guarantees are strictly weaker than those provided by DP-SGD$^+$ and our DP-FP, which are based on $\delta = 1/2D$ (note that the smallest dataset in these tasks contains $D>60$k records).}
To ensure a fair comparison with the DP-SGD method in large-scale models reported in the literature, we set the same privacy budget for Gaussian DP with CLT as that in~\citet{li2021large} for each experiment using $\epsilon$ in the set $\{1.73, 2, 2.52, 4.33, 4.75, 5.83, 5.85\}$ and $\delta = 1/2D $ with $D$ the data record number in the training set.
We then numerically calibrate $\sigma^2$ as in Line 2 of Algorithm~\ref{alg:DP-FP}.
We also report the corresponding total privacy cost documented in~\citep{li2021large} calculated by RDP and composing tradeoff function methods. Please see Section~\ref{sec:pre} and the references therein for further information.
\subsection{Main Results}
Table~\ref{table:main_results} shows the main results on four tasks.
The larger the pretrained models, the higher the accuracies for the non-DP baselines, ranging from BERT-base (110M parameters) to RoBERTa-large (355M parameters).
We compare full fine-tuning with reparameterized gradient perturbation
(RGP)~\citep{yu2021large}, and memory efficient
DP-SGD for large language model~\citep{li2021large}
as they are the state-of-the-art methods for DP fine-tuning on sentence classification at
the time of writing.
The DP-SGD scores are documented in \newcite{li2021large}, and DP-SGD hurts the baseline performance by 4-6$\%$ in average for RoBERTa-base and RoBERTa-large, even with a larger $\epsilon$ at 8 (RDP).
In particular, results of RoBERTa-base models show that DP-SGD still degrades performance by up to 8$\%$ for the SST-2 data set.
By contrast, RGP of \newcite{yu2021large} reduces the gap significantly to 1$\%$ point on SST-2 data set, but is still at least 4$\%$
far behind Non-DP BERT-base on other three tasks.
We run DP-FP on each level of the privacy budget for three epochs of steps, and report the final scores only for the model reaches the privacy budget. Note that, we use the default setting and same hyper-parameters as illustrated in the previous subsection for each experiment of our DP-FP in Table~\ref{table:main_results}.
As shown in Table~\ref{table:main_results}, DP-FP significantly improves accuracy over existing methods and closes the gap to the Non-DP counterpart. It has only a 0.58$\%$ performance drop for SST2 on RoBERTa-large with $\epsilon$ set to 1.73 under GDP with CLT in particular. We further average this DP-FP performance drops across datasets for each of the models. When $\epsilon=3$ (RDP), the average performance drops to Non-DP model are within 0.83-1.78$\%$; and when $\epsilon=8$, they are within 0.68-1.43$\%$.
Moreover, DP-FP is clearly better than DP-SGD and RGP approaches.
This significant performance advantage stems in part from the fact that DP-FP overcomes the biased gradient problem in DP-SGD
and also has a lower noise power due to micro-batch number as well as a lower noise dimension.
More interestingly, the larger the pretrained model, the smaller the gap to the Non-DP model for DP-FP.
For example, on the SST-2 task for $\epsilon=3$ (RDP), the gap between DP-FP and Non-DP is reduced from 2.41$\%$ (BERT-base) to 0.58$\%$ (RoBERTa-large).
The main reason for this, we believe, is that because our DP-FP adds noise to the latent representation before the linear classification layer, the total noise power does not scale with model size.
\begin{figure}
\centering
\includegraphics[width=.45\textwidth]{"batch_size.png"}
\abovecaptionskip=2pt
\caption{Accuracy scores for different batch sizes ($B$) on SST-2 development set.}
\label{fig:batch_heatmap}
\end{figure}
\subsection{Hyperparameter Tuning}
As suggested in~\newcite{li2021large}, DP optimization is sensitive to hyper-parameter selection.
We therfore examine the performance of DP-FP for various hyper-parameters, as our DP-FP differs significantly from DP-SGD in terms of architecture.
In this section, we report all results of RoBERTa-base with DP-FP on SST-2 data set, and we fix the total training epochs to be 3 to deplete the privacy budget, i.e., $(\epsilon,\delta)=(3,1/2D)$.
Figure~\ref{fig:batch_heatmap} shows the heat map of accuracy scores for different batch sizes with
$M = 1$ and $C = 1.0$.
Those results suggest that 1) large batch sizes, like 128 and 256, hurt performance;
2) DP-FP requires a small learning rate in order to achieve better performance.
Those two interesting observations are opposite of the findings of DP-SGD in~\newcite{li2021large}.
The intuitive explanation is given below.
By substituting $p=B/(M\cdot D)$ into Eq.~(\ref{eqn:gdp-clt-dpfp}), we have the privacy parameter
\begin{equation}
\label{eqn:exp}
\mu_{\text{tot}}
= \frac{B}{M\cdot D} \sqrt{T\cdot M
\left(\mathrm{e}^{(C/\sigma)^2}-1\right)}.
\end{equation}
Intuitively, this demonstrates that, given a fixed privacy budget, which implies a fixed $\mu_{\text{tot}}$, noise power $\sigma^2$ decreases as batch-size $B$ decreases. However, we cannot reduce $B$ too much because the model will not converge.
\begin{figure}
\centering
\includegraphics[width=.45\textwidth]{"clipping_norms.png"}
\abovecaptionskip=2pt
\caption{Accuracy scores for different clipping threshold ($C$) on SST-2 development set.}
\label{fig:clip_heatmap}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=.45\textwidth]{"micro_batch.png"}
\abovecaptionskip=2pt
\caption{Accuracy curve for different micro-batch numbers ($M$) on SST-2 development set.}
\label{fig:micro_batch_curve}
\end{figure}
Figure~\ref{fig:clip_heatmap} shows the heat map of accuracy scores for different clipping threshold ($C$) with $\epsilon = 3$,
$M = 1$, and $B = 32$.
\newcite{li2021large} show that small clipping thresholds lead to the best performance by setting $C=0.1$.
In contrast, we find different trends from that in~\newcite{li2021large} such that smaller clipping thresholds lower than 0.4 actually hurt
performance, and there is no significant changes in larger clipping thresholds (from 0.6 to 1.0).
Removing too much of the latent representation with a small $C$ in DP-FC results in too much loss of semantic information and leads to significant performance drop.
Finally, Figure~\ref{fig:micro_batch_curve} shows the curve of DP-FP with different micro-batch numbers on SST-2 development set.
In this experiment, we set the privacy budget $\epsilon$ to be 0.02 under Gaussian DP + CLT, a very strong privacy level, the batch size to be 32, and the total number of epoch to be 3.
Larger micro-batch numbers clearly lead to better performance, as the larger the $M$, the more noise power is reduced (by $\frac{1}{M}$), as shown in Proposition~\ref{prop:noise-power}.
\subsection{DP Preliminary}
\label{sec:pre}
\noindent{\textbf{DP Definition}}:
DP stabilizes the output and introduces ``noise'' or random information into the process of computing the output of a function, such as an ML model, making it nearly impossible for the adversary to determine whether or not a specific data record, such as a sentence in the training data, was used.
Individual data records face the same level of privacy risk whether or not they are included in the computation due to DP assurance.
The rigorous DP definition is as follows.
Let $X$ and $X^\prime$ be neighboring datasets, i.e., $X\sim X^\prime$, such that one can be obtained from the other by either adding or removing a data record~\citep{dwork2006calibrating}.
In an NLP task, for example, $X$ and $X^\prime$ could be two datasets with sentences,
and $X\sim X^\prime$ indicates that they differ by only one sentence.
A randomized algorithm $\mathcal M$ is $(\epsilon, \delta)$-DP if for any subsets $Y$ of $\mathcal M$, and all neighboring datasets $X \sim X^{\prime}$, we have
\begin{equation}
\label{eq:dp-def}
\Pr[{\mathcal M}(X)\in {Y}]\leq\mathrm{e}^{\epsilon }\Pr[{\mathcal M}(X^{\prime})\in {Y}]+\delta.
\end{equation}
The probability distributions of the outputs $\mathcal M(X)$ and $\mathcal M(X^\prime)$ differ only by a multiplicative factor $\mathrm{e}^{\epsilon }$ and a very small value $\delta$.
Thus, the value of $\epsilon$ determines the strength of the privacy guarantee: the lower the value of $\epsilon$, the more privacy is guaranteed, and $\delta$ can be interpreted as the likelihood of failing to achieve DP.
Note that $\mathcal M(X)$ can be any complicated random function of $X$. For instance, it could be a gradient estimate in DP-SGD as shown later in~Eq.~(\ref{eq:dpsgd}) or a token/latent representation computed by neural networks in DP-FP as shown later in~Eq.~(\ref{eqn:dp-fp}).
To quantify privacy protection, data administrators impose a privacy loss cap, also referred to as a privacy budget, i.e., $(\epsilon, \delta)$, to calibrate the amount of random noise required to design $\mathcal M(\cdot)$.
\noindent {\textbf {DP Mechanism}}:
To achieve the $(\epsilon,\delta)$-DP protection defined in Eq.~(\ref{eq:dp-def}), the random function $\mathcal M(\cdot)$ must be specified.
A simple but effective method for achieving DP protection of $X$ given $f(X)$ is obtained by first constraining the range of $f(X)$ by a clipping operator, i.e., $\mathrm{Clip}(f(X); C)$, and then randomizing the result with calibrated Gaussian noise~\citep{abadi2016deep,li2021large}.
We introduce details of $\mathrm{Clip}(\cdot; C)$ in Eq.~(\ref{eqn:clipping-dpsgd}) and Eq.~(\ref{eqn:clipping-dpfp}) with concrete examples.
The clipping threshold $C$ is also known as {\it{sensitivity}} that
guarantees the greatest output variation for any pair of $X\sim X^{\prime}$ in terms of the $\ell_2$-norm.
Mathematically, the DP mechanism is defined by
\begin{equation}
\mathcal M(f(X))=\mathrm{Clip}(f(X); C)+\mathcal{N}(0, \sigma^2)
\end{equation}
with $\sigma^2$ being calibrated according to the DP profile $(\epsilon(\delta), \delta)$ given by~\citep{balle2018improving}:
\begin{equation}
\label{eqn:compute_eps}
\delta(\epsilon ; \mu)
=\Phi\left(-\frac{\epsilon}{\mu}+\frac{\mu}{2}\right)-\mathrm{e}^{\epsilon} \Phi\left(-\frac{\epsilon}{\mu}-\frac{\mu}{2}\right),
\end{equation}
where
\begin{equation}
\label{eqn:mu}
\mu = C / \sigma,
\end{equation}
and $\Phi(t)$ is the c.d.f. of Gaussian distribution.
\noindent{\textbf{Post Processing Property}}: A key DP property is its immunity to {\it{post-processing}}~\cite{dwork2014algorithmic}, which states that a differentially private output, i.e., $\mathcal M(f(X))$, can be arbitrarily transformed using some data-independent function without compromising its privacy guarantees.
\noindent{\textbf{DP Accounting}}:
When each DP mechanism meets certain DP guarantees individually, privacy suffers as a result of the composition of these DP mechanisms. Subsampling used before a private mechanism, on the other hand, increases the guarantee of privacy. Because differentially private model training entails a series of composition of subsampling and updates, DP accounting for such a series operations is required.
There are several widely used privacy accounting methodologies to adopt, such as moments accountant~\citep{abadi2016deep}, Re\'nyi DP (RDP)~\citep{mironov2017renyi}, Gaussian DP with central limit theorem~\citep{dong2021gaussian, bu2020deep}, and numerically composing tradeoff functions~\citep{gopi2021numerical, koskela2020computing, zhu2021optimal}.
RDP, a generalization of moments accounting, provides strict upper bounds on the actual privacy cost, whereas Gaussian DP with central limit theorem (CLT), while asymptotically exact, tends to underestimate the privacy loss, and composing tradeoff functions~\citep{gopi2021numerical} provides a more precise estimate.
We use Gaussian DP with CLT for DP analysis due to its simplicity and also report the total DP cost in the experimental analysis using the various methods listed above. To make the paper self-contained, we include more information about Gaussian DP with CLT in the appendix.
\subsection{DP-SGD Challenges in Large Models}
\label{sec:dpsgd}
DP assurance has been incorporated into deep learning by appropriately randomizing conventional model updates to limit what can be breached from the training data when revealing the model~\citep{abadi2016deep}. DP-SGD, which randomizes the SGD-based optimizers such as SGD and Adam, is commonly used to achieve such a sanitized model parameter release while concealing individual training data record.
Unlike traditional off-the-shelf optimizers, DP-SGD clips the $\ell_2$-norm of {\it per-sample gradient} before adding the calibrated noise to protect the input data in each step.
In more concrete terms, the DP mechanism
$\mathcal M(\cdot)$ for estimating the gradient in each $B$-sized batch is as follows:
\begin{equation}
\label{eq:dpsgd}
{g}=\frac{1}{B} \sum_{i=1}^{B} \left(\mathrm{Clip}\left(\nabla \mathcal{L}_{i}, C\right)+ \mathcal{N}\left(0, \sigma^2I_{d}\right)\right)
\end{equation}
with $d$ the model dimension and
$\mathcal{L}_{i}$ the loss function of data record $i$. The clipping operator that shrinks the gradient
of an individual example whenever it exceeds some threshold $C$
is computed by
\begin{equation}
\label{eqn:clipping-dpsgd}
\mathrm{Clip}\left(\nabla \mathcal{L}_{i}, C\right)
\triangleq
\nabla \mathcal{L}_{i}\cdot\min\left(1, \frac{C}{\left\|{\nabla \mathcal{L}_{i}} \right\|_2}\right).
\end{equation}
The model is then updated based on ${g}$, which is known as DP-SGD\footnote{The $\text {Optimizer}(\theta, g)$ represents the DP-enhanced version of most off-the-shelf optimizers, such as DP-Adam, which, like regular Adam, performs updates and moment accumulation with privatized gradients, i.e., $g$ in Eq.~(\ref{eq:dpsgd}). In this paper, we simply refer to the DP-version optimizers as DP-SGD.}:
\begin{equation}
\theta \leftarrow \text{Optimizer}\left(\theta,
{g}\right).
\end{equation}
Since ${g}$ has been DP guaranteed with an $(\epsilon, \delta)$ privacy cost in Eq.~(\ref{eq:dpsgd}), $\theta$ on the left-hand-side is also DP protected with the same DP cost as DP is immune to post-processing.
Furthermore, for $T$-step updates, the total privacy cost can be calculated by composing $(\epsilon,\delta)$ of each steps using the methods discussed in Section~\ref{sec:pre}'s DP Accounting paragraph.
We denote the privacy budget by $(\epsilon_T, \delta_T)$ with $\epsilon_T>\epsilon$ and $\delta_T>\delta$.
Applying the above DP-SGD to large-scale models raises the following challenges:
\begin{itemize}
\item DP-SGD
requires the computation and storage of individual gradients in each step due to the clipping operation in Eq~(\ref{eq:dpsgd}). Because each individual gradient requires the same amount of memory as the model, storing individual gradients in DP-SGD consumes a significant amount of memory~\citep{li2021large}.
\item The clipping operation in Eq.~(\ref{eqn:clipping-dpsgd}) inescapably introduces a significant bias~\citep{chen2020understanding} in the update direction, preventing convergence and significantly degrading model performance in general.
\item The total noise power scales with the model size ($d$), as shown in Eq.~(\ref{eq:dpsgd}), and degrades the model performance significantly~\citep{yu2021large}. Even with small $\sigma$, the total noise power is still huge for large models, such as BERT, where $d$ scales to 110M.
\end{itemize}
|
2024-02-18T23:40:24.389Z
|
2021-12-30T02:22:14.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14430","language":"en","timestamp":1640830934000,"url":"https:\/\/arxiv.org\/abs\/2112.14430","yymm":"2112"}
|
proofpile-arXiv_000-10013
|
{"provenance":"002.jsonl.gz:10014"}
| null | null |
\section{Introduction}
\label{Introduction}
The central idea of equivalent photon approximation was originally pointed out in 1924 by Fermi~\cite{Fermi:1924tc}.
According to the fact that a fast moving charged particle carries electric fields that point radially outward and magnetic fields circling it, the field at a point some distance away from the trajectory of the particle resembles that of a real photon, Fermi replaced the electromagnetic fields from a fast-moving charged particle with an equivalent flux of photon.
The number of photons with energy $\omega$, $n(\omega)$, is given by the Fourier transform of the time-dependent electromagnetic field \cite{Phys.Rev._51_1037, Dalitz:1957dd, Nucl.Phys._23_1295}.
Therefore, the cross section of the electromagnetic interaction is approximated by the convolution of the photon flux with the relevant real photoproduction cross section.
A decade later, in order to simplify calculations of processes involving relativistic collisions of charged particles, Weizs\"{a}cker and Williams independently
conceived the technique now known as the Weizs\"{a}cker-Williams approximation (WWA) \cite{vonWeizsacker:1934nji}.
An essential advantage of WWA consists in the fact that, when using it, it is sufficient to obtain the photo-absorption cross section on the mass shell only.
Details of its off mass-shell behavior are not essential.
Subsequently, WWA as a useful technique was substantiated and successfully applied, for instance, to two-photon processes for particle production, photoproduction mechanism, meson production in electron-nucleon collisions, the determination of the nuclear parton distributions, and small-$x$ physics~\cite{Phys. Rev._104_211, Zhu:2015via, Zhu:2015qoz, Fu:2011zzf, Fu:2011zzm, Brown:1973qj, Brodsky:1971ud, Budnev:1974de, Chin.Phys.C_36_721, Yu:2017rfi, Yu:2015kva, Yu:2017pot, Drees:1989vq, Drees:1988pp, Frixione:1993yw, Nystrand:2004vn, Nystrand:2006gi,Ma:2021jes, Winther:1979zz}.
On the other hand, WWA has found application beyond the realms of QED, such as the equivalent pion method which describes the subthreshold pion production in nucleus-nucleus collision~\cite{Pirner:1980rn};
the nuclear Weizs\"{a}cker-Williams method which describes excitation processes induced by the nuclear interaction in peripheral collisions of heavy ions \cite{Feshbach:1976uu};
a non-Abelian Weizs\"{a}cker-Williams method describing the boosted gluon distribution functions in nucleus-nucleus collision~\cite{McLerran:1994vd}.
Although the tremendous successes have been achieved, the discussion about the accuracy of WWA and its applicability range is still insufficient.
A number of widely used equivalent photon fluxes are proposed beyond the WWA validity range, and some imprecise statements are given~\cite{Drees:1989vq, Drees:1988pp, Frixione:1993yw, Yu:2015kva, Yu:2017pot, Yu:2017rfi, Zhu:2015via, Zhu:2015qoz, Fu:2011zzf, Chin.Phys.C_36_721, Winther:1979zz}.
In Ref.\cite{Drees:1988pp}, Drees and Zeppenfeld studied the production of supersymmetric particles in elastic $ep$ collisions, in order to simplify the calculations from the exact computation, they derived a equivalent photon spectrum of high energy protons by using the electric dipole form factor.
They also used a exponential form factor of heavy nuclei to derive a photon spectrum for lead \cite{Drees:1989vq}.
In Ref.\cite{Nystrand:2004vn}, Nystrand derived a modified photon spectrum by considering the $m_{p}^{2}$ term which was neglected by Drees and Zeppenfeld and employed it to study the electromagnetic interactions in nucleus-nucleus collisions, as well as the ultra-peripheral collisions of heavy ions at RHIC and LHC in Ref.\cite{Nystrand:2006gi}.
The effect of including the magnetic dipole moment and the corresponding magnetic form factor of the proton has been investigated by Kniehl in Ref.\cite{Kniehl:1990iv}, where a modified WWA is formulated for the elastic $ep$ scattering.
The authors in Refs.\cite{Cahn:1990jk, Baur:1990fx} defined a semiclassical impact parameter dependent equivalent photon spectrum, which excludes the hadronic interaction easily.
Brodsky and Drees achieved two different forms of photon spectra inside a quark in Refs.\cite{Brodsky:1971ud, Terazawa:1973tb}, which are employed to study the two-photon mechanism of particle production at high energy colliding beams.
The integration of the equivalent photon spectra in all of theses above works are performed over the entire kinematically allowed region, where the WWA errors are included.
Since the equivalent photon spectrum plays the fundamental role in the calculations of photoproduction processes, in present work, we take photoproductions of photons and dileptons as examples to discuss the validity of several widely used equivalent photon spectra mentioned above.
We present the comparison between the WWA results and the exact ones to analyse the source of WWA errors.
As a result, we derive a modified photon flux of proton and apply it to the calculations of $p_{T}$ dependent cross sections.
\section{General formalism}
\label{General formalism}
A consistent analysis of the terms neglected in going from the accurate expression of diagram Fig.~\ref{fig:feyn}(a) to the WWA one permits in a natural manner to estimate the errors of equivalent photon spectra. This can be performed in a general form for every reaction.
In the process described in Fig.~\ref{fig:feyn}, the virtual photons radiated from the projectile $\alpha$ are off mass shell and no longer transversely polarized.
Therefore, the accurate form of the cross section can be derived based on the expansion of the proton or quark tensor (multiplied by $Q^{-2}$) by using the transverse and longitudinal polarization operators.
\subsection{The accurate expression of cross section}
\label{The accurate expression}
~~~~~~~~~~
\begin{figure*}
\centering
\includegraphics[width=0.42\columnwidth]{gen_coll.eps}\hspace{25mm}
\includegraphics[width=0.30\columnwidth]{gen_coll_WWA.eps}
\caption{(a): The general photoproduction processes, the virtual photon emitted from the projectile $\alpha$ interacts with parton $b$ in proton $B$. $X$ is the sum of residue of $B$ after scattering with photon. (b): photo-absorption.}
\label{fig:feyn}
\end{figure*}
The general form of the cross section for $\alpha p\rightarrow\alpha\gamma^{*}X$ described in Fig.~\ref{fig:feyn} can be written as
\begin{eqnarray}
d\sigma(\alpha+p\rightarrow \alpha+\gamma^{*}+X)
=\sum_{b}\int dx_{b}f_{b/p}(x_{b},\mu_{b}^{2})d\sigma(\alpha+b\rightarrow \alpha+\gamma^{*}+b),
\end{eqnarray}
where $x_{b}=p_{b}/p_{B}$ is the momentum fraction of parton $b$ struck by the virtual photon, $f_{b/p}(x_{b},\mu_{b}^{2})$ is the parton distribution function of massless parton $b$ in proton $B$, and $\mu_{b}$ is the factorized scale.
$d\sigma(\alpha+b\rightarrow \alpha+\gamma^{*}+b)$ is the differential cross section in the parton level and can be presented as
\begin{eqnarray}\label{dab.Gen1}
d\sigma(\alpha+b\rightarrow \alpha+\gamma^{*}+b)
=\frac{|\mathcal{M}_{\alpha b}|^{2}}{4\left[(p_{\alpha}\cdot p_{b})^{2}-p_{\alpha}^{2}p_{b}^{2}\right]^{1/2}}d\textrm{PS}_{3}(p_{\alpha}+p_{b};p_{\alpha}',p_{c},p_{b}'),
\end{eqnarray}
where we employ the short-hand notation
\begin{eqnarray}\label{dab.Gen1}
d\textrm{PS}_{n}(P;p_{1},...,p_{n})=(2\pi)^{4}\delta^{4}\left(P-\sum_{i=1}^{n}p_{i}\right)\prod_{i=1}^{n}\frac{d^{3}p_{i}}{(2\pi)^{3}2E_{i}},
\end{eqnarray}
for the Lorentz invariant N-particle phase-space element.
By decomposing the squared scattering amplitude as $|\mathcal{M}_{\alpha b}|^{2}=4\pi\alpha_{\mathrm{em}} e_{\alpha}^{2}\rho^{\mu\nu}T_{\mu\nu}/Q^{2}$ and rearranging $d\textrm{PS}_{n}$, the cross section of subprocess $\alpha+b\rightarrow \alpha+\gamma^{*}+b$ can be further expressed as follows
\begin{eqnarray}\label{dab.Gen2}
&&\!\!\!\!d\sigma(\alpha+b\rightarrow \alpha+\gamma^{*}+b)\nonumber\\
=\!\!\!\!&&\!\!\!\!4\pi e_{\alpha}^{2}\alpha_{\mathrm{em}}\frac{\rho^{\mu\nu}}{Q^2}\frac{d^{3}p'_{\alpha}}{(2\pi)^{3}2E'_{\alpha}}\left[\frac{(q\cdot p_{b})^{2}-q^{2}p_{b}^{2}}{(p_{\alpha}\cdot p_{b})^{2}-p_{\alpha}^{2}p_{b}^{2}}\right]^{1/2}T_{\mu\nu}\frac{d\mathrm{PS}_{2}(q+p_{b};p_{c},p_{b}')}{4\hat{p}_{\mathrm{CM}}\sqrt{\hat{s}}},
\end{eqnarray}
where $e_{\alpha}$ is the charge of the projectile $\alpha$, $\alpha_{\mathrm{em}}=1/137$ is the fine structure constant, and
\begin{eqnarray}\label{lambda}
\left[\frac{(q\cdot p_{b})^{2}-q^{2}p_{b}^{2}}{(p_{\alpha}\cdot p_{b})^{2}-p_{\alpha}^{2}p_{b}^{2}}\right]^{1/2}
=\frac{\hat{p}_{\mathrm{CM}}\sqrt{\hat{s}}}{p_{\mathrm{CM}}\sqrt{s_{\alpha b}}},
\end{eqnarray}
$s_{\alpha b}=(p_{\alpha}+p_{b})^{2}$ and $\hat{s}=(q+p_{b})^{2}$ are the energy square of $\alpha b$ and $\gamma^{*}b$ CM frames, respectively.
$p_{\mathrm{CM}}$ and $\hat{p}_{\mathrm{CM}}$ are the momenta of corresponding CM frames.
$T_{\mu\nu}$ is the amplitude of reaction $\gamma^{*}+b\rightarrow \gamma^{*}+b$, and the $\rho^{\mu\nu}$ quantity is the density matrix of the virtual photon produced by the projectile $\alpha$,
\begin{eqnarray}\label{Rou.Gen}
\rho^{\mu\nu}=(-g^{\mu\nu}+\frac{q^{\mu}q^{\nu}}{q^{2}})F_{2}(Q^{2})-\frac{(2p_{\alpha}-q)^{\mu}(2p_{\alpha}-q)^{\nu}}{q^{2}}F_{1}(Q^{2}),
\end{eqnarray}
$F_{1}(Q^{2})$ and $F_{2}(Q^{2})$ are the general expressions for the form factors of projectile.
It is convenient to use the following linear combinations~\cite{Budnev:1974de}
\begin{eqnarray}\label{Line.QR}
Q^{\mu}\!\!&=&\!\!\sqrt{\frac{-q^{2}}{(q\cdot p_{b})^{2}-q^{2}p_{b}^{2}}}(p_{b}-q\frac{q\cdot p_{b}}{q^{2}})^{\mu},\nonumber\\
R^{\mu\nu}\!\!&=&\!\!-g^{\mu\nu}+\frac{(q\cdot p_{b})(q^{\mu}p_{b}^{\nu}+q^{\nu}p_{b}^{\mu})-q^{2}p_{b}^{\mu}p_{b}^{\nu}
-p_{b}^{2}q^{\mu}q^{\nu}}{(q\cdot p_{b})^{2}-q^{2}p_{b}^{2}},
\end{eqnarray}
they satisfy the relations: $q_{\mu}Q^{\mu}=q_{\mu}R^{\mu\nu}=0$, $Q^{\mu}Q_{\mu}=1$, thus $\rho^{\mu\nu}$ can be expanded to
\begin{eqnarray}\label{Rou.Exp}
\rho^{\mu\nu}=\rho^{00}Q^{\mu}Q^{\nu}+\rho^{++}R^{\mu\nu},
\end{eqnarray}
where
$\rho^{++}=R^{\mu\nu}\rho_{\mu\nu}/2$, $\rho^{00}=Q^{\mu}Q^{\nu}\rho_{\mu\nu}$.
With the relations
\begin{eqnarray}\label{subab.TL}
2d\sigma_{T}(\gamma^{*}+b\rightarrow \gamma^{*}+b)
\!\!&=&\!\!R^{\mu\nu}T_{\mu\nu}\frac{d\mathrm{PS}_{2}(q+p_{b};p_{c},p_{b}')}{4\hat{p}_{\mathrm{CM}}\sqrt{\hat{s}}},\nonumber\\
d\sigma_{L}(\gamma^{*}+b\rightarrow \gamma^{*}+b)
\!\!&=&\!\!Q^{\mu}Q^{\nu}T_{\mu\nu}\frac{d\mathrm{PS}_{2}(q+p_{b};p_{c},p_{b}')}{4\hat{p}_{\mathrm{CM}}\sqrt{\hat{s}}},
\end{eqnarray}
the differential cross section of subprocess $\alpha+b\rightarrow \alpha+\gamma^{*}+b$ can finally be expressed as
\begin{eqnarray}\label{dabTL}
&&\!\!\!\!d\sigma(\alpha+b\rightarrow \alpha+\gamma^{*}+b)\nonumber\\
=\!\!\!\!&&\!\!\!\!\frac{e_{\alpha}^{2}\alpha_{\mathrm{em}}}{2\pi^{2}Q^{2}}\left[\rho^{++}d\sigma_{T}(\gamma^{*}+b\rightarrow \gamma^{*}+b)+\frac{\rho^{00}}{2}d\sigma_{L}(\gamma^{*}+b\rightarrow \gamma^{*}+b)\right]\nonumber\\
\!\!\!\!&&\!\!\!\!\times\frac{\hat{p}_{\mathrm{CM}}\sqrt{\hat{s}}}{p_{\mathrm{CM}}\sqrt{s_{\alpha b}}}\frac{d^{3}p'_{\alpha}}{E'_{\alpha}},
\end{eqnarray}
and
\begin{eqnarray}\label{Rouzz00}
\rho^{++}\!\!&=&\!\!F_{2}(Q^{2})+\frac{1}{2}\left[\frac{(2-y)^{2}}{y^{2}+Q^{2}m_{b}^{2}/(p_{\alpha}\cdot p_{b})^{2}}-\frac{4m_{\alpha}^{2}}{Q^{2}}-1\right]F_{1}(Q^{2}),\nonumber\\
\rho^{00}\!\!&=&\!\!-F_{2}(Q^{2})+\frac{(2-y)^{2}}{y^{2}+Q^{2}m_{b}^{2}/(p_{\alpha}\cdot p_{b})^{2}}F_{1}(Q^{2}).
\end{eqnarray}
Here $Q^{2}=-q^{2}$ and $d\sigma_{T (L)}/d\hat{t}$ represents the transverse (longitudinal) cross section of subprocess $\gamma^{*}+b\rightarrow\gamma^{*}+b$, its analytical expression can be found in Ref.\cite{Ma:2019mwr}.
\begin{figure*}
\centering
\includegraphics[width=0.35\columnwidth]{coh_pho.eps}\hspace{20mm}
\includegraphics[width=0.43\columnwidth]{incoh_pho.eps}
\caption{(a): Coherent emission, virtual photon is radiated coherently by the whole proton which remains intact after scattering.
(b): Incoherent emission, virtual photon is radiated incoherently by the quarks inside proton which is allowed break up after scattering.}
\label{fig:photoemi.}
\end{figure*}
We have to deal with two types of photon emission mechanisms which are described in Fig.~\ref{fig:photoemi.}: coherent emission (coh.) and incoherent emission (incoh.).
In the first type, the virtual photon is radiated coherently by the whole proton which remains intact after photon emission.
In the second type, the virtual photon is radiated incoherently by the quarks inside the proton, and the proton will dissociate or excite after photon emission.
In the case of coherent reactions in Fig.~\ref{fig:photoemi.}(a), the projectile $\alpha$ is proton: $m_{\alpha}=m_{p}$, and thus the general expressions of form factor $F_{1}(Q^{2})$ and $F_{2}(Q^{2})$ turn into the elastic proton form factors accordingly. Then the density matrix of the virtual photon in Eq.~(\ref{Rouzz00}) reads
\begin{eqnarray}\label{Rouzz00.coh.}
\rho^{++}_{\mathrm{coh}.}\!\!&=&\!\!G_{\mathrm{E}}^{2}(Q^{2})\left[7.78-\frac{1}{2}\left(\frac{4m_{\alpha}^{2}+Q^{2}}{Q^{2}}
-\frac{(2-y)^{2}}{y^{2}}\right)\frac{4m_{p}^{2}+7.78Q^{2}}{4m_{p}^{2}+Q^{2}}\right],\nonumber\\
\rho^{00}_{\mathrm{coh}.}\!\!&=&\!\!G_{\mathrm{E}}^{2}(Q^{2})\left[-7.78+\frac{(2-y)^{2}}{y^{2}}\frac{4m_{p}^{2}+7.78Q^{2}}{4m_{p}^{2}+Q^{2}}\right],
\end{eqnarray}
where the electric form factor of proton can be parameterized by a dipole form
\begin{eqnarray}\label{GE}
G_{\mathrm{E}}(Q^{2})=\frac{1}{(1+Q^{2}/0.71~\mathrm{GeV}^{2})^{2}}.
\end{eqnarray}
In the case of incoherent reactions in Fig.~\ref{fig:photoemi.}(b), the projectile $\alpha$ is the quark inside the proton: $m_{\alpha}=m_{q}=0$.
Besides, the Martin-Ryskin method is adopted for the calculations \cite{Martin:2014nqa}, where the square of the form factor is used as the coherent probability or weighting factor (WF): $\omega_{c}=G_{\mathrm{E}}^{2}(Q^2)$, and in order to avoid double counting, the "remaining" probability has to be considered in the incoherent case: $1-\omega_{c}=1-G_{\mathrm{E}}^{2}(Q^2)$.
Thus, we have $F_{1}(Q^{2})=F_{2}(Q^{2})=1-G_{\mathrm{E}}^{2}(Q^{2})$, and the density matrix of the virtual photon in Eq.~(\ref{Rouzz00}) becomes
\begin{eqnarray}\label{Rouzz00.incoh.}
&&\rho^{++}_{\mathrm{incoh.}}=\left[1-G_{\mathrm{E}}^{2}(Q^{2})\right]\left[\frac{1}{2}+\frac{(y-2)^{2}}{2y^{2}}\right],\nonumber\\
&&\rho^{00}_{\mathrm{incoh.}}=\left[1-G_{\mathrm{E}}^{2}(Q^{2})\right]\left[\frac{(y-2)^{2}}{y^{2}}-1\right].
\end{eqnarray}
~~~~~~~~~~
\subsection{The equivalent photon spectrum}
\label{The eps}
~~~~~~~~~~
The connection between the process in Fig.~\ref{fig:feyn}(a) and the one in (b) is evident.
The WWA consists in ignoring the fact that the photon in this photo-absorption amplitude [Fig.~\ref{fig:feyn}(b)] is off mass shell and no longer transversely polarized from real photo-absorption.
As a result, the photoprodution process described in Fig.~\ref{fig:feyn}(a) can be factorized in terms of the real photo-absorption cross section and the equivalent photon spectrum.
When switching to the approximate formulae of WWA, two simplifications should be performed in accurate expression [Eq.~(\ref{dabTL})].
First, the scalar photon contribution $\sigma_{L}$ is neglected; secondly, the term of $\sigma_{T}$ is substituted by its on-shell value.
This provides us a powerful approach for comparing our results with the WWA ones to study the features of the equivalent photon spectra which are widely applied in the literatures.
Taking $Q^{2}\rightarrow0$, the linear combinations in Eq.~(\ref{Line.QR}) can reduce to
\begin{eqnarray}\label{epsi.TL}
&&\lim_{Q^{2}\rightarrow0}Q^{\mu}Q^{\nu}=\frac{q^{\mu}q^{\nu}}{q^{2}},\nonumber\\
&&\lim_{Q^{2}\rightarrow0}R^{\mu\nu}=-g^{\mu\nu}+\frac{(q^{\mu}p_{b}^{\nu}+q^{\nu}p_{b}^{\mu})}{q\cdot p_{b}}.
\end{eqnarray}
According to gauge invariance $q^{\mu}T_{\mu\nu}=0$, Eq. (\ref{dabTL}) is simplified to:
\begin{eqnarray}\label{dWWA.Gen.}
&&\!\!\!\!\lim_{Q^{2}\rightarrow0}d\sigma(\alpha+b\rightarrow\alpha+\gamma^{*}+b)\nonumber\\
=\!\!\!\!&&\!\!\!\!\left(e_{\alpha}^{2}\frac{\alpha_{\mathrm{em}}}{2\pi^{2}}\frac{y\rho^{++}}{Q^{2}}\frac{d^{3}p'_{\alpha}}{E'_{\alpha}}\right)
\frac{\hat{p}_{\mathrm{CM}}\sqrt{\hat{s}}}{yp_{\mathrm{CM}}\sqrt{s_{\alpha b}}}d\sigma_{T}(\gamma^{*}+b\rightarrow \gamma^{*}+b)\bigg|_{Q^{2}=0}\nonumber\\
=\!\!\!\!&&\!\!\!\!\left[e_{\alpha}^{2}\frac{\alpha_{\mathrm{em}}}{2\pi}(y\rho^{++})dy\frac{dQ^{2}}{Q^{2}}\right]d\sigma_{T}(\gamma^{*}+b\rightarrow \gamma^{*}+b)\bigg|_{Q^{2}=0}\nonumber\\
=\!\!\!\!&&\!\!\!\!dn_{\gamma}d\sigma_{T}(\gamma^{*}+b\rightarrow \gamma^{*}+b)\bigg|_{Q^{2}=0},
\end{eqnarray}
where $y=(q\cdot p_{b})/(p_{\alpha}\cdot p_{b})$ is the relative energy loss of the projectile $\alpha$, and the general form of the equivalent photon flux $f(y)$ reads
\begin{eqnarray}\label{fgamma.Gen.}
f(y)\!\!\!\!&=&\!\!\!\!\frac{dn_{\gamma}}{dy}=\int e_{\alpha}^{2}\frac{\alpha_{\mathrm{em}}}{2\pi}y\rho^{++}\frac{dQ^{2}}{Q^{2}}\nonumber\\
\!\!\!\!&=&\!\!\!\!e_{\alpha}^{2}\frac{\alpha_{\mathrm{em}}}{2\pi}y\int\frac{dQ^{2}}{Q^{2}}\left[F_{2}(Q^{2})
+\left(\frac{2(1-y)}{y^{2}}-\frac{2m_{\alpha}^{2}}{Q^{2}}\right)F_{1}(Q^{2})\right].
\end{eqnarray}
In the case of coherent photon emission of proton, we derived a photon flux function from Eq.~(\ref{fgamma.Gen.}).
By neglecting the contributions from the magnetic form factor and adopting the dipole form of electric form factor of proton: $F_{1}(Q^{2})=F_{2}(Q^{2})=G_{\mathrm{E}}^{2}(Q^{2})$, and employing the coherent condition $Q^{2}\leq 1/R^{2}_{A}$ ($R_{A}=A^{1/3}1.2\ \textrm{fm}$ is the size of the nucleus), one obtains with $a=2m_{p}^{2}/Q^{2}_{\mathrm{max}}$ and $b=2m_{p}^{2}/0.71=2.48$,
\begin{eqnarray}\label{fgamma.MD.}
f_{\mathrm{MD}}(y)\!\!\!\!&=&\!\!\!\!\frac{\alpha_{\mathrm{em}}}{2\pi}y\left[a-2x+(2x+c_{1})d_{1}+(2x+c_{2})d_{2}\right.\nonumber\\
&&\!\!\!\!+\left.(3x+c_{3})d_{3}+(2x+c_{4})d_{4}\right],
\end{eqnarray}
where $x$ depends on $y$,
\begin{eqnarray}\label{x}
x=-\frac{1}{y}+\frac{1}{y^{2}}.
\end{eqnarray}
The factors $c_{i}$ and $d_{i}\ (i=1,...,4)$ in Eq.~(\ref{fgamma.MD.}) have the forms
\begin{eqnarray}\label{x}
&&c_{1}=1+4b\approx 10.92,\ \ \ d_{1}=\ln\frac{A}{A'},\nonumber\\
&&c_{2}=1+2b\approx 5.96,\ \ \ \ \ d_{2}=3(\frac{1}{A}-\frac{1}{A'}),\nonumber\\
&&c_{3}=\frac{3}{2}+2b\approx 6.46,\ \ \ \ d_{3}=-(\frac{1}{A^{2}}-\frac{1}{A'^{2}}),\nonumber\\
&&c_{4}=1+b\approx 3.48,\ \ \ \ \ \ d_{4}=\frac{1}{3}(\frac{1}{A^{3}}-\frac{1}{A'^{3}}),
\end{eqnarray}
where $A=(1+0.71~\mathrm{GeV}^{2}/Q^{2}_{\mathrm{min}})$, $A'=(1+0.71~\mathrm{GeV}^{2}/Q^{2}_{\mathrm{max}})$ and
\begin{eqnarray}\label{Q2lim.}
Q^{2}_{\mathrm{min}}\!\!\!\!&=&\!\!\!\!-2m_{\alpha}^{2}+\frac{1}{2s_{\alpha b}}\left[(s_{\alpha b}+m_{\alpha}^{2})(s_{\alpha b}-\hat{s}+m_{p}^{2})\right.\nonumber\\
\!\!\!\!&&\!\!\!\!\left.-(s_{\alpha b}-m_{\alpha}^{2})\sqrt{(s_{\alpha b}-\hat{s}+m_{\alpha}^{2})^{2}-4s_{\alpha b}m_{\alpha}^{2}}\right],\nonumber\\
Q^{2}_{\mathrm{max}}\!\!\!\!&=&\!\!\!\!1/R^{2}_{A}=0.027,
\end{eqnarray}
with $m_{\alpha}=m_{p}$.
Since the coherent condition is employed, $Q^{2}_{\mathrm{max}}$ is limited to very low value.
One should note that, since $y$ depends on $Q^{2}$, we can not set $y_{\mathrm{max}}=1$ directly in the calculation, instead $y_{\mathrm{max}}=0.16$ in this case.
Actually, the origin of various practically employed photon spectra is another plane wave form, which is given in Ref.\cite{Budnev:1974de} and can be presented as follows
\begin{eqnarray}\label{fgamma.Gen.V}
dn_{\gamma}=e_{\alpha}^{2}\frac{\alpha_{\mathrm{em}}}{\pi}\frac{dy}{y}\frac{dQ^{2}}{Q^{2}}\left[(1-y)\frac{Q^{2}-Q^{2}_{\mathrm{min}}}{Q^{2}}F_{1}(Q^{2})
+\frac{y^{2}}{2}F_{2}(Q^{2})\right],
\end{eqnarray}
this form can be also derived from Eq.~(\ref{fgamma.Gen.}) by assuming that $Q^{2}_{\mathrm{min}}=y^{2}m_{\alpha}^{2}/(1-y)$, which is the leading order (LO) term of complete expression Eq.~(\ref{Q2lim.}) in the expansion of $\mathcal{O}(m_{\alpha}^{2})$, and is only valuable when $m_{\alpha}^{2}\ll1\ \mathrm{GeV}^{2}$.
However, $m_{p}^{2}\approx 0.88\ \mathrm{GeV}^{2}$ does not satisfies this condition, this leads to about $10\%$ errors in various spectra.
In Ref.\cite{Drees:1988pp}, Drees and Zeppenfeld provided a approximate analytic form of Eq.~(\ref{fgamma.Gen.V}) which is widely used in the literatures \cite{Zhu:2015qoz,Yu:2015kva,Yu:2017rfi,Yu:2017pot,Fu:2011zzf,Chin.Phys.C_36_721}.
By taking $Q^{2}_{\mathrm{max}}\rightarrow\infty$ and setting $F_{1}(Q^{2})=F_{2}(Q^{2})=G_{\mathrm{E}}^{2}(Q^{2})$ and $Q^{2}-Q^{2}_{\mathrm{min}}\approx Q^{2}$, they obtained
\begin{eqnarray}\label{fgamma.DZ.}
f_{\mathrm{DZ}}(y)=\frac{\alpha_{\mathrm{em}}}{2\pi}\frac{1+(1-y)^{2}}{y}\left[\ln A-\frac{11}{6}+\frac{3}{A}-\frac{3}{2A^{2}}+\frac{1}{3A^{2}}\right].
\end{eqnarray}
Based on Eq.~(\ref{fgamma.DZ.}), Nystrand derived a modified photon spectrum which include the $Q^{2}_{\mathrm{min}}$ term in Eq.~(\ref{fgamma.Gen.V}) and can be presented as \cite{Nystrand:2004vn}
\begin{eqnarray}\label{fgamma.Ny.}
f_{\mathrm{Ny}}(y)=\frac{\alpha_{\mathrm{em}}}{2\pi}\frac{1+(1-y)^{2}}{y}\left[\frac{A+3}{A-1}\ln A-\frac{17}{6}-\frac{4}{3A}+\frac{1}{6A^{2}}\right].
\end{eqnarray}
In addition, the effect from including the magnetic dipole moment and the corresponding magnetic form factor of the proton has been estimated by Kniehl \cite{Kniehl:1990iv}.
The final expression $f_{\mathrm{Kn}}(y)$ (Eq.~(3.11) of \cite{Kniehl:1990iv}) is too long to include here, but will be discussed further below.
Another most important approach for photon spectrum is the semiclassical impact parameter description, which excludes the hadronic interaction easily.
The calculation of the semiclassical photon spectrum is explained in Ref.~\cite{CED}, and the result can be written as
\begin{eqnarray}\label{fgamma.SC}
f_{\mathrm{SC}}(y)=\frac{2Z^{2}\alpha_{\mathrm{em}}}{\pi}\left(\frac{c}{\upsilon}\right)^{2}\frac{1}{y}\left[\xi K_{0}K_{1}+\frac{\xi^{2}}{2}\left(\frac{\upsilon}{c}\right)^{2}\left(K^{2}_{0}-K^{2}_{1}\right)\right],
\end{eqnarray}
where $\upsilon$ is the velocity of the point charge $Ze$, $K_{0}(x)$ and $K_{1}(x)$ are the modified Bessel functions, and $\xi=b_{\mathrm{min}}m_{A}y/\upsilon$.
In the case of incoherent photon emission, the complete form of photon spectrum can be derived from Eq.~(\ref{fgamma.Gen.}) by setting $F_{1}(Q^{2})=F_{2}(Q^{2})=1-G_{\mathrm{E}}^{2}(Q^{2})$ and $m_{\alpha}=0$,
\begin{eqnarray}\label{fgamma.incoh.}
dn_{\gamma}(y)=e_{\alpha}^{2}\frac{\alpha_{\mathrm{em}}}{2\pi}dy\frac{dQ^{2}}{Q^{2}}\frac{1+(1-y)^{2}}{y}\left[1-G_{\mathrm{E}}^{2}(Q^{2})\right].
\end{eqnarray}
Actually, another approximate form of above equation is often used in practical calculations \cite{Zhu:2015qoz,Yu:2015kva,Yu:2017rfi,Yu:2017pot,Fu:2011zzf,Chin.Phys.C_36_721,Fu:2011zzm}, which neglect $G_{\mathrm{E}}^{2}(Q^{2})$ term of Eq.~(\ref{fgamma.incoh.}) and take $Q_{\mathrm{min}}^{2}=1~\mathrm{GeV}^{2}$ and $Q^{2}_{\mathrm{max}}=\hat{s}/4$,
\begin{eqnarray}\label{fgamma.incohI.}
f_{\mathrm{incoh}}(y)=e_{\alpha}^{2}\frac{\alpha_{\mathrm{em}}}{2\pi}\frac{1+(1-y)^{2}}{y}\ln\frac{Q_{\mathrm{max}}^{2}}{Q_{\mathrm{min}}^{2}}.
\end{eqnarray}
Finally, another important form of equivalent photon spectrum of parton is given by Brodsky, Kinoshita and Terazawa in Ref.~\cite{Brodsky:1971ud}, which can be expressed as
\begin{eqnarray}\label{fgamma.incohBKT}
&&\!\!\!\!\!\!f_{\mathrm{BKT}}(y)\nonumber\\
=&&\!\!\!\!\!\!e_{\alpha}^{2}\frac{\alpha_{\mathrm{em}}}{\pi}\Bigg\{\frac{1+(1-y)^{2}}{y}\left(\ln\frac{E}{m}-\frac{1}{2}\right)\nonumber\\
&&\!\!\!\!\!\!+\frac{y}{2}\left[\ln(\frac{2}{y}-2)+1\right]+\frac{(2-y)^{2}}{2y}\ln(\frac{2-2y}{2-y})\Bigg\}.
\end{eqnarray}
~~~~~~~~~~
\section{The $Q^{2}$, $y$ and $p_{T}$ distributions for the photoproductions of photons and dileptons}
\label{Distributions}
~~~~~~~~~~
The equivalent photon spectrum is the key function in the calculations of photoproduction processes.
Since photons and dileptons are the ideal probes of strong interaction matter (quark-gluon plasma, QGP), its photoproduction processes have received many studies within WWA.
We choose the photoproductions of photons and dileptons as examples, to discuss the features of the spectra mentioned above and illustrate the virtue of the photon flux Eq.~(\ref{fgamma.MD.}) by comparing their results with the accurate one in Eq.~(\ref{dabTL}).
In present section, we give the corresponding cross sections.
There are two types of photon contributions that should be considered: direct-photon and resolved-photon contributions~\cite{Ma:2018zzq}.
For the direct-photon process, the high-energy photon, emitted from the projectile $\alpha$, interacts with the partons $b$ of target proton $B$ by the interactions of quark-photon Compton scattering.
For the resolved-photon process, the high-energy photon can fluctuate into a color singlet state with multiple $q\bar{q}$ pairs and gluons.
Due to this fluctuation, the photon interacts with the partons in $B$ like a hadron, and the subprocesses are quark-antiquark annihilation and quark-gluon Compton scattering.
Actually, as always with photons, the situation is quite complex.
Together with the two different photon emission mechanisms mentioned earlier, we have four types of processes: coherent direct (coh.dir.), coherent resolved (coh.res.), incoherent direct (incoh.dir.) and incoherent resolved (incoh.res.) processes. These abbreviations will appear in many places of remained content
and we do not explain its meaning again.
The corresponding cross sections of the above four processes for dileptons photoproduction are
\begin{eqnarray}\label{photo.CS}
\sigma_{\mathrm{coh.dir.}}\!\!\!\!&=&\!\!\!\!\int dM^{2}dx_{b}dyf_{b/p}(x_{b},\mu^{2})f_{\gamma/p}(y)d\sigma(\gamma+b\rightarrow l^{+}l^{-}+b),\nonumber\\
\sigma_{\mathrm{coh.res.}}\!\!\!\!&=&\!\!\!\!\int dM^{2}dx_{b}dydx_{a'}f_{b/p}(x_{b},\mu^{2})f_{\gamma/p}(y)f_{a'/\gamma}(x_{a'},\mu^{2})\nonumber\\
\!\!\!\!&&\times d\sigma(a'+b\rightarrow l^{+}l^{-}+b),\nonumber\\
\sigma_{\mathrm{incoh.dir.}}\!\!\!\!&=&\!\!\!\!\int dM^{2}dx_{a}dx_{b}dyf_{a/p}(x_{a},\mu^{2})f_{b/p}(x_{b},\mu^{2})f_{\gamma/a}(y)\nonumber\\
\!\!\!\!&&\times d\sigma(\gamma+b\rightarrow l^{+}l^{-}+b),\nonumber\\
\sigma_{\mathrm{incoh.res.}}\!\!\!\!&=&\!\!\!\!\int dM^{2}dx_{a}dx_{b}dydx_{a'}f_{a/p}(x_{a},\mu^{2})f_{b/p}(x_{b},\mu^{2})f_{\gamma/a}(y)f_{a'/\gamma}(x_{a'},\mu^{2})\nonumber\\
\!\!\!\!&&\times d\sigma(a'+b\rightarrow l^{+}l^{-}+b),
\end{eqnarray}
where $f_{\gamma/p}(y)$ and $f_{\gamma/a}(y)$ are the photon spectrum of proton $A$ and its parton $a$, respectively.
$f_{a'/\gamma}(x_{a'},\mu^{2})$ is the parton distribution function in the resolved photon, $x_{a'}=p_{a'}/q$.
It should be emphasized that the above integrations are in the WWA form. In the calculations of exact result based on Eq.~(\ref{dabTL}),
there is no equivalent photon spectrum in Eq.~(\ref{photo.CS}), instead, $Q^{2}$ should be integrated out.
The partonic cross section of the subprocess $\gamma^{*}\rightarrow l^{+}l^{-}$ has the following form
\begin{eqnarray}\label{fgamma.incohBKH}
&&\!\!\!\!\!\!d\sigma(X+b\rightarrow l^{+}l^{-}+b)\nonumber\\
=&&\!\!\!\!\!\!\frac{\alpha_{\mathrm{em}}}{3\pi M^{2}}\sqrt{1-\frac{4m_{l}^{2}}{M^{2}}}\left(1+\frac{2m_{l}^{2}}
{M^{2}}\right)d\sigma(X+b\rightarrow \gamma^{*}+b)dM^{2},
\end{eqnarray}
where $M$ is the invariant mass of dileptons, $m_{l}$ is the lepton mass.
The fragmentation dilepton is also an important channel which involves a perturbative part - the bremsstrahlung of the virtual photon- and a nonperturbative part, described by the dilepton fragmentation function,
\begin{eqnarray}\label{f.frag}
D_{q_{c}}^{l^{+}l^{-}}(z_{c},M^{2},Q^{2})=\frac{\alpha_{\mathrm{em}}}{3\pi M^{2}}\sqrt{1-\frac{4m_{l}^{2}}{M^{2}}}\left(1+\frac{2m_{l}^{2}}{M^{2}}\right)D_{q_{c}}^{\gamma^{*}}(z_{c},Q^{2}),
\end{eqnarray}
where $D_{q_{c}}^{\gamma^{*}}(z_{c},Q^{2})$ is the virtual photon fragmentation function \cite{Kang:2008wv}, $z_{c}$ is the momentum fraction of the final state dileptons. The involved subprocesses in this channel are $q\gamma\rightarrow q\gamma$, $q\gamma\rightarrow q g$ and $\gamma g\rightarrow q\bar{q}$ for direct-photon contribution \cite{Ma:2019mwr}, and are $qq\rightarrow qq$, $qq'\rightarrow qq'$, $q\bar{q}\rightarrow q\bar{q}$, $q\bar{q}\rightarrow q'\bar{q}'$, $q\bar{q}'\rightarrow
q\bar{q}'$, $qg\rightarrow q\gamma$, $qg\rightarrow qg$ and $gg\rightarrow q\bar{q}$ for resolved-photon contribution \cite{Owens:1986mp}.
In the case of photons photoproduction, because a virtual photon can directly decay into a dilepton, the invariant cross sections of photons production can be easily derived from those of dileptons production if the invariant mass of dileptons is zero $(M^{2}=0)$.
Finally, the analytic expressions of distributions in $Q^{2}$, $y$ and $p_{T}$ can be found in \ref{Q2ypT}.
\section{Numerical results and discussion}
\label{Numerical results}
~~~~~~~~~~
We are now in a position to present our numerical results. First, several theoretical inputs need to be provided.
The mass of the proton is $m_{p}=0.938~\mathrm{GeV}$ \cite{Agashe:2014kda}, the strong coupling constant is taken as 1-loop form \cite{Ma:2015ykd}
\begin{eqnarray}\label{alfas}
&&\alpha_{\mathrm{s}}=\frac{12 \pi}{(33-2n_{f})\ln(\mu^{2}/\Lambda^{2})},
\end{eqnarray}
with $n_{f}=3$ and $\Lambda=0.2~\mathrm{GeV}$.
We adopt MMHT2014 NNLO set for the parton distribution function of proton \cite{Harland-Lang:2014zoa,Harland-Lang:2015nxa}, and choose the factorization scale to be $\mu=\sqrt{4p_{T}^{2}}$ \cite{Fu:2011zzm}.
The mass range of dileptons is $0.1\ \mathrm{GeV}<M<0.3\ \mathrm{GeV}$.
In our calculations the coherence condition is included for coherent reactions, which means that the wavelength of the photon is larger than the size of the nucleus, and the charged constituents inside the nucleus should act coherently.
This condition limits $Q^{2}$ and $y$ to very low values, $Q^{2}_{\mathrm{max}}=0.027\ \textrm{GeV}^{2}$ and $y_{\mathrm{max}}=0.16$ for proton.
Furthermore, the cross section for the LO initial parton hard scattering (hard.scat.) satisfies the following form
\begin{eqnarray}\label{hard.scat.}
\sigma_{\mathrm{hard.scat.}}=\int dM^{2}dx_{a}dx_{b}f_{a/p}(x_{b},\mu^{2})f_{b/p}(x_{b},\mu^{2})d\sigma(a+b\rightarrow l^{+}l^{-}+b),
\end{eqnarray}
where the subprocesses are $q\bar{q}\rightarrow g(\gamma^{*}\rightarrow l^{+}l^{-})$ and $qg\rightarrow q(\gamma^{*}\rightarrow l^{+}l^{-})$.
\begin{figure*}[htbp]
\centering
\includegraphics[width=0.45\columnwidth]{Q2_dile_7TeV.eps}
\includegraphics[width=0.45\columnwidth]{Q2_CS_dile_7TeV.eps}\\
\includegraphics[width=0.45\columnwidth]{Q2_dile_14TeV.eps}
\includegraphics[width=0.45\columnwidth]{Q2_CS_dile_14TeV.eps}
\caption{The left panels plot the ratios of differential cross sections in different forms to the exact ones, while the right panels plot the exact results of $d\sigma/dQ^{2}$.
The upper and lower panels show the results in the different LHC energies, respectively.
The abbreviation NMFF represents the exact result with no contribution of magnetic form factor.
And each results are the sum of the direct and resolved photon contributions.}
\label{fig:Q2.dile.}
\end{figure*}
In Fig.~\ref{fig:Q2.dile.}, we plot the comparison between differential cross sections in different forms and the exact ones in the left panels, and plot the exact results of $d\sigma/dQ^{2}$ in the right panels.
In the left panels, the WWA results nicely agree with the exact ones when $Q^{2}<0.1\ \mathrm{GeV}^{2}$, but the differences become evident with increasing $Q^{2}$.
The results that neglect the contribution of magnetic form factor (NMFF) almost have no difference compared to the exact ones when $Q^{2}<0.05\ \mathrm{GeV}^{2}$, the differences appear when $Q^{2}>0.1\ \mathrm{GeV}^{2}$ and become evident when $Q^{2}>1\ \mathrm{GeV}^{2}$.
At $Q^{2}=1\ \mathrm{GeV}^{2}$, the NMFF results deviate from the exact ones by about $63\%$; at $Q^{2}=10\ \mathrm{GeV}^{2}$ the deviations are about $84\%$.
Therefore, WWA is only valuable in small $Q^{2}$ region, its error is evident at large $Q^{2}$ domain and becomes rather serious in incoherent reactions.
And the contribution of magnetic form factor concentrates on the large $Q^{2}$ domain.
In the right panels, the coh. and incoh. contributions dominant the small and large $Q^{2}$ regions, respectively.
They become comparable at $Q^{2}=1\ \mathrm{GeV}^{2}$.
Comparing with the feature of WWA derived from the left panels, one can see that WWA is a good approximation for coherent reactions, and is essentially in contradiction with incoherent reactions.
\begin{figure*}[htbp]
\centering
\includegraphics[width=0.45\columnwidth]{y_dile_coh.eps}
\includegraphics[width=0.45\columnwidth]{y_CS_dile.eps}
\caption{(a): The ratios of differential cross sections of coh. in different forms to the exact ones.
(b): The exact results of $d\sigma/dy$.
And each results are the sum of direct and resolved photon contributions.}
\label{fig:y.dile.}
\end{figure*}
In Fig.~\ref{fig:y.dile.}, the results are expressed as functions of $y$.
In panel (a), the WWA results nicely agree with the exact ones when $y<0.3$, but the differences become evident with increasing $y$. Especially, when $y>0.8$ the curves show a pronounced rising.
The differences between the NMFF results and exact ones are non-negligible in the whole $y$ regions, and are the largest for large values of $y$.
At $y=0.1$, the NMFF results deviate from the exact ones by about $20\%$; at $y=0.7$ the deviations are about $86\%$.
Therefore, the error of WWA is small in small $y$ domain, while it is evident at large $y$ domain and becomes more obvious in incoh. case.
In panel (b), the curves of coh. are important when $y<0.5$ and rapidly deceased with $y$ increasing.
On the contrary, the contributions of incoh. are important in the whole $y$ regions and much higher than those of coh..
\begin{table*}[htbp]\footnotesize
\renewcommand\arraystretch{1.5}
\centering
\caption{\label{Total.CS.coh.}Total cross sections of the dilepton photoproduction in the coherent channel. }
\begin{tabular}{L{1.5cm}C{1.1cm}C{1.1cm}C{1.1cm}C{1.1cm}C{1.1cm}C{1.1cm}}
\hline
\hline
$\sigma_{\mathrm{coh}.}$ & Exact & $f_{\mathrm{DZ}}$ & $f_{\mathrm{Ny}}$ &
$f_{\mathrm{Kn}}$ & $f_{\mathrm{SC}}$ & $f_{\mathrm{MD}}$\\
\hline
$\sigma\ [\mathrm{pb}]$ $(7\ \textrm{TeV})$ & 23.14 & 38.61 & 32.56 & 35.82 & 26.12 & 23.30 \\
$\sigma/\sigma_{\mathrm{Exact}}$ $(7\ \textrm{TeV})$ & 1.00 & 1.67 & 1.41 & 1.55 & 1.13 & 1.01 \\
$\sigma\ [\mathrm{pb}]$ $(14\ \textrm{TeV})$ & 41.13 & 66.65 & 56.32 & 61.91 & 45.63 & 40.93 \\
$\sigma/\sigma_{\mathrm{Exact}}$ $(14\ \textrm{TeV})$ & 1.00 & 1.62 & 1.37 & 1.51 & 1.11 & 1.00 \\
\hline
\hline
\end{tabular}
\end{table*}
The main purpose of the present paper is studying the features of the equivalent photon spectra which are mentioned in Section \ref{The eps}, and illustrating the advantage of Eq.~(\ref{fgamma.MD.}) that we derived.
Thus, in Tables \ref{Total.CS.coh.} and \ref{Total.CS.incoh.} we calculate the total cross sections based on the different forms of spectra.
In the case of coherent reactions, we have seen that their contributions dominate the small $Q^{2}$ and $y$ regions, which is compatible with the validity conditions of WWA.
However, the WWA errors still appear at large values of $Q^{2}$ and $y$, thus the options of upper limits, $Q^{2}_{\mathrm{max}}$ and $y_{\mathrm{max}}$, become crucial to the precision of WWA. In Table \ref{Total.CS.coh.}:
\begin{enumerate}
\item We observe that the ratios of results based on $f_{\mathrm{DZ}}$, $f_{\mathrm{Ny}}$ and $f_{\mathrm{Kn}}$ to exact ones are about $1.4\sim1.7$, the common reason is that the integrations of these spectra are performed in the entire kinematical allowed regions: $Q^{2}_{\mathrm{max}}=\infty$ and $y_{\mathrm{max}}=1$ which include the large WWA errors, this verifies the views derived from Figs.~\ref{fig:Q2.dile.} and \ref{fig:y.dile.}, that WWA is only valuable in small $Q^{2}$ and $y$ domains.
Actually, in most of the physically interesting cases such a dynamical cut off $\Lambda_{\gamma}$ exists such that, the WWA errors can be effectively avoided and the photo-absorption cross sections differ only slightly from their values on the mass shell.
Thus, for the practical use of WWA, except considering the kinematically allowed regions, one should also elucidate whether there is a dynamical cut off $\Lambda^{2}_{\gamma}$, and estimate it.
However, the definite values of $\Lambda_{\gamma}^{2}$ for different processes are essentially different, and further studies are still needed.
\item For the spectrum $f_{\mathrm{DZ}}$, the advantage is that the form factor effects can be included, which properly describes the situation of the proton as photon emitter.
Since the WWA is usually used in electroproduction reactions, especially in the $ep$ collision, if one obtains the spectrum of proton from that of electron by just replacing the $m_{e}$ with $m_{p}$, it would overestimate the cross section by a factor of 2 or more.
For the spectrum $f_{\mathrm{Ny}}$, the ratios have a obvious reduction compared to those of $f_{\mathrm{DZ}}$, since $f_{\mathrm{Ny}}$ includes the $Q^{2}_{\mathrm{min}}$ term in Eq.~(\ref{fgamma.Gen.V}) which is omitted in $f_{\mathrm{DZ}}$, this factor is inversely proportional to $Q^{2}$ and thus has the obvious contribution in small $Q^{2}$ region.
Therefore, this $Q^{2}_{\mathrm{min}}$ term can not be neglected when performing the photon spectra for coherent reactions.
In addition, the ratios of $f_{\mathrm{Kn}}$ is somewhat higher than those of $f_{\mathrm{Ny}}$, since the effect of magnetic form factor of proton is included in this form compared to $f_{\mathrm{Ny}}$.
We have derived in Figs. \ref{fig:Q2.dile.} and \ref{fig:y.dile.} that the contribution of magnetic form factor concentrates on the large $Q^{2}$ domain, thus this term should essentially be excluded in the coherent case.
\item The ratios of $f_{\mathrm{SC}}$ are the smallest compared to those mentioned above, since in this form the hadronic interaction is easily excluded.
However, we can see that the deviation from the exact results still can not be neglected.
Finally, the results of $f_{\mathrm{MD}}$ nicely agree with the exact ones.
Since this form has two virtues: firstly, $f_{\mathrm{MD}}$ is derived from the complete form Eq.~(\ref{fgamma.Gen.}) which properly includes the $Q^{2}_{\mathrm{min}}$ term in Eq.~(\ref{fgamma.Gen.V}) and excludes the effects of magnetic form factor;
secondly, except considering kinematical limitations it adopts the coherence condition which limits $Q^{2}$ and $y$ to very low values, ($Q^{2}_{\mathrm{max}}=0.027\ \mathrm{GeV}^{2}$ and $y_{\mathrm{max}}=0.16$), this effectively avoid the WWA errors (which appear when $Q^{2}>0.1\ \mathrm{GeV}^{2}$ and $y>0.3$ [Figs. \ref{fig:Q2.dile.}, \ref{fig:y.dile.}]).
\end{enumerate}
\begin{table*}[htbp]\footnotesize
\renewcommand\arraystretch{1.5}
\centering
\caption{\label{Total.CS.incoh.}Total cross sections of the dilepton photoproduction in the incoherent channel.
The results of WWA are deduced from Eq.~(\ref{dWWA.Gen.}).}
\begin{tabular}{L{1.5cm}C{1.5cm}C{1.5cm}C{1.5cm}C{1.5cm}}
\hline
\hline
$\sigma_{\mathrm{incoh}.}$ & Exact & WWA & $f_{\mathrm{incoh.}}$ [Eq. (\ref{fgamma.incohI.})] & $f_{\mathrm{BKT}}$ \\
\hline
$\sigma\ [\mathrm{pb}]$ $(7\ \textrm{TeV})$ & 21.87 & 93.25 & 94.10 & 147.80 \\
$\sigma/\sigma_{\mathrm{Exact}}$ $(7\ \textrm{TeV})$ & 1.00 & 4.26 & 4.30 & 6.76 \\
$\sigma\ [\mathrm{pb}]$ $(14\ \textrm{TeV})$ & 35.14 & 170.11 & 175.27 & 266.04 \\
$\sigma/\sigma_{\mathrm{Exact}}$ $(14\ \textrm{TeV})$ & 1.00 & 4.84 & 4.99 & 7.57 \\
\hline
\hline
\end{tabular}
\end{table*}
For the incoherent reactions, we have seen that its contribution dominates the large $Q^{2}$ region and almost the whole $y$ region, this is in contradiction with the validity conditions of WWA.
In Table \ref{Total.CS.incoh.}, we observe that the ratios of the WWA parameterizations to the exact ones are prominent compared to those of the coherent cases.
This quantitatively verifies the inapplicability of WWA in incoherent reactions.
For the spectrum $f_{\mathrm{incoh}}$, the ratios should be actually much higher than the values given in Table \ref{Total.CS.incoh.}, since the term of weighting factor $1-G_{\mathrm{E}}^{2}(Q^{2})$ is omitted in this form, and will lead to unreasonable divergency in small $Q^{2}$ region.
This unphysical results are essentially caused by the serious double counting (The detailed discussion can be found in Ref.~\cite{Ma:2018zzq}).
However, an artificial cutoff $Q^{2}_{\mathrm{min}}=1\ \mathrm{GeV}^{2}$ is used in Eq.~(\ref{fgamma.incohI.}) to avoid this divergency.
But we can see that the result is still not accurate.
Furthermore, the ratios of $f_{\mathrm{BKT}}$ are the largest, since $f_{\mathrm{BKT}}$ is originally derived from $ep$ scattering, but is directly expanded to describe the probability of finding a photon in any relativistic fermion and to deal with hadronic collisions in Refs.~\cite{Kniehl:1990iv,Yu:2015kva,Yu:2017pot}, this will overestimate the cross sections.
Therefore, the accurate expression Eq.~(\ref{dabTL}) should be applied for the incoherent photon emission.
\begin{figure*}[htbp]
\centering
\includegraphics[width=0.45\columnwidth]{PT_dile_7TeV.eps}
\includegraphics[width=0.45\columnwidth]{PT_dile_14TeV.eps}
\caption{Black solid line---the exact result of coherent reactions [coh.(dir.+res.)].
Red dash line---the WWA result based on $f_{\mathrm{MD}}$ [coh.(dir.+res.)].
Blue solid line---the exact result of fragmentation dileptons produced by coherent reactions [coh.(dir.+res.)-frag.].
Magenta dash line---the WWA result based on $f_{\mathrm{MD}}$ for fragmentation dilepton photoproduction [coh.(dir.+res.)-frag.].
Dark cyan dot line---the exact result of incoherent reactions [incoh.(dir.+res.)].
Dark yellow dash dot line---the exact result of fragmentation dileptons produced by incoherent reactions [incoh.(dir.+res.)-frag.].
Wine dash dot dot line---the initial parton hard scattering processes [hard.scat.].
Olive short dash line---the sum of the above processes.
The solid lines coincide with the dash lines in the whole $p_{T}$ domain.
}
\label{fig:pT.dile.}
\end{figure*}
In Fig.~\ref{fig:pT.dile.}, we adopt the complete form Eq.~(\ref{dabTL}) and the modified equivalent photon spectrum Eq.~(\ref{fgamma.MD.}) to plot the dileptons photoproduction in $p_{T}$ distribution.
We find that the exact results of coherent photon emission are consistent with the ones of modified equivalent photon spectrum Eq.~(\ref{fgamma.MD.}) in the whole $p_{T}$ regions, this verifies again the virtue of $f_{\mathrm{MD}}$.
Since WWA is inapplicable for incoherent reactions, we only present the exact results.
One can see that the contributions of coh. and incoh. are comparable with each other, and are about two orders of magnitudes (OOM) smaller than the LO initial parton hard scattering.
This is similar to the results in Ref.\cite{Fu:2011zzm}, but is very different from the results in Ref.~\cite{Yu:2017pot} where the incoherent contributions are about two OOMs larger than the coherent ones, and the contributions of photoproduction processes are also about two OOMs larger than LO hard.scat.
Finally, it can be found that the correction of photoproduction processes to the dileptons production is about $20\%$.
\begin{figure*}[htbp]
\centering
\includegraphics[width=0.45\columnwidth]{Q2_pho_7TeV.eps}
\includegraphics[width=0.45\columnwidth]{Q2_CS_pho_7TeV.eps}\\
\includegraphics[width=0.45\columnwidth]{Q2_pho_14TeV.eps}
\includegraphics[width=0.45\columnwidth]{Q2_CS_pho_14TeV.eps}
\caption{Same as Fig.~\ref{fig:Q2.dile.} but for photons.}
\label{fig:Q2.pho.}
\end{figure*}
\begin{figure*}[htbp]
\centering
\includegraphics[width=0.45\columnwidth]{y_pho_coh.eps}
\includegraphics[width=0.45\columnwidth]{y_CS_pho.eps}
\caption{Same as Fig.~\ref{fig:y.dile.} but for photons.}
\label{fig:y.pho.}
\end{figure*}
\begin{table*}[htbp]\footnotesize
\renewcommand\arraystretch{1.5}
\centering
\caption{\label{Total.CS.coh.pho.}Total cross sections of the photoproduction of photon in the coherent channel. }
\begin{tabular}{L{1.5cm}C{1.1cm}C{1.1cm}C{1.1cm}C{1.1cm}C{1.1cm}C{1.1cm}}
\hline
\hline
$\sigma_{\mathrm{coh}.}$ & Exact & $f_{\mathrm{DZ}}$ & $f_{\mathrm{Ny}}$ &
$f_{\mathrm{Kn}}$ & $f_{\mathrm{SC}}$ & $f_{\mathrm{MD}}$\\
\hline
$\sigma\ [\mathrm{nb}]$ $(7\ \textrm{TeV})$ & 39.56 & 66.04 & 55.65 & 61.27 & 44.69 & 39.83 \\
$\sigma/\sigma_{\mathrm{Exact}}$ $(7\ \textrm{TeV})$ & 1.00 & 1.67 & 1.41 & 1.55 & 1.13 & 1.01 \\
$\sigma\ [\mathrm{nb}]$ $(14\ \textrm{TeV})$ & 70.35 & 113.65 & 96.41 & 105.57 & 79.04 & 70.65 \\
$\sigma/\sigma_{\mathrm{Exact}}$ $(14\ \textrm{TeV})$ & 1.00 & 1.62 & 1.37 & 1.50 & 1.12 & 1.00 \\
\hline
\hline
\end{tabular}
\end{table*}
\begin{table*}[htbp]\footnotesize
\renewcommand\arraystretch{1.5}
\centering
\caption{\label{Total.CS.incoh.pho.}Total cross sections of the photoproduction of photon in the incoherent channel.
The results of WWA are deduced from Eq. (\ref{dWWA.Gen.}).}
\begin{tabular}{L{1.5cm}C{1.5cm}C{1.5cm}C{1.5cm}C{1.5cm}}
\hline
\hline
$\sigma_{\mathrm{incoh}.}$ & Exact & WWA & $f_{\mathrm{incoh.}}$ [Eq. (\ref{fgamma.incohI.})] & $f_{\mathrm{BKT}}$ \\
\hline
$\sigma\ [\mathrm{nb}]$ $(7\ \textrm{TeV})$ & 37.89 & 158.67 & 160.41 & 252.06 \\
$\sigma/\sigma_{\mathrm{Exact}}$ $(7\ \textrm{TeV})$ & 1.00 & 4.19 & 4.23 & 6.65 \\
$\sigma\ [\mathrm{nb}]$ $(14\ \textrm{TeV})$ & 61.54 & 291.71 & 311.17 & 474.15 \\
$\sigma/\sigma_{\mathrm{Exact}}$ $(14\ \textrm{TeV})$ & 1.00 & 4.74 & 5.06 & 7.70 \\
\hline
\hline
\end{tabular}
\end{table*}
\begin{figure*}[htbp]
\centering
\includegraphics[width=0.45\columnwidth]{PT_pho_7TeV.eps}
\includegraphics[width=0.45\columnwidth]{PT_pho_14TeV.eps}
\caption{Same as Fig.~\ref{fig:pT.dile.} but for photons.}
\label{fig:pT.pho.}
\end{figure*}
Figs.~\ref{fig:Q2.pho.} and \ref{fig:y.pho.} present the $Q^{2}$ and $y$ distributions of real photons photoproduction.
It is shown that the differences between the WWA results and exact ones are more obvious.
The total cross sections of real photons are given by Tables \ref{Total.CS.coh.pho.} and \ref{Total.CS.incoh.pho.}, the differences between the exact results and the
ones of equivalent photon spectra are still prominent.
And Eq.~(\ref{fgamma.MD.}) is also a good approximation for the coherent photoproduction of real photons.
In Fig.~\ref{fig:pT.pho.}, the $p_{T}$ distribution of real photons production is plotted.
We also compare our results of real photons to Refs.\cite{Fu:2011zzm,Yu:2017pot}, where the inaccuracies of equivalent photon spectra are more evident.
~~~~~~~~~~
\section{Conclusion}
\label{Conclusion}
~~~~~~~~~~
In this work, we have studied the validity of several equivalent photon spectra which are widely used in previous studies and derived a modified photon flux of proton, by giving a consistent analysis of the terms neglected in going from the accurate expression to the WWA one.
Since the equivalent photon spectrum plays the fundamental role in the photoproduction processes, we have taken photoproductions of photons and dileptons as examples, to express the comparison between the WWA results and the exact ones as the distributions in $Q^{2}$ and $y$, the total cross sections were also given.
The modified equivalent photon spectrum reproduces the exact result within less than one percent.
In the sequel, we have applied the accurate expression and the modified photon flux to the $p_{T}$ dependent cross sections.
Our results are different from the results in a previous study, and show that the corrections of photoproduction processes to the dileptons and photons prolductions are about $20\%$.
~~~~~~~~~~
\section*{Acknowledgements}
~~~~~~~
This work is supported in part by National Key R \& D Program of China under grant No. 2018YFA0404204, the NSFC (China) grant 11575043.
Z. M. is supported by the fellowship of China Postdoctoral Science Foundation under grant No. 2021M692729, and by Yunnan Provincial New Academic Researcher Award for Doctoral Candidates.
~~~~~~~~~~
|
2024-02-18T23:40:24.391Z
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2021-12-30T02:20:56.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14399","language":"en","timestamp":1640830856000,"url":"https:\/\/arxiv.org\/abs\/2112.14399","yymm":"2112"}
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proofpile-arXiv_000-10014
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{"provenance":"002.jsonl.gz:10015"}
| null | null |
\section{Introduction}\label{s1}
\noindent
\subsection{Sparse Signal Reconstruction}\label{s1.1}
\noindent
In compressed sensing (CS), a crucial concern is to reconstruct a high-dimensional
signals from a relatively small number of linear measurements
\begin{equation}\label{systemequationsnoise}
\bm{ b}=\bm{ A}\bm{x}+\bm{e},
\end{equation}
where $\bm{ b}\in\mathbb{R}^m$ is a vector of measurements, $\bm{ A}\in\mathbb{R}^{m\times n}~(m\ll n)$ is a sensing matrix modeling the linear measurement process,
$\bm{x}\in \mathbb{R}^{n}$ is an unknown sparse or compressible signal and $\bm{e}\in\mathbb{R}^{m}$ is a vector of measurement errors, see, e.g., \cite{candes2006stable,donoho2006compressed}.
To reconstruct $\bm{ x}$, the most intuitive approach is to find the sparsest signal, that is, one solves
via the $\ell_{0}$ minimization problem:
\begin{equation}\label{VectorL0}
\min_{\bm{x}\in\mathbb{R}^n}\|\bm{x}\|_0~~\text{subject~ to}~~\bm{b}-\bm{ A}\bm{ x}\in\mathcal{B},
\end{equation}
where $\|\bm{x}\|_0$
is the number of nonzero coordinates of $\bm{ x}$ and $\mathcal{B}$ is a bounded set determined by the error structure.
It is known that the problem \eqref{VectorL0} is NP-hard for high dimensional signals and
faces challenges in both theoretical and computational (see, e.g., \cite{Donoho2005Stable,2006High,2006On}).
Then many fast and effective algorithms have been developed to
recover $\bm{ x}$ from \eqref{systemequationsnoise}. The Least Absolute Shrinkage and Selector Operator (Lasso) is
among the most well-known algorithms, i.e.,
\begin{equation}\label{VectorL1-Lasso}
\min_{\bm{ x}\in\mathbb{R}^n}~\lambda\|\bm{ x}\|_1+\frac{1}{2}\|\bm{ Ax} -\bm{b}\|_2^2,
\end{equation}
where $\lambda> 0$ is a parameter to balance the data fidelity term $\|\bm{Ax} -\bm{ b}\|_{2}^{2}/2$ and the regularized item $\|\bm{ x}\|_{1}$.
In the paper, we mainly consider the case that $\bm{ x}$ in numerous practical applications is not sparse itself
but is compressible with respect to some given tight frame $\bm{ D}\in\mathbb{R}^{n\times d}~(d\geq n)$.
That is,
$\bm{ x}=\bm{ D}\bm{ f}$, where $\bm{ f}\in\mathbb{R}^{d}$ is sparse.
Here the Fig. \ref{figure.Original_signal_and_tansformation_coefficents} gives an example, in which the original signal is density but is sparse under a transformation. Readers can refer to \cite{majumdar2015energy} and find such a real signal example of electroencephalography (EEG) and electrocardiography (ECG).
And we refer to \cite{han2007frames} for basic theory on tight frame.
Many recovery results for standard sparse signals
have been extended to the dictionary sparse recovery, see, e.g., \cite{bruckstein2009sparse,candes2011compressed,lin2012new} for $\ell_{1}$ and \cite{li2014compressed,lin2016restricted} for $\ell_{p}~(0<p<1)$ and references therein.
Specially, based on the idea of \eqref{VectorL1-Lasso},
the following $\ell_1$-analysis problem in \cite{candes2011compressed} is developed:
\begin{equation}\label{VectorL1-ana}
\min_{\bm{ x}\in\mathbb{R}^n}\|\bm{D}^{\top}\bm{ x}\|_1~~\text{subject~ to}
~~\bm{b}-\bm{A}\bm{x}\in\mathcal{B}.
\end{equation}
One of the most commonly used frameworks to investigate the theoretical
performance of the $\ell_{1}$-analysis method is the
restricted isometry property adapted to $\bm{D}$ ($\bm{D}$-RIP), which is first proposed by
Cand{\`e}s et al. \cite{candes2011compressed} and can be defined as follows.
\begin{definition}\label{def.DRIP}
A matrix $\bm{ A}\in\mathbb{R}^{m\times n}$
is said to obey the restricted isometry property adapted to $\bm{ D}\in\mathbb{R}^{n\times d}$
($\bm{D}$-RIP) of order $s$ with constant $\delta_{s}$ if
\begin{equation*}
(1-\delta_{s})\|\bm{Dx}\|_{2}^{2}\leq \|\bm{ A}\bm{Dx}\|_2^2\leq (1+\delta_{s})\|\bm{Dx}\|_{2}^{2}
\end{equation*}
holds for all $s$-sparse vectors $\bm{ v}\in\mathbb{R}^{d}$. The smallest constant $\delta_{s}$ is called as the the restricted isometry constant adapted to $\bm{D}$ ($\bm{D}$-RIC).
\end{definition}
Note that when $\bm{D}$ is an identity matrix, i.e., $\bm{D}=\bm{I}\in\mathbb{R}^{n\times n}$, the definition of $\bm{D}$-RIP reduces to the standard RIP in \cite{candes2005decoding,candes2006stable}.
In the recent literature \cite{lin2012new},
the restricted orthogonality constant (ROC) in \cite{candes2005decoding} for the standard
CS is extended to the CS with tight frames, which is defined as follows.
\begin{definition}\label{def.DROC}
The $\bm{D}$-restricted orthogonality constant (abbreviated as $\bm{D}$-ROC) of order $(s,t)$, is the
smallest positive number which satisfies
\begin{equation*}
|\langle \bm{ADu}, \bm{ADv}\rangle -\langle \bm{Du}, \bm{Dv}\rangle|\leq \theta_{s,t} \|\bm{u}\|_2\|\bm{v}\|_2
\end{equation*}
for all $s$-sparse and $t$-sparse vectors $\bm{u}, \bm{v}\in\mathbb{R}^{d}$.
\end{definition}
Given the definition $\bm{D}$-RIP and $\bm{D}$-ROC.
For any positive real number $\tau$ , we denote $\theta_{s,\tau t}$ and $\delta_{\tau t}$ as
$\theta_{s,\lceil \tau t \rceil}$ and $\delta_{\lceil \tau t\rceil}$, respectively. Here and below, $\lceil t \rceil$ denotes the nearest integer greater than or equal to $t$. With the new notion,
in term of $\bm{D}$-ROC, some sufficient conditions in \cite{lin2012new} have been proposed, such as
$\delta_{s}+1.25\theta_{s,s}<1$, and $\delta_{1.25s}+\theta_{s,1.25s}<1$.
\subsection{MRI Reconstruction Based on Compressed Sensing}\label{s1.2}
\noindent
On the other hand, the idea of sparse signal reconstruction (compressed sensing) also has been used in Magnetic Resonance Imaging (MRI) reconstruction. MRI is crucial in clinical applications for disease diagnosis, since its noninvasive and nonionizing radiation properties enables advanced visualization of anatomical structure. However, MRI
is limited by the scanning time for physical and physiological constraints \cite{lustig2007sparse}.
In order to reduce the scanning period, the CS method has been introduced into MRI achieving the capability of accelerating the imaging speed, which is called CS-MRI technology \cite{lustig2007sparse}.
The success of CS-MRI largely relies on sparsity representation \cite{lustig2008compressed} of the MRI images.
The image is always assumed to be sparse (or compressible) in sparse transform domain (as shown in Fig. \ref{figure.MRI_and_its_transformation}) for the sparsity-based image prior. Consequently, one can model the reconstruction process through minimizing the regularization function to promote the sparse solution. Sparse representations are confirmed to usually lead to lower reconstruction error \cite{ravishankar2010mr}.
In recent years, some scholars used redundant representation systems instead of orthogonal ones in the MRI community. Some representatives of redundant representations systems are undecimated or shift-invariant wavelet frames
\cite{baker2011translational,guerquin2011fast,liu2015balanced,vasanawala2011practical}, patch-based methods \cite{lai2016image,qu2014magnetic,zhan2015fast}, over-complete dictionaries \cite{ravishankar2010mr,huang2014bayesian}, etc. Redundancy in a redundant system obtains robust image representations and
also introduces additional benefits. For instance, the redundancy in wavelet
enables shift-invariant property. For patch-based methods, redundancy lead to better noise removal and artifact suppression. The redundant coming from over-complete dictionaries contains more atom signals than required to
represent images, which can better capture different image features, and make the representations
relatively sparser than orthogonal dictionaries do. Redundancy also makes the designing or training of such dictionaries more flexible. Even though the trained dictionary is orthogonal for image patches, the overall representation system for the whole image maybe still be redundant \cite{zhan2015fast,cai2014data} due to overlapping of patches. Most of these redundant representation systems above can be categorized as tight frame systems \cite{vetterli2012foundations}. More works about CS-MRI, readers can refer to two review papers \cite{sandilya2017compressed,ye2019compressed}.
In this paper, we focus on tight frame-based MRI image reconstruction methods.
\begin{figure*}[htbp!]
\setlength{\tabcolsep}{4.0pt}\small
\begin{tabular}{c}
\includegraphics[width=16.0cm,height=8.0cm]{Original_signal_and_tansformation_coefficents_v2.eps}
\end{tabular}
\centering
\caption{\label{figure.Original_signal_and_tansformation_coefficents} Original signal and its transformation coefficients. Upper: the original signal, which is density (not sparse); Down: the coefficients under tight frame transformation, which is sparse. }
\vspace{-0.1cm}
\end{figure*}
\begin{figure*}[htbp!]
\setlength{\tabcolsep}{4.0pt}\small
\begin{tabular}{c}
\includegraphics[width=16.0cm,height=8.0cm]{BarinMRI_and_its_transformation_v1.eps}
\end{tabular}
\centering
\caption{\label{figure.MRI_and_its_transformation} Brain MRI and its transformation coefficients. Left: the original Brain MRI, which is density (not sparse); Right: the coefficients under tight frame transformation, which is sparse. Here the black and white colors denote the pixel value $0$ and $1$, respectively.}
\vspace{-0.1cm}
\end{figure*}
\subsection{Contributions}\label{s1.3}
\noindent
In this paper, we introduce the unconstrained $\ell_{1}-\alpha\ell_{2}$-analysis minimization:
\begin{equation}\label{VectorL1-alphaL2-ASSO}
\min_{\bm{x}\in\mathbb{R}^n}~\lambda(\|\bm{ D}^{\top}\bm{x}\|_{1}-\alpha\|\bm{ D}^{\top}\bm{x}\|_{2})
+\frac{1}{2}\|\bm{Ax}-\bm{b}\|_{2}^{2}
\end{equation}
for some constant $0<\alpha\leq 1$, where $\lambda$ is a regularization parameter. Denote \eqref{VectorL1-alphaL2-ASSO} as Analysis $\ell_{1}-\alpha\ell_{2}$-Shrinkage and Selector Operator ($\ell_{1}-\alpha\ell_{2}$-ASSO).
It is rooted to the sparse signal under tight frame and
the constrained and unconstrained $\|\bm{x}\|_{1}-\alpha\|\bm{ x}\|_{2}$ minimizations, which has recently attracted a lot of attention. The constrained $\|\bm{x}\|_{1}-\alpha\|\bm{x}\|_{2}$ minimization \cite{li2020minimization,ge2021dantzig,liu2017further,lou2018fast,lou2015computing,yin2015minimization} is
\begin{equation}\label{VectorL1-alphaL2}
\min_{\bm{x}\in\mathbb{R}^n}~\|\bm{x}\|_{1}-\alpha\|\bm{x}\|_{2}
\quad \text{subject \ to} \quad \bm{b}-\bm{A}\bm{x}\in\mathcal{B}.
\end{equation}
And they have been
introduced and studied different conditions based on RIP for the recovery of $\bm{x}$.
The unconstrained $\|\bm{x}\|_{1}-\alpha\|\bm{x}\|_{2}$ minimization
\cite{liu2017further,lou2018fast,lou2015computing,yin2015minimization,geng2020Unconstrained} is
\begin{equation}\label{VectorL1-L2-SSO}
\min_{\bm{x}\in\mathbb{R}^n}~\lambda(\|\bm{x}\|_{1}-\|\bm{x}\|_{2})+\frac{1}{2}\|\bm{A}\bm{x}-\bm{b}\|_2^2.
\end{equation}
It is a key bridge for finding the solution of the constrained $\|\bm{x}\|_{1}-\alpha\|\bm{x}\|_{2}$ minimization \eqref{systemequationsnoise}.
There is an effective algorithm based on the different of convex algorithm (DCA) to solve \eqref{VectorL1-L2-SSO}, see
\cite{lou2015computing,yin2015minimization}. Numerical examples in \cite{li2020minimization,ge2021dantzig,lou2015computing,yin2015minimization} demonstrate that the $\ell_{1}-\alpha\ell_{2}$ minimization consistently outperforms the $\ell_{1}$ minimization and the $\ell_{p}$ minimization in \cite{lai2013improved} when the measurement matrix $\bm{A}$ is highly coherent.
Motivated by the smoothing and decomposition
transformations in \cite{tan2014smoothing}, the $\ell_{1}-\alpha\ell_{2}$-ASSO is written as a general nonsmooth convex optimization problem:
\begin{equation}\label{VectorL1-alphaL2-RASSO}
\min_{\bm{x}\in\mathbb{R}^n}~\lambda(\|\bm{z}\|_{1}-\alpha\|\bm{z}\|_{2})
+\frac{1}{2}\|\bm{Ax}-\bm{b}\|_{2}^{2}+\frac{\rho}{2}\|\bm{D}^{\top}\bm{x}-\bm{z}\|_{2}^{2}.
\end{equation}
We refer to it as Relaxed Analysis $\ell_{1}-\alpha\ell_{2}$-Shrinkage and Selector Operator ($\ell_{1}-\alpha\ell_{2}$-RASSO).
The main contributions are as follows:
\begin{enumerate}
\item[(i)] We develop a new elementary technique for tight frames (Propositions \ref{prop.DROC} and \ref{NonsparseROC}) and
show sufficient conditions based on $\bm{D}$-RIP and $\bm{D}$-ROC
for the recovery of $\bm{x}$ from \eqref{systemequationsnoise} via the $\ell_{1}-\alpha\ell_{2}$-ASSO \eqref{VectorL1-alphaL2-ASSO}, see to Theorem \ref{StableRecoveryviaVectorL1-alphaL2-ASSO}.
\item[(ii)] We show sufficient conditions based on $\bm{D}$-RIP frame
for the recovery of the signal $\bm{x}$ from \eqref{systemequationsnoise} via the $\ell_{1}-\alpha\ell_{2}$-RASSO \eqref{VectorL1-alphaL2-RASSO}, see to Theorem \ref{StableRecoveryviaVectorL1-alphaL2-RASSO}.
\item[(iii)]In order to find the solution of \eqref{VectorL1-alphaL2-ASSO},
we proposed an algorithm based on ADMM, which offers a noticeable boost to some sparse reconstruction algorithms for tight frames. Numerical examples based the effective algorithm is presented for both sparse signals and MRI.
The recovery performance of the propose method is highly comparable (often better) to existing state-of-the-art methods.
\end{enumerate}
\subsection{Notations and Organization}\label{s1.4}
\noindent
Throughout the paper, we use the following basic notations.
For any positive integer $n$, let $[[1,n]]$ be the set $\{1,\ldots,n\}$.
For the index set $S\subseteq [[1,n]]$, let $|S|$ be the number of entries in $S$, $\bm{x}_S$ be the vector equal to $\bm{x}$ on $S$ and to zero on $S^c$, and $\bm{D}_{S}$ be $\bm{D}$ with all but the columns indexed by $S$ set to zero vector.
For $\bm{x}\in\mathbb{R}^n$, denote
$\bm{x}_{\max(s)}$ as the vector $\bm{x}$ with all but the largest $s$ entries in absolute value set to zero,
and $\bm{x}_{-\max(s)}=\bm{x}-\bm{x}_{\max(s)}$. Let $\|\bm{D}^{\top}\bm{x}\|_{\alpha,1-2}$ be
$\|\bm{D}^{\top}\bm{x}\|_1-\alpha\|\bm{D}^{\top}\bm{x}\|_2$.
Especially,
when $\alpha=1$, denote $\|\bm{D}^{\top}\bm{x}\|_{\alpha,1-2}$ with $\|\bm{D}^{\top}\bm{x}\|_{1-2}$.
And we denote $n\times n$ identity matrix by $\bm{I}_{n}$, zeros matrix by $\bm{ O}$, and the transpose of matrix
$\bm{A}$ by $\bm{A}^{\top}$. Use the phrase ``$s$-sparse vector" to refer to vectors of sparsity at most $s$. We use boldfaced letter to denote matrix or vector. We denote the standard inner product and Euclidean norm $\langle \cdot, \cdot\rangle$ and $\|\cdot\|_{2}$, respectively.
The rest of the paper is organized as follows. In Section \ref{adds2}, we describe some preliminaries about the tight frame and the unconstraint $\ell_{1}-\alpha\ell_{2}$ analysis models. In Section \ref{s3}, we give the main conclusions and show their proof. Then we design an algorithm based on projected PISTA to solve the proposed model \eqref{VectorL1-alphaL2-ASSO} in Section \ref{s4}. And we also apply our algorithm to solve the proposed model for signal and MRI reconstruction in Section \ref{s5}. Conclusions are given in Section \ref{s6}.
\section{Preliminaries}\label{adds2}
\noindent
In this section, we first recall some significant lemmas in order to analyse
the $\ell_{1}-\alpha\ell_{2}$-ASSO \eqref{VectorL1-alphaL2-ASSO} and $\ell_{1}-\alpha\ell_{2}$-RASSO \eqref{VectorL1-alphaL2-RASSO}.
In view of the above-mentioned facts in Section \ref{s1}, the $\bm{D}$-RIP plays a key role for the
stable recovery of signal via the analysis approach. The following lemma show some basic properties in \cite{lin2012new} for the $\bm{D}$-RIP with $\delta_{s}$ and $\theta_{s,t}$.
\begin{lemma}\label{Basicproperty}
\begin{enumerate}
\item[(i)]For any $s_{1}\leq s_{2}\leq d$, we have $\delta_{s_{1}}\leq \delta_{s_{2}}$.
\item[(ii)]For any positive integers $s$ and $t_{1}\leq t_{2}$, we have $\theta_{s,t_{2}}\leq\sqrt{\frac{t_{2}}{t_{1}}}\theta_{s,t_{1}}$.
\item[(iii)]$\theta_{s_{1},t_{1}}\leq\theta_{s_{2},t_{2}}$ if $s_{1}\leq s_{2}$ and $t_{1}\leq t_{2}$ with $s_{2}+t_{2}\leq d$.
\item[(iv)]For all nonnegative integers $s,t\leq d$, we have $\theta_{s,t}\leq \delta_{s+t}$.
\end{enumerate}
\end{lemma}
Here, we proposed the standard properties for the dictionary $\bm{D}$.
\begin{proposition}\label{prop.DROC}
Let $\bar{\bm{D}}\in\mathbb{R}^{(d-n)\times d}$ be the orthogonal complement of $\bm{D}\in\mathbb{R}^{n\times d}$, i.e., $\bar{\bm{D}}\bm{D}^{\top}=\bm{0}$, $\bar{\bm{D}}\bar{\bm{D}}^{\top}=\bm{I}_{n-d}$. Then,
\begin{enumerate}
\item[(i)] For any $\bm{v}\in\mathbb{R}^{d}$, $\|\bm{v}\|_{2}^{2}=\|\bm{Dv}\|_{2}^{2}+\|\bar{\bm{D}}\bm{v}\|_{2}^{2}$.
\item[(ii)] For any $s$-sparse signal $\bm{v}\in\mathbb{R}^{d}$, we have
$$
(1-\delta_{s})\|\bm{v}\|_{2}^{2}\leq \|\bm{A}\bm{Dv}\|_2^2+\|\bar{\bm{D}}\bm{v}\|_{2}^{2}\leq (1+\delta_{s})\|\bm{v}\|_{2}^{2}.
$$
\item[(iii)]$\langle \bm{u}, \bm{v}\rangle=\langle \bar{\bm{D}}\bm{u}, \bar{\bm{D}}\bm{v}\rangle
+\langle \bm{D}\bm{u}, \bm{D}\bm{v}\rangle$ for any $\bm{u}\in\mathbb{R}^{d}$.
\item[(iv)] For any $\|\bm{u}\|_{0}\leq s$ and $\|\bm{v}\|_{0}\leq t$, we have
$$
|\langle \bm{A}\bm{Du}, \bm{A}\bm{Dv}\rangle+\langle\bar{\bm{D}}\bm{u}, \bar{\bm{D}}\bm{v}\rangle
-\langle\bm{u}, \bm{v}\rangle|
\leq\theta_{s,t}\|\bm{u}\|_{2}\|\bm{v}\|_{2}.
$$
Moreover, if $\text{\rm supp}(\bm{u})\cap\text{\rm supp}(\bm{v})=\emptyset$, we have
$$
|\langle \bm{A}\bm{Du}, \bm{A}\bm{Dv}\rangle+\langle\bar{\bm{D}}\bm{u}, \bar{\bm{D}}\bm{v}\rangle|
\leq\theta_{s,t}\|\bm{u}\|_{2}\|\bm{v}\|_{2}.
$$
\end{enumerate}
\end{proposition}
\begin{proof}
Despite the results $(i)$ and $(ii)$ in Proposition \ref{prop.DROC} are clear, we here give their proofs.
As far as we know, the results (iii) and (iv) are first proved.
$(i)$
For any $v\in\mathbb{R}^{d}$, there is
\begin{align*}
\|\bm{Dv}\|_{2}^{2}+\|\bar{\bm{D}}\bm{v}\|_{2}^{2}
&=\langle \bm{D}^{\top} \bm{D}\bm{v}, \bm{v}\rangle +\langle \bar{\bm{D}}^{\top}\bar{\bm{D}}\bm{v},\bm{v}\rangle
\nonumber\\
&=\bm{v}^{\top}\begin{bmatrix}
\bm{D}^{\top} & \bar{\bm{D}}^{\top} \\
\end{bmatrix}
\begin{bmatrix}
\bm{D} \\
\bar{\bm{D}} \\
\end{bmatrix}\bm{v}
=\|\bm{v}\|_{2}^{2},
\end{align*}
where the last equality comes from the definition of tight frame $\bm{D}\bm{D}^{\top}=\bm{I}_{n}$ and the definition of the orthogonal complement of frame $\bar{\bm{D}}\bar{\bm{D}}^{\top}=\bm{I}_{n-d}$.
$(ii)$
By the definition of $\bm{D}$-RIP, we have
\begin{align}\label{prop.DROC.eq6}
\|\bm{A}\bm{Dv}\|_2^2+\|\bar{\bm{D}}\bm{v}\|_{2}^{2}
&\leq (1+\delta_{s})\|\bm{Dv}\|_{2}^{2}+\|\bar{\bm{D}}\bm{v}\|_{2}^{2}
\leq (1+\delta_{s})\left(\|\bm{Dv}\|_{2}^{2}+\|\bar{\bm{D}}\bm{v}\|_{2}^{2}\right)\nonumber\\
&=(1+\delta_{s})\|\bm{v}\|_{2}^{2},
\end{align}
where the equality comes from the item $(i)$. Similarly, we can show the lower bound
\begin{eqnarray}\label{prop.DROC.eq7}
\|\bm{A}\bm{Dv}\|_{2}^{2}+\|\bar{\bm{D}}\bm{v}\|_{2}^{2}
\geq(1-\delta_{s})\|\bm{v}\|_{2}^{2}.
\end{eqnarray}
The combination of the upper bound \eqref{prop.DROC.eq6} and the lower bound \eqref{prop.DROC.eq7} gives the item (ii).
Next, we show our new conclusions items (iii) and (iv).
$(iii)$
According to the parallelogram identity, we have
\begin{align*}
\langle \bar{\bm{D}}\bm{u}, \bar{\bm{D}}\bm{v}\rangle
&=\frac{1}{4}\left(\|\bar{\bm{D}}(\bm{u}+\bm{v})\|_{2}^{2}
-\|\bar{\bm{D}}(\bm{u}-\bm{v})\|_{2}^{2}\right),\nonumber\\
\langle \bm{D}\bm{u}, \bm{D}\bm{v}\rangle
&=\frac{1}{4}\left(\|\bm{D}(\bm{u}+\bm{v})\|_{2}^{2}
-\|\bm{D}(\bm{u}-\bm{v})\|_{2}^{2}\right).
\end{align*}
Taking sum over both side of above two equalities, one has
\begin{align}\label{prop.DROC.eq5}
&\langle \bar{\bm{D}}\bm{u}, \bar{\bm{D}}\bm{v}\rangle
+\langle \bm{D}\bm{u}, \bm{D}\bm{v}\rangle\nonumber\\
&=\frac{1}{4}\left(\left(\|\bar{\bm{D}}(\bm{u}+\bm{v})\|_{2}^{2}+\|\bm{D}(\bm{u}+\bm{v})\|_{2}^{2}\right)
-\left(\|\bar{\bm{D}}(\bm{u}-\bm{v})\|_{2}^{2}+\|\bm{D}(\bm{u}-\bm{v})\|_{2}^{2}\right)\right)\nonumber\\
&=\frac{1}{4}\left(\|\bm{u}+\bm{v}\|_{2}^{2}-\|\bm{u}-\bm{v}\|_{2}^{2}\right),
\end{align}
where the second equality holds because of the item $(i)$. Last, by the above \eqref{prop.DROC.eq5} and the following parallelogram identity
\begin{equation*}
\langle \bm{u}, \bm{v}\rangle
=\frac{1}{4}\left(\|(\bm{u}+\bm{v})\|_{2}^{2}
-\|(\bm{u}-\bm{v})\|_{2}^{2}\right),
\end{equation*}
we get
\begin{equation*}
\langle \bm{u}, \bm{v}\rangle=\langle \bar{\bm{D}}\bm{u}, \bar{\bm{D}}\bm{v}\rangle
+\langle \bm{D}\bm{u}, \bm{D}\bm{v}\rangle.
\end{equation*}
$(iv)$
It follows from the item $(ii)$ and the definition of $\bm{D}$-ROC that
\begin{eqnarray*}
&&|\langle \bm{A}\bm{Du}, \bm{A}\bm{Dv}\rangle+\langle\bar{\bm{D}}\bm{u}, \bar{\bm{D}}\bm{v}\rangle
-\langle\bm{u}, \bm{v}\rangle|
=|\langle \bm{A}\bm{Du}, \bm{A}\bm{Dv}\rangle-\langle\bm{D}\bm{u}, \bm{D}\bm{v}\rangle|nonumber\\
&&\leq\theta_{s,t}\|\bm{u}\|_{2}\|\bm{v}\|_{2},
\end{eqnarray*}
which is our desired conclusion.
\end{proof}
Next, the following inequality is a modified cone
constraint inequality for the unconstrained $\|\bm{D}^{\top}\bm{x}\|_{1}-\alpha\|\bm{D}^{\top}\bm{x}\|_{2}$ minimization \eqref{VectorL1-alphaL2-ASSO}. Its proof is similar as that of \cite[Lemma 2.3]{geng2020Unconstrained} and we omit its proof.
\begin{lemma}\label{ConeTubeconstraint-L1-L2-ASSO}
Let $\bm{b}=\bm{Ax}+\bm{e}$ with
$\|\bm{e}\|_{2}\leq\lambda$, $T = \text{\rm supp}(\bm{D}^{\top}\bm {x}_{\max(t)})$, and $\hat{\bm{x}}$ be a
minimization solution of \eqref{VectorL1-alphaL2-ASSO}. Then
\begin{align}\label{ConeTubeconstraint-L1-L2-ASSO.eq1}
\|\bm{Ah}\|_{2}^{2}+2\lambda\|\bm{D}_{T^c}^{\top}\bm{h}\|_{1}\leq2\lambda(\|\bm{D}_{T}^{\top}\bm{h}\|_1
+\alpha\|\bm{D}^{\top}\bm{h}\|_2+2\|\bm{D}_{T^c}^{\top}\bm{x}\|_{1}
+\|\bm{Ah}\|_{2}),
\end{align}
where ${\bm h}=\hat{{\bm x}}-{\bm x}$. And moreover,
\begin{align*}
\|\bm{Ah}\|_2^2+2\lambda(\|{\bm D}_{T^c}^{\top}{\bm h}\|_1-\alpha\|{\bm D}_{T^c}^{\top}{\bm h}\|_2)
\leq2\lambda(\|{\bm D}_{T}^{\top}{\bm h}\|_1+2\|{\bm D}_{T^c}^{\top}{\bm x}\|_1
+\alpha\|{\bm D}_{T}^{\top}{\bm h}\|_2+\|\bm {Ah}\|_2),
\end{align*}
\begin{align*}
\|{\bm D}_{T^c}^{\top}{\bm h}\|_1\leq\|{\bm D}_{T}^{\top}{\bm h}\|_1+2\|{\bm D}_{T^c}^{\top}{\bm x}\|_1
+\alpha\|{\bm D}^{\top}{\bm h}\|_2+\|\bm {Ah}\|_2,
\end{align*}
and
\begin{align*}
\|{\bm Ah}\|_2^2\leq2\lambda(\|{\bm D}_{T}^{\top}{\bm h}\|_1+2\|{\bm D}_{T^c}^{\top}{\bm x}\|_1
+\alpha\|{\bm D}^{\top}{\bm h}\|_2+\|\bm {Ah}\|_2).
\end{align*}
\end{lemma}
For the $\ell_{1}-\alpha\ell_{2}$-RASSO \eqref{VectorL1-alphaL2-RASSO}, there is a similar cone
constraint inequality.
\begin{lemma}\label{ConeTubeconstraint-L1-L2-RASSO}
Let ${\bm b}={\bm {Ax}}+{\bm e}$ with
$\|\bm e\|_2\leq\lambda$, and $T = \text{\rm supp}({\bm D}^{\top}{\bm x}_{\max(s)})$. Then the
minimization solution $\hat{\bm x}$ of \eqref{VectorL1-alphaL2-RASSO} satisfies
\begin{align}\label{ConeTubeconstraint-L1-L2-RASSO.eq1}
\|\bm {Ah}\|_2^2+2\lambda\|{\bm D}_{T^c}^{\top}{\bm h}\|_1\leq2\lambda\bigg(\|{\bm D}_{T}^{\top}{\bm h}\|_1+\alpha\|{\bm D}^{\top}{\bm h}\|_2+2\|{\bm D}_{T^c}^{\top}{\bm x}\|_1
+\|\bm {Ah}\|_2+\frac{(\alpha+1)^2d}{2}\frac{\lambda}{\rho}\bigg),
\end{align}
where ${\bm h}=\hat{{\bm x}}-{\bm x}$.
\end{lemma}
\begin{proof}
Our proof is motivated by the proof of \cite[Lemma 2.3]{geng2020Unconstrained} and \cite[Lemma IV.4]{tan2014smoothing}.
Assume that $(\hat{\bm{x}},\hat{\bm{z}})$ is another solution of \eqref{VectorL1-alphaL2-RASSO}, then
\begin{align}\label{ConeTubeconstraint-L1-L2-RASSO.eq2}
0&\geq \bigg(\lambda(\|\hat{{\bm z}}\|_{1}-\alpha\|\hat{{\bm z}}\|_{2})+\frac{1}{2}\|{\bm A}\hat{{\bm x}}-{\bm b}\|_{2}^{2}+\frac{\rho}{2}\|{\bm D}^{\top}\hat{{\bm x}}-\hat{{\bm z}}\|_{2}^{2}\bigg)\nonumber\\
&\hspace{12pt}-\bigg(\lambda(\|{\bm D}^{\top}{\bm x}\|_{1}-\alpha\|{\bm D}^{\top}{\bm x}\|_{2})+\frac{1}{2}\|{\bm A}{\bm x}-{\bm b}\|_{2}^{2}\bigg)\nonumber\\
&=\frac{1}{2}\bigg(\|{\bm A}\hat{{\bm x}}-{\bm b}\|_{2}^{2}-\|{\bm A}{\bm x}-{\bm b}\|_{2}^{2}\bigg)\nonumber\\
&\hspace{12pt}+\bigg(\lambda(\|\hat{{\bm z}}\|_{1}-\alpha\|\hat{{\bm z}}\|_{2})+\frac{\rho}{2}\|{\bm D}^{\top}\hat{{\bm x}}-\hat{{\bm z}}\|_{2}^{2}-\lambda(\|{\bm D}^{\top}{\bm x}\|_{1}-\alpha\|{\bm D}^{\top}{\bm x}\|_{2})\bigg)\nonumber\\
&=:I_1+I_2.
\end{align}
Next, we deal with terms $I_1$ and $I_2$, respectively. By $\hat{{\bm x}}={\bm h}+{\bm x}$ and ${\bm Ax}-{\bm b}=-{\bm e}$,
\begin{align}\label{ConeTubeconstraint-L1-L2-RASSO.eq3}
I_1&=\frac{1}{2}\bigg(\|{\bm A}{\bm h}-{\bm e}\|_{2}^{2}-\|-{\bm e}\|_{2}^{2}\bigg)
=\frac{1}{2}\|{\bm A}{\bm h}\|_{2}^{2}-\langle {\bm A}{\bm h}, {\bm e}\rangle \nonumber\\
&\overset{(1)}{\geq} \frac{1}{2}\|{\bm A}{\bm h}\|_{2}^{2}-\|{\bm A}{\bm h}\|_{2}\|{\bm e}\|_{2}
\overset{(2)}{\geq} \frac{1}{2}\|{\bm A}{\bm h}\|_{2}^{2}-\lambda\|{\bm A}{\bm h}\|_{2},
\end{align}
where the inequality (1) comes from the Cauchy-Schwarz inequality, and the inequality (2) is from $\|{\bm e}\|_{2}\leq \lambda$.
Then, we estimate the term $I_{2}$. The subgradient optimality condition for \eqref{VectorL1-alphaL2-RASSO} is
\begin{align*
{\bm 0}&={\bm A}^{\top}({\bm A}\hat{{\bm x}}-{\bm b})+\rho{\bm D}({\bm D}^{\top}\hat{{\bm x}}-\hat{{\bm z}}),\nonumber\\
{\bm 0}&\in\lambda{\bm v}-\rho({\bm D}^{\top}\hat{{\bm x}}-\hat{{\bm z}}),
\end{align*}
where ${\bm v}$ is a Frech$\acute{e}$t sub-differential of the function $\|\hat{{\bm z}}\|_{1}-\alpha\|\hat{{\bm z}}\|_{2}$. Then, we derive that
\begin{align}\label{ConeTubeconstraint-L1-L2-RASSO.eq4}
I_2&= \lambda(\|\hat{{\bm z}}\|_{1}-\|{\bm D}^{\top}{\bm x}\|_{1})-\lambda\alpha(\|\hat{{\bm z}}\|_{2}-\|{\bm D}^{\top}{\bm x}\|_{2})
+\frac{\rho}{2}\|{\bm D}^{\top}\hat{{\bm x}}-\hat{{\bm z}}\|_{2}^{2}\nonumber\\
&= \lambda\left(\left\|{\bm D}^{\top}\hat{{\bm x}}-\frac{\lambda}{\rho}{\bm v}\right\|_{1}-\|{\bm D}^{\top}{\bm x}\|_{1}\right)
-\lambda\alpha\left(\left\|{\bm D}^{\top}\hat{{\bm x}}-\frac{\lambda}{\rho}{\bm v}\right\|_{2}-\|{\bm D}^{\top}{\bm x}\|_{2}\right)
+\frac{\rho}{2}\|\frac{\lambda}{\rho}{\bm v}\|_{2}^{2}\nonumber\\
&\overset{(a)}{\geq}\lambda\left(\|{\bm D}^{\top}\hat{{\bm x}}\|_1-\|{\bm D}^{\top}{\bm x}\|_1 -\frac{\lambda}{\rho}\|{\bm v}\|_{1}\right)
-\lambda\alpha\left(\|{\bm D}^{\top}\hat{{\bm x}}\|_2-\|{\bm D}^{\top}{\bm x}\|_2 +\frac{\lambda}{\rho}\|{\bm v}\|_{2} \right)
+\frac{\lambda^2}{2\rho}\|{\bm v}\|_{2}^{2}\nonumber\\
&=\lambda\left(\left(\|{\bm D}^{\top}\hat{{\bm x}}\|_1-\|{\bm D}^{\top}{\bm x}\|_1\right)
-\alpha\left(\|{\bm D}^{\top}\hat{{\bm x}}\|_2-\|{\bm D}^{\top}{\bm x}\|_2\right)\right)\nonumber\\
&\hspace*{12pt}+\frac{\lambda^2}{2\rho}(\|{\bm v}\|_{2}^{2}-2\|{\bm v}\|_{1}-2\alpha\|{\bm v}\|_{2})\nonumber\\
&=:I_{21}+I_{22},
\end{align}
where (a) is because of $\|{\bm u}\|_{p}-\|{\bm v}\|_{p}\leq \|\bm{u +v}\|_{p}\leq \|{\bm u}\|_{p}+\|{\bm v}\|_{p}$ for any $1\leq p\leq\infty$. Note that
\begin{align}\label{ConeTubeconstraint-L1-L2-RASSO.eq5}
I_{21}&=\lambda\left(\|{\bm D}_{T}^{\top}(\bm{ h+ x})+{\bm D}_{T^c}^{\top}(\bm{ h+ x})\|_1-\|{\bm D}_{T}^{\top}{\bm x}
+{\bm D}_{T^c}^{\top}{\bm x}\|_1\right)\nonumber\\
&\hspace*{12pt}-\lambda\alpha\left(\|{\bm D}^{\top}(\bm{ h+ x})\|_2-\|{\bm D}^{\top}{\bm x}\|_2\right)\nonumber\\
&\geq\lambda\left(\|{\bm D}_{T}^{\top}{\bm x}\|_1-\|{\bm D}_{T}^{\top}{\bm h}\|_1+\|{\bm D}_{T^c}^{\top}{\bm h}\|_1-\|{\bm D}_{T^c}^{\top}{\bm x}\|_1-
\|{\bm D}_{T}^{\top}{\bm x}\|_1-\|{\bm D}_{T^c}^{\top}{\bm x}\|_1\right)\nonumber\\
&\hspace*{12pt}-\lambda\alpha\left(\|{\bm D}^{\top}{\bm h}\|_2\right)\nonumber\\
&=\lambda(\|{\bm D}_{T^c}^{\top}{\bm h}\|_1-\|{\bm D}_{T}^{\top}{\bm h}\|_1-2\|{\bm D}_{T^c}^{\top}{\bm x}\|_1-\alpha\|{\bm D}^{\top}{\bm h}\|_2),
\end{align}
where the inequality is because of $\|\bm{u +v}\|_{1}\geq \|{\bm u}\|_{1}-\|{\bm v}\|_{1}$ and $\|\bm{u +v}\|_{1}\leq\|{\bm u}\|_{1}+\|{\bm v}\|_{1}$. On the other hand,
\begin{align}\label{ConeTubeconstraint-L1-L2-RASSO.eq6}
I_{22}&\overset{(1)}{\geq}\frac{\lambda^2}{2\rho}\left(\|{\bm v}\|_{2}^{2}-2(1+\alpha)\|{\bm v}\|_{1}\right)\nonumber\\
&=\frac{\lambda^2}{2\rho}\sum_{j=1}^{d}\left((|v|_j-(1+\alpha))^2-(1+\alpha)^2\right)
\overset{(2)}{\geq} -\frac{(1+\alpha)^2\lambda^2d}{2\rho},
\end{align}
where (1) is because of $\|{\bm v}\|_{2}\leq \|{\bm v}\|_{1}$, and the equality (2) holds if and only if $|v|_j=(1+\alpha)$ for all $j=1,\ldots,d$. Substituting the estimate \eqref{ConeTubeconstraint-L1-L2-RASSO.eq5} and \eqref{ConeTubeconstraint-L1-L2-RASSO.eq6} into \eqref{ConeTubeconstraint-L1-L2-RASSO.eq4}, one has
\begin{align}\label{ConeTubeconstraint-L1-L2-RASSO.eq7}
I_2\geq\lambda\left(\|{\bm D}_{T^c}^{\top}{\bm h}\|_1-\|{\bm D}_{T}^{\top}{\bm h}\|_1-2\|{\bm D}_{T^c}^{\top}{\bm x}\|_1-\alpha\|{\bm D}^{\top}{\bm h}\|_2-\frac{(1+\alpha)^2d\lambda }{2\rho}\right).
\end{align}
Last, substituting the estimate of $I_1$ in \eqref{ConeTubeconstraint-L1-L2-RASSO.eq3} and $I_2$ in \eqref{ConeTubeconstraint-L1-L2-RASSO.eq7} into the inequality \eqref{ConeTubeconstraint-L1-L2-RASSO.eq2}, we get \eqref{ConeTubeconstraint-L1-L2-RASSO.eq1}.
\end{proof}
The following lemma is the fundamental properties of the metric $\|{\bm x}\|_1-\alpha\|{\bm x}\|_2$ with
$0\leq\alpha\leq 1$, see \cite{yin2015minimization} and \cite{ge2021dantzig}.
\begin{lemma}\label{LocalEstimateL1-L2}
For any $\bm {x}\in \mathbb{R}^n$, the following statements hold:
\begin{description}
\item (a) Let $0 \leq \alpha\leq 1$, $T=\text{\rm supp}({\bm x})$ and $\|{\bm x}\|_0=s$. Then
\begin{align}\label{e:l12alphas}
(s-\alpha\sqrt{s})\min_{j\in T}|x_j|\leq\|{\bm x}\|_1-\alpha\|{\bm x}\|_2\leq(\sqrt{s}-\alpha)\|{\bm x}\|_2.
\end{align}
\item (b) Let $S, S_1, S_2\subseteq [n]$ satisfy $S=S_1\cup S_2$ and $S_1\cap S_2=\emptyset$. Then
\begin{align}\label{e:uplowerbound}
\|\bm {x}_{S_1}\|_{1}-\alpha\|\bm{x}_{S_1}\|_{2}+\|\bm{x}_{S_2}\|_{1}-\alpha\|\bm{x}_{S_2}\|_{2}\leq\|\bm{x}_{S}\|_{1}-
\alpha\|\bm{x}_{S}\|_{2}.
\end{align}
\end{description}
\end{lemma}
Now, we can give the key technical tool used in the main results.
It provides an estimation based on $\bm{D}$-ROC for $|\langle \bm{ ADu}, \bm{ ADv} \rangle +\langle \bar{{\bm D}}{\bm u},\bar{{\bm D}}{\bm v}\rangle|$, where
one of $\bm{u}$ or $\bm{v}$ is sparse.
Our idea is inspired by \cite[Lemma 5.1]{cai2013compressed} and \cite[Lemma 2.3]{li2019signal}.
\begin{proposition}\label{NonsparseROC}
Let $s_1, s_2\leq n$. Suppose ${\bm u},{\bm v}\in\mathbb{R}^d$ satisfy $\text{\rm supp}({\bm u})\cap \text{\rm supp}({\bm v})=\emptyset$ and $\bm u$ is $s_1$ sparse. If $\|{\bm v}\|_{1-2}\leq(s_2-\sqrt{s_2})\eta$ and $\|{\bm v}\|_\infty\leq\eta$, then
\begin{align}\label{NonsparseROC.eq1}
|\langle \bm{ ADu}, \bm{ ADv} \rangle +\langle \bar{{\bm D}}{\bm u},\bar{{\bm D}}{\bm v}\rangle|
\leq\bigg(1+\frac{\sqrt{2}}{2}\bigg)\eta \sqrt{s_2}\theta_{s_1,s_2}\|\bm{u}\|_2.
\end{align}
\end{proposition}
Before giving the proof of Proposition \ref{NonsparseROC}, we first recall
a convex combination of sparse vectors for any point based on the metric $\|\cdot\|_1-\|\cdot\|_2$.
\begin{lemma}\label{e:convexl1-2}\cite[Lemma2.2]{ge2021new}
Let a vector $\bm {\nu}\in \mathbb{R}^d$ satisfy $\|\bm {\nu}\|_{\infty}\leq \eta$, where $\alpha$ is a positive constant.
Suppose $\|\bm {\nu}\|_{1-2}\leq (s-\sqrt{s})\eta$
with a positive integer $s$ and $s\leq |\mathrm{supp}(\bm {\nu})|$. Then $\bm {\nu}$ can be represented as a convex
combination of $s$-sparse vectors $\bm {v}^{(i)}$, i.e.,
\begin{align}\label{e:com}
\bm {\nu}=\sum_{i=1}^N\lambda_i\bm {v}^{(i)},
\end{align}
where $N$ is a positive integer,
\begin{align}\label{e:thetaisum}
0<\lambda_i\leq1,\ \ \ \ \sum_{i=1}^N\lambda_i=1,
\end{align}
\begin{align}\label{e:setS}
\mathrm{supp}(\bm {v}^{(i)})\subseteq \mathrm{supp}(\bm {\nu}), \ \ \|\bm {v}^{(i)}\|_0\leq {s},\ \ \|\bm {v}^{(i)}\|_{\infty}\leq \Big(1+\frac{\sqrt{2}}{2}\Big)\eta,
\end{align}
and
\begin{align}\label{e:newinequality}
\sum_{i=1}^N\lambda_i\|\bm {v}^{(i)}\|_2^2
\leq \Big[\Big(1+\frac{\sqrt{2}}{2}\Big)^2(s-\sqrt{s})+1\Big]\theta^2.
\end{align}
\end{lemma}
Now, we give the proof of Proposition \ref{NonsparseROC} in details.
\begin{proof}
Suppose $\|\bm{v}\|_0=t$. We consider two cases as follows.
\textbf{Case I: $t\leq s_2$.}
By the item (iv) of Proposition \ref{prop.DROC} and $\|{\bm v}\|_\infty\leq\alpha$, we have
\begin{align}\label{NonsparseROC.eq2}
|\langle \bm{ ADu}, \bm{ ADv} \rangle +\langle \bar{{\bm D}}{\bm u},\bar{{\bm D}}{\bm v}\rangle|&\leq\theta_{s_1,t}\|{\bm u}\|_2\|{\bm v}\|_2\leq\theta_{s_1,t}\|{\bm u}\|_2\|{\bm v}\|_{\infty}\sqrt{\|{\bm v}\|_0}\nonumber\\
&\leq\eta\sqrt{s_2}\theta_{s_1,s_2}\|{\bm u}\|_2.
\end{align}
\textbf{Case II: $t> s_2$.}
We shall prove by induction. Assume that \eqref{NonsparseROC.eq1} holds for $t-1$.
Note that $\|{\bm v}\|_{1-2}\leq\eta (s_2-\sqrt{s_2})$ and $\|{\bm v}\|_{\infty}\leq \eta$. By Lemma \ref{e:convexl1-2}, ${\bm v}$ can be represented as the convex hull of $s_2$-sparse vectors:
$$
{\bm v}=\sum_{j=1}^N\gamma_j{\bm v}^j,
$$
where ${\bm v}^j$ is $s_2$-sparse for any $j\in[N]$ and
\begin{align*}
\sum_{j=1}^N\gamma_j=1,~~0<\gamma_j\leq 1, j\in[N].
\end{align*}
Since $\bm{v}^j$ is $s_2$-sparse and $s_2\leq t-1$, we use the induction assumption,
\begin{align}\label{NonsparseROC.eq3}
&|\langle \bm{ ADu}, \bm{ ADv} \rangle +\langle \bar{{\bm D}}{\bm u},\bar{{\bm D}}{\bm v}\rangle|
\leq\sum_{j=1}^N\gamma_j|\langle \bm{ ADu}, \bm{ AD}\bm{v}^j\rangle
+\langle \bar{{\bm D}}{\bm u},\bar{{\bm D}}{\bm v}^j\rangle|\nonumber\\
&\overset{(a)}{\leq} \sum_{j=1}^N\gamma_j\bigg(\theta_{s_1,s_2}\|\bm{v}^j\|_2\|\bm{u}\|_2\bigg)
\leq \sum_{j=1}^N\gamma_j\bigg(\theta_{s_1,s_2}\|\bm{v}^j\|_{\infty}\sqrt{\|\bm{v}^j\|_{0}}\|\bm{u}\|_2\bigg)\nonumber\\
&\overset{(b)}{\leq}\sum_{j=1}^N\gamma_j\bigg(\bigg(1+\frac{\sqrt{2}}{2}\bigg)\eta\sqrt{s_2}\theta_{s_1,s_2}\|\bm{u}\|_2\bigg)
=\bigg(1+\frac{\sqrt{2}}{2}\bigg)\eta\sqrt{s_2}\theta_{s_1,s_2}\|\bm{u}\|_2,
\end{align}
where (a) is in view of the item (iv) of Proposition \ref{prop.DROC}, and (b) is in virtue of Lemma \ref{e:convexl1-2}.
The combination of \textbf{Case I} and \textbf{Case II} is \eqref{NonsparseROC.eq1}.
\end{proof}
\section{Main Result based on D-RIC and D-ROC }\label{s3}
\noindent
In this section, we develop sufficient conditions
based on $\bm{D}$-RIC and $\bm{D}$-ROC of the
$\ell_{1}-\alpha \ell_2$-ASSO \eqref{VectorL1-alphaL2-ASSO} and $\ell_{1}-\alpha \ell_2$-RASSO \eqref{VectorL1-alphaL2-RASSO}
for the signal recovery applying the technique of the convex combination.
\subsection{Auxiliary Lemmas Under RIP Frame}\label{s3.1}
\noindent
Combining the above auxiliary results in Section \ref{adds2},
we first introduce some main inequalities, which play an important role in establishing the recovery condition of the
$\ell_{1}-\alpha \ell_2$-ASSO \eqref{VectorL1-alphaL2-ASSO} and $\ell_{1}-\alpha \ell_2$-RASSO \eqref{VectorL1-alphaL2-RASSO}.
\begin{proposition}\label{lem:CrossItem}
For positive integers $s\geq 2$ and $k\geq 1$,
let $S = \text{\rm supp}({\bm D}^{\top}{\bm h}_{\max(s)})$ and $T = \text{\rm supp}({\bm D}^{\top}{\bm x}_{\max(s)})$.
Assume that
\begin{eqnarray}\label{lem:CrossItem.eq1}
\|\bm{D}_{S^c}^{\top}\bm{h}\|_1-\alpha\|\bm{D}_{S^c}^{\top}\bm{h}\|_2
\leq a\|\bm{D}_{S}^{\top}\bm{h}\|_1+b\|\bm{D}_{S}^{\top}\bm{h}\|_2+c\|\bm{D}_{T^c}^{\top}\bm{x}\|_1+\eta\|{\bm {Ah}}\|_2
+\gamma,
\end{eqnarray}
where $a>0$ and $b,c, \eta,\gamma\geq0$ satisfy $(a-1)\sqrt{s}+(b+1)\geq0$. Then the following statements hold:
\begin{description}
\item[(i)] For the set $S$, one has
\begin{align}\label{e:CrossItem1}
&|\langle \bm{AD}\bm{D}_{S}^{\top}{\bm h},\bm{AD}\bm{D}_{S^c}^{\top}{\bm h}\rangle
|\nonumber\\
&\leq \theta_{s,s}\sqrt{s}\Big(1+\frac{\sqrt{2}}{2}\Big)\Big(\frac{a\sqrt{s}+b}{\sqrt{s}-1}\frac{\|\bm{D}_{S}^{\top}\bm{h}\|_2}{\sqrt{s}}
+\frac{c\|\bm{D}_{T^c}^{\top}\bm{x}\|_1+\eta\|{\bm Ah}\|_2+\gamma}{s-\sqrt{s}}\Big)\|\bm{D}_{S}^{\top}{\bm h}\|_2,
\end{align}
\item
\item[(ii)] Let the set
\begin{align}\label{def:S}
\tilde{S}=S\cup \Big\{i:|(\bm{D}_{S^c}^{\top}\bm{h})(i)|>\frac{1}{t-1}\frac{a\sqrt{s}+b}{\sqrt{s}-1}\frac{\|\bm{D}_{S}^{\top}\bm{h}\|_2}{\sqrt{s}}
+\frac{c\|\bm{D}_{T^c}^{\top}\bm{x}\|_1+\eta\|{\bm Ah}\|_2+\gamma}{(t-1)(s-\sqrt{s})}\Big\}
\end{align}
for any $t>3$. Then
\begin{align}\label{e:CrossItem2}
&|\langle \bm{AD}\bm{D}_{\tilde{S}}^{\top}{\bm h},\bm{AD}\bm{D}_{\tilde{S}^c}^{\top}{\bm h}\rangle
|\leq\theta_{ts, (t-1)s} \frac{\sqrt{\lceil(t-1)s\rceil}}{t-1} \Big(1+\frac{\sqrt{2}}{2}\Big)
\nonumber\\
&\times\Big(\frac{a\sqrt{s}+b}{\sqrt{s}-1}\frac{\|\bm{D}_{S}^{\top}\bm{h}\|_2}{\sqrt{s}}
+\frac{c\|\bm{D}_{T^c}^{\top}\bm{x}\|_1+\eta\|{\bm Ah}\|_2+\gamma}{s-\sqrt{s}}\Big)\|\bm{D}_{\tilde{S}}^{\top}{\bm h}\|_2.
\end{align}
\item[(iii)]
Let $\bm {A}$ satisfy $\bm D$-RIP with
\begin{align}\label{RIPConditiona1add}
\rho_{s,t}=:\delta_{ts}+\sqrt{\frac{\lceil(t-1)s\rceil}{(t-1)^2s}}\frac{(\sqrt{2}+1)(\sqrt{s}+\alpha)}{\sqrt{2}(\sqrt{s}-1)}
\theta_{st, (t-1)s}<1,
\end{align}
with $t\geq3$. Then
\begin{align}\label{prop:NSP.eq1}
\|\bm{D}_{S}^{\top}\bm{h}\|_2\leq&\frac{(\sqrt{2}+1)\theta_{ts,(t-1)s}}{\sqrt{2}(1-\rho_{s,t})}\frac{\sqrt{\lceil(t-1)s\rceil}}{(t-1)(s-\sqrt{s})}\Big(c\|\bm{D}_{T^c}^{\top}\bm{x}\|_1+\gamma\Big)
\nonumber\\
&+
\Big(\frac{(\sqrt{2}+1)\theta_{ts,(t-1)s}}{\sqrt{2}(1-\rho_{s,t})}\frac{\sqrt{(t-1)s}}{(t-1)(s-\sqrt{s})}+\frac{\sqrt{1+\delta_{ts}}}{1-\rho_{s,t}}\Big)\eta\|\bm{Ah}\|_{2},
\end{align}
where $\eta\geq 1$.
\item[(iv)]
Let $\bm {A}$ satisfy $\bm D$-RIP with
\begin{align}\label{RIPCondition1}
\rho_{s}=:\delta_{s}+\frac{(\sqrt{2}+1)(\sqrt{s}+\alpha)}{\sqrt{2}(\sqrt{s}-1)}
\theta_{s,s}<1.
\end{align}
Then
\begin{align}\label{prop:NSP.eq2}
\|\bm{D}_{S}^{\top}\bm{h}\|_2\leq&\frac{(\sqrt{2}+1)\theta_{s,s}}{\sqrt{2}(1-\rho_{s})(\sqrt{s}-1)}\Big(c\|\bm{D}_{T^c}^{\top}\bm{x}\|_1+\gamma\Big)
\nonumber\\
&+
\Big(\frac{(\sqrt{2}+1)\theta_{s,s}}{\sqrt{2}(1-\rho_{s})(\sqrt{s}-1)}+\frac{\sqrt{1+\delta_{s}}}{1-\rho_{s}}\Big)\eta\|\bm{Ah}\|_{2},
\end{align}
where $\eta\geq 1$.
\item[(v)] For the term $\|\bm{D}_{S^c}^{\top}\bm{h}\|_2$, there is
\begin{align}\label{DSC}
&\|\bm{D}_{S^c}^{\top}\bm{h}\|_2\nonumber\\
&\leq\Bigg(\sqrt{\frac{a\sqrt{s}+b}{\sqrt{s}}+\frac{\alpha^2}{4s}}+\frac{\alpha+\bar{\varepsilon}}{2\sqrt{s}}\Bigg)\|\bm{D}_{S}^{\top}\bm{h}\|_2
+\frac{1}{2\bar{\varepsilon}}\big(c\|\bm{D}_{T^c}^{\top}\bm{x}\|_1+\eta\|\bm{Ah}\|_2+\gamma\big),
\end{align}
where $\bar{\varepsilon}>0$ is a constant.
\end{description}
\end{proposition}
\begin{proof}
Please see Appendix \ref{appendx1}.
\end{proof}
\subsection{Main Result Under RIP Frame }\label{s3.2}
\noindent
Now, we show the stable recovery conditions based $\bm{D}$-RIP for
the $\ell_{1}-\alpha \ell_{2}$-ASSO (\ref{VectorL1-alphaL2-ASSO}) and
the $\ell_{1}-\alpha \ell_{2}$-RASSO (\ref{VectorL1-alphaL2-RASSO}), respectively.
\begin{theorem}\label{StableRecoveryviaVectorL1-alphaL2-ASSO}
Consider $\bm { b}=\bm { Ax}+\bm {e}$ with $\|\bm {e}\|_{2}\leq \eta$.
Let $\hat{\bm {x}}$ be the minimizer of the $\ell_{1}-\alpha \ell_{2}$-ASSO (\ref{VectorL1-alphaL2-ASSO}).
The following statements hold:
\begin{description}
\item[(i)]
If the measurement matrix $\bm {A}$ satisfies \eqref{RIPConditiona1add},
then
\begin{align*}
&\|\hat{\bm {x}}-\bm {x}\|_2\nonumber\\
\leq&
\Bigg(\left(\sqrt{\frac{\alpha+\sqrt{s}}{\sqrt{s}}+\frac{\alpha^2}{4s}}+\frac{\alpha+1}{2\sqrt{s}}+1
\right)\left(\tau+\frac{\left(\tau(1-\rho_{s,t})+\sqrt{1+\delta_{ts}}\right)C}{C(1-\rho_{s,t})+(\sqrt{s}+\alpha)\sqrt{1+\delta_{ts}}}\right)\nonumber\\
&\hspace*{12pt}+\frac{1}{2}\left(1+\frac{C(1-\rho_{s,t})}{C(1-\rho_{s,t})+(\sqrt{s}+\alpha)\sqrt{1+\delta_{ts}}}\right)\Bigg)
2\|\bm{D}_{T^c}^{\top}\bm{x}\|_1\nonumber\\
&+\left(\left(\sqrt{\frac{\alpha+\sqrt{s}}{\sqrt{s}}+\frac{\alpha^2}{4s}}+\frac{\alpha+1}{2\sqrt{s}}+1
\right)\frac{\tau(1-\rho_{s,t})+\sqrt{1+\delta_{ts}}}{1-\rho_{s,t}}+\frac{1}{2}\right)\nonumber\\
&\hspace*{12pt}\times\left(C+\frac{(\sqrt{s}+\alpha)\sqrt{1+\delta_{ts}}}{(1-\rho_{s,t})}\right)2\lambda,
\end{align*}
where the constants $\tau$ and $C$ are as follows
\begin{align}\label{Constant.t>=3}
\begin{cases}
\tau=\frac{(\sqrt{2}+1)\theta_{ts,(t-1)s}}{\sqrt{2}(1-\rho_{s,t})}\frac{\sqrt{(t-1)s}}{(t-1)(s-\sqrt{s})},&\\
C=1+\frac{(\sqrt{2}+1)\theta_{ts,(t-1)s}}{\sqrt{2}(1-\rho_{s,t})}\frac{(\sqrt{s}+\alpha)}{(s-\sqrt{s})}\frac{\sqrt{(t-1)s}}{t-1}.
\end{cases}
\end{align}
\item[(ii)]If the measurement matrix $\bm {A}$ satisfies
\eqref{RIPCondition1}, then
\begin{align*}
\|\hat{\bm {x}}-\bm {x}\|_2
\leq&
\Bigg(\left(\sqrt{\frac{\alpha+\sqrt{s}}{\sqrt{s}}+\frac{\alpha^2}{4s}}+\frac{\alpha+1}{2\sqrt{s}}+1
\right)\left(\tau+\frac{\left(\tau(1-\rho_{s})+\sqrt{1+\delta_{s}}\right)C}{C(1-\rho_{s})+(\sqrt{s}+\alpha)\sqrt{1+\delta_{s}}}\right)\nonumber\\
&\hspace*{12pt}+\frac{1}{2}\left(1+\frac{C(1-\rho_{s})}{C(1-\rho_{s})+(\sqrt{s}+\alpha)\sqrt{1+\delta_{s}}}\right)\Bigg)
2\|\bm{D}_{T^c}^{\top}\bm{x}\|_1\nonumber\\
&+\left(\left(\sqrt{\frac{\alpha+\sqrt{s}}{\sqrt{s}}+\frac{\alpha^2}{4s}}+\frac{\alpha+1}{2\sqrt{s}}+1
\right)\frac{\tau(1-\rho_{s})+\sqrt{1+\delta_{s}}}{1-\rho_{s}}+\frac{1}{2}\right)\nonumber\\
&\hspace*{12pt}\times\left(C+\frac{(\sqrt{s}+\alpha)\sqrt{1+\delta_{s}}}{(1-\rho_{s})}\right)2\lambda,
\end{align*}
where the constants $\tau$ and $C$ are as follows
\begin{align}\label{Constant.t=2}
\begin{cases}
\tau=\frac{(\sqrt{2}+1)\theta_{s,s}}{\sqrt{2}(1-\rho_{s})}\frac{1}{\sqrt{s}-1},&\\
C=1+\frac{(\sqrt{2}+1)\theta_{s,s}}{\sqrt{2}(1-\rho_{s})}\frac{\sqrt{s}+\alpha}{\sqrt{s}-1}.
\end{cases}
\end{align}
\end{description}
\end{theorem}
\begin{theorem}\label{StableRecoveryviaVectorL1-alphaL2-RASSO}
Consider $\bm { b}=\bm { Ax}+\bm {e}$ with $\|\bm {e}\|_{2}\leq \eta$. Let $\hat{\bm {x}}$ be the minimizer
of the $\ell_{1}-\alpha \ell_{2}$-RASSO (\ref{VectorL1-alphaL2-RASSO}).
The following statements hold:
\begin{description}
\item[(i)] If the measurement matrix $\bm {A}$ satisfies the $\bm D$-RIP with \eqref{RIPConditiona1add}, then
\begin{align*}
&\|\hat{\bm {x}}-\bm {x}\|_2\nonumber\\
\leq&
\Bigg(\left(\sqrt{\frac{\alpha+\sqrt{s}}{\sqrt{s}}+\frac{\alpha^2}{4s}}+\frac{\alpha+1}{2\sqrt{s}}+1
\right)\left(\tau+\frac{\left(\tau(1-\rho_{s,t})+\sqrt{1+\delta_{ts}}\right)C}{C(1-\rho_{s,t})+(\sqrt{s}+\alpha)\sqrt{1+\delta_{ts}}}\right)\nonumber\\
&\hspace*{12pt}+\frac{1}{2}\left(1+\frac{C(1-\rho_{s,t})}{C(1-\rho_{s,t})+(\sqrt{s}+\alpha)\sqrt{1+\delta_{ts}}}\right)\Bigg)\nonumber\\
&\times\left(2\|\bm{D}_{T^c}^{\top}\bm{x}\|_1+\frac{(\alpha+1)^2d}{2}\frac{\lambda}{\rho}\right)\nonumber\\
&+\left(\left(\sqrt{\frac{\alpha+\sqrt{s}}{\sqrt{s}}+\frac{\alpha^2}{4s}}+\frac{\alpha+1}{2\sqrt{s}}+1
\right)\frac{\tau(1-\rho_{s,t})+\sqrt{1+\delta_{ts}}}{1-\rho_{s,t}}+\frac{1}{2}\right)\nonumber\\
&\hspace*{12pt}\times\left(C+\frac{(\sqrt{s}+\alpha)\sqrt{1+\delta_{ts}}}{(1-\rho_{s,t})}\right)2\lambda,
\end{align*}
where the constants $\tau$ and $C$ are in \eqref{Constant.t>=3}.
\item[(ii)] If the measurement matrix $\bm {A}$ satisfies the $\bm D$-RIP with \eqref{RIPCondition1}, then
\begin{align*}
\|\hat{\bm {x}}-\bm {x}\|_2
\leq&
\Bigg(\left(\sqrt{\frac{\alpha+\sqrt{s}}{\sqrt{s}}+\frac{\alpha^2}{4s}}+\frac{\alpha+1}{2\sqrt{s}}+1
\right)\left(\tau+\frac{\left(\tau(1-\rho_{s})+\sqrt{1+\delta_{s}}\right)C}{C(1-\rho_{s})+(\sqrt{s}+\alpha)\sqrt{1+\delta_{s}}}\right)\nonumber\\
&\hspace*{12pt}+\frac{1}{2}\left(1+\frac{C(1-\rho_{s})}{C(1-\rho_{s})+(\sqrt{s}+\alpha)\sqrt{1+\delta_{s}}}\right)\Bigg)\nonumber\\
&\times\left(2\|\bm{D}_{T^c}^{\top}\bm{x}\|_1+\frac{(\alpha+1)^2d}{2}\frac{\lambda}{\rho}\right)\nonumber\\
&+\left(\left(\sqrt{\frac{\alpha+\sqrt{s}}{\sqrt{s}}+\frac{\alpha^2}{4s}}+\frac{\alpha+1}{2\sqrt{s}}+1
\right)\frac{\tau(1-\rho_{s})+\sqrt{1+\delta_{s}}}{1-\rho_{s}}+\frac{1}{2}\right)\nonumber\\
&\hspace*{12pt}\times\left(C+\frac{(\sqrt{s}+\alpha)\sqrt{1+\delta_{s}}}{(1-\rho_{s})}\right)2\lambda,
\end{align*}
where the constants $\tau$ and $C$ are in \eqref{Constant.t=2}.
\end{description}
\end{theorem}
\begin{remark}
The condition \eqref{RIPCondition1}
reduces to
\begin{equation}\label{RIPCondition2}
\delta_{2s}<\frac{1}{\frac{\bigg(1+\frac{\sqrt{2}}{2}\bigg)(\sqrt{s}+\alpha)}{\sqrt{s}-1}+1}.
\end{equation}
It is clearly weaker than the following condition in \cite[Theorem 2]{ge2021dantzig}
\begin{equation*
\delta_{2s}<\frac{1}{\sqrt{1+\frac{(\sqrt{s}+{\alpha})^2\Big(\Big(1+\frac{\sqrt{2}}{2}\Big)^2( s-\sqrt{s})+1\Big)}{s(\sqrt{s}-1)^2}}},
\end{equation*}
and
the following condition in \cite[Theorem 3.4]{ge2021new}
\begin{equation*}
\delta_{2s}<\frac{1}{\sqrt{1+\frac{(\sqrt{s}+1)^2\Big(\Big(1+\frac{\sqrt{2}}{2}\Big)^2( s-\sqrt{s})+1\Big)}{s(\sqrt{s}-1)^2}}}.
\end{equation*}
\end{remark}
\begin{remark}
We notice that Wang and Wang \cite[Equation (6)]{wang2019improved} established the following condition
\begin{equation}\label{RIPCondition.wang2019improved}
\delta_{s}+\frac{\sqrt{s}+\sqrt{2}-1}{\sqrt{s}-1}
\theta_{s,s}<1.
\end{equation}
Though it is weaker than our condition
\begin{equation}\label{RIPCondition3}
\delta_{s}+\frac{(\sqrt{2}+1)(\sqrt{s}+\alpha)}{\sqrt{2}(\sqrt{s}-1)}
\theta_{s,s}<1
\end{equation}
for {$s\geq2$}, their condition is for the constraint $\ell_1-\ell_2$ model rather than unconstraint $\ell_1-\ell_2$ model.
\end{remark}
\begin{remark}
Our condition \eqref{RIPCondition2} is weaker than that of \cite[Corollary 2]{geng2020Unconstrained}
\begin{equation}\label{CoherenceCondition.geng2020Unconstrained}
\mu<\frac{1}{3s+6},
\end{equation}
owing to $\mu=\delta_{2s}$.
\end{remark}
Proofs of Theorems $1$ and $2$ is similar, so we only present
the detail proof of Theorem \ref{StableRecoveryviaVectorL1-alphaL2-RASSO}.
\begin{proof}[Proof of Theorem \ref{StableRecoveryviaVectorL1-alphaL2-RASSO}]
We first show the conclusion $(\bm{i})$. By $\bm{D}\bm{D}^{\top}=\bm{I}_n$, we know
\begin{align}\label{decomposition}
\|\bm{h}\|_2=\|\bm{D}^{\top}\bm{h}\|_2=\sqrt{\|\bm{D}_{S}^{\top}\bm{h}\|_2^2+\|\bm{D}_{S^c}^{\top}\bm{h}\|_2^2}
\leq \|\bm{D}_{S}^{\top}\bm{h}\|_2+ \|\bm{D}_{S^c}^{\top}\bm{h}\|_2,
\end{align}
where $\|\bm{D}_{S}^{\top}\bm{h}\|_2$ and $\|\bm{D}_{S^c}^{\top}\bm{h}\|_2$ are needed to estimated, respectively.
We first estimate $\|\bm{D}_{S}^{\top}\bm{h}\|_2$.
By Lemma \ref{ConeTubeconstraint-L1-L2-RASSO}, then
the condition \eqref{lem:CrossItem.eq1} holds with $a=1,~b=\alpha,~c=2,~\eta=1,~\gamma=(\alpha+1)^{2}d\lambda/(2\rho)$.
Using Proposition \ref{lem:CrossItem} $(iii)$, one has
\begin{align}\label{prop:NSP.eq1cge}
\|\bm{D}_{S}^{\top}\bm{h}\|_2\leq&\frac{(\sqrt{2}+1)\theta_{ts,(t-1)s}}{\sqrt{2}(1-\rho_{s,t})}
\frac{\sqrt{\lceil(t-1)s\rceil}}{(t-1)(s-\sqrt{s})}
\Big(2\|\bm{D}_{T^c}^{\top}\bm{x}\|_1+\frac{(\alpha+1)^{2}d\lambda}{2\rho}\Big)
\nonumber\\
&+
\Big(\frac{(\sqrt{2}+1)\theta_{ts,(t-1)s}}{\sqrt{2}(1-\rho_{s,t})}\frac{\sqrt{\lceil(t-1)s\rceil}}
{(t-1)(s-\sqrt{s})}+\frac{\sqrt{1+\delta_{ts}}}{1-\rho_{s,t}}\Big)\|\bm{Ah}\|_{2}\nonumber\\
=&
\tau
\Big(2\|\bm{D}_{T^c}^{\top}\bm{x}\|_1+\frac{(\alpha+1)^{2}d\lambda}{2\rho}\Big)
+\bar{\tau}\|\bm{Ah}\|_{2},
\end{align}
where
\begin{align}\label{tau}
\begin{cases}
\tau=\frac{(\sqrt{2}+1)\theta_{ts,(t-1)s}}{\sqrt{2}(1-\rho_{s,t})}\frac{\sqrt{\lceil(t-1)s\rceil}}{(t-1)(s-\sqrt{s})},&\\
\bar{\tau}=\tau+\frac{\sqrt{1+\delta_{ts}}}{1-\rho_{s,t}}.
\end{cases}
\end{align}
In order to estimate $\|\bm{D}_{S}^{\top}\bm{h}\|_2$, we need an upper bound of $\|\bm{Ah}\|_{2}$. Using Lemma \ref{ConeTubeconstraint-L1-L2-RASSO}, we derive
\begin{align*}
&\|\bm{Ah}\|_2^2-2\lambda\|\bm{Ah}\|_2\nonumber\\
&\leq2\lambda\left(\|\bm{D}_{S}^{\top}\bm{h}\|_1+2\|\bm{D}_{T^c}^{\top}\bm{x}\|_1+\frac{(\alpha+1)^2d\lambda}{2\rho}+\alpha\|\bm{D}^{\top }\bm{h}\|_2-\|\bm{D}_{S^c}^{\top}\bm{h}\|_1\right)\nonumber\\
&\leq2\sqrt{s}\lambda\|\bm{D}_{S}^{\top}\bm{h}\|_2
+2\lambda\left(2\|\bm{D}_{T^c}^{\top}\bm{x}\|_1+\frac{(\alpha+1)^2d\lambda}{2\rho}+\alpha\|\bm{D}_{S}^{\top}\bm{h}\|_2+\alpha\|\bm{D}_{S^c}^{\top }\bm{h}\|_2-\|\bm{D}_{S^c}^{\top}\bm{h}\|_1\right)\nonumber\\
&\overset{(a)}{\leq} 2\lambda(\sqrt{s}+\alpha)\|\bm{D}_{S}^{\top}\bm{h}\|_2+2\lambda\left(2\|\bm{D}_{T^c}^{\top }\bm{x}\|_1+\frac{(\alpha+1)^2d\lambda}{2\rho}\right)\nonumber\\
&\overset{(b)}{\leq}2\lambda(\sqrt{s}+\alpha)\left(\bar{\tau}\|\bm{Ah}\|_2+\tau\left(2\|\bm{D}_{T^c}^{\top }\bm{x}\|_1+\frac{(\alpha+1)^2d\lambda}{2\rho}\right)\right)\nonumber\\
&\hspace*{12pt}+2\lambda\left(2\|\bm{D}_{T^c}^{\top}\bm{x}\|_1+\frac{(\alpha+1)^2d\lambda}{2\rho}\right)\nonumber\\
&=2\lambda(\sqrt{s}+\alpha)\bar{\tau}\|\bm{Ah}\|_2+2\lambda(1+(\sqrt{s}+\alpha)\tau)\left(2\|\bm{D}_{T^c}^{\top }\bm{x}\|_1+\frac{(\alpha+1)^2d\lambda}{2\rho}\right),
\end{align*}
where (a) follows from $\alpha\|\bm{D}_{S^c}^{\top}\bm{h}\|_2\leq\|\bm{D}_{S^c}^{\top}\bm{h}\|_2\leq\|\bm{D}_{S^c}^{\top}\bm{h}\|_1$,
(b) is because of \eqref{prop:NSP.eq1cge}. Thus
\begin{align*}
\|\bm{Ah}\|_2^2-2\lambda(1+(\sqrt{s}+\alpha)\bar{\tau})\|\bm{Ah}\|_2-2\lambda(1+(\sqrt{s}+\alpha)\tau )\left(2\|\bm{D}_{T^c}^{\top }\bm{x}\|_1+\frac{(\alpha+1)^2d\lambda}{2\rho}\right)\leq0.
\end{align*}
By the fact that the second order inequality $aX^2-bX-c\leq0$ for $a,b,c>0$ has the solution
$$X\leq \frac{b+\sqrt{b^2+4ac}}{2a}\leq\frac{b+\sqrt{(b+2ac/b)^2}}{2a}
=\frac{b}{a}+\frac{c}{b},$$
then
\begin{align}\label{Ah}
\|\bm{Ah}\|_2
&\leq(1+(\sqrt{s}+\alpha)\bar{\tau})2\lambda+\frac{(1+(\sqrt{s}+\alpha)\tau )\left(2\|\bm{D}_{T^c}^{\top }\bm{x}\|_1+\frac{(\alpha+1)^2d\lambda}{2\rho}\right)}{1+(\sqrt{s}+\alpha)\bar{\tau}}\nonumber\\
&=2 \bar{C}\lambda+\frac{C}{\bar{C}}\Big(2\|\bm{D}_{T^c}^{\top}\bm{x}\|_1+\frac{(\alpha+1)^2d\lambda}{2\rho}\Big),
\end{align}
where
\begin{align}\label{C.constant}
\begin{cases}
C=1+(\sqrt{s}+\alpha)\tau
=1+\frac{(\sqrt{2}+1)\theta_{ts,(t-1)s}}{\sqrt{2}(1-\rho_{s,t})}\frac{(\sqrt{s}+\alpha)}{(s-\sqrt{s})}\frac{\sqrt{(t-1)s}}{t-1},&\\
\bar{C}=1+(\sqrt{s}+\alpha)\bar{\tau}=C + \frac{(\sqrt{s}+\alpha)\sqrt{1+\delta_{ts}}}{1-\rho_{s,t}}.
\end{cases}
\end{align}
Combining \eqref{prop:NSP.eq1cge} with \eqref{Ah}, one has
\begin{eqnarray}\label{StableRecoveryviaVectorL1-alphaL2-ASSO.eq1}
&&\|\bm{D}_{S}^{\top}\bm{h}\|_2\nonumber\\
&&\leq\tau \left(2\|\bm{D}_{T^c}^{\top}\bm{x}\|_1+\frac{(\alpha+1)^2d}{2}\frac{\lambda}{\rho}\right)+\bar{\tau}
\left(2\lambda \bar{C}+\frac{C}{\bar{C}}\Big(2\|\bm{D}_{T^c}^{\top}\bm{x}\|_1+\frac{(\alpha+1)^2d\lambda}{2\rho}\Big)\right)\nonumber\\
&&=\left(\tau +\frac{C}{\bar{C}}\bar{\tau}\right)\left(2\|\bm{D}_{T^c}^{\top}\bm{x}\|_1+\frac{(\alpha+1)^2d}{2}\frac{\lambda}{\rho}\right)
+2\bar{\tau}\bar{C}\lambda.
\end{eqnarray}
Next, we consider an upper bound of $\|\bm{D}_{S^c}^{\top}\bm{h}\|_2$.
And from Proposition \eqref{lem:CrossItem} $(v)$ with $\bar{\varepsilon}=1$, it follows that
\begin{align}\label{DSCge}
&\|\bm{D}_{S^c}^{\top}\bm{h}\|_2\nonumber\\
&\leq\Bigg(\sqrt{\frac{\sqrt{s}+\alpha}{\sqrt{s}}
+\frac{\alpha^2}{4s}}+\frac{\alpha+1}{2\sqrt{s}}\Bigg)\|\bm{D}_{S}^{\top}\bm{h}\|_2
+\frac{1}{2}\Big(2\|\bm{D}_{T^c}^{\top}\bm{x}\|_1+\|\bm{Ah}\|_2+\frac{(\alpha+1)^{2}d\lambda}{2\rho}\Big).
\end{align}
Similarly, combining \eqref{Ah} with \eqref{StableRecoveryviaVectorL1-alphaL2-ASSO.eq1}, \eqref{DSCge} reduces to
\begin{align}\label{StableRecoveryviaVectorL1-alphaL2-ASSO.eq2}
&\|\bm{D}_{S^c}^{\top}\bm{h}\|_2
\leq\Bigg(\sqrt{\frac{\alpha+\sqrt{s}}{\sqrt{s}}+\frac{\alpha^2}{4s}}+\frac{\alpha+1}{2\sqrt{s}}\Bigg)\|\bm{D}_{S}^{\top}\bm{h}\|_2
+\frac{1}{2}\left(2\|\bm{D}_{T^c}^{\top}\bm{x}\|_1+\frac{(\alpha+1)^2d}{2}\frac{\lambda}{\rho}\right)\nonumber\\
&\hspace*{12pt}+\frac{1}{2}\Big(2\bar{C} \lambda +\frac{C}{\bar{C}}\Big(2\|\bm{D}_{T^c}^{\top }\bm{x}\|_1+\frac{(\alpha+1)^2d\lambda}{2\rho}\Big)\Big)\nonumber\\
&\leq\Bigg(\sqrt{\frac{\alpha+\sqrt{s}}{\sqrt{s}}+\frac{\alpha^2}{4s}}+\frac{\alpha+1}{2\sqrt{s}}
\Bigg)\|\bm{D}_{S}^{\top}\bm{h}\|_2
+\frac{1}{2}\bigg(1+\frac{C}{\bar{C}}\bigg)\left(2\|\bm{D}_{T^c}^{\top}\bm{x}\|_1+\frac{(\alpha+1)^2d}{2}\frac{\lambda}{\rho}\right)\nonumber\\
&\hspace*{12pt}+\bar{C}\lambda.
\end{align}
Therefore, substituting the estimation of $\|\bm{D}_{S}^{\top}\bm{h}\|_2$ and $\|\bm{D}_{S^c}^{\top}\bm{h}\|_2$ into \eqref{decomposition}, one has
\begin{align*}
\|\bm {h}\|_2
&\overset{(a)}{\leq}\Bigg(\sqrt{\frac{\alpha+\sqrt{s}}{\sqrt{s}}+\frac{\alpha^2}{4s}}+\frac{\alpha+1}{2\sqrt{s}}+1
\Bigg)\|\bm{D}_{S}^{\top}\bm{h}\|_2\nonumber\\
&\hspace*{12pt}+\frac{1}{2}\bigg(1+\frac{C}{\bar{C}}\bigg)\left(2\|\bm{D}_{T^c}^{\top}\bm{x}\|_1+\frac{(\alpha+1)^2d}{2}\frac{\lambda}{\rho}\right)
+\bar{C}\lambda\nonumber\\
&\overset{(b)}{\leq}\Bigg(\sqrt{\frac{\alpha+\sqrt{s}}{\sqrt{s}}+\frac{\alpha^2}{4s}}+\frac{\alpha+1}{2\sqrt{s}}+1
\Bigg)\left(\left(\tau +\bar{\tau}\frac{C}{\bar{C}}\right)\left(2\|\bm{D}_{T^c}^{\top}\bm{x}\|_1+\frac{(\alpha+1)^2d}{2}\frac{\lambda}{\rho}\right)
+\bar{\tau}\bar{C}2\lambda\right)\nonumber\\
&\hspace{24pt}+\frac{1}{2}\bigg(1+\frac{C}{\bar{C}}\bigg)\left(2\|\bm{D}_{T^c}^{\top}\bm{x}\|_1+\frac{(\alpha+1)^2d}{2}\frac{\lambda}{\rho}\right)
+\bar{C}\lambda\nonumber\\
&=
\Bigg(\Bigg(\sqrt{\frac{\alpha+\sqrt{s}}{\sqrt{s}}+\frac{\alpha^2}{4s}}+\frac{\alpha+1}{2\sqrt{s}}+1
\Bigg)\left(\tau +\bar{\tau}\frac{C}{\bar{C}}\right)+\frac{1}{2}\bigg(1+\frac{C}{\bar{C}}\bigg)\Bigg)\nonumber\\
&\hspace*{12pt}\times\left(2\|\bm{D}_{T^c}^{\top}\bm{x}\|_1+\frac{(\alpha+1)^2d}{2}\frac{\lambda}{\rho}\right)
+\left(\left(\sqrt{\frac{\alpha+\sqrt{s}}{\sqrt{s}}+\frac{\alpha^2}{4s}}+\frac{\alpha+1}{2\sqrt{s}}+1
\right)\bar{\tau}+\frac{1}{2}\right)2\bar{C}\lambda\nonumber\\
&=:E_{1}\left(2\|\bm{D}_{T^c}^{\top}\bm{x}\|_1+\frac{(\alpha+1)^2d}{2}\frac{\lambda}{\rho}\right)
+E_{2}2\lambda,
\end{align*}
where (a) is due to \eqref{StableRecoveryviaVectorL1-alphaL2-ASSO.eq2}, and (b) is from \eqref{StableRecoveryviaVectorL1-alphaL2-ASSO.eq1}.
Recall the definition of the constants $\tau,\bar{\tau}$ in \eqref{tau} and $C,\bar{C}$ in \eqref{C.constant},
we get
\begin{align}
&E_{1}
=\Bigg(\sqrt{\frac{\alpha+\sqrt{s}}{\sqrt{s}}+\frac{\alpha^2}{4s}}+\frac{\alpha+1}{2\sqrt{s}}+1\Bigg)\nonumber\\
&\hspace*{12pt}\times\left(\tau +\left(\tau+\frac{\sqrt{1+\delta_{ts}}}{1-\rho_{s,t}}\right)\left(\frac{C(1-\rho_{s,t})}{C(1-\rho_{s,t})+(\sqrt{s}+\alpha)\sqrt{1+\delta_{ts}}}\right)\right)
\nonumber\\
&+\frac{1}{2}\bigg(1+\frac{C(1-\rho_{s,t})}{C(1-\rho_{s,t})+(\sqrt{s}+\alpha)\sqrt{1+\delta_{ts}}}\bigg)\nonumber\\
&=\Bigg(\sqrt{\frac{\alpha+\sqrt{s}}{\sqrt{s}}+\frac{\alpha^2}{4s}}+\frac{\alpha+1}{2\sqrt{s}}+1
\Bigg)\left(\tau +\frac{\big(\tau(1-\rho_{s,t})+\sqrt{1+\delta_{ts}}\big)C}{C(1-\rho_{s,t})+(\sqrt{s}+\alpha)\sqrt{1+\delta_{ts}}}\right)\nonumber\\
&+\frac{1}{2}\bigg(1+\frac{C(1-\rho_{s,t})}{C(1-\rho_{s,t})+(\sqrt{s}+\alpha)\sqrt{1+\delta_{ts}}}\bigg),\nonumber\\
E_{2}&=\left(\left(\sqrt{\frac{\alpha+\sqrt{s}}{\sqrt{s}}+\frac{\alpha^2}{4s}}+\frac{\alpha+1}{2\sqrt{s}}+1
\right)\frac{\tau(1-\rho_{s,t})+\sqrt{1+\delta_{ts}}}{1-\rho_{s,t}}+\frac{1}{2}\right)\nonumber\\
&\hspace*{12pt}\times\left(C+\frac{(\sqrt{s}+\alpha)\sqrt{1+\delta_{ts}}}{1-\rho_{s,t}}\right).
\end{align}
Therefore, we complete the proof of item ($i$).
($ii$) We can prove the conclusion ($ii$) in a similar way only by replacing Proposition \ref{lem:CrossItem} (iii) with Proposition \ref{lem:CrossItem} (iv).
\end{proof}
\section{Numerical Algorithm }\label{s4}
\noindent
In this section, we develop an efficient algorithm to solve the $\ell_1-\alpha\ell_2$-ASSO
\eqref{VectorL1-alphaL2-ASSO}.
The projected fast iterative soft-thresholding algorithm (pFISTA) for tight frames in \cite{liu2016projected} is
suited solving the $\ell_1$-analysis problem \eqref{VectorL1-ana} for
MRI reconstruction.
Compared to the common iterative reconstruction methods such as iterative soft-thresholding algorithm (ISTA) in \cite{daubechies2004iterative} and fast iterative soft-thresholding algorithm (FISTA) in \cite{beck2009fast},
the pFISTA algorithm achieves better reconstruction and converges faster.
Inspired by the pFISTA algorithm,
an efficient algorithm is introduced to solve the nonconvex-ASSO problem \eqref{VectorL1-alphaL2-ASSO} in this section.
For the given frame $\bm{D}\in\mathbb{R}^{n\times d}$,
there are many dual frames.
Here, we only consider its canonical dual frame
\begin{equation}\label{canonicaldual}
\bm{\Phi}=(\bm{D}\bm{D}^{\top})^{-1}\bm{D},
\end{equation}
which satisfies
$$
\bm{\Phi}\bm{D}^{\top}=\bm{I}_{n},
$$
and is also the pseudo-inverse of $\bm{D}$ \cite{elad2007analysis}.
Taking $\bm{z}=\bm{D}^{\top}\bm{x}$,
the $\ell_1-\alpha\ell_2$-ASSO
\eqref{VectorL1-alphaL2-ASSO} is written as
\begin{equation}\label{VectorL1-alphaL2-ASSO.equi1}
\min_{\bm{ z}\in\text{Range}(\bm{D}^{\top})}~\lambda(\|\bm{z}\|_{1}-\alpha\|\bm{z}\|_{2})+\frac{1}{2}\|\bm{ A}\bm{\Phi}\bm{ z}-\bm{ b}\|_{2}^{2}.
\end{equation}
Now, we solve \eqref{VectorL1-alphaL2-ASSO.equi1} by the idea of the pFISTA algorithm.
First, we introduce an indicator function
\begin{equation*}
\chi(\bf{z})=
\begin{cases}
\bm{0},~\bm{z}\in\text{Range}(\bm{D}^{\top}),&\\
+\bm{\infty},~\text{otherwise},
\end{cases}
\end{equation*}
then an equivalent unconstrained model of \eqref{VectorL1-alphaL2-ASSO.equi1} is
\begin{equation}\label{VectorL1-alphaL2-ASSO.equi2}
\min_{\bm{ z}\in\mathbb{R}^{d}}~\lambda(\|\bm{ z}\|_{1}-\alpha\|\bm{ z}\|_{2})+\chi(\bm{z})+\frac{1}{2}\|\bm{ A}\bm{\Phi}\bm{ z}-\bm{ b}\|_{2}^{2}.
\end{equation}
Let
$$
g(\bm{z})=\lambda(\|\bm{ z}\|_{1}-\alpha\|\bm{ z}\|_{2})+\chi(\bm{z}),~~h(\bm{z})=\frac{1}{2}\|{\bm A}\bm{\Phi}{\bm z}-{\bm b}\|_{2}^{2},
$$
then \eqref{VectorL1-alphaL2-ASSO.equi2} can be rewritten as
\begin{equation}\label{OptimizationProblem}
\min_{{\bm z}\in\mathbb{R}^{d}}~g(\bm{z})+h(\bm{z}),
\end{equation}
where $g$ is a non-smooth function, and $h$ is a smooth
function with a $\ell_{h}$-Lipschitz continuous gradient ($\ell_{h}>0$), i.e
$$
\|\nabla h(\bm{z}_1)- \nabla h(\bm{z}_2)\|_{2}\leq\ell_{h}\|\bm{z}_1-\bm{z}_2\|_{2}.
$$
Next, we solve \eqref{VectorL1-alphaL2-ASSO.equi2} via ISTA by incorporating the proximal mapping
\begin{align}\label{ISTA1}
\bm{z}^{k+1}&=\text{Prox}_{\gamma g}(\bm{z}^{k}-\gamma \nabla h(\bm{z}^{k}))\nonumber\\
&=\arg\min_{\bm{ z}\in\text{Range}(\bm{D}^{\top})}\gamma\lambda(\|\bm{ z}\|_{1}-\alpha\|\bm{z}\|_{2})
+\frac{1}{2}\left\|\bm{z}-\left(\bm{z}^{k}-\gamma \nabla h(\bm{z}^{k})\right)\right\|_{2}^{2},
\end{align}
where $\gamma$ is the step size
and $\text{Prox}_{\gamma g}(\cdot)$ is the proximal operator of
the function $\gamma g$.
The proximal operator of $\mu_1\ell_1-\mu_2\ell_2$ in \cite[Proposition 7.1]{liu2017further} and \cite[Section 2]{lou2018fast}
is
\begin{equation}\label{L1L2-ProximalMap4}
\text{Prox}_{\lambda(\ell_1-\alpha\ell_2)}({\bm b})=\arg\min_{\bm{x}}\frac{1}{2}\|\bm{x}-\bm{b}\|_2^2+\lambda(\|\bm{ x}\|_1-\alpha\|\bm{ x}\|_2), \ \ \ \ 0<\alpha\leq 1,
\end{equation}
which has an explicit formula for ${\bm x}$.
And the solution in \eqref{L1L2-ProximalMap4} is unique in some special cases. Therefore the problem \eqref{ISTA1} is just as follows
\begin{align}\label{ISTA1.2}
\bm{z}^{k+1}
=\text{Proj}_{\text{Range}(\bm{D}^{\top})}\left(\text{Prox}_{\lambda(\ell_1-\alpha\ell_2)}\left(\left(\bm{z}^{k}-\gamma \nabla h(\bm{z}^{k})\right)\right)\right),
\end{align}
where $\text{Proj}_{\mathcal{C}}(\cdot)$ is a projection operator on the set $\mathcal{C}$.
So far, the original analysis model
\eqref{VectorL1-alphaL2-ASSO} has been converted into a much simpler form \eqref{ISTA1}.
However, it is a challenge to find an analytical solution of \eqref{ISTA1} since there is the constraint ${\bm z}\in\text{Range}(\bm{D}^{\top})$.
Note that the orthogonal projection operator on $\text{Range}(\bm{D}^{\top})=\{\bm{\Phi}\bm{z}:\bm{z}\in\mathbb{R}^{d}\}$ is
$$
\text{Proj}_{\text{Range}(\bm{D}^{\top})}(\bm{z})=\bm{D}^{\top}\bm{\Phi}\bm{z}.
$$
Therefore, we propose to replace \eqref{ISTA1} by
\begin{align}\label{ISTA2}
\begin{cases}
\tilde{\bm{z}}^{k+1}=\text{Prox}_{\lambda\gamma(\ell_1-\alpha\ell_2)}\left(\bm{z}^{k}-\gamma \bm{\Phi}^{\top}\bm{A}^{\top}(\bm{A}\bm{\Phi}\bm{z}^{k}-\bm{b})\right),&\\
\bm{z}^{k+1}=\text{Proj}_{\text{Range}(\bm{D}^{\top})}(\tilde{\bm{z}}^{k+1})=\bm{D}^{\top}\bm{\Phi}\tilde{\bm{z}}^{k+1}.&
\end{cases}
\end{align}
By the fact that $\bm{\Phi}\bm{D}^{\top}=\bm{I}_{n}$ and \eqref{canonicaldual}, the two steps in \eqref{ISTA2} can be recast as
\begin{align}\label{ISTA3}
\tilde{\bm{z}}^{k+1}=\text{Prox}_{\lambda\gamma(\ell_1-\alpha\ell_2)}\left(\bm{D}^{\top}\left(\bm{\Phi}\tilde{\bm{z}}^{k}-\gamma (\bm{D}\bm{D}^{\top})^{-1}\bm{A}^{\top}(\bm{A}\bm{\Phi}\tilde{\bm{z}}^{k}-\bm{b})\right)\right).
\end{align}
Now, let us turn our attention to how to get the formulation of $\bm{x}^{k+1}$.
By substituting the coefficients $\bm{x}^{k}=\bm{\Phi}\bm{z}^{k}=\bm{\Phi}\bm{D}^{\top}\bm{\Phi}\tilde{\bm{z}}^{k}=
\bm{\Phi}\tilde{\bm{z}}^{k}$ into \eqref{ISTA3}, one has
\begin{align}\label{ISTA4}
\bm{x}^{k+1}=\bm{\Phi}\text{Prox}_{\lambda\gamma(\ell_1-\alpha\ell_2)}\left(\bm{D}^{\top}\left(\bm{x}^{k}-\gamma (\bm{D}\bm{D}^{\top})^{-1}\bm{A}^{\top}(\bm{A}\bm{x}^{k}-\bm{b})\right)\right),
\end{align}
which is a solution of the $\ell_1-\alpha\ell_2$-ASSO \eqref{VectorL1-alphaL2-ASSO}.
For a tight frame, we have $\bm{\Phi}=\bm{D}$ and $\bm{D}\bm{D}^{\top}=\bm{I}_{n}$, then \eqref{ISTA4} reduces to
\begin{align}\label{ISTA5}
\bm{x}^{k+1}=\bm{D}\text{Prox}_{\lambda\gamma(\ell_1-\alpha\ell_2)}\left(\bm{D}^{\top}\left(\bm{x}^{k}-\gamma \bm{A}^{\top}(\bm{A}\bm{x}^{k}-\bm{b})\right)\right).
\end{align}
Based on all the above derivations, the
efficient algorithm of the $\ell_1-\alpha\ell_2$-ASSO \eqref{VectorL1-alphaL2-ASSO}
is proposed and summarized
in Algorithm $1$ as follows.
\medskip
\noindent\rule[0.25\baselineskip]{\textwidth}{1pt}
\label{al:pFISTA}
\centerline {\bf Algorithm $1$: the $\ell_1-\alpha\ell_2$-pFISTA for solving \eqref{VectorL1-alphaL2-ASSO}}\\
{\bf Input:}\ ${\bm A}$,${\bm D}$, ${\bm b}$, $0<\alpha\leq 1$, $\lambda$, $\gamma$. \\
{\bf Initials:}\ $\bm{ x}=\bm{x}^0$, $\bm{ y}=\bm{y}^0=\bm{x}^0$, $t=t^0=1$, $k=0$.\\
{\bf Circulate} Step 1--Step 4 until ``some stopping criterion is satisfied":
~{\bf Step 1:} Update ${\bm x}^{k+1}$ according to
\begin{equation}\label{FISTA1}
\bm{x}^{k+1}=\bm{D}~\text{Prox}_{\lambda\gamma(\ell_1-\alpha\ell_2)}\left(\bm{D}^{\top}\left(\bm{y}^{k}-\gamma \bm{A}^{\top}(\bm{A}\bm{y}^{k}-\bm{b})\right)\right).
\end{equation}
~{\bf Step 2:} Update ${\bm t}^{k+1}$ as follows
\begin{equation}\label{FISTA2}
t_{k+1}=\frac{1+\sqrt{1+4t_{k}^2}}{2}.
\end{equation}
~{\bf Step 3:} Update $\bm{y}^{k+1}$ as follows
\begin{equation}\label{FISTA3}
\bm{y}_{k+1}=\bm{x}^{k+1}+\frac{t_k-1}{t_{k+1}}(\bm{x}^{k+1}-\bm{x}^{k}).
\end{equation}
~{\bf Step 4:} Update $k$ to $k+1$.\\
{\bf Output:} $\hat{\bm {x}}$.\\
\noindent\rule[0.25\baselineskip]{\textwidth}{1pt}
\begin{remark}
In our algorithm, we set the total iterated number $K=1000$, and take the stopping criterion $\|\bm{x}^{k+1}-\bm{x}^{k}\|_{2}/\|\bm{x}^{k}\|_{2}<\epsilon$ with the tolerate error $\epsilon=10^{-6}$.
\end{remark}
\section{Numerical Experiments}\label{s5}
\noindent
In this section,
we demonstrate the performance of the $\ell_1-\alpha\ell_2$-ASSO \eqref{VectorL1-alphaL2-ASSO} via simulation experiments and compare the proposed $\ell_1-\alpha\ell_2$-ASSO \eqref{VectorL1-alphaL2-ASSO} to
the state-of-art the $\ell_1$-analysis and $\ell_p$-analysis minimization methods.
All experiments were performed under Windows
Vista Premium and MATLAB v7.8 (R2016b) running on a Huawei laptop with an Intel(R)
Core(TM)i5-8250U CPU at 1.8 GHz and 8195MB RAM of memory.
\subsection{Signal Reconstruct Under Tight Frame}\label{s5.1}
\noindent
In this subsection, we evaluate the performance of the $\ell_1-\alpha\ell_2$-ASSO \eqref{VectorL1-alphaL2-ASSO} and compared our method with the following models:
\begin{equation}\label{VectorL1-ABP}
\min_{{\bm x}\in\mathbb{R}^n}~\lambda\|{\bm D}^{\top}{\bm x}\|_{p}^{p}~\text{subject~to~}{\bm A}{\bm x}={\bm b}.
\end{equation}
When $p=1$, the method \eqref{VectorL1-ABP} is Analysis Basis Pursuit, which is solved by CVX package (see \cite{genzel2021analysis,nam2013cosparse}). When $0<p<1$, Lin and Li \cite{lin2016restricted} present an algorithm based on iteratively reweighted least
squares (IRLS) to solve the $\ell_{p}$-analysis model \eqref{VectorL1-ABP}.
Many papers have showed that IRLS method with smaller value of $p$ (for example $p= 0.1, 0.5$) perform better than that of larger value of $p$ (for example $p= 0.7, 0.9$). In addition, $p=0.5$ gave slightly higher success frequency than $p=0.1$. Please refer to \cite[Section 4]{chartrand2008restricted}, \cite[Section 8.1]{daubechies2010iteratively}, and \cite[Section 4.1]{lai2013improved}. Therefore,
we only compare our method with the $\ell_{p}$ model \eqref{VectorL1-ABP} for $p=0.5$.
First of all, we roughly follow a construction of tight random frames from \cite{nam2013cosparse}:
\begin{enumerate}
\item[(i)]First, draw a $n\times d$ Gaussian random matrix $\bm{E}$ and compute its singular value decomposition $\bm{E}=\bm{U}\bm{\Sigma}\bm{V}^{\top}$.
\item[(ii)] If $n\leq d$, we replace $\bm{\Sigma}$ by the matrix $\tilde{\bm{\Sigma}}=[\tau\bm{I}_n, \bm{0}]\in\mathbb{R}^{n\times d}$ with $\tau=\sqrt{d/n}$, which yields a tight frame $\bm{D}=\bm{U}\tilde{\bm{\Sigma}}\bm{V}^{\top}$. If $n> d$, we replace $\bm{\Sigma}$ by the matrix $\tilde{\bm{\Sigma}}=[\tau\bm{I}_d, \bm{0}]^{\top}\in\mathbb{R}^{n\times d}$ with $\tau=\sqrt{d/n}$, which yields a tight frame $\bm{D}=\bm{U}\tilde{\bm{\Sigma}}\bm{V}^{\top}$.\\
\end{enumerate}
Nam et.al. \cite{nam2013cosparse} also showed us how to generate cosparse signal $\bm{x}_{0}\in\mathbb{R}^{n}$.
We adopt their scheme and produce an $s$-cosparse signal in the following way:
\begin{enumerate}
\item[(a)] First, choose $s$ rows of the analysis operator $\bm{D}^{\top}=\bm{\Omega}\in\mathbb{R}^{n\times d}$ at random, and those are
denoted by an index set $|S|$ (thus, $|S|=s$).
\item[(b)] Second, form an arbitrary signal $\bm{y}$ in $\mathbb{R}^n$--e.g., a random vector with Gaussian i.i.d. entries.
\item[(c)] Then, project $\bm{y}$ onto the orthogonal complement of the subspace generated by the rows of $\bm{\Omega}$ that are indexed by $S$, this way getting the cosparse signal $\bm{x}_0$. Explicitly, $\bm{x}_0=\left(\bm{I}_n-\bm{\Omega}_{S}^{\top }(\bm{\Omega}_{S}\bm{\Omega}_{S}^{\top})^{-1}\bm{\Omega}_{S}\right)\bm{y}$. In fact, $\bm{D}^{\top}\bm{x}_0=[\bm{0};\bm{\Omega}_{S^c}\bm{x}_0]\in\mathbb{R}^{d}$ is $(d-s)$-sparse.
\end{enumerate}
Alternatively, one could first find a
basis for the orthogonal complement and then generate a random coefficient vector for the basis.
In the experiment, the entries of $\bm{A}\in\mathbb{R}^{m\times n}$ are drawn independently from the normal distribution. The observation is obtained by $\bm{b} =\bm{A}\bm{x}_{0}$.
Let $\hat{\bm x}$ be the reconstructed signal.
We record the success rate over $100$ independent trials. The recovery is regarded as successful if
\begin{equation}\label{rel.err}
\text{rel-err}(\hat{\bm x},{\bm x}_0)=\frac{\|\hat{\bm x}-{\bm x}_0\|_2}{\|{\bm x}_0\|_2}<\varepsilon,
\end{equation}
for $\varepsilon=10^{-2}$.
We display success rate of different algorithms to recover sparse signals over $100$ repeated trials for different cosparsity $s$.
For fairness of comparison, the key parameters of our proposed method and compared algorithms have been tuned in all experiments according to \cite{nam2013cosparse}. In all cases, the signal dimension $n$ is set to 100. We then varied the number $m$ of measurements, the cosparsity $\ell$ of the target signal, and the operator size $d$ according to the following formulae:
\begin{equation}\label{cosparsity.setup}
m=\varrho n, d=\varsigma n, \ell=n-\rho m
\end{equation}
where $0<\varrho \leq 1$, $\varsigma\geq 1$, $0<\rho\leq1$. Here we take $\varsigma=1, \rho=\{0.05,0.10,0.15,\ldots,1\}$ and $\varrho=\{0.05,0.1,0.15,\ldots,1\}$, i.e., the measurement $m=\{5,10,15,\dots,100\}$.
In Figure \ref{figure.SuccnumberPhaseTrransition-Comparion}, we plot the phase transition diagram, which characterizes sharp shifts in the success probability of reconstruction when the dimension parameter crosses a threshold. The $x$-axis and the $y$-axis represent the under-sampling ratio and co-sparsity ratio, respectively. Yellow and blue denote perfect recovery and
failure in all experiments, respectively. It can be clearly seen that Success Rate (the yellow) of the proposed $\ell_1-\alpha\ell_2$ methods are the highest in all experiments. Experimental results show that $\ell_1-\alpha\ell_2$ methods outperform $\ell_1$ method and $\ell_p$ methods.
Figure \ref{figure.CPUtimePhaseTrransition-Comparion} shows the average CPU time of all methods at different $\varrho$ and $\rho$. We can observe that the CPU time of the proposed method is significantly lower than those of $\ell_{1}$ method at whole, and higher than those of $\ell_p$ method for $p=0.5$. Thus, $\ell_1-\alpha\ell_2$-ASSO method can achieve a good balance between CPU time and recovery performance.
\begin{figure*}[t]
\begin{tabular}{ccc}
\includegraphics[width=4.5cm,height=4.5cm]{L1_Succ_v3.eps}&
\includegraphics[width=4.5cm,height=4.5cm]{IRLSp_Succ_v3.eps}&
\includegraphics[width=4.5cm,height=4.5cm]{L1L2_Succ_v2.eps}
\end{tabular}
\centering
\caption{Success percentage of the $\ell_1$-, $\ell_p$-~($p=0.5)$ and $\ell_{1}-\alpha\ell_{2}$-analysis for recover sparse signals
versus the ratios $\varrho$ and $\rho$.
}
\label{figure.SuccnumberPhaseTrransition-Comparion}
\end{figure*}
\begin{figure*}[t]
\begin{tabular}{c}
\includegraphics[width=16.0cm]{Phase_Transition_Time_v4.eps}
\end{tabular}
\centering
\caption{ CPU time of the $\ell_1$-, $\ell_p$-~($p=0.5)$ and $\ell_{1}-\alpha\ell_{2}$-analysis for the sparse signal recovery
versus the ratios $\varrho$ and $\rho$.}
\label{figure.CPUtimePhaseTrransition-Comparion}
\end{figure*}
\subsection{Reconstruction of Compressed Sensing Magnetic Resonance Imaging under Tight Frame}\label{s5.2}
\noindent
In this subsection, we consider the shift-invariant discrete wavelet transform (SIDWT) for tight frame $\bm{D}$, which
is a typical tight frame in simulation \cite{baker2011translational,baraniukrice,coifman1995translation,kayvanrad2014stationary}. And SIDWT is also called as undecimated, translation-invariant, or
fully redundant wavelets. In all the experiments, we utilize Daubechies wavelets with 4 decomposition levels in SIDWT.
In CS-MRI, the sampling operator is
$$\bm{A}=\bm{U}\mathcal{F},$$
where $\mathcal{F}$ is the discrete Fourier transform, and $\bm{U}$ is the sampling mask in the frequency
space. The matrix $\bm{U}$ is also called the undersampling matrix. We keep samples along certain radial lines passing through the center of the Fourier data ($k$-space).
We reconstruct magnetic resonance images (MRI) from incomplete spectral Fourier data: $256\times 256$ Brain MRI and $512\times 512$ Foot MRI (see \cite[Section 4.1]{li2020compressive}) via the $\ell_1-\alpha\ell_2$-ASSO \eqref{VectorL1-alphaL2-ASSO}.
Similarly, we compare the $\ell_1-\alpha\ell_2$-ASSO \eqref{VectorL1-alphaL2-ASSO} to the $\ell_1$-analysis and $\ell_p$($0<p<1$)-analysis minimization methods.
We adopt the pFISTA for tight frames in \cite{liu2016projected} to solve the $\ell_1$-analysis minimization problem.
The $\ell_p$($0<p<1$)-analysis model is
\begin{equation}\label{Lp-Analysis}
\min_{{\bm x}}~\lambda\|{\bm D}^{\top}{\bm x}\|_{p}^{p}+\frac{1}{2}\|\bm{UF}{\bm x}-{\bm b}\|_{2}^{2},
\end{equation}
which is solved by the idea of the pFISTA for tight frames. In fact, as shown in section \ref{s4},
the equality \eqref{FISTA1} is replaced by
\begin{equation}\label{FISTALp}
\bm{x}^{k+1}=\bm{D}~\text{Prox}_{\lambda\gamma\ell_p}\left(\bm{D}^{\top}\left(\bm{y}^{k}-\gamma \bm{A}^{*}(\bm{A}\bm{y}^{k}-\bm{b})\right)\right),
\end{equation}
where $0<p<1$ and the notation ${Prox}_{\lambda\ell_p}(\bf{b})$ is the proximal operator of $\ell_{p}$ norm, see \cite{marjanovic2012optimization}.
The quantitative comparison is done in terms of the relative error (RE) defined as
$$
\text{RE}=\frac{\|\hat{\bm{x}}-\bm{x}_0\|_{2}}{\|\bm{x}_0\|_{2}},
$$
where $\bm{x}_0$ is the truth image and $\hat{\bm{x}}$ is the reconstructed
image. To demonstrate how $\ell_1-\alpha\ell_2$-ASSO method compares with other methods in terms of image quality,
we show the restored versions of Brain images and Foot images and reconstruction errors in Figures
\ref{figure.Reconstruct-image-comparsion}, \ref{figure.Reconstruct-FootMRI-comparsion} and Table \ref{tab:MRI-Time}, respectively.
In Figures \ref{figure.Reconstruct-image-comparsion} and \ref{figure.Reconstruct-FootMRI-comparsion},
we show the reconstructed images of different methods for $76$ radial sampling lines
(sampling rate 30.81$\%$ and 16.17$\%$ for Brain-MRI and Foot-MRI, respectively). By inspecting the recovered images of brain, it can be seen that $\ell_1-\alpha\ell_2$ method can obtain better performance than other methods.
We also record the CPU time of all methods in Table \ref{tab:MRI-Time}.
\section{Conclusions }\label{s6}
\noindent
In this paper, we consider the signal and compressed sensing magnetic resonance imaging reconstruction under tight frame. We propose the unconstrained $\ell_{1}-\alpha\ell_{2}$-analysis model
\eqref{VectorL1-alphaL2-ASSO} and \eqref{VectorL1-alphaL2-RASSO}. Based on the restricted isometry
property and restricted orthogonality constant adapted to tight frame $\bm{D}$ ($\bm{D}$-RIP and $\bm{D}$-ROC), we develop new vital auxiliary tools (see Propositions \ref{prop.DROC} and \ref{NonsparseROC}) and sufficient conditions of stable recovery (see Theorems \ref{StableRecoveryviaVectorL1-alphaL2-ASSO} and \ref{StableRecoveryviaVectorL1-alphaL2-RASSO}). Based on the Projected FISTA \cite{liu2016projected},
we establish the fast and efficient algorithm
to solve the unconstrained $\ell_{1}-\alpha\ell_{2}$-analysis model
in Section \ref{s4}.
The proposed method has better performance than the $\ell_p$-analysis model with $0<p\leq 1$
in numerical examples for the signal and compresses sensing MRI recovery.
\begin{figure*}[htbp!]
\setlength{\tabcolsep}{4.0pt}\small
\begin{tabular}{c}
\includegraphics[width=16.0cm,height=8.0cm]{Reconstruct_image_comparsion_different_v3.eps}
\end{tabular}
\centering
\caption{\label{figure.Reconstruct-image-comparsion} Reconstructed Brain-MRI by the $\ell_1$-, $\ell_p$-~($0<p<1$) and $\ell_{1}-\alpha\ell_{2}$-analysis. From left to right in the first line: Ground truth, sample lines, $\ell_1$ reconstruction image, difference images of $\ell_1$ to the ground truth image. From left to right in the second line: $\ell_{0.1}$ reconstruction image, difference images of $\ell_{0.1}$ to the ground truth image, $\ell_{0.5}$ reconstruction image, difference images of $\ell_{0.5}$ to the ground truth image. From left to right in the third line: $\ell_{0.9}$ reconstruction image, difference images of $\ell_{0.9}$ to the ground truth image, $\ell_1-\alpha\ell_2$ reconstruction image, difference images of $\ell_1-\alpha\ell_2$ to the ground truth image.
}
\vspace{-0.1cm}
\end{figure*}
\begin{figure*}[htbp!]
\setlength{\tabcolsep}{4.0pt}\small
\begin{tabular}{c}
\includegraphics[width=16.0cm,height=8.0cm]{Reconstruct_FootMRI_comparsion_different_v1.eps}
\end{tabular}
\centering
\caption{\label{figure.Reconstruct-FootMRI-comparsion} Reconstructed Foot-MRI by the $\ell_1$-, $\ell_p$-~($0<p<1$) and $\ell_{1}-\alpha\ell_{2}$-analysis. From left to right in the first line: Ground truth, sample lines, $\ell_1$ reconstruction image, difference images of $\ell_1$ to the ground truth image. From left to right in the second line: $\ell_{0.1}$ reconstruction image, difference images of $\ell_{0.1}$ to the ground truth image, $\ell_{0.5}$ reconstruction image, difference images of $\ell_{0.5}$ to the ground truth image. From left to right in the third line: $\ell_{0.9}$ reconstruction image, difference images of $\ell_{0.9}$ to the ground truth image, $\ell_1-\alpha\ell_2$ reconstruction image, difference images of $\ell_1-\alpha\ell_2$ to the ground truth image.
}
\vspace{-0.1cm}
\end{figure*}
\begin{table}[htbp]
\setlength{\tabcolsep}{5pt}\small
\begin{center}
\caption{The CPU Time (s) of Different reconstruction Models}\label{tab:MRI-Time}
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
Image & Sampling Rate &$\ell_{1}$ &$\ell_{0.1}$ &$\ell_{0.5}$ &$\ell_{0.9}$ &$\ell_{1}-\alpha\ell_{2}$ \\\hline
Brain-MRI &30.08$\%$ &80.9519 &164.4825 &238.4073 &322.2761 &85.1567 \\\hline
Foot-MRI &16.17$\%$ &308.3546 &441.8828 &657.7717 &857.9363 &340.5498 \\\hline
\end{tabular}
\end{center}
\end{table}
\newpage
\begin{appendices}
\section{The proof of Lemma \ref{lem:CrossItem}}\label{appendx1}
\noindent
\begin{proof}
$(\bm i)$ From the condition \eqref{lem:CrossItem.eq1},
it follows that
\begin{align}\label{Coneconstraintinequality}
\|\bm{D}_{S^c}^{\top}\bm{h}\|_1-\|\bm{D}_{S^c}^{\top}\bm{h}\|_2
&\overset{(a)}{\leq}
\|\bm{D}_{S^c}^{\top}\bm{h}\|_1-\alpha\|\bm{D}_{S^c}^{\top}\bm{h}\|_2\nonumber\\
&\leq a\|\bm{D}_{S}^{\top}\bm{h}\|_1+b\|\bm{D}_{S}^{\top}\bm{h}\|_2+c\|\bm{D}_{T^c}^{\top}\bm{x}\|_1+\eta\|{\bm Ah}\|_2
+\gamma\nonumber\\
&\overset{(b)}{\leq}(a\sqrt{s}+b)\|\bm{D}_{S}^{\top}\bm{h}\|_2+c\|\bm{D}_{T^c}^{\top}\bm{x}\|_1+\eta\|{\bm Ah}\|_2
+\gamma\nonumber\\
&\leq(s-\sqrt{s})\bigg(
\frac{a\sqrt{s}+b}{\sqrt{s}-1}\frac{\|\bm{D}_{S}^{\top}\bm{h}\|_2}{\sqrt{s}}+\frac{c\|\bm{D}_{T^c}^{\top}\bm{x}\|_1+\eta\|{\bm Ah}\|_2
+\gamma}{s-\sqrt{s}}\bigg)\nonumber\\
&=:(s-\sqrt{s})\varrho,
\end{align}
where $(a)$ is due to $0<\alpha\leq 1$, and
$(b)$ follows from the fact $\|\bm{D}_{S}^{\top}\bm{h}\|_1\leq \sqrt{s}\|\bm{D}_{S}^{\top}\bm{h}\|_2$.
Furthermore, using the fact that $(a-1)\sqrt{s}+(b+1)\geq0$, i.e.,
$\frac{a\sqrt{s}+b}{\sqrt{s}-1}\geq1$,
one has
\begin{align}\label{e:etainftynew}
\|\bm{D}_{S^c}^{\top}\bm{h}\|_{\infty}\leq \frac{\|\bm{D}_{S}^{\top}\bm{h}\|_{1}}{s}\leq\frac{\|\bm{D}_{S}^{\top}\bm{h}\|_{2}}{\sqrt{s}}\leq\frac{\sqrt{s}+\alpha}{\sqrt{s}-1}\frac{\|\bm{D}_{S}^{\top}\bm{h}\|_2}{\sqrt{s}}\leq \varrho,
\end{align}
where the last inequality is due to the definition of $\varrho$ in \eqref{Coneconstraintinequality}.
By Proposition \ref{NonsparseROC} with $\bm{u}=\|\bm{D}_{S}^{\top}\bm{h}\|_1$
and $\bm v=\|\bm{D}_{S^c}^{\top}\bm{h}\|_1$, the desired inequality \eqref{e:CrossItem1} is clear.
$(\bm{ii})$
From the definition of $\tilde{S}$ in \eqref{def:S} and $\varrho$ in \eqref{Coneconstraintinequality}, it follows that
\begin{align}\label{e:etaW2infty}
\|\bm{D}_{{\tilde{S}}^c}^{\top}\bm{h}\|_{\infty}\leq\frac{\varrho}{t-1},
\end{align}
and
\begin{align}\label{e:etaW2}
\|\bm{D}_{{\tilde{S}}^c}^{\top}\bm{h}\|_1-\|\bm{D}_{{\tilde{S}}^c}^{\top}\bm{h}\|_2
=&\|\bm{D}_{S^c}^{\top}\bm{h}-\bm{D}_{\tilde{S}\setminus S}^{\top}\bm{h}\|_{1}
-\|\bm{D}_{S^c}^{\top}\bm{h}-\bm{D}_{\tilde{S}\setminus S}^{\top}\bm{h}\|_{2}\nonumber\\
\overset{(a)}{=}&\|\bm{D}_{S^c}^{\top}\bm{h}\|_1-\|\bm{D}_{\tilde{S}\setminus S}^{\top}\bm{h}\|_{1}
-\|\bm{D}_{S^c}^{\top}\bm{h}-\bm{D}_{\tilde{S}\setminus S}^{\top}\bm{h}\|_{2}\nonumber\\
\overset{(b)}{\leq}&\big(\|\bm{D}_{S^c}^{\top}\bm{h}\|_1-\|\bm{D}_{S^c}^{\top}\bm{h}\|_2\big)
-(\|\bm{D}_{\tilde{S}\setminus S}^{\top}\bm{h}\|_{1}-\|\bm{D}_{\tilde{S}\setminus S}^{\top}\bm{h}\|_{2})\nonumber\\
\overset{(c)}{\leq}&(s-\sqrt{s})\varrho-(\|\bm{D}_{\tilde{S}\setminus S}^{\top}\bm{h}\|_{1}-\|\bm{D}_{\tilde{S}\setminus S}^{\top}\bm{h}\|_{2}),
\end{align}
where $(a)$, $(b)$ and $(c)$ follow from $\tilde{S}\setminus S \subseteq S^{c}$, the triangle inequality on $\|\cdot\|_2$, and \eqref{Coneconstraintinequality}, respectively.
For the second term of the above inequality, using Lemma \ref{LocalEstimateL1-L2} (b) with $S_1=\tilde{S}\setminus S$ and $S_2=\tilde{S}^c$, we derive that
\begin{align}\label{e:eta-maxkuplowbound}
\|\bm{D}_{S^c}^{\top}\bm{h}\|_1-\|\bm{D}_{S^c}^{\top}\bm{h}\|_2
\geq&\big(\|\bm{D}_{\tilde{S}\setminus S}^{\top}\bm{h}\|_1-\|\bm{D}_{\tilde{S}\setminus S}^{\top}\bm{h}\|_2\big)
+\big(\|\bm{D}_{\tilde{S}^c}^{\top}\bm{h}\|_1-\|\bm{D}_{\tilde{S}^c}^{\top}\bm{h}\|_2\big)\nonumber\\
\geq&\|\bm{D}_{\tilde{S}\setminus S}^{\top}\bm{h}\|_1-\|\bm{D}_{\tilde{S}\setminus S}^{\top}\bm{h}\|_2\nonumber\\
\overset{(a)}{\geq}&(|\tilde{S}\setminus S|-\sqrt{|\tilde{S}\setminus S|})\min_{i\in \tilde{S}\setminus S}|(\bm{D}_{\tilde{S}\setminus S}^{\top}\bm{h})(i)|\nonumber\\
\overset{(b)}{\geq}&(|\tilde{S}\setminus S|-\sqrt{|\tilde{S}\setminus S|})\frac{\varrho}{t-1},
\end{align}
where we use Lemma \ref{LocalEstimateL1-L2} (a) and the definition of $\tilde{S}$ in $(a)$ and $(b)$, respectively.
Substituting \eqref{e:eta-maxkuplowbound} into \eqref{e:etaW2}, there is
\begin{align}\label{e:l1-2upperbounds}
\|\bm{D}_{\tilde{S}^c}^{\top}\bm{h}\|_1-\|\bm{D}_{\tilde{S}^c}^{\top}\bm{h}\|_2
\leq \Big((s(t-1)-|\tilde{S}\setminus S|)-
(\sqrt{s}(t-1)-\sqrt{|\tilde{S}\setminus S|})\Big)\frac{\varrho}{t-1}.
\end{align}
Since $t\geq3$ and $s\geq 2$,
as shown in the items (a) and (b) of
\cite[Page 18]{ge2021dantzig}, we have
$$|\tilde{S}\setminus S|< s(t-1),\ \ \ \ \sqrt{s(t-1)-|\tilde{S}\setminus S|}\leq \sqrt{s}(t-1)-\sqrt{|\tilde{S}\setminus S|}.$$
Then,
\begin{align}\label{e:etaW2upperbound}
\|\bm{D}_{\tilde{S}^c}^{\top}\bm{h}\|_1-\|\bm{D}_{\tilde{S}^c}^{\top}\bm{h}\|_2
\leq
\Big(s(t-1)-|\tilde{S}\setminus S|-\sqrt{s(t-1)-|\tilde{S}\setminus S|}\Big)\frac{\varrho}{t-1}.
\end{align}
Therefore,
from \eqref{e:etaW2infty}, \eqref{e:etaW2upperbound} and Proposition \ref{NonsparseROC} with $\bm u=\bm{D}_{\tilde{S}}^{\top}{\bm h}$,
$\bm {v}=\bm{D}_{\tilde{S}^c}^{\top}\bm{h}$,
it follows that
\begin{align}\label{e:omegal2high}
&|\langle \bm{AD}\bm{D}_{\tilde{S}}^{\top}{\bm h},\bm{AD}\bm{D}_{\tilde{S}^c}^{\top}{\bm h}\rangle
+\langle\bar{\bm{ D}}\bm{D}_{\tilde{S}}^{\top}{\bm h},\bar{\bm{ D}}\bm{D}_{\tilde{S}^c}^{\top}{\bm h} \rangle|\nonumber\\
&\leq\bigg(1+\frac{\sqrt{2}}{2}\bigg)\theta_{ts, (t-1)s-|\tilde{S}\setminus S|}\sqrt{\lceil(t-1)s\rceil-|\tilde{S}\setminus S|} \frac{\varrho}{t-1}\|\bm{D}_{\tilde{S}}^{\top}{\bm h}\|_2\nonumber\\
&\leq \bigg(1+\frac{\sqrt{2}}{2}\bigg)\theta_{ts, (t-1)s}\sqrt{\lceil(t-1)s\rceil} \frac{\varrho}{t-1}\|\bm{D}_{\tilde{S}}^{\top}{\bm h}\|_2.
\end{align}
Based on the fact $\bar{\bm{ D}}\bm{D}^{\top}=\bm{0}$, the above inequality reduces to the desired \eqref{e:CrossItem2}.
$(\bm{iii})$
For the term
$\langle \bm{Ah}, \bm{AD}\bm{D}_{\tilde{S}}^{\top}\bm{h}\rangle$, there is
\begin{align}\label{Upperbound}
&|\langle \bm{Ah}, \bm{AD}\bm{D}_{\tilde{S}}^{\top}\bm{h}\rangle\big|\leq\| \bm{Ah}\|_{2}\|\bm{AD}\bm{D}_{\tilde{S}}^{\top}\bm{h}\|_{2}
\overset{(a)}{\leq}\sqrt{1+\delta_{ts}}\|\bm{D}\bm{D}_{\tilde{S}}^{\top}\bm{h}\|_{2}\|\bm{Ah}\|_{2}\nonumber\\
&\leq\sqrt{1+\delta_{ts}}\|\bm{D}\|_{2\rightarrow 2}^{1/2}\|\bm{D}_{\tilde{S}}^{\top}\bm{h}\|_{2}\|\bm{Ah}\|_{2}
\overset{(b)}{=} \sqrt{1+\delta_{ts}}\|\bm{D}_{\tilde{S}}^{\top}\bm{h}\|_{2}\|\bm{Ah}\|_{2},
\end{align}
where $(a)$ is because of the matrix $\bm{A}$ satisfying the $\bm D$-RIP of $ts$ order,
and $(b)$ follows from $\|\bm{D}\|_{2\rightarrow 2}=\|\bm{D}\bm{D}^{\top}\|_{2\rightarrow 2}^{1/2}=1$.
Next, we will establish the lower bound of $\big|\langle \bm{Ah}, \bm{AD}\bm{D}_{\tilde{S}}^{\top}\bm{h}\rangle\big|$. Note that
\begin{align*}
\big|\langle \bm{Ah}, \bm{AD}\bm{D}_{\tilde{S}}^{\top}\bm{h}\rangle\big|\geq
\| \bm{AD}\bm{D}_{\tilde{S}}^{\top}\bm{h}\|_2^2-\big|\langle \bm{AD}\bm{D}_{\tilde{S}^c}^{\top}\bm{h}, \bm{AD}\bm{D}_{\tilde{S}}^{\top}\bm{h}\rangle\big|.
\end{align*}
From Proposition \ref{prop.DROC} with $\bm v=\bm{D}_{\tilde{S}^c}^{\top}\bm{h}$ and $\bar{\bm{ D}}\bm{D}^{\top}=\bm{0}$, it follows that
\begin{align*}
\| \bm{AD}\bm{D}_{\tilde{S}}^{\top}\bm{h}\|_2^2
=\| \bm{AD}\bm{D}_{\tilde{S}}^{\top}\bm{h}\|_2^2+\|\bar{\bm{D}}\bm{D}_{\tilde{S}}^{\top}\bm{h}\|_2^2
\geq(1-\delta_{ts})\|\bm{D}_{\tilde{S}}^{\top}\bm{h}\|_2^2.
\end{align*}
By \eqref{e:CrossItem2} in item (ii) and $\varrho$ in \eqref{Coneconstraintinequality}, we have
\begin{align*}
\big|\langle \bm{AD}\bm{D}_{\tilde{S}^c}^{\top}\bm{h}, \bm{AD}\bm{D}_{\tilde{S}}^{\top}\bm{h}\rangle\big|
\leq \theta_{ts, (t-1)s}\sqrt{\lceil(t-1)s\rceil}\bigg(1+\frac{\sqrt{2}}{2}\bigg) \frac{\varrho}{t-1}\|\bm{D}_{\tilde{S}}^{\top}\bm{h}\|_2
\end{align*}
Then,
\begin{align}\label{Lowerbound}
&\big|\langle \bm{Ah}, \bm{AD}\bm{D}_{\tilde{S}}^{\top}\bm{h}\rangle\big|\nonumber\\
&\geq(1-\delta_{ts})\|\bm{D}_{\tilde{S}}^{\top}\bm{h}\|_2^2-\theta_{ts, (t-1)s}
\bigg(1+\frac{\sqrt{2}}{2}\bigg)\frac{\sqrt{\lceil(t-1)s\rceil}}{t-1}\|\bm{D}_{\tilde{S}}^{\top}\bm{h}\|_2\nonumber\\
&\cdot\bigg(\frac{a\sqrt{s}+b}{\sqrt{s}-1}\frac{\|\bm{D}_{S}^{\top}\bm{h}\|_2}{\sqrt{s}}
+\frac{c\|\bm{D}_{T^c}^{\top}\bm{x}\|_1+\eta\|{\bm Ah}\|_2+\gamma}{s-\sqrt{s}}\bigg)\nonumber\\
&\geq\bigg( 1-\delta_{ts}-\bigg(1+\frac{\sqrt{2}}{2}\bigg)\sqrt{\frac{\lceil(t-1)s\rceil}{(t-1)^2s}}\frac{a\sqrt{s}+b}{\sqrt{s}-1}
\theta_{ts, (t-1)s }\bigg)\|\bm{D}_{\tilde{S}}^{\top}\bm{h}\|_2^2\nonumber\\
&-\theta_{ts,(t-1)s}\bigg(1+\frac{\sqrt{2}}{2}\bigg)\frac{\sqrt{\lceil(t-1)s\rceil}}{t-1}\frac{c\|\bm{D}_{T^c}^{\top }\bm{x}\|_1+\eta\|{\bm Ah}\|_2+\gamma}{s-\sqrt{s}}\|\bm{D}_{\tilde{S}}^{\top}\bm{h}\|_2,
\end{align}
where the last inequality is due to $S\subseteq \tilde{S}$.
Combining \eqref{Lowerbound} with \eqref{Upperbound}, one has
\begin{align*
&\bigg( 1-\delta_{ts}-\sqrt{\frac{\lceil(t-1)s\rceil}{(t-1)^2s}}\frac{(\sqrt{2}+1)(\sqrt{s}+\alpha)}{\sqrt{2}(\sqrt{s}-1)}
\theta_{ts,(t-1)s}\bigg)\|\bm{D}_{\tilde{S}}^{\top}\bm{h}\|_2^2\nonumber\\
&-\bigg(\theta_{ts,(t-1)s}\frac{\sqrt{2}+1}{\sqrt{2}}\frac{\sqrt{\lceil(t-1)s\rceil}}{t-1}\frac{c\|\bm{D}_{T^c}^{\top}\bm{x}\|_1+\eta\|\bm{ Ah}\|_2+\gamma}{s-\sqrt{s}}\nonumber\\
&+\sqrt{1+\delta_{ts}}\|\bm{Ah}\|_{2}\bigg)\|\bm{D}_{\tilde{S}}^{\top}\bm{h}\|_2\leq 0.
\end{align*}
Therefore, using \eqref{RIPConditiona1add} we derive that
\begin{align}\label{Estimate.hmax(s).eq1}
&\|\bm{D}_{S}^{\top}\bm{h}\|_2\leq\|\bm{D}_{\tilde{S}}^{\top}\bm{h}\|_2\nonumber\\
&\leq\frac{\theta_{ts,(t-1)s}}{(1-\rho_{s,t})}\frac{\sqrt{2}+1}{\sqrt{2}}\frac{\sqrt{\lceil(t-1)s\rceil}}{(t-1)(s-\sqrt{s})}
\left(c\|\bm{D}_{T^c}^{\top}\bm{x}\|_1+\gamma\right)\nonumber\\
&+\bigg(\frac{\theta_{ts,(t-1)s}}{(1-\rho_{s,t})}\frac{\sqrt{2}+1}{\sqrt{2}}\frac{
\sqrt{\lceil(t-1)s\rceil}}{(t-1)(s-\sqrt{s})} +\frac{\sqrt{1+\delta_{ts}}}{1-\rho_{s,t}}\bigg)\eta\|\bm{Ah}\|_{2},
\end{align}
where $\eta\geq 1$.
$(\bm{iv})$
As shown in the proof of $(iii)$, we can prove the item (iv) by item (i) and \eqref{RIPCondition1}. We here omit the detail proof.
$(\bm{v})$ The idea of the proof is the argument in \cite[ Step 2]{ge2021new}.
By \eqref{lem:CrossItem.eq1} and the fact that
$\|\bm{D}_{S^c}^{\top}\bm{h}\|_{\infty}\leq \|\bm{D}_{S}^{\top}\bm{h}\|_{1}/s\leq\|\bm{D}_{S}^{\top}\bm{h}\|_{2}/\sqrt{s}$, we have
\begin{align*}
&\|\bm{D}_{S^c}^{\top}\bm{h}\|_2^2
\leq \|\bm{D}_{S^c}^{\top}\bm{h}\|_{1}\|\bm{D}_{S^c}^{\top}\bm{h}\|_{\infty}\nonumber\\
&\leq\big(a\|\bm{D}_{S}^{\top}\bm{h}\|_1+b\|\bm{D}_{S}^{\top}\bm{h}\|_2+c\|\bm{D}_{T^c}^{\top}\bm{x}\|_1+\eta\|{\bm {Ah}}\|_2+\gamma+\alpha\|\bm{D}_{S^c}^{\top}\bm{h}\|_2\big)
\frac{\|\bm{D}_{S}^{\top}\bm{h}\|_2}{\sqrt{s}}\nonumber\\
&\leq\frac{\alpha\|\bm{D}_{S}^{\top}\bm{h}\|_2}{\sqrt{s}}\|\bm{D}_{S^c}^{\top}\bm{h}\|_2+\frac{a\sqrt{s}+b}{\sqrt{s}}\|\bm{D}_{S}^{\top}\bm{h}\|_2^2+\frac{c\|\bm{D}_{T^c}^{\top
}\bm{x}\|_1+\eta\|{\bm {Ah}}\|_2+\gamma}{\sqrt{s}}\|\bm{D}_{S}^{\top}\bm{h}\|_2 .
\end{align*}
That is,
\begin{align*}
&\bigg(\|\bm{D}_{S^c}^{\top}\bm{h}\|_2-\frac{\alpha\|\bm{D}_{S}^{\top}\bm{h}\|_2}{2\sqrt{s}}\bigg)^2\\
&\leq\bigg(\frac{\alpha^2}{4s}+\frac{a\sqrt{s}+b}{\sqrt{s}}\bigg)\|\bm{D}_{S}^{\top}\bm{h}\|_2^2
+\frac{c\|\bm{D}_{T^c}^{\top}\bm{x}\|_1+\eta\|{\bm {Ah}}\|_2+\gamma}{\sqrt{s}}\|\bm{D}_{S}^{\top}\bm{h}\|_2.
\end{align*}
Then, we obtain
\begin{align}\label{e:anotherupperbound}
&\|\bm{D}_{S^c}^{\top}\bm{h}\|_2\nonumber\\
&\leq \Bigg(\sqrt{\frac{a\sqrt{s}+b}{\sqrt{s}}+\frac{\alpha^2}{4s}}+\frac{\alpha}{2\sqrt{s}}\Bigg)\|\bm{D}_{S}^{\top}\bm{h}\|_2
+\sqrt{\frac{c\|\bm{D}_{T^c}^{\top}\bm{x}\|_1+\eta\|{\bm {Ah}}\|_2+\gamma}{\sqrt{s}}\|\bm{D}_{S}^{\top}\bm{h}\|_2}\nonumber\\
&\leq\Bigg(\sqrt{\frac{a\sqrt{s}+b}{\sqrt{s}}+\frac{\alpha^2}{4s}}+\frac{\alpha+\bar{\varepsilon}}{2\sqrt{s}}\Bigg)\|\bm{D}_{S}^{\top}\bm{h}\|_2
+\frac{1}{2\bar{\varepsilon}}\big(c\|\bm{D}_{T^c}^{\top
}\bm{x}\|_1+\eta\|\bm{Ah}\|_2+\gamma\big),
\end{align}
where the second inequality comes from the basic inequality $\sqrt{|a||b|}\leq \frac{|a|+|b|}{2}$, and the constant $\bar{\varepsilon}>0$.
\end{proof}
\end{appendices}
\section*{Acknowledgments}
The project is partially supported by the Natural Science Foundation of China (Nos. 11901037, 72071018), NSFC of Gansu Province, China (Grant No. 21JR7RA511), the NSAF (Grant No. U1830107) and the Science Challenge Project (TZ2018001).
Authors thanks Professors Xiaobo Qu for making the pFISTA code available online.
\hskip\parindent
\bibliographystyle{plain}
|
2024-02-18T23:40:24.394Z
|
2021-12-30T02:25:42.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14510","language":"en","timestamp":1640831142000,"url":"https:\/\/arxiv.org\/abs\/2112.14510","yymm":"2112"}
|
proofpile-arXiv_000-10015
|
{"provenance":"002.jsonl.gz:10016"}
| null | null |
\section{Background}
{\sc vpfit} is a comprehensive and widely used code for the analysis of absorption spectra \citep{ascl:VPFIT2014}. The theoretical methods on which it is based are described in \cite{WebbVPFIT2021}. Section 4.2 of that paper introduces a new enhancement to the code: the replacement of previous finite difference derivative calculations by analytic derivatives of the free parameters required to model a complex of absorption transitions. Derivatives of the Voigt function are needed to compute the gradient vector and Hessian matrix in non-linear least squares methods. This {\it Addendum} is concerned with column density parameters and how one can compute analytic derivatives of the Voigt function for the specific case of {\it summed} column densities, as may be required when, for example, solving for the primordial deuterium to hydrogen ratio at high redshift, averaged over an absorption complex.
Section 4.2 (iii) of \cite{WebbVPFIT2021} provides the analytic Voigt derivatives for the simple situation when all column density parameters are independent. However, in some circumstances it is advantageous to solve for a summed column density parameter i.e. the total column density of a set of absorption components within a complex. The practical method for doing this
is discussed in detail in \cite{web:VPFIT}, so further details are avoided here. When summed column density parameters are used, the associated analytic Voigt function derivatives are more complicated than those given in Equations (30) of \cite{WebbVPFIT2021} so are provided here. \\
We next explain, using numerical examples, how finite difference derivatives can fail in some cases, when using summed column density parameters, hence motivating the use of analytic expressions in order to guarantee stable algorithms.
\section{Failure of finite difference derivatives in some circumstances}
\label{sec:problem}
To illustrate the numerical instabilities that can arise, consider the following simple example. Suppose we wish to model an absorption complex comprising three adjacent components (i.e. at slightly different redshifts). Suppose further, in this illustrative example, that the model includes two atomic species, C\,{\sc iv} and Si\,{\sc iv}. Following the terminology of the main paper, one species will be a primary, the other secondary. Whilst there is only one primary, there may be multiple secondaries.
Without parameter ties, there are thus three column density parameters for each species. If the {\it total} column density for the complex is a parameter of particular interest, one can assign this quantity to the first of the column density parameters. The advantage of doing so is that the summed column density can be better constrained than the sum of the individual column densities\footnote{Depending on how the other free parameters in the model are arranged, as discussed in \cite{web:VPFIT}}.
We use a practical example similar to that given in the subsection {\it Common pattern relative ion abundances} in the {\sc vpfit} user guide \citep{web:VPFIT}, with the total C\,{\sc iv} adjusted so that the starting guesses are self-consistent\footnote{This is not a requirement for {\sc vpfit} estimates in starting to find a fit, since the program adjusts the subsidiary Si\,{\sc iv} values to make them consistent.}:
\begin{verbatim}
C IV 14.75x 2.765821aa 12.69i 0.00 1.00E+00
C IV 13.89x 2.765965ab 17.73j 0.00 1.00E+00
C IV 12.89x 2.765995c 8.31k 0.00 1.00E+00
SiIV 13.9
SiIV 13.10X 2.765965AB 17.73J 0.00 1.00E+00
SiIV 12.10X 2.765995C 8.31K 0.00 1.00E+00
\end{verbatim}
The summed column density of C\,{\sc iv} is $N_t^{\mathrm{C \,\textsc{iv}}} = 10^{14.75}$, the second and third column density components are $N_{1,2}^{\mathrm{C \,\textsc{iv}}} = 10^{13.89}, 10^{12.89}$. The column density of the first component is
\begin{equation}
\begin{aligned}
& N_1^{\mathrm{C \,\textsc{iv}}} = 10^{14.75}-10^{13.89}-10^{12.89} \,, \\
& \rightarrow \quad \log N_1^{\mathrm{C \,\textsc{iv}}} \simeq 14.68 \,.
\end{aligned}
\end{equation}
The `\%' is a marker, to indicate the start of a new group (Si\,{\sc iv} in this example), such that the first entry in the group is the total Si\,{\sc iv} column density.
The following three examples show the problem if we re-order the individual components and apply two-sided numerical derivative with finite difference derivative $fdd =0.01$.
\vspace{0.1 in}
\noindent {\bf (i) Example 1:} \vspace{0.1 in}\\
Left side:
\begin{verbatim}
C IV 14.75x -> 14.76x
C IV 13.89x
C IV 12.89x
\end{verbatim}
Right side:
\begin{verbatim}
C IV 14.75x -> 14.74x
C IV 13.89x
C IV 12.89x
\end{verbatim}
The column densities of first individual component are
\begin{equation}
\begin{aligned}
\mathrm{Left \ side}: \quad &N_1 = 10^{14.68} \rightarrow N_1 = 10^{14.69} \,, \\
\mathrm{Right \ side}: \quad &N_1 = 10^{14.68} \rightarrow N_1 = 10^{14.66} \,.
\end{aligned}
\end{equation}
In the above first example, we see that the numerator of the fdd is reasonable and the fdd itself presents no problem. However, now consider a slightly different example.
\vspace{0.1 in}
\noindent {\bf (ii) Example 2:} \vspace{0.1 in}\\
Left side:
\begin{verbatim}
C IV 14.75x -> 14.76x
C IV 14.68x
C IV 12.89x
\end{verbatim}
Right side:
\begin{verbatim}
C IV 14.75x -> 14.74x
C IV 14.68x
C IV 12.89x
\end{verbatim}
The column densities of first individual component are
\begin{equation}
\begin{aligned}
\mathrm{Left \ side}: \quad &N_1 = 10^{13.89} \rightarrow N_1 = 10^{13.95} \,, \\
\mathrm{Right \ side}: \quad &N_1 = 10^{13.89} \rightarrow N_1 = 10^{13.80} \,.
\end{aligned}
\end{equation}
In this second example, the outcome is poor because the numerator of the fdd is large (0.15) and the derivative loses accuracy.
\vspace{0.1 in}
\noindent {\bf (iii) Example 3:} \vspace{0.1 in} \\
Left side:
\begin{verbatim}
C IV 14.75x -> 14.76x
C IV 13.89x
C IV 14.68x
\end{verbatim}
Right side:
\begin{verbatim}
C IV 14.75x -> 14.74x
C IV 13.89x
C IV 14.68x
\end{verbatim}
The column densities of first individual component are
\begin{equation}
\begin{aligned}
\mathrm{Left \ side}: \quad &N_1 = 10^{12.89} \rightarrow N_1 = 10^{13.28} \,, \\
\mathrm{Right \ side}: \quad &N_1 = 10^{12.89} \rightarrow N_1 = - 10^{12.83} \,.
\end{aligned}
\end{equation}
In this third example, the result is catastrophic because one side of the fdd interval becomes negative, the fdd interval becomes essentially meaningless, and the numerical derivative fails. Of course the problem has arisen because of the ordering of the three column densities; provided the strongest component is placed first in the grouping, the problem is largely avoided. However, this is not only an undesirable solution, it is sometimes impractical, because even if the parameter guesses are ordered ``sensibly'' at the commencement of the non-linear least squares process, subsequent iterations may reduce the column density of the first component in the group such that the difficulty illustrated in example 3 arises. One can easily find, for example, blends of components where there is no obvious one strong component, relative to others in the system. Therefore, a more robust approach is needed, as discussed next.
\section{The solution to the problem -- analytic derivatives}
\label{sec:solution}
\vspace{0.1 in}
\noindent {\bf (i) Notations:}
\begin{enumerate}[leftmargin=0.5cm]
\item $N^{p}_t$: summed column density of the primary species
\item $N^{p}_i$: the column density of the $i^{th}$ component of the primary species
\item $N^{s}_t$: summed column density of the secondary species
\item $N^{s}_i$: the column density of the $i^{th}$ component of the secondary species
\end{enumerate}
The column density of the first component of each block is not an independent internal variable within {\sc vpfit}. It is necessary to calculate its derivative at iteration of the minimisation. Note that the default variables in {\sc vpfit} are $\log N^{p}_t$, $\log N^{p}_i$ $(i \neq 1)$, and $\log N^{s}_t$. The relation between the relevant variables are listed as follows.
\begin{equation}
\begin{aligned}
& N^{p}_1 = N^{p}_t - \sum_{j=2}^m N^{p}_j \,, \\
& N^{s}_1 = N^{s}_t - \sum_{j=2}^m N^{s}_j \,, \\
& \log N^{s}_j = \log N^{p}_j - \log N^{p}_t + \log N^{s}_t \,,
\end{aligned}
\end{equation}
where $m$ is the total number of component of the leading block (i.e. the primary species).
\vspace{0.1 in}
\noindent {\bf (ii) Case I: Derivative of $N^{p}_t$}
\begin{equation}
\label{eq:deriv1}
\begin{aligned}
& \frac{d \log N^{p}_1}{d \log N^{p}_t} = F_1 \,, \\
& \frac{d \log N^{p}_i}{d \log N^{p}_t} = 0 \quad (i \neq 1) \,, \\
& \frac{d \log N^{s}_1}{d \log N^{p}_t} = F_2 \,, \\
& \frac{d \log N^{s}_i}{d \log N^{p}_t} = -1 \quad (i \neq 1) \,,
\end{aligned}
\end{equation}
where
\begin{equation}
\label{eq:F1F2}
\begin{aligned}
& F_1 = \frac{N^{p}_t}{N^{p}_t - \sum_{j=2}^m N^{p}_j} \,, \\
& F_2 = \frac{N^{s}_t}{N^{s}_t - \sum_{j=2}^m N^{s}_j}-1 = F_1 - 1 \,. \\
\end{aligned}
\end{equation}
\vspace{0.1 in}
\noindent {\bf (iii) Case II: Derivative of $N^{p}_k$}
\begin{equation}
\label{eq:deriv2}
\begin{aligned}
& \frac{d \log N^{p}_1}{d \log N^{p}_k} = - F_3 \,, \\
& \frac{d \log N^{p}_i}{d \log N^{p}_k} = \delta_{ik} \quad (i \neq 1) \,, \\
& \frac{d \log N^{s}_1}{d \log N^{p}_k} = - F_3 \,, \\
& \frac{d \log N^{s}_i}{d \log N^{p}_k} = \delta_{ik} \quad (i \neq 1) \,,
\end{aligned}
\end{equation}
where $\delta_{ik}$ is the Dirac delta function and
\begin{equation}
\begin{aligned}
F_3 = \frac{ N^{p}_k}{N^{p}_t - \sum_{j=2}^m N^{p}_j} \,. \\
\end{aligned}
\end{equation}
\vspace{0.1 in}
\noindent {\bf (iv) Case III: Derivative of $N^{s}_t$}
\begin{equation}
\label{eq:deriv3}
\begin{aligned}
& \frac{d \log N^{p}_1}{d \log N^{s}_t} = 0 \,, \\
& \frac{d \log N^{p}_i}{d \log N^{s}_t} = 0 \quad (i \neq 1) \,, \\
& \frac{d \log N^{s}_1}{d \log N^{s}_t} = 1 \,, \\
& \frac{d \log N^{s}_i}{d \log N^{s}_t} = 1 \quad (i \neq 1) \,,
\end{aligned}
\end{equation}
\begin{figure}
\centering
\includegraphics[width=0.9\linewidth]{Addendum1.pdf}
\caption{The numerical errors go to zero at the right end, which means only the first component has a non-zero column density i.e. only one component is considered.}
\label{fig:err_case1}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.9\linewidth]{Addendum2.pdf}
\caption{Similar to the Fig.~\ref{fig:err_case1}; the numerical error is significant if the first component is weaker than the others.}
\label{fig:err_case2}
\end{figure}
\section{Finite difference derivative numerical errors} \label{sec:numericalerrors}
The now quantify potential fdd numerical errors using the equations given in the previous section, using the first three terms of a Taylor series expansion, which are $e^\Delta = 1+ \Delta + \frac{\Delta^2}{2} \frac{\Delta^3}{6}$ and $\log (1+\Delta) = \Delta - \frac{\Delta^2}{2} + \frac{\Delta^3}{3}$. We ignore cases where the fdd is either 0 or 1 since the fdds do not suffer from the instability described in Section \ref{sec:solution} (see the relevant parts of Eqs.~\eqref{eq:deriv1}, \eqref{eq:deriv2} and \eqref{eq:deriv3}).\\
\vspace{0.1 in}
\noindent {\bf (i) Case I: $\log N^{p}_t \rightarrow \log N^{p}_t \pm \Delta$}
Using a Taylor series expansion we obtain
\begin{equation}
\begin{aligned}
\log N^{p}_1 &\rightarrow \log N^{p}_1 + F_1 \cdot \Delta + \left( F_1 - F_1^2 \right) \frac{\ln 10 \cdot \Delta^2}{2} \\
&+ \left( F_1 - 3 F_1^2 + 2F_1^3 \right) \frac{ (\ln 10)^2 \cdot \Delta^3}{6} \,, \\
\log N^{s}_1 &\rightarrow \log N^{s}_1 + F_2 \cdot \Delta - \left( F_2 - F_2^2 \right) \frac{\ln 10 \cdot \Delta^2}{2} \\
&+ \left( F_2 + 3 F_2^2 + 2F_2^3 \right) \frac{ (\ln 10)^2 \cdot \Delta^3}{6} \,.
\end{aligned}
\end{equation}
We have the relation from Eq.~\eqref{eq:F1F2},
\begin{equation}
F_2+3F_2^2+2F_2^3=F_1-3F_1^2+2F_1^3 \,.
\end{equation}
Then, we have
\begin{equation}
\begin{aligned}
&\frac{d \log N^{p}_1}{d \log N^{p}_t} \Big\lvert_\mathrm{fdd} - \frac{d \log N^{p}_1}{d \log N^{p}_t} \Big\rvert_\mathrm{analytic} \\
& = \frac{d \log N^{s}_1}{d \log N^{s}_t} \Big\lvert_\mathrm{fdd} - \frac{d \log N^{s}_1}{d \log N^{s}_t} \Big\rvert_\mathrm{analytic} \\
& \simeq \left( F_1 - 3 F_1^2 + 2F_1^3 \right) \frac{ (\ln 10 \cdot \Delta)^2}{6} \,,
\end{aligned}
\end{equation}
\vspace{0.1 in}
\noindent {\bf (ii) Case II: $N^{p}_k \rightarrow N^{p}_k \pm \Delta$}
\begin{equation}
\label{eq:n1l-2}
\begin{aligned}
\log N^{p}_1 &\rightarrow \log N^{p}_1 - F_3 \cdot \Delta - \left( F_3 + F_3^2 \right) \frac{\ln 10 \cdot \Delta^2}{2} \\
&- \left( F_3 + 3 F_3^2 + 2F_3^3 \right) \frac{ (\ln 10)^2 \cdot \Delta^3}{6} \,.
\end{aligned}
\end{equation}
The {\sc vpfit} variable, $\log N^{s}_1$, follows the same relation as Eq.~\eqref{eq:n1l-2}. Then, we have
\begin{equation}
\begin{aligned}
&\frac{d \log N^{p}_1}{d \log N^{p}_k} \Big\lvert_\mathrm{fdd} - \frac{d \log N^{p}_1}{d \log N^{p}_k} \Big\rvert_\mathrm{analytic} \\
&= \frac{d \log N^{s}_1}{d \log N^{p}_k} \Big\lvert_\mathrm{fdd} - \frac{d \log N^{s}_1}{d \log N^{p}_k} \Big\rvert_\mathrm{analytic} \\
&\simeq - \left( F_3 + 3 F_3^2 + 2F_3^3 \right) \frac{ (\ln 10 \cdot \Delta)^2}{6} \,.
\end{aligned}
\end{equation}
Figures \ref{fig:err_case1} and \ref{fig:err_case2} illustrate the estimated fdd numerical errors as a function of the relative strength of the first component of the primary species, where
\begin{equation}
Error = \left( \frac{d \log N^{p}_1}{d \log N^{p}_t} \Big\lvert_\mathrm{fdd} \Big/ \frac{d \log N^{p}_1}{d \log N^{p}_t} \Big\rvert_\mathrm{analytic} \right) -1 \,.
\end{equation}
Both figures illustrate the basic problem i.e. that the fdd numerical error becomes large when the column density of the first component in the group is weak relative to the total column density of that group (Figure \ref{fig:err_case1}) or to a following component of that group (Figure \ref{fig:err_case2}).
|
2024-02-18T23:40:24.401Z
|
2021-12-30T02:25:11.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14490","language":"en","timestamp":1640831111000,"url":"https:\/\/arxiv.org\/abs\/2112.14490","yymm":"2112"}
|
proofpile-arXiv_000-10016
|
{"provenance":"002.jsonl.gz:10017"}
| null | null |
\section{Introduction}
The purpose of these notes is to demonstrate in
detail various approaches of constructing
supersymmetric cubic interaction vertices for
higher spin fields. To this end
we reconstruct well known cubic vertices for
$N=1$ Super Yang-Mills and for linearized
$N=1$ Supergravities. In particular, we consider the field theoretic limits of
the pure $N=1$, ${\cal D}=10$ supergravity \cite{Chamseddine:1980cp}--\cite{Bergshoeff:1981um},
of the $N=1$, ${\cal D}=4$ supergravity coupled with one chiral supermultiplet
\cite{Cremmer:1978hn},
and of the $N=1$, ${\cal D}=6$ supergravity coupled
to one $(1,0)$ tensor supermultiplet \cite{Romans:1986er}--
\cite{Nishino:1986dc}.
The reason for choosing these particular types of supergravity will become clear in the following. We shall also
comment on a generalization of these vertices
to higher spin ``Yang-Mills-like" and ``Supergravity-
like" vertices \cite{Buchbinder:2021qrg}, and on the higher spin generalizations of the corresponding free Lagrangians \cite{Sorokin:2018djm}.
First, we shall describe the covariant BRST approach,\footnote{See
\cite{Fotopoulos:2008ka} for a review of the BRST approach and \cite{Vasiliev:1999ba} --\cite{Tran:2020uqx} for reviews of
different approaches to higher spin theories.} which
is similar
to the one of Open String Field Theory.\footnote{See e.g. \cite{Erler:2019vhl}
for a recent review.}
However, unlike String Field Theory,
the BRST approach to higher spin fields is essentially a method
for the construction of free and interacting Lagrangians using gauge invariance as the only guiding principle,
without any recourse to a world-sheet description.
As the first step in this approach,
one constructs free Lagrangians
invariant under linear gauge transformations.
Because of the presence of gauge symmetry, these Lagrangians
contain
both physical and non-physical degrees of freedom.
Some of the non-physical degrees of freedom are removed using the equations of motion, the others are gauged away, and in the end one is left only with physical polarizations.
In general, these systems can contain
bosonic and fermionic fields, which are described
by Young tableaux with mixed symmetries.
However, at the free level one can consider
Lagrangians for just one or several (a finite number of)
representations of the Poincar\'e group \cite{Sagnotti:2003qa}--
\cite{Francia:2002pt}.
A further requirement of supersymmetry
singles out some particular representations of the Poincar\'e group,
both in bosonic and fermionic sectors, so the corresponding bosonic and fermionic Lagrangians are related by supersymmetry transformations.
As the second step in the BRST approach, one promotes the original gauge symmetry
to an interacting level by deforming the Lagrangian and gauge transformation
rules with nonlinear terms, in such a way that the gauge invariance is kept order by order in the coupling constant.
As in the case of free Lagrangians, supersymmetry
singles out some particular subclass
of the cubic vertices,
which were found for non-supersymmetric systems
\cite{Metsaev:2005ar}--\cite{Manvelyan:2010je}.
To describe how this approach works on the
examples of $N=1$ Super Yang-Mills and linearized Supergravities,
we start in Section
\ref{LFr}
with a description of gauge invariant free Lagrangians.
For the massless vector field the corresponding Lagrangian is the standard (Maxwell) one.
A gauge invariant Lagrangian which contains the second rank symmetric massless tensor field
as a physical component
describes
spins $2$ and $0$ simultaneously. Then we present a similar gauge invariant description for a massless spin-vector field
which contains irreducible representations of the Poincar\'e group with spins $\frac{3}{2}$
and $\frac{1}{2}$.
In Section \ref{brstfree} we reformulate the results of Section \ref{LFr}
in the BRST approach. Then we impose an additional requirement of $N=1$ supersymmetry
on these systems. Using the technique
developed in the Open Superstring Field Theory
\cite{Kazama:1986cy}
we show that $N=1$ supersymmetry requires some extra fields
(both physical and auxiliary) in the bosonic sector \cite{Sorokin:2018djm}. The obtained Lagrangian
provides a ``unified" description
of the above mentioned $N=1$ linearized supergravities in ${\cal D}=4,6$ and $10$ dimensions.
In Section \ref{cube} we turn to cubic interactions in the covariant formalism and present generic equations
which determine the cubic vertices, as well as
solutions to these equations
for three massless bosonic fields with arbitrary spins \cite{Fotopoulos:2010ay},
\cite{Metsaev:2012uy}.
We also present the generic equations for determining cubic vertices
for
two massless fermions and one massless boson \cite{Buchbinder:2021qrg}.
In Section \ref{sectionsym} we describe
$N=1$ Super Yang-Mills in this formalism and
in Section \ref{sectionsugra} we consider cubic vertices for linearized supergravities.
We comment on
the higher spin generalization
of the vertices given in Sections \ref{sectionsym}
and \ref{sectionsugra}.
These cubic vertices \cite{Buchbinder:2021qrg} are covariant
versions of the vertices for two fermionic and one bosonic
higher spin fields
in arbitrary dimensions,
first derived in the light cone formalism \cite{Metsaev:2007rn}.\footnote{Covariant cubic vertices
with two fermions
and electromagnetic and gravitational fields are given in
\cite{Henneaux:2012wg}--\cite{Henneaux:2013gba}. Supersymmetric cubic interactions on flat and $AdS$ backgrounds are given in
\cite{Alkalaev:2002rq}--\cite{Fujii:2021klx}.}
Finally,
in Section \ref{sectionlightcone}
we describe the
light cone approach to the construction of
the cubic vertices. In this approach one splits
the generators of the Poincar\'e (super)group
into dynamical and kinematical operators.
When using the field theoretic realization of the generators
one takes the kinematical operators to be quadratic in terms of superfields, whereas the dynamical generators
contain also cubic and higher order terms, i.e., cubic and higher order vertices. The requirement
that the Poincar\'e superalgebra stays intact after the nonlinear deformation
of the dynamical operators
determines these vertices order by order in the coupling constant.
We shall briefly review the construction of
\cite{Metsaev:2019dqt} (see
\cite{Metsaev:2019aig},\cite{Bengtsson:1983pg} for the case
of an arbitrary $N$)
for arbitrary spin supermultiplets
in $N=1$, ${\cal D}=1$,
and show how to obtain cubic vertices for four dimensional
$N=1$ super Yang-Mills and $N=1$
supergravity in the light cone gauge\footnote{The cubic vertices
for ${\cal D}=4$, $N=4$ super Yang-Mills in the light cone formulation are given in
\cite{Brink:1982pd}--\cite{Mandelstam:1982cb}.} as a particular example.
\section{Free Lagrangians} \setcounter{equation}0
\label{LFr}
\subsection{\texorpdfstring{$s=1$}{s=1}} \label{one}
Let us start with a massless vector field $\phi_\mu(x)$ with a standard gauge transformation rule
\begin{equation} \label{lv1-1}
\delta \phi_\mu(x) = \partial_\mu \lambda(x)
\end{equation}
The gauge invariant Klein -Gordon and transversality equations for the massless vector field can be written as
\begin{equation} \label{v1}
\Box \phi_\mu(x) = \partial_\mu C(x), \quad C(x)=
\partial^\mu \phi_\mu(x)
\end{equation}
where we introduced an auxiliary field $C(x)$, which transforms
as
\begin{equation} \label{lv1-2}
\delta C(x) = \Box \lambda(x)
\end{equation}
The equations \p{v1} can be obtained from the Lagrangian
\begin{equation} \label{lv1}
{\cal L}= - \frac{1}{2} (\partial^\mu \phi^{\nu }) (\partial_\mu \phi_{\nu })
+ C \partial^\mu \phi_{\mu }
- \frac{1}{2} C^2
\end{equation}
After eliminating the field $C(x)$ via its own equation of motion one obtains the Maxwell Lagrangian for the vector field $\phi_\mu(x)$.
\subsection{\texorpdfstring{$s=2$}{s=2} and \texorpdfstring{$s=0$}{s=0}} \label{2and0}
One can repeat a similar consideration for a second rank symmetric tensor field
$\phi_{\mu \nu}(x)$.
Now the gauge invariant Klein-Gordon equation reads,
\begin{equation} \label{s2-0}
\Box \phi_{\mu \nu}(x) = \partial_\mu C_\nu(x) + \partial_\nu C_\mu(x)
\end{equation}
where the physical field $\phi_{\mu \nu}(x)$ and the auxiliary field $C_\mu(x)$ transform as
\begin{equation} \label{s2-1-1}
\delta \phi_{\mu \nu} (x) = \partial_\mu \lambda_\nu(x) + \partial_\nu \lambda_\mu(x), \quad \delta C_\mu(x) = \Box \lambda_\mu(x)
\end{equation}
In order to write gauge invariant transversality conditions one needs
one more auxiliary field $D(x)$, which transforms as
\begin{equation} \label{s2-1}
\delta D(x) = \partial^\mu \lambda_\mu(x)
\end{equation}
Then the gauge invariant transversality equation is
\begin{equation} \label{s2-2}
\partial^\nu \phi_{\mu \nu}(x) - \partial_\mu D(x) = C_\mu(x)
\end{equation}
Finally, one can write a gauge invariant Klein-Gordon equation for the field $D(x)$
\begin{equation} \label{s2-3}
\Box D(x)= \partial^\mu C_\mu(x)
\end{equation}
and ``integrate'' the equations \p{s2-0}, \p{s2-2} and \p{s2-3} back into the Lagrangian
\begin{equation} \label{lv2}
{\cal L}= - \frac{1}{2} (\partial^\mu \phi^{\nu \rho}) (\partial_\mu \phi_{\nu \rho})
+ 2C^\mu \partial^\nu \phi_{\mu \nu}
- C^\mu C_\mu + (\partial^\mu D) (\partial_\mu D)
+ 2 D \partial^\mu C_\mu
\end{equation}
Again, the only propagating degrees of freedom are the physical components
of the field $\phi_{\mu \nu}(x)$. Its longitudinal
components and the fields $C_\mu(x)$ and $D(x)$ are either pure gauge or zero on shell.
Finally, since there is no zero trace condition involved, one obtains a gauge invariant description
simultaneously for a spin $2$ field $g_{\mu \nu}(x)$ and for a scalar $\phi(x)$, both packed in the field $\phi_{\mu \nu}(x)$.
\subsection{\texorpdfstring{$s=\frac{3}{2}$}{s=3/2} and \texorpdfstring{$s=\frac{1}{2}$}{s=1/2}}\label{32and12}
The Lagrangian describing only spin $\frac{1}{2}$
field is simply
\begin{equation}
{\cal L} = -i \, \bar \Psi \gamma^\mu \partial_\mu \Psi
\end{equation}
The next simplest example is
a spin-vector field $\Psi_\mu^a(x)$, where $"a"$ is a spinorial index (see appendix \ref{Appendix A} for the present conventions).
Gauge invariant Dirac and transversality equations can be written by introducing one auxiliary field $\chi^a(x)$
as
\begin{equation} \label{3/2-1}
\gamma^\nu \partial_\nu \Psi_\mu (x) + \partial_\mu \chi (x) =0
\end{equation}
\begin{equation} \label{3/2-2}
\partial^\mu \Psi_\mu(x)
+ \gamma^\nu \partial_\nu \chi(x) =0
\end{equation}
The equations \p{3/2-1}-- \p{3/2-1} are invariant under gauge transformations
\begin{equation}
\delta \Psi_\mu (x) = \partial_\mu \,
\lambda^\prime (x),
\quad
\delta \chi(x) = - \gamma^\nu \partial_\nu \lambda^\prime (x)
\end{equation}
and can be obtained from the Lagrangian
\begin{equation} \label{l-f-f}
{\cal L} = -i \, \bar \Psi^\nu \gamma^\mu \partial_\mu \Psi_\nu -
i \, \bar \Psi^\mu \partial_\mu \chi + i \,
\bar \chi \partial^\mu \Psi_\mu
+ i \, \bar \chi \gamma^\mu \partial_\mu \chi
\end{equation}
Similarly to the previous case, one can gauge away the auxiliary field
$\chi^a(x)$ and the non-physical polarizations of $\Psi^a_\mu(x)$. Then, one has a gauge invariant description simultaneously
of spins $\frac{3}{2}$ and $\frac{1}{2}$, the latter being the gamma-trace of the field
$ \Psi_\mu^a(x)$.
\section{BRST invariant formulation} \setcounter{equation}0 \label{brstfree}
\subsection{Set Up}
A systematic way to obtain the systems described above
is to use the BRST approach.
Let us introduce an auxiliary Fock space spanned by one set of creation and annihilation operators. The commutation relations and the vacuum are defined in the usual way
\begin{equation}
[\alpha_\mu,\alpha_\nu^+] = \eta_{\mu \nu}, \quad \alpha_\mu |0 \rangle_\alpha=0
\end{equation}
A vector $|\Phi \rangle$ in this Fock space is a series expansion in terms of the creation operators $\alpha_\mu^+$.
In the rest of this paper
we shall take the maximal number of these oscillators
to be equal to two. Using more than two oscillators will result in components of higher spin.
The divergence, gradient and d'Alembertian operators are realized as
\begin{equation} \label{ops}
l= p \cdot \alpha, \quad l^+= p \cdot \alpha^{ +}, \quad l_0= p\cdot p,
\end{equation}
with $p_\mu = -i \partial_\mu$ and $A \cdot B \equiv \eta_{\mu\nu}A^\mu B^\nu$. The operators \p{ops} form a simple algebra with
only non-zero commutator, being
\begin{equation}
[l, l^+] = l_0
\end{equation}
Following a standard procedure (see \cite{Fotopoulos:2008ka} for a review)
for each operator $l^+,l $ and $l_0$ one introduces ghosts $c, c^+, c_0$ of ghost number $+1$ and the corresponding
momenta $b^+, b, b_0$ which have ghost number $-1$ . These operators obey the anticommutation relations
\begin{equation}
\{b, c^+ \} = \{b^+, c \}=\{b_0, c_0 \}=1
\end{equation}
We define the ghost vacuum as
\begin{equation}
c|0 \rangle_{gh.}=b|0 \rangle_{gh.}=b_0|0 \rangle_{gh.}=0,
\end{equation}
the total vacuum being $| 0 \rangle = |0 \rangle_{\alpha}\otimes |0 \rangle_{gh.} $.
Using the corresponding nilpotent BRST charge
\begin{equation} \label{BRST-1}
Q = c_0 l_0 + c l^++ c^+ l - c^+ c b_0, \quad Q^2=0
\end{equation}
this set up allows one to build gauge invariant free Lagrangians in a compact form,
\begin{equation} \label{LF-1}
{\cal L}= \int dc_0 \langle \Phi |Q |\Phi \rangle
\end{equation}
where the gauge transformation is given by
\begin{equation} \delta |\Phi \rangle = Q |\Lambda \rangle \end{equation}
The Grassmann integration is carried out using the standard rule of
\begin{equation} \label{grint}
\int dc_0 \, c_0=1
\end{equation}
The requirement that the
Lagrangian \p{LF-1} has zero ghost number uniquely fixes the expansion of an arbitrary vector $|\Phi \rangle$
in terms of the ghost variables.
Noticing that the number operator
\begin{equation}
N= \alpha^+ \cdot \alpha + c^+b + b^+ c
\end{equation}
commutes with the BRST operator \p{BRST-1}, one has the following expansion
for the case of the vector field
\begin{equation} \label{v-1}
|\Phi \rangle = (\phi^\mu(x) \alpha^{+}_\mu -i C(x)c_0 b^+ ) |0 \rangle
\end{equation}
Since the BRST charge has the ghost number $+1$ and the vector $|\Phi \rangle$
has the ghost number zero, then due to \p{LF-1} the parameter of gauge transformations
has the ghost number $-1$. Therefore
\begin{equation} \label{lv-1}
|\Lambda \rangle = i b^+ \lambda(x)|0 \rangle
\end{equation}
Using the equations \p{LF-1}, \p{BRST-1} and \p{v-1} one can recover the Lagrangian
\p{lv1} after performing the normal ordering and integrating over $c_0$.
Similarly, using \p{BRST-1}, \p{v-1} and \p{lv-1} one recovers the gauge transformation rules
\p{lv1-1} and \p{lv1-2}.
One can repeat the same procedure for the system described in the subsection
\ref{2and0}. In particular, the expansion of the vector $|\Phi \rangle $
and of the parameter of gauge transformations $|\Lambda \rangle$
have the form
\begin{equation}
|\Phi \rangle = (\phi^{\mu \nu}(x) \alpha^{+}_\mu \alpha^{+}_\nu - ic_0 b^+ C^\mu(x) \alpha^{+}_\mu
+c^+ b^+ D(x)) |0 \rangle
\end{equation}
\begin{equation}
|\Lambda \rangle = i b^+ \lambda^\mu(x) \alpha^{+}_\mu|0 \rangle
\end{equation}
Using this expansion one obtains
the Lagrangian \p{lv2} and the gauge transformation
rules \p{s2-1-1} -- \p{s2-1}.
The BRST formulation
for fermions is slightly more complicated, because of the anticommuting
nature of the Dirac operator
\begin{equation}
g_0 = p \cdot \gamma
\end{equation}
As a result, one has to introduce a commuting ghost variable, which in turn
leads to an infinite expansion in its powers.
One can, however, partially fix the BRST gauge to truncate the expansion of the
vector in the Fock space to a finite form
and write the Lagrangian
\begin{eqnarray} \label{ftr-free-f}
{ \cal L}_{F}&=&\frac{1}{{\sqrt 2}}\,\,{}_a\langle \Psi_{1}|(g_0)^a{}_b|\Psi_{1}\rangle^b +
{}_a\langle \Psi_{2}|\tilde Q_{F}|\Psi_{1}\rangle^a + \\ \nonumber
&+& {}_a\langle \Psi_{1}|\tilde Q_{F}|\Psi_{2}\rangle^a +
\sqrt{2}\,\,{}_a\langle \Psi_{2}|c^+ c (g_0)^a{}_b|\Psi_{2}\rangle^b\,
\end{eqnarray}
where
\begin{equation}
\label{TQ-R}
\tilde Q_{F}=
c^+_1 \, l_1 + c_1 \, l^+_1
\end{equation}
One can check, that the Lagrangian \p{ftr-free-f} is invariant under the gauge transformations
\begin{eqnarray} \label{GTR1}
&& \delta\, |\Psi_{1}\rangle^a \ = \ \tilde Q_F |\Lambda^\prime \rangle^a \label{GT1}
\nonumber \\
&& \delta \,|\Psi_{2}\rangle^a \ = \ - \frac{1}{\sqrt{2}}(g_0)^a{}_b \, |\Lambda^\prime \rangle^b \
\end{eqnarray}
and is equivalent to \p{l-f-f}
with
\begin{equation}
|\Psi_{1}\rangle^a = \Psi^{\mu, a}(x) \alpha_\mu^+|0\rangle, \quad |\Psi_{2}\rangle^a = b^+ \chi^a (x)|0\rangle
\end{equation}
The gauge transformations are obtained by taking the gauge parameter $|\Lambda^\prime \rangle$
to be of the form
\begin{equation}
|\Lambda^\prime\rangle^a = i b^+ \lambda^{\prime a}(x) |0\rangle
\end{equation}
As in the case of the bosonic fields, the dependence on $\alpha_\mu^+$ and on ghost variables
is uniquely fixed by the choice that the field $|\Psi_{1}\rangle^a$ contains a maximal spin
equal to $\frac{3}{2}$ and the requirement that the Lagrangian \p{ftr-free-f}
has zero ghost number.
\subsection{Supersymmetry. Linearized Supergravities} \label{linsugra}
Let us notice that the systems considered in
the subsections \ref{2and0} and \ref{32and12} can not
be connected by supersymmetry transformations,
because the fields $\phi_{\mu \nu}(x)$
and $\Psi_\mu(x)$ have different numbers of physical degrees of freedom
on-shell.
In order to establish $N=1$ supersymmetry, one can take a formulation of the Open Superstring Field Theory \cite{Kazama:1986cy} as a hint
and proceed as follows \cite{Sorokin:2018djm}.
Consider two independent sets
of $\alpha$-oscillators
\begin{equation}
[\alpha_{\mu,m}, \alpha_{\nu, n}^+] = \eta_{\mu \nu} \delta_{mn}, \quad m,n=1,2
\end{equation}
The corresponding divergence and gradient operators, as well as
the ghost variables $c^\pm_m$ and $b^\mp_m$ will get the index $"m"$
as well.
Therefore, we have the algebra
\begin{equation}
[l_m, l^+_n] = \delta_{mn} \, l_0
\end{equation}
\begin{equation}
\{b_m, c^+_n \} = \{b^+_m, c_n \}=\delta_{mn}, \quad \{b_0, c_0 \}=1
\end{equation}
We take the fields in the fermionic sector to contain only the first set of oscillators.
In other words, we consider the system described in
the Subsection \ref{32and12} without changes and in the corresponding
BRST formulation in the Section \ref{brstfree}
we assume that all oscillators belong to the first set ($m=1$).
On the other hand, the vectors in the Fock space in the bosonic sector contain both types of oscillators.
Taking physical component of the field $|\Phi \rangle$ to contain
one oscillator of each type,
we get the following expansions
\begin{eqnarray} \label{MS-1
|\Phi \rangle&=&
(\phi_{\mu, \nu}(x)\alpha^{\mu, +}_1 \alpha^{\nu, +}_2 - A(x) c^+_1 b^+_2 -
B(x) c_2^+ b^+_1 \\ \nonumber
&+& ic_0 b^+_1 C_\mu(x) \alpha^{\mu, +}_2 + i c_0b^+_2 E_\mu(x) \alpha^{\mu, +}_1)
|0 \rangle.
\end{eqnarray}
and
\begin{eqnarray} \label{MS-2
|\Lambda \rangle&=&
(i b_2^+ \lambda_{\mu}(x)\alpha^{\mu, +}_1
+
i b_1^+ \rho_{\mu}(x)\alpha^{\mu, +}_2
-
c_0b_1^+ b_2^+ \tau(x))
|0 \rangle.
\end{eqnarray}
Using the corresponding nilpotent BRST charge
\begin{equation} \label{BRSTMS}
Q = c_0 l_0 + \sum_{m=1,2}(c_m l_m^++ c^+_m l_m - c^+_m c_m b_0), \quad Q^2=0
\end{equation}
it is straightforward to obtain the Lagrangian
\begin{eqnarray} \label{ms-l
L_{B}&&= - \phi^{\mu, \nu} \Box \phi_{\mu,\nu } + B\Box A + A \Box B \\ \nonumber
&& +E^\mu \partial_\mu B + C^{\nu}\partial^\mu \phi_{\nu,\mu } + C^\nu \partial_\nu A
+ E^\mu\partial^\nu
\phi_{ \nu, \mu } \\ \nonumber
&&- B \partial_\mu E^\mu - \phi^{\nu, \mu } \partial_\mu C_\nu -A \partial_\mu C^\mu - \phi^{\mu, \nu}
\partial_\mu E_\nu \\ \nonumber
&&+C^\mu C_\mu + E^\mu E_\mu\,.
\end{eqnarray}
by plugging the expressions \p{MS-1} and \p{BRSTMS}
into \p{LF-1},
performing the normal ordering of oscillators and integrating over $c_0$ according to \p{grint}.
Similarly, one can find, that Lagrangian \p{ms-l}
is invariant under the gauge transformations
\begin{eqnarray} \nonumber
&&\delta \phi_{\nu, \mu }(x) = \partial_\mu \lambda_\nu(x) + \partial_\nu \rho_\mu(x), \\ \nonumber
&&\delta A (x) = - \partial^\mu \rho_\mu(x) - \tau(x), \\ \nonumber
&&\delta B (x)= - \partial^\mu \lambda_\mu(x)+ \tau(x), \\
&&\delta C_\mu(x) =- \Box \lambda_\mu (x) + \partial_\mu \tau(x), \\ \nonumber
&&\delta E_\mu(x)=- \Box \rho_\mu(x)-\partial_\mu \tau (x).
\end{eqnarray}
The Lagrangian \p{ms-l} is analogous to the one given in
the equation \p{lv2}. However, the present Lagrangian describes
a physical field $\phi_{\mu, \nu}(x)$ with no symmetry between
the indices $\mu$ and $\nu$. As a result, the Lagrangian contains more auxiliary fields.
In particular, the fields $C_\mu(x)$ and $E_\mu(x)$ in \p{ms-l}
are analogous to the field $C_\mu(x)$ in \p{lv2}, and the fields
$A(x)$ and $B(x)$ are analogous to the field $D(x)$.
Again, after eliminating the auxiliary fields
after gauge fixing
and using the equations
of motion one is left with only physical polarizations
in the field $\phi_{\mu, \nu}(x)$. This means, that we have a description of a spin $2$ field $g_{\mu \nu}(x)$,
of an antisymmetric second rank tensor $B_{\mu \nu}(x)$
and of a scalar $\phi(x)$, all contained in the field $\phi_{\mu, \nu}(x)$.
Finally, one can check that
the total Lagrangian
\begin{eqnarray} \label{ltot}
L_{tot.}&&= - \phi^{\mu, \nu} \Box \phi_{\mu,\nu } + B\Box A + A \Box B \\ \nonumber
&& +E^\mu \partial_\mu B + C^{\nu}\partial^\mu \phi_{\nu,\mu } + C^\nu \partial_\nu A
+ E^\mu\partial^\nu
\phi_{ \nu, \mu } \\ \nonumber
&&- B \partial_\mu E^\mu - \phi^{\nu, \mu } \partial_\mu C_\nu -A \partial_\mu C^\mu - \phi^{\mu, \nu}
\partial_\mu E_\nu \\ \nonumber
&&+C^\mu C_\mu + E^\mu E_\mu \\ \nonumber
&&-i \, \bar \Psi^\nu \gamma^\mu \partial_\mu \Psi_\nu -
i \, \bar \Psi^\mu \partial_\mu \chi + i \,
\bar \chi \partial^\mu \Psi_\mu
+ i \, \bar \chi \gamma^\mu \partial_\mu \chi
\end{eqnarray}
being a sum of the Lagrangians
\p{l-f-f} and \p{ms-l},
is invariant under the supersymmetry transformations
\cite{Sorokin:2018djm}
\begin{equation} \label{susy-tr-1}
\delta \phi_{ \nu, \mu }(x) = i\, \bar \Psi_{\mu}(x) \gamma_\nu \, \epsilon, \quad
\delta C_{ \nu}(x)= -i\, \partial_\mu \bar \chi(x) \gamma^\mu \gamma_\nu\, \epsilon, \quad
\delta B(x) = -i \,\bar \chi(x) \, \epsilon,
\end{equation}
\begin{equation} \label{susy-tr-2}
\delta \Psi_\mu(x) = - \gamma^\nu\gamma^\rho\epsilon \,\partial_\nu \phi_{\rho,\mu}(x)
- \epsilon E_\mu(x),
\quad
\delta \chi(x)= - \gamma^\nu\epsilon\, C_\nu(x)\,.
\end{equation}
Let us note that we have not encountered any restriction on the number of
space-time dimensions until now. The requirement that the algebra of supersymmetry transformations
\p{susy-tr-1}--\p{susy-tr-2} closes on shell
singles out the number of space-time dimensions to be
${\cal D}=3$, 4, 6, or 10.
Decomposing the fields into irreducible representations of the Poincar\'e group as
\begin{equation} \label{dec-1}
\phi_{\mu, \nu} = \left ( \phi_{(\mu, \nu )} - \eta_{\mu \nu}\frac{1}{{\cal D}} \phi^\rho{}_\rho
\right ) + \phi_{[\mu, \nu]} +
\eta_{\mu \nu}\frac{1}{{\cal D}} \phi^{\rho}{}_\rho
\equiv
h_{\mu \nu} + B_{\mu \nu} + \frac{1}{{\cal D}}\eta_{\mu \nu}
\varphi
\end{equation}
and
\begin{equation} \label{dec-2}
\psi_\mu^a = \Psi_\mu^a +\frac{1}{{\cal D}}(\gamma_\mu)^{ab} (\gamma^\nu)_{bc} \psi_\nu^c
\equiv \Psi_\mu^a + \frac{1}{{\cal D}}(\gamma_\mu)^{ab} \Xi_b
\end{equation}
one obtains the following $N=1$ supermultiplets:
\begin{itemize}
\item In ${\cal D}=4$: a supergravity multiplet $\left ( g_{\mu \nu}(x), \psi_\mu^a(x) \right )$ and a chiral multiplet
$\left (\phi(x), a(x), \Xi(x) \right)$
where $\partial_\mu a(x) = \frac{1}{3!}\epsilon_{\mu \nu \rho \sigma} \partial^\nu B^{\rho \sigma}(x)$.
\item In ${\cal D}=6$: a supergravity multiplet
$\left ( g_{\mu \nu}(x), B_{\mu \nu}^+(x), \psi_\mu^a(x) \right )$
and a $(1,0)$ tensor multiplet $\left (\phi(x), B_{\mu \nu}^-(x), \Xi(x) \right)$,
where we decomposed $B_{\mu \nu}(x)$ into self-dual and anti self-dual parts.
\item In ${\cal D}=10$: one irreducible supergravity multiplet.
\end{itemize}
Therefore, one can say, that the Lagrangian \p{ltot} gives an ``uniform" description
of various linearized $N=1$ supergravities.
The case of ${\cal D}=3$ contains no massless propagating degrees of freedom
with spin $2$, so we shall not consider it here.
Writing the supersymmetry transformations \p{susy-tr-1}--\p{susy-tr-2}
in terms of auxiliary oscillators,
\begin{eqnarray}\label{STR1}
&& \langle 0 | \delta \, \phi_{\mu, \nu}(x) \, \alpha^{\mu}_1 \, \alpha^{\nu}_2 = i\, \langle 0 |
{\overline \Psi}_{\mu}(x) \, \alpha_{1}^\mu \, (\gamma \cdot \alpha_2) \, \epsilon
\\ \nonumber
&& \langle 0 | \delta \, C_\mu(x) \, \alpha^{\mu}_2 \, b_1 = - \langle 0 | {\overline \chi(x)} \, g_0 \, (\gamma \cdot \alpha_2)\, \epsilon \, b_1\\ \nonumber
&& \langle 0 | \delta \, B(x) \, b_1 \, c_2 = - i \,\langle 0 | {\overline \chi(x)}\, b_1 \, c_2 \, \epsilon
\end{eqnarray}
\begin{eqnarray}\label{STR2} \nonumber
&& \delta \, \Psi_{\mu}(x) \, \alpha^{\mu,+}_1|0\rangle=( -i \, g_0 \, (\gamma \cdot \alpha_2) \, \epsilon \,
\phi_{\mu, \nu}(x) \, \alpha^{\mu, +}_1 \, \alpha^{\nu, +}_2
- \epsilon \,E_\mu(x) \, \alpha^{\mu,+}_1) |0 \rangle \\
&& \delta \, \chi (x) \, b^+_1 |0\rangle =
- (\gamma \cdot \alpha_2) \, \epsilon \, b^+_1 \, C^ \mu(x) \, \alpha_{\mu, 2}^+ |0 \rangle
\end{eqnarray}
one can see that supersymmetry is "generated"
by the second set of oscillators ($m=2$). In other words, to obtain the $N=1$ supermultiplets
one can start with the fermionic sector, which contains only the first
set ($m=1$), then apply the transformations \p{STR1}--\p{STR2}
and require the closure of SUSY algebra.\footnote{The same pattern persists for the higher spin supermultiplets:
the fermionic sector contains only the first set of the oscillators,
while the bosonic sector contains the oscillators from the first set and at most one oscillator from the second set, see
\cite{Sorokin:2018djm} for the details.}
The description for the $N=1$ supersymmetric vector multiplet is similar.
Taking the fields in the bosonic and the fermionic sectors as
\begin{equation}
|\Phi \rangle=
(\phi_{\mu}(x)\alpha_{ 2}^{\mu, +} -i c_0b^+_2 E(x) )|0\rangle, \quad
|\Psi \rangle= \Psi(x)|0\rangle,
\end{equation}
one can check that the corresponding Lagrangian
\begin{equation} \label{lv1-tot}
{\cal L}= (\partial_\mu \phi^{\nu }) (\partial_\mu \phi_{\nu })
-2 E \partial^\mu \phi_{\mu }
+ E^2 - i \, \bar \Psi\gamma^\mu \partial_\mu \Psi
\end{equation}
is invariant under the supersymmetry transformations
\begin{eqnarray} \label{susyveclin}
\langle 0 | \delta \, \phi_{\mu}(x) \, \alpha^{\mu}_2 & = & i\, \langle 0 |
{\overline \Psi}(x) \, (\gamma \cdot \alpha_2) \, \epsilon
\\ \nonumber
\delta \, \Psi(x) |0\rangle &= &
( -i \, g_0 \, (\gamma \cdot \alpha_2) \, \epsilon \,
\phi_{ \mu}(x) \, \alpha^{\mu, +}_2
- \epsilon \,E(x) ) |0 \rangle
\end{eqnarray}
After eliminating the auxiliary field $E(x)$
via its own equations of motion one obtains the standard formulation
of $N=1$ vector supermultiplet in ${\cal D}= 4$, 6, or 10 with an on-shell supersymmetry.
\section{Cubic Interactions} \setcounter{equation}0
\label{cube}
\subsection{Three bosons}
In order to construct cubic interactions for the fields considered in the previous sections,
\footnote{See \cite{Bengtsson:1987jt}, \cite{Buchbinder:2006eq} for the details of construction for higher spin fields and
\cite{Neveu:1986mv} for the analogous
construction in Open Bosonic String Field Theory.}
we take three copies of the auxiliary Fock space and corresponding operators. The oscillators now obey the commutation relations
\begin{equation}\label{B4-c}
[\alpha_{\mu,m}^{(i)}, \alpha_{\nu,n}^{(j), +} ] = \delta^{ij}\delta_{mn}\eta_{\mu \nu},
\end{equation}
\begin{equation}
\{ c^{(i), +}_m, b^{(j)}_n \} = \{ c^{(i)}_m, b^{(j),+}_n \}
=
\{ c_{0,m}^{(i)} , b_{0,n}^{(j)} \} =
\delta^{ij}\delta_{mn}\,,
\end{equation}
$$
i,j=1,2,3, \quad m,n=1,2, \quad \mu,\nu=0,...,{\cal D}-1
$$
Then, we can consider the cubic Lagrangian
\begin{eqnarray} \label{LNSINT}
{ \cal L}_{3B,\text{int}} &=& \sum_{i=1}^3 \int dc_0^{(i)}
\langle \Phi^{(i)} |Q^{(i)}| \Phi^{(i)}
\rangle + \\ \nonumber
&+&g \left ( \int dc_0^{(1)} dc_0^{(2)} dc_0^{(3)} \langle \Phi^{(1)}| \langle \Phi^{(2)}| \langle \Phi^{(3)}| |V \rangle + h.c.
\right )
\end{eqnarray}
where $g $ is a coupling constant and
\begin{equation} \label{V3Bosons}
|V \rangle = V(p_\mu^{(i)}, \alpha_{\mu,m}^{(i),+}, c^{(i), +}_m, b^{(i), +}_m, b^{(i)}_{0,m} ) \,\, c_0^{(1)} c_0^{(2)} c_0^{(3)} \,\,
| 0^{(1)} \rangle \otimes
|0^{(2)} \rangle \otimes | 0^{(3)} \rangle
\end{equation}
where $V$ is a function of the creation operators that is restricted as follows. An obvious requirement is that $V$ must be Lorentz invariant. In order the Lagrangian to have the ghost number zero, the function $V$ must have the ghost number equal to zero, and finally, the requirement of the invariance of \p{LNSINT} under the non-linear gauge transformations
\begin{eqnarray} \label{GTNSINT}
\delta | \Phi^{(i)} \rangle & = & Q^{(i)} |\Lambda^{(i)} \rangle
- \\ \nonumber
&-& g \int dc_0^{(i+1)} dc_0^{(i+2)}
\left ( (\langle \Phi^{(i+1)}| \langle \Lambda^{(i+2)}|
+ \langle \Phi^{(i+2)}| \langle \Lambda^{(i+1)}|) | V \rangle \right )
\end{eqnarray}
up to the first power in $g$, implies that the vertex $| V \rangle $ is BRST invariant:
\begin{equation} \label{NSBRSTV}
(Q^{(1)} + Q^{(2)}+ Q^{(3)}) | V \rangle=0
\end{equation}
The same condition guarantees that the group structure of the gauge transformations is preserved up to the first order in $g$.
Using momentum conservation
\begin{equation} \label{mcon}
p_\mu^{(1)} + p_\mu^{(2)} + p_\mu^{(3)}=0
\end{equation}
and the commutation relations \p{B4-c},
one can show that
that the following expressions are BRST invariant for any values of the spins entering the cubic vertex
\cite{Fotopoulos:2010ay}, \cite{Metsaev:2012uy}
\begin{equation} \label{sbv-1-x}
{\cal K}^{(i)}_m =
(p^{(i+1)}- p^{(i+2)}) \cdot \alpha^{(i),+}_m +
(b_0^{(i+1)}-b_0^{(i+2)}) \, c^{(i),+}_m,
\end{equation}
\begin{equation} \label{sbv-10}
{\cal O}^{(i,i)}_{mn} = \alpha^{(i),+}_m \cdot \alpha^{(i),+}_n
+ c^{(i),+}_mb^{(i),+}_n + c^{(i),+}_nb^{(i),+}_m,
\end{equation}
\begin{equation} \label{sbv-3}
{\cal Z}_{mnp}= {\cal Q}^{(1,2)}_{mn} {\cal K}^{(3)}_p +
{\cal Q}^{(2,3)}_{np} {\cal K}^{(1)}_m +
{\cal Q}^{(3,1)}_{pm} {\cal K}^{(2)}_n,
\end{equation}
where
\begin{equation} \label{sbv-7}
{\cal Q}^{(i,i+1)}_{mn} = \alpha^{(i),+}_m \cdot \alpha^{(i+1),+}_n
+ \frac{1}{2} b^{(i),+}_m c^{(i+1),+}_n + \frac{1}{2} b^{(i+1),+}_n c^{(i),+}_m.
\end{equation}
Before turning to a description of cubic vertices between bosonic and fermionic fields,
let us note
that one can consider the cubic vertices between three bosonic fields obeying some off-shell constraints.
In particular,
for the fields considered in Subsection \ref{linsugra}
we shall impose off-shell transversality conditions
\begin{equation} \label{off-shell-1}
\partial^\mu\phi_{\mu, \nu}(x)=\partial^\nu\phi_{\mu, \nu}(x)=0
\end{equation}
These conditions in turn restrict the parameters of gauge transformations
\begin{equation} \label{off-shell-2}
\partial^\mu\lambda_{\mu}(x)=\partial^\mu\rho_{\mu}(x)=0, \quad \Box \lambda_{\mu}(x)= \Box \rho_{\mu}(x)=0, \quad \tau(x)=0.
\end{equation}
The constraints can be rewritten as
\begin{equation} l^{(i)}_1 |\phi^{(i)}\rangle = l^{(i)}_2 |\phi^{(i)}\rangle = 0\,,\quad l_1^{(i)}|\Lambda^{(i)}\rangle = l_2^{(i)}|\Lambda^{(i)}\rangle = l_0^{(i)}|\Lambda^{(i)}\rangle = 0 \label{gaugefixb} \end{equation}
As a result of these constraints, all auxiliary fields and the ghost dependence disappears in
$|\Phi^{(i)} \rangle$
and the Lagrangian
\p{LNSINT}
reduces to
\begin{equation}\label{L-bbb-gf}
{\cal L}_{\text{3B,int}} = \sum_{i=1,2,3}
\langle \phi^{(i)} |l_{0}^{(i)}| \phi^{(i)}
\rangle +
g \left ( \langle \phi^{(1)} | \,\,
\langle \phi^{(2)}| \,\,
\langle \phi^{(3)}|
| { V} \rangle
+ h.c \right )
\end{equation}
The formulation in terms of the constrained fields considerably simplifies
the consideration of supersymmetry, as we shall see below.
\subsection{Two fermions and one boson} \label{subffb}
For cubic interactions between two fermionic and one bosonic fields
the procedure is similar. Again, in order to simplify the consideration one imposes an off-shell transversality constraint on the physical field,
\begin{equation}
\partial^\mu \Psi_\mu^a(x)=0 \quad \Leftrightarrow \quad l_1 |\Psi\rangle = 0 \label{gaugefixf}
\end{equation}
thus putting to zero the auxiliary field $\chi(x)$ (see subsection \ref{32and12}).
This constraint, in turn, restricts the parameter of gauge transformations to
\begin{equation}
\gamma^\mu \partial_\mu \lambda^\prime(x)=0 \quad \Leftrightarrow \quad g_0 |\Lambda\rangle = 0 \label{gaugefixf2} \end{equation}
The corresponding cubic Lagrangian which describes interactions
between two fermionic and one bosonic fields has the form
\begin{eqnarray} \label{int.}
{\cal L}_{\text{2F1B,int}}
&=&
\sum_{i=1}^2
{}_a\langle \Psi^{(i)}|(g_0^{(i)})^a{}_b|\Psi^{(a)}\rangle^b +
{}\langle \phi^{(3)}|
l_{0}^{(3)} | \phi^{(3)} \rangle + \\ \nonumber
&&+ g\left ( \,\, \langle \phi^{(3)} | \,\,
{}_a\langle \Psi^{(1)}| \,\,
{}_b\langle \Psi^{(2)}| | {\cal V} \rangle^{ab}
+ h.c \right ).
\end{eqnarray}
The requirement that the Lagrangian \p{int.} is invariant
under the non-linear gauge transformations
\begin{eqnarray} \label{BFFG-1}
\delta | \phi^{(3)} \rangle &= &
\tilde Q_{B}^{(3)} |\Lambda_{B}^{(3)} \rangle + \\ \nonumber
&+& g ( {}_a\langle { \Psi}^{(1)}| \,\,
{}_b\langle { \Lambda_{F}}^{(2)}|
|{\cal W}_3^{1,2} \rangle^{ab}
+
{}_a\langle \Psi^{(2)}| \,\,
{}_b\langle { \Lambda}_F^{(1)}| \
|{\cal W}_{3}^{2,1} \rangle^{ab} ),
\end{eqnarray}
\begin{eqnarray} \label{BFFG-2-1}
\delta | \Psi^{(1)} \rangle^a &= &
{\tilde Q}_{F}^{(1)} |{ \Lambda_{F}}^{(1)} \rangle^a +
\\ \nonumber
&+&g (
{}_b\langle \Psi^{(2)}|\,\,
{}\langle \Lambda_{B}^{(3)}| | {\cal W}_{1}^{2,3} \rangle^{ab}
+
{ {}\langle \phi^{(3)}|\,\,
{}_b \langle \Lambda}_F^{(2)}| |
{\cal W}^{3,2}_1 \rangle^{ab}
),
\end{eqnarray}
\begin{eqnarray} \label{BFFG-2-2}
\delta | \Psi^{(2)} \rangle^a &= &
{\tilde Q}_{F}^{(2)} |{ \Lambda_{F}}^{(2)} \rangle^a + \\ \nonumber
&+&g ({}^A\langle \phi^{(3)}| \,\,
{}_b^B\langle \Lambda_{F}^{(1)}| | {\cal W}_{2}^{3,1} \rangle^{ab}
+
{}_b\langle \Psi^{(1)} |\,\,
{}\langle
{ \Lambda}_{B}^{(3)}| |
{\cal W}_2^{1,3} \rangle^{ab}
),
\end{eqnarray}
as well as the requirement of preservation of the group structure
for the gauge transformations up to the first power
in the coupling constant $g$
imposes conditions on the vertices $| {\cal V} \rangle^{ab}$ and $| {\cal W} \rangle^{ab}$ \cite{Buchbinder:2021qrg},
which are similar to \p{NSBRSTV}.
The vertex $|{\cal V}\rangle$ is again defined by a Lorentz invariant, ghost number zero function of the creation operators. Given $|{\cal V}\rangle$, gauge invariance of the Lagrangian \p{int.} holds provided one can find transformation vertices $|{\cal W}\rangle$ such that
\begin{equation} \label{xy-1b}
({ g}_{0}^{(1)})^a{}_b | {\cal W}_1^{2,3} \rangle^{bc}
-
({g}_{0}^{(2)})^c{}_b | {\cal W}_2^{1,3} \rangle^{ba}
+
{\tilde Q}_{B}^{(3)}
|{\cal V} \rangle^{ac} =0
\end{equation}
\begin{equation} \label{xy-2b}
({g}_{0}^{(1)})^a{}_b | {\cal W}_1^{3,2} \rangle^{bc}
+
l_0^{(3)}
|{\cal W}_{3}^{1,2} \rangle^{ac}+
{\tilde Q}_{F}^{(2)} | {\cal V} \rangle^{ac}
=0
\end{equation}
\begin{equation} \label{xy-3b}
({g}_{0}^{(2)})^a{}_b | {\cal W}_2^{3,1} \rangle^{bc}
+
l_0^{(3)}
|{\cal W}_{3}^{2,1} \rangle^{ac}-
{\tilde Q}_{F}^{(1)} | {\cal V} \rangle^{ca}
=0
\end{equation}
The preservation of the group structure leads to another set of equations. For consistency, there must be some functions $|{\cal X}_{i} \rangle$ such that
\begin{equation} \label{inv-w-1b}
{\tilde Q}_{F}^{(2)} | {\cal W}_1^{2,3} \rangle^{ab}
+ \tilde Q_{B}^{(3)} |
{\cal W}_{1}^{3,2} \rangle^{ab} -
{\tilde Q}_F^{(1)} | {\cal X}_1 \rangle^{ab} =0
\end{equation}
\begin{equation} \label{inv-w-2b}
{
\tilde Q}_{F}^{(1)} | {\cal W}_2^{1,3} \rangle^{ab}
+ \tilde Q_{B}^{(3)} |
{\cal W}_{2}^{3,1} \rangle^{ab} -
{\tilde Q}_F^{(2)} | {\cal X}_2 \rangle^{ab} =0
\end{equation}
\begin{equation} \label{inv-w-3b}
{\tilde Q}_{F}^{(1)}
| {\cal W}_3^{1,2} \rangle^{ab}
-
{\tilde Q}_{F}^{(2)}
|{\cal W}_3^{2,1} \rangle^{ba}
-
\tilde Q_{B}^{(3)}
|{\cal X}_{3} \rangle^{ab} =0
\end{equation}
Note that the equations \eqref{xy-1b}-\eqref{inv-w-3b} should hold only when acting on fields and transformations satisfying the constraints of equations \eqref{gaugefixb}, \eqref{gaugefixf}, and \eqref{gaugefixf2}. The generalization to the unconstrained case was written in \cite{Buchbinder:2021qrg}.
\section{\texorpdfstring{$N=1$}{N=1} Super Yang-Mills} \setcounter{equation}0
\label{sectionsym}
Let us turn to particular examples.
For the case of Super Yang-Mills
one introduces colour indices
in the equations \p{LNSINT}--\p{GTNSINT} and in
\p{int.}--\p{BFFG-2-2} and takes the fields in the bosonic and the fermionic sectors
as
\begin{equation} \label{YM-333}
| \Phi^{(i)} \rangle^A = ({\phi}_\mu^A(x) \alpha_2^{\mu (i), +}
-i { E}^A(x) c_0^{(i), + } b_2^{(i), + })|0^{(i)} \rangle,
\end{equation}
\begin{equation} \label{psi-YM-1}
| \Psi^{(i)} \rangle^{a,A} =
\Psi^{a,A} (x) |0^{(i)}\rangle
\end{equation}
the only non-zero parameter of gauge transformations being
\begin{equation}
| \Lambda^{(i)} \rangle^A = i b_2^{(i),+} \lambda^A(x) | 0^{(i)} \rangle
\end{equation}
The full interacting cubic Lagrangian is a sum of
\p{L-bbb-gf}
and of\footnote{Note that in Section \ref{subffb} we placed the boson in the third Fock space, whereas now we have three copies of the boson's Fock space in addition to three copies of the fermion's. To write the interaction in a symmetric way, we introduce a cyclic sum over the Fock space indices.}
\begin{equation}\label{LNSINT-YM-2}
{ \cal L}_{\text{int}} =\sum_{i=1}^3
{}^A\langle \Psi^{(i)} |g_{0}^{(i)}| \Psi^{(i)}
\rangle_A +
g \left ( {}^A\langle \Psi^{(1)}| {}^B\langle \Psi^{(2)}| {}^C\langle \Phi^{(3)}|
|{\cal V} \rangle_{ABC} + \,\, \text{cyclic} \right )
\end{equation}
The cubic interaction vertex between three bosons
is given by the expression \p{sbv-3} with added colour indices
\begin{equation} \label{YM-vvv}
| V \rangle_{ABC}= -\frac{i}{12} f_{ABC}
{\cal Z}_{222}
\times c_0^{(1)}c_0^{(2)}c_0^{(3)}
\,\,
| 0^{(1)} \rangle \otimes
|0^{(2)} \rangle \otimes | 0^{(3)} \rangle \,\,
\end{equation}
The cubic interaction vertex between two fermions with spins one-half and one boson with spin one is
\begin{equation} \label{YM-222}
|{\cal V} \rangle^{ab}_{ABC} = \frac{i}{3} f_{ABC}(\alpha_2^{(3),+} \cdot \gamma)^{ab} \,\,
| 0^{(1)} \rangle \otimes
|0^{(2)} \rangle \otimes | 0^{(3)} \rangle + \,\, \text{cyclic}
\end{equation}
The only non-trivial $| {\cal W} \rangle$ vertices in the solution of equations \p{xy-1b}--\p{xy-3b} are given by
\begin{equation} \label{YM-444}
| {\cal W}_1^{2,3} \rangle^{ab}_{ABC} =
| {\cal W}_2^{1,3} \rangle^{ab}_{ABC}
= f_{ABC}\, C^{ab} c_2^+\,
| 0^{(1)} \rangle \otimes
|0^{(2)} \rangle \otimes | 0^{(3)} \rangle
\end{equation}
The vertex \p{YM-222} enters the Lagrangian and determines the interaction
between the vector field and two fermions, whereas the vertices \p{YM-444}
express the nonlinear part of the gauge transformations.
After eliminating the auxiliary field $E^A(x)$ via its own equations of motion one obtains an action for $N=1$ Super Yang-Mills
up to the cubic order.
Alternatively, one could have imposed an off-shell transversality constraint
on the physical field $\phi_\mu^A(x)$ similarly to how it was
done for the supergravity multiplets (see the discussion around
the equations \p{off-shell-1} -- \p{L-bbb-gf}).
This would have put an auxiliary field $E^A(x)$ equal to zero and restricted
the parameter of gauge transformations as $\Box \lambda(x) =0$.
A higher spin generalization is given in \cite{Buchbinder:2021qrg},
by multiplying the vertex \p{YM-222} with an arbitrary function
of the BRST invariant expressions \p{sbv-1-x}--\p{sbv-3}
and then finding corresponding $| {\cal W} \rangle$ and $| {\cal X} \rangle$
vertices. These solutions are covariant versions of the vertices
found in the light-cone formalism \cite{Metsaev:2007rn}.
Because of the cubic interactions, the supersymmetry transformations
for the fermion in \p{susyveclin}
will be deformed with a nonlinear term
\begin{eqnarray} \label{SUSY-sym}
\delta^\prime | \Psi^{(i)} \rangle^{a}_A & = & g \, f_{ABC}\, \times \\ \nonumber
&\times& {}_B\langle \phi^{(i+1)}| {}_C\langle \phi^{(i+2)}|
(\gamma^{\mu \nu})^a{}_b\alpha^{(i+1),+}_{2,\mu} \alpha^{(i+2),+}_{2,\nu} \epsilon^b \,\,
| 0^{(i+1)} \rangle \otimes
|0^{(i+2)} \rangle \otimes | 0^{(i)} \rangle
\end{eqnarray}
being the standard supersymmetry transformations for the $N=1$ Yang-Mills supermultiplet.
Let us notice, however, that if one imposes the off-shell transversality condition
on the vector field, the supersymmetry transformations put the fields completely on shell.
In this way one considers on-shell cubic vertices, which transform into each other
under linear supersymmetry transformations \p{susyveclin}.
\section{\texorpdfstring{$N=1$}{N=1} Supergravities} \setcounter{equation}0
\label{sectionsugra}
In the following we show how we can write the cubic interaction vertices of $N=1$ supergravity in ${\cal D}=4,6$ and $10$
dimensions. These can be compared with the full Lagrangians of the theories which we include in Appendix \ref{AppendixB}.
The cubic vertices for $N=1$ supergravities,
which describe a nonlinear deformations of the Lagrangian \p{ltot}
can be divided into two types.
The first type is universal
in the same sense as
is the Lagrangian \p{ltot}, i.e.,
the vertices have the same form in ${\cal D}=4,6$ and $10$.
The second type of the vertices are
specific to particular dimensions.
In this section we consider the cubic interaction vertices of supergravities whose free versions were written in Section \ref{linsugra}. Namely, these are supergravity in ${\cal D}=4$ coupled to a chiral multiplet, supergravity in ${\cal D}=6$ coupled to a $(1,0)$ tensor multiplet, and pure supergravity in ${\cal D} = 10$.
In Subsection \ref{sugrauni} we write the ``universal'' vertices, which are present in ${\cal D}=4$, 6, and 10. In Subsections \ref{sugra10} and \ref{sugra6} we write specific boson-fermion-fermion vertices for ${\cal D}=10$ and 6, respectively.
\subsection{Universal vertices} \label{sugrauni}
Let us start with the first type of cubic vertices.
We impose off-shell transversality constraints
on the physical fields $\phi_{\mu,\nu}(x)$ and $\Psi_\mu^a(x)$.
Therefore, we have
\begin{equation}
| \phi^{(i)} \rangle = \phi_{\mu, \nu}(x) \,\alpha_{1}^{\mu (i),+} \alpha_2^{\nu (i),+}
|0^{ (i)} \rangle, \qquad | \Psi^{(i)}\rangle^a = \Psi_\mu^{a}(x) \, \alpha_1^{\mu (i),+} | 0^{ (i)} \rangle
\end{equation}
We take the cubic vertices for three bosons
\begin{equation} \label{sugralike3b}
-2\, g \, \langle \phi^{(1)} | \,\,
\langle \phi^{(2)}| \,\,
\langle \phi^{(3)}| {\cal Z}_{111}
{\cal Z}_{222}
| 0^{(1)} \rangle \otimes
|0^{(2)} \rangle \otimes | 0^{(3)} \rangle
\end{equation}
where the expressions for ${\cal Z}_{mnp}$ are given in
\p{sbv-3}.
For two fermions and one boson we take the vertex
\begin{equation} \label{sugralikebff}
g \,\, \langle \phi^{(3)} | \,\,
{}_a\langle \Psi^{(1)}| \,\,
{}_b\langle \Psi^{(2)}| {\cal Z}_{111}(\gamma \cdot \alpha_2^{(3),+} )^{ab}
| 0^{(1)} \rangle \otimes
|0^{(2)} \rangle \otimes | 0^{(3)} \rangle
+ \,\, \text{cyclic}.
\end{equation}
which solves the equations \p{xy-1b}--\p{xy-3b} and \p{inv-w-1b}--\p{inv-w-3b}
with
\begin{eqnarray} \label{s-w-ym-1}
({\cal W}_3^{1,2})^{ab}&=& \,c^{(2),+}_1
( \gamma \cdot \alpha_2^+)^{ab}
(\alpha^{(3), +}_1 \cdot \alpha^{(1), +}_1) \\ \nonumber
({\cal W}_3^{2,1})^{ab}&=& c^{(1),+}_1
(\gamma \cdot \alpha_2^{+} )^{ab}
(\alpha^{(2), +}_1 \cdot \alpha^{(3), +}_1)
\\ \nonumber
({\cal W}_1^{3,2})^{ab}&=& -\,c^{(2),+}_1
(p^{(1)} \cdot \gamma)^a{}_c
(\gamma \cdot \alpha_2^{+} )^{cb}
(\alpha^{(3), +}_1 \cdot \alpha^{(1), +}_1)
\\ \nonumber
({\cal W}_2^{3,1})^{ab}&=& -c^{(1),+}_1
(p^{(2)} \cdot \gamma)^a{}_c ( \gamma \cdot \alpha_2^{+})^{cb}
(\alpha^{(2), +}_1 \cdot \alpha^{(3), +}_1)
\\ \nonumber
({\cal W}_1^{2,3})^{ab}&=& -c^+_2
C^{ab} {\cal Z}_{111} +
c_1^{(3),+}
(p^{(1)} \cdot \gamma)^a{}_c
(\gamma \cdot \alpha_2^+)^{cb}
(\alpha^{(1), +}_1 \cdot \alpha^{(2), +}_1)
\\
\nonumber
({\cal W}_2^{1,3})^{ab}&=& c_2^+ C^{ab} {\cal Z}_{111}
+
c_1^{(3),+}
(p^{(2)} \cdot \gamma)^a{}_c
(\gamma \cdot \alpha_2^+)^{cb}
(\alpha^{(1), +}_1 \cdot \alpha^{(2), +}_1)
\end{eqnarray}
and
\begin{eqnarray} \label{x-sol-1} \nonumber
{\cal X}_1^{ab}&=& c_1^{(2),+} c_2^+ b_1^{(1),+} C^{ab}
(p^{(1)} \cdot \alpha_1^{(3),+}) -
c_1^{(2),+} c_1^{(3),+} b_1^{(1),+}
(p^{(1)}\cdot \gamma)^a{}_c (\gamma \cdot \alpha_2^+)^{cb}
\\
\nonumber
{\cal X}_2^{ab}&=& -c_2^+ c_1^{(1),+} b_1^{(2),+} C^{ab}
(p^{(2)} \cdot \alpha_1^{(3),+}) +
c_1^{(3),+} c_1^{(1),+} b_1^{(2),+}
(p^{(2)}\cdot \gamma)^a{}_c (\gamma \cdot \alpha_2^+)^{cb}
\\
{\cal X}_3^{ab}&=&-c_1^{(1),+} c_1^{(2),+}b_1^{(3)+}
(\gamma \cdot \alpha_2^+)^{ab}
\end{eqnarray}
The total Lagrangian, which is the sum of \p{L-bbb-gf} and
of
\begin{equation}\label{LNSINT-SG-2}
{ \cal L}_{\text{int}} =\sum_{i=1}^3
\langle \Psi^{(i)} |g_{0}^{(i)}| \Psi^{(i)}
\rangle +
g \left ( \langle \Psi^{(1)}| \langle \Psi^{(2)}| \langle \Phi^{(3)}|
|{\cal V} \rangle + \,\, \text{cyclic} \right )
\end{equation}
has the form
\begin{eqnarray} \label{t2}
{ \cal L}&=& - \phi^{\mu, \nu} \Box \phi_{\mu,\nu }
+48g ( \partial_\rho \partial_\tau \phi_{\mu, \nu}) \phi^{\mu, \nu} \phi^{\rho, \tau }
- 96 g (\partial_\rho \partial_\tau \phi_{\mu, \nu}) \phi^{\mu, \tau} \phi^{\rho, \nu} \\ \nonumber
&-&\frac12 {\bar\Psi}^\mu \gamma^\nu \partial_\nu \Psi_\mu + 12ig \phi^{\mu,\nu} {\bar \Psi}^\alpha \gamma_\nu \partial_\alpha \Psi_\mu - 6ig \phi^{\mu,\nu} {\bar \Psi}^\alpha \gamma_\nu \partial_\mu \Psi_\alpha
\end{eqnarray}
Let us note that the overall coefficients
in the cubic vertices \p{sugralike3b} and \p{sugralikebff}
are not fixed by the requirement
of the gauge invariance and this particular choice is dictated by the supersymmetry transformations.
However, due to the off-shell transversality conditions the supersymmetry transformations
\begin{equation}
\phi_{ \nu, \mu }(x) = i\, \bar \Psi_{\mu}(x) \gamma_\nu \, \epsilon, \quad
\delta \Psi_\mu(x) = - \gamma^\nu\gamma^\rho\epsilon \,\partial_\nu \phi_{\rho,\mu}(x)
\end{equation}
put the fields
completely on shell.
Similarly to the case of cubic vertices in Super Yang-Mills, the supergravity vertices of the first type
\p{sugralikebff}
can be generalized
to higher spins \cite{Buchbinder:2021qrg} by multiplying them by an arbitrary function
of the BRST invariant expressions \p{sbv-1-x}--\p{sbv-3}
and
finding corresponding $| {\cal W} \rangle$ and $| {\cal X} \rangle$
vertices.
\subsection{Vertices of {${\cal D}=10$} supergravity} \label{sugra10}
In order to consider the cubic vertices of the second type it is easier
to decompose the fermionic fields into irreducible representations of Poincar\'e group
according to \p{dec-2}.
Then in ten dimensions we have the vertex
\begin{equation} \label{hs-vsugra-3}
{}\langle \phi^{(3)} | \,\,
{}_a\langle \psi^{(1)}| \,\,
{}_b\langle \psi^{(2)}| | {\cal V}_{{\cal J}} \rangle^{ab},
\end{equation}
where
\begin{eqnarray} \label{vertex-7}
{\cal V}^{ab}_{{\cal J}}& =& (\gamma_{\mu \tau \sigma \lambda \nu})^{ab} \alpha_1^{\mu (1), +}
\alpha_1^{\nu (2), +} p^{\tau, (3)} \alpha_1^{\sigma (3),+} \alpha_2^{\lambda (3),+} +
\\ \nonumber
&+& ( \alpha_1^{(3),+} \cdot \gamma)^{ab}
[(p^{(3)} \cdot \alpha_1^{(1),+} )(\alpha_1^{(2),+} \cdot \alpha_2^{(3),+})
-
(p^{(3)} \cdot \alpha_1^{(2),+} )(\alpha_1^{(1),+} \cdot \alpha_2^{(3),+})]
- \\ \nonumber
&-& ( \alpha_2^{(3),+} \cdot \gamma)^{ab}
[(p^{(3)} \cdot \alpha_1^{(1),+} )(\alpha_1^{(2),+} \cdot \alpha_1^{(3),+})
-
(p^{(3)} \cdot \alpha_1^{(2),+} )(\alpha_1^{(1),+} \cdot \alpha_1^{(3),+})]+ \\ \nonumber
&+& (p^{(3)} \cdot \gamma )^{ab}[(\alpha_1^{(1),+} \cdot \alpha_2^{(3),+})
(\alpha_1^{(2),+} \cdot \alpha_1^{(3),+})
-
(\alpha_1^{(2),+} \cdot \alpha_2^{(3),+})
(\alpha_1^{(1),+} \cdot \alpha_1^{(3),+})]
\end{eqnarray}
This vertex corresponds to the coupling of the field $B_{\mu \nu}(x)$
with two gravitini.
The corresponding ${\cal W}$ vertices have the form
\begin{eqnarray}
({\cal W}_3^{2,1})^{ab}&=& \frac12 c^{(1),+}_1 \alpha_1^{\nu(2),+} \alpha_1^{\sigma(3),+} \alpha_2^{\lambda(3),+} (\gamma_{\sigma\lambda}\gamma_{\nu})^{ab} \\
\nonumber
({\cal W}_3^{1,2})^{ab}&=& \frac12 c^{(2),+}_1 \alpha_1^{\nu(1),+} \alpha_1^{\sigma(3),+} \alpha_2^{\lambda(3),+} (\gamma_{\sigma\lambda}\gamma_{\nu})^{ab} \\
\nonumber
({\cal W}_2^{3,1})^{ab}&=& -c^{(1),+}_1 [\frac12 (p^{(2)}\cdot \gamma)^a{}_c \alpha_1^{\nu(2),+} \alpha_1^{\sigma(3),+} \alpha_2^{\lambda(3),+} (\gamma_{\sigma\lambda}\gamma_{\nu})^{cb} + \\ \nonumber
&\quad&\quad+ p^{(1),\mu} \alpha_1^{\nu(2),+} \alpha_1^{\sigma(3),+} \alpha_2^{\lambda(3),+}(\eta_{\mu\nu}\gamma_\lambda\gamma_\nu - \eta_{\mu\lambda}\gamma_\sigma\gamma_\nu + \eta_{\mu\nu}\gamma_{\sigma\lambda})^{ab}] \\ \nonumber
({\cal W}_1^{3,2})^{ab}&=& c^{(2),+}_1 [\frac12 (p^{(2)}\cdot \gamma)^b{}_c \alpha_1^{\nu(1),+} \alpha_1^{\sigma(3),+} \alpha_2^{\lambda(3),+} (\gamma_{\sigma\lambda}\gamma_{\nu})^{ca} + \\ \nonumber
&\quad&\quad+ p^{(2),\mu} \alpha_1^{\nu(1),+} \alpha_1^{\sigma(3),+} \alpha_2^{\lambda(3),+}(\eta_{\mu\nu}\gamma_\lambda\gamma_\nu - \eta_{\mu\lambda}\gamma_\sigma\gamma_\nu + \eta_{\mu\nu}\gamma_{\sigma\lambda})^{ba}]
\end{eqnarray}
The solution for the group structure equations includes
\begin{eqnarray} \nonumber
{\cal X}_1^{ab} &=& -b_1^{(1),+} [c_1^{(3),+}c_1^{(2),+}(\gamma\cdot \alpha_2^{(3),+})^a{}_c (\gamma\cdot p^{(1)})^{cb} - c_2^{(3),+}c_1^{(2),+}(\gamma\cdot \alpha_1^{(3),+})^a{}_c (\gamma\cdot p^{(1)})^{cb}] \\ \nonumber
{\cal X}_2^{ab} &=& b_1^{(2),+} [c_1^{(3),+}c_1^{(1),+}(\gamma\cdot \alpha_2^{(3),+})^a{}_c (\gamma\cdot p^{(2)})^{cb} - c_2^{(3),+}c_1^{(1),+}(\gamma\cdot \alpha_1^{(3),+})^a{}_c (\gamma\cdot p^{(2)})^{cb}] \\
{\cal X}_3^{ab} &=& b_1^{(3),+} c_1^{(3),+}c_1^{(1),+}(\gamma\cdot \alpha_2^{(3),+})^{ab} - b_2^{(3),+} c_2^{(3),+}c_1^{(1),+}(\gamma\cdot \alpha_1^{(3),+})^{ab}
\end{eqnarray}
The remaining cubic vertices in ${\cal D}=10$ are those which correspond to the coupling of the $B_{\mu \nu}(x)$ to one gravitino and one dilatino,
\begin{equation} \label{vertex-8}
{}\langle \phi^{(3)} | \,\,
{}_a\langle \psi^{(1)}| \,\,
{}^b\langle \Xi^{(2)}| | {\cal V}_{{\cal L}} \rangle^{a}_{b},
\end{equation}
There are two such couplings. The first is
\begin{eqnarray} \label{vertex-L1}
({\cal V}_{{\cal L}_1})^a{}_b =
(\gamma_{\mu \nu \tau})^{ac}
p^{\mu, (3)} \alpha_1^{\nu (3),+} \alpha_2^{\tau (3),+} (\alpha_1^{ (1), +} \cdot \gamma )_{cb}
\end{eqnarray}
The non-trivial ${\cal W}$ vertices are
\begin{eqnarray}
({\cal W}_3^{2,1})^{ab}&=& -c^{(1),+}_1 (\gamma_{\nu\tau})^{ab} \alpha_1^{\nu(3),+}\alpha_2^{\tau(3),+}
\\ \nonumber
({\cal W}_2^{3,1})^{ab}&=& c^{(1),+}_1 \big[ (\gamma_{\nu\tau})^{a}{}_c \alpha_1^{\nu(3),+}\alpha_2^{\tau(3),+} (p_2\cdot \gamma)^{cb}\big]
\end{eqnarray}
The second vertex of this type is
\begin{eqnarray} \label{vertex-L2}
({\cal V}_{{\cal L}_2})^a{}_b &=& \delta^a{}_b
(\alpha_1^{ (3),+} \cdot \alpha_2^{ (3),+}) (\alpha_1^{ (1), +} \cdot p^{ (3)})
\end{eqnarray}
for which
\begin{eqnarray}
({\cal W}_3^{2,1})^{ab}&=&
-\frac12 c^{(1),+}_1 C^{ab} \alpha_1^{(3),+}\cdot\alpha_2^{(3),+}
\\ \nonumber
({\cal W}_2^{3,1})^{ab}&=& \frac12 c^{(1),+}_1 (p_2\cdot \gamma)^{ab} \alpha_1^{(3),+}\cdot\alpha_2^{(3),+}
\end{eqnarray}
For the last two vertices the solutions for the group structure equations are with $|{\cal X}_i\rangle = 0$.
\subsection{Vertices of ${\cal D}=6$ supergravity} \label{sugra6}
Most of the vertices of ${\cal D}=6$ supergravity have already been described above. They include the universal vertices of Subsection \ref{sugrauni}, as well as the vertices \eqref{vertex-L1} and \eqref{vertex-L2}, which are both present in ${\cal D}=6$.
The vertex that has a different form is the coupling of the $B$-field to two gravitini, which is now given by
\begin{equation} {\cal V}^{ab} = (\gamma_\tau \gamma_{\mu\nu\rho}\gamma_{\lambda})^{ab} \alpha_1^{\lambda(1),+}\alpha_1^{\tau(2),+}p^{\mu(3)}\alpha_1^{\mu(3),+}\alpha_2^{\nu(3),+} \end{equation}
The corresponding ${\cal W}$ vertices have the form
\begin{eqnarray}
({\cal W}_3^{2,1})^{ab}&=& c_1^{(1),+}\alpha_2^{\mu(2),+}\alpha_1^{\nu(3),+}\alpha_1^{\rho(3),+}(\gamma_{\nu\rho}\gamma_\mu)^{ab}\\
\nonumber
({\cal W}_3^{1,2})^{ab}&=& c_1^{(2),+}\alpha_1^{\mu(2),+}\alpha_1^{\nu(3),+}\alpha_1^{\rho(3),+}(\gamma_{\nu\rho}\gamma_\mu)^{ab}\\
\nonumber
({\cal W}_2^{3,1})^{ab}&=& -c_1^{(1),+}\big[ \alpha_1^{\nu(3),+}\alpha_2^{\rho(3),+} (\gamma_{\nu\rho})^a{}_c p^{\mu(2)}\alpha_1^{\sigma(2),+}(\gamma_\mu\gamma_\sigma)^{cb} + \\ \nonumber
&\quad&\qquad\qquad +2(p^{(1)}\cdot \alpha_1^{(2),+})\alpha_1^{\nu(3),+}\alpha_2^{\rho(3),+} (\gamma_{\nu\rho})^{ab}\big]\\ \nonumber
({\cal W}_1^{3,2})^{ab}&=& c_1^{(2),+}\big[ \alpha_1^{\nu(3),+}\alpha_2^{\rho(3),+} (\gamma_{\nu\rho})^a{}_c p^{\mu(1)}\alpha_1^{\sigma(1),+}(\gamma_\mu\gamma_\sigma)^{cb} + \\ \nonumber
&\quad&\qquad\qquad +2(p^{(2)}\cdot \alpha_1^{(1),+})\alpha_1^{\nu(3),+}\alpha_2^{\rho(3),+} (\gamma_{\nu\rho})^{ab}\big]
\end{eqnarray}
with
\begin{eqnarray} \nonumber
{\cal X}_1^{ab} &=& -b_1^{(1),+} [c_1^{(3),+}c_1^{(2),+}(\gamma\cdot \alpha_2^{(3),+})^a{}_c (\gamma\cdot p^{(1)})^{cb} - c_2^{(3),+}c_1^{(2),+}(\gamma\cdot \alpha_1^{(3),+})^a{}_c (\gamma\cdot p^{(1)})^{cb}] \\ \nonumber
{\cal X}_2^{ab} &=& b_1^{(2),+} [c_1^{(3),+}c_1^{(1),+}(\gamma\cdot \alpha_2^{(3),+})^a{}_c (\gamma\cdot p^{(2)})^{cb} - c_2^{(3),+}c_1^{(1),+}(\gamma\cdot \alpha_1^{(3),+})^a{}_c (\gamma\cdot p^{(2)})^{cb}] \\
{\cal X}_3^{ab} &=& 2b_1^{(3),+} c_1^{(1),+}c_1^{(2),+}(\gamma\cdot \alpha_2^{(3),+})^{ab} - 2b_2^{(3),+} c_1^{(1),+}c_1^{(2),+}(\gamma\cdot \alpha_1^{(3),+})^{ab}
\end{eqnarray}
solving the group structure equations.
There is also the coupling of \(B_{\mu\nu}\) to two dilatini,
\begin{equation} \label{vertex-8-1}
{}\langle \phi^{(3)} | \,\,
{}^a\langle \Xi^{(1)}| \,\,
{}^b\langle \Xi^{(2)}| | {\cal V}_{{\cal X}} \rangle_{ab},
\end{equation}
which is of the form
\begin{equation} ({\cal V}_{\cal X})_{ab} = (\gamma_{\mu\nu\rho})_{ab}p^{\mu(3)}\alpha_1^{\nu(3),+}\alpha_2^{\rho(3),+} \end{equation}
In this case all the ${\cal W}$ vertices are trivial.
\section{Light Cone Formalism}\label{sectionlightcone} \setcounter{equation}0
In this Section we describe how to construct cubic vertices of ${\cal D}=4$
$N=1$ super Yang-Mills and Supergravity in the light cone approach following \cite{Metsaev:2019dqt}
(see also \cite{Brink:2005wh}--\cite{Ponomarev:2016lrm} for a brief review of the light cone approach).
\subsection{Set Up}
To construct cubic interaction vertices in the light cone approach
let us consider a field theoretic realization
of the ${\cal D}=4$ $N=1$ super Poincar\'e algebra
\begin{align} \label{SPA}
[Q^a,Q^b]&= \frac{1}{2}(\gamma^{\mu})^{ab} P_\mu \,,\\
[Q^a,J^{\mu \nu}]&= \frac{1}{2}(\gamma^{\mu \nu})^a{}_b Q^b \,,\\
[J^{\mu \nu},P^\rho]&=P^\mu\eta^{\nu \rho}-P^\nu\eta^{\mu \rho}\,,\\
[J^{\mu \nu},J^{\rho \sigma}]&=J^{\mu \sigma}\eta^{\nu \rho}-J^{\nu \sigma}\eta^{\mu \rho}-J^{\mu \rho}\eta^{\nu \sigma}+J^{\nu \rho}\eta^{\mu \sigma}\,.
\end{align}
Here $J^{\mu \nu}$ are
generators of Lorentz transformations,
$P^\mu$ are generators of translations, and
$Q^a$ are generators of Supersymmetry transformations.
These generators are split into kinematical and dynamical generators.
Kinematical generators preserve the Cauchy surface (the light cone) and are quadratic in fields
both on free and interacting levels. The other generators are dynamical and they receive
higher order corrections in fields. These corrections are determined from the requirement
that the Poincar\'e algebra is preserved at the interacting level.
We choose the four dimensional coordinates as
\begin{equation}
x^\pm = \frac{1}{\sqrt 2} (x^3 \pm x^0), \quad z = \frac{1}{\sqrt 2} (x^1 + i x^2), \quad \bar z = \frac{1}{\sqrt 2} (x^1 - i x^2)
\end{equation}
The coordinate $x^+$ is treated as the time direction and $H=P^-$ is the Hamiltonian\footnote{ Note that $\beta$ is used instead of $p^+$
in order to simplify the form of the equations.We shall also put
$x^+=0$ for now on.}. The generators of the super Poincar\'e algebra are split according to
\begin{align}
\text{kinematical}&: && P^{+}, P^z, { P}^{\bar z}, J^{z+}, { J}^{\bar z +}, J^{+-}, J^{z \bar z}, Q^{+},{\bar Q}^{+}, &&: 9\\
\text{dynamical}&: && P^{-}, J^{z-}, { J}^{\bar z-}, Q^{-},{\bar Q}^{-} &&:5
\end{align}
It is sufficient to construct the Poincare algebra at $x^+=0$ and then evolve all the generators according to $\dot{G}=i[H,G]$. The equations to be solved are
\begin{equation}\label{hardequations}
[Q^{-},P^{-}]=[\bar Q^{-},P^{-}]=0\,, \quad
[J^{z,-},P^{-}]=[J^{\bar z,-},P^{-}]=0\,.
\end{equation}
The spectrum consists of bosonic $\phi_\lambda(x)$ and fermionic
$\psi_\lambda(x)$ fields\footnote{In this Section, unlike the previous ones, the index $\lambda$ denotes a helicity of
a field, rather than its Lorentz index.}
with the helicities $\lambda= \pm 1, \pm 2$ for bosons,
$\lambda= \pm \frac{1}{2}, \pm \frac{3}{2}$ for fermions.
It is convenient to work with partial Fourier transforms
\begin{align}
\phi_\lambda(x)&=(2\pi)^{-\tfrac{3}2} \int e^{+i(x^-\beta +z \bar p + \bar z p)} \phi_\lambda(\vec{p})\, d^{3}p\,,\\
\psi_\lambda(x)&=(2\pi)^{-\tfrac{3}2} \int e^{+i(x^-\beta + z \bar p + \bar z p)} \psi_\lambda(\vec{p})\, d^{3}p\,
\end{align}
with $d^{3}p= d \beta \, d p \, d {\bar p}$.
The fields obey the following conjugation rules
\begin{equation}
\phi^\dagger_\lambda(\vec{p}) = \phi_{-\lambda}(-\vec{p}), \quad \psi^\dagger_\lambda(\vec{p}) = \psi_{-\lambda}(-\vec{p})
\end{equation}
Introducing a Grassman momentum $p_\theta$, one can combine
the bosonic and fermionic fields into superfields
\begin{equation}
\Phi_\lambda = \phi_\lambda + \frac{p_\theta}{\beta}\psi_{\lambda -\frac{1}{2}},
\quad
\Phi_{-\lambda + \frac{1}{2}} = \psi_{-\lambda + \frac{1}{2}} + {p_\theta}\phi_{-\lambda},
\end{equation}
with conjugation properties
\begin{equation}
\overline \Phi_{-\lambda} = \phi_{-\lambda}^\dagger + \frac{p_\theta}{\beta}\psi^\dagger_{-\lambda +\frac{1}{2}},
\quad
\overline \Phi_{\lambda - \frac{1}{2}} = -\psi_{\lambda - \frac{1}{2}}^\dagger
+ {p_\theta}\phi_{\lambda}^\dagger,
\end{equation}
The equal time Poisson brackets between the fields
\begin{equation}\label{equaltime1}
[\phi_{\lambda}(\vec{p}),\phi_{\lambda^\prime}^\dagger(\vec{p}^\prime)]=\delta_{\lambda,\lambda^\prime}\frac{\delta^{3}(\vec{p}-\vec{p}^\prime)}{2\beta},
\quad
[\psi_{\lambda}(\vec{p}),\psi_{\lambda^\prime}^\dagger(\vec{p}^\prime)]=\delta_{\lambda,\lambda^\prime}\frac{\delta^{3}(\vec{p}-\vec{p}^\prime)}{2}
\end{equation}
read in terms of the superfields as
\begin{equation}\label{equaltime3}
[\Phi_{\lambda}(\vec{p},p_\theta),\overline \Phi_{\lambda^\prime}(\vec{p}^\prime,p_\theta^\prime)]=(-)^{\epsilon_{\lambda+\frac{1}{2}}}\delta_{\lambda,\lambda^\prime+\frac{1}{2}}
\frac{\delta^{3}(\vec{p}-\vec{p}^\prime)\,
\delta(p_\theta-p_\theta^\prime)}
{2\beta}
\end{equation}
where $\epsilon_\lambda$ is $0$ for integer $\lambda$ and is $1$ for half-integer $\lambda$.
The kinematical generators, which are the same both on free and interacting levels have the form
\begin{equation}
{P}^+=\beta\,, \quad {P}^z=p, \quad P^{\bar z}={\bar{p}},
\quad
{J}^{z+}=-\beta\pfrac{{\bar{p}}}, \quad {J}^{{\bar{z}}+}=-\beta\pfrac{p},
\end{equation}
\begin{equation} \nonumber
{J}^{-+}=-\frac{\partial}{\partial \beta} \beta - \frac{1}{2}p_\theta \frac{\partial}{\partial p_{\theta}} + \frac{1}{2} \epsilon_\lambda,
\quad {J}^{z{\bar{z}}}= p\partial_p-\bar p\frac{\partial}{\partial \bar p} +\lambda -p_\theta \frac{\partial}{\partial p_{\theta}}
\end{equation}
\begin{equation} \nonumber
Q^+ = (-)^{\epsilon_\lambda} \beta \frac{\partial}{\partial p_\theta}, \quad \bar Q^{+}= (-)^{\epsilon_\lambda} p_\theta
\end{equation}
The dynamical generators at the free level are
\begin{eqnarray}
H_2&=&-\frac{p{\bar{p}}}{\beta}\,, \\ \nonumber
{J}^{z-}_2&=& -\pfrac{{\bar{p}}} \frac{ p{\bar{p}}}{\beta} +p \pfrac{\beta} \nonumber
- \left (\lambda - \frac{1}{2} p_\theta \frac{\partial}{\partial p_\theta} \right ) \frac{p}{\beta}
+ \left( \frac{1}{2} p_\theta\frac{\partial}{\partial p_\theta}-\frac{1}{2}\epsilon_\lambda \right) \frac{ p}{\beta}
\\ \nonumber
{J}^{{\bar{z}}-}_2&=&- \pfrac{p} \frac{ p{\bar{p}}}{\beta} +{\bar{p}} \pfrac{\beta}
+ \left (\lambda - \frac{1}{2} p_\theta \frac{\partial}{\partial p_\theta} \right ) \frac{\bar p}{\beta}
+ \left( \frac{1}{2} p_\theta\frac{\partial}{\partial p_\theta}-\frac{1}{2}\epsilon_\lambda \right) \frac{ p}{\beta} \\ \nonumber
Q_2^- &=&(-)^{\epsilon_\lambda}\frac{p}{\beta} p_\theta \\ \nonumber
\bar Q_2^- &=&(-)^{\epsilon_\lambda} \bar p \frac{\partial}{\partial p_\theta}
\end{eqnarray}
At the level of cubic interactions one assumes the following expansion for the dynamical generators
\begin{eqnarray} \label{cubiclc}
H_3&=&H_2+\int d \Gamma_{[3]} \,
\overline \Phi^{\lambda_1 \lambda_2 \lambda_3}_{q_1 q_2 q_3} \,
h_{\lambda_1 \lambda_2 \lambda_3}^{q_1 q_2 q_3}
\\ \nonumber
Q_3^{-}&=&Q_2^{-}
+\int d \Gamma_{[3]} \,
\overline \Phi^{\lambda_1 \lambda_2 \lambda_3}_{q_1 q_2 q_3}\,
q_{\lambda_1 \lambda_2 \lambda_3}^{q_1 q_2 q_3} \\ \nonumber
{\bar Q}_3^{-}&=&{\bar Q}_2^{-}
+\int d \Gamma_{[3]}\,
\overline \Phi^{\lambda_1 \lambda_2 \lambda_3}_{q_1 q_2 q_3} \,
{\bar q}_{\lambda_1 \lambda_2 \lambda_3}^{q_1 q_2 q_3} \\ \nonumber
J^{z-}_3&=&J^{z-}_2+ \int d \Gamma_{[3]} \times \\ \nonumber
&\times&\left[
\overline \Phi^{\lambda_1 \lambda_2 \lambda_3}_{q_1 q_2 q_3} \,\,
j_{\lambda_1 \lambda_2 \lambda_3}^{q_1 q_2 q_3}-
\frac{1}{3}
\left(\sum_{k=1}^3 \frac{\partial
\overline \Phi^{\lambda_1 \lambda_2 \lambda_3}_{q_1 q_2 q_3}
}{\partial \bar{q}_k}\right)h_{\lambda_1 \lambda_2\lambda_3}^{q_1 q_2 q_3}-
\frac{1}{3}
\left(\sum_{k=1}^3
\frac{\partial
\overline \Phi^{\lambda_1 \lambda_2 \lambda_3}_{q_1 q_2 q_3}
}{\partial {q}_{\theta, k}}\right) q_{\lambda_1 \lambda_2\lambda_3}^{q_1 q_2 q_3}
\right]\,
\\ \nonumber
J^{{\bar{z}}-}_3&=&J^{\bar z-}_2+ \int d \Gamma_{[3]} \times \\ \nonumber
&\times &\left[
\overline \Phi^{\lambda_1 \lambda_2 \lambda_3}_{q_1 q_2 q_3} \,\,
{\bar{j}}_{\lambda_1 \lambda_2\lambda_3}^{q_1 q_2 q_3}-
\frac{1}{3}
\left(\sum_{k=1}^3 \frac{\partial
\overline \Phi^{\lambda_1 \lambda_2 \lambda_3}_{q_1 q_2 q_3}
}{\partial q_k}\right) h_{\lambda_1 \lambda_2\lambda_3}^{q_1 q_2 q_3}
+ \frac{1}{3}
\left(\sum_{k=1}^3 \frac{q_{\theta,k}}{\beta_k}\right) \overline \Phi^{\lambda_1 \lambda_2 \lambda_3}_{q_1 q_2 q_3}
{\bar q}_{\lambda_1 \lambda_2\lambda_3}^{q_1 q_2 q_3}
\right ]
\end{eqnarray}
where $\overline \Phi^{\lambda_1 \lambda_2 \lambda_3}_{q_1 q_2 q_3} \equiv
\overline \Phi^{\lambda_1}_{q_1}\overline \Phi^{\lambda_2}_{q_2}
\overline \Phi^{\lambda_3}_{q_3}$ and
\begin{equation}
d \Gamma_{[3]}= (2 \pi)^3 \prod_{k=1}^3 \frac{d^3 q_k}{(2 \pi)^\frac{3}{2}}
\delta^3\left(\sum_{i=1}^3 q_i\right)
\prod_{l=1}^3 dq_{ \theta,l }\,\delta\left(\sum_{j=1}^3 q_{\theta, j}\right)
\end{equation}
is an integration measure.
\subsection{${\cal D}=4$, $N=1$ Super Yang Mills and Pure Supergravity}
The cubic vertices
which are present in the interaction part of dynamical generators \p{cubiclc}
are determined from the requirement of preservation of the
algebra \p{SPA}.
A solution
which contains the $N=1$ Super Yang Mills and Supergravity vertices
has the form \cite{Metsaev:2019dqt}
\begin{equation} \label{lcs1}
h_{\lambda_1 \lambda_2 \lambda_3}^{q_1 q_2 q_3} = C^{\lambda_1 \lambda_2 \lambda_3}
({\overline{\mathbb{P}}})^{M_\lambda +1}
\prod_{i=1}^3
\beta_i^{-\lambda_i - \frac{1}{2}\epsilon_{\lambda_i}}
+
\overline C^{\lambda_1 \lambda_2 \lambda_3}
({\mathbb{P}})^{-M_\lambda -\frac{1}{2}} {\mathbb{P}}_\theta
\prod_{i=1}^3
\beta_i^{\lambda_i - \frac{1}{2}\epsilon_{\lambda_i}}
\end{equation}
\begin{equation} \label{lcs2}
q_{\lambda_1 \lambda_2 \lambda_3}^{q_1 q_2 q_3} =- C^{\lambda_1 \lambda_2 \lambda_3}
({\overline{\mathbb{P}}})^{M_\lambda } {\mathbb{P}}_\theta
\prod_{i=1}^3
\beta_i^{-\lambda_i - \frac{1}{2}\epsilon_{\lambda_i}}
\end{equation}
\begin{equation} \label{lcs3}
j_{\lambda_1 \lambda_2 \lambda_3}^{q_1 q_2 q_3} =2 C^{\lambda_1 \lambda_2 \lambda_3}
({\overline{\mathbb{P}}})^{M_\lambda } \, \chi
\prod_{i=1}^3
\beta_i^{-\lambda_i - \frac{1}{2}\epsilon_{\lambda_i}}
\end{equation}
and
\begin{equation}
M_\lambda= \lambda_1 + \lambda_2 + \lambda_3, \quad \lambda_1= s_1 - \frac{1}{2}, \quad \lambda_2= s_2 - \frac{1}{2},
\quad
\lambda_3= - s_3
\end{equation}
In these equations $C^{\lambda_1 \lambda_2 \lambda_3}$,
$\overline C^{\lambda_1 \lambda_2 \lambda_3}$
are coupling constants and
\begin{equation}
{\mathbb{P}}=
\frac13\left[ (\beta_1-\beta_2)p_3+(\beta_2-\beta_3)p_1+(\beta_3-\beta_1)p_2\right] \,,\\
\end{equation}
\begin{equation}
{\mathbb{P}}_\theta=
\frac13\left[ (\beta_1-\beta_2)p_{\theta, 3}+(\beta_2-\beta_3)p_{\theta, 1}+(\beta_3-\beta_1)p_{\theta, 2}\right] \,,\\
\end{equation}
\begin{equation}
\chi=\beta_1(\lambda_2-\lambda_3)+\beta_2(\lambda_3-\lambda_1)+\beta_3(\lambda_1-\lambda_2)\,.
\end{equation}
The momenta $\beta_i, p_i, \bar p_i$ and $p_{\theta,i}$ obey the conservation properties as in \p{mcon}.
The vertices
${\bar q}_{\lambda_1 \lambda_2 \lambda_3}^{q_1 q_2 q_3}$ and ${\bar j}_{\lambda_1 \lambda_2 \lambda_3}^{q_1 q_2 q_3}$
can be obtained from \p{lcs2}--\p{lcs3} by relevant hermitean conjugation.
The cubic vertices for $N=1$ Super Yang Mills can be recovered by choosing
$s_1= s_2=s_3=1$ in the equations above. Similarly, pure $N=1$ Supergravity
vertices can be recovered by putting $s_1= s_2=s_3=2$.
\vskip 0.5cm
\noindent {\bf Acknowledgments.} M.T. would like to thank the organizers of the online conference ``Quarks 2020"
and of the online workshop``Aspects of Symmetry" for the invitation to give talks.
M.T. would like to thank the Department of Mathematics, the University of Auckland for the hospitality
during the final stage of the project.
D.W. would like to thank the Yukawa Institute of Theoretical Physics (YITP) for its support in the period when this work was completed.
The work of I.L.B. and V.A.K. was partially supported by the Ministry of Education of Russian Federation, project FEWF-2020-0003. The work of M.T. and D.W. was supported by the Quantum Gravity Unit
of the Okinawa Institute of Science and Technology Graduate University
(OIST).
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2024-02-18T23:40:24.403Z
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2022-05-19T02:05:58.000Z
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{"arxiv_id":"2112.14419","language":"en","timestamp":1652925958000,"url":"https:\/\/arxiv.org\/abs\/2112.14419","yymm":"2112"}
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proofpile-arXiv_000-10017
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{"provenance":"002.jsonl.gz:10018"}
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\section{Introduction}
\index{Introduction}
\label{section_introduction}
The SuperKEKB asymmetric electron-positron collider \cite{PTEP:Ohnish} is currently in operation at KEK in Tsukuba, Japan, producing collision events for the Belle~II experiment \cite{Abe:2010gxa}. Its current target is a luminosity of \SI{6.5e35}{cm^{-2}s^{-1}}, a roughly 30-fold improvement over its predecessor, KEKB \cite{KEK:1995sta}. The increased luminosity is to be achieved by moving to a new ``nano-beam" scheme~\cite{Bona:2007qt}, which will reduce beams to a vertical size of \SI{50}{nm} at the interaction point (IP), one twentieth the size of its predecessor, while increasing the beam currents by approximately 50\%.
The reduced beam size, increased beam currents, and consequent increased luminosity will provide a challenging environment for physics measurements, as each improvement also increases the background at the IP. Thus, measuring and mitigating these backgrounds is of critical importance for successful physics data taking, particularly in the later stages of the experiment as SuperKEKB approaches its target luminosity.
In Section~\ref{sec:experimental_setup} we describe the detectors used for dedicated measurements of backgrounds, particularly those which were added or modified between the first and second SuperKEKB accelerator commissioning phases. In Section~\ref{sec:operational_experience} we describe the operational experience during Phase~2. Sections~\ref{sec:simulation} and~\ref{sec:exp_results} describe the simulation of relevant backgrounds and their measurements, respectively. Section~\ref{sec:collimator_study} describes the procedure for optimizing the collimator settings for the best compromise between beam lifetime and background levels. Finally, Section~\ref{sec:summary_recommendations} summarizes the results of Phase~2, extrapolations to expected levels of background for future Belle~II data taking runs, and recommendations for future operation.
\subsection{Operational Phases}
\index{Phases}
\label{sec:phases}
We divide Belle~II/SuperKEKB running into 3 major phases: Phase~1, carried out in the Spring of 2016, was the first of two dedicated commissioning phases. This phase was run without the Belle~II detector installed around the IP and also without final focusing of the accelerator and thus, no collisions. In place of Belle~II, a dedicated suite of background detectors collectively known as BEAST II was installed around the IP to directly measure rates of backgrounds produced in the accelerator. The results from Phase~1 are reported in Reference~\cite{Lewis:2018ayu}.
In Phase~2, most of the Belle~II detector assembly was installed around the IP, with the notable exception of the VerteX Detector (VXD), of which only one octant was installed. Phase~2 began in March 2018, with first electron-positron collisions in April 2018, and concluded in July 2018.
From the Belle~II point of view, the major goal of this phase was to confirm that the sensitive VXD detectors could be safely installed; for SuperKEKB, the goal was machine performance studies and achieving a target luminosity of $1 \times 10^{34} \mathrm{cm}^{-2}\mathrm{s}^{-1}$.
Phase~3 refers to the main physics data-taking run of the Belle~II experiment, encompassing the remainder of the lifetime of the project. This includes the full operation of all Belle~II subdetectors. During this phase, SuperKEKB plans to gradually improve perfomance up to its design luminosity, eventually accumulating 50 $\mathrm{ab}^{-1}$ of collision data.
\subsection{Beam-Related Backgrounds and Their Origins }
\label{sec:types}
Here, we provide an overview of the major sources of backgrounds incident onto Belle~II from the SuperKEKB beamline and their expected dependence on accelerator conditions. Using these expected dependencies, we then build a background model that we use later to extract the relative contributions to the total background rates in BEAST II and Belle~II subdetectors.
We refer to backgrounds arising from Coulomb interactions between particles in the same beam bunch as ``Touschek backgrounds''. Interacting beam particles either gain or lose energy and thus, subsequent to scattering, propagate with an energy higher or lower than the nominal bunch energy. Affected particles will deviate from the nominal orbit and eventually collide with the beam pipe wall. Showers from such collisions near the IP may reach a subdetector and produce unwanted background events. We expect the rate of Touschek scattering to be inversely proportional to the number of filled bunches in the main storage ring (MR), $n_b$, and the square of the beam current $I$, and inversely proportional to the bunch volume $\sigma_{x} \times \sigma_{y} \times \sigma_{z}$, where $x$ and $y$ refer to horizontal (in the plane of the rings) and vertical directions, respectively, and $z$ is the direction of propagation of the beam~\cite{PTEP:Ohnish}.
During operation, particles in the beams occasionally interact with residual free gas molecules inside the beam pipe. As described for Touschek scattering, resulting off-orbit beam particles can hit the beam pipe wall near the IP and cause background events, which we refer to as beam-gas events. Because the likelihood of interactions with gas nuclei increases as more gas molecules are present, the beam-gas event rate scales linearly with the beam pipe pressure.
In addition to a baseline level of gas present in the beam pipe, the pressure also tends to rise during operation as the beam causes increased outgassing from the beam pipe material. In the long term, the pressure tends to decrease as the total integral of the beam current increases; hence, the newer LER beamline exhibits significantly larger pressure, and therefore more beam-gas events, than the HER.
As electrons and positrons are accelerated around the circular MR, emission of synchrotron radiation (SR) is unavoidable. SR photons incident on Belle~II are typically in the range of a few to several tens of\,keV. To protect the innermost detectors against these photons, the inner surface of the central 2\,cm-diameter Beryllium beam pipe will be coated with a 10\,\textmu m-thick layer of gold. In order to enhance the sensitivity to these SR photons during Phase~2 the thickness of the gold layer was reduced to 6.6\,\textmu m. Dedicated studies for understanding SR backgrounds are described in Section \ref{sec:synchrotron_sim}. Because SR power is proportional to the square of both beam energy and magnetic field strength, we expect the HER to be the dominant source of SR production.
Due to the short lifetime of the circulating beams, we must inject beam even when beams are in collision and Belle~II is taking data. When beam is injected from the SuperKEKB linac into the MR, the injected bunch is of lower quality than that of the stored beam and causes a perturbation that can last several milliseconds and is observable as an increased background rate for many turns around the ring. Reducing the emittance of the injected beam is essential both to mitigate this background as well as for reaching high luminosity, so a positron damping ring was installed prior to Phase~2 to reduce emittance in the LER and provide a cleaner beam. We also use a dedicated injection background detector, sCintillation Light and Waveform Sensors (CLAWS), specifically to monitor injection background waveforms and assist in determining the necessary post-injection veto window to avoid excessive occupancy in the PiXel Detector (PXD). We describe a dedicated injection background study in Section \ref{sec:injection_backgrounds}.
We also consider backgrounds that arise from beam-beam collisions. Unlike the effects already described, these backgrounds require the beams to interact at the IP, and are hence referred to as luminosity backgrounds. Because the beam conditions during Phase~2 involved low currents and large beam sizes relative to the final design parameters, single beams were the primary sources of beam-related backgrounds. As currents are increased and the beam approaches the full nano-beam scheme, we expect luminosity backgrounds to overtake single-beam backgrounds.
\section{Experimental Setup}
\label{sec:experimental_setup}
\index{Experimental Setup}
The experimental setup for Phase~2 can be divided into three distinct systems: the SuperKEKB accelerator facility; Belle~II; and BEAST II, the collective name for the dedicated beam-background detectors.
\subsection{SuperKEKB}
SuperKEKB is the asymmetric electron-positron collider situated at the High Energy Accelerator Research Organization (KEK) in Tsukuba, Japan. It consists of an approximately 3\,km MR fed by a linear accelerator via beam transport lines, as shown in Figure~\ref{fig:skb_layout}. The MR is composed of a low energy ring providing 4\,GeV positrons and a high energy ring that provides 7\,GeV electrons. Further details concerning the physical setup of SuperKEKB can be found in Reference~\cite{PTEP:Ohnish}.
\begin{figure}
\begin{center}
\includegraphics[width=0.5\textwidth]{phase2_outline}
\caption{Overview of the SuperKEKB accelerator facility, showing the Belle~II detector installed at the interaction point.}
\label{fig:skb_layout}
\end{center}
\end{figure}
\subsection{Accelerator Parameter Monitors}
Understanding the accelerator and beam conditions is of critical importance in the simulation and measurement of beam-related backgrounds. In particular, our model of backgrounds at the IP relies on the following beam parameters, as described in Section~\ref{sec:heuristic}: vertical beam size, $\sigma_{y}$, gas pressure $P$, gas composition $Z_{\text{eff}}$, and beam current $I$. This section describes in brief the apparatus used to measure the quantities listed above. More detailed descriptions may be found in Reference~\cite{Lewis:2018ayu}.
\subsubsection{Beam Size}
Horizontal and vertical measurements of the beam sizes in both the LER and HER are performed with X-ray monitors (XRM) as well as visible light monitors, implemented in the LER and HER late in Phase~2. Beam size measurements reported here are taken from the XRM system, described in References~\cite{Mulyani:IBIC2015-TUPB025} and~\cite{IBIC2016}.
\subsubsection{Beam Pipe Gas Pressure}
\label{sec:Beam_Gas_Pressure}
As was the case in Phase~1, beam pipe gas pressure is a large uncertainty in the analysis. Gas pressure is measured with cold cathode gauges distributed approximately every 10\,m around each ring. Beam pipe gas pressure can vary substantially throughout the ring and changes far upstream of the IP can greatly influence the background rates seen in Belle~II detectors.
The beam-gas pressure is composed of a base value present in the pipe at all times, typically less than 10$^{-9}$\,Pa, and a dynamic component that appears only while the beam is in operation. The dynamic pressure component was found to increase approximately linearly with beam current.
\subsubsection{Beam-Gas Composition}
To improve the modeling of beam-gas interactions, data from three Residual Gas Analyzers (RGAs) is used to build a comprehensive model of gas species in the beam pipe. The RGAs provide partial pressures of individual mass species, which are then combined to give an effective $Z$ value, $Z_{eff}$, for the gas mixture in the pipe. In addition to the two RGAs used in Phase~1, a third RGA was added to the HER for Phase~2.
\subsection{Belle~II}
Beginning in Phase~2 the full set of Belle~II subdetectors, with the exception of most of the sensitive VXD consisting of a PXD and Silicon Vertex Detector (SVD), was installed around the IP. Surrounding the single installed octant of the VXD are a Central Drift Chamber (CDC) for particle tracking, a Time-Of-Propagation (TOP) counter and Aerogel Ring Imaging Cherenkov Device (ARICH) for particle identification, and a CsI(Tl)-based Electromagnetic CaLorimeter (ECL). This assembly is enclosed in a solenoidal magnet for generation of a 1.5\,T magnetic field. An outer detector for long-lived $K_{L}$ and $\mu$ particles consisting of a barrel with two layers of scintillators surrounded by a resistive plate counter and two endcaps composed entirely of scintillators is situated farthest outward radially.
The full Belle~II experimental setup, including the VXD, which was only partially installed for Phase~2, is shown in Figure~\ref{fig:belle_ii_detector}. More information about the Belle~II experimental setup can be found in Reference~\cite{Abe:2010gxa}. The inner detector configuration for Phase~2 is shown in detail in Figure~\ref{fig:beast:vxd_arrangement}.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=.45\textwidth]{belle_II}
\caption{Cutaway diagram of the Belle~II detector, with individual subdetector systems labeled in word balloons. The VerteX Detector (VXD) system was only partially installed, with only one octant present during SuperKEKB Phase 2.}
\label{fig:belle_ii_detector}
\end{center}
\end{figure}
\subsection{BEAST II Detectors}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=.45\textwidth]{beast_setup_v2}
\caption{Vertex detector configuration during Phase 2. Only one octant of the nominal Belle~II vertex detector, composed of the SVD and PXD, was installed. This allowed nominal Belle~II reconstruction of select, horizontally-going charged tracks from the IP. Particles going in other directions were monitored with dedicated BEAST II background monitoring detector systems (FANGS, CLAWS, and PLUME).}
\label{fig:beast:vxd_arrangement}
\end{center}
\end{figure}
During Phase~1, we also deployed a suite of background detectors collectively known as BEAST II~\cite{Lewis:2018ayu}. Each detector was designed to measure a particular type of background expected during Belle~II physics operations, including X-rays, charged particles, and fast and thermal neutrons.
In the transition from Phase~1 to Phase~2, the BEAST II support structure at the IP was removed, and replaced by the rolled-in Belle~II detector. Many of the original BEAST II detectors were re-used and moved to new locations in Phase~2, to continue the monitoring of specific background components. For example, CLAWS injection background monitors were moved very close to the IP, and BEAST fast neutron detectors were installed around the final focusing Superconducting Quadrupole (QCS) magnets on either side of the IR, where beam losses and resulting backgrounds are particularly high. Because a primary goal of Phase~2 was to determine if it was safe to install the full Belle~II VXD, only one $\phi$ (azimuthal angle) segment replica of that detector was installed. The remainder of the VXD volume was instrumented with three types of custom background detectors; FANGS, PLUME, and the new CLAWS configuration, all described below. The complete set of custom background detectors, both those remaining from Phase~1, and those new in Phase~2, will be collectively referred to as ``BEAST II detectors''. The reason for this designation is that these detectors are not integrated into the Belle~II DAQ stream, but rather monitored separately.
\subsubsection{FANGS Detector System}
The FE-I4 ATLAS Near Gamma Sensors (FANGS) detector was specifically designed for BEAST II. It is based on hybrid pixel detector modules used in the ATLAS pixel detector (FE-I4), which are sensitive to charged tracks and low-energy ($\approx$\,10\,keV) X-rays. They are designed to withstand radiation of up to 300\,Mrad and to cope with a maximum hit rate of 400\,MHz/cm$^2$~\cite{FE-I4:2012}.\par
The basic unit of the FANGS detector system is the stave, of which three are installed around the beam pipe (see \autoref{fig:fangs:staves}). Each stave contains five FE-I4 readout chips connected to 250\,\textmu m thick n-in-n planar silicon sensors, mounted on an aluminum profile. Each stave is mounted on two cooling blocks at a distance of 22\,mm from the interaction region, at azimuthal positions of $90^{\circ}$, $180^{\circ}$, and $270^{\circ}$ in $\phi$. The distance between the detector and the back-end electronics is 18\,m.\par
In BEAST II Phase~2, the system provides hit rates as well as the recorded charge spectrum from ionizing radiation. More details about the FANGS detector system can be found in Reference~\cite{Ahlburg:2016}.
\begin{figure}[phtb]
\begin{center}
\includegraphics[width=\linewidth]{fangs.JPG}
\caption{The three FE-I4 ATLAS Near Gamma Sensors (FANGS) staves, utilizing hybrid pixel sensors, installed around the beam pipe. This photo was taken before installation of other vertex detectors.}
\label{fig:fangs:staves}
\end{center}
\end{figure}
\subsubsection{CLAWS Injection Background Detector System}
The CLAWS system targets the beam background arising from injections of new particles into the main ring. Its sub-nanosecond time resolution allows measurement of the exact arrival time of minimum-ionizing particles (MIPs) emerging from bunches crossing the IP. Recording single waveforms over a period of multiple tens of milliseconds gives important insights about the time structure of injected bunches and overall beam behavior.
\par
The active part of the detector consists of plastic scintillator tiles instrumented with silicon photomultipliers (SiPMs), comparable to the CLAWS sensors used in Phase~1~\cite{Lewis:2018ayu}. The scintillator tiles are arranged in two staves with eight independent tiles each, as shown in Figure~\ref{fig:claws:ladder_example}. The sensors, which are mostly sensitive to charged particles, replace two modules of the second layer of the PXD as depicted in Figure~\ref{fig:beast:vxd_arrangement}. The switchable on-board pre-amplifiers enable a high and low gain mode. In addition, a second amplifier is located in a dockbox just outside of the VXD volume, increasing the amplitude of the analog signal before its transmission over a distance of $35\,\mathrm{m}$ to the DAQ.
\begin{figure}[phtb]
\begin{center}
\includegraphics[width=\linewidth]{DSC_0414}
\caption{Two CLAWS staves, based on plastic scintillator tiles instrumented with silicon photomultipliers, installed at $\phi = 135^{\circ}$ and 225$^{\circ}$, which correspond to the top and bottom of the figure, respectively. One FANGS stave is also visible between the two CLAWS staves at $\phi = 180^{\circ}$.}
\label{fig:claws:ladder_example}
\end{center}
\end{figure}
\subsubsection{PLUME detector system}
The Pixelated Ladder with Ultra-low Material Embedding (PLUME) detector is designed as a hit rate monitor, with a material budget low enough to minimally impact particle trajectories~\cite{Cuesta:2020bpl}. It is based on ultra-light double-sided ladders equipped with CMOS pixel sensors, initially developed as a generic concept for the vertex detector of the International Large Detector (ILD)~\cite{Nomerotski:2011zz}. Taking advantage of its very low material budget of 0.4\% of $X_0$, two PLUME ladders are installed in the VXD volume. Both units feature a sensitive area of $127\times11$~mm$^2$ (for each side), with their long axis oriented along the beam. One ladder sits parallel to the beam axis, at $\phi=125^{\circ}$, and the second one is tilted at an angle of $\theta=18^{\circ}$ with respect to the beam, at $\phi=225^{\circ}$. This arrangement, depicted in Figure~\ref{fig:beast:vxd_arrangement}, was chosen in order to monitor the beam-induced background in a range of radii from 5 to 9\,cm with only two devices.
Each ladder includes 6 MIMOSA-26 CMOS pixel sensors~\cite{Baudot:2013pca} on each side, with a total of 8$\times$10$^6$ pixels per ladder. The sensor granularity and the frame-based read-out architecture, featuring a 115~\si{\micro}s integration time, allow a counting rate in excess of $10^6$\,hits\,cm$^{-2} \cdot $s$^{-1}$. Sensors are operated with a low detection threshold, which enables sensitivity to electrons with energies greater than about 40~keV and to X-rays in the 2-10~keV energy range due to the thin sensitive layer (14~\si{\micro}m). The data acquistion (DAQ) system is designed to read out both PLUME ladders with no dead time. The system provides pixel counts for every sensor frame separated by 115\,\si{\micro}s for use in studying the background evolution of newly injected bunches, and averages the counts over 1 second for standard online monitoring. The PLUME system is shown in Figure~\ref{fig:PLUME}.
\begin{figure}
\includegraphics[width=.45\textwidth]{Plume}
\caption{BEAST II vertex detector assembly after installation of the two CMOS pixel sensor-based PLUME staves. The PLUME staves are visible at the top and left hand side of the assembly.}
\label{fig:PLUME}
\end{figure}
\subsubsection{BEAST TPC Fast Neutron Detector System}
We use a system of eight identical, compact time projection chambers (TPCs) with pixel ASIC readout to detect fast neutrons via their nuclear recoils in gas. The detectors were specifically designed for SuperKEKB commissioning~\cite{Lewis:2018ayu, Jaegle:2019jpx, Thorpe:2021qce} and provide the rate, energy spectra, and three-dimensional directions of fast neutron recoils inside the Belle~II detector. The TPCs are situated in the VXD dock spaces on either side of the IR, with four TPCs located at $z = -\SI{1.3}{m}$, as shown in Figure~\ref{fig:tpcs:geometry}, and the remaining four at $z = +\SI{1.9}{m}$. These locations are critical for neutron monitoring, as during operation the VXD docks surround a part of the beam focusing system where the radius of the incoming beam pipe decreases. This results in particularly high beam particle loss rates, leading to high rates of background showers and neutrons. These neutrons can travel radially outward and penetrate the outer Belle~II detectors from the inside. Verifying that this background component agrees with expectation is therefore important, so that extra neutron shielding can be added if required. The detectors operate using a 70:30 He:CO$_2$ gas mixture.
Detailed descriptions of gas delivery, high voltage operation, and the data acquisition system in Phase~2 can be found elsewhere~\cite{Schueler:2021}. The added detector coverage in $\phi$ allows for a measurement of spatial distributions of fast neutron recoils that compliments the fast neutron-recoil angular distribution measurements previously performed in Phase~1~\cite{Hedges:2021dgz}.
\begin{figure}[phtb]
\begin{center}
\includegraphics[width=0.5\linewidth]{TPCsPhase2}
\caption{Photograph of the VXD dock space at $z = -\SI{1.3}{m}$, an area reserved for cabling and patch panels inside the Belle~II detector, immediately outside the CDC endplate. Four BEAST TPC fast neutron detectors are installed at different azimuthal angles ($\phi$=18$^\circ$, 90$^\circ$, 198$^\circ$, and 270$^\circ$). Two $^{3}$He thermal neutron detectors are are also visible at $\phi$ = 40$^\circ$ and 230$^\circ$. (Coordinates listed are in the global accelerator system; in this figure the positive $x$ axis is to the left and $\theta$ increases clockwise.)}
\label{fig:tpcs:geometry}
\end{center}
\end{figure}
\subsubsection{He-3 Thermal Neutron Detector System}
During Phase~2 the $^3$He system monitored the thermal neutron flux and was capable of triggering an alarm upon detecting dangerous neutron levels. In preparation for Phase~2, the DAQ was upgraded to include a NIM discriminator module to improve counting efficiency, and extensive calibrations were carried out with a thermal neutron source. The detectors themselves are unchanged from Phase~1~\cite{Lewis:2018ayu}; however, they were moved to new positions to account for the presence of the Belle~II detector. Two detectors are mounted in the $+z$ dock space at 40$^{\circ}$ and 230$^{\circ}$ in $\phi$, as seen in Figure~\ref{fig:tpcs:geometry}, with a similar setup used in the $-z$ dock space with detectors at 35$^\circ$ and 215$^\circ$ in $\phi$.
\subsubsection{VXD radiation-monitor and beam-abort system}
\label{subsubsec:diamond_system}
A diamond-based detector system was designed to monitor the radiation dose rates in the interaction region, and to protect the VXD by triggering beam-abort requests in case of excessive beam losses, with a response time comparable to the revolution period of the accelerator beams.
For Phase~2, eight detectors were installed on the final version of the beam pipe, as shown in Figure~\ref{fig:diamonds_location}, forming a subset of the planned configuration of 28 diamond detectors for the VXD radiation monitoring and beam abort system of Phase~3. Their task is twofold: to contribute to the monitoring of beam losses, and to validate and tune the beam-abort function before the installation of the complete VXD in Phase~3.
After the initial dose-rate measurements at low beam currents, we performed validation tests of the abort thresholds and of the abort-signal exchanges with SuperKEKB in controlled conditions. Of the eight sensors, we devote four diamond detectors to monitoring, with the corresponding Diamond Control Unit (DCU) set to the lowest (most sensitive) current measurement range, and the remaining four detectors to the abort function, with the highest DCU current measurement range to measure large dose rates from the most severe beam losses.
We set the initial abort threshold at $1$\,rad, integrated in a moving time window of $1$\,ms (average dose rate of $1$\,krad/s in that time interval) chosen to protect the SVD silicon sensors, which could suffer localized damage from higher radiation levels.
\begin{figure}
\begin{center}
\includegraphics[width=.5\linewidth]{diamonds_package.png} \\
\includegraphics[width=\linewidth]{diamonds_BP.jpg} \\
\includegraphics[width=\linewidth]{diamonds_BP_drawing.png}
\caption{(top) Diamond sensor, used for radiation dose monitoring and beam abort triggering, mounted on its printed circuit board, without aluminum cover; (middle and bottom) diamond detectors mounted on the beam pipe: the diamond detectors are highlighted in green in the drawing.}
\label{fig:diamonds_location}
\end{center}
\end{figure}
The active part of each detector is an electronic-grade single-crystal $4.5 \times 4.5 \times 0.5$\,mm$^{3}$ diamond sensor, grown by the chemical vapor deposition technique, with two (Ti + Pt + Au) electrodes deposited on opposite faces. Each sensor is mounted on a small printed circuit board, providing mechanical support, electrical connections and screening, and is completed by a thin aluminum cover, as shown in Figure~\ref{fig:diamonds_location}.
The charge carriers, electrons and holes, produced by ionizing particles in the diamond bulk, drift in the electric field produced by the bias voltage applied to the electrodes, inducing a current proportional to the dose rate in the external circuit. Current-to-dose calibration factors were determined before installation with a pointlike $\beta$ source located at varying distances from the detector. For a measured current of $1$\,nA, the individually calibrated dose rates for the eight diamond sensors are in the $30 - 40$\,mrad/s range~\cite{Bassi:2021dno}.
The eight diamond detectors were controlled by two DCUs. The digital core of a DCU is a Field Programmable Gate Array (FPGA) that receives commands via an Ethernet interface, drives four independent high voltage modules, and accepts input data from an analog module with four input channels, including amplifiers and 16-bit analog-to-digital (ADC) converters with a $50$\,MHz sampling rate.
By adding up the input data from the ADCs, the FPGA provides monitoring data at $10$\,Hz; the Experimental Physics and Industrial Control System (EPICS)~\cite{epics} read-out software applies pedestal subtraction and conversion to dose-rate units, to make these data available; the EPICS data are archived as process variables. One of three measurement ranges could be selected at initialization: $\pm 10$\,nA, $\pm 1$\,\si{\micro}A, and $\pm 1$\,mA. The corresponding sensitivity in $10$\,Hz data was about $0.3$\,pA, $30$\,pA, and $30$\,nA, repectively. The first range could detect dose rates down to a few \si{\micro}rad/s, while the third range could measure maximum dose rates of the order of $10$\,krad/s.
An internal DCU revolving buffer-memory stores intermediate data at $100$\,kHz, with $10$\,\si{\micro}s time-granularity. Rolling sums of these data can be compared with abort thresholds at each $10$\,\si{\micro}s cycle, to provide beam-abort request signals for the HER and LER beams to the SuperKEKB beam-abort system. When activating the abort kicker magnets, SuperKEKB acknowledges the abort by sending back HER/LER abort timing signals. The DCUs use these signals to stop the writing of 100 kHz data into buffer memories; their frozen contents are read into files, which are then used for the analysis of the beam losses corresponding to each abort event.
\subsubsection{Luminosity}
LumiBelle2, a fast monitor based on diamond detectors~\cite{Pang:2019ses}, recorded luminosity measurements. LumiBelle2 detects forward radiative Bhabha events at small ($<$\,\ang{0.1}) scattering angles and provides a measurement of the relative bunch-by-bunch luminosity at 1\,Hz, with precision up to 1\%, and of the relative integrated luminosity at 1\,kHz, with precision up to 0.1\%, depending on the luminosity. In addition, the Belle~II ECL also provides luminosity data.
\subsection{Online Real-time Feedback and Offline Integration}
The decentralized nature of BEAST II and the need to unify the data for end-users, both for real-time monitoring and post-run analysis, required a dedicated infrastructure at both the hardware and software levels. To accommodate the large throughput of data from distributed sources, we use EPICS to distribute data in real-time to accelerator control, experimental operators for immediate feedback, and to a PostgreSQL database for later processing.
Several displays in the acccelerator control room provide digested summary information from BEAST II detectors, Belle~II detectors, and the SuperKEKB accelerator. All observables for real-time monitoring were shared via EPICS and the displays themselves were constructed using Control System Studio \cite{css}.
The online displays provided visual markers to indicate undesirably high background levels for operation, but did not trigger an automatic abort. During the commissioning run, we added Belle~II subdetector information to the display, indicating a relevant quantity related to the observed background in each detector which could be correlated with the values from BEAST II. This real-time monitoring feedback system has since been expanded upon and remains in use in Phase~3.
Post-run (``offline'') data integration refers to the production of a single, common output file containing SuperKEKB accelerator conditions data, BEAST II detector background levels, and relevant quantities from individual Belle~II subdetectors. In order to provide a standard format for all of the information contained in this file, we record data at the 1\,Hz level and integrate over 1\,s intervals. We include all available values from BEAST II, Belle~II, and SuperKEKB, amounting to several hundred branches and thousands of channels. ROOT~\cite{ROOT} ntuples were provided on a day-by-day basis and also as shorter summaries of particularly relevant times, such as dedicated background runs or machine studies.
\section{Background Simulation}
\label{sec:simulation}
\index{Background Simulation}
The beam background simulation methodology used here is the same as in Phase~1~\cite{Lewis:2018ayu}. The main difference is in the improvement of the simulated geometry. Simulation of synchrotron radiation is a separate effort, described in Section~\ref{sec:synchrotron_sim}.
\subsection{Single-beam backgrounds: Coulomb, Bremsstrahlung, Touschek}
Touschek and beam-gas scattering (Coulomb and Bremsstrahlung) result in beam particles that deviate from the nominal orbit. While the initial scattering can occur at any location around the ring, scattered particles tend to stop at specific locations. In particular, they are most frequently stopped by beam collimators or hit the beam pipe inside the QCS magnets nearest the interaction point where the physical aperture is narrowest with respect to the beam size.
Showers generated inside the QCS can result in secondary particles reaching Belle~II. In order to simulate these types of background accurately, a large number of components must be modeled properly: the initial scattering probability, beam optics, collimators, beam pipe shape inside the QCS, shielding material, and the Belle~II detector. We simulate the scattering, the optics, and the collimators in the Strategic Accelerator Design (SAD) beam particle tracking code~\cite{SADHP}.
SAD includes simulation of Touschek, beam-gas scattering (both Coulomb and Bremsstrahlung processes), while collimator tip scattering and injection background simulations are not implemented. Subsequent to the work presented here, SAD was embedded in a larger software framework that enables tip scattering simulation \cite{natochii2021improved}, but Phase~2 simulations with tip scattering do not exist.
The simulation gives the beam particle loss rates in the whole ring and in the interaction region, defined as $ \pm 4$\,m from the IP, for each background component. Losses in the interaction region are used as a proxy for Belle~II background rates in collimator optimization simulations, where full Geant4~\cite{AGOSTINELLI2003250} simulations would be probitive due to CPU requirements.
We consider a safe limit for background in the interaction region (IR) to be 100\,MHz including both HER and LER contributions.
For more precise Belle~II background simulations, Geant4 is used in addition to SAD. The beam pipe shape is implemented both in SAD and Geant4. If a beam particle crosses the beam pipe envelope in SAD, it is passed to Geant4 and the subsequent showering and Belle~II digitization is simulated with the core Belle~II software, basf2~\cite{Kuhr_2018,the_belle_ii_collaboration_2021_5574116}.
\subsection{Luminosity Backgrounds}
Undesirable backgrounds in the Belle~II detectors due to colliding beams are substantially easier to simulate.
Such backgrounds are produced at the IP and for that reason are largely independent of beam optics and collimator settings.
A detailed tracking of off-orbit particles around the ring is not required in this case.
We simulate luminosity backgrounds in the same way as Belle~II physics, with designated event generators followed by Geant4.
\subsection{Synchrotron Radiation}
\label{sec:synchrotron_sim}
The generation of primary charged particles and simulation of {SR} photon emission during propagation through the magnetic field requires a very large amount of CPU time to get a complete picture of the {SR} background in detectors. For this reason, we simulate SR photons using vertex and momentum information from the primary particle simulation, which performs tracking of the beam particles through the beamline.
We perform both primary and final simulations with Geant4 within the basf2 framework.
A detailed description of {SR} background generation is described in Reference~\cite{Lewis:2018ayu}.
For clarity, we present {SR} simulation results together with the measurements in Section~\ref{sec:sr_results}.
\section{Collimator Study}
\label{sec:collimator_study}
\index{Collimator Study}
SuperKEKB has a number of horizontal and vertical collimators distributed along the beamline as shown in Figure~\ref{fig:coll_early_phase3}.
As noted in Section \ref{section_introduction}, Touschek and beam-gas backgrounds are sensitive to collimation, and thus adjustment of collimator apertures can significantly reduce backgrounds.
During beam operation, collimators are widened or narrowed to remove the off-orbit beam particles away from the beam core, particularly in the IR. However, even collimators far upstream of the IP may have a significant effect on beam behavior and beam-induced backgrounds in Belle~II. To account for these effects and simultaneously determine the optimal collimator configuration during operation, we performed a number of dedicated collimator studies.
SuperKEKB collimators are implemented in the SAD simulation. In order to optimize the collimator aperture and to have a comparison between operations and simulation, a collimator simulation study was carried out prior to Phase~2 operation.
\subsection{Collimator Optimization Studies During Operations}
\begin{figure}[h]
\begin{center}
\includegraphics[width=.4\textwidth]{collimator_map_early_phase3}
\caption{Map of collimators in SuperKEKB. Collimators introduced before Phase~2 are shown in orange, and those installed between Phase~2 and Phase~3 are shown in green.}
\label{fig:coll_early_phase3}
\end{center}
\end{figure}
In keeping with the objective of determining the optimal opening width (aperture) for each collimator along the SuperKEKB beamline, collimators were systematically opened or closed in succession, after which the beam backgrounds and lifetimes were observed to determine the effect of the change.
These studies were carried out with single beams (i.e. HER or LER only), using the following procedure:
\begin{enumerate}
\item Collimators had an initial (non-optimized) setting determined during initial operations in order to protect the Belle~II detector from beam backgrounds.
\item Immediately before collimator studies, all collimators of the studied beam were slightly opened from their nominal aperture setting, so that the physical aperture was slightly larger than the narrowest point in the beam line, corresponding to the QCS components.
\item Starting from the collimator closest to the beam injection system and going toward the interaction point (IP), each collimator was closed in fixed steps. After several minutes, a top up injection was performed in order to verify the effect on injection background. After injection was completed, storage background was observed for a few minutes. This procedure was iterated for the same collimator until beam lifetime worsened with no significant improvement on the background.
\item The same procedure was applied to all collimators, until an initial optimal setting was found.
\end{enumerate}
A second goal of the study, together with the reduction of beam background in the interaction region, was to compare the optimal collimator setting found during collimator studies with that obtained in simulation. The comparison gives important indications of the accuracy of the simulation and possible improvements.
\subsection{SAD Simulation Studies}
The collimator study using SAD was carried out with the following procedure:
\begin{enumerate}
\item The starting point is the simulation with all collimators of the lattice fully open.
\item Each collimator is individually closed in steps. For each step a short version of the simulation (to save storage and computing resources) is run, to see the impact of the new collimator width on IR background and beam lifetime. The optimal width for each collimator is found when the increase in the total ring loss rate is greater than the decrease in the IR loss rate.
\item A full simulation is run with all collimators set at the optimal aperture found in the previous step, to evaluate the full impact of all collimators closed together at their optimal aperture.
\item The last collimator setting obtained is used as the starting point for a second round of the same procedure. Each collimator is individually closed further, and a short version of the simulation is used to evaluate the impact of this modification.
\item The optimization procedure concludes if no further improvement on the IR background is possible without affecting beam lifetime. If the IR background is still higher than the limit of 100\,MHz, the collimator settings are modified to reduce the IR background, even if this affects beam lifetime.
\end{enumerate}
\subsection{MC Optimization for Early Phase~3 and Comparison Against Phase~2}
\label{sec:mc_op}
For operation in early Phase~3, more collimators will be added to reduce beam backgrounds further. We perform another collimator study using SAD with accelerator parameters and collimators planned for early Phase~3. In order to directly compare simulations of Phase~2 and early Phase~3, which involve different beam parameters, the Phase~2 simulation results were scaled to the early Phase~3 parameters using Eq.~1.
As shown in Table \ref{tab:losses_table}, a comparison of simulations shows that if the optimal collimator setting found in simulation can be reproduced during early Phase~3 operations, with the installation of new collimators, the following IR background reduction factors can be achieved: LER backgrounds can be reduced by a factor of 4.4, from 346\,MHz to 78.2\,MHz, and HER backgrounds can be reduced by a factor of 3.7, from 34\,MHz to 9.3\,MHz.
\begin{table*}
\begin{center}
\begin{tabular} {| c | c | c | c | c |}
\hline
& \multicolumn{2}{|c|}{Late Phase~2} & \multicolumn{2}{|c|}{Early Phase~3}\\
\cline{2-5}
& LER & HER & LER & HER \\
\hline
IR losses - Coulomb (MHz) & 186 & 1.2 & 28.6 & 0.4\\
IR losses - Touschek (MHz) & 160 & 32.8 & 49.6 & 8.9\\
\hline
Ring losses - Coulomb (MHz) & 6116 & 1532 & 8211 & 2022\\
Ring losses - Touschek (MHz) & 75944 & 14568 & 83351 & 15127\\
\hline
Lifetime (s) & 1018 & 3589 & 815 & 3564\\
\hline
\end{tabular}
\end{center}
\caption{ Simulated beam particle loss rates for Touschek and Coulomb background components in SuperKEKB Phase~2 and Phase~3.}
\label{tab:losses_table}
\end{table*}
\section{Injection BG Study and Mitigation}
\label{sec:injection_study}
\index{Injection BG Study and Mitigation}
\subsection{Study Description}
On July 13th, 2018, a dedicated injection study using the high energy ring was carried out. Various parameter such as injection phase and beam energy of the injected bunches or septum angle and kicker height at the linear accelerator collider interface and more can be varied in order to change the particle loss induced by particle injections. A fixed injection pattern filling the high energy ring to a maximum of $200\,\mathrm{mA}$ in two steps at a fixed rate of $12\,\mathrm{Hz}$ with the lowest possible injection bunch current enables comparability among the parameter modifications. In the first step, the high energy ring is filled from $0\,\mathrm{mA}$ to $100\,\mathrm{mA}$ beam current. Injections after a current drop of $1\,\%$ keep the beam current stable over a period of around $5\,\mathrm{min}$ before the collider is filled up to $200\,\mathrm{mV}$. For the time of the study, the low energy ring was kept at zero beam current.
Reference data taken before the study and with default parameters ensure comparability.
\par
Reference data with default injection parameters was taken prior to the study. These runs are referred to as \textit{ref-inj}. Data taken during injections is indicated with \textit{inj} and non injection data is referred to as \textit{non-inj}.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.45\textwidth]{time_in_turn_herInj_fnorm=1000}
\caption{Calculated $Z_{eff}$ values for the final month of Phase~2 running.}
\label{fig:time_in_turn_herInj}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.45\textwidth]{time_in_turn_nonInj_fnorm=1000}
\caption{Calculated $Z_{eff}$ values for the final month of Phase~2 running.}
\label{fig:time_in_turn_nonInj}
\end{figure}
\subsection{Synchrotron Radiation}
\label{sec:synchrotron}
\index{Synchrotron Radiation}
\subsection{Track and Vertex Distributions}
\label{sec:trk_vtx_distributions}
\index{Track and Vertex Distributions}
\def\belletwo {Belle~II}
In addition to the reconstruction of physics events from collisions, the Belle~II tracking and vertexing software can be used to study decay vertices from beam backgrounds and $\gamma\to e^{+}e^{-}$ conversions in material away from the IP. Pairs of tracks identified by the inner tracker (PXD, SVD) and drift chamber (CDC) are assigned a pion mass hypothesis and vertexed with the RAVE algorithm \cite{rave}. The vertex positions were then examined in colliding and single beam runs to determine the location and intensity of beam particle losses and secondary interactions.
Figure \ref{fig:zvsx} shows the $(x,z)$ location of reconstructed vertices for Phase~2 physics runs, where $z$ refers to the direction along the Belle~II solenoid axis. To aid in visualization, a minimum intensity threshold has been applied and a diagram of the beamline overlaid. Reconstructed vertices are concentrated at the IP as expected from $e^{+}e^{-}$ collisions, as well as at several locations along the HER and LER beam axes. Cross-sectional $(x,y)$ diagrams taken in steps along the $z$ axis illustrate the structure of these backgrounds. Figure \ref{fig:xvsy} provides a comparative example of single HER beam background data from $-30 < z < -25$\,cm recorded under two different emittance scenarios. These studies were used to detect and investigate the aforementioned ``scraping beam'' backgrounds due to non-optimal beam steering, and to provide information for mitigation strategies such as further beam tuning and future implementation of additional shielding.
\begin{figure*}[htp]
\begin{center}
\includegraphics[width=0.98\textwidth]{overlay_suppressed.png}
\end{center}
\caption{(color online) Reconstructed vertices of charged tracks in Phase~2, plotted in the $xz$ plane. Clustering of events indicates the location and intensity of background beam-wall interactions.}
\label{fig:zvsx}
\end{figure*}
\begin{figure}[htp]
\begin{center}
\includegraphics[width=0.48\textwidth]{her0_0.png}
\includegraphics[width=0.48\textwidth]{her0_2.png}
\end{center}
\caption{(color online) Reconstructed $(x,y)$ positions of charged track vertices for HER single beam runs with differing emittance. The contour shows the envelope of the incoming HER beam. A clear change in the distribution of beam-wall interactions can be seen with the varying accelerator conditions.}
\label{fig:xvsy}
\end{figure}
\section{Belle~II Operation Experience}
\label{sec:operational_experience}
\index{Belle~II Operation Experience}
Several major changes were implemented after the conclusion of Phase~1. While the most notable was the rolling-in of Belle~II, other changes included installation of QCS magnets around the IP, implementation of the new positron damping ring and installation of new collimators in the MR.
These changes resulted in a running environment substantially different from Phase~1. The presence of sensitive Belle~II detectors around the IP necessitated real-time measurement of background levels, to allow accelerator operators to adjust beam parameters to prevent excessive radiation doses and possible damage or detector performance degradation.
Much of the Phase~2 beam time was dedicated to SuperKEKB accelerator machine tuning and studies as the accelerator team worked to reduce the vertical and horizontal beta functions at the IP, $\beta^{*}_y$ and $\beta^{*}_x$, and to improve accelerator performance, and later, to increase luminosity.
\begin{figure}
\begin{center}
\includegraphics[width=0.5\textwidth]{sigma_vs_date_Phase2}
\caption{Measured vertical beam size $\sigma_y$ at the interaction point during Phase~2 operation (2018), measured with the LumiBelle2 fast luminosity monitor~\citep{Pang:2019ses}.}
\label{fig:sigma_v_time}
\end{center}
\end{figure}
\subsection{Vacuum Baking}
A significant fraction of the SuperKEKB running time was devoted to vacuum baking to accelerate the process of outgassing from the beam pipe. During this time, beam was circulated without collisions in order to reduce the amount of residual contamination released during later data-taking runs. Belle~II was turned off during vacuum baking periods, but BEAST II always remained on.
In particular, the LER beam pipe was replaced between Phases 1 and 2 and contained considerably more residual gas than the older HER, consistently resulting in higher beam pressure levels and therefore larger beam-gas backgrounds. The dynamic pressure (slope of pressure versus current) in both the LER and HER are shown in Figure~\ref{fig:dpdi} and an example of the effect on detector backgrounds is presented in Figure~\ref{fig:baking}.
\begin{figure}
\includegraphics[width=0.45\textwidth]{dpdi_phase2}
\caption{Dynamic pressure in the LER and HER during Phase~2 running (2018). The LER shows a steady drop in gas pressure, while the well-conditioned HER has an order of magnitude lower dynamic pressure.}
\label{fig:dpdi}
\end{figure}
\begin{figure}
\includegraphics[width=.43\textwidth]{LER_baking-1}
\caption{(color online) LER backgrounds versus beam current, measured by the TOP during April and May 2018 demonstrating the effect of vacuum baking on background rates. From top to bottom, the lines correspond to beam doses of approximately 806\,A$\cdot$h, 829\,A$\cdot$h, and 905\,A$\cdot$h.}
\label{fig:baking}
\end{figure}
\subsection{First Collisions and Luminosity}
The addition of the QCS magnets and final focusing system allowed SuperKEKB to focus and collide beams. In the early hours of April 26${\mathrm{th}}$, 2018, slightly more than one month after beams first started circulating, Belle~II recorded its first electron-positron collisions. As Phase~2 progressed, the luminosity delivered to Belle~II steadily increased as a result of shrinking beam sizes and increased currents. By the end of Phase~2, SuperKEKB delivered a peak luminosity of $5.6 \times 10^{33} \text{cm}^{-2} \cdot \text{s}^{-1}$ and Belle~II had collected a total integrated luminosity of $472\,\mathrm{pb}^{-1}$.
\subsection{Creation of the BCG Group}
To prevent excessive radiation dose on the Belle~II subdetectors, close communication with the SuperKEKB accelerator team and Belle~II detector operators is necessary. To meet this need, members of Belle~II and SuperKEKB together created a new Belle~II Commissioning Group (BCG). BCG members served as liaisons between accelerator and detector operators. The introduction of real-time background displays as well as the presence of a BCG operator in the accelerator control room meant feedback could be immediately exchanged between accelerator and detector operators to ensure smooth operation and favorable beam conditions for Belle~II data collection.
\subsection{Dedicated Background Studies}
The BCG also designed and carried out dedicated background studies during the course of Phase~2. In general, these studies measured the background composition and effect of a particular change in beam parameters on the background rates in both BEAST II and Belle~II detectors. Examples of study parameters include beam size and emittance, beam steering angle, horizontal and vertical collimator aperture widths, and interaction vertex position.
As beam parameters were varied, the effect on background rates became immediately visible in the online displays, allowing for determination of optimal running conditions.
\subsection{Magnet Quenches}
A consistent issue during operation of SuperKEKB in Phase~2 was quenches occurring in the superconducting steering magnets of the final focus. During the period from April 1st to May 17th, 2018, a total of 23 magnet quenches occurred, each of which was accompanied by a beam abort and followed by an investigation. Quenches occurred in numerous areas around both the LER and HER.
We identify causes from a variety of sources, including injection kicker instabilities, beam parameter changes, machine tests, and other operational issues. Most quenches occurred during beam injections, but 6 of the 23 quenches occurred during beam storage. Because the sudden loss of a magnet affects beam trajectories unpredictably, damage to both sensitive detector components as well as the QCS magnet system near the IP is a serious concern; in addition, the recovery from a quench implies the loss of significant operation time. Therefore, we devoted a great deal of effort to reduce the frequency of such events.
For 19 quench events, the data from the DCU buffer memories with 10\,\si{\micro}s time resolution in 1\,s time windows allowed a detailed study of the beam-loss pattern preceding the quench event in the interaction region. In 15 out of 19 events a lower diamond abort-threshold would have triggered a beam abort, preventing the magnet quench.
As a result of this study, we redefined the abort thresholds: the ``fast" abort condition, aimed at detecting rapidly increasing radiation doses, kept the moving integration time-window of 1\,ms, but lowered the integral threshold value from 1\,rad to 10\,mrad; we also introduced a new ``slow" abort condition developed to avoid the long-term accumulation of slowly increasing radiation doses. For the latter signal, we set the corresponding threshold to a 200\,mrad dose integrated in a 1\,s moving time window. In both cases, a signal above threshold from at least two diamond detectors simultaneously generated an abort trigger.
This new configuration proved to be effective in preventing QCS quenches. Figure~\ref{fig:abort_example} shows two examples of moving sums of dose rates over 1\,ms intervals, reconstructed from the data stored at 100\,kHz, every 10\,\si{\micro}s, in the DCU buffer memories. The \emph{interval numbers} on the abscissa correspond to the 100\,$\times 10^{3}$ moving time windows in a 1\,s time interval. In the top panel, beam losses are small except during the injection at 5\,Hz, causing the five evident peaks. At the last peak, two moving sums exceed the ``fast" threshold and trigger the abort. In the bottom panel, the losses correlated with injection are not sufficient to trigger a ``fast" abort, but an oscillating pattern of slowly increasing losses triggers the ``slow" abort.
\begin{figure}[htb]
\begin{center}
\includegraphics[trim=0 0 0 85, clip, width=.45\textwidth]{abort_examples_1}
\includegraphics[trim=0 0 0 85,clip,width=.45\textwidth]{abort_examples_2}
\caption{Moving sums of radiation doses measured by four diamond sensors, immediately preceding two aborts. The top (bottom) panel shows an example of a fast (slow) abort. Details are provided in the text. The horizontal axis refers to the 100 $\times$ 10$^3$ moving time windows in the 1\,s time interval.}
\label{fig:abort_example}
\end{center}
\end{figure}
\subsection{Background Storms}
On several occasions in Phase~2, we observed unusually high levels of background in multiple Belle~II detectors for short periods of time. These were chiefly evident in the PXD and SVD, but also appeared in detectors farther from the IP such as the TOP.
\begin{figure}[h]
\begin{center}
\includegraphics[width=.45\textwidth]{storm_new_version}
\caption{Example of a background storm event recorded in the PXD during Belle~II operation. A sudden rise in detector activity an order of magnitude above base levels is briefly visible before returning to normal.}
\label{fig:bg_storm}
\end{center}
\end{figure}
\begin{figure}[h]
\includegraphics[trim=0 0 0 100,clip,width=.50\textwidth]{storm_1110_new}
\caption{(color online) Example of ionization observed in a TPC fast neutron detector during a background storm event, integrated over a period of 0.056\,s. The colored tracks and clusters represent individual ionization events, mostly from neutral particles. The thicker tracks are nuclear recoils initiated by fast neutrons, the smaller clusters are x-ray conversions. For comparison, during normal conditions we observe neutrons at a rate of order 1\,Hz.}
\label{fig:n_storm}
\end{figure}
Despite efforts to correlate these events, referred to as ``background storms", with accelerator conditions and particularly with magnet quenches, we find no correlation. Background storms occured in both beam storage and injection periods. An example of data recorded during a background storm is shown in Figures \ref{fig:bg_storm} and \ref{fig:n_storm}, as recorded in the PXD and TPC systems, respectively. In the latter case, we estimate that the neutron flux at the TPCs was as high as $6 \times 10^{5}$\,Hz\,cm$^{-2}$, approximately three orders of magnitude above normal levels.
\subsection{Beam Steering Corrections}
During the course of Phase~2, a persistent background contribution incompatible with the model discussed later in Section \ref{sec:heuristic} appeared in the LER. Hypothesized origins of this background include unexpectedly long horizontal beam tails, beam scraping on the IR beam pipe walls and the resulting showers, and collimator effects.
Attempts to lessen the effects of beam scraping backgrounds included beam orbit and angle tuning, along with adjustment of the collision vertex position. We determined that the HER angle at the IP was sub-optimal, and adjustment led to rapid improvements in beam conditions.
In addition, analysis reveals that the longitudinal position of the collision vertex was initially several mm away from the nominal position, causing unacceptably large numbers of tracks with origins away from the intended IP. Using iterative LER steering corrections, the real collision vertex was brought into line with the nominal position.
\subsection{Extrapolation to Early Phase~3}
\label{sec:extrapolation}
\index{Extrapolation to Early Phase~3}
During the course of Belle~II physics data taking, SuperKEKB plans to both increase beam currents and continue squeezing the beams at the IP, leading to a progressively more challenging background environment. Because we expect conditions at the target luminosity to be significantly more difficult than those in the early stages of data taking, it is most reasonable to mainly consider conditions in the first years of Phase~3, before conditions diverge significantly.
The most pressing question to answer with Phase~2 data is the safety of the sensitive VXD. We use measurements taken during Phase~2 to predict the expected level of background incident on the area around the IP, thus providing a meaningful measure of the risk for the PXD and SVD. In the event that measured and predicted levels are too high, it would be unsafe to install the VXD and physics data taking with the full set of Belle~II subdetectors would not be possible.
The reduction of LER backgrounds described in Section~\ref{sec:mc_op} is critical for keeping background rates at safe levels. Figure~\ref{fig:svd_unscaled} shows the predicted occupancy levels in the SVD, extrapolated to early Phase~3 and adjusted for updated beam optics. Absent optimization of collimators and the introduction of a vertical collimator in the upstream D06 section of the LER, these rates would likely exceed the safe operating limit of the SVD. With the recommended changes, however, we predict that occupancies will remain at the level of approximately 1\%. Likewise, predicted PXD occupancies are expected to remain under 1\%, as shown in Figure~\ref{fig:pxd_scaled}. The performance of these detectors, and of trackfinding, suffers with occupancies above 3\%. Hence, the overall predicted background rates were determined to be safe.
\begin{figure*}[htb]
\begin{center}
\includegraphics[width=0.423\textwidth]{SVD_eph3MC_raw}\,
\includegraphics[width=.4\textwidth]{SVD_eph3MC_x2018JuneDataMC}
\caption{(color online) Predicted SVD layer occupancy in early Phase~3. Left: Raw Phase~3 simulation. Right: Early Phase~3 simulation rescaled by Phase~2 data/MC ratios.}
\label{fig:svd_unscaled}
\end{center}
\end{figure*}
\begin{figure*}[h!]
\begin{center}
\includegraphics[height=.45\textwidth]{PXD_extrapolation_combined}
\caption{(color online) Predicted PXD layer occupancy in early Phase~3. Left: Raw Phase~3 simulation. Right: Early Phase~3 simulation rescaled by Phase~2 data/MC ratios.}
\label{fig:pxd_scaled}
\end{center}
\end{figure*}
\begin{figure}[h]
\begin{center}
\includegraphics[trim=05 0 55 45, clip,width=0.45\textwidth]{top_updated}
\caption{Predicted TOP detector photomultiplier hit rates for beam-gas and Touschek backgrounds, versus TOP slot position, at full design luminosity. Belle~II TOP slots are distributed azimuthally (in $\phi$) around the SuperKEKB beam axis. The goal is to reduce the sum of these background components further, to about 2.0 MHz per PMT. See text for further discussion.}
\label{fig:top_phi}
\end{center}
\end{figure}
\subsection{Extrapolation to Design Luminosity}
Finally, we extrapolate backgrounds to the full SuperKEKB design luminosity of $8 \times 10^{35}\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1}$. This is a more challenging scenario than the target luminosity of \SI{6.5e35}{cm^{-2}s^{-1}} discussed earlier, but we make the choice because simulations and collimation strategies are more mature for the design scenario. We first simulate the machine at design beam currents and optics, using a simulation-optimized collimator setup specific to those optics. Then, we conservatively re-scale the Belle~II rates of each simulated single-beam background component by the corresponding data/MC factors measured in Phase 2. As an example, we show the predicted background composition for the TOP detector in Figure~\ref{fig:top_phi}. TOP is considered particularly vulnerable to beam backgrounds due to the finite lifetime of the photomultipliers (PMTs) used, which degrade as photocathode charge is accumulated. It is therefore critical to monitor the TOP background composition and use targeted mitigation strategies such as collimation to minimize TOP backgrounds and thereby maximally extend PMT lifetimes. At design luminosity, TOP PMT rates are expected to reach order 8\,MHz per PMT from beam collisions alone; because these events arise from beam interactions at the IP, they are affected by neither collimation nor vacuum scrubbing and hence are to first order irreducible. Background events from the single beam processes shown in Figure~\ref{fig:top_phi} contribute further to the overall PMT rates.
The Belle~II goal is to further reduce these single-beam rates to a total of order 2\,MHz by utilizing vacuum scrubbing, shielding, and collimation. While HER Touschek scattering appears to be the most critical component based on Phase 2 measurements, as will be discussed below, recent improvements in the simulation of SuperKEKB-type collimators have reduced the systematic error on the magnitude of the Touschek background forecast. The dominant background and main mitigation target at the time of writing is therefore the second-largest component in Figure~\ref{fig:top_phi}, LER beam-gas. Reducing that background is one of the main SuperKEKB challenges on the road towards design luminosity in the coming years.
Recently, the pressure in the LER has reduced substantially due to vacuum scrubbing from baseline machine operation. As a result, the LER beam-gas background may be as much as one order of magnitude lower than that shown in Figure~\ref{fig:top_phi} by the time SuperKEKB reaches its target luminosity.
\section{Extrapolations and Recommendations}
\label{sec:summary_recommendations}
\index{Summary and Recommendations}
During the course of Phase~2, we observe total beam background levels larger than simulation by factors from 1 - 100, as presented in Section~\ref{sec:ratio_summary}. By refining the simulated description of the IR geometry, we have improved the agreement between simulation and measurement, as seen by comparing the top and bottom panels of Figures~\ref{fig:tous_summary} and~\ref{fig:bg_summary}.
The remaining discrepancy in beam-gas data and simulation is unsurprising, as detailed gas composition and pressure distributions are not accounted for in the simulation. In contrast, we expect better agreement for Touschek backgrounds. Instead, we observe a discrepancy of up to a factor of 100, particlularly in the HER, consistent with the results from Phase~1. After the conclusion of this work, we found that this was the result of the physical collimator model used in the beam particle tracking code, which had been accurate for the reused KEKB collimators but did not fully describe the newly installed SuperKEKB-type collimators. This has been resolved for future studies and a detailed description is available in Reference~\cite{natochii2021improved}.
We also observed unexpected backgrounds attributed to beam scraping on beam pipe material, which we reduced largely by beam-orbit tuning and collimator optimization.
Luminosity backgrounds provided a negligible contribution to the overall background level, given the low luminosities recorded during commissioning and the high levels of other backgrounds.
While ratios of data to simulation are useful in assessing simulation accuracy, we also utilize them to correct the simulation for the purposes of extrapolating backgrounds. By doing so, predictions remain robust even with large discrepancies in agreement, as observed in Phase~2.
\input{11_Extrapolation_to_Early_Phase_3/Ch_11_Extrapolation_to_Early_Phase_3}
\subsection{Summary and Recommendations}
Ultimately, based on the background measurements in Phase~2 and extrapolations forward to Phase~3 conditions, we made the recommendation to proceed as planned with the installation of the VXD and start of physics data taking runs in Phase~3.
Furthermore, we also recommend that a new collimator be installed in SuperKEKB before the start of Phase~3 to reduce the dominant LER beam-gas and Touschek backgrounds. Based on the studies described in Section~\ref{sec:collimator_study}, we determine that the most beneficial place for a new LER vertical collimator is in Section D06, near the MR injector. Following completion of Phase~2, both recommendations were implemented during the shutdown period between phases.
\section{Conclusions}
\label{sec:Conclusions}
\index{Conclusions}
The second commissioning phase of SuperKEKB saw major advances in accelerator operation, including beam luminosities of $5.6 \times 10^{33}$\,cm$^{-1}$\,s$^{-1}$ and first beam collisions, and corresponding increases in beam-induced backgrounds. To meet the demands of this more challenging environment, we successfully employed both Belle~II and dedicated BEAST II detectors to measure and mitigate backgrounds due to Touschek, beam-gas, injection, and synchrotron radiation. We carried out detailed measurements of neutron fluxes and dose rates. Finally, we performed searches for, but did not conclusively detect, luminosity backgrounds.
BEAST II successfully demonstrated numerous new technologies: FANGS, PLUME, and CLAWS detectors for measurements of incident particles and doses near the IP, directional neutron recoil monitors, and a diamond-based monitoring system that was developed to carefully track radiation doses and trigger beam aborts in the event of unacceptably high rates. Many of these systems are now used routinely in Phase~3.
The diamond sensor beam-abort system in particular has been improved to provide faster response times to high background levels, with criteria updated to include ``fast'' and ``slow'' thresholds. These improvements have proven to be effective in preventing QCS quenches and subsequent high doses of radiation on the sensitive vertex detectors during Phase~3 data taking.
We updated the Geant4 model of the Belle~II interaction area to more closely match reality, and as a result the ratio of observed to predicted single-beam background rates improved by factors of 1.12 - 14.98, indicating improved predictive ability for future conditions. Extrapolations of measurements to Phase~3 running conditions predict background levels to be within acceptable limits provided beam collimation is improved. Based on these results, we recommended proceeding with the installation of the sensitive vertex detectors along with an additional LER collimator.
As Belle~II continues in its data collection phase and the background conditions become increasingly challenging, informative studies such as those described here will continue to enable targeted mitigation for the duration of the experiment.
\section{Acknowledgments}
\label{sec:Ack}
We acknowledge support from the U.S. Department of Energy (DOE) via
Award Numbers DE-SC0007852, DE-SC0010504, DE-AC02-05CH11231, via the
U.S. Belle~II Project administered by Pacific Northwest National
Laboratory (DE-AC05-76RL01830), and via U.S. Belle~II Operations
administered by Pacific Northwest National Laboratory and Brookhaven
National Laboratory (DE-SC0012704). Pacific Northwest National Laboratory is managed and operated by the Battelle Memorial Institute. We acknowledge the financial support by the Federal Ministry of Education and Research and by the DFG Excellence Cluster “Origin and Structure of the Universe” of Germany. This work was partially supported by the European Union’s Horizon 2020 Research and Innovation programme under Grant Agreement no. 654168 and the Horizon 2020 Marie Sklodowska-Curi RISE project JENNIFER grant agreement No. 644294. The Strasbourg Group acknowledges support from L'Institut National de Physique Nucl\'eaire et de Physique des Particules (IN2P3) du CNRS (France) and Investissements d’Avenir and Université de Strasbourg (IdEx grants W15RPE12 and W17RPD30). We acknowledge the support of Grant CIDEGENT/2018/020 of Generalitat Valenciana (Spain).
\subsection{Beam-Gas and Touschek Study}
\label{sec:bg_tous_study}
\index{Beam-Gas and Touschek Study}
In the final two months of Phase~2, we carried out four major beam background studies. To avoid contamination from backgrounds arising from beam-beam interactions, each study was carried out with only a single beam (HER or LER) active. These studies are summarized in Table \ref{table_bg_studies}.
\begin{centering}
\begin{table}[h]
\begin{tabular}{| l | c | c | c | }
\hline
Date & Study Type & $\beta_y^*$ (mm) & $\beta_x^*$ (mm)\\ \hline
June 11 & HER Single Beam & 4 & 100\\
June 12 & LER Single Beam & 4 & 100\\
July 16 & HER/LER Beam Size & 3 & 100 \\ \hline
\end{tabular}
\caption{Phase~2 single-beam background studies. The July 16 study was two individual single-beam studies carried out sequentially.}
\label{table_bg_studies}
\end{table}%
\end{centering}
We use only data taken during beam storage periods in our analysis to avoid the higher background levels present during beam injections. \autoref{fig:storage} shows a snapshot of relevant accelerator data during a background study, with selected data shown in shaded areas. For each of the four studies, we fit the combined background model to experimental data in an attempt to determine the contribution of each source to the overall background level. Fits were performed individually for each BEAST II and Belle~II detector.
\begin{figure*}[h]
\begin{center}
\includegraphics[width=\textwidth]{storage_fills}
\caption{(color online) Accelerator data recorded during the LER study performed on June 12th. Data within the shaded regions correspond to beam storage fills that were included in background analyses. Vertical axis labels are colored to match the color of their corresponding plotted observables. Beam size values greater than \SI{100}{\micro\meter} are omitted.. All data are from the LER. }
\label{fig:storage}
\end{center}
\end{figure*}
\subsubsection{Measured Gas Composition}
We measure gas pressures around the ring with 300 monitors positioned in individual ring sections, as described in Section \ref{sec:Beam_Gas_Pressure}. To provide a more realistic summary of the gas conditions in the beamline, an empirical calculation to account for contributions from individual gas species is carried out using data from three RGAs instrumented along the SuperKEKB beamline. Each RGA is a mass spectrometer providing partial pressures of m/$Z$ values from 1 to 50. In principle, each m/$Z$ represents a single species of molecule in the beam-gas mix, although certain species overlap.
Using RGA data, we calculate a single effective $Z$ value for all species in the gas, denoted $Z_{\text{eff}}$, which is used when fitting the data, as described in Section~\ref{sec:heuristic}. For the period considered in Phase~2, the calculated $Z_{\text{eff}}$ consistently falls between 2 and 2.5.
In contrast, the simulation assumes a time-independent, uniform value of $Z_{\text{eff}} = 7$ around the ring.
\subsubsection{Analysis Methodology}
\label{sec:heuristic}
Based on the relationships between background types and detector rates described in Section \ref{sec:types}, we posit a combined background model to describe the contributions of Touschek and beam-gas events to the observed background rate $R$:
\begin{center}
\begin{equation}
R = B \cdot I \cdot P Z_{\text{eff}}^2 + T \cdot \frac{I^2}{\sigma_y n_\text{b}}
\label{eq:heuristic}
\end{equation}
\end{center}
where $B$ and $T$ represent the beam-gas and Touschek rates. It is convenient to rearrange Eq.~\ref{eq:heuristic} to allow for a graphical representation:
\begin{equation}
\label{eq:heur_linear}
\frac{R}{I \cdot P Z_{\text{eff}}^2} = B + T \cdot \frac{I}{PZ_{\text{eff}}^2\sigma_y n_\text{b}}
\end{equation}
If the left hand side of Eq. \ref{eq:heur_linear} is plotted versus $\frac{I}{P Z_{\text{eff}}^2 \sigma_y n_\text{b}}$, we expect a line with a slope proportional to the Touschek sensitivity, T, and an offset corresponding to the beam-gas sensitivity, B.
\subsubsection{Measurements of Background Composition}
We fit background rates observed in individual detectors to the linear background model thus described by Eq.~\ref{eq:heur_linear}, from which we obtain relative contributions of background sources. An example fit for one of the TPC neutron detectors is shown in Figure \ref{fig:heuristic}. We see clearly how varying the beam size allows us to separate observed neutron rate contributions from beam-gas and Touschek scattering. Figure~\ref{fig:tpc_phi} shows how the resulting measured background composition in each of the eight TPCs (located at four $\phi$ locations on either side of the IR) compares against the background simulation. We see that the spatial distribution of neutrons from beam-gas scattering is remarkably accurate in the simulation, for both rings. In experiment, this background is the leading source of neutrons. The beam-gas normalization is also well predicted for LER. For HER, however, observed beam-gas background exceeds predictions by one order of magnitude.
\subsubsection{Evaluation of Simulation Accuracy}
\label{sec:ratio_summary}
We evaluate the accuracy of our simulation and its consequent predictive ability by comparing predicted detector rates in all BEAST II and Belle~II detectors to those measured during dedicated background studies. Touschek- and beam-gas induced rates are calculated separately as described in Section~\ref{sec:heuristic} and divided by the simulated prediction to produce a ratio for each detector channel. In the ideal case, a ratio of 1 indicates agreement between measurement and simulation. For the detectors with high hit rates, the fitting procedure and comparison can be performed even for individual channels. This provides highly granular, position-dependent information on the simulation accuracy for individual background components, as shown in Fig.~\ref{fig:plume_ratio} for PLUME.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.45\textwidth]{combined_heuristic}
\caption{Example of using Eq. \ref{eq:heuristic} to separate beam-gas and Touschek contributions to observed background rates, for a TPC during the June 12th LER study. Filled points represent $n_\text{b}=789$, and unfilled points represent $n_\text{b}=1576$. ``ECK" (legend) refers to the setting of the emittance control knob used to tune the vertical beam size. Larger magnitudes of the knob setting value correspond to larger $\sigma_y$. Touschek contributions increase relative to beam-gas contributions with decreasing beam size.}
\label{fig:heuristic}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.5\textwidth]{tpc_summary_v2}
\caption{Summary distributions of nuclear recoil rates observed in all TPCs during the LER (top) and HER (bottom) studies with $\beta^*_y=4$\,mm optics. The TPCs are labeled by their absolute $\phi$ positions in degrees and their relative $z$ positions (BWD and FWD represent $z = -\SI{1.3}{m}$ and $z = +\SI{1.9}{m}$, respectively). Solid lines correspond to experimental data and dashed lines correspond to MC. Experimental rates are extrapolated to the accelerator conditions used in MC using the LER and HER beam-gas and Touschek fit coefficients extracted from combined heuristic fits to measured data. Beam-gas scattering dominates neutron production, and the spatial distribution of neutrons from that process appears to be accurate in the simulation. Touschek contributions not plotted are beneath the lower limits of the plot.}
\label{fig:tpc_phi}
\end{center}
\end{figure}
\begin{figure}[h]
\includegraphics[width=0.45\textwidth]{plume-Tratio-LER-June12}
\includegraphics[width=0.45\textwidth]{plume-BGratio-LER-June12}
\caption{Background data/MC ratios as measured in PLUME during the June 12th LER study. Ratios are shown separate for Touschek (top) and beam-gas (bottom). Both background components are substantially larger than predicted by simulation, with the excess having no strong position dependence.}
\label{fig:plume_ratio}
\end{figure}
Data-to-simulation Monte Carlo (Data/MC) fit results for all BEAST II and Belle~II detectors are summarized in~\autoref{fig:tous_summary} for Touschek backgrounds and~\autoref{fig:bg_summary} for beam-gas results. Values for individual detector channels or physical locations, where applicable, are shown as separate points. The top plot in each figure represents the results of the Data/MC fits using the ``old" simulation, while bottom plots show the same results using MC updated to better model the detector. Data/MC fit results are improved markedly with the new simulation, usually by orders of magnitude. Table~\ref{table:avg_of_avg} combines the individual detector results into a single overall ratio for Touschek and beam-gas backgrounds in the LER and HER. In Section~\ref{sec:summary_recommendations} we use these data to extrapolate to expected beam conditions in Phase~3.
\begin{table*}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
Ring & Background Source & October 2018 Simulation & February 2019 Simulation & October 2018/February 2019 Ratio \\
\hline
\multirow{2}{*}{HER} & Touschek & 127.82 & 113.91 & 1.12 \\
& Beam-gas & 483.50 & 32.28 & 14.98 \\
\hline
\multirow{2}{*}{LER} & Touschek & 1.62 & 0.63 & 2.57 \\
& Beam-gas & 29.39 & 2.79 & 10.53 \\
\hline
\end{tabular}
\caption{Comparison of combined detector data/MC ratios, excluding PLUME. Averages are calculated first by taking the mean of all channels in each BEAST or Belle~II detector, and then combining them into an average of averages.}
\label{table:avg_of_avg}
\end{center}
\end{table*}
\begin{figure}[h!]
\begin{center}
\includegraphics[trim=0 0 60 0,clip,width=0.45\textwidth]{Touschek_June_2018_old} \\
\includegraphics[trim=0 0 60 0, clip,width=0.45\textwidth]{Touschek_June_2018}
\caption{(color online) Ratio of observed to predicted Touschek background rates in all detectors studied with old (top) and new (bottom) simulation. Blue (Red) points represent HER (LER) results. From top to bottom, the detectors are ordered from radially outermost (TOP) to inermost (PXD).}
\label{fig:tous_summary}
\end{center}
\end{figure}
\begin{figure}[h!]
\begin{center}
\includegraphics[trim=0 0 63 0, clip,width=0.45\textwidth]{Beam_Gas_June_2018_old} \\
\includegraphics[trim=0 0 60 0, clip,width=0.45\textwidth]{Beam_Gas_June_2018}
\caption{(color online) Ratio of observed to predicted Touschek background rates in all detectors studied with old (top) and new (bottom) simulation. Blue (Red) points represent HER (LER) results. From top to bottom, the detectors are ordered from radially outermost (TOP) to inermost (PXD).}
\label{fig:bg_summary}
\end{center}
\end{figure}
\section{Experimental Results}
\label{sec:exp_results}
\input{Experimental_Results/Ch_5_Beam_Gas_and_Touschek_Study}
\subsection{Luminosity Backgrounds}
After collisions began in Phase~2, attempts were made to measure the level of luminosity backgrounds generated. However, the amounts of Touschek and beam-gas backgrounds were significantly larger than the luminosity contribution and there is no conclusive observation of luminosity-induced backgrounds.
\input{09_Track_and_Vertex_Distributions/Ch_9_Track_and_Vertex_Distributions}
\subsection{Injection Backgrounds}
\label{sec:injection_backgrounds}
This section presents time-resolved analyses of the injection-induced particle background at the IP which are the basis for the current usage of CLAWS as a beam abort system. First, we present a time resolved hit energy spectrum for each of the two main rings, illustrating the ability to detect irregularities in the beams. Subsequently, we investigate the impact of the injection bunch on neighboring bunches, demonstrating the precise timing capability of the CLAWS system.
\begin{figure}[h!]
\centering
\includegraphics[width=0.45\textwidth]{claws_hit_energy_spectrum}
\caption{(color online) Hit energy spectra in the CLAWS detector. (top) and (bottom) Time resolved hit energy spectrum of the LER and HER, respectively. (middle) Projection of the top and bottom plot with marginalized time in turn variable separated by injection and non-injection times of a revolution. The units of the color bars are equal to the vertical axis of the middle plot. The plot uses all injection data collected by CLAWS between May 25 until the end of Phase 2.}
\label{fig:claws_her_hit}
\end{figure}
\par
CLAWS signals are measured as hits with the unit of MIP, equivalent to the most probable energy deposition of a minimum-ionizing particle. The amplitude of a hit is a proxy for the number of particles crossing one CLAWS sensor simultaneously at the time of the hit. \autoref{fig:claws_her_hit} (top) and (bottom) illustrate the time-resolved hit energy distributions. The \emph{time in turn} variable displayed on the $y$ axis assigns a relative time to each circulating bunch at which it passes by the IP. This pattern repeats every revolution with $T_{rev} = \SI{10.0614}{\micro\second}$, assuming a fixed bunch fill pattern. The starting time is chosen arbitrarily. Therefore, we obtain the time in turn with \mbox{$t_{\text{turn}} = t_{\text{signal}}\,\mathrm{mod}\,T_{\text{rev}}$}. The horizontal excesses around $t_{\text{turn}} = \SI{5.5}{\micro\second}$ contain the signals of the injected bunch passing the IP. The low energy excess following the injection bunches in \autoref{fig:claws_her_hit} (top) and (bottom) are SiPM intrinsic afterpulses. All other hits emerge from stable circulating bunches. The projections of the injection and the average of all non-injection times of one revolution are illustrated in \autoref{fig:claws_her_hit} (middle).
\par
The time dependent hit energy spectra in \autoref{fig:claws_her_hit} (top) and (bottom) demonstrate the confinement of the injection induced backgrounds to a narrow time window during each turn. Highly energetic hits above $10\,\text{MIP}$ emerge almost exclusively from the newly injected bunches. Nonetheless, for each ring a second, much less pronounced, high energy excess is observable. The second excess in the HER at around $t_{\text{turn}} = \SI{2}{\micro\second}$ is the result of a single event that results in hits in multiple detectors. In contrast, the second excess in the LER around $t_{\text{turn}} = \SI{6.25}{\micro\second}$ originates from a period of approximately $5\,\text{h}$ on June 19, 2018, with signals up to $60\,\text{MIP}$ consistently observed \SI{897\pm 51}{\nano\second} after the fifth pass-by of the injection bunch. This suggests a connection of this phenomenon to injections, but it was not possible to identify its origin.
\par
\autoref{fig:claws_her_hit} (middle) demonstrates that HER \emph{(HER Inj)} injections cause a higher rate in the CLAWS sensors compared to injection bunches in the LER \emph{(LER Inj)}. For low hit energies from $1\,\text{MIP}$ to $100\,\text{MIP}$ the HER rate is around \SI{100}{\hertz\per\centi\meter\squared}, a factor of 4 higher than the respective rate from the LER, indicating larger numbers of background particles originating from LER injections. Around $300\,\text{MIP}$ the LER rate becomes higher than the HER.
In general, injections in the LER result in higher hit energies compared to injections into the HER. The slightly higher rates of the non-injection graphs of the LER compared to the HER result from the higher average beam currents in the LER during the time of data taking.
\par
\autoref{fig:claws_her_pre_post_inj_bunches} illustrates the time structure around the injection bunch of dedicated HER injections on July 13, 2018. Each peak originates from a circulating bunch with a peak-to-peak distance equal to the bunch spacing of \SI{12}{\nano\second}. This time-resolved analysis illustrates the increased background level during the passage of the injection bunch only. Adjacent bunches do not show any negative influence of the high background of the injection bunch.
\par
These results demonstrate the capabilities of CLAWS to observe even minimal irregularities in the circulating beams of both main rings. Together with its precise time resolution of \SI{317\pm4}{\pico\second} for hit energies of 10\,MIP, this makes the CLAWS detectors well suited to act as a beam abort system. For physics data taking, we will implement a CLAWS-based abort system capable of issuing a fast abort signal in case of sudden increases of background, providing the signal typically one turn faster than systems relying on the measurement of integrated doses.
\begin{figure}[h!]
\centering
\includegraphics[width=0.45\textwidth]{inj_study_her_pre_post_inj_bunches_mean-of-f15-f50}
\caption{Arrival times of signals with a minimum energy of $2\,\mathrm{MIP}$. Each peak emerges from a circulating bunch. The red lines mark exemplarily the bunch spacing of $12\,\mathrm{ns}$ around the injection bunch. The plot uses HER injection data from July 13, 2018.}
\label{fig:claws_her_pre_post_inj_bunches}
\end{figure}
\par
\input{Experimental_Results/SR_data_measurement}
\input{Experimental_Results/PXD_Dose_Estimation}
\subsection{Synchrotron Radiation in PXD}
\label{sec:sr_results}
In addition to the backgrounds described above, there is evidence for an additional synchrotron radiation background component in the PXD. In Phase~2, the presence of SR background can be inferred from study of cluster energy spectra. In particular, photons originating from SR can be identified as low energy single-pixel clusters with a characteristic spatial distribution along the $z$-direction.
The study of cluster energy spectra in the PXD was performed with physics data using the standard PXD reconstruction algorithm. A PXD hit threshold of either 5 or 7 Analog-to-Digital converter Units was applied in the front-end electronics and the highest charge in the PXD cluster was required to be above 7\,Units. The seed charge cut of 7\,Units translates to an energy threshold of around 5\,keV. These PXD clusters are not required to be matched to a charged track from the SVD or CDC, and they can come from any charged particle or photon depositing enough energy and hitting the PXD during the 20\,\si{\micro}s time window around the trigger. The resulting calibrated energy spectra are shown separately for single and multi-pixel clusters for the PXD inner forward sensor in~\autoref{fig:if_and_of_charge_5613} (top), with the corresponding simulated energy spectrum, normalized to an HER current of 250\,mA and with the same 5\,keV energy threshold (bottom). This simulated energy spectrum can be compared with measuremed data (presented in red) only qualitatively as there are two substantial differences: firstly, this simulation only considers {SR} photons originating from the HER beam, whereas the measured energy spectra contain contributions from both the HER and LER; Secondly, the measured energy spectra contain both photons and a non-negligible contribution from charged particles.
The peak around 10\,keV, which is seen both in simulated and measured energy spectra, can be explained by SR-induced fluorescence photons emitted perpendicular to the $z$ direction from the gold layer on the inner surface of the beam pipe. Such photons are capable of penetrating the beryllium beam pipe. The simulated broad distribution peaking at larger energy of 35\,keV is produced by {SR} photons which penetrate the beam pipe directly. This component is less visible in the measured energy spectra because of the much larger additional component due to charged particles, which peaks around 20\,keV.
The energy spectra can be split into a soft component ($<$10~keV) and a hard component ($>$10~keV). The hard part of the spectrum contains more multi-pixel than single pixel clusters and shows a Landau peak as expected for clusters originating from charged particles. In contrast, the soft part of the spectrum has a narrow energy peak on top of the low energy tail of the Landau distribution. The narrow peak is almost exclusively formed by single pixel clusters.
A Gaussian fit to the single pixel charge, restricted to the 7-10\,keV range, shows that the peak positions range between 8.4-8.6~keV and are nearly identical for all four PXD sensors. This last observation hints at a common origin for the narrow energy peak. The number of single pixel clusters in the energy range 7-10~keV, after subtracting a flat background, is found to be ten times higher in the first PXD layer compared to the second. This supports the hypothesis that the common origin of the soft-photon background is SR, in which the scattered photons are expected to be mostly absorbed in the first PXD layer.
\begin{figure}[htb!]
\centering
\includegraphics[trim=0 0 45 0,clip,width=.9\linewidth]{Figures/note_IF_5613_edit}
\includegraphics[width=0.45\textwidth]{EdepL1S1P2keV.png}
\caption{(color online) Measured (top) and simulated (bottom) energy spectra in the forward innermost sensor of PXD. Measured spectra are shown in blue for all clusters and in red for single pixel clusters.}
\label{fig:if_and_of_charge_5613}
\end{figure}
In addition, the flux of particles onto the PXD is measured as a function of the $z$ position and layer number during dedicated single-beam runs. The spatial profile of the soft photon flux, identified as single pixel clusters with energy $<10$ \,keV, is shown for a LER (HER) single-beam run in the top (bottom) of~\autoref{fig:soft_flux_ler_and_her}.
\begin{figure}[htb!]
\centering
\includegraphics[width=.9\linewidth]{Figures/note_flux_overlay_run_3360_Photons.png}
\includegraphics[width=0.45\textwidth]{mc_her_softphotonfluxsm.png}
\caption{(color online) Measured (top) and simulated (bottom) soft photon flux on first PXD layer (Layer 1) and second PXD layer (Layer 2) against $z$-position for single-beam runs in the HER. The average HER current in the single beam run is 250\,mA, and simulation uses the same value.}
\label{fig:soft_flux_ler_and_her}
\end{figure}
As seen in Figure~\ref{fig:soft_flux_ler_and_her}, The simulated (top) and measured (bottom) spatial profiles are found to be incompatible. Dedicated simulations of low-energy SR photons indicate the presence of an excess of events caused by photons forward scattering off the ingoing copper-coated Tantalum beam pipe and subsequently inducing fluoresence in the gold layer on the central beam pipe. For Phase~3, the impact of this enhanced low-energy component will be significantly reduced by increasing the thickness of the gold layer from 6.6 to 10\,\si{\micro}m.
\subsection{PXD-Based Total Ionizing Dose Estimate}
\label{subsec:pxd_dose_estimation}
We use the PXD system to estimate the total ionizing dose rate in the VXD during Phase~2 operation. Using simulations as reference, we perform an energy calibration of the measured PXD hits and clusters. Using this calibration we calculate the energy deposition inside the PXD, and then the dose per cluster.
\par
Taking into account the 20\,$\si{\micro}$s readout time of the PXD sensors, we calculate a dose rate with 1\,s binning and add it to the aforementioned common output files of the BEAST detectors. This technique assumes that the difference between data taken with and without selecting for physics events is negligible, which is justified as the luminosity during Phase~2 was significantly lower than is expected in the final experiment.
\par
The calculated PXD dose rate was used as a second source for the dose rate in addition to the VXD radiation-monitor and beam-abort system (see Section~\ref{subsubsec:diamond_system}) that primarily provided information about the dose rate during Phase~2 operation.
\par
Analyses show that the ratio of the PXD and diamond dose rates is not constant but depends on the accelerator beam current. While the PXD/diamond ratio was not constant over time, we find a heuristic relation between the two measurements incorporating the beam currents.
\par
In contrast to the diamond system, the PXD did not take data continuously during Phase~2 operation and only covered $\approx~12$\% of the total operation time. To fill in the gaps in the data set, we use the diamond data as well as the beam currents to infer a PXD dose rate based on the heuristic relation between the PXD and diamond data set. A more detailed description of this analysis can be found in Ref.~\cite{SchreeckPhD:2020}.
\par
Using a combination of the measured and calculated PXD dose rates we determine a total dose for all four PXD sensors as a function of time, which can be seen in Figure~\ref{fig:pxd_accum_dose}.
\begin{figure}[htb]
\centering
\includegraphics[width=0.5\textwidth]{Figures/PXD_cum_dose_totalmodule_id}
\caption{Cumulative PXD radiation dose versus time for the four individual PXD sensors. The inner sensors (top) received approximately twice the dose compared with the outer sensors (bottom).}
\label{fig:pxd_accum_dose}
\end{figure}
The DEPFETs (DEpleted P-channel Field Effect Transistors) of the PXD sensors show a threshold voltage shift when exposed to ionizing irradiation.
Figure~\ref{fig:pxd_threshold_voltage_shift} shows the measured threshold shift of two of the four PXD sensors as a function of the calculated PXD dose. In addition, the threshold shift curves recorded at two irradiation campaigns, one with a prototype \cite{RITTER201379} and one with final PXD sensors \cite{Schreeck2020}, are shown. The comparison shows good agreement between the curves.
\begin{figure}[htb]
\centering
\includegraphics[width=0.5\textwidth]{Figures/boxplot_threshold_shift_phase2_inset}
\caption{PXD threshold voltage shifts versus cumulative dose. Two of the PXD Phase~2 sensors (1.1.1 and 2.1.1) are compared with results gathered during two irradiation campaigns (PXD6 and PXD9~\cite{RITTER201379}).}
\label{fig:pxd_threshold_voltage_shift}
\end{figure}
\par
Based on the calculation described above, the inner PXD sensors received a dose of $\approx2$\,kGy, and the outer sensors $\approx1$\,kGy, between March and July 2018. We deem these levels to be acceptable for the installation of the VXD for Phase~3.
\section*{References}
|
2024-02-18T23:40:24.409Z
|
2021-12-30T02:27:13.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14537","language":"en","timestamp":1640831233000,"url":"https:\/\/arxiv.org\/abs\/2112.14537","yymm":"2112"}
|
proofpile-arXiv_000-10018
|
{"provenance":"002.jsonl.gz:10019"}
| null | null |
\section{Introduction}
Let ${\overline{G}}_n$ be given by the product of the binomial coefficients in the $n$-th row of Pascal's triangle,
\begin{equation}\label{eqn:defGn}
{\overline{G}}_n:=\prod_{k=0}^n\binom{n}{k}=\frac{(n!)^{n+1}}{\prod_{k=0}^n(k!)^2}.
\end{equation}
This sequence arises as the reciprocal of the product of all the nonzero unreduced Farey fractions of order $n$.
Its asymptotic growth is easily shown to be
\begin{equation}\label{eqn:logG-asymp}
\log{\overline{G}}_n=\frac{1}{2}n^2-\frac{1}{2}n\log n+O(n),
\end{equation}
using Stirling's formula.
The number ${\overline{G}}_n$ is $n$-smooth (i.e. having no prime factor larger than $n$), and we write its prime factorization as
\begin{equation}\label{eqn:productformula}
{\overline{G}}_n=\prod_pp^{\nu_p({\overline{G}}_n)},
\end{equation}
where $\nu_p(a)$ denotes the (additive) $p$-adic valuation of $a$.
The quantities $\nu_p({\overline{G}}_n)$ are known to equal an expression $ {\overline{\nu}}(n,p)$ defined purely in terms of the base $p$ radix
expansions of the integers up to $n$, given by
\begin{equation}\label{eqn:gnu-p}
\nu_p({\overline{G}}_n)=\frac{2}{p-1}S_p(n)-\frac{n-1}{p-1}d_p(n),
\end{equation}
where $d_p(n)$ is the sum of the base $p$ digits of $n$ and $S_p(n):=\sum_{j=1}^{n-1}d_p(j)$.
(See \cite[Theorem 5.1]{DL:20}.)
The left side of \eqref{eqn:gnu-p} is a nonnegative integer,
while examples show the two terms on the right side are sometimes not integers.
In the paper \cite{DL:20} the first and second authors studied the sizes of partial factorizations of ${\overline{G}}_n$:
\begin{equation}
G(n,x):=\prod_{p\le x}p^{\nu_p({\overline{G}}_n)},
\end{equation}
where $1 \le x \le n$, deriving estimates for the size of $\log G(n,x)$.
Theorem 1 of \cite{DL:20} obtained for
$0 < \alpha \le 1$ and for all $n \ge 2$ the estimate
\begin{equation}\label{eqn:GGnx}
\log G(n,\alpha n)=f_G(\alpha)\,n^2+O\left(\frac{1}{\alpha}n^2 \exp\left(-c\sqrt{\log n}\,\right)\right),
\end{equation}
where the limit scaling function $f_G(\alpha)$ is given by
\begin{equation}\label{eqn:fG-asymp}
f_G(\alpha)=\frac{1}{2}+\frac{1}{2}\alpha^2\left\lfloor\frac{1}{\alpha}\right\rfloor^2+\frac{1}{2}\alpha^2\left\lfloor\frac{1}{\alpha}\right\rfloor-\alpha\left\lfloor\frac{1}{\alpha}\right\rfloor
\end{equation}
and $c$ is a positive absolute constant.
The remainder term estimate in \eqref{eqn:GGnx} was improved to the power-savings estimate
$O \left( \frac{1}{\alpha} n^{7/4} (\log n)^2 \right)$
conditional on the Riemann hypothesis.
This paper studies a notion of product of generalized binomial coefficients
${\overline{\overline{G}}}_n$ defined in terms of a generalized factorization of the form
\begin{equation}\label{eqn:oHn0}
{\overline{\overline{G}}}_n := \prod_{b=2}^n b^{{\overline{\nu}}(n,b)},
\end{equation}
where the product runs over all integers $2 \le b \le n$, and the formula ${\overline{\nu}}(n,b)$
is defined purely in terms of the base $b$ radix expansion
of the integers up to $n$, by a formula
\begin{equation}\label{eqn:gnu-b}
{\overline{\nu}}(n,b):=\frac{2}{b-1}S_b(n)-\frac{n-1}{b-1}d_b(n)
\end{equation}
generalizing that of the base $p$ expansions
of integers up to $n$.
(For all $b\ge n+1$ one has ${\overline{\nu}}(n,b) =0$,
so the product in \eqref{eqn:oHn0} may be taken over all $b \ge 2$.)
We will show that all quantities ${\overline{\nu}}(n,b)$ are nonnegative integers, so that
${\overline{\overline{G}}}_n$ is an integer sequence, and in addition the integer ${\overline{\overline{G}}}_n$ is $n$-smooth;
see Theorem \ref{thm:nub}.
The expressions ${\overline{\nu}}(n,b)$ are generally {\em not } equal
to the maximal power of $b$ dividing ${\overline{G}}_n$; in fact they can sometimes be larger than, and
sometimes be smaller than, this quantity.
Here ${\overline{\overline{G}}}(n,x)$ is an integer sequence in $n$ for fixed $x$ because all
${\overline{\nu}}(n,b)$ are nonnegative integers.
We have ${\overline{\overline{G}}}_n={\overline{\overline{G}}}(n,n)$, and we have stabilization
$$
{\overline{\overline{G}}}(n,x)={\overline{\overline{G}}}(n,n)={\overline{\overline{G}}}_n\quad\mbox{for}\quad x\ge n.
$$
The main results of this paper study questions concerning the asymptotics of the partial factorizations of ${\overline{\overline{G}}}_n$,
defined by
\begin{equation}\label{eqn:Hnx-def}
{\overline{\overline{G}}}(n,x):=\prod_{2\le b\le x}b^{{\overline{\nu}}(n,b)},
\end{equation}
parallel to those in \cite{DL:20}.
The main results of this paper determine the growth rate of integer sequence ${\overline{\overline{G}}}_n$ and
more generally the growth behavior of $\log{\overline{\overline{G}}}(n,x)$ for all $n\ge 1$.
The overall approach of the proofs have parallels to that in \cite{DL:20} but have
some significant differences, as given in Section \ref{sec:13}.
There are a number of reasons for interest in the study of integer sequences like ${\overline{\overline{G}}}(n,x)$.
Binomial products single out prime bases $p$ as special via their factorization \eqref{eqn:productformula}.
But these bases do not appear special in the radix expansion formula \eqref{eqn:gnu-b}, which has
a similar form for all $b \ge 2$.
The multiplicative definition of ${\overline{\overline{G}}}_n$ in \eqref{eqn:oHn0}, which makes a product over
contributions from all radix bases $b$, is performing
an (arithmetic) integration operation in passing from prime bases $p$ to all bases $b \ge 2$.
This paper studies various properties of the sequence ${\overline{\overline{G}}}_n$ as a motivating example of the effect of this kind
of discrete integration operation.
An analogy with integration is visible in the asymptotic growth behavior of $\log {\overline{\overline{G}}}_n$
given in Theorem \ref{thm:oHn} below,
whose main term is an integrated form of the main term in the asymptotics of $\log {\overline{G}}_n$
given in \eqref{eqn:logG-asymp}, compare \eqref{eqn:main-terms-compared}.
Finally, the formula \eqref{eqn:oHn0} does not reveal any connection with binomial coefficients.
However it turns out that one can construct a set of generalized factorials and generalized
binomial coefficients that produce the sequence ${\overline{\overline{G}}}_n$ as a set of generalized binomial products,
see Section \ref{sec:8}.
\subsection{Main results: Asymptotics of ${\overline{\overline{G}}}_n$ and ${\overline{\overline{G}}}(n,x)$}\label{sec:11}
We obtain the following result for the the full sum.
\begin{thm}\label{thm:oHn}
Let ${\overline{\overline{G}}}_n$ be given by
\eqref{eqn:oHn0}.
Then for integers $n\ge2$,
\begin{equation}\label{eqn:oH-aysmp1}
\log{\overline{\overline{G}}}_n=\frac{1}{2}n^2\log n+\left(\frac{1}{2}\gamma-\frac{3}{4}\right)n^2+O\left(n^{3/2}\log n\right),
\end{equation}
where $\gamma$ is Euler's constant.
\end{thm}
This result is proved in Section \ref{sec:AH}.
Unlike \cite{DL:20} the definition \eqref{eqn:oHn0} has no known
interpretation as a product of ratios of factorials; however, see the discussion in
Section \ref{sec:14a}.
Thus we do not have Stirling's formula available to directly estimate the
size of ${\overline{\overline{G}}}_n$, nor do we have combinatorial identities and recursion formulas available in dealing
with binomial coefficients.
We note that compared to \cite{DL:20}
there are two main terms in the asymptotics, rather then one. The leading order term has
the same constant $\frac{1}{2}$ as for $\log G(n)$ in \eqref{eqn:logG-asymp}, while
Euler's constant appears in the second leading order term.
This result will be used as an initial condition in obtain estimates for partial factorizations ${\overline{\overline{G}}}(n,x)$
for general $1 \le x \le n$.
The main result of the paper
determines the size of the partial factorization function ${\overline{\overline{G}}}(n,x)$ in the range $0\le x\le n$.
It establishes the following limiting behavior as $n\to\infty$ taking $x= x(n):=\alpha n$.
\begin{thm}\label{thm:oHnx-main}
Let ${\overline{\overline{G}}}(n,x)=\prod_{b=2}^{\lfloor x\rfloor}b^{{\overline{\nu}}(n,b)}$. Then for integers $n \ge 2$ and real $\alpha\in\left[\frac{1}{\sqrt{n}},1\right]$,
\begin{equation}\label{eqn:Gnx-main}
\log{\overline{\overline{G}}}(n,\alpha n)=f_{\overline{\overline{G}}}(\alpha)\,n^2\log n+g_{\overline{\overline{G}}}(\alpha)\,n^2+O\left(n^{3/2}\log n\right),
\end{equation}
in which:
(a) $f_{{\overline{\overline{G}}}} (\alpha)$ is a function with $f_{{\overline{\overline{G}}}}(0)=0$ and defined for $\alpha>0$ by
\begin{equation}\label{eqn:oHnx-parametrized-1}
f_{{\overline{\overline{G}}}}(\alpha)=\frac{1}{2}
+\frac{1}{2}\alpha^2\left\lfloor\frac{1}{\alpha}\right\rfloor^2+\frac{1}{2}\alpha^2\left\lfloor\frac{1}{\alpha}\right\rfloor-\alpha\left\lfloor\frac{1}{\alpha}\right\rfloor;
\end{equation}
(b) $g_{{\overline{\overline{G}}}} (\alpha)$ is a function with $g_{{\overline{\overline{G}}}}(0)=0$ and defined for $\alpha>0$ by
\begin{eqnarray}
g_{{\overline{\overline{G}}}}(\alpha) &=& \bigg( \frac{1}{2} \gamma - \frac{3}{4} \bigg) -\frac{1}{2}\bigg( H_{\lfloor\frac{1}{\alpha}\rfloor}- \log \frac{1}{\alpha} \bigg)
+ \bigg(\log \frac{1}{\alpha} \bigg)\bigg( -\frac{1}{2} - \frac{1}{2} \alpha^2 \bigg\lfloor \frac{1}{\alpha} \bigg\rfloor \bigg\lfloor \frac{1}{\alpha} +1\bigg\rfloor + \alpha \bigg\lfloor \frac{1}{\alpha} \bigg\rfloor \bigg)
\nonumber \\
&&- \frac{1}{4} \alpha^2 \bigg\lfloor \frac{1}{\alpha} \bigg\rfloor \bigg\lfloor \frac{1}{\alpha} +1\bigg\rfloor + \alpha \bigg\lfloor \frac{1}{\alpha} \bigg\rfloor.\label{eqn:oHnx-parametrized-2}
\end{eqnarray}
Moreover, for integers $n \ge2$ and real $\alpha \in\left[ \frac{1}{n},\frac{1}{\sqrt{n}}\right]$,
\begin{equation}\label{eqn:oHna-bound2}
\log{\overline{\overline{G}}}(n, \alpha n) = O \left( n^{3/2} \log n\right) .
\end{equation}
\end{thm}
Theorem \ref{thm:oHnx-main} follows from Theorem \ref{thm:oHnx}, taking $\alpha= \frac{x}{n}$.
The theorem implies that $f_{{\overline{\overline{G}}}}(\alpha)$ can be defined as a limit function
$$f_{\overline{\overline{G}}}(\alpha) :=\lim_{n\to\infty}\frac{1}{n^2\log n}\log{\overline{\overline{G}}}(n,\alpha n).$$
In fact
\begin{equation} \label{eqn:main-terms-compared}
f_{{\overline{\overline{G}}}}(\alpha) = f_G(\alpha),
\end{equation}
where
$$
f_G(\alpha) :=\lim_{n\to\infty}\frac{1}{n^2}\log G(n,\alpha n)
$$
is the limit function given in \cite[Theorem 1.1]{DL:20}.
We note an alternate form for $f_{{\overline{\overline{G}}}}(\alpha)$ given by
\begin{equation}\label{eqn:alt-foH}
f_{{\overline{\overline{G}}}}(\alpha)=\frac{1}{2}\alpha^2\left(\left\lfloor\frac{1}{\alpha}\right\rfloor+\left\{\frac{1}{\alpha}\right\}^2\right),
\end{equation}
where $\{x\}=x-\lfloor x\rfloor$ is the fractional part of $x$.
It is pictured in Figure \ref{fig:AB1}.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.50]{figure1-tikz-300pt.png}
\end{center}
\caption{Graph of limit function $f_{{\overline{\overline{G}}}}(\alpha)$ in $(\alpha, \beta)$-plane, $0 \le \alpha \le 1.$ The dotted line is $\beta= \frac{1}{2} \alpha$.
Here $f_{{\overline{\overline{G}}}}(0)=0$ and $f_{{\overline{\overline{G}}}}(1)= \frac{1}{2}.$}
\label{fig:AB1}
\end{figure}
Some properties of the limit function follow from \cite[Lemma 4.2]{DL:20},
since $f_{{\overline{\overline{G}}}}(\alpha)= f_{G}(\alpha)$.
\begin{enumerate}
\item[(i)]The function $f_{\overline{\overline{G}}}(\alpha)$ is continuous on $[0,\infty)$. It is differentiable everywhere except at $\alpha=1,\frac{1}{2},\frac{1}{3},\dots$, and $\alpha=0$ (from above).
\item [(ii)]
It satisfies
\begin{equation}
f_{{\overline{\overline{G}}}} ( \alpha) \le \frac{1}{2}\alpha \quad \mbox{for} \quad 0 \le \alpha \le 1.
\end{equation}
Equality holds at $\alpha=\frac{1}{j}$ for all integer $j \ge 1$,
with $f_{{\overline{\overline{G}}}}(\frac{1}{j}) = \frac{1}{2 j}$, and at $\alpha=0$ (by convention) and at no other values.
\item[(iii)] $f_{{\overline{\overline{G}}}}(\alpha)$ is piecewise quadratic, i.e. for $j \ge1$, on each closed interval $\left[\frac{1}{j+1},\frac{1}{j}\right]$ it is given by
\begin{equation}
f_{{\overline{\overline{G}}}}(\alpha) = \frac{1}{2} - j\alpha + \frac{j(j+1)}{2} \alpha^2 \quad \mbox{for} \quad \frac{1}{j+1} \le x \le \frac{1}{j}.
\end{equation}
\end{enumerate}
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.60]{g_G_doublebar_smaller.png}
\end{center}
\caption{Graph of limit function $g_{{\overline{\overline{G}}}}(\alpha)$ in $(\alpha, \beta)$ plane $0 \le \alpha \le 1.$
Here $g_{{\overline{\overline{G}}}}(0)=0$ and $g_{{\overline{\overline{G}}}}(1) = \frac{1}{2} \gamma - \frac{3}{4} \approx - 0.46139$. }
\label{fig:AB2}
\end{figure}
The limit function $g_{{\overline{\overline{G}}}}(\alpha)$
is pictured in Figure \ref{fig:AB2}.
Some properties of this limit function are:
\begin{enumerate}
\item[(i)] $g_{{\overline{\overline{G}}}}(\alpha)$ is continuous on $[0,1].$ It is real-analytic except at points $\alpha= \frac{1}{k}$
for integer $k \ge 1$ and at $x=0$.
\item[(ii)] $g_{{\overline{\overline{G}}}}(\alpha)$ is strictly decreasing on $[0,1]$.
It has $g_{{\overline{\overline{G}}}}(1) = \frac{1}{2} \gamma - \frac{3}{4} \approx -0.46139$.
\end{enumerate}
\subsection{Results: Asymptotics of ${{\overline{A}}}(n)$ and ${{\overline{B}}}(n)$}\label{sec:13}
To obtain the main result Theorem \ref{thm:oHnx-main}, we parallel
the approach taken in \cite{DL:20}.
Taking the logarithms of both sides of the product formula \eqref{eqn:Hnx-def}
for ${\overline{\overline{G}}}(n,x)$ and substituting the formula \eqref{eqn:gnu-p}
for each $\nu_b({\overline{G}}_n)$
yields the following identity:
There holds
\begin{equation}\label{eqn:oHABx}
\log {\overline{\overline{G}}}(n,x) = {{\overline{A}}}(n, x) - {{\overline{B}}}(n, x),
\end{equation}
where
\begin{equation}\label{eqn:oA-function}
{{\overline{A}}}(n,x)=\sum_{2\le b\le x}\frac{2}{b-1}S_b(n)\log b
\end{equation}
and
\begin{equation}\label{eqn:oB-function}
{{\overline{B}}}(n,x)=\sum_{2\le b\le x}\frac{n-1}{b-1}d_b(n)\log b.
\end{equation}
The functions ${{\overline{A}}}(n, x)$ and ${{\overline{B}}}(n, x)$ are arithmetical sums
that combine
behavior of the base $b$ digits of the integer $n$, viewing $n$ as fixed, and varying the radix base $b$.
These functions are weighted averages of statistics
of the radix expansions of $n$ for varying bases $2\le b\le x$.
The interesting range of $x$ is $1 \le x \le n $ because these functions ``freeze" at $x=n$:
${{\overline{A}}}(n, x)= {{\overline{A}}}(n,n)$ for $x \ge n$ and ${{\overline{B}}}(n,x) = {{\overline{B}}}(n,n)$ for $x \ge n$.
The main part of the paper determines the asymptotics of two nonnegative arithmetic functions ${{\overline{A}}}(n,x)$ and ${{\overline{B}}}(n,x)$, from which we obtain asymptotics for $\log {\overline{\overline{G}}}(n,x)$ via \eqref{eqn:oHABx}.
Individual radix statistics have previously been extensively studied holding the radix base $b$ fixed and varying $n$,
see Section \ref{sec:15}. But here the radix sums ${{\overline{A}}}(n,x)$ and ${{\overline{B}}}(n,x)$ hold $n$ fixed and vary the radix base $b$.
The analytic details are more involved
than those of \cite{DL:20}. On the one hand many sums over all $2\le b \le n$ are
easier to handle than sums over primes in \cite{DL:20}, and the Riemann hypothesis is
not needed. However the limit functions become more complicated.
and are sums of discontinuous functions which combine into continuous limit functions.
The proofs first obtain estimates out the special case $x=n$, setting
\begin{equation}\label{eqn:oA-function-0}
{{\overline{A}}}(n):={{\overline{A}}}(n,n)=\sum_{b=2}^n\frac{2}{b-1}S_b(n)\log b,
\end{equation}
\begin{equation}\label{eqn:oB-function-0}
{{\overline{B}}}(n):={{\overline{B}}}(n,n)=\sum_{b=2}^n\frac{n-1}{b-1}d_b(n)\log b.
\end{equation}
We determine asymptotics separately for the two functions ${{\overline{A}}}(n)$ and ${{\overline{B}}}(n)$ as $n \to \infty$, giving
a main term and a bound on the remainder term. The analysis first estimates the fluctuating term ${{\overline{B}}}(n)$
depending on $d_p(n)$, and then estimates${{\overline{A}}}(n)$ using the estimates for ${{\overline{B}}}(j)$ for $ 1 \le j \le n$.
\begin{thm}\label{thm:oBn}
Let ${{\overline{B}}}(n)$ be given by \eqref{eqn:oB-function-0}.
Then for integers $n\ge2$,
\begin{equation}\label{eqn:oB-asymp1}
{{\overline{B}}}(n)=(1-\gamma)n^2\log n+\left(\gamma+\gamma_1-1\right)n^2+O\left(n^{3/2}\log n\right),
\end{equation}
where $\gamma$ is Euler's constant
and $\gamma_1$ is the first Stieltjes constant.
\end{thm}
Two constants appear in this formula, which are
the Stieltjes constants $\gamma_0$
and $\gamma_1\approx -0.07282$.
The {\em Stieltjes constants} $\gamma_n$ appear in the Laurent expansion of the Riemann zeta function at $s=1$, as
\begin{equation}
\zeta (s) = \frac{1}{s-1} + \sum_{n=0}^{\infty} \frac{ (-1)^n }{n!} \gamma_n (s-1)^n.
\end{equation}
Here $\gamma_0= \gamma \approx 0.57721$ is Euler's constant, and more generally
\begin{equation} \label{eqn:stieltjes}
\gamma_m:=\lim_{n\to\infty}\bigg(\sum_{k=1}^n\frac{(\log k)^m}{k}-\frac{(\log n)^{m+1}}{n+1}\bigg).
\end{equation}
To establish Theorem \ref{thm:oBn} we first show
the main contribution in the sum ${{\overline{B}}}(n)$ comes from those values $b$
having $b > \sqrt{n}$, whose key property is that
their base $b$ radix expansions have {\em exactly two digits}.
for some $c_0>0$. Proceeding in a similar fashion to \cite{DL:20},
summing over all two-digit patterns, we obtain a formula of shape
${{\overline{B}}}(n) = (1-\gamma)n^2 \log n +Cn^2+ O( n^{2- \delta})$ with an
unconditional power-saving remainder term
$\delta= \delta(c_0)$ possible because the sums involved are over all integers $b \le n$
rather than over all primes $p < \sqrt{n}$.
We also deduce a corresponding result for ${{\overline{A}}}(n)$.
\begin{thm}\label{thm:oAn}
Let ${{\overline{A}}}(n)$ be given by \eqref{eqn:oA-function-0}.
Then for integers $n\ge2$,
\begin{equation}\label{eqn:oA-aysmp1}
{{\overline{A}}}(n)=\left(\frac{3}{2}-\gamma\right)n^2\log n+\left(\frac{3}{2}\gamma+\gamma_1-\frac{7}{4}\right)n^2+O\left(n^{3/2}\log n\right),
\end{equation}
where $\gamma$ is Euler's constant
and $\gamma_1$ is the first Stieltjes constant.
\end{thm}
In \cite{DL:20} the analogue of Theorem \ref{thm:oAn} could be
obtained immediately from Theorem \ref{thm:oBn}
using knowledge of $\log H_n$ available using Stirling's formula.
Such information is not available for $\log {\overline{\overline{G}}}(n)$ and an entirely new method is
used in Section \ref{sec:AH} to obtain the result via Theorem \ref{thm:oBn}.
Theorem \ref{thm:oHn} follows directly from Theorem \ref{thm:oBn} and Theorem \ref{thm:oAn} ,
via
\begin{equation}\label{eqn:oHAB}
\log {\overline{\overline{G}}}_n = {{\overline{A}}}(n) - {{\overline{B}}}(n),
\end{equation}
which is a special case of \eqref{eqn:oHABx}, taking $x=n$.
The constants appearing in the main term of the asymptotics
of ${{\overline{A}}}(n)$ and ${{\overline{B}}}(n)$ in Theorem \ref{thm:oAn}
give quantitative information on
cross-correlations between the statistics $d_b(n)$ and $S_b(n)$
of the base $b$ digits of $n$ (and smaller integers) as the base $b$ varies while $n$ is held fixed.
The occurrence of Euler's constant in the main term of these asymptotic estimates
indicates subtle arithmetic behavior in these sums, even though the defining sums are taken over all $b$ and not over primes $p$; cf. the survey \cite{Lag:13}.
\subsection{Results: Asymptotics of ${{\overline{A}}}(n,x)$ and ${{\overline{B}}}(n,x)$}\label{sec:14}
We first determine asymptotics for ${{\overline{B}}}(n, \alpha n)$ for $0 \le \alpha \le 1$,
by bootstrapping the result for ${{\overline{B}}}(n)={{\overline{B}}}(n,n)$ decreasing $x$ from $x=n$.
In what follows $H_m =\sum_{j=1}^m \frac{1}{j}$
and $J_m=\sum_{j=1}^m \frac{\log j}{j}$.
\begin{thm}\label{thm:oBnx-cor}
Let ${{\overline{B}}}(n,x)=\sum_{b=2}^{\lfloor x\rfloor}\frac{n-1}{b-1}d_b(n)\log b$. Then for integers $n \ge 2$ and real
$\alpha\in \left[ \frac{1}{\sqrt{n}} , 1\right]$,
\begin{equation}\label{eqn:Bnx-main}
{{\overline{B}}}(n, \alpha n) = f_{{{\overline{B}}}}(\alpha) n^2 \log n + g_{{{\overline{B}}}}(\alpha)n^2 + O \left( n^{3/2} \log n \right),
\end{equation}
in which:
\begin{enumerate}
\item[(a)] $f_{{{\overline{B}}}}(\alpha)$ is a function with $f_{{{\overline{B}}}}(0)=0$ and defined for $\alpha>0$ by
\begin{equation}\label{eqn:oBnx-parametrized1}
f_{{{\overline{B}}}}(\alpha) = (1- \gamma)+ \left( H_{\lfloor \frac{1}{\alpha}\rfloor}- \log \frac{1}{\alpha} \right) - \alpha \left\lfloor \frac{1}{\alpha}\right\rfloor;
\end{equation}
\item[(b)] $g_{{{\overline{B}}}}(\alpha)$ is a function with $g_{{{\overline{B}}}}(0)=0$ and defined for $\alpha>0$ by
\begin{eqnarray}
g_{{{\overline{B}}}}(\alpha) &=&\left(\gamma+\gamma_1-1 \right)-
\left( H_{\lfloor \frac{1}{\alpha} \rfloor} - \log \frac{1}{\alpha} \right) - \left( J_{\lfloor\frac{1}{\alpha}\rfloor} -
\frac{1}{2} \left(\log \frac{1}{\alpha}\right)^2 \right) \nonumber\\
&&+ \left(\log \frac{1}{\alpha}\right) \left(-1+\alpha \left\lfloor \frac{1}{\alpha} \right\rfloor \right) + \alpha\left\lfloor \frac{1}{\alpha} \right\rfloor.\label{eqn:oBnx-parametrized2}
\end{eqnarray}
\end{enumerate}
Moreover, for integers $n \ge2$ and real $\alpha \in\left[ \frac{1}{n},\frac{1}{\sqrt{n}}\right]$,
\begin{equation}\label{eqn:oBna-bound2}
{{\overline{B}}}(n, \alpha n) = O \left( n^{3/2} \log n \right) .
\end{equation}
\end{thm}
The formulas for $f_{{{\overline{B}}}}(\alpha)$ and $g_{{{\overline{B}}}}(\alpha)$ are a sum of functions that are discontinuous
at the points $\alpha= \frac{1}{k}$ for $k \ge 1$. However each of the functions $f_{{{\overline{B}}}}(\alpha)$ and $g_{{{\overline{B}}}}(\alpha)$
is a continuous function of $\alpha$, extending to a limiting value at the endpoint $\alpha=0$.
The limiting values $f_{{{\overline{B}}}}(0) =0$ and $g_{{{\overline{B}}}}(0) =0$. \medskip
\begin{figure}[h]
\includegraphics[scale=0.60]{funcB_diagonal1.png}
\caption{Graph of limit function $f_{{{\overline{B}}}}(\alpha)$ in $(\alpha, \beta)$-plane, $0 \le \alpha \le 1.$ The dotted line is $\beta= (1-\gamma) \alpha$,
where $\gamma$ is Euler's constant. Here $f_{B}(0)=0$ and $f_{B}(1)= 1-\gamma$.}
\label{fig:B3}
\end{figure}
The function $f_{{{\overline{B}}}}(\alpha)$ is pictured in Figure \ref{fig:B3}. It is the same function as $f_{B}(\alpha)$ in Theorem 1.5 in \cite{DL:20}. Some properties of this limit function are:
\begin{enumerate}
\item[(i)] $f_{{{\overline{B}}}}(\alpha)$ is continuous but not differentiable on $[0,1].$
It is real-analytic except at points $\alpha= \frac{1}{k}$
for integer $k \ge 1$ and at $x=0$.
\item[(ii)] $f_{{{\overline{B}}}}(\alpha)$ is strictly increasing on $[0,1]$.
It has $f_{{{\overline{B}}}}(1) = 1-\gamma\approx 0.42278$.
\end{enumerate}
\begin{figure}[h]
\includegraphics[scale=0.60]{g_Bbar.png}
\caption{Graph of limit function $g_{{{\overline{B}}}}(\alpha)$ in $(\alpha, \beta)$-plane, $0 \le \alpha \le 1.$
Here $g_{{{\overline{B}}}}(0)=0$ and $g_{{{\overline{B}}}}(1) = \gamma+\gamma_1-1 \approx -0.49560$.
This function has kinks at $\alpha= \frac{1}{k}$ for $k=2,3, 4, ...$.}
\label{fig:B4}
\end{figure}
The function $g_{{{\overline{B}}}}(\alpha)$ is pictured in Figure \ref{fig:B4}.
Some properties of this limit function are:
\begin{enumerate}
\item[(i)] $g_{{{\overline{B}}}}(\alpha)$ is continuous but not differentiable on $[0,1].$
It is real-analytic except at points $\alpha= \frac{1}{k}$
for integer $k \ge 1$ and at $x=0$.
\item[(ii)] $g_{{{\overline{B}}}}(\alpha)$ is strictly decreasing on $[0,1]$.
It has $g_{{{\overline{B}}}}(1) = \gamma +\gamma_1 -1 \approx -0.49560$.
\end{enumerate}
Turning to ${{\overline{A}}}(n,x)$, we obtain its asymptotics using a recursion starting from ${{\overline{A}}}(n,n)$
and working downward, given by \eqref{eqn:oAnn-oAnx-formula}, which involves a
function $C(n,x)$ studied in Proposition \ref{prop:23n}.
This recursion is different from that used in \cite{DL:20}, which started from $A(x,x)$
and worked upward using $B(y,x)$ for $x < y<n$.
\begin{thm}\label{thm:oAnx-cor}
Let ${{\overline{A}}}(n, x) = \sum_{b=2}^{\lfloor x\rfloor}\frac{2}{b-1} S_b(n) \log b.$
Then for integers $n \ge 2$ and real $\alpha\in\left[\frac{1}{\sqrt{n}},1\right]$,
\begin{equation}\label{eqn:Anx-main}
{{\overline{A}}}(n, \alpha n) = f_{{{\overline{A}}}}(\alpha) n^2 \log n + g_{{{\overline{A}}}}(\alpha) n^2 + O\left( n^{3/2}\log n\right),
\end{equation}
in which:
\begin{enumerate}
\item[(a)] $f_{{{\overline{A}}}}(\alpha)$ is a function with $f_{{\overline{A}}}(0)=0$ and defined for $\alpha>0$ by
\begin{equation}\label{eqn:Anx-parametrized}
f_{{{\overline{A}}}}(\alpha) = \left(\frac{3}{2} - \gamma\right)+ \left( H_{\lfloor \frac{1}{\alpha}\rfloor}- \log \frac{1}{\alpha} \right) + \frac{1}{2} \alpha^2 \left\lfloor \frac{1}{\alpha}\right\rfloor^2 + \frac{1}{2} \alpha^2 \left\lfloor \frac{1}{\alpha}\right\rfloor
- 2 \alpha \,\left\lfloor \frac{1}{\alpha} \right\rfloor;
\end{equation}
\item[(b)] $g_{{{\overline{A}}}}(\alpha)$ is a function with $g_{{{\overline{A}}}}(0)=0$ and defined for $\alpha>0$ by
\begin{eqnarray}
g_{{{\overline{A}}}}(\alpha) &= & \left( \frac{3}{2} \gamma + \gamma_1 -\frac{7}{4}\right) - \frac{3}{2} \left( H_{\lfloor \frac{1}{\alpha} \rfloor} - \log\frac{1}{\alpha} \right)
- \left( J_{\lfloor \frac{1}{\alpha} \rfloor} \nonumber - \frac{1}{2} \left(\log \frac{1}{\alpha}\right)^2 \right)\nonumber\\
&& +\left(\log \frac{1}{\alpha}\right)\left( -\frac{3}{2} -\frac{1}{2} \alpha^2 \left\lfloor \frac{1}{\alpha}\right\rfloor \left\lfloor \frac{1}{\alpha} +1 \right\rfloor +2\alpha \left\lfloor \frac{1}{\alpha} \right\rfloor \right)
-\frac{1}{4} \alpha^2 \left\lfloor \frac{1}{\alpha}\right\rfloor \left\lfloor \frac{1}{\alpha} +1 \right\rfloor +2 \alpha \left\lfloor \frac{1}{\alpha} \right\rfloor.\label{eqn:oAnx-parametrized}
\end{eqnarray}
\end{enumerate}
Moreover, for integers $n \ge2$ and real $\alpha \in\left[ \frac{1}{n},\frac{1}{\sqrt{n}}\right]$,
\begin{equation}\label{eqn:oAna-bound2}
{{\overline{A}}}(n, \alpha n) = O \left( n^{3/2} \log n \right) .
\end{equation}
\end{thm}
The implied constant in the $O$-notation does not depend on $\alpha$.
The formulas for $f_{{{\overline{A}}}}(\alpha)$ and $g_{{{\overline{A}}}}(\alpha)$ are a sum of functions that are discontinuous
at the points $\alpha= \frac{1}{k}$ for $k \ge 1$.
However the functions $f_{{{\overline{A}}}}(\alpha)$ and $g_{{{\overline{A}}}}(\alpha)$
are continuous functions of $\alpha$, also continuous at $\alpha=0$.
The function $f_{{{\overline{A}}}}(\alpha)$ is pictured in Figure \ref{fig:A5}. It is the same function as $f_{A}(\alpha) $ in Theorem 1.6 in \cite{DL:20}. It is strictly increasing on $[0,1]$ and has $f_{{{\overline{A}}}}(1) = \frac{3}{2} -\gamma\approx0.92278$.
\begin{figure}[h]
\includegraphics[scale=0.60]{funcA_with_derivative1.png}
\caption{Graph of limit function $f_{{{\overline{A}}}}(\alpha)$ in $(\alpha, \beta)$-plane, $0 \le \alpha \le 1.$ The dotted line is $\beta= \left(\frac{3}{2}-\gamma\right) \alpha$.
Here $f_{{{\overline{A}}}}(0)=0$ and $f_{{{\overline{A}}}}(1) = \frac{3}{2}-\gamma \approx 0.92278$.
Superimposed on this graph is a plot of the derivative $f_{{{\overline{A}}}}^{'}(\alpha)$ drawn to the same scale, but with $\beta$-axis shifted so that the horizontal dotted line is $\beta=0$; the derivative has kinks at $x= \frac{1}{k}$, $k \ge 2$.}
\label{fig:A5}
\end{figure}
The function $g_{{{\overline{A}}}}(\alpha)$ is pictured in Figure \ref{fig:A6}.
Some properties of this limit function are:
\begin{enumerate}
\item[(i)] $g_{{{\overline{A}}}}(\alpha)$ is continuous but not differentiable on $[0,1].$ It is real-analytic except at points $\alpha= \frac{1}{k}$ for integer $k \ge 1$ and at $x=0$.
\item[(ii)] $g_{{{\overline{A}}}}(\alpha)$ is strictly decreasing on $[0,1]$.
It has $g_{{{\overline{A}}}}(1) = \frac{3}{2} \gamma+\gamma_1- \frac{7}{4} \approx -0.95699$.
\end{enumerate}
\begin{figure}[h]
\includegraphics[scale=0.60]{g_Abar.png}
\caption{Graph of limit function $g_{{{\overline{A}}}}(\alpha)$ in $(\alpha, \beta)$-plane, $0 \le \alpha \le 1.$
Here $g_{{{\overline{A}}}}(0)=0$ and $g_{{{\overline{A}}}}(1) = \frac{3}{2} \gamma + \gamma_1 -\frac{7}{4} \approx -0.95699$.
This function has kinks at $\alpha= \frac{1}{k}$ for $k=2,3, 4, ...$ which are not visible at this scale.}
\label{fig:A6}
\end{figure}
We obtain Theorem \ref{thm:oHnx-main} as a corollary of the
two previous theorems, substituting their estimates into the formula
$$\log {\overline{\overline{G}}}(n,x) = {{\overline{A}}}(n, x) - {{\overline{B}}}(n, x).$$
At $x=n$ Euler's constant cancels out of the main term of order $n^2 \log n$,
but it is still present in the secondary term of order $n^2$ in
Theorem \ref{thm:oHnx-main}.
\subsection{Discussion}\label{sec:14a}
We compare and contrast the main results with those for binomial products in \cite{DL:20}.
\begin{enumerate}
\item
The asymptotics of $\log{\overline{\overline{G}}}(n,x)$ has a main term of size $f(\alpha) n^2 \log n$ and
a secondary terms $g(\alpha) n^2$, where $\alpha = \frac{x}{n}$.
The main term scaling functions
$f(\alpha)$ are oscillatory functions matching those in \cite{DL:20}.
The secondary terms $g(\alpha)$ are new functions of $\alpha$
whose salient feature is that they are nonpositive.
\item
There is a remainder term with a power-savings $O(n^{3/2 + \epsilon})$, which is provable {\em unconditionally.}
Parallel results of \cite{DL:20} had conditional power-savings remainder terms, but needed to assume the Riemann hypothesis.
That paper argued (but did not prove rigorously) that the existence of a power-saving remainder term would imply the existence of
a zero-free region for the Riemann zeta function of the form $\Re(s) > 1- \delta$ for some $\delta >0$ depending on the amount
of power-saving.
For ${\overline{\overline{G}}}(n,x)$ the averaging over all integers $b\in(1,x]$
led to unconditional power savings in the
remainder term.
\end{enumerate}
\subsection{Related work}\label{sec:15}
Work of the second author and Mehta \cite{LagM:2016} in 2016
studied radix expansion statistics radix expansion statistics of integers
which hold the integer $n$ fixed,
while varying across different radix bases up to $n$
e.g. statistics $A(n,x)$ and $B(n,x)$.
The work \cite{LagM:2017} studied analogous statistics for Farey fractions.
The paper \cite{DL:20} of the first two authors studied the statistics $A(n,x)$ and $B(n,x)$
for products of binomial coefficients.
The motivation of \cite{DL:20} was study of prime number distribution from a novel direction.
There has been a great deal of study of the radix statistics $d_b(n)$ and $S_b(n)$
for a fixed base $b \ge 2$ and letting $n$ vary.
Work on $d_b(n)$ has mainly been probabilistic, for random integers
in an initial interval $[1,n]$, which is surveyed by Chen et al \cite{CHZ:2014}.
One has for all $n \ge 1$,
\begin{equation}
{\mathbb E}[ d_b(k): 0 \le k \le n-1] \le \frac{b-1}{2} \log_b n,
\end{equation}
a result which is close to sharp when $n = b^k$ for some integer $k \ge 1$.
We have
$d_b(n) \le (b-1) \log_b (n+1$,
see Lemma \ref{lem:dbn-Sbn-bound}.
It implies
$$ {{\overline{B}}}(n)\le\sum_{b=2}^n\frac{n-1}{b-1}\left(\frac{(b-1)\log(n+1)}{\log b}\right)\log b=(n-1)^2\log(n+1).$$
Work on the smoothed function $S_b(n)$
studying asymptotics of the
as $n \to \infty$ started with
Bush \cite{Bush:40} in 1940, followed by Bellman and Shapiro \cite{BelS:48},
and Mirsky \cite{Mir:49}, who in 1949 showed
that for fixed $b \ge 2$, the asymptotic formula
$$
S_b(n) = \frac{b-1}{2} n \log_b(n) +O(n).
$$
In 1952 Drazin and Griffith \cite{DG52}
deduced an inequality implying
\begin{equation}\label{eqn:S-ineq}
S_b(n) \le \frac{b-1}{2} n \log_b n,
\end{equation}
for all $b \ge 2$ and $n \ge 1$, see Lemma \ref{lem:dbn-Sbn-bound}.
This inequality is sharp: equality holds for $n=b^k$ for $k \ge 1$,
see \cite[Theorem 5.8]{LagM:2016}.
Using Drazin and Griffith's inequality \eqref{eqn:S-ineq} for $S_b(n)$ we have
\begin{equation} \label{eqn:A-inequality}
{{\overline{A}}}(n) \le {{\overline{A}}}^{\ast}(n) := \sum_{b=2}^n \frac{2}{b-1} \left(\frac{(b-1)n\log n}{2\log b}\right) \log b
= n(n-1) \log n.
\end{equation}
A formula of Trollope \cite{Tro:68} in 1968 gave an exact formula for $S_b(n)$ for base $b= 2$.
Notable work of Delange \cite{Del:1975} obtained exact formulas for $S_b(n)$ for all $b \ge 2$,
which exhibited an oscillating term in the asymptotics.
We mention later work of Flajolet et al \cite{FGKPT94} and Grabner and Hwang \cite{GH05}.
Recently Drmota and Grabner \cite{DrmGra10} surveyed this topic.
We mention other inequalities for the functions $S_b(n)$.
In 2011 Allaart \cite[Equation (4)]{Allaart:11}
showed
an approximate convexity inequality for binary expansions, showing for $0 \le \ell \le m$,
\begin{equation}
S_{2}(m+ \ell) + S_{2}(m-\ell) - 2 S_{2}(m) \le \ell.
\end{equation}
Allaart \cite[Theorem 3]{Allaart:17}
proved a generalization to any base $b$:
For integers $k$ and $m$ with $0\le k \le m$, we have
\begin{equation}
S_{b}(m+ k) + S_{b}(m-k) - 2 S_b(m) \le \left\lfloor \frac{b+1}{2} \right\rfloor k.
\end{equation}
Allaart \cite[Theorem 1]{Allaart:17}
also showed a superadditivity inequality valid for base $b$ expansions for all $m,n \ge 1$,
\begin{equation}
S_b(m+n) \ge S_b(m) + S_b(n) + \min(m,n).
\end{equation}
\subsection{Contents of paper}\label{sec:16}
Section \ref{sec:2} collects facts about digit sums and
provides estimates for a wide variety of
sums needed in later estimates. In particular we estimate the bivariate sums
\begin{equation}
\overline{C}(n,x):=\sum_{1 \le b\le x}\left\lfloor\frac{n}{b}\right\rfloor\log b.
\end{equation}
Section \ref{sec:oB} estimates ${{\overline{B}}}(n)$, proving Theorem \ref{thm:oBn}.
Section \ref{sec:AH} estimates ${{\overline{A}}}(n)$, proving Theorem \ref{thm:oAn}.
Theorem \ref{thm:oHn} for ${\overline{\overline{G}}}(n)$ then follows.
Section \ref{sec:asymp-oBnx} estimates ${{\overline{B}}}(n,x)$, proving Theorem \ref{thm:oBnx}.
Theorem \ref{thm:oBnx-cor} for ${{\overline{B}}}(n,\alpha n)$ then follows.
Section \ref{sec:asymp-oAnx} estimates ${{\overline{A}}}(n,x)$, proving Theorem \ref{thm:oAnx}.
Theorem \ref{thm:oAnx-cor} for ${{\overline{A}}}(n,\alpha n)$ then follows.
Section \ref{sec:oHnx} estimates ${\overline{\overline{G}}}(n,x)$, proving Theorem \ref{thm:oHnx}.
Theorem \ref{thm:oHnx-main} for ${\overline{\overline{G}}}(n, \alpha n)$ then follows.
Section \ref{sec:8} presents concluding remarks. These include an interpretation of
${\overline{\overline{G}}}_n$ as a product of generalized binomial coefficients, treated in \cite{LY:22},
and possible extensions of the construction of ``multiple integral" integer sequences
${\overline{G}}_n^{(-j)}$ for $j \ge 1$.
\section{Preliminaries}\label{sec:2}
The first subsection establishes properties of radix expansion statistics ${\overline{\nu}}(n, b)$, and
derives inequalities on the size of $d_b(n)$ and $S_b(n)$.
The next four subsections estimate four families of
sums for an integer $n$ and a real number $x$, treated as step functions: the harmonic numbers $H(x) = \sum_{b=1}^{\lfloor x\rfloor} \frac{1}{b},$
the sums $J(x) = \sum_{b=1}^{\lfloor x\rfloor} \frac{\log b}{b}$, the sums $\overline{C}(n,x) = \sum_{b=1}^{\lfloor x\rfloor} \lfloor \frac{n}{b} \rfloor \log b$, and
$L_i(n) = \sum_{b=2}^n b (\log b)^i$ for $ i \ge 1$.
\subsection{Radix expansion statistics} \label{subsec:21nn}
Fix an integer $b\ge2$. Let $n$ be a positive integer. Then $n$ can be written uniquely as
\begin{equation}\label{eqn:base-b
n=\sum_{i=0}^ka_i(b,n)b^i,
\end{equation}
where $a_i(b, n)\in\{0,1,2,\dots,b-1\}$ are the \emph{base-$b$ digits} of $n$ and the \emph{top} digit $a_k$ is positive. We say that $n$ has $k+1$ digits in base $b$. One has $b^k\le n<b^{k+1}$. Hence the number of base-$b$ digits of $n$ is
$$
\left\lfloor \frac{\log n}{\log b}\right\rfloor+1.
$$
Each base-$b$ digit of $n$ can also be expressed in terms of the floor function:
\begin{equation}\label{eqn:floor-recursion
a_i(b,n)=\left\lfloor\frac{n}{b^i}\right\rfloor-b\left\lfloor\frac{n}{b^{i+1}}\right\rfloor.
\end{equation}
Note that \eqref{eqn:floor-recursion} also defines $a_i(b,n)$ to be $0$ for all $i>\frac{\log n}{\log b}$. The following two statistics of the base-$b$ digits of numbers will show up frequently in this paper.
\begin{defn}
(1) The {\em sum of digits function} $d_b(n)$ is given by
\begin{equation}\label{eqn:dbn
d_b(n):=\sum_{i=0}^{\left\lfloor\frac{\log n}{\log b}\right\rfloor}a_i(b,n)=\sum_{i=0}^\infty a_i(b,n),
\end{equation}
where $a_i(b,n)$ is given by \eqref{eqn:floor-recursion}.\\
(2) The {\em running digit sum function} $S_b(n)$ is given by
\begin{equation}\label{eqn:Sbn
S_b(n) := \sum_{j=1}^{n-1} d_b(j).
\end{equation}
\end{defn}
\begin{thm}\label{thm:nub}
Let $b\ge2$ be an integer, and let the radix expansion statistic ${\overline{\nu}}(n,b)$ given for $n \ge 1$ by
\begin{equation}\label{eqn:localformula}
{\overline{\nu}}(n,b) =\frac{2}{b-1}S_b(n)-\frac{n-1}{b-1}d_b(n)
\end{equation}
Then:
\begin{enumerate}
\item[(1)] For all $n \ge 1$,
${\overline{\nu}}(n,b)$ is a nonnegative integer.
\item[(2)]
${\overline{\nu}}(n,b)=0$ if and only if $n=ab^k+b^k-1$ for some $a\in\{1,2,3,\dots,b-1\}$ and integer $k \ge0$.
\end{enumerate}
\end{thm}
\begin{proof}
To show (1), we substitute \eqref{eqn:floor-recursion} into \eqref{eqn:dbn} and obtain
\begin{equation}\label{eqn:dbn-formula
d_b(n)=\sum_{i=0}^\infty\left\lfloor\frac{n}{b^i}\right\rfloor-b\sum_{i=0}^\infty\left\lfloor\frac{n}{b^{i+1}}\right\rfloor=n-(b-1)\sum_{i=1}^\infty\left\lfloor\frac{n}{b^i}\right\rfloor.
\end{equation}
We then substitute \eqref{eqn:dbn-formula} into \eqref{eqn:Sbn} and obtain
\begin{equation}\label{eqn:Sbn-formula
S_b(n)=\sum_{j=1}^{n-1}j-(b-1)\sum_{j=1}^{n-1}\sum_{i=1}^\infty\left\lfloor\frac{j}{b^i}\right\rfloor=\frac{n(n-1)}{2}-(b-1)\sum_{i=1}^\infty\sum_{j=1}^{n-1}\left\lfloor\frac{j}{b^i}\right\rfloor.
\end{equation}
Now, we substitute \eqref{eqn:dbn-formula} and \eqref{eqn:Sbn-formula} into \eqref{eqn:localformula} and obtain
\begin{align}
{\overline{\nu}}(n,b)&=\bigg(\frac{n(n-1)}{b-1}-2\sum_{i=1}^\infty\sum_{j=1}^{n-1}\left\lfloor\frac{j}{b^i}\right\rfloor\bigg)-\bigg(\frac{n(n-1)}{b-1}-(n-1)\sum_{i=1}^\infty\left\lfloor\frac{n}{b^i}\right\rfloor\bigg)\nonumber\\
&=\sum_{i=1}^\infty\bigg((n-1)\left\lfloor\frac{n}{b^i}\right\rfloor-2\sum_{j=1}^{n-1}\left\lfloor\frac{j}{b^i}\right\rfloor\bigg)\nonumber\\
&=\sum_{i=1}^\infty\sum_{j=1}^{n-1}\left(\left\lfloor\frac{n}{b^i}\right\rfloor-\left\lfloor\frac{j}{b^i}\right\rfloor-\left\lfloor\frac{n-j}{b^i}\right\rfloor\right).\label{eqn:Gnunb-formula
\end{align}
The last quantity \eqref{eqn:Gnunb-formula} expresses ${\overline{\nu}}(n,b)$ as the sum of integers, which are all nonnegative
due to the identity valid for all real $x$ and $y$,
$$
\lfloor x +y\rfloor = \lfloor x \rfloor + \lfloor y \rfloor + \lfloor \{ x\} + \{ y\} \rfloor \ge \lfloor x \rfloor + \lfloor y \rfloor,
$$
see Graham et al \cite[Section 3.1, page 70]{GKP:94}.
Hence ${\overline{\nu}}(n,b)$ is a nonnegative integer.
We show (2). We
prove the `only if' part first. Suppose that $n$ is a positive integer not of the form $ab^k+b^k-1$, where $1\le a\le b-1$ and $k\ge0$.
Then $cb^\ell\le n\le(c+1)b^\ell-2$ for some $c\in\{1,2,3,\dots,b-1\}$ and positive integer $\ell$.
We show ${\overline{\nu}}(n,b)$ is positive.
We see that the double sum in \eqref{eqn:Gnunb-formula},
is greater than or equal to the summand with $(i,j)=\left(\ell,b^\ell-1\right)$. It follows that
$$
{\overline{\nu}}(n,b)\ge\left\lfloor\frac{n}{b^\ell}\right\rfloor-\left\lfloor\frac{b^\ell-1}{b^\ell}\right\rfloor-\left\lfloor\frac{n-b^\ell+1}{b^\ell}\right\rfloor=c-0-(c-1)=1.
$$
Thus, if ${\overline{\nu}}(n,b)=0$, then $n$ must be of the form $ab^k+b^k-1$ with $1\le a\le b-1$ and $k\ge0$.
Conversely, suppose that $n$ is of the form $ab^k+b^k-1$ with $1\le a\le b-1$ and $k\ge0$. Suppose that $j$ is an integer with $1\le j\le n-1$. For $i\le k-1$, we have $a_i(b,j)\le b-1=a_i(b,n)$.
For $i\ge k$, we also have $a_i(b,j)\le a_i(b,n)$ because $j<n$. Hence
$$
a_i(b,n-j)=a_i(b,n)-a_i(b,j)
$$
for all $i\ge0$. Summing over $i\ge0$, we obtain
$$
d_b(n-j)=d_b(n)-d_b(j).
$$
Summing over $1\le j\le n-1$, we obtain
$$
S_b(n)=(n-1)d_b(n)-S_b(n),
$$
which implies
$$
{\overline{\nu}}(n,b)=\frac{2}{b-1}S_b(n)-\frac{n-1}{b-1}d_b(n)=0.
$$
This completes the proof.
\end{proof}
\begin{rem}\label{rem:23}
In general, ${\overline{\nu}}(n,b)$ does not equal the largest integer $k$ such that $b^k$ divides ${\overline{G}}_n$, which we denote by
$\nu_b({\overline{G}}_n)$. Moreover ${\overline{\nu}}(n,b)$ can be larger or smaller than $\nu_b({\overline{G}}_n)$. For example, ${\overline{\nu}}(4,4)=3>2=\nu_4({\overline{G}}_4)$, while ${\overline{\nu}}(6,4)=1<2=\nu_4({\overline{G}}_6)$.
\end{rem}
We establish inequalities on the size of $d_b(n)$ and $S_b(n)$.
\begin{lem}\label{lem:dbn-Sbn-bound}
For integers $b\ge2$ and $n\ge1$, we have
\begin{equation}\label{eqn:dbnineq}
1\le d_b(n)\le\frac{(b-1)\log(n+1)}{\log b},
\end{equation}
\begin{equation}\label{eqn:Sbnineq}
0\le S_b(n)\le\frac{(b-1)n\log n}{2\log b}.
\end{equation}
\end{lem}
\begin{proof}
The lower bound in \eqref{eqn:dbnineq} follows from the observation that $d_b(n)$ is greater than or equal to the top (base-$b$) digit of $n$, which is at least $1$. The lower bound in \eqref{eqn:Sbnineq} then follows from the positivity of $d_b(j)$.
The upper bound in \eqref{eqn:Sbnineq} is a result of Drazin and Griffith \cite[Theorem~1]{DG52}. To prove the upper bound in \eqref{eqn:dbnineq}, we apply Theorem \ref{thm:nub}:
$$
0\le(b-1){\overline{\nu}}(n,b)=2S_b(n)-(n-1)d_b(n)=2S_b(n+1)-(n+1)d_b(n).
$$
On replacing $n$ by $n+1$ in \eqref{eqn:Sbnineq}, we obtain $S_b(n+1)\le\frac{(b-1)(n+1)\log(n+1)}{2\log b}$. Hence
$$
d_b(n)\le\frac{2}{n+1}S_b(n+1)\le\frac{(b-1)\log(n+1)}{\log b},
$$
as desired.
\end{proof}
\subsection{The harmonic numbers $H_n$} \label{subsec:22nn}
For positive real numbers $x \ge 1$, we consider the step function
$$
H(x):=\sum_{1\le b\le x}\frac{1}{b}.
$$
At integer values
$n= \lfloor x \rfloor$ we write $H(x)=H_{\lfloor x\rfloor}= H_n$, the $n$-th harmonic number.
\begin{lem}\label{lem:21n}
For positive integers $n$, we have
\begin{equation}\label{eqn:harmonic}
H_n = \log n + \gamma + \frac{1}{2n} + O\left( \frac{1}{n^2} \right),
\end{equation}
where $\gamma \approx 0.57721$ is Euler's constant.
\end{lem}
\begin{proof}
This standard result appears in Tenenbaum \cite[Chapter I.0, Theorem 5]{Ten15}.
\end{proof}
The restriction to integer $n$ is needed in Lemma \ref{lem:21n} because for positive real numbers $x$, one has
$$
H(x) - \log x - \gamma = \Omega_{\pm} \left( \frac{1}{x} \right).
$$
Indeed, using the Euler--Maclaurin summation formula \cite[Theorem~B.5]{MV07}, one can show that for real numbers $x\ge1$,
$$
H(x)=\log x+\gamma+\frac{1-2\{x\}}{2x}+O\left(\frac{1}{x^2}\right).
$$
Hence, $\limsup_{x\rightarrow\infty}x(H(x)-\log x-\gamma)=\frac{1}{2}$ and
$\liminf_{x\rightarrow\infty}x(H(x)-\log x-\gamma)=-\frac{1}{2}$.\smallskip
\subsection{Estimates: $J(x)$}\label{subsec:22nn2}
For real numbers $x \ge 1$, we consider the step function
\begin{equation}\label{eqn:Jx}
J(x):=\sum_{1 \le b\le x}\frac{\log b}{b}.
\end{equation}
At integer values
$n= \lfloor x \rfloor$ we write $J(x)=J_{\lfloor x\rfloor}=J_n$.
The asymptotics of this step function of $x$ involve the first Stieltjes constant $\gamma_1$,
defined in Section \ref{sec:13}.
\begin{lem}\label{lem:27}
For real numbers $x\ge1$, we have
\begin{equation}\label{eqn:Jx1}
J(x)=\frac{1}{2}(\log x)^2+\gamma_1+O\left(\frac{\log(x+1)}{x}\right),
\end{equation}
where $\gamma_1 \approx- 0.0728158$ is the first Stieltjes constant.
\end{lem}
\begin{proof}
By partial summation, we obtain
\begin{equation}\label{PartialSum}
J(x)=\sum_{1 \le b\le x}\frac{\log b}{b}=(\log x)H(x)-\int_1^x\frac{H(u)}{u}\,du.
\end{equation}
It is well-known that $H(u)=\log u+\gamma+R(u)$, where the remainder $R(u)\ll\frac{1}{u}$, for $u\ge1$. (See \cite[Corollary~1.15]{MV07}.)
On inserting this in \eqref{PartialSum} and rearranging, we get
\begin{align*}
J(x)&=\frac{1}{2}(\log x)^2 + (\log x) R(x) - \bigg( \int_1^\infty\frac{R(u)}{u}\,du-\int_x^\infty\frac{R(u)}{u}\,du\bigg) \\
&=\frac{1}{2}(\log x)^2+c+O\left(\frac{\log(x+1)}{x}\right),
\end{align*}
where $c:=-\int_1^\infty\frac{R(u)}{u}\,du$. By taking $x\rightarrow\infty$, we see that
$$c=\lim_{x \to \infty} \left(J(x) - \frac{1}{2} (\log x)^2\right) = \lim_{x\to \infty}\bigg(\sum_{b\le x}\frac{\log b}{b}-\frac{1}{2}(\log x)^2\bigg)=\gamma_1,$$
the first Stieltjes constant, according to \eqref{eqn:stieltjes}.
\end{proof}
\subsection{Estimates: $\overline{C}(n,x)$} \label{subsec:22nn3}
\par For real numbers $n\ge 1$ and $x \ge 1$, let
\begin{equation}\label{eqn:oC-defn}
\overline{C}(n,x):=\sum_{1 \le b\le x}\left\lfloor\frac{n}{b}\right\rfloor\log b.
\end{equation}
Here, $\overline{C}(n,x)$ is a nonnegative step function of the real variable $x$,
viewing $n$ as fixed.
This function stabilizes for $x \ge n$:
\begin{equation}\label{eqn:oCstab}
\overline{C}(n,x)=\overline{C}(n,n)\quad\mbox{for}\quad x\ge n.
\end{equation}
\begin{prop}\label{prop:23n}
(1) For real numbers $n\ge2$, we have
$$
\overline{C}(n,n)=\frac{1}{2}n(\log n)^2+(\gamma-1)n\log n+(1-\gamma)n+O\left(\sqrt{n}\log n\right).
$$
(2) For real numbers $n\ge2$ and $x$ such that $1\le x\le n$, we have
\begin{align}\label{eqn:oCn-diff}
\overline{C}(n,n)-\overline{C}(n,x)&=\int_x^n\left\lfloor\frac{n}{u}\right\rfloor\log u\,du+O\left(\frac{n\log n}{x}\right).
\end{align}
In addition,
\begin{equation} \label{eqn:oCn-integral}
\int_x^n \left\lfloor\frac{n}{u}\right\rfloor\log u\,du =
\left(H_{\left\lfloor\frac{n}{x}\right\rfloor}-\frac{x}{n}\left\lfloor\frac{n}{x}\right\rfloor\right)(n\log n-n)
- \left( J_{\left\lfloor\frac{n}{x}\right\rfloor}- \frac{x}{n}\left\lfloor\frac{n}{x}\right\rfloor \log\frac{n}{x}
\right)n.
\end{equation}
\end{prop}
To prove Proposition \ref{prop:23n} we use the following identity.
\begin{lem}\label{lem:oCfe}
For real numbers $n \ge 1$ and $x \ge 1$, we have
\begin{equation}\label{eqn:oCfe}
\overline{C}(n,n)+\overline{C}(n,x)-\overline{C}\left(n,\frac{n}{x}\right)=(\log n)\bigg(\sum_{1 \le b\le x}\left\lfloor\frac{n}{b}\right\rfloor\bigg) -\lfloor x\rfloor\log\left(\left\lfloor\frac{n}{x}\right\rfloor!\right)-nH(x)+\lfloor x\rfloor+\sum_{1 \le b\le x}\int_1^\frac{n}{b}\frac{\{u\}}{u}\,du.
\end{equation}
\end{lem}
\begin{proof}
By partial summation, we have the identity
$$
\sum_{1 \le k\le t}\log\frac{t}{k}=t-1-\int_1^t\frac{\{u\}}{u}\,du,
$$
for any $t>0$. On setting $t=\frac{n}{b}$ in this identity and summing over positive integers $b\le x$, we obtain
\begin{equation}\label{eqn:212}
\sum_{1 \le b\le x}\bigg(\sum_{1 \le k\le\frac{n}{b}}\log\frac{n}{bk}\bigg) =nH(x)-\lfloor x\rfloor-\sum_{1 \le b\le x}\int_1^\frac{n}{b}\frac{\{u\}}{u}\,du.
\end{equation}
The double sum on the left side of \eqref{eqn:212} is equal to
$$
\sum_{1 \le b\le x}\sum_{1 \le k\le\frac{n}{b}}(\log n-\log b- \log k)
=(\log n) \bigg( \sum_{1 \le b\le x}\left\lfloor\frac{n}{b}\right\rfloor\bigg) -\overline{C}(n,x)-\sum_{1 \le k\le n}\bigg(\sum_{\substack{1 \le b\le x\\ b\le\frac{n}{k}}}\log k\bigg).
$$
On substituting the right side into \eqref{eqn:212} and rearranging, we obtain
\begin{equation}\label{eqn:213}
\sum_{1\le k\le n}\bigg(\sum_{\substack{1 \le b\le x\\b\le\frac{n}{k}}}\log k\bigg) =(\log n)\bigg(\sum_{1 \le b\le x}\left\lfloor\frac{n}{b}\right\rfloor\bigg) -\overline{C}(n,x)-nH(x)+\lfloor x\rfloor+\sum_{1 \le b\le x}\int_1^\frac{n}{b}\frac{\{u\}}{u}\,du.
\end{equation}
The double sum on the left side of \eqref{eqn:213} is equal to
$$
\sum_{\frac{n}{x}<k\le n}\bigg(\sum_{1 \le b\le\frac{n}{k}}\log k\bigg) +\sum_{1 \le k\le\frac{n}{x}}\bigg(\sum_{1\le b\le x}\log k\bigg) =
\left(\overline{C}(n,n)-\overline{C}\left(n,\frac{n}{x}\right)\right)+\lfloor x\rfloor\log\left(\left\lfloor\frac{n}{x}\right\rfloor!\right).
$$
On inserting the right side into \eqref{eqn:213} and rearranging, we get \eqref{eqn:oCfe}.
\end{proof}
\begin{proof}[Proof of Proposition \ref{prop:23n}]
(1) On substituting $x=\sqrt{n}$ in Lemma \ref{lem:oCfe}, two of the terms on the left side cancel and we get
\begin{equation}\label{eq:oCnn-est}
\overline{C}(n,n)= (\log n)\bigg( \sum_{1 \le b\le\sqrt{n}}\left\lfloor\frac{n}{b}\right\rfloor \bigg)- \left\lfloor\sqrt{n}\right\rfloor\log\left(\left\lfloor\sqrt{n}\right\rfloor!\right)-nH\left(\sqrt{n}\right)+\left\lfloor\sqrt{n}\right\rfloor+\sum_{1 \le b\le\sqrt{n}}\int_1^\frac{n}{b}\frac{\{u\}}{u}\,du.
\end{equation}
Now, we estimate each term on the right of \eqref{eq:oCnn-est}. For the first term, we use $\lfloor t\rfloor=t+O(1)$, obtaining
\begin{eqnarray*}
(\log n) \bigg(\sum_{1 \le b\le\sqrt{n}}\left\lfloor\frac{n}{b}\right\rfloor \bigg)&=&
(\log n)
n H_{\lfloor\sqrt{n}\rfloor} +
O\left(\sqrt{n}\log n\right)\\
&=& n
(\log n)
\left( \log \left\lfloor \sqrt{n}\right\rfloor + \gamma +O\left(\frac{1}{\left\lfloor \sqrt{n} \right\rfloor}\right) \right) +O \left( \sqrt{n} \log n \right)\\
&=& \frac{1}{2} n (\log n)^2 +\gamma n \log n + O \left( \sqrt{n} \log n \right),
\end{eqnarray*}
where we used Lemma \ref{lem:21n} to estimate $H_{\lfloor\sqrt{n}\rfloor}$.
For the second term, Stirling's formula gives
$$
\left\lfloor\sqrt{n}\right\rfloor\log\left(\left\lfloor\sqrt{n}\right\rfloor!\right) = \frac{1}{2} n \log n - n + O \left( \sqrt{n} \log n\right).
$$
For the third term, the harmonic number estimate in Lemma \ref{lem:21n} gives
$$
nH\left(\sqrt{n}\right) = \frac{1}{2} n \log n + \gamma n + O \left(\sqrt{n} \right).
$$
The last two terms are negligible:
$$
\left\lfloor\sqrt{n}\right\rfloor+\sum_{1 \le b\le\sqrt{n}}\int_1^\frac{n}{b}\frac{\{u\}}{u}\,du \le\sqrt{n} + \sum_{1 \le b \le \sqrt{n} } \int_1^n\frac{1}{u}\,du = O \left( \sqrt{n} \log n \right).
$$
Substituting these estimates into the right side of \eqref{eq:oCnn-est} yields
$$
\overline{C}(n,n)= \frac{1}{2} n (\log n)^2 +(\gamma-1) n \log n+(1 -\gamma)n +O \left( \sqrt{n} \log n \right).
$$
(2) We will prove that for $2 \le x\le n$
\begin{equation}\label{eqn:2.16v32}
\overline{C}(n,n)-\overline{C}(n,x)=\left(H_{\left\lfloor\frac{n}{x}\right\rfloor}-\frac{x}{n}\left\lfloor\frac{n}{x}\right\rfloor\right)(n\log n-n)-\left(J_{\left\lfloor\frac{n}{x}\right\rfloor}-\frac{x}{n}\left\lfloor\frac{n}{x}\right\rfloor\log\frac{n}{x}\right)n+O\left(\frac{n\log n}{x}\right).
\end{equation}
and then deduce \eqref{eqn:oCn-integral}.\\
First, we prove \eqref{eqn:2.16v32}. We replace $x$ by $\frac{n}{x}$ in Lemma \ref{lem:oCfe}, and rearrange a term to obtain
\begin{equation}\label{eqn:2.17v32}
\overline{C}(n,n)-\overline{C}(n,x)= \overline{C}\left(n,\frac{n}{x}\right) +(\log n)\bigg(\sum_{1\le b\le\frac{n}{x}}\left\lfloor\frac{n}{b}\right\rfloor\bigg)
-\left\lfloor\frac{n}{x}\right\rfloor\log(\lfloor x\rfloor!)-nH_{\left\lfloor\frac{n}{x}\right\rfloor}+
\bigg(\left\lfloor\frac{n}{x}\right\rfloor+\sum_{1\le b\le\frac{n}{x}}\int_1^\frac{n}{b}\frac{\{u\}}{u}\,du\bigg).
\end{equation}
We estimate the terms on the right side of \eqref{eqn:2.17v32}.
For the first term, using $\lfloor t\rfloor=t+O(1)$, we see that
$$
\overline{C}\left(n,\frac{n}{x}\right)=\sum_{1\le b\le\frac{n}{x}}\left\lfloor\frac{n}{b}\right\rfloor\log b=nJ_{\left\lfloor\frac{n}{x}\right\rfloor}+
O\left(\log\left(\left\lfloor\frac{n}{x}\right\rfloor!\right)+1\right).
$$
Using the bounds
$$
0 \le \log\left(\left\lfloor\frac{n}{x}\right\rfloor!\right)\le\left\lfloor\frac{n}{x}\right\rfloor\log\left\lfloor\frac{n}{x}\right\rfloor\le\frac{n\log n}{x},
$$
we obtain the estimate
\begin{equation}\label{eqn:2.18v32}
\overline{C}\left(n,\frac{n}{x}\right)=nJ_{\left\lfloor\frac{n}{x}\right\rfloor}+O\left(\frac{n\log n}{x}\right).
\end{equation}
For the second term, again using $\lfloor t\rfloor=t+O(1)$ we obtain
\begin{equation}\label{eqn:2.19v32}
(\log n)\bigg( \sum_{1\le b\le\frac{n}{x}}\left\lfloor\frac{n}{b}\right\rfloor\bigg)=n(\log n)H_{\left\lfloor\frac{n}{x}\right\rfloor}+O\left(\frac{n\log n}{x}\right).
\end{equation}
For the third term we assert
\begin{equation}\label{eqn:2.20v32}
\left\lfloor\frac{n}{x}\right\rfloor\log(\lfloor x\rfloor!)=n(\log n)\frac{x}{n}\left\lfloor\frac{n}{x}\right\rfloor-n\frac{x}{n}\left\lfloor\frac{n}{x}\right\rfloor\left(1+\log\frac{n}{x}\right)+O\left(\frac{n\log n}{x}\right).
\end{equation}
This estimate follows using Stirling's formula with remainder in the form, for $x \ge 2$,
\begin{equation}\label{eqn:Stirling}
\log\left(\lfloor x\rfloor!\right)=x\log x-x+O(\log x),
\end{equation}
which yields
$$
\left\lfloor\frac{n}{x}\right\rfloor\log(\lfloor x\rfloor!)=\left\lfloor\frac{n}{x}\right\rfloor(x\log x-x)+O\left(\frac{n\log x}{x}\right),
$$
and \eqref{eqn:2.20v32} follows.
For the final term we have, for $n \ge 2$ and $2 \le x\le n$,
\begin{equation}\label{eqn:2.21v32}
\left\lfloor\frac{n}{x}\right\rfloor+\sum_{1\le b\le\frac{n}{x}}\int_1^\frac{n}{b}\frac{\{u\}}{u}\,du\le \frac{n}{x}+\sum_{1\le b\le\frac{n}{x}}\int_1^n\frac{1}{u}\,du = O \left( \frac{n\log n}{x} \right).
\end{equation}
On inserting \eqref{eqn:2.18v32}, \eqref{eqn:2.19v32}, \eqref{eqn:2.20v32}, and \eqref{eqn:2.21v32} into \eqref{eqn:2.17v32} and rearranging, we obtain \eqref{eqn:2.16v32}.\\
Next, we prove \eqref{eqn:oCn-integral}. By the substitution $v=\frac{n}{u}$, we get
\begin{equation}\label{eqn:2.22v32}
\int_x^n\left\lfloor\frac{n}{u}\right\rfloor\log u\,du=n\int_1^\frac{n}{x}\frac{\lfloor v\rfloor}{v^2}\log\frac{n}{v}\,dv.
\end{equation}
The integral has a closed form quadrature:
$$
\frac{d}{dv}\left(\frac{1}{v}-\frac{1}{v}\log\frac{n}{v}\right)=\frac{1}{v^2}\log\frac{n}{v}
$$
valid on unit intervals $b \le v < b+1$ where $\lfloor v \rfloor =b$.
By partial summation, the right side of \eqref{eqn:2.22v32} is then equal to
$$
n\left\lfloor\frac{n}{x}\right\rfloor\left(\frac{x}{n}-\frac{x}{n}\log x\right)-n\sum_{1\le b\le\frac{n}{x}}\left(\frac{1}{b}-\frac{1}{b}\log\frac{n}{b}\right)=x(1-\log x)\left\lfloor\frac{n}{x}\right\rfloor+(n\log n-n)H_{\left\lfloor\frac{n}{x}\right\rfloor}-nJ_{\left\lfloor\frac{n}{x}\right\rfloor}.
$$
We obtain
\begin{align*}
\int_x^n\left\lfloor\frac{n}{u}\right\rfloor\log u\,du&=x(1-\log x)\left\lfloor\frac{n}{x}\right\rfloor+(n\log n-n)H_{\left\lfloor\frac{n}{x}\right\rfloor}-nJ_{\left\lfloor\frac{n}{x}\right\rfloor}\\
&=\left(H_{\left\lfloor\frac{n}{x}\right\rfloor}-\frac{x}{n}\left\lfloor\frac{n}{x}\right\rfloor\right)(n\log n-n)-\left(J_{\left\lfloor\frac{n}{x}\right\rfloor}-\frac{x}{n}\left\lfloor\frac{n}{x}\right\rfloor\log\frac{n}{x}\right)n,
\end{align*}
completing the proof.
\end{proof}
\begin{lem}\label{cor:24}
For real numbers $n\ge2$, we have
$$
\overline{C}\left(n,\sqrt{n}\right)=\frac{1}{8}n(\log n)^2+\gamma_1 n+O\left(\sqrt{n}\log n\right).
$$
\end{lem}
\begin{proof}
By using the estimate $\lfloor t\rfloor=t+O(1)$, we see that
\begin{equation}\label{eqn:oCsqrt}
\overline{C}\left(n,\sqrt{n}\right)=\sum_{1\le b\le\sqrt{n}}\left\lfloor\frac{n}{b}\right\rfloor\log b=nJ\left(\sqrt{n}\right)+O\left(\log\left(\left\lfloor\sqrt{n}\right\rfloor!\right)\right).
\end{equation}
By applying Lemma \ref{lem:27} with $x=\sqrt{n}$, we obtain
$$
J\left(\sqrt{n}\right)=\frac{1}{8}(\log n)^2+\gamma_1+O\left(\frac{\log n}{\sqrt{n}}\right).
$$
Moreover, we have
$$
\log\left(\left\lfloor\sqrt{n}\right\rfloor!\right)\le\left\lfloor\sqrt{n}\right\rfloor\log\left\lfloor\sqrt{n}\right\rfloor\le\frac{1}{2}\sqrt{n}\log n.
$$
Inserting these estimates back into \eqref{eqn:oCsqrt} yields the lemma.
\end{proof}
\subsection{Estimates: $L_i(n)$} \label{subsec:22nn4}
For positive integers $i \ge 1$ and $n \ge 2$, we set
\begin{equation}
L_i(n) := \sum_{b=2}^n b (\log b)^i.
\end{equation}
We give formulas for all $i \ge 1$ but will only need
the cases $i=1,2$ in the sequel.
\begin{lem} \label{lem:Ln01-estimate}
For integers $i\ge1$ and $n\ge2$, we have
\begin{equation}\label{eqn:general-1}
L_i(n)=\int_1^nu(\log u)^i\,du+\theta_i(n)n(\log n)^i,
\end{equation}
where $0\le\theta_i(n)\le1$.
In particular,
\begin{equation} \label{eq:Ln1-estimate}
L_1(n) = \frac{1}{2} n^2 \log n - \frac{1}{4}n^2 + O\left( n \log n\right),
\end{equation}
\begin{equation} \label{eq:Ln2-estimate}
L_2(n) = \frac{1}{2} n^2 (\log n)^2 - \frac{1}{2} n^2 \log n + \frac{1}{4} n^2 + O\left( n (\log n)^2\right).
\end{equation}
\end{lem}
\begin{proof}
The function $u(\log u)^i$, $1\le u\le n$ is increasing. We have lower and upper bounds
$$
\int_1^nu(\log u)^i\,du\le\sum_{b=2}^nb(\log b)^i=L_i(n),
$$
$$
\int_1^nu(\log u)^i\,du\ge\sum_{b=1}^{n-1}b(\log b)^i=L_i(n)-n(\log n)^i.
$$
Thus the assertion \eqref{eqn:general-1} follows.
The assertions \eqref{eq:Ln1-estimate} and \eqref{eq:Ln2-estimate} follow from the first
assertion with the formulas
$$
\int_1^nu\log u\,du=\left[\frac{1}{2} u^2 \log u - \frac{1}{4}u^2\right]_{u=1}^n=\frac{1}{2} n^2 \log n - \frac{1}{4}n^2+\frac{1}{4},
$$
$$
\int_1^nu(\log u)^2\,du=\left[\frac{1}{2} u^2 (\log u)^2 - \frac{1}{2} u^2 \log u + \frac{1}{4} u^2\right]_{u=1}^n=\frac{1}{2} n^2 (\log n)^2 - \frac{1}{2} n^2 \log n + \frac{1}{4} n^2-\frac{1}{4},
$$
completing the proof.
\end{proof}
\begin{rem}
It can be shown by induction on $i$ that
\begin{equation}\label{eqn:general-2}
\int_1 ^n u(\log u)^i\,du=n^2\sum_{k=0}^i\frac{(-1)^kk!}{2^{k+1}}\binom{i}{k}(\log n)^{i-k}+\frac{(-1)^{i+1}i!}{2^{i+1}}.
\end{equation}
\end{rem}
\section{Estimates for ${{\overline{B}}}(n)$}\label{sec:oB}
\par In this section we obtain the estimates for ${{\overline{B}}}(n)=\sum_{b=2}^n\frac{n-1}{b-1}d_b(n)\log b$ given in Theorem \ref{thm:oBn}.
\subsection{Digit sum identity and preliminary reduction}\label{subsec:21
\par Our estimate for ${{\overline{B}}}(n)$ will be derived using the observation that $n$ has exactly $2$ digits in base $b$ when $\sqrt{n}<b\le n$.
\begin{lem}\label{lem:diff1}
Let $j$ and $n$ be positive integers. Denote by $I(j,n)$ the interval $\left(\frac{n}{j+1},\frac{n}{j}\right]\cap\left(\sqrt{n},n\right]$. Then
\begin{enumerate}
\item $I(j,n)$ is empty unless $j<\sqrt{n}$.
\item If $b\in I(j,n)$ is an integer, then $d_b(n) = n- j (b-1)$, in consequence,
\begin{equation}\label{eqn:b-digitsum}
\frac{n-1}{b-1} d_b(n) \log b = (n-1) \left( \frac{n \log b }{b-1}- j \log b\right).
\end{equation}
\end{enumerate}
\end{lem}
\begin{proof}
(1) Suppose that $x\in I(j,n)$. Then $\sqrt{n}<x\le\frac{n}{j}$, and hence $j<\sqrt{n}$.
(2) Since $\frac{n}{j+1}<b\le\frac{n}{j}$, it follows that $\left\lfloor\frac{n}{b}\right\rfloor=j$. Since $b>\sqrt{n}$, it follows that $\left\lfloor\frac{n}{b^i}\right\rfloor=0$ for all $i\ge2$. From \eqref{eqn:dbn-formula}, we have
$$
d_b(n)=n-(b-1)\sum_{i=1}^\infty\left\lfloor\frac{n}{b^i}\right\rfloor=n-j(b-1),
$$
and \eqref{eqn:b-digitsum} follows by multiplying by $\frac{n-1}{b-1}\log b$. This completes the proof.
\end{proof}
We split the sum $B(n)$ into three parts,
the third part being a cutoff term removing all $2 \le b \le \sqrt{n}$,
and the first two parts using the digit sum identity \eqref{eqn:b-digitsum}.
applied to the range $\sqrt{n} < b \le n$.
\begin{lem}\label{prop:32n}
(1) For integers $n \ge 2$, we have
\begin{equation}\label{eqn:identity}
{{\overline{B}}}(n) = {{\overline{B}}}_1(n) - {{\overline{B}}}_2(n) + {{\overline{B}}}_R(n),
\end{equation}
in which ${{\overline{B}}}_1(n)$, ${{\overline{B}}}_2(n)$, and ${{\overline{B}}}_R(n)$ are defined by
\begin{equation}
\label{eqn:oB1}
{{\overline{B}}}_{1}(n) :=n(n-1) \sum_{\sqrt{n} < b \leq n} \frac{\log b }{b-1},
\end{equation}
\begin{equation}\label{eqn:oB2}
\quad\quad\quad \,\,\, {{\overline{B}}}_{2}(n) := (n-1)\sum_{j=1}^{\lfloor \sqrt{n} \,\rfloor } \, j\Bigg(\sideset{}{'}\sum_{\frac{n}{j+1}<b \leq \frac{n}{j}} \log b\Bigg),
\end{equation}
where the prime in the inner sum in \eqref{eqn:oB2} means only $b>\sqrt{n}$ are included, and
\begin{equation}\label{eq:BR}
\quad {{\overline{B}}}_R(n):=\sum_{2\le b\le\sqrt{n}}\frac{n-1}{b-1}d_b(n)\log b.
\end{equation}
(2) For integers $n \ge 2$ the remainder term ${{\overline{B}}}_R(n)$ satisfies
\begin{equation}\label{eqn:B-remainder}
0 \le {{\overline{B}}}_{R}(n) \le \frac{3}{2}\,n^{3/2}\log n.
\end{equation}
\end{lem}
\begin{proof}
(1) Recall that ${{\overline{B}}}(n) = \sum_{b=2}^n\frac{n-1}{b-1} d_b(n) \log b.$
The remainder term ${{\overline{B}}}_R(n)$ first cuts off the terms with $2 \le b \le \sqrt{n}$ in the sum.
The other two terms ${{\overline{B}}}_1(n)$ and ${{\overline{B}}}_2(n)$ are obtained by applying
the decomposition \eqref{eqn:b-digitsum} of Lemma \ref{lem:diff1} to each index $b\in\left(\sqrt{n},n\right]$ term by term.
(2) From \eqref{eqn:dbnineq}, it follows that $0\le\frac{n-1}{b-1}d_b(n)\log b\le(n-1)\log(n+1)$. Summing from $b=2$ to $\left\lfloor\sqrt{n}\right\rfloor$, we obtain
$$
0\le {{\overline{B}}}_R(n)\le\left(\left\lfloor\sqrt{n}\right\rfloor-1\right)(n-1)\log(n+1)\le\left(\sqrt{n}\right)(n)\left(\frac{3}{2}\log n\right)=\frac{3}{2}n^{3/2}\log n
$$
as desired.
\end{proof}
The sums ${{\overline{B}}}_{1}(n)$ and ${{\overline{B}}}_{2}(n)$ are of comparable sizes, on the order of $n^2\log n$.
We estimate them separately.
\subsection{Estimate for ${{\overline{B}}}_{1}(n)$}
\label{subsec:B21asympt}
\begin{lem}\label{lem:oB1}
Let ${{\overline{B}}}_{1}(n)= n(n-1) \sum_{\sqrt{n} < b \leq n} \frac{\log b }{b-1}$.
Then for integers $n\ge2$, we have
\begin{equation}
\label{eqn:Uncond-oB1-asymp}
{{\overline{B}}}_{1}(n)=\frac{3}{8}n^2(\log n)^2+O\left(n^{3/2}\log n\right).
\end{equation}
\end{lem}
\begin{proof}
We rewrite the sum $\frac{{{\overline{B}}}_{1}(n)}{n(n-1)}$ as
\begin{equation}\label{eqn:oB1n-estimate-start}
\frac{{{\overline{B}}}_{1}(n)}{n(n-1)}=\sum_{\sqrt{n}<b\le n}\frac{\log b}{b-1}=\sum_{\sqrt{n}<b\le n}\frac{\log b}{b}+\sum_{\sqrt{n}<b\le n}\frac{\log b}{b(b-1)}.
\end{equation}
The contribution from the last sum in \eqref{eqn:oB1n-estimate-start} is negligible:
\begin{equation}\label{eqn:oB1n-estimate-1}
0\le\sum_{\sqrt{n}<b\le n}\frac{\log b}{b(b-1)}\le(\log n)\sum_{b>\sqrt{n}}\frac{1}{b(b-1)}=\frac{\log n}{\left\lfloor\sqrt{n}\right\rfloor}\le\frac{2\log n}{\sqrt{n}}.
\end{equation}
We use Lemma \ref{lem:27} to estimate the first sum on the right in \eqref{eqn:oB1n-estimate-start} and obtain
\begin{align}
\sum_{\sqrt{n}<b\le n}\frac{\log b}{b}&=J(n)-J\left(\sqrt{n}\right)\nonumber\\
&=\frac{1}{2}(\log n)^2-\frac{1}{2}\left(\log\sqrt{n}\right)^2+O\left(\frac{\log n}{\sqrt{n}}\right)\nonumber\\
&=\frac{3}{8}(\log n)^2+O\left(\frac{\log n}{\sqrt{n}}\right).\label{eqn:oB1n-estimate-2}
\end{align}
On inserting \eqref{eqn:oB1n-estimate-1} and \eqref{eqn:oB1n-estimate-2} into \eqref{eqn:oB1n-estimate-start}, we obtain
$$
\frac{{{\overline{B}}}_{1}(n)}{n(n-1)}=\frac{3}{8}(\log n)^2+O\left(\frac{\log n}{\sqrt{n}}\right).
$$
On multiplying by $n(n-1)$, we obtain \eqref{eqn:Uncond-oB1-asymp} as desired.
\end{proof}
\subsection{Estimate for ${{\overline{B}}}_{2}(n)$}\label{subsec:B12asympt}
\begin{lem} \label{lem:oB2}
Let
$${{\overline{B}}}_{2}(n) := (n-1)\sum_{j=1}^{\lfloor\sqrt{n}\rfloor }j\Bigg(\sideset{}{'}\sum_{\frac{n}{j+1}<b\leq \frac{n}{j}} \log b \Bigg),$$
where the prime in the inner sum means only $b>\sqrt{n}$ are included. Then for integers $n \ge 2$,
\begin{equation} \label{eqn:oB2-asymp}
{{\overline{B}}}_{2}(n) =\frac{3}{8} n^{2}(\log n)^2 +(\gamma-1)n^{2}\log n +\left(1- \gamma -\gamma_1\right)n^2 +O\left(n^{3/2} \log n \right),
\end{equation}
where $\gamma$ is Euler's constant and $\gamma_1$ is the first Stieltjes constant.
\end{lem}
\begin{proof}
We have
$$
\frac{{{\overline{B}}}_2(n)}{n-1}=\sum_{j=1}^{\lfloor\sqrt{n}\rfloor }\bigg( \sideset{}{'}\sum_{\frac{n}{j+1}<b\leq \frac{n}{j}} \left\lfloor\frac{n}{b}\right\rfloor\log b\bigg) =\sum_{\sqrt{n}<b\le n} \left\lfloor\frac{n}{b}\right\rfloor\log b=\overline{C}(n,n)-\overline{C}\left(n,\sqrt{n}\right).
$$
Applying Proposition \ref{prop:23n} and Lemma \ref{cor:24} to estimate $\overline{C}(n,n)$ and $\overline{C}\left(n,\sqrt{n}\right)$, we obtain
\begin{align*}
\frac{{{\overline{B}}}_2(n)}{n-1}&=\left(\frac{1}{2}n(\log n)^2+(\gamma-1)n\log n+(1-\gamma)n\right)-\left(\frac{1}{8}n(\log n)^2+\gamma_1n\right)+O\left(\sqrt{n}\log n\right)\\
&=\frac{3}{8}n(\log n)^2+(\gamma-1)n\log n+\left(1-\gamma-\gamma_1\right)n+O\left(\sqrt{n}\log n\right).
\end{align*}
On multiplying by $(n-1)$, we obtain \eqref{eqn:oB2-asymp} as desired.
\end{proof}
\subsection{Proof of Theorem \ref{thm:oBn}}
We combine the results on ${{\overline{B}}}_1(n)$ and ${{\overline{B}}}_2(n)$ to estimate ${{\overline{B}}}(n)$.
\begin{proof}[Proof of Theorem \ref{thm:oBn}]
We estimate ${{\overline{B}}}(n)$. We start from the Lemma \ref{prop:32n} decomposition
${{\overline{B}}}(n)= {{\overline{B}}}_{1}(n)- {{\overline{B}}}_{2}(n)+{{\overline{B}}}_R(n).$
By Lemma \ref{prop:32n} (2) we have ${{\overline{B}}}_R(n) = O(n^{3/2}\log n)$,
which is absorbed in the remainder term estimate in the theorem statement.
Substituting the formulas of Lemma \ref{lem:oB1} and Lemma \ref{lem:oB2}, we obtain
\begin{align*}
{{\overline{B}}}(n) &= {{\overline{B}}}_{1}(n)-{{\overline{B}}}_{2}(n)+ {{\overline{B}}}_{R}(n) \\
&= \frac{3}{8}n^{2}(\log n)^2 +O\left(n^{3/2} \log n\right) \\
& \quad\quad -\left( \frac{3}{8}n^2(\log n)^2 +(\gamma-1)n^{2}\log n + (1-\gamma -\gamma_1) n^2 +O\left(n^{3/2} \log n \right)\right) \\
&=(1-\gamma)n^{2}\log n +(\gamma+\gamma_1 -1) n^2 + O\left( n^{3/2} \log n \right),
\end{align*}
as asserted.
\end{proof}
\section{Estimates for ${{\overline{A}}}(n)$ and ${\overline{\overline{G}}}_n$}\label{sec:AH}
In this section we derive asymptotics for ${{\overline{A}}}(n) = \sum_{b=2}^n \frac{2}{b-1}S_{b}(n)\log b$
given in Theorem \ref{thm:oAn} and deduce the estimate for $\log {\overline{\overline{G}}}_n$ given in Theorem \ref{thm:oHn}.
In the case of binomial products ${\overline{G}}_n$ treated in \cite{DL:20} an asymptotic for $A(n)$ was obtained from the
relation $\log {\overline{G}}_n= A(n) - B(n)$ and the existence of a good estimate for $\log {\overline{G}}_n$ coming from
its expression as a product of factorials. Here we do not have a corresponding direct estimate for $\log {\overline{\overline{G}}}_n$, so
we must estimate ${{\overline{A}}}(n)$ directly. The proof details have some parallel with those for ${{\overline{B}}}(n)$.
\subsection{Preliminary reduction}\label{subsec:31}
Recall that ${{\overline{A}}}(n) = \sum_{b=2}^n\frac{2}{b-1} S_b(n) \log b.$
\begin{lem}\label{lem:31a}
For integers $n\ge2$, we have
\begin{equation}\label{eqn:oAn-simplified}
{{\overline{A}}}(n) = {{\overline{A}}}_1(n)+ O \left( n (\log n)^2 \right),
\end{equation}
where
\begin{equation}\label{eqn:oA1n}
{{\overline{A}}}_1(n) := \sum_{b=2}^n \sum_{j=2}^{n}\frac{2\log b}{b}d_b(j).
\end{equation}
\end{lem}
\begin{proof}
We rewrite the sum \eqref{eqn:oA-function-0}
that defines ${{\overline{A}}}(n)$ using the identity
\begin{equation}\label{eqn:identity1}
\frac{1}{b-1} = \frac{1}{b} + \frac{1}{b(b-1)}
\end{equation}
and obtain
\begin{equation} \label{eqn:oAn-start}
{{\overline{A}}}(n) = \sum_{b=2}^n \frac{2\log b}{b} S_b(n) + \sum_{b=2}^n \frac{2 \log b}{b(b-1)} S_b(n).
\end{equation}
Since $S_b(n) =\sum_{j=1}^{n-1} d_b(j)$, the first sum on the right in \eqref{eqn:oAn-start} is
\begin{equation}\label{eqn:oAn&oA1n}
\sum_{b=2}^n \frac{2\log b }{b} S_b(n) = \sum_{b=2}^n \sum_{j=1}^{n-1}\frac{2\log b}{b} d_b(j) = {{\overline{A}}}_1(n)-\sum_{b=2}^n \frac{2\log b}{b}\left(d_b(n)-1\right).
\end{equation}
By Lemma \ref{lem:dbn-Sbn-bound},
$$
0\le d_b(n)-1\le\frac{(b-1)\log(n+1)}{\log b}-1<\frac{b\log(n+1)}{\log b}.
$$
So the last sum in \eqref{eqn:oAn&oA1n} satisfies, for $n \ge 2,$
$$
0\le\sum_{b=2}^n \frac{2\log b}{b}\left(d_b(n)-1\right)<\sum_{b=2}^n2\log(n+1) \le 2n\log n.
$$
Hence
\begin{equation}\label{eqn:sumSestimate1}
\sum_{b=2}^n \frac{2\log b}{b}S_b(n)={{\overline{A}}}_1(n)+O(n\log n).
\end{equation}
Now, we treat the last sum in \eqref{eqn:oAn-start}. We apply Lemma \ref{lem:dbn-Sbn-bound} to bound $S_b(n)$, obtaining
\begin{equation}\label{eqn:sumSestimate2}
\sum_{b=2}^n \frac{2 \log b}{b(b-1)} S_b(n)\le\sum_{b=2}^n\frac{n\log n}{b}\ll n(\log n)^2.
\end{equation}
On inserting \eqref{eqn:sumSestimate1} and \eqref{eqn:sumSestimate2} into \eqref{eqn:oAn-start}, we obtain \eqref{eqn:oAn-simplified} as desired.
\end{proof}
\subsection{Estimate for ${{\overline{A}}}_{1}(n)$ reduction}\label{subsec:32}
\label{sec:oA1asympt}
\begin{lem} \label{lem:32a}
(1) For integers $n \ge 2$, the sum ${{\overline{A}}}_1(n)$ given by \eqref{eqn:oA1n} can be rewritten as
\begin{equation}\label{eqn:oAn-simplified2}
{{\overline{A}}}_1(n) = {{\overline{A}}}_{11}(n) + {{\overline{A}}}_{12}(n) - {{\overline{A}}}_{R}(n),
\end{equation}
where
\begin{equation}\label{eqn:oA11n}
{{\overline{A}}}_{11}(n) := \sum_{j=2}^n \frac{2}{j-1} {{\overline{B}}}(j),
\end{equation}
\begin{equation}\label{eqn:oA12n}
\quad\quad {{\overline{A}}}_{12} (n) := \sum_{j=2}^n \sum_{b=j+1}^n \frac{2j\log b}{b},
\end{equation}
\begin{equation}\label{eqn:oARn}
\quad\quad\quad {{\overline{A}}}_{R} (n) := \sum_{j=2}^{n} \sum_{b=2}^j \frac{2\log b}{b(b-1)} d_b(j),
\end{equation}
and ${{\overline{B}}}(n)$ is given by \eqref{eqn:oB-function-0}.\\
(2) For integers $n \ge 2$, we have
\begin{equation}
{{\overline{A}}}_R(n) \le 3 n (\log n)^2.
\end{equation}
\end{lem}
\begin{proof}
(1)
We start from \eqref{eqn:oA1n} and interchange the order of summation, obtaining
\begin{eqnarray}\label{eq:410}
{{\overline{A}}}_1(n) &=& \sum_{b=2}^n \sum_{j=2}^{n} \frac{2\log b}{b} d_b(j) \nonumber \\
& = & \sum_{j=2}^{n} \sum_{b=2}^n \frac{2\log b}{b}d_b(j) \nonumber\\
&=& \sum_{j=2}^{n} \sum_{b=2}^j \frac{2\log b}{b}d_b(j)+ \sum_{j=2}^{n} \sum_{b=j+1}^n \frac{2\log b}{b}d_b(j).
\end{eqnarray}
Recall that ${{\overline{B}}}(j) =\sum_{b=2}^j \frac{j-1}{b-1}d_{b}(j)\log b$. We next use the identity \eqref{eqn:identity1}
to rewrite the first sum on the right in \eqref{eq:410}:
$$
\sum_{j=2}^{n} \sum_{b=2}^j \frac{2\log b}{b} d_b(j) = \sum_{j=2}^{n}\sum_{b=2}^j\frac{2\log b}{b-1}d_b(j)-\sum_{j=2}^{n}\sum_{b=2}^j \frac{2\log b}{b(b-1)} d_b(j) = {{\overline{A}}}_{11}(n) - {{\overline{A}}}_{R}(n).
$$
Finally, we note that $d_b(j)=j$ for $j<b$; so the second sum on the right in \eqref{eq:410} is
$$
\sum_{j=2}^{n}\sum_{b=j+1}^n \frac{2\log b}{b}d_b(j) = \sum_{j=2}^{n} \sum_{b=j+1}^n \frac{2j\log b}{b}={{\overline{A}}}_{12}(n).
$$
(2) We first bound ${{\overline{A}}}_{R}(n)$ by
$$
0 \le {{\overline{A}}}_{R}(n) \le \sum_{j=2}^n\sum_{b=2}^{n} \frac{2\log b}{b(b-1)} d_b(j) = \sum_{b=2}^n \frac{2 \log b}{b(b-1)} \left(S_b(n+1)-1\right)
$$
Applying Lemma \ref{lem:dbn-Sbn-bound}, to bound the last quantity, we obtain for $n \ge 2$,
$$
{{\overline{A}}}_{R}(n) < \sum_{b=2}^n \frac{(n+1)\log(n+1)}{b} \le 3n(\log n)^2,
$$
as asserted.
\end{proof}
\subsection{Estimates for ${{\overline{A}}}_{11}(n)$ and ${{\overline{A}}}_{12}(n)$}\label{sec:33}
\label{subsec:oA11asympt}
\begin{lem} \label{lem:oA11}
For integers $n \ge 2,$ we have
\begin{equation}\label{eqn:oA11bound}
{{\overline{A}}}_{11}(n)=(1-\gamma)n^2\log n+\left(\frac{3}{2}\gamma+\gamma_1-\frac{3}{2}\right)n^2+O\left(n^{3/2}\log n\right).
\end{equation}
\end{lem}
\begin{proof}
We start from \eqref{eqn:oA11n} and use the identity \eqref{eqn:identity1} to rewrite ${{\overline{A}}}_{11}(n)$:
\begin{equation}\label{eqn:A11n-start}
{{\overline{A}}}_{11}(n)=\sum_{j=2}^{n} \frac{2}{j-1}{{\overline{B}}}(j)=\sum_{j=2}^n\frac{2}{j}{{\overline{B}}}(j)+\sum_{j=2}^n\frac{2}{j(j-1)}{{\overline{B}}}(j).
\end{equation}
From Theorem \ref{thm:oBn}, it follows that ${{\overline{B}}}(j)\ll j(j-1)\log j$ for $j\ge2$. As a result, the contribution from the last sum in \eqref{eqn:A11n-start} is negligible:
\begin{equation}\label{eqn:oA11n-part1}
\sum_{j=2}^n\frac{2}{j(j-1)}{{\overline{B}}}(j)\ll \sum_{j=2}^n\log j\le\sum_{j=2}^n\log n<n\log n.
\end{equation}
Now, we estimate the first sum on the right of \eqref{eqn:A11n-start} using Theorem \ref{thm:oBn}:
\begin{align*}
\sum_{j=2}^n\frac{2}{j}{{\overline{B}}}(j)&=2(1-\gamma)\sum_{j=2}^nj\log j+2\left(\gamma+\gamma_1-1\right)\sum_{j=2}^nj+O\bigg(\sum_{j=2}^n\sqrt{j}\log j\bigg)\\
&=2(1-\gamma)\sum_{j=2}^nj\log j+2\left(\gamma+\gamma_1-1\right)\left(\frac{1}{2}n^2+\frac{1}{2}n-1\right)+O\bigg(\sum_{j=2}^n\sqrt{n}\log n\bigg)\\
&=2(1-\gamma)\sum_{j=2}^nj\log j+\left(\gamma+\gamma_1-1\right)n^2+O\left(n^{3/2}\log n\right).
\end{align*}
We use Lemma \ref{lem:Ln01-estimate} to estimate $\sum_{j=2}^nj\log j$ and obtain
\begin{equation}\label{eqn:oA11n-part2}
\sum_{j=2}^n\frac{2}{j}{{\overline{B}}}(j)=(1-\gamma)n^2\log n+\left(\frac{3}{2}\gamma+\gamma_1-\frac{3}{2}\right)n^2+O\left(n^{3/2}\log n\right).
\end{equation}
On inserting \eqref{eqn:oA11n-part1} and \eqref{eqn:oA11n-part2} into \eqref{eqn:A11n-start}, we obtain \eqref{eqn:oA11bound} as desired.
\end{proof}
\begin{lem} \label{lem:oA12}
For integers $n \ge 2$, we have
$${{\overline{A}}}_{12}(n)=\frac{1}{2}n^2\log n-\frac{1}{4}n^2+O\left(n(\log n)^2\right).$$
\end{lem}
\begin{proof} We can rewrite \eqref{eqn:oA12n} in terms of $J(x)=\sum_{1\le b\le x}\frac{\log b}{b}$ as
$${{\overline{A}}}_{12} (n) = \sum_{j=2}^n 2j (J(n) - J( j)).$$
For $2\le j\le n$, it follows from Lemma \ref{lem:27} that
$$
J(n)- J(j) = \frac{1}{2} (\log n)^2- \frac{1}{2}(\log j)^2 + O \left(\frac{\log n}{j}\right).
$$
Hence
\begin{align*}
{{\overline{A}}}_{12}(n) &=\sum_{j=2}^n j (\log n)^2- \sum_{j=2}^n j(\log j)^2 +O\bigg(\sum_{b=2}^n\log n\bigg)\\
&= \left(\frac{1}{2}n^2+\frac{1}{2}n -1\right)(\log n)^2 - \sum_{j=2}^n j (\log j)^2 + O \left( n \log n \right).
\end{align*}
We use Lemma \ref{lem:Ln01-estimate} to estimate $\sum_{j=2}^nj(\log j)^2$ and obtain
$$
{{\overline{A}}}_{12}(n) = \frac{1}{2} n^2 \log n -\frac{1}{4} n^2 + O \left( n (\log n)^2 \right),
$$
as desired.
\end{proof}
\subsection{Proofs of Theorems \ref{thm:oAn} and \ref{thm:oHn} }\label{sec:35}
We derive the estimate for ${{\overline{A}}} (n)$ in Theorem \ref{thm:oAn} and that for ${\overline{\overline{G}}}_n$
in Theorem \ref{thm:oHn}.
\begin{proof}[Proof of Theorem \ref{thm:oAn}]
By Lemma \ref{lem:31a} and Lemma \ref{lem:32a},
$$
{{\overline{A}}}(n) = {{\overline{A}}}_1(n)+O \left( n (\log n)^2\right)= {{\overline{A}}}_{11}(n) + {{\overline{A}}}_{12}(n) + O \left( n (\log n)^2\right).
$$
Inserting the estimates of Lemma \ref{lem:oA11} for ${{\overline{A}}}_{11}(n)$ and Lemma \ref{lem:oA12} for ${{\overline{A}}}_{12}(n)$ yields
$$
{{\overline{A}}} (n) = \left(\frac{3}{2} -\gamma\right)n^2\log n+\left(\frac{3}{2}\gamma+\gamma_1-\frac{7}{4}\right)n^2+O\left(n^{3/2}\log n\right),
$$
as required.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:oHn}]
The estimate for ${\overline{\overline{G}}}_n$ follows from the relation $\log {\overline{\overline{G}}}_n = {{\overline{A}}}(n) - {{\overline{B}}}(n)$ using
the estimates of Theorem \ref{thm:oBn} for ${{\overline{B}}}(n)$ and Theorem \ref{thm:oAn} for ${{\overline{A}}}(n)$.
\end{proof}
\section{Estimates for the generalized partial factorization sums ${{\overline{B}}}(n,x)$}\label{sec:asymp-oBnx}
We derive estimates for ${{\overline{B}}}(n,x)$ in the interval $1 \le x \le n$
starting from the asymptotic estimates for ${{\overline{B}}}(n)= {{\overline{B}}}(n,n)$.
Let $H_m= \sum_{k=1}^m \frac{1}{k}$ denote the $m$-th harmonic number.
\begin{thm}\label{thm:oBnx}
Let ${{\overline{B}}}(n,x) = \sum_{b=2}^{\lfloor x\rfloor}\frac{n-1}{b-1} d_b(n) \log b.$
Then for integers $n \ge 2$ and real $x\in\left[\sqrt{n},n\right]$,
\begin{equation}\label{eqn:oBnx-bound}
{{\overline{B}}}(n,x) = {{\overline{B}}}_0(n,x) \, n^2 \log n + {{\overline{B}}}_1(n, x) \, n^2 + O \left( n^{3/2} \log n\right),
\end{equation}
where the functions ${{\overline{B}}}_0(n,x)$ and ${{\overline{B}}}_1(n,x)$ only depend on $\frac{x}{n}$ and are given by
\begin{equation}\label{eqn:oB0nx-bound}
{{\overline{B}}}_0(n,x) := \left(1-\gamma\right)+ \left( H_{\lfloor \frac{n}{x} \rfloor} - \log \frac{n}{x}\right)
- \frac{x}{n}\left\lfloor \frac{n}{x} \right\rfloor
\end{equation}
and
\begin{align}
{{\overline{B}}}_1(n,x) := &
\left(\gamma+ \gamma_1 -1\right)- \left( H_{\lfloor \frac{n}{x}\rfloor} - \log \frac{n}{x} \right) - \left( J_{\lfloor \frac{n}{x} \rfloor} - \frac{1}{2}\left(\log \frac{n}{x} \right)^2 \right)\nonumber\\
& - \log \frac{n}{x}+ \left(\log \frac{n}{x}\right) \frac{x}{n} \left\lfloor \frac{n}{x} \right\rfloor + \frac{x}{n}\left\lfloor \frac{n}{x} \right\rfloor.\label{eqn:oB1nx-bound}
\end{align}
Moreover, for integers $n\ge2$ and real $x\in\left[1,\sqrt{n}\right]$,
\begin{equation}\label{eqn:oBnx-bound2}
{{\overline{B}}}(n, x) = O \left( n^{3/2} \log n \right) .
\end{equation}
\end{thm}
\begin{rem}\label{rmk:52}
The functions ${{\overline{B}}}_0(n,x)$ and ${{\overline{B}}}_1(n, x)$ above are functions of a single variable $\alpha :=\frac{x}{n}$
having $0 \le \alpha \le 1.$ That is, the answer displays a {\em scale invariance}, in terms of the variables $x$
and $n$.
However various intermediate parts of the proof involve terms in $n$ and $x$ that are not scale invariant.
\end{rem}
\subsection{Preliminary reduction} \label{subsec:prelim5}
We write
\begin{equation}
{{\overline{B}}}(n, x) = {{\overline{B}}}(n) - {{\overline{B}}}^c(n,x),
\end{equation}
where $ {{\overline{B}}}^c(n,x)$ is the complement sum
\begin{equation}\label{eqn:Bcnx}
{{\overline{B}}}^c(n,x) :=
\sum_{x < b \le n} \frac{n-1}{b-1} d_b(n) \log b.
\end{equation}
The sum ${{\overline{B}}}(n)$ can be estimated by Theorem \ref{thm:oBn}.
To estimate ${{\overline{B}}}^c(n,x)$, we break it into two parts.
\begin{lem}\label{lem:51a}
For integers $n \ge 2$ and real numbers $x$ such that $\sqrt{n} \le x\le n$, we have
\begin{equation}\label{eqn:Bcc}
{{\overline{B}}}^c(n,x) = {{\overline{B}}}_{11}^c(n,x) - (n-1) \left(\overline{C}(n, n) - \overline{C}(n,x)\right),
\end{equation}
where $\overline{C}(n,x)$ is given by \eqref{eqn:oC-defn} and
\begin{equation}\label{eqn:B11c}
{{\overline{B}}}_{11}^c(n,x) := n (n-1) \sum_{x < b \le n} \frac{\log b}{b-1}.
\end{equation}
\end{lem}
\begin{proof}
Recall from \eqref{eqn:dbn-formula} that
$d_b(n)= n- (b-1)\sum_{i=1}^{\infty} \left\lfloor \frac{n}{b^i} \right\rfloor$.
Since $x > \sqrt{n}$, if $b>x$, then $b^2>x^2\ge n$, and hence $\left\lfloor\frac{n}{b^i}\right\rfloor=0$ for all $i\ge2$. In this circumstance, $d_b(n)=n-(b-1)\left\lfloor\frac{n}{b}\right\rfloor$ for $b>x$.
Inserting this formula into the definition \eqref{eqn:Bcnx}, we obtain
\begin{align}
{{\overline{B}}}^c(n,x)&=n(n-1)\sum_{x<b\le n}\frac{\log b}{b-1}-(n-1)\sum_{x<b\le n}\left\lfloor\frac{n}{b}\right\rfloor\log b\nonumber\\
&= {{\overline{B}}}_{11}^c(n,x) - (n-1) \left(\overline{C}(n, n) - \overline{C}(n,x)\right)
\end{align}
as required.
\end{proof}
\subsection{Estimate for ${{\overline{B}}}_{11}^c(n,x)$}\label{sec:51}
\begin{lem}\label{lem:52}
For real numbers $n \ge 2$ and $x$ such that $1\le x\le n$, we have
\begin{equation}\label{eqn:oBc11-estimate}
{{\overline{B}}}_{11}^c(n,x) = \frac{1}{2}n^2(\log n)^2-\frac{1}{2}n^2(\log x)^2+O\left(\frac{n^2\log n}{x}\right).
\end{equation}
\end{lem}
\begin{proof}
We start from \eqref{eqn:B11c} and use the identity \eqref{eqn:identity1} to rewrite $\frac{1}{n(n-1)}{{\overline{B}}}_{11}^c(n, x)$:
\begin{equation}\label{eqn:oBc11-start}
\frac{1}{n(n-1)}{{\overline{B}}}_{11}^c(n, x) = \sum_{x < b \le n} \frac{\log b}{b} + \sum_{x< b \le n} \frac{\log b}{b (b-1)}.
\end{equation}
The contribution from the last sum in \eqref{eqn:oBc11-start} is negligible:
\begin{equation}\label{eqn:oBc11-error}
0\le \sum_{x< b \le n} \frac{\log b}{b (b-1)}<(\log n)\sum_{b>x}\frac{1}{b(b-1)}=\frac{\log n}{\lfloor x\rfloor}<\frac{2\log n}{x}.
\end{equation}
The first sum on the right in \eqref{eqn:oBc11-start} can be estimated using Lemma \ref{lem:27}:
\begin{equation}\label{eqn:oBc11-main}
\sum_{x < b \le n} \frac{\log b}{b}=J(n)- J(x)=\frac{1}{2}(\log n)^2 - \frac{1}{2}(\log x)^2 + O\left( \frac{\log n}{x}\right).
\end{equation}
On inserting \eqref{eqn:oBc11-error} and \eqref{eqn:oBc11-main} into \eqref{eqn:oBc11-start}, we obtain
$$
\frac{1}{n(n-1)}{{\overline{B}}}_{11}^c(n,x) = \frac{1}{2}(\log n)^2-\frac{1}{2}(\log x)^2+O\left(\frac{\log n}{x}\right).
$$
On multiplying by $n(n-1)$, we obtain
$$
{{\overline{B}}}_{11}^c(n,x) = \frac{1}{2}n^2(\log n)^2-\frac{1}{2}n^2(\log x)^2-\left(\frac{1}{2}n(\log n)^2-\frac{1}{2}n(\log x)^2\right)+O\left(\frac{n^2\log n}{x}\right).
$$
Since $e^t=1+t+\frac{t^2}{2}+\dots>t$ for $t>0$, it follows that $\log\frac{n}{x}<\frac{n}{x}$ and
$$
0\le\frac{1}{2}n(\log n)^2-\frac{1}{2}n(\log x)^2=\frac{1}{2}n\left(\log\frac{n}{x}\right)(\log n+\log x)<\frac{1}{2}n\left(\frac{n}{x}\right)(2\log n)=\frac{n^2\log n}{x}.
$$
Hence \eqref{eqn:oBc11-estimate} follows.
\end{proof}
\subsection{Estimate for ${{\overline{B}}}^c(n,x)$}\label{subsec:54}
We obtain an asymptotic estimate for ${{\overline{B}}}^c(n,x)$.
\begin{prop}\label{prop:56}
For integers $n \ge 2$ and real numbers $x$ such that $\sqrt{n}\le x\le n$, we have
\begin{eqnarray}\label{eqn:oBc0}
{{\overline{B}}}^c(n,x) &=& n^2\left( J \left( \frac{n}{x} \right) - \frac{1}{2} \left(\log \frac{n}{x}\right)^2 \right) -
(n^2 \log n - n^2) \left( H_{\lfloor \frac{n}{x} \rfloor} - \log \frac{n}{x}\right) \nonumber \\
&&+\,n^2\left( 1- \left\lfloor \frac{n}{x} \right\rfloor \frac{x}{n} \right) \log \frac{n}{x}
+ (n^2\log n -n^2) \left\lfloor \frac{n}{x} \right\rfloor \frac{x}{n}
+O \left( n^{3/2} \log n\right).
\end{eqnarray}
\end{prop}
\begin{proof}
We have
\begin{equation}\label{eqn:oBnc}
{{\overline{B}}}^c(n,x) = {{\overline{B}}}_{11}^c(n,x) - (n-1)\left( \overline{C}(n,n)- \overline{C}(n, x)\right).
\end{equation}
We suppose $\sqrt{n} \le x\le n.$ From Lemma \ref{lem:52} we obtain
\begin{equation}\label{eqn:oB11c}
{{\overline{B}}}_{11}^c(n,x) = \frac{1}{2} n^2 \left( (\log n)^2 - (\log x)^2\right) + O \left( n^{3/2} \log n\right).
\end{equation}
Now Proposition \ref{prop:23n}(2) gives for $2 \le x \le n$,
\begin{equation}
\overline{C}(n,n)-\overline{C}(n,x) =
\left(H_{\left\lfloor\frac{n}{x}\right\rfloor}-\frac{x}{n}\left\lfloor\frac{n}{x}\right\rfloor\right)(n\log n-n)
- \left( J_{\left\lfloor\frac{n}{x}\right\rfloor}- \frac{x}{n}\left\lfloor\frac{n}{x}\right\rfloor \log\frac{n}{x}
\right)n +O\left(\frac{n\log n}{x}\right).
\end{equation}
Substituting these estimates into \eqref{eqn:oBnc}, and assuming $x \ge \sqrt{n}$ yields
\begin{eqnarray}\label{eq:version1}
{{\overline{B}}}^c(n,x) &= &\frac{1}{2} n^2 \left( (\log n)^2 - (\log x)^2\right) + n \left(H_{\left\lfloor\frac{n}{x}\right\rfloor}
-\frac{x}{n}\left\lfloor\frac{n}{x}\right\rfloor \right)(n\log n-n)\nonumber \\
&&- n \left( J_{\left\lfloor\frac{n}{x}\right\rfloor}- \frac{x}{n}\left\lfloor\frac{n}{x}\right\rfloor \log\frac{n}{x}
\right) + O \left( n^{3/2}( \log n)^2 \right).
\end{eqnarray}
In this formula we also replaced a factor $(n-1)$ by $n$, introducing an error $O ( n (\log n)^2)$ absorbed in the remainder term.
Our goal is to simplify this expression to obtain \eqref{eqn:oBc0}. We rewrite \eqref{eq:version1} as
\begin{equation}\label{eq:version2}
{{\overline{B}}}^c(n,x) = n^2 \left( J_{\left\lfloor\frac{n}{x}\right\rfloor} - \frac{1}{2}\left(\log \frac{n}{x}\right)^2 \right )
+(n^2 \log n -n^2 )\left( H_{\left\lfloor\frac{n}{x}\right\rfloor} - \log \frac{n}{x} \right) + {{\overline{B}}}_{21}^c(n,x) +O \left( n^{3/2} (\log n)^2\right),
\end{equation}
where we define
\begin{eqnarray} \label{eq:version3}
{{\overline{B}}}_{21}^c(n,x) &:= &\left( \frac{1}{ 2}n^2 \left(\log \frac{n}{x}\right)^2 - n^2 (\log n) \left( \log \frac{n}{x}\right) + n^2 \left(\log \frac{n}{x}\right) \right) +\frac{1}{2} n^2 \left( (\log n)^2 - (\log x)^2\right)\nonumber \\
&&
-\frac{x}{n}\left\lfloor\frac{n}{x}\right\rfloor \left(n^2\log n -n^2\right)
- \left(\frac{x}{n}\left\lfloor\frac{n}{x}\right\rfloor\right) n^2\log\frac{n}{x} .
\end{eqnarray}
Expanding $\log \frac{n}{x} = \log n - \log x$ in the first two terms in \eqref{eq:version3} gives
$$
\frac{1}{2} \left(\log \frac{n}{x}\right)^2- n^2 (\log n) \left(\log \frac{n}{x}\right) = -\frac{1}{2} n^2(\log n)^2 + \frac{1}{2} n^2 (\log x)^2,
$$
which cancels the next two terms appearing in \eqref{eq:version3}.
Rearranging the remaining uncancelled terms results in
\begin{equation} \label{eq;version4}
{{\overline{B}}}_{21}^c(n,x) = \frac{x}{n}\left\lfloor\frac{n}{x}\right\rfloor \left(n^2\log n -n^2\right)
+ \left(1- \frac{x}{n}\left\lfloor\frac{n}{x}\right\rfloor\right) n^2 \log\frac{n}{x},
\end{equation}
which when substituted in \eqref{eq:version2} yields \eqref{eqn:oBc0}.
\end{proof}
\subsection{Proof of Theorem \ref{thm:oBnx}}\label{subsec:55}
We obtain an estimate of ${{\overline{B}}}(n,x)$.
\begin{proof}[Proof of Theorem \ref{thm:oBnx}]
For $n \ge x \ge 1$ we have the decomposition
\begin{equation}
{{\overline{B}}}(n,x) = B(n) -{{\overline{B}}}^c(n,x).
\end{equation}
By Theorem \ref{thm:oBn} we have for $n \ge 2$,
$$
{{\overline{B}}}(n)=(1-\gamma)n^2\log n+\left(\gamma+\gamma_1-1\right)n^2+O\left(n^{3/2}\log n\right).
$$
By Proposition \ref{prop:56} we have for $n \ge 2$ and $\sqrt{n} \le x \le n$,
\begin{eqnarray*}
{{\overline{B}}}^c(n,x) &:=& n^2\left( J_{\lfloor \frac{n}{x} \rfloor} - \frac{1}{2} \bigg(\log \frac{n}{x}\bigg)^2 \right) -
\left(n^2 \log n - n^2\right) \left( H_{\lfloor \frac{n}{x} \rfloor} - \log \frac{n}{x} \right) \nonumber \\
&& + \, n^2\left( 1- \bigg\lfloor \frac{n}{x} \bigg\rfloor \frac{x}{n} \right) \log \frac{n}{x}
+ \left(n^2\log n -n^2\right)\bigg\lfloor \frac{n}{x} \bigg\rfloor \frac{x}{n}
+O \left( n^{3/2} \log n\right).
\end{eqnarray*}
We obtain for $n \ge 2$ and $\sqrt{n} \le x \le n$,
\begin{eqnarray}\label{eqn:oBnx-estimate51}
{{\overline{B}}}(n,x) & = & {{\overline{B}}}_0(n, x) \, n^2 \log n + {{\overline{B}}}_1 (n,x) \, n^2 + O \left(n^{3/2}\log n \right),
\end{eqnarray}
with
\begin{eqnarray*}
{{\overline{B}}}_0(n, x) &= & (1-\gamma) +\left( H_{\lfloor \frac{n}{x} \rfloor} - \log \frac{n}{x} \right) -\bigg\lfloor \frac{n}{x} \bigg\rfloor \frac{x}{n}\quad\quad\quad\quad
\end{eqnarray*}
and
\begin{eqnarray*}
{{\overline{B}}}_1(n, x) = \left(\gamma+\gamma_1-1\right)-
\left( H_{\lfloor \frac{n}{x} \rfloor} - \log \frac{n}{x} \right)- \left( J_{ \lfloor \frac{n}{x} \rfloor} - \frac{1}{2} \bigg(\log \frac{n}{x}\bigg)^2 \right)
- \left( 1- \bigg\lfloor \frac{n}{x} \bigg\rfloor \frac{x}{n} \right)\log \frac{n}{x} +\bigg\lfloor \frac{n}{x} \bigg\rfloor \frac{x}{n},
\end{eqnarray*}
which is \eqref{eqn:oBnx-bound}.
Finally, for integers $n\ge2$ and real $x\in\left[1,\sqrt{n}\right]$, we have
\begin{align*}
{{\overline{B}}}(n,x)&=\sum_{2\le b\le x}\frac{n-1}{b-1}d_b(n)\log b\\
&\le\sum_{2\le b\le x}(n-1)\log(n+1)\\
&<x(n-1)\log(n+1)\\
&<2n^{3/2}\log n,
\end{align*}
where the bound of Lemma \ref{lem:dbn-Sbn-bound} for $d_b(n)$ was used in the first inequality.
We have obtained \eqref{eqn:oBnx-bound2}.
\end{proof}
\subsection{Proof of Theorem \ref{thm:oBnx-cor}}\label{subsec:56}
\begin{proof}[Proof of Theorem \ref{thm:oBnx-cor}]
We estimate ${{\overline{B}}}(n, \alpha n)$.
The theorem follows on choosing $x= \alpha n$ in Theorem \ref{thm:oBnx} and simplifying. Now ${{\overline{B}}}_0(n, x)= f_{{{\overline{B}}}}(\alpha) $ is
a function of $\alpha= \frac{x}{n}$, with
$$
f_{{{\overline{B}}}}(\alpha) = (1-\gamma) + \left( H_{\lfloor \frac{1}{\alpha}\rfloor} - \log \frac{1}{\alpha} \right) - \alpha \bigg\lfloor \frac{1}{\alpha} \bigg\rfloor
$$
Similarly ${{\overline{B}}}_1(n,x) = g_{{{\overline{B}}}}(\alpha)$ is a function of $\alpha$ with
\begin{eqnarray*}
g_{{{\overline{B}}}}(\alpha) &=& \left(\gamma+\gamma_1-1 \right)-
\left( H_{\lfloor \frac{1}{\alpha} \rfloor} - \log \frac{1}{\alpha} \right) - \left( J_{ \lfloor \frac{1}{\alpha} \rfloor} -
\frac{1}{2} \bigg(\log \frac{1}{\alpha}\bigg)^2 \right) \\
&&\quad + \left( \alpha \bigg\lfloor \frac{1}{\alpha} \bigg\rfloor -1 \right) \log \frac{1}{\alpha} + \bigg\lfloor \frac{1}{\alpha} \bigg\rfloor \alpha.
\end{eqnarray*}
We allow $\frac{1}{\sqrt{n}} \le \alpha \le 1$, and for $n \ge 2$ the remainder term in the estimate is $O \left( n^{3/2} \log n\right)$,
independent of $\alpha$ in this range. For the range $x\in \left[1, \sqrt{n}\right]$ we use the
final estimate \eqref{eqn:oBnx-bound2}.
\end{proof}
\begin{rem}\label{rem:34}
The function $f_{{{\overline{B}}}}(\alpha)$ has $f_{{{\overline{B}}}}(1)=1-\gamma$, and has $\lim_{\alpha \to 0^+} f_{{{\overline{B}}}}(\alpha)=0$ since $ H_{\lfloor \frac{1}{\alpha}\rfloor}- \log \frac{1}{\alpha} \to \gamma$
as $\alpha \to 0^+$. Various individual terms in the formulas for $f_{{{\overline{B}}}}(\alpha)$ and $g_{{{\overline{B}}}}(\alpha)$ are discontinuous at points $\alpha=\frac{1}{k}$ for $k \ge 1$.
The function $f_{{{\overline{B}}}}(\alpha)$ was shown to be continuous on $[0,1]$ in \cite{DL:20}; the function $g_{{{\overline{B}}}}(\alpha)$ can be checked to be continuous.
\end{rem}
\section{Estimates for the generalized partial factorization sums ${{\overline{A}}}(n,x)$}\label{sec:asymp-oAnx}
The main goal of this section is to prove the following theorem.
\begin{thm}\label{thm:oAnx}
Let ${{\overline{A}}}(n,x) = \sum_{b=2}^{\lfloor x\rfloor}\frac{2}{b-1} S_b(n) \log b.$
Then for integers $n \ge 2$ and real $x\in\left[\sqrt{n},n\right]$,
\begin{equation}\label{eqn:oAnx-bound}
{{\overline{A}}}(n,x) = {{\overline{A}}}_0(n,x) n^2 \log n + {{\overline{A}}}_1(n, x) n^2 + O \left(n^{3/2} \log n \right),
\end{equation}
where the functions ${{\overline{A}}}_0(n,x)$ and ${{\overline{A}}}_1(n,x)$ only depend on $\frac{x}{n}$ and are given by
\begin{equation}\label{eqn:oA0-thm}
{{\overline{A}}}_0(n, x) := \left(\frac{3}{2} - \gamma\right) + \left(H_{\lfloor \frac{n}{x} \rfloor} - \log \frac{n}{x} \right)
+\frac{1}{2} \left(\frac{x}{n}\right)^2 \left\lfloor \frac{n}{x} \right\rfloor \left\lfloor \frac{n}{x} + 1\right\rfloor - 2 \left(\frac{x}{n}\right) \left\lfloor \frac{n}{x} \right\rfloor
\end{equation}
and
\begin{align}\label{eqn:oA1-thm}
{{\overline{A}}}_1(n, x) := &\left(\frac{3}{2} \gamma + \gamma_1 -\frac{7}{4}\right)- \frac{3}{2} \left(H_{\lfloor \frac{n}{x} \rfloor} - \log \frac{n}{x} \right) - \left( J_{\lfloor \frac{n}{x} \rfloor} - \frac{1}{2} \left(\log \frac{n}{x}\right)^2 \right) - \frac{3}{2} \log \frac{n}{x} \nonumber \\
& - \frac{1}{2} \left(\log \frac{n}{x}\right) \left(\frac{x}{n}\right)^2 \left\lfloor \frac{n}{x} \right\rfloor \left\lfloor \frac{n}{x} +1\right\rfloor + 2 \left(\log \frac{n}{x}\right) \frac{x}{n} \left\lfloor \frac{n}{x} \right\rfloor
-\frac{1}{4} \left(\frac{x}{n}\right)^2 \left\lfloor \frac{n}{x} \right\rfloor \left\lfloor \frac{n}{x}+1 \right\rfloor +
2\left(\frac{x}{n}\right) \left\lfloor \frac{n}{x} \right\rfloor.
\end{align}
Moreover, for integers $n\ge2$ and real $x\in\left[1,\sqrt{n}\right]$,
\begin{equation}\label{eqn:oAnx-bound2}
{{\overline{A}}}(n,x)=O\left(n^{3/2}\log n\right).
\end{equation}
\end{thm}
We derive estimates for ${{\overline{A}}}(n,x)$ starting from ${{\overline{A}}}(n,n)$ and working downward, via a recursion in Lemma \ref{lem:oA-diff-formula} below.
\subsection{Estimates for the complement sum ${{\overline{A}}}(n,n)-{{\overline{A}}}(n,x)$}
First, we show that ${{\overline{A}}}(n,n)-{{\overline{A}}}(n,x)$ can be written in terms of known quantities, namely ${{\overline{B}}}^c_{11}(n,x)$ and $\overline{C}(j,j)-\overline{C}(j,x)$.
\begin{lem}\label{lem:oA-diff-formula}
For integers $n\ge2$ and real numbers $x$ such that $\sqrt{n-1}\le x\le n$, we have
\begin{equation}\label{eqn:oAnn-oAnx-formula}
{{\overline{A}}}(n,n)-{{\overline{A}}}(n,x)={{\overline{B}}}^c_{11}(n,x)-2\sum_{x\le j<n}\left(\overline{C}(j,j)-\overline{C}(j,x)\right),
\end{equation}
where $\overline{C}(n,x)$ and ${{\overline{B}}}^c_{11}(n,x)$ are given by \eqref{eqn:oC-defn} and \eqref{eqn:B11c}, respectively.
\end{lem}
\begin{proof}
From \eqref{eqn:oA-function}, we have
\begin{equation}\label{eqn:oAnn-oAnx-start}
{{\overline{A}}}(n,n) ={{\overline{A}}}(n,x) + \sum_{x<b\le n}\frac{2}{b-1}S_b(n)\log b
\end{equation}
Observe that for positive integers $b>x$ and $j\le n-1$, we have $b^2>x^2\ge n-1\ge j$, and hence $\left\lfloor \frac{j}{b^i}\right\rfloor=0$ for all $i\ge2$. From \eqref{eqn:Sbn-formula}, if $b>x$, then
$$
S_b(n)=\frac{n(n-1)}{2}-(b-1)\sum_{j=1}^{n-1}\left\lfloor\frac{j}{b}\right\rfloor.
$$
On inserting this into \eqref{eqn:oAnn-oAnx-start}, we obtain
\begin{align*}
{{\overline{A}}}(n,n)-{{\overline{A}}}(n,x)&=n(n-1)\sum_{x<b\le n}\frac{\log b}{b-1}-2\sum_{j=1}^{n-1}\sum_{x<b\le n}\left\lfloor\frac{j}{b}\right\rfloor\log b\\
&={{\overline{B}}}^c_{11}(n,x)-2\sum_{j=1}^{n-1}\left(\overline{C}(j,n)-\overline{C}(j,x)\right).
\end{align*}
From \eqref{eqn:oCstab}, if $1\le j<n$, then $\overline{C}(j,n)=\overline{C}(j,j)$. Hence
$$
{{\overline{A}}}(n,n)-{{\overline{A}}}(n,x)={{\overline{B}}}^c_{11}(n,x)-2\sum_{1\le j<n}\left(\overline{C}(j,j)-\overline{C}(j,x)\right).
$$
From \eqref{eqn:oCstab}, if $1\le j<x$, then $\overline{C}(j,x)=\overline{C}(j,j)$. Hence \eqref{eqn:oAnn-oAnx-formula} follows.
\end{proof}
The next lemma gives an estimate for the sum of values of a dilated floor function. We will use this estimate to prove the main Lemma \ref{lem:oAnn-oAnx-estimate} below.
\begin{lem}\label{lem:kernel-estimate}
For real numbers $t$ and $u$ such that $1\le u\le t$, we have
$$
\sum_{j=1}^{\lfloor t\rfloor}\left\lfloor\frac{j}{u}\right\rfloor=t\left\lfloor\frac{t}{u}\right\rfloor-\frac{1}{2}u\left\lfloor\frac{t}{u}\right\rfloor^2-\frac{1}{2}u\left\lfloor\frac{t}{u}\right\rfloor+O\left(\frac{t}{u}\right).
$$
\end{lem}
\begin{proof} We write $\left\lfloor\frac{j}{u}\right\rfloor=\sum_{1\le k\le\frac{j}{u}}1$ and interchange the order of summation, obtaining
$$
\sum_{j=1}^{\lfloor t\rfloor}\left\lfloor\frac{j}{u}\right\rfloor
=\sum_{1\le j\le t}\bigg( \sum_{1\le k\le\frac{j}{u}}1\bigg) =\sum_{1\le k\le\frac{t}{u}}\bigg( \sum_{uk\le j\le t}1\bigg).
$$
The inner sum on the right counts the number of integers from $\lceil uk\rceil$ to $\lfloor t\rfloor$. Hence the above is
$$
\sum_{j=1}^{\lfloor t\rfloor}\left\lfloor\frac{j}{u}\right\rfloor=\sum_{1\le k\le\frac{t}{u}}\left(\lfloor t\rfloor-\lceil uk\rceil+1\right)=(\lfloor t\rfloor+1)\left\lfloor\frac{t}{u}\right\rfloor-\sum_{k=1}^{\left\lfloor\frac{t}{u}\right\rfloor}\lceil uk\rceil.
$$
By using the estimate $\lceil v\rceil=v+O(1)$, we obtain
\begin{align*}
\sum_{j=1}^{\lfloor t\rfloor}\left\lfloor\frac{j}{u}\right\rfloor&=(\lfloor t\rfloor+1)\left\lfloor\frac{t}{u}\right\rfloor-\sum_{k=1}^{\left\lfloor\frac{t}{u}\right\rfloor}uk+O\left(\frac{t}{u}\right)\\
&=t\left\lfloor\frac{t}{u}\right\rfloor-\frac{1}{2}u\left\lfloor\frac{t}{u}\right\rfloor^2-\frac{1}{2}u\left\lfloor\frac{t}{u}\right\rfloor+O\left(\frac{t}{u}\right)
\end{align*}
as desired.
\end{proof}
The following lemma gives an estimate for the complement sum ${{\overline{A}}}(n,n)-{{\overline{A}}}(n,x)$.
\begin{lem}\label{lem:oAnn-oAnx-estimate}
For integers $n\ge2$ and real numbers $x$ such that $\sqrt{n-1}\le x\le n$, we have
\begin{equation}\label{eqn:oAnn-oAnx-asymp}
{{\overline{A}}}(n,n)-{{\overline{A}}}(n,x)=\int_x^n\left(\left\lfloor\frac{n}{u}\right\rfloor+\left\{\frac{n}{u}\right\}^2\right)u\log u\,du+O\left(\frac{n^2\log n}{x}\right).
\end{equation}
\end{lem}
\begin{proof}
We start from \eqref{eqn:oAnn-oAnx-formula} and apply Proposition \ref{prop:23n} to estimate each
$\left(\overline{C}(j,j)-\overline{C}(j,x)\right)$:
\begin{align}\label{eqn:oAnn-oAnx-start2}
{{\overline{A}}}(n,n)-{{\overline{A}}}(n,x)&={{\overline{B}}}^c_{11}(n,x)-2\sum_{x\le j<n}\left(\overline{C}(j,j)-\overline{C}(j,x)\right)\nonumber\\
&={{\overline{B}}}^c_{11}(n,x)-2\sum_{x\le j<n}\int_x^j\left\lfloor\frac{j}{u}\right\rfloor\log u\,du+O\bigg(\frac{1}{x}\sum_{j=1}^nj\log j\bigg).
\end{align}
Now, we estimate each term on the right of \eqref{eqn:oAnn-oAnx-start}.
To simplify the error term, by Lemma \ref{lem:Ln01-estimate}, we have
\begin{equation}\label{eqn:oAnn-oAnx-error}
\frac{1}{x}\sum_{j=1}^nj\log j=O\left(\frac{n^2\log n}{x}\right).
\end{equation}
The first term can be estimated by Lemma \ref{lem:52}:
\begin{align}\label{eqn:oAnn-oAnx-first}
{{\overline{B}}}^c_{11}(n,x)&=\frac{1}{2}n^2(\log n)^2-\frac{1}{2}n^2(\log x)^2+O\left(\frac{n^2\log n}{x}\right)\nonumber\\
&=\int_x^n\frac{n^2\log u}{u}\,du++O\left(\frac{n^2\log n}{x}\right).
\end{align}
For the second term, we observe that $\left\lfloor\frac{j}{u}\right\rfloor=0$ for $0<j<u$. Hence
\begin{align}\label{eqn:oAnn-oAnx-second-1}
-2\sum_{x\le j<n}\int_x^j\left\lfloor\frac{j}{u}\right\rfloor\log u\,du&=-2\sum_{x\le j<n}\int_x^n\left\lfloor\frac{j}{u}\right\rfloor\log u\,du\nonumber\\
&=-2\int_x^n\bigg(\sum_{x\le j<n}\left\lfloor\frac{j}{u}\right\rfloor\bigg)\log u\,du.
\end{align}
The inner sum on the right of \eqref{eqn:oAnn-oAnx-second-1} can be estimated using Lemma \ref{lem:kernel-estimate}. If $1\le j<x$ and $u \ge x$, then $0<j<u$, and then $\left\lfloor\frac{j}{u}\right\rfloor=0$. Hence $\sum_{1\le j<x}\left\lfloor\frac{j}{u}\right\rfloor=0$ and
\begin{align*}
\sum_{x\le j<n}\left\lfloor\frac{j}{u}\right\rfloor&=\sum_{j=1}^n\left\lfloor\frac{j}{u}\right\rfloor-\left\lfloor\frac{n}{u}\right\rfloor\\
&=n\left\lfloor\frac{n}{u}\right\rfloor-\frac{1}{2}u\left\lfloor\frac{n}{u}\right\rfloor^2-\frac{1}{2}u\left\lfloor\frac{n}{u}\right\rfloor+O\left(\frac{n}{u}\right)\\
&=n\left\lfloor\frac{n}{u}\right\rfloor-\frac{1}{2}u\left\lfloor\frac{n}{u}\right\rfloor^2-\frac{1}{2}u\left\lfloor\frac{n}{u}\right\rfloor+O\left(\frac{n}{x}\right).
\end{align*}
On inserting this into \eqref{eqn:oAnn-oAnx-second-1}, we obtain
\begin{equation}\label{eqn:oAnn-oAnx-second-2}
-2\sum_{x\le j<n}\int_x^j\left\lfloor\frac{j}{u}\right\rfloor\log u\,du=\int_x^n\left(-2n\left\lfloor\frac{n}{u}\right\rfloor+u\left\lfloor\frac{n}{u}\right\rfloor^2+u\left\lfloor\frac{n}{u}\right\rfloor\right)\log u\,du+O\left(\frac{n^2\log n}{x}\right).
\end{equation}
On inserting \eqref{eqn:oAnn-oAnx-error}, \eqref{eqn:oAnn-oAnx-first}, and \eqref{eqn:oAnn-oAnx-second-2} into \eqref{eqn:oAnn-oAnx-start2}, we obtain
\begin{align*}
{{\overline{A}}}(n,n)-{{\overline{A}}}(n,x)&=\int_x^n\left(\frac{n^2}{u}-2n\left\lfloor\frac{n}{u}\right\rfloor+u\left\lfloor\frac{n}{u}\right\rfloor^2+u\left\lfloor\frac{n}{u}\right\rfloor\right)\log u\,du+O\left(\frac{n^2\log n}{x}\right)\\
&=\int_x^n\left(\left(\frac{n}{u}-\left\lfloor\frac{n}{u}\right\rfloor\right)^2+\left\lfloor\frac{n}{u}\right\rfloor\right)u\log u\,du+O\left(\frac{n^2\log n}{x}\right)\\
&=\int_x^n\left(\left\lfloor\frac{n}{u}\right\rfloor+\left\{\frac{n}{u}\right\}^2\right)u\log u\,du
+O\left(\frac{n^2\log n}{x}\right)
\end{align*}
as desired.
\end{proof}
The next lemma shows that the main term in \eqref{eqn:oAnn-oAnx-asymp} can be written in the form $f\left(\frac{x}{n}\right)n^2\log n+g\left(\frac{x}{n}\right)n^2$.
\begin{lem}\label{lem:sec6-integral-eval1}
For real numbers $n$ and $x$ such that $0<x\le n$, we have
\begin{equation}\label{eqn:sec6-main-term-evaluation}
\int_x^n\left(\left\lfloor\frac{n}{u}\right\rfloor+\left\{\frac{n}{u}\right\}^2\right)u\log u\,du
=n^2(\log n)\int_1^\frac{n}{x}\frac{\lfloor v\rfloor+\{v\}^2}{v^3}\,dv-n^2\int_1^\frac{n}{x}\frac{\lfloor v\rfloor+\{v\}^2}{v^3}\log v\,dv.
\end{equation}
\end{lem}
\begin{proof}
By the substitution $v=\frac{n}{u}$, we see that
\begin{align*}
\int_x^n\left(\left\lfloor\frac{n}{u}\right\rfloor+\left\{\frac{n}{u}\right\}^2\right)u\log u\,du
&=\int_\frac{n}{x}^1\left(\lfloor v\rfloor+\{v\}^2\right)\frac{n}{v}\left(\log\frac{n}{v}\right)\left(-\frac{n}{v^2}\right)\,dv\\
&=n^2\int_1^\frac{n}{x}\frac{\lfloor v\rfloor+\{v\}^2}{v^3}\log\frac{n}{v}\,dv.
\end{align*}
Since $\log\frac{n}{v}=\log n-\log v$, we obtain
$$
\int_x^n\left(\left\lfloor\frac{n}{u}\right\rfloor+\left\{\frac{n}{u}\right\}^2\right)u\log u\,du=n^2(\log n)\int_1^\frac{n}{x}\frac{\lfloor v\rfloor+\{v\}^2}{v^3}\,dv-n^2\int_1^\frac{n}{x}\frac{\lfloor v\rfloor+\{v\}^2}{v^3}\log v\,dv
$$
as desired.
\end{proof}
To evaluate the integrals on the right of \eqref{eqn:sec6-main-term-evaluation}, we use the following lemma.
\begin{lem}\label{lem:integral-calculation-EM}
Suppose that $f$ is a twice differentiable function with continuous second derivative on the interval $[1,\infty)$. Then for real numbers $\beta\ge1$,
\begin{equation}\label{eqn:integral-calculation-EM
\frac{1}{2}\int_1^\beta\left(\lfloor v\rfloor+\{v\}^2\right)f''(v)\,dv=\int_1^\beta f(v)\,dv-\sum_{b=2}^{\lfloor\beta\rfloor}f(b)-\{\beta\}f(\beta)+\frac{1}{2}\left(\lfloor\beta\rfloor+\{\beta\}^2\right)f'(\beta)-\frac{1}{2}f'(1).
\end{equation}
\end{lem}
\begin{proof}
By the Euler--Maclaurin summation formula (cf. \cite[Theorem~B.5]{MV07}),
\begin{align*}
\sum_{b=2}^{\lfloor\beta\rfloor}f(b)=&\int_1^\beta f(v)\,dv-\left(\{\beta\}-\frac{1}{2}\right)f(\beta)-\frac{1}{2}f(1)+\frac{1}{2}\left(\{\beta\}^2-\{\beta\}+\frac{1}{6}\right)f'(\beta)\\
&-\frac{1}{12}f'(1)-\frac{1}{2}\int_1^\beta\left(\{v\}^2-\{v\}+\frac{1}{6}\right)f''(v)\,dv.
\end{align*}
Rearranging the terms, we obtain
\begin{align}
\frac{1}{2}\int_1^\beta\left(\{v\}^2-\{v\}+\frac{1}{6}\right)f''(v)\,dv=&\int_1^\beta f(v)\,dv-\sum_{b=2}^{\lfloor\beta\rfloor}f(b)-\left(\{\beta\}-\frac{1}{2}\right)f(\beta)\nonumber\\
&+\frac{1}{2}\left(\{\beta\}^2-\{\beta\}+\frac{1}{6}\right)f'(\beta)-\frac{1}{2}f(1)-\frac{1}{12}f'(1).\label{eqn:Euler-Maclaurin-2
\end{align}
On the other hand, we use integration by parts to see that
\begin{align}
\frac{1}{2}\int_1^\beta\left(v-\frac{1}{6}\right)f''(v)\,dv&=\frac{1}{2}\left(v-\frac{1}{6}\right)f'(v)\bigg|_{v=1}^\beta-\frac{1}{2}\int_1^\beta f'(v)\,dv\nonumber\\
&=-\frac{1}{2}f(\beta)+\frac{1}{2}\left(\beta-\frac{1}{6}\right)f'(\beta)+\frac{1}{2}f(1)-\frac{5}{12}f'(1).\label{eqn:by-parts
\end{align}
Adding \eqref{eqn:Euler-Maclaurin-2} and \eqref{eqn:by-parts}, we obtain \eqref{eqn:integral-calculation-EM}.
\end{proof}
We apply Lemma \ref{lem:integral-calculation-EM} to evaluate the integrals on the right of \eqref{eqn:sec6-main-term-evaluation}.
\begin{lem}\label{lem:sec6-integral-eval2}
For real numbers $\alpha$ such that $0<\alpha\le1$, we have
\begin{eqnarray}\label{eqn:oAnn-oAnx-integral-f-new
\int_1^\frac{1}{\alpha}\frac{\lfloor v\rfloor+\{v\}^2}{v^3}\,dv&=&\frac{3}{2}-\left(H_{\lfloor\frac{1}{\alpha}\rfloor}-\log\frac{1}{\alpha}\right)-\alpha\left(\left\{\frac{1}{\alpha}\right\}+\frac{1}{2}\right)-\alpha^2\left(\frac{1}{2}\left\{\frac{1}{\alpha}\right\}^2-\frac{1}{2}\left\{\frac{1}{\alpha}\right\}\right),
\end{eqnarray}
\begin{eqnarray}\label{eqn:oAnn-oAnx-integral-g-new
\int_1^\frac{1}{\alpha}\frac{\lfloor v\rfloor+\{v\}^2}{v^3}\log v\,dv&=&\frac{7}{4}-\frac{3}{2}\left(H_{\lfloor\frac{1}{\alpha}\rfloor}-\log\frac{1}{\alpha}\right)-\left(J\left(\frac{1}{\alpha}\right)-\frac{1}{2}\left(\log\frac{1}{\alpha}\right)^2\right)\nonumber\\
&&-\left(\alpha\log\frac{1}{\alpha}\right)\left(\left\{\frac{1}{\alpha}\right\}+\frac{1}{2}\right)-\alpha\left(\frac{3}{2}\left\{\frac{1}{\alpha}\right\}+\frac{1}{4}\right)\nonumber\\
&&-\left(\alpha^2\log\frac{1}{\alpha}\right)\left(\frac{1}{2}\left\{\frac{1}{\alpha}\right\}^2-\frac{1}{2}\left\{\frac{1}{\alpha}\right\}\right)-\alpha^2\left(\frac{1}{4}\left\{\frac{1}{\alpha}\right\}^2-\frac{1}{4}\left\{\frac{1}{\alpha}\right\}\right).
\end{eqnarray}
\end{lem}
\begin{proof}
For \eqref{eqn:oAnn-oAnx-integral-f-new}, apply Lemma \ref{lem:integral-calculation-EM} with $f(v)=\frac{1}{v}$ and $\beta=\frac{1}{\alpha}$:
$$
\int_1^\frac{1}{\alpha}\frac{\lfloor v\rfloor+\{v\}^2}{v^3}\,dv=\log\frac{1}{\alpha}-\left(H_{\lfloor\frac{1}{\alpha}\rfloor}-1\right)-\alpha\left\{\frac{1}{\alpha}\right\}-\frac{1}{2}\alpha^2\left(\left\lfloor\frac{1}{\alpha}\right\rfloor+\left\{\frac{1}{\alpha}\right\}^2\right)+\frac{1}{2}.
$$
Replacing $\left\lfloor\frac{1}{\alpha}\right\rfloor+\left\{\frac{1}{\alpha}\right\}^2$ by $\frac{1}{\alpha}-\left\{\frac{1}{\alpha}\right\}+\left\{\frac{1}{\alpha}\right\}^2$ and rearranging the terms, we obtain \eqref{eqn:oAnn-oAnx-integral-f-new}.\\
For \eqref{eqn:oAnn-oAnx-integral-g-new}, apply Lemma \ref{lem:integral-calculation-EM} with $f(v)=\frac{3}{2v}+\frac{\log v}{v}$ and $\beta=\frac{1}{\alpha}$:
\begin{align*}
\int_1^\frac{1}{\alpha}\frac{\lfloor v\rfloor+\{v\}^2}{v^3}\log v\,dv=&\left(\frac{3}{2}\log\frac{1}{\alpha}+\frac{1}{2}\left(\log\frac{1}{\alpha}\right)^2\right)-\left(\frac{3}{2}H_{\lfloor\frac{1}{\alpha}\rfloor}+J\left(\frac{1}{\alpha}\right)-\frac{3}{2}\right)\\
&-\alpha\left\{\frac{1}{\alpha}\right\}\left(\frac{3}{2}+\log\frac{1}{\alpha}\right)-\alpha^2\left(\left\lfloor\frac{1}{\alpha}\right\rfloor+\left\{\frac{1}{\alpha}\right\}^2\right)\left(\frac{1}{4}+\frac{1}{2}\log\frac{1}{\alpha}\right)+\frac{1}{4}.
\end{align*}
Replacing $\left\lfloor\frac{1}{\alpha}\right\rfloor+\left\{\frac{1}{\alpha}\right\}^2$ by $\frac{1}{\alpha}-\left\{\frac{1}{\alpha}\right\}+\left\{\frac{1}{\alpha}\right\}^2$ and rearranging the terms, we obtain \eqref{eqn:oAnn-oAnx-integral-g-new}.
\end{proof}
\subsection{Proof of Theorem \ref{thm:oAnx}}
\label{subsec:63n}
We combine results in the previous subsection to obtain an estimate for ${{\overline{A}}}(n,x)$ as stated in Theorem \ref{thm:oAnx}.
\begin{proof}[Proof of Theorem \ref{thm:oAnx}]
Combining Theorem \ref{thm:oAn} and Lemma \ref{lem:oAnn-oAnx-estimate}, which estimate ${{\overline{A}}}(n,n)$ and ${{\overline{A}}}(n,n)-{{\overline{A}}}(n,x)$ respectively, we obtain an estimate for ${{\overline{A}}}(n,x)$:
$$
{{\overline{A}}}(n,x)=\left(\frac{3}{2}-\gamma\right)n^2\log n+\left(\frac{3}{2}\gamma+\gamma_1-\frac{7}{4}\right)n^2-\int_x^n\left(\left\lfloor\frac{n}{u}\right\rfloor+\left\{\frac{n}{u}\right\}^2\right)u\log u\,du+O\left(n^{3/2}\log n\right).
$$
The integral on the right can be evaluated using Lemma \ref{lem:sec6-integral-eval1}:
\begin{align*}
{{\overline{A}}}(n,x)=&\,\left(\frac{3}{2}-\gamma\right)n^2\log n+\left(\frac{3}{2}\gamma+\gamma_1-\frac{7}{4}\right)n^2\\
&-n^2(\log n)\int_1^\frac{n}{x}\frac{\lfloor v\rfloor+\{v\}^2}{v^3}\,dv+n^2\int_1^\frac{n}{x}\frac{\lfloor v\rfloor+\{v\}^2}{v^3}\log v\,dv+O\left(n^{3/2}\log n\right)\\
=&\,{{\overline{A}}}_0(n,x)n^2\log n+{{\overline{A}}}_1(n,x)n^2+O\left(n^{3/2}\log n\right),
\end{align*}
where the functions ${{\overline{A}}}_0(n,x)$ and ${{\overline{A}}}_1(n,x)$ are given by
\begin{equation}\label{eqn:oA0-integral}
{{\overline{A}}}_0(n,x):=\left(\frac{3}{2}-\gamma\right)-\int_1^\frac{n}{x}\frac{\lfloor v\rfloor+\{v\}^2}{v^3}\,dv,
\end{equation}
\begin{equation}\label{eqn:oA1-integral}
{{\overline{A}}}_1(n,x):=\left(\frac{3}{2}\gamma+\gamma_1-\frac{7}{4}\right)+\int_1^\frac{n}{x}\frac{\lfloor v\rfloor+\{v\}^2}{v^3}\log v\,dv.
\end{equation}
It remains to show that \eqref{eqn:oA0-integral} is equivalent to \eqref{eqn:oA0-thm} and that \eqref{eqn:oA1-integral} is equivalent to \eqref{eqn:oA1-thm}. To that end, we apply Lemma \ref{lem:sec6-integral-eval2} with $\alpha=\frac{x}{n}$ to evaluate the integrals in \eqref{eqn:oA0-integral} and \eqref{eqn:oA1-integral}.
We obtain
$$
{{\overline{A}}}_0(n,x) =\left( H_{\lfloor \frac{n}{x} \rfloor} - \log \frac{n}{x} -\gamma \right)
+ \frac{x}{n} \left( \left\{ \frac{n}{x} \right\} + \frac{1}{2} \right) -\left(\frac{x}{n}\right)^2
\left( \frac{1}{2}\left\{\frac{n}{x}\right\}^2-\frac{1}{2}\left\{\frac{n}{x}\right\}\right),
$$
\begin{eqnarray*}
{{\overline{A}}}_1(n,x) &= &
-\frac{3}{2}\left(H_{\lfloor\frac{n}{x}\rfloor}-\log\frac{n}{x}-\gamma\right)
-\left(J\left(\frac{n}{x}\right)-\frac{1}{2}\left(\log\frac{n}{x}\right)^2 -\gamma_1\right)\nonumber\\
&&-\left(\frac{x}{n}\log\frac{n}{x}\right)\left(\left\{\frac{n}{x}\right\}+\frac{1}{2}\right)
-\frac{x}{n} \left(\frac{3}{2}\left\{\frac{n}{x}\right\}+\frac{1}{4}\right)\nonumber\\
&&-\left(\frac{x}{n}\right)^2\left(\log\frac{n}{x}\right)\left(\frac{1}{2}\left\{\frac{n}{x}\right\}^2
-\frac{1}{2}\left\{\frac{n}{x}\right\}\right)
-\left(\frac{x}{n}\right)^2\left(\frac{1}{4}\left\{\frac{n}{x}\right\}^2-\frac{1}{4}\left\{\frac{n}{x}\right\}\right).
\end{eqnarray*}
Replacing $\left\{\frac{n}{x}\right\}$ by $\frac{n}{x}-\left\lfloor\frac{n}{x}\right\rfloor$ and rearranging the terms, we obtain the formulas \eqref{eqn:oA0-thm} and \eqref{eqn:oA1-thm}.\\
Finally, for integers $n\ge2$ and real $x\in\left[1,\sqrt{n}\right]$, we have
\begin{align*}
{{\overline{A}}}(n,x)&=\sum_{2\le b\le x} \frac{2}{b-1}S_b(n)\log b\\
&\le\sum_{2\le b\le x}n\log n <xn\log n\\
&\le n^{3/2}\log n,
\end{align*}
where the bound of Lemma \ref{lem:dbn-Sbn-bound} for $S_b(n)$ was used in the first inequality.
We have obtained \eqref{eqn:oAnx-bound2}.
\end{proof}
\subsection{Proof of Theorem \ref{thm:oAnx-cor}}
\label{subsec:64n}
\begin{proof}[Proof of Theorem \ref{thm:oAnx-cor}]
The result for the range $x\in [\sqrt{n}, n]$
follows from Theorem \ref{thm:oAnx} on choosing $x= \alpha n$ and simplifying.
For the range $x\in \left[1, \sqrt{n}\right]$ we use the
final estimate \eqref{eqn:oAnx-bound2}.
\end{proof}
\begin{rem}\label{rem:38}
The function $f_{{\overline{A}}}(\alpha)$ has $f_{{\overline{A}}}(1)=\frac{3}{2} -\gamma$, and has $\lim_{\alpha \to 0^+} f_{{\overline{A}}}(\alpha)=0$ since $ H_{\lfloor \frac{1}{\alpha}\rfloor}- \log \frac{1}{\alpha} \to \gamma$
as $\alpha \to 0^+$.
\end{rem}
\section{Estimates for partial factorizations ${\overline{\overline{G}}}(n,x)$}\label{sec:oHnx}
We deduce asymptotics of ${\overline{\overline{G}}}(n,x)$
\begin{thm}\label{thm:oHnx}
Let ${\overline{\overline{G}}}(n, x) = \prod_{b=2}^{\lfloor x\rfloor} b^{{\overline{\nu}}(n,b)}$. Then for integers $n\ge2$ and real $x\in\left[\sqrt{n},n\right]$,
\begin{equation}\label{eqn:oHnx-main-two-var1}
\log {\overline{\overline{G}}}(n, x ) = {\overline{\overline{G}}}_0(n,x)\,n^2\log n + {\overline{\overline{G}}}_1(n,x) n^2
+ O \left(n^{3/2} \log n \right),
\end{equation}
where the functions ${\overline{\overline{G}}}_0(n,x)$ and ${\overline{\overline{G}}}_1(n,x)$ only depend on $\frac{x}{n}$ and are given by
\begin{equation}\label{eqn:oH0nx}
{\overline{\overline{G}}}_0(n,x) := \frac{1}{2} +\frac{1}{2} \left(\frac{x}{n}\right)^2 \left\lfloor \frac{n}{x}\right\rfloor \left\lfloor \frac{n}{x}+1\right\rfloor
- \frac{x}{n} \left\lfloor \frac{n}{x} \right\rfloor
\end{equation}
and
\begin{align}
{\overline{\overline{G}}}_1(n,x) := &\left( \frac{1}{2} \gamma - \frac{3}{4} \right) - \frac{1}{2} \left( H_{\lfloor \frac{n}{x}\rfloor} - \log \frac{n}{x} \right)-\frac{1}{2}\log\frac{n}{x} -\frac{1}{2}\left(\log \frac{n}{x} \right) \left(\frac{x}{n}\right)^2 \left\lfloor \frac{n}{x} \right\rfloor \left\lfloor \frac{n}{x}+1 \right\rfloor \nonumber\\
&+\left(\log \frac{n}{x} \right)\frac{x}{n} \left\lfloor \frac{n}{x} \right\rfloor
- \frac{1}{4} \left(\frac{x}{n}\right)^2 \left\lfloor \frac{n}{x} \right\rfloor \left\lfloor \frac{n}{x} +1\right\rfloor + \frac{x}{n} \left\lfloor \frac{n}{x} \right\rfloor. \label{eqn:oH1nx}
\end{align}
Moreover, for integers $n\ge2$ and real $x\in\left[1, \sqrt{n}\right]$,
\begin{equation} \label{eqn:oHnx-main-two-var2}
\log {\overline{\overline{G}}}(n, x ) = O \left( n^{3/2} \log n\right).
\end{equation}
\end{thm}
\begin{proof}
Recall from \eqref{eqn:oHABx} the identity
$$
\log {\overline{\overline{G}}}(n, x) = {{\overline{A}}}(n, x) - {{\overline{B}}}(n,x).
$$
The result \eqref{eqn:oHnx-main-two-var1}
follows for the range $x \in [\sqrt{n}, n]$ by inserting the formulas \eqref{eqn:oAnx-bound} in Theorem \ref{thm:oAnx} and \eqref{eqn:oBnx-bound} in Theorem \ref{thm:oBnx}.
The formula \eqref{eqn:oHnx-main-two-var2} in the range $x \in \left[1, \sqrt{n}\right]$ follows from the
corresponding range bounds in Theorems \ref{thm:oAnx} and \ref{thm:oBnx}.
\end{proof}
\subsection{Proof of Theorem \ref{thm:oHnx-main}} \label{subsec:71}
\begin{proof}[Proof of Theorem \ref{thm:oHnx-main}]
The theorem follows from Theorem \ref{thm:oHnx} on choosing $x=\alpha n$ and simplifying.
The $O$-constant in the remainder term is absolute for the range $\frac{1}{\sqrt{n}} \le \alpha \le 1.$
Here $\frac{n}{x}= \frac{1}{\alpha}$.
\end{proof}
\section{Concluding remarks}\label{sec:8}
Viewing the general definition of generalized binomial products \eqref{eqn:oHn0} as a kind of integration operation (over $b \ge2$)
the smoothing aspect of the integration operation is evident in the existence
of unconditional estimates giving a power-savings remainder term; the Riemann hypothesis is
not needed.
A large class of limit functions may occur
in problems of this sort, generalizing the limit function $f_{G}(\alpha)$
in \cite{DL:20} given by \eqref{eqn:fG-asymp}. This paper
exhibited a new limit scaling functions $g_{{\overline{\overline{G}}}}(\alpha)$.
It may be of interest to determine the class of of such scaling functions
obtained by iterated integral constructions of this kind.
This definition \eqref{eqn:oHn0} did not provide any hint
whether the product possesses a sub-factorization
into analogues of binomial coefficients.
Two of the authors will show elsewhere \cite{LY:22} that
the sequence ${\overline{\overline{G}}}_n$ can alternatively
be defined as a product of
generalized binomial coefficients ${{n}\choose{k}}_{\mathbb{N}}$
which are integers, which themselves can be written in
terms of generalized factorials of a new kind
$$
{{n}\choose{k}}_{\mathbb{N}} = \frac{ [n]!_{\mathbb{N}}}{ [k]!_{\mathbb{N}} [n-k]!_{\mathbb{N}}}.
$$
These new factorials
can themselves be factorized into a product of generalized integers
$$
[n]_{\mathbb{N}} = \prod_{k=1}^n [k]_{\mathbb{N}}.
$$
These generalized integers $[n]_{\mathbb{N}}$ are not monotonically
increasing but have an internal structure driven by the prime factorization of $n$.
These generalized factorials fit in the general framework of Knuth and Wilf \cite{KW89} treating generalized
factorials and binomial coefficients as products of generalized integers (denoted $C_n$ in their paper).
The sequence of generalized integers $[n]_{\mathbb{Z}}$
is not a regularly divisible sequence as defined in \cite{KW89}.
The factorials $[n]!_{\mathbb{N}}$ above also have many of the properties of
the generalized factorials of Bhargava (\cite{Bhar:97a}, \cite{Bhar:00}, \cite{Bhar:09}).
They seem however to not be included in Bhargava's theory of $P$-orderings, but \cite{LY:22} will show they
can be covered by a generalization of this theory.\\
\noindent{\bf Acknowledgements}\\
Theorem \ref{thm:oBn} and Theorem \ref{thm:oAn} appear in the PhD. thesis of the first author,
who thanks Trevor Wooley for helpful comments. The authors thank D. Harry Richman for
helpful comments, and for plots of the scaling functions.\medskip
\noindent{\bf Funding}\\
Work of the first author was partially supported by NSF grant DMS-1701577.
Work of the second author was partly supported by NSF grant DMS-1701576 and
a Simons Fellowship in Mathematics in 2019. Work of the third author was
partially supported by NSF-grants DMS-1701576 and DMS-1701577.\medskip
|
2024-02-18T23:40:24.417Z
|
2021-12-30T02:21:50.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14422","language":"en","timestamp":1640830910000,"url":"https:\/\/arxiv.org\/abs\/2112.14422","yymm":"2112"}
|
proofpile-arXiv_000-10019
|
{"provenance":"002.jsonl.gz:10020"}
| null | null |
\section{Introduction}
Image classification is a long-standing yet important task with a wide range of applications such as autonomous driving, industrial automation, medical diagnosis, and biometric identification~\cite{autonomous_driving, industrial_automation, medical_diagnosis, biometric_identification}.
In solving the task, supervised learning (SL) techniques have been popularly used for its superiority~\cite{VGG, ResNet}.
Well-known drawback of SL is that a large number of training data are required for each and every class to be identified.
Unfortunately, in many practical scenarios, it is difficult to collect training data for certain classes (e.g., endangered species and newly observed species such as variants of COVID-19).
When there are \textit{unseen} classes where training data is unavailable, SL-based models are biased towards the \textit{seen} classes, impeding the identification of the unseen classes.
Recently, to overcome this drawback, a technique to train a classifier using manually annotated attributes (e.g., color, size, and shape; see Fig.~\ref{fig:CUB}) has been proposed~\cite{zsl_proposal, gzsl_intro}.
Key idea of this technique, dubbed as generalized zero-shot learning (GZSL), is to learn the relationship between the image and the attribute from seen classes and then use the trained model in the identification of unseen classes.
In~\cite{ALE}, for example, an approach to identify unseen classes by measuring the compatibility between the image feature and attribute has been proposed.
In~\cite{CVAE-GZSL}, a network synthesizing the image feature from the attribute has been employed to generate training data of unseen classes.
In extracting the image feature, a network trained using the classification task (e.g., ResNet~\cite{ResNet}) has been popularly used.
A potential drawback of this extraction method is that the image feature might contain attribute-irrelevant information (e.g., human fingers in Fig.~\ref{fig:CUB}), disturbing the process of learning the relationship between the image and the attribute~\cite{DLFZRL, RFF-GZSL, Disentangled-VAE}.
\begin{figure}[!t]
\centering
\centerline{\includegraphics[width=1\linewidth]{CUB.png}}
\caption{Images and attributes for different bird species sampled from the CUB dataset~\cite{CUB}.}
\label{fig:CUB}
\end{figure}
\begin{figure*}[!t]
\centering
\centerline{\includegraphics[width=15cm]{autoencoder.eps}}
\caption{Illustration of the image feature decomposition.}
\label{fig:SD}
\end{figure*}
In this paper, we propose a new GZSL technique that removes the interference caused by the attribute-irrelevant information.
Key idea of the proposed approach is to extract the \textit{semantic feature}, feature containing the attribute-related information, from the image feature and then use it in learning the relationship between the image and the attribute.
In extracting the semantic feature, we use a modified autoencoder consisting of two encoders, viz., \textit{semantic} and \textit{residual} encoders (see Fig.~\ref{fig:SD}).
In a nutshell, the semantic encoder captures all the attribute-related information in the image feature and the residual encoder catches the attribute-irrelevant information.
In the conventional autoencoder, only reconstruction loss (difference between the input and the reconstructed input) is used for the training.
In our approach, to encourage the semantic encoder to capture the attribute-related information only, we use two novel loss functions on top of the reconstruction loss.
First, we employ the mutual information (MI)-based loss to maximize (minimize) MI between the semantic (residual) encoder output and the attribute.
Since MI is a metric to measure the level of dependency between two random variables, by exploiting the MI-based loss, we can encourage the semantic encoder to capture the attribute-related information and at the same time discourage the residual encoder to capture any attribute-related information.
As a result, all the attribute-related information can be solely captured by the semantic encoder.
Second, we use the similarity-based loss to enforce the semantic encoder not to catch any attribute-irrelevant information.
For example, when a bird image contains human fingers (see Fig.~\ref{fig:CUB}), we do not want features related to the finger to be included in the semantic encoder output.
To do so, we maximize the similarity between the semantic encoder outputs of images that are belonging to the same class (bird images in our example).
Since attribute-irrelevant features are contained only in a few image samples (e.g., human fingers are included in a few bird images), by maximizing the similarity between the semantic encoder outputs of the same class, we can remove attribute-irrelevant information from the semantic encoder output.
From extensive experiments using various benchmark datasets (AwA1, AwA2, CUB, and SUN), we demonstrate that the proposed approach outperforms the conventional GZSL techniques by a large margin.
For example, for the AwA2 and CUB datasets, our model achieves 2\% improvement in the GZSL classification accuracy over the state-of-the-art techniques.
\section{Related Work and Background}
\subsection{Conventional GZSL Approaches}
The main task in GZSL is to learn the relationship between the image and the attribute from seen classes and then use it in the identification of unseen classes.
Early GZSL works have focused on the training of a network measuring the compatibility score between the image feature and the attribute~\cite{ALE, DeViSE}.
Once the network is trained properly, images can be classified by identifying the attribute achieving the maximum compatibility score.
Recently, generative model-based GZSL approaches have been proposed~\cite{CVAE-GZSL, CLSWGAN}.
Key idea of these approaches is to generate synthetic image features of unseen classes from the attributes by employing a generative model~\cite{CVAE-GZSL, CLSWGAN}.
As a generative model, conditional variational autoencoder (CVAE)~\cite{vae} and conditional Wasserstein generative adversarial network (CWGAN)~\cite{WGAN} have been popularly used.
By exploiting the generated image features of unseen classes as training data, a classification network identifying unseen classes can be trained in a supervised manner.
Over the years, many efforts have been made to improve the performance of the generative model.
In~\cite{f-vaegan-d2, CADA-VAE, Zero-VAE-GAN}, an approach to combine multiple generative models (e.g., CVAE and CWGAN) has been proposed.
In~\cite{cycle-WGAN, DASCN}, an additional network estimating the image attribute from the image feature has been used to make sure that the synthetic image features satisfy the attribute of unseen classes.
In~\cite{CLSWGAN, LsrGAN, LisGAN}, an additional image classifier has been used in the generative model training to generate distinct image features for different classes.
Our approach is conceptually similar to the generative model-based approach in the sense that we generate synthetic image features of unseen classes using the generative model.
The key distinctive point of the proposed approach over the conventional approaches is that we use the features containing only attribute-related information in the classification to remove the interference, if any, caused by the attribute-irrelevant information.
\subsection{MI for Deep Learning}
Mathematically, the MI $I(\mathbf{u}, \mathbf{v})$ between two random variables $\mathbf{u}$ and $\mathbf{v}$ is defined as
\begin{align}
I(\mathbf{u}, \mathbf{v})
&= \mathbb{E}_{p(\mathbf{u}, \mathbf{v})} \left [ \log \frac{p(\mathbf{u}, \mathbf{v})}{p(\mathbf{u}) p(\mathbf{v})} \right ] \nonumber \\
&= \int_{\mathbf{u}} \int_{\mathbf{v}} p(\mathbf{u}, \mathbf{v}) \log \frac{p(\mathbf{u}, \mathbf{v})}{p(\mathbf{u}) p(\mathbf{v})} d \mathbf{u} d \mathbf{v},
\label{eq:MI_definition}
\end{align}
where $p(\mathbf{u}, \mathbf{v})$ is the joint probability density function (PDF) of $\mathbf{u}$ and $\mathbf{v}$, and $p(\mathbf{u})$ and $p(\mathbf{v})$ are marginal PDFs of $\mathbf{u}$ and $\mathbf{v}$, respectively.
In practice, it is very difficult to compute the exact value of MI since the joint PDF $p(\mathbf{u}, \mathbf{v})$ is generally unknown and the integrals in~\eqref{eq:MI_definition} are often intractable.
To approximate MI, various MI estimators have been proposed~\cite{InfoNCE, CLUB}.
Representative estimators include InfoNCE~\cite{InfoNCE} and contrastive log-ratio upper bound (CLUB)~\cite{CLUB}, defined as
\begin{align}
I_{\text{InfoNCE}}(\mathbf{u}, \mathbf{v})
&= \mathbb{E}_{p(\mathbf{u}, \mathbf{v})} [ f(\mathbf{u}, \mathbf{v}) ] \hspace{-.5mm} \nonumber \\
&~~~- \hspace{-.5mm} \mathbb{E}_{p(\mathbf{u})} \hspace{-1mm} \left [ \log \left (
\mathbb{E}_{p(\mathbf{v})} [ \exp ( f(\mathbf{u}, \mathbf{v} ) ) ]
\right ) \right ], \label{eq:MI_lower bound_InfoNCE} \\
I_{\text{CLUB}}(\mathbf{u}, \mathbf{v})
&= \mathbb{E}_{p(\mathbf{u}, \mathbf{v})} \hspace{-1mm} \left [ \log p(\mathbf{v} | \mathbf{u} ) \right ] \hspace{-.5mm} -
\hspace{-.5mm} \mathbb{E}_{p(\mathbf{u})p(\mathbf{v})} \hspace{-1mm} \left [ \log p(\mathbf{v} | \mathbf{u} ) \right ],
\label{eq:MI_upper bound_CLUB}
\end{align}
where $f$ is a pre-defined score function measuring the compatibility between $\mathbf{u}$ and $\mathbf{v}$, and $p(\mathbf{v} | \mathbf{u})$ is the conditional PDF of $\mathbf{v}$ given $\mathbf{u}$, which is often approximated by a neural network.
The relationship between MI, InfoNCE, and CLUB is given by
\begin{align}
I_{\text{InfoNCE}}(\mathbf{u}, \mathbf{v})
\le I(\mathbf{u}, \mathbf{v})
\le I_{\text{CLUB}}(\mathbf{u}, \mathbf{v}).
\label{eq:MI inequality}
\end{align}
Recently, InfoNCE and CLUB have been used to strengthen or weaken the independence between different parts of the neural network.
For example, when one tries to enforce the independence between $\mathbf{u}$ and $\mathbf{v}$, that is, to reduce $I(\mathbf{u}, \mathbf{v})$, an approach to minimize the upper bound $I_{\text{CLUB}}(\mathbf{u}, \mathbf{v})$ of MI can be used~\cite{MI_minimization}.
Whereas, when one wants to maximize the dependence between $\mathbf{u}$ and $\mathbf{v}$, that is, to increase $I(\mathbf{u}, \mathbf{v})$, an approach to maximize the lower bound $I_{\text{InfoNCE}}(\mathbf{u}, \mathbf{v})$ of MI~\cite{MI_maximization} can be used.
\section{SE-GZSL}
In this section, we present the proposed GZSL technique called semantic feature extraction-based GZSL (SE-GZSL).
We first discuss how to extract the semantic feature from the image feature and then delve into the GZSL classification using the extracted semantic feature.
\subsection{Semantic Feature Extraction}
In extracting the semantic feature from the image feature, the proposed SE-GZSL technique uses the modified autoencoder architecture where two encoders, called semantic and residual encoders, are used in capturing the attribute-related information and the attribute-irrelevant information, respectively (see Fig~\ref{fig:SD}).
As mentioned, in the autoencoder training, we use two loss functions: 1) MI-based loss to encourage the semantic encoder to capture all attribute-related information and 2) similarity-based loss to encourage the semantic encoder not to capture attribute-irrelevant information.
In this subsection, we discuss the overall training loss with emphasis on these two.
\paragraph{MI-based Loss}
To make sure that all the attribute-related information is contained in the semantic encoder output, we use MI in the autoencoder training.
To do so, we maximize MI between the semantic encoder output and the attribute which is given by manual annotation.
At the same time, to avoid capturing of attribute-related information in the residual encoder, we minimize MI between the residual encoder output and the attribute.
Let $\mathbf{z}_{s}$ and $\mathbf{z}_{r}$ be the semantic and residual encoder outputs corresponding to the image feature $\mathbf{x}$, and $\mathbf{a}$ be the image attribute (see Fig.~\ref{fig:SD}). Then, our training objective can be expressed as
\begin{align}
\text{minimize}~~~-\lambda_{s} I(\mathbf{z}_{s}, \mathbf{a}) + \lambda_{r} I(\mathbf{z}_{r}, \mathbf{a}),
\label{eq:separation loss_MI form}
\end{align}
where $\lambda_{s}$ and $\lambda_{r}$ ($\lambda_{s}, \lambda_{r} > 0$) are weighting coefficients.
Since the computation of MI is not tractable, we use InfoNCE and CLUB (see~\eqref{eq:MI_lower bound_InfoNCE} and~\eqref{eq:MI_upper bound_CLUB}) as a surrogate of MI.
In our approach, to minimize the objective function in~\eqref{eq:separation loss_MI form}, we express its upper bound using InfoNCE and CLUB and then train the autoencoder in a way to minimize the upper bound.
Using the relationship between MI and its estimators in~\eqref{eq:MI inequality}, the upper bound $\mathcal{L}_{\text{MI}}$ of the objective function in~\eqref{eq:separation loss_MI form} is
\begin{align}
\mathcal{L}_{\text{MI}}
&= -\lambda_{s} I_{\text{InfoNCE}}(\mathbf{z}_{s}, \mathbf{a}) + \lambda_{r} I_{\text{CLUB}} (\mathbf{z}_{r}, \mathbf{a}) \nonumber \\
&= -\lambda_{s} \mathbb{E}_{p(\mathbf{z}_{s}, \mathbf{a})} [ f(\mathbf{z}_{s}, \mathbf{a}) ] \nonumber \\
&~~~ + \hspace{-.5mm} \lambda_{s} \mathbb{E}_{p(\mathbf{z}_{s})} \hspace{-1mm} \left [ \log \left (
\mathbb{E}_{p(\mathbf{a})} [ \exp ( f(\mathbf{z}_{s}, \mathbf{a}) ) ] \right ) \right ] \nonumber \\
&~~~ +\lambda_{r} \left (
\mathbb{E}_{p(\mathbf{z}_{r}, \mathbf{a})} \hspace{-1mm} \left [
\log p(\mathbf{a} | \mathbf{z}_{r} ) \right ] \hspace{-.5mm} -
\hspace{-.5mm} \mathbb{E}_{p(\mathbf{z}_{r})p(\mathbf{a})} \hspace{-1mm} \left [ \log p(\mathbf{a} | \mathbf{z}_{r} ) \right ]
\right ) \hspace{-1mm}.
\label{eq:separation loss_expectation form}
\end{align}
Let $\mathcal{Y}_{s}$ be the set of seen classes, $\mathbf{a}_{c}$ be the attribute of a seen class $c \in \mathcal{Y}_{s}$, and $\{ \mathbf{x}_{c}^{(i)} \}_{i=1}^{N_{c}}$ be the set of training image features for the class $c$.
Further, let $\mathbf{z}_{c, s}^{(i)}$ and $\mathbf{z}_{c, r}^{(i)}$ be the semantic and residual encoder outputs corresponding to the input image feature $\mathbf{x}_{c}^{(i)}$, respectively, then $\mathcal{L}_{\text{MI}}$ can be expressed as
\begin{align}
\mathcal{L}_{\text{MI}}
&= -\frac{\lambda_{s}}{N} \sum_{c \in \mathcal{Y}_{s}} \sum_{i=1}^{N_{c}} \log
\frac{\exp ( f ( \mathbf{z}_{c, s}^{(i)}, \mathbf{a}_{c} ) )}
{\frac{1}{|\mathcal{Y}_{s}|} \hspace{-.5mm} \underset{c^{\prime} \in \mathcal{Y}_{s}}{\sum} \exp ( f ( \mathbf{z}_{c, s}^{(i)}, \mathbf{a}_{c^{\prime}} ) )} \nonumber \\
&~+\frac{\lambda_{r}}{N} \hspace{-.5mm} \sum_{c \in \mathcal{Y}_{s}} \hspace{-.5mm} \sum_{i=1}^{N_{c}} \hspace{-.5mm} \left ( \log p(\mathbf{a}_{c} | \mathbf{z}_{c, r}^{(i)})
- \hspace{-2mm} \underset{c^{\prime} \in \mathcal{Y}_{s}}{\sum} \hspace{-.5mm} \frac{\log p(\mathbf{a}_{c^{\prime}} | \mathbf{z}_{c, r}^{(i)})}{|\mathcal{Y}_{s}|}
\hspace{-.5mm} \right ) \hspace{-1mm},
\label{eq:separation loss}
\end{align}
where $N = \sum_{c \in \mathcal{Y}_{s}} N_{c}$ is the total number of training image features.
\paragraph{Similarity-based Loss}
We now discuss the similarity-based loss to enforce the semantic encoder not to capture any attribute-irrelevant information.
Since images belonging to the same class have the same attribute, attribute-related image features of the same class would be more or less similar.
This means that if the semantic encoder captures attribute-related information only, then the similarity between semantic encoder outputs of the same class should be large.
Inspired by this observation, to remove the attribute-irrelevant information from the semantic encoder output, we train the semantic encoder in a way to maximize the similarity between outputs of the same class:
\begin{align}
\text{maximize}~~~\sum_{j=1}^{N_{c}} \exp (\operatornamewithlimits{sim} ( \mathbf{z}_{c, s}^{(i)}, \mathbf{z}_{c, s}^{(j)} ) ),
\label{eq:similarity loss_same class}
\end{align}
where the similarity is measured using the cosine-similarity function defined as
\begin{align*}
\operatornamewithlimits{sim}(\mathbf{u}, \mathbf{v})
&= \frac{\langle \mathbf{u}, \mathbf{v} \rangle}
{\| \mathbf{u} \|_{2} \| \mathbf{v} \|_{2}}.
\end{align*}
Also, we minimize the similarity between semantic encoder outputs of different classes to obtain sufficiently distinct semantic encoder outputs for different classes:
\begin{align}
\text{minimize}~~~\sum_{c^{\prime} \neq c} \sum_{j=1}^{N_{c^{\prime}}} \exp ( \operatornamewithlimits{sim} ( \mathbf{z}_{c, s}^{(i)}, \mathbf{z}_{c^{\prime}, s}^{(j)} ) ).
\label{eq:similarity loss_different classes}
\end{align}
Using the fact that one can maximize $A$ and minimize $B$ at the same time by minimizing $-\log \frac{1}{1+B/A} = -\log \frac{A}{A+B}$, we obtain the similarity-based loss as
\begin{align}
\mathcal{L}_{\text{sim}}
&= -\frac{1}{N} \sum_{c \in \mathcal{Y}_{s}} \sum_{i=1}^{N_{c}} \log
\frac{\underset{j=1}{\overset{N_{c}}{\sum}} \exp \hspace{-.7mm} \left ( \hspace{-.5mm} \operatornamewithlimits{sim} ( \mathbf{z}_{c, s}^{(i)}, \mathbf{z}_{c, s}^{(j)} ) \hspace{-.5mm} \right )}
{ \underset{c^{\prime} \in \mathcal{Y}_{s}}{\sum} \underset{j=1}{\overset{N_{c^{\prime}}}{\sum}} \exp \hspace{-.7mm} \left ( \hspace{-.5mm} \operatornamewithlimits{sim} ( \mathbf{z}_{c, s}^{(i)}, \mathbf{z}_{c^{\prime}, s}^{(j)} ) \hspace{-.5mm} \right ) }.
\label{eq:similarity loss}
\end{align}
\subsubsection{Overall Loss}
By adding the conventional reconstruction loss $\mathcal{L}_{\text{recon}}$ for the autoencoder, the MI-based loss $\mathcal{L}_{\text{MI}}$, and the similarity-based loss $\mathcal{L}_{\text{sim}}$, we obtain the overall loss function as
\begin{align}
\mathcal{L}_{\text{total}}
&= \mathcal{L}_{\text{recon}} + \mathcal{L}_{\text{MI}} + \lambda_{\text{sim}} \mathcal{L}_{\text{sim}},
\label{eq:decomposition loss}
\end{align}
where $\lambda_{\text{sim}}$ is a weighting coefficient and $\mathcal{L}_{\text{recon}}$ is the reconstruction loss given by
\begin{align}
\mathcal{L}_{\text{recon}}
&= \frac{1}{N} \sum_{c \in \mathcal{Y}_{s}} \sum_{i=1}^{N_{c}} \| \mathbf{x}_{c}^{(i)} - \widehat{\mathbf{x}}_{c}^{(i)} \|_{2}.
\end{align}
Here, $\widehat{\mathbf{x}}_{c}^{(i)}$ is the image feature reconstructed using the semantic and residual encoder outputs ($\mathbf{z}_{c, s}^{(i)}$ and $\mathbf{z}_{c, r}^{(i)}$) in the decoder.
When the training is finished, we only use the semantic encoder for the purpose of extracting the semantic feature.
\subsection{GZSL Classification Using Semantic Features}
So far, we have discussed how to extract the semantic feature from the image feature.
We now discuss how to perform the GZSL classification using the semantic feature.
In a nutshell, we synthesize semantic feature samples for unseen classes from their attributes.
Once the synthetic samples are generated, the semantic classifier identifying unseen classes from the semantic feature is trained in a supervised manner.
\paragraph{Semantic Feature Generation}
\begin{figure*}[h]
\centering
\centerline{\includegraphics[width=13cm]{generator.eps}}
\caption{Illustration of the synthetic semantic feature generation for unseen classes.}
\label{fig:generator}
\end{figure*}
To synthesize the semantic feature samples for unseen classes, we first generate image features from the attributes of unseen classes and then extract the semantic features from the synthetic image features using the semantic encoder (see Fig.~\ref{fig:generator}).
In synthesizing the image feature, we employ WGAN that mitigates the unstable training issue of GAN by exploiting a Wasserstein distance-based loss function~\cite{WGAN}.
The main component in WGAN is a generator $G$ synthesizing the image feature $\widetilde{\mathbf{x}}_{c}$ from a random noise vector $\boldsymbol{\epsilon} \sim \mathcal{N}(\mathbf{0}, \mathbf{I})$ and the image attribute $\mathbf{a}_{c}$ (i.e., $\widetilde{\mathbf{x}}_{c} = G(\boldsymbol{\epsilon}, \mathbf{a}_{c})$).
Conventionally, WGAN is trained to minimize the Wasserstein distance between the distributions of real image feature $\mathbf{x}_{c}$ and generated image feature $\widetilde{\mathbf{x}}_{c}$ given by
\begin{align}
\lefteqn{\hspace{-.2cm} \mathcal{L}_{G, \text{WGAN}}} \nonumber \\
&\hspace{-.3cm}= \max_{D} \hspace{-.5mm} \bigg (
\mathbb{E}_{p(\mathbf{x}_{c} | \mathbf{a}_{c})} [ D(\mathbf{x}_{c}, \mathbf{a}_{c}) ]
- \mathbb{E}_{p(\widetilde{\mathbf{x}}_{c} | \mathbf{a}_{c})} [ D(\widetilde{\mathbf{x}}_{c}, \mathbf{a}_{c}) ] \nonumber \\
&\hspace{-.3cm}~~~~~~~~~~~~~~ -\lambda_{\text{gp}} \mathbb{E}_{p(\widehat{\mathbf{x}}_{c} | \mathbf{a}_{c})} \hspace{-.8mm} \left [ \left ( \| \nabla_{\widehat{\mathbf{x}}_{c}} D(\widehat{\mathbf{x}}_{c}, \mathbf{a}_{c}) \|_{2} - 1 \right )^{2}
\right ] \hspace{-.8mm} \bigg ),
\label{eq:generator_WGAN loss}
\end{align}
where $D$ is an auxiliary network (called critic), $\widehat{\mathbf{x}}_{c} = \alpha \mathbf{x}_{c} + (1 - \alpha) \widetilde{\mathbf{x}}_{c}~(\alpha \sim \mathcal{U}(0, 1))$, and $\lambda_{\text{gp}}$ is the regularization coefficient (a.k.a., gradient penalty coefficient)~\cite{WGAN-GP}.
In our scheme, to make sure that the semantic feature $\widetilde{\mathbf{z}}_{c, s}$ obtained from $\widetilde{\mathbf{x}}_{c}$ is similar to the real semantic feature $\mathbf{z}_{c, s}$, we additionally use the following losses in the WGAN training:
\begin{align}
\mathcal{L}_{G, \text{MI}}
&= -I_{\text{InfoNCE}}(\widetilde{\mathbf{z}}_{c, s}, \mathbf{a}_{c}),
\label{eq:generator_InfoNCE loss} \\
\mathcal{L}_{G, \text{sim}}
&= -\mathbb{E}_{p(\widetilde{\mathbf{z}}_{c, s})} \hspace{-1mm} \left [
\log \frac{\underset{i=1}{\overset{N_{c}}{\sum}} \exp ( \operatornamewithlimits{sim} (\widetilde{\mathbf{z}}_{c, s}, \mathbf{z}_{c, s}^{(i)}) )}
{\underset{c^{\prime}=1}{\overset{S}{\sum}} \underset{i=1}{\overset{N_{c^{\prime}}}{\sum}} \exp ( \operatornamewithlimits{sim} (\widetilde{\mathbf{z}}_{c, s}, \mathbf{z}_{c^{\prime}, s}^{(i)}) )}
\right ].
\label{eq:generator_similarity loss}
\end{align}
We note that these losses are analogous to the losses with respect to the real semantic feature $\mathbf{z}_{c, s}$ in~\eqref{eq:separation loss_expectation form} and~\eqref{eq:similarity loss}, respectively.
By combining~\eqref{eq:generator_WGAN loss},~\eqref{eq:generator_InfoNCE loss}, and~\eqref{eq:generator_similarity loss}, we obtain the overall loss function as
\begin{align}
\mathcal{L}_{G}
&= \mathcal{L}_{G, \text{WGAN}} + \lambda_{G, \text{MI}} \mathcal{L}_{G, \text{MI}} + \lambda_{G, \text{sim}} \mathcal{L}_{G, \text{sim}}, \label{eq:generator loss}
\end{align}
where $\lambda_{G, \text{MI}}$ and $\lambda_{G, \text{sim}}$ are weighting coefficients.
After the WGAN training, we use the generator $G$ and the semantic encoder $E_{s}$ in synthesizing semantic feature samples of unseen classes.
Specifically, for each unseen class $u \in \mathcal{Y}_{u}$, we generate the semantic feature $\widetilde{\mathbf{z}}_{u, s}$ by synthesizing the image feature $\widetilde{\mathbf{x}}_{u} = G(\boldsymbol{\epsilon}, \mathbf{a}_{u})$ using the generator and then exploiting it as an input to the semantic encoder (see Fig.~\ref{fig:generator}):
\begin{align}
\widetilde{\mathbf{z}}_{u, s}
= E_{s}(\widetilde{\mathbf{x}}_{u})
= E_{s}(G(\boldsymbol{\epsilon}, \mathbf{a}_{u})).
\end{align}
By resampling the noise vector $\boldsymbol{\epsilon} \sim \mathcal{N}(\mathbf{0}, \mathbf{I})$, a sufficient number of synthetic semantic features can be generated.
\paragraph{Semantic Feature-based Classification}
After generating synthetic semantic feature samples for all unseen classes, we train the semantic feature classifier using a supervised learning model (e.g., softmax classifier, support vector machine, and nearest neighbor).
Suppose, for example, that the softmax classifier is used as a classification model.
Let $\{ \widetilde{\mathbf{z}}_{u, s}^{(i)} \}_{i=1}^{N_{u}}$ be the set of synthetic semantic feature samples for the unseen class $u$, then the semantic feature classifier is trained to minimize the cross entropy loss\footnote{We recall that $\{ \mathbf{z}_{c, s}^{(i)} \}_{i=1}^{N_{c}}$ is the set of semantic features for the seen class $c \in \mathcal{Y}_{s}$.}
\begin{align}
\mathcal{L}_{\text{CE}}
&= -\sum_{c \in \mathcal{Y}_{s}} \sum_{i=1}^{N_{c}} \log P( c | \mathbf{z}_{c, s}^{(i)} ) - \sum_{u \in \mathcal{Y}_{u}} \sum_{i=1}^{N_{u}} \log P( u | \widetilde{\mathbf{z}}_{u, s}^{(i)} ),
\end{align}
where
\begin{align}
P(y|\mathbf{z})
&= \frac{\exp (\mathbf{w}_{y}^{T} \mathbf{z} + b_{y})}{ \sum_{y^{\prime} \in \mathcal{Y}_{s} \cup \mathcal{Y}_{u}} \exp (\mathbf{w}_{y^{\prime}}^{T} \mathbf{z} + b_{y^{\prime}})}
\end{align}
and $\mathbf{w}_{y}$ and $b_{y}$ are weight and bias parameters of the softmax classifier to be learned.
\subsection{Comparison with Conventional Approaches}
There have been previous efforts to extract the semantic feature from the image feature~\cite{DLFZRL, RFF-GZSL, Disentangled-VAE, SDGZSL}.
While our approach seems to be a bit similar to~\cite{Disentangled-VAE} and~\cite{SDGZSL} in the sense that the autoencoder-based image feature decomposition method is used for the semantic feature extraction, our work is dearly distinct from those works in two respects.
First, we use different training strategy in capturing the attribute-related information.
In our approach, to make sure that the semantic encoder output contains all the attribute-related information, we use two complementary loss terms: 1) the loss term to encourage the semantic encoder to capture the attribute-related information and 2) the loss term to discourage the residual encoder to capture any attribute-related information (see~\eqref{eq:separation loss_MI form}).
Whereas, the training loss used to remove the attribute-related information from the residual encoder output has not been used in~\cite{Disentangled-VAE, SDGZSL}.
Also, we employ a new training loss $\mathcal{L}_{\text{sim}}$ to remove the attribute-irrelevant information from the semantic encoder output (see~\eqref{eq:similarity loss}), for which there is no counterpart in~\cite{Disentangled-VAE, SDGZSL}.
\section{Experiments}
\begin{table}[t]
\centering
{\resizebox{1.0\linewidth}{!}{
\begin{tabular}{c | c | c | c | c }
\toprule
Classifier input &
AwA1 &
AwA2 &
CUB &
SUN \\
\midrule
Image feature &
90.9 &
92.8 &
73.8 &
47.1 \\
Semantic feature &
\textbf{91.9} &
\textbf{93.4} &
\textbf{76.1} &
\textbf{49.3} \\
\bottomrule
\end{tabular}
}}
\caption{Top-1 accuracy of image feature-based and semantic feature-based image classifiers.}
\label{tab:result_effect of attribute-related feature extraction}
\end{table}
\begin{figure*}[!t]
\begin{minipage}[b]{0.33 \linewidth}
\centering
\centerline{\includegraphics[width=6cm]{tsne_semantic.png}}
\centerline{(a) Semantic features}
\end{minipage}
\hfill
\begin{minipage}[b]{0.33 \linewidth}
\centering
\centerline{\includegraphics[width=6cm]{tsne_visual.png}}
\centerline{(b) Image features}
\end{minipage}
\hfill
\begin{minipage}[b]{0.33 \linewidth}
\centering
\centerline{\includegraphics[width=6cm]{tsne_residual.png}}
\centerline{(c) Residual features}
\end{minipage}
\caption{t-SNE visualization of (a) semantic features, (b) image features, and (c) residual features. Samples for the same class are indicated in the same color.}
\label{fig:t-SNE}
\end{figure*}
\begin{table*}[t]
\centering
{\resizebox{1.0\linewidth}{!}{
\begin{tabular}{c | c | c c c | c c c | c c c | c c c}
\toprule
\multirow{2}{2.2cm}{\centering Method} &
\multirow{2}{2.2cm}{\centering Feature Type} &
\multicolumn{3}{c|}{AwA1} &
\multicolumn{3}{c|}{AwA2} &
\multicolumn{3}{c|}{CUB} &
\multicolumn{3}{c}{SUN} \\
\cline{3-14}
&
&
$acc_{s}$ & $acc_{u}$ & $acc_{h}$ &
$acc_{s}$ & $acc_{u}$ & $acc_{h}$ &
$acc_{s}$ & $acc_{u}$ & $acc_{h}$ &
$acc_{s}$ & $acc_{u}$ & $acc_{h}$ \\
\toprule
CVAE-GZSL &
\multirow{9}{2.2cm}{\centering ResNet} &
- & - & 47.2 &
- & - & 51.2 &
- & - & 34.5 &
- & - & 26.7 \\
f-CLSWGAN &
&
61.4 & 57.9 & 59.6 &
- & - & - &
57.7 & 43.7 & 49.7 &
36.6 & 42.6 & 39.4 \\
cycle-CLSWGAN &
&
64.0 & 56.9 & 60.2 &
- & - & - &
61.0 & 45.7 & 52.3 &
33.6 & 49.4 & 40.0 \\
f-VAEGAN-D2 &
&
- & - & - &
70.6 & 57.6 & 63.5 &
60.1 & 48.4 & 53.6 &
38.0 & 45.1 & 41.3 \\
LisGAN &
&
76.3 & 52.6 & 62.3 &
- & - & - &
57.9 & 46.5 & 51.6 &
37.8 & 42.9 & 40.2 \\
CADA-VAE &
&
72.8 & 57.3 & 64.1 &
75.0 & 55.8 & 63.9 &
53.5 & 51.6 & 52.4 &
35.7 & 47.2 & 40.6 \\
DASCN &
&
68.0 & 59.3 & 63.4 &
- & - & - &
59.0 & 45.9 & 51.6 &
38.5 & 42.4 & 40.3 \\
LsrGAN &
&
74.6 & 54.6 & 63.0 &
- & - & - &
59.1 & 48.1 & 53.0 &
37.7 & 44.8 & 40.9 \\
Zero-VAE-GAN &
&
66.8 & 58.2 & 62.3 &
70.9 & 57.1 & 62.5 &
47.9 & 43.6 & 45.5 &
30.2 & 45.2 & 36.3 \\
\midrule
DLFZRL &
\multirow{3}{2.2cm}{\centering Semantic} &
- & - & 61.2 &
- & - & 60.9 &
- & - & 51.9 &
- & - & {\underline{42.5}} \\
RFF-GZSL &
&
75.1 & 59.8 & {\underline{66.5}} &
- & - & - &
56.6 & 52.6 & {\underline{54.6}} &
38.6 & 45.7 & 41.9 \\
Disentangled-VAE &
&
72.9 & 60.7 & 66.2 &
80.2 & 56.9 & {\underline{66.6}} &
58.2 & 51.1 & 54.4 &
36.6 & 47.6 & 41.4 \\
\midrule
{\bf{SE-GZSL}} &
Semantic &
76.7 & 61.3 & {\bf{68.1}} &
80.7 & 59.9 & {\bf{68.8}} &
60.3 & 53.1 & {\bf{56.4}} &
40.7 & 45.8 & {\bf{43.1}} \\
\bottomrule
\end{tabular}
}}
\caption{GZSL classification performance of the proposed SE-GZSL technique and conventional approaches. `-' means that the result is not reported in the references. The best results are in bold, and the second best results are underlined.}
\label{tab:result_GZSL performance}
\end{table*}
\begin{table*}[t]
\centering
{\resizebox{1.0\linewidth}{!}{
\begin{tabular}{c | c c c | c c c | c c c | c c c}
\toprule
\multirow{2}{2.2cm}{\centering Loss}&
\multicolumn{3}{c|}{AwA1} &
\multicolumn{3}{c|}{AwA2} &
\multicolumn{3}{c|}{CUB} &
\multicolumn{3}{c}{SUN} \\
\cline{2-13}
&
$acc_{s}$ & $acc_{u}$ & $acc_{h}$ &
$acc_{s}$ & $acc_{u}$ & $acc_{h}$ &
$acc_{s}$ & $acc_{u}$ & $acc_{h}$ &
$acc_{s}$ & $acc_{u}$ & $acc_{h}$ \\
\toprule
$\mathcal{L}_{\text{recon}}$ &
64.6 & 53.1 & 58.3 &
68.9 & 55.7 & 61.6 &
54.5 & 46.1 & 49.9 &
38.4 & 40.6 & 39.4 \\
$\mathcal{L}_{\text{recon}}$ + $\mathcal{L}_{\text{MI}}$ &
75.0 & 57.9 & 65.4 &
74.2 & 58.6 & 65.5 &
59.4 & 51.5 & 55.1 &
37.1 & 46.5 & 41.3 \\
$\mathcal{L}_{\text{recon}}$ + $\mathcal{L}_{\text{MI}}$ + $\mathcal{L}_{\text{sim}}$ &
76.7 & 61.3 & \textbf{68.1} &
80.7 & 59.9 & \textbf{68.8} &
60.3 & 53.1 & \textbf{56.4} &
40.7 & 45.8 & \textbf{43.1} \\
\bottomrule
\end{tabular}
}}
\caption{Ablation study on the performance of SE-GZSL.}
\label{tab:result_ablation study}
\end{table*}
\subsection{Experimental Setup}
\paragraph{Datasets}
In our experiments, we evaluate the performance of our model using four benchmark datasets: AwA1, AwA2, CUB, and SUN.
The AwA1 and AwA2 datasets contain 50 classes of animal images annotated with 85 attributes~\cite{zsl_proposal, AwA2}.
The CUB dataset contains 200 species of bird images annotated with 312 attributes~\cite{CUB}.
The SUN dataset contains 717 classes of scene images annotated with 102 attributes~\cite{SUN}.
In dividing the total classes into seen and unseen classes, we adopt the conventional dataset split presented in~\cite{AwA2}.
\paragraph{Implementation Details}
As in~\cite{CLSWGAN, CADA-VAE}, we use ResNet-101~\cite{ResNet} as a pre-trained classification network and fix it in our training process.
We implement all the networks in SE-GZSL (semantic encoder, residual encoder, and decoder in the image feature decomposition network, and generator and critic in WGAN) using the multilayer perceptron (MLP) with one hidden layer as in~\cite{CLSWGAN, f-vaegan-d2}.
We set the number of hidden units to 4096 and use LeakyReLU with a negative slope of 0.02 as a nonlinear activation function.
For the output layer of the generator, the ReLU activation is used since the image feature extracted by ResNet is non-negative.
As in~\cite{InfoNCE}, we define the score function $f$ in~\eqref{eq:separation loss_expectation form} as $f(\mathbf{z}_{s}, \mathbf{a}) = \mathbf{z}_{s}^{T} \mathbf{W} \mathbf{a}$ where $\mathbf{W}$ is a weight matrix to be learned.
Also, as in~\cite{CLUB}, we approximate the conditional PDF $p(\mathbf{a} | \mathbf{z}_{r})$ in~\eqref{eq:separation loss_expectation form} using a variational encoder consisting of two hidden layers.
The gradient penalty coefficient in the WGAN loss $\mathcal{L}_{G, \text{WGAN}}$ is set to $\lambda_{\text{gp}} = 10$ as suggested in the original WGAN paper~\cite{WGAN-GP}.
We set the weighting coefficients in~\eqref{eq:separation loss}, ~\eqref{eq:decomposition loss}, and~\eqref{eq:generator loss} to $\lambda_{s}=20, \lambda_{r}=50, \lambda_{\text{sim}}=1, \lambda_{G, \text{MI}}=1, \lambda_{G, \text{sim}} = 0.025$.
\subsection{Semantic Feature-based Image Classification}
We first investigate whether the image classification performance can be improved by exploiting the semantic feature.
To this end, we train two image classifiers: the classifier exploiting the image feature and the classifier utilizing the semantic feature extracted by the semantic encoder.
To compare the semantic feature directly with the image feature, we use the simple softmax classifier as a classification model.
In Table~\ref{tab:result_effect of attribute-related feature extraction}, we summarize the top-1 classification accuracy of each classifier on test image samples for seen classes.
We observe that the semantic feature-based classifier outperforms the image feature-based classifier for all datasets.
In particular, for the SUN and CUB datasets, the semantic feature-based classifier achieves about $2\%$ improvement in the top-1 classification accuracy over the image feature-based classifier, which demonstrates that the image classification performance can be enhanced by removing the attribute-irrelevant information in the image feature.
\subsection{Visualization of Semantic Features}
In Fig.~\ref{fig:t-SNE}, we visualize semantic feature samples obtained from the CUB dataset using a t-distributed stochastic neighbor embedding (t-SNE), a tool to visualize high-dimensional data in a two-dimensional plane~\cite{t-SNE}.
For comparison, we also visualize image feature samples and residual feature samples extracted by the residual encoder.
We observe that semantic feature samples containing only attribute-related information are well-clustered, that is, samples of the same class are grouped and samples of different classes are separated (see Fig.~\ref{fig:t-SNE}(a)).
Whereas, image feature samples of different classes are not separated sufficiently (see Fig.~\ref{fig:t-SNE}(b)) and residual feature samples are scattered randomly (see Fig.~\ref{fig:t-SNE}(c)).
\subsection{Comparison with State-of-the-art}
We next evaluate the GZSL classification performance of the proposed approach using the standard evaluation protocol presented in~\cite{AwA2}.
Specifically, we measure the average top-1 classification accuracies $acc_{s}$ and $acc_{u}$ on seen and unseen classes, respectively, and then use their harmonic mean $acc_{h}$ as a metric to evaluate the performance.
In Table~\ref{tab:result_GZSL performance}, we summarize the performance of SE-GZSL on different datasets.
For comparison, we also summarize the performance of conventional methods among which DLFZRL, RFF-GZSL, and Disentangled-VAE are semantic feature-based approaches~\cite{DLFZRL, RFF-GZSL, Disentangled-VAE} and other methods are image feature-based approaches~\cite{CVAE-GZSL, CLSWGAN, cycle-WGAN, f-vaegan-d2, LisGAN, CADA-VAE, DASCN, LsrGAN, Zero-VAE-GAN}.
From the results, we observe that the proposed SE-GZSL outperforms conventional image feature-based approaches by a large margin.
For example, for the AwA2 dataset, SE-GZSL achieves about 5\% improvement in the harmonic mean accuracy over image feature-based approaches.
We also observe that SE-GZSL outperforms existing semantic feature-based approaches for all datasets.
For example, for the AwA1, AwA2, and CUB datasets, our model achieves about 2\% improvement in the harmonic mean accuracy over the state-of-the-art approaches.
\subsection{Ablation Study}
\paragraph{Effectiveness of Loss Functions}
In training the semantic feature extractor, we have used the MI-based loss $\mathcal{L}_{\text{MI}}$ and the similarity-based loss $\mathcal{L}_{\text{sim}}$.
To examine the impact of each loss function, we measure the performance of three different versions of SE-GZSL: 1) SE-GZSL trained only with the reconstruction loss $\mathcal{L}_{\text{recon}}$, 2) SE-GZSL trained with $\mathcal{L}_{\text{recon}}$ and $\mathcal{L}_{\text{MI}}$, and 3) SE-GZSL trained with $\mathcal{L}_{\text{recon}}$, $\mathcal{L}_{\text{MI}}$, and $\mathcal{L}_{\text{sim}}$.
From the results in Table~\ref{tab:result_ablation study}, we observe that the performance of SE-GZSL can be enhanced greatly by exploiting the MI-based loss $\mathcal{L}_{\text{MI}}$.
In particular, for the AwA1 and CUB datasets, we achieve more than $5\%$ improvement in the harmonic mean accuracy by utilizing $\mathcal{L}_{\text{MI}}$.
Also, for the AwA2 dataset, we achieve about $4\%$ improvement of the accuracy.
One might notice that when $\mathcal{L}_{\text{MI}}$ is not used, SE-GZSL performs worse than conventional image feature-based methods (see Table~\ref{tab:result_GZSL performance}).
This is because the semantic encoder cannot capture all the attribute-related information without $\mathcal{L}_{\text{MI}}$, and thus using the semantic encoder output in the classification incurs the loss of the attribute-related information.
We also observe that the performance of SE-GZSL can be improved further by exploiting the similarity-based loss $\mathcal{L}_{\text{sim}}$.
For example, for the AwA2 dataset, more than $3\%$ improvement in the harmonic mean accuracy can be achieved by utilizing $\mathcal{L}_{\text{sim}}$.
\paragraph{Importance of Residual Encoder}
\begin{table}[t]
\centering
{\resizebox{1.0\linewidth}{!}{
\begin{tabular}{c | c | c | c | c}
\toprule
Method &
AwA1 &
AwA2 &
CUB &
SUN \\
\toprule
SE-GZSL w/o residual encoder &
66.7 &
67.5 &
55.1 &
42.1 \\
\midrule
SE-GZSL w/ residual encoder &
{\bf{68.1}} &
{\bf{68.8}} &
{\bf{56.4}} &
{\bf{43.1}} \\
\bottomrule
\end{tabular}
}}
\caption{Harmonic mean accuracy of SE-GZSL with and without the residual encoder.}
\label{tab:rebuttal_residual encoder}
\end{table}
For the semantic feature extraction, we have decomposed the image feature into the attribute-related feature and the attribute-irrelevant feature using the semantic and residual encoders.
An astute reader might ask why the residual encoder is needed to extract the semantic feature.
To answer this question, we measure the performance of SE-GZSL without using the residual encoder.
From the results in Table~\ref{tab:rebuttal_residual encoder}, we can observe that the GZSL performance of SE-GZSL is degraded when the residual encoder is not used.
This is because if the residual encoder is removed, then the attribute-irrelevant information, required for the reconstruction of the image feature, would be contained in the semantic encoder output and therefore mess up the process to learn the relationship between the image feature and the attribute.
\section{Conclusion}
In this paper, we presented a new GZSL technique called SE-GZSL.
Key idea of the proposed SE-GZSL is to exploit the semantic feature in learning the relationship between the image and the attribute, removing the interference caused by the attribute-irrelevant information.
To extract the semantic feature, we presented the autoencoder-based image feature decomposition network consisting of semantic and residual encoders.
In a nutshell, the semantic and residual encoders capture the attribute-related information and the attribute-irrelevant information, respectively.
In training the image feature decomposition network, we used MI-based loss to encourage the semantic encoder to capture all the attribute-related information and similarity-based loss to discourage the semantic encoder to capture any attribute-irrelevant information.
Our experiments on various datasets demonstrated that the proposed SE-GZSL outperforms conventional GZSL approaches by a large margin.
\section{Acknowledgements}
This work was supported in part by the Samsung Research Funding \& Incubation Center for Future Technology of Samsung Electronics under Grant SRFC-IT1901-17 and in part by the National Research Foundation of Korea (NRF) grant funded by the
Korea government (MSIT) under Grant 2020R1A2C2102198.
{\small
\section{Introduction}
Image classification is a long-standing yet important task with a wide range of applications such as autonomous driving, industrial automation, medical diagnosis, and biometric identification~\cite{autonomous_driving, industrial_automation, medical_diagnosis, biometric_identification}.
In solving the task, supervised learning (SL) techniques have been popularly used for its superiority~\cite{VGG, ResNet}.
Well-known drawback of SL is that a large number of training data are required for each and every class to be identified.
Unfortunately, in many practical scenarios, it is difficult to collect training data for certain classes (e.g., endangered species and newly observed species such as variants of COVID-19).
When there are \textit{unseen} classes where training data is unavailable, SL-based models are biased towards the \textit{seen} classes, impeding the identification of the unseen classes.
Recently, to overcome this drawback, a technique to train a classifier using manually annotated attributes (e.g., color, size, and shape; see Fig.~\ref{fig:CUB}) has been proposed~\cite{zsl_proposal, gzsl_intro}.
Key idea of this technique, dubbed as generalized zero-shot learning (GZSL), is to learn the relationship between the image and the attribute from seen classes and then use the trained model in the identification of unseen classes.
In~\cite{ALE}, for example, an approach to identify unseen classes by measuring the compatibility between the image feature and attribute has been proposed.
In~\cite{CVAE-GZSL}, a network synthesizing the image feature from the attribute has been employed to generate training data of unseen classes.
In extracting the image feature, a network trained using the classification task (e.g., ResNet~\cite{ResNet}) has been popularly used.
A potential drawback of this extraction method is that the image feature might contain attribute-irrelevant information (e.g., human fingers in Fig.~\ref{fig:CUB}), disturbing the process of learning the relationship between the image and the attribute~\cite{DLFZRL, RFF-GZSL, Disentangled-VAE}.
\begin{figure}[!t]
\centering
\centerline{\includegraphics[width=1\linewidth]{CUB.png}}
\caption{Images and attributes for different bird species sampled from the CUB dataset~\cite{CUB}.}
\label{fig:CUB}
\end{figure}
\begin{figure*}[!t]
\centering
\centerline{\includegraphics[width=15cm]{autoencoder.eps}}
\caption{Illustration of the image feature decomposition.}
\label{fig:SD}
\end{figure*}
In this paper, we propose a new GZSL technique that removes the interference caused by the attribute-irrelevant information.
Key idea of the proposed approach is to extract the \textit{semantic feature}, feature containing the attribute-related information, from the image feature and then use it in learning the relationship between the image and the attribute.
In extracting the semantic feature, we use a modified autoencoder consisting of two encoders, viz., \textit{semantic} and \textit{residual} encoders (see Fig.~\ref{fig:SD}).
In a nutshell, the semantic encoder captures all the attribute-related information in the image feature and the residual encoder catches the attribute-irrelevant information.
In the conventional autoencoder, only reconstruction loss (difference between the input and the reconstructed input) is used for the training.
In our approach, to encourage the semantic encoder to capture the attribute-related information only, we use two novel loss functions on top of the reconstruction loss.
First, we employ the mutual information (MI)-based loss to maximize (minimize) MI between the semantic (residual) encoder output and the attribute.
Since MI is a metric to measure the level of dependency between two random variables, by exploiting the MI-based loss, we can encourage the semantic encoder to capture the attribute-related information and at the same time discourage the residual encoder to capture any attribute-related information.
As a result, all the attribute-related information can be solely captured by the semantic encoder.
Second, we use the similarity-based loss to enforce the semantic encoder not to catch any attribute-irrelevant information.
For example, when a bird image contains human fingers (see Fig.~\ref{fig:CUB}), we do not want features related to the finger to be included in the semantic encoder output.
To do so, we maximize the similarity between the semantic encoder outputs of images that are belonging to the same class (bird images in our example).
Since attribute-irrelevant features are contained only in a few image samples (e.g., human fingers are included in a few bird images), by maximizing the similarity between the semantic encoder outputs of the same class, we can remove attribute-irrelevant information from the semantic encoder output.
From extensive experiments using various benchmark datasets (AwA1, AwA2, CUB, and SUN), we demonstrate that the proposed approach outperforms the conventional GZSL techniques by a large margin.
For example, for the AwA2 and CUB datasets, our model achieves 2\% improvement in the GZSL classification accuracy over the state-of-the-art techniques.
\section{Related Work and Background}
\subsection{Conventional GZSL Approaches}
The main task in GZSL is to learn the relationship between the image and the attribute from seen classes and then use it in the identification of unseen classes.
Early GZSL works have focused on the training of a network measuring the compatibility score between the image feature and the attribute~\cite{ALE, DeViSE}.
Once the network is trained properly, images can be classified by identifying the attribute achieving the maximum compatibility score.
Recently, generative model-based GZSL approaches have been proposed~\cite{CVAE-GZSL, CLSWGAN}.
Key idea of these approaches is to generate synthetic image features of unseen classes from the attributes by employing a generative model~\cite{CVAE-GZSL, CLSWGAN}.
As a generative model, conditional variational autoencoder (CVAE)~\cite{vae} and conditional Wasserstein generative adversarial network (CWGAN)~\cite{WGAN} have been popularly used.
By exploiting the generated image features of unseen classes as training data, a classification network identifying unseen classes can be trained in a supervised manner.
Over the years, many efforts have been made to improve the performance of the generative model.
In~\cite{f-vaegan-d2, CADA-VAE, Zero-VAE-GAN}, an approach to combine multiple generative models (e.g., CVAE and CWGAN) has been proposed.
In~\cite{cycle-WGAN, DASCN}, an additional network estimating the image attribute from the image feature has been used to make sure that the synthetic image features satisfy the attribute of unseen classes.
In~\cite{CLSWGAN, LsrGAN, LisGAN}, an additional image classifier has been used in the generative model training to generate distinct image features for different classes.
Our approach is conceptually similar to the generative model-based approach in the sense that we generate synthetic image features of unseen classes using the generative model.
The key distinctive point of the proposed approach over the conventional approaches is that we use the features containing only attribute-related information in the classification to remove the interference, if any, caused by the attribute-irrelevant information.
\subsection{MI for Deep Learning}
Mathematically, the MI $I(\mathbf{u}, \mathbf{v})$ between two random variables $\mathbf{u}$ and $\mathbf{v}$ is defined as
\begin{align}
I(\mathbf{u}, \mathbf{v})
&= \mathbb{E}_{p(\mathbf{u}, \mathbf{v})} \left [ \log \frac{p(\mathbf{u}, \mathbf{v})}{p(\mathbf{u}) p(\mathbf{v})} \right ] \nonumber \\
&= \int_{\mathbf{u}} \int_{\mathbf{v}} p(\mathbf{u}, \mathbf{v}) \log \frac{p(\mathbf{u}, \mathbf{v})}{p(\mathbf{u}) p(\mathbf{v})} d \mathbf{u} d \mathbf{v},
\label{eq:MI_definition}
\end{align}
where $p(\mathbf{u}, \mathbf{v})$ is the joint probability density function (PDF) of $\mathbf{u}$ and $\mathbf{v}$, and $p(\mathbf{u})$ and $p(\mathbf{v})$ are marginal PDFs of $\mathbf{u}$ and $\mathbf{v}$, respectively.
In practice, it is very difficult to compute the exact value of MI since the joint PDF $p(\mathbf{u}, \mathbf{v})$ is generally unknown and the integrals in~\eqref{eq:MI_definition} are often intractable.
To approximate MI, various MI estimators have been proposed~\cite{InfoNCE, CLUB}.
Representative estimators include InfoNCE~\cite{InfoNCE} and contrastive log-ratio upper bound (CLUB)~\cite{CLUB}, defined as
\begin{align}
I_{\text{InfoNCE}}(\mathbf{u}, \mathbf{v})
&= \mathbb{E}_{p(\mathbf{u}, \mathbf{v})} [ f(\mathbf{u}, \mathbf{v}) ] \hspace{-.5mm} \nonumber \\
&~~~- \hspace{-.5mm} \mathbb{E}_{p(\mathbf{u})} \hspace{-1mm} \left [ \log \left (
\mathbb{E}_{p(\mathbf{v})} [ \exp ( f(\mathbf{u}, \mathbf{v} ) ) ]
\right ) \right ], \label{eq:MI_lower bound_InfoNCE} \\
I_{\text{CLUB}}(\mathbf{u}, \mathbf{v})
&= \mathbb{E}_{p(\mathbf{u}, \mathbf{v})} \hspace{-1mm} \left [ \log p(\mathbf{v} | \mathbf{u} ) \right ] \hspace{-.5mm} -
\hspace{-.5mm} \mathbb{E}_{p(\mathbf{u})p(\mathbf{v})} \hspace{-1mm} \left [ \log p(\mathbf{v} | \mathbf{u} ) \right ],
\label{eq:MI_upper bound_CLUB}
\end{align}
where $f$ is a pre-defined score function measuring the compatibility between $\mathbf{u}$ and $\mathbf{v}$, and $p(\mathbf{v} | \mathbf{u})$ is the conditional PDF of $\mathbf{v}$ given $\mathbf{u}$, which is often approximated by a neural network.
The relationship between MI, InfoNCE, and CLUB is given by
\begin{align}
I_{\text{InfoNCE}}(\mathbf{u}, \mathbf{v})
\le I(\mathbf{u}, \mathbf{v})
\le I_{\text{CLUB}}(\mathbf{u}, \mathbf{v}).
\label{eq:MI inequality}
\end{align}
Recently, InfoNCE and CLUB have been used to strengthen or weaken the independence between different parts of the neural network.
For example, when one tries to enforce the independence between $\mathbf{u}$ and $\mathbf{v}$, that is, to reduce $I(\mathbf{u}, \mathbf{v})$, an approach to minimize the upper bound $I_{\text{CLUB}}(\mathbf{u}, \mathbf{v})$ of MI can be used~\cite{MI_minimization}.
Whereas, when one wants to maximize the dependence between $\mathbf{u}$ and $\mathbf{v}$, that is, to increase $I(\mathbf{u}, \mathbf{v})$, an approach to maximize the lower bound $I_{\text{InfoNCE}}(\mathbf{u}, \mathbf{v})$ of MI~\cite{MI_maximization} can be used.
\section{SE-GZSL}
In this section, we present the proposed GZSL technique called semantic feature extraction-based GZSL (SE-GZSL).
We first discuss how to extract the semantic feature from the image feature and then delve into the GZSL classification using the extracted semantic feature.
\subsection{Semantic Feature Extraction}
In extracting the semantic feature from the image feature, the proposed SE-GZSL technique uses the modified autoencoder architecture where two encoders, called semantic and residual encoders, are used in capturing the attribute-related information and the attribute-irrelevant information, respectively (see Fig~\ref{fig:SD}).
As mentioned, in the autoencoder training, we use two loss functions: 1) MI-based loss to encourage the semantic encoder to capture all attribute-related information and 2) similarity-based loss to encourage the semantic encoder not to capture attribute-irrelevant information.
In this subsection, we discuss the overall training loss with emphasis on these two.
\paragraph{MI-based Loss}
To make sure that all the attribute-related information is contained in the semantic encoder output, we use MI in the autoencoder training.
To do so, we maximize MI between the semantic encoder output and the attribute which is given by manual annotation.
At the same time, to avoid capturing of attribute-related information in the residual encoder, we minimize MI between the residual encoder output and the attribute.
Let $\mathbf{z}_{s}$ and $\mathbf{z}_{r}$ be the semantic and residual encoder outputs corresponding to the image feature $\mathbf{x}$, and $\mathbf{a}$ be the image attribute (see Fig.~\ref{fig:SD}). Then, our training objective can be expressed as
\begin{align}
\text{minimize}~~~-\lambda_{s} I(\mathbf{z}_{s}, \mathbf{a}) + \lambda_{r} I(\mathbf{z}_{r}, \mathbf{a}),
\label{eq:separation loss_MI form}
\end{align}
where $\lambda_{s}$ and $\lambda_{r}$ ($\lambda_{s}, \lambda_{r} > 0$) are weighting coefficients.
Since the computation of MI is not tractable, we use InfoNCE and CLUB (see~\eqref{eq:MI_lower bound_InfoNCE} and~\eqref{eq:MI_upper bound_CLUB}) as a surrogate of MI.
In our approach, to minimize the objective function in~\eqref{eq:separation loss_MI form}, we express its upper bound using InfoNCE and CLUB and then train the autoencoder in a way to minimize the upper bound.
Using the relationship between MI and its estimators in~\eqref{eq:MI inequality}, the upper bound $\mathcal{L}_{\text{MI}}$ of the objective function in~\eqref{eq:separation loss_MI form} is
\begin{align}
\mathcal{L}_{\text{MI}}
&= -\lambda_{s} I_{\text{InfoNCE}}(\mathbf{z}_{s}, \mathbf{a}) + \lambda_{r} I_{\text{CLUB}} (\mathbf{z}_{r}, \mathbf{a}) \nonumber \\
&= -\lambda_{s} \mathbb{E}_{p(\mathbf{z}_{s}, \mathbf{a})} [ f(\mathbf{z}_{s}, \mathbf{a}) ] \nonumber \\
&~~~ + \hspace{-.5mm} \lambda_{s} \mathbb{E}_{p(\mathbf{z}_{s})} \hspace{-1mm} \left [ \log \left (
\mathbb{E}_{p(\mathbf{a})} [ \exp ( f(\mathbf{z}_{s}, \mathbf{a}) ) ] \right ) \right ] \nonumber \\
&~~~ +\lambda_{r} \left (
\mathbb{E}_{p(\mathbf{z}_{r}, \mathbf{a})} \hspace{-1mm} \left [
\log p(\mathbf{a} | \mathbf{z}_{r} ) \right ] \hspace{-.5mm} -
\hspace{-.5mm} \mathbb{E}_{p(\mathbf{z}_{r})p(\mathbf{a})} \hspace{-1mm} \left [ \log p(\mathbf{a} | \mathbf{z}_{r} ) \right ]
\right ) \hspace{-1mm}.
\label{eq:separation loss_expectation form}
\end{align}
Let $\mathcal{Y}_{s}$ be the set of seen classes, $\mathbf{a}_{c}$ be the attribute of a seen class $c \in \mathcal{Y}_{s}$, and $\{ \mathbf{x}_{c}^{(i)} \}_{i=1}^{N_{c}}$ be the set of training image features for the class $c$.
Further, let $\mathbf{z}_{c, s}^{(i)}$ and $\mathbf{z}_{c, r}^{(i)}$ be the semantic and residual encoder outputs corresponding to the input image feature $\mathbf{x}_{c}^{(i)}$, respectively, then $\mathcal{L}_{\text{MI}}$ can be expressed as
\begin{align}
\mathcal{L}_{\text{MI}}
&= -\frac{\lambda_{s}}{N} \sum_{c \in \mathcal{Y}_{s}} \sum_{i=1}^{N_{c}} \log
\frac{\exp ( f ( \mathbf{z}_{c, s}^{(i)}, \mathbf{a}_{c} ) )}
{\frac{1}{|\mathcal{Y}_{s}|} \hspace{-.5mm} \underset{c^{\prime} \in \mathcal{Y}_{s}}{\sum} \exp ( f ( \mathbf{z}_{c, s}^{(i)}, \mathbf{a}_{c^{\prime}} ) )} \nonumber \\
&~+\frac{\lambda_{r}}{N} \hspace{-.5mm} \sum_{c \in \mathcal{Y}_{s}} \hspace{-.5mm} \sum_{i=1}^{N_{c}} \hspace{-.5mm} \left ( \log p(\mathbf{a}_{c} | \mathbf{z}_{c, r}^{(i)})
- \hspace{-2mm} \underset{c^{\prime} \in \mathcal{Y}_{s}}{\sum} \hspace{-.5mm} \frac{\log p(\mathbf{a}_{c^{\prime}} | \mathbf{z}_{c, r}^{(i)})}{|\mathcal{Y}_{s}|}
\hspace{-.5mm} \right ) \hspace{-1mm},
\label{eq:separation loss}
\end{align}
where $N = \sum_{c \in \mathcal{Y}_{s}} N_{c}$ is the total number of training image features.
\paragraph{Similarity-based Loss}
We now discuss the similarity-based loss to enforce the semantic encoder not to capture any attribute-irrelevant information.
Since images belonging to the same class have the same attribute, attribute-related image features of the same class would be more or less similar.
This means that if the semantic encoder captures attribute-related information only, then the similarity between semantic encoder outputs of the same class should be large.
Inspired by this observation, to remove the attribute-irrelevant information from the semantic encoder output, we train the semantic encoder in a way to maximize the similarity between outputs of the same class:
\begin{align}
\text{maximize}~~~\sum_{j=1}^{N_{c}} \exp (\operatornamewithlimits{sim} ( \mathbf{z}_{c, s}^{(i)}, \mathbf{z}_{c, s}^{(j)} ) ),
\label{eq:similarity loss_same class}
\end{align}
where the similarity is measured using the cosine-similarity function defined as
\begin{align*}
\operatornamewithlimits{sim}(\mathbf{u}, \mathbf{v})
&= \frac{\langle \mathbf{u}, \mathbf{v} \rangle}
{\| \mathbf{u} \|_{2} \| \mathbf{v} \|_{2}}.
\end{align*}
Also, we minimize the similarity between semantic encoder outputs of different classes to obtain sufficiently distinct semantic encoder outputs for different classes:
\begin{align}
\text{minimize}~~~\sum_{c^{\prime} \neq c} \sum_{j=1}^{N_{c^{\prime}}} \exp ( \operatornamewithlimits{sim} ( \mathbf{z}_{c, s}^{(i)}, \mathbf{z}_{c^{\prime}, s}^{(j)} ) ).
\label{eq:similarity loss_different classes}
\end{align}
Using the fact that one can maximize $A$ and minimize $B$ at the same time by minimizing $-\log \frac{1}{1+B/A} = -\log \frac{A}{A+B}$, we obtain the similarity-based loss as
\begin{align}
\mathcal{L}_{\text{sim}}
&= -\frac{1}{N} \sum_{c \in \mathcal{Y}_{s}} \sum_{i=1}^{N_{c}} \log
\frac{\underset{j=1}{\overset{N_{c}}{\sum}} \exp \hspace{-.7mm} \left ( \hspace{-.5mm} \operatornamewithlimits{sim} ( \mathbf{z}_{c, s}^{(i)}, \mathbf{z}_{c, s}^{(j)} ) \hspace{-.5mm} \right )}
{ \underset{c^{\prime} \in \mathcal{Y}_{s}}{\sum} \underset{j=1}{\overset{N_{c^{\prime}}}{\sum}} \exp \hspace{-.7mm} \left ( \hspace{-.5mm} \operatornamewithlimits{sim} ( \mathbf{z}_{c, s}^{(i)}, \mathbf{z}_{c^{\prime}, s}^{(j)} ) \hspace{-.5mm} \right ) }.
\label{eq:similarity loss}
\end{align}
\subsubsection{Overall Loss}
By adding the conventional reconstruction loss $\mathcal{L}_{\text{recon}}$ for the autoencoder, the MI-based loss $\mathcal{L}_{\text{MI}}$, and the similarity-based loss $\mathcal{L}_{\text{sim}}$, we obtain the overall loss function as
\begin{align}
\mathcal{L}_{\text{total}}
&= \mathcal{L}_{\text{recon}} + \mathcal{L}_{\text{MI}} + \lambda_{\text{sim}} \mathcal{L}_{\text{sim}},
\label{eq:decomposition loss}
\end{align}
where $\lambda_{\text{sim}}$ is a weighting coefficient and $\mathcal{L}_{\text{recon}}$ is the reconstruction loss given by
\begin{align}
\mathcal{L}_{\text{recon}}
&= \frac{1}{N} \sum_{c \in \mathcal{Y}_{s}} \sum_{i=1}^{N_{c}} \| \mathbf{x}_{c}^{(i)} - \widehat{\mathbf{x}}_{c}^{(i)} \|_{2}.
\end{align}
Here, $\widehat{\mathbf{x}}_{c}^{(i)}$ is the image feature reconstructed using the semantic and residual encoder outputs ($\mathbf{z}_{c, s}^{(i)}$ and $\mathbf{z}_{c, r}^{(i)}$) in the decoder.
When the training is finished, we only use the semantic encoder for the purpose of extracting the semantic feature.
\subsection{GZSL Classification Using Semantic Features}
So far, we have discussed how to extract the semantic feature from the image feature.
We now discuss how to perform the GZSL classification using the semantic feature.
In a nutshell, we synthesize semantic feature samples for unseen classes from their attributes.
Once the synthetic samples are generated, the semantic classifier identifying unseen classes from the semantic feature is trained in a supervised manner.
\paragraph{Semantic Feature Generation}
\begin{figure*}[h]
\centering
\centerline{\includegraphics[width=13cm]{generator.eps}}
\caption{Illustration of the synthetic semantic feature generation for unseen classes.}
\label{fig:generator}
\end{figure*}
To synthesize the semantic feature samples for unseen classes, we first generate image features from the attributes of unseen classes and then extract the semantic features from the synthetic image features using the semantic encoder (see Fig.~\ref{fig:generator}).
In synthesizing the image feature, we employ WGAN that mitigates the unstable training issue of GAN by exploiting a Wasserstein distance-based loss function~\cite{WGAN}.
The main component in WGAN is a generator $G$ synthesizing the image feature $\widetilde{\mathbf{x}}_{c}$ from a random noise vector $\boldsymbol{\epsilon} \sim \mathcal{N}(\mathbf{0}, \mathbf{I})$ and the image attribute $\mathbf{a}_{c}$ (i.e., $\widetilde{\mathbf{x}}_{c} = G(\boldsymbol{\epsilon}, \mathbf{a}_{c})$).
Conventionally, WGAN is trained to minimize the Wasserstein distance between the distributions of real image feature $\mathbf{x}_{c}$ and generated image feature $\widetilde{\mathbf{x}}_{c}$ given by
\begin{align}
\lefteqn{\hspace{-.2cm} \mathcal{L}_{G, \text{WGAN}}} \nonumber \\
&\hspace{-.3cm}= \max_{D} \hspace{-.5mm} \bigg (
\mathbb{E}_{p(\mathbf{x}_{c} | \mathbf{a}_{c})} [ D(\mathbf{x}_{c}, \mathbf{a}_{c}) ]
- \mathbb{E}_{p(\widetilde{\mathbf{x}}_{c} | \mathbf{a}_{c})} [ D(\widetilde{\mathbf{x}}_{c}, \mathbf{a}_{c}) ] \nonumber \\
&\hspace{-.3cm}~~~~~~~~~~~~~~ -\lambda_{\text{gp}} \mathbb{E}_{p(\widehat{\mathbf{x}}_{c} | \mathbf{a}_{c})} \hspace{-.8mm} \left [ \left ( \| \nabla_{\widehat{\mathbf{x}}_{c}} D(\widehat{\mathbf{x}}_{c}, \mathbf{a}_{c}) \|_{2} - 1 \right )^{2}
\right ] \hspace{-.8mm} \bigg ),
\label{eq:generator_WGAN loss}
\end{align}
where $D$ is an auxiliary network (called critic), $\widehat{\mathbf{x}}_{c} = \alpha \mathbf{x}_{c} + (1 - \alpha) \widetilde{\mathbf{x}}_{c}~(\alpha \sim \mathcal{U}(0, 1))$, and $\lambda_{\text{gp}}$ is the regularization coefficient (a.k.a., gradient penalty coefficient)~\cite{WGAN-GP}.
In our scheme, to make sure that the semantic feature $\widetilde{\mathbf{z}}_{c, s}$ obtained from $\widetilde{\mathbf{x}}_{c}$ is similar to the real semantic feature $\mathbf{z}_{c, s}$, we additionally use the following losses in the WGAN training:
\begin{align}
\mathcal{L}_{G, \text{MI}}
&= -I_{\text{InfoNCE}}(\widetilde{\mathbf{z}}_{c, s}, \mathbf{a}_{c}),
\label{eq:generator_InfoNCE loss} \\
\mathcal{L}_{G, \text{sim}}
&= -\mathbb{E}_{p(\widetilde{\mathbf{z}}_{c, s})} \hspace{-1mm} \left [
\log \frac{\underset{i=1}{\overset{N_{c}}{\sum}} \exp ( \operatornamewithlimits{sim} (\widetilde{\mathbf{z}}_{c, s}, \mathbf{z}_{c, s}^{(i)}) )}
{\underset{c^{\prime}=1}{\overset{S}{\sum}} \underset{i=1}{\overset{N_{c^{\prime}}}{\sum}} \exp ( \operatornamewithlimits{sim} (\widetilde{\mathbf{z}}_{c, s}, \mathbf{z}_{c^{\prime}, s}^{(i)}) )}
\right ].
\label{eq:generator_similarity loss}
\end{align}
We note that these losses are analogous to the losses with respect to the real semantic feature $\mathbf{z}_{c, s}$ in~\eqref{eq:separation loss_expectation form} and~\eqref{eq:similarity loss}, respectively.
By combining~\eqref{eq:generator_WGAN loss},~\eqref{eq:generator_InfoNCE loss}, and~\eqref{eq:generator_similarity loss}, we obtain the overall loss function as
\begin{align}
\mathcal{L}_{G}
&= \mathcal{L}_{G, \text{WGAN}} + \lambda_{G, \text{MI}} \mathcal{L}_{G, \text{MI}} + \lambda_{G, \text{sim}} \mathcal{L}_{G, \text{sim}}, \label{eq:generator loss}
\end{align}
where $\lambda_{G, \text{MI}}$ and $\lambda_{G, \text{sim}}$ are weighting coefficients.
After the WGAN training, we use the generator $G$ and the semantic encoder $E_{s}$ in synthesizing semantic feature samples of unseen classes.
Specifically, for each unseen class $u \in \mathcal{Y}_{u}$, we generate the semantic feature $\widetilde{\mathbf{z}}_{u, s}$ by synthesizing the image feature $\widetilde{\mathbf{x}}_{u} = G(\boldsymbol{\epsilon}, \mathbf{a}_{u})$ using the generator and then exploiting it as an input to the semantic encoder (see Fig.~\ref{fig:generator}):
\begin{align}
\widetilde{\mathbf{z}}_{u, s}
= E_{s}(\widetilde{\mathbf{x}}_{u})
= E_{s}(G(\boldsymbol{\epsilon}, \mathbf{a}_{u})).
\end{align}
By resampling the noise vector $\boldsymbol{\epsilon} \sim \mathcal{N}(\mathbf{0}, \mathbf{I})$, a sufficient number of synthetic semantic features can be generated.
\paragraph{Semantic Feature-based Classification}
After generating synthetic semantic feature samples for all unseen classes, we train the semantic feature classifier using a supervised learning model (e.g., softmax classifier, support vector machine, and nearest neighbor).
Suppose, for example, that the softmax classifier is used as a classification model.
Let $\{ \widetilde{\mathbf{z}}_{u, s}^{(i)} \}_{i=1}^{N_{u}}$ be the set of synthetic semantic feature samples for the unseen class $u$, then the semantic feature classifier is trained to minimize the cross entropy loss\footnote{We recall that $\{ \mathbf{z}_{c, s}^{(i)} \}_{i=1}^{N_{c}}$ is the set of semantic features for the seen class $c \in \mathcal{Y}_{s}$.}
\begin{align}
\mathcal{L}_{\text{CE}}
&= -\sum_{c \in \mathcal{Y}_{s}} \sum_{i=1}^{N_{c}} \log P( c | \mathbf{z}_{c, s}^{(i)} ) - \sum_{u \in \mathcal{Y}_{u}} \sum_{i=1}^{N_{u}} \log P( u | \widetilde{\mathbf{z}}_{u, s}^{(i)} ),
\end{align}
where
\begin{align}
P(y|\mathbf{z})
&= \frac{\exp (\mathbf{w}_{y}^{T} \mathbf{z} + b_{y})}{ \sum_{y^{\prime} \in \mathcal{Y}_{s} \cup \mathcal{Y}_{u}} \exp (\mathbf{w}_{y^{\prime}}^{T} \mathbf{z} + b_{y^{\prime}})}
\end{align}
and $\mathbf{w}_{y}$ and $b_{y}$ are weight and bias parameters of the softmax classifier to be learned.
\subsection{Comparison with Conventional Approaches}
There have been previous efforts to extract the semantic feature from the image feature~\cite{DLFZRL, RFF-GZSL, Disentangled-VAE, SDGZSL}.
While our approach seems to be a bit similar to~\cite{Disentangled-VAE} and~\cite{SDGZSL} in the sense that the autoencoder-based image feature decomposition method is used for the semantic feature extraction, our work is dearly distinct from those works in two respects.
First, we use different training strategy in capturing the attribute-related information.
In our approach, to make sure that the semantic encoder output contains all the attribute-related information, we use two complementary loss terms: 1) the loss term to encourage the semantic encoder to capture the attribute-related information and 2) the loss term to discourage the residual encoder to capture any attribute-related information (see~\eqref{eq:separation loss_MI form}).
Whereas, the training loss used to remove the attribute-related information from the residual encoder output has not been used in~\cite{Disentangled-VAE, SDGZSL}.
Also, we employ a new training loss $\mathcal{L}_{\text{sim}}$ to remove the attribute-irrelevant information from the semantic encoder output (see~\eqref{eq:similarity loss}), for which there is no counterpart in~\cite{Disentangled-VAE, SDGZSL}.
\section{Experiments}
\begin{table}[t]
\centering
{\resizebox{1.0\linewidth}{!}{
\begin{tabular}{c | c | c | c | c }
\toprule
Classifier input &
AwA1 &
AwA2 &
CUB &
SUN \\
\midrule
Image feature &
90.9 &
92.8 &
73.8 &
47.1 \\
Semantic feature &
\textbf{91.9} &
\textbf{93.4} &
\textbf{76.1} &
\textbf{49.3} \\
\bottomrule
\end{tabular}
}}
\caption{Top-1 accuracy of image feature-based and semantic feature-based image classifiers.}
\label{tab:result_effect of attribute-related feature extraction}
\end{table}
\begin{figure*}[!t]
\begin{minipage}[b]{0.33 \linewidth}
\centering
\centerline{\includegraphics[width=6cm]{tsne_semantic.png}}
\centerline{(a) Semantic features}
\end{minipage}
\hfill
\begin{minipage}[b]{0.33 \linewidth}
\centering
\centerline{\includegraphics[width=6cm]{tsne_visual.png}}
\centerline{(b) Image features}
\end{minipage}
\hfill
\begin{minipage}[b]{0.33 \linewidth}
\centering
\centerline{\includegraphics[width=6cm]{tsne_residual.png}}
\centerline{(c) Residual features}
\end{minipage}
\caption{t-SNE visualization of (a) semantic features, (b) image features, and (c) residual features. Samples for the same class are indicated in the same color.}
\label{fig:t-SNE}
\end{figure*}
\begin{table*}[t]
\centering
{\resizebox{1.0\linewidth}{!}{
\begin{tabular}{c | c | c c c | c c c | c c c | c c c}
\toprule
\multirow{2}{2.2cm}{\centering Method} &
\multirow{2}{2.2cm}{\centering Feature Type} &
\multicolumn{3}{c|}{AwA1} &
\multicolumn{3}{c|}{AwA2} &
\multicolumn{3}{c|}{CUB} &
\multicolumn{3}{c}{SUN} \\
\cline{3-14}
&
&
$acc_{s}$ & $acc_{u}$ & $acc_{h}$ &
$acc_{s}$ & $acc_{u}$ & $acc_{h}$ &
$acc_{s}$ & $acc_{u}$ & $acc_{h}$ &
$acc_{s}$ & $acc_{u}$ & $acc_{h}$ \\
\toprule
CVAE-GZSL &
\multirow{9}{2.2cm}{\centering ResNet} &
- & - & 47.2 &
- & - & 51.2 &
- & - & 34.5 &
- & - & 26.7 \\
f-CLSWGAN &
&
61.4 & 57.9 & 59.6 &
- & - & - &
57.7 & 43.7 & 49.7 &
36.6 & 42.6 & 39.4 \\
cycle-CLSWGAN &
&
64.0 & 56.9 & 60.2 &
- & - & - &
61.0 & 45.7 & 52.3 &
33.6 & 49.4 & 40.0 \\
f-VAEGAN-D2 &
&
- & - & - &
70.6 & 57.6 & 63.5 &
60.1 & 48.4 & 53.6 &
38.0 & 45.1 & 41.3 \\
LisGAN &
&
76.3 & 52.6 & 62.3 &
- & - & - &
57.9 & 46.5 & 51.6 &
37.8 & 42.9 & 40.2 \\
CADA-VAE &
&
72.8 & 57.3 & 64.1 &
75.0 & 55.8 & 63.9 &
53.5 & 51.6 & 52.4 &
35.7 & 47.2 & 40.6 \\
DASCN &
&
68.0 & 59.3 & 63.4 &
- & - & - &
59.0 & 45.9 & 51.6 &
38.5 & 42.4 & 40.3 \\
LsrGAN &
&
74.6 & 54.6 & 63.0 &
- & - & - &
59.1 & 48.1 & 53.0 &
37.7 & 44.8 & 40.9 \\
Zero-VAE-GAN &
&
66.8 & 58.2 & 62.3 &
70.9 & 57.1 & 62.5 &
47.9 & 43.6 & 45.5 &
30.2 & 45.2 & 36.3 \\
\midrule
DLFZRL &
\multirow{3}{2.2cm}{\centering Semantic} &
- & - & 61.2 &
- & - & 60.9 &
- & - & 51.9 &
- & - & {\underline{42.5}} \\
RFF-GZSL &
&
75.1 & 59.8 & {\underline{66.5}} &
- & - & - &
56.6 & 52.6 & {\underline{54.6}} &
38.6 & 45.7 & 41.9 \\
Disentangled-VAE &
&
72.9 & 60.7 & 66.2 &
80.2 & 56.9 & {\underline{66.6}} &
58.2 & 51.1 & 54.4 &
36.6 & 47.6 & 41.4 \\
\midrule
{\bf{SE-GZSL}} &
Semantic &
76.7 & 61.3 & {\bf{68.1}} &
80.7 & 59.9 & {\bf{68.8}} &
60.3 & 53.1 & {\bf{56.4}} &
40.7 & 45.8 & {\bf{43.1}} \\
\bottomrule
\end{tabular}
}}
\caption{GZSL classification performance of the proposed SE-GZSL technique and conventional approaches. `-' means that the result is not reported in the references. The best results are in bold, and the second best results are underlined.}
\label{tab:result_GZSL performance}
\end{table*}
\begin{table*}[t]
\centering
{\resizebox{1.0\linewidth}{!}{
\begin{tabular}{c | c c c | c c c | c c c | c c c}
\toprule
\multirow{2}{2.2cm}{\centering Loss}&
\multicolumn{3}{c|}{AwA1} &
\multicolumn{3}{c|}{AwA2} &
\multicolumn{3}{c|}{CUB} &
\multicolumn{3}{c}{SUN} \\
\cline{2-13}
&
$acc_{s}$ & $acc_{u}$ & $acc_{h}$ &
$acc_{s}$ & $acc_{u}$ & $acc_{h}$ &
$acc_{s}$ & $acc_{u}$ & $acc_{h}$ &
$acc_{s}$ & $acc_{u}$ & $acc_{h}$ \\
\toprule
$\mathcal{L}_{\text{recon}}$ &
64.6 & 53.1 & 58.3 &
68.9 & 55.7 & 61.6 &
54.5 & 46.1 & 49.9 &
38.4 & 40.6 & 39.4 \\
$\mathcal{L}_{\text{recon}}$ + $\mathcal{L}_{\text{MI}}$ &
75.0 & 57.9 & 65.4 &
74.2 & 58.6 & 65.5 &
59.4 & 51.5 & 55.1 &
37.1 & 46.5 & 41.3 \\
$\mathcal{L}_{\text{recon}}$ + $\mathcal{L}_{\text{MI}}$ + $\mathcal{L}_{\text{sim}}$ &
76.7 & 61.3 & \textbf{68.1} &
80.7 & 59.9 & \textbf{68.8} &
60.3 & 53.1 & \textbf{56.4} &
40.7 & 45.8 & \textbf{43.1} \\
\bottomrule
\end{tabular}
}}
\caption{Ablation study on the performance of SE-GZSL.}
\label{tab:result_ablation study}
\end{table*}
\subsection{Experimental Setup}
\paragraph{Datasets}
In our experiments, we evaluate the performance of our model using four benchmark datasets: AwA1, AwA2, CUB, and SUN.
The AwA1 and AwA2 datasets contain 50 classes of animal images annotated with 85 attributes~\cite{zsl_proposal, AwA2}.
The CUB dataset contains 200 species of bird images annotated with 312 attributes~\cite{CUB}.
The SUN dataset contains 717 classes of scene images annotated with 102 attributes~\cite{SUN}.
In dividing the total classes into seen and unseen classes, we adopt the conventional dataset split presented in~\cite{AwA2}.
\paragraph{Implementation Details}
As in~\cite{CLSWGAN, CADA-VAE}, we use ResNet-101~\cite{ResNet} as a pre-trained classification network and fix it in our training process.
We implement all the networks in SE-GZSL (semantic encoder, residual encoder, and decoder in the image feature decomposition network, and generator and critic in WGAN) using the multilayer perceptron (MLP) with one hidden layer as in~\cite{CLSWGAN, f-vaegan-d2}.
We set the number of hidden units to 4096 and use LeakyReLU with a negative slope of 0.02 as a nonlinear activation function.
For the output layer of the generator, the ReLU activation is used since the image feature extracted by ResNet is non-negative.
As in~\cite{InfoNCE}, we define the score function $f$ in~\eqref{eq:separation loss_expectation form} as $f(\mathbf{z}_{s}, \mathbf{a}) = \mathbf{z}_{s}^{T} \mathbf{W} \mathbf{a}$ where $\mathbf{W}$ is a weight matrix to be learned.
Also, as in~\cite{CLUB}, we approximate the conditional PDF $p(\mathbf{a} | \mathbf{z}_{r})$ in~\eqref{eq:separation loss_expectation form} using a variational encoder consisting of two hidden layers.
The gradient penalty coefficient in the WGAN loss $\mathcal{L}_{G, \text{WGAN}}$ is set to $\lambda_{\text{gp}} = 10$ as suggested in the original WGAN paper~\cite{WGAN-GP}.
We set the weighting coefficients in~\eqref{eq:separation loss}, ~\eqref{eq:decomposition loss}, and~\eqref{eq:generator loss} to $\lambda_{s}=20, \lambda_{r}=50, \lambda_{\text{sim}}=1, \lambda_{G, \text{MI}}=1, \lambda_{G, \text{sim}} = 0.025$.
\subsection{Semantic Feature-based Image Classification}
We first investigate whether the image classification performance can be improved by exploiting the semantic feature.
To this end, we train two image classifiers: the classifier exploiting the image feature and the classifier utilizing the semantic feature extracted by the semantic encoder.
To compare the semantic feature directly with the image feature, we use the simple softmax classifier as a classification model.
In Table~\ref{tab:result_effect of attribute-related feature extraction}, we summarize the top-1 classification accuracy of each classifier on test image samples for seen classes.
We observe that the semantic feature-based classifier outperforms the image feature-based classifier for all datasets.
In particular, for the SUN and CUB datasets, the semantic feature-based classifier achieves about $2\%$ improvement in the top-1 classification accuracy over the image feature-based classifier, which demonstrates that the image classification performance can be enhanced by removing the attribute-irrelevant information in the image feature.
\subsection{Visualization of Semantic Features}
In Fig.~\ref{fig:t-SNE}, we visualize semantic feature samples obtained from the CUB dataset using a t-distributed stochastic neighbor embedding (t-SNE), a tool to visualize high-dimensional data in a two-dimensional plane~\cite{t-SNE}.
For comparison, we also visualize image feature samples and residual feature samples extracted by the residual encoder.
We observe that semantic feature samples containing only attribute-related information are well-clustered, that is, samples of the same class are grouped and samples of different classes are separated (see Fig.~\ref{fig:t-SNE}(a)).
Whereas, image feature samples of different classes are not separated sufficiently (see Fig.~\ref{fig:t-SNE}(b)) and residual feature samples are scattered randomly (see Fig.~\ref{fig:t-SNE}(c)).
\subsection{Comparison with State-of-the-art}
We next evaluate the GZSL classification performance of the proposed approach using the standard evaluation protocol presented in~\cite{AwA2}.
Specifically, we measure the average top-1 classification accuracies $acc_{s}$ and $acc_{u}$ on seen and unseen classes, respectively, and then use their harmonic mean $acc_{h}$ as a metric to evaluate the performance.
In Table~\ref{tab:result_GZSL performance}, we summarize the performance of SE-GZSL on different datasets.
For comparison, we also summarize the performance of conventional methods among which DLFZRL, RFF-GZSL, and Disentangled-VAE are semantic feature-based approaches~\cite{DLFZRL, RFF-GZSL, Disentangled-VAE} and other methods are image feature-based approaches~\cite{CVAE-GZSL, CLSWGAN, cycle-WGAN, f-vaegan-d2, LisGAN, CADA-VAE, DASCN, LsrGAN, Zero-VAE-GAN}.
From the results, we observe that the proposed SE-GZSL outperforms conventional image feature-based approaches by a large margin.
For example, for the AwA2 dataset, SE-GZSL achieves about 5\% improvement in the harmonic mean accuracy over image feature-based approaches.
We also observe that SE-GZSL outperforms existing semantic feature-based approaches for all datasets.
For example, for the AwA1, AwA2, and CUB datasets, our model achieves about 2\% improvement in the harmonic mean accuracy over the state-of-the-art approaches.
\subsection{Ablation Study}
\paragraph{Effectiveness of Loss Functions}
In training the semantic feature extractor, we have used the MI-based loss $\mathcal{L}_{\text{MI}}$ and the similarity-based loss $\mathcal{L}_{\text{sim}}$.
To examine the impact of each loss function, we measure the performance of three different versions of SE-GZSL: 1) SE-GZSL trained only with the reconstruction loss $\mathcal{L}_{\text{recon}}$, 2) SE-GZSL trained with $\mathcal{L}_{\text{recon}}$ and $\mathcal{L}_{\text{MI}}$, and 3) SE-GZSL trained with $\mathcal{L}_{\text{recon}}$, $\mathcal{L}_{\text{MI}}$, and $\mathcal{L}_{\text{sim}}$.
From the results in Table~\ref{tab:result_ablation study}, we observe that the performance of SE-GZSL can be enhanced greatly by exploiting the MI-based loss $\mathcal{L}_{\text{MI}}$.
In particular, for the AwA1 and CUB datasets, we achieve more than $5\%$ improvement in the harmonic mean accuracy by utilizing $\mathcal{L}_{\text{MI}}$.
Also, for the AwA2 dataset, we achieve about $4\%$ improvement of the accuracy.
One might notice that when $\mathcal{L}_{\text{MI}}$ is not used, SE-GZSL performs worse than conventional image feature-based methods (see Table~\ref{tab:result_GZSL performance}).
This is because the semantic encoder cannot capture all the attribute-related information without $\mathcal{L}_{\text{MI}}$, and thus using the semantic encoder output in the classification incurs the loss of the attribute-related information.
We also observe that the performance of SE-GZSL can be improved further by exploiting the similarity-based loss $\mathcal{L}_{\text{sim}}$.
For example, for the AwA2 dataset, more than $3\%$ improvement in the harmonic mean accuracy can be achieved by utilizing $\mathcal{L}_{\text{sim}}$.
\paragraph{Importance of Residual Encoder}
\begin{table}[t]
\centering
{\resizebox{1.0\linewidth}{!}{
\begin{tabular}{c | c | c | c | c}
\toprule
Method &
AwA1 &
AwA2 &
CUB &
SUN \\
\toprule
SE-GZSL w/o residual encoder &
66.7 &
67.5 &
55.1 &
42.1 \\
\midrule
SE-GZSL w/ residual encoder &
{\bf{68.1}} &
{\bf{68.8}} &
{\bf{56.4}} &
{\bf{43.1}} \\
\bottomrule
\end{tabular}
}}
\caption{Harmonic mean accuracy of SE-GZSL with and without the residual encoder.}
\label{tab:rebuttal_residual encoder}
\end{table}
For the semantic feature extraction, we have decomposed the image feature into the attribute-related feature and the attribute-irrelevant feature using the semantic and residual encoders.
An astute reader might ask why the residual encoder is needed to extract the semantic feature.
To answer this question, we measure the performance of SE-GZSL without using the residual encoder.
From the results in Table~\ref{tab:rebuttal_residual encoder}, we can observe that the GZSL performance of SE-GZSL is degraded when the residual encoder is not used.
This is because if the residual encoder is removed, then the attribute-irrelevant information, required for the reconstruction of the image feature, would be contained in the semantic encoder output and therefore mess up the process to learn the relationship between the image feature and the attribute.
\section{Conclusion}
In this paper, we presented a new GZSL technique called SE-GZSL.
Key idea of the proposed SE-GZSL is to exploit the semantic feature in learning the relationship between the image and the attribute, removing the interference caused by the attribute-irrelevant information.
To extract the semantic feature, we presented the autoencoder-based image feature decomposition network consisting of semantic and residual encoders.
In a nutshell, the semantic and residual encoders capture the attribute-related information and the attribute-irrelevant information, respectively.
In training the image feature decomposition network, we used MI-based loss to encourage the semantic encoder to capture all the attribute-related information and similarity-based loss to discourage the semantic encoder to capture any attribute-irrelevant information.
Our experiments on various datasets demonstrated that the proposed SE-GZSL outperforms conventional GZSL approaches by a large margin.
\section{Acknowledgements}
This work was supported in part by the Samsung Research Funding \& Incubation Center for Future Technology of Samsung Electronics under Grant SRFC-IT1901-17 and in part by the National Research Foundation of Korea (NRF) grant funded by the
Korea government (MSIT) under Grant 2020R1A2C2102198.
{\small
|
2024-02-18T23:40:24.425Z
|
2021-12-30T02:24:07.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14478","language":"en","timestamp":1640831047000,"url":"https:\/\/arxiv.org\/abs\/2112.14478","yymm":"2112"}
|
proofpile-arXiv_000-10020
|
{"provenance":"002.jsonl.gz:10021"}
| null | null |
\section{Three Main Project Modules}
The overall architecture of our project is composed of three fundamental modules, namely~\emph{LEarning TO Rank (LETOR)}, ~\emph{Cloud Search}, and ~\emph{Query Expansion \& Suggestion}. Note that the overall system was inspired by the Yahoo search team article that comprehensively describes not only successes but also struggles with possible
solutions in increasing Yahoo search engine's retrieval performance. The article is well-written best paper award-winner that is a good reference to understand all aspects of an end-to-end search engine. Hence, we established an overall structure of our search system using this paper as a reference~\cite{yin2016ranking}. One can find the project module details with the referenced papers that are elaborately discussed below.
\subsection{LEarning TO Rank (LETOR)}
The first module of the overall system utilizes and improves the state-of-the-art solutions in the literature to establish a successful search platform. In IR, the performance of a search engine is evaluated by measuring the retrieval performance i.e.,
retrieving most relevant documents at higher positions in search results. It has been achieved by employing traditional machine learning approaches and recently, also deep learning-based techniques has become part of this research area that is called
LETOR.
In the scope of the LETOR module, we explore the feature space by looking for the most useful features in document retrieval task. In our feature set, while some of the features exploit the textual
content, others may convey the click information in order to represent the given dataset (training data) well enough for better learning.
In the~\emph{Elasticsearch Plug-in Based Ranking} submodule, we explored textual features whereas in the ~\emph{Click Graph Based Ranking}, we delved into click-through analysis and integrated suitable features for that analysis to our feature set. In addition to these two submodules in ~\emph{Deep Learning Based Ranking}, it seems that we will not need to fullfill any feature engineering work, rather we allow the end-to-end deep learning system to extract features and learn model from the training dataset simultaneously. More details about the submodules can be found in the corresponding sections below. Furthermore in the ~\emph{Click Graph Based Ranking} which is the last submodule of LETOR, our main aim is to learn the semantic vector representations of query-document pairs without using deep learning techniques, rather a graph approach on the training dataset.
\subsubsection{Dataset}
The training dataset is composed of query-document pairs with click information as well as the relevance grades of the documents for the
given queries. For each query-document pair, we have query, document title and click information. Moreover, there exists document url information that can be used (if needed) to fetch the whole document from the website for further analysis i.e., analysing snippet, keywords (if any) or important sections in the document.
In addition to these, to evaluate our search platform we need to have relevance grades of the documents for their query pairs in the dataset. However, relevance information (ground truth) may not be available all the time and in that case we would encounter a very common problem of supervised approaches, namely the difficulty of having labelled data. Manual labelling is a costly process, therefore we may need alternative approaches such as transforming click information into relevance labels as in~\cite{joachims2002optimizing, carterette2007evaluating}. The detailed step-by-step transformation procedure can be found in our second-term progress report.
\subsubsection{Elasticsearch Plug-in Based Ranking}
In this module, initially we obtain Elasticsearch~\cite{gormley2015elasticsearch} features about query-document pairs in the click-logs. On top of this, we compute rather more complex features that are commonly used in IR systems such as tf, matched terms count, BM25, and related features that show the similarity of queries and corresponding documents in the logs. With the help of these additional features, our aim is to
exploit query-document pairs more, in this way we can obtain more information and feed the system with this enriched input for training. Hopefully, the system will be able to learn better in comparison to only Elasticsearch features case, from the click-logs (training data) and a higher accuracy for document retrieval task will be achieved.
The procedure of this submodule is as follows, each query-document pair is initially denoted as a feature vector and model is learned from the feature file by using a ranking algorithm from RankLib~\cite{dang2013lemur}. In this part of the overall system, our contribution is to propose more complex text-similarity features in addition to the relatively simple Elasticsearch features, hoping to increase the retrieval performance of our search platform.
\subsubsection{Deep Learning-Based Ranking}
Deep learning methods have become quite popular in recent years. Researchers have obtained the best accuracies so far with deep learning models in various research areas such as computer vision and NLP. Some of these computer vision and NLP tasks, on which DL-based approaches outperform traditional methods, are object detection, word embeddings, part-of-speech-tagging, finding dependencies in a given sentence as well as relatively more complex tasks like sentiment analysis and image captioning. Owing to the big achievements of deep learning on a broad range of problems, big technology companies like Google, Facebook, Amazon and Yahoo has invested in it continuously.
Additionally, deep learning methods also have been applied to increase the retrieval performance of search engines. Yahoo~\cite{yin2016ranking} and Microsoft~\cite{shen2014latent} established more successful search platforms with deep learning models in comparison to state-of-the traditional approaches. In these published works, semantic queries are supported instead of simple keyword matching by taking into account of query context, along with user intent. You can find a list of pioneering studies that employ deep learning approaches to build a search engine with high retrieval performance in more detail below. Note that all of these works utilize supervised deep learning models and in this framework, click-logs is the training set in which queries are inputs and corresponding clicked documents are seen as outputs to train the chosen deep learning architecture.
\paragraph{Paper Review: Learning Semantic Representations Using Convolutional Neural Networks for Web Search~\cite{shen2014learning}}
\paragraph{Motivation: }
Improving the modeling contextual information in click-log queries/documents and capturing it in a fine-grained manner.
\paragraph{Method: }
The paper proposes a series of new latent semantic models based on a ~\emph{convolutional neural network} to learn semantic word embeddings for search queries and Web documents. Initially, local contextual information at the word n-gram level is modeled by applying the convolution-max pooling operation. Subsequently, in order to constitute a global feature vector, salient local features in a word sequence are combined. As a final step, the high-level semantic feature vector of the input word sequence is extracted to form a global vector representation. To train the architecture, the proposed models are trained on click-through data by maximizing the conditional likelihood of clicked documents given a query, applying stochastic gradient ascent. \\
The closest work is DSSM~\cite{huang2013learning}, which is declared to outperform significantly semantic hashing and other traditional semantic models. Compared with DSSM, C-DSSM has a convolutional layer in which each word is projected within a context window to a local contextual feature vector.
\begin{figure}[!t]
\centering
{\includegraphics[scale = 0.85]{Table1.PNG}}
\caption{Superscripts $\alpha, \beta$, and $\gamma$ indicate
statistically significant improvements ($p < 0.05$) over $BM25$, $PTM$, and $DSSM$, respectively.}
\label{fig:table1}
\end{figure}
\paragraph{Experimental Results: }
The retrieval model has been evaluated on a large-scale real world data set that contains 12,071 English queries sampled from one-year period of query log files. The evaluation metric is Normalized Discounted Cumulative Gain (NDCG) and only document titles were
used for ranking. In the experiments, the click-through data used in training include 30 million of query/clicked document title pairs. \\
The proposed C-DSSM was compared with a set of baseline models, including BM25, the unigram language model (ULM), phrase-based translation model (PTR), word-based translation model (WTM), and the closest work to the current architecture, DSSM. As shown in Figure~\ref{fig:table1}, the C-DSSM outperforms all the state-of-the-art approaches with a significant margin.
\paragraph{Paper Review: A Latent Semantic Model with Convolutional-Pooling Structure for Information Retrieval~\cite{shen2014latent}}
\paragraph{Motivation: }
In spite of the notable achievements obtained in recent studies, still all the prior latent semantic models treat a query (or a document) as a BoWs. Therefore, they are not effective in detecting contextual structures of a query (or a document).
\begin{figure}[!h]
\centering
\includegraphics[scale = 0.85]{Fig1.PNG}
\caption{Sample document titles. The text is lower-cased and punctuation removed. The same word, e.g., "office", has different meanings depending on its contexts.}
\label{fig:fig1}
\end{figure}
As shown in Figure~\ref{fig:fig1} with several examples of document titles, the contextual information is very valuable in the task of semantic search and without this information, it seems that system fails to achieve high retrieval performance.
\begin{figure}[!h]
\centering
\includegraphics[scale = 0.85]{Fig3.PNG}
\caption{The CLSM maps a variable-length word sequence to a low-dimensional vector in a latent semantic space. A word contextual window size (i.e. the receptive field) of three is used in the illustration. Convolution over word sequence via learned matrix $W_c$ performed implicitly via the earlier layer's mapping with a local receptive field. The dimensionalities of the convolutional layer and the semantic layer are set to 300 and 128 in the illustration, respectively. The max operation across the sequence is applied for each of 300 feature dimensions separately. (Only the first dimension is shown to avoid figure clutter.)}
\label{fig:fig3}
\end{figure}
\begin{figure}[!h]
\centering
\includegraphics[scale = 0.85]{Fig4.PNG}
\caption{Comparative results with the previous state of the art approaches. BLTM, WTM, PTM, DSSM, and CLSM use the same click-through data for learning. Superscripts $\alpha$, $\beta$, and $\gamma$ indicate statistically significant improvements ($p < 0.05$) over $BM25$, $PTM$, and $DSSM (J = 50)$, respectively. (Models of \#1, \#2, \#11, \#12, and \#13 have been also used in the previous C-DSSM paper for comparison.)}
\label{fig:fig4}
\end{figure}
\paragraph{Method: }
In this study, a new latent semantic model that incorporates a convolutional-pooling structure over word sequences to learn low-dimensional, semantic vector representations for search queries/documents in the click-log. In order to detect the rich contextual structures, the procedure starts with each word within a sliding window (temporal context window) in a word sequence to directly capture contextual features at the word n-gram level. \\
In order to use the CLSM for IR, given a query and corresponding Web documents to be ranked, firstly the semantic vector representations for the query and all the documents using the architecture as described above. Then, a semantic relevance score is computed by measuring the cosine similarity between the semantic vectors of the query
Q and each document D in the click-log which is used for training. In this work, the underlying assumption is that a query is relevant to the documents that are clicked on for that query, and train the CLSM on the click-through data accordingly. The high-level C-DSSM architecture is depicted in Figure~\ref{fig:fig3}.
\paragraph{Experimental Results: }
The evaluation is done on a Web document ranking task using
a large-scale, real-world data set that contains. 12,071 English queries sampled from one-year query log files. Each query-document pair has a relevance label manually annotated on a 5-level relevance scale: $bad$, $fair$, $good$, $excellent$, and $perfect$, corresponding to 0 ($bad$) to 4 ($perfect$). \\
Results demonstrate that in retrieval performance, the proposed model significantly outperforms other state-of-the-art semantic models, which were prior to this work. BM25 and ULM are used as baselines and both use term vector representation. PLSA~\cite{hofmann1999probabilistic} was trained on documents only using MAP estimation with different number of topics, T. BLTM is the best performer bilingual topic model in~\cite{gao2011clickthrough}. MRF models the term dependency proposed in~\cite{metzler2005markov}. LCE is a latent concept expansion model as described in~\cite{metzler2007latent} that leverages the term-dependent information. WTM, a word-based translation model and PTM, phrase-based translation model were implemented as described in~\cite{gao2010clickthrough}. Lastly DSSM, which is the best variant of DSSM proposed in~\cite{huang2013learning}, is used for comparison by changing the number of negative samples J. Overall comparative results can be found in Figure~\ref{fig:fig4}.
\begin{figure}[!h]
\centering
\includegraphics[scale = 0.85]{Fig5.PNG}
\caption{Two types of deep matching models: (a) Representation-focused models employ a Siamese (symmetric) architecture over the text inputs; (b) Interaction-focused models employ a hierarchical deep architecture over the local interaction matrix.}
\label{fig:fig5}
\end{figure}
\paragraph{Paper Review: A Deep Relevance Matching Model for Ad-hoc Retrieval~\cite{guo2016deep}}
\paragraph{Motivation: }
Although in recent years, deep neural networks have achieved successful results in distinct research areas such as computer vision, and natural language processing (NLP) tasks as mentioned above, few positive results have been
reported in ad-hoc retrieval tasks. One of the main reasons behind this may be stemmed from the fact that many essential characteristics of the ad-hoc retrieval task have not yet been well addressed in deep models. The ad-hoc retrieval task is commonly formalized as a matching problem between textual contents of query and document in prior works using deep learning, and viewed in a similar way to many NLP asks such as paraphrase identification, question answering and automatic conversation. \\
However, researchers defend that the ad-hoc retrieval task is essentially about relevance matching while most NLP matching tasks solve semantic matching problem, and there exist some major differences between these matching tasks. Achieving high performance in relevance matching requires
proper handling of the~\textbf{exact matching signals, query term importance and diverse matching requirements}.
\paragraph{Method: }
In this paper, a novel deep relevance matching model (DRMM) for ad-hoc retrieval has been proposed. Specifically, in the proposed model, a joint deep architecture at the query term level is employed for relevance matching. By applying matching histogram mapping, a feed forward matching network, and a term gating network, the researchers can effectively incorporate the three relevance matching factors mentioned above. \\
So far, to solve the matching problem and treat ad-hoc retrieval as an NLP task, different deep matching models have been presented. These
deep learning architectures can be mainly categorized into two types based on their model architecture: a) representation-focused models, and b) interaction-focused models as depicted in Figure~\ref{fig:fig5}. \\
The first type of models, the representation-focused model, tries to build a good representation for a single text with a deep neural network, and then carries out matching between the
compositional and abstract text representations. For instance, DSSM~\cite{huang2013learning} and C-DSSM~\cite{shen2014learning} can be categorized as the representation-focused models. The other is the interaction-focused model and in this type of model, local interactions between two pieces of text are formed, and then uses deep neural networks to detect hierarchical interaction patterns for matching. Deep Match~\cite{guo2016deep} is an example for interaction-focused models and the Deep Relevance Matching Model (DRMM) proposed in this work can also be put in this category. \\
In this study, the underlying hypothesis is that semantic matching and relevance matching are not the same thing. In fact, they are quite different problems and specifically, researchers point out three fundamental differences between these two concepts. In many NLP tasks such as paraphrase
identification, question answering and automatic conversation, the matching is mainly related to~\emph{semantic matching}, i.e. identifying the semantic meaning and inferring the semantic relations between two pieces of text. The matching in ad-hoc retrieval task, on the other hand, is essentially about ~\emph{relevance matching}, i.e. identifying whether a document is relevant to a given query. These two different matching problems emphasize three distinct elements to find out solutions for NLP (semantic matching) and ad-hoc retrieval (relevance matching) respectively. In discussing these three
factors, our aim is to show why we need to differentiate ~\emph{relevance matching} from~\emph{semantic matching}.
\begin{enumerate}
\item \textbf{Similarity matching signals vs. Exact
matching signals:} In our ad-hoc retrieval task, although more complex metrics have also been proposed, the exact matching of
query terms in documents is still the most important signal. Unlikely, NLP tasks need to detect semantically related words which can convey the same meaning even if they do not share any common word or phrases. This also clarifies why some traditional IR models, which are simply based on exact matching, e.g., BM25, can work fairly well for ad-hoc
retrieval while other traditional NLP models cannot show a similar performance for NLP-related tasks.
\item \textbf{Compositional meanings vs. Query
term importance:} In the scope of NLP, it is useful to extract grammatical structures to capture compositional meaning rather than seeing sentences as a BoWs in the given text. On the other hand, in ad-hoc retrieval, queries are composed of mainly short
and keyword based phrases without complex grammatical structures. Therefore, in our case it is crucial to take into account of term importance instead of grammatical structures.
\item \textbf{Global matching requirements vs. Diverse matching requirements:} In the literature, there are various hypotheses about document length such as Verbosity Hypothesis and Scope Hypothesis. The Verbosity Hypothesis follows an assumption that a long document covers similar content but with more words. Conversely, the Scope Hypothesis considers a long document is composed of a number of unrelated appended short documents. Hence, in terms of the Verbosity Hypothesis, relevance matching might be global for the assumption that short documents have a concentrated topic, whereas based on the Scope Hypothesis, partial relevance, the relevance of different parts of the document to a query is necessary. On the other hand, semantic matching mostly requires global matching with the aim of inferring the semantic relations from the whole text. \\
Based on these important differences between relevance matching in ad-hoc retrieval and semantic matching in many NLP tasks, it is clear that we need to establish a deep model architecture which incorporates these differences into the model properly. Herein, previously proposed architectures seem to be deficient; thus a novel deep learning architecture specifically designed for relevance matching in ad-hoc retrieval, namely deep relevance matching model (DRRM), has been suggested. Note that the introduced architecture is akin to interaction-focused rather than representation-focused models since detailed matching signals are very crucial and they are inevitably lost in the latter group of models.
\end{enumerate}
\begin{figure}[!t]
\centering
\includegraphics[scale = 0.80]{Fig6.PNG}
\caption{Architecture of DRMM}
\label{fig:fig6}
\end{figure}
\underline{The Proposed Architecture:}
The introduced model applies a joint deep architecture at the
query term level over the local interactions between query and document terms for relevance matching. In this way, query term importance can be modelled and in the following steps, the
contribution of each query term to the relevance score can be measured, hence different weights can be assigned to these terms, accordingly. Note that employing a joint deep architecture at the query term level is one important difference of the introduced model from existing interaction-focused models.
Initially, based on term embeddings, local interactions between each query-document pair of terms are established. Then, for each query term, the variable-length local interactions are transformed into a fixed-length matching histogram. As a subsequent step based on the matching histogram, a feed forward matching network is employed to learn hierarchical matching patterns and a matching score is generated for each query term.
Finally, the overall matching score is produced by aggregating the scores from each single query term with a term gating network in which the aggregation weights are computed. The proposed model architecture, DRMM is depicted in Figure~\ref{fig:fig6}.
\paragraph{Experimental Results: }
In this work, given the limited number of queries for each collection, 5-fold cross validation is conducted to avoid
over-fitting. Mean Average Precision (MAP) is used for parameter optimization. For evaluation, the top-ranked 1,000 documents are compared using the three commonly used IR metrics, namely MAP, normalized discounted cumulative gain at rank 20 (nDCG@$20$), and precision at rank 20 (P@$20$). Statistical differences between state-of-
the-art models are computed using the Fisher randomization test~\cite{smucker2007comparison}.\\
\begin{figure}[!t]
\centering
\includegraphics[scale = 0.83]{Table3.PNG}
\caption{Comparison of different retrieval models over the Robust-04 collection. Significant improvement or degradation wrt QL is indicated (+/-) ($p-value < 0.05$).}
\label{fig:fig7}
\end{figure}
In the evaluation part, two TREC collections are used for evaluation. However, we report the experimental results only for one of these datasets, namely Robust-04 collection in this report. In the scope of traditional retrieval baselines, we already mentioned BM25 and QL refers to query likelihood model based on Dirichlet smoothing~\cite{zhai2004study} which is one of the best performing language models. Results are displayed in Table~\ref{fig:fig7}. It can be concluded that all the representation-focused models perform significantly worse than the traditional retrieval models which demonstrates the unsuitability of these models for relevance matching. This is a very striking result that supports the paper's claim on the relevance/semantic matching difference. Please refer to the paper itself for comprehensive results and the related discussion.
\begin{figure}[!t]
\centering
\includegraphics[scale = 0.83]{Fig7.PNG}
\caption{Architecture of DeepRank}
\label{fig:fig8}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[scale = 0.83]{Table4.PNG}
\caption{Performance comparison of different models on MQ2007. Significant improvement or degradation wrt DeepRank-CNN is denoted as (-) ($p-value < 0.05$).}
\label{fig:fig9}
\end{figure}
\paragraph{Paper Review: DeepRank: A New Deep Architecture for Relevance Ranking in Information Retrieval~\cite{pang2017deeprank}}
\paragraph{Motivation: }
Existing deep IR models such as DSSM~\cite{huang2013learning} and C-DSSM~\cite{shen2014learning} generate ranking scores
by directly applying neural networks, without a proper understanding of the relevance (i.e. differentiating semantic matching and relevance matching. Although the previous architecture, DRMM semantic matching has capability of distinguishing these two different matching problems, it does not explicitly model the relevance generation process and fails to capture important IR features such as passage retrieval intrinsic and proximity
heuristics. \\
\paragraph{Method: }
The proposed model aims to mimic relevance label generation steps applied by the
human judgement process. According to this process, the relevance label generation comprises three steps: i) relevant locations are determined, ii) local relevances are determined, iii) local relevances are aggregated to output the relevance label. Initially, to extract the relevant contexts, a detection procedure is devised. Then, to detect the local relevances by using a convolutional neural network (CNN) or two-dimensional gated recurrent units (2D-GRU) as a measure network. Finally, an aggregation network with sequential
integration and term gating mechanism is applied to produce a global relevance score. \\
Note that DeepRank as illustrated in Figure~\ref{fig:fig8}, was proposed by the same lab to alleviate the weaknesses of their previous model, DRMM. It seems that DeepRank well captures significant IR (relevance matching) characteristics that distinguish relevance matching from semantic matching, including exact/semantic matching signals, proximity heuristics, query term importance, and diverse relevance requirement.
\paragraph{Experimental Results: }
Extensive experiments are conducted to evaluate DeepRank against state-of-the-art models such as learning to rank methods, and existing deep learning models.
For evaluation NDCG, Precision, and MAP metrics are used. \\
Experiments show that LETOR4.0 (MQ2007, MQ2008) benchmark and a large scale click-through data show that DeepRank can significantly outperform all the baseline methods. More specifically, in making comparison to learning to
rank methods, DeepRank performs even better than these models, whereas other existing deep learning methods show much worse performance. Comparison results on one of the LETOR4.0 benchmark dataset, MQ2007 are displayed in Table~\ref{fig:fig9}. For comprehensive results, please refer to the evaluation part of the paper.
\subsubsection{Click Graph Based Ranking}
For LTR, in our last submodule we utilize click graph idea to obtain more and different type of information (if feasible) from the click-through logs. Note that in this submodule, we only refer to one main paper since it is the most recent and successful work in this area.
\paragraph{Paper Review: Learning Query and Document Relevance from a Web-scale Click Graph~\cite{jiang2016learning}}
\paragraph{Motivation: }
Click-through logs contain rich and valuable information. However, the click information is sometimes noisy and its coverage is limited since there is a huge number of all possible relevant query-document pairs which leads to sparsity for the click-based features. The sparsity and noise problems affect the overall click-based feature quality negatively, especially for less popular (e.g. tail queries) queries. To overcome these problems, an effective way is to use click
and content information simultaneously. For this reason, learning a vector representation for both queries and documents in the same semantic space is needed. \\
Previous state-of-the-art approaches represent queries/documents in the same space such as traditional methods, which learn low-rank vectors, or direct text matching methods like BM25 and the language models. However, prior methods have its own advantages, it has also some weaknesses. For instance, low rank embedding hurts interpretability and debuggability of the ranking function because individual dimension in the latent space is hard to interpret and direct text matching methods suffer from the lexical gap between queries and documents. Moreover, we need an approach for click-absent queries, i.e. queries which have never been observed in the search logs.
\begin{figure}[!t]
\centering
\includegraphics[scale = 0.83]{Fig8.PNG}
\caption{An example of click-through bipartite graph}
\label{fig:fig10}
\end{figure}
\paragraph{Method: }
To overcome all these challenges, the
paper proposes a propagation approach to learn
vector representation by using both content and
click information. These vector representations
can directly improve the retrieval performance for
queries and documents that exist in the click logs,
i.e. click-existing queries. A sample click graph is
shown in Figure~\ref{fig:fig10}. \\
Furthermore, for click-absent queries and documents, a two-step vector estimation algorithm is proposed which utilizes partial information of the vectors in the bipartite graph that is already created by propagation. In this way, researchers
aim to significantly improve the coverage of the vectors, which is specifically critical for long-tail queries in web-search, i.e. the queries containing
keywords that are more specific and less common than other keywords.
\begin{figure}[!t]
\centering
\includegraphics[scale = 0.83]{Fig9.PNG}
\caption{An example of unit vector generation. (The black thin lines are the edge of the click
graph, while the edges represented by the gray thick lines indicate the pseudo clicks between units
and documents.)}
\label{fig:fig11}
\end{figure}
\paragraph{Vector Propagation Algorithm: }
The goal of this propagation algorithm is to learn the vector representation of queries and documents in the same semantic space (either the query or the document space). The algorithm starts from one side (query or document) and the vectors (query or document) are initialized with the content information and the vectors are propagated to their corresponding connected nodes on the side of the click graph. \\
Using bag-of-words model to generate the vector representations using query words to represent queries, and document titles to represent documents is a more intuitive way for vector generation. However, this procedure leads to the lexical gap between queries and documents. Thus, in the paper, researchers prefer to represent queries in query semantic space and documents in document semantic space. Note that co-clicks between queries and documents show the importance level
of the propagating terms by weighting the vector representations. In this way, significant terms become more prominent while less informative terms are eventually filtered out. \\
Apart from these, in order to incorporate click-absent queries to the already established graph, the paper uses a unit vector generation algorithm that is shown in Figure~\ref{fig:fig11}. In this procedure, firstly queries and documents titles are broken down into different units (e.g., ngrams) and vector representations for each unit are learned by using the vectors that have been already learned from the click-existing queries/documents in the graph. Then, a weight score is learned
for each unit by a regression model, and finally the vectors are estimated for the click-absent queries/documents by a linear combination of the unit vectors. Owing to this procedure, the click-absent queries/documents will be connected to the click graph through the unit vectors by utilizing the partial information in the already established click graph, even if the complete version of queries do not exist in the click-logs. In this way, high-quality representations are generated also for click-absent query/documents which substantially affects the retrieval performance of search engines in practice.
\begin{figure}[!t]
\centering
\includegraphics{Table5.PNG}
\caption{As an individual ranking model (Two-tailed t-test is done for paired data where each pair is VPCG \& VG QUERY and any of the other methods, and * indicates $p-value < 0.01$ for all tests.))}
\label{fig:fig12}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics{Table6.PNG}
\caption{As a feature (Two-tailed t-test is done for paired data where each pair is "Base + VPCG \& VG" and any of the other methods, and
* indicates $p-value < 0.01$ for all tests.)))}
\label{fig:fig13}
\end{figure}
\paragraph{Experimental Results: }
Researchers establish the click-through from a major commercial search engine's search log. There are approximately 25 billion co-clicked query-document pairs,
containing about 8 billion unique queries and 3 billion unique documents. This dataset was used as training set to build the graph. Then, for evaluation (i.e. for investigating if the relevance score learned by the proposed algorithm can help to improve ranking in a learning-to-rank framework), another dataset was used which is composed of 63k queries and 775k query-document pairs as training instances and 16k queries with 243k query-document pairs as test set. \\
The relevance score of each pair (``perfect", ``excellent", ``good", ``fair", ``bad") is annotated by human annotators. The evaluation is done in two distinct ways: i) the learned relevance score can be either used directly to rank documents, or ii) added to the feature vector in a learning-to-rank framework. You can find the results in Table~\ref{fig:fig12} and~\ref{fig:fig13} below. Results show that the proposed method helps to improve ranking results in both cases.
\begin{figure}[!t]
\centering
\includegraphics{Fig10.PNG}
\caption{Framework of the search over encrypted cloud data.}
\label{fig:fig14}
\end{figure}
\subsection{Cloud Search}
Our second module is Cloud Search and for this module, we also chose a comprehensive article for reference. However, this part of our project is the module which requires the least research effort in comparison to other project modules. Therefore, we have one referenced paper for the cloud module and only give an overview of the paper without mentioning any details of the approach.
\paragraph{Paper Review: Achieving Efficient Cloud Search Services: Multi-Keyword Ranked Search over Encrypted Cloud Data Supporting Parallel Computing~\cite{fu2015achieving}}
\paragraph{Overview: }
Nowadays, cloud computing has become quite popular. A large amount of data outsourced to the cloud by data owners for the purpose of accessing the large-scale computing resources and economic savings. \\
For data protection, the sensitive data should be encrypted by the data owner before outsourcing and this leads to the fact that traditional and efficient plain-text keyword search technique be-
comes useless. Therefore, researchers investigate how to design an efficient and effective searchable encryption scheme on cloud. You can see the overall framework in Figure~\ref{fig:fig14}.
\subsection{Query Expansion \& Suggestion}
~\emph{Lexical chasm} between queries and document titles is the main obstacle for improving base relevance of a search engine. This is substantially rooted from two things: i) authors of queries and documents are different (e.g. diverse vocabulary usage), ii) insufficient knowledge of technical terms in the corresponding domain. On the light of these, we consider to alleviate the~\emph{semantic gap} problem in the~\emph{Query Expansion} sub-module to improve the retrieval performance even more, in addition to the core methods utilized in our first module, LTR.
\subsubsection{Query Expansion}
In this sub-module, we aim to expand a given query with a selected set of terms to increase the coverage of the query. In selecting the terms, we can use three proposed methods which utilize three different word embeddings, word2vec~\cite{mikolov2013efficient}, FastText~\cite{joulin2016bag}, and GloVe~\cite{pennington2014glove}. We can combine the outputs of these three methods by ensemble learning (e.g. majority voting), for instance.
\paragraph{Paper Review: Query Expansion Using Word Embeddings~\cite{kuzi2016query}}
\paragraph{Motivation: }
Query expansion may help on improving retrieval performance of a search engine. For this, suitable terms should be selected to employ query expansion in a proper way, otherwise, it may deteriorate the retrieval performance by including irrelevant documents in returning document set.
\paragraph{Method: }
In this paper, researchers propose two term scoring methods for term selection. The main idea is to choose terms that are semantically related to the query by using word2vec's cbow em-
bedding approach applied over the entire search corpus. After computing scores and candidate terms are determined, the v terms assigned the highest score by a method M are used for query
expansion. Brief description of these M scoring methods are as follows.
\begin{itemize}
\item \underline{The centroid method:}
In this approach, researchers leverage the observation of adding
word2vec vectors representing terms that constitute an expression often yields a vector that semantically corresponds to the expression. Therefore, selecting procedure is employed by comparing cosine-similarity of a specific term's word2vec score in the collection and the corresponding query as a whole.
\item \underline{Fusion-based methods:}
Differently from the Cent method, in this approach, for each query term, $q_i$, a list $L{q_i}$ of its n most similar terms
t in the corpus according to cos($q_i$; t). Then,
cosine-similarity scores are used to compute softmax-normalized probabilities. After that, the resulting term lists are fused using Comb-SUM, CombMNZ and CombMAX~\cite{fox1994combination}.
\end{itemize}
\paragraph{Experimental Results: }
In evaluation, several TREC datasets are used as benchmark datasets. Results show statistically significant results, thus
the proposed idea can be used to improve the retrieval performance of a search engine by integrating also with other word embedding methods. \\
\paragraph{Paper Review: Using Word Embeddings
for Automatic Query Expansion~\cite{roy2016using}}
\paragraph{Motivation: }
The main goal of this work is again to find suitable terms to expand a query, i.e., increasing the coverage of the query without
including irrelevant terms.
\paragraph{Method: }
For selecting terms, researchers devise a query expansion technique, where related terms to a query are picked using K-nearest neighbour approach. More specifically, in the paper,
three kNN-based term selection methods are proposed as pre-retrieval kNN, post-retrieval kNN, pre-retrieval incremental kNN approaches. For an elaborate explanation of these methods, please refer to the paper.
\paragraph{Experimental Results: }
In the experimental results, the proposed method is evaluated on the standard ad-hoc task using both TREC collection and TREC web collection. The results indicate that, as an expansion method, the incremental method is generally safe; it produces performance
improvements for most of the queries and for only a few queries, it affects the performance badly. Therefore, this method can be integrated to our query expansion scheme, as well.\\
\paragraph{Paper Review: Query Expansion with
Locally-Trained Word Embeddings~\cite{diaz2016query}}
\paragraph{Motivation: }
Commonly used word embeddings, word2vec and GloVe, when trained globally underperform corpus and query-specific embeddings in the context of query expansion and retrieval tasks. Hence, in this paper researchers investigate the effect of local embeddings on query expansion.
\paragraph{Method: }
In order to generate local word embeddings, learning embeddings on topically-constrained corpora, instead of large topically-unconstrained corpora to utilize domain-specific
information. For this purpose, in this work, a language modelling approach is adopted to produce a query-specific set of topical documents.
\paragraph{Experimental Results: }
In the evaluation part, TREC datasets and ClueWeb 2009 corpora
are used and the evaluatin metric is chosen as NDCG@$10$. The comparison is done with the baseline method of query-likelihood and results show that the proposed query expansion approach
by using local word embeddings outperform the baseline on benchmark datasets. Based on this, in our query expansion scheme, we may give a try to local word embeddings, if feasible.\\
\subsubsection{Query Suggestion}
The main goal of this submodule is to improve user experience, i.e., enabling users to reach their searching information in a more accurate and quicker way. In the context of query suggestion,
there are two papers that can be utilized in our searching platform. These two articles~\cite{liu2017query, mei2008query} differ only in small details, therefore we will only mention about the first paper that contains the core idea.
\paragraph{Query Suggestion Using Hitting Time~\cite{mei2008query}}
\paragraph{Motivation: }
This paper aims to generate query suggestions while ensuring semantic consistency with the original query not to lose context
information.
\paragraph{Method: }
In this work, finding query suggestion procedure is mainly as follows. Initially, a bipartite graph is established on click-logs and it is used for query expansion. Then on the graph, hitting time from a given query to other queries are modelled by random walking for finding candidate query suggestions.
\paragraph{Experimental Results: }
Results show that this technique is beneficial for query suggestion and can be incorporated to our platform to improve user search experience. We selected this work for query suggestion, because of two reasons: i) it is one of the most comprehensive and successful method in this area, and ii) for efficiency purposes, since we already construct a bipartite-graph on click-through logs and we can use the same graph also for query suggestion.\\
\section{Patents}
\subsection{US20160004776 A1, 2016~\cite{cloud2016}}
In this patent, cascading searching is used to locate a desired person in a social media ecosystem. The social media search system is provided on cloud and we utilize this patent to scale our
search service on a cloud platform.
\subsection{US7689520B2, 2010~\cite{burges2010machine}}
This patent proposes a machine learning system to rank a set of documents by using differentiable parameters. The ranking will be optimized according to a cost module that computes cost scores on a pair of examples. This patent helps on generating letor-based ranking models to improve our system.
|
2024-02-18T23:40:24.430Z
|
2021-12-30T02:23:06.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14444","language":"en","timestamp":1640830986000,"url":"https:\/\/arxiv.org\/abs\/2112.14444","yymm":"2112"}
|
proofpile-arXiv_000-10021
|
{"provenance":"002.jsonl.gz:10022"}
| null | null |
\section{Introduction}
\label{secIntro}
Double-$\beta$ ($\beta\beta$) decay is a rare transition process between nuclei $(A,Z)$ and $(A,Z+2)$.
There may exist two modes of this decay. One is the two-neutrino $\beta\beta$ ($2\nu\beta\beta$) decay
with the emission of two antineutrinos.
As a second-order weak process, it was first suggested by Mayer \cite{Mayer1935}
and has been first observed for $^{82}$Se \cite{Elliott1987} in 1980's and for other 10 different nuclei later \cite{Barabash2015}.
The other mode is the so-called neutrinoless $\beta\beta$ ($0\nu\beta\beta$) decay,
which is a lepton-number-violating nuclear process
and occurs only if neutrinos are their own antiparticles, i.e., the Majorana particles \cite{Haxton1984,Avignone2008}.
$0\nu\beta\beta$ decay is the only practical way to determine whether neutrinos are Majorana particles
and is helpful to address the questions of the absolute neutrino mass scale and the neutrino mass hierarchy,
provided that the corresponding nuclear matrix elements (NME) $M^{0\nu}$ are calculated exactly
\cite{Faessler1998,Suhonen1998,Engel2017,Ejiri2019,Yao2021}.
However, the deviations of $M^{0\nu}$ from different nuclear models
vary up to a factor around three times (c.f. Figure 26 in Ref. \cite{Yao2021}).
The main reason of such a divergence
stems from the difference of nuclear many-body wave functions obtained from different nuclear models.
While $0\nu\beta\beta$ decay is related to the underlying new physics, $2\nu\beta\beta$ decay is free from the unknown neutrino properties.
Also the corresponding lepton phase-space factor has been calculated with high precision \cite{Kotila2012,Stoica2013}.
Therefore, the NME of $2\nu\beta\beta$ decay, $M^{2\nu}$, provides a test ground for nuclear structure calculations.
As a second-order weak process, the $M^{2\nu}$ of $2\nu\beta\beta$ decay
involves a summation over all the virtual intermediate states of the nucleus $(A,Z+1)$,
which connects the initial nucleus $(A,Z)$ and final nucleus $(A,Z+2)$ by the $\beta$-decay like transitions.
Because of isospin symmetry, the Fermi transition is highly suppressed so that
only the NME of Gamow-Teller (GT) transition
$M^{2\nu}_{\rm GT}$ is considered in the calculations.
In 1984, Abad proposed the single-state dominance (SSD) hypothesis, which states that the $M^{2\nu}_{\rm GT}$ is dominated
by the virtual transitions through the first $1^+$ state of the intermediate nucleus $(A,Z+1)$ \cite{Abad1987}.
It can be extended to the more relaxed low-lying-states dominance (LLD) hypothesis \cite{Moreno2008}.
By analyzing the energy distribution of the emitted electrons,
the evidence of SSD for $^{82}$Se and $^{100}$Mo has been found
in CUPID \cite{Azzolini2019,Armengaud2020} and NEMO-3 \cite{Arnold2019}.
The mechanism of SSD (LLD) could be either
no contributions from highly lying $1^+$ states or the cancellations among them
\cite{Civitarese1998,Civitarese1999,Simkovic2001,Domin2005,Sarriguren2016,Simkovic2018}.
The running sum, namely, the cumulative contribution from the intermediate states,
of $M^{2\nu}_{\rm GT}$ is a useful tool in the study of SSD and LLD hypotheses \cite{Fang2010}.
It has been found that in the running sums, calculated by quasiparticle random phase approximation (QRPA) \cite{Simkovic2018}
or nuclear shell model with a spin-orbit complete model space \cite{Horoi2007},
the contributions are positive at first
and then become negative from states lying around 6-10 MeV.
Thus, the appearance of the negative contributions seems to be a universal phenomenon \cite{Simkovic2018}.
And such negative contributions could lead to the cancellations among the higher-lying $1^{+}$ states
and consequently the fulfillment of SSD or LLD hypothesis. However, the underlying reason for the emergence of such negative contributions still needs investigations.
In QRPA model, the suppression of $M^{2\nu}_{\rm GT}$ is closely related to the isoscalar (IS) pairing,
as first identified by Vogel and Zirnbauer in 1986 \cite{Vogel1986}.
By considering the particle-particle interaction,
mostly the IS pairing \cite{Rodin2011},
in the QRPA treatment of $2\nu\beta\beta$ decay,
the highly suppressed $M^{2\nu}_{\rm GT}$ of $^{130}$Te was successfully reproduced \cite{Vogel1986}.
They also found that as a function of the strength of IS pairing,
the $M^{2\nu}_{\rm GT}$ of $^{130}$Te monotonously decreases and passes through zero,
which has been found later as a common feature in many nuclei in the following studies
\cite{Faessler1998,Muto1989,AlvarezRodriguez2004,Civitarese2000,Suhonen2005,Rodin2011,Simkovic2013}.
Such a suppression of $M^{2\nu}_{\rm GT}$ when increasing IS pairing strength
is found to be related with the restoration of the Wigner's spin-isospin SU(4) symmetry
\cite{Vogel1986,Rumyantsev1998,Civitarese2000,Rodin2005,Rodin2011}.
On the other hand, the ground-state correlations introduced by the backward amplitude
in QRPA calculations compared to that of quasiparticle-Tamm-Dancoff approximation (QTDA) model
was also shown to have the suppression effects on the NME \cite{Muto1989}.
The aim of this paper is to understand the mechanism of cancellation among higher-lying $1^{+}$ states that makes SSD and LLD hypotheses valid.
Inspired by the previous works on the suppression of NME,
particular attentions will be paid on the roles of IS pairing as well as ground-state correlations on the negative contributions
in the running sums of $M^{2\nu}_{\rm GT}$.
During the past few decades, different nuclear models have been used in the study of $M^{2\nu}_{\rm GT}$
(see Refs. \cite{Faessler1998,Suhonen1998,Engel2017,Ejiri2019,Yao2021} for a review),
such as nuclear shell model \cite{Horoi2007,Caurier2012,Senkov2014,Brown2015,Li2017,Coraggio2019},
QRPA model
\cite{Vogel1986,Muto1989,AlvarezRodriguez2004,Civitarese2000,Suhonen2005,
Fang2010,Rodin2011,Suhonen2012,Sarriguren2012,Simkovic2013,Nicolas2015,Brown2015,Simkovic2018,Simkovic2018b,Terasaki2019,
Suhonen2014,Delion2015},
projected Hartree-Fock-Bogoliubov model \cite{Dixit2002,Chandra2005,Singh2007,Rath2010,Rath2019},
and interacting boson model \cite{Yoshida2013,Barea2015}.
One of the models which can avoid the closure approximation is the QRPA.
The closure approximation is not appropriate for $2\nu\beta\beta$ decay \cite{Suhonen1998}.
To study the running sum of NME, one must go beyond this approximation.
In this work, we adopt the spherical QRPA approach based on Skyrme Hartree-Fock-Bogoliubov model (Skyrme HFB + QRPA)
to systematically study the $M^{2\nu}_{\rm GT}$ of 11 nuclei with experimental NMEs.
We focus on the cancellation among higher-lying $1^{+}$ states in the running sum of $M^{2\nu}_{\rm GT}$ that leads to the fulfillment of SSD or LLD hypothesis.
Compared to QTDA model, the QRPA model introduces more ground-state correlations through the backward amplitude $Y_{\pi\nu}$.
We will study the importance of these ground-state correlations.
Their effects, entangled with the IS pairing, on the negative contributions
of the running sums for $M^{2\nu}_{\rm GT}$ are analyzed in detail. The QRPA model as well as the formalism for calculating the $M^{2\nu}_{\rm GT}$ are presented in Sec.~\ref{secTheo}.
Numerical details of our Skyrme HFB+QPRA calculations are presented in Sec.~\ref{secNume}.
Sec.~\ref{secResu} contains the results and discussions.
The summary is given in Sec.~\ref{secConclu}.
\section{Formalism}
\label{secTheo}
\subsection{Quasiparticle random phase approximation}
\label{subsecQRPA}
We first perform the Skyrme HFB calculation, the formalism of which can be found in
Refs.~\cite{Dobaczewski1984,Dobaczewski1996,Bennaceur2005}.
Then with the obtained canonical single-particle wave functions and occupation amplitudes,
the QRPA equations for the state $|nJ^{P}\rangle$ with angular momentum $J$ and parity $P$
can be constructed in the angular momentum coupled form:
\begin{equation}
\begin{aligned}
\left(
\begin{array}{cc}
{A} & {B} \\
-{B} & -{A} \\
\end{array}
\right)
\left(
\begin{array}{c}
{X}^{nJ^P} \\
{Y}^{nJ^P} \\
\end{array}
\right)
=&
\Omega^{nJ^P}
\left(
\begin{array}{c}
{X}^{nJ^P} \\
{Y}^{nJ^P} \\
\end{array}
\right),
\end{aligned}
\label{Eq_QRPA}
\end{equation}
where $X^{nJ^P}$ and $Y^{nJ^P}$ are respectively the forward and backward amplitudes, and $\Omega^{nJ^P}$ are the energy eigenvalues.
In the charge-exchange case, using the canonical basis the matrices $A$ and $B$ can be expressed as
\begin{subequations}
\begin{equation}
\begin{aligned}
{A}_{\pi\nu,\pi'\nu'}
=&
\Big(
u_{\pi} u_{\pi'} h_{\pi\pi'}
- v_{\pi} v_{\pi'} h^{T}_{\bar{\pi} \bar{\pi}'} \\ &
- u_{\pi} v_{\pi'} \Delta_{\pi \bar{\pi}'}
+ v_{\pi} u_{\pi'} \Delta^{\ast}_{\bar{\pi} \pi'}\Big) \delta_{\nu\nu'} \\
&+ \Big(
u_{\nu} u_{\nu'} h_{\nu\nu'}
- v_{\nu} v_{\nu'} h^{T}_{\bar{\nu} \bar{\nu}'} \\ &
- u_{\nu} v_{\nu'} \Delta_{\nu \bar{\nu}'}
+ v_{\nu} u_{\nu'} \Delta^{\ast}_{\bar{\nu} \nu'} \Big) \delta_{\pi\pi'}\\
&+ (u_{\pi} v_{\nu} u_{\pi'} v_{\nu'} + v_{\pi} u_{\nu} v_{\pi'} u_{\nu'})
\langle \pi \nu'| V |\nu \pi'\rangle^{ph}_J \\
&+ (u_{\pi} u_{\nu} u_{\pi'} u_{\nu'} + v_{\pi} v_{\nu} v_{\pi'} v_{\nu'} )
\langle \pi \nu | V |\pi' \nu'\rangle^{pp}_J ,
\end{aligned}
\label{Eq_QRPA_A}
\end{equation}
\begin{equation}
\begin{aligned}
{B}_{\pi\nu,\pi'\nu'}
= & (v_{\pi} u_{\nu} u_{\pi'} v_{\nu'} + u_{\pi} v_{\nu} v_{\pi'} u_{\nu'})
\langle \pi \nu'| V | \nu \pi'\rangle^{ph}_J \\
&-(v_{\pi} v_{\nu} u_{\pi'} u_{\nu'} + u_{\pi} u_{\nu} v_{\pi'} v_{\nu'})
\langle \pi \nu | V | \pi'\nu'\rangle^{pp}_J ,
\end{aligned}
\label{Eq_QRPA_B}
\end{equation}
\end{subequations}
where $u$ and $v$ are the occupation amplitudes.
$h$ and $\Delta$ are respectively the single-particle Hamiltonian and pairing field.
$|\pi\rangle$ and $|\nu\rangle$ represent respectively the proton and neutron canonical states.
Their time reversal states are denoted as $|\bar{\pi}\rangle$ and $|\bar{\nu}\rangle$.
The residual interactions $V$
are divided into the particle-hole ($ph$) channel and the particle-particle ($pp$) channel respectively.
For the $ph$ channel, the same Skyrme interaction used in the HFB calculation is adopted.
For the $pp$ channel, namely the proton-neutron paring,
the density-dependent $\delta$ force is used,
which can be further divided into the isovector (IV) part
\begin{equation}
V_{T=1}(\bm{r}_1, \bm{r}_2)
= f_{\rm IV}
\left( t'_{0 \pi\nu}
+ \frac{t'_{3 \pi\nu}}{6}\rho(\bm{r})\right)
\delta(\bm{r}_1 - \bm{r}_2)\frac{1-\hat{P}_{\sigma}}{2} ,
\label{Eq_IVpairing}
\end{equation}
and the isoscalar (IS) part
\begin{equation}
V_{T=0}(\bm{r}_1, \bm{r}_2)
= f_{\rm IS} \left( t'_{0 \pi\nu}
+ \frac{t'_{3 \pi\nu}}{6}\rho(\bm{r})\right)
\delta(\bm{r}_1 - \bm{r}_2) \frac{1+\hat{P}_{\sigma}}{2} ,
\label{Eq_ISpairing}
\end{equation}
where $\bm{r} = (\bm{r}_1+\bm{r}_2)/2$, and $\hat{P}_{\sigma}$ is the spin-exchange operator.
Parameters $t'_{0 \pi\nu}$ and $t'_{3 \pi\nu}$
are set as the average of the proton-proton and neutron-neutron pairing strengths, which are
determined from the Skyrme HFB calculations to reproduce the proton and neutron pairing gaps
from the five-point formula of experimental binding energies respectively.
Then in QRPA calculations, $f_{\rm IV}$ is fixed to make the Fermi $2\nu\beta\beta$ matrix element vanish
as it should, due to the different isospin quantum numbers of the mother nucleus ($T$) and daughter nucleus ($T-2$).
The strength of isoscalar proton-neutron paring $f_{\rm IS}$ is not well constrained \cite{Sagawa2016},
it is usually fixed through fitting the experimental NME.
We note that the isovector proton-neutron pairing $V_{T=1}$ does not affect
the Gamow-Teller $2\nu\beta\beta$ matrix element $M^{2\nu}_{\rm GT}$ \cite{Simkovic2013}.
\subsection{$2\nu\beta\beta$-decay nuclear matrix element}
\label{subsecNME}
For the $2\nu\beta\beta$ decay, due to isospin symmetry, the NME is dominated by GT transitions, $M^{2\nu}_{\rm GT}$.
For the ground-state-to-ground-state $2\nu\beta\beta$ decay, $M^{2\nu}_{\rm GT}$ can be expressed as
\begin{equation}
\begin{aligned}
M_{\rm GT}^{2\nu}
=& \sum_{n_i n_f}
\frac{ \langle 0^{+(f)}_{\rm g.s.} || \hat{O}^{-}_{\rm GT} || 1^{+}_{n_f} \rangle
\langle 1^{+}_{n_f} | 1^{+}_{n_i} \rangle
\langle 1^{+}_{n_i} || \hat{O}^{-}_{\rm GT} || 0^{+(i)}_{\rm g.s.} \rangle }
{E_{\rm int.}(n_i, n_f) + M_{\rm int.} - (M_{f}+M_{i})/2} ,
\end{aligned}
\label{Eq_NME}
\end{equation}
where $\langle 1^{+}_{n_i} || \hat{O}^{-}_{\rm GT} || 0^{+(i)}_{\rm g.s.} \rangle$
is the GT$^{-}$ transition amplitude of the initial nucleus to the intermediate nucleus. It can be obtained from QRPA model as
\begin{equation}
\begin{aligned}
\langle 1^{+}_{n_i} || \hat{O}^{-}_{\rm GT} || 0^{+(i)}_{\rm g.s.} \rangle
=& \sum_{\pi_i \nu_i} - \langle j_{\pi_i} ||\hat{O}^{-}_{\rm GT}|| j_{\nu_i} \rangle \\ & \times
\left( X^{n_i}_{\pi_i \nu_i} u_{\pi_i} v_{\nu_i} + Y^{n_i}_{\pi_i \nu_i} v_{\pi_i} u_{\nu_i} \right).
\label{Eq_GT-_i}
\end{aligned}
\end{equation}
And the transtion from the intermediate states to the final nucleus $\langle 0^{+(f)}_{\rm g.s.} || \hat{O}^{-}_{\rm GT} || 1^{+}_{n_f} \rangle$ can be expressed as
\begin{equation}
\begin{aligned}
\langle 0^{+(f)}_{\rm g.s.} || \hat{O}^{-}_{\rm GT} || 1^{+}_{n_f} \rangle
=& -\langle 1^{+}_{n_f} || \hat{O}^{+}_{\rm GT} || 0^{+(f)}_{\rm g.s.} \rangle \\
=& \sum_{\pi_f \nu_f}-\langle j_{\pi_f} ||\hat{O}^{+}_{\rm GT}|| j_{\nu_f} \rangle \\ & \times
\left( X^{n_f}_{\pi_f \nu_f} v_{\pi_f} u_{\nu_f} + Y^{n_f}_{\pi_f \nu_f} u_{\pi_f} v_{\nu_f} \right).
\end{aligned}
\label{Eq_GT+_f}
\end{equation}
The overlap factor between the intermediate states constructed from the initial and final nuclei $\langle 1^{+}_{n_f} | 1^{+}_{n_i} \rangle$ is \cite{Simkovic2004},
\begin{equation}
\begin{aligned}
\langle 1^{+}_{n_f}|1^{+}_{n_i} \rangle
=& \sum_{\pi_i \nu_i} \sum_{\pi_f \nu_f} C_{\pi_i \pi_f} C_{\nu_i \nu_f}
(X^{n_i}_{\pi_i \nu_i} X_{\pi_f \nu_f}^{n_f} - Y^{n_i}_{\pi_i \nu_i} Y_{\pi_f \nu_f}^{n_f}) \\
&~~~~~~~~ \times
(u_{\pi_i} u_{\pi_f} + v_{\pi_i} v_{\pi_f})
(u_{\nu_i} u_{\nu_f} + v_{\nu_i} v_{\nu_f}) \\
&~~~~~~~~ \times
\langle {\rm{HFB}}^{(f)}|{\rm{HFB}}^{(i)} \rangle ,
\end{aligned}
\end{equation}
where $C_{\pi_i \pi_f} = \langle \pi_i | \pi_f \rangle $
and $C_{\nu_i \nu_f} = \langle \nu_i | \nu_f \rangle $
are the overlaps of the canonical single-particle wave functions.
In the canonical basis, the overlap of the HFB ground states of initial and final nuclei reads
\begin{equation}
\begin{aligned}
\langle {\rm{HFB}}^{(f)}|{\rm{HFB}}^{(i)} \rangle
= \prod_{ \pi_i \pi_f \nu_i \nu_f}
& (u_{\pi_i} u_{\pi_f} + v_{\pi_i} v_{\pi_f}) \\ \times
& (u_{\nu_i} u_{\nu_f} + v_{\nu_i} v_{\nu_f}).
\end{aligned}
\end{equation}
The excitation energy of intermediate nucleus in the denominator is
\begin{equation}
\label{intE}
\begin{aligned}
E_{\rm int.}(n_i, n_f)
=& \frac{1}{2} \left(E_{\rm int.}^{n_i} + E_{\rm int.}^{n_f} \right) \\
=& \frac{1}{2}\Big[ (\Omega^{n_i} + \lambda^{(i)}_{\pi} - \lambda^{(i)}_{\nu} )
- (M_{\rm int.} - M_i) - \Delta m_{\nu\pi} \\
& +(\Omega^{n_f} - \lambda^{(f)}_{\pi} + \lambda^{(f)}_{\nu} )
- (M_{\rm int.} - M_f) + \Delta m_{\nu\pi} \Big] \\
=& \frac{1}{2}\Big[ (\Omega^{n_i} + \lambda^{(i)}_{\pi} - \lambda^{(i)}_{\nu} )
+(\Omega^{n_f} - \lambda^{(f)}_{\pi} + \lambda^{(f)}_{\nu} ) \\
& - (2M_{\rm int.} - M_i - M_f) \Big] ,
\end{aligned}
\end{equation}
where $\lambda$ denotes the Fermi surface and $\Delta m_{\nu\pi}$
is the mass difference between the neutron and the proton, i.e., $m_{\nu} - m_{\pi}$.
$\Omega^{n_i}$ and $\Omega^{n_f}$ are the eigenvalues of QRPA equations for mother and daughter nuclei, respectively.
With the use of Eq. \eqref{intE}, nuclear masses are eventually not needed in the calculation of NME in Eq. \eqref{Eq_NME},
only the energy with respect to the mother nucleus $(\Omega^{n_i} + \lambda^{(i)}_{\pi} - \lambda^{(i)}_{\nu} )$
and the energy with respect to the daughter nucleus $(\Omega^{n_f} - \lambda^{(f)}_{\pi} + \lambda^{(f)}_{\nu} )$ are needed,
which are obtained from QRPA calculations.
However, to calculate the energy of intermediate nucleus $E_{\rm int.}(n_i, n_f)$
the experimental mass values from AME2020 \cite{Huang2021} are used
for the masses of the initial nucleus $M_i$, the intermediate nucleus $M_{\rm int.}$, and the final nucleus $M_f$.
\section{Numerical details}
\label{secNume}
For the Skyrme HFB calculations,
the SkO$'$ interaction \cite{Reinhard1999} is used for the mean-field calculation.
For the pairing interaction, we adopt the surface density-dependent $\delta$ force, whose strength is fixed to reproduce
the experimental pairing gap obtained from the five-point formula of binding energies as stated above.
We use different cut-offs to reduce the computation burden, and obtain required accuracy.
For the mean field, we use a smooth cut-off for the pairing window with a diffuseness parameter $\mu$ being 0.1 MeV,
and the cut-off on equivalent Hartree-Fock energy $\varepsilon^{\rm{HF}}$ is set to be 80.0 MeV.
For the QRPA calculation, we use the canonical basis ($|\pi\rangle$ and $|\nu\rangle$)
with occupation amplitudes ($u$ and $v$), which are obtained by performing canonical transformation
on the quasiparticle states in HFB calculations.
The $\pi$-$\nu$ configurations are selected under the criteria $|u_{\pi} v_{\nu}|>10^{-4}$
or $|u_{\nu} v_{\pi}|>10^{-4}$, with the single-particle energies in canonical basis $\varepsilon_{\pi(\nu)}<60.0$ MeV.
For the calculations of $M^{2\nu}_{\rm GT}$ of specific intermediate states,
only the $\pi$-$\nu$ configurations with $|X^2_{\pi\nu}-Y^2_{\pi\nu}|>10^{-6}$ are taken.
The strengths of isovector proton-neutron paring $f_{\rm IV}$ are fixed by
tuning $M_{\rm F}^{2\nu}$ vanish.
But since it does not affect
the Gamow-Teller $2\nu\beta\beta$ matrix element $M^{2\nu}_{\rm GT}$,
we will not discuss it in present work.
\section{Results and discussions}
\label{secResu}
\subsection{Systematic study on $M_{\rm GT}^{2\nu}$}
\label{secSys}
\begin{figure}[t]
\centering
\includegraphics[width=0.45\textwidth]{Fig1.eps}\\
\caption{The $2\nu\beta\beta$ nuclear matrix elements $M_{\rm GT}^{2\nu}$ for 11 nuclei
as a function of isocalar pairing strength parameter (black square).
The horizontal lines represent the experimental values extracted from Ref. \cite{Barabash2015}
with $g_A = 1.00$ (red dotted), and $g_A=1.27$ (black dashed)}.
\label{Fig1}
\end{figure}
We start with systematic study of the evolution of $2\nu\beta\beta$-decay
NMEs ($M_{\rm GT}^{2\nu}$) on the IS pairing strength for 11 nuclei which have been measured.
Results are depicted in Fig.~\ref{Fig1} and compared with compiled data from
Ref. \cite{Barabash2015}.
From the figure, all these $M_{\rm GT}^{2\nu}$ decrease when IS pairing strength is large enough.
This suppression effect of IS pairing, as a common feature found in QRPA calculations,
has been well analyzed in Refs.~\cite{Vogel1986,Civitarese1987}.
The decreasing behavior of the $M_{\rm GT}^{2\nu}$ with increasing IS paring strength
enables us to find the appropriate strength parameter $f_{\rm IS}$
to reproduce the experimental values, as listed in Tab.~\ref{Tab_1}.
If one further increases the IS pairing strength, the $M^{2\nu}_{\rm GT}$ passes through zero and changes its sign.
It has been found that the broken Wigner spin-isospin SU(4) symmetry is restored
when the $2\nu\beta\beta$ closure matrix element
$M_{{\rm GT} cl}^{2\nu}= \langle f| \sum_{mn} \vec{\sigma}_m \cdot \vec{\sigma}_{n} \tau_m^{+} \tau_{n}^{+} |i\rangle=0$
\cite{Rumyantsev1998,Civitarese2000,Rodin2005,Rodin2011}.
We note that besides the IS pairing,
the nuclear deformation also leads to the suppression of the $M^{2\nu}_{\rm GT}$ \cite{Simkovic2004,AlvarezRodriguez2004},
mostly due to the difference between the shapes of initial and final nuclei,
which will not be discussed in present work.
In Tab.~\ref{Tab_1}, we list the experimental and theoretical values of $M^{2\nu}_{\rm GT}$ for these 11 nuclei.
The isoscalar pairing strength parameters $f_{\rm IS}$ used in QRPA calculations are listed in the last column.
We observe that most of the determined $f_{\rm IS}$ are around 1.2, which means the ground-state correlation
should be large enough to suppress the $M^{2\nu}_{\rm GT}$.
For the nuclei with magic numbers, $^{48}$Ca, $^{116}$Cd, and $^{136}$Xe, their $f_{\rm IS}$ values are relatively small.
This is probably caused by the overestimated small ground-state overlap factor
due to the violation of particle number in QRPA model
\cite{Yao2015} and this needs further investigation.
We further examine the SSD and the extended LLD hypotheses, so
the $M^{2\nu}_{\rm GT}$ obtained from the SSD or LLD hypothesis
$M_{\rm GT}^{2\nu}({\textrm{SSD or LLD}})$ are also listed in Tab~\ref{Tab_1}.
When $g_A=1.27$ (bare value), for $^{48}$Ca, $^{82}$Se, $^{116}$Cd, $^{128}$Te, and $^{238}$U,
the first $1^+$ states of the intermediate nuclei contribute more than $75\%$ to the total NMEs.
In this sense, the SSD hypothesis is fulfilled for these nuclei.
On the other hand, when $g_A=1.00$ (quenched value), only $^{238}$U still fulfills the SSD hypothesis.
The inconsistence on the evidence of SSD in $^{100}$Mo reported in Refs. \cite{Arnold2019,Armengaud2020}
could be the nuclear deformation that is not included in our model \cite{Moreno2008}.
We further test the LLD hypothesis for the nuclei whose NMEs do not fulfill the SSD hypothesis.
Because there is not a definite upper limit of the energy for low-lying states in intermediate nuclei,
we set it as 5 MeV.
When $g_A=1.00$, these low-lying states contribute almost the full NMEs for $^{76}$Ge, and $^{82}$Se.
In this sense, the NMEs for these nuclei fulfill the LLD hypothesis.
\begin{table}[t]
\caption{
Experimental and theoretical $2\nu\beta\beta$ nuclear matrix elements $M_{\rm GT}^{2\nu}$ for 11 nuclei with unit of MeV$^{-1}$
when $g_A=1.27$ and $g_A=1.00$.
$M_{\rm GT}^{2\nu}$ obtained by the single-state dominance (SSD)
and low-lying-states dominance (LLD) hypotheses are also listed.
The upper limit of the energy for low-lying states $E_{\rm int.}$ in LLD is set as 5 MeV.
The isocalar pairing strength parameters used in QRPA calculations are listed in the last column.
The experimental values of $M_{\rm GT}^{2\nu}$ are taken from Ref.\cite{Barabash2015}.
The results of $^{48}$Ca and $^{116}$Cd when $g_A=1.00$ are not listed,
because the experimental NMEs are always larger than the theoretical ones with this value of $g_A$.}
\centering
\resizebox{0.48\textwidth}{!}{
\begin{tabular}{ccccccc}
\hline\hline
& & Expt. & Theo. & Theo. & Theo. & \\
Nucleus &
$g_A$ &
$M_{\rm GT}^{2\nu}$ &
$M_{\rm GT}^{2\nu}$ &
$M_{\rm GT}^{2\nu}$(SSD) &
$M_{\rm GT}^{2\nu}$(LLD) &
$f_{\rm IS}$ \\ \hline
$^{48}$Ca & 1.27 & 0.046$\pm$0.004 & 0.046 & 0.036 & 0.036 & 0.7570 \\% \hline
$^{76}$Ge & 1.27 & 0.137$\pm$0.007 & 0.136 & 0.071 & 0.188 & 1.1852 \\% \hline
& 1.00 & 0.221$\pm$0.012 & 0.221 & 0.082 & 0.256 & 1.1612 \\% \hline
$^{82}$Se & 1.27 & 0.101$\pm$0.005 & 0.100 & 0.104 & 0.161 & 1.1886 \\% \hline
& 1.00 & 0.162$\pm$0.008 & 0.162 & 0.095 & 0.191 & 1.1594 \\% \hline
$^{96}$Zr & 1.27 & 0.097$\pm$0.005 & 0.099 & 0.351 & 0.252 & 1.3411 \\% \hline
& 1.00 & 0.157$\pm$0.008 & 0.158 & 0.405 & 0.309 & 1.3399 \\% \hline
$^{100}$Mo& 1.27 & 0.224$\pm$0.006 & 0.224 & 0.515 & 0.346 & 1.2824 \\% \hline
& 1.00 & 0.362$\pm$0.010 & 0.361 & 0.631 & 0.478 & 1.2768 \\% \hline
$^{116}$Cd& 1.27 & 0.127$\pm$0.004 & 0.127 & 0.112 & 0.124 & 1.0350 \\% \hline
$^{128}$Te& 1.27 & 0.056$\pm$0.007 & 0.057 & 0.043 & 0.113 & 1.1928 \\% \hline
& 1.00 & 0.090$\pm$0.012 & 0.091 & 0.047 & 0.130 & 1.1788 \\% \hline
$^{130}$Te& 1.27 & 0.038$\pm$0.005 & 0.037 & 0.020 & 0.087 & 1.1912 \\% \hline
& 1.00 & 0.061$\pm$0.008 & 0.062 & 0.022 & 0.095 & 1.1776 \\% \hline
$^{136}$Xe& 1.27 & 0.022$\pm$0.001 & 0.022 & 0.003 & 0.007 & 0.9968 \\% \hline
& 1.00 & 0.035$\pm$0.001 & 0.035 & 0.004 & 0.011 & 0.8820 \\% \hline
$^{150}$Nd& 1.27 & 0.070$\pm$0.005 & 0.070 & 0.240 & 0.257 & 1.2503 \\% \hline
& 1.00 & 0.114$\pm$0.008 & 0.116 & 0.246 & 0.289 & 1.2360 \\% \hline
$^{238}$U & 1.27 & 0.158$^{+0.109}_{-0.085}$ & 0.157 & 0.185 & 0.185 & 1.1787 \\
& 1.00 & 0.254$^{+0.176}_{-0.137}$ & 0.254 & 0.230 & 0.230 & 1.0251 \\
\hline \hline
\end{tabular}}
\label{Tab_1}
\end{table}
\begin{figure}[!t]
\centering
\includegraphics[width=0.45\textwidth]{Fig2.eps}\\
\caption{
Running sums of $M_{\rm GT}^{2\nu}$ for 11 nuclei
as a function of the excitation energy of the intermediate nucleus
with $g_A = 0.80$ (blue), $g_A = 1.00$ (red), and $g_A=1.27$ (black).
The IS pairing strength parameters $f_{\rm IS}$ used for the corresponding calculations are also shown in the figure. }
\label{Fig2}
\end{figure}
In Fig.~\ref{Fig2}, with the determined $f_{\rm IS}$ in Tab.~\ref{Tab_1},
the running sums of $M^{2\nu}_{\rm GT}$ for these 11 nuclei are depicted.
The results of $g_A=0.80$ are also shown, since this value of $g_A$ has been used in Ref. \cite{Gando2019}.
Generally, the behaviors of running sums can be divided into three types when $g_A=1.27$.
The first one is steadily increasing accompanied with small fluctuations,
as shown in $^{48}$Ca and $^{116}$Cd.
The second type is that the contributions from first few lowest states are large enough
but then the following excited states give continuous negative contributions, and
hence provide a suppression due to the cancellation between lowest states and higher-lying states.
like $^{96}$Zr, $^{100}$Mo, or $^{150}$Nd.
The last type is the most common one, observed in 6 of total 11 nuclei, i.e.,
$^{76}$Ge, $^{82}$Se, $^{128}$Te, $^{130}$Te, $^{136}$Xe, and $^{238}$U.
The cumulative contributions continuously increase up to
the excitation energy of the intermediate nucleus $E_{\rm int.}$ around 10 MeV (for $^{136}$Xe and $^{238}$U, it is 15 MeV),
then a remarkable cancellation appears due to the negative contributions of higher-lying states,
being similar as the case of the second type.
The difference between the second and the third type is how the medium energy states with $5$-$10$ MeV contribute.
The cancellation between positive contributions from the low-lying states
and negative contributions from the higher-lying states
leads to the realization of SSD for $^{82}$Se and $^{128}$Te when $g_A=1.27$, and
LLD for $^{76}$Ge and $^{82}$Se when $g_A=1.00$.
The negative contributions in the running sums seem to be universal,
which also appear in the results calculated by shell model
with complete spin-orbit partner model space \cite{Nakada1996,Horoi2007}
and QRPA models \cite{Fang2010,Suhonen2014,Delion2015,Simkovic2018}.
Compared to the nuclei of the second and third types, it is noticed that
the IS pairing strength parameters $f_{\rm IS}$ are smaller
for the nuclei of the first type.
On the other hand,
from Fig. \ref{Fig1}, when increasing $g_A$, the strength of proton-neutron isoscalar pairing
$f_{\rm IS}$ should also be increased to reproduce the experimental NME.
By comparing the running sums of $g_A=0.80, 1.00$ and $1.27$, one can find that
when $g_A$ is increasing
the negative contributions can either be induced, e.g. in $^{136}$Xe and $^{238}$U,
or be enlarged, e.g. in $^{82}$Se, $^{128}$Te, and $^{130}$Te.
Therefore, these two aspects indicate that the appearance of negative contributions in the running sums
are closely related with the magnitude of $f_{\rm IS}$, which controls
the amount of ground-state correlations introduced in QRPA \cite{Fang2010}.
So, without losing generality, we will pick up $^{128}$Te as an example and analyze the mechanism for the negative contributions of high-lying states at large $f_{\rm IS}$ in next subsections.
\subsection{Ground-state correlations: QRPA vs. QTDA}
\label{secGSC}
As stated above, the negative contributions in the running sums
are related with the ground-state correlations in the QRPA model.
In this subsection, to analyze the effects of the ground-state correlations,
as an example, we compare the QRPA model and the QTDA (quasiprticle Tamm-Dancoff approximation) model calculations of the $M^{2\nu}_{\rm GT}$ for $^{128}$Te.
Unlike the QRPA model, the ground state of the QTDA model is the quasiparticle vacuum with no ground-state correlation,
that is without the backward amplitude $Y_{\pi\nu}$ term.
In Fig.~\ref{Fig3}, the evolution of $M^{2\nu}_{\rm GT}$ as a function of
IS pairing strength $f_{\rm IS}$ from QRPA and QTDA models are shown.
These two results are different from each other.
The $M^{2\nu}_{\rm GT}$ calculated by QTDA model monotonously increases with increasing $f_{\rm IS}$
and is systematically larger than that from QRPA model.
The latter shows explicitly the suppression effect from the ground-state correlations.
This is consistent with the conclusion of Ref.~\cite{Muto1989}.
\begin{figure}[!t]
\centering
\includegraphics[width=0.4\textwidth]{Fig3.eps}\\
\caption{The $2\nu\beta\beta$ nuclear matrix elements $M_{\rm GT}^{2\nu}$ for $^{128}$Te
as a function of isocalar pairing strength parameter calculated by QRPA (black square)
and QTDA (red circle).
The horizontal lines represent the experimental values with $g_A=1.27$ (solid) and $g_A=1.00$ (dashed).}
\label{Fig3}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=0.4\textwidth]{Fig4.eps}\\
\caption{ Running sums of the $M_{\rm GT}^{2\nu}$ for $^{128}$Te
as a function of the excitation energy of the intermediate nucleus
calculated by QRPA (black solid line) and QTDA (red dashed line)
with the isoscalar pairing strength parameter $f_{\rm IS}=1.1928$.}
\label{Fig4}
\end{figure}
In Fig.~\ref{Fig4}, the running sums of $M^{2\nu}_{\rm GT}$ calculated by QTDA model and QRPA model with $f_{\rm IS}=1.1928$ which reproduce the experimental results for QRPA calcualtion, are shown respectively.
One finds that in the region of $E_{\rm int.} < 9.0$ MeV,
the running sums of these two models are different in magnitude but evolve similarly.
However, in the region of $9.0 < E_{\rm int.} < 12.0$ MeV,
the running sum of the QTDA model keeps increasing, while that of the QRPA model starts to decrease.
Therefore, the negative contributions of high-lying states are related with the ground-state correlations introduced in QRPA model.
Finally, running sum curves of both models become flat
after $E_{\rm int.}$ equals to around 12.0 MeV around the GT resonance (GTR) region,
because of large energy denominators and small transition amplitudes for these very highly excited states beyond GTR.
To understand the large difference between the $M^{2\nu}_{\rm GT}$ calculated by QRPA and QTDA models,
we further investigate GT$^{-}$ and GT$^{+}$ transition branches
involved in the calculation of $M^{2\nu}_{\rm GT}$ as shown in Eq.~(\ref{Eq_NME}).
The corresponding transition strengths $S({\rm GT}^{-})$ and $S({\rm GT}^{+})$ are shown in Fig.~\ref{Fig5}.
Since $^{128}$Te is with neutron excess, its GT$^-$ transition strength is large, so the inclusion of $Y_{\pi\nu}$ amplitude
in QRPA calculation does not make it different from QTDA calculation without $Y_{\pi\nu}$ term.
However, the GT$^+$ transition strength of $^{128}$Xe is small,
due to the blocking effects of Pauli principle with large neutron excess.
The inclusion of
ground-state isovector pairing correlations
could unblock some transition channels and increase the GT$^+$ transition strength.
The inclusion of the backward amplitude $Y_{\pi\nu}$,
the ground-state correlations introduced in QRPA model,
have also significant influence on the GT$^+$ transition strength through the relatively big $u_{\pi}v_{\nu}$ factor
in front of $Y_{\pi\nu}$.
Due to the opposite signs of $X_{\pi\nu}$ and $Y_{\pi\nu}$,
for $^{128}$Xe the GT$^+$ transition strength becomes smaller in QRPA calculation compared to that of QTDA calculation.
Through the above comparison,
it is clear the ground-state correlations,
namely the large backward amplitudes $Y_{\pi\nu}$ in QRPA induced by strong IS pairing,
play crucial roles in the calculation of $M^{2\nu}_{\rm GT}$
through their effects on the GT$^{+}$ transition of daughter nucleus.
\begin{figure}[!t]
\centering
\includegraphics[width=0.4\textwidth]{Fig5.eps}\\
\caption{The GT$^{-}$ transition strength function of $^{128}$Te $\rightarrow$ $^{128}$I
whose excitation energies $E_{\rm int.}^{(i)}$ are respect to $^{128}$I (a)
and GT$^{+}$ transition strength function of $^{128}$Xe $\rightarrow$ $^{128}$I
whose excitation energies $E_{\rm int.}^{(f)}$ are respect to $^{128}$I(b).
The black solid lines and red dashed lines are
the results calculated by QRPA and QTDA, respectively.}
\label{Fig5}
\end{figure}
\subsection{Negative contributions in running sum}
\label{secNeg}
By comparing running sums of QRPA model and QTDA model in Sec.~\ref{secGSC}, we come to the conclusion that
the negative contributions are related with ground-state correlations
introduced by $Y_{\pi \nu}$ term which can be enhanced by large IS pairing.
Actually, there is a close relation between IS pairing and ground-state correlations introduced by $Y_{\pi \nu}$ term.
With the increase of IS pairing strength, the magnitude of $B$ matrix elements in QRPA equation \eqref{Eq_QRPA_B} increases as well
due to the attractive nature of IS pairing residual interaction.
As a result, the $Y_{\pi \nu}$ amplitude will increase, as also explained in Ref. \cite{Vogel1986}.
We depict the running sums of the $M^{2\nu}_{\rm GT}$ for $^{128}$Te with different
$f_{\rm IS}$ and $\eta_{Y}$ in Fig.~\ref{Fig6}.
From the figure, the behaviors of the running sum curves are very similar when increasing $f_{\rm IS}$ and $\eta_{Y}$.
Especially, in both cases the negative contributions will be induced and further be enlarged in the 9-15 MeV (around the GTR) region.
Based on such a relation,
to simulate the increase of the IS pairing strength,
we change the magnitude of $Y_{\pi\nu}$ by replacing it with $\eta_{Y} Y_{\pi\nu}$ in the calculation of transition amplitudes,
where the factor $\eta_Y$ varies from 0.0 to 1.0.
The purpose of this study is to understand how negative contributions to $M^{2\nu}_{\rm GT} $ happen at large IS pairing strength.
One of the advantages of such a simulation, instead of varying the IS pairing strength directly, is that one can avoid
the complicated splittings and degeneracy in the excited spectra caused by the variation of the IS pairing.
The trend of the results with $\eta_{Y}=0.0$ is very similar to that of QTDA model in Fig.~\ref{Fig4}.
With the increasing $\eta_{Y}$, two features of the running sum curves are emerging.
The first one is that the contributions from the region
$E_{\rm int.} < 9.0$ MeV reduce significantly.
The second one is that the majority of these contributions in the region of
$9.0 < E_{\rm int.} < 15.0$ MeV
changes their signs from positive to negative values. It can be seen that negative contributions do appear with large enough $Y_{\pi\nu}$ amplitudes in our simulation.
Then what is the origin of these negative contributions? To answer this question, in Fig.~\ref{Fig7},
we explicitly show the contributions to $M^{2\nu}_{\rm GT}$ of intermediate states
with excitation energy ranging from 9.0-15.0 MeV as a function of $\eta_Y$.
The total contribution (black square) decrease monotonously from positive to negative values.
Then we separate the contributions which have changes in sign for the increasing $\eta_Y$,
denoted as ${M'}^{ 2\nu }_{\rm GT}$ (red circle).
From the figure, we observe that the ${M'}^{ 2\nu }_{\rm GT}$ contributes remarkably
more than 50\% to the ${M}^{ 2\nu }_{\rm GT}$ at $\eta_Y = 1.0$. So a main reason that the negative contributions appear is
the change of signs of $M^{2\nu}_{\rm GT}$ contributed by some intermediate states.
Among these intermediate states, one non-negligible contribution comes from those with an excitation energy of
$E_{\rm int.}^{(f)}=9.15$ MeV from the GT$^+$ transition side of the daughter nucleus.
We note that, the rest part of the negative contributions
do not change their signs for the increasing $\eta_Y$,
but they are small at $\eta_Y=0.0$ and become large at $\eta_Y=1.0$.
\begin{figure}[!t]
\centering
\includegraphics[width=0.4\textwidth]{Fig6.eps}\\
\caption{Running sums of the $M_{\rm GT}^{2\nu}$ for $^{128}$Te
with different $f_{\rm IS}$ (a) and $\eta_{Y}$ (b)
as a function of the excitation energy of the intermediate nucleus.}
\label{Fig6}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=0.4\textwidth]{Fig7.eps}\\
\caption{Contributions to $M_{\rm GT}^{2\nu}$ for $^{128}$Te
from excitation energies of the intermediate nucleus ranging 9.0-15.0 MeV.
Black square: the total contribution of all intermediate states in the
9.0-15.0 MeV excitation energy region.
Red circle: sum of the contributions whose signs have changed relative to $\eta_Y=0.0$ during $\eta_Y$ increasing.
Blue up triangle: sum of the contributions of those intermediate states with one coming from GT$^+$ transition of the daughter nucleus with an excitation energy of $E_{\rm int.}^{(f)}=9.15$ MeV.
Green down triangle: sum of the contributions with
$E_{\rm int.}^{(f)}=9.15$ MeV whose signs have changed relative to $\eta_Y=0.0$ during $\eta_Y$ increasing.}
\label{Fig7}
\end{figure}
To get a deeper insight for these sign changes,
we investigate the GT$^{-}$ and GT$^{+}$ transitions, respectively.
From Eq.~(\ref{Eq_NME}), the ${M}^{ 2\nu }_{\rm GT}$ for a specific intermediate state is proportional to
the GT$^-$ transition amplitude $\langle 1^{+}_{\rm int.}|| {\rm GT}^{-} || 0^{+(i)}_{\rm g.s.}\rangle$
and the GT$^+$ transition amplitude $\langle 1^{+}_{\rm int.}|| {\rm GT}^{+} || 0^{+(f)}_{\rm g.s.}\rangle$.
So we pick up several typical GT$^-$ and GT$^+$ states with excitation energies in the range of $E_{\rm int.} < 9.0$ MeV
and in the range of $9.0 < E_{\rm int.} < 15.0$ MeV,
and plot their transition amplitudes as a function of $\eta_Y$ in Fig.~\ref{Fig8}.
These six typical states are chosen, due to the following reasons:
The states at $E_{\rm int.}^{(i)} = 2.25$ MeV and $E_{\rm int.}^{(f)}=0.21$ MeV
are the first $1^+$states calculated from $^{128}$Te and $^{128}$Xe, respectively.
The state at $E_{\rm int.}^{(f)}=5.14$ MeV is with the largest GT$^+$ transition amplitude in the 0-9.0 MeV region.
For this state, the largest overlap factor $\langle 1_{n_f}^+ | 1_{n_i}^+ \rangle$
is produced by the state at $E_{\rm int.}^{(i)} =6.17$ MeV.
The state at $E_{\rm int.}^{(f)}=9.15$ MeV is with the largest GT$^+$ transition amplitude
(absolute value), among the states whose GT$^+$ transition amplitudes
have changed their signs in the 9.0-15.0 MeV region.
For this state, the largest overlap factor $\langle 1_{n_f}^+|1_{n_i}^+\rangle$
produced by the state lying in 9.0-15.0 MeV is the state at $E_{\rm int.}^{(i)} =10.39$ MeV.
From Fig.~\ref{Fig8}(a), we find that the GT$^-$ transition amplitudes are almost independent of $\eta_Y$,
which is consistent with that $B({\rm GT}^{-})$ is influenced little by the IS pairing.
On the other hand, the GT$^+$ transition amplitudes, in panel (b), show a strong dependence on $\eta_Y$,
and even change their signs for the higher excited states with $E_{\rm int.}^{(f)}=9.15$ MeV.
Such a strong dependence comes from the large magnitude of
the $u_{\pi}v_{\nu}$ factor in front of $Y_{\pi\nu}$ for neutron-rich nuclei.
For the GT$^{+}$ state of $E_{\rm int.}^{(f)}=9.15$ MeV,
one can observe that after $\eta_{Y}=0.4$, the GT$^{+}$ transition amplitude crosses zero.
Therefore, the summed contribution, coming from the
$(n_i, n_f)$ intermediate states with
GT$^+$ state of $E_{\rm int.}^{(f)}=9.15$ MeV,
change their signs and become negative,
as the blue up triangle shown in Fig. \ref{Fig7}.
On the other hand, for the GT$^+$ states with energies smaller than
9.0 MeV, such as those with $E_{\rm int.}^{(f)}=0.21$ MeV or $5.14$ MeV shown in the figure,
the absolute values of their transition amplitudes keep decreasing with increasing $\eta_Y$,
which causes the first feature observed in Fig.~\ref{Fig6}.
Therefore, the variation of the GT$^+$ transition amplitudes with $\eta_Y$ for different energy regions of $E_{\rm int.}$
give rise to the two features in Fig.~\ref{Fig6} mentioned above.
These two features are different because the GT$^+$ transitions
are unblocked by pairing correlation
for low-lying states with energies smaller than 9.0 MeV,
and hence there are non-negligible transition strengths
in the case of $\eta_{Y}=0.0$.
The increase of $\eta_{Y}$ can greatly
reduce the GT$^+$ transition amplitudes in this energy region,
but is not enough to change their signs.
On the other hand, for high-lying states with energies larger than 9.0 MeV,
the GT$^+$ transitions are almost blocked and have small strengths at $\eta_{Y}=0.0$.
In this case, the sign of GT$^+$ transition amplitude can be changed easily due to the increase of $Y_{\pi\nu}$ amplitude.
At the end, we conclude that it is the enhanced ground-state correlations tuned by strong IS pairing interaction that cause the negative contributions to ${M}^{ 2\nu }_{\rm GT}$ in the energy range of 9.0-15.0 MeV,
through their influence on the GT$^+$ transition amplitudes.
\begin{figure}[t]
\centering
\includegraphics[width=0.4\textwidth]{Fig8.eps}\\
\caption{The GT$^{-}$ transition amplitudes for 3 excited states in $^{128}$Te (a)
and GT$^{+}$ transition amplitudes for 3 excited states in $^{128}$Xe (b)
as a function of $\eta_Y$.
The values of excitation energies relative to $^{128}$I are shown in the figure.}
\label{Fig8}
\end{figure}
\section{Summary}
\label{secConclu}
In summary, we study the $2\nu\beta\beta$-decay nuclear matrix elements $M^{2\nu}_{\rm GT}$ based on the spherical Skyrme HFB+QRPA model, for 11 nuclei with experimental data and the underlying mechanism for the fulfillment of SSD or LLD hypothesis is revealed.
In our systematic investigation of $M^{2\nu}_{\rm GT}$,
the suppression effect of the IS pairing is found.
By using the IS pairing strength determined through reproducing the
experimental data with different axial-vector coupling constants $g_A$,
we investigate the SSD and LLD hypotheses.
When $g_A=1.27$, the SSD hypothesis is fulfilled by the NMEs of
$^{48}$Ca, $^{82}$Se, $^{116}$Cd, $^{128}$Te, and $^{238}$U.
When $g_A=1.00$, the SSD hypothesis is fulfilled by the NMEs of
$^{238}$U, while the LLD hypothesis is fulfilled by the NMEs of
$^{76}$Ge, and $^{82}$Se.
The realization of SSD and LLD hypotheses for $^{76}$Ge, $^{82}$Se and $^{128}$Te are to a large extent caused by
the negative contributions in the running sums of the NMEs.
Through the comparison of the running sums of different $g_A$, we find that
by increasing $g_A$, or alternatively $f_{\rm IS}$, the negative contributions can either be induced or enlarged.
We pick up $^{128}$Te as an example to further study the reason for these negative contributions in the running sum.
By comparing the running sums of QRPA and QTDA models,
the negative contributions are found related with the strong ground-state correlations induced by large IS pairing strength,
which shows the importance of QRPA calculations over the simpler QTDA calculations for NME of $2\nu\beta\beta$ decay.
Through the study of GT$^-$ and GT$^+$ transition strength functions,
it is shown that the ground-state correlations influence the NME through its crucial role in GT$^+$ transitions
for these double-$\beta$ decay nuclei with neutron excess, where many GT$^+$ transition channels are blocked because of Pauli principle.
With the inclusion of ground-state correlations in QRPA calculations,
the increase of IS pairing strength will make their contributions larger
so that they lead to the suppression of NME contributed from the energy region of intermediate states $E_{\rm int.}< 9.0$ MeV
as well as negative contributions from $9.0 < E_{\rm int.}< 15.0$ MeV.
Ground-state correlations play their roles on the above two features
by suppressing GT$^{+}$ transition amplitudes of low-lying states
and changing the signs of GT$^{+}$ transition amplitudes of higher-lying states.
\section*{Acknowledgement}
Y. F. Niu and W. L. Lv acknowledge the support of the National Natural Science
Foundation of China under Grant No. 12075104, and the Fundamental
Research Funds for the Central Universities under
Grants No. lzujbky-2021-it10.
D. L. Fang acknowledges the support of the ``Light of West'' program and the ``from zero to one" program by CAS.
C. L. Bai acknowledges the support of the National Natural Science
Foundation of China under Grants No. 11575120 and 11822504.
|
2024-02-18T23:40:24.433Z
|
2022-04-27T02:10:05.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14405","language":"en","timestamp":1651025405000,"url":"https:\/\/arxiv.org\/abs\/2112.14405","yymm":"2112"}
|
proofpile-arXiv_000-10022
|
{"provenance":"002.jsonl.gz:10023"}
| null | null |
\section*{}
\vspace{-1cm}
\section{Overview of the article series}
This article is the first of a three-article series devoted to experimental studies of barrier-free ion-molecule reactions at low collision energies. The article series addresses fundamental aspects of chemical reactivity in low-energy ion-molecule collisions which play a crucial role in the understanding and modelling of ion-molecule reactions at low temperatures.\cite{bowers79a,ausloos78a,clary85a,troe87a,clary90a,troe96a,ng92a,gerlich08a,willitsch08a,heazlewood21a} The focus is placed on the measurement and theoretical analysis of the deviations of the collision-energy-dependent reaction rates $k(E_{\rm coll})$ from the Langevin rate $k_{\rm L}$ at very low collision energies resulting from the electrostatic interaction between the charge of the ion and the electric dipole, quadrupole and octupole moments of the neutral molecule.
For these studies, we have chosen He$^+$ as a reactant ion because the high ionisation energy of He guarantees strongly exothermic barrier-free reactions with a broad range of neutral molecules. The neutral molecules we have selected are (i) ammonia (NH$_3$ and ND$_3$), which is a polar molecule with a dipole moment of $4.91\times 10^{-30}$~C\,m,\cite{marshall97a} (ii) N$_2$, which does not have a permanent dipole moment but has a quadrupole moment of $-4.65\times 10^{-40}$~C\,m$^2$,\cite{graham98} and (iii) methane (CH$_4$ and CD$_4$), which has neither a permanent dipole moment nor a quadrupole moment, but an octupole moment of $7.41\times 10^{-50}$ C\,m$^3$.\cite{birnbaum75} The reactions of all three molecules are known from previous work to proceed with high rates.\cite{albritton78,anicich86} With these three molecules, we can examine the convergence of the capture rate coefficients at low energies towards the Langevin rate as the lowest non-vanishing member of the multipole expansion series of the neutral molecule gradually increases in the sequence dipole ($n_{\rm mp}=2$), quadrupole ($n_{\rm mp}=4$), and octupole ($n_{\rm mp}=8$). This sequence also corresponds to the order of this three-article series, which presents our results on the $n_{\rm mp}=2$ case in article I, and those on the $n_{\rm mp}=4$ and $n_{\rm mp}=8$ cases in articles II and III, respectively.
The experimental observations are analysed in the realm of an adiabatic capture model inspired by the earlier theoretical work of Clary and co-workers~\cite{clary85a,clary90a,stoecklin92a} and Troe and co-workers.\cite{troe87a,troe96a,dashevskaya05a,auzinsh13a,auzinsh13b,dashevskaya16a} Because the existence or non-existence of a given $n_{\rm mp}$-pole moment is dictated by the molecular symmetry, the interpretation of the experimental observations requires consideration of the corresponding molecular-symmetry groups, in particular when determining the nuclear-spin symmetries and occupations of the rotational quantum states of the neutral molecules. We follow here earlier treatments, in particular those presented in Refs.~\cite{hougen76a,bunker06,snels00,wichmann20} and compare reactions of undeuterated and fully deuterated ammonia and methane to highlight specific aspects. The overall analysis gives insights into the dependence of the rate coefficients on the rotational quantum states of the neutral molecules and provides access to the thermal rate coefficients $k(T)$ through statistical averaging.
The overall structure of this three-article series is as follows: In this first article, we present a general introduction to fast ion-molecule reactions and detail the experimental approach and set-up we have used, which are the same for all three studies. We also summarise the main aspects of the capture model we have implemented to interpret the experimental results in a general form, adequate to treat the effects of an arbitrary $n_{\rm mp}$ moment. We then describe the experimental results we have obtained in our study of the He$^+$~+~NH$_3$ and He$^+$~+~ND$_3$ reactions and their analysis using the capture model for the $n_{\rm mp}=2$ case. In the second article, we focus on the results obtained on the $n_{\rm mp}=4$ case and present our studies of the He$^+$~+~N$_2$ reaction. To clarify the role of the sign of the quadrupole, we also compare the results obtained for N$_2$, which has a negative quadrupole moment, with those obtained for the reactions of D$_2^+$ with H$_2$, which has positive quadrupole moment. The last article is devoted to the study of the $n_{\rm mp}=8$ case with the example of the He$^+$~+~CH$_4$ and He$^+$~+~CD$_4$ reactions. It also presents our main conclusions concerning the convergence of the low-energy capture rate coefficients towards Langevin rates. This structure offers the advantage that the introductory, experimental and theoretical-modelling sections of articles II and III could be reduced to a minimum, and that the general conclusions could be condensed in the last section of article III.
\section{Introduction}
Ion-molecule reactions play a key role in the chemistry of dilute environments, such as low-density plasmas, interstellar clouds and the upper layers of planetary atmospheres. Their rate coefficients and the branching ratios for different product channels are required as input in global kinetics models designed to describe and predict the chemical compositions of these environments.\cite{herbst73a,roueff87,smith92a,herbst01a,snow08a,smith11a,wakelam10a,dishoeck17} In particular,
barrierless, exothermic ion-molecule reactions proceed with high rate coefficients even at very low temperatures and several of these reactions represent essential steps in the synthesis of molecules in the interstellar medium, enabling the formation of polyatomic molecules through reaction chains in cold ($3-50$~K) dark molecular clouds.
Measurements of the rates of ion-molecule reactions at temperatures below 50~K are challenging because ions are easily heated up by stray electric fields and space-charge effects. An electric-potential difference of only 1 mV is indeed sufficient to accelerate ions to kinetic energies corresponding to about 12~K. Studies of ion-molecule reactions in ion guides and traps~\cite{gerlich08a,wester09a,markus20a} and in supersonic flows~\cite{marquette85a,rowe95a} enable measurements of reaction rates down to $\approx 10$~K, but the range below 10~K remains largely unexplored experimentally.
Advances in the experimental studies of ion-molecule reactions near 0~K have been recently made that rely on the use of laser-cooled and sympathetically cooled ions in ion traps and Coulomb crystals.\cite{willitsch08a,willitsch17a,petralia20a,toscano20} Our own approach to study ion-molecule reactions at low energies consists of suppressing the stray-electric-field-induced heating of the ions by replacing the ion by an atom or molecule in a Rydberg state of high principal quantum number $n$.\cite{allmendinger16a,allmendinger16b,zhelyazkova20,hoeveler21a} In such states, the Rydberg electron moves on an orbit of large radius ($\propto n^2$, e.g., $\approx 50$~nm for $n= 30$), without significantly affecting the reactions involving the ion core.\cite{pratt94a,wrede05a,matsuzawa10a} The distant Rydberg electron, however, effectively shields the reaction from stray electric fields. In addition, one can exploit the large induced dipole moments of Rydberg atoms and molecules ($\mu_{\mathrm{max}}\approx3400$ D for $n=30$) to manipulate their translational motion with inhomogeneous electric fields of modest magnitude.
We have used this property to deflect, accelerate and decelerate beams of Rydberg atoms and molecules using on-chip Rydberg-Stark decelerators and deflectors.\cite{hogan12b,allmendinger14a,hogan16a,zhelyazkova19a} With such devices, supersonic beams of Rydberg atoms and molecules can be merged with supersonic beams of ground-state neutral molecules, allowing the study of ion-molecule reactions at very low collisional energies.
This merged Rydberg-packet--ground-state-beam approach exploits the large velocity dispersion of supersonic beams produced with short-pulse valves, as in recent elegant work on Penning-ionisation and associative-ionisation reactions at low collision energies by Narevicius and co-workers~\cite{shagam13a,shagam15a,bibelnik19a} and Osterwalder and co-workers,\cite{jankunas14a,jankunas15c,gordon20a} and the well-defined spatial extent and mean velocity of the deflected Rydberg-atom cloud. This combination makes it possible to achieve a collision-energy resolution as low as $\sim 100$ mK.\cite{hoeveler21a} In the last years, we have observed deviations from Langevin capture rates at low collisional energies, in particular: (a) a small enhancement ($\sim15$\%) of the $\mathrm{H}_2^+ + \mathrm{H}_2$ reaction rate constant compared to $k_{\mathrm{L}}$ at $E_{\mathrm{coll}}/k_\mathrm{B}<1$ K,\cite{allmendinger16b} and (b) a strong enhancement by more than an order of magnitude of the $\mathrm{He}^++\mathrm{CH}_3\mathrm{F}$ capture rate constant at the lowest collisional energies.\cite{zhelyazkova20} These enhancements were attributed to ion-quadrupole interactions for the reaction between H$_2^+$ and ortho-H$_2$ molecules in the $J = 1$ rotational state, and to rotational-state-dependent Stark shifts of the polar CH$_3$F molecules in the Coulomb field of the helium ion for the $\mathrm{He}^++\mathrm{CH}_3\mathrm{F}$ reaction.
Despite these advances, experimental data on the rates of ion-molecule reactions in the range below 10~K remain extremely scarce.
In contrast, theoretical models describing ion-molecule reactions are advanced and represent the main source of rate constants for these reactions at low temperatures.
At low temperatures, the dominant ion-molecule reactions are barrier-free reactions and their rates can be modelled by ion-neutral capture models. In the simplest such model, introduced by Langevin,\cite{langevin05a} the rates are determined by considering the charge-induced dipole interaction between the ion and the polarisable molecule and it is assumed that the reaction takes place with unit probability provided that the collision energy is larger than the centrifugal barrier in the effective intermolecular potential. In this regime, the rate constants are independent of the temperature or the collision energy $E_{\rm coll}$ and are given by the Langevin rate constant (in SI units)
\begin{equation}\label{eq_langevin}
k_{\rm L}=\sigma v_{\rm rel} = 2\sqrt{\frac{\pi^2\alpha^\prime (Ze)^2}{4\pi\epsilon_0\mu}}.
\end{equation}
In Eq.~(\ref{eq_langevin}), $\sigma$ is the reaction cross-section, $ v_{\rm rel}= \sqrt{2E_{\rm coll}/\mu}$ is the asymptotic relative velocity, $\alpha'$ is the polarisability volume of the neutral molecule, $Ze$ is the charge of the ion, $\mu$ is the reduced mass of the reactants and $\epsilon_0$ is the permittivity of vacuum.
Whereas this treatment is often adequate above 100~K, it fails at low temperatures because long-range forces between the charge of the ion and the dipole, quadrupole, octupole, etc. moments of the neutral molecules affect their rotational motion.\cite{bowers79a,ausloos78a,ng92a} Specifically, the orientation of the molecular axes and the alignment of the rotational angular-momentum vector tend to be locked to the collision axis, leading to strong quantum-state(index $i$)- and collision-energy-dependent rate coefficients $k_i(E_{\rm coll})$.\cite{clary90a,troe87a,troe96a} In this range, thermal capture rate constants are determined from the $k_i(E_{\rm coll})$ values by averaging over the thermally populated translational and rovibrational states $i$ of the reactants.
This article summarises the results of our studies of the reaction of \ce{He+} with \ce{NH3} and \ce{ND3}. These reactions are of astrophysical importance. Ammonia was the first polyatomic molecule to be detected in the interstellar medium.\cite{cheung68} \ce{NH3} and several of its isotopologues (\ce{NH2D}, \ce{ND2H} and \ce{ND3} and \ce{^{15}NH3}) were later found in external galaxies and interstellar ices, proto-planetary disks, and in the atmosphere of Jupiter.\cite{roueff05,hermsen85,martin79,salinas16,irwin18} Ammonia can be detected through transitions in the radio-frequency (transitions between the tunnelling components of the umbrella-inversion mode), microwave (rotational transitions), IR (vibrational transitions) and the visible/UV (electronic transitions) ranges of the electromagnetic spectrum. The ease of its detection makes it one of the most important molecules for the study of molecular clouds (ammonia is dubbed "the interstellar thermometer")\cite{danby88}. Helium is the second most abundant element in the universe ($\sim25$\% by mass)\cite{izotov10} and can be ionised by cosmic rays in the interstellar medium to form He$^+$.
The reaction between \ce{He+} and \ce{NH3} has been studied in the 1970s at temperatures at and above $\sim300$~K in flowing-afterglow,\cite{bolden70} flow-drift tube~\cite{lindinger75} and ion-cyclotron resonance~\cite{kim75} set-ups. The thermal reaction rate constant at 300 K in these studies was found to be in the range between $1.15(\pm20\%)\times10^{-15}$ m$^3$/s and $2.20(\pm10\%)\times10^{-15}$ m$^3$/s ($k_{\mathrm{L}}^{\mathrm{He^+ + NH_3}} = 1.887\times10^{-15}$ m$^3$/s). An increase of the reaction rate constant with decreasing temperature in the $300-2900$ K range was observed in Ref. \cite{lindinger75}. Marquette {\it et al.} \cite{marquette85a} have measured the thermal rate coefficient in uniform supersonic flows and reported values of $4.5\times 10^{-15}$~m$^3$/s at 27~K, $3.0\times 10^{-15}$~m$^3$/s 68~K, and $1.65\times 10^{-15}$~m$^3$/s at 300~K. Several reaction channels were identified with the following branching ratios:\cite{kim75}
\begin{align}
\ce{He^+} + \ce{NH3} &\ce{->}\; \ce{NH3+} + \ce{He} & \qquad (\sim12\%) \tag{R1}\\
&\ce{->}\; \ce{NH2^+} + \ce{H} + \ce{He} & \qquad (\sim80\%) \tag{R2}\\
&\ce{->}\; \ce{NH^+} + \ce{H2} + \ce{He} & \qquad (\sim8\%) \tag{R3}.
\end{align}
The estimated released energies in each of these reactions are: $ E_{\mathrm{R1}} = 14.401$ eV, $ E_{\mathrm{R2}} = 8.822$ eV and $ E_{\mathrm{R3}} = 6.993$ eV.\cite{ruscic05}
Low-temperature and low-collisional-energy studies of the reaction between He$^+$ and ammonia are, however, lacking, although this energy regime is particularly relevant for astrochemistry. The current work presents experimental and theoretical studies of the \ce{He+ + NH3} and \ce{He+ + ND3} reactions in the range of collisional energies $E_{\mathrm{coll}}$ between 0 and $k_{\mathrm{B}}\cdot 40$ K at a resolution of $k_{\mathrm{B}}\cdot 200$ mK at $E_{\mathrm{coll}} = 0$.
\section{The effect of the molecular multipole moment on capture rate coefficients}
\label{sec2}
\subsection{The modified Langevin interaction potential}
In the Langevin model, the interaction potential $V_{\mathrm{L}}(R)$ between an ion and a molecule is described only through the centrifugal potential and the ion-induced dipole interaction as:
\begin{equation}
V_{\mathrm{L}}(R) = \frac{L^2}{2\mu R^2} - \frac{\alpha'(Ze)^2}{8\pi\epsilon_0R^4}.
\label{eq:VintL}
\end{equation}
In Eq.~(\ref{eq:VintL}), $L$ is the angular momentum of the collision (which can be expressed as $L = \mu v_{\mathrm{rel}}b$, where $b$ is the impact parameter) and $R$ is the ion-molecule separation. If the molecule possesses multipole moments, however, additional rotational-state-dependent terms need to be added to Eq.~(\ref{eq:VintL}) to account for the Stark-shift experienced by the molecule in rotational state $i$ in the electric field of the ion. The modified potential energy is:
\begin{equation}
V_{\mathrm{int},i}(R) = V_{\mathrm{L}}(R) + \Delta E_i(R),
\label{eq:Vint}
\end{equation}
where the energy shift of state $i$, $\Delta E_i(R)$, depends on $R$ through the electric field $Ze/(4\pi\epsilon_0 R^2)$ of the ion. Eq.~(\ref{eq:Vint}) implies that the molecule adiabatically remains in its original rotational state (at $R\to\infty$) as it approaches the ion.
The electrostatic potential at a point ${\bf R}$ in space resulting from the molecular charge distribution, $\rho({\bf r}')$, can be expressed as:\cite{wong98}
\begin{equation}
\phi({\bf R}) = \frac{1}{4\pi\epsilon_0}\int_{\tau}\frac{\rho({\bf r}')}{|{\bf R}-{\bf r}'|}\mathrm{d^3}{\bf r}'.
\label{ESpot}
\end{equation}
In the region where ${\bf R} > {\bf r}'$, one can expand the $1/|{\bf R}-{\bf r}'|$ term in Eq.~(\ref{ESpot}) as:\cite{zare88a}
\begin{equation}
\frac{1}{|{\bf R}-{\bf r}'|} = \frac{1}{\sqrt{R^2 + r'^2 - 2Rr'\cos\theta}} = \sum_{\lambda=0}^{\infty}\frac{r'^{\lambda}}{R^{\lambda+1}}P_{\lambda}(\cos\theta),
\label{eq:SHexp}
\end{equation}
where $\theta$ is the angle between ${\bf R}$ and ${\bf r}'$, and $P_{\lambda}(\cos\theta)$ is a Legendre polynomial:
\begin{equation}
P_{\lambda}(\cos\theta) = \sqrt{\frac{4\pi}{2\lambda+1}}Y_{\lambda 0}(\theta,\phi),
\end{equation}
and $Y_{\lambda 0}(\theta,\phi)$ is a spherical harmonic. If one defines the multipole coefficients characterising the nuclear charge distribution, as:\cite{wong98}
\begin{equation}
Q_{\lambda} \equiv \int_{\tau} P_{\lambda}(\cos\theta)\rho({\bf r}')r'^{\lambda}\mathrm{d^3}{\bf r}',
\label{mult_coeff}
\end{equation}
the value of the electrostatic potential at any given point ${\bf R}$ can be expanded as:
\begin{equation}
\phi(R) = \frac{1}{4\pi\epsilon_0}\frac{1}{R} \sum_{\lambda=0}^{\infty}\frac{Q_{\lambda}}{R^{\lambda}}.
\label{ESexp2}
\end{equation}
For a nearly-spherical charge distribution, Eq.~(\ref{ESexp2}) is a fast-converging series, and only a few terms are needed. The terms for $\lambda = 1,2$ and $3$ correspond to dipole-, quadrupole- and octupole-moment expansion terms.
In the symmetric-top wavefunction basis, $|JKM\rangle$, the matrix elements of the multipole operator are given by:\cite{wong98}
\begin{align}
\langle J'K'M'|Q_{\lambda}|JKM \rangle & = \left[\int_{\tau}\rho({\bf r}')r'^{\lambda}\mathrm{d^3}{\bf r}'\right] \times(-1)^{M^\prime-K^\prime}\sqrt{(2J+1)(2J'+1)}\nonumber\\
& \times \tj{J}{\lambda}{J'}{M}{0}{-M'}\tj{J}{\lambda}{J'}{K}{0}{-K'},
\label{Multipole_op_ME}
\end{align}
where the integral is the multipole moment in the molecular reference frame. The symmetric-top wavefunctions, $|JKM\rangle$, are given by:\cite{zare88a}
\begin{eqnarray}
|JKM\rangle &=& \sqrt{\frac{2J+1}{8\pi^2}}\mathscr{D}^{J*}_{MK}(\theta,\phi,\chi) \nonumber\\
&=& (-1)^{M-K}\sqrt{\frac{2J+1}{8\pi^2}}\mathscr{D}^{J}_{-M-K}(\theta,\phi,\chi),
\label{SymTop_wf}
\end{eqnarray}
where $K$ and $M$ are the quantum numbers associated with the projections of the rotational angular momentum $\vec{J}$ on the molecular symmetry axis and a space-fixed axis, respectively, and $\mathscr{D}^{J*}_{MK}(\theta,\phi,\chi)$ is a Wigner rotation matrix.
The $R$-dependent Stark-shifts, $\Delta E_{i=JKM}(R)$, which describe the interaction between the ion charge and the multipole moments of the molecule, are obtained by diagonalising the Hamiltonian:
\begin{equation}
\widehat{H}(R) = \widehat{H}_{\mathrm{rot}} + \widehat{H}_{\mathrm{Stark}}(R),
\label{Stark_Ham}
\end{equation}
where $\widehat{H}_{\mathrm{rot}}$ is the rotational Hamiltonian --- a diagonal matrix consisting of the molecular rotational energies $E_{JKM}$ in zero electric field, and $\widehat{H}_{\mathrm{Stark}}(R)$ is the Stark Hamiltonian describing the multipole-moment-induced mixing of the rotational levels in the presence of an external electric field. The matrix elements of $\widehat{H}_{\mathrm{Stark}}(R)$ in the electric field of the ion are given by:
\begin{align}
\label{Stark_H_ME}
\langle J'K'M'|\widehat{H}^{\lambda}_{\mathrm{Stark}}(R)|JKM\rangle & = \frac{1}{4\pi\epsilon_0R^{\lambda+1}}\langle J'K'M'|Q_{\lambda}|JKM \rangle.
\end{align}
The rotational-level mixing is stronger for a dipole moment ($\widehat{H}_{\mathrm{Stark}}^{\lambda = 1}$ scales as $R^{-2}$) than for quadrupole ($\widehat{H}_{\mathrm{Stark}}^{\lambda=2}\propto R^{-3}$) and octupole ($\widehat{H}_{\mathrm{Stark}}^{\lambda=3}\propto R^{-4}$) moments. In addition, its strength also depends on the molecule-fixed multipole moment. After the diagonalisation of $\widehat{H}(R)$, the eigenenergies $E_i + \Delta E_i(R)$ are assigned a value of the $J$, $K$ and $M$ quantum numbers through an adiabatic correlation to the field-free $i = |JKM\rangle$ states at $R\rightarrow\infty$.
\subsection{Determination of the rotational-state-dependent capture rate coefficients}
From the total interaction potential $V_{\mathrm{int},i}(R)$ associated with a given molecular $i = |JKM\rangle$ state, one can determine for each collision energy $E_{\mathrm{coll}}=\frac{1}{2}\mu v_{\mathrm{rel}}^2$ the classically allowed maximal value of the angular momentum, $L_{\mathrm{max},i} = \mu v_{\mathrm{rel}} b_{\mathrm{max},i}$ for which the reaction can take place.
The reaction rate coefficient for state $i$ is determined through the reaction cross-section $\sigma_i = \pi b_{\mathrm{max},i}^2$ as:
\begin{align}
\label{expr_k}
k_{i}(E_{\mathrm{coll}}) & = \frac{\pi L_{\mathrm{max},i}^2}{\sqrt{2\mu^3 E_{\mathrm{coll}}}}.
\end{align}
The effect of the multipole moment is to significantly modify the interaction potential for each molecular rotational state. This is particularly true in the case of an ion-dipole interaction, as will be discussed in Sec.~\ref{sec:Vint}. The addition of the $\Delta E_{i}(R)$ terms in Eq.~(\ref{eq:Vint}) creates an interaction potential for state $i$ which is either more or less attractive than the Langevin potential, $V_{\mathrm{L}}$, so that either higher or lower values of $L_{\mathrm{max},i}$ are allowed. The exact dependence of $k_i$ on the collisional energy is thus related to the $R$ dependence of $\Delta E_i$.
Adiabatic capture models such the one described in this section and used to interpret our experimental results include the effects of long-range interactions in the entrance channel but cannot predict the branching ratios between competing reaction channels, nor the product state distributions, nor angular distributions, nor vector correlations. Moreover, they predict the total cross section, whereas our experiments only measure the part of the capture cross section that leads to reactions products, which we estimate to be about 40\% for the He$^+$ + NH$_3$ reaction, see Section~\ref{sec:concl}. Calculations of the (nonadiabatic) dynamics on the relevant multidimensional potential energy surfaces would be needed for a full interpretation of the experimental results and a full description of the reaction dynamics.
\section{Experimental approach}
\label{sec:Exp}
\subsection{The merged-beam set-up}
\begin{figure*}[!h]
\includegraphics[width = \textwidth]{Exp_set_up_v2.pdf}
\caption{\label{Fig4} Schematic overview of the merged-beam experimental setup to study the reactions of \ce{He+} and ammonia. The He and ground-state beams are produced using home-built valves generating pulses of $\approx 20$ $\mu$s duration. (i) Helium source chamber. (ii) Ground-state-beam source chamber. (iii) Region where He$^*$ is excited to a Rydberg-Stark state and the Rydberg-He beam is deflected and merged with the ammonia beam. (iv) Time-of-flight mass spectrometer (TOFMS) used to monitor the product ions. Inset (a): Timing and pulse characteristics of the potentials applied to electrodes E$_1$ and E$_2$ when the TOFMS is operated in product-ion-extraction mode. Inset (b): Pulsed-field-ionisation signal of the He$(n)$ atoms deflected by the surface deflector, after acceleration/deceleration to final velocities $v_{\mathrm{Ryd}}$ in the $750-1200$ m/s range, as a function of the time of application of the ionisation/extraction pulse, $t_{\mathrm{PFI}}$. The dash-dotted black line presents the results obtained when guiding the He$(n)$ atoms at the initial central beam velocity of $1040$ m/s.}
\end{figure*}
The experimental set-up has been briefly described in Ref.~\cite{zhelyazkova20} and is outlined schematically in Fig.~\ref{Fig4}. The merged-beam apparatus consists of two supersonic beams produced by home-built valves (nozzle-opening duration $\sim20$ $\mu$s, repetition rate of 25 Hz) - one for the He atoms subsequently excited to a Rydberg state, and the other for the ground-state (GS) beam consisting of either pure NH$_3$ or ND$_3$. The two beams propagate along axes initially separated by a 5$^{\circ}$ angle. The helium atoms are photoexcited to a Rydberg-Stark state (referred to as He$(n)$ from here on) between two planar electrodes used to generate an electric field. They are then deflected by a $50$-electrode Rydberg-Stark surface deflector and merged with the ground-state beam. The merged beams then enter a time-of-flight mass spectrometer (TOFMS), where the reaction takes place and the product ions are extracted toward a microchannel plate (MCP) detector in a direction perpendicular to the merged-beam propagation axis. The TOFMS consists of three cylindrical electrodes (labelled E$_{1-3}$ in Fig.~\ref{Fig4}, with inner radius of 35.5 mm).
The central point of the TOFMS along the beam-propagation axis defines the origin of the space-fixed coordinate system. The helium valve is positioned at ($0,-102.5,-8.2$) cm (see Fig.~\ref{Fig4}). Its copper body is tilted by a 5$^{\circ}$ angle with respect to the vertical ($z$) axis and is thermally connected by copper braids to the cold heads of a two-stage pulse-tube cryo-cooler. The He valve is temperature-stabilised to $100.0\pm0.1$ K and operated at a stagnation pressure of 2.5 bar. At the valve opening, a pulsed potential ($\sim+300$ V, pulse duration $70$ $\mu$s) applied to a ring electrode attracts a beam of electrons emitted by a tungsten filament, creating an electric discharge at the valve orifice. In this discharge, the metastable (1s)(2s) $^3$S$_1$ state of helium (lifetime of $7.859\times10^3$ s)~\cite{lach01} is populated, amongst other states.\cite{halfmann00a} The metastable helium beam (referred to as He$^*$) travels at a velocity of $\sim1040$ m/s. After passing two skimmers, the He$^*$ beam intersects a pulsed laser beam ($\lambda_{\mathrm{exc}} \cong 260$ nm, pulse duration $\sim10$ ns, pulse energy $\sim$ 800 $\mu$J, beam diameter $\sim0.2$ cm), produced by frequency tripling the output of a Nd:YAG-pumped commercial dye laser using two $\beta$-barium-borate crystals. The excitation takes place between two parallel metallic plates separated by 0.25 cm used to generate a dc electric field, $\vec{F}_{\mathrm{exc}}$. The laser is polarised parallel to $\vec{F}_{\mathrm{exc}}$; its frequency and the value of $|\vec{F}_{\mathrm{exc}}|$ are tuned such that a one-photon excitation is driven to a low-field-seeking Rydberg-Stark state $(n,k,m_{\ell}) = (30,21,0)$ just below the Inglis-Teller limit [$n$ and $m_{\ell}$ are the principal and magnetic quantum numbers, respectively, and $k = -(n-m_{\ell}-1):2:(n-m_{\ell}-1)$].\cite{gallagher94a}
After excitation, the He$(n)$ atoms are loaded into an electric quadrupole trap formed by applying time-dependent potentials to several sets of electrodes located at the surface of a $\sim5$-cm-long, $50$-electrode surface Rydberg-Stark deflector (electrode dimensions $d_y = 0.5$~mm, $d_ x = 30$~mm, separated by $\Delta d_y=0.5$~mm in the $y$-dimension). The deflector electrodes are patterned onto the surface of a printed circuit board, which is attached to a curved substrate (see Fig. 1 of Ref.~\cite{zhelyazkova20}). By applying a sinusoidally-varying potential to each set of electrodes (see Ref.~\cite{allmendinger14a,zhelyazkova19a}), the trapped cloud of Rydberg atoms can be guided in a direction parallel to the surface of the chip. By setting a frequency chirp to these sinusoidally-varying potentials, one can also accelerate or decelerate the Rydberg atoms and precisely adjust their final velocity, $v_{\mathrm{Ryd}}$, after release from the trap at the end of the decelerator. When it leaves the quadrupole trap, the Rydberg-atom cloud has a cigar shape of radius $\sim0.5$ mm~\cite{zhelyazkova20} and length $\sim3$ mm in the $x$-dimension (estimated from Monte Carlo simulations), and travels in a direction parallel to the $y$-axis, which is also the axis of propagation of the GS beam. We estimate the temperature of the Rydberg cloud from Monte Carlo simulations and from experiments to be in the $0.1-0.3$ K range.\cite{zhelyazkova20} A baffle at the end of the deflector blocks the He$^*$ beam and all Rydberg atoms which have not been captured by the quadrupole trap but have travelled over the deflector in a straight line. We estimate that approximately 100 Rydberg atoms exit the surface deflector and enter the TOFMS at every experimental cycle (repetition rate 25 Hz). The He$(n)$ atom cloud has a width of $\approx1$ mm in the $y$-direction in the centre of the reaction zone.
The orifice of the GS-beam valve is positioned at $(0,-62,0)$ cm and has an orifice diameter of $\sim0.5$ mm. It is operated with pure ammonia (either \ce{^{14}NH3} or \ce{^{14}ND3}) kept at a stagnation pressure of about 1 bar. The valve is connected to an ice-filled reservoir and temperature-stabilised to $274.0\pm0.5$ K. Two fast ionisation gauges (FIGs) positioned after the TOFMS are used to determine the velocity distribution of the GS beam. The selected velocity of the \ce{NH3} (\ce{ND3}) beam, $v^{\mathrm{GS}}$, was measured to be 1190 m/s (1170 m/s). The GS beam valve is triggered such that the molecules travelling at $v^{\mathrm{GS}}$ arrive at the centre of the TOFMS at the same time, designated as $t_{\mathrm{c}}$, as the Rydberg atoms.
The background pressures in the regions of the apparatus labelled (i)-(iv) in Fig.~\ref{Fig4} rise from about $2 \times 10^{-7}$~mbar in all regions to about $2 \times 10^{-5}$~mbar in regions (i) and (ii), $7 \times 10^{-7}$~mbar in region (iii) and $4 \times 10^{-7}$~mbar in region (iv) when the pulsed valves are operated.
We use the TOFMS in two modes: (i) a product-ion-extraction mode with mass resolution and (ii) a He$(n)$ pulsed-field ionisation mode. In mode (i), the product ions are extracted by applying potentials of 1 kV and 0.5 kV to electrodes E$_1$ and E$_2$, respectively (see Fig.~\ref{Fig4}), through a commercial (home-built) high-potential switch, with a rise time of $\sim30$ ns. These potentials correspond to an electric field of 110 V/cm at the centre of the TOFMS. The ions are extracted toward an MCP detector with a phosphor screen (Photonis, diameter 4 cm), with its centre positioned at $(0,0,14.5)$ cm. The front face of the MCP is biased at a potential of $-1.96$ kV to allow for maximal sensitivity while avoiding saturation effects caused by the strong He$^+$ signal. The potentials E$_1$ and E$_2$ are optimised to achieve maximal ion-collection efficiency and mass resolution. To generate a homogeneous extraction electric field and a field-free flight tube, three meshes (M$_{1-3}$ in Fig.~\ref{Fig4}) were installed in the ion-extraction stack. The time interval $\Delta t_{r}$ during which the ion-molecule reaction is observed ($\Delta t_{\mathrm{r}} = 7$ $\mu$s in the experiments presented here) was defined by applying a prepulse of V$_1=125$ V to electrode E$_1$ (corresponding to $\sim25$ V/cm at the TOFMS centre) to sweep out all ions located in the extraction region. The value of V$_1$ was chosen to be high enough to sweep the ions formed prior to time $t_{\mathrm{c}}-\Delta t_{\mathrm{r}}$ out of the ion-extraction zone, but not high enough to field-ionise the reactant He$(n)$ atoms.
When the TOFMS is operated in He$(n)$-detection mode [mode (ii)], a high-electric-field pulse is applied to field-ionise the Rydberg helium atoms and extract the \ce{He+} ions towards the MCP detector. The ionisation electric field [$\sim1100$ V/cm at point $(0,0,0)$] is generated by applying 5 kV to E$_1$ while keeping electrodes E$_2$ and E$_3$ grounded. This field is sufficient to ionise Rydberg atoms of principal quantum number $n\geq29$.
\subsection{Varying the velocity of the He$(n)$ beam}
We measure the dependence of the product-ion yields on the collision energy $E_{\mathrm{coll}}$ by varying the central velocity of the Rydberg atoms, $v_{\mathrm{Ryd}}$, and keeping the velocity $v^{\mathrm{GS}}$ of the GS beam constant. The short gas-pulse duration leads to a large spatial dispersion of the GS beam over the 62-cm distance between the valve orifice and the reaction zone. The particles in the GS beam thus have a well-defined velocity (with a velocity spread $\Delta v_{\mathrm{GS}} = \pm 15$ m/s) in the region where it overlaps with the He($n$) packet emerging from the chip. $v_{\mathrm{Ryd}}$ is set by applying the appropriate potentials to the surface deflector. The He$(n)$ atoms have an estimated velocity spread of full width at half maximum (FWHM) around $v_{\mathrm{Ryd}}$ of $\Delta v^{\mathrm{FWHM}}_{\mathrm{Ryd}}\approx\pm 20$ m/s.
To ensure that the same velocity class of the ammonia GS molecules interacts with the He$(n)$ atoms regardless of the selected $v_{\mathrm{Ryd}}$ value, we perform a set of measurements in which we determine the arrival time of the Rydberg-atom cloud at the centre of the reaction zone, $t_{\mathrm{c}}^{v_{\mathrm{Ryd}}}$, for every value of $v_{\mathrm{Ryd}}$. The He$(n)$ atoms can be detected if (i) the applied ionisation field is sufficiently high for pulsed field ionisation (PFI), and (ii) PFI takes place in a region inside the TOFMS such that the produced ions can reach the MCP and do not collide with the walls of the extraction stack. To obtain the data presented in inset (b) of Fig.~\ref{Fig4}, we operate the TOFMS in a He$(n)$-ionisation (single-pulse) mode and record the integrated He$^+$ signal as a function of the time of application of the PFI pulse relative to the laser excitation time, $t_{\mathrm{PFI}}$, for each set value of $v_{\mathrm{Ryd}}$. The dashed-dotted black curve represents the He$^+$ signal measured when operating the surface deflector in guiding mode, i.e., with the He$(n)$ atoms propagating at constant velocity over the deflector surface ($v^{\mathrm{guide}}_{\mathrm{Ryd}} = 1040$ m/s). This curve has an approximately Gaussian shape with a centre at $t_{\mathrm{c}}^{1040\;\mathrm{m/s}}=85.5$ $\mu$s and a FWHM of $\sim17$ $\mu$s.
By applying the appropriate frequency chirp to the time-dependent electrode potentials, we can accelerate or decelerate the He$(n)$ atoms from an initial velocity of 1040 m/s to a final velocity in the $1200-750$ m/s range during deflection. The maximum (minimum) velocity in this range corresponds to an increase (decrease) of the initial He$(n)$ kinetic energy of $\sim33$\% ($\sim48$\%). The value of $t_\mathrm{c}^{v_{\mathrm{Ryd}}}$ is determined for each of the selected values of $v_{\mathrm{Ryd}}$ by determining the centres of the corresponding distributions depicted in Fig.~\ref{Fig4}(b). In addition, the spread of the measured He$(n)$ profile increases for lower values of $v_{\mathrm{Ryd}}$ because the He($n$) packet expands over a longer flight time. The GS valve opening time is then adapted so that the centre of the GS beam overlaps with the deflected Rydberg cloud at $t_{\mathrm{c}}$.
\subsection{Normalising the measured product ion signal}
\label{subsec:norm}
In the experiment, we do not measure absolute values of the reaction rates because we do not accurately know the particle densities in the reaction zone. We measure relative values of the collision-energy-dependent rates, which we scale by a global factor for comparison with the calculated capture-rate coefficients.
The total amount of He$(n)$ that reaches the centre of the TOFMS decreases with increasing deceleration and acceleration. This decrease is illustrated by the decrease of the maximum measured height of the He$(n)$ PFI profiles
at time $t_{\mathrm{c}}$ in Fig.~\ref{Fig4}(b). For example, only 56\% and 47\% of the atoms reach the centre of the stack for $v_{\mathrm{Ryd}} = 1200$ m/s and 750 m/s, respectively, compared to $v_{\mathrm{Ryd}} = 1040$ m/s. This decrease is attributed to the opening of the quadrupole trap in the reference frame of the moving atoms,\cite{allmendinger14a} which leads to a larger fraction of the atoms escaping the trap at the highest acceleration/deceleration. For the lower values of $v_{\mathrm{Ryd}}$, additional losses result from the finite lifetimes of the Rydberg atoms. In order to correct for these effects, the integrated reaction-product signals are divided by the detected He$^+$ signal, integrated over the [$ t_{c}-\Delta t_{\mathrm{r}}/2, t_c+\Delta t_{\mathrm{r}}/2$] time interval, accounting for the times of flight of the ions to the MCP.
The different times of flight of the Rydberg cloud from the end of the deflector to the TOFMS centre correspond to different expansions in the plane perpendicular to the direction of propagation (the $xz$-plane). The ground-state-beam spatial extent is limited to 1.6 mm in the $z$-direction by the position of the baffle above the surface of the chip. The He$(n)$-expansion can thus affect the geometric overlap with the GS beam, creating a detection bias in favour of product ions generated at higher values of $v_{\mathrm{Ryd}}$. Monte-Carlo particle-trajectory simulations indicate that for the lowest values of $v_{\mathrm{Ryd}}$ ($v_{\mathrm{Ryd}}\lesssim700$ m/s), the cloud has expanded so much by the time it arrives at the stack centre that the Rydberg atoms located at its outer edge do not interact with the GS-beam molecules. This detection bias is corrected for when determining the collision-energy-dependent product yields.
\section{Experimental Results}
\label{sec:results}
Ammonia is an oblate symmetric top molecule. It has C$_{3v}$ point-group symmetry in its two equivalent \say{umbrella} equilibrium structures located at values of $\sim\pm22^{\circ}$ of the umbrella inversion angle. These structures are separated by a relatively low energy barrier ($\sim2023$ cm$^{-1}$)\cite{herzberg91a} for inversion through the D$_{3h}$ planar structure. Tunnelling through the potential-energy barrier thus leads to a splitting of $\sim0.79$ cm$^{-1}$ ($\sim0.053$ cm$^{-1}$) of the ground vibrational state rotational levels in \ce{NH3} (\ce{ND3}), the vibrational eigenfunctions being the symmetric ($s$) and antisymmetric ($a$) superpositions of the ground-state wavefunctions in each of the potential wells.
The ammonia molecule has a dipole moment of 1.468 D at the C$_{3v}$ equilibrium geometry corresponding to a partial negative charge on the nitrogen side of the molecule, and a partial positive charge on the hydrogen side.
The ammonia molecule can be described in the D$_{3h}$ point group, or the isomorphic permutation-inversion group S$_3^*$, with rovibrational energy states labelled by the irreducible representations: A$_1'$/A$_1^+$, A$_2'$/A$_2^+$, A$_1''$/A$_1^-$, A$_2''$/A$_2^-$, E$'$/E$^+$ and E$''$/E$^-$ in D$_{3h}(M)$/S$_3^*$.\cite{wichmann20,snels00,urban84,bunker06} According to the Pauli exclusion principle, the total internal wavefunction in \ce{NH3} (\ce{ND3}) must have symmetry of either A$_2^+$ or A$_2^-$ (A$_1^+$ or A$_1^-$), i.e., it must be antisymmetric (symmetric) with respect to pairwise exchange of the identical fermions (bosons). The nuclear-spin states of \ce{^{14}NH3} and \ce{^{14}ND3} span the representations:
\begin{align}
\Gamma^{\mathrm{NH_3}}_{\mathrm{ns}} &= 12\mathrm{A}_2^+\oplus 6 \mathrm{E}^+ \\
\Gamma^{\mathrm{ND_3}}_{\mathrm{ns}} &= 30\mathrm{A}_1^+\oplus 3 \mathrm{A}_2^+ \oplus 24 \mathrm{E}^+,
\label{GammaNS}
\end{align}
and the allowed rovibronic states span the representations:
\begin{align}
\Gamma^{\mathrm{NH_3,sw}}_{\mathrm{rve}} &= 12\mathrm{A}_2^+\oplus12\mathrm{A}_2^-\oplus 6 \mathrm{E}^+\oplus 6 \mathrm{E}^- \\
\Gamma^{\mathrm{ND_3,sw}}_{\mathrm{rve}} &= 30\mathrm{A}_1^+ \oplus 30\mathrm{A}_1^- \oplus 3\mathrm{A}_2^+\oplus 3\mathrm{A}_2^-\oplus 24 \mathrm{E}^+\oplus 24 \mathrm{E}^-.
\label{GammaNS2}
\end{align}
Rovibrational states of symmetries A$_1^+$ and A$_1^-$ have a statistical weight of zero and are missing in \ce{NH3}. The rovibrational statistical weights of the $(J,K)$ states in \ce{NH3} and \ce{ND3} are presented in Table~\ref{tab1} (see Refs.~ \cite{wichmann20,snels00,bunker06} for details), and the rotational-level structures are presented in Fig.~\ref{Fig1}. The two components of the inversion doublet have either A$_1^+$, A$_2^+$ or E$^+$ (positive parity) or A$_1^-$, A$_2^-$ or E$^-$ (negative parity) rovibrational symmetry. In both \ce{NH3} and \ce{ND3}, states with $K\;\mathrm{mod}\;3 = 0$ have A symmetry and states with $K\;\mathrm{mod}\;3\neq 0$ have E symmetry. The lowest states of A and E symmetry are the $(J,K) = (0,0)$ and $(J,K) = (1,1)$ states, respectively.
Because of the missing A$_1$ levels, the inversion splitting is not observed in \ce{NH3} states with $K = 0$. Instead, this manifold consists of an alternating series of negative - positive parity $J$ states (even/odd $J$ states are of A$_2^-$/A$_2^+$ symmetry).
\begin{figure*}[!h]
\includegraphics[width = 0.5\textwidth]{NH3_Rot_levels.pdf}
\includegraphics[width = 0.5\textwidth]{ND3_Rot_levels.pdf}
\caption{\label{Fig1} Rotational energy levels and symmetry labels in the lowest vibrational level of the ground electronic state of \ce{NH3} and \ce{ND3} in the S$_3^*$ permutation inversion group, for $|K|\leq2$. The numbers in parentheses indicate the spin-statistical weights $g_{\mathrm{ns}}$, the $l$ and $u$ letters indicate the lower and upper components of the inversion doublets.\cite{wichmann20} The rotational constants are taken from Ref.~\cite{urban84} (\ce{NH3}) and Ref.~\cite{daniel16} (\ce{ND3}).}
\end{figure*}
\begin{table}[]
\caption{\ Rovibronic spin statistical weights $g_{\mathrm{ns},i}$ for NH$_3$ and ND$_3$ in the S$_3^*$ permutation inversion group. Adapted from Refs.\cite{wichmann20,snels00}}
\label{tab1}
\begin{tabular*}{0.48\textwidth}{@{\extracolsep{\fill}}|lllll|}
\hline
& Symmetry (rve) & $J$ & $K$ & $g_{\mathrm{ns},i}$ \\[0.8ex] \hline
NH$_3$ & $s$; A$_1^+$ & even & 0 & 0 \\[0.5ex]
& $a$; A$_1^-$ & odd & 0 & 0 \\[0.5ex]
& $s$; A$_2^+$ & odd & 0 & 12 \\[0.5ex]
& $a$; A$_2^-$ & even & 0 & 12 \\[0.5ex]
& $s$, $a$; E$^{\pm}$ & all & 1, 2, 4, 5, $\ldots$ & 6 \\[0.5ex]
& $s$, $a$; A$_1^{\pm}$ & all & 3, 6, 9, 12, $\ldots$ & 0 \\[0.5ex]
& $s$, $a$; A$_2^{\pm}$ & all & 3, 6, 9, 12, $\ldots$ & 12 \\[0.5ex] \hline
ND$_3$ & $s$; A$_1^+$ & even & 0 & 30 \\[0.8ex]
& $a$; A$_1^-$ & odd & 0 & 30 \\[0.5ex]
& $s$; A$_2^+$ & odd & 0 & 3 \\[0.5ex]
& $a$; A$_2^-$ & even & 0 & 3 \\[0.5ex]
& $s$, $a$; E$^{\pm}$ & all & 1, 2, 4, 5, $\dots$ & 24 \\[0.5ex]
& $s$, $a$; A$_1^{\pm}$ & all & 3, 6, 9, 12, $\ldots$ & 30 \\[0.5ex]
& $s$, $a$; A$_2^{\pm}$ & all & 3, 6, 9, 12, $\ldots$ & 3 \\ [0.5ex]\hline
\end{tabular*}
\end{table}
At the typical rotational temperature of the molecules produced in our supersonic-beam source ($T_{\mathrm{rot}}\simeq 6$ K) and for the rotational constants of \ce{NH3} and \ce{ND3} ($A_{\mathrm{NH_3}} = B_{\mathrm{NH_3}} = 9.444$ cm$^{-1}$, $C_{\mathrm{NH_3}} = 6.196$ cm$^{-1}$; $A_{\mathrm{ND_3}} = B_{\mathrm{ND_3}} = 5.143$ cm$^{-1}$, $C_{\mathrm{ND_3}} = 3.125$ cm$^{-1}$),\cite{urban84,daniel16} only states with $K = 0$ and $|K| = 1$ are significantly populated. Taking into account the nuclear-spin symmetry, the ratio of states with $K = 0$ to states with $|K| = 1$ is $1:1$ and $11:16$ in \ce{NH3} and \ce{ND3}, respectively. The occupation probabilities of the different ground-state rotational levels at a rotational temperature of $T_{\mathrm{rot}}=6$ K are listed in Table~\ref{tab2}.
\begin{table}[]
\caption{Occupation probabilities of the rotational levels of the NH$_3$ and ND$_3$ at a rotational temperature of $T_{\mathrm{rot}} = 6$ K. }
\label{tab2}
\begin{tabular*}{0.48\textwidth}{@{\extracolsep{\fill}}|ll|l|l|}
\hline
& & NH$_3$ & ND$_3$ \\ \hline
$K = 0$ & $J = 0$ ($l$) & 0 & 0.295 \\
& $J = 0$ ($u$) & 0.484 & $2.94\times 10^{-2}$ \\
& $J = 1$ ($l$) & $1.57\times 10^{-2}$ & $7.58\times 10^{-3}$ \\
& $J = 1$ ($u$) & 0 & $7.42\times 10^{-2}$ \\ \hline
$|K| = 1$ & $J = 1$ ($l$) & 0.274 & 0.295 \\
& $J = 1$ ($u$) & 0.226 & 0.291 \\
& $J = 2$ ($l$) & $5.29\times 10^{-5}$ & $3.49\times 10^{-3}$ \\
& $J = 2$ ($u$) & $4.39\times 10^{-5}$ & $3.54\times 10^{-3}$ \\ \hline
\end{tabular*}
\end{table}
Displayed in Fig.~\ref{Fig5} are reaction-product time-of-flight (TOF) traces measured following reactions between He$(n)$ atoms and \ce{NH3}(a) and \ce{ND3}(b) molecules, after a reaction time of $\Delta t_{\mathrm{r}} = 7$ $\mu$s. The traces were recorded with the He$(n)$ atoms accelerated to a final velocity of $v_{\mathrm{Ryd}} = 1050$ m/s, corresponding to collisional energies $E_{\mathrm{coll}}\approx k_{\mathrm{B}}\cdot3.8$ K (a) and $k_{\mathrm{B}}\cdot2.8$ K (b). When the laser used to photoexcite He$^*$ to He$(n)$ is on (black traces in Fig.~\ref{Fig5}), a prominent peak is visible at 1.6 $\mu$s, corresponding to the \ce{He^+} ions generated by field ionisation of the He$(n)$ atoms. The extraction electric field of 110 V/cm is not sufficient to ionise the initially prepared Rydberg helium atoms with $n=30$ (this field can only ionise states with $n\gtrsim50$). However, $n$-changing blackbody-radiation-induced transitions during the $\sim85$ $\mu$s flight time of the He$(n)$ atoms from the photoexcitation region to the centre of the TOFMS populate a few higher-lying Rydberg states, which can then be field ionised with the extraction field. A constant background signal of $\sim0.7$ mV is also visible in the TOF traces in Fig.~\ref{Fig5} beyond $1.6$ $\mu$s. This background signal originates from a constant stream of He$^+$ produced either through blackbody-radiation-induced ionisation or slow tunnelling ionisation of the He$(n)$ Rydberg states in the extraction field.
In addition to the He$^+$ peak, several other peaks are visible in the mass spectrum and can be assigned to (a) \ce{NH+}, \ce{NH2+}, \ce{NH3+}/\ce{OH+}, \ce{H2O+} and \ce{N2+}, and (b) \ce{ND+}, \ce{OH+}, \ce{ND2+}/\ce{H_2O+}, \ce{ND3+} and \ce{N2+}, as indicated in Fig.~\ref{Fig5}. The main product ions observed following the He$(n)$ + \ce{NH3} (\ce{ND3}) reaction are \ce{NH2+} and \ce{NH+} (\ce{ND2+} and \ce{ND+}). The \ce{H2O+}, \ce{OH+} and \ce{N2+} ions originate from Penning-ionisation processes involving metastable He$^*$ atoms produced in the discharge and background water and nitrogen present in the GS beam. The corresponding mass peaks in the TOF spectra are indeed observed even when the excitation laser is turned off (orange traces in Fig.~\ref{Fig5}). We have verified that the \ce{NH+}, \ce{NH2+}, \ce{ND+} and \ce{ND2+} product ions detected are generated by autoionisation of the corresponding Rydberg-molecule products.
The comparison of the TOF mass spectra recorded with the excitation laser turned on and off illustrates that the peaks corresponding to the \ce{OH+}, \ce{NH3+}, \ce{ND3+}, \ce{H2O+} and \ce{N2+} ions originate mostly from Penning ionisation and that the \ce{NH^+}, \ce{ND^+}, \ce{NH_2^+} and \ce{ND_2^+} ions are only detected in significant quantities from reactions with He$(n)$.
\begin{figure}
\includegraphics[trim = 1cm 0cm 1cm 0cm, clip,width = 0.5\textwidth]{He_NH3_products.pdf}
\includegraphics[trim = 1cm 0cm 1cm 0cm, clip,width = 0.5\textwidth]{He_ND3_products.pdf}
\caption{\label{Fig5} Measured product-ion time-of-flight spectra after a $7$-$\mu$s-long reaction-observation time for the (a) He$(n)$ + \ce{NH3} and (b) He$(n)$ + \ce{ND3} reactions. The He atoms are initially excited to the $(n,k,m) = (30,21,0)$ Rydberg-Stark state. The Rydberg-atom velocity is $v_{\mathrm{Ryd}}= 1050$ m/s. The orange traces are recorded with the Rydberg-excitation laser turned off. The coloured text boxes indicate the measured product ions which are specific to the reactions between the He($n$) atoms and the ammonia molecules (see text for details). The insets show the reaction products after subtraction of the signals generated by Penning-ionisation processes.}
\end{figure}
To measure the collision-energy dependence of the product yields, we record the integrated \ce{NH+}, \ce{ND+}, \ce{NH_2+} and \ce{ND_2+} product signals for several values of the He$(n)$ velocity, $v_{\mathrm{Ryd}}$, in the $750-1200$ m/s range, keeping the GS-beam velocity constant. The results of these measurements are presented in Fig.~\ref{Fig6}(a) and (b). The range of He$(n)$ velocities corresponds to collisional energies of $E_{\mathrm{coll}}/k_{\mathrm{B}}\lesssim40$ K. In order to subtract contributions from Penning-ionisation products from the He$(n)$ + \ce{NH3}/\ce{ND3} reaction products, mass spectra were recorded without the Rydberg-excitation laser and then subtracted prior to integration (see insets of Fig.~\ref{Fig5}). The integration of the light (\ce{NH+} and \ce{ND+}) and heavy (\ce{NH2^+} and \ce{ND2^+}) reaction product ions, respectively, was performed in temporal windows indicated by the red and blue shaded areas in the insets in Fig.~\ref{Fig5}. The measured integrated product-ion signals for each value of $v_{\mathrm{Ryd}}$ were further normalised following the procedure described in Sec.~\ref{subsec:norm}. The error bars are obtained from five sets of consecutive measurements, each corresponding to an average over 1000 experimental cycles.
The integrated normalised product-ion signals $I_{\ce{NH2+}}$ and $I_{\ce{ND2+}}$ are depicted as a function of the He$(n)$ velocity using blue dots in Fig.~\ref{Fig6} (a) and (b). They show a pronounced increase as the He$(n)$ velocity, $v_{\mathrm{Ryd}}$, approaches the GS-beam velocity, which is designated by the dashed vertical lines. This increase is more pronounced for the He$(n)$ + \ce{ND3} reaction than for the He$(n)$ + \ce{NH3} reaction: for example, $I_{\ce{NH2+}}(1200\;\mathrm{m/s})/I_{\ce{NH2+}}(750\;\mathrm{m/s})\approx 2.21$, while $I_{\ce{ND2+}}(1170\;\mathrm{m/s})/I_{\ce{ND2+}}(750\;\mathrm{m/s})\approx 4.72$. The measured values of $I_{\ce{NH+}}$ and $I_{\ce{ND+}}$ [red circles in Fig.~\ref{Fig6} (a) and (b)] are smaller than the measured values of $I_{\ce{NH2+}}$ and $I_{\ce{ND2+}}$ by a factor of $f_{\ce{NH+}}=6\pm1$ and $f_{\ce{ND+}}=4.0\pm0.6$, respectively. The measured $\ce{NH2+}:\ce{NH+}$ branching ratio is slightly lower than the one reported in Ref.~\cite{kim75}. While in previous measurements the \ce{NH3+} product ions constituted approximately 12\% of the total reaction yield,\cite{kim75} in our experiments all \ce{NH3+}/\ce{ND3+} ions are primarily generated by Penning-ionisation processes and the \ce{NH3+} and \ce{ND3+} signals obtained after subtraction of the Penning-ionisation contributions are too weak and noisy for reliable estimates of the branching ratios to be carried out.
\begin{figure*}
\includegraphics[trim = 1cm 0cm 1cm 0cm, clip,width = 0.5\textwidth]{He_NH3_int_sig_v2.pdf}
\includegraphics[trim = 1cm 0cm 1cm 0cm, clip,width = 0.5\textwidth]{He_ND3_int_sig_v2.pdf}
\includegraphics[trim = 4.0cm 2cm 4.0cm 3cm, clip,width = 0.5\textwidth]{NH3_model_data.pdf}
\includegraphics[trim = 4.0cm 2cm 4.0cm 3cm, clip,width = 0.5\textwidth]{ND3_model_data.pdf}
\caption{\label{Fig6} (a) and (b) Measured integrated product ion signals, $I_{\ce{NH+}}$, $I_{\ce{NH2+}}$, $I_{\ce{ND+}}$ and $I_{\ce{ND2+}}$ (in arbitrary units) as a function of the Rydberg velocity, $v_{\mathrm{Ryd}}$, following reactions between He$(n)$ and either \ce{NH3} or \ce{ND3}. The vertical dashed lines indicate the GS-beam velocity. (c) and (d) Comparison of the scaled total measured product-ion yields for the two reactions, $I_{\ce{NH+}+\ce{NH2+}}$ and $I_{\ce{ND+}+\ce{ND2+}}$ (green circles with error bars in arbitrary units), to the state-averaged capture rate constants $k$ calculated assuming infinite resolution (orange and purple dots) and considering the finite experimental energy resolution (black lines). In panels (a) and (b), the pink data points correspond to the red data points with intensities multiplied by factors of 5.5 and 4.055, respectively. See text for details.}
\end{figure*}
\section{Calculation of the capture-rate coefficients and comparison with experimental data}
\subsection{The Stark effect in NH$_3$ and ND$_3$ and the He$^++$NH$_3$/ND$_3$ interaction potentials}
\label{sec:Vint}
The interaction between the molecular dipole moment and an electric field mixes states $|JKM\rangle\leftrightarrow |J'K'M\rangle$ with $J' = J, J\pm1$ (Eq.~(\ref{Multipole_op_ME}) with $\lambda = 1$). The electric-dipole selection rules are $\Delta J = 0, \pm1\;(0\not\leftrightarrow0)$, $\Delta K = 0$ and $\Delta M = 0$.
To describe the Stark effect in ammonia under the consideration of the inversion splitting, we expand the $|JKM\rangle$ basis set to include the inversion doublet sublevels, which have positive or negative parity. Since the rotational Hamiltonian is invariant under the parity operation, it is conventional to use the symmetrised rotational wavefunctions of well-defined parity $|J,K,M,(+/-)\rangle$. The Stark interaction mixes states of opposite parity and the non-vanishing Stark matrix elements in the symmetrised basis are:\cite{vandemeerakker12a}
\begin{equation}
\langle J'K'M'\pm|\widehat{H}_{\mathrm{Stark}}|JKM\mp\rangle = \langle J'K'M'|\widehat{H}_{\mathrm{Stark}}|JKM\rangle.
\end{equation}
The Stark effect in \ce{NH3} and \ce{ND3} was calculated by determining the eigenvalues of the Hamiltonian in Eq.~(\ref{Stark_Ham}) for values of the ion-molecule separation $R$ from 0.1 nm to 50 nm in steps of 0.1 nm (corresponding to electric fields ranging from $\sim1.4\times10^{6}$ kV/cm to $\sim5.7$ kV/cm, respectively). We use the rotational constants presented in Fig.~\ref{Fig1}.\cite{urban84,daniel16} Because only $J = 0$ and 1 levels are significantly populated in our experiments, we neglect centrifugal-distortion effects. Convergence of the eigenvalues of the $J = 0,1$ levels was reached with a basis set with a maximum value of $J_{\mathrm{max}} = 6$. The states are labelled according to their adiabatic limit for $R\rightarrow\infty$ corresponding to field-free conditions. The results of the calculation for the states with $J\leq1$, $|K|\leq1$ for electric fields in the $5-5000$ kV/cm range are presented in Fig.~\ref{Fig2}. Because of the smaller value of the tunnelling splitting and the smaller (by a factor of $\sim2$) values of the rotational constants, the Stark shifts are more pronounced in \ce{ND3} at lower fields. In addition, the Stark effect of the $J =1, |K|=1$ levels becomes linear earlier in \ce{ND3}.
We classify the Stark states of interest in ammonia in three types according to the energy shift they experience with increasing electric field: {\bf type A} -- strongly high-field-seeking states with a linear Stark shift, i.e., the lower-energy components of the inversion doublets with $J = 1$ and $|KM| = 1$; {\bf type B} -- high-field-seeking states with a quadratic Stark shift at low fields, e.g., the $|J = 0, K = 0,M=0,-\rangle$ and $|J = 1, K=0, M = 1,+ \rangle$ states; and {\bf type C} -- low-field-seeking states at relatively low fields which exhibit an energy maximum at intermediate values of the electric field before turning into high-field-seeking states at high fields, e.g., selected components of the inversion doublets with $J = 1$ and $|KM| = 1$.
States of type A comprise the lower components of the $J =1, |KM|=1$ inversion doublets in \ce{NH3} and \ce{ND3} (see Fig.~\ref{Fig2}). These states experience the strongest Stark shifts, $\Delta E_i$, in both \ce{NH3} and \ce{ND3}: $\Delta E^{\mathrm{(A)}}\sim-88$ cm$^{-1}$ in \ce{NH3} and $\sim-95$ cm$^{-1}$ in \ce{ND3} at 5000 kV/cm. States of type B with $J = 0$ (i.e., the $|0,0,0,-\rangle$ state in \ce{NH3} and the $|0,0,0,+\rangle$ and $|0,0,0,-\rangle$ states in \ce{ND3}) do not have a dipole moment at low fields, but acquire one through mixing with the $|1,0,0,\pm\rangle$ states in an electric field. In states of type C, the potential energy barriers occur at lower values of the electric field and have lower heights in \ce{ND3} than in \ce{NH3} (see Table~\ref{tab3}). At the lowest electric fields ($\lesssim80$ kV/cm), the inversion splitting of $\sim0.79$ cm$^{-1}$ in \ce{NH3} causes the $J =1, |KM| = 1$ states to undergo a quadratic Stark shift [inset of Fig.~\ref{Fig2}(a)]. In contrast, because of the much smaller inversion splitting of $\sim0.053$ cm$^{-1}$ in \ce{ND3}, the corresponding states experience a linear Stark shift, even at the lowest fields [first inset of Fig.~\ref{Fig2}(b)]. The bottom panels in Fig.~\ref{Fig2} show the electric-field-dependent dipole moments of the states depicted in the top panels.
\begin{figure*}[!h]
\includegraphics[trim = 1.75cm 0cm 1cm 0cm, clip, width = 0.5\textwidth]{NH3_StarkMap_2.pdf}
\includegraphics[trim = 1.75cm 0cm 1cm 0cm, clip, width = 0.5\textwidth]{ND3_StarkMap_2.pdf}
\caption{\label{Fig2} Top panels: The calculated Stark effect in the $J = 0,1$, $|K| = 0,1$ states of \ce{NH3} (a) and \ce{ND3} (b) in electric field of up to 5000 kV/cm. The Stark shifts at low fields are depicted on an enlarged scale in the insets. Bottom panels: The corresponding state-specific electric-field-dependent dipole moments. The top scale indicates the distance between He$^+$ and the ammonia molecule at the corresponding field values. The definition of the state categories A, B and C is provided in the text.}
\end{figure*}
\begin{table}[]
\caption{\label{tab3} Potential energy barriers in the Stark shifts of the $J = 1, |K| = 0,1$ low-field-seeking Stark states of NH$_3$ and ND$_3$.}
\begin{tabular*}{0.50\textwidth}{m{0.5cm}m{3.0cm}|m{1.0cm}m{2.75cm}}
\hline
& & Height & Position \\
&. &(cm$^{-1}$) & (kV/cm) \\ \hline
NH$_3$ & $J = 1, |KM| = 1\;(+)$ & 11.4 & 1836 ($R = 2.80$ nm) \\ [0.5ex]
& $J = 1, K = M = 0\; (+)$ & 6.03 & 1837 ($R = 2.79$ nm) \\[0.5ex] \hline
ND$_3$ & $J = 1, |KM| = 1\; (+)$ & 6.37 & 997.2 ($R = 3.80$ nm) \\[0.5ex]
& $J = 1, K = M = 0\; (+)$ & 3.32 & 1052 ($R = 3.70$ nm) \\[0.5ex]
& $J = 1, K = M = 0\; (-)$ & 3.25 & 997.2 ($R = 3.80$ nm) \\ [0.5ex] \hline\end{tabular*}
\end{table}
Because of the $R^{-2}$ dependence of the ion-dipole interaction, the $\Delta E_i(R)$ terms in Eq.~(\ref{eq:Vint}) dominate over the $R^{-4}$-dependent charge--induced-dipole interaction term for all states, as illustrated in Fig.~\ref{Fig3}. This figure compares the total interaction potentials, $V_{\mathrm{int},i}(R)$, for the states presented in Fig.~\ref{Fig2} for an s-wave collision ($L = 0$, solid lines) and a collision with $\ell = 20$ ($L^2 = \hslash^2\ell(\ell+1)$, dash-dotted lines), with $V_{\mathrm{L}}$ (black solid and dash-dotted lines, respectively).
The Stark shifts significantly modify the potential barriers, and thus affect the value of $L_{\mathrm{max},i}$ for each collisional energy (see Eq.~(\ref{expr_k})). In states of type A and B, the centrifugal barrier is substantially lowered compared to the centrifugal barrier in $V_{\mathrm{L}}$, the effect being significantly more pronounced in states of type A. For example, the $\ell = 20$ term creates a centrifugal barrier of $\sim k_{\mathrm{B}}\cdot14$ K in the Langevin interaction potential [dash-dotted black line in Fig.~\ref{Fig3}(a)], whereas the total interaction potentials for the same value of $\ell$ in the A states are strongly attractive. The Stark effect in states of type A and B thus effectively suppresses the centrifugal barriers, which leads to significantly larger value of $L_{\mathrm{max},i}^{\mathrm{(A,B)}}$ and to much higher capture rate coefficients $k^{(\mathrm{A},\mathrm{B})}_i$. As the collisional energy increases, the $E_{\mathrm{coll}}^{-1/2}$-dependence of $k_i(E_{\mathrm{coll}})$ [see Eq.~(\ref{expr_k})] leads to a decrease of the rate coefficients. States of type C, in contrast, are characterised by a non-zero potential-energy barrier even for $\ell = 0$. This $\ell = 0$ potential-energy barrier is designated as $V_{\mathrm{int}}^{\mathrm{(C),\;max}}$ in Fig.~\ref{Fig3}(a) and (b). For molecules in states of type C, the ion-molecule pair experiences a repulsive potential and the capture rate coefficients vanish for $E_{\mathrm{coll}}<V_{\mathrm{int}}^{\mathrm{(C),\;max}}$.
\begin{figure*}[!h]
\includegraphics[trim = 0.5cm 0cm 1cm 0cm, clip, width = 0.5\textwidth]{NH3_V_int.pdf}
\includegraphics[trim = 0.5cm 0cm 1cm 0cm, clip, width = 0.5\textwidth]{ND3_V_int.pdf}
\includegraphics[trim = 0.5cm 6.5cm 1cm 6.5cm, clip, width = 0.5\textwidth]{NH3_rates.pdf}
\includegraphics[trim = 0.5cm 6.5cm 1cm 6.5cm, clip, width = 0.5\textwidth]{ND3_rates.pdf}
\includegraphics[trim = 0.5cm 7.5cm 1cm 8.5cm, clip, width = 0.5\textwidth]{NH3_rates_long.pdf}
\includegraphics[trim = 0.5cm 7.5cm 1cm 8.5cm, clip, width = 0.5\textwidth]{ND3_rates_long.pdf}
\caption{\label{Fig3} Top panels: The interaction potential, $V_{\mathrm{int},i}(R)$, of the \ce{He+ + NH3}(a) and \ce{He+ + ND3}(b) reactions, for the molecular states with $J = 0,1$ and $|K| = 0,1$. The solid and dash-dotted coloured lines correspond to values of the angular momentum of the collision $L = \hslash\sqrt{\ell(\ell+1)}$ with $\ell = 0$ and $\ell = 20$, respectively. Black solid and dash-dotted lines represent the Langevin interaction potential, $V_{\mathrm{L}}(R)$. Middle and bottom panels: Calculated rotational-state-specific capture rate coefficients for $E_{\mathrm{coll}}/k_{\mathrm{B}}$ in the $0.05-40$ K (c,d) and $0.1-2.4\times10^{4}$ K (e,f) range. The horizontal black lines in (c-f) represent the value of the Langevin rate coefficient, $k_{\mathrm{L}}$, for the two reactions.}
\end{figure*}
\subsection{The rotational-state-dependent capture rate coefficients}
\label{sec:krates}
The calculated state-dependent capture rate coefficients, $k_i(E_{\mathrm{coll}})$, for the \ce{He+}+$\mathrm{NH_3}(J,K,M,\pm)$ and \ce{He+}+$\mathrm{ND_3}(J,K,M,\pm)$ reactions with $J = 0,1$ and $|K| = 0,1$ are displayed in Fig.~\ref{Fig3}(c) and (d). The calculations were performed for collisional energies $E_{\mathrm{coll}}/k_{\mathrm{B}}$ in the $0.05-40$ K range, with a step size of 50 mK. For states of type A and B, the rate coefficients are significantly higher than the Langevin rate constant $k_{\mathrm{L}}$ (black horizontal lines in the lower insets) and increase with decreasing collisional energy. The differences resulting from the quadratic Stark shift of B-type states and the linear Stark shift of the A-type states become apparent at collisional energies $E_{\mathrm{coll}}/k_{\mathrm{B}}\lesssim20$ K. Both A- and B-type states experience a gradual increase of $k$ in the $E_{\mathrm{coll}}/k_{\mathrm{B}}=20-40$ K range. However, below $E_{\mathrm{coll}}/k_{\mathrm{B}}\approx20$ K the $k^{\mathrm{(A)}}$ rate coefficients exhibit a much faster increase than the $k^{\mathrm{(B)}}$ rate coefficients. At the lowest collisional energies of $E_{\mathrm{coll}}/k_{\mathrm{B}}= 50$ mK, the rate coefficients of A-type states reach values of $k^{\mathrm{(A),He^++ND_3}} = 93.7\;k_{\mathrm{L}}$ and $k^{\mathrm{(A),He^++NH_3}} = 50.9\;k_{\mathrm{L}}$. The enhancement of the capture rate coefficients at $E_{\mathrm{coll}}/k_{\mathrm{B}}=50$ mK compared to $20$ K is $\sim1.1-1.3$ ($\sim1.2-1.6$) for type B states and $\sim4.5$ ($\sim6.8$) for type A states in \ce{NH3} (\ce{ND3}).
The $k^{\mathrm{(C)}}$ coefficients are zero for $E_{\mathrm{coll}}<V^{\mathrm{(C),max}}_{\mathrm{int}}$. For $E_{\mathrm{coll}}\geq V^{\mathrm{(C),max}}_{\mathrm{int}}$, the rate coefficients rapidly increase with $E_{\mathrm{coll}}$, and typically already exceed the value of $k_{\mathrm{L}}$ at $E_{\mathrm{coll}}/k_{\mathrm{B}}\approx10-20$ K [see the insets in Fig.~\ref{Fig3}(c) and (d)].
The behaviour of the capture rate coefficients in the high-collision-energy regime up to $E_{\mathrm{coll}}/k_{\mathrm{B}} = 2.4\times 10^4$ K is presented in Fig.~\ref{Fig3}(e) and (f). States of type A and B have rate coefficients which decrease monotonously with $E_{\mathrm{coll}}$, and approach the values of $k_{\mathrm{L}}$ for $E_{\mathrm{coll}}/k_{\mathrm{B}}\gtrsim10^4$ K. In the case of states of type C, in contrast, the rates reach a maximum at $E_{\mathrm{coll}}/k_{\mathrm{B}}\approx 100$ K and 60 K in the \ce{He+ +NH3} and \ce{He+ + ND3} reactions, respectively. Then they decrease again, eventually converging to the value of $k_{\mathrm{L}}$ at similar collision energies as the type-A and type-B states. Apparent in Fig.~\ref{Fig3}(e) and (f) is that the $k^{\mathrm{(B)}}$ rate coefficients increase with decreasing $E_{\mathrm{coll}}$, but eventually stabilise below $E_{\mathrm{coll}}/k_{\mathrm{B}}\lesssim5$ K at a value of $\sim7.5\;k_{\mathrm{L}}$ (for the $J = 1, KM = 0$ states) and $\sim13.5\;k_{\mathrm{L}}$ (for the $J=0$ states) in \ce{NH3} and $\sim10\;k_{\mathrm{L}}$ and $\sim18\;k_{\mathrm{L}}$ in \ce{ND3}, respectively. The greater increase of the $k^{\mathrm{(B)}}$ rates relative to $k_\mathrm{L}$ in \ce{ND3} is a consequence of the larger quadratic Stark-shifts of the B-type states at low fields.
In the experiments, we measure the relative collision-energy-dependent product yields, which reflect the initial molecular population distribution over the $J = 0,1$ $|K |=0,1$ rotational states of NH$_3$ and ND$_3$. However, because the A-type states experience a linear Stark shift and a large increase over $k_\mathrm{L}$ even at relatively high collisional energies (e.g., below $E_{\mathrm{coll}}/k_{\mathrm{B}}<40$ K), the overall rate coefficient is dominated by the contribution from these states, even though they are occupied only by a small fraction of the molecular population at $T_{\mathrm{rot}}=6$ K ($\sim 0.182$ in \ce{NH3} and $\sim 0.196$ in \ce{ND3}, see Fig.~\ref{Fig7} below).
\subsection{Modelling of the measured rate coefficients}
To model the experimentally measured relative product-ion signals as a function of the collision energy for the two reactions, $I_{\ce{NH+}+\ce{NH2+}}$ and $I_{\ce{ND+}+\ce{ND2+}}$, we proceed in the following way: In a first step, we calculate the weighted sum of the rotational-state-dependent capture rate constants, $k_i(E_{\mathrm{coll}})$, for a given rotational temperature, $T_{\mathrm{rot}}$, with weights determined by the corresponding state population. In a second step, we perform a convolution with a Gaussian distribution accounting for the collisional-energy-dependent energy-resolution, as described in Ref.~\cite{zhelyazkova20}, using the equation:
\begin{equation}
\frac{\Delta E_{\mathrm{coll}}}{k_{\mathrm{B}}} = \Delta T_{\mathrm{res}} + 2\sqrt{\Delta T_{\mathrm{res}}}\sqrt{\frac{E_{\mathrm{coll}}}{k_{\mathrm{B}}}},
\label{eq:ExpRes}
\end{equation}
where $\Delta E_{\mathrm{coll}}/k_{\mathrm{B}}$ is the FWHM of a Gaussian function. In Eq.~(\ref{eq:ExpRes}), $\Delta T_{\mathrm{res}}$ is an effective experimental temperature corresponding to the distribution of relative velocities of the He$(n)$ atoms and the ammonia molecules at zero nominal collisional energy (i.e., when $v_{\mathrm{Ryd}}-v_{\mathrm{c}}^{\mathrm{GS}}=0$). The value of $\Delta T_{\mathrm{res}}$ is mostly determined by the temperature of the Rydberg-atom cloud released from the quadrupole trap.\cite{zhelyazkova20} In a third step, we multiply the experimentally measured values of $I_{\ce{NH+}+\ce{NH2+}}$ and $I_{\ce{ND+}+\ce{ND2+}}$ by a global scaling factor such that the total integrated signals measured at the lowest collision energy probed experimentally matches the average rate constant determined in the second step at $E_{\mathrm{coll}}/k_{\mathrm{B}}=\Delta T_{\mathrm{res}}$. Finally, we vary $T_{\mathrm{rot}}$ and $\Delta T_{\mathrm{res}}$ until the best agreement between the experimental data and the model is obtained over the whole range of collisional energies probed.
The comparison between the experimental data and the ion-molecule capture-rate model is presented in Fig.~\ref{Fig6}(c) and (d). The best agreement was achieved for $T_{\mathrm{rot}} = 6$ K and $\Delta T_{\mathrm{res}} = 200$ mK for both reactions. The coloured (orange and purple) dots and the black curves are the rates obtained in the second and third steps of the procedure described above, respectively. The insets display the behaviour at low collisional energies ($E_{\mathrm{coll}}/k_{\mathrm{B}}\lesssim6$ K) on an enlarged scale. The agreement between the data and the model taking into account the finite resolution [black lines in Fig.~\ref{Fig6}(c) and (d)] is excellent for both reactions. The model thus quantitatively describes the experimentally observed increase of the product-ion signals with decreasing collisional energy. The overall behaviour exhibits a gradual, slow increase over the $E_{\mathrm{coll}}/k_{\mathrm{B}}\approx5-40$ K range and a much steeper increase for $E_{\mathrm{coll}}/k_{\mathrm{B}}\lesssim5$ K. The latter effect is significantly more pronounced for the \ce{He+ + ND3} reaction than for the \ce{He+ + NH3} reaction.
The calculated rate coefficients averaged over the rotational population of the molecules [purple and orange dots in Fig.~\ref{Fig6}(c) and (d)] and taking the finite experimental collision-energy resolution into account [black lines in Fig.~\ref{Fig6}(c) and (d)] are enhanced by factors of $\sim2.29$ ($\sim2.81$) at a collision energy of $k_{\mathrm{B}}\cdot200$ mK compared to the value at $E_{\mathrm{coll}}=k_{\mathrm{B}}\cdot40$ K and of $\sim1.71$ ($\sim2.09$) compared to the value at $E_{\mathrm{coll}}=k_{\mathrm{B}}\cdot 10$ K for the \ce{He+ + NH3} (\ce{He+ + ND3}) reaction.
\begin{figure*}
\includegraphics[trim = 6cm 3.4cm 6.5cm 3.7cm, clip,width = 0.5\textwidth]{NH3_st_pop.pdf}
\includegraphics[trim = 6cm 3.4cm 6.5cm 3.7cm, clip,width = 0.5\textwidth]{ND3_st_pop.pdf}
\caption{\label{Fig7} Occupation probabilities of the A-, B- and C-type of states in \ce{NH3} and \ce{ND3}, at a rotational temperature of $T_{\mathrm{rot}}=6$ K. See text for details.}
\end{figure*}
The observed larger increase with decreasing collision energy of the rates of the \ce{He+ + ND3} reaction is perfectly reproduced by the calculations and originates from (i) the larger rotational-state-dependent rate constants compared to the \ce{He+ + NH3} reaction, as discussed in Sec.~\ref{sec:krates}, and (ii) the higher contribution to the rate coefficients from type-A states in \ce{ND3} resulting from the nuclear-spin statistical factors, as is now explained.
The occupation probabilities $p$ of the A-, B- and C-type states in \ce{NH3} and \ce{ND3} at a rotational temperature of $T_{\mathrm{rot}}=6$ K are presented in Fig.~\ref{Fig7} and are $p^{\mathrm{(A),\ce{NH3}}} = 0.182$, $p^{\mathrm{(B),\ce{NH3}}} = 0.662$ and $p^{\mathrm{(C),\ce{NH3}}} = 0.156$ for \ce{NH3} and $p^{\mathrm{(A),\ce{ND3}}} = 0.196$, $p^{\mathrm{(B),\ce{ND3}}} = 0.580$ and $p^{\mathrm{(C),\ce{ND3}}} = 0.224$ for \ce{ND3}. The experimentally measured increase of product-ion signals with decreasing collisional energy can thus be almost entirely attributed to states of type A and B. The observed steeper increase below $E_{\mathrm{coll}}/k_{\mathrm{B}}\lesssim5$ K of the rate of the \ce{He+ + ND3} reaction compared to the rate of the \ce{He+ + NH3} reaction results from: (i) the more pronounced Stark effect in \ce{ND3} and (ii) the larger fraction of molecules in high-field-seeking states of type A: $p^{\mathrm{(A),\ce{ND3}}}/(p^{\mathrm{(A),\ce{ND3}}}+p^{\mathrm{(B),\ce{ND3}}}) = 0.255$ in \ce{ND3} compared to $p^{\mathrm{(A),\ce{NH3}}}/(p^{\mathrm{(A),\ce{NH3}}}+p^{\mathrm{(B),\ce{NH3}}}) = 0.216$ in \ce{NH3}.
\subsection{Thermal rate constants}
The good agreement between the experimental and calculated collision-energy-dependent rates (see Fig.~\ref{Fig6}) validates the model used to calculate the rotational-state-specific rate coefficients $k_i(E_{\rm coll})$ with $i = |JKM\pm1\rangle$. These coefficients can in turn be used to determine the thermal rate constants $k(T)$ with the relations~\cite{levine05}
\begin{equation}\label{thermal_rates1}
k(T) = \sum_i g_{{\rm ns},i} \exp{\left(-{\frac{E_{{\rm rot},i}-E_0}{k_{\rm B}T}}\right)}k_i(T),
\end{equation}
and
\begin{equation}\label{thermal_rates2}
k_i(T) = \sqrt{\frac{8k_{\rm B}T}{\pi\mu}} \int_0^\infty \sigma_i(E_{\rm coll}) \left(\frac{E_{\rm coll}}{k_{\rm B}T}\right)\exp{\left(-{\frac{E_{\rm coll}}{k_{\rm B}T}}\right)} {\rm d}\left(\frac{E_{\rm coll}}{k_{\rm B}T}\right),
\end{equation}
where the state-specific reaction cross sections $\sigma_i(E_{\rm coll})$ are given by $k_i(E_{\rm coll})/|v_{\rm rel}|$ and $v_{\rm rel}$ is the relative velocity $v_{\rm Ryd}-v^{\rm GS}$. In Eq.~(\ref{thermal_rates1}), $g_{{\rm ns},i}$ represents the nuclear-spin-statistical factors (see Table~\ref{tab1}) and $E_{{\rm rot},i}$ and $E_0$ are the rotational energy of state $i$ and the zero-point energy, respectively.
In an environment where conversion between the two nuclear-spin isomers is possible, $E_0$ is taken as the energy of the $J=0,K=0$ rotational ground state. If the nuclear-spin-symmetry is conserved, $E_0$ represents the rotational energy of the ground state of the respective nuclear-spin isomer, i.e., $E_0$ is the energy of the $(J=0,K=0)^u$ ground state for ortho NH$_3$ and ND$_3$, the energy of the lower tunnelling component of the $(J=1,K=1)$ rotational level for para
NH$_3$ and ND$_3$, and the energy of the $(J = 0, K = 0)^l$ for meta \ce{ND3} (see Fig.~\ref{Fig1}). The rate constants calculated assuming that the relative populations of the ortho, para and meta nuclear-spin isomers correspond to the nuclear-spin-statistical factors are presented in
panels (a) and (b) of Fig.~\ref{Fig8} for NH$_3$ and ND$_3$, respectively. In this figure, the full lines correspond to the case where the nuclear spin is assumed to be conserved and the dashed lines to the case where ortho-para conversion is assumed to be possible.
Whereas the nuclear-spin symmetry is known to be conserved to an excellent approximation in low-density gases and supersonic expansions,\cite{wichmann20,tanner12} conversion between the different nuclear-spin isomers may take place if the ammonia molecules are adsorbed at the surface of certain materials and dust grains. The insets in Fig.~\ref{Fig8} give the relative population of the different rotational levels for these two cases, labelled nuclear-spin-conservation (NSC) and nuclear-spin-relaxation (NSR), respectively. In both cases, the thermal capture rate is enhanced as the temperature decreases. However, the enhancement is more pronounced if the nuclear-spin symmetry is conserved because, in this case, there is always a significant population of the high-field-seeking tunnelling component of the $J=1, K=1$ state, even at 0~K.
\begin{figure*}
\includegraphics[trim = 1cm 0cm 1cm 0cm, clip,width = 0.5\textwidth]{NH3_k_thermal_v2.pdf}
\includegraphics[trim = 1cm 0cm 1cm 0cm,width = 0.5\textwidth]{ND3_k_thermal_v2.pdf}
\caption{\label{Fig8} Calculated thermal rate coefficients for the \ce{He^+ + NH3}(a) and \ce{He^+ + ND3}(b) reactions in the temperature range between 0.2~K and 10~K. The calculations with nuclear-spin-symmetry conservation (NSC) and nuclear-spin-symmetry relaxation (NSR) are presented as solid and dashed black curves, respectively. Insets: rotational states population of selected states in \ce{NH3} and \ce{ND3} as a function of the temperature, assuming nuclear-spin-symmetry conservation (solid coloured lines) and nuclear-spin-symmetry relaxation effects (dashed coloured lines).}
\end{figure*}
Equations~(\ref{thermal_rates1}) and~(\ref{thermal_rates2}) can be used to calculate the thermal capture rate coefficients at the temperatures of 27, 68, and 300~K for which the rate of the He$^+$ + NH$_3$ reaction was measured in uniform supersonic flows.\cite{marquette85a} The results are presented in Table~\ref{table4}, where the first and second rows of the body of the table list the rates calculated assuming that the nuclear-spin symmetry is conserved (NSC) or relaxed (NSR), respectively. Beyond 10~K, the state populations do not strongly depend any more on whether NSR or NSC is assumed, so that the thermal rate constant with both assumption are almost identical. The third row gives the experimental values reported by Marquette {\it et al.}\cite{marquette85a} The calculated rates are larger than the experimental ones, which is expected given that the calculated capture rates correspond to 100\% reaction probability upon close encounter of the reactants whereas the experiments only monitor the reactive collisions. The comparison of experimental and calculated values of the thermal rate coefficient suggests that about 40 to 50\% of the Langevin collisions are reactive. The last two columns of Table~\ref{table4} present the ratios $r_1$ and $r_2$ of the thermal rate coefficient obtained at 27~K to those obtained at 68 and 300~K. The calculated $r_1$ value is in perfect agreement with the experimental result, whereas the experimental and calculated values of $r_2$ differ by about 30\%.
\begin{table}[]
\caption{\label{table4} Comparison of the thermal capture rate coefficients of the He$^+$ + NH$_3$ reaction calculated with Eqs.~(\ref{thermal_rates1}) and~(\ref{thermal_rates2}) assuming nuclear-spin conservation (NSC) or nuclear-spin relaxation (NSR) during the supersonic expansion and the experimental reaction rate coefficients results of Marquette {\it et al.} \cite{marquette85a} The last two columns give the ratios $r_1$ and $r_2$ of the value obtained at 27~K to the values obtained at 68 and 300~K, respectively.}
\begin{tabular}{l| c c c c c}
& $k$(27~K) & $k$(68~K) & $k$(300~K) & $r_1$ & $r_2$\\
& (m$^3$s$^{-1}$) & (m$^3$s$^{-1}$) & (m$^3$s$^{-1}$) & & \\ \hline
NSC & $1.13\cdot 10^{-14}$ & $7.39\cdot 10^{-15}$ & $3.18\cdot 10^{-15}$ & 1.5 & 3.6 \\
NSR & $1.14\cdot 10^{-14}$ & $7.48\cdot 10^{-15}$ & $3.11\cdot 10^{-15}$ & 1.52 & 3.7 \\
Exp. \cite{marquette85a} & $4.5\cdot 10^{-15}$ & $3.0\cdot 10^{-15}$ & $1.65\cdot 10^{-15}$ & 1.5 & 2.7 \\
\end{tabular}
\end{table}
\section{Conclusions}
\label{sec:concl}
In this article, which is the first in a series of three articles dedicated to the role of long-range electrostatic interactions in ion-molecule reactions at low temperatures, we have presented the results of measurements of the collision-energy-dependent rates of the reactions between He$^+$ and ammonia (NH$_3$ and ND$_3$) in the range of collision energies between $k_{\mathrm{B}}\cdot 0.2$~K and $k_{\mathrm{B}}\cdot 40$~K. To reach such low collision energies, we used a merged-beam approach and observed the reactions within the orbit of a highly excited Rydberg electron. In this range, we observed that the rates increase strongly with decreasing collision energy, the increase being particularly pronounced below $k_{\mathrm{B}}\cdot5 $~K. We have also observed that the rates of reactions involving ND$_3$ increase more strongly than the rates of the reactions involving NH$_3$ at low collision energies. To interpret the experimental results, we have calculated the rates using an adiabatic capture model based on the calculation of the Stark shifts of the rotational levels of \ce{NH3} and \ce{ND3} in the field of the He$^+$ ion.
The calculations provided a quantitatively accurate description of the experimental observations and could be used to rationalise the observed collision-energy-dependent reaction yields in terms of capture rate coefficient for rotationally state-selected ($|JKM\pm \rangle$) ammonia molecules. These states could be classified into three categories: (A) States with $|KM|\ge 1$ correlating at low fields with the lower components of the inversion-tunnelling doublets. These states undergo linear negative Stark shifts in the field of the colliding ion, which suppresses the centrifugal barriers in the effective intermolecular potential and greatly enhances the capture rates at low collision energies; (B) States with $KM=0$, which only become sensitive to the fields at high fields through the quadratic and higher-order Stark shifts. The corresponding capture rates are only weakly enhanced at low collision energies. (C) States with positive linear Stark shifts at low fields which generate large potential energy barriers and completely suppress the reactions at low collision energies (below $\sim k_{\mathrm{B}}\cdot15$ K). At high collision energies, beyond $k_{\mathrm{B}}\cdot300$~K, the capture rates for all three categories gradually converge to the Langevin capture rates. These observations are a direct consequence of the dipolar nature of ammonia.
The model also provided an explanation for the different behaviour of the reactions involving NH$_3$ and ND$_3$ at low collision energies. The stronger enhancements of the rates observed below $k_{\mathrm{B}}\cdot5$ K in the reactions of ND$_3$ was interpreted as arising from the smaller tunnelling splittings of the inversion mode and from the larger population of the rotational states belonging to category A at the 6~K rotational temperature of the supersonic beams. The latter effect results primarily from the different nuclear-spin statistical weights in NH$_3$ and ND$_3$. From the calculated rotational-state-specific collision-energy-dependent captures rates, {\it thermal} capture rate coefficients were determined in the temperature range between 0.2 and 10~K of relevance for modelling reactions in the interstellar medium. Assuming that the relative populations of the ortho, para and meta nuclear-spin isomers reflect the nuclear-spin-statistical factors, as can be expected for dilute gases and supersonic expansions, leads to a large nonlinear increase of the thermal capture rates below 5~K. The thermal capture rate constant reach the value of $4.4\times 10^{-14}$~m$^3$/s at 0.2~K, which is more than 20 times larger than the Langevin rate constant.
Assuming that ortho-para-meta conversion processes can take place, as might be the case if the molecules have thermalised at the surface of certain materials and dust grains, leads to slightly larger rate enhancements above 2~K, but to smaller rate enhancements at the lowest temperatures (see Fig.~\ref{Fig8}).
In future work, it will be important to obtain data at even lower temperatures and improved energy resolution. Moreover, the preparation of rotationally state-selected samples would enable more extensive tests of the model predictions. The generation of pure samples of molecules in high-field-seeking states of the category A seems particularly attractive in this regard, and recent experiments have demonstrated ways of generating such samples~\cite{ploenes21}. Calculations of the (nonadiabatic) dynamics on the multidimensional potential-energy surfaces involved would be needed to interpret the observed branching ratios and the fact that only 40\% of the close collisions appear to be reactive.
\section{Acknowledgements}
We thank M. \v{Z}e\v{s}ko for his contributions to the initial phase of this project and K. H\"{o}veler and J. Deiglmayr for fruitful discussions. This work is supported financially by the Swiss National Science Foundation (Grant No. 200020B-200478) and by the European Research Council through the ERC advanced grant (Grant No. 743121) under the European Union's Horizon 2020 research and innovation programme.
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\end{document}
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{"paloma_paragraphs":[]}
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{"arxiv_id":"2112.14532","language":"en","timestamp":1640831219000,"url":"https:\/\/arxiv.org\/abs\/2112.14532","yymm":"2112"}
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proofpile-arXiv_000-10023
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{"provenance":"002.jsonl.gz:10024"}
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\section{Introduction}
The growth-optimal portfolio (GOP) is a portfolio which has a maximal expected growth rate (namely log-return) over any time horizon. As the GOP can be usually tracked to the work \cite{K1956}, it is also called the ``Kelly criterion''. The GOP can also be obtained by maximizing log-utility which has a longer history.
As the name implies, it can be used to maximize the expected growth rate of a portfolio. Indeed, it performs in some sense better than any other significantly different strategy as the time horizon increases.
Over the past half century, a lot of papers have investigated the GOP.
In theory and practice, the GOP has widely applications in a large number of areas including portfolio theory, utility theory, game theory, information theory, asset pricing theory, insurance theory. For instance, to name a few of them in the recent studies in the literature, \cite{A2000} study asset pricing problems in incomplete market; \cite{R2004} considers optimal investment problems; \cite{T2000} applies it to casino games.
We want to emphasize that the GOP ignores the relevant risk when maximizing the expected growth rate of a portfolio.
It is the seminal work \cite{markowitz1952portfolio} that takes the trade-off between the portfolio return and its risk into consideration when an investor chooses a portfolio. \cite{markowitz1952portfolio} suggests to use variance to measure the risk. Since then, the mean-variance theory becomes one of the most dominant financial theories in the realm of portfolio choice. Besides variance, alternative risk measures have been proposed to measure the risk for portfolio choice. Research along this line includes \cite{rockafellar2000optimization}, \cite{campbell2001optimal}, \cite{rockafellar2002conditional}, \cite{alexander2002economic}, \cite{alexander2004comparison},
\cite{jin2006note}, and \cite{adam2008spectral},
where the authors study single-period mean-risk portfolio selection with various risk measures, such as semi-variance, value-at-risk (VaR), expected shortfall (ES), and spectral risk measures.
There also have numerous extensions of the mean-risk portfolio optimization from the single-period setting to the dynamic, continuous-time one
\citep[e.g.][]{zhou2000continuous,bielecki2005continuous,jin2005continuous,basak2010dynamic,he2015dynamic,zhou2017dynamic,gao2017dynamic,dai2021dynamic,he2021mean}. In particular, \cite{he2015dynamic} study a continuous-time mean-risk portfolio choice when risk is measured by the weighted Value-at-Risk (WVaR) but their results are rather pessimistic. The WVaR is a quantile-based risk measure that generalizes VaR and ES, two popular risk measures in quantitative risk management.
They find that, when using WVaR (including VaR and ES) on terminal wealth to measure portfolio risk, the mean-risk model is prone to be ill-posed (i.e., the optimal value is infinite) and the investor tends to take infinite leverage on risky assets, leading to extremely risk-taking behaviors. Furthermore, the optimal risk is independent of the expected terminal target so the efficient frontier is a vertical line on the mean-WVaR plane. Their results suggest that the mean-WVaR model is an improper modeling of the trade-off between return and risk, when the WVaR is applied to the terminal wealth.
This paper proposes a continuous-time portfolio choice model with mean-WVaR criterion for portfolio log-returns, as opposed to the mean-WVaR criterion for terminal wealth in \cite{he2015dynamic}. The motivation is two-fold. First, the mean-risk criterion for log-returns is consistent with Markowitz's original idea to use mean and variance on portfolio returns. We consider a growth-optimal problem with risk control.
Moreover, many single-period mean-risk models use risk measures on portfolio returns in the literature \citep[e.g.][]{alexander2002economic,alexander2004comparison,alexander2006does,jin2006note,adam2008spectral}.
However, there is a discrepancy between single-period and dynamic portfolio choice models, as the latter typically considers mean-risk criterion for terminal wealth; an exception is \cite{dai2021dynamic} who study continuous-time mean-variance portfolio choice for portfolio log-returns. We similarly adopt the mean-WVaR criterion for log-returns which naturally generalize the single-period return when returns are continuously compounded. Second, such a criterion conquers the ill-posedness of the model in \cite{he2015dynamic}. As noted in \cite{he2015dynamic}, the mean-WVaR criterion for terminal wealth is essentially a linear program for the quantile function of terminal wealth. This linearity, in turn, leads to the optimal terminal wealth's quantile function being ``corner points'', leading to extreme risk-taking behaviors. By contrast, our mean-WVaR criterion for log-returns is not linear in the quantile function of terminal wealth, and thus conquers the ill-posedness.
In a continuous-time, complete market framework, we solve the mean-WVaR portfolio choice for log-returns with the help of the so-called quantile formulation, developed in a series of papers \citep[e.g.][]{schied2004neyman,carlier2006law,jin2008behavioral,he2011portfolio,carlier2011optimal,xia2016arrow,xu2016note}. When risk is measured by a general WVaR risk measure, we characterize the optimal terminal wealth up to the concave envelope of a certain function through a detailed and involved analysis. When risk is measured by VaR or ES, two special cases of WVaR, we derive analytical expressions for the optimal terminal wealth and portfolio policy. The optimal terminal wealth turns out to be closely related to the growth optimal portfolio: the investor classifies market scenarios into different states, in which the terminal payoff can be constant, a multiple or fraction of the growth optimal portfolio. Furthermore, we obtain the efficient frontier, which is a concave curve that connects the minimum-risk (min-risk) portfolio with the growth optimal portfolio, as opposed to the vertical line in \cite{he2015dynamic}. Our model allows for a meaningful characterization of the risk-return trade-off and may serve as a guideline for investors to set a reasonable investment target. Although \cite{he2015dynamic} provides a critique of using WVaR to measure risk, our results advocate that it is more appropriate to use WVaR, in particular, VaR and ES, on portfolio log-returns instead of terminal wealth for dynamic portfolio choice.
The rest of the paper is organized as follows. In Section \ref{sec:model}, we propose a mean-WVaR portfolio choice problem for portfolio log-returns. We solve the problem in
Section \ref{sec:solution} by quantile optimization method. Section \ref{sec:examples} presents optimal solutions and efficient frontiers when the risk is measured by VaR or ES. Some new financial insights and a comparation to the existing work are presented as well. Some concluding remarks are given in Section \ref{sec:conclusion}. Appendix \ref{sec:A1} contains three useful lemmas. All remaining proofs are placed in Appendix \ref{sec:A2}.
\section{Mean-WVaR portfolio choice model}\label{sec:model}
\subsection{Financial market}
Let $T>0$ be a given terminal time and $(\Omega,\mathcal{F}, \{ \mathcal{F}_t \}_{0 \le t \le T} ,\mathbb{P})$ be a filtered probability space, on which is defined a standard one-dimensional Brownian motion
$\{ W_t \}_{0\le t\le T}$. It is assumed that $\mathcal{F}_t=\sigma \{ W_s, 0\le s\le t \}$ augmented by all $\mathbb{P}$-null sets and that
$\mathcal{F}=\mathcal{F}_T$ is $\mathbb{P}$ complete.
We consider a Black-Scholes market in which there are a risk-free asset and a risky asset (called stock). The risk-free asset pays a constant interest rate $r>0$ and the stock price $S$ follows a geometric Brownian motion
$$\frac{dS_t}{S_t}=\mu dt+\sigma dW_t, $$
where $\mu$ and $\sigma$, the appreciation rate and volatility of the stock, are positive constants. There exists a unique positive state price density (pricing kernel) process $\xi$\footnote{With additional assumptions on $\xi_T$, our main results can be extended to a general complete market with stochastic investment opportunities.} satisfying
\begin{equation}\label{eq:xi}
\frac{d\xi_t}{\xi_t}=-r dt-\theta dW_t, \quad \xi_{0}=1,
\end{equation}
where $\theta=(\mu-r)/\sigma$ is the market price of risk in the economy. Therefore the market is complete.
Consider an economic agent with an initial endowment $x>0$ and faced an investment time horizon $[0,T]$. The agent chooses a dynamic investment strategy $\pi_t$, which represents the dollar amount invested in the stock at time $t$. Assume the trading is continuous in a self-financing fashion and there are no transaction costs. The agent's wealth process $X_t$ then follows a stochastic differential equation
\begin{equation}\label{eq:budget}
dX_t=\left[ rX_t+(\mu-r) \pi_t \right] dt+\sigma \pi_t dW_t, ~ X_0=x.
\end{equation}
The portfolio process $\pi_t$ is called an admissible portfolio if it is $\{ \mathcal{F}_t \}_{0 \le t \le T}$ progressively measurable with $\int_0^T \pi_t ^2 dt < \infty, a.s.$, and the corresponding terminal wealth satisfies $X_T \ge 0, a.s$.
Let $R_T$ be the continuously compounded return (log-return) over the horizon $[0,T]$, i.e.,
\begin{equation}\label{eq:log-return}
R_T=\frac{1}{T} \ln \frac{X_T}{x}.
\end{equation}
By convention, we define
$$\ln 0=\lim_{s \downarrow 0} \ln s=-\infty \mbox{ and } e^{-\infty}=\lim_{s \downarrow-\infty}e^s=0.$$
\subsection{Risk measure}
We now introduce a risk measure that will be used in the portfolio choice model. In this paper, we focus on the weighted VaR (WVaR) risk measure proposed by \cite{he2015dynamic}, which is a generalization of Value-at-Risk (VaR) and Expected Shortfall (ES), and encompasses many well-known risk measures that are widely used in finance and actuarial sciences, such as spectral risk measures and distortion risk measures; see \cite{wei2018risk} for a review.
For any $\mathcal{F}_T$-measurable random variable $X$, let $F_X$ denote its cumulative distribution function; and let $G_X$ denote its quantile function defined by
\begin{equation*}
G_X(z)=\inf \{x \in \mathbb{R} : F_X(x) > z \}=\sup \{x\in \mathbb{R} : F_X(x) \le z \}, ~z \in [0, 1),
\end{equation*}
with the convention $G_X(1)=\lim_{z \uparrow 1} G_X(z)$. The quantile function $G_X$ is non-decreasing, right-continuous with left limits (RCLL).
The WVaR risk measure for $X$ is defined as
\begin{equation}\label{eq:wvar}
\rho _{\Phi} (X)=-\int _{[0,1]} G_X(z) \Phi (dz),
\end{equation}
where $\Phi\in P[0,1]$ and $P[0,1]$ is the set of all probability measures on $[0,1]$.
The WVaR is a law-invariant comonotonic additive risk measure, and it covers many law-invariant coherent risk measures; see \cite{he2015dynamic} for a more detailed discussion. If $\Phi$ is the Dirac measure at $\alpha$, i.e., $\Phi (A)=\mathbf{1}_{\alpha \in A}$, for all $A \subset [0,1]$, then the corresponding WVaR measure becomes the VaR at $\alpha$, in other words,
\begin{equation*}
\rho _{\Phi} (X)=\text{VaR}_{\alpha}(X)=-G_{X}(\alpha).
\end{equation*}
If $\Phi$ admits a density $\phi (z)=\frac{1}{\alpha}\mathbf{1}_{z \le \alpha}, ~ \forall z \in [0,1]$, then the corresponding WVaR measure becomes the ES, i.e.,
\begin{equation*}
\rho _{\Phi} (X)=\text{ES}_{\alpha} (X)=-\frac{1}{\alpha} \int_0^{\alpha} G_{X} (z)dz.
\end{equation*}
In the original paper of \cite{he2015dynamic}, WVaR is applied to measure the risk of a portfolio's terminal wealth. In this paper, we propose to apply WVaR to the portfolio's log-return instead of its terminal wealth. Let $X_T$ be the terminal wealth of a portfolio and $R_T$ be the log-return of $X_T$. Due to the monotonicity of logarithm functions, the quantile function of $R_T$ is
\begin{equation*}
G_{R_T}(z)=\frac{1}{T} \ln \frac{ G_{X_T}(z)}{x}, ~ z \in [0,1].
\end{equation*}
Therefore, the WVaR of $R_T$ can be expressed as
\begin{equation}\label{eq:wvar log-return}
\rho _{\Phi} (R_T)=-\int _{[0,1]} \frac{1}{T} \ln \frac{ G_{X_T}(z)}{x} \Phi (dz)=-\frac{1}{T} \int _{[0,1]} \ln G_{X_T}(z) \Phi (dz)+\frac{1}{T} \ln x.
\end{equation}
However, the extension from terminal wealth to log-return is not straightforward as the integral in \eqref{eq:wvar log-return} may not be well-defined since $X_{T}$ may take the value of 0 with positive probability. Let
\begin{equation*}
\begin{aligned}
\mathbb{G}=\Big\{G(\cdot) \colon [0, 1] \to [0,+\infty], ~ &G\text{ is nondecreasing and RCLL on [0,1], }\\
& \text{ left-continuous at } 1, \text{ and finite-valued on } [0,1)\Big\}
\end{aligned}
\end{equation*}
be the set of quantile functions of all non-negative random variables, which include all terminal wealth of admissible portfolios. For any $G \in \mathbb{G}$ and $\Phi\in P[0,1]$, the integral $\int _{[0,1]} \ln G(z) \Phi (dz)$ is not well-defined if $G(s)=0$ for some $s \in[0,1]$ such that $\Phi ( [0,s] )>0$. Define
\begin{equation*}
\mathbb{G}_{\Phi}=\big\{G \in \mathbb{G} \colon G(s)>0 \text{ if } \Phi ( [0, s] )>0, ~ \forall s \in[0,1] \big\}.
\end{equation*}
We set
\begin{equation}\label{eq:-infty integral}
\int _{[0,1]} \ln G(z) \Phi (dz)=-\infty, ~ \forall G \in \mathbb{G} \setminus \mathbb{G}_{\Phi}.
\end{equation}
Intuitively, if the terminal wealth of a portfolio is $0$ (so that the log-return is $-\infty$) in some states, and the weighting measure $\Phi$ assigns non-zero weighs to these states, then the WVaR of the log-return is assumed to be $-\infty$. In particular, ${\mathbb{E}} [ R_T]=-\infty$ if $\mathbb{P} (X_T=0)>0$.\footnote{It is straightforward to verify ${\mathbb{E}} \left[ \max \left(R_T,0 \right) \right]<\infty$, given that $\xi_T$ is log-normally distributed. }
\subsection{Portfolio choice model}
We assume the agent chooses a dynamic portfolio strategy in the period $[0,T]$ to maximize the expected log-return while minimizing the risk of the portfolio's log-return. The risk is evaluated by a WVaR risk measure $\rho _{\Phi}$ on the portfolio's log-return $R_T$. Specifically, we consider the following dynamic portfolio choice problem
\begin{equation}\label{prob:original}
\begin{aligned}
\max _{\pi_t} ~ &~ \lambda {\mathbb{E}} [ R_T]-\rho _{\Phi} (R_T)\\
\text{subject to} ~ &~dX_t=\left[ rX_t+(\mu-r) \pi_t \right]dt+\sigma \pi_t dW_t, ~X_{T}\geq 0, ~ X_0=x,\\
&R_T=\frac{1}{T} \ln \frac{X_T}{x},
\end{aligned}
\end{equation}
where $\lambda \ge 0$ is a ``risk-tolerance" parameter that reflects the investor's tradeoff between return and risk.
This is a stochastic control problem, but not standard (namely, unlike those in \cite{yongzhou1999}) due to the existence of the nonlinear probability measure $\Phi$.
In view of the standard martingale method, e.g., \cite{karatzas1998methods}, we can first solve the following static optimization problem\footnote{This formulation implies that the optimal log-return $R_{T}$ is independent of $x$. }
\begin{equation}\label{prob:martingale}
\begin{aligned}
\max _{X_T \in \mathcal{F}_T} ~ &~ \lambda {\mathbb{E}} [ R_T]-\rho _{\Phi} (R_T) \\
\text{subject to} ~ &~{\mathbb{E}} [\xi_T X_T] \le x, ~X_{T}\geq 0, \\
&~R_T=\frac{1}{T} \ln \frac{X_T}{x},
\end{aligned}
\end{equation}
where $\xi_{T}$ is given by \eqref{eq:xi}. Then apply backward stochastic control theory to derive the corresponding optimal portfolio strategy $\pi_{t}$.
The optimization problem \eqref{prob:martingale} nests two special cases.
\begin{description}
\item[Case $\lambda=0$.] In this case the investor minimizes the risk without any consideration of the expected log-return, and solves the following minimum-risk problem
\begin{equation}\label{prob:min risk}
\begin{aligned}
\min _{X_T \in \mathcal{F}_T} ~ &~ \rho _{\Phi} (R_T)\\
\text{subject to} ~ &~{\mathbb{E}} [\xi_T X_T] \le x, ~X_{T}\geq 0, \\
&~R_T=\frac{1}{T} \ln \frac{X_T}{x}.
\end{aligned}
\end{equation}
The resulting portfolio is termed the min-risk portfolio.
\item[Case $\lambda=\infty$.] In this case the investor maximizes the expected log-return without any consideration of the risk. This is the so-called growth-optimal problem
\begin{equation}\label{prob:growth}
\begin{aligned}
\max _{X_T \in \mathcal{F}_T} ~ & {\mathbb{E}} [ R_T] \\
\text{subject to} ~ &{\mathbb{E}} [\xi_T X_T] \le x, ~X_{T}\geq 0, \\
&R_T=\frac{1}{T} \ln \frac{X_T}{x}.
\end{aligned}
\end{equation}
The optimal solution to \eqref{prob:growth} is well-known in the literature, i.e., the growth-optimal portfolio (or \cite{K1956} strategy) given by
\begin{equation}\label{eq:growth}
X_{\textrm{Kelly}}=\frac{x}{\xi_T}.
\end{equation}
The corresponding log-return is
\begin{equation*}
R_{\textrm{Kelly}}=\frac{1}{T} \ln \frac{X_{\textrm{Kelly}}}{x}=-\frac{1}{T} \ln \xi_T,
\end{equation*}
and its expected value is
\begin{equation*}
{\mathbb{E}} [R_{\textrm{Kelly}}]=-\frac{1}{T} {\mathbb{E}} [\ln \xi_T]=r+\frac{\theta^2}{2}.
\end{equation*}
\end{description}
\section{Quantile formulation and optimal solution}\label{sec:solution}
In this section, we solve the optimization problem \eqref{prob:martingale} for $0\le \lambda<\infty$.
If $\Phi (\{ 1 \})>0$, then $\rho _{\Phi} (R_{T})=-\infty$. If $\Phi$ is the the uniform measure on $[0,1]$, then $\rho _{\Phi} (R_T)=-{\mathbb{E}} [R_T]$ and the growth optimal portfolio \eqref{eq:growth} is optimal to \eqref{prob:martingale}.
To exclude these trivial cases, we make the following assumption on $\Phi$ from now on.
\begin{Assumption}\label{assumption:phi}
$\Phi (\{ 1 \})=0$ and $\Phi$ is not the uniform measure on $[0,1]$.
\end{Assumption}
The objective in \eqref{prob:martingale} is based on the quantile function of the log-return; thus, the standard convex duality method is not readily applicable. To overcome this difficulty, we employ the quantile formulation, developed in a series of papers
including \cite{schied2004neyman}, \cite{carlier2006law}, \cite{jin2008behavioral}, \cite{he2011portfolio}, \cite{carlier2011optimal}, \cite{xia2016arrow}, and \cite{xu2016note}, to change the decision variable in \eqref{prob:martingale} from the terminal wealth $X_T$ to its quantile function. This allows us to recover the hidden convexity of the problem and solve it completely.
We first show that the budget constraint in \eqref{prob:martingale} must hold with equality and the objective function is improved with a higher level of initial wealth.
\begin{lemma}\label{lemma:3.1}
If $X_T^{*}$ is an optimal solution to the problem \eqref{prob:martingale}, then $ {\mathbb{E}} [\xi_T X_T^{*}]=x$.
\end{lemma}
\noindent
All the proofs of our results are put in Appendix \ref{sec:A2}.
Denote by $F_\xi$ and $G_\xi$ the distribution and quantile functions of $\xi_T$, respectively. With slight abuse of notation, we suppress the subscript $T$ when there is no need to emphasize the dependence on $T$. Since $\xi_{T}$ is log-normally distributed, both $F_\xi$ and $G_\xi$ are $C^{\infty}$ functions.
The following lemma can be found in \cite{jin2008behavioral}.
\begin{lemma}[\cite{jin2008behavioral}]\label{lemma:3.2} We have
${\mathbb{E}} \left[ \xi_T G_X \left(1-F_{\xi}(\xi_T) \right) \right] \le {\mathbb{E}}[\xi_T X_T]$ for any lower bounded
random variable $X_T$ whose quantile function is $G_X$. Furthermore, if ${\mathbb{E}} [\xi_T G_X(1-F_{\xi}(\xi_T))] < \infty$, then the inequality
becomes equality if and only if $X_T=G_X \left(1-F_{\xi}(\xi_T) \right), ~a.s.$
\end{lemma}
From Lemmas \ref{lemma:3.1} and \ref{lemma:3.2}, we know that if $X_T$ is optimal to \eqref{prob:martingale}, then $X_T=G_X(1-F_{\xi}(\xi_T))$ where $G_X$ is the quantile function of $X_T$. Let $R_T$ be the log-return of $X_T$. We have
\begin{equation*}
{\mathbb{E}} [R_T]=\int_{[0,1)} \frac{1}{T} \ln \frac{ G_{X}(z)}{x} dz,
\end{equation*}
\begin{equation*}
\rho _{\Phi} (R_T)=-\int _{[0,1)} \frac{1}{T} \ln \frac{ G_{X}(z)}{x} \Phi (dz),
\end{equation*}
and
\begin{equation*}
{\mathbb{E}}[\xi_T X_T]=\int_{[0,1)} G_X(z) G_{\xi} (1-z)dz.
\end{equation*}
Therefore, we can consider the following quantile formulation of \eqref{prob:martingale}
\begin{equation}\label{prob:quantile}
\begin{aligned}
\max _{G \in \mathbb{G} } ~ & \lambda \int_{[0,1)} \frac{1}{T} \ln \frac{ G(z)}{x} dz+\int _{[0,1)} \frac{1}{T} \ln \frac{ G(z)}{x} \Phi (dz) \\
\text{subject to} ~
& \int_{[0,1)} G(z) G_{\xi} (1-z)dz=x,
\end{aligned}
\end{equation}
where the decision variable $G$ is the quantile function of the terminal wealth. Once we obtain the optimal solution $G^{*}$ to \eqref{prob:quantile}, then the optimal solution to \eqref{prob:martingale} is given by $$X_T^{*}=G^{*} \left( 1-F_{\xi}(\xi_T) \right).$$
Define
\begin{equation}
w(s)=\frac{ \int_{[0,s)} G_{\xi} (1-z)dz }{{\mathbb{E}} [\xi_T]}, ~ s \in [0,1].
\end{equation}
Because $\xi_{T}$ is log-normally distributed, $w$ is a $C^{\infty}$ function with $w(0)=0$, $w(1)=1$
and $w'>0$, $w''<0$ on $(0,1)$. Let $w^{-1}$ be the inverse function of $w$ and define $$H(s)=G \left( w^{-1} (s) \right), ~ s \in [0,1].$$ Then $w^{-1}$ is a $C^{\infty}$ function with $w^{-1}(0)=0$, $w^{-1}(1)=1$, and $(w^{-1})'>0$, $(w^{-1})''>0$ on $(0,1)$.
It is easy to see $G \in \mathbb{G}$ if and only if $H \in \mathbb{G}$, and $G \in \mathbb{G}_{\Phi}$ if and only if $H \in \mathbb{H}_{\Phi}$, where
\begin{equation*}
\mathbb{H}_{\Phi}=\Big\{H \in \mathbb{G} \colon H\left( w(s) \right)>0 \text{ if } \Phi ( [0,s] )>0, ~ \forall s \in[0,1] \Big\}.
\end{equation*}
In terms of new notation, we have
\begin{equation*}
\int_{[0,1)} G(z) G_{\xi} (1-z)dz={\mathbb{E}} [\xi_T] \int_{[0,1)} H(s) ds,
\end{equation*}
\begin{equation*}
\int_{[0,1)} \frac{1}{T} \ln \frac{ G(z)}{x} dz=\int_{[0,1)} \frac{1}{T} \ln \frac{ H(s)}{x} dw^{-1} (s) ,
\end{equation*}
and
\begin{equation*}
\int _{[0,1)} \frac{1}{T} \ln \frac{ G(z)}{x} \Phi (dz)=\int _{[0,1)} \frac{1}{T} \ln \frac{ H(s)}{x} d \Phi ([0,w^{-1} (s)]).
\end{equation*}
Consequently, solving \eqref{prob:martingale} reduces to solving the following quantile optimization problem (after dropping constant terms)
\begin{equation}\label{prob:quantile H}
\begin{aligned}
\max _{H \in \mathbb{G} } ~ &~ \lambda \int_{[0,1)} \ln H(s) dw^{-1} (s)+\int _{[0,1)} \ln H(s) d \Phi ([0,w^{-1} (s)]) \\
\text{subject to} ~
&~ \int_{[0,1)} H(s)ds=\frac{x}{{\mathbb{E}} [\xi_T]}.
\end{aligned}
\end{equation}
This is a concave optimization problem, so it can be tackled by the Lagrange method.
Define the Lagrangian
\begin{equation*
L(H(\cdot) ; \lambda, \eta)=\lambda \int_{[0,1)} \ln H(s) dw^{-1} (s)+\int _{[0,1)} \ln H(s) d \Phi ([0,w^{-1} (s)])-\eta \int_{[0,1)}H (s) ds,
\end{equation*}
where $\eta > 0$ is a Lagrange multiplier to fit the budget constraint in \eqref{prob:quantile H}.
Define
\begin{equation*}
\varphi ( s ; \lambda)=\frac{ \Phi ([0,w^{-1} (s)])+\lambda w^{-1} (s)}{1+\lambda}, ~ s \in [0,1],
\end{equation*}
and its left-continuous version
\begin{equation*}
\varphi ( s-; \lambda)=\frac{ \Phi ([0,w^{-1} (s)))+\lambda w^{-1} (s)}{1+\lambda}, ~ s \in (0,1].
\end{equation*}
We additionally set $\varphi ( 0-; \lambda)=0$.
We then have
\begin{equation*}
L(H(\cdot) ; \lambda, \eta)=(1+\lambda) \int _{[0,1)} \ln H(s) d\varphi ( s ; \lambda)-\eta \int_{[0,1)} H (s) ds,
\end{equation*}
and we can consider the following optimization problem
\begin{equation}\label{prob:Lagrangian}
\max_{H \in \mathbb{G} }~ L(H(\cdot) ; \lambda, \eta).
\end{equation}
Inspired by \cite{rogers2009optimal}, \cite{xu2016note}, and \cite{wei2018risk}, we introduce $\delta (s; \lambda)$, the convex envelope function of $\varphi ( s-; \lambda)$ on $[0,1]$, given by
\begin{equation}\label{eq:concave envelope}
\delta (s; \lambda)=\sup_{0 \le a \le s \le b \le 1} \frac{(b-s)\varphi (a-; \lambda)+(s-a)\varphi (b-; \lambda)}{b-a}, ~s \in [0,1].
\end{equation}
The convex envelope $\delta (s; \lambda)$ is the largest convex function dominated by $\varphi ( s-; \lambda)$, and is affine on the set $\big\{s \in (0,1) \colon \delta (s; \lambda) < \varphi ( s-; \lambda)\big\}.$
The following proposition presents the optimal solution to \eqref{prob:Lagrangian}.
\begin{proposition}\label{prop:3.1}
The optimal solution to \eqref{prob:Lagrangian} is given by $$H^{*} (s; \lambda , \eta)=\frac{1+\lambda}{\eta} \delta' (s; \lambda), ~ s \in [0,1],$$ where $ \delta' (s; \lambda)$ is the right derivative of $\delta (s; \lambda)$ with respect to $s$.
\end{proposition}
We want to find a Lagrange multiplier $\eta$ such that $H^{*} (s; \lambda , \eta)$ satisfies the budget constraint in \eqref{prob:quantile H}.
Clearly, \begin{equation*}
\int_{[0,1)} H^{*} (s; \lambda , \eta)ds=\int_{[0,1)} \frac{1+\lambda}{\eta} \delta' (s; \lambda)ds=\frac{1+\lambda}{\eta}=\frac{x}{{\mathbb{E}} [\xi_T]}.
\end{equation*}
and consequently
$$\eta=\frac{1+\lambda}{x}{\mathbb{E}} [\xi_T].$$
We are ready to state the optimal solution to \eqref{prob:quantile H}.
\begin{proposition}\label{prop:3.2}
The optimal solution to \eqref{prob:quantile H} is given by
\begin{equation*}
H^{*} \left(s; \lambda , \frac{1+\lambda}{x}{\mathbb{E}} [\xi_T] \right)=\frac{x}{{\mathbb{E}} [\xi_T]}\delta' (s; \lambda) .
\end{equation*}
\end{proposition}
Finally, the optimal solution to \eqref{prob:martingale} is given by
\begin{equation*}
X^{*}_{T,\lambda}=H^{*} \left( w(1-F_{\xi}(\xi_T); \lambda , \frac{1+\lambda}{x}{\mathbb{E}} [\xi_T] \right)=\frac{x}{{\mathbb{E}} [\xi_T]}\delta' \left(w(1-F_{\xi}(\xi_T); \lambda \right).
\end{equation*}
In particular, we can obtain the min-risk portfolio by setting $\lambda=0$:
\begin{equation*}
X_{T,0}^{*}=\frac{x}{{\mathbb{E}} [\xi_T]} \delta' (w (1-F_{\xi}(\xi_T)); 0),
\end{equation*}
which solves \eqref{prob:min risk}.
We summarize the main results of the paper in the following proposition.
\begin{proposition}[Efficient portfolio]\label{prop:efficient}
The efficient portfolio, i.e., the optimal solution to \eqref{prob:martingale} is
\begin{equation*}
X^{*}_{T,\lambda}=\frac{x}{{\mathbb{E}} [\xi_T]} \delta' (w (1-F_{\xi}(\xi_T)); \lambda ),
\end{equation*}
and the corresponding log-return is $$R^{*}_{T,\lambda}=\frac{1}{T} \ln \left( \frac{ \delta' (w (1-F_{\xi}(\xi_T)); \lambda ) }{{\mathbb{E}} [\xi_T]} \right).$$
In particular, the min-risk portfolio, i.e., the optimal solution to \eqref{prob:min risk} is
\begin{equation*}
X_{T,0}^{*}=\frac{x}{{\mathbb{E}} [\xi_T]} \delta' (w (1-F_{\xi}(\xi_T)); 0),
\end{equation*}
and the corresponding log-return is $$R^{*}_{T,0}=\frac{1}{T} \ln \left( \frac{ \delta' (w (1-F_{\xi}(\xi_T));0 ) }{{\mathbb{E}} [\xi_T]} \right).$$
\end{proposition}
\section{Examples with explicit solution}\label{sec:examples}
In this section, we present two examples to illustrate our general results. In particular, we consider the optimization problem \eqref{prob:martingale} when the WVaR risk measure is given by either VaR or ES, two popular risk measures.
\subsection{Mean-VaR efficient portfolio}
In this subsection, we specialize our setting to the mean-VaR optimization problem. In particular, we consider the optimization problem \eqref{prob:martingale} when the WVaR risk measure is given by the VaR at a confidence level $0<\alpha<1$, namely
\begin{equation*}
\rho _{\Phi} (X)=\text{VaR}_{\alpha}(X)=-G_{X}(\alpha).
\end{equation*}
In other words, $\Phi$ is given by the Dirac measure at $\alpha$.
\begin{proposition}\label{prop:4.1}
When $\Phi$ is given by the Dirac measure at $\alpha\in (0,1)$, we have the following assertions.
\begin{description}
\item[Case $\lambda=0$.]
\begin{enumerate}
\item The minimum-VaR (min-VaR) terminal wealth is
\begin{equation*}
X^{\text{VaR}}_{T,0}=
\begin{cases}
0, ~ & \xi_T > \xi_{\alpha},\\
\underline{X}_{\text{VaR}} , ~ & \xi_T \le \xi_{\alpha},
\end{cases}
\end{equation*}
where
\begin{equation*}
\begin{aligned}
\underline{X}_{\text{VaR}} &=\frac{x}{{\mathbb{E}} [\xi_T]} \cdot \frac{ 1}{ 1-w(\alpha)} ,\quad
\xi_{\alpha}=G_{\xi} (1-\alpha).
\end{aligned}
\end{equation*}
\item The optimal log-return is
$$R^{\text{VaR}}_{T,0}=\frac{1}{T} \ln \frac{X^{\text{VaR}}_{T,0}}{x}.$$
\item The expected optimal log-return is ${\mathbb{E}} [R^{\text{VaR}}_{T,0}]=-\infty.$
\item The VaR of the optimal log-return is
$$\text{VaR}_{\alpha}(R^{\text{VaR}}_{T,0})=\frac{1}{T} \ln \frac{\underline{X}_{\text{VaR}}}{x}.$$
\end{enumerate}
\item[Case $0<\lambda<\infty$.]
\begin{enumerate}
\item The mean-VaR efficient terminal wealth is
\begin{equation*}
X^{\text{VaR}}_{T,\lambda}=
\begin{cases}
\frac{\lambda}{1+\lambda} \cdot \frac{x}{\xi_T} , ~ & \xi_T > \xi_{\alpha},\\
\underline{X}_{\text{VaR}} , ~ & \underline{\xi}_{\text{VaR}} < \xi_T \le \xi_{\alpha} ,\\
\frac{\lambda}{1+\lambda} \cdot \frac{x}{\xi_T} , ~ & \xi_T \le \underline{\xi}_{\text{VaR}},
\end{cases}
\end{equation*}
where
\begin{equation*}
\begin{aligned}
\underline{X}_{\text{VaR}} &=\frac{\lambda}{1+\lambda} \cdot \frac{x}{ \underline{\xi}_{\text{VaR}}} ,\\
\xi_{\alpha} &=G_{\xi} (1-\alpha),\\
\underline{\xi}_{\text{VaR}} &=G_{\xi} (1-w^{-1}(s^{*}(\lambda) )) ,
\end{aligned}
\end{equation*}
and $s^{*}(\lambda)$ is given in Lemma \ref{lemma: f1}.
\item The optimal log-return is
$$R^{\text{VaR}}_{T,\lambda}=\frac{1}{T} \ln \frac{X^{\text{VaR}}_{T,\lambda}}{x} .$$
\item The VaR of the optimal log-return is
$$\text{VaR}_{\alpha}(R^{\text{VaR}}_{T,\lambda})=\frac{1}{T} \ln \frac{ \underline{X}_{\text{VaR}} }{x} .$$
\end{enumerate}
\end{description}
\end{proposition}
Figure \ref{figure:VaR wealth1} depicts the optimal terminal payoff of the min-VaR portfolio ($\lambda=0$), which resembles a digital option. Essentially, the investor invests all the money in a digital option that pays $\underline{X}_{\text{VaR}}$ in the good states of the market ($\xi < \xi_{\alpha}$) and 0 otherwise. The probability of winning the option depends solely on the confidence level of VaR and is given by ${\mathbb{P}} (\xi \le \xi_{\alpha})=1-\alpha$.
\begin{figure}[H]
\centering
\centering
\includegraphics[width=0.6\textwidth]{./Figures/VaR/VaR_min.eps}
\caption{The Min-VaR Efficient Terminal Wealth ($\lambda=0$)}\label{figure:VaR wealth1}
\end{figure}
Figure \ref{figure:VaR wealth2} displays the optimal terminal payoff of the mean-VaR efficient portfolio ($0<\lambda<\infty$). The investor classifies market scenarios into three subsets: in the good states ($\xi \le \underline{\xi}_{\text{VaR}}$) and in the bad states ($\xi > \xi_{\alpha}$), the terminal payoff is a fraction ($\lambda/(1+\lambda)$) of the growth optimal portfolio; in the intermediate states ($\underline{\xi}_{\text{VaR}} < \xi \le \xi_{\alpha}$), the investor receives a constant payoff $\underline{X}_{\text{VaR}}$. Moreover, the terminal wealth has a jump discontinuity at $\xi=\xi_{\alpha}$ and the corresponding log-returns are always finite (but can be extremely large or small).
\begin{figure}[H]
\centering
\includegraphics[width=0.6\textwidth]{./Figures/VaR/VaR_efficient.eps}
\caption{The Mean-VaR Efficient Terminal Wealth ($0<\lambda<\infty$)}\label{figure:VaR wealth2}
\end{figure}
Figure \ref{figure:VaR efficient} plots the mean-VaR efficient frontiers for different confidence levels $\alpha$. The efficient frontier is a concave curve that connects the growth optimal portfolio (colored dots) with the min-VaR portfolio (not shown in the graph). The growth optimal portfolio has the highest expected log-return but also the highest VaR. By contrast, the min-VaR portfolio has the smallest expected log-return (negative infinity) but also the lowest VaR. Figure \ref{figure:VaR efficient} also displays a sensitivity analysis of the efficient frontier with respect to $\alpha$, the confidence level of VaR. As $\alpha$ increases, the efficient frontier shifts to the left: for a given level of the expected log-return, the VaR of the corresponding efficient portfolio decreases as $\alpha$ increases.
\begin{figure}[H]
\centering
\includegraphics[width=0.6\textwidth]{./Figures/VaR/VaR_efficient_frontier.eps}\\
\caption{The Mean-VaR Efficient Frontier with $0<\lambda<\infty$, $T=1$, $r=0.05$, and $\theta=0.4$ }\label{figure:VaR efficient}
\end{figure}
As the optimal terminal wealth is known, we can solve for the optimal time-$t$ wealth and portfolio policy.
\begin{corollary}\label{coro:4.1}
We have the following assertions.
\begin{description}
\item[Case $\lambda=0$.]
\begin{enumerate}
\item The min-VaR efficient wealth at time $t$ is
\begin{equation*}
X^{\text{VaR}}_{t,0}=\underline{X}_{\text{VaR}} e^{-r (T-t)} N \left( d_2 ( t, \xi_t, \xi_{\alpha}) \right).
\end{equation*}
\item The optimal portfolio policy at time $t$ is
\begin{equation*}
\pi^{\text{VaR}}_{t,0}=\frac{\underline{X}_{\text{VaR}} e^{-r (T-t)} \nu \left( d_2 (t, \xi_t, \xi_{\alpha}) \right)}{ \sigma \sqrt{T-t} } .
\end{equation*}
\end{enumerate}
\item[Case $0<\lambda<\infty$.]
\begin{enumerate}
\item The mean-VaR efficient wealth at time $t$ is
\begin{equation*}
\begin{aligned}
X^{\text{VaR}}_{t,\lambda}
=&\frac{\lambda}{1+\lambda} \cdot \frac{x}{\xi_t} \left( N \left(-d_1 (t, \xi_t, \xi_{\alpha}) \right)+N \left( d_1 (t, \xi_t, \underline{\xi}_{\text{VaR}} ) \right) \right) \\
&+\underline{X}_{\text{VaR}} e^{-r(T-t)} \left( N \left( d_2 (t, \xi_t, \xi_{\alpha}) \right)-N \left( d_2 (t, \xi_t, \underline{\xi}_{\text{VaR}} ) \right) \right).
\end{aligned}
\end{equation*}
\item The optimal portfolio policy at time $t$ is
\begin{equation*}
\begin{aligned}
\pi^{\text{VaR}}_{t,\lambda}=& \frac{\lambda}{1+\lambda} \cdot \frac{x}{\xi_t} \cdot \left( N\left(-d_1(t, \xi_t, \xi_{\alpha}) \right)+N\left(d_1(t, \xi_t, \underline{\xi}_{\text{VaR}}) \right) \right) \frac{\theta}{\sigma} \\
&+\frac{ e^{-r (T-t)} \nu \left( d_2 (t, \xi_t, \xi_{\alpha}) \right) }{\sigma \sqrt{T-t} } \cdot \left( \underline{X}_{\text{VaR}}-\frac{\lambda}{1+\lambda} \cdot \frac{x}{\xi_{\alpha}} \right).
\end{aligned}
\end{equation*}
\end{enumerate}
\end{description}
Here and hereafter
\begin{equation*}
\begin{aligned}
d_1 (t, \xi_t, y) &=\frac{\ln \frac{y}{\xi _t}+(r+\frac{\theta ^2 }{2} )(T-t) }{\theta \sqrt{T-t}}, \\
d_2 (t, \xi_t, y) &=d_1 (t, \xi_t, y)-\theta \sqrt{T-t},
\end{aligned}
\end{equation*}
and $N (\cdot)$ is the standard normal distribution function, and $\nu (\cdot)$ is the standard normal probability density function.
\end{corollary}
\subsection{Mean-ES efficient portfolio}
In this subsection, we specialize our setting to the mean-ES optimization problem. In particular, we consider the optimization problem \eqref{prob:martingale} when the WVaR risk measure is given by the ES at a confidence level $0<\alpha<1$, namely
\begin{equation*}
\rho _{\Phi} (X)=\text{ES}_{\alpha} (X)=-\frac{1}{\alpha} \int_0^{\alpha} G_{X} (z)dz.
\end{equation*}
In other words, $\Phi$ admits a density $\phi (z)=\frac{1}{\alpha}\mathbf{1}_{z \le \alpha}$, for all $z \in [0,1]$.
\begin{proposition}\label{prop:4.2}
When $\Phi$ admits a density $\phi (z)=\frac{1}{\alpha}\mathbf{1}_{z \le \alpha}$ with $0<\alpha< 1$, we have the following assertions.
\begin{description}
\item[Case $\lambda=0$.] \begin{enumerate}
\item The min-ES efficient terminal wealth is
\begin{equation*}
X^{\text{ES}}_{T,0}=
\left \{
\begin{aligned}
& \frac{x}{\alpha \xi_T} , ~ & \xi_T > \overline{\xi}_{\text{ES}},\\
& \underline{X}_{\text{ES}}, ~ & \xi_T \le \overline{\xi}_{\text{ES}},
\end{aligned}
\right.
\end{equation*}
where
\begin{equation*}
\begin{aligned}
\underline{X}_{\text{ES}} &=\frac{x}{\alpha \overline{\xi}_{\text{ES}} },\\
\overline{\xi}_{\text{ES}} &=G_{\xi} (1-w^{-1}(t_0)),
\end{aligned}
\end{equation*}
and $t_0$ is given in Lemma \ref{lemma: f2}.
\item The optimal log-return is
$$R^{\text{ES}}_{T,0}=\frac{1}{T} \ln \frac{X^{\text{ES}}_{T,0}}{x}.$$
\item The ES of the optimal log-return is
\begin{align*}
\text{ES}_{\alpha}(R^{\text{ES}}_{T,0})&=\frac{1}{\alpha T} \left[ \ln \alpha \cdot N \left( - \frac{\ln \xi_{\alpha} + \left( r + \frac{\theta^2}{2} \right)T}{ \theta \sqrt{T}} \right) \right. \\
&\quad\;+ \ln \overline{\xi}_{\text{ES}} \cdot \left( N \left( \frac{\ln \overline{\xi}_{\text{ES}} + \left( r + \frac{\theta^2}{2} \right)T}{ \theta \sqrt{T}} \right) - N \left( \frac{\ln \xi_{\alpha} + \left( r + \frac{\theta^2}{2} \right)T}{ \theta \sqrt{T}} \right) \right) \\
&\quad\;\left. + \frac{\theta \sqrt{T}}{\sqrt{2 \pi}} e^{ - \frac{\left( \ln \overline{\xi}_{\text{ES}} + \left( r + \frac{\theta^2}{2} \right)T \right)^2}{2 \theta^2 T}} - \left( r + \frac{\theta^2}{2} \right) T N \left( - \frac{\ln \overline{\xi}_{\text{ES}} + \left( r + \frac{\theta^2}{2} \right)T}{ \theta \sqrt{T}} \right) \right] ,
\end{align*}
where $\xi_{\alpha} =G_{\xi} (1-\alpha).$
\end{enumerate}
\item[Case $0<\lambda<\infty$.]
\begin{enumerate}
\item The mean-ES efficient terminal wealth is
\begin{equation*}
X^{\text{ES}}_{T,\lambda}=
\begin{cases}
\frac{\frac{1}{\alpha}+\lambda}{1+\lambda} \cdot \frac{x}{\xi_T} , ~ & \xi_T > \overline{\xi}_{\text{ES}},\\
\underline{X}_{\text{ES}} , ~ & \underline{\xi}_{\text{ES}} < \xi_T \le \overline{\xi}_{\text{ES}} ,\\
\frac{\lambda}{1+\lambda} \cdot \frac{x}{\xi_T} , ~ & \xi_T \le \underline{\xi}_{\text{ES}},
\end{cases}
\end{equation*}
where
\begin{equation*}
\begin{aligned}
\underline{X}_{\text{ES}} &=\frac{\frac{1}{\alpha}+\lambda}{1+\lambda} \cdot \frac{x}{\overline{\xi}_{\text{ES}} }=\frac{\lambda}{1+\lambda} \cdot \frac{x}{\underline{\xi}_{\text{ES}}},\\
\overline{\xi}_{\text{ES}} &=G_{\xi} (1-w^{-1}( t_1(\lambda) )),\\
\underline{\xi}_{\text{ES}} &=\frac{\lambda}{\frac{1}{\alpha}+\alpha} \overline{\xi}_{\text{ES}},
\end{aligned}
\end{equation*}
and $t_1 (\lambda)$ is given in Lemma \ref{lemma: f3}.
\item The optimal log-return is
$$R^{\text{ES}}_{T,\lambda}=\frac{1}{T} \ln \frac{X^{\text{ES}}_{T,\lambda}}{x} .$$
\item The ES of the optimal log-return is
\begin{align*}
\text{ES}_{\alpha}(R^{\text{ES}}_{T,\lambda})&=\frac{1}{\alpha T} \left[ \ln \left( \frac{1+\lambda}{\frac{1}{\alpha} + \lambda} \right) \cdot N \left( - \frac{\ln \xi_{\alpha} + \left( r + \frac{\theta^2}{2} \right)T}{ \theta \sqrt{T}} \right) \right. \\
&\quad\;+ \ln \overline{\xi}_{\text{ES}} \cdot \left( N \left( \frac{\ln \overline{\xi}_{\text{ES}} + \left( r + \frac{\theta^2}{2} \right)T}{ \theta \sqrt{T}} \right) - N \left( \frac{\ln \xi_{\alpha} + \left( r + \frac{\theta^2}{2} \right)T}{ \theta \sqrt{T}} \right) \right) \\
&\quad\;\left. + \frac{\theta \sqrt{T}}{\sqrt{2 \pi}} e^{ - \frac{\left( \ln \overline{\xi}_{\text{ES}} + \left( r + \frac{\theta^2}{2} \right)T \right)^2}{2 \theta^2 T}} - \left( r + \frac{\theta^2}{2} \right) T N \left( - \frac{\ln \overline{\xi}_{\text{ES}} + \left( r + \frac{\theta^2}{2} \right)T}{ \theta \sqrt{T}} \right) \right],
\end{align*}
where $\xi_{\alpha} =G_{\xi} (1-\alpha).$
\end{enumerate}
\end{description}
\end{proposition}
Figure \ref{figure:ES wealth1} depicts the optimal terminal payoff of the min-ES portfolio ($\lambda=0$). The investor classifies market scenarios into two subsets: in the good states ($\xi \le \overline{\xi}_{\text{ES}}$), the investor receives a constant payoff $\underline{X}_{\text{ES}}$; in the bad states ($\xi > \overline{\xi}_{\text{ES}}$), the payoff is a multiple ($1/\alpha$) of the growth optimal portfolio.
\begin{figure}[H]
\centering
\centering
\includegraphics[width=0.6\textwidth]{./Figures/ES/ES_min.eps}
\caption{The Min-ES Efficient Terminal Wealth ($\lambda=0$)}\label{figure:ES wealth1}
\end{figure}
Figure \ref{figure:ES wealth2} displays the optimal terminal payoff of the mean-ES efficient portfolio ($0<\lambda<\infty$). The investor classifies market scenarios into three subsets: in the good states ($\xi \le \underline{\xi}_{\text{ES}}$), the terminal payoff is a fraction ($\lambda/(1+\lambda)$) of the growth optimal portfolio; in the intermediate states ($\underline{\xi}_{\text{ES}} < \xi \le \overline{\xi}_{\text{ES}}$), the investor receives a constant payoff $\underline{X}_{\text{ES}}$; in the bad states ($\xi > \overline{\xi}_{\text{ES}}$), the terminal payoff is a multiple ($(\frac{1}{\alpha}+\lambda)/(1+\lambda)$) of the growth optimal portfolio. In contrast to the mean-VaR efficient portfolio, the terminal payoff of the mean-ES efficient portfolio is continuous in the state price density.
\begin{figure}[H]
\centering
\centering
\includegraphics[width=0.6\textwidth]{./Figures/ES/ES_efficient.eps}
\caption{The Mean-ES Efficient Terminal Wealth ($0<\lambda<\infty$)}\label{figure:ES wealth2}
\end{figure}
Figure \ref{figure:ES efficient} shows the mean-ES efficient frontiers for different confidence levels $\alpha$. The efficient frontier is a concave curve that connects the growth optimal portfolio (colored dots) with the min-ES portfolio (colored crosses). The growth optimal portfolio has the highest expected log-return but also the highest ES. By contrast, the min-ES portfolio has the smallest expected log-return but also the lowest ES. In contrast to the min-VaR portfolio, the risk of the min-ES portfolio is finite and thus the mean-ES efficient frontier is a finite curve. Figure \ref{figure:ES efficient} also displays a sensitivity analysis of the efficient frontier with respect to $\alpha$, the confidence level of ES. As $\alpha$ increases, the efficient frontier shifts to the left: for a given level of the expected log-return, the ES of the corresponding efficient portfolio decreases as $\alpha$ increases. In particular, the minimum ES that the investor can achieve is decreasing in $\alpha$.
\begin{figure}[H]
\centering
\includegraphics[width=0.6\textwidth]{./Figures/ES/ES_efficient_frontier.eps}
\caption{The Mean-ES Efficient Frontier with $0<\lambda<\infty$, $T=1$, $r=0.05$, and $\theta=0.4$. }\label{figure:ES efficient}
\end{figure}
The following corollary presents the optimal time-$t$ wealth and portfolio policy. The proof is similar to that of Corollary \ref{coro:4.1} and thus we omit it.
\begin{corollary}
We have the following assertions.
\begin{description}
\item[Case $\lambda=0$.] \begin{enumerate}
\item The min-ES efficient wealth at time $t$ is
\begin{equation*}
X^{\text{ES}}_{t,0}=\frac{x}{\alpha \xi_t} N \left(-d_1 (t, \xi_t, \overline{\xi}_{\text{ES}} ) \right)+\underline{X}_{\text{ES}} e^{-r (T-t)} N \left( d_2 (t, \xi_t, \overline{\xi}_{\text{ES}} ) \right).
\end{equation*}
\item The efficient portfolio policy at time $t$ is
\begin{equation*}
\pi^{\text{ES}}_{t,0}=\frac{x}{\alpha \xi_t} N \left(-d_1 (t, \xi_t, \overline{\xi}_{\text{ES}} ) \right) \frac{\theta}{\sigma} .
\end{equation*}
\end{enumerate}
\item[Case $0<\lambda<\infty$.]
\begin{enumerate}
\item The mean-ES efficient wealth at time $t$ is
\begin{align*}
X^{\text{ES}}_{t,\lambda}
&= \frac{\frac{1}{\alpha}+\lambda}{1+\lambda} \cdot \frac{x}{\xi_t} N \left(-d_1 ( t, \xi_t, \overline{\xi}_{\text{ES}} ) \right)\\
&\quad\;+\underline{X}_{\text{ES}} e^{-r(T-t)} \left( N \left( d_2 (t, \xi_t, \overline{\xi}_{\text{ES}} ) \right)-N \left( d_2 (t, \xi_t, \underline{\xi}_{\text{ES}} ) \right) \right)\\
&\quad\;+\frac{\lambda}{1+\lambda} \cdot \frac{x}{\xi_t} N \left( d_1 (t, \xi_t, \underline{\xi}_{\text{ES}} ) \right).
\end{align*}
\item The efficient portfolio policy at time $t$ is
\begin{align*}
\pi^{\text{ES}}_{t,\lambda}& =\frac{x}{\xi_t} \cdot \frac{\theta}{\sigma} \left(\frac{\frac{1}{\alpha}+\lambda}{1+\lambda} N \left(-d_1 (t, \xi_t, \overline{\xi}_{\text{ES}} ) \right)+\frac{\lambda}{1+\lambda} N \left( d_1 (t, \xi_t, \underline{\xi}_{\text{ES}} ) \right) \right).
\end{align*}
\end{enumerate}
\end{description}
\end{corollary}
\subsection{Comparison with \cite{he2015dynamic}}
\cite{he2015dynamic} consider a continuous-time mean-risk portfolio choice problem in which the risk is measured by WVaR. They assume the decision-maker minimizes the risk of terminal wealth, while maintaining the expected terminal wealth above a prescribed target. They find that the model can lead to extreme risk-taking behaviors. When bankruptcy is allowed, the optimal terminal wealth is binary, i.e., the investor invests a small amount of money in an extremely risky digital option and saves the rest of the money in the risk-free asset. When bankruptcy is prohibited, the terminal wealth can be three-valued and the optimal strategy is to invest a small amount of money in an extremely risky digital option and put the rest in an asset with moderate risk. These strategies are not commonly seen in practice and are not appropriate for many investors. Furthermore, the optimal value (the risk) is independent of the expected terminal wealth target. Therefore the efficient frontier is a vertical line in the mean-risk plane and there is no explicit trade-off between risk and return. They conclude that using the WVaR on terminal wealth is not an appropriate model of risk for portfolio choice.
In contrast to \cite{he2015dynamic}, our model uses the expected target and risk measure on log-returns instead of terminal wealth. When the risk is evaluated by the VaR or ES, two popular risk measures, we find that the investor classifies market scenarios into different states, in which the terminal payoff is a multiple or fraction of the growth optimal portfolio, or constant. Furthermore, the efficient frontier is a concave curve that connects the min-risk portfolio with the growth optimal portfolio. Our model allows for an explicit characterization of the risk-return trade-off and may serve as a guideline for investors to set reasonable investment targets. Our results demonstrate that it is more appropriate to use the WVaR, in particular, the VaR and ES, on the log-return instead of the terminal wealth for portfolio choice.
\section{Conclusion}\label{sec:conclusion}
We have proposed and solved a dynamic mean-WVaR portfolio choice problem with risk measured to log-returns, as opposed to terminal wealth in \cite{he2015dynamic}. Our model conquers the ill-posedness of the mean-WVaR criterion for terminal wealth in \cite{he2015dynamic}, and allows for an explicit and meaningful characterization of the trade-off between return and risk. We have demonstrated that our proposed mean-WVaR criterion for log-returns is more appropriate and tractable than the mean-WVaR criterion for terminal wealth in serving as a guideline for dynamic portfolio choice.
\newpage
|
2024-02-18T23:40:24.446Z
|
2021-12-30T02:23:20.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14451","language":"en","timestamp":1640831000000,"url":"https:\/\/arxiv.org\/abs\/2112.14451","yymm":"2112"}
|
proofpile-arXiv_000-10024
|
{"provenance":"002.jsonl.gz:10025"}
| null | null |
\section{Introduction}
\label{sec:intro}
\qquad General circulation models (GCMs) require high accuracy solutions of radiative transfer equation at every grid step (spatial and temporal) in order to reproduce peculiar mesoscale structures that are notably presented in Venus atmosphere. However, it is unfeasible to include detailed line-by-line calculations to GCMs, subsequently some ways of parameterization should be involved.
\qquad By this time there is a sufficient lack of self-consistent parameterizations that whether are not too simplistic (e.g. double-gray radiative transfer or Newtonian approximation) or not too much constrained to various atmospheric parameters, specific GCM or way of solving radiative transfer equation. Thus, our motivation is to offer the fresh way to accelerate radiative calculations suitable to climate modeling. The earlier approach is based on look-up-tables of heating and cooling rates generated by \citet{Haus2017} and utilized in IPSL GCM by \citet{Garate-Lopez2018}. However, we adhere to the different concept of speeding up the radiation block of GCM.
\qquad In our approach we follow original k-distribution technique described in \citep{Fomin2005} instead of familiar correlated-k method. We construct effective cross-sections for absorption by gaseous constituents and unknown UV absorber regardless of its chemical composition. Mentioned parameterizations depend only on initial absorption cross-sections. In addition, we also suggest spectral points for calculation of Rayleigh scattering and clouds optical features (extinction coefficient and scattering indicatrix). Thus, we propose a universal tool to accelerate radiative transfer calculations. One needs to solve radiative transfer equation just eight times using our effective cross-sections to accurately reproduce fluxes and heating rates in UV region, while other blocks in atmospheric modeling scheme can be unchanged. We validate the efficiency of our parameterizations through line-by-line procedures with reference Monte-Carlo model. Construction of parameterizations of molecular absorption in other spectral regions will be the subject of future works.
\qquad The paper is organized as follows. In section \ref{sec:model} we explain the modeling part of the work: the technique for constructing the parameterizations and description of the reference Monte-Carlo model used in validation procedures. Section \ref{sec:result} refers to the dataset with parameterizations and the result of validation procedures. We provide some conclusive remarks in section \ref{sec:conclusion}.
\section{Methodology}
\label{sec:model}
\large{
\subsection{Construction of parameterizations}
}
\qquad The way we construct the parameterizations essentially corresponds to the one used in \citep{Fomin2005, Fomin2017} for shortwave region of Earth's atmosphere with some revision. In the case of Venus atmosphere in UV spectral region this method is simplified due to ignorance of absorption cross-sections temperature dependence \citep{Haus2015}. The idea of creating parameterizations is the following. Let us consider two modeling atmospheres: ``simple'' atmosphere with no scattering and no reflection from surface (A) and full-consistent Venus atmosphere (used in reference line-by-line calculations) (B). And suppose that one needs to find (``retrieve'') absorption cross-section of fixed gaseous species from given radiative fluxes obtained in the atmospheres A and B. For fixed wavelength $\lambda^*$ ``retrieval'' procedures for both atmospheres obviously will give the same value: $\sigma(\lambda^*)$, where $\sigma(\lambda)$ is the initial absorption cross-section. However, radiation calculations for atmosphere A are much faster and simpler than for atmosphere B, because they are based on the trivial Beer-Lambert law. We can follow the same procedure in simple atmosphere A but for the given spectral interval $\Delta\lambda$ and instead of monochromatic cross-sections ``retrieve'' \textit{effective cross-sections}: $\sigma_{\text{eff}}(\Delta\lambda)$. Effective cross-sections represent profile of initial absorption cross-sections averaged with solar spectrum in the interval $\Delta\lambda$. We define effective cross-section in the i-th atmospheric layer through the following formula:
\begin{equation*}
\sigma_{\text{eff},\,i}=\frac{\text{ln}(F_{i+1}^{\,\downarrow}/F_{i}^{\,\downarrow})}{\Delta\zeta_i},
\end{equation*}
where $F_i^{\,\downarrow}$ and $F_{i+1}^{\,\downarrow}$ are the downward fluxes at the bottom and at the top of the $i$-th discrete height layer. $\Delta\zeta_i = \zeta_{i+1}-\zeta_{i}$ \, is the number of absorber molecules along the direct solar radiation in $i$-th layer. As the absorber amount along the radiation path strongly correlates with the zenith angle, it is more useful to introduce dependence of the effective cross-sections from the value $\zeta$, instead of height $h$. Thus defined values $\sigma_{\text{eff},\,i}$ can be used in one-time solution of radiative transfer equation to find radiative fluxes and heating rates in the given spectral interval $\Delta\lambda$.
\qquad More wider the interval $\Delta\lambda$ -- more bigger the difference between radiative fluxes and heating rates calculated for simple atmosphere A and ``real'' atmosphere B. Nevertheless, we found out that spectral intervals where effective cross-sections obtained in simple atmosphere A yield good accuracy, appear to be very large. Thus, we divide 125-400 nm region into eight \textit{k-intervals}, where effective cross-sections yield good accuracy in fluxes and heating rates. For providing stable justification of our approach, we validate our parameterizations with reference Monte Carlo model. During the validation procedures for each k-interval we compare radiative fluxes and heating rates calculated in two ways: as a result of one-time solution of radiative transfer equation using effective cross-section for this k-interval and as a result of line-by-line calculations in this k-interval with reference Monte Carlo model.
\qquad After thorough manual inspection of the whole UV spectral interval, we divided it into 8 k-intervals to calculate effective cross-sections for each of them. In each k-interval we selected wavelength points for calculation of Rayleigh scattering in order to obtain best fits with reference simulations. For convenience, optical features of the Venus clouds (extinction coefficient and scattering indicatrix) could be taken at the same point since they don't significantly depend on wavelength in UV region. We also specify the incident solar flux at the top of the atmosphere (TOA) for each k-interval as this value relates to the way of choosing solar spectrum data. All detailed information about the k-intervals is summarized in Table \ref{tab:k-intervals}.
\bigskip
\begin{minipage}{\linewidth}
\centering
\captionof{table}{Detailed information about k-intervals for accelerating radiative calculations in UV region}
\label{tab:k-intervals}
\begin{tabular}{ C{1.0in} C{1.4in} C{1.2in} C{0.8in} C{1.4in}}\toprule[1.5pt]
\bf k-interval & \bf Spectral region, $\bf 10^4\text{cm}^{-1}$ & \bf Superior absorbers & \bf ${\bf \nu_R}$, $\bf \text{cm}^{-1}$ & \bf Incident solar flux, W/m$^{\bf 2}$ \\\midrule
F-UV & 5 - 8 & CO$_2$ & 66000 & 0.20\\
M-UV & 3.(3) -- 5 & SO$_2$, CO$_2$ & 36000 & 27.74\\
N-UVa & 3.1 -- 3.(3) & SO$_2$, UVA & 32000 & 28.19\\
N-UVb & 3.0 -- 3.1 & SO$_2$, UVA & 30000 & 19.91\\
N-UVc & 2.9 -- 3.0 & UVA, SO$_2$ & 30000 & 20.20\\
N-UVd & 2.7 -- 2.9 & UVA, SO$_2$ & 28000 & 52.40\\
N-UVe & 2.55 - 2.7 & UVA, SO$_2$ & 26250 & 17.64\\
N-UVf & 2.5 - 2.55 & UVA, SO$_2$ & 25250 & 47.27\\
\bottomrule[1.25pt]
\end {tabular}\par
\bigskip
\normalsize{
\begin{itemize}
\item $\nu_R$ -- is the recommended wavenumber for calculation of Rayleigh scattering and clouds optical features during one-time radiative transfer simulation
\item UVA relates to the unknown ultraviolet absorber
\item The left border of M-UV region was chosen for convenience (300 nm)
\item More dominant absorber goes first. Other active absorbers can be neglected while considering the whole UV-region (more in Section \ref{subsec:validation}).
\end{itemize}
}
\end{minipage}
\subsection{Monte-Carlo reference model}
\qquad Our reference Monte-Carlo model was previously successfully exploited for Earth' conditions in UV region by \citet{Sukhodolov2016}. It is a high-resolution (0.25 cm$^{-1}$) plane-parallel short-wave version of FLBLM model \citep{Forster2011, Fomin2012}. For the present studies it was adopted to the Venus conditions according to the baseline work by \citet{Haus2015}, in which authors performed line-by-line calculations along with sensitivity analysis to various spectroscopic and atmospheric parameters from 125 nm to 1000 $\mu$m. Hereafter we list the key parameters of the atmospheric model.
\subsubsection{UV absorption cross-sections}
\qquad Photoabsorption spectra of gaseous components in the atmosphere are relatively smooth in comparison with visible and IR-spectra. This fact is largely determined by the nature of absorption of UV rays: it drives the course of the chemical reaction of photodissociation. Thus, to incorporate UV spectroscopy into simulations, there is no need to trace the contours of individual lines. Relevant sources on absorption cross-sections of atmospheric gaseous constituents are summarized in Table \ref{tab:photoabsorption}. Note that in some experiments extinction coefficients or absorption coefficients were measured, which are not so convenient in our studies. We refer to MPI-Mainz spectral atlas \citep{Keller-Rudek2013} to gain datasets in unique format: absorption cross-section in cm$^2$/molecule from wavelength in nm. Figure \ref{fig:photoabsorption} shows absorption cross-sections as functions from wavelength for all gaseous constituents which are considered to be active in UV-region. The strongest absorption for all gaseous components is manifested in the far ultraviolet in 125-200 nm, where the intensity of solar radiation is insignificant. In present paper cross-sections of carbon dioxide are represented as a combination of data from \citet{Shemansky1972}, \citet{Parkinson2003}, \citet{Yoshino1996} in different spectral intervals. As \citet{Haus2015} we also neglected the temperature dependence of the cross-sections for all gases. Nonetheless, \citet{Marcq2019} consider temperature dependent absorption cross-sections for CO$_2$ and SO$_2$. We address this possible issue in Section \ref{subsec:validation}.
\begin{figure}
\centering
\includegraphics[scale=1.0]{ACS.png}
\caption{Absorption cross-sections of gaseous constituents in Venus atmosphere in UV region}
\label{fig:photoabsorption}
\end{figure}
\begin{table}[!htbp]
\caption{\bf Sources of wavelength-dependent absorption cross-sections}
\centering
\begin{tabular}{llll}
Species & Spectral region & Source paper & Temperature\\
\midrule
CO$_2$ & 125 -- 163 nm & \citep{Yoshino1996} & 295K\\
CO$_2$ & 163 -- 200 nm & \citep{Parkinson2003} & 295K\\
CO$_2$ & 200 -- 300 nm & \citep{Shemansky1972} & 298K \\
SO$_2$ & 125 -- 400 nm & \citep{Manatt1993} & 293K\\
SO & 125 -- 330 nm & \citep{Phillips1981, Belyaev2012} & 293K\\
O$_3$ & 125 -- 186 nm & \citep{Mason1996} & 298K\\
O$_3$ & 186 -- 400 nm & \citep{JPL2011} & 293-298K\\
H$_2$O & 125 -- 198 nm & \citep{JPL2011} & 298K \\
HCl & 140 -- 220 nm & \citep{Inn1975} & 298K \\
OCS & 185 -- 300 nm & \citep{Molina1981} & 295K \\
\bottomrule
\end{tabular}
\label{tab:photoabsorption}
\end{table}
\subsubsection{Unknown UV absorber}
\qquad The unknown UV absorber (UVA) is considered to be an additional source of radiation absorption in the spectral range 320-400 nm \citep{Pollack1980}. We include the UVA to our atmospheric model according to one of the latest parameterizations by \citet{Haus2015} (we use in simulations both high-altitude and low-altitude models, fig. 18) in contrast to \citet{Crisp1986, Marcq2019} who effectively considered it by depressing single scattering albedo of cloud mode 1. \citet{Haus2015} extracted (``retrieved'') the absorption cross-sections and concentration in such a way that the simulation results were in adequate agreement with the experimental Bond albedo data by \citet{Moroz1981}. This approach allows to treat unknown UV absorber as another species with its own concentration profile and absorption cross-section without specifying a concrete molecular formula. Though nominal high-altitude model given by \citet{Haus2015} features only 10 particles per cm$^3$ in 58-72 km region, huge absorption cross-sections conduce effective optical depths of UV absorber comparable to corresponding ones of SO$_2$. The active debate about the chemical composition of UVA is still going and the cycle of sulfur species in the upper clouds layer is not yet fully understood. Because the frontier OSSO parameterization by \citet{Frandsen2016} didn't yield realistic radiance factors in radiation scheme in \citep{Marcq2019}, we decided to keep ``retrieved'' cross-section by \citet{Haus2015} regardless of chemical composition.
\subsubsection{Other parameters}
\label{sec:other_param}
\qquad According to \citet{Haus2015, Marcq2019}, CO$_2$, SO$_2$, SO, H$_2$O, OCS, HCl and O$_3$ are the only active gases in UV-region which may possibly affect the outcome of radiation simulations. CO$_2$ abundance is assumed to be uniform in height and equals to 0.965. Due to the strong variability of SO$_2$ concentrations we incorporate two vertical profiles in radiative transfer simulations. The first one is taken from \citet{Haus2015} fig. 5, that corresponds to 150 ppmv below the clouds, 15\% larger than recommended by \citet{Bezard1993}. The second profiles features abundance of SO$_2$ below the clouds in according with \citet{Vandaele2017}, fig 1. Meanwhile, concentrations above the clouds in this profile is taken from theoretical models of \citet{Zhang2012}. SO:SO$_2$ ratio is set to be constant and equal to 10\% \citep{Marcq2019, Marcq2020}. The volume mixing ratio of water vapor is assumed to be 32.5 ppmv below 50 km \citep{Marcq2009,Arney2014} and 3 ppmv above 70 km \citep{Cottini2012, Fedorova2016}. Recently discovered ozone is accounted in radiative transfer simulations via profile from theoretical model by \citet{Krasnopolsky2013}, which features typical abundance of 10 ppbv at 90 km. OCS concentrations are significant only below clouds, while HCl volume mixing ratio is set to be 0.5 ppmv for all heights. However, during validation we show that most radiation redistribution occurs only up above the clouds layer, so the influence of H$_2$O, OCS and HCl is negligibly small.
\qquad Strength of the Rayleigh scattering is proportional to fourth power of wavenumber. Thus, it is assumed to be a crucial ingredient in radiation simulations in UV region. We take Rayleigh scattering into consideration in the way similar that \citet{Haus2015} do. They made slight correction to formula of \citet{Hansen1974} for pure CO2 atmosphere, changing the surface pressure. We use the same value of $p_0=92.1$ bar in agreement with VIRA model. Despite Rayleigh scattering optical depth surges while going to the depth of the atmosphere, it will be shown that in UV region taking this scattering into account is minor compared to gaseous absorption. For example in F-UV region it can be completely neglected as it manifests itself in heights where nearly all the radiation is already absorbed.
\qquad Solar spectrum in the UV region may vary slightly depending on solar activity. We utilized the COSI spectrum from \citet{Shapiro2010, Shapiro2011}, which was recommended in \citep{Sukhodolov2016}. As shown by numerical experiments, usage of a various spectra does not significantly affect the resulting parameterizations of gaseous absorption. It is only required to change accordingly the solar fluxes in each k-interval falling on the upper boundary of the atmosphere.
\qquad We used fixed VIRA \citep{Zasova2006} thermal profile for equatorial region on the night side from 0 to 150 km in our modeling purposes. With this approximation we abide by \citet{Marcq2019} due to above mentioned assumption of temperature independence of absorption cross-sections. So, the present parameterization could not be affected by applying different temperature profiles. However the heating rates depend on pressure, we expect that they predominantly are influenced by absorption features and incident solar flux.
\qquad Our simulating scheme uses recent cloud model that consists of four modes of 75\% H$_2$SO$_4$ spherical aerosols \citep{Pollack1993, Zasova2007, Haus2013}. To find microphysical parameters we apply Mie theory code \citep{Fomin1998}. All four modes are log-normally distributed with parameters of modal radii of 0.3, 1.0, 1.4, 3.65 $\mu$m and unitless dispersions of 1.56, 1.29, 1.23, 1.28, respectively \citep{Pollack1993, Haus2013}. Refractive indices data were taken from \citet{Palmer1975}. Scattering properties do not significantly alter with wavelength in UV region. Thus, any other clouds parameters may be easily inserted in atmospheric model, as our gaseous absorption parameterization is obtained regardless of clouds features.
\section{Result and validation with the reference model}
\label{sec:result}
\qquad Original data of this study (mainly including resulting effective cross-sections) are available at Mendeley Data (\href{https://dx.doi.org/10.17632/97rskzggbj.1}{https://dx.doi.org/10.17632/97rskzggbj.1})
\subsection{Effective cross-sections}
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{CO2_F-UV.png}
\hfill
\includegraphics[width=0.48\textwidth]{SO2_M-UV.png}
\caption{CO$_2$ (left) and SO$_2$ (right) effective cross-sections in F-UV and M-UV k-intervals respectively}
\label{fig:cross_sec}
\end{figure}
\qquad In this section we present some information about the resulting parameterizations. In all tables $\sigma$ is an effective cross-section (in cm$^2$ per one molecule) and $\zeta$ is the absorber amount along the direct solar radiation from TOA to the given point (in molecules per cm$^2$). Tables are given for eight k-intervals defined in Table \ref{tab:k-intervals}. Fig.\ref{fig:cross_sec} shows effective cross-sections for CO$_2$ and SO$_2$ in F-UV and M-UV k-intervals respectively. Obtained effective cross-sections are literally the result of averaging the initial absorption cross-sections over the solar spectrum reached the current atmospheric level. At the upper levels, solar radiation is still weakly transformed and the averaging is actually carried out over the initial solar spectrum. While penetrating in the atmosphere, the solar spectrum changes so that the radiation remains only in those spectral regions where the absorption is weak. This explains the decrease in effective cross-sections with increasing absorber amount.
\subsection{Validation}
\label{subsec:validation}
\qquad Validation of the presented parameterizations was performed by comparison of downward and upward solar fluxes and heating rates obtained through fast and reference simulations. Figs. \ref{fig:validation} and \ref{fig:validation2} show upward and downward fluxes for different atmospheric profiles for 0\degree \, zenith angle. Fig. \ref{fig:validation} displays results of calculations with two different SO$_2$ profiles (see \ref{sec:other_param}), keeping UVA profile fixed (high-altitude model by \citet{Haus2015}). On fig. \ref{fig:validation2} we present results of calculations with high-altitude and low altitude models by \citet{Haus2015}, keeping fixed SO$_2$ profile from the latter study. For verification the sustainability of our approach we also checked the parameterizations for two solar zenith angles: 0\degree \,and\, 75\degree\, (see fig. \ref{fig:valid_heating}). Featured heights lie in range below 100 km. Validation displays good fits, resulting with relative discrepancies of less than 3\% for fluxes and less than 1\% for heating rates. During this procedure we also could analyze the contribution of each absorber. Atmospheric O$_3$ and SO play minor role in UV interval, less than 1 K/day contribution to the heating rate. However, we added cross-sections of this species to the dataset. Thus, we state that only CO$_2$, SO$_2$ and unknown UV absorber are relevant in terms of examination the integrated UV fluxes and heating rates.
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{Hauss_ET-KD_0deg.25-80_fig.png}
\hfill
\includegraphics[width=0.48\textwidth]{Zhang_ET-KD_0deg.25-80_fig.png}
\hfill
\caption{Downward and upward fluxes obtained from Monte-Carlo line-by-line simulations and from fast simulations with employing k-distribution (KD) technique with effective cross-sections. Profiles of SO$_2$ are taken from \citet{Haus2015} (left) and from \citet{Zhang2012,Vandaele2017} (right). Spectral interval 125 -- 400 nm.}
\label{fig:validation}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{XXXold_ET-KD_0deg.25-33_fig.png}
\hfill
\includegraphics[width=0.48\textwidth]{XXXold_ET-KD_0deg.25-33_fig.png}
\hfill
\caption{Downward and upward fluxes obtained from Monte-Carlo line-by-line simulations and from fast simulations with employing k-distribution (KD) technique with effective cross-sections. Profiles of unknown ultraviolet absorber are: high-altitude model (left) and low-altitude model (right) by \citet{Haus2015}. Spectral interval 300 -- 400 nm.}
\label{fig:validation2}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[scale=0.65]{heating_rates.png}
\caption{Solar heating rates for zenith angles $\theta=0\,\degree$\, and $\theta=75\,\degree$\, for Venus middle atmosphere. SO$_2$ and UVA profiles are taken from \citet{Haus2015}. Spectral interval 125 -- 400 nm. Dashed lines for the fast (KD) calculations are almost invisible due to good fit.}
\label{fig:valid_heating}
\end{figure}
\qquad We investigated temperature dependence of absorption cross-sections of CO$_2$ in F-UV region taking data of Parkinson for 195 and 295 K \citep{Yoshino1996, Parkinson2003} and linearly extrapolating it between those temperatures. Resulting heating rates discrepancies appeared to be less than 1\%. We expect that for climate modeling purposes the temperature dependence could be ignored as the relevant atmosphere is from 50 -- 100 km where cross-sections doesn't change dramatically. However, we do not dismiss the interest that future research should be conducted on this issue.
\section{Conclusion}
\label{sec:conclusion}
\qquad In this work we present parameterizations of absorption by gaseous species and unknown UV absorber in 25000 - 80000 cm$^{-1}$ spectral interval. Such parameterizations are determined only by wavelength-dependent absorption cross-sections. Thus, it easily can be inserted in any radiative transfer scheme of any GCM. The result was validated with the help of original high-resolution line-by-line Monte-Carlo radiative transfer model. We expect that such parameterizations can facilitate Venus climate modeling studies. Future works will be dedicated to creating effective parameterizations of molecular and clouds absorption in visible and infrared spectral interval.
\newpage
|
2024-02-18T23:40:24.449Z
|
2022-04-11T02:15:27.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14487","language":"en","timestamp":1649643327000,"url":"https:\/\/arxiv.org\/abs\/2112.14487","yymm":"2112"}
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proofpile-arXiv_000-10025
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{"provenance":"002.jsonl.gz:10026"}
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\section{Introduction}
Duality map is a profound and widely used concept in physics, such as quark-hadron duality which bridge the gap between theoretical
predictions and experimentally observable quantities~\cite{Shifman:2000jv}, and the AdS/CFT by which string theory could quantitatively explain features of strongly coupled Quantum Chromodynamics (QCD)~\cite{Maldacena:1998zhr}. In this paper, we are interested in the renowned fermion-boson duality~\cite{Girardeau:1960}, and fermion-bit duality~\cite{Wetterich:2016yaw}. The former shows the equivalence of fermionic and bosonic particle systems, the latter indicates a map between fermions and Ising spins. Inspired by them, we propose a map between nucleons which bound in nuclei and Ising spins in Ising model, it is labeled as ``SRC-bit duality'', here SRC is abbreviation of short-range correlation. In the rest of this section, introduction of these dualities are presented in order.
The fermion-boson duality stems from the following question --- can bosons and fermions transform into each other? It is possible in supersymmetry, a hot candidate for solving the hierarchy problem, which is still waiting to be examined by high energy experiment. However, in low dimensional non-relativistic systems, the equivalence of bosonic and fermionic systems are reported long time ago~\cite{Girardeau:1960,Girardeau:1965,Mattis1965,Coleman:1974bu,Tomonaga:1950zz,Luttinger:1963zz,schmidt:1998,crescimanno:2001,cheon:pla,Cheon:1998iy}. Massless boson and fermion theories in $1+1$ dimensional Minkowski and curved space-time are proved to be equivalent~\cite{freundlich:1972,davies:1978}. Because the spin-statistics relation is based on relativity, non-relativistic system can escape from the spin-statistics relation, thus boson can be spinning and fermion be spinless. The relation between boson and spinless fermion may shed light on the general properties of boson-fermion duality.
Fermion-bit duality is proposed from the fact that models for fermions and Ising spins share the same type of observables. In the case of fermions, the observables are occupation numbers $n(x)$ that take values zero or one. For $n(x)=1$ a fermion is present at $x$, while for $n(x)=0$ no fermion located in $x$. In the case of Ising spins, $s(x)$ can take values $\pm 1$ which can be understood as magnetic dipole moments of atomic ``spins'' in ferromagnetism. A relation between occupation numbers and Ising spins can be readily established, $n(x)=(s(x)+1)/2$. Based on these simple observations a map between fermions and Ising models has been proposed~\cite{Wetterich:2009tq,Wetterich:2010eh}. Since Ising spins can be associated to bits of information, this duality is under the name of ``fermion-bit duality''. It is worth noting that if Ising spins are considered as ``discrete bosons'', this fermion-bit duality establishes a general equivalence of fermions and bosons, which indicates the three dualities, ie., fermion-boson duality, fermion-bit duality and boson-bit duality are closely related.
Inspired by these duality maps, we propose a new one between nucleons and Ising spins. One may think of the stable nucleus as a tight ball of neutrons and protons (collectively called nucleons), held together by the strong nuclear force. This basic picture has been worked very well. Afterwards, deep inelastic scattering (DIS) led to the discovery that the nucleon is made of quarks. However, due to the small nuclear binding energy and the idea of quark-gluon confinement, it was thought that quarks had no explicit role in the nucleus, hence nuclei could still be described in terms of nucleons and mesons. In 1982, this understanding was changed by measurements performed by the European Muon Collaboration (EMC)~\cite{Aubert:1983xm}. The initial expectation was that physics at GeV scale would be insensitive to the nuclear binding effects which are typically on the order of several MeV scale. However, the collaboration discovered the per-nucleon deep inelastic structure function in iron is smaller than that of deuterium in the region $0.3<x_B<0.7$, here $x_B$ is the Bjorken variable. This phenomenon is known as EMC effect and has been observed for a wide range of nuclei \cite{Arneodo:1988aa,Arneodo:1989sy,Allasia:1990nt,Gomez:1993ri,Seely:2009gt}. Although the understanding of how the quark-gluon structure of a nucleon is modified by the surrounding nucleons has been brought to a whole new level, there is still no consensus as to the underlying dynamics that drives this effect. Currently, one of the leading approaches for describing the EMC effect is: nucleons bound in nuclei are unmodified, same as ``free'' nucleons most of the time, but are modified substantially when they fluctuate into SRC pairs. The connection between SRC and EMC effects has been extensively investigated in nuclear structure function measurements~\cite{Egiyan:2005hs,Hen:2012fm,Hen:2014nza,Duer:2018sby,Weinstein:2010rt,Chen:2016bde,Lynn:2019vwp,Xu:2019wso,Hatta:2019ocp,Huang:2021cac,Hen:2016kwk}.
SRC pairs are conventionally defined in momentum space as a pair of nucleons with high relative momentum and low center-of-mass (c.m.) momentum, where high and low are relative to the Fermi momentum of medium and heavy nuclei. In this paper, we will emphasize the similarities between the descriptions of SRCs and Ising spins. Based on these similarities, a new kind of map has been proposed in this paper, it is labeled as ``SRC-bit duality''.
The present manuscript is arranged as follows. In Sect.~\ref{duality_s_i}, we will discuss the map between SRCs in nucleons and Ising spins. One longstanding theoretical model which is used to depict the nucleons is presented as a correspondence to an explicit Ising model. Sect.~\ref{Simulation} will devoted to simulations of nucleon states in terms of Ising model, our preliminary results support the proposed map between nucleons and Ising spins. Finally, we summarize our work and comment on the future developments in Sect.~\ref{conclusions}.
\section{Map between SRCs and Ising spins}
\label{duality_s_i}
\subsection{Notations and definitions}
\label{duality_s_i_1}
The description of SRCs in nucleons and Ising spins in Ising model could share the same type of observables, as in line with implementations of fermions within Ising model. Ising spin $s(x)$ can take values $\pm 1$, each represents one of two spin states (spin up or down). Similarly, we can use $s_{src}(x)$ to represent the state of a nucleon. In nucleus, nucleons behave approximately as independent particles in a mean field, but occasionally ($20\%-25\%$ in medium or heavy nuclei) two nucleons get close enough to each other so that temporarily their singular short-range interaction cannot be well described by a mean-field approximation. These are the two-nucleon short-range correlations (2N-SRC). For $s_{src}(x)=1$ a nucleon which belongs to SRC pair is presented, while for $s_{src}(x)=0$ the corresponding nucleon can be regarded as independent particle. The relation between these two ``spins'' is
\begin{eqnarray}\label{relation}
s(x)=2s_{src}(x)-1 \,.
\end{eqnarray}
In addition to this simple relation, there are other common features between these two systems. In Ising model, the spins are arranged in a lattice, allowing each spin to interact with its neighbors. Consider a set $\Lambda$ of lattice sites, for each lattice site $x\in \Lambda$ there is a discrete variable $s(x)$ such that $s(x)\in\{+1,-1\}$, representing the site's spin. For any two adjacent sites $x,y\in\Lambda$ there is an interaction $J$. Besides, every site $y\in\Lambda$ is influenced by an external magnetic field $h$, the corresponding Hamiltonian is
\begin{eqnarray}\label{Ising_H}
H = -\frac{J}{2}\sum_{\langle x y\rangle} s_{x} s_{y}- \mu h \sum_{y} s_{y} \,,
\end{eqnarray}
where the first sum is over pairs of adjacent spins and the second term represents the universal interaction with external magnetic field, the magnetic moment is given by $\mu$. The physical quantities describing nucleons can also be divided into two parts, one for short-range and the other for long-range. The pedagogically sketched diagrams for Ising model and structure of nucleus are presented in Fig.~\ref{picsIS}. For instance, the nucleon spectral function $P(\mathbf{p},E)$ which is the joint probability to find a nucleon in a nucleus with momentum $\mathbf{p}$ and removal energy $E$ can be modeled as~\cite{CiofidegliAtti:1995qe}
\begin{eqnarray}
P(\mathbf{p}, E)= P_{1}(\mathbf{p}, E) + P_{0}(\mathbf{p}, E) \,,
\end{eqnarray}
where the subscript $1$ refers to high-lying continuum states that are caused by the short-range correlations and the subscript $0$ refers to values of $E$ corresponding to low-lying intermediate excited states.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.49\columnwidth]{pics1.eps}
\centering
\includegraphics[width=0.49\columnwidth]{pics2.eps}
\centering
\caption{Schematic diagrams for Ising model in ferromagnetism (left) and nucleus structure (right), both of them can be divided into two parts. For Ising model, they are adjacent interaction and universal external magnetic field interaction. For nucleus structure, they are SRCs and interaction with mean field which is created by nucleon's average interaction with the other nucleons.}
\label{picsIS}
\end{figure}
Another example is nuclear gluon distribution, we can parameterize the nuclear gluon distribution in the EMC region as that for the structure function~\cite{Xu:2019wso,Frankfurt:1993sp,Segarra:2019gbp}
\begin{eqnarray}
g_{A}\left(x_B, Q^{2}\right)= 2 n_{src}^{A} \delta \tilde{g}\left(x_B, Q^{2}\right) +A g_{p}\left(x_B, Q^{2}\right) \,,
\end{eqnarray}
where $n_{src}^{A}$ represents number of SRC pairs in nucleus $A$. Here we have made an approximation that all nuclear modifications originate from the nucleon-nucleon SRCs in the EMC region. $\delta \tilde{g}\left(x_B, Q^{2}\right)$ represents the difference between the gluon distribution in the SRC pair or in the free nucleon.
Inspired by the term ``fermion-bit duality'' presented in Ref.~\cite{Wetterich:2016yaw}. We propose a new ``SRC-bit duality'' which represents the map between the state of each nucleon in nucleus and the state of each lattice site in Ising model. An obvious benefit of this duality is that it allows to describe properties of nucleons in terms of classical statistical systems for Ising spins, with many highly developed methods available. The two-nucleon short-range correlations are defined operationally in experiments as having small c.m. momentum and large relative momentum, to which there are approximately $20\%$ nucleons belong. This means in a medium or heavy nuclei, the average $\bar{s}_{src}=0\times 80\%+1\times 20\%=0.2$, which corresponds to average Ising spin $\bar{s}=-0.6$ in terms of Eq.~(\ref{relation}).
For a Hamiltonian shown in Eq.~(\ref{Ising_H}), the partition function is
\begin{eqnarray}
Z=\sum_{\left\{s(x)\right\}} e^{-\beta H} \,,
\end{eqnarray}
where $\beta=(k_B T)^{-1}$ and $\left\{s(x)\right\}$ means sum over all possible configurations of spins. If the approach of mean field approximation utilized, the Hamiltonian can be further simplified,
\begin{eqnarray}
H_{M F}=-\sum_{x} \mu s(x)\left(h+\bar{h}\right) \,,
\end{eqnarray}
here $\bar{h}\equiv (\mathcal{Z} J/\mu)\bar{s}$, $\mathcal{Z}$ is known as coordination number and $\bar s$ is average Ising spin. Therefore, the partition function reads
\begin{eqnarray}
Z_{MF} &=& \prod_{x=1}^{N}\left(\sum_{s(x)=\pm 1} e^{-\beta \mu(h+\bar{h}) s(x)}\right) \nn\\
&=& \left[2 \cosh \left(\frac{\mu h}{k_{B} T}+\frac{\mathcal{Z} J }{k_{B} T}\bar{s}\right)\right]^{N} \,.
\end{eqnarray}
The magnetization can be written as
\begin{eqnarray}\label{M12}
\left\{\begin{array}{l}
M=N \mu \bar{s} \,,\\
M=-(\partial F/\partial h)=N\mu \tanh \left( \frac{\mu h}{k_{B} T}+\frac{\mathcal{Z} J }{k_{B} T}\bar{s} \right) \,,
\end{array}\right.
\end{eqnarray}
here $F$ is the Helmholtz free energy. $\bar s$ can be deduced from Eq.~(\ref{M12})
\begin{eqnarray}\label{bars}
\bar{s}=\tanh \left(\frac{\mu h}{k_{B} T}+\frac{\mathcal{Z} J}{k_{B} T} \bar{s}\right) \,.
\end{eqnarray}
Recall that $\bar{s}=-0.6$ is the counterpart of $\bar{s}_{src}=0.2$. It is interesting to find out under which condition can we get $\bar{s}=-0.6$. First, in the case of Ising model with no external magnetic field, Eq.~(\ref{bars}) simplifies
\begin{eqnarray}\label{nohsimp}
\bar{s}=\tanh \left(\frac{\mathcal{Z} J}{k_{B} T} \bar{s}\right) \,.
\end{eqnarray}
It is the coefficient $\frac{\mathcal{Z} J}{k_{B} T}$ determines the solution. The left panel of Fig.~\ref{pics12} shows shapes of r.h.s. of Eq.~(\ref{nohsimp}) with different coefficients numerically, one can see that the wanted solution $\bar{s}=-0.6$ can be achieved when $\frac{\mathcal{Z} J}{k_{B} T}\approx 1.16$. Second, in the case of Ising model with external magnetic field, both $\frac{\mu h}{k_B T}$ and $\frac{\mathcal{Z} J}{k_B T}$ will contribute to the result, the right panel of Fig.~\ref{pics12} presents the typical shape of r.h.s. of Eq.~(\ref{bars}) with desired solution $\bar{s}=-0.6$.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.35\columnwidth]{noh.eps}
\centering
\hspace{15mm}
\includegraphics[width=0.35\columnwidth]{yesh.eps}
\centering
\caption{Left panel shows the shapes of r.h.s. of Eq.~(\ref{nohsimp}) with different coefficients, the brown one is $f(\bar{s})=\bar{s}$. Desired solution (orange point) could only acquired with appropriate choice of coefficient $\frac{\mathcal{Z} J}{k_B T}$. Right panel shows the typical shape of r.h.s. of Eq.~(\ref{bars}) with desired solution (orange point).}
\label{pics12}
\end{figure}
We would like to give a few remarks here:
\begin{enumerate}
\item All these discussions are independent of the particular dynamics of the systems. The Sect.~\ref{duality_s_i} in our paper therefore constitutes a very general map from a discrete classical statistical ensemble to structure of nucleus which is formed by strong interaction. In this respect it is in line with earlier implementations of fermions within bosonic systems under gauge transformation.
\item According to the simple relation in Eq.~(\ref{relation}), there must be phase transition in Ising model in order to map the SRC phenomenon in nucleus (since $\bar{s}=0$ leads to $\bar{s}_{src}=0.5$, this is not the correct number measured in experiments). In one dimension, the solution of Ising model admits no phase transition, thus one should go to higher dimension to delve into this subject rigorously.
\item There are some arguments that single-particle correlations such as momentum distributions and single-particle spectral densities are not forced to be identical between bosons and fermions in Ref.~\cite{Sekino:2020urg}. Similarly, whether SRC-bit duality would help people understand parton distribution functions such as gluon distribution needs further investigation.
\item SRCs of more than two nucleons such as 3N-SRC probably also exist in nuclei although their probability is expected to be significantly smaller than the 2N-SRC. Thus there would be three states of a nucleon --- does not belongs to SRC, belongs to 2N-SRC or 3N-SRC. The 3-state Potts model is a natural extension of the Ising model where the spin on a lattice takes one of three possible values \cite{Wu:1982ra}. It would be intriguing to generalize the SRC-bit duality into these three states situations.
\end{enumerate}
The important aspect of Ising model is that a variety of problems can be investigated by the similar kind of modeling. Undoubtedly, with many highly developed methods available for Ising model, the SRC-bit duality map provides a new insight into the study of EMC effect in nuclear physics.
\subsection{Potential theoretical model which can be simulated with Ising model}
\label{duality_s_i_2}
In the rest of this section, we will discuss a theoretical model as a potential candidate for possessing an explicit Ising model and thus could be investigated in terms of many highly developed statistical methods. In this theory, the nucleon can be regarded as a superposition of two different configurations where one is ``bloblike'' configuration (BLC) with the normal nucleon size and the other is ``pointlike'' configuration (PLC) \cite{Frankfurt:1985cv}. The BLC can be thought of as an object that is similar to a nucleon. The PLC represents a three-quark system of small size which dominates the high-$x_B$ behavior of parton distribution function.
The ideal that different constituents of the nucleon have different sizes is directly related to EMC effects \cite{Frank:1995pv}. The Hamiltonian is given by the matrix
\begin{eqnarray}
H_{0}=\left[\begin{array}{cc}
E_{B} & V \\
V & E_{P}
\end{array}\right] \,,
\end{eqnarray}
where $E_P$ and $E_B$ are energies of PLC and BLC respectively, $V$ is the hard-interaction potential that connects the two components. We choose $E_P \gg E_B$ and $|V| \ll E_{P}-E_{B}$, so that the nucleon is mainly BLC. When placed in a nucelus, the BLC component of a nucleon feels an attractive nuclear potential Hamiltonian $H_1$:
\begin{eqnarray}\label{introU}
H_{1}=\left[\begin{array}{ll}
U & 0 \\
0 & 0
\end{array}\right] \,.
\end{eqnarray}
Therefore the complete Hamiltonian $H=H_0+H_1$ is presented as
\begin{eqnarray}
H=\left[\begin{array}{cc}
E_{B}-|U| & V \\
V & E_{P}
\end{array}\right] \,.
\end{eqnarray}
It is worth noting the inclusion of $U$ increase the energy difference between the BLC and the PLC components, which decreases the PLC probability. The eigenstates of $H$ are labeled as $| N \rangle_M$ and $| N^* \rangle_M$, here the subscript $M$ means medium-modified, they are approximately
\begin{eqnarray}\label{decom_NM1}
|N\rangle_{M} &=& |B\rangle+\epsilon_{M}|P\rangle \,, \nn\\
\left|N^{*}\right\rangle_{M} &=& -\epsilon_{M}|B\rangle+|P\rangle \,,
\end{eqnarray}
where $\epsilon_{M}=V/(E_{B}-|U|-E_{P})$, $|B\rangle$ stands for BLC state and $|P\rangle$ for PLC state. One can also write down the eigenstates of $H_0$,
\begin{eqnarray}
|N\rangle &=& |B\rangle+\epsilon|P\rangle \,,\nn\\
\left|N^{*}\right\rangle &=& -\epsilon|B\rangle+|P\rangle \,,
\end{eqnarray}
with $\epsilon=V/(E_B-E_P)$. Therefore the medium-modified nucleon $|N\rangle_{M}$ could be expressed in terms of the unmodified eigenstates $|N\rangle$ and $|N^*\rangle$ as
\begin{eqnarray}
|N\rangle_{M} \approx |N\rangle+\left(\epsilon_{M}-\epsilon\right)\left|N^{*}\right\rangle \,.
\end{eqnarray}
It is the second term whose functionality resembles the SRC pair described above which dominates the high-$x_B$ behavior of structure function, ie., the EMC effect measured in DIS experiments. By adjusting the amount of excited state $\left|N^{*}\right\rangle$ contained in nucleon $|N\rangle_{M}$, the deviation of the EMC ratio from unity could be predicted \cite{Frank:1995pv,CiofidegliAtti:2007ork}. In this theory, the degree of deviation is controlled by $U$, $V$ and $E_P-E_B$. We use $s_{src}=1$ to represent the excited state $\left|N^{*}\right\rangle$ and $s_{src}=0$ for the ground state $\left|N\right\rangle$. According to the simple relation in Eq.~(\ref{relation}), their correspondences are lattice sites with spin $s=1$ and $s=-1$ respectively. For an Ising model in certain dimension, the variables which determine the final magnetization state are temperature $T$, coupling $J$ and external magnetic field $h$.
One can envision that before bound in a nucleus, the nucleons can be regarded as a collection of "free particles" whose components are $|N\rangle = |B\rangle+\epsilon|P\rangle$. It is the potential $V$ that connects the two components, the amount of PLC decreases with the increase of $V^{-1}$ in which case most would be BLC. This is very similar to Ising model in ferromagnetism without external magnetic field, the states of lattices tends to be the same with the increase of coupling $J$. The energy difference $\Delta E=E_P-E_B$ is also an important factor, the number of BLC and PLC would be approximately equal when $\Delta E^{-1} \to \infty$, analogous to the case of no phase transition when $T\to\infty$ which is provided as an acceptance criteria for different spin states in Ising model. For $\Delta E^{-1} \to 0$, the huge energy difference indicates there is few PLC in nucleon, corresponding to no SRC pair. Similarly, when $T\to 0$, nearly all of the spin states are the same in Ising model.
Now suppose the nucleons are bound in a nucleus, they would feel an attractive nuclear potential $U$, which further decreases the PLC probability according to Eq.~(\ref{decom_NM1}). Similar situation occurs on paramagnetics when we add an downward external magnetic field $h$, this would convert more spin states of lattice sites to $-1$. From this point of view the general properties of variables in the models for nucleons and Ising spins are the same.
\section{Monte Carlo simulations of nucleon states}
\label{Simulation}
\subsection{Notations and definitions}
We will explore the issue of simulation on nucleon states in nucleus in terms of mature methods available for two-dimensional square lattice Ising model in more detail. In general, a Monte Carlo simulation processes a subset of configurations in the configuration space of a given
system, according to a predefined probability distribution. Here we would set the stats of all lattice sites to $-1$ as our predefined distribution. Eq.(\ref{relation}) allows one to relate the $s=-1$ to $s_{src}=0$ of nucleon state, which corresponds to the situation where all the nucleons don't belong to SRC.
The Metropolis algorithm will be utilized to perform importance sampling of the configuration space \cite{CasquilhoCamp}. In this method, a Markov chain of configurations is generated in which each configuration $C_{\ell+1}$ is obtained from the previous one $C_{\ell}$ with a suitably chosen transition probability $\omega_{(\ell \to \ell+1)}$ which is determined by Metropolis function
\begin{eqnarray}\label{MProb}
\omega_{(\ell \to \ell+1)}=\min \left[1, \exp \left(-\frac{\Delta E}{k_{B} T}\right)\right] \,,
\end{eqnarray}
ie.,
\begin{eqnarray}
\omega_{(\ell \to \ell+1)} &=& \exp \left(-\frac{\Delta E}{k_{B} T}\right)\,, \quad \text { if } \Delta E>0 \,,\nn\\
\omega_{(\ell \to \ell+1)} &=& 1\,, \quad \text { if } \Delta E<0 \,,
\end{eqnarray}
here $\Delta E = E(C_{\ell+1}) - E(C_{\ell})$. The process of $C_{\ell} \to C_{\ell+1}$ constitutes one Monte Carlo step (MCS), which may be taken as our unit of computational ``time''. We will take the time to $1\times 10^6$ in our simulation.
\subsection{Modified Hamiltonian of Ising model}
The Hamiltonian of Ising model has already been shown in Eq.~(\ref{Ising_H}). We will modify this Hamiltonian to make it more suitable for describing nucleons. The states of nucleons are simulated on a $400\times400$ lattice and these $1.6\times 10^5$ lattice sites are grouped into $8\times 10^4$ pairs taking into account the SRC always appears in pair. The spin of any pair of sites is either $+1$ or $-1$, the value of coupling $J$ depends on the magnetization state of the system.
We divide the first term in Eq.~(\ref{Ising_H}) which describes interaction between adjacent spins into two parts, one is responsible for the adjacent interaction between a pair of $+1$ states, the other accounts for the interaction between a pair of $-1$ states. Therefore, the modified Hamiltonian of Ising model is
\begin{eqnarray}\label{M_Ising_H}
H = -2C \, J_{up}\sum_{\langle i \rangle} s_{i} + 2C \, J_{down}\sum_{\langle j \rangle} s_{j} - \mu h \sum_{\langle j' \rangle} s_{j'} \,, \quad \text { for any } s_{i}=+1 ~\text{and}~ s_{j,j'}=-1 \,. \nn\\
\end{eqnarray}
The first sum runs over all pairs of nucleons with $s=+1$, and the second sum is over all pairs with $s=-1$. The factor $2$ is introduced to remind that there are two lattice sites with same spin in one pair, the dimension of coefficient $C$ is $E$, it is used to characterize the interaction strength relative to the external magnetic field. The third sum depicts the universal interaction with external magnetic field, we take $h$ to a negative value whose functionality only act on the $-1$ states, reducing the energy of this system. This is consistent with the function of $U$ in the theoretical model introduced at previous section in Eq.~(\ref{introU}).
One important feature of Metropolis algorithm shown in Eq.~(\ref{MProb}) indicates it allows MCS which increases the energy, albeit with a low probability if they increase the energy by a large amount. The variation of energy in every MCS are influenced by the couplings $J_{up}$ and $J_{down}$ which are expressed in terms of average Ising spin in this modified Ising model,
\begin{eqnarray}
J_{up} &=& -\frac{1}{L^2} \sum_{\mu} s_\mu = -\bar s \,,\\
J_{down} &=& 1+\frac{1}{L^2} \sum_{\mu} s_\mu = 1+\bar s \,.
\end{eqnarray}
Before simulation, we also need to specify the initial configuration, here is all spins point down (i.e., $s_i=-1$ for all $i$) at $t=0$. This configuration corresponds to a bunch of ``free'' nucleons without SRC pairs. Then we add an external magnetic field and evolve this system until it attains the equilibrium distribution. The simulation results would be shown in next section.
\subsection{Final results}
Here we present the final simulation results for the modified Ising model which are utilized to mimic the nucleons bound in a medium or heavy nuclei. Fig.~\ref{simulation} presents the evolution of the system from initial state ($s_i=-1$ for all $i$) to equilibrium state. After completion of $1\times10^6$ MCS, the ratio of the two components (nucleon belongs to SRC or not) remains at $20:80$ at $T=2.5$, which is consistent with experimental data. In this simulation, we take the coefficient $C=2$ and the external field $h=-4$. Fig.~\ref{svstime} shows the stability of this simulation, after $1\times 10^5$ MCS, $\bar s$ remains at about $-0.6$.
\begin{figure}[htbp]
\begin{minipage}{0.5\columnwidth}
\centerline{\includegraphics[width=1.\columnwidth]{MCS1.eps}}
\vspace{0pt}
\caption*{(a)}
\centerline{\includegraphics[width=1.\columnwidth]{MCS3.eps}}
\vspace{0pt}
\caption*{(c)}
\end{minipage}
\begin{minipage}{0.5\columnwidth}
\centerline{\includegraphics[width=1.\columnwidth]{MCS2.eps}}
\vspace{0pt}
\caption*{(b)}
\centerline{\includegraphics[width=1.\columnwidth]{MCS4.eps}}
\vspace{0pt}
\caption*{(d)}
\end{minipage}
\caption{Simulation of bound nucleons in terms of two-dimensional lattice of size $400\times 400$ sites, here we only show a small part of the simulation micrograph ($50\times 50$) for better view. Every pair of red balls indicates a pair of SRC, other lattice sites which are tinted in yellow represent nucleons which are nearly free. The micrograph (a) is taken after completion of $1\times 10^3$ MCS in which the memory of initial configuration has not been lost. Micrograph (b) is taken after $1\times 10^4$ MCS where the variation of the system tends to be gentle. Micrograph (c) and (d) are taken after $5\times 10^5$ and $1\times 10^6$ MCS respectively, one can tell from these two diagrams that the system has reached equilibrium state. When $T=2.5$, the ratio of the two components remains at $20:80$, which is consistent with experimental data.}
\label{simulation}
\end{figure}
Average spin $\bar s$ vs temperature $T$ curve is plotted in Fig.~\ref{svsT}. As is evident from this figure, the system tends to more disordered (i.e., $\bar s \to 0$) as the temperature increases, whose corresponding situation in nucleus has been described in last section as $\Delta E^{-1} \to \infty$. Fig.~\ref{svsh} presents the influence of external magnetic field $h$ on $\bar s$. When reaching the stable state, the system is more orderly (i.e., $\bar s \to -1$) as $|h|$ increases, whose corresponding situation has been also discussed in previous section.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.65\columnwidth]{Stime.eps}
\caption{Average spin vs time, it clearly shows the average spin of this system stabilizes at about $\bar s=-0.6$ after $1\times 10^5$ MCS, which is consistent with the micrographs in Fig.~\ref{simulation}.}
\label{svstime}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.65\columnwidth]{svsT.eps}
\caption{Average spin vs temperature, the values are taken after $1\times 10^6$ MCS, here the $C$ and $h$ are fixed at $2$ and $-4$ respectively. $|\bar s|$ becomes smaller the higher the temperature.}
\label{svsT}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.65\columnwidth]{svsh.eps}
\caption{Average spin vs external magnetic field, the values are taken after $1\times 10^6$ MCS, here we take $C=2$ and $h=-4$. $|\bar s|$ becomes bigger the stronger the strength of $h$.}
\label{svsh}
\end{figure}
The results shown in this section have qualitatively confirmed the correctness of utilizing Ising model to describe the states of nucleons which are bound in nucleus. This is an encouraging conclusion which means we can investigate nucleons in terms of many highly developed methods in normal and modified Ising model. However, one should note that the simulation in this manuscript is rather rough. To fulfill the power of Ising model, systematic study on parameter selection is called for. Besides, the estimation of $20\%$ nucleons which belong to SRC in medium or heavy nuclei needs to be explored in more detail, this Ising model based simulation should be able to describe a series of explicit nuclei and reproduce the linear relation between the magnitude of the EMC effect and SRC scale factor \cite{Weinstein:2010rt}.
\section{Conclusions}
\label{conclusions}
Based on the fact that the SRCs in nucleons and Ising spins in Ising model could share the same type of observables, a new SRC-bit duality map is proposed in this work. As a powerful tool, this map connects the state of each nucleon and state of each lattice site. We have considered a nuclear theory as a correspondence to an explicit Ising model and implemented a simulations of nucleon states in terms of Ising model, our preliminary results support the proposed map between nucleons and Ising spins.
Duality is of great use to physicists, it serves as a bridge to connect physical quantities on both sides. Apparently, the investigation of nucleons by SRC-bit duality is at the nascent stage, but with the advancement in computer hardware and efficient algorithms, it's applications in research areas related to nuclear structure appears to be bright. More rigorous investigations on the related issue are urgently called for. Besides conceptual advances, the treatment of classical statistics and quantum particles in a common formalism could lead to unexpected cross-fertilization on both sides.
\section*{Acknowledgements}
We thank Prof. Wei Wang, Dr. Shuai Zhao and Jian-Ping Dai for valuable discussions. J.X. is supported in part by National Natural Science Foundation of China under Grant No. 12105247 and 12047545, the China Postdoctoral Science Foundation under Grant No. 2021M702957. Y.S.L is supported in part by National Natural Science Foundation of China under Grant No. 12002209.
\par\vskip40pt
|
2024-02-18T23:40:24.451Z
|
2022-03-01T02:45:40.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14549","language":"en","timestamp":1646102740000,"url":"https:\/\/arxiv.org\/abs\/2112.14549","yymm":"2112"}
|
proofpile-arXiv_000-10026
|
{"provenance":"002.jsonl.gz:10027"}
| null | null |
\section{Introduction}
\vspace{-2mm}
\label{sec:intro}
Generative Adversarial Networks (GANs) \citep{goodfellow2014generative} are an extremely popular class of generative models used for text and image generation in various fields of science and engineering, including biomedical imaging~\citep{yi2019generative,nie2018medical,wolterink2017generative}, autonomous driving~\citep{hoffman2018cycada,zhang2018deeproad}, and robotics~\citep{rao2020rl,bousmalis2018using}. However, GANs are widely known to be prone to \textit{mode collapse}, which refers to a situation where the generator only samples a few modes of the real data, failing to faithfully capture other more complex or less frequent categories. While the mode collapse problem is often overlooked in text and image generation tasks, and even traded off for higher realism of individual samples~\citep{karras2019style,brock2018large}, dropping infrequent classes may cause serious problems in real-world problems, in which the infrequent classes represent important anomalies. For example, a collapsed GAN can produce racial/gender biased images~\citep{Menon_2020_CVPR}.\blfootnote{* Equal technical contribution. $\dagger$ Work done during internship.}
Moreover, mode collapse causes instability in optimization, which can damage both diversity and the realism of individual samples of the final results. As an example, we visualized the training progression of the vanilla GAN~\citep{goodfellow2014generative} for a simple bimodal distribution in the top row of Figure~\ref{fig:oscillation}. At collapse, the discriminator conveniently assigns high realism to the region unoccupied by the generator, regardless of the true density of real data. This produces a strong gradient for the generator to move its samples toward the dropped mode, swaying mode collapse to the other side. So, the discriminator loses its ability to detect fake samples it was previously able to, such as point \textbf{X}\tikz\draw[red,fill=red] (0,5) circle (.3ex);. The oscillation continues without convergence.
We observe that the mode collapse problem is closely related to Catastrophic Forgetting~\citep{mccloskey1989catastrophic, mcclelland1995there, ratcliff1990connectionist} in continual learning.
A promising line of works~\citep{sidetuning2019, NIPS2019_9429, rusu2016progressive, fernando2017pathnet} tackle the problem in the supervised learning setting by instantiating multiple predictors, each of which takes charge in a particular subset of the whole distribution. We also tackle the problem of mode collapse in GAN by tracking the severity of Catastrophic Forgetting by storing a few exemplar data during training, spawning an additional discriminator if forgetting is detected, Figure~\ref{fig:oscillation}. The key idea is that the added discriminator is left intact unless the generator recovers from mode dropping of that sample, essentially sidestepping catastrophic forgetting.
We show that our proposed approach based on adaptive addition of discriminators can be added to any of the existing GAN frameworks, and is most effective in preventing mode collapse. Furthermore, the improved stability of training boosts the standard metrics on popular GAN frameworks. To summarize, our contributions are: \emph{First}, we propose a novel GAN framework, named Adaptive Multi Adversarial Training (AMAT), that effectively prevents Catastrophic Forgetting in GANs by spawning additional discriminators during training. \emph{Second}, we also propose a computationally efficient synthetic data generation procedure for studying mode collapse in GANs that allows visualizing high dimensional data using normalizing flows. We show that mode collapse occurs even in the recent robust GAN formulations. \emph{Third}, our method can be plugged into any state-of-the-art GAN frameworks and still improve the quality and coverage of the generated samples.
\begin{figure*}
\centering
\includegraphics[trim={0 0 0 10},clip,width=0.90\textwidth]{figs/totalcomb2.png}
\vspace{3mm}
\caption{\textbf{Visualizing training trajectories}: Distribution of real (green dots) and fake (blue dots) over the course of vanilla GAN (top row) and our method (the second row and below). The background color indicates the prediction heatmap of the discriminator with blue being fake and warm yellow being real. Once the vanilla GAN falls into mode collapse (top row), it ends up oscillating between the two modes without convergence. Also, the discriminator's prediction at point X oscillates, indicating catastrophic forgetting in the discriminator. AMAT algorithm adapts to the need, and a new discriminator is spawned during training which effectively learns the forgotten mode, guiding the GAN optimization toward convergence. }
\label{fig:oscillation}
\vspace{-3mm}
\end{figure*}
\vspace{-5mm}
\section{Related Works}
\vspace{-2mm}
\label{sec:related}
Previous works have focused on independently solving either catastrophic forgetting in supervised learning or mode collapse during GAN training. In this section we review these works in detail and discuss our commonalities and differences.
\vspace{-2mm}
\subsection{Mitigating Mode Collapse in GANs}
Along with advancement in the perceptual quality of images generated by GAN~\citep{miyato2018spectral,karras2019style,brock2018large,karras2020analyzing}, a large number of papers~\citep{durugkar2016generative,metz2016unrolled,arjovsky2017wasserstein,srivastava2017veegan,nguyen2017dual,lin2018pacgan,MeschederICML2018,karras2019style} identify the problem of mode collapse in GANs and aim to mitigate it. However mode collapse was seen as a secondary symptom that would be naturally solved as the stability of GAN optimization progresses~\citep{arjovsky2017wasserstein,MeschederICML2018,bau2019seeing}.
To explicitly address mode collapse, Unrolled GAN~\citep{metz2016unrolled} proposes an unrolled optimization of the discriminator to optimally match the generator objective, thus preventing mode collapse. VEEGAN~\citep{srivastava2017veegan} utilizes the reconstruction loss on the latent space. PacGAN~\citep{lin2018pacgan} feeds multiple samples of the same class to the discriminator when making the decisions about real/fake. In contrast, our approach can be plugged into existing state-of-the-art GAN frameworks to yield additional performance boost.
\vspace{-4mm}
\subsection{Multi-adversarial Approaches}
\vspace{-1mm}
The idea of employing more than one adversarial network in GANs to improve results has been explored by several previous works independent of the connection to continual learning and catastrophic forgetting. MGAN~\citep{hoang2018mgan} uses multiple generators, while D2GAN~\citep{nguyen2017dual} uses two discriminators, and GMAN~\citep{durugkar2016generative} and MicrobatchGAN~\citep{mordido2020microbatchgan} proposed a method with more than two discriminators that can be specified as a training hyperparameter beforehand.
However, all previous works require the number of discriminators to be fixed beforehand, which is a major drawback since it depends on several intricate factors such as training dynamics, data distribution complexity, model architecture, initialization hyper-parameters etc. and is expensive and difficult to approximate even with several runs of the algorithm. In contrast, noting by the connection of multi-adversarial training to parameter expansion approaches to catastrophic forgetting, we propose an \textit{adaptive} method that can add discriminators incrementally during training thus achieving superior performance than existing works both on data quality metrics as well as overall computational effort.
\begin{table*}[t]
\resizebox{\textwidth}{!}{
\begin{tabular}{c|c|c|c|c|c||c}
\begin{tabular}{c}
$g(\mathbf{z}) = $
\end{tabular}
& ${1}$
& \begin{tabular}[c]{@{}c@{}}
$\mathbf{A}_{392 \times 2}$
\\ \end{tabular} &
\begin{tabular}[c]{@{}c@{}}
$\mathbf{z}$
\end{tabular}
& \begin{tabular}[c]{@{}c@{}}
MLP
\end{tabular}
& \begin{tabular}[c]{@{}c@{}}
MLP, $\mathbf{A}_{392 \times 2}$
\end{tabular}
& \begin{tabular}{@{}c@{}}
\small{MNIST} \end{tabular}
\\[1ex]
\hline
\begin{tabular}[c]{@{}c@{}}
Label\\ \end{tabular}
& {\fontfamily{lmtt}\selectfont Level I}
& {\fontfamily{lmtt}\selectfont Level II}
& {\fontfamily{lmtt}\selectfont Level III}
& {\fontfamily{lmtt}\selectfont Level IV}
& {\fontfamily{lmtt}\selectfont Level V}
& -
\\
\hline
\begin{tabular}[c]{@{}c@{}}
\small{GAN-NS \citep{goodfellow2014generative}}\\ \end{tabular}
& \cmark \enskip \big\vert \enskip \xmark
& \cmark \enskip \big\vert \enskip \xmark
& \cmark \enskip \big\vert \enskip \cmark
& \cmark \enskip \big\vert \enskip \cmark
& \cmark \quad \big\vert \quad \cmark
& \xmark
\\
\begin{tabular}[c]{@{}c@{}}
\small{WGAN} \citep{arjovsky2017wasserstein} \\ \end{tabular}
& \xmark \enskip \big\vert \enskip \xmark
& \cmark \enskip \big\vert \enskip \xmark
& \cmark \enskip \big\vert \enskip \xmark
& \cmark \enskip \big\vert \enskip \cmark
& \cmark \quad \big\vert \quad \cmark
& \xmark
\\
\begin{tabular}[c]{@{}c@{}}
\small{Unrolled GAN} \citep{metz2016unrolled} \end{tabular}
& \xmark \enskip \big\vert \enskip \xmark
& \xmark \enskip \big\vert \enskip \xmark
& \xmark \enskip \big\vert \enskip \xmark
& \xmark \enskip \big\vert \enskip \xmark
& \cmark \quad \big\vert \quad \cmark
& \xmark
\\
\begin{tabular}[c]{@{}c@{}}
\small{D2GAN} \citep{nguyen2017dual} \end{tabular}
& \xmark \enskip \big\vert \enskip \xmark
& \xmark \enskip \big\vert \enskip \xmark
& \xmark \enskip \big\vert \enskip \xmark
& \xmark \enskip \big\vert \enskip \xmark
& \cmark \quad \big\vert \quad \cmark
& \xmark
\\
\begin{tabular}[c]{@{}c@{}}
\small{GAN-NS + AMAT} \end{tabular}
& \xmark \enskip \big\vert \enskip \xmark
& \xmark \enskip \big\vert \enskip \xmark
& \xmark \enskip \big\vert \enskip \xmark
& \xmark \enskip \big\vert \enskip \xmark
& \cmark \quad \big\vert \quad \cmark
& \xmark
\\
\hline
\end{tabular}
}
\vspace{2mm}
\caption{\xmark \hspace{0.1mm} indicates that the generator could effectively learn all the data modes, while \cmark \hspace{0.1mm} means \textit{despite best efforts with tuning} the training suffers from mode collapse (more than a quarter of data modes dropped). We show results with the SGD (left) \& ADAM (right) optimizers. MNIST results with ADAM optimizer are provided for reference. We observe that MNIST is a relatively easy dataset, falling between {\fontfamily{lmtt}\selectfont Level I} and {\fontfamily{lmtt}\selectfont II} in terms of complexity.}
\label{tab:synthetic}
\vspace{-4mm}
\end{table*}
\vspace{-4mm}
\subsection{Overcoming Catastrophic Forgetting in GAN}
\vspace{-1mm}
Methods to mitigate catastrophic forgetting can be categorized into three groups: a) regularization based methods~\citep{kirkpatrick2017overcoming} b) memory replay based methods~\citep{rebuffi2017icarl} c) network expansion based methods~\citep{zhang2018deeproad, NIPS2019_9429}. Our work is closely related to the third category of methods, which dynamically adds more capacity to the network, when faced with novel tasks. This type of methods, adds \emph{plasticity} to the network from new weights (fast-weights) while keeping the \emph{stability} of the network by freezing the past-weights (slow-weights).
Additionally, we enforce stability by letting a discriminator to focus on a few set of classes, not by freezing its weights.
The issue of catastrophic forgetting in GANs has been sparsely explored before.
\citet{chen2018self} and \citet{tran2019self} propose a self-supervised learning objective to prevent catastrophic forgetting by adding new loss terms. \citet{liang2018generative} proposes an online EWC based solution to tackle catastrophic forgetting in the discriminator. We propose a prominently different approach based on parameter expansion rather than regularization. While the regularization based approaches such as \citet{liang2018generative} attempt to retain the previously learnt knowledge by constrained weight updates, the parameter expansion approaches effectively sidestep catastrophic forgetting by freezing previously encoded knowledge. \citet{thanhcatastrophic} also discuss the possibility of catastrophic forgetting in GAN training but their solution is limited to theoretical analyses with simplistic proposals such as assigning larger weights to real samples and optimizing the GAN objective with momentum. Practically, we observed that their method performs worse than a plain vanilla DCGAN on simple real world datasets like CIFAR10. In contrast, our method leverages insights from continual learning and has a direct connections to prevalent parameter expansion approaches in supervised learning. We benchmark extensively on several datasets and state-of-the-art GAN approaches where our method consistently achieves superior results to the existing methods.
\vspace{-7mm}
\section{Proposed Method}
\vspace{-2mm}
In this section, we first describe our proposed data generation procedure that we use as a petri dish for studying mode collapse in GANs.
The procedure uses random normalizing flows for simultaneously allowing training on complex high dimensional distributions yet being perfectly amenable to 2D visualizations.
Next, we describe our proposed Adaptive Multi Adversarial Training (AMAT) algorithm that effectively detects catastrophic forgetting and spawns a new discriminator to prevent mode collapse.
\vspace{-5mm}
\subsection{Synthetic Data Generation with Normalizing flows}
\vspace{-1mm}
Mode dropping in GANs in the context of catastrophic forgetting of the discriminator is a difficult problem to investigate using real datasets. This is because the number of classes in the dataset cannot be easily increased, the classes of fake samples are often ambiguous, and the predictions of the discriminator cannot be easily visualized across the whole input space.
In this regard, we present a simple yet powerful data synthesis procedure that can generate complex high dimensional multi-modal distributions, yet maintaining perfect 2-D visualization capabilities. Samples from a 2-D Gaussian distribution are augmented with biases and subjected to an invertible normalizing flow~\citep{karami2019invertible} parameterized by well conditioned functions $g_i: \mathbb{R}^{d^0_i} \rightarrow \mathbb{R}^{d^1_i}$. This function can be followed by a linear upsampling transformation parameterized by a $d^1_i \times d^0_{i+1}$ dimensional matrix $A^i$ (Algorithm \ref{algo:synthetic}).
The entire transform is deliberately constructed to be a bijective function so that every generated sample in $\hat{y} \in \mathbb{R}^D$ can be analytically mapped to $\mathbb{R}^2$, allowing perfect visualization on 2D space. Furthermore, by evaluating a dense grid of points in $\mathbb{R}^2$, we can understand discriminator's learned probability distribution on $\mathbf{z}$ manifold as a heatmap on the 2D plane.
This synthetic data generation procedure enables studying mode collapse in a controlled setting. This also gives practitioners the capability to train models on a chosen data complexity with clean two-dimensional visualizations of both the generated data and the discriminator's learnt distribution.
This tool can be used for debugging new algorithms using insights from the visualizations.
In the case of mode collapse, a quick visual inspection would give the details of which modes face mode collapse or get dropped from discriminator's learnt distribution.
\vspace{-2mm}
\subsection{Adaptive Multi Adversarial Training}
\vspace{-1mm}
Building upon the insight on relating catastrophic forgetting in discriminator to mode collapse in generator, we propose a multi adversarial generative adversarial network training procedure.
The key intuition is that the interplay of catastrophic forgetting in the discriminator with the GAN minimax game, leads to an oscillation generator. Thus, as the generator shifts to a new set of modes the discriminator forgets the learnt features on the previous modes.
\begin{minipage}[b]{.46\textwidth}
\begin{algorithm}[H]
\DontPrintSemicolon
\SetNoFillComment
\begin{algorithmic}
\STATE \textbf{Input:} Mean $\{\mu_i\}_{i=1}^K$ and standard deviation $\{\sigma_i\}_{i=1}^K$ for initialization, $\{g_i\}_{i=1}^{L}$ well conditioned $\mathbb{R}^2 \rightarrow \mathbb{R}^2$ functions
\STATE Sample weights ${\bm{w}} \sim \text{Dirichlet}(K)$ \\
\tcc{Sample from 2D gaussian mixture}
\STATE $\mathbf{x}_{2D} \sim \Sigma_{i = 1}^N w_i \mathcal{N}( \mu_i, \sigma_i)$
\STATE $\mathbf{x}^0_{2D} = \Big[[x^0_{2D}; {1}], [x^1_{2D}; {1}] \Big] $ \newline
\tcc*{Randomly Init Normalizing Flow}
\FOR{$k = 1$ to $k = K$}
\IF{k is even}
\STATE ${\bm{x}}^k = \big[{\bm{x}}^{k}_0, {\bm{x}}^{k}_1 \cdot g_k({\bm{x}}^k_0)\big] $
\ELSE
\STATE ${\bm{x}}^k = \big[{\bm{x}}^{k}_0 \cdot g_k({\bm{x}}^k_1), {\bm{x}}^{k}_1 \big] $
\ENDIF
\ENDFOR
\end{algorithmic}
\caption{Synthetic Data Generation}
\label{algo:synthetic}
\end{algorithm}
\vspace{-4mm}
\begin{algorithm}[H]
\DontPrintSemicolon
\SetNoFillComment
\caption{DSPAWN: Discriminator \mbox{Spawning} Routine}
\label{algo:spawn}
\begin{algorithmic}
\STATE \textbf{Require:} Exemplar Data $\{{\bm{e}}\}_{i=1}^m$
\STATE \textbf{Input:} Discriminator set $\displaystyle {\mathbb{D}} = \{f_w^{i}\}_{i=1}^K$ \newline \tcc*{Check forgetting on exemplars}
\FOR{$i = 1$ to $i = m$}
\STATE ${\bm{s}}[k] \leftarrow f_w^{k}({\bm{e}}_i) \ \forall\ k\ \in \{1 \dots K\}$\;
\IF{ $K *\max({\bm{s}}) > \alpha_t * \sum_k{\bm{s}}[k]$}
\STATE Initialize $f_w^{K+1}$ with random weights $w$ \newline \tcc*{Spawn a new discriminator}
\STATE Initialize random weight $w^{K+1}$
\STATE $\displaystyle {\mathbb{D}} \leftarrow \{f_w^{i}\}_{i=1}^K \bigcup f_w^{K+1}$
\STATE \textbf{break}
\ENDIF
\ENDFOR
\STATE \textbf{return} Discriminator Set $\displaystyle {\mathbb{D}}$
\end{algorithmic}
\end{algorithm}
\end{minipage}\hfill
\begin{minipage}[b]{.48\textwidth}
\begin{algorithm}[H]
\caption{A-MAT: Adaptive Multi-Adversarial Training}
\label{algo:multi}
\DontPrintSemicolon
\begin{algorithmic}
\STATE \textbf{Require: }${\bm{w}}^{i}_0$, ${\bm{\theta}}_0$ initial discriminator \& generator params, greediness param $\epsilon$, $\{T_k\}$ spawn warmup iteration schedule
\STATE $\displaystyle {\mathbb{D}} \leftarrow \{f_w^{0}\}$\;
\WHILE{${\bm{\theta}}$ has not converged}
\STATE Sample $\{{\bm{z}}^{(i)}\}_{i=1}^B \sim p(z)$
\STATE Sample $\{{\bm{x}}^{(i)}\}_{i=1}^B \sim \mathbb{P}_r$
\STATE Sample $\{\sigma_1(i)\}_{i=1}^B \sim \text{Uniform}(1, K)$
\STATE Sample $\{\alpha(i)\}_{i=1}^B \sim \text{Bernoulli}(\epsilon)$ \newline \tcc{Loss weights over discriminators}
\STATE Sample weights ${\bm{m}} \sim \text{Dirichlet}(K)$
\STATE $\hat{\bm{x}}^{(i)} \leftarrow g_\theta({\bm{z}}^{(i)}) $
\STATE $\sigma_2(i) \leftarrow \argmin_{k}f_w^{k}(\hat{\bm{x}}^{(i)})$ \newline \tcc*{Discriminator responsible for $\hat{x}^{(i)}$}
\STATE $\sigma_{z}(i) \leftarrow \alpha(i) \sigma_1(i) + (1 - \alpha(i)) \sigma_2(i)$ \newline \tcc*{Discriminator responsible for $x^{(i)}$}
\STATE $\sigma_x(i) \leftarrow \sigma_1(i)$ \newline \tcc*{Training Discriminators}
$L_{w} \leftarrow \sum_{i=1}^B [f_w^{\sigma_{x}(i)}({\bm{x}}_i) - 1]^{-} - [f_w^{\sigma_{z}(i)}(\hat{\bm{x}}_i) + 1]^+$
\FOR{$k = 1$ to $k = \vert \displaystyle {\mathbb{D}} \vert$}
\STATE $w^k \leftarrow \text{ADAM}(L_{w})$
\ENDFOR \newline \tcc*{Training Generator}
\STATE $s[k] \leftarrow \sum_{i=1}^B f_w^{k}(\hat{\bm{x}}^{(i)}) \ \forall\ k\ \in \{1 \dots \vert \displaystyle {\mathbb{D}} \vert\}$\newline
\tcc*{Weighed mean over discriminators}
\STATE $L_{\theta} \leftarrow \text{sort}({\bm{m}}) \cdot \text{sort}(s) $
\STATE $\theta \leftarrow \text{ADAM}(L_{\theta})$\;
\IF{more than $T_t$ warm-up iterations since the last spawn}
\STATE $\displaystyle {\mathbb{D}} \leftarrow \text{DSPAWN}(\{f_w^{i}\})$
\ENDIF
\ENDWHILE
\end{algorithmic}
\end{algorithm}
\vspace{-3mm}
\end{minipage}
However if there are multiple discriminators available, each discriminator can implicitly \emph{specialize} on a subset of modes. Thus even if the generator oscillates, each discriminator can remember their own set of modes, and they will not need to move to different set of modes. This way we can effectively \emph{sidestep} forgetting and ensure the networks do not face significant distribution shift. A detailed version of our proposed method is presented in Algorithm \ref{algo:multi}.
\newline
\newline
\textbf{Spawning new discriminators}: We initialize the AMAT training Algorithm~\ref{algo:multi} with a regular GAN using just one discriminator. We also sample a few randomly chosen exemplar data points with a maximum of one real sample per mode, depending on dataset complexity. The exemplar data points are used to detect the presence of catastrophic forgetting in the currently active set of discriminators $\displaystyle {\mathbb{D}}$ and spawn a new discriminator if needed. Specifically (Algorithm \ref{algo:spawn}), we propose that if \emph{any} discriminator among $\displaystyle {\mathbb{D}}$ has an unusually high score over an exemplar data point ${\bm{e}}_i$, this is because the mode corresponding to ${\bm{e}}_i$ has either very poor generated samples or has been entirely dropped.
In such a situation, if training were to continue we risk catastrophic forgetting in the active set $\displaystyle {\mathbb{D}}$, if the generator oscillate to near ${\bm{e}}_i$. This is implemented by comparing the $\max$ score over $\displaystyle sD$ at ${\bm{e}}_i$ to the average score over $\displaystyle {\mathbb{D}}$ and spawning a new discriminator when the ratio exceeds $\alpha_t (> 1)$. Further, we propose to have $\alpha_t (> 1)$ a monotonically increasing function of $| \displaystyle {\mathbb{D}}|$, thus successively making it harder to spawn each new discriminator. Additionally, we use a warm-up period $T_t$ after spawning each new discriminator from scratch to let the spawned discriminator train before starting the check over exemplar data-points.
\newline
\newline
\noindent \textbf{Multi-Discriminator Training: } We evaluate all discriminators in $\displaystyle {\mathbb{D}}$ on the fake samples but do not update all of them for all the samples. Instead, we use the discriminator scores to assign responsibility of each data point to only one discriminator.
\newline
\newline
\noindent \textbf{Training over fake samples}: We use an $\epsilon$-greedy approach for fake samples where the discriminator with the lowest output score is assigned responsibility with a probability $1 - \epsilon$ and a random discriminator is chosen with probability $\epsilon$.
\newline
\newline
\noindent \textbf{Training over real samples}: The discriminator is always chosen uniformly randomly thus we slightly prefer to assign the same discriminator to the fake datapoints from around the same mode to ensure that they do not forget the already learnt modes and switch to another mode. The random assignment of real points ensure that the same preferentially treated discriminator also gets updated on real samples.
Further for optimization stability, we ensure that the real and fake sample loss incurred by each discriminator is roughly equal in each back-propagation step by dynamically reweighing them by the number of data points the discriminator is responsible for. We only update the discriminator on the losses of the samples they are responsible for.
While it may seem that adding multiple discriminators makes the procedure expensive, in practice the total number of added discriminator networks never surpass three for the best results.
It is possible to change the hyperparameters to allow a large number of discriminators but that results in sub-optimal results and incomplete training. The optimal hyperparameter selection is explained in the Appendix for each dataset.
Further, the additional discriminators get added during later training stages and are not trained from the start, saving compute in comparison to prior multi-adversarial works which train all the networks from the beginning.
Also, unlike AdaGAN \citep{tolstikhin2017adagan} and similar Boosted GAN models that need to store multiple Generators post training, the final number of parameters required during inference remains unchanged under AMAT . Thus the inference time remains the same, but with enhanced mode coverage and sample diversity. Unlike \citep{tolstikhin2017adagan}, our discriminator addition is adaptive, i.e. discriminators are added during the training thus being more efficient.
\newline
\newline
\noindent \textbf{Generator Training:} We take a weighted mean over the discriminators scores on the fake datapoints for calculating the generator loss. At each step, the weights each discriminator in $\displaystyle {\mathbb{D}}$ gets is in decreasing order of its score on the fake sample. Hence, the discriminator with the lowest score is given the most weight since it is the one that is currently specializing on the mode the fake sample is related to. In practice, we sample weights randomly from a Dirichlet distribution (and hence implicitly they sum to $1$) and sort according to discriminator scores to achieve this. We choose soft weighing over hard binary weights because since the discriminators are updated in an $\epsilon$ greedy fashion, the discriminators other than the one with the best judgment on the fake sample might also hold useful information. Further, we choose the weights randomly instead of fixing a chosen set to ensure AMAT is more deadset agnostic since the number of discriminator used changes with the dataset complexity so does the number of weights needed. While a suitable function for generating weights can work well on a particular dataset, we found random weights to work as well across different settings.
\vspace{-3mm}
\section{Results}
\vspace{-3mm}
\label{sec:results}
We test our proposed method on several synthetic and real datasets \& report a consistent increase in performance on GAN evaluation metrics such as
Inception Score \citep{salimans2016improved} and Frech\'et Inception Distance \citep{heusel2017gans} with our proposed AMAT. We also showcase our performance in the GAN fine-tuning regime with samples on the CUB200 dataset \citep{WelinderEtal2010} which qualitatively are more colorful and diverse than an identical BigGAN without AMAT (Figure \ref{fig:cub200}).
\begin{table*}[t]
\resizebox{\textwidth}{!}
\begin{tabular}{ccccc|c c}
\toprule
& GAN & UnrolledGAN & D2GAN & RegGAN & DCGAN & with AMAT \\ \midrule
\# Modes covered & $628.0 \pm 140.9$ & $817.4 \pm 37.9$ & $1000 \pm 0.0$ & $ 955.5\pm18.7$ & $849.6\pm 62.7$ & $\mathbf{1000 \pm 0.0}$ \\
KL (samples $\Vert$ data) & $2.58\pm 0.75$ & $1.43 \pm 0.12$ & $0.080 \pm 0.01$ & $0.64 \pm 0.05$ & $0.73\pm0.09$ & $0.078\pm0.01$ \\ \bottomrule
\end{tabular}
}
\vspace{3mm}
\caption{\textbf{Quantitative Results on the Stacked MNIST dataset}: Applying our proposed adaptive multi adversarial training (AMAT) procedure to a simple DCGAN achieves perfect mode coverage, better than many existing methods for mode collapse.}
\label{tab:stacked}
\end{table*}
\begin{table*}[t]
\smaller
\resizebox{\textwidth}{!}{
\begin{tabular}{c| cc|cc|cc}
\toprule
& & & GAN-NS & AMAT + & & AMAT + \\
Model & D2GAN & MicrobatchGAN & w/ ResNet & GAN-NS & DCGAN & DCGAN \\ \midrule
IS & $7.15 \pm 0.07$ & $6.77$ & $6.7 \pm 0.06$ & $\mathbf{8.1 \pm 0.04}$ & $6.03\pm0.05$ & $\mathbf{6.32\pm0.06}$ \\
FID & - & - & $28.91$ & $\mathbf{16.35}$ & $33.42$ & $\mathbf{30.14}$ \\ \midrule
& WGAN-GP & AMAT + & &AMAT + & & AMAT + \\
Model & w/ ResNet & WGAN-GP & SN-GAN & SN-GAN & BigGAN & BigGAN \\ \midrule
IS & $7.59\pm0.10$ & $\mathbf{7.80\pm0.07}$ &$ 8.22\pm0.05$ &$\mathbf{8.34\pm0.04}$& $9.22$ & $\mathbf{9.51\pm0.06}$ \\
FID & $19.2$ & $\mathbf{17.2}$ &$14.21$ &$\mathbf{13.8}$ & $8.94$ & $\mathbf{6.11}$ \\
\bottomrule
\end{tabular}
}
\vspace{3mm}
\centering
\caption{\textbf{Quantitative Results on CIFAR10}: We benchmark AMAT against several other multi-adversarial baselines as well as on several GAN architectures across all of which we observe a consistent performance increase.}
\label{tab:cifar}
\end{table*}
\vspace{-3mm}
\subsection{Synthetic Data}
\vspace{-1mm}
We utilize the proposed synthetic data generation procedure with randomly initialized normalizing flows to visualize the training process of a simple DCGAN \citep{radford2015unsupervised}. Figure \ref{fig:oscillation} visualizes such a training process for a simple bimodal distribution. Observing the pattern of generated samples over the training iteration and the shifting discriminator landscape, we note a clear mode oscillation issue present in the generated samples driven by the shifting discriminator output distribution. Focusing on a single fixed real point in space at any of the modes, we see a clear oscillation in the discriminator output probabilities strongly indicating the presence of catastrophic forgetting in the discriminator network.
\noindent \textbf{Effect of Data Complexity on Mode Collapse}: We use the flexibility in choosing transformations $g_i$ to generate datasets of various data distribution complexities as presented in Table $\ref{tab:synthetic}$. Choosing $g(z)$ with successively more complicated transformations can produce synthetic datasets of increasing complexity, the first five of which we roughly classify as {\fontfamily{lmtt}\selectfont Levels}. The {\fontfamily{lmtt}\selectfont Levels} are generated by using simple transforms such as identitym constant mapping, small Multi layer perceptrons and well conditioned linear transforms ($\mathbf{A}$).
On this benchmark, we investigate mode collapse across different optimizers such as SGD \& ADAM \citep{kingma2014adam} on several popular GAN variants such as the non-saturating GAN Loss (GAN-NS) \citep{goodfellow2014generative}, WGAN \citep{arjovsky2017wasserstein} and also methods targeting mitigating mode collapse specifically such as Unrolled GAN \citep{metz2016unrolled} and D2GAN \citep{nguyen2017dual}.
We show results of our proposed AMAT training procedure with a simple GAN-NS, which matches performance with other more complicated mode collapse specific GAN architectures, all of which are robust to mode collapse up to {\fontfamily{lmtt}\selectfont Level IV}.
In practice we find all benchmarked methods to collapse at {\fontfamily{lmtt}\selectfont Level V}. Thus, in contrast to other simple datasets like MNIST~\citep{lecun1998mnist}, Gaussian ring, or Stacked MNIST~\citep{metz2016unrolled}, the complexity of our synthetic dataset can be arbitrarily tuned up or down to gain insight into the training and debugging of GAN via visualizations.
\begin{table*}[t]
\vspace{-2mm}
\centering
\resizebox{\textwidth}{!}{
\begin{tabular}{c| ccccc| c}
\toprule
Effect & Large $|\displaystyle {\mathbb{D}}|$ & Spawn too late & Greedy $\displaystyle \nabla$D & Random for fake & $\mathbf{1}$-hot weight & Proposed \\
Ablation & Small $\alpha$, Short $T_t$ & Long $T_t$ schedule & $\epsilon = 0$ & $\epsilon$-greedy for real & vector ${\bm{m}}$ & Method \\ \midrule
IS & $8.83 \pm 0.04$ & $9.28 \pm 0.08$ & $9.31 \pm 0.06$ & $8.95 \pm 0.04$ & $9.25 \pm 0.05$ & $9.51 \pm 0.06$ \\
FID & $14.23$ & $9.37$ & $8.6$ & $12.5$ & $9.25$ & $6.11$ \\
\bottomrule
\end{tabular}
}
\vspace{3mm}
\caption{\textbf{BigGAN + AMAT Ablations on CIFAR10} (A) A spawning condition with small $\alpha$ and short warmup schedule that leads to large number of discriminators (>$7$) (B) Long warm-up schedules that spawn new discriminators late into training (C) A greedy strategy for assigning responsibility of fake samples ($\epsilon = 0$) (D) Flipping the data splitting logic with responsibilities of fake samples being random and of real being $\epsilon$-greedy (E) Choosing the discriminator with lowest score for updating Generator instead of soft random weighting.}
\vspace{-2mm}
\label{tab:ablations}
\end{table*}
\begin{figure*}
\centering
\begin{subfigure
\centering
\includegraphics[width=0.48\textwidth]{figs/dis_modes.pdf}
\end{subfigure}
\begin{subfigure
\centering
\includegraphics[width=0.48\textwidth]{figs/dis_test_acc.pdf}
\end{subfigure}
\vspace{1mm}
\caption{{\textbf{Investigating the forgetting-collapse interplay:} We investigate our hypothesis that catastrophic forgetting is associated with mode collapse. On the left pane, we plot the magnitude of mode collapse by counting the number of modes produced by the generator. On the right pane, we assess the quality of the discriminator features by plotting the accuracy of linear classifier on top of the discriminator features at each epoch. In the original model, the coverage of modes and the quality of discriminator features are both low and decreasing. In particular, the test accuracy from the discriminator's features drops almost to randomly initialized weights (shown as \textit{control}). On the other hand, adding AMAT (\textit{MultiD}) dramatically improves both mode coverage and the discriminator test accuracy.
}}
\label{fig:classification}
\vspace{-4mm}
\end{figure*}
\vspace{-3mm}
\subsection{Stacked MNIST}
\vspace{-2mm}
We also benchmark several models on the Stacked MNIST dataset following \citep{metz2016unrolled, srivastava2017veegan}. Stacked MNIST is an extension of the popular MNIST dataset~\citep{lecun1998gradient} where each image is expanded in the channel dimension to $28 \times 28 \times 3$ by concatenating $3$ single channel images. The resulting dataset has a $1000$ overall modes. We measure the number of modes covered by the generator as the number of classes that are generated at least once within a pool of $25,600$ sampled images. The class of the generated sample is identified with a pretrained MNIST classifier operating channel wise on the original stacked MNIST image.
\newline
\newline
\noindent \textbf{Understanding the forgetting-collapse interplay}: In Section \ref{sec:intro}, we discuss our motivation for studying catastrophic forgetting for mitigating mode collapse. We also design an investigative experiment to explicitly observe this interplay by comparing the number of modes the generator learns against the quality of features the discriminator learns throughout GAN training on the stacked MNIST dataset. We measure the number of modes captured by the generator through a pre-trained classification network trained in a supervised learning fashion and frozen throughout GAN training. To measure the amount of \emph{`forgetting`} in discriminator, we extract features of real samples from the penultimate layer of the discriminator and train a small classifier on the real features for detecting real data mode. This implicitly indicates the quality and information contained in the the discriminator extracted features. However, the performance of classification network on top of discriminator features is confounded by the capacity of the classification network itself. Hence we do a control experiment, where we train the same classifier on features extracted from a randomly initialized discriminator, hence fixing a lower-bound to the classifier accuracy.
\begin{table*}[t]
\centering
\resizebox{\textwidth}{!}{
\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c|c}
\toprule
Classes & Plane & Car & Bird & Cat & Deer & Dog & Frog & Horse & Ship & Truck & Avg \\ \midrule
BigGAN & $24.23$ & $12.32$ & $24.85$ & $21.21$ & $12.81$ & $22.74$ & $17.95$ & $13.16$ & $12.11$ & $18.39$ & $8.94$ \\
$+$ AMAT & $20.50$ & $10.30$ & $23.48$ & $18.48$ & $11.51$ & $19.41$ & $11.50$ & $12.24$ & $10.69$ & $12.94$ & $6.11$ \\
$\Delta \%$ & $18.2$ & $19.6$ & $5.8$ & $14.8$ & $11.3$ & $17.2$ & $\mathbf{56.1}$ & $7.5$ & $11.7$ & $\mathbf{42.1}$ & $\mathbf{46.3}$
\\ \bottomrule
\end{tabular}
}
\vspace{3mm}
\caption{\textbf{Per-class FID on CIFAR10}: FID improves consistently across all classes.}
\label{tab:classwise}
\vspace{-3mm}
\end{table*}
Referring to Figure \ref{fig:classification}, we observe a clear relation between the number of modes the generator covers at an iteration and the accuracy of the classification network trained on the discriminator features at the same iteration. In the vanilla single discriminator scenario, the classification accuracy drops significantly, indicating a direct degradation of the discriminative features which is followed by a complete collapse of G. In the collapse phase, the discriminator's learnt features are close to random with the classification accuracy being close to that of the control experiment. This indicates the presence of significant catastrophic forgetting in the the discriminator network.
In contrast, training the same generator with the proposed AMAT procedure leads to stable training with almost all the modes being covered. The classification accuracy increasing before saturation. Catastrophic forgetting is \textit{effectively sidestepped} by adaptive multi adversarial training which produces stable discriminative features during training that provide a consistent training signal to the generator thereby covering all the modes with little degradation.
\vspace{-1mm}
\subsection{CIFAR10}
\vspace{-1mm}
\begin{figure*}[t!]
\centering
\begin{subfigure
\centering
\includegraphics[width=0.45\textwidth]{figs/gans.pdf}
\end{subfigure} \hfill
\begin{subfigure
\centering
\includegraphics[width=0.45\textwidth]{figs/gans_1d.pdf}
\end{subfigure}
\caption{\textbf{Sample Diversity on CUB200:} We showcase samples from a BigGAN pretrained on imagenet \& finetuned on CUB200 with the AMAT procedure (left) and from an identical BigGAN finetuned without AMAT (right). Observe that while the sample quality is good for both, the samples generated with AMAT are more colorful \& diverse, with bright reds and yellow against several backgrounds. While the samples from vanilla fine-tuning are restricted to whites/grays \& only a hint of color.
}
\label{fig:cub200}
\vspace{-6mm}
\end{figure*}
We extensively benchmark AMAT on several GAN variants including unconditional methods such as DCGAN \citep{radford2015unsupervised}, ResNet-WGAN-GP \citep{gulrajani2017improved,he2016deep} \& SNGAN \citep{miyato2018spectral} and also conditional models such as BigGAN \citep{brock2018large}. Table \ref{tab:cifar} shows the performance gain on standard GAN evaluation metrics such as Inception Score and Fr\'echet distance of several architectures when trained with AMAT procedure. The performance gains indicate effective curbing of catastrophic forgetting in the discriminator with multi adversarial training. We use the public evaluation code from SNGAN \citep{miyato2018spectral} for evaluation. Despite having components such as spectral normalization, diversity promoting loss functions, additional R1 losses \& other stable training tricks that might affect catastrophic forgetting to different extents, we observe a consistent increase in performance across all models. Notably the ResNet GAN benefits greatly with AMAT despite a powerful backbone -- with IS improving from $6.7$ to $8.1$, indicating that the mode oscillation problem is not mitigated by simply using a better model.
AMAT improves performance by over $35\%$ even on a well performing baseline such as BigGAN (Table \ref{tab:cifar}). We investigate classwise FID scores of a vanilla BigGAN and an identical BigGAN + AMAT on CIFAR10 and report the results in Table $\ref{tab:classwise}$. Performance improves across all classes with previously poor performing classes such as `Frog' \& `Truck' experiencing the most gains. Further, we ablate several key components of AMAT procedure on the BigGAN architecture with results reported in Table \ref{tab:ablations}. We observe all elements to be critical to overall performance. Specifically, having a moderate $\alpha$ schedule to avoid adding too many discriminators is critical. Another viable design choice is to effectively flip the algorithm's logic and instead choose the fake points randomly while being $\epsilon$ greedy on the real points. This strategy performs well on simple datasets but loses performance with BigGAN on CIFAR10 (Table \ref{tab:ablations}). In all experiments, the computational time during inference is the same as the base model, \textit{irrespective of the number of discriminators} added during the training, since only a single generator is trained with AMAT and all the discriminators are discarded.
\vspace{-4mm}
\section{Conclusion}
\vspace{-3mm}
In summary, motivated from the observation of catastrophic forgetting in the discriminator, we propose a new adaptive GAN training framework that adds additional discriminators to prevent mode collapse. We show that our method can be added to existing GAN frameworks to prevent mode collapse, generate more diverse samples and improve FID \& IS. In future, we plan to apply AMAT to fight mode collapse in high resolution image generation settings.
\vspace{-3mm}
\subsubsection*{Acknowledgements}
\vspace{-2mm}
We thank Jathushan Rajasegaran for his helping with the forgetting-collapse interplay experiments and Taesung Park for feedback and comments on the early drafts of this paper.
\clearpage
\section*{Appendix}
\section*{CIFAR10 Experiments}
\label{appendix:cifarexp}
\subsection*{BigGAN + AMAT Experiments}
For the baseline we use the author's official PyTorch implementation \footnote{\href{https://github.com/ajbrock/BigGAN-PyTorch}{https://github.com/ajbrock/BigGAN-PyTorch}}. For our experiments on AMAT $+$ BigGAN, we kept the optimizer as Adam \citep{kingma2014adam} and used the hyperparameters $\beta_1 = 0.0, \beta_2 = 0.9$. We did not change the model architecture parameters in any way. The best performance was achieved with learning rate $=0.0002$ for both the Generator and all the Discriminators. The batch size for the $G$ and all $D$ is $50$, and the latent dimension is chosen as $128$. The initial value of $T_t = 5$ epochs, and after the first discriminator is added, $T_t$ is increased by 5 epochs every $T_t$ epochs. The initial value of $\alpha_t = 1.5$, and it is increased by a factor of $3.5$ every time a discriminator is added. To check whether to add another discriminator or not, we use 10 exemplar images, 1 from each CIFAR10 class. While assigning datapoints to each discriminator, we use an epsilon greedy approach. We chose $\epsilon = 0.25$, where the datapoint is assigned to a random discriminator with a probability $\epsilon$. The number of discriminator(s) updates per generator update is fixed at $4$. We also use exponential moving average for the generator weights with a decay of $0.9999$.
\subsection*{SN-GAN + AMAT Experiments}
We used the SN-GAN implementation from \href{https://github.com/GongXinyuu/sngan.pytorch}{https://github.com/GongXinyuu/sngan.pytorch}, which is the PyTorch version of the authors' Chainer implementation \href{https://github.com/pfnet-research/sngan_projection}{https://github.com/pfnet-research/sngan\_projection}. We kept the optimiser as Adam and used the hyperparameters $\beta_1 = 0.0, \beta_2 = 0.9$. The batch size for generator is $128$ and for the discriminators is $64$, and the latent dimension is $128$. The initial learning rate is $0.0002$ for both generator and the discriminators. The number of discriminator(s) updates per generator update is fixed at $7$. The initial value of $T_t = 2$ epochs, and is increased by $1$ epoch after every discriminator is added. $\alpha_t$ is initialized as $1.5$, and is increased by a factor of $1.3$ after a discriminator is added, till $20$ epochs, after which it is increased by a factor of $3.0$. These larger increases in $\alpha_t$ are required to prevent too many discriminators from being added over all iterations. We chose $\epsilon = 0.3$, where the datapoint is assigned to a random discriminator with a probability $\epsilon$. We use 10 exemplar images, 1 from each CIFAR10 class.
\textbf{ResNet GAN:} We use the same ResNet architecture as above, but remove the spectral normalization from the model. The optimizer parameters, learning rate and batch sizes remain the same as well. The number of discriminator(s) updates per generator update is fixed at $5$. The initial value of $T_t = 10$ epochs, and is increased by $5$ epochs after every discriminator is added. $\alpha_t$ is initialized as $1.5$, and is increased by a factor of $2.0$ after a discriminator is added. We chose $\epsilon = 0.2$, where the datapoint is assigned to a random discriminator with a probability $\epsilon$. We use 10 exemplar images, 1 from each CIFAR10 class.
\textbf{ResNet WGAN-GP:} In the above model, the hinge loss is replaced by the Wasserstein loss with gradient penalty. The optimizer parameters, learning rate and batch sizes remain the same as well.The number of discriminator(s) updates per generator update is fixed at $2$. The initial value of $T_t = 5$ epochs, and is increased by $5$ epochs after a discriminator is added. $\alpha_t$ is initialized as $1.5$, and is increased by a factor of $3.0$ after a discriminator is added. We chose $\epsilon = 0.2$, where the datapoint is assigned to a random discriminator with a probability $\epsilon$. We use 10 exemplar images, 1 from each CIFAR10 class. These images are chosen randomly from each class, and may not be the same as the ones for other CIFAR10 experiments.
\subsection*{DCGAN + AMAT Experiments}
We used standard CNN models for our DCGAN as shown in Table \ref{tab:dcganArch}. We use Adam optimizer with hyperparameters $\beta_1 = 0.0, \beta_2 = 0.9$. The learning rate for generator was $0.0002$, and the learning rate for the discriminator(s) was $0.0001$. The number of discriminators updates per generator was fixed at 1. The initial value of $T_t = 4$ epochs, and is increased $5$ epochs after a discriminator is added. $\alpha_t $ is initialized as $1.5$ and is increased by a factor of $1.5$ every time a discriminator is added. We chose $\epsilon = 0.3$, where the datapoint is assigned to a random discriminator with a probability $\epsilon$. We use 10 exemplar images, 1 from each CIFAR10 class.
\section*{Stacked MNIST Experiments}
Stacked MNIST provides us a test-bed to measure mode collapse. A three channel image is generated by stacking randomly sampled MNIST classes, thus creating a data distribution if 1000 modes. We use this dataset to show that, when generator oscillates to a different set of modes, catastrophic forgetting is induced in discriminator and this prevents the generator to recover previous modes. To study this phenomenon, we need to measure the correlation between number of modes the generator covered and the catastrophic forgetting in the discriminator. Measuring the number of modes is straight forward, we can by simply classify each channels of the generated images using a MNIST pretrained classifier to find its corresponding mode. However, to measure catastrophic forgetting in the discriminator, we use a proxy setting, where we take the high-level features of the real images from the discriminator and train a simple classifier on top of that. The discriminative quality of the features taken from the discriminator indirectly measure the ability of the network to remember the modes. Finally, as a control experiment we randomize the weights of the discriminator, and train a classifiers on the feature taken from randomized discriminator. This is to show that, extra parameters in the classifier does not interfere our proxy measure for the catastrophic forgetting. Finally, we train a DCGAN with a single discriminator, and a similar DCGAN architecture with our proposed AMAT procedure, and measure the number of modes covered by the generator and the accuracy of the discriminator.
\section*{A Fair comparison on discriminator capacity}
Our AMAT approach incrementally adds new discriminators to the GAN frameworks, and its overall capacity increases over time. Therefore, it is not fair to compare a model with AMAT training procedure with its corresponding the single discriminator model. As a fair comparison to our AMAT algorithm, we ran single discriminator model with approximately matching its discriminator capacity to the final AMAT model. For example, SN-GAN with AMAT learning scheme uses 4 discriminators at the end of its training. Therefore we use a discriminator with four times more parameters for the single discriminator SN-GAN model. This is done by increasing the convolutional fillters in the discriminator. Table~\ref{tab:fairD} shows that, even after matching the network capacity, the single discriminator models do not perform well as compared to our AMAT learning.
\begin{table*}[!t]
\centering
\resizebox{\textwidth}{!}{
\begin{tabular}{c|c| ccccc}
\toprule
& Scores & DCGAN & ResNetGAN & WGAN-GP & SN-GAN & BigGAN \\ \midrule
& \#of Param of D & 1.10 M & 3.22 M & 2.06 M & 4.20 M & 8.42 M \\
w/o AMAT & IS & 5.97 $\pm$ 0.08 & 6.59 $\pm$ 0.09 & 7.72 $\pm$ 0.06 & 8.24 $\pm$ 0.05 & 9.14 $\pm$ 0.05 \\
& FID & 34.7 & 36.4 & 19.1 & 14.5 & 10.5 \\ \midrule
& \#of Param of D & 1.02 M & 3 $\times$ 1.05 M & 2 $\times$ 1.05 M & 4 $\times$ 1.05 M & 8.50 M \\
+ AMAT & IS & 6.32 $\pm$ 0.06 & 8.1 $\pm$ 0.04 & 7.80 $\pm$ 0.07 & 8.34 $\pm$ 0.04 & 9.51 $\pm$ 0.06 \\
& FID & 30.14 & 16.35 & 17.2 & 13.8 & 6.11 \\ \midrule
\bottomrule
\end{tabular}
}
\vspace{2mm}
\caption{Increasing network capacity alone does not capture more modes. Even after the discriminator capacity is matched, single discriminator GANs do not perform as well as multi-adversarial GANs with AMAT learning}
\label{tab:fairD}
\end{table*}
\section*{Synthetic Data Experiments}
\label{appendix:synthetic}
We add flow-based non-linearity (Algorithm \ref{algo:synthetic}) to a synthetic 8-Gaussian ring dataset. We chose $K=5$ as our non-linearity depth and chose a randomly initiated 5 layer MLP as our non-linear functions. We use an MLP as our GAN generator and discriminator (Table \ref{tab:mlpArch}). We use the Adam optimizer with hyperparameters $\beta_1 = 0.0, \beta_2 = 0.9$. The learning rate for the generator and discriminator was $0.0002$. The number of discriminator updates per generator update is fixed to 1, and the batch size is kept 64. The initial value of $T_t = 5$ epochs, and is increased by $10$ every time a discriminator is added. $\alpha_t$ is initialized as $1.5$, and is increased by a factor of $1.5$ after a discriminator is added for the first 50 epochs. After that $\alpha_t$ is increased by a factor of $3$. We chose $\epsilon = 0.25$, where the datapoint is assigned to a random discriminator with a probability $\epsilon$. 1 random datapoint from each of the 8 modes is selected as the exemplar image.
Figure \ref{fig:eightgaussian} shows the difference in performance of a standard MLP GAN (\ref{tab:mlpArch} and the same MLP GAN with AMAT. The GIF on the lefts shows a cyclic mode collapse due the discriminator suffering from catastrophic forgetting. The same GAN with is able to completely mitigate catastrophic forgetting with just 2 discriminators added during training, on a 728-dimensional synthetic data.
\begin{table}[!htb]
\parbox{0.45\linewidth}{\centering
\begin{tabular}{c}
\toprule
\midrule
$z \in \mathbb{R}^{128} \sim \mathcal{N}(0,I) $\\
\midrule
dense $\rightarrow 4 \times 4 \times 512$\\
\midrule
$4\times4$, stride$=$2 deconv. BN 256 ReLU\\
\midrule
$4\times4$, stride$=$2 deconv. BN 128 ReLU\\
\midrule
$4\times4$, stride$=$2 deconv. BN 64 ReLU\\
\midrule
$3\times3$, stride$=$1 conv. 3 Tanh\\
\midrule
\bottomrule
\end{tabular}
\vspace{3mm}
\caption*{Generator}
}
\vspace{1mm}
\parbox{0.45\linewidth}{\centering
\begin{tabular}{c}
\toprule
\midrule
$x \in \mathbb{R}^{32 \times 32 \times 3}$\\
\midrule
$3\times3$, stride$=$1 conv. 64 lReLU\\
$4\times4$, stride$=$2 conv. 64 lReLU\\
\midrule
$3\times3$, stride$=$1 conv. 128 lReLU\\
$4\times4$, stride$=$2 conv. 128 lReLU\\
\midrule
$3\times3$, stride$=$1 conv. 256 lReLU\\
$4\times4$, stride$=$2 conv. 256 lReLU\\
\midrule
$3\times3$, stride$=$1 conv. 512 lReLU\\
\midrule
dense $\rightarrow 1$\\
\midrule
\bottomrule
\end{tabular}
\vspace{3mm}
\caption*{Discriminator}
}
\vspace{3mm}
\caption{DCGAN Architecture for CIFAR10}
\label{tab:dcganArch}
\end{table}
\begin{table*}[!t]
\centering
\parbox{0.45\linewidth}{\centering
\begin{tabular}{c}
\toprule
\midrule
$z \in \mathbb{R}^{25} \sim \mathcal{N}(0,I) $\\
\midrule
dense $\rightarrow 128$, BN 128 ReLU\\
\midrule
dense $\rightarrow 128$, BN 128 ReLU\\
\midrule
dense $\rightarrow 512$, BN 512 ReLU\\
\midrule
dense $\rightarrow 1024$, BN 1024 ReLU\\
\midrule
dense $\rightarrow 2$, Tanh\\
\midrule
\bottomrule
\end{tabular}
\vspace{2mm}
\caption*{Generator}
}
\parbox{0.45\linewidth}{\centering
\begin{tabular}{c}
\toprule
\midrule
$x \in \mathbb{R}^{2}$\\
\midrule
dense $\rightarrow 128$ ReLU\\
\midrule
dense $\rightarrow 512$ ReLU\\
\midrule
dense $\rightarrow 1$ Sigmoid\\
\midrule
\bottomrule
\end{tabular}
\vspace{2mm}
\caption*{Discriminator}
}
\vspace{1mm}
\caption{MLP architecture for Synthetic Dataset}
\label{tab:mlpArch}
\end{table*}
\begin{figure*}
\begin{subfigure
\centering
\animategraphics[loop,autoplay,width=0.24\textwidth]{5}{figs/gif0/}{0}{47}
\end{subfigure}
\begin{subfigure
\centering
\animategraphics[loop,autoplay,width=0.24\textwidth]{5}{figs/gen/frame_}{1}{48}
\end{subfigure}
\begin{subfigure
\centering
\animategraphics[loop,autoplay,width=0.24\textwidth]{5}{figs/d0/frame_}{1}{48}
\end{subfigure}
\begin{subfigure
\centering
\animategraphics[loop,autoplay,width=0.24\textwidth]{5}{figs/d1/frame_}{1}{48}
\end{subfigure}
\caption{ \small{\textbf{GAN training visualization}: (Figure contains animated graphics, better viewed in Adobe Acrobat Reader) Training trajectories of an MLP in table \ref{tab:mlpArch} (leftmost panel) and an MLP trained with our AMAT procedure (Algorithm \ref{algo:multi}) (rest three panels) on a $784$-dimensional synthetic dataset. Green dots represent real samples and the blue dots represent the generated samples. The vanilla GAN samples are overlayed against discriminator's output heatmap where the warm yellow color indicates a high probability of being real and cold violet indicates fake. In the AMAT + GAN panels, the discriminator landscapes are shown separately for both discriminators with the second discriminator being spawned at iteration $4000$ (Algorithm \ref{algo:spawn}). The $2$D visualizations of the $784$D data space is facilitated by our synthetic data generation procedure (Algorithm \ref{algo:synthetic}). }}
\label{fig:eightgaussian}
\end{figure*}
\clearpage
\section{Introduction}
\label{sec:intro}
Generative Adversarial Networks (GANs) \citep{goodfellow2014generative} are an extremely popular class of generative models used for text and image generation in various fields of science and engineering, including biomedical imaging~\citep{yi2019generative,nie2018medical,wolterink2017generative}, autonomous driving~\citep{hoffman2018cycada,zhang2018deeproad}, and robotics~\citep{rao2020rl,bousmalis2018using}. However, GANs are widely known to be prone to \textit{mode collapse}, which refers to a situation where the generator only samples a few modes of the real data, failing to faithfully capture other more complex or less frequent categories. While the mode collapse problem is often overlooked in text and image generation tasks, and even traded off for higher realism of individual samples~\citep{karras2019style,brock2018large}, dropping infrequent classes may cause serious problems in real-world problems, in which the infrequent classes represent important anomalies. For example, a collapsed GAN can produce racial/gender biased images~\citep{Menon_2020_CVPR}.
Moreover, mode collapse causes instability in optimization, which can damage not only the diversity but also the realism of individual samples of the final results. As an example, we visualized the training progression of the vanilla GAN~\citep{goodfellow2014generative} for a simple bimodal distribution in the top row of Figure~\ref{fig:oscillation}. At collapse, the discriminator conveniently assigns high realism to the region unoccupied by the generator, regardless of the true density of the real data. This produces a strong gradient for the generator to move its samples toward the dropped mode, swaying mode collapse to the opposite side. In particular, the discriminator loses its ability to detect fake samples it was previously able to, such as point \textbf{X}\tikz\draw[red,fill=red] (0,5) circle (.3ex);. The oscillation continues without convergence.
We observe that the mode collapse problem in GAN training is closely related to Catastrophic Forgetting~\citep{mccloskey1989catastrophic, mcclelland1995there, ratcliff1990connectionist} in continual learning.
A promising line of works~\citep{sidetuning2019, NIPS2019_9429, rusu2016progressive, fernando2017pathnet} tackle the problem in the supervised learning setting by instantiating multiple predictors, each of which takes charge in a particular subset of the whole distribution. Likewise, we also tackle the problem of mode collapse in GAN by tracking the severity of Catastrophic Forgetting by storing a few exemplar data during training, and spawning an additional discriminator if forgetting is detected, Figure~\ref{fig:oscillation}. The key idea is that the added discriminator is left intact unless the generator recovers from mode dropping of that sample, essentially sidestepping catastrophic forgetting.
We show that our proposed approach based on adaptive addition of discriminators can be added to any of the existing GAN frameworks, and is most effective in preventing mode collapse. Furthermore, the improved stability of training boosts the standard metrics on popular GAN frameworks. To summarize, our contributions are: \emph{First}, we propose a novel GAN framework, named Adaptive Multi Adversarial Training (AMAT), that effectively prevents Catastrophic Forgetting in GANs by spawning additional discriminators during training. \emph{Second}, we also propose a computationally efficient synthetic data generation procedure for studying mode collapse in GANs that allows visualizing high dimensional data using normalizing flows. We show that mode collapse occurs even in the recent robust GAN formulations. \emph{Third}, our method can be plugged into any state-of-the-art GAN frameworks and still improve the quality and coverage of the generated samples.
\begin{figure*}
\centering
\includegraphics[trim={0 0 0 10},clip,width=0.90\textwidth]{figs/totalcomb2.png}
\vspace{3mm}
\caption{\textbf{Visualizing training trajectories}: Distribution of real (green dots) and fake (blue dots) over the course of vanilla GAN (top row) and our method (the second row and below). The background color indicates the prediction heatmap of the discriminator with blue being fake and warm yellow being real. Once the vanilla GAN falls into mode collapse (top row), it ends up oscillating between the two modes without convergence. Also, the discriminator's prediction at point X oscillates, indicating catastrophic forgetting in the discriminator. AMAT algorithm adapts to the need, and a new discriminator is spawned during training which effectively learns the forgotten mode, guiding the GAN optimization toward convergence. }
\label{fig:oscillation}
\vspace{-3mm}
\end{figure*}
\vspace{-5mm}
\section{Related Works}
\vspace{-2mm}
\label{sec:related}
Previous works have focused on independently solving either catastrophic forgetting in supervised learning or mode collapse during GAN training. In this section we review these works in detail and discuss our commonalities and differences.
\vspace{-4mm}
\subsection{Mitigating Mode Collapse in GANs}
Along with advancement in the perceptual quality of images generated by GAN~\citep{miyato2018spectral,karras2019style,brock2018large,karras2020analyzing}, a large number of papers~\citep{durugkar2016generative,metz2016unrolled,arjovsky2017wasserstein,srivastava2017veegan,nguyen2017dual,lin2018pacgan,MeschederICML2018,karras2019style} identify the problem of mode collapse in GANs and aim to mitigate it. However mode collapse was seen as a secondary symptom that would be naturally solved as the stability of GAN optimization progresses~\citep{arjovsky2017wasserstein,MeschederICML2018,bau2019seeing}.
To explicitly address mode collapse, Unrolled GAN~\citep{metz2016unrolled} proposes an unrolled optimization of the discriminator to optimally match the generator objective, thus preventing mode collapse. VEEGAN~\citep{srivastava2017veegan} utilizes the reconstruction loss on the latent space. PacGAN~\citep{lin2018pacgan} feeds multiple samples of the same class to the discriminator when making the decisions about real/fake. In contrast, our approach can be plugged into existing state-of-the-art GAN frameworks to yield additional performance boost.
\vspace{-4mm}
\subsection{Multi-adversarial Approaches}
\vspace{-1mm}
The idea of employing more than one adversarial network in GANs to improve results has been explored by several previous works independent of the connection to continual learning and catastrophic forgetting. MGAN~\citep{hoang2018mgan} uses multiple generators, while D2GAN~\citep{nguyen2017dual} uses two discriminators, and GMAN~\citep{durugkar2016generative} and MicrobatchGAN~\citep{mordido2020microbatchgan} proposed a method with more than two discriminators that can be specified as a training hyperparameter beforehand.
However, all previous works require the number of discriminators to be fixed beforehand, which is a major drawback since it depends on several intricate factors such as training dynamics, data distribution complexity, model architecture, initialization hyper-parameters etc. and is expensive and difficult to approximate even with several runs of the algorithm. In contrast, noting by the connection of multi-adversarial training to parameter expansion approaches to catastrophic forgetting, we propose an \textit{adaptive} method that can add discriminators incrementally during training thus achieving superior performance than existing works both on data quality metrics as well as overall computational effort.
\begin{table*}[t]
\resizebox{\textwidth}{!}{
\begin{tabular}{c|c|c|c|c|c||c}
\begin{tabular}{c}
$g(\mathbf{z}) = $
\end{tabular}
& $\mathbbm{1}$
& \begin{tabular}[c]{@{}c@{}}
$\mathbf{A}_{392 \times 2}$
\\ \end{tabular} &
\begin{tabular}[c]{@{}c@{}}
$\mathbf{z}$
\end{tabular}
& \begin{tabular}[c]{@{}c@{}}
MLP
\end{tabular}
& \begin{tabular}[c]{@{}c@{}}
MLP, $\mathbf{A}_{392 \times 2}$
\end{tabular}
& \begin{tabular}{@{}c@{}}
\small{MNIST} \end{tabular}
\\[1ex]
\hline
\begin{tabular}[c]{@{}c@{}}
Label\\ \end{tabular}
& {\fontfamily{lmtt}\selectfont Level I}
& {\fontfamily{lmtt}\selectfont Level II}
& {\fontfamily{lmtt}\selectfont Level III}
& {\fontfamily{lmtt}\selectfont Level IV}
& {\fontfamily{lmtt}\selectfont Level V}
& -
\\
\hline
\begin{tabular}[c]{@{}c@{}}
\small{GAN-NS \citep{goodfellow2014generative}}\\ \end{tabular}
& \cmark \enskip \big\vert \enskip \xmark
& \cmark \enskip \big\vert \enskip \xmark
& \cmark \enskip \big\vert \enskip \cmark
& \cmark \enskip \big\vert \enskip \cmark
& \cmark \quad \big\vert \quad \cmark
& \xmark
\\
\begin{tabular}[c]{@{}c@{}}
\small{WGAN} \citep{arjovsky2017wasserstein} \\ \end{tabular}
& \xmark \enskip \big\vert \enskip \xmark
& \cmark \enskip \big\vert \enskip \xmark
& \cmark \enskip \big\vert \enskip \xmark
& \cmark \enskip \big\vert \enskip \cmark
& \cmark \quad \big\vert \quad \cmark
& \xmark
\\
\begin{tabular}[c]{@{}c@{}}
\small{Unrolled GAN} \citep{metz2016unrolled} \end{tabular}
& \xmark \enskip \big\vert \enskip \xmark
& \xmark \enskip \big\vert \enskip \xmark
& \xmark \enskip \big\vert \enskip \xmark
& \xmark \enskip \big\vert \enskip \xmark
& \cmark \quad \big\vert \quad \cmark
& \xmark
\\
\begin{tabular}[c]{@{}c@{}}
\small{D2GAN} \citep{nguyen2017dual} \end{tabular}
& \xmark \enskip \big\vert \enskip \xmark
& \xmark \enskip \big\vert \enskip \xmark
& \xmark \enskip \big\vert \enskip \xmark
& \xmark \enskip \big\vert \enskip \xmark
& \cmark \quad \big\vert \quad \cmark
& \xmark
\\
\begin{tabular}[c]{@{}c@{}}
\small{GAN-NS + AMAT} \end{tabular}
& \xmark \enskip \big\vert \enskip \xmark
& \xmark \enskip \big\vert \enskip \xmark
& \xmark \enskip \big\vert \enskip \xmark
& \xmark \enskip \big\vert \enskip \xmark
& \cmark \quad \big\vert \quad \cmark
& \xmark
\\
\end{tabular}
}
\hline
\vspace{1mm}
\caption{\xmark \hspace{0.1mm} indicates that the generator could effectively learn all the data modes, while \cmark \hspace{0.1mm} means \textit{despite best efforts with tuning} the training suffers from mode collapse (more than a quarter of data modes dropped). We show results with the SGD (left) \& ADAM (right) optimizers. MNIST results with ADAM optimizer are provided for reference. We observe that MNIST is a relatively easy dataset, falling between {\fontfamily{lmtt}\selectfont Level I} and {\fontfamily{lmtt}\selectfont II} in terms of complexity.}
\label{tab:synthetic}
\vspace{-4mm}
\end{table*}
\vspace{-4mm}
\subsection{Overcoming Catastrophic Forgetting in GAN}
\vspace{-1mm}
Methods to mitigate catastrophic forgetting can be categorized into three groups: a) regularization based methods~\citep{kirkpatrick2017overcoming} b) memory replay based methods~\citep{rebuffi2017icarl} c) network expansion based methods~\citep{zhang2018deeproad, NIPS2019_9429}. Our work is closely related to the third category of methods, which dynamically adds more capacity to the network, when faced with novel tasks. This type of methods, adds \emph{plasticity} to the network from new weights (fast-weights) while keeping the \emph{stability} of the network by freezing the past-weights (slow-weights).
Additionally, we enforce stability by letting a discriminator to focus on a few set of classes, not by freezing its weights.
The issue of catastrophic forgetting in GANs has been sparsely explored before.
\citet{chen2018self} and \citet{tran2019self} propose a self-supervised learning objective to prevent catastrophic forgetting by adding new loss terms. \citet{liang2018generative} proposes an online EWC based solution to tackle catastrophic forgetting in the discriminator. We propose a prominently different approach based on parameter expansion rather than regularization. While the regularization based approaches such as \citet{liang2018generative} attempt to retain the previously learnt knowledge by constrained weight updates, the parameter expansion approaches effectively sidestep catastrophic forgetting by freezing previously encoded knowledge. \citet{thanhcatastrophic} also discuss the possibility of catastrophic forgetting in GAN training but their solution is limited to theoretical analyses with simplistic proposals such as assigning larger weights to real samples and optimizing the GAN objective with momentum. Practically, we observed that their method performs worse than a plain vanilla DCGAN on simple real world datasets like CIFAR10. In contrast, our method leverages insights from continual learning and has a direct connections to prevalent parameter expansion approaches in supervised learning. We benchmark extensively on several datasets and state-of-the-art GAN approaches where our method consistently achieves superior results to the existing methods.
\vspace{-5mm}
\section{Proposed Method}
\vspace{-2mm}
In this section, we first describe our proposed data generation procedure that we use as a petri dish for studying mode collapse in GANs.
The procedure uses random normalizing flows for simultaneously allowing training on complex high dimensional distributions yet being perfectly amenable to 2D visualizations.
Next, we describe our proposed Adaptive Multi Adversarial Training (AMAT) algorithm that effectively detects catastrophic forgetting and spawns a new discriminator to prevent mode collapse.
\vspace{-3mm}
\subsection{Synthetic Data Generation with Normalizing flows}
Mode dropping in GANs in the context of catastrophic forgetting of the discriminator is a difficult problem to investigate using real datasets. This is because the number of classes in the dataset cannot be easily increased, the classes of fake samples are often ambiguous, and the predictions of the discriminator cannot be easily visualized across the whole input space.
In this regard, we present a simple yet powerful data synthesis procedure that can generate complex high dimensional multi-modal distributions, yet maintaining perfect 2-D visualization capabilities. Samples from a 2-D Gaussian distribution are augmented with biases and subjected to an invertible normalizing flow~\citep{karami2019invertible} parameterized by well conditioned functions $g_i: \mathbb{R}^{d^0_i} \rightarrow \mathbb{R}^{d^1_i}$. This function can be followed by a linear upsampling transformation parameterized by a $d^1_i \times d^0_{i+1}$ dimensional matrix $A^i$ (Algorithm \ref{algo:synthetic}).
The entire transform is deliberately constructed to be a bijective function so that every generated sample in $\hat{y} \in \mathbb{R}^D$ can be analytically mapped to $\mathbb{R}^2$, allowing perfect visualization on 2D space. Furthermore, by evaluating a dense grid of points in $\mathbb{R}^2$, we can understand discriminator's learned probability distribution on $\mathbf{z}$ manifold as a heatmap on the 2D plane.
This synthetic data generation procedure enables studying mode collapse in a controlled setting. This also gives practitioners the capability to train models on a chosen data complexity with clean two-dimensional visualizations of both the generated data and the discriminator's learnt distribution.
This tool can be used for debugging new algorithms using insights from the visualizations.
In the case of mode collapse, a quick visual inspection would give the details of which modes face mode collapse or get dropped from discriminator's learnt distribution.
\vspace{-4mm}
\subsection{Adaptive Multi Adversarial Training}
Building upon the insight on relating catastrophic forgetting in discriminator to mode collapse in generator, we propose a multi adversarial generative adversarial network training procedure.
The key intuition is that the interplay of catastrophic forgetting in the discriminator with the GAN minimax game, leads to an oscillation generator. Thus, as the generator shifts to a new set of modes the discriminator forgets the learnt features on the previous modes.
\input{algos}
However if there are multiple discriminators available, each discriminator can implicitly \emph{specialize} on a subset of modes. Thus even if the generator oscillates, each discriminator can remember their own set of modes, and they will not need to move to different set of modes. This way we can effectively \emph{sidestep} forgetting and ensure the networks do not face significant distribution shift. A detailed version of our proposed method is presented in Algorithm \ref{algo:multi}.
\newline
\newline
\textbf{Spawning new discriminators}: We initialize the AMAT training Algorithm~\ref{algo:multi} with a regular GAN using just one discriminator. We also sample a few randomly chosen exemplar data points with a maximum of one real sample per mode, depending on dataset complexity. The exemplar data points are used to detect the presence of catastrophic forgetting in the currently active set of discriminators $\displaystyle {\mathbb{D}}$ and spawn a new discriminator if needed. Specifically (Algorithm \ref{algo:spawn}), we propose that if \emph{any} discriminator among $\displaystyle {\mathbb{D}}$ has an unusually high score over an exemplar data point ${\bm{e}}_i$, this is because the mode corresponding to ${\bm{e}}_i$ has either very poor generated samples or has been entirely dropped.
In such a situation, if training were to continue we risk catastrophic forgetting in the active set $\displaystyle {\mathbb{D}}$, if the generator oscillate to near ${\bm{e}}_i$. This is implemented by comparing the $\max$ score over $\displaystyle \vD$ at ${\bm{e}}_i$ to the average score over $\displaystyle \vD$ and spawning a new discriminator when the ratio exceeds $\alpha_t (> 1)$. Further, we propose to have $\alpha_t (> 1)$ a monotonically increasing function of $| \displaystyle {\mathbb{D}}|$, thus successively making it harder to spawn each new discriminator. Additionally, we use a warmup period $T_t$ after spawning each new discriminator from scratch to let the spawned discriminator train before starting the check over exemplar data-points.
\newline
\newline
\noindent \textbf{Multi-Discriminator Training: } We evaluate all discriminators in $\displaystyle {\mathbb{D}}$ on the fake samples but do not update all of them for all the samples. Instead, we use the discriminator scores to assign responsibility of each data point to only one discriminator.
\newline
\newline
\noindent \textbf{Training over fake samples}: We use an $\epsilon$-greedy approach for fake samples where the discriminator with the lowest output score is assigned responsibility with a probability $1 - \epsilon$ and a random discriminator is chosen with probability $\epsilon$.
\newline
\newline
\noindent \textbf{Training over real samples}: The discriminator is always chosen uniformly randomly thus we slightly prefer to assign the same discriminator to the fake datapoints from around the same mode to ensure that they do not forget the already learnt modes and switch to another mode. The random assignment of real points ensure that the same preferentially treated discriminator also gets updated on real samples.
Further for optimization stability, we ensure that the real and fake sample loss incurred by each discriminator is roughly equal in each back-propagation step by dynamically reweighing them by the number of data points the discriminator is responsible for. We only update the discriminator on the losses of the samples they are responsible for.
While it may seem that adding multiple discriminators makes the procedure expensive, in practice the total number of added discriminator networks never surpass three for the best results.
It is possible to change the hyperparameters to allow a large number of discriminators but that results in sub-optimal results and incomplete training. The optimal hyperparameter selection is explained in the Appendix for each dataset.
Further, the additional discriminators get added during later training stages and are not trained from the start, saving compute in comparison to prior multi-adversarial works which train all the networks from the beginning.
Also, unlike AdaGAN \citep{tolstikhin2017adagan} and similar Boosted GAN models that need to store multiple Generators post training, the final number of parameters required during inference remains unchanged under AMAT . Thus the inference time remains the same, but with enhanced mode coverage and sample diversity. Unlike \citep{tolstikhin2017adagan}, our discriminator addition is adaptive, i.e. discriminators are added during the training thus being more efficient.
\newline
\newline
\noindent \textbf{Generator Training:} We take a weighted mean over the discriminators scores on the fake datapoints for calculating the generator loss. At each step, the weights each discriminator in $\displaystyle {\mathbb{D}}$ gets is in decreasing order of its score on the fake sample. Hence, the discriminator with the lowest score is given the most weight since it is the one that is currently specializing on the mode the fake sample is related to. In practice, we sample weights randomly from a Dirichlet distribution (and hence implicitly they sum to $1$) and sort according to discriminator scores to achieve this. We choose soft weighing over hard binary weights because since the discriminators are updated in an $\epsilon$ greedy fashion, the discriminators other than the one with the best judgment on the fake sample might also hold useful information. Further, we choose the weights randomly rather than fixing a chosen set to ensure AMAT is more deadset agnostic since the number of discriminator used changes with the dataset complexity so does the number of weights needed. While a suitably chosen function for generating weights can work well on a particular dataset, we found random weights to work as well across different settings.
\vspace{-5mm}
\section{Results}
\vspace{-3mm}
\label{sec:results}
We test our proposed method on several synthetic and real datasets \& report a consistent increase in performance on GAN evaluation metrics such as
Inception Score \citep{salimans2016improved} and Frech\'et Inception Distance \citep{heusel2017gans} with our proposed AMAT. We also showcase our performance in the GAN fine-tuning regime with samples on the CUB200 dataset \citep{WelinderEtal2010} which qualitatively are more colorful and diverse than an identical BigGAN without AMAT (Figure \ref{fig:cub200}).
\begin{table*}[t]
\resizebox{\textwidth}{!}
\begin{tabular}{ccccc|c c}
\toprule
& GAN & UnrolledGAN & D2GAN & RegGAN & DCGAN & with AMAT \\ \midrule
\# Modes covered & $628.0 \pm 140.9$ & $817.4 \pm 37.9$ & $1000 \pm 0.0$ & $ 955.5\pm18.7$ & $849.6\pm 62.7$ & $\mathbf{1000 \pm 0.0}$ \\
KL (samples $\Vert$ data) & $2.58\pm 0.75$ & $1.43 \pm 0.12$ & $0.080 \pm 0.01$ & $0.64 \pm 0.05$ & $0.73\pm0.09$ & $0.078\pm0.01$ \\ \bottomrule
\end{tabular}
}
\vspace{3mm}
\caption{\textbf{Quantitative Results on the Stacked MNIST dataset}: Applying our proposed adaptive multi adversarial training (AMAT) procedure to a simple DCGAN achieves perfect mode coverage, better than many existing methods for mode collapse.}
\label{tab:stacked}
\end{table*}
\begin{table*}[t]
\smaller
\resizebox{\textwidth}{!}{
\begin{tabular}{c| cc|cc|cc}
\toprule
& & & GAN-NS & AMAT + & & AMAT + \\
Model & D2GAN & MicrobatchGAN & w/ ResNet & GAN-NS & DCGAN & DCGAN \\ \midrule
IS & $7.15 \pm 0.07$ & $6.77$ & $6.7 \pm 0.06$ & $\mathbf{8.1 \pm 0.04}$ & $6.03\pm0.05$ & $\mathbf{6.32\pm0.06}$ \\
FID & - & - & $28.91$ & $\mathbf{16.35}$ & $33.42$ & $\mathbf{30.14}$ \\ \midrule
& WGAN-GP & AMAT + & &AMAT + & & AMAT + \\
Model & w/ ResNet & WGAN-GP & SN-GAN & SN-GAN & BigGAN & BigGAN \\ \midrule
IS & $7.59\pm0.10$ & $\mathbf{7.80\pm0.07}$ &$ 8.22\pm0.05$ &$\mathbf{8.34\pm0.04}$& $9.22$ & $\mathbf{9.51\pm0.06}$ \\
FID & $19.2$ & $\mathbf{17.2}$ &$14.21$ &$\mathbf{13.8}$ & $8.94$ & $\mathbf{6.11}$ \\
\bottomrule
\end{tabular}
}
\vspace{3mm}
\centering
\caption{\textbf{Quantitative Results on CIFAR10}: We benchmark AMAT against several other multi-adversarial baselines as well as on several GAN architectures across all of which we observe a consistent performance increase.}
\label{tab:cifar}
\end{table*}
\subsection{Synthetic Data}
We utilize the proposed synthetic data generation procedure with randomly initialized normalizing flows to visualize the training process of a simple DCGAN \citep{radford2015unsupervised}. Figure \ref{fig:oscillation} visualizes such a training process for a simple bimodal distribution. Observing the pattern of generated samples over the training iteration and the shifting discriminator landscape, we note a clear mode oscillation issue present in the generated samples driven by the shifting discriminator output distribution. Focusing on a single fixed real point in space at any of the modes, we see a clear oscillation in the discriminator output probabilities strongly indicating the presence of catastrophic forgetting in the discriminator network.
\noindent \textbf{Effect of Data Complexity on Mode Collapse}: We use the flexibility in choosing transformations $g_i$ to generate datasets of various data distribution complexities as presented in Table $\ref{tab:synthetic}$. Choosing $g(z)$ with successively more complicated transformations can produce synthetic datasets of increasing complexity, the first five of which we roughly classify as {\fontfamily{lmtt}\selectfont Levels}. The {\fontfamily{lmtt}\selectfont Levels} are generated by using simple transforms such as identitym constant mapping, small Multi layer perceptrons and well conditioned linear transforms ($\mathbf{A}$).
On this benchmark, we investigate mode collapse across different optimizers such as SGD \& ADAM \citep{kingma2014adam} on several popular GAN variants such as the non-saturating GAN Loss (GAN-NS) \citep{goodfellow2014generative}, WGAN \citep{arjovsky2017wasserstein} and also methods targeting mitigating mode collapse specifically such as Unrolled GAN \citep{metz2016unrolled} and D2GAN \citep{nguyen2017dual}.
We show results of our proposed AMAT training procedure with a simple GAN-NS, which matches performance with other more complicated mode collapse specific GAN architectures, all of which are robust to mode collapse up to {\fontfamily{lmtt}\selectfont Level IV}.
In practice we find all benchmarked methods to collapse at {\fontfamily{lmtt}\selectfont Level V}. Thus, in contrast to other simple datasets like MNIST~\citep{lecun1998mnist}, Gaussian ring, or Stacked MNIST~\citep{metz2016unrolled}, the complexity of our synthetic dataset can be arbitrarily tuned up or down to gain insight into the training and debugging of GAN via visualizations.
\begin{table*}[t]
\vspace{-2mm}
\centering
\resizebox{\textwidth}{!}{
\begin{tabular}{c| ccccc| c}
\toprule
Effect & Large $|\displaystyle {\mathbb{D}}|$ & Spawn too late & Greedy $\displaystyle \nabla$D & Random for fake & $\mathbf{1}$-hot weight & Proposed \\
Ablation & Small $\alpha$, Short $T_t$ & Long $T_t$ schedule & $\epsilon = 0$ & $\epsilon$-greedy for real & vector ${\bm{m}}$ & Method \\ \midrule
IS & $8.83 \pm 0.04$ & $9.28 \pm 0.08$ & $9.31 \pm 0.06$ & $8.95 \pm 0.04$ & $9.25 \pm 0.05$ & $9.51 \pm 0.06$ \\
FID & $14.23$ & $9.37$ & $8.6$ & $12.5$ & $9.25$ & $6.11$ \\
\bottomrule
\end{tabular}
}
\vspace{3mm}
\caption{\textbf{BigGAN + AMAT Ablations on CIFAR10} (A) A spawning condition with small $\alpha$ and short warmup schedule that leads to large number of discriminators (>$7$) (B) Long warm-up schedules that spawn new discriminators late into training (C) A greedy strategy for assigning responsibility of fake samples ($\epsilon = 0$) (D) Flipping the data splitting logic with responsibilities of fake samples being random and of real being $\epsilon$-greedy (E) Choosing the discriminator with lowest score for updating Generator instead of soft random weighting.}
\vspace{-2mm}
\label{tab:ablations}
\end{table*}
\begin{figure*}
\centering
\begin{subfigure
\centering
\includegraphics[width=0.48\textwidth]{figs/dis_modes.pdf}
\end{subfigure}
\begin{subfigure
\centering
\includegraphics[width=0.48\textwidth]{figs/dis_test_acc.pdf}
\end{subfigure}
\vspace{1mm}
\caption{{\textbf{Investigating the forgetting-collapse interplay:} We investigate our hypothesis that catastrophic forgetting is associated with mode collapse. On the left pane, we plot the magnitude of mode collapse by counting the number of modes produced by the generator. On the right pane, we assess the quality of the discriminator features by plotting the accuracy of linear classifier on top of the discriminator features at each epoch. In the original model, the coverage of modes and the quality of discriminator features are both low and decreasing. In particular, the test accuracy from the discriminator's features drops almost to randomly initialized weights (shown as \textit{control}). On the other hand, adding AMAT (\textit{MultiD}) dramatically improves both mode coverage and the discriminator test accuracy.
}}
\label{fig:classification}
\vspace{-4mm}
\end{figure*}
\vspace{-5mm}
\subsection{Stacked MNIST}
\vspace{-2mm}
We also benchmark several models on the Stacked MNIST dataset following \citep{metz2016unrolled, srivastava2017veegan}. Stacked MNIST is an extension of the popular MNIST dataset~\citep{lecun1998gradient} where each image is expanded in the channel dimension to $28 \times 28 \times 3$ by concatenating $3$ single channel images. The resulting dataset has a $1000$ overall modes. We measure the number of modes covered by the generator as the number of classes that are generated at least once within a pool of $25,600$ sampled images. The class of the generated sample is identified with a pretrained MNIST classifier operating channel wise on the original stacked MNIST image. We also measure the KL divergence between the label distribution predicted by the MNIST classifier in the previous experiment and the expected data distribution.
\newline
\newline
\noindent \textbf{Understanding the forgetting-collapse interplay}: In Section \ref{sec:intro}, we discuss our motivation for studying catastrophic forgetting for mitigating mode collapse. We also design an investigative experiment to explicitly observe this interplay by comparing the number of modes the generator learns against the quality of features the discriminator learns throughout GAN training on the stacked MNIST dataset. We measure the number of modes captured by the generator through a pre-trained classification network trained in a supervised learning fashion and frozen throughout GAN training. To measure the amount of \emph{`forgetting`} in discriminator, we extract features of real samples from the penultimate layer of the discriminator and train a small classifier on the real features for detecting real data mode. This implicitly indicates the quality and information contained in the the discriminator extracted features. However, the performance of classification network on top of discriminator features is confounded by the capacity of the classification network itself. Hence we do a control experiment, where we train the same classifier on features extracted from a randomly initialized discriminator, hence fixing a lower-bound to the classifier accuracy.
\begin{table*}[t]
\centering
\resizebox{\textwidth}{!}{
\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c|c}
\toprule
Classes & Plane & Car & Bird & Cat & Deer & Dog & Frog & Horse & Ship & Truck & Avg \\ \midrule
BigGAN & $24.23$ & $12.32$ & $24.85$ & $21.21$ & $12.81$ & $22.74$ & $17.95$ & $13.16$ & $12.11$ & $18.39$ & $8.94$ \\
$+$ AMAT & $20.50$ & $10.30$ & $23.48$ & $18.48$ & $11.51$ & $19.41$ & $11.50$ & $12.24$ & $10.69$ & $12.94$ & $6.11$ \\
$\Delta \%$ & $18.2$ & $19.6$ & $5.8$ & $14.8$ & $11.3$ & $17.2$ & $\mathbf{56.1}$ & $7.5$ & $11.7$ & $\mathbf{42.1}$ & $\mathbf{46.3}$
\\ \bottomrule
\end{tabular}
}
\vspace{3mm}
\caption{\textbf{Per-class FID on CIFAR10}: FID improves consistently across all classes.}
\label{tab:classwise}
\vspace{-3mm}
\end{table*}
Referring to Figure \ref{fig:classification}, we observe a clear relation between the number of modes the generator covers at an iteration and the accuracy of the classification network trained on the discriminator features at the same iteration. In the vanilla single discriminator scenario, the classification accuracy drops significantly, indicating a direct degradation of the discriminative features which is followed by a complete collapse of G. In the collapse phase, the discriminator's learnt features are close to random with the classification accuracy being close to that of the control experiment. This indicates the presence of significant catastrophic forgetting in the the discriminator network.
In contrast, training the same generator with the proposed AMAT procedure leads to stable training with almost all the modes being covered. The classification accuracy increasing before saturation. Catastrophic forgetting is \textit{effectively sidestepped} by adaptive multi adversarial training which produces stable discriminative features during training that provide a consistent training signal to the generator thereby covering all the modes with little degradation.
\vspace{-2mm}
\subsection{CIFAR10}
\begin{figure*}[t!]
\centering
\begin{subfigure
\centering
\includegraphics[width=0.45\textwidth]{figs/gans.pdf}
\end{subfigure} \hfill
\begin{subfigure
\centering
\includegraphics[width=0.45\textwidth]{figs/gans_1d.pdf}
\end{subfigure}
\caption{\textbf{Sample Diversity on CUB200:} We showcase samples from a BigGAN pretrained on imagenet \& finetuned on CUB200 with the AMAT procedure (left) and from an identical BigGAN finetuned without AMAT (right). Observe that while the sample quality is good for both, the samples generated with AMAT are more colorful \& diverse, with bright reds and yellow against several backgrounds. While the samples from vanilla fine-tuning are restricted to whites/grays \& only a hint of color.
}
\label{fig:cub200}
\vspace{-6mm}
\end{figure*}
We extensively benchmark AMAT on several GAN variants including unconditional methods such as DCGAN \citep{radford2015unsupervised}, ResNet-WGAN-GP \citep{gulrajani2017improved,he2016deep} \& SNGAN \citep{miyato2018spectral} and also conditional models such as BigGAN \citep{brock2018large}. Table \ref{tab:cifar} shows the performance gain on standard GAN evaluation metrics such as Inception Score and Fr\'echet distance of several architectures when trained with AMAT procedure. The performance gains indicate effective curbing of catastrophic forgetting in the discriminator with multi adversarial training. We use the public evaluation code from SNGAN \citep{miyato2018spectral} for evaluation. Despite having components such as spectral normalization, diversity promoting loss functions, additional R1 losses \& other stable training tricks that might affect catastrophic forgetting to different extents, we observe a consistent increase in performance across all models. Notably the ResNet GAN benefits greatly with AMAT despite a powerful backbone -- with IS improving from $6.7$ to $8.1$, indicating that the mode oscillation problem is not mitigated by simply using a better model.
AMAT improves performance by over $35\%$ even on a well performing baseline such as BigGAN (Table \ref{tab:cifar}). We also investigate the classwise FID scores of a vanilla BigGAN and an identical BigGAN + AMAT on CIFAR10 and report the results in Table $\ref{tab:classwise}$. Performance improves across all classes with previously poor performing classes such as `Frog' \& `Truck' experiencing the most gains. Further, we also ablate several key components of the AMAT procedure on the BigGAN architecture with results reported in Table \ref{tab:ablations}. We observe all elements to be critical to overall performance. Specifically, having a moderate $\alpha$ schedule to avoid adding too many discriminators is critical. Also, another viable design choice is to effectively flip the algorithm's logic and instead choose the fake points randomly while being $\epsilon$ greedy on the real points. This strategy performs well on simple datasets but loses performance with BigGAN on CIFAR10 (Table \ref{tab:ablations}). In all the above experiments, the computational time during inference is the same as the base model, \textit{irrespective of the number of discriminators} added during the training, since only a single generator is trained with AMAT and all the discriminators are discarded.
\vspace{-4mm}
\section{Conclusion}
\vspace{-3mm}
In summary, motivated from the observation of catastrophic forgetting in the discriminator, we propose a new adaptive GAN training framework that adds additional discriminators to prevent mode collapse. We show that our method can be added to existing GAN frameworks to prevent mode collapse, generate more diverse samples and improve FID \& IS. In future, we plan to apply AMAT to fight mode collapse in high resolution image generation settings.
\nocite{langley00}
|
2024-02-18T23:40:24.453Z
|
2021-12-30T02:21:23.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14406","language":"en","timestamp":1640830883000,"url":"https:\/\/arxiv.org\/abs\/2112.14406","yymm":"2112"}
|
proofpile-arXiv_000-10027
|
{"provenance":"002.jsonl.gz:10028"}
| null | null |
\section{Introduction}
Wireless multiple-input multiple-output (MIMO) technology is the subject of extensive theoretical and practical research for next-generation cellular systems \cite{wannstrom2013lte, huh2012achieving}, which consider multiuser MIMO as one of the core technologies \cite{huh2011network}. A considerable research effort has been dedicated to performance evaluation of MIMO systems in realistic cellular environments \cite{farajidana20093gpp}. One of the most important features in mobile systems is to provide much higher rate data services to a large number of users without corresponding increase in transmitter power and bandwidth \cite{wang2013spectral}. The efficiency of communication systems is traditionally measured in terms of spectral efficiency (SE), which is directly related to the channel capacity in bit/s. This metric indicates how efficiently a limited frequency spectrum is utilized \cite{heliot2012energy}.
Evaluation of the channel capacity for the MIMO system in terms of SE has attracted considerable research interest in the past decades \cite{foschini1998limits, telatar1999capacity}. The cell averaged SE is an important parameter for evaluating performance of cellular systems, and it is often obtained by using sophisticated system-level simulations. For traditional cellular systems, the cell-wide SE was studied \cite{alouini1999area}. The problem of user scheduling in 5G systems requires fast SE computations \cite{chataut2019channel, liu2016efficient}, which can be enabled using machine learning (ML) methods \cite{ullah2020machine}.
Advanced ML techniques are capable of providing simple and efficient solutions, given that complicated phases of design and training are completed. ML has recently been applied to power-control problems in wireless communications. In \cite{sun2017learning} a deep neural network for optimizing the averaging SE is constructed for a system serving a few dozens of users. The neural network structure can be reused \cite{zappone2018model} for solving an energy-efficiency problem. In \cite{van2019sum} the joint data and pilot non-convex power optimization for maximum SE in multi-cell Massive MIMO systems is studied. In \cite{sanguinetti2018deep} deep learning (DL) is used to solve max-min and max-prod power allocation (PA) problems in downlink of Massive MIMO. In \cite{zhao2020power, d2019uplink} the PA problem in cell-free Massive MIMO is solved using DL, which is closely related to the SE evaluation.
Recently the application of neural networks to massive MIMO got significant attention in the literature. Neural networks can be applied to channel decoding, detection, channel estimation, and resource management \cite{xia20}. A supervised neural-network-based approach for precoding (predicting precoding matrix $W$ using the dataset of (channel matrix $H$, precoding matrix $W$) pairs) for the multiple-input and single-output (MISO) systems is proposed. Also, a mixed problem statement is considered where precoding is computed via a conventional iterative scheme, while the neural network predicts optimal per-user power allocation, which is utilized in the iterative scheme. In \cite{huang20} an unsupervised neural-network-based approach for precoding in a MIMO system is proposed, where a neural network predicts the precoding based on a channel matrix, and spectral efficiency is used as the loss function directly, so there is no need to provide precoding as targets during training. There are several attempts \cite{guo2020regression, ullah2020machine, rozenblit2018machine} to predict signal to interference-and-noise-ratio (SINR), which is closely related to predicting SE, but these studies are very limited and take into account only the power distributions of users, but not a channel, precoding, detection matrices.
Although the aforementioned studies deal with problem statements that are somewhat comparable to the problem we solve, there is still a lot of potential for research. Firstly, these works mostly do not describe data generation, so it is unclear whether their results are applicable in practice. In contrast, we study the applicability of ML models in a wider range of Quadriga scenarios. Secondly, the previous research only describes a basic approach for solving the problem and does not study the influence of different neural network architectures, i. e. transformers, and does not consider other efficient machine learning algorithms, e. g. gradient boosting or linear models, that may contribute a lot to the quality of the solution. We include the mentioned algorithms and architectures in our experiments. Finally, all of the works above do not focus on SE prediction, while we consider it as one of our goals. To our knowledge, the prediction of SE using ML for a detailed MIMO model has not been previously studied.
The remainder of the paper is organized as follows. In Section~\ref{sec:MIMO Background} we describe a Massive MIMO system model, particular precoding methods, quality measures, types of detection matrices, and power constraints. In Section~\ref{sec:SE Problem} we consider the problem of spectral efficiency estimation using machine learning methods. In Section~\ref{sec:se_approach} we describe channel dataset, algorithm features, and standard machine learning approaches including linear models, neural networks, and gradient boosting algorithms. Section~\ref{sec:experiments} contains numerical results and Section~\ref{sec:conclusion} concludes the paper. Appendix Section~\ref{sec:appendix} contains results for a transformer-based method.
\section{MIMO System Background}\label{sec:MIMO Background}
We consider a precoding problem in multi-user massive MIMO communication in 5G cellular networks. In such system, a base station has multiple transmitting antennas that emit signals to several users simultaneously. Each user also has multiple receiving antennas. The base station measures the quality of channel between each transmitter and receiver. This is known as channel state information. The precoding problem is to find an appropriate weighting (phase and gain) of the signal transmission in such a way that the spectral efficiency is maximum at the receiver. We consider the following downlink multi-user linear channel:
\begin{equation}\label{eq:system_model}
r_k = G_k ( H_k W x + n_k), \quad k = 1 \dots K.
\end{equation}
In this model, we have $K$ users and we would like to transmit $L_k$ symbols to $k$-th user. Hence in total we would like to transmit a vector $x\in\mathbb{C}^L$, where $L=L_1+\dots+L_K$. First, we multiply the vector being transmitted by a precoding matrix $W\in\mathbb{C}^{T{\times}L}$, where $T$ is the total number of transmitting antennas on the base station. Then we transmit the precoded signal to all users. Suppose that $k$-th user has $R_k$ receiving antennas and $H_k\in\mathbb{C}^{R_k{\times}T}$ is a channel between $k$-th user antennas and antennas on the base station. Then $k$-th user receives $H_kWx + n_k$, where $n_k$ is Gaussian noise. Finally, $k$-th user applies a detection for transmitted symbols by multiplying the received vector by a detection matrix $G_k\in\mathbb{C}^{L_k{\times}R_k}$. The vector of detected symbols for the $k$-th users is denoted by $r_k\in\mathbb{C}^{L_k}$. The whole process of symbol transmission is presented in Fig.~\ref{fig:system_model} \cite{Conjugate}. Usually, in downlink the numbers of symbols being transmitted, user antennas, and base station antennas are related as $L_k \leqslant R_k \leqslant T$.
\begin{figure}
\centering
\includegraphics[scale=0.8]{MIMOExample}
\includegraphics[width=\linewidth]{LinearMIMO}
\caption{System model. Multi-User Precoding allows to transmit different information to different users simultaneously. The problem is to predict spectral efficiency function for a given precoding matrix $W$.}
\label{fig:system_model}
\end{figure}
\subsection{Precoding Methods}
We denote a concatenation of user channel matrices as $H=[H_1,\dots,H_K]\in\mathbb{C}^{R{\times}T}$. Then we make a singular-value decomposition of each matrix as $H_k=U_k^\mathrm{H}S_kV_k$, where $U_k\in\mathbb{C}^{R_k{\times}R_k}$ is a unitary matrix, $S_k\in\mathbb{R}^{R_k{\times}R_k}$ is a diagonal matrix and $V_k \in\mathbb{C}^{R_k{\times}T}$ is a semi-unitary matrix. In such way we obtain the decomposition: $H=[U_1^\mathrm{H}S_1V_1,\dots,U_K^\mathrm{H}S_KV_K]\in\mathbb{C}^{R{\times}T}$. For each $V_k$ we denote a sub-matrix $\widetilde{V}_k \in \mathbb C ^ {L_k{\times}T}$ with rows corresponding to the $L_k$ largest singular values from $S_k$. We denote a concatenation of all $\widetilde{V}_k$ as $\widetilde{V}=[\widetilde{V}_1,\dots,\widetilde{V}_K]\in\mathbb{C}^{L{\times}T}$ .
There are some well-known heuristic precoding algorithms \cite{MRT, RZF, ZF, RZF2, RZF19}:
\begin{align}
&W_{\mathrm{MRT}} = \mu \widetilde{V}^\mathrm{H}P \in \mathbb C^{T \times L} \mbox{ -- Maximum Ratio Transmission (MRT)}, \\
&W_{\mathrm{ZF}} = \mu \widetilde{V}^\mathrm{H}(\widetilde{V}\widetilde{V}^\mathrm{H})^{-1}P \in \mathbb C ^ {T \times L} \mbox{ -- Zero-Forcing (ZF)}.
\end{align}
The diagonal matrix $P\in\mathbb{C}^{L{\times}L}$ is a column-wise power normalization of precoding matrix, $\mu $ is a scalar power normalization constant for meeting per-antenna power constraints~\eqref{eq:power_constraints}. Normalization by constant $\mu$ is done as the last step, after normalization by $P$. The value $\sigma^2$ is a variance of Gaussian noise $n_k$ during transmission \eqref{eq:system_model}. We also consider LBFGS precoding optimization scheme \cite{bobrov2021study}.
\subsection{Quality Functions}
The base station chooses an optimal precoding matrix $W$ based on measured channel $H$, which maximizes the so-called \textit{Spectral Efficiency} (\textit{SE}) \cite{SE, Gotoh01, Benson06}. Function of SE is closely related to \textit{Signal-to-Interference-and-Noise Ratio} and is expressed as :
\begin{equation}
\mathrm{SE}(W, H, G, \sigma^2) = \frac{1}{K} \sum_{k=1}^K L_k \log_2 (1 + \mathrm{SINR}_k^{eff}(W, H_k, G_k, \sigma^2)).
\label{eq:spectral_efficiency}
\end{equation}
The set of symbol indexes targeted to $k$-th user is denoted as $\mathcal{L}_k$:
\begin{align}
&\mathrm{SINR}_k^{eff}(W, H_k, G_k, \sigma^2) = \Big({\prod\nolimits_{l \in \mathcal{L}_k} \mathrm{SINR}_l(W, H_k, g_l, \sigma^2) } \Big)^{\frac{1}{L_k}},\\
&\mathrm{SINR}_l(W, H_k, g_l, \sigma^2) = \dfrac{| g_l H_k w_l |^2}{\sum_{i \ne l}^{L} | g_l H_k w_i |^2 + \sigma^2 \| g_l\|^2}, \quad \forall l \in \mathcal{L}_k,
\end{align}
where $w_l \in \mathbb C ^ T$ is a precoding for the $l$-th symbol, $g_l \in \mathbb C ^ {R_k}$ is a detection vector of the $l$-th symbol, $\sigma^2$ is a variance of Gaussian noise \eqref{eq:system_model}. The total power of transmitting antennas without loss of generality is assumed to be equal to one.
Additionally, we consider the function of \textit{Single-User SINR} (\textit{SUSINR}) :
\begin{equation}\label{SUSINR}
\mathrm{SUSINR}(\widetilde{S}, \sigma^2) = \frac{1}{\sigma^2} \bigg(\prod_{k=1}^K \frac{1}{L_k} \bigg(\prod\nolimits_{l \in \mathcal{L}_k} s_l^2 \bigg)^{\frac{1}{L_k}}\bigg)^{\frac 1 K}.
\end{equation}
The formula (\ref{SUSINR}) describes channel quality for the specified user without taking into account the others. The matrix $\widetilde{ S} \in \mathbb C^{L \times L}$ contains all singular values of $\widetilde{S}_k$ on the main diagonal: $\widetilde{S} = \text{diag} (\widetilde{ S}_1 \dots \widetilde{S}_K) \in \mathbb C^{L \times L}$, and $\widetilde{S}_k \in \mathbb R ^ {L_k \times L_k}$ is a diagonal sub-matrix of $S_k$ consisting of $L_k$ largest singular values of the $k$-th user, and $s_l$ are the corresponding elements of $\widetilde{S}$ \cite{Conjugate}.
\subsection{Power Constraints}
It is important to note that each transmitting antenna has a restriction on power of the transmitter. Assuming all the symbols being transmitted are properly normalized, this results in the following constraints on the precoding matrix $W$:
\begin{equation}
\|w^i\|^2 = \sum_k|w^i_k|^2\le\frac{1}{T}\ \ \forall i=1,\dots,T.
\label{eq:power_constraints}
\end{equation}
The total power is assumed to be equal to one.
\subsection{Detection Matrices}
A detection matrix $G_k$ is specific to each user. Below we consider the main detection algorithms: MMSE and MMSE-IRC. In the case of MMSE, the detection matrix $G_k$ is calculated as \cite{MMSE}:
\begin{equation}
G_k = (H_kW_k)^\mathrm{H}(H_kW_k(H_kW_k)^\mathrm{H}+\sigma^2 I)^{-1},
\label{eq:MMSE}
\end{equation}
where matrix $H_kW_k$ is estimated using pilot signals on the user side.
\section{Spectral Efficiency Prediction Problem}\label{sec:SE Problem}
\subsection{Machine Learning Background}
In this subsection, we give a background on machine learning (ML) tasks and methods. In ML, we are given a training dataset $\mathcal{D}^{tr}=\{(x^{tr}_i, y^{tr}_i)\}_{i=1}^{N^{tr}}$ of $N^{tr}$ pairs (input object $x$, target $y$). The problem is to learn an algorithm $a(x)$ (also called prediction function) that predicts target $y$ for any new object $x$. Usually $x \in \mathbb{R}^d$, however, complex inputs could also be used. Conventionally used targets include $y \in \mathbb{R}$ (regression problem) or $y \in \{1, \dots, C\}$ (classification problem, $C$ is the number of classes). We are also given a set of $N^{te}$ testing objects, $\mathcal{D}^{te}=\{(x^{te}_i, y^{te}_i)\}_{i=1}^{N^{te}}$, and a metric $Q(a)=\sum_{i=1}^{N^{te}} q(y_i, a(x_i))$ that measures the quality of predicting target for new objects. This metric should satisfy the problem-specific requirements and could be non-differentiable. Summarizing, in order to use machine learning methods we need to specify objects, targets, metrics, and collect the dataset of (object, target) pairs.
\label{sec:se}
\subsection{Objects and Targets} Firstly, we consider \textbf{the problem of predicting the SE:} given $T$ antennas at the base station, $K$ users with $R_k$ antennas and $L_k$ symbols each, and set of channel matrices $\{H_k\}_{k=1}^K$ (input object), the problem is to predict the spectral efficiency SE \eqref{eq:spectral_efficiency} that could be achieved with some precoding algorithm (precoding is not modeled in this problem). We assume that $T=64$, $R_k=4$, $L_k=2 $ for all $k = 1 \dots K$ are fixed, while the number of users $K$ could be variable. Since $R_k$ is fixed, each object $\{H_k \in \mathbb{C}^{R_k \times T}\}_{k=1}^K$, which serves as an input for our ML algorithms, can be represented by three dimensional tensor $\mathcal{H} \in \mathbb{C}^{K \times 4 \times 64}$.
\subsection{Metric} Since spectral efficiency prediction is a regression problem, we can use metrics known from this class of tasks. Namely, we use Mean Average Percentage Error (MAPE):
\begin{equation}\label{eq:MAPE}
\mathrm{MAPE(a)} = \sum_{i=1}^{N^{te}} \biggl| \frac{\mathrm{SE}_i - a(\mathcal{H}_i)}{\mathrm{SE}_i} \biggr|,
\end{equation}
where $\mathrm{SE}_i$ is the spectral efficiency computed using specific precoding, and $a(\mathcal{H}_i)$ is spectral efficiency predicted by the algorithm $a$ for the $i$-th object $\mathcal{H}_i$.
\subsection{Problem Statement and Research Questions}
Given the dataset $\{\mathcal{H}_i, \mathrm{SE}_i\}_{i=1}^{N^{tr}}$, our goal is to train an algorithm $a(\mathcal{H}_i)$ that predicts spectral efficiency SE based on the object $\mathcal{H}$. The algorithm may also take additional information as input, e. g. the level of noise $\sigma^2$ or SUSINR value. We also consider an alternative problem statement with user-wise spectral efficiency prediction. Our research questions are as follows:
\begin{itemize}
\item Can we predict the spectral efficiency (average or user-wise) with acceptable quality, e.\,g.\, with error on the test dataset, which is an order of magnitude less than the value we predict, i.\,e.\, $\mathrm{MAPE}<10\%$~\eqref{eq:MAPE}?
\item Which machine learning algorithm out of the considered classes (linear models, gradient boosting, neural networks) is the most suitable for working with channel data?
\end{itemize}
\section{Proposed Methods}\label{sec:se_approach}
\subsection{Channel Dataset}
To obtain a dataset for this problem, we generate input channel matrices $H$, find precoding $W$ for each case using a certain precoding scheme, fixed for this particular dataset, and compute the target spectral efficiency SE using~\eqref{eq:spectral_efficiency}. To generate channel coefficients, we use Quadriga~\cite{Quadriga}, an open-source software for generating realistic radio channel matrices. We consider two Quadriga scenarios, namely Urban and Rural. For each scenario, we generate random sets of user positions and compute channel matrices for the obtained user configurations. We describe the generation process in detail in our work \cite{Conjugate}.
In the majority of the experiments, we consider three precoding algorithms: Maximum Ratio Transmission (MRT) \cite{MRT}, Zero Forcing (ZF) \cite{ZF} and LBFGS Optimization \cite{bobrov2021study}. The first two classic algorithms are quite simple, while the last one achieves higher spectral efficiency, however being slower. We provide proof-of-concept results for Interference Rejection Combiner (IRC) SE \eqref{eq:IRC1} \cite{IRC}.
\subsection{Algorithm Features}
We consider well-known machine learning algorithms for regression task, namely linear models, gradient boosting, and fully connected neural networks. We rely on the following pipeline:
\begin{enumerate}
\item Take an object $\mathcal{H}$ as input (and possibly other inputs, e. g. SUSINR).
\item Extract a feature vector representation $x = f(\mathcal{H})$ based on the raw input $\mathcal{H}$.
\item Apply the algorithm $a$ to feature vector $x$ to obtain $a(x) = \mathrm{SE}$.
\end{enumerate}
Further, we discuss how to obtain feature extraction procedure $f$ and prediction function $a$. Motivated by the exact formulas for the case of Maximum Ratio and Zero-Forcing precoding, we select the following \textit{default} features from the Singular Value Decomposition (SVD) of the $H = U^\mathrm{H} S V$ matrix, where $K$ is the number of users, and $L_i$ is the $i$-th user number of layers:
\begin{itemize}
\item $\{s^2_{ij}\}_{i,j=1}^{K, L_i}$ -- singular values;
\item $\{c_{ijkm}\}_{\underset{i \ne j}{i,j,k,m=1}}^{K,K,L_i, L_j}$ - pairwise layer correlations, $c_{ijkm} = corr(V_{ik}, V_{jm}) = | V_{ik} V_{jm}|^2$.
\end{itemize}
It can be shown that spectral efficiency~\eqref{eq:spectral_efficiency}, which is the value being predicted, depends only on the squares of the first $L_i$ singular values of $H_i$ of all users $i = 1 \dots K$ and the correlations between the first $L_i$ layers of all possible user pairs, which justifies this choice of \textit{default} features.
There are several issues with these features: 1) the number of features is variable, as the number of users $K$ varies among different objects $\mathcal{H} \in \mathbb{C}^{K \times 4 \times 64}$ in the dataset, while the aforementioned ML models take a feature vector of fixed size as input 2) the target objective, SE, is invariant to permutation of users, hence there should be a symmetry in feature representations. To address these issues, we propose two modifications of the \textit{default} features.
Firstly, we introduce symmetry with respect to user permutations by sorting the feature values along with user indices. For singular values, we sort users by the largest singular value. For pairwise correlations, we sort pairs of users by the largest correlation value of their layers.
Secondly, we alleviate the issue with a variable number of users by introducing \textit{polynomial} features. For a sequence $x = (x_1, x_2, \ldots, x_n)$ of features, we can apply symmetric polynomial transformation, obtaining fixed number of features $poly_k(x) = (e_1(x), e_2(x), \ldots, e_k(x))$:
\[
e_1(\{x_1, x_2, \ldots, x_n\}) = x_1 + \ldots + x_n
\]
\[
e_2(\{x_1, x_2, \ldots, x_n\}) = \sum_{1 \leqslant j_1 < j_2 \leqslant n} x_{j_1} x_{j_2}
\]
\[
\dots
\]
\[
e_k(\{x_1, x_2, \ldots, x_n\}) = \sum_{1 \leqslant j_1 < j_2 < \ \dotsm \ < j_k \leqslant n} x_{j_1} \dots x_{j_k}
\]
Therefore, in the following experiments, we consider 3 types of features: \textit{default}, \textit{sorted}, \textit{poly}$_{k}$. In some experiments, we also use a SUSINR as an additional feature. Models which are trained on \textit{default} and \textit{sorted} features require a dataset with a fixed number of users $K$=const, while \textit{poly}$_k$ features allow us to train models on a dataset with variable number of users $K$ between objects $\mathcal{H}_i$, which is a significant advantage. After extracting features, we use three ML algorithms: linear models, gradient boosting, and fully-connected neural networks. These algorithms are described in Section~\ref{sec:ml}.
\subsection{Machine Learning Methods Used for Predicting SE}\label{sec:ml}
In this work, we rely on the three most commonly used machine learning algorithms: linear models, gradient boosting, and neural networks. We discuss the positive and negative sides of these algorithms along with the description of how we apply them to 5G data. We compare all the results obtained by these models in Sec.~\ref{sec:experiments}.
\subsubsection{Linear Models}
We begin by evaluating simple \textit{linear regression models}. Linear models are very fast and memory-efficient and, being applied over a well-chosen feature space, could achieve high results in practice. Inspired by the fact that SUSINR is linear with respect to squares of singular values, we suppose that considering linear models is reasonable in our setting. We use the Scikit-learn implementation of linear regression and particularly, class \verb|LassoCV| which tunes the regularization hyperparameter using cross-validation.
\textit{Linear models} are one of the simplest and fastest supervised learning algorithms. Linear models work with data represented in matrix form where each object $x$ is represented by a feature vector: $x \in \mathbb{R}^d$. The dimensionality $d$ is fixed and the same for all objects. A linear model predicts the target for the input object $x$ with a linear function $f(x, w) = \sum_{j=1}^d w_j x_j$. In this prediction function, \textit{parameters} $w$ (also called \textit{weights}) are unknown and need to be found based on the training data $\mathcal{D}^{tr}$. Usually, this is achieved by optimizing a loss function $\ell(y, a(x))$ that measures the penalty of predicting target $y$ with algorithm $f(x, w)$
\subsubsection{Gradient Boosting}
We proceed by evaluating an out-of-the-box \textit{gradient boosting model}, a simple, but efficient nonlinear model. Gradient boosting is considered in as one of the best-performing algorithms for regression tasks. Moreover, gradient boosting is fast and does not require a lot of memory to store trained models. We use the CatBoost~\cite{Catboost} implementation of gradient boosting. The hyperparameters of our model are listed in Tab. \ref{tab:hyp_catboost_se_prediction}.
Gradient boosting processes input data and combines the predictions of several \textit{base} algorithms $b_1, \dots, b_N$, which are usually decision trees. The final prediction function is a weighted sum of base learners $a(x) = \sum_{n=1}^N \gamma_n b_n(x)$. Such algorithms are trained one by one, and each following algorithm tries to improve the prediction of the already built composition.
Boosting models, as well as linear models, are suitable only for the data with a fixed feature vector dimensionality. That is always the case for \textit{poly}$_k$ features, while \textit{default} and \textit{sorted} features require a fixed number of users $K$ in pairing.
\subsubsection{Neural Networks}
Finally, we consider \textit{fully-connected neural networks}. Their strength is the ability to capture complex nonlinear dependencies in data. In \textit{deep learning}, the prediction function $a$, called \textit{neural network}, is constructed as a composition of differentiable parametric functions: $a(x) = f(x, w)$ where \textit{parameters} $w \in \mathbb{R}^p$ are learned via the optimization of criteria $\ell(y, f(x, w))$. The distinctive feature of neural networks is the wide range of architectures, i. e. different compositions $f(x, w)$, that can be chosen given the specifics of the particular problem.
However, the drawback of neural networks is that they require careful hyperparameter tuning, i. e. they usually achieve low results in out-of-the-box configuration. Thus, in practice, gradient boosting is often preferred over neural networks. Still, we conduct experiments using neural networks as well, to estimate their capabilities of processing channel data and obtaining reasonable prediction quality. One more drawback of neural networks is that they are slower than classic ML algorithms such as linear models or boosting, and require more memory to store parameters. But there are several techniques aimed at reducing the time and memory complexity of the trained models \cite{Sparcification, Sparcification2, Vardrop, Distillation}.
Another important issue regarding neural networks is tuning their hyperparameters, which is essential for their performance. We train our fully-connected neural networks using SGD with momentum as it performed better than another popular optimization algorithm, Adam \cite{kingma2014adam}. The mini-batch has a size 32. We choose the learning rate using a search on the grid $\{10^{-p}\}$, $p=2, \dots, 5$ minimizing the train mean squared error. This is how we ensure the network does not underfit, i. e. is capable of recognizing train objects. However, it could overfit, i. e. memorize train data without learning actual dependencies in the data. To avoid it, we utilize standard regularization techniques: weight decay and dropout. We tune their hyperparameters using grid search, minimizing MAPE on the held-out set.
For neural networks, we normalize input data, both features, and targets, as it is essential for training convergence. Specifically, for each feature and target, we compute the mean and standard deviation over the training data, subtract the mean values from the elements of all the feature vectors, and divide by the standard deviation. This procedure needs to be done for both training and testing data, with the mean and standard deviation being computed over training data. For computing metrics, we scale the target values back to the original scale.
\section{Experimental Results}\label{sec:experiments}
\begin{figure*}
\begin{center}
\includegraphics[height=8cm]{MAPE_Urban_Scenario_Linear_Regression_Features}
\includegraphics[height=8cm]{MAPE_Urban_Scenario_Gradient_Boosting_Features}
\caption{Comparison of the SE prediction algorithms: linear methods with different features and boosting with different features. The \textbf{urban} scenario is considered; and three different precoding methods -- Zero-Forcing (Blue), Maximum Ratio (Orange), LBFGS (Grey). The lower the MAPE value, the better.}
\label{fig:lin_boost_urban_setups}
\end{center}
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics[height=8cm]{MAPE_Rural_Scenario_Linear_Regression_Features}
\includegraphics[height=8cm]{MAPE_Rural_Scenario_Gradient_Boosting_Features}
\caption{Comparison of the SE prediction algorithms: linear methods with different features and boosting with different features. The \textbf{rural} scenario is considered; and three different precoding methods -- Zero-Forcing (Blue), Maximum Ratio (Orange), LBFGS (Grey). The lower the MAPE value, the better.}
\label{fig:lin_boost_rural_setups}
\end{center}
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics[height=8cm]{MAPE_Mixed_Scenario_Linear_Regression_Features}
\includegraphics[height=8cm]{MAPE_Mixed_Scenario_Gradient_Boosting_Features}
\caption{Comparison of the SE prediction algorithms: linear methods with different features and boosting with different features. The \textbf{mixed} scenario (urban and rural together) is considered; and three different precoding methods -- Zero-Forcing (Blue), Maximum Ratio (Orange), LBFGS (Grey). The lower the MAPE value, the better.}
\label{fig:lin_boost_mixed_setups}
\end{center}
\end{figure*}
This section contains the results for linear models, gradient boosting and neural networks. Firstly, we fix the number of users $K$ and analyze how our ML algorithms work for different precoding methods in different scenarios. Then we show how the results change for different values of $K$ and experiment with the solutions based on polynomial features in the case of variable number of users. We also measure time and memory complexity of the proposed algorithms. Finally, we apply the proposed approaches to the task of user-wise SE prediction.
In all of our experiments, train data size $N^{tr}=1.6 \cdot 10^4$ and test data size $N^{te}=3.6 \cdot 10^3$.
The results for linear models and gradient boosting for different precoding methods and in different scenarios are shown in Figs.~(\ref{fig:lin_boost_urban_setups}, \ref{fig:lin_boost_rural_setups}, \ref{fig:lin_boost_mixed_setups}). Linear models demonstrate reasonable prediction quality. On the other hand, gradient boosting solutions provide significantly more accurate predictions. In terms of features, the best solutions are obtained using sorted features and polynomial features with the degree three. The results are only shown for a fixed number of users, $K=4$, but all the results hold for other considered values of $K$ (2 and 8 users).
\begin{figure*}
\begin{center}
\includegraphics[height=8cm]{MAPE_Urban_Scenario_Boosting_Sorted}
\caption{Comparison of the SE prediction algorithms for different numbers of users -- 2, 4, and 8. All models are trained on sorted features datasets and a fixed number of users and validated on the test data with the same number of users. The lower the MAPE value, the better.}
\label{fig:diff_users}
\end{center}
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics[height=8cm]{MAPE_Urban_Scenario_Linear_Regression}
\includegraphics[height=8cm]{MAPE_Urban_Scenario_Gradient_Boosting}
\caption{Comparison of the SE prediction algorithms for different numbers of users -- 2, 4, and 8. All models are trained on a combined train dataset with the variable number of users (2, 4, and 8 together) and then tested on the test data with the fixed number of users. Polynomial features with the degree three are used. The lower the MAPE value, the better.}
\label{fig:all_users}
\end{center}
\end{figure*}
In Fig.~\ref{fig:diff_users}, we compare the results of the best-proposed models (boosting on sorted and polynomial features with the degree three) for different values of $K$. In this experiment, all models are trained on datasets with a fixed number of users and then are tested with the same number of users. The results show that the proposed algorithms perform well for all the considered numbers of users.
The dimensionality of polynomial features is independent of the number of users $K$, therefore we can train all methods with polynomial features on the data with variable $K$. In Fig.~\ref{fig:all_users} we show the results of such an experiment. We train all models on a combined train dataset with a variable number of users (2, 4, and 8) and then test them on the test data with the fixed number of users. From the results we conclude that gradient boosting successfully tackles this problem for all precoding methods
\subsection{Time and Memory Complexity}
\begin{table}
\caption{Time complexity comparison. The average time for computing SE for one object is given in milliseconds. The computation time of the Zero-Forcing true SE values is given. Then, we compare the best-proposed models -- gradient boosting on sorted and polynomial features with the degree three. All these methods also use the same preprocessing, time for which is shown in the third column. The preprocessing consists of the computation of correlation matrix and squared singular values. }
\label{time}
\centering
\vspace{0.2in}
\begin{tabular}{c|c|ccc}
\textbf{Number of} & \textbf{Zero-Forcing }& \textbf{Preprocessing} & \textbf{Boosting} & \textbf{Boosting}
\\
\textbf{Users} & \textbf{ground truth} & & $+sorted$ \textbf{features} & $+poly_3$ \textbf{features} \\
\hline
\{2\} & 0.42 & 0.02 & 0.021 & 0.091
\\
\{4\} & 0.78 & 0.05 & 0.023 & 0.092
\\
\{8\} & 2.17 & 0.174 & 0.032 & 0.11
\\
\{2, 4, 8\} & 1.12 & 0.08 & - & 0.097
\\
\end{tabular}
\end{table}
In Tab.~\ref{time}, we report the inference time of our best models, namely boosting algorithms with sorted and polynomial features, and of several reference algorithms, e. g. the computational time of true SE values for the Zero-Forcing precoding method. We observe that the time complexity of gradient boosting (including preprocessing) is an order smaller than the time of computing true SE values for the Zero-Forcing algorithm.
\subsection{User-Wise SE Prediction}
In the previous experiments for SE prediction, we predicted SE for the whole pairing of users, i.e. averaged SE. Now, we wish to verify whether it is also possible to obtain reasonable prediction quality for each user separately since it can be useful in practice for selecting modulation coding scheme for each user \cite{bobrov2021massive}. We define the target SE$_u$ for each user as SE before averaging by the users: $\mathrm{SE}_k = \log_2(1 + \mathrm{SINR}^{eff}_k)$.
The features for user-wise SE prediction are the same as for the case of average SE prediction. For each user, we once again use the corresponding singular values, correlations between the chosen user's layers and the layers of other users, and additional features, such as noise power or equivalently SUSINR. These features characterize the relation of one particular user to all other users.
The results for the Urban scenario, 2/4/8 users (separate models) are shown in Fig.~\ref{fig:per_user}. Note that MAPE values in these experiments are not comparable to the values from previous sections, because of different target spaces. The results show that the gradient boosting approach outperforms the linear regression approach, for all considered user counts and precoding algorithms.
\begin{figure*}
\begin{center}
\includegraphics[height=8cm]{MAPE_Urban_Scenario_Boosting_Model_2}
\caption{Comparison of the SE prediction algorithms for user-wise SE prediction. The results for the Urban scenario, 2/4/8 users (separate models) are shown. Sorted features are used for boosting models. The lower the MAPE value, the better.}
\label{fig:per_user}
\end{center}
\end{figure*}
\subsection{SE-IRC Prediction}
We also provide the results for \textit{SE-IRC} \cite{IRC} prediction based on MMSE-IRC detection \eqref{eq:IRC1}, to verify that our method is robust to the change of the detection. For the case of MMSE-IRC, the detection matrix is calculated as:
\begin{equation}\label{eq:IRC1}
G_k = \left( H_{k}W_{k} \right) ^\mathrm{H} \left( H_{k}W_{k} \left( H_{k}W_{k} \right) ^\mathrm{H} + R_{uu}^{k} + \sigma^2 I \right) ^{-1},
\end{equation}
where the matrix \(R_{uu}^{k} \) is related to unit symbol variance and is calculated as follows:
\begin{equation}\label{eq:IRC}
R_{uu}^{k}=H_{k} \left(W W^\mathrm{H}-W_{k}W_{k}^\mathrm{H} \right) H_{k}^\mathrm{H}=H_{k} \left( \sum _{u=1,u \neq k}^{K}W_{u}W_{u}^\mathrm{H} \right) H_{k}^\mathrm{H}.
\end{equation}
The results for the Urban scenario, 2/4/8 users (separate models) are shown in Fig.~\ref{fig:per_user}. From this results we conclude that the LBFGS-IRC composition of precoding and detection allows us to obtain better prediction quality with gradient boosting in comparison with other precoding schemes except MRT for all considered user counts.
\subsection{Results for Fully-Connected Neural Networks}
To test whether neural networks could improve feature-based spectral efficiency prediction, we train a fully-connected neural network on the \textit{default} features, for the Urban scenario, 4 users, Zero Forcing precoding method. We consider one and three hidden layers configurations with 200 neurons at each layer. Our results show that while linear models provide MAPE of 0.0679, and gradient boosting -- of 0.0383, the one-/three-layer neural networks achieve MAPE of 0.0371 / 0.0372 (see Fig. \ref{fig:fcn}).
Thus, we conclude that using fully-connected neural networks is comparable to using gradient boosting in terms of prediction quality, while the tuning of network hyperparameters and the training procedure itself require significantly more time and memory than those of the other methods. However, these experiments showed that neural networks are in general applicable to channel data.
\begin{figure*}
\begin{center}
\includegraphics[height=8cm]{MAPE_Urban_Scenario_Zero_Forcing_4_Users_Default_Features}
\caption{Comparison of the SE prediction algorithms: linear regression, gradient boosting, one-layer neural network (NN), three-layers NN with default features. One scenario is considered -- urban; and one precoding method -- Zero-Forcing. The lower the MAPE value, the better.}
\label{fig:fcn}
\end{center}
\end{figure*}
\section{Conclusion}\label{sec:conclusion}
To summarize, in this paper we consider the problem of spectral efficiency prediction using machine learning methods. We looked at three methods of forming feature vector representations for user channel data. We compared several machine learning algorithms, namely linear models, gradient boosting, and fully connected neural networks. We found that gradient boosting applied to sorted objects provides the best results, while linear models achieve lower quality. The neural networks perform similarly to boosting, but require more time and effort to set up. In almost all cases, prediction quality reaches MAPE below 10\% using gradient boosting and neural networks. This valuable result will allow us to significantly improve the quality of MIMO wireless communication in the future.
\section*{Acknowledgements}
Authors are grateful to Dmitry Kovkov and Irina Basieva for discussions.
\section*{Funding}
The work is funded by Huawei Technologies.
\bibliographystyle{tfs}
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2024-02-18T23:40:24.460Z
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2021-12-30T02:21:50.000Z
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{"paloma_paragraphs":[]}
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{"arxiv_id":"2112.14423","language":"en","timestamp":1640830910000,"url":"https:\/\/arxiv.org\/abs\/2112.14423","yymm":"2112"}
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proofpile-arXiv_000-10028
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{"provenance":"002.jsonl.gz:10029"}
| null | null |
\section{Introduction}
\subsection{Essential manifolds, incompressible hypersurface and positive scalar curvature} Recall that a closed manifold is called {\it aspherical} if its universal covering is contractible, i.e. homotopy equivalent to a point. The {\it aspherical conjecture} \cite{Rosenberg1983} asserts that any closed aspherical manifold cannot admit smooth metrics with positive scalar curvature. When the dimension is two, this conjecture is just a direct consequence from the Gauss-Bonnet formula and the classfication for closed surfaces. In higher dimensions, this conjecture becomes more difficult and we briefly introduce the researches on that. In 1983, Gromov and Lawson \cite{GL83} first verified this conjecture in three dimensional case. Shortly after that, Schoen and Yau wrote a short survey \cite{SY87} with a sketchy proof for the aspherical conjecture in dimension four. But until recently, Gromov \cite{Gromov2020} as well as Chodosh and Li \cite{CL2020} verified this conjecture up to dimension five independently. All of these works actually lead to the consideration for the following enhanced aspherical conjecture. {\it That is, any closed essential manifold admits no smooth metric with positive scalar curvature, where essential manifolds here mean those admitting a non-zero degree map to aspherical ones.} Just recently, the enhanced aspherical conjecture up to dimension five was proven in detail by Chodosh, Li and Liokumovich in \cite{CLL2021} although this seems to be already asserted by Gromov in his four lectures \cite{Gromov2021}.
Instead of the manifold itself being essential, the existence of an essential incompressible closed hypersurface also provides a topological obstruction for positive scalar curvature metrics. Here an incompressible hypersurface in an $n$-manifold $M^n$ simply means a continuous map $i:\Sigma\to M$ such that $\Sigma$ is an essential closed $(n-1)$-manifold and the induced map $i_*:\pi_1(\Sigma) \to \pi_1(M)$ between fundamental groups is injective. This kind of result was first proven by Schoen and Yau in their papers \cite{SY1979} and \cite{SY1982}, where they showed that if an orientable complete $3$-manifold contains an orientable incompressible closed surface with positive genus, then it admits no smooth metric with positive scalar curvature. Later in \cite{GL83}, Gromov and Lawson made the following generalization: if $M$ is a compact manifold with enlargeable\footnote{A manifold $M^n$ is called enlargeable if for any $\epsilon>0$ we can find a covering of $M$ which is spin and admits a non-zero degree map to $\mathbf S^n(1)$ with Lipschitz contant less than $\epsilon$. Typical examples are Cantan-Hardmard manifolds. The use of this requirement here is the same as our class $\mathcal C_{deg}$ defined next.} essential boundary such that the inclusion map $\pi_1(\partial M)\to \pi_1(M)$ is injective, then it cannot admit any smooth metric with positive scalar curvature and mean convex boundary. We point out that one can even obtain a stronger conclusion from their proof. That is, if $M^n$ is a closed $n$-manifold containing an incompressible enlargeable essential closed hypersurface, then it cannot admit any smooth metric with positive scalar curvature. Based on the latter statement the original result by Gromov and Lawson follows easily from a doubling trick.
In this paper, we would like to strengthen Gromov-Lawson's result along two directions. First we are able to drop the requirement on the closeness of underlying manifold with the help of soap bubble (also called $\mu$-bubble) method. Recently, soap bubbles were used in various situations to establish results on non-compact manifolds and the audience can refer to \cite{Zhu2021}, \cite{Zhu2020}, \cite{Thomas2020}, \cite{LUY2021} and \cite{CL2020}. Second we can relax the incompressible condition to some {\it relatively incompressible condition} that allows part of the fundamental group mapped to zero. As we shall see later, the first improvement enables us to establish positive mass theorems with an incompressible condition and arbitrary ends.
Before the statement of our main theorem, let us introduce some notions for convenience. In the following, we denote $\mathcal C_{deg}$ to be the collection of those closed orientable aspherical manifolds $M$ satisfying the following property: any essential closed manifold over\footnote{A closed essential manifold over $M$ means that it admits a non-zero degree map to $M$.} $M$ admits no positive scalar curvature metric. Once the enhanced aspherical conjecture is proven, the class $\mathcal C_{deg}$ is nothing but all aspherical closed manifolds. At this stage, the class $\mathcal C_{deg}$ is known to include the following manifolds:
\begin{itemize}
\item[(a)] closed aspherical manifolds with dimension no greater than five \cite{CLL2021};
\item[(b)] the $n$-torus $T^n$ with dimension $n\leq 8$ \cite{SY2017} (No dimensional restriction is needed in \cite{SY2017}, but we focus on the case $n\leq 8$ for safety);
\item[(c)] closed Cantan-Hardmard $n$-manifolds with dimension $n\leq 8$ \cite{Gromov2018}.
\end{itemize}
We point out that the dimensional restriction $n\leq 8$ comes from the regularity theory for minimizing currents in geometric measure theory. It is well-knwon that no singularities occur when the ambient manifold has dimension no greater than $7$ and singularities can be perturbed away in eight dimensional manifolds through Nathan Smale's work in \cite{Nathan1993}.
With the class $\mathcal C_{deg}$ we can state our theorem as following
\begin{theorem}\label{Thm: main 1}
Let $\Sigma_0$ be an orientable closed aspherical manifold in $\mathcal C_{deg}$, and $\Sigma$ be an orientable essential closed manifold over $\Sigma_0$ associated with a non-zero degree map $f:\Sigma\to \Sigma_0$. Assume that $M$ is a differentiable $n$-manifold ($n\leq 7$, compact or non-compact) with an orientable essential closed hypersurface $i:\Sigma\to M$ such that $\ker i_*\subset \ker f_*$ in $\pi_1(\Sigma)$, where $i_*$ and $f_*$ are the corresponding induced maps of $i$ and $f$ between fundamental groups respectively. Then $M$ admits no complete positive scalar curvature metric. Moreover, if $g$ is a complete smooth metric on $M$ with nonnegative scalar curvature, then $(M,g)$ is flat.
\end{theorem}
\subsection{Generalized connected sum along submanifolds}
In the second part of this paper, we are going to make a discussion on the relationship between the generalized connected sum and positive scalar curvature. First let us recall the definition of the generalized connected sum considered in \cite{SY79} as well as \cite{GL80} . Let $\Sigma^k$ be a closed $k$-manifold and define the class
\begin{equation*}
\mathcal M^n_\Sigma=\left\{(M,i)\left|
\begin{array}{c}
\text{$M$ is a differentiable $n$-manifold and $i:\Sigma\to M$}\\
\text{is an embedding with trivial normal bundle}
\end{array}\right.\right\}.
\end{equation*}
For convenience, a pair $(M,i)$ in $\mathcal M_{\Sigma}^n$ will be called a marked manifold in the following. We emphasize that there is no requirement on the compactness of marked manifolds in the definition of $\mathcal M^n_\Sigma$. Given two marked manifolds $(M_1,i_1)$ and $(M_2,i_2)$ in $\mathcal M_{\Sigma}^n$, the generalized connected sum
$$(M_1,i_1)\#_{\Sigma}(M_2,i_2)$$
is defined to be the manifold obtained from the surgery as follows. Take a tubular neighborhood $U_l$ of $i_l(\Sigma)$ in each $M_l$ for $l=1,2$. Clearly $\partial U_1$ and $\partial U_2$ are isomorphic sphere bundles over $\Sigma$ (actually over $i_1(\Sigma)$ and $i_2(\Sigma)$). Pick up $\Phi$ to be a {\it fiber preserving diffeomorphism} between $\partial U_1$ and $\partial U_2$. Then $(M_1,i_1)\#_{\Sigma}(M_2,i_2)$ is defined to be the gluing space
$$
\left(M_1-U_1\right)\sqcup_\Phi \left(M_2-U_2\right).
$$
When $\Sigma$ is a single point, it is clear that the generalized connected sum reduces to the classical one.
The researches on the relationship between the generalized connected sum and positive scalar curvature mainly focus on two opposite directions: the generalized connected sum preserving admission or obstruction for positive scalar curvature. Along the first direction, Schoen and Yau \cite{SY79} as well as Gromov and Lawson \cite{GL80} proved that: given any closed manifold $\Sigma^k$ with $k+3\leq n$, if $(M_1,i_1)$ and $(M_2,i_2)$ are two marked manifolds in $\mathcal M^n_{\Sigma}$ admitting complete metrics with positive scalar curvature, the generalized connected sum $(M_1,i_1)\#_{\Sigma}(M_2,i_2)$ also admits a complete positive scalar curvature metric. On the other hand, based on the work of Schoen and Yau \cite{SY79}\cite{SY2017} as well as the soap bubble method, Chodosh and Li \cite{CL2020} proved that $(T^n,i)\#_{\Sigma}(M_2,i_2)$ with $n\leq 7$ cannot admit any complete metric with positive scalar curvature in the case when $\Sigma$ is a single point.
Here we would like to further generalize Chodosh-Li's work and this serves as a preparation for our positive mass theorems with an incompressible conditions and more general asymptotics. For our purpose, let us limit our attention to the following special case. Namely we take $\Sigma$ to be the $k$-torus $T^k$ and $(T^n,i)$ to be the $n$-torus associated with a linear embedding $i:T^k\to T^n$. Clearly the marked manifold $(T^n,i)$ is an element in $\mathcal M^n_{T^k}$ and we can consider the generalized connected sum $(T^n,i)\#_{T^k}(M_2,i_2)$ for any $(M_2,i_2)$ in $\mathcal M^n_{T^k}$.
Given any marked manifold $(M,i)$ in $\mathcal M^n_{\Sigma}$, we say that $(M,i)$ satisfies the incompressible condition if the inclusion map $i_*:\pi_1(\Sigma)\to \pi_1(M)$ is injective. When $\Sigma$ is $\mathbf S^1$, $(M,i)$ is said to satisfy the homotopially non-trivial condition if the map $i:\mathbf S^1\to M$ is not homotopic to a point.
As an application of our main Theorem \ref{Thm: main 1}, we can establish
\begin{proposition}\label{Prop: main 2}
Let $n\leq 7$ and $k\leq n-2$. If $(M_2,i_2)$ is a marked $n$-manifold in $\mathcal M^n_{T^k}$ satisfying the incompressible condition, then the generalized connected sum $(T^n,i)\#_{T^k}(M_2,i_2)$ admits no complete metric with positive scalar curvature. Moreover, if $g$ is a complete metric on $(T^n,i)\#_{T^k}(M_2,i_2)$ with nonnegative scalar curvature, then the metric $g$ must be flat.
\end{proposition}
When $k=1$, we can further strengthen above result as following
\begin{proposition}\label{Prop: main 3}
Let $3\leq n\leq 7$. If an orientable marked manifold $(M_2,i_2)$ in $\mathcal M^n_{\mathbf S^1}$ satisfies the homotopically non-trivial condition, then the generalized connected sum $(T^n,i)\#_{\mathbf S^1}(M_2,i_2)$ admits no complete metric with positive scalar curvature. Moreover, if $g$ is a complete metric on $(T^n,i)\#_{\mathbf S^1}(M_2,i_2)$ with nonnegative scalar curvature, then the metric $g$ must be flat.
\end{proposition}
We point out that Proposition \ref{Prop: main 3} is closely related to the class of Schoen-Yau-Schick manifolds. Let us recall from \cite{Gromov2018} that an orientable closed manifold $M^n$ is called a Schoen-Yau-Schick manifold if there are cohomology classes $\beta_1,\,\beta_2,\ldots,\beta_{n-2}$ in $H^1(M,\mathbf Z)$ such that the homology class
$$
[M]\frown(\beta_1\smile\beta_2\smile\cdots\smile \beta_{n-2})
$$
in $H_2(M,\mathbf Z)$ is non-spherical, that is, it does not lie in the image of the Hurewicz homomorphism $\pi_2(M)\to H_2(M,\mathbf Z)$. In their work \cite{SY2017}, Schoen and Yau proved that every orientable closed Schoen-Yau-Schick manifold admits no smooth metric with nonnegative scalar curvature unless it is flat. Actually, the generalized connected sum $(T^n,i)\#_{\mathbf S^1}(M_2,i_2)$ turns out to be a Schoen-Yau-Schick manifold if the marked manifold $(M_2,i_2)$ is a closed orientable manifold satisfying the homotopiclly non-trivial condition (see the proof of Proposition \ref{Prop: main 3}). Even though, an extra effort needs to be made in our proof for Proposition \ref{Prop: main 3} to deal with the possible non-compactness of $M_2$.
Recall that Chodosh, Li and Liokumovich \cite{CLL2021} proved that any essential closed manifold admits no smooth metric with positive scalar curvature. In particular, the connected sum of an aspherical closed manifold with any other closed manifold preserves the topological obstruction for positive scalar curvature. For further generalizations of this fact it is natural to consider the generalized connected sum of an aspherical closed manifold with other closed or open manifolds. Since this paper is mainly devoted to establishing a class of positive mass theorems with an incompressible condition, we shall leave the discussion on this interesting topic to another paper.
In our later discussion on positive mass theorems with an incompressible condition, we will use the idea of Lohkamp compactification \cite{Lohkamp1999} so that the positive mass theorem can be reduced to a topological obstruction problem for complete metrics with positive scalar curvature, where Proposition \ref{Prop: main 2} and \ref{Prop: main 3} are ready to apply.
\subsection{Positive mass theorems with an incompressible condition}
The positive mass theorem for asymptotically flat manifolds \cite{SY79PMT}\cite{SY2017} (see also \cite{Witten1981} for the spin case) appears to be one of the most beautiful results in both geometry of scalar curvature and general relativity. It states that the ADM mass has to be nonnegative for any asymptotically flat manifold with nonnegative scalar curvature, and the mass vanishes exactly when the manifold is isometric to the Euclidean space.
From quantum gravity theory and string theory, gravitational instantons provide more examples of ``asymptotically flat'' manifolds with more general asymptotics at infinity. By definition a gravitational instanton means a non-compact hyperk\"ahler $4$-manifolds with decaying curvature at infinity, and various examples were discussed in \cite{K1989}, \cite{CK1998}, \cite{CH2005} as well as \cite{CK2002}. Given these plentiful examples it is a rather difficult problem to classify all possible gravitational instantons and it seems a nice idea to focus only on several special classes. In particular, Cherkis and Kapustin conjectured a classification scheme (see \cite{EJ2008}), which involves a consideration on the following four special families of gravitational instantons:
\begin{itemize}
\item[(i)] Asymptotically Locally Euclidean (ALE): asymptotical to $\mathbf R^4/\Gamma$ at infinity for some discrete finite group $\Gamma\subset O(4)$;
\item[(ii)] Asymptotically Locally Flat (ALF): asymptotical to the total space of a circle bundle over $\mathbf R^3$ or $\mathbf R^3/\mathbf Z_2$ at infinity;
\item[(iii)] ALG (no explicit meaning): asymptotical to the total space of $T^2$-bundle over $\mathbf R^2$ at infinity;
\item[(iv)] ALH (no explicit meaning): asymptotical to the total space of $F^3$-bundle over $\mathbf R$, where $F$ is a flat closed $3$-manifold.
\end{itemize}
Given these interesting examples, it is a natural question whether positive mass theorems hold for ``asymptotically flat'' manifolds with more general asymptotics. In their work \cite{HP78}, Hawking and Pope proposed the {\it generalized positive action conjecture}: any ALE 4-manifold with vanishing scalar curvature has nonnegative ADM mass, which vanishes if and only if the manifold is Ricci flat with self-dual Weyl curvature. Unfortunately, this conjecture turns out to be false in general. Actually, LeBrun \cite{Lebrun1988} constructed a family of counter-examples from the resolution of quotient spaces $\mathbf C^2/\mathbf Z_k$ with $k\geq 3$. On the other hand, counter-examples vialating the philosophy of positive mass theorems can be also found in the class of ALF manifolds, where there is a well-known example given by the product manifold $\mathbf R^2\times \mathbf S^2$ equipped with the Reissnet-Nordstr\"om metric (see \cite{Minerbe2008} for instance). We point out that this is a complete Riemannian manifold asymptotic to $\mathbf R^3\times \mathbf S^1$ at infinity with vanishing scalar curvature and negative total mass. The common feature of these counter-examples is that their ends are not incompressible.
Over the decades, attempts have been made to establish positive mass theorems with necessary additional conditions. Inspired from Witten's proof on the positive mass theorem for asymptotically flat manifolds, various authors proved positive mass theorems with an additional spin compactible condition in different scenarios. For further information, the audience can refer to \cite{Dahl1997}, \cite{Minerbe2008} and \cite{Dai2004}. Roughly speaking, the spin compactible condition can be used to guarantee the existence of spinors from a perturbation of classical ones, and then the Witten's proof can be applied in these cases without difficulty. Despite of its effectiveness, the spin compactible condition is usually not easy to verify and so this motivates us to search for other convenient conditions to guarantee the validity of positive mass theorems.
In the third part of this paper, we will research on positive mass theorems with some type of incompressible condition. In the following, we would like to consider the following class of manifolds.
\begin{definition}\label{Defn: AF with fiber F}
Let $(F,g_F)$ be a closed flat manifold. A triple $(M,g,\mathcal E)$ is called an asymptotically flat manifold with fiber $F$ if
\begin{itemize}
\item $(M,g)$ is a complete Riemannian manifold with $d:=\dim M-\dim F\geq 3$;
\item $\mathcal E$ is an end of $M$ diffeomorphic to $(\mathbf R^d-B)\times F$;
\item Denote $g_0=g_{euc}\oplus g_F$. The metric $g$ on $\mathcal E$ satisfies
\begin{equation}\label{Eq: decay 1}
|g-g_0|_{g_0}+r|\nabla_{g_0}(g-g_0)|_{g_0}+r^2|\nabla_{g_0}^2(g-g_0)|_{g_0}=O(r^{-\mu}),\quad\mu>\frac{d-2}{2},
\end{equation}
where $\nabla_{g_0}$ is the covariant derivative with respect to $g_0$, and $r$ is the distance function on $\mathbf R^d$;
\item $R(g)\in L^1(\mathcal E,g)$.
\end{itemize}
\end{definition}
\begin{remark}
We emphasize that $(M,g)$ can have more than one ends but no additional requirement other than the completeness is imposed for ends other than $\mathcal E$. It may be better to call $(M,g,\mathcal E)$ asymptotically flat manifold with fiber $F$ and arbitrary ends as in \cite{LUY2021}, but we just omit ``arbitrary ends'' for short since the name is already very long.
\end{remark}
The definition above includes the following interesting examples:
\begin{itemize}
\item[(i)] {\it Schwarzschild-like manifolds with fiber $F$.} This class consists of the manifold $(\mathbf R^d-O)\times F^k$ equipped with the metrics
$$
g=\left(1+\frac{A}{r^{d-2}}\right)^{\frac{4}{d+k-2}}(g_{euc}\oplus g_F),\quad A\geq 0,
$$
where $O$ is the origin of $\mathbf R^d$ and $g_F$ is a flat metric on $F$. This provides a scalar-flat family of asymptotically flat manifolds with fiber $F$.
\item[(ii)] {\it Schwarzschild-product manifolds with fiber $F$.} Similarly, we can equip $(\mathbf R^d-O)\times F^k$ with the metrics
$$
g=\left(\left(1+\frac{A}{r^{d-2}}\right)^{\frac{4}{d-2}}g_{euc}\right)\oplus g_F,\quad A\geq 0.
$$
Clearly this provides another scalar-flat family of asymptotically flat manifolds with fiber $F$.
\item[(iii)] {\it Schwarzschild manifolds.} Fix $A\geq 0$. We can equip $(\mathbf R^d-B_{r_0})\times \mathbf S^1$, $r_0=A^{\frac{1}{d-2}}$, with the metric
$$
g=\left(\left(1+\frac{A}{r^{d-2}}\right)^{\frac{4}{d-2}}g_{euc}\right)\oplus \left(\left(\frac{1-Ar^{2-d}}{1+Ar^{2-d}}\right)^2\mathrm d\theta^2\right).
$$
When $A>0$, this provides a family of Ricci-flat metrics on $\mathbf S^{d-1}\times \mathbf R^2$ (after completion), which is asymptotically flat with fiber $\mathbf S^1$.
\item[(iv)] {\it Reissnet-Nordstr\"om metrics.} Fix real numbers $A$ and $B\neq 0$. Let us define the metric
$$
g=\left(\left(1+\frac{A}{r}+\frac{A^2+B^2}{4r^2}\right)^{2}g_{euc}\right)\oplus \left(\left(\frac{1-\frac{1}{4}r^{-2}(A^2+B^2)}{1+r^{-1}A+\frac{1}{4}r^{-2}(A^2+B^2)}\right)^2\mathrm d\theta^2\right)
$$
on $(\mathbf R^3-B_{r_0})\times \mathbf S^1$ with $r_0=\frac{1}{2}\sqrt{A^2+B^2}$. After completion this gives a family of scalar-flat metrics on $\mathbf S^2\times \mathbf R^2$, which is asymptotically flat with fiber $\mathbf S^1$.
\end{itemize}
For asymptotically flat manifolds with fiber $F$ we can introduce a total mass in the same spirit of the ADM mass for asymptotically flat manifolds.
\begin{definition}
Let $(M,g,\mathcal E)$ be an asymptotically flat manifold with fiber $F$. The total mass of $(M,g,\mathcal E)$ is defined to be
\begin{equation*}
\begin{split}
m(M,g,\mathcal E)=&\frac{1}{2|\mathbf S^{d-1}|\vol(F,g_F)}\\
&\qquad\cdot\lim_{\rho\to+\infty}\int_{S_\rho\times F}\ast_{g_0}\left(\Div_{g_0}g-d\tr_{g_0}g\right),
\end{split}
\end{equation*}
where $S_\rho$ is denoted to be the $\rho$-sphere in $\mathbf R^d$ centered at the origin and $\ast_{g_0}$ is the Hodge star operator with respect to the metric $g_0$.
\end{definition}
\begin{remark}
One easily checks that $m(M,g,\mathcal E)$ conincides with the ADM mass up to a positive scale if $(M,g,\mathcal E)$ is asymptotically flat in the usual sense.
\end{remark}
We are able to prove the following
\begin{theorem}\label{Thm: main 4}
If $(M,g,\mathcal E)$, $\dim M\leq 7$, is an asymptotically flat manifold with fiber $F$ such that its scalar curvature $R(g)\geq 0$ and one of the following holds:
\begin{itemize}
\item $i_*:\pi_1(\mathcal E)\to \pi_1(M)$ is injective;
\item or $F=\mathbf S^1$ and $i_*:\pi_1(\mathcal E)\to \pi_1(M)$ is non-zero,
\end{itemize}
then we have $m(M,g,\mathcal E)\geq 0$. In the first case, $m(M,g,\mathcal E)=0$ yields that $(M,g)$ is flat. In the second case, if the metric $g$ further satisfies
$$
\sum_{i=0}^3 r^i|\nabla^i_{g_0}(g-g_0)|_{g_0}=O(r^{-\mu}),\quad\mu>\frac{d-2}{2},
$$
then $m(M,g,\mathcal E)=0$ yields that $(M,g)$ splits as $\mathbf R^d\times \mathbf S^1$.
\end{theorem}
Even though we establish the class of positive mass theorems with an incompressible condition, it does not rule out any possibility for a positive mass theorem without incompressible conditions. For example, the Eguchi-Hason gravitational instanton \cite{EH1979} provides an example of complete manifolds with zero mass but it has a compressible end diffeomorphic to $\mathbf R^4/\mathbf Z_2$. So it might be possible to prove a positive mass theorem for ALE manifolds with its end diffeomorphic to $\mathbf R^4/\mathbf Z_2$ without any additional conditions.
It follows from Bartnik's work \cite{Bartnik1986} that the total mass is actually a modified integral of scalar curvature. From the Gauss-Bonnet formula, the counterpart for a complete surface turns out to be the angle at infinity. This leads us to consider the following class of manifolds.
\begin{definition}
Let $(F,g_F)$ be a closed flat manifold. A triple $(M,g,\mathcal E)$ is called an asymptotically conical manifold with fiber $F$ and (arbitrary ends) if
\begin{itemize}
\item $(M,g)$ is a complete Riemannian manifold with $\dim M-\dim F= 2$;
\item $\mathcal E$ is an end of $M$ diffeomorphic to $(\mathbf R^2-B)\times F$;
\item there is a positive constant $\beta$ such that the metric $g$ on $\mathcal E$ satisfies
\begin{equation}\label{Eq: decay 2}
|g-g_\beta|_{g_\beta}+r|\nabla_{g_\beta}(g-g_\beta)|_{g_\beta}+r^2|\nabla_{g_\beta}^2(g-g_\beta)|_{g_\beta}=O(r^{-\mu}),\quad \mu>0,
\end{equation}
where $g_\beta=\mathrm dr^2+\beta^2r^2\mathrm d\theta^2+g_F$, $\nabla_{g_\beta}$ is its corresponding covariant derivative, and $r$ is the distance function on $\mathbf R^2$;
\end{itemize}
For any asymptotically conical manifold $(M,g,\mathcal E)$, the constant $2\pi\beta$ will be called the angle of $(M,g,\mathcal E)$ at infinity.
\end{definition}
Denote $i_F:F\to M$ to be the natural inclusion map from the composition of the inclusions $i_1:F\to \mathcal E$ and $i_2:\mathcal E\to M$. Similarly we have
\begin{theorem}\label{Thm: main 5}
If $(M,g,\mathcal E)$, $\dim M\leq 7$, is an asymptotically conical manifold with fiber $T^{n-2}$ such that its scalar curvature $R(g)\geq 0$ and one of the following holds:
\begin{itemize}
\item $F=T^{n-2}$ and $(i_F)_*:\pi_1(F)\to \pi_1(M)$ is injective;
\item or $F=\mathbf S^1$ and $(i_F)_*:\pi_1(F)\to \pi_1(M)$ is non-zero,
\end{itemize}
then the angle of $(M,g,\mathcal E)$ at infinity is no greater than $2\pi$, where the equality implies that $(M,g)$ is flat.
\end{theorem}
We point out that Theorem \ref{Thm: main 4} is a generalization of the classical positive mass theorem for asymptotically flat manifolds and the idea of the proof is similar to the original one. However, there are still additional works to be done.
First, the analysis involved turns out to be much more delicate due to the existence of the fiber $F$ and arbitrary ends. In order to solve PDEs on asymptotically flat manifolds with fiber $F$, we have to establish a weighted Sobolev inequality (see Proposition \ref{Prop: Sobolev}) on $(\mathbf R^d-B)\times F$, which allows tests function to be non-zero on the inner boundary. Also, some extra efforts need to be devoted to our construction of appropriate conformal factors due to the existence of arbitrary ends other than $\mathcal E$ (compare Proposition \ref{3.2likeSY} with \cite[Lemma 3.2]{SY79PMT}).
Second, the volume comparison theorem cannot be used to prove the flatness of $(M,g)$ from its Ricci-flatness if $F=\mathbf S^1$ and $(i_F)_*:\pi_1(F)\to \pi_1(M)$ is only non-zero. As a brand new observation, we find out that the fast decay Ricci curvature implies the following
result.
\begin{proposition}
If $(M,g,\mathcal E)$ is an asymptotically flat manifold with fiber $F=\mathbf S^{1}$ and the unique end $\mathcal E$ such that
$$\sum\limits^3_{k=0}r^k|\nabla_{g_0}^k (g-g_{0})|_{g_0}=O(r^{-\mu}), \quad\mu>\frac{d-2}{2},$$
and
$$|\nabla_{g_0} Ric|_{g_0}+r|\nabla_{g_0} Ric|_{g_0}=O(r^{-n-\epsilon})\quad \text{for some}\quad\epsilon>0 .$$
Let $\{x^{1},\cdots,x^{d},\theta\}$ be the given coordinate system of $(M,g,\mathcal E)$ at infinity, where $\theta$ is the parameter of $S^{1}$. Then there exists a coordinate system $\{y^{1},\cdots,y^{d},\theta\}$ outside a compact subset such that
$\Delta_{g}y^{i}=0,\,i=1,\cdots,d$.
Moreover, if we denote $\alpha^{i}=dy^{i}$, then we have
\begin{equation}\label{Eq: GB mass}
\sum\limits^{d}_{i=1}\int_{M}|\nabla_g\alpha^{i}|_g^{2}+\Ric(\alpha^{i},\alpha^{i})\mathrm{d}V_{g}=c(d,S^{1})m(M,g,\mathcal{E}),
\end{equation}
where $c(d,S^{1})=d|\mathbf S^{d-1}|\vol(S^{1},g_{S^{1}})$.
\end{proposition}
\begin{remark}
The left hand side of \eqref{Eq: GB mass} is called the Gauss-Bonnet mass by Minerbe in his work \cite{Minerbe2008}. This proposition says that if the Ricci curvature decay is fast enough, then the Gauss-Bonnet mass is equal to the total mass up to a positive scale (independent of the metric $g$).
\end{remark}
In the proof of Theorem \ref{Thm: main 5}, the Lohkamp compactification is no longer valid since the density theorem like Proposition \ref{conformal end} is hard to prove due to the possible blow-up of the conformal factors on asymptotically conical manifolds with fiber $F$. As an alternative, we develop a new compactification method based on the quasi-spherical metric (see Proposition \ref{Prop: PMT to PSC 2}).
\subsection{Brown-York mass and the fill-in problem} Other than the total mass, another central topic in general relativity is the definition of a quantity measuring the mass contained in a bounded region, which is called {\it quasi-local mass}. Brown and York \cite{BY1991} \cite{BY1993} raised the so called Brown-York mass as a candidate for quasi-local mass.
Given a compact
spacelike hypersurface $\Omega$ in a spacetime, assuming its boundary $\partial\Omega$ is a $2$-sphere with
positive Gauss curvature, the Brown–York mass of $\partial\Omega$ is given by
$$
m_{BY}(\partial\Omega,\Omega)=\frac{1}{8\pi}\int_{\partial\Omega}(H_0-H)\,\mathrm d\sigma.
$$
Here $\mathrm d\sigma$ is the induced area element on $\partial\Omega$, $H$ is the mean curvature of $\partial\Omega$ in $\Omega$, and $H_0$ is the mean curvature of $\partial\Omega$ after isometrically embedded into the Euclidean $3$-space. This definition makes sense since the solution of Weyl's embedding problem \cite{Nirenberg1953} \cite{Pogorelov1964} guarantees the existence and uniqueness of an isometric embedding from $2$-spheres with positive Gaussian curvature to the Euclidean space. It is also reasonable since the third named author and Tam \cite{ST2002} proved the nonnegativity of the Brown-York mass when $\Omega$ has nonnegative scalar curvature and mean convex boundary. But without doubt the definition is too restrictive if the boundary $\partial\Omega$ of the region $\Omega$ has to be $2$-spheres with positive Gaussian curvature.
In order to generalize the notion of Brown-York mass to more general surfaces, Mantoulidis and Miao \cite{MM2017} introduced the $\Lambda$-invariant as follows. Given a connected closed surface $(\Sigma,g_\Sigma)$, one first makes the collection $\mathcal F$ of all {\it admissible fill-ins} of $(\Sigma,g_\Sigma)$, i.e. compact $3$-manifolds $(\Omega,g)$ such that
\begin{itemize}
\item $(\partial\Omega,g|_{\partial\Omega})$ is isometric to $(\Sigma,g_\Sigma)$;
\item $\partial\Omega$ is mean-convex with respect to the outer unit normal;
\item the scalar curvature $R(g)\geq 0$.
\end{itemize}
Then the $\Lambda$-invariant is defined to be
$$
\Lambda(\Sigma,g_\Sigma)=\max_{\Omega\in\mathcal F}\int_{\partial\Omega}H\,\mathrm d\sigma.
$$
The essential consideration for $\Lambda$-invariant is the finiteness since if the $\Lambda$-invariant is proven to be finite all the time, then one can introduce the (generalized) Brown-York mass to be
\begin{equation}\label{Eq: BY}
m_{BY}(\partial\Omega,\Omega)=\Lambda(\partial\Omega,g|_{\partial\Omega})-\int_{\partial\Omega}H\,\mathrm d\sigma.
\end{equation}
By proving the finiteness of $\Lambda$-invariant for $2$-spheres with arbitrary smooth metrics, Mantoulidis and Miao successfully got rid of the positive Gaussian curvature condition in the definition of Brown-York mass. But it still leaves a problem whether the Brown-York mass given by \eqref{Eq: BY} is well-defined for closed surfaces with positive genus.
As another applications of our Proposition \ref{Prop: main 2} and \ref{Prop: main 3}, we can generalize the Brown-York mass to the flat $2$-torus case. Namely, we can show
\begin{proposition}\label{Prop: main 7}
For any flat metric $g_{flat}$ on $T^2$, we have
$$
\Lambda(T^2,g_{flat})<+\infty.
$$
Futhermore, if $(T^2,g_{flat})$ splits as $\mathbf S^1(a)\times \mathbf S^1(b)$, then
$$
\Lambda(T^2,g_{flat})=4\pi^2\max\{a,b\}.
$$
\end{proposition}
This proposition (as well as the more general Proposition \ref{Prop: main 6}) also gives a partial answer to the following fill-in problem raised by Gromov. In his work \cite{Gromov2019b}, Gromov suggested a systematic discussion on the following fill-in problem: {\it given a closed Riemannian $n$-manifold $(\Sigma,g_\Sigma)$ associated with a smooth function $h:\Sigma\to \mathbf R$, can one find a compact Riemannian $(n+1)$-manifold $(\Omega,g)$ such that
\begin{itemize}
\item[(i)] $(\partial\Omega,g|_{\partial\Omega})$ is isometric to $(\Sigma,g_\Sigma)$ through a diffeomorphism $\phi:\partial\Omega\to\Sigma$;
\item[(ii)] the mean curvature function $H$ of $\partial\Omega$ with respect to the unit outer normal equals to $h\circ \phi$;
\item[(iii)] the scalar curvature $R(g)$ is nonnegative?
\end{itemize} }
Other than those partial results in \cite{Gromov2019b}, \cite{SWWZ2021}, \cite{SWW2020}, we mention two complete results related to the fill-in problem above. Without the prescribed function $h$, Shi, Wang and Wei \cite{SWW2020} proved that $(\Sigma,g_\Sigma)$ always admits a compact Riemannian manifold $(\Omega,g)$ satisfying (i) and (iii) if $\Sigma$ can be realized as the boundary of a compact manifold. Based on this result, Miao \cite{Miao2021} proved that there is a universal constant $C=C(\Sigma,g_\Sigma)$ such that $(\Omega,g)$ satisfying (i), (ii) and (iii) does not exist if the prescribed function $h\geq C$.
Related to the estimate for $\Lambda$-invariant, Gromov proposed the following conjecture:
\begin{conjecture}
There is a universal constant $C=C(\Sigma,g_\Sigma)$ such that any $(\Omega,g)$ with properties (i), (ii) and (iii) satisfies
$$
\int_{\partial\Omega} H\,\mathrm d\sigma\leq C.
$$
\end{conjecture}
This conjecture is now largely open if no additional condition is imposed. Under the further assumption that $\Omega$ has mean convex boundary, the works in \cite{ST2002}, \cite{MM2017}, \cite{SWWZ2021}, \cite{SWW2020} as well as Proposition \ref{Prop: main 7} give several partial answers.
\subsection{The arrangement of this paper}
The rest of this paper is organized as follows. In Section 2, we give the proof for Theorem \ref{Thm: main 1}. In Section 3, we present proofs for Proposition \ref{Prop: main 2} and Proposition \ref{Prop: main 3}. Section 4 is devoted to proving Theorem \ref{Thm: main 4} and Theorem \ref{Thm: main 5} with different compactification arguments. As a preparation, we also establish weighted Sobolev inequalities on $(\mathbf R^n-B)\times T^k$ (see Proposition \ref{Prop: Sobolev} and Corollary \ref{Cor: Sobolev Lp}). Finally we include a discussion on the fill-in problem and prove Proposition \ref{Prop: main 7}.
\medskip
{\it Acknowledgements.} We would like to thank Professor Shing-Tung Yau for drawing our attention to positive mass theorems on manifolds with general asymptotic structure at the infinity. We are also grateful to Dr. Chao Li for many inspiring discussions with the third named author on relationships between the positive mass theorem for ALF manifolds and the fill-in problem.
\section{Proof for Theorem \ref{Thm: main 1}}
Our proof for Theorem \ref{Thm: main 1} is mainly based on the idea from Gromov and Lawson \cite{GL83} but some modifications are needed in order to use the soap bubble method.
It would be nice if we can just deal with an embedding $i:\Sigma\to M$, and so we start with the following reduction lemma.
\begin{lemma}\label{Lem: reduction}
Under the same assumption of Theorem \ref{Thm: main 1}, we can find an essential closed manifold $\Sigma'$ over $\Sigma_0$ and a manifold $M'$ such that there are an embedding $i':\Sigma'\to M'$, a projection map $p':M'\to \Sigma_0$, and a non-zero degree map $f':\Sigma'\to \Sigma_0$
such that the following diagram
\begin{equation*}
\xymatrix{ \Sigma'\ar[rr]^{i'}\ar[dr]_{f'} & &M'\ar[dl]^{p'}\\
& \Sigma_0&}
\end{equation*}
is commutative.
\end{lemma}
\begin{proof}
Fix a marked point $\sigma$ in $\Sigma$ and denote $m=i(\sigma)$ in $M$. First let us take the covering $p:(\hat M,\hat m)\to (M,m)$ such that
$$p_*(\pi_1(\hat M,\hat m))=i_*(\pi_1(\Sigma,\sigma))\subset \pi_1(M,m).$$
From lifting we can find a continuous map $\hat i:(\Sigma,\sigma)\to (\hat M,\hat m)$ such that the following diagram
\begin{equation*}
\xymatrix{
& (\hat M,\hat m)\ar[d]^{p}\\
(\Sigma,\sigma)\ar[ur]^{\hat i}\ar[r]_{i}& (M,m)}
\end{equation*}
is commutative.
Since the induced map $p_*:\pi_1(\hat M,\hat m)\to (M,m)$ is injective, we have
$
\ker \hat i_*=\ker i_*\subset \ker f_*
$ in $\pi_1(\Sigma)$. As a result, we can find a homomorphism $\hat J:\pi_1(\hat M,\hat m)\to \pi_1(\Sigma_0,\sigma_0)$ with $\sigma_0=f(\sigma)$ such that $f_*=\hat J\circ \hat i_*$. Since $\Sigma_0$ is aspherical, from the algebraic topology (see \cite[Proposition 1B.9]{Hatcher2002} for instance) we can construct a continuous map $\hat j:(\hat M,\hat m)\to (\Sigma_0,\sigma_0)$ such that the induced map $\hat j_*:\pi_1(\hat M,\hat m)\to \pi_1(\Sigma_0,\sigma_0)$ equals to $\hat J$. Furthermore, the map $\hat j\circ \hat i$ is homotopic to $f$ and so they induce the same correspondance between homology groups of $\Sigma$ and $\Sigma_0$. In particular, the fundamental class $[\Sigma]$ is mapped to a non-trivial homology class $\beta$ in $H_{n-1}(\hat M)$ under the map $\hat i$. From Lemma \ref{Lem: realize} we can find an embedded orientable hypersurface $\Sigma'$ to represent the homology class $\beta$. Notice that we have
$$\hat j_*(\beta)=f_*([\Sigma])=(\deg f)\cdot[\Sigma_0]\neq 0\in H_{n-1}(\Sigma_0,\mathbf Z).$$
Therefore, we can find one component of $\Sigma'$ such that the map $\hat j$ restricted to this component gives a non-zero degree map to $\Sigma_0$. For simplicity we still denote it by $\Sigma'$. Let $i':\Sigma'\to \hat M$ be the embedding of $\Sigma'$ and $f'=\hat j\circ i'$. Then we have the following commutative diagram \begin{equation*}
\xymatrix{ \Sigma'\ar[rr]^{i'}\ar[dr]_{f'} & &\hat M \ar[dl]^{\hat j}\\
& \Sigma_0&}.
\end{equation*}
The proof is now completed by taking $M'=\hat M$ and $p'=\hat p$.
\end{proof}
Now we are ready to prove Theorem \ref{Thm: main 1}.
\begin{proof}[Proof of Theorem \ref{Thm: main 1}]
From Lemma \ref{Lem: reduction} we can assume that $i:\Sigma\to M$ is an embedding and that there is a map $j:M\to \Sigma_0$ such that the following diagram
\begin{equation*}
\xymatrix{ \Sigma\ar[rr]^{i}\ar[dr]_{f} & &M \ar[dl]^{j}\\
& \Sigma_0&}
\end{equation*}
is commutative. Possibly passing to a two-sheeted covering, we can assume that both $\Sigma$ and $M$ are orientable and so it makes sense to consider homology groups of $\Sigma$ and $M$ with integer coefficients. Fix a marked point $\sigma$ in $\Sigma$ and we denote $m=i(\sigma)$ and $\sigma_0=f(\sigma)$.
In the following, we consider the covering $\tilde p:(\tilde M,\tilde m)\to (M,m)$ such that
\begin{equation}\label{Eq: lift fundamental group}
\tilde p_*(\pi_1(\tilde M,\tilde m))=i_*(\pi_1(\Sigma,\sigma))\subset \pi_1(M,m).
\end{equation}
By lifting we can construct an embedding $\tilde i:(\Sigma,\sigma)\to (\tilde M,\tilde m)$ such that the diagram
\begin{equation}\label{Diagram: 1}
\xymatrix{
& (\tilde M,\tilde m)\ar[d]^{\tilde p}\ar[dr]^{\tilde j=j\circ \tilde p}&\\
(\Sigma,\sigma)\ar[ur]^{\tilde i}\ar[r]_{i}& (M,m)\ar[r]^{j}&(\Sigma_0,\sigma_0)}.
\end{equation}
is commutative.
In order to apply the soap bubble method, we need to show that the embedded hypersurface $\tilde \Sigma=\tilde i(\Sigma)$ separates $\tilde M$ into two unbounded components. This can be seen from the following contradiction argument. Assume that $\tilde M-\tilde \Sigma$ has only one component. Then we can find a closed curve $\tilde\gamma$ which intersects $\tilde \Sigma$ once. From \eqref{Eq: lift fundamental group} and the injectivity of $\tilde p_*$, it is clear that the map $\tilde i_*:\pi_1(\Sigma,\sigma)\to \pi_1(\tilde M,\tilde m)$ is surjective. In particular, we can find a closed curve $\gamma$ in $\Sigma$ such that $\tilde i\circ \gamma$ is a closed curve in $\tilde\Sigma$ which is homotopic to $\tilde \gamma$. Notice that $\tilde \Sigma$ has trivial normal bundle and so we can push $\tilde i\circ \gamma$ to one side of $\tilde \Sigma$ such that the new closed curve has no intersection with $\tilde \Sigma$. This is impossible since the intersection number keeps invariant under homotopic. Since $\tilde\Sigma$ is homologically non-trivial, the components of $\tilde M-\tilde\Sigma$ are both unbounded.
Now we are going to deduce a contradiction if there is a complete metric $g$ on $M$ with positive scalar curvature. Lift this metric to $\tilde g$ on $\tilde M$. The idea is to find an orientable hypersurface in $\tilde M$ homologous to $\tilde\Sigma$, which admits a smooth metric with positive scalar curvature, based on the soap bubble method. Since $\tilde\Sigma$ separates $\tilde M$ into two unbounded components, the signed distance function to $\tilde\Sigma$ is well-defined and surjective onto $\mathbf R$. Through mollification we can construct a smooth proper function $\rho:\tilde M\to\mathbf R$ with $\Lip \rho <1$ and $\rho^{-1}(0)=\tilde\Sigma$. Given any small positive constant $\epsilon$, from a simple construction (see \cite{Zhu2020} for instance) we can find a smooth function
$$h_\epsilon:\left(-\frac{1}{n\epsilon},\frac{1}{n\epsilon} \right)\to\mathbf R$$
satisfying $h_\epsilon'<0$,
\begin{equation}\label{Eq: positive outside}
\frac{n}{n-1}h_\epsilon^2+2h_\epsilon'=n(n-1)\epsilon^2\quad \text{in}\quad \left(-\frac{1}{n\epsilon},-\frac{1}{2n}\right]\cup\left[\frac{1}{2n}, \frac{1}{n\epsilon}\right),
\end{equation}
\begin{equation}\label{Eq: error inside}
\sup_{-\frac{1}{2n}\leq t\leq \frac{1}{2n}} \left|\frac{n}{n-1}h_\epsilon^2+2h_\epsilon'\right|\leq C(n)\epsilon,
\end{equation}
and
\begin{equation}\label{Eq: barrier}
\lim_{t\to\mp \frac{1}{n\epsilon}} h_\epsilon(t)=\pm\infty.
\end{equation}
Denote $\Omega_0=\{\rho <0\}$. We consider the functional
$$
\mathcal B_\epsilon(\Omega)=\mathcal H^{n-1}(\partial^*\Omega)-\int_{ M}(\chi_\Omega-\chi_{\Omega_0})h_\epsilon\circ\rho\,\mathrm d\mathcal H^n
$$
defined on
$$
\mathcal C_\epsilon=\left\{\text{Caccipoli set}\,\Omega\subset \tilde M:\Omega\Delta\Omega_0\Subset\phi^{-1}\left(- \frac{1}{n\epsilon},\frac{1}{n\epsilon}\right)\right\}.
$$
Notice that the subset $K=\rho^{-1}\left([-\frac{1}{2n},\frac{1}{2n}]\right)$ is compact and so the scalar curvature $R(\tilde g)$ in $K$ is bounded below by a positive constant. Combined with \eqref{Eq: positive outside} and \eqref{Eq: error inside}, we can take $\epsilon$ to be small enough such that the quantity
$$
R(\tilde g)+\left(\frac{n}{n-1}h_\epsilon^2+2h_\epsilon'\right)\circ \rho
$$
is positive everywhere. Since the dimension of $\tilde M$ is no greater than $7$, we can find a smooth minimizer $\Omega$ in $\mathcal C_\epsilon$ for the functional $\mathcal B_\epsilon$ with \eqref{Eq: barrier} as a barrier condition (refer to \cite{Zhu2020} for details). It is not difficult to compute that $\partial\Omega$ satisfies the following stability inequality
\begin{equation}\label{Eq: stability}
\int_{\partial\Omega}|\nabla\phi|^2\mathrm d\sigma\geq \frac{1}{2}\int_{\partial\Omega}\left(R(\tilde g)-R_{\partial\Omega}+\left(\frac{n}{n-1}h_\epsilon^2+2h_\epsilon'\right)\circ \rho\right)\phi^2\,\mathrm d\sigma
\end{equation}
for all $\phi$ in $C^\infty(M)$, where $R_{\partial\Omega}$ is the scalar curvature of $\partial\Omega$ with the induced metric from $(\tilde M,\tilde g)$. From \eqref{Eq: stability} one can deduce that the conformal Laplace operator of $\partial\Omega$ is positive and so $\partial\Omega$ admits a smooth conformal metric with positive scalar curvature. From the definition of class $\mathcal C_\epsilon$, we have
$$\partial\Omega=\tilde\Sigma+\partial(\Omega-\Omega_0)$$
in the chain level, which implies $\partial\Omega$ is homologous to $\tilde\Sigma$. At this stage, the desired contradiction is close at hand. Since $\partial\Omega$ is homologous to $\tilde\Sigma$, the map $\tilde j:\tilde M\to \Sigma_0$ in diagram \eqref{Diagram: 1} restricted to $\partial\Omega$ gives a non-zero degree map from $\partial\Omega$ to $\Sigma_0$. This yields that $\partial\Omega$ is an essential closed manifold over $\Sigma_0$. From our assumption $\Sigma_0\in\mathcal C_{deg}$ we conclude that $\partial\Omega$ cannot admit any smooth metric with positive scalar curvature and we obtain a contradiction.
{Finally let us prove the rigidity part. Given a complete metric $g$ on $M$ with nonnegative scalar curvature, the obstruction for positive scalar curvature metric combined with Kazdan's deformation theorem \cite{Kazdan82} yields that the metric $g$ is Ricci flat. From previous proof the covering $\tilde M$ has two ends and the splitting theorem \cite{CG1971} yields that $\tilde M$ splits as the Riemannian product $\tilde\Sigma'\times \mathbf R$, where $\tilde \Sigma'$ is a Ricci-flat essential closed manifold over $\Sigma_0$. From \cite[Theorem 1.4]{FW1974}, we can assume $\tilde\Sigma'=\tilde\Sigma_1'\times T^k$ after passing to a covering without loss of generality, where $\tilde\Sigma_1'$ is closed and simply connected. All we need to show is $k=n-1$. Denote $f:(\tilde\Sigma',\tilde\sigma')\to (\Sigma_0,\sigma_0)$ to be the non-zero degree map associated to $\tilde\Sigma$ and consider the covering $\tilde p:(\tilde\Sigma_0,\tilde\sigma_0)\to (\Sigma_0,\sigma_0)$ such that
$$f_*(\pi_1(\tilde\Sigma',\tilde\sigma'))=\tilde p_*(\pi_1(\tilde\Sigma_0,\tilde\sigma_0)).$$
Since we can lift $f$ to a non-zero degree map $\tilde f:\tilde\Sigma'\to \tilde\Sigma_0$, we see that $\tilde\Sigma_0$ is closed and so the top dimensional homology group $H_{n-1}(\tilde\Sigma_0)\neq 0$. On the other hand, since aspherical manifolds have no torsion elements in their fundamental groups, the fundamental group of $\tilde\Sigma_0$ has to be $\mathbf Z^l$ with $l\leq k$. It is a basic topological fact that the homotopy type of an aspherical manifolds is uniquely determined by its fundamental group. Therefore, $\tilde\Sigma_0$ must be homotopic to the $l$-torus $T^l$ with $l=n-1$ and this implies $k=n-1$.}
\end{proof}
\section{Proof for Proposition \ref{Prop: main 2} and \ref{Prop: main 3}}\label{Sec: 3}
This section devotes to a detailed proof for Proposition \ref{Prop: main 2} and \ref{Prop: main 3}.
\begin{proof}[Proof of Proposition \ref{Prop: main 2}]
From the definition of the generalized connected sum, we have
$$(T^n,i)\#_{T^k}(M_2,i_2)=T^k\times (T^{n-k}-B)\sqcup_{\Phi} \left(M_2-U_2\right),$$
where $U_2$ is the tubular neighborhood of $i_2(\Sigma)$ in $M_2$ and the gluing map $\Phi$ is a fiber preserving diffeomorphism between $\partial(T^k\times B)$ and $\partial U_2$.
Notice that each $\mathbf S^1$ is in good pairing with some $T^{n-k-1}$ in $T^{n-k}-B$ as shown in Figure \ref{Fig: 1}.
\begin{figure}[htbp]
\centering
\includegraphics[width=5cm]{1.eps}
\caption{$T^{n-k}-B$ can be viewed as the quotient space of punctured cube $I^{n-k}-B$ by identifying opposite faces. Faces and edges has nice intersection property. For example, the right face represents a linear $T^{n-k-1}$ in $T^k-B$, which has intersection number one with the circle $\mathbf S^1$ represented by the horizon edge.}
\label{Fig: 1}
\end{figure}
Fix one such $T^{n-k-1}$ and denote it by $T^{n-k-1}_0$. We would like to show the incompressiblity of the hypersurface $\Sigma =T^k\times T^{n-k-1}_0$ in $(T^n,i)\#_{T^k}(M_2,i_2)$. Fix a point $p$ in $\Sigma$. Then the fundamental group $\pi_1(\Sigma,p)$ is isometric to $\mathbf Z^{n-1}$ in the sense that every closed curve $\gamma:(\mathbf S^1,1)\to (\Sigma,p)$ is homotopic to a closed curve in the form of
$$
\gamma_1^{c_1}\ast \gamma_2^{c_2}\ast \cdots\ast \gamma_{n-1}^{c_{n-1}},\quad c_i \in \mathbf Z,
$$
where $\gamma_i$ are coordinate circles in $\Sigma$ passing through $p$.
In the following, we are going to show that $\gamma$ is homotopically trivial in $\Sigma$ under the assumption that $\gamma$ is homotopic to a point in $(T^n,i)\#_{T^k}(M_2,i_2)$. Based on the nice pairing property above, for each $i\geq k+1$ we can find $T^{n-k-1}_i$ in $T^{n-k}-B$ such that
$$
c_i=[\gamma]\cdot [T^k\times T^{n-k-1}_i]=0.
$$
This means that the closed curve $\gamma$ can be homotopic to a closed curve in the $T^k$-component of $\Sigma$, still denoted by $\gamma$. Notice that $\gamma$ can be homotopic to a closed curve $\bar \gamma$ in the boundary $\partial\left(T^k\times B\right)$. Denote $c_1:T^k\times B\to i(T^k)$ and $c_2:U_2\to i_2(T^k)$ to be the contraction maps associated to tubular neighborhoods. Since the diffeomorphism $\Phi$ is fiber-preserving, we have the following commutative diagram
\begin{equation*}
\xymatrix{\partial(T^k\times B)\ar[r]^{\Phi}\ar[d]^{\partial c_1}&\partial U_2\ar[d]^{\partial c_2}\\
i(T^k)\ar[r]^{i_2\circ i^{-1}}&i_2(T^k).}
\end{equation*}
If the closed curve $\gamma$ cannot be homotopic to a point in the $T^k$-component of $\Sigma$,
then the closed curve $\bar\gamma$ is homotopically non-trivial in the $T^k$-component of $\partial(T^k\times B)$ and the same thing holds for the closed curve $(\partial c_1)\circ \bar \gamma$ in $i(T^k)$. From above commutative diagram,
we conclude that the closed curve $\Phi\circ \bar\gamma$ is homotopic to a homotopically non-trivial closed curve in $i_2(T^k)$. Now
the incompressible property implies that $\Phi\circ \bar\gamma$ is also homotopically non-trivial in $M_2$, let alone $M_2-U_2$. It then follows from Lemma \ref{Lem: no contraction extension} and Lemma \ref{Lem: lifting property example} that $\Phi\circ\bar \gamma$ cannot be homotopically trivial in $(T^n,i)\#_{T^k}(M_2,i_2)$, but this is impossible since the closed curve $\Phi\circ \bar\gamma$ after gluing is homotopic to $\gamma$, which has been assumed to be null homotopic in $(T^n,i)\#_{T^k}(M_2,i_2)$. As a consequence, the closed curve $\gamma$ is homotopic to a point in $T^k$ as well as in $\Sigma$. Now our main Theorem \ref{Thm: main 1} can be applied to obtain the desired consequence.
\end{proof}
Next let us give a detailed proof for Proposition \ref{Prop: main 3}.
\begin{proof}[Proof of Proposition \ref{Prop: main 3}]
Let us start with the following special case.
{\it Case 1. $M_2$ is closed and orientable.} From Figure \ref{Fig: 1}, we can find $(n-1)$ pairs of $T^{n-2}$ and $\mathbf S^1$ in $T^{n-1}-B$ with intersection number one. Let us label them as $T^{n-2}_l$ and $\mathbf S^1_l$ for $l=1,2,\ldots,n-1$ such that $\mathbf S^1_l$ is exactly the $l$-th circle component of $T^{n-1}-B$. Denote $\beta_l$ to be the cohomology class coming from the Poincar\'e dual of $\mathbf S^1\times T^{n-2}_l$ in the generalized connected sum
$$(T^n,i)\#_{\mathbf S^1}(M_2,i_2)=\mathbf S^1\times (T^{n-1}-B)\sqcup_\Phi (M_2-U_2).$$
Notice that $\mathbf S^1\times T^{n-2}_l$ has intersection number one with $\mathbf S^1_l$. So every class $\beta_l$ is non-trivial in $H^1(\bar M,\mathbf Z)$. Here and in the sequel, we will write $\bar M=(T^n,i)\#_{\mathbf S^1}(M_2,i_2)$ for short.
Now we would like to show that the homology class
$$
\tau:=[M]\frown(\beta_1\smile\beta_2\smile\cdots\smile\beta_{n-2})
$$
in $H_2(\bar M,\mathbf Z)$ does not lie in the image of Hurewicz homomorphism $\pi_2(\bar M)\to H_2(\bar M,\mathbf Z)$. Otherwise, we can find a cycle $C$ in $\bar M$ consisting of $2$-spherical components such that it represents the class $\tau$. Now let us investigate the intersection of the cycle $C$ and the hypersurface $\Sigma:=\mathbf S^1\times T^{n-2}_{n-1}$. From basic algebraic topology, the intersection $C\cap \Sigma$ is a union of closed curves in $\Sigma$, which is homologous to the $\mathbf S^1$-component of $\Sigma$. Clearly, each of these curves is homotopic to a point through a deformation in some $2$-spherical component of $\Sigma$. Notice that $\Sigma$ is a $(n-1)$-torus and so the Hurewicz homomorphism $\pi_1(\Sigma)\to H_1(\Sigma,\mathbf Z)$ is an isomorphism. As a result, the $\mathbf S^1$-component of $\Sigma$ is null homotopic in $\bar M$, and the same thing holds for the $\mathbf S^1$-component of $\partial \left(\mathbf S^1\times B\right)$. Denote $\gamma$ to be a closed curve in $\partial \left(\mathbf S^1\times B\right)$ homotopic to the $\mathbf S^1$-component. It follows from Lemma \ref{Lem: no contraction extension} and Lemma \ref{Lem: lifting property example} that the closed curve $\Phi\circ\gamma$ must shrink to a point in $M_2-U_2$.
From the commutative diagram
\begin{equation*}
\xymatrix{\partial(\mathbf S^1\times B)\ar[r]^{\Phi}\ar[d]^{\partial c_1}&\partial U_2\ar[d]^{\partial c_2}\\
i(\mathbf S^1)\ar[r]^{i_2\circ i^{-1}}&i_2(\mathbf S^1).}
\end{equation*}
we see that $\Phi\circ \gamma$ is homotopic to $i_2:\mathbf S^1\to M_2$ and so the map $i_2:\mathbf S^1\to M_2$ is homotopically trivial. This leads to a contradiction.
The above discussion yields that $\bar M$ is a closed Schoen-Yau-Schick manifold and then the desired consequence comes from Schoen-Yau's work in \cite{SY2017}.
{\it General case when $M_2$ may be non-compact.} We still use the soap bubble method to overcome the difficulty from non-compactness. As before, we can find $(n-1)$ pairs of $T^{n-2}$ and $\mathbf S^1$ in $T^{n-1}-B$ with intersection number one, labelled by $T^{n-2}_l$ and $\mathbf S^l_l$ with $l=1,2,\ldots,n-1$. Let $\bar M$ be the covering of $(T^n,i)\#_{\mathbf S^1}(M_2,i_2)$ from an unwinding along $\mathbf S^1_1$. Intuitively, one can imagine $\bar M$ as follows. First we take the covering $\mathbf R\times T^{n-2}$ of $T^{n-1}$. After taking away infinitely many disjoint balls $\{B_i\}_{i\in\mathbf Z}$ and production with the circle, the manifold
$$
\mathbf S^1\times\left(\mathbf R\times T^{n-2}-\bigcup_{i\in\mathbf Z} B_i\right)
$$
provides a covering of $\mathbf S^1\times (T^{n-1}-B)$. Next we glue infinitely many copies of $M_2-U_2$ to this manifold along each boundary component $\mathbf S^1\times \partial B_i$ and then the resulting manifold is our desired covering $\bar M$. If we lift the $(n-1)$-torus $\mathbf S^1\times T^{n-2}_1$ to $\bar M$, we can obtain an embedded hypersurface, denoted by $\bar T^{n-1}_1$, in $\bar M$ that seperates $\bar M$ into two unbounded components.
Now let us deduce a contradiction under the assumption that the generalized connected sum $(T^n,i)\#_{\mathbf S^1}(M_2,i_2)$ admits a complete metric with positive metric. By lifting, we obtain a complete metric with positive scalar curvature on $\bar M$. With the exactly same argument as in the proof of Theorem \ref{Thm: main 1}, we can find an embedded hypersurface $\bar\Sigma$ homologous to $\bar T^{n-1}_1$ in $\bar M$ such that $\bar\Sigma$ admits a smooth metric with positive scalar curvature. The image of $\bar\Sigma$ under the covering map is now a hypersurface $\Sigma$ in $(T^n,i)\#_{\mathbf S^1}(M_2,i_2)$ homologous to $\mathbf S^1\times T^{n-2}_1$ with the same property. Since $\Sigma$ is compact, we can proceed our discussion in a large compact subset $K$ of $(T^n,i)\#_{\mathbf S^1}(M_2,i_2)$ containing $\Sigma$. In this compact subset $K$, it still makes sense to take the Poincar\'e dual $\beta_l$ of $\mathbf S^1\times T^{n-2}_l$. Through the same argument as in Case 1, one can conclude that $\Sigma$ is a closed orientable Schoen-Yau-Schick manifold. This leads to a contradiction since $\Sigma$ admits a smooth metric with positive scalar curvature.
\end{proof}
\section{Positive mass theorems with an incompressible condition}
\subsection{A weighted Sobolev Inequality}\label{Sec: Sobolev}
In this subsection, we denote $\mathbf R^n$ to be the Euclidean $n$-space with $n\geq 2$ and $B$ to be the {\it open} unit ball in $\mathbf R^n$. Also we denote $r$ to be the radical distance function on $\mathbf R^n$. Given positive constants $d_i$, $i=1,2,\ldots,k$, we use $T^k$ to denote the product $k$-torus
$$
\mathbf S^1(d_1)\times \mathbf S^1(d_2)\times \cdots \times \mathbf S^1(d_k).
$$
The purpose of this section is to prove the following weighted Sobolev inequality.
\begin{proposition}\label{Prop: Sobolev}
There is a universal constant $C$ depending only on $n$, $k$ and $d_i$, $i=1,2,\ldots,k$, such that the inequality
\begin{equation}\label{Eq: sobolev weighted}
\left(\int_{(\mathbf R^n-B)\times T^k}|f|^{\frac{n+k}{n+k-1}}r^{\frac{-k}{n+k-1}}\,\mathrm dx\mathrm dt\right)^{\frac{n+k-1}{n+k}}\leq C\int_{(\mathbf R^n-B)\times T^k}|\nabla f|\,\mathrm dx\mathrm dt
\end{equation}
holds for any $f$ in $C_c^1((\mathbf R^n-B)\times T^k)$. Here the function $f$ is allowed to take non-zero values on the inner boundary $\partial B\times T^k$.
\end{proposition}
Before the proof we need the following lemma for preparation.
\begin{lemma}\label{Lem: sobolev average}
If $f$ is a function in $C_c^1((\mathbf R^n-B)\times T^k)$ satisfying
\begin{equation}\label{Eq: zero average}
\int_{\mathbf S^1(d_i)} f(x,t_1,t_2,\ldots,t_k)\,\mathrm dt_i=0,\quad i=1,2,\ldots,k,
\end{equation}
then we have
\begin{equation}\label{Eq: sobolev no weight}
\left(\int_{(\mathbf R^n-B)\times T^k}|f|^{\frac{n+k}{n+k-1}}\,\mathrm dx\mathrm dt\right)^{\frac{n+k-1}{n+k}}\leq \int_{(\mathbf R^n-B)\times T^k}|\nabla f|\,\mathrm dx\mathrm dt.
\end{equation}
\end{lemma}
\begin{proof}
The proof is standard and it can be found in \cite{Evans2010}. Here we just include the arguments for completeness. First let us do zero extension of $f$ and $\nabla f$ into $B\times T^k$. Although this breaks the continuity of these functions, it doesn't matter in our discussion below. For convenience, we view the extended $f$ and $\nabla f$ as functions over $\mathbf R^n\times [0,d_1]\times\cdots\times [0,d_k]$, and we denote
\begin{equation*}
I_j=\left\{
\begin{array}{cc}
\mathbf R, & 1\leq j\leq n; \\
{[0,d_{j-n}]}, & n+1\leq j\leq n+k.
\end{array}
\right.
\end{equation*}
Since $f$ has compact support and satisfies \eqref{Eq: zero average}, it holds for any $1\leq j\leq n+k$ that
\begin{equation}\label{Eq: basic inequality}
|f(x_1,\ldots, x_{n+k})|\leq \int_{I_j}|\nabla f(x_1,\ldots,y_j,\ldots,x_{n+k})|\mathrm dy_j.
\end{equation}
As a result, we have
$$
|f(x_1,\ldots, x_{n+k})|^{\frac{n+k}{n+k-1}}\leq \prod_{j=1}^{n+k}\left(\int_{I_j}|\nabla f(x_1,\ldots,y_j,\ldots,x_{n+k})|\mathrm dy_j\right)^{\frac{1}{n+k-1}}.
$$
Integrating this inequality over $I_1$ and applying the H\"older inequality, we can obtain
\begin{equation*}
\begin{split}
\int_{I_1}|f(x_1,\ldots, x_{n+k})&|^{\frac{n+k}{n+k-1}}\mathrm dx_1
\leq\left(\int_{I_1}|\nabla f(x_1,\ldots,y_j,\ldots,x_{n+k})|\mathrm dy_1\right)^{\frac{1}{n+k-1}}\\
&\cdot \prod_{j=2}^{n+k}\left(\int_{I_j}\int_{I_1}|\nabla f(x_1,\ldots,y_j,\ldots,x_{n+k})|\mathrm dy_j\mathrm dx_1\right)^{\frac{1}{n+k-1}}.
\end{split}
\end{equation*}
Now we continue to integrate above inequality over the rest $I_j$ step by step with the use of the H\"older inequality. Eventually we will arrive at
\begin{equation*}
\begin{split}
\left(\int_{\mathbf R^n\times [0,d_1]\times\cdots\times [0,d_k]}|f|^{\frac{n+k}{n+k-1}}\,\mathrm dx\right)^{\frac{n+k-1}{n+k}}\leq \int_{\mathbf R^n\times [0,d_1]\times\cdots\times [0,d_k]}|\nabla f|\,\mathrm dx,
\end{split}
\end{equation*}
which is exactly the desired inequality \eqref{Eq: sobolev no weight}.
\end{proof}
Next we are going to prove Proposition \ref{Prop: Sobolev} inductively with an averaging trick. In fact, we can establish the following more general result.
\begin{proposition}
If $f$ is a function in $C_c^1((\mathbf R^n-B)\times T^k)$ satisfying
\begin{equation*}
\int_{\mathbf S^1(d_i)} f(x,t_1,t_2,\ldots,t_k)\,\mathrm dt_i=0,\quad i=1,2,\ldots,l,
\end{equation*}
with $0\leq l\leq k$, then there is a universal constant $C$ depending only on $n$, $k$, $l$ and $d_i$, $i=1,2,\ldots,k$, such that
\begin{equation}
\left(\int_{(\mathbf R^n-B)\times T^k}|f|^{\frac{n+k}{n+k-1}}r^{\frac{l-k}{n+k-1}}\,\mathrm dx\mathrm dt\right)^{\frac{n+k-1}{n+k}}\leq C\int_{(\mathbf R^n-B)\times T^k}|\nabla f|\,\mathrm dx\mathrm dt.
\end{equation}
\end{proposition}
\begin{proof}
The proof will be completed from an induction on the difference $k-l$. It follows from Lemma \ref{Lem: sobolev average} that above proposition holds when $k-l=0$. From induction we can assume that above proposition holds when $k-l=m$ and then deal with the case $k-l=m+1$. For any $f$ in $C_c^1((\mathbf R^n-B)\times T^k)$ satisfying
\begin{equation*}
\int_{\mathbf S^1(d_i)} f(x,t_1,t_2,\ldots,t_k)\,\mathrm dt_i=0,\quad i=1,2,\ldots,l,
\end{equation*}
we write it as
$
f=f_1+f_2
$
with
$$
f_1=\frac{1}{d_{l+1}}\int_{\mathbf S^1(d_{l+1})}f(x,t_1,t_2,\ldots,t_k)\,\mathrm dt_{l+1}.
$$
Clearly, $f_2$ is a function in $C_c^1((\mathbf R^n-B)\times T^{k-1})$ satisfying
\begin{equation*}
\int_{\mathbf S^1(d_i)} f_2(x,t_1,t_2,\ldots,t_k)\,\mathrm dt_i=0,\quad i=1,2,\ldots,l+1.
\end{equation*}
Notice that the gradients of $f_1$ and $f_2$ are both controlled by the gradient of $f$. So we have
\begin{equation}\label{Eq: estimate f2}
\begin{split}
& \left(\int_{(\mathbf R^n-B)\times T^k}|f_2|^{\frac{n+k}{n+k-1}}r^{-\frac{m+1}{n+k-1}}\,\mathrm dx\mathrm dt\right)^{\frac{n+k-1}{n+k}}\\\leq& \left(\int_{(\mathbf R^n-B)\times T^k}|f_2|^{\frac{n+k}{n+k-1}}r^{-\frac{m}{n+k-1}}\,\mathrm dx\mathrm dt\right)^{\frac{n+k-1}{n+k}} \\
\leq&C\int_{(\mathbf R^n-B)\times T^k}|\nabla f_2|\,\mathrm dx\mathrm dt
\leq C\int_{(\mathbf R^n-B)\times T^k}|\nabla f|\,\mathrm dx\mathrm dt.
\end{split}
\end{equation}
Here and in the sequel, the symbol $C$ is always denoted to be a universal constant depending only on $n$, $k$, $l$ and $d_i$, $i=1,2,\ldots,k$.
On the other hand, the function $f_1$ can be viewed as a function in $C_c^1((\mathbf R^n-B)\times T^{k-1})$ and it satisfies
\begin{equation*}
\int_{\mathbf S^1(d_i)} f_1(x,t_1,t_2,\ldots,t_{k-1})\,\mathrm dt_i=0,\quad i=1,2,\ldots,l.
\end{equation*}
From inductive assumption we see
\begin{equation*}
\int_{(\mathbf R^n-B)\times T^k}|f_1|^{\frac{n+k}{n+k-1}}r^{-\frac{m+1}{n+k-1}}\mathrm dx\mathrm dt
= d_{l+1}\int_{(\mathbf R^n-B)\times T^{k-1}}|f_1|^{\frac{n+k}{n+k-1}}r^{-\frac{m+1}{n+k-1}}\mathrm dx\mathrm dt.
\end{equation*}
Now we have to divide the discussion into two cases:
{\it Case 1.} $l>0$. Since we have $m<k-1$ in this case, it follows from H\"older inequality that
\begin{equation*}
\begin{split}
&\left(\int_{(\mathbf R^n-B)\times T^{k-1}}|f_1|^{\frac{n+k}{n+k-1}}r^{-\frac{m+1}{n+k-1}}\mathrm dx\mathrm dt\right)^{\frac{n+k-1}{n+k}}\\
\leq&\left(\int_{(\mathbf R^n-B)\times T^{k-1}}|f_1|^{\frac{n+k-1}{n+k-2}}r^{-\frac{m}{n-k-2}}\mathrm dx\mathrm d t\right)^{\frac{n+k-2}{n+k-1}}\\
&\qquad\qquad\cdot\left(\int_{(\mathbf R^n-B)\times T^{k-1}}r^{-n-k+m+1}\mathrm dx\mathrm dt\right)^{\frac{1}{(n+k)(n+k-1)}}\\
\leq & C\int_{(\mathbf R^n-B)\times T^{k-1}}|\nabla f_1|\mathrm dx\mathrm dt
\leq C\int_{(\mathbf R^n-B)\times T^{k}}|\nabla f_1|\mathrm dx\mathrm dt.
\end{split}
\end{equation*}
This yields
$$
\left(\int_{(\mathbf R^n-B)\times T^k}|f_1|^{\frac{n+k}{n+k-1}}r^{-\frac{m+1}{n+k-1}}\mathrm dx\mathrm dt\right)^{\frac{n+k-1}{n+k}}\leq C\int_{(\mathbf R^n-B)\times T^k}|\nabla f|\mathrm dx\mathrm dt.
$$
{\it Case 2.} $l=0$. A more delicate analysis is involved in this case since $r^{-n}$ is not integrable in $\mathbf R^n-B$. The basic idea is to do further averaging procedure to $f_1$. Write $f_1=f_{11}+f_{12}$ with
$$
f_{11}=\frac{1}{d_{l+2}}\int_{\mathbf S^1(d_{l+2})}f_1(x,t_1,\cdots,t_k)\mathrm dt_{l+2}.
$$
It is easy to check that we can obtain
$$
\left(\int_{(\mathbf R^n-B)\times T^k}|f_{12}|^{\frac{n+k}{n+k-1}}r^{-\frac{k}{n+k-1}}\mathrm dx\mathrm dt\right)^{\frac{n+k-1}{n+k}}\leq C\int_{(\mathbf R^n-B)\times T^k}|\nabla f_{12}|\mathrm dx\mathrm dt
$$
with the help of the inductive assumption and the H\"older inequality. However, the critical exponent appears when we try to deal with the function $f_{11}$. So we continue to do averaging procedure to $f_{11}$, and finally we reduce the desired estimate to a function $\bar f$ in $C_c^1((\mathbf R^n-B)\times T^k)$ satisfying
$$
\bar f(x,t_1,\ldots,t_k)=g(x).
$$
It suffices to show
$$
\left(\int_{\mathbf R^n-B}|g|^{\frac{n+k}{n+k-1}}r^{-\frac{k}{n+k-1}}\mathrm dx\right)^{\frac{n+k-1}{n+k}}\leq C\int_{\mathbf R^n-B}|\nabla g|\,\mathrm dx.
$$
Actually we have the following Hardy inequality
\begin{equation}\label{Eq: hardy}
\int_{\mathbf R^n-B}|g|r^{-1}\,\mathrm dx\leq \frac{1}{n-1}\int_{\mathbf R^n-B}|\nabla g|\,\mathrm dx.
\end{equation}
To see this, let us do some calculation.
\begin{equation*}
\begin{split}
\int_{\mathbf R^n-B}|g|r^{-1}\,\mathrm dx&=\int_{\mathbf S^{n-1}}\mathrm d\sigma \int_1^{+\infty}|g|r^{n-2}\,\mathrm dr\\
&\leq\int_{\mathbf S^{n-1}}\mathrm d\sigma \int_1^{+\infty}\mathrm dr\int_{r}^{+\infty}|\nabla g|(s,\theta)r^{n-2}\,\mathrm ds\\
&=\int_{\mathbf S^{n-1}}\mathrm d\sigma \int_1^{+\infty}\mathrm ds\int_{1}^s|\nabla g|(s,\theta)r^{n-2}\,\mathrm dr\\
&\leq\frac{1}{n-1}\int_{\mathbf S^{n-1}}\mathrm d\sigma\int_1^{+\infty}|\nabla g|(s,\theta)s^{n-1}\,\mathrm ds\\
&=
\frac{1}{n-1}\int_{\mathbf R^n-B}|\nabla g|\,\mathrm dx.
\end{split}
\end{equation*}
Now the desired inequality comes from a simple interpolation of \eqref{Eq: sobolev no weight} and \eqref{Eq: hardy}. Namely, we have
\begin{equation*}
\begin{split}
&\left(\int_{\mathbf R^n-B}|g|^{\frac{n+k}{n+k-1}}r^{-\frac{k}{n+k-1}}\mathrm dx\right)^{\frac{n+k-1}{n+k}}\\
\leq &\left(\int_{\mathbf R^n-B}|g|^{\frac{n}{n-1}}\mathrm dx\right)^{\frac{n}{n+k}}\left(\int_{\mathbf R^n-B}|g|r^{-1}\,\mathrm dx\right)^{\frac{k}{n+k}}\\
\leq &C\int_{\mathbf R^n-B}|\nabla g|\,\mathrm dx.
\end{split}
\end{equation*}
Now we complete the proof.
\end{proof}
For further generalization of Proposition \ref{Prop: Sobolev}, we point out that the validity of Proposition \ref{Prop: Sobolev} essentially relies on
\begin{itemize}
\item[(i)] the no-weighted Sobolev inequality
\begin{equation*}
\left(\int_{(\mathbf R^n-B)\times T^k}|f|^{\frac{n+k}{n+k-1}}\,\mathrm dx\mathrm dt\right)^{\frac{n+k-1}{n+k}}\leq \int_{(\mathbf R^n-B)\times T^k}|\nabla f|\,\mathrm dx\mathrm dt
\end{equation*}
for all $f$ in $C_c^1((\mathbf R^n-B)\times T^k)$ with
$$
\int_{\mathbf S^1(d_i)} f(x,t_1,t_2,\ldots,t_k)\,\mathrm dt_i=0,\quad i=1,2,\ldots,k.
$$
\item[(ii)] the Hardy inequality
$$
\int_{\mathbf R^n-B}|f|r^{-1}\mathrm dx\leq \frac{1}{n-1}\int_{\mathbf R^n-B}|\nabla f|\,\mathrm dx
$$
for all $f$ in $C_c^1(\mathbf R^n-B)$.
\end{itemize}
It is easy to generalize above inequalities to $L^p$-version. Namely we have
\begin{lemma}
If $f$ is a function in $C_c^1((\mathbf R^n-B)\times T^k)$ satisfying
\begin{equation*}
\int_{\mathbf S^1(d_i)} f(x,t_1,t_2,\ldots,t_k)\,\mathrm dt_i=0,\quad i=1,2,\ldots,k,
\end{equation*}
then for any $1\leq p<n+k$ there is a universal constant $C$ depending only on $n$, $k$, $p$ and $d_i$, $i=1,2,\ldots,k$, such that
\begin{equation*}
\left(\int_{(\mathbf R^n-B)\times T^k}|f|^{\frac{(n+k)p}{n+k-p}}\,\mathrm dx\mathrm dt\right)^{\frac{n+k-p}{(n+k)p}}\leq C\left(\int_{(\mathbf R^n-B)\times T^k}|\nabla f|^p\,\mathrm dx\mathrm dt\right)^{\frac{1}{p}}.
\end{equation*}
\end{lemma}
\begin{proof}
The proof is standard. First we point out that the key to inequality \eqref{Eq: sobolev no weight} is the validity of \eqref{Eq: basic inequality}, which only requires that $f$ has zero in all $\mathbf S^1$ slices. Therefore, if $f$ is a function in $C_c^1((\mathbf R^n-B)\times T^k)$ satisfying
\begin{equation*}
\int_{\mathbf S^1(d_i)} f(x,t_1,t_2,\ldots,t_k)\,\mathrm dt_i=0,\quad i=1,2,\ldots,k,
\end{equation*}
then \eqref{Eq: sobolev no weight} can be applied to the function $|f|^{\frac{p(n+k-1)}{n+k-p}}$. Then we have
\begin{equation*}
\begin{split}
&\left(\int_{(\mathbf R^n-B)\times T^k}|f|^{\frac{p(n+k)}{n+k-p}}\,\mathrm dx\mathrm dt\right)^{\frac{n+k-1}{n+k}}\\
\leq &\frac{p(n+k-1)}{n+k-p}\int_{(\mathbf R^n-B)\times T^k}|f|^{\frac{(n+k)(p-1)}{n+k-p}}|\nabla f|\,\mathrm dx\mathrm dt\\
\leq &\frac{p(n+k-1)}{n+k-p}\left(\int_{(\mathbf R^n-B)\times T^k}|f|^{\frac{p(n+k)}{n+k-p}}\,\mathrm dx\mathrm dt\right)^{\frac{p-1}{p}}\left(\int_{(\mathbf R^n-B)\times T^k}|\nabla f|^p\,\mathrm dx\mathrm dt\right)^{\frac{1}{p}}.
\end{split}
\end{equation*}
This gives
\begin{equation*}
\left(\int_{(\mathbf R^n-B)\times T^k}|f|^{\frac{(n+k)p}{n+k-p}}\,\mathrm dx\mathrm dt\right)^{\frac{n+k-p}{(n+k)p}}\leq \frac{p(n+k-1)}{n+k-p}\left(\int_{(\mathbf R^n-B)\times T^k}|\nabla f|^p\,\mathrm dx\mathrm dt\right)^{\frac{1}{p}}.
\end{equation*}
This is just the desired inequality.
\end{proof}
\begin{lemma}
Let $f$ be a function in $C_c^1(\mathbf R^n-B)$. Then for any $1\leq p<n$ we have
$$
\int_{\mathbf R^n-B}|f|^pr^{-p}\,\mathrm dx\leq\left(\frac{p}{n-p}\right)^{p}\int_{\mathbf R^n-B}|\nabla f|^p.
$$
\end{lemma}
\begin{proof}
The proof comes from a direct calculation. We have
\begin{equation*}
\begin{split}
\int_{\mathbf R^n-B}|f|^pr^{-p}\,\mathrm dx&=\int_{\mathbf S^{n-1}}\mathrm d\sigma \int_1^{+\infty}|f|^pr^{n-1-p}\,\mathrm dr\\
&\leq p\int_{\mathbf S^{n-1}}\mathrm d\sigma \int_1^{+\infty}\mathrm dr\int_{r}^{+\infty}|f|^{p-1}(s,\theta)|\nabla f|(s,\theta)r^{n-1-p}\,\mathrm ds\\
&=p\int_{\mathbf S^{n-1}}\mathrm d\sigma \int_1^{+\infty}\mathrm ds\int_{1}^s|f|^{p-1}(s,\theta)|\nabla f|(s,\theta)r^{n-1-p}\,\mathrm dr\\
&\leq\frac{p}{n-p}\int_{\mathbf S^{n-1}}\mathrm d\sigma\int_1^{+\infty}|f|^{p-1}(s,\theta)|\nabla f|(s,\theta)s^{n-p}\,\mathrm ds\\
&=
\frac{p}{n-p}\left(\int_{\mathbf S^{n-1}}\mathrm d\sigma\int_1^{+\infty}|f|^{p}(s,\theta)s^{n-1-p}\,\mathrm ds\right)^{\frac{p-1}{p}}\\
&\quad\cdot\left(\int_{\mathbf S^{n-1}}\mathrm d\sigma\int_1^{+\infty}|\nabla f|^{p}(s,\theta)s^{n-1}\,\mathrm ds\right)^{\frac{1}{p}}\\
&=\frac{p}{n-p}\left(\int_{\mathbf R^n-B}|f|^pr^{-p}\,\mathrm dx\right)^{\frac{p-1}{p}}\left(\int_{\mathbf R^n-B}|\nabla f|^p\,\mathrm dx\right)^{\frac{1}{p}}.
\end{split}
\end{equation*}
This gives the desired inequality.
\end{proof}
Combining above two lemmas with the averaging trick, we can obtain the following
\begin{proposition}\label{Prop: Sobolev Lp}
For any $1\leq p<n$ there is a universal constant $C$ depending only on $n$, $k$, $p$ and $d_i$, $i=1,2,\ldots,k$, such that the inequality
\begin{equation*}
\left(\int_{(\mathbf R^n-B)\times T^k}|f|^{\frac{(n+k)p}{n+k-p}}r^{\frac{-kp}{n+k-p}}\,\mathrm dx\mathrm dt\right)^{\frac{n+k-p}{(n+k)p}}\leq C\left(\int_{(\mathbf R^n-B)\times T^k}|\nabla f|^p\,\mathrm dx\mathrm dt\right)^{\frac{1}{p}}
\end{equation*}
holds for any $f$ in $C_c^1((\mathbf R^n-B)\times T^k)$. Here the function $f$ is allowed to take non-zero values on the inner boundary $\partial B\times T^k$.
\end{proposition}
Clearly the Sobolev inequality still holds if one passes to an equivalent metric or a quotient space. Since every closed flat manifold is covered by a flat torus \cite{B1911} \cite{B1912}, we conclude that
\begin{corollary}\label{Cor: Sobolev Lp}
Let $(F,g_{flat})$ be a flat closed $k$-manifold and $g$ be a smooth metric on $(\mathbf R^n-B)\times F$ equivalent to $g_{euc}\oplus g_{flat}$. For any $1\leq p<n$ there is a universal constant $C$ depending only on $n$, $k$, $p$ and $F$, such that the inequality
\begin{equation*}
\left(\int_{(\mathbf R^n-B)\times F}|f|^{\frac{(n+k)p}{n+k-p}}r^{\frac{-kp}{n+k-p}}\,\mathrm d\mu_{ g}\right)^{\frac{n+k-p}{(n+k)p}}\leq C\left(\int_{(\mathbf R^n-B)\times F}|\nabla f|^p\,\mathrm d\mu_{ g}\right)^{\frac{1}{p}}
\end{equation*}
holds for any $f$ in $C_c^1((\mathbf R^n-B)\times F)$.
\end{corollary}
\subsection{Lohkamp compactification and Proof for Theorem \ref{Thm: main 4}} In this subsection, we are going to prove our main Theorem \ref{Thm: main 4} for asymptotically flat manifolds with fiber $F$ (recall from Definition \ref{Defn: AF with fiber F}). First we point out that the fiber $F$ can be assumed to be $T^k$, $k=\dim F$, from a lifting argument without loss of generality, and this is always the case in this subsection. In the following discussion, $n$ is denoted to be $\dim M$, $k$ is denoted to be $\dim F$ and $d=n-k$.
Let us start with the following lemma, which provides a convenient way to compute the mass $m(M,g,\mathcal E)$.
\begin{lemma}\label{defn of mass}
Let $(M, g, \mathcal{E})$ be an asymptotically flat manifold with fiber $F$ and arbitrary ends. Fix an orthornormal basis $\{e_{a}\}$ with respect to $g_{0}=g_{euc}\oplus g_F$ consisting of $\{\frac{\partial }{\partial x_{i}},f_{\alpha}\}$ such that $g$ has the expression $g=g_{ab}e_a\otimes e_b$. Then the mass of $(M, g, \mathcal{E})$ is given by
$$
\begin{aligned}
m(M, g, \mathcal{E})
&=\frac{1}{2\left|\mathbf{S}^{d-1}\right| \operatorname{Vol}\left(F, g_{F}\right)} \lim _{\rho \rightarrow+\infty}\int_{S_{\rho}\times F}(\partial_{i}g_{ij}-\partial_{j}g_{aa})\ast\mathrm{d}x^{j}\mathrm{d}\gamma.
\end{aligned}
$$
In above expression, the symbol $a$ runs over all index, $i$ runs over the index on Euclidean space $\mathbf R^d$, $\alpha$ runs over index on $F$, $d\gamma$ is volume form of $g_{F}$ and $\ast$ is the Hodge star operator corresponding to the Euclidean metric $\mathrm{d}x^{2}$ in $\mathbf R^{d}$.
\end{lemma}
\begin{proof}
Let $\{e_a^*\}$ be the dual frame of $\{e_a\}$. From the definition of the total mass we have
\[
\begin{split}
m(M, g, \mathcal{E})
&=\frac{1}{2\left|\mathbf{S}^{d-1}\right| \operatorname{Vol}\left(F, g_{F}\right)} \lim _{\rho \rightarrow+\infty}\int_{S_{\rho}\times F}(\partial_{a}g_{ab}-\partial_{b}g_{aa})\ast e_b^*\\
&=\frac{1}{2\left|\mathbf{S}^{d-1}\right| \operatorname{Vol}\left(F, g_{F}\right)} \lim _{\rho \rightarrow+\infty}\int_{S_{\rho}\times F}(\partial_{a}g_{aj}-\partial_{j}g_{aa})\ast\mathrm{d}x^{j}\mathrm{d}\gamma.
\end{split}
\]
We obtain the desired formula after applying the divergence theorem to the integral of the first term along $F$.
\end{proof}
Denote $U$ to be an open set such that $\mathcal{E}\subset U $ and $\overline{U-\mathcal{E}}$ is compact. We have
\begin{lemma}\label{3}For $p=2$, there exist a constant $C_{U}$ depending only on $g$, $U$, $n$ and $k$ such that for $f \in C_{c}^{1}\left( \overline{U}\right)$ ,
\begin{equation}\label{sobelov on U}
\left(\int_{U}|f|^{\frac{2(d+k)}{d+k-2}} r^{\frac{-2k}{d+k-2}} \mathrm{~d} \mu_{g}\right)^{\frac{d+k-2}{2(d+k)}} \leq C_{U}\left(\int_{U}|\nabla f|^{2} \mathrm{~d} \mu_{g}\right)^{\frac{1}{2}}
\end{equation}
\end{lemma}
\begin{proof}
Since the metric $g$ is euivalent to $g_0=g_{euc}\oplus g_F$, Corollary \ref{Prop: Sobolev Lp} also holds for $(\mathcal E,g)$. To extend the Sobolev inequality on $(U,g)$, the argument is the same as in \cite[Lemma 3.1]{SY1979}.
\end{proof}
For a function, it can be written as the difference of its positive part and negetive part, i.e.
$$f=f_{+}-f_{-},f_{+}=\max\{f,0\},f_{-}=\max\{-f,0\}.$$
\begin{proposition}\label{3.2likeSY}
Assume that $f$ is a smooth function on $M$ with compact support in $U$ such that
the negative part $f_{-}$ of $f$ satisfies
\begin{equation}\label{f_{-}control}
\left(\int_{U}\left|f_{-}\right|^{\frac{d+k}{2}} r^{k}\,\mathrm{d} \mu_{g}\right)^{\frac{2}{d+k}}\leq \frac{1}{2C_{U}^{2}},
\end{equation}
here $C_{U}$ is the sobelov constant comes from Lemma $\ref{3}$, and
\begin{equation}\label{f control}
\left(\int_{U}|f|^{\frac{2 (d+k)}{d+k+2}} r^{\frac{2k(d+k-2)}{(d+k+2)^{2}}} \mathrm{d} \mu_{g}\right)^{\frac{d+k+2}{2(d+k)}} \leq C_{0}
\end{equation}
holds for some constants $C_{0}$.
Then the equation
\begin{equation}\label{main equation}
\Delta_{g} u-f u=0
\end{equation}
has a positive solution $u$ on $M$ such that $u$ has the expansion
\begin{equation}\label{expansion}
u=1+\frac{A}{ r^{d-2}}+\omega
\end{equation}
with
\begin{equation}\label{A}
A=-\frac{1}{(d-2)\left|\mathbf{S}^{d-1}\right||T^{k}|} \int_{U} f u \,\mathrm{d} \mu_{g}
\end{equation}
and for any $\epsilon>0$ small, $|\omega|+r|\partial \omega|+r^{2}\left|\partial^{2} \omega\right| = O(r^{-(d-1-\epsilon)})$ in $\mathcal{E}$ as $r\rightarrow \infty$. Moreover, we have
\begin{equation}\label{integral on boundary}
\int_{\partial U} \frac{\partial u}{\partial \vec{n}} \mathrm{~d} \sigma_{g}=0 \quad \text { and } \quad \int_{\partial U} u \frac{\partial u}{\partial \vec{n}} \mathrm{~d} \sigma_{g} \leq 0
\end{equation}
where $\vec{n}$ is the outward unit normal vector of $\partial U$.
\end{proposition}
\begin{proof}
The proof is divided into two steps.
\textbf{Step 1: the existence of $u$.}
Let us take a smooth exhaustion
$$
U=U_{0} \subset U_{1} \subset U_{2} \subset \cdots
$$
such that $U_{i}-\mathcal{E}$ has a compact closure for all $i$ and
$$
M=\bigcup_{i=0}^{\infty} U_{i}
$$
such exhaustion can be construucted from collecting points no greater than a certain distance to $U$. Fix some $U_{i}$ and consider the following equation
\begin{equation}\label{v_{i}in U_{i}}
\Delta_{g} v_{i}-f v_{i}=f \quad\text { in } \quad U_{i}; \quad \frac{\partial v_{i}}{\partial \vec{n}}=0 \quad\text { on } \quad\partial U_{i}
\end{equation}
where $\vec{n}$ is denoted to be the outward unit normal of $\partial U_{i}$. It is standard that equation \eqref{v_{i}in U_{i}} can be solved by exhaustion method. Let us extend the radical function $r$ by defining $r=1$ outside $\mathcal{E}$. For any $R>1$, we take
$$
U_{i, R}=\left\{x \in U_{i}: r(x) \leq R\right\}
$$
We have $\partial U_{i, R}=\partial U_{i} \cup \partial B_{R}$, where $\partial B_{R}=\{x \in M: r(x)=R\} \subset \mathcal{E}$. With this setting, we consider the following equation
\begin{equation}\label{v_{i,R}in U_{i,R}}
\left\{\begin{array}{clc}
\Delta_{g} v_{i, R}-f v_{i, R}=f & \text { in } & U_{i, R} \\
\frac{\partial v_{i, R}}{\partial n}=0 & \text { on } & \partial U_{i} \\
v_{i, R}=0 & \text { on } & \partial B_{R}
\end{array}\right.
\end{equation}
From Fredholm alternative \cite[Theorem 5.3]{GT2001}, equation \eqref{v_{i,R}in U_{i,R}} has a smooth solution $v_{i, R}$ once the corresponding homogeneous equation to \eqref{v_{i,R}in U_{i,R}} has only zero solution, so we verify this shortly. Let $\zeta$ be a solution of the homogeneous equation. From integral by parts we see
$$
\int_{U_{i, R}}\left|\nabla_{g} \zeta\right|^{2} \mathrm{~d} \mu_{g}=-\int_{U_{i, R}} f \zeta^{2} \mathrm{~d} \mu_{g}
$$
From inequality \eqref{sobelov on U}, the left hand side is greater than or equal to
$$
\frac{1}{C_{U}^{2}}\left(\int_{U_{R}}|\zeta|^{\frac{2(d+k) }{d+k-2}} r^{\frac{-2k}{d+k-2}} \mathrm{~d} \mu_{g}\right)^{\frac{d+k-2}{d+k }}
$$
however, the right hand side is less than or equal to
$$
\left(\int_{U_{R}}\left|f_{-}\right|^{\frac{d+k}{2}} r^{k}\,\mathrm{d} \mu_{g}\right)^{\frac{2}{d+k}}\left(\int_{U_{R}}|\zeta|^{\frac{2(d+k) }{d+k-2}} r^{\frac{-2k}{d+k-2}} \,\mathrm{d} \mu_{g}\right)^{\frac{d+k-2}{d+k}}
$$
where we used H\"{o}lder inequality and fact that $f$ has compact support in $U$. Combined with \eqref{f_{-}control} we obtain
$$
\int_{U_{R}}|\zeta|^{\frac{2(d+k) }{d+k-2}} r^{\frac{-2k}{d+k-2}} \,\mathrm{d} \mu_{g}=0
$$
hence $\zeta$ vanishes in $U_{R}$. Notice that $\zeta$ is a harmonic function in $U_{i, R}-U_{R}$ and a small neighbourhood of $\partial U$, then the unique continuation yields $\zeta \equiv 0$ on $U_{i,R}$, so \eqref{v_{i,R}in U_{i,R}} has a smooth solution $v_{i, R}$. Multiplying $v_{i, R}$ to both side of \eqref{v_{i,R}in U_{i,R}}, then integrating and using H\"{o}lder inequality, we have
$$
\begin{aligned}
\int_{U_{i, R}}\left|\nabla_{g} v_{i, R}\right|^{2} \mathrm{~d} \mu_{g} \leq
&\left(\int_{U_{R}}\left|f_{-}\right|^{\frac{d+k}{2}} r^{k}\,\mathrm{d} \mu_{g}\right)^{\frac{2}{d+k}}\left(\int_{U_{R}}|v_{i, R}|^{\frac{2(d+k) }{d+k-2}} r^{\frac{-2k}{d+k-2}}\, \mathrm{d} \mu_{g}\right)^{\frac{d+k-2}{d+k}}\\
+&\left(\int_{U_{R}}|f|^{\frac{2 (d+k)}{d+k+2}} r^{\frac{2k(d+k-2)}{(d+k+2)^{2}}}\,\mathrm{d} \mu_{g}\right)^{\frac{d+k+2}{2(d+k)}}\left(\int_{U_{R}}\left|v_{i, R}\right|^{\frac{2(d+k)}{d+k-2}} r^{\frac{-2k}{d+k-2}}\, \mathrm{d} \mu_{g}\right)^{\frac{d+k-2}{2 (d+k)}}.
\end{aligned}
$$
Combined with \eqref{sobelov on U}, \eqref{f_{-}control} and \eqref{f control}, it follows
\begin{equation}\label{estimate of v_{i,R}}
\left(\int_{U_{R}}\left|v_{i, R}\right|^{\frac{2(d+k)}{d+k-2}} r^{\frac{-2k}{d+k-2}} \,\mathrm{d} \mu_{g}\right)^{\frac{d+k-2}{2 (d+k)}} \leq 2 C_{U}^{2} C_{0}.
\end{equation}
Take an increasing sequence $R_{j}$ such that $R_{j} \rightarrow+\infty$ as $j \rightarrow \infty$. Since the weight function $r^{\frac{-2k}{d+k-2}}$ is bounded from below in any compact set, from the $L^{p}$-estimate \cite[Theorem 9.11]{GT2001}
and the Schauder estimate \cite[Theorem 6.2]{GT2001}, we have
\begin{equation}\label{K in U}
\sup _{K}\left|\nabla_{g}^{k} v_{i, R_{j}}\right| \leq C(k, K)
\end{equation}
for any compact subset $K$ of $U$, where $C(k, K)$ is a constant depending on $k$ and $K$, but independent of $i$ and $R_{j}$. Fix a positive constant $r_{0}>1$.
Now we claim that
\begin{equation}\label{C(i,r_{0})}
\sup _{U_{i,r_{0}}}\left|\nabla_{g}v_{i, R_{j}}\right| \leq C(i, r_{0})
\end{equation}
holds for some constant $C(i,r_{0})$ independent of $R_{j}$. Otherwise, there is a subsequence (still denoted by $v_{i, R_{j}}$) such that
$$
\sup _{U_{i, r_{0}}}\left|v_{i, R_{j}}\right| \rightarrow+\infty \quad \text { but } \quad\left(\int_{U_{r_{0}}}\left|v_{i, R_{j}}\right|^{\frac{2 (d+k)}{d+k-2}} \,\mathrm{d} \mu_{g}\right)^{\frac{d+k-2}{2 (d+k)}} \leq 2 C_{U}^{2} C_{0}C_{r_{0}},
$$
where $C_{r_{0}}$ is a constant depending on $r_{0}$, which comes from the weight $r^{\frac{-2k}{d+k-2}}$.
Let
$$
w_{i, R_{j}}=\left(\sup _{U_{i, r_{0}}}\left|v_{i, R_{j}}\right|\right)^{-1} v_{i, R_{j}}.
$$
Clearly $\left|w_{i, R_{j}}\right| \leq 1$ in $U_{i, r_{0}}$ and so it follows from the Schauder estimates \cite[ Theorem 6.2 and 6.30]{GT2001} that up to a subsequence $w_{i, R_{j}}$ converges uniformly to a limit function $w_{i}$ solving
$$
\Delta_{g} w_{i}-f w_{i}=0 \quad \text { in } \quad U_{i, r_{0}} ; \quad \frac{\partial w_{i}}{\partial \vec{n}}=0 \quad \text { on } \quad \partial U_{i},
$$
and satisfying
$$
\sup _{U_{i, r_{0}}}\left|w_{i}\right|=1 \quad \text { and } \quad w_{i}=0 \quad \text { in } \quad U_{r_{0}},
$$
where the latter comes from \eqref{K in U}. As before, $w_{i}$ is a harmonic function in $U_{i, r_{0}}-U_{r_{0}}$ and a small neighbourhood of $\partial U$, then the unique continuation yields $w_{i} \equiv 0$ in $U_{i, r_{0}}$, which leads to a contradiction.
Now the uniform bound for $v_{i, R_{j}}$ in $U_{i, r_{0}}$ combined with Schauder estimate as well as
estimate \eqref{C(i,r_{0})} implies that a subsequence of $v_{i, R_{j}}$ converges to a smooth solution $v_{i}$ of \eqref{v_{i}in U_{i}}. Clearly we have
\begin{equation}\label{v_{i}estimate}
\left(\int_{U}\left|v_{i}\right|^{\frac{2 (d+k)}{d+k-2}} r^{\frac{-2k}{d+k-2}}\,\mathrm{d} \mu_{g}\right)^{\frac{d+k-2}{2 (d+k)}} \leq 2 C_{U}^{2} C_{0}.
\end{equation}
Next we claim that for any small $\epsilon>0$ there are constants $r_0$ and $C'$ independent of $i$ such that
\begin{equation}\label{rough decay}
|v_{i}(x)|\leq C'r^{-(d-2-\epsilon)}, \quad r(x)\geq r_{0}.
\end{equation}
To prove this, let us construct a comparison function $z(r)$ in the form of $Cr^{-(d-2-\epsilon)}$ with $C>0$ on the region $\{r\geq r_0\}$. From simple calculation we see
$$\Delta_{g} (r^{-(d-2-\epsilon)})=-\epsilon(d-2-\epsilon)r^{-(d-\epsilon)}+O(r^{-(d-\epsilon+\mu)}).$$
So we can take $r_0$ large enough such that $z(r)$ is superharmonic in $\{r\geq r_0\}$. Note that from \eqref{estimate of v_{i,R}} we have the uniform estimate
$$\sup _{r(x)=r_{0}} |v_{i,R_{j}}(x)|\leq \Lambda$$
for a universal constant $\Lambda$ independent of $i$ and $j$. Also, the function $v_{i, R_{j}}$ vanishes on $\partial B_{R_{j}}$.
After
choosing a large $C$, then we have $z(r)\geq v_{i,R_{j}}$ on boundary of $B_{R_{j}}-B_{r_{0}}$. Since $v_{i,R_{j}}$ is harmonic in $\{r\geq r_0\}$, we have $\Delta_{g} (z(r)-v_{i,R_{j}})\leq 0$. It follows from maximum principle that $v_{i,R_{j}}\leq z(r)$ in $\{r\geq r_0\}$. By letting $j\to +\infty$, we conclude that $v_i\leq z(r)$ in $\{r\geq r_0\}$. Similarly we also have $v_{i}\geq -z(r)$ and this completes the proof of \eqref{rough decay}.
Notice that $v_{i}$ is a harmonic function in $U_{i}-U$. Then the divergence theorem yields
$$
\int_{\partial U} \frac{\partial v_{i}}{\partial \vec{n}} \mathrm{~d} \sigma_{g}=\int_{\partial U_{i}} \frac{\partial v_{i}}{\partial \vec{n}} \mathrm{~d} \sigma_{g}=0
$$
and
$$
\int_{\partial U} v_{i} \frac{\partial v_{i}}{\partial \vec{n}} \mathrm{~d} \sigma_{g}=-\int_{U_{i}-U}\left|\nabla_{g} v_{i}\right|^{2} \mathrm{~d} \sigma_{g} \leq 0.
$$
Next we consider the function $u_{i}=1+v_{i}$ instead. Clearly, $u_{i}$ solves
$$
\Delta_{g} u_{i}-f u_{i}=0 \quad \text { in } \quad U_{i}; \quad \frac{\partial u_{i}}{\partial \vec{n}}=0 \quad \text { on } \quad \partial U_{i}.
$$
To estimete $u_{i}$, we first claim that $u_{i}$ is positive everywhere in $U_{i}$. Otherwise, the set $\Omega_{i,-}=$ $\left\{u_{i}<0\right\}$ is non-empty. Observe that $u_{i}$ is positive at the infinity of $\mathcal{E}$ due to \eqref{rough decay} and so $\Omega_{i,-}$ is compact. From integral by parts as well as the boundary condition of $u_{i}$, we have
$$
\int_{\Omega_{i,-}}\left|\nabla_{g} u_{i}\right|^{2} \,\mathrm{d} \mu_{g}=-\int_{\Omega_{i,-}} f u_{i}^{2}\, \mathrm{d} \mu_{g}.
$$
As before, we conclude from \eqref{sobelov on U} and \eqref{f_{-}control} that $u_{i}$ vanishes in $\Omega_{i,-}$, which is obviously impossible since the unique continuation leads to a contradiction. Now, we are able to apply the Harnack inequality \cite[Theorem 8.20]{GT2001} to obtain local smooth convergence of $u_{i}$. In fact, functions $u_{i}$ have a uniform bound in any compact subset of $M$ from \eqref{v_{i}estimate}. Combined with the Schauder estimates, it implies that $u_{i}$ converges smoothly to a nonnegative limit function $u$ up to a subsequence. Clearly $u$ solves the equation $\Delta_{g} u-f u=0$ in $M .$
\textbf{Step 2: the expansion of $u$ at the infinity of $\mathcal{E}$.}
Denote $v=u-1$. Then $v$ is harmonic when $r\geq r_0$. From \eqref{rough decay} we also have
\begin{equation*}
|v(x)|\leq C'r^{-(d-2-\epsilon)}, \quad r(x)\geq r_0.
\end{equation*}
Note that $\mathcal E$ is covered by $(\mathbf R^d-B)\times\mathbf R^k$. Lift $g$ to a smooth metric $\tilde{g}$ on $(\mathbf R^d-B)\times\mathbf R^k$ and $v$ to a function $\tilde v$ on $(\mathbf R^d-B)\times\mathbf R^k$. After applying \cite[Theorem 6.2]{GT2001} to the function $\tilde v$ in balls $B_r^{n+k}(x_0)\subset (\mathbf R^d-B)\times\mathbf R^k$ with $x_0=(a_0,b_0)$ and $|a_0|=r$, we see
$$
|\partial \tilde v|(x_{0})+r|\partial^{2} \tilde v|(x_{0})+r^2|\partial^{3} \tilde v|(x_{0})\leq Cr^{-(d-1-\epsilon)}
$$
for some constant $C$. Hence we have
\begin{equation}\label{decay of v}
|v(x)|+r|\partial v(x)|+r^{2}|\partial^{2} v(x)|+r^3|\partial^3 v(x)|=O(r^{-(d-2-\epsilon)}).
\end{equation}
Next we work with the metric $g_{0}=g_{\text {euc }}+g_{T^{k}}$ on $\mathcal{E}$. Let us consider the decomposition $v=\Pi_{0}v+\Pi_{1}v$ with
$$
\Pi_{0}v(a_{0},\cdot)=\frac{1}{|\vol(T^{k},g_{T^k})|}\int_{a_{0}\times T^{k}}v(a_{0},\theta)\,\mathrm{d}\theta\quad \text{and}\quad\Pi_{1}v=v-\Pi_{0}v,
$$
where $a_{0}\in \mathbf R^{d}-B$ and $\theta\in T^k$. Note that $\Pi_{0}v$ can be viewed as a function on $\mathbf R^{d}-B$. So we can compute
$$
\begin{aligned}
\Delta_{g_{\text {euc }}}\Pi_{0}v&=\Delta_{g_{0}}\frac{1}{|T^{k}|}\int_{a_{0}\times T^{k}}v(a_{0},\theta)d\theta\\&=\frac{1}{|T^{k}|}\int_{a\times T^{k}}\{(\delta^{ij}-g^{ij})\partial_{ij}v(a_{0},\theta)+g^{ij}\Gamma_{ij}^{k}\partial_{k}v(a_{0},\theta)\}d\theta\\
&=O(r^{-(d-\epsilon+\mu)})
\end{aligned}$$
and $\partial(\Delta_{g_{\text {euc}}}\Pi_{0}v)=O(r^{-(d-1-\epsilon+\mu)})$.
From \cite[Lemma 3.2]{SY79PMT} there exists a constant $A$ such that
$$\Pi_{0}v=\frac{A}{r^{d-2}}+\omega_{0},\quad\text{where}\quad|\omega_{0}|+r|\partial\omega_{0}|+r^{2}|\partial^{2}\omega_{0}|=O(r^{-(d-1)}).$$
From the definition of $\Pi_{1}v$, for each $a_0\in \mathbf R^d-B$ there always exists a $\theta_{0}\in T^k$ such that $\Pi_{1}v(a_{0},\theta_{0})=0$. For any $\theta\in T^k$ we take a geodesic curve $\gamma$ from $\theta_{0}$ to $\theta$ with respect to the given flat metric on $T^{k}$. From \eqref{decay of v} we have
$$|\Pi_{1}v(a_{0},\theta)|\leq\int_{\{a_{0}\}\times \gamma}\left|\frac{\partial v}{\partial s}\right|\,\mathrm{d}s=O(r^{-(d-1-\epsilon)}).$$
Notice that there always exists a vanishing point on $T^k$ for derivatives $\partial_{\theta_\alpha}v$ and $\partial^2_{\theta_\alpha\theta_\beta}v$. The same argument leads to
$$
r|\partial(\Pi_{1}v)(a_{0},\theta)|+r^2|\partial^2(\Pi_{1}v)(a_{0},\theta)|=O(r^{-(d-1-\epsilon)}).
$$
Denote $\omega=\omega_{0}+\Pi_{1}v$. Then we have
$$
u=1+\frac{A}{r^{d-2}}+\omega\quad,\quad|\omega|+r|\partial\omega|+r^{2}|\partial^2\omega|=O(r^{-(d-1-\epsilon)}).
$$
The expression \eqref{A} of $A$ follows from
integrating \eqref{main equation} on $U_{R}$ and taking $R\rightarrow\infty$.
\end{proof}
In the following, we will call $(M, \bar{g}, \mathcal{E})$ an {\it asymptotically Schwarzschild-like manifold with fiber $F$}, if it is an asymptotically flat manifold with fiber $F$ and there exist constants $A$ and $q >n-2$ such that
$$\bar g=\left(1+\frac{A}{r^{d-2}}\right)^{\frac{4}{d+k-2}}(g_{\text {euc }}+g_{F})+\omega,$$
where
$$|w|+r|\partial w|+r^{2}|\partial^{2}w|=O(r^{-q})$$
on the end $\mathcal{E}$. In this case, the total mass is expressed by
$$
m(M,\bar g,\mathcal E)=\frac{2(d+k-1)(d-2)}{d+k-2}A.
$$
\begin{proposition}\label{conformal end} Assume that $(M, g, \mathcal{E})$ is an asymptotically flat manifold with fiber $T^{k}$ with nonnegative scalar curvature and total mass $m$. Then for any $\tilde\epsilon>0$, we can construct a new complete metric $\bar{g}$ on $M$ such that $(M, \bar{g}, \mathcal{E})$ is an asymptotically Schwarzschild-like manifold with fiber $T^{k}$, which has nonnegative scalar curvature and mass $\bar{m}$ satisfying $|\bar{m}-m| \leq \tilde\epsilon$.
\end{proposition}
\begin{proof}First we can write the metric $g$ as
$$
g=\left(1+\frac{m_{1}}{r^{d-2}}\right)^{\frac{4}{d+k-2}} (g_{euc}+g_{T^{k}})+\tilde{g}$$
with
$$ m_1=\frac{d+k-2}{2(d+k-1)(d-2)}m.
$$
From Lemma \ref{defn of mass} we have
\begin{equation}\label{zeromasslimit}
\lim _{\rho \rightarrow+\infty}\int_{S_{\rho}\times T^{k}}(\partial_{i}\tilde{g}_{ij}-\partial_{j}\tilde{g}_{aa})\ast\mathrm{d}x^{j}\mathrm{d}\gamma=0.
\end{equation}
Take a fixed nonnegative cutoff function $\zeta: \mathbf{R} \rightarrow[0,1]$ such that $\zeta \equiv 0$ in $(-\infty, 2], \zeta \equiv 1$ in $[3,+\infty)$. With $s>1$ is a constant to be determined later, we define
$$
\hat{g}^{s}=\left(1+\frac{m_{1}}{r^{d-2}}\right)^{\frac{4}{d+k-2}} (g_{euc}+g_{T^{k}})+\left(1-\zeta\left(\frac{r}{s}\right)\right) \tilde{g}.
$$
Without loss of generality, we can assume that the asymptotic order in \eqref{Eq: decay 1} satisfies $\mu\leq d-2$. Clearly we have
$$
\left|\hat{g}^{s}-(g_{euc}+g_{T^{k}})\right|+r\left|\partial \hat{g}^{s}\right|+r^{2}\left|\partial ^{2}\hat{g}^{s}\right| \leq C r^{-\mu},
$$
where $C$ is always denoted to be a universal constant independent of $s$ here and in the sequel. In particular, the metrics $\hat{g}^{s}$ are uniformly equivalent to $g$ in some neighborhood $U$ of $\mathcal{E}$, and so the Sobolev inequality \eqref{sobelov on U} holds for all metrics $\hat{g}^{s}$ with a unifrom constant $C_{U}$ independent of $s$. It is easy to see that
$$R\left(\hat{g}^{s}\right) \geq 0 \quad \text{in}\quad \{r \leq 2 s\}; \quad R\left(\hat{g}^{s}\right) \equiv 0 \quad \text{in}\quad \{r \geq 3 s\} $$
and that $\left|R\left(\hat{g}^{s}\right)\right| \leq C s^{-(\mu+2)}$ in $ \{s \leq r \leq 4 s\}$. Then we have
\begin{equation}\label{R integral estimate}
\left(\int_{\{s \leq r \leq 4 s\}}\left|R\left(\hat{g}^{s}\right)\right|^{\frac{2 (d+k)}{d+k+2}}r^{\frac{2k(d+k-2)}{(d+k+2)^{2}}} \mathrm{~d} \mu_{\hat{g}^{s}}\right)^{\frac{d+k+2}{2 (d+k)}} \leq Cs^{-\frac{4k}{(d+k)(d+k+2)}}
\end{equation}
and
\begin{equation}\label{R_{-}integral}
\left(\int_{U}\left|R\left(\hat{g}^{s}\right)_{-}\right|^{\frac{d+k}{2}} r^{k}\mathrm{~d} \mu_{g}\right)^{\frac{2}{d+k}}\leq C s^{-\frac{d-2}{2}}\leq \frac{1}{4C_{U}^{2}}
\end{equation}
for sufficiently large $s$.
Now we take another nonnegative cutoff function $\eta: \mathbf{R} \rightarrow[0,1]$ such that $\eta \equiv 0$ in $(-\infty, 1] \cup[4,+\infty)$ and $\eta \equiv 1$ in $[2,3] .$ Let $\eta_{s}(x)=\eta(r(x) / s) .$ It follows from \eqref{R_{-}integral}that for any $s$ we can take a small constant $\delta_{s}>0$ such that
$$
\left(\int_{U}\left|\left(\eta_{s} R\left(\hat{g}^{s}\right)- \eta_{s}\delta_{s}\right)_{-}\right|^{\frac{d+k}{2}}r^{k}\, \mathrm{d} \mu_{\hat{g}^{s}}\right)^{\frac{2}{d+k}} \leq \frac{1}{2C_{U}^{2}}.
$$
From \eqref{R integral estimate} and taking $\delta_{s}$ small enough, we also have
\begin{equation}
\left(\int_{U}|\eta_{s} R\left(\hat{g}^{s}\right)- \eta_{s}\delta_{s}|^{\frac{2 (d+k)}{d+k+2}} r^{\frac{2k(d+k-2)}{(d+k+2)^{2}}} \,\mathrm{d} \mu_{g}\right)^{\frac{d+k+2}{2(d+k)}} \leq Cs^{-\frac{4k}{(d+k)(d+k+2)}}
\end{equation}
Based on Proposition \ref{3.2likeSY} we can construct a solution $u_{s}$ of the following equation
$$
\Delta_{\hat{g}^{s}} u_{s}-C_{d,k}\left(\eta_{s} R\left(\hat{g}^{s}\right)-\delta_{s} \eta_{s}\right) u_{s}=0,\quad C_{d,k}=\frac{d+k-2}{4(d+k-1)}.
$$
Moreover, $u_{s}$ has the expansion $u_{s}=1+A_{s}r^{-(d-2)}+O(r^{-(d-1-\epsilon)})$ for any small $\epsilon>0$, where
$$
A_{s}=-\frac{C_{d,k}}{(d-2)\left|\mathbf{S}^{d-1}\right||T^{k}|} \int_{U}\left(\eta_{s} R\left(\hat{g}^{s}\right)-\eta_{s}\delta_{s} \right) u_{s} \,\mathrm{d} \mu_{\hat{g}^{s}}
$$
We now show that $A_{s}$ is small if $s$ is large. To prove this, we consider the function $v_{s}=u_{s}-1$. From H\"{o}lder inequality, we have
\begin{equation}\label{A_{s}term1}
\begin{aligned}
\left|\int_{U} \eta_{s} R\left(\hat{g}^{s}\right) u_{s} \,\mathrm{d} \mu_{\hat{g}^{s}}\right|
&\leq\left(\int_{\{s \leq r\leq 4 s\}}\left|R\left(\hat{g}^{s}\right)\right|^{\frac{2(d+k)}{d+k+2}}r^{\frac{2k(d+k-2)}{(d+k+2)^{2}}} \,\mathrm{d} \mu_{\hat{g}^{s}}\right)^{\frac{d+k+2}{2 (d+k)}}\\
&\cdot\left(\int_{\{s \leq r \leq 4 s\}}\left|v_{s}\right|^{\frac{2 (d+k)}{d+k-2}} r^{\frac{-2k}{d+k-2}}\,\mathrm{d} \mu_{\hat{g}^{s}}\right)^{\frac{d+k-2}{2 (d+k)}} \\
&\qquad+\left|\int_{\{s \leq r \leq 4 s\}} \eta_{s} R\left(\hat{g}^{s}\right) \mathrm{d} \mu_{\hat{g}^{s}}\right|
\end{aligned}
\end{equation}
and
\begin{equation}\label{A_{s}term2}
\begin{split}
\left|\int_{U} \delta_{s} \eta_{s} u_{s} \,\mathrm{d} \mu_{\hat{g}_{s}}\right| &\leq C\delta_{s}s^{d}
+C\delta_{s}s^{\frac{2k(d+k-2)+d(d+k+2)^{2}}{2(d+k)(d+k+2)}}\\
&\cdot\left(\int_{\{s \leq r \leq 4 s\}}\left|v_{s}\right|^{\frac{2 (d+k)}{d+k-2}}r^{\frac{-2k}{d+k-2}}\, \mathrm{d} \mu_{\hat{g}^{s}}\right)^{\frac{d+k-2}{2 (d+k)}}.
\end{split}
\end{equation}
Recall from the proof of \eqref{estimate of v_{i,R}}, we have
$$
\begin{aligned}
&\left(\int_{\{s \leq r \leq 4 s\}}\left|v_{s}\right|^{\frac{2 (d+k)}{d+k-2}} r^{\frac{-2k}{d+k-2}}\,\mathrm{d} \mu_{\hat{g}^{s}}\right)^{\frac{d+k-2}{2 (d+k)}}\\ \leq& 2 C_{U}^{2}\left(\int_{U}\left|\eta_{s} R\left(\hat{g}^{s}\right)-\eta_{s}\delta_{s} \right|^{\frac{2 (d+k)}{d+k+2}}r^{\frac{2k(d+k-2)}{(d+k+2)^{2}}} \,\mathrm{d} \mu_{\hat{g}^{s}}\right)^{\frac{d+k+2}{2 (d+k)}}\\
\leq& 2 C_{U}^{2}Cs^{-\frac{4k}{(d+k)(d+k+2)}} \rightarrow 0, \quad \text{as }\quad s\rightarrow \infty.
\end{aligned}
$$
Hence if we choose $\delta_{s}$ small enough, we see that the left hand side of \eqref{A_{s}term2} converges to zero as $s\rightarrow \infty$. Due to \eqref{R integral estimate}, we only need to estimate the left hand side of \eqref{A_{s}term1} to conclude $A_s\to 0$ as $s\to+\infty$.
Recall the divergence structure of scalar curvature up to higher order error (see \cite{Bartnik1986}). That is, with the coordinate system from Lemma \ref{defn of mass} we have
$$R(g)=|g|^{-\frac{1}{2}}\partial_{a}(g_{ab,b}-g_{bb,a})+O(r^{-(2+2\mu)}).$$
For the nonnegativity of $R\left(\hat{g}^{s}\right)$ outside $\{2 s \leq r \leq 3 s\}$, we conclude that there are two possibilities for the left hand side of \eqref{A_{s}term1}:
(i) if the integral of $(\eta_{s} R)_{+}$ larger than integral of $(\eta_{s} R)_{-}$, then
$$
\begin{aligned}
\left|\int_{\{s \leq r \leq 4 s\}} \eta_{s} R\left(\hat{g}^{s}\right) \mathrm{d} \mu_{\hat{g}^{s}}\right|&\leq
\left|\int_{\{s\leq r\leq 4s\}} R\left(\hat{g}^{s}\right) \mathrm{d} \mu_{\hat{g}^{s}}\right|\\
&\leq \left|\int_{\{r=4s\}}(\hat{g}^{s}_{ij,j}-\hat{g}^{s}_{aa,i})\ast\mathrm{d}x^{j}\mathrm{d}\gamma\right.\\
&\qquad-\left.\int_{\{r=s\}}(\hat{g}^{s}_{ij,j}-\hat{g}^{s}_{aa,i})\ast\mathrm{d}x^{j}\mathrm{d}\gamma\right|+ o(1).
\end{aligned}
$$
(ii) if the integral of $(\eta_{s} R)_{+}$ not larger than integral of $(\eta_{s} R)_{-}$, then
$$
\begin{aligned}
\left|\int_{\{s \leq r \leq 4 s\}} \eta_{s} R\left(\hat{g}^{s}\right) \mathrm{d} \mu_{\hat{g}^{s}}\right|&\leq
\left|\int_{\{2s\leq r\leq 3s\}} R\left(\hat{g}^{s}\right) \mathrm{d} \mu_{\hat{g}^{s}}\right|\\
&\leq \left|\int_{r=3s}(\hat{g}^{s}_{ij,j}-\hat{g}^{s}_{aa,i})\ast\mathrm{d}x^{j}\mathrm{d}\gamma\right.
\\
&-\left.\int_{r=2s}(\hat{g}^{s}_{ij,j}-\hat{g}^{s}_{aa,i})\ast\mathrm{d}x^{j}\mathrm{d}\gamma\right|+ o(1).
\end{aligned}
$$
Here the indices are used in the same way as in Lemma \ref{defn of mass}. In both cases, it follows from \eqref{zeromasslimit} and the definition of the total mass that
$$
\left|\int_{\{s \leq r \leq 4 s\}} \eta_{s} R\left(\hat{g}^{s}\right) \mathrm{d} \mu_{\hat{g}^{s}}\right|=o(1), \quad \text { as } \quad s \rightarrow \infty.
$$
Finally we conclude
$$
\left|\int_{U} \eta_{s} R\left(\hat{g}^{s}\right) u_{s} \mathrm{~d} \mu_{\hat{g}^{s}}\right|+\left|\int_{U} \delta_{s} \eta_{s} u_{s} \mathrm{~d} \mu_{\hat{g}_{s}}\right|=o(1), \quad \text { as } \quad s \rightarrow \infty.
$$
By taking $s$ large enough, we can guarantee $\left|A_{s}\right| \leq \frac{d+k-2}{2(d-2)(n-1+k)}\tilde\epsilon$ for any given $\tilde\epsilon>0$. Fix such an $s$ below.
Take
$$
u_{s, \tau}=\frac{u_{s}+\tau}{1+\tau}
$$
with $\tau$ a positive constant to be determined later. We consider the conformal deformation $\bar{g}=\left(u_{s, \tau}\right)^{\frac{4}{d+k-2}} \hat{g}^{s}$. A straightforward computation gives
$$
\begin{aligned}
R(\bar{g}) &=\left(u_{s, \tau}\right)^{-\frac{d+k+2}{d+k-2}}\left(-C_{d,k} \Delta_{\hat{g}^{s}} u_{s, \tau}+R\left(\hat{g}^{s}\right) u_{s, \tau}\right) \\
&=\left(u_{s, \tau}\right)^{-\frac{d+k+2}{d+k-2}}(1+\tau)^{-1}\left(\left(\left(1-\eta_{s}\right) R\left(\hat{g}^{s}\right)+\delta_{s} \eta_{s}\right) u_{s}+R\left(\hat{g}^{s}\right) \tau\right)
\end{aligned}
$$
Note that $R\left(\hat{g}^{s}\right)$ takes possible negative values only in $\{2 s \leq r \leq 3 s\}$, where the term $\left(\left(1-\eta_{s}\right) R\left(\hat{g}^{s}\right)+\delta_{s} \eta_{s}\right) u_{s}$ has a positive lower bound, so we can take $\tau$ small enough such that $R(\bar{g})$ is nonnegative everywhere. Since the function $u_{s, \tau}$ is no less than $\tau(1+\tau)^{-1}$, the metric $\bar{g}$ is still complete. In the region $\{r \geq 3 s\}$, the metric $\bar{g}$ can be expressed as
$$
\bar{g}=\left(u_{s, \tau}\right)^{\frac{4}{d+k-2}}\left(1+\frac{m_{1}}{r^{d-2}}\right)^{\frac{4}{d+k-2}} (g_{euc}+g_{T^{k}})=\left(1+\frac{m_{2}}{r^{d-2}}\right)^{\frac{4}{d+k-2}}(g_{euc}+g_{T^{k}})+w,
$$
where $|w|+r|\partial w|+r^{2}|\partial^{2}w|=O(r^{-(d-1-\epsilon)})$ for any small $\epsilon>0$ and $m_{2}=m_{1}+{A_{s}}({1+\tau})^{-1}$. So we have
$$|m(M,g,\mathcal E)-m(M,\bar{g},\mathcal E)|=\frac{2(d-2)(d-1+k)}{(d+k-2)(1+\tau)}|A_{s}|\leq \tilde\epsilon.$$
This competes the proof.
\end{proof}
\begin{proof}[Proof for Theorem \ref{Thm: main 4}]
For the inequality part, we argue by contradiction. Suppose that $(M, g, \mathcal{E})$ has negative total mass $m$. Set $\tilde\epsilon=-m / 2$. From Proposition \ref{conformal end} above we can construct a new complete metric $\bar{g}$ on $M$ such that $(M, \bar{g}, \mathcal{E})$ is an asymptotically Schwarzschild-like manifold with fiber $T^k$, which has nonnegative scalar curvature and total mass $\bar{m}$ no greater than $-m / 2 .$ Moreover, the metric $\bar{g}$ can be expressed as $\bar{g}=u^{\frac{4}{d+k-2}}(g_{euc}+g_{T^{k}})$ near the infinity, which is also scalar flat. Hence the function $u$ is a harmonic function near infinity with respect to the metric $g_{euc}+g_{T^{k}}$. Notice that $u$ has the expansion
$$
u=1+\frac{m_{2}}{r^{d-2}}+O(r^{-(d-1-\epsilon)}), \quad m_{2}<0,
$$
for any small $\epsilon>0$ be small, so we can take $s_{1}$ large enough such that $u<1$ on $\left\{r=s_{1}\right\}$. Denote
$$
\bar\epsilon
=1-\sup _{r(x)=s_{1}} u(x)
$$
It is clear that $u>1-\bar\epsilon / 4$ in $\left\{r \geq s_{2}\right\}$ for sufficiently large $s_{2}>s_{1}$. Take a cutoff function $\zeta:[0,+\infty) \rightarrow[0,1-\bar\epsilon / 2]$ such that $\zeta(t)=t$ when $t \leq 1-3 \bar\epsilon / 4$ and $\zeta(t)=1-\bar\epsilon / 2$ when $t \geq 1-\bar\epsilon / 4$. Moreover, we can also require $\zeta^{\prime} \geq 0$ and $\zeta^{\prime \prime} \leq 0$ in $[0,+\infty)$ as well as $\zeta^{\prime \prime}<0$ in $(1-3 \bar\epsilon/ 4,1-\bar\epsilon / 4)$. Let
$$
v=\left\{\begin{array}{cc}
\zeta \circ u, & r \geq s_{1} \\
u & r \leq s_{1}
\end{array}\right.
$$
Clearly $v$ is a smooth function defined on entire $M$ since $v$ equals to $u$ around $\left\{r=s_{1}\right\}$. A direct computation shows
$$
\Delta v=\zeta^{\prime \prime}|\nabla u|^{2}+\zeta^{\prime} \Delta u \leq 0 \quad \text { in } \quad\left\{r \geq s_{1}\right\}
$$
and further $\Delta v<0$ at some point in $\left\{s_{1}<r<s_{2}\right\}$, where $\Delta$ and $\nabla$ are Laplace and gradient operators with respect to the metric $g_{euc}+g_{T^{k}}$. Define
$$
\tilde{g}=\left(\frac{v}{u}\right)^{\frac{4}{d+k-2}} \bar{g}
$$
It is easy to verify that $\tilde{g}$ is a complete metric on $M$ with nonnegative scalar curvature (positive somewhere), which is exactly the metric $g_{euc}+g_{T^{k}}$ near the infinity of $\mathcal{E}$. In particular, we can close $(M, \tilde{g})$ around the infinity of $\mathcal{E}$ to construct a smooth complete metric on $T^{n} \#_{T^k} \bar M$ with nonnegative scalar curvature (positive somewhere) for some compactification $\bar M$ of $M$ by adding an inifnity $T^k$ to $M$. From our topological assumption such manifold cannot admit any complete metric with positive scalar curvature (see a further explanation in the proof of Theorem \ref{Thm: main 5} in next subsection) and we obtain a contradiction.
Now let us focus on the rigidity part under the assumption that the total mass of $(M, g, \mathcal{E})$ is zero. First we show that the scalar curvature of $g$ has to vanish everywhere . Otherwise, we can take a neighborhood $U$ of $\mathcal{E}$ such that $R(g)$ is positive at some point $p$ in $U$. Take a nonnegative cutoff function $\eta: M \rightarrow[0,1]$ with small compact support in $U$ such that $\eta \equiv 1$ around
point $p$. It follows from Proposition \ref{3.2likeSY} that there is a positive function $u$ solving
$$
\Delta_{g} u-C_{d,k} \eta R(g) u=0,
$$
which has the expansion $u=1+A r^{-(d-2)}+O\left(r^{-(d-1-\epsilon)}\right)$ with $A<0$.
Define
$$
\bar{g}=\left(\frac{u+1}{2}\right)^{\frac{4}{d+k-2}} g
$$
It is not difficult to verify that $(M, \bar{g}, \mathcal{E})$ is an asymptotically flat manifold with fiber $T^k$, which has nonnegative scalar curvature but negative total mass.
Next we prove the Ricci flatness of the metric $g$. Otherwise we take a neighborhood $U$ of $\mathcal{E}$ such that $\operatorname{Ric}(g)$ does not vanish at some point $p$ in $U$. Take a nonnegative cutoff function $\eta: M \rightarrow[0,1]$ with compact support $S$ in $U$ such that $\eta \equiv 1$ around point $p$. After applying a variation argument to the first Neumann eigenvalue of conformal Laplace operator \cite[Lemma 3.3]{Kazdan82}, we can pick up a positive constant $\varepsilon$ small enough such that the metric $\bar{g}=g-\varepsilon \eta \operatorname{Ric}(g)$ satisfies
\begin{equation}\label{kazdan}\int_{S}\left|\nabla_{\bar{g}} \zeta\right|^{2}+C_{d,k} R(\bar{g}) \zeta^{2} \mathrm{~d} \mu_{\bar{g}}>c \int_{S} \zeta^{2} \mathrm{~d} \mu_{\bar{g}}, \quad \forall \zeta \neq 0 \in C^{\infty}(S),
\end{equation}
for some positive constant $c$. For $\varepsilon$ small, we still have a uniform Sobolev constant of $U$ with respect to the metric $\bar{g}$, still denoted by $C_{U}$. Since $\bar{g}$ is almost equal to $g$, we can also require
$$
\left(\int_{U}\left|(R(\bar{g}))_{-}\right|^{\frac{d+k}{2}} r^{k}\,\mathrm{d} \mu_{\bar{g}}\right)^{\frac{2}{d+k}} \leq \frac{1}{4C_{U}^{2}}
$$
by further decreasing the value of $\varepsilon$. Let $\tilde{\eta}$ be another nonnegative cutoff function with compact support $\tilde{S}$ in $U$, which has a positive lower bound in the support of $\eta$. For $\delta$ be a constant small enough, we can construct a smooth positive function $u$ from Proposition \ref{3.2likeSY} solving
$$
\Delta_{\bar{g}} u-C_{d,k}(R(\bar{g})-\delta \tilde{\eta}) u=0
$$
The function $u$ has the expansion $u=1+A r^{-(d-2)}+O\left(r^{-(d-1-\epsilon)}\right)$ for any $\epsilon>0$ be small, and we expect $A$ to be negative when $\delta$ is sufficiently small. From integral by parts we have
$$
\begin{aligned}
(d-2)\left|\mathbf{S}^{d-1}\right| \vol(T^k,g_{T^k})A &=-\left(\int_{U}\left|\nabla_{\bar{g}} u\right|^{2}+C_{d,k}(R(\bar{g})-\delta \tilde{\eta}) u^{2} \,\mathrm{d} \mu_{\bar{g}}\right) \\
& \leq-\left(\int_{U}\left|\nabla_{\bar{g}} u\right|^{2}+C_{d,k} R(\bar{g}) u^{2}\, \mathrm{d} \mu_{\bar{g}}\right)+C_{d,k}\delta \int_{\tilde{S}} u^{2} \,\mathrm{d} \mu_{\bar{g}}.
\end{aligned}
$$
From \eqref{kazdan} and the fact $R(\bar{g}) \equiv 0$ outside $\tilde{S}$, it follows from \cite[Lemma 2.1]{Kazdan82} that
$$
\int_{U}\left|\nabla_{\bar{g}} \zeta\right|^{2}+C_{d,k} R(\bar{g}) \zeta^{2} \mathrm{~d} \mu_{\bar{g}}>\tilde{c} \int_{\tilde{S}} \zeta^{2} \mathrm{~d} \mu_{\bar{g}}, \quad \forall \zeta \neq 0 \in C^{\infty}(U),
$$
for some positive constant $\tilde{c}$ independent of $\delta$. So we can guarantee $A<0$ by taking $\delta$ small enough. As before, we consider the conformal metric
$$
\tilde{g}=\left(\frac{u+\tau}{1+\tau}\right)^{\frac{4}{d+k-2}} \bar{g}
$$
with $\tau$ a positive constant to be determined later. A straightfrward computation shows
$$
R(\tilde{g})=\left(\frac{u+\tau}{1+\tau}\right)^{-\frac{d+k+2}{d+k-2}}(1+\tau)^{-1}(\delta \tilde{\eta} u+\tau R(\bar{g})).
$$
In particular, we have $R(\tilde{g}) \geq 0$ if $\tau$ is chosen to be small enough. However, it is easy to verify that $(M, \tilde{g}, \mathcal{E})$ is an asymptotically flat manifold with fiber $T^k$ whose total mass is negative, which contradicts to the discussion above.
Now we can deduce that $M$ has only one end $\mathcal{E}$. Otherwise, the Cheeger-Gromoll splitting theorem (see \cite{CG1971}) yields that $(M, g)$ must be isometric to a Riemannian product manifold $N \times \mathbf{R}$ for a closed manifold $N$. In particular, each $N$-slice is totally geodesic. However, this is impossible since if such slice appears at the infinity of the asymptotically flat end $\mathcal{E}$, we can take a $S_r\times T^{k}$ touching this slice from outside, which contradicts to the maximum principle. Consider the universal covering $(\hat M,\hat g)$.
It remains to show the flatness of $(M,g)$ and the discussion will be divided into two cases. If $k=\dim F\geq 2$, then we consider the universal covering $(\hat M,\hat g)$ of $(M,g)$. From the incompressible condition $(\hat M,\hat g)$ is a complete Ricci-flat manifold with one of its end diffeomorphic to $(\mathbf R^d-B)\times \mathbf R^k$, where the metric is asymptotic to the Euclidean metric at infinity. The standard volume comparison theorem tells us that $(M,g)$ must be flat.
In the case when $F=\mathbf S^1$, the lifting of $(M,g)$ may not lift the end $\mathcal E$ entirely to $(\mathbf R^d-B)\times \mathbf R$ and so the argument above cannot be applied. Instead we are going to use the Bochner method with a nice observation formulated as the following Proposition \ref{Ricci decay}. From Proposition \ref{Ricci decay} we see that there exist $d$ $g$-parallel one-forms $\alpha^{1},\cdots,\alpha^{d}$, which further yields that $(M,g)$ splits as the standard $\mathbf R^{d}\times \mathbf S^{1}$ (see for instance \cite[Page 945]{Minerbe2008}).
\end{proof}
\begin{proposition}\label{Ricci decay}
If $(M,g,\mathcal E)$ is an asymptotically flat manifold with fiber $F=S^{1}$ and the unique end $\mathcal E$ such that
\begin{equation}\label{Eq: metric decay}
\sum\limits^3_{k=0}r^k|\nabla_{g_0}^k (g-g_{0})|_{g_0}=O(r^{-\mu}), \quad\mu>\frac{d-2}{2},
\end{equation}
and
\begin{equation}\label{Eq: Ricci decay}
|\nabla_{g_0} Ric|_{g_0}+r|\nabla_{g_0} Ric|_{g_0}=O(r^{-n-\epsilon})\quad \text{for some}\quad\epsilon>0 .
\end{equation}
Let $\{x^{1},\cdots,x^{d},\theta\}$ be the given coordinate system of $(M,g,\mathcal E)$ at infinity, where $\theta$ is the parameter of $S^{1}$. Then there exists a coordinate system $\{y^{1},\cdots,y^{d},\theta\}$ outside a compact set such that
$\Delta_{g}y^{i}=0,\,i=1,\cdots,d$.
Moreover, if we denote $\alpha^{i}=dy^{i}$, then we have
\begin{equation}\label{Eq: gb mass}
\sum\limits^{d}_{i=1}\int_{M}|\nabla_g\alpha^{i}|_g^{2}+\Ric(\alpha^{i},\alpha^{i})\mathrm{d}V_{g}=c(d,S^{1})m(M,g,\mathcal{E}),
\end{equation}
where $c(d,S^{1})=d|\mathbf S^{d-1}|\vol(S^{1},g_{S^{1}})$.
\end{proposition}
\begin{proof}
The proof is based on the theory of weighted spaces established in \cite{Minerbe2008} and here we just sketch the analysis part but emphasize on the difference. Without loss of generality, we can assume $\mu\leq d-2$. As in \cite{Minerbe2008}, to find desired functions $y^i$, we try to solve the equation $\Delta_g u^i=\Delta_g(\chi x^i)$ for some suitable cut-off function $\chi$ and then take $y^i=\chi x^i-u^i$.
From the metric decay \eqref{Eq: metric decay}, we compute $\Delta_gx^i=O(r^{-\mu-1})$ and this yields $\Delta_{g}(\chi x^{i})\in L_{\delta'-2}^{2}(M)$ for any $\delta'>1+\frac{d}{2}-\mu$.
It then follows from \cite[Corollary 2]{Minerbe2008} that there exists a function $u^{i}\in H_{\delta'}^{2}(M)$ such that $\Delta_{g}u^{i}=\Delta_{g}(\chi x^{i})$. The Moser iteration (see \cite[Page 942, (23)]{Minerbe2008}) yieds
$$|u^{i}|= O(r^{1-\mu'}), \quad\mu'= \min\left\{\frac{d}{2}-\delta'+1,\mu\right\}.$$
Fix $\epsilon_{1}>0$ small enough such that if we take $\delta'=1+\frac{d}{2}-\mu+\epsilon_{1}$, then $\mu'=\mu-\epsilon_{1}>\frac{d-2}{2}$. Similar as in step 2 of the proof of Proposition \ref{3.2likeSY}, the Schauder estimate yields
$$r|\partial u^i|+r^{2}|\partial^{2} u^i|= O(r^{1-\mu'})$$
With a similar analysis on the derivative of $\Delta_{g}u^{i}=\Delta_{g}x^{i}$, we further obtain
$$r^{3}|\partial^{3}u^{i}|= O(r^{1-\mu'}),$$
here and above the partial derivative is taken with respect to the coordinate system $\{x^{1},\cdots,x^{d},\theta\}$.
Now let us consider $y^{i}=\chi x^{i}-u^{i}$. These estimates imply that $\{y^{1},\cdots,y^{d},\theta\}$ form a coordinate system outside a large compact subset. A direct calculation shows that with respec to the coordinate system $\{y^{1},\cdots,y^{d},\theta\}$ the metric $g$ has the form
$
g=g_{0}'+\omega,
$
where $g_{0}'=dy^{2}\oplus g_{S^{1}}$ and $\omega$ satisfies
$$\sum\limits^2_{k=0}r^k|\nabla_{g'_0}^k \omega|_{g_0'}=O(r^{-\mu}), \quad\mu>\frac{d-2}{2}$$
In the following, we work in this new coordinate system and the partial derivative will be taken with respect to $\{y^{1},\cdots,y^{d},\theta\}$. In particular, $\partial_i$ means $\partial_{y_i}$. Let us also denote $g_{ij}=g(\frac{\partial}{\partial y^{i}},\frac{\partial}{\partial y^{j}})$ and $r=|y|$.
It remains to show \eqref{Eq: gb mass}. From the Bochner formula as well as $\Delta_{g}y^{i}=0$, we have
\begin{equation}\label{Bochner0}
\Delta_g\left(\frac{1}{2}|dy^{i}|_g^{2}\right)=|\nabla_g dy^{i}|_g^{2}+\Ric(dy^{i},dy^{i})
\end{equation}
and
$$
\frac{1}{2}\Delta_{g}g^{ij}=g(\nabla_g dy^{i},\nabla_g dy^{j})+\Ric(dy^{i},dy^{j}).
$$
From the Ricci decay \eqref{Eq: Ricci decay} we see $\Delta_{g}(g^{ij}-\delta_{ij})=O(r^{-d-\epsilon_{2}})$, where $\epsilon_{2}=\min\{\epsilon,2\mu'+2-d\}>0$. From above discussion we conclude that
$$g^{ij}-\delta_{ij}\in L^{2}_{\delta}(M) \quad\text{for any}\quad \delta>\frac{d}{2}-\mu',$$
and
$$\Delta_{g}(g^{ij}-\delta_{ij})\in L^{2}_{\delta''-2}(M) \quad\text{for any}\quad\delta''>2-\frac{d}{2}-\epsilon_{2}.$$
It follows from \cite[Proposition 4]{Minerbe2008} and \cite[Proposition 4]{Minerbe2008} that
if we can choose $\delta$ and $\delta''$ such that
\begin{equation}\label{Eq: order picking}
1-\frac{d}{2}\leq\delta''<2-\frac{d}{2}<\delta\leq \frac{d}{2},
\end{equation}
then we have the expansion
$$g^{ij}=\delta_{ij}-c_{ij}r^{2-d}+v^{ij},$$
where $c_{ij}$ are constants with $c_{ij}=c_{ji}$ and the functions $v^{ij}$ belongs to $ H^{2}_{\delta''}(M)$. Recall that $\mu'\leq \mu\leq d-2$ and $\delta''$ can be chosen to be arbitrarily close to $2-\frac{d}{2}$, the inequality \eqref{Eq: order picking} is fairly easy to hold. After repeating the argument in estimates for $u^i$, we see that $v^{ij}$ are higher-order error terms satisfying
$$\sum_{k=0}^2r^k|\partial^kv^{ij}|=O(r^{2-d-\epsilon_{3}}),\quad\text{where}\quad \epsilon_{3}=2-\frac{d}{2}-\delta''>0.$$
After a possible orthogonal transformation of $\{y^{1},\cdots,y^{d}\}$, we can assume $c_{ij}=c_{i}\delta_{ij}$ without loss of generality. Finally we have
$$
g_{ij}=\delta_{ij}+c_{i}\delta_{ij}r^{2-d}+w_{ij},\quad g_{i\theta}=w_{i\theta},\quad g_{\theta\theta}=1+w_{\theta\theta},
$$
such that
$$\sum\limits^2_{k=0}r^{k}|\partial^{k}w_{ij}|=O(r^{2-d-\epsilon_{3}})$$
and
$$ \sum\limits^2_{k=0}r^{k}|\partial^{k}w_{i\theta}|+r^{k}|\partial^{k}w_{\theta\theta}|=O(r^{-\mu'}).$$
It follows from \cite[Theorem 4.2]{Bartnik1986} that the total mass $m(M,g,\mathcal E)$ calculated with respect to $\{y^{1},\cdots,y^{d},\theta\}$ is the same as that with respect to $\{x^{1},\cdots,x^{d},\theta\}$. Recall that we have $\Delta_{g}y^{i}=0$. A straightforward computation yields
\begin{equation}\label{theta expression}
(d-2)\left(c_{i}-\frac{1}{2}\mathop{\Sigma}\limits_{j=1}^{d}c_{j}\right)\frac{y^{i}}{|y|^{d}}+\partial_{\theta}g_{i\theta}+\frac{1}{2}\partial_{i}g_{\theta\theta}+O(r^{-\mu''})=0,
\end{equation}
where $\mu''=\min\{n-1+\epsilon_{3},2\mu'+1\}>d-1$.
Integrating \eqref{theta expression} and suming over $i$, we obtain
$$\lim_{\rho\to+\infty}\int_{S_\rho\times S^{1}}(-\partial_{i}g_{\theta\theta})\ast\mathrm{d}y^{i}\mathrm{d}\theta=\lim_{\rho\to+\infty}\int_{S_\rho\times S^{1}}-\frac{(d-2)^{2}}{d}\left(\sum\limits_{i=1}^{d}c_{i}\right)\rho^{1-d}\mathrm{d}S_{\rho}\mathrm{d}\theta.$$
Therefore we have
\begin{displaymath}
\begin{aligned}
m(M,g,\mathcal E)&=\frac{1}{2|\mathbf S^{d-1}|\vol(S^{1},g_{S^{1}})}\lim_{\rho\to+\infty}\int_{S_\rho\times S^{1}}(\partial_{j}g_{ij}-\partial_{i}g_{aa})\ast\mathrm{d}y^{i}\mathrm{d}\theta\\
&=\frac{1}{d-1}\cdot\frac{1}{2|\mathbf S^{d-1}|\vol(S^{1},g_{S^{1}})}\lim_{\rho\to+\infty}\int_{S_\rho\times S^{1}}(\partial_{j}g_{ij}-\partial_{i}g_{jj})\ast\mathrm{d}y^{i}\mathrm{d}\theta.
\end{aligned}
\end{displaymath}
Now let us integrate both sides of \eqref{Bochner0} and take the sum over $i$. This gives
\begin{displaymath}
\begin{aligned}
\sum\limits^{d}_{i=1}\int_{M}|\nabla_g\alpha^{i}|_g^{2}+\Ric(\alpha^{i},\alpha^{i})\mathrm{d}V_{g}&=\sum\limits^{d}_{i=1}\lim_{\rho\to+\infty}\int_{B_\rho\times S^{1}}\frac{1}{2}\Delta_g(|dy^{i}|_g^{2})\mathrm{d}V_{g}\\
&=\sum\limits^{d}_{i=1}\lim_{\rho\to+\infty}\int_{S_\rho\times S^{1}}\frac{1}{2}\partial_{j}g^{ii}\ast \mathrm{d}y^{i}\mathrm{d}\theta\\
&=\frac{d-2}{2}\left(\sum\limits^{d}_{i=1}c_{i}\right)|\mathbf S^{d-1}|\vol(S^{1},g_{S^{1}})\\
&=d\cdot|\mathbf S^{d-1}|\vol(S^{1},g_{S^{1}}) m(M,g,\mathcal E),
\end{aligned}
\end{displaymath}
and we complete the proof.
\end{proof}
\subsection{Compactification with quasi-spherical metric and Proof for Theorem \ref{Thm: main 5}} For asymptotically conical manifolds with fiber $F$, Lohkamp compactification based on the conformal deformation is no longer valid since the expansion of a harmonic function on $\mathbf R^2$ is rather difficult due to its blow-up at infinity. As an alternative, we will use the quasi-spherical metric to complete the compactification. Quasi-spherical metric was first introduced by Bartnik in \cite{Bartnik1993} and later used by the third named author and Tam \cite{ST2002} to prove the nonnegativity of the Brown-York mass. Its use in the proof of positive mass theorems (angle estimates) is a new observation in this paper.
Let $(M,g)$ be an asymptotically conical manifold with fiber $T^{n-2}$. Notice that the end $\mathcal E$ (diffeomorphic to $(\mathbf R^2-B)\times T^{n-2}$) can be compactified to be $(\mathbf S^2-B)\times T^{n-2}$ by adding a $T^{n-2}$ at infinity. So we can denote $\bar M$ to be the corresponding compactification of $M$ and $i_2:T^{n-2}\to \bar M$ to be the embedding mapping onto the infinity $T^{n-2}$ in $\bar M$. Let us start with the following
\begin{proposition}\label{Prop: PMT to PSC 2}
If the asymptotically conical manifold $(M,g,\mathcal E)$ with fiber $T^{n-2}$ has nonnegative scalar curvature and its angle at inifnity is (strictly) greater than $2\pi$, then the generalized connected sum $(T^n,i)\#_{T^{n-2}}(\bar M,i_2)$ admits a complete metric with positive scalar curvature.
\end{proposition}
\begin{proof}
Let $r$ be the radical distance function on $\mathbf R^2$ and we denote $S_r$ to be the hypersurface $C_r\times T^{n-2}$ in $M$ for $r >1$, where $C_r$ is the circle in $\mathbf R^2$ with radius $r$ centered at the origin. Denote $g_{flat}$ to be the flat metric on $T^{n-2}$. We can write
$$
g_\beta|_{S_r}=\beta^2r^2\mathrm d\theta^2+g_{flat}+\eta_r,
$$
where the decay condition \eqref{Eq: decay 2} implies
\begin{equation}\label{Eq: eta r}
|\eta_r|_{g_\beta}+r|\nabla_{g_\beta}\eta_r|_{g_\beta}+r^2|\nabla_{g_\beta}^2\eta_r|_{g_\beta}=O(r^{-\mu}).
\end{equation}
Fix a large positive constant $r_0$ to be determined later and pick up a cut-off function $\zeta:[r_0,2r_0]\to [0,1]$ such that it takes value $1$ around $r_0$ and value $0$ around $2r_0$ as well that its derivative satisfies $|\zeta'|\leq 4r_0^{-1}$. For $r_0\leq r\leq 2r_0$, we define
$$
\bar g_r=\beta^2r^2\mathrm d\theta^2+g_{flat}+\zeta(r)\eta_r.
$$
With $u$ a positive smooth function to be determined later, we introduce the following metric
\begin{equation*}
\bar g=\left\{
\begin{array}{cc}
g_\beta,&r\leq r_0;\\
u^2\mathrm dr^2+\bar g_r,&r_0\leq r\leq 2r_0;\\
\beta^2g_{euc}\oplus g_{flat}, &r\geq 2r_0.
\end{array}
\right.
\end{equation*}
Actually we hope that this metric $\bar g$ is a piecewisely smooth metric on $M$ with nonnegative scalar curvature in the distribution sense of Miao \cite{Miao2002}, which has exactly two corners along the hypersurfaces $S_{r_0}$ and $S_{2r_0}$. This motivates us to solve the following quasi-spherical metric equation
\begin{equation}\label{Eq: QS equation}
H_r\frac{\partial u}{\partial r}=u^2\Delta_{\bar g_r}u+\frac{1}{2}R(\bar g_r)(u-u^3)-\frac{1}{2}R(\bar g_{ref})u,\quad r_0\leq r\leq 2r_0
\end{equation}
with boundary value
\begin{equation}\label{Eq: initial condition}
u(r_0,\cdot)=\frac{H_{r_0}}{H_{r_0,g_\beta}}\quad\text{on}\quad S_{r_0},
\end{equation}
where $\bar g_{ref}$ is the reference metric $\mathrm dr^2+\bar g_r$, $H_r$ is the mean curvature of $S_r$ with respect to $\bar g_{ref}$ and $H_{r_0,g_\beta}$ is the mean curvature of $S_{r_0}$ with respect to the metric $g_\beta$. Geometrically, the equation \eqref{Eq: QS equation} with condition \eqref{Eq: initial condition} matches the mean curvature on two sides of the corner along $S_{r_0}$ and guarantees that $\bar g$ has vanishing scalar curvature when $r_0\leq r\leq 2r_0$. In order to control the mean curvature at the second corner, we need to make a careful analysis on the equation above. From \eqref{Eq: eta r} we can compute
$$
H_r=r^{-1}+O(r^{-1-\mu}),\quad R(\bar g_r)=O(r^{-2-\mu})\quad \text{and}\quad R(\bar g_{ref})=O(r^{-2-\mu}).
$$
Notice that both $H_{r_0}$ and $H_{r_0,g_\beta}$ have the order $r^{-1}_0+O(r_0^{-1-\mu})$. So the initial value satisfies $u(r_0,\cdot)=1+O(r_0^{-\mu})$. Denote $v(r)$ to be the solution of the following ordinary differential equation
$$
\frac{\mathrm dv}{\mathrm dr}=\frac{1}{2}f(r)(v-v^3)-\frac{1}{2}g(r)v,\quad v(r_0)=\max_{S_{r_0}}u(r_0,\cdot).
$$
where
$$
f(r)=\max_{S_r}\frac{R(\bar g_r)}{H_r}\quad\text{and}\quad g(r)=\min_{S_r}\frac{R(\bar g_{ref})}{H_r}.
$$
It is not difficult to solve this equation and obtain
$$
v(r)=\left[\left(v^{-2}(r_0)+\int_{r_0}^rf(\tau)e^{-\int_{r_0}^\tau(g(s)-f(s))\mathrm ds}\mathrm d\tau\right)e^{\int_{r_0}^r(g(s)-f(s))\mathrm ds}\right]^{-2}.
$$
Based on the estimates above we conclude $v=1+O(r_0^{-\mu})$, and then it follows from the maximum principle that
$
u\leq 1+O(r^{-\mu}_{0}).
$ Now we focus on the mean curvatures on two sides of the second corner along $S_{2r_0}$. Clearly the mean curvature of $S_{2r_0}$ with respect to the product metric $\beta^2g_{euc}\oplus g_{flat}$ is
$$
H_{2r_0,prod}=\frac{1}{2r_0\beta},
$$
and the mean curvature of $S_{2r_0}$ with respect to the reference metric $\bar g_{ref}$ satisfies
$$
H_{2r_0}=\frac{1}{2r_0 u}\geq \frac{1}{2r_0}+O(r_0^{-1-\mu}).
$$
Since the angle of $(M,g,\mathcal E)$ at infinity is greater than $2\pi$, i.e. $\beta >1$, we can take $r_0$ to be large enough such that $H_{2r_0}>H_{2r_0,prod}$. So $(M,\bar g,\mathcal E)$ has nonnegative scalar curvature in the distribution sense.
Notice that $(M,\bar g,\mathcal E)$ is isometric to the Riemannian product of the standard Euclidean $2$-space and a flat $T^{n-2}$. We can glue the opposite faces of $C\times T^{n-2}$ for a large cube $C\subset \mathbf R^2$ to obtain a piecewisely smooth metric $\tilde g$ on $(T^n,i)\#_{T^{n-2}}(\bar M,i_2)$ with nonnegative scalar curvature in the distribution sense. Combining Miao's smoothing trick in \cite{Miao2002} and Kazdan's deformation in \cite{Kazdan82} we can construct a complete metric on $(T^n,i)\#_{T^{n-2}}(\bar M,i_2)$ with positive scalar curvature. In detail, it follows from \cite[Proposition 3.1]{Miao2002} and \cite[Lemma 4.1]{LM2019} that we can find a smooth perturbed metric such that the first Neumann eigenvalue of the conformal Laplace operator is positive in a large compact region. Now we can use \cite[Theorem A]{Kazdan82} to construct a complete conformal metric on $(T^n,i)\#_{T^{n-2}}(\bar M,i_2)$ with positive scalar curvature.
\end{proof}
In the following, we focus on asymptotically conical manifolds with angle $2\pi$ at infinity. In order to deal with the rigidity case, we need some more analysis.
\begin{lemma}\label{Lem: comparison function}
Given an asymptotically conical manifold $(M,g,\mathcal E)$ with $\beta=1$ and fixed a constant $0<\alpha<1$, there is a universal constant $r^*_0=r^*_0(M,g,\mathcal E,\alpha)$ such that the function
$
w_1=\log r+\log^{\alpha} r
$ and $w_2=\log r-r^{-\mu/2}$
are superharmonic when $r\geq r^*_0$.
\end{lemma}
\begin{proof}
This comes from a direct calculation. From the decay condition \eqref{Eq: decay 2} we see
\begin{equation*}
\begin{split}
\Delta_g w_1&=\left(1+O(r^{-\mu})\right)\frac{\partial^2w_1}{\partial r^2}+\left(r^{-1}+O(r^{-1-\mu})\right)\frac{\partial w_1}{\partial r}\\
&=\alpha(\alpha-1)r^{-2}\log^{\alpha-2}r+O(r^{-2-\mu}).
\end{split}
\end{equation*}
and
$$
\Delta_g w_2=-\frac{\mu^2}{4}r^{-2-\frac{\mu}{2}}+O(r^{-2-\mu}).
$$
Clearly there is a universal constant $r^*_0$ such that $\Delta_g w_1<0$ and $\Delta_g w_2<0$ when $r\geq r^*_0$. This completes the proof.
\end{proof}
Let $\tilde g$ be a smooth metric on $M$ satisfying the asymptotical behavior \eqref{Eq: decay 2}. In the following, we consider the conformal Laplace operator $\tilde\Delta_{conf}$ with respect to the metric $\tilde g$ given by
$$
\tilde\Delta_{conf}=\Delta_{\tilde g}-\frac{n-2}{4(n-1)}R(\tilde g).
$$
For any compact region $U$, we use $\mu_1(\tilde\Delta_{conf},U)$ to denote the first Neumann eigenvalue of $\tilde\Delta_{conf}$ in $U$.
For convenience, we also extend the radical function $r$ from the end $\mathcal E$ (diffeomorphic to $(\mathbf R^2-B)\times F$) to the whole $M$ by defining it to be one outside $\mathcal E$. For any $\rho>1$ we use $B_\rho$ to denote the region $\{r<\rho\}$.
The following lemma provides nice comformal factors for later use.
\begin{lemma}\label{Lem: conformal factor}
Assume
\begin{itemize}
\item $\mu_1(\tilde\Delta_{conf},U)>0$ for some compact region $U$;
\item $R(\tilde g)\geq 0$ outside $U$.
\end{itemize}
Then we can find a sequence of smooth positive functions $u_k$ such that
\begin{itemize}
\item $-\tilde\Delta_{conf}u_k$ is quasi-positive\footnote{A function is said to be quasi-positive if it is nonnegative everywhere and positive somewhere.} in $B_{\rho_k}$ and $u_k=1$ on $\partial B_{\rho_k}$, where $\rho_k\to+\infty$ as $k\to+\infty$;
\item $u_k$ is bounded below by a positive constant;
\item the normal derivative
$$
\frac{\partial u_k}{\partial r}\geq \frac{\delta}{\rho_k\log\rho_k}
$$
for some positive universal constant independent of $k$.
\end{itemize}
\end{lemma}
\begin{proof}
Denote $\tilde U$ to be a larger compact region containing $U$. Since the first Neumann eigenvalue $\mu_1(\tilde\Delta_{conf}, U)$ is positive, there is a positive constant $c_*$ such that $\mu_1(\tilde\Delta_{conf},\tilde U)\geq c_*$. Namely, it holds
$$
\int_{\tilde U}|\tilde\nabla_{conf}\phi|^2\,\mathrm d\mu_{\tilde g}\geq c_*\int_{\tilde U}\phi^2\,\mathrm d\mu_{\tilde g}
$$
for all $\phi\in C^\infty(\tilde U)$. Here and in the sequel, let us formally write
$$
|\tilde\nabla_{conf}\phi|^2:=|\nabla_{\tilde g}\phi|^2+\frac{n-2}{4(n-1)}R(\tilde g)\phi^2
$$
for short. For later use, we fix a smooth cut-off function $\eta:M\to [0,1]$ with compact support such that $\eta\equiv 1$ in $U$ and $\eta\equiv 0$ outside $\tilde U$.
Take a sequence $\{\rho_k\}$ with $\rho_k\to+\infty$ as $k\to+\infty$ and let us construct the desired function $u_k$ from the exhaustion method. Let
$$
\mathcal E\subset \mathcal V_1\subset\mathcal V_2\subset\cdots
$$
be an exhaustion of $M$ such that the closure of $\mathcal V_j-\mathcal E$ is compact. Without loss of generality, we can assume that the region $U$ is contained in $\mathcal V_1$ and $B_{\rho_1}$. First we point out that the homogenous equation
$$
-\tilde\Delta_{conf}u=\frac{1}{2}\eta c_* u\quad\text{in}\quad \mathcal V_j\cap B_{\rho_k}
$$
with mixed boundary condition
$$
\frac{\partial u}{\partial\vec n_-}=0\quad \text{on}\quad \partial \mathcal V_j,\quad u=0\quad\text{on}\quad \partial B_{\rho_k}
$$
only has zero solution, where $\vec n_-$ is denoted to be the unit normal vector of $\partial\mathcal V_k$ in $M$ opposite to the end $\mathcal E$. To see this, we just need to integrate by parts and see
\[
\begin{split}
0&=\int_{\mathcal V_j\cap B_{\rho_k}}|\tilde\nabla_{conf}u|^2-\frac{1}{2}\eta c_* u^2\,\mathrm d\mu_{\tilde g}\geq \frac{1}{2}c_*\int_{\tilde U}u^2\,\mathrm d\mu_{\tilde g}.
\end{split}
\]
This yields $u\equiv 0$ in $\tilde U$. Since $u$ is a harmonic function outside $\tilde U$, it follows from the maximum principle that $u$ vanishes in the whole $\mathcal V_k\cap B_{\rho_k}$. From the Fredholm alternative, we see that the equation
$$
-\tilde\Delta_{conf}u_{kj}=\frac{1}{2}\eta c_* u_{kj}\quad\text{in}\quad \mathcal V_j\cap B_{\rho_k}
$$
with mixed boundary condition
$$
\frac{\partial u_{kj}}{\partial\vec n_-}=0\quad \text{on}\quad \partial \mathcal V_j,\quad u_{kj}=1\quad\text{on}\quad \partial B_{\rho_k}
$$
is solvable.
Next let us show that up to a subsequence the functions $u_{kj}$ converge to a smooth positive function $u_k$ satisfying the equation
$$
-\tilde\Delta_{conf}u_{k}=\frac{1}{2}\eta c_* u_{k}\quad\text{in}\quad B_{\rho_k}
$$
with $u_k=1$ on $\partial B_{\rho_k}$. Denote $v_{kj}=u_{kj}-1$. After integrating by parts and using the fact $\mu_1(\tilde\Delta_{conf},\tilde U)\geq c_*$ as well as $R(\tilde g)\geq 0$ outside $\tilde U$, we see that
\[
\begin{split}
\frac{1}{2}c_*\int_{\tilde U}v_{kj}^2\,\mathrm d\mu_{\tilde g}&\leq\int_{\mathcal V_j\cap B_{\rho_k}}|\tilde\nabla_{conf}v_{kj}|^2-\frac{1}{2}\eta c_*v_{kj}^2\,\mathrm d\mu_{\tilde g}\\
&=\frac{1}{2}c_*\int_{\mathcal V_j\cap B_{\rho_k}}\eta v_{kj}\,\mathrm d\mu_{\tilde g}.
\end{split}
\]
Combined with the H\"older inequality, we conclude
$$
\left(\int_{\tilde U}v_{kj}^2\,\mathrm d\mu_{\tilde g}\right)^{\frac{1}{2}}\leq \vol_{\tilde g}(\tilde U).
$$
From the standard elliptic theory, the function $v_{kj}$ as well as $u_{kj}$ has a uniform $C^0$-norm in $U$ independent of $k$ and $j$. Since all functions $u_{kj}$ are positive, the Harnack inequality combined with the maximum principle yields that for any compact subset $K$ of $M$ there is a universal constant $C$ depending on $K$ such that $\|u_{kj}\|_{C^0(K)}\leq C$ for all large enough $j$. From the standard elliptic theory the functions $u_{kj}$ converge to a smooth positive function $u_k$ up to a subsequence and $u_k$ also satisfies the estimate
\begin{equation}\label{Eq: C0 norm uk}
\|u_k\|_{C^0(K)}\leq C.
\end{equation}
At this stage, let us point out some extra useful estimates for the functions $u_k$. From a straightforward calculation we see
$$
\int_{\partial \mathcal V_1}\frac{\partial u_{kj}}{\partial \vec n_-}\,\mathrm d\sigma_{\tilde g}=\int_{\partial \mathcal V_j}\frac{\partial u_{kj}}{\partial \vec n_-}\,\mathrm d\sigma_{\tilde g}=0
$$
and
$$
\int_{\partial \mathcal V_1}u_{kj}\frac{\partial u_{kj}}{\partial \vec n_-}\,\mathrm d\sigma_{\tilde g}=\int_{\partial \mathcal V_j}u_{kj}\frac{\partial u_{kj}}{\partial \vec n_-}\,\mathrm d\sigma_{\tilde g}-\int_{\mathcal V_j-\mathcal V_1}|\nabla_{\tilde g} v_{kj}|^2\,\mathrm d\mu_{\tilde g}\leq 0
$$
As a result, the function $u_k$ satisfies
\begin{equation}\label{Eq: uk estimates}
\int_{\partial \mathcal V_1}\frac{\partial u_k}{\partial \vec n_-}\,\mathrm d\sigma_{\tilde g}=0\quad\text{and}\quad \int_{\partial \mathcal V_1}u_k\frac{\partial u_k}{\partial \vec n_-}\,\mathrm d\sigma_{\tilde g}\leq 0.
\end{equation}
In the following, we devote to show the lower bound estimate for normal derivative $\partial_ru_k$ on $\partial B_{\rho_k}$. From previous discussion the functions $u_k$ has a uniform $C^0$-norm in each compact subsets and so $u_k$ converges to a nonnegative function $u_\infty$ up to a subsequence. From the Harnack inequality there are two possibilities:
{\it Case 1. $u_\infty\equiv 0$.} In this case, we can obtain the desired estimate from a comparison argument. Let $r_0^*$ and $w_2$ be the same as in Lemma \ref{Lem: comparison function}. Notice that the value of $u_k$ on $\{r=r_0^*\}$ will be less than $1/3$ when $k$ is large enough. So we have
$$
u_{k}\leq \frac{1}{3}+\frac{2}{3}\frac{w_2}{\log \rho_k}\quad \text{on}\quad\partial B_{r_0^*}
$$
and
$$u_k=\frac{1}{3}+\frac{2}{3}\frac{w_2}{\log \rho_k}=1\quad \text{on}\quad \partial B_{\rho_k}.$$
Since $w_2$ is super-harmonic when $r_0^*\leq r\leq \rho_k$, we conclude
$$
u_k\leq\frac{1}{3}+\frac{2}{3}\frac{w_2}{\log \rho_k}\quad\text{when}\quad r_0^*\leq r\leq \rho_k,
$$
and
$$\frac{\partial u_k}{\partial r}\geq \left(\frac{2}{3}+o(1)\right)\frac{1}{\rho_k\log\rho_k}\geq \frac{1}{2\rho_k\log\rho_k} \quad \text{on}\quad \partial B_{\rho_k}
$$
for $k$ large enough.
{\it Case 2. $u_\infty$ is a positive function.} From \eqref{Eq: uk estimates} we have
$$
\int_{\partial \mathcal V_1}u_\infty\frac{\partial u_\infty}{\partial \vec n_-}\,\mathrm d\sigma_{\tilde g}\leq 0
$$
and so for any $\rho>1$ we have
\[
\begin{split}
\int_{\partial B_\rho}u_\infty\frac{\partial u_\infty}{\partial \vec n_+}\,\mathrm d\sigma_{\tilde g}&=\int_{\mathcal V_1\cap B_\rho}|\tilde\nabla_{conf}u_\infty|^2-\frac{1}{2}\eta c_*u_\infty^2\,\mathrm d\mu_{\tilde g}-\int_{\partial\mathcal V_1}u_\infty\frac{\partial u_\infty}{\partial \vec n_-}\,\mathrm d\sigma_{\tilde g}\\
&\geq \frac{1}{2}c_*\int_{\tilde U}u_\infty^2\,\mathrm d\mu_{\tilde g}>0,
\end{split}
\]
where $\vec n_+$ is denoted to be the unit normal of $\partial B_\rho$ in $M$ pointing to the end $\mathcal E$. From the maximum principle as well as \eqref{Eq: C0 norm uk} we know
$$
u_k\leq\max\left\{\max_{\partial B_1}u_k,1\right\} \leq C
$$
for some universal constant $C$ independent of $k$ and the same estimate holds for $u_\infty$. It then follows from \cite[Theorem 5]{Moser1961} that $u$ has a finite limit as $r\to+\infty$, which then yields
$$
u_\infty=O(1)\quad\text{and}\quad |\nabla_{\tilde g}u_\infty|=o(r^{-1}),\quad \text{as}\quad r\to+\infty.
$$
This implies
$$
\int_{\partial B_\rho}u_\infty\frac{\partial u_\infty}{\partial \vec n_+}\,\mathrm d\sigma_{\tilde g}=o(1),\quad\text{as}\quad \rho\to+\infty,
$$
which is impossible.
Finally, we do some modification for $u_k$ such that it is bounded below by a positive constant. Notice that $-\tilde\Delta_{conf}$ is quasi-positive in $B_{\rho_k}$ and bounded below by a positive constant in $U$. Since the scalar curvature $R(\tilde g)$ can only be negative in $U$, we can replaced $u_k$ by
$$
\frac{u_k+\epsilon_k}{1+\epsilon_k}
$$
with some small positive constant $\epsilon_k$ such that all desired properties are satisfied.
\end{proof}
Now we are ready to prove Theorem \ref{Thm: main 5}.
\begin{proof}[Proof for Theorem \ref{Thm: main 5}]
First we show that the angle of $(M,g,\mathcal E)$ at infinity is no greater than $2\pi$. Otherwise, Proposition \ref{Prop: PMT to PSC 2} yields that the generalized connected sum $(T^n,i)\#_{T^{n-2}}(\bar M,i_2)$ admits a complete metric with positive scalar curvature. In the following, we will investigate the topological restrictions of $(T^n,i)\#_{T^{n-2}}(\bar M,i_2)$ from our assumption on $M$. Recall that the gluing procedure is illustrated by the following Figure \ref{Fig: 7}.
\begin{figure}[htbp]
\centering
\includegraphics[width=12cm]{7.eps}
\caption{The gluing procedure}
\label{Fig: 7}
\end{figure}
With the help of the coordinate system on the end $\mathcal E$, the manifold $T^{n-2}\times (T^2-B)$ is foliated by $T^{n-2}$-slices and this induces a canonical perturbation $s:T^{n-2}\to \partial(T^{n-2}\times B)$ of the embedding $i:T^{n-2}\to T^n$, which maps $T^{n-2}$ to some $T^{n-2}$-slice in $\partial(T^{n-2}\times B)$. Denote
$$\Phi:\partial\left(T^{n-2}\times(T^2-B)\right)\to \partial\left(\bar M-T^{n-2}\times B\right) $$
to be the gluing map. From the assumption on $M$, we see that
\begin{itemize}
\item the map $\Phi\circ s:T^{n-2}\to \bar M-T^{n-2}\times B$ is incompressible for $n\geq 3$;
\item or $n=3$ and the map $\Phi\circ s:\mathbf S^1\to \bar M-\mathbf S^1\times B$ is homotopically non-trivial.
\end{itemize}
With the exactly same argument in the proof of Proposition \ref{Prop: main 2} and \ref{Prop: main 3}, this condition\footnote{This is enough since in the proof of Proposition \ref{Prop: main 2} and Proposition \ref{Prop: main 3} the contradiction is deduced by showing non-contractibility of certain curves in $M_2-U_2$.} can be used to rule out the existence of complete metrics with positive scalar curvature on
$(T^n,i)\#_{T^{n-2}}(\bar M,i_2)$, but this contradicts to the implication from Proposition \ref{Prop: PMT to PSC 2}.
Next let us show the flatness of $(M,g)$ under the assumption $\beta=1$. The proof will be divided into three steps.
{\it Step 1. $(M,g)$ is scalar flat.}
Otherwise, the scalar curvature $R(g_\beta)$ must be positive at some point $p$ in $M$. Let $\tilde g=g_\beta$ and $U$ be a small geodesic centered at the point $p$. Then it is easy to verify that $\mu_1(\tilde\Delta_{conf},U)>0$ and that $R(\tilde g)$ is nonnegative outside $U$. From Lemma \ref{Lem: conformal factor} we can find a sequence of positive smooth solutions $u_k$ such that
\begin{itemize}
\item $-\tilde\Delta_{conf}u_k$ is quasi-positive in $B_{\rho_k}$ and $u_k=1$ on $\partial B_{\rho_k}$, where $\rho_k\to+\infty$ as $k\to+\infty$;
\item $u_k$ is bounded below by a positive constant;
\item the normal derivative
$$
\frac{\partial u_k}{\partial r}\geq \frac{\delta}{\rho_k\log\rho_k}
$$
for some positive universal constant independent of $k$.
\end{itemize}
Now we would like to follow along the proof of Proposition \ref{Prop: PMT to PSC 2} with some modifications. In the definition of the piecewisely smooth metric $\bar g$, we add a conformal deformation. Namely, we let
\begin{equation*}
\bar g=\left\{
\begin{array}{cc}
\bar g_\beta:=u_k^{\frac{4}{n-2}} \tilde g,&r\leq \rho_k;\\
u^2\mathrm dr^2+\bar g_r,&\rho_k\leq r\leq 2\rho_k;\\
\beta^2g_{euc}\oplus g_{flat}, &r\geq 2\rho_k.
\end{array}
\right.
\end{equation*}
We point out that the metric $\bar g$ is complete due to the positive lower bound for $u_k$. In the following, the analysis is similar as before. The mean curvature of $S_{\rho_k}$ with respect to $\bar g_\beta$ satisfies
$$
H_{\rho_k,\bar g_\beta}\geq\frac{1}{\rho_k}+\frac{\delta'}{\rho_k\log\rho_k}
$$
for some positive constant $\delta'$ independent of $\rho_k$. As a result, the initial value of $u$ can be chosen to satisfy
$$u(\rho_k,\cdot)\leq 1-\delta'\log^{-1}\rho_k+O(\rho_k^{-\epsilon}).$$ Doing the same analysis as in the proof of Proposition \ref{Prop: PMT to PSC 2}, we have
$$u(2\rho_k,\cdot)\leq 1-\delta'\log^{-1}\rho_k +O(\rho_k^{-\epsilon}).$$ This yields
$$
H_{2\rho_k}=\frac{1}{2\rho_k u}\geq \frac{1}{2\rho_k}+\frac{\delta'}{2\rho_k\log\rho_k}+O(\rho_k^{-\epsilon}).
$$
On the other hand, we have
$$
H_{2\rho_k,prod}=\frac{1}{2\rho_k}.
$$
By taking $\rho_k$ to be large enough, we still have the strict inequality $H_{2\rho_k}>H_{2\rho_k,prod}$ and this further implies the existence of a complete metric on the generalized connected sum $(T^n,i)\#_{T^{n-2}}(\bar M,i_2)$ with positive scalar curvature, which is impossible from the previous discussion.
{\it Step 2. $(M,g)$ is Ricci flat.}
This comes from a standard deformation argument. Assume otherwise that the Ricci curvature of the metric $g$ does not vanish everywhere. From \cite[Lemma 3.3]{Kazdan82} we can pertube the metric $g$ inside a small geodesic ball $U$ such that the purtubed metric (denoted by $\tilde g$) satisfies
$
\mu_1(\tilde\Delta_{conf},U)>0.
$
Since the metric $g$ is unchanged outside $U$, we see $R(\tilde g)\geq 0$ outside $U$. With the same argument as in Step 1, we can deduce the same contradiction. As a result, $(M,g)$ has to be Ricci flat.
{\it Step 3. $(M,g)$ is flat.}
In dimension three, flatness is equivalent to Ricci-flatness and we are already done. So we just need to deal with the case when the dimension of $M$ is no less than four. We try to use the volume comparison theorem for Ricci nonnegative Riemannian manifolds. Let us denote $\tilde {\mathcal E}$ to be the universal covering of $\mathcal E$. Clearly, $\tilde {\mathcal E}$ is diffeomorphic to $[1,+\infty)\times\mathbf R\times \mathbf R^{n-2}$ and the covering map $\pi:\tilde{\mathcal E}\to \mathcal E\approx (\mathbf R^2-B)\times T^{n-2}$ is given by
$$
(r,\theta,x)\mapsto (r,\theta,\bar x)
$$
with the help of the polar coordinate system at infinity, where $x\mapsto \bar x$ is the quotient map from $\mathbf R^{n-2}$ to $T^{n-2}$. Denote $\pi:(\tilde M,\tilde g)\to (M,g)$ to be the universal covering of $(M,g)$ and $e:\mathcal E\to M$ to be the canonical embedding of the end $\mathcal E$. We can lift $e$ to a map $\tilde e:\tilde{\mathcal E}\to \tilde M$ such that we have the following commutative diagram
\begin{equation}\label{Eq: lift diagram}
\xymatrix{\tilde{\mathcal E}\ar[r]^{\tilde e}\ar[d]^{\pi}&\tilde M\ar[d]^{\pi}\\
\mathcal E\ar[r]^{e}&M.}
\end{equation}
Denote
$$
\Omega=\{(r,\theta,x):r\geq 1,\,0<\theta<2\pi\}\subset \tilde{\mathcal E}.
$$
Let us show that $\tilde e|_{\Omega}:\Omega\to \tilde M$ is an embedding. The only thing needs to prove is that the map $\tilde e|_{\Omega}$ is injective. To see this, we pick up two different points $p$ and $q$ in $\tilde{\mathcal E}$ and try to show $\tilde e(p)\neq \tilde e(q)$. Otherwise, a path $\gamma:[0,1]\to \Omega$ with $\gamma(0)=p$ and $\gamma(1)=q$ maps to a closed curve $\pi(\gamma)$ in $\mathcal E$, whose image $(e\circ \pi)(\gamma)$ in $M$ is homotopically trivial due to the commutative diagram \ref{Eq: lift diagram}. From the definition of $\Omega$, the closed curve $(e\circ\pi)(\gamma)$ lies in $(i_{T^{n-2}})_*(\pi_1(T^{n-2}))$. From the assumption we know that
$$
(i_{T^{n-2}})_*:\pi_1(T^{n-2})\to \pi_1(M)
$$
is injective and so $\pi(\gamma)$ can be homotopic to a point in $\mathcal E$. This is impossible since $\mathcal E$ is the universal covering and the path $\gamma$ has different end-points.
Fix a point $\tilde p$ in $\tilde M$. From the comparison theorem the flatness of $(M,g)$ will follow from the estimate
$$
\liminf_{\rho\to+\infty}\frac{\vol_{\tilde g}(B_{\tilde g}(\tilde p,\rho))}{\omega_n\rho^n}\geq 1.
$$
For convenience, let us denote $\Omega_{r_0}=\Omega\cap \{r\geq r_0\}$ for any $r_0>1$. It is clear that for any $\epsilon>0$ there is a $r_0>1$ such that
$$
(1-\epsilon)\tilde g_{euc}\leq \tilde g\leq (1+\epsilon)\tilde g_{euc}\quad\text{on}\quad \Omega_{r_0},
$$
where $\tilde g_{euc}$ is the metric on $\Omega$ given by $\tilde g_{euc}=\mathrm dr^2+r^2\mathrm d\theta^2+\mathrm dx^2$. The figure \ref{Fig: 9} below illustrates how $\Omega_{r_0}$ looks like $\mathbf R^n$ with certain parts removed.
\begin{figure}[htbp]
\centering
\includegraphics[width=9cm]{9.eps}
\caption{The region $\Omega_{r_0}$}
\label{Fig: 9}
\end{figure}
In the following, we will calculate the volume of the intersection $B_{\tilde g}(\tilde p,\rho)\cap\Omega$. For convenience, we denote $\tilde q=(r_0,\pi,0)$ and $\tilde C=\{(r_0,\theta,0):0<\theta<2\pi\}$. Clearly we have $\diam_{\tilde g}\tilde C\leq (1+\epsilon)2\pi r_0$. We also let $\tilde r=\sqrt{r^2+|x|^2}$ be the radical distance function on $\Omega_{r_0}$. For any point $\tilde q'=(r,\theta,x)$ with $\tilde r(\tilde q')=\rho$, we can find the path
$$
\tilde \gamma(t)=((1-t)r_0+tr,\theta,tx),\quad t\in[0,1],
$$
connecting $\tilde C$ and $\tilde q'$. A direct computation shows that the $\tilde g$-length of the curve $\gamma$ does not exceed $(1+\epsilon)\rho$. Denote $\Lambda=\dist(\tilde p,\tilde q)+(1+\epsilon)2\pi r_0$. Now we see
\[
\begin{split}
\vol_{\tilde g}(B_{\tilde g}(\tilde p,\rho+\Lambda))&\geq\vol_{\tilde g}(B_{\tilde g}(\tilde C,\rho)\subset \Omega)\\
&\geq (1-\epsilon)^n\vol_{\tilde g_{euc}}\left(\{\tilde r\leq (1+\epsilon)^{-1}\rho\}\subset \Omega\right)\\
&=\left(\frac{1-\epsilon}{1+\epsilon}\right)^n\omega_n\rho^n+O(\rho^{n-2}),\quad \text{as}\quad \rho\to+\infty.
\end{split}
\]
This implies $$
\liminf_{\rho\to+\infty}\frac{\vol_{\tilde g}(B_{\tilde g}(\tilde p,\rho))}{\omega_n\rho^n}\geq \left(\frac{1-\epsilon}{1+\epsilon}\right)^n,
$$
and the proof is completed by letting $\epsilon\to 0$.
\end{proof}
\section{Proof for Proposition \ref{Prop: main 7}}
This section is devoted to a detailed proof for Proposition \ref{Prop: main 7}. Instead of a direct proof, we are going to show the following stronger result first.
\begin{proposition}\label{Prop: main 6}
Let $(\Sigma,g_\Sigma)$, $\dim\Sigma \geq 2$, be a convex hypersurface or closed curve in Euclidean space and $(F,g_F)$ be a circle $\mathbf S^1$ or a flat $k$-torus $(T^k,g_{flat})$. If $(\Omega,g)$, $\dim \Omega \leq 7$, is a compact manifold with $R(g)\geq 0$ and mean convex boundary such that
\begin{itemize}
\item $\partial\Omega$ with the induced metric is isometric to the Riemannian product $\Sigma\times F$ through a diffeomorphism $\phi=(\phi_1,\phi_2): \partial \Omega\to\Sigma\times F$;
\item the induced map $(\phi_{2}^{-1})_*:\pi_1( F) \to \pi_1(\Omega)$ is injective when $F=T^k$
or $(\phi_{2}^{-1})_*:\pi_1(F) \to \pi_1(\Omega)$ is non-zero when $F=\mathbf S^1$,
\end{itemize}
then
\begin{equation}\label{Eq: total curvature estimates}
\int_{\partial\Omega} H\,\mathrm d\sigma_g\leq \Lambda(\Sigma,g_{\Sigma})\cdot \vol(F,g_F),
\end{equation}
where $H$ is the mean curvature of $\partial\Omega$ in $\Omega$ with respect to the unit outer normal and $\Lambda(\Sigma,g_\Sigma)$ is denoted to be the total mean curvature of $\Sigma$ in the Euclidean space. Moreover, if we have the equality in \eqref{Eq: total curvature estimates}, then the metric $g$ is flat.
\end{proposition}
\begin{proof}
The proof is based on the quasi-spherical metric. In \cite{ST2002}, it was shown that the quasi-spherical metric over the exterior region of Euclidean space is always asymptotically flat at infinity such that the positive mass theorem comes into play. Here we take a different way combining the quasi-spherical metric with the compactification trick such that Proposition \ref{Prop: main 2} and \ref{Prop: main 3} can be applied.
Let us divide our discussion into two cases.
{\it Case 1. $\dim \Sigma\geq 2$.} Denote $\Omega_{ext}$ to be the exterior region outside of $\Sigma$ in the Euclidean space. Since $\Sigma$ is convex, we can find an equidistant foliation
$$
\Psi:\Sigma\times [0,+\infty)\to \Omega_{ext}
$$
and the Euclidean metric on $\Omega_{ext}$ can be written as
$
g_{euc}=\mathrm dt^2+g_t
$,
where $g_t$ is the induced metric of the hypersurface $\Sigma_t=\Psi(\Sigma\times\{t\})$. After taking the $T^k$-component into consideration, let us denote $\bar\Omega_{ext}=\Omega_{ext}\times T^k$ and $\bar\Sigma_t=\Sigma_t\times T^k$. We also write $\bar g_t=g_t+g_{flat}$ and $\bar g=\mathrm dt^2+\bar g_t$. As before, we consider the following quasi-spherical metric equation
\begin{equation}\label{Eq: QS}
\bar H_t\frac{\partial u}{\partial t}=u^2\Delta_{\bar g_t}u+\frac{1}{2}R(\bar g_t)(u-u^3),\quad u(0,\cdot)=u_0>0,
\end{equation}
which guarantees the quasi-spherical metric $\bar g_{qs}=u^2\mathrm dt^2+\bar g_t$ has vanishing scalar curvature. Here $\bar H_t$ and $R(\bar g_t)$ are denoted to be the mean curvature and the scalar curvature of the hypersurface $\bar\Sigma_t$ and $u_0:\Sigma\to \mathbf R$ is a positive function to be determined later.
First we point out that the equation \eqref{Eq: QS} is solvable no matter what initial value $u_0$ is prescribed. To see this, notice that $\Sigma_t$ is convex all the time and so the scalar curvature $R(\bar g_t)$ is nonnegative. Then the parabolic comparison principle yields the following a priori $C^0$-estimate
$$
\min\left\{1,\min_\Sigma u_0\right\}\leq u\leq \max\left\{1,\max_\Sigma u_0\right\}.
$$
This means that $u$ cannot blow up in finite time and so the solution $u$ exists on $\bar\Omega_{ext}$.
Now we analyze the behavior of the solution $u$ at infinity. From the same comparison argument as in \cite[Lemma 2.2]{ST2002}, it is not difficult to deduce the basic estimate
\begin{equation}\label{Eq: estimate u}
|u-1|\leq Ct^{2-m},
\end{equation}
where $C$ is a universal constant independent of $t$ and $m=\dim\Sigma+1$. Next let us investigate the function $v=t^{m-2}(u-1)$. From above estimate we see that $v$ is uniformly bounded. From the equation \eqref{Eq: QS} the function $v$ satisfies
\begin{equation}\label{Eq: v}
\bar H_t\frac{\partial v}{\partial t}=u^2\Delta_{\bar g_t}v+\left((m-2)\frac{\bar H_t}{t}-\frac{1}{2}R(\bar g_t)u(u+1)\right)v.
\end{equation}
A flat $k$-torus $(T^k,g_{flat})$ can be viewed as the quotient space $(\mathbf R^k,g_{euc})/\Gamma$ for some lattice $\Gamma$ in $\mathbf R^k$. This induces a covering map
$$
\pi:\Omega_{ext}\times \mathbf R^k\to \bar\Omega_{ext}.
$$
For convenience, we will lift the functions $u$ and $v$ on $\Omega_{ext}\times \mathbf R^k$. Namely, we define $\tilde u=u\circ \pi$ and $\tilde v=v\circ \pi$. Clearly both $\tilde u$ and $\tilde v$ are $\Gamma$-invariant. For any positive integer $l$, we denote the scaling
\begin{equation}\label{Eq: parabolic scaling}
\Phi_l:\Sigma\times \mathbf R^k\times [0,+\infty)\to\Sigma\times \mathbf R^k\times[0,+\infty),\quad (\sigma,x,t)\mapsto (\sigma,lx,lt).
\end{equation}
Now let us consider the function
$
\tilde v_l=\tilde v\circ \Phi_l.
$
Denote $\tilde u_l=\tilde u\circ \Phi_l$. Let
$$\tilde H_{t,l}=l\cdot(\bar H_t\circ \pi\circ \Phi_l)$$
and
$$
\tilde g_{t,l}=l^{-2}\cdot(\pi\circ\Phi_l)^*(\bar g_{lt}).
$$
It is not difficult to check that $\tilde v_l$ satisfies the equation
\begin{equation}
\tilde H_{t,l}\frac{\partial \tilde v_l}{\partial t}=\tilde u_l^2\Delta_{\tilde g_{t,l}}\tilde v_l+\left((m-2)\frac{\tilde H_{t,l}}{t}-\frac{1}{2}R(\tilde g_{t,l})\tilde u_l(\tilde u_l+1)\right)\tilde v_l.
\end{equation}
Finally we point out that all $\tilde v_l$ are $\Gamma$-invariant.
Now we would like to investigate the limit of functions $\tilde v_l$. First we need to understand the behavior of $\tilde v$ as $t\to+\infty$. Define
$$
\beta(t)=\min_{\Sigma\times \mathbf R^k} \tilde v(t,\cdot).
$$
Let us show that $\beta(t)$ has a limit as $t\to+\infty$. Notice that
$$
\beta(t)=\min_{\Sigma\times T^k} v(t,\cdot).
$$
From the equation \eqref{Eq: v} and the estimate \eqref{Eq: estimate u} as well as the estimates for $\bar H_t$ and $R(\bar g_t)$ from \cite[Lemma 2.1]{ST2002}, the parabolic comparison principle yields
$$
\beta(t_2)\geq \beta(t_1)-Ct_1^{-1},\quad \text{for all}\quad 1\leq t_1\leq t_2,
$$
where $C$ is a universal constant independent of $t_1$ and $t_2$. This implies that $\beta(t)$ has a limit as $t\to+\infty$, denoted by $\beta_0$. Based on this we can show that the functions $\tilde v_l$ converge smoothly to the constant function $\tilde v_\infty\equiv \beta_0$ in compact subsets of $\Sigma\times \mathbf R^k\times (0,+\infty)$. To see this, first notice that all $\tilde v_l$ is uniformly bounded by the bound for $v$. It follows from \cite[Lemma 2.1]{ST2002} that
$
\tilde H_{t,l}
$
converges to $(m-1)t^{-1}$ smoothly and that $\tilde g_{t,l}$ converges to $\tilde g_{t,\infty}=t^2g_{round}+g_{euc}$ smoothly as $l\to +\infty$, where $g_{round}$ is a smooth metric on $\Sigma$ with constant curvature $1$. As a result, $\tilde v_{l}$ has uniformly bounded $C^k$-estimate for any $k$. Up to a subsequence, $\tilde v_l$ converges to a limit smooth function $\tilde v_\infty$ satisfying
$$
\frac{m-1}{t}\frac{\partial \tilde v_\infty}{\partial t}=\Delta_{\tilde g_{t,\infty}}\tilde v_\infty.
$$
Notice that $\beta(t)$ is always attained by $\tilde v(t,\cdot)$ in a fixed bounded compact region in $\Sigma\times \mathbf R^k$. From previous discussion, $\tilde v_\infty$ attains its minimum $\beta_0$ in the interior of $\Sigma\times \mathbf R^k\times(0,+\infty)$. The strong maximum principle yields that $\tilde v_\infty\equiv \beta_0$ and so
\begin{equation}
\lim_{t\to+\infty}v(t,\cdot)=\beta_0.
\end{equation}
Now let us deduce a contradiction under the assumption
\begin{equation}\label{Eq: contradiction}
\int_{\partial\Omega} H\,\mathrm d\sigma_g> \Lambda(\Sigma,g_{\Sigma})\cdot \vol(F,g_F).
\end{equation}
Correspondingly let us set
$$
u_0=\frac{\bar H_0}{H}.
$$
From a similar calculation as in \cite[Lemma 4.2]{ST2002}, it follows that the integral
\begin{equation}\label{Eq: total mean curvature slice}
\int_{\bar\Sigma_t}\bar H_t\left(u^{-1}(t,\cdot)-1\right)\mathrm d\sigma_{\bar g_t}
\end{equation}
is monotone increasing as $t$ increases. From the inequality \eqref{Eq: contradiction} we conclude that the constant $\beta_0$ is negative and so for $t$ large enough the mean curvature of $\Sigma_t$ with respect to the quasi-spherical metric $\bar g_{qs}$ is greater than that with respect to $\bar g$. From this we can obtain a contradiction with the same gluing argument as in the proof of Proposition \ref{Prop: PMT to PSC 2}.
For the equality case, we observe that the integral \eqref{Eq: total mean curvature slice} has to vanish all the time. After taking the derivative with respect to $t$ and using the equation of $u$, we see
$$
\int_{\bar \Sigma_t}R(\bar g_t)u^{-1}(t,\cdot)\left(u(t,\cdot)-1\right)^2\mathrm d\sigma_{\bar g_t}\equiv 0
$$
and so $u$ must be one identically. This yields that we can glue $(\Omega,g)$ with $(\bar\Omega_{ext},\bar g)$ along their boundaries with the same mean curvatures on both sides. Now the gluing argument is still valid except that we cannot construct a smooth metric with positive scalar curvature simply from smoothing and conformal tricks. In this case, we can use the Ricci-DeTurk flow (see \cite{ST2018} for instance) to construct a family of smooth metrics with nonnegative scalar curvature (Notice that the underlying space is closed now). From Proposition \ref{Prop: main 2} and \ref{Prop: main 3} it follows that these metrics are flat and so the metric $g$ has to be flat.
{\it Case 2. $\dim\Sigma=1$, i.e. $\Sigma$ is a closed curve.} The proof follows from the same idea as above but some analysis are different. Since all closed curves with the same length are isometric, we can assume $\Sigma$ to be some round curve $C_{r_0}$ without loss of generality.
Let us adopt the same notations as above and consider the following quasi-spherical metric equation
\begin{equation}\label{Eq: QS 2}
\frac{1}{r_0+t}\frac{\partial u}{\partial t}=u^2\Delta_{\bar g_t}u,\quad u(0,\cdot)=u_0>0.
\end{equation}
First it follows from parabolic comparison principle that
$$
\min_{\Sigma\times T^k} u\leq u\leq \max_{\Sigma\times T^k} u.
$$
From this we conclude that $u$ exists on $\bar\Omega_{ext}$. We are going to show that $u(t,\cdot)$ converges to a constant as $t\to+\infty$. As before, we lift $u$ to a function $\tilde u$ on $\Omega_{ext}\times \mathbf R^k$ and denote $\tilde u_l=\tilde u\circ \Phi_l$ with $\Phi_l$ the scaling given by \eqref{Eq: parabolic scaling}. Denote
$$
\beta(t)=\min_{\Sigma\times \mathbf R^k} \tilde u(t,\cdot).
$$
Based on the parabolic maximum principle, we know that $\beta(t)$ must be monotone when $t$ is large enough and so it has a limit $\beta_0$ as $t\to+\infty$. Similar as before, we can show that the function $\tilde u_l$ converges to $\beta_0$ as $l\to+\infty$ and so we have
$$
\lim_{t\to+\infty}u(t,\cdot)=\beta_0.
$$
From the equation \eqref{Eq: QS 2} we have
$$
\frac{\mathrm d}{\mathrm dt}\int_{\bar\Sigma_t}u^{-1}(t,\cdot)\,\mathrm d\sigma_{\bar g_t}=\frac{1}{r_0+t}\int_{\bar\Sigma_t}u^{-1}(t,\cdot)\,\mathrm d\sigma_{\bar g_t}.
$$
It then follows
$$
\beta_0=2\pi r_0\vol(T^k,g_{flat})\left(\int_{\bar\Sigma_0}u_0^{-1}\,\mathrm d\sigma_{\bar g_0}\right)^{-1}.
$$
Now let us deduce the contradiction under the assumption
$$
\int_{\partial\Omega}H\,\mathrm d\sigma_g>2\pi\vol(T^k,g_{flat}).
$$
Correspondingly we set $u_0=(r_0 H)^{-1}$ and so $\beta_0<1$. As a result, for $t$ large enough the mean curvature of $\Sigma_t$ with respect to the quasi-spherical metric $u^2\mathrm dt^2+\bar g_t$ is greater than that with respect to $\bar g$. Again we can obtain a contradiction with the same gluing argument as in the proof of Proposition \ref{Prop: PMT to PSC 2}.
For the equality case, the scalar flatness and the Ricci flatness of the metric $g$ come from the deformation arguments in \cite[Theorem 4.2]{SWY2019} and \cite[Corollary 2.1]{MST2010} respectively. In the following, we would like to show that the metric $g$ is flat. From the proof of Theorem \ref{Thm: main 5} this is true if the regions $(\Omega,g)$ and $(\bar\Omega_{ext},\bar g_{qs})$ are glued in a smooth manner. So the main difficuly lies in the potential singularity along the corner and we plan to deal with this issue by Ricci-DeTurk flow. We hope that the desired flatness can be preserved during the Ricci flow and there are several things to be done.
First let us check that the quasi-spherical metric $\bar g_{qs}$ is actually asymptotically conical with angle $2\pi$ at infinity. Let $v=u-1$. This is equivalent to the following estimate
\begin{equation}\label{Eq: decay qs metric}
|v|+r|\nabla_{\bar g}v|+r^2|\nabla^2_{\bar g}v|=O(r^{-\mu}),\quad \text{as}\quad r\to +\infty,
\end{equation}
for some $\mu>0$, where $r=r_0+t$ is the radical distance of $\mathbf R^2$. To analyze the behavior of the function $v$, we lift it to a function $\tilde v$ on $\Omega_{ext}\times \mathbf R^k$ and investigate its scaling $\tilde v_l=\tilde v\circ\Phi_l$ as before. From a direct computation, $\tilde v_l$ satisfies the following equation
\begin{equation}\label{Eq: uniform parabolic}
\frac{l}{r_0+l t}\frac{\partial \tilde v_l}{\partial t}=(\tilde v_l+1)^2\Delta_{\tilde g_{t,l}}\tilde v_l\quad\text{in}\quad \Sigma\times \mathbf R^k\times[0,+\infty),
\end{equation}
where $\tilde g_{t,l}$ is the smooth metric on $\Sigma\times \mathbf R^k$ given by
$$
\tilde g_{t,l}=\frac{(r_0+l t)^2}{l^2}\mathrm d\theta^2+g_{euc}.
$$
We point out that the equation $\eqref{Eq: uniform parabolic}$ gives a sequence of locally uniform parabolic equations. Denote
$$
M_l(t)=\max_{\Sigma\times \mathbf R^k}\tilde v_l(t,\cdot)\quad\text{and}\quad m_l(t)=\min_{\Sigma\times \mathbf R^k}\tilde v_l(t,\cdot).
$$
From parabolic maximum principle, the function $M_l(t)$ is decreasing and the function $m_l(t)$ is increasing as $t$ increases.
Recall that $\tilde v_l$ is $\Gamma$-invariant, where $\Gamma$ is the descrete group such that $(T^k,g_{flat})$ is the quotient $(\mathbf R^k,g_{euc})/\Gamma$.
From \cite[Theorem 1.1]{GS2021} and a covering argument we can apply the Harnack inequality to the functions $M_l(1)-\tilde v_l$ and $\tilde v_l-m_l(1)$ in $\Sigma\times \mathbf R^k\times[1,4]$. As a result, we obtain
\[
\begin{split}
M_l(1)-m_l(2)&=\max_{2\leq t\leq 3}(M_l(1)-\tilde v_l)\\
&\leq C\min_{2\leq t\leq 3}(M_l(1)-\tilde v_l)=C(M_l(1)-M_l(2))
\end{split}
\]
and
\[
\begin{split}
M_l(2)-m_l(1)&=\max_{2\leq t\leq 3}(\tilde v_l-m_l(1))\\
&\leq C\min_{2\leq t\leq 3}(\tilde v_l-m_l(1))=C(M_l(2)-m_l(1)),
\end{split}
\]
for some universal constant $C$ independent of $\tilde v_l$. Let us denote
$$\Osc_l(t)=M_l(t)-m_l(t)$$ and then we have $\Osc_l(2)\leq c\Osc_l(1)$ for some universal constant $0<c<1$ independent of $l$. From the construction of $\tilde v_l$, this further implies
$$\Osc(2l)\leq c\Osc(l),\quad \forall\,l\geq 1,$$
where $\Osc(t)$ is denoted to be the oscillation of $v(t,\cdot)$ on $\Sigma\times T^k$. Now it is standard to deduce $\Osc(t)=O(t^{-\mu})$ for some $\mu>0$ as $t\to +\infty$. Since we have
$$
\int_{\bar\Sigma_t}u^{-1}(t,\cdot)\,\mathrm d\sigma_{\bar g_t}\equiv 1,
$$
the function $v$ either is identical to zero or changes sign on $\bar\Sigma_t$. As a result, its $C^0$-norm on $\bar\Sigma_t$ is well controlled by its oscillation and so we have the estimate $|v|=O(r^{-\mu})$ with $r=r_0+t$. As a consequence, we see that the function $\tilde v_l$ satisfies $|\tilde v_l(t,\cdot)|=O(l^{-\mu})$ when $1\leq t\leq 5$. With a similar argument as in the proof of \cite[Lemma 2.5]{ST2002}, we conclude that
$$
|\tilde v_l(t,\cdot)|+|\partial \tilde v_l(t,\cdot)|+|\partial^2\tilde v_l(t,\cdot)|=O(l^{-\mu}),\quad 2\leq t\leq 4.
$$
From the definition of the map $\Phi_l$ as well as the fact that the length of $\partial_\theta$ is comparable to $l$ when $2l\leq t\leq 4l$, it is easy to deduce
$$
|v|+t|\nabla_{\bar g}v|+t^2|\nabla^2_{\bar g}v|=O(l^{-\mu}),\quad 2l\leq t\leq 4l.
$$
This yields the desired estimate \eqref{Eq: decay qs metric}.
Denote $(\tilde M_{glue},\tilde g_{glue})$ to be the Riemannian manifold from the gluing of $(\Omega,g)$ and $(\bar\Omega_{ext},\bar g_{qs})$ along their boundaries. Next we would like to use a deformation argument to verify the Ricci flatness of $(\tilde M_{glue},\tilde g_{glue})$ in its smooth part as well as the coincidence of second fundamental forms of $\partial\Omega$ in $(\tilde M_{glue},\tilde g_{glue})$ from both sides with respect to the outer unit normal pointing to $(\bar \Omega_{ext},\bar g_{qs})$. Let $U$ be a region in $\tilde M_{glue}$ containing $\Omega$ with compact closure and $h$ be a smooth $(0,2)$-tensor in $U$ with compact support such that $h$ has the form of $h_{\alpha\beta}\mathrm dx^\alpha\otimes \mathrm dx^\beta$ in some Fermi coordinate $(s,x^\alpha)$ around $\partial\Omega$. In the following, we
consider the metrics $\tilde g_{\tau}=\tilde g_{glue}+\tau h$ for $\tau$ small. From a similar discussion as in \cite[P. 423-426]{Kato1995}, the quadratic form
$$
Q_\tau(u)=\int_U|\nabla_{\tilde g_\tau}\phi|^2+cR_{\tilde g_\tau}\phi^2\,\mathrm d\mu_{\tilde g_\tau}, \quad \phi\in C^\infty(U),\quad c=\frac{\dim\Omega-2}{4(\dim\Omega-1)},
$$
is analytic with respect to $\tau$ and so its first Neumann eigenvalue $\tilde \lambda_\tau$ and its first eigenfunction $\tilde u_\tau$ with $\|\tilde u_\tau\|_{L^2(U,\tilde g_\tau)}=\vol(U,\tilde g_{glue})$ are analytic with respect to $\tau$ as well. Since the scalar curvature $R(\tilde g_{glue})$ vanishes everywhere, we see $\tilde\lambda_0=0$ and $\tilde u_0\equiv 1$. From the choise of $h$, we also know that the mean curvatures of $\partial\Omega$ in $(\tilde M,\tilde g_\tau)$ are the same on two sides with respect to the outer unit normal, denoted by $\tilde H_\tau$. From a similar computation as in \cite[Lemma 2.1]{BC2019} (see also \cite[Lemma 2.3]{GZ2021}), we have
\[
\begin{split}
&\vol(U,\tilde g_{glue})\left.\frac{\mathrm d}{\mathrm d\tau}\right|_{\tau =0}\tilde \lambda_\tau\\
=&c\int_{U}\left.\frac{\partial}{\partial \tau}\right|_{\tau=0}R(\tilde g_\tau)\,\mathrm d\mu_{\tilde g_{glue}}\\
=&c\int_{\Omega}\left.\frac{\partial}{\partial \tau}\right|_{\tau=0}R(\tilde g_\tau)\,\mathrm d\mu_{\tilde g_{glue}}+2c\int_{\partial\Omega}\left.\frac{\partial}{\partial \tau}\right|_{\tau=0}\tilde H_\tau\,\mathrm d\sigma_{\tilde g_{glue}}\\
&+c\int_{U-\Omega}\left.\frac{\partial}{\partial \tau}\right|_{\tau=0}R(\tilde g_\tau)\,\mathrm d\mu_{\tilde g_{glue}}+2c\int_{\partial\Omega}\left.\frac{\partial}{\partial \tau}\right|_{\tau=0}(-\tilde H_\tau)\,\mathrm d\sigma_{\tilde g_{glue}}\\
=&2c\int_{U}\langle h,\Ric(\tilde g_{glue})\rangle_{\tilde g_{glue}}\,\mathrm d\mu_{\tilde g_{glue}}+2c\int_{\partial\Omega}\langle A_--A_+,h\rangle_{\tilde g_{glue}}\,\mathrm d\sigma_{\tilde g_{glue}},
\end{split}
\]
where $A_-$ and $A_+$ are second fundamental forms of $\partial\Omega$ in $\Omega$ and $\tilde M-\Omega$ with respect to the outer unit normal respectively. If the smooth part of $\tilde g_{glue}$ is not Ricci-flat or the second fundamental forms $A_-$ and $A_+$ are different, we can pick up a suitable $(0,2)$-tensor $h$ such that the first Neumann eigenvalue $\tilde \lambda_\tau$ becomes positive for some $\tau$ and then a contradiction can be derived from Lemma \ref{Lem: conformal factor} as well as the conformal deformation and compactification arguments in the proof of Theorem \ref{Thm: main 5}.
Now we are ready to investigate the Ricci-DeTurk flow from $(M,\tilde g_{glue})$. The argument here is very similar to that in \cite{MS2012}. Fix a background metric $h$ on $M$ such that $h=g_{euc}\oplus g_{flat}$ around infinity and
$$
(1+\epsilon)^{-1}h\leq \tilde g_{glue}\leq (1+\epsilon)h
$$
for sufficiently small $\epsilon$. M. Simon \cite{Simon2002} proved that there is a Ricci-DeTurk $h$-flow $\{\tilde g(s)\}_{0\leq s\leq T}$ from the metric $\tilde g_{glue}$ satisfying
$$
(1+2\epsilon)^{-1}h\leq \tilde g(s)\leq (1+2\epsilon)h,\quad \forall\,s\in[0,T].
$$
The above $h$-flow $\tilde g_h(s)$ is constructed by first taking a sequence of smoothings $\tilde g^\delta$ converging to $\tilde g_{glue}$ as $\delta\to 0$, then running $h$-flows $\{\tilde g_h^\delta(s)\}_{0\leq s\leq T}$ from $\tilde g^\delta$ on a uniform interval $[0,T]$ and finally taking the limit of $h$-flows $\tilde g_h^\delta(s)$ as $\delta\to 0$. We will have a careful analysis on this procedure. Since the metric $\tilde g_{glue}$ is now $C^{1,1}$, as in \cite{Miao2002} we can construct a family of smooth metrics $\tilde g^\delta$ on $M$ converging to $\tilde g_{glue}$ as $\delta\to 0$ such that
\begin{itemize}
\item $\tilde g^\delta$ differs from $\tilde g_{glue}$ in a fixed compact subset of $M$ containing $\partial\Omega$;
\item $\tilde g^\delta$ satisfies the decay estimate
\begin{equation}\label{Eq: C2 epsilon decay}
|\tilde g^\delta-h|_h+r|\nabla_h(\tilde g^\delta-h)|_h+r^2|\nabla_h^2(\tilde g^\delta-h)|\leq Cr^{-\mu}
\end{equation}
for a universal constant $C$ independent of $\delta$;
\item there are universal positive constants $C_1$, $C_2$ and $C_3$ independent of $\delta$ such that
$
R(\tilde g^\delta)\geq -C_1
$
and
$$
\int_M|R(\tilde g^\delta)|\,\mathrm d\mu_{\tilde g^\delta}\leq C_2,\quad \int_{M}R(\tilde g^\delta)_-\,\mathrm d\mu_{\tilde g^\delta}\leq C_3\delta,
$$
where $R(\tilde g^\delta)_-$ is denoted to be the negative part of $R(\tilde g^\delta)$.
\end{itemize}
In the following, a metric is said to be asymptotically flat in $C^2_\mu$ if it satisfies \eqref{Eq: C2 epsilon decay}. From the work of Shi in \cite{Shi1989}, we can run Ricci flows $\{\tilde g^\delta(s)\}_{0\leq s\leq T}$ from $\tilde g^\delta$ on a unifrom interval $[0,T]$. Based on weighted spaces or the maximum principle, it is not difficult to show that $\tilde g^\delta(s)$ is still asymptotically flat in $C^2_\mu$. Now arguing as in \cite[Section 3]{MS2012} we can show that
\begin{equation}\label{Eq: almost nonnegative scalar curvature}
\int_{M}R(\tilde g^\delta(s))_-\,\mathrm d\mu_{\tilde g^\delta(s)}\leq C_3'\delta,
\end{equation}
where $C_3'$ is a universal constant depending on $s$ but independent of $\delta$. Now let us turn to the $h$-flow $\{\tilde g^\delta_h(s)\}_{0\leq s\leq T}$ from the smoothing metric $\tilde g^\delta$. The $h$-flow $\{\tilde g^\delta_h(s)\}_{0\leq s\leq T}$ differs from the Ricci flow $\{\tilde g^\delta(s)\}_{0\leq s\leq T}$ by a family of diffeomorphisms, but it has the advantage that we can take the limit $h$-flow $\tilde g_h(s)$ by letting $\delta\to 0$. From the maximum principle we argue as in \cite[Appendix]{MS2012} to conclude that $\tilde g^\delta_h(s)$ is asymptotically flat in $C^2_\mu$ with a uniform constant in the decay estimate \ref{Eq: C2 epsilon decay} (Notice that our starting metric is in $C^2_\mu$ which is better than the original situation in \cite[Appendix]{MS2012}). Notice that the estimate \eqref{Eq: almost nonnegative scalar curvature} also holds for $\tilde g^\delta_h$ and we conclude that the limit $h$-flow $\tilde g_h(s)$ has nonnegative scalar curvature. It follows from Theorem \ref{Thm: main 5} that the metric $\tilde g_h(s)$ has to be flat. Away from the corner $\partial\Omega$, the limit $h$-flow $\tilde g_h(s)$ converges smoothly to $\tilde g_{glue}$ and so $(\Omega,g)$ is flat.
\end{proof}
\begin{proof}[Proof for Proposition \ref{Prop: main 7}]
The proof will be divided into two cases:
{\it Case 1. $(T^2,g_{flat})$ is isometric to $\mathbf S^1(a)\times \mathbf S^1(b)$.} From Proposition \ref{Prop: main 6}, all we need to show is that if $\Omega$ is a compact $3$-manifold with boundary diffeomorphic to $\mathbf S^1(a)\times \mathbf S^1(b)$, then one of $\mathbf S^1(a)$ and $\mathbf S^1(b)$ is homotopically non-trivial in $\Omega$. This is actually a simple fact in topology and the reasoning is as follows. If $\mathbf S^1(a)$ is homotopically non-trivial, then we are done. Now let us assume that $\mathbf S^1(a)$ is homotopic to a point in $\Omega$. From Dehn's lemma we can find an embedded disk $D$ in $\Omega$ with boundary to be $\mathbf S^1(a)$. Notice that $\partial\Omega\cup D$ is homeomorphic to a solid torus minus a ball. So $\Omega$ is homeomorphic to a compact $3$-manifold in the form of $B^2\times \mathbf S^1(b)\#N$, where $N$ is a closed $3$-manifold. Now the circle $\mathbf S^1(b)$ must be homotopically non-trivial in $\Omega$.
From above discussion we can apply Proposition \ref{Prop: main 6} to conclude
\[
\begin{split}
\Lambda(T^2,g_{flat})&\leq \max\left\{2\pi b\cdot\Lambda(\mathbf S^1(a)),2\pi a\cdot\Lambda(\mathbf S^1(b))\right\}\\
&=4\pi^2\max\{a,b\}.
\end{split}
\]
{\it General case.} The proof is the same as in \cite{SWWZ2021} which is based on the quasi-spherical metric and a gluing procedure. First observe that all flat metrics on $T^2$ actually form a connected space. To see this, we start with an arbitrary flat metric $g_{flat}$ on $T^2$. It is well-known that $( T^2,g_{flat})$ can be viewed as a quotient space $\mathbf R^2/\Gamma$ for some lattice $\Gamma$ in $\mathbf R^2$. We can find an orientation-preserving affine transformation $\Phi:\mathbf R^2\to \mathbf R^2$ such that $\Phi(\mathbf Z^2)=\Gamma$. From \cite[Corollary 3.6]{GM2018} we can pick up a smooth family of orientation-preserving affine transformations
$\{\Phi_t\}_{0\leq t\leq 1}$ with $\Phi_0=\Phi$ and $\Phi_1=\id$. Then the pullback metric $\Phi_t^*(g_{euc})$ induces a smooth family of flat metrics $\{g_t\}_{0\leq t\leq 1}$ on $T^2$ with $g_0=g_{flat}$ and $g_1$ to be the product metric $g_{prod}=\mathrm d\theta_1^2+\mathrm d\theta_2^2$.
Now the basic idea is to extend any admissible fill-in of $(T^2,g_{flat})$ to an admissible fill-in of $(T^2,\lambda^2 g_{prod})$ for some $\lambda>0$ and track the change of mean curvature. The key turns out to be the construction of the neck region illustrated in Figure \ref{Fig: 8}.
\begin{figure}[htbp]
\centering
\includegraphics[width=7cm]{8.eps}
\caption{The extension of admissible fill-in}
\label{Fig: 8}
\end{figure}
Notice that there is a positive constant $k$ depending only on $\{\Phi_t\}_{0\leq t\leq 1}$ such that the metric $\bar g=\mathrm ds^2+s^2g_{\frac{s-1}{k}}$ on $[1,k+1]\times T^2$ satisfying the estimate
\begin{equation}\label{Eq: second fundamental form}
\left|\bar A_s-\frac{1}{s}\bar g_{\frac{s-1}{k}}\right|_{g_{\frac{s-1}{k}}}\leq \frac{1}{2s},\quad \text{where}\quad \bar g_{\frac{s-1}{k}}=s^2g_{\frac{s-1}{k}},
\end{equation}
where $\bar A_s$ is the second fundamental form of $\{s\}\times T^2$ with respect to $\partial_s$. We now investigate the following quasi-spherical equation
$$
\bar H_s\frac{\partial u}{\partial s}=u^2\Delta_{\bar g_{\frac{s-1}{k}}}u-\frac{1}{2}R(\bar g)u,\quad u(0,\cdot)=u_0>0.
$$
The solution $u$ exists on $[1,k+1]\times T^2$ for any positive initial value $u_0$ due to the same argument as in the proof of Proposition \ref{Prop: main 6}.
A direct computation combined with \eqref{Eq: second fundamental form} yields
\begin{equation}\label{Eq: change of mean curvature}
\begin{split}
\frac{\mathrm d}{\mathrm ds}\int_{\{s\}\times T^2}\bar H_s u^{-1}\,\mathrm d\sigma_s &=\frac{1}{2}\int_{\{s\}\times T^2}\left(\bar H_s^2-|\bar A_s|^2\right)u^{-1}\,\mathrm d\sigma_s\\
&\geq \frac{1}{6s}\int_{\{s\}\times T^2}\bar H_s u^{-1}\,\mathrm d\sigma_s.
\end{split}
\end{equation}
Let us start with an admissible fill-in $(\Omega,g)$ of $(T^2,g_{flat})$, whose boundary has total mean curvature $T_0$. After setting suitable initial value $u_0$, we can construct a quasi-spherical metric $\tilde g=u^2\mathrm ds^2+s^2\gamma_{\frac{s-1}{k}}$ on the neck region such that the mean curvature on two sides of the corner coincide. With a handle of corners as in \cite{Miao2002}, we can construct an admissible fill-in $(\Omega',g')$ of $\left(T^2,(k+1)^2g_{prod}\right)$. From \eqref{Eq: change of mean curvature} it follows that the total mean curvature of $\partial\Omega'$ is no less than $C(k)T_0$. From the discussion of Case 1, we conclude
$$\Lambda(T^2,g_{flat})\leq 4\pi^2(k+1)C(k)^{-1}<+\infty.$$
This completes the proof.
\end{proof}
\newpage
|
2024-02-18T23:40:24.463Z
|
2021-12-30T02:23:05.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14442","language":"en","timestamp":1640830985000,"url":"https:\/\/arxiv.org\/abs\/2112.14442","yymm":"2112"}
|
proofpile-arXiv_000-10029
|
{"provenance":"002.jsonl.gz:10030"}
| null | null |
\section{Introduction}
Causality is one of the most fundamental concepts in science and deeply embedded in the concept of computation. In ordinary quantum algorithms, represented within the quantum circuit model, the gates act in a fixed order on the systems. However, the study of causality at the intersection between quantum mechanics and gravity within the last two decades \cite{hardy2005probability, Zych_2019} suggested that quantum computation can be extended to more general scenarios, in which the order of the gates is controlled with an additional quantum state \cite{Chiribella_2013, Oreshkov_2012}. The use of indefinite causal structures provide numerous advantages in the field of quantum information. For instance, they lead to an exponential reduction for certain communication tasks \cite{Guerin_2016} and offer advantages in channel discrimination tasks \cite{bavaresco2020strict}. Moreover, they allow to transfer information through zero-capacity channels \cite{PhysRevLett.120.120502, salek2018quantum, chiribella2018indefinite, Guo2020, goswami2020}, although the same effect appears in causal circuits \cite{abbott2018communication, Gu_rin_2019, Rubino_2021}. Beside the theoretical interest of indefinite causal structures, including the study of the computational complexity \cite{Ara_jo_2017, Baumeler_2018}, they were experimentally demonstrated in enhanced tabletop experiments~\cite{Procopio_2015, Rubino_2017, rubino2017experimental, Goswami_2018, Guerin_experiment, taddei2020experimental}.
The simplest example of an indefinite causal structure is the quantum-$n$-switch. Here, any permutation of the $n$ unitaries can be applied on the target system but the order in which these unitaries are applied depends on the state of an additional quantum system. For example, in the case of the quantum-2-switch, a qubit controls whether the gate $U_0$ is applied before or after another gate $U_1$. It is known that using these structures one can decide whether the two gates $U_0$ and $U_1$ commute or anticommute with a single call to each gate. Solving the same task within the standard quantum circuit model, however, requires to call at least one gate twice \cite{Chiribella_2012}. This effect has also been experimentally demonstrated by Procopio et al. \cite{Procopio_2015}. In this way, the use of indefinite causal structures allows for an advantage in the number of gates that has to be called (queries).
A generalization of this task to $n$ unitary gates, originally introduced in Araújo et al. \cite{1} and often called Fourier promise problems (FPP), can be solved with the quantum-$n$-switch and a single call to each gate. At the same time, the best known solution with a causal algorithm calls $O(n\log_2{(n)})$ gates \cite{renner2021reassessing}. This result suggests that a quantum computer with a quantum-controlled ordering of gates require asymptotically fewer resources than a quantum computer with a fixed gate ordering to solve the same task. Unfortunately, the physical conditions to achieve this advantage are very demanding: for the tasks with $n$ unitaries the dimension of the control and target systems must be at least $n!$. This makes it virtually impossible to demonstrate this computational advantage experimentally. For this reason, another generalization of the task to more unitary gates has been proposed and experimentally demonstrated (for $n=4$) by Taddei et al. \cite{taddei2020experimental}. These problems, called Hadamard promise problems (HPP), offer an advantage by using the quantum-$n$-switch compared to causal circuits as well, but most importantly require only qubits. However, so far only one task of this class with four gates is explicitly known, and it remained open whether this advantage is preserved in the limiting case of a large number of black-box gates.
Here we generalize these tasks to an arbitrary number of unitary gates and show that they (1) provide a provable gap in query complexity between a quantum-controlled ordering of gates and causal quantum circuits in the asymptotic case, and (2) require only qubit gates. In fact, while all of these tasks can be solved with the quantum-$n$-switch and a single call to each gate, we prove that a causal algorithm requires at least $2n-1$ calls to the gates. Furthermore, we show that the best known techniques with a fixed gate ordering require $O(n\log_2{(n)})$ queries and conjecture that no better causal solution exists. Our findings allow to verify experimentally the scalable computational advantage of indefinite causal structures.
\section{The Hadamard Promise Problem} \label{HPP}
In the Hadamard promise problem, originally introduced in Ref.~\cite{taddei2020experimental}, a set of $d$-dimensional unitary gates $\{U_i\}_0^{n-1}$ is given and certain permutations of these unitaries are chosen. These permutations are denoted by $\Pi_x$ where the index $x$ ranges from $0$ to $n_x-1$ and $n_x\leq n!$ is the number of selected permutations. It is promised that for some $y\in \{0,1,...,n_x-1\}$ the following relations hold:
\begin{align}
\forall x\in \{0,1,...,n_x-1\}: \ \Pi_x=s(x,y)\cdot \Pi_0 \, .
\label{promise}
\end{align}
Here, the coefficients $s(x,y)$ form a $n_x \times n_x$ Hadamard matrix, an orthogonal matrix whose entries are either $+1$ or $-1$. More formally, $s(x,y)\in \{+1,-1\}$ and the rows are pairwise orthogonal to each other:\footnote{To avoid confusion, we want to mention that we label the columns with $x$ and the rows with $y$.}
\begin{align}
\begin{split}
\forall y,y'&\in\{0,1,...,n_x-1\}:\\ &\sum^{n_x-1}_{x=0} s(x,y)\cdot s(x,y')=n_x\cdot \delta_{y, y'} \, .
\end{split}
\end{align}
The task is to find the value~$y$ for which these promises are satisfied.
The simplest HPP involves two black-box unitaries $U_0$ and $U_1$. For the two permutations $\Pi_0=U_1U_0$ and $\Pi_1=U_0U_1$ it is promised that $\Pi_x=s(x,y)\ \Pi_0$ where $s(x,y)=(-1)^{x\cdot y}$. While the promise for $x=0$ becomes $\Pi_0=\Pi_0$, which is trivially satisfied, for $x=1$ it translates into:
\begin{align}
U_0U_1=(-1)^{y}\cdot U_1U_0 \, .
\end{align}
Hence, the two gates either commute ($y=0$) or anticommute ($y=1$) and the task is to find out which property is the correct one. As already mentioned in the introduction, it is known that this task can be solved with the quantum-2-switch by calling each gate only once, while in any causal quantum algorithm at least one gate has to be called twice \cite{Chiribella_2012}.\\
\begin{table}[H]
\centering
\begin{tabular}{|c||c|c||cc|cc|}\hline
\multirow{2}{*}{\backslashbox{$y$}{$x$}}
&$x=0$&$x=1$&\multicolumn{3}{c}{Examples}&\\
&$(\Pi_{0}=\Pi_{0})$&$(\Pi_{1}=(-1)^{y}\ \Pi_{0})$&$U_0$&&$U_1$&\\\hline\hline
$y=0$ & 1 & 1 &$\sigma_x$&&$\sigma_x$&\\ \hline
$y=1$ & 1 & -1 &$\sigma_y$&&$\sigma_x$&\\ \hline
\end{tabular}
\caption{The Hadamard matrix for the simplest HPP in which two unitaries either commute ($y=0$) or anticommute ($y=1$). The task is to find the correct value of~$y$.}
\label{tabex1}
\end{table}
\begin{comment}
\begin{table}[H]
\centering
\begin{tabular}{cc|c||cc}
$U_0$&$U_1$&$y$&
\begin{sideways}$\Pi_{0}$\end{sideways}&
\begin{sideways}$\Pi_{1}=(-1)^{y}\Pi_{0}$\end{sideways}\\\hline \hline
$X$&$X$& 0 & 1 & 1\\
$Y$&$Z$& 1 & 1 & -1\\
\end{tabular}
\caption{The Hadamard matrix for the simplest HPP in which two unitaries either commute or anticommute. The task is to find the correct property.}
\end{table}
\begin{table}[H]
\centering
\begin{tabular}{|c||cc||cc|}\hline
&\begin{sideways}$\Pi_{0}=\Pi_{0}$\end{sideways}&\begin{sideways}$\Pi_{1}=(-1)^{y}\Pi_{0}$\end{sideways}&\begin{sideways}Unitaries for\end{sideways}&\begin{sideways}this promise\end{sideways}\\\hline
\backslashbox{y}{x} & 0 & 1& $U_0$& $U_1$ \\\hline\hline
0 & 1 & 1 &$\sigma_x$&$\sigma_x$ \\
1 & 1 & -1 &$\sigma_y$&$\sigma_z$\\\hline
\end{tabular}
\caption{Caption}
\label{tab:my_label}
\end{table}
\end{comment}
\section{Generalizing HPPs}\label{secourmethod}
For higher $n$ only a few explicit HPPs are known. In this work, we will introduce a procedure that allows us to find a HPP for any number of involved black-box gates. The main idea is that we can combine two HPPs each with $m$ and $n$ ($d$-dimensional) unitary gates into another HPP with $m+n-1$ ($d$-dimensional) unitary gates. To do so, we denote the $m_x$ permutations of the $m$ unitaries in the first HPP with $\Pi^{(1)}_{x_1}$ such that they satisfy the following promises:
\begin{align}
\forall x_1\in \{0,1,...,m_x-1\}: \ \Pi^{(1)}_{x_1}&=s_1(x_1, y_1)\cdot \Pi^{(1)}_{0} \, .
\end{align}
In the second HPP there are $n$ involved $d$-dimensional black-box unitaries and the $n_x$ permutations, denoted as $\Pi^{(2)}_{x_2}$, satisfy the following promises:
\begin{align}
\forall x_2\in \{0,1,...,n_x-1\}: \ \Pi^{(2)}_{x_2}&=s_2(x_2, y_2)\cdot \Pi^{(2)}_{0} \, .
\end{align}
Now we choose one of the $m$ unitaries from the first HPP and replace this unitary in each of the permutations $\Pi^{(1)}_{x_1}$ with $\Pi^{(2)}_{x_2}$. In this way, we obtain $n_x\cdot m_x$ new permutations that we label with $\Pi_{(x_1,x_2)}$. One can observe that these new permutations satisfy the following relations:
\begin{align}
\Pi_{(x_1,x_2)}&=s_2(x_2,y_2)\cdot \Pi_{(x_1,0)}\\
&=s_2(x_2,y_2)\cdot s_1(x_1,y_1)\cdot \Pi_{(0,0)}\, .
\end{align}
Since $s_1(x_1,y_1)$ and $s_2(x_2,y_2)$ form an $m_x \times m_x$ and $n_x \times n_x$ Hadamard matrix, respectively, the resulting matrix with entries $s((x_1,x_2),(y_1,y_2)):=s_2(x_2,y_2)\cdot s_1(x_1,y_1)$ is a $(m_x\cdot n_x)\times (m_x\cdot n_x)$ Hadamard matrix. We prove this formally in Appendix~\ref{appA}. Hence, we have obtained another HPP with $m+n-1$ involved ($d$-dimensional) unitary black-box gates.
To give an example, we can consider the simplest HPP in Table~\ref{tabex1} with two involved unitaries. Let $U_0$ and $\tilde{U}_1$ be the unitaries for which it is promised that they either commute ($y_1=0$) or anticommute ($y_1=1$). The permutations $\Pi^{(1)}_{x_1}$ read then:
\begin{align}
\Pi^{(1)}_{x_1=0}&=\tilde{U}_1U_0\\
\Pi^{(1)}_{x_1=1}&=U_0\tilde{U}_1=(-1)^{y_1}\cdot \tilde{U}_1U_0 \, .
\end{align}
Now we can take another instance of the same HPP with $\Pi^{(2)}_{x_2=0}=U_2U_1$ and $\Pi^{(2)}_{x_2=1}=U_1U_2$ such that the two unitaries $U_1$ and $U_2$ again either commute ($y_2=0$) or anticommute ($y_2=1$):
\begin{align}
U_1U_2=(-1)^{y_2}\ U_2U_1 \, .
\end{align}
Replacing now $\tilde{U}_1$ in both of the permutations $\Pi^{(1)}_{x_1=0}=\tilde{U}_1U_0$ and $\Pi^{(1)}_{x_1=1}=U_0\tilde{U}_1$ once with $\Pi^{(2)}_{x_2=0}=U_2U_1$ and once with $\Pi^{(2)}_{x_2=1}=U_1U_2$, we obtain in total four permutations for which the following promises hold:
\begin{align}
\Pi_{(0,0)}=U_2U_1U_0& \, ,&\label{exam31} \\
\Pi_{(0,1)}=U_1U_2U_0&=(-1)^{y_2}&U_2U_1U_0 \, ,\\
\Pi_{(1,0)}=U_0U_2U_1&=(-1)^{y_1}&U_2U_1U_0 \, ,\\
\Pi_{(1,1)}=U_0U_1U_2&=(-1)^{y_1+y_2}&U_2U_1U_0 \, .\label{exam34}
\end{align}
We illustrate in Table~\ref{tab3} that these relations form indeed a $4\times 4$ Hadamard matrix. In a next step, one could split one of these three unitaries into another pair of either commuting or anticommuting unitaries. In this way, one would obtain an HPP with four unitaries and eight permutations.
\begin{comment}
For example if we split $U_2$ into $U'_2$ and $U'_3$ such that $U'_2U'_3=(-1)^{y_3}\ U'_3U'_2$ with $y_3\in\{0,1\}$ (and replace the prime afterwards), they read as follows:
\begin{align}
\Pi_{0,0,0}=U_3U_2U_1U_0&&&\\
\Pi_{1,0,0}=U_3U_2U_0U_1&=(-1)^{y_1}&U_3U_2U_1U_0&\\
\Pi_{0,1,0}=U_1U_0U_3U_2&=(-1)^{y_2}&U_3U_2U_1U_0&\\
\Pi_{1,1,0}=U_0U_1U_3U_2&=(-1)^{y_1+y_2}&U_3U_2U_1U_0&\\\\
\Pi_{0,0,1}=U_2U_3U_1U_0&=(-1)^{y_3}&U_3U_2U_1U_0&\\
\Pi_{1,0,1}=U_2U_3U_0U_1&=(-1)^{y_1+y_3}&U_3U_2U_1U_0&\\
\Pi_{0,1,1}=U_1U_0U_2U_3&=(-1)^{y_2+y_3}&U_3U_2U_1U_0&\\
\Pi_{1,1,1}=U_0U_1U_2U_3&=(-1)^{y_1+y_2+y_3}&U_3U_2U_1U_0& \, .
\end{align}
Note that one could also have split either $U_0$ or $U_1$ into two commuting or anticommuting unitaries, which would lead to another inequivalent HPP.
\end{comment}
Following this, we obtain a HPP for every number of unitary black-box gates $n$ with $n_x=2^{n-1}$ permutations and therefore a Hadamard matrix of dimension $2^{n-1}\times 2^{n-1}$. Note, however, that we are not restricted to split a unitary into a pair of commuting or anticommuting unitaries, but replacing a unitary by any set of permutations that form a HPP by themselves is possible.
\begin{table}[H]
\centering
\begin{tabular}{|c||c|c|c|c||c|c|cc|}\hline
\multirow{2}{*}{\backslashbox{$(y_1,y_2)$}{$(x_1,x_2)$}} &$x=$&$x=$&$x=$&$x=$& \multicolumn{3}{c}{Examples}&\\
& $(0,0)$& $(1,0)$& $(0,1)$& $(1,1)$&$U_0$& $U_1$& $U_2$& \\\hline\hline
$y=(0,0)$ & 1 & 1 & 1 & 1 &$\sigma_x$&$\sigma_x$&$\mathds{1}$& \\\hline
$y=(0,1)$ & 1 & 1 & -1 & -1 &$\sigma_x$&$\frac{\sigma_y+\sigma_z}{\sqrt{2}}$&$\frac{\sigma_y-\sigma_z}{\sqrt{2}}$& \\\hline
$y=(1,0)$ & 1 & -1 & 1 & -1 &$\sigma_y$&$\sigma_x$&$\mathds{1}$& \\\hline
$y=(1,1)$ & 1 & -1 & -1 & 1 &$\sigma_y$&$\frac{\sigma_y+\sigma_z}{\sqrt{2}}$&$\frac{\sigma_y-\sigma_z}{\sqrt{2}}$ &\\ \hline
\end{tabular}
\caption{The Hadamard matrix for the HPP given in \eqref{exam31}-\eqref{exam34} (for short: $\Pi_{(x_1,x_2)}=(-1)^{x_1\cdot y_1+x_2\cdot y_2}\ \Pi_{(0,0)}$). For every possible combination of the parameters $y=(y_1,y_2)$ a set of unitaries that satisfy the promise is given.
}
\label{tab3}
\end{table}
To show that these tasks are indeed realisable, one has to prove that unitaries that satisfy these promises exist. It turns out that for many tasks of this class this can be done by a straightforward approach. For instance, we obtained the examples in Table~\ref{tab3} by simply replacing the examples of $U_1=\sigma_x$ in Table~\ref{tabex1} with a pair of unitaries that either commute (if $y_2=0$) or anticommute (if $y_2=1$) and whose product is proportional to the original unitary~$U_1=\sigma_x$:
\begin{align}
U_1&=\sigma_x\ \xrightarrow{y_2=0}\ U_1=\sigma_x&&U_2=\mathds{1} \\
U_1&=\sigma_x\ \xrightarrow{y_2=1}\ U_1=\frac{\sigma_y+\sigma_z}{\sqrt{2}}&&U_2=\frac{\sigma_y-\sigma_z}{\sqrt{2}}
\end{align}
In this sense, we obtain the examples for the task with $n+1$ unitaries from the examples for the task with $n$ unitaries. Since there are some subtleties with this procedure, we discuss this further in Appendix~\ref{secexistence}.
\begin{comment}
\begin{table}[H]
\centering
\begin{tabular}{ccc|cc||cccc|cccc}
$U_0$&$U_1$&$U_2$&$y_1$&$y_2$&
\begin{sideways}$\Pi_{0,0}$\end{sideways}&
\begin{sideways}$\Pi_{1,0}=(-1)^{y_1}\Pi_{0,0}$\end{sideways}&
\begin{sideways}$\Pi_{0,1}=(-1)^{y_2}\Pi_{0,0}$\end{sideways}&
\begin{sideways}$\Pi_{1,1}=(-1)^{y_1+y_2}\Pi_{0,0}$\end{sideways}\\\hline \hline
$X$&$X$&$\mathds{1}$& 0 & 0 & 1 & 1 & 1 & 1 \\
$X$&$\frac{Y+Z}{\sqrt{2}}$&$\frac{Y-Z}{\sqrt{2}}$& 0 & 1 & 1 & 1 & -1 & -1\\
$Y$&$Z$&$\mathds{1}$& 1 & 0 & 1 & -1 & 1 & -1\\
$Y$&$\frac{X+Y}{\sqrt{2}}$&$\frac{X-Y}{\sqrt{2}}$& 1 & 1 & 1 & -1 & -1 & 1
\end{tabular}
\caption{The Hadamard matrix for the HPP given in .... For every possible combination of the $y_i$ a set of unitaries is given that satisfy these promises. Note, that they are not only rotations of Pauli matrices.}
\end{table}
\begin{table}[H]
\centering
\begin{tabular}{|c||cccc||ccc|}\hline
&\begin{sideways}$\Pi_{0,0}=\Pi_{0,0}$\end{sideways}&\begin{sideways}$\Pi_{1,0}=(-1)^{y_1}\Pi_{0,0}$\end{sideways}&\begin{sideways}$\Pi_{0,1}=(-1)^{y_2}\Pi_{0,0}$\end{sideways}&\begin{sideways}$\Pi_{1,1}=(-1)^{y_1+y_2}\Pi_{0,0}$\end{sideways}&\multicolumn{3}{c}{Unitaries for this promise}\\\hline
\backslashbox{$y=(y_1,y_2)$}{$x=(x_1,x_2)$} & \begin{sideways}(0,0)\end{sideways} & \begin{sideways}(1,0)\end{sideways}& \begin{sideways}(0,1)\end{sideways} & \begin{sideways}(1,1)\end{sideways}& $U_0$& $U_1$& $U_2$ \\\hline\hline
(0,0) & 1 & 1 & 1 & 1 &$\sigma_x$&$\sigma_x$&$\mathds{1}$ \\
(0,1) & 1 & 1 & -1 & -1 &$\sigma_x$&$\frac{\sigma_y+\sigma_z}{\sqrt{2}}$&$\frac{\sigma_y-\sigma_z}{\sqrt{2}}$ \\
(1,0) & 1 & -1 & 1 & -1 &$\sigma_y$&$\sigma_z$&$\mathds{1}$ \\
(1,1) & 1 & -1 & -1 & 1 &$\sigma_y$&$\frac{\sigma_x+\sigma_y}{\sqrt{2}}$&$\frac{\sigma_x-\sigma_y}{\sqrt{2}}$ \\ \hline
\end{tabular}
\caption{The Hadamard matrix for the HPP given in .... For every possible combination of the $y_i$ a set of unitaries is given that satisfy these promises. Note, that they are not only rotations of Pauli matrices.}
\label{tab:my_label}
\end{table}
\end{comment}
\section{Solution with the quantum-n-switch}
\begin{figure}[hbt!]
\centering
\includegraphics[width=0.5\textwidth]{switch.pdf}
\caption{Solving the HPP in Table~\ref{tab3} with the 3-switch: The state of the control system $\ket{x}_c=\ket{x_1}_{c_1} \ket{x_2}_{c_2}$ determines in which order the gates are applied on the target system. If the control system is initialized in a superposition, the quantum-$3$-switch can be used to solve this HPP by calling each unitary $U_i$ only once.
}
\label{fig:my_label}
\end{figure}
As pointed out in Ref.~\cite{taddei2020experimental}, every HPP (independent of whether it is constructed using our method or otherwise) can be solved with the quantum-$n$-switch and a single call to each gate. The quantum-$n$-switch is denoted here as $S_n$. It is the quantum gate that applies the permutation $\Pi_x$ on the target system~$\ket{\Psi_t}$ whenever the control system is in the state $\ket{x}$:
\begin{align}
\forall x\in\{0,1,...,n_x-1\}:\ S_n\ket{x}_c\otimes\ket{\Psi_t}=\ket{x}_c\otimes \Pi_x\ket{\Psi_t} \, .
\end{align}
Moreover, to every Hadamard matrix $s(x,y)$ we associate the corresponding unitary transformation $H_{n_x}$ that is defined as:
\begin{align}
\forall y\in\{0,1,...,n_x-1\}:\ H_{n_x}\ket{y}=\frac{1}{\sqrt{n_x}}\sum_{x=0}^{n_x-1}s(x,y)\ket{x} \, .
\end{align}
To solve HPPs, the $n_x$-dimensional control system is first transformed into an equal superposition of all states $x\in\{0,1,...,n_x-1\}$, usually by applying a Hadamard transformation to all control qubits.
Meanwhile, the target system $\ket{\Psi_t}$ is initialized in an arbitrary $d$-dimensional state:
\begin{align}
\left(\frac{1}{\sqrt{n_x}}\sum_{x=0}^{n_x-1}\ket{x}_c\right)\otimes\ket{\Psi_t} \, .
\end{align}
\begin{comment}
\begin{align}
\left(H_{n_x}\ket{0}_c\right)\otimes \ket{\Psi_t}=\left(\frac{1}{\sqrt{n_x}}\sum_{x=0}^{n_x-1}\ket{x}_c\right)\otimes\ket{\Psi_t} \, .
\end{align}
\end{comment}
Now, if the $n$-switch is applied, depending on the state $\ket{x}$ of the control system, the permutation $\Pi_x$ is applied on the target system~$\ket{\Psi_t}$ (see Fig.~\ref{fig:my_label} for an illustration of the map for the case of $n=3$):
\begin{align}
S_n\left(\frac{1}{\sqrt{n_x}}\sum_{x=0}^{n_x-1}\ket{x}_c\right)\otimes\ket{\Psi_t}=\frac{1}{\sqrt{n_x}}\sum_{x=0}^{n_x-1}\ket{x}_c\otimes\Pi_x\ket{\Psi_t} \, .
\end{align}
With the promise $\Pi_x=s(x,y)\cdot \Pi_0$, this state can be rewritten into:
\begin{align}
\begin{split}
\frac{1}{\sqrt{n_x}}&\sum_{x=0}^{n_x-1}\ket{x}_c\otimes\Pi_x\ket{\Psi_t}\\
&=\left(\frac{1}{\sqrt{n_x}}\sum_{x=0}^{n_x-1}s(x,y)\ket{x}_c\right)\otimes\Pi_0\ket{\Psi_t} \, .
\label{switcheq1}
\end{split}
\end{align}
In this way, the target system always ends up in the state $\Pi_0\ket{\Psi_t}$ (independent of~$x$) and factorizes out. Observe that the final state of the control system is precisely $H_{n_x}\ket{y}_c$. Hence, applying the inverse (transposed) Hadamard transform $H^{-1}_{n_x}$ on the control system, we obtain:
\begin{align}
H_{n_x}^{-1}\left(\frac{1}{\sqrt{n_x}}\sum_{x=0}^{n_x-1}s(x,y)\ket{x}_c\right)\otimes\Pi_0\ket{\Psi_t}=\ket{y}_c\otimes\Pi_0\ket{\Psi_t} \, .\label{switcheq2}
\end{align}
In this way, the solution~$y$ can be read out by a measurement of the control system in the computational basis. In the $n$-switch each unitary is called exactly once. Hence, the total query complexity of this algorithm is $n$.
\begin{comment}
\newcommand{S_n}{S_n}
\begin{figure}
\begin{center}
$\Qcircuit @C=0.5em @R=1em {
\lstick{\ket{0}_c} & \gate{H_{n_x}} & \multigate{1}{S_n} & \qw & \gate{H^{-1}_{n_x}} & \qw & \rstick{\ket{y}_c} \\
\lstick{\ket{\Psi_t}} & \qw & \ghost{S_n} & \qw & \qw & \qw & \rstick{\Pi_0 \ket{\Psi_t}} }$
\end{center}
\caption{The solution of every HPP with the quantum-$n$-switch: The Hadamard transform is used to prepare the control system in an equal superposition of all states $x$. With the quantum-$n$-switch $S_n$ the permutation $\Pi_x$, depending on the state of the control system $\ket{x}$, is applied on the target system. After the inverse Hadamard transform $H^{-1}_{n_x}$ acts on the control system, the solution~$y$ can be read out with a measurement in the computational basis.}
\label{figsimswitch}
\end{figure}
\begin{align}
\begin{split}
F_{n_x}^{-1}S_n&F_{n_x}\ket{0}_c\otimes \ket{\Psi_t}\\
&=F_{n_x}^{-1}S_n\left(\frac{1}{\sqrt{n_x}}\sum_{x=0}^{n_x-1}\ket{x}_c\right)\otimes\ket{\Psi_t}\\
&=F_{n_x}^{-1}\left(\frac{1}{\sqrt{n_x}}\sum_{x=0}^{n_x-1}\ket{x}_c\otimes\Pi_x\ket{\Psi_t}\right)\\
&=F_{n_x}^{-1}\left(\frac{1}{\sqrt{n_x}}\sum_{x=0}^{n_x-1}\omega^{x\cdot y}\ket{x}_c\right)\otimes\Pi_0\ket{\Psi_t}\\
&=\ket{y}_c\otimes\Pi_0\ket{\Psi_t} \, . \label{eqswitch}
\end{split}
\end{align}
\end{comment}
\section{Solution with causal quantum algorithms}\label{secsimswitch}
\begin{figure}[hbt!]
\smaller[1]
\begin{center}
$\Qcircuit @C=0.5em @R=1em {
\lstick{\ket{0}_{c_1}} & \gate{H} & \ctrl{3} & \qw & \ctrl{3} & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \ctrlo{3} & \qw & \ctrlo{3} & \gate{H} & \qw & \rstick{\ket{y_1}_{c_1}} \\
\lstick{\ket{0}_{c_2}} & \gate{H} & \qw & \qw & \qw & \ctrl{3} & \qw & \ctrl{3} & \qw & \ctrlo{3} & \qw & \ctrlo{3} & \qw & \qw & \qw & \gate{H} & \qw & \rstick{\ket{y_2}_{c_2}} \\
\lstick{\ket{\Psi_t}} & \qw & \qswap & \gate{U_0} & \qswap & \qswap & \gate{U_1} & \qswap & \gate{U_2} & \qswap & \gate{U_1} & \qswap & \qswap & \gate{U_0} & \qswap & \qw & \qw & \rstick{\Pi_0\ket{\Psi_t}} \\
\lstick{\ket{a_0}} & \qw & \qswap & \qw & \qswap & \qw & \qw & \qw & \qw & \qw & \qw & \qw & \qswap & \qw & \qswap & \qw & \qw & \rstick{U_0\ket{a_0}} \\
\lstick{\ket{a_1}} & \qw & \qw & \qw & \qw & \qswap & \qw & \qswap & \qw & \qswap & \qw & \qswap & \qw & \qw & \qw & \qw & \qw & \rstick{U_1\ket{a_1}}}$
\end{center}
\normalsize
\caption{Simulation of the four permutations $U_2U_1U_0$, $U_1U_2U_0$, $U_0U_2U_1$, $U_0U_1U_2$ involved in the HPP given in Table~\ref{tab3} with the smallest possible number of used black-box gates. A measurement of the control qubits at the end reveals the solution $y=(y_1, y_2)$.}
\label{soln3}
\end{figure}
It is possible to simulate the quantum-$n$-switch with a causal algorithm and $O(n^2)$ calls to the black-box gates. Since every HPP can be solved with the quantum-$n$-switch, every simulation thereof (or more precisely the simulation of all involved permutations) can solve the same task as well. For a detailed study of the simulation of the quantum-$n$-switch we refer to Ref.~\cite{Facchini_2015} (but also Ref.~\cite{Colnaghi_2012, 1, renner2021reassessing}). For example, all permutations involved in the HPP given in Table~\ref{tab3} can be simulated with the algorithm in Fig.~\ref{soln3}. This is also the shortest possible solution since such an algorithm can be used to determine for each pair of the unitaries $U_0$, $U_1$ and $U_2$ whether the pair of unitaries commute or anticommute (by setting the remaining gate to $\mathds{1}$). Such a causal algorithm requires to call at least two of the three unitaries twice, hence at least five gates are called in total.
The idea can be extended to HPPs with a set of $n$ unitary gates. Each such problem contains as a subproblem the task of deciding for each pair of gates whether that pair commutes or anticommutes. This later problem requires a minimum number of queries and thus also determines a lower bound on the number of queries for the original problem. This is specified by the following lemma.
\begin{lemma}
Consider the class of all problems that can be generated from the HPP in Table~\ref{tabex1} with the method introduced in Section~\ref{secourmethod}. For every HPP (with a set of $n$ different gates) in that class a solution with a causal quantum algorithm has to call at least $2n-1$ unitary gates.
\end{lemma}
\begin{proof}
We can show by induction, that every solution to that task must be able to determine for every pair of unitary gates whether that pair commutes or anticommutes, when we set all remaining gates to $\mathds{1}$. For the base case of $n=2$, we note that there is only the HPP given by Table~\ref{tabex1} itself for which the statement is clearly correct.
For the induction step, remember that any task with $n+1$ unitary gates is obtained by replacing one unitary $U_i$ from a task with $n$ gates with two unitaries that we denote here as $U^{(1)}_{i}$ and $U^{(2)}_{i}$. We can check that a solution to the new task must be able to determine for every pair of gates whether that pair commutes or anticommutes: (1) If the solution for the task with $n$ gates is able to determine for every pair of unitaries whether they commute or anticommute, a solution to the new task with $n+1$ gates is able to determine for every pair $U_j$ and $U_k$ with $j,k\neq i$ this property when we set $U^{(1)}_{i}=U^{(2)}_{i}=\mathds{1}$. (2) Similar, a solution to the new task is able to determine whether $U^{(1)}_{i}$ and $U_j$ (for every $j\neq i$) commute or anticommute when we set $U^{(2)}_{i}=\mathds{1}$. The analog argument holds for $U^{(2)}_{i}$ and every $U_j$ with $j\neq i$. (3) For the remaining pair of $U^{(1)}_{i}$ and $U^{(2)}_{i}$, this follows by construction of the task since part of the solution of the new task is exactly to determine whether $U^{(1)}_{i}$ and $U^{(2)}_{i}$ commute or anticommute. This proves the induction hypothesis.
Since a causal algorithm that is able to determine whether two gates commute or anticommute has to call at least one of the two gates twice \cite{Chiribella_2012}, this requires, in total, to call at least $n-1$ gates twice. Therefore, at least $2n-1$ gates have to be called in total.
\end{proof}
However, we believe that for most tasks in that class a causal solution has to call more than $O(n)$ gates. To motivate our conjecture, we want to point out that a similar argument as above holds for a very simple HPP that contains only two permutations and is defined by:
\begin{align}
\Pi_0&:=U_{n-1}U_{n-2}...U_{2}U_1U_0 \, ,\\
\Pi_1&:=U_{0}U_{1}U_{2}...U_{n-2}U_{n-1} \, .
\end{align}
It is promised that $\Pi_1=(-1)^y\ \Pi_0$ and the task is to determine $y$. A solution to that HPP is able to determine for every pair of unitaries $U_j$ and $U_k$ whether they commute or anticommute. More precisely, if we set all remaining unitaries to $\mathds{1}$, the two permutations reduce to $\Pi_0=U_kU_j$ and $\Pi_1=U_jU_k$ (given that w.l.o.g. $j<k$) from which the statement follows. In general, however, a HPP with $n$ gates that is generated with our method contains many more permutations (in fact $2^{n-1}$) and is able to determine much more structure between the unitaries.
Therefore, we conjecture that, for small $n$, a simulation of all involved permutations is the most efficient causal solution. For larger $n$, methods similar to the ones introduced in Ref.~\cite{renner2021reassessing} can be used to find more efficient solutions. Indeed, we show in Appendix~\ref{secsimswitchgeneral} that all HPPs that we can generate with our method can be solved with a causal quantum algorithm and $O(n\log_2{(n)})$ calls to the black-box gates.\footnote{There are other known techniques to solve the same tasks. They are discussed in Ref.~\cite{taddei2020experimental} and it is argued there that they require more calls to the black-box gates than a simulation of all permutations.} While we conjecture that this is the most efficient causal solution, we want to mention that there might be other problems in this class (obtained different than with our method) that offer a larger advantage.
\section{Conclusion}
Indefinite causal structures can be used to solve certain tasks more efficiently than any causally ordered quantum algorithm. In this work, we generalized a specific class of problems that provide an advantage of using a superposition of different gate orderings in the asymptotic limit. These tasks are constructed for an arbitrary number of gates and are suitable for an experimental demonstration of this computational advantage as they only involve low dimensional target systems (qubits). We showed that, while all of these tasks can be solved with the quantum-$n$-switch and a single call to each gate, causal algorithms require more calls to the black-box unitaries. We want to mention that the simplest HPP with two commuting or anticommuting unitary gates can be translated to an exponential advantage for certain communication tasks in Ref.~\cite{Guerin_2016}. We believe that our generalization of that task leads to advantages for (multipartite) communication tasks as well.
Furthermore, we found that all of these tasks can be solved with a causal algorithm and $O(n \log_2{(n)})$ calls to the black-box gates. We want to point out that currently there is no known task for which the advantage in the number of gates that has to be called is larger then $O(n)$ (for indefinite causal structures) versus $O(n \log_2{(n)})$ (for causal quantum circuits). This raises the important challenge of finding computational tasks for which indefinite causal structures provide a more significant advantage.
\section*{Acknowledgements}
We acknowledge financial support from the Austrian Science Fund (FWF) through BeyondC (F7103-N38), the project no. I-2906, as well as support by the John Templeton Foundation through grant 61466, The Quantum Information Structure of Spacetime (qiss.fr), the Foundational Questions Institute (FQXi) and the research platform TURIS. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation. Furthermore, we are thankful for the tutorial on Q-circuit \cite{eastin2004qcircuit} that helped a lot to create the quantum circuits in \LaTeX.
|
2024-02-18T23:40:24.473Z
|
2021-12-30T02:27:19.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14541","language":"en","timestamp":1640831239000,"url":"https:\/\/arxiv.org\/abs\/2112.14541","yymm":"2112"}
|
proofpile-arXiv_000-10030
|
{"provenance":"002.jsonl.gz:10031"}
| null | null |
\section{Introduction}\label{sec:introduction}}
The use of unmanned aerial vehicles (UAVs) as base stations has emerged as a solution to address the demand for high-speed and reliable wireless communication in various scenarios. UAV base stations can be deployed easily, provide connectivity during emergency situations, and improve coverage and quality of service if needed.
One of the challenging problems related to aerial base stations is the placement of the base station. The optimal base station altitude can be defined as the one that maximizes the coverage, which requires the prediction of path loss distribution in the target area. There are various models that can be used to estimate the path loss distribution, including Okumura \cite{sarkar2003survey}, Hata \cite{hata1980empirical} and Walfisch-Ikegami \cite{ChichonKurner1995} models. These models require a general classification of the area, such as "urban," "sub-urban," and "rural." Since such a rough classification of an area may not yield accurate predictions in all cases, there have been efforts to improve the results using additional information, such as the building density between transmitter and receiver \cite{5401041,6169225}. When the 3D model of an area is available, it is possible to apply ray tracing simulation techniques to get accurate results \cite{7913702, 6206329}. In addition to the requirement of a 3D model of the area, simulation based approaches have the disadvantage of high computational cost.
There are articles specifically aiming to optimize the altitude of an aerial base station. For example, in \cite{al2014optimal}, air-to-ground channel is modeled as the addition of line-of-sight (LoS) and non-line-of-sight (NLoS) terms, where the probability of LoS depends on three parameters based on the ITU recommendation \cite{data2003prediction}: the ratio of building area to total area, the average number of buildings per unit area, and the height distribution of buildings. The formulation derived in \cite{al2014optimal} is used in other work, including \cite{alzenad20173}, where horizontal and vertical placements are decoupled, and \cite{kalantari2016number}, where the problem of placing multiple aerial base stations is addressed. In \cite{mozaffari2015drone}, using a probabilistic model for LoS, the downlink coverage performance of aerial base stations is investigated, and the optimal base station altitude, which leads to a maximum ground coverage and minimum required transmit power for a single base station, is derived. The work is extended for the placement of multiple aerial base stations in \cite{mozaffari2016efficient}. In \cite{8764728}, air-to-ground channels characteristics, including height-dependent path loss models, root mean square (RMS) delay spread and the number of multi-path components, at 3.9GHz for suburban areas and for UAV altitudes up to 40m are investigated and compared against ray tracing simulations. In \cite{8770066}, large-scale and small-scale channel parameters are extracted for LoS and NLoS cases at 1GHz and 4GHz from actual measurements; and an altitude-dependent path loss model is proposed. In \cite{7835273}, extensive measurements are taken to model air-to-ground channel characteristics, including path loss, RMS delay spread, Rician K-factor, and inter-antenna correlations for sub-urban and near-urban environments.
In recent years, the use of machine learning techniques, including deep neural networks, in wireless communication applications has become increasingly popular \cite{zhang2019deep}. Machine learning based methods have been proposed to estimate path loss \cite{Zhang2019}, classify the area model selection \cite{5624542}, identify wireless technology \cite{8292183}, identify interference \cite{schmidt2017wireless}, and allocate resources \cite{8011311}. Deep neural networks have been used to predict path loss exponent and shadowing factor \cite{ates2019} and path loss distribution \cite{Omar2020}.
In this paper, we propose a deep learning based approach to determine path loss distribution in an area and optimize aerial base station (transmitter) altitude. Motivated by the fact that regional characteristics, such as building densities and heights, play a major role in path loss values, we aim to predict the path loss distribution of an area directly from its satellite image, which captures all regional characteristics visually. Given a dataset of satellite images and corresponding path loss distributions, a deep neural network is trained to convert an image to the corresponding path loss distribution.
Once the path loss distributions are estimated, the coverage, i.e., the percentage of the transmitters with path loss values less than a threshold in the region, is calculated. We design our network such that it produces path loss distributions for multiple base station altitudes. Therefore, we choose the altitude which results in the maximum coverage.
Our work is unique in the sense that it allows determining the optimal base station altitude directly from a satellite image, which is input to a deep neural network to predict path loss distribution. The path loss distribution portion of our method has similarities to the work done in \cite{Omar2020}, but there are major differences. Other than the fact that \cite{Omar2020} does not involve any altitude optimization, the network in \cite{Omar2020} produces path loss distribution for a single transmitter altitude. In this paper, the network is designed to produce multiple path loss distributions, where each distribution corresponds to a different transmitter altitude. The work in \cite{Omar2020} can be extended for multiple altitudes; however, a separate network has to be trained and stored for each transmitter altitude. In this paper, the training process involves a single network, sharing common layers for different altitudes. This reduces the overall number of parameters to be learned, which is critical in deep learning to achieve better performance for a given dataset size.
To train the proposed network, we first generate a dataset by running ray tracing simulations on 3D models of regions for which we have the satellite images as well. The ray tracing simulations produce path loss distributions, which are coupled with satellite images to form the dataset. A deep network is trained to produce path loss distributions for multiple altitudes from each satellite image. While the training process takes time, once trained, the network simply takes a satellite image to instantly produce multiple path loss distributions.
\begin{figure*}[ht]
\centering
\includegraphics[width=16cm,page=1]{Figures/lastfig-min.pdf}
\caption{A satellite image of a region, the corresponding path loss map obtained through ray tracing simulations, and the path loss distribution.}
\label{powermap}
\end{figure*}
The paper is organized as follows. In Section~\ref{sec:approach}, the proposed approach is briefly presented. The details, including the dataset generation procedure and the deep neural network architecture, are given in Section~\ref{sec:dataset} and Section~\ref{sec:prediction}, respectively. The performance of the deep neural network, the details of altitude optimization process, and the experimental results are discussed in Section~\ref{sec:results}. Finally, in Section~\ref{sec:conclusion}, the paper is concluded with a summary and possible future work.
\begin{figure*}[ht]
\centering
\includegraphics[width=16.4cm]{Figures/approach4.pdf} %
\caption{(a) A satellite image of the target region is input to a deep neural network to produce path loss distributions at multiple altitudes. The network consists of a convolutional neural network (CNN) layer, followed by several fully connected layers (fc). The output layer is designed such that the path loss distribution for a specific transmitter height (${h}_i$) is produced at a specific part of the output. The initial portion of the network is taken from the VGG16 network \cite{vgg16}; the entire network is re-trained for our dataset. (b) A threshold value is chosen to determine the coverage. Receivers with path loss value greater than the threshold are not covered by the transmitter. Path loss distributions are passed through the threshold to determine the ratio of covered receivers. The altitude that has the maximum coverage ratio is the optimum altitude.}
\label{fig:PredictionModel}
\end{figure*}
\section{Overview of the Proposed Approach}
\label{sec:approach}
The base station altitude optimization problem is defined as determining the altitude which results in the maximum number of serviceable receivers. Received signal strength, which is a function of the operation frequency, transmit power of the base station, antenna gains of the transmitter and receiver, and path loss, determines if a receiver is in the coverage area or not. Assuming there is only one transmitter and the receivers are of same type, the path loss becomes the only variable determining the coverage. The maximum allowable path loss, which we denote $PL_{th}$, can be set as the threshold; any receiver with path loss higher than this threshold is considered out of coverage.
Fig. \ref{powermap} gives an illustration of path loss map and path loss distribution of an area. On the left of the figure, we have a satellite image of a region. The 3D model, which is not shown, of the region is input to a ray tracing software to obtain the path loss values of the receivers, which are uniformly distributed over the entire area. The path loss values are shown as a color map in the middle of the figure. The path loss values are quantized to 26 bins (with 3dB bin widths) to form the path loss distribution, which is shown on the right of the figure. The coverage (the ratio of serviceable receivers with path loss value less than a specific threshold) is calculated by summing up the path loss values less than the threshold.
To determine the optimal altitude which yields the maximum coverage, we need to have path loss distributions corresponding to different base station altitudes. We, therefore, design our path loss estimation network to generate path loss distributions for multiple altitudes in a single inference.
Our approach is illustrated in Fig.~\ref{fig:PredictionModel}. An image is input to a deep neural network, which consists of convolutional layers and fully connected layers. The convolutional layers extracts high and low level features, which are then flattened passed through several fully connected layers. The output of the network is a vector, which has path loss distributions for different transmitter altitudes in its designated parts. The path loss distributions are then passed through a specified threshold value to calculate coverage for each altitude. The altitude maximizing the coverage is set as the optimum altitude.
Training a deep network requires a large dataset. In cases where the dataset size is limited, a common practice is to take a pre-trained network and re-train it with the available data for the desired application. We use this technique in our application. Specifically, we took a portion of the pre-trained VGG16 network \cite{vgg16}, appended two fully connected layers, and trained the entire network with our data. The VGG16 is pre-trained on the ImageNet dataset \cite{imagenet}, and it extracts high and low level image features, representing the regional characteristics. The fully connected layers convert these features path loss distributions.
\section{Dataset Generation}
\label{sec:dataset}
The data requirement for training a deep neural network is extremely large. Generating the dataset using actual measurements were not feasible due to time, cost and legal issues of flying UAVs in urban areas. Therefore, we use the dataset generation method presented in \cite{ates2019}. Using areas with known 3D models, ray tracing simulations are used to obtain the path loss at each receiver point for each transmitter altitude. The received path loss values are then converted to path loss distributions.
Specifically, 500 geographical regions, each with a size of $1.8 \times 1.8$ km, are extracted as in \cite{ates2019}. For each region, we have the satellite image as well as the 3D model. Each 3D model is imported to a ray tracing simulation software.\footnote{https://www.remcom.com/wireless-insite-em-propagation-software} The transmitter is placed in the center of the region at a specific height. We placed 12,100 ($110 \times 110$) receivers uniformly distributed over the region at 1.5 meters above ground. We assume having a flat terrain with dry earth material and concrete buildings. We run the simulations for each transmitter altitude and each region separately. We choose 40m, 80m, 120m, and 300m as the representative set of transmitter altitudes in our experiments. The parameters used in the dataset generation are listed in Table \ref{params}.
The resulting path loss values at the receivers are processed to generate the path loss distributions. The path loss values corresponding to receivers that are inside a building are removed by utilizing the 3D models, which indicate the building locations. The remaining path loss values are quantized to 26 bins, starting from 55dB to 130dB bin centers, and bin widths of 3dB. The distributions are normalized so that the bin values add up to one for each distribution.
We repeat the simulations for 4 different altitudes. The path loss distributions for each altitude are concatenated to construct a target vector of size 104-by-1. By repeating the simulations for different target areas, we construct a dataset of 500 images and corresponding target vectors.
\begin{table}[ht]
\centering
\resizebox{7cm}{!}{\begin{tabular}
{|l|l|}
\hline
Transmission frequency & 900 MHz \\ \hline
Transmission altitudes & 40, 80, 120 and 300 m \\ \hline
Transmitted power & +43 dBm \\ \hline
Receiver sensitivity & -85 dBm \\ \hline
Antenna radiation pattern & Omni-directional \\ \hline
Receiver antenna height & 1.5 m \\ \hline
Antenna polarization & Vertical \\ \hline
Transmitted signal & Sinusoid \\ \hline
Bandwidth & 8 MHz \\ \hline
\end{tabular}}
\caption{Ray tracing simulation parameters.}
\label{params}
\end{table}
\section{Network Architecture}
\label{sec:prediction}
The input to the network is of size $224 \times 224 \times 3$. The network produces 104 values, corresponding to path loss distributions for four transmitter heights.
The network consists of a common pre-trained network (VGG16~\cite{vgg16}) followed by fully connected layers. The VGG16 network is a well-known architecture used for image classification; and it is pre-trained on the ImageNet dataset \cite{imagenet}. This portion of the network extracts the common features associated with the region. These features include low-level features such as lines, edges, corners, and other primitive features, as well as high-level features such as geometric structures and shapes with different colors and texture. Our intuition is that these features well describe the regional characteristics that directly affect the communication channel properties of the area. The last layers allow specialization/prediction of distributions for different altitudes by properly weighting the features extracted from the common layers. The final layer is a 104 unit layer with a softmax function at the end.
The dataset is divided into 400 images for training and 100 images for testing. During training, the stochastic gradient descent optimizer \cite{bottou1991stochastic} is used with a learning rate of 0.0001, a momentum of 0.7, a batch size of 8, and cross-entropy as the loss function.
The network can be modified (by reducing the size of the bin width and therefore increasing the number of bins) to increase the resolution of path loss distributions. This would increase the number of parameters to be learned during training; and we would need more data to successfully train the network parameters. Similarly, we can add more altitudes to the network, which would again require more training data.
\section{Results and Discussions}
\label{sec:results}
\subsection{Deep learning model performance}
The mean squared error (MSE) between the true and predicted path loss distribution values is used to evaluate the network performance. Table \ref{tab:MSE} shows the MSE for each altitude. By comparing the MSE results to the variance of the true distribution values in the test set, it can be seen that the network is predicting the true distributions well. From the table, we notice that the prediction performance is better for higher altitudes. This improvement is due to the higher probability of LoS links for high altitudes. When the UAV is at a low altitude, such as $40$m, the communication link may suffer from shadowing due to high building, making it more difficult to predict path loss.
\begin{table}[ht]
\centering
\begin{tabular}{|c|c|c|}
\hline
Altitude & MSE [$10^{-3}$] & Test set variance [$10^{-3}$] \\ \hline
$40$ m & 0.65 & 4.29 \\ \hline
$80$ m & 0.32 & 2.61 \\ \hline
$120$ m & 0.21 & 2.44 \\ \hline
$300$ m & 0.11 & 4.42 \\ \hline
\end{tabular}
\caption{MSE between the true and predicted distribution values, and the variance of the true distribution values in the test set.}
\label{tab:MSE}
\end{table}
In Fig.~\ref{regression_plot}, the true versus predicted path loss distribution values are shown for the entire test set. We note that the predicted path loss values match the true values well, except for a relatively small number of outliers.
In Fig.~\ref{tab:Output_Samples}, we provide some examples of the test set showing the true versus predicted distributions with the corresponding satellite images. The regions are arranged from best to worst according to the average MSE of the four altitudes; the average MSE is written above each image. We notice that regions with low MSE values correspond to sub-urban areas. The MSE increases in the regions where there are high-rise buildings, causing extreme shadowing and multi-path reflections. In the last example, there is a high-rise building in the middle of the region, blocking the transmitter signal and resulting in high path loss at almost all receiver locations. As the transmitter altitude is increased, the receivers finally start get the signal from the transmitter. Even in such challenging situations, the network predicts the distributions well.
\begin{figure}
\centering
\includegraphics[page=2,scale=0.5]{Figures/vgg16_output3.pdf}
\caption{True versus predicted path loss distribution values.}
\label{regression_plot}
\end{figure}
Fig.~\ref{tab:Output_Samples} also includes the results of two baseline methods, i.e., the free-space path loss model \cite{sarkar2003survey} and the Okumura-Hata model \cite{hata1980empirical}. The distributions predicted with the free-space path loss and Okumura-Hata models do not match well with the true distributions as these prediction methods are generic models with few categories for the regional characteristics and can not handle all sorts of variations. On the contrary, our approach can extract the regional characteristics and provide more accurate predictions.
\begin{figure*}
\begin{tabular}{llll}
& \begin{minipage}{1in}
\includegraphics[page=13,scale=0.5,trim=0.5cm 0cm 0cm 0cm]{Figures/vgg16_output2_compressed.pdf}
\end{minipage} &
\begin{minipage}{1in}
\includegraphics[page=103,scale=0.5,trim=0.5cm 0cm 0cm 0cm]{Figures/vgg16_output2_compressed.pdf}
\end{minipage} &
\begin{minipage}{1in}
\includegraphics[page=55,scale=0.5,trim=0.5cm 0cm 0cm 0cm]{Figures/vgg16_output2_compressed.pdf}
\end{minipage} \\
&
\begin{minipage}{2in}
\includegraphics[page=14,scale=0.35]{Figures/vgg16_okufree_rural.pdf}
\end{minipage}
& \begin{minipage}{2in}
\includegraphics[page=104,scale=0.35]{Figures/vgg16_okufree_suburban.pdf}
\end{minipage} & \begin{minipage}{2in}
\includegraphics[page=56,scale=0.35]{Figures/vgg16_okufree_urban.pdf}
\end{minipage} \\
& \begin{minipage}{1in}
\includegraphics[page=99,scale=0.5,trim=0.5cm 1cm 1cm 0.1cm]{Figures/vgg16_output2_compressed.pdf}
\end{minipage} &
\begin{minipage}{1in}
\includegraphics[page=159,scale=0.5,trim=0.5cm 1cm 1cm 0.1cm]{Figures/vgg16_output2_compressed.pdf}
\end{minipage} &
\begin{minipage}{1in}
\includegraphics[page=201,scale=0.5,trim=0.5cm 1cm 1cm 0.1cm]{Figures/vgg16_output2_compressed.pdf}
\end{minipage} \\
&
\begin{minipage}{2in}
\includegraphics[page=100,scale=0.35]{Figures/vgg16_okufree_rural.pdf}
\end{minipage}
& \begin{minipage}{2in}
\includegraphics[page=160,scale=0.35]{Figures/vgg16_okufree_suburban.pdf}
\end{minipage} & \begin{minipage}{2in}
\includegraphics[page=202,scale=0.35]{Figures/vgg16_okufree_urban.pdf}
\end{minipage} \\
\end{tabular}
\caption{Sample results of the proposed path loss distribution method, compared to the true distributions, and against the free-space and Okumura-Hata models for different regions and transmitter altitudes.}
\label{tab:Output_Samples}
\end{figure*}
\begin{figure*}
\centering
\begin{tabular}{cc}
\includegraphics[page=1,scale=0.42]{Figures/covplot2dtrlog.pdf} \\
\includegraphics[page=9,scale=0.42]{Figures/covplot2dtrlog.pdf} \\
\includegraphics[page=8,scale=0.42]{Figures/covplot2dtrlog.pdf}
\end{tabular}
\caption{Coverage percentages as a function of UAV altitude and path loss threshold levels for three sample regions.}
\label{Output_Samples_true}
\end{figure*}
\subsection{Altitude optimization}
UAV altitude optimization can be done by choosing the altitude that has the maximum coverage, which is defined as the area under the path loss distribution curve below a specific threshold value. Since the distributions are quantized, the coverage is calculated by summing the values that are below the threshold value.
To investigate altitude optimization, we performed additional simulations for different altitudes, for a few regions. Fig.~\ref{Output_Samples_true} shows the coverage curves for three different regions at seven different altitudes ($40$m, $80$m, $120$m, $160$m, $300$m, $600$m, $1500$m), and for five different path loss threshold values ($116$dB, $119$dB, $122$dB, $125$dB, $128$dB). In each row of the figure, we have the satellite image of the region, the coverage as a function of transmitter altitude, and the coverage as a function of path loss threshold.
\begin{table*}
\centering
\resizebox{\textwidth}{!}{\begin{tabular}{|l|l||*{8}{c|}}
\cline{3-10}
\multicolumn{2}{c|}{}&\multicolumn{2}{c|}{\textbf{40 m}}& \multicolumn{2}{c|}{\textbf{80 m}}&\multicolumn{2}{c|}{ \textbf{120 m}}& \multicolumn{2}{c|}{\textbf{300 m}} \\ \hline
Satellite image &$PL_{th}$&True&Predicted&True&Predicted&True&Predicted&True&Predicted \\ \hline \hline
\multirow{4}{*}{\includegraphics[page=55,scale=0.14,trim=1cm 1cm 1cm 1cm,clip]{Figures/vgg16_output2_compressed.pdf}}& 128 dB
& 0.8849 &0.9153
&0.9643&0.9701
& \textcolor{red}{\textbf{0.9657}} &\textcolor{red}{\textbf{0.9802}}
&0.9532&0.978 \\
& 125 dB
& 0.8614 &0.8933
& 0.9479 &0.9598
& \textcolor{red}{\textbf{0.9512}} &\textcolor{red}{\textbf{0.9713}}
& 0.9337&0.966 \\
& 122 dB
& 0.8316 &0.864
& 0.925 &0.9429
&\textcolor{red}{\textbf{0.9308}} &\textcolor{red}{\textbf{0.9582}}
&0.9092 & 0.9487 \\
& 119 dB
& 0.7949 &0.8294
& 0.8898 &0.9222
& \textcolor{red}{\textbf{0.9052}}&\textcolor{red}{\textbf{0.9417}}
& 0.8799& 0.9262
\\\hline
\multirow{4}{*}{\includegraphics[page=77,scale=0.14,trim=1cm 1cm 1cm 1cm,clip]{Figures/vgg16_output2_compressed.pdf}}& 128 dB
&0.962 &0.9209
& 0.9681 &\textcolor{red}{\textbf{0.9808}}
&\textcolor{red}{\textbf{0.969}} &0.9786
&0.9659 &0.9805 \\
& 125 dB
& 0.952 & 0.902
& \textcolor{red}{\textbf{0.9614}} & \textcolor{red}{\textbf{0.9722}}
& 0.9609 & 0.9705
&0.9538 & 0.9688\\
& 122 dB
& 0.9371 &0.8753
& \textcolor{red}{\textbf{0.9531}} & 0.9575
& 0.953 & \textcolor{red}{\textbf{0.9593}}
&0.9402 & 0.9502 \\
& 119 dB
& 0.9186 &0.845
& 0.9405 &0.9379
& \textcolor{red}{\textbf{0.9426}}&\textcolor{red}{\textbf{0.9431}}
& 0.9183 &0.924
\\\hline
\multirow{4}{*}{\includegraphics[page=199,scale=0.14,trim=1cm 1cm 1cm 1cm,clip]{Figures/vgg16_output2_compressed.pdf}}& 128 dB
&0.1776 & 0.6299
&0.7695 &0.8105
&0.8787 & 0.8743
&\textcolor{red}{\textbf{0.9687}} & \textcolor{red}{\textbf{0.9988}}\\
& 125 dB
&0.131 & 0.5816
&0.7338 &0.7786
&0.8537 &0.8492
&\textcolor{red}{\textbf{0.9503}} & \textcolor{red}{\textbf{0.9827}}\\
& 122 dB
& 0.0796&0.5468
& 0.6988&0.7437
& 0.829&0.8217
& \textcolor{red}{\textbf{0.9308}}&\textcolor{red}{\textbf{0.9625}} \\
& 119 dB
& 0.0415&0.5017
&0.665 &0.7019
& 0.7968&0.7908
& \textcolor{red}{\textbf{0.9102}}&\textcolor{red}{\textbf{0.9366}} \\
\hline
\end{tabular}}
\caption{Coverage values calculated from true distributions and predicted distributions at different path loss thresholds $PL_{th}$ for various regions. Highest true and predicted coverage values are highlighted in "red" color.}
\label{tab:compare}
\end{table*}
In the first region, there are relatively few buildings. This means, the region can be covered well with LoS links even when the UAV altitude is low. As the altitude becomes higher, the coverage drops because of the reduction in signal strength. (See the coverage-altitude plot.) As the path loss threshold value is increased, we get more coverage as expected.
In the second and the third regions, there are more buildings; therefore, low UAV altitudes result in less coverage. Better coverage can be achieved by increasing the UAV altitude. As the UAV altitude is increased, the coverage eventually starts to decay due to the increasing distance between the transmitter and receiver. The optimum altitude for the second region is higher than the optimum altitude for the third region because of the existence of high-rise buildings in the second region. For a path loss threshold of $128$dB, a coverage of about $95\%$ is achieved when the transmitter is at $600$m for the second region; whereas, the optimum coverage is achieved when the transmitter is at $300$m for the third region.
We can also make further deductions by investigating the coverage-altitude and coverage-threshold plots. For example, in the first region, where we mostly have LoS links, the coverage changes little for $40$m to $600$m. The effect of altitude is more when there are buildings in the region, as in the cases of second and third images.
Combining the path loss distribution estimations from the deep network and the altitude optimization process, we have a complete flow from an image to an optimum altitude. We tested the process with the path loss distributions obtained from the deep network. In Table \ref{tab:compare}, we exemplify coverage values obtained from true distributions and predicted distributions. We show two correct and one incorrect altitude selection. In the first and third regions, both the true distributions and predicted distributions result in the same altitude, 120m and 300m, respectively. In the second example, predicted distributions resulted in incorrect optimum altitudes for two threshold values. Even in those few incorrect cases the coverage values are close to each other at different altitudes, meaning that even an incorrect altitude is selected, similar coverage would be achieved.
\section{Conclusions}
\label{sec:conclusion}
In this paper, we present a deep learning based approach to estimate the path loss distributions in an area for multiple altitudes in a single inference from a 2D satellite image, and then use the predicted path loss distributions to optimize the altitude of a UAV, serving as a base station in an air-to-ground communication systems.
The training process of the deep neural network is computational demanding; but once the network is trained, the inference can be done accurately in real time. This is a significant advantage over ray tracing simulations.
While we demonstrated the idea for four different altitudes due to the extensive simulation requirements to obtain the training dataset for each altitude, the approach can be extended for more altitudes. The main challenge is the dataset generation process, which require many satellite images and ray tracing simulations on the 3D models corresponding to the satellite images. Similarly, the method can be extended for other transmission frequencies.
Another possible extension of this work is to move the transmitter position in large satellite image, crop a target region centered around the transmitter position, and predict the path loss distributions for each transmitter position using the cropped target region. In this way, both the optimum altitude and position can be determined.
\bibliographystyle{IEEEtran}
|
2024-02-18T23:40:24.475Z
|
2021-12-30T02:27:31.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14551","language":"en","timestamp":1640831251000,"url":"https:\/\/arxiv.org\/abs\/2112.14551","yymm":"2112"}
|
proofpile-arXiv_000-10031
|
{"provenance":"002.jsonl.gz:10032"}
| null | null |
\section{Introduction}\label{sec:introduction}
\renewcommand{\thethm}{\arabic{thm}}
This paper is devoted to the study of face rings of simplicial spheres, especially PL-spheres. The face ring $\kk[\Delta]$ of a simplicial complex $\Delta$ is a quotient of a polynomial ring over a field $\kk$. Its properties were investigated by Hochster \cite{H75}, Reisner \cite{Rei76} and Stanley \cite{Sta75} in 1970's. Face rings play a key role in combinatorial commutative algebra, because their algebraic properties reflect many surprising combinatorial and topological properties of the associated simplicial complexes.
For example, the hard Lefschetz theorem for face rings of simplicial polytope, which was originally proved by Stanley \cite{S80}, and later by McMullen \cite{Mc93,Mc96}, verifies the necessity part of McMullen's conjecture \cite{Mc71} that posited a complete characterization of face numbers of simplicial polytopes. The sufficiency part was proved by Billera and Lee \cite{BL81}. Since then, one of the most important problems in the realm of face enumeration is whether or not McMullen's
conditions extend to simplicial spheres or even homology spheres. This question has become known as the $g$-conjecture. The algebraic version of this conjecture, which implies the $g$-conjecture, posits that all homology spheres satisfy the hard Lefschetz theorem.
Recently, a proof of the algebraic $g$-conjecture, due to Adiprasito, has appeared \cite{A18}. Adiprasito's proof is quite technical, because it involves many complicated geometric constructions. More recently, Papadakis and Petrotou \cite{PP20} gave another purely algebraic proof of this conjecture for the case that $\kk$ has characteristic $2$.
Then, Adiprasito, Papadakis, and Petrotou posted a new preprint \cite{APP21}, which provides far reaching generalizations of these algebraic results to much
more general simplicial complexes such as normal pseudomanifolds.
A more surprising result is that Papadakis and Petrotou \cite{PP20} established an essentially stronger algebraic properties than the Lefschetz property for simplicial spheres, which says that every simplicial sphere is generically anisotropic over a filed of characteristic $2$ (see Subsection \ref{subsec:anisotropy}).
In this paper, we mainly follow the idea in \cite{PP20}, and focus on the face rings of PL-spheres.
This paper is organized as follows: in Section \ref{sec:preliminaries}, we introduce the basic notions and collect the previous results about face rings. In Section \ref{sec:equivalence}, we establish an equivalent but simpler condition of being generically anisotropic (Lemma \ref{lem:aniso equiv}), and use it to generalize a result in \cite{PP20} that $1$-dimensional simplicial spheres are generically anisotropic over any field $\mathbb{F}$, to stacked spheres (Theorem \ref{thm:0-move}). Section \ref{sec:char 2} is devoted to giving a simple proof of Papadakis-Petrotou theorem (Theorem \ref{thm:PP}) for PL-spheres by using a bistellar move argument (Theorem \ref{thm:bistellar}). In Section \ref{sec:2-sphere}, we prove that the generic anisotropy of odd dimensional PL-spheres implies the generic anisotropy of even dimensional PL-spheres of one higher dimension (Theorem \ref{thm:general char}). As a consequence, we see that every $2$-dimensional simplicial sphere is generically anisotropic over any field $\mathbb{F}$ (Corollary \ref{cor:2-sphere}), solving a conjecture in \cite{PP20,APP21} in this special case.
\section{Preliminaries}\label{sec:preliminaries}
\renewcommand{\thethm}{\thesection.\arabic{thm}}
\subsection{Basic concepts}\label{subsec:notation}
Throughout this paper, we assume that $\Delta$ is a simplicial complex.
If $\Delta$ has $m$ vertices, we usually identify the vertices of $\Delta$ with $[m]=\{1,\dots,m\}$.
We refer to $i$-dimensional faces as \emph{$i$-faces}, and denote by $\FF_i(\Delta)$ the set of $i$-faces of $\Delta$.
A simplicial complex is \emph{pure} if all of its facets (maximal faces) have the same dimension.
By $\Delta^{m-1}$ we denote the simplicial complex consisting of all subsets of $[m]$; its boundary $\partial\Delta^{m-1}$ will be the subcomplex of all proper subsets of $[m]$. By abuse of notation, sometimes the symbol $\sigma$ will be used ambiguously to denote a face $\sigma\in\Delta$ and also the simplicial complex consisting of $\sigma$ and all its faces.
For a $(d-1)$-dimensional simplicial complex $\Delta$, the \emph{$f$-vector} of $\Delta$ is
\[(f_0,f_1,\dots,f_{d-1}),\]
where $f_i$ is the number of the $i$-dimensional faces of $\Delta$.
Sometimes it is convenient to set $f_{-1}=1$ corresponding to the empty set. The \emph{$h$-vector} of $\Delta$ is the integer vector
$(h_0,h_1,\dots,h_d)$ defined from the equation
\[
h_0t^d+\cdots+h_{d-1}t+h_d=f_{-1}(t-1)^d+f_0(t-1)^{d-1}+\cdots+f_{d-1}.
\]
The \emph{link} and \emph{star} of a face $\sigma\in\Delta$ are respectively the subcomplexes
\[\begin{split}
\mathrm{lk}_\sigma\Delta=&\{\tau\in\Delta:\tau\cup\sigma\in\Delta,\tau\cap\sigma=\emptyset\};\\
\mathrm{st}_\sigma\Delta=&\{\tau\in\Delta:\tau\cup\sigma\in\Delta\}.
\end{split}\]
The \emph{join} of two simplicial complexes $\Delta$ and $\Delta'$, where the vertex set $\FF_0(\Delta)$ is disjoint from $\FF_0(\Delta')$, is the simplicial complex
\[\Delta*\Delta'=\{\sigma\cup\sigma':\sigma\in\Delta, \sigma'\in\Delta'\}.\]
Let $\Delta$ be a pure simplicial complex of dimension $d$ and $\sigma\in\Delta$ a $(d-i)$-face such that $\mathrm{lk}_\sigma\Delta=\partial\Delta^i$ and the subset $\tau=\FF_0(\Delta^i)\subset \FF_0(\Delta)$ is not a face of $\Delta$. Then the operation $\chi_\sigma$ on $\Delta$ defined by
\[\chi_\sigma\Delta=(\Delta\setminus\mathrm{st}_\sigma\Delta)\cup(\partial\sigma*\tau)\]
is called a \emph{bistellar $i$-move}. Obviously we have $\chi_\tau\chi_\sigma\Delta=\Delta$. Two pure simplicial complexes are \emph{bistellarly equivalent} if one is transformed to another by a finite sequence of bistellar moves.
A \emph{piecewise linear (PL for short) $d$-sphere} is a simplicial complex which has a common subdivision with $\partial\Delta^{d+1}$. A \emph{PL $d$-manifold} is a simplicial complex $\Delta$ of dimension $d$ such that $\mathrm{lk}_\sigma\Delta$ is a PL-sphere of dimension $d-|\sigma|$ for every nonempty face $\sigma\in\Delta$, where $|\sigma|$ denotes the cardinality of $\sigma$.
It is obvious that two bistellarly equivalent PL-manifolds are PL homeomorphic, i.e., they have a common subdivision. The following fundamental result shows that the converse is also true.
\begin{thm}[Pachner {\cite[(5.5)]{P91}}]\label{thm:pachner}
Two PL-manifolds are bistellarly equivalent if and only if they are PL homeomorphic.
\end{thm}
A simplicial complex $\Delta$ is called a \emph{$\kk$-homology $d$-sphere} ($\kk$ is a field) if
\[H_*(\mathrm{lk}_\sigma\Delta;\kk)=H_*(S^{d-|\sigma|};\kk)\quad \text{for all }\sigma\in\Delta \text{ (including $\sigma=\emptyset$)}.\]
(Remark: Usually, the terminology ``homology sphere" means a manifold having the homology of a sphere. Here we take it in a more relaxed sense than its usual meaning.)
\subsection{Face rings and l.s.o.p}\label{subsec:l.s.o.p.}
For a field $\kk$, let $S=\kk[x_1,\dots,x_m]$ be the polynomial algebra with one generator for each
vertex in $\Delta$. It is a graded algebra by setting $\deg x_i=1$.
The \emph{Stanley-Reisner ideal} of $\Delta$ is
\[I_\Delta:=(x_{i_1}x_{i_2}\cdots x_{i_k}:\{i_1,i_2,\dots,i_k\}\not\in\Delta)
\]
The \emph{Stanley-Reisner ring} (or \emph{face ring}) of $\Delta$ is the quotient \[\kk[\Delta]:=S/I_\Delta.\]
Since $I_\Delta$ is a monomial ideal, the quotient ring $\kk[\Delta]$ is graded by degree.
For a face $\sigma=\{i_1,\dots,i_k\}\in\FF_{k-1}(\Delta)$, denote by $\xx_\sigma=x_{i_1}\cdots x_{i_k}\in\kk[\Delta]$ the face monomial corresponding to $\sigma$.
A sequence $\Theta=(\theta_1,\dots,\theta_d)$ of $d=\dim\Delta+1$ linear forms in $S$ is called a \emph{linear
system of parameters} (or \emph{l.s.o.p.} for short) if
\[\kk(\Delta;\Theta):=\kk[\Delta]/(\Theta)\]
has Krull dimension zero, i.e., it is a finite-dimensional $\kk$-space.
The quotient ring $\kk(\Delta;\Theta)$ is an \emph{Artinian reduction} of $\kk[\Delta]$. We will use the simplified notation $\kk(\Delta)$ for $\kk(\Delta;\Theta)$ whenever it creates no confusion, and write the component
of degree $i$ of $\kk(\Delta)$ as $\kk(\Delta)_i$. For a subcomplex $\Delta'\subset\Delta$, let $I$ be the ideal of $\kk[\Delta]$ generated by faces in $\Delta\setminus\Delta'$, and denote $I/I\Theta$ by $\kk(\Delta,\Delta';\Theta)$ or simply $\kk(\Delta,\Delta')$.
A linear sequence $\Theta=(\theta_1,\dots,\theta_d)$ is an l.s.o.p if and only if the restriction $\Theta_\sigma=r_\sigma(\Theta)$ to each face $\sigma\in\Delta$ generates the polynomial algebra $\kk[x_i:i\in\sigma]$; here $r_\sigma:\kk[\Delta]\to\kk[x_i:i\in\sigma]$ is the projection homomorphism. If $\Theta$ is an l.s.o.p. for $\kk[\Delta]$, then $\kk(\Delta)$ is spanned by the face monomials (see \cite[II. Theorem 5.1.16]{BH98}). It is a general fact that if $\kk$ is an infinite field, then $\kk[\Delta]$ admits an l.s.o.p. (Noether normalization lemma).
Suppose $\Theta=(\theta_i=\sum_{j=1}^m a_{ij}x_j)_{i=1}^d$ is an l.s.o.p. for $\kk[\Delta]$. Then there is an associated $d\times m$ matrix $M_\Theta=(a_{ij})$.
Let $\boldsymbol{\lambda}_i=(a_{1i},a_{2i},\dots,a_{di})^T$ denote the column vector corresponding to the vertex $i\in[m]$. For any ordered subset $I=(i_1,\dots,i_k)\subset [m]$, the submatrix $M_\Theta(I)$ of $M_\Theta$ is defined to be
\[M_\Theta(I)=(\boldsymbol{\lambda}_{i_1},\dots,\boldsymbol{\lambda}_{i_k}).\]
\subsection{Cohen-Macaulay and Gorenstein complexes}
Throughout this subsection, $\kk$ is an infinite field of arbitrary characteristic.
Let $\Delta$ be a simplicial complex of dimension $d-1$. The face ring $\kk[\Delta]$ is a \emph{Cohen-Macaulay ring} if for any l.s.o.p $\Theta=(\theta_1,\dots,\theta_d)$, $\kk[\Delta]$ is a free $\kk[\theta_1,\cdots,\theta_d]$ module. In this case, $\Delta$ is called a \emph{Cohen-Macaulay complex over $\kk$}.
If $\Delta$ is Cohen-Macaulay, the following result of Stanley shows that the $h$-vector of $\Delta$ has a pure algebraic description.
\begin{thm}[Stanley \cite{Sta75}]\label{thm:stanley}
Let $\Delta$ be a $(d-1)$-dimensional Cohen-Macaulay complex and let $\Theta=
(\theta_1,\dots,\theta_d)$ be any l.s.o.p. for $\kk[\Delta]$. Then
\[\dim_\kk\kk(\Delta)_i=h_i(\Delta),\quad \text{for all } 0\leq i\leq d.\]
\end{thm}
The face ring $\kk[\Delta]$ is a \emph{Gorenstein ring} if $\kk(\Delta;\Theta)$ is a Poincar\'e duality $\kk$-algebra for any l.s.o.p $\Theta$.
In this case, $\Delta$ is called a \emph{Gorenstein complex over $\kk$}.
Further, $\Delta$ is called \emph{Gorenstein*} if $\kk[\Delta]$ is Gorenstein and $\Delta$ is not a cone, i.e., $\Delta\neq\Delta^0*\Delta'$ for any $\Delta'$.
These algebraic properties of face rings have combinatorial-topological characterisations as follows.
\begin{thm}\label{thm:algebraic property}
Let $\Delta$ be a simplicial complex. Then
\begin{enumerate}[(a)]
\item {\rm(Reisner \cite{Rei76})} $\Delta$ is Cohen-Macaulay (over $\kk$) if and only if for all faces $\sigma\in\Delta$ (including $\sigma=\emptyset$)
and $i<\dim\mathrm{lk}_\sigma\Delta$, we have $\w H_i(\mathrm{lk}_\sigma\Delta;\kk)=0$.\vspace{8pt}
\item {\rm(Stanley \cite[II. Theorem 5.1]{S96})} $\Delta$ is Gorenstein* (over $\kk$) if and only if it is a $\kk$-homology sphere.\vspace{8pt}
\end{enumerate}
\end{thm}
\subsection{Lefschetz property of face rings}\label{subsec:lefschetz}
The Lefschetz property of face rings is strongly connected to many topics in algebraic geometry, commutative algebra and combinatorics. For instance, the Lefschetz property for homology spheres is just the algebraic $g$-conjecture, and then implies the $g$-conjecture. Its general definition is as follows.
\begin{Def}\label{def:lefschetz}
Let $\Delta$ be a Cohen-Macaulay complex (over $\kk$). We say that $\kk[\Delta]$ (or simply $\Delta$) has the \emph{Lefschetz property}
if there is an Artinian reduction $\kk(\Delta;\Theta)$ of $\kk[\Delta]$ and a linear form
$\omega\in\kk[\Delta]_1$ such that the multiplication map
\[\cdot\omega^{i}:\kk(\Delta;\Theta)_{j}\to\kk(\Delta;\Theta)_{j+i}\]
is either surjective or injective for all $i\geq 1$ and $j\geq 0$.
\end{Def}
In the case of a $\kk$-homology $(d-1)$-sphere $\Delta$, the Lefschetz property for $\Delta$ is equivalent to the property that there exists $(\Theta,\omega)$ such that the multiplication map \[\cdot\omega^{d-2i}:\kk(\Delta;\Theta)_{i}\to\kk(\Delta;\Theta)_{d-i}\]
is an isomorphism for all $i\leq d/2$, known as the \emph{hard Lefschetz property}.
Note that the set of $(\Theta,\omega)$ in definition \ref{def:lefschetz} is
Zariski open, but it may be empty. The recent striking result by Adiprasito \cite{A18}, and by Adiprasito, Papadakis, and Petrotou \cite{APP21} shows that this set is nonempty for homology spheres.
\begin{thm}[Hard Lefschetz \cite{A18,APP21}]\label{thm:hard lefschetz}
Let $\kk$ be an infinite field. Then every $\kk$-homology sphere has the Lefschetz property.
\end{thm}
\subsection{Anisotropy of face rings}\label{subsec:anisotropy}
As we mentioned in Section \ref{sec:introduction}, in order to prove the Lefschetz property for simplicial spheres, Papadakis and Petrotou \cite{PP20} established a stronger property for face rings of simplicial spheres.
Here is the formal definition.
\begin{Def}\label{def:anisotropy}
Let $\Delta$ be a $\kk$-homology sphere of dimension $d-1$. An Artinian reduction $\kk(\Delta)$ of $\kk[\Delta]$ is said to be \emph{anisotropic} if for every nonzero element $u\in\kk(\Delta)_i$ with $i\leq d/2$, the square $u^2$ is also nonzero in $\kk(\Delta)_{2i}$. We call $\Delta$ anisotropic over $\kk$ if such an Artinian reduction exists.
\end{Def}
\begin{thm}[{\cite[Papadakis-Petrotou]{PP20}}]\label{thm:PP}
If $\mathbb{F}$ is a field of characteristic $2$ and $\Delta$ is a $\mathbb{F}$-homology sphere, then there exists a purely transcendental field extension $\kk$ of $\mathbb{F}$ such that $\Delta$ is anisotropic over $\kk$.
\end{thm}
It turns out that being anisotropic is stronger than the Lefschetz property in the sence that if every simplicial sphere is anisotropic over $\kk$, then every simplicial sphere has the Lefschetz property over any infinite field $\kk'$ of the same characteristic as $\kk$ (see \cite[Section 9]{PP20}).
The transcendental field extension $\kk$ and the Artinian reduction $\kk(\Delta)$ in Theorem \ref{thm:PP} can be chosen as follows. Suppose that $\dim \Delta=d-1$, $\FF_0(\Delta)=[m]$. Set \[\kk=\mathbb{F}(a_{ij}:1\leq i\leq d,\,1\leq j\leq m),\]
the field of rational functions on variables $a_{ij}$,
and set $\theta_i=\sum_{i=1}^m a_{ij}$ for $1\leq i\leq d$. Then $\Theta=(\theta_i:1\leq i\leq d)$ is clearly an l.s.o.p. for $\kk[\Delta]$. For such $\kk$ and $\Theta$, \cite{PP20} shows that $\kk(\Delta;\Theta)$ is anisotropic. In this situation, $\Delta$ is also refered to be \emph{generically anisotropic over $\mathbb{F}$}.
\subsection{Canonical modules for homology balls} In this subsection we recall some results about the \emph{canonical module} (see \cite[I.12]{S96} for the definition) of $\kk[\Delta]$ when $\Delta$ is a homology ball.
Let $\Delta$ be a $\kk$-homology $(d-1)$-ball with boundary $\partial \Delta$. Then there is an exact sequence
\begin{equation}\label{eq:exact}
0\to I\to \kk[\Delta]\to\kk[\partial\Delta]\to 0,
\end{equation}
where $I$ is the ideal of $\kk[\Delta]$ generated by all faces in $\Delta\setminus\partial\Delta$. By a theorem of Hochster \cite[Theorem II.7.3]{S96} $I$ is the canonical module of $\kk[\Delta]$. Then from \cite[Theorem I.3.3.4 (d) and Theorem I.3.3.5 (a)]{BH98} we have the following proposition.
\begin{prop}\label{prop:pairing}
Let $\Delta$ be a $\kk$-homology $(d-1)$-ball. Then there is a perfect bilinear pairing between the Artinian reductions
\[
\kk(\Delta)_i\times \kk(\Delta,\partial\Delta)_{d-i}\to \kk(\Delta,\partial\Delta)_d=\kk.
\]
\end{prop}
\subsection{Lee's formula}\label{subsec:Lee's}
In this subsection, we introduce a formula due to Lee that expresses non-square-free monomials in $\kk(\Delta)$ in terms of face monomials. First, we recall a useful result in \cite{PP20}.
\begin{lem}[{\cite[Corollary 4.5]{PP20}}]\label{lem:generator}
Let $\Delta$ be a $(d-1)$-dimensional $\kk$-homology sphere or ball, $\Theta$ be an l.s.o.p. for $\kk[\Delta]$. Suppose that $\sigma_1$ and $\sigma_2$ are two ordered facets of $\Delta$, which have the same orientation in $\Delta$. Then \[\det(M_\Theta(\sigma_1))\cdot\xx_{\sigma_1}=\det(M_\Theta(\sigma_2))\cdot\xx_{\sigma_2}\] in $\kk(\Delta)_d$ or in $\kk(\Delta,\partial\Delta)_d$, respectively.
\end{lem}
When $\Delta$ is a $(d-1)$-dimensional $\kk$-homology sphere or ball, $\kk(\Delta)_d$ or $\kk(\Delta,\partial\Delta)_d$ respectively, is an $1$-dimensional $\kk$-vector space, which is spanned by a facet. So each facet $\sigma\in\Delta$ defines a map \[\Psi_\sigma:\kk(\Delta)_d\text{ or } \kk(\Delta,\partial\Delta)_d\to \kk\]
such that for all $\alpha$ in $\kk(\Delta)_d$ or $\kk(\Delta,\partial\Delta)_d$,
\[\alpha=\Psi_\sigma(\alpha)\det(M_\Theta(\sigma))\xx_{\sigma}.\]
Lemma \ref{lem:generator} says that $\Psi_\sigma=\pm\Psi_\tau$ for any two facets $\sigma,\tau\in\Delta$.
If we fix an orientation on $\Delta$, this map is independent of the choice of the oriented facet, giving a \emph{canonical function} $\Psi_\Delta:\kk(\Delta)_d\text{ or } \kk(\Delta,\partial\Delta)_d\to \kk$ (see \cite[Remark 4.6]{PP20}). In particular, if $\sigma$ is a facet of $\Delta$, then $\Psi_\Delta(\xx_\sigma)=1/\det(M_\Theta(\sigma))$.
To state Lee's formula, we will need the following notation. Under the assumption of Lemma \ref{lem:generator}, let $\mathbf{a}=(a_1,\dots,a_d)^T\in\kk^d$ be a vector such that every $d\times d$ minor of the matrix $(M_\Theta\mid\mathbf{a})$ is nonsingular. For any ordered subset $I\subset[m]$ with $|I|=d$, let $A_I=\det(M_\Theta(I))$, and for any $i\in I$, denote by $A_I(i)$ the determinant of the matrix obtained from $M_\Theta(I)$ by replacing the column vector $\boldsymbol{\lambda}_i$ with $\mathbf{a}$.
\begin{thm}[Lee {\cite[Theorem 11]{Lee96}}]\label{thm:Lee}
Let $\Delta$ be a $(d-1)$-dimensional $\kk$-homology sphere (resp. $\kk$-homology ball), and fix an orientation on $\Delta$. Then for a monomial $x_{i_1}^{r_1}\cdots x_{k}^{r_k}\in\kk(\Delta)_d$ (resp. $x_{i_1}^{r_1}\cdots x_{k}^{r_k}\in\kk(\Delta,\partial \Delta)_d$), $r_i>0$, we have
\[\Psi_\Delta(x_{i_1}^{r_1}\cdots x_{k}^{r_k})=\sum_{F\in\FF_{d-1}(\mathrm{st}_\sigma\Delta)}\frac{\prod_{i\in\sigma}A_F(i)^{r_i-1}}{A_F\prod_{i\in F\setminus\sigma}A_F(i)},\]
where $\sigma=\{i_1,\dots,i_k\}$ and the sum is over all ordered facets of $\mathrm{st}_\sigma\Delta$, which are compatible with the given orientation.
\end{thm}
\section{Equivalence of anisotropy}\label{sec:equivalence}
As we have seen in subsection \ref{subsec:anisotropy}, if $\mathbb{F}$ is a field of characteristic $2$ and $\Delta$ is a $\mathbb{F}$-homology $(d-1)$-sphere with $m$ vertices, then $\kk(\Delta;\Theta)$ is anisotropic for the field extension
\[\kk:=\mathbb{F}(a_{ij}:1\leq i\leq d,\,1\leq j\leq m)\]
and the l.s.o.p. $\Theta=(\theta_i=\sum_{i=1}^m a_{ij})_{i=1}^d$.
In fact, the field extension in Theorem \ref{thm:PP} can be chosen to be smaller than $\kk$, as we will see below.
Let
\[\kk'=\mathbb{F}(a_{ij}:1\leq i\leq d,\,d+1\leq j\leq m),\]
and denote by $A$ the $d\times (m-d)$ matrix $(a_{ij})$. One easily sees that there is an l.s.o.p. $\Theta'$ for $\kk'[\Delta]$ such that $M_{\Theta'}=( I_d\mid A)$, where $I_d$ is the $d\times d$ identity matrix.
\begin{lem}\label{lem:aniso equiv}
Suppose that $\Delta$ is a $\mathbb{F}$-homology ($\mathrm{char}\,\mathbb{F}$ is arbitrary) $(d-1)$-sphere with $m$ vertices. Let $\kk$, $\Theta$ and $\kk'$, $\Theta'$ be as above.
Then $\kk(\Delta;\Theta)$ is anisotropic if and only if $\kk'(\Delta;\Theta')$ is anisotropic.
\end{lem}
\begin{proof}
``$\Rightarrow$". There exists a matrix $N\in GL(d,\kk)$ such that $N M_\Theta=(I_d\mid B)$, where $B=(b_{ij})$ ($1\leq i\leq d,\,d+1\leq j\leq m$) is a $d\times (m-d)$ matrix with entries $b_{ij}\in\kk$. Denote by $\Theta_0$ the l.s.o.p. corresponding to $(I_d\mid B)$. Clearly, the two ideals generated by $\Theta$ and $\Theta_0$ are the same. Let
\[\kk_0=\mathbb{F}(b_{ij}:1\leq i\leq d,\,d+1\leq j\leq m).\]
Then $\kk_0$ is a subfield of $\kk$. One easily sees that $b_{ij}$ are algebraically independent elements over $\mathbb{F}$, so there is an isomorphism $\kk'\cong\kk_0$ given by $a_{ij}\mapsto b_{ij}$, and then an induced isomorphism $\kk'(\Delta;\Theta')\cong \kk_0(\Delta;\Theta_0)$. Since $\kk_0(\Delta;\Theta_0)\subset \kk(\Delta;\Theta_0)=\kk(\Delta;\Theta)$
and $\kk(\Delta;\Theta)$ is anisotropic, any nonzero element
$u\in\kk_0(\Delta;\Theta_0)_i$ with $i\leq d/2$ satisfies $u^2\neq0$. Hence $\kk'(\Delta;\Theta')$ is anisotropic.
``$\Rightarrow$". Pick an arbitrary order on the variables $a_{ij}$ for $1\leq i\leq d$, $1\leq j\leq d$, and rewrite them as $a_1,a_2,\dots,a_{d^2}$. Let $\kk_0=\kk'$, and recursively define $\kk_i=\kk_{i-1}(a_i)$, i.e. the field of fractions of $\kk_{i-1}[a_i]$, for $1\leq i\leq d^2$. Hence there is a sequence of field extension \[\kk'=\kk_0\subset\kk_1\subset\cdots\subset\kk_{d^2}=\kk.\]
Let $\Theta_0=\Theta'$ be the l.s.o.p. for $\kk_0[\Delta]$. For $1\leq i\leq d^2$, if $a_{i}=a_{jk}$, then define an l.s.o.p. $\Theta_i$ for $\kk_i[\Delta]$ such that $M_{\Theta_i}$ is obtained from $M_{\Theta_{i-1}}$ by replacing the $(j,k)$-entry by $a_{jk}$.
We will prove that $\kk_{i}(\Delta;\Theta_i)$ are all anisotropic for $0\leq i\leq d^2$ by induction on $i$. The base case $i=0$ is just the assumption. For the induction step, set $R_i=\mathbb{F}[a_1,\dots,a_i]$, and denote by $\mathfrak{p}_i$ the prime ideal
\[\mathfrak{p}_i=\begin{cases}
(a_i-1)\ &\text{if $a_i=a_{kk}$ for some $1\leq k\leq d$},\\
(a_i)\ &\text{otherwise.}
\end{cases}\]
Then there is a ring homomorphism $\eta_i:(R_i)_{\mathfrak{p}_i}\to \kk_{i-1}$, where $(R_i)_{\mathfrak{p}_i}\subset\kk_i$ denote the localization of $R_i$ at $R-\mathfrak{p}_i$, given by $\eta_i(a_j)=a_j$ for $1\leq j\leq i-1$, and
\[\eta_i(a_i)=\begin{cases}
1\ &\text{ if $a_i=a_{kk}$ for some $k$},\\
0\ &\text{ otherwise.}
\end{cases}\]
Given a nonzero element $\alpha\in\kk_{i}(\Delta;\Theta_i)_j$ with $j\leq d/2$, there exists a nonzero element $t\in\mathfrak{p}_i$ such that
\[t\alpha\in(R_i)_{\mathfrak{p}_i}(\Delta;\Theta_i)\ \text{ and }\ 0\neq\eta_i(t\alpha)\in\kk_{i-1}(\Delta;\Theta_{i-1}).\]
Since $\kk_{i-1}(\Delta;\Theta_{i-1})$ is anisotropic by induction, $(\eta_i(t\alpha))^2\neq0$. It follows that $t^2\alpha^2$ is not zero in $(R_i)_{\mathfrak{p}_i}(\Delta;\Theta_i)$, and then $0\neq\alpha^2\in\kk_{i}(\Delta;\Theta_i)$. So $\kk_{i}(\Delta;\Theta_i)$ is anisotropic.
\end{proof}
As an application of Lemma \ref{lem:aniso equiv}, we can obtain a result in \cite{PP20}. That is, every simplicial $1$-sphere is generically anisotropic over any field. Note that every simplicial $1$-sphere is obtained from $\partial \Delta^2$ by a sequence of bistellar $0$-moves, so this result is a corollary of the following more general theorem.
\begin{thm}\label{thm:0-move}
Let $\Delta$ be a $\mathbb{F}$-homology $(d-1)$-sphere and suppose that $\Delta'$ is obtained from $\Delta$ via a bistellar $0$-move. Then
$\Delta$ is generically anisotropic over $\mathbb{F}$ if and only if $\Delta'$ is generically anisotropic over $\mathbb{F}$.
\end{thm}
\begin{proof}
Suppose that $\FF_0(\Delta)=[m-1]$, $\FF_0(\Delta')=[m]$, and without loss of generality assume that $\Delta'=\chi_\sigma\Delta$ for a facet $\sigma=[d]$. Set
\begin{gather*}
\kk=\mathbb{F}(a_{ij}:1\leq i\leq d,\,d+1\leq j\leq m-1),\\
\kk'=\mathbb{F}(a_{ij}:1\leq i\leq d,\,d+1\leq j\leq m).
\end{gather*}
Denote by $\Theta$ the l.s.o.p. for $\kk'[\Delta']$ such that $M_{\Theta}=(I_d\mid A)$, where $A=(a_{ij})$ is the $d\times(m-d)$ matrix. The restriction of $\Theta$ to $\kk[\Delta]$ is also denoted $\Theta$, and we will omit it from the notation of artinian reductions of face rings.
According to Lemma \ref{lem:aniso equiv}, it suffices to show that $\kk(\Delta)$ is anisotropic if and only if $\kk'(\Delta')$ is anisotropic.
One direction is easy, since
\[\kk(\Delta)_{\geq 1}=\kk(\Delta,\sigma)_{\geq 1}=\kk(\Delta',\mathrm{st}_{\{m\}}\Delta')_{\geq 1}\subset \kk'(\Delta')_{\geq 1}.\]
The other direction takes more thought. Suppose that $\kk(\Delta)$ is anisotropic. Let $\kk_0=\kk$, and recursively define $\kk_i=\kk_{i-1}(a_{m,i})$ for $1\leq i\leq d$. Then $\kk=\kk_0\subset\kk_1\subset\cdots\subset\kk_{d}=\kk'$ is a sequence of field extension. Arguing as in the proof of Lemma \ref{lem:aniso equiv}, one can show that $\kk_i(\Delta)$ are all anisotropic for $0\leq i\leq d$.
We first consider the case that $\dim\Delta=d-1$ is odd. Set $n=d/2$ and $\sigma_1=[n-1]\cup\{m\}$. Then $\kk'(\Delta')_n$ has a basis of the form: $\{\sigma_1,\sigma_2,\dots,\sigma_s\}$, where $s=h_n(\Delta')$, $\sigma_i\in\Delta'\setminus\mathrm{st}_{\{m\}}\Delta'$ for $2\leq i\leq s$. This follows from the following short exact sequence:
\[0\to\kk'(\Delta',\mathrm{st}_{\{m\}}\Delta')\to\kk'(\Delta')\to\kk'(\mathrm{st}_{\{m\}}\Delta')\to0.\]
Write a nonzero element $\alpha\in\kk'(\Delta')_n$ as $\alpha=\sum_{i=1}^s l_i\xx_{\sigma_i}$, $l_i\in\kk'$. If $l_1=0$, then $\alpha\in\kk'(\Delta',\mathrm{st}_{\{m\}}\Delta')=\kk'(\Delta,\sigma)\subset\kk'(\Delta)$, thus $\alpha^2\neq 0$ since $\kk'(\Delta)$ is anisotropic by the previous paragraph.
On the other hand, if $l_1\neq 0$, we may assume $l_1=1$. Then, since $\xx_{\sigma_1}\xx_{\sigma_i}=0$ for all $2\leq i\leq s$, we have
\begin{equation}\label{eq:square}
\alpha^2=\xx_{\sigma_1}^2+(l_2\xx_{\sigma_2}+\cdots+l_s\xx_{\sigma_s})^2.
\end{equation}
An easy calculation shows that in $\kk'(\Delta')_d$, \[\xx_{\sigma_1}^2=-\frac{\prod_{i=1}^{n-1}a_{m,i}}{\prod_{i=n}^{d-1}a_{m,i}}x_1x_2\cdots x_{d-1}x_{m}.\]
Hence $\Psi_{\Delta'}(\xx_{\sigma_1}^2)=\pm \prod_{i=1}^{n-1}a_{m,i}/\prod_{i=n}^{d}a_{m,i}$ by the definition of canonical function.
Set $\beta=l_2\xx_{\sigma_2}+\cdots+l_s\xx_{\sigma_s}$. If $\alpha^2=0$, then by \eqref{eq:square},
\begin{equation}\label{eq:fraction}
\Psi_{\Delta'}(\beta^2)=\Psi_{\Delta}(\beta^2)=\pm\frac{\prod_{i=1}^{n-1}a_{m,i}}{\prod_{i=n}^{d}a_{m,i}}.
\end{equation}
Let $R=\kk[a_{m,1},\dots,a_{m,d}]$, and let $\mathfrak{p}$ be the prime ideal $(a_{m,d})\subset R$. Then as in the proof of Lemma \ref{lem:aniso equiv}, there is a ring homomorphism $\eta:R_\mathfrak{p}\to \kk_{d-1}$ given by $\eta(a_{m,i})=a_{m,i}$ for $1\leq i\leq d-1$ and $\eta(a_{m,d})=0$.
Since $\Psi_{\Delta'}(\xx_{\sigma_i}\xx_{\sigma_j})\in\kk$ for all $2\leq i,j\leq s$, it follows from \eqref{eq:fraction} that there exits a power $b=a_{m,d}^k$ ($k\in\Zz^+$) such that $bl_i\in R_\mathfrak{p}$ for all $2\leq i\leq s$ and $\eta(bl_j)\neq 0$ for some $j$. So \[0\neq\eta(b\beta)\in\kk_{d-1}(\Delta',\mathrm{st}_{\{m\}}\Delta')\subset\kk_{d-1}(\Delta),\]
and then $\eta(b^2\beta^2)=(\eta(b\beta))^2\neq0$, since $\kk_{d-1}(\Delta)$ is anisotropic. This implies that
$\Psi_\Delta\eta(b^2\beta^2)=\eta(b^2\Psi_\Delta(\beta^2))\neq 0$. However, the equation \eqref{eq:fraction}, together with the fact that $a_{m,d}^2\mid b^2$, gives $\eta(b^2\Psi_\Delta(\beta^2))=0$, a contradiction.
Thus we get the anisotropy of $\kk'(\Delta')$ in degree $n$.
If $0\neq\alpha\in\kk'(\Delta')_i$ for $i<n$, then there exits $\alpha'\in\kk'(\Delta')_{n-i}$ such that $0\neq\alpha\alpha'\in\kk'(\Delta')_n$.
This is because $\kk'(\Delta')$ is Gorenstein. So we reduce to the case when $i=n$.
For the case that $\dim\Delta=d-1$ is even, set $n=(d-1)/2$. Then one can similarly show that $\kk'(\Delta')_n$ has a basis of the form: $\{\sigma_1,\sigma_2,\dots,\sigma_s\}$, where $s=h_n(\Delta')$, $\sigma_1=[n-1]\cup\{m\}$ and $\sigma_i\in\Delta'\setminus\mathrm{st}_{\{m\}}\Delta'$ for $2\leq i\leq s$.
As before, it suffices to verify that $\alpha^2\neq0$ for any nonzero element $\alpha=\sum_{i=1}^{s}l_i\xx_{\sigma_i}\in\kk'(\Delta')_n$ with $l_1=1$. Note that $x_m\alpha^2=x_m\xx_{\sigma_1}^2$, and
\[\Psi_{\Delta'}(x_m\xx_{\sigma_1}^2)=\pm\frac{\prod_{i=1}^{n-1}a_{m,i}}{\prod_{i=n}^{d}a_{m,i}}\neq 0.\]
Then the statement follows immediately.
\end{proof}
Here is another application of Lemma \ref{lem:aniso equiv}, which shows that Theorem \ref{thm:PP} reduces to odd dimensional spheres.
\begin{thm}\label{thm:odd}
Let $\Delta$ be a $\mathbb{F}$-homology sphere of dimension $2(n-1)$, and denote by $S\Delta:=\partial\Delta^1*\Delta$ the suspension of $\Delta$.
If $S\Delta$ is generically anisotropic over $\mathbb{F}$, then $\Delta$ is also generically anisotropic over $\mathbb{F}$.
\end{thm}
\begin{proof}
Suppose that $\FF_0(\Delta)=[m]$, and write $S\Delta=K\cup K'$, where $K=\{v\}*\Delta$ and $K'=\{v'\}*\Delta$. Let $\kk$ be the rational function field
\[\mathbb{F}(a_{ij}:1\leq i\leq 2n,\,1\leq j\leq m),\]
and let $\kk_0\subset\kk$ be the subfield
\[\mathbb{F}(a_{ij}:2\leq i\leq 2n,\,1\leq j\leq m).\]
Choose an l.s.o.p. $\Theta=(\theta_1,\theta_2,\dots,\theta_{2n})$ for $\kk[S\Delta]$ such that $M_\Theta=(A\mid\boldsymbol{\lambda}_v,\boldsymbol{\lambda}_{v'})$, where $A=(a_{ij})$, and $(\boldsymbol{\lambda}_v,\boldsymbol{\lambda}_{v'})=(I_2\mid 0)^T$.
Let $\Theta_0=(\theta_2,\theta_3,\dots,\theta_{2n})$. Clearly, $\Theta_0$ restricted to $\Delta$ is an l.s.o.p. for $\kk[\Delta]$.
It is known that there are two isomorphisms:
\begin{gather*}
\kk(K;\Theta)\cong\kk(\Delta;\Theta_0),\ x_i\mapsto x_i\text{ for }1\leq i\leq m,\ x_v\mapsto x_v-\theta_{1};\\
\kk(\Delta;\Theta_0)_*\cong\kk(K,\Delta;\Theta)_{*+1},\ \alpha\mapsto x_v\alpha
\end{gather*}
(see e.g. \cite[Lemma 3.2 and 3.3]{A18}). Hence for a nonzero element $\alpha\in\kk(\Delta;\Theta_0)$, we have $0\neq x_v\alpha\in\kk(K,\Delta;\Theta)$. Note that $K$ is a homology ball with boundary $\Delta$.
Assume $S\Delta$ is generically anisotropic over $\mathbb{F}$. Then $\kk(S\Delta;\Theta)$ is anisotropic by the proof of Lemma \ref{lem:aniso equiv}. For any nonzero element $\alpha\in\kk_0(\Delta;\Theta_0)_i\subset \kk(\Delta;\Theta_0)_i$, $i\leq n-1$, the second isomorphism above shows that $0\neq x_v\alpha\in\kk(K,\Delta;\Theta)_{i+1}$. Hence we have
$0\neq(x_v\alpha)^2\in\kk(K,\Delta;\Theta)$, since $\kk(K,\Delta;\Theta)=\kk(S\Delta,K';\Theta)\subset\kk(S\Delta;\Theta)$ and $\kk(S\Delta;\Theta)$ is anisotropic.
By Proposition \ref{prop:pairing}, this means that $x_v\alpha^2$ is not zero in $\kk(K;\Theta)$, and then $0\neq\alpha^2\in\kk_0(\Delta;\Theta_0)\subset \kk(\Delta;\Theta_0)$ because of the above isomorphism $\kk(K;\Theta)\cong\kk(\Delta;\Theta_0)$. So $\kk_0(\Delta;\Theta_0)$ is anisotropic. This is equivalent to saying that $\Delta$ is generically anisotropic over $\mathbb{F}$.
\end{proof}
\section{A proof of Theorem \ref{thm:PP} for PL-spheres}\label{sec:char 2}
In this section $\mathbb{F}$ denotes a field of characteristic $2$.
By Theorem \ref{thm:odd} and Theorem \ref{thm:pachner}, if we can show that the characteristic $2$ anisotropy of odd dimensional PL-spheres is preserved by bistellar moves, then Theorem \ref{thm:PP} holds for all PL-spheres.
\begin{thm}\label{thm:bistellar}
Let $\Delta$ be a $\mathbb{F}$-homology $(2n-1)$-sphere and suppose that $\Delta'$ is obtained from $\Delta$ via a bistellar $q$-move. Then
$\Delta$ is generically anisotropic over $\mathbb{F}$ if and only if $\Delta'$ is generically anisotropic over $\mathbb{F}$.
\end{thm}
\begin{proof}
The case $q=0$ or $2n-1$ is Theorem \ref{thm:0-move}. So we assume $q\neq0,\,2n-1$. Let $\kk$ be the field extension of $\mathbb{F}$, and $\Theta$ be the l.s.o.p. for $\kk[\Delta]$ and $\kk[\Delta']$, as defined at the beginning of section \ref{sec:equivalence}. Assume $\kk(\Delta;\Theta)$ is anisotropic. We need to show that $\kk(\Delta';\Theta)$ is also anisotropic. As we have seen in the proof of Theorem \ref{thm:0-move}, it suffices to show that $\kk(\Delta';\Theta)$ is anisotropic in degree $n$.
Suppose that $\FF_0(\Delta)=\FF_0(\Delta')=[m]$, and $\Delta'=\chi_\sigma\Delta$ for some $\sigma\in\Delta$ with $\mathrm{lk}_\sigma\Delta=\partial\Delta^q=\partial\tau$. Then there are short exact sequences:
\begin{gather}
0\to \kk(\Delta,\mathrm{st}_\sigma\Delta)\to\kk(\Delta)\to\kk(\mathrm{st}_\sigma\Delta)\to0,\label{eq:delta}\\
0\to \kk(\Delta',\mathrm{st}_\tau\Delta')\to\kk(\Delta')\to\kk(\mathrm{st}_\tau\Delta')\to0.
\label{eq:delta'}
\end{gather}
Since $\kk(\mathrm{st}_\tau\Delta')_i=0$ for $i\geq\dim\sigma+1=2n-q$,
it follows from \eqref{eq:delta} and \eqref{eq:delta'} that if $q\geq n$, then
\[\kk(\Delta')_{\geq n}=\kk(\Delta',\mathrm{st}_\tau\Delta')_{\geq n}=\kk(\Delta,\mathrm{st}_\sigma\Delta)_{\geq n}\subset\kk(\Delta)_{\geq n}.\]
Hence $\kk(\Delta')$ is automatically anisotropic when $q\geq n$. It remains to consider the case when $0<q<n$.
Without loss of generality we assume that $\tau=[q+1]$, $\sigma=[2n+1]\setminus[q+1]$.
Let $\kk_0$ be the rational function field
\[\mathbb{F}(a_{ij}:1\leq i\leq 2n,\,2n+2\leq j\leq m),\]
and let $\kk_1$ be the rational function field $\kk_0(b_1,\dots,b_{2n})$.
Suppose that $\Theta_1=(\theta_1,\dots,\theta_{2n})$ is a sequence of linear forms of $\kk_1[x_1,\dots,x_m]$ such that the column vectors of $M_\Theta$ are given by
\[\boldsymbol{\lambda}_{j}=
\begin{cases}
(a_{1,j},a_{2,j}\dots,a_{2n,j})^T &\text{ for } 2n+2\leq j\leq m,\\
(b_1,\dots,b_{2n-q-1},0,\dots,0)^T &\text{ for } j=2n+1,\\
(\lambda_{1,j},\lambda_{2,j},\dots,\lambda_{2n,j})^T &\text{ for } 1\leq j\leq 2n,
\end{cases}\]
where
\[\lambda_{ij}=
\begin{cases}
1 &\text{ if } i=j,\\
b_i &\text{ if } i-j=2n-q-1,\\
0 &\text{ otherwise.}
\end{cases}\]
It is easy to verify the following facts:
\begin{enumerate}[(a)]
\item $\Theta_1$ is an l.s.o.p. for both $\kk_1[\Delta]$ and $\kk_1[\Delta']$;
\item\label{it:b} $\kk_1(\mathrm{st}_\tau\Delta';\Theta_1)_n=\kk_1$ is spanned by the face monomial $\xx_{\sigma_1}:=x_1\cdots x_n$;
\item\label{it:c} \[\xx_{\sigma_1}^2=\frac{\prod_{i=q+2}^nb_i}{\prod_{i=n+1}^{2n}b_i}\cdot\prod_{i=1}^{2n}x_i\in\kk_1(\Delta';\Theta_1)_{2n}.\]
\end{enumerate}
Here we set $\prod_{i=q+2}^nb_i=1$ if $q=n-1$.
Similar to Lemma \ref{lem:aniso equiv}, one can show that $\kk(\Delta;\Theta)$ (resp. $\kk(\Delta';\Theta)$) is anisotropic if and only if $\kk_1(\Delta;\Theta_1)$ (resp. $\kk_1(\Delta';\Theta_1)$) is anisotropic. The rest of the proof is to derive the anisotropy of $\kk_1(\Delta';\Theta_1)$ from the anisotropy of $\kk_1(\Delta;\Theta_1)$.
By \eqref{eq:delta'} and fact \eqref{it:b}, $\kk_1(\Delta')_n$ has a basis $\{\sigma_1,\sigma_2,\dots,\sigma_s\}$ ($s=h_n(\Delta')$) with $\sigma_i\in \Delta'\setminus\mathrm{st}_\tau\Delta'$ for $2\leq i\leq s$.
For a nonzero element $\alpha\in\kk_1(\Delta')_n$, write \[\alpha=\sum_{i=1}^sl_i\xx_{\sigma_i},\ l_i\in\kk_1.\]
If $l_1=0$, then $\alpha\in\kk_1(\Delta',\mathrm{st}_\tau\Delta')=\kk_1(\Delta,\mathrm{st}_\sigma\Delta)\subset \kk_1(\Delta)$. Since $\kk_1(\Delta)$ is anisotropic, we have $0\neq\alpha^2\in\kk_1(\Delta',\mathrm{st}_\tau\Delta')\subset \kk_1(\Delta')$ in this case.
To deal with the case $l_1\neq0$, define a partial differential operator
\[\PP:=\frac{\partial^n}{\partial b_{2n-q}\cdots\partial b_{2n}}.\]
Then, we have
\[\PP\Psi_{\Delta'}(\alpha^2)=\PP\sum_{i=1}^s l_i^2\Psi_{\Delta'}(\xx_{\sigma_i}^2)=\sum_{i=1}^s l_i^2\PP\Psi_{\Delta'}(\xx_{\sigma_i}^2),\]
where the first and second equality both come from the assumption that $\mathrm{char}\,\mathbb{F}=2$.
Since $\tau=[q+1]\not\in \Delta'\setminus\mathrm{st}_\tau\Delta'$, it follows that for any $2\leq i\leq s$, there exist $1\leq n_i\leq q+1$ and a facet $F_i$ in $\mathrm{st}_{\sigma_i}\Delta'$ such that $n_i\not\in F_i$. This implies that the variable $b_{m_i}$, where $m_i=n_i+2n-q-1$, dose not appear in the rational functions $A_{F_i}$ and $A_{F_i}(j)$ for any $2\leq i\leq s$ and $j\in F_i$ (see the discussion preceding Theorem \ref{thm:Lee} for the definitions of $A_{F_i}$ and $A_{F_i}(j)$).
Note that $2n-q\leq m_i\leq 2n$.
Therefore $\PP\Psi_{\Delta'}(\xx_{\sigma_i}^2)=0$ for all $2\leq i\leq s$ by Theorem \ref{thm:Lee}, and then $\PP\Psi_{\Delta'}(\alpha^2)=l_1^2\PP\Psi_{\Delta'}(\xx_{\sigma_1}^2)$.
Combining this with fact \eqref{it:c} and the fact that \[\Psi_{\Delta'}(x_1\cdots x_{2n})=\frac{1}{\det(M_{\Theta_1}([2n]))}=1,\]
we immediately have $\PP\Psi_{\Delta'}(\alpha^2)\neq 0$, and so $\alpha^2\neq 0$.
Thus for any nonzero element $\alpha\in\kk_1(\Delta')_n$, we have $\alpha^2\neq0$, finishing the proof of the theorem.
\end{proof}
\section{Anisotropy and Lefschetz property of simplicial 2-spheres}\label{sec:2-sphere}
In this section $\mathbb{F}$ denotes a field of arbitrary characteristic.
In section \ref{sec:equivalence}, we have seen that the generic anisotropy of odd dimensional spheres implies the generic anisotropy of even dimensional spheres of one lower dimension.
A natural question is that whether the same condition implies the generic anisotropy of even dimensional spheres of one higher dimension. In this section we will show that the answer to this question is yes for PL-spheres.
\begin{thm}\label{thm:general char}
If every PL-sphere of dimension $2n-1$ is generically anisotropic over $\mathbb{F}$, then every PL-sphere of dimension $2n$ is also generically anisotropic over $\mathbb{F}$, and has the Lefschetz property over any infinite field of the same characteristic as $\mathbb{F}$.
\end{thm}
Since every simplical $1$-sphere is generically anisotropic over $\mathbb{F}$ by Theorem \ref{thm:0-move} and all simplicial $2$-sphere are PL, the following corollary is an immediate consequence of Theorem \ref{thm:general char}.
\begin{cor}\label{cor:2-sphere}
Every simplicial $2$-sphere is generically anisotropic over any field $\mathbb{F}$, and has the Lefschetz property over any infinite field $\kk$.
\end{cor}
Note that the Lefschetz property of simplicial $2$-spheres is also implied by the result in \cite{Murai10}.
Before giving the proof of Theorem \ref{thm:general char}, we state an easy result about rational functions without proof.
Let $\kk$ be a field, and let $\kk(x)$ be the field of rational functions over $\kk$. For a nonzero element $\phi=f/g\in\kk(x)$ with $f,g\in\kk[x]$,
define the degree of $\phi$ by $\deg(\phi)=\deg(f)-\deg(g)$, and define the leading coefficient of $\phi$ as $L(\phi):=L(f)/L(g)$, where $L(f)$ and $L(g)$ are the leading coefficient of $f$ and $g$ respectively. Moreover, we assume $\deg(0)=-\infty$ and $L(0)=0$ in $\kk(x)$.
\begin{lem}\label{lem:leading}
Let $\kk(x)$ be as above. Then for a nonzero element $\alpha=\sum_{i\in I}\phi_i$ with $\phi_i\in\kk(x)$, we have
\[\deg(\alpha)\leq M:=\max\{\deg(\phi_i):i\in I\},\]
where equality holds if and only if $\sum_{\deg(\phi_i)=M}L(\phi_i)\neq0$.
\end{lem}
\begin{proof}[Proof of Theorem \ref{thm:general char}]
Suppose that $\Delta$ is a $2n$-dimensional PL-sphere with vertex set $[m]$, and set $K=\Delta^0*\Delta=\{v\}*\Delta$. Let $\kk$ be the rational function field
\[\mathbb{F}(a_{ij}:1\leq i\leq 2n+2,\,1\leq j\leq m).\]
Choose an l.s.o.p. $\Theta$ for $\kk[K]$ such that $M_\Theta=(A\mid\boldsymbol{\lambda}_v)$, where $A=(a_{ij})$ is the $(2n+2)\times m$ matrix, and $\boldsymbol{\lambda}_v=(1,0\dots,0)^T$.
If we can show that $\kk(K,\Delta;\Theta)$ is anisotropic, then $\Delta$ is generically anisotropic over $\mathbb{F}$ by the argument in the proof of Theorem \ref{thm:odd}, and the second statement of the theorem will follow from the result in \cite[Section 9]{PP20}.
As we have seen before, it suffices to verify that this property of $\kk(K,\Delta;\Theta)$ is preserved by bistellar moves on $\Delta$. A result similar to Theorem \ref{thm:0-move} reduces us to considering bistellar $q$-moves for $q\neq 0,\,2n$.
Suppose that $\Delta'=\chi_\sigma\Delta$ for some $\sigma\in\Delta$ ($\dim\sigma\neq 0,\,2n$) with $\mathrm{lk}_\sigma\Delta=\partial\tau$, and without loss of generality assume that $\sigma\cup\tau=[2n+2]$ and $1\in\sigma$.
Let $K'=\{v\}*\Delta'$.
Define two subfield of $\kk$ as
\[\kk_0=\mathbb{F}(a_{ij}:1\leq i\leq 2n+2,\,2\leq j\leq m),\ \ \kk_1=\kk_0(a_{1,1}),\]
and let $\Theta_1=(\theta_1,\theta_2,\dots,\theta_{2n+2})$ be the l.s.o.p. for both $\kk_1[K]$ and $\kk_1[K']$, such that $M_{\Theta_1}$ is obtained from $M_\Theta$ by replacing the column vector $\boldsymbol{\lambda}_1$ with $(a_{1,1},1,0,\dots,0)^T$.
Then by the proof of Lemma \ref{lem:aniso equiv}, $\kk(K,\Delta;\Theta)$ (resp. $\kk(K',\Delta';\Theta)$) is anisotropic if and only if $\kk_1(K,\Delta;\Theta_1)$ (resp. $\kk_1(K',\Delta';\Theta_1)$) is anisotropic. For this reason, we only consider the anisotropy of $\kk_1(K,\Delta;\Theta_1)$ and $\kk_1(K',\Delta';\Theta_1)$ in what follows, and for simplicity, we omit $\Theta_1$ from the notation.
Assume that $\kk_1(K',\Delta')$ is anisotropic. We need to show that $\kk_1(K,\Delta)$ is also anisotropic, i.e., for any nonzero element $\alpha\in\kk_1(K,\Delta)_{n+1}$, $\alpha^2\neq0$. We first construct a basis for $\kk_1(K,\Delta)_{n+1}$.
Suppose that $\Aa_0=\{\rho_1,\dots,\rho_r\}\subset\mathrm{lk}_{\{1\}}\Delta$, where $r=h_n(\mathrm{lk}_{\{1\}}\Delta)$, forms a basis for $\kk(\mathrm{st}_{\{1\}}\Delta)_n$. Then the short exact sequence
\[0\to\kk_1(\Delta,\mathrm{st}_{\{1\}}\Delta)\to\kk_1(\Delta)\to\kk_1(\mathrm{st}_{\{1\}}\Delta)\to0\]
implies that $\Aa_0$ can be extended to a basis $\Aa_0\cup\BB_0$ for $\kk_1(\Delta)_n$, where $\BB_0=\{\pi_1,\dots,\pi_s\}$ with $\pi_i\in \Delta\setminus\mathrm{st}_{\{1\}}\Delta$, $s=h_n(\Delta)-r$. Let $\sigma_i=\{v\}\cup\rho_i$, $\tau_i=\{v\}\cup\pi_i$, and $\Aa=\{\sigma_1,\dots,\sigma_r\}$, $\BB=\{\tau_1\dots,\tau_s\}$. Then the isomorphism $\kk(\Delta)\xr{\cdot x_v}\kk(K,\Delta)$ shows that $\Aa\cup\BB$ forms a basis for $\kk(K,\Delta)_{n+1}$.
For a nonzero element $\alpha\in\kk_1(K,\Delta)_{n+1}$, write $\alpha=\alpha_1+\alpha_2$, where
\[\alpha_1=\sum_{i=1}^rl_i\xx_{\sigma_i},\ l_i\in\kk_1,\ \text{ and }\ \alpha_2=\sum_{j=1}^sk_j\xx_{\tau_j},\ k_j\in\kk_1.\]
Now consider the value $\Psi_K(\alpha^2)$.
Since $x_v=-\sum_{i=1}^m a_{1,i}x_i$ in $\kk_1(K,\Delta)$, we have
\[\xx_{\sigma_i}\xx_{\sigma_j}=-\sum_{k=1}^m a_{1,k}\cdot x_vx_k\xx_{\rho_i}\xx_{\rho_j} \text{ for } 1\leq i,j\leq r.\]
Thus, setting $L_1=\mathrm{lk}_{\{1\}}\Delta$ and $\Theta_0=(\theta_3,\dots,\theta_{2n+2})$, and using Theorem \ref{thm:Lee} to compute the canonical functions $\Psi_K$ on $\kk_1(K,\Delta)_{2n+2}$ and $\Psi_{L_1}$ on $\kk_0(L_1;\Theta_0)_{2n}$ respectively, we see that
\begin{equation}\label{eq:degree1}
\Psi_K(\xx_{\sigma_i}\xx_{\sigma_j})=\pm\Psi_{L_1}(\xx_{\rho_i}\xx_{\rho_j})a_{1,1}+b,\ \text{ for some}\ b\in\kk_0.
\end{equation}
Since $\tau_i\not\in\mathrm{st}_{\{1\}} K$ for all $\tau_i\in\BB$, a similar computation shows that
\begin{equation}\label{eq:degree2} \Psi_K(\xx_{\tau_i}\xx_{\tau_j}),\,\Psi_K(\xx_{\sigma_i}\xx_{\tau_j})\in\kk_0\ \text{ for all } i,j.
\end{equation}
Thinking of $l_i,\,k_j$ as rational functions with coefficients in $\kk_0$, define \begin{gather*}
m_1:=\max\{\deg(l_i):1\leq i\leq r\},\ \ m_2:=\max\{\deg(k_j):1\leq j\leq s\},\\
\beta_1=\sum_{\deg(l_i)=m_1}L(l_i)\xx_{\sigma_i},\ \beta_2=\sum_{\deg(k_j)=m_2}L(k_j)\xx_{\tau_j}.
\end{gather*}
Here $\deg(\cdot)$, $L(\cdot)$ are defined in the discussion preceding Lemma \ref{lem:leading}.
If $m_1<m_2$, then $\beta_2\neq 0$.
The assumption that $\Delta'=\chi_\sigma\Delta$ and $1\in\sigma$ implies that $K\setminus(\mathrm{st}_{\{1\}} K\cup\Delta)\subset K'\setminus(\mathrm{st}_{\{1\}} K'\cup\Delta')$. Hence
\[\beta_2\in\kk_1(K,\mathrm{st}_{\{1\}} K\cup\Delta)\subset \kk_1(K',\mathrm{st}_{\{1\}} K'\cup\Delta')\subset \kk_1(K',\Delta').\]
It follows that $\Psi_K(\beta_2^2)=\Psi_{K'}(\beta_2^2)\neq 0$, since $\kk_1(K',\Delta')$ is anisotropic. By Lemma \ref{lem:leading}, this means that $\deg(\Psi_K(\alpha_2^2))=2m_2$. Furthermore, \eqref{eq:degree1} and \eqref{eq:degree2} give
\[
\deg(\Psi_K(\alpha_1^2))\leq 2m_1+1<2m_2,\ \ \deg(\Psi_K(\alpha_1\alpha_2))\leq m_1+m_2<2m_2.
\]
Thus $\deg(\Psi_K(\alpha^2))=2m_2$, and so $\alpha^2\neq0$ in this case.
On the other hand, if $m_1\geq m_2$, then $\beta_1\neq 0$. It follows that \[0\neq\gamma_1:=\sum_{\deg(l_i)=m_1}L(l_i)\xx_{\rho_i}\in \kk_0(L_1;\Theta_0)_{n}.\]
Hence, if the condition in the theorem holds, then $\Psi_{L_1}(\gamma_1^2)\neq0$, since $L_1$ is a $(2n-1)$-dimensional PL-sphere.
By \eqref{eq:degree1}, \eqref{eq:degree2} and Lemma \ref{lem:leading}, it follows that \[\deg(\Psi_K(\alpha^2))=\deg(\Psi_K(\alpha_1^2))=2m_1+1.\]
So $\alpha^2\neq0$ still holds in this case, and the proof is complete.
\end{proof}
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2024-02-18T23:40:24.478Z
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2022-02-02T02:15:40.000Z
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{"paloma_paragraphs":[]}
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{"arxiv_id":"2112.14493","language":"en","timestamp":1643768140000,"url":"https:\/\/arxiv.org\/abs\/2112.14493","yymm":"2112"}
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proofpile-arXiv_000-10032
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{"provenance":"002.jsonl.gz:10033"}
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\section{Introduction}
Symmetry is one of the oldest and most important central concepts used in nearly every branch of science. Classically, symmetry has been modelled by group actions but a more general paradigm of Hopf algebras or quantum groups is also quite common by now (see, e.g. (\cite{drinfeld1986icm}), (\cite{MR901157}), (\cite{MR1358358}) and references therein, just to mention a few). In this article, we want to consider quantum group symmetry of a very popular class of physical model, namely the Potts model. Typically, Potts models are studied on infinite lattices or infinite graphs and thermodynamic properties are analysed. However, we will consider only finite graphs for a simpler mathematical treatment. There is another important departure from the conventional framework. Instead of looking at phase transition in terms of break of continuity or smoothness of some suitable thermodynamic term, we'll take a symmetry viewpoint, following the classical ideas of Landau (\cite{landau1969statistical}), which says that a change of the group of symmetry of the underlying physical system signifies a change of phase. For example, gaseous phase has lot more symmetry than liquid which is constrained to have a fixed volume. Similarly, liquid has more symmetry than solid. We will consider quantum group symmetry generalising group symmetry. We give a few examples where a slight perturbation of the hamiltonian of the model leads to a significant change of the (quantum) symmetry group, which may be interpreted as a phase change. It is perhaps interesting to note that in several models of condensed matter physics,
there are theoretical and experimental explanations of effect of doping-induced phase transition in terms of change of the point symmetry group of the underlying crystal structures (see for instance, (\cite{perez2012symmetry})). We propose extension of such ideas to the realm of quantum group symmetry. However, we must admit that our mathematical examples are in not the realistic models taken from physics or any other natural science. All we want in this article is to illustrate our novel perspective through a few kind of `toy-models' to encourage an in-depth analysis of real-life Potts model based on quantum group symmetry. The simplicity of the models in our paper help us compute the exact quantum symmetry of them, using which we could show instance of phase transition with the change of symmetry.
\par
It may be noted that the interplay of quantum groups and operator algebras with the models (including Potts model) of statistical mechanics goes back to the seminal work of Jones, (see for example, \cite{MR990215}) leading to a theory of work connecting subfactor theory, quantum field theory and so on with such physical models. Later Banica showed (see for example, \cite{MR1654119}) that the spin and vertex models of Jones do come from quantum groups.
\par
However, an important difference between our approach to (quantum) symmetry from the works mentioned above is that we consider a (quantum) symmetry group (co)acting on the vertices of the lattice or graph commuting with the hamiltonian instead of an on-site symmetry coming from the permutation of the set of states or the commutator with the site-to-site transfer matrices as in (\cite{MR1317365}). Nevertheless, as the transfer matrices are closely related to the hamiltonian of the underlying model, there may be an interesting connection between the two approaches, which we will try to explore later.
\par
Let us now give a brief sketch of the paper. We consider Potts model on a finite graph with a finite state space, identified with elements of a suitable cyclic group and formulate a notion of (co)action of compact quantum groups on the space of functions on the vertex set of the graph such that the coaction commutes with the hamiltonian (i.e. it gives a symmetry of the underlying model). Following the line of (\cite{MR2174219}) we prove the existence of a universal compact quantum group which gives symmetry of the Potts model in the above sense, calling it the quantum group of symmetry of the model. Then we compute this quantum group in a few examples and show how a slight change of the parameters of the model can result in a rather remarkable change of quantum symmetry. More interestingly, we give an example where classical symmetry group remains the same even after the change of the model parameters but quantum symmetry group changes. This makes a strong case for studying quantum group symmetry of physical models.
\section{Prelimineries}
\subsection{Compact quantum group}
We start with a brief description of compact quantum group and few related concepts associated with it. For more details, see (\cite{MR1645264}), (\cite{MR1358358}), (\cite{MR901157}) and (\cite{MR3559897}). All C* algebras are assumed to be unital in this paper and the tensor product considered is the minimal tensor product among C* algebras.
\begin{defn}
A compact quantum group is a pair $(\mathcal{S},\Delta)$ where $\mathcal{S}$ is a unital C* algebra and $\Delta:\mathcal{S}\rightarrow \mathcal{S}\otimes \mathcal{S}$ is a homomorphism of C* algebras satisfying the following conditions:
\begin{enumerate}
\item $(\Delta\otimes id)\Delta=(id\otimes\Delta)\Delta$ (Coassociativity).
\item Each of the linear spans of $\Delta(\mathcal{S})(1\otimes \mathcal{S})$ and $\Delta(\mathcal{S})(\mathcal{S}\otimes 1)$ is norm-dense in $\mathcal{S}\otimes \mathcal{S}$.
\end{enumerate}
\end{defn}
It is known that there exists a unique Haar state on a compact quantum group which is the non-commutative analogue of Haar measure on a classical compact group.
\begin{defn}
The Haar state $h$ on a compact quantum group is the unique state on $\mathcal{S}$ which satisfies the following conditions:
\begin{equation*}
(h\otimes id)\Delta(a)=h(a)1_{S} \quad \text{and}\quad (id\otimes h)\Delta(a)=h(a)1_{S}
\end{equation*}
for all $a\in \mathcal{S}$.
\end{defn}
\begin{defn}
A quantum group homomorphism $\Phi$ between two compact quantum groups $(\mathcal{S}_1,\Delta_1)$ and $(\mathcal{S}_2,\Delta_2)$ is a C* algebra homomorphism $\Phi:\mathcal{S}_1\rightarrow\mathcal{S}_2$ satisfying the following condition:
\begin{equation*}
(\Phi\otimes\Phi)\circ\Delta_1=\Delta_2\circ\Phi
\end{equation*}
\end{defn}
Now we breifly state few facts from the co-representation theory of compact quantum groups.
\begin{defn}
Let $n\in\mathbb{N}$. An n dimensional co-representation of a compact quantum group $(\mathcal{S},\Delta)$ is an $\mathcal{S}$ valued $n\times n$ matrix $(v_{ij})_{n\times n}$ of satisfying the following property:
\begin{equation*}
\Delta(v_{ij})=\sum_{k=1}^n v_{ik} \otimes v _{kj}
\end{equation*}
$v_{ij}$'s are called matrix elements of the n dimensional co-representation of $(\mathcal{S},\Delta)$.
\end{defn}
It is known that for a compact quantum group $(\mathcal{S},\Delta)$, there is a dense subalgebra $\mathcal{S}_0$ generated by the matrix elements of finite dimensional co-representations of $(\mathcal{S}_0,\Delta)$ which forms a Hopf * algebra in its own right. $h$ is faithful when restricted to $\mathcal{S}_0$ i.e. for $a$ in $\mathcal{S}_0$, $a=0$ whenever $h(a^*a)=0$.
\subsection{Co-actions and quantum automorphism groups }
\begin{defn}
Let $\mathcal{A}$ be a unital C* algebra. A co-action of a compact quantum group $(\mathcal{S},\Delta)$ on $\mathcal{A}$ is a * homomorphism $\mathcal{A}\rightarrow \mathcal{A}\otimes\mathcal{S}$ satisfying the following conditions:
\begin{enumerate}
\item $(\alpha\otimes id)\alpha=(id\otimes\Delta)\alpha$.
\item Linear span of $\alpha(\mathcal{A})(1_{\mathcal{A}}\otimes\mathcal{S})$ is norm-dense in $\mathcal{A}\otimes\mathcal{S}$.
\end{enumerate}
\end{defn}
\begin{defn}
A co-action of $(\mathcal{S},\Delta)$ on $\mathcal{A}$ is said to be faithful if there does not exist any proper Woronowicz C* subalgebra $\mathcal{S}_1$ of $\mathcal{S}$ such that $\alpha$ is a coaction of $\mathcal{S}_1$ on $\mathcal{A}$. A continuous linear functional $\tau$ on $\mathcal{A}$ is said to be invariant under $\alpha$ if the following holds:
\begin{equation*}
(\tau\otimes id)\alpha(a)=\tau(a)1_{\mathcal{S}}
\end{equation*}
for all $a$ in $\mathcal{S}$.
\end{defn}
In this paper we will only be considering faithful co-actions.
\par
For a unital C*algebra $\mathcal{A}$ the $\textbf{category of quatum transformation groups}$ is the category whose objects are the compact quantum groups co-acting on $\mathcal{A}$ and morphisms are quantum group homomorphisms intertwining such co-actions, that is, for any morphism $\Phi$ between two compact quantum transformation groups $(\mathcal{S},\Delta)$ and $(\mathcal{S}',\Delta')$ the following diagram commutes:
\begin{center}
\includegraphics[]{commdiag}
\end{center}
where $\alpha$ and $\alpha'$ are co-actions of $(\mathcal{S},\Delta)$ and $(\mathcal{S}',\Delta')$ on $\mathcal{A}$ respectively. The universal object in this category, if it exists, is said to be the $\textbf{quantum automorphism}$ $\textbf{group}$ of $\mathcal{A}$.
\begin{rem}
For an arbitrary unital C* algebra the quantum automorphism group might not exist. The algebra of $n\times n$ complex matrices is such an example. However the quantum automorphism group does exist if we restrict the category to a smaller category where co-actions are trace preserving. For details, see (\cite{MR1637425}).
\end{rem}
\subsection{Quantum permutation group on a finite set}
Let $X_n=\{1,2,...,n\}$. Consider the algebra $C(X_n)$. We note that
\begin{equation*}
C(X_n):=span\{\chi_i, i=1,2,...,n\}.
\end{equation*}
where $\chi_i$ is the characteristic function on $i$, that is, $\chi_i(k)=1$ if $k=i$ and $\chi_i(k)=0$ otherwise.
The universal object in the $\textbf{category of quantum transformation groups}$ for the algebra $C(X_n)$ exists and Wang described it in (\cite{MR1637425}). It is called the quantum permutation group on $X_n$ and is denoted by $(S_n^+,\Delta_n)$. $S_n^+$ is the universal C* algebra generated by generators $\{u_{ij},i,j=1,2,...,n\}$ satisfying the following relations:
\begin{enumerate}
\item $u_{ij}^2=u_{ij}=u_{ij}^*$ for $i,j=1,2,...,n$
\item $\sum_{i=1}^n u_{ij}=\sum_{j=1}^n u_{ij}=1$
\end{enumerate}
The co-product $\Delta:{{S_n}^+}\rightarrow{{S_n}^+}\otimes{{S_n}^+}$ is given by $\Delta(u_{ij})=\sum_{k=1}^n u_{ik}\otimes u_{kj}$
The cannonical co-action $\alpha$ of $(S_n^+,\Delta)$ on $C(X_n)$ is given by
\begin{equation}
\alpha(\chi_{i})=\sum_{j=1}^n \chi_j \otimes u_{ji}; \quad i=1,2,..,n
\end{equation}
\par
It is also worth mentioning that $(S_n^+,\Delta_n)$ is a compact quantum group of Kac type (see (\cite{MR1637425}) and references within) which makes the corrsponding Haar state $h:S_n^+\rightarrow\mathbb{C}$ tracial, that is, $h(ab)=h(ba)$ for all $a,b\in S_n^+$.
\begin{nota}
let $\alpha$ be a co-action of a compact quantum group $(S,\Delta)$ on $C(X_n)$. The co-representation matrix of the co-action $\alpha$ is an $S$-valued $n\times n$ matrix $Q=(q_{ij})_{n\times n}$ such that the following holds:
\begin{equation*}
\alpha(\chi_i)=\sum_{j=1}^n \chi_j \otimes q_{ji}, \quad i=1,2,..,n
\end{equation*}
As our co-actions are faithful, $S$ is generated by $q_{ij}$'s as a C* algebra.
\end{nota}
\subsection{Quantum automorphism group of a finite graph}
let $(V,E)$ be an undirected finite graph with vertex set $V$ and edge set $E$. We also assume that $(V,E)$ has no loops and no multiple edges among any two points. Let $A=(a_{ij})_{n\times n}$ denote the adjacency matrix of $(V,E)$, that is, $a_{ij}=1$ if $(i,j)\in E$ and $a_{ij}=0$ otherwise.
\begin{defn}
A co-action $\alpha$ of $(S,\Delta)$ on $C(V)$ is said to be co-acting by preserving the quantum symmetry of $(V,E)$ if $Q$ commutes with $A$ where $Q$ is the co-representation matrix of $\alpha$.
\end{defn}
Consider the category whose objects are compact quantum groups co-acting on $(V,E)$ by preserving the quantum symmetry of the graph $(V,E)$ and morphisms are quantum group morphisms intertwining two such co-actions. The following result is due to Banica in 2005. See (\cite{MR2146039}).
\begin{thm}\label{uni.obj.finite.graph}
The universal object in the above mentioned category exists and is given by $S_n^+/UA-AU$ where $U$ denotes the corepresentation matrix of the cannonical co-action of $S_n^+$ on $C(V)$. It is called the quantum automorphism group of $(V,E)$.
\end{thm}
In fact, the adjacency matrix $A$ does not have any special role in the proof of existence of the universal object in (\cite{MR2146039}) and by arguments similar to those used in (\cite{MR2146039}) and (\cite{MR2174219}) one can obtain a version of theorem (\ref{uni.obj.finite.graph}) for any complex valued matrix $A=(a_{ij})_{n\times n}$ indexed by the vertices. This, in particular covers the case of weighted graphs and finite metric spaces. For the convenience of the reader, let us state this as a theorem:
\begin{thm}\label{uni.obj.complex.matrix}
Let $X_n$ be a finite set with $n$ elements and $A\in M_n(\mathbb{C})$. Let us consider the category whose objects are compact quantum groups co-acting on $C(X_n)$ in such a way that the corepresentation matrix $Q$ commutes with $A$. Then the universal object in this category exists and is given by $S_n^+/UA-AU$ where $U$ is the co-representation matrix of the cannonical co-action of $(S,\Delta)$ on $C(X_n)$.
\end{thm}
\section{Potts Model and its quantum symmetry}
Let us say a few words about the basics of statistical mechanics for the readers without any physics background. The fundamental idea of statistical mechanics is to consider an "ensemble" or totality of all possible states of a physical system in equilibrium and assign a probability distribution, usually the so-called Boltzmann (or Gibbs) distribution. The probability distribution (or the probability density) of a state depends on the corresponding energy level (typically given by the hamiltonian, say $H$) and the absolute temparature of the system ($T$). Usually, the probability $P(\rho)$ of a state $\rho$ is taken as a multiple of $e^{-\beta H(\rho)}$ where $\beta=1/T$. Then one considers the average expected value of the states calculated according to the above distribution. The partition function is given by
\begin{equation*}
Z=\sum_{\rho}e^{-\beta H(\rho)}
\end{equation*}
where $\rho$ varies over all states. This is of central importance and its behavior with respect to the inverse temparature $\beta$ is studied. Any break of analyticity in $\beta$ is thought as a sign of a phase transition. However, as remarked earlier we have taken a different approach to phase transition in this paper.
\par
Potts model, or more general vertex and spin models are some of the most popular and useful models arising primarily in statistical (including quantum statistical) mechanics, but they have found wide applications in many other areas of physics and even other scientific (including social sciences) disciplines. Usually, the physical picture of a Potts model considers atoms occupying the vertices of a lattice or more general graphs, each of which can be in one of a specified set of physical states. The edges joining two such atoms are thought of as bonds between them and an atom interacts only with the nearest neighbours.
\par
We will be using a simpler version of Potts model for simpler mathematical treatment. For more details on Potts model, see (\cite{martin1991}).
\par
Let $(V,E)$ be an undirected finite graph with no loops and multiple edges among two vertices. A q-state Potts model ($q\in\mathbb{N}$ and $q\geq 2$) on $(V,E)$ consists of a set of configurations $\Omega_P$ and a hamiltonian $H_P:\Omega_P\rightarrow\mathbb{C}$ defined as follows:
\begin{defn}
A configuration $\omega$ for a q-state Potts Model on $(V,E)$ is a function from $V$ to the set $X_q$. The hamiltonian $H_P$ is defined to be:
\begin{equation}\label{1}
H_P(\omega) := \sum_{(i,j)\in E} J_{ij}\delta_{\omega(i),\omega(j)} \qquad \text{for all}\quad \omega \in \Omega_P
\end{equation}
where $J_{ij}\in\mathbb{C}$ and $J_{ij}=J_{ji}$ for all $i,j\in V$. The expression $\delta_{\omega(i),\omega(j)}$ is equal to $1$ if $\omega(i)=\omega(j)$ and $0$ otherwise.
\end{defn}
By taking $J'_{ij}=J_{ij}a_{ij}$ we get,
\begin{equation*}
H_P(\omega)=\sum_{i,j\in V} J'_{ij}\delta_{\omega(i),\omega(j)} \qquad \text{for all}\quad \omega \in \Omega_P
\end{equation*}
Now we discuss the notion of quantum symmetry on Potts model.
\par
Let $\alpha$ be a co-action of a compact quantum group $(\mathcal{S},\Delta)$ on $C(V)$. We want to describe what it means for $\alpha$ to preserve the hamiltonian $H_P$. Such a co-action can be described to preserve the quantum symmetry of the q-state Potts model on $(V,E)$. For our purpose, it is convenient to see a configuration $\omega$ as an element of $C(V) \otimes C^*(\mathbb{Z}_q)$ such that,
\begin{equation*}
\omega(i)=\chi_{g_i}\qquad \text{for some}\quad g_i \in \mathbb{Z}_q.
\end{equation*}
\par
Let $\tau:C^*(\mathbb{Z}_q) \rightarrow \mathbb{C}$ be a linear functional defined by $\tau(f)=f(e)$, where $e$ is the identity of the cyclic group $\mathbb{Z}_q$.
Let us define a bilinear form $<,>_{H_P}$ on $C(V)\otimes C^*(\mathbb{Z}_q)$ by
$$<f,h>_{H_P} = \sum_{i,j \in V} J'_{ij} \tau ({f(i)}^**h(j)) $$
where $f$ and $h$ are arbitrary elements in $C(V) \otimes C^*(\mathbb{Z}_q)$ and ${f(i)}^*(g)=\overline{f(i)(g^{-1})}$.
We Observe that,
\begin{align*}
<f,h>_{H_P}&=\sum_{i,j \in V} J'_{ij} \tau (({f(i)}^**h(j))\\
&=\sum_{i,j\in V}J'_{ij}\tau\big((\sum_{g_1\in \mathbb{Z}_q}\overline{f(i)(g_1)}\chi_{g_1}^*)*(\sum_{g_2\in \mathbb{Z}_q}h(j)(g_2)\chi_{g_2})\big)\\
&=\sum_{\substack{i,j\in V\\g_1,g_2 \in \mathbb{Z}_q}}J'_{ij}\overline{f(i)(g_1)}h(j)(g_2)\tau\big(\chi_{g_1^{-1}g_2}\big)\\
&=\sum_{\substack{i,j\in V\\g\in \mathbb{Z}_q}}J'_{ij}\overline{f(i)(g)}h(j)(g)\qquad (\text{as $\tau(\chi_g)=1$ iff $g=e$})\\
\end{align*}
Let $\omega \in \Omega_P$. We observe that,
\begin{align}
\notag<\omega,\omega>_{H_P}&=\sum_{\substack{i,j\in V\\g\in \mathbb{Z}_q}}J'_{ij}\overline{\omega(i)(g)}\omega(j)(g)\\\notag
&=\sum_{i,j\in V}J'_{ij}(\sum_{g\in\mathbb{Z}_q}\overline{\omega(i)(g)}\omega(j)(g))\\\notag
&=\sum_{i,j\in V}J'_{ij}\delta_{g_i,g_j}\qquad (\text{as $\omega(i)(g)=\chi_{g_i}$})\\\label{1.9}
&=H_P(\omega).
\end{align}
$<,>_{H_P}$ induces a $S$ valued bilinear form $\widetilde{<,>_{H_P}}$ on $C(V)\otimes C^*(\mathbb{Z}_q)\otimes S$ given by
\begin{equation*}
\widetilde{<f\otimes a,h\otimes b>_{H_P}} := <f,h>_{H_P} a^*b
\end{equation*}
\par
Let $\alpha'$ be the coaction on $C(V)\otimes C^*(\mathbb{Z}_q)$ induced by $\alpha$. It is given by $\alpha'=(id\otimes \sigma_{23})(\alpha \otimes id)$ where $\sigma_{23}$ is the standard flip between 2nd and 3rd coordinates.
\par
\begin{defn}\label{a.0}
$\alpha$ is said to preserve the hamiltonian of q-state Potts model on $(V,E)$ if the following holds:
\begin{equation}\label{2.00}
<\omega,\omega>_{H_P}=\widetilde{<\alpha'(\omega),\alpha'(\omega)>_{H_P}}
\end{equation}
\end{defn}
\begin{nota}\label{b}
We introduce some notations for our convenience. Let $\beta,\gamma\in V$.
\begin{equation*}
Q^{\beta \gamma} := \sum_{k,l \in V} J'_{kl} q_{k\beta } q_{l\gamma }.
\end{equation*}
Let $f\in C(V)$ be defined by,
\begin{equation*}
f(\beta)=\sum_{j \in V} J'_{\beta j} = \sum_{i \in V} J'_{i \beta} \hspace{5mm} \forall \beta \in V.
\end{equation*}
\end{nota}
\par
By evaluationg right hand side of equation $\eqref{2.00}$ we get,
\begin{align}
\widetilde{<\alpha'(\omega), \alpha'(\omega)>_{H_P}}&=<\sum_{i \in V}\alpha'(\chi_i\otimes\omega(i)),\sum_{j \in V}\alpha'(\chi_j\otimes\omega(j))>_{H_P}\notag\\
&=\sum_{i,j,k,l\in V}<\chi_k \otimes\omega(i), \chi_l\otimes \omega(j)>_{H_P} q_{ki}q_{lj}\notag\\
&=\sum_{\substack{i,j,k,l\in V\\g\in \mathbb{Z}_q}}J'_{kl}\overline{\omega(i)(g)}\omega(j)(g)q_{ki}q_{lj}\notag\\
&=\sum_{\substack{i,j\in V\\g\in \mathbb{Z}_q}}\overline{\omega(i)(g)}\omega(j)(g) Q^{ij} \label{2.01}
\end{align}
\begin{rem} \label{c}
From ($\ref{1.9}$) and ($\ref{2.01}$) it follows that $\alpha$ preserves the hamiltonian of q-state Potts model on $(V,E)$ iff the following holds:
\begin{equation}\label{2.1}
\sum_{\substack{i,j\in V\\g\in \mathbb{Z}_q}}\overline{\omega(i)(g)}\omega(j)(g)J'_{ij}1=\sum_{\substack{i,j\in V\\g\in \mathbb{Z}_q}}\overline{\omega(i)(g)}\omega(j)(g)Q^{ij} \qquad \text{for all}\quad \omega \in \Omega_P
\end{equation}
\end{rem}
\begin{lem}\label{d}
If the co-representation matrix $Q$ corresponding to $\alpha$ commutes with $J'$, then $\alpha$ preserves the hamiltonian $H_P$ in the sense of definition ($\ref{a.0}$).
\end{lem}
\begin{proof}
Let $i,j\in V$. We observe that,
\begin{equation*}
\begin{aligned}
Q^{ij}=\sum_{k,l \in V} J'_{kl} q_{ki} q_{lj} &= \sum_{k\in V} q_{ki} (\sum_{l\in V}J'_{kl} q_{lj}) \\
&=\sum_{k\in V} \sum_{l\in V} J'_{lj} q_{ki} q_{kl} \\
&=\sum_{k\in V} J'_{ij} q_{ki}=J'_{ij}1\\
\end{aligned}
\end{equation*}
Hence from ($\ref{2.1}$), the result follows.
\end{proof}
\par
It is interesting to ask, whether the converse is true, that is, the corepresentation matrix $Q$ of a hamiltonian preserving coaction $\alpha$ commutes with $J'$ or not. It turns out to be true. To show that, we will need two following lemmas.
\begin{lem}\label{e}
Let $\alpha$ be the co-action of $(S,\Delta)$ on $C(V)$ as before and $h$ be the haar functional on $(S, \Delta)$. If $h(Q^{\beta\gamma})=J'_{\beta\gamma}$ for all $\beta,\gamma \in V$, then $QJ'=J'Q$, where $Q$ is the co-representation matrix of $\alpha$.
\end{lem}
\begin{proof}
Let $\beta, \gamma \in V$. Then
\begin{align}
\Delta(Q^{\beta\gamma}) &=\Delta(\sum_{k,l \in V}J'_{kl}q_{k\beta }q_{l\gamma })\notag \\
&=\sum_{k,l \in V}J'_{kl}\Delta(q_{k\beta })\Delta(q_{l\gamma })\notag \\
&=\sum_{k,l \in V}J'_{kl}(\sum_{k'\in V}q_{k k'}\otimes q_{k'\beta})(\sum_{l'\in V}q_{ll'}\otimes q_{l'\gamma}) \notag \\
&=\sum_{k,l,k',l' \in V}J'_{kl}(q_{ kk'}q_{ll'} \otimes q_{k'\beta}q_{l'\gamma}) \label{2.2}
\end{align}
As $h$ is the Haar functional, we have,
\begin{equation}
(h \otimes Id)\Delta(s)=h(s)1 \hspace{5mm} \forall s \in S
\end{equation}
From equation$\eqref{2.2}$ we get,
\begin{align*}
(h \otimes Id)\Delta(Q^{\beta\gamma}) &=\sum_{k,l,k',l' \in V} J'_{kl}h(q_{kk'}q_{ll'})q_{k'\beta }q_{l'\gamma} \\
&=\sum_{k',l'\in V}(h(\sum_{k,l\in V}J'_{kl}q_{kk'}q_{ll'}))q_{k'\beta }q_{l'\gamma} \\
&=\sum_{k',l'\in V}q_{k'\beta}q_{l'\gamma }h(Q^{k'l'}) \\
&=\sum_{k',l'\in V}q_{k'\beta}q_{l'\gamma}J'_{k'l'}=Q^{\beta \gamma}
\end{align*}
From our hypothesis it follows that
\begin{equation}\label{2.3}
Q^{\beta\gamma}=J'_{\beta\gamma}1.
\end{equation}
Finally we observe,
\begin{align*}
(QJ')_{ij}=\sum_{k\in V}q_{ik}J'_{kj}
&=\sum_{k\in V}q_{ik}(\sum_{k',l\in V}J'_{k'l}q_{k'k}q_{lj}) \quad \text{(from $\eqref{2.3}$)} \\
&=\sum_{k,l\in V}J'_{il}q_{ik}q_{lj} \\
&=\sum_{l\in V}J'_{il}q_{lj}=(J'Q)_{ij}
\end{align*}
Hence we get that $Q$ and $J'$ commutes.
\end{proof}
From lemma ($\ref{d}$) and lemma ($\ref{e}$), we get the following result:
\begin{thm}\label{f}
Let $(S,\Delta)$ be a compact quantum group co-acting on $C(V)$. The co-representation matrix $Q$ commutes with $J'$ if and only if $Q^{\beta\gamma}=J'_{\beta\gamma}1$ for all $\beta, \gamma \in V$.
\end{thm}
\par
\begin{lem}\label{g}
Let $\alpha$ be a coaction of $(S,\Delta)$ on $C(V)$. If $\alpha$ preserves the hamiltonian for a q-state Potts Model on $(V,E)$, then $\alpha (f)=f\otimes 1$ where $f$ is described in notation($\ref{b}$)
\end{lem}
\begin{proof}
To show that $\alpha(f) = f\otimes 1$, it is enough to show
\begin{equation}\label{2.31}
\sum_{i\in V}f(i)q_{\beta i}= f(\beta)1
\end{equation}
for all $\beta\in V$.
\par
Let us fix $\beta$ in $V$ and $g_0 \in \mathbb{Z}_q$ such that $g_{0} \neq e$. We define $\omega:V \rightarrow C^*(\mathbb{Z}_q)$ by $\omega(\beta)=\chi_{g_0}$ and $\omega(i) = \chi_{e}$ for $i\neq \beta$.
For $\omega$, we evaluate right hand side of equation $\eqref{2.1}$ as follows:
\begin{align*}
\sum_{\substack{i,j\in V\\g\in \mathbb{Z}_q}}\overline{\omega(i)(g)}\omega(j)(g)Q^{ij}&=\sum_{\substack{i\in V\\g\in \mathbb{Z}_q}}\overline{\omega(i)(g)}\omega(\beta)(g)Q^{i\beta}+\sum_{\substack{j\in V\\g\in \mathbb{Z}_q}}\overline{\omega(\beta)(g)}\omega(j)(g)Q^{\beta j}\\&\:\:\:\:\:+\sum_{\substack{i,j\neq \beta\\g\in \mathbb{Z}_q}}\overline{\omega(i)(g)}\omega(j)(g)Q^{ij}\\
&=\sum_{i,j\neq \beta}Q^{ij}\\
&=\sum_{i,j\neq \beta}\sum_{k,l\in V}J'_{kl}q_{ki}q_{lj}\\
&= \sum_{k,l\in V} J'_{kl}(1 - q_{k\beta })(1-q_{l\beta }) \\
&=\sum_{k,l\in V}J'_{kl}(1-q_{k\beta }-q_{l\beta }) \\
&=\sum_{k\in V} f(k)-2\sum_{l\in V} f(l)q_{l\beta }.
\end{align*}
For $\omega$, evaluating left hand side of equation$\eqref{2.1}$ we get,
\begin{align*}
\sum_{\substack{i,j\in V\\g\in \mathbb{Z}_q}}J'_{ij}\overline{\omega(i)(g)}\omega(j)(g)&=\sum_{\substack{i\in V\\g\in \mathbb{Z}_q}}J'_{i\beta}\overline{\omega(i)(g)}\omega(\beta)(g)+\sum_{\substack{j\in V\\g\in \mathbb{Z}_q}}J'_{\beta j}\overline{\omega(\beta)(g)}\omega(j)(g)\\
&\:\:\:\: +\sum_{\substack{i,j\neq\beta\\g\in \mathbb{Z}_q}}J'_{ij}\overline{\omega(i)(g)}\omega(j)(g)\\
&=\sum_{i,j \neq \beta}J'_{ij}\\
&=\sum_{j\neq\beta}(f(j)-J'_{\beta j})\\
&=\sum_{j\in V}f(j)-2f(\beta)\\
\end{align*}
By our hypothesis and remark ($\ref{c}$) we know that equation (\ref{2.1}) holds. Hence, we conclude that (\ref{2.31}) is true.
\end{proof}
Now we are in a position to prove the main result of this paper.
\begin{thm}\label{h}
Let $\alpha$ is a coaction of $(S,\Delta)$ on $C(V)$. If $\alpha$ preserves the hamiltonian of q-state Potts model on $(V,E)$, then the co-representation matrix $Q$ of $\alpha$ commutes with $J'$.
\end{thm}
\begin{proof}
We fix $\beta,\gamma \in V$ such that $\beta\neq\gamma$ and $g_0\in\mathbb{Z}_q$ which is not $e$. We define a configuration $\omega:V \rightarrow C^*(\mathbb{Z}_q)$ by
\begin{equation*}
\begin{aligned}
\omega(\beta)&=\chi_{g_0},\\ \omega(\gamma)&=\chi_{g_0},\\ \omega(i)&=\chi_e\quad \text{for}\quad i\neq\beta,\gamma.\\
\end{aligned}
\end{equation*}.
\par
Evaluating right hand side of equation$\eqref{2.1}$ for $\omega$ we get,
\
\begin{align}
\notag\sum_{\substack{i,j\in V\\g\in \mathbb{Z}_q}}\overline{\omega(i)(g)}\omega(j)(g)Q^{ij}=&\sum_{\substack{j\neq\beta,\gamma\\g\in\mathbb{Z}_q}}\overline{\omega(\beta)(g)}\omega(j)(g)Q^{\beta j}+\sum_{\substack{j\neq\beta,\gamma\\g\in\mathbb{Z}_q}}\overline{\omega(\gamma)(g)}\omega(j)(g)Q^{\gamma j}\\\notag
&+\sum_{\substack{i\neq\beta,\gamma\\g\in\mathbb{Z}_q}}\overline{\omega(i)(g)}\omega(\beta)(g)Q^{i\beta}+\sum_{\substack{i\neq\beta,\gamma\\g\in\mathbb{Z}_q}}\overline{\omega(i)(g)}\omega(\gamma)(g)Q^{i\gamma}\\\notag
&+\sum_{\substack{i\neq\beta,\gamma\\j\neq\beta,\gamma\\ g\in\mathbb{Z}_q}}\overline{\omega(i)(g)}\omega(j)(g)Q^{ij}+Q^{\beta\gamma}+Q^{\gamma\beta}\\\notag
=&\sum_{\substack{i\neq\beta,\gamma\\j\neq\beta,\gamma}}Q^{ij}+Q^{\beta\gamma}+Q^{\gamma\beta}\\\notag
=&\sum_{k,l\in V}J'_{kl}(\sum_{\substack{i\neq\beta,\gamma\\j\neq\beta,\gamma}}q_{ki}q_{lj})+Q^{\beta\gamma}+Q^{\gamma\beta}\\\notag
=&\sum_{k,l\in V}J'_{kl}(1-q_{k\beta}-q_{k\gamma })(1-q_{l\beta}-q_{l\gamma })+Q^{\beta\gamma}+Q^{\gamma\beta}\\\notag
=&\sum_{k,l \in V}J'_{kl}(1-q_{k\beta}-q_{k\gamma }-q_{l\beta}+q_{k\gamma }q_{l\beta }-q_{l\gamma }+q_{k\beta}q_{l\gamma })\\\notag&\:\:+Q^{\beta\gamma}+Q^{\gamma\beta}\\\notag
=&\sum_{k\in V} f(k)-f(\beta)1-f(\gamma)1-f(\beta)1+Q^{\gamma\beta}-f(\gamma)1\\\notag&\:\:+Q^{\beta\gamma}+Q^{\beta\gamma}+Q^{\gamma\beta}\\\label{2.4}
=&\sum_{k\in V}f(k)-2f(\beta)1-2f(\gamma)1+2Q^{\beta\gamma}+2Q^{\gamma\beta}
\end{align}
By evaluating left hand side of equation (\ref{2.1}) for $\omega$ we get,
\begin{align}
\notag\sum_{\substack{i,j\in V\\g\in \mathbb{Z}_q}}\overline{\omega(i)(g)}\omega(j)(g)J'_{ij}=&\sum_{\substack{j\neq\beta,\gamma\\g\in\mathbb{Z}_q}}\overline{\omega(\beta)(g)}\omega(j)J'_{\beta j}+\sum_{\substack{j\neq\beta,\gamma\\g\in\mathbb{Z}_q}}\overline{\omega(\gamma)(g)}\omega(j)(g)J'_{\gamma j}\\\notag
&+\sum_{\substack{i\neq\beta,\gamma\\g\in\mathbb{Z}_q}}\overline{\omega(i)(g)}\omega(\beta)(g)J'_{i\beta}+\sum_{\substack{i\neq\beta,\gamma\\g\in\mathbb{Z}_q}}\overline{\omega(i)(g)}\omega(\gamma)(g)J'_{i\gamma}\\\notag
&+\sum_{\substack{i\neq\beta,\gamma\\j\neq\beta,\gamma\\\notag g\in\mathbb{Z}_q}}\overline{\omega(i)(g)}\omega(j)(g)J'_{ij}+J'_{\beta\gamma}+J'_{\gamma\beta}\\\notag
=&\sum_{\substack{i\neq\beta,\gamma\\j\neq\beta,\gamma}}J'_{ij}+J'_{\beta\gamma}+J'_{\gamma\beta}\\\notag
=&\sum_{i\neq\beta,\gamma}(f(i)-J'_{i\beta}-J'_{i\gamma})+J'_{\beta\gamma}+J'_{\gamma\beta}\\\notag
=&\sum_{i\neq\beta,\gamma}f(i)-\sum_{i\neq\beta,\gamma}J'_{i\beta}-\sum_{i\neq\beta,\gamma}J'_{i\gamma}+J'_{\beta\gamma}+J'_{\gamma\beta}\\\notag
=&\sum_{i\in V}f(i)-f(\beta)-f(\gamma)-f(\beta)+J'_{\gamma\beta}-f(\gamma)+J'_{\beta\gamma}\\\notag&\:\:+J'_{\beta\gamma}+J'_{\gamma\beta}\\\label{2.41}
=&\sum_{i\in V}f(i)-2f(\beta)-2f(\gamma)+2J'_{\beta\gamma}+2J'_{\gamma\beta}
\end{align}
From our hypothesis and remark (\ref{c}) we know that equation ($\ref{2.1}$) holds. Hence, from $\eqref{2.4}$ and $\eqref{2.41}$ and lemma (\ref{g}) we get,
\begin{align*}
Q^{\beta\gamma}+Q^{\gamma\beta}&=(J'_{\beta\gamma}+J'_{\gamma\beta})1\\
\text{which implies},\qquad h(Q^{\beta\gamma})&=J'_{\beta\gamma}1.\qquad (\text{as $h$ is tracial on $S$})
\end{align*}
Since our choice of $\beta$, $\gamma$ was arbitrary, from Lemma($\ref{e}$) the theorem follows.
\end{proof}
\par
\begin{lem}\label{h.1}
let $\alpha$ be a co-action of a compact quantum group $(S,\Delta)$ on $C(V)$ which preserves the hamiltonian of a q-state Potts model on $(V,E)$. Let $k\in\mathbb{C}$ and $K:=span \{\chi_v\in C(V)| f(v)=k \}$. Then $\alpha(K)\subseteq K\otimes S$.
\end{lem}
\begin{proof}
From lemma (\ref{g}), it follows that $\alpha(f)=f\otimes 1$ where $f$ is described in notation (\ref{b}). Hence we have,
\begin{equation*}\label{x}
\sum_{i\in V}f(i)q_{ji}=f(j)1 \qquad \text{for all}\quad j\in V.
\end{equation*}
Now we choose $v\in V$ such that $f(v)=k$ and $w\in V$ such that $f(w)\neq k$. To prove our claim it is enough to show that $q_{vw}=0$.
We observe that,
\begin{equation*}
q_{vw}f(v) = q_{vw}\sum_{k\in V}f(k)q_{vk}=q_{vw}f(w)\\
\end{equation*}
As $f(v)\neq f(w)$, it follows that $q_{vw}=0$.
\end{proof}
\begin{thm}\label{i}
Let $(S,\Delta)$ be a compact quantum group which co-acts on $C(V)$ via $\alpha$ by preserving the hamiltonian of the q-state Potts Model on $(V,E)$. We define $\alpha^{(2)}$ as
\begin{equation*}
\alpha ^{(2)}= (id\otimes id\otimes m)(id\otimes\sigma_{23}\otimes id)(\alpha\otimes\alpha)
\end{equation*}
where $m$ is the multiplication on $S$ and $\sigma_{23}$ is the standard flip on second and third coordinate. Let $K_c:= Span\{\chi_k \otimes \chi_l| J'_{kl}=c\}$. Then $\alpha^{(2)}(K_c) \subseteq K_c \otimes S$.
\end{thm}
\begin{proof}
Let $i,j \in V$. We have
\begin{equation*}
\alpha^{(2)}(\chi_i \otimes \chi_j)= \sum_{k,l \in V}\chi_k\otimes \chi_l \otimes q_{ki}q_{lj}
\end{equation*}
Our claim is equivalent to the statement that
\begin{equation}\label{3}
q_{ki}q_{lj}=0 \quad\text{if}\quad J'_{kl}\neq J'_{ij}.
\end{equation}
\par
As $\alpha$ preserves the hamiltonian on $(V,E)$, from theorem($\ref{h}$) and remark(${\ref{f}}$) it follows that, \begin{equation*}
Q^{ij}=J'_{ij}1 \qquad \text{for all $i,j$ in $V$}.
\end{equation*}
\par
Let us first assume that $J'$ to be a real matrix. Let $\phi : V \times V \rightarrow \mathbb{R}$ be defined by $\phi(i,j)=J'_{ij}$. Let $Image(\phi)=\{k_1,k_2,...,k_r\}$ where $k_1<k_2<...<k_r \in \mathbb{R}$. Let $(i_0,j_0)\in V\times V$ such that $J'_{i_0,j_0}=k_r$. We note that,
\begin{equation}\label{4}
\sum_{k,l \in V}(J'_{i_0j_0}-J'_{kl})h(q_{ki_0}q_{lj_0})
=h(J'_{i_0j_0}1-Q^{i_0j_0})=0
\end{equation}
where $h$ is the Haar state on $(S,\Delta)$.We Observe that
\begin{equation}\label{5}
h(q_{ki}q_{lj})=h(q_{ki}q_{lj}q_{ki}) \geq 0 \qquad \text{for all}\quad i,j,k,l \in V.
\end{equation}
As $(J'_{i_0j_0}-J'_{kl})\geq 0$, from equation (\ref{4}) and equation (\ref{5}) it follows that, for any pair $(k,l)\in V \times V$ with $J'_{i_0j_0}\neq J'_{kl} $, $h(q_{ki_0}q_{lj_0})=0$. As $h$ is faithful on the underlying dense Hopf * Algebra of $(S,\Delta)$, it follows that $q_{ki_0}q_{lj_0}=0$ for any pair $(k,l)\in V \times V$ with $J'_{i_0j_0}\neq J'_{kl} $.
\par
We have proved equation (\ref{3}) to be true for a pair $(i,j)\in V\times V$ with $J'_{ij}=k_r$. For other $k_n$'s we proceed by induction. We fix $k_n\in Image(\Phi)$ and assume that,
\begin{equation*}
q_{ki}q_{lj}=0 \quad \text{where}\quad J'_{ij}=k_n\quad\text{and}\quad J'_{ij}\neq J'_{kl}.
\end{equation*}. We want to see whether the above statement holds true for $k_{n-1}$. We choose $i,j\in V$ such that $J'_{ij}=k_{n-1}$. For $k,l\in V$, if $J'_{kl}>J'_{ij}$, then $q_{ki}q_{lj}=0$ by our induction hypothesis. As before, we observe that,
\begin{equation}\label{6}
\sum_{\substack{k,l\in V\\\text{such that} J'_{kl}\leq J'_{ij}}}(J'_{ij}-J'_{kl})h(q_{ki}q_{lj})=\sum_{k,l \in V}(J'_{ij}-J'_{kl})h(q_{ki}q_{lj})
=h(J'_{ij}1-Q^{ij})=0
\end{equation}
From (\ref{6}), it follows that, for $k,l\in V$ with $J'_{kl}< J'_{ij} $, $h(q_{ki}q_{lj})=0$, which in turn implies that $q_{ki}q_{lj}=0$. The theorem is proved when the matrix $J'$ is real.
\par
Now let us assume that $J'=J'_R + iJ'_I$ where $J'_R$ and $J'_I$ are real matrices. As the co-representation matrix $Q$ consists of positive elements in $S$, it follows that $Q$ commutes with $J'$ iff $Q$ commutes with both $J'_R$ and $J'_I$. Using the real case we conclude that for $i,j,k,l \in V$,
\begin{align*}
q_{ki}q_{lj}&=0 \qquad \text{if}\quad J'_{R_{ij}}\neq J'_{R_{kl}}\\
q_{ki}q_{lj}&=0 \qquad \text{if}\quad J'_{I_{ij}}\neq J'_{I_{kl}}
\end{align*}
As $J'_{ij}=J'_{R_{ij}}+iJ'_{I_{ij}}$, it follows that $q_{ki}q_{lj}=0$ if $J'_{ij}\neq J'_{kl}$ for $i,j,k,l\in V$.
\end{proof}
\begin{lem}\label{j}
Let $\alpha$ be a coaction of $S$ on $C(V)$ preserving the hamiltonian $H_P$ on $(V,E)$. It is immidiate that $a_{ij}=0$ implies $J'_{ij}=0$. If we further assume that $J'_{ij}=0$ implies $a_{ij}=0$, then $\alpha$ also preserves the quantum symmetry of the underlying graph $(V,E)$, that is, the co-representation matrix $Q$ commutes with the adjacency matrix $A$ of $(V,E)$.
\end{lem}
\begin{proof}
We define $R^{ij}=\sum_{k,l\in V}a_{kl}q_{ki}q_{lj}$. From theorem ($\ref{f}$), we know that $Q$ commutes with $A$ iff $R^{ij}=a_{ij}1$ for all $i,j\in V$. To prove the result, it is enough to show
\begin{equation*}
R^{ij}=a_{ij}1\quad\text{for all}\quad i,j\in V.
\end{equation*} From our hypothesis and theorem ($\ref{i}$) it follows that
\begin{equation}\label{6.1}
q_{ki}q_{lj}=0 \qquad \text{if $a_{ij}=1$ but $a_{kl}=0$}
\end{equation}
Let $i,j\in V$ be such that $a_{ij}=1$. We observe that,
\begin{equation*}
1=\sum_{k,l\in V}q_{ki}q_{lj}=\sum_{\substack{k,l\in V\\\text{with} \hspace{1mm} a_{kl}=1}}q_{ki}q_{lj}+\sum_{\substack{k,l\in V\\\text{with}\hspace{1mm} a_{kl}=0}}q_{ki}q_{lj}=R^{ij}+0\quad \text{(from $\eqref{6.1}$)}
\end{equation*}
The other case where $a_{ij}=0$ similarly follows.
\end{proof}
\begin{thm}
There exists a unique universal object in the category of compact quantum groups co-acting on $(V,E)$ by preserving the hamiltonian of q-state Potts model.
\end{thm}
\begin{proof}
From lemma (\ref{d}) and theorem (\ref{h}), it follows that a compact quantum group $(S,\Delta)$ co-acts on $(V,E)$ via preserving the hamiltonian iff the co-representation matrix $Q$ commutes with $J'$. Hence, from theorem (\ref{uni.obj.complex.matrix}) our claim follows.
\end{proof}
Let us call this unique universal object quantum symmetry group of Potts model on $(V,E)$. Furthermore, if the hypothesis in lemma ($\ref{j}$) is satisfied, then this object turns out to be a quantum subgroup of the quantum automorphsim group of the underlying graph $(V,E)$. .
\section{Examples}
\subsection{Example 1}
Here we look at an example of Potts model where we observe that a slight fluctuation in hamiltonian can destroy the quantum symmetry present in system and turn it into a classical one. This change in quantum symmetry can indicate towards a kind of phase transition in the system.
\par
We start with the graph of a cube as shown in figure (\ref{ex1.}). Let the set of vertices be $V=\{1,2,3,4,1',2',3',4'\}$ and the edges $E$ are as shown in the picture.
Let $\lambda \in \mathbb{C}$. Consider the hamiltonian $H_{\lambda}$ given by,
\begin{equation}
H_{\lambda}(\omega)=\sum_{i,j\in V}J'_{ij}\delta_{\omega(i),\omega(j)} \quad \forall \omega\in\Omega_P
\end{equation}
where $J'_{ij}$ is given by,
\begin{equation*}
\begin{aligned}
J'_{ij}&=1\quad\text{if}\quad (i,j)\neq (4,4'),(4',4).\\
J'_{44'}&=J'_{4'4}=\lambda\\
\end{aligned}
\end{equation*}
\begin{figure}
\begin{center}
\includegraphics[]{Example1fig}
\end{center}
\caption{The graph of a cube with a specified hamiltonian on it. }\label{ex1.}
\end{figure}
\begin{rem}
When $\lambda=1$, The quantum symmetry group of Potts Model is the quantum automoprhism group for the graph $(V,E)$ which is a non-classical compact quantum group. See for instance (\cite{MR2336835}).
\end{rem}
\begin{lem} When $\lambda$ is not $1$, the quantum symmetry group of Potts model on $(V,E)$ becomes commutative.
\end{lem}
\begin{proof}Let $\lambda$ be a complex number which is not $1$. Let $\alpha$ be a coaction of a compact quantum group $(S,\Delta)$ on $C(V)$ preserving the hamiltonian $H_{\lambda}$. From theorem($\ref{h}$), the co-representation matrix $Q$ of $\alpha$ commutes with $J'$.
\par
From lemma ($\ref{h.1}$) it follows that $q_{i4}=q_{4j}=q_{i4'}=q_{4'j}=0$ when $i\neq 4,4'$ and $j\neq 4,4'$. Hence it follows that $$q_{44'}=q_{4'4} \quad\text{and}\quad q_{44}=q_{4'4'}$$
\par
Using $QJ'=J'Q$, we get the following commutation relations:
\begin{equation}\label{y}
\begin{aligned}
q_{44}=&(J'Q)_{14}=(QJ')_{14}=q_{11}+q_{13}\\
0=&(J'Q)_{24}=(QJ')_{24}=q_{21}+q_{23}\\
q_{4'4}=&(J'Q)_{1'4}=(QJ')_{1'4}=q_{1'1}+q_{1'3}\\
0=&(J'Q)_{2'4}=(QJ')_{2'4}=q_{2'1}+q_{2'3}\\
q_{4'4'}=&(J'Q)_{1'4'}=(QJ')_{1'4'}=q_{1'1'}+q_{1'3'}\\
0=&(J'Q)_{2'4'}=(QJ')_{2'4'}=q_{2'1'}+q_{2'3'}\\
\end{aligned}
\end{equation}
From equations (\ref{y}) We note that $q_{21}+q_{23}=0$, which implies $q_{21}=0$ and $q_{23}=0$. Similarly $q_{2'1}=q_{2'3}=q_{21'}=q_{23'}=q_{2'1'}=q_{2'3'}=0$. The co-representation matrix $Q$ becomes,
$$
\begin{bmatrix}
q_{11}&0&q_{13}&0&q_{11'}&0&q_{13'}&0\\
0&q_{22}&0&0&0&q_{22'}&0&0\\
q_{31}&0&q_{33}&0&q_{31'}&0&q_{33'}&0\\
0&0&0&q_{44}&0&0&0&q_{44'}\\
q_{1'1}&0&q_{1'3}&0&q_{1'1'}&0&q_{1'3'}&0\\
0&q_{2'2}&0&0&0&q_{2'2'}&0&0\\
q_{3'1}&0&q_{3'3}&0&q_{3'1'}&0&q_{3'3'}&0\\
0&0&0&q_{4'4}&0&0&0&q_{4'4'}\\
\end{bmatrix}
$$
By equating 1st row of $(J'Q)$ and $(QJ')$ we get,
\begin{equation}\label{z}
\begin{aligned}
q_{1'1}=(J'Q)_{11}&=(QJ')_{11}=q_{11'}\\
q_{22}=(J'Q)_{12}&=(QJ')_{12}=q_{11}+q_{13}=q_{44}\\
q_{1'3}=(J'Q)_{13}&=(QJ')_{13}=q_{13'}\\
q_{1'1'}=(J'Q)_{11'}&=(QJ')_{11'}=q_{11}\\
q_{22'}=(J'Q)_{12'}&=(QJ')_{12'}=q_{11'}+q_{13'}=q_{44'}\\
q_{1'3'}=(J'Q)_{13'}&=(QJ')_{13'}=q_{13}\\
\end{aligned}
\end{equation}
Finally, from (\ref{z}) and (\ref{y}) we observe that,
\begin{equation}
q_{i'j}=q_{ij'}\quad \text{and}\quad q_{i'j'}=q_{ij}\quad \text{for all}\quad i,j\in \{1,2,3,4\}.
\end{equation}
These are enough relations to conclude that the entries of $Q$ commute with each other. Hence the quantum symmetry group of Potts model on $(V,E)$ is commutative.
\end{proof}
\subsection{Example 2}
We look at an example of Potts model where slight fluctuation of hamiltonian changes the quantum symmetry of the system but does not affect its classical symmetry.
\par
We consider the graph $(V,E)$ shown in figure (\ref{ex2.}). Let $\lambda\in \mathbb{C}$. Consider the following hamiltonian $H_{\lambda}$ on $(V,E)$ given by
\begin{equation*}
H_{\lambda}(\omega)=\sum_{i,j\in V}J'_{ij}\delta_{\omega(i),\omega(j)} \quad \forall \omega\in\Omega_P
\end{equation*}
\begin{figure}
\begin{center}
\includegraphics[]{Example2fig}
\end{center}
\caption{The graph in example 2 with the corresponding hamiltonian.}\label{ex2.}
\end{figure}
where $J'_{78}=J_{17}=J'_{18}=J'_{25}=J'_{26}=\lambda$ and $J'_{ij}=1$ otherwise. The underlying graph $(V,E)$ does not have any quantum symmetry (see section (4.4) of \cite{schmidt2020thesis}). In light of theorem (\ref{i}), we observe that,
\begin{rem}
When $\lambda\neq 0$, the quantum symmetry group of Potts model is $\mathbb{C}(\mathbb{Z}_2)\otimes\mathbb{C}(\mathbb{Z}_2)$.
\end{rem}
\begin{lem}
When $\lambda=0$, the quantum symmetry group of Potts model on $(V,E)$ becomes non-classical.
\end{lem}
\begin{proof}
let $(S,\Delta)$ be the quantum symmetry group for Potts model on $(V,E)$ co-acting on $C(V)$ via $\alpha$. As before, from theorem ($\ref{h}$) it follows that the co-representation matrix $Q$ commutes with $J'$.
\par
Note that from lemma (\ref{h.1}) it follows that
\begin{equation}\label{e.2.1}
\begin{aligned}
q_{3j}=q_{4j}&=0\quad \text{for}\quad j\neq 3,4.\\
q_{1j}=q_{2j}&=0\quad \text{for}\quad j\in\{3,4,5,6\}.\\
q_{7j}=q_{8j}&=0 \quad \text{for} \quad j\in\{3,4,5,6\}.
\end{aligned}
\end{equation}
We observe that,
\begin{equation}\label{e.2.2}
\begin{aligned}
q_{34}=(J'Q)_{74}&=(QJ')_{74}=q_{78}\\
q_{34}=(J'Q)_{54}&=(QJ')_{54}=q_{56}\\
0=(J'Q)_{13}&=(QJ')_{13}=q_{17}\\
0=(J'Q)_{14}&=(QJ')_{14}=q_{18}\\
0=(J'Q)_{23}&=(QJ')_{23}=q_{27}\\
0=(J'Q)_{24}&=(QJ')_{24}=q_{28}\\
\end{aligned}
\end{equation}
In light of equations (\ref{e.2.1}) and (\ref{e.2.2}), the co-representation matrix $Q$ becomes,
$$
\begin{bmatrix}
1-p&p&0&0&0&0&0&0\\
p&1-p&0&0&0&0&0&0\\
0&0&1-q&q&0&0&0&0\\
0&0&q&1-q&0&0&0&0\\
0&0&0&0&1-q&q&0&0\\
0&0&0&0&q&1-q&0&0\\
0&0&0&0&0&0&1-q&q\\
0&0&0&0&0&0&q&1-q
\end{bmatrix}
$$
where two projections $p$ and $q$ do not need to commute. Hence we conclude that the quantum symmetry group of Potts model on $(V,E)$ is $\mathbb{C}(\mathbb{Z}_2)*\mathbb{C}(\mathbb{Z}_2)$.
\begin{rem}
For $\lambda=0$, the quantum symmetry group for Potts model is $\mathbb{C}(\mathbb{Z}_2)*\mathbb{C}(\mathbb{Z}_2)$ which is indeed a non-classical comapct quantum group. On the other hand, the classical symmetry group for Potts model is $\mathbb{C}(\mathbb{Z}_2)\otimes\mathbb{C}(\mathbb{Z}_2)$, which is same as the case when $\lambda\neq 0$. Hence we observe that, for $\lambda=0$ classical symmetry in the system remains same but quantum symmety changes drastically.
\end{rem}
\textbf{Acknowledgement}:
\begin{itemize}
\item We thank the referee for his/her insightful comments and poinitng out some more references related to our work.
\item It is also to be noted that Debashish Goswami is partially supported by J.C. Bose national fellowship awarded by D.S.T., Government of India.
\end{itemize}
\end{proof}
\par
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\end{document}
|
2024-02-18T23:40:24.482Z
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2022-07-05T02:17:49.000Z
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{"paloma_paragraphs":[]}
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{"arxiv_id":"2112.14485","language":"en","timestamp":1656987469000,"url":"https:\/\/arxiv.org\/abs\/2112.14485","yymm":"2112"}
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proofpile-arXiv_000-10033
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{"provenance":"002.jsonl.gz:10034"}
| null | null |
\section{Introduction}\label{sectionIntroduction}
We study existence and asymptotic behavior of minimizers for the minimization problem
\begin{equation}\label{minimizationGeneralizedFunctional}
E_{s,g}[m] \coloneqq \inf\left\{\capE_{s,g}(E) \mid \text{$E \subset \mathR^N:$ measurable, $|E| = m$} \right\}
\end{equation}
for any $m>0$, where we define the functional $\capE_{s,g}$ as
\begin{equation}\label{generalizedFunctional}
\capE_{s,g}(E) \coloneqq P_s(E) + \, V_g(E)
\end{equation}
for any measurable set $E \subset \mathR^N$. Note that the first term $P_s$ of \eqref{generalizedFunctional} is the \textit{fractional $s$-perimeter} with $s \in (0,\,1)$ defined by
\begin{equation}\nonumber
P_s(E) \coloneqq \int_{E}\int_{E^c}\frac{1}{|x-y|^{N+s}}\,dx\,dy
\end{equation}
for any measurable set $E \subset \mathR^N$, and the second term $V_g$ of \eqref{generalizedFunctional} is the generalized Riesz potential, defined by
\begin{equation}\nonumber
V_g(E) \coloneqq \int_{E}\int_{E}g(x-y)\,dx\,dy
\end{equation}
for any measurable set $E \subset \mathR^N$, where $g: \mathR^N \setminus \{0\} \to \mathR$ is a non-negative, measurable, and radially symmetric function. The precise assumptions on $g$ will be given in Section \ref{sectionMainResults}.
Problem \eqref{minimizationGeneralizedFunctional} can be regarded as a nonlocal counterpart of the minimization problem
\begin{equation}\label{classicalLiquidDropModel}
E_{g}[m] \coloneqq \inf\left\{ \capE_g(E) \coloneqq P(E) + V_{g}(E) \mid \text{$E \subset \mathR^N$, $|E|=m$}\right\}
\end{equation}
where we let $P(E)$ be the classical perimeter of a set $E$.
A relevant physical case of Problem \eqref{classicalLiquidDropModel} is when $N=3$ and $g(x) = |x|^{-1}$ for $x\in\mathR^N\setminus\{0\}$. In this case, this problem is referred as Gamow's liquid drop model and was firstly investigated by George Gamow in \cite{Gamow} to reveal some basic properties of atoms and provide a simple model of the nuclear fission. In this model, an atomic nucleus can be regarded as nucleons (protons and neutrons) contained in a set $E \subset \mathR^N$. The nucleons are assumed to be concentrated with constant density and implies the number of nucleons is proportional to $|E|$. From a physical point of view, the classical perimeter term corresponds to surface tension, which is minimised by spherical nuclei. On the other hand, the Riesz potential corresponds to a Coulomb repulsion, which tends to drive nuclei away from each other. Due to these properties, the competition between the perimeter term and Riesz potential occurs. By rescaling, one can easily observe this phenomenon. Indeed, using the dilation $\lambda \mapsto \lambda \,E$ for a set $E$, we have that
\begin{equation}\nonumber
P(\lambda E) + V_{g}(\lambda E) = \lambda^{N-1}\,P(E) + \int_{E}\int_{E} \lambda^{2N}\,g(\lambda(x-y))\,dx\,dy
\end{equation}
for any $\lambda>1$ and measurable set $E \subset \mathR^N$. Then, if the kernel $g$ satisfies $g(x) \thickapprox |x|^{-(N+\delta)}$ as $|x| \to \infty$ for $\delta < 1$, we have that $\lambda^{2N}\,g(\lambda\,x) \thickapprox \lambda^{N-\delta}$. Thus, the Riesz potential dominates the perimeter as $\lambda \to \infty$ since $\delta < 1$. On the other hand, if $\lambda \to 0$, then the perimeter dominates the Riesz potential. In Problem \eqref{minimizationGeneralizedFunctional}, the nonlocal perimeter $P_s$ with $s \in (0,\,1)$ behaves like the classical perimeter when $s$ approaches to 1 (see the asymptotic behavior of the $s$-fractional perimeter and more general results in \cite{ADPM, BBM, CaVa, Davila, LeSp01, LeSp02, Ponce}). The authors in \cite{CMT} published a survey on this model and the historical background and some references are therein.
Now let us briefly review the previous works on the classical liquid drop model. Recently, the authors in \cite{FrNa} revisited this model and some references are also therein. The main interest from the mathematical point of view is to investigate the following three topics: the existence of minimizer, the non-existence of minimizer, and the minimality of the ball. Kn\"upfer and Muratov in \cite{KnMu01, KnMu02} considered when $g$ is equal to $|x|^{-\alpha}$ for $\alpha \in (0,\,N)$ with $N \geq 2$ and proved that there exists constants $0< m_0 \leq m_1 \leq m_2 <\infty$ such that the following three things hold: if $N \geq 2$, $\alpha \in (0,\,N)$, and $m \leq m_1$ , then Problem \eqref{classicalLiquidDropModel} admits a minimizer; if $N \geq 2$, $\alpha \in (0,\,2)$, and $m > m_2$, then Problem \eqref{classicalLiquidDropModel} does not admit a minimizer; finally, if $m \leq m_0$, then the ball is the unique minimizer whenever either $N = 2$ and $\alpha \in (0,\,2)$, or $3 \leq N \leq 7$ and $\alpha \in (0,\,N-1)$. Later, Julin in \cite{Julin} proved that, if $N \geq 3$ and $g(x) = |x|^{-(N-2)}$, the ball is the unique minimizer of $\capE_g$ whenever $m$ is sufficiently small. Bonacini and Cristoferi in \cite{BoCr} studied the case of the full parameter range $N \geq 2$ and $\alpha \in (0,\,N-1)$ when $g(x) = |x|^{-\alpha}$. Moreover, for a small parameter $\alpha$, the authors in \cite{BoCr} gave a complete characterization of the ground state. Namely, they showed that, if $\alpha$ is sufficiently small, there exists a constant $m_c$ such that the ball is the unique minimizer of $\capE_g$ for $m \leq m_c$ and $\capE_g$ does not have minimizers for $m > m_c$. In a slightly different context, Lu and Otto in \cite{LuOt} showed the non-existence of minimizers for large volumes and that the ball is the unique minimizer for small volumes when $N = 3$ and $g(x) = |x|^{-1}$. The authors were motivated by the ionization conjecture and the energy that the authors studied includes background potential, which behaves like an attractive term. In the similar context to \cite{LuOt}, the authors in \cite{FNB} showed the non-existence of minimizers for large volumes. In contrast, the authors in \cite{ABCT} proved that a variant of Gamow's model including the background potential admits minimizers for any volume, due to the effects from the background potential against the Riesz potential. Very recently, Novaga and Pratelli in \cite{NoPr} showed the existence of generalized minimizers for the energy associated with $\capE_g$ for any volume. After this work, Carazzao, Fusco, and Pratelli in \cite{CFP} showed that the ball is the unique minimizer for small volumes in any dimensions and for a general function $g$. Concerning the behavior of (generalized) minimizers for large volumes, Pegon in \cite{Pegon} showed that, if the kernel $g$ decays sufficiently fast at infinity and if the volume is sufficiently large, then minimizers exist and converge to a ball, up to rescaling, when the volume goes to infinity. Shortly after, Merlet and Pegon in \cite{MePe} proved that, in dimension $N=2$, minimizers are actually balls for large enough volumes.
One remarkable feature of our results is that some nonlocal effect from the fractional perimeter of $\capE_{s,g}$ enables us to obtain minimizers of Problem \eqref{minimizationGeneralizedFunctional} for any volumes. As we mentioned above, it is known that Problem \eqref{classicalLiquidDropModel} admits the ball as the unique minimizer for sufficiently small volumes if the kernel $g$ is a Riesz kernel. Moreover, the author in \cite{Rigot} proved the existence of minimizers of $\capE_g$ for any volumes if the kernel $g$ has a compact support. Even if the kernel $g$ does not have a compact support but, if $g$ decays sufficiently fast, the author in \cite{Pegon} recently showed the existence of minimizers of $\capE_g$ for sufficiently large volumes. On the other hand, in our problem, we reveal that, if the kernel $g$ does not have a compact support but decays sufficiently fast, then minimizers of $\capE_{s,g}$ exist for any volumes. Hence, unlike the cases studied in \cite{Rigot} and \cite{Pegon}, a sort of nonlocal contribution of the fractional perimeter can ensure the existence of minimizers for any volumes.
By a heuristic argument, one can observe that, if $g$ decays sufficiently fast, the fractional perimeter dominates the Riesz potential even if the volume is sufficiently large. Indeed, if $g(x) \lesssim |x|^{-(N+s')}$ and $s' > s$, then we obtain that
\begin{align}
\capE_{s,g}(\lambda E) &= \lambda^{N-s}\,P_s(E) + \lambda^{2N}\,\int_{E}\int_{E}g(\lambda(x-y))\,dx\,dy \nonumber\\
&= \lambda^{N-s} \left(P_s(E) + \lambda^{s-s'}\,\int_{E}\int_{E}\lambda^{N+s'}g(\lambda(x-y))\,dx\,dy \right)\nonumber
\end{align}
for any set $E \subset \mathR^N$ and $\lambda >0$. Since we assume that $s'>s$, the Riesz potential could be dominated by the nonlocal perimeter term as $\lambda \to \infty$. Thus, one natural question is what would be the behavior of the energy like in the case that the kernel $g$ behaves like the kernel $|x|^{-(N+s)}$ of the fractional perimeter $P_s$.
In this paper, we answer this question. More precisely, we obtain the existence of minimizers for any volume and we characterize the asymptotic behavior of minimizers as the volume goes to infinity, assuming that the kernel $g$ decays faster than the kernel of the fractional perimeter $P_s$. More precisely, we first prove the existence of minimizers of $\capE_{s,g}$ for any volume. To see this, we assume that $g$ is symmetric with respect to the origin, radially non-increasing, and decays faster than the kernel of the fractional perimeter $P_s$. For the details, we refer to Section \ref{sectionPreliminary}. The strategy of the proof is inspired by the concentration-compactness lemma by Lions \cite{Lions01, Lions02} and has been adapted by many authors (see, for instance, \cite{GoNo, dCNRV, CeNo} for topics closely related to ours). We will give some intuitive explanation of the strategy before proving the claim in Section \ref{sectionExisMiniFastDecayGAnyVol}.
Secondly, we prove the existence of generalized minimizers of a generalized functional $\widetilde{\capE}_{s,g}$, which we will define later, under the assumption that the kernel $g$ vanishes at infinity. It is easy to see that this assumption is weaker than the assumption that $g$ decays faster than the kernel of the nonlocal perimeter, which we imposed to prove the first result. For convenience, we here give the definitions of the generalized functional and generalized minimizers. For any $m>0$, we define a \textit{generalized functional} of $\capE_{s,g}$ over the family of sequences of the sets $\{E^k\}_{k\in\mathN}$ with $\sum_{k=1}^{\infty}|E^k| = m$ as
\begin{equation}\label{defiGeneralziedFunctional}
\widetilde{\capE}_{s,g}\left(\{E^k\}_{k\in\mathN}\right) \coloneqq \sum_{k=1}^{\infty}\capE_{s,g}(E^k).
\end{equation}
Then we consider the minimization problem
\begin{equation}\label{minimizationGeneralizedMinimizerFunctional}
\inf\left\{\widetilde{\capE}_{s,g}\left(\{E^k\}_{k\in\mathN}\right) \mid \text{$E^k$: measurable for any $k$, $\sum_{k}|E^k| = m$} \right\}
\end{equation}
and show the existence of a minimizer of Problem \eqref{minimizationGeneralizedMinimizerFunctional} for any $m>0$. We call such a minimizer a \textit{generalized minimizer} of $\capE_{s,g}$. The precise statement will be given Theorem \ref{theoremExistGeneralizedMiniAnyVolume} in Section \ref{sectionMainResults}. The idea to prove our second result is to show the identity
\begin{equation}\nonumber
\inf\left\{\capE_{s,g}(E) \mid |E|=m \right\} = \inf\left\{\widetilde{\capE}_{s,g}(\{E^k\}_k) \mid \sum_{k=1}^{\infty} |E^k| = m \right\}
\end{equation}
for any $m>0$ and apply the same method which we use in the proof of our first result.
Finally, we investigate the asymptotic behavior of minimizers as the volume goes to infinity, under the assumption that $g$ decays faster at infinity than the kernel $|x|^{-(N+s)}$ of the fractional perimeter $P_s$. Here we require an assumption on $g$ which is stronger than the one we assume in the existence result. To study the asymptotic behavior, we consider an equivalent minimization problem. More precisely, one can have two problems equivalent to $E_{s,g}[m]$ for $m>0$ under a proper decay assumption on $g$. Indeed, since the kernel $g$ is integrable over $\mathR^N$ under some proper assumptions, one can rewrite the Riesz potential as
\begin{equation}\nonumber
\int_{E}\int_{E} g(x-y)\,dx\,dy = |E|\,\|g\|_{L^1(\mathR^N)} - \int_{E}\int_{E^c} g(x-y)\,dx\,dy
\end{equation}
for any measurable set $E \subset \mathR^N$ with $|E| < \infty$. Hence, the minimization problem \eqref{minimizationGeneralizedFunctional} becomes
\begin{equation}\label{minimizationModifiedProblem}
\widehat{E}_{s,g}[m] \coloneqq \inf\left\{ P_s(E) - \int_{E}\int_{E^c} g(x-y)\,dx\,dy \mid |E|=m \right\}
\end{equation}
for any $m>0$. Moreover, by rescaling, one can further modify the minimization problem \eqref{minimizationModifiedProblem} into the equivalent problem
\begin{equation}\label{minimizationScalingModifiedProbelm}
\widehat{E}^{\lambda}_{s,g}(B_1) \coloneqq \inf\left\{ \widehat{\capE}^{\lambda}_{s,g}(F) \coloneqq P_s(F) - \int_{E}\int_{E^c} \lambda^{N+s}g(\lambda(x-y))\,dx\,dy \mid |F|=|B_1| \right\}
\end{equation}
for any $\lambda>0$. Note that we will revisit more precisely the notations \eqref{minimizationModifiedProblem} and \eqref{minimizationScalingModifiedProbelm} in Section \ref{sectionMainResults}. With this notation, our last theorem is as follows; suppose that $\{F_n\}_{n}$ is any sequence of the minimizers of $\widehat{\capE}^{\lambda_n}_{s,g}$ such that $\lambda_n \to \infty$ and $|F_n|=|B_1|$ for any $n$. Then we have that the full sequence satisfies
\begin{equation}\nonumber
|F_n \Delta B_1| \xrightarrow[n \to \infty]{} 0
\end{equation}
up to translations.
The organization of this paper is as follows: in Section \ref{sectionMainResults}, we will state our main results, namely, the existence of minimizers, the existence of generalized minimizers, and the convergence of any sequence of rescaled minimizers to the ball. In Section \ref{sectionPreliminary}, we will give several preliminary properties of minimizers of our energy. In Section \ref{sectionExisMiniFastDecayGAnyVol}, we will prove the existence of minimizers for any volumes and, in Section \ref{sectionExistGeneMiniAnyVol}, we will prove the existence of generalized minimizers for any volumes. In Section \ref{sectionAsymptoticMiniLargeVol}, we will study the asymptotic behavior of rescaled minimizers as the volume goes to infinity. We will also give the $\Gamma$-convergence result for our energy.
\begin{center}
\textbf{{\small Acknowledgments}}
\end{center}
The authors would like to thank Marc Pegon for fruitful discussions on this work and several comments on the first draft of our manuscript.
The authors were supported by the INDAM-GNAMPA and by the PRIN Project 2019/24 {\it Variational methods for stationary and evolution problems with singularities and interfaces}.
\section{Main reuslts}\label{sectionMainResults}
We start with the assumptions on the kernel $g$ of the Riesz potential in the energy $\capE_{s,g}$. Throughout this paper, we assume that $s \in (0,\,1)$ and $g:\mathR^N \setminus \{0\} \to \mathR$ is in $L^1_{loc}(\mathR^N)$ and not identically equal to zero. We consider the following conditions on $g$:
\begin{itemize}
\item[(g1)] $g$ is non-negative and radially non-increasing, namely,
\begin{equation}\nonumber
g(\lambda\,x) \leq g(x) \quad \text{for $x \in \mathR^N \setminus \{0\}$ and $\lambda \geq 1$}.
\end{equation}
\item[(g2)] $g$ is symmetric with respect to the origin, namely, $g(-x) = g(x)$ for any $x\in\mathR^N \setminus \{0\}$.
\end{itemize}
When we prove the existence of minimizers of $\capE_{s,g}$ in Section \ref{sectionExisMiniFastDecayGAnyVol}, we further assume the following condition on $g$:
\begin{itemize}
\item[(g3)] There exist constants $R_0>1$ and $\beta \in (0,\,1)$ such that
\begin{equation}\nonumber
g(x) \leq \frac{\beta}{|x|^{N+s}} \quad \text{for any $|x| \geq R_0$}.
\end{equation}
($g$ decays faster than the kernel of $P_s$ far away from the origin.)
\end{itemize}
On the other hand, when we prove the existence of generalized minimizers of $\widetilde{\capE}_{s,g}$ in Section \ref{sectionExistGeneMiniAnyVol}, we assume the following condition, weaker than $(\mathrm{g}3)$:
\begin{itemize}
\item[(g4)] $g$ vanishes at infinity, namely, $g(x) \to 0$ as $|x| \to \infty$.
\end{itemize}
Moreover, when we study the asymptotic behavior of rescaled minimizers with large volumes in Section \ref{sectionAsymptoticMiniLargeVol}, we further impose the following assumption on $g$:
\begin{itemize}
\item[(g5)] There exists a constant $\gamma \in (0,\,1)$ such that
\begin{equation}\nonumber
g(x) \leq \frac{\gamma}{|x|^{N+s}} \quad
\text{for any $x \in \mathR^N \setminus \{0\}$}, \quad g(x) = o\left(\frac{1}{|x|^{N+s}}\right) \quad \text{as $|x| \to \infty$}.
\end{equation}
\end{itemize}
\begin{remark}
From the assumption that $g \in L^1_{loc}(\mathR^N)$, we can easily show that $V_g(B) < +\infty$ for any ball $B \subset\ \mathR^N$. Indeed, one may compute
\begin{equation}\nonumber
V_g(B) \leq \int_{B}\int_{2B(y)} g(x-y)\,dx\,dy = |B|\int_{2B(0)}g(x)\,dx < \infty.
\end{equation}
Moreover, if we assume $(\mathrm{g}3)$, we actually have that $g$ is integrable in $\mathR^N$. Indeed, since $g \in L^1_{loc}(\mathR^N)$, we have that $\|g\|_{L^1(B_{R_0})} < \infty$. On the other hand, from $(\mathrm{g}3)$ and the integrability of $|x|^{-(N+s)}$ in $B^c_{R_0}$, we also have that $\|g\|_{L^1(B^c_{R_0})} < \infty$. Hence, the claim holds true.
\end{remark}
\begin{remark}
A condition ensuring assumption $(\mathrm{g}5)$ is the existence of constants $R_0>1$, $\gamma \in (0,\,1)$, and $t > s$ such that
\begin{align}\label{assumptionGDecayInfinity}
g(x) \leq
\begin{cases}
\displaystyle
\frac{\gamma}{|x|^{N+s}} & \quad \text{if $0< |x| < R_0$}\\
\displaystyle
\frac{1}{|x|^{N+t}} & \quad \text{if $|x| \geq R_0$}.
\end{cases}
\end{align}
Notice that this assumption is stronger than assumption $(\mathrm{g}3)$. To show that \eqref{assumptionGDecayInfinity} implies $(\mathrm{g}5)$, we first take any $\varepsilon>0$ and, without loss of generality, assume that $\varepsilon < R_0^{-(t-s)}$ where $R_0$ is as in \eqref{assumptionGDecayInfinity}. Then it holds that
\begin{equation}\label{assumptionGDecayInfinity02}
\frac{1}{|x|^{N+t}} \leq \frac{\varepsilon}{|x|^{N+s}} \quad \text{for any $|x| \geq \varepsilon^{-\frac{1}{t-s}}$}
\end{equation}
and thus, from \eqref{assumptionGDecayInfinity} and \eqref{assumptionGDecayInfinity02}, we obtain that
\begin{equation}\nonumber
g(x) \leq \frac{\varepsilon}{|x|^{N+s}}
\end{equation}
for any $|x| \geq \varepsilon^{-\frac{1}{t-s}}$. Note that we have used the assumption $t > s$. From \eqref{assumptionGDecayInfinity}, we can easily show that $g(x) \leq \gamma|x|^{-(N+s)}$ for any $x \neq 0$. Hence, this completes the proof of the claim.
\end{remark}
\begin{remark}
In the case of Problem \eqref{minimizationScalingModifiedProbelm} for large volumes, the author in \cite{Pegon} assumed that the kernel $g$ satisfies
\begin{equation}\label{assumptionKernelG}
g \in L^1(\mathR^N), \quad \int_{\mathR^N}|x| \,g(x)\,dx < + \infty.
\end{equation}
This condition with the radial symmetry of $g$ implies that $g$ satisfies
\begin{equation}
g(x) \leq \frac{1}{|x|^{N+s}} \quad \text{for $|x|<1$}, \quad g(x) \leq \frac{c(g)}{|x|^{N+1}} \quad
\text{for $|x|>1$}
\end{equation}
where $c(g)>0$ is some constant. One may find the proof of this implication, for instance, in \cite{Carazzato}. Hence, it is easy to see that the assumption \eqref{assumptionKernelG} implies our assumption $(\mathrm{g}5)$.
\end{remark}
Now we can state the main results of this paper. In the first result, we show the existence of minimizers of $\capE_{s,g}$ for any volume under the assumption that $g$ decays faster than the kernel of $P_s$ at infinity.
\begin{theorem}\label{theoremExistMiniAnyVolumeFasterDecay}
Assume that the kernel $g: \mathR^N \setminus \{0\} \to \mathR$ satisfies the assumptions $(\mathrm{g}1)$, $(\mathrm{g}2)$, and $(\mathrm{g}3)$. Then, there exists a minimizer of $\capE_{s,g}$ with the volume $m$ for any $m>0$.
Moreover, the boundary of every minimizer has the regularity of class $C^{1,\alpha}$ with $\alpha \in (0,\,1)$ except a closed set of Hausdorff dimension $N-3$.
\end{theorem}
The proof is inspired by so-called the ``concentration-compactness" lemma by Lions in \cite{Lions01, Lions02} and we apply the same idea shown in \cite{dCNRV}. We will roughly explain the idea of the proof in Section \ref{sectionExisMiniFastDecayGAnyVol}.
In the second theorem, we show the existence of generalized minimizer of $\widetilde{\capE}_{s,g}$ for any volume, under the assumption that $g$ vanishes at infinity. Notice that this assumption is weaker than the one we impose in Theorem \ref{theoremExistMiniAnyVolumeFasterDecay}.
\begin{theorem}\label{theoremExistGeneralizedMiniAnyVolume}
Assume that the kernel $g: \mathR^N \setminus \{0\} \to \mathR$ satisfies the assumptions $(\mathrm{g}1)$, $(\mathrm{g}2)$, and $(\mathrm{g}4)$. Then, there exists a generalized minimizer of $\widetilde{\capE}_{s,g}$ for any $m>0$, namely, there exist a number $M \in \mathN$ and a sequence of sets $\{E^k\}_{k\in\mathN}$ such that
\begin{equation}\nonumber
\sum_{k=1}^{M} \capE_{s,g}(E^k) = \inf\left\{\widetilde{\capE}_{s,g}(\{E^k\}_k) \mid \sum_{k=1}^{M}|E^k| = m \right\},
\end{equation}
and $E^k$ is also a minimizer of $\capE_{s,g}$ among sets of volume $|E^k|$ for every $k\in\mathN$.
\end{theorem}
As we mentioned in Section \ref{sectionIntroduction}, the idea of the proof is based on the observation that Problem \eqref{minimizationGeneralizedMinimizerFunctional} can be reduced into Problem \eqref{minimizationGeneralizedFunctional}.
Finally, we study the asymptotic behavior of minimizers of $\capE_{s,g}$ when the volume goes to infinity, under the assumption that $g$ decays much faster than the kernel $|x|^{-(N+s)}$ of $P_s$ far away from the origin.
Before stating the theorem, in order to study the behavior of the minimizers of the minimization problem $E_{s,g}[m]$ for any $m>0$, it is convenient to lift the volume constraint onto the functional itself and work with fixed volume $|B_1|$. To see this, we first define a rescaled kernel by
\begin{equation}\label{rescaledKernel}
g_{\lambda}(x) \coloneqq \lambda^{N+s} \, g(\lambda\,x)
\end{equation}
for any $x \neq 0$ and $\lambda>0$. Then we show the equivalence of the rescaled problem in the following proposition.
\begin{proposition}[Equivalent problem]\label{propositionRescaledProblem}
Let $m>0$. Assume that the kernel $g : \mathR^N \setminus \{0\} \to \mathR$ is in $L^1_{loc}(\mathR^N)$. Then, setting $\lambda^N \coloneqq m\,|B_1|^{-1}$, we have that the problem $E_{s,g}[m]$ is equivalent to
\begin{equation}\nonumber
E^{\lambda}_{s,g}(B_1) \coloneqq \inf\left\{ P_s(F) + V_{g_\lambda}(F) \mid \text{$F \subset \mathR^N:$ measurable, $|F| = |B_1|$} \right\}
\end{equation}
where $g_{\lambda}$ is given in \eqref{rescaledKernel}.
Moreover, under the assumption that $g$ is integrable on $\mathR^N$, the minimization problem $E_{s,g}[m]$ is also equivalent to Problem \eqref{minimizationScalingModifiedProbelm}.
\end{proposition}
\begin{proof}
Given any $E$ with $|E|=m$ and setting $F \coloneqq \lambda^{-1}\,E$ where $\lambda^N = m\,|B_1|^{-1}$, we have that $|F| = |B_1|$ and
\begin{align}\label{rescaledIdenityEnergy}
\capE_{s,g}(E) &= \lambda^{N-s}\,P_s(F) + \lambda^{2N}\,\int_{F}\int_{F}g(\lambda(x-y))\,dx\,dy \nonumber\\
&= \lambda^{N-s} \left( P_s(F) + \int_{F}\int_{F}\lambda^{N+s}g(\lambda(x-y))\,dx\,dy \right) \nonumber\\
&= \lambda^{N-s} \left( P_s(F) + V_{g_\lambda}(F) \right)
\end{align}
where $g_{\lambda}(x) \coloneqq \lambda^{N+s}\,g(\lambda\,x)$ as in \eqref{rescaledKernel}. For the latter part of the claim, we first recall the equivalent minimization problem
\begin{equation}\nonumber
\widehat{E}_{s,g}[m] \coloneqq \inf \left\{P_s(E) - \int_{E}\int_{E^c}g(x-y)\,dx\,dy \right\},
\end{equation}
which is equivalent to the problem $E_{s,g}[m]$ for any $m>0$. Thus, from \eqref{rescaledIdenityEnergy}, we obtain that
\begin{equation}\nonumber
\capE_{s,g}(E) = \lambda^{N-s} \left( P_s(F) - \int_{F}\int_{F^c}g_{\lambda}(x-y)\,dx\,dy + m\,\|g\|_{L^1(\mathR^N)} \right).
\end{equation}
Hence, we conclude that the claim is valid.
\end{proof}
Now we are prepared to state the last theorem of this present paper.
\begin{theorem}\label{theoremAsympMiniLargeVolume}
Let $s\in(0,\,1)$ and $\{\lambda_n\}_{n\in\mathN} \subset (1,\,\infty)$ with $\lambda_n \to \infty$ as $n \to \infty$. Let $\{F_n\}_{n\in\mathN}$ be a sequence of minimizers for $\widehat{\capE}^{\lambda_n}_{s,g}$ with $|F_n|=|B_1|$ for each $n\in\mathN$. Assume that the kernel $g: \mathR^N \setminus \{0\} \to \mathR$ is radially symmetric and satisfies the assumptions $(\mathrm{g}1)$, $(\mathrm{g}2)$, and $(\mathrm{g}5)$. Then, the sequence $\{F_n\}_{n\in\mathN}$ converges to the unit ball $B_1$, up to translations, in the sense of $L^1$-topology, namely,
\begin{equation}\nonumber
|F_n \Delta B_1| \xrightarrow[n \to \infty]{} 0.
\end{equation}
\end{theorem}
\begin{remark}
In this paper, we basically assume that the kernel $g$ is locally integrable in $\mathR^N$, especially near the origin; however, Theorem \ref{theoremAsympMiniLargeVolume} is still valid even if $g$ is not integrable in the ball centred at the origin. This is because the assumption that $g(x) \leq |x|^{-(N+s)}$ for $x \neq 0$ is sufficient enough for the nonlocal perimeter $P_g$ to be finite for any ball $B$. We emphasize that the local integrability of $g$ ensures that Problem \eqref{minimizationScalingModifiedProbelm} is equivalent to Problem \eqref{minimizationGeneralizedFunctional}, which is a nonlocal and generalized version of the liquid drop model by Gamow.
\end{remark}
The idea of the proof is based on the same argument of Theorem \ref{theoremExistMiniAnyVolumeFasterDecay} and the $\Gamma$-convergence result on the energy $\widehat{\capE}^{\lambda}_{s,g}$ as $\lambda \to \infty$. We will give the precise strategy of the proof in Section \ref{sectionAsymptoticMiniLargeVol} and, for the $\Gamma$-convergence result, the readers should refer to Proposition \ref{propositionGammaConvergenceNonlocalEnergy} in Section \ref{sectionAsymptoticMiniLargeVol}.
\section{Preliminary results for minimizers of $\capE_{s,g}$}\label{sectionPreliminary}
In this section, we collect several properties for minimizers of $\capE_{s,g}$ under the assumptions on $g$ in Section \ref{sectionMainResults}
First of all, we recall one important property on the fractional perimeter $P_s$ with $0<s<1$.
\begin{proposition}\label{propositionIntersectionConvexSmaller}
For any $s \in (0,\,1)$ and measurable set $E \subset \mathR^N$ with $|E|<\infty$, it follows that $P_s(E \cap K) \leq P_s(E)$ for every convex set $K \subset \mathR^N$.
\end{proposition}
The proof can be found in \cite[Lemma B.1]{FFMMM} and we do not give a proof of this proposition here. We also refer to \cite[Corollary 5.3]{CRS} and \cite{ADPM} for related properties to Proposition \ref{propositionIntersectionConvexSmaller}.
The assumption that $g$ is radially non-increasing enables us to show the scaling property of $\capE_{s,g}$ by simple computations.
\begin{lemma}[Scaling lemma]\label{lemmaScalingEnergy}
Let $E \subset \mathR^N$ be a measurable set with $|E| <\infty$. Assume that the kernel $g:\mathR^N \setminus \to \mathR$ satisfies $(\mathrm{g}1)$ and $(\mathrm{g}2)$. Then, for any $\lambda \geq 1$, it follows that
\begin{equation}\nonumber
\capE_{s,g}(\lambda \,E) \leq \lambda^{2N} \capE_{s,g}(E).
\end{equation}
\end{lemma}
\begin{proof}
From the change of variables and the choice of $\lambda>1$, we have
\begin{equation}\label{scalingEsti01}
P_s(\lambda\,E) = \int_{\lambda\,E}\int_{\lambda\,E^c}\frac{dx\,dy}{|x-y|^{N+s}} = \lambda^{N-s} \int_{E}\int_{E^c} \frac{dx\,dy}{|x-y|^{N+s}} = \lambda^{N-s} P_s(E) \leq \lambda^{2N} P_s(E)
\end{equation}
for any $E \subset \mathR^N$ and $\lambda >1$. From the assumptions on $g$ and the change of variables again, we can compute the Riesz potential as follows:
\begin{equation}\label{scalingEsti02}
V_g(\lambda\,E) = \lambda^{2N} \int_{E}\int_{E} g(\lambda(x-y))\,dx\,dy \leq \lambda^{2N} \int_{E}\int_{E} g(x-y)\,dx\,dy = \lambda^{2N} V_g(E)
\end{equation}
for any $E \subset \mathR^N$ with $|E| <\infty$ and $\lambda>1$. Therefore, from \eqref{scalingEsti01} and \eqref{scalingEsti02}, we obtain
\begin{equation}\nonumber
\capE_{s,g}(\lambda \,E) \leq \lambda^{2N} (P_s(E) + V_g(E)) \leq \lambda^{2N} \capE_{s,g}(E)
\end{equation}
and this completes the proof.
\end{proof}
We next prove the boundedness of minimizers of $\capE_{s,g}$ among sets of volume $m$.
\begin{lemma}[Boundedness of minimizers] \label{lemmaBoundednessMinimizers}
Let $m>0$. Assume that the kernel $g$ satisfies the conditions $(\mathrm{g}1)$ and $(\mathrm{g}2)$. If $E \subset \mathR^N$ is a minimizer of $\capE_{s,g}$ with the volume $m$, then $E$ is bounded up to negligible sets, namely, there exists a constant $\hat{R}>0$ such that $|E \setminus B_{\hat{R}}(0)| = 0$.
\end{lemma}
\begin{proof}
Let $E$ be a minimizer of $\capE_{s,g}$ with $|E|=m$. By setting $\phi(r) \coloneqq |E \setminus B_r(0)|$ for any $r>0$, we have that $\phi^{\prime}(r) = -\capH^{N-1}( E \cap \partial B_r(0) )$ for a.e. $r>0$. In order to prove the claim, we suppose by contradiction that $\phi(r) > 0$ for any $r>0$. Setting $E_r \coloneqq E \cap B_r(0)$ for any $r>0$ and $\lambda_r \coloneqq \frac{m}{m - \phi(r)}$ for any $r>0$, then we choose $\lambda_r E_r$ as the competitor of $E$ if $\phi(r) < m$ and thus, we have that
\begin{align}\label{estimateMinimalityForBoundedness}
\capE_{s,g}(E) \leq \capE_{s,g}(\lambda_r E_r) &\leq (\lambda_r)^{N-s} P_s(E_r) + (\lambda_r)^{2N} V_{g_{\lambda_r}}(E_r) \nonumber\\
&\leq \capE_{s,g}(E_r) + \left((\lambda_r)^{N-s} - 1\right)P_s(E_r) + \left((\lambda_r)^{2N} - 1\right) V_g(E_r).
\end{align}
Since $\phi(r) \to 0$ as $r \to \infty$, we can choose a constant $R_0>0$ such that $\phi(r) \leq m/2$ for any $r \geq R_0$ and thus, we may assume that
\begin{equation}\label{estimateMinimalityForBoundedness02}
(\lambda_r)^{N-s} -1 \leq c_0\,\phi(r), \quad (\lambda_r)^{2N} - 1 \leq c'_0\,\phi(r)
\end{equation}
for any $r \geq R_0$ where $c_0$ and $c'_0$ are some positive constants depending only on $N$, $s$, and $m$. Then, by using the decomposition property of $P_s$ and $V_g$ and combining \eqref{estimateMinimalityForBoundedness02} with \eqref{estimateMinimalityForBoundedness}, we have that
\begin{align}\label{estimateMinimalityForBoundedness03}
P_s(E \setminus B_r(0)) &\leq P_s(E \setminus B_r(0)) + V_g(E \setminus B_r(0)) \nonumber\\
&\leq 2\int_{E \cap B_r(0)}\int_{E \setminus B_r(0)}\frac{dx\,dy}{|x-y|^{N+s}} + c_0\,\phi(r)\,P_s(E_r) + c'_0\,\phi(r)\,V_g(E_r)
\end{align}
for any $r \geq R_0$. From Proposition \ref{propositionIntersectionConvexSmaller} and the definition of $V_g$, we have that, for any $r>0$,
\begin{equation}\nonumber
P_s(E_r) + V_g(E_r) \leq P_s(E) + V_g(E) = E_{s,g}[m].
\end{equation}
Thus, from \eqref{estimateMinimalityForBoundedness03}, we obtain that
\begin{equation}\nonumber
P_s(E \setminus B_r(0)) \leq 2\int_{E \cap B_r(0)}\int_{E \setminus B_r(0)}\frac{dx\,dy}{|x-y|^{N+s}} + (c_0+c'_0)E_{s,g}[m]\,\phi(r)
\end{equation}
for any $r \geq R_0$. Now using the isoperimetric inequality of $P_s$ and the fact that $E \cap B_r(0) \subset B^c_{|y|-r}(y)$ for any $y \in E \setminus B_r(0)$, we obtain
\begin{align}\label{estimateMinimality}
\frac{P_s(B_1)}{|B_1|^{\frac{N-s}{N}}}\,\phi(r)^{\frac{N-s}{N}} &\leq 2\int_{E \setminus B_r(0)}\int_{B^c_{r-|y|}(y)}\frac{dx\,dy}{|x-y|^{N+s}} + (c_0+c'_0)E_{s,g}[m]\,\phi(r) \nonumber\\
&= \frac{2|\partial B_1|}{s}\int_{E \setminus B_r(0)}\int_{|y|-r}^{\infty}\frac{1}{t^{1+s}}\,dt\,dy + (c_0+c'_0)E_{s,g}[m]\,\phi(r) \nonumber\\
&= \frac{2|\partial B_1|}{s}\int_{E \setminus B_r(0)}\frac{1}{(|y|-r)^s}\,dy + (c_0+c'_0)E_{s,g}[m]\,\phi(r) \nonumber\\
&= \frac{2|\partial B_1|}{s} \int_{r}^{\infty}\frac{-\phi^{\prime}(\sigma)}{(\sigma - r)^s}\,d\sigma + (c_0+c'_0)E_{s,g}[m]\,\phi(r)
\end{align}
for any $r>0$. Here we have used the co-area formula in the last equality. Since $\phi$ is non-increasing, there exists a constant $R'_0=R'_0(N,s,m)>0$ such that
\begin{equation}\label{absorptionTerm}
(c_0+c'_0)E_{s,g}[m]\,\phi(r) \leq \frac{P_s(B_1)}{2|B_1|^{\frac{N-s}{N}}}\,\phi(r)^{\frac{N-s}{N}}
\end{equation}
for any $r \geq \max\{R_0, \, R'_0\}$. From \eqref{estimateMinimality} and \eqref{absorptionTerm}, we obtain
\begin{equation}\label{estimateMinimalityKey}
c_1\,\phi(r)^{\frac{N-s}{N}} \leq c_2\,\int_{r}^{\infty}\frac{-\phi^{\prime}(\sigma)}{(\sigma - r)^s}\,d\sigma
\end{equation}
for any $r \geq \max\{R_0, \, R'_0\}$ where we set $c_1 \coloneqq (2|B_1|^{\frac{N-s}{N}})^{-1}\,P_s(B_1)$ and $c_2 \coloneqq 2s^{-1}|\partial B_1|$. By integrating the both sides in \eqref{estimateMinimalityKey} over $r \in [R,\,\infty)$ for any fixed constant $R \geq \max\{R_0, \, R'_0\}$ and changing the order of the integration, we obtain
\begin{align}\label{keyEstiBoundednessMini}
c_1\,\int_{R}^{\infty}\phi(r)^{\frac{N-s}{N}}\,dr \leq c_2\,\int_{R}^{\infty}\int_{r}^{\infty}\frac{-\phi^{\prime}(\sigma)}{(\sigma - r)^s}\,d\sigma\,dr &= c_2\,\int_{R}^{\infty}\int_{R}^{\sigma} \frac{-\phi^{\prime}(\sigma)}{(\sigma - r)^s}\,dr\,d\sigma \nonumber\\
&= -\frac{c_2}{1-s}\, \int_{R}^{\infty}\phi^{\prime}(\sigma)\,(\sigma - R)^{1-s}\,d\sigma.
\end{align}
Hence, by employing the same argument shown in \cite[Lemma 4.1]{dCNRV} and \cite[Proposition 3.2]{CeNo} together with \eqref{keyEstiBoundednessMini}, we obtain that $\phi(R) = 0$, which contradicts the assumption that $\phi(r)>0$ for any $r>0$. Therefore, we conclude the existence of the constant $\hat{R}>0$ such that $|E \setminus B_{\hat{R}}| = 0$.
\end{proof}
Next, by using assumption $(\mathrm{g}4)$, we show the sub-additivity result of the function $m \mapsto E_{s,g}[m]$. We recall that $E_{s,g}[m]$ is defined by
\begin{equation}\nonumber
\inf\left\{ \capE_{s,g}(E) \mid \text{$E \subset \mathR^N$: measurable, $|E|=m$} \right\}.
\end{equation}
for any $m>0$.
\begin{lemma}[Sub-additivity of $E_{s,g}$]\label{lemmaSubadditivityEnergy}
Let $m>0$ be any number. Assume that the kernel $g:\mathR^N \setminus \{0\} \to \mathR$ satisfies $(\mathrm{g}1)$, $(\mathrm{g}2)$, and $(\mathrm{g}4)$. Then, for any $m_1 \in (0,\,m]$, it holds
\begin{equation}\nonumber
E_{s,g}[m] \leq E_{s,g}[m_1] + E_{s,g}[m - m_1].
\end{equation}
\end{lemma}
\begin{proof}
The idea is in the same spirit as the one shown in \cite[Lemma 3]{LuOt} (see also \cite{Onoue}).
Let $m > 0$ be any constant and we take any $m_1 \in (0,\,m)$. By definition, for any $\eta>0$, there exist measurable sets $E_1,\,E_2 \subset \mathbb{R}^N$ with the volume constraints $|E_1|=m_1$ and $|E_2|=m - m_1$ such that
\begin{equation}\label{minimizingSeqInequality}
\capE_{s,g}(E_1) + \capE_{s,g}(E_2) \leq E_{s,g}[m_1]+ E_{s,g}[m_2] + \eta.
\end{equation}
Now we may assume that $E_1$ and $E_2$ are bounded. Indeed, we can observe that the minimum of $\capE_{s,g}$ among unbounded sets of volume $m$ is not smaller than the minimum of $\capE_{s,g}$ among bounded sets of volume $m$. To see this, for any unbounded set $E$ with $|E|=m$, we can choose sufficiently large $R>1$ in such a way that $|E \setminus B_R(0)|$ is as small as possible. Then, setting $\widehat{E} \coloneqq \lambda(R)\,(E \cap B_R(0))$ where $\lambda(R)^N \coloneqq \frac{m}{m - |E \setminus B_R(0)|} \geq 1$, we obtain, from Lemma \ref{lemmaScalingEnergy}, that
\begin{equation}\nonumber
|\widehat{E}| = \lambda(R)^N\,(m - |E \setminus B_R(0)|) = m
\end{equation}
and
\begin{align}\label{keyEstiEnergyModifiedSet}
\capE_{s,g}(\widehat{E}) &\leq \lambda(R)^{2N} \capE_{s,g}(E \cap B_R(0)) \nonumber\\
&\leq \lambda(R)^{2N} \capE_{s,g}(E) - P_s(E \setminus B_R(0)) + 2\int_{E \cap B_R(0)}\int_{E \setminus B_R(0)}\frac{dx\,dy}{|x-y|^{N+s}}.
\end{align}
Here we have used the following identity of the nonlocal perimeter:
\begin{equation}\nonumber
P_s(E \cup F) = P_s(E) + P_s(F) - 2\int_{E}\int_{F}\frac{1}{|x-y|^{N+s}}\,dx\,dy \nonumber
\end{equation}
for any measurable sets $E,\,F \subset \mathR^N$. From the isoperimetric inequality and the computation in \eqref{estimateMinimality} in Lemma \ref{lemmaBoundednessMinimizers}, we have that
\begin{equation}\label{keyEstiEnergyModifiedSet02}
\capE_{s,g}(\widehat{E}) \leq \lambda(R)^{2N} \capE_{s,g}(E) - C_1\,|E \setminus B_R(0)|^{\frac{N-s}{N}} + C_2\, \int_{R}^{\infty}\frac{\capH^{N-1}(E \cap \partial B_{\sigma}(0))}{(\sigma - R)^s}\,d\sigma
\end{equation}
where we set $C_1 \coloneqq P_s(B_1)\,|B_1|^{-\frac{N-s}{N}}$ and $C_2 \coloneqq 2s^{-1}|\partial B_1|$. Since $E$ is unbounded, we have that the function $R \mapsto |E \setminus B_R(0)|$ is non-increasing and not equal to zero for any $R>0$. Thus, by applying the same argument in Lemma \ref{lemmaBoundednessMinimizers}, we can find that there exists a sequence $\{R_i\}_{i\in\mathN}$ such that $R_i \to \infty$ as $i \to \infty$ and
\begin{equation}\label{negativityRemainderTerm}
- C_1\,|E \setminus B_{R_i}(0)|^{\frac{N-s}{N}} + C_2\, \int_{R_i}^{\infty}\frac{\capH^{N-1}(E \cap \partial B_{\sigma}(0))}{(\sigma - R_i)^s}\,d\sigma < 0
\end{equation}
for any $i\in\mathN$. Hence, from \eqref{keyEstiEnergyModifiedSet02} and \eqref{negativityRemainderTerm}, it follows that
\begin{equation}\nonumber
\inf\{\capE_{s,g}(E) \mid \text{$E$: measurable \& bounded, $|E|=m$}\} \leq \capE_{s,g}(\widehat{E})
< \lambda(R_i)^{2N} \capE_{s,g}(E)
\end{equation}
for any $i\in\mathN$. From the fact that $\lambda(R_i) \to 1$ as $i \to \infty$, the arbitrariness of $E$ and by letting $i \to \infty$, we finally obtain that
\begin{equation}\nonumber
\inf\{\capE_{s,g}(E) \mid \text{$E$: bounded, $|E|=m$}\} \leq \inf\{\capE_{s,g}(E) \mid \text{$E$: unbounded, $|E|=m$}\},
\end{equation}
as we desired.
Now we focus on the case that both $E_1$ and $E_2$ are bounded. Since $E_1,\,E_2$ are bounded, we can find a vector $e \in \mathS^{N-1}$ such that it follows that
\begin{equation}\nonumber
\dist(E_1,\,(E_2 + d\,e)) \xrightarrow[d \to \infty]{} \infty.
\end{equation}
Then we may compute the energy as follows:
\begin{align}
\capE_{s,g}(E_1\cup(E_2+d\,e)) &= P_s(E_1\cup(E_2+d\,e)) + V_g(E_1\cup(E_2+d\,e)) \nonumber\\
&\leq P_s(E_1) + P_s(E_2+d\,e) \nonumber\\
&\qquad + V_g(E_1) + V_g(E_2+d\,e) + 2\int_{E_1}\int_{E_2+d\,e} g(x-y)\,dx\,dy \nonumber\\
&\leq \capE_{s,g}(E_1) +\capE_{s,g}(E_2) + 2\int_{E_1}\int_{E_2+d\,e} g(x-y)\,dx\,dy. \nonumber
\end{align}
Here we have used the translation invariance of $P_s$ and $V_g$. From assumption $(\mathrm{g}4)$, which says that $g$ vanishes at infinity, we can show that
\begin{equation}\nonumber
\int_{E_1}\int_{E_2+d\,e} g(x-y)\,dx\,dy \xrightarrow[d \to \infty]{} 0.
\end{equation}
Since $|E_1 \sqcup (E_2 + d\,e)| = |E_1| + |E_2| = m$ for sufficiently large $d>0$ and from \eqref{minimizingSeqInequality}, we obtain
\begin{equation}\nonumber
E_{s,g}[m_1 + m - m_1] \leq E_{s,g}[m_1]+ E_{s,g}[m - m_1] + \eta + o(1).
\end{equation}
Letting $d \to \infty$ and then $\eta \to 0$, we conclude that the lemma is valid.
\end{proof}
Next we prove the uniform density estimate of minimizers of $\capE_{s,g}$ for large $m>0$.
\begin{lemma}[Uniform density estimate]\label{lemmaUniformDensity}
Let $m \geq 1$. We assume that the kernel $g :\mathR^N \setminus \{0\} \to \mathR$ is integrable in $\mathR^N$ and satisfies the assumptions $(\mathrm{g}1)$ and $(\mathrm{g}2)$. Then there exist constants $c_0>0$ and $r_0>0$, depending only on $N$, $s$, and $g$, such that, if $E$ is a minimizer of $\capE_{s,g}$ with $|E| = m$, then it holds that
\begin{equation}\label{uniformDensityEstimate}
|E \cap B_r(x)| \geq c_0\, r^{N}
\end{equation}
for any $r\in(0,\,r_0]$ and $x\in\mathR^N$ with $|E \cap B_r(x)|>0$ for any $r>0$.
\end{lemma}
\begin{remark}
Notice that, directly from Lemma \ref{lemmaUniformDensity}, we fail to obtain the uniform density estimates of minimizers of $\widehat{\capE}_{s,g}$. Indeed, when we use the dilation $\lambda \mapsto \lambda \,E$ of a minimizer $E$ in \eqref{uniformDensityEstimate} in such a way that $|\lambda \, E| = |B_1|$, the constant $r_0$ in Lemma \ref{lemmaUniformDensity} vanishes as $\lambda \to \infty$.
\end{remark}
\begin{proof}
Let $E$ be a minimizer of $\capE_{s,g}$ with $|E|=m$ and $x_0 \in E$ be any point such that $|E \cap B_r(x_0)| > 0$ for any $r>0$. We set $\lambda_r^N \coloneqq \frac{m}{m - |E \cap B_r(x_0)|} \geq 1$ for any $r>0$. We may assume that $|E \cap B_r(x_0)| < m$ for any $0<r<1$. Then, from the minimality of $E$, we have the inequality
\begin{equation}\nonumber
\capE_{s,g}(E) \leq \capE_{s,g}\left(\lambda_r (E \setminus B_r(x_0)) \right)
\end{equation}
for any $r >0$. We now recall the following identity on the nonlocal perimeter:
\begin{equation}\label{decompositionNonlocalPeri}
P_s(E \cup F) = P_s(E) + P_s(F) - 2\int_{E}\int_{F}\frac{1}{|x-y|^{N+s}}\,dx\,dy
\end{equation}
for any measurable sets $E,\,F\subset \mathR^N$ with $E \cap F = \emptyset$. Then, from Lemma \ref{lemmaScalingEnergy}, \eqref{decompositionNonlocalPeri}, and the fact that $V_g(E \setminus B_r(x_0)) \leq V_g(E)$, we have that
\begin{align}\label{keyDensityEstimate}
\capE_{s,g}(E) &\leq \capE_{s,g}(E \setminus B_r(x_0)) + (\lambda_r^{2N}-1)\,\capE_{s,g}(E \setminus B_r(x_0)) \nonumber\\
&\leq \capE_{s,g}(E \setminus B_r(x_0)) + (\lambda_r^{2N}-1)\,\Big(\capE_{s,g}(E) - P_s(E \cap B_r(x_0)) \nonumber\\
&\qquad \left. + \int_{E \cap B_r(x_0)}\int_{E \setminus B_r(x_0)} \frac{2\,dx\,dy}{|x-y|^{N+s}} \right)
\end{align}
for $0<r<1$. Similarly to \eqref{decompositionNonlocalPeri}, we also have the following identity on the Riesz potential:
\begin{equation}\label{decompositionRiesz}
V_g(E \cup F) = V_g(E) + V_g(F) + 2\int_{E}\int_{F}g(x-y)\,dx\,dy
\end{equation}
for any measurable sets $E,\,F\subset \mathR^N$ with $E \cap F = \emptyset$. Thus, from \eqref{decompositionNonlocalPeri}, \eqref{keyDensityEstimate}, and \eqref{decompositionRiesz}, we further have that
\begin{align}\label{keyDensityEstimate02}
\lambda_r^{2N} \, P_s(E \cap B_r(x_0))
&\leq \lambda_r^{2N} \, \int_{E \cap B_r(x_0)}\int_{E \setminus B_r(x_0)} \frac{2\,dx\,dy}{|x-y|^{N+s}} + (\lambda_r^{2N} - 1)\,E_{s,g}[m]
\end{align}
for any $0 < r < 1$. Recalling the definition of $\lambda_r$, we have that
\begin{equation}\label{defLambda}
\lambda_r^{2N} = \frac{m^2}{(m - |E \cap B_r|)^2}, \quad \lambda_r^{2N} - 1 = \frac{|E \cap B_r|}{m - |E \cap B_r|} \left( 2 + \frac{|E \cap B_r|}{m - |E \cap B_r|} \right)
\end{equation}
for any $0 < r < 1$. From \eqref{keyDensityEstimate02} and \eqref{defLambda}, we finally obtain
\begin{equation}\nonumber
P_s(E \cap B_r(x_0))
\leq \int_{E \cap B_r(x_0)}\int_{E \setminus B_r(x_0)} \frac{2\,dx\,dy}{|x-y|^{N+s}} + \frac{2E_{s,g}[m]}{m} |E \cap B_r(x_0)|
\end{equation}
for any $0 < r <1$. Hence, from the nonlocal isoperimetric inequality, we have that
\begin{align}\label{densityEstimate01}
&\frac{P_s(B_1)}{|B_1|^{\frac{N-s}{N}}} |E
\cap B_r(x_0)|^{\frac{N-s}{N}} \nonumber\\
&\leq P_s(E \cap B_r(x_0)) + V_g(E \cap B_r(x_0)) + 2\int_{E \cap B_r(x_0)}\int_{E \setminus B_r(x_0)}g(x-y)\,dx\,dy \nonumber\\
&\leq 2 \int_{E \cap B_r(x_0)}\int_{E \setminus B_r(x_0)} \frac{1}{|x-y|^{N+s}}\,dx\,dy + \frac{2 \, E_{s,g}[m]}{m}|E \cap B_r(x_0)|
\end{align}
for any small $r>0$. Noticing that $E \setminus B_r(x_0) \subset
B^c_{r-|y-x_0|}(y)$ for any $y \in E\cap B_r(x_0)$ and from the co-area formula, we have the following estimate:
\begin{align}\label{densityEstimate02}
\int_{E \cap B_r(x_0)}\int_{E \setminus B_r(x_0)} \frac{1}{|x-y|^{N+s}}\,dx\,dy &\leq \int_{E \cap B_r(x_0)}\int_{B^c_{r-|y-x_0|}(y)} \frac{1}{|x-y|^{N+s}}\,dx\,dy \nonumber\\
&\leq \frac{|\partial B_1|}{s}\int_{E \cap B_r(x_0)} \frac{1}{(r-|y-x_0|)^s}\,dy \nonumber\\
&= \frac{|\partial B_1|}{s} \int_{0}^{r}\frac{\capH^{N-1}(E \cap \partial B_{\sigma})}{(r-\sigma)^s}\,d\sigma.
\end{align}
Now we set $\phi(r) \coloneqq | E \cap B_r(x_0) |$ for any $r>0$ and we have that, for a.e. $r>0$, $\phi^{\prime}(r) = \capH^{N-1}(E \cap \partial B_{r})$. Thus we obtain from \eqref{densityEstimate01} and \eqref{densityEstimate02}, that
\begin{align}\label{densityEstimate03}
C(N,s)\,\phi(r)^{\frac{N-s}{N}} \leq \frac{2|\partial B_1|}{s} \int_{0}^{r}\frac{\phi^{\prime}(\sigma)}{(r-\sigma)^s}\,d\sigma + \frac{2\,E_{s,g}[m]}{m}\,\phi(r)
\end{align}
for any small $r>0$ where $C(N,s) \coloneqq |B_1|^{-\frac{N-s}{N}}\,P_s(B_1)$. Now we show that $m^{-1}E_{s,g}[m]$ is bounded by the constant independent of $m \geq 1$. Indeed, from the definition of $E_{s,g}[m]$ and by changing the variable $x \mapsto r_m\,x$, we first have that
\begin{align}\label{estimateMinimumUpperBound}
E_{s,g}[m] \leq \capE_{s,g}(B_{r_m}) &= P_s(B_{r_m}) + V_g(B_{r_m}) \nonumber\\
&\leq \left(\frac{m}{|B_1|}\right)^{\frac{N-s}{N}}P_s(B_1) + \left(\frac{m}{|B_1|}\right)^{2}\,2\int_{B_1}\int_{B_1}g(r_m(x-y))\,dx\,dy,
\end{align}
where $r_m>0$ is the constant satisfying $|B_{r_m}|=m$. Moreover, from the assumptions on $g$ and by changing the variable again, we have that
\begin{align}\label{estimateMinimumUpperBound02}
\int_{B_1}\int_{B_1}g(r_m(x-y))\,dx\,dy &\leq \int_{B_1(0)}\sup_{y\in\mathR^N}\int_{B_1(0)}g(r_m\,(x-y))\,dx \,dy \nonumber\\
&= |B_1|\,r_m^{-N}\int_{B_{r_m}(0)}g(x)\,dx \leq |B_1|^2\|g\|_{L^1(\mathR^N)}\,m^{-1} .
\end{align}
Thus, from \eqref{estimateMinimumUpperBound}, \eqref{estimateMinimumUpperBound02} and the assumption that $m \geq 1$, we obtain
\begin{equation}\nonumber
m^{-1}E_{s,g}[m] \leq \frac{P_s(B_1)}{|B_1|^{\frac{N-s}{N}}}m^{-\frac{s}{N}} + 2\|g\|_{L^1(\mathR^N)} \leq \frac{P_s(B_1)}{|B_1|^{\frac{N-s}{N}}} + 2\|g\|_{L^1(\mathR^N)} \eqqcolon \tilde{C}(N,s,g)
\end{equation}
and this completes the proof of the claim. Since $\phi$ is non-decreasing and $\phi(r) \leq |B_1|\,r^N$ for any $r>0$, we have that
\begin{equation}\nonumber
4\,\tilde{C}(N,s,g)\,\phi(r) \leq C(N,s)\,\phi(r)^{\frac{N-s}{N}}, \quad r_0 \coloneqq \left(\frac{P_s(B_1)}{4\tilde{C}(N,s,g)|B_1|}\right)^{\frac{1}{s}}
\end{equation}
for any $r \in (0,\,r_0]$. notice that $r_0$ is independent of $m$. Then integrating the both side of \eqref{densityEstimate03} over $r \in [0,\,r']$ for any $r' \in (0,\,r_0]$, we obtain that
\begin{equation}\nonumber
\frac{C(N,s)}{2}\,\int_{0}^{r'}\phi(r)^{\frac{N-s}{N}}\,dr \leq \frac{2|\partial B_1|}{s} \int_{0}^{r'}\int_{0}^{r}\frac{\phi^{\prime}(\sigma)}{(r-\sigma)^s}\,d\sigma \,dr.
\end{equation}
By changing the order of the integral, we have that
\begin{align}
C'(N,s)\,\int_{0}^{r'}\phi(r)^{\frac{N-s}{N}}\,dr &\leq \int_{0}^{r'}\int_{\sigma}^{r'}\frac{\phi^{\prime}(\sigma)}{(r-\sigma)^s} \,dr\,d\sigma \nonumber\\
&= \frac{1}{1-s}\int_{0}^{r'}\phi^{\prime}(r)\,(r'-r)^{1-s}\,dr \nonumber\\
&\leq \frac{(r')^{1-s}}{1-s}\phi(r') \nonumber
\end{align}
for any $r' \in (0,\,r_0]$ where we set $C'(N,s) \coloneqq 4^{-1}\,|\partial B_1|^{-1}\,s\,C(N,s)$. Now in order to prove the uniform density estimate, we suppose by contradiction that there exists a constant $r_1 \in (0,\,r_0]$ such that
\begin{equation}\nonumber
|E \cap B_{r_1}(x_0)| \leq c_0^{-\frac{N}{s}}\,r_1^N, \quad c_0 \coloneqq \frac{s(1-s)\,P_s(B_1)}{16|\partial B_1|\,|B_1|^{\frac{N-s}{N}}}.
\end{equation}
Then by applying the same argument in \cite[Lemma 3.1]{FFMMM}, we can obtain $|E \cap B_{\frac{r_1}{2}}(x_0)| = 0$, which is a contradiction to the choice of $x_0$. Notice that the constants $c_0$ and $r_0$ are independent of $E$, $x_0$, and $r$.
\end{proof}
\section{Existence of minimizers for $\capE_{s,g}$ for any volumes}\label{sectionExisMiniFastDecayGAnyVol}
In this section, we prove Theorem \ref{theoremExistMiniAnyVolumeFasterDecay}, namely, the existence of minimizers of the functional $\capE_{s,g}$ for any volume $m>0$ under the assumption that the kernel $g$ of the Riesz potential decays faster than the kernel of the fractional perimeter $P_s$, namely, $x \mapsto |x|^{-(N+s)}$.
Before proving Theorem \ref{theoremExistMiniAnyVolumeFasterDecay}, we show an auxiliary lemma, which states the existence of minimizers in a given bounded set of $\capE_{s,g}$ with volume $m>0$.
\begin{lemma}\label{lemmaExistMiniInBoundedSet}
Let $m>0$. Assume that the kernel $g: \mathR^N \setminus \{0\} \to \mathR$ is in $L^1_{loc}(\mathR^N)$. Then, for any $R \geq m^{\frac{1}{N}}$, there exists a minimizer $E^R$ of $\capE_{s,g}$ with $|E|=m$ such that $E^R \subset Q_R$ where $Q_R \subset \mathR^N$ is the cube of size $R$.
\end{lemma}
\begin{remark}
We can observe that, if $E_n$ is a minimizer of $\capE_{s,g}$ among subsets of $Q_n$ with $|E|=m$ for every $n$ and $\chi_{E_n} \to \chi_{E_{\infty}}$ as $n\to\infty$ in $L^1_{loc}$ with $|E_{\infty}|=m$, then it follows that $E_{\infty}$ is a minimizer of $\capE_{s,g}$ among sets of volume $m$. Indeed, we take any $F \subset \mathR^N$ with $|F|=m$ and set $\mu_n(F) \coloneqq \frac{m}{m-|F \setminus Q_{n-1}|} \geq 1$ for any large $n\in\mathN$. Then, we can readily see that
\begin{equation}\nonumber
\mu_n(F) \xrightarrow[n\to\infty]{} 1, \quad |\mu_n(F)\,(F \cap Q_{n-1})| = m, \quad \mu_n(F)\,(F \cap Q_{n-1}) \subset Q_n \quad
\text{for large $n\in\mathN$}
\end{equation}
Thus, by choosing the set $\mu_n(F)\,(F \cap Q_{n-1})$ as a competitor against the minimizer $E_n$ and from Lemma \ref{lemmaScalingEnergy} and Proposition \ref{propositionIntersectionConvexSmaller}, we have
\begin{equation}\nonumber
\capE_{s,g}(E_n) \leq \mu_n(F)^{2N}\capE_{s,g}(F \cap Q_{n-1}) \leq \mu_n(F)^{2N} \capE_{s,g}(F).
\end{equation}
Hence, from the lower semi-continuity, we obtain that the claim is valid.
\end{remark}
\begin{proof}
The proof is followed by the direct method of the calculus of variations and the compact embedding $W^{s,1} \hookrightarrow L^1$ for $s \in (0,\,1)$ in a bounded set (see \cite{dNPV} for the compactness). Let $m>0$ and $R \geq m^{\frac{1}{N}}$ and let $\{E^R_n\}_{n\in\mathN}$ be a minimizing sequence of $\capE_{s,g}$ with $|E^R_n|=m$ and $E^R_n \subset Q_R$. Then, by definition, we have
\begin{equation}\nonumber
\sup_{n\in\mathN} P_s(E^R_n; Q_R) = \sup_{n\in\mathN} P_s(E^R_n; Q_R) \leq \inf\left\{ \capE_{s,g}(E) \mid \text{$E \subset Q_R$, $|E|=m$} \right\} < \infty
\end{equation}
and thus, from the compactness and the uniformly boundedness of $\{E^R_n\}_{n\in\mathN}$, we can choose a set $G^R \subset Q_R$ such that, up to extracting a subsequence, $\chi_{E^R_n} \to \chi_{G^R}$ in $L^1(Q_R)$. Moreover, we obtain that $|G^R| = \lim_{n \to \infty}|E^R_n| = m$ and thus, $G^R$ with $|G^R|=m$. Finally, from the lower semi-continuous of the fractional perimeter and Fatou's lemma, we obtain that
\begin{equation}\nonumber
\capE_{s,g}(G^R) \leq \liminf_{n\to\infty} P_s(E^R_n) + \liminf_{n\to\infty} V_g(E^R) = \inf\left\{ \capE_{s,g}(E) \mid \text{$E \subset Q_R$, $|E|=m$} \right\}
\end{equation}
with $|G^R| = m$. This implies that $G^R$ is a minimizer.
\end{proof}
Now we are prepared to show Theorem \ref{theoremExistMiniAnyVolumeFasterDecay}. The idea to prove the existence is based on the argument by Di Castro, et al. in \cite{dCNRV} (see also \cite{GoNo, CeNo}). As we mentioned in the introduction, the idea was originally inspired by the so-called ``concentration-compactness" principle introduced by P.~L. Lions in \cite{Lions01, Lions02}. When one studies the variational problems in unbounded domain, the possible loss of compactness may occur from the vanishing or splitting into many pieces of minimizing sequences.
Although the proof of this method may be technical, we briefly explain the strategy of it in the following; when we obtain the existence of minimizers of the minimization problems of isoperimetric type, we usually apply the direct method in the calculus of variations. More precisely, we first take any minimizing sequence; then we try to construct another sequence of the minimizing sequence in such a way that the new elements are uniformly bounded and the energy of the new sequence is smaller than that of the original sequence (one may often refer to this procedure as ``truncation"); thus, by some compactness, we can extract a convergent subsequence in proper topology; finally, we may conclude that, by lower semi-continuity, the limit of the subsequence should be a minimizer as desired.
Unfortunately, in our problem, we might not be able to easily construct another sequence, which satisfies ``good" properties we want, from the original minimizing sequence. One possible reason is as follows; as is well-known, the fractional perimeter $P_s$ behaves like an attracting term, while the Riesz potential associated with the kernel $g$ could disaggregate minimizers into many different components. Moreover, in general, as the volume of a minimizer gets larger, the effect that separates minimizers into pieces from the Riesz potential may get stronger. However it is not obvious whether or not the nonlocal perimeter term can overcome such an effect from the Riesz potential because we cannot easily capture the precise behavior of a general kernel $g$. Therefore, we select the following strategy to handle the problems: first, taking any minimizing sequence $\{E_n\}_{n}$ of $\capE_{s,g}$ with $|E_n|=m>0$, we decompose each element $E_n$ into many pieces with the cubes $\{Q_n^i\}_{i}$ in such a way that each piece has non-negligible volumes. Then we ``properly" collect all the components $\{E_n \cap Q_n^i\}_{i}$ of $E_n$ such that $\dist(Q_n^i,\,Q_n^j) \to c^{ij} <\infty$ as $n\to\infty$ for $i \neq j$ (the case that $c^{ij}=\infty$ for $i \neq j$ is called the ``dichotomy" in the sense of Lions'). Thanks to the uniformly boundedness of $\{P_s(E_n)\}_{n}$ and the isoperimetric inequality of $P_s$, we can obtain a sequence of the limit sets $\{G^i\}_{i}$ of the components of $E_n$ that we have ``properly" collected such that $\{G^i\}_{i}$ is the collection with $c^{ij} = \infty$ for any $i \neq j$. Now we need to show that the amount of the volume of $\{G^i\}_{i}$ is equal to $m$ (this means that we exclude the ``vanishing phenomena" in the sense of Lions'). Once we have shown that $\sum_{i}|G^i|=m$, the faster decay of the kernel $g$ in Riesz potential enables us to prove that the only one element in $\{G_i\}_i$ should be the true minimizer of $\capE_{s,g}$ among sets of volume $m$.
\begin{proof}[Proof of Theorem \ref{theoremExistMiniAnyVolumeFasterDecay}]
Let $m>0$ be any number and let $\{E_n\}_{n\in\mathN}$ be a sequence of minimizers of $\capE_{s,g}$ with $|E_n|=m$ and $E_n \subset Q_n$. Notice that, as we see in Lemma \ref{lemmaExistMiniInBoundedSet}, the existence of the minimizer is guaranteed. We first decompose $Q_n$ into the unit cubes and denote by $\{Q_n^i\}_{i=1}^{I_n}$, where we choose a number $I_n \in \{1,\cdots,\,n^N\}$ in such a way that $|E_n \cap Q_n^i|>0$ for any $i \in \{1,\cdots,\,I_n\}$. We set $x_n^i \coloneqq |E_n \cap Q_n^i|$ and, since $E_n \subset Q_n$ for any $n\in\mathN$, we have that
\begin{equation}\label{keyTechnical01}
\sum_{i=1}^{I_n} x_n^i = |E_n| = m
\end{equation}
for any $n\in\mathN$. Since $E_n$ is a minimizer with $|E_n|=m$ for any $n$, we can choose a ball with the volume $m$ as a competitor and then, from the local integrability of $g$, have
\begin{equation}\label{uniformBoundednessNonlocPeri}
\sup_{n\in\mathN}P_s(E_n) \leq P_s(B_m) + V_g(B_m) \leq \left(\frac{m}{|B_1|}\right)^{\frac{N-s}{N}}\,P_s(B_1) + m\,\|g\|_{L^1(2B_m)} < \infty
\end{equation}
where $B_m$ is the open ball with the volume $m$ for each $m>0$. From \eqref{uniformBoundednessNonlocPeri} and the isoperimetric inequality shown in \cite[Lemma 2.5]{dCNRV}, we obtain
\begin{equation}\label{keyTechnical02}
\sum_{i=1}^{I_n} (x_n^i)^{\frac{N-s}{N}} \leq C\sum_{i=1}^{I_n}P_s(E_n; Q_n^i) \leq 2CP_s(E_n) \leq C_1 < \infty
\end{equation}
for any $n\in\mathN$, where $C$ and $C_1$ are the positive constants independent of $n$. Up to reordering the cubes $\{Q_n^i\}_{i}$, we may assume that $\{x_n^i\}_{i}$ is a non-increasing sequence for any $n\in\mathN$. Thus, applying the technical result shown in \cite[Lemma 4.2]{GoNo} or \cite[Lemma 7.4]{dCNRV} with \eqref{keyTechnical01} and \eqref{keyTechnical02}, we obtain that
\begin{equation}\label{keyTechnical03}
\sum_{i=k+1}^{\infty} x_n^i \leq \frac{C_2}{k^{\frac{s}{N}}}
\end{equation}
for any $k\in\mathN$, where we set $x_n^i \coloneqq 0$ for any $i > I_n$ and $C_2$ is the positive constant independent of $n$ and $k$.
Hence, by using the diagonal argument, we have that, up to extracting a subsequence, $x_n^i \to \alpha^i \in [0,\,m]$ as $n \to \infty$ for every $i\in\mathN$ and, from \eqref{keyTechnical01} and \eqref{keyTechnical03},
\begin{equation}\label{identityLimitMeasu}
\sum_{i=1}^{\infty} \alpha^i = m.
\end{equation}
Now we fix the centre of the cube $z_n^i \in Q_n^i$ for each $i$ and $n$. Up to extracting a further subsequence, we may assume that $|z_n^i - z_j^i| \to c^{ij} \in [0,\,\infty]$ as $n \to \infty$ for each $i,\,j \in \mathN$ and, since we have, from \eqref{uniformBoundednessNonlocPeri}, the uniform bound of the sequence $\{P_s(E_n-z_n^i)\}_{n\in\mathN}$ and its upper-bound is independent of $i$, there exists a measurable set $G^i \subset \mathR^N$ such that, up to a subsequence,
\begin{equation}\nonumber
\chi_{E_n-z_n^i} \xrightarrow[n \to \infty]{} \chi_{G^i} \quad \text{in $L^1_{loc}$-topology}.
\end{equation}
We define the relation $i \sim j$ for every $i,\,j\in\mathN$ as $c^{ij} < \infty$ and we denote by $[i]$ the equivalent class of $i$. Moreover, we define the set of the equivalent class by $\capI$. Then, in the following, we show a sort of lower semi-continuity, More precisely,
\begin{equation}\label{lowerSemicontiConcentration}
\sum_{[i] \in \capI}P_s(G^i) \leq \liminf_{n \to \infty}P_s(E_n), \quad \sum_{[i] \in \capI}V_g(G^i) \leq \liminf_{n \to \infty}V_g(E_n).
\end{equation}
Indeed, we first fix $M \in \mathN$ and $R>0$ and we take the equivalent classes $i_1,\cdots,\,i_M$. Notice that, if $p \neq q$, then $|z_n^{i_p} - z_n^{i_q}| \to \infty$ as $n\to\infty$ and thus we have that $\{z_n^{i_p}+Q_R\}_{p}$ are disjoint sets for large $n$ and
\begin{equation}\nonumber
\int_{z_n^{i_p} + Q_R}\int_{z_n^{i_q} + Q_R} \frac{1}{|x-y|^{N+s}}\,dx\,dy \xrightarrow[n \to \infty]{} 0
\end{equation}
where $Q_R$ is the cube of side $R$. We recall the inequality of the nonlocal perimeter;
\begin{equation}\nonumber
P_s(E; A) + P_s(E; B) \leq P_s(E; A \sqcup B) + 2\int_{A}\int_{B}\frac{dx\,dy}{|x-y|^{N+s}}
\end{equation}
for any measurable disjoint sets $A,\,B\subset \mathR^N$. As a consequence, from the lower semi-continuity of $P_s$, we obtain
\begin{align}
\sum_{p=1}^{M} P_s(G^{i_p} ; Q_R) &\leq \liminf_{n \to \infty} \sum_{p=1}^{M} P_s(E_n-z_n^{i_p} ; Q_R) \nonumber\\
&= \liminf_{n \to \infty} \sum_{p=1}^{M} P_s(E_n ; z_n^{i_p}+Q_R) \nonumber\\
&\leq \liminf_{n \to \infty} P_s\left(E_n; \bigcup_{p=1}\left(z_n^{i_p}+Q_R\right) \right) \nonumber\\
&\qquad + \liminf_{n \to \infty}2\sum_{p \neq q} \int_{z_n^{i_p} + Q_R}\int_{z_n^{i_q} + Q_R} \frac{dx\,dy}{|x-y|^{N+s}} \nonumber\\
&\leq \liminf_{n \to \infty} P_s(E_n). \nonumber
\end{align}
Letting $R \to \infty$ and then $M \to \infty$, we obtain the first claim of \eqref{lowerSemicontiConcentration}. For the second claim, we again take any $M\in\mathN$ and $R>0$. We recall the identity
\begin{equation}\nonumber
V_g(A) + V_g(B) = V_g(A \sqcup B) - 2\int_{A}\int_{B}g(x-y)\,dx\,dy
\end{equation}
for any measurable disjoint set $A,\,B \subset \mathR^N$. Then, in the same way as we have observed in the first claim, we have, from Fatou's lemma and the non-negativity of $g$, that
\begin{align}
\sum_{p=1}^{M} V_g(G^{i_p} \cap Q_R) &\leq \liminf_{n \to \infty} \sum_{p=1}^{M} V_g \left( \left(E_n-z_n^{i_p}\right) \cap Q_R \right) \nonumber\\
&= \liminf_{n \to \infty} \sum_{p=1}^{M} V_g \left(E_n \cap \left(z_n^{i_p}+ Q_R\right) \right) \nonumber\\
&\leq \liminf_{n \to \infty} V_g\left(E_n \cap \bigcup_{p=1}^{M} \left(z_n^{i_p}+Q_R\right)\right) \nonumber\\
&\leq \liminf_{n \to \infty} V_g(E_n). \nonumber
\end{align}
Here we have used the fact that the sets $\{z_n^{i_p} + Q_R\}_{p=1}^{M}$ are disjoint if $n$ is sufficiently large from the choice of the points $\{z_n^{i_p}\}_{p=1}^{M}$. Thus, letting $R \to \infty$ and then $M\to\infty$, we obtain the second claim.
Now we show that
\begin{equation}\nonumber
\sum_{[i] \in \capI} |G^i| = m.
\end{equation}
Indeed, from the $L^1_{loc}$-convergence of $\{\chi_{E_n - z_n^i}\}_{n\in\mathN}$ for any $i$, we have that, for any $R>0$ sufficiently large,
\begin{equation}\label{estimateFromBelowLimitMeasu}
|G^i| \geq |G^i \cap Q_R| = \lim_{n \to \infty}|(E_n-z_n^i) \cap Q_R|.
\end{equation}
If $j\in\mathN$ is such that $j \sim i$ and $c^{ij}<\frac{R}{100}$, then we have that $Q_n^i - z_n^i \subset Q_R$ for large $R>0$ and all $n$. Thus, from \eqref{estimateFromBelowLimitMeasu}, it follows
\begin{align}\label{estimateParticleVolume}
|(E_n-z_n^i) \cap Q_R| &= \sum_{i=1}^{I_n} |(E_n-z_n^i) \cap Q_R \cap \left(Q_n^i - z_n^i\right)| \nonumber\\
&\geq \sum_{j:\,c^{ij}<\frac{R}{100}}|(E_n-z_n^i) \cap Q_R \cap \left(Q_n^i - z_n^i\right)| \nonumber\\
&= \sum_{j:\,c^{ij}<\frac{R}{100}}|E_n \cap Q_n^i|
\end{align}
for all $n$ and large $R>0$. Therefore, combining \eqref{estimateParticleVolume} with \eqref{estimateFromBelowLimitMeasu}, we obtain
\begin{equation}\nonumber
|G^i| \geq \sum_{j:\,c^{ij}<\frac{R}{100}} \alpha^i
\end{equation}
and, letting $R \to \infty$, we have
\begin{equation}\nonumber
|G^i| \geq \sum_{j: \, c^{ij}<\infty} \alpha^i = \sum_{j \in [i]} \alpha^i.
\end{equation}
Hence, recalling \eqref{identityLimitMeasu}, we have
\begin{equation}\label{keyVolumeInequality}
\sum_{[i] \in \capI} |G^i| \geq \sum_{[i] \in \capI}\sum_{j \in [i]} \alpha^i = m.
\end{equation}
For the other inequality, we can easily obtain from the choice of $\{G^i\}_{i}$ in the following manner; for any $M\in\mathN$ and $R>0$, we take the equivalent classes $i_1,\cdots,\,i_M$ and then have that
\begin{align}\label{keyVolumeEstimateOtherDirec}
\sum_{p=1}^{M} |G^{i_p} \cap Q_R| &= \lim_{n \to \infty} \sum_{p=1}^{M} \left|\left(E_n - z_n^{i_p}\right) \cap Q_R \right| \nonumber\\
&= \lim_{n \to \infty} \sum_{p=1}^{M} \left|E_n \cap \left(z_n^{i_p} + Q_R\right) \right|.
\end{align}
Recalling the condition that $|z_n^{i_p} - z_n^{i_q}| \to \infty$ as $n \to \infty$ if $p \neq q$, we have that, for sufficiently large $n\in\mathN$, $\left(z_n^{i_p} + Q_R\right) \cap \left(z_n^{i_q} + Q_R\right) = \emptyset$ for any $p \neq q$. From \eqref{keyVolumeEstimateOtherDirec}, we have that
\begin{equation}\nonumber
\sum_{p=1}^{M} |G^{i_p} \cap Q_R| = \lim_{n \to \infty} \left|E_n \cap \bigcup_{p=1}^{M}\left(z_n^{i_p} + Q_R\right) \right| \leq m
\end{equation}
and thus, letting $R \to \infty$ and then $M \to \infty$, we obtain that
\begin{equation}\nonumber
\sum_{[i] \in \capI} |G^{i}| = \sum_{p=1}^{\infty} |G^{i_p}| \leq m.
\end{equation}
This completes the proof of the claim. Taking into account all the above arguments, we obtain the existence of sets $\{G^i\}_{[i]\in\capI}$ satisfying the properties that
\begin{equation}\label{keyPropertylowSemicontiVolumeEqualParticles}
\sum_{[i] \in \capI} \capE_{s,g}(G^i) \leq \liminf_{n \to \infty} \capE_{s,g}(E_n), \quad \sum_{[i] \in \capI} |G^{i}| = m.
\end{equation}
Now we claim that each particle $G^i$ for $[i] \in \capI$ is a minimizer of $\capE_{s,g}$ among sets of volume of $|G^i|$. Moreover, we show that $G^i$ is bounded for each $[i] \in \capI$. Indeed, we first recall the definition of $E_{s,g}$, which says that
\begin{equation}\nonumber
E_{s,g}[m] \coloneqq \inf\left\{ \capE_{s,g}(E) \mid |E|=m \right\}
\end{equation}
for any $m>0$, and the sub-additivity result of the function $m \mapsto E_{s,g}[m]$ as is shown in Lemma \ref{lemmaSubadditivityEnergy}. Notice that, in this theorem, we impose assumption $(\mathrm{g}3)$ as we show in Section \ref{sectionPreliminary}, which is stronger than $(\mathrm{g}4)$. Thus, we can apply Lemma \ref{lemmaSubadditivityEnergy} to the case in the present proof. Then, from \eqref{keyPropertylowSemicontiVolumeEqualParticles}, we have that
\begin{align}\label{energyMinimalityEachPrticle}
\sum_{p=1}^{M} \left(\capE_{s,g}(G^{i_p}) - E_{s,g}[|G^{i_p}|] \right) &\leq E_{s,g}[m] - \sum_{p=1}^{M} E_{s,g}[|G^{i_p}|] \nonumber\\
&\leq E_{s,g}\left[ \sum_{p=M+1}^{\infty}|G^{i_p}| \right] + E_{s,g}\left[ \sum_{p=1}^{M}|G^{i_p}| \right] - \sum_{p=1}^{M} E_{s,g}[|G^{i_p}|] \nonumber\\
&\leq E_{s,g}\left[ \sum_{p=M+1}^{\infty}|G^{i_p}| \right]
\end{align}
for any $M\in\mathN$. We can observe that $E_{s,g}[m] \to E_{s,g}[0] = 0$ as $m \to 0$ because $E_{s,g}[m]$ can be bounded by the quantity $C_1\,m^{\frac{N-s}{N}} + C_2\,m$ for small $m>0$, where $C_1$ and $C_2$ are the constants depending only on $N$, $s$, and $g$. Hence, letting $M \to \infty$ in \eqref{energyMinimalityEachPrticle}, we obtain that
\begin{equation}\nonumber
\sum_{[i] \in \capI} \left(\capE_{s,g}(G^{i_p}) - E_{s,g}[|G^{i_p}|] \right) = \sum_{p=1}^{\infty} \left(\capE_{s,g}(G^{i_p}) - E_{s,g}[|G^{i_p}|] \right) \leq 0
\end{equation}
and, from the fact that each term of the series is non-negative, we conclude that each term of the series is equal to zero. This implies that, for every $[i]\in\capI$, $G^i$ is a minimizer of $\capE_{s,g}$ among sets of volume of $|G^i|$. To see the boundedness of $\{G^i\}_{[i]\in\capI}$, it is sufficient to apply Lemma \ref{lemmaBoundednessMinimizers} to $G^i$ for each $[i]\in\capI$.
Now we show that the set $\capI$ of the equivalent classes is actually a finite set. Indeed, we first set $m^{i_p} \coloneqq |G^{i_p}|$ for any $p \in \mathN$ and, since $\sum_{p=1}^{\infty} m^{i_p} = m$, we can observe that $m^{i_p} \to 0$ as $p \to \infty$ and, moreover, $\mu_{\ell} \coloneqq \sum_{p=\ell+1}^{\infty} m^{i_p} \to 0$ as $\ell \to \infty$. Then, we can choose $\widetilde{p} \in \mathN$ such that $m^{i_{\widetilde{p}}} \geq \frac{m}{2^{\widetilde{p}+1}}$. Now using the sets $\{G^{i_p}\}_{p=1}^{\infty}$, we construct a new family of sets $\{\widetilde{G}^{i_p}\}_{p=1}^{H}$ for some $H \in \mathN$, depending only on $N$, $s$, and $m$, in the following manner; we choose $H \in \mathN$ so large that $H \geq \widetilde{p}$ and set $\widetilde{G}^{i_p} \coloneqq G^{i_p}$ for any $p \in \{1,\cdots ,\,H\}$ with $p \neq \widetilde{p}$ and $\widetilde{G}^{i_{\widetilde{p}}} \coloneqq \lambda\,G^{i_{\widetilde{p}}}$ where $\lambda^{N} \coloneqq \frac{m^{i_{\widetilde{p}}} + \mu_{H}}{m^{i_{\widetilde{p}}}}$. Then, we have the volume identity that
\begin{equation}\label{volumeIdentityReorganized}
\sum_{p=1}^{H} \left| \widetilde{G}^{i_p} \right| = \sum_{p=1, \,p\neq\widetilde{p}}^{H} \left| G^{i_p} \right| + \lambda^N\, |G^{i_{\widetilde{p}}}| = \sum_{p=1,\,p\neq\widetilde{p}}^{H} m^{i_p} + m^{i_{\widetilde{p}}} + \mu_{H} = m.
\end{equation}
Now we compute the energy for $\{\widetilde{G}^{i_p}\}_{p=1}^{H}$ as follows to show that the total energy of each elements of $\{\widetilde{G}^{i_p}\}_{p=1}^{H}$ is more efficient than that of $\{G^{i_p}\}_{p=1}^{\infty}$; from the definition of $\lambda \geq 1$ and Lemma \ref{lemmaScalingEnergy}, we have that
\begin{align}\label{comparisonNewSetsEnergy}
\sum_{p=1}^{H} \capE_{s,g}(\widetilde{G}^{i_p}) &\leq \sum_{p=1,\,p\neq\widetilde{p}}^{H} \capE_{s,g}(G^{i_p}) + \lambda^{2N}\,\capE_{s,g}(G^{i_{\widetilde{p}}}) \nonumber\\
&= \sum_{p=1}^{\infty} \capE_{s,g}(G^{i_p}) + \left(\lambda^{2N} - 1\right)\,\capE_{s,g}(G^{i_{\widetilde{p}}}) - \sum_{p=H+1}^{\infty}\capE_{s,g}(G^{i_p}) \nonumber\\
&\leq \sum_{[i] \in \capI} \capE_{s,g}(G^{i_p}) + \frac{2^{\widetilde{p}+1}\,E_{s,g}[m]}{m}\,\mu_{H} - \sum_{p=H+1}^{\infty} P_s(G^{i_p}).
\end{align}
Here, in the last inequality, we have also used \eqref{keyPropertylowSemicontiVolumeEqualParticles}. From the isoperimetric inequality of $P_s$ and \eqref{comparisonNewSetsEnergy}, we further obtain that
\begin{align}
\sum_{p=1}^{H} \capE_{s,g}(\widetilde{G}^{i_p}) &\leq \sum_{[i] \in \capI} \capE_{s,g}(G^i) + \frac{2^{\widetilde{p}+1}\,E_{s,g}[m]}{m}\, \mu_{H} - C\sum_{p=H+1}^{\infty} \left(m^{i_p}\right)^{\frac{N-s}{N}} \nonumber\\
&\leq \sum_{[i] \in \capI} \capE_{s,g}(G^i) + \frac{2^{\widetilde{p}+1}\,E_{s,g}[m]}{m}\, \mu_{H} - C \left(\sum_{p=H+1}^{\infty} m^{i_p}\right)^{\frac{N-s}{N}} \nonumber\\
&= \sum_{[i] \in \capI} \capE_{s,g}(G^i) + \frac{2^{\widetilde{p}+1}\,E_{s,g}[m]}{m}\, \mu_{H} - C \left( \mu_{H} \right)^{\frac{N-s}{N}}. \nonumber
\end{align}
Taking the number $H$ so large that $H \geq \widetilde{p}$ and
\begin{equation}\nonumber
\frac{2^{\widetilde{p}+1}\,E_{s,g}[m]}{m}\, \mu_{H} - C \left( \mu_{H} \right)^{\frac{N-s}{N}} \leq 0,
\end{equation}
then we finally obtain that
\begin{equation}\label{reductionFiniteElementsMinimizer}
\sum_{p=1}^{H} \capE_{s,g}(\widetilde{G}^{i_p}) \leq \sum_{[i] \in \capI} \capE_{s,g}(G^i) \leq \liminf_{n \to \infty}\capE_{s,g}(E_n).
\end{equation}
This completes the proof of the claim.
Finally, we show that there exists one number $i_0 \in \mathN$ such that $|G^i|=0$ for any $[i] \in \capI$ with $i \neq i_0$, using the assumption $(\mathrm{g}3)$. This proves the existence of minimizers of $\capE_{s,g}$, namely, Theorem \ref{theoremExistMiniAnyVolumeFasterDecay}. Indeed, if the claim is true, then from \eqref{lowerSemicontiConcentration} and \eqref{keyVolumeInequality}, we obtain
\begin{equation}\nonumber
\capE_{s,g}(G^{i_0}) = P_s(G^{i_0}) + V_g(G^{i_0}) = \sum_{[i] \in \capI}\left(P_s(G^i) + V_g(G^i) \right) \leq \liminf_{n \to \infty} \capE_{s,g}(E_n) = E_{s,g}[m]
\end{equation}
and
\begin{equation}\nonumber
|G^{i_0}| = \sum_{[i]\in\capI}|G^i| =m.
\end{equation}
Therefore, $G^{i_0}$ is a minimizer of $\capE_{s,g}$ with $|G^{i_0}| = m$ for any $m>0$.
From \eqref{keyVolumeInequality}, there exists at least one number $p'\in\mathN$ such that $|G^{i_{p'}}| > 0$. Then we claim that, if $q \neq p'$, then it holds that $|G^{i_q}|=0$. Indeed, from the previous claim, we can restrict ourselves to consider a finite number of sets $\{\widetilde{G}^{i_p}\}_{p=1}^{H}$, which satisfies \eqref{reductionFiniteElementsMinimizer} and $\sum_{p=1}^{H}|\widetilde{G}^{i_p}| = m$, instead of $\{G^i\}_{[i]\in\capI}$. Moreover, we may assume that $H \geq p'$. Since we have shown that the sets $\{\widetilde{G}^{i_p}\}_{p=1}^{H}$ are bounded, we can choose the points $\{z^{i_p}\}_{p=1,\,p \neq p'}^{H}$ such that each set $\widetilde{G}^{i_p} + R\,z^{i_p}$ is far away from the others for large $R>1$. We can thus compute the energy as follows; from the translation invariance of $\capE_{s,g}$, it holds that
\begin{align}
\sum_{p=1}^{H} \capE_{s,g}(\widetilde{G}^{i_p}) &= \sum_{p=1\,p \neq p',q}^{H} \capE_{s,g}(\widetilde{G}^{i_p}) + \capE_{s,g}(\widetilde{G}^{i_{p'}}) + \capE_{s,g}(\widetilde{G}^{i_q}) \nonumber\\
&= \sum_{p=1\,p \neq p',q}^{H} \capE_{s,g}(\widetilde{G}^{i_p}) + \capE_{s,g}(\widetilde{G}^{i_{p'}}) + \capE_{s,g}(\widetilde{G}^{i_q} + R\,z^{i_q}) \nonumber\\
&= \sum_{p=1\,p \neq p',q}^{H} \capE_{s,g}(\widetilde{G}^{i_p}) + \capE_{s,g}(\widetilde{G}^{i_{p'}} \cup (\widetilde{G}^{i_q} + R\,z^{i_q})) \nonumber\\
&\qquad + 2\int_{\widetilde{G}^{i_{p'}}} \int_{\widetilde{G}^{i_q} + R\,z^{i_q}} \frac{dx\,dy}{|x-y|^{N+s}} - 2\int_{\widetilde{G}^{i_{p'}}} \int_{\widetilde{G}^{i_q} + R\,z^{i_q}}g(x-y)\,dx\,dy \nonumber
\end{align}
for any $q \in \{1,\cdots,\,H\}$ with $q \neq p'$ and sufficiently large $R>1$. Recalling the assumption $(\mathrm{g}3)$ that $g(x) \leq \beta|x|^{-(N+s)}$ for any $|x| \geq R_0$ and some $\beta\in(0,\,1)$, and choosing $R>1$ in such a way that the set $\widetilde{G}^{i_q} + R\,z^{i_q}$ has the distance of more than $R_0$ from $\widetilde{G}^{i_{p'}}$, we obtain that
\begin{align}\label{keyEstimateReductionOneElement}
\sum_{p=1}^{H} \capE_{s,g}(\widetilde{G}^{i_p}) &\geq \sum_{p=1,\,p \neq p',\,q}^{H} \capE_{s,g}(\widetilde{G}^{i_p}) + \capE_{s,g}(\widetilde{G}^{i_{p'}} \cup (\widetilde{G}^{i_q} + R\,z^{i_q})) \nonumber\\
&\qquad + 2(1-\beta)\int_{\widetilde{G}^{i_{p'}}} \int_{\widetilde{G}^{i_q} + R\,z^{i_q}} \frac{dx\,dy}{|x-y|^{N+s}}.
\end{align}
By repeating the same argument finite times for the rest of the sets $\{\widetilde{G}^{i_p}\}_{p=1,\,p \neq p',q}^{H}$ with sufficiently large $R>1$, we obtain the similar inequalities to \eqref{keyEstimateReductionOneElement} and, finally, we can derive the inequality that
\begin{align}\label{keyEstimateReductionOneElement02}
\sum_{p=1}^{H} \capE_{s,g}(\widetilde{G}^{i_p}) &\geq \capE_{s,g}\left( \widetilde{G}^{i_{p'}} \cup \bigcup_{p=1,\,p \neq p'}^{H}\left( \widetilde{G}^{i_p} + R\,z^{i_p} \right) \right) \nonumber\\
&\qquad + 2(1-\beta)\sum_{p=1,\,p \neq p'}^{H} \int_{\widetilde{G}^{i_{p'}}} \int_{\widetilde{G}^{i_p} + R\,z^{i_p}} \frac{dx\,dy}{|x-y|^{N+s}}.
\end{align}
Since $\widetilde{G}^{i_{p'}} \cup \bigcup_{p=1,\,p \neq p'}^{H}\left( \widetilde{G}^{i_p} + R\,z^{i_p} \right)$ are the union of disjoint sets, we have, from \eqref{volumeIdentityReorganized}, that
\begin{equation}\nonumber
\left| \widetilde{G}^{i_{p'}} \cup \bigcup_{p=1,\,p \neq p'}^{H}\left( \widetilde{G}^{i_p} + R\,z^{i_p} \right)\right| = \sum_{p=1}^{H} |\widetilde{G}^{i_{p}}| = m.
\end{equation}
Thus, from \eqref{keyEstimateReductionOneElement02}, we obtain
\begin{align}
2(1-\beta)&\sum_{p=1,\,p \neq p'}^{H} \int_{\widetilde{G}^{i_{p'}}} \int_{\widetilde{G}^{i_p} + R\,z^{i_p}} \frac{dx\,dy}{|x-y|^{N+s}} + E_{s,g}[m] \nonumber\\
&\leq \sum_{p=1,\,p \neq p'}^{H} \int_{\widetilde{G}^{i_{p'}}} \int_{\widetilde{G}^{i_p} + R\,z^{i_p}} \frac{dx\,dy}{|x-y|^{N+s}} + \capE_{s,g}\left( \widetilde{G}^{i_{p'}} \cup \bigcup_{p=1,\,p \neq p'}^{H}\left( \widetilde{G}^{i_p} + R\,z^{i_p} \right) \right) \nonumber\\
&\leq \sum_{p=1}^{H} \capE_{s,g}(\widetilde{G}^{i_p}) \leq E_{s,g}[m] \nonumber
\end{align}
and it follows that
\begin{equation}\nonumber
2(1-\beta)\sum_{p=1,\,p \neq p'}^{H} \int_{\widetilde{G}^{i_{p'}}} \int_{\widetilde{G}^{i_p} + R\,z^{i_p}} \frac{dx\,dy}{|x-y|^{N+s}} \leq 0
\end{equation}
for large $R>1$. Since each term of the sum is non-negative, $\beta < 1$, and $|\widetilde{G}^{i_{p'}}|>0$, we conclude that $|\widetilde{G}^{i_p}| = 0$ for all $p \neq p'$. Therefore, the final claim is valid and this completes the proof of Theorem \ref{theoremExistMiniAnyVolumeFasterDecay}.
\end{proof}
\subsection{Regularity of the boundaries of minimizers}
In this subsection, we consider the regularity of the boundary of a minimizer of $\capE_{s,g}$ under suitable conditions on the kernel $g$. To see this, we recall several results on the regularity of the so-called \textit{almost $s$-fractional minimal surfaces}. The first one is on the $C^1$-regularity of the $s$-fractional almost minimal surfaces, which was shown by Caputo and Guillen in \cite{CaGu}.
\begin{theorem}[\cite{CaGu}] \label{regularityCaputoGuillen}
Let $s \in (0,\,1)$ and $\delta>0$, and let $\Omega \subset \mathR^N$ be any bounded domain with Lipschitz boundary. Suppose $E$ is a $(P_s,\rho,\delta,)$-minimal in $\Omega$, where $\rho: (0,\,\delta) \to \mathR$ is a non-decreasing and bounded function with some growth condition. Here we mean by $(P_s,\rho,\delta,)$-minimal in $B_R$ for some $R>0$ that for any $x_0\in\partial E$, a measurable set $F \subset \mathR^N$, and $0 < r < \min\{ \delta, \, \dist(x_0,\partial B_R)\}$ with $E \bigtriangleup F \subset B_r(x_0)$, we have
\begin{equation}\label{almostMinimizerDef}
P_s(E;B_R) \leq P_s(F;B_R) + \rho(r)\,r^{N-s}.
\end{equation}
Then $\partial E$ is of class $C^1$ in $B_{\frac{R}{2}}$, except a closed set of $\capH^{N-2}$-dimension.
\end{theorem}
\begin{remark}
We remark that we can choose, for instance, the function $r \mapsto C\,r^{\delta}$ with $0< \delta \leq s$ and some constant $C>0$ as the function $\rho$ in Theorem \ref{regularityCaputoGuillen}. Hence, we are allowed to consider the exponent of the growth term in \eqref{almostMinimizerDef} up to $N$.
\end{remark}
Now we recall another result of the regularity of the boundary of minimizers. Namely, we show the improvement of flatness statement proved in \cite[Theorem 3.4]{FFMMM} by using the method developed in \cite{CaGu}. This result implies $C^{1,\alpha}$-regularity of almost minimal surfaces by using a standard argument.
\begin{theorem}[\cite{CaGu, FFMMM}]\label{improvementFlatnessFFMMM}
Let $s_0 \in (0,\,1)$ and $\Lambda >0$. Then there exist $\tau,\,\eta,\,q \in (0,\,1)$ depending only on $N$, $s_0$, and $\Lambda$ with the following property: we assume that $E$ is a $\Lambda$-minimizer of the $s$-perimeter for some $s \in [s_0,\,1)$ with $x_0 \in \partial E$. Here we mean by $\Lambda$-minimizer that $E$ is a bounded measurable set in $\mathR^N$ satisfying the condition that, for any bounded set $F \subset \mathR^N$,
\begin{equation}\nonumber
P_s(E) \leq P_s(F) + \frac{\Lambda}{1-s}|E \Delta F|.
\end{equation}
Then, if
\begin{equation}\nonumber
\partial E \cap B_1(x_0) \subset \{y \mid |(y-x_0)\cdot e | < \tau \}
\end{equation}
for some $e \in \mathS^{N-1}$, then there exists $e_0 \in \mathS^{N-1}$ such that
\begin{equation}\nonumber
\partial E \cap B_\eta(x_0) \subset \{y \mid |(y-x_0)\cdot e_0 | < q\,\tau,\eta \}.
\end{equation}
\end{theorem}
Originally, the regularity of nonlocal minimal surfaces was obtained by Caffarelli, Roquejoffre, and Savin in \cite{CRS}, which states that every $s$-minimal surface is locally $C^{1,\alpha}$ except the singular sets of $\capH^{N-2}$-dimension. More importantly, thanks to the result by Savin and Valdinoci in \cite{SaVa}, the singular set of $s$-minimal surfaces has the Hausdorff dimension up to $N-3$. Hence one can have that $s$-minimal surfaces in $\mathR^2$ are fully $C^{1,\alpha}$-regular.
As a consequence of these regularity results, we obtain the regularity of the minimizers of $\capE_{s,g}$. Before stating the regularity result, we reduce the minimization problem $E_{s,g}[m]$ for any $m>0$ to another minimization problem because that reduction may allow us to consider more easily. More precisely, we show that any solutions of the minimization problem $E_{s,g}[m]$ are also the solutions of the unconstrained minimization problem
\begin{equation}\nonumber
\inf\left\{ \capE_{s,g,\mu_0}(E) \mid \text{$E \subset \mathR^N$: measurable} \right\}
\end{equation}
for some constant $\mu_0>0$ and any $m>0$, where we define $\capE_{s,g,\mu_0}$ as
\begin{equation}\nonumber
\capE_{s,g,\mu}(F) \coloneqq\capE_{s,g}(F)+ \mu\,\left| |F|-m \right|
\end{equation}
for any $F \subset \mathR^N$ and $\mu>0$.
\begin{proposition}[Reduction to an unconstrained problem]\label{propositionEquivalenceProblem}
Let $m>0$. Assume that the kernel $g$ satisfies the conditions $(\mathrm{g}1)$ and $(\mathrm{g}2)$. Then there exists a constant $\mu_0=\mu_0(N,s,g,m)>0$ such that, if $E$ is a minimizer of $\capE_{s,g}$ with $|E|=m$, then $E$ is also a minimizer of $\capE_{s,g,\mu}$ among sets in $\mathR^N$ for any $\mu \geq \mu_0$.
\end{proposition}
\begin{proof}
Suppose by contradiction that, for any $n \in \mathN$, there exist a minimizer $E_n$ of $\capE_{s,g}$ with $|E_n|=m$ and a constant $\mu(n) \geq n$ such that $E_n$ is not a minimizer of $\capE_{s,g,\mu(n)}$. Then, by assumption, we can choose a sequence $\{F_n\}_{n\in\mathN}$ such that
\begin{equation}\label{estimateFromSBC}
\capE_{s,g,\mu(n)}(F_n) < \capE_{s,g,\mu(n)}(E_n)
\end{equation}
for any $n \in \mathN$. First of all, we show that $|F_n| \xrightarrow[n\to\infty]{} m$. Indeed, we set $B_m$ as a open ball in $\mathR^N$ whose volume is equal to $m$. Then from \eqref{estimateFromSBC} and the minimality of $E_n$ with $|E_n|=m$ for any $n\in\mathN$, we have that
\begin{equation}\label{minimalityInequality}
\capE_{s,g,\mu(n)}(F_n) < \capE_{s,g,\mu(n)}(E_n) = \capE_{s,g}(E_n) = E_{s,g}[m].
\end{equation}
Thus, denoting $r_m$ by the radius of the ball $B_m$ and using the change of variables, we obtain
\begin{equation}\label{convergenceofF_n}
\mu(n)\,||F_n| - m| < E_{s,g}[m] < \infty
\end{equation}
for any $n\in\mathN$. From the definition of $r_m$, the right-hand side in \eqref{convergenceofF_n} is finite and independent of $n$. Hence, letting $n \to \infty$ in \eqref{convergenceofF_n}, we obtain the claim that $|F_n| \rightarrow m$ as $n\to\infty$. Finally, we derive a contradiction in the following manner. We first define $\widetilde{F}_n$ as $\widetilde{F}_n \coloneqq \lambda_n \, F_n$ where $\lambda_n^N \coloneqq m\,|F_n|^{-1}$ and, by definition, we can observe that $|\widetilde{F}_n| = m$. In the sequel, we may assume that, up to extracting a subsequence, $|F_n| \leq m$ for $n\in\mathN$. Indeed, we suppose by contradiction that, for any subsequence $\{F_{n_k}\}_{k\in\mathN}$ of $\{F_n\}_{n\in\mathN}$, we always have that $|F_{n_k}| > m$ for any $k\in\mathN$. From the continuity of the Lebesgue measure, for each $k\in\mathN$, there exists a constant $R_k>0$ such that $|F_{n_k} \cap B_{R_{n_k}}(0)| = m$ for every $k\in\mathN$. Thus, from the minimality of $E_n$ for any $n\in\mathN$ and Proposition \ref{propositionIntersectionConvexSmaller}, we have the estimate that
\begin{equation}\nonumber
\capE_{s,g,\mu(n)}(E_{n_k}) = \capE_{s,g}(E_{n_k}) \leq \capE_{s,g}(F_{n_k} \cap B_{R_{n_k}}(0)) \leq P_s(F_{n_k}) + V_g(F_{n_k}) = \capE_{s,g}(F_{n_k})
\end{equation}
for any $k\in\mathN$, which contradicts the estimate \eqref{estimateFromSBC} since $\capE_{s,g}(F_{n_k}) \leq \capE_{s,g,\mu(n)}(F_{n_k})$ for any $k\in\mathN$. Hence, from \eqref{estimateFromSBC}, the minimality of $E_n$, the assumption that $\lambda_n \geq 1$ for any $n \in \mathN$, and Lemma \ref{lemmaScalingEnergy}, we have
\begin{equation}\label{secondEstiMinimality}
\capE_{s,g,\mu(n)}(F_n) < \capE_{s,g}(E_n) \leq \capE_{s,g}(\widetilde{F}_n) \leq \lambda_n^{2N} \capE_{s,g}(F_n).
\end{equation}
From the definition, we notice that $||F_n| - m| = |\lambda_n^{-N} \, m - m| = m\,\lambda_n^{-N} \, |\lambda_n^N - 1|$ for any $n$. Hence, from \eqref{secondEstiMinimality} and dividing the both side of \eqref{secondEstiMinimality} by $||F_n| - m|$, we obtain
\begin{equation}\label{secondEstiMinimality03}
\mu(n) \leq m^{-1}\,\lambda_n^N \frac{|\lambda_n^{2N}-1|}{|\lambda_n^N-1|} P_s(F_n) + m^{-1}\,\lambda_n^N \frac{|\lambda_n^{2N}-1|}{|\lambda_n^N-1|} V_g(F_n)
\end{equation}
for any $n\in\mathN$. Recalling \eqref{estimateFromSBC} and \eqref{minimalityInequality}, we have that $P_s(F_n) + V_g(F_n) < E_{s,g}[m] < \infty$. Moreover, we observe that $\frac{|\lambda_n^{2N}-1|}{|\lambda_n^N-1|} \leq 2$ for sufficiently large $n \in \mathN$. Therefore, from \eqref{secondEstiMinimality03}, we obtain
\begin{equation}\label{keyEstimateContradictionArgument}
\mu(n) \leq 6m^{-1}\,E_{s,g}[m]
\end{equation}
for sufficiently large $n \in \mathN$ and thus obtain a contradiction.
\end{proof}
Now we are prepared to show the regularity of minimizers for $\capE_{s,g}$
\begin{lemma}[Regularity of minimizers]\label{lemmaRegularityMinimizers}
Let $s \in (0,\,1)$ and let $m>0$. Assume that the kernel $g: \mathR^N \setminus \{0\} \to \mathR$ satisfies $(\mathrm{g}1)$, $(\mathrm{g}2)$, and $(\mathrm{g}3)$. If $E \subset \mathR^N$ is a minimizer of $\capE_{s,g}$ among sets of volume $m$, then $\partial E$ is of class $C^{1,\alpha}$ with $0 < \alpha < 1$, except a closed set of $\capH^{N-3}$-dimension.
\end{lemma}
\begin{proof}
First of all, from Lemma \ref{lemmaBoundednessMinimizers}, we have the essential boundedness of the minimizer $E \subset \mathR^N$, namely, $E \subset B_{R_1}(0)$ up to negligible sets for some $R_1>0$. Without loss of generality, we may assume that $R_1 \geq R_0$ where $R_0$ is given in assumption $(\mathrm{g}3)$ in Section \ref{sectionPreliminary}. In order to apply the regularity result of Theorem \ref{improvementFlatnessFFMMM} to our case, it is sufficient to show that the set $E$ is $\Lambda$-minimizer in the sense of Theorem \ref{improvementFlatnessFFMMM} for some constant $\Lambda>0$ independent of $E$. From Proposition \ref{propositionEquivalenceProblem}, we know that $E$ with $|E|=m$ is also a solution to the minimization problem
\begin{equation}\nonumber
\min\{\capE_{s,g,\mu_0}(E) \mid \text{$E\subset\mathR^N$: measurable}\}.
\end{equation}
where $\mu_0>0$ is as in Proposition \ref{propositionEquivalenceProblem} and is independent of $E$. Hence, from the minimality of $E$, we have that
\begin{align}\label{minimalitySetE}
\capE_{s,g,\mu_0}(E) \leq \capE_{s,g,\mu_0}(F)
\end{align}
for any bounded measurable set $F \subset \mathR^n$. We may assume that $F$ is finite with respect to the $s$-fractional perimeter; otherwise the inequality \eqref{minimalitySetE} is obviously valid. Then from the fact that $|E|=m$, we have
\begin{align}\label{almostMinimizersEstimate}
P_s(E) &\leq P_s(F) + V_g(F) - V_g(E) + \mu_0\,||F|-|E|| \nonumber\\
&\leq P_s(F) + V_g(F) - V_g(E) + \mu_0\,|F \Delta E|.
\end{align}
Regarding the Riesz potential, we can compute the difference $ V_g(F) - V_g(E)$ as follows:
\begin{align}\label{estimateRiesz}
| V_g(F) - V_g(E)| &\leq \left| \int_{F}\int_{F \cup E}g(x-y) \,dx\,dy - \int_{E}\int_{F \cup E}g(x-y) \,dx\,dy \right| \nonumber\\
&\leq 2 \int_{F \Delta E}\int_{F \cup E}g(x-y) \,dx\,dy \nonumber\\
&\leq 2|F \Delta E| \,\int_{\mathR^N} g(x)\,dx.
\end{align}
Note that, from the local integrability of $g$ and assumption $(\mathrm{g}3)$, the kernel $g$ is integrable in $\mathR^N$ and thus, the right-hand side in \eqref{estimateRiesz} is finite. Hence, by combining \eqref{estimateRiesz} with \eqref{almostMinimizersEstimate}, we obtain that
\begin{equation}\nonumber
P_s(E) \leq P_s(F) + \left( 2\|g\|_{L^1(\mathR^N)} + \mu_0\right)\,|F \Delta E|
\end{equation}
for any measurable set $F \subset \mathR^N$. Therefore, by employing Theorem \ref{regularityCaputoGuillen} and \ref{improvementFlatnessFFMMM}, we can conclude that the claim is valid.
\end{proof}
\section{Existence of generalized minimizers for $\widetilde{\capE}_{s,g}$ for any volumes}\label{sectionExistGeneMiniAnyVol}
In this section, we prove Theorem \ref{theoremExistGeneralizedMiniAnyVolume}, namely, the existence of generalized minimizers for the generalized functional $\widetilde{\capE}_{s,g}$ given as \eqref{defiGeneralziedFunctional} for any volumes. To see this, we impose slightly more general assumptions on $g$ than we do to prove the existence of minimizers of $\capE_{s,g}$ for any volumes in Section \ref{sectionExisMiniFastDecayGAnyVol}. More precisely, we assume that the kernel $g \in L^1_{loc}(\mathR^N)$ satisfies the assumptions $(\mathrm{g}1)$, $(\mathrm{g}2)$, and $(\mathrm{g}4)$ in Section \ref{sectionPreliminary}.
Before proving the main theorem, we show one lemma, saying that one can modify a ``generalized" minimizing sequence for the generalized functional $\widetilde{\capE}_{s,g}$ into a ``usual" minimizing sequence for the functional $\capE_{s,g}$. More precisely, we prove
\begin{lemma}\label{lemmaModifyMinimizingSeq}
Let $s\in(0,\,1)$. Assume that the kernel $g : \mathR^N \setminus \{0\} \to \mathR$ satisfies the assumptions $(\mathrm{g}1)$, $(\mathrm{g}2)$, and $(\mathrm{g}4)$. Then, for any $m>0$, it follows that
\begin{equation}\nonumber
\inf\left\{\capE_{s,g}(E) \mid |E|=m \right\} = \inf\left\{\widetilde{\capE}_{s,g}(\{E^k\}_k) \mid \sum_{k=1}^{\infty} |E^k| = m \right\}.
\end{equation}
\end{lemma}
\begin{proof}
The proof of this lemma proceeds in a similar manner to the method in the proof of Theorem \ref{theoremExistMiniAnyVolumeFasterDecay}; however, it seems a little technical and thus we do not omit the detail.
First of all, we can readily observe that the inequality
\begin{equation}\nonumber
\inf\left\{\capE_{s,g}(E) \mid |E|=m\right\} \geq \inf\left\{\widetilde{\capE}_{s,g}(\{E^k\}_k) \mid \sum_{k=1}^{\infty} |E^k| = m \right\}
\end{equation}
holds true. Hence, it remains for us to prove the other inequality. To see this, we take any minimizing sequence $\{\{E^k_n\}_{k}\}_{n}$ for the generalized functional $\widetilde{\capE}_{s,g}$. Then it follows that, for any $\varepsilon>0$, there exists a number $n_0 \in \mathN$ such that
\begin{equation}\label{minimizingInequality}
\widetilde{\capE}_{s,g}(\{E^k_n\}_k) \leq \widetilde{E}_{s,g}[m] + \varepsilon
\end{equation}
for any $n \geq n_0$. Since the minimum is attained with a minimizing sequence of which each element is bounded, we may assume that $E^k_n$ is bounded for each $k,\,n\in\mathN$. In the sequel, we fix one $n \in \mathN$ with $n \geq n_0$ until we give another remark.
First of all, we want to show that each element $\{E^k_n\}_{k}$ of the minimizing sequence $\{\{E^k_n\}_k\}_{n}$ can be chosen as an element of finitely many sets. More precisely, we will show that there exist a number $K_n\in\mathN$ and a sequence $\{\widetilde{E}^k_n\}_{k=1}^{K_n}$ such that the energy of $\{\widetilde{E}^k_n\}_{k=1}^{K_n}$ is smaller than that of $\{E^k_n\}_{k}$ and it converges to $\widetilde{E}_{s,g}[m]$ as $n \to \infty$. We set $m^{k}_n \coloneqq |E^{k}_n|$ for any $k \in \mathN$ and, since $\sum_{k=1}^{\infty} m^{k}_n = m <\infty$, we can observe that $m^{k}_n \to 0$ as $k \to \infty$ and, moreover, $\mu_{\ell} \coloneqq \sum_{k=\ell+1}^{\infty} m^{k}_n \to 0$ as $\ell \to \infty$. Then, we can choose $\widetilde{k} \in \mathN$ such that $m^{\widetilde{k}}_n \geq \frac{m}{2^{\widetilde{k}+1}}$. Indeed, if not, it follows that $m^k_n < \frac{m}{2^{k+1}}$ for any $k\in\mathN$. Then, we have that $m = \sum_{k=1}^{\infty}m^k < \sum_{k=1}^{\infty} \frac{m}{2^{k+1}} = \frac{m}{2}$, which is a contradiction. Now using the sets $\{E^{k}_n\}_{k=1}^{\infty}$, we construct a new family of sets $\{\widetilde{E}^{k}_n\}_{k=1}^{K_n}$ for some $K_n \in \mathN$, depending on $n$, in the following manner; we first choose $K_n \in \mathN$ so large that $K_n \geq \widetilde{k}$ and set $\widetilde{E}^{k}_n \coloneqq E^{k}_n$ for any $k \in \{1,\cdots ,\,K_n\}$ with $k \neq \widetilde{k}$ and $\widetilde{E}^{\widetilde{k}}_n \coloneqq \lambda_n\,E^{\widetilde{k}}_n$ where $\lambda^{N}_n \coloneqq \frac{m^{\widetilde{k}}_n + \mu_{K_n}}{m^{\widetilde{k}}_n}$. Then, we have the volume identity that
\begin{equation}\label{minimizingSeqFinitenessExisGeneMini01}
\sum_{k=1}^{K_n} \left| \widetilde{E}^{k}_n \right| = \sum_{k=1, \,k\neq\widetilde{k}}^{K_n} \left| E^{k}_n \right| + \lambda^N_n\, |E^{\widetilde{k}}_n| = \sum_{k=1,\,k\neq\widetilde{k}}^{K_n} m^{k}_n + m^{\widetilde{k}}_n + \mu_{K_n} = m.
\end{equation}
Now we compute the energy for $\{\widetilde{E}^{k}_n\}_{k=1}^{K_n}$ in order to show that the total energy of each elements of $\{\widetilde{E}^{k}_n\}_{k=1}^{K_n}$ is more efficient than that of the original sequence $\{E^{k}_n\}_{k=1}^{\infty}$. From the definition of $\lambda_n \geq 1$ and Lemma \ref{lemmaScalingEnergy}, we have that
\begin{align}\label{minimizingSeqFinitenessExisGeneMini02}
\sum_{k=1}^{K_n} \capE_{s,g}(\widetilde{E}^{k}_n) &\leq \sum_{k=1,\,k\neq\widetilde{k}}^{K_n} \capE_{s,g}(E^{k}_n) + \lambda^{2N}_n\,\capE_{s,g}(E^{\widetilde{k}}_n) \nonumber\\
&= \sum_{k=1}^{\infty} \capE_{s,g}(E^{k}_n) + \left(\lambda^{2N}_n - 1\right)\,\capE_{s,g}(E^{\widetilde{k}}_n) - \sum_{k=K_n+1}^{\infty}\capE_{s,g}(E^{k}_n) \nonumber\\
&\leq \sum_{k=1}^{\infty} \capE_{s,g}(E^{k}_n) + \frac{2^{\widetilde{k}+1}\,E_{s,g}[m]}{m}\,\mu_{K_n} - \sum_{k=K_n+1}^{\infty} P_s(E^k_n).
\end{align}
Here, in the last inequality, we have also used \eqref{minimizingSeqFinitenessExisGeneMini01}. From the isoperimetric inequality of $P_s$, \eqref{minimizingSeqFinitenessExisGeneMini02}, and \eqref{minimizingInequality}, we further obtain that
\begin{align}
\sum_{k=1}^{K_n} \capE_{s,g}(\widetilde{E}^{k}_n) &\leq \sum_{k=1}^{\infty} \capE_{s,g}(E^k_n) + \frac{2^{\widetilde{k}+1}\,E_{s,g}[m]}{m}\, \mu_{K_n} - C\sum_{k=K_n+1}^{\infty} \left(m^{k}_n\right)^{\frac{N-s}{N}} \nonumber\\
&\leq \sum_{k=1}^{\infty} \capE_{s,g}(E^k_n) + \frac{2^{\widetilde{k}+1}\,E_{s,g}[m]}{m}\, \mu_{K_n} - C \left(\sum_{k=K_n+1}^{\infty} m^{k}_n\right)^{\frac{N-s}{N}} \nonumber\\
&= \sum_{k=1}^{\infty} \capE_{s,g}(E^k_n) + \frac{2^{\widetilde{k}+1}\,E_{s,g}[m]}{m}\, \mu_{K_n} - C \left( \mu_{K_n} \right)^{\frac{N-s}{N}}. \nonumber
\end{align}
Recalling the vanishing property of $\mu_{\ell}$ as $\ell \to \infty$ and taking the number $K_n$ so large that $K_n \geq \widetilde{k}$ and
\begin{equation}\nonumber
\frac{2^{\widetilde{k}+1}\,E_{s,g}[m]}{m}\, \mu_{K_n} - C \left( \mu_{K_n} \right)^{\frac{N-s}{N}} < 0,
\end{equation}
then we finally obtain, from \eqref{minimizingInequality} and \eqref{minimizingSeqFinitenessExisGeneMini01}, that
\begin{equation}\nonumber
\widetilde{E}_{s,g}[m] \leq \sum_{k=1}^{K_n} \capE_{s,g}(\widetilde{E}^{k}_n) < \sum_{k=1}^{\infty} \capE_{s,g}(E^k_n) \leq \widetilde{E}_{s,g}[m] + \varepsilon
\end{equation}
for large $n\in\mathN$. Finally letting $n \to \infty$, we obtain that the sum $\sum_{k=1}^{K_n} \capE_{s,g}(\widetilde{E}^{k}_n)$ converges to $\widetilde{E}_{s,g}[m]$ as $n \to \infty$. This completes the proof of the claim.
Now we are prepared to prove the other inequality of the claim by using assumption $(\mathrm{g}4)$ saying that $g$ vanishes at infinity. From the fact that $\sum_{k=1}^{K_n}|E^k_n| = m$, we can choose one $k'\in\mathN$ with $|E^{k'}_n| > 0$. Since we have assumed that the sets $\{E^k_n\}_{k=1}^{K_n}$ are bounded, we can choose the points $\{z^{k}_n\}_{k=1,\,k \neq k'}^{K_n}$ such that each set $E^k_n + R\,z^{k}_n$ is far away from the others for sufficiently large $R>1$. We can thus compute the energy as follows; from the translation invariance of $\capE_{s,g}$, it holds that
\begin{align}
\sum_{k=1}^{K_n} \capE_{s,g}(E^k_n) &= \sum_{k=1,\,k\neq k', \ell}^{K_n} \capE_{s,g}(E^k_n) + \capE_{s,g}(E^{k'}_n) + \capE_{s,g}(E^{\ell}_n) \nonumber\\
&= \sum_{k=1,\,k\neq k', \ell}^{K_n} \capE_{s,g}(E^k_n) + \capE_{s,g}(E^{k'}_n) + \capE_{s,g}(E^{\ell}_n + R\,z^{\ell}_n) \nonumber\\
&= \sum_{k=1,\,k\neq k', \ell}^{K_n} \capE_{s,g}(E^k_n) + \capE_{s,g}(E^{k'}_n \cup (E^{\ell}_n + R\,z^{\ell}_n)) \nonumber\\
&\qquad + 2\int_{E^{k'}_n} \int_{E^{\ell}_n + R\,z^{\ell}_n} \frac{dx\,dy}{|x-y|^{N+s}} - 2\int_{E^{k'}_n} \int_{E^{\ell}_n + R\,z^{\ell}_n}g(x-y)\,dx\,dy \nonumber
\end{align}
for any $\ell \in \{1,\cdots,\,K_n\}$ with $\ell \neq k'$ and sufficiently large $R>1$. Thus, we obtain that
\begin{align}\nonumber
\sum_{k=1,\,k\neq k', \ell}^{K_n} \capE_{s,g}(E^k_n) + \capE_{s,g}(E^{k'}_n \cup (E^{\ell}_n + R\,z^{\ell}_n)) &\leq \sum_{k=1}^{K_n} \capE_{s,g}(E^k_n) \nonumber\\
&\qquad + 2\int_{E^{k'}_n} \int_{E^{\ell}_n + R\,z^{\ell}_n}g(x-y)\,dx\,dy
\end{align}
for any $\ell \in \{1,\cdots,\,K_n\}$ with $\ell \neq k'$ and sufficiently large $R>1$. By repeating the same argument finite times for the rest of the sets $\{E^k_n\}_{k=1,\,k \neq k',\ell}^{K_n}$ with sufficiently large $R>1$ and from the translation invariance of $\capE_{s,g}$, we can derive the inequality
\begin{align}\label{keyEstimateReductionOneElementGeneMini02}
\capE_{s,g}\left( E^{k'}_n \cup \bigcup_{k=1,\,k\neq k'}^{K_n}\left( E^k_n + R\,z^{k}_n \right) \right) &\leq \sum_{k=1}^{K_n} \capE_{s,g}(E^k_n) \nonumber\\
&\qquad + 2\sum_{k=1}^{K_n-1}\sum_{\ell=k+1}^{K_n} \int_{F^k_n(R)} \int_{F^{\ell}_n(R)} g(x-y)\,dx\,dy
\end{align}
where we define the sets $\{F^k_n(R)\}_{k=1}^{K_n}$ in such a way that $F^k_n(R) \coloneqq E^{k}_n + R\,z^{k}_n$ if $k \neq k'$ and $F^{k'}_n(R) \coloneqq E^{k'}_n$. Note that the sets $\{F^k_n(R)\}_{k=1}^{K_n}$ satisfy
\begin{equation}\label{mutuallyFarAway}
\dist(F^k_n(R), \,F^{\ell}_n(R)) \xrightarrow[R \to \infty]{} \infty
\end{equation}
for any $k, \, \ell \in \{1,\cdots,\,K_n\}$ with $k \neq \ell$. Since $\sum_{k=1}^{K_n}|E^k_n| = m$ and $E^{k'}_n \cup \bigcup_{k=1,\,k\neq k'}^{K_n}\left( E^k_n + R\,z^{i_p} \right)$ are the union of disjoint sets, we have that
\begin{equation}\nonumber
\left| E^{k'}_n \cup \bigcup_{k=1,\,k\neq k'}^{K_n}\left( E^k_n + R\,z^{i_p} \right)\right| = \sum_{k=1}^{K_n} |E^{k}_n| = m.
\end{equation}
Thus, from \eqref{minimizingInequality} and \eqref{keyEstimateReductionOneElementGeneMini02}, we obtain
\begin{align}\label{keyEstimateReductionOneElementGeneMini03}
E_{s,g}[m] &\leq \capE_{s,g}\left( E^{k'}_n \cup \bigcup_{k=1,\,k\neq k'}^{K_n}\left( E^k_n + R\,z^{k}_n \right) \right) \nonumber\\
&\leq \sum_{k=1}^{K_n} \capE_{s,g}(E^k_n) \nonumber\\
&\qquad + 2\sum_{k=1}^{K_n}\sum_{\ell=k+1}^{K_n} \int_{F^k_n(R)} \int_{F^{\ell}_n(R)} g(x-y)\,dx\,dy \nonumber\\
&\leq \widetilde{E}_{s,g}[m] + \varepsilon \nonumber\\
&\qquad + 2\sum_{k=1}^{K_n}\sum_{\ell=k+1}^{K_n} \int_{F^k_n(R)} \int_{F^{\ell}_n(R)} g(x-y)\,dx\,dy
\end{align}
Hence, if we show that the last term of the right-hand side in \eqref{keyEstimateReductionOneElementGeneMini03} converges to zero as $R \to \infty$ for each $n \geq n_0$, then we conclude that the inequality
\begin{equation}\nonumber
\inf\left\{\capE_{s,g}(E) \mid |E|=m \right\} = E_{s,g}[m] \leq \widetilde{E}_{s,g}[m] = \inf\left\{\widetilde{\capE}_{s,g}(\{E^k\}_k) \mid \sum_{k=1}^{\infty} |E^k| = m \right\}
\end{equation}
holds and this completes the proof of the lemma. To conclude the proof of the lemma, it is sufficient to show that, under assumption $(\mathrm{g}4)$, the convergence
\begin{equation}\nonumber
\sum_{k=1}^{K_n}\sum_{\ell=k+1}^{K_n} \int_{F^k_n(R)} \int_{F^{\ell}_n(R)} g(x-y)\,dx\,dy \xrightarrow[R \to \infty]{} 0
\end{equation}
holds for each $n \geq n_0$. We fix $n \geq n_0$. From assumption $(\mathrm{g}4)$, we have that, for any $\varepsilon>0$, there exists a constant $R(\varepsilon)>0$ such that $g(z) < \varepsilon$ for any $|z| \geq R(\varepsilon)$. On the other hand, from \eqref{mutuallyFarAway}, we can also choose a constant $R'(\varepsilon)>0$ such that $|x-y| \geq R(\varepsilon)$ for any $R>R'(\varepsilon)$, $x \in F^k_n(R)$, $y \in F^{\ell}_n(R)$, and $k,\,\ell \in \{1,\cdots,\,K_n\}$ with $k \neq \ell$. Thus, taking these into account, we obtain that, for any $R > R'(\varepsilon)$,
\begin{equation}\nonumber
\sum_{k=1}^{K_n}\sum_{\ell=k+1}^{K_n} \int_{F^k_n(R)} \int_{F^{\ell}_n(R)} g(x-y)\,dx\,dy < \varepsilon \,\sum_{k=1}^{K_n}|F^k_n(R)| \sum_{\ell=1}^{K_n} |F^\ell_n(R)|.
\end{equation}
Recalling the definition of the sets $\{F^k_n(R)\}_{k}$, we have that $\sum_{k=1}^{K_n}|F^k_n(R)| \leq m$. Therefore, we obtain that
\begin{equation}\nonumber
\sum_{k=1}^{K_n}\sum_{\ell=k+1}^{K_n} \int_{F^k_n(R)} \int_{F^{\ell}_n(R)} g(x-y)\,dx\,dy < m^2\,\varepsilon
\end{equation}
for any $R > R'(\varepsilon)$ and this completes the proof of the claim.
\end{proof}
Now we prove Theorem \ref{theoremExistGeneralizedMiniAnyVolume}, namely, the existence of generalized minimizers of $\widetilde{\capE}_{s,g}$ under the assumptions $(\mathrm{g}1)$, $(\mathrm{g}2)$, and $(\mathrm{g}4)$ in Section \ref{sectionPreliminary}.
\begin{proof}[Proof of Theorem \ref{theoremExistGeneralizedMiniAnyVolume}]
Let $m>0$. Thanks to Lemma \ref{lemmaModifyMinimizingSeq}, it is sufficient to take any sequence $\{E_n\}_{n\in\mathN}$ such that $|E_n| = m$ for any $n\in\mathN$ and
\begin{equation}\label{minimizingSequenceGeneMini}
\lim_{n \to \infty} \capE_{s,g}(E_n) = \widetilde{E}_{s,g}[m]
\end{equation}
instead of taking a minimizing sequence for $\widetilde{E}_{s,g}[m]$. Thus, we can now apply the same argument as in the proof of the existence of minimizers of $\capE_{s,g}$ in Theorem \ref{theoremExistMiniAnyVolumeFasterDecay} because we assume that $g$ satisfies the assumptions $(\mathrm{g}1)$ and $(\mathrm{g}2)$. This enables us to choose a sequence of a finite number of measurable sets $\{G^i\}_{i=1}^H$ with $H \in \mathN$ such that
\begin{equation}\label{limitMinimizerManyComponents}
\sum_{i=1}^{H}\capE_{s,g}(G^i) \leq \liminf_{n \to \infty}\capE_{s,g}(E_n), \quad \sum_{i=1}^{H} |G^i| = m.
\end{equation}
Moreover, each set $G^i$ is a minimizer of $\capE_{s,g}$ among sets of volume $|G^i|$. Therefore, from \eqref{minimizingSequenceGeneMini} and \eqref{limitMinimizerManyComponents}, we conclude that the sequence $\{G^i\}_{i=1}^H$ is a generalized minimizer of $\widetilde{\capE}_{s,g}$ with $\sum_{i=1}^{H}|G^i| = m$ as we desired.
\end{proof}
\section{Asymptotic behavior of minimizers for large volumes}\label{sectionAsymptoticMiniLargeVol}
In this section, we study the asymptotic behavior of minimizers of $\capE_{s,g}$ with large volumes under the assumption that the kernel $g$ decays sufficiently fast. To see this, we first prove the $\Gamma$-convergence in $L^1$-topology of the functional associated with Problem \eqref{minimizationScalingModifiedProbelm} to the fractional perimeter $P_s$ as $m$ goes to infinity. Since it is well-known that a sequence of minimizers for a functional converges to a minimizer of its $\Gamma$-limit, we can derive the convergence of a sequence of the minimizers to the unit ball, by rescaling, up to translations.
\subsection{$\Gamma$-convergence of $\widehat{\capE}^{\lambda}_{s,g}$ to the fractional perimeter as $\lambda \to \infty$}
Before proving Theorem \ref{theoremAsympMiniLargeVolume}, we establish the $\Gamma$-convergence result for the energy $\capE^{\lambda}_{s,g}$ under the assumption that the kernel $g$ decays sufficiently fast. Before stating the claim, we give several notations and the definition of the functional $\capF^{\lambda}_{s,g}$ on $L^1(\mathR^N)$. First, we recall the definition of the $s$-fractional Sobolev semi-norm $[f]_{W^{s,1}}(\mathR^N)$ as follows:
\begin{equation}\nonumber
[f]_{W^{s,1}}(\mathR^N) = \frac{1}{2}\int_{\mathR^N} \int_{\mathR^N}\frac{|f(x) - f(y)|}{|x-y|^{N+s}}\,dx\,dy
\end{equation}
for $f \in L^1$. Note that $[\chi_E]_{W^{s,1}}(\mathR^N) = P_s(E)$ for any measurable set $E \subset \mathR^N$. As is shown in \cite[Proposition 4.2 and Corollary 4.4]{BLP}, it follows that any integrable function of bounded variation is also finite with respect to the fractional semi-norm $[\cdot]_{W^{s,1}}$. Secondly, in order to study the $\Gamma$-convergence of the sequence $\{\widehat{\capE}^{\lambda}_{s,g}\}_{\lambda>1}$ given in Proposition \ref{propositionRescaledProblem}, we define the functional $\widehat{\capF}^{\lambda_n}_{s,g}$ as
\begin{align}\label{functionalGammaConv}
\widehat{\capF}^{\lambda_n}_{s,g}(f) \coloneqq
\left\{
\begin{array}{ll}
[f]_{W^{s,1}}(\mathR^N) - & \displaystyle \frac{1}{2}\int_{\mathR^N}\int_{\mathR^N} |f(x) - f(y)|\, g_{\lambda}(x-y) \,dx\,dy \\
\\
& \text{if $f=\chi_{F}$ for some bounded set $F \subset \mathR^N$ with $P_s(F) < \infty$,} \\
\\
+\infty \quad &\text{otherwise.}
\end{array}
\right.
\end{align}
Note that the functional $\widehat{\capF}^{\lambda}_{s,g}(f)$ for any $\lambda>0$ is well-defined. Moreover, if $f =\chi_E$ for some bounded set $E$ with $P_s(E) < \infty$, then we can easily see that $\widehat{\capF}^{\lambda}_{s,g}(f) = \widehat{\capE}^{\lambda}_{s,g}(E)$.
Now we prove the $\Gamma$-convergence of the functional $\widehat{\capF}^{\lambda_n}_{s,g}$ to $\widehat{\capF}^{\infty}_{s}$ (we give the definition of $\widehat{\capF}^{\infty}_{s}$ in the following proposition) as $n \to \infty$ in the $L^1$-topology.
\begin{proposition}[$\Gamma$-convergence to the $s$-fractional semi-norm] \label{propositionGammaConvergenceNonlocalEnergy}
Let $\{\lambda_n\}_{n\in\mathN}$ be any sequence of positive real numbers going to infinity as $n\to\infty$. Assume that the kernel $g : \mathR^N \setminus \{0\} \to \mathR$ is radially symmetric and satisfies the assumptions $(\mathrm{g}1)$, $(\mathrm{g}2)$, and $(\mathrm{g}5)$ in Section \ref{sectionPreliminary}. Then the sequence $\{\widehat{\capF}^{\lambda_n}_{s,g}\}_{n\in\mathN}$ $\Gamma$-converges, with respect to $L^1$-topology, to the functional $\widehat{\capF}^{\infty}_{s}$ defined by
\begin{align}
\widehat{\capF}^{\infty}_{s}(f) \coloneqq
\left\{
\begin{array}{ll}
\displaystyle [f]_{W^{s,1}}(\mathR^N) \quad &\text{if $f=\chi_{F}$ for some bounded $F \subset \mathR^N$ with $P_s(F) < \infty$,} \nonumber\\
\\
+\infty \quad &\text{otherwise.} \nonumber
\end{array}
\right.
\end{align}
\end{proposition}
\begin{remark}
Recall that, in this paper, we assume that the kernel $g$ is locally integrable in $\mathR^N$, especially near the origin; however, Proposition \ref{propositionGammaConvergenceNonlocalEnergy} is still valid even if $g$ is not integrable in the ball centred at the origin. This is because the assumption that $g(x) \leq |x|^{-(N+s)}$ for $x \neq 0$ is sufficient enough for the functional \eqref{functionalGammaConv} to be well-defined.
\end{remark}
\begin{proof}
We recall the definition of the $\Gamma$-convergence. We say that $\{\widehat{\capF}^{\lambda_n}_{s,g}\}_{n\in\mathN}$ $\Gamma$-converges to $\widehat{\capF}^{\infty}_{s}$ with respect to $L^1$-topology if the two estimates hold
\begin{equation}\nonumber
\Gamma_{L^1-}\limsup_{n\to\infty}\widehat{\capF}^{\lambda_n}_{s,g}(f) \leq \widehat{\capF}^{\infty}_{s}(f), \quad \widehat{\capF}^{\infty}_{s}(f) \leq \Gamma_{L^1-}\liminf_{n\to\infty}\widehat{\capF}^{\lambda_n}_{s,g}(f)
\end{equation}
for any $f \in L^1(\mathR^N)$, where we set
\begin{equation}\label{defgammaConveSup}
\Gamma_{L^1-}\limsup_{n\to\infty}\widehat{\capF}^{\lambda_n}_{s,g}(f) \coloneqq \min\left\{\limsup_{n\to\infty} \widehat{\capF}^{\lambda_n}_{s,g}(f_n) \mid \text{$f_n \xrightarrow[n\to\infty]{} f$ in $L^1(\mathR^N)$} \right\}
\end{equation}
and
\begin{equation}\label{defgammaConveInf}
\Gamma_{L^1-}\liminf_{n\to\infty}\widehat{\capF}^{\lambda_n}_{s,g}(f) \coloneqq \min\left\{\liminf_{n\to\infty} \widehat{\capF}^{\lambda_n}_{s,g}(f_n) \mid \text{$f_n \xrightarrow[n\to\infty]{} f$ in $L^1(\mathR^N)$} \right\}.
\end{equation}
Note that the minimum in \eqref{defgammaConveSup} and \eqref{defgammaConveInf} is attained by the diagonal argument.
First of all, we prove that $\Gamma_{L^1-}\limsup_{n\to\infty}\widehat{\capF}^{\lambda_n}_{s,g}(f) \leq \widehat{\capF}^{\infty}_{s}(f)$ for any $f \in L^1(\mathR^N)$. In the case that $f$ is not a characteristic function of some bounded set with a finite nonlocal perimeter, we obviously have that $\widehat{\capF}^{\infty}_{s}(f) = \infty$ and the inequality holds. Thus, we may assume that $f = \chi_F$ for a bounded set $F \subset \mathR^N$ with $P_s(F) < \infty$. Setting a sequence $\{f_n\}_{n\in\mathN}$ as $f_n = f = \chi_F$ for any $n\in\mathN$, we obtain, from the non-negativity of $g$, that
\begin{equation}\nonumber
\widehat{\capF}^{\lambda_n}_{s,g}(f_n) \leq \widehat{\capF}^{\infty}_{s}(f)
\end{equation}
for any $n\in\mathN$ and thus, it follows that $\Gamma_{L^1-}\limsup_{n\to\infty}\widehat{\capF}^{\lambda_n}_{s,g}(f) \leq \widehat{\capF}^{\infty}_{s}(f)$.
Next we prove that $\widehat{\capF}^{\infty}_{s}(f) \leq \Gamma_{L^1-}\liminf_{n\to\infty}\widehat{\capF}^{\lambda_n}_{s,g}(f)$ for any $f \in L^1(\mathR^N)$. We take any sequence $\{f_n\}_{n\in\mathN} \subset L^1(\mathR^N)$ such that $f_n \to f$ in $L^1$ as $n \to \infty$. In the case that $f$ is not a characteristic function of some bounded set with a finite nonlocal perimeter, we claim that there exists a number $n_0 \in \mathN$ such that $f_n$ is also not a characteristic function of a measurable set for any $n \geq n_0$. Indeed, we suppose by contradiction that there exists a subsequence $\{f_{n_k}\}_{k\in\mathN}$ such that $f_{n_k} = \chi_{F_{n_k}}$ for some measurable set $F_{n_k} \subset \mathR^N$ for any $k \in \mathN$. Since $f_{n_k} \to f$ in $L^1$ as $k \to \infty$ and $f_{n_k} \in \{0,\,1\}$ for any $k\in\mathN$, we obtain that $f \in \{0,\,1\}$ a.e. in $\mathR^N$ and $f$ can be written as $f = \chi_F$ for some measurable $F\ \subset \mathR^N$. This contradicts the assumption that $f$ is not a characteristic function. Hence, we conclude that, for large $n \in \mathN$, $\widehat{\capF}^{\lambda_n}_{s,g}(f_n) = \infty$ and the claim holds true. Thus, in the sequel, we may assume that $f = \chi_F$ for some bounded set $F \subset \mathR^N$ with $P_s(F) < \infty$.
Under the above assumption, we first compute the second term of the functional $\widehat{\capF}^{\lambda_n}_{s,g}$ in \eqref{functionalGammaConv}. Let $\varepsilon \in (0,\,1)$. From assumption $(\mathrm{g}5)$, we can choose a constant $R_\varepsilon > 1$ such that $g(x) \leq \frac{\varepsilon}{|x|^{N+s}}$ for $|x| \geq R_\varepsilon$. Then, from the definition of $g_{\lambda_n}$ for any $n\in\mathN$, we have that
\begin{align}\label{splittingSecondTermEnergy}
&\int_{\mathR^N}\int_{\mathR^N} |f(x) - f(y)| \, g_{\lambda_n}(x-y) \,dx\,dy \nonumber\\
&= \iint_{\{(x,\,y) \mid \, \lambda_n|x-y| < R_\varepsilon \}} |f(x) - f(y)| \, g_{\lambda_n}(x-y) \,dx\,dy \nonumber\\
&\qquad + \iint_{\{(x,\,y) \mid \, \lambda_n|x-y| \geq R_\varepsilon \}} |f(x) - f(y)| \, g_{\lambda_n}(x-y) \,dx\,dy \nonumber\\
&\leq \iint_{\{(x,\,y) \mid \, \lambda_n|x-y| < R_\varepsilon \}} \frac{|f(x) - f(y)|}{|x-y|^{N+s}} \,dx\,dy \nonumber\\
&\qquad + \varepsilon \iint_{\{(x,\,y) \mid \, \lambda_n|x-y| \geq R_\varepsilon \}} \frac{|f(x) - f(y)|}{|x-y|^{N+s}} \,dx\,dy
\end{align}
for any $n \in \mathN$. Thus, from the definition of $\widehat{\capE}^{\lambda_n}_{s,g}$ and \eqref{splittingSecondTermEnergy}, we can obtain
\begin{align}\label{splittingSecondTermEnergy02}
\widehat{\capF}^{\lambda_n}_{s,g}(f_n) &\geq [f_n]_{W^{s,1}}(\mathR^N) - \frac{1}{2}\iint_{\{(x,\,y) \mid \, \lambda_n|x-y| < R_\varepsilon \}} \frac{|f_n(x) - f_n(y)|}{|x-y|^{N+s}} \,dx\,dy \nonumber\\
&\qquad - \frac{\varepsilon}{2}\iint_{\{(x,\,y) \mid \, \lambda_n|x-y| \geq R_\varepsilon \}} \frac{|f_n(x) - f_n(y)|}{|x-y|^{N+s}} \,dx\,dy \nonumber\\
&\geq \frac{1-\varepsilon}{2}\iint_{\{(x,\,y) \mid \, \lambda_n|x-y| \geq R_\varepsilon \}} \frac{|f_n(x) - f_n(y)|}{|x-y|^{N+s}} \,dx\,dy
\end{align}
for any $\varepsilon \in (0,\,1)$ and $n\in\mathN$. Thus, letting first $n \to \infty$ and then $\varepsilon \to 0$ with Fatou's lemma and the monotone convergence theorem, we finally obtain
\begin{align}
\liminf_{n \to \infty}\widehat{\capF}^{\lambda_n}_{s,g}(f_n) &\geq \limsup_{\varepsilon \to 0}\frac{1-\varepsilon}{2}\iint_{\mathR^N \times \mathR^N} \frac{|f(x) - f(y)|}{|x-y|^{N+s}} \,dx\,dy \nonumber\\
&= \frac{1}{2}\int_{\mathR^N}\int_{\mathR^N}\frac{|f(x) - f(y)|}{|x-y|^{N+s}} \,dx\,dy = [f]_{W^{s,1}}(\mathR^N). \nonumber
\end{align}
Therefore, from the above arguments, we complete the proof.
\end{proof}
\subsection{Convergence of minimizers of $\widehat{\capE}^{\lambda}_{s,g}$ to the ball as $\lambda \to \infty$}
Now we are prepared to prove Theorem \ref{theoremAsympMiniLargeVolume}. In this theorem, we impose on $g$ the assumptions that $g$ is radially symmetric and decays sufficiently fast. One important difference between the assumptions on $g$ of Theorem \ref{theoremExistMiniAnyVolumeFasterDecay} and \ref{theoremAsympMiniLargeVolume} is that the decay of the kernel $g$ in Theorem \ref{theoremAsympMiniLargeVolume} is strictly faster than that in Theorem \ref{theoremExistMiniAnyVolumeFasterDecay}. To observe the convergence, we consider Problem \ref{minimizationScalingModifiedProbelm} and finally we take the limit $\lambda \to \infty$ instead of Problem \ref{minimizationGeneralizedFunctional} with the limit $m \to \infty$ for convenience.
The strategy for the proof of the asymptotic behavior of the minimizers is as follows; in contrast to the idea for studying the asymptotic behavior of minimizers, for instance, in \cite{FFMMM, Pegon, Carazzato}, we may not be able to employ directly the quantitative isoperimetric inequality for the nonlocal perimeter $P_s$ to use a Fuglede-type argument. This is because we may have the volume of the symmetric difference between minimizers and the balls naively bounded by the volume and perimeter of the symmetric difference. Since we are dealing with minimizers with large volumes, it is not obvious that the bound of the symmetric difference could give us the $L^1$-convergence of minimizers to the ball. Therefore, we adopt another strategy in the following way; we first take any sequence $\{F_n\}_{n}$ of the minimizers for $\widehat{\capE}^{\lambda_n}_{s,g}$ with $|F_n|=|B_1|$. Then we apply so-called ``concentration-compactness" lemma that we use to show the existence of minimizers in Section \ref{sectionExisMiniFastDecayGAnyVol} and \ref{sectionExistGeneMiniAnyVol}. As a consequence of the lemma, we can choose a sequence of sets $\{G^i\}_{i}$ and points $\{z_n^i\}_{i,n}$ such that, up to extracting a subsequence,
\begin{equation}\label{keyInfoAsymptoticLargeMini}
\sum_{i}P_s(G^i) \leq \liminf_{n \to \infty}\widehat{\capE}^{\lambda_n}_{s,g}(F_n), \quad F_n - z_n^i \xrightarrow[n \to \infty]{} G^i \quad \text{in $L^1_{loc}$}, \quad \sum_{i}|G^i| = |B_1|
\end{equation}
thanks to the assumptions on $g$. Then, from the isoperimetric inequality of $P_s$ and the minimality of $F_n$, we can actually obtain that each $G^i$ coincides with the Euclidean ball, up to translations and negligible sets, whenever $|G^i|>0$. Finally, from \eqref{keyInfoAsymptoticLargeMini}, we can show that the only one element in $\{G^i\}_{i}$ has a positive volume and its volume is equal to $|B_1|$. From Brezis-Lieb lemma, the convergence in \eqref{keyInfoAsymptoticLargeMini} is improved to the $L^1$-convergence. Combining the $\Gamma$-convergence result, we conclude the proof.
\begin{proof}[Proof of Theorem \ref{theoremAsympMiniLargeVolume}]
Let $\{\lambda_n\}_{n\in\mathN}$ be any sequence going to infinity as $n \in \mathN$ and we take any sequence $\{F_n\}_{n\in\mathN}$ of the minimizers for $\widehat{\capE}^{\lambda_n}_{s,g}$ with $|F_n| = |B_1|$ for any $n \in \mathN$. From the assumption $(\mathrm{g}5)$, we can choose a constant $\gamma \in (0,\,1)$ such that $g_{\lambda_n}(x) \leq \gamma|x|^{-(N+s)}$ for any $|x| \neq 0$. From the minimality of $F_n$ for each $n\in\mathN$, we have that
\begin{equation}\nonumber
P_s(F_n) \leq P_s(B_1) + P_{g_{\lambda_n}}(F_n) = P_s(B_1) + \gamma \, P_s(F_n)
\end{equation}
for any $n \in \mathN$ and thus, we obtain that $\{P_s(F_n)\}_{n}$ is uniformly bounded with respect to $n$, namely, $\sup_{n\in\mathN}P_s(F_n) \leq (1-\gamma)^{-1}P_s(B_1) < \infty$. As a consequence of the uniform bound of $\{P_s(F_n)\}_{n}$, we can now apply the same method as in the proof of Theorem \ref{theoremExistMiniAnyVolumeFasterDecay} (see also \cite{dCNRV}) to the sequence $\{F_n\}_{n}$. Although we discuss in the proof of Theorem \ref{theoremExistMiniAnyVolumeFasterDecay}, we rewrite the argument in the sequel for convenience.
First of all, we decompose $\mathR^N$ into the unit cubes and denote by $\{Q_n^i\}_{i=1}^{\infty}$. We set $x_n^i \coloneqq |F_n \cap Q_n^i|$ and have that
\begin{equation}\label{keyTechnicalAsymptotic01}
\sum_{i=1}^{\infty} x_n^i = |F_n| = |B_1|
\end{equation}
for any $n\in\mathN$. Moreover, from the isoperimetric inequality shown in \cite[Lemma 2.5]{dCNRV}, we obtain
\begin{equation}\label{keyTechnicalAsymptotic02}
\sum_{i=1}^{\infty} (x_n^i)^{\frac{N-s}{N}} \leq C\sum_{i=1}^{\infty}P_s(F_n; Q_n^i) \leq 2CP_s(F_n) \leq C_1 < \infty
\end{equation}
for any $n\in\mathN$, where $C$ and $C_1$ are the positive constants independent of $n$. Up to reordering the cubes $\{Q_n^i\}_{i}$, we may assume that $\{x_n^i\}_{i}$ is a non-increasing sequence for any $n\in\mathN$. Thus, applying the technical result shown in \cite[Lemma 4.2]{GoNo} or \cite[Lemma 7.4]{dCNRV} with \eqref{keyTechnicalAsymptotic01} and \eqref{keyTechnicalAsymptotic02}, we obtain that
\begin{equation}\label{keyTechnicalAsymptotic03}
\sum_{i=k+1}^{\infty} x_n^i \leq \frac{C_2}{k^{\frac{s}{N}}}
\end{equation}
for any $k\in\mathN$ where $C_2$ is the positive constant independent of $n$ and $k$. Hence, by using the diagonal argument, we have that, up to extracting a subsequence, $x_n^i \to \alpha^i \in [0,\,|B_1|]$ as $n \to \infty$ for every $i\in\mathN$ and, from \eqref{keyTechnicalAsymptotic01} and \eqref{keyTechnicalAsymptotic03},
\begin{equation}\label{identityLimitMeasuAsymptotic}
\sum_{i=1}^{\infty} \alpha^i = |B_1|.
\end{equation}
Now we fix the centre of the cube $z_n^i \in Q_n^i$ for each $i$ and $n$. Up to extracting a further subsequence, we may assume that $|z_n^i - z_j^i| \to c^{ij} \in [0,\,\infty]$ as $n \to \infty$ for each $i,\,j \in \mathN$. As already seen in the above, we have the uniform bound of the sequence $\{P_s(F_n-z_n^i)\}_{n\in\mathN}$ and its upper-bound is independent of $i$ and thus, from the compactness, there exists a measurable set $G^i \subset \mathR^N$ such that, up to extracting a further subsequence,
\begin{equation}\nonumber
\chi_{F_n-z_n^i} \xrightarrow[n \to \infty]{} \chi_{G^i} \quad \text{in $L^1_{loc}$-topology}.
\end{equation}
We define the relation $i \sim j$ for every $i,\,j\in\mathN$ as $c^{ij} < \infty$ and we denote by $[i]$ the equivalent class of $i$. Moreover, we define the set of the equivalent class by $\capI$. Then, by applying the same argument as in the proof of Theorem \ref{theoremExistMiniAnyVolumeFasterDecay}, it is easy to show that
\begin{equation}\nonumber
\sum_{[i] \in \capI} |G^i| = |B_1|.
\end{equation}
As a first step, we want to show a sort of lower semi-continuity of the energy, more precisely, we will prove the following inequality;
\begin{equation}\label{lowerSemicontiConcentrationAsymptotic}
\sum_{[i] \in \capI}P_s(G^i) \leq \liminf_{n \to \infty}\widehat{\capE}^{\lambda_n}_{s,g}(F_n) = \liminf_{n \to \infty}\left(P_s(F_n) - P_{g_{\lambda_n}}(F_n) \right).
\end{equation}
Indeed, we first fix $M \in \mathN$ and $R>0$ and we take the equivalent classes $i_1,\cdots,\,i_M$. Notice that, if $p \neq q$, then $|z_n^{i_p} - z_n^{i_q}| \to \infty$ as $n\to\infty$ and thus we have that $\{z_n^{i_p}+Q_R\}_{p}$ are disjoint sets for large $n$ and
\begin{equation}\label{vanishingEnergyGeneralizedMini}
\int_{z_n^{i_p} + Q_R}\int_{z_n^{i_q} + Q_R} \frac{1}{|x-y|^{N+s}}\,dx\,dy \xrightarrow[n \to \infty]{} 0
\end{equation}
where $Q_R$ is the cube of side $R$. Then, by using a similar argument shown in the proof of the $\Gamma$-liminf inequality in Proposition \ref{propositionGammaConvergenceNonlocalEnergy} with \eqref{vanishingEnergyGeneralizedMini}, we can conduct the following similar argument: let $\varepsilon \in (0,\,1)$ and, from $(\mathrm{g}5)$, we can choose a constant $R_\varepsilon>1$ such that $g(x) \leq \frac{\varepsilon}{|x|^{N+s}}$ for any $|x| \geq R_\varepsilon$. Then it holds that
\begin{align}\label{liminfInequalityAsymptotic}
&\liminf_{n \to \infty}\left(P_s(F_n) - P_{g_{\lambda_n}}(F_n) \right) \nonumber\\
&\geq (1-\varepsilon) \liminf_{n \to \infty}\left(\int_{F_n \cap A^{M,R}_n}\int_{F^c_n}\frac{\chi_{\{|x-y| \geq r_n^{\varepsilon}\}}(x,\,y)}{|x-y|^{N+s}} \,dx\,dy \right) \nonumber\\
&\qquad + (1-\varepsilon) \liminf_{n \to \infty}\left(\int_{F_n \setminus A^{M,R}_n}\int_{A^{M,R}_n \setminus F_n}\frac{\chi_{\{|x-y| \geq r_n^{\varepsilon}\}}(x,\,y)}{|x-y|^{N+s}} \,dx\,dy \right) \nonumber\\
&\qquad \quad + (1-\varepsilon)\liminf_{n \to \infty}2\sum_{p \neq q} \int_{z^{i_p}_n + Q_R}\int_{z^{i_q}_n + Q_R}\frac{\chi_{\{|x-y| \geq r_n^{\varepsilon}\}}(x,\,y)}{|x-y|^{N+s}}\,dx\,dy
\end{align}
for any $\varepsilon \in (0,\,1)$ where we set $r_n^{\varepsilon} \coloneqq \lambda^{-1}_n\,\varepsilon^{-\frac{1}{1-s}}$ for each $n$ and $A^{M,R}_n \coloneqq \cup_{p=1}^{M}(z^{i_p}_n + Q_R)$. Now we recall the inequality of double integrals;
\begin{align}\label{subadditivityNonlocalPeri}
&P_s(E;A) + P_s(E;B) \leq P_s(E; A \sqcup B) + 2\int_{A}\int_{B}\frac{dx\,dy}{|x-y|^{N+s}}
\end{align}
for any measurable disjoint sets $A,\,B\subset \mathR^N$, where we define $P_s(E;A) \coloneqq \int_{E \cap A}\int_{E^c} + \int_{E \setminus A}\int_{A \setminus E}$ for measurable sets $E,\,A \subset \mathR^N$ (we omit the integrand for simplicity). Hence, from \eqref{liminfInequalityAsymptotic}, \eqref{subadditivityNonlocalPeri}, and the lower semi-continuity of $P_s$ in $L^1_{loc}$-topology with Fatou's lemma, we obtain
\begin{align}
&\liminf_{n \to \infty}\left(P_s(F_n) - P_{g_{\lambda_n}}(F_n) \right) \nonumber\\
&\geq (1-\varepsilon)\liminf_{n \to \infty} \sum_{p=1}^{M} \left(\int_{F_n \cap (z^{i_p}_n + Q_R)} \int_{F_n^c} \frac{\chi_{\{|x-y|\geq r_n^{\varepsilon}\}}(x,\,y)}{|x-y|^{N+s}}\,dx\,dy \right. \nonumber\\
&\qquad \left. + \int_{F_n \setminus (z^{i_p}_n + Q_R)} \int_{(z^{i_p}_n + Q_R) \setminus F_n} \frac{\chi_{\{|x-y|\geq r_n^{\varepsilon}\}}(x,\,y)}{|x-y|^{N+s}}\,dx\,dy \right) \nonumber\\
&\geq (1-\varepsilon)\sum_{p=1}^{M} P_s(G^{i_p} ; Q_R) \nonumber
\end{align}
for any $\varepsilon \in (0,\,1)$. Letting $R \to \infty$, $M \to \infty$, and $\varepsilon \to 0$, we finally conclude that the inequality \eqref{lowerSemicontiConcentrationAsymptotic} holds true. Taking into account all of the above arguments, we obtain the existence of sets $\{G^i\}_{[i]\in\capI}$ satisfying the properties that
\begin{equation}\label{keyPropertylowSemicontiVolumeEqualParticlesAsymptotic}
\sum_{[i] \in \capI} P_s(G^i) \leq \liminf_{n \to \infty} \widehat{\capE}^{\lambda_n}_{s,g}(F_n), \quad \sum_{[i] \in \capI} |G^{i}| = |B_1|.
\end{equation}
Secondly, we want to show that each $G^{i}$ actually coincides, up to translations and negligible sets, with the Euclidean ball with the volume $|G^{i}|$, whenever $|G^{i}|>0$. Indeed, we first set $B_i$ as the ball of radius $r_i \coloneqq |B_1|^{-1/N}\,|G^i|^{1/N}$ for each $[i] \in \capI$. Then, from \eqref{keyPropertylowSemicontiVolumeEqualParticlesAsymptotic} and the minimality of $F_n$, we have that
\begin{align}\label{energyMinimalityEachPrticleAsymptotic}
\sum_{[i] \in \capI} \left( P_s(G^{i}) - P_s(B_i) \right) &\leq \liminf_{n \to \infty}\widehat{\capE}^{\lambda_n}_{s,g}(F_n) - \sum_{p=1}^{M} P_s(B_i) \nonumber\\
&\leq P_s(B_1) - \sum_{[i] \in \capI} \left(\frac{|G^i|}{|B_1|}\right)^{\frac{N-s}{N}} P_s(B_1) \nonumber\\
&\leq P_s(B_1) - P_s(B_1) \left(\sum_{[i]\in\capI}\frac{|G^i|}{|B_1|}\right)^{\frac{N-s}{N}} = 0.
\end{align}
From the isoperimetric inequality of $P_s$, we know that $P_s(B_i) \leq P_s(G^i)$ for any $[i] \in \capI$ and the equality holds if and only if $G^i = B_i$ up to translation and negligible sets. Hence, from \eqref{energyMinimalityEachPrticleAsymptotic}, we conclude that the claim holds true.
Next we show that the set $\capI$ of the equivalent classes is actually a finite set. Indeed, we first set $m^{i_p} \coloneqq |G^{i_p}|$ for any $p \in \mathN$ and, since $\sum_{p=1}^{\infty} m^{i_p} = |B_1|$, we can observe that $m^{i_p} \to 0$ as $p \to \infty$ and, moreover, $\mu_{\ell} \coloneqq \sum_{p=\ell+1}^{\infty} m^{i_p} \to 0$ as $\ell \to \infty$. Then, we can choose $\widetilde{p} \in \mathN$ such that $m^{i_{\widetilde{p}}} \geq 2^{-(\widetilde{p}+1)}|B_1|$. Now using the sets $\{G^{i_p}\}_{p=1}^{\infty}$, we construct a new family of sets $\{\widetilde{G}^{i_p}\}_{p=1}^{H}$ for some $H \in \mathN$, depending only on $N$ and $s$, in the following manner; we choose $H \in \mathN$ so large that $H \geq \widetilde{p}$ and set $\widetilde{G}^{i_p} \coloneqq G^{i_p}$ for any $p \in \{1,\cdots ,\,H\}$ with $p \neq \widetilde{p}$ and $\widetilde{G}^{i_{\widetilde{p}}} \coloneqq \eta\,G^{i_{\widetilde{p}}}$ where $\eta^{N} \coloneqq m^{-i_{\widetilde{p}}}(m^{i_{\widetilde{p}}} + \mu_{H})$. Then, we have the volume identity that
\begin{equation}\label{volumeIdentityReorganizedAsymptotic}
\sum_{p=1}^{H} \left| \widetilde{G}^{i_p} \right| = \sum_{p=1, \,p\neq\widetilde{p}}^{H} \left| G^{i_p} \right| + \eta^N\, |G^{i_{\widetilde{p}}}| = \sum_{p=1,\,p\neq\widetilde{p}}^{H} m^{i_p} + m^{i_{\widetilde{p}}} + \mu_{H} = |B_1|.
\end{equation}
Now we compute the energy for $\{\widetilde{G}^{i_p}\}_{p=1}^{H}$ as follows to show that the total energy of each elements of $\{\widetilde{G}^{i_p}\}_{p=1}^{H}$ is more efficient than that of $\{G^{i_p}\}_{p=1}^{\infty}$; from the definition of $\eta \geq 1$ and Lemma \ref{lemmaScalingEnergy}, we have that
\begin{align}\label{comparisonNewSetsEnergyAsymptotic}
\sum_{p=1}^{H} P_s(\widetilde{G}^{i_p}) &= \sum_{p=1,\,p\neq\widetilde{p}}^{H} P_s(G^{i_p}) + \eta^{N-s}\,P_s(G^{i_{\widetilde{p}}}) \nonumber\\
&= \sum_{p=1}^{\infty} P_s(G^{i_p}) + \left(\eta^{N-s} - 1\right)\,P_s(G^{i_{\widetilde{p}}}) - \sum_{p=H+1}^{\infty}P_s(G^{i_p}) \nonumber\\
&\leq \sum_{[i] \in \capI} P_s(G^{i_p}) + \frac{2^{\widetilde{p}+1}c_{N,s}\,P_s(B_1)}{|B_1|}\,\mu_{H} - \sum_{p=H+1}^{\infty} P_s(G^{i_p})
\end{align}
where $c_{N,s}>0$ is some constant depending only on $N$ and $s$. Here, in the last inequality, we have also used \eqref{keyPropertylowSemicontiVolumeEqualParticlesAsymptotic}. From the isoperimetric inequality of $P_s$ and \eqref{comparisonNewSetsEnergyAsymptotic}, we further obtain that
\begin{align}
\sum_{p=1}^{H} P_s(\widetilde{G}^{i_p}) &\leq \sum_{[i] \in \capI} P_s(G^i) + \frac{2^{\widetilde{p}+1}c_{N,s}\,P_s(B_1)}{|B_1|}\, \mu_{H} - C\sum_{p=H+1}^{\infty} \left(m^{i_p}\right)^{\frac{N-s}{N}} \nonumber\\
&\leq \sum_{[i] \in \capI} P_s(G^i) + \frac{2^{\widetilde{p}+1}c_{N,s}\,P_s(B_1)}{|B_1|}\, \mu_{H} - C \left(\sum_{p=H+1}^{\infty} m^{i_p}\right)^{\frac{N-s}{N}} \nonumber\\
&= \sum_{[i] \in \capI} P_s(G^i) + \frac{2^{\widetilde{p}+1}c_{N,s}\,P_s(B_1)}{|B_1|}\, \mu_{H} - C \left( \mu_{H} \right)^{\frac{N-s}{N}}. \nonumber
\end{align}
Taking the number $H$ so large that $H \geq \widetilde{p}$ and
\begin{equation}\nonumber
\frac{2^{\widetilde{p}+1}c_{N,s}\,P_s(B_1)}{|B_1|}\, \mu_{H} - C \left( \mu_{H} \right)^{\frac{N-s}{N}} \leq 0,
\end{equation}
then we finally obtain that
\begin{equation}\label{reductionFiniteElementsMinimizerAsymptotic}
\sum_{p=1}^{H} P_s(\widetilde{G}^{i_p}) \leq \sum_{[i] \in \capI} P_s(G^i) \leq \liminf_{n \to \infty}\widehat{\capE}^{\lambda_n}_{s,g}(F_n) = \liminf_{n \to \infty}\left(P_s(F_n) - P_{g_{\lambda_n}}(F_n) \right).
\end{equation}
This completes the proof of the claim that the set $\capI$ is finite.
Finally, we show that there exists one number $i_0 \in \mathN$ such that $|G^i|=0$ for any $[i] \in \capI$ with $i \neq i_0$. From \eqref{keyPropertylowSemicontiVolumeEqualParticlesAsymptotic}, there exists at least one number $p'\in\mathN$ such that $|G^{i_{p'}}| > 0$. Then we claim that, if $q \neq p'$, then it holds that $|G^{i_q}|=0$. Indeed, from the previous claim, we can restrict ourselves to consider a finite number of sets $\{\widetilde{G}^{i_p}\}_{p=1}^{H}$, which satisfies \eqref{reductionFiniteElementsMinimizerAsymptotic} and $\sum_{p=1}^{H}|\widetilde{G}^{i_p}| = |B_1|$, instead of $\{G^i\}_{[i]\in\capI}$. Moreover, we may assume that $H \geq p'$. Since we have shown that the sets $\{\widetilde{G}^{i_p}\}_{p=1}^{H}$ are identified with the balls of the volume $|G^{i_p}|$ whenever $|G^{i_p}|>0$ for $p \in \{1,\cdots,H\}$, we can choose the points $\{z^{i_p}\}_{p=1,\,p \neq p'}^{H}$ such that each set $\widetilde{G}^{i_p} + R\,z^{i_p}$ is far away from the others for large $R>1$. We can thus compute the energy as follows; by translation invariance, we have that
\begin{align}
\sum_{p=1}^{H} P_s(\widetilde{G}^{i_p}) &= \sum_{p=1\,p \neq p',q}^{H} P_s(\widetilde{G}^{i_p}) + P_s(\widetilde{G}^{i_{p'}}) + P_s(\widetilde{G}^{i_q}) \nonumber\\
&= \sum_{p=1\,p \neq p',q}^{H} P_s(\widetilde{G}^{i_p}) + P_s(\widetilde{G}^{i_{p'}}) + P_s(\widetilde{G}^{i_q} + R\,z^{i_q}) \nonumber\\
&= \sum_{p=1\,p \neq p',q}^{H} P_s(\widetilde{G}^{i_p}) + P_s(\widetilde{G}^{i_{p'}} \cup (\widetilde{G}^{i_q} + R\,z^{i_q})) \nonumber\\
&\qquad + 2\int_{\widetilde{G}^{i_{p'}}} \int_{\widetilde{G}^{i_q} + R\,z^{i_q}} \frac{dx\,dy}{|x-y|^{N+s}} \nonumber
\end{align}
for any $q \in \{1,\cdots,\,H\}$ with $q \neq p'$ and sufficiently large $R>1$. By repeating the same argument finite times for the rest of the sets $\{\widetilde{G}^{i_p}\}_{p=1,\,p \neq p',q}^{H}$ with sufficiently large $R>1$, we obtain the inequality that
\begin{align}\label{keyEstimateReductionOneElement02Asymptotic}
\sum_{p=1}^{H} P_s(\widetilde{G}^{i_p}) &\geq P_s\left( \widetilde{G}^{i_{p'}} \cup \bigcup_{p=1,\,p \neq p'}^{H}\left( \widetilde{G}^{i_p} + R\,z^{i_p} \right) \right) \nonumber\\
&\qquad + \sum_{p=1,\,p \neq p'}^{H} \int_{\widetilde{G}^{i_{p'}}} \int_{\widetilde{G}^{i_p} + R\,z^{i_p}} \frac{dx\,dy}{|x-y|^{N+s}}.
\end{align}
Since $\widetilde{G}^{i_{p'}} \cup \bigcup_{p=1,\,p \neq p'}^{H}\left( \widetilde{G}^{i_p} + R\,z^{i_p} \right)$ are the union of disjoint sets, we have, from \eqref{volumeIdentityReorganizedAsymptotic}, that
\begin{equation}\nonumber
\left| \widetilde{G}^{i_{p'}} \cup \bigcup_{p=1,\,p \neq p'}^{H}\left( \widetilde{G}^{i_p} + R\,z^{i_p} \right)\right| = \sum_{p=1}^{H} |\widetilde{G}^{i_{p}}| = |B_1|.
\end{equation}
Thus, from \eqref{reductionFiniteElementsMinimizerAsymptotic}, \eqref{keyEstimateReductionOneElement02Asymptotic}, and the minimality of $F_n$, we obtain
\begin{align}
\sum_{p=1,\,p \neq p'}^{H} \int_{\widetilde{G}^{i_{p'}}} \int_{\widetilde{G}^{i_p} + R\,z^{i_p}} \frac{dx\,dy}{|x-y|^{N+s}} + P_s(B_1) &\leq \sum_{p=1,\,p \neq p'}^{H} \int_{\widetilde{G}^{i_{p'}}} \int_{\widetilde{G}^{i_p} + R\,z^{i_p}} \frac{dx\,dy}{|x-y|^{N+s}} \nonumber\\
&\qquad + P_s\left( \widetilde{G}^{i_{p'}} \cup \bigcup_{p=1,\,p \neq p'}^{H}\left( \widetilde{G}^{i_p} + R\,z^{i_p} \right) \right) \nonumber\\
&\leq P_s(B_1) \nonumber
\end{align}
and it follows that
\begin{equation}\nonumber
\sum_{p=1,\,p \neq p'}^{H} \int_{\widetilde{G}^{i_{p'}}} \int_{\widetilde{G}^{i_p} + R\,z^{i_p}} \frac{dx\,dy}{|x-y|^{N+s}} \leq 0
\end{equation}
for large $R>1$. Since each term of the sum is non-negative and $|\widetilde{G}^{i_{p'}}|>0$, we conclude that $|\widetilde{G}^{i_p}| = 0$ for all $p \neq p'$.
Therefore, taking into account all of the above arguments, we may conclude that there exist a set $G' \subset \mathR^N$ and points $\{z'_n\}_{n\in\mathN} \subset \mathR^N$ such that, up to extracting a subsequence, we have
\begin{equation}\nonumber
\chi_{F_n - z'_n} \xrightarrow[n \to \infty]{} \chi_{G'} \quad \text{in $L^1_{loc}$}, \quad |G'|=|B_1|.
\end{equation}
From Brezis-Lieb lemma in \cite{BrLi} and the fact that $|G'| = |B_1|$, we obtain that the convergence
\begin{equation}\nonumber
\chi_{F_n - z'_n} \xrightarrow[n \to \infty]{} \chi_{G'} \quad \text{in $L^1_{loc}$}
\end{equation}
holds in $L^1$ sense. As a consequence, by applying the $\Gamma$-convergence result for the energy $\widehat{\capE}_{s,g}$ as shown in Proposition \ref{propositionGammaConvergenceNonlocalEnergy}, we obtain that $G'$ is a minimizer of the nonlocal perimeter $P_s$, up to translations, because each element of $\{F_n\}_{n}$ is a minimizer of $\widehat{\capE}_{s,g}$. Thus, from the isoperimetric inequality, we conclude that $G'$ coincides with the unit ball up to negligible sets. Finally, we may repeat the above argument for any subsequence of $\{F_n\}_{n\in\mathN}$ and therefore, we conclude that Theorem \ref{theoremAsympMiniLargeVolume} is valid.
\end{proof}
\begin{remark}
We mention that we are not able to obtain a better convergence of minimizers for Problem \eqref{minimizationScalingModifiedProbelm} than $L^1$-convergence. In general, once we have the $L^1$-convergence and uniform density estimate for minimizers, we can obtain the Hausdorff (possibly $C^1$) convergence of the boundaries of the minimizers to the boundary of the unit ball (see, for instance, \cite{GoNo}). However, we do not know whether the density estimates of minimizers for Problem \eqref{minimizationScalingModifiedProbelm} are valid, while we have the uniform density estimates of minimizers for Problem \eqref{minimizationGeneralizedFunctional}, as shown in Lemma \ref{lemmaUniformDensity}. We might hope that, under some stronger assumptions on $g$ (for instance, some control of the gradient of $g$), the uniform density estimates of minimizers for Problem \eqref{minimizationScalingModifiedProbelm} could be valid.
\end{remark}
|
2024-02-18T23:40:24.485Z
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2021-12-30T02:25:37.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14505","language":"en","timestamp":1640831137000,"url":"https:\/\/arxiv.org\/abs\/2112.14505","yymm":"2112"}
|
proofpile-arXiv_000-10034
|
{"provenance":"002.jsonl.gz:10035"}
| null | null |
\section{Introduction}
With the development of information technology, most of the engineering and science records are currently stored in computers with the electronic forms, such as earthquake records \cite{1ogata1981lewis}, medical records \cite{2johnson2016mimic}, public safety records \cite{3mohler2018improving}, failure records \cite{4zhang2020survival}, and so on. Meanwhile, with the increase of internet applications, these diverse applications also lead to a large number of records, like IPTV records \cite{5luo2014you} and social network records \cite{6zhao2015seismic}. In general, these records are in the form of asynchronous sequence data, which contain the event occurring time and type of event.
These event sequence data contain a lot of valuable knowledge, which can be used to understand the relationship between events, predict possible future events, as well as the time of occurrence, and so on. How to mine the knowledge from these asynchronous sequences is a subject worthy of continued attention, and one of the most widely used methods is the point process model \cite{7daley2007introduction}, and the most commonly used model in the point process model is the Hawkes process model \cite{8hawkes1971spectra}.
In previous years, because the Hawkes process with self-excited characteristics can well model the trigger mode between events, the traditional Hawkes process is widely applied in many fields. For instance, Ogata utilizes space-time Hawkes process to analyze the earthquake and corresponding aftershock \cite{1ogata1981lewis}. Zhao et al. make use of the Hawkes process to predict the popularity of tweets on Internet \cite{6zhao2015seismic}. Reynaud-Bouret et al. provide a new way to detect the distances between genomic events on the DNA (Deoxyribonucleic Acid) sequences \cite{9reynaud2010adaptive}. Kobayashi and Lamhbiotte \cite{10kobayashi2016tideh} present a new framework of the time-dependent Hawkes process, whose impact functions are time-dependent. They make use of this model to reveal the periodic characteristics of Twitter reposting. Xu et al. utilize the Hawkes process to uncover the Granger causality between users choosing to watch TV programs \cite{11xu2016learning}. Zhou et al. present ADMM (Alternating Direction Method of Multipliers)-based algorithm to train the Hawkes process to find out the relationship between the social media user \cite{12zhou2013learning}.
Unfortunately, there are two critical inherent shortcomings for the traditional Hawkes process, one shortcoming is that the traditional Hawkes process ignores the influence of mutual inhibition between events, which does not conform to the actual situation to a certain extent. Another shortcoming is the lack of strong nonlinear fitting capability in the traditional Hawkes process, which also limits the expressive ability of the model.
Therefore, to mitigate the above problem, and with the development of neural networks and deep learning, also due to the ability to effectively model sequence data of recurrent neural network (RNN), the Hawkes process models based on RNN are proposed. For example, Du et al. embed sequence data information (including time-stamp and event type) into RNN and propose recurrent marked temporal point processes (RMTPP) to model the conditional intensity function considering the historical nonlinear dependence \cite{13du2016recurrent}. Similar to RMTPP, Mei et al. propose a continuous-time LSTM (Long Short-Term Memory) to simulate the conditional intensity of point processes, which is called neural Hawkes processes \cite{14mei2017neural}. In this continuous neural Hawkes process, the effect of the previous event is decreasing with time. And in \cite{15xiao2017modeling} two RNNs are used to model the conditional intensity function, one is for processing time-stamp information and the other for processing historical event information.
The goals of these models are effectively model the sequence data, and accurately predict the types of events that will occur in the future and the moments of event occurrence. For instance, for electronic medical records, we can predict the types and times of the next attack based on the patient’s medical history, and provide more effective and timely help for the patient's treatment. We can also predict the next possible failure time and type for a large-scale production system based on the failure sequence, and perform maintenance and prevention in advance to improve safety and economic benefits for the production process.
However, these models based on RNN inevitably inherit the disadvantages of RNN. For example, for certain chronic sequelae, it may take a long time for patients to develop symptoms. These sequelae may have obvious long-term characteristics, such as diabetes, cancer, and other chronic diseases. These RNN-based models are difficult to describe the long-term dependence between events with a long sequence distance \cite{16bengio1994learning}, while the ideal point process model should be able to attack these problems. Moreover, when training RNN-based models, problems such as gradient vanishing and explosion \cite{17pascanu2013difficulty} often appear, this will necessarily impact the performance of the model.
Fortunately, RNN is not the only choice of sequential modeling now, with the advent of self-attention mechanism \cite{18DBLP:journals/corr/BahdanauCB14}, sequence models in literatures have become vast and are growing rapidly, among them, the most striking one, transformer \cite{19vaswani2017attention}, are developed and applied to speech recognition \cite{20yu2016automatic}, machine translation \cite{21koehn2009statistical}, video recognition \cite{22girdhar2019video} and other fields. Inspired by these successes, Zhang et al. present the self-attention Hawkes process \cite{23zhang2020self}, furthermore, Zuo et al. propose transformer Hawkes process based on the attention mechanism and encoder structure in transformer \cite{24DBLP:conf/icml/ZuoJLZZ20}.
But for the existing attention Hawkes model, the input feeding into the transformer is a simple stacking of the event encoding and the temporal position encoding, while we tend to extract the existing underlying intrinsic information in the event type and the time stamp in the event sequence. Therefore, inspired by the relative encoding Hawkes process \cite{25dai2019transformer,26al2019character}, we modify the traditional dot product attention and propose a new dot product attention mechanism, which additionally introduces temporal encoding as input to the attention mechanism, we dub this model as Temporal Augmented Attention Transformer Hawkes Process (TAA-THP).
Our paper is organized as follows. In section 2, we systematically introduce the related work of our methods, and next, we describe our model in detail, which is related to the modified dot-product temporal augmented attention structure, and the acquisition of conditional intensity function based on the hidden representation of events sequences. In section 4, we instruct the prediction method and the objective function of TAA-THP. Then, we conduct the experiments to confirm the effectiveness of the TAA-THP model, we also perform the ablation study to figure out the impact of the introduction of temporal attention. At last, we summarize our research work and look forward to future research directions.
\section{Related work}
\subsection{Traditional Hawkes process}
Hawkes process is one of the most widely applied point process models in sequential modeling, which is proposed by Alan Hawkes in 1971 \cite{8hawkes1971spectra}. The general paradigm of Hawkes process is shown as Eq. \ref{eq1}:
\begin{equation}
\label{eq1}
\lambda (t) = \mu (t) + \sum\limits_{i:t_i < t} {\phi (t - t_i )}
\end{equation}
where $\mu (t)$ is the background intensity function, describes the base possibility of event occurrence over time, $\phi (t)$ is the impact function, which is used to represents the historical event impact, and $\sum\limits_{i:t_i < t} {\phi (t - t_i )} $ records all the impact of history events to this current instant $t$. The traditional Hawkes process model in Eq. \ref{eq1} ignores the inhibition effect between the events, and until now, there are still countless variants based on this model in various application fields.
Mohler proposes a novel modulated Hawkes process, to quickly identify risks to protect communities from a range of social harm events, such as crime, drug abuse, traffic accident and medical emergencies \cite{3mohler2018improving}. Zhang et al. \cite{4zhang2020survival} use the Weibull background intensity instead of the constant background intensity, which can better model the trend of failure occurrence over time, Zhang et al. verify the effectiveness of the Weibull-Hawkes process, and reveal the trend of background probability of failures in the compressor station over time and the Granger causality between the failures. Xu et al. \cite{5luo2014you} use the non-parametric Hawkes process model, and a corresponding learning algorithm to obtain the Granger causality on of IPTV users on watching the program.
Kobayashi and Lamhbiotte use the time-dependent Hawkes process to prove that Twitter reposts have a clear tidal effect \cite{10kobayashi2016tideh}. Yang et al. \cite{27yang2017online} come up with an online learning method of Hawkes process based on the nonparametric method. In 2018, Alan reviews and summarizes the application of the Hawkes process in the financial field \cite{28hawkes2018hawkes}. Hansen et al. show powerful expressive ability in describing the neural excitation process of Hawkes process in neuroscience with the Lasso (Least absolute shrinkage and selection operator) method \cite{29hansen2015lasso}.
\subsection{Neural Hawkes Process}
Du et al. present the RMTPP models \cite{12zhou2013learning}, which can learn the history effect through RNN, including history event type and time-stamp. This model abandons the restrictions of the Hawkes process and other point process models firstly and achieves improvements than the traditional Hawkes process. Xiao et al. \cite{15xiao2017modeling} utilize two RNN to model the event sequence, one of them is used to model the background intensity and the other is used to model the impact of historical events. Mei and Eisner \cite{14mei2017neural} propose a new LSTM, i.e., the continuous-time LSTM, whose state can decay with time, and based on this LSTM, they come up with the neural Hawkes process to model the asynchronous event sequence.
With the mature self-attention mechanism, the self-attention-based neural Hawkes processes are proposed, the self-attention Hawkes process is the first model which utilizes the self-attention mechanism \cite{23zhang2020self}. Based on the success of the transformer, Zuo et al. utilize the encoder structure in the transformer to get the hidden representation of sequences data, and then convert it to the continuous conditional intensity functions \cite{24DBLP:conf/icml/ZuoJLZZ20}. In recent researches about the neural Hawkes process, the self-attention Hawkes process and transformer Hawkes process achieve great success, which are on the foundation of self-attention mechanism, thus we also focus on the self-attention in our proposed TAA-THP.
\subsection{Transformer models}
In 2017, Vaswani et al. propose the transformer model \cite{19vaswani2017attention}, which makes full use of self-attention mechanism \cite{18DBLP:journals/corr/BahdanauCB14}, and achieves great success in sequence learning. The architecture of transformer is shown as Fig.\ref{fig1}.
\begin{figure}[!htbp]
\centering
\includegraphics[scale=1]{transformer.pdf}
\caption{The architecture of transformer model}
\label{fig1}
\end{figure}
The transformer model is made up of multiple encoder and decoder modules and there are the only self-attention mechanism and position-wise-feed-forward structure without using the recurrence neural network architecture. Meanwhile, recent researches shows that recurrent learning may be more important beyond imagination in sequential learning, thus, Dehghani et al. propose the universal transformer \cite{30DBLP:conf/iclr/DehghaniGVUK19}, which combine the recurrent learning and self-attention mechanism, moreover, in order to better allocate model computing resources, the Adaptive Computation Time (ACT) mechanism \cite{31graves2016adaptive} is introduced into models, then, this model achieves better results than transformers and also achieves Turing completeness. Dai et al. come up with a new transformer called Transformer-XL \cite{25dai2019transformer}, which consists of a segment-level recurrence mechanism and a novel positional encoding scheme, not only does the model perform better, the learning speed is also faster than previous models.
\section{Proposed Model}
In this section, we are going to introduce the details of the temporal attention augmented transformer Hawkes process, including its model structure and corresponding continuous conditional intensity function. We list the notations used in this paper, which is shown in Table 1.
\begin{table}[!htbp]
\centering
\caption{Nomenclature}
\label{tb1}
\begin{tabular}{ccc}
\hline
Symbols & Description & Size\\ \hline
$S_e$ & The dataset of sequences & / \\
$s_n$ & The $n$-th sequence & / \\
$I_n$ & The length of the $n$-th sequence & $\mathbb{N}^+$ \\
$N$ & The total number of sequences& $\mathbb{N}^+$ \\
$C$ & The total number of type of events in sequences & $\mathbb{N}^+$ \\
$t_i$ & The time stamp of \textit{i}-th event & $\mathbb{R}$ \\
$c_i$ & The event type of \textit{i}-th event & $\mathbb{N}^+$ \\
$D$ & The model dimension of transformer & $\mathbb{N}^+$ \\
$D_H$ & The dimension of position-wise-feed-forward part of transformer & $\mathbb{N}^+$ \\
$\bm{x}$ & The temporal position encoding & $\mathbb{R}^D$ \\
$\bm{E}$ & The embedding matrix of event type & $\mathbb{R}^{D\times C}$ \\
$\bm{c_i}$ & one-hot encoding vector of each event & $\mathbb{R}^C$ \\
$L$ & The number of multi-head attention & $\mathbb{R}$ \\
$\left\{ {{\mathbf{A}}_l } \right\}_{l = 1}^L $ & The attention variable of multi-head attention & ${\mathbb{R}}^{D_V}$ \\
$D_K$ & The dimension of query and key vector & $\mathbb{N}^+$ \\
$D_V$ & The dimension of value vector & $\mathbb{N}^+$ \\
$\bm{S}$ & State & $\mathbb{R}^{D}$ \\
$\left\{ {{\bm{Q}}_l } \right\}_{l = 1}^L $ & The query variable of multi-head attention & $
\mathbb{R}^{I_n \times D_K } $ \\
$\left\{ {{\bm{K}}_l } \right\}_{l = 1}^L $ & The key variable of multi-head attention & $
\mathbb{R}^{I_n \times D_K } $ \\
$\left\{ {{\bm{V}}_l } \right\}_{l = 1}^L $ & The value variable of multi-head attention & $
\mathbb{R}^{I_n \times D_V } $ \\
$\left\{ {{\bm{W}}_Q^l } \right\}_{l = 1}^L$ & The query matrix of multi-head attention & $\mathbb{R}^{D \times D_K } $ \\
$\left\{ {{\bm{W}}_K^l } \right\}_{l = 1}^L$ & The key matrix of multi-head attention & $\mathbb{R}^{D \times D_K } $ \\
$\left\{ {{\bm{W}}_V^l } \right\}_{l = 1}^L$ & The value matrix of multi-head attention & $\mathbb{R}^{D \times D_V } $ \\
$\left\{ {{\bm{W}}_{Tem}^l } \right\}_{l = 1}^L$ & The temporal attention variable of multi-head attention & $\mathbb{R}^{D \times D_K } $\\
$\left\{ {{\bm{b}}_{lq} } \right\}_{l = 1}^L $ & The bias of dot-product attention & $\mathbb{R}^{D_K } $ \\
$\left\{ {{\bm{b}}_{lq} } \right\}_{l = 1}^L $ & The bias of temporal attention & $\mathbb{R}^{D_K } $ \\
$\bm{W}_multi$ & The aggregation matrix of multi-head attention & $\mathbb{R}^{L{D_V}\times D } $ \\
$\left\{ {{\bm{W}}_i^{FC} ,{\bm{b}}_i^{FC} } \right\}_{i = 1}^4$
& The parameters of fully connected neural network & / \\
$\bm{H}$ & The hidden representation of sequence & $\mathbb{R}^{I_n\times D}$ \\
$\bm{h}(t_i)$ & The hidden representation of \textit{i}-th event & $\mathbb{R}^{D}$ \\
$\mathcal{H}_t$ & The previous history at time \textit{t} & \\
$b_c$ & The background intensity of event type \textit{c} & $\mathbb{R}$ \\
${\bm{w}}_{\alpha _c } $ & The weight of continuous parameter of conditional intensity function of event type & $\mathbb{R}^{1\times D}$ \\
${\bm{w}}_c^T $& Historical weight parameter of conditional intensity function of event type & $\mathbb{R}^{1\times D}$ \\
$\bm{W}_{time}$ & The prediction parameter of time-stamp & $\mathbb{R}^{1\times D}$ \\
$\bm{W}_{type}$ &The prediction parameter of event type & $\mathbb{R}^{C\times D}$ \\ \hline
\end{tabular}
\end{table}
\subsection{Temporal Attention Augmented Transformer Hawkes Process}
Generally speaking, transformer-based Hawkes process model utilizes the encoder structure of different kinds of the transformer to get the hidden representation of event sequence. Assuming there are sequences in the dataset $S_e$ , and the $n$-th sequence is $s_n = \{ t_i ,c_i \} _{i = 1}^{I_n } $ , whose length is $I_n$ . Each pair in $S_n$ is composed with two parts, e.g., $t_i$ and $c_i$ . Among them, $c_i$ is the type of event that occurred and $t_i$ is the corresponding time-stamp. Following point process theory, time-stamp is the event occurring instant in tandem along the time axis, which is consistent with the position encoding of the transformer-based model for asynchronous sequences. Thus, we can make use of this general method to encode the timestamp as a positional encoding, similar to other transformer-based models \cite{19vaswani2017attention}, and \cite{24DBLP:conf/icml/ZuoJLZZ20}, which is shown in Eq.\ref{eq2}:
\begin{equation}
\label{eq2}
[{\bm{x}}(t_i )]_j = \left\{ {\begin{array}{*{20}c}
{\cos \left( {t_i /10000^{\frac{{j - 1}}{D}} } \right),{\rm{if}}\:j\:{\rm{is}}\:{\rm{odd}},} \\
{\sin \left( {t_i /10000^{\frac{j}{D}} } \right),{\rm{if}}\:j\:{\rm{is}}\:{\rm{even}}.} \\
\end{array}} \right.
\end{equation}
where $\bm{x}$ is the position encoding for transformer model, throughout the latter part of the paper, we will call it as temporal encoding, and $D$ is the dimension of transformer model. For the event encoding, we utilize one-hot encoding vector of each event ${\mathbf{c}}_i \in \mathbb{R}^C $, and embedding matrix ${\mathbf{E}} \in \mathbb{R}^{D \times C}$ to get it. In this way, we can get ${\bm{X}}^T$ and $({\bm{EC}}_n )^T $, which are the corresponding temporal encoding and event encoding of the sequence, where ${\bm{X}} = \{ {\bm{x}}(t_1 ),{\bm{x}}(t_2 ),...,{\bm{x}}(t_{I_n } )\} \in \mathbb{R}^{D \times I_n }$ and $
{\bm{C}}_n = [{\bm{c}}_1 ,{\bm{c}}_2 ,...,{\bm{c}}_{I_n } ] \in \mathbb{R}^{C \times I_n } $ . So as to obtain the hidden representation of event sequence, we input the ${\bm{X}}^T$ and $({\bm{EC}}_n )^T $ to the model we proposed, whose schematic diagram is shown in Fig.\ref{fig2}.
\begin{figure}[!htbp]
\centering
\includegraphics[scale=1]{2.pdf}
\caption{The schematic diagram of Temporal Attention Augmented Transformer Hawkes Process (TAA-THP), in TAA-THP, we modify the traditional attention mechanism and highlight the use of temporal encoding information of event sequence.}
\label{fig2}
\end{figure}
As shown in Fig.\ref{fig2}, in initial, event and temporal encoding are inputted to the model, and the input of each encoding layer is the output of the previous layer plus temporal encoding. Which means in the first layer of TAA-THP, we can calculate . Meanwhile, we modify the multi-head self-attention and add temporal encoding as an additional auxiliary input. The tricks, such as layer normalization, residual connection, and dropout, are also involved in TAA-THP, to utilize the information of temporal encoding iteratively, the traditional dot-product multi-head self-attention in encoding layers are modified by us, the new dot-product multi-head self-attention with temporal augmented attention is written as follows:
\begin{equation}
\label{eq3}
{\bm{A}}_l = Softmax\left[ {mask\left( {\frac{{\left( {{\bm{Q}}_l + {\bm{b}}_{lq} } \right){\bm{K}}_l^T + \left( {{\bm{Q}}_l + {\bm{b}}_{lt} } \right)\left( {{\bm{X}}^T {\bm{W}}_{Tem}^l } \right)^T }}
{{\sqrt {D_K } }}} \right)} \right]{\bm{V}}_l
\end{equation}
In Eq.3, ${\bm{b}}_{lq}$ and ${\bm{b}}_{lq }$ are the bias vectors of query matrix , these biases will help the model to calculate the attention more flexible. The term $\left( {{\mathbf{Q}}_l + {\mathbf{b}}_{lq} } \right){\mathbf{K}}_l^T $ is the modified dot-product attention operation, which just adds a bias term in traditional dot-product attention operation. We focus on another term, i.e., $\left( {{\mathbf{Q}}_l + {\mathbf{b}}_{lt} } \right)\left( {{\mathbf{X}}^T {\mathbf{W}}_{Tem}^l } \right)^T$ , this term re-introduces the temporal information into the dot-product, rather than just add the temporal encoding to $\bm{S}$ . In this term, ${\bm{X}}^T {\bm{W}}_{Tem}^l$ is the linear transformation of temporal encoding ${\bm{X}}^T$ ,${\bm{W}}_{Tem}^l \in \mathbb{R}^{D \times D_K } $ is the linear transformation matrix of the $l$-th head attention, and for the $l$-th query, key and value matrix are obtained from Eq.\ref{eq4}:
\begin{equation}
\label{eq4}
{\bm{Q}}_l = {\bm{SW}}_Q^l ,{\bm{K}}_l = {\bm{SW}}_K^l ,{\bm{V}}_l = {\bm{SW}}_V^l
\end{equation}
where $\bm{S}$ , as described in following Algorithm 1, is the input of each encoding layer.${\bm{W}}_Q^l ,{\bm{W}}_K^l \in \mathbb{R}^{D \times D_K }$ and ${\bm{W}}_V \in \mathbb{R}^{D \times D_V }$ are a linear transformation of $\bm{S}$ . And in general, for our proposed transformer model TAA-THP, we impose the constraint $D_K = D_V $, which is similar to the usual conventions. Meanwhile, in order to shield the impact of future events on current events (one-way characteristics of time) we utilize the masked self-attention mechanism similar to \cite{18DBLP:journals/corr/BahdanauCB14}, which set the elements above the main diagonal of the matrix to be negative infinity, the function $mask( \cdot )$ is used to ensure that future events in the matrix will not affect the attention weights of current events.
By means of Eq.\ref{eq3} and Eq.\ref{eq4}, we can obtain , which is the $l$-th attention matrix. To improve the expressive ability of the model, most of the existing attention mechanism models use multi-head attention. Assuming there are $L$ self-attention heads, then, we can get $L$ outputs ${\bm{A}}_1 ,{\bm{A}}_2 ,...,{\bm{A}}_L $ based on Eq.\ref{eq3} and Eq.\ref{eq4}, the corresponding parameters are $\left\{ {{\bm{W}}_Q^l ,{\bm{W}}_K^l ,{\bm{W}}_{Tem}^l ,{\bm{W}}_V^l } \right\}_{l = 1}^L $ . The formula for combining multi-head attention into one output is described as Eq.\ref{eq5}:
\begin{equation}
\label{eq5}
{\mathbf{A}} = \left[ {{\mathbf{A}}_1 ,{\mathbf{A}}_2 ,...,{\mathbf{A}}_L } \right]{\mathbf{W}}_{multi}
\end{equation}
where ${\bm{W}}_{multi} \in \mathbb{R}^{LD_V \times D}$ is the aggregation matrix. In the position-wise-feed-forward part of each encoding layer, a convolutional neural network (CNN) module is added between the two fully connected layers, the CNN implement the following calculation:
\begin{equation}
\label{eq6}
{\bm{S}} = {\rm{CNN}}\left( {{\rm{ReLU}}({\bm{AW}}_1^{FC} + {\bm{b}}_1 )} \right){\bm{W}}_2^{FC} + {\bm{b}}_2
\end{equation}
The CNN module consists of a one-dimension convolutional layer, a nonlinear activation function (ReLU function) and a one-dimension max-pooling layer. Based on the thumb rule and the cross-validation results, we set the convolution kernel have 1 input channel, 4 output channels, the size of the convolution kernel is 3, the stride is 2, and the padding is 0, and the size and stride of max-pooling layer are 2.
Our aim for incorporating the CNN module is to improve the local perception of the model so that the model can better learn the dependencies between events. In summary, the algorithm for calculating the implicit representation of the event sequence through asynchronous event sequence data is shown in Algorithm \ref{alg1}.
\begin{algorithm}
\renewcommand{\algorithmicrequire}{\textbf{Input:}}
\renewcommand{\algorithmicensure}{\textbf{Output:}}
\caption{Temporal Attention Augmented Transformer Hawkes Process (TAA-THP)}
\label{alg1}
\begin{algorithmic}[2]
\REQUIRE The number of encoding layers: $n$, event-type encoding $({\bm{EC}}_n )^T$
. Temporal encoding (position encoding) ${\bm{X}}^T$ .
\ENSURE Hidden representation (State) $ {\mathbf{H}} \in \mathbb{R}^{D \times I_N } $of event sequence
\STATE Initialize state $ {\bm{S}} \leftarrow ({\bm{EC}}_n )^T $ .
\STATE \textbf{for} $i$ \textbf{in} $n$ ,
\STATE ${\bm{S}} \leftarrow {\bm{S}} + {{ }}{\bm{X}}^T $
,
\STATE ${\bm{S}} \leftarrow output\;of\;{\rm{the}}\;i{\rm{ - th }}\;{\rm{encoding\;layer}}({\bm{S}},{\bm{X}}^T )$
\STATE \textbf{return} ${\bm{H}} \leftarrow {\bm{S}}$
\end{algorithmic}
\end{algorithm}
And inspired by \cite{24DBLP:conf/icml/ZuoJLZZ20} and \cite{32wang2019language}, using Algorithm \ref{alg1}, we will get the output $\bm{H}$ , and $\bm{H}$ will be utilized as the input of a postprocessing part, which consists of a fully connected layer, RNN layer, and another fully connected layer. The structure of the postprocessing part is summed up in Eq. \ref{eq7}:
\begin{equation}
\label{eq7}
{\bm{H}} = {\rm{RNN}}\left( {{\rm{ReLU}}({\bm{HW}}_3^{FC} + {\bm{b}}_3 )} \right){\mathbf{W}}_4^{FC} + {\bm{b}}_4
\end{equation}
Based on cross-validation and other related researches, we set the network dimension of RNN to zero (which means no postprocessing part) or 64. The types of RNN networks that can be used include but are not limited to LSTM and GRU, after $\bm{H}$ passes through this part, the dimension of $\bm{H}$ remains unchanged. Whether the post-processing part in the proposed TAA-THP can improve the performance of the model depends on the characteristics of the dataset.
\subsection{Conditional Intensity Function}
The relationship between the probability of occurrence and occurring time of each type of event can be described by its corresponding conditional intensity function, after getting the hidden representation of sequence, we can utilize it to calculate the conditional intensity function $\lambda _c (t\left| {\mathcal{H}_t } \right.)$, where $\mathcal{H}_t = (t_j ,c_j ):t_j < t$, which is the history before time , we adopt the similar approach in [24] to calculate the conditional intensity function, which is shown as Eq.\ref{eq8}:
\begin{equation}
\label{eq8}
\lambda _c (t\left| {\mathcal{H}_t } \right.) = f(b_c + \alpha _c \frac{{t - t_i }}
{{t_i }} + {\bm{w}}_c^T {\bm{h}}(t_i ))
\end{equation}
where $t_i$ is the time-stamp of the last event which its occurrence time is the closest to $t$, and $\bm{h}_t$ is the hidden representation corresponding to this event. The overall background intensity is $b_c + \alpha_c {\bm{h}}(t_i )\frac{{t - t_i }}
{{t_i }}$ , here $t \in [t_i ,t_{i + 1} )$ , $b_c$ is the constant background intensity, $\alpha_c$ is the coefficient of time-dependent background intensity. In \cite{24DBLP:conf/icml/ZuoJLZZ20}, the $\alpha_c$ is fixed to -0.1, which indicates that when no event occurs, the conditional intensity function will inevitably decay over time. This assumption does not completely conform to reality and limits the expressive ability of the model. In TAA-THP, to make up for this drawback, we let $\alpha_c = {\bm{w}}_{\alpha_c } {\bm{h}}(t_i )$, i.e., the background intensity change with the past history, which improves the model's ability to fit the conditional intensity function. Compared with the traditional Hawkes process model, the time-dependent background intensity can better fit the basic probability of event occurrence. $f(x) = \frac{1} {\beta }\log (1 + e^{\beta x} ) $ is softplus function, and $\beta$ is the softness parameter of softplus function. Softplus function is an improved version of the ReLU function, compared with ReLU, this nonlinear function is smoother, and the activity of most neurons is guaranteed, which makes the model learning ability stronger.
Finally, the overall conditional intensity function is the sum of the conditional intensity functions of all types of events, as shown in Eq.\ref{eq9}:
\begin{equation}
\label{eq9}
\lambda (t\left| {\mathcal{H}_t } \right.){\rm{ = }}\sum\limits_{c = 1}^C {\lambda _c (t\left| {\mathcal{H}_t } \right.)}
\end{equation}
\section{Prediction Task and Model Training}
In general, the objective function of the point process model is constructed from the likelihood function \cite{7daley2007introduction}, more specifically, for a sequence $s_n = \{ t_i ,c_i \} _{i = 1}^{I_n }$ , the log-likelihood function of $s_n$ is shown as Eq.\ref{eq10}:
\begin{equation}
\label{eq10}
L(s_n ) = \sum\limits_{i = 1}^{I_n } {\log } \,\lambda (\left. {t_i } \right|\mathcal{H}_i ) - \int_{t_1 }^{t_{I_n } } {\lambda (\left. t \right|\mathcal{H}_t )} dt
\end{equation}
and assuming there are $N$ sequences in the dataset, then the model parameters can be solved by the maximum log-likelihood principle:
\[
\max \sum\nolimits_{n = 1}^N {L(s_n )}
\]
However, let $\Lambda = \int_{t_1 }^{t_{I_n } } {\lambda \left( {\left. t \right|\mathcal{H}_t } \right)} dt
$ , due to $\lambda (\left. t \right|\mathcal{H}_t )$ is derived from the neural network, the closed-form of is hard to get, and we have two approximate methods to solve it, Monte Carlo sampling method \cite{33robert2013monte} and the numerical analysis method \cite{34stoer2013introduction}. The Monte Carlo sampling method \cite{33robert2013monte}, the approximate value of $\Lambda$ is shown in Eq.\ref{eq11}:
\begin{equation}
\label{eq11}
\hat \Lambda _{MC} = \sum\limits_{i = 2}^L {\left( {t_i - t_{i - 1} } \right)} (\frac{1}
{M}\sum\limits_{m = 1}^M {\lambda (u_m )} )
\end{equation}
where $u_m$ is sampled from the uniform distribution $U(t_{i - 1} ,t_i )$ . It is worth noting that $\hat \Lambda _{MC} $ calculated by this method is an unbiased estimate of $\Lambda$. And for the numerical analysis method \cite{34stoer2013introduction}, based on the trapezoidal rule, we can get the estimate of $\Lambda$ as shown in Eq.\ref{eq12}:
\begin{equation}
\label{eq12}
\hat \Lambda _{NA} = \sum\limits_{i = 2}^{I_n } {\frac{{t_i - t_{i - 1} }}
{2}} (\lambda (\left. {t_i } \right|\mathcal{H}_i ) + \lambda (\left. {t_{i - 1} } \right|\mathcal{H}_{i - 1} ))
\end{equation}
The computational complexity of the numerical analysis method is significantly less than the Monte Carlo sampling method, but the error is larger. It is a reasonable alternative when the accuracy requirements are not high.
Below we give a more scrutinizing discussion. Recall that after obtaining the conditional intensity function of each type of event through Eq.\ref{eq8}, we can predict the occurrence instant of events in the future. Traditionally \cite{7daley2007introduction}, the dominant paradigm to calculate the conditional intensity function can be gotten from Eq.\ref{eq13}:
\begin{equation}
\label{eq13}
\begin{array}{l}
p(\left. t \right|{\cal H}_t ) = \lambda (\left. t \right|{\cal H}_t )\exp t( - \int_{t_i }^t {\lambda (\left. s \right|{\cal H}_t )} ds) \\
\hat t_{i + 1} = \int_{t_i }^\infty {t\cdot p(\left. t \right|{\cal H}_t )dt} \\
\hat c_{i + 1} = \mathop {\arg \max }\limits_c \frac{{\lambda _c (\hat t_{i + 1} |{\cal H}_{i + 1} )}}{{\lambda (\hat t_{i + 1} |{\cal H}_{i + 1} )}} \\
\end{array}
\end{equation}
However, this paradigms are flexible, but highly intractable, which have some limitations, for instance, $\lambda (\left. t \right|{\cal H}_t )$ is derived from the neural network, which is hard to get the closed-form solution of $\int {t\cdot p(\left. t \right|{\cal H}_t )dt} $ , moreover, we need to calculate the improper integral $\int_{t_i }^\infty {t\cdot p(\left. t \right|{\cal H}_t )dt} $ based on Monte Carlo sampling \cite{33robert2013monte}, generally speaking, the upper limit of integration can be replaced by a large enough number when we take advantage of Monte Carlo sampling from the uniform distribution to approximately calculate the integral, but this still has a larger error compared with the upper limit of integration being infinite, Monte Carlo sampling under the exponential distribution can take the upper limit of integration to infinity, while high computational complexity limits the utility of this approach. Therefore, we desire to use the powerful fitting ability of the neural network and the extracted hidden representation from the sequence of events, to predict the type and time of events that may occur in the future. The prediction formula is presented as Eq.\ref{eq14}:
\begin{equation}
\label{eq14}
\begin{array}{l}
\hat t_{i + 1} = {\bm{W}}_{time} {\bf{h}}(t_i ) \\
\widehat{\bf{p}}_{i + 1} = Softmax({\bm{W}}_{type} {\bf{h}}(t_i )) \\
\hat c_{i + 1} = \mathop {\arg \max }\limits_c {\bf{\hat p}}_{i + 1} (c) \\
\end{array}
\end{equation}
In addition, the prediction parameter, ${\bm{W}}_{time}$ and ${\bm{W}}_{type}$ , should also be optimized, we define the prediction loss of occurring time of each type of event and type of events as Eq.\ref{eq15} and Eq.\ref{eq16}:
\begin{equation}
\label{eq15}
L_{time} (s_n ) = \sum\nolimits_{i = 2}^{I_n } {(t_i - \hat t_i )^2 }
\end{equation}
\begin{equation}
\label{eq16}
L_{type} (s_n ) = \sum\nolimits_{i = 2}^{I_n } { - {\bf{c}}} _i^T \log (\widehat{\bf{p}}_i )
\end{equation}
Thus, the overall objective function of TAA-THP is shown as Eq.\ref{eq17}:
\begin{equation}
\label{eq17}
\min \sum\limits_{n = 1}^N { - L(s_n )} + \alpha _{type} L_{type} (s_n ) + \alpha _{time} L_{time} (s_n )
\end{equation}
where $\alpha _{type} $ and $\alpha _{time}$ are hyperparameters, which will help keep the stability of model
training, the objective function in Eq. 17, can be optimized by stochastic gradient descent algorithm
including but not limited to momentum method and ADAM (Adaptive moment estimation)\cite{35kingma2014adam},
we use ADAM optimizer with default hyperparameter i.e.,
we set learning rate=0.0001, betas are 0.9 and 0.999, eps =$10^8$, weight\_decay = 0.
\section{Experiments}
In this section, we are going to compare the model performance of TAA-THP and numerous baselines on a wide range of synthetic and real-life datasets, and figure out the effect of the introduction of temporal augmented attention. Firstly, we introduce the details of datasets and baselines, then, we compare the model performance on the foundation of experimental results, at last, we explore the effect of the presence or absence of temporal augmented attention on the performance of the model in the ablation study.
\subsection{Datasets}
In this subsection, we make use of two artificial datasets, and four real-world datasets of event sequence to conduct the experiments, Table 2 introduces the characteristics of each dataset:
\begin{table}[!htbp]
\centering
\label{tb2}
\caption{Characteristics of datasets used in experiments.}
\begin{tabular}{cccccc}
\hline
\multirow{2}{*}{Dataset} & \multirow{2}{*}{C} & \multicolumn{3}{c}{Sequence Length} & \multirow{2}{*}{Events} \\ \cline{3-5}
& & Min & Aver. & Max & \\ \hline
Synthetic & 5 & 20 & 60 & 100 & 602,697 \\
NeuralHawkes & 5 & 20 & 60 & 100 & 602,984 \\
Retweets & 3 & 50 & 109 & 264 & 2,173,533 \\
MIMIC-II & 75 & 2 & 4 & 33 & 2,419 \\
StackOverflow & 22 & 41 & 72 & 736 & 480,413 \\
Financial & 2 & 829 & 2074 & 3319 & 414,800 \\ \hline
\end{tabular}
\end{table}
\textbf{Synthetic and NeuralHawkes} \cite{14mei2017neural}: Mei generates two sets of artificial sequences based on the thinning algorithm, including Hawkes process model and continues-time LSTM neural Hawkes process model, the parameters are randomly sampled.
\textbf{Retweets} \cite{6zhao2015seismic}:This dataset contains a large number of sequences of tweets, every sequence includes an original tweet (the original content is some user post) and its following retweets. The time and label of the user of each retweet are recorded in the sequence, and labels are divided into three classes based on the number of their followers: “small”, “medium”, and “large”.
\textbf{StackOverflow} \cite{36leskovec2014snap}: StackOverflow is one of the most famous programming Q\&A websites. The website rewards users with various badges to encourage them to take part in community activities. It is worth noting that the same badge can be donated to the same user, and this dataset includes lots of users’ reward histories in the range of two years, each user’s history is treated as a sequence, and each event in sequences represents the acquisition of badge.
\textbf{Electrical Medical Records} \cite{2johnson2016mimic}: MIMIC-II dataset includes patients’ records of visitation to a hospital’s ICU during seven years. Each patient’s record is treated as an independent sequence, and each event contains the corresponding time-stamp and diagnosis of the patient.
\textbf{Financial Transactions} \cite{12zhou2013learning}: The financial dataset involves a lot of short-term transaction history of a stock in one day. The operation of each transaction is recorded as the event, and this dataset only has two kinds of events: “buy” and “sell”, and the unit of the time-stamp in milliseconds.
Based on the cross-validation results on validation dataset, we obtain the hyper-parameters of our model on different datasets, which are shown as Table 3.
\begin{table}[!htbp]
\centering
\label{tb3}
\caption{The hyper-parameters of Temporal Attention Augmented Transformer Hawkes Process}
\begin{tabular}{ccccccccc}
\hline
Dataset & Batch Size & $D$ & $D_H$ & $D_K=D_V$ & Heads of attention & Layers of model & $D_{\rm{RNN}}$ & Dropout \\ \hline
Synthetic & 16 & 64 & 256 & 64 & 3 & 3 & 64 & 0.1 \\
NeuralHawkes & 16 & 64 & 256 & 64 & 3 & 3 & 64 & 0.1 \\
Retweets & 16 & 64 & 256 & 64 & 3 & 3 & 64 & 0.1 \\
MIMIC-II & 1 & 128 & 256 & 256 & 5 & 5 & 0 & 0.1 \\
StackOverflow & 4 & 128 & 512 & 256 & 4 & 4 & 64 & 0.1 \\
Financial & 1 & 128 & 512 & 512 & 4 & 4 & 64 & 0.1 \\ \hline
\end{tabular}
\end{table}
\subsection{Baselines}
\textbf{RMTPP} \cite{12zhou2013learning}: Du et al. come up with a recurrent network point process model, which effectively models the event types and time-stamp in the sequence by embedding history to the vector.
\textbf{NHP} \cite{14mei2017neural}: Mei and Eisner firstly present a novel neural Hawkes process, which based on the continuous-time LSTM, the continuous-time LSTM has decay property of impact of historical events.
\textbf{SAHP} \cite{23zhang2020self}: Zhang et al. firstly come up with the sequence encoding through the self-attention mechanism, which utilizes the position encoding method to encode the time-stamp, and after the implicit representation of the sequence is obtained, the implicit representation of the sequence is used to calculate the model parameters of the Hawkes process.
\textbf{THP} \cite{24DBLP:conf/icml/ZuoJLZZ20}: Base on the performance improvement of the transformer, Zuo et al. propose the transformer Hawkes process, which achieves state-of-the-art performance. We can obtain the following experimental results to verify the performance of THP based on the model and the hyper-parameter they provide.
\subsection{Experimental results and comparison}
In this subsection, in order to validate the performance of TAA-THP, we compare the TAA-THP and baselines’ performance. First of all, we compare the loglikelihood value of TAA-THP and baselines on the test dataset, and for simplicity, we named it Loglike. The Loglike is the most basic model measurement for the point process model, which indicates how well the model fits in the data. In our experimental process, we set $\alpha_time $ and $\alpha_type $ in TAA-THP to zero. The loglike of baselines and TAA-THP on different test datasets are shown in Table 4.
\begin{table}[!htbp]
\centering
\caption{The value of log-likelihood function of different models on the test datasets.}
\begin{tabular}{cccccc}
\hline
Datasets & RMTPP & NHP & SAHP & THP & TAA-THP \\ \hline
Synthetic & \textbackslash{} & -1.33 & 0.520 & 0.834 & \textbf{1.66} \\
NeuralHawkes & \textbackslash{} & -1.02 & 0.241 & 0.966 & \textbf{1.77 } \\
Retweets & -5.99 & -5.06 & -5.85 & -4.69 & \textbf{-1.04} \\
StackOverflow & -2.60 & -2.55 & -1.86 & -0.559 & \textbf{-0.545} \\
MIMIC-II & -1.35 & -1.38 & -0.520 & -0.143 & \textbf{-0.111} \\
Financial & -3.89 & -3.60 & \textbackslash{} & -1.388 & \textbf{-1.18} \\ \hline
\end{tabular}
\end{table}
As we can find out in Table 4, TAA-THP outperforms existing baselines in terms of the value of the log-likelihood function on all test datasets. This phenomenon proves that TAA-THP can more effectively model the event sequences than existing baselines. However, the acquisition of Loglike can’t provide effective help for the practical application of the model, because loglike is only an abstract measurement, it does not have practical meaning.
Thus, we tend to compare the prediction accuracies in terms of the occurring time of each type of event and type of events for existing baselines and our proposed TAA-THP model, which have very important practical application significance. For instance, in terms of the electronic medical records, based on the patient's medical history, we can predict the type of illness and the occurring time of the next attack, and provide more effective and timely help for the patient's treatment. We can also predict the next possible failure time and type for a large-scale system based on the failure sequence, and perform maintenance and prevention in advance to improve safety and economic benefits in production.
It is worth noting that, for any sequence of events $s_n$ , assuming the length is $I_n$ , we don’t predict the possible type and time of the first event, in other words, for the sequence $s_n$ , we make n-1 predictions, and we set $\alpha_time=0.01 $ and $\alpha_type=1 $ . The prediction accuracies of event types are shown in Table 5 and Fig. 3.
\begin{table}[]
\centering
\caption{Predict accuracies for different models on various datasets.}
\begin{tabular}{ccccc}
\hline
Dataset & RMTPP & NHP & THP & TAA-THP \\ \hline
StackOverflow & 45.9 & 46.3 & 46.79 & 46.83 \\
MIMIC-II & 81.2 & 83.2 & 83.2 & 84.4 \\
Financial & 61.95 & 62.20 & 62.23 & 62.43 \\ \hline
\end{tabular}
\end{table}
\begin{figure}[!htbp]
\label{fig3}
\centering
\subfigure[]{
\includegraphics[width=4.5cm]{soacc.pdf}
}
\quad
\subfigure[]{
\includegraphics[width=4.5cm]{mimicacc.pdf}
}
\quad
\subfigure[]{
\includegraphics[width=4.5cm]{finacc.pdf}
}
\caption{Predictive accuracies for existing baselines and TAA-THP. Based on the five times train-dev-test partition, five experiments are performed on each dataset, and then the mean and standard deviation of different models are obtained.}
\end{figure}
From Fig. 3 and Table 5, on these real-world datasets, we can see that the accuracies of event prediction of our TAA-THP model have been improved compared with the existing baselines. In our point of view, we think these results are caused by the introduction of temporal attention, which decouples the simple stacking of event encoding and temporal position encoding in the traditional dot product attention mechanism. And we will verify this hypothesis on the ablation study in subsection 5.4.
However, we can see that the improvement of prediction accuracies is relatively small, however, The experimental results of Loglike have confirmed that TAA-THP can better model the sequence data, and then we will test the prediction ability of occurrences of the events for different models.
As for the prediction error of the occurring time of each type of event, we use RMSE (Root Mean Squared Error) as uniform metric evaluation criteria, RMSE of the existing baselines and TAA-THP are shown in Table 6 and Fig. 4.
\begin{table}[!htbp]
\centering
\caption{RMSE of different models on various datasets}
\begin{tabular}{ccccc}
\hline
Dataset & RMTPP & NHP & THP & TAA-THP \\ \hline
StackOverflow & 9.78 & 9.83 & 4.99 & 3.91 \\
MIMIC-II & 6.12 & 6.13 & 0.859 & 0.868 \\
Financial & 1.56 & 1.56 & 0.02575 & 0.02570 \\ \hline
\end{tabular}
\end{table}
\begin{figure}[!htbp]
\label{fig3}
\centering
\subfigure[]{
\includegraphics[width=4.5cm]{rmseso.pdf}
}
\quad
\subfigure[]{
\includegraphics[width=4.5cm]{mcrmse.pdf}
}
\quad
\subfigure[]{
\includegraphics[width=4.5cm]{fcrmse.pdf}
}
\subfigure[]{
\includegraphics[width=4.5cm]{rmsezoom.pdf}
}
\caption{RMSE for existing baselines and TAA-THP. Based on the five times train-dev-test partition, five experiments are performed on each dataset, and then the mean and standard deviation of different models are obtained.}
\end{figure}
From Fig. 4 and Table 6, we can find out that the RMSE of TAA-THP is generally smaller than the ones of the existing baselines. The experimental results for Loglike, accuracy, and RMSE has the obvious improvement than the existing baselines on the different scenes, for example, for the MIMIC-II dataset the number of the event types is 75, and the average length of sequences is only 4, for the Retweets dataset the number of the event types is 3, and the average length of sequences is 264, while for StackOverflow dataset the number of the event types is 22, and the average length of sequences is 736, both of them are relatively big, and for Financial dataset, the number of event types is only 2, and the average length of sequences is 2074. The characteristics of these datasets are obvious differences, our model shows a certain degree of robustness, and we can conclude that the TAA-THP can effectively capture the short-term and long-term dependencies between events.
\subsection{Ablation study}
So as to figure out the specific influence of the introduction of temporal augmented attention, we conduct the ablation study on StackOverflow, MIMIC-II, and Financial datasets. If we remove the temporal augmented attention in TAA-THP network architecture, the only difference between it and traditional dot-product attention is the bias ${\bm{b}}_{lq} $ . Therefore, here, we temporarily name this attention structure as biased attention, and the dot-product operation is shown as Eq.\ref{eq18}:
\begin{equation}
\label{eq18}
{\bf{A}}_l = Softmax\left[ {mask\left( {\frac{{\left( {{\bf{Q}}_l + {\bf{b}}_{lq} } \right){\bf{K}}_l^T }}{{\sqrt {D_K } }}} \right)} \right]{\bf{V}}_l
\end{equation}
Compare with Eq.3, the term $\left( {{\bf{Q}}_l + {\bf{b}}_{lt} } \right)\left( {{\bf{X}}^T {\bf{W}}_{Tem}^l } \right)^T $ is removed, which is the temporal attention, we compare the three evaluation criteria, e.g., Accuracy, RMSE, and Loglike on three datasets, the results are shown in Table 7. And to more intuitively compare the impact of temporal attention, we visualize the performance of models in Fig. 5, Fig. 6 and Fig. 7.
\begin{table}[!htbp]
\centering
\caption{The ablation experimental results of temporal attention.}
\begin{tabular}{cccc}
\hline
Dataset & Evaluation criteria & Biased attention & TAA-THP \\ \hline
\multirow{3}{*}{StackOverflow} & Accuracy & 46.80 & 46.83 \\
& RMSE & 4.04 & 3.91 \\
& Loglike & -0.547 & -0.545 \\ \hline
\multirow{3}{*}{MIMIC-II} & Accuracy & 84.00 & 84.41 \\
& RMSE & 0.872 & 0.868 \\
& Loglike & -0.129 & -0.111 \\ \hline
\multirow{3}{*}{Financial} & Accuracy & 62.38 & 62.44 \\
& RMSE & 0.02573 & 0.02570 \\
& Loglike & -1.03 & -1.08 \\ \hline
\end{tabular}
\end{table}
\begin{figure}[!htbp]
\centering
\subfigure[]{
\includegraphics[width=4.5cm]{soabrmse.pdf}
}
\quad
\subfigure[]{
\includegraphics[width=4.5cm]{mcabrmse.pdf}
}
\quad
\subfigure[]{
\includegraphics[width=4.5cm]{fiabrmse.pdf}
}
\caption{in terms of Loglike, visualization of ablation experimental results with and without temporal attention.}
\end{figure}
\begin{figure}[!htbp]
\centering
\subfigure[]{
\includegraphics[width=4.5cm]{soabacc.pdf}
}
\quad
\subfigure[]{
\includegraphics[width=4.5cm]{mcabacc.pdf}
}
\quad
\subfigure[]{
\includegraphics[width=4.5cm]{fiabacc.pdf}
}
\caption{in terms of RMSE criteria, visualization of ablation experimental results with and without temporal attention. }
\end{figure}
\begin{figure}[!htbp]
\centering
\subfigure[]{
\includegraphics[width=4.5cm]{sologab.pdf}
}
\quad
\subfigure[]{
\includegraphics[width=4.5cm]{milogab.pdf}
}
\quad
\subfigure[]{
\includegraphics[width=4.5cm]{filogab.pdf}
}
\caption{in terms of prediction accuracies, visualization of ablation experimental results with and without temporal attention}
\end{figure}
From Fig. 5 and Table 7, we can see that with removing of temporal attention, the Loglikes on the datasets become smaller overall, which verifies the effectiveness of introducing temporal attention mechanism. Meanwhile, we can find out that the Loglike and accuracies of Biased attention are still higher than the existing baselines, we guess that this is the effect of bias ${\bm{b}}_{lq}$, which can keep most of the neurons in the model activated.
Fig. 6, Fig. 7, and Table 7 show the prediction performance of models, we find that the lack of temporal attention reduces the prediction performance of event occurring time and type of event. This result confirms the merit of introducing temporal attention. Moreover, after removing temporal attention from the TAA-THP model, the prediction accuracies of the event will decrease, which can test and verify the hypothesis we put forward in subsection 5.3, the additional temporal attention indeed helps to decouple the simple concatenating of event encoding and time position encoding. In our standpoint, when we introduce temporal attention, it passes through a linear transformation ${\bm{X}}^T {\bm{W}}_{Tem}^l $ which is depicted in Eq.\ref{eq3}, which is similar to the training mechanism of the residual network. During the model training, the model can spontaneously decide whether the temporal information will feed into the multi-head attention, how much information is actually sent over the multi-head attention, it depends on the value of elements of $
{\bm{W}}_{Tem}^l $
, through the experimental results, we see that the model with temporal attention performs better, which confirms that temporal attention does exist in the model in its own right, and it plays an effective and positive role.
\section{Conclusions and future works}
In this paper, we come up with a new structure of the transformer Hawkes process. We improve the traditional dot-product attention mechanism. We propose to introduce temporal attention to the encoder of the transformer. The experimental results confirm that the additional introduction of temporal attention significantly improves the performance, especially in the predicting error for the occurring time and type of next event, and the overall likelihood function value on the sequence of events. In addition, through additional ablation study, we verify that the introduction of temporal attention is indeed effective. And in the future, we hope to propose a more valuable model based on the transformer and incorporate another generative neural network model to acquire better performance.
|
2024-02-18T23:40:24.493Z
|
2021-12-30T02:24:03.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14472","language":"en","timestamp":1640831043000,"url":"https:\/\/arxiv.org\/abs\/2112.14472","yymm":"2112"}
|
proofpile-arXiv_000-10035
|
{"provenance":"002.jsonl.gz:10036"}
| null | null |
\section{Introduction}
A detailed description of QCD dynamics at current and future collider
experiments is vital to discriminate Standard Model background from
signals of New Physics using the most advanced methods ranging from
traditional event shape variables to jet substructure methods to
machine learning techniques. The comparison to detailed theory
predictions is vital and a large range of analytic predictions up to
the most versatile Monte Carlo event generators are used in comparison
to experimental data. Specifically, Monte Carlo event generators have
recently seen a tremendous development regarding the simulation of the
hard scattering process, and the attention has shifted to the core,
parton shower, component among other developments on non-perturbative
modelling. The question of how the accuracy of parton branching
algorithms can be quantified, and how they can eventually be extended
beyond their current approximations -- both regarding resummed
perturbation theory in Quantum Chromodynamics (QCD), as well as beyond
the customary limit of a large number of colour charges $N$ -- has
received significant attention which we will briefly review in this
introduction to motivate the present work.
Parton branching algorithms have long been used in the context of
analytic resummation \cite{Catani:1989ne}, including up to
next-to-leading logarithmic (NLL) accuracy for a large class of event
shape variables \cite{Catani:1992ua} and properties of hard objects
created in hadron-hadron collisions \cite{Catani:1990rr}. These
techniques notably exploit the coherence properties of QCD and the
structure of the observable such that angular ordering can be used
upon azimuthal averaging to simplify the resummation procedure. They
have also been the key ingredient behind designing the Herwig
angular-ordered parton shower algorithm
\cite{Marchesini:1987cf,Gieseke:2003rz,Bahr:2008pv,Bellm:2019zci},
which is thus capable of describing global event shape variables at
NLL accuracy. Due to ambiguities in how recoil of the emissions is
distributed at the end of the evolution, however, failures of this
naive expectation may be introduced and need careful investigation
\cite{Bewick:2019rbu}. Even in the case of NLL accurate coherent
branching evolution, a number of shortcomings are present in these
algorithms, and have motivated additional work, notably spin and
colour correlations \cite{Richardson:2018pvo} as well as mass effects
\cite{Cormier:2018tog}.
The biggest shortcoming, however, is the inability of coherent
branching algorithms to describe observables which are sensitive to a
change in colour structure from subsequent emissions beyond the
dynamics accounted for by QCD coherence, and this applies already in
the large-$N$ limit. Crucially, it is the lack of a dipole-type
picture which disallows coherent branching from describing these
correlations, and hence there is no chance for such an evolution to
account for any of the logarithmic enhancements in non-global
observables. It is clear though, that non-global observables
\cite{Dasgupta:2001sh, Becher:2016mmh, Martinez:2018ffw} should be
setting the level of complexity which needs to be addressed for the
plethora of sophisticated analysis methods employed at the currently
operating hadron colliders, and projected to be used even at future
$e^+e^-$ colliders. Note however that coherent branching is in fact
able to describe the exact-$N$ structure of global event shapes at
least at leading logarithmic (LL) level.
Dipole shower algorithms \cite{Gleisberg:2008ta, Platzer:2011bc,
Dinsdale:2007mf, Giele:2013ema}, on the other hand, do offer the
possibility to properly resum non-global effects at leading colour
(and in fact have been used, in a specialized form, for specifically
this task \cite{Dasgupta:2001sh, Becher:2016mmh}). Despite the fact
that they have been used to try and extend parton branching beyond the
leading order \cite{Dulat:2018vuy}, their initial accuracy remained
unclear and has been proven to be wrong at various levels for global
observables
\cite{Dasgupta:2018nvj,Dasgupta:2020fwr,Forshaw:2020wrq}. Recent work
has therefore focused on trying to combine the best of dipole and
coherent branching algorithms such as to maintain the accuracy for
global event shapes, and a proper (\textit{i.e.}\ at least leading-colour and
leading logarithmic accurate) description of non-global observables
\cite{Hamilton:2020rcu,Holguin:2020joq}. Within this context it turned
out that recoils can cause severe accuracy problems, and that there is
a delicate interplay between ordering, recoil and the question of how
the soft behaviour is distributed across the partonic systems
competing to emit the next parton, a mechanism typically referred to
as `partitioning'. Notice that a dipole-type picture, or extensions
thereof, is also required to have accurate control over recoils
distributed emission by emission, something which is extremely
important when trying to combine with fixed-order QCD calculations,
either through matching or merging. Most of the time, the development
of parton showers within this context has been closely tied to the
development of fixed-order subtraction terms reflecting the
singularity structure of QCD {\it cross sections} for the emission of
one, or possibly two, unresolved partons.
The shortcomings of existing parton shower algorithms, and possible
improvements, have been addressed in \cite{Dasgupta:2018nvj} by
checking how well they reproduce the soft radiation pattern for two
subsequent emissions, while \cite{Forshaw:2020wrq} has been taking a
different point of view in starting from formulating parton branching
algorithms at the amplitude level \cite{Forshaw:2019ver}. This is a
theoretical framework which addresses the iterative build-up of an
amplitude (and its conjugate) describing the emission of multiple
unresolved partons, and is thus the prime framework to analyse how
cross sections can be described in the limit of a large number of
unresolved partons. A systematic expansion of the resulting
expressions around the large-$N$ limit, for example, then leads to
commonly used parton branching algorithms which can be extended beyond
the large-$N$ limit. The same formalism can also be used to derive
further limiting cases such as coherent branching, which has
ultimately led to the proposal of an improved dipole shower
algorithm. We stress that only within such a framework, and similar
approaches pursued by Nagy and Soper \cite{Nagy:2020rmk}, is it
possible to analyse the entire set of correlations in between multiple
emissions, and to systematically obtain an approximation of the
iterations, rather than iterating a single-emission approximation of
the cross section. The parton branching at amplitude level was
designed around the soft gluon evolution algorithm presented earlier,
and included the hard-collinear contributions by extending the dipole
structure of the leading order soft evolution. Work is underway to
extend this algorithm to the next order, including an expansion around
the colour-diagonal contributions \cite{Platzer:2020lbr} corresponding
to the $d'$-expansions advocated in \cite{DeAngelis:2020rvq}.
In this work we focus on extending the parton branching in amplitude
level algorithms to account for more than singly-unresolved limits,
concentrating on corrections due to iterating the emission of two or
more simultaneously unresolved partons. We work at a finite number of
colours, and include the full spin dependence aiming to establish
factorization at the level of the 'density operator', \textit{i.e.}\ the
amplitude and its conjugate, for which we can devise the definition of
splitting kernels. These encompass all enhanced configurations of
both soft, collinear, or any combination of unresolved limits. We also
highlight the connection of the underlying power counting to the way
recoil is distributed and how kinematics are parametrized.
This paper is structured as follows: In Sec.~\ref{sec:Factorization}
we will outline the general framework in which we establish the
iterative factorization of the density operator. We will highlight the
connection to the light-cone gauge we employed, as well as the
distribution of recoil of emissions among the hard jet axes, which
allows us to establish a diagrammatic framework in which the splitting
kernels can be calculated. After the projection onto different
collinearly enhanced contributions, which we refer to as partitioning,
we will be able to establish a systematic power counting and a final
calculational recipe for the emission operators. In
Sec.~\ref{sec:Paritioning}, the partitioning is set up in an
algorithmic manner and is suited to be carried out to an in principle
arbitrary number of emissions from a hard process.
In Sec.~\ref{sec:Mapping} we detail one particular instance of a
suitable momentum mapping and discuss its properties in the unresolved
limit, as well as how it can be used to construct an actual parton
branching algorithm concerning the real emission contributions. In
combination with singling out collinear limits, and the proper choice
of a backward vector $n$, we can determine the splitting kernels. We
discuss a few examples in Sec.~\ref{sec:Kernels}, also addressing how
the underlying amplitude can be taken on-shell from the final state
considered (before emissions are added) and which power suppressed
effects we neglect in doing so.
In Sec.~\ref{sec:Applications} we conclude by discussing applications
of our formalism, and how our framework can be generalized to the case
of virtual corrections. The determination of a full set of QCD
splitting kernels is devoted to a future publication, however we do
discuss several examples, as well as the possibility to link our
formalism to a partitioning of the known singular behaviour of two
emissions, and a systematic removal of overlap between the soft and
collinear limits. A number of technical details and material for
future reference is devoted to several appendices, which do not need
to be considered to understand the main findings of our current work.
\section{Construction of splitting kernels}
\label{sec:Factorization}
While traditional parton branching algorithms use probabilistic
paradigms based on the factorization of cross sections, our aim is to
address parton branching algorithms at the amplitude level, as
recently theoretically developed and implemented in
\cite{Martinez:2018ffw,Forshaw:2019ver,Forshaw:2020wrq,DeAngelis:2020rvq}. Only
within such an approach is it possible to iteratively approximate
amplitudes with many legs and to derive Markovian algorithms which
then might be able to perform multiple emissions at the probabilistic
level; iterating approximations of a fixed-order cross section will
yield the right singularity structure, but otherwise not guarantee the
right resummation properties. In the present work, we focus on the
systematic construction of real emission splitting kernels within such
a branching algorithm, addressing the simultaneous emission of more
than one unresolved parton. The goal of the present work is to not
perform this specifically in one unresolved limit, but to obtain a
combination of splitting operators which smoothly cover the different
singular limits. Some aspects of virtual corrections have recently
also been addressed in \cite{Platzer:2020lbr} and will be
combined with the present effort in future work. We can of course use
our results to obtain splitting kernels which can also serve as
subtraction terms in fixed-order corrections, though this is not our
primary goal.
To set the notation, we consider scattering amplitudes $|{\cal
M}\rangle$ to be vectors in colour and spin space (see
\cite{Catani:1996vz} for an outline of this formalism), from which
cross sections originate as
\begin{equation}
\label{eqs:crosssec}
\sigma[u] = \sum_n \int {\rm Tr} \left[|{\cal M}(1,...,n)\rangle
\langle{\cal M}(1,...,n)|\right] u(1,...,n) {\rm d}\phi(1,...,n|Q) \ ,
\end{equation}
where ${\rm d}\phi(1,...,n|Q)$ commonly refers to the $n$-parton phase
space given a total momentum $Q$ and the arguments of the amplitude
vector and phase space measure refer to both parton momenta, as well
as other degrees of freedom needed to identify a particular final
state. $u$ is a generic observable with the same convention on
arguments understood, \textit{i.e.}\ $u(1,...,n) = u(p_1,...,p_n)$ for
specific momenta. In the present work we also only consider massless
partons. The trace in \eqref{eqs:crosssec} refers to summing over
the colour and spin degrees of freedom in the amplitude and its
conjugate, seen as an operator in colour and spin space, the cross
section density operator
\begin{equation}
{\mathbf A}_n = |{\cal M}(1,...,n)\rangle \langle{\cal M}(1,...,n)| \ ,
\end{equation}
which will be the central object of our investigation. The trace can
typically be expressed using the completeness of external wave
functions, and we shall decompose the respective numerator of a cut
propagator in such a way that we can find a similar decomposition of
internal lines, with suitably parametrized kinematics. With this in
mind, we will then formulate a diagrammatic construction of the relevant
splitting kernels, directly involving a power counting to identify the
leading contributions of interest. Working in a physical gauge with
gauge fixing $C^a = n\cdot A^a$, $n^2=0$, we in particular note that
we can express the propagator numerators for cut gluon lines of
on-shell momentum $q$ as
\begin{equation}
d_{\mu\nu}(q) = -\eta^{\mu\nu} + \frac{n^\mu q^\mu + n^\nu q^\mu}{n\cdot q} \ ,
\end{equation}
such that we can use
\begin{equation} \label{eq:projector-gluon}
d_{\mu\nu}(q) = d_{\mu\rho}(q)\ P^{\rho\sigma}(p) \ d_{\sigma\nu}(q),
\qquad P^{\rho\sigma}(p) = d^{\rho\sigma}(p) \ ,
\end{equation}
which holds for $q^2=0$, while for quark lines we can employ
\begin{equation}\label{eq:projector-fermion}
\slashed{q} =
\sqrt{\frac{n\cdot q}{n\cdot p}}\ \frac{ n\cdot p}{n\cdot q} \slashed{q} \
\slashed{P}(p)\ \slashed{q} \frac{n\cdot p}{n\cdot q}\ \sqrt{\frac{n\cdot q}{n\cdot p}},
\qquad \slashed{P}(p) = \frac{\slashed{n}}{2n\cdot p} \ ,
\end{equation}
where $p$ is the forward component we associate to the external
momentum $q$ in a decomposition
\begin{equation}
q^\mu = \alpha\ p^\mu + \beta\ n^\mu + q_\perp^\mu \ ,\qquad q_\perp\cdot n
= q_\perp\cdot p = 0 \ .
\end{equation}
The reason for the awkward decomposition of the quark numerator as in
\eqref{eq:projector-fermion} is to associate the square-root factors
with the vertex the quark couples into (or leave it as an explicit
factor if such a coupling is not considered), while the factors
$n\cdot p/n\cdot q$ provide a convenient normalization in the power
counting such that quark and gluon lines have a uniform scaling. The
role of soft quarks in the factorization can then be tracked
separately from more general considerations on factorization of the
amplitude. This is discussed in Sec.~\ref{sec:decompositions-intro}.
In a more abstract notation we thus define
\begin{align}\label{eq:projectors-general}
\mathbf{P}(q) \equiv
\begin{cases}
P^{\rho \sigma}(p) = d^{\rho \sigma} (p), & \text{for gluons}, \\
\slashed{P}(p) = \frac{\slashed{n}}{2n \cdot p}, & \text{for quarks},
\end{cases}
\end{align}
where it should always be clear which forward momentum $p$ we
associate to a certain external momentum $q$, subject to a
decomposition of the density operator into different collinearly
singular configurations to be discussed in Sec.~\ref{sec:decompositions-counting}. Moreover, we
can use polarisation sums to represent these operators, \textit{i.e.}\
\begin{align}
d^{\mu\nu}(p) &= \epsilon^\mu_+ (p,n) \epsilon ^\nu_- (p,n) + (\mu \leftrightarrow \nu), \\
\slashed{n} &= \sum\limits_\lambda u_\lambda (n) \bar{u}_\lambda (n),
\end{align}
with
\begin{align}
\epsilon_{\pm}^2 = 0, \quad \epsilon_{\pm}\cdot \epsilon_{\mp} = -1, \quad \epsilon_{\pm} \cdot p = \epsilon_\pm \cdot n = 0.
\end{align}
Also note that for
internal lines carrying a sum of momenta, we can decompose the
respective numerators via
\begin{eqnarray}
\label{eqs:linearity}
d^{\mu\nu}(q_I+q_J) &=& \frac{n\cdot q_I}{n\cdot q_I + n\cdot q_J}\times d^{\mu\nu}(q_I)
+ (I \leftrightarrow J), \\ \nonumber
\slashed{q}_I+\slashed{q}_J &=&
\sqrt{\frac{n\cdot q_I}{n\cdot p_i}}
\left( \frac{n\cdot p_i}{n\cdot q_I} \slashed{q}_I\right) \sqrt{\frac{n\cdot q_I}{n\cdot p_i}}
+ (I
\leftrightarrow J),
\end{eqnarray}
where $q_I$ and $q_J$ themselves can be sums of momenta in distinct
sets of partons $I$ and $J$. Moreover, the square root factors in the
second line are of the same use as in \eqref{eq:projector-fermion}.
Similarly, we can decompose the three-gluon vertex due to its
linearity in the momenta, \textit{i.e.}\
\input{diagrams/soft-three-vertex-decomp} If now all of the partons in
the set $J$ become soft, then only the first terms in
\eqref{eqs:linearity} and \eqref{eq:linearity-three-gluon-vertex} will
contribute in a leading soft limit. This paves the way to establish a
universal set of rules to decompose the cross section with many
external, unresolved, legs as follows: Using
\eqref{eq:projectors-general}, we can write
\begin{equation}\label{eq:trace-amplitude}
{\rm Tr} \left[{\mathbf A}_n\right] = {\rm Tr}_c
\left[\tilde{{\mathbf A}}_n\times {\mathbf P}_n \right] \ ,
\end{equation}
where ${\rm Tr}_c$ now solely refers to a trace in colour space,
$\tilde{{\mathbf A}}_n$ is the density operator obtained from
${\mathbf A}_n$ by removing all external wave functions. The sum over
external polarisations is implicitly carried out by associating the
normalized completeness relations of external wave functions to each
external line, \textit{i.e.}\ applying a factor $d^{\mu\nu}(q)$ for each
external gluon line and a factor of $(\slashed{q}\ n\cdot p / n\cdot
q)\times \sqrt{n\cdot q / n\cdot p}$ to each external fermion line,
together with the insertion of the respective operator ${\mathbf P}$.
This setup allows us to treat internal and external lines for quarks
and gluons alike on the same footing and devise universal power
counting rules in the following parts of this section. Those will
eventually be used to dissect numerator structures of emission
amplitudes in terms of their soft or collinear scaling and allows for
a precise association of which parts of an amplitude contribute in
singular limits at a given power. The results of these considerations
for one and two emissions are given in
Sec.~\ref{sec:single-emission-splitting-kernel} and
Sec.~\ref{sec:two-emission-splitting-kernel} respectively. Also note
that it is in principle possible to include a polarized measurement
into the definition of the ${\mathbf P}$ operator by appropriately
restricting the sums of external wave functions, though this is beyond
the scope of the present work.
Starting from the amplitude, we consider a certain subset of diagrams
contributing to additional emissions from a general amplitude $|{\cal
M}\rangle$, which we consider to be a vector in spin and colour
space. In particular, we are interested in the emission of several, $k\ge
1$ additional particles, for which we consider the diagrams
factorizing in an $(n+k)$-particle amplitude in terms of emission
diagrams and an underlying, $n$-particle hard amplitude as
\begin{multline}
|{\cal M}_{n+k}(1,...,{n+k})\rangle =
\sum_{p=1}^k \sum_{r\in S_{n,p,k}}
\mathbf{Sp}_{(r_{11}|...|r_{1\ell_1})}...\mathbf{Sp}_{(r_{p1}|...|r_{p\ell_p})}\\
|\tilde{{\cal M}}_{n}(1,...,{(r_{11}|...|r_{1\ell_1})},...,
{(r_{p1}|...|r_{p\ell_p})},...,{n+k} )\rangle,
\end{multline}
or in a diagrammatic representation
\input{diagrams/master-diagram-splitting}
$S_{n,p,k}$ is the set of possible splitting assignments involving $p$
emitters and $k$ emissions out of $n+k$ external particles, and
\begin{equation}
{(r_{i1}|...|r_{i\ell_i})} = {r_{i1}}+...+{r_{i\ell_i}}
\end{equation}
are the off-shell momenta of the branching internal lines, where each
of the $r_{ik}$ denotes a certain external momentum $q_j$ involved in
the splitting process. We are interested in the unresolved limits of
the $k$ emitted partons, in which all of the off-shell momenta
${(r_{i1}|...|r_{i\ell_i})}$ will become on-shell and the splitting
amplitudes are expected to factorize in a universal manner, within
which the underlying amplitude $\tilde{\mathcal{M}}$ in fact becomes an on-shell
amplitude. The leading singular behaviour of the cross section is then
obtained by considering the square of the amplitude, or equivalently a
density-operator type object, such that none of the partons
participating in an unresolved limit are connected to an internal
line. To be more precise we can write
\begin{multline}\label{eqs:density-operator-general}
|{\cal M}_{n+k}(1,...,{n+k})\rangle\langle {\cal M}_{n+k}(1,...,{n+k})| = \\
\sum_{p=1}^k \sum_{\bar{p}=1}^k \sum_{r\in S_{n,p,k}}\sum_{\bar{r}\in S_{n,\bar{p},k}}
\mathbf{Sp}_{(r_1)}...\mathbf{Sp}_{(r_p)}
|{\cal M}_{n}(1,...,{(r_1)},...,
{(r_p)},...,{n+k} )\rangle\\ \langle {\cal M}_{n}(1,...,{(\bar{r}_1)},...,
{(\bar{r}_{\bar{p}})},...,{n+k} ) | \mathbf{Sp}^\dagger_{(\bar{r}_1)}...
\mathbf{Sp}^\dagger_{(\bar{r}_{\bar{p}})} \ \times \hat\Delta^{r}_{\bar{r}}\ +\\
\text{(subleading)}
\end{multline}
where we have introduced short-hands for the splitting labels, $(r_i) =
(r_{i1}|...|r_{i\ell_i})$ and similar. The tensor
$\hat\Delta^{r}_{\bar{r}}$ ensures that we only include those diagram topologies
which give rise to leading singularities. See
Appendix~\ref{sec:delta-tensor} for its explicit construction.
This extraction of leading singularities is valid as long as we work
in a physical gauge. The diagrammatic representation of one such leading
singular contribution to the density operator is:
\input{diagrams/splitting-general-algorithm-squared}
\input{diagrams/density-operator-boxes}
This procedure needs to be understood as follows: First we apply our
identities to decompose the square of the matrix element into the
operators ${\mathbf P}$ and a set of cut lines, where a dashed cut
line indicates a use of
$$
\slashed{q} \frac{\ n\cdot p}{n\cdot q}\quad \text{or}\quad d^{\mu\nu}(q)
$$
for each on-shell external quark or gluon line,
respectively. Internal lines carrying a sum of different momenta will
be decomposed as indicated by Eqs.~(\ref{eqs:linearity}), and a factor
of $\sqrt{n\cdot q_a/n\cdot p_a}\sqrt{n\cdot q_b/n\cdot p_b}$ will be
applied to each quark-gluon vertex with incoming fermion momentum
$q_a$ (with forward momentum $p_a$) and each outgoing fermion momentum
$q_b$ (with forward momentum $p_b$). We then carry out the
decomposition of an amplitude's numerator in terms of its forward (along
jet directions $p_i^\mu$), backward (along the gauge vector $n^\mu$)
and transverse ($n_\perp^\mu$) components, indicated by the grey boxes
in the last diagram of the workflow depicted above. Forward
directions are chosen with respect to jets emerging from a hard
interaction, such that we can properly trace collinear and/or soft
configurations through several emitted partons in the diagram, as well
as iterating kernels describing a particular unresolved configuration,
however a universal factorization onto an on-shell density operator
will generally only happen in very specific configurations of
unresolved emissions. To this extend we shall re-interpret
Eq.~(\ref{eqs:density-operator-general}) by writing it as a sum over
different topologies of diagrams in the amplitude and conjugate
amplitude, which we label $\rho$ and $\bar{\rho}$, respectively:
\begin{equation}\label{eqs:diagrams}
\tilde{{\mathbf A}}_{n+k} =
\sum_{\rho,\bar{\rho}} \widetilde{\mathbf{Sp}}^{(k)}_{n,\rho}\ {\mathbf B}_{n,n+k,\rho,\bar{\rho}}\
\widetilde{\mathbf{Sp}}^{(k),\dagger}_{n,\bar{\rho}} \ ,
\end{equation}
where ${\mathbf B}_{n,n+k,\rho,\bar{\rho}}$ is obtained by removing
the splitting diagrams labelled by $\rho$ and $\bar{\rho}$, and still
constitutes the outer product of two amplitudes with different
compositions of on-shell and off-shell external lines.
Crucially, the
off-shell lines at which we will attempt to factor off emission
subdiagrams will be nearly on-shell in the unresolved configurations,
yet we need to restore overall energy momentum conservation. In our
approach we choose to do this by a global Lorentz transformation
combined with a scaling of each of the final state momenta, such that
momenta $q_i$ after emission, and with momentum conservation
maintained, $\sum_i q_i = Q$ are parametrised as
\begin{equation}
\label{eq:globalrecoil}
q_i^\mu = \frac{1}{\hat{\alpha}} \Lambda^\mu {}_\nu \hat{q}_i^\nu \ ,
\end{equation}
where the $\hat{q}_i$ are decomposed into the jet directions and do
not obey momentum conservation, $\sum_i \hat{q}_i = Q+ N$. The scaling
factor $\hat{\alpha}$ then simply encodes the correction of the different
mass shells $(Q+N)^2$ and $Q^2$. The recoil momentum $N$ can thus
easily be absorbed through the Lorentz transformation. The advantage
of such an approach is that recoil effects are now completely removed
from considerations about factorizing the amplitude, as we can use
Lorentz invariance and the known mass dimension of the amplitude to
remove this effect. More precisely,
\begin{equation}
\label{eq:amplitudeglobalrecoil}
|{\cal M}(q_1,...,q_n)\rangle = \frac{1}{\hat{\alpha}^{2n-4}}
|{\cal M}(\hat{q}_1,...,\hat{q}_n)\rangle \ .
\end{equation}
In the following we will, unless stated otherwise, therefore treat the
transformed and untransformed momenta as equivalent. Notice that for
additional, massive particles involved, we can easily extend such a
recoil scheme which will not change the argument given here. Notably,
when working in a physical gauge, as we choose below, the gauge vector
$n$ will also have to be transformed by the recoil prescription in
order for the argument to stay valid. We will link this to how the
recoil transformation should be iterated in
Sec.~\ref{sec:decompositions-counting}.
An expansion around a certain limit in which the off-shell lines
become on-shell in \eqref{eqs:diagrams} is possible only if we can
expose this particular limit by a weighting factor, ensuring that no
other singular limit can contribute by virtue of either the weighting
(or partitioning) factor we introduce or the physical gauge we
choose. We therefore write \eqref{eqs:diagrams} as
\begin{multline}\label{eqs:paritioning-origin}
\tilde{{\mathbf A}}_{n+k} =
\sum_{\rho}\left[
\sum_{\lambda}\left(\frac{w_{\rho,\lambda;\rho}}{w_{\rho,\lambda}}
\widetilde{\mathbf{Sp}}^{(k)}_{n,\rho}\ {\mathbf B}_{n,n+k,\rho,\lambda}\
\widetilde{\mathbf{Sp}}^{(k),\dagger}_{n,\lambda} +
\frac{w_{\lambda,\rho;\rho}}{w_{\lambda,\rho}}
\widetilde{\mathbf{Sp}}^{(k)}_{n,\lambda}\ {\mathbf B}_{n,n+k,\lambda,\rho}\
\widetilde{\mathbf{Sp}}^{(k),\dagger}_{n,\rho}
\right)\right. +\\\left.
\sum_{\lambda\ne \rho}\sum_{\sigma\ne \rho} \frac{w_{\sigma,\lambda;\rho}}{w_{\sigma,\lambda}}
\widetilde{\mathbf{Sp}}^{(k)}_{n,\sigma}\ {\mathbf B}_{n,n+k,\sigma,\lambda}\
\widetilde{\mathbf{Sp}}^{(k),\dagger}_{n,\lambda}
\right] \ ,
\end{multline}
with $w_{\rho,\bar{\rho}} = \sum_\sigma w_{\rho,\bar{\rho};\sigma}$
and where $w_{\rho,\bar{\rho};\sigma}$ ensures that a diagram with
propagator structures identified by $\rho$ and $\bar{\rho}$ contains a
collinear enhancement solely as dictated by the topology $\sigma$, but
not by those identified through $\rho$ and $\bar{\rho}$ (though all of
these might contain common configurations of collinear enhancement).
An index $\rho$ can be visualized as part of a set of distinct
emission patterns, which can in turn be mapped to ordered tuples of
the set of all partons involved in an $(n+k)$-splitting. This
procedure will allow us to uniquely identify one parametrization of
the external momenta in terms of forward and backward directions which
we can then use to set up a systematic power counting in each singular
limit.
As an example to illustrate the contributions in
\eqref{eqs:paritioning-origin}, we examine it for fixed $\rho =
(ijk)$, meaning a splitting with a hard line $i$ and two sequential
emissions $j$ and $k$ off of it. Then the first line in
\eqref{eqs:paritioning-origin} represents all two emission topologies
containing this emission pattern, such as the self energy $E^{(1)}$
for $\lambda = \rho$ or topology $B^{(1)}$ for $\lambda=(ij)(lk)$ and
their conjugates (see App.~\ref{app:two-emission-diagrams} for a list
of topologies). The third term represents all topologies that do not
contain this emission pattern, such as the $X$-topology. Each
contribution is multiplied by their respective weighting factor which
cancels all singular factors not belonging to the $(ijk)$-collinear
sector. These are precisely the partitioning factors of
Sec.~\ref{sec:Paritioning}. In this way of a decomposition, we are
able to collect all possible contributions to a specific splitting
pattern. The collection of such terms for fixed $\rho$ is what we
call \emph{splitting kernels} which are defined in
Sec.~\ref{sec:Kernels}.
\subsection{Decomposition of amplitudes in a physical gauge}
\label{sec:decompositions-intro}
Once we have identified a certain splitting process by the emission
subdiagrams contributing in the amplitude factor, \textit{i.e.}\ one term
of fixed $\rho$ in \eqref{eqs:paritioning-origin}, we decompose all
of the momenta involved in each splitting $(r_i)$ as
\begin{equation}\label{eq:Sudakov-decomp-single}
r_{ik}^\mu = z_{ik} \, p_i^\mu +
\frac{p_{\perp,ik}^2}{z_{ik}\ 2 p_i\cdot n}\, n^\mu + k_{\perp,ik}^\mu \ ,
\end{equation}
where $k_{\perp,ik}^2=-p_{\perp,ik}^2$, $p_i\cdot k_{\perp,ik} =
n\cdot k_{\perp,ik}=0$ and $p_i$ is the momentum of a progenitor of
the splitting. An off-shell internal line $I$ within a splitting
subdiagram $(r_i)$ in the amplitude carries momenta of the form
\begin{align}\label{eq:Sudakov-decomp-combined}
q_I^\mu = \sum_{k\in I} r_{ik} = z_I \, p_i^\mu +
\frac{S_I + p_{\perp,I}^2}{ z_I\ 2 p_i\!\cdot\! n}\, n^\mu + k_{\perp,I}^\mu \ ,
\end{align}
where $q_I^2=S_I>0$ and the transverse components as well as momentum
fractions are additive
\begin{equation}
k_{\perp,I}^\mu = \sum_{k\in I} k_{\perp,ik}^\mu,\qquad z_I = \sum_{k\in I} z_{ik},
\end{equation}
while
\begin{equation}
S_I = \Big(\sum_{k\in I} z_{ik}\Big) \sum_{l\in I} \frac{p_{\perp,il}^2}{z_{il}} +
\sum_{k,l\in I} k_{\perp,ik}\cdot k_{\perp,il}, \qquad p_{\perp,I}^2 =
\Big(\sum\limits_{k \in I} k_{\perp,ik}^\mu\Big)^2,
\end{equation}
such that
\begin{equation}
\beta_I \equiv \frac{S_I + p_{\perp,I}^2}{2 z_I\ p_i\!\cdot\! n} =\frac{1}{2p_i\cdot n} \sum_{k\in I} \frac{p_{\perp,ik}^2}{z_{ik}},
\end{equation}
consistent with momentum conservation. Notice that, if all lines
combining into $I$ are collinear w.r.t.\ $p_i$, meaning
$p_{\perp,ik}\to \lambda p_{\perp,ik}$, we obtain $S_I\sim \lambda^2$;
if all lines are soft, $p_{\perp,ik}\to \lambda p_{\perp,ik}$ and
$z_{ik}\to \lambda z_{ik}$, we obtain the same. However, if one line
$k$ is hard collinear in the sense that it has an $z_{ik} = {\cal
O}(1)$, we find $S_I\sim \lambda^2$ only in the limit of all lines
becoming collinear, while $S_I\sim \lambda$ in the combined
soft/collinear limit. For the backward component this implies that it
is ${\cal O}(\lambda)$ for all soft, or a soft/collinear combination,
and ${\cal O}(\lambda^2)$ for an all-collinear configuration. An
overview of the line- and vertex-scalings in all possible collinear
and soft settings is shown in Sec.~\ref{sec:scaling-of-lines}.
We can use \eqref{eq:Sudakov-decomp-combined} for decomposing the
scaled fermion numerators $(n\cdot p_i/n\cdot q_I) \slashed{q}_I$ as
\input{diagrams/fermion-lines-sudakov-scaling}
and for gluon lines $d^{\mu\nu}(q_I)$ as
\input{diagrams/gluon-lines-sudakov-scaling}
These decompositions hold both for the numerators we obtain from the
sum over external wave functions, and internal lines. Comparing
\eqref{eq:power-counting-lines-fermion} with
\eqref{eq:power-counting-lines-gluon} shows that the components in
these decompositions give the same soft and collinear scaling
behaviour for quarks and gluons. To this extent we can 'colour' all
internal lines in the amplitude and the conjugate amplitude, as well
as the external lines by the different contributions in the
decomposition above. Note that it suffices both for external on-shell
gluon and quark lines to use the projectors with only the forward
momentum component $p_i$ as an argument, due to
\begin{subequations}
\begin{align}
\slashed{r}_{ik} \, \frac{\slashed{n}}{2 n \!\cdot\! p_i} \,
\slashed{r}_{ik} &= z_{ik} \slashed{r}_{ik},
\\
d^{\mu \rho}
(r_{ik}) \, d_{\rho \sigma} (p_i) \, d^{\sigma \nu} (r_{ik}) &=
d^{\mu \nu} (r_{ik}).
\end{align}
\end{subequations}
Moreover, we find the following rules for connecting on-shell lines
via the projectors, \textit{viz.}\
\input{diagrams/external-line-rules}%
which hold both for quarks and gluons. Due to
\eqref{eq:line-rules-black} and \eqref{eq:line-rules-perp-perp}, we
note that the backwards ($n$) components on external lines do not need
to be taken into account in the decomposition of an amplitude. Via
these rules, we are able to translate our density operator-like
discussion to the one of cut diagrams or vice versa.
Let us first discuss the vertex structures coupling these different
internal lines. We find that, making use of the fact that $n_\mu
d^{\mu\nu}=0$, the quark gluon vertex and three gluon vertex can
effectively be decomposed into two components depending only on the
longitudinal and only on the transverse components,
\input{diagrams/definition-col-perp-vertices}
We also find that certain combinations of lines always vanish,
irrespective of the progenitor momentum and/or the off-shellness of the
lines. In particular this applies to the cases in which more than one
line with a black square connects to a vertex, indicating that a
backward component can only propagate upon exchanging at least some
transverse or longitudinal momentum with another parton,
\input{diagrams/boxed-vertices}
both for quark and gluon insertions. The following vertices also
vanish but only in the gluon case:
\input{diagrams/boxed-gluon-vertices}
Further simplifications occur if all lines which are coupled together
share the same forward direction; in this case we also have that
vertices coupling three forward components vanish identically, if the
participating momenta have been decomposed with respect to the same
forward direction $p_i$.
\subsection{Decomposing around singular limits}
\label{sec:decompositions-counting}
Let us first focus on those terms in \eqref{eqs:paritioning-origin}
for which we can uniquely assign a splitting configuration to the
diagram in the amplitude or conjugate amplitude, \textit{i.e.}\ the terms
in the first line of \eqref{eqs:paritioning-origin}.
\input{diagrams/notation}
To illustrate our terminology let us focus on a topology like the one
depicted in Fig.~\ref{fig:notation}, where we have identified a
certain set of splitting amplitudes in the amplitude side (the left
hand part of the diagram) which we refer to as `splitter lines', and
possibly different combinations of splitting amplitudes in the
conjugate (right hand) side of the density operator diagram. Splittings
of this kind, which solely involve momenta which are attached to a
single splitting blob on the left hand side are referred to as
`conjugate splitting lines', while otherwise we refer to them as
`interferer lines'. Fig.~\ref{fig:notation} contains an example of our
notation.
\input{diagrams/collinear-directions}
Our power counting in the singular limits will be organized around the
assumption that we can prevent those momenta attached to a splitter
line from developing collinear singularities with respect to any other
than their progenitor direction. This can be imposed by the
restrictions of a certain observable which \textit{e.g.}\ is requiring a fixed
number of jets, or by an algorithmic construction of a `partitioning',
an example of which we will outline in Sec.~\ref{sec:Paritioning} for
one and two emissions. We illustrate this idea in
Fig.~\ref{fig:coll-directions}, where double lines represent different
hard directions and the colours signify which lines are connected in a
corresponding cut-diagram. Using the weighting factors $\omega_{\rho
\bar{\rho}; \sigma}$ of \eqref{eqs:paritioning-origin}, one can
extract the collinear behaviour to some subset of hard lines, \textit{e.g.}\ the
yellow ones in Fig.~\ref{fig:coll-directions}, via the choice of the
respective splitting configuration $\sigma$. For a fixed-order
analysis such a partitioning can serve as the basis to construct
subtraction terms, or we can view it in an integrated way to obtain
evolution kernels for a full-fledged parton branching algorithm. A
further assumption we make is that each `splitter line' has one hard
momentum flowing through it, and that each `conjugate splitter' and
`interferer line' do so as well. We therefore consider to decompose
all momenta which are assigned to the `splitter' lines to be
decomposed with one progenitor momentum per splitter line, both in the
amplitude as well as its conjugate, \textit{i.e.}\ we shall {\it not} decompose
the momenta attaching to interferer lines in terms of the hard
momentum defining the interferer direction. Conversely, the hard
momentum in the interferer direction shall be parametrized taking into
account the interferer direction. Couplings between different
collinear sectors, hence, only need to be considered for conjugate
splitter lines, or for vertices coupling lines from different splitter
lines.
Let us stress that for conjugate splitter or interferer lines which
have internal lines from different collinear sectors we can employ
\eqref{eqs:linearity} and \eqref{eq:linearity-three-gluon-vertex}
to decouple the different sectors. Such a line, say one carrying
$q_I+q_J$, will then split into one with a numerator according to
$q_I$ and one according to $q_J$, and the net enhancement to be
studied is the partitioning's effect along with the combination
\begin{equation}\nonumber
\frac{z_I}{z_I+z_J (n\cdot p_j/n\cdot p_i)} \times \frac{1}{(q_I+q_J)^2}
\times \text{(partitioning/observable)} \ ,
\end{equation}
and a similar one for exchanging $I$ with $J$. If the partitioning
and/or the observable thus guarantees that there will not be a
singularity arising from $q_I$ becoming collinear to $q_J$, the above
configuration is finite also in the limit of $q_I$ becoming soft,
$z_I\to 0$ as long as a hard momentum flow asserts $z_J={\cal
O}(1)$. Notice that we still might need to consider those
contributions in expanding to a certain power of $\lambda$, however in
the present work we shall only be concerned with extracting the
singular behaviour.
The contributions entering in the second line of
\eqref{eqs:paritioning-origin} are slightly more complicated in the
sense that there is no one-to-one correspondence between a collinear
sector and a sub-diagram.
However, since the partitioning factor $w_{\lambda,\sigma;\rho}$
guarantees that there will only be a collinear enhancement as dictated by the
topology $\rho$ and this topology being excluded from the ones summed over,
emission lines in $\lambda$ and $\sigma$ are forced to become
soft for the contribution to be enhanced.
Then, our discussion on the
decomposition of internal lines applies, and the coupling within
common collinear sectors applies otherwise.
Therefore, we can think of these
contributions consisting of interferer lines in both the amplitude and
conjugate amplitude.
\subsection{Iterating the recoil transformation}
\label{sec:recoil-iteration}
Our discussion on the power counting can also be applied to iterating
emission kernels in a Markovian manner, as long as we make sure that
the recoil transformation does not change the general
decomposition. To be precise, we consider, in a $k+1$st emission
step, a decomposition\footnote{We suppress the label for the
collinear sector in this discussion.}
\begin{equation}\label{eq:qkplus1-iteration}
q_{(k+1)}^{\mu} = \frac{1}{\hat{\alpha}_{(k+1)}}(\Lambda_{(k+1)})^\mu {}_\nu
\left(z_{(k+1)} p^{\nu}_{(k)} + \beta_{(k+1)} n_{(k+1)}^{\nu} +
k_{\perp,(k+1)}^\nu\right) \, ,
\end{equation}
where the forward direction has been changed due to the previous
emission and itself admits a decomposition (we do here not distinguish
what branching it belonged to)
\begin{equation}\label{eq:pk-iteration}
p_{(k)}^{\mu} = \frac{1}{\hat{\alpha}_{(k)}}(\Lambda_{(k)})^\mu {}_\nu
\left(z_{(k)} p^{\nu}_{(k-1)} + \beta_{(k)} n_{(k)}^{\nu} +
k_{\perp,(k)}^\nu\right) \, .
\end{equation}
It is therefore tempting to redefine the backward direction as
\begin{equation}
\label{eqs:nredef}
n^\mu_{(k+1)} = \frac{1}{\hat{\alpha}_{(k)}}
(\Lambda_{(k)})^\mu {}_\nu \, n^\nu_{(k)},
\end{equation}
which leads to
\begin{align}
q_{(k+1)}^{\mu} = \frac{1}{\hat{\alpha}_{(k+1)} \hat{\alpha}_{(k)} }(\Lambda_{(k+1)})^\mu {}_\nu (\Lambda_{(k)})^\nu {}_\rho
&\left[z_{(k+1)} z_{(k)} p^{\rho}_{(k-1)}
+ \left(\beta_{(k+1)} + z_{(k+1)} \beta_{(k)} \right) n_{(k)}^{\rho} \right. \nonumber \\
& {}+ \left.z_{(k+1)} k_{\perp,(k)} +\hat{\alpha}_{(k)} (\Lambda_{(k)}^{-1})^\rho {}_\sigma \, k_{\perp,(k+1)}^\sigma \right] \, ,
\end{align}
However, this might actually not be sufficient to guarantee
that the mapping iterates the momentum parametrization in the way that
would not change our general power counting arguments.
In particular
this happens if, for an emission off of a hard line, we allow a
transverse component assigned through the emission process. In this
case, iterating the above to the second emission will generate a
transverse momentum which, despite being still orthogonal to the
backward direction chosen initially by virtue of \eqref{eqs:nredef},
will not be orthogonal to the forward direction anymore, \textit{i.e.}\
\begin{equation}
p_{(k-1)} \cdot (\Lambda^{-1}_{(k)} k_{\perp,(k+1)}) = -\frac{\hat{\alpha}_{(k)} }{z_{(k)}} k_{\perp,(k)} \cdot (\Lambda^{-1}_{(k)} k_{\perp,(k+1)}) \neq 0,
\end{equation}
potentially introducing azimuthal correlations between the emissions
and spoiling our power counting arguments. As already mentioned, this
can only be avoided by not introducing transverse momentum components
for hard lines in the first place, which we will call ``unbalanced
mapping'', or by parametrizing two or more emissions in one step. In
Sec.~\ref{sec:Mapping}, we explore mappings of this kind in detail. We
believe that this discussion intimately connects to the findings one
of the authors and others have been highlighting in \cite{Forshaw:2020wrq,Gieseke:2003rz,Dasgupta:2018nvj,Dasgupta:2020fwr}. We
therefore do not limit the discussion of our mappings to include
balance of transverse recoil, and leave this open to an algorithmic
choice which can, however, crucially impact the accuracy of algorithms
build on top of our calculations. Similar remarks apply to the
ordering variable, which should then be chosen, in line with the
partitioning which has been used to separate different collinear
sectors, to not spill the configurations we have been discussing
within our power counting arguments. We also stress that the above way
of re-defining the backward (gauge) vector is crucial to establish
that the recoil transformation factors out of the amplitude in the
same homogeneous way as discussed earlier, since the amplitude
additionally depends on the backward direction $n$. In the presence
of non-coloured (or coloured, massive) objects, the recoil scheme
above can easily be generalized and still be implemented by virtue of
a single Lorentz transformation; however the amplitude will not easily
satisfy a scaling property since the massive momenta cannot be
rescaled in order to remain on their definite, non-vanishing mass
shell. However, in the singular regions, the scaling factor
$\hat{\alpha}$ will in any case tend to unity such that no additional
complication should arise in this case.
Depending on how the transverse recoil has been chosen, different
diagrams then contribute after decomposing the internal lines into the
Sudakov decomposition and applying the vertex rules above. We will
highlight a few examples below, and defer a more detailed analysis to
Sec.~\ref{sec:Kernels}, where we also discuss the relation to splitting functions
and the soft limit, as well as the strategy employed for the dipole
subtraction terms in \cite{Catani:1996vz}.
\subsection{Local backward direction}\label{sec:local-backward}
Provided we have decomposed the momenta into a component towards a certain hard
direction, but with an arbitrary backward direction,
\begin{equation}
q^\mu = \tilde{z}\, p^\mu + \frac{\tilde{p}_\perp^2}{\tilde{z} \, 2p\!\cdot\!\tilde{n}} \tilde{n}^\mu + \tilde{k}_\perp \ ,
\end{equation}
with $\tilde{k}_\perp\cdot p = \tilde{k}_\perp\cdot \tilde{n} = 0$,
$\tilde{k}_\perp^2 = -\tilde{p}_\perp^2$ and $p\cdot \tilde{n}>0$ as
usual, we can uniquely relate this to our global parametrization with
respect to $p$, $n$ and an appropriate transverse momentum $k_\perp$
as
\begin{equation}
p_\perp^2 = \tilde{p}_\perp^2 R \ , \qquad
z = \tilde{z} R \ , \qquad
R = 1 + \frac{1}{\tilde{z}} \frac{n\cdot \tilde{k}_\perp}{n\cdot p}
+ \frac{\tilde{p}_\perp^2}{2\tilde{z}^2} \frac{n\cdot \tilde{n}}{n\cdot p\ p\cdot \tilde{n}} \ ,
\end{equation}
and
\begin{equation}\label{eq:kperp-redef-local-ni}
k_\perp^\mu = \frac{\tilde{p}_\perp^2}{2\tilde{z}}
\left(\frac{\tilde{n}^\mu}{\tilde{n}\cdot p} - \frac{n^\mu}{n\cdot
p}\right) +\tilde{z}(1-R)p^\mu + \tilde{k}_\perp^\mu \ .
\end{equation}
Hence the leading scaling in the soft and collinear limits is the same
in between the different parametrizations: the $R$ factor is of ${\cal
O}(1)$ in each of the soft and collinear limits, and the transverse
momentum has the same leading scaling in both parametrizations, as
well. The above result will also allows us to formulate local recoil
schemes in which we choose a backward direction per collinear
configuration, and to properly link this parametrization to our power
counting rules.
\subsection{Scaling of individual internal lines and vertices}
\label{sec:scaling-of-lines}
In order to analyse how the individual diagrams scale in the relevant
limits, we need to consider external lines carrying a hard momentum
(which we refer to as `resolved' in the sense of a strong ordering),
internal lines with a hard momentum and a momentum composed of soft,
collinear or soft and collinear momenta. On top of this we consider
`unresolved' lines, which can become arbitrarily soft and/or collinear
and for which both external as well as internal lines deliver the same
scaling. It is clear that this distinction is only making sense if we
separate the contributions in such a way that we need to consider
collinear singular configurations only with respect to one line
carrying a hard momentum, and soft contributions otherwise. Achieving
such a separation will be discussed in
Sec.~\ref{sec:partitioning-algorithm}, where we introduce one example
of partitioning factors allowing for such a separation.
From the general form of the Sudakov decomposition
\eqref{eq:Sudakov-decomp-combined} we then conclude that we need to
assign scaling factors as outlined in Tables~\ref{tab:power-counting1}
and \ref{tab:power-counting2}, respectively, where we refer to the
different momentum flows as:
\begin{itemize}
\item h = external line with all momenta hard, possibly perturbed by a
transverse momentum with respect to the hard direction, though we
also consider the case in which the original direction is kept;
\item h+c = a sum of hard and collinear momenta;
\item h+s = a sum of hard and soft momenta; and
\item h+c+s = a sum of hard, collinear and soft momenta, for which,
owing to the lower scaling power, it mostly is the soft limit which
determines the properties.
\end{itemize}
For lines which are allowed to become arbitrarily unresolved, we refer to
\begin{itemize}
\item s = a sum of purely soft momenta, or a single soft momentum;
\item c = a sum of purely collinear momenta, or a single collinear momentum; and
\item s+c = a combination of soft and collinear momenta.
\end{itemize}
Notice that the lines with the white square need to be considered as
${\cal O}(1)$ in any of the unresolved limits.
\input{diagrams/scaling-table-1}
In \tabref{tab:power-counting1}, we show the scaling of hard lines (emitters or spectators) in all possible combinations of hard, soft and collinear momenta shown above.
The first rows in each block correspond to the case where the $k_\perp$
recoil is included in the emitter mapping.
The second rows correspond to the
case without a $k_\perp$ component which applies to spectator lines
in general and can apply to emitters depending on the mapping used.
Both of these mapping types are discussed in
\secref{sec:Mapping}. The hard lines are signified by their horizontal orientation.
In \tabref{tab:power-counting2}, vertical lines are used to refer to unresolved (soft or collinear) lines and the scaling is shown for the two different `box' contributions.
For both the hard and unresolved lines, the white
box contributions do not have a scaling.
Notice that we need to
complete this picture by propagator factors which scale as
$1/\lambda^2$ in the hard+collinear configuration, $1/\lambda$ in the
hard+soft configuration as well as $1/\lambda$ in the
hard+collinear+soft configuration. Completely unresolved propagators
scale as $1/\lambda^2$ in a collinear or soft configuration, and as
$1/\lambda$ in a soft+collinear configuration.
\input{diagrams/scaling-table-2}
\section{Partitioning}
\label{sec:Paritioning}
An intrinsic assumption of our power counting is that we envisage to
employ a partitioning of the soft behaviour into mutually exclusive
collinear limits. This is important in order to organize a given
observable into deviations from an ideal jet topology, but it will
also give us a key to parametrize a kinematic mapping suited to
collinear branching along a given direction and is thus of crucial
importance to set up a parton shower algorithm. In this section we
generalize both of the typically exploited partitionings to the
multi-emission case. With the case of multiple emissions there are
often multiple collinear combinations which can also be of different
orders \textit{i.e.}\ double or triple collinear. Whether a diagram contributes
to a collinear limit or not is determined by the presence of internal
propagators going on-shell in the respective limit. For $k$ emissions,
each diagram carries $2k$ internal propagators which can independently
become singular in different collinear limits. It is the goal of our
algorithm to partition these propagator factors into a set of
splitting kernels, such that each kernel is only singular in the
\emph{one} collinear configuration it addresses. The partitioning
factors themselves only contain soft singularities and for a given
diagram add up to one across all collinear configurations. After
partitioning, a specific momentum mapping can be applied to the
amplitudes which contribute to a kernel, as the emitter and emission
momenta can be labelled.
\subsection{Basic purpose}\label{sec:basic-example-partitioning}
The approach we use is best explained in a simple example of a
rational function $f$ singular in two independent variables $x_1$ and
$x_2$.
\begin{align}
f(x_1,x_2) = \frac{n(x_1,x_2)}{S_1(x_1,x_2) S_2(x_1,x_2)},
\end{align}
where $n(x_1,x_2)$ is a polynomial. We are firstly interested in the
leading collinear singular behaviour for some specific configuration
which corresponds to one of the $x_i \rightarrow 0$. Here, this is
parametrized by $S_1\rightarrow 0$ for $x_1\rightarrow 0$ while $
S_2\neq 0$ and vice versa for $x_2\rightarrow 0$. By defining the
partitioning factors
\begin{align}
\mathbb{P}^{(f)}_{(x_1)} = \frac{S_2}{S_1+S_2}, \quad \mathbb{P}^{(f)}_{(x_2)} = \frac{S_1}{S_1+S_2},
\end{align}
we can decompose $f$ into
\begin{align}\label{eq:partitioning-ex-1}
f = \left[\mathbb{P}^{(f)}_{(x_1)} + \mathbb{P}^{(f)}_{(x_2)} \right] f = \frac{n(x_1,x_2) }{S_1(S_1+S_2)} + \frac{n(x_1,x_2) }{S_2(S_1+S_2)}.
\end{align}
This allows us to define the splitting kernels
\begin{align}\label{kernel-example}
\mathbb{U}_{(x_i)} = \mathbb{P}^{(f)}_{(x_i)} f
= \frac{1}{S_i} \frac{n(x_1,x_2)}{S_1+S_2}.
\end{align}
What we have achieved is a decomposition into objects which are
singular solely in one of the variables. These objects will later be
identified with so called splitting kernels. Moreover, the second
factor on the right hand side of \eqref{kernel-example} is
non-singular in any single $x_i\rightarrow 0$ collinear limit and
therefore only contains soft singularities.
Note that in this simple case, it is possible to set $x_i=0$ in the
numerator polynomial $n(x_1,x_2)$ without loss of information for the
leading singular behaviour. Nevertheless, this is not possible in
more complicated cases where the $x_i$ can appear to some power $>1$
in the denominator. Lastly, by only using the singular variables
$x_i$ in the partitioning, we make sure that we smoothly approach the
original collinear singular behaviour of the function when both
variables go to zero simultaneously. In the actual partitioning
algorithm, this corresponds to keeping the correct soft-singular
behaviour.
A second interesting option for partitioning is given by means of subtractions
instead of partitioning factors of \eqref{eq:partitioning-ex-1}. It
reads
\begin{equation}
f = \frac{1}{2}\left[f -\Delta_2 + \Delta_1\right] + (1 \leftrightarrow 2),
\end{equation}
where
\begin{equation}
\Delta_1 = \frac{1}{S_1 \, S_2\vert_{x_1 \to 0} }, \quad \Delta_2 = \frac{1}{S_2 \, S_1\vert_{x_2 \to 0} }.
\end{equation}
Here we use the fact that $S_1(x_2 \to 0)$ and $S_2(x_1 \to 0)$ are non-zero.
In this case, a
splitting kernel can be defined as
\begin{equation}
\mathbb{U}_{(x_1)} = \frac{1}{2}\left[\frac{1}{S_1 S_2} - \Delta_2 + \Delta_1\right].
\end{equation}
This again gives a function which shows no singularity when $x_2
\rightarrow 0$, but reproduces the original singular behaviour when
$x_1\rightarrow 0$. We discuss this type of partitioning in
Sec.~\ref{sec:AOPartitioning}.
\subsection{Partial fractioning partitioning algorithm}\label{sec:partitioning-algorithm}
As a first non-trivial example for our algorithm, we present the
approach for the case of two emissions in this section. Firstly, it is
expedient to define the various limits we are interested in. These
can be represented by the set of collinear configurations $\mathbf{C}$
for a given number of partons and emissions. In the two emission
case, this set reads
\begin{equation}
\mathbf{C} = \{ (i\parallel j \parallel k), (i \parallel j \parallel l), \dots, (i \parallel j)(k \parallel l),\dots \},
\end{equation}
where the notation $(i \parallel j)$ stands for two different external
partons with momenta $q_i$ and $q_j$ becoming collinear. It consists
of all configurations contributing to a triple collinear limit, \textit{i.e.}\
all triplets and pairs-pairs one can build from the set of external
partons.
Next, we define
the set $\mathbf{C}^{(d)}$ which contains only the configurations in which
diagram $d$ can become singular. As an example, we take the two
emission interference diagram $A^{(1)}$ (see
Fig.~\ref{fig:triplet-triplet-diagrams} in
App.~\ref{app:two-emission-diagrams}), of which the propagator factors
$\mathcal{P}(A_1)$ are given by
\begin{align}\label{eq:propagators-A1}
\mathcal{P}(A_1) = \frac{1}{S_{ij} \, S_{ijk} \, S_{kl} \, S_{jkl}}.
\end{align}
Here, we have used the notation
\begin{align}
S_{ij} = S(q_i, q_j) \equiv (q_i + q_j)^2.
\end{align}
The singular collinear configurations and respective factors are collected in Table~\ref{tab:singular-factors-A1}.
Then, $\mathbf{C}^{(A_1)}$ is represented by the first column of the table.
\begin{table}[h]
\begin{center}
\begin{tabular}{c|cc}
Configuration & Vanishing & Non-vanishing \\
\hline
$(i \parallel j \parallel k)$ & $S_{ij}S_{ijk}$ & $ S_{kl} S_{jkl} $ \\
$(i \parallel j \parallel l)$ & $S_{ij}$ & $ S_{ijk}S_{kl}S_{jkl} $ \\
$(i \parallel k \parallel l)$ & $S_{kl}$ & $ S_{ij}S_{ijk}S_{jkl} $ \\
$(j \parallel k \parallel l)$ & $S_{kl} S_{jkl}$ & $ S_{ij} S_{ijk} $ \\
$(i \parallel j), (k \parallel l)$ & $S_{ij} S_{kl}$ & $ S_{ijk} S_{jkl} $ \\
\end{tabular}
\end{center}
\caption{Singular configurations and the relevant propagator factors for the two emission diagram $A_1$.}
\label{tab:singular-factors-A1}
\end{table}
Then we can define $\mathbf{S}_c^d$, the set of vanishing S-invariants
contained in diagram $d$ for configuration $c$, which corresponds to
the second column in Table~\ref{tab:singular-factors-A1}. The
partitioning factors for some singular configuration $c$ and diagram
$d$ can then generally be defined as
\begin{align}\label{partitioning-definition}
\mathbb{P}_c^{(d)} \equiv \frac{ F_c^{(d)} }{\mathbb{F}^{(d)} },
\end{align}
where the cancelling factors $F_c^{(d)}$ are given by
\begin{align} \label{cancelling-factors}
F^{(d)}_{c} = \Big( \prod\limits_{S_{c'} \in \mathbf{S}_c^{(d)}} S_{c'} \, \mathcal{P}(\mathcal{A}^{(d)}) \Big)^{-1} \varsigma^{p},
\end{align}
and
\begin{equation}\label{sum-cancelling-factors}
\mathbb{F}^{(d)} \equiv \sum\limits_{c \in \mathbf{C}^{(d)}} F_c^{(d)}.
\end{equation}
The cancelling factors $F^{(d)}_c$ correspond to the last column in Table~\ref{tab:singular-factors-A1}.
Lastly, we have introduced the scale $\varsigma^p$ in order to keep
the partitioning factors dimensionless.
The power $p$ needs to be
chosen such that all cancelling factors carry the same mass
dimension.\footnote{Note that this scale could be avoided by only
partitioning into the \emph{leading} singular behaviours. The caveat
is that there are diagrams for which it is \emph{a priori} unclear
which of the singular structures will give leading contributions to
some kernel. Therefore, it is safer in this sense to simply partition
into all possible singular structures.}
In the following, we assume that $i$ and $l$ are hard partons. For
topology $A^{(1)}$, this provokes that we only need to deal with the three
leading singular configurations of
Table~\ref{tab:singular-factors-A1}. Its cancelling factors are
\begin{subequations}
\begin{eqnarray}
F^{(A^{(1)})}_{(ijk)} &=& \left(S_{ij}S_{ijk} A^{(1)} \right)^{-1} = S_{kl} S_{jkl}, \\
F^{(A^{(1)})}_{(jkl)} &=& \left(S_{kl}S_{jkl} A^{(1)} \right)^{-1} = S_{ij} S_{ijk}, \\
F^{(A^{(1)})}_{(ij)(kl)} &=& \left(S_{ij}S_{kl} A^{(1)} \right)^{-1} = S_{ijk} S_{jkl},
\end{eqnarray}
\end{subequations}
which corresponds to the entries of the third column of Table~\ref{tab:singular-factors-A1}.
Then, the amplitude's fully partitioned propagator is given by
\begin{align} \label{eq:A1-partitioned}
\sum\limits_{c\in \mathbf{C}_{A^{(1)}}} \mathbb{P}^{(A^{(1)})}_c \mathcal{P}( A^{(1)})
= \frac{1}{S_{kl}S_{jkl} + S_{ij}S_{ijk} + S_{ijk}S_{jkl} } \times \left(\frac{1}{S_{ij}S_{ijk}} + \frac{1}{S_{kl}S_{jkl}} + \frac{1}{S_{ij}S_{kl}} \right).
\end{align}
The first,second and third term will give leading singular
contributions to splitting kernels $\mathbb{U}_{(ijk)}$,
$\mathbb{U}_{(jkl)}$ and $\mathbb{U}_{(ij)(kl)}$, respectively.
Note that topologies with only one hard line, \textit{i.e.}\ the self-energy like ones need no partitioning due to the absence of a recoiler that the emissions could become collinear to.
In \tabref{tab:scaling-two-emission}, we show the collinear and soft scalings for the two-emission topologies of App.~\ref{app:two-emission-diagrams} together with their fractional partitioning factors for the $(i\parallel j \parallel k)$ configuration.
\begin{table}[h]
\begin{center}
\begin{tabular}{c|cccc}
& CC & CS & SC & SS \\
\hline
$ A^{(1)} $ & $1/\lambda^4$ & $1/\lambda^4$ & $1/\lambda^2$ & $1/\lambda^4$\\
$ A^{(2)} $ & $1/\lambda^4$ & $1/\lambda^3$ & $1/\lambda^3$ & $1/\lambda^4$ \\
$ A^{(3)} $ & $1/\lambda^4$ & $1/\lambda^3$ & $1/\lambda^2$ & $1/\lambda^4$ \\
$ A^{(4)} $ & $1/\lambda^6$ & $1/\lambda^4$& $1/\lambda^3$ & $1/\lambda^5$ \\
$ A^{(5)} $ & $1/\lambda^6$ & $1/\lambda^3$ & $1/\lambda^3$ & $1/\lambda^6$ \\
$ B^{(1)} $ & $1/\lambda^6$ & $1/\lambda^6$& $1/\lambda^3$ & $1/\lambda^4$ \\
$ B^{(2)} $ & $1/\lambda^6$ & $1/\lambda^4$ & $1/\lambda^5$ & $1/\lambda^4$ \\
$ B^{(3)} $ & $1/\lambda^6$ & $1/\lambda^5$ & $1/\lambda^3$ & $1/\lambda^5$ \\
$ B^{(4)} $ & $1/\lambda^6$ & $1/\lambda^6$& $1/\lambda^3$ & $1/\lambda^4$ \\
$ B^{(5)} $ & $1/\lambda^6$ & $1/\lambda^4$ & $1/\lambda^5$ & $1/\lambda^4$ \\
$ B^{(6)} $ & $1/\lambda^6$ & $1/\lambda^5$ & $1/\lambda^3$ & $1/\lambda^5$ \\
$ X^{(1)} $ & $1/\lambda^4$ & $1/\lambda^4$ & $1/\lambda^4$ & $1/\lambda^4$ \\
$ X^{(2)} $ & $1/\lambda^4$ & $1/\lambda^4$ & $1/\lambda^2$ & $1/\lambda^2$ \\
$ E^{(1)} $ & $1/\lambda^8$& $1/\lambda^6$ & $1/\lambda^4$ & $1/\lambda^4$ \\
$ E^{(2)} $ & $1/\lambda^8$ & $1/\lambda^5$ &$1/\lambda^5$ & $1/\lambda^4$ \\
$ E^{(3)} $ & $1/\lambda^8$& $1/\lambda^5$ & $1/\lambda^4$& $1/\lambda^5$ \\
$ E^{(4)} $ & $1/\lambda^8$& $1/\lambda^5$ & $1/\lambda^4$& $1/\lambda^5$ \\
$ E^{(5)} $ & $1/\lambda^8$& $1/\lambda^4$ & $1/\lambda^4$& $1/\lambda^6$ \\
\end{tabular}
\end{center}
\caption{Scaling for propagator times partitioning factor of two emission single emitter topologies when partitioned to $(i\parallel j \parallel k)$. Here, `CC' refers to the triple collinear limit where $i||j||k$, `CS' refers to $(i \parallel j)$ with soft $k$, `SC' to $(i \parallel k)$ and $j$ soft and `SS' is the double soft limit where both $j$ and $k$ are soft.}
\label{tab:scaling-two-emission}
\end{table}
Eventually, these scalings need to be combined with the respective numerators scalings discussed in Sec.~\ref{sec:scaling-of-lines} in order to determine which topology contributes to a splitting kernel at a given power in $\lambda$.
We show the resulting table in Sec.~\ref{sec:Applications}.
The partitioning factors as defined in \eqref{partitioning-definition}
fulfil two important requirements as already mentioned in
Sec.~\ref{sec:basic-example-partitioning}., \textit{viz.}\
\begin{enumerate}
\item Non-singular behaviour in any collinear limit,
\item Soft-collinear and purely soft limits are reproduced correctly.
\end{enumerate}
The first property is guaranteed by the fact that the sum in
\eqref{sum-cancelling-factors} does not vanish in \emph{any} collinear
limit. One can show this fact by realising that for all given
collinear configurations $c \in \mathbf{C}^{(d)}$, the partitioning
factors $F^{(d)}_{c} \neq 0$ per definition. An example is
$c=(i||j)(k||l)$ for which $F^{(A^{(1)})}_{c_1} = \varsigma \, S_{ijk}
S_{jkl} \neq 0$ in said configuration. It can be that $F^{(d)}_{c}$
vanishes in another configuration $c'$, but not all of them
simultaneously. The reason is that the in the case where partitioning
is carried out, there are necessarily recoiler partons present leading
to the fact that not all propagator factors of the respective diagram
vanish simultaneously. In all other configurations $c
\notin \mathbf{C}^{(d)}$, we know that $(A^{(d)})^{-1} \neq 0$ and
therefore, neither the partitioning factors $F^{(A_d)}_c$ can vanish
here. Thus, there is no configuration where all $F^{(A_d)}_{c} = 0$
simultaneously for a given diagram $d$.
Secondly, the property of the correct scaling behaviour in soft- and
soft-collinear limits is guaranteed by the fact that for a given
diagram, we only use the $S$-invariants appearing in the respective
diagram to construct the partitioning factors, which form a partition
of unity. Therefore, there is no spurious finite remainder in
$\mathbb{F}^{(d)}$ that could spoil the approach of the original soft
singularity.
We stress that the analysis in this section has focused on the case
that we have identified certain partons as those carrying the hard
momentum. Additional combinatorics are thus needed if several partons
in the same jet could carry the hard momentum, but in an algorithmic
implementation, see Sec.~\ref{sec:Applications}, we will be able to
decide at each branching which of the daughter particles is to carry
the hard momentum, and whether that particular configuration will then
contribute to the leading behaviour, \textit{e.g.}\ our power counting directly
implies that a hard quark transitioning into a soft quark and a hard
gluon will be immediately suppressed by the $\sqrt{z_i z_j}$ factor we
associate to the effective quark-gluon vertex.
\subsection{Subtraction (angular ordered) partitioning algorithm}
\label{sec:AOPartitioning}
An alternative approach to partitioning is to isolate and subtract
collinear divergencies in a way that a single collinear divergence
remains in the diagram just considered. For one emission, this
procedure leads to angular ordering, essentially using
\begin{equation}
\label{eq:angular-ordering}
\frac{1}{S_{ij}\, S_{jk}} = \frac{1}{S_{ik}} \frac{1}{2 E_j^2}
\left(\frac{n_i\cdot n_k}{n_i\cdot n_j\, n_j\cdot n_k} -
\frac{1}{n_j\cdot n_k} + \frac{1}{n_i\cdot n_j}\right) +
(i\leftrightarrow k) \ ,
\end{equation}
where we have decomposed the momenta into energies and directions,
$p_j = E_j\, n_j$;
in this case, any measure of energy could have been
used, \textit{i.e.}\ in general we consider $E_i = T\cdot p_i$ with some
timelike vector $T$. In the above expression, the first term is
singular only in the $(i||j)$ configuration, while the second one is
singular in the complementary $(k||j)$ limit.
In this section we show
how this paradigm can be generalized to a larger number of emissions
and we find that the resulting partitioning shows the same scaling
behaviour as the partial fractioning variant, thus allowing to use the same power counting.
The basic idea of the subtraction partitioning now starts from a
slight re-write of the angular ordering logic; to this extent rewrite
\eqref{eq:angular-ordering} as
\begin{equation}
\frac{1}{S_{ij}\, S_{jk}} = \frac{1}{2}\left(\frac{1}{S_{ij}S_{jk}} -
\frac{E_k}{E_j\, S_{ik}} \frac{1}{S_{jk}} +
\frac{E_i}{E_j\, S_{ik}}\frac{1}{S_{ij}}\right) + (i\leftrightarrow
k) \ .
\end{equation}
The second term in the parentheses has been constructed to subtract
off the $1/S_{jk}$ divergence upon replacing $1/S_{ij}$ by its
limiting expression for $(j||k)$, \textit{i.e.}\
\begin{equation}
S_{ij} \xrightarrow{(j\parallel k)} E_i E_j \, n_i \!\cdot\! n_k = \frac{E_j}{ E_k} S_{ik}
\end{equation}
and the second term adds back what
has been subtracted in the term with $i$ and $k$ interchanged.
This decomposition allows to define the partitioned propagator
\begin{equation}
{\mathbb P}_{(i \parallel j)}\left[\frac{1}{S_{ij}\, S_{jk}}\right] =
\frac{1}{2} \left(\frac{1}{S_{ij}\, S_{jk}} - \Delta_{(j\parallel k)} + \Delta_{(i \parallel j)} \right),
\end{equation}
with
\begin{equation}
\Delta_{(i\parallel j)} = \frac{E_i}{E_j} \frac{1}{S_{ik}S_{ij}}, \quad \Delta_{(j\parallel k)} = \frac{E_k}{E_j} \frac{1}{S_{ik}S_{jk}}.
\end{equation}
It fulfils the expectation of giving back the original singular
scaling behaviour in the $(i \parallel j)$ case while being
non-singular when $(j \parallel k)$.
In fact, this construction is algorithmic and can be generalized to
more than one emission; the only complication which arises is in the
fact that several collinear limits might be overlapping, \textit{i.e.}\
some of the $\Delta$ factors themselves contain sub-singularities, and
in turn these need to be subtracted out of the possible subtraction
terms to guarantee that one resulting partition summand truly only
reflects one collinearly singular configuration.
In particular we can write the procedure to construct subtraction
terms as
\begin{equation}
\Delta_{\tau_1;\tau_2,...,\tau_m}[P] =
{\mathbb F}_{\tau_1}[P]\left({\mathbb S}_{\tau_1}[P] -
\overline{\sum\limits_{\mathcal{S}/ \tau_1}}
\Delta_{\tau_{i_1};\tau_{i_2},...,\tau_{i_{m-1}}}\left[{\mathbb S}_{\tau_1}[P]\right] \right) \ ,
\end{equation}
where ${\mathbb F}_{\tau_1}[P]$ indicates to collect those factors
from the propagator factor $P$ which are non-singular in the limit
identified by $\tau_1$ and to replace them by their limiting
expressions in this limit. Conversely, ${\mathbb S}_{\tau_1}[P]$
indicates to take the singular factors. Moreover, $\mathcal{S}$ is
the set of singular configurations of ${\mathbb S}_{\tau_1}[P]$ and
the summation symbol stands for dividing by the number of terms of the
sum. The indices after the semicolon in $\Delta$ signify the order of
the limits taken beforehand which can play a role if these limits do
not commute in terms of the partitioning $\Delta$'s. We then proceed
to subtract the remaining overlapping singular limits, and we define
\begin{equation}
\Delta_{\tau_1;\tau_2,...,\tau_m}[P] = \Delta_{\tau_2;\tau_3,...,\tau_m}[P] \qquad \text{if }P \text{ not }\tau_1\text{ collinear singular,}
\end{equation}
$\Delta_{\sigma;}(P) = 0$ if $P$ is non-singular in $\sigma$, and
$\Delta_{\sigma;}(P) = {\mathbb F}_{\sigma}[P]{\mathbb S}_{\sigma}[P]$
otherwise. The resulting partitioned propagator then is
\begin{equation}\label{eq:AO-partitioned-prop}
{\mathbb P}_{\sigma}[P] = \frac{1}{m} \left( P +(m-1)\Delta_{\sigma;\tau_1,...,\tau_{m-1}}[P] -
\sum_{i=1}^{m-1} \Delta_{\tau_i;\tau_1,...,\tau_{i-1},\sigma,\tau_{i+1},...,\tau_{m-1}}[P]\right) \ ,
\end{equation}
where $\sigma,\tau_1,...,\tau_{m-1}$ denote the $m$ configurations in
which $P$ can develop collinear singularities.
As an example, we show this version of the partitioning algorithm
again for the amplitude $A^{(1)}$ with propagators factors of
\eqref{eq:propagators-A1}. The configurations where
$\mathcal{P}(A^{(1)})$ has a leading singularity are
\begin{equation}
\begin{split}
\tau_1 &= (i\parallel j \parallel k),\\
\tau_2 &= (j \parallel k \parallel l), \\
\tau_3 &= (i\parallel j) (k \parallel l).
\end{split}
\end{equation}
Therefore, $m=3$ in this case. First, we collect the limiting
expressions in these configurations, \textit{i.e.}\
\begin{equation}
\begin{split}
\mathbb{F}_{\tau_1} [\mathcal{P}(A^{(1)})]
&= \frac{1}{S_{kl} S_{jkl}}\bigg\rvert_{\tau_1} = \frac{E_i^2}{E_k(E_j + E_k)} \frac{1}{S_{il}^2}, \\
\mathbb{F}_{\tau_2} [\mathcal{P}(A^{(1)})]
&= \frac{1}{S_{ij} S_{ijk}}\bigg\rvert_{\tau_2} = \frac{E_l^2}{E_j(E_j + E_k)} \frac{1}{S_{il}^2}, \\
\mathbb{F}_{\tau_3} [\mathcal{P}(A^{(1)})]
&= \frac{1}{S_{ijk} S_{jkl}}\bigg\rvert_{\tau_3} = \frac{E_i^2 E_l^2}{E_k(E_i + E_j) \, E_j(E_k +E_l)} \frac{1}{S_{il}^2}.
\end{split}
\end{equation}
Note that a single index specification is sufficient here, because the
order of taking the limits does not matter in this example. Then, we
can construct the first subtraction term, \textit{viz.}\
\begin{equation}\label{eq:AO-examples-A1-Delta1}
\Delta_{\tau_1} [\mathcal{P}(A^{(1)})] = \mathbb{F}_{\tau_1} [\mathcal{P}(A^{(1)})]
\sum\limits_{\tau \in \mathbf{S}/\tau_1} \left( \frac{1}{S_{ij}S_{ijk}} - \Delta_{\tau}\left[ \frac{1}{S_{ij}S_{ijk}} \right]\right).
\end{equation}
This term is supposed to only cancel $(i\parallel j \parallel
k)$-singularities. Therefore, we subtract off the overlapping
sub-singularity in the parentheses via
\begin{equation}
\Delta_{\tau_3}\left[\frac{1}{S_{ij}S_{ijk}} \right]
= \mathbb{F}_{\tau_3}\left[\frac{1}{S_{ijk}}\right] \frac{1}{S_{ij}},
\end{equation}
where $\tau_3$ is the only relevant configuration in the sum of
\eqref{eq:AO-examples-A1-Delta1}, because this sub-propagator is
non-singular in $\tau_2$. The same procedure applies to the other two
subtraction terms. Eventually, we find
\begin{equation}
\begin{split}
\Delta_{\tau_1} [\mathcal{P}(A^{(1)})]
&= \frac{E_i^2}{ E_k(E_j+E_k)} \frac{1}{S_{il}^2} \left(\frac{1}{S_{ij}S_{ijk}} - \frac{E_i E_l }{E_k (E_i + E_j)} \frac{1}{S_{il} S_{ij}} \right), \\
\Delta_{\tau_2} [\mathcal{P}(A^{(1)})]
&= \frac{E_l^2}{ E_j(E_j+E_k)} \frac{1}{S_{il}^2} \left(\frac{1}{S_{kl}S_{jkl}} - \frac{E_i E_l }{E_j (E_k + E_l)} \frac{1}{S_{il} S_{kl}} \right), \\
\Delta_{\tau_3} [\mathcal{P}(A^{(1)})]
&= \frac{E_i^2 E_l^2}{ E_k(E_i + E_j) \, E_j(E_k +E_l)} \frac{1}{S_{il}^2} \left(\frac{1}{S_{ij}S_{kl}} - \frac{E_i }{E_k} \frac{1}{S_{il} S_{ij}} - \frac{E_l }{E_j} \frac{1}{S_{il} S_{kl}} \right).
\end{split}
\end{equation}
Then, using \eqref{eq:AO-partitioned-prop}, the partitioned propagator
of $A^{(1)}$ is given by
\begin{equation}
\begin{split}
\mathcal{P}(A^{(1)})
&= \frac{1}{3} \left(\frac{1}{S_{ij} S_{ijk} S_{kl} S_{jkl}} + 2 \Delta_{\tau_1}[\mathcal{P}(A^{(1)})] - \Delta_{\tau_2}[\mathcal{P}(A^{(1)})] -\Delta_{\tau_3}[\mathcal{P}(A^{(1)})] \right), \\
&+ \frac{1}{3} \left(\frac{1}{S_{ij} S_{ijk} S_{kl} S_{jkl}} - \Delta_{\tau_1}[\mathcal{P}(A^{(1)})] + 2 \Delta_{\tau_2}[\mathcal{P}(A^{(1)})] -\Delta_{\tau_3}[\mathcal{P}(A^{(1)})] \right), \\
&+ \frac{1}{3} \left(\frac{1}{S_{ij} S_{ijk} S_{kl} S_{jkl}} - \Delta_{\tau_1}[\mathcal{P}(A^{(1)})] - \Delta_{\tau_2}[\mathcal{P}(A^{(1)})] +2 \Delta_{\tau_3}[\mathcal{P}(A^{(1)})] \right),
\end{split}
\end{equation}
where the first, second and third line are ${\mathbb
P}_{\tau_1}[\mathcal{P}(A^{(1)})]$, ${\mathbb
P}_{\tau_2}[\mathcal{P}(A^{(1)})]$ and ${\mathbb
P}_{\tau_3}[\mathcal{P}(A^{(1)})]$, respectively.
\section{Momentum mapping}
\label{sec:Mapping}
In this section, we define two instances of a momentum mapping in
terms of the Sudakov-like decomposition
of~\eqref{eq:Sudakov-decomp-single} where momentum conservation
between emitters and emissions is manifest and a global
Lorentz-transformation is introduced for momentum conservation when
emissions are added to a given process, as already advertised in
Sec.~\ref{sec:recoil-iteration}. Using the dictionary of
Sec.~\ref{sec:local-backward}, both versions of the mapping are
compatible with the power counting rules introduced in
Sec.~\ref{sec:scaling-of-lines}.
The momentum mapping is set up in a language inspired by parton
showers. In a parton shower setup, one starts from a set of massless
on-shell momenta of a hard process which are then dressed up with
emissions, which again are massless and on-shell. Diagrammatically,
this can be represented by \input{diagrams/adding-emissions.tex}
where
the $p_i$ are the momenta to which emissions are added with $i\in
\mathbf{S}$ (``splitters'') and the $p_r$ are available for recoil
absorption with $r\in \mathbf{R}$ (``recoilers''). This leads us to
define the momenta after emissions in terms of the forward ($p_i$),
backward ($n_i$) and transverse components ($n_{\perp,l}^{(i)}$) as
\begin{subequations}\label{mapping}
\begin{align}
q_r &\equiv \frac{1}{\hat{\alpha}} \Lambda p_r \ ,
\\
\bar{q}_i &\equiv \frac{1}{\hat{\alpha}} \Lambda \left[ (1 - A_i)p_i
\right] \quad{\text{(unbalanced)}} \ ,
\\
q_i &\equiv \frac{1}{\hat{\alpha}} \Lambda \left[ (1 - A_i)p_i + \big(y_i - \tilde{B}_i\big) n_i - \tilde{n}^{(i)}_\perp \right] \quad{\text{(balanced)}},
\\
\label{eq:kil-mapped}
k_{il} &\equiv \frac{1}{\hat{\alpha}} \Lambda \left[ \alpha_{il} \, p_i + \tilde{\beta}_{il}\, n_i + \sqrt{\alpha_{il} \tilde{\beta}_{il}\, } n^{(i)}_{\perp,l}\right] \ ,
\end{align}
\end{subequations}
where we show one version with balanced ($q_i$) and one with unbalanced ($\bar{q}_i$) transverse components and use the shorthands
\begin{align}
A_i \equiv \sum\limits_{l \in \mathbf{E}_i}
\alpha_{il} \ , \quad
\tilde{B}_i \equiv \sum\limits_{l \in \mathbf{E}_i} \tilde{\beta}_{il} \ , \quad
\tilde{\beta}_{il} = (1-A_i)\beta_{il} \ , \quad
\tilde{n}^{(i)}_\perp \equiv \sum\limits_{l \in
\mathbf{E}_i} \sqrt{\alpha_{il} \tilde{\beta}_{il}}
\, n^{(i)}_{\perp,l} \ ,
\end{align}
Note the inclusion of a Lorentz transformation $\Lambda$ together with
a scaling $\hat{\alpha}$ which are needed for the non-trivial global
recoil and momentum conservation. The latter leads to
\begin{equation}\label{mom-conservation-1}
\sum\limits_{i\in \mathbf{S}} \Big( q_i + \sum\limits_{l \in \mathbf{E}_i} k_{il} \Big) + \sum\limits_{r \in \mathbf{R}} q_r = Q \ ,
\end{equation}
where $Q$ is the original overall momentum transfer
\begin{equation}\label{mom-transfer-Q}
Q \equiv \sum\limits_{i\in \mathbf{S}} p_i + \sum\limits_{r\in \mathbf{R}} p_r \ .
\end{equation}
Inserting \eqref{mapping} into \eqref{mom-conservation-1} gives
\begin{equation}\label{mom-conservation-2}
Q = \frac{1}{\hat{\alpha}} \Lambda \Big[ \sum\limits_{r\in \mathbf{R}} p_r + \Big(\sum\limits_{i \in \mathbf{S}} p_i + y_i n_i \Big) \Big] \ .
\end{equation}
Squaring this equation fixes the scaling to
\begin{equation}
\hat{\alpha}^2 = \frac{(Q + N)^2}{Q^2} \ ,
\end{equation}
where
\begin{equation}
N\equiv \sum\limits_{i \in \mathbf{S}} y_i \, n_i \ .
\end{equation}
Using Lorentz-invariance of amplitudes, $\hat{\alpha}$ provides the
means to implement a global recoil as shown
in~\eqref{eq:amplitudeglobalrecoil}.
The numbers $\alpha_{il}$ and $\beta_{il}$ take values in $(0,1)$ and
parametrize the soft and collinear behaviour of the emission momenta
$k_{il}$, as well as the emitter momenta $q_i$.
In the balanced version of the mapping, $y_i$
quantifies the off-shellness of the emitter-emission system. This can
be seen by squaring the momentum sum for emissions and the emitter,
\textit{i.e.}\
\begin{align}\label{overall-mom-transfer}
\Big(q_i + \sum\limits_l k_{il}\Big)^2 &= y_i \, 2 p_i\!\cdot\! n_i \ .
\end{align}
Therefore, both the soft and collinear limits are parametrized by the
$y_i\rightarrow 0$ limit. The $y_i$ are fixed via the on-shell
relations for the $q_i$, \textit{i.e.}\
\begin{align}\label{eq:yi-fixed}
y_i = (1-A_i) B_i - \frac{(\tilde{n}_\perp^{(i)})^2}{ 2 p_i \cdotp n_i (1-A_i)} \ ,
\end{align}
which shows that a scaling in $y_i$ is fixed by scaling the
$\alpha_{il}$ and $\beta_{il}$. Instead, for the unbalanced version
of the mapping, we formally have $y_i=\tilde{B_i}$ and $\tilde
n_\perp^{(i)} = 0$ and overall momentum transfer is given by
\begin{equation}
\Big(\overline{q}_i + \sum\limits_l k_{il} \Big)^2 = \tilde{B}_i \, 2 p_i \cdot n_i + (\tilde{n}_\perp^{(i)})^2 \ .
\end{equation}
The on-shellness of the emission momenta $k_{il}$ fixes the virtuality
of the transverse components as
\begin{equation}\label{eq:n-perp-square}
\left(n^{(i)}_{\perp,l}\right)^2 = -2 p_i \cdotp n_i \ .
\end{equation}
These components are further determined by the transversality relations
\begin{equation}\label{eq:orthogonality}
p_i \cdotp n^{(i)}_{\perp,l} = 0 \quad \text{and} \quad n_i \cdotp n^{(i)}_{\perp,l} = 0,
\quad \forall i \in \mathbf{S} \text{ and } \forall l \in \mathbf{E}_i \ .
\end{equation}
We can choose $n_i$ locally w.r.t.\ jet directions while
simultaneously using the gauge vector $n$ to expand in a global basis
of spinors and polarization vectors. The effect of such a choice is
discussed at the end of Sec.~\ref{sec:recoil-iteration}. A suitable
choice for the (in principle arbitrary) lightlike backwards components
$n_i$ is
\begin{equation}
n_i^\mu = Q^\mu - \frac{Q^2}{2 p_i \!\cdot\! Q}\, p_i^\mu \ .
\end{equation}
In App.~\ref{app:phase-space}, we discuss the phase space
factorisation in terms using this type of momentum mapping.
The momentum mapping allows us to parametrise the scaling of the
momentum components in the style of SCET \cite{Becher:2014oda}
according to the following table
\begin{center}
\begin{tabular}{c|c c}
$k_{il}$ & $(p_i, n_i , n^{(i)}_{\perp,l})$ & $(\alpha_{il}, y_i, \beta_{il})$ \\ \hline
(forward) collinear & $Q( 1, \lambda^2, \lambda)$ & $(1,\lambda^2,\lambda^2)$\\
soft & $Q(\lambda, \lambda, \lambda)$ & $(\lambda,\lambda,\lambda)$.
\end{tabular}
\end{center}
The overall scaling in $\lambda$ can subsequently be used to study the
IR singular behaviour of an emission amplitude. For this purpose, we
present all relevant dot-products which can appear in squared emission
amplitudes in terms of the momentum mapping of \eqref{mapping}, \textit{viz.}\
\begin{subequations}
\begin{align} \label{eq:S-invariants-new-mapping}
S(q_i,K_i) &= y_i \, 2 p_i \cdotp n_i,\\
S(q_i, k_{il}) &= \frac{2}{\hat{\alpha}^2} \left[ \left( \alpha_{il} (y_i - \tilde{B}_i) + \tilde\beta_{il}(1-A_i) \right) p_i\cdotp n_i - \sqrt{(1-A_i)\alpha_{il} \tilde\beta_{il}}\, n^{(i)}_{\perp,l} \cdotp \tilde{n}^{(i)}_{\perp} \right] \ ,
\\
S(k_{il}, k_{il'}) &= \frac{2}{\hat{\alpha}^2} \left[ (\alpha_{il} \tilde\beta_{il'} + \alpha_{il'} \tilde\beta_{il}) \, p_i \cdotp n_i + \sqrt{\alpha_{il} \tilde\beta_{il} \alpha_{il'} \tilde\beta_{il'}}\, n^{(i)}_{\perp,l} \cdotp n^{(i)}_{\perp,l'} \right] \ ,
\\
S(k_{il}, q_r) &=
\frac{2}{\hat{\alpha}^2} \left[ \alpha_{il}\, p_i \cdotp p_r + \tilde\beta_{il}\, n_i \cdotp p_r
+ \sqrt{\alpha_{il} \tilde\beta_{il}}\, n^{(i)}_{\perp,l} \cdotp p_r
\right] \ , \\
S(q_i, q_r) &=
\frac{2}{\hat{\alpha}^2} \left[ \left(1-A_i\right) p_i \cdotp p_r
+ \left(y_i - \tilde{B}_i\right) n_i \cdotp p_r
- \sqrt{1-A_i}\,\tilde{n}^{(i)}_\perp \cdotp p_r
\right] \ ,
\end{align}
\end{subequations}
where we have used the shorthand
\begin{align}
K_i \equiv \sum\limits_{l \in \mathbf{E}_i} k_{il} \ .
\end{align}
In the unbalanced version of the mapping, the following $S$-invariants change:
\begin{subequations}
\begin{align} \label{eq:S-invariants-new-mapping-unbalanced}
S(\bar{q}_i,K_i) &= \tilde{B}_i \, 2 p_i \cdot n_i + (\tilde{n}_\perp^{(i)})^2 \ ,
\\
S(\bar{q}_i, k_{il}) &= \frac{2 p_i\cdotp n_i}{\hat{\alpha}^2} (1-A_i) \tilde{\beta}_{il} \ ,
\\
S(\bar{q}_i, q_r) &=
\frac{2}{\hat{\alpha}^2} \left[ \left(1-A_i\right) p_i \cdotp p_r
\right] \ ,
\end{align}
\end{subequations}
The $\lambda$-scaling of these invariants for both variants of the mapping can be summarized as
\begin{center}
\begin{tabular}{c|c c c c c}
$k_{il}$ & $S(q_i,K_i)$ & $S(q_i, k_{il})$ & $S(k_{il}, k_{il'})$ & $S(k_{il}, q_r) $ & $S(q_i, q_r) $ \\ \hline
collinear & $\lambda^2$ & $\lambda^2$ & $\lambda^2$ & $1$ & $1$ \\
soft & $\lambda$ & $\lambda$ & $\lambda^2 $ & $\lambda$ & $1$
\end{tabular}.
\end{center}
This information is helpful in determining the set of diagrams that
contributes to a given soft, collinear or mixed limit and it is, of course,
compatible with the scaling we have used in deriving our effective
Feynman rules.
\section{Splitting kernels}
\label{sec:Kernels}
In this section, we will discuss how we can built up full {\it
splitting kernels} in combination of our findings on the power
counting, the partitioning and the underlying momentum mapping. Using
the partitioning factors of Sec.~\ref{sec:partitioning-algorithm},
they can be defined as follows:
\begin{align}\label{kernel-definition}
\mathbb{U}_{c} \equiv \sum\limits_d \left[\mathbb{P}^{(d)}_{c} \mathcal{A}^{(d)} \right].
\end{align}
$\mathcal{A}^{(d)} $ represents a certain topology under consideration,
\textit{i.e.}\ a configuration with a fixed set of propagators in the
amplitude and the conjugate amplitude. We find it convenient to
represent these still as cut diagrams, however it should be clear
that their meaning is to be taken in the sense of the density
operator as discussed in Sec.~\ref{sec:Factorization}.
We choose the latter language with the intent of making an iterative
procedure in treating emissions more tangible. The subscript $c$
stands for the collinear configuration that the kernel is addressing.
An example is the triple-collinear configuration $c = (i \parallel j
\parallel k)$ for the two emission case, where $\mathbb{U}_{(i j k)}$ contains
all leading singular contributions for the momenta $q_i$, $q_j$ and
$q_k$ becoming collinear. The fact that $\mathbb{U}_{(i j k)}$ contains no
other leading singular structures is assured via the partitioning
factors $\mathbb{P}^{(d)}_{ijk}$. Schematically,
\eqref{kernel-definition} can be paraphrased as
\vspace{5pt}
\input{diagrams/splitting-kernel-master}
In the language of cut diagrams, the first term can be read as
\emph{self-energy like} contributions, \textit{i.e.}\ contributions where the
partons on the amplitude and conjugate amplitude side originate from
the same splitter parton. In lightcone gauge, these contributions
contain the leading collinear singularities. The second term in
\eqref{splitting-kernel-master-diagram} refers to contributions where
partons on the amplitude side identified with partons from several
different splitting groups on the conjugate side. These will be
relevant for \emph{mixed} soft-collinear limits. Lastly, the third
term stands for diagrams where none of the splitter partons on the
conjugate side are identified with the splitter on the amplitude side.
They correspond to \emph{interference} diagrams which in lightcone
gauge are relevant for soft limits only.
In addition to the contributions depicted in
\eqref{splitting-kernel-master-diagram}, one can have multiple
emitters on the amplitude side. For two emissions, this corresponds
to configurations contributing to double-unresolved limits such as
$c'=(i \parallel j)(k \parallel l)$ where two sets of partons exhibit
independent collinearities. The construction of the respective
splitting kernels follow the same logic as for the single emitter
case, the main difference being that one can not categorize the
various contributions into self-energy like, mixed and interference
diagrams as easily.
As a next step, we can insert the momentum mapping of
Sec.~\ref{sec:Mapping} for a given configuration $c$ into the
amplitudes entering the splitting kernel $\mathbb{U}_c$. This step
essentially fixes the notion of which partons are considered as
emitters, recoilers and emissions. Having made sure that there are no
other singular configurations that need addressing here, we can be
certain that all leading singular contributions can be captured via
the insertion of the respective momentum mapping.
\subsection{Power counting algorithm}
In order to provide a general rule for which contributions to keep in
terms of our power counting in splitting kernels, several remarks are
in order. First of all, it is easy to show the fact that interference
contributions do not contribute to leading collinear limits using our
formalism. The most important rule to make use of here is the fact
that amplitudes which only contain forward components of the same
collinear sector vanish exactly (this was already mentioned at the end
of Sec.~\ref{sec:decompositions-intro}), \textit{i.e.}\
\input{diagrams/boxed-vertex-whites.tex}
Now, in order for a diagram to contribute in the leading collinear
limit, it must have a scaling of $1/\lambda^{2k}$, where $k$ is the
number of emissions. In interference contributions, only the
propagator factors of the splitter side give a collinear scaling as
long as we use either version of our partitioning algorithm to get rid
of collinearities on the interferer legs. Then, the propagator
factors of the diagram shown in \eqref{eq:k-emmission-interference}
give a scaling of $1/\lambda^{2k}$. This means that such
contributions can only give rise to leading collinearities when the
numerator does not scale, meaning that one would need only the forward
components of each line, or white boxes respectively. Knowing that
the respective vertices exactly vanish on the splitter side, we find
\vspace{4pt}
\input{diagrams/k-emission-interference.tex}
The above holds generically also for appearances of the collinear
three-gluon vertex of \eqref{eq:three-gluon-col-vertex}. This shows
that interferences do not contribute to the leading collinear limits.
For contributions that are relevant for mixed soft-collinear
behaviour, a general discussion is more involved. We start here by
discussing of two emissions, but expect this pattern to hold also for
$k>2$. The leading singular amplitudes will have at least two
$(\perp)$-boxes on the splitter and one on the conjugate side. An
example is
\vspace{4pt}
\input{diagrams/mixed-amp-2-emissions}
The numerator of these leading amplitudes goes as $\lambda_c^3$ while
the denominator goes as $\lambda_c^6$ in the purely collinear limit.
Therefore, soft-collinear mixed contributions do not contribute
leadingly here.
Naively, the collinearly leading power amplitude on the splitter side
is the one with just one internal $(\perp)$-component, but an explicit
check of all possible gluon and quark insertions shows that it
vanishes exactly, \textit{i.e.}\
\smallskip
\input{diagrams/internal-perp}%
This holds when both vertices are of the same type, \textit{i.e.}\ either both
collinear or both with $(\perp)$-only components.
With neither the interference, nor the mixed topologies contributing
to the purely collinear limits, we can now use the fact that any
interferer leg connecting to an emission will contribute only if this
emission becomes soft. Thus we can additionally carry out a soft
decomposition for the interferer legs here using
\eqref{eqs:linearity}, neglecting the terms which scale as ${\cal
O}(\lambda)$ in this limit.
Finally, we come to the self-energy type amplitudes. For $k$
emissions, the respective propagator factors will induce a scaling of
$1/\lambda_c^{4k}$ while from phase space considerations, we again
expect a leading scaling of $1/\lambda_c^{2k}$. This shows that the
numerators will give a scaling of $\lambda_c^{2k}$ to arrive at the
expected leading scaling. For two emissions, we have shown this fact
already due to vanishing of \eqref{eq:vanishing-internal-perp}. This
means that the numerator for self-energy like contributions for two
emissions starts at $\lambda_c^4$. We expect this pattern to hold for
any number of emissions. An argument in favour of this is the
vanishing of the amplitude with the naively lowest scaling, \textit{i.e.}\
\input{diagrams/internal-perp-general.tex}%
In conclusion, we can formulate the power counting algorithm as
follows: we begin by using the collinear power counting for the
numerator scaling from Tab.~\ref{tab:power-counting1} and
\ref{tab:power-counting2} to find the a leading collinear
contribution. This includes finding and using rules such as
\eqref{eq:vanishing-vertices-all-white} or
\eqref{eq:vanishing-internal-perp}. Next, knowing that interferer
lines do not contribute in leading collinear limits, we can carry out
an additional soft decomposition using \eqref{eqs:linearity}. The
result is a splitting kernel which yields the correct leading power
soft and collinear behaviour. We will discuss this in more detail for
one and two emissions in
Sec.~\ref{sec:single-emission-splitting-kernel} and
\ref{sec:two-emission-splitting-kernel}.
\subsection{Factorisation to hard amplitude}
In this section, we want to discuss the factorisation properties of
the splitting kernels. For this purpose, we discuss the following
splitting kernel where we omit non-splitter lines for clarity, \textit{i.e.}\
\input{diagrams/splitting-kernel-factorisation.tex}%
Here, we denote hard partons providing different forward directions by
$h_1$ and $h_2$. The existence of these hard partons can be
guaranteed via the respective choice of an observable, such as a
two-jet observable in this example. In other words, phase space
configurations where the latter would become soft or collinear are cut
off by the observable. All other partons in the process, \textit{i.e.}\ the
\emph{emissions}, can become soft or collinear w.r.t.\ the hard ones.
Now, the partitioning factor $\mathbb{P}^{(d)}_{c \parallel h_1}$
works such that only configurations with collinearities to parton
$h_1$ give rise to a leading singular behaviour of the kernel.
Therefore, parton $h_2$ can be treated without loss of generality as a
\emph{recoiler} which will only be of importance for soft limits. Let
us denote the set of partons on the amplitude side by $I$, the the
ones connected to $h_1$ on the conjugate side by $\bar{I}$ and the
ones connected to $h_2$ by $\bar{J}$. The observable guarantees that
$I$ contains a hard parton.
We can use the projector rules of \eqref{eq:line-rules} to establish
factorisation in the following sense:
\input{diagrams/splitting-kernel-factorisation-3.tex}%
For the splitter line on the amplitude side, there are no
$(\perp)$-components coming from the hard amplitude when transverse
momenta are balanced between the emitter and emissions. Then, the
leading contribution from this side comes just from the forward
momentum component.\footnote{Note that a few of the two emission
amplitudes have a black box appearing on the hard amplitude
line. Nevertheless, Tab.~\ref{tab:two-emissions-sp1} shows that these
could only contribute to the leading soft-collinear limits. Moreover,
one has to check whether such contributions vanish identically on a case by case basis.} The
leading contributions on recoiler lines come from its forward
components which are from different collinear sectors and therefore,
\eqref{eq:vanishing-vertices-all-white} does not apply here. The
situation is more complicated for the conjugate side splitter line.
Here, the $(\perp)$-components entering the hard amplitude are in
general unbalanced and could give contributions to the leading
singular limits. Nevertheless, we suspect that only the combination
shown in \eqref{splitting-kernel-factorisation-3} could give such a
contribution. Then, by including the projection operators in the
splitting amplitude, one can neglect all but the forward momenta
components from the hard amplitude.
The reason for our suspicion comes from the one and two emission
examples. For one emission, there are no $(\perp)$-components coming
from the hard amplitude at all by choice of a respective recoil
scheme. For two emissions, there are contributions from such
components, but we can use \eqref{eq:line-rules} to decompose them,
\textit{i.e.}\
\input{diagrams/perp-components-hard-amp.tex}%
The last diagram vanishes due to
\eqref{eq:vanishing-vertices-all-white}, making all momenta components
from the hard amplitude other than the forward one obsolete.
Therefore, in the sense of \eqref{splitting-kernel-factorisation-3},
we can establish the factorisation of the leading soft and collinear
singularities to the hard amplitude with only its forward components.
Note that the projector insertions for the splitter and recoiler line
are redundant here, but essential for the conjugate splitter line.
The discussion above is also relevant for recoil schemes that include
unbalanced $(\perp)$-components coming from the hard amplitude. Even
in this case, we find that the leading $\lambda_c$-contributions come
solely from diagrams of the type of the middle one in
\eqref{eq:perp-components-hard-amp}. The reason is that the same
logic as in the balanced $(\perp)$-momentum case applies when the
splitting amplitudes are factored using the projector, as in
\begin{fmffile}{perp-components-hard-amp-2}
\fmfset{thin}{.7pt}
\fmfset{arrow_len}{1.8mm}
\fmfset{curly_len}{1.8mm}
\fmfset{dot_len}{1.2mm}
\begin{align}
\begin{gathered}
\begin{fmfgraph*}(100,50)
\fmfstraight
\fmfleft{lb,lm,lt}
\fmfright{rb,rm,rt}
\fmf{phantom}{lm,b1,v1,b2,d1,d2,b3,v2,b4,p,b5,rm}
\fmf{phantom}{lb,b1b,v1b,b2b,d1b,d2b,b3b,v2b,b4b,pb,b5b,rb}
\fmffreeze
\fmf{plain}{lm,d1}
\fmf{dots}{d1,d2}
\fmf{plain}{d2,b4}
\fmf{plain}{b5,rm}
\fmf{plain}{v1,bb1,v1b}
\fmf{plain}{v2,bb2,v2b}
\fmfv{decor.shape=square,decor.filled=30,decor.angle=0,decor.size=6}{b1}
\fmfv{decor.shape=square,decor.filled=30,decor.angle=0,decor.size=6}{b2}
\fmfv{decor.shape=square,decor.filled=30,decor.angle=0,decor.size=6}{b3}
\fmfv{decor.shape=square,decor.filled=empty,decor.angle=0,decor.size=6}{b4}
\fmfv{decor.shape=square,decor.filled=empty,decor.angle=0,decor.size=6,label=\tiny{$\perp$},label.dist=0}{b5}
\fmfv{decor.shape=square,decor.filled=30,decor.angle=0,decor.size=6}{bb1}
\fmfv{decor.shape=square,decor.filled=30,decor.angle=0,decor.size=6}{bb2}
%
\fmfv{label=\small{$\mathbf{P}$},label.dist=0}{p}
\fmfrectangle{5}{7}{rm}
\end{fmfgraph*}
\end{gathered}
\end{align}
\end{fmffile}%
Here, the grey boxes represent any component of the momentum
decomposition. The left part of the diagram follows the same rules as
in the balanced case. Then, the $(\perp)$-box on the hard amplitudes
side will add a power of $\lambda_c$ to any contribution and will
therefore be subleading.
\subsection{Single emission case}
\label{sec:single-emission-splitting-kernel}
In this section, we discuss explicit results for the one-emission
quark and gluon splitting kernels. The relevant subamplitudes are
listed in Tab.~\ref{tab:one-emission-pc}.\footnote{ Note that those
with a backward component on an external leg, \textit{i.e.}\ a black box, do not
contribute because they identically vanish when contracted with the
projectors of \eqref{eq:projector-gluon} and
\eqref{eq:projector-fermion}.}
\input{one-emission-PC.tex} From these, we construct the splitting
kernel $\mathbb{U}_{(ij)}$ with the lowest possible overall collinear
power scaling, \textit{i.e.}\ the leading singular contributions when an emission
$j$ becomes collinear to a hard parton $i$. Inserting splitter lines
into the amplitude and conjugate amplitude side, we generate
self-energy like contributions. Their propagator factors give a power
scaling of $\mathcal{O}(1/\lambda_c^4)$ in the collinear and
$\mathcal{O}(1/\lambda_s^2)$ in the soft limit. The lowest numerator
scaling is of $\mathcal{O}(\lambda_c^2)$, giving an overall scaling of
$\mathcal{O}(1/\lambda_c^2)$ in this case. Simultaneously, the
amplitudes with the lowest soft scaling contained in here, namely the
ones with a $(\perp)$-momentum component on the emission lines.
Combining a splitter and interferer line with hard partons $i$ and
$k$, respectively, gives interference-like contributions. These have
propagator factors from two different collinear sectors. We extract
the $(ij)$-collinear behaviour by applying the partitioning
$\mathbb{P}_{(ij)}$ which allows us to use the collinear power
counting of Tab.~\ref{tab:one-emission-pc} in the first place.
Otherwise, additional subamplitudes with the correct collinear scaling
w.r.t. to parton $k$ would have to be taken into account. In this
setting, both the leading collinear and soft scaling is of
$\mathcal{O}(1/\lambda_{c/s}^2)$. The lowest collinear numerator
scaling if of $\mathcal{O}(\lambda_{c})$ which shows again that the
interferences do not contribute in the leading collinear limit.
Eventually, we find that the full one emission $(ij)$-splitting kernel
consists of
\vspace{4pt}
\input{diagrams/one-emission-kernel.tex}%
We now want to exhibit the interplay between soft and collinear
contributions in this splitting kernel and base this on a general
discussion of Eikonal currents. First, we note that the emission of a
soft gluon with momentum $q_j$ off of a hard quark line $i$ leads to
the Eikonal vertex rule \cite{BASSETTO1983201}
\begin{fmffile}{soft-qqg-vertex}
\fmfset{thin}{.7pt}
\fmfset{arrow_len}{2.5mm}
\fmfset{curly_len}{1.5mm}
\begin{align}\label{eq:soft-qqg-vertex}
\begin{gathered}
\begin{fmfgraph*}(50,40)
\fmfleft{l}
\fmfright{rb,rm,rt}
\fmf{fermion}{l,v,rm}
\fmffreeze
\fmf{phantom,label=\tiny{$q_i$},label.side=left}{v,rm}
\fmf{curly,label=\tiny{$q_j$},label.side=right}{v,rb}
\fmfv{label=\tiny{$\mu$},label.dist=5}{rb}
\fmfv{decor.shape=circle,decor.filled=full,decor.size=2}{v}
\end{fmfgraph*}
\end{gathered}
\;
\propto \frac{q_i^\mu}{q_i \!\cdot\! q_j} +\mathcal{O}\left(1\right),
\end{align}
\end{fmffile}%
where $Q$ is the hard scale of the full process.
Using this vertex rule in a self-energy like contribution, we find
\input{diagrams/one-emission-self-energy-eikonal.tex}%
where $\mathbf{T}_i$ is the colour charge operator associated with
parton $i$. We will shortly see that this is just the soft-divergent
part of the splitting function $\hat{P}_{qg}$ (\textit{i.e.}\ the second term of
\eqref{eq:one-emission-coll}). Now we compare this to the
contribution from the corresponding exchange diagram. Inserting the
Eikonal couplings here, we find
\input{diagrams/one-emission-interference-eikonal.tex}%
The first term of this contribution is the squared Eikonal current
representing the leading soft singular behaviour. The second (third)
term is independent of the parton momentum $q_k$ ($q_i$). This allows
the use of colour conservation when summing over all partons, \textit{i.e.}\
\begin{equation}
\sum\limits_{k\neq i} \mathbf{T}_k = -\mathbf{T}_i \ .
\end{equation}
Therefore, the last two terms in \eqref{eq:eikonal-calc-interference}
cancel exactly against the soft singular contribution from
\eqref{eq:eikonal-self-energy} for parton $i$ and $k$ and one is left
only with the (gauge-invariant) squared Eikonal current from the first
term of \eqref{eq:eikonal-calc-interference} in the soft limit. This
analysis in the one emission case highlights a fundamental difference
of our approach as compared to constructing fixed-order subtraction
terms as in \cite{Catani:1996vz}: Our formalism would keep the
structure of the interference diagrams, which are collinear finite due
to the presence of the gauge-vector dependent terms as in
Eq.~(\ref{eq:eikonal-calc-interference}). At the same time,
Eq.~(\ref{eq:eikonal-self-energy}) will contain the entire collinear
splitting function. Approaches like the dipole formalism start from
the known soft and collinear behaviours, and explicitly remove the
overlap in between them through partitioning the soft behaviour,
including the soft-collinear singularity, in between different kernels
for collinear sectors. Especially in view of the more complicated
colour structures pertaining to the interference diagrams, our
formalism might be beneficial as it explicitly appreciates the fact
that the colour correlations are collinear finite.
Our partitioning algorithms allow us, however, to apply a similar
logic to the interplay of soft and collinear limits at the level of
cross section factorization: we further discuss this idea in
Sec.~\ref{sec:soft-collinear-functions}, which can serve as an
additional starting point for parton branching algorithms beyond the
leading order\footnote{In fact, while we were finalizing the present
work, Ref.~\cite{Gellersen:2021eci} which discusses a similar
question}.
\subsubsection{Quark-gluon splitting}
In order to discuss these results in terms of our power counting
algorithm, we employ the Sudakov decomposition of
\eqref{eq:Sudakov-decomp-single}, \textit{i.e.}\
\begin{align}\label{eq:mapping-alpha-1E}
q_i^\mu &= z_i\, p_i^\mu + \frac{p_{\perp,i}^2}{z_i \, 2p_i \!\cdot\! n}n^\mu + k_{\perp,i}^\mu, \nonumber\\
q_j^\mu &= z_j\, p_i^\mu + \frac{p_{\perp,j}^2}{z_j \, 2p_i \!\cdot\! n}n^\mu + k_{\perp,j}^\mu,\nonumber \\
q_k^\mu &= z_k \, p_k^\mu.
\end{align}
Using this mapping, we find vertex rules for splitter lines, namely
\input{diagrams/one-emission-amps.tex}%
The second diagram represents the Eikonal coupling in terms of our
mapping. In order to reproduce the splitting function and soft limits,
we instate one specific version of the momentum mapping,\textit{i.e.}\
\begin{align}\label{eq:to-z-mapping}
z_i &= z, \quad z_j = 1-z,\quad z_k=1, \nonumber\\
k_{\perp,i} &= - k_{\perp,k} = k_{\perp}, \nonumber \\
p_{\perp,i}^2 &= p_{\perp,j}^2 = p_{\perp}^2 \ .
\end{align}
Using these rules together with the projectors of
\eqref{eq:projectors-general} for connecting the amplitude and
conjugate amplitude, the self-energy like contributions of
\eqref{eq:one-emission-kernel} give
\input{diagrams/one-emission-self-energy-decomp-density.tex}%
where the soft-singular term in square brackets solely comes about via
the second diagram above. In total, we reproduce the well-known spin
averaged quark gluon splitting function in $d=4-2\epsilon$ dimensions
\cite{Catani:1996vz}
\begin{equation}\label{eq:Pqg-CS}
\langle \hat{P}_{qg} (z)\rangle = C_F \left[ \frac{1+z^2}{1-z} - \epsilon (1-z)\right].
\end{equation}
For the interference contributions of \eqref{eq:one-emission-kernel},
we find
\vspace{4pt}
\input{diagrams/qg-exchange-diagrams.tex}%
Note that one has to include a factor of $\sqrt{z}$ coming from parton
$i$ on the conjugate side. Inserting the mapping of
\eqref{eq:mapping-alpha-1E} in the partitioned version of
\eqref{eq:eikonal-calc-interference}, \textit{i.e.}\
\begin{align*}
&\mathbb{P}_{(ij)} \, 8 \pi \alpha_s (-\mathbf{T}_i \!\cdot\! \mathbf{T}_k) \frac{4}{S_{ij} S_{jk}}\left[ q_i \!\cdot\! q_k - q_k \!\cdot\! q_j \frac{q_i \!\cdot\! n}{q_j \!\cdot\! n} - q_i \!\cdot\! q_j \frac{q_k \!\cdot\! n}{q_j \!\cdot\! n}\right].
\end{align*}
This has the same soft limit as \eqref{eq:interference-qg-1E} (modulo
a factor of two from using two times the real part in
\eqref{eq:eikonal-calc-interference}) This represents a non-trivial
check of our power counting rules. Also notice the similarity between
\eqref{eq:interference-two-perp} and the soft singular term in
\eqref{eq:one-emission-coll}. We do not have an immediate
cancellation between both contributions as was the case for
\eqref{eq:eikonal-self-energy}
vs.\ \eqref{eq:eikonal-calc-interference}. This is due to the fact
that we implement the partitioning before carrying out these
cancellations and because with each of the three terms in the
partitioned Eikonal above containing various powers of soft
contributions, these terms mix and we can not isolate each term via a
power counting. Nevertheless, when adding up the kernels
$\mathbb{U}_{(ij)}$ and $\mathbb{U}_{(jk)}$,
\eqref{eq:eikonal-calc-interference} is recovered in the soft limit
and the soft divergent parts of the splitting functions are cancelled
against the respective contributions from the interferences (with the
sum over partitioning factors collapsing to 1), leaving only soft
interference contributions of the kind in
\eqref{eq:interference-one-perp}.
Notice that we can, of course, acquire the same behaviour from a
different momentum parametrization. In order to show this, it is
interesting to look at a momentum mapping with unbalanced
($\perp$)-component of the emission (in \eqref{eq:mapping-alpha-1E},
these components are balanced between emittee end emission), because
it shows how different contributions in terms of our power counting
algorithm lead to the same results. This version of the mapping reads
\begin{align}
q_i^\mu &= z p_i^\mu \ , \nonumber \\
q_j^\mu &= (1-z)p_i^\mu + \frac{p_\perp^2}{(1-z) 2 p_i \!\cdot\! n} n^\mu - k_\perp^\mu \ , \nonumber \\
q_k^\mu &= p_k^\mu \ .
\end{align}
With this choice, momentum conservation implies the shift of the ($\perp$)-component to the momentum of the splitter line, \textit{i.e.}\
\begin{equation}
q_{I}^\mu = q_i^\mu + q_j^\mu = p_i^\mu + \frac{p_\perp^2}{(1-z) 2 p_i \!\cdot\! n} n^\mu - k_\perp^\mu \ .
\end{equation}
This is consistent with the Sudakov decomposition of \eqref{eq:Sudakov-decomp-combined} when
\begin{equation}
S_{I} = \frac{z}{1-z} p_\perp^2
\end{equation}
is inserted.
Now, we need to take into account the ($\perp$)-components in the internal emitter line instead of the external one as compared to \eqref{eq:one-emission-coll} when carrying out the power counting.
Therefore, the splitting function now comes about via
\input{diagrams/one-emission-self-energy-decomp-density-kT-emitter.tex}%
Note how the individual contributions differ from
\eqref{eq:one-emission-coll}, yet the splitting function of
\eqref{eq:Pqg-CS} is reproduced again when combining the terms. The
interference contributions become
\vspace{5pt}
\input{diagrams/qg-exchange-diagrams-2.tex}%
The Eikonal contribution of \eqref{eq:interference-one-perp-2}
coincides with the one from before while the soft-singular one
differs. Nevertheless, we find the same correspondence of
soft-singular terms between interference- and self-energy like
contributions when comparing \eqref{eq:one-emission-coll-kT-emitter}
to \eqref{eq:interference-two-perp-2}.
\subsubsection{Gluon-gluon splitting}
The same argumentation holds for the emission from hard gluon. In
this case, one reproduces \eqref{eq:eikonal-self-energy} and
\eqref{eq:eikonal-calc-interference} by using the vertex rule
\begin{fmffile}{soft-ggg-vertex}
\fmfset{thin}{.7pt}
\fmfset{arrow_len}{2.5mm}
\fmfset{curly_len}{1.5mm}
\begin{align}\label{eq:soft-ggg-vertex}
\begin{gathered}
\begin{fmfgraph*}(50,40)
\fmfleft{l}
\fmfright{rb,rm,rt}
\fmfbottom{b}
\fmf{curly}{l,v,rm}
\fmffreeze
\fmf{phantom,label=\tiny{$q_i$},label.side=left}{v,rm}
\fmf{curly,label=\tiny{$q_j$},label.side=right}{v,b}
\fmfv{label=\tiny{$\mu$},label.dist=5}{rm}
\fmfv{label=\tiny{$\nu$},label.dist=5}{l}
\fmfv{label=\tiny{$\rho$},label.dist=5}{b}
\fmfv{decor.shape=circle,decor.filled=full,decor.size=2}{v}
\end{fmfgraph*}
\end{gathered}
\quad
\propto \frac{q_i^\rho}{q_i \!\cdot\! q_j} \eta^{ \mu \nu} +\mathcal{O}\left(1\right),
\end{align}
\end{fmffile}%
and by realizing that longitudinal contributions proportional to
$q_i^\mu$ vanish due to gauge invariance of the underlying hard
amplitude \cite{PhysRevD.55.6819}. Then, we again need to show that
our power-counting rules reproduce this soft behaviour leading to the
same cancellations between interference and self-energy like
contributions as in the quark case. One complication arises here due
to the three-gluon vertex carrying a momentum dependence which we need
to account for using \eqref{eq:vertex-rules}. The respective rules
for a splitter line are
\vspace{5pt}
\input{diagrams/one-emission-amps-ggg.tex}%
The rules for interferer vertices are
\vspace{5pt}
\input{diagrams/one-emission-amps-ggg-interferer.tex}%
Note that these vertices are only relevant for interferer-like
amplitudes and therefore only contribute in soft limits. This is why
we include the soft partitioning factors and vertex decomposition of
\eqref{eqs:linearity} and \eqref{eq:linearity-three-gluon-vertex} here
which greatly simplifies the vertex rule. Note also that in the
strict soft limit, $z_j \sim \mathcal{O}(\lambda)$ while $z_k = 1$ and
the soft partitioning factor tends to unity.
Then, after applying \eqref{eq:to-z-mapping} again, the splitting
function $P_{gg}$ comes about via
\input{diagrams/one-emission-gluon-self-energy-decomp-density.tex}%
The first term of this expression shows the soft-divergent part of
$P_{gg}$. Turning to the interferences, we have fewer diagrams to
take care of because the $(\perp)$-vertices give soft-subleading
contributions. The Eikonal contribution is then given by
\vspace{10pt}
\input{diagrams/gg-exchange-diagrams-eikonal.tex}%
Here, we have used the fact that the hard amplitudes are transverse
meaning that terms with a $p^{\mu_i/\nu_i}$ and $p^{\mu_k/\nu_k}$
vanish. By the same logic, the leading-soft divergent part is given
by
\vspace{10pt}
\input{diagrams/gg-exchange-diagrams.tex}%
\subsection{Two emission case}
\label{sec:two-emission-splitting-kernel}
\input{table6.tex} \input{table7.tex} \input{table-interferers.tex} In
\tabref{tab:two-emissions-sp1}, we show the leading splitter
amplitudes with collinear vertex insertions for all relevant two
emission singular configurations, while \tabref{tab:two-emissions-sp2}
shows $(\perp)$-vertex insertions. In
\tabref{tab:two-emissions-interferer-table} we show the respective
interferer amplitudes. From here, one can deduce the numerator
scaling for all relevant two-emission topologies.
These tables feature several noteworthy aspects we want to mention
here. Concerning the triple collinear limit $(i \parallel j \parallel
k)$, we see that many of the splitter amplitudes shown minimally scale
as $\lambda^2$. All of these need to be taken into account for this
limit while for interferer lines, there is only one relevant
amplitude, namely the first one shown in
\tabref{tab:two-emissions-interferer-table}. For self-energy
topologies, where one finds a propagator scaling of $1/\lambda^8$ for
$E^{(1)}$, $E^{(2)}$ and $E^{(3)}$ this means that the overall scaling
drops to $1/\lambda^4$ as one would expect for a contribution to a
splitting function. Moreover, we find that these are the \emph{only}
relevant topologies for two emission splitting functions as can be
seen in \eqref{splitting-abelian} and \eqref{splitting-non-abelian}.
Instead, for the double soft limit, there is only one amplitude whose
numerator does not scale, namely the one with $(\perp)$-momentum
components on the emission lines. This will be the only relevant
numerator structure in said limit on the amplitude side. On the
interferer side, shown in
Tab.~\ref{tab:two-emissions-interferer-table} all amplitudes shown are
relevant, because none of them scale in the double soft-limit.
Nevertheless, none of the amplitudes with a $(\perp)$-vertex shown in
\tabref{tab:two-emissions-sp2} can contribute on the interferer line
in a double soft limit, because the inclusion of such a vertex always
adds a power of $\lambda$ to the scaling. This can already be seen in
the one emission example, where there are only two relevant
interference contributions of \eqref{eq:interference-gg-1E} and
\eqref{eq:1E-eikonal} for the same reasons.
Another interesting finding is the appearance of amplitudes with a
black box connecting to the hard amplitude. If such an amplitude were
to contribute in a leading singular limit, it would be a sign of a
factorisation breakdown with the reason being that the backwards
components of momenta from the hard amplitude can not be neglected.
Nevertheless, a closer inspection of said amplitude shows that in
conjunction with the scaling of the relevant partitioning and
propagator factors, they do not contribute in any leading singular
limit for two emissions.
In general, it is algorithmically an easy task to collect all
contributions relevant for a specific leading singularity. One can
first check the scaling of partitioning times propagator factors to
find potentially relevant topologies (\textit{i.e.}\ the ones which scale as
$1/\lambda^{4}$ or worse) and then pick the corresponding amplitudes
from our lists that feature the lowest scaling in the respective
limit. As mentioned in the one emission example, one can finally
apply the decomposition of \eqref{eqs:linearity} for interferer lines
to single out leading soft-singular contributions.
We show the scaling of the partitioned propagator factors together with their respective numerator scaling in Tab.~\ref{tab:scaling-two-emission-with-num}.
All contributions with a scaling of $1/\lambda^4$ should enter the kernel for the collinear configuration in question, in this case the $\mathbb{U}_{(ijk)}$-kernel.
Several remarks for this collection of data are in order.
Firstly, we see that only the self-energy like topologies $E$ contribute in the triple collinear limit.
For soft-collinear configurations, only a handful of topologies give rise to leading contributions, while almost all of them contribute in the double-soft limit.
An interesting difference between the partitioning types is that topology $A^{(3)}$ does not contribute in the double-soft limit for fractional partitioning while it does in the angular ordered version.
Both partitionings give rise to the same leading contributions otherwise.
\begin{table}[h]
\begin{center}
\begin{tabular}{c|cccc}
& CC & CS & SC & SS \\
\hline
$ A^{(1)} $ & $1/\lambda^2$ & $1/\lambda^3$ & $1/\lambda$ & \red{$1/\lambda^4$}\\
$ A^{(2)} $ & $1/\lambda^2$ & $1/\lambda^2$ & $1/\lambda^2$ & \red{$1/\lambda^4$} \\
$ A^{(3)} $ & $1/\lambda^2$ & $1/\lambda^2$ & $1/\lambda$ & $1/\lambda^3$ \\
$ A^{(4)} $ & $1/\lambda^3$ & $1/\lambda^2$& $1/\lambda$ & \red{$1/\lambda^4$} \\
$ A^{(5)} $ & $1/\lambda^3$ & $1/\lambda$ & $1/\lambda$ & \red{$1/\lambda^4$} \\
$ B^{(1)} $ & $1/\lambda^3$ & \red{$1/\lambda^4$}& $1/\lambda^2$ & \red{$1/\lambda^4$} \\
$ B^{(2)} $ & $1/\lambda^3$ & $1/\lambda^3$ & $1/\lambda^3$ & \red{$1/\lambda^4$} \\
$ B^{(3)} $ & $1/\lambda^3$ & $1/\lambda^3$ & $1/\lambda^2$ & \red{$1/\lambda^4$} \\
$ B^{(4)} $ & $1/\lambda^3$ & \red{$1/\lambda^4$}& $1/\lambda^2$ & \red{$1/\lambda^4$} \\
$ B^{(5)} $ & $1/\lambda^3$ & $1/\lambda^3$ & $1/\lambda^3$ & \red{$1/\lambda^4$} \\
$ B^{(6)} $ & $1/\lambda^3$ & $1/\lambda^3$ & $1/\lambda^2$ & \red{$1/\lambda^4$} \\
$ X^{(1)} $ & $1/\lambda^2$ & $1/\lambda^2$ & \red{$1/\lambda^4$} & \red{$1/\lambda^4$} \\
$ X^{(2)} $ & $1/\lambda^2$ & $1/\lambda^2$ & $1/\lambda^2$ & $1/\lambda^2$ \\
$ E^{(1)} $ & \red{$1/\lambda^4$}& \red{$1/\lambda^4$} & $1/\lambda^2$ & \red{$1/\lambda^4$} \\
$ E^{(2)} $ & \red{$1/\lambda^4$} & $1/\lambda^3$ &$1/\lambda^3$ & \red{$1/\lambda^4$} \\
$ E^{(3)} $ & \red{$1/\lambda^4$}& $1/\lambda^3$ & $1/\lambda^2$& \red{$1/\lambda^4$} \\
$ E^{(4)} $ & \red{$1/\lambda^4$}& $1/\lambda^3$ & $1/\lambda^2$& \red{$1/\lambda^4$} \\
$ E^{(5)} $ & \red{$1/\lambda^4$}& $1/\lambda^2$ & $1/\lambda^2$& \red{$1/\lambda^4$} \\
\end{tabular}
\end{center}
\caption{Scaling for propagator times partitioning factor of two emission single emitter topologies when partitioned to $(i\parallel j \parallel k)$ together with the respective numerator scaling. Here, `CC' refers to the triple collinear limit where $i||j||k$, `CS' refers to $(i \parallel j)$ with soft $k$, `SC' to $(i \parallel k)$ and $j$ soft and `SS' is the double soft limit where both $j$ and $k$ are soft. The leading terms are marked in red.}
\label{tab:scaling-two-emission-with-num}
\end{table}
As an illustrative example for a single diagram, we discuss the power counting for the
$B^{(1)}$-topology for the emission of two gluons, \textit{i.e.}\
\input{diagrams/B1-density-operator-like.tex}%
First, we check the scaling of the diagram's propagator times the
partitioning factors for all limits where the partons $j,k$ can become
unresolved explicitly for the partial fractioning version of the
partitioning. An example is the triple collinear $(i\parallel j
\parallel k)$-limit. The respective partitioning factor which
eliminates the collinear singularity w.r.t.\ parton $l$ is
\begin{align}
\mathbb{P}_{(ijk)}^{B^{(1)}} =
\frac{\varsigma^4 S_{k l}}{\varsigma^2 S_{k l} S_{i j k}+\varsigma^4 S_{i j k}+2 S_{i j}^2 S_{i j k}+\varsigma^4 S_{k l}} \ .
\end{align}
The configurations $c$ where $\mathbb{P}_{c}^{B^{(1)}} \times B^{(1)}$
could give rise to a leading singular contribution are shown in
\tabref{tab:PtimesB1-scaling}.
\bgroup \def\arraystretch{1.5}%
\begin{table}[h]
\begin{center}
\begin{tabular}{l|cccc}
& $(i\parallel j \parallel k)$ & $(i\parallel j) (k \parallel l)$ & $(i \parallel j), k$ soft & $j,k$ soft
\\[5pt] \hline
$\mathbb{P}_{(ijk)}^{B^{(1)}}$ & $\frac{1}{\lambda^6}$ & $\frac{1}{\lambda^4}$ & $\frac{1}{\lambda^6}$ & $\frac{1}{\lambda^4}$
\\
$\mathbb{P}_{(ij)(kl)}^{B^{(1)}}$ & $\frac{1}{\lambda^4}$ & $\frac{1}{\lambda^6}$ & $\frac{1}{\lambda^6}$ & $\frac{1}{\lambda^4}$
\end{tabular}
\caption{Leading propagator scalings of $\mathbb{P}_{c}^{B^{(1)}} \times B^{(1)}$ for four different configurations where the partons $j,k$ can become unresolved.}
\label{tab:PtimesB1-scaling}
\end{center}
\end{table}
\egroup Next, we check the numerator scaling, \textit{i.e.}\ the possible
insertions of subamplitudes from \tabref{tab:two-emissions-sp1} for
the amplitude side and \tabref{tab:one-emission-pc} for the conjugate
side. In each case, the $(ij)$-collinearity leads to a scaling of at
least $\lambda^2$ for the numerator. Additionally, the $(ik)$- and
$(kl)$-collinearities give an additional power of $\lambda$ in the
configurations where $k$ is not soft. Then, non of the purely
collinear settings give rise to a leading singular contribution (\textit{i.e.}\
the partitioned amplitude scales as $1/\lambda^n$ with $n<4$). The
only relevant contributions come in the cases where $k$ is soft and
$j$ is either collinear to $i$ or also soft from a partitioning into
either $(i \parallel j \parallel k)$ or $(i\parallel j) (k\parallel
l)$ (meaning the soft limits in \tabref{tab:PtimesB1-scaling}). When
applying $\mathbb{P}_{(ijk)}^{B^{(1)}}$ in the double soft limit,
there are only two numerator structures which have no
$\lambda$-scaling and will therefore give rise to a leading
contribution, \textit{viz.}\
\vspace{8pt}
\input{diagrams/B2-density-operator-like-leading-soft.tex}
The only other leading contributions appear in the
$(i\parallel j)$ and $k$ soft configuration. The relevant numerator
structures here are the ones from above and the combinations where the
$(\perp)$-boxes of emission $j$ are moved to one of the nearest
neighbour boxes.
In conclusion, we find that, as apparent already from its topology,
$B^{(1)}$ contributes leadingly only in the soft-collinear and double
soft limits with its only relevant partitions being
$\mathbb{P}^{B^{(1)}}_{(ijk)}$ and $\mathbb{P}^{B^{(1)}}_{(ij)(kl)}$.
Two other interesting examples are the $B^{(2)}_{ijkl}$- and
$X^{(1)}_{ijkl}$-topologies, \textit{i.e.}\
\input{diagrams/B2-density-operator-like.tex}%
\vspace{5pt}
and
\input{diagrams/X-density-operator-like.tex}
For $B^{(2)}_{ijkl}$, the same numerator power counting
applies as for $B^{(1)}_{ijkl}$. By checking the scaling of its
propagator factors times the partitioning, we find that \emph{only}
leading singular contribution appears in the double soft limit in
$\mathbb{P}^{B^{(2)}}_{(ijk)} B^{(2)}_{ijkl}$. By the same logic, we
find that $X^{(1)}_{ijkl}$ exclusively contributes in the double soft limit.
The reason in this case is that each partition of $X^{(1)}_{ijkl}$ is at
most $1/\lambda^4$-singular. In any collinear or soft-collinear
configuration, the numerator will contribute at least one power of
$\lambda$, making these contributions subleading. In the double-soft
limit though, each partition (except for the one into $(il)(jk)$) will
contribute equally with the respective leading numerator structures
being of $\mathcal{O}(1)$.
\section{Applications and Outlook}
\label{sec:Applications}
In the previous section we have been outlining how one can, given a
partitioning of the soft radiation into different collinear sectors,
systematically factor the leading behaviour of multi-leg
amplitudes. Taking this as a starting point, several applications are
in reach: At a fixed order in perturbation theory, one can simply
square the amplitude thus obtained, and derive an interpolating
formula for the singular behaviour of a fixed number of emissions.
While it was not our primary goal to use this as a subtraction
term (and we have thus possibly not created functions which are most
easily integrated analytically), one can still use the results to
migrate certain contributions in between real emission and virtual
corrections if there is no immediate need for an analytic integration
\textit{e.g.}\ within the context of the loop-tree duality local subtractions
\cite{Bierenbaum:2010xg,Driencourt-Mangin:2019yhu} and similar
approaches \cite{Platzer:2020lbr}.
Within the context of a specific (class of) observables, which exhibit
a definite perturbation around a set of hard jets, our findings are a
vital input to a resummation programme. In particular, when using
azimuthal averaging, and our partitioning which generalizes the
subtractions behind angular ordering, we expect that we can obtain a
generalization of the coherent branching formalism beyond the usual
next-to-leading logarithmic accurate algorithm for global event
shapes. Colour correlations can directly be addressed within our
analysis and more recent approaches to simplify the structure of
colour in the soft limit might be vital to supplement such a
formalism, which we will address in future work. An immediate
difference of our approach is that we have deliberately not chosen to
analytically move soft-collinear contributions in between different
classes of diagrams. This implies that the colour-diagonal
contributions which involve the same colour charge acting in the
amplitude and its conjugate, will still deliver the full soft- and
hard-collinear singularity. They do not need to rely on the soft,
colour-correlated contribution to collapse into the collinear
singularity upon relying on colour conservation. This might be
advantageous in a numerical implementation, but also implies that our
non-trivially colour correlated contributions truly describe the
effect of large-angle soft radiation, which could thus be separated in
an analytic way; this should be confronted with the approaches of
collinear subtractions outlined in \cite{Forshaw:2019ver}. Within the
same parton-branching at the amplitude level approach, our analysis
can be exploited to derive splitting kernels beyond the limit of
iterated singly-unresolved emissions in order to build up the leading
behaviour of an amplitude with many legs. In this case, we will be
able to extend the approaches of doubly-unresolved soft radiation
\cite{Platzer:2020lbr} to include soft- and hard-collinear limits and
thus arrive at a more general algorithm. This is the primary scope of
our work, together with the generalized angular-ordered partitioning,
as well as the flexible recoil schemes. These allow us to analyse the
effect in comparison to previous work in this direction at the level
of existing parton branching algorithms \cite{Forshaw:2020wrq} for
which we are convinced that there will be a deeper link established
between recoil, partitioning, the form of the evolution kernel and
the accuracy of an overall resulting parton branching algorithm.
To be definite, we here give expressions for the factoring vertices in
terms of cutting apart the internal lines using the definition of our
projector operators in \eqref{eq:projectors-general}, \textit{i.e.}\ we do
contract the vertices which are dressed up with the respective hard,
transverse or backward propagator components with polarization vectors
and spinors in the respective collinear sectors. From this we obtain
(complex) weights for each colour structure, which we can then iterate
in a amplitude-level Monte Carlo algorithm. This would in turn,
besides the colour quantum numbers, also sample spin quantum
numbers. The procedure is sketched below in the case of a gluon
emission attaching to a quark splitter and quark spectator line,
where the $\lambda$ and $\bar{\lambda}$ refer to the helicities
on the amplitude and conjugate amplitude side, respectively:
\vspace{5pt}
\input{diagrams/projector-insertion-example.tex}
At the end of an evolution built up this way, the amplitude would then
be squared by evaluating the matrix element of the final projector
corresponding to the most external lines we have been starting with in
the first place, and our decomposition and normalization guarantees
that there is no need to take into account additional factors. The
other vertices we encounter in the case of a single and double
emission are outlined below. Here and in the following, the symbol
``$\simeq$'' stands for only showing the scalar emission quantity that
factorizes to the hard amplitude while leaving the rest of the
polarization sum and the hard amplitude itself implicit.
The relevant expressions for a quark-gluon splitting are
\vspace{5pt}
\input{diagrams/vertex-rules-projectors.tex}
The amplitudes for a gluon-gluon splitting read
\vspace{5pt}
\input{diagrams/vertex-rules-projectors-gluons.tex}
The combination of these vertices together with one choice of
partitioning then constitutes a complete set of splitting kernels in
the single emission case. Also note that one will encounter iterations
of these when building up the double emission case. Differences to
solely iterating single emissions mostly arise because of the
partitioning and additional two-emission diagrams which can not be
acquired by iterations. The only other changes to the kernels are
then encoded by the difference of the full double emission mapping to
an iteration of the single emission one and how these relate to the
global parametrization we have chosen to derive the splitting
amplitudes. Details of this will be subject to a follow-up
publication.
\section{Summary and Conclusions}
\label{sec:Summary}
The construction of parton branching algorithms requires a command of
the singular limits of QCD amplitudes which give rise to
logarithmically enhanced contributions. While the extraction of a
definitive singular limit is an established task, parton branching
algorithms, much like the development of subtraction terms, require a
removal of overlapping singularities in between the various soft and
collinear limits. The most general parton branching algorithms will
need to address the iteration of emissions at the amplitude level in
order to reflect the full set of correlations or to set the framework
to derive improved algorithms. In the present work we have set out a
framework which starts from the structure of QCD amplitudes using a
physical gauge in order to systematically factorize emissions at the
level of the density operator, organizing emission kernels through
various ``key'' topologies of collinear splittings. This enabled us to
use a power counting and effective set of Feynman rules to determine
the singularity structures without resorting to analysing the overlap
in between different limits: our algorithm will directly provide an
interpolating formula to extract the respective splitting kernels. We
have found that the way the kinematics is parametrized and how recoil
is handled, has a significant impact on the form of the final kernels
and more notably on the possibility to iterate them. This fact is
crucial, since one needs to remove iterated strongly ordered emissions
from a full kernel addressing the doubly unresolved limits. We have
also developed more general partitioning algorithms, which are able to
distribute the various interference contributions among leading
collinear limits, generalizing both the Catani-Seymour partial
fractioning idea as well as a differential version of angular
ordering, which effectively subtracts out collinear divergences. We
have demonstrated how our partitioning algorithm can be used to
distribute the known double-soft behaviour between different classes
of contributions, however we stress that this is not our primary
strategy to attack the problem of constructing splitting kernels for
parton branching algorithms.
Our formalism leads us to branching amplitudes, which carry full spin
and colour information and can be used within existing Monte Carlo
efforts such as the CVolver \cite{Platzer:2013fha,DeAngelis:2020rvq}
framework. One particular fact, which is worth highlighting, is that
we keep the hard-collinear behaviour separate in terms of the full
splitting function and find that our interference contributions are
manifestly suppressed in the collinear limit. We are able to perform
this distinction at the expense of an explicit dependence of the gauge
vector, for which we have explicitly shown how we can translate this
dependence into a backward direction local to each collinear
sector. In an upcoming publication we will use the present formalism
to outline the full set of double-emission kernels \cite{double:22}. We also anticipate
that a similar formalism, extended to virtual corrections and combined
with the techniques presented in \cite{Platzer:2020lbr} will allow us
to construct a full second-order evolution at the amplitude level which
can be used as a rigorous starting point for improved parton branching
algorithms.
\section*{Acknowledgements}
The work of ML is supported partially by the DFG Collaborative
Research Center TRR 257 ``Particle Physics Phenomenology after the
Higgs Discovery''. ESD has been supported by the Marie
Skłodowska-Curie Innovative Training Network MCnetITN3 (grant
agreement no. 722104). This work has also been supported in part by
the COST actions CA16201 ``PARTICLEFACE'' and CA16108 ``VBSCAN''. We
are grateful to the Erwin Schr\"odinger Institute Vienna for
hospitality and support while significant parts of this work have been
achieved within the Research in Teams programmes ``Higher-order
Corrections to Parton Branching at the Amplitude Level'' (RIT2020) and
``Amplitude Level Evolution I: Initial State Evolution.'' (RIT0421).
We are deeply obliged to Stefan Gieseke for his trust and support, and
we would like to thank Jeffrey Forshaw, Jack Holguin, Kirill Melnikov,
Ines Ruffa and Malin Sj\"odahl for fruitful discussions.
\newpage
|
2024-02-18T23:40:24.497Z
|
2021-12-30T02:23:34.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14454","language":"en","timestamp":1640831014000,"url":"https:\/\/arxiv.org\/abs\/2112.14454","yymm":"2112"}
|
proofpile-arXiv_000-10036
|
{"provenance":"002.jsonl.gz:10037"}
| null | null |
\section{Introduction}\label{sec:intro}
DNA-sequencing aims to measure the genetic makeup of individuals.
Without going into details about the many different technologies, these
processes determine (fragments of) the genetic sequence. Commonly, the
primary data analysis consists, among other steps, of: 1) alignment
against a reference genome, e.g., GRCh38 for human samples, and 2)
variant calling. The primary result is a list of variants, i.e., a set of
differences that is specific for the measured individual (sample), often
reported in a tabular file like the Variant Call Format~(VCF)~\cite{vcf}.
These variants are used in subsequent applications ranging from
fundamental and association research studies to clinical diagnostics. It
is advantageous to look only at differences (with regard to some
reference) as the genome is usually large~(ca.~$3 \cdot 10^9$ nucleotides
for humans), but the individual differences (between two genomes) are
relatively small~(ca.~$0.6\%$~\cite{variation}).
When variants are associated with phenotypic traits they are reported in
literature and stored (with their annotation) in (locus specific)
databases. Usually, the representation of the variant in VCF is refined
to a representation more suitable for reporting.
For this, many (domain specific) languages exist. Most notable are: the
recommendations of the Human Genome Variation Society~(HGVS)~\cite{hgvs},
SPDI~\cite{spdi}; the internal data model for variants used by the
National Center for Biotechnology Information (NCBI) and the Variant
Representation Specification~(VRS)~\cite{vrs}; developed by the Global
Alliance for Genomic Health (GA4GH). These languages attempt to represent
the observed differences in a human-understandable and/or machine
interpretable manner and whereas VCF is implicitly tied to a specific
process, these representations are process agnostic and universally
interpretable.
Within the domain of variant recording, some simplifications are common.
First, small (local) variants originating from the same allele are
recorded separately because this is convenient when storing large numbers
of variants in databases. Phasing information, i.e., how small variants
relate to each other when properly distributed over alleles, is often
lost or incomplete. This is, partially, a direct consequence of the
sequencing technology and partially because this information is removed.
Second, in some representations (notably, HGVS) uncertainties might be
expressed. Usually, the uncertainties relate to the positioning of the
variant within the reference genome, but also the exact makeup of larger
insertions might by unknown.
Finally, unchanged regions may be implicit. During the primary data
analysis, in particular the alignment step, the sequence from the
reference genome is assumed to be present even when direct evidence,
e.g., coverage information from the sequencing process, is lacking.
For the remainder of this paper, we adopt a strict view on the nature of
variants: 1) We see a combination of (many) smaller variants originating
from the same allele as just another variant. This closely follows the
interpretation of allele descriptions in HGVS. Considering small variants
in isolation is possible by trivially constructing an artificial allele
containing only the variant(s) of interest. 2) We consider only
\emph{interpretable} variants, i.e., given a \emph{reference sequence}
there is a deterministic and unambiguous way of ``applying'' the variants
such that the result is the (originally) measured
\emph{observed sequence} cf. the Unix~\texttt{diff} and \texttt{patch}
utilities.
As is already observed within the various variant representation
languages, it is often possible to have multiple representations
describing the same observed sequence. These possibilities can originate
from the choice of ``operator'', e.g., a single nucleotide variant (SNV)
can also be represented by a deletion (of one nucleotide) followed by an
insertion (of again one nucleotide). Another source contributing to the
number of possibilities is the structure of the reference sequence.
Consider the reference sequence~\texttt{ATTTA} and the observed
sequence~\texttt{ATTA}. One of the symbols~\texttt{T} is removed; to say
which one specifically yields a number ($3$) of possibilities.
To determine a universally accepted representation of a variant, most
variant representation languages employ a \emph{normalization} procedure.
Normalization chooses a \emph{canonical} representation from the set of
possibilities. Unfortunately, this procedure is not standardized over the
various languages, e.g., the $3'$-rule in HGVS vs. the $5'$-rule in VCF.
Within a certain language, however, proper normalization solves the
problem of identifying \emph{equivalent} variant representations. The
implications of using non-normalized variants representations have been
reviewed in~\cite{variant_name, variant_calling, curation, litvar}.
Solutions to this problem are presented
in~\cite{unified, improved_vcf, vmc_spec, smash, plyranges, gql, ask2me}.
Often dedicated tooling~\cite{varsome, validator, mutalyzer, vde} is
needed to rigorously apply the proposed normalization procedure.
Normalized variant representations can be textually compared using
standard string matching.
\begin{figure}[ht!]
\begin{center}
\begin{tikzpicture}[yscale=-.8, minimum size=4.5mm]
\node[rectangle, fill=black!15, thick] (al0_ref0) at (0.0, 0) {\texttt{G}};
\node[rectangle, fill=black!15, thick] (al0_ref1) at (0.5, 0) {\texttt{A}};
\node[rectangle, fill=black!15, thick] (al0_ref2) at (1.0, 0) {\texttt{A}};
\node[rectangle, fill=black!15, thick] (al0_ref3) at (1.5, 0) {\texttt{T}};
\node[rectangle, fill=black!15, thick] (al0_ref4) at (2.0, 0) {\texttt{C}};
\node[rectangle, fill=black!15, thick] (al0_ref5) at (2.5, 0) {\texttt{-}};
\node[rectangle, fill=black!15, thick] (al0_ref6) at (3.0, 0) {\texttt{-}};
\node[rectangle, fill=black!15, thick] (al0_ref7) at (3.5, 0) {\texttt{G}};
\node[rectangle, fill=black!15, thick] (al0_obs0) at (0.0, 1) {\texttt{G}};
\node[rectangle, fill=black!15, thick] (al0_obs1) at (0.5, 1) {\texttt{-}};
\node[rectangle, fill=black!15, thick] (al0_obs2) at (1.0, 1) {\texttt{A}};
\node[rectangle, fill=black!15, thick] (al0_obs3) at (1.5, 1) {\texttt{T}};
\node[rectangle, fill=black!15, thick] (al0_obs4) at (2.0, 1) {\texttt{C}};
\node[rectangle, fill=black!15, thick] (al0_obs5) at (2.5, 1) {\texttt{C}};
\node[rectangle, fill=black!15, thick] (al0_obs6) at (3.0, 1) {\texttt{T}};
\node[rectangle, fill=black!15, thick] (al0_obs7) at (3.5, 1) {\texttt{G}};
\draw[semithick] (al0_ref0) -- (al0_obs0);
\draw[semithick] (al0_ref2) -- (al0_obs2);
\draw[semithick] (al0_ref3) -- (al0_obs3);
\draw[semithick] (al0_ref4) -- (al0_obs4);
\draw[semithick] (al0_ref7) -- (al0_obs7);
\path (al0_ref0.west) -- (al0_ref7.east) node[midway, above=1.5em] {$\texttt{[2delA;5\_6insCT]}$};
\node[rectangle, fill=black!15, thick] (al1_ref0) at (0.0, 2.5) {\texttt{G}};
\node[rectangle, fill=black!15, thick] (al1_ref1) at (0.5, 2.5) {\texttt{A}};
\node[rectangle, fill=black!15, thick] (al1_ref2) at (1.0, 2.5) {\texttt{A}};
\node[rectangle, fill=black!15, thick] (al1_ref3) at (1.5, 2.5) {\texttt{T}};
\node[rectangle, fill=black!15, thick] (al1_ref4) at (2.0, 2.5) {\texttt{-}};
\node[rectangle, fill=black!15, thick] (al1_ref5) at (2.5, 2.5) {\texttt{C}};
\node[rectangle, fill=black!15, thick] (al1_ref6) at (3.0, 2.5) {\texttt{-}};
\node[rectangle, fill=black!15, thick] (al1_ref7) at (3.5, 2.5) {\texttt{G}};
\node[rectangle, fill=black!15, thick] (al1_obs0) at (0.0, 3.5) {\texttt{G}};
\node[rectangle, fill=black!15, thick] (al1_obs1) at (0.5, 3.5) {\texttt{-}};
\node[rectangle, fill=black!15, thick] (al1_obs2) at (1.0, 3.5) {\texttt{A}};
\node[rectangle, fill=black!15, thick] (al1_obs3) at (1.5, 3.5) {\texttt{T}};
\node[rectangle, fill=black!15, thick] (al1_obs4) at (2.0, 3.5) {\texttt{C}};
\node[rectangle, fill=black!15, thick] (al1_obs5) at (2.5, 3.5) {\texttt{C}};
\node[rectangle, fill=black!15, thick] (al1_obs6) at (3.0, 3.5) {\texttt{T}};
\node[rectangle, fill=black!15, thick] (al1_obs7) at (3.5, 3.5) {\texttt{G}};
\draw[semithick] (al1_ref0) -- (al1_obs0);
\draw[semithick] (al1_ref2) -- (al1_obs2);
\draw[semithick] (al1_ref3) -- (al1_obs3);
\draw[semithick] (al1_ref5) -- (al1_obs5);
\draw[semithick] (al1_ref7) -- (al1_obs7);
\node[rectangle, fill=black!15, thick] (al4_ref0) at (6.0, 0) {\texttt{G}};
\node[rectangle, fill=black!15, thick] (al4_ref1) at (6.5, 0) {\texttt{A}};
\node[rectangle, fill=black!15, thick] (al4_ref2) at (7.0, 0) {\texttt{A}};
\node[rectangle, fill=black!15, thick] (al4_ref3) at (7.5, 0) {\texttt{T}};
\node[rectangle, fill=black!15, thick] (al4_ref4) at (8.0, 0) {\texttt{C}};
\node[rectangle, fill=black!15, thick] (al4_ref5) at (8.5, 0) {\texttt{-}};
\node[rectangle, fill=black!15, thick] (al4_ref6) at (9.0, 0) {\texttt{G}};
\node[rectangle, fill=black!15, thick] (al4_obs0) at (6.0, 1) {\texttt{G}};
\node[rectangle, fill=black!15, thick] (al4_obs1) at (6.5, 1) {\texttt{A}};
\node[rectangle, fill=black!15, thick] (al4_obs2) at (7.0, 1) {\texttt{-}};
\node[rectangle, fill=black!15, thick] (al4_obs3) at (7.5, 1) {\texttt{T}};
\node[rectangle, fill=black!15, thick] (al4_obs4) at (8.0, 1) {\texttt{C}};
\node[rectangle, fill=black!15, thick] (al4_obs5) at (8.5, 1) {\texttt{T}};
\node[rectangle, fill=black!15, thick] (al4_obs6) at (9.0, 1) {\texttt{G}};
\draw[semithick] (al4_ref0) -- (al4_obs0);
\draw[semithick] (al4_ref1) -- (al4_obs1);
\draw[semithick] (al4_ref3) -- (al4_obs3);
\draw[semithick] (al4_ref4) -- (al4_obs4);
\draw[semithick] (al4_ref6) -- (al4_obs6);
\path (al4_ref0.west) -- (al4_ref6.east) node[midway, above=1.5em] {$\texttt{[3delA;5\_6insT]}$};
\node[rectangle, fill=black!15, thick] (al5_ref0) at (6.0, 2.5) {\texttt{G}};
\node[rectangle, fill=black!15, thick] (al5_ref1) at (6.5, 2.5) {\texttt{A}};
\node[rectangle, fill=black!15, thick] (al5_ref2) at (7.0, 2.5) {\texttt{A}};
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\node[rectangle, fill=black!15, thick] (al5_ref4) at (8.0, 2.5) {\texttt{C}};
\node[rectangle, fill=black!15, thick] (al5_ref5) at (8.5, 2.5) {\texttt{-}};
\node[rectangle, fill=black!15, thick] (al5_ref6) at (9.0, 2.5) {\texttt{G}};
\node[rectangle, fill=black!15, thick] (al5_obs0) at (6.0, 3.5) {\texttt{G}};
\node[rectangle, fill=black!15, thick] (al5_obs1) at (6.5, 3.5) {\texttt{-}};
\node[rectangle, fill=black!15, thick] (al5_obs2) at (7.0, 3.5) {\texttt{A}};
\node[rectangle, fill=black!15, thick] (al5_obs3) at (7.5, 3.5) {\texttt{T}};
\node[rectangle, fill=black!15, thick] (al5_obs4) at (8.0, 3.5) {\texttt{C}};
\node[rectangle, fill=black!15, thick] (al5_obs5) at (8.5, 3.5) {\texttt{T}};
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\node[draw, fit=(al0_ref0) (al4_obs6)] {};
\node[draw, fit=(al1_ref0) (al5_obs6)] {};
\begin{scope}[on background layer]
\fill[orange!30] ([xshift=.4pt, yshift=.5pt]al0_ref1.north west) rectangle ([xshift=-.4pt, yshift=-.5pt]al0_obs1.south east);
\path (al0_ref1) -- (al0_obs1) node[midway] {\footnotesize $\dagger$};
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\path (al0_ref6) -- (al0_obs6) node[midway] {\footnotesize $\star$};
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\path (al1_ref1) -- (al1_obs1) node[midway] {\footnotesize $\star$};
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\path (al1_ref6) -- (al1_obs6) node[midway] {\footnotesize $\star$};
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\path (al4_ref5) -- (al4_obs5) node[midway] {\footnotesize $\star$};
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\path (al5_ref5) -- (al5_obs5) node[midway] {\footnotesize $\star$};
\end{scope}
\end{tikzpicture}
\caption{Two pairs (top and bottom) of minimal sequence level alignments
for two variants (left and right) with respect to the same reference
sequence~\texttt{GAATCG}. In the top pair the highlighted \emph{changes}
show one matching ($\star$) change between the variants, but also
non-matching ($\dagger$) changes. If we would only consider this pair,
the conclusion would be that the variants overlap.
However, taking \emph{all} minimal alignments into account, we will
eventually encounter the bottom pair. Here, the right variant has no
non-matching changes, while the left variant has one non-matching change.
This leads to the conclusion that the left variant contains the right
variant. Note that there are more minimal alignments for the left variant
that are not shown for brevity, but they do not influence the
conclusion.}\label{fig:example}
\end{center}
\end{figure}
\noindent
Arguably, identification of equivalent variant representations, i.e.,
determining whether two variant descriptions result in the same observed
sequence, is currently the most interesting query in the variant domain,
as it allows for the grouping and matching of equivalent variants and
their annotations. With the advent of long read single molecule
sequencing technologies (provided by platforms such as PacBio and Oxford
Nanopore), which are capable of providing direct evidence of numerous
small variants that originate from the same allele, a richer set of
questions arises.
In this paper we explore the relations of variants in an exhaustive
manner. In addition to the equivalence relation, we partition the domain
of binary variant relations into five Boolean relations: equivalence;
containment, i.e., either a variant is fully contained in another or a
variant fully contains another; overlap, i.e., two variants have (at
least) one common element; and disjoint, i.e., no common elements.
Because of this partitioning, exactly one of the five aforementioned
relations is true for every pair of variants. For determining the
relation, we consider all (minimal) variant representations
simultaneously. An introductory example is given in
Figure~\ref{fig:example}.
\section{Formalization}\label{sec:formal}
Formally, a variant representation is a pair $(R, \varphi)$, where $R$ is
a \emph{string}; a finite sequence of \emph{symbols} from a non-empty
finite \emph{alphabet}, e.g.,
$\Sigma = \{\texttt{A}, \texttt{C}, \texttt{G}, \texttt{T}\}$
called the reference sequence, and $\varphi$ is a finite set of
\emph{operations} transforming the string~$R$ into the string~$O$, the
observed sequence. The \emph{length} of a string~$S$, denoted by $|S|$,
is the number of symbols in $S$. We refer to the symbol on position~$i$
of string~$S$ as $S_i$ with $1 \le i \le |S|$. This notation is extended
in the natural way for substrings of $S$, i.e., $S_{i\ldots j}$
represents the string containing the contiguous
symbols~$S_i,\ldots, S_j$, with $1 \le i < j \le |S|$.
Note that the set of operations is dependent on the variant
representation language used. Without loss of generality, we assume that
all operations are in the form of a deletion of zero or more symbols
followed by an insertion of zero or more symbols. The actual problem of
transforming a reference sequence into an observed sequence is, for
instance, handled in~\cite{mutalyzer}.
The difference between the reference sequence~($R$) and the observed
sequence~($O$) is the ``actual'' variant, which is, to some extend,
independent from the original representation ($\varphi$) as we take
\emph{all} minimal representations into account. To this end we perform a
global pairwise alignment between $R$ and $O$. In contrast to the
specialized alignment methods used in, for instance, short read
sequencing, we use an elementary form of alignment which is close to a
commonly used distance metric, the
\emph{Levenshtein distance}~\cite{levenshtein}.
The \emph{simple edit distance}, i.e., the Levenshtein distance without
substitutions and weighing both deletions and insertions as $1$, from
string~$R$ to string~$O$ is given by $d(R, O) = D(|R|, |O|)$ defined by
the recurrence relation with $1 \le i \le |R|$ and $1 \le j \le |O|$:
\begin{equation}\label{eq:edit}
\begin{aligned}
D(0, 0) &= 0, \\
D(i, 0) &= i, \\
D(0, j) &= j, \\
D(i, j) &=
\begin{cases}
D(i - 1, j - 1) & \text{if } R_i = O_j, \\
\min
\begin{cases}
D(i - 1, j) + 1, \\
D(i, j - 1) + 1 \\
\end{cases} & \text{otherwise}.
\end{cases}
\end{aligned}
\end{equation}
\bigskip
\noindent
The simple edit distance is related to the
\emph{Longest Common Subsequence}~(LCS) problem~\cite{survey}:
\begin{equation}\label{eq:lcs}
D(i, j) = i + j - 2\cdot|\text{LCS}(R_{1\ldots i}, O_{1\ldots j})|.
\end{equation}
\noindent
Commonly, the recurrence relation is computed using a dynamic programming
approach by filling a matrix containing the solutions to
Equation~\ref{eq:edit} in a bottom-up fashion~\cite{string_to_string}.
Consider the computation of the simple edit distance between
$R = \texttt{CATATATCG}$ and $O = \texttt{CTTATAGCAT}$ in
Figure~\ref{fig:edit}.
\begin{figure}[ht!]
\[
\begin{array}{rc|ccccccccccc}
\renewcommand*{\arraystretch}{1.2}
& & \phantom{\circled{0}} & \phantom{\circled{1}} & \phantom{\circled{2}} & \phantom{\circled{3}} & \phantom{\circled{4}} & \phantom{\circled{5}} & \phantom{\circled{6}} & \phantom{\circled{7}} & \phantom{\circled{8}} & \phantom{\circled{9}} & \phantom{\circled{10}} \\
& & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
& & \texttt{.} & \texttt{C} & \texttt{T} & \texttt{T} & \texttt{A} & \texttt{T} & \texttt{A} & \texttt{G} & \texttt{C} & \texttt{A} & \texttt{T} \\
\hline
0 & \texttt{.} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
1 & \texttt{C} & 1 & \circled{0} & 1 & 2 & 3 & 4 & 5 & 6 & \circled{7} & 8 & 9 \\
2 & \texttt{A} & 2 & 1 & 2 & 3 & \circled{2} & 3 & \circled{4} & 5 & 6 & \circled{7} & 8 \\
3 & \texttt{T} & 3 & 2 & \circled{1} & \circled{2} & 3 & \circled{2} & 3 & 4 & 5 & 6 & \circled{7} \\
4 & \texttt{A} & 4 & 3 & 2 & 3 & \circled{2} & 3 & \circled{2} & 3 & 4 & \circled{5} & 6 \\
5 & \texttt{T} & 5 & 4 & \circled{3} & \circled{2} & 3 & \circled{2} & 3 & 4 & 5 & 6 & \circled{5} \\
6 & \texttt{A} & 6 & 5 & 4 & 3 & \circled{2} & 3 & \circled{2} & 3 & 4 & \circled{5} & 6 \\
7 & \texttt{T} & 7 & 6 & \circled{5} & \circled{4} & 3 & \circled{2} & 3 & 4 & 5 & 6 & \circled{5} \\
8 & \texttt{C} & 8 & \circled{7} & 6 & 5 & 4 & 3 & 4 & 5 & \circled{4} & 5 & 6 \\
9 & \texttt{G} & 9 & 8 & 7 & 6 & 5 & 4 & 5 & \circled{4} & 5 & 6 & 7 \\
\end{array}
\]
\caption{Computation of the simple edit distance between
$R = \texttt{CATATATCG}$ and $O = \texttt{CTTATAGCAT}$. Matching symbols
are annotated with a circle. The distance~(7) is given by the
bottom-right element.}\label{fig:edit}
\end{figure}
The simple edit distance $D(|R|, |O|) = 7$ is given by the bottom-right
element. Matching symbols are annotated with a circle. From this matrix
a \emph{minimal representation} can be computed by tracing back from the
bottom-right element to the top-left element while doing only orthogonal
(left and up) steps for non-matching elements if the next element has a
lower value than the current one. Vertical steps correspond to deletions,
which are fully characterized by their position~$1 \le i \le |R|$. We
annotate deletions with \texttt{del} and the deleted symbol from $R$ for
convenience. Horizontal steps correspond to insertions, which occur
between two consecutive positions~$i$ and $i + 1$, with $0 \le i \le |R|$
and listing the inserted symbol (taken from $O$). Insertions are
annotated with \texttt{ins}. Special care is needed for consecutive
insertions, i.e., multiple insertions between $i$ and $i + 1$. These
insertions are \emph{ordered} and are often abbreviated as a single
insertion inserting multiple symbols.
For matching elements (circled) a diagonal step (up and left) is allowed,
keeping the current value.
The computation complexity of the simple edit distance is
$\mathcal{O}(|R| \cdot |O|)$~\cite{lcs_complex}, although many tailored
algorithms exist that have an improved bound for specific classes of
strings~\cite{survey, tour, efficient_alllcs, optimal_lcs}. In practice
this means that only a subset of the elements of the matrix needs to be
computed, in particular if only one solution (or just the distance value)
is required.
In general, the number of equivalent trace backs, called
LCS~\emph{embeddings} in~\cite{fast_alllcs, bounds}, is exponentially
bounded by $\binom{|R| + |O|}{|R|}$. We call the set of all minimal
representations~$\Phi(R, O)$ and we formalize the relations between
variants with regard to reference sequence~$R$ by using their
respective~$O$~and~$P$~observed sequences as generic representations as
follows. In the remainder of this paper we use the HGVS~syntax (not
the full nomenclature) to describe a particular variant representation.
\begin{definition}[Equivalence]\label{def:equivalence}
Two variants~$\varphi_O$~and~$\varphi_P$ are \emph{equivalent} if and
only if $\Phi(R, O) = \Phi(R, P)$.
\end{definition}
\noindent
As all minimal representations are equivalent, the set of all minimal
representations must be equal. Consequently, $O = P$. Note that these
variants are equivalent regardless of the reference sequence~$R$.
\paragraph{Example:}
$R = \texttt{TTTTTT}, \quad \varphi_O = \texttt{1delT}, \quad \varphi_P = \texttt{6delT}$ \\
\texttt{1delT} and \texttt{6delT} are equivalent because their respective
observed sequences are equal. Classic normalization procedures followed
by exact string matching are sufficient to draw the same conclusion. This
does not hold for the remaining relations as they rely on checking all
combinations of all minimal alignments.
\begin{definition}[Containment]\label{def:containment}
The variant~$\varphi_O$ \emph{contains} the variant~$\varphi_P$ if and
only if $\varphi_O' \varsupsetneq \varphi_P'$ for some
$\varphi_O' \in \Phi(R, O)$ and $\varphi_P' \in \Phi(R, P)$ and
$\varphi_O$ is not equivalent to $\varphi_P$.
\end{definition}
\noindent
We find a representation within the set of minimal representations
for~$O$ that is a proper subset of a representation within the set of
minimal representations for~$P$.
\paragraph{Example:}
$R = \texttt{TTTTTT}, \quad \varphi_O = \texttt{2\_5delinsGGG}, \quad \varphi_P = \texttt{3T>G}$ \\
\texttt{2\_5delinsGGG} contains \texttt{3T>G} and conversely,
\texttt{3T>G} is contained by \texttt{2\_5delinsGGG}.
\bigskip
\noindent
Notable examples of this relation can be found by comparing multiple
alleles of polymorphic simple tandem repeats, i.e., a long repeat
expansion contains all shorter ones.
\begin{definition}[Overlap]\label{def:overlap}
Two non-equivalent variants~$\varphi_O$~and~$\varphi_P$ \emph{overlap} if
and only if $\varphi_O' \cap \varphi_P' \neq \varnothing$ for some
$\varphi_O' \in \Phi(R, O)$ and $\varphi_P' \in \Phi(R, P)$ while neither
$\varphi_O$ contains $\varphi_P$ nor $\varphi_P$ contains $\varphi_O$.
\end{definition}
\noindent
A proper subset of a representation within the set of minimal
representations for~$O$ is shared with a proper subset of a
representation within the set of minimal representations for~$P$.
\paragraph{Example:}
$R = \texttt{TTTTTT}, \quad \varphi_O = \texttt{2\_4delinsGG}, \quad \varphi_P = \texttt{3T>A}$ \\
\texttt{2\_4delinsGG} has overlap with \texttt{3T>A} (the converse
is also true).
\bigskip
\noindent
Polymorphic SNVs are a notable example of the overlap relation, as they
share the deleted nucleotide, and the inserted nucleotide is different by
definition.
\begin{definition}[Disjoint]\label{def:disjoint}
Two variants~$\varphi_O$~and~$\varphi_P$ are \emph{disjoint} if they are
not equivalent, are not contained in one another, and do not overlap.
\end{definition}
\noindent
None of the minimal representations of~$O$ share anything with any of the
minimal representations of~$P$.
\paragraph{Example:}
$R = \texttt{TTTTTT}, \quad \varphi_O = \texttt{2\_3insA}, \quad \varphi_P = \texttt{4\_5insA}$ \\
\texttt{2\_3insA} and \texttt{4\_5insA} \emph{are disjoint}. Although
both insert the same symbol (\texttt{A}), this cannot occur at a common
position within~$R$.
\bigskip
\noindent
The properties of the Boolean relations given in
Table~\ref{tab:properties} follow directly from the aforementioned
definitions. The table is provided for completeness and future reference,
and throughout this paper we use these properties to reason about
relations.
\begin{table}[ht!]
\caption{Properties of the Boolean relations. The converse of
``contains'' is ``is contained'' and \textit{vice versa}.}%
\label{tab:properties}
\begin{center}
\begin{tabular}{l | lll}
\textbf{relation} & \textbf{symmetry} & \textbf{reflexivity} & \textbf{transitivity} \\
\hline
equivalent & symmetric & reflexive & transitive \\
contains & asymmetric & irreflexive & transitive \\
is contained & asymmetric & irreflexive & transitive \\
overlap & symmetric & irreflexive & intransitive \\
disjoint & symmetric & irreflexive & intransitive \\
\end{tabular}
\end{center}
\end{table}
\section{An Efficient Algorithm}\label{sec:algorithm}
The formal definitions of the Boolean relations presented in
Section~\ref{sec:formal} depend on the enumeration of all minimal variant
representations. As explained in~\cite{bounds}, the number of
representations is bounded exponentially by the length of strings~$R$
and~$O$. For large strings (such as whole human chromosomes up to
ca.~$250 \cdot 10^6$) this approach is infeasible. In this section we
present an alternative and efficient way for the computation of each of
the relations.
\paragraph{Equivalence} As follows directly from
Definition~\ref{def:equivalence}, equivalence can be computed by a string
matching over $O$ and $P$ in $\mathcal{O}(\min(|O|, |P|))$ time and
$\mathcal{O}(|O| + |P|)$ space (storing both strings). This is optimal.
Alternatively, we can compute \emph{metric}~$d$ for $O$ and $P$:
$d(O, P) = 0 \iff \varphi_O$ is equivalent to $\varphi_P$.
\paragraph{Containment} By using the properties of the metric~$d$, we
observe that computing the minimal distances is sufficient. Using the
\emph{triangle inequality}:
$d(R, O) - d(R, P) = d(O, P) \land d(O, P) > 0 \iff \varphi_O$
contains~$\varphi_P$.
\paragraph{Disjoint} Again, using the triangle inequality for metric~$d$:
$d(R, O) + d(R, P) = d(O, P) \land d(O, P) > 0 \implies \varphi_O$
and~$\varphi_P$ are disjoint. Unfortunately, the converse is not true.
Consider the counterexample $R = \texttt{CT}$, $O = \texttt{TG}$, and
$P = \texttt{GC}$. $O$ and $P$ are disjoint despite their simple edit
distances being: $d(R, O) = 2,\, d(R, P) = 2,\, d(O, P) = 2$. Their
representations, however, have no common elements:
$\Phi(R, O) = \{\texttt{[1delC;2\_3insG]}\}$ and
$\Phi(R, P) = \{\texttt{[0\_1insG;2delT]}\}$.
\bigskip
\noindent
The aforementioned distance-based approach can be efficiently computed
using any LCS distance algorithm tailored for similar strings, e.g.,
\cite{lcs_wu}. However, to separate the disjoint and overlap relations we
need to consider all minimal representations. With the notable exception
of the naive dynamic programming approach introduced in
Section~\ref{sec:formal}, existing algorithms typically do not compute
all representations. The naive approach suffers from a
$\mathcal{O}(|R| \cdot |O|)$ space complexity rendering it infeasible
for whole human chromosomes.
\subsection{Computing all Minimal Variant Representations}\label{sec:lcs}
Here, we present an efficient algorithm to compute the relevant elements
of the recurrence relation (Equation~\ref{eq:edit}) to be able to
reconstruct \emph{all} minimal representations (alignments) within the
theoretical complexity bounds: $\mathcal{O}(|R| \cdot |O|)$ time and
using $\mathcal{O}(|R| + |O|)$ temporary space (excluding storing
the solution). In practice, because of the high similarity between $R$
and $O$ the expected run-time is linear. The output of this algorithm is
an LCS-graph as introduced by~\cite{efficient_alllcs}.
We use the generic A* search algorithm~\cite{astar} which uses a
heuristic to guide the search. In general, the space requirements of
A*~search might be of concern. However, in our case, the space is
quadratically bounded by the number of elements in the matrix.
Furthermore, we demonstrate that by expanding partial solutions in a
particular order, it is possible to bound the space requirements
linearly: $\mathcal{O}(|R| + |O|)$.
\bigskip
\noindent
We introduce the \emph{admissible} heuristic:
\begin{equation}\label{eq:heuristic}
h(R, O, i, j) = \left\lvert\left(|R| - i\right) - \left(|O| - j\right)\right\rvert.
\end{equation}
\noindent
The heuristic~$h$ represents a best-case guess for the minimal distance
from the current element~$(i, j)$ to the bottom-right element of the
matrix (hoping to match as many symbols as possible). A* minimizes the
total cost function for each solution:
\begin{equation}\label{eq:astar}
f(R, O, i, j) = D(i, j) + h(R, O, i, j),
\end{equation}
\noindent
by taking into account the actual cost to reach element~$(i, j)$, given
by $D(i, j)$ (see Equation~\ref{eq:edit}), and the projected minimal
cost~$h$.
A*~search iteratively expands partial solutions, also called the
\emph{frontier}, based on the lowest $f$~value until the target element
is expanded. In our case the progression of $f$-values is determined by
the heuristic value of the first element $h(0, 0) = ||R| - |O||$
increasing with steps of~$2$ as the cost function increases by $1$ for
each orthogonal step and the heuristic changes with either $+1$ or $-1$
for each orthogonal step. Diagonal steps, i.e., matching symbols, do not
incur a change in $f$-value. This results in a constant parity for the
$f$-values. The simple edit distance is given by the $f$-value of the
target element~$(|R|, |O|)$. Constructing all minimal variant
representations is analogous to the naive approach detailed in
Section~\ref{sec:formal}.
In typical A* implementations, the frontier is implemented as a priority
queue. In our case, we observe that we can keep track of the elements in
the frontier by describing a ``convex'' shape in the matrix. We use two
arrays~$\mathit{rows}$ and~$\mathit{cols}$ that store the right-most
element for a given column and the bottom-most element for a given row
respectively.
In Figure~\ref{fig:astar} we present the progression of the expansion of
the matrix elements for the example: $R = \texttt{CATATATCG}$ and
$O = \texttt{CTTATAGCAT}$. We use $\mathcal{O}(|R| + |O|)$~space
(excluding the output) and we expand at most
$\mathcal{O}(|R| \cdot |O|)$~elements.
\begin{figure}[ht!]
\subfloat[Expanded elements for $f = 1$.]{
\begin{minipage}[c][1\width]{
0.45\textwidth}
\centering
\[
\scalemath{0.62}{
\begin{array}{rc|ccccccccccc}
\renewcommand*{\arraystretch}{1.2}
& & \phantom{\circled{0}} & \phantom{\circled{1}} & \phantom{\circled{2}} & \phantom{\circled{3}} & \phantom{\circled{4}} & \phantom{\circled{5}} & \phantom{\circled{6}} & \phantom{\circled{7}} & \phantom{\circled{8}} & \phantom{\circled{9}} & \phantom{\circled{10}} \\
& & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
& & \texttt{.} & \texttt{C} & \texttt{T} & \texttt{T} & \texttt{A} & \texttt{T} & \texttt{A} & \texttt{G} & \texttt{C} & \texttt{A} & \texttt{T} \\
\hline
0 & \texttt{.} & 1 & 1 & & & & & & & & & \\
1 & \texttt{C} & & \circled{1} & 1 & & & & & & & & \\
2 & \texttt{A} & & & & & & & & & & & \\
3 & \texttt{T} & & & & & & & & & & & \\
4 & \texttt{A} & & & & & & & & & & & \\
5 & \texttt{T} & & & & & & & & & & & \\
6 & \texttt{A} & & & & & & & & & & & \\
7 & \texttt{T} & & & & & & & & & & & \\
8 & \texttt{C} & & & & & & & & & & & \\
9 & \texttt{G} & & & & & & & & & & & \\
\end{array}
}
\]
\vspace{-2em}
{\small
\begin{align*}
\mathit{rows} &= [1, 2], \\
\mathit{cols} &= [0, 1, 1]
\end{align*}}%
\end{minipage}}
\hfill
\subfloat[Expanded elements for $f = 3$.]{
\begin{minipage}[c][1\width]{
0.45\textwidth}
\centering
\[
\scalemath{0.62}{
\begin{array}{rc|ccccccccccc}
\renewcommand*{\arraystretch}{1.2}
& & \phantom{\circled{0}} & \phantom{\circled{1}} & \phantom{\circled{2}} & \phantom{\circled{3}} & \phantom{\circled{4}} & \phantom{\circled{5}} & \phantom{\circled{6}} & \phantom{\circled{7}} & \phantom{\circled{8}} & \phantom{\circled{9}} & \phantom{\circled{10}} \\
& & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
& & \texttt{.} & \texttt{C} & \texttt{T} & \texttt{T} & \texttt{A} & \texttt{T} & \texttt{A} & \texttt{G} & \texttt{C} & \texttt{A} & \texttt{T} \\
\hline
0 & \texttt{.} & 1 & 1 & 3 & & & & & & & & \\
1 & \texttt{C} & 3 & \circled{1} & 1 & 3 & & & & & & & \\
2 & \texttt{A} & & 3 & 3 & 3 & \circled{3} & & & & & & \\
3 & \texttt{T} & & & \circled{3} & \circled{3} & 3 & \circled{3} & & & & & \\
4 & \texttt{A} & & & & & \circled{3} & 3 & \circled{3} & & & & \\
5 & \texttt{T} & & & & & & \circled{3} & 3 & & & & \\
6 & \texttt{A} & & & & & & & \circled{3} & 3 & & & \\
7 & \texttt{T} & & & & & & & & & & & \\
8 & \texttt{C} & & & & & & & & & & & \\
9 & \texttt{G} & & & & & & & & & & & \\
\end{array}
}
\]
\vspace{-2em}
{\small
\begin{align*}
\mathit{rows} &= [2, 3, 4, 5, 6, 6, 7], \\
\mathit{cols} &= [1, 2, 3, 3, 4, 5, 6, 6]
\end{align*}}%
\end{minipage}}
\subfloat[Expanded elements for $f = 5$.]{
\begin{minipage}[c][1\width]{
0.45\textwidth}
\centering
\[
\scalemath{0.62}{
\begin{array}{rc|ccccccccccc}
\renewcommand*{\arraystretch}{1.2}
& & \phantom{\circled{0}} & \phantom{\circled{1}} & \phantom{\circled{2}} & \phantom{\circled{3}} & \phantom{\circled{4}} & \phantom{\circled{5}} & \phantom{\circled{6}} & \phantom{\circled{7}} & \phantom{\circled{8}} & \phantom{\circled{9}} & \phantom{\circled{10}} \\
& & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
& & \texttt{.} & \texttt{C} & \texttt{T} & \texttt{T} & \texttt{A} & \texttt{T} & \texttt{A} & \texttt{G} & \texttt{C} & \texttt{A} & \texttt{T} \\
\hline
0 & \texttt{.} & 1 & 1 & 3 & 5 & & & & & & & \\
1 & \texttt{C} & 3 & \circled{1} & 1 & 3 & 5 & & & & & & \\
2 & \texttt{A} & 5 & 3 & 3 & 3 & \circled{3} & 5 & & & & & \\
3 & \texttt{T} & & 5 & \circled{3} & \circled{3} & 3 & \circled{3} & 5 & & & & \\
4 & \texttt{A} & & & 5 & 5 & \circled{3} & 3 & \circled{3} & 5 & & & \\
5 & \texttt{T} & & & & \circled{5} & 5 & \circled{3} & 3 & 5 & & & \\
6 & \texttt{A} & & & & & \circled{5} & 5 & \circled{3} & 3 & 5 & & \\
7 & \texttt{T} & & & & & & \circled{5} & 5 & 5 & 5 & & \\
8 & \texttt{C} & & & & & & & & & \circled{5} & 5 & \\
9 & \texttt{G} & & & & & & & & & & & \\
\end{array}
}
\]
\vspace{-2em}
{\small
\begin{align*}
\mathit{rows} &= [3, 4, 5, 6, 7, 7, 8, 8, 9], \\
\mathit{cols} &= [2, 3, 4, 5, 6, 7, 7, 7, 8, 8]
\end{align*}}%
\end{minipage}}
\hfill
\subfloat[Expanded elements for $f = 7$.]{
\begin{minipage}[c][1\width]{
0.45\textwidth}
\centering
\[
\scalemath{0.62}{
\begin{array}{rc|ccccccccccc}
\renewcommand*{\arraystretch}{1.2}
& & \phantom{\circled{0}} & \phantom{\circled{1}} & \phantom{\circled{2}} & \phantom{\circled{3}} & \phantom{\circled{4}} & \phantom{\circled{5}} & \phantom{\circled{6}} & \phantom{\circled{7}} & \phantom{\circled{8}} & \phantom{\circled{9}} & \phantom{\circled{10}} \\
& & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
& & \texttt{.} & \texttt{C} & \texttt{T} & \texttt{T} & \texttt{A} & \texttt{T} & \texttt{A} & \texttt{G} & \texttt{C} & \texttt{A} & \texttt{T} \\
\hline
0 & \texttt{.} & 1 & 1 & 3 & 5 & 7 & & & & & & \\
1 & \texttt{C} & 3 & \circled{1} & 1 & 3 & 5 & 7 & & & & & \\
2 & \texttt{A} & 5 & 3 & 3 & 3 & \circled{3} & 5 & \circled{7} & & & & \\
3 & \texttt{T} & 7 & 5 & \circled{3} & \circled{3} & 3 & \circled{3} & 5 & 7 & & & \\
4 & \texttt{A} & & 7 & 5 & 5 & \circled{3} & 3 & \circled{3} & 5 & 7 & & \\
5 & \texttt{T} & & & \circled{7} & \circled{5} & 5 & \circled{3} & 3 & 5 & 7 & & \\
6 & \texttt{A} & & & & 7 & \circled{5} & 5 & \circled{3} & 3 & 5 & \circled{7} & \\
7 & \texttt{T} & & & & & 7 & \circled{5} & 5 & 5 & 5 & 7 & \circled{7} \\
8 & \texttt{C} & & & & & & 7 & 7 & 7 & \circled{5} & 5 & 7 \\
9 & \texttt{G} & & & & & & & & \circled{7} & 7 & 7 & 7 \\
\end{array}
}
\]
\vspace{-2em}
{\small
\begin{align*}
\mathit{rows} &= [4, 5, 6, 7, 8, 8, 9, 10, 10, 10], \\
\mathit{cols} &= [3, 4, 5, 6, 7, 8, 8, 9, 9, 9, 9]
\end{align*}}%
\end{minipage}}
\caption{Computing the relevant elements of Equation~\ref{eq:edit} to
efficiently reconstruct the set of all elements of the set of all minimal
variant representations.}\label{fig:astar}
\end{figure}
\noindent
The non-filled elements are not part of any minimal representation. The
circled elements are needed to create the LCS-graph and therefore stored.
The remaining elements are expanded, but not stored. For each circled
element we determine its place in the LCS by:
\begin{equation}
\left\lfloor\frac{i + j - D(i, j)}{2}\right\rfloor - 1.
\end{equation}
\noindent
This allows us to construct the LCS-graph efficiently. The LCS-graph for
the example in Figure~\ref{fig:astar} is given in Figure~\ref{fig:graph}.
The nodes in the LCS-graph are ordered on their position in the LCS. To
construct the variant representations, edges are added for each node on
level~$n$ to each node on level~$n + 1$ if
$\mathit{i}_{n+1} > \mathit{i}_n$ and
$\mathit{j}_{n+1} > \mathit{j}_n$, where each node is given by its
position~$(i, j)$ in $D$.
For instance, there is an edge from node~$(3, 2)$ on level~$1$ to
node~$(5, 3)$ on level~$2$ (\texttt{4delA}). All orthogonal steps
(Manhattan paths) within such a rectangle contribute to the set of
elements of all minimal variant representations ($\bigcup\Phi$). To
distinguish between the relations disjoint and overlap, it is sufficient
to determine whether the two sets~($\bigcup\Phi(R, O)$
and~$\bigcup\Phi(R, P)$) of these elements is disjoint. Note that the
number of elements in each set is bounded quadratically as opposed to
enumerating all, exponentially bounded, minimal representations. Within
the genetic variant domain, the reference sequence and both observed
sequences are expected to be highly similar, resulting in expected linear
time. Some practical implementation enhancements can also be applied,
notably, reducing the number of elements to be added to the set by taking
(partially) overlapping rectangles (edges in the LCS-graph) into account.
For small alphabets, e.g., DNA nucleotides, an efficient bit string can
be used in lieu of a proper set implementation.
\begin{figure}[ht!]
\begin{center}
\resizebox{\textwidth}{!}
{
\begin{tikzpicture}
\node[align=center, draw, circle, fill] (00) at (0, 0) {};
\node[align=center, draw, ellipse, thick, minimum width=4.8em] (11) at (2, 0) {\texttt{C}\\ $(1, 1)$};
\node[align=center, draw, ellipse, thick, minimum width=4.8em] (32) at (5, -2) {\texttt{T}\\ $(3, 2)$};
\node[align=center, draw, ellipse, thick, minimum width=4.8em] (33) at (5, 0) {\texttt{T}\\ $(3, 3)$};
\node[align=center, draw, ellipse, thick, minimum width=4.8em] (24) at (5, 2) {\texttt{A}\\ $(2, 4)$};
\node[align=center, draw, ellipse, thick, minimum width=4.8em] (53) at (8, -2) {\texttt{T}\\ $(5, 3)$};
\node[align=center, draw, ellipse, thick, minimum width=4.8em] (44) at (8, 0) {\texttt{A}\\ $(4, 4)$};
\node[align=center, draw, ellipse, thick, minimum width=4.8em] (35) at (8, 2) {\texttt{T}\\ $(3, 5)$};
\node[align=center, draw, ellipse, thick, minimum width=4.8em] (64) at (11, -2) {\texttt{A}\\ $(6, 4)$};
\node[align=center, draw, ellipse, thick, minimum width=4.8em] (55) at (11, 0) {\texttt{T}\\ $(5, 5)$};
\node[align=center, draw, ellipse, thick, minimum width=4.8em] (46) at (11, 2) {\texttt{A}\\ $(4, 6)$};
\node[align=center, draw, ellipse, thick, minimum width=4.8em] (75) at (14, -2) {\texttt{T}\\ $(7, 5)$};
\node[align=center, draw, ellipse, thick, minimum width=4.8em] (66) at (14, 0) {\texttt{A}\\ $(6, 6)$};
\node[align=center, draw, ellipse, thick, minimum width=4.8em] (69) at (14, 2) {\texttt{A}\\ $(6, 9)$};
\node[align=center, draw, ellipse, thick, minimum width=4.8em] (88) at (17, -2) {\texttt{C}\\ $(8, 8)$};
\node[align=center, draw, ellipse, thick, minimum width=4.8em] (97) at (17, 0) {\texttt{G}\\ $(9, 7)$};
\node[align=center, draw, ellipse, thick, minimum width=4.8em] (710) at (17, 2) {\texttt{T}\\ $(7, 10)$};
\node[align=center, draw, circle, fill] (1011) at (20, 0) {};
\draw[-{Latex[width=.5em]}, semithick] (00) -- (11);
\draw[-{Latex[width=.5em]}, semithick] (11) -- (24) node[above, left, midway] {\footnotesize \texttt{1\_2insTT}};
\draw[-{Latex[width=.5em]}, semithick] (11) -- (33) node[above, midway] {\footnotesize \texttt{2A>T}};
\draw[-{Latex[width=.5em]}, semithick] (11) -- (32) node[below, left, midway] {\footnotesize \texttt{2delA}};
\draw[-{Latex[width=.5em]}, semithick] (24) -- (35);
\draw[-{Latex[width=.5em]}, semithick] (33) -- (44);
\draw[-{Latex[width=.5em]}, semithick] (32) -- (53) node[above, midway] {\footnotesize \texttt{4delA}};
\draw[-{Latex[width=.5em]}, semithick] (32) -- (44) node[above, left, midway] {\footnotesize \texttt{3\_4insT}};
\draw[-{Latex[width=.5em]}, semithick] (35) -- (46);
\draw[-{Latex[width=.5em]}, semithick] (44) -- (55);
\draw[-{Latex[width=.5em]}, semithick] (53) -- (64);
\draw[-{Latex[width=.5em]}, semithick] (46) -- (69) node[above, midway] {\footnotesize \texttt{5delinsGC}};
\draw[-{Latex[width=.5em]}, semithick] (55) -- (66);
\draw[-{Latex[width=.5em]}, semithick] (55) -- (69) node[above, left, midway] {\footnotesize \texttt{5\_6insAGC}};
\draw[-{Latex[width=.5em]}, semithick] (64) -- (75);
\draw[-{Latex[width=.5em]}, semithick] (66) -- (710) node[above, left, midway] {\footnotesize \texttt{6\_7insGCA}};
\draw[-{Latex[width=.5em]}, semithick] (66) -- (88) node[below, near end, xshift=-.5em] {\footnotesize \texttt{7T>G}};
\draw[-{Latex[width=.5em]}, semithick] (66) -- (97) node[above, midway] {\footnotesize \texttt{7\_8delTC}};
\draw[-{Latex[width=.5em]}, semithick] (69) -- (710);
\draw[-{Latex[width=.5em]}, semithick] (75) -- (88) node[below, midway] {\footnotesize \texttt{7\_8insAG}};
\draw[-{Latex[width=.5em]}, semithick] (75) -- (97) node[above, near end, xshift=-.5em] {\footnotesize \texttt{8C>A}};
\draw[-{Latex[width=.5em]}, semithick] (710) -- (1011) node[above, right, midway] {\footnotesize \texttt{8\_9delCG}};
\draw[-{Latex[width=.5em]}, semithick] (88) -- (1011) node[below, right, midway] {\footnotesize \texttt{9delinsAT}};
\draw[-{Latex[width=.5em]}, semithick] (97) -- (1011) node[above, midway, xshift=-.5em] {\footnotesize \texttt{9\_10insCAT}};
\end{tikzpicture}}
\end{center}
\caption{The LCS-graph for $R = \texttt{CATATATCG}$ and
$O = \texttt{CTTATAGCAT}$. The coordinates refer to the coordinates of
the matching characters in Figure~\ref{fig:astar}. Unlabeled edges
indicate consecutive matches and do not contribute to the set of
elements of all minimal variant representations.}\label{fig:graph}
\end{figure}
\subsection{Maximal Influence Interval}\label{sec:influence}
Given any pair of variants (within the context of the same reference
sequence), it is likely that their relation is disjoint purely based on
their respective positions in the reference sequence. These disjoint
relations can be determined efficiently at the cost of some
pre-computation for individual variants (n.b. not pairs of variants).
For each variant the \emph{maximal influence interval}, i.e., the
interval given by the lowest column index for a deletion or an insertion
in~$D$ and the highest column index for a deletion or an insertion
in~$D$. This interval gives the extreme bounds, as positions in the
reference sequence, of possible changes due to this variant. A pair of
variants can only be non-disjoint when their influence intervals
intersect. The pre-computing of the influence intervals of individual
variants is specifically worthwhile in the context of repeated querying,
e.g., a (locus specific) database and VCF annotation.
\section{Experiments}\label{sec:experiments}
To obtain an intuition of the impact of the proposed approach, we
analyzed the well-studied \textit{CFTR}~gene (NG\_016465.4 with
$257,\!188$bp) that provides instructions for making the cystic
fibrosis transmembrane conductance regulator protein).
In dbSNP~\cite{dbsnp} there are $62,\!215$~interpretable variants for the
\textit{CFTR}~gene which would lead to $1,\!935,\!322,\!005$~pairs of
variants to analyze. Using the pre-filtering method described in
Section~\ref{sec:influence}, only $92,\!251$~eligible pairs of
variants with a potential non-disjoint relation remain.
\begin{table}[ht!]
\caption{Relation counts for the pairwise comparison of variants in the
\textit{CFTR}~gene. The counts are given based on the upper triangular
matrix, so the converse relations are not included.}\label{tab:relcounts}
\begin{center}
\begin{tabular}{l | r}
\textbf{relation} & \textbf{count} \\
\hline
equivalent & $0$ \\
contains & $5,\!491$ \\
is contained & $4,\!629$ \\
overlap & $37,\!690$ \\
disjoint & $44,\!441$ \\
\end{tabular}
\end{center}
\end{table}
\noindent
When the algebra is applied to the remaining pairs we obtain the results
in Table~\ref{tab:relcounts}. We observe that there are no equivalent
pairs, meaning that there are no duplicate variant entries for
\textit{CFTR} in dbSNP (indicating a correct application of the
standard normalization techniques). There are $10,\!120$~containment
relations (either contains or is contained), $37,\!690$~pairs have some
form of overlap, and $44,\!441$~pairs are completely disjoint.
\begin{figure}[ht!]
\begin{center}
\begin{tikzpicture}
\begin{axis}[%
height=7cm,
width=10cm,
axis y line*=left,
axis x line*=bottom,
ticklabel style={font=\small},
xlabel=length of influence interval,
ylabel=average number of relations
]
\addplot[blue!50,fill=blue!30,only marks] table [%
x=length,
y=count]{\influencelength};
\end{axis}
\end{tikzpicture}
\caption{Scatterplot of the average number of relations for all variants
in \textit{CFTR} with a certain influence interval length.}%
\label{fig:influence_scatter}
\end{center}
\end{figure}
Zooming in to individual variant level, we find that $16,\!939$~variants
are disjoint with all other variants based on pre-filtering alone and
$45,\!276$ are potentially involved in a non-disjoint relation with
another variant. $16,\!814$ of which turn out to be disjoint with all
other variants. In total, $33,\!753$~variants are completely disjoint
with all other variants. The remaining $28,\!462$~variants have a
non-disjoint relation to some other variant(s).
\noindent
In Table~\ref{tab:examples}, we see a selection of variants in
\textit{CFTR} that, at first sight, have a counter-intuitive relation
with another variant. For pair~$1$, the left hand side (LHS) variant
contains the right hand side (RHS) variant because the former can be
left-justified to \texttt{11402\_11406del} to incorporate the deletion of
region $11,\!402$ to $11,\!403$. For pair~$2$, the containment is less
obvious, the LHS needs to be rewritten to
\texttt{[151240\_151241insTATA;151270\_151271insCA]} to make this
containment relation intuitively clear. For pair~$3$, the LHS can be
written as \texttt{[151242\_151243del;151271\_151278del]} to make the
overlap relation between the two variants clear. For pair~$4$,
left-justification of the LHS to \texttt{112270\_112271insCTCTCTC} and
rewriting the RHS to
\texttt{[112269\_112270insCC; 112270\_112271insCTCT]} makes the overlap
relation obvious. Finally, we can see from both pair~$2$ and~$3$ that in
practice, variants that are reported to be well separated, still may have
something in common.
\begin{table}[hb!]
\caption{Examples of non-trivial relations between variants in
\textit{CFTR}. The variants are described using the HGVS nomenclature
with respect to reference sequence NG\_016465.4 using the genomic~(g.)
coordinate system.}\label{tab:examples}
\begin{center}
\begin{tabular}{l|l|l|l}
\textbf{pair} & \textbf{LHS variant} & \textbf{relation} & \textbf{RHS variant} \\
\hline
$1$ & \texttt{11404\_11408del} & contains & \texttt{11402\_11403del} \\
$2$ & \texttt{151270\_151271insTATACA} & contains & \texttt{151240\_151241insAT} \\
$3$ & \texttt{151271\_151280del} & overlap & \texttt{151240\_151255del} \\
$4$ & \texttt{112274\_112275insCTCTCTC} & overlap & \texttt{112269\_112270insCCTCTC} \\
\end{tabular}
\end{center}
\end{table}
\noindent
The ratio between the length of the influence interval and the number of
relations a variant has on average is shown in
Figure~\ref{fig:influence_scatter}. The length of the influence interval
correlates strongly with the number of relations of a variant as
expected. The variants with the largest influence interval lengths
($>\!150$) all happen to be large deletions.
\begin{figure}[ht!]
\begin{center}
\begin{tikzpicture}
\begin{axis}[%
height=7cm,
width=10cm,
xlabel=number of relations,
axis y line*=left,
axis x line*=bottom,
ticklabel style={font=\small},
xticklabels={$1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$, $>\!10$},
xtick={1,...,11},
yticklabel style={/pgf/number format/fixed,},
scaled y ticks=false,
ylabel=number of variants,
ybar]
\addplot[blue!50,fill=blue!30] table[%
x=relations,
y=count] {\relationscount};
\end{axis}
\end{tikzpicture}
\caption{The distribution of the number of relations per variant. The
long tail of counts of $11$ and above are aggregated. The most relations
a single variant has is $435$.}\label{fig:rel_count}
\end{center}
\end{figure}
\noindent
The distribution of the number of relations per variant is shown in
Figure~\ref{fig:rel_count}. More than half of all variants ($16,\!735$)
have a single non-trivial relation with another variant, the remaining
$11,\!727$~variants have a non-trivial relation with multiple variants.
The distributions for both overlap and inclusion relations, are nearly
identical.
\section{Discussion}\label{sec:discussion}
The relation between a pair of variants is only well-defined when both
variants are described in the context of the same reference sequence. In
general, we can extend the definitions to include variants on different
reference sequences, the natural interpretation of which would be to
consider two variants on different reference sequences to be disjoint,
e.g., a variant on human chromosome~1 has nothing in common with a
variant on human chromosome~2. This interpretation is sensible as long as
the reference sequences are unrelated. In practice however, many
reference sequences are actually referring to the same (or a strongly
related) genetic locus, e.g., genes on chromosomes, different transcripts
for the same gene and chromosomes in different reference genomes.
Arguably, variants described in the context of these reference sequences
could be seen as having potentially a non-disjoint relation. To properly
compare these variants on sequence level, the differences between the
reference sequences should also be taken into account.
Structural variants are often reported in non-exact manner, i.e., not on
sequence level precise. These representations are unsuitable for our
method. Even if an exact structural variant representation is given, it
unlikely to yield meaningful results; as the exact positions are not the
same across samples. Instead, e.g., gene copies can be analyzed by the
algebra when they are provided individually.
The choice of relations presented here follows the ones from set theory,
commonly used in a wide range of domains. For some specific domains more
refined relations exists as well, e.g., for intervals the relations:
``starts with'', ``ends with'' and ``is directly adjacent'' are useful
extensions~\cite{interval}. The set of relations could be further
partitioned using these, or other, refinements.
Unfortunately, the set of relations (see Table~\ref{tab:properties}) does
not contain a relation that implies an ordering of variants, i.e.,
$\Phi(R, O) \le \Phi(R, P)$. A partial order of variants would require a
relation with the properties: reflexive, antisymmetric and transitive.
Sorting variants or storing variants in a particular order in a database
(indexing) is meaningless in the context of this algebra. The interval
ordering based on the pre-computed influence intervals described in
Section~\ref{sec:influence} mitigates this problem.
\subsection{Maximal Overlap}
The actual make-up of the common changes between two variants is never
computed. For all relations except the overlap relation the common
changes can be trivially given: none for disjoint variants, either of the
variants for equivalence, and the ``smaller'' variant for containment,
i.e., the one that is contained within the other. This leaves, however,
the overlapping variants. In general, there are many different sets of
common changes between overlapping variants, some of which, especially
the larger ones, may be more (biologically) relevant than others. The
algorithm described in Section~\ref{sec:algorithm} determines whether
there is at least one common change. Computing the maximal size of the
overlap requires enumerating an exponential number of possible
alignments, which is infeasible for all but extremely short sequences.
\subsection{General Normalization}\label{sec:general}
The current practice of normalizing variant representations is
sufficiently powerful to cater for the equivalence relation (also
illustrated in Section~\ref{sec:experiments}). Determining other
relations is, in general, impossible when given a single normalized
representation. Even SNVs, often regarded as trivially normalized, are
problematic when querying for containment. Consider
reference~$R = \texttt{CACAT}$ and the SNV~\texttt{3C>T} to obtain the
observed sequence~$O = \texttt{CATAT}$. In the classical sense no
normalization is necessary. When we consider a second
variant~\texttt{3\_4insT} (\texttt{CACTAT}) we might draw the conclusion
that this insertion is contained within the SNV based on the normalized
position. A possible third variant~\texttt{2\_3insT} (\texttt{CATCAT})
has the same relation, but is less trivially found. When substrings
adjacent to the variant match subsequences of the deleted or inserted
string, the number of alignments increases exponentially, so regardless
of which normalization procedure is used, however sophisticated, counter
examples like this can always be constructed. Therefore procedures that
rely on normalization will, in general, lead to wrong conclusions and
cannot be employed to determine relations between variants.
Within the domain specific languages for variant representations
different normalization schemes are used, where arbitrary choices
influence the normalized representation, e.g., the $3'$ and $5'$-rules.
From the alignment matrix~$D$, it is also possible to choose a canonical
path that represents a normalized representation.
Sensible choices are either a bottom-most or top-most path. This
corresponds to favoring either deletions over insertions at the beginning
of a variant (or vice versa). Note that for all minimal variant
descriptions in any of the domain specific languages, corresponding
alignments can be found. It could be worthwhile to investigate whether a
comprehensive set of deterministic rules exist to find these alignments
as this can be used in the formalization of these languages.
\subsection{Non-minimal Variant Representations}
So far, we assumed that all variant representations are minimal with
regard to Equation~\ref{eq:edit}. In practice, this is not always the
case, nor is it necessary for our approach to work as the only constraint
on the variant representation is its interpretability (see
Section~\ref{sec:intro}). The relations are computed on all minimal
alignments, where a non-minimal representation is minimized as part of
the procedure. Interpreting the relations based on non-minimal
representations yield surprising results. When we consider the reference
$R = \texttt{GCTTT}$ with variant~$\varphi_O = \texttt{[1G>A;2C>G;3T>C]}$
($O = \texttt{AGCTT}$) and variant~$\varphi_P = \texttt{[1G>A;2C>G]}$
($P = \texttt{AGTTT}$). The naive conclusion, based on the non-minimal
representation, would be $\varphi_O$ contains $\varphi_P$. However, both
$\varphi_O$ and $\varphi_P$ are not minimal. The minimal alignments for
$\Phi(R, O) = \{\texttt{[0\_1insA;3delT]}, \texttt{[0\_1insA;4delT]},
\texttt{[0\_1insA;5delT]}\}$ and the minimal alignment for
$\Phi(R, P) = \{\texttt{[0\_1insA;2delC]}\}$ show that that the actual
relation is overlap instead of containment.
\bigskip
\noindent
A variant representation (in the classical sense) that covers all
possible minimal alignments simultaneously is impossible to find in the
general case because of potential mutual exclusivity of sub-alignments.
A trivial solution is the full listing of the observed sequence.
This, however, offsets the benefits of a representation that is
humanly understandable and furthermore it introduces a huge amount of
redundant information for larger sequences.
\noindent
However, based on the influence intervals introduced in
Section~\ref{sec:influence}, a normalized \emph{maximal} variant
representation can be defined. These take the
form of a deletion insertion where the deletion spans the entire
influence interval and the insertion potentially contains redundant
reference information. For the SNV example in Section~\ref{sec:general}
the maximal representation is~\texttt{2\_3delinsAT}, where first
an~\texttt{A} is deleted and inserted again. SPDI (and consequently VRS)
prescribes a normalization procedure that follows a similar
approach~\cite{spdi} by extending the variant in both directions using a
rolling procedure. We note that such a procedure, in general, does not
result in all minimal alignments (nor the extreme bounds) being contained
in the representation for all variants.
Arguably, a maximal representation is not suitable in all contexts, e.g.,
reporting clinical results, but within the context of storing large
quantities of variants in, for instance a database, the proposed maximal
representations are appealing as the variants can be properly ordered and
indexed on their deleted interval. Furthermore, these representations
contain all information needed to determine the relations with other
variants in the database without the need to use the reference sequence.
The drawback, however, is that potentially larger inserted sequences are
stored (\texttt{AT} in the example). In practice however, the influence
intervals are small compared to the length of the reference sequence.
\section{Conclusions}\label{sec:conclusion}
Looking beyond the identification of equivalent variants, we introduced a
comprehensive set of Boolean relations: equivalence, containment, overlap
and disjoint, that partitions the domain of binary variant relations.
Using these relations, additional variants of interest, i.e., variants
with a specific relation to the queried variant can be identified. We
determine these relations by taking all minimal alignments (on sequence
level) into account. The relations can be computed efficiently using a
novel algorithm that computes all minimal alignments. We have shown that
these relations occur frequently in existing datasets, notably large ones
like dbSNP. Approximately half of the variants in the \textit{CFTR}~gene
in dbSNP has at least one non-disjoint relation with another variant
within the same gene. We have shown that normalization of variant
representations is not powerful enough to answer any but the trivial
relation queries. Inspired by the alignment matrix, we introduced a
maximal variant representation that allows for querying all relations
while the variants are efficiently indexed in a database setting.
In the case where phased variants (alleles) are available, directly
querying on other (combinations of) variants is possible, e.g., is a
variant contained within a given allele.
The quantification and the make-up of the overlap relation remains an
open problem. Locus specific databases can, without changing their
internal representation of variants, use our algebra to query on these
relations. Because our method is not tied to a particular representation,
it can also be applied in VCF annotation tools.
\noindent
A Python implementation is available at
\path{https://github.com/mutalyzer/algebra} as well as an
interface at \path{https://v3.mutalyzer.nl/algebra}.
\subsection{Future Work}
The current Python implementation is suitable for sequences with the
length of that of an average gene. An implementation in a more
performance oriented language allows for the use of whole human
chromosomes.
Although, from the algebra perspective, a single canonical (or
normalized) representation is insufficient, we see advantages of having
such a representation in different contexts (especially for human
interpretation). By looking at patterns within all the minimal
alignments, we can potentially construct a canonical representation that
reflects these patterns on sequence level in the variant, e.g., repeated
elements can be separated from larger variants or a sequence level
argument can be given for why close by SNVs should be (or not be)
combined. These observations could be combined in a new implementation of
a variant description extractor~\cite{vde}.
Dealing with variants in an algebraic way can possibly be extended to
higher-level calculations such as union and subtraction. The ability to
mathematically construct larger alleles from smaller variants seems
appealing in many domains. These techniques would also enable a proper
sequence level remapping of variants onto other reference sequences,
which is a recurring problem with the publication of every new reference
genome.
\printbibliography
\end{document}
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2021-12-30T02:25:23.000Z
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{"paloma_paragraphs":[]}
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{"arxiv_id":"2112.14494","language":"en","timestamp":1640831123000,"url":"https:\/\/arxiv.org\/abs\/2112.14494","yymm":"2112"}
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proofpile-arXiv_000-10037
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{"provenance":"002.jsonl.gz:10038"}
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\section*{Acknowledgements}
Edith Cohen is supported by Israel Science Foundation grant no.\ 1595-19.
Haim Kaplan is supported by Israel Science Foundation grant no.\ 1595-19, and the Blavatnik Family Foundation.
Yishay Mansour has received funding from the European Research Council (ERC) under the European Union’sHorizon 2020 research and innovation program (grant agreement No. 882396), by the Israel Science Foundation (grant number 993/17) and the Yandex Initiative for Machine Learning at Tel Aviv University.
Uri Stemmer is partially supported by the Israel Science Foundation (grant 1871/19) and by the Cyber Security Research Center at Ben-Gurion University of the Negev.
\printbibliography
\section{Empirical Results}
We implemented in Python our two main algorithms for $k$-tuple clustering: $\mathsf{PrivatekAverages}$ and $\mathsf{PrivatekNoisyCenters}$. We compared the two algorithms in terms of the sample complexity that is needed to privately separate the samples from a given mixture of Gaussians. Namely, how many $k$-tuples we need to sample
such that, when executing $\mathsf{PrivatekAverages}$ or $\mathsf{PrivatekNoisyCenters}$, the resulting $k$-tuple $Y = \set{\py_1,\ldots,\py_k}$ satisfies the following requirement: For every $i \in [k]$, there exists a point in $Y$ (call it $\py_i$), such that for every sample $\px$ that was drawn from the $i$'th Gaussian, it holds that $i = \operatorname*{argmin}_{j \in [k]} \norm{\px - \py_j}$.
We perform three tests, where in each test we considered a uniform mixture of $k$ standard spherical Gaussians
around the means
$\set{R\cdot \pt{e}_i, -R\cdot \pt{e}_i}_{i=1}^{k/2}$, where $\pt{e}_i$ is the $i$'th standard basis vector.
In all the tests, we generated each $k$-tuple by running algorithm k-means++ \cite{kmeansplusplus} over enough samples.
In Test1 (\cref{Test1}) we examined the sample complexity in the case $d=1$, $k=2$, for $R \in \set{2^5,2^6,\ldots,2^{9}}$. In Test2 (\cref{Test2}) we examined the case $d=4$, $R = 512 \cdot k$, for $k \in \set{2,4,6,8}$. In Test3 (\cref{Test3}) we examined the case $k=2$, $R=256\sqrt{d}$, for $d \in \set{4,8,12,16}$. In all the experiments we used privacy parameters $\varepsilon = 1$ and $\delta = e^{-28}$, and used $\beta = 0.05$. In all the tests of $\mathsf{PrivatekNoisyCenters}$, we chose $\Delta = \frac{10}{\varepsilon}\cdot k \log(k/\delta) \sqrt{\log(k/\beta)}$, the number of $k$-tuples that we generated was exactly $3781$ (the minimal value that is required for privacy), but the number of samples per $k$-tuple varied from test to test. In the tests of $\mathsf{PrivatekAverages}$, we chose $\Lambda = 2^{10} \cdot k \sqrt{d}$ and $r_{\min} = 0.1$, we generated each $k$-tuple using $\approx 15 \cdot k$ samples, but the number of $k$-tuples varied from test to test.\footnote{By using $\tilde{\Omega}(kd)$ samples for creating each $k$-tuple, in Test3 (\cref{Test3}) we could avoid the dependency of $R$ in $\sqrt{d}$ (see \cref{sec:Gaussians:remarks} for more details). However, since we only used $O(k)$ samples for each $k$-tuple when testing $\mathsf{PrivatekAverages}$, then we could not avoid this dependency.}
All the experiments were tested in a MacBook Pro Laptop with 4-core Intel i7 CPU with 2.8GHz, and with 16GB RAM.
\begin{figure}[t]
\centerline{\includegraphics[scale=.5]{Test1.png}}
\caption{The case $d=1$ and $k=2$, for varies $R$.}
\label{Test1}
\end{figure}
\begin{figure}[t]
\centerline{\includegraphics[scale=.5]{Test2.png}}
\caption{The case $d=4$ and $R=512 \cdot k$, for varies $k$.}
\label{Test2}
\end{figure}
\begin{figure}[b]
\centerline{\includegraphics[scale=.5]{Test3.png}}
\caption{The case $k=2$, $R = 256 \sqrt{d}$, for varies $d$.}
\label{Test3}
\end{figure}
The graphs show the main bottleneck of Algorithm $\mathsf{PrivatekAverages}$ in practice. It requires only $O_{\varepsilon,\delta}(k d)$ tuples (or $O_{\varepsilon,\delta}(k \sqrt{d})$ for large values of $d$) in order to succeed, but the hidden constant is $\approx 500,000$ for our choice of $\varepsilon$ and $\delta$, and this does not improve even when the assumed separation $R$ is very large. The cause of this large constant is the group privacy of size $O(k \ell)$ that we do in Step~\ref{step:computing-noisy-bound-aver}, where recall that $\ell = O\paren{\frac{\log^2(1/\delta)}{\varepsilon \log n}}$ (\cref{def:ell}). While in theory this $\ell$ is relatively small, with our choice of parameters we get $\ell \approx 1000$. This means that we need to execute the private average algorithm with $\hat{\varepsilon} \approx \frac{\varepsilon}{4000 k}$. Internally, this $\hat{\varepsilon}$ is shared between other private algorithms, and in particular, with an Interior Point algorithm that is one of the internal components of the average algorithm from \cref{prop:approx-aver-Rd}. This algorithm is implemented using the exponential mechanism \cite{MT07}, which simply outputs a random noise when the number of points is too small.
We remark that prior work on differentially-private clustering, including in "easy" settings, is primarily theoretical. In particular, we are not aware of implemented methods that we could use as a baseline.\footnote{\Enote{Added:}We remark that in different settings, such as node, edge or weight-differential privacy, there exist some available implementations (e.g., \cite{PinotMYGA18}).}
As a sanity check, we did consider the following naive baseline:
For every sample point, add a Gaussian noise to make it private. Now, the resulting noisy samples are just samples from a new Gaussian mixture. Then, run an off-the-shelf non-private method to learn the parameters of this mixture.
We tested this naive method on the simple case $d=1$ and $k=2$, where we generated samples from a mixture of standard Gaussians that are separated by $R = 512$. By the Gaussian mechanism, the noise magnitude that we need to add to each point for guaranteeing $(\varepsilon,\delta)$-differential privacy, is $\sigma \approx \frac{\Lambda}{\varepsilon}\sqrt{\log(1/\delta)} \gg 1$ for some $\Lambda > R$, meaning that the resulting mixture consists of very close Gaussians. We applied GaussianMixture from the package sklearn.mixture to learn this mixture, but it failed even when we used $100 M$ samples, as this method is not intended for learning such close Gaussians. We remark that there are other non-private methods that are designed to learn any mixture of Gaussians (even very weakly separated ones) using enough samples (e.g., \cite{suresh2014near}).
The sample complexity and running time of these methods, however, are much worse than ours even asymptotically (e.g., the running time of \cite{suresh2014near} is exponential in $k$), and moreover, we are not aware of any implementation we could use.\footnote{Asymptotically, \cite{suresh2014near} requires at least $\tilde{\Omega}(d k^9)$ samples, and runs in time $\tilde{\Omega}(n^2d + d^2 (k^7 \log d)^{k^2})$. For the setting of learning a mixture of $k$ well-separated Gaussians, the approach of first adding noise to each point and then applying a non-private method such as \cite{suresh2014near}, results with much worse parameters than our result, which only requires $\tilde{O}(dk)$ samples and runs in time $\tilde{O}(dk^2n)$.}
\section{Conclusion}
We developed an approach to bridge the \remove{gaping }gap between the theory and practice of differentially private clustering methods. For future, we hope to further optimize the "constants" in the $k$-tuple clustering algorithms, making the approach practical for instances with lower separation. Tangentially, the inherent limitations of private versus non-private clustering suggest exploring different rigorous notions of privacy in the context of clustering.
\section{Introduction}
Differential privacy \cite{DworkMNS06} is a mathematical definition of privacy, that aims to enable statistical analyses of databases while providing strong guarantees that individual-level information does not leak. Privacy is achieved in differentially private algorithms through randomization and the introduction of ``noise'' to obscure the effect of each individual, and thus differentially private algorithms can be less accurate than their non-private analogues. In most cases, this loss in accuracy is studied theoretically, using asymptotic tools. As a result, there is currently a significant gap between what is known to be possible {\em theoretically} and what can be done {\em in practice} with differential privacy. In this work we take an important step towards bridging this gap
in the context of {\em clustering related tasks}.
The construction of differentially private clustering algorithms has attracted a lot of attention over the last decade, and many different algorithms have been suggested.\footnote{\cite{BDMN05,NRS07,FFKN09,
McSherry09,GuptaLMRT10,Mohan2012,Wang2015,NockCBN16,
Su2016,NSV16,DannyPrivatekMeans,Balcan17a,
NS18_1Cluster,HuangL18,KaplanSt18,Stemmer20,ShechnerSS20,Ghazi0M20,Nguyen20}} However, to the best of our knowledge, none of these algorithms have been implemented: They are not particularly simple and suffer from large hidden constants that translate to a significant loss in utility, compared to non-private implementations.
\begin{question}
How hard is it to cluster privately with a practical implementation?
\end{question}
We take an important step in this direction using the following approach. Instead of directly tackling ``standard'' clustering tasks, such as $k$-means clustering, we begin by identifying a very simple clustering problem that still seems to capture many of the challenges of practical implementations (we remark that this problem is completely trivial without privacy requirements). We then design effective (private) algorithms for this simple problem. Finally, we reduce ``standard'' clustering tasks to this simple problem, thereby obtaining private algorithms for other tasks.
In more detail, we introduce the following problem, called the {\em $k$-tuple clustering} problem.
\begin{definition}[informal, revised in Definition~\ref{def:ktupleclustering}]
An instance of the {\em $k$-tuple clustering} problem is a collection of $k$-tuples.
Assuming that the input tuples can be partitioned into $k$ ``obvious clusters'', each consisting of one point of each tuple, then the goal is to report $k$ ``cluster-centers'' that correctly partition the input tuples into clusters.
If this assumption on the input structure does not hold, then the outcome is not restricted.
\end{definition}
\begin{remark}\;
\begin{enumerate}
\item By ``obvious clusters'' we mean clusters which are far away from each other.\vspace{-7pt}
\item The input tuples are {\em unordered}. This means, e.g.,\ that the ``correct'' clustering might place the first point of one tuple with the fifth point of another tuple.\vspace{-7pt}
\item Of course, we want to solve this problem while guaranteeing differential privacy. Intuitively, this means that the outcome of our algorithm should not be significantly effected when arbitrarily modifying one of the input tuples.
\end{enumerate}
\end{remark}
Observe that without the privacy requirement this task is trivial: We can just take one arbitrary input tuple $(x_1,...,x_k)$ and report it. With the privacy requirement, this task turns out to be non-trivial. It’s not that this problem cannot be solved with differential privacy. It can. It’s not even that the problem requires large amounts of data asymptotically. It does not. However, it turns out that
designing an implementation with a practical privacy-utility tradeoff, that is effective on
finite datasets (of reasonable size), is quite challenging.
\subsection{Our algorithms for the ${\boldsymbol k}$-tuple problem}
We present two (differentially private) algorithms for the $k$-tuple clustering problem, which we call $\mathsf{PrivatekAverages}$ and $\mathsf{PrivatekNoisyCenters}$.
Both algorithms first privately test if indeed the input is partitioned into $k$ obvious clusters and quit otherwise.
They differ by the way they compute the centers in case this test passes.
Algorithm $\mathsf{PrivatekAverages}$ privately averages each identified cluster. Algorithm $\mathsf{PrivatekNoisyCenters}$, on the other hand, does not operate by averaging clusters. Instead, it selects one of the input $k$-tuples, and then adds a (relatively small) Gaussian noise to every point in this tuple. We prove that this is private if indeed there are $k$ obvious clusters in the input.
We evaluate these two algorithms empirically, and show that, while algorithm $\mathsf{PrivatekAverages}$ is ``better in theory'', algorithm $\mathsf{PrivatekNoisyCenters}$ is much more practical for some interesting regimes of parameters.
We now give a simplified overview of the ideas behind our algorithms. For concreteness, we focus here on $\mathsf{PrivatekAverages}$. Recall that in the $k$-tuple clustering problem, we are only required to produce a good output assuming the data is ``nice'' in the sense that the input tuples can be clustered into $k$ ``far clusters'' such that every cluster contains exactly one point from every tuple. However, with differential privacy we are ``forced'' to produce good outputs even when this niceness assumption does not hold. This happens because if the input data is ``almost nice'' (in the sense that modifying a small number of tuples makes it nice) then differential privacy states that the outcome of the computation should be close to what it is when the input data is nice.
So, the definition of differential privacy forces us to cope with ``almost nice'' datasets. Therefore, the niceness test that we start with has to be a bit clever and ``soft'' and
succeed with some probability also for
data which is ``almost nice''.
Then, in order to achieve good performances, we have to utilize the assumption that the data is ``almost nice'' when we compute the private centers.
To compute these centers, Algorithm $\mathsf{PrivatekAverages}$ determines ({\em non-privately}) a clustering of the input tuples, and then averages (with noise) each of the clusters. The conceptual challenge here is to show that even though the clustering of the data is done non-privately, it is stable enough such that the outcome of this algorithm still preserves privacy.
\subsection{Applications}\label{sec:introApplications}
The significance of algorithms $\mathsf{PrivatekAverages}$ and $\mathsf{PrivatekNoisyCenters}$ is that
many clustering related tasks can be privately solved by a reduction to the $k$-tuple clustering problem. In this work we explore two important use-cases: (1) Privately approximating the $k$-means under stability assumption, and (2) Privately learning the parameters of a mixture of well-separated Gaussians.
\smallskip
\noindent
{\bf ${\boldsymbol k}$-Means Clustering}
In $k$-means clustering, we are given a database ${\cal P}$ of $n$ input points in ${\mathbb R}^d$, and the goal is to identify a set $C$ of $k$ {\em centers} in ${\mathbb R}^d$ that minimizes the sum of squared distances from each input point to its nearest center. This problem is NP-hard to solve exactly, and even NP-hard to approximate to within a multiplicative factor smaller than $1.0013$ \citep{lee2017improved}. The current (non-private) state-of-the-art algorithm achieves a multiplicative error of $6.357$ \citep{ahmadian2019better}.
One avenue that has been very fruitful in obtaining more accurate algorithms (non-privately) is to look beyond worst-case analysis \cite{OstrovskyRSS12,awasthi2010stability,ABS12,balcan2009approximate,bilu2012stable,kumar2010clustering}. In more details, instead of constructing algorithms which are guaranteed to produce an approximate clustering for any instance, works in this vain give stronger accuracy guarantees by focusing only on instances that adhere to certain ``nice'' properties (sometimes called stability assumptions or separation conditions). The above mentioned works showed that such ``nice'' inputs can be clustered much better than what
is possible in the worst-case (i.e., without assumptions on the data).
Given the success of non-private stability-based clustering, it is not surprising that such stability assumptions were also utilized in the privacy literature, specifically by \citet{NRS07,Wang2015,HuangL18,ShechnerSS20}. While several interesting concepts arise from these four works, none of their algorithms have been implemented, their algorithms are relatively complex, and their practicability on finite datasets is not clear.
We show that the problem of stability-based clustering (with privacy) can be reduced to the $k$-tuple clustering problem. Instantiating this reduction with our algorithms for the $k$-tuple clustering problem, we obtain a simple and practical algorithm for clustering ``nice'' $k$-means instances privately.
\smallskip
\noindent
{\bf Learning Mixtures of Gaussians.}
Consider the task of {\em privately} learning the parameters of an unknown mixtures of Gaussians given i.i.d.\ samples from it. By now, there are various private algorithms that learn the parameters of a {\em single} Gaussian \cite{KV18,KLSU19,CWZ19,BS19,KSU20,BDKU20}. Recently, \cite{KSSU19} presented a private algorithm for learning mixtures of well-separated (and bounded) Gaussians.
We remark, however, that besides the result of \cite{BDKU20}, which is a practical algorithm for learning a single Gaussian, all the other results are primarily theoretical.
By a reduction to the $k$-tuples clustering problem, we present a simple algorithm that privately learns the parameters of a separated (and bounded) {\em mixture} of $k$ Gaussians.
From a practical perspective, compared with the construction of the main algorithm of \cite{KSSU19}, our algorithm is simple and implementable. From a theoretical perspective, our algorithm offers reduced sample complexity, weaker separation assumption, and modularity. See \cref{sec:Gauss:comparison} for the full comparison.
\subsection{Other Related Work}
The work of \citet{NRS07} presented the sample-and-aggregate method to convert a non-private algorithm into a private algorithm, and applied it to easy clustering problems. However, their results are far from being tight, and they did not explore certain considerations (e.g., how to minimize the impact of a large domain in learning mixture of Gaussians).
Another work by \citet{BKSW19} provides a general method to convert from a cover of a class of distributions to a private learning algorithm for the same class. The work gets a near-optimal sample complexity, but the algorithms have exponential running time in both $k$ and $d$ and their learning guarantees are incomparable to ours (they perform proper learning, while we provide clustering and parameter estimation).
In the work of \cite{KSSU19}, they presented an alternative algorithm for learning mixtures of Gaussians, which optimizes the sample-and-aggregate approach of \cite{NRS07}, and is somewhat similar to our approach. That is, their algorithm executes a non-private algorithm several times, each time for obtaining a new ``$k$-tuple'' of means estimations, and then aggregates the findings by privately determine a new $k$-tuple of means estimation.
But their approach has two drawbacks. First, in order to privately do that, their algorithm ignores the special $k$-tuples structure, and apply a more wasteful and complicated ``minimal enclosing ball'' algorithm from \cite{NS17_clustering,NSV16}. Second, in contrast to them, for creating a $k$-tuple, our algorithm only applies a non-private algorithm for \emph{separating} the samples in the mixture (i.e., for determine which samples belong to the same Gaussian), and not for estimating their parameters. This yields that we need less samples per invocation of the non-private algorithm for creating a single $k$-tuple, which results with an improved sample complexity (each $k$-tuple in our case is just the averages of each set of samples, which might not necessarily be very close to the true means, but is close enough for our setting where the Gaussians are well-separated). Finally, given a private separation of the sample, we just apply some private algorithm for estimating the parameters of each (single) Gaussian (e.g., \cite{KV18,KLSU19,CWZ19,BS19,KSU20,BDKU20}).
For more details about our construction, see \cref{sec:mixture-of-gaus}.
Furthermore, there are many differentially-private algorithms that are related to learning mixture of Gaussians (notably PCA) \cite{BlumDwMcNi05,KT13,CSS13,DTTZ14}, and differentially-private algorithms for clustering \cite{NRS07,GuptaLMRT10,NSV16,NS17_clustering,Balcan17a,KaplanSt18,HuangL18,Ghazi0M20}. We remark that for the learning Gaussians mixtures problem, applying these algorithms naively would introduce a polynomial dependence on the range of the data, which we seek to avoid.
\remove{
\Enote{Old intro:}
\section{Introduction}
\ECnote{Add general text about what is privacy and its importance ... }
put text
\ECnote{Clustering-type problems, define the DP requirement (central model) in that context}
In $k$-means clustering, we are given a database ${\cal P}$ of $n$ points in ${\mathbb R}^d$, and the goal is to identify a set $C$ of $k$ centers in ${\mathbb R}^d$ that approximately minimizes ${\rm COST}_{{\cal P}}(C) = \sum_{\px \in {\cal P}} \min_{\pc \in C}\norm{\px - \pc}^2$. We denote the lowest possible cost as ${\rm OPT}_{k}({\cal P})$. Since the task of minimizing the $k$-means is NP-hard, the literature has focused on approximation algorithms, with the current (non-private) state-of-the-art achieving multiplicative error of $6.357$ \cite{ahmadian2019better}.
\ECnote{Mention practical solutions to the problem. Cite, EM, $k$-means++, bi-criteria. Also cite earlier results on multiplicative factors theoretical. Make a distinction between theory (very large running times, asymptotically) and practice.}
\ECnote{Condense the below to precisely define neighboring and DP}
\begin{definition}[Neighboring databases]\label{def:neighboring}
Let ${\cal D} = \set{x_1,\ldots,x_n}$ and ${\cal D}' = \set{x_1',\ldots,x_n'}$ be two databases over a domain ${\cal X}$. We say that ${\cal D}$ and ${\cal D}'$ are \textbf{neighboring} if there is exactly one index $i \in [n]$ with $x_i \neq x_i'$.
\end{definition}
\begin{definition}[$(\varepsilon,\delta)$-indistinguishable]\label{def:indis}
Two random variable $X,X'$ over a domain ${\cal X}$ are called $(\varepsilon,\delta)$-indistinguishable, if for any event $T \subseteq {\cal X}$, it holds that
$\pr{X \in T} \leq e^{\varepsilon} \cdot \pr{Y \in T} + \delta$. If $\delta = 0$, we say that $X$ and $X'$ are $\varepsilon$-indistinguishable.
\end{definition}
\begin{definition}[$(\varepsilon,\delta)$-differential privacy \cite{DworkMNS06}]\label{def:DP}
An algorithm $\cA$ is called $(\varepsilon,\delta)$-differentially private, if for any two neighboring databases ${\cal D},{\cal D}'$ it holds that $\cA({\cal D})$ and $\cA({\cal D}')$ are $(\varepsilon,\delta)$-indistinguishable. If $\delta = 0$ (i.e., pure privacy), we say that $\cA$ is called $\varepsilon$-differentially private.
\end{definition}
\ECnote{Add text, borrowing from below, on results for DP clustering in the central model, difficulties, gap from practice. Why these problems so far defied practical solutions. Also on mixtures of Gaussians.}
Unlike in the non-private literature, it is known that every private algorithm for approximating the $k$-means cost must have an additive error (even computationally unbounded algorithms), which scales with the diameter of the input space. Hence, it is standard to assume an upper bound $\Lambda$ on the $\ell_2$ norm of all the input points ${\cal P}$. Typically (though not always), one aims to minimize the multiplicative error while keeping the additive error at most polylogarithmic in the size of the database. The current state-of-the-art construction \Enote{ .....}
\ECnote{cite line of prior work on DP clustering (without assumptions on input). Culminating in Ravi/Nguyen. stressing not practical but illuminating}
\ECnote{Meta question: Can we get closer to practice when the data is "Easy."?}
\ECnote{Discuss easy or stable instances, what is known without privacy. In particular discuss $k$-means clustering and separating mixtures.}
The non-private literature is ripe with results geared for efficient solutions on easier instances. Including dodging the NP hardness of unrestricted $k$-means clustering.
{\bf edit below}
As a common concrete example of such ${\cal P}$ and ${\cal A}$ that satisfy the above stability property is when ${\cal A}$ is
a good (non-private) approximation algorithm for $k$-means, and ${\cal P}$ is well-separated for $k$-means, a notion that was first introduced in \citet{OstrovskyRSS12}. Formally, ${\cal P}$ is called $\phi$-separated if ${\rm OPT}_{k}({\cal P}) \leq \phi^2 {\rm OPT}_{k-1}({\cal P})$. It turns out that if ${\cal P}$ is $\phi$-separated for $k$-means, and ${\cal A}$ is a $\omega$-approximation algorithm for $k$-means, where $\phi^2(1 + \omega)$ is sufficiently small, then ${\cal P}$ and ${\cal A}$ satisfy our required stability property, yielding that our algorithm $\mathsf{PrivatekMeans}$ succeed well over ${\cal P}$ using ${\cal A}$.
Following the work of \citet{OstrovskyRSS12}, several other works have related other notions of input-stability to clustering \cite{awasthi2010stability,awasthi2012center,balcan2009approximate,bilu2012stable,kumar2010clustering}.
\ECnote{Discuss easy or stable instances, what is known with privacy. }
{\bf edit below}
In addition, several works gave differentially-private algorithms for approximating the minimal Wasserstein distance (\cite{Wasserstein1969}) of well-separated instances \cite{NRS07,wang2015differentially,huang2018optimal}, and in a recent work, \citet{ShechnerSS20} presented a differentially-private algorithm for approximating the $k$-means of such well-separated instances. The main result of \cite{ShechnerSS20} is a very simple transformation, called Private-Stable-$k$-Means, from any private $\omega$-approximation algorithm to a private $(1 + O(\phi^2))$-one, that works over $\phi$-separated database ${\cal P}$ with sufficiently small $\phi^2(1 + \omega)$. In addition, they presented Algorithm SampleAggregate-$k$-means that transform any (non-private) $k$-means approximation algorithm ${\cal A}$ into a private one, with very similar guarantees to our construction. Their algorithm also randomly partitions ${\cal P}$ into $T$ subsets (for large enough $T \ll n$), and executes ${\cal A}$ on each subset to obtain $k$ centers that are close to the optimal centers. But then, in order to privately estimating the averages of each cluster of the resulting partitioned multiset of $k$-tuples, their algorithm ignores the $k$-tuples structure and uses the (complicated) \cite{NS17_clustering}'s minimal enclosing ball algorithm for determine the clusters, while our algorithm uses Algorithm $\mathsf{PrivatekAverages}$ which is much more simpler and practical.
\ECnote{Perhaps here discuss mixture of Gaussians, what is known, what would we consider stable.}
\ECnote{Our approach and contributions at the high level}
We define a basic problem of \emph{$k$-tuple clustering}.
The input to $k$-tuple clustering is a set ${\cal T}$ of unordered $k$-tuples of vectors in ${\mathbb R}^d$. The desired output is a single $k$-tuple that (informally) perfectly separates a maximum number of tuples from ${\cal T}$.
We design differentially private algorithms for $k$-tuple clustering that (informally) provide good utility on datasets for which a good solution exist.
The algorithms we design for $k$-tuple clustering are relatively simple and friendlier to implement compared with alternative solutions based on existing differentially private tools and at the same time also provide tighter asymptotic bounds.
We propose a framework to tackle private solutions for $k$-clustering type problems, where the input is a dataset of points in ${\mathbb R}^d$ and the output a set of $k$ points (a $k$-tuple).
Our framework uses
private $k$-tuple clustering as a tool and also assumes a non-private algorithm for the original problem that inputs a set of points and generates a $k$-tuple (e.g., $k$ cluster centers).
The framework applies the non-private algorithm multiple times, to subsets (random samples) of the data. These runs yield a set ${\cal T}$ of $k$-tuples. Our private $k$-tuple clustering algorithm is then applies to ${\cal T}$ to generate a private $k$-tuple.
The utility we obtain is conditioned on the instance being "stable" with respect to the original problem: Informally, that the $k$-tuples generated by the multiple applications of the non-private algorithm cluster well.
We apply our framework to two well-studied $k$-clustering type problems:
Our first application is to $k$-means clustering on well-separated instances. We obtain a differentially-private algorithm that is considerably simpler and more efficient than the state-of-the-art result of \citet{ShechnerSS20}.
\ECnote{Put here more direct comparison to Shechner... (what we have that is not there)}
The second application is to privately learn the parameters of mixtures of well-separated Gaussians.
Our solution is both simpler and provides an asymptotic improvement over the state-of-the-art result of [Kamath et al. NeurIPS 2019].
Finally, we present empirical results for learning mixtures of very separated Gaussians, making our work the first that present a practical differentially-private algorithm for this task.
\section{The tuple clustering problem}
\ECnote{Precise definition, discussion on non-private and alternative private approaches and what is obtained, pseudocode, statement of properties}
A $k$-tuple
$X = \set{\px_1,\ldots,\px_k}$
is an unordered set of $k$ vectors $\px_i \in {\mathbb R}^d$.
For a $k$-tuple $X$ and parameter $\Delta >0$
we define radii $(r_i^{X})_{i=1}^k$ so that
$r_i^X = \min_{j \neq i} \norm{\px_i - \px_j}$.
\begin{definition}[$\Delta$-partition of a tuple by a tuple]
We say that a $k$-tuple $X$ $\Delta$-partitions another $k$-tuple $Y$
if each entry $\px_i$ of $X$ can be uniquely matched to entry $\py_{i_j}$ of $Y$ so that
\[
\norm{\px_i - \py_{i_j}} \leq \frac1{\Delta} \cdot r_i^X(\Delta) .\]
\end{definition}
\Hnote{ Why $r_i^X(\Delta)$ ? and not $r_i^X$}
\begin{definition}[$\Delta$-far balls]
For a $k$-tuple $X$ and $\Delta$ we can consider a set of $k$ balls
${\cal B} = \set{B_1,\ldots,B_k}$, where $B_i(\px_i,\frac{1}{\Delta} r^X_i)$ has center $\px_i$ and radius, $\frac{1}{\Delta} r^X_i$.
We refer to such balls as $\Delta$-far balls.
\end{definition}
Equivalently, A tuple $Y$ is $\Delta$-partitioned by $X$ if each ball in ${\cal B}$ contains exactly one entry from $Y$.
\Hnote{Don't we want to parameterize the balls by X ? ${\cal B}^X$ ?}
\begin{definition}[$k$-tuple clustering]
The input to the problem is database ${\cal T}$ of $k$-tuples of size $n$ and $\Delta>1$. The output is a tuple that $\Delta$-partitions the maximum number of other tuples.
\end{definition}
We define "easy" instances to be those with a perfect solution, that is,
we are promised that there is a tuple that $\Delta$-partition all other tuples.
Without the requirement of privacy, the "easy" problem is quite easy to solve exactly.
One can just pick an arbitrary $k$-tuple $\set{\px_1,\ldots,\px_k} \in {\cal T}$, construct the cluster ${\cal P}_1,\ldots,{\cal P}_k$ where ${\cal P}_i = \set{\px \in {\rm Points}({\cal T}) \colon i = \operatorname*{argmin}_{j \in [k]} \norm{\px - \px_j}}$, and then compute the average $\pa_i$ of each ${\cal P}_i$. \Hnote{why not just spit an arbitrary tuple ?}
However, this algorithm is not differentially private. To be differentially private, any outcome should be obtained with approximately the same probability when we change a single $k$-tuple (possibly moving to a neighboring database with no perfect solution \Hnote{We also have to deal with databases which are not easy ?}). In this algorithm, changing a single $k$-tuple may completely change the resulting partition (e.g., if the arbitrary tuple that we choose at the first step is exactly the different tuple), and hence, change the resulting averages.
We therefore must consider approximate solutions and probabilistic outputs.
Even for $k=1$ (which is a task of finding a point close to all other points), we need to assume some upper bound $\Lambda$ on the $\ell_2$ norm of the points in each tuple in ${\cal T}$. Second, we have to assume $r_{\min} > 0$
\ECnote{We need to explain $r_{\min}$ this better. Perhaps we can factor in $r_{\min}$ to the approximation. }
We say that an algorithm is an $(\alpha,\beta)$-approximation to a perfect $k$-tuple clustering instance ${\cal T}$ if with probability $1-\beta$, it outputs $\set{pa_1,\ldots,\pa_k}$ that $\alpha\Delta$ partitions all tuples.
\ECnote{I did some editing. Is the above ok? }
\ECnote{discuss approaches with existing components and why they are not satisfactory. Not simple, large constants. }
A direct approach to a private solution is through the Minimal Enclosing Ball algorithm of \citet{NS17_clustering}. This algorithm is given $n$ points and a (large enough) parameter $t \leq n$, and privately outputs a ball of radius $O(r_{opt})$ that contains (almost) $t$ points, where $r_{opt}$ is the radius of the smallest ball that contains $t$ points. To apply this algorithm to our problem, we ignore the $k$-tuple structure and execute \cite{NS17_clustering}'s algorithm $k$ times on the database ${\rm Points}({\cal T})$. Each time it would privately determine a ball that contains (almost) all the points of one of the ${\cal P}_i$'s, and remove them from ${\rm Points}({\cal T})$ before the next call to the algorithm. In order to privately compute an estimate $\tpa_i$ of each average $\pa_i$ such that $\norm{\pa_i - \tpa_i}$ is proportional to the (minimal) radius of a ball that contains ${\cal P}_i$, one can execute the private average algorithm of \citet{NSV16}. It can be shown that with this process, we can achieve $(\varepsilon,\delta)$-differential privacy for $(\alpha,\beta)$-estimating the $k$ averages of ${\cal P}$, where $\alpha = O\paren{\frac{\sqrt{d}}{\varepsilon n} \log \paren{\frac{nd}{\beta}} \log\paren{\frac{1}{\delta}}}$.
However, the downside of this approach is that \cite{NS17_clustering}'s algorithm is not simple, and does not exploits the special $k$-tuples structure of the database.\Enote{maybe also explain why it is not simple. E.g., that it uses LSH, RecConcave, and more ...}\ECnote{Uri mentioned polynomial super-linear dependence on $kd$ also. We need to make this paragraph concise and crisp. We also use averaging (in one of the approaches), we just have a different method of creating the subsets.}
\ECnote{
Present our solution taking things from section 9 and hiding all the unnecessary. Yes to Pseudocode. Precise statement of properties. just say enough so that we can precisely state the results and explain the algorithm. No proofs but intutions. yes to pictures. }
\section{Application to $k$-means clustering}
\section{Application to separating mixtures of Gaussians}
\section{Empirical results}
\section{Conclusion}
\subsection{tuple clustering}\label{sec:intro:kAver}
\ECnote{
Previous: Estimating the Averages of Partitioned $k$-Tuples
}
Suppose that we are given a database ${\cal T}$ of size $n$, each element in ${\cal T}$ is an (unordered) $k$-tuple over ${\mathbb R}^d$, and we are promised that ${\cal T}$ can be partitioned by $k$ far balls in ${\mathbb R}^d$. Namely, there exist $k$ balls $B_1,\ldots,B_k$ over ${\mathbb R}^d$, each $B_i$ is centered at some point $\pc_i \in {\mathbb R}^d$ and has radius $r_i \geq 0$, and the following holds: (1) For every $i \neq j: \text{} \norm{\pc_i - \pc_j} > 16 \cdot \max\set{r_i,r_j}$ (i.e., the balls are very far from each other),\footnote{Throughout this work, we denote by $\norm{\cdot}$ the standard $\ell_2$ norm.} and (2) For every $i \in [k]$ and a $k$-tuple $X = \set{\px_1,\ldots,\px_k} \in {\cal T}:\text{ }\size{X \cap B_i} = 1$ (i.e., each tuple in ${\cal T}$ contains exactly one point in each ball).\Enote{add picture?}
Given this promise, let ${\rm Points}({\cal T})$ be the multiset (of size $kn$) of all the points in all the $k$-tuples in ${\cal T}$, and consider the task of estimating the average $\pa_i$ of each cluster ${\cal P}_i = {\rm Points}({\cal T}) \cap B_i$. Specifically, we say that an algorithm $(\alpha,\beta)$-estimates the averages of partitioned points, if given such a database ${\cal T}$ as input, then with probability $1-\beta$, it outputs $\set{\tilde{\pa}_1,\ldots,\tilde{\pa}_k}$ such that, for each $i \in [k]$, there exists an average (call it $\pa_i$) with $\norm{\tpa_i - \pa_i} \leq \alpha \cdot r_i$.
Without privacy, it is quite easy to $(0,0)$-estimate the $k$ averages.
One can just pick an arbitrary $k$-tuple $\set{\px_1,\ldots,\px_k} \in {\cal T}$, construct the cluster ${\cal P}_1,\ldots,{\cal P}_k$ where ${\cal P}_i = \set{\px \in {\rm Points}({\cal T}) \colon i = \operatorname*{argmin}_{j \in [k]} \norm{\px - \px_j}}$, and then compute the average $\pa_i$ of each ${\cal P}_i$.
However, this algorithm is not differentially private. To be differentially private, any outcome should be obtained with approximately the same probability when we change a single $k$-tuple (possibly moving to a neighboring database that is not partitioned by far balls). In this algorithm, changing a single $k$-tuple may completely change the resulting partition (e.g., if the arbitrary tuple that we choose at the first step is exactly the different tuple), and hence, change the resulting averages.
In order to privately estimate the $k$ averages, we first need to make some assumptions on the database ${\cal T}$, even for $k=1$ (which is just the task of estimating the average of all points). First, we need to assume some upper bound $\Lambda$ on the $\ell_2$ norm of the points in each tuple in ${\cal T}$. Second, since we are interested in an average estimation that is proportional to the (minimal) radius of a ball that contains them, we also need to assume some lower bound $r_{\min} > 0$ on the radii of the (minimal) $k$ balls that partition ${\cal T}$, or alternatively, assume that the points in ${\rm Points}({\cal T})$ belong to a finite grid
\Enote{Ref?}\Hnote{From ``Second, ...'': I do not follow this, is it still related to $k=1$ ?
if all the points are at exactly the same place then we can report this place, i think ?
For $k>1$ maybe you can say that there is a reduction from finding an interior point, but we need to be careful}
One way to privately estimate the $k$ averages is to use the minimal enclosing ball algorithm of \citet{NS17_clustering}. This algorithm is given $n$ points and a (large enough) parameter $t \leq n$, and privately outputs a ball of radius $O(r_{opt})$ that contains (almost) $t$ points, where $r_{opt}$ is the radius of the smallest ball that contains $t$ points. In our case, one can ignore the $k$-tuple structure and execute \cite{NS17_clustering}'s algorithm $k$ times on the database ${\rm Points}({\cal T})$. Each time it would privately determine a ball that contains (almost) all the points of one of the ${\cal P}_i$'s, and remove them from ${\rm Points}({\cal T})$ before the next call to the algorithm. In order to privately compute an estimation $\tpa_i$ of each average $\pa_i$ such that $\norm{\pa_i - \tpa_i}$ is proportional to the (minimal) radius of a ball that contains ${\cal P}_i$, one can execute the private average algorithm of \citet{NSV16}. It can be shown that with this process, we can achieve $(\varepsilon,\delta)$-differential privacy for $(\alpha,\beta)$-estimating the $k$ averages of ${\cal P}$, where $\alpha = O\paren{\frac{\sqrt{d}}{\varepsilon n} \log \paren{\frac{nd}{\beta}} \log\paren{\frac{1}{\delta}}}$.
However, the downside of this approach is that \cite{NS17_clustering}'s algorithm is not simple, and does not exploits the special $k$-tuples structure of the database.\Enote{maybe also explain why it is not simple. E.g., that it uses LSH, RecConcave, and more ...}
In this work we present a much more simpler and practical $(\varepsilon,\delta)$-differentially private algorithm that $(\alpha,\beta)$-estimates the averages of partitioned $k$-tuples, with essentially the same parameter $\alpha$. In order to understand the main idea of our algorithm,
suppose we are given an $n$-size database ${\cal T}$ of $k$-tuples, and a set of far balls ${\cal B} = \set{B_1,\ldots,B_k}$ around the centers $\set{\pc_1,\ldots,\pc_k}$ (respectively), that ``almost'' partitions ${\cal T}$. By ``almost'' we mean that, there exists some $\ell \ll n$ such that, by removing $\ell$ tuples from ${\cal T}$, we obtain a database that is partitioned by $\set{B_1,\ldots,B_k}$. Throughout this work, we call it $\ell$-partitioning.
Now consider the following process $\mathsf{EstimateAverages}({\cal T},{\cal B})$: (1) Partition ${\rm Points}({\cal T})$ into $k$ multisets ${\cal P}_1,\ldots,{\cal P}_k$, where each ${\cal P}_i$ consists of the points in ${\cal P}$ that $\pc_i$ is closest center to them, and (2) Compute a noisy average $\tpa_i$ of each ${\cal P}_i$ using \cite{NSV16}'s algorithm with privacy parameters $\varepsilon,\delta \in (0,1]$, and output the set $\set{\tpa_1,\ldots,\tpa_k}$.
Our main observation is that, for any neighboring databases ${\cal T},{\cal T}'$, and any sets of far balls ${\cal B}$ and ${\cal B}'$ that $\ell$-partition ${\cal P}$ and ${\cal P}'$ (respectively), the distributions of the outputs $\mathsf{EstimateAverages}({\cal P},{\cal B})$ and $\mathsf{EstimateAverages}({\cal P}',{\cal B}')$ are $(O( k \ell \varepsilon), O(k \ell \delta))$-indistinguishable.\Enote{define ind} The intuition is that, since ${\cal T}$ and ${\cal T}'$ are neighboring, the partition $\set{{\cal T}_1,\ldots,{\cal T}_k}$ in $\mathsf{EstimateAverages}({\cal P},{\cal B})$ and $\set{{\cal T}_1',\ldots,{\cal T}_k'}$ in $\mathsf{EstimateAverages}({\cal P}',{\cal B}')$ must be almost the same (in fact, we show that there are at most $2\ell + 1$ tuples that the two partitions do not agree on).
The first attempt for translating the above observation into a differentially private algorithm that estimates the $k$ averages of ${\cal T}$, is to fix some small $\ell$, compute (non-privately) a set of far balls ${\cal B} = \set{B_1,\ldots,B_k}$ that $\ell$-partitioned ${\cal T}$ (fail if such balls do not exist), and then output $\mathsf{EstimateAverages}({\cal P},{\cal B})$. The main problem is that differential privacy is a worst-case guarantee. If we execute the above process with two neighboring databases ${\cal T},{\cal T}'$ such that ${\cal T}$ is $\ell$-partitioned by far balls and ${\cal T}'$ is not $\ell$-partitioned by any far balls, then the execution over ${\cal T}'$ will fail, meaning that the output over ${\cal T}$ is very distinguishable from the output over ${\cal T}'$.
In order the overcome the above obstacle, we present a simple and efficient algorithm $\mathsf{PrivateTestPartition}$ that privately tests whether the input database ${\cal T}$ is well partitioned or not.
For pedagogical reasons, assume that $\mathsf{PrivateTestPartition}$ gets as input two different databases of $k$-tuples ${\cal P}$ and ${\cal Q}$ with the promise that ${\cal P} \cup {\cal Q}$ is partitioned by far balls (later, given a multiset ${\cal T}$, we apply $\mathsf{PrivateTestPartition}$ with ${\cal P} = {\cal Q} = {\cal T}$ and slightly pay a group privacy of size $2$).
Roughly, given ${\cal P}$ and ${\cal Q}$ as input, $\mathsf{PrivateTestPartition}$ chooses a random $J=\tilde{O}\paren{1/\varepsilon}$-size sub-multiset ${\cal R} \subset {\cal P}$. For each tuple $X = \set{\px_1,\ldots,\px_k} \in {\cal R}$, it defines a set of far balls ${\cal B}_X$ around the centers $\px_1,\ldots,\px_k$, privately checks (by adding Laplace noise of magnitude $\tilde{O}(J/\varepsilon)$) if the value of $\ell_X = \min\set{\ell \colon {\cal Q}\text{ is }\ell\text{-partitioned by }{\cal B}_{\px}}$ does not exceed some (small) threshold (denote this decision bit by ${\rm pass}_X$), and then applies a private counting algorithm on all the $J$ decisions bits do decide whether the test succeed or failed. On success, it outputs one of the balls ${\cal B}_X$ that had ${\rm pass}_X = 1$ (if ${\rm pass}_X = 0$ for all $X \in {\cal R}$, it outputs empty balls). For proving utility, we note that when the test succeed, then with high probability there exists at least one $X \in {\cal R}$ with ${\rm pass}_X = 1$, yielding that with high probability, any choice of such $X$ results with balls $B_X$ that $\ell$-partitions ${\cal Q}$ with some small $\ell$.
For proving that the test is private, consider two executions of $\mathsf{PrivateTestPartition}$ over neighboring databases $({\cal P},{\cal Q})$ and $({\cal P}',{\cal Q}')$. If ${\cal P} \neq {\cal P}'$ (and ${\cal Q} = {\cal Q}'$), then the privacy holds since at most one decision bit can be affected. If ${\cal Q} \neq {\cal Q}'$ (and ${\cal P} = {\cal P}'$), then each estimation of $\ell_X$ is $\varepsilon/J$-differentially private, and by basic composition, all of them together are $\varepsilon$-differentially private. Hence, the privacy of this case now holds by simple post-processing argument (we remark that, since ${\cal R}$ is just a small subset of ${\cal P}$, we actually reduce its value using sub-sampling argument, which is significant for practice).
In summary, our algorithm $\mathsf{PrivatekAverages}$ on input ${\cal T}$, executes $\mathsf{PrivateTestPartition}$ with ${\cal P} = {\cal Q} = {\cal T}$ which outputs $({ Status},{\cal B})$. If ${ Status} = ``Success"$, it executes $\mathsf{EstimateAverages}({\cal T},{\cal B})$ and outputs the resulting $k$ average estimations $\set{\tpa_1,\ldots,\tpa_k}$.
\subsection{Applications}
The importance of Algorithm $\mathsf{PrivatekAverages}$ is that
many classical clustering-based tasks can be privately solved by a reduction to the problem of estimating the averages of partitioned tuples. In this work, we chose to present two important use-cases: (1) Privately approximating the $k$-means under stability assumption, and (2) Privately learning the parameters of a mixture of well-separated Gaussians.
\paragraph{$k$-Means Clustering}
In $k$-means clustering, we are given a database ${\cal P}$ of $n$ points in ${\mathbb R}^d$, and the goal is to identify a set $C$ of $k$ centers in ${\mathbb R}^d$ that approximately minimizes ${\rm COST}_{{\cal P}}(C) = \sum_{\px \in {\cal P}} \min_{\pc \in C}\norm{\px - \pc}^2$. We denote the lowest possible cost as ${\rm OPT}_{k}({\cal P})$. Since the task of minimizing the $k$-means is NP-hard, the literature has focused on approximation algorithms, with the current (non-private) state-of-the-art achieving multiplicative error of $6.357$ \cite{ahmadian2019better}.
Unlike in the non-private literature, it is known that every private algorithm for approximating the $k$-means cost must have an additive error (even computationally unbounded algorithms), which scales with the diameter of the input space. Hence, it is standard to assume an upper bound $\Lambda$ on the $\ell_2$ norm of all the input points ${\cal P}$. Typically (though not always), one aims to minimize the multiplicative error while keeping the additive error at most polylogarithmic in the size of the database. The current state-of-the-art construction \Enote{ .....}
Now, suppose that we are given a database ${\cal P}$ and a (non-private) $k$-means approximation algorithm ${\cal A}$, such that the following stability property holds: When executing ${\cal A}$ on a (large enough) random subset of ${\cal P}$, then with high probability, the output $\pc_1,\ldots,\pc_k$ is very close to the optimal $k$-means $\pc_1^*,\ldots,\pc_k^*$ of ${\cal P}$. Then using this property, we present our private algorithm $\mathsf{PrivatekMeans}$ that easily approximate the $k$-means of ${\cal P}$. Roughly, Algorithm $\mathsf{PrivatekMeans}$ randomly partitions ${\cal P}$ into $T$ subsets (for large enough $T \ll n$), and execute ${\cal A}$ on each subset to obtain $k$ centers that are close to the optimal centers. Then, it collects all the $T$ $k$-tuples of centers and executes Algorithm $\mathsf{PrivatekAverages}$ over them. The privacy follows since, for any given partition, two neighboring databases differ by only one of the subsets, and therefore, the resulting $T$ databases of $k$-tuples differ by at most $1$ tuple. For utility, note that by the stability property, w.h.p. all the $T$ $k$-tuples are partitioned by small $k$ balls around the optimal centers $\pc_1^*,\ldots,\pc_k^*$. Therefore, when executing $\mathsf{PrivatekAverages}$, we expect to get $k$ centers $\tilde{\pc}_1,\ldots,\tilde{\pc}_k$ that are close to $\pc_1^*,\ldots,\pc_k^*$.
We then show that by performing a single (noisy) Lloyd step that preserve privacy, the resulting $k$ centers well approximate the $k$-means cost.
As a common concrete example of such ${\cal P}$ and ${\cal A}$ that satisfy the above stability property is when ${\cal A}$ is
a good (non-private) approximation algorithm for $k$-means, and ${\cal P}$ is well-separated for $k$-means, a notion that was first introduced in \citet{OstrovskyRSS12}. Formally, ${\cal P}$ is called $\phi$-separated if ${\rm OPT}_{k}({\cal P}) \leq \phi^2 {\rm OPT}_{k-1}({\cal P})$. It turns out that if ${\cal P}$ is $\phi$-separated for $k$-means, and ${\cal A}$ is a $\omega$-approximation algorithm for $k$-means, where $\phi^2(1 + \omega)$ is sufficiently small, then ${\cal P}$ and ${\cal A}$ satisfy our required stability property, yielding that our algorithm $\mathsf{PrivatekMeans}$ succeed well over ${\cal P}$ using ${\cal A}$.
Following the work of \citet{OstrovskyRSS12}, several other works have related other notions of input-stability to clustering \cite{awasthi2010stability,awasthi2012center,balcan2009approximate,bilu2012stable,kumar2010clustering}. In addition, several works gave differentially-private algorithms for approximating the minimal Wasserstein distance (\cite{Wasserstein1969}) of well-separated instances \cite{NRS07,wang2015differentially,huang2018optimal}, and in a recent work, \citet{ShechnerSS20} presented a differentially-private algorithm for approximating the $k$-means of such well-separated instances. The main result of \cite{ShechnerSS20} is a very simple transformation, called Private-Stable-$k$-Means, from any private $\omega$-approximation algorithm to a private $(1 + O(\phi^2))$-one, that works over $\phi$-separated database ${\cal P}$ with sufficiently small $\phi^2(1 + \omega)$. In addition, they presented Algorithm SampleAggregate-$k$-means that transform any (non-private) $k$-means approximation algorithm ${\cal A}$ into a private one, with very similar guarantees to our construction. Their algorithm also randomly partitions ${\cal P}$ into $T$ subsets (for large enough $T \ll n$), and executes ${\cal A}$ on each subset to obtain $k$ centers that are close to the optimal centers. But then, in order to privately estimating the averages of each cluster of the resulting partitioned multiset of $k$-tuples, their algorithm ignores the $k$-tuples structure and uses the (complicated) \cite{NS17_clustering}'s minimal enclosing ball algorithm for determine the clusters, while our algorithm uses Algorithm $\mathsf{PrivatekAverages}$ which is much more simpler and practical.
\paragraph{Learning Mixtures of Gaussians}
In the learning mixtures of Gaussians problem,
we are given samples for a mixture ${\cal D}$ of $k$ Gaussians over ${\mathbb R}^d$. The mixture is specified by $k$ components $G_1,\ldots,G_k$, where each $G_i$ is selected with probability $\omega_i \in [0,1]$ and is distributed as a Gaussian with mean $\mu_i \in {\mathbb R}^d$ and a covariance matrix $\Sigma_i \in {\mathbb R}^{d \times d}$. The goal is to recover the parameters $\set{(\omega_i, \mu_i, \Sigma_i)}_{i=1}^k$ from the samples. That is, we would like to output a mixture $\hat{{\cal D}} = \set{(\hat{\omega}_i, \hat{\mu}_i, \hat{\Sigma}_i)}_{i=1}^k$ such that ${\cal D}$ and $\hat{{\cal D}}$ are close in total-variation distance (which we denote by ${\rm d}_{\rm TV}(\cdot)$). We say that an algorithm $(\alpha,\beta)$-learns a mixture ${\cal D}$ with sample complexity $n$, if given $n$ sample from ${\cal D}$, with probability $1-\beta$ it outputs $\hat{{\cal D}}$ such that ${\rm d}_{\rm TV}({\cal D},\hat{{\cal D}}) \leq \alpha$.
\Enote{cite non-private works here? I guess that there are plenty }
It turns out that in order to privately learn a mixture, even for $d = k=1$ (i.e., learn a single univariable Gaussian), we must assume a priori bounds $R, \sigma_{\min},\sigma_{\max}$ such that $\forall i:\text{}\norm{\mu_i} \leq R\text{ and } \sigma_{\min} \leq \norm{\Sigma_i} \leq \sigma_{\max}$ \cite{BS16,KV18} (where by $\norm{\cdot}$ we denote the standard $\ell_2$ norm of a matrix).
There are various private algorithms that learns the parameters of a single Gaussian \cite{KV18,KLSU19,CWZ19,BS19,KSU20,BDKU20}. Recently, \cite{KSSU19} presented a private algorithm for learning mixtures of well-separated (and bounded) Gaussians. Specifically, under the separation assumption $\forall i \neq j: \text{ } \norm{\mu_i - \mu_j} \geq \Omega\paren{\sqrt{k \log(nk/\beta)} + \sqrt{\frac1{w_i} + \frac1{w_j}}}\cdot \max \set{\norm{\Sigma_i}^{1/2}, \norm{\Sigma_j}^{1/2}}$,
they presented an $(\varepsilon,\delta)$-differentially private algorithm that $(\alpha,\beta)$-learns such a separated and $(R,\sigma_{\min},\sigma_{\max})$-bounded mixture of $k$ Gaussians with sample complexity $n = \paren{\frac{d^2}{\alpha^2 w_{\min}} + \frac{d^2}{\varepsilon \alpha w_{\min}} + \frac{{\rm poly}(k) d^{3/2}}{w_{\min} \varepsilon}} \cdot {\rm polylog}\paren{\frac{d k R \sigma_{\max}}{\alpha \beta \varepsilon \delta \sigma_{\min}}}$. We remark, however, that besides the result of \cite{BDKU20}, which is a practical algorithm for learning a single Gaussian, all the other results are primarily theoretical. \Enote{maybe here is the place to explain why the construction of \cite{KSSU19} is far from being practical}
In this work, we present a simple and practical algorithm $\mathsf{PrivatekGMM}$ that privately $(\alpha,\beta)$-learns the parameters of an $(R,\sigma_{\min},\sigma_{\max})$-bounded mixture of $k$ Gaussians under a weaker separation assumption from the one made by \cite{KSSU19}. Roughly, Algorithm $\mathsf{PrivatekGMM}$ can use any non-private algorithm ${\cal A}$ that learns the means of a Gaussian mixture. Given such ${\cal A}$, it executes it $T$ times (for large enough $T$), each time with enough new samples for guaranteeing w.h.p. that the means estimations $\tilde{\mu}_1,\ldots,\tilde{\mu}_k$ of ${\cal A}$ will be close enough to the actual means $\mu_1,\ldots,\mu_k$ (specifically, we need that $\forall i:\text{ }\norm{\mu_i - \tilde{\mu}_i} < \frac1{16} \cdot \min_{j \neq i} \norm{\mu_i - \mu_j}$), and then it executes our algorithm $\mathsf{PrivatekAverages}$ on the resulting $T$-size database of all the $k$-tuples means estimations. Since w.h.p. all the $T$ $k$-tuples are partitioned by small balls around the actuals means $\mu_1,\ldots,\mu_k$, then for large enough $T$ (but much smaller than $n$), we are guaranteed to obtain $k$ avegares estimations $\tilde{\pa}_1,\ldots,\tilde{\pa}_k$ that are very close to the actual means.
In the next step, Algorithm $\mathsf{PrivatekGMM}$ partitions a fresh set of $n$ samples
according to $\tilde{\pa}_1,\ldots,\tilde{\pa}_k$, where each sample $\px$ belongs to the $i$'th set if $\pa_i$ is the closest point to it among $\tilde{\pa}_1,\ldots,\tilde{\pa}_k$. Using a standard projection argument, we show that if we assume a separation of the form $\forall i \neq j: \text{ } \norm{\mu_i - \mu_j} > 2\sqrt{2\log(\beta/n)} \cdot \max \set{\norm{\Sigma_i}^{1/2}, \norm{\Sigma_j}^{1/2}}$ (which is independent of the dimension $d$), then with probability $1-\beta$, the above partition perfectly classifiy the points correctly (i.e., two points belong to the same set iff they both were sampled from the same Gaussian). Finally, we apply a private algorithm ${\cal A}'$ for learning the parameters of each single Gaussian (e.g., \cite{KLSU19} or \cite{BDKU20}) in each of the $k$ sets in the partition.
Overall, our algorithm $\mathsf{PrivatekGMM}$ wraps (using algorithm $\mathsf{PrivatekAverages}$) any given non-private algorithm ${\cal A}$ that learns a mixture of Gaussians, along with any given private algorithm ${\cal A}'$ that learns a single Gaussian, in order to learn the parameters of a mixture of separated (and bounded) Gaussians under a weaker separation assumption than the one used by \cite{KSSU19}. The construction uses only black-box accesses to ${\cal A}$ and ${\cal A}'$, and also can be parallelized very easily. Furthermore, we show that using an additional step of sub-sampling, our algorithm improves the sample complexity of \cite{KSSU19}. See \cref{sec:Gauss:comparison} for the full comparison.
\Enote{add discussion why JL is not usefull when we apply it directly on the learning mixtures of Gaussians problem}
\paragraph{Other Problems}
\Enote{TBD}
}
\section{Missing Proofs}\label{sec:missing-proofs}
\subsection{Proving \cref{prop:cost-of-sample-is-good}}\label{missing-proof:cost-of-sample-is-good}
In this section we prove \cref{prop:cost-of-sample-is-good}, restated below
\begin{proposition}[Restatement of \cref{prop:cost-of-sample-is-good}]
\propCostOfSampleIsGood
\end{proposition}
In the following, fix values of $s$ and $\beta$, let $\xi = \xi(s,\beta)$ and $M = M(s,\beta)$.
The following event and claims are with respect to the random process in \cref{prop:cost-of-sample-is-good}.
\begin{claim}[Event $E$ \cite{ShechnerSS20}]\label{claim:E}
Let $E$ be the event that for every $C \in B(0,\Lambda)^k$, we have that
\begin{align*}
\size{\frac{n}s \cdot {\rm COST}_{{\cal S}}(C) - {\rm COST}_{{\cal P}}(C)} \leq \sqrt{M \cdot {\rm COST}_{{\cal P}}(C)} := \Delta(C)
\end{align*}
Then it holds that $\pr{E} \geq 1-\beta$.
\end{claim}
We next prove some useful facts that holds when event $E$ occurs.
\begin{claim}\label{claim:cost-tC}
Conditioned on event $E$, it holds that
\begin{align*}
{\rm COST}_{{\cal P}}(\tilde{C}) \leq \omega\cdot {\rm OPT}_k({\cal P}) + \Delta(C^{*}_{{\cal P}}) + \Delta(\tilde{C}),
\end{align*}
letting $\tilde{C}$ be the set from \cref{prop:cost-of-sample-is-good}, and letting $C^{*}_{{\cal P}}$ be the optimal $k$-means of ${\cal P}$.
\end{claim}
\begin{proof}
Let $C^{*}_{{\cal S}}$ be the optimal $k$-means of ${\cal S}$.
By the assumption on the algorithm $\mathsf{A}$, the set $\tilde{C}$ satisfies ${\rm COST}_{{\cal S}}(\tilde{C}) \leq \omega \cdot {\rm OPT}_k({\cal S})$. The proof follows by the following calculation
\begin{align*}
{\rm COST}_{{\cal P}}(\tilde{C})
&\leq \frac{n}{s} \cdot {\rm COST}_{{\cal S}}(\tilde{C}) + \Delta(\tilde{C})\\
&\leq \omega\cdot \frac{n}{s} \cdot {\rm COST}_{{\cal S}}(C^{*}_{{\cal S}}) + \Delta(\tilde{C})\\
&\leq \omega\cdot \frac{n}{s} \cdot {\rm COST}_{{\cal S}}(C^{*}_{{\cal P}}) + \Delta(\tilde{C})\\
&\leq \omega \cdot \frac{n}{s} \cdot \paren{\frac{m}{n}\cdot {\rm COST}_{{\cal P}}(C^{*}_{{\cal P}}) + \frac{s}{n} \cdot \Delta(C^{*}_{{\cal P}})} + \Delta(\tilde{C})\\
&= \omega\cdot {\rm OPT}_k({\cal P}) + \Delta(C^{*}_{{\cal P}}) + \Delta(\tilde{C}),
\end{align*}
where the third inequality holds by event $E$,
\end{proof}
We now prove a corollary of \cref{claim:cost-tC}.
\begin{corollary}\label{cor:cost-Ct:1}
Conditioned on event $E$, it holds that
\begin{align*}
\Delta(\tilde{C}) \leq 2\paren{M + \sqrt{M \omega {\rm OPT}_k({\cal P})}}
\end{align*}
\end{corollary}
\begin{proof}
Let $x = \Delta(\tilde{C}) = \sqrt{M \cdot {\rm COST}_{{\cal P}}(\tilde{C})}$.
By \cref{claim:cost-tC}, it holds that
\begin{align*}
\frac{x^2}{M} - x \leq \omega\cdot {\rm OPT}_k({\cal P}) + \sqrt{M \cdot {\rm OPT}_k({\cal P})}.
\end{align*}
Since $x \geq 0$, we conclude that
\begin{align}\label{eq:bounding-x}
x
&\leq \frac12\cdot \paren{M + \sqrt{M^2 + 4M \omega {\rm OPT}_k({\cal P}) + 4M^{1.5} \sqrt{{\rm OPT}_k({\cal P})}}}\nonumber\\
&\leq M + \sqrt{M \omega {\rm OPT}_k({\cal P})} + M^{0.75} \cdot {\rm OPT}_k({\cal P})^{1/4}\\
&\leq 2\paren{M + \sqrt{M \omega {\rm OPT}_k({\cal P})}},\nonumber
\end{align}
where the second inequality holds by the fact that $\sqrt{a+b} \leq \sqrt{a} + \sqrt{b}$ for $a,b \geq 0$, and the last inequality holds since the third term in (\ref{eq:bounding-x}) is either smaller than the first term, or smaller than the second one (recall that $M \geq 1$).
\end{proof}
The proof of \cref{prop:cost-of-sample-is-good} now immediately follows by \cref{claim:cost-tC} and \cref{cor:cost-Ct:1}.
\subsection{Proving \cref{prop:close-centers-have-similar-cost}}\label{missing-proof:close-centers-have-similar-cost}
\begin{proposition}[Restatement of \cref{prop:close-centers-have-similar-cost}]
\propCloseCentersHaveSimilarCost
\end{proposition}
\begin{proof}
In the following, for $\px \in {\cal P}$ let $i_{\px} = \operatorname*{argmin}_{i}\set{\norm{\px - \pc_{i}}}$ (i.e., the index of the closest center to $\px$ in $C$), and let $j_{\px} = \operatorname*{argmin}_{j}\set{\norm{\px - \pc_{j}'}}$ (i.e., the index of the closest center to $\px$ in $C'$).
It holds that
\begin{align*}
\sum_{i=1}^k {\rm OPT}_1({\cal P}_i)
&\leq \sum_{i=1}^k \sum_{\px \in {\cal P}_i} \norm{\px - \pc_i}^2\\
&= \sum_{\px \in {\cal P}} \norm{\px - \pc_{j_{\px}}}^2\\
&= \sum_{\px \in {\cal P}} \norm{\px - \pc_{i_{\px}}}^2 + \sum_{\px \in {\cal P}} \paren{\norm{\px - \pc_{j_{\px}}}^2 - \norm{\px - \pc_{i_{\px}}}^2}\\
&= {\rm COST}_{{\cal P}}(C) + \sum_{\px \in {\cal P}} \paren{\norm{\px - \pc_{j_{\px}}}^2 - \norm{\px - \pc_{i_{\px}}}^2}
\end{align*}
In the following, fix $\px \in {\cal P}$.
We now bound
\begin{align*}
\norm{\px - \pc_{j_{\px}}}^2-\norm{\px -\pc_{i_{\px}}}^2
= \left(\norm{\px - \pc_{j_{\px}}}-\norm{\px
-\pc_{i_{\px}}}\right)\left(\norm{\px
-\pc_{j_{\px}}}+\norm{\px -\pc_{i_{\px}}}\right)
\end{align*}
%
First, since $\norm{\px - \pc_{j_{\px}}'} \leq \norm{\px - \pc_{i_{\px}}'} $ it holds that
\begin{align}\label{eq:x-cjx}
\norm{\px - \pc_{j_{\px}}} \leq \norm{\px - \pc_{j_{\px}}'} + \norm{\pc_{j_{\px}}' - \pc_{j_{\px}}} \leq \norm{\px - \pc_{i_{\px}}'} + \gamma \norm{\pc_{i_{\px}} - \pc_{j_{\px}}}
\end{align}
Second,
\begin{align*}
\norm{\px - \pc_{i_{\px}}} \geq \norm{\px - \pc_{i_{\px}}'} - \norm{\pc_{i_{\px}}' - \pc_{i_{\px}}} \geq \norm{\px - \pc_{i_{\px}}'} - \gamma \norm{\pc_{i_{\px}} - \pc_{j_{\px}}}
\end{align*}
Therefore
\begin{align*}
\norm{\px - \pc_{j_{\px}}} - \norm{\px - \pc_{i_{\px}}} \leq 2\gamma \norm{\pc_{i_{\px}} - \pc_{j_{\px}}}
\end{align*}
Now, $\norm{\px -\pc_{i_{\px}}} \leq \norm{\px -\pc_{j_{\px}}}$ and therefore
\begin{align*}
\norm{\px -\pc_{j_{\px}}}+\norm{\px -\pc_{i_{\px}}}
&\leq 2 \norm{\px -\pc_{j_{\px}}}\\
&\leq 2\norm{\px - \pc_{i_{\px}}'} + 2\gamma \norm{\pc_{i_{\px}} - \pc_{j_{\px}}}\\
&\leq 2\paren{\norm{\px - \pc_{i_{\px}}} + \norm{\pc_{i_{\px}}' - \pc_{i_{\px}}}}+ 2\gamma \norm{\pc_{i_{\px}} - \pc_{j_{\px}}}\\
&\leq 2\norm{\px - \pc_{i_{\px}}} + 4\gamma \norm{\pc_{i_{\px}} - \pc_{j_{\px}}},
\end{align*}
where the second inequality holds by \cref{eq:x-cjx}.
We now like to bound $\norm{\pc_{i_{\px}} - \pc_{j_{\px}}}$ as
a function of $\norm{\px - \pc_{i_{\px}}}$.
We first bound $\norm{\pc_{i_{\px}} - \pc_{j_{\px}}}$ as a
function of $\norm{\px - \pc_{i_{\px}}'}$.
\begin{align}\label{eq:px-pci-lowbound-ym-2}
2\norm{\px - \pc_{i_{\px}}'}
&\geq \norm{\px - \pc_{i_{\px}}'} + \norm{\px - \pc_{j_{\px}}'}\nonumber\\
&\geq \norm{\pc_{i_{\px}}' - \pc_{j_{\px}}'}\nonumber\\
&\geq \norm{\pc_{i_{\px}} - \pc_{j_{\px}}} - \norm{\pc_{i_{\px}} - \pc_{i_{\px}}'} - \norm{\pc_{j_{\px}} - \pc_{j_{\px}}'}\nonumber\\
&\geq (1-2\gamma) \norm{\pc_{i_{\px}} - \pc_{j_{\px}}},
\end{align}
In addition
\[
\norm{\px - \pc_{i_{\px}}'}\leq \norm{\px - \pc_{i_{\px}}} +
\norm{\pc_{i_{\px}} - \pc_{i_{\px}}'} \leq \norm{\px -
\pc_{i_{\px}}} + \gamma \norm{\pc_{i_{\px}} - \pc_{j_{\px}}}
\]
Therefore,
\[
2\norm{\px - \pc_{i_{\px}}} \geq
(1-4\gamma)
\norm{\pc_{i_{\px}} - \pc_{j_{\px}}}
\]
We have that
\begin{align*}
\norm{\px - \pc_{j_{\px}}}^2-\norm{\px -\pc_{i_{\px}}}^2 &=
\left(\norm{\px - \pc_{j_{\px}}}-\norm{\px
-\pc_{i_{\px}}}\right)\left(\norm{\px
-\pc_{j_{\px}}}+\norm{\px -\pc_{i_{\px}}}\right)\\
&\leq \left(2\gamma \norm{\pc_{i_{\px}} -
\pc_{j_{\px}}}\right)\left(2 \norm{\px - \pc_{i_{\px}}}
+4\gamma \norm{\pc_{i_{\px}} - \pc_{j_{\px}}}\right)\\
&\leq \left(\frac{4\gamma }{1-4\gamma}\norm{\px - \pc_{i_{\px}}}
\right)\left((2+\frac{8\gamma }{1-4\gamma}) \norm{\px -
\pc_{i_{\px}}} \right)\\
&\leq 32\gamma \norm{\px - \pc_{i_{\px}}}^2,
\end{align*}
where the least inequality holds since $\gamma \leq 1/8$.
Now we can get the bound on the summation:
\begin{align*}
\sum_{\px \in {\cal P}} \paren{\norm{\px - \pc_{j_{\px}}}^2 - \norm{\px - \pc_{i_{\px}}}^2}
\leq \sum_{\px \in {\cal P}} 32\gamma \norm{\px - \pc_{i_{\px}}}^2
\leq 32\gamma {\rm COST}_{{\cal P}}(C)
\end{align*}
\end{proof}
\subsection{Proving \cref{thm:kGauss-utility}}\label{missing-proof:thm:kGauss-utility}
In this section we prove the utility guarantee of $\mathsf{PrivatekGMM}$.
We first by proving the following proposition that states the following: Assume that $\pX \sim {\cal N}(\mu,\Sigma)$ with $\norm{\Sigma} = \sigma^2$,
and let $\py,\pz \in {\mathbb R}^d$ such that (1) $\norm{\py - \mu}$ is ``large enough'' (larger than $\Omega\paren{\sigma \sqrt{\log(1/\beta)}}$ ) , and (2) $\norm{\pz - \mu}$ is ``small enough''. Then with probability $1-\beta$ (over $\pX$) it holds that $\norm{\pX - \pz} < \norm{\pX - \py}$. Note that such an argument is trivial when $\norm{\py - \mu}$ is at least $\Omega(\sigma \sqrt{d \log(1/\beta)})$, but using a standard projection argument, we can avoid the dependency in $d$.
\begin{proposition}\label{prop:separation}
Let $\pX \sim {\cal N}(\mu,\Sigma)$ where $\norm{\Sigma} = \sigma^2$, let $\py \in {\mathbb R}^d$ with $\norm{\py - \mu} \geq 2(1+\gamma)\sqrt{2\log\paren{1/\beta}} \cdot \sigma$ for some $\gamma > 0$, and let $\pz \in {\mathbb R}^d$ with $\norm{\pz - \mu} \leq \frac{\gamma}{3(1+\gamma)} \norm{\py - \mu}$. Then with probability $1-\beta$ (over the choice of $\pX$), it holds that $\norm{\pX - \pz} < \norm{\pX-\py}$.
\end{proposition}
\begin{proof}
Assume w.l.o.g. that $\mu = \pt{0}$.
Let $\pW = \pz + \frac{\iprod{\pX-\pz,\py-\pz}}{\norm{\py-\pz}^2} (\py-\pz)$ be the projection of $\pX$ onto the line between $\py$ and $\pz$. In the following we bound the probability that $\frac{\iprod{\pX-\pz,\py-\pz}}{\norm{\py-\pz}^2} < \frac12$, which implies that $\norm{\pW - \pz} < \norm{\pW-\py}$, and therefore, $\norm{\pX - \pz} < \norm{\pX-\py}$. Note that $\rm ip{\pX,\py-\pz}$ is distributed according to the (one dimensional) Gaussian ${\cal N}(\pt{0}, (\py-\pz)^T \Sigma (\py-\pz))$ and it holds that $(\py-\pz)^T \Sigma (\py -\pz) \leq \sigma^2 \norm{\py-\pz}$. Therefore, by \cref{fact:one-Gaus-concet} we obtain that with probability $1-\beta$ it holds that $\rm ip{\pX,\py-\pz} < \sigma \norm{\py-\pz} \sqrt{2 \log(1/\beta)}$, and in the following we continue with the analysis assuming that this occurs. The proposition now follows by the following calculation.
\begin{align*}
\frac{\iprod{\pX-\pz,\py-\pz}}{\norm{\py-\pz}^2}
&= \frac{\rm ip{\pX,\py-\pz} - \rm ip{\pz,\py-\pz}}{\norm{\py-\pz}^2}\\
&< \frac{\sigma \norm{\py-\pz} \sqrt{2 \log(1/\beta)} + \norm{\pz} \norm{\py -\pz}}{\norm{\py-\pz}^2}\\
&\leq \frac{\sigma \sqrt{2 \log(1/\beta)}}{\paren{1 - \frac{\gamma}{3(1+\gamma)}} \norm{y}} + \frac{\frac{\gamma}{3(1+\gamma)}}{1 - \frac{\gamma}{3(1+\gamma)}}\\
&\leq \frac{1}{2(1+\gamma) \paren{1 - \frac{\gamma}{3(1+\gamma)}}} + \frac{\frac{\gamma}{3(1+\gamma)}}{1 - \frac{\gamma}{3(1+\gamma)}}\\
&= \frac{1 + \frac{2 \gamma}{3}}{2(1 + \gamma)\frac{3 + 2\gamma}{3(1+\gamma)}}\\
&= \frac12,
\end{align*}
where in the second inequality holds since $\norm{\py-\pz} \geq \norm{\py} - \norm{\pz} \geq \paren{1 - \frac{\gamma}{3(1+\gamma)}} \norm{y}$, and the third inequality holds by the assumption on $\norm{\py}$.
\end{proof}
In addition, we use the following fact.
\begin{fact}\label{fact:dTV-of-mixtures}
Let ${\cal D} = \sum_{i=1}^k w_i {\cal D}_i$ be a mixture of the $k$ distributions ${\cal D}_1, \ldots, {\cal D}_k$, and let ${\cal D}' = \sum_{i=1}^k w_i' {\cal D}_i'$ be a mixture of the $k$ distributions ${\cal D}_1', \ldots, {\cal D}_k'$. Assume that for every $i \in [k]$ it holds that ${\rm d}_{\rm TV}({\cal D}_i,{\cal D}_i') \leq \frac{\alpha}{2}$ and $\size{w_i - w_i'} \leq \frac{\alpha}{k}$. Then ${\rm d}_{\rm TV}({\cal D},{\cal D}) \leq \alpha$.
\end{fact}
We now ready to prove \cref{thm:kGauss-utility}, stated for convenient below.
\begin{theorem}[Restatement of \cref{thm:kGauss-utility}]
\thmKGaussUtility
\end{theorem}
\begin{proof}
Let $E_1 = \bigwedge_{j \in [t], i \in [k]} E_1^{t,i}$ where $E_1^{j,i}$ is the event that the $s$-size set ${\cal S}_j$ in Step~\ref{step:sample-gaus} of $\mathsf{GenEmpiricalMeans}$ contains at least $\frac{w_i s}{2}$ samples from the $i$'th Gaussian. Note that for every $j \in [t]$ and $i \in [k]$, it holds that
\begin{align*}
\pr{E_1^{j,i}}
&= \pr{{\rm Bin}(s,w_i) \geq \frac{s w_i}{2}}\\
&\geq 1 - \pr{{\rm Bin}(s,w_{\min}) < \frac{s w_{\min}}{2}}\\
&\geq 1 - e^{-\frac{w_{\min} s}{4}}
\end{align*}
where the last inequality holds by \cref{fact:binom_concentration}. Therefore, we obtain that $\pr{E_1^{j,i}} \geq 1 - \frac{\beta}{8 k t}$ whenever $s \geq \frac{4}{w_{\min}} \log\paren{8 k t/\beta}$, which holds by the assumption on $s$. By the union bound, we deduce that
\begin{align}\label{eq:E1}
\pr{E_1} \geq 1 - \beta/8
\end{align}
In the following, assume that event $E_1$ occurs. For $j \in [t]$ and $i \in [k]$ let
$\hat{{\cal S}}_j^i$ be all the points in ${\cal S}_j$ that have been drawn from the $i$'th Gaussian ${\cal N}(\mu_{i},\Sigma_{i})$, and let $\hat{\mu}_{j,i} = {\rm Avg}\paren{\hat{{\cal S}}_j^i}$. Let $E_2 = \bigwedge_{j \in [t], i \in [k]} E_2^{j,i}$, where $E_2^{j,i}$ is the event that $\norm{\hat{\mu}_{j,i} - \mu_i} \leq \frac{\gamma h}{16} \cdot \sigma_{i}$. Since $\hat{\mu}_{j,i}$ is the average of at least $\frac{w_{i} s}{2}$ samples from the Gaussian ${\cal N}(\mu_i,\Sigma_i)$, we obtain by \cref{fact:gaus-avg} that with probability $1 - \frac{\beta}{8 k t}$ it holds that
\begin{align}\label{eq:hmu_i-to-mu_i}
\norm{\hat{\mu}_{j,i} - \mu_i}
\leq \frac{\sqrt{2d} + 2\sqrt{\log \paren{\frac{8 k t}{\beta}}}}{\sqrt{w_i s}} \cdot \sigma_i
\leq \frac{(1+\gamma) h}{\Delta} \cdot \sigma_i,
\end{align}
where the last inequality holds by the assumption on $s$ (by the second term in the maximum). Therefore, event $E_2^{j,i}$ occurs with probability at least $1 - \frac{\beta}{8 k t}$, and we conclude by the union bound that
\begin{align}\label{eq:E2}
\pr{E_2 \mid E_1} \geq 1 - \beta/8
\end{align}
Let $E_3 = \bigwedge_{j=1}^t E_3^j$, where $E_3^j$ is the event that the resulting labeling function $L_j$ in Step~\ref{step:labeling} of the $j$'th iteration in $\mathsf{GenEmpiricalMeans}$ satisfies:
\begin{align*}
\forall \px,\px' \in {\cal S}_j:\quad \px,\px' \text{ were drawn from the same Gaussian } \iff L_j(\px) = L_j(\px').
\end{align*}
Since $\mathsf{A}$ is a $(s,\frac{\beta}{8 t})$-labeling algorithm for ${\cal D}$, it holds that $\pr{E_3^j} \geq 1 - \frac{\beta}{8 t}$ for every $j \in [t]$, and we deduce by the union bound that
\begin{align}\label{eq:E3}
\pr{E_3} \geq 1 - \beta/8
\end{align}
In the rest of the analysis we assume that event $E_1 \land E_2 \land E_3$ occurs. This means that for every $j \in [t]$ there exists a permutation $\pi_j$ over $[k]$ such that for each $i \in [k]$, the set of all points in ${\cal S}_j$ that have been drawn from the $i$'th Gaussian (which we denoted by ${\cal S}_j^i$) equals to $\set{\px \in {\cal S}_j \colon L_j(\px) = \pi_j(i)}$, and assume without loss of generality that for all $j \in [t]$, $\pi_j$ is the identity (i.e., $\pi_j(i) = i$). Therefore, for all $j \in [t]$ and $i \in [k]$ it holds that $\hat{\mu}_{j,i} = \bar{\mu}_{j,i}$, where $\bar{\mu}_{j,i}$ is the empirical mean from Step~\ref{step:compute-emp-mean}. Namely, we obtained that
\begin{align}\label{eq:bar_mu-vs-mu}
\forall j \in [t], i \in [k]: \quad \norm{\bar{\mu}_{j,i} - \mu_i} \leq \frac{(1+\gamma) h}{\Delta} \cdot \sigma_{i},
\end{align}
and in particular, it holds that
\begin{align}\label{eq:bar_mu-le-Lambda}
\forall j \in [t], i \in [k]: \quad \norm{\bar{\mu}_{j,i}} \leq \norm{\mu_{j,i}} + \frac{(1+\gamma) h}{\Delta} \cdot \sigma_{i} \leq \Lambda
\end{align}
Therefore, we deduce that ${\cal T}$ from Step~\ref{step:kAveragesOnM} of $\mathsf{PrivatekGMM}$ is contained in $(B(\pt{0},\Lambda)^k)^*$, and is partitioned by the $\Delta$-far balls ${\cal B} = \set{B_i(\mu_i, r_i = \frac{(1+\gamma) h}{\Delta} \cdot \sigma_i )}_{i=1}^k$ (\cref{def:sep-balls}), where ${\rm Partition}({\cal T})$ is exactly $\set{{\cal P}_1 =\set{ \bar{\mu}_{j,1}}_{j=1}^t,\ldots,{\cal P}_k = \set{\bar{\mu}_{j,k}}_{j=1}^t}$ (note that the balls are indeed $\Delta$-far by the separation assumption that $\norm{\mu_i - \mu_j} \geq (1+\gamma) h \max\set{\sigma_i,\sigma_j}$).
Therefore, since $\mathsf{B}$ is an $(t,\text{ }\alpha=1,\text{ }r_{\min}=\frac{(1+\gamma) h}{\Delta}\cdot \sigma_{\min},\text{ }\beta/8, \text{ }\Delta,\text{ }\Lambda=R + \frac{(1+\gamma) h}{\Delta}\cdot \sigma_{\max})$-averages-estimator, we obtain that the output $\set{\hat{\pa}_1,\ldots,\hat{\pa}_k}$ of $\mathsf{B}({\cal T})$ in Step~\ref{step:kAveragesOnM} satisfy w.p. $1-\beta/8$ that:
\begin{align}\label{eq:a-to-avg}
\forall i \in [k]: \quad \norm{\hpa_i - {\rm Avg}\paren{{\cal P}_i}} \leq \max\set{r_i,r_{\min}} \leq \frac{(1+\gamma)h}{\Delta}\cdot \sigma_i
\end{align}
In the following, we denote by $E_4$ the event that \cref{eq:a-to-avg} occurs, where recall that we proved that
\begin{align}
\pr{E_4 \mid E_1 \land E_2 \land E_3} \geq 1 - \beta/8
\end{align}
In the following, we also assume that event $E_4$ occurs.
Recall that by \cref{eq:bar_mu-vs-mu}, for each $j \in [t]$ and $i \in [k]$ it holds that
\begin{align}\label{eq:avg-to-mu_i}
\norm{{\rm Avg}\paren{{\cal P}_i} - \mu_i}
\leq \frac{(1+\gamma)h}{\Delta} \cdot \sigma_i,
\end{align}
and we deduce by \cref{eq:a-to-avg,eq:avg-to-mu_i} that for all $i \in [k]$ it holds that
\begin{align}\label{eq:hpa_i-to-mu_i}
\norm{\hpa_i - \mu_i} \leq \frac{2(1+\gamma)h}{\Delta} \cdot \sigma_i.
\end{align}
Therefore, for all $i \neq j$ it holds that
\begin{align}\label{eq:hpa_i-to-mu_j}
\norm{\hpa_j - \mu_i}
\geq \norm{\mu_i - \mu_j} - \norm{\hpa_j - \mu_j}
&\geq \paren{1-2/\Delta} (1+\gamma)h \cdot \max\set{\sigma_i,\sigma_j}\\
&\geq (1+\gamma/2)h \cdot \max\set{\sigma_i,\sigma_j}\nonumber
\end{align}
where the second inequality holds by the separation assumption along with \cref{eq:avg-to-mu_i}, and the last one holds by the choice of $\Delta$.
Hence, we deduce from \cref{eq:hpa_i-to-mu_i,eq:hpa_i-to-mu_j} that for each $i \neq j$ it holds that
\begin{align}\label{eq:hpa_i-to-mu_i-sec}
\norm{\hpa_i - \mu_i} \leq \frac{2}{\Delta - 2} \cdot \norm{\hpa_j - \mu_i} = \frac{\gamma/2}{3(1+\gamma/2)}\cdot \norm{\hpa_j - \mu_i},
\end{align}
where the last inequality holds since recall that $\Delta = 8 + 12/\gamma$.
Since $h \geq 2 \sqrt{2 \log\paren{1/\beta'}}$ for $\beta' = \frac{\beta}{8n}$, then by \cref{prop:separation} (when setting $\py = \hpa_j$ and $\pz = \hpa_i$) along with \cref{eq:hpa_i-to-mu_j,eq:hpa_i-to-mu_i-sec}, for every $i \neq j$, when sampling a point $\px$ from the $i$'th Gaussian ${\cal N}(\mu_i,\sigma_i)$, then with probability $1 - \frac{\beta}{8n}$ it holds that $\norm{\px - \hpa_i} < \norm{\px - \hpa_j}$. Therefore, let $E_5$ be the event that for all $i \in [k]$ and all $\px \in {\cal P}''$ that have been sampled from the $i$'th Gaussian ${\cal N}(\mu_i,\Sigma_i)$, it holds that $\hpa_i$ is the closest point to each of them among $\set{\hat{\pa}_1,\ldots,\hat{\pa}_k}$. Then by the union bound it holds that
\begin{align}\label{eq:E_5}
\pr{E_5 \mid E_1 \land E_2 \land E_3 \land E_4} \geq 1 - \beta/8
\end{align}
In the following we also assume that event $E_5$ occurs. Let $E_6 = \bigwedge_{i \in [k]} E_6^{i}$ where $E_6^{i}$ is the event that ${\cal P}''$ contains at least $\frac{w_i n}{2}$ samples from the $i$'th Gaussian (namely, $\size{{\cal P}''_i} \geq \frac{w_i n}{4}$). Similar calculation to bounding $\pr{E_1}$, it holds that
\begin{align}\label{eq:E_6}
\pr{E_6 \mid E_1 \land \ldots \land E_5} \geq 1 - \beta/8
\end{align}
provided that $n \geq \frac{4}{w_{\min}} \log\paren{8 k/\beta}$, which holds by the assumption on $n$.
In the following we assume that event $E_6$ occurs, and let $E_7 = \land_{i=1}^k E_7^i$, where $E_7^i$ is the event that the output $(\hat{\mu}_i, \hat{\Sigma}_i)$ of the private algorithm $\mathsf{A}'$ in Step~\ref{step:priv-single-Gauss-est} of the $i$'th iteration satisfies ${\rm d}_{\rm TV}({\cal N}(\mu_i,\Sigma_i), {\cal N}(\hat{\mu}_i, \hat{\Sigma}_i)) \leq \eta/2$. By the assumption on algorithm $\mathsf{A}'$, we obtain that
$\pr{E_7^i} \geq 1 - \frac{\beta}{16k}$ whenever $\size{{\cal P}''_i} \geq \upsilon$, which holds w.p. $1-\frac{\beta}{16}$ when $n \geq \frac{2\upsilon + \log(16k/\beta)}{w_i}$ (follows by \cref{fact:binom_concentration} for ${\rm Bin}(n,w_i)$ and $t = \sqrt{w_i n \log(16k/\beta)}$). Since $n \geq \frac{2\upsilon+ \log(16k/\beta)}{w_{\min}}$ by assumption, we obtain by the union bound that
\begin{align}\label{eq:E_7}
\pr{E_7 \mid E_1 \land \ldots \land E_6} \geq 1 - \beta/8.
\end{align}
In the following, for $i \in [k]$ let $L_i$ be the value of the Laplace noise in Step~\ref{step:Lap} of the $i$'th iteration, let $E_8^i$ be the event that $\size{L_i} \leq \frac2{\varepsilon} \log\paren{16 k/\beta}$, and let $E_8 = \land_{i=1}^k E_8^i$. By \cref{fact:laplace-concent}, for any fixing of $i \in [k]$ it holds that $\pr{E_8^i} \geq 1 - \frac{\beta}{8 k}$, and therefore, by the union bound it holds that
\begin{align}\label{eq:E_8}
\pr{E_8} \geq 1 - \beta/8.
\end{align}
In the following we also assume that $E_8$ occurs. It is left to show that when event $E_1 \land \ldots \land E_8$ occurs, then for every $i \in [k]$ it holds that
\begin{align}\label{eq:hw_i-w_i}
\forall i \in [k]: \quad \size{\hat{w}_i - w_i} \leq \eta/k.
\end{align}
Indeed, given \cref{eq:hw_i-w_i} and event $E_7$, we deduce by \cref{fact:dTV-of-mixtures} that ${\rm d}_{\rm TV}({\cal D},\hat{{\cal D}}) \leq \eta$, which holds with probability at least $\pr{E_1 \land \ldots \land E_8} \geq 1-\beta$ (holds by \cref{eq:E1} to \cref{eq:E_8}).
We now prove that \cref{eq:hw_i-w_i} holds when $E_1 \land \ldots \land E_8$ occurs.
Fix $i \in [k]$, let $L = \sum_{j=1}^k L_j$, and compute
\begin{align*}
\size{\hat{w}_i - w_i}
&= \size{\frac{\hat{n}_i}{\hat{n}} - \frac{n_i}{n}}
= \size{\frac{n_i + L_i}{n + L} - \frac{n_i}{n}}\\
&= \size{\frac{n L_i - n_i L}{n(n+L)}}
= \size{\frac{(n-n_i)L_i - n_i \sum_{j \neq i} L_j}{n(n+L)}}\\
&\leq \frac{\frac{2k}{\varepsilon} \log\paren{8 k/\beta}}{n - \frac{2k}{\varepsilon} \log\paren{8 k/\beta}}\\
&\leq \eta/k,
\end{align*}
where the first inequality holds by event $E_8$, and the last one holds whenever $n \geq \frac{4 k^2}{\varepsilon \eta} \cdot \log\paren{8 k/\beta}$, which holds by the assumption on $n$.
\end{proof}
\section{Mixture of Gaussians}\label{sec:mixture-of-gaus}
In this section we present our second application of $k$-tuple clustering, which is an $(\varepsilon,\delta)$-differentially private algorithm $\mathsf{PrivatekGMM}$ for learning a mixture of well separated and bounded $k$ Gaussians. We first start with relevant preliminaries for this section.
\subsection{Preliminaries}
The total variation distance between two distributions $P$ and $Q$ over a universe ${\cal U}$ is defined by ${\rm d}_{\rm TV}(P,Q) = \sup_{{\cal S} \subseteq {\cal U}} \size{P({\cal S}) - Q({\cal S})}$. Given a matrix $A = (a_{i,j})_{i,j \in [d]} \in {\mathbb R}^{d \times d}$, we let $\norm{A} = \sup_{\norm{\px} = 1} \norm{A \px}$ be its $\ell_2$ norm.
\subsubsection{Gaussians}
Let ${\cal N}(0,1)$ be the standard Gaussian distribution over ${\mathbb R}$ with probability density function $p(z) = \frac1{\sqrt{2\pi}} e^{-\frac{z^2}2}$.
In ${\mathbb R}^d$, let ${\cal N}(\pt{0},{\mathbb I}_{d \times d})$ be the standard multivariate Gaussian distribution. That is, if $\pZ \sim {\cal N}(\pt{0},{\mathbb I}_{d \times d})$ then $\pZ = (Z_1,\ldots,Z_d)$ where $Z_1,\ldots,Z_d$ are i.i.d. according to $N(0,1)$. Other Gaussian distributions over ${\mathbb R}^d$ arise by applying (invertible) linear maps on ${\cal N}(\pt{0},{\mathbb I}_{d \times d})$. That is, the distribution $\pX \sim {\cal N}(\mathbf{\mu}, \Sigma = AA^T)$ for $\mu \in {\mathbb R}^d$ and (invertible) $A \in {\mathbb R}^{d \times d}$ is defined by $\pX = A \pZ + \mathbf{\mu}$, where $\pZ \sim N(\pt{0},{\mathbb I}_{d \times d})$, and it holds that $\ex{\pX} = \mu$ and ${\rm Cov}(\pX) = \paren{{\rm Cov}(X_i,X_j)}_{i,j}$ (the covariance matrix) equals to $\Sigma$. The contours of equal density are ellipsoids around $\mu$: $\set{\px \in {\mathbb R}^d \colon (\px - \mu)^T \Sigma^{-1}(\px - \mu) = r^2}$. We let ${\cal G}(d)$ be the family of all $d$-dimensional Gaussian --- that is, the set of all distribution ${\cal N}(\mu,\Sigma)$ where $\mu \in {\mathbb R}^d$ and $\Sigma$ is a $d \times d$ positive semidefinite (PSD) matrix.
\begin{definition}[Bounded Gaussian]\label{def:bounded-gaus}
For $R,\sigma_{\max},\sigma_{\min} > 0$, a Gaussian $\pG = {\cal N}(\mu,\Sigma) \in {\cal G}(d)$ is $(R,\sigma_{\max},\sigma_{\min})$-bounded if $\norm{\mu} \leq R$ and $\sigma_{\min}^2 \leq \norm{\Sigma} \leq \sigma_{\max}^2$.
\end{definition}
We next define the properties of a general algorithm that learns the parameters of a (single) bounded Gaussian.
\begin{definition}[Learner for Bounded Gaussians]\label{def:LearnAlgSingleGaus}
Let $\mathsf{A}$ be an algorithm that gets as input a database ${\cal P} \in ({\mathbb R}^d)^*$ and outputs $(\hat{\mu},\hat{\Sigma})$.
We say that $\mathsf{A}$ is an $(\upsilon,\eta,\beta)$-learner for $(R,\sigma_{\max},\sigma_{\min})$-bounded Gaussians, if for any
such bounded Gaussian ${\cal N}(\mu,\Sigma)$, algorithm $\mathsf{A}$ given $\upsilon$ i.i.d.\ samples from it as input, outputs w.p. $1-\beta$ a pair $(\hat{\mu},\hat{\Sigma})$ with ${\rm d}_{\rm TV}({\cal N}(\mu,\Sigma), {\cal N}(\hat{\mu},\hat{\Sigma})) \leq \eta$.
\end{definition}
\remove{
\begin{definition}[Private Algorithm for Learning a Bounded Gaussian]\label{def:privLearnAlgSingleGaus}
Let $\mathsf{A}$ be an algorithm that gets as input a database ${\cal P} \in ({\mathbb R}^d)^*$ and parameters $d, \varepsilon,\delta,\eta,\beta,R,\sigma_{\max},\sigma_{\min}$, and outputs $(\hat{\mu},\hat{\Sigma})$. Let $s = s(d, \varepsilon,\delta,\eta,\beta,R,\sigma_{\max},\sigma_{\min})$ be a function.
We say that $\mathsf{A}$ is a \textbf{private algorithm for learning a bounded Gaussian with sample complexity $\upsilon = \upsilon(d, \varepsilon,\delta,\eta,\beta,R,\sigma_{\max},\sigma_{\min})$} if given the above parameters, $\mathsf{A}$ is an $(\varepsilon,\delta)$-differentially private algorithm that satisfy the following utility guarantee: If ${\cal N}(\mu,\Sigma)$ is a $(R,\sigma_{\max},\sigma_{\min})$-bounded Gaussian, and ${\cal P}$ consists of at least $\upsilon$ i.i.d.\ samples from ${\cal N}(\mu,\Sigma)$, then with probability at least $1-\beta$ it holds that ${\rm d}_{\rm TV}({\cal N}(\mu,\Sigma), {\cal N}(\hat{\mu},\hat{\Sigma})) \leq \eta$.
\end{definition}
}
In our construction, we would like to use a \emph{differentially private} algorithm that learns the parameters of single (bounded) Gaussians.
The best known examples for such algorithms are the constructions of \cite{KLSU19} and \cite{BDKU20} in the \emph{zero Concentrated DP} model (zCDP \cite{BS16}). For instance, the algorithm of \cite{KLSU19} is $\frac{\varepsilon^2}2$-zCDP $(\upsilon,\eta,\beta)$-learner for $(R,\sigma_{\max},\sigma_{\min})$-bounded Gaussians, for
$\upsilon = \tilde{O}\paren{\paren{\frac{d^2}{\eta^2} + \frac{d^2}{\varepsilon \eta} + \frac{d^{3/2} \sqrt{\log \paren{\frac{\sigma_{\max}}{\sigma_{\min}}}} + \sqrt{d \log R}}{\varepsilon}}\cdot \log(1/\beta)}$.
We first remark that $\varepsilon$-DP implies $\frac{\varepsilon^2}{2}$-zCDP, and the latter implies $(\varepsilon\sqrt{\log(1/\delta)},\delta)$ for every $\delta > 0$. We also remark that
without privacy, the required sample complexity is $\Theta\paren{\frac{d^2 \log(1/\beta)}{\eta^2}}$, which means that privacy comes almost for free unless $\frac1{\varepsilon}, \frac{\sigma_{\max}}{\sigma_{\min}}$ or $R$ are quite large.
\subsubsection{Gaussian Mixtures}
The class of Gaussian $k$-mixtures in ${\mathbb R}^d$ is
\begin{align*}
{\cal G}(d,k) := \set{\sum_{i=1}^k w_i \pG_i \colon \pG_1,\ldots,\pG_k \in {\cal G}(d), w_1,\ldots,w_k > 0, \sum_{i=1}^k w_i = 1}
\end{align*}
A Gaussian mixture can be specified by a set of $k$ triplets: $\set{(\mu_1,\Sigma_1,w_1), \ldots, (\mu_k,\Sigma_k,w_k)}$, where each triplet represents the mean, covariance matrix, and mixing weight of one of its components.
\begin{definition}[Bounded Mixture of Gaussians]
For $R,\sigma_{\max},\sigma_{\min},w_{\min} > 0$, a Gaussian mixture ${\cal D} = \set{(\mu_1,\Sigma_1,w_1), \ldots, (\mu_k,\Sigma_k,w_k)} \in {\cal G}(d,k)$ is $(R,\sigma_{\max},\sigma_{\min},w_{\min})$-bounded if for all $i \in [k]$, the Gaussian ${\cal N}(\mu_i,\Sigma_i)$ is $(R,\sigma_{\max},\sigma_{\min})$-bounded and $w_i \geq w_{\min}$.
\end{definition}
\begin{definition}[Separated Mixture of Gaussians]
Let ${\cal D} = \set{(\mu_1,\Sigma_1,w_1), \ldots, (\mu_k,\Sigma_k,w_k)}$ be a mixture of $k$ Gaussians over ${\mathbb R}^d$, for $i \in [k]$ let $\sigma_i^2 = \norm{\Sigma_i}$, and let $h > 0$.
We say that ${\cal D}$ is $h$-separated if
\begin{align*}
\forall 1 \leq i < j \leq k: \text{ }\norm{\mu_i-\mu_j} \geq h \cdot \max\set{\sigma_i,\sigma_j}.
\end{align*}
\end{definition}
We next define a labeling algorithm for a mixture ${\cal D}$.
\begin{definition}[Labeling Algorithm for a Mixture of Gaussians]\label{def:LabelingAlg}
Let $s,k \in {\mathbb N}$, $\beta \in (0,1)$ and let
${\cal D} = \set{(\mu_1,\Sigma_1,w_1), \ldots, (\mu_k,\Sigma_k,w_k)}$ be a mixture of $k$ Gaussians. We say that an Algorithm $\mathsf{A}$ is an \textbf{$(s,\beta)$-labeling algorithm for the mixture ${\cal D}$} if with probability $1-\beta$, when sampling a database ${\cal P}$ of $s$ i.i.d.\ samples from ${\cal D}$, algorithm $\mathsf{A}$ on inputs ${\cal P},k$, outputs a labeling function $L \colon {\cal P} \rightarrow [k]$ such that for all $\px, \px' \in {\cal P}$: $\quad L(\px) = L(\px')$ $\iff$ $\px$ and $\px'$ were drawn from the same Gaussian.
\end{definition}
There are various examples of non-private algorithms that learns the parameters of mixtures of Gaussian under different separations assumptions, and most of them can be easily converted into a labeling algorithm.
For instance, \cite{DS00,AK01} showed how to learn mixtures with separation that is only proportional to $d^{1/4}$. Moreover, there is a wide line of works that show how to handle mixtures with separation that is independent of $d$: Separation that is proportional to $\sqrt{k}$ \cite{AM05}, $k^{1/4}$ \cite{VW04}, $k^{\varepsilon}$ \cite{HSJ18,KPSJS18,diakonikolas18}, or even $\sqrt{\log k}$ \cite{RV17}. In \cref{sec:GausAlg} we show that our algorithm can transform each such non-private algorithm into a private one, as long as we are given $n$ points from a mixture that is at least $\tilde{\Omega}(\log n)-$separated.
\subsubsection{Concentration Bounds}
\begin{fact}[One-dimensional Gaussian]\label{fact:one-Gaus-concet}
Let $\pX \sim {\cal N}(0,\sigma^2)$. Then for any $\beta > 0$ it holds that
\begin{align*}
\pr{\pX \geq \sigma \sqrt{2 \log(1/\beta)}} \leq \beta.
\end{align*}
\end{fact}
\begin{fact}[follows by the Hanson-Wright inequality \cite{HV71}]\label{fact:gaus-concent-new}
If $\pX \sim {\cal N}(\mu,\Sigma)$ then with probability at least $1-\beta$ it holds that
\begin{align*}
\norm{\pX - \mu} \leq \paren{\sqrt{d} + \sqrt{2 \log(1/\beta)}} \cdot \sqrt{\norm{\Sigma}}.
\end{align*}
\end{fact}
The following fact is an immediate corollary of \cref{fact:gaus-concent-new}.
\begin{fact}\label{fact:gaus-avg}
Let $X_1,\ldots,X_m$ be i.i.d. random variables distributed according to a $d$-dimensional Gaussian ${\cal N}(\mu,\Sigma)$, and let $\sigma^2 = \norm{\Sigma}$. Then with $1-\beta$ it holds that
\begin{align*}
\norm{{\rm Avg}(X_1,\ldots,X_m) - \mu} \leq \frac{\sqrt{d} + \sqrt{2 \log (1/\beta)}}{\sqrt{m}} \cdot \sigma,
\end{align*}
\end{fact}
\begin{proof}
Follows by \cref{fact:gaus-concent-new} since ${\rm Avg}(X_1,\ldots,X_m)$ is distributed according to ${\cal N}(\mu,\frac1{m}\cdot \Sigma)$.
\end{proof}
\subsection{Algorithm $\mathsf{PrivatekGMM}$}\label{sec:GausAlg}
In this section we describe our algorithm $\mathsf{PrivatekGMM}$ (\cref{alg:kGaus}) that privately learns a mixture of separated and bounded $k$ Gaussians ${\cal D} = \set{(\mu_1,\Sigma_1,w_1), \ldots, (\mu_k,\Sigma_k,w_k)}$.
\begin{algorithm}[$\mathsf{GenEmpiricalMeans}$]\label{alg:genBalancedSamples}
\item Input: A database ${\cal P}' = \set{\px_1,\ldots,\px_{n}}$ and parameters $k, s,t \in {\mathbb N}$, where $n \geq st$.
\item Oracle: a (non-private) labeling algorithm $\mathsf{A}$ for a mixture of Gaussians.
\item Operation:~
\begin{enumerate}
\item For each $j \in [t]$:
\begin{enumerate}
\item Let ${\cal S}_j = \set{\px_{(j-1) s + 1},\ldots,\px_{j s}}$.\label{step:sample-gaus}
\item Execute $\mathsf{A}$ on inputs $\tilde{{\cal P}} = {\cal S}_j, \tilde{k} =k$, and let $L_j \colon {\cal S}_j \rightarrow [k]$ be the resulting labeling function.\label{step:labeling}
\item For each $i \in [k]:$
\begin{itemize}
\item Compute $\bar{\mu}_{j,i} = {\rm Avg}\paren{\set{\px \in {\cal S}_j \colon L_j(\px) = i}}$.\label{step:compute-emp-mean}
\end{itemize}
\item Set $M_j = \set{\bar{\mu}_{j,1}, \ldots,\bar{\mu}_{j,k}} \in ({\mathbb R}^d)^k$.\label{step:balanced-samples}
\end{enumerate}
\item Output ${\cal T} = \set{M_1, \ldots, M_t} \in (({\mathbb R}^d)^k)^t$.\label{step:new-multisets-gaus}
\end{enumerate}
\end{algorithm}
\begin{algorithm}[$\mathsf{PrivatekGMM}$]\label{alg:kGaus}
\item Input: A database ${\cal P} = \set{\px_1,\ldots,\px_{2n}} \in \paren{{\mathbb R}^d}^{2n}$, parameters $k,s,t \in {\mathbb N}$ s.t. $n \geq s t$,, and privacy parameter $\varepsilon,\delta > 0$.
\item Oracles: A (non-private) labeling algorithm $\mathsf{A}$, an $(\varepsilon/4,\delta/2)$-DP learner $\mathsf{A}'$ for (single) Gaussians, and an $(\varepsilon,\delta)$-DP $k$-tuple clustering algorithm $\mathsf{B}$.
\item Operation:~
\begin{enumerate}
\item Let ${\cal P}' = \set{\px_1,\ldots,\px_{n}}$ and ${\cal P}'' = \set{\px_{n+1},\ldots,\px_{2n}}$.\label{step:split-database}
\item Compute ${\cal T} = \mathsf{GenEmpiricalMeans}^{\mathsf{A}}({\cal P}',k,s,t)$.\label{step:cM}
\item Compute $\set{\hat{\pa}_1,\ldots,\hat{\pa}_k} = \mathsf{B}({\cal T})$.\label{step:kAveragesOnM}
\item For each $i \in [k]:$\label{step:Gaussian:forloop}
\begin{enumerate}
\item Let ${\cal P}_i''$ be the points in ${\cal P}''$ that $\hpa_i$ is the closest point to them among $\set{\hat{\pa}_1,\ldots,\hat{\pa}_k}$. \label{step:compute-clusters-kmeans-gaus}
\item Compute $(\hat{\mu}_i,\hat{\Sigma}_i) = \mathsf{A}'({\cal P}_i'')$.\label{step:priv-single-Gauss-est}
\item Let $\hat{n}_i \gets \size{{\cal P}_i''} + {\rm Lap}(4/\varepsilon)$.\label{step:Lap}
\end{enumerate}
\item For each $i \in [k]: \quad $ Set $\hat{w}_i = \frac{\hat{n}_i}{\sum_j \hat{n}_j}$.\label{step:wi}
\item Output $\hat{{\cal D}} = \set{(\hat{\mu}_1,\hat{\Sigma}_1,\hat{w}_1),\ldots,(\hat{\mu}_k,\hat{\Sigma}_k,\hat{w}_k)}$.
\end{enumerate}
\end{algorithm}
\subsubsection{Properties of $\mathsf{PrivatekGMM}$}
The following theorem summarizes the privacy guarantee of $\mathsf{PrivatekGMM}$.
As a running example, fix the following target parameters: An accuracy parameter $\eta > 0$, a confidence parameter $\beta > 0$, privacy parameters $\varepsilon,\delta \in (0,1)$ and bounding parameters $R,\sigma_{\max},\sigma_{\min} > 0$.
Also, think of the (non-private) labeling algorithm $\mathsf{A}$ as the one of \cite{AM05} that needs $s = \tilde{O}\paren{\frac{dk}{w_{\min}}}$ samples, think of the private (single) Gaussian learner $\mathsf{A}'$ as the $(\varepsilon/2,\delta/4)$-DP variant of the algorithm of \cite{KLSU19} that needs $\upsilon = \tilde{O}\paren{\frac{d^2}{\eta^2} + \paren{\frac{d^2}{\varepsilon \eta} + \frac{d^{3/2} \sqrt{\log \paren{\frac{\sigma_{\max}}{\sigma_{\min}}}} + \sqrt{d \log R}}{\varepsilon}} \cdot \sqrt{\log(1/\delta)}}$ samples,
and think of algorithm $\mathsf{B}$ as our averages-estimator $\mathsf{PrivatekAverages}$ (\cref{alg:FindAverages}) that needs $t =\tilde{O}\paren{\frac{d k \cdot \log^{2.5}(1/\delta)\paren{\sqrt{\log(1/\delta)} + \log\paren{\Lambda/r_{\min}}}}{\varepsilon^2}}$ tuples. The values of $r_{\min}$ and $\Lambda$ that we should use are determined by our utility guarantee (\cref{thm:kGauss-utility}). This means that for the oracle $\mathsf{B}$ we actually should wrap $\mathsf{PrivatekAverages}$ such that if there is a tuple that contains a point which is not in $B(\pt{0},\Lambda)$, then $\mathsf{B}$ replaces it with some arbitrary fixed tuple (e.g., the all-zero tuple $\pt{0}^k$)
We first state the privacy guarantee of $\mathsf{PrivatekGMM}$.
\begin{theorem}[Privacy of $\mathsf{PrivatekGMM}$]\label{thm:kGauss-privacy}
Let $\mathsf{A}$ be an (arbitrary, non-private) labeling algorithm,
let $\mathsf{A}'$ be an $(\varepsilon/4,\delta/2)$-DP algorithm, and let $\mathsf{B}$ be an $(\varepsilon,\delta)$-differentially private algorithm for databases ${\cal T} \in (({\mathbb R}^d)^k)^t$ (i.e., of size $t$).
Then for every $d,k,R,\sigma_{\max},\sigma_{\min},w_{\min} > 0$ and $\eta,\beta,\varepsilon,\delta, \gamma \in (0,1)$, algorithm $\mathsf{PrivatekGMM}^{\mathsf{A},\mathsf{A}',\mathsf{B}}(\cdot, k, s,t,\eta,\beta,\varepsilon, \delta,R, \sigma_{\max}, \sigma_{\min})$ is $(\varepsilon,\delta)$-differentially private for databases ${\cal P} \in \paren{{\mathbb R}^d}^*$.
\end{theorem}
\begin{proof}
Assume for simplicity (and without loss of generality) that the input algorithm $\mathsf{A}$ is deterministic, let ${\cal P},\tilde{{\cal P}} \in \paren{{\mathbb R}^d}^{2n}$ be two neigboring databases, and consider two executions $\mathsf{PrivatekGMM}({\cal P})$ and $\mathsf{PrivatekGMM}(\tilde{{\cal P}})$ (both with the same input parameters and oracles). Let ${\cal P}'$, ${\cal P}''$, ${\cal T}$ be the multisets from the execution $\mathsf{PrivatekGMM}({\cal P})$, and let $\tilde{{\cal P}}'$, $\tilde{{\cal P}}''$, $\tilde{{\cal T}}$ be the corresponding multisets in the execution $\mathsf{PrivatekGMM}(\tilde{{\cal P}})$.
If ${\cal P}'\neq \tilde{{\cal P}}'$ (i.e., neighboring), then assume w.l.o.g. that the two executions share the same randomness in Steps \ref{step:cM} (i.e., in the execution of $\mathsf{GenEmpiricalMeans}$), but use independent randomness in the execution of $\mathsf{B}$ in Step \ref{step:kAveragesOnM} and in the next steps of the algorithm. Therefore, ${\cal T}$ and ${\cal T}'$ differ by at most one $k$-tuple.
Hence, by the privacy guarantee of $\mathsf{B}$ along with group privacy (\cref{fact:group-priv}) we obtain that the resulting outcome $\set{\hpa_1,\ldots,\hpa_k}$ in Step~\ref{step:kAveragesOnM} of both executions is $(\varepsilon,\delta)$-indistinguishable. Since ${\cal P}' \neq \tilde{{\cal P}}'$ implies that ${\cal P}'' = \tilde{{\cal P}}''$, we conclude by post-processing (\cref{fact:post-processing}) that the final outcome $\hat{{\cal D}}$ is also $(\varepsilon,\delta)$-indistinguishable.
In the rest of the analysis we focus on the case that ${\cal P}' = \tilde{{\cal P}}'$ and ${\cal P}'' \neq \tilde{{\cal P}}''$ (i.e., neighboring). In this case,
we assume that both executions share the same randomness up to (and including) Step~\ref{step:kAveragesOnM}, and use independent randomness from \ref{step:Gaussian:forloop} till the end.
Therefore, the value of $\set{\hpa_1,\ldots,\hpa_k}$ in Step~\ref{step:kAveragesOnM} is identical in both executions. Let ${\cal P}''_1,\ldots,{\cal P}''_k$ be the multisets from Step~\ref{step:compute-clusters-kmeans-gaus} in the execution $\mathsf{PrivatekGMM}({\cal P})$, and let ${\tilde{{\cal P}}}''_1,\ldots,{\tilde{{\cal P}}}''_k$ be these multisets in the execution $\mathsf{PrivatekGMM}({\tilde{{\cal P}}})$.
Since ${\cal P}''$ and ${\tilde{{\cal P}}}''$ are neighboring, there exists at most two values $i,j \in [k]$ such that ${\cal P}''_i \neq {\tilde{{\cal P}}}_i''$ and ${\cal P}''_j \neq {\tilde{{\cal P}}}_j''$, and in both cases the multisets are neighboring (in the other indices the multisets are equal). By the properties of the private algorithm $\mathsf{A}'$ and basic composition (\cref{thm:composition1}), the vector $((\hat{\mu}_1,\hat{\Sigma}_1),\ldots,(\hat{\mu}_k,\hat{\Sigma}_k))$ computed in Step~\ref{step:priv-single-Gauss-est} of both executions is $(2\cdot \frac{\varepsilon}{4},2 \cdot \frac{\delta}{2})$-indistinguishable. Moreover, by the properties of the Laplace Mechanism along with basic composition, the vector $(\hat{n}_1,\ldots,\hat{n}_k)$ is $(2\cdot \frac{\varepsilon}{4}, 0)$-indistinguishable. By applying again basic composition we deduce that both vectors together are $(\varepsilon,\delta)$-indistinguishable, and therefore we conclude by post-processing (\cref{fact:post-processing}) that the resulting $\hat{{\cal D}}$ in both execution is $(\varepsilon,\delta)$-indistinguishable.
\end{proof}
The following theorem summarizes the utility guarantee of $\mathsf{PrivatekGMM}$.
\def\thmKGaussUtility{
Let $n,d,k,s,t,\upsilon \in {\mathbb N}$, $R,\sigma_{\max},\sigma_{\min},w_{\min}, \gamma > 0$, $\eta,\beta,\varepsilon,\delta \in (0,1)$, let $\Delta = 8 + 12/\gamma$, and let ${\cal D} = \set{(\mu_1,\Sigma_1,w_1), \ldots, (\mu_k,\Sigma_k,w_k)}$ be an $(R,\sigma_{\max},\sigma_{\min},w_{\min})$-bounded $(1+\gamma)h$-separated
mixture of $k$ Gaussians in ${\mathbb R}^d$, for $h \geq 2\sqrt{2\log\paren{8n/\beta}}$.
In addition, let $\mathsf{A}$ be a (non-private) $\paren{s,\frac{\beta}{8t}}$-labeling algorithm for ${\cal D}$ (\cref{def:LabelingAlg}), let $\mathsf{A}'$ be an $(\upsilon,\frac{\eta}2,\frac{\beta}{16k})$-learner for $(R,\sigma_{\max},\sigma_{\min})$-bounded Gaussians (\cref{def:LearnAlgSingleGaus}), and let $\mathsf{B}$ be an $(t,\text{ }\alpha=1,\text{ }r_{\min}=\frac{(1+\gamma) h}{\Delta}\cdot \sigma_{\min},\text{ } \beta/8,\text{ }\Delta,\text{ }\Lambda=R + \frac{(1+\gamma) h}{\Delta}\cdot \sigma_{\max})$-averages-estimator for $k$-tuple clustering (\cref{def:averages-estimator}).
Assume that
\begin{align*}
s \geq \frac{4}{w_{\min}}\cdot \max\biggl\{\log\paren{8 k t/\beta}, \quad \frac{\Delta^2\paren{d + 2 \log\paren{16 k t/\beta}}}{(1+\gamma^2) h^2} \biggr\},
\end{align*}
and that
\begin{align*}
n \geq \max \biggl\{s\cdot t, \quad \frac{2 \upsilon + \log(16k/\beta)}{w_{\min}}, \quad \frac{4k^2}{\varepsilon \eta} \cdot \log\paren{8k/\beta} \biggr\}
\end{align*}
Then with probability $1-\beta$, when sampling a database ${\cal P}$ of $2n$ i.i.d.\ samples from ${\cal D}$, the execution $\mathsf{PrivatekGMM}^{\mathsf{A},\mathsf{A}',\mathsf{B}}({\cal P},k,s,t,\varepsilon)$ outputs $\hat{{\cal D}}$ such that ${\rm d}_{\rm TV}({\cal D},\hat{{\cal D}}) \leq \eta$.
}
\begin{theorem}[Utility of $\mathsf{PrivatekGMM}$]\label{thm:kGauss-utility}
\thmKGaussUtility
\end{theorem}
The proof of the theorem appears at \cref{missing-proof:thm:kGauss-utility}. Very roughly,
the first term in the bound on $n$ is because $\mathsf{GenEmpiricalMeans}$ splits the $n$ samples into $t$ pieces, each contains $s$ samples.
The second and third terms in the bound on $n$ are the number of samples that are needed for guaranteeing that with probability $1-\frac{\beta}4$, for each $i\in [k]$, the resulting $(\hat{\mu}_i,\hat{\Sigma}_i)$ in Step~\ref{step:priv-single-Gauss-est} satisfy ${\rm d}_{\rm TV}({\cal N}(\hat{\mu}_i,\hat{\Sigma}_i), {\cal N}(\mu_i,\Sigma_i)) \leq \frac{\eta}2$ and the resulting $\hat{w}_i$ in Step~\ref{step:wi} satisfy $\size{\hat{w}_i - w_i} \leq \frac{\eta}{k}$, which yields that ${\rm d}_{\rm TV}(\hat{{\cal D}},{\cal D}) \leq \eta$ (see \cref{fact:dTV-of-mixtures}). We remark that regardless of the non-private algorithm $\mathsf{A}$ that we are using and its assumption on ${\cal D}$, we only require that ${\cal D}$ is more than $2\sqrt{2\log\paren{\frac{8n}{\beta}}}$-separated, which follows by the projection argument in \cref{prop:separation}.
\remove{
\begin{remark}
\Enote{Add running time analysis}
\end{remark}
}
\subsection{Remarks}\label{sec:Gaussians:remarks}
It is tempting to think that our approach, which relies on the algorithm $\mathsf{B} = \mathsf{PrivatekAverages}$ for aggregating the non-private findings by a reduction to $k$-tuple clustering, requires that the distance between the means should be proportional to $\sqrt{d}$, because this is the distance of the samples from their means. However, recall that $\mathsf{PrivatekGMM}$ do not set the $k$-tuple to be some arbitrary $k$ samples from different Gaussians. Rather, it sets it to the \emph{averages} of the samples in each set (See Step~\ref{step:compute-emp-mean} in \cref{alg:genBalancedSamples}), which decreases the distance from the actual means. In particular, when there are $O(d)$ samples in each such set, the dependency in $d$ is eliminated and the reduction to the $k$-tuple clustering follows (even when the distance between the means is much smaller than $\sqrt{d}$, as we consider).
Furthermore, note that our algorithm $\mathsf{PrivatekGMM}$ in Step~\ref{step:compute-clusters-kmeans-gaus} relies on the fact that the output $\set{\hat{\pa}_1,\ldots,\hat{\pa}_k}$ of $\mathsf{PrivatekAverages}$ separates correctly fresh samples from the mixture.
This might seem strange since even if $\set{\hat{\pa}_1,\ldots,\hat{\pa}_k}$ is very close to the actual means $\set{\mu_1,\ldots,\mu_k}$, the distance of each sample from its mean is proportional to $\sqrt{d}$, while the assumed separation between the means is independent of $d$. This yields that when $d$ is large, then the samples are much far from their means compared to the distance between the means. Namely,
if $\px$ is sampled from the $i$'th Gaussian and $\norm{\mu_i-\mu_j}$ is independent of $d$ (for large $d$), then $\norm{\px - \mu_i} \gg \norm{\mu_i - \mu_j}$. Yet, in our analysis we use a projection argument (see \cref{prop:separation}) which yields that w.h.p. it holds that $\norm{\px - \mu_i} < \norm{\px- \mu_j}$, even though $\norm{\px - \mu_i} \gg \norm{\mu_i - \mu_j}$.
\subsection{Comparison to the Main Algorithm of \cite{KSSU19}}\label{sec:Gauss:comparison}
The main private algorithm of \cite{KSSU19} mimics the approach of the (non-private) algorithm of \cite{AM05}, which is to use PCA to project the data into a low-dimensional space, and then clustering the data points in that low-dimensional space.
This projection enable both algorithms to learn mixtures that have the following separation
\begin{align}\label{eq:AM05-sep}
\forall i,j\colon \quad \norm{\mu_i-\mu_j} \geq C \paren{\sqrt{k \log(nk/\beta)} + \frac1{\sqrt{w_i}} + \frac1{\sqrt{w_j}}} \cdot \max\set{\sigma_i,\sigma_j},
\end{align}
for some constant $C > 0$ (albeit that the constant of \cite{KSSU19} is much larger, say $C=100$ instead of $C=4$ as in \cite{AM05}).
But while \cite{AM05} use a simple Kruskal-based clustering method,
\cite{KSSU19} developed alternative (and much more complicated) clustering methods that are more amenable to privacy. Finally, after the clustering phase, \cite{KSSU19} use a variant of the private algorithm of \cite{KLSU19} to learn the parameters of each Gaussian. Overall, the algorithm of \cite{KSSU19} learns an $(R,\sigma_{\max},\sigma_{\min},w_{\min})$-bounded mixture of Gaussian that is separated as in \cref{eq:AM05-sep}, with sample complexity
\begin{align*}
n \geq \paren{\frac{d^2}{\eta^2 w_{\min}} + \frac{d^2}{\varepsilon \eta w_{\min}} + \frac{{\rm poly}(k) d^{3/2}}{w_{\min} \varepsilon}} \cdot {\rm polylog}\paren{\frac{d k R \sigma_{\max}}{\eta \beta \varepsilon \delta \sigma_{\min}}}
\end{align*}
In the following, we compare between \cite{KSSU19}'s algorithm and ours (Algorithm $\mathsf{PrivatekGMM}$) in two different aspects: separation assumption and sample complexity.
\subsubsection{Separation Assumption}
The utility guarantee of $\mathsf{PrivatekGMM}$ (\cref{thm:kGauss-utility}) only requires a separation of slightly more than $h =2\sqrt{2 \log(8n/\beta)}$. Therefore, our algorithm can transform any non-private algorithm (in a modular way) that learns mixtures with separation $X$ into a private algorithm that learns with separation $\max\set{X, h}$. In particular, we can use \cite{AM05} as our non-private labeling algorithm $\mathsf{A}$ to learn mixtures with separation as in \cref{eq:AM05-sep} (with the small constant $C=4$), and we can also use any other non-private algorithm (like \cite{VW04,HSJ18,KPSJS18,diakonikolas18,RV17}) and inherent their separation assumption.
In contrast, the approach of the main algorithm of \cite{KSSU19} may only be extended to methods that use statistical properties of the data (like PCA), and not to other algorithmic machineries such as the sum-of-squares that are used for reducing the separation assumption.
\subsubsection{Sample Complexity}
The main algorithm of \cite{KSSU19} learns an $(R,\sigma_{\max},\sigma_{\min},w_{\min})$-bounded mixture of Gaussians that is separated as in \cref{eq:AM05-sep}, with sample complexity (roughly) $\tilde{O}\paren{\frac{\upsilon}{w_{\min}} + \frac{k^9 d^{3/2}}{w_{\min} \varepsilon}}$ (ignoring logarithmic factors), where $\upsilon = \upsilon(d, \varepsilon,\delta,\eta,\beta,R,\sigma_{\max},\sigma_{\min}) = \tilde{O}\paren{\frac{d^2}{\eta^2} + \frac{d^2}{\varepsilon \eta}}$ is the sample complexity of \cite{KLSU19} for learning the parameters of a single Gaussian (ignoring logarithmic factors in $R,\sigma_{\max}/\sigma_{\min},1/\delta,1/\beta$).
By \cref{thm:kGauss-utility},
the sample complexity of our algorithm is $\tilde{O}\paren{s \cdot t + \frac{t \cdot d}{w_{\min}} + \frac{\upsilon}{w_{\min}} + \frac{4k^2}{\varepsilon \eta}}$ (ignoring logarithmic factors), where $s$ is the sample complexity needed by the non-private algorithm $\mathsf{A}$ for labeling correctly the samples with confidence $\leq \frac{\beta}{8 t}$ (e.g., if we use the algorithm of \cite{AM05}, then $s = \tilde{O}\paren{\frac{dk}{w_{\min}}}$, and for simplifying the comparison, we assume that this is indeed the algorithm that we use). Since $t = \tilde{O}\paren{\frac{d k}{\varepsilon^2}}$, we obtain a sample complexity of (roughly) $\tilde{O}\paren{\frac{k^2 d^2}{\varepsilon^2 w_{\min}} + \frac{\upsilon}{w_{\min}} + \frac{4k^2}{\varepsilon \eta}}$, which might me larger than the one of \cite{KSSU19} if $d$ or $1/\varepsilon$ are very large (compared to $k$). Yet, we can easily improve the dependency in both $d$ and $\varepsilon$.
Using sub-sampling, we can execute Step~\ref{step:cM} of $\mathsf{PrivatekGMM}$ on an $\varepsilon n$-size random subset of ${\cal P}'$ (for the small desired $\varepsilon$), but now we only need a constant $\varepsilon$ for these steps. This immediately reduces the $1/\varepsilon^2$ in our sample complexity into $1/\varepsilon$.
In addition, as mentioned in \cref{sec:reducing-by-JL}, using the average algorithm of \cite{NSV16} in $\mathsf{PrivatekAverages}$ (instead of the average algorithm from \cref{prop:approx-aver-Rd}), we can reduce a factor of $\sqrt{d}$.
For summary, using sub-sampling and the algorithm of \cite{NSV16}, we obtain an improved sample complexity of $\tilde{O}\paren{\frac{k^2 d^{3/2}}{\varepsilon w_{\min}} + \frac{\upsilon}{w_{\min}} + \frac{4k^2}{\varepsilon \eta}}$, which strictly improves the sample complexity of \cite{KSSU19}.
\remove{
\subsubsection{Running-Time and Practicality}
\Enote{TBD. add running time analysis to all our algorithms, and check whether \cite{KSSU19} provide concrete running time analysis. Also, mention that our algorithm uses in a black-box way the the labeling algorithm and private learning algorithm of a single Gaussian. And also, that our algorithm can be easily parallelized}
}
\remove{
\Enote{Consider mentioning also that our algorithm is much more simpler, probabily more efficient and practical, and also can be easily parallelized.}
\Enote{
Consider adding the following discussion to the intro:
Can we use JL transform for learning a mixture of Gaussian? Suppose we have two spherical Gaussians ${\cal G}_1 = {\cal N}(\mu_1,\sigma^2 {\mathbb I}_{d \times d})$ and ${\cal G}_2 ={\cal N}(\mu_2,\sigma^2 {\mathbb I}_{d \times d})$ that we would like to separate, where $\norm{\mu_1 - \mu_2}$ is large enough (but independent of $d$). In the JL transform, we choose $k = \log n$ vectors $\py_1,\ldots,\py_k \sim {\cal N}(0,{\mathbb I}_{d \times d})$ and we transform each $\px \in {\mathbb R}^d$ into $(\rm ip{\px,\py_1},\ldots,\rm ip{\px,\py_k})$. Note that each $\rm ip{\px,\py_i}$ is distributed according to the $1$-dimensional Gaussian ${\cal N}(0,d(\sigma^2+1))$, which yields that if $\pX \sim {\cal N}(\mu,\sigma^2 {\mathbb I}_{d \times d})$ then $f(\pX) \sim {\cal N}(\mu,\sigma^2 (d+1) {\mathbb I}_{k \times k})$. So the problem is that the variance of the new Gaussian depends on $d$, while the new separation between $\mu_i$ and $\mu_j$ is independent of $d$ (only increases by a factor of $\sqrt{k}$)!
}
}
\subsection{Private $k$-Means under Separation Assumption}\label{sec:application}
In this section we show that our stability assumption holds with high probability when the multiset ${\cal P}$ is separated according to \citet{OstrovskyRSS12}.
Formally, a multiset of points ${\cal P}$ is \emph{$\phi$-separated} for $k$-means if ${\rm OPT}_k({\cal P}) \leq \phi^2 {\rm OPT}_{k-1}({\cal P})$.
In \cref{def:sep-ost} we strength this definition of \cite{OstrovskyRSS12} to include also an additive separating term $\xi$.
\begin{definition}[$(\phi,\xi)$-separated]\label{def:sep-ost}
A multiset ${\cal P} \in ({\mathbb R}^d)^*$ is $(\phi,\xi)$-separated for $k$-means if
\noindent ${\rm OPT}_k({\cal P}) + \xi \leq \phi^2\cdot {\rm OPT}_{k-1}({\cal P})$. Note that ${\cal P}$ is $\phi$-separated iff it is $(\phi,0)$-separated.
\end{definition}
We use the following theorem from \cite{OstrovskyRSS12} which states that when ${\cal P}$ is $\phi$-separated for $k$-means for sufficiently small $\phi$, then any set of $k$ centers that well approximate the $k$ means cost, must have the property that each of its centers is relatively close to an optimal center.
\begin{theorem}[\cite{OstrovskyRSS12}]\footnote{The statement of this theorem was taken from \cite{ShechnerSS20}.}\label{thm:Ostrovsky}
Let $\nu$ and $\phi$ be such that $\frac{\nu + \phi^2}{1-\phi^2} < \frac1{16}$. Suppose that ${\cal P} \in ({\mathbb R}^d)^*$ is $\phi$-separated for $k$-means. Let $C^* = \set{\pc_1^*,\ldots,\pc_k^*}$ be a set of \emph{optimal} centers for ${\cal P}$, and let $C= \set{\pc_1,\ldots,\pc_k}$ be centers such that ${\rm COST}_{{\cal P}}(C) \leq \nu \cdot {\rm OPT}_{k-1}({\cal P})$. Then for each $\pc_i$ there is a distinct optimal center, call it $\pc_i^*$, such that $\norm{\pc_i - \pc_i^*} \leq 2 \cdot \frac{\nu + \phi}{1 - \phi}\cdot D_i$, where $D_i = \min_{j \neq i}\norm{\pc_i^* - \pc_j^*}$.
\end{theorem}
The following lemma states that for suitable choices of $\phi$ and $\xi$, if ${\cal P}$ is $(\phi,\xi)$-separated for $k$-means, then with high probability, the event $E_{C^*}^{\gamma}$ over a random execution of $\mathsf{GenCenters}$ (\cref{def:event-ECgamma}) occurs, where $C^*$ is the optimal $k$-means for ${\cal P}$.
\begin{lemma}[Bounding the stability probability]\label{lem:bounding-stability-under-sep}
Let $n,d,k,s,t \in {\mathbb N}$, $\beta, \phi \in (0,1)$, $\gamma \in (0,1/16]$, be values such that $\frac{(1 + \omega) \phi^2}{1-\phi^2} < \frac1{16}$ and $\gamma \geq 2\cdot \frac{\omega \phi^2 + \phi}{1 - \phi}$. Let $\mathsf{A}$ be a (non-private) $\omega$-approximation algorithm for $k$-means, let ${\cal P} \in (B(\pt{0},\Lambda))^n$ and let $C^* = \set{\pc_1^*,\ldots,\pc_k^*} \in ({\mathbb R}^d)^k$ be the \emph{optimal} $k$-means for ${\cal P}$. Assume that ${\cal P}$ is $(\phi,\xi)$-separated for $k$-means, where $$\xi = \xi\paren{s, \beta/t} = \tilde{O}\paren{\Lambda^2 k d \log(n t/\beta) \cdot \frac{n}{s} + \Lambda \sqrt{k d \log(n t/\beta) \cdot \omega {\rm OPT}_{k}({\cal P}) \cdot \frac{n}{s}}}$$
is the function from \cref{prop:cost-of-sample-is-good}.
Then when executing $\mathsf{GenCenters}^{\mathsf{A}}({\cal P},k,s,t)$, the event $E_{C^*}^{\gamma}$ (\cref{def:event-ECgamma}) occurs with probability at least $1-\beta$.
\end{lemma}
\begin{proof}
For $j \in [t]$, let ${\cal S}_j$ and $\tilde{C}_j$ be the values in steps \ref{step:sample} and \ref{step:non-priv-centers} of $\mathsf{GenCenters}$ (respectively). Note that by
\cref{prop:cost-of-sample-is-good} and the union bound, with probability at least $1-\beta$ it holds that
\begin{align}\label{eq:small-cost}
\forall j \in [t]: \quad {\rm COST}_{{\cal P}}(\tilde{C}_j) \leq \omega\cdot {\rm OPT}_k({\cal P}) + \xi \leq \omega \phi^2 {\rm OPT}_{k-1}({\cal P}),
\end{align}
where the last inequality holds by the assumption that ${\cal P}$ is $(\phi,\xi)$-separated for $k$-means and that $\omega \geq 1$.
In the following, assume that (\ref{eq:small-cost}) occurs. Since ${\cal P}$ is (in particular) $\phi$-separated, and since the conditions of \cref{thm:Ostrovsky} hold with $\nu = \omega \phi^2$, we obtain from \cref{thm:Ostrovsky} that for
every $i \in [k]$ and $j \in [t]$, there exists $\tpc_i^j \in \tilde{C}_j$ such that $\norm{\pc_i^* - \tpc_i^j} \leq \gamma D_i$, meaning that event $E_{C^*}^{\gamma}$ occurs, as required.
\end{proof}
As a corollary of \cref{claim:kMeans-utility,lem:bounding-stability-under-sep}, we obtain our main application of algorithm $\mathsf{PrivatekMeans}$.
\begin{theorem}\label{cor:util-result}
Let $n,s,t,k,d \in {\mathbb N}$, $\varepsilon,\delta,\beta \in (0,1]$, $\Lambda > 0$, let $\phi, {\cal P}, \mathsf{A}, \omega, \gamma$ as in \cref{lem:bounding-stability-under-sep}, and let $\zeta, \mathsf{A}',\mathsf{B}$ as in \cref{claim:kMeans-utility}.
Then when executing $\mathsf{PrivatekMeans}^{\mathsf{A},\mathsf{A}',\mathsf{B}}({\cal P},s,t,k,\varepsilon,\delta,\beta,\gamma)$, with probability $1-2\beta$, the resulting centers $\hat{C} = \set{\hpc_1,\ldots,\hpc_k}$ satisfy
\begin{align*}
{\rm COST}_{{\cal P}}(\hat{C}) \leq (1 + 64\gamma) {\rm OPT}_{k}({\cal P}) + \zeta k (\zeta + 2\Lambda)
\end{align*}
\end{theorem}
\begin{proof}
The proof almost immediately holds by \cref{claim:kMeans-utility,lem:bounding-stability-under-sep} when applying them to the optimal $k$-means of ${\cal P}$, which we denote by $C^* = \set{\pc_1^*,\ldots,\pc_k^*}$. The only missing requirement is to show that $D^* := \min_{i \neq j} \norm{\pc_i^* - \pc_j^*} \geq 1/n$, as required by \cref{claim:kMeans-utility}. For proving this, note that on the one hand it holds that ${\rm OPT}_{k-1}({\cal P}) \leq D^* n + {\rm OPT}_{k}({\cal P})$, and on the other hand, since we assume that ${\cal P}$ is $(\phi,\xi)$-separated for $\phi \leq 1$ and $\xi \geq 1$ then it holds that ${\rm OPT}_k({\cal P}) + 1 \leq {\rm OPT}_{k-1}({\cal P})$. From the two inequalities we conclude that $D^* \geq 1/n$ and the corollary follows.
\end{proof}
We note that by the utility guarantee (\cref{claim:utility-kAverg}) of our $k$-tuple clustering $\mathsf{PrivatekAverages}$ (\cref{alg:FindAverages}), by choosing $\mathsf{B} = \mathsf{PrivatekAverages}(\cdot, \varepsilon/6, \delta/(4 e^{\varepsilon}), \beta/2, r_{\min}=\gamma/n)$ we obtain that $\mathsf{B}$ is an $\left(\frac{\varepsilon}{6},\frac{\delta}{4 e^{\varepsilon}}\right)$-differentially private $(t, \alpha=1, \: r_{\min}=\gamma/n, \: \beta/2, \: \Delta = 1/\gamma, \: \Lambda)$-averages-estimator for $k$-tuple clustering, where (ignoring ${\rm polylog}(n,d,k)$ factors)
\begin{align*}
t =\tilde{\Omega}\paren{\frac{d k \cdot \log^{2.5}(1/\delta)\paren{\sqrt{\log(1/\delta)} + \log\paren{\Lambda/r_{\min}}}}{\varepsilon^2}}.
\end{align*}
Hence, by taking $n = 2st$ we conclude that $\mathsf{PrivatekMeans}^{\mathsf{A}, \mathsf{A}', \mathsf{B}}(\cdot, k, s, t, \varepsilon,\delta,\beta,\gamma)$ is an $(\varepsilon,\delta)$-differentially private algorithm (\cref{claim:kMeans-privacy}) with the utility guarantee stated in \cref{cor:util-result}. In particular, when taking $n = 2st$ we obtain that our theorem holds for an additive error $\xi$ in the separation (see \cref{lem:bounding-stability-under-sep} for the definition of $\xi$), where (ignoring logarithmic factors)
\begin{align*}
\xi
= \tilde{\Omega}\paren{\Lambda^2 k d t + \Lambda \sqrt{k d t \omega \cdot {\rm OPT}_{k}({\cal P})}}
\end{align*}
\section{Additional Preliminaries}
\subsection{Additional Facts About Differential Privacy}
\subsubsection{The Exponential Mechanism}
We next describe the Exponential Mechanism of \citet{MT07}. Let ${\cal X}$ be a domain and ${\cal H}$ a set of solutions. Given a database ${\cal S} \in {\cal X}^*$, the Exponential Mechanism privately chooses a “good” solution $h$ out of the possible set of solutions ${\cal H}$. This “goodness” is quantified using a quality function
that matches solutions to scores.
\begin{definition}(Quality function)
A quality function is a function $q\colon {\cal X}^* \times {\cal H} \mapsto {\mathbb R}$ that maps a database ${\cal S} \in {\cal X}^*$ and a solution $h \in {\cal H}$ to a real number, identified as the score of the solution $h$ w.r.t the database ${\cal S}$.
\end{definition}
Given a quality function $q$ and a database ${\cal S}$, the goal is to chooses a solution $h$ approximately maximizing $q({\cal S},h)$. The Exponential Mechanism chooses a solution probabilistically, where the probability mass that is assigned to each solution $h$ increases exponentially with its quality $q({\cal S},h)$:
\begin{definition}(The Exponential Mechanism)\label{def:exp-mech}
Given input parameter $\varepsilon$, finite solution set ${\cal H}$, database ${\cal S} \in {\cal X}^m$, and a sensitivity $1$ quality function $q$, choose randomly $h \in {\cal H}$ with probability proportional to $\exp(\varepsilon\cdot q({\cal S},h)/2)$.
\end{definition}
\begin{proposition}(Properties of the Exponential Mechanism)\label{prop:exp-mech}
(i) The Exponential Mechanism is $\varepsilon$-differentially private. (ii) Let $\hat{e} := \max_{f \in {\cal H}}\set{q({\cal S},f)}$ and $\Delta > 0$. The Exponential Mechanism outputs a solution $h$ such that $q({\cal S},h) \leq \hat{e} - \Delta$ with probability at most $\size{{\cal H}}\cdot \exp\paren{-\varepsilon\Delta/2}$.
\end{proposition}
\subsubsection{Private Interior Point and Bounding Segment in ${\mathbb R}$}\label{sec:interior-point}
\begin{proposition}[Finding an Interior Point in ${\mathbb R}$]\label{prop:interior-point}
Let $\varepsilon \in (0,1)$, $\Lambda > 0$ and $g \in [0,\Lambda]$. There exists an efficient $\varepsilon$-differentially private algorithm that takes an $n$-size database ${\cal S}$ of numbers in the segment $[-\Lambda,\Lambda]$ and outputs a number $z \in [-\Lambda,\Lambda]$ that with probability $1 - 2(\Lambda/g + 1) \cdot \exp\paren{-\varepsilon n/4}$ it holds that $z \in [\min({\cal S}) - g, \max({\cal S}) + g]$. The algorithm runs in time $\tilde{O}\paren{n}$ (ignoring $\log\paren{\frac{n \Delta}{g}}$ factors).
\end{proposition}
\begin{proof}
Define the grid $G = \set{-\Lambda, -\Lambda+g, \ldots, -\Lambda + \ceil{\frac{2\Lambda}{g}}\cdot g}$, and for every $x \in G$ let ${\rm left}(x) = - \Lambda + \floor{\frac{x + \Lambda}{g}} \cdot g$ (i.e., the closest grid point to $x$ from the left side) and ${\rm right}(x) = - \Lambda + \ceil{\frac{x + \Lambda}{g}} \cdot g$ (i.e., the closest grid point to $x$ from the right side). Now, apply the exponential mechanism (\ref{def:exp-mech}) with the quality function
\begin{align*}
\forall y \in G\colon \quad q({\cal S},y) = \min\set{\size{\set{x \in {\cal S} \colon {\rm left}(x) \leq y}}, \size{\set{x \in {\cal S} \colon {\rm right}(x) \geq y}}}
\end{align*}
For the utility analysis, let $m$ be the median of ${\cal S}$, and note that $q({\cal S},{\rm left}(m)), q({\cal S},{\rm right}(m)) \geq n/2$.
Therefore, by \cref{prop:exp-mech}, with probability $\geq 1 - \size{G}\cdot \exp\paren{-\varepsilon n/4} \geq 1 - 2(\Lambda/g + 1) \cdot \exp\paren{-\varepsilon n/4}$, the mechanism outputs a point $z$ with $q(S,z) > 0$, which yields in particular that $z \in [\min({\cal S}) - g, \max({\cal S}) + g]$.
For the running time analysis, we implement the sampling as follows: For $x \in {\cal S}$ we let $A_x = \set{{\rm left}(x)-g,{\rm left}(x),{\rm right}(x), {\rm right}(x) + g}$, and let $A = \cup_{x \in {\cal S}} A_x$. Note that for every consecutive grid points $y, y'=y+g \in G$ with $q({\cal S}, y) \neq q({\cal S}, y')$, it holds that $y,y' \in A$: If $q({\cal S}, y) > q({\cal S}, y')$, there must exist $x \in {\cal S}$ such that $x \in (y-g,y]$, yielding that $y \in [x,x+g) \implies y = {\rm right}(x), y' = {\rm right}(x) + g$. Otherwise (i.e., $q({\cal S}, y) < q({\cal S}, y')$), there must exist $x \in {\cal S}$ such that $x \in [y',y'+g)$, yielding that $y' \in (x-g,x] \implies y' = {\rm left}(x), y = {\rm left}(x) - g$.
Then, we sort $A$ in time $\tilde{O}(n)$, and let $a_1 \leq \ldots \leq a_m$ be the sorted elements in $A$ (recall that $m = \size{A} \leq 4n$). For each $i \in [m+1]$, we compute $w({\cal S},a_i) = q({\cal S},a_i) \cdot \size{G \cap (a_{i-1},a_i]}$ (i.e., $w({\cal S},a_i)$ is the the original quality of $a_i$ times the number of grid points in $(a_{i-1},a_i]$, where $a_0 = -\Lambda-g$ and $a_{m+1} = \Lambda+g$), and choose a value $a_i$ with probability $\propto w({\cal S},a_i)$. Note that the computation of each $w({\cal S},a_i)$ can be done in time $\tilde{O}(1)$ using simple binary searches over the (sorted) multisets ${\cal S}_{{\rm left}} = \cup_{x \in {\cal S}} \set{{\rm left}(x)}$ and ${\cal S}_{{\rm right}} = \cup_{x \in {\cal S}} \set{{\rm right}(x)}$ (a ``multiset'' union, that includes duplications).
Finally, given the chosen $a_i$ from the mechanism, it is left to sample a uniform point in $G \cap (a_{i-1},a_i]$ (since we know, by the property of $A$, that all the point there have the same value of $q({\cal S},\cdot)$). This can be easily implemented in time $O(\log \size{G}) = \tilde{O}(1)$.
\end{proof}
\begin{proposition}[Finding a Bounding Segment of Points in ${\mathbb R}$]\label{prop:bounding-seg}
Let $\beta, \varepsilon \in (0,1)$, $\Lambda > 0$ and $g \in [0,\Lambda]$. There exists an efficient $\varepsilon$-differentially private algorithm that takes an $n$-size database ${\cal S}$ of numbers in the segment $[-\Lambda,\Lambda]$ and outputs a segment $[x,y]$ such that with probability at least $1-\beta$ the following holds:
\begin{itemize}
\item $\size{{\cal S} \cap [x,y]} \geq n - \frac{8}{\varepsilon} \log \paren{\frac{4\Lambda}{g \beta}} - 2$ (i.e., the segment contain most of the points in ${\cal S}$), and
\item $y - x \leq \max({\cal S})-\min({\cal S})+ 4g$.
\end{itemize}
The algorithm runs in time $\tilde{O}\paren{n}$ (ignoring $\log\paren{\frac{n \Delta}{\varepsilon \beta g}}$ factors).
\end{proposition}
\begin{proof}
In the following assume that $n \geq \frac{8}{\varepsilon} \log \paren{\frac{4\Lambda}{g \beta}} + 2$ (otherwise the proof trivially holds for any segment $[x,x]$). Let ${\cal S}_0$ be the smallest $\frac{4}{\varepsilon} \log \paren{\frac{4\Lambda}{g \beta}} + 1$ points in ${\cal S}$, and let ${\cal S}_1$ be the largest $\frac{4}{\varepsilon} \log \paren{\frac{4\Lambda}{g \beta}} + 1$ points in ${\cal S}$. For each $b \in \set{0,1}$ apply \cref{prop:interior-point} (interior point) on ${\cal S}_b$ for finding a number $z_b \in [-\Lambda,\Lambda]$ that belongs to $[\min({\cal S}_{b}) - g, \max({\cal S}_b) + g]$ with probability at least $1 - 2(\Lambda/g + 1) \cdot \exp\paren{-\varepsilon \size{{\cal S}_b}/4} \geq 1-\beta/2$. Therefore, by setting $x = z_0-g$ and $y = z_1+g$ we get that with probability $1-\beta$ it holds that: (1) $[\max({\cal S}_0), \min({\cal S}_1)] \subseteq [x,y]$ and that (2) $[x,y] \subseteq [\min({\cal S}_0)-2g, \max({\cal S}_1)+2g] = [\min({\cal S})-2g, \max({\cal S})+2g]$. By (1) we get that all points in ${\cal S}$ expect (at most) $(\size{{\cal S}_0} - 1) + (\size{{\cal S}_1} - 1) \leq \frac{8}{\varepsilon} \log \paren{\frac{4\Lambda}{g \beta}}$ are inside $[x,y]$, and by (2) we get that $y-x \geq \max({\cal S})-\min({\cal S}) + 4g$, as required.
For the running time analysis, note that by sorting ${\cal S}$ we can determine ${\cal S}_0$ and ${\cal S}_1$ in time $\tilde{O}(n)$, and the cost of executing the algorithm from \cref{prop:interior-point} on each ${\cal S}_b$ is $\tilde{O}\paren{\frac1{\varepsilon}} = \tilde{O}\paren{n}$.
\end{proof}
\subsubsection{Estimating the Average of Points}\label{sec:approx-aver}
\begin{proposition}[Estimating the Average of Bounded Points in ${\mathbb R}$]\label{prop:approx-aver-R}
Let $\beta, \varepsilon,\delta \in (0,1)$, $\Lambda > 0$ and $r_{\min} \in [0,\Lambda]$. There exists an efficient $(\varepsilon,\delta)$-differentially private algorithm that takes an $n$-size database ${\cal S}$ of numbers in the segment $[-\Lambda,\Lambda]$ and satisfy the following utility guarantee: If $n \geq \frac{16}{\varepsilon} \log \paren{\frac{4\Lambda}{r_{\min} \beta}} +4$, then with probability $1-\beta$, the algorithm outputs a number $\hat{a} \in {\mathbb R}$ such that
\begin{align*}
\size{\hat{a} - {\rm Avg}({\cal S})} \leq O\paren{\frac{\max\set{r,r_{\min}}}{\varepsilon n} \paren{\sqrt{\log(1/\delta) \log(1/\beta)} + \log \paren{\frac{\Lambda}{r_{\min} \beta}}}},
\end{align*}
where $r = \max({\cal S}) - \min({\cal S})$.
The algorithm runs in time $\tilde{O}(n)$ (ignoring $\log\paren{\frac{n \Delta}{r_{\min} \varepsilon \beta}}$ factors).
\end{proposition}
\begin{proof}
The algorithm does the following: (1) Privately find a bounding segment $[x,y]$ using \cref{prop:bounding-seg} with parameters $\beta/2,\varepsilon/2,g = r_{\min},\Lambda$, let $\hat{r} = y-x$ and let ${\cal S}' = {\cal S} \cap [x,y]$ (2) Use the ($1$-dimensional) Gaussian mechanism (\cref{fact:Gaus}) with $\lambda = \frac{\hat{r}}{\size{{\cal S}'}}$ and parameters $\beta/2,\varepsilon/2, \delta$ for computing a noisy average $\hat{a}$ of ${\cal S}'$ (see \cref{obs:Gaus-aver}). By the properties of the Gaussian mechanism (see \cref{remark:Gaus-add-del}) along with basic composition it holds that the above algorithm is $(\varepsilon,\delta)$-differentially private. For the utility analysis, note that with probability $1-\beta$, the segment $[x,y]$ satisfies the conditions of \cref{prop:bounding-seg} and the noise added to the average in the second step is at most $O\paren{\frac{\hat{r}}{\varepsilon \size{{\cal S}'}} \sqrt{\log(1/\delta) \log(1/\beta)}}$. In the rest of the analysis we assume that this event occurs. Now, by definition of $r$, it holds that
\begin{align*}
\size{{\rm Avg}(S) - {\rm Avg}({\cal S}')} \leq \frac{r \size{{\cal S} \setminus {\cal S}'}}{n} \leq \frac{8 r}{\varepsilon n} \log \paren{\frac{2\Lambda}{r_{\min} \beta}} +\frac{2 r}{n}
\end{align*}
Moreover, it holds that
\begin{align*}
\size{\hat{a} - {\rm Avg}({\cal S}')} \leq O\paren{\frac{\hat{r}}{\varepsilon \size{{\cal S}'}} \sqrt{\log(1/\delta) \log(1/\beta)}} \leq
O\paren{\frac{\max\set{r,r_{\min}}}{\varepsilon n} \sqrt{\log(1/\delta) \log(1/\beta)}},
\end{align*}
where the second inequality holds since $\hat{r} \leq r + 4r_{\min}$ and $\size{{\cal S}'} \geq n/2$ by the assumption on $n$. The proof now follow by the above two inequalities.
For the running time analysis, step (1) takes $\tilde{O}(n)$ time (\cref{prop:bounding-seg}). Step (2) that executes the Gaussian Mechanism, takes $\tilde{O}(n)$ time for computing the average, and $\tilde{O}(1)$ for sampling a number from a single one-dimensional.
\end{proof}
\begin{proposition}[Estimating the Average of Bounded Points in ${\mathbb R}^d$ (Restatement of \cref{prop:approx-aver-Rd})]
\propEstAvgInRd
\end{proposition}
\begin{proof}
The algorithm does the following: For each $i \in [d]$, let ${\cal S}_i = \set{x_i \colon (x_1,\ldots,x_d) \in {\cal S}}$ and compute an estimation $\hat{a}_i$ of ${\rm Avg}({\cal S}_i)$ (in time $\tilde{O}(n)$) by applying \cref{prop:approx-aver-R} with parameters $r_{\min},\Lambda$, $\tilde{\varepsilon} = \frac{\varepsilon}{2\sqrt{2d \log(2/\delta)}}$, $\tilde{\delta} = \frac{\delta}{d}$, $\tilde{\beta} = \frac{\beta}{d}$. Finally, output $\hpa = (\hat{a}_1,\ldots, \hat{a}_d)$. It is clear by advanced composition (\cref{thm:composition2}) that the algorithm is $(\varepsilon,\delta)$-differentially private. For the utility guarantee, note that with probability at least $1-\beta$, for every $i \in [d]$ it holds that
\begin{align*}
\size{\hat{a}_i - {\rm Avg}({\cal S}_i)}
&\leq O\paren{\frac{r}{\tilde{\varepsilon} n} \paren{\sqrt{\log(1/\tilde{\delta}) \log(1/\tilde{\beta})} + \log \paren{\frac{\Lambda}{r_{\min} \tilde{\beta}}}}}\\
&= O\paren{\frac{r \sqrt{d \log(1/\delta)}}{\varepsilon n} \paren{\sqrt{\log(d/\delta) \log(d/\beta)} + \log \paren{\frac{\Lambda d}{r_{\min} \beta}}}},
\end{align*}
and hence
\begin{align*}
\norm{\hpa - {\rm Avg}({\cal S})}
&= \sqrt{\sum_{i=1}^d (\hat{a}_i - {\rm Avg}({\cal S}_i))^2}\\
&\leq O\paren{\frac{r d \sqrt{\log(1/\delta)}}{\varepsilon n} \paren{\sqrt{\log(d/\delta) \log(d/\beta)} + \log \paren{\frac{\Lambda d}{r_{\min} \beta}}}}
\end{align*}
\end{proof}
\begin{remark}\label{remark:bound-aver-add-del}
The above two algorithms guarantee differential-privacy whenever two neighboring databases have equal size. However, they can be easily extended to a more general case in which the privacy guarantee also holds in cases of addition and deletion of a point, by extending the Gaussian mechanism used in \cref{prop:approx-aver-R} (see \cref{remark:Gaus-add-del}) with essentially the same noise magnitude.
\end{remark}
\section{$k$-Tuples Clustering}
We first introduce a new property of a collection of (unordered) $k$-tuples \Hnote{k-sets (shorter and more accurate) \Enote{Do we really want to change ``tuples'' to ``sets''?}} $\set{\px_1,\ldots,\px_k} \in ({\mathbb R}^d)^k$, which we call \emph{partitioned by $\Delta$-far balls}.
\begin{definition}[$\Delta$-far balls]\label{def:far-balls}
A set of $k$ balls ${\cal B} = \set{B_i = B(\pc_i,r_i)}_{i=1}^k$ over ${\mathbb R}^d$ is called \textbf{$\Delta$-far balls}, if for every $i \in [k]$ it holds that $\norm{\pc_i - \pc_j} \geq \Delta \cdot \max\set{r_i,r_j}$ (i.e., the balls are relatively far from each other).
\end{definition}
\begin{definition}[partitioned by $\Delta$-far balls]\label{def:sep-balls}
\Hnote{add picture}
A $k$-tuple $X \in ({\mathbb R}^d)^k$ is partitioned by a given set of $k$ $\Delta$-far balls ${\cal B} = \set{B_1,\ldots,B_k}$, if for every $i \in [k]$ it holds that $\size{X \cap B_i} = 1$. A multiset of $k$-tuples ${\cal T} \in (({\mathbb R}^d)^k)^*$ is \textbf{partitioned by} ${\cal B}$, if each $X \in {\cal T}$ is partitioned by ${\cal B}$. We say that ${\cal T}$ is \textbf{partitioned by $\Delta$-far balls} if such a set ${\cal B}$ of $k$ $\Delta$-far balls exists.
\end{definition}
In some cases we want to use a notion of a multiset ${\cal T}$ \emph{almost} partitioned by $\Delta$-far balls. This is defined below using the additional parameter $\ell$.
\begin{definition}[$\ell$-nearly partitioned by $\Delta$-far balls]
A multiset ${\cal T} \in (({\mathbb R}^d)^k)^*$ is \textbf{$\ell$-nearly partitioned by} a given set of $\Delta$-far balls ${\cal B} = \set{B_1,\ldots,B_k}$, if there are at most $\ell$ tuples in ${\cal T}$ that are not partitioned by ${\cal B}$. We say that ${\cal T}$ is \textbf{$\ell$-nearly partitioned by $\Delta$-far balls} if such a set of $\Delta$-far balls ${\cal B} = \set{B_1,\ldots,B_k}$ exists.
\end{definition}
For a database of $n$ $k$-tuples ${\cal T} \in (({\mathbb R}^d)^k)^*$, we let ${\rm Points}({\cal T})$ be the collection of all the points in all the $k$-tuples in ${\cal T}$.
\begin{definition}[The points in a collection of $k$-tuples]
For ${\cal T} = \set{\set{\px_{1,j}}_{j=1}^k,\ldots,\set{\px_{n,j}}_{j=1}^k} \in (({\mathbb R}^d)^k)^n$, we define ${\rm Points}({\cal T}) = \set{\px_{i,j}}_{i\in [n], j \in [k]} \in ({\mathbb R}^d)^{kn}$.
\end{definition}
The following proposition states that if ${\cal T}$ is partitioned by $\Delta$-far balls for $\Delta > 3$, then each choice of $\Delta$-far balls that partitions ${\cal T}$ induces the same partition.
\begin{proposition}\label{prop:unique-partition}
Let ${\cal T} \in (({\mathbb R}^d)^k)^*$ be a multiset that is partitioned by a set of $\Delta$-far balls ${\cal B} = \set{B_1,\ldots,B_k}$ for $\Delta > 3$. Then for every $k$-tuple $X = \set{\px_1,\ldots,\px_k} \in {\cal T}$ and for every $i \in [k]$, there exists a ball in ${\cal B}$ (call it $B_i$), such that ${\rm Points}({\cal T}) \cap B_i = \set{\px \in {\rm Points}({\cal T}) \colon i = \operatorname*{argmin}_{j \in [k]} \norm{\px - \px_j}}$.
\end{proposition}
\begin{proof}
Let $X = \set{\px_1,\ldots,\px_k} \in {\cal T}$, and for every $i \in [k]$ let $B_i = B(\pc_i,r_i) \in {\cal B}$ be the ball that contains $\px_i$. We prove the proposition by showing that for every $i$ and every $\px \in {\rm Points}({\cal T}) \cap B_i$, it holds that $i = \operatorname*{argmin}_{j \in [k]} \norm{\px - \px_j}$.
In the following, fix $\px \in {\rm Points}({\cal T}) \cap B_i$. On the one hand, since $\px \in B_i$, it holds that $\norm{\px - \px_i} \leq r_i$. On the other hand, for any $j \neq i$ it holds that
\begin{align*}
\norm{\px - \px_j} \geq \norm{\pc_i - \pc_j} - \norm{\pc_i - \px} - \norm{\pc_j - \px_j} > 3\max\set{r_i,r_j} - r_i - r_j \geq r_i,
\end{align*}
where the strict inequality holds since $B_i,B_j$ are $\Delta$-far balls for $\Delta > 3$. Namely, we deduce that $\norm{\px - \px_i} < \norm{\px - \px_j}$, as required.
\end{proof}
\remove{
\ECnote{Perhaps there is even a simpler proof with $k$-tuples? Isn't it the case that in any partition to far balls, each point from one tuple is in the same ball with the closest point to it from each other tuple? Also, for this purpose it might be enough to use "4" instead of "6" in the far balls def (not sure what we need later).}
\begin{proof}
Fix a $k$-tuple $X = \set{\px_1,\ldots,\px_k} \in {\cal T}$, and let ${\cal P}_i = \set{\px \in {\rm Points}({\cal T}) \colon i = \operatorname*{argmin}_{j}\norm{\px-\px_j}}$.
The proof trivially holds for $k=1$. Therefore, in the following we assume that $k \geq 2$.
Assume towards a contradiction that there exist $\px,\py \in {\rm Points}({\cal T})$ such that $\px, \py \in B_i = B(\pc_i,r_i)$ (i.e., belong to the same ball in ${\cal B}$) but $\px \in B_s' = B(\pc_s',r_s')$ and $\py \in B_{t}' = B(\pc_t',r_t')$ for $s \neq t$ (i.e., belong to different balls in ${\cal B}'$). We next fix $\pz \in \paren{B_{s}' \cup B_{t}'} \cap {\rm Points}({\cal T})$ and prove that $\pz \in B_i$ (see \cref{fig1}).
\begin{figure}[htbp]
\centerline{\includegraphics[scale=.2]{figure1.png}}
\caption{An example of $\px,\py,\pz \in {\cal P}$ such that $\px, \py \in B_i$, $\px \in B_s'$, $\py \in B_{t}'$ and $\pz \in B_{s}' \cup B_{t}'$.}
\label{fig1}
\end{figure}
\noindent Since both ${\cal B}$ and ${\cal B}'$ are sets of far balls, the assumptions yield that
\begin{align*}
4 \max\set{r_s',r_t'} < \norm{\pc_s' - \pc_t'} - \norm{\px - \pc_s'} - \norm{\py - \pc_t'} < \norm{\px - \py} \leq 2 r_i.
\end{align*}
Therefore, it holds that
\begin{align}\label{eq:pz-px-py}
\min\set{\norm{\pz - \px}, \norm{\pz - \py}} \leq 2\max\set{r_s',r_t'} < r_i,
\end{align}
which yields that
\begin{align}\label{eq:pz-pc_i}
\norm{\pz - \pc_i}
&\leq \min\set{\norm{\pz - \px} + \norm{\px - \pc_i}, \norm{\pz - \py} + \norm{\py - \pc_i}}\nonumber\\
&\leq \min\set{\norm{\pz - \px}, \norm{\pz - \py}} + r_i\nonumber\\
&< 2r_i,
\end{align}
where the second inequality holds since $\px,\py \in B_i$ and the last one by \cref{eq:pz-px-py}. Note that \cref{eq:pz-pc_i} immediately yields that $\pz \in B_i \cap {\rm Points}({\cal T})$ since ${\cal T}$ is partitioned by ${\cal B}$, and therefore, $\pz$ cannot belong to a different ball in ${\cal B}$ since it is too close to $\pc_i$.
In summary, we proved that
\begin{align}\label{eq:contain-two-balls}
{\rm Points}({\cal T}) \cap \paren{B_{s}' \cup B_{t}'} \subseteq {\rm Points}({\cal T}) \cap B_i.
\end{align}
Now fix some $k$-tuple $\set{\px_1,\ldots,\px_k} \in {\cal T}$. Since ${\cal B}'$ partitions ${\cal T}$, there exists a permutation $\pi$ of $[k]$ such that $\px_{\pi(s)} \in B_{s}'$ and $\px_{\pi(t)} \in B_{t}'$. Therefore, we deduce by \cref{eq:contain-two-balls} that both $\px_{\pi(s)}$ and $\px_{\pi(t)}$ belong to the same ball $B_i$ in ${\cal B}$, in contradiction to the assumption that ${\cal B}$ partitions ${\cal T}$.
\end{proof}
}
We now formally define the partition of a database ${\cal T} \in (({\mathbb R}^d)^k)^*$ which is partitioned by $\Delta$-far balls for $\Delta > 3$.
\begin{definition}[${\rm Partition}({\cal T})$]\label{def:clusters-rel}
Given a multiset ${\cal T} \in (({\mathbb R}^d)^k)^*$ which is partitioned by $\Delta$-far balls for $\Delta > 3$,
we define the partition of ${\cal T}$, which we denote by ${\rm Partition}({\cal T}) = \set{{\cal P}_1,\ldots,{\cal P}_k}$, by fixing an (arbitrary) $k$-tuple $X = \set{\px_1,\ldots,\px_k} \in {\cal T}$ and setting ${\cal P}_i = \set{\px \in {\rm Points}({\cal T}) \colon i = \operatorname*{argmin}_{j \in [k]} \norm{\px - \px_j}}$.
\end{definition}
By \cref{prop:unique-partition}, this partition is well defined (i.e., is independent of the choice of the $k$-tuple $X$).
We now define the $k$-tuple clustering problem.
\begin{definition}[$k$-tuple clustering]\label{def:ktupleclustering}
The input to the problem is a database ${\cal T} \in (({\mathbb R}^d)^k)^n$ and a parameter $\Delta > 3$. The goal is to output a $k$-tuple $Y = \set{\py_1,\ldots,\py_k} \in ({\mathbb R}^d)^k$ such that the following holds: If ${\cal T}$ is partitioned by $\Delta$-far balls, then for every $i \in [k]$, there exists a cluster in ${\rm Partition}({\cal T})$ (call it ${\cal P}_i$) such that ${\cal P}_i = \set{\px \in {\rm Points}({\cal T}) \colon i = \operatorname*{argmin}_{j \in [k]} \norm{\px - \py_j}}$.
\end{definition}
Namely, in the $k$-tuple clustering problem, the goal is to output a $k$-tuple $Y$ that partitions ${\cal T}$ correctly.
We remark that without privacy, the problem is completely trivial, since any $k$-tuple $X \in {\cal T}$ is a good solution by definition.
We also remark that for applications, we are also interested in the quality of the solution. Namely, how small is the distance between $\py_i$ and ${\cal P}_i$, compared to the other clusters in ${\rm Partition}({\cal T})$. This is captured in the following definition.
\begin{definition}[good and good-averages solutions]\label{def:gamma-good}
\Hnote{Better to separate the "goodness" and the requirement that all radii are at least some minimum value}\Enote{I'm not sure it's better}
Let ${\cal T} \in (({\mathbb R}^d)^k)^n$ and $\alpha, r_{\min} \geq 0$. We say that a $k$-tuple $Y = \set{\py_1,\ldots,\py_k} \in ({\mathbb R}^d)^k$ is an \emph{$(\alpha,r_{\min})$-good solution} for clustering ${\cal T}$, if there exists a set of $\Delta$-far balls (for $\Delta > 3$) ${\cal B} = \set{B_i = B(\pc_i,r_i)}_{i=1}^k$ that partitions ${\cal T}$ such that for every $i \in [k]$ it holds that
\begin{align*}
\norm{\py_i - \pc_i} \leq \alpha \cdot \max\set{r_i,r_{\min}}
\end{align*}
If such ${\cal B}$ exists with $r_i \geq r_{\min}$ for every $i \in [k]$, we say that $Y$ is an \emph{$\alpha$-good-average} solution.\Hnote{How does this relate to average ?
it is also still related to $r_{\min}$}
In a special case where such set of balls exists for $\pc_i = {\rm Avg}({\cal P}_i)$ where $\set{{\cal P}_1,\ldots,{\cal P}_k} = {\rm Partition}({\cal P})$, we say that $Y$ is an \emph{$(\alpha,r_{\min})$-good-averages} solution
\end{definition}
We remark that the additional parameter $r_{\min}$ is usually needed when considering differentially private algorithms, even for simpler problems like estimating the average of points (e.g., see \cref{prop:approx-aver-Rd}).
In addition, note that the quality of the solution is measured by how small is $\alpha$.
The following claim states that if ${\cal T}$ is partitioned by $\Delta$-far balls for $\Delta > 3$, then any $\alpha$-good solution according to \cref{def:gamma-good} for $\alpha < \Delta/2 - 1$ is also a $k$-tuples clustering solution according to \cref{def:ktupleclustering}.
\begin{claim}\label{claim:good-sol-is-valid}
If ${\cal T} \in (({\mathbb R}^d)^k)^n$ is partitioned by $\Delta$-far balls for $\Delta > 3$, and $Y \in ({\mathbb R}^d)^k$ is an $\alpha$-good solution for clustering ${\cal T}$ for $\alpha < \Delta/2 - 1$, then for every $i \in [k]$, there exists a cluster in ${\rm Partition}({\cal T})$ (call it ${\cal P}_i$) such that ${\cal P}_i = \set{\px \in {\rm Points}({\cal T}) \colon i = \operatorname*{argmin}_{j \in [k]} \norm{\px - \py_j}}$.
\end{claim}
\begin{proof}
Since $Y$ is an $\alpha$-good solution, there exists a set of $\Delta$-far balls ${\cal B} = \set{B_i = B(\pc_i,r_i)}_{i=1}^k$ such that for every $i \in [k]$ it holds that
\begin{align*}
\norm{\py_i - \pc_i} \leq \alpha \cdot r_i.
\end{align*}
Now denote by ${\cal P}_i$ the cluster ${\rm Points}({\cal T}) \cap B_i \in {\rm Partition}({\cal T})$.
It remains to prove that for every $\px \in {\cal P}_i$ and $j \neq i$ it holds that $\norm{\px - \py_i} < \norm{\px - \py_j}$. In the following, fix such $\px$ and $j$.
On the one hand, it holds that
\begin{align*}
\norm{\px - \py_i} \leq \norm{\px - \pc_i} + \norm{\py_i - \pc_i}
\leq (1 + \alpha) r_i.
\end{align*}
On the other hand, we have
\begin{align*}
\norm{\px - \py_j} \geq \norm{\pc_i - \pc_j} - \norm{\px-\pc_i} - \norm{\py_j - \pc_j}
\geq \Delta \cdot \max\set{r_i,r_j} - r_i - \alpha \cdot r_j
\geq (\Delta - 1 - \alpha) r_i.
\end{align*}
Hence, we conclude that $\norm{\px - \py_i} < \norm{\px - \py_j}$ whenever $\alpha < \Delta/2 - 1$, as required.
\end{proof}
For applications, we focus on a specific type of algorithms for the $k$-tuple clustering problems, that outputs a good-averages solution.
\begin{definition}[averages-estimator for $k$-tuple clustering]\label{def:averages-estimator}
Let $\mathsf{A}$ be an algorithm that gets as input a database in $(({\mathbb R}^d)^k)^*$. We say that $\mathsf{A}$ is an \emph{$(n,\alpha,r_{\min},\beta,\Delta,\Lambda)$-averages-estimator\Hnote{I do not think these carrying all these parameters all the time is good.
You can fix n to denote the size of the dataset and the diameter of the domain
maybe even $\Delta$. Do it once and don't carry them in the notation if the are fixed throughout} for $k$-tuple clustering}, if for every ${\cal T} \in
(B(\pt{0},\Lambda)^k)^n \subseteq (({\mathbb R}^d)^k)^n$ that is partitioned by $\Delta$-far balls, $\mathsf{A}({\cal T})$ outputs w.p. $1-\beta$ an $(\alpha,r_{\min})$-good-averages solution $Y \in ({\mathbb R}^d)^k$ for clustering ${\cal T}$.
\end{definition}
Note that we allow the algorithm to handle only tuples over $B(\pt{0},\Lambda)$. If the algorithm can handle arbitrary tuples over ${\mathbb R}^d$, we omit the last parameter $\Lambda$.
\subsection{Additional Facts}
In this section we prove some facts about $\Delta$-far balls for $\Delta > 6$.
The following proposition states that if ${\cal B}=\set{B_i}_{i=1}^k$ and ${\cal B}' = \set{B_i'}_{i=1}^k$ are two sets of $\Delta$-far balls for $\Delta > 6$, and $\px$ is a point that belongs to either $B_i$ or $B_i'$ for some $i \in [k]$, such that $B_i$ and $B_i'$ intersect each other, then the center of $B_i$ is closest to $\px$ among all the centers of all the balls in ${\cal B}$ (and the same holds w.r.t. $B_i'$ and ${\cal B}'$).
\begin{proposition}\label{prop:close-sets-of-far-balls}
Let ${\cal B} = \set{B_i = B(\pc_i,r_i)}_{i=1}^k$ and ${\cal B}' = \set{B_i' = B(\pc_i',r_i')}_{i=1}^k$ be two sets of $\Delta$-far balls for $\Delta > 6$ s.t. for every $i \in [k]$ it holds that $B_i \cap B_i' \neq \emptyset$.
Then for every $i \in [k]$ and every $\px \in B_i \cup B_i'$, it holds that $i = \operatorname*{argmin}_{j \in [k]} \norm{\px - \pc_j}$.
\end{proposition}
\begin{proof}
Fix $i \in [k]$ and $\px \in B_i \cup B_i'$.
If $\px \in B_i$, the proof trivially follows. Therefore, in the following we assume that $\px \notin B_i$, and therefore, $\px \in B_i'$.
Note that on the one hand, it holds that
\begin{align}\label{eq:upbound-px-pc_i}
\norm{\px - \pc_i}
\leq \norm{\px - \pc_i'} + \norm{\pc_i' - \pc_i}
\leq r_i' + (r_i + r_i')
= 2r_i' + r_i
\end{align}
On the other hand, fix $j \neq i$, and note that
\begin{align}\label{eq:lowbound-px-pc_j-1}
\norm{\px - \pc_j}
&\geq \norm{\pc_i - \pc_j} - \norm{\px - \pc_i}\\
&> 6\max\set{r_i,r_j} - (2r_i' + r_i)\nonumber\\
&\geq 5\max\set{r_i,r_j} - 2r_i',\nonumber
\end{align}
where the second inequality holds by \cref{eq:upbound-px-pc_i} along with the fact that ${\cal B}$ are $\Delta$-far balls for $\Delta > 6$.
Therefore, if $\max\set{r_i,r_j} \geq r_i'$, we deduce by \cref{eq:upbound-px-pc_i,eq:lowbound-px-pc_j-1} that $\norm{\px - \pc_i} < \norm{\px - \pc_j}$. Otherwise (i.e., $\max\set{r_i,r_j} < r_i'$), note that
\begin{align}\label{eq:lowbound-px-pc_j-2}
\norm{\px - \pc_j}
&\geq \norm{\pc_i' - \pc_j'} - \norm{\px - \pc_i'} - \norm{\pc_j' - \pc_j}\\
&> 6\max\set{r_i',r_j'} - r_i' - (r_i' + r_i)\nonumber\\
&> 3r_i'.\nonumber
\end{align}
Hence, we deduce by \cref{eq:upbound-px-pc_i,eq:lowbound-px-pc_j-2} that $\norm{\px - \pc_i} < \norm{\px - \pc_j}$ also in this case, which concludes the proof of the proposition.
\end{proof}
We next prove that if ${\cal T}$ is partitioned by $\Delta$-far balls for $\Delta > 6$, and ${\cal B}$, a set of $\Delta$-far balls, partitions at least one tuple in ${\cal T}$, then by partitioning the points in ${\rm Points}({\cal T})$ w.r.t. the centers of the balls in ${\cal B}$, we obtain exactly ${\rm Partition}({\cal T})$.
\begin{proposition}\label{prop:from-almost-par-to-evenly-par}
Let ${\cal T} \in (({\mathbb R}^d)^k)^n$ be a multiset that is partitioned by $\Delta$-far balls for $\Delta > 6$, let ${\cal B} = \set{B_1,\ldots,B_k}$ be a set of $\Delta$-far balls that partitions at least one $k$-tuple of ${\cal T}$, and let $\pc_1,\ldots,\pc_k$ be the centers of $B_1,\ldots,B_k$, respectively. In addition, for every $i \in [k]$ let ${\cal Q}_i = \set{\px \in {\rm Points}({\cal T}) \colon i = \operatorname*{argmin}_{j \in [k]} \norm{\px - \pc_j}}$. Then $\set{{\cal Q}_1,\ldots,{\cal Q}_k} = {\rm Partition}({\cal T})$.
\end{proposition}
\begin{proof}
Let $X = \set{\px_1,\ldots,\px_k} \in {\cal T}$ be the assumed $k$-tuple that is partitioned by ${\cal B}$, let ${\cal B}^* = \set{B_1^*,\ldots,B_k^*}$ be a set of $\Delta$-far balls that partitions (all of) ${\cal T}$, and assume w.l.o.g. that $\px_i \in B_i \cap B_i^*$ for every $i \in [k]$. \cref{prop:close-sets-of-far-balls} yields that for every $i \in [k]$ and $\px \in B_i^*$ it holds that $\px \in {\cal Q}_i$, yielding that $B_i^* \cap {\rm Points}({\cal T}) \subseteq {\cal Q}_i$. Since both sets $\set{B_i^*}_{i=1}^k$ and $\set{{\cal Q}_i}_{i=1}^k$ consist of disjoints sets that cover all the points in ${\rm Points}({\cal T})$, we conclude that $\set{{\cal Q}_1,\ldots,{\cal Q}_k} = {\rm Partition}({\cal T})$.
\end{proof}
\remove{
In the following we describe a simple algorithm $\Algg{\rm BBBBBBBBBBBBBBBB}$ that takes a multiset ${\cal T} \in (({\mathbb R}^d)^k)^n$ and distinguishes between the case that ${\cal T}$ is partitioned by \textbf{very} far balls, and the case that ${\cal T}$ is \textbf{not} partitioned by $k$ far balls. In the former case it also outputs a set of $k$ far balls that partitions ${\cal T}$. The algorithm is described in \cref{fig:IsPartitioned}.
\begin{figure}[thb!]
\begin{center}
\noindent\fbox{
\parbox{.95\columnwidth}{
\begin{center}{ \bf Algorithm $\Algg{\rm IsWeaklyPartitioned}$}\end{center}
\textbf{Input:} A multiset ${\cal T} \in (({\mathbb R}^d)^k)^*$.
\begin{enumerate}
\item Choose an arbitrary tuple $(\px_1,\ldots,\px_k) \in {\cal T}$, and initialize ${\cal T}_1,\ldots,{\cal T}_k = \emptyset$.
\item For $(\px_1',\ldots,\px_k') \in {\cal T}$:\label{step:for-loop-tuple}
\begin{enumerate}
\item For $i=1$ to $k$:\label{step:for-loop-i}
\begin{enumerate}
\item Define $\pi(i) = \operatorname*{argmin}_{j \in [k]} \norm{\px_i - \px_j'}$.
\item ${\cal T}_i = {\cal T}_i \cup \set{\px_{\pi(i)}'}$ (a multiset union).
\end{enumerate}
\item If $\pi$ is not a permutation of $[k]$, output $(\text{``NO''},\perp)$.
\end{enumerate}
\item Output $(\text{``YES''},(\px_1,\ldots,\px_k))$.
\end{enumerate}
}}
\end{center}
\caption{A non-private algorithm that tests whether ${\cal T}$ can be partitioned.\label{fig:IsPartitioned}}
\end{figure}
The following proposition summarizes the properties of $\Algg{\rm BBBBBBBBBBBBBBBB}$.
\def\propIsPartitioned{
On inputs ${\cal T} \in (({\mathbb R}^d)^k)^n$, Algorithm $\Algg{\rm BBBBBBBBBBBBBBBB}$ satisfies the following guarantees:
\begin{itemize}
\item If ${\cal T}$ is partitioned by \textbf{very} far balls, then the algorithm outputs $(\text{``YES''},{\cal B})$ where ${\cal B}$ is a set of $k$ far balls that partitions ${\cal T}$.
\item If ${\cal T}$ is \textbf{not} partitioned by far balls, then the algorithm outputs $(\text{``NO''},\perp)$.
\end{itemize}
The algorithm runs in time $O(d n k^2)$.
}
\begin{proposition}\label{prop:IsPartitioned}
\propIsPartitioned
\end{proposition}
\begin{proof}
Consider an execution of $\Algg{\rm BBBBBBBBBBBBBBBB}({\cal T})$. If ${\cal T}$ is \textbf{not} partitioned by far balls, then by definition, the output of the execution is $(\text{``NO''},\perp)$. In the rest of the analysis we assume that ${\cal P}$ is partitioned by $k$ \textbf{very} far balls.
By construction, the resulting multisets $\set{{\cal T}_1,\ldots,{\cal T}_k}$ in the for-loop (Step~\ref{step:for-loop-tuple}) are exactly ${\rm Partition}({\cal T})$.
Let ${\cal B} = \set{B_i = B(\px_i, r_i)}_{i=1}^k$ be the set of $k$ balls from Step~\ref{step:resulting-balls}, and let ${\cal B}^* = \set{B_i^* = B(\px_i^*, r_i^*)}_{i=1}^k$ be a set of $k$ \textbf{very} far balls that partitions ${\cal T}$. The proof of the proposition now follows since for every $i \neq j$ it holds that
\begin{align*}
\norm{\px_i - \px_j}
&\geq \norm{\px_i^* - \px_j^*} - \norm{\px_i - \px_i^*} - \norm{\px_j - \px_j^*}\\
&> 14 \max\set{r_i^*,r_j^*} - r_i^* - r_j^*\\
&\geq 12 \max\set{r_i^*,r_j^*}\\
&\geq 6 \max\set{r_i,r_j},
\end{align*}
where the last inequality holds since $r_i \leq 2 r_i^*$ and $r_j \leq 2 r_j^*$ by construction.
\end{proof}
}
\section{Preliminaries}
\subsection{Notation}
In this work, a $k$-tuple
$X = \set{\px_1,\ldots,\px_k}$
is an \emph{unordered} set of $k$ vectors $\px_i \in {\mathbb R}^d$.
For $\px \in {\mathbb R}^d$, we denote by $\norm{\px}$ the $\ell_2$ norm of $\px$. For $\pc \in {\mathbb R}^d$ and $r > 0$, we denote $B(\pc,r) := \set{\px \in {\mathbb R}^d \colon \norm{\px - \pc} \leq r}$.
For a multiset ${\cal P} \in ({\mathbb R}^d)^*$ we denote by ${\rm Avg}({\cal P}) := \frac1{\size{{\cal P}}}\cdot \sum_{\px \in {\cal P}} \px$ the average of all points in ${\cal P}$. Throughout this work, a database ${\cal D}$ is a multiset. For two multisets ${\cal D} = \set{x_1,\ldots,x_n}$ and ${\cal D}' = \set{x_1',\ldots,x_m'}$, we let ${\cal D} \cup {\cal D}'$ be the multiset $\set{x_1,\ldots,x_n,x_1',\ldots,x_m'}$. For a multiset ${\cal D} = \set{x_1,\ldots,x_n}$ and a set $S$, we let ${\cal D} \cap S$ be the multiset $\set{x_i}_{i \in {\cal I}}$ where ${\cal I} = \set{i \in [n] \colon x_i \in S}$.
All logarithms considered here are natural logarithms (i.e., in base $e$).
\subsection{Indistinguishability and Differential Privacy}
\begin{definition}[Neighboring databases]\label{def:neighboring}
Let ${\cal D} = \set{x_1,\ldots,x_n}$ and ${\cal D}' = \set{x_1',\ldots,x_n'}$ be two databases over a domain ${\cal X}$. We say that ${\cal D}$ and ${\cal D}'$ are \textbf{neighboring} if there is exactly one index $i \in [n]$ with $x_i \neq x_i'$.
\end{definition}
\begin{definition}[$(\varepsilon,\delta)$-indistinguishable]\label{def:indis}
Two random variable $X,X'$ over a domain ${\cal X}$ are called $(\varepsilon,\delta)$-indistinguishable, iff for any event $T \subseteq {\cal X}$, it holds that
$\pr{X \in T} \leq e^{\varepsilon} \cdot \pr{X' \in T} + \delta$. If $\delta = 0$, we say that $X$ and $X'$ are $\varepsilon$-indistinguishable
\end{definition}
\begin{definition}[$(\varepsilon,\delta)$-differential privacy \cite{DworkMNS06}]\label{def:DP}
An algorithm $\cA$ is called $(\varepsilon,\delta)$-differentially private, if for any two neighboring databases ${\cal D},{\cal D}'$ it holds that $\cA({\cal D})$ and $\cA({\cal D}')$ are $(\varepsilon,\delta)$-indistinguishable. If $\delta = 0$ (i.e., pure privacy), we say that $\cA$ is $\varepsilon$-differentially private.
\end{definition}
\begin{lemma}[\cite{BS16}]\label{lem:indis}
Two random variable $X,X'$ over a domain ${\cal X}$ are $(\varepsilon,\delta)$-indistinguishable, iff there exist events $E, E' \subseteq {\cal X}$ with $\pr{X \in E},\pr{X' \in E'} \geq 1-\delta$ such that $X|_{E}$ and $X'|_{E'}$ are $\varepsilon$-indistinguishable.
\end{lemma}
\subsubsection{Basic Facts}
The following fact is a corollary of \cref{lem:indis}.
\begin{fact}\label{fact:indis-cor}
Let $X,X'$ be two random variables over a domain ${\cal X}$, and let $E, E' \subseteq {\cal X}$ be two events. If $X|_E$ and $X'|_{E'}$ are $(\varepsilon,\delta_1)$-indistinguishable and $\pr{X \in E},\pr{X' \in E'} \geq 1 - \delta_2$, then $X$ and $X'$ are $(\varepsilon,\delta_1 + \delta_2)$-indistinguishable.
\end{fact}
\begin{proof}
Since $X|_E$ and $X'|_{E'}$ are $(\varepsilon,\delta_1)$-indistinguishable, we deduce by \cref{lem:indis} that there exists events $F \subseteq E$ and $F' \subseteq E'$ with $\pr{X \in F \mid E}, \pr{X' \in F' \mid E'} \geq 1-\delta_1$ such that $X|_F$ and $X'|_{F'}$ are $(\varepsilon,0)$-indistinguishable. In addition, note that
\begin{align*}
\pr{X \in F} = \pr{X \in E}\cdot \pr{X \in F \mid E} \geq (1-\delta_2)(1-\delta_1) \geq 1 - (\delta_1 + \delta_2).
\end{align*}
Similarly, it holds that $ \pr{X' \in F'} \geq 1 - (\delta_1 + \delta_2)$. Therefore, by applying the opposite direction of \cref{lem:indis} on the events $F$ and $F'$, we deduce that $X$ and $X'$ are $(\varepsilon, \delta_1 + \delta_2)$-indistinguishable.
\end{proof}
In addition, we use the following facts.
\begin{fact}\label{fact:conditioning}
Let $X,X'$ be two $\varepsilon$-indistinguishable random variables over a domain ${\cal X}$, and let $E, E' \subseteq {\cal X}$ be two events with $\pr{X \in E},\pr{X' \in E'} \geq 1-\delta$. Then $X|_{E}$ and $X'|_{E'}$ are $(\varepsilon - \ln(1-\delta), \: \frac{e^{\varepsilon} \delta }{1-\delta})$-indistinguishable.
\end{fact}
\begin{proof}
Fix a subset $T \subseteq {\cal X}$ and compute
\begin{align*}
\pr{X \in T \mid E}
\leq \frac{\pr{X \in T }}{\pr{E}}
\leq \frac{e^{\varepsilon} \cdot \pr{X' \in T}}{1-\delta}
&\leq \frac{e^{\varepsilon}}{1-\delta} \cdot \paren{\pr{X' \in T \mid E'} + \pr{X' \notin E'}}\\
&\leq e^{\varepsilon -\ln(1-\delta)} \cdot \pr{X' \in T \mid E'} + \frac{e^{\varepsilon} \delta}{1-\delta}
\end{align*}
where the last inequality holds since $\pr{X' \notin E'} \leq \delta$ by assumption.
\end{proof}
\begin{fact}\label{prop:similar-E}
Let $X,X'$ be two random variables over a domain ${\cal X}$. Assume there exist events $E,E' \subseteq {\cal X}$ such that the following holds:
\begin{itemize}
\item $\pr{X \in E} \in e^{\pm \varepsilon}\cdot \pr{X' \in E'}$, and
\item $X|_{E}$ and $X'|_{E'}$ are $(\varepsilon^*,\delta)$-indistinguishable, and
\item $X|_{\neg E}$ and $X'|_{\neg E'}$ are $(\varepsilon^*,\delta)$-indistinguishable.
\end{itemize}
Then $X,X'$ are $(\varepsilon + \varepsilon^*,\delta e^{\varepsilon})$-indistinguishable.
\end{fact}
\begin{proof}
Fix an event $T \subseteq {\cal X}$ and compute
\begin{align*}
\lefteqn{\pr{X \in T}
= \pr{X \in T \mid E}\cdot \pr{X \in E} + \pr{X \in T \mid \neg E}\cdot \pr{X \notin E}}\\
&\leq \paren{e^{\varepsilon^*}\cdot \pr{X' \in T \mid E'} + \delta}\cdot e^{\varepsilon}\cdot \pr{X' \in E'} + \paren{e^{\varepsilon^*}\cdot \pr{X' \in T \mid \neg E'} + \delta}\cdot e^{\varepsilon}\cdot \pr{X' \notin E'}\\
&= e^{\varepsilon + \varepsilon^*} \cdot \pr{X' \in T} + \delta e^{\varepsilon}.
\end{align*}
\end{proof}
\subsubsection{Group Privacy and Post-Processing}
\begin{fact}[Group Privacy]\label{fact:group-priv}
If $\cA$ is $(\varepsilon,\delta)$-differentially private, then for all pairs of databases ${\cal S}$ and ${\cal S}'$ that differ by $k$ points it holds that $\cA({\cal S})$ and $\cA({\cal S}')$ are $(k\varepsilon, k e^{k\varepsilon} \delta)$-indistinguishable.
\end{fact}
\begin{fact}[Post-processing]\label{fact:post-processing}
If $\cA$ is $(\varepsilon,\delta)$-differentially private, then for every (randomized) function $F$ it holds that $F \circ \cA$ is $(\varepsilon,\delta)$-differentially private.
\end{fact}
\subsubsection{Composition}
\begin{theorem}[Basic composition, adaptive case \cite{DRV10}]\label{thm:composition1}
If $\cA_1$ and $\cA_2$ satisfy $(\varepsilon_1,\delta_1)$ and $(\varepsilon_2,\delta_2)$ differential privacy (respectively), then any algorithm that adaptively uses $\cA_1$ and $\cA_2$ (and does not access the database otherwise) ensures $(\varepsilon_1+\varepsilon_2,\delta_1+\delta_2)$-differential privacy.
\end{theorem}
\begin{theorem}[Advanced composition~\cite{DRV10}]\label{thm:composition2}
Let $0<\varepsilon_0,\delta'\leq1$, and let $\delta_0\in[0,1]$. An algorithm that adaptively uses $k$ algorithms that preserve $(\varepsilon_0,\delta_0)$-differential privacy (and does not access the database otherwise) ensures $(\varepsilon,\delta)$-differential privacy, where $\varepsilon=\sqrt{2k\ln(1/\delta')}\cdot\varepsilon_0+2k\varepsilon_0^2$ and $\delta = k\delta_0+\delta'$.
\end{theorem}
\subsubsection{The Laplace Mechanism}
\begin{definition}[Laplace distribution]
For $\sigma \geq 0$, let ${\rm Lap}(\sigma)$ be the Laplace distribution over ${\mathbb R}$ with probability density function $p(z) = \frac1{2 \sigma} \exp\paren{-\frac{\size{z}}{\sigma}}$.
\end{definition}
\begin{fact}\label{fact:laplace-concent}
Let $\varepsilon > 0$. If $X \sim {\rm Lap}(1/\varepsilon)$ then for all $t > 0: \quad \pr{\size{X} > t/\varepsilon} \leq e^{-t}$.
\end{fact}
\begin{definition}[Sensitivity]
We say that a function $f \colon {\cal U}^n \rightarrow {\mathbb R}$ has sensitivity $\lambda$ if for all neigboring databases ${\cal S}, {\cal S}'$ it holds that $\size{f({\cal S}) - f({\cal S}')} \leq \lambda$.
\end{definition}
\begin{theorem}[The Laplace Mechanism \cite{DworkMNS06}]\label{fact:laplace}
Let $\varepsilon > 0$, and assume $f \colon {\cal U}^n \rightarrow {\mathbb R}$ has sensitivity $\lambda$. Then the mechanism that on input ${\cal S} \in {\cal U}^n$ outputs $f({\cal S}) + {\rm Lap}(\lambda/\varepsilon)$ is $\varepsilon$-differentially private.
\end{theorem}
\subsubsection{The Gaussian Mechanism}
\begin{definition}[Gaussian distribution]
For $\mu \in {\mathbb R}$ and $\sigma \geq 0$, let ${\cal N}(\mu,\sigma^2)$ be the Gaussian distribution over ${\mathbb R}$ with probability density function $p(z) = \frac1{\sqrt{2\pi}} \exp\paren{-\frac{(z-\mu)^2}{2 \sigma^2}}$.
\end{definition}
\begin{fact}\label{fact:one-gaus-concent}
Let $\pX = (X_1,\ldots,X_d)$, where the $X_i$'s are i.i.d. random variables, distributed according to $ {\cal N}(0,\sigma^2)$.
Then for all
$\beta > 0: \quad \pr{\norm{\pX} \leq \paren{\sqrt{d} + \sqrt{2 \log(1/\beta)}} \cdot \sigma} \geq 1-\beta$.
\end{fact}
\begin{definition}[$\ell_2$-sensitivity]
We say that a function $f \colon {\cal U}^n \rightarrow {\mathbb R}^d$ has $\ell_2$-sensitivity $\lambda$ if for all neigboring databases ${\cal S}, {\cal S}'$ it holds that $\norm{f({\cal S}) - f({\cal S}')} \leq \lambda$.
\end{definition}
\begin{theorem}[The Gaussian Mechanism \cite{DKMMN06}]\label{fact:Gaus}
Let $\varepsilon,\delta \in (0,1)$, and assume $f \colon {\cal U}^n \rightarrow {\mathbb R}^d$ has $\ell_2$-sensitivity $\lambda$. Let $\sigma \geq \frac{\lambda}{\varepsilon}\sqrt{2 \log(1.25/\delta)}$. Then the mechanism that on input ${\cal S} \in {\cal U}^n$ outputs $f({\cal S}) + \paren{{\cal N}(0,\sigma^2)}^d$ is $(\varepsilon,\delta)$-differentially private.
\end{theorem}
\begin{observation}\label{obs:Gaus-aver}
For the case that ${\cal S} \in ({\mathbb R}^d)^n$ and $f({\cal S}) = {\rm Avg}({\cal S})$, if we are promised that each coordinate of the points is bounded by a segment of length $\Lambda$, then the sensitivity is bounded by $\lambda = \Lambda/n$, and therefore, by taking $\sigma = O(\frac{\Lambda}{\varepsilon n} \sqrt{\log(1/\delta)})$ we get by \cref{fact:one-gaus-concent} that with probability $1-\beta$, the resulting point $\pz$ of the mechanism satisfies
$\norm{\pz - {\rm Avg}({\cal S})} \leq \frac{\Lambda \sqrt{\log(1.25/\delta)}}{\varepsilon n} \paren{\sqrt{d} + \sqrt{2\log(1/\beta)}}$.
\end{observation}
\begin{remark}\label{remark:Gaus-add-del}
\cref{fact:Gaus} guarantees differential-privacy whenever two neighboring databases have equal size. However, it can be easily extended to a more general case in which the privacy guarantee also holds in cases of addition and deletion of a point, with essentially the same noise magnitude (e.g., see Appendix A in \cite{NSV16}).
\end{remark}
The following proposition states the following: Assume that $\pX \sim \mu + ({\cal N}(0,\sigma^2))^d$ for some $\mu \in {\mathbb R}^d$,
and let $\py \in {\mathbb R}^d$ such that $\norm{\py - \mu}$ is ``large enough'' (i.e., larger than $\Omega\paren{\sigma \sqrt{\log(1/\beta)}}$). Then with probability $1-\beta$ (over $\pX$) it holds that $\norm{\pX - \mu} < \norm{\pX - \py}$. Note that such an argument is trivial when $\norm{\py - \mu}$ is at least $\Omega(\sigma \sqrt{d \log(1/\beta)})$, but here we are aiming for a distance that is independent of $d$. The proof of the proposition, which appears at \cref{missing-proof:thm:kGauss-utility} as a special case of \cref{prop:separation}, is based on a standard projection argument.
\begin{proposition}\label{prop:separation-spherical-case}
Let $\pX \sim \mu + ({\cal N}(0,\sigma^2))^d$ and let $\py \in {\mathbb R}^d$ with $\norm{\py - \mu} > 2\sqrt{2\log\paren{\frac1{\beta}}} \cdot \sigma$. Then with probability $1-\beta$ (over the choice of $\pX$), it holds that $\norm{\pX - \mu} < \norm{\pX-\py}$.
\end{proposition}
\subsubsection{Estimating the Average of Points}\label{sec:prelim:est-aver}
As mentioned in \cref{obs:Gaus-aver}, the Gaussian mechanism (\cref{fact:Gaus}) allows for privately estimating the average of points in $B(\pt{0},\Lambda) \subseteq {\mathbb R}^d$ within $\ell_2$ error of $\approx \frac{\Lambda \sqrt{d}}{\varepsilon n}$. In some cases, we could relax the dependency on $\Lambda$. For example, using the following proposition.
\def\propEstAvgInRd{
Let $\varepsilon \in (0,1)$, $d, \Lambda > 0$ and let $r_{\min} \in [0,\Lambda]$. There exists an efficient $(\varepsilon,\delta)$-differentially private algorithm that takes an $n$-size database ${\cal S}$ of points inside the ball $B(\pt{0},\Lambda)$ in ${\mathbb R}^d$ and satisfy the following utility guarantee: Assume that $n \geq \frac{32\sqrt{2d \log(2/\delta)}}{\varepsilon} \log\paren{\frac{4d\Lambda}{r_{\min} \beta}} + 4$, and
let $r >0$ be the minimal radius of a $d$-dimensional ball that contains all points in ${\cal S}$. Then with probability $1-\beta$, the algorithm outputs $\hpa \in {\mathbb R}^d$ such that
\begin{align*}
\norm{\hpa - {\rm Avg}({\cal S})} \leq O\paren{\max\set{r,r_{\min}}\cdot \frac{d \sqrt{\log(1/\delta)}}{\varepsilon n} \paren{\sqrt{\log(d/\delta) \log(d/\beta)} + \log \paren{\frac{\Lambda d}{r_{\min} \beta}}}}.
\end{align*}
The algorithm runs in time $\tilde{O}(d n)$ (ignoring logarithmic factors).
}
\begin{proposition}[Estimating the Average of Bounded Points in ${\mathbb R}^d$]\label{prop:approx-aver-Rd}
\propEstAvgInRd
\end{proposition}
\cref{prop:approx-aver-Rd} can be seen as a simplified variant of \cite{NSV16}'s private average algorithm. The main difference is that \cite{NSV16} first uses the Johnson Lindenstrauss (JL) transform \cite{JL84} to randomly embed the input points in ${\mathbb R}^{d'}$ for $d' \approx \log n$, and then estimates the average of the points in each axis of ${\mathbb R}^{d'}$. As a result, they manage to save a factor of $\sqrt{d}$ upon \cref{prop:approx-aver-Rd} (at the cost of paying a factor of $\log n$ instead). However, for simplifying the construction and the implementation, we chose to omit the JL transform step, and we directly estimate the average along each axis of ${\mathbb R}^{d}$. For completeness, we present the full details of \cref{prop:approx-aver-Rd} in \cref{sec:approx-aver}.
\Enote{maybe mention other methods like CoinPress, KV, etc, that are developed for Gaussians, but can be used here as well.}
\subsubsection{Sub-Sampling}
\begin{lemma}[\cite{BKN10,KLNRS11}]\label{lem:subsampling}
Let ${\cal A}$ be an $(\varepsilon^*,\delta^*)$-differentially private algorithm operating on databases of size $m$. Fix $\varepsilon \leq 1$, and denote $n = \frac{m}{\varepsilon}(3 + \exp(\varepsilon^*))$. Construct an algorithm ${\cal B}$ that on an input database ${\cal D} = (z_i)_{i=1}^n$, uniformly at random selects a subset ${\cal I} \subseteq [n]$ of size $m$, and executes ${\cal A}$ on the multiset ${\cal D}_{{\cal I}} = (z_i)_{i \in {\cal I}}$. Then ${\cal B}$ is $(\varepsilon,\delta)$-differentially private, where $\delta = \frac{n}{4m}\cdot \delta^*$.
\end{lemma}
The following lemma states that switching between sampling
with replacement and without replacement has only a small effect on privacy.
\begin{lemma}[\cite{BNSV15}]\label{lem:DP-with-replacement}
Fix $\varepsilon \leq 1$ and let ${\cal A}$ be an $(\varepsilon,\delta)$-differentially private algorithm operating on
databases of size $m$. For $n \geq 2m$, construct an algorithm ${\cal A}'$ that on input a database ${\cal D}$ of size $n$, subsamples (with replacement) $m$ rows from ${\cal D}$, and runs ${\cal A}$ on the result. Then ${\cal A}'$ is $(\varepsilon',\delta')$-differentially
private for $\varepsilon' = 6 \varepsilon m/n$ and $\delta' = \exp\paren{6 \varepsilon m/n} \cdot \frac{4m}{n}\cdot \delta$.
\end{lemma}
\remove{
\subsection{Preliminaries from Learning Theory}
A concept $c:{\cal X}\rightarrow \{0,1\}$ is a predicate that labels {\em examples} taken from the domain ${\cal X}$ by either 0 or 1. A \emph{concept class} ${\cal C}$ over ${\cal X}$ is a set of concepts (predicates) mapping ${\cal X}$ to $\{0,1\}$.
\begin{definition}[VC dimension]
Let ${\cal C}$ be a concept class over ${\cal X}$ and let ${\cal S} = \set{x_1,\ldots,x_m} \subseteq {\cal X}$. We say that ${\cal S}$ is shattered by ${\cal C}$ if $\size{\set{(c(x_1),\ldots,c(x_m)) \colon c \in {\cal C}}} = 2^m$. The VC dimension of ${\cal C}$, which is denoted by ${\rm VC}({\cal C})$, is the maximal size of a set ${\cal S}$ that is shattered by ${\cal C}$.
\end{definition}
\begin{theorem}[VC bounds \cite{VC}]\label{thm:vc-bounds}
Let ${\cal H} = \set{h \colon {\cal X} \rightarrow \set{0,1}}$ be a concept class, let $c \colon {\cal X} \rightarrow \set{0,1}$ be a function,
let $\mu$ be a distribution over ${\cal X}$ and let ${\cal S}$ be a set of $m$ i.i.d. samples from $\mu$.
Then with probability $1-\delta$ (over the choice of ${\cal S}$), for any $h \in {\cal H}$ with ${\rm error}_{{\cal S}}(c,h) = \ppr{x \gets {\cal S}}{h(x) \neq c(x)} = 0$ (realizable case) it holds that
\begin{align*}
{\rm error}_{\mu}(c,h) \leq \frac{8 {\rm VC}({\cal H}) \log \frac{2m}{{\rm VC}({\cal H})} + 4\log \frac{4}{\delta}}{m}
\end{align*}
where ${\rm error}_{\mu}(c,h) = \ppr{x \sim \mu}{h(x) \neq c(x)}$.
In general (agnostic case), with probability $1-\delta$, for every $h \in {\cal H}$ it holds that
\begin{align*}
{\rm error}_{\mu}(c,h) \leq {\rm error}_{{\cal S}}(c,h) + \sqrt{\frac{{\rm VC}({\cal H})\paren{\log \frac{2m}{{\rm VC}({\cal H})}+1}+ \log \frac{4}{\delta}}{m}},
\end{align*}
\end{theorem}
In the following, fix a parameter $d \in {\mathbb N}$ and let $\cC_{\text{ball}}$ be the class of all $d$-dimensional balls in ${\mathbb R}^d$. Namely, $\cC_{\text{ball}} = \set{h_{\pt{c},r} \colon {\mathbb R}^d \rightarrow \set{0,1}}_{\pc \in {\mathbb R}^d, r \geq 0}$, where $h_{\pt{c},r}(\px) = 1 \iff \px \in B(\pc,r)$.
\begin{fact}[\cite{Dud79}]\label{fact:vc-of-ball}
The VC-dimension of $\cC_{\text{ball}}$ is $d+1$.
\end{fact}
\begin{fact}[\cite{BlumerEhHaWa89}]\label{fact:k-fold}
For any concept class ${\cal C}$, the VC-dimension of ${\cal C}^{k \cup}$, the $k$-fold union of ${\cal C}$, is at most $3 {\rm VC}({\cal C}) k \log (3k)$, where ${\cal C}^{k \cup} := \set{c_1 \cup \ldots \cup c_k \colon c_1,\ldots,c_k \in {\cal C}}$.
\end{fact}
\cref{fact:vc-of-ball,fact:k-fold} imply the following corollary.
\begin{corollary}\label{cor:vc-of-k-ball}
The VC-dimension of $\cC_{k\text{-balls}}$, the class of all $k$-fold union of balls in ${\mathbb R}^d$, is at most $3 (d+1) k \log (3k)$.
\end{corollary}
}
\remove{
\subsection{The Johnson Lindenstrauss transform}\label{sec:JL}
\begin{definition}[The JL random projection from ${\mathbb R}^d$ to ${\mathbb R}^m$]\label{def:JL}
Let $f \colon {\mathbb R}^d \rightarrow {\mathbb R}^m$ be the projection that works as follows: Pick $m$ vectors $\pz_1,\ldots,\pz_m$ from a standard $d$-dimensional Gaussian distribution with density $p(\px) = \frac{1}{(2 \pi)^{d/2}} \exp(-\norm{\px}^2/2)$. For any vector $\px \in {\mathbb R}^d$, define $f(\px) = (\rm ip{\px,\pz_1}, \ldots, \rm ip{\px,\pz_m})$ (where $\rm ip{\px,\py} = \sum_{i=1}^d x_i y_i$).
\end{definition}
\begin{theorem}[\cite{JL84}]\label{thm:JL}
Let $f \colon {\mathbb R}^d \rightarrow {\mathbb R}^m$ be the random projection from \cref{def:JL}, and let $n \in {\mathbb N}$, $\varepsilon > 0$. Assuming that $m = \Omega\paren{\frac{1}{\varepsilon^2} \cdot \log\paren{n/\beta}}$, then with probability $1-\beta$ it holds that
\begin{align*}
\forall \px,\py \in {\cal P}: \quad \norm{f(\px)-f(\py)} \in (1 \pm \varepsilon) \sqrt{m} \norm{\px - \py}
\end{align*}
\end{theorem}
}
\subsection{Concentration Bounds}
\begin{fact}[Hoeffding's inequality]\label{fact:Hoeffding}
Let $X_1,\ldots,X_n$ be independent random variables, each $X_i$ is strictly bounded by the interval $[a_i,b_i]$, and let $\bar{X} = \frac1n\sum_{i=1}^n X_i$. Then for every $t \geq 0$:
\begin{align*}
\pr{\size{\bar{X} - \ex{\bar{X}}} \geq t} \leq 2\exp\paren{-\frac{2n^2 t^2}{\sum_{i=1}^n (b_i-a_i)^2}}
\end{align*}
\end{fact}
\begin{fact}[{\cite[Theorem 5.3]{AminCO}}]\label{fact:binom_concentration}
Let $X \sim {\rm Bin}(n,p)$, then for all $t \geq 0$:
\begin{enumerate}
\item $\pr{X \geq \ex{X} + t} \leq \exp\paren{-\frac{t^2}{2\left(np + t/3\right)}}$.
\item $\pr{X \leq \ex{X} - t} \leq \exp\paren{-\frac{t^2}{2np}}$.
\end{enumerate}
\end{fact}
\section{Introduction}
Related work:
Mean estimation:
\url{https://arxiv.org/abs/2006.06618}
We do (some of) this non-parametrically. No need to know or privately estimate spread.
Need to read carefully also for relevant work.
Enclosing ball:
\url{https://arxiv.org/abs/1707.04766}
Somewhat different problem. Large constants. More complex. Worst asymptotics for our setting??
\section{The friendly elements problem}
The input is a set of elements $T$ of size $|T|=n$.
We have a symmetric predicate $\text{like}$ $x \leftrightarrow y$ on pairs of elements.
We are interested in an algorithm with the following properties:
(i) the output is either a "fail" or "success". If "success," the output also includes a set $T_G$ of "good elements." The set $T_G$ must include all {\em perfect} elements (a perfect element likes all elements) and may contain only elements that like more than half of elements.
(ii) When all elements are perfect, the success probability is $1$.
In addition, we have the following requirements for $(\varepsilon,\delta)$ and $\Delta>1$.
We say that two datasets $T$ and $T'$ are neighbors if they differ in at most one element.
(iii) The fail/success bit is $(\varepsilon,\delta)$-indistinguishable for any two neighboring datasets. Namely, if $p$ and $p'$ are the success probability of $T$ and $T'$ then $p \leq e^\varepsilon p' + \delta$ and
$(1-p) \leq e^\varepsilon (1-p') + \delta$.
(iv) The algorithm is randomized so that each element in $T$ has an independent probability to be included in $T_G$ (perfect elements have probability $1$ and elments with $n/2$ or fewer likes have probability $0$). If $p$ and $p'$ are the vectors of probabilities of two neighboring datasets then $\norm{p-p'}_1 \leq \Delta$.
We present two solutions for friendly elements. The first one performs ${n \choose 2}\approx n^2$ like queries. The second
(that provides high probability guarantees for requirements (ii) and (iv)) performs $O(n\log n)$ like queries.
Note that in terms of asymptotic dependence on $n$, this is the best we can hope for simply satisfying properties (i)-(ii).
The effect of requirements (iii)-(iv) and the privacy parameters $(\varepsilon,\delta)$ and $\Delta$ is that they imply a minimum value on $n$ that is needed to support these specifications.
[To do: For the more efficient $O(n\log n)$ method, we need to refine the hard bound in (iv) on $\norm{p-p'}_1$ with allowing for higher values and lower success probability.]
We first describe how we apply friendly-elements within a sample-and-aggregate framework.
We then proceed with solutions, starting with some basic tools that we use.
\section{Applications}
Elements are points in a metric space (or pseudo metric). We are interested in a "center" point. With DP, typically there is an additive error that depends on the diameter of the space $\Lambda$. For example with DP averaging the error is proportional to $\Lambda \sqrt{d}/(n \varepsilon)$.
But suppose we are only interested in a "center" when all points are within a distance $r\ll \Lambda$ except perhaps for a few outliers.
The friendly elements is a way to do this. Two elements are friends if their "distance" is at most $r$. The output (if "success") are elements $T_G$ with guaranteed max distance $2r$ between any pair. Therefore, we can apply any "centering" method with $\Lambda=r$.
For applications we may need to also return $r$ that is within say a small factor from the smallest one that yields "success." So the output space can only include $2^i r_{\min}$ for some integral $i$.
A naive way to do this is by sequentially doubling $r$ until there is success. This results in $\log(r/r_{\min})$ checks. A better way is to do a binary search by doubling $i$ first. This gives us $\log\log(r/r_{\min})$ since we search over $\log(r/r_{\min})$ values. Sparse vector might improve the asymptotics further.
[Comment: On real datasets we can expect $\log\log(r/r_{\min}) \leq 8$ but still this is a factor on the number of elements. There should be a better way. ]
\Enote{I didn't follow it...}\ECnote{Yes, you are right. I erased What was there as it does not work. There is also a lower bound of $\log^* |X|$...) }.
\subsection{Sample-and-aggregate framework}
Sample-and-aggregate \cite{} is a framework for DP solution of problems on instances where a solution for a small random sample of the data is very likely to be a good solution for the full dataset.
We use friendly-elements as a component in a sample-and-aggregate framework.
For a dataset of size $N$, the smallest sample size $s$ that is needed for this to hold is a property of the dataset. When this holds, we can get $n = N/s$ disjoint random samples. The framework applies a non-private algorithm to each of the $n$ samples and then privately "aggregates" the results. Solutions that work with smaller $n$ are better as they apply to more datasets.
To apply the frameworks we need
\begin{itemize}
\item
A non-private algorithm $A$ that inputs a dataset $D$ of points and returns a solution $x \gets A(D)$.
\item
A similarity predicate between solutions $x_i \leftrightarrow x_j$. Similarity is reflexive -- we have $x \leftrightarrow x$ for all $x$. Informally, similarity means that the solutions are interchangeable (one is almost as good as the other with respect to $D$).
\item
An $(\varepsilon,\delta)$-DP aggregator $C(X)$: Takes an input a set $T_G$ of at least $|T_G|\geq 0.9n$ solutions so that for any $x,y \in T_G$ either $x \leftrightarrow y$ or there exists $z$ such that $x \leftrightarrow z$ and $z \leftrightarrow y$.
Aggregator returns a private solution $x^* \gets C(X)$. Informally, the interpretation is that $x^*$ is 2-hops "similar" to any of the $x_i$'s.
\end{itemize}
Note that in the original sample-and-aggregate formulation, the aggregator needs to be "robust." It is applied directly to the solutions from all samples. That set is allowed to contain some "bad" solutions. Our friendly-elements module allows for the use of a simpler aggregator when all solutions are "good."
\begin{enumerate}
\item Set $D$ of $N$ data points, $n$
\item
Randomly partition $D$ into $n$ equal-size sets $D_i$.
\item
Apply the non-private $A$ to each part to obtain $x_i \gets A(D_i)$. Note that from neighboring inputs $D$ and $D'$ (differ on one data point) we obtain neighboring sets $T=\{x_i\}$ and $T'=\{x'_i\}$.
\item
Apply friendly-elements to the set $T$. We either fail or obtain a subset $T_G$ with almost all of $T$ that satisfies the input requirements of the aggregator. Recall (property of friendly-elements) that this probabilistic output $T_G$ is stable.
\item
Return $C(T_G)$
\end{enumerate}
When the dataset has the property that the $n$ solutions are similar, we succeed and return a private solution that is similar to them.
\section{Preliminaries: Tools}
\subsection{Tool: optimal DP thresholding}
Let $A$ be a function that inputs a dataset $X$ and returns an integer $A(X)$.
We have the property that $A$ has sensitivity $1$: On ``neighboring'' datasets $X$ and $X'$, $|A(X)-A(X')| \leq 1$.
We seek an $(\epsilon,\delta)$-DP thresholding predicate as follows: When $A(X) \geq \tau$ the output is $1$. Otherwise, when $A(x) <\tau$, the probability of $1$ is minimum.
[Comment: The problem can be solved with truncated Laplace noise, but not optimally. This roughly by adding $(1/\varespilon) \ln(1/\delta)$ plus Laplace $[1/\varespilon]$ noise (truncated to remove $\delta$ tails).]
The optimal DP thresholding scheme \cite{} is as follows:
Let
\begin{equation}\label{Ldef:eq}
L(\epsilon,\delta) := \frac{1}{\varepsilon} \ln\left(
\frac{e^\varepsilon -1 +2\delta}{\delta(e^\varepsilon +1)}
\right) \approx \frac{1}{\varepsilon} \ln\left(\frac{\min\{1,\varepsilon\}}{2\delta} \right)
\end{equation}
we assume that $\epsilon$ and $\delta$ are such that $L$ is an integer.
We define$(\pi^*_i)_{i\geq 1}$ as follows: \Enote{So the $L$ is chosen to be the value such that $\pi_{L+1}^*$ is the same in both cases? i.e., $\delta \frac{e^{\varepsilon (L+1)}-1 }{e^\varepsilon -1} = 1- \delta \frac{e^{\varepsilon (2L+2-(L+1))} -1}{e^{\varepsilon} -1}$?}
\begin{equation} \label{pistar:eq}
\pi^*_i = \left\{ \begin{array}{ll}
0 &\; \text{\small $ i \leq 0$ }\\
\delta \frac{e^{\varepsilon i}-1 }{e^\varepsilon -1} &\; \text{\small $0\leq i\leq L+1$ }\\
1- \delta \frac{e^{\varepsilon (2L+2-i)} -1}{e^{\varepsilon} -1} &\; \text{\small $L+1 \leq i \leq 2L+1$ }\\
1 &\; \text{\small $i \geq 2L+2$ }
\end{array}\right.
\end{equation}
Optimal DP thresholding for $\tau, \varepsilon, \delta$:
\begin{enumerate}
\item $i \gets A(x)$
\item Return $1$ ("pass") with probability
$p(i) :=\pi^*_{i-\tau +2L+2}$. Return $0$ otherwise (probability $q(i) := 1-p(i)$.
\end{enumerate}
Note that the DP thresholding always returns $0$ when $A(X) \leq \tau-2L-2$ and as required always returns $1$ when $A(X) \geq \tau$.
We will use the following property of $\pi^*_i$:
\begin{equation} \label{ratioprop:eq}
\max\left\{\frac{\pi^*_i - \delta}{\pi^*_{i-1}}, \frac{1-\pi^*_{i-1} - \delta}{1-\pi^*_{i}}\right\} \leq e^\varepsilon
\end{equation}
\paragraph{Reverse threshold:}
The mechanism can be flipped to provide
a "reversed" guarantee of passing with probability $1$ when $A(x) \leq \tau$. This is provided with pass probabilities:
\[
p(i) := \pi^*_{\tau - i +2L+2}\ .
\]
\paragraph{Probabilisitc $A(x)$:}
The formulation generalizes to when $A(X)$ returns values according to a distribution. In this case sensitivity $1$ informally means that for any $T$,
$\Pr[A(X')\leq T-1]\leq \Pr[A(X)\leq T]\leq \Pr[A(X')\leq T+1] $. That is,
we can map outcomes of $A(X)$ to outcomes of $A(X')$ with difference at most $1$.
Note that in this case the threshold mechanism may no longer be optimal, as it does not account for the randomization in $A$. For example, that randomization might already provide a private threshold. For our purposes however this suffices.
\paragraph{Continuous $i$:}
Formulation works when $i$ is continuous. We still require sensitivity $1$.
\subsection{Tool: DP Thresholding with a varying local sensitivitiy bound}
We now consider a situation where we have
a function $B(X) \geq 0$ with local sensitivity bound of the form
$B(X') \leq c + e^\nu B(X) $.
Equivalently,
$B(X')-B(X) \leq c + (e^\nu-1) B(X)$.
We want to apply DP thresholding so that if $B(X)=0$, pass probability is $1$ and otherwise pass probability is minimum.
We compute a monotone mapping $b^{-1}()$ of values
so that the function $b^{-1}\circ B(X)$ has sensitivity at most $1$.
\begin{lemma}
Let $B()\geq 0$ be such that on any two neighboring datasets $X$ and $X'$,
$B(X') \leq c + e^\nu B(X)$.
Define
\[
b^{-1}(y) := \frac{1}{\nu}\ln \left(1+ y \frac{e^\nu -1}{c}\right)\ .
\]
Then
the function $b^{-1}\circ B(X)$ has sensitivity 1. That is, on any two neighboring datasets,
$b^{-1}\circ B(X') \leq 1 + b^{-1}\circ B(X)$.
Also, $B(X)= 0$ if and only if $b^{-1}\circ B(X)=0$.
\end{lemma}
\begin{proof}
\ECnote{Replaced my "proof" with Eliad's proof.}
\begin{align*}
b^{-1} \circ B(X')
&= \frac1{\nu} \ln\paren{1 + B(X')\cdot \frac{e^{\nu}-1}{c}}\\
&\leq \frac1{\nu} \ln\paren{1 + (c+e^{\nu}B(X))\cdot \frac{e^{\nu}-1}{c}}\\
&= \frac1{\nu} \ln\paren{e^{\nu} + e^{\nu}B(X) \cdot \frac{e^{\nu}-1}{c}}\\
&= 1 + \frac1{\nu} \ln\paren{1 + B(X)\cdot \frac{e^{\nu}-1}{c}}\\
&= 1 + b^{-1} \circ B(X)
\end{align*}
\end{proof}
To DP threshold $B(X) \leq 0$, we apply
$(\varepsilon,\delta)$-DP-thresholding $\leq 0$ to $b^{-1}\circ B(X)$.
When $B(X)=0$ (and $b^{-1}\circ B(X)=0$), we pass with probability $1$ and otherwise the pass probability is minimum.
We now ask what is the maximum possible $B(X)$ that yields a nonzero probability of passing.
The maximum pass value in terms of
$b^{-1}\circ B$ is $2L(\epsilon,\delta)+2$, where
$L(\epsilon,\delta) \approx \frac{1}{\varepsilon} \ln\left(\frac{\varepsilon}{2\delta} \right)$. We then have
\[ B_{\max} \gets b(2 L(\epsilon,\delta) +2)
\leq c \frac{e^{y \nu}-1}{e^\nu -1},
\] where
$y\leq 2(1+ \frac{1}{\varepsilon} \ln\left(\frac{\varepsilon}{2\delta} \right))$.
In the regime $y \nu < 1$ we get
\begin{equation} \label{Bmax:eq}
B_{\max}\leq \approx c y \approx
2 c(1+ \frac{1}{\varepsilon} \ln\left(\frac{\varepsilon}{2\delta} \right))
\end{equation}
In our application we will use
$\nu = \frac{2\ln(1/(2\eta))}{n}$, where $\eta \ll 1/n$ (but does not need to be smaller than $1/n^2$) and is discussed later.
The condition $y\nu < 1$ gives as a bound on $n$: \Enote{So, we assume in this part that $\nu = \frac{2\ln(1/(2\delta))}{n}$?}\ECnote{I replaced one $\delta$ with $\eta$ that is between $1/n$ and $1/n^2$ and is sufficient. Essentially we want $n\eta$ to be very small since $c=1+n\eta$. This is different than the requirements dictating choice of $\delta$. In principle like in the tuples, the $n$ (number of elements) is much smaller than the number of points in the dataset. And the $\delta$ is determined by dataset size. So it makes sense to separate $\delta$ and $\eta$}
\[
\frac{2\ln(1/(2\eta))}{n} 2(1+ \frac{1}{\varepsilon} \ln\left(\frac{\varepsilon}{2\delta} \right) < 1
\]
To make this happen we need roughly \[ \frac{n}{\ln n} > 4 \frac{1}{\varepsilon} \ln (1/(2\delta)\]
So for $\delta \ll 1/n$ we get
$n> 4 \frac{1}{\varepsilon} \ln (1/(2\delta) (\ln(4/\varepsilon) + \ln\ln (1/\delta) $
[Comment: for a more refined analysis we need to consider the probability of passing with each $y$
$p(y) := \pi^*_{\tau - y +2L+2}$. This allows us to account for larger $B$'s that occur with lower probability. ]
\subsection{Tool: Generalized group privacy}
The following is a sketch of a generalized group privacy lemma.
{\bf To Do:} For the $O(n\log n)$, for pure DP, or for tighter analysis we need to extend it to allow larger $\Delta$ mitigated with lower success probability (rather than fixed bound on $\Delta$).
\begin{lemma}
Let $A(X)$ be a randomized algorithm on sets $X$ such that when
$X$ and $X'$ differ in one element then $A(X)$ and $A(X')$ are
$(\varepsilon,\delta)$-indistinguishable.
[We use a discrete universe of elements of size $n$, to simplify presentation]
Consider now a vector of probabilities
$(p_1,\ldots,p_n)$ and the algorithm $A^*(p)$ that constructs $X$ by drawing each $i\in [n]$ independently with probability $p_i$ and returns $A(X)$.
Consider $p$ and $p'$ such that $\norm{p-p'}_1 = \Delta$. Then
$A^*(p)$ and $A^*(p')$ are $(\Delta\varepsilon, \Delta e^{\Delta\varepsilon} \delta)$-indistinguishable.
[Comment the above is not precise, we lose a little more, but not much more. This with respect to $p$ being a 0/1 vector (standard group privacy).]
\end{lemma}
\begin{proof}
Sketch:
Without loss of generality we can treat $p$ as the $0$ vector and $p'$ to be a vector with L1 norm $\Delta$. This because the set difference can be viewed as independent of the particular outcome of $p$. More formally,
we can consider all randomizations (say draw a random value for each element $i$ and include if below $p_i$) that resulted in a fixed draw with $p$. Then and the respective possibilities from $p'$ and the respective distribution on set differences size.
If the $L1$ norm is $\leq \Delta$ then, roughly, the probability that we get a set difference of $i\Delta$ is bounded by $1/(e i!)$ using the Poisson approximation to binomial distribution (good bound when $\Delta=1$), noting that binomial with $pn=1$ is worst case for the spread. \Enote{so again, you assume that each element is chosen w.p. $p = \frac{\Delta}{n}$ (since this should be the worst case), and that $Bin(n,p = \frac{\Delta}{n}) \approx Poisson(\Delta)$.}
[note: The above will be helpful also when we incorporate success probabilities. as set difference dist holds when conditioned on a fixed outcome of $p$, including outcomes that result in "success"]
We now do group privacy analysis. First fixing the outcome on $p$. Then considering the possible set differences and their probabilities. For each set difference, we can apply the known standard group privacy bound.
The calculation below is for pure differential privacy $\delta=0$. For that the standard bound is that $\varepsilon$ increases by a multiplicative factor of set difference.
We get the effective privacy parameter $\varepsilon^*$ so that
\[
e^{\varepsilon^*} \leq \frac{1}{e} \sum_{i\geq 0} \frac{1}{i!}e^{i\Delta\varepsilon}
\]
(summing over probability of set difference $i\Delta$ times the group privacy bound for that set difference (multiplier $e^{i\Delta\varepsilon}$))
To Do: Do the calculation with $\delta$. Note that as the function $e^x$ is concave, so we are losing a little over standard group privacy with $\Delta$. But since the values are fairly concentrated -- due to independence -- and the $1/i!$ drop -- we should not lose much.
Without independence we would be forced to use Markov and would get $1/i$ that does not drop fast enough.
To Do: Some extension needed for scenario with sampling. We don't have a hard bound $\Delta$. We have possibility of larger $\Delta$
but combined with lower success probabilities.
So $\Delta$ is not bounded. But the probability of reporting decreases sharply with $\Delta$. This should also similarly work.
\end{proof}
\section{$O(n^2)$ Good elements algorithm}
The good elements algorithms has two components. The first is to identify a set $T_G$ of good elements by testing each element. We also compute the expected value $B:= E[|T_G|]$ of the number of good tuples. The second is to decide, based on the $B$, if to fail (report nothing) or succeed and report $T_G$.
\paragraph{Computing set of good elements $T_G$}
The input is a set of elements $T$ and a "like" predicate $x\sim y$.
For each element $x$ we consider
the number of elements in $T$ that like $x$:
\[A(x) := \sum_{y\in T} I_{x\sim y}\ .\]
\begin{enumerate}
\item Input: A set of elements $T$ with $|T|=n$. A predicate on elements $x\sim y$.
\item Output: A set of good elements $T_G\subset T$ that must include all perfect elements $A(x)=n$, and may not include any bad element $A(x)\leq n/2$. Also compute the value $B :- E[|T_G|]$ for that data set.
\item $B\gets 0$ (initialize value of $E[|T_G|]$)
\item Set $L'$ such that $2L'+2 = n/2$
\item Set $\eta \ll 1/n$ and choose $\nu$ to be smallest such that $L(\nu,\eta) = L'$ (We have $\nu \approx \leq 2 \ln(1/(2\eta))/n$.)
\item For each tuple $x \in T$:
\begin{enumerate}
\item $A(x) \gets \sum_{y\in T} I_{x\sim y}$.
\item If $(\nu,\eta)$-DP thresholding of $A(x) \geq \tau=n$ then $T_G \gets T_G \cup \{x\}$ (element $x$ is good).
\item $B \gets B+ \pi^*_{n - A(x) +2L+2}$ (probability of $x$ being labeled good by DP-threshold)
\end{enumerate}
\item
$T_B \gets T\setminus T_G$. Return $T_G$ and $B$
\end{enumerate}
\subsection{Local sensitivity analysis}
The good elements set $T_G$ is drawn according to a vector of probabilities
\[
r(T) := (p(A(x))_x = (\pi^*_{n - A(x) +2L+2})_{x}\ .\]
Recall that the probabilities and $L$ depend on $(\nu,\eta)$ used in the DP thresholding algorithm.
(For convenience we use vector notation while treating these vectors as sparse over the universe of elements. Keeping in mind that the universe might be continuous)
Recall that $E[|T_G|] = \norm{r(T)}_1$.
For two neighboring datasets $T$ and $T'$ ($|T\setminus T'|\leq 1$ and vice versa). We can consider the vectors $r(T)$ and $r(T')$.
We bound the extent in which the expected number of bad elements can increase between neighboring datasets. We also bound the L1 distance of the probability vectors.
\begin{lemma}
Let $T$ and $T'$ be neighboring.
Then
\begin{align}
\norm{r(T')}_1 &\leq 1+e^\nu \norm{r(T)}_1+ n\eta \\
\norm{r(T') - r(T)}_1 &\leq 1+ (e^\nu-1) \norm{r(T)}_1+ n\eta
\end{align}
\end{lemma}
\begin{proof}
Consider $x\in T\cap T'$. We have $A_T(x) = A_{T'}(x) \pm 1$ and
$q(A_{T'}(x)) \leq q(A_T(x)) e^\nu + \eta$ and
$p(A_{T'}(x)) \leq p(A_T(x)) e^\nu + \eta$ (holds by \cref{ratioprop:eq}).
\begin{align*}
E[|T'_B|] &= \sum_{x\in T'} q(A_{T'}(x)) \leq 1 + \sum_{x\in T'\cap T} q(A_{T'}(x)) \leq
1 + \sum_{x\in T'\cap T} q(A_{T}(x)-1) \\
&\leq 1+ e^\nu \sum_{x\in T'\cap T} q(A_T(x)) + n\eta \leq 1 + e^\nu E[|T_B|] + n\eta
\end{align*}
This because the element that is different can contribute at most $1$ and for all other tuples $A(x)$ changed by at most $1$ and we use~\eqref{ratioprop:eq}.
Similarly
\begin{align*}
\norm{r(T') - r(T)}_1 &\leq
1+ \sum_{x\in T\cap T'}
q(A_{T'}(x))-q(A_{T}(x)) \\
&\leq 1+ (e^\nu-1)\sum_{x\in T} q(A_{T}(x)) + n \eta =
1+ (e^\nu-1) \norm{r(T)}_1+ n\eta
\end{align*}
Comment: Note that for $q > 1/2$, we can use a tighter bound. This is relevant if we use $T_G$, that is large. We can bound
$E[|T'_G|] - E[|T_G]]$
it in terms of the smaller of $T_B$ or $T_G$ or more finely, by considering $q$. We get
\[
E[|T'_B|] - E[|T_B]] \leq 1+n\eta + (e^\nu-1)\min\{E[|T_B|],E[|T_G|] \}
\]
\end{proof}
[Comment. We would want to balance $\eta$ and $\nu$ given $L$ as to minimize the L1 distance in our regime of interest]
\subsection{putting it together}
Comment: Need to set privacy parameters so it composes to what we want.
We apply DP thresholding with varying local sensitivity to $E[|T_B|]$ , with $c=1+n\eta$.
\begin{enumerate}
\item Apply the good elements subroutine to compute the set $T_G$ and $y \gets E[|T_B|] = \sum_{x\in T} q(A_{T}(x)$
[With that subroutine we need to use $\eta \ll 1/n$ and choose $\nu$ as low as we can given that and $n$. Recall that we get $\nu \approx\leq 2 \ln(1/(2\eta))/n$]
\item Recall that the local sensitivity of $E[|T_B|]$ is $(1+n\eta) +e^\nu E[|T_B|]$. We apply thresholding with varying local sensitivity, using $\nu$ and $c=1+n\eta$: We set $m \gets b^{-1}(y)$ \Enote{has sensitivity $1$}, apply $(\varepsilon,\delta)$ DP thresholding with $\leq 0$. If we fail, we return $\bot$. Otherwise, we return $T_G$.\Enote{So by thresholding you mean that we compute $\hat{m} = m + {\rm Lap}(1/\varepsilon)$, and we pass the test if $\hat{m} \leq 0$?}\ECnote{Yes. Can use Laplace or truncated Laplace. Only that this is lossy and wanted to squeeze the constants. So we use a precise optimal thresholding}
By design, the threshold "pass" bit is $(\epsilon,\delta)$ DP.\Enote{Do we need the $\delta$ here?}\ECnote{We could remove it. It seems. I did not check carefully. But then we need to work with high probability. Which we will need to anayway for the sampling.... And the analysis is not completely clean 0/1. So it is cleaner to first use $\delta$.}
\item
We obtain that a pass is possible (see \eqref{Bmax:eq}) only if $y\leq B_{\max} = 2(1+n\eta)(1+ \frac{1}{\varepsilon}\ln (\varepsilon/(2\delta))$.
This means that the $L1$ distance when passing between neighboring datasets is at most
\[\Delta \leq c + (e^\nu-1) B_{\max} \approx 1+ n\eta + \nu B_{\max} .\]
(approximation holds when $\nu\ll 1$, that is, $2\ln(1/(2\eta) \ll n$ which holds for small values of $n$).
Now note that $\Delta$ should be close to $1$ when $\eta \ll 1/n$ and
$\nu B_{\max} \ll 1$. Substituting the bounds we have on $\nu$ and $B_{\max}$ we get
\[
n \gg 2(1+n\eta)(1+ \frac{1}{\varepsilon}\ln (\min\{1,\varepsilon\}/(2\delta)) 2 \ln(1/(2\eta))
\approx
4 \frac{1}{\varepsilon}\ln n \ln(1/(2\delta)
\]
(for say $\eta = 1/(n\ln n)$.
We apply the generalized group privacy with $\Delta$ to obtain the claimed property.
\end{enumerate}
Comment: We can optimize a bit more the minimum $n$ (factor of 2) by not using the maximum $B_{\max}$ and accounting in the privacy analysis for the lower reporting probability when $b^{-1} B$ increases.
Comment: We can use coordinated sampling techniques to have a non-negatively correlated instead of independent choice so that the number of bad elements is very close to its expectation and the actual difference between neighboring datasets is even more concentrated around the L1 distance. But not sure it is worth the work.
\section{$O(n\log n)$ solution for friendly elements }
Incorporating sampling:
Now we want to be efficient. Instead of counting exactly the number of liking tuples, for each $x$ we sample $S$ tuples. We then compute $A(x)$ as the number of tuples in $S$ that like $x$. (better with different sample for each $x$ for independence).
\paragraph{Sample size:}
The sample size $s$ should be large enough so that say $2/3$ likes on the sample imply with very high probability that there are $> n/2$ likes on the full set.
We need the sample to be at least $a \log(n)$. We then do DP thresholding on the stretch with $2L''+2 = s/3$ . And then $\nu'$ the solution of $L(\nu,\delta) = L''$. We get $\nu' \approx 3\ln(1/2\delta)/s$.
The sampling will decrease the ratio of neighboring $q(i)$ by $s/n$, so we get the guarantees we need.
So the only constraint on sample size is ensuring that $2/3$ filters out tuples with $\leq n/2$ friends.
With the crude multiplicative Chernoff bound $\delta=1/3$ and we get $s \approx 72\ln(n)$ to guarantee that probability $1/n^2$ of passing a tuple with $\leq n/2$ friends. But this constant can be tightened by using the raw Chernoff bounds.
\paragraph{thresholding by $|T_B|$ and not its expectation}
Thresholding the actual number of bad tuples $T_B$ and not its expectation.
Now L1 norm of neighboring and also worst-case local sensitivity can increase. But we don't need to account for worst case as this is mitigated with higher fail probability when $E[|T_B|]$ is larger. We need to do the computation but it should balance out.
Actually might also reduce the factor 2 due to "largest" $E[|T_B|]$ that can pass the threshold. Instead when mitigating by likelihood it should help.
\section{Comments on application of private friendly elements}
In the applications for clustering, each element is a $k$-tuple obtained by applying a clustering algorithm on a part of the data.
Comments:
If we can assume that the algorithm that returns the tuples exposes privacy of points at random (return a random member of the cluster) we can integrate this into the privacy analysis which will allow for a larger $\delta$ (in terms of number of inverse (poly) number of tuples rather than numbers of points). The number of tuples needed does not grow with the number of points in the datasets.
We bound sensitivity in terms of neighboring tuples. If the algorithm that generates tuples has the property that one member of the tuple changes on neighboring point sets then we can save in the group privacy argument.
The solution can be modified to problem specs where elements are considered perfect if the like all except for $a$ elements and require success when at most $b$ elements are not perfect. This extends the applications.
In application to clustering. We still can proabbly avoid solving $n$ small problems. In the non-private setting, one small problem suffices.
What we can do is emulate. Solve one problem. Then process and "emulate" output of $n$ problems. This involves discarding all ambiguous points. Then aggregate.
\printbibliography
\section{Introduction}
Points for intro:
\begin{itemize}
\item
Friendly Elements Problem
\item
Tradeoff between input size and "stability"
\item
Explain stability in terms of an L1 norm between output probabilities.
We extend group privacy analysis to this setting (may have other applications)
\item
Structure of Friendly Elements solution: Returned elements and
Private Success bit: Allows for a very high stability and also guarantee that when success we remove only few points, the minimum needed for private success bit.
\item
Sample complexity (input size): The bottleneck is the private success bit. There is a counting "lower bound"
\[
\ell(\varepsilon,\delta,\beta) := \frac{1}{\varepsilon}\ln(\frac{\varepsilon}{2\delta\max\{\beta,\delta\}})\]
The sample complexity we obtain for FriendlyCore is a small constant factor on that. In particular, no dependencies on dimension or particulars of the metric space.
\item
Two common input models: Swap model (points get replaced but their number is known non-privately) and Add/Delete.
The literature has generic "reductions" between the models, but these reductions are lossy in terms of constants. We analyse both to avoid that.
The set we return contains almost all point except for few as needed for DP
\item
Instead of specifying $r$, can search for "right" $r$ cheaply....(quantify) Some applications need that. Input size bottleneck then come from that search.
\item
Applications: Friendly Elements paradigm. Averaging.
Elaborate: Sample-and-aggregate (e.g. clustering, we get tuples).
\item
Important point:
\end{itemize}
Extensions:
\begin{itemize}
\item Computation. Current presented for $n^2$ "$f$" computations. Can be reduced to $n\log n$.
\item Can be used in other regimes. Specified number of "outliers" (with DP counting below that). This for at most $n/2$ outliers. Also can be used without "success" bit: No restriction on size of output set $T_G$. Set is still stable but L1 can grow to a small constant.
\end{itemize}
\subsection{Related work}
{\bf Enclosing/densest ball methods:}
\url{https://arxiv.org/abs/1707.04766}
Solves our problem by computing a bounding ball that fits the input. The private ball can then be used to return a "stable" set of points (all points inside the ball).
But these methods are impractical due to algorithmic complexity and large constant factors and the approach of computing a private ball implies additional dependencies on the dimension. Some methods are limited to Euclidean spaces in that they integrate dimensionality reduction and locality sensitive hashing.
The downstream tasks may not exhibit dependencies on the dimension at all or may use other metric spaces (e.g. within some sample-and-aggregate methods) or may do it with much lower constant factors overheads (e.g., averaging).
The computation of a center of a ball is a more complex task with additional asymptotic dependencies on the dimension. The methods are limited to spaces such as Euclidean that support dimensionality reduction or locality sensitive hashing. While the downstream applications may tasks in applications sometimes dependence in the dimension is necessary for downsteam aggregation tasks, it can be
by also performing an aggregation (finding a Euclidean ball that contains many points).
The ball can be then used to return a stable set. But the problem of returning a stable set can be solved more efficiently and as we show, without dependence on the dimension.
Somewhat different problem. Large constants. More complex. Worst asymptotics for our setting??
In practice, clipping norms and winsorizing are used to limit the diameter of the input domain. These are classical techniques in statistics to control empirical variance.
But to apply clipping effectively we also need to know a "center" point, which is sometimes the problem we are out to solve.
{\bf Mean estimation}, where it is assumed that data originates from a distribution.
Karwa and Vadhan
\url{https://arxiv.org/abs/1711.03908} proposed methods for mean estimation. In particular, for Gaussians with known veriance they proposed partitioning the interval to segments of length $\sigma$ and using a private sparse histogram to identify the densest bin. The scaling to multi-variate setting (higher dimentiosn) is not efficient.
Steinke and Bun proposed a trimmed mean approach through smooth sensitivity in concentrated DP
\url{https://arxiv.org/abs/1906.02830}. It is not comparable.
CoinPress \url{https://arxiv.org/pdf/2006.06618.pdf}
based on
\url{https://arxiv.org/abs/2001.02285}
considers multi-variate Gaussians and propose and iterative search that (again) computes an enclosing ball, trimms points, and computes a private average using the smaller radius. Coinpress is practical and outperforms prior methods. It demonstrates little privacy loss for estimating Gaussian parameters.
Coinpress performs the diameter reduction in an entangled way with the harder problem of mean estimation. This makes sense when the latter is the only application. We show that diameter reduction is an easier problem with a faster solution and a helpful tool for a variety of downstream tasks that may or may not be dimension-dependent. Moreover, our solution applies in general (pseudo) metric spaces.
We circumvent the need to compute a "center" by using pairwise distances. Our approach surprisingly reduces diameter reduction to a one dimensional problem. \ECnote{Need to see whether or not we obtain better results than CoinPress for Gaussian mean estimation}
\section{The friendly elements problem}
The input is a set of elements $T$ of size $|T|=n$.
We have a symmetric predicate $\text{like}$ $x \leftrightarrow y$ on pairs of elements.
We are interested in an algorithm with the following properties:
(i) the output is either a "fail" or "success". If "success," the output also includes a set $T_G$ of "good elements." The set $T_G$ must include all {\em perfect} elements (a perfect element likes all elements) and may contain only elements that like more than half of elements.
(ii) When all elements are perfect, the success probability is $1$.
In addition, we have the following requirements for $(\varepsilon,\delta)$ and $\Delta>1$.
We say that two datasets $T$ and $T'$ are neighbors if they differ in at most one element.
(iii) The fail/success bit is $(\varepsilon,\delta)$-indistinguishable for any two neighboring datasets. Namely, if $p$ and $p'$ are the success probability of $T$ and $T'$ then $p \leq e^\varepsilon p' + \delta$ and
$(1-p) \leq e^\varepsilon (1-p') + \delta$.
(iv) The algorithm is randomized so that each element in $T$ has an independent probability to be included in $T_G$ (perfect elements have probability $1$ and elments with $n/2$ or fewer likes have probability $0$). If $p$ and $p'$ are the vectors of probabilities of two neighboring datasets then $\norm{p-p'}_1 \leq \Delta$.
We present two solutions for friendly elements. The first one performs ${n \choose 2}\approx n^2$ like queries. The second
(that provides high probability guarantees for requirements (ii) and (iv)) performs $O(n\log n)$ like queries.
Note that in terms of asymptotic dependence on $n$, this is the best we can hope for simply satisfying properties (i)-(ii).
The effect of requirements (iii)-(iv) and the privacy parameters $(\varepsilon,\delta)$ and $\Delta$ is that they imply a minimum value on $n$ that is needed to support these specifications.
Our goal is to understand this limit and design a method that works for small $n$.
[To do: For the more efficient $O(n\log n)$ method, we need to refine the hard bound in (iv) on $\norm{p-p'}_1$ with allowing for higher values and lower success probability.
Specifically, we consider the distribution of $\Delta$ and respective probabilities of success with this $\Delta$. Larger $\Delta$ is allowed with smaller success probabilities.
]
We first describe potential applications of friendly-elements.
We then proceed with solutions, starting with some basic tools that we use.
\section{Applications}
Elements $T$ are points in a metric space (or pseudo metric).
1. DP outlier removal: For given $r$, test if "almost all" pairs are within distance $r$. Return a "stable core" $T_g\subset T$ of points with diameter $2r$ or fail. For this applications we might want to tweak requirements to imply success probability $1$ also with few outliers.
2. More accurate DP "center" point. With DP, typically there is an additive error that depends on the diameter of the space $\Lambda$. For example with DP averaging the error is proportional to $\Lambda \sqrt{d}/(n \varepsilon)$.
But suppose we are only interested in a "center" when all points are within a distance $r\ll \Lambda$ except perhaps for a few outliers.
The friendly elements is a way to do this. Two elements are friends if their "distance" is at most $r$. The output (if "success") are elements $T_G$ with guaranteed max distance $2r$ between any pair. Therefore, to the output $T_G$ we can apply any "centering" method with $\Lambda=2r$.
3. Some applications, we do not have a target $r$. We are interested in returning $r_{eff}$ that is within say a small factor than the smallest one that yields "success." (imprecise).
So the output space can only include $2^i r_{\min}$ for some integral $i$. A naive way to do this is by sequentially testing $2^i r$ until there is success. Naively using group privacy, this results in $\log(r_{eff}/r_{\min})$ checks and loss.
A better way is to do a binary search over $i$. We first double $i$ until "success" to determine essentially $\log(r_{eff}/r_{\min})$ within a factor of $2$. We then do a binary search on that domain. Overall, the group privacy overhead is $\log\log(r_{eff}/r_{\min})$.
We still did not specify how the "decisions" are made in the search. Also, group privacy here seems too coarse as there is a fine structure to exploit.
[Comment: On real datasets we can expect $\log\log(r/r_{\min}) \leq 8$ but still this is a factor on the number of elements. There should be a better way. ]
\Enote{I didn't follow it...}\ECnote{Yes, you are right. I erased What was there as it does not work. There is also a lower bound of $\log^* |X|$...) }.
\subsection{Sample-and-aggregate framework}
Sample-and-aggregate \cite{} is a framework for DP solution of problems on instances where a solution for a small random sample of the data is very likely to be a good solution for the full dataset.
We use friendly-elements as a component in a sample-and-aggregate framework.
For a dataset of size $N$, the smallest sample size $s$ that is needed for this to hold is a property of the dataset. When this holds, we can get $n = N/s$ disjoint random samples. The framework applies a non-private algorithm to each of the $n$ samples and then privately "aggregates" the results. Solutions that work with smaller $n$ are better as they apply to more datasets.
To apply the frameworks we need
\begin{itemize}
\item
A non-private algorithm $A$ that inputs a dataset $D$ of points and returns a solution $x \gets A(D)$.
\item
A similarity predicate between solutions $x_i \leftrightarrow x_j$. Similarity is reflexive -- we have $x \leftrightarrow x$ for all $x$. Informally, similarity means that the solutions are interchangeable (one is almost as good as the other with respect to $D$).
\item
An $(\varepsilon,\delta)$-DP aggregator $C(X)$: Takes an input a set $T_G$ of at least $|T_G|\geq 0.9n$ solutions so that for any $x,y \in T_G$ either $x \leftrightarrow y$ or there exists $z$ such that $x \leftrightarrow z$ and $z \leftrightarrow y$.
Aggregator returns a private solution $x^* \gets C(X)$. Informally, the interpretation is that $x^*$ is 2-hops "similar" to any of the $x_i$'s.
\end{itemize}
Note that in the original sample-and-aggregate formulation, the aggregator needs to be "robust." It is applied directly to the solutions from all samples. That set is allowed to contain some "bad" solutions. Our friendly-elements module allows for the use of a simpler aggregator when all solutions are "good."
\begin{enumerate}
\item Set $D$ of $N$ data points, $n$
\item
Randomly partition $D$ into $n$ equal-size sets $D_i$.
\item
Apply the non-private $A$ to each part to obtain $x_i \gets A(D_i)$. Note that from neighboring inputs $D$ and $D'$ (differ on one data point) we obtain neighboring sets $T=\{x_i\}$ and $T'=\{x'_i\}$.
\item
Apply friendly-elements to the set $T$. We either fail or obtain a subset $T_G$ with almost all of $T$ that satisfies the input requirements of the aggregator. Recall (property of friendly-elements) that this probabilistic output $T_G$ is stable.
\item
Return $C(T_G)$
\end{enumerate}
When the dataset has the property that the $n$ solutions are similar, we succeed and return a private solution that is similar to them.
\section{Preliminaries: Tools}
\subsection{Tool: optimal DP thresholding}
Let $A$ be a function that inputs a dataset $X$ and returns a real number $A(X)$.
We have the property that $A$ has sensitivity $1$: On ``neighboring'' datasets $X$ and $X'$, $|A(X)-A(X')| \leq 1$.
We seek an $(\epsilon,\delta)$-DP thresholding predicate as follows: When $A(X) \geq \tau$ the output is "pass". Otherwise, when $A(x) <\tau$, the probability of "pass" is minimum.
[Comment: The problem can be solved with truncated Laplace noise, but not optimally. This roughly by adding $(1/\varespilon) \ln(1/\delta)$ plus Laplace $[1/\varepsilon]$ noise (truncated to remove $\delta$ tails).]
The optimal DP thresholding scheme \cite{} is as follows:
Let
\begin{equation}\label{Ldef:eq}
L(\epsilon,\delta) := \frac{1}{\varepsilon} \ln\left(
\frac{e^\varepsilon -1 +2\delta}{\delta(e^\varepsilon +1)}
\right) \approx \frac{1}{\varepsilon} \ln\left(\frac{\min\{1,\varepsilon\}}{2\delta} \right)
\end{equation}
we assume that $\epsilon$ and $\delta$ are such that $L$ is an integer.
We define$(\pi^*_i)_{i\geq 1}$ as follows: \Enote{So the $L$ is chosen to be the value such that $\pi_{L+1}^*$ is the same in both cases? i.e., $\delta \frac{e^{\varepsilon (L+1)}-1 }{e^\varepsilon -1} = 1- \delta \frac{e^{\varepsilon (2L+2-(L+1))} -1}{e^{\varepsilon} -1}$?}\ECnote{This works for continuous $i$. There was also some typo $+2$ to $+1$ in the exponent that I fixed. On the points $i = L,L+1$ $\pi^*_L = 1 - \pi^*_{L+1}$.}\ECnote{This is simply the optimal (maximum) solution of the DP constraintes $\pi^*_i \leq e^\varepsilon \pi^*_{i-1} + \delta$ and $(1-\pi^*_i) \leq e^\varepsilon (1-\pi^*_{i+1}) + \delta$}\Enote{Why is it the optimal solution?}\ECnote{It is the optimal solution when $i$ is restricted to be an integer (prior paper). Making values as large as possible given that $\pi^*_0=0$. It is just solving the system of equations $\pi^*_i = \min\{e^\epsilon \pi^*_{i-1}+\delta, e^{-varepsilon}(\pi^*_{i-1}+\delta-1)+1 \}$. We can then extend it to continuous. This is because of the perhaps unclean definition of neighboring $<1$. If we require that closer datasets than $1$ distance are more similar (depends on distance) we get the nicer continuous opt. But for our purposes we can work with the step form. }
\begin{equation} \label{pistar:eq}
\pi^*_i = \left\{ \begin{array}{ll}
0 &\; \text{\small $ i \leq 0$ }\\
\delta \frac{e^{\varepsilon i}-1 }{e^\varepsilon -1} &\; \text{\small $0\leq i\leq L+1$ }\\
1- \delta \frac{e^{\varepsilon (2L+1-i)} -1}{e^{\varepsilon} -1} &\; \text{\small $L+1 \leq i \leq 2L+1$ }\\
1 &\; \text{\small $i \geq 2L+1$ }
\end{array}\right.
\end{equation}
{\bf Extension to continuous (discrete)}
The solution extends to $\pi^{c*}_x$ for a continuous $x\geq 0$: For $x\in [0,2L+1]$ we define $\pi^{c*}_x = \pi^*_{\lceil x \rceil}$.
This is the (simultaneous) pointwise maximum solution at $x\geq 0$ for the constraints
\begin{align*}
\pi*_0 & = 0\\
\pi^*_x &\leq e^\varepsilon \pi^*_{x'} + \delta\, \; \forall |x-x'|\leq 1 \\
(1-\pi^*_x) &\leq e^\varepsilon (1-\pi^*_{x'}) + \delta\, \; \forall |x-x'|\leq 1
\end{align*}
Explanation: since $\pi^*$ is non-decreasing wlog, it suffices to consider
$\pi*_x = 0$ for $x\leq 0$, $\pi^*_x \leq e^\varepsilon \pi^*_{x-1} + \delta$ and $(1-\pi^*_i) \leq e^\varepsilon (1-\pi^*_{i+1}) + \delta$, solve only for integral values to get $\pi^*_i$, and verify that the extended solution holds for real values.
{\bf Extension to continuous (smooth)}
An alternative solutions that is continuous and differentiable and does not make assumption on integrality of $L$ is the following. This is also nicer to work with when $n$ is not given and we work in the deletion/addition model. This is not optimal at all points \Enote{What do we actually lose here? Why \cref{ratioprop:eq} holds around the middle (say, $x = M - 1/2$?)}\ECnote{I just verified it}
\Enote{Ok. Another question: Last meeting we wanted to consider a stronger privacy. That is, the neighboring databases may have different size (by at most 1). In that case, the resulting $M$, $\varepsilon$ and $\delta$, which are a function of $n$, may vary in the two executions. Do you have any hint how we should handle it? Should we estimate the databse size by computing $\tilde{n} = n + {\rm Lap}(1/\varepsilon)$ and working with it? Or is there a better solution?}
\ECnote{It seems that we do not need to noise $n$ even in the deletion/addition model. We are not explicitly using $n$ in the output. What really matters is how $n-A(x)$ changes between neighboring datasets. And for all but the "different" element it should still be $1$ (there might be a tiny diff in $\pi$ because $n/2$ changes also). }
\begin{equation}\label{M:eq}
M(\varepsilon,\delta) := \frac{1}{\varepsilon} \ln(\frac{1}{2\delta} (e^\varepsilon-1) +1)
\end{equation}
\begin{equation} \label{pistarcont:eq}
\pi^*_x = \left\{ \begin{array}{ll}
0 &\; \text{\small $ x \leq 0$ }\\
\delta \frac{e^{\varepsilon x}-1 }{e^\varepsilon -1} &\; \text{\small $ x \in [0,M]$ }\\
1- \delta \frac{e^{\varepsilon (2M-x)}-1 }{e^\varepsilon -1} &\; \text{\small $x \in [M,2M]$}\\
1 &\; \text{\small $x \geq 2M$}
\end{array}\right.
\end{equation}
The various $\pi^*$ (discrete, continuous) satisfy the following
\begin{equation} \label{ratioprop:eq}
\max\left\{\frac{\pi^*_i - \delta}{\pi^*_{i-1}}, \frac{1-\pi^*_{i-1} - \delta}{1-\pi^*_{i}}\right\} \leq e^\varepsilon
\end{equation}
{\bf Application: Specifying $M$}
In our main application we specify $M$.
We parametrize with $\gamma$ so that $\delta = 1/(\gamma M)$. We then solve \eqref{M:eq} for $\varepsilon$.
We get
$\varepsilon = C/M$ where
$C$ is the solution of $C = \ln\left(\frac{\gamma}{2} C +1\right)$.
This approximation holds for all $M$ such that $M \gg C$, that is, $\varepsilon \ll 1$.
\ECnote{In the application we use $n = 4M$ so we get that we need $n \gg 4C$ from this requirement (say $n>50$)}
Some values of $\gamma$ and corresponding $C$:
\begin{equation}\label{gammaC:eq}
\begin{array}{rr}
\gamma & C \\
\hline
3.4 & 1 \\
10 & 2.6 \\
10^2 & 6 \\
10^3 & 8 \\
10^4 & 11\\
10^5 & 13.5
\end{array}
\end{equation}
{\bf Application: Optimal DP thresholding for $\tau, \varepsilon, \delta$}:
\begin{enumerate}
\item $y \gets A(x)$
\item Return "pass" with probability
$p(y) :=\pi^*_{y-\tau +2M(\varepsilon,\delta)}$. Return "fail" otherwise (probability $q(y) := 1-p(y)$).
\end{enumerate}
Note that the DP thresholding always returns "fail" when $A(x) \leq \tau-2M$ and as required always returns "pass" when $A(X) \geq \tau$.
DP follows from property \eqref{ratioprop:eq}.
\paragraph{Reverse threshold:}
The mechanism can be flipped to provide
a "reversed" guarantee of passing with probability $1$ when $A(x) \leq \tau$. This is provided with pass probabilities:
\[
p(i) := \pi^*_{\tau - i +2M}\ .
\]
\paragraph{Probabilisitc $A(x)$:}
The formulation generalizes to when $A(X)$ returns values according to a distribution. In this case sensitivity $1$ informally means that for any $T$,
$\Pr[A(X')\leq T-1]\leq \Pr[A(X)\leq T]\leq \Pr[A(X')\leq T+1] $. That is,
we can map outcomes of $A(X)$ to outcomes of $A(X')$ with difference at most $1$.
Note that in this case the threshold mechanism may no longer be optimal, as it does not account for the randomization in $A$. For example, that randomization might already provide a private threshold. For our purposes however this suffices.
\subsection{Tool: DP Thresholding with a varying local sensitivitiy bound}
We now consider a situation where we have
a function $B(X) \geq 0$ with local sensitivity bound of the form
$B(X') \leq c + e^\nu B(X) $.
Equivalently,
$B(X')-B(X) \leq c + (e^\nu-1) B(X)$.
We want to apply DP thresholding so that if $B(X)=0$, pass probability is $1$ and otherwise pass probability is minimum.
We compute a monotone mapping $b^{-1}()$ of values
so that the function $b^{-1}\circ B(X)$ has sensitivity at most $1$.
\begin{lemma}
Let $B()\geq 0$ be such that on any two neighboring datasets $X$ and $X'$,
$B(X') \leq c + e^\nu B(X)$.
Define
\[
b^{-1}(y) := \frac{1}{\nu}\ln \left(1+ y \frac{e^\nu -1}{c}\right)\ .
\]
Then
the function $b^{-1}\circ B(X)$ has sensitivity 1. That is, on any two neighboring datasets,
$b^{-1}\circ B(X') \leq 1 + b^{-1}\circ B(X)$.
Also, $B(X)= 0$ if and only if $b^{-1}\circ B(X)=0$.
\end{lemma}
\begin{proof}
\ECnote{Replaced my "proof" with Eliad's proof.}
\begin{align*}
b^{-1} \circ B(X')
&= \frac1{\nu} \ln\paren{1 + B(X')\cdot \frac{e^{\nu}-1}{c}}\\
&\leq \frac1{\nu} \ln\paren{1 + (c+e^{\nu}B(X))\cdot \frac{e^{\nu}-1}{c}}\\
&= \frac1{\nu} \ln\paren{e^{\nu} + e^{\nu}B(X) \cdot \frac{e^{\nu}-1}{c}}\\
&= 1 + \frac1{\nu} \ln\paren{1 + B(X)\cdot \frac{e^{\nu}-1}{c}}\\
&= 1 + b^{-1} \circ B(X)
\end{align*}
\end{proof}
To DP threshold $B(X) \leq 0$, we apply
$(\varepsilon,\delta)$-DP-thresholding $\leq 0$ to $b^{-1}\circ B(X)$.
When $B(X)=0$ (and $b^{-1}\circ B(X)=0$), we pass with probability $1$ and otherwise the pass probability is minimum.
For the analysis we consider the following:
\begin{itemize}
\item
We now ask what is the maximum possible $B(X)$ that yields a nonzero probability of passing.
The maximum pass value in terms of
$b^{-1}\circ B$ is $2M(\epsilon,\delta)$, where
$M(\epsilon,\delta) \approx \frac{1}{\varepsilon} \ln\left(\frac{\varepsilon}{2\delta} +1\right)$. We then have
\[ B_{\max} \gets b(2 M(\epsilon,\delta))
\leq c \frac{e^{2M(\epsilon,\delta) \nu}-1}{e^\nu -1}.
\]
In the regime $2 M(\epsilon,\delta) \nu < 1$ we get
\begin{equation} \label{Bmax:eq}
B_{\max}\leq \approx 2 c M(\epsilon,\delta)
\end{equation}
The approximation $2 M(\epsilon,\delta) \nu < 1$ gives as a bound on $n$ (when we need this approximation). In our application $n\nu = 3C$ ($4C$ in the swap neighboring model). And $C$ is small (say $3$--$10$ see Table \eqref{gammaC:eq}). So we get $n> 6C M(\varepsilon,\delta)$. This is $20\times$ to $60\times$ $M(\varespsilon,\delta)$.
\item
The probability of "pass" for each value of $B\in[0, B_{\max}]$. This is needed for a more refined privacy analysis. For larger $B$ the probability of pass is lower. When $y = b^{-1}\circ B$ it is
$p(y) := \pi^*_{\tau - y +2M}$. We want to use these lower probabilities to tighten the analysis.
In the generalized group privacy analysis. The value of $B$ impacts the probability difference between neighboring datasets. But larger $B$ also has a lower pass probability. In particular, the pass probability with $y=M$ ($\approx B = B_{\max}/2$) is $0.5$. \ECnote{Potentially we can save up to a factor $\times 2$ in $n$ in this route.}
\end{itemize}
\subsubsection{Addition/deletion model}
Here we have the relations (Eliad's notation).
$\mu' \leq e^{c/m}\mu + \frac{2c}{\lambda}1.05 +1$. (this assuming $m>10c$ so $e^{c/m}-1 \leq 1.05 c/m +1$
We use $k := \frac{2c}{\lambda}1.05 +1$. \ECnote{Per below, we will have $c$ "small" and $m$ at least as large as $s_{\max}/2$, so with too small an $m$ we "fail".}
The growth in $\mu$ is largest when we increase $m$. When we go the other way around (a point is deleted)
and $m$ decreases, the mutiplicative factor is lower and $k$ is smaller by $1$. So to minimize steps always want to start with lower $m$ for same $q$.
We are interested in $\omega(q,m)$. This is the smallest number of hops (through neighbors) that we can get to dataset $\omega(q,m)$ from a dataset with $q=0$ (and some $m_0$) (where $m_0$ has "success" with probability clost to $1$ (or $1-\beta$).
This function by its definition and the properties above has sensitivity $1$.
So the relation we use allows for the smallest possible number of hops (allowing $\mu$ to grow as fast as possible).
We can treat this in a discrete way
\[
\mu_{i+1} = e^{\frac{c}{m_0 +i/2}} \mu_i +k\ . \]
This is an upper bound on $\mu$ that can be reached in $i$ steps when we start with data with $q=0$ and $m=m_0$.
We can use the differential equation
\[
\frac{\partial \mu}{\partial s} = (e^{c/(m_0 + s/2} -1)) \mu + k
\]
The solution has no closed form (uses exponential integrals) but can be approximated numerically.
In our use, we would want to DP threshold according to $\omega$ with $0$ having probability $1$.
We can upper bound $\omega(q,m)$.
One way to grossly approximate it (upper bound) is by "fixing" $m_0$ on its initial value, which makes for larger increase. We get
$\mu_{i+1} = e^{\frac{c}{m_0}} \mu_i +k$
The solution is
\[
\mu_s = k \frac{e^{s c/m_0} -1}{ e^{c/m_0} -1}
\]
The smallest $m_0$ can be for $s$ steps is $m-s/2$.
So we get
\[
\mu_s = k \frac{e^{s c/(m-s/2)} -1}{ e^{c/(m-s/2)} -1}
\]
So we set $\omega(q,m)$ to be the solution for $s$ of
\[
q = k \frac{e^{s c/(m-s/2)} -1}{ e^{c/(m - s/2)} -1}
\]
Let $s_{\max}$ be the maximum possible number of steps, roughly,
\[s_{\max} = \frac{1}{\varepsilon}\ln(1/(2\beta\delta))\ .\]
An even coarser bound for $s$ is the solution for $s$ of
\[
q = k \frac{e^{s c/(m-s_{\max}/2)} -1}{ e^{c/(m - s_{\max}/2)} -1}
\]
That is,
\[
s = \frac{m-s_{\max}/2}{c}\ln\left( \frac{1}{k} q \left(e^{c/(m - s_{\max}/2)} - 1\right)+1\right)
\]
The regime
\[ s_{\max} c \ll m - s_{\max}/2 \implies m \gg S_{\max}(c+1/2)} \]
Means that $s \approx q/k$. If $m$ is large we should not loose "privacy budget" to test that and then we apply success probability according to $s$.
But here we pay multiplicative $c$ on $s_{\max}$ for the size of $m$. Can we do better?
Perhaps we can handle smaller $m$ that is closer to $s_{\max}/2$. With larger $q$.
\subsection{Tool: Generalized group privacy}
The following is a sketch of a generalized group privacy lemma.
{\bf To Do:} For the $O(n\log n)$, for pure DP, or for tighter analysis we need to extend it to allow larger $\Delta$ mitigated with lower success probability (rather than fixed bound on $\Delta$).
\begin{lemma}
Let $A(X)$ be a randomized algorithm on sets $X$ such that when
$X$ and $X'$ differ in one element then $A(X)$ and $A(X')$ are
$(\varepsilon,\delta)$-indistinguishable.
[We use a discrete universe of elements of size $n$, to simplify presentation]
Consider now a vector of probabilities
$(p_1,\ldots,p_n)$ and the algorithm $A^*(p)$ that constructs $X$ by drawing each $i\in [n]$ independently with probability $p_i$ and returns $A(X)$.
Consider $p$ and $p'$ such that $\norm{p-p'}_1 = \Delta$. Then
$A^*(p)$ and $A^*(p')$ are $(\Delta (e^\varepsilon-1)), \Delta e^{\Delta(e^\varepsilon-1)} \delta)$-indistinguishable.
\end{lemma}
\begin{proof}
Let $V$ be drawn from Binomial $p$ and $V'$ from binomial $p'$. Let $\Delta = \| p-p'\|_1$.
Consider a joint probability space of $V$ and $V'$ as follows.
The joint space is a product space of the following over coordinates $i$.
We draw $u_i\sim U[0,1]$ and set $V_i=1$ if and only if $u_i \leq p_i$ and similarly $V'_i=1$ iff $u_i \leq p'_i$.
Note that when $p_i \leq p'_i$ there are three possible outcomes for $V_i V'_i$ that are $11$, $01$, $00$. And symmetrically outcomes $00$, $11$, and $10$ when $p_i > p'_i$. The probability of outcomes where $V_i\not= V'_i$ is $\Delta_i = |p_i-p'_i$.
We now consider a partition of the joint probability space.
The partition is also product over independent coordinates. Each coordinate $i$ has two possible subparts.
In a subpart for coordinate $i$, a piece has a fixed outcome for either $V_i$ or $V'_i$ as follows.
If $p_i < p'_i$ there are two possible subparts with probabilities $\tau_i,1-\tau_i$, where
$\tau_i := \frac{p}{1-(p'-p)}$.
\[
\begin{array}{cc}
V'_i = 1 & u_i \leq \tau_i\\
V_i = 0 & u_i > \tau_i
\end{array}
\]
Note that $\tau_i < p'$ and hence $V'_i=1$ for all outcomes in the first subpart (and possibly some outcomes in the second component....).
This because $p < p'$ iff $p(1-p') < p' (1-p')$ iff $p < p'- p'^2 + pp'$ iff $p < p'(1-p'+p)$.
Also note that trivially $\tau_i > p_i$ and hence $V_i=0$ for all outcomes in the second subpart.
Symmetrically, if $p_i > p'_i$
we use $\tau_i := \frac{p'}{1-(p-p')}$ and
there are two possibly subparts
\[
\begin{array}{cc}
V_i = 1 & u_i \leq \tau_i \\
V'_i = 0 & u_i > \tau_i
\end{array}
\]
The subparts have the property that there is a "fixed point" either the value of $V_i$ or the value of $V'_i$ are the same for all outcomes in the subpart.
\begin{lemma}
For each $i$ and subpart, the probability of $V_i\not= V_i$, conditioned on the subpart, is exactly $\Delta_i = |p'_i-p_i|$.
\end{lemma}
\begin{proof}
Consider the first subpart above with $V'_i=1$. We have $V_i=1$ when $u_i<p$. The conditional probability that $V_i = V'_i = 1$ is $\frac{p_i}{\frac{p_i}{1-(p_i'-p_i)} }= 1 - (p_i'-p_i)$. Hence the conditional probability that $V_i \not= V'_i$ is $\Delta_i = p'_i-p_i$
\end{proof}
We work with the partition of the joint probability space that is a product space of one of the two subparts from each coordinate.
We now get to the group privacy analysis.
For possible outputs $T$ of Algorithm $A$ we relate the probabilities that $A^*(p)\in T$ and
that of $A^*(p')\in T$.
Note that
\[
\Pr[A^*(p)\in T] = \sum_{F\in parts} \Pr[F] \Pr_{(V,V') \sim F} [A(V) \in T]\ .
\]
And symmetrically for $p'$.
\[
\Pr[A^*(p')\in T] = \sum_{F\in parts} \Pr[F] \Pr_{(V,V') \sim F} [A(V') \in T]\ .
\]
The following Claim will complete the proof
\begin{claim}
Within each part $F$,
\[ \Pr_{(V,V') \sim F} [A(V') \in T] \leq
e^{\Delta(e^\varepsilon-1)}} \Pr_{(V,V') \sim F} [A(V) \in T] + \Delta e^{\Delta(e^\varepsilon-1)} \delta
\]
\end{claim}
\begin{proof}
We consider part $F$. Let $C$ be the center vector of the part. It has some fixed coordinates $I\subset [n]$ of $V$ and some fixed coordinates $I' = [n]\setminus I$ of $V'$.
We now relate the probability $\Pr_{(V,V')\sim F} A(V) \in T$ and $\Pr[A(C)\in T$. This depends on the coordinates $I'$ where $V$ is different than $C$. Note that on all coordinates where the fixed point is from $V$, there is no difference that is, no contribution to hamming distance. On coordinates $I'$ where the fixed point is from $V'$ the difference is Binomial from $\Delta_i$. The relevant L1 distance is $\Delta' := \sum_{i\in I'} \Delta_i$.
Using the argument for L1, we get
For pure something like $\Pr[A(V)\in T] \leq \Pr[A(C)\in T] e^{\Delta'(e^\varepsilon-1)}$.
Note that the reverse inequality also holds. We should have
$\Pr[A(C)\in T] \leq \Pr[A(V)\in T] e^{\Delta'(e^\varepsilon-1)}$.
We similarly relate the probability that a vector $V'$ from the piece has $A(V') \in T$ to the probability that $A(C)\in T$. We similarly consider its hamming distance from $C$. Similarly, the distribution behaves like Binomials of $\Delta_i$ on the coordinates $I$. We denote $\Delta'' = \sum_{i\in I} \Delta_i$.
We get $\Pr[A(V')\in T] \leq \Pr[A(C)\in T] e^{\Delta''(e^\varepsilon-1)}$.
Note that $\Delta'+\Delta'' = \Delta$.
Now we combine the two inequalities to obtain
\[\Pr[A(V')\in T] \leq \Pr[A(C)\in T] e^{\Delta''(e^\varepsilon-1)} \leq \Pr[A(V)\in T] e^{\Delta'(e^\varepsilon-1)} e^{\Delta''(e^\varepsilon-1)}
= \Pr[A(V)\in T] e^{\Delta(e^\varepsilon-1)}
\]
\end{proof}
We sum this over all the pieces $F$ of the joint space
with the respective probabilities of the pieces to get the group privacy claim.
\end{proof}
The following is analysis with respect to a fixed vector. Group privacy within a piece.
\begin{proof}
Sketch:
Without loss of generality we can treat $p$ as the $0$ vector and $p'$ to be a vector with L1 norm $\Delta$. This because the set difference can be viewed as independent of the particular outcome of $p$. More formally,
we can consider all randomizations (say draw a random value for each element $i$ and include if below $p_i$) that resulted in a fixed draw with $p$. Then and the respective possibilities from $p'$ and the respective distribution on set differences size.
If the $L1$ norm is $\leq \Delta$ then, roughly, the probability that we get a set difference of $i\Delta$ is bounded by $1/(e i!)$ using the Poisson approximation to binomial distribution (good bound when $\Delta=1$), noting that binomial with $pn=1$ is worst case for the spread. \Enote{so again, you assume that each element is chosen w.p. $p = \frac{\Delta}{n}$ (since this should be the worst case), and that $Bin(n,p = \frac{\Delta}{n}) \approx Poisson(\Delta)$.}
[note: The above will be helpful also when we incorporate success probabilities. as set difference dist holds when conditioned on a fixed outcome of $p$, including outcomes that result in "success"]
We now do group privacy analysis. First fixing the outcome on $p$. Then considering the possible set differences and their probabilities. For each set difference, we can apply the known standard group privacy bound.
The calculation below is for pure differential privacy $\delta=0$. For that the standard bound is that $\varepsilon$ increases by a multiplicative factor of set difference.
We get the effective privacy parameter $\varepsilon^*$ so that
\[
e^{\varepsilon^*} \leq \frac{1}{e} \sum_{i\geq 0} \frac{1}{i!}e^{i\Delta\varepsilon}
\]
(summing over probability of set difference $i\Delta$ times the group privacy bound for that set difference (multiplier $e^{i\Delta\varepsilon}$))
To Do: Do the calculation with $\delta$. Note that as the function $e^x$ is concave, so we are losing a little over standard group privacy with $\Delta$. But since the values are fairly concentrated -- due to independence -- and the $1/i!$ drop -- we should not lose much.
Without independence we would be forced to use Markov and would get $1/i$ that does not drop fast enough.
To Do: Some extension needed for scenario with sampling. We don't have a hard bound $\Delta$. We have possibility of larger $\Delta$
but combined with lower success probabilities.
So $\Delta$ is not bounded. But the probability of reporting decreases sharply with $\Delta$. This should also similarly work.
\end{proof}
\section{Bounding change in probabilities}
\subsection{Replacement model}
In this model the number $n$ of elements is fixed for all datasets and neighboring datasets $T$ and $T'$ differ by substituting one element with another element.
We denote by $A_T(x)$ the number of friends in $T$ of element $x$.
We would like $x$ to be good with probability $1$ if $A_T(x)=|T|=n$ and with probability $0$ when $A_T(X) \leq |T|/2$.
We use $M=n/4$. Let $z_i = A_T(x_i) - n/2$. We use the probabilities
$p_T(x) = \pi^*_{M,z}$.
Consider an element $x\in T \cap T'$.
We have $A_{T'}(x) = A_T(x) +\{0,1,-1\}$. Hence $z' = z +\{0,1,-1\}$.
We can use \eqref{ratioprop:eq} to get the bound
\begin{equation} \label{diffbound:eq}
|p_T(x) - p_{T'}(x)|
\leq (e^\varepsilon -1) \min\{ \min\{ p_T(x), p_{T'}(x) \}, 1- \max\{ p_T(x), p_{T'}(x) \}\} + \delta
\end{equation}
\subsection{Addition/Deletion model}
Two neighboring datasets have $|T\setminus T' \cup T'\setminus T|=1$. In particular, the number of elements has
$||T|-|T'||=1$.
Consider an element in the intersection $x\in T\cap T'$. If $|T'|=|T|+1$ we have $A_{T'}(x)-A_T(x) \in \{0,1\} $.
We use $z= A_T(x_i) - |T|/2$ and $M = |T|/4$. We fix $\gammma$ and use $\delta = 1/(\gamma M)$ and $\varepsilon = C/M$. We assume $M \gg C$ so $e^\varepsilon -1$ is very closely approximated by $\varepsilon$.
We then have $p_T(x) \gets \pi^*_{M,z}$.
\begin{equation} \label{pistarcont:eq}
p_T(x) \gets \pi^*_{M,z} = \left\{ \begin{array}{ll}
0 &\; \text{\small $ z \leq 0$ }\\
\frac{1}{\gamma C} \left( e^{C z/M}-1\right) &\; \text{\small $ z \in [0,M]$ }\\
1- \frac{1}{\gamma C} \left( e^{C (2M-z)/M}-1 \right) &\; \text{\small $z \in [M,2M]$}\\
1 &\; \text{\small $z \geq 2M$}
\end{array}\right.
\end{equation}
For neighboring datasets we have $z' -z = \pm 1/2$ and $M' - M = \pm 1/4$ (all four combinations possible).
We want to bound $|p_T(x) - p_{T'}(x)|$ similarly to \eqref{diffbound:eq}.
We consider $\pi^*_{M+1/4,z-1/2} - \pi^*_{M,z}$ (The other combinations need to be considered and also differences of $(1-\pi^*)$).
We bound separately the variation in $z$ and in $M$. For the variation in $z$ we have
\begin{align*}
\pi^*_{M,z+1/2} - \pi^*_{M,z} &\leq (e^{\varepsilon/2}-1) \min\{\pi^*_{M,z}, 1- \pi^*_{M,z+1/2}\} + \delta \frac{1}{e^{\varepsilon/2} +1} \\
&\leq \approx \frac{C}{2M} \min\{\pi^*_{M,z}, 1- \pi^*_{M,z+1/2}\} +\frac{1}{2M\gamma}
\end{align*}
(noting that $\pi^*_{M,z+1/2} \geq \pi^*_{M,z}$)
We now consider the variation in $M$.
Note that
$\frac{\partial e^{\frac{Cz}{M} } }{\partial M} < 0$ and that
$\frac{\partial^2 e^{\frac{Cz}{M} } }{\partial^2 M} > 0$
Therefore $\pi^*_{M,z}$ is decreasing in $M$ but the rate of decrease is decreasing.
Thus for $z\in [0,M]$
\[ \pi^*_{M,z} - \pi^*_{M+1/4,z} \leq -\frac{1}{4} \frac{\partial \pi^*_{y,z}}{\partial y}(y=M) =
-\frac{1}{4 C\gamma} \frac{\partial e^{\frac{C}{M} z} }{\partial M} = \frac{z}{4M^2\gamma}e^{\frac{C}{M} z}
\]
\[
= \frac{z}{4M^2\gamma} (\gamma C \pi^*_{M,z} +1) = \frac{C}{4M}\frac{z}{M} \pi^*_{M,z} + \frac{z}{M} \frac{1}{4M\gamma}
\]
\ECnote{Need to also consider $z\in [M,2M]$ and the patch area}
For $z\in [0,M]$,
\[\pi^*_{M,z} = \frac{1}{\gamma M} \frac{e^{\frac{C}{M} z} -1}{e^{\frac{C}{M}} -1} \approx \frac{1}{C\gamma}(e^{\frac{C}{M} z} -1) \]
Combining we obtain (we can use this for all combinations)
\[ \pi^*_{M,z+1/2} - \pi^*_{M+1/4,z} \leq \frac{3}{4} \frac{C}{M} \pi^*_{M,z} + \frac{3}{4}\frac{1}{\gamma M}
\]
\ECnote{We should be able to replace $\pi^*_{M,z}$ in the r.h.s. with its minimum with $1-\pi^*_{M,z}$}
\ignore{
*****************
Note that $(A_T(x)-|T|/2) - (A_{T'}(x)-|T'|/2) \in \pm\tfrac{1}{2}$.
In the good elements algorithm, we use
$\eta = 1/(\gamma n)$, set $M(\eta,\nu)= n/2$ and get that $n \nu = C(\gamma)$.
Where $C(\gamma)$ is the solution of $C = 2\ln(\frac{\gamma C}{2} +1)$.
Consider the respective $\pi^*_z$ with $(\nu,\delta)$ \eqref{pistarcont:eq}.
With respect to fixed $\gamma$ and varying $n$ we use the notation
$\pi^*_{n,z}$.
An element $x$ gets probability
\[ p_T(x) = \pi^*_{2A(x)-n} .\]
We now consider two neighboring datasets and $x\in T\cap T'$ and $p_T(x)$ and $p_{T'}(x)$.
We consider thw two values that can change separately: $2A(x)-n$ and $n$. The latter also determines $\eta,\nu,M$.
Both values can change by $\pm 1$ (all combinations are possible).
\ECnote{Trying to gain the X2. But might not be able to and resort if change due to $n$ is not of a smaller order.}
Suppose if only $2A(x)-n$ changes (this is hypothetical. $n$ must change also... but for analysis of variation).
Then we can use \eqref{ratioprop:eq} as is.
We get $p' \leq e^\nu p + \eta$ and $(1-p') \leq e^\nu (1-p) + \eta$.
$\pi^*_{n,z} - \pi^*_{n,z+1} \leq \nu \pi^*_{n,z} + \eta$ (and same for complements etc).
Suppose $2(A(x))-n$ stays the same and $n$ changes by $1$. We get $\eta'= \frac{n}{n+1}\eta$ and
$\nu'= \frac{n}{n+1}\nu$.
We can assume small $\nu$ so
\[\pi^*_z \approx \frac{\eta}{\nu} (e^{\nu z}-1) = \frac{1}{C \gamma} (e^{Cz/n}-1)\]
($z = 2A(x)-n$ varies from $0$ to $n$).
\[ \pi^*_{n,z} - \pi^*_{n+1,z} = \frac{1}{C\gamma}(e^{Cz/n} - e^{Cz/(n+1)}) \approx -\frac{1}{C\gamma} \frac{\partial e^{Cz/n}}{\partial{n}} = \frac{z}{n^2 \gamma} e^{Cz/n} \]
\[
= \frac{z}{n^2 \gamma} \left( C\gamma \pi^*_{z,n} +1\right) = \frac{z}{n} \nu \pi^*_{z,n} + \frac{z}{n} \eta
\]
}
\section{$O(n^2)$ Good elements algorithm}
The good elements algorithms has two components. The first is to identify a set $T_G$ of good elements by testing each element. We also compute the expected value $B:= E[|T_G|]$ of the number of good tuples. The second is to decide, based on the $B$, if to fail (report nothing) or succeed and report $T_G$.
\paragraph{Computing set of good elements $T_G$}
The input is a set of elements $T$ and a "like" predicate $x\sim y$.
For each element $x$ we consider
the number of elements in $T$ that like $x$:
\[A(x) := \sum_{y\in T} I_{x\sim y}\ .\]
\begin{enumerate}
\item Input: A set of elements $T$ with $|T|=n$. A predicate on elements $x\sim y$.
\item Output: A set of good elements $T_G\subset T$ that must include all perfect elements $A(x)=n$, and may not include any bad element $A(x)\leq n/2$. Also compute the value $B := E[|T_G|]$ for that data set.
\item $B\gets 0$ (initialize value of $E[|T_G|]$)
\item Set $M\gets n/4$ and $\gamma$.
\item For each element $x \in T$:
\begin{enumerate}
\item $A(x) \gets \sum_{y\in T} I_{x\sim y}$.
\item
Include $x$ in $T_G$ with probability $\pi^*_{M,A(x)-n/2}$ (probability is $1$ when $A(x)=n$ and is $0$ when $A(x)\leq n$).
\item $B \gets B+ \pi^*_{M, A(x) -n/2}$ (probability of $x$ being labeled good by DP-threshold)
\end{enumerate}
\item
$T_B \gets T\setminus T_G$. Return $T_G$ and $B$
\end{enumerate}
\subsection{Local sensitivity analysis}
The good elements set $T_G$ is drawn according to a vector of probabilities
\[
r(T) := (p(A(x))_x = (\pi^*_{M, A(x)-n/2})_{x}\ .\]
(For convenience we use vector notation while treating these vectors as sparse over the universe of elements. Keeping in mind that the universe might be continuous)
Recall that $E[|T_G|] = \norm{r(T)}_1$.
For two neighboring datasets $T$ and $T'$ ($|T\setminus T'|\leq 1$ and vice versa). We can consider the vectors $r(T)$ and $r(T')$.
We bound the extent in which the expected number of bad elements can increase between neighboring datasets. We also bound the L1 distance of the probability vectors.
\begin{lemma} \label{localsenseitivity:lemma}
Let $T$ and $T'$ be neighboring.
Then
\begin{align}
\norm{r(T') - r(T)}_1 &\leq 1+ (3C/n) \norm{r(T)}_1+ 3/\gamma
\end{align}
in the addition/deletion model.
The bound is $1+ (4C/n) \norm{r(T)}_1+ 4/\gamma$ in the swap model.
\end{lemma}
\ECnote{This can be tightened to $\min\{r, r', 1-r, 1-r'\}$ in the r.h.s. Entries over $1/2$ change proportionally to $1-r()$}
\ignore{
\begin{proof}
\ECnote{Needs to be changed to new notation}
Consider $x\in T\cap T'$. We have $A_T(x) = A_{T'}(x) \pm 1$ and
$q(A_{T'}(x)) \leq q(A_T(x)) e^\nu + \eta$ and
$p(A_{T'}(x)) \leq p(A_T(x)) e^\nu + \eta$ (holds by \cref{ratioprop:eq}).
\begin{align*}
E[|T'_B|] &= \sum_{x\in T'} q(A_{T'}(x)) \leq 1 + \sum_{x\in T'\cap T} q(A_{T'}(x)) \leq
1 + \sum_{x\in T'\cap T} q(A_{T}(x)-1) \\
&\leq 1+ e^\nu \sum_{x\in T'\cap T} q(A_T(x)) + n\eta \leq 1 + e^\nu E[|T_B|] + n\eta
\end{align*}
This because the element that is different can contribute at most $1$ and for all other tuples $A(x)$ changed by at most $1$ and we use~\eqref{ratioprop:eq}.
\ECnote{
The argument is nearly identical for deletion/addition. We consider the difference with the different element (not in $x \not\in T \cap T'$) removed. We then see that for the common element $x \in T \cap T'$
$|T|-A_T(x)$ and $|T'|-A_{T'}(x)$ differ by at most $1$. Then everything else is the same.
}
Similarly
\begin{align*}
\norm{r(T') - r(T)}_1 &\leq
1+ \sum_{x\in T\cap T'}
q(A_{T'}(x))-q(A_{T}(x)) \\
&\leq 1+ (e^\nu-1)\sum_{x\in T} q(A_{T}(x)) + n \eta =
1+ (e^\nu-1) \norm{r(T)}_1+ n\eta
\end{align*}
Comment: Note that for $q > 1/2$, we can use a tighter bound. This is relevant if we use $T_G$, that is large. We can bound
$E[|T'_G|] - E[|T_G]]$
it in terms of the smaller of $T_B$ or $T_G$ or more finely, by considering $q$. We get
\[
E[|T'_B|] - E[|T_B]] \leq 1+n\eta + (e^\nu-1)\min\{E[|T_B|],E[|T_G|] \}
\]
\end{proof}
[Comment. We would want to balance $\eta$ and $\nu$ given $L$ as to minimize the L1 distance in our regime of interest]
}
\subsection{putting it together}
Comment: Need to set privacy parameters so it composes to what we want.
We apply DP thresholding with varying local sensitivity to $E[|T_B|]$ , with $c=1+n\eta$.
\begin{enumerate}
\item Apply the good elements subroutine to compute the set $T_G$ and $y \gets E[|T_B|] = \sum_{x\in T} (1- p_T(x))$
\item Recall that the local sensitivity of $E[|T_B|]$ is $(1+3/\gamma) +e^{3C/n} E[|T_B|]$ (Lemma~\ref{localsensitivity:lemma}). (there is "$4$" instead of "$3$" in swap model).
We apply thresholding with varying local sensitivity, using $\nu = 3C/n$ and $c=1+3/\gamma$: We set $m \gets b^{-1}(y)$ which has sensitivity $1$. We apply $(\varepsilon,\delta)$ DP thresholding with $\leq 0$. If we fail, we return $\bot$. Otherwise, we return $T_G$.\Enote{So by thresholding you mean that we compute $\hat{m} = m + {\rm Lap}(1/\varepsilon)$, and we pass the test if $\hat{m} \leq 0$?}\ECnote{Yes. Can use Laplace or truncated Laplace. Only that this is lossy and wanted to squeeze the constants. So we use a precise optimal thresholding}
By design, the threshold "pass" bit is $(\epsilon,\delta)$ DP.\Enote{Do we need the $\delta$ here?}\ECnote{We could remove it. It seems. I did not check carefully. But then we need to work with high probability. Which we will need to anayway for the sampling.... And the analysis is not completely clean 0/1. So it is cleaner to first use $\delta$.}
\item
We obtain that a pass is possible (see \eqref{Bmax:eq}) only if $E[|T_B|] \leq B_{\max} \approx 2(1+ 3/\gamma) M(\varepsilon,\delta).
The $L1$ distance between two neighboring datasets is at most
\[
1 + 3\frac{C}{n} B + 3/\gamma
\]
If we use $B_{\max}$, we get L1 distance
\[
1 + 6C (1+3/\gamma)\frac{1}{n} M(\varepsilon,\delta) + 3/\gamma
\]
\ECnote{The "$1$" comes from the different element. In the others we have an undertanding on the difference per attribute}
Using values from Table \eqref{gammaC:eq} we get that with
$n \geq 3C B_{\max} = 6C(1+3/\gamma) M(\varepsilon,\delta)$ the distance is at most $2+3/\gamma$.
So $\gamma,C = 3.4,1$ we get distance $<3$ and $n=12 M(\varepsilon,\delta)$.
With $\gamma,C = 10 ,2.6$ we get distance $2.3$ for $n= 90 M(\varepsilon,\delta)$ and
distance $1.8$ for $n= 180 M(\varepsilon,\delta)$
\ignore{
\[\Delta \leq c + (e^\nu-1) B_{\max} \approx 1+ n\eta + \nu B_{\max} .\]
(approximation holds when $\nu\ll 1$. We have that $n \nu < 30$ (for $\gamma = 100$), so this holds for small values of $n$, say $n<100$).
We want $\Delta$ close to $1$ but also small $n$.
We use the relation between $\gamma = 1/(\eta n)$ and $(n\nu)$ as a function of $\gamma$ (see \eqref{gammannu:eq}).
Define \[\ell(\varepsilon,\delta) = (1+ \frac{1}{\varepsilon} \ln (\min\{1,\varepsilon\}/(2\delta))
\]
We get
\[
\Delta \leq 1 + 1/\gamma + 2(1+1/\gamma)(n\nu) \frac{\ell}{n}\ .
\]
If we use $\gamma=1$ we get $\Delta = 2 + 20 \frac{\ell}{n}$. So for say, $n = 20\ell$ we get
$\Delta=3$.
If we use $\gamma=100$ we get $\Delta = 1.01 + 2.02 * 29 * \frac{\ell}{n}$.
So for $n=120 \ell$ we get $\Delta \approx 1.5$ and for $n=240 \ell$ we get $\Delta \approx 1.25$.
}
\ECnote{Recall that this is all for worst case $B$. But this $B$ has $\delta$ probability only and $\tfrac{1}{2} B_\max$ has probability about $1/2$. This needs to be integrated in the group privacy analysis.}
We apply the generalized group privacy with $\Delta$ to obtain the claimed property.
\end{enumerate}
\ECnote{We can optimize a bit more the minimum $n$ (factor of 2) by not using the maximum $B_{\max}$ and accounting in the privacy analysis for the lower reporting probability when $b^{-1} B$ increases. }
\ECnote{Comment: Potentially we can use correlated sampling techniques to have a non-negatively correlated instead of independent choice so that the number of bad elements is very close to its expectation and the actual difference between neighboring datasets is even more concentrated around the L1 distance. But not sure it is worth the work.}
\section{Refined analysis}
Group privacy:
Let $G$ be a randomized algorithms that is applied to a set $T$ outputs a set $D=G(T)$. Let $A$ be $(\varepsilon,\delta)$-DP.
Suppose that for any two neighboring datasets $T$ and $T'$ we can partition the output distributions of $G(T)$ and $G(T')$ to matching events $B(T,i)$ and $B(T',i)$ so that the following holds: For all $i$,
$\Pr_T[B(T,i)] = \Pr_{T'}[B(T',i)]$. For all $i$, $B(T,i)$ contains exactly one set $D$.
Let $A$ be an algorithm that on neighboring datasets produces outputs that are $(\varepsilon,\delta)$-indistinguishable. Consider $A|p(D)$ that outputs $A(D)$ with probability $p$ and $\bot$ otherwise. Then the outputs of $A|p(D)$ on neighboring dataset are $(\varepsilon, p\delta)$-indistinguishable (check).
Let $A$ be an algorithm that on neighboring datasets produces outputs that are $(\varepsilon,\delta)$-indistinguishable.
Let $S$ be a predicate that is $(\varepsilon',\delta')$-indistinguishable on neighboring datasets.
Let $A|S (D)$, which outputs $A(D)$ if $S(D)$. {\bf say something that gains from the sampling by $S$.}
\section{$O(n\log n)$ solution for friendly elements }
Incorporating sampling:
Now we want to be efficient. Instead of counting exactly the number of liking tuples, for each $x$ we sample $S$ tuples. We then compute $A(x)$ as the number of tuples in $S$ that like $x$. (better with different sample for each $x$ for independence).
\paragraph{Sample size:}
The sample size $s$ should be large enough so that say $2/3$ likes on the sample imply with very high probability that there are $> n/2$ likes on the full set.
We need the sample to be at least $a \log(n)$. We then do DP thresholding on the stretch with $2L''+2 = s/3$ . And then $\nu'$ the solution of $L(\nu,\delta) = L''$. We get $\nu' \approx 3\ln(1/2\delta)/s$.
The sampling will decrease the ratio of neighboring $q(i)$ by $s/n$, so we get the guarantees we need.
So the only constraint on sample size is ensuring that $2/3$ filters out tuples with $\leq n/2$ friends.
With the crude multiplicative Chernoff bound $\delta=1/3$ and we get $s \approx 72\ln(n)$ to guarantee that probability $1/n^2$ of passing a tuple with $\leq n/2$ friends. But this constant can be tightened by using the raw Chernoff bounds.
\paragraph{thresholding by $|T_B|$ and not its expectation}
Thresholding the actual number of bad tuples $T_B$ and not its expectation.
Now L1 norm of neighboring and also worst-case local sensitivity can increase. But we don't need to account for worst case as this is mitigated with higher fail probability when $E[|T_B|]$ is larger. We need to do the computation but it should balance out.
Actually might also reduce the factor 2 due to "largest" $E[|T_B|]$ that can pass the threshold. Instead when mitigating by likelihood it should help.
\section{Comments on application of private friendly elements}
In the applications for clustering, each element is a $k$-tuple obtained by applying a clustering algorithm on a part of the data.
Comments:
If we can assume that the algorithm that returns the tuples exposes privacy of points at random (return a random member of the cluster) we can integrate this into the privacy analysis which will allow for a larger $\delta$ (in terms of number of inverse (poly) number of tuples rather than numbers of points). The number of tuples needed does not grow with the number of points in the datasets.
We bound sensitivity in terms of neighboring tuples. If the algorithm that generates tuples has the property that one member of the tuple changes on neighboring point sets then we can save in the group privacy argument.
The solution can be modified to problem specs where elements are considered perfect if the like all except for $a$ elements and require success when at most $b$ elements are not perfect. This extends the applications.
In application to clustering. We still can proabbly avoid solving $n$ small problems. In the non-private setting, one small problem suffices.
What we can do is emulate. Solve one problem. Then process and "emulate" output of $n$ problems. This involves discarding all ambiguous points. Then aggregate.
\printbibliography
\section{Our Algorithms}\label{sec:finding-averages}
In this section we present two $(\varepsilon,\delta)$-differentially private algorithms for the $k$-tuple clustering problem: $\mathsf{PrivatekAverages}$ and $\mathsf{PrivatekNoisyCenters}$. Algorithm $\mathsf{PrivatekAverages}$ attempts to solve the problem by determining the clusters in ${\rm Partition}({\cal T})$ and then privately estimating the average of each cluster using the algorithm from \cref{prop:approx-aver-Rd}. Algorithm $\mathsf{PrivatekNoisyCenters}$, on the other hand, does not operate by averaging clusters. Instead, it first selects one of the input tuples $X \in {\cal T}$ (in a special way), and then adds a (relatively small) Gaussian noise to this tuple.\footnote{We remind that all the tuples in this work are \emph{unordered}, and indeed the privacy analysis of our algorithms relies on it (i.e., the domain of outputs is all the unordered $k$-tuples, and $(\varepsilon,\delta)$-indistinguishability holds for each subset of this domain).
Both algorithms share the same first step, which is to call $\mathsf{PrivateTestPartition}$ (\cref{alg:TestPar}) that privately decides whether ${\cal T}$ is $\ell$-nearly partitioned by $\Delta$-far balls or not (for small $\ell$), and if so, determines (non-privately) a set of $\Delta$-far balls ${\cal B} = \set{B_1,\ldots,B_k}$ that $\ell$-nearly partitions ${\cal T}$. In \cref{sec:alg-test-close-tuples} we describe Algorithm $\mathsf{PrivateTestCloseTuples}$, which is the main component of $\mathsf{PrivateTestPartition}$.
In \cref{sec:alg-test} we describe $\mathsf{PrivateTestPartition}$ and state its properties. Then, in \cref{sec:alg-find-averages} we describe $\mathsf{PrivatekAverages}$ and prove its guarantees, and in \cref{sec:alg-prac-find-averages} we describe $\mathsf{PrivatekNoisyCenters}$ and prove its guarantees.
\subsection{Algorithm $\mathsf{PrivateTestCloseTuples}$}\label{sec:alg-test-close-tuples}
In this section we describe $\mathsf{PrivateTestCloseTuples}$ (\cref{alg:TestCloseTuples}), which given two multisets of $k$-tuples ${\cal T}_1$ and ${\cal T}_2$, privately checks whether the tuples in ${\cal T}_1$ are close to the tuples in ${\cal T}_2$.
\begin{algorithm}[$\mathsf{PrivateTestCloseTuples}$]\label{alg:TestCloseTuples}
\item Input: Multisets ${\cal T}_1 \in (({\mathbb R}^d)^k)^m$ and ${\cal T}_2 \in (({\mathbb R}^d)^k)^n$, a privacy parameter $\varepsilon_1 \in (0,1]$ for ${\cal T}_1$, a privacy parameters $\varepsilon_2 \in (0,1]$ for ${\cal T}_2$, a confidence parameter $\beta \in (0,1]$, and a separation parameter $\Delta > 6$.
\item Operation:~
\begin{enumerate}
\item For each $X = \set{\px_1,\ldots,\px_k} \in {\cal T}_1$:\label{step:loop-over-R}
\begin{enumerate}
\item Let ${\cal B}_{X} = \set{B_i^{X} = B(\px_i, r_i^{X})}_{i=1}^k$, where $r_i^X = \frac1{\Delta}\cdot \min_{j \neq i} \norm{\px_i - \px_j}$.\label{step:B_x}
\item Let $\ell_{X} = \size{\set{Y \in {\cal T}_2 \colon Y\text{ is \textbf{not} partitioned by }{\cal B}_{X}}}$.\label{step:l_x}
\item Let $\hat{\ell}_{X} = \ell_{X} + {\rm Lap}\paren{\frac{m}{\varepsilon_2}}$.\label{step:hl_x}
\item Set ${\rm pass}_{X} = \begin{cases} 1 & \hat{\ell}_{X} \leq \frac{m}{\varepsilon_2} \cdot \log\paren{\frac{m}{\beta}}\\ 0 &\text{otherwise} \end{cases}$.\label{step:pass}
\end{enumerate}
\item Let $s = \sum_{X \in {\cal T}_1} {\rm pass}_{X}$ and compute $\hat{s} \gets s + {\rm Lap}\paren{\frac1{\varepsilon_1}}$.\label{step:s}
\item If $\hat{s} < m - \frac1{\varepsilon_1} \log\paren{\frac1{\beta}}$, set ${ Status}
= \text{"Failure"}$. Otherwise, set ${ Status}
= \text{"Success"}$.\label{step:status}
\item If ${ Status}
= \text{"Success"}$ and ${\rm pass}_{X} = 1$ for at least one $X \in {\cal T}_1$,
let $X^*$ be the first tuple in ${\cal T}_1$ with ${\rm pass}_{X^*} = 1$ and set ${\cal B} = {\cal B}_{X^*}$.
Otherwise, set ${\cal B}$ to be a set of $k$ empty balls.\label{step:x-star}
\item Output $({ Status},{\cal B})$.
\end{enumerate}
\end{algorithm}
\subsubsection{Properties of $\mathsf{PrivateTestCloseTuples}$}
The properties of $\mathsf{PrivateTestCloseTuples}$ are summarized by the following claims.
\begin{claim}[Correctness]\label{claim:correctness-closeTuples}
Assume that ${\cal T} = {\cal T}_1 \cup {\cal T}_2$ is partitioned by $(2\Delta+2)$-far balls. Then with probability $1-\beta$, when executing $\mathsf{PrivateTestCloseTuples}$ on input ${\cal T}_1,{\cal T}_2,\varepsilon_1,\varepsilon_2,\beta$,$\Delta$, it outputs $(\text{``Success"},{\cal B})$, where ${\cal B}$ is a set of $\Delta$-far balls that partitions ${\cal T}$.
\end{claim}
\begin{proof}
We first prove that for every $X \in {\cal T}_1$, the set of balls ${\cal B}_{X} = \set{B_i^{X} = B(\px_i, r_i^{X})}_{i=1}^k$ from Step~\ref{step:B_x} is a set of $\Delta$-far balls that partitions ${\cal T}_2$. Fix $X = \set{\px_1,\ldots,\px_k} \in {\cal T}_1$, let ${\cal B} = \set{B_i = B(\pc_i, r_i)}_{i=1}^k$ be a set of $(2\Delta+2)$-far balls that partitions ${\cal T}$ (such a set exists by assumption), and assume w.l.o.g. that $\forall i \in [k] \colon \px_i \in B_i$.
In addition, recall that $r_i^{X} = \frac1{\Delta}\cdot \min_{j \neq i} \norm{\px_i - \px_j}$ (Step~\ref{step:B_x}), and therefore, by definition it holds that ${\cal B}_{X}$ is a set of $\Delta$-far balls. It is left to prove that it partitions ${\cal T}$. Note that for every $i \neq j$ it holds that
\begin{align*}
\norm{\px_i - \px_j}
&\geq \norm{\pc_i - \pc_j} - \norm{\px_i - \pc_i} - \norm{\px_j - \pc_j}\\
&> (2\Delta+2) \cdot \max\set{r_i,r_j} - r_i - r_j\\
&\geq 2\Delta \cdot \max\set{r_i,r_j}
\end{align*}
Therefore, for every $i \in [k]$, $r_i^{X} = \frac1{\Delta}\cdot \min_{j \neq i} \norm{\px_i - \px_j} > 2\cdot r_i$. Since $\px_i \in B_i$, we conclude that $B_i \subseteq B_i^{X}$, which yields that ${\cal B}_{X}$ partitions ${\cal T}$.
Therefore, for every $X = \set{\px_1,\ldots,\px_k} \in {\cal T}_1$ it holds that $\ell_{X}$, the value from Step~\ref{step:l_x}, is $0$. Hence, by \cref{fact:laplace-concent} and the union bound, with probability $1-\frac{\beta}2$ it holds that $\forall X \in {\cal T}_1:\text{ }{\rm pass}_{X} = 1$, which yields that $s = m$ (where $m = \size{{\cal T}_1}$). When $s = m$, we obtain by \cref{fact:laplace-concent} that with probability $1- \frac{\beta}{2}$ it holds that $\hat{s} \geq s - \frac1{\varepsilon_1} \log(1/\beta) = m - \frac1{\varepsilon_1} \log(1/\beta)$, i.e., ${ Status} = \text{``Success''}$. This concludes the proof of the claim.
\end{proof}
\begin{claim}[${ Status}$ is $\varepsilon_1$-DP w.r.t. ${\cal T}_1$]\label{claim:status-is-private-T1}
Let ${\cal T}_1,{\cal T}_1' \in (({\mathbb R}^d)^k)^m$ be two neighboring databases, let ${\cal T}_2 \in (({\mathbb R}^d)^k)^n$, and consider two independent executions $\mathsf{PrivateTestCloseTuples}({\cal T}_1, {\cal T}_2)$ and $\mathsf{PrivateTestCloseTuples}({\cal T}_1', {\cal T}_2)$ (with the same parameters $\varepsilon_1,\varepsilon_2,\beta, \Delta$). Let ${ Status}$ and ${ Status}'$ be the status outcomes of the two executions (respectively). Then ${ Status}$ and ${ Status}'$ are $\varepsilon_1$-indistinguishable.
\end{claim}
\begin{proof}
Note that each $k$-tuple $X \in {\cal T}_1$ can affect only the bit ${\rm pass}_{X}$. Therefore,
by the properties of the Laplace mechanism (\cref{fact:laplace}) and post-processing (\cref{fact:post-processing}), it holds that ${ Status}$ and ${ Status}'$ are $\varepsilon_1$-indistinguishable.
\end{proof}
\begin{claim}[${ Status}$ is $\varepsilon_2$-DP w.r.t. ${\cal T}_2$]\label{claim:status-is-private-T2}
Let ${\cal T}_2,{\cal T}_2' \in (({\mathbb R}^d)^k)^n$ be two neighboring databases, let ${\cal T}_1 \in (({\mathbb R}^d)^k)^m$, and consider two independent executions $\mathsf{PrivateTestCloseTuples}({\cal T}_1, {\cal T}_2)$ and $\mathsf{PrivateTestCloseTuples}({\cal T}_1, {\cal T}_2')$ (with the same parameters $\varepsilon_1,\varepsilon_2,\beta$). Let ${ Status}$ and ${ Status}'$ be the status outcomes of the two executions (respectively). Then ${ Status}$ and ${ Status}'$ are $\varepsilon_2$-indistinguishable.
\end{claim}
\begin{proof}
For each $X \in {\cal T}_1$, let $\ell_{X}, {\rm pass}_{X}$ and $\ell_{X}', {\rm pass}_{X}'$ be the values computed in the loop \ref{step:loop-over-R} in the two executions (respectively). Since $\size{\ell_{X} - \ell_{X}'} \leq 1$, we obtain by the properties of the Laplace mechanism, along with post-processing, that ${\rm pass}_{X}$ and ${\rm pass}_{X}'$ are $\frac{\varepsilon_2}{m}$-indistinguishable. Hence, by basic composition (\cref{thm:composition1}) we deduce that $\set{{\rm pass}_{X}}_{X \in {\cal T}_1}$ and $\set{{\rm pass}_{X}'}_{X \in {\cal T}_1}$ are $\varepsilon_2$-indistinguishable, and we conclude by post-processing that ${ Status}$ and ${ Status}'$ are $\varepsilon_2$-indistinguishable.
\end{proof}
The following claim states that when $\mathsf{PrivateTestCloseTuples}({\cal T}_1,{\cal T}_2)$ outputs $(\text{``Success''},{\cal B})$, then with high probability, ${\cal T}_2$ is almost partitioned by ${\cal B}$.
\begin{claim}[On success, ${\cal B}$ almost partitions ${\cal T}_2$]\label{claim:main-T2}
Let $\delta > 0$, let ${\cal T}_1 \in (({\mathbb R}^d)^k)^m$ and ${\cal T}_1 \in (({\mathbb R}^d)^k)^n$, and assume that $m > \frac1{\varepsilon_1}\cdot \paren{2 \log(1/\delta) + \log(1/\beta)}$.
Consider a random execution of \\$\mathsf{PrivateTestCloseTuples}({\cal T}_1,{\cal T}_2,\varepsilon_1,\varepsilon_2,\beta)$, and let $({ Status},{\cal B})$ be the outcome of the execution. Let $S$ be the event that ${ Status} = \text{"Success"}$,
and let $E \subseteq S$ be the event that ${\cal T}_2$ is $\ell$-nearly partitioned by ${\cal B}$, where $\ell = \frac{m}{\varepsilon_2} \cdot \log\paren{\frac{m}{\beta \delta}}$. Then the following holds:
If $\pr{S} \geq \delta$, then $\pr{E \mid S} \geq 1 - \delta$.
\end{claim}
\begin{proof}
Let $\set{{\rm pass}_{X}}_{X \in {\cal T}_1}$ be the values from \cref{alg:TestPar} in the execution $\mathsf{PrivateTestCloseTuples}({\cal T}_1,{\cal T}_2,\varepsilon_1,\varepsilon_2,\beta)$, and let $W$ be the event that there exists $X \in {\cal T}_1$ with ${\rm pass}_{X} = 1$. Note that
\begin{align*}
\pr{\neg W \mid S}
\leq \frac{\pr{S \mid \neg W}}{\pr{S}}
\leq \frac{\pr{{\rm Lap}(1/\varepsilon_1) > \frac{2}{\varepsilon_1}\cdot \log\paren{\frac1{\delta}}}}{\delta}
\leq \frac{\delta^2}{2 \delta}
\leq \frac{\delta}2,
\end{align*}
where the second inequality holds since $\pr{S} \geq \delta$ and since $m - \frac1{\varepsilon_1} \log\paren{\frac1{\beta}} > \frac{2}{\varepsilon_1}\cdot \log\paren{\frac1{\delta}}$, and the third one holds by \cref{fact:laplace-concent}.
Therefore, in the following we prove the claim by showing that
\begin{align}\label{eq:E-mid-W-goal}
\pr{E \mid W \land S} \geq 1 -\frac{\delta}2
\end{align}
Let $X^*$ be the tuple from Step~\ref{step:x-star} (it exists when $W \land S$ occurs), and recall that ${\cal B} = {\cal B}_{X^*}$ and that $\ell_{X^*}$ is the minimal value such that ${\cal T}_2$ is $\ell_{X^*}$-nearly partitioned by ${\cal B}$. Since ${\rm pass}_{X^*} = 1$, it holds that $\hat{\ell}_{X^*} = \ell_{X^*} + {\rm Lap}(m/\varepsilon_2) \leq \frac{m}{\varepsilon_2}\cdot \log\paren{\frac{m}{\beta}}$. \cref{eq:E-mid-W-goal} now follows by the following calculation.
\begin{align*}
\pr{E \mid W \land S}
&= \pr{\ell_{X^*} > \frac{m}{\varepsilon_2} \cdot \log\paren{\frac{m}{\beta \delta}} \mid \hat{\ell}_{X^*} \leq \frac{m}{\varepsilon_2}\cdot \log\paren{\frac{m}{\beta}}}\\
&\leq \pr{{\rm Lap}(m/\varepsilon_2) < -\frac{m}{\varepsilon_2}\cdot \log\paren{\frac{1}{\delta}}}\\
&\leq \frac{\delta}{2},
\end{align*}
where the last inequality holds by \cref{fact:laplace-concent}.
\end{proof}
\subsection{Algorithm $\mathsf{PrivateTestPartition}$}\label{sec:alg-test}
In this section we describe $\mathsf{PrivateTestPartition}$ (\cref{alg:TestPar}) and state its properties.
In the following, we define $m$ and $\varepsilon_1$ (functions of $n,\varepsilon,\delta,\beta$) that are used by $\mathsf{PrivateTestPartition}$.
\begin{definition}\label{def:m}
Let $m = m(n,\varepsilon,\delta,\beta)$ be the smallest integer that satisfies $m > \frac1{\varepsilon_1} \cdot \paren{2 \log(1/\delta) + \log(1/\beta)}$, where $\varepsilon_1 = \log(\frac{\varepsilon n}{2 m} - 3)$.
\end{definition}
The dependence between $m$ and $\varepsilon_1$ for Algorithm $\mathsf{PrivateTestPartition}$ is due to the choice of ${\cal T}_1$ as an $m$-size random sample of ${\cal T}$. A smaller $m$ allows for a larger value of $\varepsilon_1$ for the same overall privacy, by a sub-sampling argument (e.g., \cref{lem:subsampling}).
We note that for $n \gg 1/\varepsilon$ and $\beta,\delta \geq \frac{1}{{\rm poly}(n)}$, we have $\varepsilon_1 = \Theta(\log n)$, which yields that $m = O(1)$. For smaller values of $\delta$, we obtain that $m = O\paren{\frac{\log(1/\delta)}{\log n}}$.
\begin{algorithm}[$\mathsf{PrivateTestPartition}$]\label{alg:TestPar}
\item Input: A multiset ${\cal T} \in (({\mathbb R}^d)^k)^n$, privacy parameters $\varepsilon,\delta \in (0,1]$, confidence parameter $\beta \in (0,1]$, and separation parameter $\Delta > 6$.
\item Operation:~
\begin{enumerate}
\item Let $m$ and $\varepsilon_1$ be the values from \cref{def:m} w.r.t. $n,\varepsilon,\delta,\beta$, and let $\varepsilon_2 = \varepsilon/2$.
\item Let ${\cal T}_1$ be a uniform sample of $m$ $k$-tuples from ${\cal T}$ (without replacement), and let ${\cal T}_2 = {\cal T}$.
\item Output $({ Status},{\cal B}) = \mathsf{PrivateTestCloseTuples}({\cal T}_1,{\cal T}_2,\varepsilon_1,\varepsilon_2,\beta,\Delta)$.
\end{enumerate}
\end{algorithm}
\subsubsection{Properties of $\mathsf{PrivateTestPartition}$}
The following claim is an immediate corollary of \cref{claim:correctness-closeTuples}
\begin{claim}[Correctness]\label{claim:correctness}
Assume that ${\cal T}$ is partitioned by $(2\Delta+2)$-far balls. Then with probability $1-\beta$, when executing $\mathsf{PrivateTestCloseTuples}$ on input ${\cal T},\varepsilon,\delta,\beta,\Delta$, it outputs $(\text{``Success"},{\cal B})$, where ${\cal B}$ is a set of $\Delta$-far balls that partitions ${\cal T}$.
\end{claim}
The following claim is a corollary of \cref{claim:status-is-private-T1,claim:status-is-private-T2}.
\begin{claim}[${ Status}$ is private]\label{eq:status-is-private}
Let ${\cal T}$ and ${\cal T}'$ be two neighboring databases, and consider two independent executions $\mathsf{PrivateTestPartition}({\cal T})$ and $\mathsf{PrivateTestPartition}({\cal T}')$ (with the same parameters $\varepsilon,\delta,\beta$). Let ${ Status}$ and ${ Status}'$ be the status outcomes of the two executions (respectively). Then ${ Status}$ and ${ Status}'$ are $\varepsilon$-indistinguishable.
\end{claim}
\begin{proof}
As a first step, assume that we have two (different) copies of ${\cal T}$, call them $\tilde{{\cal T}}_1$ and $\tilde{{\cal T}}_2$, where ${\cal T}_1$ is chosen from the copy $\tilde{{\cal T}}_1$, and ${\cal T}_2$ is chosen from the copy $\tilde{{\cal T}}_2$, and let $(\tilde{{\cal T}}_1',\tilde{{\cal T}}_2')$ be a neighboring database of $(\tilde{{\cal T}}_1,\tilde{{\cal T}}_2)$. If $\tilde{{\cal T}}_2$ and $\tilde{{\cal T}}_2'$ are neighboring (and $\tilde{{\cal T}}_1 = \tilde{{\cal T}}_1'$), we obtain by \cref{claim:status-is-private-T2} that ${ Status}$ and ${ Status}'$ are $\varepsilon/2$-indistinguishable. Therefore, assume that $\tilde{{\cal T}}_1$ and $\tilde{{\cal T}}_1'$ are neighboring (and $\tilde{{\cal T}}_2 = \tilde{{\cal T}}_2'$). By \cref{claim:status-is-private-T1}, ${ Status}$ and ${ Status}'$ are $\varepsilon_1$-indistinguishable if the resulting samples ${\cal T}_1$ and ${\cal T}_1'$ in the two executions are neighboring. Since ${\cal T}_1$ is just an $m$-size sample from $\tilde{{\cal T}}_1$, and since $\varepsilon_1 = \log(\frac{\varepsilon n}{2 m} - 3)$, we obtain by subsampling argument (\cref{lem:subsampling}) that ${ Status}$ and ${ Status}'$ are $\varepsilon/2$-indistinguishable also in this case.
Finally, going back to our case where $\tilde{{\cal T}}_1 = \tilde{{\cal T}}_2 = {\cal T}$, we deduce by the above analysis along with group privacy (of $2$) that ${ Status}$ and ${ Status}'$ are $\varepsilon$-indistinguishable.
\end{proof}
The following claim is an immediate corollary of \cref{claim:main-T2}. It states that when the tests succeed, then w.h.p., ${\cal T}$ is $\ell$-nearly partitioned by ${\cal B}$, for the value of $\ell$ defined below.
\begin{definition}\label{def:ell}
Let $\ell = \ell(n,\varepsilon,\delta,\beta) = \frac{2 m}{\varepsilon} \cdot \log\paren{\frac{m}{\beta \delta}}$, where $m = m(n,\varepsilon,\delta,\beta)$ is the value from \cref{def:m}.
\end{definition}
We note that $\ell = O\paren{\frac{\log^2(1/\delta)}{\varepsilon \log n}}$. When $\beta,\delta \geq 1/{\rm poly}(n)$, we have that $\ell = O\paren{\frac1{\varepsilon} \log n}$.
\begin{claim}[On success, ${\cal B}$ almost partitions ${\cal T}$]\label{claim:main}
Let ${\cal T} \in (({\mathbb R}^d)^k)^n$ and $\delta > 0$.
Consider a random execution of $\mathsf{PrivateTestPartition}({\cal T},\varepsilon,\delta,\beta,\Delta)$, and let $({ Status},{\cal B})$ be the outcome of the execution. Let $S$ be the event that ${ Status} = \text{"Success"}$,
and let $E \subseteq S$ be the event that ${\cal T}$ is $\ell$-nearly partitioned by ${\cal B}$, where $\ell = \ell(n,\varepsilon,\delta,\beta)$ is the value from \cref{def:ell}. Then the following holds:
If $\pr{S} \geq \delta$, then $\pr{E \mid S} \geq 1 - \delta$.
\end{claim}
\begin{proof}
Immediately holds by \cref{claim:main-T2} since $\ell = \frac{m}{\varepsilon_2}\cdot \log\paren{\frac{m}{\beta \delta}}$, and since it holds that $m > \frac1{\varepsilon_1}\cdot \paren{2 \log(1/\delta) + \log(1/\beta)}$ (by definition), as required by \cref{claim:main-T2}.
\end{proof}
Recall that Algorithm $\mathsf{PrivateTestPartition}$ has two outputs: A bit $Status$ and a set of balls ${\cal B}$.
As we stated in Claim~\ref{eq:status-is-private}, the bit $Status$ preserves privacy. The set of balls ${\cal B}$, however, does {\em not}. Still, in the following sections we use Algorithm $\mathsf{PrivateTestPartition}$ as a subroutine in our two main algorithms $\mathsf{PrivatekAverages}$ and $\mathsf{PrivatekNoisyCenters}$. To argue about the privacy properties of these algorithms, we rely on the following key property of algorithm $\mathsf{PrivateTestPartition}$.
\begin{claim}\label{claim:privacy-framework}
Let $\mathsf{A}^*$ be an algorithm that gets as input a multiset ${\cal T} \in (({\mathbb R}^d)^k)^n$ and a set of balls ${\cal B} = \set{B_1,\ldots,B_k}$, and let $\ell = \ell(n,\varepsilon/2,\delta/4,\beta/2)$ be the value from \cref{def:ell}. Assume that $\mathsf{A}^*$
has the property that for any neighboring multisets ${\cal T},{\cal T}'$ and any sets of $\Delta$-far balls ${\cal B},{\cal B}'$ that $\ell$-nearly partitions ${\cal T}$ and ${\cal T}'$ (respectively), it holds that $\mathsf{A}^*({\cal T},{\cal B})$ and $\mathsf{A}^*({\cal T}',{\cal B}')$ are $(\varepsilon^*,\delta/4)$-indistinguishable. Let $\mathsf{A}$ be the algorithm that on input ${\cal T}$, does the following steps: (1) Compute $({ Status},{\cal B}) = \mathsf{PrivateTestPartition}\paren{{\cal T},\varepsilon/2,\delta/4,\beta/2,\Delta}$, and (2) If ${ Status} = \text{``Failure''}$, output $\perp$ and abort, and otherwise output $\mathsf{A}^*({\cal T},{\cal B})$. Then $\mathsf{A}$ is $(\varepsilon/2 + \varepsilon^*,\delta)$-differentially private.
\end{claim}
\begin{proof}
Let ${\cal T}$ and ${\cal T}'$ be two neighboring multisets of size $n$. In the following we consider two independent executions: $\mathsf{A}({\cal T})$ and $\mathsf{A}({\cal T}')$. In $\mathsf{A}({\cal T})$, let $O$ be the outcome, let $S,E$ be the events from \cref{claim:main} w.r.t. the execution of $\mathsf{PrivateTestPartition}$ in step (1), and let $({ Status},{\cal B})$ be the resulting output of $\mathsf{PrivateTestPartition}$. Similarly, let $O',S',E',{ Status}',{\cal B}'$ be the events and random variables w.r.t. the execution $\mathsf{A}({\cal T}')$. Let $q = \pr{S}$ and $q' = \pr{S'}$. By \cref{eq:status-is-private} and by group privacy (\cref{fact:group-priv}), ${ Status}$ and ${ Status}'$ are $\frac{\varepsilon}{2}$-indistinguishable. Therefore, $q \in e^{\pm \varepsilon/2} \cdot q'$. Recall that $\mathsf{A}$ outputs $\perp$ and aborts whenever ${ Status} = \text{"Failure"}$, and therefore, $\pr{O = \perp} = 1-q$ and $\pr{O' = \perp} = 1-q'$. If $q < \frac{\delta}2$ then $q' < e^{\varepsilon/2} \cdot \frac{\delta}2 \leq \delta$ (recall that $\varepsilon\leq 1$), and therefore, $\pr{O = \perp}, \pr{O' = \perp} \geq 1 - \delta$. This means that $O$ and $O'$ are $(0,\delta)$-indistinguishable in the case that $q < \frac{\delta}2$ (by \cref{lem:indis}). Similarly, it holds that $O$ and $O'$ are $(0,\delta)$-indistinguishable when $q' < \frac{\delta}2$.
Hence, in the rest of the analysis we assume that $q,q' \geq \frac{\delta}2$.
By \cref{prop:similar-E}, since $O|_{\neg S} \equiv O'|_{\neg S'}$ (both outcomes equal to $\perp$ when ${ Status} = { Status}' = "\text{Failure}"$)
and since $\pr{S} \in e^{\pm \varepsilon/2} \cdot \pr{S'}$, it is enough to prove that $O|_S$ and $O'|_{S'}$ are $(\varepsilon^*,\frac{\delta}2)$-indistinguishable.
Furthermore, since $\pr{E \mid S},\pr{E' \mid S'} \geq 1 - \frac{\delta}{4}$ (by \cref{claim:main}), we deduce by \cref{fact:indis-cor} that it is enough to prove that $O|_{E}$ and $O'|_{E'}$ are $(\varepsilon^*,\frac{\delta}4)$-indistinguishable, meaning that we only need to prove indistinguishability in the case that ${\cal T}$ and ${\cal T}'$ are $\ell$-nearly partitioned by ${\cal B}$ and ${\cal B}'$, respectively. The proof of the claim now follows since $\mathsf{A}^*({\cal T},{\cal B})|_{E}$ and $\mathsf{A}^*({\cal T}',{\cal B}')|_{E'}$ are $(\varepsilon^*,\delta/4)$-indistinguishable by the assumption on the algorithm $\mathsf{A}^*$.
\end{proof}
\begin{remark}\label{remark:Test-runtime}
Note that $\mathsf{PrivateTestPartition}$ runs in time $O(m d k^2 n) = \tilde{O}(d k^2 n)$ since for each iteration $X \in {\cal T}_1$ in $\mathsf{PrivateTestCloseTuples}$, Step~\ref{step:B_x} takes $O(dk^2)$ time, and Step~\ref{step:l_x} takes $O(d k^2 n)$ times.
\end{remark}
\subsection{Algorithm $\mathsf{PrivatekAverages}$}\label{sec:alg-find-averages}
In this section we describe and state the properties of $\mathsf{PrivatekAverages}$ (\cref{alg:FindAverages}) which is our first algorithm for $k$-tuple clustering.
\begin{algorithm}[$\mathsf{PrivatekAverages}$]\label{alg:FindAverages}
\item Input: A multiset ${\cal T} \in \paren{B(0,\Lambda)^k}^n \subseteq (({\mathbb R}^d)^k)^n$, privacy parameters $\varepsilon,\delta \in (0,1]$, a confidence parameter $\beta \in (0,1]$, and a lower bound on the radii $r_{\min} \in [0,\Lambda]$.
\item Operation:~
\begin{enumerate}
\item Compute $({ Status},{\cal B} = \set{B_1,\ldots,B_k}) = \mathsf{PrivateTestPartition}({\cal T},\varepsilon/2,\delta/4,\beta/2, \Delta)$ for $\Delta = 7$.\label{step:call-testSeparation}
\item If ${ Status} = \text{"Failure"}$, output $\perp$ and abort.
\item Let $\pc_1,\ldots,\pc_k$ be the centers of $B_1,\ldots,B_k$ (respectively), and let
${\cal Q}_i = \set{\px \in {\rm Points}({\cal T}) \colon i = \operatorname*{argmin}_{j \in [k]} \norm{\px - \pc_j}}$.\label{step:compute-clusters}
\item Let $\ell = \ell(n,\varepsilon/2,\delta/4,\beta/2)$ be the value from \cref{def:ell}.
\item For $i=1$ to $k$:
\begin{enumerate}
\item Compute a noisy average $\hat{\pa}_i$ of ${\cal Q}_i$ by executing the algorithm from \cref{prop:approx-aver-Rd} with parameters $\Lambda, r_{\min}, \hat{\beta} = \frac{\beta}{2k}, \hat{\varepsilon} = \frac{\varepsilon}{4k(\ell+1)}, \hat{\delta} = \frac{\delta}{8 k \exp(\varepsilon/2)(\ell + 1)}$.\label{step:computing-noisy-bound-aver}
\end{enumerate}
\item Output $\hat{A} = \set{\hat{\pa}_1,\ldots,\hat{\pa}_k}$.\label{step:kAverages:output}
\end{enumerate}
\end{algorithm}
\subsubsection{Properties of $\mathsf{PrivatekAverages}$}
The properties of $\mathsf{PrivatekAverages}$ are given in the following theorems.
\begin{theorem}[Privacy of $\mathsf{PrivatekAverages}$]\label{claim:privacy}
Let $d, k, \Lambda > 0$, $r_{\min} \in [0,\Lambda]$, $\varepsilon,\delta, \beta \in (0,1]$. Then for any integer $n \geq 2\cdot \ell(n,\varepsilon/2,\delta/4,\beta/2) + 2$ (where $\ell$ is the function from \cref{def:ell}),
algorithm $\mathsf{PrivatekAverages}(\cdot,\varepsilon,\delta,\beta,r_{\min})$ is $(\varepsilon,\delta)$-differentially private for databases ${\cal T} \in (B(\pt{0},\Lambda)^k)^n \subseteq (({\mathbb R}^d)^k)^n$.
\end{theorem}
\begin{proof}
Let ${\cal T}$ and ${\cal T}'$ be two neighboring multisets of size $n$. In the following we consider two independent executions: $\mathsf{PrivatekAverages}({\cal T})$ and $\mathsf{PrivatekAverages}({\cal T}')$ (both with the same parameters $r_{\min},\varepsilon, \delta, \beta$). In $\mathsf{PrivatekAverages}({\cal T})$, let $O$ be the output, and let ${\cal B} = \set{B_1,\ldots,B_k},{\cal Q}_1,\ldots,{\cal Q}_k$ be the values from the execution $\mathsf{PrivatekAverages}({\cal T})$. Similarly, we let
$O', {\cal B}' = \set{B_1',\ldots,B_k'},{\cal Q}_1',\ldots,{\cal Q}_k'$ be the these values w.r.t. the execution $\mathsf{PrivatekAverages}({\cal T}')$. By \cref{claim:privacy-framework}, if we treat Step~\ref{step:compute-clusters} to \ref{step:kAverages:output} as algorithm $\mathsf{A}^*$ of the claim, it is enough to prove that $O = \hat{A}$ and $O'= \hat{A}'$ are $(\varepsilon/2,\delta/4)$-indistinguishable only in the case that ${\cal T}$ and ${\cal T}'$ are $\ell$-nearly partitioned by ${\cal B}$ and ${\cal B}'$, respectively. In addition, note that since ${\cal T}$ and ${\cal T}'$ are neighboring, and since $n \geq 2\ell + 2$, there exists at least one $k$-tuple that is partitioned by both ${\cal B}$ and ${\cal B}'$, yielding that for each ball $B_i \in {\cal B}$, there exists a balls in ${\cal B}'$ (call it $B_i'$), such that $B_i \cap B_i' \neq \emptyset$. Since ${\cal B}$ and ${\cal B}'$ are sets of $\Delta$-far balls for $\Delta = 7$, \cref{prop:close-sets-of-far-balls} yields that for every $\px \in B_i$ (or $B_i'$), it holds that $i = \operatorname*{argmin}_{j \in [k]}\norm{\px - \pc_j} = \operatorname*{argmin}_{j \in [k]}\norm{\px - \pc_j'}$. Therefore, in the two executions, $\set{{\cal Q}_1,\ldots,{\cal Q}_k}$ and $\set{{\cal Q}_1',\ldots,{\cal Q}_k'}$ agree on all the points of all the common $(n-1)$ $k$-tuples of ${\cal T}$ and ${\cal T}'$ that are partitioned by ${\cal B}$ or ${\cal B}'$. Since there are at least $k \cdot (n-1-\ell)$ such points, we deduce that there are at most $k (\ell + 1)$ points that the partitions $\set{{\cal Q}_1,\ldots,{\cal Q}_k}$ and $\set{{\cal Q}_1',\ldots,{\cal Q}_k'}$ disagree on.
In the following, let $s_i$ be the number of points that the multisets ${\cal Q}_i$ and ${\cal Q}_i'$ differ by. Note that each point that the partitions disagree on contributes at most $1$ to at most two of the $s_i$'s. Hence, $\sum_{i=1}^k s_i \leq 2k(\ell + 1)$.
By the privacy guarantee of \cref{prop:approx-aver-Rd} (see \cref{remark:bound-aver-add-del}) along with group privacy (\cref{fact:group-priv}), for each $i \in [k]$, the resulting noisy averages $\hpa_i$ of the execution $\mathsf{PrivatekAverages}({\cal P})$, and the resulting $\hpa_i'$ of the execution $\mathsf{PrivatekAverages}({\cal P}')$, which computed in Step~\ref{step:computing-noisy-bound-aver}, are
$(\frac{\varepsilon s_i}{4k(\ell + 1)},\frac{\delta s_i}{8k(\ell + 1)})$-indistinguishable. Thus, by basic composition (\cref{thm:composition1}) we deduce that $\set{\hpa_1,\ldots,\hpa_k}$ and $\set{\hpa_1'\ldots,\hpa_k'}$ are $( \frac{\varepsilon \sum_{i=1}^k s_i}{4k(\ell + 1)},\frac{\delta \sum_{i=1}^ks_i}{8 k(\ell + 1)}) = (\frac{\varepsilon}{2}, \frac{\delta}{4})$-indistinguishable, as required.
\end{proof}
\begin{theorem}[Utility of $\mathsf{PrivatekAverages}$]\label{claim:utility-kAverg}
There exists a universal constant $\lambda > 0$ such that the following holds:
Let $n,d, k \in {\mathbb N}$, $\Lambda > 0$, $r_{\min} \in [0,\Lambda]$, $\varepsilon,\delta,\beta \in (0,1]$, let $\ell = \ell(n,\frac{\varepsilon}2,\frac{\delta}4,\frac{\beta}2)$ be the value from \cref{def:ell}. If $n \geq \frac{\lambda k \ell \sqrt{d \log(d k \ell/\delta)} \log\paren{\frac{\Lambda d k}{r_{\min} \beta}}}{\varepsilon}$, then algorithm $\mathsf{PrivatekAverages}(\cdot, \varepsilon, \delta, \beta, r_{\min})$ is an $(\alpha,r_{\min},\beta,\Delta=16,\Lambda)$-averages-estimator for $k$-tuple clustering (\cref{def:averages-estimator}), for
\begin{align*}
\alpha = \alpha(n,d,k,\varepsilon,\delta,\beta,\Lambda,r_{\min}) := \frac{ \lambda d k \ell \sqrt{\log\paren{\frac{ k \ell}{\delta}}}}{\varepsilon n} \paren{\sqrt{\log\paren{\frac{d k \ell}{\delta}}\log\paren{\frac{d k \ell}{\beta}}} + \log \paren{\frac{\Lambda d k}{r_{\min} \beta}}}.
\end{align*}
\remove{
and let ${\cal T} \in \paren{B(0,\Lambda)^k}^n \subseteq (({\mathbb R}^d)^k)^n$.
Assume that $n \geq \frac{\lambda k \ell \sqrt{d \log(d k \ell/\delta)} \log\paren{\frac{\Lambda d k}{r_{\min} \beta}}}{\varepsilon}$
and that
${\cal T}$ is partitioned by $\Delta$-far balls for $\Delta = 16$.
Then w.p. $\geq 1-\beta$ over a random execution of $\mathsf{PrivatekAverages}({\cal T}, \varepsilon, \delta, \beta, r_{\min})$, the output $\hat{A} = \set{\hat{\pa}_1,\ldots,\hat{\pa}_k}$ of the execution is an \emph{$(\alpha,r_{\min})$-good-average} solution for clustering ${\cal T}$ (according to \cref{def:gamma-good}), where
\begin{align*}
\alpha = \alpha(n,d,k,\varepsilon,\delta,\beta,\Lambda,r_{\min}) := \frac{ \lambda d k \ell \sqrt{\log\paren{\frac{ k \ell}{\delta}}}}{\varepsilon n} \paren{\sqrt{\log\paren{\frac{d k \ell}{\delta}}\log\paren{\frac{d k \ell}{\beta}}} + \log \paren{\frac{\Lambda d k}{r_{\min} \beta}}}.
\end{align*}
}
\end{theorem}
We remind that $\ell = O\paren{\frac{\log^2(1/\delta)}{\varepsilon \log n}}$. Therefore, by ignoring ${\rm polylog}(n,d,k,\ell)$ factors, we obtain that for $n = \tilde{\Omega}\paren{\frac{d k \cdot \log^{2.5}(1/\delta)\paren{\sqrt{\log(1/\delta)} + \log\paren{\Lambda/r_{\min}}}}{\varepsilon^2}}$, it holds that $\alpha = O(1)$.
\begin{proof}
Let ${\cal T} \in ({\cal B}(\pt{0},\Lambda)^k)^n \subset (({\mathbb R}^d)^k)^n$ that is partitioned by $(\Delta=16)$-far balls. Consider
a random execution of $\mathsf{PrivatekAverages}({\cal T},\varepsilon,\delta,\beta,r_{\min})$, and let $\tilde{\beta} = \frac{\beta}{2}$ be the value from Step \ref{step:call-testSeparation}.
Since ${\cal T}$ is partitioned by $(2\cdot 7 + 2)$-far balls, \cref{claim:correctness} yields that with probability $1 - \beta/2$,
the set ${\cal B} = \set{B_1,\ldots,B_k}$ (computed in Step~\ref{step:call-testSeparation}) partitions ${\cal T}$. In the following we assume that this event occurs. Let $\set{{\cal Q}_1,\ldots,{\cal Q}_k}$ be the clusters that were computed in Step~\ref{step:compute-clusters} of $\mathsf{PrivatekAverages}$. By \cref{prop:from-almost-par-to-evenly-par}, it holds that $\set{{\cal Q}_1,\ldots,{\cal Q}_k} = {\rm Partition}({\cal T})$. The proof now follows by the utility guarantee of \cref{prop:approx-aver-Rd} for each $i \in [k]$ with the parameters defined in Step~\ref{step:computing-noisy-bound-aver} of the algorithm.
\end{proof}
\begin{remark}[Run time of $\mathsf{PrivatekAverages}$]\label{remark:kAver-runtime}
Step~\ref{step:call-testSeparation} of $\mathsf{PrivatekAverages}$ takes $\tilde{O}(dk^2 n)$ time (see \cref{remark:Test-runtime}). By \cref{prop:approx-aver-Rd}, the $k$ executions of Step~\ref{step:computing-noisy-bound-aver} takes time $\sum_{i=1}^k \tilde{O}(\size{{\cal T}_i}) = \tilde{O}(dkn)$ (ignoring logarithmic factors). Overall, the running time of $\mathsf{PrivatekAverages}$ is $\tilde{O}(dk^2n)$.
\end{remark}
\subsubsection{Reducing the dependency in the dimension $d$}\label{sec:reducing-by-JL}
When the dimension $d$ is large, algorithm $\mathsf{PrivatekAverages}$ is an averages-estimator for $\alpha = \tilde{O}\paren{d/n}$
(ignoring ${\rm poly}(k,1/\varepsilon)$ and ${\rm polylog}(n,\delta,\beta,\Lambda,1/r_{\min})$ factors). This means that if we aim for $\alpha = O(1)$, we must take $n = \tilde{\Omega}(d)$, and in some settings, such a dependency in the dimension might be expensive. Yet, we can easily reduce the $d$ into $\sqrt{d}$ by replacing in Step~\ref{step:computing-noisy-bound-aver} the average algorithm of \cref{prop:approx-aver-Rd} by the average algorithm of \cite{NSV16} that uses the JL transform for saving a factor of $\sqrt{d}$ (see the last paragraph in \cref{sec:prelim:est-aver} for more details).
\subsection{Algorithm $\mathsf{PrivatekNoisyCenters}$}\label{sec:alg-prac-find-averages}
In this section we describe Algorithm $\mathsf{PrivatekNoisyCenters}$ (\cref{alg:noisy-tuple}) which is our second algorithm for $k$-tuple clustering.
\begin{algorithm}[$\mathsf{PrivatekNoisyCenters}$]\label{alg:noisy-tuple}
\item Input: A multiset ${\cal T} \in (({\mathbb R}^d)^k)^n$, privacy parameters $\varepsilon \in (0,1]$, $\delta \in (0,1/2]$, confidence parameter $\beta \in (0,1]$, and a separation parameter $\Delta \gg 6$.
\item Operation:~
\begin{enumerate}
\item Compute $({ Status},{\cal B} = \set{B_1,\ldots,B_k}) = \mathsf{PrivateTestPartition}({\cal T},\varepsilon/2,\delta/4,\beta/2, \Delta)$.\label{step:call-testSeparation-prac
\item If ${ Status} = \text{"Failure"}$, output $\perp$ and abort.
\item Let $\pc_1,\ldots,\pc_k$ be the centers of $B_1,\ldots,B_k$ (respectively).\label{step:centers}
\item For $i=1$ to $k$:\label{step:for-loop-prac}
\begin{enumerate}
\item Let $\lambda_i = \frac{2}{\Delta} (1+\gamma_i) \min_{j \neq i} \norm{\pc_i - \pc_j}$ where $\gamma_i = \frac4{\Delta-2} \cdot \paren{{\rm Lap}(4k/\varepsilon) + \frac{4k}{\varepsilon} \log(4k/\delta) + 1}$.
\item Let $\hat{\pc}_i = \pc_i + ({\cal N}(\pt{0}, \sigma_i^2))^d$, where $\sigma_i = \frac{4 k \lambda_i}{\varepsilon} \sqrt{2 \log(10k/\delta)}$.
\end{enumerate}
\item Output $\hat{C} = \set{\hat{\pc}_1,\ldots,\hat{\pc}_k}$.\label{step:kAverages:output-prac}
\end{enumerate}
\end{algorithm}
\subsubsection{Properties of $\mathsf{PrivatekNoisyCenters}$}
The properties of $\mathsf{PrivatekNoisyCenters}$ are given in the following theorems.
\begin{theorem}[Utility of $\mathsf{PrivatekNoisyCenters}$]\label{thm:util-NoisykCen}
Let $d,k > 0$, $\varepsilon,\beta,\delta \in (0,1]$ with $\delta < \beta$, let ${\cal T} \in (({\mathbb R}^d)^k)^n$, and assume that ${\cal T}$ is partitioned by $(2\Delta+2)$-far balls, for
$\Delta = \Omega\paren{\frac{k \log\paren{k/\delta} \sqrt{\log(k/\beta)}}{\varepsilon}}$. Then when executing $\mathsf{PrivatekNoisyCenters}({\cal T},\varepsilon,\delta,\beta,\Delta)$, with probability $1-\beta$, the output $\hat{C} = \set{\hat{\pc}_1,\ldots,\hat{\pc}_k}$
satisfy for every $i$ and $j \neq i$ that
$\norm{\hat{\pc}_i - \pc_i} < \norm{\hat{\pc}_i - \pc_j}$, where $\pc_1,\ldots,\pc_n$ are the centers from Step~\ref{step:centers}.
\end{theorem}
We remark that the
$k$ factor in the $\Delta$
in \cref{thm:util-NoisykCen}, comes from applying basic composition (\cref{thm:composition1}) over the $k$ noisy centers $\hat{C}$. This however can be reduced to $\tilde{O}(\sqrt{k})$ factor by applying advanced composition (\cref{thm:composition2}).
\begin{proof}
By the union bound on all the choices of $\gamma_i$, w.p. $1-\delta/8 \geq 1 - \beta/8$, for each $i \in [k]$ it holds that $\min_{j \neq i} \norm{\pc_i - \pc_j} \geq \Omega\paren{\sqrt{\log(k/\beta)}} \cdot \sigma_i$. Therefore, for every $i\neq j$ we can apply \cref{prop:separation-spherical-case} with $\mu = \pc_i$ and $\py = \pc_j$ to obtain that with proper choices of the constants in $\Delta$, with probability $1-\frac{\beta}{2k^2}$ it holds that $\norm{\hat{\pc}_i - \pc_i} < \norm{\hat{\pc}_i - \pc_j}$. By the union bound over all $i \neq j$ we deduce that with probability $1 - \beta/2$ this holds for every $i \neq j$, as required.
\end{proof}
\begin{theorem}[Privacy of $\mathsf{PrivatekNoisyCenters}$]\label{thm:priv-NoisykCen}
Let $d,k > 0$, $\varepsilon,\beta \in (0,1]$, $\delta \in (0,1/2]$, $\Delta > 6$. Then for any integer $n \geq 2\cdot \ell(n,\varepsilon/2,\delta/4,\beta/2) + 2$ (where $\ell$ is the function from \cref{def:ell}), $\mathsf{PrivatekNoisyCenters}(\cdot,\varepsilon,\delta,\beta,\Delta)$ is $(\varepsilon + \delta/4, \delta)$-differentially private for databases ${\cal T} \in (({\mathbb R}^d)^k)^n$.
\end{theorem}
\begin{proof}
Let ${\cal T}$ and ${\cal T}'$ be two neighboring multisets of size $n$. In the following we consider two independent executions: $\mathsf{PrivatekNoisyCenters}({\cal T})$ and $\mathsf{PrivatekNoisyCenters}({\cal T}')$ (both with the same parameters $r_{\min},\varepsilon, \delta, \beta$). In $\mathsf{PrivatekNoisyCenters}({\cal T})$, let $O$ be the output, and let ${\cal B} = \set{B_i = B(\pc_i,r_i)}_{i=1}^k$ be the $\Delta$-far balls in the execution $\mathsf{PrivatekNoisyCenters}({\cal T})$. Similarly, we let
$O', {\cal B}' = \set{B_i' = B(\pc_i',r_i')}_{i=1}^k$ be the these values w.r.t. the execution $\mathsf{PrivatekNoisyCenters}({\cal T}')$. By \cref{claim:privacy-framework}, it is enough to prove that the resulting outputs $O = \tilde{C}$ and $O' = \tilde{C}'$ of Steps \ref{step:centers} to \ref{step:kAverages:output-prac}
are $(\varepsilon/2 + \delta/4,\delta/4)$-indistinguishable only in the case that ${\cal T}$ and ${\cal T}'$ are $\ell$-nearly partitioned by ${\cal B}$ and ${\cal B}'$, respectively. Since $2 \ell \leq n-2$ and since ${\cal T}$ and ${\cal T}'$ are neighboring, there must exists a $k$-tuple $X = \set{\px_1,\ldots,\px_k} \in {\cal T}$ that is partitioned by both ${\cal B}$ and ${\cal B}'$. In the rest of the analysis we assume (w.l.o.g.) that $\px_i \in B_i \cap B_i'$ for every $i \in [k]$.
In the following, we prove that for every $i \in [k]$ it holds that $\min_{j \neq i} \norm{\pc_i - \pc_j}$ is close to $\min_{j \neq i} \norm{\pc_i' - \pc_j'}$. For every $i\neq j$ it holds that
\begin{align*}
\norm{\pc_i - \pc_j}
&\leq \norm{\pc_i - \pc_i'} + \norm{\pc_j - \pc_j'} + \norm{\pc_i' - \pc_j'}\\
&\leq \norm{\pc_i - \px_i} + \norm{\pc_i' - \px_i} + \norm{\pc_j - \px_j} + \norm{\pc_j' - \px_j} + \norm{\pc_i' - \pc_j'}\\
&\leq r_i + r_i' + r_j + r_j' + \norm{\pc_i' - \pc_j'}\\
&\leq \frac2{\Delta} \norm{\pc_i - \pc_j} + \frac2{\Delta} \norm{\pc_i' - \pc_j'} + \norm{\pc_i' - \pc_j'}.
\end{align*}
Therefore,
\begin{align*}
\norm{\pc_i - \pc_j} \leq \frac{\Delta+2}{\Delta-2} \norm{\pc_i' - \pc_j'} = \paren{1 + \frac4{\Delta-2}} \norm{\pc_i' - \pc_j'}.
\end{align*}
Now let $i \in [k]$, and let $s = \operatorname*{argmin}_{j \neq i} \norm{\pc_i - \pc_j}$ and $t = \operatorname*{argmin}_{j \neq i} \norm{\pc_i' - \pc_j'}$. We deduce that
\begin{align}\label{eq:compare-minimums}
\min_{j \neq i} \norm{\pc_i - \pc_j} = \norm{\pc_i - \pc_s} \leq \norm{\pc_i - \pc_t} \leq \paren{1 + \frac4{\Delta-2}} \norm{\pc_i' - \pc_t'} = \paren{1 + \frac4{\Delta-2}} \cdot \min_{j \neq i} \norm{\pc_i' - \pc_j'}
\end{align}
Similarly, it holds that $\min_{j \neq i} \norm{\pc_i' - \pc_j'} \leq \paren{1 + \frac4{\Delta-2}} \cdot \min_{j \neq i} \norm{\pc_i - \pc_j}$.
Therefore, by the properties of the laplace mechanism, we deduce that for each $i$, the values of $\lambda_i$ and $\lambda_i'$ are $\frac{\varepsilon}{4k}$-indistinguishable, and by basic composition we deduce that $\set{\lambda_i}_{i=1}^k$ and $\set{\lambda_i'}_{i=1}^k$ are all together $\varepsilon/4$-indistinguishable.
In the following, let $L$ be the event that $\forall i \in [k]:\text{ }\gamma_i \geq \frac{4}{\Delta-2}$, and $L'$ be the event that $\forall i \in [k]:\text{ }\gamma_i' \geq \frac{4}{\Delta-2}$. By \cref{fact:laplace-concent} and the union bound, it holds that $\pr{L},\pr{L'} \leq \delta/8$. Therefore, by \cref{fact:indis-cor}, it is enough to prove that $\tilde{C}|_{L}$ and $\tilde{C}'|_{L'}$
are $(\varepsilon/2 + \delta/4,\delta/8)$-indistinguishable.
First, by \cref{fact:conditioning}, we deduce that $\set{\lambda_i}_{i=1}^k|_{L}$ and $\set{\lambda_i'}_{i=1}^k|_{L'}$ are $(\varepsilon/4 + \delta/4)$-indistinguishable\Enote{Missing here a delta error of $e^{\varepsilon} \delta/(8-\delta) \leq \delta/4$.}. We now continue with the analysis assuming that $\lambda_i = \lambda_i'$ for all $i \in [k]$. Note that for every $i$ it holds that
\begin{align*}
\norm{\pc_i - \pc_i'}
&\leq \norm{\pc_i - \px_i} + \norm{\pc_i' - \px_i}\\
&\leq r_i + r_i'\\
&\leq \frac1{\Delta} \cdot \paren{\min_{j \neq i} \norm{\pc_i - \pc_j} + \min_{j \neq i} \norm{\pc_i' - \pc_j'}}\\
&\leq \lambda_i,
\end{align*}
where the last inequality holds by \cref{eq:compare-minimums} (assuming that $L$ occurs). Therefore, by the properties of the Gaussian Mechanism (\cref{fact:Gaus}), we deduce that for each $i$, $\hpc_i$ and $\hpc_i'$ are $(\frac{\varepsilon}{4k},\frac{\delta}{8k})$-indistinguishable, and by basic composition (\cref{thm:composition1}) we deduce that $\hat{C}$ and $\hat{C}'$ are $(\frac{\varepsilon}{4},\frac{\delta}{8})$-indistinguishable (assuming that $\lambda_i = \lambda_i'$ for all $i \in [k]$). Finally, recall that $\set{\lambda_i}_{i=1}^k|_{L}$ and $\set{\lambda_i'}_{i=1}^k|_{L'}$ are $(\varepsilon/4 + \delta/4)$-indistinguishable, and therefore, we conclude by adaptive composition (\cref{thm:composition1}) that $\hat{C}$ and $\hat{C}'$ are $(\varepsilon/2 + \delta/4,\delta/8)$-indistinguishable.
\end{proof}
\begin{remark}[Run time of $\mathsf{PrivatekNoisyCenters}$]\label{remark:kNoisyCenters-runtime}
Step~\ref{step:call-testSeparation-prac} of $\mathsf{PrivatekNoisyCenters}$ takes $\tilde{O}(dk^2 n)$ time (see \cref{remark:Test-runtime}). The foor-loop in Step~\ref{step:for-loop-prac} only takes $O(d k n)$ time. Overall, the running time of $\mathsf{PrivatekNoisyCenters}$ is $\tilde{O}(dk^2n)$.
\end{remark}
\section{$k$-Means Clustering}\label{sec:kMeans}
In this section we present our first application of $k$-tuples clustering, which is an $(\varepsilon,\delta)$-differentially private $k$-means approximation algorithm $\mathsf{PrivatekMeans}$ with utility guarantee that holds when the input is stable in the sense that we will define.
We first start with preliminaries about $k$-means clustering.
\subsection{Preliminaries}\label{sec:kMeans:Preliminaries}
For a multiset ${\cal P} \in ({\mathbb R}^d)^*$ and a $k$-tuple of centers $C = \set{\pc_1,\ldots,\pc_k} \in ({\mathbb R}^d)^k$, we denote ${\rm COST}_{{\cal P}}(C) := \sum_{\px \in {\cal P}} \min_{i \in [k]}\norm{\px - \pc_i}^2$ and denote ${\rm OPT}_k({\cal P}) := \min_{C \in ({\mathbb R}^d)^k} {\rm COST}_{{\cal P}}(C)$.
The following proposition states that given a multiset ${\cal P} \in ({\mathbb R}^d)^n$ and an $\omega$-approximation algorithm $\mathsf{A}$ for $k$-means, then when sampling $s$ i.i.d.\ points from ${\cal P}$ and executing $\mathsf{A}$ on these points, then with probability $1-\beta$ we obtain $k$ centers with cost $\approx \omega {\rm OPT}_{k}({\cal P})$ (up to a small additive error that depends on $s$ and $\beta$). The proof appears at \cref{missing-proof:cost-of-sample-is-good}.
\def\propCostOfSampleIsGood{
Let ${\cal P}$ be a multiset of $n$ points in ${\cal B}(\pt{0},\Lambda) \subseteq {\mathbb R}^d$ and let ${\cal A}$ be an $\omega$-approximation algorithm for $k$-means. Consider the following random execution: (1) Construct a multiset ${\cal S}$ of $s$ i.i.d.\ samples from ${\cal P}$, (2) Compute $\tilde{C} = {\cal A}({\cal S},k)$. Then for every $\beta > 0$, with probability $1-\beta$ it holds that
\begin{align*}
{\rm COST}_{{\cal P}}(\tilde{C}) \leq \omega\cdot {\rm OPT}_k({\cal P}) + \xi(s,\beta),
\end{align*}
where $\xi(s,\beta) := 4\paren{M(s,\beta)+ \sqrt{M(s,\beta) \cdot \omega {\rm OPT}_k({\cal P})}}$ for $M(s,\beta) := 25 \Lambda^2 k d \log \paren{\frac{2nd}{\beta}} \cdot \frac{n}{s}$.
}
\begin{proposition}\label{prop:cost-of-sample-is-good}
\propCostOfSampleIsGood
\end{proposition}
The following proposition states that given a multiset of points ${\cal P}$ and given two $k$-tuples of centers $C = \set{\pc_1,\ldots,\pc_k}$ and $C' = \set{\pc_{1}', \ldots, \pc_{k}'}$ such that each $\pc_i'$ is relatively close to a unique center $\pc_i$ in $C$, then by clustering the points according to $C'$ and performing a single Lloyd step, we get new centers whose $k$-means cost is almost bounded by ${\rm COST}_{{\cal P}}(C)$. The proof appears at \cref{missing-proof:close-centers-have-similar-cost}.
\def\propCloseCentersHaveSimilarCost{
Let $k \in {\mathbb{N}} $ and $\gamma \in [0,1/8]$.
Let ${\cal P} \in ({\mathbb R}^d)^*$, let $C = \set{\pc_1,\ldots,\pc_k}$ and $C' = \set{\pc_{1}', \ldots, \pc_{k}'}$ be two $k$-tuples of centers in ${\mathbb R}^d$ such that
for every $i \in [k]$ it holds that $\norm{\pc_i' - \pc_i} \leq \gamma\cdot D_i$, where $D_i = \min_{j \neq i}\norm{\pc_i - \pc_j}$. In addition, for every $i \in [k]$ let ${\cal P}_i$ be the multiset of all points in ${\cal P}$ that $\pc_i'$ is closest to them in $C'$.
Then
$$\sum_{i=1}^k {\rm OPT}_1({\cal P}_i) \leq (1 + 32\gamma) {\rm COST}_{{\cal P}}(C).$$
}
\begin{proposition}\label{prop:close-centers-have-similar-cost}
\propCloseCentersHaveSimilarCost
\end{proposition}
\subsection{Private $k$-Means Under Stability Assumption}\label{sec:kMeans:Alg}
In this section we describe our private algorithm $\mathsf{PrivatekMeans}$ for approximation the $k$-means when the input is stable in the sense that we will define next.
The idea is the following: Fix a database ${\cal P} \in ({\mathbb R}^d)^n$, parameters $s,t \in {\mathbb N}$ and a (non-private) $k$-means approximation algorithm $\mathsf{A}$. Now execute $\mathsf{A}$ on $s$ i.i.d.\ samples from ${\cal P}$, and repeat this process $t$ times. Consider the event (over this process) that all the $t$ sets of $k$ centers are almost located at the same positions.
More formally, consider a random execution of $\mathsf{GenCenters}^{\mathsf{A}}({\cal P},k,s,t)$ (\cref{alg:generateCenters}). For a $k$-tuple of centers $C = \set{\pc_1,\ldots,\pc_k} \in ({\mathbb R}^d)^k$ and a small stability parameter $\gamma > 0$ (say, $\gamma = 0.01$), let $E_C^{\gamma}$ be the event that is defined below.
\begin{definition}[Event $E_{C}^{\gamma}$ over a random sampling of $\mathsf{GenCenters}$]\label{def:event-ECgamma}
Let $E_{C}^{\gamma}$ be the event that for every $j \in [t]$ and $i \in [k]$, there exists a center in $\tilde{C}_j$ (call it $\tpc_i^j$) such that $\norm{\tpc_i^j - \pc_i} \leq \gamma\cdot D_i$, where $D_i = \min_{j \neq i}\norm{\pc_i - \pc_j}$.
\end{definition}
Namely, event $E_{C}^{\gamma}$ implies that the output $\tilde{{\cal C}} \in (({\mathbb R}^d)^k)^t$ of $\mathsf{GenCenters}$ is partitioned by $\Delta$-far balls for $\Delta = 1/\gamma$, where ${\rm Partition}(\tilde{{\cal C}})$ (according to \cref{def:clusters-rel}) is exactly $\set{\set{\tpc_1^j}_{j=1}^t, \ldots, \set{\tpc_k^j}_{j=1}^t}$ (i.e., for each $i \in [k]$, the centers $\set{\tpc_i^j}_{j=1}^t$ are very close to each other, compared to the distance from the other centers).
In this section, we describe our general $(\varepsilon,\delta)$-differentially private algorithm $\mathsf{PrivatekMeans}$, that uses oracle accesses to a non-private $k$-means algorithm $\mathsf{A}$ and a private good-average $k$-tuple clustering algorithm $\mathsf{B}$, such that the following utility guarantee it achieved:
For any $k$-centers $C$ and a small enough $\gamma$,
when the event $E_{C}^{\gamma}$ occurs over $\mathsf{GenCenters}^{\mathsf{A}}({\cal P},k,s,t)$, then with probability $1-\beta$, algorithm $\mathsf{PrivatekMeans}$ outputs $\hat{C} = \set{\hat{\pc}_1,\ldots,\hat{\pc}_k}$ with ${\rm COST}_{{\cal P}}(\hat{C}) \leq (1 + O(\gamma)) {\rm COST}_{{\cal P}}(C)$ (plus some small additive error). $\mathsf{PrivatekMeans}$ is described in \cref{alg:kMeans} and its properties are proven in \cref{sec:kMeans:properties}. In \cref{sec:application} we show that a variant of the separation assumption in \cite{OstrovskyRSS12} implies that event $E_{C^*}^{\gamma}$ holds with high probability, where $C^*$ are the optimal $k$ means for ${\cal P}$.
\begin{algorithm}[$\mathsf{GenCenters}$]\label{alg:generateCenters}
\item Input: A multiset ${\cal P}$ of points in $B(\pt{0},\Lambda) \subseteq {\mathbb R}^d$ and parameters $k, s,t \in {\mathbb N}$.
\item Oracle: A (non-private) $k$-means algorithm $\mathsf{A}$.
\item Operation:~
\begin{enumerate}
\item For each $j \in [t]$:
\begin{enumerate}
\item Let ${\cal S}_j$ be a database containing $s$ i.i.d.\ samples from ${\cal P}$ (with replacement).\label{step:sample}
\item Compute the $k$-tuple of centers $\tilde{C}_j = \mathsf{A}({\cal S}_j)$.\label{step:non-priv-centers}
\end{enumerate}
\item Output ${\cal T} = \set{\tilde{C}_1,\ldots,\tilde{C}_t}$.\label{step:new-multisets}
\end{enumerate}
\end{algorithm}
\begin{algorithm}[$\mathsf{PrivatekMeans}$]\label{alg:kMeans}
\item Input: A multiset ${\cal P}$ of $n$ points in $B(\pt{0},\Lambda) \subseteq {\mathbb R}^d$, parameters $k, s, t \in {\mathbb N}$, privacy parameters $\varepsilon,\delta \in (0,1]$, confidence parameter $\beta \in (0,1]$, and a stability parameter $\gamma > 0$.
\item Oracle: A (non-private) $k$-means algorithm $\mathsf{A}$, a private average algorithm $\mathsf{A}'$,
and a private $k$-tuple clustering algorithm $\mathsf{B}$.
\item Operation:~
\begin{enumerate}
\item Compute ${\cal T} = \mathsf{GenCenters}^{\mathsf{A}}({\cal P},k,s,t)$.\label{step:gen}
\item Compute $\set{\hat{\pa}_1,\ldots,\hat{\pa}_k} = \mathsf{B}({\cal T})$.\label{step:the-aver-estim}
\item For each $i \in [k]:$
\begin{itemize}
\item Let ${\cal P}_i$ be the points in ${\cal P}$ that $\hpa_i$ is the closest point to them among $\set{\hat{\pa}_1,\ldots,\hat{\pa}_k}$. \label{step:compute-clusters-kmeans}
\item Compute $\hpc_i = \mathsf{A}'({\cal P}_i)$.\label{step:add-gaus-noise}
\end{itemize}
\item Output $\set{\hat{\pc}_1,\ldots,\hat{\pc}_k}$.
\end{enumerate}
\end{algorithm}
\subsection{Properties of $\mathsf{PrivatekMeans}$}\label{sec:kMeans:properties}
The following theorem captures the privacy guarantee of $\mathsf{PrivatekMeans}$.
\begin{theorem}[Privacy of $\mathsf{PrivatekMeans}$]\label{claim:kMeans-privacy}
Let $n,s,t,d,k \in {\mathbb N}$, $\Lambda > 0$, $\beta,\varepsilon,\delta, \gamma \in (0,1]$,
let $\mathsf{A}$ be an (arbitrary) algorithm that outputs $k$ centers in $B(\pt{0},\Lambda) \subset {\mathbb R}^d$, let $\mathsf{A}'$ be an $(\frac{\varepsilon}{12},\frac{\delta}{8 e^{\varepsilon}})$-DP algorithm for databases over $B(\pt{0},\Lambda)$, and let
$\mathsf{B}$ be an $\left(\frac{\varepsilon}{6},\frac{\delta}{4 e^{\varepsilon}}\right)$-DP algorithm for databases ${\cal T} \in (B(\pt{0},\Lambda)^k)^t$ (i.e., of size $t$). If $n \geq 2st$, then algorithm $\mathsf{PrivatekMeans}^{\mathsf{A}, \mathsf{A}', \mathsf{B}}(\cdot, k,s,t, \varepsilon,\delta,\beta,\gamma)$
is $(\varepsilon,\delta)$-differentially private for databases ${\cal P}$ over $B(\pt{0},\Lambda) \subset {\mathbb R}^d$.
\end{theorem}
\begin{proof}
The proof builds on the fact that switching between sampling
with replacement and without replacement has only a small effect on
the privacy, as stated in \cref{lem:DP-with-replacement}.
Consider a different variant $\widetilde{\mathsf{Gen}}{\mathsf{Centers}}$ of the procedure $\mathsf{GenCenters}$, in which the sampling of the $\approx s \cdot t$ points in all the iterations of Step~\ref{step:sample} is done without replacement, and consider a variant $\widetilde{\mathsf{Priv}}{\mathsf{atekMeans}}$ of $\mathsf{PrivatekMeans}$ in which it executes $\widetilde{\mathsf{Gen}}{\mathsf{Centers}}$ in Step~\ref{step:gen} rather than $\mathsf{GenCenters}$.
Let ${\cal P} = \set{\px_1,\ldots,\px_n}$ and ${\cal P}' = \set{\px_1',\ldots,\px_n'}$ be two neighboring databases of points. In the following we consider two independent executions $\widetilde{\mathsf{Priv}}{\mathsf{atekMeans}}({\cal P})$ and $\widetilde{\mathsf{Priv}}{\mathsf{atekMeans}}({\cal P}')$ (both with the same parameters $k,\varepsilon,\delta,\beta$ and oracles $\mathsf{A},\mathsf{B}$). For $j \in [t]$ let ${\cal J}_j \subseteq [n]$ be the $s$ chosen indices of the points ${\cal S}_j$ in Step~\ref{step:sample} of $\widetilde{\mathsf{Gen}}{\mathsf{Centers}}$ (i.e., ${\cal S}_j = \set{\px_i}_{i \in {\cal J}_j}$), and let ${\cal J}_j'$ be the same indices in the execution $\widetilde{\mathsf{Priv}}{\mathsf{atekMeans}}({\cal P}')$. Since ${\cal J}_j$ and ${\cal J}_j'$ only depend on $n$ and not on the content of ${\cal P}$ and ${\cal P}'$, it is enough to prove that the output of both executions is $(\varepsilon,\delta)$-indistinguishable conditioned on the event that ${\cal J}_j = {\cal J}_j'$ for every $j \in [t]$. In the following, we assume that this event occurs.
Since ${\cal P}$ and ${\cal P}'$ are neighboring, there exists at most one index $j \in [t]$ such that ${\cal S}_j$ of the execution $\widetilde{\mathsf{Priv}}{\mathsf{atekMeans}}({\cal P})$ is different than the corresponding set in $\widetilde{\mathsf{Priv}}{\mathsf{atekMeans}}({\cal P}')$, and therefore, the outputs $\hat{{\cal C}}$ of $\widetilde{\mathsf{Gen}}{\mathsf{Centers}}$ are different by at most one $k$-tuple.
Therefore, by the assumption over algorithm $\mathsf{B}$, we deduce that the outcome of Step~\ref{step:the-aver-estim} is $(\frac{\varepsilon}{6}, \frac{\delta}{4 e^{\varepsilon}})$-differentially private.
In the following, we prove that for any fixing of $k$ averages $\tpa_1,\ldots,\tpa_k$, Step~\ref{step:add-gaus-noise} is $(\frac{\varepsilon}{6}, \frac{\delta}{4 e^{\varepsilon}})$-differentially private. Given that, we deduce that $\widetilde{\mathsf{Priv}}{\mathsf{atekMeans}}$ is $(\frac{\varepsilon}{3}, \frac{\delta}{2 e^{\varepsilon}})$-differentially private by (adaptive) composition of Steps~\ref{step:the-aver-estim} and \ref{step:add-gaus-noise} (\cref{thm:composition1}). Hence, we conclude that the original algorithm $\mathsf{PrivatekMeans}$, that chooses the points with replacement, is $(\varepsilon,\delta)$-differentially private by applying \cref{lem:DP-with-replacement} with $m = s t \leq \frac{n}2, \frac{\varepsilon}{3}, \frac{\delta}{2 e^{\varepsilon}}$.
It is left to prove the privacy guarantee of Step~\ref{step:add-gaus-noise}. For that, fix $k$ averages $\hat{A}=\set{\tpa_1,\ldots,\tpa_k}$, let ${\cal P}_1,\ldots,{\cal P}_k$ be the $k$ multisets in Step~\ref{step:add-gaus-noise} w.r.t ${\cal P}$ and $\hat{A}$, and let ${\cal P}_1',\ldots,{\cal P}_k'$ be the same multisets w.r.t ${\cal P}'$ and $\hat{A}$. Since ${\cal P}$ and ${\cal P}'$ are neighboring, there exist at most two indices $i \in [k]$ such that ${\cal P}_i \neq {\cal P}_i'$, and for each one of them, ${\cal P}_i$ and ${\cal P}_i'$ are neighboring. Therefore, by the privacy guarantee of $\mathsf{A}'$ along with basic composition (\cref{thm:composition1}), Step~\ref{step:add-gaus-noise} is $(2 \cdot \frac{\varepsilon}{12}, 2\cdot \frac{\delta}{8 e^{\varepsilon}})$-differentially private, as required.
\end{proof}
The following theorem, which captures the utility guarantee of $\mathsf{PrivatekMeans}$, states that when event $E^{\gamma}_C$ (\cref{def:event-ECgamma}) occurs by $\mathsf{GenCenters}$ in Step~\ref{step:gen}, then with probability at least $1-\beta$, the output $\hat{C} = \set{\hat{\pc}_1,\ldots,\hat{\pc}_k}$ has ${\rm COST}_{{\cal P}}(\hat{C}) \leq (1 + O(\gamma)) {\rm COST}_{{\cal P}}(C) + O\paren{\zeta^2 k}$, where $\zeta/m$ is the additive error of $\mathsf{A}'$ over database of size $m$ with confidence $1- O(\beta/k)$.
We remark that by setting $\mathsf{A}'$ as the Gaussian mechanism (\cref{fact:Gaus}) with privacy parameter $\paren{O(\varepsilon),O(\delta)}$, we obtain that $\zeta = O\paren{\frac{\Lambda \sqrt{\log(1/\delta)}}{\varepsilon} \paren{\sqrt{d} + \sqrt{\log(k/\beta)}}}$.
\begin{theorem}[Utility of $\mathsf{PrivatekMeans}$]\label{claim:kMeans-utility}
Let $n,s,t,d,k,\Lambda > 0$, $\beta,\varepsilon,\delta \in (0,1]$, $\gamma \in (0,\frac1{16}]$, and let ${\cal P}$ be a multiset of $n$ points in $B(\pt{0},\Lambda) \subseteq {\mathbb R}^d$. Let $\mathsf{A}$ be an algorithm that outputs $k$ centers in ${\mathbb R}^d$. Let $\mathsf{A}'$ be an algorithm that given a multiset ${\cal S}$ of points over $B(\pt{0},\Lambda)$, w.p. $1-\frac{\beta}{2k}$ estimates ${\rm Avg}({\cal S})$ up to an additive error $\zeta/\size{{\cal S}}$ for $\zeta = \zeta(d, \Lambda, \frac{\beta}{2k})$.
Let $\mathsf{B}$ be an $(t, \: \alpha=1,\text{ }r_{\min}=\gamma/n,\: \beta/2, \: \Delta=1/\gamma, \: \Lambda)$-averages-estimator $k$-tuple clustering algorithm (\cref{def:averages-estimator}). Finally, let $C = \set{\pc_1,\ldots,\pc_k} \in ({\mathbb R}^d)^k$ with $\min_{i \neq j} \norm{\pc_i - \pc_j} \geq 1/n$. Consider a random execution of $\mathsf{PrivatekMeans}^{\mathsf{A},\mathsf{A}',\mathsf{B}}({\cal P},t,k,\varepsilon,\delta,\beta,\gamma)$ conditioned that the event $E_{C}^{\gamma}$ occurs by $\mathsf{GenCenters}$ in Step~\ref{step:gen} of the execution.
Then with probability $1-\beta$ (over the above conditional execution), the output $\hat{C} = \set{\hpc_1,\ldots,\hpc_k}$ of $\mathsf{PrivatekMeans}$ satisfies
\begin{align*}
COST_{{\cal P}}(\set{\hpc_1,\ldots,\hpc_k}) \leq (1 + 64\gamma) {\rm COST}_{{\cal P}}(C) + \zeta k (\zeta + 2\Delta)
\end{align*}
\end{theorem}
\begin{proof}
Consider a random execution of $\mathsf{PrivatekMeans}^{\mathsf{A},\mathsf{A}',\mathsf{B}}({\cal P},t,k,\varepsilon,\delta,\beta,\gamma)$ conditioned on the event $E^{\gamma}_C$. For $j \in [t]$, let $\tilde{C}_j = \set{\tpc_1^j,\ldots,\tpc_k^j}$ be the value from Step~\ref{step:non-priv-centers} of the $j$'th iteration of $\mathsf{GenCenters}$, where we denote by $\tpc_i^j$ the center that is close to $\pc_i$, i.e., $\norm{\tpc_i^j - \pc_i} \leq \gamma \cdot D_i$, where $D_i = \min_{j \neq i}\norm{\pc_i- \pc_j}$ (such center exists by event $E_C^{\gamma}$). In addition, for $i \in [k]$, let $\pa_i = {\rm Avg}(\set{\tpc_{i}^j}_{j=1}^t)$ and note that
\begin{align}\label{eq:dist-pai-pci}
\forall i \in [k]:\text{}\norm{\pa_i - \pc_i}
&= \norm{\frac1{t} \sum_{j=1}^t \tpc_{i}^j - \pc_i}\\
&\leq \frac1{t} \sum_{j=1}^t \norm{\tpc_{i}^j - \pc_i}\nonumber\\
&\leq \gamma \cdot D_i.\nonumber
\end{align}
Now, let ${\cal T}=\set{\tilde{C}_1 = \set{\tpc_i^1}_{i=1}^k, \ldots,\tilde{C}_t = \set{\tpc_i^t}_{i=1}^k}$ be the output of $\mathsf{GenCenters}$ in Step~\ref{step:gen} of $\mathsf{PrivatekMeans}$, and note that ${\cal T}$ is partitioned by the set of $(\Delta = 1/\gamma)$-far balls $\set{B(\pc_i, r_i = \gamma D_i)}_{i=1}^k$ where ${\rm Partition}({\cal T}) = \set{\set{\tpc_{1}^j}_{j=1}^t, \ldots, \set{\tpc_{k}^j}_{j=1}^t}$. Therefore, when executing algorithm $\mathsf{B}$ in Step~\ref{step:the-aver-estim}, we obtain by assumption a set of $k$ points $\set{\hpa_1,\ldots,\hpa_k}$ such that with probability $1 - \frac{\beta}{2}$ it holds that
\begin{align}\label{eq:dist-pai-hpai}
\forall i \in [k]:\text{ }
\text{}\norm{\pa_i - \hpa_i} \leq \max\set{r_i,r_{\min}} \leq \gamma\cdot D_i,
\end{align}
where in the second inequality we used the fact that $r_i \leq \gamma D_i$ and that $r_{\min} = \gamma/n \leq \gamma D_i$. Therefore, we deduce by \cref{eq:dist-pai-pci,eq:dist-pai-hpai} that with probability $1 - \frac{\beta}2$ it holds that
\begin{align}\label{eq:hpa-pc-dist}
\forall i \in [k]:\text{}\norm{\hpa_i - \pc_i} \leq 2\gamma \cdot D_i.
\end{align}
Let ${\cal P}_1,\ldots,{\cal P}_k$ be the clusters from Step~\ref{step:compute-clusters-kmeans} of the algorithm. If \cref{eq:hpa-pc-dist} occurs, then by \cref{prop:close-centers-have-similar-cost} we get that
\begin{align}\label{eq:avg-cost}
\sum_{i=1}^k \sum_{\px \in {\cal P}_i} \norm{\px - {\rm Avg}({\cal P}_i)}^2 \leq (1 + 64\gamma) {\rm COST}_{{\cal P}}(C).
\end{align}
Since the algorithm computes a noisy estimation $\hpc_i$ of each ${\rm Avg}({\cal P}_i)$ using the oracle $\mathsf{A}'$, we get that with probability $1-k\hat{\beta} = 1 - \frac{\beta}{2}$ it holds that
\begin{align}\label{eq:dist-from-avg}
\forall i \in [k]:\quad \norm{\hpc_i - {\rm Avg}({\cal P}_i)} \leq \zeta/\size{{\cal P}_i}
\end{align}
Finally, since \cref{eq:avg-cost} occurs with probability $1 - \frac{\beta}{2}$, and \cref{eq:dist-from-avg} occurs with probability $1 - \frac{\beta}{2}$, we conculde that with probability $1-\beta$ both of them occurs, which implies that
\begin{align*}
\lefteqn{{\rm COST}_{{\cal P}}(\set{\hpc_1,\ldots,\hpc_k})}\\
&\leq \sum_{i=1}^k \sum_{\px \in {\cal P}_i} \norm{\px - \hpc_i}^2\\
&= \sum_{i=1}^k \sum_{\px \in {\cal P}_i} \paren{\norm{\px - {\rm Avg}({\cal P}_i)}^2 + \norm{\hpc_i - {\rm Avg}({\cal P}_i)}^2 + 2 \norm{\px - {\rm Avg}({\cal P}_i)}\cdot \norm{\hpc_i - {\rm Avg}({\cal P}_i)}}\\
&\leq (1 + 64\gamma) {\rm COST}_{{\cal P}}(C) + k \zeta^2 + 2 \Lambda k \cdot \zeta.
\end{align*}
where in the last term of the second inequality we used the fact that $\norm{\px - {\rm Avg}({\cal P}_i)} \leq \Lambda$ for all $i \in [k]$ and $\px \in {\cal P}_i$.\Hnote{We just discard $\size{{\cal P}_i}^2$ in the denominator in the second term of the second inequality ?}\Enote{Yes}\Hnote{can't we gain anything for this ? mention this ?}\Enote{I guess we can and we should!! In general, I believe we can also improve the theoretical bounds here if we do the analysis more carefully.}
\end{proof}
\remove{
\begin{remark}[Run time of $\mathsf{PrivatekMeans}$]
The algorithm performs $t$ oracle queries to the non-private algorithm ${\cal A}$, each time over a collection of points of size $s = O(n/t)$. Then, the most expensive step is executing $\mathsf{PrivatekAverages}$, which takes $\tilde{O}(d k^2 n)$ time.
\end{remark}
}
\remove{
}
\remove{
}
\remove{
\Enote{Delete the following event}
}
\section{Private $k$-Means Under Stability Assumption}\label{sec:kMeans}
In this section we present an $(\varepsilon,\delta)$-differential private $k$-means approximation algorithm $\mathsf{PrivatekMeans}$ with utility guarantee that holds when the input is stable in the sense that we will define. We do so by reducing the problem of approximating the $k$-means of stable data into the problem from \cref{sec:finding-averages} of approximating the averages of clusters of points that are fully-partitioned by $k$ far balls. The idea is the following: Fix a database ${\cal P} \in ({\mathbb R}^d)^n$, a parameter $m < n$ and a (non-private) $k$-means approximation algorithm ${\cal A}$. Now assume that when executing ${\cal A}$ on $m$ i.i.d. samples from ${\cal P}$, then with high probability all these sets of $k$ centers are almost located at the same positions. More formally, if we repeate the above process $T$ times (for some parameter $T$) and denote by $\tilde{C}_t = \set{\pc_1^t, \ldots, \pc_k^t}$ the centers of iteration $t \in [T]$, then our stability assumption requires that with high probability, the multiset of all resulting centers ${\cal C} = \bigcup_{t \in [T]} \tilde{C}_t$ is fully-partitioned by $k$ far balls, where ${\rm Partition}({\cal C})$ (according to \cref{def:clusters-rel}) is exactly $\set{\set{\pc_1^t}_{t=1}^T, \ldots, \set{\pc_k^t}_{t=1}^T}$ (i.e., for each $i \in [k]$, the $i$'th centers of each execution are very close to each other, compared to the distance from the other centers). Then under this assumption, \Hnote{Don't we want to formalize this assumption in some environment and give it a name/number so it is easier to refer to and the text is less vague ?} we show in this section how to use algorithm $\mathsf{PrivatekAverages}$ for approximating the $k$-means of ${\cal P}$. The full details of $\mathsf{PrivatekMeans}$ are described in \cref{fig:kMeans}. The privacy guarantee is proven in \cref{sec:priv-kMeans}, the utility guarantee (under the stability assumption) is proven in \cref{sec:util-kMeans}. In \cref{sec:application} we show that a variant of the separation assumption in \cite{OstrovskyRSS12} implies our stability assumption.
\begin{figure}[thb!]
\begin{center}
\noindent\fbox{
\parbox{.95\columnwidth}{
\begin{center}{ \bf Algorithm $\mathsf{PrivatekMeans}$}\end{center}
\textbf{Input:} A multiset ${\cal P}$ of points in $B(\pt{0},\Lambda) \subseteq {\mathbb R}^d$, parameter $k \in {\mathbb N}$, privacy parameters $\varepsilon,\delta >0$ and a confidence parameter $\beta > 0$. Let $n = \size{{\cal P}}$.
\textbf{Additional input:} A (non-private) $k$-means algorithm ${\cal A}$.
\begin{enumerate}
\item Let $T \in {\mathbb N}$ be a value to be determined by the analysis.\label{step:T}
\item For each $t \in [T]$:
\begin{enumerate}
\item Let ${\cal S}_t$ be a database containing $m= \floor{\frac{n}{2T}}$ i.i.d. samples from ${\cal P}$ (with replacement).\label{step:sample}
\item Compute $\tilde{C}_t \gets {\cal A}({\cal S}_t)$.\label{step:non-priv-centers}
\end{enumerate}
\item Let $\tilde{{\cal C}} = \bigcup_{t \in [T]} \tilde{C}_t$ (a ``multiset union'', that includes duplications).\label{step:new-multisets}
\item Execute $\mathsf{PrivatekAverages}$ (\cref{fig:FindAverages}) on the $kT$-size multiset $\tilde{{\cal C}}$ with input parameters
\noindent $\tilde{\alpha} = \frac1{n}, \tilde{\varepsilon} = \frac{\varepsilon}{6 k}$, $\tilde{\delta} = \frac{\min\set{\delta,\beta}}{4 e^{\varepsilon} k}$, $\tilde{k} = k$. Let $\set{\hat{\pa}_1,\ldots,\hat{\pa}_k}$ be the outcome of the execution.\label{step:the-aver-estim}
\item For each $i \in [k]:$
\begin{itemize}
\item Let $\hat{{\cal P}}_i$ be the points in ${\cal P}$ that $\hpa_i$ is the closest point to them among $\set{\hat{\pa}_1,\ldots,\hat{\pa}_k}$. \label{step:compute-clusters-kmeans}
\item Use the Gaussian Mechanism (\ref{fact:Gaus}) with parameters $\Lambda, \hat{\varepsilon} = \frac{\varepsilon}{6k}, \hat{\delta} = \frac{\delta}{8 e^{2\varepsilon} k}, \hat{\beta} = \frac{\beta}{2k}$ to compute a noisy average $\hpc_i$ of $\hat{{\cal P}}_i$.\label{step:add-gaus-noise
\end{itemize}
\item Output $\set{\hat{\pc}_1,\ldots,\hat{\pc}_k}$.
\end{enumerate}
}}
\end{center}
\caption{A private $k$-means approximation algorithm $\mathsf{PrivatekMeans}$ under stability assumption.\label{fig:kMeans}}
\end{figure}
\subsection{Privacy of $\mathsf{PrivatekMeans}$}\label{sec:priv-kMeans}
The following claim states that for a suitable choice of the parameter $T$ in Step~\ref{step:T}, then the algorithm $\mathsf{PrivatekMeans}$ is differential-private.
\begin{claim}[Privacy]\label{claim:kMeans-privacy}
Let $N = N(d,k,\varepsilon,\delta)= \tilde{O}\paren{k^4 d \log^4(1/\delta)/\varepsilon^2}$ be the value from \cref{claim:main}, and assume that $T \geq \ceil{N/k}$, where $T$ is the value in Step~\ref{step:T} of $\mathsf{PrivatekMeans}$.
Then Algorithm $\mathsf{PrivatekMeans}$ is $(\varepsilon ,\delta)$-differential private.
\end{claim}
\begin{proof}
The proof builds on the fact that switching between sampling
with replacement and without replacement has only a small effect on
the privacy, as stated in \cref{fact:DP-with-replacement}.
Consider a different variant of $\mathsf{PrivatekMeans}'$, in which the sampling of the $\approx n/2$ points in all Steps~\ref{step:sample} ($m = \floor{n/(2 T)}$ points in $T$ iterations) is done without replacement.
Let ${\cal P} = \set{\px_1,\ldots,\px_n}$ and ${\cal P}' = \set{\px_1',\ldots,\px_n'}$ be two neighboring databases of points \Enote{define}. In the following we consider two independent executions $\mathsf{PrivatekMeans}'({\cal P})$ and $\mathsf{PrivatekMeans}'({\cal P}')$ (both with the same parameters $k,\varepsilon,\delta,\beta$). In the execution $\mathsf{PrivatekMeans}'({\cal P})$, for $t \in [T]$ let ${\cal J}_t \subseteq [n]$ be the $m$ chosen indices of the points ${\cal S}_t$ in Step~\ref{step:sample} (i.e., ${\cal S}_t = \set{\px_j}_{j \in {\cal J}_t}$), and let ${\cal J}_t'$ be the indices w.r.t the execution $\mathsf{PrivatekMeans}'({\cal P}')$. Since ${\cal J}_t$ and ${\cal J}_t'$ only depend on $n$ and not on the content of ${\cal P}$ and ${\cal P}'$, it is enough to prove that the output of both executions is $(\varepsilon,\delta)$-indistinguishable conditioned on the event that ${\cal J}_t = {\cal J}_t'$ for every $t \in [T]$. In the following, we assume that this event occurs.
Since ${\cal P}$ and ${\cal P}'$ are neighboring, there exists at most one index $t \in [T]$ such that ${\cal S}_t$ of the execution $\mathsf{PrivatekMeans}'({\cal P})$ is different than the corresponding set in $\mathsf{PrivatekMeans}'({\cal P}')$, and therefore, the resulting $\hat{{\cal C}}$ in Step~\ref{step:new-multisets} is different by at most $k$ points. Since $\mathsf{PrivatekAverages}$ is $(\tilde{\varepsilon},\tilde{\delta})$-differential private for multisets of size $\geq N$ (\cref{claim:privacy}), we deduce that the resulting $k$ averages $\set{\tpa_1,\ldots,\tpa_k}$ in Step~\ref{step:the-aver-estim} are $(k\tilde{\varepsilon},k\tilde{\delta})$- indistinguishable, i.e., Step~\ref{step:the-aver-estim} is $(\frac{\varepsilon}{6}, \frac{\delta}{4 e^{\varepsilon}})$-differential private. Note that for any fixing of $\set{\tpa_1,\ldots,\tpa_k}$, Step~\ref{step:add-gaus-noise} is a composition of $k$ Gaussian mechanisms, and by basic composition (\cref{thm:composition1}) it is $(\frac{\varepsilon}{6}, \frac{\delta}{8 e^{2\varepsilon}})$-differential private. By iterative composition \Enote{is iterative is the right word?} of Step~\ref{step:the-aver-estim} and Step~\ref{step:add-gaus-noise} (\cref{fact:iter-comp}), we deduce that $\mathsf{PrivatekMeans}'$ is $(\frac{\varepsilon}{3}, \frac{\delta}{2 e^{\varepsilon}})$-differential private. Finally, we conclude that the original $\mathsf{PrivatekMeans}$, that chooses the points with replacement, is $(\varepsilon,\delta)$-differential private by applying \cref{fact:DP-with-replacement} with $m = n/2, \frac{\varepsilon}{3}, \frac{\delta}{2 e^{\varepsilon}}$.
\end{proof}
\subsection{Utility of $\mathsf{PrivatekMeans}$}\label{sec:util-kMeans}
\Enote{Continue from here}
In the following, we formalize our stability assumption by fixing some arbitrary set of $k$ centers $C= \set{\pc_1,\ldots,\pc_k}$ and a parameter $\gamma \in [0,1]$ and require that all resulting centers $\tilde{C}_t$ in Step~\ref{step:non-priv-centers} are "$\gamma$-close" to $C$:
\begin{definition}[Event $E_{C}^{\gamma}$ (over a random execution of $\mathsf{PrivatekMeans}$)]\label{def:event-ECgamma}
Let $\gamma \in [0,1]$, $C = \set{\pc_1,\ldots,\pc_k} \subseteq {\mathbb R}^d$ and let $D_i = \min_{j \neq i}\norm{\pc_i - \pc_j}$. We define $E_{C}^{\gamma}$ to be the event that for every $t \in [T]$ and $i \in [k]$, there exists $\tpc_i^t \in \tilde{C}_t$ such that $\norm{\tpc_i^t - \pc_i} \leq \gamma\cdot D_i$.
\end{definition}
Assuming that for each $i \in [k]$ the centers $\set{\pc_i^t}_{t=1}^T$ are close to each other, then for each $i$ any choice of a point $\pc_i$ that is close to $\set{\pc_i^t}_{t=1}^T$ will make the event $E_{C}^{\gamma}$ to occur (for suitable $\gamma$). Therefore, we can think of $C = \set{\pc_1,\ldots,\pc_k}$ as the choice of such centers that minimize ${\rm COST}_{{\cal P}}(C)$. If ${\cal A}$ is an $\omega$-approximation algorithm for $k$-means, it holds that ${\rm COST}_{{\cal P}}(C) \leq \min\set{{\rm COST}_{{\cal P}}(\tilde{C}_t)}_{t=1}^T \approx \min\set{ \omega \cdot {\rm OPT}_k({\cal P})}_{t=1}^T$ (the ``$\approx$'' holds by \cref{prop:cost-of-sample-is-good} for large enough $m$). In \cref{sec:application} we even see that when ${\cal P}$ is separated for $k$-means according to \cite{OstrovskyRSS12}, then with high probability the event $E_{C^*}^{\gamma}$ occur, where $C^*$ is an optimal $k$-means for ${\cal P}$.
The following lemma summarizes the utility guarantee of $\mathsf{PrivatekMeans}$. Roughly speaking, the lemma states that when event $E_{C}^{\gamma}$ occurs, then with high probability $\mathsf{PrivatekMeans}$ outputs $k$ centers $\hat{C} = \set{\hat{\pc}_1,\ldots,\hat{\pc}_k}$ such that ${\rm COST}_{{\cal P}}(\hat{C})$ is bounded by $(1 + O(\gamma)) {\rm COST}_{{\cal P}}(C)$ (plus some small additive error).
\begin{lemma}[Utility]\label{lem:kMeans-utility}
Let $\varepsilon,\delta,\beta \in (0,1)$, let ${\cal A}$ be a (non-private) algorithm, let ${\cal P} \in (B(\pt{0},\Lambda))^n$, let $C = \set{\pc_1,\ldots,\pc_k} \subseteq {\mathbb R}^d$ with $\min_{i \neq j} \norm{\pc_i - \pc_j} \geq 1/n$ and let $\gamma \in [0,\frac1{28})$. Consider a random execution of $\mathsf{PrivatekMeans}$ on inputs ${\cal P},k,\varepsilon,\delta,\beta,{\cal A}$ with
\begin{align*}
T = \Omega\paren{\frac{d k^2 \ell \sqrt{\log\paren{\frac{k \ell}{\min\set{\beta,\delta}}}}}{\gamma} \cdot \log\paren{\frac{d k \ell n \Lambda}{\min\set{\beta,\delta}}}},
\end{align*}
letting $\ell = \tilde{O}\paren{(dk \log(T/\delta) + \log^2(1/\delta))/\varepsilon}$ be the value from \cref{claim:main} with respect to $\tilde{n} = k T, d, k, \tilde{\varepsilon} = \frac{\varepsilon}k, \tilde{\delta} = \frac{\min\set{\delta,\beta}}{k}$.
Assume that event $E_{C}^{\gamma}$ occurs, and let $\hat{C} = \set{\hpc_1,\ldots,\hpc_k}$ be output of the execution. Then with probability $1-\beta/2$ it holds that
\begin{align*}
COST_{{\cal P}}(\set{\hpc_1,\ldots,\hpc_k}) \leq (1 + 64\gamma) {\rm COST}_{{\cal P}}(C) + O\paren{\frac{\Lambda^2 k^4 d}{\varepsilon^2} \log(k/\delta) \log(k/\beta)}
\end{align*}
\end{lemma}
\begin{proof}
In the following, for $t \in [T]$ let $\hat{C}_t = \set{\hpc_1^t,\ldots,\hpc_k^t}$ where we denote by $\hpc_i^t$ the center that is close to $\pc_i$, i.e., $\norm{\hpc_i^t - \pc_i} \leq \gamma \cdot D_i$, where $D_i = \min_{j \neq i}\norm{\pc_i- \pc_j}$ (holds by event $E_C^{\gamma}$). In addition, for $i \in [k]$, let $\pa_i = {\rm Avg}(\set{\hpc_{i}^t}_{t=1}^T)$ and note that
\begin{align}\label{eq:dist-pai-pci}
\forall i \in [k]:\text{}\norm{\pa_i - \pc_i} \leq \gamma \cdot D_i
\end{align}
Now, let ${\cal C}$ be the multiset from Step~\ref{step:new-multisets} of the algorithm, and note that ${\cal C}$ is fully-partitioned by the set of balls $\set{B(\pc_i, r_i = 2\gamma D_i)}_{i=1}^k$ which are also \textbf{very} far balls (according to \cref{def:sep-balls}) since $\gamma < \frac1{28}$, and the clusters of ${\cal C}$ are $\set{\set{\hpc_{1}^t}_{t=1}^T, \ldots, \set{\hpc_{k}^t}_{t=1}^T}$. Therefore, when applying algorithm $\mathsf{PrivatekAverages}$ in Step~\ref{step:the-aver-estim}, we obtain by \cref{claim:utility-kAverg} a set of $k$ points $\set{\hpa_1,\ldots,\hpa_k}$ such that
\begin{align}\label{eq:dist-pai-hpai}
\forall i \in [k]:\text{ }
&\text{}\norm{\pa_i - \hpa_i}\nonumber\\
&\leq O\paren{\frac{r_i \cdot d k^2 \ell \sqrt{\log(k \ell/\tilde{\delta})}}{\tilde{\varepsilon} \tilde{n}} \paren{(1 + \tilde{\alpha}/r_i) \log(d k \ell/\tilde{\delta}) + \log \paren{\frac{\Lambda d k}{\tilde{\alpha} \tilde{\delta}}}}}\nonumber\\
&\leq O\paren{\frac{d k^2 \ell \sqrt{\log\paren{\frac{k \ell}{\min\set{\beta,\delta}}}}}{T} \cdot \log\paren{\frac{d k \ell n \Lambda}{\min\set{\beta,\delta}}}} \cdot D_i\\
&\leq \gamma\cdot D_i,\nonumber
\end{align}
where in the second inequality we used the fact that $r_i \leq D_i$ and that $D_i \geq 1/n = \tilde{\alpha}$, and in the second inequality we used the lower bound on $T$. Therefore, we deduce by \cref{eq:dist-pai-pci,eq:dist-pai-hpai} that
\begin{align}
\forall i \in [k]:\text{}\norm{\hpa_i - \pc_i} \leq 2\gamma \cdot D_i
\end{align}
Now, let $\hat{{\cal P}}_1,\ldots,\hat{{\cal P}}_k$ be the clusters from Step~\ref{step:compute-clusters-kmeans} of the algorithm. By \cref{prop:close-centers-have-similar-cost} we get that
\begin{align}\label{eq:avg-cost}
\sum_{i=1}^k \sum_{\px \in \hat{{\cal P}}_i} \norm{\px - {\rm Avg}(\hat{{\cal P}}_i)}^2 \leq (1 + 64\gamma) {\rm COST}_{{\cal P}}(C)
\end{align}
Since the algorithm compute a noisy estimation $\hpc_i$ of each ${\rm Avg}(\hat{{\cal P}}_i)$, we get by the properties of the Gaussian mechanism (see \cref{obs:Gaus-aver}) that with probability $1-\beta$ it holds that
\begin{align}\label{eq:dist-from-avg}
\forall i \in [k]:\text{}\norm{\hpc_i - {\rm Avg}(\hat{{\cal P}}_i)}
&\leq O\paren{\frac{\Lambda}{\hat{\varepsilon} \size{\hat{{\cal P}}_i}} \sqrt{d \log(1/\hat{\delta}) \log(1/\hat{\beta})}}\\
&= O\paren{\frac{\Lambda k^2}{\varepsilon \size{\hat{{\cal P}}_i}} \sqrt{d \log(k/\delta) \log(k/\beta)}}\nonumber
\end{align}
We conclude from \cref{eq:avg-cost,eq:dist-from-avg} that
\begin{align*}
\lefteqn{{\rm COST}_{{\cal P}}(\set{\hpc_1,\ldots,\hpc_k})}\\
&\leq \sum_{i=1}^k \sum_{\px \in \hat{{\cal P}}_i} \norm{\px - \hpc_i}^2\\
&= \sum_{i=1}^k \sum_{\px \in \hat{{\cal P}}_i} \paren{\norm{\px - {\rm Avg}(\hat{{\cal P}}_i)}^2 + \norm{\hpc_i - {\rm Avg}(\hat{{\cal P}}_i)}^2 + 2 \norm{\px - {\rm Avg}(\hat{{\cal P}}_i)}\cdot \norm{\hpc_i - {\rm Avg}(\hat{{\cal P}}_i)}}\\
&\leq (1 + 64\gamma) {\rm COST}_{{\cal P}}(C) + O\paren{\frac{\Lambda^2 k^4 d}{\varepsilon^2} \log(k/\delta) \log(k/\beta)} + 2 \Delta \cdot O\paren{\frac{\Lambda k^3}{\varepsilon} \sqrt{d \log(k/\delta) \log(k/\beta)}}\\
&= (1 + 64\gamma) {\rm COST}_{{\cal P}}(C) + O\paren{\frac{\Lambda^2 k^4 d}{\varepsilon^2} \log(k/\delta) \log(k/\beta)}
\end{align*}
\end{proof}
\remove{
}
\remove{
}
\remove{
\Enote{Delete the following event}
}
|
2024-02-18T23:40:24.512Z
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2021-12-30T02:23:06.000Z
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{"paloma_paragraphs":[]}
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{"arxiv_id":"2112.14445","language":"en","timestamp":1640830986000,"url":"https:\/\/arxiv.org\/abs\/2112.14445","yymm":"2112"}
|
proofpile-arXiv_000-10038
|
{"provenance":"002.jsonl.gz:10039"}
| null | null |
\section{Introduction}
\label{sect:intro}
In recent years, deep learning has enjoyed tremendous success in many
real-world application areas such as computer vision (e.g., image recognition,
image segmentation, medical image analysis), natural language processing
(e.g., speech recognition,
machine translation, information retrieval),
finance (e.g., fraud detection, risk management),
and many more.
In the basic setting of supervised learning, where the goal
is to find an approximation of a target function given a limited amount
of training data, a major appeal of deep learning methods is their
apparent ability to scale to very high-dimensional domains.
In these settings, more traditional approximation schemes frequently suffer from the
so-called \emph{curse of dimensionality}: The number of computational
steps necessary to achieve a given approximation accuracy grows
exponentially with the dimension of the domain of the
function that is to be approximated
(cf., e.g., Bellman \cite{Bellman1957}, Novak \& Wo\'zniakowski \cite{NovakWozniakowski2008,NovakWozniakowski2010},
and Novak \& Ritter \cite{novak1997curse}).
By contrast,
deep learning methods
appear to achieve good approximation accuracy in applications
even for functions
on very high dimensional domains in a manageable amount of time.
Empirical studies and simulations further corroborate
this impression that deep learning methods, in many settings,
are able to overcome the curse of dimensionality
in the sense that the number of computational steps
necessary to achieve a given approximation accuracy
grows at most polynomially with the dimension $d\in\N=\{1,2,3,\dots\}$ of the
domain of the target function and the reciprocal $\nicefrac1\eps$ of the approximation
accuracy $\eps\in(0,\infty)$.
This latter property is sometimes called \emph{polynomial tractability};
cf., e.g., Novak \& Wo\'zniakowski \cite{NovakWozniakowski2008,NovakWozniakowski2010}.
Note, however, that lower bounds have also been established, proving that
general classes of algorithms (including deep learning methods)
cannot overcome the curse of dimensionality for all
reasonable classes of target functions; see, e.g.,
Heinrich \& Sindambiwe \cite{heinrich1999monte},
Heinrich \cite{heinrich2006randomized},
Grohs \& Voigtlaender \cite{grohs2021proof},
Petersen \& Voigtlaender \cite{petersen2018optimal},
and Yarotsky \cite{yarotsky2017error}.
While there is not yet a comprehensive mathematical theory
explaining in a rigorous manner the encouraging empirical results
obtained so far, there
is by now a substantial body of literature -- of which we will provide a
brief overview below --
shedding light on the \emph{expressive power}
of artificial neural networks (ANNs).
For a deep learning approximation scheme to be able to overcome the curse of dimensionality,
the class of ANNs used for approximation must be
sufficiently expressive in the sense that the number of parameters
needed to describe the approximating ANNs
grows at most polynomially
in the dimension $d$ of the domain and the reciprocal $\nicefrac1\eps$
of the prescribed approximation accuracy $\eps$.
The present paper is a contribution to this line of research,
investigating the expressiveness of deep ANNs and
in particular the question for which classes of target functions,
deep ANNs possess sufficient expressive power to
achieve approximations without the curse of dimensionality,
in the sense described above.
\paragraph{Previous research}
Initial research on the expressive power of ANNs
focused on universal approximation results, showing that
even shallow ANNs, i.e., those with a single hidden layer,
can approximate very large classes of functions
to an arbitrary degree of accuracy as long as the
number of neurons is allowed to grow arbitrarily large. We refer
to \cite{gallant1988there,carroll1989construction
Cybenko1989,funahashi1989approximate
hornik1989multilayer} for universal approximation results
using sigmoidal activation functions (see also
\cite{nguyen1999approximation,blum1991approximation}),
we refer to \cite{chen1995approximation,mhaskar1996neural,park1991universal}
for universal approximation results using radial basis functions
as activation functions, and
we refer to
\cite{hornik1990universal,hornik1991approximation,hornik1993some,leshno1993multilayer}
for universal approximation results using more general classes
of activation functions (see also \cite{irie1988capabilities,kidger2020universal}).
We also refer to \cite{Hanin2017,kidger2020universal}
for universal approximation results for deep ANNs
of bounded width.
First results which demonstrated that
even shallow ANNs with sigmoidal activation functions
have the expressive power to break the curse of dimensionality
in the approximation of certain classes of functions
were obtained in \cite{barron1992neural,barron1993universal
barron1994approximation,jones1992simple} in the 1990s. This approach was subsequently
extended in several ways, see, e.g., \cite{donahue1997rates,KurKaiKre1997
makovoz1996random,makovoz1998uniform,kurkova2002comparison
BurgerNeubauer2001,KaiKurSang2012} for further results
in this mold using sigmoidal activation functions
and \cite{breiman1993hinging,klusowski2018approximation
kainen2009complexity,MhaskarMicchelli1994,MhaskarMicchelli1995
caragea2020neural}
for similar results using other activation functions. The surveys
\cite{Ellacott1994,pinkus1999approximation} provide an overview
of the state-of-the-art of this research in the 1990s.
We also refer, e.g., to
\cite{lavretsky2002geometric,kurkova2008geometric
elbraechter2021deep,perekrestenko2018universal,wang2018exponential}
for upper bounds on the number of parameters necessary to achieve
particular approximations with shallow ANNs that are polylogarithmic in the
reciprocal of the desired approximation accuracy.
Furthermore,
\cite{maiorov2000near,devore1997approximation,guhring2019error,shen2019nonlinear}
prove upper bounds on the number
of parameters necessary to achieve particular
approximations with shallow ANNs
that suffer from the curse of dimensionality.
In the opposite direction, even in the 1990s, several authors
studied the limitations of shallow ANNs, see, e.g.,
\cite{ChuiMhaskar1994,candes1998ridgelets,maiorov2000near,schmitt2000lower}.
More recently, a large number of results have demonstrated that
deep ANNs overcome certain limitations
of shallow ANNs and in particular,
in various circumstances have the capacity to achieve
approximations
using significantly fewer parameters than shallow ANNs with the
same accuracy would need. We refer, e.g., to
\cite{daniely2017depth,almira2021negative,poggio2017why
chui2019deep,eldan2016power,safran17a} for results
comparing the expressive power of single-hidden layer
ANNs to that of ANNs with two or more
hidden layers and we refer, e.g., to \cite{elbraechter2021deep
GrohsIbrgimovJentzen2021,ChenWu2019,MhaskarPoggio2016,safran17a
cohen2016expressive}
for more general results demonstrating the superiority
of deeper ANNs in certain circumstances.
Perhaps surprisingly, it has been shown in Maiorov \& Pinkus \cite{MaioPinkus1999}
that there
exists an analytic sigmoidal activation function such that any
Lebesgue square integrable function on the unit cube in $\R^d$ can be approximated
to an arbitrary degree of accuracy with respect to the
$L^2$-norm with a network
with two layers and at most $6d+3$ neurons in each layer
using this special activation function (for similar results,
see also, e.g., \cite{GuliIsm2018a,GULIYEV2018296}).
While this breaks the curse of dimensionality in a certain sense, the employed activation
function is pathological and not useful for practical purposes.
In recent years, much research has thus focused on finding
upper bounds on the number of parameters necessary to achieve a
particular approximation using deep ANNs
with various practically relevant activation functions.
We refer, e.g., to \cite{LuShenYangZhang2020,guhring2019error
shen2020deep,shen2019nonlinear,petersen2018optimal,voigtlaender2019approximation
mhaskar1993} for approximation results using deep ANNs
that suffer from the curse of dimensionality
(see also \cite{BeckJentzenKuckuck2019
BolcskeiGrohsKutyniokPetersen2019OptimalApproximation}).
For results showing that deep ANNs
have the expressive power to overcome the curse of dimensionality,
the approximation of solutions of various classes of PDEs
has been a particularly active area in recent years.
For results showing that deep ANNs
with rectified linear unit (ReLU) activations have the expressive power
to approximate solutions of certain PDEs without the
curse of dimensionality, we refer, e.g., to
\cite{beneventano2020highdimensional,BernerGrohsJentzen2018
ElbraechterSchwab2018,GononGrohsEtAl2019,gonon2020deep
GrohsHerrmann2020,GrohsWurstemberger2018,GrohsHornungJentzen2019
GrohsJentzenSalimova2019,JentzenSalimovaWelti2018
HornungJentzenSalimova2020,HutzenthalerJentzenKruseNguyen2019
ReisingerZhang2019,SchwabZech2019}. We also refer to
\cite{EHanJentzen2017,HanJentzenE2018,Jiequn2020AlgorithmsPDEs
BeckBeckerCheriditoJentzenNeufeld2019} and the
surveys \cite{Kuckuck2020overview,Jiequn2020AlgorithmsPDEs} for more practically
oriented results giving empirical indications
that deep ANNs can approximate
solutions to certain PDEs without the curse of dimensionality.
There are also a number of recent results demonstrating that deep ANNs have the
expressive power to approximate more general classes of functions,
not related to PDEs, without the curse of dimensionality;
we refer, e.g., to \cite{lee2017ability,bach2017breaking
cheridito2021efficient,ShahamCloningerCoifman2018,wang2018exponential
caragea2020neural} for such
results using deep ANNs with ReLU activations
and we refer to \cite{li2020better,lee2017ability} for results
using deep ANNs with other activation functions.
Lower bounds on the number of parameters necessary for a deep ANN
with ReLU activation functions to achieve a particular approximation
have been demonstrated, e.g., in \cite{almira2021negative,ChenWu2019
perekrestenko2018universal,yarotsky2017error,petersen2018optimal,GrohsIbrgimovJentzen2021}.
In particular, \cite{yarotsky2017error,petersen2018optimal} show that
there are natural classes of functions which deep ANNs with
ReLU activations cannot approximate without suffering from the curse of
dimensionality (see also \cite{grohs2021proof,heinrich1999monte,heinrich2006randomized,GrohsIbrgimovJentzen2021}).
Finally, we refer to the recent survey \cite{guehring2020expressivity}
for a much more complete overview of results on the expressivity
of deep ANNs than we can give here.
\paragraph{Our result}
It is the key purpose of this article to develop
a machinery to study the high-dimensional approximation capacities of ANNs and, in particular,
to show that deep ANNs have the expressive power to overcome
the curse of dimensionality in the approximation of a suitable large class of functions.
In that respect, our contribution fits in with some of the recent research
mentioned above, cf., e.g., \cite{lee2017ability,bach2017breaking,cheridito2021efficient,
ShahamCloningerCoifman2018,wang2018exponential}.
In particular, the character of our results is similar to those found in
Cheridito et al.\ \cite{cheridito2021efficient} in many respects.
However, our approach is quite different.
While the setup in Cheridito et al.\ \cite{cheridito2021efficient} is more general
and uses the concept of catalog networks, some of our arguments are more direct and
therefore, allow us to derive approximation results for certain target functions
that go beyond those shown in \cite{cheridito2021efficient}.
The present article also has certain ideas and methods in common with
Beneventano et al.~\cite{beneventano2020highdimensional}.
Before we present, in \cref{Theo:introduction} below,
a slightly simplified version of our main result,
let us briefly explain the statement and
introduce some of the notions
used therein.
The class of approximating functions used throughout this
article consists of the realizations of (fully connected feed-forward)
ANNs. These are alternating compositions of
affine linear functions and fixed, non-linear \emph{activation functions}.
In our case, the activation functions will always be chosen as the
multidimensional
rectified linear unit (ReLU) function
$A\colon\prb{\bigcup_{d\in\N}\R^d}\to\prb{\bigcup_{d\in\N}\R^d}$, which satisfies
for all $d\in\N$ and all
$x=(x_1,\dots,x_d)\in\R^d$ that
$A(x)=(\max\{x_1,0\},\dots,\max\{x_d,0\})$.
Since the activation functions are fixed,
an ANN is determined by the matrices and vectors used to specify
all of the affine linear functions appearing in its realization.
More precisely, in our formalization (cf., e.g.,
Petersen \& Voigtlaender \cite[Definition~2.1]{petersen2018optimal}
and Beck et al.\ \cite[Definition~2.1]{BeckJentzenKuckuck2019}),
the set of neural networks is given as
\begin{equation}
\mathbf{N}
=
\bigcup_{L\in\N}\bigcup_{l_0,l_1,\dots,l_L\in\N}
\prbbb{\bigtimes_{k=1}^L (\R^{l_k\times l_{k-1}}\times\R^{l_k})}
.
\end{equation}
For every $L\in\N$, $l_0,l_1,\dots,l_L\in\N$,
$\mathscr f\in\prb{\bigtimes_{k=1}^L(\R^{l_k\times l_{k-1}}\times\R^{l_k})}$,
we think of $L$ as the \emph{length}\footnote{What we call the length is
also sometimes
called the depth of the neural network in the scientific literature.} of the
neural network $\mathscr f$
and we think for every $k\in\{0,1,\dots,L\}$ of $l_k$ as the
\emph{dimension of} (or the \emph{number of neurons in}) \emph{the $k$-th layer}
of $\mathscr f$.
Furthermore, for every
$L\in\N$, $l_0,l_1,\dots,l_L\in\N$,
$\mathscr f=((W_1,B_1),(W_2,B_2),\dots,(W_L,B_L))\in\prb{\bigtimes_{k=1}^L(\R^{l_k\times l_{k-1}}\times\R^{l_k})}$,
we define the \emph{number of parameters}
of $\mathscr f$ as $\mathcal P(\mathscr f)=\sum_{k=1}^Ll_k(l_{k-1}+1)$
(this is the total number of entries in the \emph{weight
matrices} $W_1,W_2,\dots,W_L$ and
\emph{bias vectors} $B_1,B_2,\dots,B_L$),
we define for every
$k\in\{1,2,\dots,L\}$
the \emph{$k$-th layer affine transformation} as the function
$\mathfrak L_k^{\mathscr f}\colon \R^{l_{k-1}}\to\R^{l_k}$
which satisfies for all $x\in\R^{l_{k-1}}$ that $\mathfrak L_k^{\mathscr f}(x)=W_kx+B_k$,
and we define the \emph{realization} of
$\mathscr f$ as the composition
\begin{equation}
\realisation(\mathscr f)
=
\mathfrak L_{L}^{\mathscr f}\circ A\circ \mathfrak L_{L-1}^{\mathscr f}
\circ A\circ\dots\circ A\circ \mathfrak L_1^{\mathscr f}
.
\end{equation}
For an illustration of such
a neural network we refer to \cref{figure_1}.
\def\layersep{4cm}
\begin{figure}
\centering
\begin{adjustbox}{width=\textwidth}
\begin{tikzpicture}[shorten >=1pt,-latex,draw=black!100, node distance=\layersep,auto]
\tikzstyle{every pin edge}=[<-,shorten <=1pt]
\tikzstyle{neuron}=[circle,fill=black!25,draw=black!75,minimum size=20pt,inner sep=0pt,thick]
\tikzstyle{input neuron}=[neuron,draw={rgb:red,1;black,1}, fill={rgb:red,1;white,5}];
\tikzstyle{output neuron}=[neuron, draw={rgb:blue,1;black,1}, fill={rgb:blue,1;white,5}];
\tikzstyle{hidden neuron}=[neuron,draw={rgb:green,1;black,1}, fill={rgb:green,1;white,5}];
\tikzstyle{annot} = [text width=9em, text centered]
\tikzstyle{annot2} = [text width=4em, text centered]
\foreach \name / \y in {1,...,3}
\node[input neuron,label=$x_{\y}$] (I-\name) at (0,-3*\y+2.75) {};
\path[yshift=1.5cm]
node[hidden neuron,label=${A}(x_1)$] (H-1) at (\layersep,-1 cm) {}
node[hidden neuron] (Q-1) at (\layersep,-2.5 cm) {}
node[hidden neuron,label=${A}(x_2)$] (H-2) at (\layersep,-4 cm) {}
node[hidden neuron,label=${A}(-x_2)$] (Q-2) at (\layersep,-5.5 cm) {}
node[hidden neuron,label=${A}(x_3)$] (H-3) at (\layersep,-7 cm) {}
node[hidden neuron] (Q-3) at (\layersep,-8.5 cm) {};
\node[hidden neuron,label=${A}({A}(x_{1})+{A}(-x_{1}))$] (H2-1) at (2*\layersep,-3*1+2.75) {};
\node[hidden neuron] (H2-2) at (2*\layersep,-3*2+2.75) {};
\node[hidden neuron,label=below:${A}({A}(x_{3})+{A}(-x_{3}))$] (H2-3) at (2*\layersep,-3*3+2.75) {};
\node[output neuron] (H3-1) at (3*\layersep,-3.25 cm) {};
\node[annot,right of=H3-1, node distance=1.7cm, align=center] () {$\vass{x_1}+\vass{x_2}+\vass{x_3}$};
\foreach \y in {1,...,3}
\foreach \target in {H,Q}
\foreach \source in {1,...,3}
\path (I-\source) edge [draw=black!30] (\target-\y);
\foreach \y in {1,...,3}
\foreach \source in {H,Q}
\foreach \target in {1,...,3}
\path (\source-\y) edge [draw=black!30] (H2-\target);
\foreach \source in {1,...,3}
\path (I-\source) edge node{$1$} (H-\source);
\foreach \source in {1,...,3}
\path (I-\source) edge [swap] node{$-1$} (Q-\source);
\foreach \source in {1,...,3}
\path (H-\source) edge node{$1$} (H2-\source);
\foreach \source in {1,...,3}
\path (Q-\source) edge [swap] node{$1$} (H2-\source);
\path (H2-1) edge node{$1$} (H3-1);
\path (H2-2) edge node{$1$} (H3-1);
\path (H2-3) edge [swap] node{$1$} (H3-1);
\path[yshift=1.5cm]
node[hidden neuron,label=${A}(-x_1)$] (Q-1) at (\layersep,-2.5 cm) {}
node[hidden neuron,label=${A}(-x_3)$] (Q-3) at (\layersep,-8.5 cm) {};
\node[hidden neuron,label=${A}({A}(x_{2})+{A}(-x_{2}))$] (H2-2) at (2*\layersep,-3*2+2.75) {};
\node[annot,above of=H-1, node distance=1.5cm, align=center] (hl) {1st hidden layer\\(2nd layer)};
\node[annot,above of=H2-1, node distance=2.25cm, align=center] (hl2) {2nd hidden layer\\(3rd layer)};
\node[annot,above of=H3-1, node distance=5.25cm, align=center] (hl3) {Output layer\\(4th layer)};
\node[annot,left of=hl, align=center] {Input layer\\ (1st layer)};
\node[annot2,below of=Q-3, node distance=1cm, align=center] (sl) {${l}_1=6$};
\node[annot2,below of=H2-3, node distance=1.7cm, align=center] (sl2) {${l}_2=3$};
\node[annot2,left of=sl, align=center] {${l}_0=3$};
\node[annot2,right of=sl2, align=center] {${l}_3=1$};
\end{tikzpicture}
\end{adjustbox}
\caption{\label{figure_1}Graphical illustration of an example neural network
which has as realizaton the $\ell^1$-norm on $\R^3$. The neural network
has $2$ hidden layers and length $L=3$
with $3$ neurons in the input layer (corresponding to ${l}_0 = 3$),
$6$ neurons in the first hidden layer (corresponding to ${l}_1 = 6$),
$3$ neurons in the second hidden layer (corresponding to ${l}_2 = 3$),
and one neuron in the output layer (corresponding to ${l}_3 = 1$).
In this situation we have an ANN with $39$ weights and $10$ biases
adding up to $\param(\mathscr{f})=49$ parameters overall of which $15$
are nonzero (indicated by black arrows).
The realization function
$\realisation(\mathscr f)\in C(\R^3,\R)$
of the considered deep ANN maps each $3$-dimensional input vector
$x = ( x_1, x_2, x_3 ) \in \R^3$ to the 1-dimensional
output $(\realisation(\mathscr f))(x)=\vass{x_1}+\vass{x_2}+\vass{x_3}$.}
\end{figure}
Our goal is to show that ANNs have sufficient expressiveness to
approximate certain sequences of functions
without the curse of dimensionality.
To formalize this, let
$\norm{\cdot}\colon\pr{\bigcup_{d\in\N}\R^d}\to\R$ be the function which
satisfies
for all $d\in\N$, $x\in\R^d$ that $\norm{x}=\PR{\sum_{i=1}^d (x_i)^2}^{\nicefrac12}$.
Given a sequence $(\mathfrak d_d)_{d\in\N}\subseteq\N$ of natural numbers,
a sequence $f=(f_d)_{d\in\N}\in\prb{\bigtimes_{d\in\N} C(\R^d,\R^{\mathfrak d_d})}$
of functions, and a radius $r\in[0,\infty)$,
we consider ANNs to have sufficient expressiveness
to approximate the sequence $f$ on balls of
radius $r$ without the curse of dimensionality
if and only if
there exists a constant $c\in\R$ such that for every
dimension $d\in\N$
and every approximation accuracy $\eps\in(0,1]$ there exists a neural network
$\mathscr f\in\ANNs$ such that
\begin{equation}
\label{eq:approx_without_cod}
\paramANN(\mathscr f)\leq cd^c\eps^{-c},
\quad
\realisation(\mathscr f)\in C(\R^d,\R^{\mathfrak d_{d}}),
\quad\text{and}\quad
\sup\nolimits_{[-r,r]^d}\norm{f_d(x)-(\realisation(\mathscr f))(x)}\leq \eps
.
\end{equation}
The main result of this article,
\cref{Theo:example_multiple_composition_loclip}
in \cref{Section:7} below, demonstrates, roughly speaking, that
ANNs have sufficient expressiveness to approximate
certain sequences of functions which can be constructed by
means of a finite number of compositions using
locally Lipschitz continuous functions, maxima, and products
without the curse of dimensionality on arbitrarily large
balls with respect to the standard norm.
In order to make this statement more precise, we present in
\cref{Theo:introduction} below a slightly simplified version
of our main result.
\begin{samepage}
\begin{athm}{theorem}{Theo:introduction}
Let
$\norm{\cdot}\colon \prb{\bigcup_{d \in \N}\R^d} \to \R$
and $A \colon \prb{\bigcup_{d \in \N}\R^d} \to \prb{\bigcup_{d \in \N}\R^d}$
sa\-tis\-fy for all $d \in \N$, $x=(x_1
\ldots, x_d) \in \R^d$ that
\begin{equation}
\norm{x}= \PRb{\ssssum_{i=1}^d\vass{x_i}^2 }^{\nicefrac{1}{2}}
\qquad\text{and}\qquad
A(x) = (\max\{x_1,0\}
\allowbreak \ldots, \max\{x_d,0\}),
\end{equation}
let
$
\ANNs
=
\bigcup_{L \in \N}
\bigcup_{l_0
\ldots, l_L \in \N }
\prb{
\bigtimes_{k = 1}^L (\R^{l_k \times l_{k-1}} \times \R^{l_k})
}
$,
let
$\realisation \colon \ANNs \to \prb{\bigcup_{k,l\in\N}\,C(\R^k,\R^l)}$ and
$\paramANN \colon \ANNs \to \N$
satisfy for all
$ L\in\N$,
$l_0,l_1,
\ldots, l_L \in \N$,
$
\mathscr{f}
=
((W_1, B_1)
\allowbreak \ldots, (W_L,\allowbreak B_L))
\in \allowbreak
\prb{ \bigtimes_{k = 1}^L\allowbreak(\R^{l_k \times l_{k-1}} \times \R^{l_k})}
$,
$x_0 \in \R^{l_0},\, x_1 \in \R^{l_1},
\ldots,\, \allowbreak x_{L} \in \R^{l_{L}}$
with
$\forall \, k \in \{1
\ldots,L\} \colon\allowbreak x_k =A(W_k x_{k-1} + B_k)$
that
\begin{align}
\paramANN(\mathscr{f}) = \ssssum_{k = 1}^L l_k(l_{k-1} + 1),\;\;
\realisation(\mathscr{f}) \in C(\R^{l_0},\R^{l_L}),\;\;\text{and}\;\;
( \realisation(\mathscr{f}) ) (x_0) = W_L x_{L-1} + B_L,
\end{align}
let $r,n \in \N$, $a_1,a_2,
\ldots, a_n \in \N_0 \cup \{-1\}$, and let
$f_{k,d} \colon \R \to \R$, $k, d \in \N$, and
$F_{k} \colon \prb{\bigcup_{d \in \N} \R^d} \to \prb{\bigcup_{d \in \N} \R^d}$, $k \in \N_0 \cup \{-1\}$,
satisfy for all
$k, d \in \N$,
$x,y \in \R$,
$v = (v_1
\ldots, v_d) \in \R^d$
that
$\vass{f_{k,d}(x)} \leq 1 \leq a_1$,
$\vass{f_{k,d}(x) - f_{k,d}(y)} \leq r(1 + \vass{x} + \vass{y})^{r} \vass{x-y}$,
$F_{-1}(v) = \prb{v_1, \allowbreak \max\{v_1,v_2\}, \allowbreak \ldots, \allowbreak \max\{v_1,
\ldots, v_d\} }$,
$F_{0} (v) = \prb{v_1, \allowbreak v_1 v_2, \allowbreak \ldots, \allowbreak v_1 v_2\cdots v_d }$,
and
$F_{k}(v) = \prb{f_{k,1}(v_1),
\allowbreak \ldots, f_{k,d}(v_d)}$.
Then there exist
$(\mathscr{F}_{d,\varepsilon})_{(d, \varepsilon) \in \N \times (0,1]} \subseteq \ANNs$
and $c \in \R$
such that for all
$d \in \N$,
$\varepsilon \in (0,1]$
it holds that
$\paramANN(\mathscr{F}_{d,\varepsilon}) \leq cd^c\varepsilon^{-c}$,
$\realisation(\mathscr{F}_{d,\varepsilon}) \in C(\R^d, \R^d)$, and
\begin{equation}
\label{intro:1}
\textstyle
\sup_{x \in [-r,r]^d}\norm{ (F_{a_n} \circ \ldots
\circ F_{a_1})(x) - (\realisation(\mathscr{F}_{d,\varepsilon}))(x) }
\leq
\varepsilon
.
\end{equation}
\end{athm}
\end{samepage}
\cref{Theo:introduction} above is an immediate consequence of
\cref{Coro:example_multiple_composition_loclip_2bis}
in \cref{Subsection:7.2} below.
\cref{Coro:example_multiple_composition_loclip_2bis} follows from
\cref{Theo:example_multiple_composition_loclip}
in \cref{Subsection:7.2} below, the main result of this article.
As described above, \cref{Theo:introduction} concerns the approximation of
certain sequences of functions by (realizations of) ANNs.
For convenience of notation, these sequences of functions are
formalized as continuous functions from the union $\bigcup_{d\in\N}\R^d$
to itself, where $\bigcup_{d\in\N}\R^d$ is equipped with the topology of
the disjoint union.
In this setting, the functions that are to be approximated
can be written as compositions of three types of functions.
The first type consists of the \emph{maximum function}
from $\bigcup_{d\in\N}\R^d$ to itself
which for every $d\in\N$ maps $v=(v_1,\dots,v_d)\in\R^d$ to
$(v_1,\max\{v_1,v_2\},\dots,\max\{v_1,v_2,\dots,v_d\})$;
the second type consists of the \emph{product function}
from $\bigcup_{d\in\N}\R^d$ to itself
which for every $d\in\N$ maps $v=(v_1,\dots,v_d)\in\R^d$ to
$(v_1,v_1v_2,\dots,v_1v_2\cdots v_d)$;
and the third type consists of functions
that are componentwise applications of
certain bounded, locally Lipschitz continuous
functions from $\R$ to $\R$.
In order to exhibit the scope of this result,
let us illustrate \cref{Theo:introduction}
by means of several examples.
We note that \cref{Theo:introduction} implies that
deep ANNs have sufficient expressivity to
approximate all of the following sequences of functions\footnote{Note that
$\arcsin\in C([-1,1],\R)$, $\arctan\in C(\R,\R)$,
and
$\tanh\in C(\R,\R)$ are the unique continuous functions which
satisfy for all $x\in(-\tfrac\pi2,\tfrac\pi2)$, $y\in\R$ that
$\arcsin(\sin(x))=x$,
$\arctan(\tan(x))=x$, and
$\tanh(y)=\frac{e^y-e^{-y}}{e^y+e^{-y}}$.}
without the curse of dimensionality on arbitrarily
large balls in the sense of \eqref{eq:approx_without_cod} above:
\begin{gather}
\R^d \ni ( x_1, x_2, \dots, x_d ) \mapsto
\sin\prb{\textstyle{\prod\nolimits_{j=1}^{d}} \sin( x_j )} \in \R , \quad d \in \N ,
\\
\R^d \ni ( x_1, x_2, \dots, x_d ) \mapsto
\prb{\sssprod_{j=1}^{d} e^{-\nicefrac{\vass{x_j}^2}{2}}} \in \R , \quad d \in \N ,
\\
\R^d \ni ( x_1, x_2, \dots, x_d ) \mapsto
\tanh\prb{ \sssprod_{j=1}^{d} \arcsin\prb{ \tfrac{x_j}{1+\vass{x_j}^2} } } \in \R, \quad d \in \N,
\\
\R^d \ni ( x_1, x_2, \dots, x_d ) \mapsto
\prb{ \sssprod_{j=1}^{\lfloor d/2 \rfloor} \PRb{ \tfrac{x_{2j} \arctan(x_{2j-1})}{1+\vass{x_{2j}}^2} } } \in \R, \quad d \in \N,
\\
\R^d \ni ( x_1, x_2, \dots, x_d ) \mapsto
\max\pR{\cos\pr{\vass{x_1}^2},\cos\pr{\vass{x_2}^2},\ldots,\cos\pr{\vass{x_d}^2}} \in \R, \quad d \in \N,
\\
\R^d \ni ( x_1, x_2, \dots, x_d ) \mapsto
\sssprod_{j=1}^{d}\max\{\sin(x_1),\sin(\tfrac{x_2}2),\ldots,\sin(\tfrac{x_j}j)\} \in \R, \quad d \in \N,
\\
\R^d \ni ( x_1, x_2, \dots, x_d ) \mapsto
\prb{\sssprod_{j=1}^{d}(1+e^{(-1)^jx_j})}^{-1} \in \R, \quad d \in \N,
\\
\R^d\ni(x_1,x_2,\dots,x_d)\mapsto
\max\nolimits_{j\in\{1,2,\dots,d\}}\cos\prb{\sssprod_{i=1}^j \tanh(x_i)}
\in \R,\quad d\in\N,
\\
\R^d\ni(x_1,x_2,\dots,x_d)\mapsto
\max\nolimits_{j\in\{1,2,\dots,d\}}\PRb{\sssprod_{i=1}^j\prb{\max\nolimits_{k\in\{1,2,\dots,i\}} \tfrac{1-\abs{x_k}^4}{1+\abs{x_k}^4}}}
\in\R,\quad d\in\N,
\\
\R^d\ni(x_1,x_2,\dots,x_d)\mapsto
\sssprod_{j=1}^d\PRb{\max\nolimits_{i\in\{1,2,\dots,j\}}\prb{\sssprod_{k=1}^i \tfrac{(x_k)^3}{1+\abs{x_k}^3}}}
\in\R,\quad d\in\N,
\\
\R^d\ni(x_1,x_2,\dots,x_d)\mapsto
\max\nolimits_{j\in\{1,2,\dots,d\}}\PRb{\pr{-1}^j\prb{\sssprod_{i=1}^{j} \cos(\vass{x_i}^4)}}
\in \R,\quad d\in\N,
\\
\R^d\ni(x_1,x_2,\dots,x_d)\mapsto
\sssprod_{k=1}^{d}\PRb{\max\nolimits_{j\in\{1,2,\dots,k\}}\prb{\PRb{\sin\prb{\sssprod_{i=1}^{j} \tfrac{x_i}{1+\vass{x_i}}}}^3}}
\in \R,\quad d\in\N
.
\end{gather}
For all of these examples, the fact that deep ANNs have sufficient expressivity to
approximate the sequence in question
without the curse of dimensionality is a direct consequence of
\cref{Coro:example_multiple_composition_loclip_proj}
in \cref{Subsection:7.2} below,
which in
turn follows easily from \cref{Theo:introduction}.
The remainder of this article is organized as follows.
In \cref{Section:2} we introduce our formalization of ANNs and
operations on ANNs and prove or recall from the literature
the relevant fundamental results needed later.
In \cref{Section:3} we define the cost of an ANN approximation
and ANN approximation spaces and we show how the cost
behaves under certain operations on functions, including composition.
In \cref{Section:4} we construct ANN approximations for locally Lipschitz
continuous function from $\R$ to $\R$.
In \cref{Section:5} we show how to represent the multi-dimensional maximum functions
efficiently with ReLU networks.
In \cref{Section:6} we develop efficient ANN approximations for the
multi-dimensional product functions.
In \cref{Section:7} we combine the results from
\cref{Section:3,Section:4,Section:5,Section:6}
to obtain the central results of this article,
including \cref{Theo:introduction} above.
\section{Artificial neural network (ANN) calculus}
\label{Section:2}
In this section we present the formalism and theory of ANNs that we use
throughout this article.
Though most of this material has appeared, sometimes in slightly
different form, in previous works (cf., e.g,
\cite{petersen2018optimal,GrohsHornungJentzen2019,GrohsJentzenSalimova2019
BeckJentzenKuckuck2019,ElbraechterSchwab2018,cheridito2021efficient})
and the results are essentially elementary,
we recall them here, since these definitions and results are
used extensively throughout the
rest of this article.
In particular,
\cref{def:ANN} below is a slightly shortened version of,
e.g., Grohs et al.\ \cite[Definition~2.1]{GrohsHornungJentzen2019},
\cref{def:multidim_version} is, e.g.,
Grohs et al.\ \cite[Definition~2.2]{GrohsHornungJentzen2019},
\cref{def:ANNrealization} is, e.g.,
Grohs et al.\ \cite[Definition~2.3]{GrohsHornungJentzen2019},
\cref{def:ReLU} is, e.g.,
Beck et al.\ \cite[Definition~2.4]{BeckJentzenKuckuck2019},
\cref{subsec:parallelization} corresponds to
Grohs et al.~\cite[Subsection~2.3.1]{GrohsHornungJentzen2019},
\cref{def:ANNcomposition} is, e.g.,
Petersen \& Voigtlaender \cite[Definition~2.2]{petersen2018optimal},
\cref{Lemma:CompositionAssociative} is, e.g.,
Grohs et al.\ \cite[Lemma~2.8]{GrohsHornungJentzen2019},
\cref{Lemma:PropertiesOfCompositions_n1} is an extension of,
e.g., Grohs et al.\ \cite[Lemma~2.6]{GrohsHornungJentzen2019},
\cref{Subsection:2.5} corresponds to
Grohs et al.\ \cite[Subsection~2.2.4]{GrohsHornungJentzen2019}, and
\cref{subsection:2.6} is based on
Grohs et al.\ \cite[Subsection~2.3.2]{GrohsHornungJentzen2019}.
\subsection{Definition of ANNs}
\begin{definition}[Set of ANNs]
\label{def:ANN}
We denote by $\ANNs$ the set given by
\begin{equation}
\begin{split}
\ANNs
&=
\adjustlimits
\bigcup_{L \in \N}
\bigcup_{ (l_0,l_1,\ldots, l_L) \in \N^{L+1} }
\prbbb{
\bigtimes_{k = 1}^L \pr{\R^{l_k \times l_{k-1}} \times \R^{l_k}}
},
\end{split}
\end{equation}
we denote by
$\paramANN \colon \ANNs \to \N$,
$\lengthANN \colon \ANNs \to \N$,
$\inDimANN \colon \ANNs \to \N$,
$\outDimANN \colon \ANNs \to \N$,
$\hidLengthAnn\colon \ANNs \to \N_0$,
$\dims\colon\ANNs\to \prb{\bigcup_{L=2}^\infty\, \N^{L}}$,
and $\singledims_n \colon \ANNs \to \N_0$, $n \in \N_0$,
the functions which satisfy
for all $L\in\N$, $l_0,l_1,\ldots,\allowbreak l_L \in \N$,
$
\mathscr{f}
\in \allowbreak
\prb{ \bigtimes_{k = 1}^L\allowbreak(\R^{l_k \times l_{k-1}} \times \R^{l_k})}$,
$n \in \N_0$
that
$\paramANN(\mathscr{f}) =
\sum_{k = 1}^L l_k(l_{k-1} + 1) $,
$\lengthANN(\mathscr{f})=L$,
$\inDimANN(\mathscr{f})=l_0$,
$\outDimANN(\mathscr{f})=l_L$,
$\hidLengthAnn(\mathscr{f})=L-1$,
$\dims(\mathscr{f})= (l_0,l_1,\ldots, l_L)$, and
\begin{equation}
\singledims_n(\mathscr{f})= \begin{cases}
l_n &\colon n\leq L\\
0 &\colon n>L.
\end{cases}
\end{equation}
\end{definition}
\cfclear
\begin{definition}[ANNs]
\label{def:neuralnetwork}
We say that $\Phi$ is an artificial neural network (we say that $\Phi$
is an ANN) if and only
if it holds that $\Phi\in\ANNs$.
\end{definition}
\subsection{Realizations of ANNs}
\begin{definition}[Multidimensional versions]\label{def:multidim_version}
Let $\psi \colon \R \to \R$ be a function.
Then we denote by
$\Mult{\psi} \colon \prb{\bigcup_{d \in \N}\R^d} \to \prb{\bigcup_{d \in \N}\R^d}$
the function which satisfies for all
$d \in \N$,
$ x = ( x_1, x_2, \dots, x_{d} ) \in \R^{d} $
that
\begin{equation}\label{multidim_version:Equation}
\Mult{\psi}(x)
=
\pr*{
\psi(x_1),
\psi(x_2)
,
\ldots
,
\psi(x_d)
}.
\end{equation}
\end{definition}
\cfclear
\begin{definition}[Realizations associated to ANNs]
\label{def:ANNrealization}
\cfconsiderloaded{def:ANNrealization}
Let $a\in C(\R,\R)$.
Then we denote by
$
\functionANN \colon \ANNs \to \prb{\bigcup_{k,l\in\N}\,C(\R^k,\R^l)}
$
the function which satisfies
for all $ L\in\N$, $l_0,l_1,\ldots, l_L \in \N$,
$
\mathscr{f}
=
((W_1, B_1),(W_2, B_2),\allowbreak \ldots, (W_L,\allowbreak B_L))
\in \allowbreak
\prb{\bigtimes_{k = 1}^L\allowbreak(\R^{l_k \times l_{k-1}} \times \R^{l_k})}
$,
$x_0 \in \R^{l_0}, x_1 \in \R^{l_1}, \ldots, x_{L} \in \R^{l_{L}}$
with $\forall \, k \in \{1,2,\ldots,L\} \colon x_k =\Mult{a}(W_k x_{k-1} + B_k)$
that
\begin{equation}
\label{ANNrealization:ass2}
\functionANN(\mathscr{f}) \in C(\R^{l_0},\R^{l_L})\qandq
( \functionANN(\mathscr{f}) ) (x_0) = W_L x_{L-1} + B_L
\end{equation}
\cfload.
\end{definition}
\cfclear
\begin{definition}[ReLU activation function]
\label{def:ReLU}
\cfconsiderloaded{def:ReLU}
We denote by $\ReLU \colon \R \to \R$ the function which satisfies for all $x \in \R$ that $\ReLU(x) = \max\{x,0\}$.
\end{definition}
\cfclear
\subsection{Parallelizations of ANNs with the same length}
\label{subsec:parallelization}
\begin{definition}[Parallelization of ANNs with the same length]
\label{def:simpleParallelization}
\cfconsiderloaded{def:simpleParallelization}
Let $n\in\N$. Then we denote by
\begin{equation}
\parallelizationSpecial_{n}\colon \pRb{(\mathscr{f}_1,\mathscr{f}_2,\dots, \mathscr{f}_n)\in\ANNs^n\colon \lengthANN(\mathscr{f}_1)= \lengthANN(\mathscr{f}_2)=\ldots =\lengthANN(\mathscr{f}_n) }\to \ANNs
\end{equation}
the function which satisfies for all $L\in\N$,
$(l_{1,0},l_{1,1},\dots, l_{1,L}), (l_{2,0},l_{2,1},\dots, l_{2,L}),\dots,\allowbreak (l_{n,0},\allowbreak l_{n,1},\allowbreak\dots, l_{n,L})\in\N^{L+1}$,
$\mathscr{f}_1=((W_{1,1}, B_{1,1}),(W_{1,2}, B_{1,2}),\allowbreak \ldots, (W_{1,L},\allowbreak B_{1,L}))\in \prb{ \bigtimes_{k = 1}^L\allowbreak(\R^{l_{1,k} \times l_{1,k-1}} \times \R^{l_{1,k}})}$,
$\mathscr{f}_2=((W_{2,1}, B_{2,1}),(W_{2,2}, B_{2,2}),\allowbreak \ldots, (W_{2,L},\allowbreak B_{2,L}))\in \prb{ \bigtimes_{k = 1}^L\allowbreak(\R^{l_{2,k} \times l_{2,k-1}} \times \R^{l_{2,k}})}$,
\dots,
$\mathscr{f}_n=((W_{n,1}, B_{n,1}),(W_{n,2}, B_{n,2}),\allowbreak \ldots, (W_{n,L},\allowbreak B_{n,L}))\in \prb{ \bigtimes_{k = 1}^L\allowbreak(\R^{l_{n,k} \times l_{n,k-1}} \times \R^{l_{n,k}})}$
that
\begin{equation}\label{parallelisationSameLengthDef}
\begin{alignedat}{2}
\parallelizationSpecial_{n}(\mathscr{f}_1,\mathscr{f}_2,\dots,\mathscr{f}_n)
&=
\Vast(
&&\pr*{\begin{pmatrix}
W_{1,1}& 0& 0& \cdots& 0\\
0& W_{2,1}& 0&\cdots& 0\\
0& 0& W_{3,1}&\cdots& 0\\
\vdots& \vdots&\vdots& \ddots& \vdots\\
0& 0& 0&\cdots& W_{n,1}
\end{pmatrix} ,\begin{pmatrix}B_{1,1}\\B_{2,1}\\B_{3,1}\\\vdots\\ B_{n,1}\end{pmatrix}},
\\&&&
\pr*{\begin{pmatrix}
W_{1,2}& 0& 0& \cdots& 0\\
0& W_{2,2}& 0&\cdots& 0\\
0& 0& W_{3,2}&\cdots& 0\\
\vdots& \vdots&\vdots& \ddots& \vdots\\
0& 0& 0&\cdots& W_{n,2}
\end{pmatrix} ,\begin{pmatrix}B_{1,2}\\B_{2,2}\\B_{3,2}\\\vdots\\ B_{n,2}\end{pmatrix}}
,\dots,
\\&&&
\pr*{\begin{pmatrix}
W_{1,L}& 0& 0& \cdots& 0\\
0& W_{2,L}& 0&\cdots& 0\\
0& 0& W_{3,L}&\cdots& 0\\
\vdots& \vdots&\vdots& \ddots& \vdots\\
0& 0& 0&\cdots& W_{n,L}
\end{pmatrix} ,\begin{pmatrix}B_{1,L}\\B_{2,L}\\B_{3,L}\\\vdots\\ B_{n,L}\end{pmatrix}}
\Vast)
\end{alignedat}
\end{equation}
\cfout[.]
\end{definition}
\cfclear
\begin{athm}{lemma}{Lemma:ParallelizationElementary}
\cfconsiderloaded{Lemma:ParallelizationElementary}
Let $n,L\in\N$, $\mathscr{f}_1,\mathscr{f}_2,\ldots,\mathscr{f}_n\in\ANNs$ satisfy for all $j\in\{1,2,\dots,n\}$ that $\lengthANN(\mathscr{f}_j)=L$
\cfload[.]Then
\begin{equation}\label{ParallelizationElementary:Display}
\parallelizationSpecial_{n}(\mathscr{f}_1,\mathscr{f}_2,\dots,\mathscr{f}_n)\in
\prbbb{\bigtimes_{k = 1}^L\allowbreak\prb{\R^{\PR{\sum_{j=1}^n\singledims_k(\mathscr{f}_j)} \times \PR{\sum_{j=1}^n\singledims_{k-1}(\mathscr{f}_j)}} \times \R^{\PR{\sum_{j=1}^n\singledims_k(\mathscr{f}_j)}}}}
\end{equation}
\cfout[.]
\end{athm}
\begin{proof}[Proof of \cref{Lemma:ParallelizationElementary}]
Note that \eqref{parallelisationSameLengthDef} ensures that for all $k\in\{0,1,\dots, L\}$ it holds that
$\singledims_k(\parallelizationSpecial_{n}(\mathscr{f}_1,\mathscr{f}_2,\dots,\mathscr{f}_n))=\sum_{j=1}^n\singledims_k(\mathscr f_j)$ \cfload. Hence, we obtain \eqref{ParallelizationElementary:Display}.
This completes the proof of \cref{Lemma:ParallelizationElementary}.
\end{proof}
\cfclear
\begin{athm}{prop}{Lemma:PropertiesOfParallelizationEqualLength}
Let $a\in C(\R,\R)$,
$n,L\in\N$,
$\mathscr{f}=(\mathscr{f}_1,\mathscr{f}_2,\allowbreak\dots,\allowbreak \mathscr{f}_n)\in\ANNs^n$
satisfy for all $j\in\{1,2,\dots,n\}$ that $\lengthANN(\mathscr{f}_j)=L$
\cfload.
Then
\begin{enumerate}[(i)]
\item\label{PropertiesOfParallelizationEqualLength:ItemOne} it holds that
\begin{equation}
\functionANN(\parallelizationSpecial_{n}(\mathscr{f}))\in C\prb{\R^{[\sum_{j=1}^n \inDimANN(\mathscr{f}_j)]},\R^{[\sum_{j=1}^n \outDimANN(\mathscr{f}_j)]}}
\end{equation}
and
\item\label{PropertiesOfParallelizationEqualLength:ItemTwo} it holds for all $x_1\in\R^{\inDimANN(\mathscr{f}_1)},x_2\in\R^{\inDimANN(\mathscr{f}_2)},\dots, x_n\in\R^{\inDimANN(\mathscr{f}_n)}$ that
\begin{equation}\label{PropertiesOfParallelizationEqualLengthFunction}
\begin{split}
&\prb{\functionANN\prb{\parallelizationSpecial_{n}(\mathscr{f}) }}(x_1,x_2,\dots, x_n)
\\&=\prb{(\functionANN(\mathscr{f}_1))(x_1), (\functionANN(\mathscr{f}_2))(x_2),\dots,
(\functionANN(\mathscr{f}_n))(x_n) }
\end{split}
\end{equation}
\end{enumerate}
\cfout.
\end{athm}
\begin{proof}[Proof of \cref{Lemma:PropertiesOfParallelizationEqualLength}]
Note that \eqref{ANNrealization:ass2} and \cref{Lemma:ParallelizationElementary} imply
\cref{PropertiesOfParallelizationEqualLength:ItemOne}.
Observe that \eqref{ANNrealization:ass2} and \eqref{parallelisationSameLengthDef} establish
\cref{PropertiesOfParallelizationEqualLength:ItemTwo}.
The proof of \cref{Lemma:PropertiesOfParallelizationEqualLength} is thus complete.
\end{proof}
\cfclear
\begin{athm}{prop}{Lemma:PropertiesOfParallelizationEqualLengthDims}
Let
$n,L\in\N$,
$\mathscr{f}=(\mathscr{f}_1,\mathscr{f}_2,\allowbreak\dots,\allowbreak \mathscr{f}_n)\in\ANNs^n$,
$(l_{1,0},l_{1,1},\dots, l_{1,L}),\allowbreak (l_{2,0},l_{2,1},\allowbreak\dots,\allowbreak l_{2,L}),\allowbreak \dots, (l_{n,0},l_{n,1},\dots, l_{n,L})
\in\N^{L+1}$ satisfy for all $j\in\{1,2,\dots,n\}$ that
$\dims(\mathscr{f}_j)=(l_{j,0},l_{j,1},\dots,\allowbreak l_{j,L})$
\cfload.
Then
\begin{enumerate}[(i)]
\item \label{PropertiesOfParallelizationEqualLengthDims:Dims} it holds that
$\dims\prb{\parallelizationSpecial_{n}(\mathscr{f})}=\smallsum_{j=1}^n \dims(\mathscr{f}_j)=\prb{\smallsum_{j=1}^n l_{j,0}, \smallsum_{j=1}^n l_{j,1},\dots, \smallsum_{j=1}^n l_{j,L}}$ and
\item \label{PropertiesOfParallelizationEqualLengthDims:Params}
it holds that
\begin{equation}
\paramANN(\parallelizationSpecial_{n}(\mathscr{f}))\le \tfrac{1}{2} \PRb{\smallsum\nolimits_{j=1}^n \paramANN(\mathscr{f}_j)}^2
\end{equation}
\end{enumerate}
\cfout.
\end{athm}
\begin{proof}[Proof of \cref{Lemma:PropertiesOfParallelizationEqualLengthDims}]
Note that the assumption that for all $ j\in\{1,2,\dots,\allowbreak n\}$ it holds that $\dims(\mathscr{f}_j) \allowbreak =(l_{j,0},l_{j,1},\dots, l_{j,L})$ and \cref{Lemma:ParallelizationElementary} establish \cref{PropertiesOfParallelizationEqualLengthDims:Dims}. Observe that \cref{PropertiesOfParallelizationEqualLengthDims:Dims}
demonstrates that
\begin{equation}
\begin{split}
&\paramANN(\parallelizationSpecial_{n}(\mathscr{f}))
=\sum_{k=1}^L \PRb{\smallsum\nolimits_{i=1}^n l_{i,k}} \PRb{\prb{\smallsum\nolimits_{i=1}^n l_{i,k-1}}+1}
\\&\le \sum_{i=1}^n \sum_{j=1}^n\sum_{k=1}^L l_{i,k} (l_{j,k-1}+1)
\le \sum_{i=1}^n \sum_{j=1}^n\sum_{k,\ell=1}^L l_{i,k} (l_{j,\ell-1}+1)
\\&= \sum_{i=1}^n \sum_{j=1}^n\PR*{\smallsum\nolimits_{k=1}^L l_{i,k}}
\PR*{\smallsum\nolimits_{\ell=1}^L (l_{j,\ell-1}+1)}
\\&\le \sum_{i=1}^n \sum_{j=1}^n\PR*{\smallsum\nolimits_{k=1}^L \tfrac{1}{2}l_{i,k} (l_{i,k-1}+1)}
\PR*{\smallsum\nolimits_{\ell=1}^L l_{j,\ell}(l_{j,\ell-1}+1)}
\\&= \sum_{i=1}^n \sum_{j=1}^n \tfrac{1}{2}\paramANN(\mathscr{f}_i) \paramANN(\mathscr{f}_j)
=\tfrac{1}{2}\PRb{\smallsum\nolimits_{i=1}^n \paramANN(\mathscr{f}_i)}^2
\cfload.
\end{split}
\end{equation}
This establishes \cref{PropertiesOfParallelizationEqualLengthDims:Params}.
The proof of \cref{Lemma:PropertiesOfParallelizationEqualLengthDims} is thus complete.
\end{proof}
\cfclear
\begin{athm}{cor}{Lemma:ParallelizationImprovedBoundsOne}
Let $n\in\N$, $\mathscr{f}=( \mathscr{f}_1,\mathscr{f}_2,\dots,\allowbreak \mathscr{f}_n)\in\ANNs^n$ satisfy
$\dims(\mathscr{f}_1)=\dims(\mathscr{f}_2)=\ldots=\dims(\mathscr{f}_n)$
\cfload.
Then $\paramANN(\parallelizationSpecial_{n}(\mathscr{f}))\le n^2 \paramANN(\mathscr{f}_1)$
\cfout.
\end{athm}
\begin{proof}[Proof of \cref{Lemma:ParallelizationImprovedBoundsOne}]
Throughout this proof
let
$L\in\N$,
$l_0,l_1,\dots, l_L\in\N$
satisfy for all
$j\in\{1,2,\dots,n\}$
that
$\dims(\mathscr{f}_j)=(l_0,l_1,\dots, l_{L})$.
\Nobs that
\cref{PropertiesOfParallelizationEqualLengthDims:Dims} in \cref{Lemma:PropertiesOfParallelizationEqualLengthDims}
demonstrates that
\begin{equation}
\begin{split}
\paramANN(\parallelizationSpecial_{n}(\mathscr{f}_1,\mathscr{f}_2,\dots,\mathscr{f}_n))
&=
\smallsum\limits_{j=1}^L (nl_j)\prb{(nl_{j-1})+1}
\le
\smallsum\limits_{j=1}^L (nl_j)\prb{(nl_{j-1})+n}
\\&=
n^2 \PRbbb{\smallsum\limits_{j=1}^L l_j(l_{j-1}+1)}
=
n^2 \paramANN(\mathscr{f}_1)
\end{split}
\end{equation}
\cfout.
This completes the proof of \cref{Lemma:ParallelizationImprovedBoundsOne}.
\end{proof}
\cfclear
\subsection{Compositions of ANNs}
\begin{definition}[Standard composition of ANNs]
\label{def:ANNcomposition}
\cfconsiderloaded{def:ANNcomposition}
We denote by $\compANN{(\cdot)}{(\cdot)}\colon\allowbreak \{(\mathscr{f}_1,\mathscr{f}_2)\allowbreak\in\ANNs\times \ANNs\colon \inDimANN(\mathscr{f}_1)=\outDimANN(\mathscr{f}_2)\}\allowbreak\to\ANNs$ the function which satisfies for all
$ L,\mathfrak{L}\in\N$, $l_0,l_1,\ldots, l_L, \allowbreak\mathfrak{l}_0,\mathfrak{l}_1,\allowbreak\ldots, \mathfrak{l}_\mathfrak{L} \in \N$,
$
\mathscr{f}_1
=
((W_1, B_1),(W_2, B_2),\allowbreak \ldots, (W_L,\allowbreak B_L))
\in \allowbreak
\prb{ \bigtimes_{k = 1}^L\allowbreak(\R^{l_k \times l_{k-1}} \allowbreak \times \R^{l_k})}
$,
$
\mathscr{f}_2
=
((\mathfrak{W}_1, \mathfrak{B}_1),\allowbreak(\mathfrak{W}_2, \mathfrak{B}_2),\allowbreak \ldots, (\mathfrak{W}_\mathfrak{L},\allowbreak \mathfrak{B}_\mathfrak{L}))
\in \allowbreak
\prb{ \bigtimes_{k = 1}^\mathfrak{L}\allowbreak(\R^{\mathfrak{l}_k \times \mathfrak{l}_{k-1}} \times \R^{\mathfrak{l}_k})}
$
with $l_0=\inDimANN(\mathscr{f}_1)=\outDimANN(\mathscr{f}_2)=\mathfrak{l}_{\mathfrak{L}}$
that
\begin{equation}\label{ANNoperations:Composition}
\begin{split}
&\compANN{\mathscr{f}_1}{\mathscr{f}_2}=\\&
\begin{cases}
\begin{array}{r}
\prb{(\mathfrak{W}_1, \mathfrak{B}_1),(\mathfrak{W}_2, \mathfrak{B}_2),\ldots, (\mathfrak{W}_{\mathfrak{L}-1},\allowbreak \mathfrak{B}_{\mathfrak{L}-1}),
(W_1 \mathfrak{W}_{\mathfrak{L}}, W_1 \mathfrak{B}_{\mathfrak{L}}+B_{1}),\\ (W_2, B_2), (W_3, B_3),\ldots,(W_{L},\allowbreak B_{L})}
\end{array}
&: L>1<\mathfrak{L} \\[3ex]
\prb{ (W_1 \mathfrak{W}_{1}, W_1 \mathfrak{B}_1+B_{1}), (W_2, B_2), (W_3, B_3),\ldots,(W_{L},\allowbreak B_{L}) }
&: L>1=\mathfrak{L}\\[1ex]
\prb{(\mathfrak{W}_1, \mathfrak{B}_1),(\mathfrak{W}_2, \mathfrak{B}_2),\allowbreak \ldots, (\mathfrak{W}_{\mathfrak{L}-1},\allowbreak \mathfrak{B}_{\mathfrak{L}-1}),(W_1 \mathfrak{W}_{\mathfrak{L}}, W_1 \mathfrak{B}_{\mathfrak{L}}+B_{1}) }
&: L=1<\mathfrak{L} \\[1ex]
(W_1 \mathfrak{W}_{1}, W_1 \mathfrak{B}_1+B_{1})
&: L=1=\mathfrak{L}
\end{cases}
\end{split}
\end{equation}
\cfload.
\end{definition}
\cfclear
\begin{definition}
[ReLU identity networks]
\label{def:ReLU_identity}
\cfconsiderloaded{def:ReLU_identity}
We denote by
$(\ReLUidANN{d})_{d \in \N} \subseteq \ANNs$
the \cfadd{def:neuralnetwork}ANNs which satisfy
for all
$d \in \{2,3,\dots\}$
that
\begin{equation}
\label{def:ReLU_identity_d_is_one}
\begin{split}
\ReLUidANN{1} = \pr*{ \!\pr*{\!\begin{pmatrix}
1\\
-1
\end{pmatrix},
\begin{pmatrix}
0\\
0
\end{pmatrix}\! },
\prbb{ \begin{pmatrix}
1& -1
\end{pmatrix},
0 } \!
} \in \prb{(\R^{2 \times 1} \times \R^{2}) \times (\R^{1 \times 2} \times \R^1) }
\end{split}
\end{equation}
and $\ReLUidANN{d} = \parallelizationSpecial_{d} (\ReLUidANN{1},\ReLUidANN{1},\dots, \ReLUidANN{1})$ \cfload.
\end{definition}
\cfclear
\begin{athm}{prop}{Prop:identity_representation}
Let $d \in \N$. Then
\begin{enumerate}[(i)]
\item \label{identity_representation:1} it holds that $\dims(\ReLUidANN{d})= (d, 2d, d)$,
\item \label{identity_representation:2} it holds for all $x \in \R^d$ that $(\realisation_\ReLU(\ReLUidANN{d}))(x) = x$, and
\item \label{identity_representation:3} it holds that $\paramANN(\ReLUidANN{d}) = 4d^2+3d$
\end{enumerate}
\cfout.
\end{athm}
\begin{proof}
[Proof of \cref{Prop:identity_representation}]
Note that \eqref{def:ReLU_identity_d_is_one} and \cref{Lemma:PropertiesOfParallelizationEqualLengthDims} prove \cref{identity_representation:1}.
Observe that \eqref{def:ReLU_identity_d_is_one} and \cref{Lemma:PropertiesOfParallelizationEqualLength} establish \cref{identity_representation:2}.
Note that \cref{identity_representation:1} implies \cref{identity_representation:3}. The proof of \cref{Prop:identity_representation} is thus complete.
\end{proof}
\cfclear
\cfclear
\begingroup
\newcommand{\f}{\mathscr{f}}
\newcommand{\g}{\mathscr{g}}
\begin{athm}{lemma}{lem:comp2}
Let $\f,\g\in\ANNs$ satisfy
$\inDimANN(\f)=\outDimANN(\g)$ \cfload.
Then
\begin{enumerate}[(i)]
\item \llabel{it:1}
it holds that $\inDimANN(\compANN{\f}{\g})=\inDimANN(\g)$ and
\item \llabel{it:2}
it holds that $\outDimANN(\compANN{\f}{\g})=\outDimANN(\f)$
\end{enumerate}
\cfout.
\end{athm}
\begin{aproof}
\Nobs that \cref{ANNoperations:Composition} establishes
\lref{it:1} and \lref{it:2}.
\finishproofthus
\end{aproof}
\endgroup
\cfclear
\begin{athm}{lemma}{Lemma:CompositionAssociative}
Let
$\mathscr{f}_1,\mathscr{f}_2,\mathscr{f}_3\in\ANNs$
satisfy
$\inDimANN(\mathscr{f}_1)=\outDimANN(\mathscr{f}_2)$ and
$\inDimANN(\mathscr{f}_2)=\outDimANN(\mathscr{f}_3)$
\cfload.
Then $
\compANN{(\compANN{\mathscr{f}_1}{\mathscr{f}_2})}{\mathscr{f}_3}=\compANN{\mathscr{f}_1}{(\compANN{\mathscr{f}_2}{\mathscr{f}_3})} $
\cfadd{lem:comp2}\cfout.
\end{athm}
\begin{aproof}
\Nobs that \cite[Lemma~2.8]{GrohsHornungJentzen2019} establishes
that $
\compANN{(\compANN{\mathscr{f}_1}{\mathscr{f}_2})}{\mathscr{f}_3}=\compANN{\mathscr{f}_1}{(\compANN{\mathscr{f}_2}{\mathscr{f}_3})} $
\finishproofthus
\end{aproof}
\cfclear
\begin{athm}{prop}{Lemma:PropertiesOfCompositions_n2}
Let $n \in \{2,3,\dots\}$,
$\mathscr{f}_1, \mathscr{f}_2, \ldots, \mathscr{f}_n \in \ANNs$
satisfy for all $k\in \{2,3, \ldots, n\}$ that
$\inDimANN(\mathscr{f}_{k-1})=\outDimANN(\mathscr{f}_{k})$
\cfload.
Then
\begin{enumerate}[(i)]
\item \label{PropertiesOfCompositions_n:Length} it holds that
\begin{equation}
[\lengthANN(\compANN{\mathscr{f}_1}{{\mathscr{f}_2}}\bullet \ldots \bullet \mathscr{f}_n)-1]=[\lengthANN(\mathscr{f}_1)-1]+[\lengthANN(\mathscr{f}_2)-1]+ \ldots + [\lengthANN(\mathscr{f}_n)-1],
\end{equation}
\cfadd{Lemma:CompositionAssociative}
\item \label{PropertiesOfCompositions_n:HiddenLenght} it holds that
$\hidLengthAnn(\mathscr{f}_1 \bullet \mathscr{f}_2\bullet \ldots \bullet \mathscr{f}_n)=\hidLengthAnn(\mathscr{f}_1)+\hidLengthAnn(\mathscr{f}_2)+ \ldots + \hidLengthAnn(\mathscr{f}_n)$,
\item \label{PropertiesOfCompositions_n:Input}
it holds that
$\inDimANN(\mathscr{f}_1 \bullet \mathscr{f}_2\bullet \ldots \bullet \mathscr{f}_n)=\inDimANN(\mathscr{f}_n)$,
\item \label{PropertiesOfCompositions_n:Output}
it holds that
$\outDimANN(\mathscr{f}_1 \bullet \mathscr{f}_2\bullet \ldots \bullet \mathscr{f}_n)=\outDimANN(\mathscr{f}_1),
$
and
\item \label{PropertiesOfCompositions_n:Realization}
it holds for all $\activation\in C(\R,\R)$ that
\begin{equation}\label{PropertiesOfCompositions:RealizationEquation_n}
\functionANN({\mathscr{f}_1}\bullet{\mathscr{f}_2}\bullet \ldots \bullet \mathscr{f}_n)=[\functionANN(\mathscr{f}_1)]\circ [\functionANN(\mathscr{f}_2)]\circ \ldots \circ [\functionANN(\mathscr{f}_n)]
\end{equation}
\end{enumerate}
\cfout.
\end{athm}
\begin{proof}
[Proof of \cref{Lemma:PropertiesOfCompositions_n2}]
Note that \cite[Proposition~2.6]{GrohsHornungJentzen2019}
and induction prove \cref{PropertiesOfCompositions_n:Length,PropertiesOfCompositions_n:HiddenLenght,PropertiesOfCompositions_n:Input,PropertiesOfCompositions_n:Output,PropertiesOfCompositions_n:Realization}.
This completes the proof of \cref{Lemma:PropertiesOfCompositions_n2}.
\end{proof}
\cfclear
\begingroup
\newcommand{\f}{\mathscr f}
\newcommand{\g}{\mathscr g}
\begin{athm}{lemma}{lem:dimcomp}
Let $\f,\g\in\ANNs$ satisfy $\inDimANN(\f)=\outDimANN(\g)$ \cfload. Then
\begin{enumerate}[(i)]
\item \label{it:dimcomp.1}
it holds that
\begin{equation}
\dims(\compANN{\f}{\g})
=
(\singledims_0(\g),\singledims_1(\g),\dots,\singledims_{\lengthANN(\g)-1}(\g),
\singledims_1(\f),\singledims_2(\f),\dots,\singledims_{\lengthANN(\f)}(\f))
\end{equation}
and
\item \label{it:dimcomp.2}
it holds that
\begin{equation}
\begin{split}
&\dims(\compANN{\f}{\compANN{\ReLUidANN{\outDimANN(\g)}}\g})
\\&=
(\singledims_0(\g),\singledims_1(\g),\dots,\singledims_{\lengthANN(\g)-1}(\g),
2\singledims_{\lengthANN(\g)}(\g),
\singledims_1(\f),\singledims_2(\f),\dots,\singledims_{\lengthANN(\f)}(\f))
\end{split}
\end{equation}
\end{enumerate}
\cfout.
\end{athm}
\begin{aproof}
Note that \cite[Proposition~2.6]{GrohsHornungJentzen2019} proves \cref{it:dimcomp.1,it:dimcomp.2}.
\finishproofthus
\end{aproof}
\endgroup
\cfclear
\begin{athm}{prop}{Lemma:PropertiesOfCompositions_n1}
Let $n \in \{2,3,\dots\}$,
$\mathscr{f}_1, \mathscr{f}_2, \ldots, \mathscr{f}_n \in \ANNs$ and let $l_{k,j}\in\N$, $j\in\{1,2,\ldots,\lengthANN({\mathscr{f}_k})\}$, $k\in\{1,2, \ldots, n\}$,
satisfy for all $k\in \{1,2, \ldots, n\}$ that
$\inDimANN(\mathscr{f}_{\min\{k,n-1\}})=\outDimANN(\mathscr{f}_{\min\{k+1,n\}})$ and $\dims(\mathscr{f}_k)=(l_{k,0},l_{k,1},\dots, l_{k,\lengthANN(\mathscr{f}_k)})$
\cfload.
Then
\begin{enumerate}[(i)]
\item \label{PropertiesOfCompositions_n:Params} it holds that
\begin{equation}
\begin{split}
\paramANN(\compANN{\mathscr{f}_1}{\mathscr{f}_2}\bullet \ldots \bullet \mathscr{f}_n)&
= \PR*{\sum_{k=1}^n\paramANN(\mathscr{f}_k) }
+ \PR*{\sum_{k = 1}^{n-1}l_{k,1}( l_{k+1,\lengthANN(\mathscr{f}_{k+1})-1}+1) }
\\&\quad
-\PR*{\sum_{k = 1}^{n-1}l_{k,1}(l_{k,0} + 1) }
-\PR*{\sum_{k = 2}^{n}l_{k,\lengthANN(\mathscr{f}_{k})}(l_{k,\lengthANN(\mathscr{f}_{k})-1} + 1) }
\\&\leq
\PR*{ \sum_{k=1}^n\paramANN(\mathscr{f}_k) }
+ \PR*{\sum_{k = 1}^{n-1}l_{k,1}( l_{k+1,\lengthANN(\mathscr{f}_{k+1})-1}+1)},
\end{split}
\end{equation}
and
\item \label{PropertiesOfCompositions_id_n:Params} it holds that
\begin{equation}
\begin{split}
&\paramANN({\mathscr{f}_1} \bullet \ReLUidANN{\outDimANN(\mathscr{f}_2)} \bullet {\mathscr{f}_2} \bullet \ReLUidANN{\outDimANN(\mathscr{f}_3)} \bullet
\ldots \bullet \ReLUidANN{\outDimANN(\mathscr{f}_n)} \bullet \mathscr{f}_n)
\\&= \PR*{\sum_{k=1}^n\paramANN(\mathscr{f}_k) }
-\PR*{\sum_{k = 1}^{n-1}l_{k,1}(l_{k,0} + 1) }
-\PR*{\sum_{k = 2}^{n}l_{k,\lengthANN(\mathscr{f}_{k})}(l_{k,\lengthANN(\mathscr{f}_{k})-1} + 1) }
\\ &\quad
+ \PR*{\sum_{k = 1}^{n-1}l_{k,1}(2l_{k,0} + 1) }
+2\PR*{\sum_{k = 2}^{n}l_{k,\lengthANN(\mathscr{f}_{k})}(l_{k,\lengthANN(\mathscr{f}_{k})-1} + 1) }
\\&\leq
\PR*{ \sum_{k=1}^n\paramANN(\mathscr{f}_k) }
+\PR*{\sum_{k = 1}^{n-1}l_{k,1}(l_{k,0} + 1) }
+\PR*{\sum_{k = 2}^{n}l_{k,\lengthANN(\mathscr{f}_{k})}(l_{k,\lengthANN(\mathscr{f}_{k})-1} + 1) }
\end{split}
\end{equation}
\end{enumerate}
\cfout.
\end{athm}
\begin{proof}
[Proof of \cref{Lemma:PropertiesOfCompositions_n1}]
Observe that \cref{lem:dimcomp} and induction establish \cref{PropertiesOfCompositions_n:Params,PropertiesOfCompositions_id_n:Params}.
The proof of \cref{Lemma:PropertiesOfCompositions_n1} is thus complete.
\end{proof}
\cfclear
\begin{athm}{prop}{Lemma:PropertiesOfCompositions_n3}
Let $n \in \{2,3,\dots\}$,
$\mathscr{f}_1, \mathscr{f}_2, \ldots, \mathscr{f}_n \in \ANNs$
satisfy for all $k\in \{2,3, \ldots, n\}$ that
$\inDimANN(\mathscr{f}_{k-1})=\outDimANN(\mathscr{f}_{k})$
\cfload.
Then
\begin{enumerate}[(i)]
\item \label{PropertiesOfCompositions_n:Params2}
it holds that
\begin{equation}
\paramANN(\compANN{\mathscr{f}_1}{\mathscr{f}_2}\bullet \ldots \bullet \mathscr{f}_n)
\leq 2 \PR*{\sum_{k=1}^{n-1}\paramANN(\mathscr{f}_k)\paramANN(\mathscr{f}_{k+1})},
\end{equation}
\item \label{PropertiesOfCompositions_id_n:Params2} it holds that
\begin{equation}
\paramANN({\mathscr{f}_1} \bullet \ReLUidANN{\outDimANN(\mathscr{f}_2)} \bullet {\mathscr{f}_2} \bullet \ReLUidANN{\outDimANN(\mathscr{f}_3)} \bullet
\ldots \bullet \ReLUidANN{\outDimANN(\mathscr{f}_n)} \bullet \mathscr{f}_n)
\leq 3\PR*{ \sum_{k=1}^{n}\paramANN(\mathscr{f}_k) } - \paramANN(\mathscr f_1)-\paramANN(\mathscr f_n),
\end{equation}
and
\item \label{PropertiesOfCompositions_n:comparison} it holds that
\begin{equation}
\sum_{k=1}^{n}\paramANN(\mathscr{f}_k) \leq \sum_{k=1}^{n-1}\paramANN(\mathscr{f}_k)\paramANN(\mathscr{f}_{k+1})
\end{equation}
\end{enumerate}
\cfout.
\end{athm}
\begin{proof}[Proof of \cref{Lemma:PropertiesOfCompositions_n3}]
Throughout this proof let $l_{k,j}\in\N$, $j\in\{1,2,\ldots,\lengthANN({\mathscr{f}_k})\}$, $k\in\{1,2, \ldots, n\}$,
satisfy for all $k\in\{1,2,\ldots,n\}$ that $\dims(\mathscr{f}_k)=(l_{k,0},l_{k,1},\dots, l_{k,\lengthANN(\mathscr{f}_k)})$ \cfload.
Observe that the fact that for all $a,b \in [2, \infty)$ it holds that
$
a+b \leq 2\max\{a,b\} \leq \min\{a,b\}\max\{a,b\}
= ab
$
and the fact that
$\min_{i \in \{1,2, \ldots, n\}}\paramANN(\mathscr{f}_i) \geq 2$ ensure that
\begin{equation}
\begin{split}
\sum_{k=1}^{n}\paramANN(\mathscr{f}_k)
&\leq
\paramANN(\mathscr{f}_1)+\paramANN(\mathscr{f}_n)+2\PR*{ \sum_{k=2}^{n-1}\paramANN(\mathscr{f}_k) }
\\&=
\PR*{\paramANN(\mathscr{f}_1)+\paramANN(\mathscr{f}_2)}
+ \PR*{\paramANN(\mathscr{f}_2)+\paramANN(\mathscr{f}_3)}
+ \ldots
+ \PR*{\paramANN(\mathscr{f}_{n-1})+\paramANN(\mathscr{f}_n)}
\\&\leq
\sum_{k=1}^{n-1}\paramANN(\mathscr{f}_k)\paramANN(\mathscr{f}_{k+1}) \ifnocf.
\end{split}
\end{equation}
\cfload[. ]This establishes \cref{PropertiesOfCompositions_n:comparison}.
Note that \cref{Lemma:PropertiesOfCompositions_n1} and \cref{PropertiesOfCompositions_n:comparison} imply that
\begin{equation}
\begin{split}
\paramANN(\compANN{\mathscr{f}_1}{\mathscr{f}_2} \bullet \ldots \bullet \mathscr{f}_n)
&\leq
\PR*{ \sum_{k=1}^n\paramANN(\mathscr{f}_k) }
+ \PR*{ \sum_{k = 1}^{n-1}l_{k,1}( l_{k+1,\lengthANN(\mathscr{f}_{k+1})-1}+1)}
\\ & \leq
\PR*{\sum_{k=1}^{n}\paramANN(\mathscr{f}_k)} + \PR*{
\sum_{k=1}^{n-1}\paramANN(\mathscr{f}_k)\paramANN(\mathscr{f}_{k+1}) }
\\ & \leq 2 \PR*{
\sum_{k=1}^{n-1}\paramANN(\mathscr{f}_k)\paramANN(\mathscr{f}_{k+1})}
\end{split}
\end{equation}
\cfload. This establishes \cref{PropertiesOfCompositions_n:Params2}.
Observe that \cref{Lemma:PropertiesOfCompositions_n1} ensures that
\begin{equation}
\begin{split}
&
\paramANN({\mathscr{f}_1} \bullet \ReLUidANN{\outDimANN(\mathscr{f}_2)} \bullet {\mathscr{f}_2} \bullet \ReLUidANN{\outDimANN(\mathscr{f}_3)} \bullet
\ldots \bullet \ReLUidANN{\outDimANN(\mathscr{f}_n)} \bullet \mathscr{f}_n)
\\&\leq
\PR*{ \sum_{k=1}^n\paramANN(\mathscr{f}_k) }
+\PR*{\sum_{k = 1}^{n-1}l_{k,1}(l_{k,0} + 1)}
+\PR*{\sum_{k = 2}^{n}l_{k,\lengthANN(\mathscr{f}_{k})}(l_{k,\lengthANN(\mathscr{f}_{k})-1} + 1) }
\\&\leq
\PR*{ \sum_{k=1}^n\paramANN(\mathscr{f}_k) }
+\PR*{\sum_{k = 1}^{n-1}\paramANN(\mathscr{f}_k)}
+\PR*{\sum_{k = 2}^{n}\paramANN(\mathscr{f}_k) }
\\&=
3\PR*{ \sum_{k=1}^{n}\paramANN(\mathscr{f}_k) }
-\paramANN(\mathscr f_1)-\paramANN(\mathscr f_n)
\ifnocf.
\end{split}
\end{equation}
\cfload[.
Hence,
we obtain
\cref{PropertiesOfCompositions_id_n:Params2}.
The proof of \cref{Lemma:PropertiesOfCompositions_n3} is thus complete.
\end{proof}
\subsection{Powers and extensions of ANNs}
\label{Subsection:2.5}
\begin{definition}
[Identity matrices]
\label{def:identityMatrix}
\cfconsiderloaded{def:identityMatrix}
Let $d\in\N$. Then we denote by $\idMatrix_{d}\in \R^{d\times d}$ the identity matrix in $\R^{d\times d}$.
\end{definition}
\begin{definition}[Affine linear transformation ANNs]
\label{def:ANN:affine}
Let $m,n\in\N$, $W\in\R^{m\times n}$, $B\in\R^m$. Then we denote by
$\affineANN_{W,B} \in (\R^{m\times n}\times \R^m) \subseteq \ANNs$ the \cfadd{def:neuralnetwork}ANN
given by $\affineANN_{W,B} = ((W,B))$ (cf.~\cref{def:ANN}).
\end{definition}
\cfclear
\begin{definition}
[Iterated compositions of ANNs]
\label{def:iteratedANNcomposition}
\cfconsiderloaded{def:iteratedANNcomposition}
We denote by $(\cdot)^{\bullet n}\colon \{\mathscr{f}\in \ANNs\colon \inDimANN(\mathscr{f})=\outDimANN(\mathscr{f})\}\allowbreak\to\ANNs$, $n\in\N_0$, the functions
which satisfy for all $n\in\N_0$, $\mathscr{f}\in\ANNs$ with $\inDimANN(\mathscr{f})=\outDimANN(\mathscr{f})$ that
\begin{equation}\label{iteratedANNcomposition:equation}
\begin{split}
\mathscr{f}^{\bullet n}=
\begin{cases} \affineANN_{\idMatrix_{\outDimANN(\mathscr{f})},0}
&: n=0 \\
\,\compANN{\mathscr{f}}{(\mathscr{f}^{\bullet (n-1)})} &: n\in\N
\end{cases}
\end{split}
\end{equation}
\cfload.
\end{definition}
\cfclear
\begin{athm}{lemma}{lem:dimensionsiteratedcompositions}
Let $\mathscr{f}\in\ANNs$ satisfy $\inDimANN(\mathscr{f})=\outDimANN(\mathscr{f})$ \cfload. Then it holds for all $n\in\N_0$ that
\begin{equation}
\label{lem:dimensionsiteratedcompositions:equation}
\cfadd{def:iteratedANNcomposition}
\inDimANN(\mathscr{f}^{\bullet n})=\outDimANN(\mathscr{f}^{\bullet n})=\inDimANN(\mathscr{f})
\end{equation}
\cfout.
\end{athm}
\begin{proof}[Proof of \cref{lem:dimensionsiteratedcompositions}]
First, note that \eqref{iteratedANNcomposition:equation} assures that
\begin{equation}
\cfadd{def:iteratedANNcomposition}
\label{lem:dimensionsiteratedcompositions:eq1}
\inDimANN(\mathscr{f}^{\bullet 0})=\outDimANN(\mathscr{f}^{\bullet 0})=\outDimANN(\mathscr{f})=\inDimANN(\mathscr{f})
.
\end{equation}
\cfload[.]Furthermore, observe that \cref{Lemma:PropertiesOfCompositions_n2} and \eqref{iteratedANNcomposition:equation} ensure that for all $n\in\N$ it holds that $\outDimANN(\mathscr{f}^{\bullet n})=\outDimANN(\compANN{\mathscr{f}}{(\mathscr{f}^{\bullet (n-1)})})=\outDimANN(\mathscr{f})$ and
\begin{equation}
\inDimANN(\mathscr{f}^{\bullet n})=\inDimANN(\compANN{\mathscr{f}}{(\mathscr{f}^{\bullet (n-1)})})=\inDimANN(\mathscr{f}^{\bullet (n-1)}).
\end{equation}
This, \eqref{lem:dimensionsiteratedcompositions:eq1}, and induction establish
\eqref{lem:dimensionsiteratedcompositions:equation}. The proof of \cref{lem:dimensionsiteratedcompositions} is thus complete.
\end{proof}
\cfclear
\begin{definition}[Extensions of ANNs]
\label{def:ANNenlargement}
\cfconsiderloaded{def:ANNenlargement}
Let $L\in\N$, $\mathscr{g}\in \ANNs$ satisfy $\inDimANN(\mathscr{g})=\outDimANN(\mathscr{g})$ \cfload.
Then
we denote by $\longerANN{L,\mathscr{g}}\colon \{\mathscr{f}\in\ANNs\colon (\lengthANN(\mathscr{f})\le L \andShort \outDimANN(\mathscr{f})=\inDimANN(\mathscr{g})) \}\to \ANNs$ the function which satisfies for all $\mathscr{f}\in\ANNs$ with $\lengthANN(\mathscr{f})\le L$ and $\outDimANN(\mathscr{f})=\inDimANN(\mathscr{g})$ that
\begin{equation}\label{ANNenlargement:Equation}
\longerANN{L,\mathscr{g}}(\mathscr{f})= \compANN{\prb{\mathscr{g}^{\bullet (L-\lengthANN(\mathscr{f}))}}}{\mathscr{f}}
\cfadd{def:iteratedANNcomposition}
\cfadd{lem:dimensionsiteratedcompositions}
\end{equation}
\cfout.
\end{definition}
\cfclear
\begin{athm}{lemma}{Lemma:PropertiesOfANNenlargementGeometry}
Let $d,{i}\in\N$, $\mathscr{f},\mathscr{g}\in \ANNs$ satisfy $\outDimANN(\mathscr{f})=d$ and $\dims(\mathscr{g})=(d,{i},d)$
\cfload.
Then
\begin{enumerate}[(i)]
\item \label{PropertiesOfANNenlargementGeometry:BulletPower} it holds for all $n\in\N_0$ that
$\lengthANN(\mathscr{g}^{\bullet n})=n+1$, $\dims(\mathscr{g}^{\bullet n})\in \N^{n+2}$, and
\begin{equation}\label{BulletPower:Dimensions}
\dims(\mathscr{g}^{\bullet n}) =
\begin{cases}
(d,d) &: n=0\\
(d,{i},{i},\dots,{i},d) &: n\in\N
\cfadd{def:iteratedANNcomposition}
\end{cases}
\end{equation}
and
\item \label{PropertiesOfANNenlargementGeometry:ItemLonger}
it holds
for all
$L\in\N\cap [\lengthANN(\mathscr{f}),\infty)$
that $\lengthANN\prb{\longerANN{L,\mathscr{g}}(\mathscr{f})}=L$ and
\begin{equation}
\label{PropertiesOfANNenlargementGeometry:ParamsLonger}
\begin{split}
&\paramANN(\longerANN{L,\mathscr{g}}(\mathscr{f}))
\\&\le
\begin{cases} \paramANN(\mathscr{f})
&: \lengthANN(\mathscr{f})=L \\
\PR*{\prb{\max\pRb{1,\tfrac{i}{d}}}\paramANN(\mathscr{f})+
\prb{(L-\lengthANN(\mathscr{f})-1) \,{i}+d}({i}+1)
} &: \lengthANN(\mathscr{f})<L
\end{cases}
\end{split}
\end{equation}
\end{enumerate}
\cfout.
\end{athm}
\begin{proof}[Proof of \cref{Lemma:PropertiesOfANNenlargementGeometry}]
Throughout this proof let $l_0, l_1,\dots, l_{\lengthANN(\mathscr{f})}\in\N$ satisfy
$\dims(\mathscr{f})=(l_0,l_1,\dots, l_{\lengthANN(\mathscr{f})})\in \N^{\lengthANN(\mathscr{f})+1}$
and let $a_{L,k}\in \N$, $k\in \{0,1,\ldots,L\}$, $L\in \N\cap [\lengthANN(\mathscr{f}),\infty)$, satisfy
for all $L\in \N\cap [\lengthANN(\mathscr{f}),\infty)$, $k\in \{0,1,\ldots,L\}$ that
\begin{equation}\label{PropertiesOfANNenlargementGeometry:extendedDimensions}
a_{L,k}=
\begin{cases}
l_k &: k<\lengthANN(\mathscr{f}) \\
{i} &: \lengthANN(\mathscr{f})\le k<L\\
d &: k= L\ifnocf.
\end{cases}
\end{equation}
Note that the fact that $\mathscr{g}^{\bullet 0}=(\idMatrix_{d},0)\in \R^{d\times d}\times \R^d$
establishes that
\begin{equation}
\label{BulletPower:DimensionsProofbasecase}
\cfadd{def:iteratedANNcomposition}
\lengthANN(\mathscr{g}^{\bullet 0})=1\qquad\text{and}\qquad
\dims(\mathscr{g}^{\bullet 0})=(d,d)\in\N^2
\end{equation}
\cfload.
Next observe that \eqref{ANNoperations:Composition}
and \eqref{iteratedANNcomposition:equation}
imply that $\mathscr{g}^{\bullet 1}=\mathscr{g}$. The assumption that $\dims(\mathscr{g})=(d,i,d)$ therefore ensures that
\begin{equation}
\label{BulletPower:DimensionsProofbasecase1}
\cfadd{def:iteratedANNcomposition}
\lengthANN(\mathscr{g}^{\bullet 1})=1+1\qquad\text{and}\qquad
\dims(\mathscr{g}^{\bullet 1})=(d,i,d)\in\N^{1+2}.
\end{equation}
Furthermore, note that \cref{Lemma:PropertiesOfCompositions_n2},
\cref{lem:dimcomp}, \eqref{iteratedANNcomposition:equation},
the fact that $\lengthANN(\mathscr{g})=2$,
and the assumption that $\dims(\mathscr{g}) = (d,{i},d)$ prove that for all $n\in\N$ with $\lengthANN(\mathscr{g}^{\bullet n})=n+1$ and $\dims(\mathscr{g}^{\bullet n})=(d,i,i,\ldots,i,d)\in\N^{n+2}$
it holds that
\begin{equation}\label{BulletPower:DimensionsInduction2}
\begin{split}
\lengthANN(\mathscr{g}^{\bullet (n+1)})&=\lengthANN(\compANN{\mathscr{g}}{(\mathscr{g}^{\bullet n})})=\lengthANN(\mathscr{g})+\lengthANN(\mathscr{g}^{\bullet n})-1=\lengthANN(\mathscr{g}^{\bullet n})+1=(n+1)+1
\end{split}
\end{equation}
and
\begin{equation}
\begin{split}
\dims(\mathscr{g}^{\bullet (n+1)})=\dims(\compANN{\mathscr{g}}{(\mathscr{g}^{\bullet n})})=(d,i,i,\ldots,i,d)\in\N^{(n+1)+2}\ifnocf.
\end{split}
\end{equation}
\cfload. This,
\eqref{BulletPower:DimensionsProofbasecase}, \eqref{BulletPower:DimensionsProofbasecase1},
and induction establish \cref{PropertiesOfANNenlargementGeometry:BulletPower}.
Observe that \cref{Lemma:PropertiesOfCompositions_n2}, \cref{lem:dimcomp}, \cref{PropertiesOfANNenlargementGeometry:BulletPower}, \eqref{ANNenlargement:Equation}, and \eqref{PropertiesOfANNenlargementGeometry:extendedDimensions} show that
for all $L\in \N\cap [\lengthANN(\mathscr{f}),\infty)$ it holds
that
\begin{equation}\label{PropertiesOfANNenlargementGeometryLengthLonger}
\begin{split}
\lengthANN\prb{\longerANN{L,\mathscr{g}}(\mathscr{f})}
&= \lengthANN\prb{\compANN{\prb{\mathscr{g}^{\bullet (L-\lengthANN(\mathscr{f}))}}}{\mathscr{f}}}
=\lengthANN\prb{ \mathscr{g}^{\bullet (L-\lengthANN(\mathscr{f}))} }+\lengthANN(\mathscr{f})-1
\\&=(L-\lengthANN(\mathscr{f})+1)+\lengthANN(\mathscr{f})-1
=L
\end{split}
\end{equation}
and
\begin{equation}\label{PropertiesOfANNenlargementGeometryDimsLonger}
\begin{split}
\dims\prb{\longerANN{L,\mathscr{g}}(\mathscr{f})}
&=\dims\prb{\compANN{\prb{\mathscr{g}^{\bullet (L-\lengthANN(\mathscr{f}))}}}{\mathscr{f}}}
=(a_{L,0},a_{L,1},\dots, a_{L,L})\ifnocf.
\end{split}
\end{equation}
\cfload[.]Combining this with \eqref{PropertiesOfANNenlargementGeometry:extendedDimensions} demonstrates that $\lengthANN\prb{\longerANN{\lengthANN(\mathscr{f}),\mathscr{g}}(\mathscr{f})}
=\lengthANN(\mathscr{f})$ and
\begin{equation}
\begin{split}
\dims\prb{\longerANN{\lengthANN(\mathscr{f}),\mathscr{g}}(\mathscr{f})}
&=(a_{\lengthANN(\mathscr{f}),0},a_{\lengthANN(\mathscr{f}),1},\dots, a_{\lengthANN(\mathscr{f}),\lengthANN(\mathscr{f})})=(l_0,l_1,\dots, l_{\lengthANN(\mathscr{f})})=\dims(\mathscr{f})\ifnocf.
\end{split}
\end{equation}
\cfload[.]Hence, we obtain that
\begin{equation}\label{PropertiesOfANNenlargementGeometry:NotLongerParams}
\paramANN\prb{\longerANN{\lengthANN(\mathscr{f}),\mathscr{g}}(\mathscr{f})}=\paramANN(\mathscr{f}).
\end{equation}
Next note that \eqref{PropertiesOfANNenlargementGeometry:extendedDimensions}, \eqref{PropertiesOfANNenlargementGeometryDimsLonger}, and the fact that $l_{\lengthANN(\mathscr{f})}=\outDimANN(\mathscr{f})=d$ imply that for all $L\in\N\cap (\lengthANN(\mathscr{f}),\infty)$ it holds that
\begin{equation}
\begin{split}
&\paramANN\prb{\longerANN{L,\mathscr{g}}(\mathscr{f})}
=\smallsum\limits_{k = 1}^{L} a_{L,k}(a_{L,k-1} + 1)
\\&=\PR*{\smallsum\limits_{k = 1}^{\lengthANN(\mathscr{f})-1} a_{L,k}(a_{L,k-1} + 1)}+ a_{L,\lengthANN(\mathscr{f})}(a_{L,\lengthANN(\mathscr{f})-1} + 1)
\\&\quad+\PR*{\smallsum\limits_{k = \lengthANN(\mathscr{f})+1}^{L-1} a_{L,k}(a_{L,k-1} + 1)}+a_{L,L}(a_{L,L-1} + 1)
\\&=\PR*{\smallsum\limits_{k = 1}^{\lengthANN(\mathscr{f})-1} l_{k}(l_{k-1} + 1)}+ {i}(l_{\lengthANN(\mathscr{f})-1} + 1)
\\&\quad+ \prb{L-1-(\lengthANN(\mathscr{f})+1)+1}i(i+1)
+ d(i+1)
\\&=\PR*{\smallsum\limits_{k = 1}^{\lengthANN(\mathscr{f})-1} l_{k}(l_{k-1} + 1)}+ \tfrac{i}{d} \PRb{l_{\lengthANN(\mathscr{f})}(l_{\lengthANN(\mathscr{f})-1} + 1)}
+ \prb{L-\lengthANN(\mathscr{f})-1}i(i+1)
+ d(i+1)
\\&\le \PR*{\max\pRb{1,\tfrac{i}{d}}}\PR*{\smallsum\limits_{k = 1}^{\lengthANN(\mathscr{f})} l_{k}(l_{k-1} + 1)}
+ \prb{L-\lengthANN(\mathscr{f})-1}i(i+1)
+ d(i+1)
\\&=\PR*{\max\pRb{1,\tfrac{i}{d}}}\paramANN(\mathscr{f})
+ \prb{L-\lengthANN(\mathscr{f})-1}i(i+1)
+ d(i+1).
\end{split}
\end{equation}
Combining this with \eqref{PropertiesOfANNenlargementGeometryLengthLonger} and
\eqref{PropertiesOfANNenlargementGeometry:NotLongerParams}
establishes \cref{PropertiesOfANNenlargementGeometry:ItemLonger}.
The proof of \cref{Lemma:PropertiesOfANNenlargementGeometry} is thus complete.
\end{proof}
\cfclear
\begin{athm}{lemma}{Lemma:PropertiesOfANNenlargementRealization}
Let $\activation\in C(\R,\R)$, $ \mathscr f,\mathfrak{I}\in \ANNs$ satisfy
for all $x\in\R^{\inDimANN(\mathfrak{I})}$ that
$\outDimANN(\mathscr f)=\inDimANN(\mathfrak{I})=\outDimANN(\mathfrak{I})$
and $(\functionANN( \mathfrak{I}))(x)=x$
\cfload.
Then
\begin{enumerate}[(i)]
\item \label{PropertiesOfANNenlargementRealization: itemOne} it holds for all $n\in\N_0$, $x\in\R^{\inDimANN(\mathfrak{I})}$ that
\begin{equation}\label{BulletPowerRealization:Dimensions}
\functionANN(\mathfrak{I}^{\bullet n})\in C(\R^{\inDimANN(\mathfrak{I})},\R^{\inDimANN(\mathfrak{I})}) \qandq (\functionANN(\mathfrak{I}^{\bullet n}))(x)=x
\cfadd{def:iteratedANNcomposition}
\end{equation}
and
\item \label{PropertiesOfANNenlargementRealization:ItemIdentityLonger}
it holds
for all
$L\in\N\cap [\lengthANN(\mathscr{f}),\infty)$,
$x\in\R^{\inDimANN(\mathscr{f})}$
that
\begin{equation}
\functionANN(\longerANN{L,\mathfrak{I}}(\mathscr{f}))\in C(\R^{\inDimANN(\mathscr{f})},\R^{\outDimANN(\mathscr{f})})
\qandq
\prb{\functionANN(\longerANN{L,\mathfrak{I}}(\mathscr{f}))}(x)=\prb{\functionANN(\mathscr{f})}(x)
\end{equation}
\end{enumerate}
\cfout.
\end{athm}
\begin{proof}[Proof of \cref{Lemma:PropertiesOfANNenlargementRealization}]
First, note that \eqref{iteratedANNcomposition:equation} and the assumption that $\inDimANN(\mathfrak{I})=\outDimANN(\mathfrak{I})$ demonstrate that
\begin{equation}
\label{Lemma:PropertiesOfANNenlargementRealization:basecase}
\functionANN(\mathfrak{I}^{\bullet 0})\in C(\R^{\inDimANN(\mathfrak{I})},\R^{\inDimANN(\mathfrak{I})})\qquad\text{and}\qquad
\forall\,x\in\R^{\inDimANN(\mathfrak{I})}\colon
(\functionANN(\mathfrak{I}^{\bullet 0}))(x)
=\idMatrix_{\outDimANN(\mathfrak I)}x+0
=x.
\end{equation}
Moreover, observe that
\cref{Lemma:PropertiesOfCompositions_n2},
the assumption that
for all
$x\in\R^{\inDimANN(\mathfrak{I})}$
it holds that
$(\functionANN(\mathfrak{I}))(x)=x$, and
\eqref{iteratedANNcomposition:equation}
ensure that for all
$n\in\N_0$
with
$\functionANN(\mathfrak{I}^{\bullet n})\in C(\R^{\inDimANN(\mathfrak{I})},\R^{\inDimANN(\mathfrak{I})})$ and
$\forall\,x\in\R^{\inDimANN(\mathfrak{I})}\colon(\functionANN(\mathfrak{I}^{\bullet n}))(x)=x$
it holds that
\begin{equation}
\functionANN\prb{\mathfrak{I}^{\bullet (n+1)}}=\functionANN(\compANN{\mathfrak{I}}{(\mathfrak{I}^{\bullet n})})
=\PR{\functionANN(\mathfrak{I})}\circ \PR{\functionANN(\mathfrak{I}^{\bullet n})}\in C(\R^{\inDimANN(\mathfrak{I})},\R^{\inDimANN(\mathfrak{I})})
\end{equation}
and
\begin{equation}
\begin{split}
\forall\, x\in\R^{\inDimANN(\mathfrak{I})}\colon\,
\prb{\functionANN\prb{\mathfrak{I}^{\bullet (n+1)}}}(x)
&=
\prb{[\functionANN(\mathfrak{I})]\circ [\functionANN(\mathfrak{I}^{\bullet n})]}(x)
\\&=
(\functionANN(\mathfrak{I}))\prb{\pr{\functionANN(\mathfrak{I}^{\bullet n})}(x)}
=
(\functionANN(\mathfrak{I}))(x)
=
x.
\end{split}
\end{equation}
This,
\eqref{Lemma:PropertiesOfANNenlargementRealization:basecase},
and induction
establish \cref{PropertiesOfANNenlargementRealization: itemOne}.
Note that \cref{Lemma:PropertiesOfCompositions_n2},
\eqref{ANNenlargement:Equation},
and the assumption that $\outDimANN(\mathscr f)=\inDimANN(\mathfrak I)$
imply that
for all $L\in\N\cap [\lengthANN(\mathscr{f}),\infty)$
it holds that
\begin{equation}
\label{conditionforitem2}
\begin{split}
\functionANN(\longerANN{L,\mathfrak{I}}(\mathscr{f}))= \functionANN\prb{ \compANN{\prb{\mathfrak{I}^{\bullet (L-\lengthANN(\mathscr{f}))}}}{\mathscr{f}}}&=\prb{\functionANN\prb{\mathfrak{I}^{\bullet (L-\lengthANN(\mathscr{f}))}}}\circ(\functionANN( \mathscr{f}))
\\&\in C(\R^{\inDimANN(\mathscr{f})},\R^{\inDimANN(\mathfrak{I})})=C(\R^{\inDimANN(\mathscr{f})},\R^{\outDimANN(\mathscr{f})}).
\end{split}
\end{equation}
\Cref{PropertiesOfANNenlargementRealization: itemOne} therefore shows that
for all $L\in\N\cap [\lengthANN(\mathscr{f}),\infty)$,
$x\in\R^{\inDimANN(\mathscr{f})}$ it holds that
\begin{equation}
\begin{split}
\prb{\functionANN(\longerANN{L,\mathfrak{I}}(\mathscr{f}))}(x)&=\prb{\functionANN\prb{\mathfrak{I}^{\bullet (L-\lengthANN(\mathscr{f}))}}}\prb{(\functionANN(\mathscr{f}))(x)}=(\functionANN(\mathscr{f}))(x).
\end{split}
\end{equation}
This and \eqref{conditionforitem2} establish \cref{PropertiesOfANNenlargementRealization:ItemIdentityLonger}.
The proof of \cref{Lemma:PropertiesOfANNenlargementRealization} is thus complete.
\end{proof}
\subsection{Parallelizations of ANNs with different length}
\label{subsection:2.6}
\cfclear
\begin{definition}[Parallelization of ANNs with different length]\label{def:generalParallelization}
\cfconsiderloaded{def:generalParallelization}
Let $n\in\N$, $\mathscr{g}=(\mathscr{g}_1,\mathscr{g}_2,\dots, \mathscr{g}_n)\in \ANNs^n$ satisfy for all $j\in\{1,2,\dots, n\}$ that $\hidLengthAnn(\mathscr{g}_j)=1$ and $\inDimANN(\mathscr{g}_j)=\outDimANN(\mathscr{g}_j)$ \cfload.
Then
we denote by
\begin{equation}
\parallelization_{n,\mathscr{g}}\colon \pRb{(\mathscr{f}_1,\mathscr{f}_2,\dots,\mathscr{f}_n)\in\ANNs^n\colon\smallsum_{j=1}^{n}\vass{\outDimANN(\mathscr{f}_j)-\inDimANN(\mathscr{g}_j)}=0}\to \ANNs
\end{equation}
the function which satisfies for all $\mathscr{f}=(\mathscr{f}_1,\mathscr{f}_2,\dots, \mathscr{f}_n)\in\ANNs^n$ with
$\sum_{j=1}^{n}\vass{\outDimANN(\mathscr{f}_j)- \inDimANN(\mathscr{g}_j)}=0$
that
\begin{multline}\label{generalParallelization:DefinitionFormula}
\parallelization_{n,\mathscr{g}}(\mathscr{f})=\parallelizationSpecial_{n}\prb{\longerANN{\max_{k\in\{1,2,\dots,n\}}\lengthANN(\mathscr{f}_k),\mathscr{g}_1}({\mathscr{f}_1}),\longerANN{\max_{k\in\{1,2,\dots,n\}}\lengthANN(\mathscr{f}_k),\mathscr{g}_2}({\mathscr{f}_2}),\\\dots,\longerANN{\max_{k\in\{1,2,\dots,n\}}\lengthANN(\mathscr{f}_k),\mathscr{g}_n}({\mathscr{f}_n})}
\end{multline}
\cfadd{Lemma:PropertiesOfANNenlargementGeometry}
\cfload.
\end{definition}
\cfclear
\begin{athm}{prop}{Lemma:PropertiesOfParallelizationRealization}
Let $n\in\N$, $\mathfrak{I}=(\idANNshort{1},\idANNshort{2},\dots, \idANNshort{n})$, $\mathscr{f}=(\mathscr{f}_1,\mathscr{f}_2,\dots,\allowbreak \mathscr{f}_n)\in\ANNs^n$
satisfy for all $j\in\{1,2,\dots, n\}$, $x\in\R^{\outDimANN(\mathscr{f}_j)}$ that $\hidLengthAnn(\idANNshort{j}) =1$, $ \inDimANN(\mathfrak{I}_j)=\outDimANN(\mathfrak{I}_j)=\outDimANN(\mathscr{f}_j)$, and
$(\functionANN(\idANNshort{j}))(x)=x$
\cfload.
Then
\begin{enumerate}[(i)]
\item \label{PropertiesOfParallelizationRealization:ItemOne} it holds that
\begin{equation}
\functionANN\prb{\parallelization_{n,\mathfrak{I}}(\mathscr{f})}\in C\prb{\R^{[\sum_{j=1}^n \inDimANN(\mathscr{f}_j)]},\R^{[\sum_{j=1}^n \outDimANN(\mathscr{f}_j)]}}
\end{equation}
and
\item \label{PropertiesOfParallelizationRealization:ItemTwo}
it holds for all $x_1\in\R^{\inDimANN(\mathscr{f}_1)},x_2\in\R^{\inDimANN(\mathscr{f}_2)},\dots, x_n\in\R^{\inDimANN(\mathscr{f}_n)}$ that
\begin{equation}\label{PropertiesOfParallelizationRealizationEqualLengthFunction}
\begin{split}
&\prb{ \functionANN\prb{\parallelization_{n,\mathfrak{I}}(\mathscr{f})} } (x_1,x_2,\dots, x_n)
\\&=\prb{(\functionANN(\mathscr{f}_1))(x_1), (\functionANN(\mathscr{f}_2))(x_2),\dots,
(\functionANN(\mathscr{f}_n))(x_n) }
\end{split}
\end{equation}
\end{enumerate}
\cfout.
\end{athm}
\begin{proof}[Proof of \cref{Lemma:PropertiesOfParallelizationRealization}]
Throughout this proof
let
$L\in\N$
satisfy
$L= \allowbreak\max_{j\in\{1,2,\allowbreak\dots,n\}} \allowbreak\lengthANN(\mathscr{f}_j)$.
Note that
\cref{PropertiesOfANNenlargementGeometry:ItemLonger} in \cref{Lemma:PropertiesOfANNenlargementGeometry},
the assumption that
for all
$j\in\{1,2,\dots, n\}$
it holds that
$\hidLengthAnn(\idANNshort{j}) =1$,
and \cref{PropertiesOfANNenlargementRealization:ItemIdentityLonger} in \cref{Lemma:PropertiesOfANNenlargementRealization}
demonstrate that
\begin{enumerate}[(I)]
\item
it holds for all
$j\in\{1,2,\dots, n\}$
that
$\lengthANN(\longerANN{L,\mathfrak{I}_j}(\mathscr{f}_j))=L$
and
\item
it holds for all
$x_1\in\R^{\inDimANN(\mathscr{f}_1)}$,
$x_2\in\R^{\inDimANN(\mathscr{f}_2)}$,
$\dots$,
$x_n\in\R^{\inDimANN(\mathscr{f}_n)}$,
$j\in\{1,2,\dots, n\}$
that
$\functionANN(\longerANN{L,\mathfrak{I}_j}(\mathscr{f}_j))\in C(\R^{\inDimANN(\mathscr{f}_j)},\R^{\outDimANN(\mathscr{f}_j)})$
and
\begin{equation}
\prb{\functionANN(\longerANN{L,\mathfrak{I}_j}(\mathscr{f}_j))}(x_j)
=
(\functionANN(\mathscr{f}_j))(x_j)
\end{equation}
\end{enumerate}
\cfload.
\cref{Lemma:PropertiesOfParallelizationEqualLength}
therefore
implies that
\begin{enumerate}[(A)]
\item
it holds that
\begin{equation}
\functionANN\prb{\parallelizationSpecial_n\prb{\longerANN{L,\mathfrak{I}_1}({\mathscr{f}_1}),\longerANN{L,\mathfrak{I}_2}({\mathscr{f}_2}),\dots,\longerANN{L,\mathfrak{I}_n}({\mathscr{f}_n})}}
\in C\prb{\R^{[\sum_{j=1}^n \inDimANN(\mathscr{f}_j)]},\R^{[\sum_{j=1}^n \outDimANN(\mathscr{f}_j)]}}
\end{equation}
and
\item
it holds for all
$x_1\in\R^{\inDimANN(\mathscr{f}_1)}$,
$x_2\in\R^{\inDimANN(\mathscr{f}_2)}$,
$\dots$,
$x_n\in\R^{\inDimANN(\mathscr{f}_n)}$
that
\begin{equation}
\begin{split}
&
\prb{\functionANN\prb{\parallelizationSpecial_n(\longerANN{L,\mathfrak{I}_1}({\mathscr{f}_1}),\longerANN{L,\mathfrak{I}_2}({\mathscr{f}_2}),\dots,\longerANN{L,\mathfrak{I}_n}({\mathscr{f}_n}))}}(x_1,x_2,\dots, x_n)
\\&=
\prbb{\prb{\functionANN(\longerANN{L,\mathfrak{I}_1}({\mathscr{f}_1}))}(x_1), \prb{\functionANN(\longerANN{L,\mathfrak{I}_2}({\mathscr{f}_2}))}(x_2),\dots,
\prb{\functionANN(\longerANN{L,\mathfrak{I}_n}({\mathscr{f}_n}))}(x_n) }
\\&=
\prb{(\functionANN(\mathscr{f}_1))(x_1),(\functionANN(\mathscr{f}_2))(x_2),\dots, (\functionANN(\mathscr{f}_n))(x_n)}
\end{split}
\end{equation}
\end{enumerate}
\cfload.
Combining this with \eqref{generalParallelization:DefinitionFormula}
and the fact that
$L=\allowbreak\max_{j\in\{1,2,\dots,n\}} \allowbreak\lengthANN(\mathscr{f}_j)$
establishes \cref{PropertiesOfParallelizationRealization:ItemOne,PropertiesOfParallelizationRealization:ItemTwo}.
The proof of \cref{Lemma:PropertiesOfParallelizationRealization} is thus complete.
\end{proof}
\cfclear
\begin{athm}{cor}{Lemma:PropertiesOfParallelization}
Let $a\in C(\R,\R)$, $n\in\N$, ${l}_1,{l}_2,\dots, {l}_n\in\N$, $\mathscr{g}=(\mathscr{g}_1,\mathscr{g}_2,\dots,\allowbreak \mathscr{g}_n)$, $\mathscr{f}=(\mathscr{f}_1,\mathscr{f}_2,\allowbreak\dots, \mathscr{f}_n)\in \ANNs^n$
satisfy for all $j\in\{1,2,\dots, n\}$ that $\dims(\mathscr{g}_j) = (\outDimANN(\mathscr{f}_j),{l}_j,\allowbreak\outDimANN(\mathscr{f}_j))$
\cfload.
Then
\begin{equation}
\begin{split}
\paramANN\prb{\parallelization_{n,\mathscr{g}}(\mathscr{f})}
&\le
\frac{1}{2} \biggl[\ssum_{j=1}^n\Bigl(
\PR[\big]{\max\bigl\{1,\tfrac{l_j}{\outDimANN(\mathscr{f}_j)}\bigr\}}\,\paramANN(\mathscr{f}_j)
\\&\qquad
+
\prbb{\PRbb{\max_{k\in\{1,2,\dots,n\}} \lengthANN(\mathscr{f}_k)} {l}_j +\outDimANN(\mathscr{f}_j)}({l}_j+1)\Bigl)\biggl]^2
\end{split}
\end{equation}
\cfout.
\end{athm}
\begin{proof}[Proof of \cref{Lemma:PropertiesOfParallelization}]
Throughout this proof
let
$L\in\N$
satisfy
$L=\max_{k\in\{1,2,\dots,n\}} \lengthANN(\mathscr{f}_k)$.
Observe that
\eqref{generalParallelization:DefinitionFormula},
\cref{PropertiesOfParallelizationEqualLengthDims:Params} in \cref{Lemma:PropertiesOfParallelizationEqualLengthDims}, and
\cref{PropertiesOfANNenlargementGeometry:ItemLonger} in \cref{Lemma:PropertiesOfANNenlargementGeometry}
assure that
\begin{equation}
\begin{split}
\paramANN\prb{\parallelization_{n,\mathscr{g}}(\mathscr{f})}
&=
\paramANN\prb{\parallelizationSpecial_n\prb{\longerANN{L,\mathscr{g}_1}({\mathscr{f}_1}),\longerANN{L,\mathscr{g}_2}({\mathscr{f}_2}),\dots,\longerANN{L,\mathscr{g}_n}({\mathscr{f}_n})}}
\\&\le
\tfrac{1}{2} \PR*{\smallsum\nolimits_{j=1}^n \paramANN(\longerANN{L,\mathscr{g}_j}({\mathscr{f}_j}))}^2
\\&\le
\tfrac{1}{2} \prbb{\PR*{\smallsum\nolimits_{j=1}^n
\PRb{\max\bigl\{1,\tfrac{{l}_j}{\outDimANN(\mathscr{f}_j)}\bigr\}}\,\paramANN(\mathscr{f}_j)
\,\indicator{(\lengthANN(\mathscr{f}_j),\infty)}(L)}
\\&\qquad+\PR*{\smallsum\nolimits_{j=1}^n\prb{
(L-\lengthANN(\mathscr{f}_j)-1) \,{l}_j
+\outDimANN(\mathscr{f}_j)}({l}_j+1)
\,\indicator{(\lengthANN(\mathscr{f}_j),\infty)}(L)}
\\&\qquad+\PR*{\smallsum\nolimits_{j=1}^n \paramANN(\mathscr{f}_j)\,\indicator{\{\lengthANN(\mathscr{f}_j)\}}(L)}}^{\!2}
\\&\le
\tfrac{1}{2} \prbb{\PR*{\smallsum\nolimits_{j=1}^n
\PR[\big]{\max\bigl\{1,\tfrac{{l}_j}{\outDimANN(\mathscr{f}_j)}\bigr\}}\,\paramANN(\mathscr{f}_j)
\,\indicator{(\lengthANN(\mathscr{f}_j),\infty)}(L)}
\\&\qquad+\PR*{\smallsum\nolimits_{j=1}^n\prb{
L {l}_j
+\outDimANN(\mathscr{f}_j)}({l}_j+1)
\,\indicator{(\lengthANN(\mathscr{f}_j),\infty)}(L)}
\\&\qquad+\PR*{\smallsum\nolimits_{j=1}^n \PR[\big]{\max\bigl\{1,\tfrac{{l}_j}{\outDimANN(\mathscr{f}_j)}\bigr\}}\paramANN(\mathscr{f}_j)\,\indicator{\{\lengthANN(\mathscr{f}_j)\}}(L)}}^{\!2}
\\&\le
\tfrac{1}{2} \PR*{\smallsum\nolimits_{j=1}^n
\PR[\big]{\max\bigl\{1,\tfrac{{l}_j}{\outDimANN(\mathscr{f}_j)}\bigr\}}\,\paramANN(\mathscr{f}_j)
+\prb{L{l}_j+\outDimANN(\mathscr{f}_j)}({l}_j+1)}^2
\end{split}
\end{equation}
\cfload.
This completes the proof of \cref{Lemma:PropertiesOfParallelization}.
\end{proof}
\subsection{Clipping functions as ANNs}
\label{subsection:2.7}
\begin{definition}[Clipping function]
\label{def:clipping_function}
Let
$ u \in \R $,
$ v \in [ u, \infty ) $.
Then
we denote by
$ \clip{u}{v} \colon \R \to \R $
the function which satisfies for all
$ x \in \R $
that
\begin{equation}
\clip{u}{v}( x )
=
\max\{ u, \min\{ x, v \} \}
.
\end{equation}
\end{definition}
\cfclear
\begin{definition}[Multidimensional clipping functions]
\label{def:clip}
\cfconsiderloaded{def:clip}
Let
$n \in \N$,
$u\in \R$,
$v\in [u,\infty)$.
Then we denote by
$ \Clip uvn \colon \R^{n} \to \R^{n} $
the function which satisfies for all
$ x \in \R^n $
that
\begin{equation}
\Clip uvn(x)
=
\Mult{\clip{u}{v}}(x)
\end{equation}
\cfload.
\end{definition}
\cfclear
\begin{athm}{lemma}{lem:clipping_function}
Let
$n\in\N$,
$ u \in \R $,
$ v \in [ u, \infty ) $.
Then there exists
$\mathscr{f}\in\ANNs$
such that
\begin{enumerate}[(i)]
\item\label{lem:clipping_function:item_0}
it holds that
$\realisation_\mathfrak{r}(\mathscr{f})\in C(\R^n,\R^n)$,
\item\label{lem:clipping_function:item_1}
it holds for all
$x\in\R^n$
that
$(\realisation_\ReLU(\mathscr{f}))(x)=\Clip uvn( x )$, and
\item\label{lem:clipping_function:item_2}
it holds that
$\dims(\mathscr{f})=(n,n,n,n)$
\item\label{lem:clipping_function:item_3}
it holds that
$\paramANN(\mathscr{f})=3n^2+3n$
\end{enumerate}
\cfout.
\end{athm}
\begin{proof}[Proof of \cref{lem:clipping_function}]
Throughout this proof let $\mathscr{g}_1,\mathscr{g}_2\in\ANNs$ satisfy
\begin{equation}
\begin{split}
\mathscr{g}_1=\compANN{\affineANN_{1,u}}{((\idMatrix_n,0),(\idMatrix_n,0))}\bullet\affineANN_{1,-u}\quad \text{and}\quad\mathscr{g}_2=\affineANN_{-1,v}\bullet((\idMatrix_n,0),(\idMatrix_n,0))\bullet\affineANN_{-1,v}
\end{split}
\end{equation}
\cfload. Observe that \cref{Lemma:PropertiesOfCompositions_n2} shows that for all $x\in\R$ it holds that
\begin{equation}
\begin{split}
\pr[\big]{\realisation_\ReLU(\mathscr{g}_1)}(x)=\max\{x-u,0\}+u=\max\{u,x\}
\end{split}
\end{equation}
and
\begin{equation}
\begin{split}
\pr[\big]{\realisation_\ReLU(\mathscr{g}_2)}(x)=-\mathord{\max}\{-x+v,0\}+v=\min\{v,x\}
\end{split}
\end{equation}
\cfload. This and \cref{Lemma:PropertiesOfCompositions_n2} imply that for all $x\in\R$ it holds that
\begin{equation}
\begin{split}
\pr[\big]{\realisation_\mathfrak{r}(\mathscr{g}_1\bullet\mathscr{g}_2)}(x)
&=
\pr[\big]{[\realisation_\mathfrak{r}(\mathscr{g}_1)]\circ[\realisation_\mathfrak{r}(\mathscr{g}_2)]}(x)=\max\{u,\min\{v,x\}\}=\clip{u}{v}( x )
\cfadd{def:clipping_function}
\end{split}
\end{equation}
\cfload. \cref{Lemma:PropertiesOfParallelizationEqualLength} hence ensures that for all $x=(x_1,x_2,\dots,x_n)\in\R^n$ it holds that
\begin{equation}
\begin{split}
\label{lem:clipping_function:eq1}
&\pr[\big]{\realisation_\mathfrak{r}(\parallelizationSpecial_{n}(\mathscr{g}_1\bullet\mathscr{g}_2,\mathscr{g}_1\bullet\mathscr{g}_2,\dots,\mathscr{g}_1\bullet\mathscr{g}_2))}(x)
\\&=\pr[\Big]{(\realisation_\mathfrak{r}(\mathscr{g}_1\bullet\mathscr{g}_2))(x_1),(\realisation_\mathfrak{r}(\mathscr{g}_1\bullet\mathscr{g}_2))(x_2),\dots,
(\realisation_\mathfrak{r}(\mathscr{g}_1\bullet\mathscr{g}_2))(x_n)}=\mathfrak{C}_{u,v,n}(x)
\cfadd{def:clip}
\end{split}
\end{equation}
\cfload.
Furthermore, note that \cref{Lemma:PropertiesOfCompositions_n2}
and the fact that for all $i\in\{1,2\}$ it holds that $\dims(\mathscr{g}_i)=(1,1,1)$
imply that $\dims(\mathscr{g}_1\bullet\mathscr{g}_2)=(1,1,1,1)$.
\cref{Lemma:PropertiesOfParallelizationEqualLengthDims} therefore shows
that $\dims(\parallelizationSpecial_{n}(\mathscr{g}_1\bullet\mathscr{g}_2,\mathscr{g}_1\bullet\mathscr{g}_2,\dots,\mathscr{g}_1\bullet\mathscr{g}_2))=(n,n,n,n)$.
Combining this with \eqref{lem:clipping_function:eq1} establishes
\cref{lem:clipping_function:item_0,lem:clipping_function:item_1,lem:clipping_function:item_2,lem:clipping_function:item_3}.
The proof of \cref{lem:clipping_function} is thus complete.
\end{proof}
\section{Approximability of functions}
\label{Section:3}
This section contains essential definitions and general results on
the approximation of high-dimensional functions by ANNs, which will
be crucial for the later results in this article and in particular
for proving \cref{Theo:introduction} in the introduction.
In \cref{Subsection:3.1} below we define the cost of an ANN approximation,
which is the number of parameters necessary to approximate
a function by a neural network up to a given accuracy and with the approximation
satisfying a specified Lip\-schitz condition.
This definition is inspired by Cheridito et al.~\cite[Definition~9]{cheridito2021efficient}.
Also in \cref{Subsection:3.1}, we collect basic results on this
notion of cost.
In \cref{Subsection:3.3} we introduce ANN approximation spaces,
which are, roughly speaking, sets of sequences of functions
which can be approximated by ANNs with the ReLU activation function
such that the number of parameters grows at most
polynomially in the dimension of the domain, in the diameter of the compact set
on which we approximate, and in the reciprocal of the approximation accuracy.
\Cref{Subsection:3.4,Subsection:3.5} deal with ANN approximations of
compositions of functions. In \cref{Subsection:3.4} we analyze how
the cost of ANN approximations behaves under composition,
while the goal of \cref{Subsection:3.5} is to prove that the approximation
spaces defined in \cref{Subsection:3.3} are closed under composition.
Finally, \cref{Subsection:3.6} concerns parallelizations of
functions, i.e., functions which are componentwise applications
of univariate functions. Again we analyze how the cost of ANN approximations
behaves under this operation and then prove a certain result
about closedness of the approximation spaces defined in \cref{Subsection:3.3}.
We refer the reader to Beneventano et al.~\cite[Subsection~3.3]{beneventano2020highdimensional}
for related notions of approximation spaces of ANN approximable
functions. We also note that Gribonval et al.~\cite{gribonval2019approximation}
introduces a concept of approximation spaces of functions in the context
of ANN approximations, which is, however, only distantly related
to the notion presented here.
\subsection{Costs for ANN approximations}
\label{Subsection:3.1}
\begin{definition}
[Standard norm]
\label{def:Euclidean_norm}
We denote by $\norm{\cdot} \colon \prb{\bigcup_{n \in \N} \R^n} \rightarrow [0,\infty)$ the function which satisfies for all $n \in \N$, $x = (x_1,x_2,\dots,x_n) \in \R^n$ that
$
\norm{x} = \PRb{ \textstyle{\sum}_{j=1}^{n} \vass{x_j}^2 }^{\nicefrac{1}{2}}
$.
\end{definition}
\begin{definition}[Costs for ANN approximations]
\label{def:cost_Lipschitz_approx_set}
Let $a \in C(\R,\R)$.
Then we denote by $\CostLipwo{a}{} \colon \prb{\bigcup_{m,n \in \N}\bigcup_{D \subseteq \R^m} C(D, \R^n)} \times [0, \infty]^2 \rightarrow [1,\infty]$ the function which satisfies for all $m,n \in \N$, $D \subseteq \R^m$, $f \in C(D,\R^n)$, $L,\varepsilon \in [0, \infty]$ that
\begin{equation}
\begin{split}
\label{Cost_Lipschitz_set}
&\CostLip{a}{A}{f}{L}{\varepsilon} = \\
&\min \pr*{ \pR*{ \mathfrak{p} \in \N \colon \PR*{\!\!
\begin{array}{c}
\exists\, \mathfrak{m} \in \N, \mathscr{f} \in \ANNs \colon
\PRb{(\realisation_a(\mathscr{f}) \in C(\R^\mathfrak{m},\R^n))\land{}\\
(\mathfrak{p}=\paramANN(\mathscr{f}))\land (D \subseteq \R^\mathfrak{m})\land (\fa{x,y} D \text{ with } x \ne y \colon\\
\mednorm{ \functionANNbis{\mathscr{f}}(x)-\functionANNbis{\mathscr{f}}(y)} \leq L\norm{x-y})\land{}\\
(\fa{x} D \colon
\mednorm{f(x) - \functionANNbis{\mathscr{f}}(x)} \leq \varepsilon)}\\
\end{array} \!\! }
} \cup \{\infty\} }
\end{split}
\end{equation}
(cf.\ \cref{def:ANN,def:ANNrealization,def:Euclidean_norm}).
\end{definition}
\cfclear
\begin{athm}{lemma}{lem_cost_of_Lip_approx_set_equivalence_0}
Let $a \in C(\R,\R)$, $m,n \in \N$, $D\subseteq\R^m$, $L,\varepsilon \in [0,\infty]$, $f \in C(D,\R^n)$.
Then $\CostLip{a}{\emptyset}{f}{L}{\varepsilon}\geq 2n$ \cfload.
\end{athm}
\begin{proof}[Proof of \cref{lem_cost_of_Lip_approx_set_equivalence_0}.]
Observe that for all
$\mathfrak{m}\in\N$,
$\mathscr{f}\in\ANNs$
with $\realisation_a(\mathscr{f}) \in C(\R^\mathfrak{m},\R^n)$
it holds that
$\singledims_{\lengthANN(\mathscr f)}(\mathscr{f})=n$
\cfload.
Hence, we obtain that for all
$\mathfrak{m}\in\N$,
$\mathscr{f}\in\ANNs$
with $\realisation_a(\mathscr{f}) \in C(\R^\mathfrak{m},\R^n)$
it holds that
\begin{equation}
\paramANN(\mathscr{f})
\geq
[\singledims_{\lengthANN(\mathscr f)}(\mathscr{f})](\singledims_{\lengthANN(\mathscr f)-1}(\mathscr{f})+1)
\geq
n(1+1)= 2n.
\end{equation}
The proof of \cref{lem_cost_of_Lip_approx_set_equivalence_0} is thus complete.
\end{proof}
\cfclear
\begin{athm}{lemma}{lem_cost_of_Lip_approx_set_equivalence_1}
Let $a \in C(\R,\R)$, $n \in \N$, $L,\varepsilon \in [0,\infty]$, $f \in C(\emptyset,\R^n)$.
Then $\CostLip{a}{\emptyset}{f}{L}{\varepsilon}=2n$ \cfout.
\end{athm}
\begin{proof}[Proof of \cref{lem_cost_of_Lip_approx_set_equivalence_1}.]
Throughout this proof let $W\in\R^{n\times 1}$.
Note that \eqref{ANNrealization:ass2} shows that
\begin{enumerate}[(I)]
\item\label{lem_cost_Lip_app_set_equ_1_item_1_proof} it holds that $\realisation_a(\affineANN_{W,0}) \in C(\R,\R^n)$,
\item\label{lem_cost_Lip_app_set_equ_1_item_2_proof} it holds that $\paramANN(\affineANN_{W,0}) = 2n$,
\item \label{lem_cost_Lip_app_set_equ_1_item_3_proof} it holds for all $x \in \emptyset$ that $ \mednorm{f(x) - \functionANNbis{\affineANN_{W,0}}(x)} \leq \varepsilon$, and
\item \label{lem_cost_Lip_app_set_equ_1_item_4_proof} it holds for all $x,y \in \emptyset$ with $x \ne y$ that $\mednorm{ \functionANNbis{\affineANN_{W,0}}(x)-\functionANNbis{\affineANN_{W,0}}(y)} \leq L\norm{x-y}$
\end{enumerate}
\cfload[.
Hence, we obtain that $\CostLip aAfL\eps\leq 2n$.
Combining this with \cref{lem_cost_of_Lip_approx_set_equivalence_0} ensures that $\CostLip{a}{\emptyset}{f}{L}{\varepsilon}= 2n$ \cfload.
This completes the proof of \cref{lem_cost_of_Lip_approx_set_equivalence_1}.
\end{proof}
\cfclear
\begin{athm}{lemma}{lem_cost_of_Lip_approx_set_equivalence}
Let $a \in C(\R,\R)$, $m,n \in \N$, $D \subseteq \R^m $, $f \in C(D,\R^n)$, $L,\varepsilon \in [0,\infty]$\cfload.
Then it holds that $\CostLip{a}{A}{f}{L}{\varepsilon} < \infty$ if and only if there exists $\mathscr{f} \in \ANNs$ such that
\begin{enumerate}[(i)]
\item\label{lem_cost_Lip_app_set_equ_item_1} there exists $\mathfrak{m}\in\N$ such that $D\subseteq\R^\mathfrak{m}$ and $\realisation_a(\mathscr{f}) \in C(\R^\mathfrak{m},\R^n)$,
\item\label{lem_cost_Lip_app_set_equ_item_2} it holds for all $x \in D$ that $\mednorm{f(x) - \functionANNbis{\mathscr{f}}(x)} \leq \varepsilon$,
\item\label{lem_cost_Lip_app_set_equ_item_3} it holds for all $x,y \in D$ with $x \ne y$ that
$\mednorm{ \functionANNbis{\mathscr{f}}(x)-\functionANNbis{\mathscr{f}}(y)} \leq L \mednorm{x-y}$, and
\item\label{lem_cost_Lip_app_set_equ_item_4} it holds that $\paramANN(\mathscr{f}) = \CostLip{a}{A}{f}{L}{\varepsilon}$
\end{enumerate}
\cfout.
\end{athm}
\begin{proof}[Proof of \cref{lem_cost_of_Lip_approx_set_equivalence}.]
Observe that the fact that $\paramANN(\ANNs)\subseteq\N$ ensures that for all $\mathscr{f}\in\ANNs$ with $\paramANN(\mathscr{f})=\CostLip{a}{A}{f}{L}{\varepsilon}$ it holds that
$
\CostLip{a}{A}{f}{L}{\varepsilon}=\paramANN(\mathscr{f})<\infty
$
\cfload.
Combining this with \eqref{Cost_Lipschitz_set} completes the proof of \cref{lem_cost_of_Lip_approx_set_equivalence}.
\end{proof}
\cfclear
\begin{athm}{cor}{cor_cost_of_Lip_approx_set_equivalence}
Let $a \in C(\R,\R)$, $m,n \in \N$, $D \subseteq \R^m $, $f \in C(D,\R^n)$, $L,\varepsilon \in [0,\infty]$ satisfy $D \neq \emptyset$ \cfload.
Then it holds that $\CostLip{a}{A}{f}{L}{\varepsilon} < \infty$ if and only if there exists $\mathscr{f} \in \ANNs$ such that
\begin{enumerate}[(i)]
\item\label{cor_cost_Lip_app_set_equ_item_1} it holds that $\realisation_a(\mathscr{f}) \in C(\R^m,\R^n)$,
\item\label{cor_cost_Lip_app_set_equ_item_2} it holds for all $x \in D$ that $\mednorm{f(x) - \functionANNbis{\mathscr{f}}(x)} \leq \varepsilon$,
\item\label{cor_cost_Lip_app_set_equ_item_3} it holds for all $x,y \in D$ with $x \ne y$ that
$\mednorm{ \functionANNbis{\mathscr{f}}(x)-\functionANNbis{\mathscr{f}}(y)} \leq L \mednorm{x-y}$, and
\item\label{cor_cost_Lip_app_set_equ_item_4} it holds that $\paramANN(\mathscr{f}) = \CostLip{a}{A}{f}{L}{\varepsilon}$
\end{enumerate}
\cfout.
\end{athm}
\begin{proof}[Proof of \cref{cor_cost_of_Lip_approx_set_equivalence}.]
Observe that for all $\mathscr{f}\in\ANNs$ with $\realisation_a(\mathscr{f})\in C(\R^m,\R^n)$ there exists $\mathfrak{m}\in\N$ such that
\begin{equation}
\label{cor_cost_of_Lip_approx_set_equivalence:eq1}
D\subseteq\R^\mathfrak{m}\qquad\text{and}\qquad\realisation_a(\mathscr{f})\in C(\R^\mathfrak{m},\R^n)
\end{equation}
\cfload[.]Furthermore, note that the fact that $\emptyset\neq D\subseteq\R^m$ ensures that for all $\mathfrak{m}\in\N$ with $D\subseteq\R^\mathfrak{m}$ it holds that $\mathfrak{m}=m$. Combining this and \eqref{cor_cost_of_Lip_approx_set_equivalence:eq1} with \cref{lem_cost_of_Lip_approx_set_equivalence} completes the proof of \cref{cor_cost_of_Lip_approx_set_equivalence}.
\end{proof}
\cfclear
\begin{athm}{lemma}{lem_cost_of_Lip_approx_set_equivalence_incodomain}
Let
$m,n \in \N$,
$D\subseteq\R^m$,
$R \in [0,\infty)$,
$f \in C(D,\R^n)$,
$L,\varepsilon \in [0,\infty]$
satisfy
$f(D)\subseteq [-R,R]^n$ and
$\CostLip{\ReLU}{A}{f}{L}{\varepsilon} < \infty$
\cfload.
Then there exists
$\mathscr{f} \in \ANNs$
such that
\begin{enumerate}[(i)]
\item\label{lem_cost_Lip_app_set_equ_incodomain_item_1}
it holds that
$\realisation_{\ReLU}(\mathscr{f}) \in C(\R^m,\R^n)$,
\item\label{lem_cost_Lip_app_set_equ_incodomain_item_1a}
it holds for all
$x \in D$
that
$(\realisation_\ReLU(\mathscr{f}))(x) \in [-R,R]^n$,
\item\label{lem_cost_Lip_app_set_equ_incodomain_item_2}
it holds for all
$x \in D$
that
$\mednorm{f(x) -(\realisation_\ReLU(\mathscr{f}))(x)} \leq \varepsilon$,
\item\label{lem_cost_Lip_app_set_equ_incodomain_item_3}
it holds for all
$x,y \in D$
with $x \ne y$
that
$\mednorm{ (\realisation_\ReLU(\mathscr{f}))(x)-(\realisation_\ReLU(\mathscr{f}))(y)} \leq L \mednorm{x-y}$, and
\item\label{lem_cost_Lip_app_set_equ_incodomain_item_4}
it holds that
$\paramANN(\mathscr{f}) = \CostLip{\mathfrak{r}}{A}{f}{L}{\varepsilon}+2n(n+1)$
\end{enumerate}
\cfout.
\end{athm}
\begin{aproof}
\Nobs that
\cref{lem:clipping_function}
ensures that there exists
$\mathscr{g}\in\ANNs$
which satisfies that
\begin{enumerate}[(I)]
\item\label{proof:lem_cost_of_Lip_approx_set_equivalence_incodomain:item_0}
it holds that
$\realisation_\mathfrak{r}(\mathscr{g})\in C(\R^n,\R^n)$,
\item\label{proof:lem_cost_of_Lip_approx_set_equivalence_incodomain:item_1}
it holds for all
$x\in\R^n$
that
$(\realisation_\mathfrak{r}(\mathscr{g}))(x)=\Clip{-R}Rn( x )$, and
\item\label{proof:lem_cost_of_Lip_approx_set_equivalence_incodomain:item_2}
it holds that
$\dims(\mathscr{g})=(n,n,n,n)$
\end{enumerate}
\cfload.
Furthermore, \nobs that
\cref{cor_cost_of_Lip_approx_set_equivalence} and
the assumption that
$\CostLip{\ReLU}{A}{f}{L}{\varepsilon} < \infty$
show that there exists
$\mathscr{h}\in\ANNs$
which satisfies that
\begin{enumerate}[(A)]
\item\label{lem_cost_Lip_app_set_equ_incodomain_proof_item_1}
it holds that
$\realisation_\mathfrak{r}(\mathscr{h}) \in C(\R^m,\R^n)$,
\item\label{lem_cost_Lip_app_set_equ_incodomain_proof_item_2}
it holds for all
$x \in D$
that
$\mednorm{f(x) - (\realisation_\mathfrak{r}(\mathscr{h}))(x)} \leq \varepsilon$,
\item\label{lem_cost_Lip_app_set_equ_incodomain_proof_item_3}
it holds for all
$x,y \in D$
with $x \ne y$
that
$\mednorm{ (\realisation_\mathfrak{r}(\mathscr{h}))(x)-(\realisation_\mathfrak{r}(\mathscr{h}))(y)} \leq L \mednorm{x-y}$, and
\item\label{lem_cost_Lip_app_set_equ_incodomain_proof_item_4}
it holds that
$\paramANN(\mathscr{h}) = \CostLip{\mathfrak{r}}{A}{f}{L}{\varepsilon}$
\end{enumerate}
\cfload.
Next \nobs that
\cref{Lemma:PropertiesOfCompositions_n2},
\cref{proof:lem_cost_of_Lip_approx_set_equivalence_incodomain:item_0},
and \cref{lem_cost_Lip_app_set_equ_incodomain_proof_item_1}
establish that
\begin{equation}
\eqlabel{i1}
\realisation_{\ReLU}(\compANN{\mathscr g}{\mathscr h})
\in C(\R^m,\R^n)
\ifnocf.
\end{equation}
\cfload[.
Moreover, \nobs that
\cref{Lemma:PropertiesOfCompositions_n2},
\cref{proof:lem_cost_of_Lip_approx_set_equivalence_incodomain:item_1},
and the fact that $\Clip{-R}Rn(\R^n)\subseteq[-R,R]^n$
imply that for all
$x\in D$
it holds that
\begin{equation}
\eqlabel{i2}
(\realisation_{\mathfrak{r}}(\compANN{\mathscr g}{\mathscr h}))(x)
=
\prb{[\realisation_{\mathfrak{r}}(\mathscr{g})]\circ[\realisation_{\mathfrak{r}}(\mathscr{h})]}(x)
=
\Clip{-R}Rn\prb{(\realisation_{\mathfrak{r}}(\mathscr{h}))(x)}
\in[-R,R]^n
.
\end{equation}
\Cref{lem_cost_Lip_app_set_equ_incodomain_proof_item_2},
the assumption that $f(D)\subseteq[-R,R]^n$,
and the fact that
for all
$x,y\in\R^n$
it holds that
$\mednorm{\Clip{-R}Rn(x)-\Clip{-R}Rn(y)}\leq \mednorm{x-y}$
hence
ensure that for all
$x\in D$
it holds that
\begin{equation}
\eqlabel{i3}
\begin{split}
\mednorm{f(x)-(\realisation_{\mathfrak{r}}(\compANN{\mathscr g}{\mathscr h}))(x)}
&=
\norm[\big]{\Clip{-R}Rn(f(x))-\Clip{-R}Rn\prb{(\realisation_{\mathfrak{r}}(\mathscr{h}))(x)}}
\\&\leq
\mednorm{f(x)-(\realisation_{\mathfrak{r}}(\mathscr{h}))(x)}\leq \varepsilon
.
\end{split}
\end{equation}
In the next step \nobs that
\cref{lem_cost_Lip_app_set_equ_incodomain_proof_item_3}
and the fact that
for all
$x,y\in\R^n$
it holds that
$\norm{\Clip{-R}Rn(x)-\Clip{-R}Rn(y)}\leq\norm{x-y}$
imply that for all
$x,y\in D$ with $x\neq y$
it holds that
\begin{equation}
\eqlabel{i4}
\begin{split}
&\mednorm{(\realisation_{\mathfrak{r}}(\compANN{\mathscr g}{\mathscr h}))(x)-(\realisation_{\mathfrak{r}}(\compANN{\mathscr g}{\mathscr h}))(y)}
\\&=
\norm[\big]{\Clip{-R}Rn\prb{(\realisation_{\mathfrak{r}}(\mathscr{h}))(x)}-\Clip{-R}Rn\prb{(\realisation_{\mathfrak{r}}(\mathscr{h}))(y)}}
\\&\leq
\mednorm{(\realisation_{\mathfrak{r}}(\mathscr{h}))(x)-(\realisation_{\mathfrak{r}}(\mathscr{h}))(y)}\leq L\mednorm{x-y}.
\end{split}
\end{equation}
Furthermore, \nobs that
\cref{Lemma:PropertiesOfCompositions_n1},
\cref{proof:lem_cost_of_Lip_approx_set_equivalence_incodomain:item_2},
and \cref{lem_cost_Lip_app_set_equ_incodomain_proof_item_4}
prove that
\begin{equation}
\eqlabel{i5}
\begin{split}
&\paramANN(\mathscr{g}\bullet\mathscr{h})
\\&=
\paramANN(\mathscr{g})+\paramANN(\mathscr{h})+\PR{\singledims_1(\mathscr{g})}(\singledims_{\lengthANN(\mathscr{h})-1}(\mathscr{h})+1)
\\&\quad-\PR{\singledims_1(\mathscr{g})}(\singledims_{0}(\mathscr{g})+1)-\PR{\singledims_{\lengthANN(\mathscr{h})}(\mathscr{h})}(\singledims_{\lengthANN(\mathscr{h})-1}(\mathscr{h})+1)
\\&=3n(n+1)+\paramANN(\mathscr{h})+n(\singledims_{\lengthANN(\mathscr{h})-1}(\mathscr{h})+1)-n(n+1)-n(\singledims_{\lengthANN(\mathscr{h})-1}(\mathscr{h})+1)
\\&=\CostLip{\ReLU}{A}{f}{L}{\varepsilon}+2n(n+1).
\end{split}
\end{equation}
Combining this with \eqqref{i1}, \eqqref{i2}, \eqqref{i3}, and \eqqref{i4}
establishes \cref{lem_cost_Lip_app_set_equ_incodomain_item_1,lem_cost_Lip_app_set_equ_incodomain_item_1a,lem_cost_Lip_app_set_equ_incodomain_item_2,lem_cost_Lip_app_set_equ_incodomain_item_3,lem_cost_Lip_app_set_equ_incodomain_item_4}.
The proof of \cref{lem_cost_of_Lip_approx_set_equivalence_incodomain} is thus complete.
\end{aproof}
\cfclear
\begin{athm}{lemma}{Lemma:Monotonicity_of_Cost}
Let $a \in C(\R,\R)$, $m,n \in \N$, $D,E \subseteq \R^m$, $f \in C(D,\R^n)$, $\varepsilon, L \in [0,\infty]$, $\epsilon \in [\varepsilon, \infty]$, $\mathfrak{L} \in [L, \infty]$ satisfy $E\subseteq D$.
Then
$
\CostLip{a}{B}{f|_{E}}{\mathfrak{L}}{\epsilon}
\leq
\CostLip{a}{B}{f}{L}{\varepsilon}
$
\cfout.
\end{athm}
\begin{proof}
[Proof of \cref{Lemma:Monotonicity_of_Cost}]
Throughout this proof assume w.l.o.g.\ that $\CostLip{a}{B}{f}{L}{\varepsilon}<\infty$ and $D \neq \emptyset$ \cfadd{lem_cost_of_Lip_approx_set_equivalence_1}\cfload.
Note that \cref{cor_cost_of_Lip_approx_set_equivalence} ensures that there exists $\mathscr{f} \in \ANNs$ which satisfies that
\begin{enumerate}[(i)]
\item it holds that $\realisation_a(\mathscr{f}) \in C(\R^m,\R^n)$,
\item it holds for all $x \in D$ that $\mednorm{f(x) - \functionANNbis{\mathscr{f}}(x)} \leq \varepsilon$,
\item it holds for all $x,y \in D$ with $x \ne y$ that
$\mednorm{ \functionANNbis{\mathscr{f}}(x)-\functionANNbis{\mathscr{f}}(y)} \leq L \mednorm{x-y}$, and
\item it holds that $\paramANN(\mathscr{f}) = \CostLip{a}{D}{f}{L}{\varepsilon}$
\end{enumerate}
\cfload.
\Nobs that the assumption that $\varepsilon \leq \epsilon$, $L\leq\mathfrak{L}$, and $E\subseteq D$ demonstrates that
\begin{equation}\begin{split}
&\CostLip{a}{B}{f|_E}{\mathfrak{L}}{\epsilon}
\\&=
\min \pr*{ \pR*{ \mathfrak{p} \in \N \colon \PR*{
\begin{array}{cc}
\exists\, \mathscr{g} \in \ANNs, \mathfrak m\in\N \colon
\PRb{(\realisation_a(\mathscr{g}) \in C(\R^{\mathfrak m},\R^n))\land{}\\
(\mathfrak{p}=\paramANN(\mathscr{g}))\land(\fa{x,y} E \text{ with } x \ne y \colon\\
\mednorm{ \functionANNbis{\mathscr{g}}(x)-\functionANNbis{\mathscr{g}}(y)} \leq \mathfrak{L}\norm{x-y})\land{}\\
(\fa{x} E \colon
\mednorm{f(x) - \functionANNbis{\mathscr{g}}(x)} \leq \epsilon)}\\
\end{array} }
} \cup \{\infty\} }
\\&\leq
\paramANN(\mathscr{f})
=
\CostLip{a}{B}{f}{L}{\varepsilon}.
\end{split}\end{equation}
This completes the proof of \cref{Lemma:Monotonicity_of_Cost}.
\end{proof}
\cfclear
\cfclear
\begin{athm}{lemma}{lemma_Lip_catalogs_are_Lip_set}
Let $a \in C(\R,\R)$, $m,n \in \N$, $D \subseteq \R^m$, $f \in C(D,\R^n)$, $L \in [0,\infty]$, $c \in (0, \infty)$ satisfy for all $\varepsilon \in (0,c)$ that $\CostLip{a}{D}{f}{L}{\varepsilon} < \infty$ \cfload.
Then it holds for all $x,y \in D$ with $x \ne y$ that $\mednorm{f(x)-f(y)} \leq L \mednorm{x-y}$ \cfout.
\end{athm}
\begin{proof}[Proof of \cref{lemma_Lip_catalogs_are_Lip_set}.]
Throughout this proof assume w.l.o.g.\ that $D \neq \emptyset$. Observe that the assumption that for all $\varepsilon \in (0,c)$ it holds that $\CostLip{a}{D}{f}{L}{\varepsilon} < \infty$ and \cref{cor_cost_of_Lip_approx_set_equivalence} imply that there exist $ (\mathscr{f}_{\varepsilon})_{\varepsilon \in (0,c)} \subseteq \ANNs$
which satisfy that
\begin{enumerate}[(i)]
\item \label{lemma_Lip_catalogs_are_Lip:1}
it holds for all
$\varepsilon \in (0,c)$
that
$\realisation_{a}(\mathscr{f}_{\varepsilon}) \in C(\R^{m},\R^{n})$,
\item \label{lemma_Lip_catalogs_are_Lip:3}
it holds for all
$\varepsilon \in (0,c)$,
$x \in D$
that
$\mednorm{f(x) - \functionANNbis{\mathscr{f}_{\varepsilon}}(x)} \leq \varepsilon$, and
\item \label{lemma_Lip_catalogs_are_Lip:2}
it holds for all
$\varepsilon \in (0,c)$,
$x,y \in D$
with $x \ne y$
that
$\mednorm{\functionANNbis{\mathscr{f}_{\varepsilon}}(x) - \functionANNbis{\mathscr{f}_{\varepsilon}}(y)} \leq L \mednorm{x-y}$
\end{enumerate}
\cfload.
Note that \cref{lemma_Lip_catalogs_are_Lip:3} and \cref{lemma_Lip_catalogs_are_Lip:2} ensure that for all $\varepsilon \in (0,c)$, $x,y \in D$ with $x \ne y$ it holds that
\begin{equation}\begin{split}
&\mednorm{f(x) - f(y)}
\\&\leq
\norm{f(x) - \functionANNbis{\mathscr{f}_{\varepsilon}}(x)}
+ \norm{\functionANNbis{\mathscr{f}_{\varepsilon}}(x) - \functionANNbis{\mathscr{f}_{\varepsilon}}(y)} + \norm{\functionANNbis{\mathscr{f}_{\varepsilon}}(y) - f(y)}
\\&\leq
\varepsilon + L \mednorm{x-y} + \varepsilon
.
\end{split}\end{equation}
Hence, we obtain that for all $x,y \in D$ with $x \ne y$ it holds that
\begin{equation}
\begin{split}
\mednorm{f(x) - f(y)} &= \limsup_{\varepsilon \searrow 0} \mednorm{f(x) - f(y)}
\leq \limsup_{\varepsilon \searrow 0} \PR{ 2\varepsilon + L \mednorm{x-y} } = L \mednorm{x-y}.
\end{split}
\end{equation}
This completes the proof of \cref{lemma_Lip_catalogs_are_Lip_set}.
\end{proof}
\subsection{Spaces of ANN approximable functions}
\label{Subsection:3.3}
\begin{definition}
[ANN approximation spaces for one-dimensional functions]
\label{def:polyC}
Let $c,r \in [0, \infty)$. Then we denote by $\Capprox{c}{r}$ the set given by
\begin{multline}
\label{polyC:1}
\Capprox{c}{r}=
\pRb{
f \in C(\R,\R) \colon \PRb{
\forall \, \radius \in [r, \infty),\varepsilon \in (0,1] \colon
\CostLip{\ReLU}{[-\radius, \radius]}{f|_{\PR{-R,R}}} {c\radius^{c}}{\varepsilon} \leq c\radius^c \varepsilon^{-c} }
}
\end{multline}
(cf.\ \cref{def:ReLU,def:cost_Lipschitz_approx_set}).
\end{definition}
\cfclear
\begin{athm}{cor}{cor:polyC}
Let $c,r \in [0, \infty)$\cfload. Then
\begin{enumerate}[(i)]
\item\label{cor:polyC_item_1b} it holds for all $R\in[r,\infty)$ that $\Capprox{c}{r}\subseteq \Capprox{c}{R}$,
\item\label{cor:polyC_item_1a} it holds for all $C\in[c,\infty)$ that $\Capprox{c}{r}\subseteq \Capprox{C}{\max\{1,r\}}$, and
\item\label{cor:polyC_item_2} it holds that $\Capprox{c}{r}\subseteq \Capprox{\max\{c,cr^c\}}{1}$
\end{enumerate}
\cfout.
\end{athm}
\begin{proof}
[Proof of \cref{cor:polyC}]
First, \nobs that \eqref{polyC:1} establishes \cref{cor:polyC_item_1b}.
Next \nobs that \cref{Lemma:Monotonicity_of_Cost} shows that for all $\rho\in[1,\infty)$, $C\in[c,\infty)$, $f\in \Capprox{c}{\rho}$, $\radius\in[\rho,\infty)$, $\varepsilon\in(0,1]$ it holds that
\begin{equation}
\begin{split}
\CostLip{\ReLU}{[-\radius,\radius] }{f|_{[-R,R]}}{C\radius^C}{\varepsilon}\leq\CostLip{\ReLU}{[-\radius,\radius] }{f|_{[-R,R]}}{c\radius^c}{\varepsilon}\leq c\radius^c\varepsilon^{-c}\leq C\radius^C\varepsilon^{-C}
\end{split}
\end{equation}
\cfload. Hence, we obtain that for all $\rho\in[1,\infty)$, $C\in[c,\infty)$ it holds that $\Capprox{c}{\rho}\subseteq\Capprox{C}{\rho}$. Combining this with \cref{cor:polyC_item_1b} ensures that for all $C\in[c,\infty)$ it holds that
\begin{equation}
\begin{split}
\Capprox{c}{r}\subseteq\Capprox{c}{\max\{1,r\}}\subseteq\Capprox{C}{\max\{1,r\}}.
\end{split}
\end{equation}
This establishes \cref{cor:polyC_item_1a}.
\Nobs that
the assumption that $c\geq 0$
implies that
for all $\radius \in [1, \infty)$ it holds that $\radius\leq \max\{1,r\}\radius\in [r, \infty)$ and
\begin{equation}
c(\max\{1,r\}\radius)^c= \max\{c,cr^c\}\radius^{c}\leq \max\{c,cr^c\} \radius^{\max\{c,cr^c\}}.
\end{equation}
This, \eqref{polyC:1}, and \cref{Lemma:Monotonicity_of_Cost} (applied with
$a \curvearrowleft \ReLU$,
$m \curvearrowleft 1$,
$n \curvearrowleft 1$,
$D \curvearrowleft [-\mathord{\max}\{1,r\}\radius,\allowbreak\max\{1,r\}\radius]$,
$E \curvearrowleft [-\radius,\radius]$,
$f\is f|_{[-\mathord{\max}\{1,r\}\radius,\max\{1,r\}\radius]}$,
$\varepsilon \curvearrowleft \varepsilon$,
$L \curvearrowleft c\pr{\max\{1,r\}\radius}^c$,
$\epsilon \curvearrowleft \varepsilon$,
$\mathfrak{L} \curvearrowleft \max\{c,cr^c\}\radius^{\max\{c,cr^c\}}$
for $f\in\Capprox{c}{r}$, $\radius\in[1,\infty)$, $\varepsilon\in(0,1]$
in the notation of \cref{Lemma:Monotonicity_of_Cost}) demonstrate that for all $f\in \Capprox{c}{r}$, $\radius \in [1, \infty)$, $\varepsilon \in (0,1]$ it holds that
\begin{equation}
\begin{split}
&\CostLipB{\ReLU}{[-\radius,\radius] }{f|_{[-R,R]}}{\max\{c,cr^c\} \radius^{\max\{c,cr^c\}}}{\varepsilon}
\\&\leq
\CostLipB{\ReLU}{[-\mathord{\max}\{1,r\}\radius,\max\{1,r\}\radius] }{ f|_{[-\mathord{\max}\{1,r\}\radius,\max\{1,r\}\radius]}}{c(\max\{1,r\}\radius)^c}{\varepsilon}
\\&\leq c(\max\{1,r\}\radius)^c \varepsilon^{-c}\leq \max\{c,cr^c\}\radius^{\max\{c,cr^c\}}\varepsilon^{-\mathord{\max}\{c,cr^c\}}.
\end{split}
\end{equation}
Therefore, we obtain \cref{cor:polyC_item_2}.
The proof of \cref{cor:polyC} is thus complete.
\end{proof}
\cfclear
\begin{athm}{lemma}{Lemma:codomain}
Let\footnote{Here and in the rest of the article, $\bigcup_{n\in\N}\R^n$ is equipped with the
topology of the disjoint union.} $f \in C\prb{\bigcup_{n \in \N}\R^n, \bigcup_{n \in \N}\R^n}$, $d \in \N$. Then there exists a unique $\delta \in \N$ such that $f(\R^d) \subseteq \R^{\delta}$.
\end{athm}
\begin{proof}
[Proof of \cref{Lemma:codomain}]
Throughout this proof let $\delta \in \N$ satisfy $f(1,2, \ldots, d) \allowbreak \in \R^{\delta}$.
Observe that
the assumption that
$f \in C\prb{\bigcup_{n \in \N}\R^n, \bigcup_{n \in \N}\R^n}$
and the fact that
$\R^d$ is connected
imply that
$f(\R^d)$ is connected.
The fact that
\begin{equation}
\label{Lemma:codomain:disjoint_decomposition}
\begin{split}
f\pr{\R^d}
=
\pr{f\pr{\R^d}\cap\R^{\delta}}\cup\prb{f\pr{\R^d}\cap\prb{\textstyle\bigcup_{n \in \N\setminus\{\delta\}}\R^n}},
\end{split}
\end{equation}
the fact that
$\pr{f\pr{\R^d}\cap\R^{\delta}}$, $\prb{f\pr{\R^d}\cap\prb{\bigcup_{n \in \N\setminus\{\delta\}}\R^n}}\subseteq f\pr{\R^d}$ are open sets with respect to the subspace topology on $f(\R^d)$,
the fact that
$\pr{f\pr{\R^d}\cap\R^{\delta}}\cap\prb{f\pr{\R^d}\cap\prb{\bigcup_{n \in \N\setminus\{\delta\}}\R^n}}=\emptyset$,
and the fact that
$f(1,2, \ldots, d)\in(f(\R^d)\cap\R^{\delta})$
hence
ensure that
\begin{equation}
f(\R^d)\cap\R^{\delta}=f(\R^d)
\qquad\text{and}\qquad
f\pr{\R^d}\cap\prb{\textstyle\bigcup_{n \in \N\setminus\{\delta\}}\R^n}=\emptyset
.
\end{equation}
The proof of \cref{Lemma:codomain} is thus complete.
\end{proof}
\begin{definition}
[Image dimensions]
\label{def:smart_restriction}\cfadd{Lemma:codomain}\cfconsiderloaded{def:smart_restriction}
Let $f \in C\prb{\bigcup_{d \in \N}\R^d, \bigcup_{d \in \N}\R^d}$, $d \in \N$. Then we denote by $\imdim{f}{d} \in \N$ the unique natural number which satisfies $f(\R^d) \subseteq \R^{\imdim{f}{d}}$ \cfload.
\end{definition}
\cfclear
\begin{definition}
[Ceiling of real numbers]
\label{def:ceiling}
We denote by $\ceil{\cdot} \colon \R \to \Z$ the function which satisfies for all $x \in \R$ that $\ceil{x} = \min(\Z \cap [x, \infty))$.
\end{definition}
\begingroup
\newcommand{\const}{C}
\cfclear
\begin{athm}{lemma}{Lemma:extension_on_R_poly_a}
Let
$a\in C(\R,\R)$,
$m,n\in\N$,
$f \in C( \R^m,\R^n)$,
$c,\const ,r \in [0,\infty)$,
$\eps\in(0,1]$
satisfy for all
$\radius \in \N \cap [r, \infty)$
that
$\const =\max\{1,c(\max\{2,2\ceil{r}\})^{\max\{1,c\}}\}$
and
\begin{equation}
\label{extension_on_R_poly_a:1}
\CostLipA{a}{[-\radius,\radius]^m }
{f|_{[-\radius,\radius]^m}}
{cm^c \! \radius^c}
{\varepsilon}
\leq
cm^c \! \radius^c \varepsilon^{-c}
\end{equation}
\cfload.
Then it holds for all
$\radius \in [1, \infty)$
that
\begin{equation}
\CostLipA{a}{[-\radius,\radius]^m }
{f|_{[-\radius,\radius]^m}}
{\const m^\const \! \radius^\const }
{\varepsilon}
\leq
\const m^\const \! \radius^\const \varepsilon^{-\const }.
\end{equation}
\end{athm}
\begin{aproof}
Throughout this proof assume w.l.o.g.\ that $r\geq 1$.
Note that
the fact that $C\geq c2\ceil r^{\max\{1,c\}}$
ensures that for all
$\radius \in [1, \infty)$
it holds that
$\radius\leq\ceil{r\radius}\in \N \cap [r, \infty)$ and
\begin{equation}
cm^c\ceil{r\radius}^c
\leq
c\ceil{r}^{\max\{1,c\}}m^c\ceil{\radius}^{\max\{1,c\}}
\leq
c\pr{2\ceil{r}}^{\max\{1,c\}}m^c\!\radius^{\max\{1,c\}}
\leq
\const m^{\const }\!\radius^{\const }
.
\end{equation}
This,
\eqref{extension_on_R_poly_a:1}, and
\cref{Lemma:Monotonicity_of_Cost}
show that for all
$\radius \in [1, \infty)$
it holds that
\begin{equation}
\begin{split}
\CostLipA{a}{[-\radius,\radius]^m }
{f|_{[-\radius,\radius]^m}}
{\const m^\const \! \radius^\const }
{\varepsilon}
&\leq
\CostLipA{a}{[-\ceil{r\radius},\ceil{r\radius}]^m }
{f|_{[-\ceil{r\radius},\ceil{r\radius}]^m}}
{cm^c\ceil{r\radius}^c}
{\varepsilon}
\\&\leq
cm^c\ceil{r\radius}^c \varepsilon^{-c}
\leq
\const m^\const \!\radius^{\const }\varepsilon^{-\const }.
\end{split}
\end{equation}
\finishproofthis
\end{aproof}
\endgroup
\cfclear
\begin{athm}{lemma}{Lemma:extension_on_R_poly}
Let $a\in C(\R,\R)$, $f \in C\prb{\bigcup_{d \in \N}\R^d, \bigcup_{d \in \N}\R^d}$.
Then the following three statements are equivalent:
\begin{enumerate}[(i)]
\item \label{extension_on_R_poly:item_1}
There exists $c \in [0,\infty)$ such that for all $d,\radius \in \N$, $\varepsilon \in (0,1]$ it holds that
\begin{equation}
\CostLipA{a}{[-\radius,\radius]^d }{\pr{[-\radius,\radius]^d \ni x \mapsto f(x) \in \R^{\imdim{f}{d}}}}{cd^c \! \radius^c}{\varepsilon} \leq c d^c \! \radius^c \varepsilon^{-c}
\end{equation}
\cfload.
\item \label{extension_on_R_poly:item_2}
There exist $c,r \in [0,\infty)$ such that for all $d \in \N$, $\radius \in \N \cap [r, \infty)$, $\varepsilon \in (0,1]$ it holds that
\begin{equation}
\CostLipA{a}{[-\radius,\radius]^d }{\pr{[-\radius,\radius]^d \ni x \mapsto f(x) \in \R^{\imdim{f}{d}}}}{cd^c \! \radius^c}{\varepsilon} \leq c d^c \! \radius^c \varepsilon^{-c}.
\end{equation}
\item \label{extension_on_R_poly:item_3}
There exists $c \in [0,\infty)$ such that for all $d \in \N$, $\radius \in [1, \infty)$, $\varepsilon \in (0,1]$ it holds that
\begin{equation}
\CostLipA{a}{[-\radius,\radius]^d }{\pr{[-\radius,\radius]^d \ni x \mapsto f(x) \in \R^{\imdim{f}{d}}}}{cd^c \! \radius^c}{\varepsilon} \leq c d^c \! \radius^c \varepsilon^{-c}.
\end{equation}
\end{enumerate}
\end{athm}
\begin{aproof}
First observe that
\cref{extension_on_R_poly:item_3} implies \cref{extension_on_R_poly:item_1}
and
\cref{extension_on_R_poly:item_1} implies \cref{extension_on_R_poly:item_2}.
Moreover, note that
\cref{Lemma:extension_on_R_poly_a}
(applied with
$m\is d$,
$n\is \imdim fd$,
$f\is (\R^d\ni x\mapsto f(x)\in\R^{\imdim fd})$,
$c\is c$,
$r\is r$,
$\eps\is\eps$,
for $d\in\N$, $R\in[1,\infty)$, $\eps\in(0,1]$
in the notation of \cref{Lemma:extension_on_R_poly_a})
shows that
\cref{extension_on_R_poly:item_2} implies \cref{extension_on_R_poly:item_3}.
The proof of \cref{Lemma:extension_on_R_poly} is thus complete.
\end{aproof}
\cfclear
\begin{definition}
[ANN approximation space for multidimensional functions]
\label{def:polyD_mult}
\cfconsiderloaded{def:polyD_mult}
We denote by $\Dapprox$ the set given by
\begin{equation}
\begin{split}
\label{polyD_mult:1}
&\Dapprox=
\pRbbb{f \in C\prb{ \textstyle{\bigcup_{d \in \N}\R^d, \bigcup_{d \in \N}\R^d}} \colon
\\
&\PR*{
\begin{array}{c}
\exists \, c \in [0, \infty)\colon \forall \, d, \radius \in \N, \varepsilon \in (0,1] \colon \PRb{
\prb{f([-\radius,\radius]^d)\subseteq[-cd^c \! \radius^c,cd^c \! \radius^c]^{\imdim{f}{d}}}\land{}
\\
(\imdim{f}{d}\leq cd^c)\land
\prb{\CostLipA{\ReLU}{[-\radius,\radius]^d }{\pr{[-\radius,\radius]^d \ni x \mapsto f(x) \in \R^{\imdim{f}{d}}}}{cd^c \! \radius^c}{\varepsilon}{}
\leq{} c d^c \! \radius^c \varepsilon^{-c}}}
\end{array}
}{}}
\end{split}
\end{equation}
\cfload.
\end{definition}
\cfclear
\subsection{ANN approximations for compositions of functions}
\label{Subsection:3.4}
\begin{athm}{prop}{Prop:composition_cost_ReLU}
Let $d_1,d_2,d_3 \in \N$,
$\varepsilon, L_1, L_2, R_1, R_2 \in [0, \infty)$ and
let
$f_1 \colon [-R_1,R_1]^{d_1} \to \R^{d_2}$ and
$f_2 \colon [-R_2,R_2]^{d_2} \to \R^{d_3}$
satisfy $f_1([-R_1,R_1]^{d_1})\subseteq[-R_2,R_2]^{d_2}$.
Then\footnote{Note that for all sets $A, B, C, D$ and all
functions $f\colon A\to B$ and $g\colon C\to D$
such that $f(A)\subseteq C$ we denote by $g\circ f\colon A\to D$
the function which satisfies for all $a\in A$ that
$(g\circ f)(a)=g(f(a))$.}
\begin{equation}
\begin{split}
\label{Prop:composition_cost_ReLU:eq}
&\CostLip{\ReLU}{A}{f_2 \circ f_1}{L_2L_1}{\varepsilon}
\leq 4d_2(d_2+1)+\smallsum\limits_{k=1}^{2}2\,\CostLipB{\ReLU}{B}{f_k}{L_k}{\tfrac{\varepsilon}{2(L_2)^{2-k}}}
\end{split}
\end{equation}
\cfout.
\end{athm}
\begin{proof}
[Proof of \cref{Prop:composition_cost_ReLU}]
Throughout this proof let $a_1,a_2 \in \R$ satisfy $a_1 = 2L_2$ and $a_2 = 2$ and
assume w.l.o.g.\ that
$\CostLipB{\ReLU}{A}{f_2}{L_2}{\tfrac{\varepsilon}{a_2}} + \CostLipB{\ReLU}{B}{f_1}{L_1}{\tfrac{\varepsilon}{a_1}} < \infty$ \cfload.
Observe that \cref{cor_cost_of_Lip_approx_set_equivalence}, \cref{lem_cost_of_Lip_approx_set_equivalence_incodomain},
and the assumption that $\CostLipB{\ReLU}{A}{f_2}{L_2}{\tfrac{\varepsilon}{a_2}} + \CostLipB{\ReLU}{B}{f_1}{L_1}{\tfrac{\varepsilon}{a_1}} < \infty$
imply that there exist
$\mathscr{f}_1, \mathscr{f}_2 \in \ANNs$ which satisfy that
\begin{enumerate}[(I)]
\item \label{composition_cost_ReLU:item1} it holds for all $k \in \{1,2\}$ that $ \realisation_\ReLU(\mathscr{f}_k) \in C(\R^{d_k},\R^{d_{k+1}})$,
\item \label{composition_cost_ReLU:item1a} it holds for all $x \in [-R_1,R_1]^{d_1}$ that $ (\realisation_\ReLU(\mathscr{f}_1))(x) \in [-R_2,R_2]^{d_2}$,
\item \label{composition_cost_ReLU:item2} it holds for all $k \in \{1,2\}$, $x,y \in [-R_k,R_k]^{d_k}$ that $\mednorm{\functionnbReLUANN{\mathscr{f}_k} (x) - \functionnbReLUANN{\mathscr{f}_k} (y)} \leq L_k \mednorm{x-y}$,
\item \label{composition_cost_ReLU:item3} it holds for all $k \in \{1,2\}$, $x \in [-R_k,R_k]^{d_k}$ that $\mednorm{f_k(x) - \functionnbReLUANN{\mathscr{f}_k} (x)} \leq \frac{\varepsilon}{a_k}$, and
\item \label{composition_cost_ReLU:item4} it holds for all $k \in \{1,2\}$ that $\paramANN(\mathscr{f}_k) \leq \CostLipB{\ReLU}{A}{f_k}{L_k}{\tfrac{\varepsilon}{a_k}}+(2-k)2d_2(d_2+1)$
\end{enumerate}
\cfload. Note that \cref{composition_cost_ReLU:item1}, \cref{Prop:identity_representation}, and \cref{Lemma:PropertiesOfCompositions_n2} ensure that for all $x \in \R^{d_1}$ it holds that $\realisation_{\ReLU}(\mathscr{f}_2\bullet\mathbb{I}_{d_2}\bullet \mathscr{f}_1)\in C(\R^{d_1},\R^{d_{3}})$ and
\begin{equation}
\label{composition_cost_ReLU:a}
\begin{split}
(\realisation_{\ReLU}(\compANN{\mathscr{f}_2 }{\ReLUidANN{d_2}}\bullet \mathscr{f}_1))(x) = \prb{[\realisation_{\ReLU}(\mathscr{f}_2)] \circ [\realisation_{\ReLU}(\mathscr{f}_1)]}(x)
\end{split}
\end{equation}
\cfload.
This,
the triangle inequality,
\cref{composition_cost_ReLU:item1a},
\cref{composition_cost_ReLU:item2}, and
\cref{composition_cost_ReLU:item3}
show that
\begin{equation}
\label{composition_cost_ReLU:1}
\begin{split}
& \sup_{x \in [-R_1,R_1]^{d_1}} \mednorm{(f_2 \circ f_1)(x) - (\realisation_{\ReLU}(\mathscr{f}_2 \bullet\mathbb{I}_{d_2}\bullet \mathscr{f}_1))(x) }
\\&= \sup_{x \in [-R_1,R_1]^{d_1}} \bignorm{\prb{[\realisation_{\ReLU}(\mathscr{f}_2)] \circ [\realisation_{\ReLU}(\mathscr{f}_1)]}(x)-(f_2 \circ f_1)(x) }
\\&\leq \PRbbbb{\sup_{x \in [-R_1,R_1]^{d_1}} \bignorm{\prb{[\realisation_{\ReLU}(\mathscr{f}_2)] \circ [\realisation_{\ReLU}(\mathscr{f}_1)]}(x)-([\realisation_{\ReLU}(\mathscr{f}_2)] \circ f_1)(x) }}
\\&\quad +
\PRbbbb{\sup_{x \in [-R_1,R_1]^{d_1}} \bignorm{\prb{[\realisation_{\ReLU}(\mathscr{f}_2)] \circ f_1}(x)-(f_2 \circ f_1)(x) } }
\\&\leq \PRbbbb{\sup_{x \in [-R_1,R_1]^{d_1}} L_2\bignorm{\pr{\realisation_{\ReLU}(\mathscr{f}_1)}(x)-f_1(x) }}
+
\PRbbbb{\sup_{y \in [-R_2,R_2]^{d_2}} \bignorm{\pr{\realisation_{\ReLU}(\mathscr{f}_2)} (y)-f_2 (y) } }
\\&\leq \tfrac{L_2\varepsilon}{a_1} + \tfrac{\varepsilon}{a_2}
\leq \tfrac{\varepsilon}{2} + \tfrac{\varepsilon}{2}
= \varepsilon
.
\end{split}
\end{equation}
Furthermore, note that
\cref{composition_cost_ReLU:item2} and
\eqref{composition_cost_ReLU:a}
imply that for all
$x,y \in [-R_1,R_1]^{d_1}$
it holds that
\begin{equation}
\label{composition_cost_ReLU:2}
\begin{split}
&\mednorm{(\realisation_{\ReLU}(\mathscr{f}_2 \bullet\mathbb{I}_{d_2}\bullet \mathscr{f}_1))(x) - (\realisation_{\ReLU}(\mathscr{f}_2 \bullet\mathbb{I}_{d_2}\bullet \mathscr{f}_1))(y)}
\\&=
\bignorm{\prb{[\realisation_{\ReLU}(\mathscr{f}_2)] \circ [\realisation_{\ReLU}(\mathscr{f}_1)]}(x)
-\prb{[\realisation_{\ReLU}(\mathscr{f}_2)] \circ [\realisation_{\ReLU}(\mathscr{f}_1)]}(y)}
\\&\leq
L_2\mednorm{(\realisation_{\ReLU}(\mathscr{f}_1))(x)-(\realisation_{\ReLU}(\mathscr{f}_1))(y)} \leq L_2L_1 \mednorm{x-y}.
\end{split}
\end{equation}
Combining this with
\eqref{Cost_Lipschitz_set}
and \eqref{composition_cost_ReLU:1}
ensures that
\begin{equation}
\begin{split}
\label{cost_bounded_by_example}
\CostLip{\ReLU}{A}{f_2 \circ f_1}{L_2L_1}{\varepsilon}
&\leq \paramANN(\mathscr{f}_2 \bullet\mathbb{I}_{d_2}\bullet \mathscr{f}_1)
.
\end{split}
\end{equation}
Moreover, observe that \cref{PropertiesOfCompositions_id_n:Params2} of \cref{Lemma:PropertiesOfCompositions_n3} (applied with $n \curvearrowleft 2$, $\mathscr{f}_1 \curvearrowleft \mathscr{f}_2$, $\mathscr{f}_2 \curvearrowleft \mathscr{f}_1$ in the notation of \cref{Lemma:PropertiesOfCompositions_n3}) and \cref{composition_cost_ReLU:item4} show that
\begin{equation}
\begin{split}
\paramANN(\mathscr{f}_2 \bullet\mathbb{I}_{d_2}\bullet \mathscr{f}_1)
&\leq
2\,\paramANN(\mathscr{f}_1)+2\,\paramANN(\mathscr{f}_2)
\\&\leq
2 \, \CostLipB{\ReLU}{B}{f_1}{L_1}{\tfrac{\varepsilon}{a_1}}
+ 2\prb{\CostLipB{\ReLU}{A}{f_2}{L_2}{\tfrac{\varepsilon}{a_2}}+2d_2(d_2+1)}
\\&=
2\, \CostLipB{\ReLU}{B}{f_1}{L_1}{\tfrac{\varepsilon}{a_1}}
+2 \, \CostLipB{\ReLU}{A}{f_2}{L_2}{\tfrac{\varepsilon}{a_2}}
+4d_2(d_2+1).
\end{split}
\end{equation}
This and \eqref{cost_bounded_by_example} establish \eqref{Prop:composition_cost_ReLU:eq}. The proof of \cref{Prop:composition_cost_ReLU} is thus complete.
\end{proof}
\subsection{Compositions of ANN approximable functions}
\label{Subsection:3.5}
\cfclear
\begingroup
\newcommand{\df}{{m}}
\newcommand{\dfp}{{m}}
\newcommand{\dg}{{n}}
\newcommand{\dgp}{{n}}
\newcommand{\ig}{{o}}
\newcommand{\f}{f}
\newcommand{\g}{g}
\renewcommand{\c}{c}
\newcommand{\C}{C}
\newcommand{\cf}{{\mathfrak f}}
\newcommand{\cg}{{\mathfrak g}}
\renewcommand{\r}{R}
\begin{athm}{lemma}{Lemma:Comp_theory_1}
Let
$\df,\dg,\ig,\c,\C\in\N$,
$\cf,\cg\in(0,\infty)$,
$\eps\in(0,1]$,
let
$\f\colon \R^{\df}\to\R^{\dg}$
and $\g\colon \R^{\dg}\to\R^{\ig}$
be functions,
and assume for all $\r \in \N$, $\delta\in(0,\eps]$ that
$\c=\ceil{\max\{\cf,\cg,2\}}$,
$\C =2^{\c+2}\c^{(2\c^2+\c+1)}$,
$\dg\leq \cf\dfp^\cf$,
$\f([-\r,\r]^\df)\subseteq [-\cf \dfp^{\cf }\!\r^{\cf },\cf \dfp^{\cf }\!\r^{\cf }]^{\dg}$,
$\CostLipB{\ReLU}{[-\r,\r]^\df }{\f|_{[-\r,\r]^\df}}{\cf \dfp^{\cf } \! \r^{\cf }}{\delta}\leq \cf \dfp^{\cf } \! \r^{\cf }\delta^{-\cf }$,
and $\CostLipB{\ReLU}{[-\r,\r]^d }{\g|_{[-\r,\r]^\dg}}{\cg \dgp^{\cg } \! \r^{\cg }}{\delta}\leq \cg \dgp^{\cg } \! \r^{\cg }\delta^{-\cg }$
\cfload.
Then it holds for all $\r \in \N$ that
\begin{equation}
\label{Lemma:Comp_theory_1:statement}
\CostLipA{\ReLU}{[-\r,\r]^d }
{(\g \circ \f)|_{[-\r,\r]^\df}}
{\C \dfp^{\C} \! \r^{\C}}{\varepsilon}\leq \C \dfp^\C \! \r^\C\varepsilon^{-\C}.
\end{equation}
\end{athm}
\begin{aproof}
First, \nobs that
\cref{Lemma:Monotonicity_of_Cost}
ensures that for all
$\r \in \N$,
$\delta\in(0,\eps]$
it holds that
\begin{equation}
\label{Comp_theory_1:a}
\begin{split}
\CostLipA{\ReLU}{[-\r,\r]^\df}{\f|_{[-\r,\r]^\df}}{\c\dfp^\c \! \r^\c}{\delta}
\leq
\CostLipA{\ReLU}{[-\r,\r]^\df}{\f|_{[-\r,\r]^\df}}{\cf\dfp^\cf \! \r^\cf}{\delta}
\leq
\cf \dfp^\cf \! \r^\cf \delta^{-\cf}
\leq
\c \dfp^\c \! \r^\c \delta^{-\c}\ifnocf.
\end{split}
\end{equation}
\cfload[.
In addition, \nobs that
\cref{Lemma:Monotonicity_of_Cost}
implies that for all
$\r \in \N$,
$\delta\in(0,\eps]$
it holds that
\begin{equation}
\label{Comp_theory_1:b}
\begin{split}
\CostLipA{\ReLU}{[-\r,\r]^\dg }{\g|_{[-\r,\r]^\dg}}{\c\dgp^\c \! \r^\c}{\delta}
\leq
\CostLipA{\ReLU}{[-\r,\r]^\dg }{\g|_{[-\r,\r]^\dg}}{\cg\dgp^\cg \! \r^\cg}{\delta}
\leq
\cg \dgp^\cg \! \r^\cg \delta^{-\cg}
\leq
\c \dgp^\c \! \r^\c \delta^{-\c}.
\end{split}
\end{equation}
Next \nobs that
the assumption that $\C = 2^{\c+2}\c^{(2\c^2+\c+1)}$
and the fact that $c\geq 2$
show that
\begin{equation}
\max\{\c^{2\c+2},2\c^2+\c,\c^2+\c\}\leq 4\c^{4\c}\leq\C.
\end{equation}
This
and the fact that $\dg\leq\cf\dfp^\cf\leq\c\dfp^\c$
demonstrate that for all
$\r \in \N$
it holds that
\begin{equation}
\label{Comp_theory_1:1}
\begin{split}
\c\dgp^\c(\c\dfp^\c\!\r^\c)^\c
( \c\dfp^\c \! \r^\c)
\leq
\c(\c\dfp^\c)^\c (\c\dfp^\c\!\r^\c)^\c
( \c\dfp^\c \! \r^\c)
=
\c^{2\c+2}\dfp^{(2\c^2+\c)}\!\r^{(\c^2+\c)}
\leq
\C \dfp^{\C} \! \r^{\C}
.
\end{split}
\end{equation}
\Cref{Lemma:Monotonicity_of_Cost}
hence
ensures that for all
$\r \in \N$
it holds that
\begin{equation}
\label{Comp_theory_1:3}
\begin{split}
\CostLipA{\ReLU}{[-\r,\r]^\df }{(\g \circ \f)|_{[-\r,\r]^{\df}}}
{\C \dfp^{\C} \! \r^{\C}}{\varepsilon}
\leq
\CostLipA{\ReLU}{[-\r,\r]^\df }{(\g \circ \f)|_{[-\r,\r]^\df}}
{\c\dgp^\c (\c\dfp^\c\!\r^\c)^\c (\c\dfp^\c \! \r^\c)}{\varepsilon}.
\end{split}
\end{equation}
This,
the fact that $f([-\r,\r]^{\df})\subseteq[-\cf\dfp^\cf\!\r^\cf,\cf\dfp^\cf\!\r^\cf]^\dg\subseteq[-\c\dfp^\c\!\r^\c,\c\dfp^\c\!\r^\c]^\dg$,
and \cref{Prop:composition_cost_ReLU}
(applied with
$d_1 \curvearrowleft \df$,
$d_2 \curvearrowleft \dg$,
$d_3 \curvearrowleft \ig$,
$\varepsilon \curvearrowleft \eps$,
$L_1 \curvearrowleft \c \dfp^\c \! \r^\c$,
$L_2 \curvearrowleft \c\dgp^\c (\c\dfp^\c\!\r^\c)^\c$,
$R_1 \curvearrowleft \r$,
$R_2 \curvearrowleft \c\dfp^\c\!\r^\c$,
$f_1 \curvearrowleft \f|_{[-\r,\r]^{\df}}$,
$f_2 \curvearrowleft \g|_{[-\c\dfp^\c\!\r^\c,\c\dfp^\c\!\r^\c]^{\dg}}$
for $\r \in \N$
in the notation of \cref{Prop:composition_cost_ReLU})
prove that for all
$\r \in \N$
it holds that
\begin{equation}
\label{Comp_theory_1:4}
\begin{split}
&
\CostLipA{\ReLU}{[-\r,\r]^\df }
{(\g \circ \f)|_{[-\r,\r]^{\df}}}
{\C \dfp^{\C} \! \r^{\C}}{\eps}
\\&\leq
2\,\CostLipB{\ReLU}{}
{f|_{[-\r,\r]^\df}}
{\c\dfp^\c\!\r^\c}
{\tfrac{\eps}{2\c\dgp^\c (\c\dfp^\c\!\r^\c)^\c}}
\\&\quad+2\,\CostLipA{\ReLU}{}
{g|_{[-\c\dfp^\c\!\r^\c ,\c\dfp^\c\!\r^\c ]^{\dg}}}
{\c\dgp^\c (\c\dfp^\c\!\r^\c)^\c}
{\tfrac{\eps}{2}}
+4\dg(\dg+1)
.
\end{split}
\end{equation}
Combining
this
with
\eqref{Comp_theory_1:a}
and \eqref{Comp_theory_1:b}
shows that for all
$\r \in \N$
it holds that
\begin{equation}
\begin{split}
&
\CostLipB{\ReLU}{[-\r,\r]^d }
{(\g \circ \f)|_{[-\r,\r]^{\df}}}
{\C \dfp^{\C} \! \r^{\C}}{\eps}
\\&\leq
2\c\dfp^\c\!\r^\c\PR*{\tfrac{2\c\dgp^\c (\c\dfp^\c\!\r^\c)^\c}{\eps}}^\
+2\c\dgp^\c (\c\dfp^\c\!\r^\c)^\c \PR*{\tfrac{2}{\eps}}^
+4\dg(\dg+1)
\\&=
2\c\dfp^\c\!\r^\c2^\c\c^\c\dgp^{(\c^2)}(\c\dfp^\c\!\r^\c)^{(\c^2)}\eps^{-\c
+2\c\dgp^\c (\c\dfp^\c\!\r^\c)^\c 2^\c\eps^{-\c
+4\dg(\dg+1)
\\&=
2^{\c+1}\c^{(\c^2+\c+1)}\dfp^{(\c^3+\c)}\!\r^{(\c^3+\c)}\dgp^{(\c^2)}\eps^{-\c
+2^{\c+1}\c^{\c+1}\dgp^\c\dfp^{(\c^2)}\!\r^{(\c^2)} \eps^{-\c
+4\dg(\dg+1)
\\&\leq
2^{\c+1}\c^{(\c^2+\c+1)}\dfp^{(\c^3+\c)}\!\r^{(\c^3+\c)}\dgp^{(\c^2)}\eps^{-\c
+2^{\c+1}\c^{\c+1}\dgp^\c\dfp^{(\c^2)}\!\r^{(\c^2)} \eps^{-\c
+8\dgp^2
\\&\leq
2^{\c+2}\c^{(\c^2+\c+1)}\dfp^{(\c^3+\c)}\!\r^{(\c^3+\c)}\dgp^{(\c^2)}\eps^{-\c}
.
\end{split}
\end{equation}
The fact that $\dg\leq \cf\dfp^\cf\leq \c\dfp^\c$
therefore
implies that for all
$\r\in\N$
it holds that
\begin{equation}
\begin{split}
\CostLipB{\ReLU}{[-\r,\r]^d }
{(\g \circ \f)|_{[-\r,\r]^{\df}}}
{\C \dfp^{\C} \! \r^{\C}}{\eps}
&\leq
2^{\c+2}\c^{(\c^2+\c+1)}\dfp^{(\c^3+\c)}\!\r^{(\c^3+\c)}(\c\dfp^\c)^{(\c^2)}\eps^{-\c}
\\&=
2^{\c+2}\c^{(2\c^2+\c+1)}\dfp^{(2\c^3+\c)}\!\r^{(\c^3+\c)}\eps^{-\c}
.
\end{split}
\end{equation}
Combining
this
with
the fact that
\begin{equation}
\max\{2^{(\c+2)}\c^{(2\c^2+\c+1)},2\c^3+\c,\c^3+\c,\c\}
=
2^{\c+2}\c^{(2\c^2+\c+1)}
=
\C
\end{equation}
establishes that for all
$\r \in \N$
it holds that
\begin{equation}
\CostLipB{\ReLU}{[-\r,\r]^d }
{(\g \circ \f)|_{[-\r,\r]^{\df}}}
{\C \dfp^{\C} \! \r^{\C}}{\eps}
\leq
\C\dfp^\C\!\r^\C\eps^{-\C}
.
\end{equation}
\finishproofthis
\end{aproof}
\endgroup
\begingroup
\newcommand{\const}{c}
\cfclear
\begin{athm}{lemma}{lemma:catching_image}
Let $f,g \in C\prb{\bigcup_{d \in \N}\R^d, \bigcup_{d \in \N}\R^d}$, $\const , \fconstant ,\gconstant \in(0,\infty)$ satisfy for all $d,\radius\in\N$ that
$f([-R,R]^d)\subseteq [-\fconstant d^{\fconstant }\!\radius^{\fconstant },\fconstant d^{\fconstant }\!\radius^{\fconstant }]^{\imdim{f}{d}}$,
$g([-R,R]^d)\subseteq [-\gconstant d^{\gconstant }\!\radius^{\gconstant },\gconstant d^{\gconstant }\!\radius^{\gconstant }]^{\imdim{g}{d}}$,
$\imdim{f}{d}\leq \fconstant d^{\fconstant }$,
$\imdim{g}{d}\leq \gconstant d^{\gconstant }$,
and $\const =\max\{2\gconstant\ceil{\fconstant },\allowbreak \gconstant \ceil{\fconstant }^{2\gconstant } \}$ \cfload.
Then it holds for all $d,\radius\in\N$ that
$\imdim{g\circ f}{d}\leq \const d^{\const }$ and
$(g\circ f)([-R,R]^d)\subseteq[-\const d^\const \!\radius^\const ,\const d^\const \!\radius^\const ]^{\imdim{g\circ f}{d}}$.
\end{athm}
\begin{proof}
[Proof of \cref{lemma:catching_image}]
Note that the fact that for all $x\in[0,\infty)$ it holds that $x\leq\ceil{x}$ and the assumption that for all $d,\radius\in\N$ it holds that $f([-R,R]^d)\subseteq [-\fconstant d^{\fconstant }\!\radius^{\fconstant },\fconstant d^{\fconstant }\!\radius^{\fconstant }]^{\imdim{f}{d}}$
show that for all $d,\radius\in\N$ it holds that
\begin{equation}
\begin{split}
\label{Lemma:catching_image:eq2}
(g\circ f)([-R,R]^d)&\subseteq g\prb{\PRb{-\ceil{\fconstant d^{\fconstant }\!\radius^{\fconstant }},\ceil{\fconstant d^{\fconstant }\!\radius^{\fconstant }}}^{\imdim{f}{d}}}
.
\end{split}
\end{equation}
This, the assumption that for all $d,\radius\in\N$ it holds that $g([-R,R]^d)\subseteq [-\gconstant d^{\gconstant }\!\radius^{\gconstant },\gconstant d^{\gconstant }\!\radius^{\gconstant }]^{\imdim{g}{d}}$, and the fact that for all $d\in\N$ it holds that $\imdim{g}{\imdim{f}{d}}=\imdim{g\circ f}{d}$ demonstrate that for all $d,\radius\in\N$ it holds that
\begin{equation}
\begin{split}
\label{Lemma:catching_image:eq3}
(g\circ f)([-R,R]^d)&\subseteq \PRb{-\gconstant (\imdim{f}{d})^{\gconstant }\ceil{\fconstant d^{\fconstant }\!\radius^{\fconstant }}^{\gconstant },\gconstant (\imdim{f}{d})^{\gconstant }\ceil{\fconstant d^{\fconstant }\!\radius^{\fconstant }}^{\gconstant }}^{\imdim{g\circ f}{d}}
.
\end{split}
\end{equation}
In addition, observe that
the assumption that for all $d\in\N$ it holds that $\imdim{f}{d}\leq\fconstant d^{\fconstant }$,
the fact that for all $x,y\in(0,\infty)$ it holds that $\ceil{xy}\leq\ceil{\ceil{x}\ceil{y}}=\ceil{x}\ceil{y}$,
and the fact that for all $n\in\N$, $x\in(0,\infty)$ it holds that $\ceil{n^x}=\ceil{n^{\ceil{x}}n^{x-\ceil{x}}}\leq \ceil{n^{\ceil{x}}}=n^{\ceil{x}}$ ensure that for all $d,\radius\in\N$ it holds that
\begin{equation}
\begin{split}
\gconstant (\imdim{f}{d})^{\gconstant }\ceil{\fconstant d^{\fconstant }\!\radius^{\fconstant }}^{\gconstant }
\leq &~\gconstant (\fconstant d^{\fconstant })^{\gconstant }\ceil{\fconstant }^{\gconstant }\ceil{d^{\fconstant }}^{\gconstant }\ceil{\radius^{\fconstant }}^{\gconstant }\\
\leq &~\gconstant (\ceil{\fconstant} d^{\fconstant })^{\gconstant }\ceil{\fconstant }^{\gconstant }d^{\ceil{\fconstant }\gconstant }\radius^{\ceil{\fconstant }\gconstant }\\
\leq &~\gconstant \ceil{\fconstant }^{2\gconstant }d^{2\ceil{\fconstant }\gconstant}\radius^{\ceil{\fconstant }\gconstant }\\
\leq &~\const d^{\const }\!\radius^{\const }.
\end{split}
\end{equation}
This and \eqref{Lemma:catching_image:eq3} imply that for all $d,\radius\in\N$ it holds that
\begin{equation}
\label{lemma:catching_image:eq1}
\begin{split}
(g\circ f)([-R,R]^d)&\subseteq [-\const d^{\const }\!\radius^{\const },\const d^{\const }\!\radius^{\const }]^{\imdim{g\circ f}{d}}.
\end{split}
\end{equation}
Furthermore, note that the assumption that for all $d\in\N$ it holds that $\imdim{f}{d}\leq\fconstant d^{\fconstant }$ and $\imdim{g}{d}\leq\gconstant d^{\gconstant }$ demonstrates that for all $d\in\N$ it holds that
\begin{equation}
\begin{split}
\imdim{g\circ f}{d}=\imdim{g}{\imdim{f}{d}}
\leq \gconstant (\imdim{f}{d})^{\gconstant }
\leq \gconstant (\fconstant d^{\fconstant })^{\gconstant }
= \gconstant \fconstant^{\gconstant }d^{\fconstant\gconstant }
\leq\const d^{\const }.
\end{split}
\end{equation}
The proof of \cref{lemma:catching_image} is thus complete.
\end{proof}
\endgroup
\begingroup
\newcommand{\const}{C}
\newcommand{\cONST}{c}
\newcommand{\constf}{\mathfrak f}
\newcommand{\constg}{\mathfrak g}
\cfclear
\begin{athm}{lemma}{lemma:D_closed}
Let $f,g\in\Dapprox$ \cfload. Then $(g\circ f)\in\Dapprox$.
\end{athm}
\begin{aproof}
\Nobs that
the assumption that $f,g\in\Dapprox$
shows that there exist
$\constf ,\constg \in[0,\infty)$
which satisfy that
\begin{enumerate}[(I)]
\item\llabel{it1}
for all
$d,R\in\N$
it holds that
$f([-R,R]^d)\subseteq[-\constf d^{\constf }\!R^{\constf },\constf d^{\constf }\!R^{\constf }]^{\imdim fd}$
and $g([-R,R]^d)\subseteq[-\constg d^{\constg }\!R^{\constg },\constg d^{\constg }\!R^{\constg }]^{\imdim gd}$,
\item\llabel{it2}
for all
$d\in\N$
it holds that
$\imdim fd\leq \constf d^{\constf }$
and $\imdim gd\leq \constg d^{\constg }$,
and
\item\llabel{it3}
for all
$d,R\in\N$,
$\eps\in(0,1]$
it holds that
\begin{equation}
\begin{split}
\CostLipA{\ReLU}{}{\pr{[-R,R]^d\ni x\mapsto f(x)\in\R^{\imdim fd}}}{\constf d^{\constf }\!R^{\constf }}{\eps}
&\leq
\constf d^{\constf }\!R^{\constf }\eps^{-\constf }
\\\text{and}\qquad
\CostLipA{\ReLU}{}{\pr{[-R,R]^d\ni x\mapsto g(x)\in\R^{\imdim gd}}}{\constg d^{\constg }\!R^{\constg }}{\eps}
&\leq
\constg d^{\constg }\!R^{\constg }\eps^{-\constg }
\end{split}
\end{equation}
\end{enumerate}
\cfload.
\Nobs that
\lref{it1},
\lref{it2},
\lref{it3}, and
\cref{Lemma:Comp_theory_1}
(applied with
$m\is d$,
$n\is \imdim fd$,
$o\is \imdim{g\circ f}d$,
$\mathfrak f\is \constf $,
$\mathfrak g\is \constg $,
$\eps\is\eps$,
$f\is(\R^d\ni x\mapsto f(x)\in\R^{\imdim fd})$,
$g\is(\R^{\imdim fd}\ni x\mapsto g(x)\in\R^{\imdim {g\circ f}d})$
for $d\in\N$, $\eps\in(0,1]$
in the notation of \cref{Lemma:Comp_theory_1})
prove that there exists
$\cONST\in(0,\infty)$
such that for all
$d,R\in\N$,
$\eps\in(0,1]$
it holds that
\begin{equation}
\eqlabel{1}
\CostLipA{\ReLU}{}
{\pr{[-R,R]^d\ni x\mapsto (g\circ f)(x)\in \R^{\imdim{g\circ f}d}}}
{\cONST d^\cONST\!R^\cONST}
\eps
\leq
\cONST d^\cONST\!R^\cONST\eps^{-\cONST}
.
\end{equation}
Moreover, \nobs that
\cref{lemma:catching_image}
ensures that there exists
$\const \in(0,\infty)$
such that for all
$d,\radius\in\N$
it holds that
$\imdim{g\circ f}{d}\leq\const d^{\const }$
and $(g\circ f)([-R,R]^d)\subseteq[-\const d^\const \!\radius^\const ,\const d^\const \!\radius^\const ]^{\imdim{g\circ f}{d}}$.
Combining
this,
\eqqref{1}, and
\cref{Lemma:Monotonicity_of_Cost}
with
\eqref{polyD_mult:1}
establishes that
$(g\circ f)\in\Dapprox$.
This completes the proof of \cref{lemma:D_closed}.
\end{aproof}
\endgroup
\subsection{Parallelizations of ANN approximable functions}
\label{Subsection:3.6}
\cfclear
\begin{athm}{lemma}{lemma:Lipschitz_error}
Let $L \in \R$, $d \in \N$, $m_1,m_2,\ldots,m_d\in\N$, let $g_i \in C(\R^{m_i},\R)$, $i \in \{1, 2, \ldots, d\}$, satisfy for all $i \in \{1, 2, \ldots, d\}$, $x,y\in\R^{m_i}$ that $\vass{g_i(x) - g_i(y)} \leq L\norm{x - y}$, and let $f \in C\prb{\R^{\PR{\sum_{i=1}^dm_i}},\R^d}$ satisfy for all $x=(x_1,x_2,\ldots,x_d)\in\prb{\bigtimes_{i=1}^d\R^{m_i}}$ that $f(x_1,x_2,\ldots,x_d)=(g_1(x_1), g_2(x_2), \ldots, g_d(x_d))$.
Then it holds for all $x,y\in\R^{\PR{\sum_{i=1}^dm_i}}$ that
\begin{equation}
\mednorm{f(x) - f(y)} \leq L \mednorm{x-y}
\end{equation}
\cfload.
\end{athm}
\begin{aproof}
Observe that for all $x=(x_1,x_2,\ldots,x_d)$, $y=(y_1,y_2,\ldots,y_d)\in\prb{\bigtimes_{i=1}^d\R^{m_i}}$ it holds that
\begin{equation}
\begin{split}
\mednorm{f(x) - f(y)}
= \PR*{\ssum_{i=1}^d \vass*{g_i(x_i) - g_i(y_i)}^2}^{\frac{1}{2}}
\leq
L \PR*{\ssum_{i=1}^d \norm{x_i-y_i}^2}^{\frac{1}{2}}
\leq L \mednorm{x - y}.
\end{split}
\end{equation}
This completes the proof of \cref{lemma:Lipschitz_error}.
\end{aproof}
\begingroup
\newcommand{\const}{\kappa}
\newcommand{\cONST}{C}
\cfclear
\begin{athm}{lemma}{Lemma:Comp_theory_2.1}
Let
$d,\radius\in\N$,
$c,\const ,\cONST ,r \in [0, \infty)$,
$\varepsilon \in (0,1]$,
$f_1,f_2,\ldots,f_d \in \Capprox{c}{r}$,
$F \in C(\R^d, \R^d)$
satisfy for all
$x = (x_1, x_2, \ldots, x_d) \in \R^d$
that $F(x) = (f_1(x_1), f_2(x_2), \ldots, f_d(x_d))$,
$\const =\max\{c,cr^c\}$, and
$\cONST = \max\{61\const ^2, 2\const +2\}$
\cfload.
Then
\begin{equation}
\begin{split}
\CostLipA{\ReLU}{\indicator{[-\radius,\radius]^d}}{F|_{[-\radius,\radius]^d}}{\cONST d^{\cONST } \! \radius^{\cONST }}{\varepsilon}
\leq
\cONST d^{\cONST } \! \radius^{\cONST } \varepsilon^{-\cONST }
\end{split}
\end{equation}
\cfout.
\end{athm}
\begin{aproof}
\Nobs that \cref{cor:polyC} ensures that
$\Capprox{c}{r}\subseteq \Capprox{\const }{1}$.
The assumption that $f_1,f_2,\ldots,f_d \in \Capprox{c}{r}$
therefore
implies that for all
$m\in\{1,2,\ldots,d\}$
it holds that
\begin{equation}
\CostLipB{\ReLU}{\indicator{[-\radius,\radius]^d}}{(f_m)|_{[-\radius,\radius]}}{\const \radius^{\const }}{d^{-\frac{1}{2}}\varepsilon}
\leq
\const \radius^{\const }\prb{d^{-\frac{1}{2}}\varepsilon}^{-\const }
=
\const \radius^{\const }d^{\frac{\const }{2}}\varepsilon^{-\const }
\ifnocf.
\end{equation}
\cfload[.
This
and \cref{cor_cost_of_Lip_approx_set_equivalence}
assure that there exist
$\mathscr{f}_1,\mathscr{f}_2,\ldots,\mathscr{f}_d\in\ANNs$
which satisfy that
\begin{enumerate}[(I)]
\item \label{Comp_theory_3:item_1}
it holds for all
$m\in\{1,2,\ldots,d\}$
that
$\realisation_{\ReLU}( \mathscr{f}_m) \in C(\R,\R)$,
\item \label{Comp_theory_3:item_2}
it holds for all
$m\in\{1,2,\ldots,d\}$,
$x \in [-\radius,\radius]$
that
$\vass{f_m (x) - (\realisation_{\ReLU}(\mathscr{f}_m))(x)}
\leq d^{-\frac{1}{2}}\varepsilon$,
\item \label{Comp_theory_3:item_3}
it holds for all
$m\in\{1,2,\ldots,d\}$,
$x,y \in [-\radius,\radius]$
that
\begin{equation}
\vass{(\realisation_{\ReLU}( \mathscr{f}_m ))(x) - (\realisation_{\ReLU}( \mathscr{f}_m ))(y)} \leq \const \radius^\const \vass{x-y},
\end{equation}
and
\item \label{Comp_theory_3:item_4}
it holds for all
$m\in\{1,2,\ldots,d\}$
that
$\paramANN(\mathscr{f}_m) \leq \const \radius^\const d^{\frac{\const }{2}}\varepsilon^{-\const }$
\end{enumerate}
\cfload.
Next let $\mathscr{g} \in \ANNs$ satisfy
\begin{equation}
\label{Comp_theory_3:1}
\mathscr{g}
=
\parallelization_{d, (\ReLUidANN{1},\ReLUidANN{1},\ldots,\ReLUidANN{1})}\prb{\mathscr{f}_{1}, \mathscr{f}_{2}, \ldots, \mathscr{f}_{d} }
\end{equation}
\cfload.
\Nobs that
\eqref{Comp_theory_3:1},
\cref{Comp_theory_3:item_1},
\cref{Comp_theory_3:item_2},
\cref{Prop:identity_representation}, and
\cref{Lemma:PropertiesOfParallelizationRealization}
ensure that for all
$x=(x_1,x_2,\ldots,x_d)\in[-\radius,\radius]^d$
it holds that
$\realisation_{\ReLU}(\mathscr{g}) \in C(\R^d,\R^d)$ and
\begin{equation}
\label{Lemma:Comp_theory_2.1:prop:approx}
\begin{split}
\norm{F(x) - \pr{\realisation_{\ReLU}(\mathscr{g})}(x)}
&=
\PR*{\ssum_{m=1}^{d}\vass*{f_m(x_m) - \pr{\realisation_{\ReLU}(\mathscr{f}_m)}(x_m)}^2}^\frac{1}{2}
\leq
\PRb{d\prb{d^{-1}\varepsilon^2}}^\frac{1}{2}
=
\varepsilon
\end{split}
\end{equation}
\cfload.
In addition, \nobs that
\eqref{Comp_theory_3:1},
\cref{Comp_theory_3:item_3},
\cref{Lemma:PropertiesOfParallelizationRealization}, and
\cref{lemma:Lipschitz_error}
demonstrate that for all
$x,y\in[-\radius,\radius]^d$
it holds that
\begin{equation}
\label{Lemma:Comp_theory_2.1:prop:lip}
\norm{\pr{\realisation_{\ReLU}(\mathscr{g})}(x) - \pr{\realisation_{\ReLU}(\mathscr{g})}(y)}
\leq \const \radius^\const \norm{x-y}.
\end{equation}
In the next step, \nobs that
\cref{Prop:identity_representation}
implies that
$\dims(\ReLUidANN{1})=(1,2,1)$.
Combining
this,
\eqref{Comp_theory_3:1}, and
\cref{Comp_theory_3:item_4}
with
\cref{Lemma:PropertiesOfParallelization}
(applied with
$n \curvearrowleft d$,
$(\mathscr{g}_1,\mathscr{g}_2,\allowbreak\dots, \mathscr{g}_n) \curvearrowleft (\ReLUidANN{1},\ReLUidANN{1},\ldots,\ReLUidANN{1})$,
$\mathscr{f} \curvearrowleft (\mathscr{f}_1,\mathscr{f}_2,\allowbreak\dots, \mathscr{f}_d)$
in the notation of \cref{Lemma:PropertiesOfParallelization}), and
the fact that $\max_{m\in\{1,2,\dots,d\}}\lengthANN(\mathscr{f}_m)\leq \max_{m\in\{1,2,\dots,d\}}\param(\mathscr{f}_m)$
shows that
\begin{equation}
\begin{split}
\label{Comp_theory_3:2}
\paramANN\pr{\mathscr{g}}
&\leq
\frac{1}{2} \PRbbb{\ssum_{j=1}^d
\prb{2\,\paramANN(\mathscr{f}_j)
+
\prb{2\PRb{\max\nolimits_{m\in\{1,2,\dots,d\}} \lengthANN(\mathscr f_m)}+1}(2+1)}}^2
\\&\leq
\tfrac{1}{2}
\prb{2d \const \radius^\const d^{\frac{\const }{2}} \varepsilon^{-\const }
+9d \const \radius^\const d^{\frac{\const }{2}} \varepsilon^{-\const }
}^{2}
\\&\leq
61\const ^2 d^{2+\const } \! \radius^{2\const } \varepsilon^{-2\const }\leq
\cONST d^{\cONST } \! \radius^{\cONST } \varepsilon^{-\cONST }.
\end{split}
\end{equation}
This, \eqref{Lemma:Comp_theory_2.1:prop:approx}, \eqref{Lemma:Comp_theory_2.1:prop:lip}, \cref{Lemma:Monotonicity_of_Cost}, and the fact that $\cONST \geq\const $ establish that
\begin{equation}
\begin{split}
&
\CostLipA{\ReLU}{\indicator{[-\radius,\radius]^d}}{F|_{[-\radius,\radius]^d}}{\cONST d^{\cONST } \! \radius^{\cONST }}{\varepsilon}
\leq
\CostLipB{\ReLU}{\indicator{[-\radius,\radius]^d}}{F|_{[-\radius,\radius]^d}}{\const \radius^\const }{\varepsilon}
\leq
\cONST d^{\cONST } \! \radius^{\cONST } \varepsilon^{-\cONST }.
\end{split}
\end{equation}
This completes the proof of \cref{Lemma:Comp_theory_2.1}\cfload.
\end{aproof}
\endgroup
\begingroup
\newcommand{\const}{C}
\cfclear
\begin{athm}{lemma}{Lemma:Comp_theory_3}
Let
$c,r \in [0, \infty)$,
$(f_n)_{n\in \N} \subseteq \Capprox{c}{r}$,
$F \in C\prb{\bigcup_{d \in \N}\R^d, \bigcup_{d \in \N}\R^d}$
satisfy for all
$d,\radius \in \N$,
$x = (x_1, x_2, \ldots, x_d) \in \R^d$,
$y\in[-\radius,\radius]$
that
$F(x) = (f_1(x_1), f_2(x_2), \ldots, f_d(x_d))$
and $\vass{f_{d}(y)} \leq cd^c\!R^c$
\cfload.
Then $F \in \Dapprox$
\cfout.
\end{athm}
\begin{proof}[Proof of \cref{Lemma:Comp_theory_3}]
\setnote
\Nobs that
\cref{Lemma:Comp_theory_2.1}
ensures that there exists
$\const \in[\max\{1,c\},\infty)$
which satisfies that for all
$d, \radius \in \N$,
$\varepsilon \in (0,1]$
it holds that
\begin{equation}
\label{Lemma:Comp_theory_3:eq1}
\begin{split}
\CostLipA{\ReLU}{\indicator{[-\radius,\radius]^d}}{\pr{[-\radius,\radius]^d \ni x \mapsto F(x) \in \R^{d}}}{\const d^{\const } \! \radius^{\const }}{\varepsilon}
\leq
\const d^{\const } \! \radius^{\const } \varepsilon^{-\const }
\end{split}
\end{equation}
\cfload.
Moreover, \nobs that
the assumption that
for all
$d,\radius \in \N$,
$y\in[-R,R]$
it holds that
$\vass{f_{d}(y)} \leq cd^c\!R^c$
and the fact that
$c\leq \const $
imply that for all
$d,\radius \in \N$
it holds that
\begin{equation}
\label{Lemma:Comp_theory_3:eq2}
F([-\radius,\radius]^d)
\subseteq
\prb{\sssbigtimes_{k=1}^{d}[-ck^c\! R^c,ck^c\! R^c]}
\subseteq
[-cd^c\! R^c,cd^c\! R^c]^d
\subseteq
[-\const d^{\const } \! \radius^{\const },\const d^{\const } \! \radius^{\const }]^d
.
\end{equation}
Furthermore, \nobs that
the fact that
$\const \geq 1$
implies that for all
$d\in\N$
it holds that
$\imdim{F}{d}=d\leq \const d^\const $
\cfload.
This,
\eqref{Lemma:Comp_theory_3:eq1}, and
\eqref{Lemma:Comp_theory_3:eq2}
ensure that
$F \in \Dapprox$ \cfload.
This completes the proof of \cref{Lemma:Comp_theory_3}.
\end{proof}
\endgroup
\section{ANN approximations for Lipschitz continuous functions}
\label{Section:4}
In this section we present results on the approximation of
locally Lipschitz continuous functions from $\R$ to $\R$
by ANNs with the ReLU activation function.
The results in this section are essentially well-known
and only for completeness we also include detailed proofs here.
In \cref{Subsection:4.1} below we introduce ANNs for
piecewise linear interpolation
and in \cref{Subsection:4.3} we use these ANNs
to show that locally Lipschitz continuous functions
from $\R$ to $\R$ belong to the approximation spaces
introduced in the previous section.
\subsection{Linear interpolations with ANNs}
\label{Subsection:4.1}
\cfclear
\begin{definition}[Interpolating ANNs]
\label{def:interpolatingDNN}
\cfconsiderloaded{def:interpolatingANN}
Let
$N \in \N$,
$h_0,h_1,\ldots, h_{N},\gp_0,\gp_1,\ldots,\gp_N \in \R$
satisfy $\gp_0<\gp_1<\ldots<\gp_N$.
Then we denote by
$\interpolatingNN{h_0,h_1,\ldots, h_{N}}{\gp_0,\gp_1,\ldots,\gp_N } \in \ANNs$ the \cfadd{def:neuralnetwork}ANN given by
\begin{equation}
\label{def:interpolatingDNN:eq1}
\begin{split}
&\interpolatingNN{h_0,h_1,\ldots, h_{N}}{\gp_0,\gp_1,\ldots,\gp_N } =
\mleft(\!
\pr*{\!
\begin{pmatrix} 1\\1\\\vdots\\1 \end{pmatrix},\!
\begin{pmatrix} -\gp_0\\-\gp_1\\\vdots\\-\gp_N \end{pmatrix}
\!}\mright.
,\\&\qquad\mleft.\vphantom{\begin{pmatrix} 1\\1\\\vdots\\1 \end{pmatrix}}
\pr*{
\begin{pmatrix}
\tfrac{h_1-h_0}{\gp_1 - \gp_0} &
\tfrac{h_2-h_1}{\gp_2-\gp_1} - \tfrac{h_1-h_0}{\gp_1-\gp_0} &
\ldots &
\tfrac{h_N-h_{N-1}}{\gp_N-\gp_{N-1}} - \tfrac{h_{N-1}-h_{N-2}}{\gp_{N-1}-\gp_{N-2}} &
-\tfrac{h_N-h_{N-1}}{\gp_N-\gp_{N-1}}
\end{pmatrix}
,
h_0
}
\!\mright) \\
& \in (\R^{(N+1) \times 1} \times \R^{N+1}) \times (\R^{1 \times (N+1)} \times \R)
\end{split}
\end{equation}
\cfload.
\end{definition}
\cfclear
\begin{athm}{lemma}{DNN_interpolation}
Let
$N \in \N$,
$h_0,h_1,\ldots, h_{N},\gp_0,\gp_1,\ldots,\gp_N \in \R$
satisfy $\gp_0<\gp_1<\ldots\allowbreak<\gp_N$.
Then
\begin{enumerate}[(i)]
\item \label{DNN_interpolation:item1}
it holds that
$\dims\prb{\interpolatingNN{h_0,h_1,\ldots, h_{N}}{\gp_0,\gp_1,\ldots,\gp_N } }
=
(1, N+1, 1)$,
\item \label{DNN_interpolation:realization}
it holds that
$\functionReLUANN\prb{\interpolatingNN{h_0,h_1,\ldots, h_{N}}{\gp_0,\gp_1,\ldots,\gp_N } }
\in C(\R,\R)$,
\item \label{DNN_interpolation:item2}
it holds for all
$n\in \{0, 1, \ldots, N\}$
that $\prb{ \functionReLUANN \prb{ \interpolatingNN{h_0,h_1,\ldots, h_{N}}{\gp_0,\gp_1,\ldots,\gp_N }}} (\gp_n)=h_n$,
\item \label{DNN_interpolation:item3}
it holds for all
$n\in \{0, 1, \ldots, N-1 \}$, $x \in (-\infty,\xi_0]$, $y\in\PR{\xi_n,\xi_{n+1}}$, $z\in[\xi_N,\infty)$
that $\prb{ \functionReLUANN \prb{\interpolatingNN{h_0,h_1,\ldots, h_{N}}{\gp_0,\gp_1,\ldots,\gp_N }}}(x)=h_0$, $\prb{ \functionReLUANN \prb{\interpolatingNN{h_0,h_1,\ldots, h_{N}}{\gp_0,\gp_1,\ldots,\gp_N }}}(z)=h_N$, and
\begin{equation}
\prb{ \functionReLUANN \prb{\interpolatingNN{h_0,h_1,\ldots, h_{N}}{\gp_0,\gp_1,\ldots,\gp_N }}}(y)=h_n + \prb{\tfrac{h_{n+1} - h_n}{\gp_{n+1} - \gp_n} }(y - \gp_n),\quad
\end{equation}
\item \label{DNN_interpolation:item4}
it holds for all
$n\in \{ 1,2, \ldots, N \}$, $x \in [\gp_{n-1}, \gp_n]$
that
\begin{equation}
\prb{ \functionReLUANN \prb{ \interpolatingNN{h_0,h_1,\ldots, h_{N}}{\gp_0,\gp_1,\ldots,\gp_N }}}(x) \in \PRb{ \min\{h_{n-1}, h_n\}, \max\{h_{n-1}, h_n\} },
\end{equation}
and
\item \label{DNN_interpolation:item5}
it holds for all
$x \in \R$
that
\begin{equation}
\prb{ \functionReLUANN \prb{ \interpolatingNN{h_0,h_1,\ldots, h_{N}}{\gp_0,\gp_1,\ldots,\gp_N } }}(x)
\in \PRb{ \min\nolimits_{n \in \{0, 1, \ldots, N \}} h_n, \max\nolimits_{n \in \{0, 1, \ldots, N \}} h_n }
\end{equation}
\end{enumerate}
\cfout.
\end{athm}
\begin{proof}[Proof of \cref{DNN_interpolation}]
Throughout this proof
let $c_n \in \R$, $n\in \{0, 1, \ldots, N\}$, satisfy
for all
$
n\in \N\cap(0,N)$ that $c_0=\tfrac{h_1-h_0}{\gp_1 - \gp_0}$, $c_N=-\tfrac{h_N-h_{N-1}}{\gp_N-\gp_{N-1}}$, and
\begin{equation}
c_n= \tfrac{h_{n+1}-h_n}{\gp_{n+1}-\gp_n} - \tfrac{h_n-h_{n-1}}{\gp_n-\gp_{n-1}}
\ifnocf.
\end{equation}
\cfload[.
Observe that
\cref{def:interpolatingDNN:eq1}
implies
\cref{DNN_interpolation:item1}.
Note that
\cref{ANNrealization:ass2} and
\eqref{def:interpolatingDNN:eq1}
prove that for all
$x \in \R$
it holds that
$\functionReLUANN\prb{\interpolatingNN{h_0,h_1,\ldots, h_{N}}{\gp_0,\gp_1,\ldots,\gp_N } }
\in C(\R,\R)$
and
\begin{equation}
\begin{split}
\label{DNN_interpolation_explicit:eq}
\prb{\functionReLUANN \prb{\interpolatingNN{h_0,h_1,\ldots, h_{N}}{\gp_0,\gp_1,\ldots,\gp_N }}}(x)
&=
\begin{pmatrix}
c_0&c_1&\ldots&c_N
\end{pmatrix}
\begin{pmatrix}
\max\{x-\gp_0,0\}\\
\max\{x-\gp_1,0\}\\
\vdots\\
\max\{x-\gp_N,0\}
\end{pmatrix}
+
h_0
\\&=
h_0+\sum_{k=0}^{N}c_k\max\{x-\gp_k,0\}\ifnocf.
\end{split}
\end{equation}
\cfload[.
Hence,
we obtain
\cref{DNN_interpolation:realization}.
This and
the assumption that
$\xi_0<\xi_1<\ldots<\xi_N$
ensure that for all
$x \in (-\infty,\gp_0]$
it holds that
\begin{equation}
\label{DNN_interpolation:eq1}
\begin{split}
\prb{\functionReLUANN \prb{\interpolatingNN{h_0,h_1,\ldots, h_{N}}{\gp_0,\gp_1,\ldots,\gp_N }}}(x)
=
h_0+0
=
h_0.
\end{split}
\end{equation}
In addition, observe that the assumption that $\xi_0<\xi_1<\ldots<\xi_N$ and the fact that
for all
$n\in \{0, 1, \ldots, N-1\}$ it holds that $\sum_{k=0}^{n}c_k = \tfrac{h_{n+1}-h_n}{\gp_{n+1}-\gp_n}$
show that for all
$n\in \{0, 1, \ldots, N-1 \}$,
$x\in [\gp_n,\gp_{n+1}]$
it holds that
\begin{equation}
\label{DNN_interpolation:eq2}
\begin{split}
&
\prb{\functionReLUANN \prb{\interpolatingNN{h_0,h_1,\ldots,h_{N}}{\gp_0,\gp_1,\ldots,\gp_N }}}(x)
-
\prb{\functionReLUANN \prb{\interpolatingNN{h_0,h_1,\ldots, h_{N}}{\gp_0,\gp_1,\ldots,\gp_N }}}(\gp_n)
\\&=
\sum_{k=0}^{N}c_k\PRb{\max\{x-\gp_k,0\}-\max\{\gp_{n}-\gp_k,0\}}
\\&=
\sum_{k=0}^{n}c_k \PR{(x-\gp_k)-(\gp_{n}-\gp_k)}
\\&=
\sum_{k=0}^{n}c_k (x-\gp_{n})
=
\prb{\tfrac{h_{n+1}-h_n}{\gp_{n+1}-\gp_n}}(x-\gp_n)
.
\end{split}
\end{equation}
This and \eqref{DNN_interpolation:eq1} demonstrate that for all
$x\in [\gp_0,\gp_{1}]$
it holds that
\begin{equation}
\label{DNN_interpolation:eqn0}
\begin{split}
& \prb{\functionReLUANN \prb{\interpolatingNN{h_0,h_1,\ldots, h_{N}}{\gp_0,\gp_1,\ldots,\gp_N }}}(x)\\
&=\prb{\functionReLUANN \prb{\interpolatingNN{h_0,h_1,\ldots, h_{N}}{\gp_0,\gp_1,\ldots,\gp_N }}}(\gp_0)
+
\prb{\functionReLUANN \prb{\interpolatingNN{h_0,h_1,\ldots, h_{N}}{\gp_0,\gp_1,\ldots,\gp_N }}}(x) - \prb{\functionReLUANN \prb{\interpolatingNN{h_0,h_1,\ldots, h_{N}}{\gp_0,\gp_1,\ldots,\gp_N }}}(\gp_0) \\
&=h_0
+
\prb{\tfrac{h_{1}-h_0}{\gp_{1}-\gp_0}}(x-\gp_0)\ifnocf.
\end{split}
\end{equation}
Moreover, note that \eqref{DNN_interpolation:eq2} implies that
for all
$n \in \N\cap(0,N)$,
$x\in [\gp_n,\gp_{n+1}]$
with
$\forall \, y\in [\gp_{n-1},\gp_{n}] \colon\allowbreak
\prb{ \functionReLUANN \allowbreak \prb{\interpolatingNN{h_0,h_1,\ldots, h_{N}}{\gp_0,\gp_1,\ldots,\gp_N }}}(y)
\allowbreak =
h_{n-1} + \prb{\tfrac{h_{n} - h_{n-1}}{\gp_{n} - \gp_{n-1}} }(y- \gp_{n-1})
$
it holds that
\begin{equation}
\begin{split}
\label{DNN_interpolation:eqinductionstep}
&\prb{\functionReLUANN \prb{\interpolatingNN{h_0,h_1,\ldots, h_{N}}{\gp_0,\gp_1,\ldots,\gp_N }}}(x)
\\&=
\prb{\functionReLUANN \prb{\interpolatingNN{h_0,h_1,\ldots, h_{N}}{\gp_0,\gp_1,\ldots,\gp_N }}}(\gp_n)
+
\prb{\functionReLUANN \prb{\interpolatingNN{h_0,h_1,\ldots, h_{N}}{\gp_0,\gp_1,\ldots,\gp_N }}}(x) - \prb{\functionReLUANN \prb{\interpolatingNN{h_0,h_1,\ldots, h_{N}}{\gp_0,\gp_1,\ldots,\gp_N }}}(\gp_n) \\
&=
h_{n-1} + \prb{\tfrac{h_{n} - h_{n-1}}{\gp_{n} - \gp_{n-1}} }(\gp_{n}- \gp_{n-1})
+
\prb{\tfrac{h_{n+1}-h_n}{\gp_{n+1}-\gp_n}}(x-\gp_n)\\
&=
h_{n}
+
\prb{\tfrac{h_{n+1}-h_n}{\gp_{n+1}-\gp_n}}(x-\gp_n)\ifnocf.
\end{split}
\end{equation}
\cfload[.
Combining this and \eqref{DNN_interpolation:eqn0} with induction proves that for all
$n\in \{0, 1, \ldots, N-1 \}$,
$x\in [\gp_n,\gp_{n+1}]$
it holds that
\begin{equation}
\label{DNN_interpolation:eq3}
\prb{\functionReLUANN \prb{\interpolatingNN{h_0,h_1,\ldots, h_{N}}{\gp_0,\gp_1,\ldots,\gp_N }}}(x)
=
h_n + \prb{\tfrac{h_{n+1} - h_n}{\gp_{n+1} - \gp_n} }(x - \gp_n).
\end{equation}
The fact that for all
$n\in \{0, 1, \ldots, N\}$ it holds that $\gp_n\leq \gp_{N}$, the fact that
$\sum_{k=0}^{N}c_k=0$, and \eqref{DNN_interpolation_explicit:eq} therefore
imply that for all
$x\in [\gp_N,\infty)$
it holds that
\begin{align}\begin{split}
& \prb{\functionReLUANN \prb{\interpolatingNN{h_0,h_1,\ldots, h_{N}}{\gp_0,\gp_1,\ldots,\gp_N }}}(x)
-
\prb{\functionReLUANN \prb{\interpolatingNN{h_0,h_1,\ldots, h_{N}}{\gp_0,\gp_1,\ldots,\gp_N }}}(\gp_N)\\
&= \sum_{k=0}^{N}c_k\PR{\max\{x-\gp_k,0\}-\max\{\gp_{N}-\gp_k,0\}}\\
&=\sum_{k=0}^{N}c_k [(x-\gp_k)-(\gp_{N}-\gp_k)]= \sum_{k=0}^{N}c_k (x-\gp_{N})=0.\end{split}
\end{align}
This and \eqref{DNN_interpolation:eq3} show that for all
$x\in [\gp_N,\infty)$
it holds that
\begin{equation}
\begin{split}
\label{DNN_interpolation:eqlast}
\prb{\functionReLUANN \prb{\interpolatingNN{h_0,h_1,\ldots, h_{N}}{\gp_0,\gp_1,\ldots,\gp_N }}}(x)
&=
\prb{\functionReLUANN \prb{\interpolatingNN{h_0,h_1,\ldots, h_{N}}{\gp_0,\gp_1,\ldots,\gp_N }}}(\gp_N)\\
&=
h_{N-1} + \prb{\tfrac{h_{N} - h_{N-1}}{\gp_{N} - \gp_{N-1}} }(\gp_N - \gp_{N-1})
=
h_N.
\end{split}
\end{equation}
Combining this, \eqref{DNN_interpolation:eq1}, and \eqref{DNN_interpolation:eq3} establishes \cref{DNN_interpolation:item3}.
Note that \cref{DNN_interpolation:item3} implies \cref{DNN_interpolation:item2,DNN_interpolation:item4}. Observe that \eqref{DNN_interpolation:eq1}, \eqref{DNN_interpolation:eqlast}, and \cref{DNN_interpolation:item4} ensure \cref{DNN_interpolation:item5}.
The proof of \cref{DNN_interpolation} is thus complete.
\end{proof}
\subsection{ANN approximations for locally Lipschitz continuous functions}
\label{Subsection:4.3}
\begingroup
\newcommand{\newconstant}{a}
\cfclear
\begin{athm}{lemma}{Prop:interpolation_3}
Let
$\radius, c,\varepsilon \in (0, \infty)$,
$\newconstant \in [0, \infty)$,
$N \in \N$,
$(\gp_{n})_{n \in \Z} \subseteq \R$
satisfy for all
$n \in \Z$
that
$\gp_{n} = \clip{-R}{R}\pr{n(2c)^{-1}(1+2\radius)^{-\newconstant} \varepsilon}$ and
$N = \min\{k \in \N \colon \gp_{k} = \radius\}$,
let
$f \in C([-\radius,\radius],\R)$
satisfy for all
$x,y\in[-\radius,\radius]$
that
\begin{align}\label{interpolation_302}
\abs{f(x)-f(y)}\leq c(1+\abs{x}+\abs{y})^{\newconstant}\vass{x-y},
\end{align}
and let
$g \colon \R \to \R$
satisfy for all
$n\in\Z\cap[-N,N)$,
$x\in (\gp_{n},\gp_{n+1}]$,
$y\in[R,\infty)$
that
$g(y)=f(R)$,
$g(-y)=f(-R)$,
and
\begin{equation}\label{interpolation_303}
g(x)
=
f(\gp_{n}) + \pr*{\frac{f(\gp_{n+1}) - f(\gp_{n})}{\gp_{n+1}-\gp_{n}}} (x-\gp_{n})
\end{equation}
\cfload.
Then
\begin{enumerate}[(i)]
\item \label{interpolation_309}
it holds for all
$n\in\Z$
that
$
\gp_{n+1}-\gp_{n}
\leq
\frac{\varepsilon}{2c (1 + 2\radius)^{\newconstant}}
$,
\item \label{interpolation_308b}
it holds for all
$n\in\Z$
that
$g (\gp_{n})=f(\gp_{n})$,
\item \label{interpolation_307}
it holds for all
$x,y\in\R$
that
$
\abs{g(x)-g(y)}
\leq
c(1+2\radius)^{\newconstant}\abs{x-y}$,
and
\item \label{interpolation_308}
it holds that
$\sup_{x\in[-\radius,\radius]}\abs{f(x)-g(x)}\leq \varepsilon$.
\end{enumerate}
\end{athm}
\begin{proof}[Proof of \cref{Prop:interpolation_3}]
\Nobs that
the fact that
for all
$x,y\in\R$
it holds that
$\abs{\clip{-R}R(x)-\clip{-R}R(y)}\leq\abs{x-y}$
shows that for all
$n\in\Z$
it holds that
\begin{equation}
\begin{split}
\gp_{n+1}-\gp_n
&=
\abs{\gp_{n+1}-\gp_n}
\leq
\abs{(n+1)(2c)^{-1}(1+2\radius)^{-\newconstant} \varepsilon-n(2c)^{-1}(1+2\radius)^{-\newconstant} \varepsilon}
\\&\leq
\abs{(2c)^{-1}(1+2\radius)^{-\newconstant} \varepsilon}
=
(2c)^{-1}(1+2\radius)^{-\newconstant} \varepsilon
.
\end{split}
\end{equation}
This
establishes
\cref{interpolation_309}.
\Nobs that
the fact that
for all
$x\in\R$
it holds that
$\clip{-R}{R}(-x)=-\clip{-R}{R}(x)$
implies that for all
$n\in\Z$
it holds that
$\gp_{-n}=-\gp_n$
\cfload.
This
ensures that for all
$M\in\Z\cap[N,\infty)$
it holds that
\begin{equation}
\gp_{M}
=
R
\qandq
\gp_{-M}
=
-R
.
\end{equation}
Hence,
we obtain that for all
$M\in\Z\cap[N,\infty)$
it holds that
\begin{equation}
g(\gp_{M})
=
g(R)
=
f(R)
=
f(\gp_{M})
\qandq
g(\gp_{-M})
=
g(-R)
=
f(-R)
=
f(\gp_{-M})
.
\end{equation}
Combining
this
with
\eqref{interpolation_303}
proves that for all
$n\in\Z\cap[-N,N)$,
$x\in[\gp_{n},\gp_{n+1}]$
it holds that
\begin{equation}
\label{interpolation_303_boundary}
g(x)
=
f(\gp_{n}) + \pr*{\frac{f(\gp_{n+1}) - f(\gp_{n})}{\gp_{n+1}-\gp_{n}}} (x-\gp_{n})
.
\end{equation}
This
establishes
\cref{interpolation_308b}.
\Nobs that
\eqref{interpolation_302} and
\eqref{interpolation_303_boundary}
imply that for all
$n\in\Z\cap[-N,N)$,
$x,y \in [\gp_{n}, \gp_{n+1}]$
it holds that
\begin{equation}
\label{interpolation_306}
\begin{split}
\abs{g(x)-g(y)}
&=
\vass*{\pr*{\frac{f(\gp_{n+1}) - f(\gp_{n})}{\gp_{n+1}-\gp_{n}}} (x-y)}
\\&\leq
c(1 + \vass*{\gp_{n+1}} + \vass*{\gp_{n}})^{\newconstant} \abs{x-y}
\\&\leq
c(1+2\radius)^{\newconstant} \vass{x-y}
.
\end{split}
\end{equation}
This,
\cref{interpolation_308b}, and
\eqref{interpolation_302}
demonstrate that for all
$n,m \in \Z\cap[-N,N)$,
$x \in [\gp_{n}, \gp_{n+1}]$,
$y \in [\gp_m, \gp_{m+1}]$
with $n < m$
it holds that
\begin{equation}\begin{split}
\label{interpolation_310}
\vass{g(x)-g(y)}
&\leq
\vass{g(x)-g(\gp_{n+1})}+\vass{g(\gp_{n+1})-g(\gp_m)}+\vass{g(\gp_m)-g(y)}
\\&=
\vass{g(x)-g(\gp_{n+1})}+\vass{f(\gp_{n+1})-f(\gp_m)}+\vass{g(\gp_m)-g(y)}
\\&\leq
c(1 + \vass*{\gp_{n+1}} + \vass*{\gp_{n}})^{\newconstant} (\gp_{n+1}-x)
\\&\quad+ c(1 + \vass*{\gp_{n+1}} + \vass*{\gp_{m}})^{\newconstant}(\gp_m-\gp_{n+1})
\\&\quad + c(1 + \vass*{\gp_{m}} + \vass*{\gp_{m+1}})^{\newconstant} (y-\gp_m)
\\&\leq
c(1+2\radius)^{\newconstant} (y-x)
=
c(1+2\radius)^{\newconstant} \vass{y-x}
.
\end{split}
\end{equation}
Combining
this
with
\eqref{interpolation_306}
demonstrates that for all
$x,y\in[-R,R]$
it holds that
\begin{equation}
\label{interpolation_310.1}
\vass{g(x)-g(y)}
\leq
c (1+2\radius)^{\newconstant}\vass{x-y}
.
\end{equation}
Moreover, \nobs that
the assumption that
for all
$y\in[R,\infty)$
it holds that
$g(y)=f(R)$ and
$g(-y)=f(-R)$
ensures that for all
$x\in\R$
it holds that
$g(x)=g\pr*{\clip{-R}{R}(x)}$.
This and
\eqref{interpolation_310.1}
show that for all
$x,y\in\R$
it holds that
\begin{equation}
\label{interpolation_310.2}
\begin{split}
\vass{g(x)-g(y)}
&=
\vass*{g\pr*{\clip{-R}{R}(x)}-g\pr*{\clip{-R}{R}(y)}}
\\&\leq
c (1+2\radius)^{\newconstant}\vass{\clip{-R}{R}(x)-\clip{-R}{R}(y)}
\\&\leq
c (1+2\radius)^{\newconstant}\vass{x-y}
.
\end{split}
\end{equation}
This
establishes
\cref{interpolation_307}.
\Nobs that
\eqref{interpolation_302},
\cref{interpolation_308b}, and
\cref{interpolation_307}
imply that for all
$n\in \Z\cap[-N,N)$,
$x\in [\gp_{n}, \gp_{n+1}]$
it holds that
\begin{equation}
\label{Prop:interpolation_3:eq1}
\begin{split}
\abs{f(x) - g(x)}
&=
\abs{f(x)- f(\gp_{n})+f(\gp_{n})-g(x)}
\\&=
\abs{f(x)-f(\gp_{n})+g(\gp_{n})-g(x)}
\\&\leq
\abs{f(x)-f(\gp_{n})}-\abs{g(\gp_{n})-g(x)}
\\&\leq
c((1 + \vass*{\gp_{n}} + \vass{x})^{\newconstant}+(1 + 2\radius)^{\newconstant}) \abs{x-\gp_{n}}
\\&\leq
2c(1 + 2\radius)^{\newconstant} \abs{\gp_{n+1}-\gp_{n}}
\\&\leq
\frac{2c(1 + 2\radius)^{\newconstant}\varepsilon}{2c (1+2\radius)^{\newconstant}}
=
\varepsilon
.
\end{split}
\end{equation}
This
establishes
\cref{interpolation_308}.
The proof of \cref{Prop:interpolation_3} is thus complete.
\end{proof}
\endgroup
\begingroup
\cfclear
\newcommand{\newconstant}{a}
\begin{athm}{prop}{Prop:locally_Lipschitz_approx_2}
Let
$\radius \in [1, \infty)$,
$c\in(0,\infty)$,
$\newconstant \in [0, \infty)$,
$\varepsilon \in (0, 1]$,
$N \in \N$,
$(\gp_{n})_{n \in \Z} \subseteq \R$
satisfy for all
$n \in \Z$
that
$\gp_{n} = \clip{-R}{R}\pr{n(2c)^{-1}(1+2\radius)^{-\newconstant} \varepsilon}$ and
$N = \min\{k \in \N \colon \gp_{k} = \radius\}$,
let
$f \in C([-\radius,\radius],\R)$
satisfy for all
$x,y\in[-\radius,\radius]$
that
\begin{align}\label{locally_Lipschitz_approx_2_02}
\abs{f(x)-f(y)}
\leq
c(1+\abs{x}+\abs{y})^{\newconstant}\abs{x-y}
,
\end{align}
and let
$\mathscr{f}\in\ANNs$
satisfy
$\mathscr{f} = \interpolatingNN{f(\gp_{-N}),f(\gp_{-N+1}),\ldots, f(\gp_{N})}{\gp_{-N}, \gp_{-N+1},\ldots,\gp_{N} }$
\cfload.
Then
\begin{enumerate}[(i)]
\item \label{locally_Lipschitz_approx_2:item0}
it holds that
$
\dims(\mathscr{f})
=
(1, 2N+1, 1)
$,
\item \label{locally_Lipschitz_approx_2:realization}
it holds that
$
\functionReLUANN(\mathscr{f})
\in
C(\R,\R)
$,
\item \label{locally_Lipschitz_approx_2:item6}
it holds for all
$n\in \Z\cap[-N,N]$
that
$(\functionReLUANN (\mathscr{f}) )(\gp_n)=f(\gp_n)$,
\item \label{locally_Lipschitz_approx_2:item1}
it holds for all
$x,y\in\R$
that
$
\vass{(\functionReLUANN (\mathscr{f}) )(x)-(\functionReLUANN (\mathscr{f}) )(y)}
\leq
c(1+2\radius)^{\newconstant} \vass{x-y}
$,
\item \label{locally_Lipschitz_approx_2:item2}
it holds that
$
\sup_{x \in [-\radius,\radius]}\vass{f(x)-(\functionReLUANN (\mathscr{f}) )(x)}
\leq
\varepsilon
$, and
\item \label{locally_Lipschitz_approx_2:item3}
it holds that
$
\paramANN(\mathscr{f})
\leq
12c (1+2\radius)^{\newconstant} \radius \varepsilon^{-1} + 10
\leq
(12c + 10) 3^{\newconstant} \!\radius^{\newconstant+1} \varepsilon^{-1}
$
\end{enumerate}
\cfout.
\end{athm}
\begin{proof}[Proof of \cref{Prop:locally_Lipschitz_approx_2}]
\Nobs that
\cref{DNN_interpolation} and
\cref{Prop:interpolation_3}
prove
\cref{locally_Lipschitz_approx_2:item0
locally_Lipschitz_approx_2:realization
locally_Lipschitz_approx_2:item6
locally_Lipschitz_approx_2:item2
locally_Lipschitz_approx_2:item1}.
\Nobs that
the assumption that
$N=\min\{k \in \N \colon \gp_{k} = \radius\}$
ensures that
$\gp_{N-1}<\radius$.
This
implies that
$N-1 \leq \frac{2c(1+2\radius)^{\newconstant}\! \radius}{\varepsilon}$.
\Cref{locally_Lipschitz_approx_2:item0}
hence
assures that
\begin{equation}
\begin{split}
\paramANN(\mathscr{f})
&=
3(2N) + 4
\leq
6\prb{\tfrac{2c (1+2\radius)^{\newconstant}\! \radius}{\varepsilon} + 1}+4
\\&=
12c (1+2\radius)^{\newconstant}\! \radius\varepsilon^{-1} + 10
\\&\leq
12c 3^{\newconstant} \!\radius^{\newconstant+1} \varepsilon^{-1} + 10
\leq
(12c + 10) 3^{\newconstant} \!\radius^{\newconstant+1} \varepsilon^{-1}
.
\end{split}
\end{equation}
This
establishes
\cref{locally_Lipschitz_approx_2:item3}.
The proof of \cref{Prop:locally_Lipschitz_approx_2} is thus complete.
\end{proof}
\endgroup
\begingroup
\newcommand{\newconstant}{a}
\cfclear
\begin{athm}{cor}{Coro:loc_Lipschitz_cost_constant_targetfct}
Let ${c}\in\R$, $f \in C(\R, \R)$ satisfy for all $x\in \R$ that $f(x)={c}$\cfload.
Then
\begin{enumerate}[(i)]
\item\label{Coro:loc_Lipschitz_cost_constant_targetfct_item_0}
it holds that
$\realisation_\ReLU(\affineANN_{0,{c}}) \in C(\R,\R)$,
\item\label{Coro:loc_Lipschitz_cost_constant_targetfct_item_1}
it holds for all
$x \in \R$
that
$ (\realisation_\ReLU(\affineANN_{0,{c}}))(x)=f(x)$,
\item\label{Coro:loc_Lipschitz_cost_constant_targetfct_item_2}
it holds for all
$x,y \in \R$
that
$\vass{ (\realisation_\ReLU(\affineANN_{0,{c}}))(x)-(\realisation_\ReLU(\affineANN_{0,{c}}))(y)} =0$,
and
\item\label{Coro:loc_Lipschitz_cost_constant_targetfct_item_3}
it holds that
$\paramANN(\affineANN_{0,{c}}) = 2$
\end{enumerate}
\cfout.
\end{athm}
\begin{proof}[Proof of \cref{Coro:loc_Lipschitz_cost_constant_targetfct}]
\Nobs[observe] that
\eqref{ANNrealization:ass2} and
the fact that
$\dims(\affineANN_{0,{c}})=(1,1)$
establish
\cref{Coro:loc_Lipschitz_cost_constant_targetfct_item_0
Coro:loc_Lipschitz_cost_constant_targetfct_item_1
Coro:loc_Lipschitz_cost_constant_targetfct_item_2
Coro:loc_Lipschitz_cost_constant_targetfct_item_3}.
The proof of \cref{Coro:loc_Lipschitz_cost_constant_targetfct} is thus complete.
\end{proof}
\endgroup
\begingroup
\newcommand{\newconstant}{a}
\cfclear
\begin{athm}{cor}{Coro:loc_Lipschitz_cost_bis}
Let
$c, \newconstant \in [0, \infty)$,
$f \in C(\R,\R)$
satisfy for all
$x,y\in \R$
that
$\vass{f(x)-f(y)} \leq c(1+\abs{x}+\abs{y})^{\newconstant} \vass{x-y}$.
Then it holds for all
$\radius \in [1, \infty)$,
$\varepsilon \in (0,1]$
that
\begin{equation}
\CostLip{\ReLU}{[-\radius, \radius]}{f|_{[-\radius, \radius]} }{c(1+2\radius)^{\newconstant} }{\varepsilon}
\leq
12c \radius (1+2\radius)^{\newconstant} \varepsilon^{-1} + 10
\end{equation}
\cfout.
\end{athm}
\begin{proof}[Proof of \cref{Coro:loc_Lipschitz_cost_bis}]
\Nobs that
\cref{Prop:locally_Lipschitz_approx_2} and
\cref{Coro:loc_Lipschitz_cost_constant_targetfct}
establish that for all
$\radius \in [1, \infty)$,
$\varepsilon \in (0,1]$
there exists
$\mathscr{f} \in \ANNs$
such that
\begin{enumerate}[(I)]
\item
it holds that
$\functionReLUANN (\mathscr{f})\in C(\R,\R)$,
\item
it holds for all
$x,y\in\R$
that
\begin{equation}
\vass{(\functionReLUANN (\mathscr{f}) )(x)-(\functionReLUANN (\mathscr{f}) )(y)}
\leq
c(1+2\radius)^{\newconstant} \vass{x-y}
,
\end{equation}
\item
it holds that
$
\sup_{x \in [-\radius,\radius]}\vass{f(x)-(\functionReLUANN (\mathscr{f}) )(x)}
\leq
\varepsilon
$,
and
\item
it holds that
$
\paramANN(\mathscr{f})
\leq
12c (1+2\radius)^{\newconstant}\!\radius \varepsilon^{-1} + 10
$
\end{enumerate}
\cfload.
This and
\cref{lem_cost_of_Lip_approx_set_equivalence}
establish that for all
$\radius \in [1, \infty)$,
$\varepsilon \in (0,1]$
it holds that
$
\CostLip{\ReLU}{[-\radius, \radius]}{f}{c(1+2\radius)^{\newconstant} }{\varepsilon}
\leq
12c (1+2\radius)^{\newconstant}\!\radius \varepsilon^{-1} + 10
$ \cfload.
This completes the proof of \cref{Coro:loc_Lipschitz_cost_bis}.
\end{proof}
\endgroup
\begingroup
\newcommand{\newconstant}{a}
\cfclear
\begin{athm}{cor}{Lemma:loc_Comp_theory_15_sum}
Let
$c, \newconstant \in [0, \infty)$,
$f \in C(\R,\R)$
satisfy for all
$x,y\in \R$
that
$
\vass{f(x)-f(y)}
\leq
c(1+\abs{x}+\abs{y})^{\newconstant} \vass{x-y}
$.
Then
$f \in \Capprox{(12c+10) 3^{\newconstant}}{1}$ \cfout.
\end{athm}
\begin{aproof}
Note that
\cref{Lemma:Monotonicity_of_Cost} and
\cref{Coro:loc_Lipschitz_cost_bis}
establish that for all
$\radius \in [1, \infty)$, $\varepsilon \in (0,1]$
it holds that
\begin{equation}
\begin{split}
\CostLip{\ReLU}{[-\radius, \radius]}{f_{[-\radius, \radius]}}{c 3^{\newconstant} \!\radius^{\newconstant} }{\varepsilon}
&\leq
\CostLip{\ReLU}{[-\radius, \radius]}{f_{[-\radius, \radius]}}{c (1+2\radius)^{\newconstant} }{\varepsilon}
\\&\leq
12c (1+2\radius)^{\newconstant}\!\radius \varepsilon^{-1} + 10
\\&\leq
(12c+10)3^{\newconstant} \!\radius^{\newconstant+1} \varepsilon^{-1}
\end{split}
\end{equation}
\cfload.
This,
the fact that
$\max\{c 3^{\newconstant}, \newconstant, (12c + 10) 3^{\newconstant}, \newconstant+1, 1\} = \max\{(12c + 10) 3^{\newconstant}, \newconstant+1\} = (12c + 10) 3^{\newconstant}$,
and \cref{Lemma:Monotonicity_of_Cost}
ensure that for all
$\radius \in [1, \infty)$,
$\varepsilon \in (0,1]$
it holds that
\begin{equation}
\begin{split}
\CostLipB{\ReLU}{[-\radius, \radius]}{f|_{[-\radius, \radius]}}{(12c + 10) 3^{\newconstant} \!\radius^{(12c + 10) 3^{\newconstant}} }{\varepsilon}
&\leq
\CostLip{\ReLU}{[-\radius, \radius]}{f|_{[-\radius, \radius]}}{c 3^{\newconstant} \!\radius^{\newconstant} }{\varepsilon}
\\&\leq
(12c + 10) 3^{\newconstant} \!\radius^{(12c + 10) 3^{\newconstant}} \varepsilon^{-(12c + 10) 3^{\newconstant}}
.
\end{split}
\end{equation}
The proof of \cref{Lemma:loc_Comp_theory_15_sum} is thus complete.
\end{aproof}
\endgroup
\section{ANN representations for maxima}
\label{Section:5}
In this section we show
for every $d\in\N$ how to exactly represent the $d$-dimensional maximum function
as the realization of a ReLU ANN with a bound on the
number of parameters that is quadratic in $d$.
As a consequence, we prove in \cref{Coro:max_d_cost} in \cref{Subsection:5.2}
below that the vector-valued multi-dimensional maximum functions
are in the approximation spaces defined in
\cref{Section:3} above.
The results in this section are essentially well-known.
In particular, we refer, e.g., to Beck et al.~\cite[Subsection~3.1.2]{BeckJentzenKuckuck2019}
and to Jentzen \& Riekert \cite[Subsection~3.2]{jentzen2020strong}
for closely related results and only for completeness we include
full proofs here.
\subsection{Explicit ANN representations for maxima}
\label{Subsection:5.1}
\cfclear
\begin{athm}{lemma}{Lemma:max_d_welldefined}
There exist unique $\mathscr{f}_d\in \ANNs$, $d \in \{2,3,\ldots\}$, such that
\begin{enumerate}[(i)]
\item \label{max_d_welldefined:item_1} it holds for all $d \in \{2,3,\ldots\}$ that
$\mathcal{I}(\mathscr{f}_d) = d$,
\item \label{max_d_welldefined:item_2} it holds for all $d \in \{2,3,\ldots\}$ that
$\mathcal{O}(\mathscr{f}_d) = 1$, and
\item \label{max_d_welldefined:item_4} it holds for all $d \in \{2,3,\ldots\}$ that
\begin{equation}
\mathscr{f}_d =
\begin{dcases}
\!\! \pr*{ \!\! \pr*{ \!\!
\begin{pmatrix}
1 & -1 \\
0 & 1 \\
0 & -1
\end{pmatrix},
\begin{pmatrix}
0 \\
0 \\
0
\end{pmatrix} \! \! }, \pr*{
\begin{pmatrix}
1 & 1 & -1
\end{pmatrix}, 0 } \! \!}
& \colon d=2 \\
\mathscr{f}_{\frac{d+1}{2}} \bullet \prb{\parallelizationSpecial_{\frac{d+1}{2}}( \mathscr{f}_2, \mathscr{f}_2 , \ldots, \mathscr{f}_2, \ReLUidANN{1}) }
& \colon d \in \{3, 5, 7, \ldots\} \\
\compANN{\mathscr{f}_{\frac{d}{2}} }{ \prb{\parallelizationSpecial_{\frac{d}{2}}( \mathscr{f}_2, \mathscr{f}_2, \ldots, \mathscr{f}_2) }}
& \colon d \in \{4, 6, 8, \ldots\}
\end{dcases}
\end{equation}
\end{enumerate}
\cfadd{Lemma:PropertiesOfParallelizationEqualLengthDims}
\cfout.
\end{athm}
\begin{proof}[Proof of \cref{Lemma:max_d_welldefined}]
Throughout this proof let
$\mathscr{g}\in ((\R^{3 \times 2} \times \R^{3}) \times (\R^{1 \times 3} \times \R^1))\subseteq\ANNs$
satisfy
\begin{equation}
\label{max_d_welldefined:basis}
\mathscr{g}=\pr*{ \!\! \pr*{ \!\!
\begin{pmatrix}
1 & -1 \\
0 & 1 \\
0 & -1
\end{pmatrix},
\begin{pmatrix}
0 \\
0 \\
0
\end{pmatrix} \! \! }, \pr*{
\begin{pmatrix}
1 & 1 & -1
\end{pmatrix}, 0 } \! \!}\ifnocf.
\end{equation}
\cfload[.
\Nobs that \eqref{max_d_welldefined:basis} ensures that
\begin{equation}
\label{max_d_welldefined:0}
\mathcal{I}(\mathscr{g})
=
2
\qquad \text{and}\qquad
\mathcal{O}(\mathscr{g})
=
1
.
\end{equation}
\Cref{Lemma:PropertiesOfParallelizationEqualLengthDims},
\cref{Prop:identity_representation}, and
\cref{lem:comp2}
hence
show that for all
$d\in\{3,5,7,\ldots\}$,
$\mathscr{h}\in\ANNs$
with
$\mathcal{I}(\mathscr{h}) = \frac{d+1}{2}$ and
$\mathcal{O}(\mathscr{h}) = 1$
it holds that
$
\outDimANN\prb{\mathscr{h}\bullet\prb{\parallelizationSpecial_{\frac{d+1}{2}} (\mathscr{g}, \mathscr{g}, \ldots,\allowbreak \mathscr{g}, \ReLUidANN{1})}}
=
\outDimANN(\mathscr{h})
=
1
$
and
\begin{equation}
\begin{split}
\label{max_d_welldefined:1}
&\inDimANN\prb{\mathscr{h}\bullet\prb{\parallelizationSpecial_{\frac{d+1}{2}} (\mathscr{g}, \mathscr{g}, \ldots\allowbreak, \mathscr{g}, \ReLUidANN{1})}}
=
\inDimANN\prb{\parallelizationSpecial_{\frac{d+1}{2}} (\mathscr{g}, \mathscr{g}, \ldots\allowbreak, \mathscr{g}, \ReLUidANN{1})}
\\&=
\PRb{\tfrac{d-1}{2}}\inDimANN(\mathscr{g})+\inDimANN(\ReLUidANN{1})=(d-1)+1
=
d
\ifnocf.
\end{split}
\end{equation}
\cfload[.
\Moreover
\eqref{max_d_welldefined:0},
\cref{Lemma:PropertiesOfParallelizationEqualLengthDims}, and
\cref{lem:comp2}
prove that for all
$d\in\{4,6,8,\ldots\}$,
$\mathscr{h}\in\ANNs$
with
$\mathcal{I}(\mathscr{h}) = \frac{d}{2}$ and
$\mathcal{O}(\mathscr{h}) = 1$
it holds that
$
\outDimANN\prb{\mathscr{h}\bullet\prb{\parallelizationSpecial_{\frac{d}{2}} (\mathscr{g}, \mathscr{g}, \ldots,\allowbreak \mathscr{g})}}
=
\outDimANN(\mathscr{h})
=
1
$
and
\begin{equation}
\inDimANN\prb{\mathscr{h}\bullet\prb{\parallelizationSpecial_{\frac{d}{2}} (\mathscr{g}, \mathscr{g}, \ldots\allowbreak, \mathscr{g})}}
=
\inDimANN\prb{\parallelizationSpecial_{\frac{d}{2}} (\mathscr{g}, \mathscr{g}, \ldots\allowbreak, \mathscr{g})}
=
\PRb{\tfrac{d}{2}}\inDimANN(\mathscr{g})
=
d
.
\end{equation}
Combining
this,
\eqref{max_d_welldefined:0}, and
\eqref{max_d_welldefined:1}
with
induction
establishes
\cref{max_d_welldefined:item_1
max_d_welldefined:item_2
max_d_welldefined:item_4}.
The proof of \cref{Lemma:max_d_welldefined} is thus complete.
\end{proof}
\cfclear
\begin{definition}
[ANN representations of maximum functions]
\label{def:max_d}\cfconsiderloaded{def:max_d}
We denote by $\maxANN_d\in\ANNs$, $d \in \{2,3,\ldots\}$, the \cfadd{def:neuralnetwork}ANNs which satisfy that
\begin{enumerate}[(i)]
\item \label{max_d_def:1} it holds for all $d \in \{2,3,\ldots\}$ that
$\mathcal{I}(\maxANN_d) = d$,
\item \label{max_d_def:2} it holds for all $d \in \{2,3,\ldots\}$ that
$\mathcal{O}(\maxANN_d) = 1$, and
\item \label{max_d_def:4} it holds for all $d \in \{2,3,\ldots\}$ that
\begin{equation}
\label{max_d_def:4eq}
\maxANN_d =
\begin{dcases}
\!\! \mleft( \!\! \mleft( \!\!
\begin{pmatrix}
1 & -1 \\
0 & 1 \\
0 & -1
\end{pmatrix},
\begin{pmatrix}
0 \\
0 \\
0
\end{pmatrix} \! \! \mright), \mleft(
\begin{pmatrix}
1 & 1 & -1
\end{pmatrix}, 0 \mright) \! \! \mright)
& \colon d=2 \\
\maxANN_{\frac{d+1}{2}} \bullet \prb{\parallelizationSpecial_{\frac{d+1}{2}}( \maxANN_2, \maxANN_2 , \ldots, \maxANN_2, \ReLUidANN{1}) }
& \colon d \in \{3, 5, 7, \ldots\} \\
\compANN{\maxANN_{\frac{d}{2}} }{ \prb{\parallelizationSpecial_{\frac{d}{2}}( \maxANN_2, \maxANN_2, \ldots, \maxANN_2) }}
& \colon d \in \{4, 6, 8, \ldots\}
\end{dcases}
\end{equation}
\end{enumerate}
\cfadd{Lemma:max_d_welldefined}
\cfload.
\end{definition}
\cfclear
\begin{athm}{lemma}{lem:max_d_lipschitz}
Let
$d \in \N$,
$x=(x_1,x_2,\ldots,x_d),\,\allowbreak y=(y_1,y_2,\ldots,y_d)\in\R^d$.
Then
\begin{equation}
\begin{split}
&\vass{\max\{x_1,x_2,\ldots,x_d\}-\max\{y_1,y_2,\ldots,y_d\}}
\\&\leq
\max\{\vass{x_1-y_1},\vass{x_2-y_2},\ldots,\vass{x_d-y_d}\}
\leq
\norm{x-y}
\end{split}
\end{equation}
\cfout.
\end{athm}
\begin{proof}[Proof of \cref{lem:max_d_lipschitz}]
Throughout this proof
assume w.l.o.g.\ that
$\max\{x_1,x_2,\ldots,x_d\}\geq\max\{y_1,y_2,\ldots,y_d\}$ and
let
$m\in\{1,2,\ldots,d\}$
satisfy
$x_m=\max\{x_1,x_2,\ldots,x_d\}$.
\Nobs that
\begin{equation}
\begin{split}
&\vass{\max\{x_1,x_2,\ldots,x_d\}-\max\{y_1,y_2,\ldots,y_d\}}
\\&=
x_m-\max\{y_1,y_2,\ldots,y_d\}
\leq
x_m-y_m
=
\vass{x_m-y_m}
\\&\leq
\max\{\vass{x_1-y_1},\vass{x_2-y_2},\ldots,\vass{x_d-y_d}\}
\\&=
\PRb{\max\pRb{\vass{x_1-y_1}^2,\vass{x_2-y_2}^2,\ldots,\vass{x_d-y_d}^2}}^{\frac12}
\\&\leq
\PRbbb{\ssum_{i=1}^d\abs{x_i-y_i}^2}^{\frac12}
=
\norm{x-y}
\end{split}
\end{equation}
\cfload.
This completes the proof of \cref{lem:max_d_lipschitz}.
\end{proof}
\cfclear
\begin{athm}{prop}{Prop:max_d}
Let
$d \in \{2,3,\ldots\}$.
Then
\begin{enumerate}[(i)]
\item \label{max_d:item_1}
it holds that
$\realisation_{\ReLU}\pr*{{\maxANN_d}} \in C(\R^d,\R)$,
\item \label{max_d:item_2}
it holds for all
$x = (x_1,x_2, \ldots, x_d)$,
$y=(y_1,y_2, \ldots, y_d) \in \R^d$
that
$
\abs{(\realisation_{\ReLU}({\maxANN_d}))(x)
- (\realisation_{\ReLU}({\maxANN_d}))(y)}
\leq
\max\{\abs{x_1-y_1}, \abs{x_2-y_2}, \ldots, \abs{x_d-y_d}\}
\leq
\mednorm{x-y}
$,
\item \label{max_d:item_3}
it holds for all
$x = (x_1,x_2, \ldots, x_d) \in \R^d$
that
$(\realisation_{\ReLU}({\maxANN_d}))(x) = \max\{x_1, x_2, \ldots, x_d\}$,
\item \label{max_d:item_4}
it holds that
$\lengthANN(\maxANN_d) = \ceil{\log_2(d)}+1$,
\item \label{max_d:item_5}
it holds for all
$i \in \N_0$
that
$\mathbb D_i(\maxANN_d) \leq 3\ceil[\big]{\tfrac{d}{2^i}}$,
and
\item \label{max_d:item_6}
it holds that
$\paramANN(\maxANN_d) \leq 3d^2 + 18d + 12\ceil{\log_2(d)} - \tfrac{13}{2}$
\end{enumerate}
\cfout.
\end{athm}
\begin{aproof}
\Nobs that
the fact that $\inDimANN(\maxANN_d)=d$ and $\outDimANN(\maxANN_d)=1$
implies
\cref{max_d:item_1} \cfload.
\Nobs that
\eqref{max_d_def:4eq}
ensures that for all
$x=(x_1,x_2)\in\R^2$
it holds that
\begin{equation}
\label{Prop:max_d:0}
(\realisation_{\ReLU}(\maxANN_2))(x)
=
\max\{x_1-x_2,0\}+\max\{x_2,0\}-\max\{-x_2,0\}
=
\max\{x_1,x_2\}
\end{equation}
\cfload.
This,
\cref{Lemma:PropertiesOfParallelizationEqualLength},
\cref{Prop:identity_representation}, and
\cref{Lemma:PropertiesOfCompositions_n2}
show that for all
$k\in\{3,5,7,\ldots\}$,
$x=(x_1,x_2,\ldots,x_k)\in\R^k$
with $\forall\, y=\prb{y_1,y_2,\ldots,y_{\frac{k+1}{2}}}\in\R^{\nicefrac{(k+1)}{2}}\colon\prb{\realisation_{\ReLU}\prb{\maxANN_{\frac{k+1}{2}}}}(y)=\max\pRb{y_1,y_2,\ldots,y_{\frac{k+1}{2}}}$
it holds that
\begin{equation}
\begin{split}
\label{Prop:max_d:1}
&(\realisation_{\ReLU}\pr{\maxANN_{k}})(x)
\\&=\prb{\realisation_{\ReLU}\prb{\compANN{\maxANN_{\frac{k+1}{2}}}{\PRb{\parallelizationSpecial_{\frac{k+1}{2}}( \maxANN_2, \maxANN_2 , \ldots, \maxANN_2, \ReLUidANN{1})}}}}(x)
\\&=
\prb{\PRb{\realisation_{\ReLU}\prb{\maxANN_{\frac{k+1}{2}}}}\circ\PRb{\realisation_{\ReLU}\prb{\parallelizationSpecial_{\frac{k+1}{2}}( \maxANN_2, \maxANN_2 , \ldots, \maxANN_2, \ReLUidANN{1})}}}(x)
\\&=
\prb{\realisation_{\ReLU}\prb{\maxANN_{\frac{k+1}{2}}}}\prb{\maxANN_{2}(x_1,x_2),\maxANN_{2}(x_3,x_4),\ldots,\maxANN_{2}(x_{k-2},x_{k-1}),\ReLUidANN{1}(x_k)}
\\&=
\prb{\realisation_{\ReLU}\prb{\maxANN_{\frac{k+1}{2}}}}\prb{\max\{x_1,x_2\},\max\{x_3,x_4\},\ldots,\max\{x_{k-2},x_{k-1}\},x_k}
\\&=
\max\pRb{\max\{x_1,x_2\},\max\{x_3,x_4\},\ldots,\max\{x_{k-2},x_{k-1}\},x_k}
\\&=
\max\{x_1,x_2,\ldots,x_k\}
\end{split}
\end{equation}
\cfload.
\Moreover
\eqref{Prop:max_d:0},
\cref{Lemma:PropertiesOfParallelizationEqualLength},
\cref{Prop:identity_representation}, and
\cref{Lemma:PropertiesOfCompositions_n2}
demonstrate that for all
$k\in\{4,6,8,\ldots\}$,
$x=(x_1,x_2,\ldots,x_k)\in\R^k$
with
$\forall\, y=\prb{y_1,y_2,\ldots,y_{\frac{k}{2}}}\in\R^{\nicefrac{k}{2}}\colon\prb{\realisation_{\ReLU}\prb{\maxANN_{\frac{k}{2}}}}(y)=\max\pRb{y_1,y_2,\ldots,\allowbreak y_{\frac{k}{2}}}$
it holds that
\begin{equation}
\begin{split}
\label{Prop:max_d:2}
(\realisation_{\ReLU}\pr{\maxANN_{k}})(x)&=\prb{\realisation_{\ReLU}\prb{\compANN{\maxANN_{\frac{k}{2}}}{\PRb{\parallelizationSpecial_{\frac{k}{2}}( \maxANN_2, \maxANN_2 , \ldots, \maxANN_2)}}}}(x)
\\&=
\prb{\PRb{\realisation_{\ReLU}\prb{\maxANN_{\frac{k}{2}}}}\circ\PRb{\realisation_{\ReLU}\prb{\parallelizationSpecial_{\frac{k}{2}}( \maxANN_2, \maxANN_2 , \ldots, \maxANN_2)}}}(x)
\\&=
\prb{\realisation_{\ReLU}\prb{\maxANN_{\frac{k}{2}}}}\prb{\maxANN_{2}(x_1,x_2),\maxANN_{2}(x_3,x_4),\ldots,\maxANN_{2}(x_{k-1},x_{k})}
\\&=
\prb{\realisation_{\ReLU}\prb{\maxANN_{\frac{k}{2}}}}\prb{\max\{x_1,x_2\},\max\{x_3,x_4\},\ldots,\max\{x_{k-1},x_{k}\}}
\\&=
\max\pRb{\max\{x_1,x_2\},\max\{x_3,x_4\},\ldots,\max\{x_{k-1},x_{k}\}}
\\&=
\max\{x_1,x_2,\ldots,x_k\}
.
\end{split}
\end{equation}
Combining
this,
\eqref{Prop:max_d:0}, and
\eqref{Prop:max_d:1}
with induction
proves that for all
$k\in\{2,3,\ldots\}$, $x=(x_1,x_2,\dots,x_k)\in\R^k$
it holds that
$\pr{\realisation_{\ReLU}\pr{{\maxANN_k}}}(x) = \max\{x_1,x_2,\dots,x_k\}$.
Therefore,
we obtain
\cref{max_d:item_3}.
\Nobs that
\cref{max_d:item_3} and
\cref{lem:max_d_lipschitz}
imply
\cref{max_d:item_2}.
\Nobs that
\eqref{max_d_def:4eq}
ensures that
\begin{equation}
\llabel{eq:length2}
\lengthANN(\maxANN_2)
=
2
=
\ceil{\log_2(2)}+1
\qandq
\dims(\maxANN_2)
=
(2,3,1)
.
\ifnocf.
\end{equation}
\cfload[.
Combining
\eqref{max_d_def:4eq}
with
\Cref{Prop:identity_representation} and
\cref{Lemma:PropertiesOfCompositions_n2}
hence
shows that for all
$k\in\{3,5,7,\ldots\}$
with
$\lengthANN\prb{\maxANN_{\frac{k+1}2}} = \ceil[\big]{\log_2\prb{\frac{k+1}{2}}}+1$
it holds that
\begin{equation}\llabel{eq:lengthodd}
\begin{split}
&
\lengthANN(\maxANN_k)
=
\lengthANN\prb{\maxANN_{\frac{k+1}2}}+\lengthANN\prb{\parallelizationSpecial_{\frac{k+1}{2}} (\maxANN_2, \maxANN_2, \ldots\allowbreak, \maxANN_2, \ReLUidANN{1})}-1
\\&=
\ceil[\big]{\log_2\prb{\tfrac{k+1}{2}}}+1+\lengthANN(\maxANN_2)-1
=
\ceil{\log_2\pr{k+1}}+1
=
\ceil{\log_2\pr{k}}+1
.
\end{split}
\end{equation}
\Moreover
\eqref{max_d_welldefined:0},
\eqref{max_d_def:4eq}, and
\cref{Lemma:PropertiesOfCompositions_n2}
prove that for all
$k\in\{4,6,8,\ldots\}$
with
$\lengthANN\prb{\maxANN_{\frac k2}} = \ceil[\big]{\log_2\prb{\frac{k}{2}}}+1$
it holds that
\begin{equation}
\begin{split}
&
\lengthANN(\maxANN_k)
=
\lengthANN\prb{\maxANN_{\frac k2}}+\lengthANN\prb{\parallelizationSpecial_{\frac{k}{2}} (\maxANN_2, \maxANN_2, \ldots\allowbreak, \maxANN_2)}-1
\\&=
\ceil[\big]{\log_2\prb{\tfrac{k}{2}}}+1+\lengthANN(\maxANN_2)-1
=
\ceil{\log_2\pr{k}}+1
.
\end{split}
\end{equation}
Combining
this
with
\lref{eq:length2} and
\lref{eq:lengthodd}
establishes
\cref{max_d:item_4}.
\Nobs that \lref{eq:length2}
implies that for all
$i\in\N_0$
it holds that
\begin{equation}\llabel{eq:dimM2}
\singledims_i(\maxANN_2)
\leq
3\leq 3\ceil[\big]{\tfrac{2}{2^{i}}}
.
\end{equation}
\Moreover
\lref{eq:length2},
\cref{Lemma:PropertiesOfParallelizationEqualLengthDims}, and
\cref{Prop:identity_representation}
show that for all
$k\in\{3,5,7,\dots\}$
it holds that
\begin{equation}\llabel{eq:para_length1}
\begin{split}
&\dims\prb{\parallelizationSpecial_{\frac{k+1}{2}}( \maxANN_2, \maxANN_2 , \ldots, \maxANN_2, \ReLUidANN{1})}
=
\PRb{\tfrac{k-1}2}\dims(\maxANN_2)+\dims(\ReLUidANN{1})
\\&=
\prb{k-1,\tfrac{3(k-1)}2,\tfrac{k-1}2}+(1,2,1)
=
\prb{k,\tfrac{3k+1}2,\tfrac{k+1}2}
.
\end{split}
\end{equation}
Combining
this
with
\eqref{max_d_def:4eq},
\cref{Lemma:PropertiesOfParallelizationEqualLengthDims},
\cref{Prop:identity_representation}, and
\cref{Lemma:PropertiesOfCompositions_n1}
demonstrates that for all
$k\in\{3,5,7,\ldots\}$,
$i\in\{2,3,\dots\}$
with
$\forall\, j\in\N_0\colon\singledims_j\prb{\maxANN_{\frac{k+1}{2}}}\leq 3\ceil[\big]{\tfrac{k+1}{2^{j+1}}}$
it holds that
\begin{equation}\llabel{eq:dim1Modd}
\begin{split}
\singledims_1\pr{\maxANN_{k}}
&=
\tfrac{3k+1}{2}
\leq
3\ceil[\big]{\tfrac{k}2}
\qquad\text{and}\qquad
\singledims_i\pr{\maxANN_{k}}
=
\singledims_{i-1}\prb{\maxANN_{\frac{k+1}{2}}}
\leq
3\ceil[\big]{\tfrac{k+1}{2^i}}
=
3\ceil[\big]{\tfrac{k}{2^i}}
.
\end{split}
\end{equation}
\Moreover
\lref{eq:length2},
\cref{Lemma:PropertiesOfParallelizationEqualLengthDims}, and
\cref{Prop:identity_representation}
imply that for all
$k\in\{2,4,6,\dots\}$
it holds that
\begin{equation}\llabel{eq:para_length2}
\dims\prb{\parallelizationSpecial_{\frac{k}{2}}( \maxANN_2, \maxANN_2 , \ldots, \maxANN_2)}
=
\PRb{\tfrac{k}2}\dims(\maxANN_2)
=
\prb{k,\tfrac{3k}2,\tfrac{k}2}
.
\end{equation}
This,
\eqref{max_d_def:4eq},
\cref{Lemma:PropertiesOfParallelizationEqualLengthDims},
\cref{Prop:identity_representation}, and
\cref{Lemma:PropertiesOfCompositions_n1}
prove that for all
$k\in\{4,6,8,\ldots\}$,
$i\in\{2,3,\dots\}$
with
$\forall\, j\in\N_0\colon\singledims_j\prb{\maxANN_{\frac{k}{2}}}\leq 3\ceil[\big]{\tfrac{k}{2^{j+1}}}$
it holds that
\begin{equation}\llabel{eq:dim1Meven}
\begin{split}
\singledims_1\pr{\maxANN_{k}}
&=
\tfrac{3k}{2}
=
3\ceil[\big]{\tfrac{k}2}
\qquad\text{and}\qquad
\singledims_i\pr{\maxANN_{k}}
=
\singledims_{i-1}\prb{\maxANN_{\frac{k}{2}}}
\leq
3\ceil[\big]{\tfrac{k}{2^i}}
.
\end{split}
\end{equation}
Combining this,
\lref{eq:dimM2}, and
\lref{eq:dim1Modd}
with induction
establishes that for all $k\in\{2,3,\ldots\}$, $i\in\N$ it holds that $\singledims_i(\maxANN_{k})\leq 3\ceil[\big]{\tfrac{k}{2^i}}$.
This and the fact that $\singledims_0(\maxANN_{d})=\inDimANN(\maxANN_{d})=d\leq 3\ceil[\big]{\tfrac{d}{2^0}}$ imply \cref{max_d:item_5}.
\Nobs that
the fact that
$\sum_{i=1}^{\infty}\tfrac{1}{4^i}=\tfrac{1}{3}$
and \cref{max_d:item_5}
show that for all
$s \in \{3,4,\ldots\}$
it holds that
\begin{equation}
\begin{split}
\paramANN(\maxANN_s)
&=
\sum_{i=0}^{\lengthANN\pr*{\maxANN_s}-1}\!\!\pr*{\singledims_{i}\pr*{\maxANN_s}+1}\singledims_{i+1}\pr*{\maxANN_s}
\\&=
\PR*{\sum_{i=1}^{\lengthANN\pr*{\maxANN_s}-2}\!\!\pr*{\singledims_{i}\pr*{\maxANN_s}+1}\singledims_{i+1}\pr*{\maxANN_s}}+\pr*{\singledims_{0}\pr*{\maxANN_s}+1}\singledims_{1}\pr*{\maxANN_s}
\\&\quad+\pr*{\singledims_{\lengthANN\pr*{\maxANN_s}-1}\pr*{\maxANN_s}+1}\singledims_{\lengthANN\pr*{\maxANN_s}}\pr*{\maxANN_s}
\\&\leq
\PR*{\sum_{i=1}^{\ceil{\log_2(s)}-1} \!\!\!\! \pr*{ 3 \ceil[\big]{\tfrac{s}{2^i}}+1} 3 \ceil[\big]{\tfrac{s}{2^{i+1}}}}
+ (s+1) 3 \ceil[\big]{\tfrac{s}{2}} + 4
\\&\leq
\PR*{ \sum_{i=1}^{\ceil{\log_2(s)}-1} \!\!\!\! \pr*{ 3 \prb{\tfrac{s}{2^i}+1}+1} 3 \pr*{\tfrac{s}{2^{i+1}}+1}} + (s+1)\prb{\tfrac{3s}{2}+\tfrac{3}{2}} + 4
\\&=
\PR*{\sum_{i=1}^{\ceil{\log_2(s)}-1} \!\!\!\!
\prb{\tfrac{9s^2}{2^{2i+1}}+\tfrac{15s}{2^i}+12}}
+ \tfrac{3s^2}{2}+3s + \tfrac{11}{2}
\\&\leq
\tfrac{9s^2}{2}\PR*{\sum_{i=1}^{\ceil{\log_2(s)}-1} \!\!\!\! \tfrac{1}{4^{i}}}
+ 15s\PR*{\sum_{i=1}^{\ceil{\log_2(s)}-1} \!\!\!\! \tfrac{1}{2^{i}}}
+ \PR*{\sum_{i=1}^{\ceil{\log_2(s)}-1} \!\!\! 12}
+ \tfrac{3s^2}{2}+3s + \tfrac{11}{2}
\\&\leq
\tfrac{9s^2}{6}
+ 15s
+ 12\ceil{\log_2(s)}- 12
+ \tfrac{3s^2}{2}+3s + \tfrac{11}{2}
\\&=
3s^2+ 18s+ 12\ceil{\log_2(s)}- \tfrac{13}{2}
\ifnocf.
\end{split}
\end{equation}
\cfload[.
Combining
this
with
the fact that
$\paramANN(\maxANN_2)=13\leq 12+ 36+ 12- \tfrac{13}{2}$
establishes
\cref{max_d:item_6}.
\finishproofthus
\end{aproof}
\subsection{Maxima in the spaces of ANN approximable functions}
\label{Subsection:5.2}
\cfclear
\begin{athm}{cor}{Coro:max_d_cost}
Let $d \in \N$ and let $F \colon \R^d \to \R^d$ satisfy for all $x = (x_1, x_2, \ldots, x_d) \in \R^d$ that $F(x) = \prb{x_1, \max\{x_1,x_2\}, \ldots, \max\{x_1,x_2,\dots,x_d\}}$.
Then $\CostLipB{\ReLU}{\R^d}{F}{\sqrt{d}}{0} \leq 3d^4+30d^3$
\cfout.
\end{athm}
\begin{proof}[Proof of \cref{Coro:max_d_cost}]
Throughout this proof
assume w.l.o.g.\ that
$d\geq 2$,
let
$A \in \R^{d^2 \times d}$
satisfy
for all
$x=(x_1,x_2, \ldots, x_d) \in \R^d$ that
\begin{equation}
\begin{split}
Ax
=
(\underbrace{x_1, x_1, \ldots, x_1}_d,
\underbrace{x_1,x_2, x_2, \ldots, x_2}_d,
\ldots,
\underbrace{x_1, x_2, \ldots, x_d}_d)
,
\end{split}
\end{equation}
and let
$\mathscr{g} \in \ANNs$
satisfy
$\mathscr{g} = \compANN{\PR{\parallelizationSpecial_{d}(\maxANN_d, \maxANN_d, \ldots, \maxANN_d)}}{ \affineANN_{A,0}}$
\cfadd{Prop:identity_representation}\cfload.
\Nobs that
\cref{Lemma:PropertiesOfParallelizationEqualLength},
\cref{Lemma:PropertiesOfCompositions_n2}, and
\cref{Prop:max_d}
ensure that for all
$x \in \R^d$
it holds that
\begin{equation}\eqlabel{eq:F}
\realisation_{\ReLU}(\mathscr{g})\in C(\R^d,\R^d)
\qquad\text{and}\qquad
(\realisation_{\ReLU}(\mathscr{g}))(x) = F(x)
\ifnocf.
\end{equation}
\cfload[.
This and
\cref{lem:max_d_lipschitz}
imply that for all
$x=(x_1,x_2,\ldots,x_d)$, $y=(y_1,y_2,\ldots,y_d) \in \R^d$
it holds that
\begin{equation}
\label{cor:max_d_lipschitz}
\begin{split}
\norm{(\realisation_{\ReLU}(\mathscr{g}))(x) - (\realisation_{\ReLU}(\mathscr{g}))(y)}
&=
\PR*{\ssum_{i=1}^{d} \vass{\max\{x_1,x_2,\dots,x_i\}-\max\{y_1,y_2,\dots,y_i\}}^2}^{\frac{1}{2}}
\\&\leq
\PR*{\ssum_{i=1}^d \norm{(x_1,x_2,\dots,x_i)-(y_1,y_2,\dots,y_i)}^2}^{\frac12}
\\&\leq
\PR*{\ssum_{i=1}^{d} \norm{x-y}^2}^{\frac{1}{2}}
=
\sqrt{d} \norm{x-y}
\end{split}
\end{equation}
\cfload.
Observe that
\cref{lem:dimcomp}
demonstrates that for all
$i\in\N$
it holds that
$
\singledims_0(\mathscr{g})
=
d\leq d^2
=
\singledims_0\pr*{\parallelizationSpecial_{d}(\maxANN_d, \maxANN_d, \ldots, \maxANN_d)}
$ and
\begin{equation}
\singledims_i(\mathscr{g})
=
\singledims_i\pr*{\parallelizationSpecial_{d}(\maxANN_d, \maxANN_d, \ldots, \maxANN_d)}
.
\end{equation}
Combining
this,
\cref{Lemma:ParallelizationImprovedBoundsOne}, and
\cref{Prop:max_d}
with
the fact that
for all
$n\in\N$
it holds that
$\log_2(n)\leq n$
ensures that
\begin{equation}
\begin{split}
\paramANN(\mathscr{g})
&\leq
\paramANN\pr*{\parallelizationSpecial_{d}(\maxANN_d, \maxANN_d, \ldots, \maxANN_d)}
\\&\leq
d^2\,\paramANN(\maxANN_d)
\\&\leq
d^2\prb{3d^2 + 18d + 12\ceil{\log_2(d)} - \tfrac{13}{2}}
\\&\leq
d^2\prb{3d^2 + 18d + 12d}
\\&=
3d^4+30d^3
\end{split}
\end{equation}
\cfload.
This,
\eqref{cor:max_d_lipschitz}, and
\eqqref{eq:F}
prove that
$\CostLipB{\ReLU}{{\R^d}}{F}{\sqrt{d}}{0} \leq 3d^4+30d^3$ \cfload.
This completes the proof of \cref{Coro:max_d_cost}.
\end{proof}
\cfclear
\begin{athm}{cor}{Coro:max_d_classes}
Let
$F \colon \prb{\bigcup_{d \in \N}\R^d} \to \prb{\bigcup_{d \in \N}\R^d}$
satisfy for all
$d \in \N$,
$x = (x_1, x_2, \allowbreak\ldots, x_d) \in \R^d$
that
$F(x) = \prb{x_1, \max\{x_1,x_2\}, \ldots, \max\{x_1,x_2,\dots,x_d\}}$
\cfload.
Then
$F \in \Dapprox$
\cfout.
\end{athm}
\begin{proof}[Proof of \cref{Coro:max_d_classes}]
\Nobs that
\cref{Lemma:Monotonicity_of_Cost} and
\cref{Coro:max_d_cost}
ensure that for all
$d,\radius \in \N$,
$\varepsilon \in (0,1]$
it holds that
\begin{equation}
\begin{split}
&\CostLipB{\ReLU}{{[-\radius,\radius]^d}}{\pr{[-\radius,\radius]^d \ni x \mapsto F(x) \in \R^d}}{33d^{33}\! \radius^{33}}{\varepsilon}
\\&\leq
\CostLipB{\ReLU}{{[-\radius,\radius]^d}}{\pr{\R^d \ni x \mapsto F(x) \in \R^d}}{\sqrt{d}}{0}
\\&\leq
3d^4+30d^3
\leq
33d^{4}\!\radius \varepsilon^{-1}
\leq
33d^{33}\!\radius^{33} \varepsilon^{-33}
\ifnocf.
\end{split}
\end{equation}
\cfload[.
Combining
this and
the fact that
for all
$d,\radius\in\N$
it holds that
$F([-\radius,\radius]^d)\subseteq[-\radius,\radius]^d$
with
\eqref{polyD_mult:1}
assures that
$F \in \Dapprox$ \cfload.
This completes the proof of \cref{Coro:max_d_classes}.
\end{proof}
\cfclear
\begin{athm}{cor}{Coro:max_d_1bis}
Let
$d \in \N$.
Then
$\CostLip{\ReLU}{\R^d}{\Clip{-1}1d}{1}{0} \leq 3d^2+3d$
\cfout.
\end{athm}
\begin{aproof}
\Nobs that
\cref{lem:clipping_function}
implies that
$\CostLip{\ReLU}{\R^d}{\Clip{-1}1d}{1}{0} \leq 3d^2+3d$ \cfload.
\finishproofthus
\end{aproof}
\section{ANN approximations for products}
\label{Section:6}
In this section we establish
in \cref{Prop:new_product_d}
in \cref{Subsection:6.3} below and
in \cref{Coro:prod_d_cost}
in \cref{Subsection:6.4}
new bounds for the cost
of deep ReLU ANN approximations of the multi-dimensional product functions.
Results related to these findings can be found, e.g., in
Yarotsky~\cite[Subsection~3.1]{yarotsky2017error},
Schwab \& Zech~\cite[Proposition~3.3]{SchwabZech2019}, and
Cheridito et al.~\cite[Proposition~33]{cheridito2021efficient}.
In \cref{Subsection:6.1} we introduce ReLU ANN approximations of the square
function and in \cref{Subsection:6.2} we present ReLU ANN approximations
for the two-dimensional product function. These results are essentially
well-known.
In particular, \cref{Lemma:square_01} is
proved in Grohs et al.~\cite[Section~5]{GrohsIbrgimovJentzen2021}
(see also Yarotsky \cite[Proposition~2]{yarotsky2017error},
Grohs et al.~\cite[Section~3]{GrohsHornungJentzen2019},
and Cheridito et al.~\cite[Proposition~16]{cheridito2021efficient}).
Finally, in \cref{Subsection:6.4} we use these results
to show that certain sequences of functions involving products
are in the approximation spaces introduced in
\cref{Section:3} above.
\subsection{ANN approximations for the square function on compact intervals}
\label{Subsection:6.1}
\begin{athm}{lemma}{Lemma:square_01}
Let $\varepsilon \in (0,\infty)$.
Then there exists $\mathscr{f} \in \ANNs$ such that
\begin{enumerate}[(i)]
\item\label{square_01:item_1} it holds that $\functionReLUANN{(\mathscr{f})} \in C(\R,\R)$,
\item\label{square_01:item_4} it holds for all $x,y \in \R$ that $\vass{\functionnbReLUANN{\mathscr{f}}(x) - \functionnbReLUANN{\mathscr{f}}(y)} \leq 2\vass{x-y}$,
\item\label{square_01:item_2} it holds that $\sup_{x \in [0,1]} \vass{x^2 - \functionnbReLUANN{\mathscr{f}}(x)} \leq \varepsilon$,
\item\label{square_01:item_7} it holds that $\lengthANN(\mathscr{f}) = \max\pRb{2, \ceil[\big]{\tfrac{1}{2} \log_2(\varepsilon^{-1})} }$,
\item\label{square_01:item_6} it holds that $\dims(\mathscr{f}) = (1, 4, 4, \ldots, 4, 1)$,
and
\item\label{square_01:item_5} it holds that $\paramANN(\mathscr{f}) \leq \max\{13,10\log_2(\varepsilon^{-1}) - 7\}$
\end{enumerate}
(cf.\ \cref{def:ANN,def:ANNrealization,def:ReLU}).
\end{athm}
\begin{aproof}
\Nobs that
Lemmas 5.1 and 5.2 in Grohs et al.\ \cite{GrohsIbrgimovJentzen2021}
prove \cref{square_01:item_1,square_01:item_2
square_01:item_5,square_01:item_6,square_01:item_7,square_01:item_4}.
\finishproofthus
\end{aproof}
\cfclear
\begin{athm}{lemma}{Lemma:square_r}
Let
$\radius, \varepsilon \in (0,\infty)$.
Then there exists
$\mathscr{f} \in \ANNs$
such that
\begin{enumerate}[(i)]
\item\label{Square_r:item_1}
it holds that
$\functionReLUANN{(\mathscr{f})} \in C(\R,\R)$,
\item\label{Square_r:item_4}
it holds for all
$x,y \in \R$
that
$
\vass{\functionnbReLUANN{\mathscr{f}}(x) - \functionnbReLUANN{\mathscr{f}}(y)}
\leq
2\radius \vass{x-y}
$,
\item\label{Square_r:item_2}
it holds that
$
\sup_{x \in [-\radius, \radius]} \vass{x^2 - \functionnbReLUANN{\mathscr{f}}(x)}
\leq
\varepsilon
$,
\item\label{Square_r:item_7}
it holds that
$
\lengthANN(\mathscr{f})
=
\max\pRb{3, \ceil[\big]{ \log_2 (\radius)-\tfrac{1}{2} \log_2(\varepsilon) + 1}}
$,
\item\label{Square_r:item_6}
it holds that
$\dims(\mathscr{f}) = (1, 2, 4, 4, \ldots, 4, 1)$,
and
\item\label{Square_r:item_5}
it holds that
$
\paramANN(\mathscr{f})
\leq
\max\{21, 20\log_2 (\radius) - 10\log_2(\varepsilon) +1\}
$
\end{enumerate}
\cfout.
\end{athm}
\begin{aproof}
Throughout this proof
let
$\varphi_1\in\pr*{\pr*{\R^{2\times 1}\times\R^2}\times\pr*{\R^{1\times 2}\times\R}}\subseteq\ANNs$,
$\varphi_2 \in\pr*{\R^{1\times 1}\times\R}\subseteq \ANNs$
satisfy
\begin{equation}
\label{Lemma:square_r:networks}
\varphi_1
=
\pr*{\!\pr*{\!
\begin{pmatrix}
\radius^{-1}\\
-\radius^{-1}
\end{pmatrix}
,
\begin{pmatrix}
0\\
0
\end{pmatrix}
\!}
,
\pr*{
\begin{pmatrix}
1 & 1
\end{pmatrix}
,
0
}\!}\qandq
\varphi_2
=
\affineANN_{\radius^2,0}
\end{equation}
\cfload.
\Nobs that
\eqref{Lemma:square_r:networks}
implies that
\begin{equation}
\llabel{eq:dims}
\dims(\varphi_1)
=
(1,2,1)
\qandq
\dims(\varphi_2)
=
(1,1)
.
\end{equation}
\Moreover
\eqref{Lemma:square_r:networks}
ensures that for all
$x \in \R$
it holds that
\begin{equation}
\label{Lemma:square_r:eq:varphi}
\functionReLUANN(\varphi_1), \functionReLUANN{(\varphi_2)} \in C(\R,\R),
\quad
\functionnbReLUANN{\varphi_1}(x)
=
\frac{\vass{x}}{\radius},
\quad\text{and}\quad
\functionnbReLUANN{\varphi_2}(x)
=
\radius^2x
\end{equation}
\cfload.
\Moreover
\cref{Lemma:square_01} (applied with
$\varepsilon \curvearrowleft \frac{\varepsilon}{\radius^2}$
in the notation of \cref{Lemma:square_01})
shows that there exists
$\mathscr{g} \in \ANNs$
which satisfies that
\begin{enumerate}[(I)]
\item
it holds that
$\functionReLUANN{(\mathscr{g})} \in C(\R,\R)$,
\item\label{Square_r:Lip}
it holds for all
$x,y \in \R$
that
$
\abs{\functionnbReLUANN{\mathscr{g}}(x) - \functionnbReLUANN{\mathscr{g}}(y)}
\leq
2\abs{x-y}
$,
\item\label{Square_r:approx}
it holds that
$
\sup_{x \in [0,1]} \vass{x^2 - \functionnbReLUANN{\mathscr{g}}(x)}
\leq
\frac{\varepsilon}{\radius^2}
$,
\item\label{square_t:7}
it holds that
$
\lengthANN(\mathscr{f})
=
\max\pRb{2, \ceil[\big]{ \log_2 (\radius)-\tfrac{1}{2} \log_2(\varepsilon)}}
$,
\item\label{Square_r:6}
it holds that
$\dims(\mathscr{g}) = (1, 4, 4, \ldots, 4, 1)$,
and
\item\label{Square_r:param}
it holds that
$
\paramANN(\mathscr{g})
\leq
\max\{13,20\log_2(\radius)-10\log_2(\varepsilon)-7\}
$\ifnocf.
\end{enumerate}
\cfload[.
Next let $\mathscr{f} \in \ANNs$ satisfy
\begin{equation}
\label{Square_r:phi}
\mathscr{f}
=
\compANN{\compANN{\varphi_2}{\mathscr{g}}}{\varphi_1}
\end{equation}
\cfload.
\Nobs that
\lref{eq:dims},
\eqref{Square_r:phi},
\cref{square_t:7},
\cref{Square_r:6}, and
\cref{lem:dimcomp}
assure that
\begin{equation}
\label{Square_r:eq_pr_-1}
\lengthANN(\mathscr{f})
=
\max\pRb{3, \ceil[\big]{ \log_2 (\radius)-\tfrac{1}{2} \log_2(\varepsilon) + 1}}
\qandq
\dims(\mathscr{f})
=
(1, 2, 4, 4, \ldots, 4, 1)
.
\end{equation}
\Moreover
\eqref{Lemma:square_r:eq:varphi},
\eqref{Square_r:phi},
and
\cref{Lemma:PropertiesOfCompositions_n2}
prove that for all
$x \in \R$
it holds that
\begin{equation}
\label{Square_r:phi_reduced}
\functionReLUANN{(\mathscr{f})} \in C(\R,\R)
\qandq
\functionnbReLUANN{\mathscr{f}}(x)
=
\radius^2 \PRb{\functionnbReLUANN{\mathscr{g}}\prb{\tfrac{\vass{x}}{\radius}}}
\ifnocf.
\end{equation}
\cfload[.
Combining
this
with
\cref{Square_r:approx}
demonstrates that
\begin{equation}
\begin{split}
\label{Square_r:eq_pr_0}
&
\sup\nolimits_{x \in [-\radius, \radius]} \vass{x^2 - \functionnbReLUANN{\mathscr{f}}(x)} = \sup\nolimits_{x \in [-\radius, \radius]} \vass[\big]{\radius^2\PRb{\tfrac{\vass{x}}{\radius}}^2 - \radius^2 \PRb{\functionnbReLUANN{\mathscr{g}}\prb{\tfrac{\vass{x}}{\radius}}}}
\\&=
\sup\nolimits_{y \in [0,1]} \vass[\big]{\radius^2 y^2 - \radius^2 \PRb{\functionnbReLUANN{\mathscr{g}}(y)}} = \radius^2 \PRb{ \sup\nolimits_{x \in [0,1]} \vass{x^2 - \functionnbReLUANN{\mathscr{g}}(x)} }
\leq
\varepsilon
.
\end{split}
\end{equation}
\Moreover
\eqref{Square_r:phi_reduced} and
\cref{Square_r:Lip}
imply that for all
$x,y\in\R$
it holds that
\begin{equation}
\begin{split}
\label{Square_r:eq_pr_2}
&
\vass*{\functionnbReLUANN{\mathscr{f}}(x)-\functionnbReLUANN{\mathscr{f}}(y) }
\\&=
\radius^2\vass[\big]{\functionnbReLUANN{\mathscr{g}}\prb{\tfrac{\vass{x}}{\radius}}-\functionnbReLUANN{\mathscr{g}}\prb{\tfrac{\vass{y}}{\radius}}}
\leq
2\radius^2\vass[\big]{\tfrac{\vass{x}}{\radius}-\tfrac{\vass{y}}{\radius}}
\leq
2\radius\vass{x-y}.
\end{split}
\end{equation}
\Moreover
\eqref{Lemma:square_r:eq:varphi},
\cref{Square_r:param}, and
\cref{Lemma:PropertiesOfCompositions_n1}
show that
\begin{equation}
\begin{split}
\paramANN(\mathscr{f})
&\leq
\paramANN(\mathscr{g}) + 2(1+1)+1(2+1)+4(2+1) - 1(2+1) - 4(1+1)
\\&\leq
\max\{13+8,20\log_2 (\radius) - 10\log_2(\varepsilon) -7 + 8\}
\\&\leq
\max\{21,20\log_2 (\radius) - 10\log_2(\varepsilon) +1\}
\ifnocf.
\end{split}
\end{equation}
\cfload[.
Combining
this
with
\eqref{Square_r:eq_pr_-1},
\eqref{Square_r:phi_reduced},
\eqref{Square_r:eq_pr_0}, and
\eqref{Square_r:eq_pr_2}
establishes
\cref{Square_r:item_1,Square_r:item_2
Square_r:item_4,Square_r:item_5,Square_r:item_6,Square_r:item_7}.
\finishproofthus
\end{aproof}
\subsection{ANN approximations for two-dimensional products}
\label{Subsection:6.2}
\cfclear
\begin{athm}{lemma}{Lemma:prod_r}
Let $\radius, \varepsilon \in (0,\infty)$.
Then there exists $\mathscr{f} \in \ANNs$ such that
\begin{enumerate}[(i)]
\item\label{prod_r:item_well_def}
it holds that
$\functionReLUANN{(\mathscr{f})} \in C(\R^2,\R)$,
\item\label{prod_r:item_Lip}
it holds for all
$x,y \in \R^2$
that
$\vass{\functionnbReLUANN{\mathscr{f}}(x) - \functionnbReLUANN{\mathscr{f}}(y)} \leq \sqrt{32}\radius \norm{x-y}$,
\item\label{prod_r:item_approx_int}
it holds that
$\sup_{x,y \in [-\radius, \radius]} \vass{xy - \functionnbReLUANN{\mathscr{f}}(x,y)} \leq \varepsilon$,
\item\label{prod_r:item_7}
it holds that
$\lengthANN(\mathscr{f}) = \max\pRb{3, \ceil[\big]{\log_2 (\radius)-\tfrac{1}{2} \log_2(\varepsilon) + \tfrac{3}{2} + \tfrac{1}{2}\log_2(3)}}$,
\item\label{prod_r:item_6}
it holds that
$\dims(\mathscr{f}) = (2, 6, 12, 12, \ldots, 12, 1)$,
and
\item\label{prod_r:item_cost}
it holds that
$\paramANN(\mathscr{f}) \leq \max\{157 , 211 + 180\log_2 (\radius) - 90\log_2(\varepsilon)\}$
\end{enumerate}
\cfout.
\end{athm}
\begin{proof}
[Proof of \cref{Lemma:prod_r}.]
Throughout this proof let $\varphi_1 \in\pr*{\R^{3\times 2}\times\R^3}\subseteq \ANNs$, $\varphi_2 \in\pr*{\R^{1\times 3}\times\R}\subseteq \ANNs$ satisfy
\begin{equation}
\label{Lemma:prod_r:eq:networks}
\varphi_1 = \pr*{ \!\begin{pmatrix} 1 & 1 \\ 1 & 0 \\ 0 & 1 \end{pmatrix}, \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}\!}\qandq
\varphi_2 = \pr*{ \tfrac{1}{2}\begin{pmatrix} 1 & -1 & -1 \end{pmatrix}, 0}
\end{equation}
\cfload. \Nobs that \eqref{Lemma:prod_r:eq:networks} ensures that
\begin{equation}
\label{Lemma:prod_r:eq:varphi_new}
\dims(\varphi_1) = (2, 3),\;\;
\dims(\varphi_2) = (3,1),\;\;
\functionReLUANN{(\varphi_1)} \in C(\R^2,\R^3),
\;\;\text{and}\;\;
\functionReLUANN{(\varphi_2)} \in C(\R^3,\R)
\ifnocf.
\end{equation}
\cfload[.
\Moreover \cref{Lemma:prod_r:eq:networks} implies that for all $x,y,z \in \R$ it holds that
\begin{equation}
\label{Lemma:prod_r:eq:varphi}
\functionnbReLUANN{\varphi_1}(x,y) = (x+y,x,y),\qandq\functionnbReLUANN{\varphi_2}(x,y,z) = \frac{x-y-z}{2}
\ifnocf.
\end{equation}
\cfload[.]\Nobs that \cref{Lemma:square_r} (applied with
$\radius \curvearrowleft 2\radius$,
$\varepsilon \curvearrowleft \tfrac{2\varepsilon}{3}$
in the notation of \cref{Lemma:square_r}) shows that there exists $\mathscr{g} \in \ANNs$ which satisfies that
\begin{enumerate}[(I)]
\item \label{prod_r:1} it holds that $\functionReLUANN{(\mathscr{g})} \in C(\R,\R)$,
\item \label{prod_r:3} it holds for all $x,y \in \R$ that $ \vass{ \functionnbReLUANN{\mathscr{g}}(x) - \functionnbReLUANN{\mathscr{g}}(y) } \leq 4\radius \vass{x-y}$,
\item \label{prod_r:2} it holds that $\sup_{x \in [-2\radius,2\radius]} \vass{x^2 - \functionnbReLUANN{\mathscr{g}}(x)} \leq \frac{2\varepsilon}{3}$,
\item \label{prod_r:4} it holds that $\lengthANN(\mathscr{g}) = \max\pRb{3, \ceil[\big]{ \log_2 (\radius)-\tfrac{1}{2} \log_2(\varepsilon) + \tfrac{3}{2} + \tfrac{1}{2}\log_2(3)}}$,
\item \label{prod_r:6} it holds that $\dims(\mathscr{g}) = (1,2,4, 4, \ldots, 4, 1)$,
and
\item\label{prod_r:param} it holds that $\paramANN(\mathscr{g}) \leq \max\{21,20\log_2 (\radius) - 10\log_2(\varepsilon) + 10\log_2(3) +11\}$.
\end{enumerate}
Next let $\mathscr{f} \in \ANNs$ satisfy
\begin{equation}
\label{prod_r:phi}
\mathscr{f} = \compANN{\compANN{\varphi_2}{(\parallelizationSpecial_{3}(\mathscr{g},\mathscr{g},\mathscr{g}))}}{\varphi_1
\end{equation}
\cfload.
\Nobs that
\eqref{Lemma:prod_r:eq:varphi_new},
\eqref{prod_r:phi},
\cref{prod_r:6},
\cref{Lemma:PropertiesOfParallelizationEqualLengthDims},
\cref{Lemma:PropertiesOfCompositions_n2}, and
\cref{lem:dimcomp}
ensure that
\begin{equation}
\label{prod_r:phi:eq1}
\lengthANN(\mathscr{f}) = \lengthANN(\mathscr{g})
\qandq
\dims(\mathscr{f}) = (2, 6, 12, 12, \ldots, 12, 1)
.
\end{equation}
Next \nobs that \cref{Lemma:PropertiesOfCompositions_n2} and \eqref{Lemma:prod_r:eq:varphi} prove that for all $x,y\in\R$ it holds that $\functionReLUANN{(\mathscr{f})} \in C(\R^2,\R)$ and
\begin{equation}
\label{prod_r:phi:prep}
\functionnbReLUANN{\mathscr{f}}(x,y)
=
\tfrac{1}{2}\PRb{\functionnbReLUANN{\mathscr{g}}(x+y)
-\functionnbReLUANN{\mathscr{g}}(x)
-\functionnbReLUANN{\mathscr{g}}(y)}.
\end{equation}
This, \cref{prod_r:2}, and \cref{Lemma:PropertiesOfCompositions_n2} demonstrate that for all $x,y \in [-\radius, \radius]$ it holds that
\begin{equation}
\begin{split}
\label{prod_r:phi:eq2}
&\vass{xy - \functionnbReLUANN{\mathscr{f}}(x,y)}
\\&= \tfrac{1}{2}\abs[\big]{(x+y)^2 - x^2 - y^2 - \functionnbReLUANN{\mathscr{g}}(x+y) + \functionnbReLUANN{\mathscr{g}}(x) + \functionnbReLUANN{\mathscr{g}}(y)} \\
&\leq \tfrac{1}{2} \vass{(x+y)^2 - \functionnbReLUANN{\mathscr{g}}(x+y)} + \tfrac{1}{2}\vass{x^2 - \functionnbReLUANN{\mathscr{g}}(x)}
+ \tfrac{1}{2}\vass{y^2 - \functionnbReLUANN{\mathscr{g}}(y)} \leq \varepsilon.
\end{split}
\end{equation}
\Moreover
\eqref{prod_r:phi:prep} and
\cref{prod_r:3}
show that for all
$x_1,x_2,y_1,y_2\in \R$
it holds that
\begin{equation}
\begin{split}
\label{prod_r:phi:eq3}
&\vass{\functionnbReLUANN{\mathscr{f}}(x_1,x_2) - \functionnbReLUANN{\mathscr{f}}(y_1,y_2)}
\\&\leq
\tfrac{1}{2}\prb{ \vass{\functionnbReLUANN{\mathscr{g}}(x_1+x_2) - \functionnbReLUANN{\mathscr{g}}(y_1+y_2)} \\
&\quad+ \vass{\functionnbReLUANN{\mathscr{g}}(x_1) - \functionnbReLUANN{\mathscr{g}}(y_1)} + \vass{\functionnbReLUANN{\mathscr{g}}(x_2) - \functionnbReLUANN{\mathscr{g}}(y_2)} } \\
&\leq
2\radius \prb{ \vass{(x_1+x_2)-(y_1+y_2)} + \vass{x_1-y_1} + \vass{x_2-y_2} } \\
&\leq
4\radius \pr*{ \vass{x_1-y_1} + \vass{x_2-y_2} }
\leq
\sqrt{32}\radius \norm{(x_1-y_1,x_2-y_2)}
\end{split}
\end{equation}
\cfload.
\Moreover
the fact that
$3\leq 256^{\nicefrac{1}{5}}=2^{\nicefrac{8}{5}}$
implies that
$\log_2(3)\leq\tfrac{8}{5}$.
Combining
\eqref{prod_r:phi},
\cref{prod_r:param},
and \cref{Lemma:ParallelizationImprovedBoundsOne}
with
\cref{Lemma:PropertiesOfCompositions_n1}
therefore
ensures that
\begin{equation}
\begin{split}
\paramANN({\mathscr{f}})
&\leq
\paramANN(\varphi_2)+ 9 \paramANN(\mathscr{g}) +\paramANN(\varphi_1)+1(12+1)+6(2+1) - 1(3+1)-6(3+1)
\\&\quad -3(12+1)-3(2+1)
\\&= 9 \paramANN(\mathscr{g}) +44 -76
\\&\leq
\max\{189-32, 9(11 + 10\log_2(3)) -32 + 180\log_2 (\radius) - 90\log_2(\varepsilon)\}
\\&\leq
\max\{157 , 211 + 180\log_2 (\radius) - 90\log_2(\varepsilon)\}
.
\end{split}
\end{equation}
This,
\eqref{prod_r:phi:eq1},
\eqref{prod_r:phi:prep},
\eqref{prod_r:phi:eq2},
\eqref{prod_r:phi:eq3}, and
\cref{prod_r:4}
establish
\cref{prod_r:item_Lip,prod_r:item_7,prod_r:item_cost,prod_r:item_6,prod_r:item_well_def,prod_r:item_approx_int}.
The proof of \cref{Lemma:prod_r} is thus complete\cfload.
\end{proof}
\subsection{ANN approximations for multi-dimensional products}
\label{Subsection:6.3}
\cfclear
\begingroup
\begin{athm}{lemma}{Lem:product_d}
Let
$d \in \N $,
$\radius, \varepsilon \in (0, \infty)$.
Then there exists
$\mathscr{f} \in \ANNs$
such that
\begin{enumerate}[(i)]
\item \llabel{it:realization}
it holds that
$\realisation_\ReLU(\mathscr{f}) \in C(\R^{2d},\R^d)$,
\item \llabel{it:lipschitz}
it holds for all
$x,y \in \R^{2d}$
that
$
\norm{\functionnbReLUANN{\mathscr{f}} (x) - \functionnbReLUANN{\mathscr{f}} (y)}
\leq
\sqrt{32}\radius\mednorm{x-y}
$,
\item \llabel{it:approx}
it holds for all
$x = (x_1,x_2, \ldots, x_{2d}) \in [-\radius, \radius]^{2d}$
that
\begin{equation}
\norm*{\pr*{x_1x_2,x_3x_4,\ldots,x_{2d-1}x_{2d}} - \functionnbReLUANN{\mathscr{f}} (x)}
\leq
\varepsilon,
\end{equation}
\item \llabel{it:length}
it holds that
$
\lengthANN(\mathscr{f})
=
\max\pRb{3, \ceil[\big]{\log_2(\radius)-\tfrac12 \log_2(\varepsilon) + \tfrac32 + \tfrac12\log_2(3)}}
$,
\item \llabel{it:dims}
it holds that
$\dims(\mathscr{f}) = (2d, 6d, 12d, 12d, \ldots, 12d, d)$, and
\item \llabel{it:params}
it holds that
$
\paramANN(\mathscr{f})
\leq
d^2\max\{157 , 211 + 45\pr{4\log_2(R) - 2\log_2(\varepsilon)+\log_2(d)}\}
$
\end{enumerate}
\cfout.
\end{athm}
\newcommand{\g}{\mathscr{g}}
\begin{aproof}
\Nobs that
\cref{Lemma:prod_r}
(applied with
$\radius \curvearrowleft \radius$,
$\varepsilon \curvearrowleft d^{-\frac{1}{2}}\varepsilon$
in the notation of \cref{Lemma:prod_r})
proves that there exists
$\g\in \ANNs$
which satisfies that
\begin{enumerate}[(I)]
\item \llabel{it:single_realization}
it holds that
$\realisation_{\ReLU}({\g}) \in C(\R^2,\R)$,
\item \llabel{it:single_lipschitz}
it holds for all
$x,y\in\R^{2}$
that
$
\vass{\functionnbReLUANN{\g}(x) - \functionnbReLUANN{\g}(y)}
\leq
\sqrt{32} R \mednorm{x-y}
$,
\item \llabel{it:single_approx}
it holds that
$
\sup_{x,y \in [-R, R] } \vass{xy - \functionnbReLUANN{\g}(x,y)}
\leq
d^{-\frac{1}{2}}\varepsilon
$,
\item \llabel{it:single_length}
it holds that
$
\lengthANN(\g)
=
\max\pRb{3, \ceil[\big]{\log_2(\radius)-\tfrac12 \log_2(\varepsilon) + \tfrac32 + \tfrac12\log_2(3)}}
$,
\item \llabel{it:single_dims}
it holds that
$\dims(\g) = (2, 6, 12, 12, \ldots, 12, 1)$,
and
\item \llabel{it:single_params}
it holds that
$
\paramANN(\g)
\leq
\max\{157 , 211 + 180\log_2(R) - 90\log_2(\varepsilon)+45\log_2(d)\}
$
\end{enumerate}
\cfload.
Next let
$\mathscr{f} \in \ANNs$
satisfy
\begin{equation}
\begin{split}
\llabel{eq:fdef}
\mathscr{f}
&=
\parallelizationSpecial_{d} \pr*{\g,\g,\ldots,\g}
\ifnocf.
\end{split}
\end{equation}
\cfload[.
\Nobs that
\lref{eq:fdef},
\lref{it:single_realization},
\lref{it:single_dims}, and
\cref{Lemma:PropertiesOfParallelizationEqualLengthDims}
ensure that
\begin{equation}
\begin{split}
\llabel{eq:realdims}
\lengthANN(\mathscr f)=\lengthANN(\g)
\qandq
\dims(\mathscr{f})
=
d\dims(\g)=(2d, 6d, 12d, 12d, \ldots, 12d, d)
.
\end{split}
\end{equation}
\Moreover
\lref{eq:fdef},
\lref{it:single_params}, and
\cref{Lemma:ParallelizationImprovedBoundsOne}
imply that
\begin{equation}
\begin{split}
\llabel{eq:params}
\paramANN(\mathscr{f})
\leq
d^2\paramANN(\g)
\leq
d^2\max\{157 , 211 + 180\log_2(R) - 90\log_2(\varepsilon)+45\log_2(d)\}
.
\end{split}
\end{equation}
\Moreover
\lref{eq:fdef},
\lref{it:single_approx}, and
\cref{Lemma:PropertiesOfParallelizationEqualLength}
show that for all
$x=(x_1,x_2,\ldots,\allowbreak x_{2d})\in[-R,R]^{2d}$
it holds that
$\realisation_\ReLU(\mathscr{f}) \in C(\R^{2d},\R^d)$
and
\begin{equation}
\begin{split}
\llabel{eq:approx}
&
\norm*{\pr*{x_1x_2,x_3x_4,\ldots,\allowbreak x_{2d-1}x_{2d}} - \functionnbReLUANN{\mathscr{f}} (x)}
\\&=
\PR*{\ssum_{i=1}^d\vass*{x_{2i-1}x_{2i} - \functionnbReLUANN{\g} (x_{2i-1},x_{2i})}^2}^{\frac{1}{2}}
\leq
\PR*{\ssum_{i=1}^d d^{-1}\varepsilon^2}^{\frac{1}{2}}
=
\varepsilon
.
\end{split}
\end{equation}
Next we combine
\lref{eq:fdef},
\lref{it:single_lipschitz}, and
\cref{Lemma:PropertiesOfParallelizationEqualLength}
with
\cref{lemma:Lipschitz_error}
(applied with
$L\is \sqrt{32}R$,
$d\is d$,
$(g_1,g_2,\dots,g_d)\is(\realisation_{\ReLU}({\g}),\realisation_{\ReLU}({\g}),\dots,\realisation_{\ReLU}({\g}))$,
$f\is \realisation_{\ReLU}(\mathscr f)$
in the notation of \cref{lemma:Lipschitz_error})
to obtain that for all
$x,y\in\R^{2d}$
it holds that
\begin{equation}
\norm{\functionnbReLUANN{\mathscr f}(x) - \functionnbReLUANN{\mathscr f}(y)}
\leq
\sqrt{32} R \mednorm{x-y}
.
\end{equation}
This,
\lref{eq:realdims},
\lref{eq:params},
\lref{eq:approx}, and
\lref{it:single_length}
establish
\cref{\loc.it:realization,\loc.it:lipschitz,\loc.it:approx,\loc.it:length,\loc.it:dims,\loc.it:params}.
\finishproofthus
\end{aproof}
\endgroup
\cfclear
\begin{athm}{lemma}{lem:comp_lipschitz_approx}
Let
$n\in\N$,
$d_0,d_1,\ldots,d_n\in\N$,
$L_1,L_2,\ldots,L_n,\varepsilon_1,\varepsilon_2,\ldots,\varepsilon_n\in[0,\infty)$,
let $D_i\subseteq\R^{d_{i-1}}$, $i\in\{1,2,\ldots,n\}$, be sets,
for every $i\in\{1,2,\ldots,n\}$ let
$f_i\colon D_i\to\R^{d_i}$ and
$g_i\colon \R^{d_{i-1}}\to\R^{d_i}$
satisfy for all $x\in D_i$ that
\begin{equation}
\label{lem:comp_lipschitz_approx:eq1}
\norm{f_i(x)-g_i(x)}\leq\varepsilon_i,
\end{equation}
and assume for all $j\in\N\cap(0,n)$, $x,y\in\R^{d_j}$ that
\begin{equation}
\label{lem:comp_lipschitz_approx:eq1.1}
f_j(D_j)\subseteq D_{j+1}
\qquad\text{and}\qquad
\norm{g_{j+1}(x)-g_{j+1}(y)}\leq L_{j+1}\norm{x-y}
\ifnocf.
\end{equation}
\cfload[.]Then it holds for all $x\in D_1$ that
\begin{equation}
\begin{split}
\label{lem:comp_lipschitz_approx:eq4}
&\norm{\pr*{f_n\circ f_{n-1}\circ\ldots\circ f_1}(x)-\pr*{g_n\circ g_{n-1}\circ\ldots\circ g_1}(x)}\leq\ssum_{i=1}^n\PRbbb{\prbbb{\sprod_{j=i+1}^n L_j}\varepsilon_i}.
\end{split}
\end{equation}
\end{athm}
\begin{proof}[Proof of \cref{lem:comp_lipschitz_approx}]
Throughout this proof
assume w.l.o.g.\ that
$n\geq 2$,
let
$F_i\colon D_1\to\R^{d_{i}}$, $i\in\{1,2,\ldots,n\}$,
satisfy
for all
$i\in\{1,2,\ldots,n\}$, $x\in D_1$
that
\begin{equation}
\label{lem:comp_lipschitz_approx:eq2}
F_i(x)=\pr*{f_i\circ f_{i-1}\circ\ldots\circ f_1}(x),
\end{equation}
and let
$G_i\colon \R^{d_{i-1}}\to\R^{d_n}$, $i\in\{1,2,\ldots,n\}$,
satisfy
for all
$i\in\{1,2,\ldots,n\}$,
$x\in \R^{d_{i-1}}$
that
\begin{equation}
\label{lem:comp_lipschitz_approx:eq3}
G_i(x)=\pr*{g_n\circ g_{n-1}\circ\ldots\circ g_i}(x).
\end{equation}
\Nobs that
\cref{lem:comp_lipschitz_approx:eq1.1} and
\eqref{lem:comp_lipschitz_approx:eq3}
imply that for all
$i\in\{2,3,\ldots,n\}$,
$x,y\in\R^{d_{i-1}}$
it holds that
\begin{equation}
\label{lem:comp_lipschitz_approx:eq5}
\norm{G_i(x)-G_i(y)}
\leq
\prbbb{\sprod_{j=i}^n L_j}\norm{x-y}
.
\end{equation}
Combining this with \cref{lem:comp_lipschitz_approx:eq1}, \cref{lem:comp_lipschitz_approx:eq2}, and \cref{lem:comp_lipschitz_approx:eq3} ensures that for all $x\in D_1$ it holds that
\begin{align}
&\nonumber\norm{F_n(x)-G_n(x)}
\\&\nonumber=\norm*{F_n(x)-G_n(F_{n-1}(x))+\PR*{\ssum_{i=2}^{n-1}\pr*{G_{i+1}\pr*{F_{i}(x)}-G_{i}\pr*{F_{i-1}(x)}}}+G_2(F_{1}(x))-G_{1}(x)}
\\&\nonumber\leq\norm*{F_n(x)-G_n(F_{n-1}(x))}+\PR*{\ssum_{i=2}^{n-1}\norm*{G_{i+1}\pr*{F_{i}(x)}-G_{i}\pr*{F_{i-1}(x)}}}+\norm*{G_2(F_{1}(x))-G_{1}(x)}
\\&\nonumber=\norm*{f_n(F_{n-1}(x))-g_n(F_{n-1}(x))}+\PR*{\ssum_{i=2}^{n-1}\norm*{G_{i+1}\pr*{f_i\pr*{F_{i-1}(x)}}-G_{i+1}\pr*{g_i\pr*{F_{i-1}(x)}}}}
\\&\quad+\norm*{G_2(f_{1}(x))-G_{2}(g_{1}(x))}
\\&\nonumber\leq\varepsilon_n+\PRbbb{\ssum_{i=2}^{n-1}\prbbb{\sprod_{j=i+1}^n L_j}\norm*{f_i\pr*{F_{i-1}(x)}-g_i\pr*{F_{i-1}(x)}}}+\prbbb{\sprod_{j=2}^n L_j}\norm*{f_{1}(x)-g_{1}(x)}
\\&\nonumber\leq\varepsilon_n+\PRbbb{\ssum_{i=2}^{n-1}\prbbb{\sprod_{j=i+1}^n L_i}\varepsilon_i}+\prbbb{\sprod_{j=2}^n L_j}\varepsilon_1
.
\end{align}
This completes the proof of \cref{lem:comp_lipschitz_approx}.
\end{proof}
\cfclear
\begingroup
\begin{athm}{lemma}{lem:log2_est}
Let
$d\in\N$.
Then
$\log_2\pr{d}\leq \frac{5d}{9}$.
\end{athm}
\begin{aproof}
Throughout this proof let
$f\in C((0,\infty),\R)$
satisfy for all
$x\in(0,\infty)$
that
$f(x)=\frac{x}{\log_2(x)}=\frac{\ln(2)x}{\ln(x)}$.
\Nobs that
the fact that
for all
$x\in[3,\infty)$
it holds that
$\ln(x)\geq 1$
ensures that for all
$x\in[3,\infty)$
it holds that
\begin{equation}
\llabel{log2_estimate:0}
\begin{split}
f'(x)
=
\ln(2)\PR*{\frac{1}{\ln(x)}-\frac{x}{x[\ln(x)]^2}}
=
\frac{\ln(2)}{\ln(x)}\PR*{1-\frac{1}{\ln(x)}}
\geq
0
.
\end{split}
\end{equation}
Combining
this
with
the fact that
$\log_2(3)\leq\log_2(32^{\nicefrac{1}{3}})=\frac{5}{3}$
and the fundamental theorem of calculus
shows that for all
$x\in[3,\infty)$
it holds that
\begin{equation}
\llabel{log2_estimate:1}
\begin{split}
f(x)
=
f(3)+\int_3^x f'(y)\,\mathrm{d}y
\geq f(3)
=
\frac{3}{\log_2(3)}
&\geq
\frac{9}{5}
.
\end{split}
\end{equation}
This,
the fact that
$\log_2(1)=0$,
and the fact that
$\log_2(2)=1\leq\frac{10}{9}$
establish that
$
\log_2\pr{d}
\leq
\frac{5d}{9}
$.
\finishproofthus
\end{aproof}
\endgroup
\cfclear
\begingroup
\newcommand{\f}{\mathscr f}
\begin{athm}{lemma}{lem:prod_pow2}
Let
$d\in\N$,
$R,\eps\in(0,\infty)$
\cfload.
Then there exists $\f\in\ANNs$ such that
\begin{enumerate}[(i)]
\item
\label{functiontoshow}
it holds that
$\realisation_\ReLU(\f)\in C\prb{\R^{(2^d)},\R}$,
\item
\label{lipschitztoshow}
it holds for all
$x,y\in\R^{(2^d)}$
that
$
\abs{(\realisation_\ReLU(\f))(x)-(\realisation_\ReLU(\f))(y)}
\leq
2^{\frac{5d}{2}}\! R^{(2^{d}-1)}
\norm{x-y}
$,
\item
\label{approxtoshow}
it holds for all
$x=(x_1,x_2,\dots,x_{2^d})\in[-R,R]^{(2^d)}$
that
\begin{equation}
\abs[\bigg]{\PRbbb{\sprod_{i=1}^{{2^d}} x_i}-(\realisation_\ReLU(\f))(x)}
\leq
\eps,
\end{equation}
and
\item
\label{paramtoshow}
it holds that
$\paramANN(\f)\leq 426d4^{d}+90\ReLU\pr*{\log_2(R)}8^{d}+90\ReLU\pr*{\log_2\pr*{\eps^{-1}}}4^{d}$
\end{enumerate}
\cfout.
\end{athm}
\newcommand{\h}[1]{\mathscr{h}_{#1}}
\begin{aproof}
Throughout this proof
assume w.l.o.g.\ that
$d\geq 2$,
for every $i\in\{1,2,\ldots,d\}$
let $D_i\subseteq\R^{\pr{2^{d-i+1}}}$
satisfy
$D_i=\PRb{-R^{\pr{2^{i-1}}},R^{\pr{2^{i-1}}}}^{\pr{2^{d-i+1}}}$, and
for every $i\in\{1,2,\ldots,d\}$
let $p_i\colon D_i\to\R^{\pr{2^{d-i}}}$
satisfy for all
$x=(x_1,x_2,\dots,x_{2^{d-i+1}})\in D_i$
that
\begin{equation}
p_i(x)
=
(x_1x_2,x_3x_4,\ldots,x_{2^{d-i+1}-1}x_{2^{d-i+1}}).
\end{equation}
\Nobs that
\cref{Lem:product_d}
(applied with
$d \curvearrowleft 2^{d-i}$,
$R \curvearrowleft R^{(2^{i-1})}$,
$\varepsilon \curvearrowleft 2^{\frac{5i-5d}{2}}\! R^{(2^{i}-2^{d})}d^{-1}\eps$
for $i\in\{1,2,\ldots,d\}$
in the notation of \cref{Lem:product_d})
shows that for every $i\in\{1,2,\dots,d\}$ there exists
$\h i\in\ANNs$
which satisfies that
\begin{enumerate}[(I)]
\item
\llabel{function}
it holds
that
$\realisation_\ReLU(\h i)\in C\prb{\R^{(2^{d-i+1})},\R^{(2^{d-i})}}$,
\item
\llabel{lipschitz}
it holds for all
$x,y\in\R^{(2^{d-i+1})}$
that
$
\norm{(\realisation_\ReLU(\h i))(x)-(\realisation_\ReLU(\h i))(y)}
\leq
\sqrt{32}R^{(2^{i-1})}\norm{x-y}
$,
\item
\llabel{approx}
it holds for all
$x\in D_i$
that
\begin{equation}
\norm{p_i(x)-(\realisation_\ReLU(\h i))(x)}
\leq
2^{\frac{5i-5d}{2}}\! R^{(2^{i}-2^{d})}d^{-1}\eps
,
\end{equation}
and
\item
\llabel{param}
it holds
that
\begin{align}
&\paramANN(\h i)
\\&\leq\nonumber
4^{d-i}\max\pRb{157 , 211 + 45\prb{4\log_2\prb{R^{(2^{i-1})}} - 2\log_2\prb{2^{\frac{5i-5d}{2}}\! R^{(2^{i}-2^{d})}d^{-1}\eps}\allowbreak+d-i}}
\ifnocf.
\end{align}
\end{enumerate}
\cfload[.
Next let
$\f\in\ANNs$
satisfy
\begin{equation}
\llabel{fdef}
\f
=
\compANN{\mathscr{h}_d}{\ReLUidANN{2}}\bullet\compANN{\mathscr{h}_{d-1}}{\ReLUidANN{2^{2}}}\bullet\ldots\bullet\mathscr{h}_2\bullet\ReLUidANN{2^{d-1}}\bullet\mathscr{h}_1
\ifnocf.
\end{equation}
\cfload[.
\Nobs that
\lref{fdef},
\lref{function},
\cref{Prop:identity_representation}, and
\cref{Lemma:PropertiesOfCompositions_n2}
ensure that
\begin{equation}
\llabel{functioncheck}
\realisation_\ReLU(\f)
=
\PR*{\realisation_\ReLU(\mathscr{h}_d)}\circ\PR*{\realisation_\ReLU(\mathscr{h}_{d-1})}\circ\ldots\circ\PR*{\realisation_\ReLU(\mathscr{h}_1)}
\in C\prb{\R^{(2^d)},\R}
\ifnocf.
\end{equation}
\cfload[.
\Lref{lipschitz}, and
induction
therefore
imply that for all
$x,y\in\R^{(2^d)}$
it holds that
\begin{equation}
\llabel{eq:lipschitz}
\begin{split}
\vass[\big]{
(\realisation_\ReLU(\f))(x)
-
(\realisation_\ReLU(\f))(y)
}
&\leq
\prb{\sqrt{32}}^{d} R^{(2^{d-1}+2^{d-2}+\ldots+2^0)}\norm{x-y}
\\&=
2^{\frac{5d}{2}}\! R^{(2^{d}-1)}\norm{x-y}
.
\end{split}
\end{equation}
Next \nobs that
the fact that
for all
$i\in\{1,2,\dots,d\}$,
$x,y\in[-R^{(2^{i-1})},R^{(2^{i-1})}]$
it holds that
$xy\in[-R^{(2^{i})},R^{(2^{i})}]$
demonstrates that for all
$i\in\N\cap(0,d)$
it holds that
$p_i(D_i)\subseteq D_{i+1}$.
Combining
this,
\lref{functioncheck},
\lref{lipschitz}, and
\lref{approx}
with
\cref{lem:comp_lipschitz_approx}
(applied with
$n \is d$,
$(d_0,\allowbreak d_1,\dots,d_n)\is(2^d,\allowbreak 2^{d-1},\dots,2^0)$,
$(L_i)_{i\in\{1,2,\ldots,n\}} \is \prb{\sqrt{32} R^{(2^{i-1})}}{}_{i\in\{1,2,\ldots,d\}}$,
$(\varepsilon_i)_{i\in\{1,2,\ldots,n\}} \is \prb{2^{\frac{5i-5d}{2}}\! R^{(2^{i}-2^{d})}d^{-1}\eps}{}_{i\in\{1,2,\ldots,d\}}$,
$(D_1,\allowbreak D_2,\dots,D_n)\is(D_1,\allowbreak D_2,\dots,D_d)$,
$(f_1,\allowbreak f_2,\dots,f_n)\is(p_1,\allowbreak p_2,\dots,p_d)$,
$(g_1,\allowbreak g_2,\dots,g_n)\is \pr{\realisation_\ReLU(\mathscr{h}_1),\allowbreak \realisation_\ReLU(\mathscr{h}_2),\dots,\realisation_\ReLU(\mathscr{h}_d)}$
in the notation of \cref{lem:comp_lipschitz_approx})
ensures that for all
$x=(x_1,x_2,\dots,x_{2^d})\in[-R,R]^{(2^d)}$
it holds that
\begin{equation}
\begin{split}
\llabel{approxcheck}
&\abs[\bigg]{\PRbbb{\sprod_{i=1}^{{2^d}} x_i} - (\realisation_\ReLU(\f))(x)}
\\&=
\abs[\big]{
\pr*{p_d\circ p_{d-1}\circ \ldots\circ p_1}(x)
-
\prb{\PR*{\realisation_\ReLU(\mathscr{h}_d)}\circ\PR*{\realisation_\ReLU(\mathscr{h}_{d-1})}\circ\ldots\circ\PR*{\realisation_\ReLU(\mathscr{h}_1)}}(x)
}
\\&\leq
\ssum_{i=1}^d\PRbbb{\prbbb{\sprod_{j=i+1}^d \sqrt{32} R^{(2^{j-1})}}2^{\frac{5i-5d}{2}}\! R^{(2^{i}-2^{d})}d^{-1}\eps}
\\&=
d^{-1}\varepsilon\PRbbb{\ssum_{i=1}^{d}\pr*{2^{\frac{5d-5i}{2}}\! R^{(2^{d}-2^{i})}2^{\frac{5i-5d}{2}}R^{(2^{i}-2^{d})}}}
=
\eps
.
\end{split}
\end{equation}
\Moreover
\lref{param} and
the fact that
for all
$i\in\{1,2,\dots,d\}$
it holds that
\begin{equation}
\begin{split}
&4\log_2\prb{R^{(2^{i-1})}} - 2\log_2\prb{2^{\frac{5i-5d}{2}}\! R^{(2^{i}-2^{d})}d^{-1}\eps}+d-i
\\&=
2^{i+1}\log_2(R)-2\prb{\tfrac{5i-5d}2+(2^i-2^d)\log_2(R)-\log_2(d)+\log_2(\eps)}+d-i
\\&=
2\pr*{2^{d}\log_2(R) - \log_2\pr{\eps}+\log_2\pr{d}+3d-3i}
\\&\leq
2\pr*{2^{d}\ReLU\pr{\log_2(R)} + \ReLU\pr{\log_2\pr{\eps^{-1}}}+\log_2\pr{d}+3d-3i}
\end{split}
\end{equation}
imply that for all
$i\in\{1,2,\dots,d\}$
it holds that
\begin{equation}
\begin{split}
\llabel{param_h_i}
&\paramANN(\h i)
\\&\leq
4^{d-i}\max\pRb{157 , 211 + 90\pr*{2^{d}\ReLU\pr{\log_2(R)} + \ReLU\pr{\log_2\pr{\eps^{-1}}}+\log_2\pr{d}+3d-3i}}
\\&\leq
4^{d-i}\PRb{ 90\prb{2^{d}\ReLU\pr{\log_2(R)} + \ReLU\pr{\log_2\pr{\eps^{-1}}}+\log_2\pr{d}}+211+270d}
.
\end{split}
\end{equation}
Combining
this,
\cref{Lemma:PropertiesOfCompositions_n3}, and
\cref{lem:log2_est}
with
the fact that
$\sum_{i=1}^{d}4^{d-i}=\sum_{i=0}^{d-1}4^{i}= \frac{4^d-1}{3}$
shows that
\begin{equation}
\begin{split}
\llabel{paramcheck}
\paramANN(\f)
&\leq
3\PR*{\ssum_{i=1}^d\paramANN(\h i)}
\\&\leq
(4^{d}-1)\PRb{ 90\pr*{2^{d}\ReLU\pr*{\log_2(R)} + \ReLU\pr{\log_2\pr{\eps^{-1}}}+\log_2\pr{d}}+211+270d}
\\&\leq
4^d\PRb{ 90\ReLU\pr*{\log_2(R)}2^d + 90\ReLU\pr{\log_2\pr{\eps^{-1}}}+50d+106d+270d}
\\&\leq
426d4^{d}+90\ReLU\pr*{\log_2(R)}8^{d}+90\ReLU\pr{\log_2\pr{\eps^{-1}}}4^{d}
.
\end{split}
\end{equation}
Combining
this
with
\lref{functioncheck},
\lref{eq:lipschitz}, and
\lref{approxcheck}
establishes
\cref{functiontoshow,lipschitztoshow,approxtoshow,paramtoshow}\cfload.
\finishproofthus
\end{aproof}
\endgroup
\cfclear
\begingroup
\begin{athm}{prop}{Prop:new_product_d}
Let $d \in \N$, $\radius\in[1,\infty)$, $\varepsilon \in (0, \infty)$.
Then there exists $\mathscr{f} \in \ANNs$ such that
\begin{enumerate}[(i)]
\item \label{new_continuous_d} it holds
that $\realisation_\ReLU(\mathscr{f}) \in C(\R^d,\R)$,
\item \label{new_Lipschitz_d} it holds for all
$x,y \in \R^d$ that
$\vass{\functionnbReLUANN{\mathscr{f}} (x)
- \functionnbReLUANN{\mathscr{f}} (y)}
\leq \sqrt{32}d^{\frac{5}{2}}\! R^{2d-1} \mednorm{x-y}$,
\item\label{new_approx_d} it holds for all
$x = (x_1,x_2, \ldots, x_d) \in [-\radius, \radius]^d$ that
$\vass[\big]{\PRb{\sprod_{i = 1}^d x_i} - \functionnbReLUANN{\mathscr{f}} (x)} \leq \varepsilon$, and
\item\label{new_cost_d} it holds
that
$\paramANN(\mathscr{f}) \leq 1896 d^3+720 \log_2(R)d^3 + 360\ReLU\pr{\log_2\pr{\eps^{-1}}}d^2
$
\end{enumerate}
\cfout.
\end{athm}
\newcommand{\D}{D}
\begin{proof}
[Proof of \cref{Prop:new_product_d}]
Throughout this proof
assume w.l.o.g.\ that\cfadd{Prop:identity_representation}
$d\geq 2$,
let
$\D\in\N$
satisfy
$\D=2^{\ceil{\log_2(d)}}$
and let
$A\in\R^{\D\times d}$, $B\in\R^{\D}$
satisfy for all
$x=(x_1,x_2,\ldots,x_d)\in\R^d$
that
\begin{equation}
\llabel{affinedef}
Ax+B=(x_1,x_2,\ldots,x_d,1,1,\ldots,1)
\end{equation}
\cfload.
\Nobs that \cref{lem:prod_pow2} (applied with
$d \curvearrowleft \ceil{\log_2(d)}$,
$R \curvearrowleft R$,
$\eps \curvearrowleft \eps$
in the notation of \cref{lem:prod_pow2}) ensures that there exists $\mathscr{g}\in\ANNs$ which satisfies that
\begin{enumerate}[(I)]
\item
\llabel{functionprop}
it holds that
$\realisation_\ReLU(\mathscr{g})\in C(\R^{\D},\R)$,
\item
\llabel{lipschitzprop}
it holds for all
$x,y\in\R^{\D}$
that
$
\abs{(\realisation_\ReLU(\mathscr{g}))(x)-(\realisation_\ReLU(\mathscr{g}))(y)}
\leq
D^{\frac{5}{2}}\! R^{\D-1}
\norm{x-y}
$,
\item
\llabel{approxprop}
it holds for all
$x=(x_1,x_2,\dots,x_{\D})\in[-R,R]^{\D}$
that
$
\abs[\big]{\PRb{\sprod_{i=1}^{\D} x_i} - (\realisation_\ReLU(\mathscr{g}))(x)}
\leq
\eps
$,
and
\item
\llabel{paramprop}
it holds that
\begin{equation}
\paramANN(\mathscr{g})
\leq 426\log_2(\D)\D^2+ 90\ReLU\pr*{\log_2(R)}\D^3+ 90\ReLU\pr{\log_2\pr{\eps^{-1}}}\D^2
\end{equation}
\end{enumerate}
\cfload[.
\Nobs that
\cref{Lemma:PropertiesOfCompositions_n2},
\lref{functionprop},
and the fact that
$\realisation_\ReLU(\affineANN_{A,B})\in C(\R^d,\R^\D)$
imply that
\begin{equation}
\llabel{functioncheck}
\realisation_\ReLU(\compANN{\mathscr{g}}{\affineANN_{A,B}})=\PR{\realisation_\ReLU(\mathscr{g})}\circ\PR{\realisation_\ReLU(\affineANN_{A,B})}\in C(\R^{d},\R)\ifnocf.
\end{equation}
\cfload[.
\Moreover
\lref{lipschitzprop},
\lref{affinedef},
the fact that
$D\leq 2d$,
and the assumption that $R\geq 1$
show that for all
$x,y\in\R^d$
it holds that
\begin{equation}
\llabel{lipschitzcheck}
\begin{split}
&\vass[\big]{(\realisation_\ReLU(\compANN{\mathscr{g}}{\affineANN_{A,B}}))(x)-(\realisation_\ReLU(\compANN{\mathscr{g}}{\affineANN_{A,B}}))(y)}
\\&=
\vass*{\prb{\PR{\realisation_\ReLU(\mathscr{g})}\circ\PR{\realisation_\ReLU(\affineANN_{A,B})}}(x)-\prb{\PR{\realisation_\ReLU(\mathscr{g})}\circ\PR{\realisation_\ReLU(\affineANN_{A,B})}}(y)}
\\&\leq
D^{\frac{5}{2}}\! R^{\D-1}\norm{\pr{\realisation_\ReLU(\affineANN_{A,B})}(x)-\pr{\realisation_\ReLU(\affineANN_{A,B})}(y)}
\\&=
D^{\frac{5}{2}}\! R^{\D-1}\norm{Ax+B-(Ay+B)}
\\&=
D^{\frac{5}{2}}\! R^{\D-1}\norm{x-y}
\leq
\sqrt{32}d^{\frac{5}{2}}\! R^{2d-1}\norm{x-y}
.
\end{split}
\end{equation}
\Moreover
\lref{affinedef} and
the assumption that
$R\geq 1$
ensure that for all
$x\in[-R,R]^d$
it holds that
$Ax+B\in[-R,R]^D$.
\Lref{approxprop}
therefore
demonstrates that for all
$x=(x_1,x_2,\ldots,x_d)\in[-R,R]^d$
it holds that
\begin{equation}
\llabel{approxcheck}
\begin{split}
&\vass*{\PRbbb{\sprod_{i=1}^{d} x_i} - (\realisation_\ReLU(\compANN{\mathscr{g}}{\affineANN_{A,B}}))(x)}
\\&=
\vass*{\PRbbb{\sprod_{i=1}^{d} x_i}- \prb{\PR{\realisation_\ReLU(\mathscr{g})}\circ\PR{\realisation_\ReLU(\affineANN_{A,B})}}(x)}
\\&=
\vass*{\PRbbb{\sprod_{i=1}^{d} x_i} - \pr{\realisation_\ReLU(\mathscr{g})}(x_1,x_2,\ldots,x_d,1,1,\ldots,1)}\leq\varepsilon
.
\end{split}
\end{equation}
In the next step \nobs that
\cref{Lemma:PropertiesOfCompositions_n2} and
\cref{lem:dimcomp}
imply that for all
$i\in\{1,2,\ldots,\lengthANN(\mathscr{g})\}$
it holds that
$\lengthANN(\compANN{\mathscr{g}}{\affineANN_{A,B}})=\lengthANN(\mathscr g)$,
$\singledims_0(\compANN{\mathscr{g}}{\affineANN_{A,B}})=\singledims_0(\affineANN_{A,B})=d\leq D=\singledims_0(\mathscr{g})$, and
$\singledims_i(\compANN{\mathscr{g}}{\affineANN_{A,B}})=\singledims_i(\mathscr{g})$.
Hence, we obtain that
$
\paramANN(\compANN{\mathscr{g}}{\affineANN_{A,B}})
\leq
\paramANN(\mathscr{g})
$.
Combining
this,
\lref{paramprop},
and \cref{lem:log2_est}
with
the fact that
$D\leq 2d$
and
the assumption that $R\geq 1$
shows that
\begin{equation}
\begin{split}
\llabel{paramcheck}
\paramANN(\compANN{\mathscr{g}}{\affineANN_{A,B}})
&\leq
426\log_2(\D)\D^2+ 90\log_2(R)\D^3+ 90\ReLU\pr{\log_2\pr{\eps^{-1}}}\D^2
\\&\leq
237 \D^3+90 \log_2(R)\D^3 + 90\ReLU\pr{\log_2\pr{\eps^{-1}}}\D^2
\\&\leq
1896 d^3+720 \log_2(R)d^3 + 360\ReLU\pr{\log_2\pr{\eps^{-1}}}d^2
.
\end{split}
\end{equation}
This, \lref{functioncheck}, \lref{lipschitzcheck}, and \lref{approxcheck} establish \cref{new_cost_d,new_approx_d,new_Lipschitz_d,new_continuous_d}. The proof of \cref{Prop:new_product_d} is thus complete\cfload.
\end{proof}
\endgroup
\subsection{Products in the spaces of ANN approximable functions}
\label{Subsection:6.4}
\cfclear
\begin{athm}{cor}{Coro:prod_d_cost}
Let
$d \in \N$,
$R\in[1,\infty)$,
$\eps\in(0,\infty)$
and let
$F \colon \R^d \to \R^d$
satisfy for all
$x = (x_1, x_2, \ldots, x_d) \in \R^d$
that
$F(x) = \prb{x_1, x_1x_2, \ldots, x_1x_2 \cdots x_d}$.
Then
\begin{equation}
\label{Coro:prod_d_cost_ineq}
\begin{split}
&\CostLipB{\ReLU}{[-\radius, \radius]^d}{F|_{[-\radius, \radius]^d}}{\sqrt{32}d^{3}\! R^{2d-1}}{\varepsilon}
\leq
2296 d^5+720 \log_2(R)d^5 + 360\ReLU\pr{\log_2\pr{\eps^{-1}}}d^4
\end{split}
\end{equation}
\cfout.
\end{athm}
\begin{aproof}
Throughout this proof
assume w.l.o.g.\ that\cfadd{Prop:identity_representation}
$d\geq 2$
and let
$A \in \R^{d^2 \times d}$,
$B \in \R^{d^2}$
satisfy
for all
$x=(x_1,x_2, \ldots, x_d) \in \R^d$
that
\begin{equation}
\label{Coro:prod_d_cost:eq:linear}
Ax+B
=
(\underbrace{x_1, 1, 1, \ldots, 1}_d, \underbrace{x_1,x_2, 1, \ldots, 1}_d, \ldots, \underbrace{x_1, x_2, \ldots, x_d}_d)
\ifnocf.
\end{equation}
\cfload[.
\Nobs that
\eqref{Coro:prod_d_cost:eq:linear}
implies that for all
$x=(x_1,x_2, \ldots, x_d)$, $y=(y_1,y_2, \ldots, y_d)\in\R^d$
it holds that
\begin{equation}
\label{Comp_theory_14:1.1}
\begin{split}
&\norm*{\pr*{\realisation_{\ReLU}\pr*{\affineANN_{A,B}}}(x)-\pr*{\realisation_{\ReLU}\pr*{\affineANN_{A,B}}}(y)}
=
\norm*{Ax+B-(Ay+B)}
\\&=
\PR*{\sum_{i=1}^d\sum_{j=1}^i\vass{x_j-y_j}^2}^{\frac{1}{2}}
\leq
\PR*{\sum_{i=1}^d\sum_{j=1}^d\vass{x_j-y_j}^2}^{\frac{1}{2}}
=
\sqrt{d}\norm{x-y}
\ifnocf.
\end{split}
\end{equation}
\cfload[.
\Moreover
\cref{Prop:new_product_d} (applied with
$d \curvearrowleft d$,
$\radius \curvearrowleft \radius$,
$\varepsilon \curvearrowleft d^{-\frac12}\varepsilon$
in the notation of \cref{Prop:new_product_d})
proves that there exists
$\mathscr f\in \ANNs$
which satisfies that
\begin{enumerate}[(I)]
\item \label{Coro:prod_d_cost:i1}
it holds that
$\realisation_\ReLU{(\mathscr{f})} \in C(\R^d,\R)$,
\item \label{Coro:prod_d_cost:i2}
it holds for all $x,y\in\R^d$ that
$
\vass*{
\functionnbReLUANN{\mathscr{f}}(x)
-
\functionnbReLUANN{\mathscr{f}}(y)
}
\leq
\sqrt{32}d^{\frac{5}{2}}\! R^{2d-1}\mednorm{x-y}
$,
\item \label{Coro:prod_d_cost:i3}
it holds for all $x=(x_1,x_2,\ldots,x_d)\in[-R,R]^d$ that
$
\abs[\big]{
\PRb{\sprod_{i = 1}^d x_i}
-
\functionnbReLUANN{\mathscr{f}}(x)
}
\leq
d^{-\frac12}\varepsilon
$,
and
\item \label{Coro:prod_d_cost:i4}
it holds that
$
\paramANN(\mathscr{f})
\leq
1896 d^3+720 \log_2(R)d^3 + 360\ReLU\pr{\log_2\pr{d^2\eps^{-1}}}d^2
$
\end{enumerate}
\cfload.
Next let
$\mathscr g\in \ANNs$
satisfy
\begin{equation}
\label{Comp_theory_14:1}
\mathscr{g} = \compANN{\parallelizationSpecial_{d}\pr*{\mathscr{f}, \mathscr{f}, \ldots, \mathscr{f} } }{ \affineANN_{A,B}}
\end{equation}
\cfload.
\Nobs that
\eqref{Comp_theory_14:1},
\cref{Coro:prod_d_cost:i1},
\cref{Lemma:PropertiesOfParallelizationEqualLength}, and
\cref{Lemma:PropertiesOfCompositions_n2} ensure that
\begin{equation}
\begin{split}
\label{Comp_theory_14:item_bis_1}
\realisation_{\ReLU}( \mathscr{g})=\PR*{\realisation_\ReLU\pr*{\parallelizationSpecial_{d}\pr*{\mathscr{f}, \mathscr{f}, \ldots, \mathscr{f} }}}\circ\PR{\realisation_\ReLU(\affineANN_{A,B})} \in C(\R^d,\R^d)
\ifnocf.
\end{split}
\end{equation}
\cfload[.
This,
\eqref{Coro:prod_d_cost:eq:linear},
\cref{Coro:prod_d_cost:i3},
\cref{Lemma:PropertiesOfParallelizationEqualLength}, and
the assumption that $R\geq 1$
show that for all
$x=(x_1,x_2, \ldots, x_d) \in [-\radius, \radius]^d$
it holds that
\begin{equation}
\label{Comp_theory_14:item_bis_2}
\begin{split}
\mednorm{F(x) - (\realisation_{\ReLU}( \mathscr{g}))(x)}&= \PR*{\sum_{j=1}^d\vass*{\PR[\big]{\sprod\nolimits_{i = 1}^j x_i}
-
\functionnbReLUANN{\mathscr{f}}(x_1,x_2, \ldots, x_j,1,1,\ldots,1) }^2}^{\frac{1}{2}}
\\&\leq
\PRb{d\prb{d^{-\frac12}\varepsilon}^2}^{\frac{1}{2}}=\varepsilon.
\end{split}
\end{equation}
\Moreover
\eqref{Comp_theory_14:1.1},
\eqref{Comp_theory_14:item_bis_1},
\cref{Coro:prod_d_cost:i2},
\cref{Lemma:PropertiesOfParallelizationEqualLength}, and
\cref{lemma:Lipschitz_error} demonstrate that for all
$x ,y \in \R^d$
it holds that
\begin{equation}
\label{Comp_theory_14:item_bis_3}
\mednorm{(\realisation_{\ReLU}( \mathscr{g}))(x) - (\realisation_{\ReLU}( \mathscr{g}))(y)}
\leq
\sqrt{32}d^{\frac{5}{2}}\! \radius^{2d -1}\prb{\sqrt{d}\mednorm{x-y}}
=
\sqrt{32}d^{3}\! R^{2d-1}\mednorm{x-y}
.
\end{equation}
In the next step \nobs that
\cref{Lemma:PropertiesOfCompositions_n2} and
\cref{lem:dimcomp}
imply that for all
$i\in\{1,2,\ldots,\lengthANN(\mathscr{g})\}$
it holds that
$\lengthANN(\mathscr g)=\lengthANN(\parallelizationSpecial_{d}\pr*{\mathscr{f}, \mathscr{f}, \ldots, \mathscr{f} })$,
$\singledims_0(\mathscr{g})=\singledims_0(\affineANN_{A,B})=d\leq d^2=\singledims_0(\parallelizationSpecial_{d}\pr*{\mathscr{f}, \mathscr{f}, \ldots, \mathscr{f} })$, and
$\singledims_i(\mathscr{g})=\singledims_i(\parallelizationSpecial_{d}\pr*{\mathscr{f}, \mathscr{f}, \ldots, \mathscr{f} })$.
Hence, we obtain that
$
\paramANN(\mathscr{g})
\leq
\paramANN(\parallelizationSpecial_{d}\pr*{\mathscr{f}, \mathscr{f}, \ldots, \mathscr{f} })
$.
This,
\cref{Coro:prod_d_cost:i4},
and
\cref{Lemma:ParallelizationImprovedBoundsOne}
demonstrate that
\begin{equation}
\label{Comp_theory_14:0}
\begin{split}
\paramANN(\mathscr{g})
&\leq
d^2\prb{1896 d^3+720 \log_2(R)d^3 + 360\ReLU\pr{\log_2\pr{d^2\eps^{-1}}}d^2}
\\&=
d^2\prb{1896 d^3+720 \log_2(R)d^3 + 360\ReLU\pr{2\log_2(d)+\log_2\pr{\eps^{-1}}}d^2}
\\&\leq
d^2\prb{1896 d^3+720 \log_2(R)d^3 + 360\ReLU\pr{\log_2\pr{\eps^{-1}}}d^2+720\log_2(d)d^2}
.
\end{split}
\end{equation}
\Cref{lem:log2_est}
hence
shows that
\begin{equation}
\begin{split}
\paramANN(\mathscr{g})
&\leq
d^2\prb{1896 d^3+720 \log_2(R)d^3 + 360\ReLU\pr{\log_2\pr{\eps^{-1}}}d^2+400d^3}
\\&=
2296 d^5+720 \log_2(R)d^5 + 360\ReLU\pr{\log_2\pr{\eps^{-1}}}d^4
.
\end{split}
\end{equation}
Combining
this,
\cref{Comp_theory_14:item_bis_1},
\cref{Comp_theory_14:item_bis_2}, and
\cref{Comp_theory_14:item_bis_3}
ensures that
\begin{equation}
\begin{split}
&\CostLipB{\ReLU}{[-\radius, \radius]^d}{F|_{[-\radius, \radius]^d}}{\sqrt{32}d^3\! \radius^{2d-1}}{\varepsilon}
\leq
2296 d^5+720 \log_2(R)d^5 + 360\ReLU\pr{\log_2\pr{\eps^{-1}}}d^4
\end{split}
\end{equation}
\cfload[.
\finishproofthis
\end{aproof}
\cfclear
\begin{athm}{cor}{Coro:prod_d_classes_1}
Let
$F \in C\prb{\bigcup_{d \in \N}\R^d, \bigcup_{d \in \N}\R^d}$
satisfy
for all
$d \in \N$,
$x = (x_1, x_2, \ldots, x_d) \allowbreak \in \R^d$
that
$F(x) = \prb{\clip{-1}{1}(x_1), \allowbreak \prod_{i = 1}^2 \clip{-1}{1}(x_i), \allowbreak \ldots, \allowbreak \prod_{i = 1}^d \clip{-1}{1}(x_i)}$
\cfload.
Then
$F \in \Dapprox$
\cfout.
\end{athm}
\begin{aproof}
Observe that
\cref{Lemma:Monotonicity_of_Cost} and
\cref{Coro:max_d_1bis}
ensure that for all
$d \in \N$,
$\radius \in [1,\infty)$
it holds that
\begin{equation}
\label{Coro:prod_d_classes_1:eq1}
\begin{split}
&\CostLipA{\ReLU}{\R^d}{
(\Clip{-1}1d)|_{[-\radius,\radius]^d}}{1}{0}
\leq
3d^2+3d
\end{split}
\end{equation}
\cfload.
This,
\cref{Lemma:Monotonicity_of_Cost}, and
\cref{Prop:composition_cost_ReLU}
(applied with
$d_1 \is d$,
$d_2 \is d$,
$d_3 \is d$,
$\varepsilon \is \varepsilon$,
$L_1 \is 1$,
$L_2 \is \sqrt{32}d^3$,
$R_1 \is R$,
$R_2 \is 1$,
$f_1 \is (\Clip{-1}1d)|_{[-R,R]^d}$,
$f_2 \is \prb{[-1,1]^d\ni x\mapsto F(x)\in\R^d}$
for
$d \in \{2,3,\ldots\}$,
$\radius \in[1, \infty)$,
$\varepsilon \in (0,1]$
in the notation of \cref{Prop:composition_cost_ReLU})
show that for all
$d \in \{2,3,\ldots\}$,
$\radius \in[1, \infty)$,
$\varepsilon \in (0,1]$
it holds that
\begin{equation}
\begin{split}
&\CostLipB{\ReLU}{[-\radius, \radius]^d}{\pr{[-\radius, \radius]^d \ni x \mapsto F(x) \in \R^d}}{\sqrt{32}d^3}{\varepsilon}
\\&=
\CostLipB{\ReLU}{[-\radius, \radius]^d}{\pr{[-\radius, \radius]^d \ni x \mapsto (F\circ \Clip{-1}1d)(x) \in \R^d}}{\sqrt{32}d^3}{\varepsilon}
\\&
\leq
4d(d+1)+2\,\CostLipB{\ReLU}{[-\radius, \radius]^d}{(\Clip{-1}1d)|_{[-\radius,\radius]^d}}{1}{0}
\\&
\quad
+2\,\CostLipB{\ReLU}{[-1, 1]^d}{\pr{[-1, 1]^d \ni x \mapsto F(x) \in \R^d}}{\sqrt{32}d^3}{\tfrac{\varepsilon}{2}}
\\&
\leq
4d(d+1)+2(3d^2+3d)+2\,\CostLipB{\ReLU}{[-1, 1]^d}{\pr{[-1, 1]^d \ni x \mapsto F(x) \in \R^d}}{\sqrt{32}d^3}{\tfrac{\varepsilon}{2}}
\\&
=10d^2+10d+2\,\CostLipB{\ReLU}{[-1, 1]^d}{\pr{[-1, 1]^d \ni x \mapsto F(x) \in \R^d}}{\sqrt{32}d^3}{\tfrac{\varepsilon}{2}}
.
\end{split}
\end{equation}
This,
\cref{Lemma:Monotonicity_of_Cost},
\cref{Coro:prod_d_cost},
the fact that
for all
$d\in\N$,
$x=(x_1,x_2,\ldots,x_d)\in[-1,1]^d$
it holds that
$F(x)=\pr{x_1, x_1x_2, \ldots, x_1x_2 \cdots x_d}$,
and the fact that
for all
$\varepsilon \in (0,1]$
it holds that
$\ReLU(\log_2(2\varepsilon^{-1}))=\max\{1+\log_2(\varepsilon^{-1}),0\}\leq 1+\varepsilon^{-1}$
imply that for all
$d \in \{2,3,\ldots\}$,
$\radius \in[1, \infty)$,
$\varepsilon \in (0,1]$
it holds that
\begin{equation}
\label{Coro:prod_d_classes_1:eq5}
\begin{split}
&\CostLipB{\ReLU}{[-\radius, \radius]^d}{\pr{[-\radius, \radius]^d \ni x \mapsto F(x) \in \R^d}}{14 d^{14}\!\radius^{14}}{\varepsilon}
\\&\leq\CostLipB{\ReLU}{[-\radius, \radius]^d}{\pr{[-\radius, \radius]^d \ni x \mapsto F(x) \in \R^d}}{\sqrt{32}d^3}{\varepsilon}
\\&\leq
10d^2+10d + 2\prb{2296 d^5 + 360\ReLU\pr{\log_2\pr{2\eps^{-1}}}d^4}
\\&\leq
10d^2+10d + 4592 d^5 + 720d^4+720d^4\varepsilon^{-1}
\\&\leq
4955 d^5 + 360d^5\varepsilon^{-1}
\\&\leq
5315 d^5\varepsilon^{-1}
\leq
14 d^{14}\!\radius^{14}\varepsilon^{-14}
.
\end{split}
\end{equation}
\Moreover
\eqref{Coro:prod_d_classes_1:eq1},
\cref{Lemma:Monotonicity_of_Cost}, and
the fact that
for all
$x\in\R$
it holds that
$F(x)=\Clip{-1}11(x)$
ensure that for all
$\radius \in[1, \infty)$,
$\varepsilon \in (0,1]$
it holds that
\begin{equation}
\begin{split}
\CostLipA{\ReLU}{\R^1}{
[-\radius,\radius]\ni x\mapsto F(x)\in \R}{14\radius^{14}}{\varepsilon}
&\leq
\CostLipA{\ReLU}{\R^1}{
(\Clip{-1}11)|_{[-\radius,\radius]}}{1}{0}
\\&\leq
6
\leq 14 \radius^{14}\varepsilon^{-14}.
\end{split}
\end{equation}
Combining
this and
\eqref{Coro:prod_d_classes_1:eq5}
with
the fact that
for all
$d,\radius\in\N$
it holds that
$\imdim{F}{d}=d$ and
$F([-\radius,\radius]^d)\subseteq[-1,1]^d$
establishes that
$F \in \Dapprox$ \cfload.
This completes the proof of \cref{Coro:prod_d_classes_1}\cfload.
\end{aproof}
\section{ANN approximations for high-dimensional functions}
\label{Section:7}
In this section we combine the results of
\cref{Section:4,Section:5,Section:6} above
on the approximation capabilities of ANNs regarding certain concrete classes
of functions with the properties of approximation spaces
proved in \cref{Section:3} above to establish
the main result of this article,
\cref{Theo:example_multiple_composition_loclip} in \cref{Subsection:7.2} below.
We provide a weaker variant of this result
in \cref{Coro:example_multiple_composition_loclip_2bis}
in \cref{Subsection:7.2} below,
where the constant appearing in the
bound on the parameters may depend
in an arbitrary way -- instead of polynomially -- on the radius of the region
where the approximation property holds,
thereby simplifying the statement.
\cref{Theo:introduction} in the introduction is
a direct consequence of
\cref{Coro:example_multiple_composition_loclip_2bis}.
\subsection{Compositions involving maxima, products, and approx\-imable functions}
\label{Subsection:7.1}
\cfclear
\begin{athm}{prop}{Prop:example_multiple_composition}
Let $c,r \in [0, \infty)$, $n \in \N$, $a_1, a_2, \ldots, a_n \in \N_0 \cup \{-1\}$, $(f_{k,d})_{(k,d) \in \N^2} \subseteq \Capprox{c}{r}$,
and let
$F_{k} \colon \prb{\bigcup_{d \in \N} \R^d} \to \prb{\bigcup_{d \in \N} \R^d}$, $k \in \N_0 \cup \{-1\}$,
satisfy for all $k,d,\radius \in \N$, $x\in[-R,R]$, $v = (v_1, v_2, \ldots, v_d) \in \R^d$ that
$\vass{f_{k,d}(x)} \allowbreak\leq cd^c\! R^c$,
$F_{-1}(v) = \prb{v_1, \allowbreak \max\{v_1,v_2\}, \allowbreak \ldots, \allowbreak \max\{v_1, v_2, \ldots, v_d\} }$,
$F_{0} (v) = \prb{\clip{-1}{1}(v_1), \allowbreak \prod_{i = 1}^2 \clip{-1}{1}(v_i), \allowbreak \ldots, \allowbreak \prod_{i = 1}^d \clip{-1}{1}(v_i)}$, and
$F_{k}(v) = \prb{f_{k,1}(v_1), f_{k,2}(v_2), \allowbreak \ldots,\allowbreak f_{k,d}(v_d)}$
\cfload.
Then $
(F_{a_n} \circ \ldots \circ F_{a_2} \circ F_{a_1}) \in \Dapprox
$
\cfout.
\end{athm}
\begin{proof}
[Proof of \cref{Prop:example_multiple_composition}]
Note that \cref{Lemma:Comp_theory_3} ensures that
$\{ F_1, F_2, \ldots \}\subseteq \Dapprox$ \cfload.
Moreover, observe that \cref{Coro:max_d_classes} shows that $F_{-1} \in \Dapprox$.
Furthermore, note that \cref{Coro:prod_d_classes_1} proves that $F_{0} \in \Dapprox$.
Combining this, the fact that $F_{-1} \in \Dapprox$, and the fact that $\{ F_1, F_2, \ldots\} \subseteq \Dapprox$ with \cref{lemma:D_closed} and induction establishes that
$
(F_{a_n} \circ \ldots \circ F_{a_2} \circ F_{a_1}) \in \Dapprox
$.
This completes the proof of \cref{Prop:example_multiple_composition}.
\end{proof}
\cfclear
\begin{athm}{cor}{Coro:example_multiple_composition_bis}
Let
$c,r \in [0, \infty)$,
$n \in \N$,
let
$a \colon \{1,2, \ldots, n\} \to (\N_0 \cup \{-1\})$
satisfy
$\inf\prb{[a^{-1}(\N)]\cup \{\infty\}} \leq \inf\prb{[a^{-1}(\{0\})]\cup \{\infty\}}$,
let
$F_{k} \colon \prb{\bigcup_{d \in \N} \R^d} \to \prb{\bigcup_{d \in \N} \R^d}$, $k \in \N_0 \cup \{-1\}$,
and let
$(f_{k,d})_{(k,d) \in \N^2} \subseteq \Capprox{c}{r}$,
satisfy
for all
$k, d \in \N$,
$x \in \R$,
$v = (v_1, v_2, \ldots, v_d) \in \R^d$
that
$\vass{f_{k,d}(x)} \leq 1$,
$F_{-1}(v) = \prb{v_1, \allowbreak \max\{v_1,v_2\}, \allowbreak \ldots, \allowbreak \max\{v_1, v_2, \ldots, v_d\} }$,
$F_{0} (v) = \prb{v_1, \allowbreak v_1 v_2, \allowbreak \ldots, \allowbreak v_1 v_2\cdots v_d }$,
and $F_{k}(v) = \prb{f_{k,1}(v_1), f_{k,2}(v_2), \allowbreak \ldots, f_{k,d}(v_d)}$
\cfload.
Then $
(F_{a(n)} \circ \ldots \allowbreak \circ F_{a(2)} \circ F_{a(1)}) \in \Dapprox
$
\cfout.
\end{athm}
\begin{aproof}
Throughout this proof
let
$G_k\colon\prb{\bigcup_{d \in \N} \R^d} \to \prb{\bigcup_{d \in \N} \R^d}$, $k \in \N_0 \cup \{-1\}$,
satisfy
for all
$k\in\N\cup\{-1\}$,
$d\in\N$,
$x=(x_1,x_2,\dots,x_d)\in\R^d$
that
$G_k=F_k$
and $G_0(x)=\prb{\clip{-1}{1}(x_1), \allowbreak \prod_{i = 1}^2 \clip{-1}{1}(x_i), \allowbreak \ldots, \allowbreak \prod_{i = 1}^d \clip{-1}{1}(x_i)}$
\cfload.
\Nobs that
the assumption that
$\inf\prb{[a^{-1}(\N)]\cup \{\infty\}} \leq \inf\prb{[a^{-1}(\{0\})]\cup \{\infty\}}$,
the fact that
for all
$d\in\N$
it holds that
$(G_{0})|_{[-1,1]^d}=(F_{0})|_{[-1,1]^d}$,
and the fact that
for all
$d\in\N$,
$i\in\N_0\cup\{-1\}$
it holds that
$F_i([-1,1]^d)\subseteq[-1,1]^d$
ensure that
$(F_{a(n)} \circ \ldots \circ F_{a(2)} \circ F_{a(1)}) = (G_{a(n)} \circ \ldots \circ G_{a(2)} \circ G_{a(1)})$.
Combining this with
\cref{Prop:example_multiple_composition}
establishes that
$(F_{a(n)} \circ \ldots \circ F_{a(2)} \circ F_{a(1)}) \in \Dapprox$ \cfload.
\finishproofthis
\end{aproof}
\subsection{Compositions involving maxima, products, and regular functions}
\label{Subsection:7.2}
\cfclear
\begin{athm}{theorem}{Theo:example_multiple_composition_loclip}
Let $r \in [0, \infty)$, $n \in \N$, let $a \colon \{1,2, \ldots, n\} \to (\N_0 \cup \{-1\})$ satisfy $\inf\prb{[a^{-1}(\N)]\cup \{\infty\}} \leq \inf\prb{[a^{-1}(\{0\})]\cup \{\infty\}}$, let
$f_{k,d} \colon \R \to \R$, $k,d \in \N$,
satisfy for all $k, d \in \N$, $x,y \in \R$ that $\vass{f_{k,d}(x)} \leq 1$ and
$\vass{f_{k,d}(x) - f_{k,d}(y)} \leq r(1 + \abs{x} + \abs{y})^{r} \abs{x-y}$,
and let
$F_{k} \colon \prb{\bigcup_{d \in \N} \R^d} \to \prb{\bigcup_{d \in \N} \R^d}$, $k \in \N_0 \cup \{-1\}$,
satisfy for all $d,k \in \N$, $v = (v_1, v_2, \ldots, v_d) \in \R^d$ that
$F_{-1}(v) = \prb{v_1, \allowbreak \max\{v_1,v_2\}, \allowbreak \ldots, \allowbreak \max\{v_1, v_2, \ldots, v_d\} }$,
$F_{0} (v) = \prb{v_1, \allowbreak v_1 v_2, \allowbreak \ldots, \allowbreak v_1 v_2 \cdots v_d }$,
and
$F_{k}(v) = \prb{f_{k,1}(v_1), f_{k,2}(v_2), \allowbreak \ldots, f_{k,d}(v_d)}$\cfload.
Then $
(F_{a(n)} \circ \ldots \allowbreak\circ F_{a(2)} \circ F_{a(1)}) \in \Dapprox
$
\cfout.
\end{athm}
\begin{proof}
[Proof of \cref{Theo:example_multiple_composition_loclip}]
Note that \cref{Lemma:loc_Comp_theory_15_sum} ensures that for all $k,d \in \N$ it holds that $f_{k,d} \in \Capprox{(12r+10) 3^{r}}{1}$ \cfload.
Combining this with \cref{Coro:example_multiple_composition_bis} establishes that
$ (F_{a(n)} \circ \ldots \circ F_{a(2)} \circ F_{a(1)}) \in \Dapprox$ \cfload.
This completes the proof of \cref{Theo:example_multiple_composition_loclip}.
\end{proof}
\cfclear
\begin{athm}{cor}{Coro:example_multiple_composition_loclip_2}
Let
$r \in [0, \infty)$,
$n \in \N$,
let
$a \colon \{1,2, \ldots, n\} \to (\N_0 \cup \{-1\})$
satisfy
$\inf\prb{[a^{-1}(\N)]\cup \{\infty\}} \leq \inf\prb{[a^{-1}(\{0\})]\cup \{\infty\}}$,
let
$f_{k,d} \colon \R \to \R$, $k,d \in \N$,
satisfy
for all
$k, d \in \N$,
$x,y \in \R$
that
$\vass{f_{k,d}(x)} \leq 1$ and
$\vass{f_{k,d}(x) - f_{k,d}(y)} \leq r(1 + \abs{x} + \abs{y})^{r} \abs{x-y}$,
and let
$F_{k} \colon \prb{\bigcup_{d \in \N} \R^d} \to \prb{\bigcup_{d \in \N} \R^d}$, $k \in \N_0 \cup \{-1\}$,
satisfy for all
$d,k \in \N$,
$v = (v_1, v_2, \ldots, v_d) \in \R^d$
that
$F_{-1}(v) = \prb{v_1, \allowbreak \max\{v_1,v_2\}, \allowbreak \ldots, \allowbreak \max\{v_1, v_2, \ldots, v_d\} }$,
$F_{0} (v) = \prb{v_1, \allowbreak v_1 v_2, \allowbreak \ldots, \allowbreak v_1 v_2 \cdots v_d }$,
and $F_{k}(v) = \prb{f_{k,1}(v_1), f_{k,2}(v_2), \allowbreak \ldots, f_{k,d}(v_d)}$
\cfload.
Then there exist
$(\mathscr{F}_{d,\radius,\varepsilon})_{(d,\radius, \varepsilon) \in \N \times [1, \infty) \times (0,1]} \subseteq \ANNs$
and $c \in \R$
such that for all
$d \in \N$,
$\radius \in [1, \infty)$,
$\varepsilon \in (0,1]$
it holds that
$\paramANN(\mathscr{F}_{d,\radius,\varepsilon}) \leq cd^c\!\radius^c\varepsilon^{-c}$,
$\realisation_{\ReLU}(\mathscr{F}_{d,\radius,\varepsilon}) \in C( \R^d, \R^d)$, and
\begin{equation}
\sup_{x \in [-\radius,\radius]^d}\norm{ (F_{a(n)} \circ \ldots \circ F_{a(2)} \circ F_{a(1)})(x) - (\realisation_{\ReLU}(\mathscr{F}_{d,\radius,\varepsilon}))(x) } \leq \varepsilon
\end{equation}
\cfout.
\end{athm}
\begin{proof}
[Proof of \cref{Coro:example_multiple_composition_loclip_2}]
Note that \cref{Theo:example_multiple_composition_loclip} assures that
$
(F_{a(n)} \circ \ldots \circ F_{a(2)} \circ F_{a(1)}) \in \Dapprox\ifnocf.
$
\cfload[.]This and \cref{Lemma:extension_on_R_poly} imply that
there exists $c \in [0, \infty)$ which satisfies that for all $d \in \N$, $\radius \in [1, \infty)$, $\varepsilon \in (0, 1]$
it holds that
$\CostLipB{\ReLU}{[-\radius,\radius]^d}{\pr{[-\radius,\radius]^d \ni x \mapsto (F_{a(n)} \circ \ldots \circ F_{a(2)} \circ F_{a(1)})(x) \in \R^d}}{cd^c \! \radius^c \allowbreak}{\varepsilon} \leq c d^c \! \radius^c \varepsilon^{-c}$ \cfload.
Combining this with \cref{cor_cost_of_Lip_approx_set_equivalence} establishes that there exist $(\mathscr{F}_{d,\radius,\varepsilon})_{(d,\radius, \varepsilon) \in \N \times [1, \infty) \times (0,1]} \subseteq \ANNs$ such that
\begin{enumerate}[(I)]
\item\label{example_multiple_composition_loclip_2:item_1} it holds for all $d \in \N$, $\radius \in [1, \infty)$, $\varepsilon \in (0,1]$ that $\realisation_{\ReLU}(\mathscr{F}_{d,\radius,\varepsilon}) \in C(\R^d,\R^d)$,
\item\label{example_multiple_composition_loclip_2:item_2} it holds for all $d \in \N$, $\radius \in [1, \infty)$, $\varepsilon \in (0,1]$ that
\begin{equation}
\sup_{x \in [-r,r]^d}\norm{ (F_{a(n)} \circ \ldots \circ F_{a(2)} \circ F_{a(1)})(x) - (\realisation_{\ReLU}(\mathscr{F}_{d,\radius,\varepsilon}))(x)} \leq \varepsilon,
\end{equation}
\item\label{example_multiple_composition_loclip_2:item_3} it holds for all $d \in \N$, $\radius \in [1, \infty)$, $\varepsilon \in (0,1]$, $x,y \in \R^d$ with $x \ne y$ that $\mednorm{(\realisation_{\ReLU}(\mathscr{F}_{d,\radius,\varepsilon}))(x) - (\realisation_{\ReLU}(\mathscr{F}_{d,\radius,\varepsilon}))(y)} \leq cd^c \!\radius^c \mednorm{x-y}$, and
\item\label{example_multiple_composition_loclip_2:item_4} it holds for all $d \in \N$, $\radius \in [1, \infty)$, $\varepsilon \in (0,1]$ that $\paramANN(\mathscr{F}_{d,\radius,\varepsilon}) \leq cd^c \!\radius^c \varepsilon^{-c}$\ifnocf.
\end{enumerate}
\cfload[.]This completes the proof of \cref{Coro:example_multiple_composition_loclip_2}.
\end{proof}
\cfclear
\begin{athm}{cor}{Coro:example_multiple_composition_loclip_2bis}
Let $r \in [0, \infty)$, $n \in \N$, $a_1,a_2,\ldots,a_n \in \N_0 \cup \{-1\}$, let
$f_{k,d} \colon \R \to \R$, $k,d \in \N$, satisfy for all $k, d \in \N$, $x,y \in \R$ that $\vass{f_{k,d}(x)} \leq 1 \leq a_1$ and
$\vass{f_{k,d}(x) - f_{k,d}(y)} \leq r(1 + \abs{x} + \abs{y})^{r} \abs{x-y}$,
and let
$F_{k} \colon \prb{\bigcup_{d \in \N} \R^d} \to \prb{\bigcup_{d \in \N} \R^d}$, $k \in \N_0 \cup \{-1\}$,
satisfy for all $k,d \in \N$, $v = (v_1, v_2, \ldots, v_d) \in \R^d$ that
$F_{-1}(v) = \prb{v_1, \allowbreak \max\{v_1,v_2\}, \allowbreak \ldots, \allowbreak \max\{v_1, v_2, \ldots, v_d\} }$,
$F_{0} (v) = \prb{v_1, \allowbreak v_1 v_2, \allowbreak \ldots, \allowbreak v_1 v_2{}\cdots{} v_d }$,
and
$F_{k}(v) = \prb{f_{k,1}(v_1), f_{k,2}(v_2), \allowbreak \ldots, f_{k,d}(v_d)}$\cfload.
Then there exist $(\mathscr{F}_{d,\varepsilon})_{(d, \varepsilon) \in \N \times (0,1]} \subseteq \ANNs$ and $c \in \R$ such that for all $d \in \N$, $\varepsilon \in (0,1]$ it holds that
$\paramANN(\mathscr{F}_{d,\varepsilon}) \leq cd^c\varepsilon^{-c}$, $\realisation_{\ReLU}(\mathscr{F}_{d,\varepsilon}) \in C( \R^d, \R^d)$, and
\begin{equation}
\sup_{x \in [-r,r]^d}\norm{ (F_{a_n} \circ \ldots \circ F_{a_2} \circ F_{a_1})(x) - (\realisation_{\ReLU}(\mathscr{F}_{d,\varepsilon}))(x) } \leq \varepsilon
\end{equation}
\cfout.
\end{athm}
\begin{proof}
[Proof of \cref{Coro:example_multiple_composition_loclip_2bis}]
Note that the assumption that $a_1\geq 1$ implies that
$
\inf\pr{\{i\in\{1,2,\ldots,n\}\colon a_i\in\N\}\cup\{\infty\}}=1\leq \inf\pr{\{i\in\{1,2,\ldots,n\}\colon a_i=0\}\cup\{\infty\}}.
$
\cref{Coro:example_multiple_composition_loclip_2} hence ensures that there exist $(\mathscr{F}_{d,\varepsilon})_{(d, \varepsilon) \in \N \times (0,1]} \subseteq \ANNs$ and $c \in \R$ such that for all $d \in \N$, $\varepsilon \in (0,1]$ it holds that
$\paramANN(\mathscr{F}_{d,\varepsilon}) \leq cd^c\varepsilon^{-c}$, $\realisation_{\ReLU}(\mathscr{F}_{d,\varepsilon}) \in C( \R^d, \R^d)$, and
\begin{equation}
\sup_{x \in [-r,r]^d}\norm{ (F_{a_n} \circ \ldots \circ F_{a_2} \circ F_{a_1})(x) - (\realisation_{\ReLU}(\mathscr{F}_{d,\varepsilon}))(x) } \leq \varepsilon\ifnocf.
\end{equation}
\cfload[.]This completes the proof of \cref{Coro:example_multiple_composition_loclip_2bis}.
\end{proof}
\cfclear
\begin{athm}{cor}{Coro:example_multiple_composition_loclip_proj}
Let
$r \in [0, \infty)$,
$n \in \N$,
$a_1,a_2,\ldots,a_n \in \N_0 \cup \{-1\}$,
let
$p_d\colon\R^d\to\R$, $d\in\N$,
satisfy
for all
$d\in\N$,
$x=(x_1,x_2,\dots,x_d)\in\R^d$
that
$p_d(x)=x_d$,
let
$f_{k,d} \colon \R \to \R$, $k,d \in \N$,
satisfy
for all
$k, d \in \N$,
$x,y \in \R$
that
$\vass{f_{k,d}(x)} \leq 1 \leq a_1$ and
$\vass{f_{k,d}(x) - f_{k,d}(y)} \leq r(1 + \abs{x} + \abs{y})^{r} \abs{x-y}$,
and let
$F_{k} \colon \prb{\bigcup_{d \in \N} \R^d} \to \prb{\bigcup_{d \in \N} \R^d}$, $k \in \N_0 \cup \{-1\}$,
satisfy
for all
$k,d \in \N$,
$v = (v_1, v_2, \ldots, v_d) \in \R^d$
that
$F_{-1}(v) = \prb{v_1, \allowbreak \max\{v_1,v_2\}, \allowbreak \ldots, \allowbreak \max\{v_1, v_2, \ldots, v_d\} }$,
$F_{0} (v) = \prb{v_1, \allowbreak v_1 v_2, \allowbreak \ldots, \allowbreak v_1 v_2{}\cdots{} v_d }$,
and
$F_{k}(v) = \prb{f_{k,1}(v_1), f_{k,2}(v_2), \allowbreak \ldots, f_{k,d}(v_d)}$\cfload.
Then there exist
$(\mathscr{F}_{d,\varepsilon})_{(d, \varepsilon) \in \N \times (0,1]} \subseteq \ANNs$
and $c \in \R$
such that
for all
$d \in \N$,
$\varepsilon \in (0,1]$
it holds that
$\paramANN(\mathscr{F}_{d,\varepsilon}) \leq cd^c\varepsilon^{-c}$,
$\realisation_{\ReLU}(\mathscr{F}_{d,\varepsilon}) \in C( \R^d, \R)$,
and
\begin{equation}
\llabel{claim}
\sup_{x \in [-r,r]^d}\abs{ (p_d\circ F_{a_n} \circ \ldots \circ F_{a_2} \circ F_{a_1})(x) - (\realisation_{\ReLU}(\mathscr{F}_{d,\varepsilon}))(x) } \leq \varepsilon
\end{equation}
\cfout.
\end{athm}
\begin{aproof}
Throughout this proof
let
$P_d\in\R^{1\times d}$, $d\in\N$,
satisfy for all
$d\in\{2,3,\dots\}$
that
$P_1 = (1)\in\R^{1\times 1}$ and
$P_d=\begin{pmatrix}0&0&\cdots&0&1\end{pmatrix}\in\R^{1\times d}$.
\Nobs that
\cref{Coro:example_multiple_composition_loclip_2bis}
establishes that there exist
$(\mathscr{G}_{d,\varepsilon})_{(d, \varepsilon) \in \N \times (0,1]} \subseteq \ANNs$
and
$c \in \R$
which satisfy that for all
$d \in \N$,
$\varepsilon \in (0,1]$
it holds that
$\paramANN(\mathscr{G}_{d,\varepsilon}) \leq cd^c\varepsilon^{-c}$,
$\realisation_{\ReLU}(\mathscr{G}_{d,\varepsilon}) \in C( \R^d, \R^d)$, and
\begin{equation}
\llabel{eq:approx}
\sup_{x \in [-r,r]^d}\norm{ (F_{a_n} \circ \ldots \circ F_{a_2} \circ F_{a_1})(x) - (\realisation_{\ReLU}(\mathscr{G}_{d,\varepsilon}))(x) }
\leq
\varepsilon
\end{equation}
\cfload[.
\Moreover
\cref{lem:dimcomp}
implies that for all
$d\in\N$,
$\eps\in(0,1]$,
$n\in\{0,1,\dots,\lengthANN(\mathscr{G}_{d,\varepsilon})-1\}$
it holds that
$\lengthANN(\compANN{\affineANN_{P_d,0}}{\mathscr{G}_{d,\varepsilon}})=\lengthANN(\mathscr{G}_{d,\varepsilon})$,
$\singledims_n(\compANN{\affineANN_{P_d,0}}{\mathscr{G}_{d,\varepsilon}})=\singledims_n(\mathscr{G}_{d,\varepsilon})$,
and
$\singledims_{\lengthANN(\mathscr{G}_{d,\varepsilon})}(\compANN{\affineANN_{P_d,0}}{\mathscr{G}_{d,\varepsilon}})=1\leq \singledims_{\lengthANN(\mathscr{G}_{d,\varepsilon})}(\mathscr{G}_{d,\varepsilon})$
\cfload.
The fact that
for all
$d\in\N$,
$\eps\in(0,1]$
it holds that
$\paramANN(\mathscr{G}_{d,\varepsilon}) \leq cd^c\varepsilon^{-c}$
\hence
demonstrates that for all
$d\in\N$,
$\eps\in(0,1]$
it holds that
\begin{equation}
\llabel{eq:1}
\paramANN(\compANN{\affineANN_{P_d,0}}{\mathscr{G}_{d,\varepsilon}})
\leq
\paramANN(\mathscr{G}_{d,\varepsilon})
\leq
cd^c\varepsilon^{-c}
.
\end{equation}
Next \nobs that
\cref{Lemma:PropertiesOfCompositions_n2} and
the fact that
for all
$d\in\N$,
$x\in\R^d$
it holds that
$p_d(x)=P_dx=(\realisation_\ReLU(\affineANN_{P_d,0}))(x)$
shows that for all
$d\in\N$,
$\varepsilon \in (0,1]$
it holds that
\begin{equation}
\llabel{eq:2}
\realisation_{\ReLU}(\compANN{\affineANN_{P_d,0}}{\mathscr{G}_{d,\varepsilon}})
=
[\realisation_\ReLU(\affineANN_{P_d,0})]\circ [\realisation_{\ReLU}(\mathscr{G}_{d,\varepsilon})]
=
p_d\circ [\realisation_{\ReLU}(\mathscr{G}_{d,\varepsilon})]
\in C(\R^d,\R)
.
\end{equation}
This,
\lref{eq:approx},
and the fact that for all
$d\in\N$,
$x=(x_1,x_2,\dots,x_d)\in\R^d$
it holds that
$\abs{p_d(x)}=\abs{x_d}\leq \norm{x}$
prove that for all
$d \in \N$,
$\varepsilon \in (0,1]$
it holds that
\begin{equation}
\begin{split}
&\sup_{x \in [-r,r]^d}\abs[\big]{ (p_d\circ F_{a_n} \circ \ldots \circ F_{a_2} \circ F_{a_1})(x) - (\realisation_{\ReLU}(\compANN{\affineANN_{P_d,0}}{\mathscr{G}_{d,\varepsilon}}))(x) }
\\&=
\sup_{x \in [-r,r]^d}\abs[\big]{ p_d\prb{(F_{a_n} \circ \ldots \circ F_{a_2} \circ F_{a_1})(x)} - p_d\prb{(\realisation_{\ReLU}(\mathscr{G}_{d,\varepsilon}))(x)} }
\\&=
\sup_{x \in [-r,r]^d}\abs[\big]{ p_d\prb{(F_{a_n} \circ \ldots \circ F_{a_2} \circ F_{a_1})(x) - (\realisation_{\ReLU}(\mathscr{G}_{d,\varepsilon}))(x)} }
\\&\leq
\sup_{x \in [-r,r]^d}\norm{ (F_{a_n} \circ \ldots \circ F_{a_2} \circ F_{a_1})(x) - (\realisation_{\ReLU}(\mathscr{G}_{d,\varepsilon}))(x) }
\leq
\varepsilon
.
\end{split}
\end{equation}
Combining
this
with
\lref{eq:1} and
\lref{eq:2}
establishes
\lref{claim}.
\finishproofthis
\end{aproof}
\subsection*{Acknowledgements}
This project is based on the master thesis of PB written from
January 2020 to June 2020 at ETH Zurich under the supervision
of AJ and PC.
This work has been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) under Germany's Excellence Strategy EXC 2044-390685587, Mathematics Münster:
Dynamics--Geometry--Structure.
\bibliographystyle{acm}
|
2024-02-18T23:40:24.537Z
|
2021-12-30T02:26:39.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14523","language":"en","timestamp":1640831199000,"url":"https:\/\/arxiv.org\/abs\/2112.14523","yymm":"2112"}
|
proofpile-arXiv_000-10039
|
{"provenance":"002.jsonl.gz:10040"}
| null | null |
\section{Great Expectations -- Explainable Machines\label{ch:intro-sec:intro}}%
Transparency, interpretability and explainability promote understanding and confidence. %
As a society, we strive for transparent governance and justified actions that can be scrutinised and contested. %
Such a strong foundation provides a principled mechanism for reasoning about fairness and accountability, which we have come to expect in many areas. %
Artificial Intelligence (AI) systems, however, are not always held to the same standards. %
This becomes problematic when data-driven algorithms power applications that either implicitly or explicitly affect people's lives, for example in banking, justice, job screenings or school admissions~\cite{o2016weapons}. %
In such cases, creating explainable predictive models or retrofitting transparency into pre-existing systems is usually expected by the affected individuals, or simply enforced by law. %
A number of techniques and algorithms are being proposed to this end; however, as a relatively young research area, there is no consensus within the AI discipline on a suite of technology to address these challenges.%
Building intelligible AI systems is oftentimes problematic, especially given their varied, and sometimes ambiguous, audience~\cite{kirsch2017explain,preece2018stakeholders}, purpose~\cite{hall2019systematic} and application domain~\cite{bhatt2020explainable}. %
While intelligent systems are frequently deemed (unconditionally) opaque, it is not a definitive property and it largely depends on all of the aforementioned factors, some of which fall beyond the consideration of a standard AI development lifecycle. %
Without clearly defined explainability desiderata~\cite{sokol2020explainability} addressing such diverse aspects can be challenging, in contrast to designing AI systems purely based on their predictive performance, which is often treated as a quality proxy that can be universally measured, reported and compared. %
In view of this disparity, many engineers (incorrectly) consider these two objectives as inherently incompatible~\cite{rudin2019stop}, thus choosing to pursue high predictive performance at the expense of opaqueness, which may be incentivised by business opportunities.%
While high predictive power of an AI system might make it useful, its explainability %
determines its acceptability. %
The pervasiveness of automated decision-making in our everyday life, some of it bearing social consequences, requires striking a balance between the two that is appropriate for what is at stake; for example, approaching differently a car autopilot and an automated food recommendation algorithm. %
Another domain that could benefit from powerful and explainable AI is (scientific) discovery~\cite{roscher2020explainable} -- intelligent systems may achieve super-human performance, e.g., AlphaGo~\cite{silver2017mastering}, however a lack of transparency renders their mastery and ingenuity largely unattainable. %
Such observations have prompted the Defense Advanced Research Projects Agency (DARPA) to announce the eXplainable AI (XAI) programme~\cite{gunning2016broad,gunning2017explainable} that promotes building a suite of techniques to %
(i) create more explainable models while preserving their high predictive performance and %
(ii) enable humans to understand, trust and effectively manage intelligent systems.%
To address these challenges, AI explainability and Machine Learning (ML) interpretability solutions are developed at breakneck speed, giving a perception of a chaotic field that may seem difficult to navigate at times. %
Despite these considerable efforts, a universally agreed terminology and evaluation criteria are still elusive, with many methods introduced to solve a commonly acknowledged but under-specified problem, and their success judged based on ad-hoc measures. %
In this paper we take a step back and re-evaluate the foundations of this research to organise and reinforce its most prominent findings that are essential for advancing this field, with the aim of providing a well-grounded platform for tackling DARPA's XAI mission. %
Our work thus reconciles and complements the already vast body of technical contributions, philosophical and social treatises, as well as literature surveys by bringing to light the interdependence of interdisciplinary and multifaceted concepts upon which explainable AI and interpretable ML are built. %
Our discussion manoeuvres through this incoherent landscape by connecting numerous open questions and challenges -- rather than attempting to solve any individual issue -- which is achieved through a comprehensive review of published work that acknowledges any difficulties or disagreements pertaining to these research topics.%
In particular, we review the notions of a \emph{black box} and \emph{opaqueness} in the context of artificial intelligence, and formalise \emph{explainability} -- the preferred term in our treatment of the subject (Section~\ref{sec:bbnes}). %
We discuss the meaning and purpose of explanations as well as identify theoretical and practical prerequisites for lifting unintelligibility of predictive systems, based on which we define explainability as a socially-grounded technology providing insights that lead to \emph{understanding}, which both conceptualises such techniques and fixes their evaluation criterion. %
Furthermore, we show how transparency, and other terms often used to denote similar ideas, can be differentiated from explainability -- both overcome opaqueness, but only the latter leads to understanding -- which we illustrate with decision trees of different sizes. %
While the premise of our definition is clear, understanding largely depends upon the explanation recipients, who come with a diverse range of background knowledge, experience, mental models and expectations~\cite{gregor1999explanations}. %
Therefore, in addition to technical requirements, explainability tools should also embody various social traits as their output is predominantly aimed at humans. %
We discuss these aspects of AI and ML explainers in Section~\ref{ch:intro-sec:humans}, which considers the role of an \emph{explanation audience} and people's preference for \emph{contrastive statements} -- XAI insights inspired by explainability research in the social sciences~\cite{miller2019explanation}. %
We also examine the social and bi-directional \emph{explanatory process} underlying conversational explanations between humans, highlighting the desire of explainees (i.e., the audience of explanations) to customise, personalise and contest various aspects of explanatory artefacts provided for opaque systems within a congruent interaction~\cite{sokol2018glass,sokol2020one}. %
We then connect these desiderata to validation and evaluation approaches for explainable AI and Interpretable ML (IML) techniques, arguing for studies that concentrate on assessing the understanding gained by the target audience in favour of other metrics.%
Next, in Section~\ref{sec:tradeoff}, we take a closer look at explainability by design (ante-hoc) and techniques devised to remove opaqueness from pre-existing black boxes (post-hoc and, often, model-agnostic), focusing on the latter type given that such methods are universally applicable to a wide variety models, which increases their potential reach and impact. %
While these explainers are appealing, their modus operandi can be an unintended cause of low-fidelity explanations that lack truthfulness with respect to the underlying black box~\cite{rudin2019stop}. %
Furthermore, their flexibility means that, from a technical perspective, they can be applied to any predictive model, however they may not necessarily be equally well suited to the intricacies of each and every one of them. %
Creating a faithful post-hoc explainer requires navigating multiple trade-offs reflected in choosing specific components of otherwise highly-modular explainability framework and parameterising these building blocks based on the specific use case~\cite{sokol2019blimey,sokol2020towards,sokol2020limetree,sokol2020tut}. %
These observations %
prompt us to revisit the disputed \emph{transparency--predictive power trade-off} and assess \emph{benefits} of interpretability that go beyond understanding of predictive algorithms and their decisions, such as their \emph{fairness} and \emph{accountability}.%
We continue our investigation in Section~\ref{sec:elephant} by assessing %
\emph{explainability needs for various parts of predictive systems} -- data, models and predictions -- as well as multiplicity and diversity of these, sometimes incompatible, insights. %
To this end, we use an \emph{explainability taxonomy} derived from the Explainability Fact Sheets~\cite{sokol2020explainability} to reason about such systems within a well-defined framework that considers both their social and technical requirements. %
Notably, it covers human aspects of explanations, thus giving us a platform to examine the audience (explainees), explanation complexity and fidelity, as well as the interaction mode, among many others. %
This discussion leads us to a high-level overview of landmark XAI and IML literature that highlights the interplay between various (often interdisciplinary and multifaceted) concepts popular in the field, thus painting a coherent perspective. %
Section~\ref{sec:summary} then summarises our main observations and contributions:%
\begin{itemize}
\item we formally define explainability and catalogue other relevant nomenclature;%
\item we establish a spectrum of opaqueness determined by the desired level of transparency and interpretability;%
\item we identify important gaps in human-centred explainability from the perspective of current technology;%
\item we dispute universality of post-hoc explainers given their complexity and high degree of modularity; and%
\item we address explanation multiplicity through explanatory protocols for data, models and predictions.%
\end{itemize}
These insights pave the way for the development of more intelligible and robust machine learning and artificial intelligence explainers.%
\section{Defining Black-Box and Explainable Artificial Intelligence\label{sec:bbnes}}%
To avoid a common criticism of explainability research, we begin by discussing the concept of interpretability. %
To this end, we identify causes of opaqueness when dealing with intelligent systems and assess prerequisites for their understanding. %
In this setting we observe \textbf{shades of black-boxiness}: an interpretability spectrum determined by the extent of understanding exhibited by explainees, which, among others, is conditioned upon their mental capacity and background knowledge. %
We link this finding with various notions used in XAI and IML literature, a connection that helps us to fix the nomenclature and \textbf{define explainability} (our preferred term).%
The term \textbf{black box} can be used to describe a system whose internal workings are opaque to the observer -- its operation may only be traced by analysing its inputs and outputs~\cite{beizer1995black,bunge1963general}. %
Similarly, in computer science (including AI and ML) a black box is a (data-driven) algorithm that can be understood as an automated process that we cannot reason about beyond observing its behaviour. %
For AI in particular, \citet{rudin2019stop} points out two main sources of opaqueness: %
(i) a \emph{proprietary} system, which may be transparent to its creators, but operates as a black box; and %
(ii) a system that is too \emph{complex} to be comprehend by \emph{any human}. %
While the latter case concerns entities that are universally opaque for the \emph{entire population}, we argue that -- in contrast to a binary classification~\cite{dawkins2011tyranny} -- this definition of black boxes essentially establishes a (continuous) \emph{spectrum of understanding}. %
Notably, different levels and degrees of transparency and understandability have previously been pointed out and discussed in relation to individual elements of predictive systems, explainees' background knowledge and complexity of information conveyed by explanations, however these observations are often limited to multiple, hand-crafted discrete categories~\cite{marr1982vision,lipton2016mythos,arrieta2020explainable,roscher2020explainable,kim2021multi}.%
For example, a seminal inquiry into opaqueness of visual perception systems by \citet{marr1982vision} suggested three different levels at which information processing devices can be understood. %
The top tier is \emph{computational theory}, which concerns abstract specification of the problem at hand and the overall goal. %
It is followed by \emph{representation and algorithm}, which deals with implementation details and selection of an appropriate representation. %
The final level is \emph{hardware implementation}, which simply establishes physical realisation of the explained problem. %
To illustrate his framework, \citet{marr1982vision} argued that understanding why birds fly cannot be achieved by only studying their feathers: ``%
In order to understand bird flight, we have to understand aerodynamics; only then do the structure of feathers and the different shapes of birds' wings make sense.%
'' %
Nonetheless, he points out that these three tiers are only loosely related and some phenomena may be explained at only one or two of them, therefore it is important to identify which of these levels need to be covered in each individual case to arrive at understanding.%
\citeauthor{lipton2016mythos}'s categorisation of transparency~\cite{lipton2016mythos} -- which he defines as the ability of a human to comprehend the (ante-hoc) mechanism employed by a predictive model -- may be roughly seen as a modern interpretation of \citeauthor{marr1982vision}'s levels of understanding~\cite{marr1982vision}. %
The first of \citeauthor{lipton2016mythos}'s dimensions is \emph{decomposability}, which entails appreciation of individual components that constitute a predictive system, namely: input, parameterisation and computation; %
it can be compared to \citeauthor{marr1982vision}'s \emph{computational theory}. %
Next, \emph{algorithmic transparency} involves understanding the modelling process embodied by a predictive algorithm, which relates to \citeauthor{marr1982vision}'s \emph{representation and algorithm}. %
Finally, \emph{simulatability} enables humans to simulate a decisive process in vivo at the level of the entire model, capturing a concept similar to \citeauthor{marr1982vision}'s \emph{hardware implementation}. %
These three levels of \citeauthor{lipton2016mythos}'s notion of transparency span diverse processes fundamental to predictive modelling, and their understanding can offer a comprehensive, albeit predominantly technical, view of such systems.%
While not universally recognised, knowledge, perception and comprehension of a phenomenon undeniably depend upon the observer's cognitive capabilities and mental model, the latter of which is an internal representation of this phenomenon built on real-world experiences~\cite{kulesza2013too}. %
For example, \citet{kulesza2013too} outline a \emph{fidelity}-based understanding spectrum spanning two dimensions:%
\begin{description}[labelindent=2\parindent]%
\item[completeness] captures how truthful the understanding is overall (\emph{generality}); and%
\item[soundness] determines how accurate the understanding is for a particular phenomenon (\emph{specificity}).%
\end{description}
Therefore, a \emph{complete} understanding of an event from a certain domain is equivalently applicable to other, possibly unrelated, events from the same domain; for example, gravity in relation to a pencil falling from a desk. %
A \emph{sound} understanding, on the other hand, accurately describes an event without (over-)simplifications, which may result in misconceptions; for example, leaving a pencil on a slanted surface results in it falling to the ground. %
Striking the right balance between the two depends upon the observer and may be challenging to achieve: completeness without soundness is likely to be too broad, hence uninformative; and the opposite can be too specific to the same effect.%
\footnote{%
Note that comparable distinctions can be found across literature. %
For example, a differentiation between \emph{functional} and \emph{mechanistic} \emph{understanding}, where the former concerns ``functions, goals, and purpose[s]'' and the latter relies on ``parts, processes, and proximate causal mechanisms''~\cite{lombrozo2014explanation,lombrozo2019mechanistic}. %
A similar categorisation is also pertinent to \emph{knowledge}, which can either be \emph{declarative} or \emph{procedural}~\cite{gregor1999explanations} -- the former allows to recall facts (i.e., ``knowing that'') and the latter translates to skills that enable performing a cognitive task (i.e., ``knowing how'').%
}%
Within this space, \citet{kulesza2013too} identify two particularly appealing types of a mental model%
:%
\begin{description}[labelindent=2\parindent,leftmargin=4\parindent]
\item[functional] which is enough to operationalise a concept but does not necessarily entail the understanding of its underlying mechanism (akin to The Chinese Room Argument~\cite{searle1980minds,penrose1989emperor}); and%
\item[structural] which warrants a detailed understanding of how and why a concept operates.%
\end{description}
For example, a functional understanding of a switch and a light bulb circuit can be captured by the dependency between flipping the switch and the bulb lighting up. %
A structural understanding of the same phenomenon, on the other hand, may focus on the underlying physical processes, e.g., closing an electrical circuit allows electrons to ``move'', which heats up the bulb's filament, thus emitting light (note simplifications employed by this explanation). %
The former understanding is confined to operating a light switch, while the latter can be generalised to many other electrical circuits. %
Each one is aimed at a different audience and their complexity should be fine-tuned for the intended purpose as explanations misdirected towards an inappropriate audience may be incomprehensible. %
These observations lead us to argue that such a spectrum of understanding in human explainability can constitute a yardstick for determining explanatory qualities of predictive algorithms -- a link that has mostly been neglected in the literature, but which can help us to explicitly define popular XAI and IML terminology.%
A considerable amount of research into explainable AI and interpretable ML published in recent years appears to suggest that it is a freshly established field; however, in reality it is more of a renaissance~\cite{gregor1999explanations}. %
While work in this area indeed picked up the pace in the past decade, interest in creating transparent and explainable, data-driven algorithms dates back at least to the 1990s~\cite{rudin2019stop}, and further back to the 1970s if expert systems are taken into account~\cite{leondes2001expert,gregor1999explanations}. %
With such a rich history and the increased publication velocity attributed to the more recent re-establishment of the field, one may think that this research area has clearly defined objectives and a widely shared and adopted \textbf{terminology}. %
However, with an abundance of keywords that are often used interchangeably in the literature -- without precisely defining their meaning -- this is not yet the case. %
The most common terms include, but are not limited to:%
\begin{tasks}[label=\labelitemi,item-indent=3\parindent,label-offset=0pt,after-item-skip={\dimexpr\itemsep+\parsep}](4)%
\task explainability,
\task observability,
\task transparency,
\task explicability,
\task intelligibility,
\task comprehensibility,
\task understandability,
\task interpretability,
\task simulatability,
\task explicitness,
\task justification,
\task rationalisation,
\task sensemaking,%
\task insight,
\task evidence,
\task reason, and
\task cause.
\end{tasks}
Other keywords -- such as function (of), purpose (of), excuse, consequence, effect, implication and meaning -- can also be found in non-technical explainability research~\cite[page 32]{achinstein1983nature}. %
Additionally, \emph{explanandum} or \emph{explicandum} often appear in XAI and IML literature, however these terms, which are borrowed from philosophy of science, denote the concept to be explained.%
While early XAI and IML research might have missed out on an opportunity to clearly define its goals and nomenclature, recent work has attempted to rectify this problem~\cite{%
biran2014justification,biran2017explanation,rudin2019stop,gilpin2018explaining,alvarez2019weight,mohseni2021multidisciplinary,chen2022machine,lipton2016mythos,vilone2020explainable,markus2021role,guidotti2018survey,langer2021we,rosenfeld2019explainability,bibal2016interpretability,papenmeier2022complicated,montavon2018methods,yao2021explanatory,furnkranz2020cognitive,murdoch2019definitions,roscher2020explainable,kim2021multi,adadi2018peeking,offert2017know,arrieta2020explainable,marcinkevivcs2020interpretability,kaur2022sensible,stepin2021survey,palacio2021xai%
}. %
Within this spaces, some authors simply list the relevant terms without assigning any meaning to them~\cite{marcinkevivcs2020interpretability} while others cite dictionary definitions~\cite{roscher2020explainable,arrieta2020explainable,palacio2021xai} -- e.g., to interpret is ``to explain [\ldots] the meaning of'' or ``present in understandable terms'' according to Merriam-Webster -- or suggest that many such keywords are synonymous~\cite{bibal2016interpretability}. %
It is also common to find circular or tautological definitions, which use one term from the list shown above to specify another~\cite{adadi2018peeking}; %
for example, ``something is explainable when we can interpret it'', ``interpretability is making sense of ML models'', ``interpretable systems are explainable if their operations can be understood by humans'' or ``intelligibility is the possibility to comprehended something''. %
Hierarchical and ontological definitions of these terms also appear in the literature~\cite{rosenfeld2019explainability,bibal2016interpretability,stepin2021survey}, often creating a web of connections that is difficult to parse, follow and apply. %
Another counterproductive approach to defining these concepts assumes that their meaning is inherently intuitive or can only be captured by tacit knowledge -- viewpoints that can be summarised by ``I know it when I see it'' phrase~\cite{offert2017know}.%
A different route to specifying these terms is binding keywords to particular components of a predictive pipeline or associating them with the technical and social properties of such a system~\cite{markus2021role,lipton2016mythos,roscher2020explainable}; however, the former is just a labelling strategy and the latter assumes that we can achieve explainability by simply fulfilling a set of requirements. %
A fictitious scenario under this purview could determine that data are understandable, models are transparent and predictions are explainable; %
the overall interpretability of predictive pipelines, on the other hand, is determined by the fidelity, brevity and relevance of the insights produced by the designated method. %
More specifically, transparency is oftentimes associated with ante-hoc methods (the internal mechanisms of a predictive model) and interpretability with post-hoc approaches (the observable behaviour a system)~\cite{lipton2016mythos}; %
alternatively, interpretability is designated for models and explainability for their outputs (more precisely, reasons behind predictions)~\cite{kim2021multi}. %
Similarly, simulatability may be linked to an entire model, decomposability to its individual components and algorithmic transparency to the underlying training algorithm~\cite{lipton2016mythos}. %
Interpretability has also been defined as a domain-specific notion that imposes ``a set of application-specific constraints on the model'', thus making this concept only applicable to predictive models that can provide their own explanations (i.e., ante-hoc interpretability) and should not, along with the term explainability, be used to refer to ``approximations to black box model predictions'' (i.e., post-hoc explainability)~\cite{rudin2019stop}. %
In this particular view, therefore, a predictive model is interpretable if it ``obeys structural knowledge of the domain, such as monotonicity, causality, structural (generative) constraints, additivity or physical constraints that come from domain knowledge''.%
Alternatively, interpretability could be used as an umbrella term and refer to data, models and post-hoc approaches as long as it facilitates extracting helpful information from these components or produces insights into them~\cite{murdoch2019definitions}. %
This designation, however, may not be universal and, to complicate matters even more, certain terms can be used with respect to multiple elements of a predictive system, causing confusion. %
For example, transparency may relate to predictive models, interpretability to input data and models, and explainability to data, models and human recipients~\cite{roscher2020explainable}. %
While transparency may be uniquely linked to predictive models, it can still carry multiple meanings and apply to operations of a model as well as its design, components and the underlying algorithmic process~\cite{roscher2020explainable,lipton2016mythos}. %
Desiderata-based definitions appearing in the literature, on the other hand, specify explainability through a mixture of properties such as interpretability (determined by clarity \& parsimony) and fidelity (consisting of completeness \& soundness)~\cite{markus2021role,rosenfeld2019explainability}; nonetheless, note that achieving both does not necessarily guarantee better comprehension of the explained system. %
In like manner, \citet{alvarez2019weight} used the \emph{weight of evidence} idea from information theory to mathematically define AI and ML explainability and outline a precise list of its desiderata. %
While appealing, the complexities of the real world make their conceptualisation difficult to apply at large. %
\citet{murdoch2019definitions} followed a similar route and captured interpretation, understanding and evaluation of data-driven systems in the predictive--descriptive--relevant (PDR) framework, which spans: predictive performance (accuracy) of a model; descriptive capabilities of its (post-hoc) explainer quantified via fidelity; and relevancy of the resulting explanatory insights determined by their ability to provide ``insight[s] for a particular audience into a chosen problem''.%
Within this chaotic landscape some researchers propose flexible definitions that are inspired by interdisciplinary work and can accommodate a variety of contexts while maintaining a precise and coherent meaning. %
\citet{gilpin2018explaining} offered definitions of ``explanation'', ``interpretability'' and ``explainability'' drawing from a broad body of literature in an effort to standardise XAI and IML findings. %
While their notions appear somewhat vague -- explanations should answer ``Why?'' and ``Why-should?'' questions until such questions can no longer be asked -- they argue for making explanations \emph{interpretable} and \emph{complete}, striving towards human \emph{understanding} that depends on the explainee's cognition, knowledge and biases. %
Similarly, \citet{biran2014justification} were concerned with \emph{explanations}, which they characterised as ``giving a reason for a prediction'' and answering ``how a system arrives at its prediction''. %
They also defined \emph{justifications} as ``putting an explanation in a context'' and conveying ``why we should believe that the prediction is correct'', which, they note, do not necessarily have to correspond to how the predictive system actually works. %
Notably, many of these observations reappear across diverse publications, with the shared theme indicating that explanations should always answer an implicit or explicit ``Why?'' question~\cite{koura1988approach}, in addition to addressing ``What?'' and ``How?''~\cite{miller2019explanation,palacio2021xai}. %
In a later piece of work, \citet{biran2017explanation} defined explainability as ``the degree to which an observer can understand the cause of a decision'' (also adopted by \citet{miller2019explanation}), thus making it much more explainee-centred. %
While many authors use the term \emph{cause} rather loosely in XAI and IML research, we argue against such practice -- it is important to reserve it exclusively for insights extracted from \emph{causal} models~\cite{pearl2018book}.%
More recently, based on an extensive review of literature in computer science and related disciplines, \citet{mohseni2021multidisciplinary} provided a collection of definitions for the most common terms in explainable AI and interpretable ML, nonetheless the underlying rationale is predominantly qualitative making them difficult to operationalise. %
Similarly, \citet{arrieta2020explainable} differentiated between the following terms: understandability/intelligibility, comprehensibility, interpretability/transparency and explainability. %
Specifically, they defined interpretability or transparency as a \emph{passive} characteristic of a model that allows humans to make sense of it on different levels -- e.g., its internal mechanisms and derivation of predictions -- therefore relating it to the cognitive skills, capacities and limitations of individual explainees. %
Explainability, on the other hand, was described as an \emph{active} characteristic of a model that is achieved through actions and procedures employed (by the model) to clarify its functioning for a certain audience. %
\citet{montavon2018methods} also offered definitions of these two terms -- interpretability and explainability -- from a perspective of functional understanding~\cite{lombrozo2014explanation,lombrozo2019mechanistic}. %
They characterised interpretability as a mapping of an abstract concept, e.g., a predicted class, onto a domain that can be comprehended by a human; %
explainability, in their view, is responsible for providing a collection of factors -- expressed in an interpretable domain -- that contribute to an automated decision of a particular instance. %
In summary, their goal is to study and understand how inputs are mapped to outputs, possibly via a human-comprehensible representation of relevant concepts.%
Each definition conveys a more or less precise meaning that can be used to label relevant techniques, however they do not necessarily clarify and help to navigate the complex landscape of IML and XAI research. %
To organise this space, we categorise the underlying terminology based on three criteria:%
\begin{itemize}
\item \emph{properties} of systems;%
\item \emph{functions} and \emph{roles} that they serve; and%
\item \emph{actions} required to process, assimilate and internalise information elicited by them.%
\end{itemize}
The core concept around which we build our nomenclature is \textbf{explainability}; we define it as \textbf{insights that lead to understanding} (the \textbf{role} of an explanation) -- a popular and widely accepted rationale in the social sciences~\cite{koehler1991explanation,baumeister1994self,lombrozo2006structure,paez2019pragmatic,woodward1979scientific,achinstein1983nature}. %
While it may seem abstract, understanding can be assessed with questioning dialogues~\cite{walton2007dialogical,walton2011dialogue,walton2016dialogue,arioua2015formalizing,madumal2019grounded} -- e.g., a machine interrogating the explainees to verify their understanding of the phenomenon being explained at the desired level of detail -- which are the opposite of explanatory dialogues. %
Such a process reflects how understanding is tested in education, where the quality of tuition as well as knowledge and skills of pupils are evaluated through standardised tests and exams (albeit not without criticism~\cite{mead2015teacher}). %
Furthermore, encouraging people to explain a phenomenon helps them to realise the extent of their ignorance and confront the complexity of the problem, which are important factors in exposing The Illusion of Explanatory Depth~\cite{rozenblit2002misunderstood} -- a belief that one understands more than one actually does.%
This notion of explainability and the three building blocks of XAI and IML terminology allow us to precisely define the other popular terms. %
Therefore,%
\begin{tasks}[label=\labelitemi,item-indent=3\parindent,label-offset=0pt,after-item-skip={\dimexpr\itemsep+\parsep}](3)%
\task \emph{observability},%
\task \emph{transparency},%
\task \emph{explicability},%
\task \emph{intelligibility},%
\task \emph{comprehensibility},%
\task \emph{understandability},%
\task \emph{interpretability},%
\task \emph{simulatability}, and%
\task \emph{explicitness}%
\end{tasks}
are \textbf{properties} of an AI or ML system %
that enable it to %
directly (ante-hoc) or indirectly (post-hoc) %
convey information of varied complexity, the \emph{understanding} of which depends upon the cognitive capabilities and (domain) expertise of the explainee. %
For example, observing an object falling from a table is a transparent phenomenon per se, but the level of its understanding, if any, is based upon the depth of the observer's physical knowledge. %
Such characteristics provide%
\begin{tasks}[label=\labelitemi,item-indent=3\parindent,label-offset=0pt,after-item-skip={\dimexpr\itemsep+\parsep}](3)%
\task \emph{justification},%
\task \emph{rationalisation},%
\task \emph{insight},%
\task \emph{evidence}, and%
\task \emph{reason}%
\end{tasks}
(\textbf{roles}) that can be used to%
\begin{tasks}[label=\labelitemi,item-indent=3\parindent,label-offset=0pt,after-item-skip={\dimexpr\itemsep+\parsep}](3)%
\task \emph{reason} about,%
\task \emph{make sense} of,%
\task \emph{rationalise},%
\task \emph{justify},%
\task \emph{interpret}, or%
\task \emph{comprehend}%
\end{tasks}
(note that here these are used as verbs) behaviour of a -- black-box or glass-box -- predictive system, all of which are \textbf{actions} that under the right circumstances lead to \emph{understanding}. %
While \emph{simulatability} is also based upon observing a transparent process and replicating it, such an \textbf{action} does not necessarily imply understanding of the underlying phenomenon -- recall the difference between declarative and procedural knowledge~\cite{gregor1999explanations}, structural and functional mental models~\cite{kulesza2013too}, functional and mechanistic understanding~\cite{lombrozo2014explanation,lombrozo2019mechanistic} and The Chinese Room Argument~\cite{searle1980minds} discussed earlier. %
Lastly, a \emph{cause} has a similar meaning to a \emph{reason}, but the first one is derived from a causal model, whereas the latter is based purely on observations of the behaviour of a (black-box) model.%
Such a setting makes a welcome connection between the XAI and IML terminology synthesised by the equation%
\[%
\texttt{Explainability} \; = \; %
\underbrace{%
\texttt{Reasoning} \left( \texttt{Transparency} \; | \; \texttt{Background Knowledge} \right)%
}_{\textit{understanding}}%
\text{,}%
\]%
which defines \texttt{Explainability} as the \textbf{process} of deriving \emph{understanding} -- i.e., extracting meaning -- through \texttt{Rea\-son\-ing} applied to \texttt{Transparent} insights distilled from a data-driven predictive system that are adjusted to the explainee's \texttt{Background Knowledge}. %
In this process, the \texttt{Reasoning} can either be done by the explainer or the explainee, and there is an implicit assumption that the explainee's \texttt{Background Knowledge} aligns with the \texttt{Transparent} representation of the predictive model. %
If the latter does not hold, mitigation techniques such as employing an \emph{interpretable representation} can be used to communicate concepts that are otherwise incomprehensible~\cite{ribeiro2016why,sokol2019blimey,sokol2020towards}. %
\texttt{Reasoning} also comes in many different shapes and sizes depending on the underlying system (\texttt{Transparency}) as well as the explainer and the explainee (\texttt{Background Knowledge}); for example, %
logical reasoning with facts, %
causal reasoning over a causal graph, %
case-based reasoning with a fixed similarity metric, and %
artificial neuron activation analysis for a \emph{shallow} neural network.%
Therefore, linear models are transparent given a reasonable number of features; additionally, with the right ML and domain background knowledge -- requirement of normalised features, effect of feature correlation and the meaning of coefficients -- the explainee can reason about their properties, leading to an explanation based on understanding. %
Similarly, a visualisation of a \emph{shallow} decision tree can be considered both transparent and explainable assuming that the explainee understands how to navigate its structure (ML background knowledge) and the features are meaningful (domain background knowledge); again, it is up to the explainee to reason about these insights. %
When the size of a tree increases, however, its visualisation loses the explanatory power because many explainees become unable to process and reason about its structure. %
Restoring the explainability of a deep tree requires delegating the reasoning process to an algorithm that can digest its structure and output sought after insights in a concise representation. %
For example, when explaining a prediction, the tree structure can be traversed to identify a similar instance with a different prediction, e.g., as encoded by two neighbouring leaves with a shared parent, thus demystifying the automated decision~\cite{sokol2019desiderata,sokol2021towards}.%
While understanding and applying the newly acquired knowledge to unseen tasks are recurring themes in XAI and IML literature~\cite{doshi2017towards,bibal2016interpretability,kim2021multi,biran2017explanation,furnkranz2020cognitive,chen2022machine,rosenfeld2019explainability} -- with a few notable exceptions~\cite{palacio2021xai,yao2021explanatory,langer2021we} -- they rarely ever play the central role. %
For example, \citet{palacio2021xai} define an explanation as ``the process of describing one or more facts, such that it facilitates the understanding of aspects related to said facts (by a human consumer)''. %
Similarly, \citet{yao2021explanatory} ``highlight[s] one important feature of explanations: they elicit understanding (in humans)''; %
\citeauthor{yao2021explanatory} proceeds to suggest that the three levels of analysis proposed by \citet{marr1982vision}, and discussed earlier, should be extended with a \emph{social} level to reflect that AI models, especially the ones of concern to XAI and IML researchers, do not operate in isolation from humans.%
Furthermore, some researchers appear to converge towards a similar definition to ours. %
\citet{papenmeier2022complicated} characterise transparency as ``the extent to which information is exposed to a system's inner workings'' and interpretability as ``the extent to which transparency is meaningful to a human'', both of which lead them to formalise explanations as ``the mechanisms by which a system communicates information about its inner workings (transparency) to the user''. %
In like manner, \citet{roscher2020explainable} posit that ``the scientist is using the data, the transparency of the method, and its interpretation to explain the output results (or the data) using domain knowledge and thereby [\ldots] obtain[s] a scientific outcome'', which process should lead to understanding by presenting properties of ML models in humans-comprehensible terms; %
they further suggest that ``explainability usually cannot be achieved purely algorithmically'', which resonates with the role of human \emph{reasoning} in our definition. %
Similarly, \citet{langer2021we} identify a ``(given) context'' as the element moderating ``explanatory information'' to facilitate ``stakeholders' understanding'' (of a subset of components present in a complex system). %
Additionally, while not explicitly stated, one interpretation of \citeauthor{rosenfeld2019explainability}'s notion of explanations~ \cite{rosenfeld2019explainability} is a collection of human-centred processes that allow explainees to understand a predictive model by presenting them with a suitable representation of its logic (a concept that they call \emph{explicitness}), however based on a graphical representation of their framework explainability is achieved through interpretability, only one component of which is transparency. %
Adjacent to XAI and IML, \citet{bohlender2019towards} discussed explanations in software systems, which are meant to ``resolve [a] lack of understanding'' pertaining to a particular aspect (explanandum) of a system for a specific audience and ``the processing of [an] explanation [\ldots] is what makes [an] agent [\ldots] understand the explanandum'' -- an action that may require the use of cognitive or computational resources and that may only be operationalised in specific contexts determined, for example, by the background knowledge of the explainees.%
Given the importance of \emph{understanding} in our definition, as well as explainability research outside of XAI and IML~\cite[pages 23, 42, 57]{achinstein1983nature}, it is crucial to review its acquisition and operationalisation together with how it is distinct from and more desirable than not only declarative but also procedural \emph{knowledge}. %
For example, consider justifications that can be seen to communicate why a decision is correct without necessarily providing the exact logic behind it~\cite{biran2017explanation}, therefore preventing the explainee from internalising and reapplying these insights to other scenarios. %
Similarly, knowledge can be acquired and recalled thus giving the impression of understanding but recitation in itself does not imply comprehension or operationalisation (i.e., an ability to apply it) -- an observation that follows from The Chinese Room Argument~\cite{searle1980minds}. %
For example, memorising a textbook or answers to a set of questions may suffice to pass an exam but falls short of effectively resolving related yet distinct problems~\cite{gregor1999explanations}. %
This is an important insight for evaluating explainability -- as defined in this paper -- since we first have to specify what it means to attain understanding before we can assess effectiveness of the explanatory process~\cite{paez2019pragmatic}. %
Furthermore, internalising knowledge into understanding and correctness thereof are conditioned on the reasoning capabilities and background knowledge of the explainee~\cite{achinstein1983nature,gregor1999explanations}. %
The detrimental effects of misalignments in this space were shown by \citet{bell2022its}, who reported that explainability mechanisms pertinent to inherently interpretable models may be confusing, especially so for a lay audience; %
for example, ante-hoc transparency of decision trees achieved though a visualisation of their structure misleads people (lacking technical expertise) into believing that the feature used by the root-node split is the most important attribute.%
\section{Humans and Explanations -- Two Sides of the Same Coin\label{ch:intro-sec:humans}}
Defining explainability as leading to understanding and our categorisation into \emph{properties}, \emph{functions} and \emph{actions} highlight an important aspect of this research topic: explanations do not operate in a vacuum, they are highly contextual and directed at some autonomous agent, either a human or machine, who is as important as the explainability algorithm itself. %
Notably, up until recently XAI and IML research has been undertaken mostly within the computer science realm~\cite{miller2017explainable}, thus bringing in various biases and implicit assumptions from this predominantly technical field. %
While some explainability research has found its way into other scientific disciplines, e.g., law~\cite{wachter2017counterfactual}, the majority gravitated around technical properties. %
This research agenda was disrupted by \citet{miller2017explainable}, who observed that the function of an explanation and its recipients are largely neglected -- a phenomenon which they dubbed ``inmates running the asylum'' -- leading to a substantial paradigm shift. %
\citeauthor{miller2019explanation}'s follow-on work~\cite{miller2019explanation} grounded this observation in (human) explainability research from the social sciences, where this topic has been studied for decades, thus providing invaluable insights that can benefit XAI and IML.%
\citeauthor{miller2019explanation}'s findings~\cite{miller2019explanation} have arguably reshaped the field, with a substantial proportion of the ensuing research acknowledging the \textbf{explainees} -- their autonomy, goals, expectations, intentions and interactions. %
While explainability of data-driven systems has various benefits, it is usually in focus when an AI agent fails, behaves anomalously or operates inconsistently with the explainee's beliefs, expectations or mental model, e.g., an unexpected ML prediction causing a disagreement; alternatively, an explanation may be requested to support learning or provide information needed to solve a problem or complete a task~\cite{gregor1999explanations}. %
In such cases, explainees' preferences, needs and goals should be addressed to maximise the effectiveness of an explanation, for example by appropriately adjusting its complexity~\cite{miller2019explanation,gregor1999explanations}. %
This step can be further improved by treating explainability as a process instead of one-off information offloading~\cite{miller2019explanation,gregor1999explanations}; by satisfying the explainees' natural desire to \textbf{interact} and communicate with the explainer within a predictable protocol, they are provided with an opportunity to seamlessly customise and personalise the explanation~\cite{sokol2020one}. %
Perhaps the most influential of \citeauthor{miller2019explanation}'s observations is the humans' preference for \textbf{contrastive} explanations given their prominence in everyday life. %
We discuss these three fundamental aspects of human-centred explainability in more detail below.%
Understanding can be an elusive objective when it comes to explaining intelligent systems since each unique \textbf{explanation audience} may expect to receive different insights, e.g., a medical diagnosis can be communicated in terms of test results or observable symptoms depending on whether it is directed towards medical staff or patients. %
While in our considerations we implicitly assume that the explanation recipient is a human, it may as well be another algorithm that further processes such insights, in which case other, more suitable, properties would be of interest. %
When taken into account, the \emph{purpose} of explainability and the explainee's goal also influence the explanation composition~\cite{gregor1999explanations}. %
For example, an explanation will look different when it helps to debug an ML model and when it justifies a negative outcome of a loan application; note that the target audience also differs, with the former aimed at ML engineers and the latter at lay people. %
A complementary view, based on a \emph{means-end account} of XAI~\cite{buchholz2022means}, argues to examine ``\emph{what} should be explained (topic), \emph{to whom} something should be explained (stakeholder), \emph{why} something should be explained (goal), and \emph{how} something should be explained (instrument)''. %
Addressing such desiderata by accounting for the explainee's cognitive capabilities and skill level, however, is challenging as it requires access to the explainee's background knowledge and mental model, which are vague and often undefined concepts that cannot be easily extracted and modelled.%
Nonetheless, just by considering the audience and purpose of an explanation, we can identify (and deliver) a collection of relevant properties. %
In certain cases, such as the aforementioned loan application, the \emph{actionability} of explanatory insights is crucial, e.g., suggesting that an individual would receive a loan had he or she been 10 years younger is futile. %
Multiplicity of apparently indistinguishable arguments can also decrease the perceived quality of an explanation when one is chosen at random without a user-centred heuristic in place, which, again, depends on the application domain and audience. %
For example, research suggests~\cite{miller2019explanation} that if one of multiple, otherwise equivalent, \emph{time}-ordered events has to be chosen as an explanation, the most recent one will best resonate with the explainee; additionally, prioritising explanations by their \emph{novelty} will keep the explainee engaged and attentive, and distinguishing between \emph{sufficient} and \emph{necessary} conditions for a given outcome can help to reduce cognitive load. %
While desirable, \emph{brevity} of an explanation can sometimes be at odds with its comprehensiveness and \emph{completeness} -- sacrificing the big picture (which in itself may be too convoluted to understand) for concise communication~\cite{kulesza2015principles}. %
Explanatory minimalism, nonetheless, bears the danger of oversimplification; however, when it %
is a strict requirement, explanation \emph{soundness} can be favoured to focus on factors pertinent to the explained instance and discard more general reasons that are largely irrelevant. %
Such an approach can introduce inaccuracies with respect to the overall data-driven system, but the explanations remain truthful for the individual instance. %
Striking the right balance between generality and specificity of an explanation -- as well as achieving all the other aforementioned desiderata -- is notoriously challenging and often requires tuning its soundness and completeness for the intended audience and application, which itself may be impractical when done manually, and prohibitively difficult through capturing the explainee's mental model.%
While posing problems for AI explainers, satisfying this wide range of diverse assumptions and expectations comes naturally to humans when they engage in an \textbf{explanatory process} among themselves. %
This is partly due to shared background knowledge, and is further amplified by interactive communication that allows to rapidly iterate over questions, exchange informations and refine answers to arrive at understanding. %
One explanation does not fit all and treating explainability as a bi-directional process provides a platform to appreciate uniqueness of each explainee through personalised explanations~\cite{sokol2020one} that enable transfer of knowledge and help to develop understanding~\cite{gregor1999explanations}. %
While these topics have received relatively little attention in the XAI and IML literature, we can draw design insights and inspirations from research on \emph{explanatory debugging} of predictive models~\cite{kulesza2015principles}. %
Therefore, an interactive explanatory process should be \emph{iterative}, enabling the explainee to learn, provide feedback and receive updated insights until reaching a satisfactory conclusion; %
the explainer ought to always \emph{honour user-provided feedback} by incorporating it into the explanation generation process, or clearly communicate a precise reason if that is impossible; %
the communication should be \emph{reversible} to %
allow the explainee to retract a requested change or annul a piece of feedback when it was provided by mistake, or to explore an interesting part of the predictive model through a speculative enquiry; and, finally, %
the whole process should be \emph{incremental} to easily attribute each piece of feedback to an explanation change, thereby showing up-to-date results regardless of how small the tweaks are.%
Even though dialogue is fundamental to human explainability, it is largely absent in XAI and IML techniques~\cite{sokol2020one}, which are often based on one-way communication, where the user receives a description of a data-driven system without an opportunity to request more details or contest it. %
A similar interaction in a form of the aforementioned questioning dialogues can also be used to judge the explainee's understanding of the explained concept, thus be a proxy for assessing effectiveness of the explainer. %
Notably, human dialogue tends to be verbal or written, both of which are based on the natural language. %
While ubiquitous, this form of communication is not equally effective in conveying all types of information, requiring humans to augment it with visual aids, which are especially helpful when the interaction serves explanatory purposes. %
The same strategy can be adopted in explainable AI and interpretable ML, where the explainer would switch between various explanatory artefacts -- such as (written and spoken) text, images, plots, mathematical formulation, numbers and tables -- depending on which one is best suited for the type of information being communicated in a given context~\cite{gregor1999explanations}. %
Mixing and matching them is also possible, e.g., a numerical table or a plot complemented with a caption, and may be beneficial as the whole can be greater than the sum of its parts, especially that certain explanation types may require a specific communication medium or explanatory artefact to be effective. %
Using visualisation, textualisation and (statistical) summarisation, however, does not guarantee a coherent relation, structure or story conveyed by these communication media alone, which could possibly be achieved by grounding them in a shared context through \emph{logical reasoning} or \emph{formal argumentation}~\cite{dung2009assumption}; %
additional inspiration can be found in \emph{sensemaking theory}, which was recently adapted to XAI and IML applications~\cite{kaur2022sensible}.%
\textbf{Contrastive explanations} -- more specifically, counterfactuals -- dominate the human explanatory process and are considered the gold standard in explainability of data-driven systems~\cite{miller2019explanation}. %
They juxtapose a hypothetical situation (foil) next to the factual account %
with the aim to emphasise the consequences of or ``would be'' change in the outcome. %
Counterfactuals statements can be characterised by their lineage: \emph{model-driven} explanations are represented by artificial data points (akin to centroids), whereas \emph{data-driven} explanations are instances recovered from a (training) data set (similar to medoids). %
Furthermore, the contrast can either be implicit -- i.e., ``Why class \(X\)?'' (hence not any other class) -- or explicit -- i.e., ``Why class \(X\) and not \(Y\)?'' %
Counterfactuals are appropriate for lay audiences and domain experts alike, can use concepts of varying difficulty and be expressed in different media such as text and images. %
They are parsimonious as the foil tends to be based on a single factor, but, if desired, can account for an arbitrary degree of feature covariance. %
They support interaction, customisation and personalisation, e.g., a foil built around a user-selected feature provided in an explanatory dialogue, which can be used to restrict their search space, possibly making them easier to retrieve. %
When deployed in a user-centred application, they can provide the explainees with appealing insights by building the foil only around actionable features. %
However, their effectiveness may be problematic when explaining a proprietary predictive system, e.g., built as a black box with the intention to protect a trade secret, since counterfactual explanations can leak sensitive information, thereby allowing the explainee to steal or game the underlying model. %
In an open world, they also suffer from vaguely defined or imprecise notions known as \emph{non-concepts}~\cite{offert2017know}, e.g., ``What is not-a-dog?''%
These idealised properties make counterfactual statements appealing, but some may get lost in practice, e.g., an imperfect implementation, resulting in subpar explanations. %
On the face of it, these explanatory artefacts resemble causal insights, but unless they are generated with a causal model~\cite{pearl2016causal}, they should not be treated as such and instead be interpreted as descriptors of a decision boundary used by a predictive system. %
If they are model-driven, as opposed to data-driven, they may not necessarily come from the data manifold, yielding (out-of-distribution) explanations that are neither feasible nor actionable in the real life, e.g., ``Had you been 200 years old, \ldots'' %
Even if they are consistent with the data distribution, the foil may still come from a sparse region, thus prescribing possible but improbable feature values~\cite{poyiadzi2020face}. %
Counterfactual explanations are often specific to a single data point, although humans are known to generalise such insights to unseen and possibly unrelated cases -- recall The Illusion of Explanatory Depth effect~\cite{rozenblit2002misunderstood} -- which may result in overconfidence.%
Fulfilling all of these desiderata can help in developing an explainability system that enables the explainees to (better) \emph{understand} an automated decision-making process, which, as we noted earlier, offers a yardstick by which success of such techniques can be quantified~\cite{kim2021multi,paez2019pragmatic,bohlender2019towards,langer2021we}. %
To be effective, however, the approach to assess, evaluate and measure understanding has to account for the aforementioned contextual parameters such as the application domain (and its sensitivity), function of the explanation, intended audience (and its background knowledge) as well as (technical) caveats of the XAI or IML algorithm~\cite{arrieta2020explainable,murdoch2019definitions}. %
Just like beauty, which is in the eye of the beholder, the (perceived) quality, helpfulness and success of an explanation are judged by the recipient~\cite{kim2021multi}. %
Therefore, producing explanations is a necessary but insufficient condition of engendering understanding, which additionally requires them to be relevant to the stakeholders, comprehensible by them and compatible with their desiderata~\cite{langer2021we,gregor1999explanations}, %
for example, by aligning the type of insights and their level of transparency towards the chosen use case and audience (in view of the anticipated background knowledge and skills). %
Specifically, consider the truthfulness of explanatory information, which may be reduced for a particular purpose without harming the audience's ability to derive understanding; %
the simplification used in depicting underground lines and stops is a case in point as it conveys the cues necessary to navigate such a transportation system despite foregoing an accurate representation of scale and distance~\cite{paez2019pragmatic}. %
Notably, a transparent representation that is universal and satisfies all the different needs and expectations of diverse explainee groups may not exist, just like an agreement between these individuals on the objective nature of understanding. %
Given this strong dependence on human perception, the effectiveness of explanations should be evaluated empirically~\cite{bohlender2019towards} to combat The Illusion of Explanatory Depth~\cite{rozenblit2002misunderstood}, and, in view of The Chinese Room Argument~\cite{searle1980minds}, the studies should go beyond assessing simple task completion to capture the difference between knowledge and understanding within a well-defined, real-life context expected in the deployment.%
\section{The Discord over Sacrificing Explainability for Predictive Power\label{sec:tradeoff}}%
Theoretical desiderata do not always align with the operationalisation and practicalities of XAI and IML algorithms, and the latter are what ends up affecting our lives. %
For example, explainability is an inherently social process that usually involves bi-directional communication, but most implementations -- even the ones using contrastive statements~\cite{wachter2017counterfactual,waa2018contrastive} -- output a single explanation that is optimised according to some predefined metric, not necessarily addressing concerns of an individual explainee~\cite{sokol2020one}. %
Similarly, while inherently transparent predictive models and ante-hoc explainers may be preferred~\cite{rudin2019stop}, such solutions are often model-dependent, labour-intensive and tend to be application-specific, which limits their scope as well as wider applicability and adoption. %
Instead, post-hoc and model-agnostic explainers dominate the field~\cite{ribeiro2016why,lundberg2017unified,ribeiro2018anchors,sokol2019blimey} since they are considered one-stop solutions -- a unified explainability experience without a cross-domain adaptation overhead. %
This silver bullet framework, however, comes at a cost: subpar fidelity that can result in misleading or outright incorrect explanations. %
While increasingly such considerations find their way into publications, they are often limited to acknowledging the method's shortcomings, stopping short of offering a %
\begin{wrapfigure}[12]{i}{.28\textwidth}%
\centering
\vspace{-.25\baselineskip}
\includegraphics[width=.225\textwidth]{fig/explainability-vs-performance}%
\caption{%
Fictitious depiction of the anecdotal trade-off between transparency and predictive power of AI systems~\cite{gunning2016broad}.%
\label{fig:ch1:tradeoff}%
}
\end{wrapfigure}
viable solution.%
A common belief motivating many methods published in the explainability literature is the perceived \emph{dichotomy} between transparency and predictive power of AI and ML systems. %
A popular example supporting this theory is the unprecedented effectiveness of deep neural networks on certain tasks, whose ever increasing complexity, e.g., the number of layers %
and hidden units, improves their performance at the expense of transparency. %
This trade-off has been reiterated in the DARPA XAI program's Broad Agency Announcement~\cite{gunning2016broad} and supported by an appealing graph reproduced in Figure~\ref{fig:ch1:tradeoff}. %
However, at the time of publication it has been a \emph{theory} based mostly on anecdotal evidence, with \citet{rudin2019stop} criticising plots like this given their lack of scale, transparency or precise performance metrics, and supporting data. %
Notably, \citeauthor{rudin2019stop} argues that investing more effort into feature engineering and data modelling can help to build inherently explainable AI and ML systems that perform on a par with their black-box alternatives~\cite{chen2018interpretable}.%
This anecdotal trade-off and a tendency to prioritise predictive power mean that explainability is often only an afterthought. %
Such a mindset %
contributes to a landscape with an abundance of well-performing but inherently opaque algorithms that are in need of explainability, thus creating a demand for universal explainers that are post-hoc and model-agnostic, such as surrogates~\cite{craven1996extracting,ribeiro2016why,sokol2019blimey}. %
This seemingly uncompromising development approach -- where state-of-the-art performance remains the main objective, later complemented with a post-hoc explainer -- offers an attractive alternative (and rebuttal) to designing inherently explainable AI and ML systems, whose creation arguably requires more effort. %
While such explainers are compatible with any black-box model, they are not necessarily equally well suited for every one of them~\cite{sokol2020tut} -- after all the computer science folklore of ``no free lunch'' (a single, universal algorithm cannot outperform all the others across the board) applies here as well -- which is reflected in the continuous stream of novel XAI and IML techniques being proposed in the literature, many of whom report competing or contradictory findings (likely because of diverging operational contexts). %
Some post-hoc and model-agnostic explainers boast appealing properties and guarantees, however upon closer inspection one often encounters caveats and assumptions required for these to hold, such as the underlying ``black box'' being a linear model~\cite{lundberg2017unified}. %
Additionally, making an explainer model-agnostic introduces an extra layer of complexity that usually entails a degree of randomness and decreased fidelity~\cite{zhang2019should,sokol2019blimey}, so that using them may become a stopgap to claim explainability of an inherently opaque predictive system instead of addressing genuine explainability needs. %
Correctly interpreting the insights produced by XAI and IML methods may therefore be challenging as it requires a sufficient level of (technical) expertise and alignment with the explainees' background knowledge for the recipients to understand the capabilities and limitations of the explainer, thus avoid drawing incorrect conclusions -- %
this is especially relevant to post-hoc and model-agnostic approaches given their higher complexity~\cite{mittelstadt2019explaining,sokol2020limetree,sokol2020towards,sokol2020tut}.%
In \citeauthor{rudin2019stop}'s view~\cite{rudin2019stop}, many high-stakes AI and ML systems can be made explainable by design with enough effort put towards data pre-processing, feature engineering and modelling (which otherwise, e.g., for neural networks, may go into architecture search and parameter tuning). %
Such ante-hoc explainers are usually domain-specific and after the initial engineering endeavour they are easy to manage and maintain. %
While this approach should be championed for structured (tabular) data where it has been shown to perform on a par with state-of-the-art black boxes~\cite{chen2018interpretable}, the same may be unachievable for sensory data such as images and sounds, for which opaque models, e.g., deep neural networks, have the upper hand. %
In addition to black boxes modelling sensory data, pre-existing, inaccessible or legacy predictive systems may require interpretability, in which case they can only be retrofitted with post-hoc explainers. %
Such techniques are also helpful to engineers and developers working with predictive models since they enable inspection and debugging of data-driven systems. %
However, falling back on off-the-shelf solutions may not guarantee acceptable fidelity~\cite{sokol2019blimey,sokol2020limetree,sokol2020tut} (specifically, soundness and completeness), which is of particular importance and may require tailor-made explainers and transparent communication of their limitations.%
While composing a predictive pipeline, we have an abundance of pre-processing as well as modelling tools and techniques at our disposal, a selection of which will end up in the final system. %
The XAI and IML landscape, on the other hand, is quite different, especially for post-hoc and model-agnostic approaches: explainers tend to be end-to-end tools with only a handful of parameters exposed to the user. %
In view of ``no free lunch'', this is undesirable as despite being model-agnostic, i.e., compatible with any model type, these monolithic algorithms cannot perform equally well for every one of them~\cite{sokol2019blimey,sokol2020tut}. %
This variability in their behaviour can often be attributed to a misalignment between the assumptions baked into an explainer and the properties of the explained system, which manifests itself in low fidelity.%
Model-specific or ante-hoc explainers as advocated by \citet{rudin2019stop} can be used to address this issue; %
however, as discussed earlier, such a solution may have limited applicability and cannot be retrofitted to pre-existing predictive systems. %
Resolving a similar challenge in machine learning and data mining often comes down to a series of investigative steps to guide algorithmic choices down the line, which can be operationalised within a standardised process for knowledge discovery such as KDD~\cite{fayyad1996data}, CRISP-DM~\cite{chapman2000crisp,martinez2019crisp} or BigData~\cite{agrawal2012challenges}. %
For example, by analysing feature correlation, data uniformity and class imbalance, we can account for these phenomena when engineering features and training models, thereby making the resulting systems more accountable and robust. %
Nonetheless, while we may have a set of universal properties expected of XAI and IML systems~\cite{sokol2020explainability}, we lack a dedicated process that could guide the development and assessment of explainers -- their practical requirements and needs -- which likely hinders adherence to best practice. %
Although one can imagine a generic workflow for designing inherently interpretable (ante-hoc) systems~\cite{rudin2019stop}, a similar endeavour should not be neglected for model-agnostic and post-hoc explainers that could be adapted to individual predictive black boxes by capitalising on their flexibility and modularity~\cite{sokol2019blimey}, possibly overcoming low fidelity~\cite{sokol2020limetree}.%
More recently, in response to \citeauthor{rudin2019stop}'s call to action~\cite{rudin2019stop}, the disputed trade-off between \emph{transparency} and \emph{predictive power} has been revisited with a greater scientific rigour~\cite{herm2021don,bell2022its}. %
\citet{herm2021don} used explainability as a tool to aid people in problem-solving, investigating it under the aforementioned two dimensions complemented with \emph{comprehensibility}, which should be a direct consequence of a model being recognised as explainable. %
Their preliminary findings are somewhat at odds with \citeauthor{rudin2019stop}'s postulate, especially so for high-stakes scenarios for which the user studies suggest that an artificial neural network enhanced with SHAP explanations boasts high predictive performance and is also perceived as explainable. %
Similarly, \citet{bell2022its} performed empirical quantification of this trade-off in two public policy domains and found that (inherently) interpretable models may not necessarily be more explainable than black boxes. %
The results of their experiments show that even opaque models may be recognised as explainable -- nonetheless, the authors emphasise the importance of ante-hoc explainability in mission-critical domains -- hinting that the trade-off may be more nuanced than acknowledged in the literature. %
Notably, while the explainee's perception may suggest that black boxes accompanied by post-hoc explainers are up to the task, the fidelity, correctness and veracity of such insights remain contestable, especially in view of the recipient's susceptibility to be convinced by sheer presence of ``explanations'' that themselves may not necessarily be truthful~\cite{herman2017promise} or meaningful (as famously shown by The Copy Machine study~\cite{langer1978mindlessness}). %
With just a few such inquiries available, the topic requires further investigation to offer a clear view on the possible trade-off between transparency and predictive power in XAI and IML. %
We postulate that -- in accordance with our definition -- particular focus should be put on the \emph{reasoning} employed by explainees to extract meaning and create understanding based on transparent insights into black boxes given that a misconception of how to interpret them may result in a false sense of explainability~\cite{mittelstadt2019explaining}.%
A distinct viewpoint on this matter manifests itself in %
claims that we should not expect machine learning algorithms, such as deep neural networks, to be explainable and instead regulate them purely based on their real-life performance~\cite{simonite2019google} and behaviour~\cite[minute 29]{norvig2017google}, however it is not a widely shared belief~\cite{jones2018geoff}. %
This insight comes from the alleged inability of humans to explain their actions since such justifications are post-factum stories that are concocted and retrofitted for the benefit of the audience. %
Certifying autonomous agents based on their output, on the other hand, is consistent with human values as one can hypothesise about committing a crime, but one cannot be punished unless such a thought is acted upon. %
While the origin and nature of human thought processes may be shrouded in mystery, its formulation is expected to follow the reason of logic to be (socially) acceptable. %
In particular, \citet{miller2019but} refutes performance-based validation by arguing that explainability stemming from regulatory requirements is secondary to concerns arising from societal values such as ethics and trust. %
Importantly, making data-driven systems understandable can instil confidence into the public as it allows the creators of such technologies to justify, control and improve them as well as lead to new discoveries~\cite{adadi2018peeking}. %
An appropriate and comprehensive explainability solution can also become a technological springboard to reducing or eliminating bias~\cite{saxena2019perceptions,saxena2019fairness}, unfairness~\cite{buolamwini2018gender,olteanu2019social,kusner2017counterfactual} and lack of accountability (to the benefit of robustness~\cite{ackerman2019three,goodfellow2015explaining}, safety~\cite{angwin2016machine,grzywaczewski2017training} and security) from data-driven predictive models, thus improving their trustworthiness~\cite{sokol2019fairness}.%
\begin{figure}[!t]
\centering
\includegraphics[trim={6pt 87pt 6pt 87pt},clip,width=.65\textwidth]{fig/elephant}%
\caption{%
Depiction of The Blind Men and the Elephant parable~\cite{saxe2016blind} illustrating that any complex subject can be studied in many ways. %
It also symbolises that individual pieces of evidence may often be contradictory and insufficient to understand the bigger picture without first being aggregated and grounded within a shared context.\label{fig:ch1:elephant}}%
\end{figure}
\section{Explanation Diversity and Multiplicity -- What to Explain and How to Explain It\label{sec:elephant}}
So far we have primarily focused on explaining predictions and actions of intelligent systems since they are observable and can be related to by a wide range of explainees regardless of their background. %
However, automated \textbf{predictions} are just artefacts of a more elaborate %
artificial intelligence or machine learning predictive process, which manipulates \textbf{data} to infer \textbf{models} that generalise well, thus are capable of predicting (previously unseen) instances~\cite{flach2012machine}. %
Since any element of this workflow can be opaque~\cite{sokol2017role,sokol2017roleARW}, %
comprehensive explanations may need to consist of insights pertaining to the entire predictive pipeline, discussing diverse topics such as data collection and (pre-)processing, modelling caveats and assumptions, and the meaning and interpretation of predictions, %
all of which can be %
bundled together in a shared user interface to provide a multi-faceted view of the investigated system~\cite{krause2016interacting,krause2016using,weld2019challenge}.%
Additionally, as each explanation may provide just a small, and quite possibly distorted, reflection of the true behaviour of a data-driven model, achieving the desired level of transparency (and understanding) might require communicating multiple, complementary insights for each unintelligible step or observation, which in turn bears the danger of overwhelming and confusing the explainee. %
This multitude of explanatory information has to be navigated carefully and can be understood as unique probing and inspection techniques that without a shared context may yield competing or even contradictory evidence akin to the parable of The Blind Men and the Elephant~\cite{saxe2016blind} %
illustrated in Figure~\ref{fig:ch1:elephant}. %
Note that this phenomenon is not unique to explainability; %
multiplicity of data-driven models all of whom exhibit comparable predictive performance despite intrinsic differences, sometimes called the Rashomon effect of statistics, is well documented~\cite{breiman2001statistical,marx2020predictive,fisher2019all,sokol2022ethical}. %
Furthermore, as AI and ML processes are directional -- from data, through models, to predictions -- the latter components depend on the former, which also applies to their respective explanations. %
For example, if data attributes are incomprehensible, explanations of models and predictions expressed in terms of these features will also be opaque.%
\textbf{Explaining data} may be challenging without any modelling assumptions, hence %
there may not necessarily exist a pure data explanation method beyond simple \emph{summary statistics} (e.g., class ratio or per-class feature distribution) and \emph{descriptors} (e.g., ``the classes are balanced'', ``the data are bimodal'' or ``these features are highly correlated''). %
Note that the former simply state well-defined properties and may not be considered explanations, whereas the latter can be contrastive and lead to understanding. %
Importantly, data are already a model -- they express a (subjective and partial) view of a phenomenon and come with certain assumptions, measurement errors or even embedded cultural biases (e.g., ``How much is a lot?''). %
Data statements~\cite{bender2018data}, data sheets~\cite{gebru2018datasheets} and nutrition labels~\cite{holland2018dataset} attempt to address such concerns by capturing these (often implicit) assumptions. %
As a form of data explanations, they characterise important aspects of data and their collection process in a coherent format, e.g., experimental setup, collection methodology (by whom and for what purpose), pre-processing (cleaning and aggregation), privacy aspects, data ownership, and so on.%
\textbf{Explaining models} %
in whole or in parts (e.g., specific sub-spaces or cohorts) %
should engender a general, truthful and accurate understanding of their functioning. %
While some predictive systems may be inherently transparent, e.g., shallow decision trees, their simulatability~\cite{lipton2016mythos} -- the explainee's ability to simulate their decision process mentally \emph{in vivo} -- may not produce understanding (see Section~\ref{sec:bbnes}). %
Popular model explanations include feature importance~\cite{breiman2001random,fisher2019all}, feature influence on predictions~\cite{friedman2001greedy}, presenting the model in cognitively-digestible portions~\cite{krause2016using,smilkov2016embedding} and model simplification~\cite{craven1996extracting} (e.g., mimicking its behaviour or a global surrogate). %
Since not all models operate directly on the input features, an \emph{interpretable representation} may be necessary to convey an explanation, e.g., a super-pixel segmentation of an image~\cite{ribeiro2016why}; alternatively, if the data are comprehensible, landmark exemplars can be used to explain the behaviour of a model or its parts~\cite{kim2014bayesian,kim2016examples}.%
\begin{wrapfigure}[15]{i}{.43\textwidth}%
\centering
\vspace{-.5\baselineskip}
\includegraphics[width=.4\textwidth]{fig/ice-pd_cropped}%
\caption{%
Explanation of a model predicting the probability of the \emph{versicolor} class when varying the \emph{petal length} attribute for the Iris data set~\cite{fisher1936use}. %
Individual Conditional Expectation of a selected instance is plotted in red; %
the orange curve is the Partial Dependence of the model computed by averaging all individual ICEs (displayed in grey).%
\label{fig:ch1:ice-pd}}%
\end{wrapfigure}
\textbf{Predictions} are explained to communicate a rationale behind a particular decision of a model. %
Depending on the explanation type, a range of diverse aspects concerning the model's decisive process can be provided to the explainee. %
For example, the user may be interested in feature importance~\cite{ribeiro2016why}, feature influence~\cite{lundberg2017unified}, relevant data examples~\cite{kim2015ibcm} and training instances~\cite{koh2017understanding}, or contrastive statements~\cite{wachter2017counterfactual,poyiadzi2020face}, to name a few. %
Note that while some of these explanation types are similar to model explanations, here they are explicitly generated with respect to a single data point and may not necessarily generalise beyond this particular case, whereas for model explanations they convey similar information for all data (i.e., the entire modelled space). %
A good example of this duality is information communicated by Individual Conditional Expectation (ICE~\cite{goldstein2015peeking}) and Partial Dependence (PD~\cite{friedman2001greedy}), both of which are feature influence explanations -- the first with respect to a single data point and the latter concerning a model -- as shown in Figure~\ref{fig:ch1:ice-pd}. %
Akin to model explanations, the information can be conveyed in the raw feature space or using an interpretable representation.%
With such a diverse range (and possibly large quantity) of explanations, their presentation requirements -- \textbf{content}, \textbf{delivery format}, \textbf{communication medium} and \textbf{provision protocol} or \textbf{mechanism}~\cite{sokol2020explainability,gregor1999explanations} -- will naturally vary~\cite{sokol2017role,sokol2017roleARW}. %
A simple approach to characterise an AI component is (statistical) \emph{summarisation} -- it is commonly used for describing properties of data with numerical tables and vectors, which can be difficult to digest for non-experts. %
\emph{Visualisation} -- a graphical representation of a phenomenon -- is a more advanced, insightful and flexible analytical tool. %
Static figures communicate information in one direction, akin to summarisation; however, creating interactive plots can facilitate a ``dialogue'' with an explainee, thereby catering to a more diverse audience. %
Visualisations are often supported by short narratives in the form of captions, which increase their informativeness. %
\emph{Textualisation} -- a natural language description of a phenomenon -- can express concepts of higher complexity and dimensionality than plots, which can help to overcome the curse of dimensionality and the inherent limitations of the human visual system. %
Communicating with text enables a true dialogue and has been shown to be more insightful and effective than presenting raw, numerical and visual data~\cite{portet2009automatic}, which can accompany the narrative to improve its expressiveness. %
A further refinement of textualisation is formal \emph{argumentation}~\cite{dung2009assumption} -- a structured and logically-coherent dialogue accounting for every disputable statement and giving the explainee an opportunity to contest the narrative, thus providing explanations leading to understanding rather than informative descriptions. %
Finally, such explanatory processes can either be triggered automatically, invoked (and driven) by the users or offered contextually whenever a need for a clarification is detected by the explainer~\cite{gregor1999explanations}.%
Thus far we have been mainly concerned with AI and ML explainability on a relatively abstract level, all of which constitute just a small portion of XAI and IML research. %
In an ideal world, relevant publications would consider many of the aforementioned factors and build their mechanisms around them, however it has only recently become a trend and numerous early pieces of work lack such a reflection. %
To complement the viewpoint presented in the preceding sections and bridge the \emph{theoretical} (foundational and social) and \emph{technical} (algorithmic and engineering) aspects of explainers we briefly traverse through \textbf{practical explainability research}. %
Without aiming to be exhaustive -- given the availability of several comprehensive surveys~\cite{guidotti2018survey,linardatos2021explainable} -- we finish this section by identifying a number of landmark contributions that have influenced the entire research field. %
We also omit topics adjacent to explainability, such as interactive exploratory user interfaces~\cite{wexler2017facets,tensorboard}, creative visualisations of explainability approaches~\cite{krause2016interacting} and systems combining multiple explainability techniques within a single tool~\cite{weld2019challenge}.%
The most popular explainers are \emph{model-agnostic} and \emph{post-hoc} as they can be retrofitted into any predictive system (at the expense of adding a modelling layer that may negatively impact explanation fidelity). %
These include RuleFit~\cite{friedman2008predictive}, Local Interpretable Model-agnostic Explanations (LIME~\cite{ribeiro2016why}), anchors~\cite{ribeiro2018anchors}, SHapley Additive exPlanations (SHAP~\cite{lundberg2017unified}), Black-box Explanations through Transparent Approximations (BETA~\cite{lakkaraju2016interpretable,lakkaraju2017approximations}), PD~\cite{friedman2001greedy}, ICE~\cite{goldstein2015peeking} and Permutation Importance (PI~\cite{breiman2001random}), among many others. %
Most of these methods operate directly on raw data, with the exception of LIME and anchors, which use interpretable representations to improve intelligibility of explanations composed for complex data domains such as text and images. %
Another attractive avenue of explainability research, which partly overlaps with post-hoc methods, is opening up (deep) neural networks by designing tools and techniques \emph{specific to these approaches} or, more broadly, compatible with differentiable predictors. %
These models are notoriously opaque, however their superior predictive performance for a wide spectrum of applications increases their popularity and accelerates their proliferation~\cite{lecun2015deep}. %
Relevant explainability techniques include global surrogates~\cite{craven1996extracting}, saliency maps~\cite{zintgraf2017visualizing}, influential training instances~\cite{koh2017understanding}, counterfactuals~\cite{wachter2017counterfactual} (which are surprisingly similar to the problematic adversarial examples~\cite{goodfellow2015explaining}), and influential high-level, human-intelligible insights based on Testing with Concept Activation Vectors (TCAV~\cite{kim2018interpretability}). %
An alternative XAI and IML research agenda concentrates on \emph{inherently explainable} predictive models, and \emph{ante-hoc} explainers designed for popular data-driven systems. %
Examples of the former are generalised additive models~\cite{lou2013accurate} and falling rule list~\cite{wang2015falling}; whereas the latter include global and local explanations of na\"ive Bayes classifiers~\cite{kulesza2015principles}, and clustering insights based on prominent exemplars and dominating features~\cite{kim2014bayesian}.%
\section{Towards Understanding Facilitated Through Intelligible and Robust Explainers\label{sec:summary}}%
In this paper we explored the relatively recent and still evolving domains of artificial intelligence explainability and machine learning interpretability. %
We introduced the main topics and provided the philosophical, theoretical and technical background needed to appreciate the depth and complexity of this research. %
In particular, we highlighted two different mental models: \emph{functional} -- enough understanding to operationalise a concept; and \emph{structural} -- in-depth, theoretical appreciation of underlying processes. %
We further argued that the former -- a shallow form of understanding -- aligns with The Chinese Room Argument~\cite{searle1980minds,penrose1989emperor} and the notion of simulatability~\cite{lipton2016mythos}. %
We also reviewed diverse notions of explainability, interpretability, transparency, intelligibility and many other terms that are often used interchangeably in the literature, and argued in favour of \emph{explainability}. %
We defined this concept as (logical) \emph{reasoning} applied to transparent XAI and IML insights interpreted under specific \emph{background knowledge} within a given context -- a process that engenders \emph{understanding} in explainees. %
We used these observations to challenge the popular view that decision trees are explainable just because they are transparent. %
Deep or wide trees lack interpretability, which can be restored by applying a suitable form of logical reasoning -- a prerequisite of explainability -- undertaken by either an algorithm or a human investigator.%
While the most visible aspect of XAI and IML research is the technology that enables it, explainees -- the recipients of such explanations who tend to be humans -- are just as important (and ought to be treated as first-class citizens) since their \emph{understanding} of the underlying predictive system and its behaviour determines the ultimate success of an explainer. %
We explored this topic by looking at human-centred explainability and various desiderata that this concept entails, in particular focusing on explicitly acknowledging presence of humans and projecting the explanations directly at them. %
To this end, we pursued important insights from the social sciences that prescribe how to adapt machine explainability to fulfil expectations of the explainees, hence achieve seamless explanatory interaction. %
The two crucial observations in this space are: (i) a preference for (meaningful) \emph{contrastive} explanations, which form the cornerstone of human-centred explainability; and (ii) facilitating an interactive, dialogue-like, bi-directional explanatory \emph{process} -- akin to a conversation -- as opposed to delivering a one-off ``take it or leave it'' explanation %
to ensure coherence with people's expectations regardless of their background knowledge and prior experience with this sort of technology. %
Notably, the explanation type and delivery medium should also be adapted to the circumstances. %
This is particularly important when the audience is diverse as one predefined type of an explanation may be insufficient since it is unlikely to address all the possible concerns, questions and unique perspectives. %
An XAI or IML explainer that communicates through contrastive explanations and provides the explainees with an opportunity to interactively customise and personalise them~\cite{madumal2019grounded} -- offering a chance to contest and rebut them in case of a disagreement -- should therefore be considered the gold standard~\cite{wachter2017counterfactual,miller2019explanation}.%
In addition to enhancing explainee satisfaction, operating within this purview has other, far-reaching benefits such as enabling algorithmic fairness evaluation, accountability assessment and debugging of predictive models. %
It is also compatible with all the elements of the artificial intelligence or machine learning workflow -- which consists of data, models and predictions -- as each of these components may be in need of interpretability. %
In view of a variety of explainability approaches, each operating in a unique way, we also looked at the disputed trade-off between explainability and predictive power, the existence of which has mostly been supported by anecdotal evidence thus far, albeit recent studies show that this dependency may be more nuanced than previously expected. %
We then connected this debate to the distinction between inherent (ante-hoc) and retrofitted (post-hoc) explainability: the former provides explanations of superior quality but requires extensive engineering effort to be built, whereas the latter is flexible and universal at the expense of fidelity. %
While the former may be shunned due to the required work, we argued that building trustworthy post-hoc explainers may be just as complicated and demand just as much commitment since these seemingly easy to use tools %
conceal a complex process governing their composition and influencing their quality behind the facade of universality~\cite{sokol2019blimey,sokol2020limetree,sokol2020towards,sokol2020tut}. %
This considerable effort required to set them up, therefore, illuminates a crucial question: Is it better to spend time and effort on configuring post-hoc explainers or instead invest these resources into building inherently explainable predictive models? %
Unsurprisingly, there is no definitive answer given the uniqueness of each individual case, e.g., legacy systems and predictors built from scratch.%
Regardless of the particular implementation and operationalisation details, explainers of automated decision-making systems should adopt and embody as many of these findings as possible to engender trust in data-driven predictive models. %
Since each explanation reveals just a fragment of the modelling process and only the right mixture of evidence can paint the full picture, XAI and IML approaches need to be responsive and adapt seamlessly to the users' requests and expectations. %
Such an engaging algorithmic interlocutor should build logically consistent narratives and serve more as a guide and a teacher than a facts reporter. %
To this end, we need to develop an explanatory process built on top of a system that enables logical reasoning between intelligent agents: human--machine or machine--machine. %
An appropriate foundation -- managing the dialogue as well as tracking and storing the evolving knowledge base of the involved parties -- should benefit and encourage an interdisciplinary research agenda drawing from multiple areas of computer and social sciences. %
In the end, nonetheless, the explainee needs to be a savvy interrogator, asking the right questions and firmly navigating the entire process to understand the behaviour of such data-driven oracles. %
After all, in Arthur C.\ Clarke's words: %
``Any sufficiently advanced technology is indistinguishable from magic.'' %
While this view may partially reflect a broader perception of artificial intelligence and machine learning applications, the work presented here reconciles XAI and IML research published to date to establish a solid foundation for addressing open questions in an effort to demystify predictive algorithms and harness their full potential. %
The logical next step in this pursuit is development of a comprehensive framework, flexible protocol and suite of (quantitative \& qualitative) metrics to meaningfully evaluate the quality and effectiveness of explainable AI and interpretable ML techniques, allowing us to choose the best possible solution for each unique problem.%
\renewcommand{\acksname}{Acknowledgements}
\begin{acks}
This research was partially supported by %
the TAILOR project, funded by EU Horizon 2020 research and innovation programme under GA No 952215; and %
the ARC Centre of Excellence for Automated Decision-Making and Society, funded by the Australian Government through the Australian Research Council (project number CE200100005).%
\end{acks}
\section*{Author Contributions}
Conceptualisation, K.S.; Methodology, K.S.; Investigation, K.S.; Writing -- Original Draft, K.S.; Writing -- Review \& Editing, K.S.\ and P.F.; Supervision, P.F.; Funding Acquisition, P.F.%
\section*{Declaration of Interests}
The authors declare no competing interests.%
\bibliographystyle{ACM-Reference-Format}
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2024-02-18T23:40:24.564Z
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2022-09-12T02:05:29.000Z
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{"paloma_paragraphs":[]}
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{"arxiv_id":"2112.14466","language":"en","timestamp":1662948329000,"url":"https:\/\/arxiv.org\/abs\/2112.14466","yymm":"2112"}
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proofpile-arXiv_000-10040
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{"provenance":"002.jsonl.gz:10041"}
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\section{Introduction}
State estimation of a dynamical system plays an indispensable role in the correct functionality of a wide variety of applications such as energy management systems \cite{9088966}, smart grids \cite{8546788}, robotics \cite{barfoot2017state}, and intelligent transportation systems \cite{7775094}. For linear state-space models (SSMs) with additive White Gaussian noise (AWGN), Kalman Filter (KF) is the optimal state estimator in terms of the minimum mean square error (MSE) criterion \cite{kalman1960new}. For nonlinear SSMs, the filtering becomes more challenging owing to the underlying functional nonlinearities. For such systems, several filtering approaches have been presented in the literature including the Extended Kalman Filter (EKF) \cite{grewal2014kalman}, the Unscented Kalman Filter (UKF)\cite{julier1997new}, Particle Filters (PFs) \cite{gordon1993novel} etc. For a survey of nonlinear state estimators, the reader is referred to \cite{norgaard2000new}.
The performances of standard filtering approaches rely on the assumption that the statistics of \textit{nominal} noise entering the system are perfectly known \textit{apriori}. However, the assumption can easily be violated in practice when the measurements are disturbed by noise not described by the known statistics. \textcolor{black}{In addition to the nominal noise, the observations can further be corrupted by other abnormalities commonly referred to as \textit{gross errors} which include outliers and biases \cite{narasimhan1999data}. Generally modeled with zero-mean statistics, outliers are sudden sporadic surges in the measurements. Their occurrence can be attributed to factors like sensor degradation, communication failures, environmental influences, etc. \cite{9050910}. On the other hand, biases manifest in the data with non-zero mean noise statistics \cite{9050891}. Multiple factors in various applications lead to the appearance of biased observations. Examples include miscalibrations of sensors, other configuration aberrations like errors in sensor location or alignment, clock errors, or malfunctioning \cite{belfadel2019single}. In this work, we keep our focus only on the appearance of \textcolor{black}{measurement biases} and how these can be compensated within the filtering framework.}
Since the magnitude of biases in the data, the instances of their occurrence, and the particular measurement dimensions which get affected are unknown and only partial statistics describing such corruptions are available, the problem of filtering during their presence is nontrivial. The challenge is further compounded by the functional non-linearities appearing in the SSMs. Given the significance of dealing with \textcolor{black}{biases} in the data during filtering, the topic has historically garnered the attention of various investigators. The approaches for devising bias-robust filters differ in the way such anomalies are neutralized in the filtering process.
Conventionally, biases are catered by assuming that the affected measurement dimensions are known in advance. Moreover, the bias evolution models are assumed to be simplistic or completely ignored during filtering. A straightforward approach is to jointly consider the state vector and bias vector for inference supposing the biases are described in a simple Markovian manner. With computational limitations at that time, earlier works attempted to reduce the processing overhead for such formulations \cite{1099223}. In a similar vein, Schmidt aimed to simplify the joint state and bias estimation in the SKF formulation resulting in the celebrated Schmidt Kalman Filter (SKF) \cite{schmidt1966application}. Interestingly, the bias is not estimated at each time step and only its correlations with the state are updated instead, making the SKF \textit{suboptimal} even if the bias transition can perfectly be modeled \cite{grewal2014kalman}. Ideas similar to the SKF have also been proposed to cater for biased measurements, in terms of exploiting partial information e.g. positivity of biases \cite{filtermobile}. These kinds of \textit{suboptimal} approaches are more useful in scenarios where it is safe to ignore the information regarding the evolution of \textcolor{black}{measurement biases}. With the advances in available processing power, the joint state and parameter estimation approach, in the KF framework, remains the standard go-to approach for catering \textcolor{black}{biased observations}\cite{zhang2021extended}.
Conventional methods are more relevant when the bias manifests in the observations consistently throughout the entire duration of system operation. However, these methods inherently assume prior knowledge regarding the affected measurement dimensions. For example, the authors do not consider bias estimation in the angle of arrival (AOA) measurements for filtering \cite{1326705}. Such information may be available beforehand for some applications. However, it must generally be obtained from some \textit{detection} mechanism for the algorithms to work properly especially if the compromised dimensions vary over time.
As a result, more sophisticated schemes have been proposed by integrating the detection process with the filtering framework. To this end, following two possibilities exist: 1) use some external/separate detectors, 2) incorporate the detection/compensating process within a unified filtering framework. Both of these approaches have their merits and drawbacks. External detectors are particularly advantageous in terms of their off-the-shelf accessibility to several options. However, the performance of robust filtering is highly dependent on the functionality of these detectors and the way they integrate. On the other hand, internal detection methodologies are harder to design but obviate dependence on external algorithms.
In the literature, several kinds of external bias detectors sometimes called fault detection and identification algorithms (FDI), have been reported for several applications like tracking using UWB, GPS, and UMTS measurements. For example, in \cite{886790}, biased measurements are identified simply by comparison of the standard deviation of range measurements with a detection threshold. A similar approach is to use normalized residuals to detect the presence of \textcolor{black}{any bias}\cite{hu2020robust}. The method proposed in \cite{562692} uses a historical record of sequential observations and performs a hypothesis test for detection. For bias detection, the use of classical statistical hypothesis tests like likelihood ratio test (LRT) and other probability ratio tests, has also been documented \cite{686556,7080484,6549130, hu2020robust}. In addition, other methods resort to deep learning for the determination of affected measurements \cite{9108193}. There are several bias compensating filtering methods that rely on these kinds of separate detectors \cite{8805384,filtermobile,1583910}.
The other approach based on inherent bias detection and compensation for filtering is more challenging due to two underlying reasons. First, modeling bias inside the SSM, in a Markovian fashion, is tricky. As noted in \cite{jourdan2005monte} the bias transition cannot be simply modeled as a Gaussian centered at the current bias value. In \cite{jourdan2005monte}, the authors model the bias stochastically by assuming it remains clamped to the previous value with certain predefined probability and jumps with the remaining probability within a set range represented as a uniform distribution. In \cite{gonzalez2009mobile}, a similar model for describing bias is used. The use of random bias model has been proposed in \cite{chughtai2020robust} where the bias is represented with Gaussian density and subsequently the mean of the distribution is described in a Markovian manner. The use of Bernoulli random vectors is proposed to switch between multiple models catering for the presence and absence of \textcolor{black}{biases}. These kinds of models describe \textcolor{black}{measurement biases} effectively but make the use of KFs variants difficult since the underlying distributions for modeling do not remain Gaussian. Furthermore, the sole use of Gaussian approximations for inference no longer remains suitable. Therefore, the authors in \cite{jourdan2005monte,gonzalez2009mobile,chughtai2020robust}, opt for the powerful PFs for the inference that can effectively handle arbitrary probability densities. The use of a large number of particles ranging from 15000 to 20000 is reported in these works. Therefore, the computational overhead of PF remains a concern and can be prohibitive in different applications.
Literature survey indicates the need for computationally efficient methods, with inherent detection mechanisms, to deal with \textcolor{black}{measurement biases}. Though PFs, with reduced processing burden, can be devised and several researchers are concerned with this direction of investigation \cite{1036890,5443441}, an alternate is to use the Kalman filtering results for devising tractable robust filters. Recently, the use of Variational Bayes methods has gained traction in this regard and we focus on this class of approach. Though several robust methods belonging to this class exist that consider \textcolor{black}{outliers in data} \cite{8398426,6349794,8869835,chughtai2021outlier}, however, generally there is a shortage of such methods that deal with \textcolor{black}{biased measurements}. Recent attempts in this direction include the work in \cite{9050891} where the authors consider the Student's-t-inverse-Wishart distribution to handle time-varying bias. However, only linear systems are considered in the derivation.
Given the backdrop, we present a novel robust filtering method to deal with \textcolor{black}{biases} in the measurements for nonlinear systems. {The main contributions of this work are as follows.}
\begin{itemize}
\item {We present the bias detecting and mitigating (BDM) filter to deal with \textcolor{black}{measurements bias} using the Gaussian filtering framework. We resort to Variational inference for designing the filter with an internal bias detection mechanism as opposed to schemes requiring external detectors.}
\item {For a given model we evaluate the Posterior Cramer-Rao Bound (PCRB) to determine the theoretical benchmark for the error performance.}
\item {For two different scenarios i.e. persistent and temporary bias presence we evaluate the performance of the BDM filter as compared to different estimators in the literature indicating the advantages of the proposed method.}
\end{itemize}
The way we have organized this article is as follows. Section \ref{modelling_sec} describes our modeling choice for incorporation of \textcolor{black}{biases} inside the SSM. In Section \ref{inference}, the derivation of the filter is provided. Subsequently, the performance evaluation results have been discussed in Section \ref{simulation_sec}. Lastly, conclusive comments are given in Section \ref{Conc}.
\textcolor{black}{\subsection*{Related Works}}
\textcolor{black}{Robust filtering theory has been extensively investigated in the literature. Researchers have been interested in various dimensions of the problem including the design and evaluation of filters robust to different effects like anomalies (outliers and biases) in the process and measurement models, missing observations, system modeling errors, adversarial attacks on sensors, etc. Several robust filters have been proposed and evaluated for different applications resorting to techniques from diverse areas. In this subsection, we briefly highlight some other robust filtering methods.}
\textcolor{black}{The classical Wiener filter \cite{wiener1949extrapolation} has been successfully extended to its robust counterparts in the literature. The basic idea is to consider the least favorable power spectral densities (PSDs) for any specific uncertainty model assumed for the signal of interest and the noise. Subsequently, the optimal (using min-max MSE criterion) solution is sought to devise such filters \cite{1056875}.}
\textcolor{black}{Similarly, the KF has numerous robust extensions apart from the ones discussed in the main introduction section. Some of its basic outlier-robust derivatives include $3\sigma$-rejection and score function type KFs \cite{zoubir2018robust}. These formulations use score functions applied to the residuals of the observed and predicted measurements to minimize the effect of outliers. Another robust KF derivative is the approximate conditional mean (ACM) filter which is based on the approximation of the conditional observation density prior to updating \cite{1100882}. Some other variations rely on a bank of KFs to gain robustness \cite{schick1994robust}. The cause of robust filtering has also been well served by the theory of M-estimation which has helped develop many of these methods \cite{zoubir2018robust}. In addition, regression-based KFs have also been proposed for achieving robustness \cite{5371933}. Similarly, guaranteed cost-based methods \cite{317138}, Krein space methods \cite{lee2004robust} and linearly constrained KFs (LCKFs) have also been proposed in this regard \cite{9638328}. Ambiguity sets for catering model distributional uncertainties have also been employed to this end \cite{shafieezadeh2018wasserstein}. Besides different information theoretic criteria have been used to devise robust KFs \cite{chen2017maximum}. KFs extensions to deal with system parametric uncertainties are also well-documented \cite{lewis2017optimal}.}
\textcolor{black}{Other types of robust filtering approaches have also been reported in the literature. $\text{H}_{\infty}$ is a popular approach that aims to minimize the worst-case estimation error by formulating a min-max problem using a smartly chosen objective function \cite{simon2006optimal}. Similarly, mixed Kalman/$\text{H}_{\infty}$ approaches have also been proposed leveraging the merits of both the methods \cite{7555348}. The use of finite impulse response (FIR) filters can also be found in this regard \cite{5428832,8355704,8744320}. Similarly, robust recursive estimators for SSMs resorting to sensitivity penalization-based methods have been devised \cite{zhou2010sensitivity}. The use of nonparametric techniques can also be found in the robust filtering literature as well \cite{zoubir2018robust}.} \\
\section{\textcolor{black}{Bias} Modeling}\label{modelling_sec}
\textcolor{black}{As the standard SSM does not consider the possibility of \textcolor{black}{measurement biases} in its \textit{generative} structure \cite{6266757} it needs to be modified}. At the same time, the model should remain amenable for VB inference. To this end, we choose the inference model from our previous work \cite{chughtai2020robust}, with a few modifications. For a discrete time SSM, the process and measurement equations are given as follows
\begin{align}
\mathbf{x}_k= & \mathbf{f}(\mathbf{x}_{k\text{-}1})+\mathbf{q}_{k\text{-}1}\label{eqn1}\\
\mathbf{y}_k =& \mathbf{h}(\mathbf{x}_{k})+\mathbf{r}_k + \boldsymbol{{\mathcal{I}}}_{k}{\mathbf{\Theta}}_k \label{eqn2}
\end{align}
where $k$ denotes the time-index, $\mathbf{x}_k\in\mathbb{R}^n$ and $\mathbf{y}_k\in\mathbb{R}^m$ are the state and measurement vectors respectively, $\mathbf{q}_{k\text{-}1}\in\mathbb{R}^n$ and $\mathbf{r}_k\in\mathbb{R}^m$ are white process and measurement noise vectors, $\mathbf{f}(.)$ and $\mathbf{h}(.)$ represent nonlinear process and measurement dynamics respectively, $\mathbf{\Theta}_k\in\mathbb{R}^m$ models the effect of biases in the measurements and $\boldsymbol{\mathcal{I}}_{k} \in \mathbb{R}^{m\times m}$ is a diagonal matrix with Bernoulli elements ${\mathcal{I}}^i_{k}$ used to indicate the occurrence of bias in different dimensions. We assume the following noise distributions: $\mathbf{q}_{k\text{-}1} \sim \mathcal{N}(\mathbf{0},\mathbf{Q}_{k\text{-}1})$ and $\mathbf{r}_k \sim \mathcal{N}(\mathbf{0}, \mathbf{R}_k)$. We assume that measurements are obtained from independent sensors making $\mathbf{R}_k$ diagonal. For inferential tractability, the model as originally reported is simplified by ignoring the added randomness in the bias magnitude in \eqref{eqn2}. The bias evolution is expressed as follows where the modeling rationale remains the same as originally reported in \cite{chughtai2020robust}.
\begin{equation}
{\mathbf{\Theta}}_k = (\mathbf{I}-\boldsymbol{{\mathcal{I}}}_{k\text{-}1}){\widetilde{\mathbf{\Theta}}}_{k} +\boldsymbol{{\mathcal{I}}}_{k\text{-}1}({\mathbf{\Theta}}_{k\text{-}1} + {\Delta_k})\label{eqn3}
\end{equation}
In \eqref{eqn3}, each entry of ${\Delta_k}$ allows for any drifts/changes in the bias value over time, in the corresponding dimension, given bias was present at the previous time step. On the contrary, if no bias occurred in any given dimension, at the preceding instant, it can possibly occur with a very large variance ${\sigma^2_{\widetilde{{\Theta}}}}$ (assuming an uninformative prior) described by the respective entries of a zero mean random vector ${\widetilde{\mathbf{\Theta}}}_{k}$. The distributions of ${\Delta_k}$ and ${\widetilde{\mathbf{\Theta}}}_{k}$ are supposed to be white and normally distributed given as
\begin{align}
{\Delta_k} &\sim \mathcal{N}(\mathbf{0},\breve{\mathbf{\Sigma}}_{k}) \text{ with } \breve{\mathbf{\Sigma}}_{k}= \mathrm{diag}\left(\sigma^2_{\vartriangle{1}}, \cdots, \sigma^2_{\vartriangle{m}} \right) \label{PPFeqn30}&\\
{\widetilde{\mathbf{\Theta}}}_{k} &\sim \mathcal{N}(\mathbf{0},\widetilde{\mathbf{\Sigma}}_{k}) \text{ with } \widetilde{\mathbf{\Sigma}}_{k} = \mathrm{diag}\left({\sigma^2_{\widetilde{{\Theta}}}}, \cdots, {\sigma^2_{\widetilde{{\Theta}}}} \right) \label{PPFeqn31}
\end{align}
Note that for tractability, we have modified ${\widetilde{\mathbf{\Theta}}}_{k}$ to be normally distributed instead of obeying a uniform distribution. However, assuming a very large variance does not make a practical difference. Also note that for simplicity, we do not take any transition model for $\boldsymbol{\mathcal{I}}_{k}$ and assume its elements occur independently at each instance. Remaining modeling assumptions are kept the same as originally reported.
\section{Recursive Bayesian Inference}\label{inference}
Considering the inference model in \eqref{eqn1}-\eqref{eqn3}, the Bayes rule can be employed recursively to express the joint posterior distribution of $\mathbf{x}_k$, $\bm{\mathcal{I}}_k$ (considering only the random entries ${\mathcal{I}}^i_k$) and ${\mathbf{\Theta}}_k$ conditioned on the set of all the observations $\mathbf{y}_{1:{k}}$ analytically as \begin{equation}
p(\mathbf{x}_k,\bm{\mathcal{I}}_k,{\mathbf{\Theta}}_k|\mathbf{y}_{1:{k}})=\frac{p(\mathbf{y}_k|\mathbf{x}_{k},\bm{\mathcal{I}}_k,{\mathbf{\Theta}}_k) p(\mathbf{x}_k,\bm{\mathcal{I}}_k,{\mathbf{\Theta}}_k|\mathbf{y}_{1:{k\text{-}1}})}{p(\mathbf{y}_k|\mathbf{y}_{1:{k\text{-}1}})}
\label{eqn_vb_1}
\end{equation}
Theoretically, the joint posterior can be marginalized to obtain the expression for $p(\mathbf{x}_k|\mathbf{y}_{1:{k}})$. With this approach, the exact sequential Bayesian processing becomes computationally infeasible. Therefore, we adopt the VB method \cite{vsmidl2006variational} for inference where the product of VB marginals is conveniently used to approximate the joint posterior as
\begin{equation}
p(\mathbf{x}_k,\bm{\mathcal{I}}_k,{\mathbf{\Theta}}_k|\mathbf{y}_{1:{k}})\approx q(\mathbf{x}_k)q(\bm{\mathcal{I}}_k)q({\mathbf{\Theta}}_k)
\label{eqn_vb_2}
\end{equation}
With an objective to minimize the Kullback-Leibler divergence (KLD) between the marginal product and the true posterior, the VB method leads to the following marginals
\begin{align}
q(\mathbf{x}_k)&\propto \exp( \big\langle\mathrm{ln}(p(\mathbf{x}_k,\bm{\mathcal{I}}_k,{\mathbf{\Theta}}_k|\mathbf{y}_{1:{k}})\big\rangle_{ q({{\bm{\mathcal{I}}}_k}) {q(\mathbf{\Theta}}_k)})\label{eqn_vb_3}\\
q(\bm{\mathcal{I}}_k)&\propto \exp ( \big\langle\mathrm{ln}(p(\mathbf{x}_k,\bm{\mathcal{I}}_k,{\mathbf{\Theta}}_k|\mathbf{y}_{1:{k}})\big\rangle_{{q(\mathbf{x}_k){q(\mathbf{\Theta}}_k)}})\label{eqn_vb_4}\\
q({\mathbf{\Theta}}_k)&\propto \exp ( \big\langle\mathrm{ln}(p(\mathbf{x}_k,\bm{\mathcal{I}}_k,{\mathbf{\Theta}}_k|\mathbf{y}_{1:{k}})\big\rangle_{q(\mathbf{x}_k)q(\bm{\mathcal{I}}_k)})\label{eqn_vb_5}
\end{align}
where $ \langle.\rangle_{q(\bm{\psi}_k)}$ denotes the expectation of the argument with respect to a distribution $q(\bm{\psi}_k)$. The VB marginals can be updated iteratively until convergence, using \eqref{eqn_vb_3}-\eqref{eqn_vb_5} in turn. The procedure provides a convenient way to approximate the true marginals of the joint posterior by approximating these as $ p(\mathbf{x}_k|\mathbf{y}_{1:{k}})$$\approx$$ {q^c(\mathbf{x}_k)}$, $ p(\bm{\mathcal{I}}_k|\mathbf{y}_{1:{k}})$$\approx$${q^c(\bm{\mathcal{I}}_k)}$ and $ p({\mathbf{\Theta}}_k|\mathbf{y}_{1:{k}})$$\approx$$ {q^c({\mathbf{\Theta}}_k)}$ where ${q^c(.)}$ denotes the VB marginals obtained after convergence.
\subsection{Prediction}
Assuming that at each time step the posterior is approximated with a product of marginals, the predictive density can be \textcolor{black}{approximated} as
\begin{align}
&p(\mathbf{x}_k,\bm{\mathcal{I}}_k,{\mathbf{\Theta}}_k|\mathbf{y}_{1:{k\text{-}1}})\approx p(\bm{\mathcal{I}}_k)p(\mathbf{x}_k|\mathbf{y}_{{1:k\text{-}1}}) p({\mathbf{\Theta}}_k|\mathbf{y}_{1:k\text{-}1}) \label{eqn_vb_8}
\end{align}
with
\begin{align}
p(\mathbf{x}_k|\mathbf{y}_{1:{k\text{-}1}})&= \int p(\mathbf{x}_k|\mathbf{x}_{k\text{-}1}) p(\mathbf{x}_{k\text{-}1}|\mathbf{y}_{1:{k\text{-}1}}) d\mathbf{x}_{k\text{-}1} \label{eqn_vb_9}\\
p({\mathbf{\Theta}}_k|\mathbf{y}_{1:k\text{-}1}) &\approx \int\int p({\mathbf{\Theta}}_k|\bm{\mathcal{I}}_{k\text{-}1},{\mathbf{\Theta}}_{k\text{-}1}) p(\bm{\mathcal{I}}_{k\text{-}1}|\mathbf{y}_{1:{k\text{-}1}})\nonumber\\ &\hspace{.5cm}p({\mathbf{\Theta}}_{k\text{-}1}|\mathbf{y}_{1:{k\text{-}1}}) d\bm{\mathcal{I}}_{k\text{-}1}d{\mathbf{\Theta}}_{k\text{-}1}\label{eqn_vb_10}
\end{align}
We assume that the occurrence of bias is independent for each dimension and its historical existence. Using $\theta^i_k$ to denote the prior probability of occurrence of bias in the $i$th observation, the distribution of $\bm{\mathcal{I}}_k$ is defined as product of independent Bernoulli distributions of each element
\begin{equation}
p(\bm{\mathcal{I}}_k)=\prod_{i=1}^{m}p({{\mathcal{I}}}^i_k)=\prod_{i=1}^{m} (1-{\theta^i_k}) \delta({{{\mathcal{I}}}^i_k})+{\theta^i_k}\delta( {{{\mathcal{I}}}^i_k}-1)
\label{eqn_model_11}
\end{equation}
where $\delta(.)$ denotes the delta function.
To evaluate \eqref{eqn_vb_9}-\eqref{eqn_vb_10}, suppose the following distributions for the posterior marginals at time step $k-1$
\begin{align}
&p(\mathbf{x}_{k\text{-}1}|\mathbf{y}_{1:{k\text{-}1}})\approx {q^c(\mathbf{x}_{k\text{-}1})}\approx \mathcal{N}(\mathbf{x}_{k\text{-}1}|\mathbf{\hat{x}}^+_{k\text{-}1},\mathbf{{P}}^+_{k\text{-}1}) \label{eqn_vb_11}\\
&p(\bm{\mathcal{I}}_{k\text{-}1}|\mathbf{y}_{1:{k\text{-}1}})\approx {q^c(\bm{\mathcal{I}}_{k\text{-}1})}=\prod_{i=1}^{m}p({{\mathcal{I}}}^i_{k\text{-}1}|\mathbf{y}_{1:{k\text{-}1}})\nonumber \\
&=\prod_{i=1}^{m} (1-{\Omega^i_{k\text{-}1}}) \delta({{{\mathcal{I}}}^i_{k\text{-}1}})+{\Omega^i_{k\text{-}1}}\delta( {{{\mathcal{I}}}^i_{k\text{-}1}}-1) \label{eqn_vb_12} \\
&p({\mathbf{\Theta}}_{k\text{-}1}|\mathbf{y}_{1:{k\text{-}1}})\approx {q^c({\mathbf{\Theta}}_{k\text{-}1})} \approx\mathcal{N}(\mathbf{\Theta}_{k\text{-}1}|\mathbf{\hat{\Theta}}^+_{k\text{-}1},\mathbf{{\Sigma}}^+_{k\text{-}1})\label{eqn_vb_13}
\end{align}
where ${\Omega^i_{k}}$ denotes the posterior probability of bias occurrence in the $i$th dimension. The notation $\mathcal{N}(\mathbf{x}|\mathbf{m},\mathbf{{\Sigma}})$ represents a multivariate normal distribution with mean $\mathbf{m}$ and covariance $\mathbf{{\Sigma}}$, evaluated at $\mathbf{x}$. The verification of the functional forms of the distributions and the expressions of their parameters are provided in the subsequent update step of the Bayesian filter.
Since $\mathbf{f}(.)$ is assumed to \textcolor{black}{be} nonlinear, $p(\mathbf{x}_k|\mathbf{y}_{1:k\text{-}1})$ can be approximated, using general Gaussian filtering results \cite{sarkka2013bayesian}, as $\mathcal{N}(\mathbf{x}_{k}|\mathbf{\hat{x}}^-_{k},\mathbf{{P}}^-_{k})$ with the parameters predicted as follows
\begin{flalign}
\mathbf{\hat{x}}^{-}_{k}&=\big\langle \mathbf{f}(\mathbf{x}_{k\text{-}1})\big\rangle_{p(\mathbf{x}_{k\text{-}1}|\mathbf{y}_{1:k\text{-}1})}&\label{eqn_vb_14}\\
\mathbf{P}^{-}_{k}&=\big\langle(\mathbf{f}(\mathbf{x}_{k\text{-}1})-\mathbf{\hat{x}}^{-}_{k})(\mathbf{f}(\mathbf{x}_{k\text{-}1})-\mathbf{\hat{x}}^{-}_{k})^{\top}\big\rangle_{p(\mathbf{x}_{k\text{-}1}|\mathbf{y}_{1:{k\text{-}1}})}+\mathbf{Q}_{k\text{-}1}\label{eqn_vb_15}
\end{flalign}
The remaining term required to approximate the predictive density recursively in \eqref{eqn_vb_8} is $p({\mathbf{\Theta}}_k|\mathbf{y}_{k\text{-}1})$. Observing \eqref{eqn3} and \eqref{eqn_vb_10}, it is evident that $p({\mathbf{\Theta}}_k|\mathbf{y}_{k\text{-}1})$ is a sum of $2^m$ Gaussian densities scaled by the probabilities of combinations of bias occurrence at previous time instance. Obviously this makes recursive inference intractable, so we propose to approximate this distribution with a single Gaussian density $\mathcal{N}(\mathbf{\Theta}_{k}|\mathbf{\hat{\Theta}}^-_{k},\mathbf{{\Sigma}}^-_{k})$ using moment matching \cite{sarkka2013bayesian}. The parameters of the distribution are updated as
\begin{flalign}
\mathbf{\hat{\Theta}}^{-}_{k}&=\mathbf{\Omega}_{k\text{-}1}\mathbf{\hat{\Theta}}^{+}_{k\text{-}1}\label{eqn_vb_16}\\
\mathbf{\Sigma}^{-}_{k}&=(\mathbf{I}-\mathbf{\Omega}_{k\text{-}1}) \widetilde{\mathbf{\Sigma}}_{k}+\mathbf{\Omega}_{k\text{-}1}\breve{\mathbf{\Sigma}}_{k}& \nonumber\\
&+\mathbf{{\Sigma}}^+_{k\text{-}1}\odot(\mathrm{diag}(\mathbf{\Omega}_{k\text{-}1}){\mathrm{diag}(\mathbf{\Omega}_{k\text{-}1})}^{\top}+\mathbf{\Omega}_{k\text{-}1}(\mathbf{I}-\mathbf{\Omega}_{k\text{-}1}))&\nonumber \\
&+\mathbf{\Omega}_{k\text{-}1}(\mathbf{I}-\mathbf{\Omega}_{k\text{-}1})(\mathrm{diag}(\mathbf{\hat{\Theta}}^{+}_{k\text{-}1}))^2 &
\label{eqn_vb_17}
\end{flalign}
where $\mathbf{\Omega}_{k\text{-}1}$ is a diagonal matrix with entries ${\Omega}^i_{k\text{-}1}$ denoting the posterior probability of bias occurrence at time step $k-1$. The operator $\odot$ is the Hadamard product and $\mathrm{diag}$ is used for vector to diagonal matrix conversion and vice versa. The reader is referred to the Appendix for detailed derivations of \eqref{eqn_vb_16}-\eqref{eqn_vb_17}.
\subsubsection*{Remarks}
We note the following in \eqref{eqn_vb_16}-\eqref{eqn_vb_17}
\begin{itemize}
\item $\mathbf{\Omega}_{k\text{-}1}$ dictates how the parameters $\mathbf{\hat{\Theta}}^-_{k}$ and $\mathbf{{\Sigma}}^-_{k}$ are predicted.
\item For the case when $\mathbf{\Omega}_{k\text{-}1}=\mathbf{I}$, i.e. bias is inferred in each dimension at time step $k{-}1$ with probability 1, $ \mathbf{\hat{\Theta}}^{-}_{k}=\mathbf{\hat{\Theta}}^{+}_{k\text{-}1}$ and $\mathbf{\Sigma}^{-}_{k}=\mathbf{{\Sigma}}^+_{k\text{-}1}+\breve{\mathbf{\Sigma}}_{k}$. In other words, the mean of the bias prediction (for each dimension) is retained and its covariance is predicted as sum of previous covariance and the covariance considered for the amount of drift/change in the bias.
\item For the case when $\mathbf{\Omega}_{k\text{-}1}=\mathbf{0}$, i.e. no bias is inferred in each dimension at time step $k{-}1$ with probability 1, $ \mathbf{\hat{\Theta}}^{-}_{k}=\mathbf{0}$ and $\mathbf{\Sigma}^{-}_{k}=\widetilde{\mathbf{\Sigma}}_{k}$. In other words, the mean of the bias prediction (for each dimension) is $\mathbf{0}$ and its covariance is predicted with very large entries.
\item Similarly, the prediction mechanism can be understood if only some dimensions are inferred to be disturbed with probability 1. The bias in the particular dimensions are predicted with the mean retained and covariance updated as addition of previous covariance plus the covariance allowed for the drift/change.
\item Lastly, if there is partial confidence on the occurrence of bias in any dimension at $k{-}1$, the predicted Gaussian distribution is shifted to the mean of bias estimate at $k{-}1$ scaled with a factor of ${\Omega}^i_{k\text{-}1}$. In addition, the covariance gets inflated by addition of scaled elements of $\widetilde{\mathbf{\Sigma}}_{k}$ and squared terms of mean at $k{-}1$. In other words, it can be interpreted in a sense that unless there is a very high confidence of occurrence of bias at the previous time instance, the bias would be predicted with a large covariance.
\end{itemize}
\subsection{Update}
For the update step, we resort to \eqref{eqn_vb_1}-\eqref{eqn_vb_5} and use \eqref{eqn_vb_8} for approximating the predictive density. For detailed derivations, the reader is referred to the Appendix.
Parameters of $q(\mathbf{x}_k)$ are updated iteratively as
\begin{flalign}
\hat {\mathbf{x}} _{k} ^{+} &= \hat {\mathbf{x}}_{k}^{-} + \mathbf{K}_k(\mathbf{y}_{k} - {\boldsymbol{\Omega}} _{k} \mathbf{\hat{\Theta}}^+_{k} - \bm{\mu}_k) &\label{eqn_vb_18} \\
\bm{\mu}_k&=\langle \mathbf{h}(\mathbf{x}_{k}) \rangle_{p(\mathbf{x}_k|\mathbf{y}_{k\text{-}1})}& \label{eqn_vb_19}\\
\mathbf{P}_{k}^{+} &= \mathbf{P}_{k}^{-} - \mathbf{C}_{k}\mathbf{K}_{k}^{\top}& \label{eqn_vb_20}\\
\mathbf{K}_{k} &= \mathbf{C}_{k}\mathbf{S}_{k}^{-1}& \label{eqn_vb_21}\\
\mathbf{C}_{k}&=\big \langle(\mathbf{x}_{k} - \hat {\mathbf{x}}_{k}^{-} )(\mathbf{h}(\mathbf{x}_{k}) - \bm{\mu}_k) \rangle_{p(\mathbf{x}_k|\mathbf{y}_{k\text{-}1})}& \label{eqn_vb_22}\\
\mathbf{S}_{k}&=\big \langle(\mathbf{h}(\mathbf{x}_{k}) - \bm{\mu}_k)(\mathbf{h}(\mathbf{x}_{k}) - \bm{\mu}_k)^{\top} \rangle_{p(\mathbf{x}_k|\mathbf{y}_{k\text{-}1})} + \mathbf{R}_{k}& \label{eqn_vb_23}
\end{flalign}
Parameters of ${q(\bm{\mathcal{I}}_k)}$ are updated iteratively as
\begin{flalign}
\Omega_{k}^{i} &= {\Pr({\mathcal{I}}_{k}^{i} = 1)}/({{\Pr}({\mathcal{I}}_{k}^{i} = 1) + \Pr({\mathcal{I}}_{k}^{i} = 0)})&\label{eqn_vb_29}
\end{flalign}
where denoting $k_1$ as the proportionality constant
\begin{flalign}
&\Pr({\mathcal{I}}^{i}_{k} = 0) = k_1 (1 - \theta_{k}^{i}) \exp{\big({-}\frac {1}{2} \big(\frac {(y^{i}_{k} - {\nu}^i_k )^{2}}{R_{k}^{i}} + \bar{h}^2_k\big)\big)}& \label{eqn_vb_24}\\
&\Pr({\mathcal{I}}_{k}^{i} = 1) = k_1 \theta_{k}^{i} \exp{ \big({-}\frac{1}{2} \frac{\bar{h}^2_k + \bar{\Theta}^2_k + ( {\nu}^i_k + \hat{\Theta}_{k}^{+^i} - y_{k}^{i})^{2}}{R_{k}^{i}}\big)}&\label{eqn_vb_25}\\
&\bm{\nu}_k= \langle {\mathbf{h}}(\mathbf{x}_{k}) \rangle_{q(\mathbf{x}_k)}&\label{eqn_vb_26}
\end{flalign}
\begin{flalign}
&\bar{h}^2_k = \langle(h^{i}(\textbf{x}_{k}) - {\nu}^i_k )^{2}\rangle_{q(\mathbf{x}_k)}& \label{eqn_vb_27}\\
&\bar{\Theta}^2_k = \langle({\Theta}_{k}^{i} - {{\hat{\Theta}}_k^{+i}})^2\rangle_{q(\mathbf{\Theta}_k)}&\label{eqn_vb_28}
\end{flalign}
Parameters of $q(\mathbf{\Theta}_k)$ are updated iteratively as
\begin{flalign}
\hat {\mathbf{\Theta}}_{k}^{*} &= \hat{\mathbf{\Theta}}_{k}^{-} + \bm{\mathcal{K}}_{k}(\mathbf{y}_{k} - ( \bm{\nu}_k + \boldsymbol{\Omega}_{k} \hat{\mathbf{\Theta}}_{k}^{-}))&\label{eqn_vb_30}\\
\mathbf{{\Sigma}}^*_{k} &= \mathbf{{\Sigma}}^-_{k} - \bm{\mathcal{C}}_{k}\bm{\mathcal{K}}_{k}^{\top}&\label{eqn_vb_31}\\
\bm{\mathcal{K}}_{k} &= \bm{\mathcal{C}}_{k}\bm{\mathcal{S}}_{k}^{-1}&\label{eqn_vb_32}\\
\bm{\mathcal{C}}_{k}&=\mathbf{{\Sigma}}^-_{k}\boldsymbol{\Omega}_{k}^{\top}&\label{eqn_vb_33}
\\
\bm{\mathcal{S}}_{k} &= \boldsymbol{\Omega}_{k}\mathbf{{\Sigma}}^-_{k}\boldsymbol{\Omega}_{k}^{\top} + \mathbf{R}_{k}&\label{eqn_vb_34}\\
\hat{\mathbf{\Theta}}_{k}^{+}&=\mathbf{\Sigma}_{k}^{+}{\mathbf{{\Sigma}}^*_{k}}^{-1}\hat{\mathbf{\Theta}}_{k}^{*}&\label{eqn_vb_35}\\
\mathbf{\Sigma}_{k}^{+} &= \big(\boldsymbol{\Omega}_{k}(\mathbf{I}-\boldsymbol{\Omega}_{k})\mathbf{R}_{k}^{-1} + {\mathbf{\Sigma}_{k}^*}^{-1}\big)^{-1}&\label{eqn_vb_36}
\end{flalign}
\begin{algorithm}[ht!]
\SetAlgoLined
Initialize\ $\hat {\mathbf{x}} _{0} ^{+},\mathbf{P}^{+}_0,\hat{\mathbf{\Theta}}_{0}^{+},
\mathbf{\Sigma}_{0}^{+}
,\widetilde{\mathbf{\Sigma}}_{k},\breve{\mathbf{\Sigma}}_{k},\mathbf{Q}_k,\mathbf{R}_k$;
\For{$k=1,2...K$}{
Evaluate $\hat {\mathbf{x}} _{k}^{-},\mathbf{P}^{-}_k$ with \eqref{eqn_vb_14} and \eqref{eqn_vb_15}\;
Evaluate $\mathbf{\hat{\Theta}}^{-}_{k},\mathbf{\Sigma}^{-}_{k}$ with \eqref{eqn_vb_16} and \eqref{eqn_vb_17}\;
Initialize $\theta^i_k$, $\Omega^{i(0)}_k$,$\hat {\mathbf{\Theta}} _{k} ^{+(0)}$ the convergence threshold $\tau$, \textcolor{black}{$\gamma$}$=\tau+1,\text{the iteration index} \ l=1$\;
Evaluate $\hat {\mathbf{x}} _{k} ^{+(0)}$ and ${\mathbf{P}^{+ (0)}_k}$ with \eqref{eqn_vb_18}-\eqref{eqn_vb_23}\;
\While{\textcolor{black}{$\gamma$}$>\tau$}{
Update $\Omega^{i(l)}_k\ \forall\ i$ with \eqref{eqn_vb_29}-\eqref{eqn_vb_28}\;
Update $\hat {\mathbf{\Theta}} _{k} ^{+(l)}$ and ${\mathbf{\Sigma}^{+ (l)}_k}$ with \eqref{eqn_vb_30}-\eqref{eqn_vb_36}\;
Update $\hat {\mathbf{x}} _{k} ^{+(l)}$ and ${\mathbf{P}^{+ (l)}_k}$ with \eqref{eqn_vb_18}-\eqref{eqn_vb_23}\;
Evaluate \textcolor{black}{$\gamma$}$={\|\hat {\mathbf{x}} _{k} ^{+(l)}-{\mathbf{x}} _{k} ^{+(l-1)} \|}/{\|{\mathbf{x}} _{k} ^{+(l-1)}\|}$\;
$l=l+1$\;
}
$\hat {\mathbf{x}} _{k} ^{+}=\hat {\mathbf{x}} _{k} ^{+(l-1)}$ and ${\mathbf{P}^{+}_k}={\mathbf{P}^{+(l-1)}_k}$\;
$\hat{\mathbf{\Theta}}_{k}^{+}=\hat{\mathbf{\Theta}}_{k}^{+(l-1)}$and $\mathbf{\Sigma}_{k}^{+}=\mathbf{\Sigma}_{k}^{+(l-1)}$\;
}
\caption{The proposed BDM filter}
\label{Algo1}
\end{algorithm}
\subsection{BDM Filter}
Using the proposed approximations, in the prediction and update steps, we have devised a recursive filter referred as the BDM filter. \textcolor{black}{Unless real-world experiments reveal any prior information regarding the occurrence of bias in each dimension, we propose using an uninformative prior for ${\mathcal{I}}^i_{k}\ \forall\ i$ which is commonly adopted for such cases. The Bayes-Laplace and the maximum entropy methods for obtaining uninformative prior for a parameter with finite values lead to the uniform prior distribution \cite{martz199414,turkman2019computational}. We adopt this choice of prior for our case i.e. $\theta^i_k=0.5\ \forall\ i$ which has been advocated in the literature for designing robust filters \cite{8869835,chughtai2021outlier}.} Algorithm \ref{Algo1} outlines the devised BDM filter.
\section{Numerical Experiments}\label{simulation_sec}
To evaluate the comparative performance of the devised algorithm, numerical experiments have been conducted on an Apple MacBook Air with a 3.2 GHz M1 Processor and 8 GBs of unified RAM using Matlab R2021a.
For comparative fairness, we consider methods based on the Unscented Kalman Filter (UKF) as their basic algorithmic workhorse. The following filters have been taken into account for comparisons: the standard UKF, the selective observations rejecting UKF (SOR-UKF) \cite{chughtai2021outlier}, the Unscented Schmidt Kalman Filter (USKF)\cite{stauch2015unscented} and the constrained Unscented Kalman Filter (CUKF). The CUKF is devised by modifying the CSRUKF \cite{filtermobile} by using the standard UKF instead of SRUKF as its core algorithm.
In terms of handling data corruption, the standard UKF has no bias compensation means in its construction. By contrast, the SOR-UKF is an outlier-robust filter, with inherent data anomaly detection mechanism, which discards the observations found to be corrupted. Lastly, the USKF and CUKF both compensate for the bias partially and require an external bias detection mechanism. Bias mitigating filters with inherent detection mechanism are generally scant in the literature and most of these are based on the PFs. The USKF is a modified version of the SKF, adapted for nonlinear systems, where the bias is not exactly estimated rather its correlations with the state are updated. The CUKF is based on two major functional components. First, it resorts to the UKF for estimation. Subsequently, it draws sigma points based on these estimates which are projected onto a region, by solving an optimization problem, supposing a constraint that the measurements can only be positively biased. In our numerical evaluation, we assume perfect detection for these two algorithms. Note that in the implementation of the USKF and CUKF, we switch to the standard UKF when no bias is detected. Also note that the proposed BDM filter has no limitations in terms of whether any bias positively or negatively disturbs the measurements. However, since the CUKF assumes a positive bias we keep this restriction in our simulations. In particular, bias \textcolor{black}{in each dimension }is added as a shifted Gaussian $\mathcal{N}(\mu,\sigma^2)$ where $\mu \geq 0$ \cite{filtermobile}.
\textcolor{black}{For performance evaluation, we resort to a target tracking problem in a wireless network where the range measurements are typically biased \cite{weiss2008network}. The observations get biased owing to the transitions between line-of-sight (LOS) and non-line-of-sight (NLOS) conditions \cite{4027766}. Such tracking systems find applications in emergency services, fleet management, intelligent transportation, etc. \cite{4960267}.}
\textcolor{black}{We consider the process equation for the target assuming an unknown turning rate as \cite{8398426}}
\begin{flalign}
\mathbf{x}_k=\mathbf{f}(\mathbf{x}_{k-1})+\mathbf{q}_{k-1}\label{eqn_res1a}
\end{flalign}
with
\begin{align}
\mathbf{f}(\mathbf{x}_{k-1}) &= \begin{pmatrix} \text{1} & \frac{\text{sin}(\omega_{k}{\textcolor{black}{\zeta_t}})}{\omega_{k}} & \text{0} & \frac{\text{cos}(\omega_{k}{\textcolor{black}{\zeta_t}})-\text{1}}{\omega_{k}} & \text{0} \\ \text{0} & \text{cos}(\omega_{k}{\textcolor{black}{\zeta_t}}) & \text{0} & -\text{sin}(\omega_{k}{\textcolor{black}{\zeta_t}}) & \text{0}\\ \text{0} & \frac{\text{1}-\text{cos}(\omega_{k}{\textcolor{black}{\zeta_t}})}{\omega_{k}} & \text{1} &\frac{\text{sin}(\omega_{k}{\textcolor{black}{\zeta_t}})}{\omega_{k}} & \text{0} \\ \text{0} & \text{\text{sin}}(\omega_{k}{\textcolor{black}{\zeta_t}}) & \text{0} & \text{cos}(\omega_{k}{\textcolor{black}{\zeta_t}}) & \text{0}\\ \text{0} & \text{0} & \text{0} & \text{0} &\text{1} \end{pmatrix} \mathbf{x}_{k-\text{1}}\label{eqn_res1}
\end{align}
where the state vector $\mathbf{x}_k= [a_k,\dot{{a_k}},b_k,\dot{{b_k}},\omega_{k}]^{\top}$
contains the 2D position coordinates \(({a_k} , {b_k} )\), the respective velocities \((\dot{{a_k}} , \dot{{b_k}} )\), the angular velocity $\omega_{k}$ of the target at time instant $k$, \( {\textcolor{black}{\zeta_t}} \) is the sampling period, and $\mathbf{q}_{k-\text{1}} \sim N\left(0,\mathbf{Q}_{k-\text{1}}.\right)$. $\mathbf{Q}_{k-\text{1}}$ is given in terms of scaling parameters $\eta_1$ and $\eta_2$ as \cite{8398426}
\begin{equation}
\mathbf{Q}_{k-\text{1}}=\begin{pmatrix} \eta_1 \mathbf{M} & 0 & 0\\0 &\eta_1 \mathbf{M}&0\\0&0&\eta_2
\end{pmatrix}, \mathbf{M}=\begin{pmatrix} {\textcolor{black}{\zeta_t}}^3/3 & {\textcolor{black}{\zeta_t}}^2/2\\{\textcolor{black}{\zeta_t}}^2/2 &{\textcolor{black}{\zeta_t}}
\end{pmatrix}\nonumber
\end{equation}
Range readings are obtained from $m$ sensors installed around a rectangular area where the $i$th sensor is located at $\big(a^{\rho_i}=350(i-1),b^{\rho_i}=350\ ((i-1)\mod2)\big)$. The nominal measurement equation can therefore be expressed as
\begin{equation}
\mathbf{y}_k = \mathbf{h}(\mathbf{x}_{k})+\mathbf{r}_k\label{eqn_res2}
\end{equation}
with
\begin{equation}
{h^i(\mathbf{x}_k )} = \sqrt{\big( (a_{k} - a^{\rho_i})^{2} + (b_{k} - b^{\rho_i})^{2} \big)}\label{eqn_res2b}
\end{equation}
{For the duration of bias presence the following observation equations, based on the random bias model \cite{park2022robust}, are assumed}
\begin{align}
\mathbf{y}_k &= \mathbf{h}(\mathbf{x}_{k})+\mathbf{r}_k+ \overbrace{\boldsymbol{{\mathcal{J}}}_{k} (\mathbf{o}_k+\triangle \mathbf{o}_k)}^{\mathbf{b}_k}\label{eqn_res2c}\\
\mathbf{y}_{k+1} &= \mathbf{h}(\mathbf{x}_{k+1})+\mathbf{r}_{k+1}+ \boldsymbol{{\mathcal{J}}}_{k} (\mathbf{o}_k+\triangle \mathbf{o}_{k+1})\label{eqn_res2d}
\end{align}
{where $\mathbf{o}_k\in\mathbb{R}^m$ models the effect of statistically independent biases in the measurements and $\boldsymbol{\mathcal{J}}_{k} \in \mathbb{R}^{m\times m}$ is a diagonal matrix with independent Bernoulli elements with parameter $\lambda$.} \textcolor{black}{Our measurement model for evaluation is inspired by the LOS and NLOS models in \cite{4027766,4960267} where shifted Gaussian distributions are used to model the biased measurements. In effect, bias with magnitude ${o}^i_k$ with some additional uncertainty $\triangle o^i_k \sim \mathcal{N}({0},\Sigma_\mathbf{o}^i)$ affects the \textit{i}th dimension at time step $k$ if ${\mathcal{J}}^i_{k}=1$. Since the exact magnitudes of biases are not generally known apriori and can occur randomly in a given range we assume $o^i_k$ to be uniformly distributed i.e. ${o}^i_k\sim\mathcal{U}(0,\Lambda^i)$. This is in contrast to the approaches in \cite{4027766,4960267} where the magnitudes of biases are assumed to be known perfectly. Similarly at time step $k+1$, the bias sustains with the same magnitude of the previous time step $k$ in the \textit{i}th dimension i.e. ${o}^i_k$ with some uncertainty $\triangle o^i_{k+1}$ if the \textit{i}th dimension at the previous time step $k$ was affected i.e. ${\mathcal{J}}^i_{k}=1$. Note that the evaluation model allows us to compare different methods under extreme conditions varying from the case of no biased dimension to the case where every observation can possibly get biased.}
For evaluation we assign the following values to different parameters: ${\Sigma_\mathbf{o}^i} = {0.4}, \Lambda^i=90, \mathbf{x}_{0} = [0, 10, 0, -5, \frac{3 \pi}{180}]^{\top}, \mathbf{R}_k = 4\mathbf{I}, {\textcolor{black}{\zeta_t}} = 1, \eta_1=0.1, \eta_2=1.75\times10^{-4}$. We assume $m=4$, since for higher-dimensional problems even rejection-based methods like the SOR-UKF can have acceptable performance for a larger probability of errors owing to the redundancy of useful information in other uncorrupted dimensions. However, this does not limit the applicability of the proposed method for large $m$. Our point is to emphasize how properly utilizing information from an affected dimension, a characteristic of {analytical} redundancy approaches, is more useful in contrast to completely rejecting the information, a characteristic of {hardware} redundancy approaches \cite{chughtai2020robust}. In addition, we evaluate the relative performance of the proposed filter with similar {analytical} redundancy approaches.
\begin{figure*}[ht!]
\centering
\begin{subfigure}[h!]{0.22\textwidth}
\centering
\includegraphics[width=\linewidth,trim=0 0 20cm 0,clip=true]{boxCB02.eps}
\caption{$\lambda = 0.2$}
\label{Box11}
\end{subfigure}
\begin{subfigure}[h!]{0.22\textwidth}
\centering
\includegraphics[width=\linewidth,trim=0 0 20cm 0,clip=true]{boxCB04.eps}
\caption{$\lambda = 0.4$}
\label{Box12}
\end{subfigure}
\begin{subfigure}[h!]{0.22\textwidth}
\centering
\includegraphics[width=\linewidth,trim=0 0 20cm 0,clip=true]{boxCB06.eps}
\caption{$\lambda = 0.6$}
\label{Box13}
\end{subfigure}
\begin{subfigure}[h!]{0.22\textwidth}
\centering
\includegraphics[width=\linewidth,trim=0 0 20cm 0,clip=true]{boxCB08.eps}
\caption{$\lambda = 0.8$}
\label{Box14}
\end{subfigure}
\caption{Box plots of state RMSE for Case 1 with increasing $\lambda$} \label{Box1}
\end{figure*}
\begin{figure}[ht!]
\centering
\includegraphics[width=\linewidth]{Case1MC.eps}
\caption{Tracking performance of algorithms over time for an example MC run for Case 1}
\label{fig:Case1_run}
\end{figure}
For each method, the UKF parameters are set as $\alpha=1$, $\beta=2$ and $\kappa=0$. We consider $T = 400$ time steps and carry out 100 Monte Carlo (MC) simulations for each case. All the filters are initialized with a state estimate equal to $\mathbf{x}_{0}$ its error covariance as $\mathbf{Q}_{k}$ and the bias estimate is set equal to zero initially. The other parameters supposed for the BDM-UKF are: $\mathbf{\Sigma}_{0}^{+}=0.001\mathbf{I},\widetilde{\mathbf{\Sigma}}_{k}=1000\mathbf{R}_k,\breve{\mathbf{\Sigma}}_{k}=0.1\mathbf{R}_k$ and $\theta^i_k=0.5~\forall~i$ and $\tau=10^{-4}$. In CUKF the auxiliary parameters are assigned values as: $\epsilon=1,\sigma_n=2$. Lastly, the USKF is initialized with a zero mean bias with a covariance of $500\mathbf{I}$ having zero cross-covariance with the state. Other parameters for all the filters are kept the same as originally reported.
\subsection*{Performance bounds}
To benchmark the relative performance of the considered algorithms, we evaluate the PCRB for the target tracking model in \eqref{eqn_res1a}-\eqref{eqn_res2c}. The bound remains valid where system dynamics can be modeled as \eqref{eqn_res1a} and where the measurement model can be considered to switch between \eqref{eqn_res2} and \eqref{eqn_res2c}-\eqref{eqn_res2d} since we use a generalized model in our derivation. We evaluate the PCRB assuming perfect apriori knowledge of the occurrence time, duration of the bias and $\lambda$, resorting to the approach presented in \cite{668800,van2013detection} for a generalized nonlinear dynamical system, with white process and measurement noise, given as follows
\begin{align}
\mathbf{x}_{k+1}&=\mathbf{f}\left(\mathbf{x}_{k}, \mathbf{q}_{k}\right)\\
\mathbf{y}_{k}&=\mathbf{h}\left(\mathbf{x}_{k}, \mathbf{r}_{k},\mathbf{y}_{k-1},\cdots,\mathbf{y}_{k-z}\right)
\end{align}
where $z$ is a positive integer.
The PCRB matrix for the estimation error of $\mathbf{x}_k$ can be written as
\begin{equation}
\text{PCRB}_k\triangleq{\mathbf{J}_k}^{-1}\label{PCRB_1}
\end{equation}
where $\mathbf{J}_k$ can be expressed recursively as
\begin{align}
\mathbf{J}_{k+1}&=\mathbf{D}_{k}^{22}-\mathbf{D}_{k}^{21}\left(\mathbf{J}_{k}+\mathbf{D}_{k}^{11}\right)^{-1} \mathbf{D}_{k}^{12}\label{PCRB_2}
\end{align}
with
\begin{align}
&\mathbf{D}_{k}^{11}=\langle-\Delta_{\mathbf{x}_{k}}^{\mathbf{x}_{k}} \log p\left(\mathbf{x}_{k+1} \mid \mathbf{x}_{k}\right)\rangle_{p(\mathbf{x}_{k+1},\mathbf{x}_{k})}\label{PCRB_3}\\
&\mathbf{D}_{k}^{12}=\langle-\Delta_{\mathbf{x}_{k}}^{\mathbf{x}_{k+1}} \log p\left(\mathbf{x}_{k+1} \mid \mathbf{x}_{k}\right)\rangle_{p(\mathbf{x}_{k+1},\mathbf{x}_{k})}\label{PCRB_4}\\
&\mathbf{D}_{k}^{21}=\langle-\Delta_{\mathbf{x}_{k+1}}^{\mathbf{x}_{k}} \log p\left(\mathbf{x}_{k+1} \mid \mathbf{x}_{k}\right)\rangle_{p(\mathbf{x}_{k+1},\mathbf{x}_{k})}=\left[\mathbf{D}_{k}^{12}\right]^{\top}\label{PCRB_5}\\
&\mathbf{D}_{k}^{22}= \mathbf{D}_{k}^{22}(1)+\mathbf{D}_{k}^{22}(2)\\
&\mathbf{D}_{k}^{22}(1)=\langle-\Delta_{\mathbf{x}_{k+1}}^{\mathbf{x}_{k+1}} \log p\left(\mathbf{x}_{k+1} \mid \mathbf{x}_{k}\right)\rangle_{p(\mathbf{x}_{k+1},\mathbf{x}_{k})} \nonumber\\
&\mathbf{D}_{k}^{22}(2)=\nonumber\\
&\langle-\Delta_{\mathbf{x}_{k+1}}^{\mathbf{x}_{k+1}} \log p\left(\mathbf{y}_{k+1} \mid \mathbf{x}_{k+1},\mathbf{y}_{k},\cdots,\mathbf{y}_{k-z+1}\right)\rangle_{p(\mathbf{y}_{k+1},\mathbf{x}_{k+1},\cdots)} \label{PCRB_6}
\end{align}
where
\begin{align}
\Delta_{\Psi}^{\Theta}&=\nabla_{\Psi} \nabla_{\Theta}^{\top} \label{PCRB_8}\\
\nabla_{\Theta}&=\left[\frac{\partial}{\partial \Theta_{1}}, \cdots, \frac{\partial}{\partial \Theta_{r}}\right]^{\top}\label{PCRB_9}
\end{align}
The bound is valid given the existence of derivatives and expectations terms in \eqref{PCRB_1}-\eqref{PCRB_9} for an asymptotically unbiased estimator \cite{668800}. Using results from \cite{668800,van2013detection} we obtain
\begin{align}
&\mathbf{D}_{k}^{11}=\left[\nabla_{\mathbf{x}_{k}} \mathbf{f}^{\top}\left(\mathbf{x}_{k}\right)\right] \mathbf{Q}_{k}^{-1}\left[\nabla_{\mathbf{x}_{k}} \mathbf{f}^{\top}\left(\mathbf{x}_{k}\right)\right]^{\top}\label{PCRB_10}\\
&\mathbf{D}_{k}^{12}=-\nabla_{\mathbf{x}_{k}} \mathbf{f}^{\top}\left(\mathbf{x}_{k}\right) \mathbf{Q}_{k}^{-1}\label{PCRB_11}\\
&\mathbf{D}_{k}^{22}(1)=\mathbf{Q}_{k}^{-1}
\end{align}
For the period where the nominal equation \eqref{eqn_res2} remains applicable, $\mathbf{D}_{k}^{22}(2)$ can be expressed as \cite{van2013detection}
\begin{align}
&\mathbf{D}_{k}^{22}(2)=\nonumber
\\&-\mathbf{Q}_{k}^{-1} \langle \tilde{\mathbf{F}}_{k} \rangle_{p(\mathbf{x}_{k})} \left[\mathbf{J}_{k}+ \langle \tilde{\mathbf{F}}_{k}^{\top} \mathbf{Q}_{k}^{-1} \tilde{\mathbf{F}}_{k} \rangle_{p(\mathbf{x}_{k})} \right]^{-1} \langle \tilde{\mathbf{F}}_{k}^{\top}\rangle_{p(\mathbf{x}_{k})}\mathbf{Q}_{k}^{-1} \nonumber \\
&+\langle \tilde{\mathbf{H}}_{k+1}^{\top} \mathbf{R}_{k+1}^{-1} \tilde{\mathbf{H}}_{k+1}\rangle_{p(\mathbf{x}_{k+1})}
\end{align}
where $\tilde{\mathbf{F}}_{k}$ and $\tilde{\mathbf{H}}_{k}$ are the Jacobians of $\mathbf{f}(\mathbf{x}_{k})$ and $\mathbf{h}(\mathbf{x}_{k})$ respectively.
For the duration of bias occurrence, $\mathbf{D}_{k}^{22}(2)$ can be evaluated using \eqref{PCRB_6}, equivalently written as follows using a technique similar to \cite{6916255}
\begin{equation}
\mathbf{D}_{k}^{22}(2)=\big\langle \frac{[\nabla_{\mathbf{x}_{k+1}} p\left(\mathbf{y}_{k+1} \mid \mathbf{x}_{k+1},\mathbf{y}_{k},\mathbf{x}_{k}\right)][.]^\top}{[p\left(\mathbf{y}_{k+1} \mid \mathbf{x}_{k+1},\mathbf{y}_{k},\mathbf{x}_{k}\right)]^2} \big\rangle_{p(\mathbf{y}_{k+1},\mathbf{x}_{k+1},\cdots)}
\end{equation}
First consider \eqref{eqn_res2c} to evaluate $\mathbf{D}_{k}^{22}(2)$ at the instance of bias occurrence for which $p\left(\mathbf{y}_{k+1} \mid \mathbf{x}_{k+1},\mathbf{y}_{k},\mathbf{x}_{k}\right) $ can be approximated using Monte Carlo method as
\begin{align}
&p\left(\mathbf{y}_{k+1} \mid \mathbf{x}_{k+1},\mathbf{y}_{k},\mathbf{x}_{k}\right)= p\left(\mathbf{y}_{k+1} \mid \mathbf{x}_{k+1}\right) \\
&=\int p\left(\mathbf{y}_{k+1} \mid \mathbf{x}_{k+1},\mathbf{b}_{k+1}\right)p(\mathbf{b}_{k+1})d\mathbf{b}_{k+1}\\
&=\int \mathcal{N}\left(\mathbf{y}_{k+1}|\mathbf{h}(\mathbf{x}_{k+1})+\mathbf{b}_{k+1},\mathbf{R}_k\right) p(\mathbf{b}_{k+1})d\mathbf{b}_{k+1}\\
&\approx \frac{1}{N_{mc1}}\sum_i\mathcal{N}\left(\mathbf{y}_{k+1}|\mathbf{h}(\mathbf{x}_{k+1})+\mathbf{b}^{(i)}_{k+1},\mathbf{R}_k\right)
\end{align}
where $\mathbf{b}^{(i)}_{k+1}$, $i=1,\cdots,N_{mc1}$, are independent and identically distributed (i.i.d.) samples such that $\mathbf{b}^{(i)}_{k+1}\sim p(\mathbf{b}_{k+1})$.
Accordingly, $\mathbf{D}_{k}^{22}(2)$ can be approximated as
\begin{equation}
\mathbf{D}_{k}^{22}(2)=\big\langle \frac{[\nabla_{\mathbf{x}_{k+1}} \sum_i \exp(\phi^{(i)})][.]^\top}{[\sum_i \exp(\phi^{(i)})]^2} \big\rangle_{p(\mathbf{y}_{k+1},\mathbf{x}_{k+1})}
\end{equation}
where $\phi^{(i)}=-0.5(\mathbf{y}_{k+1}-(\mathbf{h}(\mathbf{x}_{k+1})+\mathbf{b}^{(i)}_{k+1}))^{\top}\mathbf{R}_{k+1}^{-1}(\mathbf{y}_{k+1}-(\mathbf{h}(\mathbf{x}_{k+1})+\mathbf{b}^{(i)}_{k+1}))$. We can further write
\begin{equation}
\mathbf{D}_{k}^{22}(2)=\big\langle \frac{\sum_i [\exp(\phi^{(i)})\nabla_{\mathbf{x}_{k+1}}\phi^{(i)} ][.]^\top}{[\sum_i \exp(\phi^{(i)})]^2} \big\rangle_{p(\mathbf{y}_{k+1},\mathbf{x}_{k+1})}
\end{equation}
where $\nabla_{\mathbf{x}_{k+1}}\phi^{(i)}=\tilde{\mathbf{H}}_{k+1}^{\top}\mathbf{R}_{k+1}^{-1}(\mathbf{y}_{k+1}-\mathbf{h}(\mathbf{x}_{k+1})-\mathbf{b}^{(i)}_{k+1})$
Resultingly, $\mathbf{D}_{k}^{22}(2)$ is approximated as
\begin{equation}
\mathbf{D}_{k}^{22}(2)\approx\frac{1}{N_{mc2}}\sum_j \frac{\sum_i [\exp(\phi^{(i,j)})\nabla_{\mathbf{x}_{k+1}}\phi^{(i,j)} ][.]^\top}{[\sum_i \exp(\phi^{(i,j)})]^2}
\end{equation}
with $\phi^{(i,j)}=-0.5(\mathbf{y}^{(j)}_{k+1}-(\mathbf{h}(\mathbf{x}^{(j)}_{k+1})+\mathbf{b}^{(i)}_{k+1}))^{\top}\mathbf{R}_{k+1}^{-1}(\mathbf{y}^{(j)}_{k+1}-(\mathbf{h}(\mathbf{x}^{(j)}_{k+1})+\mathbf{b}^{(i)}_{k+1}))$ and $\nabla_{\mathbf{x}_{k+1}}\phi^{(i,j)}=\tilde{\mathbf{H}}_{k+1}^{\top}\mathbf{R}_{k+1}^{-1}(\mathbf{y}^{(j)}_{k+1}-\mathbf{h}(\mathbf{x}^{(j)}_{k+1})-\mathbf{b}^{(i)}_{k+1})$. $\mathbf{y}^{(j)}_{k+1},\mathbf{x}^{(j)}_{k+1}$, $j=1,\cdots,N_{mc2}$, are i.i.d. samples such that $(\mathbf{y}^{(j)}_{k+1},\mathbf{x}^{(j)}_{k+1})\sim p(\mathbf{y}_{k+1},\mathbf{x}_{k+1})$.
Lastly, to evaluate $\mathbf{D}_{k}^{22}(2)$ for the bias persistence period we consider \eqref{eqn_res2d} and the difference of \eqref{eqn_res2c}-\eqref{eqn_res2d}.
Therefore, $p\left(\mathbf{y}_{k+1} \mid \mathbf{x}_{k+1},\mathbf{y}_{k},\mathbf{x}_{k}\right) $ can be approximated for this case using Monte Carlo method as
\begin{align}
&p\left(\mathbf{y}_{k+1} \mid \mathbf{x}_{k+1},\mathbf{y}_{k},\mathbf{x}_{k}\right)\\
&=\int p\left(\mathbf{y}_{k+1} \mid \mathbf{x}_{k+1},\mathbf{y}_{k},\mathbf{x}_{k},\boldsymbol{{\mathcal{J}}}_{k}\right)p(\boldsymbol{{\mathcal{J}}}_{k})d\boldsymbol{{\mathcal{J}}}_{k}\\
&\approx \frac{1}{N_{mc3}}\sum_i\mathcal{N}(\mathbf{y}_{k+1}|\mathbf{h}(\mathbf{x}_{k+1})+\boldsymbol{{\mathcal{J}}}^{(i)}_{k}(\mathbf{y}_k-\mathbf{h}(\mathbf{x}_{k}))\nonumber \\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , \mathbf{R}_{k+1}+\boldsymbol{{\mathcal{J}}}^{(i)}_{k}(\mathbf{R}_{k}+2\Sigma_\mathbf{o}))
\end{align}
where $\boldsymbol{{\mathcal{J}}}^{(i)}_{k}$, $i=1,\cdots,N_{mc3}$, are i.i.d. samples such that $\boldsymbol{{\mathcal{J}}}^{(i)}_{k}\sim p(\boldsymbol{{\mathcal{J}}}_{k})$. Resultingly, $\mathbf{D}_{k}^{22}(2)$ is approximated as
\begin{equation}
\mathbf{D}_{k}^{22}(2)=\big\langle \frac{[\nabla_{\mathbf{x}_{k+1}} \sum_i \exp(\theta^{(i)})][.]^\top}{[\sum_i \exp(\theta^{(i)})]^2} \big\rangle_{p(\mathbf{y}_{k+1},\mathbf{x}_{k+1},\cdots)}
\end{equation}
where $\theta^{(i)}=-0.5(\mathbf{y}_{k+1}-(\mathbf{h}(\mathbf{x}_{k+1})+\boldsymbol{{\mathcal{J}}}^{(i)}_{k}(\mathbf{y}_k-\mathbf{h}(\mathbf{x}_{k})))^{\top}{(\mathbf{R}_{k+1}+\boldsymbol{{\mathcal{J}}}^{(i)}_{k}(\mathbf{R}_{k}+2\Sigma_\mathbf{o}))}^{-1}(\mathbf{y}_{k+1}-(\mathbf{h}(\mathbf{x}_{k+1})+\boldsymbol{{\mathcal{J}}}^{(i)}_{k}(\mathbf{y}_k-\mathbf{h}(\mathbf{x}_{k})))$.
Furthermore
\begin{equation}
\mathbf{D}_{k}^{22}(2)=\big\langle \frac{\sum_i [\exp(\theta^{(i)})\nabla_{\mathbf{x}_{k+1}}\theta^{(i)} ][.]^\top}{[\sum_i \exp(\theta^{(i)})]^2} \big\rangle_{p(\mathbf{y}_{k+1},\mathbf{x}_{k+1},\cdots)}
\end{equation}
where $\nabla_{\mathbf{x}_{k+1}}\theta^{(i)}=\tilde{\mathbf{H}}_{k+1}^{\top}{(\mathbf{R}_{k+1}+\boldsymbol{{\mathcal{J}}}^{(i)}_{k}(\mathbf{R}_{k}+2\Sigma_\mathbf{o}))}^{-1}(\mathbf{y}_{k+1}-(\mathbf{h}(\mathbf{x}_{k+1})+\boldsymbol{{\mathcal{J}}}^{(i)}_{k}(\mathbf{y}_k -\mathbf{h}(\mathbf{x}_{k}) )))$. Resultingly, $\mathbf{D}_{k}^{22}(2)$ is approximated as
\begin{equation}
\mathbf{D}_{k}^{22}(2)\approx\frac{1}{N_{mc4}}\sum_j \frac{\sum_i [\exp(\theta^{(i,j)})\nabla_{\mathbf{x}_{k+1}}\theta^{(i,j)} ][.]^\top}{[\sum_i \exp(\theta^{(i,j)})]^2}
\end{equation}
with $\theta^{(i,j)}=-0.5(\mathbf{y}^{(j)}_{k+1}-(\mathbf{h}(\mathbf{x}^{(j)}_{k+1})+\boldsymbol{{\mathcal{J}}}^{(i)}_{k}(\mathbf{y}^{(j)}_k-\mathbf{h}(\mathbf{x}^{(j)}_{k})))^{\top}{(\mathbf{R}_{k+1}+\boldsymbol{{\mathcal{J}}}^{(i)}_{k}(\mathbf{R}_{k}+2\Sigma_\mathbf{o}))}^{-1}(\mathbf{y}^{(j)}_{k+1}-(\mathbf{h}(\mathbf{x}^{(j)}_{k+1})+\boldsymbol{{\mathcal{J}}}^{(i)}_{k}(\mathbf{y}^{(j)}_k-\mathbf{h}(\mathbf{x}^{(j)}_{k})))$ and
$\nabla_{\mathbf{x}_{k+1}}\theta^{(i,j)}=\tilde{\mathbf{H}}_{k+1}^{\top}{(\mathbf{R}_{k+1}+\boldsymbol{{\mathcal{J}}}^{(i)}_{k}(\mathbf{R}_{k}+2\Sigma_\mathbf{o}))}^{-1}(\mathbf{y}^{(j)}_{k+1}-(\mathbf{h}(\mathbf{x}^{(j)}_{k+1})+\boldsymbol{{\mathcal{J}}}^{(i)}_{k}(\mathbf{y}^{(j)}_k -\mathbf{h}(\mathbf{x}^{(j)}_{k}) )))$.
$\mathbf{y}^{(j)}_{k+1},\mathbf{x}^{(j)}_{k+1},\mathbf{y}^{(j)}_{k},\mathbf{x}^{(j)}_{k}$, $j=1,\cdots,N_{mc4}$, are i.i.d. samples such that $(\mathbf{y}^{(j)}_{k+1},\mathbf{x}^{(j)}_{k+1},\mathbf{y}^{(j)}_{k},\mathbf{x}^{(j)}_{k})\sim p(\mathbf{y}_{k+1},\mathbf{x}_{k+1},\mathbf{y}_{k},\mathbf{x}_{k})$.
\begin{figure*}[t!]
\centering
\begin{subfigure}[t!]{0.22\textwidth}
\centering
\includegraphics[width=\linewidth,trim=0 0 20cm 0,clip=true]{box02.eps}
\caption{$\lambda = 0.2$}
\label{Box21}
\end{subfigure}
\begin{subfigure}[t!]{0.22\textwidth}
\centering
\includegraphics[width=\linewidth,trim=0 0 20cm 0,clip=true]{box04.eps}
\caption{$\lambda = 0.4$}
\label{Box22}
\end{subfigure}
\begin{subfigure}[t!]{0.22\textwidth}
\centering
\includegraphics[width=\linewidth,trim=0 0 20cm 0,clip=true]{box06.eps}
\caption{$\lambda = 0.6$}
\label{Box23}
\end{subfigure}
\begin{subfigure}[t!]{0.22\textwidth}
\centering
\includegraphics[width=\linewidth,trim=0 0 20cm 0,clip=true]{box08.eps}
\caption{$\lambda = 0.8$}
\label{Box24}
\end{subfigure}
\caption{Box plots of state RMSE for Case 2 with increasing $\lambda$}\label{Box2}
\end{figure*}
\begin{figure}[ht!]
\centering
\vspace{.07cm}
\includegraphics[width=\linewidth]{Case2MC.eps}
\caption{Tracking performance of algorithms over time for an example MC run for Case 2}
\label{fig:Case2_run}
\end{figure}
\subsection{Case 1: Consistent bias presence}
\begin{table}[b!]
\centering
\begin{tabular}{|c c c c c c|}
\hline
$\boldsymbol{\lambda}$ &\textbf{UKF}&\textbf{USKF}& \textbf{BDM-UKF} & \textbf{SOR-UKF} & \textbf{CUKF} \\
\hline\hline
0.2 &0.0469&0.0781& 0.0866 & 0.0821& 0.5762 \\
0.4 &0.0470&0.0849& 0.1000 & 0.1199& 1.2499 \\
0.6 &0.0471&0.0883& 0.1121 & 0.1416& 1.8012 \\
0.8 &0.0470&0.0903& 0.1492 & 0.1637& 2.2305 \\
\hline
\end{tabular}
\caption{Average time for 100 MC runs with constant bias at different values of $\lambda$ .}
\label{table:1}
\end{table}
First, we consider the case of biases consistently appearing in the measurements representative of real-world scenarios where the observations are systematically biased. We consider that the bias corrupts each observation according to \eqref{eqn_res2c}-\eqref{eqn_res2d} and bias occurs in each dimension from the start of the simulation with probability $\lambda$ and sustains for the complete run time.
Fig.~\ref{Box1} shows the distribution of the root mean squared error (RMSE) calculated over 100 MC runs for the algorithms under consideration at different values of $\lambda$. {The lower bounds based on the diagonal entries of $\text{PCRB}_k$ depict the benchmark performance for different methods theoretically achievable considering different values of $\lambda$. For evaluation of $\text{PCRB}_k$ we assume $N_{mc1}=N_{mc2}=N_{mc3}=N_{mc4}=100$.} We can observe an intuitive trend in the relative performance of different methods. The standard UKF exhibits the worst estimation quality since it has no means for bias detection and compensation. For the data rejection-based method i.e. the SOR-UKF, we see that for lower probabilities of existence of bias discounting the observations performs satisfactorily. However, for higher values of $\lambda$ the rejection scheme does not work well since consistently rejecting large number of the measurements leads to loss of essential information. The CUKF and the USKF generally perform better than the SOR-UKF. \textcolor{black}{As compared to the other robust filters} the USKF is found to have more error at lower values of $\lambda$ since it does not treat each dimension selectively. It rather uses the entire vector of measurements for compensation even if one of the dimensions is corrupted unlike the CUKF which offers a selective treatment. Importantly, it can be observed that BDM-UKF results in the least error among all the methods.
Fig.~\ref{fig:Case1_run} shows the state RMSE of the algorithms over time for one MC run depictive of the general trend in Fig.\ref{Box1} for large values of $\lambda$ for this case. The BDM-UKF deals with biased observations more effectively followed by the partially compensating methods namely the CUKF and the USKF. The SOR-UKF, on the other hand, loses track of the ground truth due to permanent rejection of essential information. Similarly, the standard UKF with its inability to deal with \textcolor{black}{biases} also exhibits large errors.
Lastly, we evaluate the computational overhead of each algorithm for Case 1. The mean processing time, considering 100 MC runs, for each method has been summarized in Table \ref{table:1}. We find the standard UKF to be the most economical and does not exhibit any significant change with different values of $\lambda$. The USKF takes more time since it involves updating the state and bias correlation terms. We observe a rise in the processing time of the USKF with increasing $\lambda$ since the USKF is invoked more frequently than the standard UKF we utilize when no bias is detected. The BDM-UKF and the SOR-UKF, both having an inbuilt detection mechanism, have a similar processing burden. Lastly, we find the CUKF to be the most computationally expensive algorithm. This can be owed to the presence of a convex quadratically constrained quadratic program (QCQP) which we solve using the MATLAB optimization toolbox.
\subsection{Case 2: Momentary bias presence}
We also consider the case where bias randomly appears for a short duration, characterizing practical scenarios e.g. where ambient effects disturb the data signals briefly. We again use \eqref{eqn_res2c}-\eqref{eqn_res2d} for each observation contamination and suppose bias occurs in each dimension with probability $\lambda$ at instant $t=100$ and sustains till $t=130$ before disappearing.
Fig.~\eqref{Box2} depicts the spread of the RMSEs for 100 MC runs for each algorithm with varying $\lambda$ {along with lower bounds based on the PCRB using 100 MC samples for all the calculations. The plotting scale in this case allows us to clearly see a rise in the lower bound with increasing values of $\lambda$ which has also been observed for the previous case. This can be attributed to more chances of occurrence of \textcolor{black}{biased observations} with increasing $\lambda$ leading to increased estimation uncertainty}. In terms of the relative performance of different methods we see a similar pattern as for the previous case, owing to the same rationale regarding the functionality of the methods. The UKF generally has the largest RMSE, followed by the USKF, the SOR-UKF and the CUKF. The momentary appearance of bias does not degrade the performance of SOR-UKF as compared to the last case, except for $\lambda=0.8$ where it mostly diverges. SUKF is found to be relatively less effective at lower values of $\lambda$ due to its non-selective nature. Lastly, the BDM-UKF results in the lowest RMSE. Fig.~\ref{fig:Case2_run} shows the state RMSE of the algorithms over time for one MC run representative of the general trend in Fig.~\ref{Box2} for large values of $\lambda$ for this case. We can observe similar results as in the preceding case.
\begin{table}[h!]
\centering
\begin{tabular}{|c c c c c c|}
\hline
$\boldsymbol{\lambda}$ &\textbf{UKF}&\textbf{USKF}& \textbf{BDM-UKF} & \textbf{SOR-UKF} & \textbf{CUKF} \\
\hline\hline
0.2 &0.0458&0.0515& 0.0682 & 0.0669& 0.09114 \\
0.4 &0.0489&0.0521& 0.0687 & 0.0672& 0.1110 \\
0.6 &0.0482&0.0520& 0.0687 & 0.0828& 0.1470 \\
0.8 &0.0478&0.0520& 0.0690 & 0.0902& 0.2059 \\
\hline
\end{tabular}
\caption{Average time for 100 MC runs \textcolor{black}{with momentary bias} at different values of $\lambda$ .}
\label{table:2}
\end{table}
Lastly, the mean processing overhead of each algorithm for this case is presented in Table \ref{table:2}. We can see that the UKF takes approximately similar times as for Case 1. The overhead of remaining methods is reduced since the bias duration has now decreased. The order in which different algorithms appear in terms of the relative computational expense remains the same as observed in the previous case following from the same reasoning.
\section{Conclusion}\label{Conc}
The performances of standard filtering approaches degrade when the measurements are disturbed by noise with unknown statistics. Focusing on the presence of measurement \textcolor{black}{biases}, we devise the BDM filter with inherent error detection and mitigation functionality. Performance evaluation reveals the efficacy of the BDM in dealing with both persistently and temporarily present biases. We find the BDM filter more accurate compared to rejection-based KF methods i.e. SOR-UKF. Moreover, owing to better utilization of the measurements, the BDM filter has lower estimation errors as compared to the methods with similar KF based approaches, the USKF and the CUKF, aiming to exploit information from the corrupted dimensions. The BDM filter is easier to employ as it avoids the use of external detectors and any optimization solver. The gains come at the expense of increased computational overhead which is comparatively higher compared to the UKF and USKF. However, it is comparable to SOR-UKF and lower than the CUKF which requires an additional optimization solver.
|
2024-02-18T23:40:24.571Z
|
2022-10-25T02:15:00.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14432","language":"en","timestamp":1666664100000,"url":"https:\/\/arxiv.org\/abs\/2112.14432","yymm":"2112"}
|
proofpile-arXiv_000-10041
|
{"provenance":"002.jsonl.gz:10042"}
| null | null |
\section{Introduction}
In \cite{sjostrand1,sjostrand2}, Sj\"ostrand studied a class of pseudo-differential operators continuous on $L^2(\RR^n)$ which is closed under composition and taking inverses (when they exist). The corresponding space of Weyl symbols, nowadays known as the Sj\"ostrand algebra, is the modulation space \cite{F2,F3,Gr1}
$$
M^{\infty,1}(\RR^{2n})=\{a\in\SSS'(\RR^{2n})\,|\, V_{\varphi}a\in L^{\infty,1}(\RR^{4n})\}\footnote{Sj\"ostrand's original definition is different but amounts to the same space.}
$$
where $L^{\infty,1}(\RR^{4n})$ is the standard mixed-norm Lebesgue space and $V_{\varphi}$ is the short-time Fourier transform (from now, abbreviated as STFT) with window $\varphi\in\SSS(\RR^{2n})\backslash\{0\}$, i.e. $V_{\varphi}a(X,\Xi)=\langle a, e^{-2\pi i \Xi\, \cdot}\,\overline{\varphi(\cdot-X)}\rangle$, $X,\Xi\in\RR^{2n}$. Sj\"ostrand later generalised the theory in \cite{sjostrand3} to broader classes of symbols with limited regularity. These articles have sprung a line of research concerning $\Psi$DOs and FIOs with symbols in modulation and Wiener amalgam spaces and their application to PDEs; see \cite{ACT,ben-oko,bmntt,CN,CR,CGNR,CNT,fei-gn,Gr2,gro-rz,GT,kob-sug-t,T-1,0T} and the references therein. In \cite{sjostrand2}, Sj\"ostrand proved the following characterisation of $M^{\infty,1}(\RR^{2n})$ (see the proof of \cite[Proposition 3.1]{sjostrand2}). Let $a\in\SSS'(\RR^{2n})$ and let $\chi(x)=e^{-\pi |x|^2}$, $x\in\RR^n$. Then $a\in M^{\infty,1}(\RR^{2n})$ if and only
$$
\exists f\in L^1(\RR^{2n})\; \mbox{such that}\; |\langle a^w \pi(X)\chi,\overline{\pi(\Xi)\chi}\rangle|\leq f(\Xi-X),\quad X,\Xi\in\RR^{2n};
$$
here $\pi(X)$, $X\in\RR^{2n}$, stands for the time-frequency shift $\pi(X)\phi(y)=e^{2\pi i \xi y}\phi(y-x)$, $X=(x,\xi)\in\RR^{2n}$. To be precise, this result is not part of the claim of \cite[Proposition 3.1]{sjostrand2} but it is shown in its proof. The proof employs so called families of uniformly confined symbols introduced by Bony and Lerner \cite{bon-ler,bony1} and they will play a significant role in this article as well; discrete version of such types of families were previously employed by H\"ormander \cite{hormander}. Similar constructions were also considered by Feichtinger \cite{fei4} in the context of Wiener amalgam and decomposition spaces. Later, Gr\"ochenig and Rzeszotnik \cite{gro-rz} (see also \cite{Gr2}) employed time-frequency and Banach algebra techniques to generalise the results of \cite{sjostrand2}. As a byproduct of their analysis, they proved the following characterisation of the H\"ormander algebra $S^0_{0,0}(\RR^{2n})$ \cite[Theorem 6.2]{gro-rz}. Let $\chi\in\SSS(\RR^n)\backslash\{0\}$ be such that $\{\pi(\lambda)\chi\,|\, \lambda\in\Lambda\}$ is a tight Gabor frame for $L^2(\RR^n)$ on the lattice $\Lambda$ in $\RR^{2n}$. Then for $a\in\SSS'(\RR^{2n})$, the following conditions are equivalent:
\begin{itemize}
\item[$(i)$] $a\in \SSS^0_{0,0}(\RR^{2n})$;
\item[$(ii)$] for every $s\geq 0$ there is $C_s\geq 1$ such that $|\langle a^w \pi(X)\chi,\overline{\pi(\Xi)\chi}\rangle|\leq C_s(1+|\Xi-X|)^{-s}$, $X,\Xi\in\RR^{2n}$;
\item[$(iii)$] for every $s\geq 0$ there is $C_s\geq 1$ such that $|\langle a^w \pi(\lambda_1)\chi,\overline{\pi(\lambda_2)\chi}\rangle|\leq C_s(1+|\lambda_2-\lambda_1|)^{-s}$, $\lambda_1,\lambda_2\in\Lambda$.
\end{itemize}
The assumption that $\{\pi(\lambda)\chi\,|\, \lambda\in\Lambda\}$ is a tight Gabor frame for $L^2(\RR^n)$ is only needed for the equivalence with $(iii)$. The property $(iii)$ is usually called almost diagonalisation as it shows that the matrix of the pseudodifferential operator with respect to the Gabor frame is rapidly decaying outside of the diagonal; the property $(ii)$ can be viewed as a continuous version of this. Of course, the important part is that the symbol class $S^0_{0,0}(\RR^{2n})$ (resp., $M^{\infty,1}(\RR^{2n})$ in the above result of Sj\"ostrand) is characterised by this property; see \cite[Section 5.2]{CR}, \cite{CGNR,CNT,CGN} and the references therein for other such characterisations of modulation and Wiener amalgam spaces and almost diagonalisation of $\Psi$DOs and FIOs. It is natural to ask whether such characterisation can be obtained for the symbol classes of the frequently employed global calculi. In the main result of this article (Theorem \ref{main-theorem-dia}) we provide a characterisation in the spirit of $(ii)$ of the general Weyl-H\"ormander classes $S(M,g)$ \cite{hormander,horm3}. They include as special cases the symbol classes of almost all global calculi that appear in the literature, like the Shubin and SG-calculus \cite{NR,Shubin}, the Beals-Fefferman calculus \cite{bf1}, the H\"ormander $S_{\rho,\delta}$-calculus \cite{horm3,Shubin}, etc. Our main result covers as a special case the characterisation $(ii)$ of $S^0_{0,0}(\RR^{2n})$. A discrete characterisation in the spirit of $(iii)$ still remains as an open problem.\\
\indent The general idea in the proof of the main result of the article is to extend the time-frequency analysis to the Weyl-H\"ormander classes. However, the techniques from \cite{Gr2,gro-rz} are unlikely to work in other cases apart from the classes $S^m_{0,0}$. The reason for this is that the STFT is well-suited for analysing distributions when the underlying space carries the Euclidean metric. But, for the purposes of analysing the classes $S(M,g)$, the phases space is generally not Euclidean: it carries a metric $g$ that defines $S(M,g)$ which is never Euclidian except in the case of $S^m_{0,0}$. For this purpose, we generalise the STFT by replacing the window with a uniformly confined family of symbols; such families were introduced by Bony and Lerner in their seminal paper \cite{bon-ler} (cf. \cite{lernerB,unterberger}). These families keep track of the growth of the metric as the translates of the window in the classical STFT do for the Euclidian metric. Because of this, the generalised STFT which we introduce is better suited for the analysis of the general Weyl-H\"ormander classes. With its help, we define new modulation spaces. Our goal then is to show that $S(M,g)$ can be represented as an intersection of such modulation spaces and prove a characterisation of each of them in the same spirit as $(ii)$; this will give the desired characterisation of $S(M,g)$ as well.\\
\indent The paper is organised as follows. In Subsection \ref{properties} we collect several technical results concerning H\"ormander metrics. Sections \ref{C-family} and \ref{gms} are the core of the article. We start Section \ref{C-family} by recalling the notion of uniformly confined families of symbols and show several technical results about them. Then we define our generalisation of the STFT and study its mapping properties. Section \ref{gms} is devoted to the generalised modulation spaces and their properties. Section \ref{sec dia} contains the main result of the article: the characterisation of the Weyl-H\"ormander classes via the action of the corresponding Weyl operators on time-frequency shifts. However, these are no longer of the form $\pi(X)\chi$ but also carry encoded information about the metric. When $g$ is symplectic (a typical example is the metric that generates the classes $S^m_{\rho,\rho}(\RR^{2n})$, $\rho\in[0,1)$), the time-frequency shifts reduce to $\pi(X)$ but are intertwined with metaplectic operators that come from $g$ (Corollary \ref{cor-for-sym-metric-metplectope}). In the special case of the Euclidian metric, these metaplectic operators boil down to the identity operator. In Subsection \ref{ex111}, we specialise the main result to the frequently employed global calculi. At the very end, there are two appendices containing the proofs of several technical results stated in the previous sections.\\
\indent As a showcase, we present here the characterisation of $S^0_{\rho,\delta}(\RR^{2n})$, $0\leq \delta\leq \rho\leq 1$, $\delta<1$. Let $\theta_0\in\mathcal{C}^{\infty}([0,\infty))\backslash\{0\}$ be a non-negative, non-increasing function satisfying $\supp\theta_0\subseteq [0,1]$. Pick $r\in(0,1/2]$ and set
$$
\theta_{(x,\xi)}(y,\eta):=\theta_0(r^{-2}\langle \xi\rangle^{2\delta}|x-y|^2+r^{-2}\langle \xi\rangle^{-2\rho}|\xi-\eta|^2),\quad (x,\xi),(y,\eta)\in\RR^{2n}.
$$
For each $\xi\in \RR^n$, denote by $Q_{\xi}$ the diagonal $2n\times 2n$ matrix whose first $n$ entries along the diagonal are $\langle \xi\rangle^{\delta}$ while the second $n$ entries are $\langle\xi\rangle^{-\rho}$ and let $\Psi_{\xi}$ be the operator
$$
\Psi_{\xi}f(y,\eta):=f(Q_{\xi}^{-1}(y,\eta))=f(\langle \xi\rangle^{-\delta}y,\langle\xi\rangle^{\rho}\eta).
$$
Pick any $\chi\in\SSS(\RR^n)$ such that its Wigner transform is not $0$ at the origin (i.e., $W(\chi,\chi)(0)\neq 0$). Then, $a\in\SSS'(\RR^{2n})$ belongs to $S^0_{\rho,\delta}(\RR^{2n})$ if and only if for every $N>0$ there is $C_N>0$ such that
\begin{multline*}
\left|\left\langle \left(\Psi_{\frac{\xi+\eta}{2}} \left(a\theta_{\left(\frac{x+y}{2},\frac{\xi+\eta}{2}\right)}\right)\right)^w \pi\left(Q_{\frac{\xi+\eta}{2}}(x,\xi)\right)\chi, \overline{\pi\left(Q_{\frac{\xi+\eta}{2}}(y,\eta)\right)\chi}\right\rangle\right|\\
\leq C_N(1+\langle\xi+\eta\rangle^{\delta}|x-y|+\langle \xi+\eta\rangle^{-\rho}|\xi-\eta|)^{-N},\quad (x,\xi),(y,\eta)\in\RR^{2n}.
\end{multline*}
When $\rho=\delta\in[0,1)$, $\theta_{(x,\xi)}$ can be dropped from these bounds and they still characterise the elements of $S^0_{\rho,\rho}(\RR^{2n})$. Written in this form, when $\rho=0$ they reduce to the condition $(ii)$ we mentioned above. Furthermore, one can rewrite the latter as follows. The tempered distribution $a$ belongs to $S^0_{\rho,\rho}(\RR^{2n})$, $0\leq \rho<1$, if and only if for every $N>0$ there is $C_N>0$ such that
\begin{multline*}
\left|\left\langle a^w \pi(x,\xi)\Phi_{\frac{\xi+\eta}{2}}\chi, \overline{\pi(y,\eta)\Phi_{\frac{\xi+\eta}{2}}\chi}\right\rangle\right|\\
\leq C_N(1+\langle\xi+\eta\rangle^{\rho}|x-y|+\langle \xi+\eta\rangle^{-\rho}|\xi-\eta|)^{-N},\quad (x,\xi),(y,\eta)\in\RR^{2n},
\end{multline*}
where $\Phi_{\xi}$, $\xi\in\RR^n$, is the metaplectic operator $\Phi_{\xi}\chi(t)=\langle \xi\rangle^{\rho n/2}\chi(\langle \xi\rangle^{\rho}t)$.
\section{Preliminaries}
Given a smooth finite dimensional manifold\footnote{Manifolds are always assumed to be second-countable.} $\mathcal{N}$, one can unambiguously define the $\sigma$-algebra of Lebesgue measurable sets on $\mathcal{N}$ by declaring a set $E$ to be Lebesgue measurable if $\phi(E\cap U)$ is Lebesgue measurable for each chart $(\phi,U)$. The notion of a negligible set (nullset) in $\mathcal{N}$ is unambiguous since they are diffeomorphism invariant. If $F$ is a topological space, we say that $f:\mathcal{N}\rightarrow F$ is Lebesgue measurable (from now, always abbreviated as measurable) if $f$ is measurable with respect to the Borel $\sigma$-algebra on $F$ and the Lebesgue $\sigma$-algebra on $\mathcal{N}$, while we call $f$ Borel measurable if it is measurable with respect to the Borel $\sigma$-algebras on $\mathcal{N}$ and $F$.\\
\indent Let $V$ be an $n$-dimensional real vector space with $V'$ being its dual and set $W:=V\times V'$. We will always denote the points in $W$ with capital letters $X,Y,Z,\ldots$. Throughout the article, unless otherwise stated, the \underline{only} regularity assumption we impose on any Riemannian metric will be measurability\footnote{We emphasise this since, in the literature, it is customary to impose stronger regularity assumptions like continuity or smoothness.}; i.e. a Riemannian metric $g$ on $W$ is a Lebesgue measurable section of the 2-covariant tensor bundle $T^2T^*W$ that is symmetric and positive-definite at every point. We will always denote the corresponding quadratic forms by the same symbol: $g_X(T):=g_X(T,T)$, $T\in T_XW$. As standard, for each $X\in W$, we employ the canonical identification of $W$ with $T_XW$ that sends every $Y\in W$ to the directional derivative in direction $Y$ at $X$. For $T\in W\backslash\{0\}$, we denote by $\partial_T$ the vector field on $W$ given by the directional derivative in direction $T$ at every point $X\in W$. If $\partial_T$ acts on a function of two or more variables we will always emphasises in which variable it acts by writing $\partial_{T;X}, \partial_{T;Y},\ldots$. The space $W$ is symplectic when equipped with the canonical symplectic form $[(x,\xi),(y,\eta)]=\langle \xi,y\rangle -\langle \eta,x\rangle$, where $\langle \cdot,\cdot\rangle$ is the dual pairing for $V$ and $V'$. We denote by $\sigma:W\rightarrow W'$ the isomorphism induced by the symplectic form; notice that ${}^t\sigma=-\sigma$. Let $g$ be a Riemannian metric on $W$ and, for $X\in W$, denote by $Q_X:W\rightarrow W'$ the isomorphism induced by $g_X$. For $X\in W$, set $Q^{\sigma}_X:={}^t\sigma Q_X^{-1}\sigma:W\rightarrow W'$ and let $g^{\sigma}_X(T,S):=\langle Q^{\sigma}_XT,S\rangle$, $T,S\in W$. Then $g^{\sigma}$ is again a Riemannian metric on $W$ called the symplectic dual of $g$; it is also given by $g^{\sigma}_X(T)=\sup_{S\in W\backslash\{0\}} [T,S]^2/g_X(S)$. The Riemannian metric $g$ is said to be a H\"ormander metric \cite{hormander}, i.e., an admissible metric in the terminology of \cite{bony,lernerB}, if the following three conditions are satisfied:
\begin{itemize}
\item[$(i)$] slow variation: there exist $C_0\geq 1$ and $r_0>0$ such that
$$
g_X(X-Y)\leq r^2_0\Rightarrow C^{-1}_0g_Y(T)\leq g_X(T)\leq C_0g_Y(T),\quad X,Y,T\in W;
$$
\item[$(ii)$] temperance: there exist $C_0\geq 1$ and $N_0\geq 0$ such that
$$
\left(g_X(T)/g_Y(T)\right)^{\pm 1}\leq C_0(1+g^{\sigma}_X(X-Y))^{N_0},\quad X,Y,T\in W;
$$
\item[$(iii)$] the uncertainty principle: $g_X(T)\leq g^{\sigma}_X(T),\; X,T\in W$.
\end{itemize}
The constants $C_0\geq 1$, $r_0>0$ and $N_0\geq 0$ for which all of the above conditions hold true will be called \textit{structure constants} for $g$. The H\"ormander metric $g$ is said to be symplectic if $g=g^{\sigma}$.\\
\indent A positive measurable function $M$ on $W$ is said to be $g$-admissible if there are $C\geq 1$, $r>0$ and $N\geq 0$ such that
\begin{gather*}
g_X(X-Y)\leq r^2\Rightarrow C^{-1}M(Y)\leq M(X)\leq CM(Y),\quad X,Y\in W;\\
\left(M(X)/M(Y)\right)^{\pm1}\leq C(1+g^{\sigma}_X(X-Y))^N,\quad X,Y\in W.
\end{gather*}
The constants $r>0$ and $N\geq 0$ for which these conditions hold true are called \textit{admissibility constants} for $M$. We will also call $r$ a \textit{slow variation constant} and $N$ a \textit{temperance constant} for $M$.\\
\indent Given a $g$-admissible weight $M$, the space of symbols $S(M,g)$ is defined as the space of all $a\in\mathcal{C}^{\infty}(W)$ for which
$$
\|a\|^{(k)}_{S(M,g)}=\sup_{l\leq k}\sup_{\substack{X\in W\\ T_1,\ldots, T_l\in W\backslash\{0\}}}\frac{|a^{(l)}(X;T_1,\ldots,T_l)|} {M(X)\prod_{j=1}^lg_X(T_j)^{1/2}}<\infty,\quad k\in\NN.
$$
With this system of seminorms, $S(M,g)$ becomes a Fr\'echet space. One can always regularise the metric making it smooth without changing the notion of $g$-admissibility of a weight and the space $S(M,g)$; the same can be done for any $g$-admissible weight (see \cite{hormander}). We point out that $S(M,g)\subseteq \mathcal{O}_{\mathcal{M}}(W)$, where $\mathcal{O}_{\mathcal{M}}(W)$ is the space of multipliers for $\SSS(W)$.\\
\indent Given $a\in\SSS(W)$, the Weyl quantisation $a^w$ is the operator
$$
a^w\varphi(x)=\int_{V'}\int_V e^{2\pi i\langle x-y,\xi\rangle}a((x+y)/2,\xi)\varphi(y)dyd\xi,\quad \varphi\in \SSS(V),
$$
where $dy$ is a left-right Haar measure on $V$ with $d\xi$ being its dual measure on $V'$ so that the Fourier inversion formula holds with the standard constants. Consequently, $a^w$ as well as the completion of the product measure $dyd\xi$ on $W$ are unambiguously defined. We denote this Haar measure by $d\lambda$ and employ it to include test functions in spaces of distributions on $W$; we will also denote it by $dX,dY,dZ,\ldots$ when it appears in integrals. We point out that $d\lambda$ is exactly the measure induced by the symplectic volume form. To avoid working with distribution densities, we fix a left-right Haar measure on $V$ with its dual measure on $V'$ and we employ these to include test functions into the spaces of distributions over $V$ and $V'$. Then, for any $a\in\SSS(W)$, $a^w$ extends to a continuous operator from $\SSS'(V)$ into $\SSS(V)$. The definition of the Weyl quantisation extends to symbols in $\SSS'(W)$ and in this case $a^w:\SSS(V)\rightarrow \SSS'(V)$ is continuous. When $a\in S(M,g)$ for a $g$-admissible weight $M$, $a^w$ is continuous as an operator on $\SSS(V)$ and it uniquely extends to a continuous operator on $\SSS'(V)$ (see \cite{hormander}). Furthermore, if $a\in S(M_1,g)$ and $b\in S(M_2,g)$, then $a^wb^w=(a\#b)^w$, where $a\# b\in S(M_1M_2,g)$ and the bilinear map $\#: S(M_1,g)\times S(M_2,g)\rightarrow S(M_1M_2,g)$ is continuous (see \cite{hormander}).\\
\indent If $F_1$ and $F_2$ are two locally convex spaces (from now, always abbreviated as l.c.s.), we denote by $\mathcal{L}(F_1,F_2)$ the space of continuous linear mappings from $F_1$ into $F_2$, while $\mathcal{L}_b(F_1,F_2)$ stands for this space equipped with the topology of uniform convergence on all bounded sets. When $F_1=F_2=F$, we abbreviate these notations and simply use $\mathcal{L}(F)$ and $\mathcal{L}_b(F),$ respectively.\\
\indent Given a Fr\'echet space $F$, recall that a map $\mathbf{f}:W\rightarrow F$ is said to be strongly measurable if there exists a sequence of Lebesgue measurable simple functions on $W$ with values in $F$ which converges pointwise a.e. to $\mathbf{f}$. The map $\mathbf{f}:W\rightarrow F$ is said to be weakly measurable if for each $f'\in F'$, the function $W\rightarrow \CC$, $X\mapsto \langle f', \mathbf{f}(X)\rangle$, is measurable.
\begin{remark}\label{rem-for-meas-strong-weak}
If $F$ is a separable Fr\'echet space, the mapping $\mathbf{f}:W\rightarrow F$ is strongly measurable if and only if it is weakly measurable. This is also equivalent to the requirement that the preimage of every open set is measurable (i.e. $\mathbf{f}$ is measurable). Furthermore, there exists a sequence of Lebesgue measurable simple functions which converges pointwise everywhere to $\mathbf{f}$. These facts follow from \cite[Theorem 1]{thomas} and the remark following it since every separable Fr\'echet space is a Polish space.
\end{remark}
\indent Given a Riemannian metric $g$ on $W$, we denote by $|g_X|$, $X\in W$, the determinant of $g_X$ computed in a symplectic basis. We point out that $|g_X|$ is the same in any symplectic basis of $W$; we will always tacitly apply this fact throughout the rest of the article. To verify it, let $E_j$, $j=1,\ldots,2n$, and $\tilde{E}_j$, $j=1,\ldots,2n$, be two symplectic bases. Let $A,\tilde{A}:W\rightarrow W'$ be the isomorphisms that send these bases to their respective dual bases and let $P:W\rightarrow W$ be the symplectic map given by $P(E_j)=\tilde{E}_j$, $j=1,\ldots,2n$. Since $\det(P)=1$ and $\det(\tilde{A}^{-1}({}^tP)^{-1}\tilde{A})=1$, the claim follows from
\begin{align*}
\det((g_X(E_j,E_k))_{j,k})&=\det(A^{-1}Q_X)=\det(P^{-1}\tilde{A}^{-1}({}^tP)^{-1}\tilde{A} \tilde{A}^{-1}Q_X)\\
&=\det(\tilde{A}^{-1}Q_X) =\det((g_X(\tilde{E}_j,\tilde{E}_k))_{j,k}).
\end{align*}
From now, we denote by $|g|$ the measurable function $W\rightarrow (0,\infty)$, $X\mapsto |g_X|$. If $g$ is a H\"ormander metric, we denote by $dv_g$ the measure on $W$ induced by the volume form of $g$. Because of temperance, $dv_g$ is a complete regular Borel measure defined on the Lebesgue $\sigma$-algebra of $W$ which takes finite values on compact sets and its nullsets are exactly the Lebesgue nullsets in $W$ (see the proof of \cite[Theorem 3.11, p. 59]{grigoryan}). Notice that $dv_g=|g|^{1/2}d\lambda$.\\
\indent Let $g$ be a H\"ormander metric on $W$ and let $w:W\times W\rightarrow (0,\infty)$ be a measurable function such that both $w$ and $1/w$ are polynomially bounded, i.e. there are $C>0$ and $\tau_1,\tau_2\geq 0$ such that
\begin{align}\label{pol-b-d-wei-s}
C^{-1}(1+|X|+|Y|)^{-\tau_1}\leq w(X,Y)\leq C(1+|X|+|Y|)^{\tau_2},\quad X,Y\in W,
\end{align}
where $|\cdot|$ is a norm on $W$; notice that if $w$ satisfies \eqref{pol-b-d-wei-s} with one norm on $W$ then it satisfies it with any other norm with the same $\tau_1$ and $\tau_2$ and possibly different $C>0$. For $1\leq p\leq \infty$, we denote by $L^p_w(W\times W,dv_gd\lambda)$ the Banach space of all measurable functions $f$ on $W\times W$ such that $fw\in L^p(W\times W,dv_gd\lambda)$; when $g$ is a Euclidian metric, we will simply denote it by $L^p_w(W\times W)$. Clearly, $\SSS(W\times W)\subseteq L^p_w(W\times W,dv_gd\lambda)\subseteq \SSS'(W\times W)$ and the inclusions are continuous; when $p<\infty$, they are also dense. We point out that $L^{\infty}_w(W\times W,dv_gd\lambda)=L^{\infty}_w(W\times W)$. If $w$ is a positive measurable function on $W$ that satisfies bounds of the form \eqref{pol-b-d-wei-s} on $W$, the weighted space $L^p_w(W)$, $1\leq p\leq \infty$, is defined analogously.
\subsection{Several technical results about H\"ormander metrics}\label{properties}
Let $g$ be a H\"ormander metric with structure constants $C_0\geq 1$, $r_0>0$ and $N_0\geq 0$. Given $X\in W$ and $r>0$, denote $U_{X,r}:=\{Y\in W\,|\, g_X(X-Y)\leq r^2\}$. Throughout the article, the functions
\begin{equation}\label{funco-for-met-meas-cont-whe-gisco}
(X,Y)\mapsto g^{\sigma}_X(Y-U_{X,r})\quad \mbox{and}\quad (X,Y)\mapsto g^{\sigma}_X(U_{Y,r}-U_{X,r})
\end{equation}
will often appear in integrals in $X$ or in $Y$. We point out that these functions are always measurable and, when $g$ is continuous, \eqref{funco-for-met-meas-cont-whe-gisco} are also continuous. The proof of these facts can be found in Appendix \ref{appendix1-proof-subwithhor-metfa}: see Lemma \ref{lemma-meas-cont-metr-obtf} and the remarks following it. Here we collect a number of inequalities which we will frequently employ throughout the rest of the article. Some of these can be found in \cite{bon-che,bon-ler,hormander,lernerB}; for the sake of completeness, we give proofs for all of them in Appendix \ref{appendix1-proof-subwithhor-metfa}.
\begin{lemma}\label{tec-res-ine-for-hor-metr-ini}
Let $0<r\leq r_0$. The following inequalities hold true:
\begin{gather}
(g_X(T)/g_Y(T))^{\pm 1}\leq C_0^{N_0+2}(1+g^{\sigma}_X(Y-U_{X,r}))^{N_0},\;\; X,Y\in W,\, T\in W\backslash\{0\};\label{ineq-for-metric-p-1}\\
1+g_Y(X-Y)\leq 2(1+r^2)C_0^{N_0+2}(1+g^{\sigma}_X(Y-U_{X,r}))^{N_0+1},\;\; X,Y\in W;\label{ineq-for-metric-p-2}\\
g^{\sigma}_Y(Y-U_{X,r})\leq C_0^{N_0+2}g^{\sigma}_X(Y-U_{X,r})(1+g^{\sigma}_X(Y-U_{X,r}))^{N_0},\;\; X,Y\in W;\label{ineq-for-metric-p-3}\\
\sup_{Y\in W}\int_W (1+g^{\sigma}_X(Y-U_{X,r}))^{-(N_0+1)(n+1)-nN_0} dv_g(X)<\infty.\label{ineq-for-metric-p-3-1}
\end{gather}
Fixing any symplectic basis on $W$ to evaluate $|g_X|$ and $|g^{\sigma}_X|$, the following hold true:
\begin{gather}
|g_X|\leq 1\leq |g^{\sigma}_X|,\quad |g_X||g^{\sigma}_X|=1,\quad X\in W;\label{ineq-for-metric-p-4}\\
g_X(X-Y)\leq r_0^2\Rightarrow (|g_X|/|g_Y|)^{\pm 1}\leq C_0^{2n}\;\; \mbox{and}\;\; (|g^{\sigma}_X|/|g^{\sigma}_Y|)^{\pm 1}\leq C_0^{2n};\label{ineq-for-metric-p-5}\\
\max\{(|g_X|/|g_Y|)^{\pm 1},(|g^{\sigma}_X|/|g^{\sigma}_Y|)^{\pm 1}\}\leq C_0^{2nN_0+4n}(1+g^{\sigma}_X(Y-U_{X,r}))^{2nN_0},\;\; X,Y\in W.\label{ineq-for-metric-p-6}
\end{gather}
Let $|\cdot|$ be a norm on $W$. There exists $C\geq 1$ such that
\begin{equation}\label{ineq-for-metric-p-7}
C^{-1}(1+|X|)^{-4nN_0}\leq |g_X|\leq |g^{\sigma}_X|\leq C(1+|X|)^{4nN_0},\quad X\in W.
\end{equation}
\end{lemma}
\begin{remark}\label{equ-for-sym-metr-meas-lebesmes}
When $g$ is symplectic, \eqref{ineq-for-metric-p-4} implies $|g_X|=1$, $X\in W$, and hence $dv_g=d\lambda$.
\end{remark}
\section{A generalisation of the short-time Fourier transform}\label{C-family}
In this section, we introduce the main tool that we need for the proof of the main result: a geometric version of the symplectic short-time Fourier transform.\\
\indent From now on, $g$ is a fixed H\"ormander metric on $W$ with structure constants $C_0\geq 1$, $r_0>0$ and $N_0\geq 0$; as before, for each $X\in W$, $Q_X,Q^{\sigma}_X:W\rightarrow W'$ are the isomorphisms induced by $g_X$ and $g^{\sigma}_X$.
\subsection{Essentially uniformly confined families of symbols}
We introduce the key ingredient for the geometric generalisation of the symplectic short-time Fourier transform. It consists of families of Schwartz functions which ``keep track of the growth of the metric''; they are going to be the analogue of the windows in the classical STFT.\\
\indent We start with two technical results. Their proofs are straightforward and we omit them (cf. Remark \ref{rem-for-meas-strong-weak}).
\begin{lemma}\label{lemma-regularity-sta}
Let $A:W\rightarrow \operatorname{GL}(W)$ and $f:W\rightarrow W$ be measurable and let $\psi\in\SSS(W)$. For each $X\in W$, define $\tilde{\psi}_X(Y):=\psi(A(X)Y+f(X))$, $Y\in W$. Then $W\rightarrow \SSS(W)$, $X\mapsto \tilde{\psi}_X$, is well-defined and strongly measurable.
\end{lemma}
\begin{lemma}\label{lemma-regularity-sta1}
Let $\tilde{g}$ be a Riemannian metric on $W$ and let $\chi_0\in\SSS(\RR)$. For each $X\in W$, define $\tilde{\psi}_X(Y):=\chi_0(\tilde{g}_X(X-Y,X-Y))$, $Y\in W$. Then $W\rightarrow \SSS(W)$, $X\mapsto \tilde{\psi}_X$, is well-defined and strongly measurable.
\end{lemma}
Let $X\in W$ and $r>0$ be arbitrary but fixed. Following Bony and Lerner \cite[Definition 2.1.1]{bon-ler} (cf. \cite[Definition 2.3.1, p. 84]{lernerB}), we say that $\varphi\in\mathcal{C}^{\infty}(W)$ is $g_X$-confined in $U_{X,r}$ if\footnote{Here and throughout the rest of the article we employ the principle of vacuous (empty) product for numbers and operators, i.e. $\prod_{j=1}^0r_j= \prod_{j\in\emptyset}r_j=1$ and $\prod_{j=1}^0A_j= \prod_{j\in\emptyset}A_j=\operatorname{Id}$.}
$$
\|\varphi\|^{(k)}_{g_X,U_{X,r}}:=\sup_{l\leq k} \sup_{\substack{Y\in W\\ T_1,\ldots,T_l\in W\backslash\{0\}}} \frac{|\varphi^{(l)}(Y;T_1,\ldots,T_l)| (1+g^{\sigma}_X(Y-U_{X,r}))^{k/2}}{\prod_{j=1}^l g_X(T_j)^{1/2}}<\infty,\; k\in\NN.
$$
For fixed $X$ and $r$, the space of $g_X$-confined symbols in $U_{X,r}$ coincides with $\SSS(W)$ and the above norms generate the topology of $\SSS(W)$. The authors of \cite{bon-che,bon-ler,lernerB} considered families of functions $\{\varphi_X\in\SSS(W)\, |\, X\in W\}$ which are uniformly $g_X$-confined in $U_{X,r}$, i.e. $\sup_{X\in W} \|\varphi_X\|^{(k)}_{g_X,U_{X,r}}<\infty$, $k\in\NN$, since these are a convenient tool for studying the symbol classes $S(M,g)$ and the corresponding pseudo-differential operators (see also \cite{pil-pra1}). However, we will be interested in such families which are essentially uniformly confined.
\begin{definition}
For $r\in(0,r_0]$, we denote by $\Conf_g(W;r)$ the vector space of all equivalence classes of strongly measurable maps $\boldsymbol{\varphi}:W\rightarrow \SSS(W)$ which satisfy the following condition:
\begin{equation}\label{family-seminorms-confsym}
\|\boldsymbol{\varphi}\|^{(k)}_{g,r}:=\esssup_{X\in W}\|\boldsymbol{\varphi}(X)\|^{(k)}_{g_X,U_{X,r}}<\infty,\quad k\in\NN.
\end{equation}
\end{definition}
The right-hand side of \eqref{family-seminorms-confsym} makes sense since the function $W\rightarrow [0,\infty)$, $X\mapsto \|\boldsymbol{\varphi}(X)\|^{(k)}_{g_X,U_{X,r}}$, is measurable. To see this, set $\varphi_X:=\boldsymbol{\varphi}(X)\in \SSS(W)$, $X\in W$, and let $l,k\in\NN$ with $l\leq k$. The required measurability follows from the fact that the function
\begin{equation}\label{fun-in-norm-for-conf-sym}
W\times W\times (W\backslash\{0\})^l\rightarrow\CC,\; (X,Y,T_1,\ldots,T_l)\mapsto \frac{\varphi_X^{(l)}(Y;T_1,\ldots,T_l) (1+g^{\sigma}_X(Y-U_{X,r}))^{k/2}}{\prod_{j=1}^l g_X(T_j)^{1/2}},
\end{equation}
is measurable (see Remark \ref{rem-for-measura-cont-when-met-ssh}) and, for each fixed $X\in W$, this is a continuous function of $(Y,T_1,\ldots, T_l)\in W\times (W\backslash\{0\})^l$ (so the supremum over $W\times (W\backslash\{0\})^l$ in the definition of $\|\boldsymbol{\varphi}(X)\|^{(k)}_{g_X,U_{X,r}}$ can be replaced with essential supremum).
\begin{remark}\label{remark-for-equi-for-strong-weak-borel-meas-for-s}
In view of Remark \ref{rem-for-meas-strong-weak}, $\boldsymbol{\varphi}:W\rightarrow \SSS(W)$ is strongly measurable if and only if it is weakly measurable. We will frequently tacitly apply this fact throughout the rest of the article.
\end{remark}
It is straightforward to verify that with the system of seminorms \eqref{family-seminorms-confsym}, $\Conf_g(W;r)$ becomes a Fr\'echet space. We say that $\boldsymbol{\varphi}\in \Conf_g(W;r)$ is of class $\mathcal{C}^k$, $0\leq k\leq\infty$, if $\boldsymbol{\varphi}\in\mathcal{C}^k(W;\SSS(W))$; when $k=0$ and $k=\infty$ we also say that $\boldsymbol{\varphi}$ is continuous and smooth, respectively.
\begin{remark}
If the metric $g$ is continuous and $\boldsymbol{\varphi}\in\Conf_g(W;r)$ is continuous, then \eqref{fun-in-norm-for-conf-sym} is also continuous (see Remark \ref{rem-for-measura-cont-when-met-ssh}). Hence, the essential supremum in \eqref{family-seminorms-confsym} can be replaced by supremum.
\end{remark}
The following class of elements in $\Conf_g(W;r)$ will play an important role throughout the rest of the article.
\begin{definition}
We say that $\boldsymbol{\varphi}\in\Conf_g(W;r)$ is \textit{non-degenerate} if
\begin{equation}\label{non-degen-ele-cond-int-cottt}
\inf_{Y\in W}\int_W |\varphi_X(Y)|^2dv_g(X)>0,\quad \mbox{where}\;\; \varphi_X:=\boldsymbol{\varphi}(X),\; X\in W.
\end{equation}
\end{definition}
We collect the properties of the elements of $\Conf_g(W;r)$ that we need in the following lemma.
\begin{lemma}\label{rem-about-part-of-unity}
${}$
\begin{itemize}
\item[$(i)$] Let $0<r_1\leq r_2\leq r_0$. Then $\Conf_g(W;r_1)\subseteq \Conf_g(W;r_2)$ and the inclusion is continuous.
\item [$(ii)$] Let $\boldsymbol{\varphi},\boldsymbol{\psi}\in\Conf_g(W;r)$ and $a\in S(1,g)$. For each $X\in W$, set
$$
(\boldsymbol{\varphi}\boldsymbol{\psi})(X):=\boldsymbol{\varphi}(X)\boldsymbol{\psi}(X), \quad (a\boldsymbol{\varphi})(X):=a\boldsymbol{\varphi}(X),\quad \overline{\boldsymbol{\varphi}}(X):=\overline{\boldsymbol{\varphi}(X)}.
$$
Then $\boldsymbol{\varphi}\boldsymbol{\psi}, a\boldsymbol{\varphi},\overline{\boldsymbol{\varphi}}\in\Conf_g(W;r)$. With the multiplications defined above, $\Conf_g(W;r)$ is a Fr\'echet algebra and a topological module over the Fr\'echet algebra $S(1,g)$ where the latter has the ordinary pointwise multiplication\footnote{We emphasis this to avoid confusion since $S(1,g)$ is also a Fr\'echet algebra with multiplication given by the sharp product $\#$.}.
\item[$(iii)$] For each $\boldsymbol{\varphi}\in\Conf_g(W;r)$, set
\begin{equation}\label{func-for-com-famil-bou}
I_{\boldsymbol{\varphi}}(Y):=\int_W \varphi_X(Y) dv_g(X),\;\; Y\in W,\quad \mbox{with}\;\; \varphi_X:=\boldsymbol{\varphi}(X),\; X\in W.
\end{equation}
Then the mapping $\Conf_g(W;r)\rightarrow S(1,g)$, $\boldsymbol{\varphi}\mapsto I_{\boldsymbol{\varphi}}$, is well-defined and continuous. If $\boldsymbol{\varphi}$ is non-degenerate, then $1/I_{|\boldsymbol{\varphi}|^2}\in S(1,g)$, where $|\boldsymbol{\varphi}|^2:=\boldsymbol{\varphi}\overline{\boldsymbol{\varphi}}$.
\end{itemize}
\end{lemma}
\begin{proof} The proof of $(i)$ and $(ii)$ is straightforward; for the proof concerning $a\boldsymbol{\varphi}$ one employs \eqref{ineq-for-metric-p-1}. The fact that $\Conf_g(W;r)\rightarrow S(1,g)$, $\boldsymbol{\varphi}\mapsto I_{\boldsymbol{\varphi}}$, is well-defined and continuous follows from \eqref{ineq-for-metric-p-1} and \eqref{ineq-for-metric-p-3-1}. When $\boldsymbol{\varphi}$ is non-degenerate, \cite[Lemma 2.4]{hormander} implies that $1/I_{|\boldsymbol{\varphi}|^2}\in S(1,g)$.
\end{proof}
\begin{example}\label{exi-of-good-par-off}
We give three examples of non-degenerate elements of $\Conf_g(W;r)$ which we are going to use on several occasions throughout the rest of the article.\\
\\
\noindent $(i)$ The following construction is due to Bony and Lerner \cite[Theorem 3.1.3]{bon-ler} (see also \cite[Theorem 2.2.7, p. 70]{lernerB}); we outline the main ideas and refer to the proofs of \cite[Theorem 2.2.7, p. 70]{lernerB} and \cite[Theorem 3.1.3]{bon-ler} for the details. Pick non-negative and non-increasing $\chi_0\in\mathcal{C}^{\infty}([0,\infty))$ such that $\chi_0=1$ on $[0,1/2]$ and $\supp\chi_0\subseteq [0,1]$. Let $0<r\leq r_0$ and set $\tilde{\psi}_X:=\chi_0(r^{-2}g_X(X-\cdot))$, $X\in W$. Then $X\mapsto \tilde{\boldsymbol{\psi}}(X):=\tilde{\psi}_X$ belongs to $\Conf_g(W;r)$ (cf. Lemma \ref{lemma-regularity-sta1}), $\inf_{Y\in W} I_{\tilde{\boldsymbol{\psi}}}(Y)>0$ and $1/I_{\tilde{\boldsymbol{\psi}}}\in S(1,g)$; see the proof of \cite[Theorem 2.2.7, p. 70]{lernerB}. Define $\boldsymbol{\psi}:=\tilde{\boldsymbol{\psi}}/I_{\tilde{\boldsymbol{\psi}}}$. Then $\supp\boldsymbol{\psi}(X)\subseteq U_{X,r}$, for all $X\in W$, and $\boldsymbol{\psi}$ is non-degenerate. The latter holds true since, for each $X\in W$, $\psi_X:=\boldsymbol{\psi}(X)$ satisfies the bounds
$$
1/\|I_{\tilde{\boldsymbol{\psi}}}\|_{L^{\infty}(W)}\leq \psi_X(Y)\leq \|1/I_{\tilde{\boldsymbol{\psi}}}\|_{L^{\infty}(W)},\quad Y\in U_{X,r/2}.
$$
Furthermore, $I_{\boldsymbol{\psi}}(Y)=\int_W \psi_X(Y) dv_g(X)=1$, $Y\in W$.\\
\\
\noindent $(ii)$ We slightly modify the above construction so that the resulting element of $\Conf_g(W;r)$ is smooth. Let $0<r\leq r_0$. First we regularise the metric with the help of $\boldsymbol{\psi}$ constructed above for any fixed $r'\in(0,r_0]$. Set $\psi_X:=\boldsymbol{\psi}(X)$, $X\in W$, and define
$$
\tilde{\tilde{g}}_X(T,S):=\int_W g_Y(T,S)\psi_Y(X)dv_g(Y),\quad X,T,S\in W.
$$
Then $\tilde{\tilde{g}}$ is a smooth Riemannian metric on $W$ and, setting
$\tilde{\tilde{g}}_X(T):=\tilde{\tilde{g}}_X(T,T),$ there exists $C'\geq 1$ such that $C'^{-1}\tilde{\tilde{g}}_X(T)\leq g_X(T)\leq C'\tilde{\tilde{g}}_X(T)$, $X,T\in W$; see \cite[Remark 2.2.8, p. 71]{lernerB}. Let $\tilde{g}_X(T,S):=C'^{-1}\tilde{\tilde{g}}_X(T,S)$, $X,T,S\in W$. Then $\tilde{g}$ is a smooth H\"ormander metric that satisfies $\tilde{g}_X\leq g_X\leq C'^2\tilde{g}_X$. Notice that the constant $N_0$ for $g$ is the same as the corresponding constant for $\tilde{g}$ and, for the constant $\tilde{r}_0$ from the slow variation of $\tilde{g}$, we can take $\tilde{r}_0:=r_0/C'$. Let $\chi_0$ be as in $(i)$ and set $\tilde{\varphi}_X:=\chi_0(C'^2r^{-2}\tilde{g}_X(X-\cdot))$, $X\in W$. Then, similarly as above, $X\mapsto \tilde{\boldsymbol{\varphi}}(X):=\tilde{\varphi}_X$ belongs to $\Conf_g(W;r)\cap \mathcal{C}^{\infty}(W;\DD(W))$, $\inf_{Y\in W} I_{\tilde{\boldsymbol{\varphi}}}(Y)>0$ and $1/I_{\tilde{\boldsymbol{\varphi}}}\in S(1,g)$. Define $\boldsymbol{\varphi}:=\tilde{\boldsymbol{\varphi}}/I_{\tilde{\boldsymbol{\varphi}}}$. Then $\boldsymbol{\varphi}$ satisfies the following properties:
\begin{itemize}
\item[$(a)$] $\boldsymbol{\varphi}(X)$ is non-negative and $\supp\boldsymbol{\varphi}(X)\subseteq U_{X,r}$, $X\in W$;
\item[$(b)$] $\boldsymbol{\varphi}\in\Conf_g(W;r)\cap \mathcal{C}^{\infty}(W;\DD(W))$ and $\boldsymbol{\varphi}$ is non-degenerate;
\item[$(c)$] setting $\varphi_X:=\boldsymbol{\varphi}(X)$, $X\in W$, it holds that $I_{\boldsymbol{\varphi}}(Y)=\int_W \varphi_X(Y) dv_g(X)=1$, $Y\in W$;
\item[$(d)$] for all $k,l\in\NN$ and $N>0$ it holds that
\begin{equation*}
\sup_{\substack{k'\leq k\\ l'\leq l}} \sup_{X,Y\in W} \sup_{\substack{T_1,\ldots, T_{k'}\in W\backslash\{0\}\\ S_1,\ldots, S_{l'}\in W\backslash\{0\}}} \frac{\left|\left(\prod_{j=1}^{k'}\partial_{T_j;X}\right) \left(\prod_{j=1}^{l'}\partial_{S_j;Y}\right) \varphi_X(Y)\right|(1+g^{\sigma}_X(Y-U_{X,r}))^N}{\left(\prod_{j=1}^{k'} g_X(T_j)^{1/2}\right)\left(\prod_{j=1}^{l'} g_X(S_j)^{1/2}\right)}<\infty.
\end{equation*}
\end{itemize}
The fact that $\boldsymbol{\varphi}$ is non-degenerate follows from the bounds
\begin{equation}\label{est-from-belo-and-above-part-unit}
1/\|I_{\tilde{\boldsymbol{\varphi}}}\|_{L^{\infty}(W)}\leq \varphi_X(Y)\leq \|1/I_{\tilde{\boldsymbol{\varphi}}}\|_{L^{\infty}(W)},\quad Y\in U_{X,r/(2C')},
\end{equation}
and the proof of the rest of the properties in $(a)$, $(b)$ and $(c)$ is straightforward. Since the proof of $(d)$ is rather lengthy and technical, it is moved to Appendix \ref{appendix2-proof-dexm}.\\
\\
\noindent $(iii)$ Let $g$ be the Euclidean metric on $\RR^{2n}$; notice that one can take any $r_0>0$ as a slow variation constant for $g$. Given any $\varphi\in\SSS(\RR^{2n})\backslash\{0\}$, the mapping $\boldsymbol{\varphi}:\RR^{2n}\rightarrow \SSS(\RR^{2n})$, $\boldsymbol{\varphi}(X):=\varphi(\cdot-X)$, is a smooth non-degenerate element of $\Conf_g(\RR^{2n};r)$ for any $r>0$; in this case, the quantity \eqref{non-degen-ele-cond-int-cottt} equals $\|\varphi\|^2_{L^2(\RR^{2n})}$.
\end{example}
\begin{remark}
If $g$ is smooth and satisfies the bounds \eqref{ine-for-met-der-on-all-var}, one does not need to regularise it as in Example \ref{exi-of-good-par-off} $(ii)$ in order for $\boldsymbol{\varphi}$ to satisfy the properties $(a)-(d)$; it suffices to define $\boldsymbol{\varphi}$ via $g$. The H\"ormander metrics of the commonly used calculi almost always satisfy \eqref{ine-for-met-der-on-all-var}. In fact, if $g_{(x,\xi)}=f(x,\xi)^{-2}|dx|^2+F(x,\xi)^{-2}|d\xi|^2$ is a H\"ormander metric on $\RR^{2n}$ with $f$ and $F$ smooth positive and $f\in S(f,g)$ and $F\in S(F,g)$ (this holds true for the Shubin calculus, the SG-calculus, the H\"ormander $S_{\rho,\delta}$-calculus), then it is straightforward to check that $g$ satisfies the bounds \eqref{ine-for-met-der-on-all-var}.
\end{remark}
\subsection{A geometric version of the short-time Fourier transform}\label{GSTFT}
We denote by $\mathcal{F}_{\sigma}$ the symplectic Fourier transform on $W$:
$$
\mathcal{F}_{\sigma}f(X)=\int_W e^{-2\pi i [X,Y]} f(Y)dY,\quad f\in L^1(W).
$$
Recall that $\mathcal{F}_{\sigma}\mathcal{F}_{\sigma}=\operatorname{Id}$.\\
\indent Let $\boldsymbol{\varphi}\in\Conf_g(W;r)$ and set $\varphi_X:=\boldsymbol{\varphi}(X)$, $X\in W$. We define the \textit{geometric short-time Fourier transform} (GSTFT) $\VV_{\boldsymbol{\varphi}}f$ of $f\in \SSS'(W)$ with respect to $\boldsymbol{\varphi}$ as
$$
\VV_{\boldsymbol{\varphi}}f(X,\Xi):=\mathcal{F}_{\sigma}(f\overline{\varphi_X})(\Xi)=\langle f,e^{-2\pi i [\Xi,\cdot]}\overline{\varphi_X}\rangle,
\quad X,\Xi\in W.
$$
When $f\in L^1_{(1+|\cdot|)^{-s}}(W)$, for some $s\geq 0$, where $|\cdot|$ is (any) norm on $W$, we have
$$
\VV_{\boldsymbol{\varphi}}f(X,\Xi)=\int_W e^{-2\pi i[\Xi,Y]} f(Y)\overline{\varphi_X(Y)} dY,\quad X,\Xi\in W.
$$
Given any norm $|\cdot|$ on $W$, for each $k\in\NN$, the mapping $W\rightarrow \DD_{L^{\infty}_{(1+|\cdot|)^{-k-1}}}(W)$, $\Xi\mapsto e^{-2\pi i [\Xi,\cdot]}$, is of class $\mathcal{C}^k$. Hence, $W\times W\rightarrow \SSS(W)$, $(X,\Xi)\mapsto e^{-2\pi i[\Xi,\cdot]}\overline{\varphi}_X$, is strongly measurable. If $\boldsymbol{\varphi}$ is of class $\mathcal{C}^k$, $0\leq k\leq\infty$, then this mapping is also of class $\mathcal{C}^k$. Consequently, the function $W\times W\rightarrow \CC$, $(X,\Xi)\mapsto \VV_{\boldsymbol{\varphi}}f(X,\Xi)$, is always measurable and, if $\boldsymbol{\varphi}$ is of class $\mathcal{C}^k$, $0\leq k\leq \infty$, then $\VV_{\boldsymbol{\varphi}}f\in\mathcal{C}^k(W\times W)$, $0\leq k\leq\infty$.
\begin{remark}\label{rem-for-euc-metric-stand}
Let $g$ be the standard Euclidian metric on $\RR^{2n}$ and $\boldsymbol{\varphi}\in\Conf_g(\RR^{2n};r)$ as constructed in Example \eqref{exi-of-good-par-off} $(iii)$. Then $\VV_{\boldsymbol{\varphi}}f(X,\Xi)=V_{\varphi}f(X,\sigma \Xi)$, $X,\Xi\in\RR^{2n}$, where $V_{\varphi}$ is the standard STFT on $\RR^{2n}$.
\end{remark}
Our immediate goal is to study the mapping properties of $\VV_{\boldsymbol{\varphi}}$. For this purpose, we consider the Fr\'echet space $\ds\lim_{\substack{\longleftarrow\\ s\rightarrow \infty}}L^{\infty}_{(1+|\cdot|)^s}(W\times W)$ and the $(LB)$-spaces $\ds\lim_{\substack{\longrightarrow\\ s\rightarrow \infty}} L^{\infty}_{(1+|\cdot|)^{-s}}(W\times W)$ and $\ds\lim_{\substack{\longrightarrow\\ s\rightarrow \infty}} L^1_{(1+|\cdot|)^{-s}}(W\times W)$, where $|\cdot|$ is any norm on $W\times W$ (the linking mappings in the projective and inductive limits are the canonical inclusion); neither of these spaces depends on the particular choice of the norm $|\cdot|$. From \cite[Theorem 1.4]{reiher} (cf. \cite{bar-n-o,dierolf}), it follows that both inductive limits are regular and complete; for the completeness of $\displaystyle \lim_{\substack{\longrightarrow\\ s\rightarrow \infty}} L^{\infty}_{(1+|\cdot|)^{-s}}(W\times W)$, one employs \cite[Theorem 1.4]{reiher} and a standard argument to show that the space is quasi-complete and consequently complete in view of \cite[Theorem 3, p. 402]{kothe1}. We have the following continuous inclusions:
\begin{multline*}
\SSS(W\times W)\subseteq \lim_{\substack{\longleftarrow\\ s\rightarrow \infty}} L^{\infty}_{(1+|\cdot|)^s}(W\times W) \subseteq\lim_{\substack{\longrightarrow\\ s\rightarrow \infty}} L^{\infty}_{(1+|\cdot|)^{-s}}(W\times W)\subseteq\\
\subseteq \lim_{\substack{\longrightarrow\\ s\rightarrow \infty}} L^1_{(1+|\cdot|)^{-s}}(W\times W)\subseteq \SSS'(W\times W).
\end{multline*}
Furthermore, $\SSS(W\times W)$ is dense in $\displaystyle\lim_{\substack{\longrightarrow\\ s\rightarrow \infty}} L^1_{(1+|\cdot|)^{-s}}(W\times W)$ and the following topological isomorphism holds true \cite[Theorem 1.4]{reiher}:
\begin{equation}\label{top-iso-l1linf-spa-reg-lim-topplastr}
\left(\lim_{\substack{\longrightarrow\\ s\rightarrow \infty}} L^1_{(1+|\cdot|)^{-s}}(W\times W)\right)'_b=\lim_{\substack{\longleftarrow\\ s\rightarrow \infty}} L^{\infty}_{(1+|\cdot|)^s}(W\times W),
\end{equation}
where the index $b$ stands for the strong dual topology.\\
\indent The ensuing proposition collects the continuity properties of the GSTFT.
\begin{proposition}\label{lemma-for-conti-of-stft-sympl}
Let $0<r\leq r_0$.
\begin{itemize}
\item[$(i)$] The sesquilinear mapping
\begin{equation}\label{ses-lin-mapp-stft-for-s'}
\SSS'(W)\times \Conf_g(W;r)\rightarrow \lim_{\substack{\longrightarrow \\ s\rightarrow \infty}} L^{\infty}_{(1+|\cdot|)^{-s}}(W\times W),\quad (f,\boldsymbol{\varphi})\mapsto \VV_{\boldsymbol{\varphi}}f,
\end{equation}
is well-defined and hypocontinuous. Furthermore, for any bounded subset $B$ of $\SSS'(W)$ there is $s>0$ such that $\VV_{\boldsymbol{\varphi}}f\in L^{\infty}_{(1+|\cdot|)^{-s}}(W\times W)$, for all $f\in B$, $\boldsymbol{\varphi}\in\Conf_g(W;r)$, and the set of linear mappings
\begin{equation*}
\Conf_g(W;r)\rightarrow L^{\infty}_{(1+|\cdot|)^{-s}}(W\times W),\quad\boldsymbol{\varphi}\mapsto \VV_{\overline{\boldsymbol{\varphi}}}f,\qquad f\in B,
\end{equation*}
is an equicontinuous subset of $\mathcal{L}(\Conf_g(W;r),L^{\infty}_{(1+|\cdot|)^{-s}}(W\times W))$.
\item[$(ii)$] The sesquilinear mapping
$$
\SSS(W)\times \Conf_g(W;r)\rightarrow \lim_{\substack{\longleftarrow \\ s\rightarrow \infty}} L^{\infty}_{(1+|\cdot|)^s}(W\times W),\quad (\psi,\boldsymbol{\varphi})\mapsto \VV_{\boldsymbol{\varphi}}\psi,
$$
is well-defined and continuous.
\item[$(iii)$] Let $\boldsymbol{\varphi}\in\Conf_g(W;r)$ be of class $\mathcal{C}^{\infty}$. Assume that the function $W\times W\rightarrow \CC$, $(X,Y)\mapsto \varphi(X,Y):=\boldsymbol{\varphi}(X)(Y)$, satisfies the following: for every $k\in\NN$, there is $s\geq 0$ such that for all $m>0$
$$
\sup_{\substack{l'\leq k\\ l''\leq k}} \sup_{X,Y\in W}\sup_{\substack{T_1,\ldots,T_{l'}\in W\backslash\{0\}\\ S_1,\ldots,S_{l''}\in W\backslash\{0\}}} \frac{\left|(\prod_{j=1}^{l'}\partial_{T_j;X}) (\prod_{j=1}^{l''}\partial_{S_j;Y})\varphi(X,Y)\right| (1+g^{\sigma}_X(Y-U_{X,r}))^{m/2}} {(\prod_{j=1}^{l'} g_X(T_j)^{1/2}) (\prod_{j=1}^{l''} g_X(S_j)^{1/2})(1+|X|)^s(1+|Y|)^s}<\infty,
$$
where $|\cdot|$ is (any) norm on $W$. Then the mapping $\SSS(W)\rightarrow \SSS(W\times W)$, $\psi\mapsto \VV_{\boldsymbol{\varphi}}\psi$, is well-defined and continuous.
\end{itemize}
\end{proposition}
\begin{remark}
The condition in $(iii)$ is satisfied when $\boldsymbol{\varphi}$ is as in Example \ref{exi-of-good-par-off} $(ii)$; in fact, it is satisfied with $s=0$ for all $k\in\NN$.
\end{remark}
\begin{proof} We first address $(i)$. We claim that the second part of $(i)$ implies the first one. To see this, notice first that the second part implies that \eqref{ses-lin-mapp-stft-for-s'} is separately continuous. Indeed, it implies that for each fixed $\boldsymbol{\varphi}\in\Conf_g(W;r)$, the linear map
$$
\SSS'(W)\rightarrow \ds\lim_{\substack{\longrightarrow \\ s\rightarrow \infty}} L^{\infty}_{(1+|\cdot|)^{-s}}(W\times W),\quad f\mapsto \VV_{\boldsymbol{\varphi}}f,
$$
is well-defined and maps bounded sets into bounded sets and consequently, it is continuous since $\SSS'(W)$ is bornological. The continuity in $\boldsymbol{\varphi}\in\Conf_g(W;r)$ for fixed $f\in\SSS'(W)$ is immediate. As both $\SSS'(W)$ and $\Conf_g(W;r)$ are barrelled spaces, \cite[Theorem 5, p. 159]{kothe2} implies that \eqref{ses-lin-mapp-stft-for-s'} is hypocontinuous. It remains to prove the second part. Let $B$ be a bounded subset of $\SSS'(W)$. Then $B$ is equicontinuous. Hence, there exist $C'\geq 1$ and $k\in\ZZ_+$ such that
\begin{equation}\label{equicon-subset-s'-bou-for-dualit}
|\langle f,\psi\rangle|\leq C' \sup_{k'\leq k}\sup_{\substack{Y\in W\\ T_1,\ldots, T_{k'}\in W\backslash\{0\}}} \frac{|\partial_{T_1}\ldots\partial_{T_{k'}}\psi(Y)|(1+g_0(Y))^{k/2}} {\prod_{j=1}^{k'} g_0(T_j)^{1/2}},\;\; \psi\in\SSS(W),\, f\in B.
\end{equation}
Let $\boldsymbol{\varphi}\in\Conf_g(W;r)$ and $\varphi_X:=\boldsymbol{\varphi}(X)$, $X\in W$. For $k'\leq k$, $\partial_{T_1;Y}\ldots\partial_{T_{k'};Y}(e^{-2\pi i[\Xi,Y]}\varphi_X(Y))$ is a sum of $2^{k'}$ terms of the form
\begin{equation}\label{est-par-forfirst-part-cont-s'}
e^{-2\pi i[\Xi,Y]}\left(\prod_{j\in K'_1}(-2\pi i)[\Xi,T_j]\right)\left(\left(\prod_{j\in K'_2}\partial_{T_j;Y}\right)\varphi_X(Y)\right),
\end{equation}
where the sets $K'_1$ and $K'_2$ are disjoint and their union is $\{1,\ldots,k'\}$. Observe that \eqref{est-par-forfirst-part-cont-s'} is bounded by
\begin{align*}
&\frac{(2\pi)^{k'}\|\varphi_X\|^{(k)}_{g_X,U_{X,r}}}{(1+g^{\sigma}_X(Y-U_{X,r}))^{k/2}} \left(\prod_{j\in K'_1}g_0(T_j)^{1/2}\right)\left(\prod_{j\in K'_1} g^{\sigma}_0(\Xi)^{1/2}\right)\left(\prod_{j\in K'_2}g_X(T_j)^{1/2}\right)\\
&\leq \frac{C'_1\|\varphi_X\|^{(k)}_{g_X,U_{X,r}} (1+g^{\sigma}_0(\Xi))^{k/2} (1+g^{\sigma}_0(X))^{kN_0/2} \prod_{j=1}^{k'}g_0(T_j)^{1/2}}{(1+g^{\sigma}_X(Y-U_{X,r}))^{k/2}}.
\end{align*}
Since $g_0(Y)\leq 2g_0(X-Y)+2g_0(X)$, we deduce
\begin{align}
g_0(Y)&\leq 2C_0g_X(X-Y)(1+g^{\sigma}_0(X))^{N_0}+2g_0(X)\leq 2C_0(1+g_X(X-Y))(1+g^{\sigma}_0(X))^{N_0+1}\nonumber \\
&\leq 2C_0(1+2r^2+2g^{\sigma}_X(Y-U_{X,r})) (1+g^{\sigma}_0(X))^{N_0+1}.\label{est-for-met-at-poi-sing}
\end{align}
In view of \eqref{equicon-subset-s'-bou-for-dualit}, this implies
$$
|\VV_{\overline{\boldsymbol{\varphi}}}f(X,\Xi)|\leq C'_2 \|\varphi_X\|^{(k)}_{g_X,U_{X,r}} (1+g^{\sigma}_0(\Xi))^{k/2} (1+g^{\sigma}_0(X))^{k(2N_0+1)/2},\;\; X,\Xi\in W,
$$
which, in turn, proves the second part of $(i)$.\\
\indent We turn our attention to $(ii)$. For each $k\in\ZZ_+$, denote by $\|\psi\|_k$ the norm of $\psi\in \SSS(W)$ given on the right-hand side of \eqref{equicon-subset-s'-bou-for-dualit}. Let $k\in\ZZ_+$ be arbitrary but fixed. Pick $N,N_1\in\ZZ_+$ such that $N\geq kN_0+k$ and $N_1\geq 2kN_0+k+2n+2$. For each fixed $\Xi\in W$ choose $\theta=\theta(\Xi)\in W$ such that $g_0(\theta)=1$ and $[\Xi,\theta]=g^{\sigma}_0(\Xi)^{1/2}$. Let $\psi\in \SSS(W)$ and $\boldsymbol{\varphi}\in\Conf_g(W;r)$. Employing
\begin{equation}\label{equ-for-chn-est-der-fun}
(1-(2\pi i)^{-1}\partial_{\theta;Y})^ke^{-2\pi i[\Xi, Y]}=(1+g^{\sigma}_0(\Xi)^{1/2})^ke^{-2\pi i[\Xi,Y]},
\end{equation}
we integrate by parts in the integral defining $\VV_{\boldsymbol{\varphi}}\psi(X,\Xi)$ and infer
\begin{align*}
(1+&g^{\sigma}_0(\Xi)^{1/2})^k|\VV_{\boldsymbol{\varphi}}\psi(X,\Xi)|\\
&\leq C'_1\sum_{k'+k''\leq k}\int_W |\partial^{k''}_{\theta}\psi(Y)||\partial^{k'}_{\theta;Y}\varphi_X(Y)|dY\\
&\leq C'_2\|\psi\|_{N_1}\|\varphi_X\|^{(2N)}_{g_X,U_{X,r}} \sum_{k'+k''\leq k}\int_W \frac{g_X(\theta)^{k'/2}dY}{(1+g_0(Y))^{N_1/2}(1+g^{\sigma}_X(Y-U_{X,r}))^N}.
\end{align*}
As $g^{\sigma}_0(T)\leq C''g_0(T)$, $ T\in W$, for some $C''\geq 1$, employing \eqref{ineq-for-metric-p-1} we infer (recall $g_0(\theta)=1$)
\begin{align}
g_X(\theta)^{1/2}&\leq C'_3g_Y(\theta)^{1/2} (1+g^{\sigma}_X(Y-U_{X,r}))^{N_0/2}\nonumber\\
&\leq C'_4(1+g_0(Y))^{N_0/2} (1+g^{\sigma}_X(Y-U_{X,r}))^{N_0/2}.\label{ine-for-part-of-est-s'-cont}
\end{align}
In view of \eqref{ineq-for-metric-p-1}, we have
\begin{align*}
1+g_0(X-Y)^{1/2}&\leq C'_5(1+g_Y(X-Y)^{1/2})(1+g_0(Y))^{N_0/2}\\
&\leq C'_6 (1+g_X(X-Y)^{1/2}) (1+g^{\sigma}_X(Y-U_{X,r}))^{N_0/2}(1+g_0(Y))^{N_0/2}\\
&\leq C'_7 (1+g^{\sigma}_X(Y-U_{X,r}))^{1+N_0/2}(1+g_0(Y))^{N_0/2}.
\end{align*}
Consequently, as $1+g_0(X)^{1/2}\leq (1+g_0(X-Y)^{1/2})(1+g_0(Y)^{1/2})$, we have
\begin{equation}\label{est-for-cont-of-s'-with-stft-metr}
1+g_0(X)^{1/2}\leq C'_8 (1+g^{\sigma}_X(Y-U_{X,r}))^{1+N_0/2}(1+g_0(Y))^{(N_0+1)/2}.
\end{equation}
Employing \eqref{ine-for-part-of-est-s'-cont} and \eqref{est-for-cont-of-s'-with-stft-metr} in the above estimate for $\VV_{\boldsymbol{\varphi}}\psi$, we deduce
\begin{equation*}
(1+g^{\sigma}_0(X)^{1/2})^k(1+g^{\sigma}_0(\Xi)^{1/2})^k |\VV_{\boldsymbol{\varphi}}\psi(X,\Xi)|\leq C'_9 \|\psi\|_{N_1}\|\varphi_X\|^{(2N)}_{g_X,U_{X,r}},\;\; X,\Xi\in W,
\end{equation*}
which completes the proof of $(ii)$.\\
\indent It remains to prove $(iii)$. As before, we denote by $\|\cdot\|_k$, $k\in\NN$, the norms on $\SSS(W)$ given by the right-hand side of \eqref{equicon-subset-s'-bou-for-dualit}. Let $\psi\in \SSS(W)$ and $k\in\ZZ_+$ be arbitrary but fixed. Pick $s\geq 0$ for which the condition in $(iii)$ holds for this $k$; we employ the norm $|X|:=g_0(X)^{1/2}$ on $W$ in this condition. Set $m:= (2+N_0)(2kN_0+s+k)$ and pick $N\in\ZZ_+$ such that $N\geq k+s+(N_0+1)(2kN_0+s+k)+2n+2$. For $l',l''\leq k$, $l',l''\in\NN$, we employ \eqref{equ-for-chn-est-der-fun} with $\theta\in W$ chosen as in $(ii)$ to infer
\begin{multline*}
(1+g^{\sigma}_0(\Xi)^{1/2})^k \left|\left(\prod_{j=1}^{l'}\partial_{T_j;X}\right)\left(\prod_{j=1}^{l''} \partial_{S_j;\Xi}\right) \VV_{\boldsymbol{\varphi}}\psi(X,\Xi)\right|\\
\leq C_1\sum_{\substack{k_1+k_2+k_3\leq k\\ k_1\leq l''}} \int_W \left|\partial_{\theta;Y}^{k_1}\left(\prod_{j=1}^{l''}[S_j,Y]\right)\right| |\partial_{\theta;Y}^{k_2}\psi(Y)| \left|\left(\prod_{j=1}^{l'}\partial_{T_j;X}\right) \partial_{\theta;Y}^{k_3}\varphi_X(Y)\right|dY.
\end{multline*}
Notice that
$$
\left|\partial_{\theta;Y}^{k_1}\left(\prod_{j=1}^{l''}[S_j,Y]\right)\right|\leq C_2 g_0(Y)^{(l''-k_1)/2}\prod_{j=1}^{l''}g^{\sigma}_0(S_j)^{1/2}\leq C_3 (1+g_0(Y))^{k/2}\prod_{j=1}^{l''}g_0(S_j)^{1/2}.
$$
Consequently, in view of \eqref{est-for-cont-of-s'-with-stft-metr}, we deduce
\begin{align*}
&(1+g^{\sigma}_0(X)^{1/2})^k(1+g^{\sigma}_0(\Xi)^{1/2})^k \left|\left(\prod_{j=1}^{l'}\partial_{T_j;X}\right)\left(\prod_{j=1}^{l''} \partial_{S_j;\Xi}\right) \VV_{\boldsymbol{\varphi}}\psi(X,\Xi)\right|\\
&\leq C_4\|\psi\|_N\left(\prod_{j=1}^{l''}g_0(S_j)^{1/2}\right) \sum_{k_1+k_2+k_3\leq k} \int_W \frac{g_X(\theta)^{k_3/2}(1+g_0(X))^{(s+k)/2}\prod_{j=1}^{l'}g_X(T_j)^{1/2} dY}{(1+g_0(Y))^{(N-k-s)/2} (1+g^{\sigma}_X(Y-U_{X,r}))^{m/2}}\\
&\leq C_5\|\psi\|_N\left(\prod_{j=1}^{l'}g_0(T_j)^{1/2}\right) \left(\prod_{j=1}^{l''}g_0(S_j)^{1/2}\right) \\
&{}\quad \cdot\sum_{k_1+k_2+k_3\leq k} \int_W \frac{(1+g_0(X))^{(k_3N_0+l'N_0+s+k)/2} dY}{(1+g_0(Y))^{(N-k-s)/2} (1+g^{\sigma}_X(Y-U_{X,r}))^{m/2}}\\
&\leq C_6\|\psi\|_N\left(\prod_{j=1}^{l'}g_0(T_j)^{1/2}\right) \left(\prod_{j=1}^{l''}g_0(S_j)^{1/2}\right) \int_W (1+g_0(Y))^{-n-1}dY.
\end{align*}
As the very last integral is a positive constant, the proof is complete.
\end{proof}
If $r<r_0$ is sufficiently small, one can always regularise the elements of $\Conf_g(W;r)$ in the following way such that they satisfy the condition in Proposition \ref{lemma-for-conti-of-stft-sympl} $(iii)$.
\begin{lemma}\label{lem-for-smooth-tem-part-of-unity-in-both-coord}
Let $r',r>0$ satisfy $0<r'\sqrt{C_0}<r\leq r_0$ and let $0<r'_0\leq C_0^{-1/2}r-r'$. Let $\boldsymbol{\varphi}\in\Conf_g(W;r'_0)$ be such that $\supp\varphi_X\subseteq U_{X,r'_0}$, $ X\in W$, where $\varphi_X:=\boldsymbol{\varphi}(X)$. For $\boldsymbol{\psi}\in\Conf_g(W;r')$, set $\psi_X:=\boldsymbol{\psi}(X)$, $X\in W$, and
\begin{equation}\label{eqs11-1}
\widetilde{\boldsymbol{\psi}}(X):=\widetilde{\psi}_X,\quad \mbox{where}\quad \widetilde{\psi}_X(Y):=\int_W \psi_Z(Y)\varphi_Z(X)dv_g(Z),\; X,Y\in W.
\end{equation}
Then $\widetilde{\boldsymbol{\psi}}\in\Conf_g(W;r)$, it is of class $\mathcal{C}^{\infty}$ and it satisfies the condition in Proposition \ref{lemma-for-conti-of-stft-sympl} $(iii)$ with $s=0$ for all $k\in\NN$. Moreover, the mapping $\Conf_g(W;r')\rightarrow \Conf_g(W;r)$, $\boldsymbol{\psi}\mapsto \widetilde{\boldsymbol{\psi}}$, with $\widetilde{\boldsymbol{\psi}}$ given by \eqref{eqs11-1}, is continuous.\\
\indent Furthermore, if there exists $r''>0$ such that $\supp\psi_X\subseteq U_{X,r''}$, $X\in W$, then $\supp\tilde{\psi}_X\subseteq U_{X,(r''+r'_0)\sqrt{C_0}}$, $X\in W$.
\end{lemma}
\begin{proof} Let $k\in \NN$ be arbitrary but fixed. Pick $N\in\ZZ_+$ such that $N\geq 2(N_0+1)(n+1)+2nN_0+k$. For $l',l''\leq k$, $l',l''\in\NN$, \eqref{eqs11-1} gives
\begin{multline*}
\frac{\left|(\prod_{j=1}^{l'}\partial_{T_j;X}) (\prod_{j=1}^{l''}\partial_{S_j;Y})\widetilde{\psi}_X(Y)\right|} {(\prod_{j=1}^{l'} g_X(T_j)^{1/2}) (\prod_{j=1}^{l''} g_X(S_j)^{1/2})}\\
\leq C'_1\|\boldsymbol{\psi}\|^{(N)}_{g,r'}\int_W \frac{\left|\left(\prod_{j=1}^{l'}\partial_{T_j;X}\right)\varphi_Z(X)\right|} {(1+g^{\sigma}_Z(Y-U_{Z,r'}))^{N/2}\prod_{j=1}^{l'} g_Z(T_j)^{1/2}}dv_g(Z).
\end{multline*}
We estimate the integrand as follows. When $Z\in W$ is such that $g_Z(X-Z)> r'^2_0$, the integrand is $0$. If $g_Z(X-Z)\leq r'^2_0$, then $g^{\sigma}_Z(Y-U_{Z,r'})\geq g^{\sigma}_X(Y-U_{Z,r'})/C_0$ and $U_{Z,r'}\subseteq U_{X,r}$. To verify the latter, let $Z'\in U_{Z,r'}$. Then
\begin{align*}
g_X(X-Z')^{1/2}&\leq C_0^{1/2}g_Z(X-Z')^{1/2}\leq C_0^{1/2}(g_Z(X-Z)^{1/2}+g_Z(Z-Z')^{1/2})\\
&\leq C_0^{1/2}(r'_0+r')\leq r,
\end{align*}
which proves that $U_{Z,r'}\subseteq U_{X,r}$. Hence, in view of \eqref{ineq-for-metric-p-3-1}, we deduce
$$
\frac{\left|(\prod_{j=1}^{l'}\partial_{T_j;X}) (\prod_{j=1}^{l''}\partial_{S_j;Y})\widetilde{\psi}_X(Y)\right|} {(\prod_{j=1}^{l'} g_X(T_j)^{1/2}) (\prod_{j=1}^{l''} g_X(S_j)^{1/2})}
\leq C'_2\|\boldsymbol{\psi}\|^{(N)}_{g,r'}(1+g^{\sigma}_X(Y-U_{X,r}))^{-k/2}.
$$
Employing standard arguments and these bounds, one shows that $\widetilde{\boldsymbol{\psi}}\in\mathcal{C}^{\infty}(W;\SSS(W))$. The rest of the claimed properties of $\widetilde{\boldsymbol{\psi}}$ are an immediate consequence of the above bounds. The proof of the very last statement is straightforward and we omit it.
\end{proof}
\begin{remark}
The above lemma is applicable with $\boldsymbol{\varphi}\in\Conf_g(W;r)$ constructed in Example \ref{exi-of-good-par-off} $(i)$ and Example \ref{exi-of-good-par-off} $(ii)$.
\end{remark}
Similarly as the classical STFT, $\VV_{\boldsymbol{\varphi}}$ satisfies the following orthogonality relation.
\begin{proposition}
Let $\boldsymbol{\varphi},\boldsymbol{\psi}\in\Conf_g(W;r)$ and $f_1,f_2\in L^2(W)$. Then $\VV_{\boldsymbol{\varphi}}f_1, \VV_{\boldsymbol{\psi}}f_2\in L^2(W\times W, dv_gd\lambda)$ and
\begin{equation}\label{equality-orth-for-fam}
(\VV_{\boldsymbol{\varphi}}f_1,\VV_{\boldsymbol{\psi}}f_2)_{L^2(W\times W, dv_gd\lambda)}= (f_1,I_{\boldsymbol{\varphi}\overline{\boldsymbol{\psi}}}f_2)_{L^2(W)}.
\end{equation}
\end{proposition}
\begin{proof} It is enough to show \eqref{equality-orth-for-fam} for $f_1,f_2\in\SSS(W)$ as the rest will follow by density (cf. Proposition \ref{lemma-for-conti-of-stft-sympl}). Setting $\varphi_X:=\boldsymbol{\varphi}(X)$, $\psi_X:=\boldsymbol{\psi}(X)$, $X\in W$, Parseval's identity gives
\begin{align*}
(\VV_{\boldsymbol{\varphi}}f_1,\VV_{\boldsymbol{\psi}}f_2)_{L^2(W\times W, dv_gd\lambda)}&= \int_W\left(\int_W \mathcal{F}_{\sigma}(f_1\overline{\varphi_X})(\Xi) \overline{\mathcal{F}_{\sigma}(f_2\overline{\psi_X})(\Xi)}d\Xi\right)dv_g(X)\\
&=\int_W\left(\int_W f_1(Y)\overline{\varphi_X(Y)}\, \overline{f_2(Y)}\psi_X(Y)dY\right)dv_g(X)\\
&=(f_1,I_{\boldsymbol{\varphi}\overline{\boldsymbol{\psi}}}f_2)_{L^2(W)}.
\end{align*}
\end{proof}
In view of Lemma \ref{rem-about-part-of-unity} $(iii)$, we have the following consequence.
\begin{corollary}\label{cor-for-thestft-sympl}
Let $\boldsymbol{\varphi}\in\Conf_g(W;r)$. Then
\begin{equation}\label{shot-time-onl2-sympl}
\VV_{\boldsymbol{\varphi}}:L^2(W)\rightarrow L^2(W\times W, dv_gd\lambda)
\end{equation}
is continuous. If in addition $\boldsymbol{\varphi}$ is non-degenerate, then \eqref{shot-time-onl2-sympl} is a topological imbedding.
\end{corollary}
In view of Proposition \ref{lemma-for-conti-of-stft-sympl} $(i)$, for any $\boldsymbol{\varphi}\in\Conf_g(W;r)$, it holds that
$$
\VV_{\boldsymbol{\varphi}}:\SSS'(W)\rightarrow \lim_{\substack{\longrightarrow\\ s\rightarrow \infty}} L^1_{(1+|\cdot|)^{-s}}(W\times W)\,\, \mbox{is well-defined and continuous.}
$$
For $\ds G\in \lim_{\substack{\longrightarrow\\ s\rightarrow \infty}} L^1_{(1+|\cdot|)^{-s}}(W\times W)$ and $\boldsymbol{\varphi}\in\Conf_g(W;r)$, we define $\VV^*_{\boldsymbol{\varphi}}G\in \SSS'(W)$ by
$$
\langle \VV^*_{\boldsymbol{\varphi}}G,\chi\rangle:=\int_{W\times W} G(X,\Xi)\overline{\VV_{\boldsymbol{\varphi}}\overline{\chi}(X,\Xi)}dv_g(X)d\Xi=\langle G,|g|^{1/2}\overline{\VV_{\boldsymbol{\varphi}}\overline{\chi}}\rangle,\;\; \chi\in\SSS(W),
$$
where the last dual pairing is in the sense of \eqref{top-iso-l1linf-spa-reg-lim-topplastr}. The right-hand side makes sense in view of Proposition \ref{lemma-for-conti-of-stft-sympl} $(ii)$ and \eqref{ineq-for-metric-p-7}. Furthermore, Proposition \ref{lemma-for-conti-of-stft-sympl} $(ii)$ implies that $\VV^*_{\boldsymbol{\varphi}}G$ is indeed a well-defined element of $\SSS'(W)$. We collect the properties which we need for $\VV^*_{\boldsymbol{\varphi}}$ in the following proposition.
\begin{proposition}\label{lemma-for-the-adj-of-stft-sympl}
Let $0<r\leq r_0$.
\begin{itemize}
\item[$(i)$] The bilinear mapping
\begin{equation}\label{bil-map-adj-stft-gen-def}
\lim_{\substack{\longrightarrow\\ s\rightarrow\infty}} L^1_{(1+|\cdot|)^{-s}}(W\times W) \times \Conf_g(W;r)\rightarrow \SSS'(W),\quad (G,\boldsymbol{\varphi})\mapsto \VV^*_{\boldsymbol{\varphi}}G,
\end{equation}
is hypocontinuous. Furthermore,
\begin{equation}\label{compo-stft-with-adj-symp}
\VV^*_{\boldsymbol{\varphi}}\VV_{\boldsymbol{\psi}}f=I_{\boldsymbol{\varphi} \overline{\boldsymbol{\psi}}}f,\quad f\in \SSS'(W),\, \boldsymbol{\varphi},\boldsymbol{\psi}\in\Conf_g(W;r).
\end{equation}
\item[$(ii)$] Let $\boldsymbol{\varphi}\in\Conf_g(W;r)$. For each $G\in L^2(W\times W, dv_gd\lambda)$, the mapping
\begin{equation}\label{mapp-for-theadj-ofthe-stft-sympl}
W\times W\rightarrow L^2(W),\quad (X,\Xi)\mapsto G(X,\Xi) e^{2\pi i[\Xi,\cdot]} \boldsymbol{\varphi}(X),
\end{equation}
is strongly measurable and Pettis integrable on $W\times W$ with respect to $dv_gd\lambda$ and
$$
\VV^*_{\boldsymbol{\varphi}}G=\int_{W\times W} e^{2\pi i[\Xi,\cdot]}G(X,\Xi)\boldsymbol{\varphi}(X) dv_g(X)d\Xi.
$$
Furthermore, the mapping
$$
L^2(W\times W,dv_gd\lambda)\rightarrow L^2(W),\quad G\mapsto \VV^*_{\boldsymbol{\varphi}}G,
$$
is the adjoint to \eqref{shot-time-onl2-sympl}.
\item[$(iii)$] The bilinear mapping \eqref{bil-map-adj-stft-gen-def} restricts to a well-defined and continuous bilinear mapping
$$
\lim_{\substack{\longleftarrow\\ s\rightarrow\infty}} L^1_{(1+|\cdot|)^s}(W\times W) \times \Conf_g(W;r)\rightarrow \SSS(W),\quad (G,\boldsymbol{\varphi})\mapsto \VV^*_{\boldsymbol{\varphi}}G.
$$
\item[$(iv)$] Assume that $g$ is smooth and satisfies the following condition: for every $S\in W$ there exists a $g$-admissible weight $M_S$ such that the function $X\mapsto g_X(S)$ belongs to $S(M_S,g)$. If $\boldsymbol{\varphi}\in \Conf_g(W;r)$ satisfies the condition in Proposition \ref{lemma-for-conti-of-stft-sympl} $(iii)$, then $\VV^*_{\boldsymbol{\varphi}}$ uniquely extends to a continuous mapping $\VV^*_{\boldsymbol{\varphi}}:\SSS'(W\times W)\rightarrow \SSS'(W)$.
\end{itemize}
\end{proposition}
\begin{proof} We first address $(i)$. Proposition \ref{lemma-for-conti-of-stft-sympl} $(ii)$ verifies that \eqref{bil-map-adj-stft-gen-def} is separately continuous. Since both spaces in the domain of \eqref{bil-map-adj-stft-gen-def} are barrelled, \cite[Theorem 5, p. 159]{kothe2} verifies that \eqref{bil-map-adj-stft-gen-def} is hypocontinuous. The validity of \eqref{compo-stft-with-adj-symp} for $f\in\SSS(W)$ is a consequence of \eqref{equality-orth-for-fam} and the general case follows from the density of $\SSS(W)$ in $\SSS'(W)$ and Proposition \ref{lemma-for-conti-of-stft-sympl} $(i)$. We turn our attention to $(ii)$. The fact that \eqref{mapp-for-theadj-ofthe-stft-sympl} is strongly measurable follows from the comments before Remark \ref{rem-for-euc-metric-stand}. The rest of $(ii)$ follows from Corollary \ref{cor-for-thestft-sympl} and a straightforward computation. To prove $(iii)$, let $k,l\in\NN$ be arbitrary but fixed. For $\boldsymbol{\varphi}\in \Conf_g(W;r)$, setting $\varphi_X:=\boldsymbol{\varphi}(X)$, $X\in W$, and employing the same technique as in the proof of Proposition \ref{lemma-for-conti-of-stft-sympl} $(i)$, one can show that
\begin{multline*}
\left|\left(\prod_{j=1}^k\partial_{T_j;Y}\right)(e^{2\pi i [\Xi,Y]}\varphi_X(Y))\right|\\
\leq \frac{C_1\|\varphi_X\|^{(k+l)}_{g_X,U_{X,r}} (1+g^{\sigma}_0(\Xi))^{k/2}(1+g^{\sigma}_0(X))^{kN_0/2} \prod_{j=1}^k g_0(T_j)^{1/2}} {(1+g^{\sigma}_X(Y-U_{X,r}))^{l/2}}.
\end{multline*}
Let $\ds G\in \lim_{\substack{\longleftarrow\\ s\rightarrow\infty}} L^1_{(1+|\cdot|)^s}(W\times W)$. Employing \eqref{inequ-for-met-sim-vol-der} and \eqref{est-for-met-at-poi-sing}, we deduce
\begin{multline*}
\frac{(1+g_0(Y))^{l/2}\left|\left(\prod_{j=1}^k\partial_{T_j}\right) \VV^*_{\boldsymbol{\varphi}}G(Y)\right|}{\prod_{j=1}^k g_0(T_j)^{1/2}}\\
\leq C_2\|\boldsymbol{\varphi}\|^{(k+l)}_{g,r}\int_{W\times W} |G(X,\Xi)| (1+g^{\sigma}_0(\Xi))^{k/2}(1+g^{\sigma}_0(X))^{(kN_0+l(N_0+1)+2nN_0)/2}dXd\Xi,
\end{multline*}
which implies the claim in $(iii)$.\\
\indent We now address $(iv)$. For every $S_1,S_2\in W$, it holds that $g_X(S_1,S_2)=(g_X(S_1+S_2)-g_X(S_1)-g_X(S_2))/2$. Let $E_j$, $j=1,\ldots,2n$, be a symplectic basis for $W$. For $k\in\NN$, the condition in $(iv)$ gives the following bound for some $m_0\geq 0$:
\begin{align*}
|\partial^k_Tg_X(E_j,E_l)|&\leq C'g_X(T)^{k/2}(M_{E_j+E_l}(X)+M_{E_j}(X)+M_{E_l}(X))\\
&\leq C''g_X(T)^{k/2}(1+g^{\sigma}_0(X))^{m_0},
\end{align*}
for all $X\in W$, $j,l=1,\ldots,2n$. Consequently,
$$
|\partial^k_T|g_X||\leq \tilde{C}'g_X(T)^{k/2}(1+g^{\sigma}_0(X))^{2nm_0},\quad X,T\in W.
$$
The Fa\'a di Bruno formula (see \cite[Section 4.3.1]{lernerB}) applied to the composition of $X\mapsto |g_X|$ with $t\mapsto \sqrt{t}$ together with \eqref{inequ-for-met-sim-vol-der} yield (for $k\in\ZZ_+$)
\begin{align*}
|\partial^k_T(|g_X|^{1/2})|&\leq C'_1\sum_{l=1}^k|g_X|^{-(2l-1)/2}\sum_{\substack{k_1+\ldots +k_l=k\\ k_j\geq 1}}\prod_{j=1}^l |\partial^{k_j}_T|g_X||\\
&\leq C'_2 g_X(T)^{k/2}(1+g^{\sigma}_0(X))^{2nm_0k}\sum_{l=1}^k(1+g^{\sigma}_0(X))^{(2l-1)nN_0}\\
&\leq C'_3 g_X(T)^{k/2}(1+g^{\sigma}_0(X))^{2nm_0k+2knN_0}.
\end{align*}
Now, \cite[Lemma 4.2.3, p. 302]{lernerB} verifies the following bound for $k\in\ZZ_+$:
$$
\left|\left(\prod_{j=1}^k\partial_{T_j}\right)|g_X|^{1/2}\right|\leq C'_3(1+g^{\sigma}_0(X))^{2nm_0k+2knN_0}\prod_{j=1}^kg_X(T_j)^{1/2},\;\; X,T_1,\ldots,T_k\in W.
$$
Let $B$ be a bounded subset of $\SSS(W)$. The above estimate together with Proposition \ref{lemma-for-conti-of-stft-sympl} $(iii)$ imply that $B_1=\{|g|^{1/2} \overline{\VV_{\boldsymbol{\varphi}}\overline{\chi}}\,|\, \chi\in B\}$ is a bounded subset of $\SSS(W\times W)$. For any $G\in \SSS(W\times W)$, we have $\sup_{\chi\in B}|\langle \VV^*_{\boldsymbol{\varphi}}G,\chi\rangle|=\sup_{\phi\in B_1}|\langle G, \phi\rangle|$, which proves the claim in $(iv)$.
\end{proof}
\begin{remark}
Given any H\"ormander metric $g$, one can always find a smooth H\"ormander metric $\tilde{g}$ which satisfies the condition in Proposition \ref{lemma-for-the-adj-of-stft-sympl} $(iv)$ and is equivalent to $g$ (i.e., there exists $C\geq 1$ such that $C^{-1}g_X(T)\leq \tilde{g}_X(T)\leq Cg_X(T)$, $X,T\in W$). In fact, the smooth H\"ormander metric $\tilde{g}$ constructed in Example \ref{exi-of-good-par-off} $(ii)$ is equivalent to $g$ and \eqref{ine-for-met-der-on-all-var} proves that for each $S\in W\backslash\{0\}$, the function $X\mapsto \tilde{g}_X(S)$ belongs to $S(M_S,\tilde{g})$ with $M_S(X):=\tilde{g}_X(S)$, $X\in W$; clearly $M_S$ is $\tilde{g}$-admissible.
\end{remark}
\begin{remark}
The H\"ormander metrics of the commonly used calculi almost always satisfy the assumption in Proposition \ref{lemma-for-the-adj-of-stft-sympl} $(iv)$ (like the Shubin calculus, the SG-calculus, the H\"ormander $S_{\rho,\delta}$-calculus etc.). In fact, if $g_{x,\xi}=f(x,\xi)^{-2}|dx|^2+F(x,\xi)^{-2}|d\xi|^2$ is a H\"ormander metric on $\RR^{2n}$ such that $f$ and $F$ are smooth, positive and $f\in S(f,g)$ and $F\in S(F,g)$, then $g$ satisfies the assumption in Proposition \ref{lemma-for-the-adj-of-stft-sympl} $(iv)$.
\end{remark}
\section{Geometric modulation spaces}\label{gms}
In this section, we define a class of generalised modulation spaces with the help of the geometric short-time Fourier transform which we introduced in Subsection \ref{GSTFT}. Besides being of independent interest, they will play a crucial role in the proof of the main result.\\
\indent We start by introducing the following class of admissible weights. A positive measurable function $\eta:W\times W\rightarrow (0,\infty)$ is called a \textit{uniformly admissible weight with respect to the metric} $g$ if there are constants $\tilde{C}\geq 1$, $\tilde{r}>0$ and $\tilde{\tau}\geq 0$ such that
\begin{gather}
g_X(X-Y)\leq\tilde{r}^2\Rightarrow \tilde{C}^{-1}\eta(Y,\Xi)\leq \eta(X,\Xi)\leq \tilde{C}\eta(Y,\Xi),\quad X,Y,\Xi\in W,\label{slow-variation-with-resp-to-the-metric-uni}\\
\eta(X+X',Y)\leq \tilde{C}\eta(X,Y)(1+g^{\sigma}_X(X'))^{\tilde{\tau}},\quad X,X',Y\in W,\label{tem-with-res-to-first}\\
\eta(X,Y+Y')\leq \tilde{C}\eta(X,Y)(1+g^{\sigma}_X(Y'))^{\tilde{\tau}},\quad X,Y,Y'\in W.\label{tem-with-res-to-second}
\end{gather}
We call any number $\tilde{r}>0$ for which \eqref{slow-variation-with-resp-to-the-metric-uni} holds true a \textit{slow variation constant} for $\eta$ and we call any number $\tilde{\tau}\geq0$ for which \eqref{tem-with-res-to-first} and \eqref{tem-with-res-to-second} hold true a \textit{temperance constant} for $\eta$. In such case, we call the pair $(\tilde{r},\tilde{\tau})$ \textit{admissibility constants} for $\eta$.
\begin{remark}\label{rem-for-adm-func-for-fromad}
The reason for this notation is the following. If $M$ is a $g$-admissible weight with admissibility constants $r$ and $N$, then the function $(X,Y)\mapsto M(X)$ is uniformly admissible with respect to $g$ with admissibility constants $(r,N)$.\\
\indent Notice that \eqref{ineq-for-metric-p-5} and \eqref{ineq-for-metric-p-6} imply that the functions $(X,Y)\mapsto |g_X|$ and $(X,Y)\mapsto |g^{\sigma}_X|$ are uniformly admissible with respect to $g$ with admissibility constants $(r_0,2nN_0)$.
\end{remark}
\begin{remark}\label{uni-adm-for-stand-euc-metr}
If $g$ is the standard Euclidean metric on $\RR^{2n}$, then $\eta:\RR^{4n}\rightarrow (0,\infty)$ is uniformly admissible for $g$ if and only if $\eta$ is moderate with respect to the Beurling weight $(1+|\cdot|)^{\tau}$, for some $\tau\geq 0$ (cf. \cite[Definition 11.1.1, p. 217]{Gr1}).
\end{remark}
\begin{lemma}\label{admisibil-weig-mul}
Let $\eta$ and $\eta_1$ be two uniformly admissible weights with respect to $g$ with admissibility constants $(\tilde{r},\tilde{\tau})$ and $(\tilde{r}_1,\tilde{\tau}_1)$, respectively. Then $\eta\eta_1$ is uniformly admissible weight with respect to $g$ with admissibility constants $(\min\{\tilde{r},\tilde{r}_1\},\tilde{\tau}+\tilde{\tau}_1)$. Furthermore, $\eta^s$, $s\in\RR$, is also uniformly admissible with respect to $g$ with admissibility constants $(\tilde{r}, s\tilde{\tau})$ when $s\geq 0$ and $(\tilde{r}, |s|\tilde{\tau}(N_0+1))$ when $s<0$.
\end{lemma}
\begin{proof} The only non-trivial part is to verify \eqref{tem-with-res-to-first} for $\eta^{s}$, $s<0$, with the constant $|s|\tilde{\tau}(N_0+1)$ in place of $\tilde{\tau}$. This follows from the following chain of inequalities:
\begin{align*}
\eta(X,Y)&\leq \tilde{C}\eta(X+X',Y)(1+g^{\sigma}_{X+X'}(X'))^{\tilde{\tau}}\\
&\leq \tilde{C}\eta(X+X',Y)(1+C_0g^{\sigma}_X(X')(1+g^{\sigma}_X(X'))^{N_0})^{\tilde{\tau}}\\
&\leq \tilde{C}C_0^{\tilde{\tau}}\eta(X+X',Y)(1+g^{\sigma}_X(X'))^{\tilde{\tau}(N_0+1)}.
\end{align*}
\end{proof}
Let $w$ be a positive measurable function on $W\times W$ which satisfies \eqref{pol-b-d-wei-s}. For $1\leq p,q\leq\infty$, we denote by $L^{p,q}_w(W\times W, dv_gd\lambda)$ the Banach space of all measurable functions $f$ on $W\times W$ which satisfy
$$
\left(\int_W\left(\int_W |f(X,\Xi)|^p w(X,\Xi)^p dv_g(X)\right)^{q/p}d\Xi\right)^{1/q}<\infty
$$
(with the obvious modifications when $p=\infty$ or $q=\infty$). Similarly, we denote by $\widetilde{L}^{p,q}_w(W\times W, dv_gd\lambda)$ the Banach space of all measurable functions $f$ on $W\times W$ which satisfy
$$
\left(\int_W\left(\int_W |f(X,\Xi)|^q w(X,\Xi)^q d\Xi\right)^{p/q}dv_g(X)\right)^{1/p}<\infty
$$
(again, with the obvious modifications when $p=\infty$ or $q=\infty$). When $g$ is the Euclidean metric, we will simply denote these spaces by $L^{p,q}_w(W\times W)$ and $\widetilde{L}^{p,q}_w(W\times W)$, respectively. Clearly,
$$
L^{p,p}_w(W\times W,dv_gd\lambda)=\widetilde{L}^{p,p}_w(W\times W,dv_gd\lambda)= L^p_w(W\times W,dv_gd\lambda),\quad p\in[1,\infty].
$$
We have the following continuous inclusions for all $1\leq p,q\leq \infty$:
\begin{gather*}
\lim_{\substack{\longleftarrow \\ s\rightarrow\infty}} L^{\infty}_{(1+|\cdot|)^s}(W\times W)\subseteq L^{p,q}_w(W\times W,dv_g d\lambda)\subseteq \lim_{\substack{\longrightarrow \\ s\rightarrow\infty}} L^1_{(1+|\cdot|)^{-s}}(W\times W),\\
\lim_{\substack{\longleftarrow \\ s\rightarrow\infty}} L^{\infty}_{(1+|\cdot|)^s}(W\times W)\subseteq \widetilde{L}^{p,q}_w(W\times W,dv_g d\lambda)\subseteq \lim_{\substack{\longrightarrow \\ s\rightarrow\infty}} L^1_{(1+|\cdot|)^{-s}}(W\times W).
\end{gather*}
When $p<\infty$ and $q<\infty$, the inclusions are also dense. If $\eta$ is uniformly admissible weight with respect to $g$, than (cf. Lemma \ref{admisibil-weig-mul}),
$$
\eta(X,Y)^{\pm 1}\leq C\eta(0,0)^{\pm1}(1+g^{\sigma}_0(X))^N(1+g^{\sigma}_0(Y))^N,\quad X,Y\in W,
$$
for some $C,N>0$; consequently, it satisfies \eqref{pol-b-d-wei-s}.\\
\indent The generalised modulation spaces which we are going to define will be modelled on $\widetilde{L}^{p,q}_{\eta}(W\times W,dv_gd\lambda)$ rather than on $L^{p,q}_{\eta}(W\times W,dv_gd\lambda)$. As it turns out, the former spaces are better suited as co-domains and domains for the GSTFT and its adjoint.\\
\indent Anticipating what follows, we prove the following result.
\begin{theorem}\label{com-adj-stft-sympl1}
Let $\eta$ be a uniformly admissible weight with respect to $g$ with slow variation constant $\tilde{r}$ and let $0<r'_0\leq\min\{r_0,\tilde{r}\}$. For any $p,q\in[1,\infty]$ and any $\boldsymbol{\varphi},\boldsymbol{\psi}\in\Conf_g(W;r'_0)$, the mapping
$$
\widetilde{L}^{p,q}_{\eta}(W\times W,dv_gd\lambda)\rightarrow \widetilde{L}^{p,q}_{\eta}(W\times W,dv_gd\lambda),\quad G\mapsto \VV_{\boldsymbol{\psi}}\VV^*_{\boldsymbol{\varphi}} G,
$$
is well-defined and continuous. Furthermore, for each $p,q\in[1,\infty]$ there exist $C\geq 1$ and $k\in\ZZ_+$ such that
\begin{equation}\label{uni-bound-for-cont-mml}
\|\VV_{\boldsymbol{\psi}}\VV^*_{\boldsymbol{\varphi}} G\|_{\widetilde{L}^{p,q}_{\eta}(W\times W,dv_gd\lambda)}\leq C \|\boldsymbol{\psi}\|^{(k)}_{g,r'_0}\|\boldsymbol{\varphi}\|^{(k)}_{g,r'_0} \|G\|_{\widetilde{L}^{p,q}_{\eta}(W\times W,dv_gd\lambda)},
\end{equation}
for all $G\in \widetilde{L}^{p,q}_{\eta}(W\times W,dv_gd\lambda)$, $\boldsymbol{\varphi}, \boldsymbol{\psi}\in \Conf_g(W;r'_0)$.
\end{theorem}
Before we prove Theorem \ref{com-adj-stft-sympl1}, we show the following special case.
\begin{proposition}\label{com-adj-stft-sympl}
Let $\eta$, $\tilde{r}$ and $r'_0$ be as in Theorem \ref{com-adj-stft-sympl1}. For any $p\in[1,\infty]$ and any $\boldsymbol{\varphi},\boldsymbol{\psi}\in\Conf_g(W;r'_0)$, the mapping
$$
L^p_{\eta}(W\times W,dv_gd\lambda)\rightarrow L^p_{\eta}(W\times W,dv_gd\lambda),\quad G\mapsto \VV_{\boldsymbol{\psi}}\VV^*_{\boldsymbol{\varphi}} G,
$$
is well-defined and continuous. Furthermore, for each $p\in[1,\infty]$ there exist $C\geq 1$ and $k\in\ZZ_+$ such that
\begin{equation}\label{uni-bound-for-cont-mml2}
\|\VV_{\boldsymbol{\psi}}\VV^*_{\boldsymbol{\varphi}} G\|_{L^p_{\eta}(W\times W,dv_gd\lambda)}\leq C \|\boldsymbol{\psi}\|^{(k)}_{g,r'_0}\|\boldsymbol{\varphi}\|^{(k)}_{g,r'_0} \|G\|_{L^p_{\eta}(W\times W,dv_gd\lambda)},
\end{equation}
for all $G\in L^p_{\eta}(W\times W,dv_gd\lambda)$, $\boldsymbol{\varphi}, \boldsymbol{\psi}\in \Conf_g(W;r'_0)$.
\end{proposition}
\begin{proof} Let $\tilde{\tau}\geq 0$ be a temperance constant for $\eta$. Let $\boldsymbol{\varphi},\boldsymbol{\psi}\in\Conf_g(W;r'_0)$ and set $\varphi_X:=\boldsymbol{\varphi}(X)$, $\psi_X:=\boldsymbol{\psi}(X)$, $X\in W$. In view of Proposition \ref{lemma-for-conti-of-stft-sympl} $(i)$ and Proposition \ref{lemma-for-the-adj-of-stft-sympl} $(i)$,
\begin{align}\label{conti-com-st-adj}
\VV_{\boldsymbol{\psi}}\VV^*_{\boldsymbol{\varphi}}: \lim_{\substack{\longrightarrow \\ s\rightarrow\infty}} L^1_{(1+|\cdot|)^{-s}}(W\times W)\rightarrow \lim_{\substack{\longrightarrow \\ s\rightarrow \infty}} L^{\infty}_{(1+|\cdot|)^{-s}}(W\times W)
\end{align}
is well-defined and continuous. For $G\in \ds\lim_{\substack{\longrightarrow \\ s\rightarrow\infty}} L^1_{(1+|\cdot|)^{-s}}(W\times W)$, we infer
$$
\VV_{\boldsymbol{\psi}}\VV^*_{\boldsymbol{\varphi}}G(X,\Xi)=\langle G, |g|^{1/2}\overline{\VV_{\boldsymbol{\varphi}}(e^{2\pi i[\Xi,\cdot]}\psi_X)}\rangle.
$$
Notice that $\overline{\VV_{\boldsymbol{\varphi}}(e^{2\pi i[\Xi,\cdot]}\psi_X)(Z_1,Z_2)}=\VV_{\boldsymbol{\psi}}\varphi_{Z_1}(X,\Xi-Z_2)$ and consequently,
$$
|\VV_{\boldsymbol{\psi}}\VV^*_{\boldsymbol{\varphi}}G(X,\Xi)|\leq \int_{W\times W} |G(Z_1,\Xi-Z_2)||\VV_{\boldsymbol{\psi}}\varphi_{Z_1}(X,Z_2)||g_{Z_1}|^{1/2}dZ_1dZ_2,\;\; X,\Xi\in W.
$$
We make the following\\
\\
\noindent \textbf{Claim.} For every $N>0$ there are $C\geq 1$ and $k\in\ZZ_+$ such that
\begin{multline*}
|\VV_{\boldsymbol{\psi}}\varphi_{Z_1}(X,Z_2)||g_{Z_1}|^{1/2}\leq C \|\varphi_{Z_1}\|^{(k)}_{g_{Z_1}, U_{Z_1,r'_0}}\|\psi_X\|^{(k)}_{g_X,U_{X,r'_0}} (1+g^{\sigma}_{Z_1}(Z_2))^{-N}(1+g^{\sigma}_X(Z_2))^{-N}\\
\cdot (1+g^{\sigma}_{Z_1}(U_{X,r'_0}-U_{Z_1,r'_0}))^{-N} (1+g^{\sigma}_X(U_{X,r'_0}-U_{Z_1,r'_0}))^{-N},
\end{multline*}
for all $X,Z_1,Z_2\in W$, $\boldsymbol{\varphi},\boldsymbol{\psi}\in\Conf_g(W;r'_0)$.\\
\\
We postpone its proof for later and continue with the proof of the proposition. Let $X'\in U_{X,r'_0}$ and $Z'_1\in U_{Z_1,r'_0}$ be arbitrary. Then
\begin{align*}
\eta(X,\Xi)&\leq \tilde{C}\eta(X',\Xi)\leq \tilde{C}^2\eta(Z'_1,\Xi)(1+g^{\sigma}_{Z'_1}(X'-Z'_1))^{\tilde{\tau}}\\
&\leq C_0^{\tilde{\tau}}\tilde{C}^3 \eta(Z_1,\Xi)(1+g^{\sigma}_{Z_1}(X'-Z'_1))^{\tilde{\tau}}\\
&\leq C_0^{\tilde{\tau}}\tilde{C}^4 \eta(Z_1,\Xi-Z_2)(1+g^{\sigma}_{Z_1}(Z_2))^{\tilde{\tau}} (1+g^{\sigma}_{Z_1}(X'-Z'_1))^{\tilde{\tau}}.
\end{align*}
As $X'\in U_{X,r'_0}$ and $Z'_1\in U_{Z_1,r'_0}$ are arbitrary, we deduce
\begin{equation*}
\eta(X,\Xi)\leq C_0^{\tilde{\tau}}\tilde{C}^4 \eta(Z_1,\Xi-Z_2)(1+g^{\sigma}_{Z_1}(Z_2))^{\tilde{\tau}} (1+g^{\sigma}_{Z_1}(U_{X,r'_0}-U_{Z_1,r'_0}))^{\tilde{\tau}}.
\end{equation*}
We employ the bounds in the Claim to infer that there exists $k\in\ZZ_+$ such that
\begin{align}
|&\VV_{\boldsymbol{\psi}}\VV^*_{\boldsymbol{\varphi}}G(X,\Xi)|\eta(X,\Xi)\nonumber\\
&\leq C_1\|\boldsymbol{\varphi}\|^{(k)}_{g,r'_0}\|\psi_X\|^{(k)}_{g_X,U_{X,r'_0}} \int_{W\times W} |G(Z_1,\Xi-Z_2)|\eta(Z_1,\Xi-Z_2)\nonumber\\
&\qquad\cdot (1+g^{\sigma}_{Z_1}(Z_2))^{-n-1}(1+g^{\sigma}_X(Z_2))^{-n-1} (1+g^{\sigma}_{Z_1}(U_{X,r'_0}-U_{Z_1,r'_0}))^{-(2n+1)(N_0+1)}\nonumber \\
&\qquad \cdot (1+g^{\sigma}_X(U_{X,r'_0}-U_{Z_1,r'_0}))^{-(2n+1)(N_0+1)}dZ_1dZ_2,\label{bounds-for-gen}
\end{align}
for all $X,\Xi\in W$. Let $X'\in U_{X,r'_0}$ and $Z'_1\in U_{Z_1,r'_0}$ be arbitrary. We have
\begin{align*}
g_X(X-Z_1)&\leq 3g_X(X-X')+3g_X(Z'_1-Z_1)+3g_X(X'-Z'_1)\\
&\leq 3r'^2_0+3C_0g_{X'}(Z'_1-Z_1)+3g^{\sigma}_X(X'-Z'_1)\\
&\leq 3r'^2_0+3C_0^2g_{Z'_1}(Z'_1-Z_1)(1+g^{\sigma}_{X'}(X'-Z'_1))^{N_0} +3g^{\sigma}_X(X'-Z'_1)\\
&\leq 3r'^2_0+3r'^2_0C_0^{N_0+3}(1+g^{\sigma}_X(X'-Z'_1))^{N_0} +3g^{\sigma}_X(X'-Z'_1),
\end{align*}
and consequently,
\begin{align}\label{bounds-metric-x}
g_X(X-Z_1)\leq 3(2r'^2_0+1)C_0^{N_0+3}(1+g^{\sigma}_X(U_{X,r'_0}-U_{Z_1,r'_0}))^{N_0+1}.
\end{align}
Analogously,
\begin{align}\label{bounds-metric-z1}
g_{Z_1}(X-Z_1)\leq 3(2r'^2_0+1)C_0^{N_0+3}(1+g^{\sigma}_{Z_1}(U_{X,r'_0}-U_{Z_1,r'_0}))^{N_0+1}.
\end{align}
First we prove the claim in the proposition when $p=1$ and $p=\infty$. When $p=\infty$, \eqref{bounds-for-gen} and \eqref{bounds-metric-x} together with the identity $|g_X||g^{\sigma}_X|=1$ imply
\begin{align*}
|&\VV_{\boldsymbol{\psi}}\VV^*_{\boldsymbol{\varphi}}G(X,\Xi)|\eta(X,\Xi)\\
&\leq C_2\|\boldsymbol{\varphi}\|^{(k)}_{g,r'_0}\|\psi_X\|^{(k)}_{g_X,U_{X,r'_0}} \|G\|_{L^{\infty}_{\eta}(W\times W)} \int_{W\times W} \frac{|g_X|^{1/2}|g^{\sigma}_X|^{1/2}dZ_1dZ_2} {(1+g^{\sigma}_X(Z_2))^{n+1}(1+g_X(X-Z_1))^{n+1}}\\
&= C_2\|\boldsymbol{\varphi}\|^{(k)}_{g,r'_0}\|\psi_X\|^{(k)}_{g_X,U_{X,r'_0}} \|G\|_{L^{\infty}_{\eta}(W\times W)} \int_W\frac{|g_X|^{1/2}dZ_1}{(1+g_X(Z_1))^{n+1}} \int_W\frac{|g^{\sigma}_X|^{1/2}dZ_2}{(1+g^{\sigma}_X(Z_2))^{n+1}}\\
&\leq C_3 \|\boldsymbol{\varphi}\|^{(k)}_{g,r'_0}\|\psi_X\|^{(k)}_{g_X,U_{X,r'_0}} \|G\|_{L^{\infty}_{\eta}(W\times W)},
\end{align*}
which proves the claim in the proposition when $p=\infty$. Next, we consider the case $p=1$. For $X'\in U_{X,r'_0}$, in view of \eqref{ineq-for-metric-p-5} and \eqref{ineq-for-metric-p-6}, we have
$$
|g_X|\leq C_0^{6n+2nN_0}|g_{Z_1}|(1+g^{\sigma}_{Z_1}(X'-U_{Z_1,r'_0}))^{2nN_0},
$$
hence
\begin{equation}\label{bound-for-v-jj}
|g_X|\leq C_0^{2nN_0+6n} |g_{Z_1}|(1+g^{\sigma}_{Z_1}(U_{X,r'_0}-U_{Z_1,r'_0}))^{2nN_0}.
\end{equation}
Now, \eqref{bounds-for-gen} and \eqref{bounds-metric-z1} imply
\begin{multline}\label{ine-for-the-case-1-b}
\|\VV_{\boldsymbol{\psi}}\VV^*_{\boldsymbol{\varphi}}G\|_{L^1_{\eta}(W\times W,dv_gd\lambda)}\\
\leq C'_2\|\boldsymbol{\varphi}\|^{(k)}_{g,r'_0}\|\boldsymbol{\psi}\|^{(k)}_{g,r'_0} \int_{W\times W} |G(Z_1,Z_2)|\eta(Z_1,Z_2)F(Z_1,Z_2)|g_{Z_1}|^{1/2}dZ_1dZ_2,
\end{multline}
where
$$
F(Z_1,Z_2)=\int_{W\times W} (1+g^{\sigma}_{Z_1}(\Xi-Z_2))^{-n-1}(1+g_{Z_1}(X-Z_1))^{-n-1}dXd\Xi.
$$
Employing $|g_{Z_1}||g^{\sigma}_{Z_1}|=1$, similarly as above, we infer that $F\in L^{\infty}(W\times W)$ and the bound \eqref{ine-for-the-case-1-b} proves the claim in the proposition when $p=1$. We prove the case when $p\in (1,\infty)$ by interpolation. Fix $p\in(1,\infty)$ and notice that the above implies that
\begin{align}
&\VV_{\boldsymbol{\psi}}\VV^*_{\boldsymbol{\varphi}}: L^1(W\times W,d\mu_p)\rightarrow L^1(W\times W,d\mu_p)\quad \mbox{and}\label{boun-ope-for-int}\\
&\VV_{\boldsymbol{\psi}}\VV^*_{\boldsymbol{\varphi}}: L^{\infty}(W\times W,d\mu_p)\rightarrow L^{\infty}(W\times W,d\mu_p)\label{boun-ope-for-int-anoth}
\end{align}
are well-defined and continuous, where $d\mu_p$ is the measure $\eta^pdv_gd\lambda$ (in view of Lemma \ref{admisibil-weig-mul}, $\eta^p$ is uniformly admissible with a slow variation constant $\tilde{r}$); notice that $L^{\infty}(W\times W,d\mu_p)=L^{\infty}(W\times W,dv_gd\lambda)=L^{\infty}(W\times W)$. Now, in view of \eqref{conti-com-st-adj}, the claim in the proposition follows from the Riesz-Thorin interpolation theorem \cite[Theorem 1.3.4, p. 37]{Grafakos}; \eqref{uni-bound-for-cont-mml2} follows from the bounds we showed for \eqref{boun-ope-for-int} and \eqref{boun-ope-for-int-anoth} and the bounds in the Riesz-Thorin theorem.\\
\indent It remains to prove the Claim. Let $N\in\ZZ_+$ be arbitrary and pick $N_1\in\ZZ_+$ such that $N_1\geq N(2N_0+1)+n+1$. For every fixed $Z_1,Z_2\in W$, there exists $\theta=\theta(Z_1,Z_2)\in W$ such that $g_{Z_1}(\theta)=1$ and $[Z_2,\theta]=g^{\sigma}_{Z_1}(Z_2)^{1/2}$. Then
$$
(1-(2\pi i)^{-1}\partial_{\theta;Y})^{2N}e^{-2\pi i[Z_2,Y]}= (1+g^{\sigma}_{Z_1}(Z_2)^{1/2})^{2N}e^{-2\pi i[Z_2,Y]}.
$$
We estimate as follows
\begin{align*}
(1+&g^{\sigma}_{Z_1}(Z_2))^N|\VV_{\boldsymbol{\psi}}\varphi_{Z_1}(X,Z_2)||g_{Z_1}|^{1/2}\\
&\leq C'_1|g_{Z_1}|^{1/2}\sum_{N'+N''\leq 2N}\int_W |\partial^{N'}_{\theta;Y}\varphi_{Z_1}(Y)||\partial^{N''}_{\theta;Y}\psi_X(Y)| dY\\
&\leq C'_1\|\varphi_{Z_1}\|^{(2N_1)}_{g_{Z_1}, U_{Z_1,r'_0}}\|\psi_X\|^{(2N_1)}_{g_X,U_{X,r'_0}}\\
&{}\quad\cdot\sum_{N'+N''\leq 2N}\int_W \frac{g_X(\theta)^{N''/2}|g_{Z_1}|^{1/2}} {(1+g^{\sigma}_{Z_1}(Y-U_{Z_1,r'_0}))^{N_1}(1+g^{\sigma}_X(Y-U_{X,r'_0}))^{N_1}}dY.
\end{align*}
Applying \eqref{ineq-for-metric-p-1} twice, we infer (recall, $g_{Z_1}(\theta)=1$)
\begin{align*}
g_X(\theta)&\leq C_0^{N_0+2} g_Y(\theta)(1+g^{\sigma}_X(Y-U_{X,r'_0}))^{N_0}\\
&\leq C_0^{2N_0+4} (1+g^{\sigma}_{Z_1}(Y-U_{Z_1,r'_0}))^{N_0}(1+g^{\sigma}_X(Y-U_{X,r'_0}))^{N_0}.
\end{align*}
Hence,
\begin{multline}\label{bounds-for-int-metrr}
(1+g^{\sigma}_{Z_1}(Z_2))^N|\VV_{\boldsymbol{\psi}}\varphi_{Z_1}(X,Z_2)||g_{Z_1}|^{1/2} \leq C'_2 \|\varphi_{Z_1}\|^{(2N_1)}_{g_{Z_1}, U_{Z_1,r'_0}}\|\psi_X\|^{(2N_1)}_{g_X,U_{X,r'_0}}\\
\cdot\int_W (1+g^{\sigma}_{Z_1}(Y-U_{Z_1,r'_0}))^{-N_1+NN_0} (1+g^{\sigma}_X(Y-U_{X,r'_0}))^{-N(N_0+1)}|g_{Z_1}|^{1/2}dY.
\end{multline}
Let $X'\in U_{X,r'_0}$ and $Z'_1\in U_{Z_1,r'_0}$. Then
\begin{align*}
g^{\sigma}_{Z_1}(X'-Z'_1)\leq 2C_0g^{\sigma}_{Z'_1}(X'-Y)+2g^{\sigma}_{Z_1}(Y-Z'_1).
\end{align*}
We estimate the first term as follows
\begin{align*}
g^{\sigma}_{Z'_1}(X'-Y)&\leq C_0g^{\sigma}_Y(X'-Y)(1+g^{\sigma}_{Z'_1}(Y-Z'_1))^{N_0}\\
&\leq C_0^2g^{\sigma}_{X'}(X'-Y)(1+g^{\sigma}_{X'}(X'-Y))^{N_0} (1+g^{\sigma}_{Z'_1}(Y-Z'_1))^{N_0}\\
&\leq C_0^{2N_0+3}g^{\sigma}_X(X'-Y)(1+g^{\sigma}_X(X'-Y))^{N_0} (1+g^{\sigma}_{Z_1}(Y-Z'_1))^{N_0}.
\end{align*}
Consequently,
$$
1+g^{\sigma}_{Z_1}(X'-Z'_1)\leq 2C_0^{2N_0+4}(1+g^{\sigma}_X(X'-Y))^{N_0+1}(1+g^{\sigma}_{Z_1}(Y-Z'_1))^{N_0+1}.
$$
As $X'\in U_{X,r'_0}$ and $Z'_1\in U_{Z_1,r'_0}$ are arbitrary, we deduce
\begin{multline*}
(1+g^{\sigma}_{Z_1}(U_{X,r'_0}-U_{Z_1,r'_0}))^N\\
\leq 2^NC_0^{N(2N_0+4)}(1+g^{\sigma}_X(U_{X,r'_0}-Y))^{N(N_0+1)} (1+g^{\sigma}_{Z_1}(Y-U_{Z_1,r'_0}))^{N(N_0+1)}.
\end{multline*}
Employing this bound in \eqref{bounds-for-int-metrr}, we infer
\begin{align*}
(1+&g^{\sigma}_{Z_1}(Z_2))^N(1+g^{\sigma}_{Z_1}(U_{X,r'_0}-U_{Z_1,r'_0}))^N |\VV_{\boldsymbol{\psi}}\varphi_{Z_1}(X,Z_2)||g_{Z_1}|^{1/2}\\
&\leq C'_3 \|\varphi_{Z_1}\|^{(2N_1)}_{g_{Z_1}, U_{Z_1,r'_0}}\|\psi_X\|^{(2N_1)}_{g_X,U_{X,r'_0}} \int_W \frac{|g_{Z_1}|^{1/2}dY} {(1+g^{\sigma}_{Z_1}(Y-U_{Z_1,r'_0}))^{n+1}}\\
& \leq C'_4\|\varphi_{Z_1}\|^{(2N_1)}_{g_{Z_1}, U_{Z_1,r'_0}}\|\psi_X\|^{(2N_1)}_{g_X,U_{X,r'_0}}\int_W \frac{|g_{Z_1}|^{1/2}dY} {(1+g_{Z_1}(Y-Z_1))^{n+1}}.
\end{align*}
Since the very last integral is uniformly bounded by a constant for all $Z_1\in W$, we deduce the following bounds: for every $N>0$ there are $C\geq 1$ and $k\in\ZZ_+$ such that
\begin{multline}\label{intme-bound-for-theshorttime-sympl}
|\VV_{\boldsymbol{\psi}}\varphi_{Z_1}(X,Z_2)||g_{Z_1}|^{1/2}\leq C \|\varphi_{Z_1}\|^{(k)}_{g_{Z_1}, U_{Z_1,r'_0}}\|\psi_X\|^{(k)}_{g_X,U_{X,r'_0}}\\
(1+g^{\sigma}_{Z_1}(Z_2))^{-N}(1+g^{\sigma}_{Z_1}(U_{X,r'_0}-U_{Z_1,r'_0}))^{-N},
\end{multline}
for all $X,Z_1,Z_2\in W$, $\boldsymbol{\varphi},\boldsymbol{\psi}\in\Conf_g(W;r'_0)$. We claim that \eqref{intme-bound-for-theshorttime-sympl} implies the desired bounds. To see this, let $X'\in U_{X,r'_0}$ and $Z'_1\in U_{Z_1,r'_0}$ and notice that
\begin{align*}
1+g^{\sigma}_X(Z_2)&\leq 1+C_0g^{\sigma}_{X'}(Z_2)\leq 1+C_0^2g^{\sigma}_{Z'_1}(Z_2)(1+g^{\sigma}_{Z'_1}(X'-Z'_1))^{N_0}\\
&\leq 1+C_0^{N_0+3}g^{\sigma}_{Z_1}(Z_2)(1+g^{\sigma}_{Z_1}(X'-Z'_1))^{N_0},
\end{align*}
which gives
\begin{equation}\label{inequ-met-1}
1+g^{\sigma}_X(Z_2)\leq C_0^{N_0+3}(1+g^{\sigma}_{Z_1}(Z_2))(1+g^{\sigma}_{Z_1}(U_{X,r'_0}-U_{Z_1,r'_0}))^{N_0}.
\end{equation}
Similarly, for $X'\in U_{X,r'_0}$ and $Z'_1\in U_{Z_1,r'_0}$, we infer
\begin{align*}
1+g^{\sigma}_X(X'-Z'_1)&\leq 1+C_0g^{\sigma}_{X'}(X'-Z'_1)\leq 1+C^2_0g^{\sigma}_{Z'_1}(X'-Z'_1)(1+g^{\sigma}_{Z'_1}(X'-Z'_1))^{N_0}\\
&\leq 1+C^{N_0+3}_0g^{\sigma}_{Z_1}(X'-Z'_1)(1+g^{\sigma}_{Z_1}(X'-Z'_1))^{N_0},
\end{align*}
whence
\begin{equation}\label{inequ-met-2}
1+g^{\sigma}_X(U_{X,r'_0}-U_{Z_1,r'_0})\leq C^{N_0+3}_0(1+g^{\sigma}_{Z_1}(U_{X,r'_0}-U_{Z_1,r'_0}))^{N_0+1}.
\end{equation}
Combining \eqref{inequ-met-1} and \eqref{inequ-met-2} with \eqref{intme-bound-for-theshorttime-sympl}, we deduce the desired bounds in the Claim.
\end{proof}
\begin{proof}[Proof of Theorem \ref{com-adj-stft-sympl1}] The case when $p=q\in[1,\infty]$ follows from Proposition \ref{com-adj-stft-sympl}. We divide the proof of the rest in seven cases. Let $\boldsymbol{\varphi},\boldsymbol{\psi}\in\Conf_g(W;r'_0)$ and set $\varphi_X:=\boldsymbol{\varphi}(X)$, $\psi_X:=\boldsymbol{\psi}(X)$, $X\in W$.\\
\indent \underline{Case 1: $p=\infty$, $q\in[1,\infty)$.} Let $G\in \widetilde{L}^{\infty,q}_{\eta}(W\times W,dv_gd\lambda)$. Then, \eqref{bounds-for-gen} together with the Minkowski integral inequality implies
\begin{multline*}
\left(\int_W|\VV_{\boldsymbol{\psi}}\VV^*_{\boldsymbol{\varphi}}G(X,\Xi)|^q \eta(X,\Xi)^qd\Xi\right)^{1/q}\leq C_1 \|\boldsymbol{\varphi}\|^{(k)}_{g,r'_0}\|\psi_X\|^{(k)}_{g_X,U_{X,r'_0}} \|G\|_{\widetilde{L}^{\infty,q}_{\eta}(W\times W,dv_gd\lambda)}\\
\cdot\int_{W\times W} (1+g^{\sigma}_X(Z_2))^{-n-1} (1+g^{\sigma}_X(U_{X,r'_0}-U_{Z_1,r'_0}))^{-(2n+1)(N_0+1)}dZ_1dZ_2,
\end{multline*}
a.a. $X$. Employing $|g_X||g^{\sigma}_X|=1$ and \eqref{bounds-metric-x}, by the same technique as in the proof of Proposition \ref{com-adj-stft-sympl} one can show that the integral on the right is uniformly bounded by a constant for all $X\in W$, which completes the proof of Case 1.\\
\indent \underline{Case 2: $p=1$, $q=\infty$.} Let $G\in \widetilde{L}^{1,\infty}_{\eta}(W\times W, dv_gd\lambda)$. Then \eqref{bounds-for-gen} together with $|g_{Z_1}||g^{\sigma}_{Z_1}|=1$ implies
\begin{multline*}
|\VV_{\boldsymbol{\psi}}\VV^*_{\boldsymbol{\varphi}}G(X,\Xi)|\eta(X,\Xi)\leq C_1\|\boldsymbol{\varphi}\|^{(k)}_{g,r'_0}\|\psi_X\|^{(k)}_{g_X,U_{X,r'_0}}\\
\cdot \int_{W\times W} \frac{|G(Z_1,\Xi-Z_2)|\eta(Z_1,\Xi-Z_2)|g_{Z_1}|^{1/2} |g^{\sigma}_{Z_1}|^{1/2}} {(1+g^{\sigma}_{Z_1}(Z_2))^{n+1} (1+g^{\sigma}_{Z_1}(U_{X,r'_0}-U_{Z_1,r'_0}))^{(2n+1)(N_0+1)}}dZ_1dZ_2.
\end{multline*}
Set $F(Z_1):=\esssup_{Z_2\in W}|G(Z_1,Z_2)|\eta(Z_1,Z_2)$. Notice that \eqref{bound-for-v-jj} implies
$$
|g^{\sigma}_{Z_1}|\leq C_0^{2nN_0+6n}|g^{\sigma}_X|(1+g^{\sigma}_{Z_1}(U_{X,r'_0}-U_{Z_1,r'_0}))^{2nN_0}.
$$
In view of \eqref{bounds-metric-z1}, we deduce
\begin{multline*}
\esssup_{\Xi\in W} |\VV_{\boldsymbol{\psi}}\VV^*_{\boldsymbol{\varphi}}G(X,\Xi)|\eta(X,\Xi)\leq C'_2 \|\boldsymbol{\varphi}\|^{(k)}_{g,r'_0}\|\psi_X\|^{(k)}_{g_X,U_{X,r'_0}}\\
\cdot |g^{\sigma}_X|^{1/2}\int_W \frac{F(Z_1)|g_{Z_1}|}{(1+g_{Z_1}(X-Z_1))^{n+1}}\left(\int_W \frac{|g^{\sigma}_{Z_1}|^{1/2}dZ_2} {(1+g^{\sigma}_{Z_1}(Z_2))^{n+1}}\right) dZ_1,
\end{multline*}
a.a. $X$. Consequently,
\begin{align*}
\|&\VV_{\boldsymbol{\psi}}\VV^*_{\boldsymbol{\varphi}} G\|_{\widetilde{L}^{1,\infty}_{\eta}(W\times W,dv_gd\lambda)}\\
&\leq C'_3\|\boldsymbol{\varphi}\|^{(k)}_{g,r'_0}\|\boldsymbol{\psi}\|^{(k)}_{g,r'_0} \int_W F(Z_1)|g_{Z_1}|^{1/2}\left(\int_W\frac{|g_{Z_1}|^{1/2}dX} {(1+g_{Z_1}(X-Z_1))^{n+1}}\right)dZ_1\\
&\leq C'_4 \|\boldsymbol{\varphi}\|^{(k)}_{g,r'_0}\|\boldsymbol{\psi}\|^{(k)}_{g,r'_0} \|G\|_{\widetilde{L}^{1,\infty}_{\eta}(W\times W,dv_gd\lambda)},
\end{align*}
which completes the proof of Case 2.\\
\indent Before we consider the remaining cases, we introduce the following notations. For any positive measurable functions $w$ on $W\times W$ which satisfies \eqref{pol-b-d-wei-s}, denote by $P_w$ the topological isomorphism
$$
P_w:\lim_{\substack{\longrightarrow \\ s\rightarrow\infty}} L^1_{(1+|\cdot|)^{-s}}(W\times W)\rightarrow \lim_{\substack{\longrightarrow \\ s\rightarrow\infty}} L^1_{(1+|\cdot|)^{-s}}(W\times W),\quad P_w(F)=Fw;
$$
its inverse is $P_{1/w}$. Furthermore, we denote by $Q$ the topological isomorphism
$$
Q:\lim_{\substack{\longrightarrow \\ s\rightarrow\infty}} L^1_{(1+|\cdot|)^{-s}}(W\times W)\rightarrow \lim_{\substack{\longrightarrow \\ s\rightarrow\infty}} L^1_{(1+|\cdot|)^{-s}}(W\times W),\quad Q(F)(X,\Xi)=F(\Xi,X);
$$
clearly $Q\circ Q=\operatorname{Id}$. For any measurable $w_1:W\times W\rightarrow (0,\infty)$ which satisfies \eqref{pol-b-d-wei-s}, $P_w$ and $Q$ restrict to the following bijective isometries for all $p,q\in[1,\infty]$:
\begin{gather*}
P_w: L^{p,q}_{w_1}(W\times W)\rightarrow L^{p,q}_{w_1/w}(W\times W),\quad P_w: \widetilde{L}^{p,q}_{w_1}(W\times W)\rightarrow \widetilde{L}^{p,q}_{w_1/w}(W\times W),\\
Q:L^{p,q}_{w_1}(W\times W)\rightarrow \widetilde{L}^{q,p}_{Qw_1}(W\times W).
\end{gather*}
\indent \underline{Case 3: $p\in(1,\infty)$, $q=1$.} The proof is by interpolation. Set
$$
\eta_1(X,\Xi):=|g_X|^{\frac{1}{2p}-\frac{1}{2}}\eta(X,\Xi),\quad \eta_2(X,\Xi):=|g_X|^{1/(2p)}\eta(X,\Xi),\qquad X,\Xi\in W.
$$
Lemma \ref{admisibil-weig-mul} and Remark \ref{rem-for-adm-func-for-fromad} verify that $\eta_1$ and $\eta_2$ are uniformly admissible with slow variation constant $\min\{r_0,\tilde{r}\}$. In view of Case 1 and Proposition \ref{com-adj-stft-sympl},
\begin{align}
&\VV_{\boldsymbol{\psi}}\VV^*_{\boldsymbol{\varphi}}:\widetilde{L}^{1,1}_{\eta_1}(W\times W, dv_gd\lambda)\rightarrow \widetilde{L}^{1,1}_{\eta_1}(W\times W, dv_gd\lambda)\quad \mbox{and} \label{boun-interpol-firs}\\
&\VV_{\boldsymbol{\psi}}\VV^*_{\boldsymbol{\varphi}}: \widetilde{L}^{\infty,1}_{\eta_2}(W\times W, dv_gd\lambda)\rightarrow \widetilde{L}^{\infty,1}_{\eta_2}(W\times W, dv_gd\lambda)\label{bound-interpol-anoth-map}
\end{align}
are well-defined and continuous. Consequently,
\begin{align*}
&QP_{\eta_2}(\VV_{\boldsymbol{\psi}}\VV^*_{\boldsymbol{\varphi}})P_{1/\eta_2}Q: L^{1,1}(W\times W)\rightarrow L^{1,1}(W\times W)\quad \mbox{and}\\
&QP_{\eta_2}(\VV_{\boldsymbol{\psi}}\VV^*_{\boldsymbol{\varphi}})P_{1/\eta_2}Q: L^{1,\infty}(W\times W)\rightarrow L^{1,\infty}(W\times W)
\end{align*}
are well-defined and continuous. Now, the Riesz-Thorin interpolation theorem for $L^{p,q}$-spaces \cite[Section 7, Theorem 2]{ben-pen} implies that $QP_{\eta_2}(\VV_{\boldsymbol{\psi}}\VV^*_{\boldsymbol{\varphi}})P_{1/\eta_2}Q: L^{1,p}(W\times W)\rightarrow L^{1,p}(W\times W)$ is well-defined and continuous, which, in turn, implies the claim in the theorem in view of the identity
\begin{align}\label{equ-for-com-via-m}
\VV_{\boldsymbol{\psi}}\VV^*_{\boldsymbol{\varphi}}= P_{1/\eta_2}Q(QP_{\eta_2}(\VV_{\boldsymbol{\psi}} \VV^*_{\boldsymbol{\varphi}})P_{1/\eta_2}Q)QP_{\eta_2};
\end{align}
\eqref{uni-bound-for-cont-mml} follows from the bounds that we proved for \eqref{boun-interpol-firs} and \eqref{bound-interpol-anoth-map} in Proposition \ref{com-adj-stft-sympl} and Case 1 and the bounds in the Riesz-Thorin theorem \cite[Section 7, Theorem 2]{ben-pen}.
\indent \underline{Case 4: $p=1$, $q\in(1,\infty)$.} The proof is similar to the proof of Case 3.\\
\indent \underline{Case 5: $1<q<p<\infty$.} Let $\eta_2$ be as in Case 3. In view of Case 3 and Proposition \ref{com-adj-stft-sympl}, we infer that
\begin{align}
&\VV_{\boldsymbol{\psi}}\VV^*_{\boldsymbol{\varphi}}:\widetilde{L}^{p,1}_{\eta}(W\times W, dv_gd\lambda)\rightarrow \widetilde{L}^{p,1}_{\eta}(W\times W, dv_gd\lambda)\quad \mbox{and} \label{inter-bound-first-map-adv}\\
&\VV_{\boldsymbol{\psi}}\VV^*_{\boldsymbol{\varphi}}:\widetilde{L}^{p,p}_{\eta}(W\times W, dv_gd\lambda)\rightarrow \widetilde{L}^{p,p}_{\eta}(W\times W, dv_gd\lambda)\label{interp-bound-secon-map-adit-elm}
\end{align}
are well-defined and continuous and hence,
\begin{align*}
&QP_{\eta_2}(\VV_{\boldsymbol{\psi}}\VV^*_{\boldsymbol{\varphi}})P_{1/\eta_2}Q: L^{1,p}(W\times W)\rightarrow L^{1,p}(W\times W)\quad \mbox{and}\\
&QP_{\eta_2}(\VV_{\boldsymbol{\psi}}\VV^*_{\boldsymbol{\varphi}})P_{1/\eta_2}Q: L^{p,p}(W\times W)\rightarrow L^{p,p}(W\times W)
\end{align*}
are well-defined and continuous. The Riesz-Thorin interpolation theorem \cite[Section 7, Theorem 2]{ben-pen} implies that $QP_{\eta_2}(\VV_{\boldsymbol{\psi}}\VV^*_{\boldsymbol{\varphi}})P_{1/\eta_2}Q: L^{q,p}(W\times W)\rightarrow L^{q,p}(W\times W)$ is well-defined and continuous. In view of \eqref{equ-for-com-via-m}, this yields the claim; \eqref{uni-bound-for-cont-mml} follows from the bounds we proved for \eqref{inter-bound-first-map-adv} and \eqref{interp-bound-secon-map-adit-elm} in Case 3 and Proposition \ref{com-adj-stft-sympl} and the bounds in the Riesz-Thorin theorem \cite[Section 7, Theorem 2]{ben-pen}.\\
\indent \underline{Case 6: $1<p<q<\infty$.} In view of Case 4 and Proposition \ref{com-adj-stft-sympl},
\begin{align*}
&\VV_{\boldsymbol{\psi}}\VV^*_{\boldsymbol{\varphi}}:\widetilde{L}^{1,q}_{\eta_1}(W\times W, dv_gd\lambda)\rightarrow \widetilde{L}^{1,q}_{\eta_1}(W\times W, dv_gd\lambda)\quad \mbox{and}\\
&\VV_{\boldsymbol{\psi}}\VV^*_{\boldsymbol{\varphi}}:\widetilde{L}^{q,q}_{\eta_3}(W\times W, dv_gd\lambda)\rightarrow \widetilde{L}^{q,q}_{\eta_3}(W\times W, dv_gd\lambda)
\end{align*}
are well-defined and continuous with $\eta_1$ as in Case 3 and $\eta_3(X,\Xi):=|g_X|^{\frac{1}{2p}-\frac{1}{2q}}\eta(X,\Xi)$, $X,\Xi\in W$ (cf. Lemma \ref{admisibil-weig-mul} and Remark \ref{rem-for-adm-func-for-fromad}). The rest of the proof is analogous to the proof of Case 5; of course, now one interpolates between $L^{q,1}$ and $L^{q,q}$.\\
\indent \underline{Case 7: $p\in(1,\infty)$, $q=\infty$.} We claim that
\begin{align}\label{st-ad-dual-a-a}
\langle \VV_{\boldsymbol{\psi}}\VV^*_{\boldsymbol{\varphi}}G,\chi\rangle=\int_{W\times W} G(X,\Xi)\overline{\VV_{\boldsymbol{\varphi}}\VV^*_{\boldsymbol{\psi}} (|g|^{-1/2}\overline{\chi})(X,\Xi)} dv_g(X)d\Xi,
\end{align}
for all $G\in \ds\lim_{\substack{\longrightarrow \\ s\rightarrow\infty}} L^1_{(1+|\cdot|)^{-s}}(W\times W)$, $\chi\in\SSS(W\times W)$. Notice that the integral on the right is absolutely convergent in view of Proposition \ref{lemma-for-conti-of-stft-sympl} $(ii)$ and Proposition \ref{lemma-for-the-adj-of-stft-sympl} $(iii)$. It is straightforward to check that \eqref{st-ad-dual-a-a} holds when $G\in \SSS(W\times W)$ and the general case follows by density. Let $G\in \widetilde{L}^{p,\infty}_{\eta}(W\times W,dv_gd\lambda)$. Denote by $p'\in(1,\infty)$ the H\"older conjugate index to $p$ and let $\eta_2$ be as in Case 3. In view of Case 3 and \eqref{st-ad-dual-a-a}, we have the following chain of inequalities for all $\chi\in\SSS(W\times W)$ (Lemma \ref{admisibil-weig-mul} verifies that $1/\eta$ is uniformly admissible with slow variation constant $\tilde{r}$):
\begin{align*}
|\langle \VV_{\boldsymbol{\psi}}\VV^*_{\boldsymbol{\varphi}}G,\chi\rangle|&\leq \|G\|_{\widetilde{L}^{p,\infty}_{\eta}(W\times W,dv_gd\lambda)} \|\VV_{\boldsymbol{\varphi}}\VV^*_{\boldsymbol{\psi}} (|g|^{-1/2}\overline{\chi})\|_{\widetilde{L}^{p',1}_{1/\eta} (W\times W,dv_gd\lambda)}\\
&\leq C''\|\boldsymbol{\varphi}\|^{(k)}_{g,r'_0}\|\boldsymbol{\psi}\|^{(k)}_{g,r'_0} \|G\|_{\widetilde{L}^{p,\infty}_{\eta}(W\times W,dv_gd\lambda)} \||g|^{-1/2}\chi\|_{\widetilde{L}^{p',1}_{1/\eta} (W\times W,dv_gd\lambda)}\\
&= C''\|\boldsymbol{\varphi}\|^{(k)}_{g,r'_0}\|\boldsymbol{\psi}\|^{(k)}_{g,r'_0} \|G\|_{\widetilde{L}^{p,\infty}_{\eta}(W\times W,dv_gd\lambda)} \|\chi\|_{\widetilde{L}^{p',1}_{1/\eta_2} (W\times W)}.
\end{align*}
Since the strong dual of $\widetilde{L}^{p',1}_{1/\eta_2}(W\times W)$ is $\widetilde{L}^{p,\infty}_{\eta_2}(W\times W)= \widetilde{L}^{p,\infty}_{\eta}(W\times W,dv_gd\lambda)$, the claim follows.
\end{proof}
We are now ready to define the generalised modulation spaces. Let $\eta$ be a uniformly admissible weight with respect to $g$ with slow variation constant $\tilde{r}>0$ and let $\boldsymbol{\varphi}$ be a non-degenerate element of $\Conf_g(W;\min\{r_0,\tilde{r}\})$. For $p,q\in[1,\infty]$, we define the geometric modulation space $\widetilde{\MM}^{p,q}_{\eta}(W)$ by
$$
\widetilde{\MM}^{p,q}_{\eta}(W):= \{f\in\SSS'(W)\, |\, \VV_{\boldsymbol{\varphi}}f\in \widetilde{L}^{p,q}_{\eta}(W\times W, dv_gd\lambda)\}
$$
and we equip it with the norm $\|f\|_{\widetilde{\MM}^{p,q}_{\eta}}:= \|\VV_{\boldsymbol{\varphi}}f\|_{\widetilde{L}^{p,q}_{\eta}(W\times W,dv_gd\lambda)}$; the latter is indeed a norm in view of \eqref{compo-stft-with-adj-symp} and Lemma \ref{rem-about-part-of-unity} $(iii)$.
\begin{theorem}\label{mod-spa-donot-depend-on-partuni}
Let $p,q\in[1,\infty]$ and let $\eta$ be a uniformly admissible weight with respect to $g$ with slow variation constant $\tilde{r}>0$. Different choices of non-degenerate elements of $\Conf_g(W;\min\{r_0,\tilde{r}\})$ produce the same space $\widetilde{\MM}^{p,q}_{\eta}$ with equivalent norms. Furthermore, $\widetilde{\MM}^{p,q}_{\eta}$ is a Banach space and for any $\boldsymbol{\theta}\in\Conf_g(W;\min\{r_0,\tilde{r}\})$ the mappings
\begin{equation}\label{mappings-for-mod-spaces-sym-bbb}
\VV_{\boldsymbol{\theta}}: \widetilde{\MM}^{p,q}_{\eta}\rightarrow \widetilde{L}^{p,q}_{\eta}(W\times W,dv_gd\lambda)\quad \mbox{and}\quad \VV^*_{\boldsymbol{\theta}}: \widetilde{L}^{p,q}_{\eta}(W\times W,dv_gd\lambda)\rightarrow \widetilde{\MM}^{p,q}_{\eta}
\end{equation}
are well-defined and continuous.
\end{theorem}
\begin{proof} Set $r'_0:=\min\{r_0,\tilde{r}\}$. Let $\boldsymbol{\varphi}$ and $\boldsymbol{\psi}$ be two non-degenerate elements of $\Conf_g(W;r'_0)$. Lemma \ref{rem-about-part-of-unity} implies that $I_{|\boldsymbol{\varphi}|^2},1/I_{|\boldsymbol{\varphi}|^2}\in S(1,g)$ and $\boldsymbol{\psi}/I_{|\boldsymbol{\varphi}|^2}\in\Conf_g(W;r'_0)$. In view of \eqref{compo-stft-with-adj-symp}, we infer
\begin{equation}\label{equ-for-com-mod-inv}
\VV_{\boldsymbol{\psi}/I_{|\boldsymbol{\varphi}|^2}} \VV^*_{\boldsymbol{\varphi}}\VV_{\boldsymbol{\varphi}}f= \VV_{\boldsymbol{\psi}/I_{|\boldsymbol{\varphi}|^2}}(I_{|\boldsymbol{\varphi}|^2}f)= \VV_{\boldsymbol{\psi}}f,\quad f\in\SSS'(W).
\end{equation}
Hence, if $f\in\SSS'(W)$ is such that $\VV_{\boldsymbol{\varphi}}f\in \widetilde{L}^{p,q}_{\eta}(W\times W,dv_gd\lambda)$, then Theorem \ref{com-adj-stft-sympl1} together with \eqref{equ-for-com-mod-inv} imply that $\VV_{\boldsymbol{\psi}}f\in \widetilde{L}^{p,q}_{\eta}(W\times W,dv_gd\lambda)$ and
$$
\|\VV_{\boldsymbol{\psi}}f\|_{\widetilde{L}^{p,q}_{\eta}(W\times W,dv_gd\lambda)}\leq C' \|\VV_{\boldsymbol{\varphi}} f\|_{\widetilde{L}^{p,q}_{\eta}(W\times W,dv_gd\lambda)}.
$$
This proves the independence of $\widetilde{\MM}^{p,q}_{\eta}$ of the non-degenerate element of $\Conf_g(W;r'_0)$ employed as well as the equivalence of the norms induced on $\widetilde{\MM}^{p,q}_{\eta}$. Now, the second mapping in \eqref{mappings-for-mod-spaces-sym-bbb} is well-defined and continuous in view of Theorem \ref{com-adj-stft-sympl1}. To prove this for the first mapping in \eqref{mappings-for-mod-spaces-sym-bbb}, pick a non-degenerate $\boldsymbol{\varphi}\in\Conf_g(W;r'_0)$. Then, in the same way as for \eqref{equ-for-com-mod-inv}, we infer $\VV_{\boldsymbol{\theta}}f=\VV_{\boldsymbol{\theta}/I_{|\boldsymbol{\varphi}|^2}} \VV^*_{\boldsymbol{\varphi}}\VV_{\boldsymbol{\varphi}}f$, $ f\in\SSS'(W)$. Consequently, Theorem \ref{com-adj-stft-sympl1} implies that the first mapping in \eqref{mappings-for-mod-spaces-sym-bbb} is also well-defined and continuous. It remains to prove that $\widetilde{\MM}^{p,q}_{\eta}$ is a Banach space. Let $f_j$, $j\in\ZZ_+$, be a Cauchy sequence in $\widetilde{\MM}^{p,q}_{\eta}$. There exists $G\in \widetilde{L}^{p,q}_{\eta}(W\times W,dv_gd\lambda)$ such that $\VV_{\boldsymbol{\varphi}}f_j\rightarrow G$ in $\widetilde{L}^{p,q}_{\eta}(W\times W,dv_gd\lambda)$. In view of \eqref{compo-stft-with-adj-symp}, we infer
$$
f_j=\VV^*_{\boldsymbol{\varphi}/I_{|\boldsymbol{\varphi}|^2}}\VV_{\boldsymbol{\varphi}}f_j \rightarrow \VV^*_{\boldsymbol{\varphi}/I_{|\boldsymbol{\varphi}|^2}}G\in \widetilde{\MM}^{p,q}_{\eta}\quad \mbox{in}\,\, \widetilde{\MM}^{p,q}_{\eta},
$$
which completes the proof.
\end{proof}
\begin{remark}\label{con-incl-of-s-in-genmodspace}
If $\eta_1$ and $\eta_2$ are uniformly admissible and $\eta_1/\eta_2\in L^{\infty}(W\times W)$, then
$$
\SSS(W)\subseteq \widetilde{\MM}^{p,q}_{\eta_2}(W)\subseteq \widetilde{\MM}^{p,q}_{\eta_1} \subseteq \SSS'(W)\quad \mbox{continuously, for all}\,\, p,q\in[1,\infty].
$$
\end{remark}
\begin{remark}\label{rem-for-mod-spa-clas-and-geo-sympl-trs}
Let $p,q\in[1,\infty]$ and let $\eta:\RR^{4n}\rightarrow (0,\infty)$ be a measurable function which is moderate with respect to the Beurling weight $(1+|\cdot|)^{\tau}$, for some $\tau\geq 0$. Similarly as in the case of the classical modulation space $M^{p,q}_{\eta}(\RR^{2n})$, one can define the modulation space $\widetilde{M}^{p,q}_{\eta}(\RR^{2n})$ associated to $\widetilde{L}^{p,q}_{\eta}(\RR^{2n}\times \RR^{2n})$ (see \cite[Section 4.2]{D-P-P-V}); of course $M^{p,p}_{\eta}(\RR^{2n})= \widetilde{M}^{p,p}_{\eta}(\RR^{2n})$. If $g$ is the standard Euclidean metric on $\RR^{2n}$, setting $\widetilde{\eta}(X,\Xi):=\eta(X,-\sigma\Xi)$, $X,\Xi\in\RR^{2n}$, Remark \ref{rem-for-euc-metric-stand} and Remark \ref{uni-adm-for-stand-euc-metr} give
$$
\widetilde{\MM}^{p,q}_{\eta}(\RR^{2n})= \widetilde{M}^{p,q}_{\widetilde{\eta}}(\RR^{2n}),\, p,q\in[1,\infty],\quad \mbox{and}\quad \widetilde{\MM}^{p,p}_{\eta}(\RR^{2n})= M^{p,p}_{\widetilde{\eta}}(\RR^{2n}),\, p\in[1,\infty].
$$
\end{remark}
\section{Almost diagonalisation}\label{sec dia}
\subsection{Structural properties of $S(M,g)$}\label{structur}
As a first benefit of the above construction, we show that the Weyl-H\"ormander symbol class $S(M,g)$ can be represented as an intersection of weighted geometric modulation spaces. Besides being of independent interest, this is one of the key ingredients in the proof of our main result. We start by introducing the following notation. Notice that the function $W\times W\rightarrow (0,\infty)$, $(X,\Xi)\mapsto 1+g^{\sigma}_X(\Xi)$, is uniformly admissible with slow variation constant $r_0$. For $1\leq p\leq \infty$ and $s\geq 0$, define
$$
u_{p,s}:W\times W\rightarrow (0,\infty),\quad u_{p,s}(X,\Xi):=|g_X|^{\frac{1}{2}(1-\frac{1}{p})}(1+g^{\sigma}_X(\Xi))^s.
$$
In view of Remark \ref{rem-for-adm-func-for-fromad} and Lemma \ref{admisibil-weig-mul}, $u_{p,s}$ is uniformly admissible with slow variation constant $r_0$. Consequently, if $M$ is a $g$-admissible weight with slow variation constant $r$, Remark \ref{rem-for-adm-func-for-fromad} and Lemma \ref{admisibil-weig-mul} verify that $(X,\Xi)\mapsto u_{p,s}(X,\Xi)/ M(X)$ is uniformly admissible with slow variation constant $\min\{r,r_0\}$. Notice that
\begin{equation}\label{inclusion-for-mod-space-for-weight}
\widetilde{\MM}^{\infty,p}_{u_{p,s_2}/M}\subseteq \widetilde{\MM}^{\infty,p}_{u_{p,s_1}/M}\,\, \mbox{continuously, when}\,\, s_2\geq s_1.
\end{equation}
\begin{proposition}\label{rep-res-hor}
Let $M$ be a $g$-admissible weight and $1\leq p\leq \infty$. Then
\begin{equation}\label{ident-hormand-class-modspac-prl}
S(M,g)=\lim_{\substack{\longleftarrow\\ s\rightarrow \infty}}\widetilde{\MM}^{\infty,p}_{u_{p,s}/M}\quad \mbox{topologically},
\end{equation}
where the linking mappings in the projective limit are the canonical inclusions \eqref{inclusion-for-mod-space-for-weight}.
\end{proposition}
\begin{remark}\label{new151}
In view of Remark \ref{rem-for-mod-spa-clas-and-geo-sympl-trs}, this generalises \cite[Lemma 6.1]{gro-rz}.
\end{remark}
\begin{proof} For simplicity, denote the projective limit on the right in \eqref{ident-hormand-class-modspac-prl} by $\mathfrak{M}_p$. In view of Remark \ref{con-incl-of-s-in-genmodspace}, $\SSS(W)\subseteq \mathfrak{M}_p\subseteq \SSS'(W)$ continuously. Let $r'_0\leq \min\{r_0,1\}$ be a slow variation constant for $u_{p,s}/M$, $s\geq 0$, and let $\boldsymbol{\varphi}$ be the smooth non-degenerate element of $\Conf_g(W;r'_0)$ constructed in Example \ref{exi-of-good-par-off} $(ii)$; recall that $I_{\boldsymbol{\varphi}}(Y)=1$, $Y\in W$, and $\supp\varphi_X\subseteq U_{X,r'_0}$, $X\in W$, where we set $\varphi_X:=\boldsymbol{\varphi}(X)$, $X\in W$. We employ $\boldsymbol{\varphi}$ to define the norm on $\widetilde{\MM}^{\infty,p}_{u_{p,s}/M}$, $s\geq 0$, $p\in[1,\infty]$. We claim that $\mathfrak{M}_{p_2}\subseteq \mathfrak{M}_{p_1}$ continuously when $1\leq p_1\leq p_2\leq \infty$. To see this, assume that $1\leq p_1< p_2<\infty$; the proof of the case $p_2=\infty$ is analogous. Let $f\in \mathfrak{M}_{p_2}$. Employing H\"older's inequality with $\widetilde{p}=p_2/p_1$ and $\widetilde{q}=p_2/(p_2-p_1)$ together with the fact $|g_X||g^{\sigma}_X|=1$, we infer (for any $s\geq 0$)
\begin{align*}
\frac{|g_X|^{\frac{1}{2}(1-\frac{1}{p_1})}}{M(X)}&\left(\int_W |\VV_{\boldsymbol{\varphi}}f(X,\Xi)|^{p_1}(1+g^{\sigma}_X(\Xi))^{sp_1}d\Xi\right)^{1/p_1}\\
&\leq\frac{|g_X|^{\frac{1}{2}(1-\frac{1}{p_2})}}{M(X)}\left(\int_W |\VV_{\boldsymbol{\varphi}}f(X,\Xi)|^{p_2} (1+g^{\sigma}_X(\Xi))^{(s+n+1)p_2}d\Xi\right)^{1/p_2}\\
&\quad\cdot\left(\int_W (1+g^{\sigma}_X(\Xi))^{-(n+1)p_1p_2/(p_2-p_1)} |g^{\sigma}_X|^{1/2}d\Xi\right)^{(p_2-p_1)/(p_1p_2)}.
\end{align*}
Since the very last integral is uniformly bounded by a single constant for all $X\in W$, the claim follows. Consequently, to prove the claim in the proposition, it suffices to show that
\begin{align}\label{inc-to-be-proved-for-the-por-thssm}
S(M,g)\subseteq \mathfrak{M}_{\infty}\quad \mbox{and}\quad \mathfrak{M}_1\subseteq S(M,g),\quad \mbox{continuously}.
\end{align}
\indent To prove the first inclusion in \eqref{inc-to-be-proved-for-the-por-thssm}, let $f\in S(M,g)$ and $k\in \ZZ_+$ be arbitrary. Write
$$
\VV_{\boldsymbol{\varphi}}f(X,\Xi)=\int_W e^{-2\pi i[\Xi,Y]}f(Y)\varphi_X(Y)dY.
$$
For every $X,\Xi\in W$ fixed, there exists $\theta=\theta(X,\Xi)\in W$ such that $g_X(\theta)=1$ and $[\Xi,\theta]=g^{\sigma}_X(\Xi)^{1/2}$. Notice that
$$
(1-(2\pi i)^{-1}\partial_{\theta;Y})^{2k}e^{-2\pi i[\Xi,Y]}=(1+g^{\sigma}_X(\Xi)^{1/2})^{2k}e^{-2\pi i[\Xi,Y]}.
$$
We estimate as follows:
\begin{align*}
|\VV_{\boldsymbol{\varphi}}&f(X,\Xi)|u_{\infty,k}(X,\Xi)/M(X)\\
&\leq C'_1|g_X|^{1/2}\sum_{k'+k''\leq 2k}\int_W M(Y)^{-1}|\partial_{\theta}^{k'}f(Y)||\partial_{\theta;Y}^{k''}\varphi_X(Y)|dY\\
&\leq C'_1\|f\|^{(2k)}_{S(M,g)}|g_X|^{1/2}\sum_{k'+k''\leq 2k}\int_W g_Y(\theta)^{k'/2} |\partial_{\theta;Y}^{k''}\varphi_X(Y)| dY\\
&\leq C'_2\|f\|^{(2k)}_{S(M,g)}\|\varphi_X\|^{(N)}_{g_X,U_{X,r'_0}}\int_W (1+g^{\sigma}_X(Y-U_{X,r'_0}))^{-n-1}|g_X|^{1/2}dY,
\end{align*}
for sufficiently large $N\in\ZZ_+$. Since $1+g_X(X-Y)\leq C'_3(1+g^{\sigma}_X(Y-U_{X,r'_0}))$, the very last integral is uniformly bounded by a single constant for all $X\in W$ and the validity of the first part of \eqref{inc-to-be-proved-for-the-por-thssm} follows.\\
\indent We turn our attention to the second inclusion in \eqref{inc-to-be-proved-for-the-por-thssm}. Let $f\in \mathfrak{M}_1$. For $T_1,\ldots, T_k\in W\backslash\{0\}$, $k\in\NN$, we infer
$$
\mathcal{F}_{\sigma}(\partial_{T_1}\ldots\partial_{T_k}(f\varphi_X))=(2\pi i)^k[\cdot,T_1]\cdot\ldots \cdot[\cdot,T_k]\mathcal{F}_{\sigma}(f\varphi_X)\in L^1(W),\;\; \mbox{a.a.}\; X,
$$
and consequently $\partial_{T_1}\ldots\partial_{T_k}(f\varphi_X)\in\mathcal{F}_{\sigma}L^1(W)\subseteq \mathcal{C}_0(W)$, a.a. $X$. Thus, in view of \eqref{est-from-belo-and-above-part-unit}, $\partial_{T_1}\ldots \partial_{T_k}(f\psi)\in\mathcal{C}(W)$, for all $\psi\in\DD(W)$ (since the interiors of the sets in \eqref{est-from-belo-and-above-part-unit} cover $W$ when $X$ varies in $W\backslash(\mbox{a nullset})$). We deduce $f\in\mathcal{C}^{\infty}(W)$. Let $k\in \NN$ and $l\leq k$ be arbitrary but fixed and set $N:=(N_0+1)(n+1)+N_0n+N_0k/2$. Since $\supp\varphi_X\subseteq U_{X,r'_0}$, for $T_1,\ldots,T_l\in W\backslash\{0\}$, we infer
\begin{align}
|\partial_{T_1}\ldots\partial_{T_l}f(Y)|/M(Y)&\leq C'_1\int_W |\partial_{T_1;Y}\ldots\partial_{T_l;Y}(f(Y)\varphi_X(Y))|/M(X)dv_g(X)\nonumber\\
&= C'_1\int_W \frac{|\partial_{T_1;Y}\ldots\partial_{T_l;Y}(f(Y)\varphi_X(Y))|} {(1+g^{\sigma}_X(Y-U_{X,r'_0}))^NM(X)} dv_g(X).\label{est-for-the-opp-incls}
\end{align}
Employing $\mathcal{F}_{\sigma}\mathcal{F}_{\sigma}=\operatorname{Id}$ and \eqref{ineq-for-metric-p-1}, we estimate as follows:
\begin{align*}
|\partial_{T_1;Y}&\ldots\partial_{T_l;Y}(f(Y)\varphi_X(Y))|/M(X)\\
&= \frac{(2\pi)^l}{M(X)}\left|\int_W e^{-2\pi i[Y,\Xi]}\mathcal{F}_{\sigma}(f\varphi_X)(\Xi)\prod_{j=1}^l[\Xi,T_j]d\Xi\right|\\
&\leq \frac{(2\pi)^l\prod_{j=1}^lg_Y(T_j)^{1/2}}{M(X)}\int_W |\VV_{\boldsymbol{\varphi}}f(X,\Xi)|g^{\sigma}_Y(\Xi)^{l/2}d\Xi\\
&\leq \frac{C'_2(1+g^{\sigma}_X(Y-U_{X,r'_0}))^{kN_0/2} \prod_{j=1}^lg_Y(T_j)^{1/2}}{M(X)}\int_W |\VV_{\boldsymbol{\varphi}}f(X,\Xi)|g^{\sigma}_X(\Xi)^{l/2}d\Xi.
\end{align*}
Plugging this estimate into \eqref{est-for-the-opp-incls}, we deduce
$$
\frac{|\partial_{T_1}\ldots \partial_{T_l}f(Y)|}{M(Y)\prod_{j=1}^lg_Y(T_j)^{1/2}}\leq C'_3\|f\|_{\widetilde{\MM}^{\infty,1}_{u_{1,k/2}/M}} \int_W \frac{dv_g(X)}{(1+g^{\sigma}_X(Y-U_{X,r'_0}))^{(N_0+1)(n+1)+N_0n}}.
$$
In view of \eqref{ineq-for-metric-p-3-1}, the integral is uniformly bounded for all $Y\in W$ by a constant and the proof of the second part of \eqref{inc-to-be-proved-for-the-por-thssm} is complete.
\end{proof}
\subsection{Characterisation of $S(M,g)$}\label{char-M,g}
Before we state the main result of the article, we introduce the following notation. As standard, we denote by $\Sp(W)$ the Lie group of symplectic maps on $W$ and we denote by $\Mp(W)$ the metaplectic group as defined by Weil \cite{weil}; i.e. it is a two-fold cover of $\Sp(W)$ (the Weil representation).\footnote{Some authors construct $\Mp(W)$ as a circle cover of $\Sp(W)$.} Let $(\cdot,\cdot)_W$ be an inner product on $W$ and let $L:W\rightarrow W'$ be the induced isomorphism. Notice that $\langle LX,Y\rangle=(X,Y)_W$, for all $X,Y\in W$. Denote by $\Sym(W,(\cdot,\cdot)_W)$ the vector space of all symmetric operators on the Hilbert space $(W,(\cdot,\cdot)_W)$ and let $\Sym_+(W,(\cdot,\cdot)_W)$ be the subset of $\Sym(W,(\cdot,\cdot)_W)$ consisting of all symmetric positive-definite operators; $\Sym_+(W,(\cdot,\cdot)_W)$ is an open convex cone in $\Sym(W,(\cdot,\cdot)_W)$. For any $A\in\Sym_+(W,(\cdot,\cdot)_W)$ we denote by $A^{1/2}$ the unique symmetric positive-definite square root of $A$. Recall that the mapping $\Sym_+(W,(\cdot,\cdot)_W)\rightarrow \Sym_+(W,(\cdot,\cdot)_W)$, $A\mapsto A^{1/2}$, is smooth. If $\gamma$ is any positive definite quadratic form on $W$ with $Q:W\rightarrow W'$ being the induced linear map, then $L^{-1}Q\in\Sym_+(W,(\cdot,\cdot)_W)$ and hence its square root $(L^{-1}Q)^{1/2}$ is a well-defined element of $\Sym_+(W,(\cdot,\cdot)_W)$.
\begin{lemma}\label{lema-for-sym-root-quadr}${}$
\begin{itemize}
\item[$(i)$] Let $A:W\rightarrow W'$ be an isomorphism which satisfies ${}^tA=A$. Then $\sigma^{-1}A$ is symplectic if and only if $\sigma^{-1}A\sigma^{-1}A=-I$.
\item[$(ii)$] Let $(\cdot,\cdot)_W$ be an inner product on $W$ with $L:W\rightarrow W'$ the induced isomorphism. Assume that $\sigma^{-1}L$ is symplectic (or, equivalently, $\sigma^{-1}L\sigma^{-1}L=-I$ in view of $(i)$). Let $\gamma$ be a positive-definite quadratic form on $W$ with $Q:W\rightarrow W'$ the induced isomorphism. Then
\begin{equation}\label{equ-for-square-root-of-sympl-dual}
(L^{-1}Q^{\sigma})^{1/2}=-L^{-1}\sigma(L^{-1}Q)^{-1/2}L^{-1}\sigma.
\end{equation}
Furthermore, for all $X,Y\in W$, it holds that
\begin{equation}\label{equ-for-movin-accrr-sym-form-quad-root}
[(L^{-1}Q)^{1/2}X,Y]=[X,(L^{-1}Q^{\sigma})^{-1/2}Y]\quad \mbox{and}\quad [(L^{-1}Q^{\sigma})^{1/2}X,Y]=[X,(L^{-1}Q)^{-1/2}Y].
\end{equation}
\end{itemize}
\end{lemma}
\begin{proof} To verify $(i)$, notice that
$$
[\sigma^{-1}AX,\sigma^{-1}AY]=-\langle A\sigma^{-1}AX,Y\rangle=-[\sigma^{-1}A\sigma^{-1}AX,Y],\quad X,Y\in W.
$$
The last term is equal to $[X,Y]$, for all $X,Y\in W$, if and only if $\sigma^{-1}A\sigma^{-1}A=-I$.\\
\indent We turn our attention to $(ii)$. It is straightforward to verify that the right-hand side of \eqref{equ-for-square-root-of-sympl-dual} is symmetric positive-definite operator on $W$ and that the composition of this operator with itself is $L^{-1}Q^{\sigma}$ (recall $Q^{\sigma}=-\sigma Q^{-1}\sigma$). Consequently, \eqref{equ-for-square-root-of-sympl-dual} holds true. To prove \eqref{equ-for-movin-accrr-sym-form-quad-root} it suffices to show the second equality as then the first will follow from this and the fact $(Q^{\sigma})^{\sigma}=Q$. For $X,Y\in W$, in view of $(i)$, we infer
\begin{align*}
[(L^{-1}Q^{\sigma})^{1/2}X,Y]&=-\langle (L^{-1}Q)^{-1/2}L^{-1}\sigma X, \sigma L^{-1} \sigma Y\rangle\\
&=((L^{-1}Q)^{-1/2}L^{-1}\sigma X,Y)_W= (L^{-1}\sigma X,(L^{-1}Q)^{-1/2}Y)_W\\
&=[X,(L^{-1}Q)^{-1/2}Y].
\end{align*}
\end{proof}
\begin{remark}
If $(\cdot,\cdot)_V$ is any inner product on $V$ with $(\cdot,\cdot)_{V'}$ being the dual inner product on $V'$ then
$$
((x,\xi),(y,\eta))_W:=(x,y)_V+(\xi,\eta)_{V'},\quad (x,\xi),(y,\eta)\in W,
$$
satisfies the assumption in Lemma \ref{lema-for-sym-root-quadr} $(ii)$.
\end{remark}
\begin{remark}\label{rem-for-det-of-root-of-qua}
If $(\cdot,\cdot)_W$ is such that $\sigma^{-1}L:W\rightarrow W$ has determinant $1$, then $\det (L^{-1}Q_X)=|g_X|$. To see this, let $E_j$, $j=1,\ldots,2n$, be a symplectic basis on $W$ with $E'_j$, $j=1,\ldots,2n$, being the dual basis on $W'$ and denote by $P:W\rightarrow W'$ the isomorphism that sends $E_j$ to $E'_j$, $j=1,\ldots,2n$; clearly ${}^tP=P$. Then
\begin{align*}
|g_X|&=\det((\langle E_j,Q_XE_k\rangle)_{j,k})= \det((\langle E'_j,P^{-1}Q_XE_k\rangle)_{j,k})=\det(P^{-1}Q_X)\\
&= \det(P^{-1}\sigma)\det(\sigma^{-1}L)\det(L^{-1}Q_X)=\det(L^{-1}Q_X);
\end{align*}
the very last equality follows from the fact $\det(P^{-1}\sigma)=1$ since $P^{-1}\sigma\in\Sp(W)$. Similarly, in this case we also have $|g^{\sigma}_X|=\det(L^{-1}Q^{\sigma}_X)$. In particular, $\det (L^{-1}Q_X)=|g_X|$ and $\det(L^{-1}Q^{\sigma}_X)=|g^{\sigma}_X|$ when $\sigma^{-1}L$ is symplectic, i.e. when $\sigma^{-1}L\sigma^{-1}L=-I$ (cf. Lemma \ref{lema-for-sym-root-quadr} $(i)$).
\end{remark}
Given an inner product $(\cdot,\cdot)_W$ on $W$ with $L:W\rightarrow W'$ the induced isomorphism, for each $X\in W$, we denote by $\Psi^{g,L}_X$ the topological isomorphism
\begin{align*}
\Psi^{g,L}_X:\SSS(W)\rightarrow \SSS(W),\quad (\Psi^{g,L}_X\varphi)(Y)=\varphi((L^{-1}Q_X)^{-1/2}Y),\; Y\in W,
\end{align*}
and we extend it by duality to the topological isomorphism
\begin{align}\label{change-of-var-for-dualit}
\Psi^{g,L}_X:\SSS'(W)\rightarrow \SSS'(W),\quad \langle\Psi^{g,L}_Xf,\varphi\rangle=|\det (L^{-1}Q_X)|^{1/2} \langle f,\varphi\circ(L^{-1}Q_X)^{1/2}\rangle.
\end{align}
As standard, we denote by $\pi(x,\xi)$, $(x,\xi)\in W$, the time-frequency shift $\pi(x,\xi)f=e^{2\pi i \langle \xi,\cdot\rangle} f(\cdot-x)$, $f\in\SSS'(V)$. For $\chi_1,\chi_2\in\SSS(V)$, $W(\chi_1,\chi_2)$ stands for the Wigner function
$$
W(\chi_1,\chi_2)(x,\xi)=\int_V e^{-2\pi i \langle \xi,y\rangle}\chi_1(x+y/2)\overline{\chi_2(x-y/2)}dy,\quad (x,\xi)\in W;
$$
recall that $W(\chi_1,\chi_2)\in \SSS(W)$.\\
\indent We are ready to state and prove the main result of the article.
\begin{theorem}\label{main-theorem-dia}
Let $a\in \SSS'(W)$ and $M$ be a $g$-admissible weight with slow variation constant $r$. Let $(\cdot,\cdot)_W$ be an inner product on $W$ whose induced isomorphism $L:W\rightarrow W'$ is such that $\sigma^{-1}L$ is a symplectic map and let $\Psi^{g,L}_X$, $X\in W$, be the isomorphisms defined in \eqref{change-of-var-for-dualit}. Let $\{\theta_X\,|\, X\in W\}\subseteq \mathcal{O}_{\mathcal{M}}(W)$ and $\chi\in\SSS(V)$. Consider the following conditions:
\begin{itemize}
\item[$(i)$] $a\in S(M,g)$;
\item[$(ii)$] for each $N\in\NN$, the function
\begin{multline*}
(X,\Xi)\mapsto M((X+\Xi)/2)^{-1}(1+g_{\frac{X+\Xi}{2}}(X-\Xi))^N\\
\cdot\left\langle \left(\Psi^{g,L}_{\frac{X+\Xi}{2}}(a\theta_{\frac{X+\Xi}{2}})\right)^w \pi\left((L^{-1}Q_{\frac{X+\Xi}{2}})^{1/2}X\right)\chi, \overline{\pi\left((L^{-1}Q_{\frac{X+\Xi}{2}})^{1/2}\Xi\right)\chi}\right\rangle
\end{multline*}
belongs to $L^{\infty}(W\times W)$.
\end{itemize}
Set
\begin{equation}\label{def-for-var-for-non-deg-ele-confmainthe}
\varphi_X(Y):=\overline{\theta_X(Y)} W(\chi,\chi)((L^{-1}Q_X)^{1/2}(Y-X)),\quad X,Y\in W.
\end{equation}
If $\boldsymbol{\varphi}:W\mapsto \SSS(W)$, $\boldsymbol{\varphi}(X)=\varphi_X$, belongs to $\Conf_g(W;\min\{r_0,r\})$, then $(i)$ implies $(ii)$. If, furthermore, $\boldsymbol{\varphi}$ is non-degenerate, then $(ii)$ implies $(i)$.
\end{theorem}
\begin{remark}
Since $\theta_X\in \mathcal{O}_{\mathcal{M}}(W)$, $X\in W$, we have $a\theta_X\in \SSS'(W)$, for all $a\in \SSS'(W)$ and $X\in W$. Thus, the right hand side of $(ii)$ is well-defined for each $X,\Xi\in W$.
\end{remark}
\begin{remark}
In Lemma \ref{fam-sat-con-maithforvar} below we give a class of families $\{\theta_X\,|\, X\in W\}$ which satisfy all of the assumptions in Theorem \ref{main-theorem-dia} for all $\chi\in\SSS(V)$ such that $W(\chi,\chi)(0)\neq 0$ (see also Remark \ref{rem-for-exa-famisatisallofassummainthe}).
\end{remark}
\begin{proof} Set $\tilde{Q}_X:=L^{-1}Q_X$ and $\tilde{Q}^{\sigma}_X=L^{-1}Q^{\sigma}_X$, $X\in W$. Assume that $\boldsymbol{\varphi}\in\Conf_g(W;\min\{r_0,r\})$. First we show that for each $a\in \SSS'(W)$, the function
\begin{equation}\label{fun-for-measur}
W\times W\rightarrow \CC,\quad (X,\Xi)\mapsto \left\langle\left(\Psi^{g,L}_{\frac{X+\Xi}{2}}(a\theta_{\frac{X+\Xi}{2}})\right)^w \pi\left(\tilde{Q}^{1/2}_{\frac{X+\Xi}{2}}X\right)\chi, \overline{\pi\left(\tilde{Q}^{1/2}_{\frac{X+\Xi}{2}}\Xi\right)\chi}\right\rangle,
\end{equation}
is measurable. Assume first that $a\in\SSS(W)$. Then \cite[Lemma 3.1]{Gr2} (especially its proof; cf. \cite[Equation (4.4)]{gro-rz}) gives
\begin{equation}\label{eq2-ss11}
\langle a^w\pi(X)\chi,\overline{\pi(\Xi)\chi}\rangle=e^{if(X,\Xi)}\int_W e^{-2\pi i[Y,X-\Xi]}a(Y) W(\chi,\chi)\left(Y-\frac{X+\Xi}{2}\right) dY,
\end{equation}
where $f$ is a real-valued smooth function on $W\times W$ which does not depend on $a$ and $\chi$. In view of Remark \ref{rem-for-det-of-root-of-qua} and Lemma \ref{lema-for-sym-root-quadr} $(ii)$, we infer ($W(\chi,\chi)$ is real-valued)
\begin{align}
&\left\langle\left(\Psi^{g,L}_{\frac{X+\Xi}{2}}(a\theta_{\frac{X+\Xi}{2}})\right)^w \pi\left(\tilde{Q}^{1/2}_{\frac{X+\Xi}{2}}X\right)\chi, \overline{\pi\left(\tilde{Q}^{1/2}_{\frac{X+\Xi}{2}}\Xi\right)\chi}\right\rangle\nonumber\\
&=e^{if\left(\tilde{Q}^{1/2}_{\frac{X+\Xi}{2}}X,\tilde{Q}^{1/2}_{\frac{X+\Xi}{2}}\Xi\right)} |g_{\frac{X+\Xi}{2}}|^{1/2}\int_W e^{-2\pi i\left[Y\, , \,\tilde{Q}^{\sigma\, -1/2}_{\frac{X+\Xi}{2}}\tilde{Q}^{1/2}_{\frac{X+\Xi}{2}}(X-\Xi)\right]}a(Y) \overline{\varphi_{\frac{X+\Xi}{2}}(Y)}dY\nonumber\\
&=e^{if\left(\tilde{Q}^{1/2}_{\frac{X+\Xi}{2}}X,\tilde{Q}^{1/2}_{\frac{X+\Xi}{2}}\Xi\right)} |g_{\frac{X+\Xi}{2}}|^{1/2} \left\langle a,e^{-2\pi i\left[\,\cdot\, ,\,\tilde{Q}^{\sigma\, -1/2}_{\frac{X+\Xi}{2}}\tilde{Q}^{1/2}_{\frac{X+\Xi}{2}}(X-\Xi)\right]} \overline{\varphi_{\frac{X+\Xi}{2}}}\right\rangle.\label{equ-for-the-mapping-for-main-res}
\end{align}
Since $X\mapsto \tilde{Q}_X^{1/2}$ and $X\mapsto \tilde{Q}^{\sigma\, -1/2}_X$ are measurable maps from $W$ into $\Sym_+(W,(\cdot,\cdot)_W)$ (recall, the square root is smooth on $\Sym_+(W,(\cdot,\cdot)_W)$), this integral representation together with the fact that $W\rightarrow \SSS(W)$, $X\mapsto \varphi_X$, is strongly measurable implies that the function \eqref{fun-for-measur} is measurable when $a\in \SSS(W)$. Now, the case when $a\in \SSS'(W)$ follows from the fact that $\SSS(W)$ is sequentially dense in $\SSS'(W)$. In view of \eqref{equ-for-the-mapping-for-main-res}, this also verifies the following key identity:
\begin{multline}\label{ident-sympstft-actpsid}
\left|\left\langle\left(\Psi^{g,L}_{\frac{X+\Xi}{2}}(a\theta_{\frac{X+\Xi}{2}})\right)^w \pi\left(\tilde{Q}^{1/2}_{\frac{X+\Xi}{2}}X\right)\chi, \overline{\pi\left(\tilde{Q}^{1/2}_{\frac{X+\Xi}{2}}\Xi\right)\chi}\right\rangle\right|\\
=|g_{\frac{X+\Xi}{2}}|^{1/2}\left|\VV_{\boldsymbol{\varphi}}a\left(\frac{X+\Xi}{2}, \tilde{Q}^{\sigma\, -1/2}_{\frac{X+\Xi}{2}}\tilde{Q}^{1/2}_{\frac{X+\Xi}{2}}(\Xi-X)\right)\right|,\;\; X,\Xi\in W,
\end{multline}
for all $a\in \SSS'(W)$ (\eqref{ident-sympstft-actpsid} transfers the problem from $\Psi$DOs to the theory of the generalised modulation spaces which we developed in the previous sections). Notice that $g^{\sigma}_X(\tilde{Q}^{\sigma\, -1/2}_X\tilde{Q}^{1/2}_X\Xi)=g_X(\Xi)$, $X,\Xi\in W$. Consequently, the condition $(ii)$ is equivalent to the following: for every $N\in\NN$ there exist $C_N>0$ and a nullset $K_N\subseteq W\times W$ such that
\begin{multline}\label{condi-equiv-toc-in-theore}
\left|\VV_{\boldsymbol{\varphi}}a\left(\frac{X+\Xi}{2}, \tilde{Q}^{\sigma\, -1/2}_{\frac{X+\Xi}{2}}\tilde{Q}^{1/2}_{\frac{X+\Xi}{2}}(\Xi-X)\right)\right|\\
\leq \frac{C_NM((X+\Xi)/2)}{|g_{\frac{X+\Xi}{2}}|^{1/2} \left(1+g^{\sigma}_{\frac{X+\Xi}{2}}\left(\tilde{Q}^{\sigma\, -1/2}_{\frac{X+\Xi}{2}} \tilde{Q}^{1/2}_{\frac{X+\Xi}{2}}(\Xi-X)\right)\right)^N},\;\; (X,\Xi)\in (W\times W)\backslash K_N.
\end{multline}
Consider the bijection $A:W\times W\rightarrow W\times W$, $A(Z_1,Z_2)=(Z_1,\tilde{Q}^{\sigma\, -1/2}_{Z_1}\tilde{Q}^{1/2}_{Z_1}Z_2)$ with inverse $A^{-1}(Z_1,Z_2)=(Z_1,\tilde{Q}^{-1/2}_{Z_1}\tilde{Q}^{\sigma\, 1/2}_{Z_1}Z_2)$. Both $A$ and $A^{-1}$ are measurable maps. We claim that the images of every nullset $\tilde{K}\subseteq W\times W$ under $A$ and $A^{-1}$ are again nullsets. To see this, pick a Borel nullset $K\supseteq \tilde{K}$ and denote by $\kappa_K$ the characteristic function of $K$. Since $\kappa_K\circ A^{-1}$ is the characteristic function of $A(K)$, it follows that $\kappa_K\circ A^{-1}$ is measurable. Remark \ref{rem-for-det-of-root-of-qua}, Fubini's theorem and the fact $|g_X||g^{\sigma}_X|=1$ give
\begin{align*}
(\lambda\times\lambda)(A(K))&=\int_W\left(\int_W \kappa_K(A^{-1}(Z_1,Z_2)) dZ_2\right)dZ_1\\
&= \int_W|g_{Z_1}|\left(\int_W \kappa_K(Z_1,Z_2) dZ_2\right)dZ_1=\int_K |g_{Z_1}|dZ_1dZ_2=0.
\end{align*}
Consequently, $A(\tilde{K})$ is a nullset; the proof that $A^{-1}(\tilde{K})$ is a nullset is analogous. Thus, in view of \eqref{condi-equiv-toc-in-theore}, the condition $(ii)$ is equivalent to
$$
(Z_1,Z_2)\mapsto M(Z_1)^{-1}|g_{Z_1}|^{1/2}(1+g^{\sigma}_{Z_1}(Z_2))^N |\VV_{\boldsymbol{\varphi}}a(Z_1,Z_2)|\quad \mbox{belongs to}\,\, L^{\infty}(W\times W),
$$
for all $N\in\NN$. Now, both claims in the theorem follow from Theorem \ref{mod-spa-donot-depend-on-partuni} and Proposition \ref{rep-res-hor}.
\end{proof}
\begin{remark}\label{rem-for-con-smooth-metr-fun}
If both $g$ and $\boldsymbol{\varphi}$ are of class $\mathcal{C}^k$, $0\leq k\leq\infty$, then \eqref{equ-for-the-mapping-for-main-res} implies that the function \eqref{fun-for-measur} is also of class $\mathcal{C}^k$, $0\leq k\leq\infty$, for each $a\in\SSS'(W)$ (see the comments before Remark \ref{rem-for-euc-metric-stand}).
\end{remark}
\begin{remark}\label{sim-con-iim-for-evewo}
Introducing the change of variables $(X,\Xi)\mapsto (X+\Xi,X-\Xi)$, one verifies that the condition $(ii)$ is equivalent to the following:
\begin{itemize}
\item[$(ii)'$] for each $N\in\NN$, the function
\begin{multline*}
(X,\Xi)\mapsto M(X)^{-1}(1+g_X(\Xi))^N\\
\cdot\left\langle \left(\Psi^{g,L}_X(a\theta_X)\right)^w \pi\left((L^{-1}Q_X)^{1/2}(X+\Xi)\right)\chi, \overline{\pi\left((L^{-1}Q_X)^{1/2}(X-\Xi)\right)\chi}\right\rangle
\end{multline*}
belongs to $L^{\infty}(W\times W)$.
\end{itemize}
\end{remark}
Employing analogous technique as in the proof of Theorem \ref{main-theorem-dia}, we can characterise the symbols in $\widetilde{\MM}^{p,p}_{\eta}$ in similar fashion.
\begin{theorem}\label{charc-of-symplec-mod-space-with-growth}
Let $L$ and $\Psi^{g,L}_X$, $X\in W$, be as in Theorem \ref{main-theorem-dia}. Let $\eta$ be a uniformly admissible weight with respect to $g$ with slow variation constant $r$ and let $1\leq p\leq \infty$. Let $\{\theta_X\,|\, X\in W\}\subseteq \mathcal{O}_{\mathcal{M}}(W)$ and $\chi\in\SSS(V)$. Consider the following conditions for $a\in\SSS'(W)$:
\begin{itemize}
\item[$(i)$] $a\in \widetilde{\MM}^{p,p}_{\eta}$;
\item[$(ii)$] the function
\begin{multline*}
(X,\Xi)\mapsto |g_{\frac{X+\Xi}{2}}|^{-\frac{1}{2}+\frac{3}{2p}}\, \eta\left(\frac{X+\Xi}{2}, (L^{-1}Q^{\sigma}_{\frac{X+\Xi}{2}})^{-1/2}(L^{-1}Q_{\frac{X+\Xi}{2}})^{1/2} (\Xi-X)\right)\\
\cdot\left\langle \left(\Psi^{g,L}_{\frac{X+\Xi}{2}}(a\theta_{\frac{X+\Xi}{2}})\right)^w \pi\left((L^{-1}Q_{\frac{X+\Xi}{2}})^{1/2}X\right)\chi, \overline{\pi\left((L^{-1}Q_{\frac{X+\Xi}{2}})^{1/2}\Xi\right)\chi}\right\rangle
\end{multline*}
belongs to $L^p(W\times W)$.
\end{itemize}
If $\boldsymbol{\varphi}:W\mapsto \SSS(W)$, $\boldsymbol{\varphi}(X)=\varphi_X$, with $\varphi_X$ given by \eqref{def-for-var-for-non-deg-ele-confmainthe}, belongs to $\Conf_g(W;\min\{r_0,r\}),$ then $(i)$ implies $(ii)$. If, furthermore, $\boldsymbol{\varphi}$ is non-degenerate, then $(ii)$ implies $(i)$.
\end{theorem}
\begin{proof} We claim that for each $a\in\SSS'(W)$, the function in $(ii)$ is measurable. Since \eqref{fun-for-measur} is measurable, we only need to prove that the part with the $\eta$ function is measurable; notice that it equals $\eta\circ A((X+\Xi)/2,\Xi-X)$ with $A$ being the measurable bijection we defined in the proof of Theorem \ref{main-theorem-dia}. To verify the measurability of the latter, it suffices to show that $(X,\Xi)\mapsto \eta\circ A(X,\Xi)$ is measurable. This follows from the fact that the image under $A^{-1}$ of every nullset is again a nullset (see the proof of Theorem \ref{main-theorem-dia}).\\
\indent When $p=\infty$, the proof of the theorem is the same as the proof of Theorem \ref{main-theorem-dia}. Assume that $1\leq p<\infty$. The identity \eqref{ident-sympstft-actpsid} together with a change of variables (cf. Remark \ref{rem-for-det-of-root-of-qua}) verifies that $(ii)$ is equivalent to $\VV_{\boldsymbol{\varphi}}a\in L^p_{\eta}(W\times W,dv_gd\lambda)$. Now, both assertions in the theorem follow from Theorem \ref{mod-spa-donot-depend-on-partuni}.
\end{proof}
One can simplify the condition $(ii)$ by introducing the change of variables from Remark \ref{sim-con-iim-for-evewo}.
\subsection{Examples and consequences of Theorem \ref{main-theorem-dia}}\label{ex111}
Our first goal is to give examples of $\chi\in\SSS(V)$ and $\{\theta_X\,|\, X\in W\}\subseteq \mathcal{O}_{\mathcal{M}}(W)$ which satisfy the assumptions of Theorem \ref{main-theorem-dia}. We start with the following technical result.
\begin{lemma}\label{lemma-for-wig-gen-dec}
Let $(\cdot,\cdot)_W$ be an inner product on $W$ with $L:W\rightarrow W'$ the induced isomorphism. Let $\psi\in\SSS(W)$ and, for each $X\in W$, define $\tilde{\psi}_X:=\psi((L^{-1}Q_X)^{1/2}(\cdot -X))$. Then $\tilde{\psi}_X\in\SSS(W)$, $X\in W$, and the map $W\rightarrow \SSS(W)$, $X\mapsto \tilde{\psi}_X$, is strongly measurable. Furthermore,
\begin{equation}\label{est-for-con-par-als}
\sup_{l\leq k}\sup_{\substack{X,Y\in W\\ T_1,\ldots, T_l\in W\backslash \{0\}}}\frac{|\partial_{T_1;Y}\ldots \partial_{T_l;Y}\tilde{\psi}_X(Y)|(1+g_X(X-Y))^N}{\prod_{j=1}^l g_X(T_j)^{1/2}}<\infty, \;\; k,N\in\NN.
\end{equation}
In particular, if $g$ is symplectic, then $W\rightarrow \SSS(W)$, $X\mapsto \tilde{\psi}_X$, belongs to $\Conf_g(W;r)$, for all $r\in(0,r_0]$. If, in addition, $\psi(0)\neq 0,$ then $X\mapsto \tilde{\psi}_X$ is non-degenerate.
\end{lemma}
\begin{proof} Denote by $|\cdot|$ the norm on $W$ induced by $(\cdot,\cdot)_W$ and set $\tilde{Q}_X:=L^{-1}Q_X$, $X\in W$. Clearly $\tilde{\psi}_X\in\SSS(W)$, $X\in W$, and the strong measurability of $W\rightarrow \SSS(W)$, $X\mapsto \tilde{\psi}_X$, follows from Lemma \ref{lemma-regularity-sta}. To prove the bounds \eqref{est-for-con-par-als}, let $T_1,\ldots,T_l\in W\backslash\{0\}$ and $N\in\NN$. Then
$$
|\partial_{T_1;Y}\ldots \partial_{T_l;Y}\tilde{\psi}_X(Y)|= |\psi^{(l)}(\tilde{Q}_X^{1/2}(Y-X);\tilde{Q}_X^{1/2}T_1,\ldots,\tilde{Q}^{1/2}_XT_l)|\leq \frac{C\prod_{j=1}^l|\tilde{Q}^{1/2}_XT_j|}{(1+|\tilde{Q}^{1/2}_X(Y-X)|)^N}.
$$
Since $|\tilde{Q}^{1/2}_XT|=g_X(T)^{1/2}$, $ T\in W$, the bounds \eqref{est-for-con-par-als} immediately follow.\\
\indent Assume now that $g$ is symplectic. The fact that $W\rightarrow \SSS(W)$, $X\mapsto \tilde{\psi}_X$, belongs to $\Conf_g(W;r)$, for all $r\in(0,r_0]$, is an immediate consequence of \eqref{est-for-con-par-als}. To prove that it is non-degenerate when $\psi(0)\neq 0$, pick $r_1,c_1>0$ such that $|\psi(X)|\geq c_1$, for all $|X|\leq r_1$. Set $\varepsilon:=\min\{r_0,r_1/\sqrt{C_0}\}$. Let $Y\in W$ be arbitrary but fixed. Then $|\psi(\tilde{Q}^{1/2}_X(Y-X))|\geq c_1$, $ X\in U_{Y,\varepsilon}$. To verify this, notice that when $X\in U_{Y,\varepsilon}$ we have
$$
|\tilde{Q}^{1/2}_X(Y-X)|^2=g_X(Y-X)\leq C_0g_Y(Y-X)\leq r_1^2,
$$
which implies the claim. In view of Remark \ref{equ-for-sym-metr-meas-lebesmes}, we infer
\begin{align*}
\int_W|\tilde{\psi}_X(Y)|^2dv_g(X)&\geq \int_{U_{Y,\varepsilon}} |\psi(\tilde{Q}^{1/2}_X(Y-X))|^2 |g_Y|^{1/2}dX\\
&\geq c_1^2\int_{U_{Y,\varepsilon}} |g_Y|^{1/2}dX= c'.
\end{align*}
Consequently, $X\mapsto \tilde{\psi}_X$ is non-degenerate.
\end{proof}
\begin{remark}\label{cont-metr-par-co}
If $g$ is continuous (not necessarily symplectic), it is straightforward to show that $W\rightarrow \SSS(W)$, $X\mapsto \tilde{\psi}_X$, is also continuous (by employing Taylor expansion).
\end{remark}
The following lemma gives an example of a family $\{\theta_X\,|\, X\in W\}$ which satisfies all of the assumptions in Theorem \ref{main-theorem-dia}.
\begin{lemma}\label{fam-sat-con-maithforvar}
Let $(\cdot,\cdot)_W$ and $L$ be as in Theorem \ref{main-theorem-dia}. Let $r>0$ and let $\theta_0\in \mathcal{C}^{\infty}([0,\infty))\backslash\{0\}$ be any non-negative and non-increasing function which satisfies $\supp \theta_0\subseteq [0,1]$. Set $\tilde{r}:=\min\{r_0,r\}$ and define $\theta_X(Y):=\theta_0(\tilde{r}^{-2}g_X(X-Y))$, $X,Y\in W$. For any $\chi\in\SSS(V)$ satisfying
\begin{equation}\label{equ-for-con-on-thefun-tobe-fam}
W(\chi,\chi)(0)=2^n\int_V \chi(x)\overline{\chi(-x)}dx\neq 0,
\end{equation}
the map $\boldsymbol{\varphi}:W\rightarrow \SSS(W)$, $\boldsymbol{\varphi}(X)=\varphi_X$, with $\varphi_X$ given by \eqref{def-for-var-for-non-deg-ele-confmainthe}, is a non-degenerate element of $\Conf_g(W;\min\{r_0,r\})$.
\end{lemma}
\begin{remark}\label{rem-for-exa-famisatisallofassummainthe}
Given a $g$-admissible weight $M$ (resp., a uniformly admissible weight $\eta$ with respect to $g$) with a slow variation constant $r$, the lemma shows that $\{\theta_X\,|\, X\in W\}$ satisfies all of the assumptions in Theorem \ref{main-theorem-dia} (resp., in Theorem \ref{charc-of-symplec-mod-space-with-growth}) for any $\chi\in\SSS(V)$ that satisfies \eqref{equ-for-con-on-thefun-tobe-fam}.
\end{remark}
\begin{proof} Denote by $|\cdot|$ the norm on $W$ induced by $(\cdot,\cdot)_W$. Set $\tilde{Q}_X:=L^{-1}Q_X$, $X\in W$. The map $W\rightarrow \SSS(W)$, $X\mapsto \theta_X$, is strongly measurable in view of Lemma \ref{lemma-regularity-sta1} and, similarly as in the proof of \cite[Theorem 2.2.7, p. 70]{lernerB}, one can show that it belongs to $\Conf_g(W;\tilde{r})$. Consequently, Lemma \ref{lemma-for-wig-gen-dec} implies that $\boldsymbol{\varphi}\in\Conf_g(W;\tilde{r})$. It remains to show that $\boldsymbol{\varphi}$ is non-degenerate. There is $0<\varepsilon'<1$ such that $\theta_0\geq \varepsilon'$ on $[0,\varepsilon']$. Pick $r_1,c_1>0$ such that $|W(\chi,\chi)(X)|\geq c_1$, for all $|X|\leq r_1$, and set $\varepsilon:=\sqrt{\varepsilon'/C_0}\min\{\tilde{r},r_1\}$. For $Y\in W$ fixed, we employ \eqref{ineq-for-metric-p-5} and estimate as follows:
\begin{align*}
\int_W|\varphi_X(Y)|^2dv_g(X)&\geq \int_{U_{Y,\varepsilon}}\theta_0(\tilde{r}^{-2}g_X(X-Y))^2 |W(\chi,\chi)(\tilde{Q}^{1/2}_X(Y-X))|^2 |g_X|^{1/2}dX\\
&\geq C^{-n}_0\int_{U_{Y,\varepsilon}} \theta_0(\tilde{r}^{-2}C_0g_Y(X-Y))^2|W(\chi,\chi)(\tilde{Q}^{1/2}_X(Y-X))|^2 |g_Y|^{1/2}dX.
\end{align*}
When $X\in U_{Y,\varepsilon}$, we have $\tilde{r}^{-2}C_0g_Y(X-Y)\leq \varepsilon'$ and
\begin{equation*}
|\tilde{Q}^{1/2}_X(Y-X)|=g_X(X-Y)^{1/2}\leq \sqrt{C_0}g_Y(X-Y)^{1/2}\leq r_1.
\end{equation*}
Consequently,
$$
\int_W|\varphi_X(Y)|^2dv_g(X)\geq C^{-n}_0\varepsilon'^2c_1^2\int_{U_{Y,\varepsilon}}|g_Y|^{1/2}dX= c'',\quad Y\in W,
$$
which verifies that $\boldsymbol{\varphi}$ is non-degenerate.
\end{proof}
A typical example for $\chi$ is to take a Gaussian. To be precise, let $e_1,\ldots,e_n$ be a basis for $V$ such that the volume with respect to the measure $dx$ of the parallelepiped formed with these vectors is $1$, and let $e'_1,\ldots,e'_n$ be the dual basis of $V'$. If we set
$$
\chi(x):=e^{-\pi\sum_{j=1}^n|\langle e'_j,x\rangle|^2},\quad x\in V,
$$
then $W(\chi,\chi)(x,\xi)=2^{n/2}e^{-2\pi\sum_{j=1}^n(|\langle e'_j,x\rangle|^2+|\langle \xi,e_j\rangle|^2)}$ (see \cite[Equation (4.20), p. 72]{Gr1}).\\
\indent Assume now that $g$ is a symplectic H\"ormander metric on $W$ and let $(\cdot,\cdot)_W$ and $L$ be as in Theorem \ref{main-theorem-dia}. As an immediate consequence of Lemma \ref{lemma-for-wig-gen-dec}, we can even take $\theta_X(Y)=1$ and $X\mapsto \varphi_X:=W(\chi,\chi)((L^{-1}Q_X)^{1/2}(\cdot-X))$ will still belong to $\Conf_g(W;r)$ for any $r\in(0,r_0]$ and any $\chi\in \SSS(V)$. Moreover, if $\chi$ satisfies \eqref{equ-for-con-on-thefun-tobe-fam}, then Lemma \ref{lemma-for-wig-gen-dec} also yields that $X\mapsto \varphi_X$ is non-degenerate. Furthermore, Lemma \ref{lema-for-sym-root-quadr} $(ii)$ implies that $\tilde{Q}_X^{1/2}:=(L^{-1}Q_X)^{1/2}\in \Sp(W)$, $X\in W$. Denote by $\Pi$ the surjective homomorphism $\Pi:\Mp(W)\rightarrow \Sp(W)$. The metaplectic covariance of the Weyl quantisation \cite[Theorem 7.13, p. 205]{deGosson} (cf. \cite[Theorem 4.3]{hormander}) yields that there is $\Phi^{g,L}_X\in\Mp(W)$ such that
$$
\Pi(\Phi^{g,L}_X)=\tilde{Q}^{-1/2}_X\quad \mbox{and}\quad (\Psi^{g,L}_Xa)^w=(\Phi^{g,L}_X)^*a^w\Phi^{g,L}_X,\quad X\in W,\, a\in\SSS'(W)
$$
(there are two such operators and they only differ by a sign). Let $\tau_X$, $X\in W$, be the unitary operator $\tau_{(x,\xi)}\chi(y)=e^{2\pi i\langle \xi,y-x/2\rangle}\chi(y-x)$, $\chi\in \SSS(V)$, $(x,\xi)\in W$. Clearly, $\tau_{(x,\xi)}=e^{-\pi i\langle \xi,x\rangle}\pi(x,\xi)$. Since $\Omega\tau_X\Omega^*=\tau_{\Pi(\Omega)X}$, $\Omega\in\Mp(W)$, $X\in W$
(see \cite[Theorem 7.13, p. 205]{deGosson}), we infer that for each $X,Y\in W$, $\pi(\tilde{Q}^{1/2}_Y X)$ is equal to $(\Phi^{g,L}_Y)^*\pi(X)\Phi^{g,L}_Y$ up to a constant of modulus $1$. Consequently, the above discussion gives
\begin{multline}\label{equ-sym-metaplect-operators-for-weylq}
\left|\left\langle \left(\Psi^{g,L}_{\frac{X+\Xi}{2}}a\right)^w \pi\left(\tilde{Q}_{\frac{X+\Xi}{2}}^{1/2}X\right)\chi, \overline{\pi\left(\tilde{Q}_{\frac{X+\Xi}{2}}^{1/2}\Xi\right)\chi}\right\rangle\right|\\
=\left|\left\langle a^w\pi(X) \Phi^{g,L}_{\frac{X+\Xi}{2}}\chi, \overline{\pi(\Xi)\Phi^{g,L}_{\frac{X+\Xi}{2}}\chi}\right\rangle\right|, \; X,\Xi\in W,
\end{multline}
for all $a\in\SSS'(W)$, $\chi\in\SSS(V)$. We proved the following consequence of Theorem \ref{main-theorem-dia}.
\begin{corollary}\label{cor-for-sym-metric-metplectope}
Assume that $g$ is a symplectic H\"ormander metric and let $a\in \SSS'(W)$ and $\chi\in\SSS(V)$. Let $M$, $r$, $L$ and $\Psi^{g,L}_X$, $X\in W$, be as in Theorem \ref{main-theorem-dia}. Then $(L^{-1}Q_X)^{1/2}\in\Sp(W)$, for all $X\in W$. For each $X\in W$, choose $\Phi^{g,L}_X\in\Mp(W)$ such that $\Pi(\Phi^{g,L}_X)=(L^{-1}Q_X)^{-1/2}$ and consider the following conditions:
\begin{itemize}
\item[$(i)$] $a\in S(M,g)$;
\item[$(ii)$] for each $N\in\NN$, the function
\begin{multline*}
(X,\Xi)\mapsto M((X+\Xi)/2)^{-1}(1+g_{\frac{X+\Xi}{2}}(X-\Xi))^N\\
\cdot\left\langle \left(\Psi^{g,L}_{\frac{X+\Xi}{2}}a\right)^w \pi\left((L^{-1}Q_{\frac{X+\Xi}{2}})^{1/2}X\right)\chi, \overline{\pi\left((L^{-1}Q_{\frac{X+\Xi}{2}})^{1/2}\Xi\right)\chi}\right\rangle
\end{multline*}
belongs to $L^{\infty}(W\times W)$;
\item[$(iii)$] for each $N\in\NN$, the function
\begin{equation*}
(X,\Xi)\mapsto M((X+\Xi)/2)^{-1}(1+g_{\frac{X+\Xi}{2}}(X-\Xi))^N \left|\left\langle a^w\pi(X) \Phi^{g,L}_{\frac{X+\Xi}{2}}\chi, \overline{\pi(\Xi)\Phi^{g,L}_{\frac{X+\Xi}{2}}\chi}\right\rangle\right|
\end{equation*}
belongs to $L^{\infty}(W\times W)$.
\end{itemize}
Then $(i)\Rightarrow (ii)\Leftrightarrow(iii)$. If, furthermore, $\chi$ satisfies \eqref{equ-for-con-on-thefun-tobe-fam}, then $(i)\Leftrightarrow (ii)\Leftrightarrow(iii)$.
\end{corollary}
\begin{remark}
In view of \eqref{fun-for-measur} and \eqref{equ-sym-metaplect-operators-for-weylq}, the functions in $(ii)$ and $(iii)$ are measurable for all $a\in\SSS'(W)$ and $\chi\in\SSS(V)$. If both $g$ and $M$ are continuous, then these functions are also continuous for all $a\in\SSS'(W)$ and $\chi\in\SSS(V)$ (cf. Remark \ref{rem-for-con-smooth-metr-fun} and Remark \ref{cont-metr-par-co}).
\end{remark}
The above discussion also proves the following consequence of Theorem \ref{charc-of-symplec-mod-space-with-growth} (cf. Remark \ref{equ-for-sym-metr-meas-lebesmes}).
\begin{corollary}
Assume that $g$ is a symplectic H\"ormander metric and let $a\in \SSS'(W)$, $\chi\in\SSS(V)$ and $1\leq p\leq \infty$. Let $\eta$, $r$, $L$ and $\Psi^{g,L}_X$, $X\in W$, be as in Theorem \ref{charc-of-symplec-mod-space-with-growth}. Then $(L^{-1}Q_X)^{1/2}\in\Sp(W)$, for all $X\in W$. For each $X\in W$, choose $\Phi^{g,L}_X\in\Mp(W)$ such that $\Pi(\Phi^{g,L}_X)=(L^{-1}Q_X)^{-1/2}$ and consider the following conditions:
\begin{itemize}
\item[$(i)$] $a\in \widetilde{\MM}^{p,p}_{\eta}$;
\item[$(ii)$] the function
\begin{multline*}
(X,\Xi)\mapsto \eta((X+\Xi)/2, \Xi-X)\\
\cdot\left\langle \left(\Psi^{g,L}_{\frac{X+\Xi}{2}}a\right)^w \pi\left((L^{-1}Q_{\frac{X+\Xi}{2}})^{1/2}X\right)\chi, \overline{\pi\left((L^{-1}Q_{\frac{X+\Xi}{2}})^{1/2}\Xi\right)\chi}\right\rangle
\end{multline*}
belongs to $L^p(W\times W)$;
\item[$(iii)$] the function
\begin{equation*}
(X,\Xi)\mapsto \eta((X+\Xi)/2, \Xi-X)\left|\left\langle a^w\pi(X) \Phi^{g,L}_{\frac{X+\Xi}{2}}\chi, \overline{\pi(\Xi)\Phi^{g,L}_{\frac{X+\Xi}{2}}\chi}\right\rangle\right|
\end{equation*}
belongs to $L^p(W\times W)$.
\end{itemize}
Then $(i)\Rightarrow (ii)\Leftrightarrow(iii)$. If, furthermore, $\chi$ satisfies \eqref{equ-for-con-on-thefun-tobe-fam}, then $(i)\Leftrightarrow (ii)\Leftrightarrow(iii)$.
\end{corollary}
We now specialise the main results to the commonly used calculi. These are all generated by H\"ormander metrics on $\RR^{2n}$ of the form
\begin{equation}\label{beals-fferf-metric-gen}
g_{(x,\xi)}=f(x,\xi)^{-2}|dx|^2+F(x,\xi)^{-2}|d\xi|^2,
\end{equation}
with $f$ and $F$ positive measurable functions on $\RR^{2n}$ (notice that \eqref{beals-fferf-metric-gen} is symplectic if and only if $F=1/f$). In this case, employing the standard inner product on $\RR^{2n}$, $\tilde{Q}^{1/2}_{(x,\xi)}$ is just a diagonal matrix whose first $n$ entries along the diagonal are $1/f(x,\xi)$ and the second $n$ entries are $1/F(x,\xi)$. The construction in Lemma \ref{fam-sat-con-maithforvar} amounts to
\begin{equation}\label{def-of-tht-for-ord-cal}
\theta_{(x,\xi)}(y,\eta)=\theta_0\left(\frac{|x-y|^2}{\tilde{r}^2f(x,\xi)^2}+ \frac{|\xi-\eta|^2}{\tilde{r}^2F(x,\xi)^2}\right),\quad (x,\xi),(y,\eta)\in\RR^{2n},
\end{equation}
and the isomorphism \eqref{change-of-var-for-dualit} is given by
\begin{equation}\label{iso-ind-by-the-met-in-cas-ordmecal}
\Psi_{(x,\xi)}\psi(y,\eta)=\psi(f(x,\xi)y,F(x,\xi)\eta),\quad \psi\in\SSS(\RR^{2n}).
\end{equation}
If $g$ is a symplectic H\"ormander metric on $\RR^{2n}$ of the form \eqref{beals-fferf-metric-gen} (hence, $F=1/f$), then the metaplectic operators $\Phi_{(x,\xi)}$, $(x,\xi)\in\RR^{2n}$, which satisfy $\Pi(\Phi_{(x,\xi)})=\tilde{Q}^{-1/2}_{(x,\xi)}$ are given by \cite[Proposition 7.8, p. 202]{deGosson}
\begin{equation}\label{metap-op-for-sym-hor-met-onr}
\Phi_{(x,\xi)}\chi(y)=\pm f(x,\xi)^{-n/2}\chi(y/f(x,\xi)).
\end{equation}
Combining Theorem \ref{main-theorem-dia}, Corollary \ref{cor-for-sym-metric-metplectope}, Remark \ref{sim-con-iim-for-evewo}, Lemma \ref{lemma-for-wig-gen-dec} and Lemma \ref{fam-sat-con-maithforvar}, we deduce the following result.
\begin{corollary}
Let $g$ be a H\"ormander metric on $\RR^{2n}$ given by \eqref{beals-fferf-metric-gen} with slow variation constant $r_0$ and let $M$ be a $g$-admissible weight with slow variation constant $r$. For each $(x,\xi)\in\RR^{2n}$, let $\Psi_{(x,\xi)}:\SSS'(\RR^{2n})\rightarrow\SSS'(\RR^{2n})$ be the isomorphism given by \eqref{iso-ind-by-the-met-in-cas-ordmecal} and define $\theta_{(x,\xi)}\in\DD(\RR^{2n})$ by \eqref{def-of-tht-for-ord-cal} with $\tilde{r}:=\min\{r_0,r\}$ and $\theta_0$ as in Lemma \ref{fam-sat-con-maithforvar}. Let $\chi\in\SSS(\RR^n)$ be such that $W(\chi,\chi)(0)\neq 0$. For any $a\in\SSS'(\RR^{2n})$, the following statements are equivalent:
\begin{itemize}
\item[$(i)$] $a\in S(M,g)$, i.e. $a\in\mathcal{C}^{\infty}(\RR^{2n})$ and $\|M^{-1} f^{|\beta|}F^{|\alpha|}\partial^{\alpha}_{\xi} \partial^{\beta}_xa\|_{L^{\infty}(\RR^{2n})}<\infty$, $\alpha,\beta\in\NN$;
\item[$(ii)$] for each $N\in\NN$, the function
\begin{multline*}
(x,\xi,y,\eta)\mapsto M(x,\xi)^{-1} \left(1+\frac{|y|}{f(x,\xi)}+ \frac{|\eta|}{F(x,\xi)}\right)^N\\
\left\langle \left(\Psi_{(x,\xi)}(a \theta_{(x,\xi)})\right)^w \pi\left(\frac{x+y}{f(x,\xi)}, \frac{\xi+\eta}{F(x,\xi)}\right) \chi, \overline{\pi\left(\frac{x-y}{f(x,\xi)}, \frac{\xi-\eta}{F(x,\xi)}\right) \chi}\right\rangle
\end{multline*}
belongs to $L^{\infty}(\RR^{4n})$.
\end{itemize}
If, in addition, $g$ is symplectic (i.e., $F=1/f$) then $(i)$ (and consequently, also $(ii)$) is equivalent to each of the following statements:
\begin{itemize}
\item[$(iii)$] for each $N\in\NN$, the function
\begin{multline*}
(x,\xi,y,\eta)\mapsto M(x,\xi)^{-1} (1+|y|f(x,\xi)^{-1}+ |\eta|f(x,\xi))^N\\
\left\langle (\Psi_{(x,\xi)}a)^w \pi\left(\frac{x+y}{f(x,\xi)}, (\xi+\eta)f(x,\xi)\right) \chi, \overline{\pi\left(\frac{x-y}{f(x,\xi)}, (\xi-\eta)f(x,\xi)\right) \chi}\right\rangle
\end{multline*}
belongs to $L^{\infty}(\RR^{4n})$;
\item[$(iv)$] for each $N\in\NN$, the function
\begin{multline*}
(x,\xi,y,\eta)\mapsto M(x,\xi)^{-1} (1+|y|f(x,\xi)^{-1}+ |\eta|f(x,\xi))^N\\
|\langle a^w\pi(x+y,\xi+\eta) \Phi_{(x,\xi)}\chi, \overline{\pi(x-y,\xi-\eta)\Phi_{(x,\xi)}\chi}\rangle|
\end{multline*}
belongs to $L^{\infty}(\RR^{4n})$, where $\Phi_{(x,\xi)}$ is the metaplectic operator \eqref{metap-op-for-sym-hor-met-onr}.
\end{itemize}
\end{corollary}
The corollary characterises the symbol classes of all of the commonly used calculi. Taking $g$ to be the Euclidian metric on $\RR^{2n}$ (i.e. $f(x,\xi)=F(x,\xi)=1$) and $M(x,\xi)=1$, both $(iii)$ and $(iv)$ reduce to the characterisation of $S^0_{0,0}(\RR^{2n})$ given in \cite[Theorem 6.2 $(i)$-$(ii)$]{gro-rz} (after changing variables with the transformation inverse to the one employed in Remark \ref{sim-con-iim-for-evewo}).
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2024-02-18T23:40:24.576Z
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2022-10-05T02:03:37.000Z
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{"paloma_paragraphs":[]}
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{"arxiv_id":"2112.14461","language":"en","timestamp":1664935417000,"url":"https:\/\/arxiv.org\/abs\/2112.14461","yymm":"2112"}
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proofpile-arXiv_000-10042
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{"provenance":"002.jsonl.gz:10043"}
| null | null |
\section{Introduction}
Let $\mathbb{N}$ be the set of nonnegative integers, $\mathbb{Z}$ the set of integers, $\mathbb{Z}^+ \vcentcolon = \mathbb{N} \setminus \set{0}$,
and, for $\alpha_1, \dots, \alpha_n \in \mathbb{N}$, $MN(\alpha_1, \dots, \alpha_n) \vcentcolon= \{\sum_{i=1}^n \lambda_i \alpha_i : \lambda_1, \dots ,\lambda_n \in \mathbb N\}$.
A \textit{list} in a set $S$ is a finite sequence of elements from $S$. If $n \in \mathbb{Z}^+$ and $(\alpha_1, \dots, \alpha_n)$ is a list in $\mathbb{N}$,
we will say $\alpha_1, \dots, \alpha_n$ are \ti{coprime} to mean $\gcd (\alpha_1, \dots, \alpha_n) = 1$. If we assume $\gcd(\alpha_1, \dots, \alpha_n) \neq 1$ when $\alpha_1 = \dots = \alpha_n = 0$,
then a list in $\mathbb{N}$ is coprime if and only if the list contains positive integers and the positive integers in the list are coprime. For an additive group $G$, if $S \subseteq G$ and $g \in G$, then let $g + S \vcentcolon= \set{g + s : s \in S}$.
Here is a 19th century theorem, due to Sylvester and Frobenius and others, restated to suit our purpose:
\begin{theorem}
If $\alpha_1, \dots, \alpha_n\in \mathbb{N}$ are coprime, then for some $w\in \mathbb N$, $w+\mathbb N\subseteq MN(\alpha_1, \dots, \alpha_n)$.
\end{theorem}
\noindent Note that because $0\in \mathbb N$, if $w+\mathbb N\subseteq MN(\alpha_1, \dots, \alpha_n)$ then \\$w\in MN(\alpha_1, \dots, \alpha_n)$.
Note also that because $\mathbb N$ is closed under addition, if $w+\mathbb N\subseteq MN(\alpha_1, \dots, \alpha_n)$ and $b\in \mathbb N$, then $w+b+\mathbb N\subseteq w+\mathbb N\subseteq MN(\alpha_1, \dots, \alpha_n)$.
Thus, if we define, for $\alpha_1, \dots, \alpha_n\in \mathbb Z^+$, $\frob{\alpha_1, \dots, \alpha_n} \vcentcolon= \{w\in \mathbb N:w+\mathbb N\subseteq MN(\alpha_1, \dots, \alpha_n)\}$,
we have that either $\frob{\alpha_1, \dots, \alpha_n}=\emptyset$ or \\$\frob{\alpha_1, \dots, \alpha_n}=\chi (\alpha_1, \dots, \alpha_n)+\mathbb N$, where $\chi(\alpha_1, \dots, \alpha_n)$ is the least element of $\frob{\alpha_1, \dots, \alpha_n}$.
The classical Frobenius problem \cite{brauer}, sometimes called the ``coin problem,'' is to evaluate $\chi(\alpha_1, \dots, \alpha_n)$ at coprime lists $(\alpha_1, \dots, \alpha_n)$ in $\mathbb{N}$.
Frobenius used to discuss this problem in his lectures \cite{alfonsin2005diophantine}, so this area of number theory happens to be named after him.
For $n=2$, there is a formula: if $\alpha, \beta \in \mathbb{N}$ and $\gcd (\alpha, \beta)=1$, then $\chi(\alpha, \beta)=(\alpha-1)(\beta-1)$.
For $n>2$ there are no known general formulas, although there are formulas for special cases and there are algorithms; see \cite{trimm}. The classical Frobenius problem inspired the results in \cite{gaussian}, which in turn inspired Nicole Looper \cite{looper} to generalize the classical Frobenius problem by introducing the notion of a Frobenius template (defined below), allowing one to pose similar problems in rings other than $\mathbb{Z}$.
All of our rings will contain a multiplicative identity, 1, and most will be commutative.
An \textit{additive monoid} in a ring $R$, is a subset of $R$ that is closed under addition and contains the identity $0$; monoids, unlike groups, do not have to contain inverses. A \textit{Frobenius template} in a ring $R$ is a triple $(A, C, U)$ such that: \begin{enumerate}[label=(\roman*)]
\item $A$ is a nonempty subset of $R$,
\item $C$ and $U$ are additive monoids in $R$, and
\item for all lists $(\alpha_1, \dots, \alpha_n)$ in $A$, \\
$MN(\alpha_1, \dots ,\alpha_n)=\{\sum_{i=1}^n \lambda_i\alpha_i:\lambda_1, \dots ,\lambda_n\in C\}$ is a subset of $U$.
\end{enumerate} For any valid template, the properties above guarantee that $MN(\alpha_1, \dots ,\alpha_n)$ will also be an additive monoid in $R$.
The \textit{Frobenius set} of a list $\alpha_1, \dots ,\alpha_n\in A$ is
$\frob{\alpha_1, \dots, \alpha_n} \vcentcolon= \set{w\in R:w+U\subseteq MN(\alpha_1, \dots ,\alpha_n)}$. Notice that $w \in \frob{\alpha_1, \dots, \alpha_n}$ implies that $w \in MN(\alpha_1, \dots, \alpha_n)$ since $0 \in U$, so $\frob{\alpha_1, \dots, \alpha_n} \subseteq MN(\alpha_1, \dots, \alpha_n) \subseteq U$. Notice also that the assumption that $U$ is closed under addition guarantees that if $w \in \frob{\alpha_1, \dots, \alpha_n}$, then $w + U \subseteq \frob{\alpha_1, \dots, \alpha_n}$.
For a given template $(A, C, U)$, the corresponding \textit{Frobenius problem} is the following pair of tasks:
\begin{enumerate}
\item Determine for which lists $\alpha_1, \dots ,\alpha_n\in A$ it is true that $\frob{\alpha_1, \dots, \alpha_n}\neq \emptyset$.
\item For lists $\alpha_1, \dots ,\alpha_n$ such that $\frob{\alpha_1, \dots, \alpha_n}\neq \emptyset$, describe the set $\frob{\alpha_1, \dots, \alpha_n}$.
\end{enumerate}
Note that in all the Frobenius templates that have been studied so far, it has always been the case that $\frob{\alpha_1, \dots, \alpha_n}$, when nonempty, is a finite union of sets of the form $w + U$ for some $w \in R$.
The classical Frobenius problem revolves around the template $(\mathbb{N}, \mathbb{N}, \mathbb{N})$ in the ring $\mathbb{Z}$. As discussed earlier, if $n = 2$ and $\alpha_1, \alpha_2 \in \mathbb{N}$, then $\frob{\alpha_1, \alpha_2}$ is nonempty if (and only if) $\alpha_1, \alpha_2$ are coprime, and $\frob{\alpha_1, \alpha_2} =\chi(\alpha_1, \alpha_2)+ \mathbb{N} = (\alpha_1 - 1)(\alpha_2 - 1) + \mathbb{N}$ in such cases. So the classical Frobenius problem has been completely solved when $n = 2$.
Our definition above of a Frobenius template is slightly less general than Looper's definition in \cite{looper}. The difference is that in Looper's templates, $U=U(\alpha_1, \dots ,\alpha_n)$ is allowed to vary with the list $\alpha_1, \dots ,\alpha_n$.
This is absolutely necessary in order to get interesting and nontrivial results when the ring involved is a subring of $\mathbb C$, the complex numbers, that is not contained in $\mathbb R$, the real numbers.
The ring of Gaussian integers, $\mathbb Z[i]=\{a+bi:a, b\in \mathbb Z\}$, is such a ring, a particularly famous member of the family of rings $\{\mathbb Z[\sqrt{m}i] : m\in \mathbb Z^+ \}$.
To see why we must let the element $U$ of a Frobenius template in such a ring depend on the list $\alpha_1, \dots ,\alpha_n\in A$, here is an example in $\mathbb Z[\sqrt{3}i]$.
Suppose $C=\{a+b\sqrt{3}i:a, b\in \mathbb N\} = \set{re^{i\theta} \in \mathbb{Z}[\sqrt{3}i] : 0 \leq r \text{ and } 0 \leq \theta \leq \frac{\pi}{2}}$ and $A=C \setminus \{0\}$.
Then consider $\alpha= 2e^{i \frac{\pi}{3}} = 1+\sqrt{3}i \in A$ and $\beta = 1 \in A$.
What should $U$ be for this list?
We set $U(\alpha, \beta)=\{re^{i\theta}\in \mathbb Z[\sqrt{3}i] : 0 \leq r \text{ and } 0\leq \theta \leq \frac{5\pi}{6}\}$, an angular sector in $\mathbb Z[\sqrt{3}i]$.
We so choose $U(\alpha, \beta)$ because $MN(\alpha, \beta)=\set{\lambda_1 (2e^{i \frac{\pi}{3}}) +\lambda_2 : \lambda_1, \lambda_2 \in C} \subseteq U(\alpha, \beta)$,
and no angular sector in $\mathbb Z[\sqrt{3}i]$ properly contained in $U(\alpha,\beta)$ contains $MN(\alpha, \beta)$.
Further, if $\hat{U}$ is an additive monoid in $\mathbb Z[\sqrt{3}i]$ which properly contains $U(\alpha, \beta)$, then no translate $w+\hat{U}$, $w\in \mathbb C$, is contained in $MN(\alpha, \beta)$. Thus $U(\alpha, \beta)$ is the only possible additive monoid in $\mathbb Z[\sqrt{3}i]$ containing $MN(\alpha, \beta)$ such that $\frob{\alpha, \beta} = \{w:w+U(\alpha, \beta)\subseteq MN(\alpha, \beta)\}$ might possibly be nonempty.
We leave to the reader the verification of the claims in the previous paragraph.
We hope that the main point is clear:
letting $U$ vary with the list $\alpha_1, \dots ,\alpha_n\in A\subseteq \mathbb Z[\sqrt{m}i]$ in the formulation of Frobenius problems in $\mathbb Z[\sqrt{m}i]$ is necessitated by the nature of multiplication in the complex numbers.
For an appreciation of Frobenius problems in such settings, see \cite{gaussian} and \cite{splitprimes}.
We do not take on similar difficulties here.
In the next section we give some more or less obvious results in various Frobenius templates, then review previous results concerning templates in the rings $\mathbb Z[\sqrt{m}]$ with $m\in \mathbb Z^+ \setminus \{n^2:n\in \mathbb Z^+\}$. In the third section, we classify the Frobenius set in a pleasing modification of the classical Frobenius template. In the fourth section, we solve Frobenius problems in rings of $2 \times 2$ (upper) triangular matrices with constant diagonal and entries from a ring $Q$, where different choices of $Q$ allow different templates. In the last section, we further generalize the idea of a Frobenius template and explore an example of this generalization.
\section{Some fundamentals and known results}
Throughout this section, $R$ will be a ring with multiplicative identity 1. A list $(\alpha_1, \dots ,\alpha_n)$ in $R$ \textit{spans unity} in $R$ if and only if $1=\lambda_1\alpha_1+...+\lambda_n\alpha_n$ for some $\lambda_1, \dots ,\lambda_n\in R$.
\begin{proposition}
Let $(A, C, U)$ be a Frobenius template in $R$ such that $1 \in U$.
If $(\alpha_1, \dots ,\alpha_n)$ is a list in $A$ such that $\frob{\alpha_1, \dots, \alpha_n}\neq \emptyset$, then $(\alpha_1, \dots ,\alpha_n)$ spans unity in $R$.
\end{proposition}
\begin{proof}
Let $w \in \frob{\alpha_1, \dots, \alpha_n}$, i.e. $w\in R$ and $w+U \subseteq MN(\alpha_1, \dots ,\alpha_n)$.
Since $0 \in U$ and $1 \in U$, it follows that $w, w+1 \in MN(\alpha_1, \dots ,\alpha_n)$.
Therefore, for some $\lambda_1, \dots ,\lambda,\gamma_1, \dots ,\gamma_n\in C$,
$w=\sum_{i=1}^n \lambda_i \alpha_i$ and
$w+1=\sum_{i=1}^n \gamma_i\alpha_i$, so $1=\sum_{i=1}^n (\gamma_i - \lambda_i) \alpha_i$.
\end{proof}
As a warm-up, here are some valid Frobenius templates $(A, C, U)$ with, in most cases, trivial solutions to the corresponding Frobenius problems. In each example, $(\alpha_1, \dots, \alpha_n)$ denotes a list in $A$. Check that $A$ is nonempty, $C$ and $U$ are additive monoids in $R$, and $U$ always contains $MN(\alpha_1, \dots, \alpha_n)$.
\begin{enumerate}
\item In the template $(R, \set{0}, \set{0})$, we have $MN(\alpha_1, \dots, \alpha_n) = \set{0}$ and\\ $\frob{\alpha_1, \dots, \alpha_n} = \set{0}$. This result extends to any template $(A, C, U)$ with $C = U = \set{0}$.
\item Similar to example 1, templates of the form $(\set{0}, C, \set{0})$ always have $MN(0) = \set{0}$ and $\frob{0} = \set{0}$.
\item In the template $(R, R, R)$, Proposition 2.1 shows that $(\alpha_1, \dots, \alpha_n)$ spans unity in $R$ if $\frob{\alpha_1, \dots, \alpha_n} \neq \emptyset$. Conversely, if $(\alpha_1, \dots, \alpha_n)$ spans unity, then $\frob{\alpha_1, \dots, \alpha_n} = R$.
\item Suppose that $I$ is an ideal of $R$ and the template is $(\{1\}, I, I)$.
Clearly $MN(1) = I = U$, so $U \subseteq \frob{1}$. Recall that $\frob{\alpha_1, \dots, \alpha_n}$ is a subset of $MN(\alpha_1, \dots, \alpha_n)$; thus $\frob{1} = U$.
\item Again, let $I$ be an ideal of $R$, and now consider the template $(I, R, I)$.
When $I = R$, this is example 3.
In contrast, let $I$ be a proper ideal. Then no list $(\alpha_1, \dots, \alpha_n)$ spans unity in $R$ --- but this is precisely because $1\notin I$, so no facile conclusion based on Proposition 2.1 presents itself.
Frobenius problems for this class of templates could be interesting.
For instance, if the template is $(2 \mathbb{Z}, \mathbb{Z}, 2 \mathbb{Z})$ in $\mathbb Z$, where $2\mathbb Z$ is the ideal of even integers, it is straightforward to prove that $\frob{\alpha_1, \dots, \alpha_n}\neq \emptyset$ if and only if the integers $\abs{\frac{\alpha_1}{2}}, \dots , \abs{\frac{\alpha_n}{2}}$ are coprime.
In this case, $MN(\alpha_1, \dots ,\alpha_n)=2\mathbb Z$ and $\frob{\alpha_1, \dots, \alpha_n} = 2 \text{Frob}'\paren{\frac {\alpha_1}{2}, \dots, \frac {\alpha_n}{2}}$, where $\text{Frob}'$ denotes Frobenius sets with respect to the classical template $(\mathbb{N}, \mathbb{N}, \mathbb{N})$.
\item If the template is $((0, \infty), [0, \infty), [0, \infty))$ in $\mathbb{R}$, then $MN(\alpha_1, \dots, \alpha_n) = [0, \infty) = \frob{\alpha_1, \dots, \alpha_n}$ for every list $(\alpha_1, \dots, \alpha_n)$. On the other hand, if we restrict the coefficients to the set of rational numbers $\mathbb{Q}$ with the template $((0, \infty), \mathbb{Q} \cap [0, \infty), [0, \infty))$ in $\mathbb{R}$, then $\frob{\alpha_1, \dots, \alpha_n}$ is always empty, as $MN(\alpha_1, \dots, \alpha_n)$ is countable and any translate of $[0, \infty)$ is uncountable.
\item Suppose that $R_1$ and $R_2$ are rings with associated Frobenius templates $(A_1, C_1, U_1)$ and $(A_2, C_2, U_2)$, respectively.
Let $A_1 \times A_2$ be the Cartesian product of $A_1$ and $A_2$. The same goes for $C_1 \times C_2$ and $U_1 \times U_2$,
which are additive monoids (under componentwise addition) in the product ring $R_1 \times R_2$. It is simple to see that $(A_1 \times A_2, C_1 \times C_2, U_1 \times U_2)$ is a valid Frobenius template in $R_1 \times R_2$. Let $(\beta_1, \dots ,\beta_n)$ be a list in $A_1 \times A_2$ with each $\beta_i=(\gamma_i, \mu_i)$ for $i=1, \dots, n$.
If $\frob{\beta_1, \dots, \beta_n} \neq \emptyset$ in $(A_1 \times A_2, C_1 \times C_2, U_1 \times U_2)$, then\\ $\frob{\beta_1, \dots ,\beta_n} = \textnormal{Frob}_1(\gamma_1, \dots, \gamma_n) \times \textnormal{Frob}_2(\mu, \dots, \mu_2)$, where $\textnormal{Frob}_1$ and $\textnormal{Frob}_2$ are Frobenius sets in $(A_1, C_1, U_1)$ and $(A_2, C_2, U_2)$, respectively. For $m \in \mathbb{Z}^+$, this result extends to the product ring $R_1 \times \dots \times R_m$.
\end{enumerate}
Besides the Gaussian integers \cite{gaussian, splitprimes}, prior work on generalized Frobenius problems has concentrated on templates in the real subring $\mathbb Z[\sqrt{m}] \vcentcolon= \{a+b\sqrt{m}:a, b\in \mathbb Z\}$, where $m \in \mathbb{Z}^+$ is not a perfect square.
Below, we showcase some highlights from this work.
If $m$ is a positive integer with irrational square root, then $\mathbb{Z}[\sqrt{m}]$ is a subring of $\mathbb{R}$.
Let $\mathbb{N}[\sqrt{m}] \vcentcolon=\{a+b\sqrt{m} : a, b\in \mathbb{N}\}$ and $\mathbb{Z}[\sqrt{m}]^+ \vcentcolon=\mathbb{Z}[\sqrt{m}]\cap [0, \infty)$, which are both additive monoids in $\mathbb{Z}[\sqrt{m}]$ that contain $1$.
\begin{enumerate}
\item In the template $(\mathbb N[\sqrt{m}], \mathbb N[\sqrt{m}], \mathbb N[\sqrt{m}])$, the result below is proven in \cite{looper}:
{\theorem If $\alpha_1, \dots ,\alpha_n \in \mathbb N[\sqrt{m}]$, $\alpha_i=a_i+b_i\sqrt{m}$, $a_i, b_i\in \mathbb N$, $i=1, \dots ,n$,
then $\frob{\alpha_1, \dots, \alpha_n}\neq \emptyset$ if and only if $(\alpha_1, \dots, \alpha_n)$ spans unity in $\mathbb Z[\sqrt{m}]$ and at least one of $a_1, \dots ,a_n$, $b_1, \dots ,b_n$ is zero.}
The result stated in \cite{looper}, Theorem 3, is slightly weaker than the formulation above, but needlessly so, since the proof in \cite{looper} proves the statement given here.
\item In the template $(\mathbb Z[\sqrt{m}]^+, \mathbb Z[\sqrt{m}]^+, \mathbb Z[\sqrt{m}]^+)$, the following is nearly proven in \cite{beneish}:
{\theorem
If $\alpha_1, \dots ,\alpha_n\in \mathbb Z[\sqrt{m}]^+$, then the following are equivalent:
\begin{enumerate}[label=(\roman*)]
\item $\frob{\alpha_1, \dots, \alpha_n}\neq \emptyset$;
\item $\frob{\alpha_1, \dots, \alpha_n}=\mathbb Z[\sqrt{m}]^+$;
\item $(\alpha_1, \dots ,\alpha_n)$ spans unity in $\mathbb Z[\sqrt{m}]$.
\end{enumerate}
}
Since Theorem 2 of \cite{beneish} proves $(iii) \implies (ii)$, we simply appeal to Proposition 2.1 to prove the statement above, which is somewhat stronger than Theorem 2 in \cite{beneish}.
Also in \cite{beneish}, with reference to the template $(\mathbb N[\sqrt{m}], \mathbb N[\sqrt{m}], \mathbb N[\sqrt{m}])$,\\ $\frob{\alpha_1, \dots, \alpha_n}$ is determined in a very special class of cases. Recall that for coprime positive integers $c_1, \dots, c_n$, $\cond{c_1, \dots, c_n}$ is the smallest $w \in \mathbb{Z}^+$ such that $w + \mathbb{N} \subseteq \set{\sum_{i=1}^n \lambda_i c_i : \lambda_1, \dots ,\lambda_n\in \mathbb{N} }$.
{\theorem
For $a_1, \dots ,a_r, b_1, \dots ,b_s\in \mathbb{N}$, \\ $(a_1, \dots ,a_r, b_1\sqrt{m}, \dots ,b_s\sqrt{m})$ is a list in $\mathbb{N}[\sqrt{m}]$.
We have
\[
\begin{split}
&\frob{a_1, \dots ,a_r, b_1\sqrt{m}, \dots ,b_s\sqrt{m}} \neq \emptyset\\ &\iff a_1, \dots ,a_r, b_1m, \dots ,b_sm \text{ are coprime,}
\end{split}
\]
and, in such cases,
\[
\begin{split}
&\frob{a_1, \dots ,a_r, b_1\sqrt{m}, \dots ,b_s\sqrt{m}}\\
&= \chi(a_1, \dots ,a_r, b_1m, \dots ,b_sm) + \chi(a_1, \dots ,a_r, b_1, \dots ,b_s)\sqrt{m} + \mathbb N[\sqrt{m}] \text{.}
\end{split}
\] } Note that the case $s=0$ is allowed in this result. Also, if the integers $a_1, \dots ,a_r, b_1m, \dots ,b_sm$ are coprime, then so are $a_1, \dots ,a_r, b_1, \dots ,b_s$, so the second part of Theorem 2.3 uses well-defined expressions.
\item With $\chi(a_1, \dots, a_n)$ as above, recall that in the $n = 2$ case of the classical template $(\mathbb{N}, \mathbb{N}, \mathbb{N})$ in $\mathbb{Z}$, $\chi(a_1, a_2) = (a_1 - 1)(a_2 - 1)$ for coprime $a_1, a_2$. In a tour de force in \cite{doyon}, Kim proves a similar formula for the template $(\mathbb N[\sqrt{m}], \mathbb N[\sqrt{m}], \mathbb N[\sqrt{m}])$ in $\mathbb Z[\sqrt{m}]$:
{\theorem
Suppose $\alpha=a+b\sqrt{m}, \beta=c+d\sqrt{m}$, $a, b, c, d\in \mathbb N$, $a+b, c+d>0$, $abcd=0$
(see Theorem 2.1, above), and suppose that $\alpha, \beta$ span unity in $\mathbb Z[\sqrt{m}]$.
Then $\frob{\alpha, \beta}=(\alpha-1)(\beta-1)(1+\sqrt{m})+\mathbb N[\sqrt{m}]$.}
\end{enumerate}
What's next? We propose these categories of rings where one might discover results of interest of the Frobenius type.
\begin{enumerate}
\item \textit{Subrings of algebraic extensions of }$\mathbb{Q}$.
Prior work on the Gaussian integers and the rings $\mathbb Z[\sqrt{m}]$ has put us into the foothills of a mountain range in this area.
Besides the extensions of $\mathbb{Q}$ of finite degree, what about the ring of algebraic integers?
Or, a better choice to start with, the ring of real algebraic integers?
\item \textit{Polynomial rings.}
Somebody we know is working on this, but we haven't heard from him for a year or so.
\item \textit{Rings of square matrices.}
Did we say our rings have to be commutative?
No, we did not. Keep in mind that in noncommutative rings, the precise definition of\\ $MN(\alpha_1, \dots, \alpha_n)$ becomes important, since for $\alpha_1, \dots, \alpha_n$ from $A$ and $\lambda_1, \dots, \lambda_n$ from $C$, the linear combinations $\sum_{i = 1}^n \lambda_i \alpha_i$ and $\sum_{i = 1}^n \alpha_i \lambda_i$ are not necessarily equal.
\end{enumerate}
\section{Modifying the classical template}
The classical template $(\mathbb{N}, \mathbb{N}, \mathbb{N})$ uses nonnegative integer coefficients. What happens when we modify the available coefficients? Consider the template $(\mathbb{N}, (n + \mathbb{N}) \cup \set{0}, \mathbb{N})$ for some $n \in \mathbb{Z}^+$. When $n = 1$, this is the classical template.
Remember that when $a_1, \dots, a_k \in \mathbb{N}$ are coprime, $\cond{a_1, \dots, a_k}$ is the unique integer such that $\frob{a_1 \dots, a_k} = \cond{a_1, \dots, a_k} + \mathbb{N}$ with respect to the classical template. In the results that follow, $\cond{a_1, \dots, a_k}$ retains this meaning, whereas $\text{Frob}$ and $MN$ reference the modified template $(\mathbb{N}, (n + \mathbb{N}) \cup \set{0}, \mathbb{N})$.
{\proposition If $a_1, \dots, a_k \in \mathbb{N}$ are coprime, then
\[ (a_1 + \dots + a_k)n + \cond{a_1, \dots, a_k} + \mathbb{N} \subseteq \frob{a_1, \dots, a_k}\text. \]}
\begin{proof}
Suppose $\omega \in \mathbb{N}$ and $\omega \geq (a_1 + \dots + a_k)n + \cond{a_1, \dots, a_k}$. We will show $\omega \in \frob{a_1, \dots, a_k}$, i.e.
\[ \omega + \mathbb{N} \subseteq MN(a_1, \dots, a_k) = \sett{ \sum_{i=1}^k \lambda_i a_i : \lambda_1, \dots, \lambda_k \in (n + \mathbb{N}) \cup \set{0} }\text. \]
Suppose $f \in \mathbb{N}$, $f \geq \omega$. It suffices to show that $f \in MN(a_1, \dots, a_k)$. Since $f \geq \omega$, we have that $f - (a_1 + \dots + a_k)n \geq \cond{a_1, \dots, a_k}$, so there exist coefficients $\gamma_1, \dots, \gamma_k \in \mathbb{N}$ such that $f = (a_1 + \dots + a_k)n + \sum_{i=1}^k \gamma_i a_i = \sum_{i=1}^k (\gamma_i + n)a_i$. Taking $\lambda_i = \gamma_i + n \geq n$ for $i =1, \dots, k$ yields the desired result.
\end{proof}
While reading the next proof, remember that $\cond{a, b} = (a - 1)(b - 1)$ for coprime $a, b \in \mathbb{N}$, so $\cond{a, b} - 1 = ab - a - b$.
{\proposition Let $a, b \in \mathbb{N}$ and $n \in \mathbb{Z}^+$. If $a, b$ are coprime and $n - 1$ is not divisible by $a$ or $b$, then $\frob{a, b} = (a + b)n + \cond{a, b} + \mathbb{N}$. }
\begin{proof}
By the preceding proposition, it suffices to show that $\frob{a, b} \subseteq (a + b)n + \cond{a, b} + \mathbb{N}$. So let $t \in \mathbb{N} \setminus \paren{(a + b)n + \cond{a, b} + \mathbb{N}}$, and suppose that $t \in \frob{a, b}$. We will find a contradiction, which will complete the proof.
Since $t \in \frob{a, b}$, we have $t + \mathbb{N} \subseteq MN(a, b)$, so $an + bn + \cond{a, b} - 1 \geq t$ is an element of $MN(a, b)$, so there exist $\lambda_1, \lambda_2 \in (n + \mathbb{N}) \cup \set{0}$ such that
\[ \cond{a, b} - 1 = \lambda_1 a + \lambda_2 b - an - bn = (\lambda_1 - n) a + (\lambda_2 - n) b \text. \]
Now, if both $\lambda_1$ and $\lambda_2$ are $\geq n$, then $\cond{a, b} - 1$ equals a linear combination of $a$ and $b$ with nonnegative integer coefficients, which is impossible. We also cannot have $\lambda_1 = 0 = \lambda_2$ since $\cond{a, b} + an + bn > 1$. So consider the case that $\lambda_1 = 0$ and $\lambda_2 \geq n$. In that case, $(\lambda_2 - n)b = \cond{a, b} - 1 + an = ab - b + a(n - 1)$, so $(\lambda_2 - n - a + 1)b = a(n - 1)$, so $b$ divides $a(n - 1)$. But $a$ and $b$ are coprime, so this implies that $b$ divides $n - 1$, which we have assumed to be false. Similarly $\lambda_1 \geq n$ and $\lambda_2 = 0$ would imply that $(\lambda_1 - n)a = \cond{a, b} - 1 + bn = ab - a + b(n - 1)$, yielding the contradiction that $a$ divides $n - 1$. Thus, all cases yield a contradiction, so our supposition that $t \in \frob{a, b}$ must be false.
\end{proof}
\section{\texorpdfstring{$\mathbf{2 \times 2}$}{Lg} triangular matrices with constant diagonal}
For a ring $Q$ with multiplicative identity $1$, let $R$ be the set $Q^2 = Q \times Q$ under coordinatewise addition, with multiplication in $R$ defined by $(a, b) \cdot (c, d) = (ac, ad + bc)$. $R$ is a ring with multiplicative identity $(1, 0)$, isomorphic to the ring of upper triangular $2 \times 2$ matrices over $Q$ with constant diagonal: $(a, b) \sim \begin{bmatrix} a & b \\ 0 & a \end{bmatrix}$. If $Q$ is commutative, then $R$ is commutative.
We will concentrate on the cases when $Q$ is a subring of the real field $\mathbb{R}$ containing $1$, and the Frobenius template is
\[
\begin{split}
&\paren{ A(Q), C(Q), U(Q) } \\
&= \paren{ \paren{Q \cap (0, \infty)} \times \paren{Q \cap [0, \infty)}, \paren{Q \cap [0, \infty)}^2, \paren{Q \cap [0, \infty)}^2}.
\end{split} \]
{\proposition Let $Q$ be a subring of $\mathbb{R}$ containing $1$. For any positive integer $n$, let $(\alpha_1, \dots, \alpha_n)$ be a list in $A(Q) \setminus \paren{ Q \times \set{ 0 } }$. Then $\frob{\alpha_1, \dots, \alpha_n} = \emptyset$.}
\begin{proof}
Let $\alpha_i = (a_i, b_i)$, $i = 1, \dots, n$. Elements of $MN(\alpha_1, \dots, \alpha_n)$ look like
\[
\begin{split}
(t, u)
&= \sum_{i=1}^n (a_i,b_i) \cdot (c_i,d_i) \\
&= \sum_{i=1}^n (a_ic_i, a_id_i + b_ic_i) \\
&= \paren{\sum_{i=1}^n a_ic_i, \sum_{i=1}^n \paren{ a_id_i + b_ic_i} },
\end{split}
\]
where $c_i \geq 0$ and $d_i \geq 0$ for all $i = 1, \dots, n$.
Suppose that $\frob{\alpha_1, \dots, \alpha_n}$ is nonempty, so there is a tuple $(t, u) \in Q^2$ such that $(t, u) + (Q \cap [0, \infty))^2 \subseteq MN(\alpha_1, \dots, \alpha_n)$. Therefore, for all $q \geq 0 $, we have $(t, u) + (q, 0) = (t + q, u) \in MN(\alpha_1, \dots, \alpha_n)$. So for larger and larger $q$, $t + q = \sum_{i=1}^n a_i c_i(q)$ and $u = \sum_{i = 1}^n ( a_i d_i(q) + b_i c_i(q) )$ for some nonnegative integers $c_i(q)$ and $d_i(q)$ ($i = 1, \dots, n$) that vary with $q$. Clearly $ \sum_{i = 1}^n a_i c_i(q) = t + q \to \infty$ as $q \to \infty$. Because each fixed $a_i, b_i > 0$, and each varying $c_i(q), d_i(q) \geq 0$, it follows that $\max_{1 \leq i \leq n} c_i(q) \to \infty$ as $q \to \infty$, and consequently $u = \sum_{i = 1}^n (a_i d_i(q) + b_i c_i(q)) \to \infty$ as $q \to \infty$. But $u$ is a given constant; it does not vary with $q$. Therefore, our supposition must be false, so $\frob{\alpha_1, \dots, \alpha_n} = \emptyset$.
\end{proof}
Moving on, we now know a sufficient condition for $\frob{\alpha_1, \dots, \alpha_n} = \emptyset$ in a large swath of templates. The natural question is whether $\frob{\alpha_1, \dots, \alpha_n}$ is ever nonempty, and the answer is yes.
{\proposition Let $Q$ be a subfield of $\mathbb{R}$. For any positive integer $n$, let\\ $(\alpha_1, \dots, \alpha_n)$ be a sequence of $2$-tuples $\alpha_i = (a_i, b_i) \in A(Q)$ (for $i = 1, \dots, n$) satisfying $b_1 = 0$. Then $\frob{\alpha_1, \dots, \alpha_n} = MN(\alpha_1,\dots, \alpha_n) = U(Q)$.}
\begin{proof}
We have
\[ MN(\alpha_1, \dots, \alpha_n) = \set{ \paren{ \sum_{i = 1}^n a_i c_i, \sum_{i = 1}^n a_i d_i + \sum_{i = 2}^n b_i c_i } : c_i, d_i \in Q \cap [0, \infty) } \text. \]
It will suffice to show that $(0, 0) \in \frob{\alpha_1, \dots, \alpha_n}$. Let $f, g \geq 0$, $f, g \in Q$, so that $f, g \in (0, 0) + U(Q)$. Then use the coefficients $c_1 = \frac{f}{a_1}$, $d_1 = \frac{g}{a_1}$, and $c_j = d_j = 0$ for $j = 2, \dots, n$ to show that $(f, g) \in MN(\alpha_1, \dots, \alpha_n)$, so $(0, 0) \in \frob{\alpha_1, \dots, \alpha_n}$.
\end{proof}
Proposition 4.1 can be generalized, \ti{mutatis mutandis,} to $Q$ being any linearly ordered commutative ring with unity. The same holds for Proposition 4.2, with the additional stipulation that $a_1$ is a unit. In both cases, the ordering is compatible with addition and multiplication.
In other words, setting $Q$ as a subfield of $\mathbb{R}$ yields a boring Frobenius template, since we can shrink elements of $\mathbb{R}$ using coefficients in $c_i, d_i \in Q \cap (0, 1)$. If we prohibit such shrinking, the template becomes much more interesting. Notice that this modification is similar to Section 3's modification of the classical template.
To make the following result fit on the page, we will temporarily adopt the convention that for a tuple $(a, b)$, the notation $(a, b)_+$ stands for $(a, b) + [0, \infty)^2$.
{\proposition Let $Q = \mathbb{R}$, and change $C(\mathbb{R})$ to be $C = \set{(0, 0)} \cup [1, \infty)^2$. Let $\alpha_1 = (a_1, 0), \alpha_2 = (a_2, b_2)$ form a list $(\alpha_1, \alpha_2)$ of tuples in $A$. If $b_2 = 0$, then $\frob{\alpha_1, \alpha_2} = (\min(a_1, a_2), \min(a_1, a_2))_+$. If $b_2 > 0$, then
\[
\begin{split}
&\frob{\alpha_1, \alpha_2}
= \\
&\begin{cases}
(a_1, a_1)_+ & a_1 \leq a_2 \text{ and } a_1 \leq b_2 \\
\paren{a_1, a_1}_+ \cup \paren{\frac{a_1 a_2}{b_2} + a_1, b_2}_+ & b_2 < a_1 \leq a_2 \\
\paren{a_1, a_2}_+ \cup \paren{ a_2, \frac{a_1 b_2}{a_2} + a_2}_+ & a_1 > a_2 \text{ and } a_2 \leq b_2 \\
\paren{a_1, a_2}_+ \cup \paren{ a_2, \frac{a_1 b_2}{a_2} + a_2}_+ \cup \paren{ \frac{(a_2)^2}{b_2} + a_1, b_2}_+ & b_2 < a_2 < a_1
\end{cases}
\end{split}
\] }
Simple modifications to the proof of Proposition 4.1 will prove that Proposition 4.1 holds in this template. Since $\frob{\alpha_1, \alpha_2} \subseteq \frob{\alpha_1, \dots, \alpha_n}$, Proposition 4.3 combines with Proposition 4.1 to prove that if $\alpha_i = (a_i, b_i) \in (0, \infty) \times [0, \infty)$ for $i = 1, \dots, n$ and $n > 1$, then $\frob{\alpha_1, \dots, \alpha_n}$ is nonempty if and only if some $b_i = 0$.
\begin{proof}
\tb{Case 0}: Suppose $b_2 = 0$, so elements of $MN(\alpha_1, \alpha_2)$ look like $(a_1 c_1 + a_2 c_2, a_1 d_1 + a_2 d_2)$ for $c_1, c_2, d_1, d_2 \in [0, \infty) \setminus (0, 1)$. Notice that $f \in (0, \min(a_1,$\\$ a_2))$ or $g \in (0, \min(a_1, a_2))$ implies $(f, g) \notin MN(\alpha_1, \alpha_2)$. Therefore, if $(t, u) \in \frob{\alpha_1, \alpha_2}$, then $t, u \geq \min(a_1, a_2)$. The converse is trivial, so this case is done.
The set $MN(\alpha_1, \alpha_2)$ is
\[ \sett{ \paren{a_1 c_1 + a_2 c_2, a_1 d_1 + a_2 d_2 + b_2 c_2} : c_1, c_2, d_1, d_2 \in [0, \infty) \setminus (0, 1) } \text{.} \]
The trickiness of the remaining cases is that both coordinates share the coefficient $c_2$. Similar to the last paragraph, notice that $f \in (0, \min(a_1, a_2))$ or $g \in (0, \min(a_1, a_2, b_2))$ implies $(f, g) \notin MN(\alpha_1, \alpha_2)$; therefore, $(t, u) \in \frob{\alpha_1, \alpha_2}$ implies $t \geq \min(a_1, a_2)$ and $u \geq \min(a_1, a_2, b_2)$.
\tb{Case 1}: $a_1 \leq a_2$ and $a_1 \leq b_2$. By remarks above, in this case $\frob{\alpha_1, \alpha_2} \subseteq (a_1, a_1) \cup [0, \infty)^2$. On the other hand, if $f, g \geq a_1$, then $(f, g) = (\frac f {a_1}, \frac g {a_1}) \alpha_1 + (0, 0) \alpha_2$, so $(f, g) \in MN(\alpha_1, \alpha_2)$. Therefore, $(a_1, a_1) + [0, \infty)^2 \subseteq \frob{\alpha_1, \alpha_2}$, so $\frob{\alpha_1, \alpha_2} = (a_1, a_1) + [0, \infty)^2$.
\tb{Case 2}: $0 < b_2 < a_1 \leq a_2$. Then $\frob{\alpha_1, \alpha_2} \subseteq (a_1, b_2) + [0, \infty)^2$ and, by the proof in Case 1,
\[ (a_1, a_1) + [0, \infty)^2 = [a_1, \infty)^2 \subseteq \frob{\alpha_1, \alpha_2} \text. \]
Therefore, to determine $\frob{\alpha_1, \alpha_2}$ in this case, it is sufficient to determine which $(t, u) \in [a_1, \infty) \times [b_2, a_1)$ are in $\frob{\alpha_1, \alpha_2}$.
If $(t, u) \in \frob{\alpha_1, \alpha_2} \subseteq MN(\alpha_1, \alpha_2)$, then for some $c_1, c_2, d_1, d_2 \in \set{0} \cup [1, \infty)$, $t = a_1 c_1 + a_2 c_2$ and $u = a_1 d_1 + a_2 d_2 + c_2 b_2$. Then $b_2 \leq u < a_1 \leq a_2$ implies that $d_1 = d_2 = 0$ and $c_2 = \frac u {b_2}$. Then we have that $t = a_1 c_1 + a_2 \frac u { b_2}$. If $t < a_1 + a_2 \frac u {b_2}$, then $c_1 = 0$ and $t = a_2 \frac u {b_2}$. But this would imply that for any $t'$ such that $t < t' < a_1 + a_2 \frac u {b_2}$, $(t', u) \notin MN(\alpha_1, \alpha_2)$, and this would contradict the assumption that $(t, u) \in \frob{\alpha_1, \alpha_2}$.
Therefore, if $(t, u) \in \frob{\alpha_1, \alpha_2}$ and $b_2 \leq u < a_1$, then $t \geq a_1 + a_2 \frac u {b_2}$. But $(t, u) \in \frob{\alpha_1, \alpha_2}$ implies that $(t, u') \in \frob{\alpha_1, \alpha_2}$ for every $u' \geq u$. Therefore, $t \geq a_1 + a_2 \frac {u'} {b_2}$ for every $u'$ satisfying $u \leq u' < a_1$. Therefore, $t \geq a_1 + \frac{a_1 a_2}{b_2}$.
Thus \begin{align*}
(a_1, a_1) + [0, \infty)^2 &\subseteq \frob{\alpha_1, \alpha_2} \\
&\subseteq \paren{(a_1, a_1) + [0, \infty)^2} \cup \paren{ \bigg[\frac{a_1 a_2}{b_2} + a_1, \infty \bigg) \times [b_2, a_1)} \\
&\subseteq \paren{(a_1, a_1) + [0, \infty)^2} \cup \paren{\paren{\frac{a_1 a_2}{b_2} + a_1, b_2} + [0, \infty)^2 } \text,
\end{align*}
so the proof in this case will be over if we show that $\paren{\frac {a_1 a_2}{b_2} + a_1, b_2}$ is in $\frob{\alpha_1, \alpha_2}$. Suppose that $f \geq \frac {a_1 a_2}{b_2} + a_1$ and $g \geq b_2$. We will see that $(f, g) \in MN(\alpha_1, \alpha_2)$. We may as well assume that $g < a_1$. Then $(f, g) = \paren{\frac 1 {a_1} \paren{f - \frac{g a_2}{b_2}}, 0} \alpha_1 + \paren{ \frac g {b_2}, 0} \alpha_2$, so $(f, g) \in MN(\alpha_1, \alpha_2)$ because $\frac g {b_2} \geq 1$ and $\frac 1 {a_1}\paren{f - \frac{g a_2}{b_2}} \geq \frac{1}{a_1} \paren{\frac{a_1 a_2}{b_2} + a_1 - \frac{a_1 a_2}{b_2} } = 1$.
\tb{Case 3}: $a_2 < a_1$ and $a_2 \leq b_2$. By arguments on display above, $(a_1, a_1) + [0, \infty)^2 \subseteq \frob{\alpha_1, \alpha_2} \subseteq (a_2, a_2) + [0, \infty)^2$. But it is also easy to see that $(a_1, a_2) \in \frob{\alpha_1, \alpha_2}$ ($\implies (a_1, a_2) + [0, \infty)^2 \in \frob{\alpha_1, \alpha_2}$): if $f \geq a_1$ and $g \geq a_2$, then $(f, g) = \paren{\frac f {a_1}, 0} \alpha_1 + \paren{0, \frac g {a_2}} \alpha_2 \in MN(\alpha_1, \alpha_2)$.
It remains to determine $\frob{\alpha_1, \alpha_2} \cap \paren{[a_2, a_1) \times [a_2, \infty) }$. Suppose that $(t, u) \in \frob{\alpha_1, \alpha_2}$ and $a_2 \leq t < a_1$. Let $c_1, c_2, d_1, d_2 \in \set{0} \cup [1, \infty)$ satisfy $t = a_1 c_1 + a_2 c_2$, $u = a_1 d_1 + a_2 d_2 + c_2 b_2$. Then $a_2 \leq t < a_1$ implies that $c_1 = 0$ and $c_2 = \frac t {a_2}$. If $u < a_2 + \frac{t b_2}{a_2}$ then $d_1 = d_2 = 0$ and $u = \frac{tb_2}{a_2}$. But then if $u < u' < a_2 + \frac {t b_2}{a_2}$, $(t, u') \notin MN(\alpha_1, \alpha_2)$, contradicting the assumption that $(t, u) \in \frob{\alpha_1, \alpha_2}$.
Therefore, $u \geq a_2 + \frac{tb_2}{a_2}$. But $(t, u) \in \frob{\alpha_1, \alpha_2}$ implies that $(t', u) \in \frob{\alpha_1, \alpha_2}$ for all $t'$ such that $t < t' < a_1$. Therefore, $u \geq a_2 + \frac{t' b_2}{a_2}$ for all such $t'$. Therefore, $u \geq a_2 + \frac{a_1 b_2}{a_2}$.
To finish the proof in this case, it will suffice to show that $[a_2, a_1) \times [a_2 + \frac{a_1 b_2}{a_2}, \infty) \subseteq \frob{\alpha_1, \alpha_2}$, and for that it will suffice to show that if $a_2 \leq f < a_1$ and $a_2 + \frac{a_1 b_2}{a_2} \leq g$ then $(f, g) \in MN(\alpha_1, \alpha_2)$. For such $(f, g)$, $(f, g) = \paren{\frac f {a_2}, \frac 1 {a_2} \paren{g - \frac {f b_2}{a_2}}} \alpha_2$, so $(f, g) \in MN(\alpha_1, \alpha_2)$ since $\frac f {a_2} \geq 1$ and $\frac 1{a_2}\paren{g - \frac {f b_2}{a_2}} \geq \frac 1 {a_2} \paren{a_2 + \frac{a_1 b_2}{a_2} - \frac{a_1 b_2}{a_2}} = 1$.
\tb{Case 4}: $0 < b_2 < a_2 < a_1$. Clearly $\frob{\alpha_1, \alpha_2} \subseteq (a_2, b_2) + [0, \infty)^2$. If $a_1 \leq f$ and $a_2 \leq g$ then $(f, g) = \paren{\frac f{a_1}, 0} \alpha_1 + \paren{0, \frac g{a_2}} \alpha_2$, so $(f, g) \in MN(\alpha_1, \alpha_2)$; therefore, $(a_1, a_2) + [0, \infty)^2 \subseteq \frob{\alpha_1, \alpha_2}$.
Next, we shall show that
\[ \frob{\alpha_1, \alpha_2} \cap \paren{[a_2, a_1) \times [b_2, \infty)} = [a_2, a_1) \times \bigg[a_2 + \frac{a_1 b_2}{a_2}, \infty\bigg) \text, \]
by an argument that will be familiar to anyone who has read the proofs in cases 2 and 3.
Suppose that $(t, u) \in \frob{\alpha_1, \alpha_2}$ and $a_2 \leq t < a_1$. Then for some $c_2, d_1, d_2 \in \set{0} \cup [1, \infty)$, $t = c_2 a_2$ and $u = a_1 d_1 + a_2 d_2 + \frac t{a_2} b_2$. If $u < a_2 + \frac{tb_2}{a_2}$, then $d_1 = d_2 = 0$ and $u = \frac{tb_2}{a_2}$. But then for all $u' \in \paren{u, \frac{t b_2}{a_2}}$, $(t, u') \notin MN(\alpha_1, \alpha_2)$, which contradicts the assumption that $(t, u) \in \frob{\alpha_1, \alpha_2}$.
Therefore, $u \geq a_2 + \frac{tb_2}{a_2}$. But then the fact that $(t', u) \in \frob{\alpha_1, \alpha_2}$ for all $t'$ satisfying $t < t' < a_1$ implies that $u \geq a_2 + \frac{t' b_2}{a_2}$ for all such $t'$. Therefore $u \geq a_2 + \frac{a_1 b_2}{a_2}$, which shows that
\[ \frob{\alpha_1, \alpha_2} \cap \paren{[a_2, a_1) \times [b_2, \infty) } \subseteq [a_2, a_1) \times \bigg[a_2 + \frac{a_1 b_2}{a_2}, \infty\bigg) \text. \]
On the other hand, if $a_2 \leq f < a_1$ and $g \geq a_2 + \frac{a_1b_2}{a_2}$, then $(f, g) = \paren{\frac{f}{a_2}, \frac{1}{a_2}\paren{g - \frac{fb_2}{a_2}}} \alpha_2$.
From $\frac{f}{a_2} \geq 0$ and
\[\frac{1}{a_2} \paren{g - \frac{f b_2}{a_2}}
\geq \frac 1 {a_2}\paren{a_2 + \frac{a_1 b_2}{a_2} - \frac{a_1 b_2}{a_2}} = 1,\]
we conclude that $(f, g) \in MN(\alpha_1, \alpha_2)$. Thus \[\frob{\alpha_1, \alpha_2} \cap \paren{[a_2, a_1) \times [b_2, \infty) } = [a_2, a_1) \times \bigg[a_2 + \frac{a_1 b_2}{a_2}, \infty\bigg).\]
To finish Case 4, we will determine $\frob{\alpha_1, \alpha_2} \cap \paren{[a_1, \infty) \times [b_2, a_2)}$. Suppose that $(t, u) \in \frob{\alpha_1, \alpha_2}$, with $t \geq a_1$ and $b_2 \leq u < a_2$. Then for some $c_1, c_2 \in \set{0} \cup [1, \infty)$, $t = a_1 c_1 + a_2 c_2$ and $u = b_2 c_2$. Then $t = a_1 c_1 + \frac{a_2 u}{b_2}$. If $t < a_1 + \frac{u a_2}{b_2}$, then $c_1 = 0$ and $t = \frac{u a_2}{b_2}$.
Then for all $t'$ such that $t < t' < a_1 + \frac{u a_2}{b_2}$, $(t', u) \notin MN(\alpha_1, \alpha_2)$, which contradicts the assumption that $(t, u) \in \frob{\alpha_1, \alpha_2}$. Therefore, $t \geq a_1 + \frac{u a_2}{ b_2}$. But then $(t, u) \in \frob{\alpha_1, \alpha_2}$ implies that $(t, u') \in \frob{\alpha_1, \alpha_2}$ for all $u'$ satisfying $u < u' < a_2$. Therefore, $t \geq a_1 + \frac{u' a_2}{b_2}$ for each such $u'$. Therefore, $t \geq a_1 + \frac{a_2^2}{b_2}$. Thus $\frob{\alpha_1, \alpha_2} \cap \paren{[a_1, \infty) \times [b_2, a_2)} \subseteq [a_1 + \frac{a_2^2}{b_2}, \infty) \times [b_2, a_2)$. On the other hand, suppose that $a_1 + \frac{a_2^2}{b_2} \leq f$ and $b_2 \leq g < a_2$. Then $(f, g) = \paren{\frac{1}{a_1}\paren{f - \frac{g a_2}{b_2}}, 0} \alpha_1 + \paren{\frac{g}{b_2}, 0} \alpha_2$. Thus, since $\frac{g}{b_2} \geq 1$ and $\frac{1}{a_1} \paren{f - \frac{g a_2}{b_2}} \geq \frac{1}{a_1} \paren{a_1 + \frac{a_2^2}{b_2} - \frac{a_2^2}{b_2}} = 1$, $(f, g) \in MN(\alpha_1, \alpha_2)$. Thus $\frob{\alpha_1, \alpha_2} \cap [a_1, \infty) \times [b_2, a_2) = [a_1 + \frac{a_2^2}{b_2}, \infty) \times [b_2, a_2)$. This, together with previous results, proves the claim in Case 4.
\end{proof}
We now shift focus to $Q = \mathbb{Z}$. Recall that $\cond{a_1, \dots, a_n}$ is well-defined for coprime positive integers $a_1, \dots, a_n$, with respect to the classical Frobenius template $(\mathbb{N}, \mathbb{N}, \mathbb{N})$.
{\proposition For $n\geq 2$, let $(\alpha_1, \dots, \alpha_n)$ be a list of $2$-tuples $\alpha_i = (a_i,b_i) \in \mathbb{Z}^+ \times \mathbb{N}$ (for $i = 1, \dots, n$) satisfying $\gcd(a_1, \dots, a_n) = 1$ and $b_1 = 0$. Then \[ \paren{ \cond{a_1, \dots, a_n}, \cond{a_1, \dots, a_n} + (a_1 - 1) \sum_{i = 2}^n b_i } + \mathbb{N}^2 \subseteq \frob{\alpha_1, \dots, \alpha_n}.\]}
\begin{proof}
Let $t = \cond{a_1, \dots, a_n}$ and $u = \cond{a_1, \dots, a_n} + (a_1 - 1) (\sum_{j = 2}^n b_j)$. It's sufficient to show that $(t, u) \in \frob{\alpha_1, \dots, \alpha_n}$, i.e.\\ $(t, u) + \mathbb{N}^2 \subseteq MN(\alpha_1, \dots, \alpha_n)$. Pick an arbitrary $(f,g) \in (t,u) + \mathbb{N}^2$, i.e. choose integers $f \geq \cond{a_1, \dots, a_n}$ and $g \geq \cond{a_1, \dots, a_n} + \sum_{j = 2}^n b_j(a_1 - 1)$. To complete the proof, we will show that
\[ (f,g) \in MN(\alpha_1, \dots, \alpha_n) = \sett{ \paren{\sum_{i=1}^n a_i c_i, \sum_{i=1}^n a_i d_i + \sum_{i=2}^n b_i c_i } : c_i, d_i \in \mathbb{N}}. \]
Since $f \geq \cond{a_1, \dots, a_n}$, we can write $f = \sum_{i = 1}^n a_i c_i$ for some nonnegative integers $c_1, \dots, c_n$. For each $i \in \set{2, \dots, n}$, use Euclidean division to write $c_i = a_1 q_i + \tilde{c}_i$ for some new nonnegative integers $q_i$ and $\tilde{c}_i \leq (a_1 - 1)$, then set $\tilde{c}_1 = c_1 + \sum_{i=2}^n a_i q_i$ so that $\sum_{i = 1}^n a_i c_i = f = \sum_{i = 1}^n a_i \tilde{c}_i$. Can we find nonnegative $d_1, \dots, d_n$ such that $g = \sum_{i=1}^n a_i d_i + \sum_{i=2}^n b_i \tilde{c}_i$? Well, $g - \sum_{i=2}^n b_i \tilde{c}_i \geq g - \sum_{i=2}^n b_i (a_1 - 1) \geq \cond{a_1, \dots, a_n}$, so yes. Therefore, $(f,g) \in MN(\alpha_1, \dots, \alpha_n)$.
\end{proof}
{\corollary For $n\geq 2$, let $(\alpha_1, \dots, \alpha_n)$ be a list of $2$-tuples $\alpha_i = (a_i,b_i) \in \mathbb{Z}^+ \times \mathbb{N}$ (for $i = 1, \dots, n$) satisfying $\gcd(a_1, \dots, a_n) = 1$. $\frob{\alpha_1, \dots, \alpha_n}$ is nonempty if and only if at least one $b_i = 0$.}
\begin{proof}
Combine propositions 4.1 and 4.4.
\end{proof}
We have seen results like this before, such as Theorem 2.1 of Section 2 and the paragraph preceding the proof of Proposition 4.3. Combining this corollary with the next result completely solves the Frobenius problem in the case $n = 2$ and $Q = \mathbb{Z}$.
{\proposition Suppose that $a_1, a_2 \in \mathbb{Z}^+, b \in \mathbb{N}$, $a_1$ and $a_2$ are coprime, and $\alpha_1 = (a_1, 0), \alpha_2 = (a_2, b)$. Then \[\frob{\alpha_1, \alpha_2} = \paren{ \cond{a_1, a_2}, \cond{a_1, a_2} + b(a_1 - 1) } + \mathbb{N}^2.\]}
\vspace{-20pt}
\begin{proof}
Let $(t, u) \in \mathbb{N} \times \mathbb{N}$. By Proposition 4.4, $t \geq \cond{a_1, a_2}$ and $u \geq \cond{a_1, a_2} + b( a_1 - 1)$ implies that $(t, u) \in \frob{\alpha_1, \alpha_2}$. For the converse, we will prove the contrapositive. Suppose that $t < \cond{a_1, a_2}$. Then $\cond{a_1, a_2} - 1 \geq t$ is not in $\set{a_1 c_1 + a_2 c_2 : c_1, c_2 \in \mathbb{N}}$, so $(\cond{a_1, a_2} - 1, u)$ is not in $\{ ( a_1 c_1 + a_2 c_2, a_1 d_1 + a_2 d_2 + b c_2) : c_1, c_2, d_1, d_2 \in \mathbb{N} \} = MN(\alpha_1, \alpha_2)$, yet $(\cond{a_1, a_2} - 1,u) \in (t,u) + \mathbb{N}^2$. Hence $(t,u) \notin \frob{\alpha_1, \alpha_2}$.
Now assume that $u < \cond{a_1, a_2} + b(a_1 - 1)$. Set $f = a_1 c_1 + a_2 (a_1 - 1)$ for some nonnegative $c_1$ large enough such that $f \geq t$. Consider any alternative expression of $f$ as $f = a_1 \tilde{c}_1 + a_2 \tilde{c}_2$ using nonnegative coefficients $\tilde{c}_1, \tilde{c}_2$. Recall that $f = a_1 c_1 + a_2 (a_1 - 1)$, so $a_1(c_1 - \tilde{c}_1) = a_2(\tilde{c}_2 - (a_1 - 1))$. Then $a_1 | (\tilde{c}_2 - (a_1 - 1))$ since $a_1$ and $a_2$ are coprime, so $\tilde{c}_2 \equiv a_1 - 1 \pmod{a_1}$. And $\tilde{c}_2$ is nonnegative, so $\tilde{c}_2 \geq (a_1 - 1)$, so $\tilde{c}_2 = (a_1 - 1) + a_1 k$ for some nonnegative integer $k$. Set $g = \cond{a_1, a_2} + b(a_1 - 1) - 1$ so that $g \geq u$; hence $(f,g) \in (t,u) + \mathbb{N}^2$.
Suppose that $(f,g) \in MN(\alpha_1, \alpha_2)$; there are nonnegative coefficients $d_1$ and $d_2$ such that $g = a_1d_1 + a_2d_2 + b \tilde{c}_2 = a_1 d_1 + a_2 d_2 + b ((a_1 - 1) + a_1 k)$. Consequently, $g - b(a_1 - 1) = a_1(d_1 + bk) + a_2d_2$ for nonnegative integers $(d_1 + b k)$ and $d_2$, so $g - b(a_1 - 1) \in MN(a_1, a_2)$. But this is impossible, because $g - b(a_1 - 1) = \cond{a_1,a_2} - 1 \notin MN(a_1, a_2)$, so our supposition must be false; $(f, g) \notin MN(\alpha_1, \alpha_2)$. Therefore, $(t, u) \notin \frob{\alpha_1, \alpha_2}$.
\end{proof}
Proposition 4.5 shows that when $n = 2$, the set inclusion in the conclusion of Proposition 4.4 is an equality. However, for lists of length $> 2$, there are counterexamples to the reverse set inclusion of Proposition 4.4, so Proposition 4.5 does not generalize to longer lists of tuples. The following is a counterexample: Let $\alpha_1=(3,0)$, $\alpha_2=(5,2)$, and $\alpha_3=(7,4)$. Then it can be shown that $(5,16) \in \frob{\alpha_1, \alpha_2, \alpha_3}+\mathbb{N}^2$, but $(5,16) \notin \{\cond{a_1, \dots, a_n} + (a_1 - 1) \paren{\sum_{i = 2}^n b_i}\} + \mathbb{N}^2$. In fact, it can be shown that $\frob{\alpha_1, \alpha_2, \alpha_3}= (5,9) + \mathbb{N}^2$. Furthermore, for $\beta_1 = (3, 0)$, $\beta_2 = (5, 1)$, and $\beta_3 = (7, 4)$ we have
\[ \frob{\beta_1, \beta_2, \beta_3} = \paren{(5, 9) + \mathbb{N}^2} \cup \paren{(8, 7) + \mathbb{N}^2} \text, \]
so the Frobenius set might even be a union of two sets, each of the form $(a, b) + \mathbb{N}^2$, neither contained in the other.
This situation resembles that of the classical template, in the sense that the Frobenius problem is completely solved for lists of length $2$, but not for lists of length $> 2$.
\section{A more general template}
Here we broaden our horizons and pass from the templates $(A, C, U)$ where $A$ and $C$ are thought of as subsets of the same overlying ring, to templates $(A', C', U')$ where $A'$ is a monoid and $C'$ is a set of functions acting on $A'$. The first kind of template, i.e. the only kind hitherto discussed in this paper, can be considered a special case of the second; furthermore, templates of the second kind cannot in general be interpreted as examples of the first kind. The different entries in the new kind of template have the same roles as the corresponding entries in the original kind of template.
For the sake of presentation, we will showcase a certain example of this new kind of template and leave the precise definitions to the reader. Let $A = U = \mathbb{N} \times \mathbb{N} \times \mathbb{N},$ and let $C$ be the set of upper triangular matrices in $M_3(\mathbb{N})$.
In the ring $\mathbb{Z}^3$ with coordinate addition and multiplication, $A = U = \mathbb{N}^3$ is a monoid. However, $C$ is not contained in $\mathbb{Z}^3,$ so this template is different from those considered previously. Note that $C$ contains an isomorphic copy of $\mathbb{N}^3$ via the semiring embedding given by
$$
\begin{bmatrix}
x \\
y \\
z \\
\end{bmatrix}
\mapsto
\begin{bmatrix}
x & 0 & 0 \\
0 & y & 0 \\
0 & 0 & z \\
\end{bmatrix}.
$$
Consider a general pair of tuples $(a, b, c)^T$ and $(d, e, f)^T \in A$. A general member of
$$
MN\left(\begin{bmatrix}
a\\ b\\ c\\
\end{bmatrix},
\begin{bmatrix}
d\\e\\f\\
\end{bmatrix}\right)
$$
has the form
$$
\begin{bmatrix}
u & v & w \\
0 & x & y \\
0 & 0 & z \\
\end{bmatrix}
\begin{bmatrix}
a\\ b\\ c\\
\end{bmatrix}
+
\begin{bmatrix}
u' & v' & w' \\
0 & x' & y' \\
0 & 0 & z' \\
\end{bmatrix}
\begin{bmatrix}
d \\ e \\ f \\
\end{bmatrix} \text,
$$
with all entries coming from $\mathbb{N}$. As expected,
$\frob{(a, b, c)^T, (d, e, f)^T}$ is defined to be
\[
\sett{\mb{w}\in MN\left(
\begin{bmatrix}
a\\ b\\ c\\
\end{bmatrix},
\begin{bmatrix}
d\\e\\f\\
\end{bmatrix}\right):\mb{w}+\mathbb{N}^3
\subseteq MN\left(
\begin{bmatrix}
a\\ b\\ c\\
\end{bmatrix},
\begin{bmatrix}
d\\e\\f\\
\end{bmatrix}\right)} \text.
\]
Hence, by our understanding of the classical Frobenius problem, we see that
$$
\frob{\begin{bmatrix}
a \\ b\\ c\\
\end{bmatrix},
\begin{bmatrix}
d \\ e\\ f\\
\end{bmatrix}}
=
\begin{bmatrix}
\chi(a,b,c,d,e,f)\\
\chi(b,c,e,f) \\
\chi(c,f) \\
\end{bmatrix} + \mathbb{N}^3,
$$
when nonempty, which is true if and only if $\gcd(c,f) = 1$.
From the case $k = 2$ it is straightforward to see what $\frob{\alpha_1,...,\alpha_k}$ is for arbitrary $k \in \mathbb{Z}^+$ and $\alpha_1,...,\alpha_k \in A$. It is not difficult to generalize these results to $m \times 1$ column vectors and $m \times m$ matrices for $m > 3$. Therefore these cases are no longer interesting, except for the connection between them and the classical Frobenius problem.
We can get more challenging problems by restricting the matrices in $C$. For instance, we could require the matrices to be symmetric, or upper triangular with constant diagonal.
These examples point to a generalized Frobenius template $(A', C', U')$ in which $A'$ (or perhaps $A' \cup \set{0}$) and $U'$ are monoids in a ring $R$, and $C'$ is a monoid in the ring of endomorphisms of $(R,+)$ such that each $\varphi \in C$ maps $A$ into $U.$
\medskip
\bibliographystyle{plain}
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2021-12-30T02:21:32.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14411","language":"en","timestamp":1640830892000,"url":"https:\/\/arxiv.org\/abs\/2112.14411","yymm":"2112"}
|
proofpile-arXiv_000-10043
|
{"provenance":"002.jsonl.gz:10044"}
| null | null |
\section*{Abstract}
We reexamine the the classical multidimensional scaling (MDS). We study some special cases, in particular, the exact solution for the sub-space formed by the 3 dimensional principal coordinates is derived. Also we give the extreme case when the points are collinear. Some insight into the effect on the MDS solution of the excluded eigenvalues (could be both positive as well as negative) of the doubly centered matrix is provided. As an illustration, we work through an example to understand the distortion in the MDS construction with positive and negative eigenvalues.
\section{Basics of the classical MDS}
We recall in this section some basics of the classical MDS from Mardia et al (1979, Section 14.2).
Let us denote the $n\times n$ distance matrix as $\bD =(d_{ij})$ and form the matrix $\bA$
\begin{align}
\bA = \{a_{ij}\}, a_{ij} &= -\frac{1}{2}d_{ij}^2,
\end{align}
and define the corresponding doubly centered matrix $\bB$,
\beq
\bB = \bH\bA\bH,
\eeq
where $\bH = \mathbf{I_n}-\frac{\vect{1}_n \vect{1}_n^T}{n}$ is the centering matrix in the standard notation. We can rewrite $\bB$ as
\beq
\bB = \bX\bX^T,
\eeq
and the $n \times n$ matrix $\bX$ of the principal coordinates in the Euclidean space
(assuming that $\bB$ is semi-positive) is given by
\beq \label{X}
\bX =\mathbf{\Gamma}\mathbf{\Lambda}^{\half}=({\lambda_1}^{\frac{1}{2}}{\gamma_1},{\lambda_2}^{\frac{1}{2}}{\gamma_2}\ldots, {\lambda_n}^{\frac{1}{2}}{\gamma_n})= (\vect{x}_{(1)},\vect{x}_{(2)}, \ldots, \vect{x}_{(n)}),
\eeq
where
$$\vect{x}_{(i)}={\lambda_i}^{\frac{1}{2}}{\vect{\gamma}_i}, i=1,\ldots,n ,$$
with the spectral decomposition of $\bB$
\beq \label{Bspectral}
\bB = \mathbf{\Gamma} \mathbf{\Lambda} \mathbf{\Gamma}^T,
\eeq
and $$\mathbf{\Gamma}= \{\gamma_{ij}\}=(\vect{\gamma}_1,\ldots, \vect{\gamma}_n)$$
is the orthogonal matrix of eigenvectors and $\mathbf{\Lambda}$ is the diagonal matrix of eigenvalues,
$$\mathbf{\Lambda} = \text{Diag}\{\lambda_{1},\ldots,\lambda_{n} \}.$$
Indeed, the $n$ principal coordinates $\vect{x}_i, i=1,2, \ldots,n $ are the rows of $\bX$, namely,
\beq \label{Xcoord}
\bX=
\begin{pmatrix}
\vect{x}^T_1 \\
\vect{x}^T_2 \\
\vdots \\
\vect{x}^T_n
\end{pmatrix}
, \vect{x}_i^T=({\lambda_1}^{\frac{1}{2}}{\gamma}_{i1},\ldots,{\lambda_n}^{\frac{1}{2}}{\gamma}_{in}), i=1,\ldots, n.
\eeq
We can use any ``subpart" of $\bX$ to define the principal coordinates of a low dimensional space as our MDS solution. Note that the last eigenvalue $\lambda_{n}$ is zero so at least $\vect{x}_{(n)}=0$ so we can work on the remaining $n-1$ dimensional coordinates.
Note that, for simplicity, we have taken $\bX$ as the $n \times n$ matrix rather than $n \times p$
matrix. We now show that, for any dissimilarity matrix $\bD$ with real entries but not necessarily semi-positive definite $\bB$ as in above, $\lambda_1$ will be always positive. We have
$$ tr(\bB) = tr (\bH^2\bA)=tr (\bH\bA)= tr (\bA)-tr(\vect{1}^T \bA \vect{1})/n$$
so that
\beq \label{tr}
\sum_{i=1}^{n} \lambda_i = \sum_{i<j}d_{i,j}^2/n >0.
\eeq
Hence $\lambda_1>0$. Thus implying that we can always "fit" one-dimensional configuration for any distance/ dissimilarity matrix.
Let us now consider the case when the n points lie on a line then we will have only one non-zero eigenvalue $\lambda$ of $\bB$ so from \eqref{tr}, it is given by
$$\lambda = \sum_{i<j}d_{i,j}^2/n. $$
\begin{itemize}
\item For n=3 with points on the line with the inter-point distances as $a,b,c$, we have
$\lambda =(a^2+b^2+c^2)/3,$ where if $AB=a, BC=b,AC=c$ with the points $A,B,C$ in that order then $a+b=c.$
\item If the points are $1,2,\ldots,n$ then we find that
$\lambda = n(n^2-1)/12$
so with the distances scaled to (0,1), we have $\lambda$ = $n(n^2-1)/12(n-1)^2$ and $\lambda$=O(n).
\end{itemize}
\section{The MDS solution for $2 \times 2$ distance matrix}
Let $X=(\vect{x}_{(1)},\vect{x}_{(2)})$ where
$\vect{x}_k,k=1,2$ are the coordinates in 2 dimensions. Suppose the two points are separated by a distance $d$.
We have
\begin{align}
D =
\begin{pmatrix}
0 & d \\
d & 0
\end{pmatrix} , \hspace{3pt}
B =
\begin{pmatrix}
\frac{d^{2}}{4} & \frac{-d^{2}}{4} \\
\frac{-d^{2}}{4} & \frac{d^{2}}{4}
\end{pmatrix}. \nn
\intertext{The eigenvalues of $\bB$ are given by}
|\bB-\lambda\bI_{2}| &= \lambda^{2}-2 \lambda \frac{d^{2}}{4}. \label{DecompersionTwo}
\intertext{Solving \eqref{DecompersionTwo} gives the eigenvalues}
\lambda_{1} = \frac{d^{2}}{2} \ &\text{and} \ \lambda_{2} = 0, \nn
& \nn
\intertext{with the corresponding eigenvectors }
\vect{\gamma}_{1} = \left(\frac{1}{\sqrt{2}}, \frac{-1}{\sqrt{2}}\right)^{T}
\ &\text{and} \
\vect{\gamma}_{2} = \left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)^{T}. \nn
\intertext{Now using \hspace{3pt}$\vect{x}_{(k)}=\lambda_{k}^{\frac{1}{2}}\vect{\gamma}_{k}$ from \eqref{X}, and we get for $\bX $}
\vect{x}_{(1)} = \left(\frac{d}{2}, \frac{-d}{2}\right)^{T} \hspace{3pt} &\text{and} \hspace{3pt}
\vect{x}_{(2)} = \left(0,0 \right)^{T} \label{TwoSol}.
\end{align}
So the principal coordinates from \eqref{Xcoord} are
\beq \label{coord2D}
\vect{x}_1 = \left(\frac{d}{2}, 0\right)^{T} \hspace{3pt} \text{and} \hspace{3pt}
\vect{x}_2 = \left(-\frac{d}{2},0 \right)^{T}.
\eeq
To get a lower dimensional coordinate space (one dimensional), we can simply use, along the line ( $x$- axis), the following two points:
$$x_1=\frac{d}{2},\quad x_2=-\frac{d}{2}.$$
We can now shift $x $ conveniently by using $ x^*$ = $ x +\frac{d}{2} $
so we have the new coordinates $$x^*_1=d,\quad x^*_2=0 $$ along the $x^*$- axis in one dimension with the origin at $x^*_2$.
The solution \eqref{coord2D} is trivial as the points only require placing a distance $d$ apart to be recovered, although it does serve as a pointer for the $3 \times 3$ distance matrix in the next section.
\section{The MDS solution for $3 \times 3$ distance matrix}
We now extend the last section of the $2 \times 2$ distance matrix to the $3 \times 3$ distance matrix where in principle, we need to follow the same steps.
Let now $\bX=(\vect{x}_{(1)},\vect{x}_{(2)},\vect{x}_{(3)})$, where $\vect{x}_k, k=1,2,3$ give the coordinates of points in three dimensions . Let
\begin{align}
\bD &=
\begin{pmatrix}
0 & a & b \\
a & 0 & c \\
b & c & 0
\end{pmatrix}.\\ \text{Then it can be seen that} \nn \\
\bB & = \frac{1}{18}
\begin{pmatrix}
4a^2+4b^2-2c^2 & -5a^2+b^2+c^2 & a^2-5b^2+c^2 \\
-5a^2+b^2+c^2 & 4a^2-2b^2+4c^2 & a^2+b^2-5c^2 \\
a^2-5b^2+c^2 & a^2+b^2-5c^2 & -2a^2+4b^2+4c^2 \end{pmatrix}. \nn
\intertext{and}
|\bB-\lambda\bI_{3}| &= \left(\frac{-1}{6}(a^{2}b^{2}+a^{2}c^{2}+b^{2}c^{2}) +\frac{1}{12}(a^{4}+b^{4}+c^{4})\right)\lambda \nn \\
& \quad \quad +\frac{1}{3}(a^{2}+b^{2}+c^{2})\lambda^{2}-\lambda^{3}. \label{DecompersionThree}\\
\intertext{Let}
\Delta&=\sqrt{a^{4}+b^{4}+c^{4}-a^{2}b^{2}-a^{2}c^{2}-b^{2}c^{2}} \label{Delta}.\\
\intertext{Solving \eqref{DecompersionThree} we find that the eigenvalues are }
\lambda_1 &=\frac{1}{6}(a^{2}+b^{2}+c^{2} +2\Delta ), \nn \\
\lambda_2 &=\frac{1}{6}(a^{2}+b^{2}+c^{2} -2\Delta ), \nn \\
\text{and} \ \lambda_3 &=0. \label{EigenvaluesThree}
\intertext{We show below in the proof of Theorem 1 that $\Delta$ is always non-negative.
To find the corresponding eigenvectors, $\bB$ is rotated using a Helmert rotation matrix $\bR$}
\bR &= \begin{pmatrix}
\frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \\
\frac{-1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\
\frac{1}{\sqrt{6}} & \frac{-2}{\sqrt{6}} & \frac{1}{\sqrt{6}}
\end{pmatrix}. \nn \\
\intertext{That is}
\bR\bB\bR^{T} &=
\begin{pmatrix}
0 & 0 & 0 \\
0 & \frac{b^2}{2} & \frac{(-a^2+c^2)}{2\sqrt{3}} \\
0 & \frac{(-a^2+c^2)}{2\sqrt{3}} & \frac{(2a^2 - b^2 +2c^2)}{6}
\end{pmatrix}
=\begin{pmatrix}
0 & 0 & 0 \\
0 & \sigma^{2}_{1,1} & \sigma^{2}_{1,2} \\
0 & \sigma^{2}_{1,2} & \sigma^{2}_{2,2}
\end{pmatrix} \label{NullMat1}
\end{align}
which has a $2 \times 2$ symmetric matrix nested within a $3 \times 3$ null matrix. We first give the
eigenvectors of $\bR^{T}\bB\bR$ by using a result of Mardia et al (1979, page 246, Exercise 8.1.1) ,
\begin{align}
\vect{\phi}_1 &=
\begin{pmatrix}
0 \\
\sigma^{2}_{2,2}-\sigma^{2}_{1,1}+\Theta \\
-2\sigma^{2}_{1,2}
\end{pmatrix}
\hspace{3pt} \text{and} \hspace{3pt}
\vect{\phi}_2 =
\begin{pmatrix}
0 \\
2\sigma^{2}_{1,2} \\
\sigma^{2}_{2,2}-\sigma^{2}_{1,1}+\Theta
\end{pmatrix} \label{NullMatVect}
\intertext{where $\Theta = \sqrt{(\sigma^{2}_{1,1}-\sigma^{2}_{2,2})^{2}+4\sigma^4_{1,2}}$. Next, the rotation is reversed by pre-multiplying the {\bf unnormalized} eigenvectors \eqref{NullMatVect} by $\bR$ to deduce the eigenvectors of $\bB$}
\vect{\gamma}_{1} &=
\begin{pmatrix}
b^{2}-c^{2}+\Delta \\
-a^{2}+c^{2} \\
a^{2}-b^{2}-\Delta
\end{pmatrix},\
\vect{\gamma}_{2} =
\begin{pmatrix}
2a^{2}-b^{2}-c^{2}-\Delta \\
-a^{2}+2b^{2}-c^{2}+2\Delta \\
-a^{2}-b^{2}+2c^{2}-\Delta
\end{pmatrix}
, \
\vect{\gamma}_{3} =
\begin{pmatrix}
1 \\
1 \\
1
\end{pmatrix} \label{EigenvectorsThree}
\intertext{ where $\Delta$ is given by \eqref{Delta}. Now using \hspace{3pt}$\vect{x}_{(k)}=\lambda_{k}^{\frac{1}{2}}\vect{\gamma}_{k}$ from \eqref{X}, and we get for $\bX $}
\vect{x}_{(1)} &= w_{1}
\begin{pmatrix}
b^{2}-c^{2}+\Delta \\
-a^{2}+c^{2} \\
a^{2}-b^{2} -\Delta
\end{pmatrix} \label{x1}
\text{where} \nn \\
w_{1} &= \sqrt{(a^{2}+b^{2}+c^{2}+2 \Delta )/(12\delta )},\\
\delta &=\Delta (2\Delta-a^{2}+2 b^{2}-c^{2})\nn
\\
\intertext{and}
\vect{x}_{(2)} &= w_{2}
\begin{pmatrix}
2a^{2}-b^{2}-c^{2}-\Delta \\
-a^{2}+2b^{2}-c^{2}+2\Delta \\
-a^{2}-b^{2}+2c^{2}-\Delta
\end{pmatrix} \label{x2}
\text{where}\nn \\
w_{2} &=
\sqrt{(a^{2}+b^{2}+c^{2}-2 \Delta )/(12\delta )}, \\
\delta &=\Delta (2\Delta-a^{2}+2 b^{2}-c^{2})\nn \\
\intertext{and}
\vect{x}_{(3)} &=(0,0,0)^{T} \label{x3}\\
\text{since} \hspace{3pt} \lambda_{3}=0. \nn
\end{align}
The constants $w_{1}$ and $w_{2}$ are a product of $\lambda_{k}^{\frac{1}{2}}$ and the eigenvector normalization constant.
Let A, B and C be the vertices of the triangle then say $AB=a, BC=b, AC=c $ .
Further, let $x,y,z $ be the three axes then with $n=3$ the principal coordinates from \eqref{Xcoord} can be written down using $\vect{x}_{(1)},\vect{x}_{(2)},\vect{x}_{(3)}$ from \eqref{x1}, \eqref{x2}, and \eqref{x3} respectively. As the triangle lies in the $x-y$ plane , we have the following theorem with $(x_i,y_i), i=1,2,3$ of A, B, C respectively by ignoring the $z-$ coordinates.\\
{\bf Theorem 1.}
Let $a^{2}+b^{2}+c^{2} \geq 2 \Delta $ where $\Delta$ is given by \eqref{Delta} , we have
\beq \label {A}
x_1= w_{1} ( b^{2}-c^{2}+\Delta ),\quad y_1= w_{2}(2a^{2}-b^{2}-c^{2}-\Delta ),
\eeq
\beq \label {B}
x_2= w_{1} ( -a^{2}+c^{2} ),\quad y_2= w_{2}(-a^{2}+2b^{2}-c^{2}+2\Delta ),
\eeq
\beq \label {C}
x_3= w_{1} (a^{2}-b^{2} -\Delta ),\quad y_3= w_{2}(-a^{2}-b^{2}+2c^{2}-\Delta ),
\eeq
where
$$w_{1} = \sqrt{(a^{2}+b^{2}+c^{2}+2 \Delta )/(12\delta )},\quad w_{2}=
\sqrt{(a^{2}+b^{2}+c^{2}-2 \Delta )/(12\delta )},$$
with $\delta =\Delta (2\Delta-a^{2}+2 b^{2}-c^{2}).$
Further, the center of gravity of the triangle is at $(0,0)$.
\\
{\bf Proof.} Most of the results are already proved above. Note that $\Delta$ and $\delta$ are non-negative using the following inequality of the Geometric Mean and Arithmetic Mean given below successively.
$$ x^2y^2 \leq (x^4+y^4)/2.$$
Alternatively, we see easily $ \Delta >0$ on noting that
$$ 2\Delta ^2 = (a^2 -b^2)^2 + (a^2 -c^2)^2 + (b^2 -c^2)^2. $$
{\bf Corollary 1.} Let $a=c, a < b < 2a $ then for this isosceles case, we have
\beq \label {Iso}
x_1= b/2, y_1= - (4a^{2}-b^{2})/6;\quad
x_2= 0,y_2= -2 y_1 ;\quad
x_3= -x_1, y_3= y_1.
\eeq
{\bf Proof.} From the equations \eqref{EigenvaluesThree}, \eqref{x1} and \eqref{x2}, we find that for $a<b<2a$ we have
$$\lambda_1 = b^2/2, \lambda_2 =(4 a^2 - b^2)/6, \quad w_{1} = 1/(4\Delta ),\quad w_{2}=
\sqrt{(4 a^{2}-b^{2})}/(12\Delta ),$$
where $\Delta =b^{2}-a^{2}.$ Using these results in Theorem 1, our proof follows.
\\
We now consider a wide range of particular isosceles triangles
\begin{itemize}
\item If $b=a$ , we have an equilateral triangle.
\item If $b=2a$ , we have a flat triangle as $\lambda_2 = \lambda_3 =0.$
\item If $b>2a$ then $\lambda_1>0, \lambda_3 =0$ but $\lambda_2$ is imaginary so we can have a real solution only in one dimension.
\item If $a$ is very large and $b$ is fixed then we have a peaked isosceles triangle.
\end{itemize}
Note that for the isosceles triangle, without any loss of generalities by rescaling, we can write the coordinates of $A,B,C$ as
$$ A=(1,-e),B=(0,2e),C=(-1,-e)$$ where $ e= \sqrt{(4 a^{2}-b^{2})}/(3 b).$
It allows the equilateral case with $a=b$ ( as a limit) leading to the coordinates
$$ A(1,-1/\sqrt{3}),B(0,2/\sqrt{3}),C(-1,-1/\sqrt{3}).$$
{\bf Remark 1.} Equation \eqref{EigenvaluesThree}, which gives the eigenvalues of $\bB$, can be used to determine if the desired Euclidean properties of $\bD$ are violated. Rearranging the equation for the second eigenvalues \eqref{EigenvaluesThree} or $w_2 \geq 0$ gives the condition ( for $\bB$ to be semi- positive definite)
\begin{align}
a^{2}+b^{2}+c^{2} &\geq 2 \Delta.
\end{align}
Hence, if this inequality holds then $\bD$ is Euclidean.
{\bf Remark 2.} For visualization, we can shift the origin (and rotate if so desired) for the points $A, B, C $. For example, in
\eqref{A}, \eqref{B}, \eqref{C}, we can use the transformation (as in the $2x2$ case)
$$x_i^*= x_i-x_1, y_i^*=y_i-y_1$$ so we have $$x_1^*= 0, y_1^*=0$$ which helps in visualizing the isosceles case, in particular.
\section{Effect of excluding eigenvalues in the MDS solution}
When the distance /dissimilarity matrix is very general, the corresponding matrix $\bB$ can have some negative eigenvalues, which can distort the Euclidean fitted configuration. We now give some insight into this possible effect.
Let $\bD=(\delta_{ij})$ be an $n \times n$ dissimilarity matrix. We are using slightly different notation than in the first section to emphasize that we are working now on a dissimilarity matrix and with the fitted distances $(d_{ij})$ for the MDS solution .
Suppose as in \eqref{Bspectral} the corresponding matrix $\bB$ has spectral decomposition $\bB = \mathbf{\Gamma} \mathbf{\Lambda} \mathbf{\Gamma}^T$, with the eigenvalues in decreasing order (there is always
at least one zero, and perhaps some negative eigenvalues) where as in Section 1, $\mathbf{\Lambda}$ is the diagonal matrix with the eigenvalues $\lambda_\ell, \ell=1,\ldots,n, $ and $\mathbf{\Gamma}$ is the matrix of the eigenvectors.
Write
\beq \label{gij}
g_{ij}^{(\ell)} = \lambda_\ell (\gamma_{i,\ell} - \gamma_{j,\ell})^2.
\eeq
Then from \eqref{Xcoord}, the distance between the points $\vect{x}_i^T=({\lambda_1}^{\frac{1}{2}}{\gamma}_{i1},\ldots,{\lambda_n}^{\frac{1}{2}}{\gamma}_{in} ) $ and $\vect{x}_j^T=({\lambda_1}^{\frac{1}{2}}{\gamma}_{j1},\ldots,{\lambda_n}^{\frac{1}{2}}{\gamma}_{jn})$ is given by
\beq \label{deltaij}
\delta^2_{ij} = \sum_{\ell=1}^n g_{ij}^{(\ell)}
\eeq
which is an exact identity. If the MDS solution uses the first $p $ eigenvalues
(assumed to be nonnegative, $p \leq n $), then the squared Euclidean distances for
this MDS solution are given by
\beq \label{dij}
d^2_{ij} = \sum_{\ell=1}^p g_{ij}^{(\ell)}.
\eeq
The difference between \eqref{deltaij} and \eqref{dij} is
\beq \label{diffij}
\delta^2_{ij} - d^2_{ij} = \sum_{\ell=p+1}^n g_{ij}^{(\ell)}
\eeq
and measures the extent at sites $i,j$ to which the MDS solution fails
to recover the starting dissimilarities.
Let us fix 2 sites $i,j$ and consider two mutually exclusive possibilities (of
course more complicated situations can occur):
(a) Suppose $g_{ij}^{(\ell)}$ is near 0 for all $\ell = p+1, \ldots, n$
except for one value $\ell = \ell_1$, say. Further suppose that
$\lambda_{\ell_1} > 0$. Then from \eqref{gij} and \eqref{diffij}, we have
$$\delta^2_{ij} - d^2_{ij} > 0; $$
(b) Suppose $g_{ij}^{(\ell)}$ is near 0 for all $\ell = p+1, \ldots, n$
except for one value $\ell = \ell_2$, say. Further suppose that
$\lambda_{\ell_2} < 0$. Then again from \eqref{gij} and \eqref{diffij},
$$\delta^2_{ij} - d^2_{ij} < 0 .$$
Hence, if $g_{ij}$ given by \eqref{gij} is positive (negative), the Euclidean distance will be smaller than (greater than) the dissimilarity.
We now give a numerical example.
{\bf Example.} We look at the journey times between a selection of 5 rail
stations in Yorkshire (UK) to understand how the eigenvectors of $\bB$ can help to understand the behaviour of a solution of $\bB$ with some negative eigenvalues.
There are two rail lines between Leeds and
York; a fast line with direct trains, and a slow line that stops
at various intermediate stations including Headingley, Horsforth
and Harrogate.
Here the ``journey time'' is defined as the time taken to reach the
destination station for a passenger who begins a journey at the
starting station at 12:00 noon. For example, consider a passenger
beginning a journey at Leeds station at 12:00. If the next train for York
leaves at 12:08 and arrives in York at 12:31, then the journey
time is 31 minutes (8 minutes waiting in Leeds plus 23 minutes on
the train). The times here are taken from a standard weekday
timetable.
Table \ref{table:rail} below gives the dissimilarities between all pairs of
stations, where the dissimilarity between two stations $S1$ and $S2$ is
defined as the smaller of two times: the journey time from $S1$ to $S2$
and the journey time from $S2$ to $S1$. Further the dissimilarity
between a station and itself is taken to be 0.
\begin{table}[h]
\begin{center}
\caption{ Dissimilarity matrix $\bD$ for train journey
times between 5 rail stations in Yorkshire.}
\begin{tabular}{lrrrrr}
& A & B & C & D & E\\
(1) A: Leeds & 0 & 23 & 23 & 53 & 31 \\
(2) B: Headingley & 23 & 0 & 11 & 34 & 71\\
(3) C: Horsforth & 23 & 11 & 0 & 34 & 67\\
(4) D: Harrogate & 53 & 34 & 34 & 0 & 44\\
(5) E: York & 31 & 71 & 67 & 44 & 0
\end{tabular}
\label{table:rail}
\end{center} \end{table}
\begin{figure}[h]
\begin{center}
\includegraphics[width=4in,height=4in]{q4f1.eps}
\caption{ Two-dimensional MDS solution for train
journey times between 5 rail stations in Yorkshire. A: Leeds, B: Headingley, C: Horsforth, D: Harrogate, E: York.}
\label{fig:rail}
\end{center} \end{figure}
\newpage
The eigenvalues of $\bB$ are
$$\lambda_1= 3210, \lambda_2= 1439, \lambda_3=61,
\lambda_4=0,
\lambda_5= -964, $$
and the corresponding eigenvectors in $\mathbf{\Gamma}$ are
\begin{verbatim}
Eigenvectors
[,1] [,2] [,3] [,4] [,5]
[1,] 0.08 0.63 -0.06 -0.45 0.63
[2,] -0.48 0.06 -0.66 -0.45 -0.38
[3,] -0.41 0.06 0.75 -0.45 -0.26
[4,] 0.03 -0.77 -0.04 -0.45 0.45
[5,] 0.77 0.02 0.00 -0.45 -0.45
\end{verbatim}
We take our MDS solution to be the two dimensional principal coordinates. Figure \ref{fig:rail} plots these two dimensional principal coordinates. Obviously
as seen by the eigenvalues, $\bD$ is not a distance matrix. Also we can check that
$$71 = \delta_{25} > \delta_{12}+ \delta_{15} = 23 + 31 = 54,$$ which violates
the triangle inequality.
The eigenvalues are 3210, 1439, 61, 0, -964. The first two are
considerably larger than the rest in absolute value, suggesting
the 2D MDS solutions should be
a good representation. In particular,
$\lambda_3=61$ seems negligible,
$\lambda_4=0$ is an eigenvalue that always appears with eigenvector $\vect{1}_n$,
$\lambda_5= -964$ is smaller than the first two eigenvalues, but not entirely
negligible and may cause some distortion in the reconstruction as we now examine.
Figure 1 shows that the stations lie roughly on a circle (not surprising since
there are two lines between Leeds and York). Also, Headingley and
Horsforth are close together, and Leeds is further from Harrogate than
from York in terms of the dissimilarity though geographically Harrogate is nearer to Leeds than York.
In the MDS solution, the Euclidean distance between Headingley and
Horsforth is 7.1, which is smaller than the dissimilarity value
11. On the other hand, in the MDS solution the Euclidean distance between
Leeds and York is 45.5, which is larger than the dissimilarity
value 31. We can now explain this behaviour using the spectral decomposition of $\bB$, and using the result derived in this section. Let us now denote the stations A, $\ldots$, E by $1,\ldots ,5 $ respectively.
Eigenvector entries for selected stations (and the corresponding eigenvalues 61 and -964 respectively)\\
\begin{tabular}{lrr}
Station & $j=3$ & $j=5$ \\
Headingley (2) & -0.66 & -0.38\\
Horsforth (3) & 0.75 & -0.26\\
absolute difference & {\bf 1.41} & 0.12 \\
\\
Leeds (1) & -0.06 & 0.63\\
York (5) & 0.00 & -0.45\\
absolute difference & 0.06 & {\bf 1.08}
\end{tabular}
Hence, the difference between Headingley and Horsforth is dominated by
the eigenvector $j=3$ (with positive eigenvalue, 61), whereas the
difference between Leeds and York is dominated by the
eigenvector $j=5$ (with negative eigenvalue, -964). In fact, the numerical values of the terms \eqref{gij} in the difference between the two distances given by \eqref{diffij} are
$$ g_{23}^{(3)} = 121.3, \quad g_{23}^{(4)}= 0,\quad g_{23}^{(5)} = -13.9, $$
and
$$ g_{15}^{(3)} = 0.2,\quad g_{15}^{(4)} = 0,\quad g_{15}^{(5)} = -1124.4, $$
so the dominated contributions $g_{23}^{(3)}$ and $g_{15}^{(5)}$ are clearly seen. This discussion explains why in the MDS solution, the Euclidean distance between Headingley (2) and
Horsforth (3) is smaller than the dissimilarity value
whereas the Euclidean distance between
Leeds (1) and York (5) is larger than the dissimilarity
value.
\section{Acknowledgment} We wish to express our thanks to Wally Gilks and John Kent for their helpful comments and to the University of Leeds for the Example in Section 4 from an Examination paper. The first author would also like to thank the Leverhulme Trust for the Emeritus Fellowship.
\section{References}
Mardia, K. V., Kent, J. T., and Bibby, J. M. (1979). {\it Multivariate Analysis}. Academic press.
\end{document}
\begin{align}
\bR =
\begin{pmatrix}
\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\
-\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}
|
2024-02-18T23:40:24.588Z
|
2021-12-30T02:25:30.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14503","language":"en","timestamp":1640831130000,"url":"https:\/\/arxiv.org\/abs\/2112.14503","yymm":"2112"}
|
proofpile-arXiv_000-10044
|
{"provenance":"002.jsonl.gz:10045"}
| null | null |
\section{Introduction}\label{introduction}
Software systems evolve and are typically developed through program evolution cycles that involve frequent code modifications \cite{handsOnDevOps}. Therefore, when software evolves, the program modifications need to be tested to avoid introducing faults and ensure the expected program behavior. To test an evolving program, developers need to perform
\textit{regression testing}, i.e., assessing the impact of the change on the program by generating additional test cases targeting the change and its dependencies~\cite{YooH12}. Typically, developers have to write or generate test cases that exercise the changes, stress their dependencies, and check that the program changes behave as intended \cite{ApiwattanapongSCOH06}.
Mutation testing is an established software testing technique~\cite{PapadakisK00TH19}. It is typically applied to reveal faults in a program by modifying the program (aka injecting mutants) and generating tests to reveal the faults (i.e., kill the mutants) in the modified program. Mutation testing is an effective approach to improve the test suite's \revise{strengths} by ensuring that it is adequate and diverse enough to kill all injected mutants. In the last decade, mutation testing has focused on selecting or reducing the number of executed mutants to ensure that mutation testing is feasible and scales in practice. To this end, researchers have proposed mutation testing with a specific type of mutants \cite{offut1993}, mutant reduction by detecting equivalent mutants \cite{KintisPJMTH18, ChekamPBTS20} or by focusing on a particular category of mutants \revise{such as subsuming mutants\footnote{Subsuming mutants \cite{jia2009} or disjoint mutants \cite{KintisPM10} is a set of mutants that has no mutant that is killed by a proper subset of tests that kill another mutant.}} or hard-to-kill mutants \cite{PapadakisCT18, KurtzAODKG16, KintisPM10}.
Traditional mutation testing involves injecting mutants into the entire code base of the software.
However, mutation testing of evolving programs is challenging due to the \textit{scale of the required mutation analysis, the complexity of the program, and the difficulty of determining the impact of the dependencies of the program changes}. The sub-field of mutation testing addressing these issues by targeting the mutation testing program changes is referred to as \textit{commit-aware mutation testing} \cite{ma2020commit}.
A few commit-aware mutation testing approaches have been proposed to tackle the challenges of mutation testing of evolving software systems~\cite{petrovic2018, ma2020commit, ma2021mudelta, CACHIA2013}. These approaches suggest that mutation testing of evolving systems should focus on the program changes rather than the entire program. Recent studies have also indicated that commit-relevant mutants can be found on unchanged code due to unforeseen interactions between changed and unchanged code~\cite{ma2020commit, ma2021mudelta}. However, these studies do not provide scientific insights into the nature and properties of commit-relevant mutants and their utility over time. For instance, it is necessary to understand the distribution and program location of commit-relevant mutants to effectively identify, select, or predict commit-relevant mutants.
In this paper, we address this challenge by conducting an exploratory empirical study to investigate the properties of commit-relevant mutants.
Specifically, we examined the distribution, location, prevalence, predictability, and utility of commit-relevant mutants, as well as subsuming commit-relevant mutants\footnote{\revise{Subsuming commit-relevant mutants is a set of commit-relevant mutants that has no commit-relevant mutant killed by a proper subset of tests that kill another commit-relevant mutant}}. To achieve this, we propose an experimental approach for identifying commit-relevant mutants using the notion of observational slicing \cite{BinkleyGHIKY14}, i.e., the relevance of an instruction to a program point of interest (such as a program state or variable(s)) can be determined by mutating instructions and observing their impact to the point of interest (changes on the target program state or variable). Since we aim to identify mutants relevant to changed instructions, we check the impact of mutants located on the changed code, as performed by observational slicing, on mutants located on unchanged code. In essence, with this approach, we measure the impact of second-order mutants on the first-order ones \cite{KintisPM12, KintisPM15}, which captures the existence of implicit interactions between the changed and unchanged code parts.
Overall, our formulation of the commit-aware mutation testing addresses the limitations and challenges of the state of the art~\cite{ma2020commit, ma2021mudelta}, in particular, making it more general and applicable for most evolving systems (\textit{see Section \ref{exp-goals}}). Using this approach, we elicit the properties of commit-relevant mutants and study the advantage of commit-relevant mutant selection in comparison to random mutant selection or mutants located on program changes.
To the best of our knowledge, this is the most extensive empirical study of commit-relevant mutants. Specifically, our evaluation setup contains 10,071,875 mutants and 288 commits extracted from five (5) mature open-source software repositories. Our experiments took over 68,213 CPU days of computation. The main objective of this work is to provide scientific insights concerning the application of mutation analysis in testing evolving software systems. The main findings of this paper are summarized as follows:
\begin{itemize}
\item \textit{Commit-relevant mutants are prevalent}. In our evaluation, \textit{30\%} of mutants are commit-relevant, on average. Hence, \revise{by reducing the number of mutants (by around 70\%) and concentrating merely on those representing change-aware test requirements, considerable cost reductions can be achieved}.
\item \textit{Selecting subsuming commit-relevant mutants significantly reduces \revise{the number of mutants}}. \revise{Selection of \textit{subsuming commit-relevant mutants} reduces even further the number of mutants, by about 93\%, on average.}
\item \textit{A large proportion of commit-relevant mutants are located outside of the program changes}. The majority of the commit-relevant mutants are located outside the changed methods (\textit{69\%}).
\item \textit{Several evaluated commit or mutant related features can not reliably predict (subsuming) commit-relevant mutants}. For instance, (the number of) commit-relevant mutants cannot be reliably predicted by features such as the commit size or mutant operator types.
\item \textit{State of the art mutant selection approaches miss a large portion of commit-relevant mutants.} As an example, random mutant selection techniques miss approximately 45\% of subsuming commit-relevant mutants when analyzing the scope of 20 mutants.
\item \textit{Commit-relevant mutation testing significantly reduces the text executions in comparison to the state of the art mutant selection methods}. \revise{Specifically, commit-relevant mutation testing reduces the number of test executions by about 16 times compared to random mutant selection.}
\end{itemize}
\section{Commit-aware Mutation Testing}
\label{sec:approach}
\subsection{Definition}
\revise{
Intuitively, \textit{commit-relevant} mutants are those that are linked with (capture) changed program behaviour, by the committed changes.
}
\revise{
These mutants are those that a) are killable and are located on the changed lines, because they capture behaviour relevant to the committed changes, and b) those that are killable, are located on unchanged lines and affect the changed, by the commit, program behaviour, because they capture the interaction of the changed and unchanged code. This is approximated by a special form of observational slicing that uses higher order mutants. The idea is that mutants, located on unmodified code, that impact the behavior of mutants located on modified code, are commit-relevant because they interact/depend with the changed code. Consider two first-order mutants $M_X$ and $M_Y$, such that $M_X$ is located on changed code and $M_Y$ is located within the changed code. Then, the higher order mutant ($M_{XY}$) is the one created by combining $M_X$ and $M_Y$. We say that $M_X$ is \textit{commit-relevant } if the higher order mutant ($M_{XY}$) has a different program behaviour from the first order mutants $M_X$ and $M_Y$. That is, $M_X$ is commit-relevant if ($M_{XY}$ != $M_{Y}$) and ($M_{XY}$ != $M_{X}$).
Formally, the definition of \textit{commit-relevant mutant} can be formed as:
}
\begin{definition}[Commit-relevant mutants]\label{def:relevant-mutant}
\revise{
A mutant \textit{$M_X$} is relevant to a commit-change, if a) it is killable and is located on the changed code, or b) there is a second order mutant \textit{$M_{XY}$} (formed by the mutant pair of \textit{$M_X$}, located outside the change, and \textit{$M_Y$}, located on the change) that has different behaviour from the two first-order mutants \textit{$M_X$} and \textit{$M_Y$} that it is composed of.
}
\end{definition}
\begin{figure*}[bt]
\begin{center}
\includegraphics[width=1.0\textwidth,trim={0cm 4cm 0cm 4cm},clip]{figures/Demonstrating_example_v3.pdf}
\caption{Example of relevant and not-relevant mutants. Left Sub-Figure: Mutant $M_X^1$ is relevant as mutant \textit{$M_Y$} impacts its behavior. Center Sub-Figure: Mutant $M_X^2$ is non-relevant as mutant \textit{$M_Y$} does not impacts its behavior. Right Sub-Figure: mutant $M_X^3$ is not relevant since there is no behavioral difference for every possible $M_Y$.}
\label{fig:Demonstrating_example}
\end{center}
\end{figure*}
\subsection{\revise{Motivating Examples}}
\subsubsection*{\textbf{\revise{Simple Example}}}
\revise{\autoref{fig:Demonstrating_example} describes three simple scenarios illustrating commit-relevant mutants on a toy code example.}
In the code snippet on the left, we observe the example function \textit{fun} that takes two arguments (integer arrays of size \textit{3}).
It starts by sorting the arrays' elements, then makes computations, and returns an integer as a result.
The {\color{green}green} rectangle on line seven (\textit{7}) represents the line that has been modified in the code.
Using Java comments (symbols ``\textit{//}'') on line three (\textit{3}) we represent mutant outside the change $M_X$, and the mutant on the change $M_Y$ on line seven (\textit{7}).
Mutant $M_X$ changes the value of variable ``\textit{R}'' to zero (\textit{0}), while the mutant \textit{$M_Y$} changes the value of variable ``\text{L}'' to one (\textit{1}).
Consider that our test suite is confirmed just by one test that invokes function \textit{fun} with the following arguments: \textit{z = \{0, 3, 3\} } and \textit{k = \{0, 2, 3\}}.
The output of the corresponding input value is observed from the inside of an atomic assertion as the input's actual value.
We can see that after comparing obtained values after running each mutant in isolation, given the same test input, the mutant's behavior is different.
Following our definition, this suggests that \textit{$M_X$} is relevant to the modification, since the actual execution value output (\textit{fun(z, k)}) for mutant \textit{$M_X$} is \textit{0}, which is different from mutant \textit{$M_Y$} whereas \textit{fun(z, k) = 1}, and \textit{$M_{XY}$} \textit{fun(z, k) = -1}.
The code snippet in the middle of the figure presents the scenario in which a mutant is not relevant to the modification.
Precisely, let us consider the same mutant $M_Y$ on the change on line \textit{5.+} as before, but now mutant $M_X$ located outside the change is on line \textit{3}.
Mutant $M_X$ modifies the assignment statement into \textit{R = 0}.
Given the same test input as before (i.e., \textit{z = \{0, 3, 3\} } and \textit{k = \{0, 2, 3\}}), and following the mutants execution behavior, we can
observe that mutants show no observable interaction.
Therefore, mutant \textit{$M_X$} is not considered relevant for this particular change.
The code snippet on the right side shows an additional example of a non-relevant mutant.
However, in this example, we observe two mutants that are unreachable from each other.
These two mutants, for any test input, do not show observable differential interaction.
Therefore, mutant \textit{$M_X$} is considered to be non-relevant to test the corresponding change.
\begin{minipage}{\textwidth}
\begin{minipage}[b]{\textwidth}
\begin{figure}[H]
\centering
\includegraphics[scale=0.5,trim={0cm 3.4cm 0cm 3.3cm},clip]{figures/Real_case_example.pdf}
\caption{\revise{Method (\texttt{read()}) excerpted from the \texttt{BoundedReader.java} program in the
Apache commons-io project (version \texttt{81210eb})}}
\label{fig:Real_case_example}
\end{figure}
\end{minipage}
\end{minipage}
\subsubsection*{\textbf{\revise{Real Example}}}
\revise{
\autoref{fig:Real_case_example} presents an excerpt of a program from the Apache commons-io\footnote{https://github.com/apache/commons-io} project, version \texttt{81210eb}. The figure shows an evolution of the program in which the function \texttt{read} was modified in line 142 (from \texttt{org.apache.commons.io.input.BoundedReader.java}). The program change adds a constraints on the function's return value, suggesting that it should return negative one (-1) in case the buffer does not contain any more values, and otherwise it should return the index of the current iteration (i.e., \texttt{i}). Specifically, the previous version of the program always returned \texttt{i} in line 142, but the modified version either returns -1, when \texttt{i} is equals to 0, or \texttt{i} otherwise.}
\revise{
Function \texttt{read} takes three parameters, namely, an array of chars \texttt{cbuf}, and two integers \texttt{off} and \texttt{len}. Intuitively, function \texttt{read} aims at modifying a certain number of characters (\texttt{len}) of array \texttt{cbuf}, starting from the given offset position \texttt{off}.
The function starts by reading a new character from a different buffer (see built-in \texttt{read()} invocation in line 140), then it proceeds to update \texttt{cbuf} array with the new character, and finally it returns the number of updated characters\footnote{For more information about the implementation of this method, please refer to official implementation documentation page:\cite{junitAPIlink}}
Notice that the \texttt{read()} invocation (line 140) returns the fed character as an integer in the range between zero (0) to 65535 (0x00-0xffff), or it returns negative one (-1) if the end of the buffer has been reached.
}
\revise{
As an example, consider a testing scenario that executes function \texttt{read} with the following inputs:
}
\noindent
\revise{
\hspace*{2cm}\texttt{read([`X', `X', `X', `X'], 1, 2);}
}
\revise{
\noindent
and the buffer accessed by the \texttt{read()} call in line 140 is as:
}
\noindent
\revise{
\hspace*{2cm}\texttt{[`0', `1', `2', `3', `4', `5', `6', `7', `8', `9', `0'];}
}
\noindent
\revise{
For this test case, both versions of the program, the previous one and the recently changed version, will return the same output (i.e., \texttt{len = 2}).
Moreover, both versions of the program will produce the same modifications into array \texttt{cbuf} given as input, resulting in:
\begin{center}
\texttt{[`X', `0', `1', `X']}
\end{center}
This is an example of a test case that does not exercise the program changes, since the change (line 142) is never executed for this test case. Hence, the test does not show any behavioral difference between the previous version of the program and the current modified version of it.
}
\begin{minipage}{\textwidth}
\begin{minipage}[b]{\textwidth}
\begin{table}[H]
\caption{\revise{
Test output observation for \autoref{fig:Real_case_example} showing the program behavior (outputs) of original program, the changed program, and the first and second order mutants of the program. The test observations are performed using input \texttt{read([`X',`X',`X',`X'], 1, 2)} and an empty buffer (\texttt{[]}) fed to the built-in function \texttt{read} (in line 140)}}
\label{tab:case-study-table}
\centering
\scalebox{0.9}{
\begin{tabular}{l c c c c c}
\toprule
& \textbf{\scriptsize{Program Versions}} & \textbf{\scriptsize{Program Changes}} & \textbf{\scriptsize{Code line}} & \textbf{\scriptsize{Test output}} & \textbf{\scriptsize{Commit-Relevance}}\tabularnewline
\midrule
\scriptsize{Pre-commit} & \scriptsize{Old version \texttt{51f13c84}} & \scriptsize{\texttt{i}} & \scriptsize{\textit{142}} & \scriptsize{0} & \scriptsize{N/A} \tabularnewline
\scriptsize{Post-commit} & \scriptsize{New version \texttt{81210eb}} & \scriptsize{\texttt{i == 0 ? -1 : i}} & \scriptsize{\textit{142}} & \scriptsize{-1} & \scriptsize{N/A} \tabularnewline
\scriptsize{First-order mutant} & \scriptsize{M1} & \scriptsize{\texttt{== $\Rightarrow$ !=}} & \scriptsize{\textit{142}} & \scriptsize{0} & \scriptsize{\textit{Relevant}} \tabularnewline
\scriptsize{First-order mutant} & \scriptsize{M2} & \scriptsize{\texttt{len $\Rightarrow$ 0}} & \scriptsize{\textit{139}} & \scriptsize{2} & \scriptsize{\textit{Non-Relevant}} \tabularnewline
\scriptsize{First-order mutant} & \scriptsize{M3} & \scriptsize{i $\Rightarrow$ ++i} & \scriptsize{\textit{139}} & \scriptsize{1} & \scriptsize{\textit{Relevant}} \tabularnewline
\scriptsize{First-order mutant} & \scriptsize{M4} & \scriptsize{\texttt{delete statement}} & \scriptsize{\textit{140}} & \scriptsize{2} & \scriptsize{\textit{Non-Relevant}} \tabularnewline
\scriptsize{Second-order mutant} & \scriptsize{M12} & \scriptsize{\texttt{== $\Rightarrow$ != $\land$ len $\Rightarrow$ 0}} & \scriptsize{\textit{142 $\land$ 139}} & \scriptsize{2} & \scriptsize{N/A} \tabularnewline
\scriptsize{Second-order mutant} & \scriptsize{M13} & \scriptsize{\texttt{== $\Rightarrow$ != $\land$ i $\Rightarrow$ ++i}} & \scriptsize{\textit{142 $\land$ 139}} & \scriptsize{-1} & \scriptsize{N/A} \tabularnewline
\scriptsize{Second-order mutant} & \scriptsize{M14} & \scriptsize{\texttt{== $\Rightarrow$ != $\land$ delete statement}} & \scriptsize{\textit{142 $\land$ 140}} & \scriptsize{2} & \scriptsize{N/A} \tabularnewline
\bottomrule
\end{tabular}
}
\end{table}
\end{minipage}
\end{minipage}
\subsubsection*{Commit-aware Mutation Testing}
\revise{Now, consider that during the mutation testing analysis, four mutants ($M1$ to $M4$) are injected into the function\texttt{read}, as it is shown in \autoref{fig:Real_case_example} via Java comments (``\texttt{//}'').
Particularly, \textit{Mutant $M1$} is located on the modified statement in line 142, i.e., it is a mutant withing the program change, and it replaces the condition \texttt{==} (equal) with \texttt{!=} (not equal).
\textit{Mutant $M2$} is located on line 139 (outside the change) and replaces the variable \texttt{len} with a constant value zero (0), mutating the condition of the \texttt{for} loop \texttt{i < len} to \texttt{i < 0}.
\textit{Mutant $M3$} is also injected on the same statement (line 139)
but it uses an unary insertion (\texttt{++i}) to update variable $i$ within the condition check of the \texttt{for} loop, such that condition (\texttt{i < len}) is mutated to (\texttt{++i < len}).
Finally, \textit{Mutant $M4$} removes the statement located on line 140 (i.e., \texttt{c = read()}).
}
\revise{Then, by using our HOM-based approach, we can create higher-order mutants by pairing all four mutants. Precisely, we pair the mutants located outside the change with the mutants on the commit-change (line 142), and we obtain three higher-order mutants $M12$, $M13$, and $M14$. \autoref{tab:case-study-table} illustrates the behavior (outputs) of function \texttt{read} under its previous version, its current changed version, and all the mutants.
}
\revise{
Consider now a different testing scenario in which the input buffer accessed by the built-in function \texttt{read} in line 140 is empty (i.e., \texttt{[]}).
This testing scenario shows the behavioral difference between the previous version and the modified version of the program, since it executes the change (in line 142). We observe that, while the execution of the previous version of the program returns zero (0), the modified version returns negative one (-1).
}
\revise{
\autoref{tab:case-study-table} highlights the behavior (i.e., output) of each mutant. First, according to the traditional definition of commit-relevant mutants, $M1$ is a commit-relevant mutant, since it is located on the program change~\cite{CACHIA2013}. Additionally, according to our extension of the definition of commit-aware mutation (\textit{see} \autoref{sec:approach}), we compare the output of the second-order mutants and their isolated first-order mutants. We observe that the second-order mutant $M13$ is also a commit relevant mutant.
This is because the second order mutant ($M3$) has a different behavior from the isolated first-order mutants (i.e., $M13$ != $M3$, and $M13$ != $M1$). Meanwhile, the other second-order mutants, i.e., $M12$ and $M14$ are not commit-relevant because they have similar behaviors as the isolated first-order mutants (i.e., $M12$ == $M2$, and $M14$ == $M4$).
}
\subsubsection*{Commit-aware Criteria}
\revise{
Let us illustrate the importance of the strict constraint employed in our approach to compare the behaviors of first-order and second-order mutants (e.g., the need to ensure $M13$ != $M3$ \textit{AND} $M13$ != $M1$) using a counter-example.
Specifically, we will discuss the rationale for this constraint and why considering a less strict constraint does not suffice (and mutants like, for instance, $M2$ are not commit-relevant, despite the fact that $M12$ != $M1$ but $M12$ == $M2$).
Let us consider the example program in \autoref{fig:Real_case_example} and the output behavior observed in \autoref{tab:case-study-table}. Even though $M1$ and $M12$ have different output, inspecting the behavior of $M12$ on the program change using the provided test cases, we observe
that the behavior of the second-order mutant $M12$ is not different from that of $M2$. In fact, $M2$ does not execute the program change nor the mutant within the change. Indeed there is no input that can force mutant $M2$ to execute the changed line (in line 142), since the \texttt{for} loop condition will always evaluate to false.
This implies that using a less strict constraint (e.g., an ``OR'' operation) in our check, will lead to such mis-identification of commit-relevant mutants, implying that mutants that can never lead to the execution of the program change (e.,g., $M2$) can be mis-classified as relevant to the program change. Thus, it is important to ensure that the behavior of the second-order mutant and that of the isolated first-order mutants are indeed different.
}
\subsubsection*{Subsumption Relation}
\revise{Let us illustrate the subsumption relation of commit-relevant mutants.
Consider the two commit-relevant mutants in our example, i.e.,
\textit{mutant $M3$} that we have identified as commit-relevant, and
\textit{mutant $M1$}
located on the changed statement (commit-relevant by default).
In our example (\autoref{fig:Real_case_example}), both mutants are killed by the initial test input (in \autoref{tab:case-study-table}).
Let us consider that the test suite has one additional test, in particular the following test:
\begin{center}
\texttt{read([`X',`X',`X',`X'], 1, 1)}
\end{center}
\noindent
Given this new test input, we observe that \textit{mutant $M1$} is killed by this test input (output zero (0)),
but \textit{mutant $M3$} is not killed by the test (output one (1)).
Following the definition of mutant subsumption \cite{ammann_establishing_2014}, we can observe that a test that distinguishes \textit{mutant $M3$}, will also distinguish \textit{mutant $M1$}, but
\textit{mutant $M1$} can be distinguished by a test that cannot distinguish \textit{mutant $M3$}.
In this situation, we say that \textit{$M3$ subsumes $M1$}, making $M3$ a \textit{subsuming commit-relevant mutant}.
This example illustrates a scenario where a subsuming commit-relevant mutants is located outside the program change.
The mutant residing outside the change subsumes a mutant residing on the program change, which makes the test requirement of the mutant on the change redundant.
We can satisfy both mutants ($M1$ and $M3$) by writing test requirements to identify the subsuming commit-relevant mutant located outside of the committed change (i.e., $M3$), which is the commit-relevant mutant also identified by our approach.}
\subsection{State of the Art}
Let us provide background on the state of the art in commit-aware mutation testing. There are very few research papers on commit-aware mutation testing. In particular, we highlight the papers on the formalization of the concept as well as techniques for selecting commit-aware mutants.
\begin{itemize}
\item \textbf{\textit{Formalization of Commit-aware Mutation Testing:}}
Ma \textit{et al. }~\cite{ma2020commit} defines and formalizes the concept of commit-aware mutation testing.
\revise{
The paper defines commit-relevant mutants as a set of mutants capturing the interactions that affect the changed program behaviors.
The paper further shows
that there is only a weak correlation between commit-relevant mutation scores and traditional mutation scores, thus, highlighting the need for a commit-aware test assessment metric.
In its evaluation, the authors demonstrate
the strength of commit-aware mutation in revealing commit-related faults by showing that traditional mutation
has about 30\% less chances of revealing commit-introducing faults, in comparison to commit-aware mutation testing.}
\revise{
In this work, we propose an alternative approach to identify commit-relevant mutants.
In comparison to \cite{ma2020commit}, our approach
removes the strict test contract assumption
that requires test suites being executable across program versions, i.e., there are not contract changes.
In our setup, we address this concern by employing only the post-commit version and the committed change for commit-aware mutation testing, with no restriction on the evolution of the test suite (\textit{see Section \ref{exp-goals}}).
}
\item \textbf{\textit{Diff-based Commit-aware Mutant Selection:}} Cachia et al.~\cite{CACHIA2013} proposed an incremental mutation testing that limits the scope of mutant generation to strictly changed code regions since the last mutation run. Similarly, Petrovic et al.~\cite{petrovic2018} proposed a diff-based probabilistic mutant selection technique that focuses only on the mutants located within the program changes. These approaches ignore the program dependencies between the committed changes and the unmodified code by design. Hence, in this work, we present and employ an experimental approach that accounts for the dependencies between program changes and unmodified code when identifying commit-relevant mutants to demonstrate the extent to which using mutant from modified code can help. Our approach analyses the effect of second-order mutants, one within the program change and the other in the unmodified code, on the evolving program behavior.
\item \textbf{\textit{Machine Learning based Commit-aware Mutant Selection:}}
\texttt{Mudelta}~\cite{ma2021mudelta} presents a machine learning and static analysis-based approach for predicting commit-relevant mutants.
\revise{This paper illustrates the importance of commit-aware mutation testing, particularly its ability to reduce mutation testing effort and reveal commit-related faults.
In comparison to random mutant selection, \texttt{Mudelta} reveals 45\% more commit-relevant mutants, and
achieves 27\% higher fault revealing ability in fault introducing commits.
In this work, we propose a complementary dynamic analysis approach for commit-aware mutation testing. Specifically, we propose a new HOM-based observation slicing approach for commit-aware mutation testing, that is applicable in the absence of static code features or (large) training data.
}
\item \textbf{\textit{\revise{Interface Mutation:}}} \revise{
\citet{delamaro2001interface} proposed an inter-procedural mutation analysis approach for generating mutants that are relevant for integration testing, i.e., suitable for testing the interactions between program units. In particular, since interface mutation aims at testing component integrations, it injects mutants on the component contracts (interfaces) and pairs of them at the component call sites and related uses of the interface parameter inside the components bodies in order to capture potential interactions between the caller and called components. This mechanism, of capturing dependences through pairs of mutants, is somehow similar to our approach but more restrictive as it targets interfaces (call sites and method parameter uses). In contrast commit-aware mutation testing aims at identifying relevant dependencies between changed and unchanged code and not between components.
}
\end{itemize}
\subsection{Design Requirements}
\label{exp-goals}
Our commit-relevant mutation approach aims to fulfill certain requirements to ensure we gather and study a vast number of commits and commit-relevant mutants. These design requirements address some of the limitations and challenges of state of the art ~\cite{ma2020commit, ma2021mudelta, petrovic2018}. In particular, we address the following:
\begin{itemize}
\item \textbf{\textit{Location of Commit-relevant Mutants:}}
In this work, we focus on identifying commit-relevant mutants within and outside the program changes, i.e., within the commit-change as done in prior work~\cite{petrovic2018}, and the unmodified program code. In particular, we are interested in revealing behavioral interactions induced by the program changes on the rest of the unmodified program code. We achieve this by identifying the commit-relevant mutants outside of the program changes. In particular, we employ second-order mutants; we analyze the impact of second-order mutants on the behavior of the evolving program (\textit{see Section \ref{sec:approach-overview}}).
\item \textbf{\textit{Test Contract:}}
Our experimental approach employs only the test suite from the post-commit program version. In previous work~\cite{ma2020commit}, the experimental design requires the execution of test suites across the pre-commit and post-commit versions of the program. Plus, it implies that the number of tests does not increase or decreases across versions, i.e., the test contract is intact. Therefore, the approach observes the delta between versions by comparing mutants test suites from pre- and post-change commit. This assumption is impractical in several cases since the test suite also evolves as the program evolves, e.g., when implementing new features or fixing bugs. In our work, we observed that this assumption is not common in practice. In particular, in our study, the proportion of commits where the test contract is preserved is less than 40\%.
\item \textbf{\textit{\revise{Commit Patches and Hunks}}}:
\revise{In our approach for commit-aware mutation testing, we require the commit patches and commit hunks for empirical evaluation and analysis.
Indeed, commit properties are vital for commit-aware mutation testing and commonly used
by the state of the art techniques~\cite{ma2020commit,ma2021mudelta}. In our work, we employ commit properties (i.e., patches and commit hunks) for both commit-relevant mutant detection and
experimental analysis. Particularly, our approach requires the commit patches (i.e., the delta between pre-commit and post-commit versions) to identify the interaction of the mutants within the change and the mutants outside the change.
We also employ commit hunks in our experimental analysis, to identify
the number of individual code-chunks structure present in a commit.
Applying commit hunks in our analysis sheds light on the relationship between altered statements in the commit hunk and the mutants residing outside the commit hunk.
}
\item \textbf{\textit{Post-Commit Version:}} Our experimental approach requires \textit{the post-commit version of the program to be compilable, executable, and testable}. \revise{These requirements are vital for the dynamic analysis of our approach and they only apply to the \textit{post-commit version} of the program.
In contrast, previous work~\cite{ma2020commit} requires two program versions (pre-commit and post-commit) and \textit{assumes a green test suite, no build failures and no compilation errors for both program versions}.
In our evaluation setup, we find that these conditions are uncommon (less than 40\% of the cases).
To address this concern, we ensure our approach requires the post-commit version of the program, without need for the pre-commit version. This allows collecting significantly more commits for our study, and allows to evaluate a vast amount of commit-relevant mutants. }
\item \textbf{\textit{Test Oracle: }} \revise{In this work, we employ test assertions of system units as our test oracle. This is a fine-grained test oracle used by unit tests. Here it is important to note that when we refer to assertions these are \textit{test assertions}, and \textit{not program assertions} and are used for checking the observable behaviour of the program units, as mandated by strong mutation.
In practice, we use test assertions to define the mutant behaviour
study the impact of mutants and changes on program behaviour.
}
\item \textbf{\textit{Number of Commits:}} Our empirical study characterized commit-relevant mutants and required a substantial number of commits and commit-relevant mutants. Dues to the flexibility of our experimental approach, in this study, we analyzed significantly (10x) more commits and mutants than previous work~\cite{ma2020commit}. We addressed the significant limitations and assumptions of prior work, which could prevent gathering a sufficient number of commits and commit-relevant mutants. For instance, as stated in the paper \revise{Ma et al.}~\cite{ma2020commit}, it is challenging to find commits in open-source projects that do not break the test contract, i.e., keep the test suite intact. This challenge further inhibits our goal of automatically studying the characteristics of relevant mutants. Addressing the concerns above allows us to gather and study more commits than previous studies.
\end{itemize}
Our experimental approach aims to target the requirements above to ensure that we gather many commits and cover several realistic corner cases for evolving software systems. Overall, fulfilling these requirements and addressing these concerns enables us to collect significantly more commits and identify significantly more commit-relevant mutants for our study. In particular, our study involved 10x more commits and 6x more commit-relevant mutants than previous studies.
\subsection{Approach Overview}
\label{sec:approach-overview}
Our study aims at investigating the existence and distribution of commit-relevant mutants in evolving software systems.
Specifically, we study the relationship between the lines of code changed in a commit hunk and the mutants residing on program locations outside the commit hunk under consideration.
Intuitively, we want to study the interaction between two program locations, where one location is part of the commit hunk, and the other is outside the change.
We plan to employ high-order mutants (second-order to be more precise) and simulate potential changes in a commit hunk and the mutants outside the commit hunk.
This study aims at providing scientific evidence of the relationship and relevance of mutants (test requirements) outside commit hunks that need to be taken into account when testing evolving systems.
To determine if a mutant is relevant for a commit hunk, we plan to observe whether the commit changes affect mutants' behavior.
Intuitively, suppose a change in a location in the commit hunk (produced by a mutation) affects the outcome of the mutant outside the commit. In that case, we have evidence that there exists an interaction between these two locations, indicating that the mutant is \emph{relevant} for the commit. The absence of interactions indicates either the existence of equivalent mutants \cite{KintisPM12, KintisPM15} or the absence of dependence/relevance. To account for the case of equivalent mutants and ensure the relevance of observations, we sample multiple mutants per statement.
Mutants' behavior is (partially) determined by observing their covering test set.
Implying that if we want to observe the interaction between mutants on different locations, the test set should make any difference in the mutant's behavior whether they are run in isolation or combined both.
More precisely, if we can observe that the behavior of two mutants $M_X$ and $M_Y$ run in isolation differs from the behavior of the second-order mutant $M_{XY}$ (obtained by combining both mutations $M_X$ and $M_Y$), then we can conclude that mutants $M_X$ and $M_Y$ influence each other.
Figure \ref{fig:Approach_intro} depicts such a situation, where the test set $\{t_0, t_1\}$ is able to observe that $M_{XY}$'s behavior differs from $M_X$ and $M_Y$'s behavior. For instance, test $t_0$ passes on mutants $M_X$ and $M_Y$ but fails on mutant $M_{XY}$.
Thus, we can conclude that locations in which mutations $M_X$ and $M_Y$ were applied to interact with each other.
Following a similar idea, consider generating one of the mutants outside the change ($M_X$) and the other one on the change ($M_Y$), and their combination makes a second-order mutant ($M_{XY}$) suitable for observing if there exists an interaction between them.
To determine if mutant $M_X$ is relevant for the commit change, we can iterate this process by exploring different high-order mutants $M_{XY}$ by varying mutant on the change $M_Y$, with the aim at finding one combination that evidence their interaction.
To compare mutants' behaviors, first, we need an intersection set of tests covering mutants $M_X$, $M_Y$, and $M_{XY}$.
Second, we proceed to run these tests to observe a difference between the mutants.
Instead of considering only passing and failing output as a standard unit level testing oracle, we instrument tests and contained assertions to obtain and compare actual assertion values.
For instance, an assertion like \texttt{assertEquals(0, Z)} can be violated by a (potentially) infinite number of values for \texttt{Z}, all of them violating the assertion. Suppose after executing mutants $M_X$, $M_Y$ and $M_{XY}$, the value of \texttt{Z} is different. In that case, we can observe a difference in their execution, allowing us to determine if there exists an interaction between these mutants, concluding that mutant $M_X$ is relevant for the commit change.
Section~\ref{sec:pitest-assert-tool} describes the implementation details on how we instrument test executions to obtain actual assertion values.
Figure \ref{fig:Approach_figure_example} illustrates our approach to detect interactions between mutants by comparing their behavioral assertion values.
It depicts that after executing each first-order mutant $M_X$ and $M_Y$ in isolation (assertions that cover them), we compare the output values with the value obtained after running second-order mutant $M_{XY}$.
If running $M_X$ and $M_Y$ in isolation differs from running $M_{XY}$, we determine that mutant $M_X$ is relevant for the commit change.
\begin{figure*}[htp!]
\begin{center}
\includegraphics[width=0.8\textwidth,trim={0cm 5cm 0cm 4cm},clip]{figures/Approach_figure_example_v3.pdf}
\caption{A mutant \textit{$M_X$} is relevant to a commit-change, if any higher order mutant \textit{$M_{XY}$}, shows different behavior from \textit{$M_X$} and \textit{$M_Y$} executed in isolation}
\label{fig:Approach_figure_example}
\end{center}
\end{figure*}
\subsection{Algorithm}
\label{sec:approach-algorithm}
To perform an empirical study toward distinguishing relevant mutants, we generate the first order mutants located around and on the commit change (i.e., $M_X$ and $M_Y$, respectively). The second-order mutants (i.e., $M_{XY}$) are a combination of the previous two.
The \emph{mutant-assertion matrices} were obtained by executing the mutants against developer-written and automatically generated test pools. Note that test run status is pass/fall for Java programs; therefore, to observe behavioral differences produced by mutants, we need to focus on test assertions and record assertion execution actual value output of each test on every mutant.
Precisely, for every mutant and every test assertion, a mutant-assertion matrix stores the assertion values obtained after running a mutant against a test.
As noted, this study performs mutation analysis on commits from Java programs, using Pitest\footnote{http://pitest.org/} as the Java mutation testing tool, and EvoSuite\footnote{https://www.evosuite.org/} as the state of the art test case generation tool.
Section \ref{analysis_procedure} provides further details regarding mutants test case generation and test assertions instrumentation.
After computing mutant-assertion matrices, we proceed to approximate which mutants are relevant to the change, according to our Definition~\ref{def:relevant-mutant} following the steps incorporated in Algorithm ~\ref{algo:relevant}.
The algorithm summarises previously described process, where functions \emph{MutantsonChangeMutantOuput}, \emph{aroundChangeMutantOutput}, \emph{highOrderMutantOutput} return the output of a specific test \textit{assertion} execution per specific \textit{mutant}.
Finally, Algorithm~\ref{algo:relevant} returns a set of relevant mutants for a particular commit change.
\revise{
This algorithm has a worst-case polynomial time complexity of $O(n^4)$, due to the four nested \texttt{for} loops ($O(n*n*n*n)$). For each of the three inputs fed to the algorithm (\emph{TestSuite}, \emph{MutantsOnChange} and \emph{MutantsAroundChange}),
there is a linear-time complexity ($O(n)$). Additionally, there is a linear-time complexity ($O(n)$) for evaluating each test assertion corresponding to the test cases.
Overall, the performance of the algorithm depends on the number of mutants in the change, the number of mutants injected in the modified code, the size of the test suite and the number of assertions in each test. Specifically, to derive higher-order mutants, we consider every pair of mutants within and outside the change, we also execute all test cases corresponding to these mutants, and evaluate all test assertions in each test case. This algorithm can be optimized by improving the number of evaluated tests, assertions or pairs of mutants.}
\revise{The complexity of this algorithm can be reduced to ($O(log(n) * n^3)$ via a binary search on the pair of mutants (outside the change) that exposes a behavioral difference. Likewise, the complexity can be reduced to cubic complexity ($O(n^3)$) by executing a constant number of test cases/assertions ($O(1)$). For instance, an improvement is achievable by selecting and executing only the most relevant tests for the changes, e.g., from historical test executions in the CI. A reduction is also achievable if only one test assertion is evaluated for each test case, e.g., executing only the assertion that captures the interaction between the pair of mutants
has a constant time complexity ($O(1)$).
}
\setlength{\algomargin}{0.99em}
\setlength{\textfloatsep}{13pt}
\begin{algorithm}
\SetAlgoLined
\LinesNumbered
\KwData{TestSuite, MutantsOnChange, MutantsAroundChange}
\KwResult{Relevant Mutants}
$RelevantMuts \leftarrow \emptyset$\;
\For{$X \in MutantsAroundChange$}{
\For{$Y \in MutantsOnChange$}{
\For{$test \in TestSuite$}{
\For{$assertion \in test$}{
$Yval \leftarrow onChangeMutantOuput(assertion, Y)$\;
$Xval \leftarrow aroundChangeMutantOutput(assertion, X)$\;
$XYval \leftarrow highOrderMutantOutput(assertion, Y, X)$\;
\If{$Yval \neq XYval \land Xval \neq XYval$}{
$RelevantMuts \leftarrow RelevantMuts \cup \{X\}$\;
\textbf{jump to line 2 and take next mutant $X$}\;
}
}
}
}
}
\Return $RelevantMuts$ \;
\caption{Approximate Commit-relevant Mutants Set}
\label{algo:relevant}
\end{algorithm}
\section{Background}\label{mutation_testing}
\subsection{Mutation Testing}
Mutation is a test adequacy criterion in which test requirements are characterized by mean of \emph{mutants} obtained by performing slight syntactic modifications to the original program (for instance, the relational expression $\texttt{a > b}$ can be mutated into $\texttt{a < b}$).
Intuitively, these mutants aim at representing artificially injected faults that can be used to assess the effectiveness and thoroughness of a test suite in detecting these seeded faults.
Then, the tester starts by analyzing the mutants and proceeds to design test cases to \emph{kill} them, i.e., to distinguish the observable behavior between the mutant and the original program.
Hence, the adequacy of a test suite concerning the mutation criterion, called \emph{mutation score}, is computed as the ratio of killed mutants over the total number of mutants.
Notice that the number of mutants not necessarily represent the number of test cases required to cover all of them since several mutants can be redundant.
On the one hand, there may exist mutants that cannot be killed by any test since they are functionally \emph{equivalent} to the original program.
On the other hand, one test may kill other mutants at the same time.
Thus, the effort put into analyzing and executing redundant mutants is wasted; hence it is desirable to analyze only the mutants that add value.
\subsection{Subsuming Mutants}
\label{sec:subsuming_mutants}
Subsuming relations aims at finding the minimal set of mutants required to cover all (killable) mutants~\cite{ammann_establishing_2014}. Intuitively, this set of mutants has minimal redundancies and represents a nearly optimal mutation testing process with respect to cost \cite{PapadakisHHJT16, PapadakisCT18}. More formally, let us consider that $M_1$, $M_2$, and $T$ be two mutants and a test suite, respectively.
Consider also that $T_1 \subseteq T$ and $T_2 \subseteq T$ are the set of tests from $T$ that kill mutants $M_1$ and $M_2$, respectively, where $T_1 \neq \emptyset$ and $T_2 \neq \emptyset$ indicating that both $M_1$ and $M_2$ are killable mutants.
We say that mutant $M_1$ subsumes mutant $M_2$, if and only if, $T_1 \subseteq T_2$.
In case $T_1=T_2$, we say that mutants $M_1$ and $M_2$ are indistinguishable.
The set of mutants that are both killable and subsumed only by indistinguishable mutants are called \emph{subsuming mutants}.
For instance, assuming that $T_1=\{t_1,t_2\}$ and $T_2=\{t_1,t_2,t_3\}$, one can notice that every time we run a test to kill mutant $M_1$ (i.e., $t_1$ or $t_2$) we will also kill mutant $M_2$.
While the vice versa does not hold since if we kill mutant $M_2$ by $t_3$, we will not kill mutant $M_1$.
In this case we say that $M_1$ subsumes $M_2$.
\revise{Several researchers have studied the impact and prevalence of subsuming mutants for traditional mutation testing~\cite{PapadakisHHJT16, PapadakisCT18, alipour2016evaluating, guimaraes2020optimizing}.
For instance, \citet{alipour2016evaluating} demonstrated that subsuming mutants can reduce traditional mutation testing effort by up to 80\%. In particular, in their empirical study on mutation test reduction, found that subsuming mutants can reduce the number of mutants requiring analysis by up to 80\%. Their study demonstrated the importance of subsuming mutants in traditional mutation testing, emphasizing that there is strong inter-dependency among mutants. In their empirical evaluation of \textit{traditional mutation testing} (involving four C projects and thousands of mutants), the paper found that test case reduction based on a single mutant can reduce mutation testing effort (in terms of the number of mutant test executions) by 33 to 80\%~\cite{alipour2016evaluating}.
Likewise, \citet{guimaraes2020optimizing} empirically demonstrated that identifying dynamic subsumption relations among mutants reduces traditional mutation test execution time by 53\%. \citet{delamaro2001interface} also demonstrated that identifying the inter-procedural relation among mutants in two program units helps to identify interface mutants, i.e., the mutants that are
relevant for
mutation testing during system integration.
}
\revise{However, despite the evidence of the impact of subsuming mutants on traditional mutation testing and integration testing, their impact
on commit-aware mutation testing remains unknown. Thus, \textit{in this paper, we study the prevalence and distribution of mutants relevant for a committed change and the extent to which subsuming relations are maintained.} }
\subsection{High-Order Mutants}
Depending on the number of mutation operators we apply to the original program, we can categorize the obtained mutants by the number of simple changes one has to introduce to form them.
That is, \emph{first-order mutants} (FOM) is obtained by making only one simple syntactic change to the original program.
Second-order mutants (SOM) are obtained by making two syntactic changes to the original program (or applying one mutation to first-order mutants). In the general case, higher-order mutants (HOM)~\cite{jia2009} are produced after the successful application of \textit{n} mutations to the original program.
At the very beginning, using higher-order mutants in mutation testing was not considered viable because of the \emph{Coupling Effect} proposed by DeMillo et al. \cite{demillo1978}. It stated that ``Test data that distinguishes all programs differing from a correct one by only simple errors is so sensitive that it also implicitly distinguishes more complex errors''.
However, later on, \citet{offut1992} defined first-order mutants as simple faults while characterizing higher-order mutants as complex artificial defects.
In this study, we plan to use second-order mutants as the means for studying whether mutants located outside the commit change interacts with the mutants located within the change. Then we use this information to determine if mutants are relevant or not for given commit changes. Details are presented later in Section~\ref{sec:approach}.
\begin{figure}[bt]
\centering
\includegraphics[scale=0.30,trim={0cm 3.5cm 0cm 3.5cm},clip]{figures/Figure_program_evolution_v_3.pdf}
\caption{\revise{Typical evolution of a software and its test suite showing three versions ($v_1$ to $v_3$) of a program (\texttt{main}) and and its test suite (\texttt{test}). The {\color{green} green} portions of the program (\texttt{main}) symbolize the \textit{program changes} (e.g., a commit), and the explosions symbolize the \textit{mutants} injected into the program.
In the test suite (\texttt{test}), $t_i$ symbolizes a \textit{test case} $i$, and the {\color{green} green} rectangles represent changes in the test suite (i.e., addition and modification of tests). The test suite and source code evolve as the program evolves through versions. As the size of the program increases, we can observe that the number of mutants increases as well. This eventually leads to a substantial number of irrelevant mutants that result in waste of efforts. In the figure, with {\color{red} red} the mutants that are commit-relevant and with {\color{yellow} yellow} the irrelevant ones. Focusing only on commit-relevant mutants reduces the number of mutants requiring attention and leads to significant cost reductions. Additionally, the set of commit-relevant mutants quantifies the extent to which practitioners have tested the program behaviors affected by the change. } }
\label{fig:testrig}
\end{figure}
\subsection{Testing Evolving Systems}
Software systems evolve frequently, hence, it is pertinent to provide methods and tools to analyze the impact of the program changes.
\emph{Regression testing} helps in this respect by re-running the test suite on the new version of the code to ensure that the previously developed functionality behaves as expected.
Software evolves for many reasons (e.g., due to bug fixes, code refactoring or new features).
Therefore it is important to understand how to
test the program change, if it is enough to test only the changed lines, as well as how many test requirements and test cases will need to be analyzed. \revise{Notably, developers are burdened with the challenge of testing evolving systems, specifically,
how to effectively analyse
the difference in the program behaviors induced by their changes. These are the main challenges of regression testing, and in this work, we aim to study these challenges via the lens of mutation testing.}
\revise{Generally, as the software evolves, the test suite also evolves. Concretely, as the program changes (e.g., due to new features or bug fixes), new tests are added or old tests may be modified to exercise those changes.
Figure \ref{fig:testrig} illustrates the evolution of a program and its test suite during a typical software development process,
showing changes in four versions of the program (\texttt{main}) and the test suite (\texttt{test}).
In this example, we illustrate that analyzing all mutants is costly, as the number of mutants (both red and yellow in \autoref{fig:testrig}) is independent of the program changes (is actually depended on the size of the programs) and increases as the program size increases.
Hence, traditional mutation testing will be costly, since it uses more mutants than required. More importantly, by doing so, developers will have to analyze mutants that are not relevant to what they actually committed. In this work, we study how to address these challenges using commit-aware mutation testing.
}
A common approach to address these challenges is to leverage code coverage information, i.e., analyzing the test coverage of a particular change, to decide if the change needs further test cases or not.
However, previous works~\cite{Fowler,KurtzAODKG16} have shown that many severe integration issues arise from unforeseen interactions triggered between introduced change and the rest of the software.
\revise{Therefore, there is a need for change-aware test metrics to guide effective regression testing
and allow developers to quantify the extent to which they
tested the error-prone program behaviors affected by their changes.}
We plan to use mutation testing to address these interactions by targeting suitable mutants that demonstrate an (implicit) interaction between the changed lines and the unmodified part of the program (i.e., the code outside the change).
\revise{These mutants form the change-relevant requirements and should be used to determine whether test suites are adequate and provide guidance in improving the test suite.}
\section{Conclusion}\label{sec:conclusion}
We presented an empirical evaluation of the characteristics of commit-relevant mutants. In particular, we have studied the prevalence, location, effectiveness, and efficiency of commit-relevant mutants. We have also examined the comparative advantage of commit-relevant mutants compared to two baseline methods, i.e., random mutant selection and selecting mutants within program changes. Notably, we found that commit-relevant mutants are \textit{highly prevalent} (30\%), and primarily located outside of program changes (81\%).
In addition, we observed that effective selection of commit-relevant mutants affords a significant testing advantage. Specifically, it has the potential of significantly reducing the cost of mutation, and it is significantly more effective and efficient than random mutant selection and analysis of only mutants within the program change.
We also investigate the predictability of commit-relevant mutants by considering typical proxy variables (such as the number of mutants within a change, mutant type, and commit size) that may correlate with commit-relevant mutants. However, our empirical findings show that these candidate proxy features do not reliably predict commit-relevant mutants, indicating that more research is required to develop tools that successfully detect this kind of mutants.
\section{Discussion}\label{Discussion}
\subsection{Summary of Findings}
Commit-relevant mutation testing allows developers to identify and select the mutants necessary for testing the program changes to avoid regression bugs and newly introduced failures. This paper presents an empirical study that examines the prevalence and characteristics of commit-relevant mutants and provides scientific insights concerning the mutation testing of evolving software systems. Our main empirical findings include the following:
\begin{enumerate}
\item
Commit-relevant mutants, at unit level, are \textit{highly prevalent} (30\%) and most commit-relevant mutants (81\%) are \textit{located outside of program commit changes}. Hence, it is important to conduct mutation analysis of evolving systems to determine the influence of the program changes on the rest of the unmodified code.
\item
Adequate selection of (subsuming) commit-relevant mutants significantly reduces \revise{the number of mutants involved (approximately 93\%)}; thus, there is a huge benefit to developing effective and practical techniques for the selection of (subsuming) commit-relevant mutants in evolving systems.
\item
Predicting (subsuming) commit-relevant mutants is not a trivial task. In our evaluation, we studied several candidate \textit{proxy variables} that \emph{do not reliably predict} commit-relevant mutants, including the number of mutants within a change, mutant type, and commit size. Hence, we encourage the development of statistical or machine learning approaches and program analysis techniques to predict or identify commit-relevant mutants automatically.
\item
Selecting commit relevant mutants is \textit{significantly more effective and efficient than random mutant selection and the analysis of only mutants within the program change}. Commit-relevant mutation testing can reduce testing effort \revise{(i.e., number of test executions)} by up to \revise{16} times, and by half, compared to random mutant selection and mutants within a change, respectively.
\end{enumerate}
Firstly, our evaluation results show that most commit-relevant mutants located outside of the commit changes due to the interaction of changes with the unmodified program code. However, in our evaluation, commit-relevant mutants that capture evolving software behavior are located all around the program changes. Besides, we observe that effective selection of commit-relevant mutants significantly reduces the number of mutants requiring analysis. Thus, we encourage researchers to investigate automated methods for identifying and selecting commit-relevant mutants, for instance, using statistical analysis or program analysis.
In addition, we observed that commit-relevant prediction and selection is a challenging task. For example, many proxy variables could not reliably predict commit-relevant mutants in our analysis (\RQ2 to \RQ4). To buttress this, we further conducted a correlation analysis of the features of commit-relevant and non-relevant mutants using control and data flow features selected from Chekam \textit{et al. }~\cite{ChekamPBTS20}.
The goal is to determine if mutants' features previously used for other prediction tasks, for instance, for selecting fault revealing mutants ~\citet{ChekamPBTS20}, can also distinguish commit-relevant mutants.
\autoref{fig:RQ4-features_correlation} presents our findings using a heat map, where each map coordinate represents Spearman correlation coefficient calculated between two features on the coordinates.
These features characterize relevant and not relevant mutants, labeled with suffix "R" or "N", respectively.
Notably, we observe that there are no strong positive or negative correlations among these features.
This implies that these features can not directly help in distinguishing between commit-relevant and non-relevant mutants.
However, we can observe two cases of a medium positive correlation between the same class features, in particular, \textit{CfgDepth} and \textit{NumInDataD} between both classes show correlation.\footnote{\textit{CfgDepth} means the depth of a mutant in the control flow graph, i.e., the number of basic blocks to follow to reach the mutant, and \textit{NumOutDataD} refers to the number of mutants on expressions on which a mutant $m$ is data-dependent.} This phenomenon is expected since there will be more data-dependent expressions as the depth of a mutant in the control flow graph increases.
\begin{figure*}[bt!]
\begin{center}
\includegraphics[width=1.\textwidth]{figures/Heatmap_Mutants_properties_spearman.pdf}
\caption{Correlation between features of relevant and non-relevant mutants labeled with suffixes ``R" and ``N", respectively. The features examined include the following:
\texttt{CfgDepth} - \textit{Depth of a mutant in Control Flow Graph, i.e., the number of basic blocks to follow in order to reach the mutant};
\texttt{NumOutDataD} - \textit{Number of mutants on expressions data-dependent on a mutant expression};
\texttt{NumInDataD} - \textit{Number of mutants on expressions on which a mutant $m$ is data-dependent};
\texttt{NumOutCtrlD} - \textit{Number of mutants on expressions control-dependent on a mutant}; and
\texttt{NumInCtrlD} - \textit{Number of mutants on expressions on which $m$ is control-dependent}.
}
\label{fig:RQ4-features_correlation}
\end{center}
\end{figure*}
Furthermore, we found that commit-relevant mutant selection considerably improves the effectiveness and efficiency of testing evolving systems, especially in comparison to the random mutant selection, and using the mutants within the program changes (\RQ5 and \RQ6). Overall, these empirical findings shed more light on the challenge of mutation testing of evolving systems and provide directions for future research into the selection and prediction of commit-relevant mutants.
\subsection{Implications}
\revise{
The main insight of our study is
\textit{the need to pay attention to the effective identification, selection or prioritization of commit-relevant mutants}. This is particularly important to reduce the effort required for mutation-based regression testing. Notably, an effective commit-aware mutant selection method can significantly reduce the number of mutants involved. We also shown that \textit{commit-relevant mutants are located both within and outside program changes}. Precisely, we demonstrate that beyond the committed changes, other program locations are also important for commit-aware mutation testing. Hence, it is important to identify the relevant program locations for commit-aware mutant injection. To achieve this, we encourage the use of program analysis techniques (e.g., slicing) that determines the program dependencies between changes and the rest of the program, such that mutant injection is focused on selecting such dependencies to effectively reduces the search space and cost for mutation testing. It is also pertinent to note that \textit{the subsumption relation of mutants can help in reducing considerably the effort during commit-aware mutation testing}. Indeed, it is important to prioritize subsuming mutants during mutation testing of evolving systems. }
\revise{
To achieve the aforementioned goals, i.e., automate the identification and selection of commit-relevant mutants to aid developers,
we turn to the research community to develop and investigate the techniques required for effective commit-aware mutation testing.
Thus, the takeaway of this study is the need to develop: a) novel techniques for \textit{selecting, prioritizing and predicting commit-relevant mutants}; and b) \textit{commit-aware test metrics} to determine the adequacy of commit-aware mutation testing. Although the problem of mutant selection/identification of the relevant mutants is active for traditional mutation testing, this is hardly well-studied for commit-aware mutation testing.
This is an important problem since several studies~\cite{ma2020commit, ma2021mudelta} (including this study) have demonstrated that traditional (random) mutation testing
is significantly costly for evolving software.
}
\revise{
This paper has further illustrated that dynamic approaches (like observation slicing) can complement
static or machine learning based
approaches in effectively identifying commit-relevant mutants.
We have also observed that commit-relevant mutants
cannot be predicted using only the committed changes or program dependence properties.
This implies that
the current state-of-the-art is not generally applicable for commit-aware mutation testing in practice. Thus, for more effective approaches, we believe researchers need to consolidate the knowledge from several sources, including the commit difference, mutant properties, the semantic behavior of mutants, and the semantic divergence produced by the change.
To this end, we encourage further investigation of the effectiveness of such techniques for commit-aware mutation testing, and the development of newer program analysis based approached (e.g., symbolic execution or search-based techniques) for identifying commit-aware mutants.
}
\revise{
Finally, previous research~\cite{ma2021mudelta} has shown that commit-aware mutation testing requires different test metrics from traditional mutation testing. Thus, we encourage the researchers to define new test metrics targeting the changes and their dependencies, and investigate their effectiveness for commit-aware mutation testing. Overall, we expect that addressing these challenges will reduce the performance gap between the state-of-the-art in traditional mutation testing and commit-aware mutation testing.
}
\section{Experimental Setup}\label{Setup}
\subsection{\revise{Goals}}\label{exp-goals}
\revise{
The main goal of our study is
to investigate the prevalence and characteristics of commit-relevant mutants in evolving software systems in terms of their program location and relationship to \textit{commit hunks} and \textit{mutant types}. We also study their effectiveness and efficiency in testing evolving systems in comparison to the state-of-the-art. Specifically, our empirical goal is to achieve the following three main goals:
\begin{enumerate}
\item study the \textit{properties of commit-relevant mutants}, in terms of their prevalence, mutant types, location and proportions, as well as the\textit{ subsumption relation of commit-relevant mutants} (RQ1, RQ2 and RQ4);
\item examine the \textit{relationship between commit-relevant mutants and commit properties} (e.g., commit size) (RQ3);
\item investigate the benefit of commit-relevant mutation testing, in terms of their \textit{effectiveness} and \textit{efficiency} in comparison to the baselines (RQ5 and RQ6).
\end{enumerate}
}
\revise{
Overall, our study aims at providing insights on the properties of commit-relevant mutants and at demonstrating their importance and effectiveness in testing evolving systems.
}
\subsection{Research Questions}\label{rqs}
As we aim at assessing the potential of mutation testing in evolving systems, we investigate the following research questions (\RQ s).
\begin{description}
\item[\RQ1] \textbf{Commit-Relevant Mutants:} What is the \emph{prevalence} of ``commit-relevant mutants'' among the whole set of mutants?
\revise{
\begin{description}
\item[\RQ1.1] How are commit-relevant mutants distributed in the program?
\item[\RQ1.2] Are commit-relevant mutants located within or outside the developers' committed changes?
\item[\RQ1.3] Is there any correlation between the number of commit-relevant mutants located within program changes and the number of commit-relevant mutants outside the changes?
\end{description}
}
\item[\RQ2] \textbf{Subsuming Commit-Relevant Mutants:} What is the \emph{proportion} of ``subsuming commit-relevant mutants'', i.e., the number of commit-relevant mutants that subsumes other commit-relevant mutants, such that testing only these subsuming mutants is sufficient to test all other commit-relevant mutants?
\item[\RQ3] \textbf{Commit Size:} Is there a relationship between the \textit{size of the commit} (i.e., number of commit hunks) and the number of (subsuming) commit-relevant mutants?
\item[\RQ4] \textbf{Commit-Relevant Mutant Types:} What is the \emph{distribution of mutant types} in commit-relevant mutants?
\item[\RQ5] \textbf{Comparative Effectiveness: } How \textit{effective} are (subsuming) commit-relevant mutants, in comparison to the \textit{baselines} (i.e., random mutation and ``commit-only mutation'')?
\item[\RQ6] \textbf{Test Executions: } What is the \textit{performance} of (subsuming) commit-relevant mutants in comparison to the \textit{baselines}, in terms of the number of \textit{required test executions}?
\end{description}
RQ1 aims at improving our understanding of the locations, prevalence, and number of relevant mutants in relation to committed changes. The answer to the question allows having a rough view of the relevant mutant's distribution within and outside committed code. Answering RQ2 will show the extent to which the relevant mutant sets have redundancies. Previous work \cite{PapadakisHHJT16} has shown that redundant mutants inflate mutation scores with the unfortunate effect of obscuring its utility. We, therefore, would like to validate whether relevant mutant sets also suffer from such inflation effects. RQs 3 and 4 analyze the relation between commit size and prevalence of mutant types in relevant mutant sets to check whether there are levels/thresholds at which relevant mutants do not yield much benefits. Finally, RQs 5 and 6 aim at quantifying the potential benefits of using relevant mutants during project evolutions concerning cost and effectiveness.
\subsection{Analysis Procedure}\label{analysis_procedure}
We focus our empirical study on commits of Java programs as selected subjects.
To perform the mutation analysis, we employ Pitest\footnote{http://pitest.org/}~\cite{pitest}, one of the state-of-the-art Java mutation testing tools.
We approximate the set of commit-relevant mutants by following the algorithm introduced in Section~\ref{sec:approach}.
Besides the approximated set of commit-relevant mutants located outside of commit-change, we also record and consider as commit-relevant all those mutants residing on the location of commit-change (in our approach, \textit{$M_Y$} mutants).
This corresponds to work done by \cite{petrovic2018}, whereas the commit-relevant mutants set is made out of mutants located on the commit diff, i.e., statements modified or added by commit.
To make our approximation robust, we follow previous studies process steps \cite{KurtzAODKG16,ammann_establishing_2014,PapadakisK00TH19}.
Our approach uses mutant-assertion matrices to identify mutants interactions that constitute, up to our knowledge, the first study conducted on test assertion level for Java programming language (bypassing standard tests passing/failing mutation behavior for Java programs).
Mutant-assertion matrices were computed by running large test pools built by considering developer tests and adding automatically generated tests using EvoSuite\footnote{https://www.evosuite.org/}~\cite{FraserZ12}, a state of the art test case generation tool.
From the computed mutant-assertion matrices, we obtain three sets of mutants: \textit{mutants on a change}, \textit{mutants relevant to a change} and \textit{mutants not relevant to a change}.
To answer \RQ1 we study the prevalence and location of commit-relevant mutants in every commit by analyzing the average number of relevant/non-relevant mutants and their distribution.
We address \RQ2 by studying the proportion of subsuming commit-relevant mutants among all commit-relevant mutants and all subsuming mutants. This will estimate an extra possible reduction we can achieve if we focus only on subsuming mutants.
We consider traditional passing/failing test behavior to compute the set of subsuming mutants per subject (notice that this information is also captured when mutant-assertion matrices were built).
To address \RQ3 and \RQ4, we perform a similar statistical analysis. Still, we study any correlation between the number of commit-relevant mutants and the size of commit hunks, and the type of mutants. In \RQ5 and \RQ6, we simulate a mutation testing scenario where the tester starts by picking a mutant for analysis for which a test to kill it is developed.
During this simulation, for each analyzed mutant, we randomly pick the test to kill it from the pool and compute which other mutants are collaterally killed by the same test.
The process proceeds by picking a survived mutant until every mutant has been killed.
We consider a mutant as equivalent if there is no test in the pool that kills it.
This kind of simulation has been used in various related works to assess the effectiveness of mutation testing techniques~\cite{KurtzAODKG16,ammann_establishing_2014,PapadakisK00TH19,ma2020commit}.
We consider four different mutant selection techniques when answering \RQ5 and \RQ6. Two of them we use as \emph{baselines}, where one consists of \emph{randomly} selecting \revise{from the set of all} mutants, and the other one consists of selecting only the mutants on the change.
Another selection technique consists of selecting from the pool of commit-relevant mutants, while the last technique consists of selecting subsuming commit-relevant mutants.
We aim to obtain the best-effort evaluation by maximizing effectiveness and minimizing the effort. We focus on the first 20 mutants picked by a tester to test commit changes, while we measure effectiveness in terms of the \emph{commit-relevant mutation score} reached by the selected mutants that guide the testing process.
Simultaneously, we measure the computational effort in terms of the number of \emph{test executions} required to accomplish the same effect over the different baselines (different mutants pools).
\revise{In this simulation, we are interested in the test executions with the tests derived by the analysed mutants. The dependent variable is the test sets, while the independent variable is the test executions.}
We iterate the process (killing all selected mutants) 100 times and compute the relevant mutation score and computation effort.
\subsection{Subject Programs and Commits}\label{subject_programs}
We focus our empirical study on commits of a set of well-known, well-tested, and matured Java open-source projects taken from Apache Commons Proper repository\footnote{https://commons.apache.org}.
The process of mining repositories, data analysis, and collection, was performed as follows:
\begin{enumerate}
\item Our study focuses on the following projects: \texttt{commons-collections, commons-lang, comm\\ons-net, commons-io, commons-csv}.
These projects differ in size while having the most extended history of evolution.
We extracted commits from the year 2005 to 2020. \revise{To extract commit patches and hunks in our setup, we employ \texttt{PyDriller}\footnote{\url{https://pydriller.readthedocs.io/en/latest/intro.html}} (\texttt{V1.15}) to mine commits from the selected projects\footnote{\texttt{PyDriller} is an open-source Python framework that helps developers mine software repositories and extract information given the GIT URL of the repository of interest.}. We applied \texttt{PyDriller} to query the project's information such as commits hash id, modifications date, modified source code, modification operation, and hunks of the commits and quickly exported such information into a JSON file.}
\item We kept only commits that use JUnit4+\footnote{https://junit.org/junit4/} as a framework to write repeatable tests since it is required by
EvoSuite~\cite{FraserZ12}, the test generation tool we use for automatically augmenting test suites.
\item We filtered out those commits that do not compile, do not have a green test suite (i.e., some of the tests are failing), or do not affect a program's source code (i.e., commits that only change configuration files).
Some commits with failing tests are filtered out since Pitest requires a green test suite to perform mutation testing analysis.
\item Due to the significant execution time for commits containing several files, we set a limit for 72h of execution on a High-Performance Computer to generate and execute mutants per commit.
Please note that the test suites contain developer-written and automatically generated tests, where both are used to create mutation matrices.
All experiments were conducted on two nodes with 20 physical cores and 256GB of RAM. Specifically on Intel Skylake Xeon Gold 2.6GHz processors, running on Linux Ubuntu OS across four threads.
\end{enumerate}
Overall, we generated 9,368,052 high-order mutants and 260,051 first-order mutants, over 288 commits, that required 68,213 CPUs days of execution.
Table \ref{tab:subjects} summarises the details of the mined commits. Column ``\# Commits'' reports the number of commits mined per project, column ``\# LOC'' (Lines Of Code) indicates a subject scope in terms of lines of code, ``Maturity" reports on the date of first commit, column ``\# FOM'' (First-Order Mutant) indicates the total number of First Order Mutants generated for those commits, ``\#Mutants on Change'' indicates the number of First Order Mutants generated on the changed lines, column ``\#HOM'' (High-Order Mutant) indicates the total number of High Order Mutants generated, column ``\# Dev. Tests'' (Developer written Tests) reports on the number of developer written test cases, and column ``\# Evosuite Tests'' reports on the number of automatically generated tests.
{\Large
\begin{table}[!bt]
\centering
\caption{Details of Subjects Programs and Studied Commits}
\label{tab:subjects}
\scalebox{.9}{
\begin{tabular}{l r r r r r r r r r r}
\toprule
\textbf{\scriptsize{Commons Projects}} & \textbf{\scriptsize{\# LOC}} & \textbf{\scriptsize{\# Maturity}} & \textbf{\scriptsize{\# Commits}} & \textbf{\scriptsize{\# FOM}} & \textbf{\scriptsize{\# Mutants on Change}} & \textbf{\scriptsize{\# HOM}}& \textbf{\scriptsize{\# Dev. Tests}} & \textbf{\scriptsize{\# EvoSuite Tests}} \tabularnewline
\midrule
\scriptsize{collections} & \scriptsize{74,170} & \scriptsize{14/04/2001} & \scriptsize{45} & \scriptsize{27,417} & \scriptsize{2,026} & \scriptsize{1,192,188} & \scriptsize{4,797} & \scriptsize{1,285} \tabularnewline\midrule
\scriptsize{io} & \scriptsize{29,193} & \scriptsize{25/01/2002} & \scriptsize{30} & \scriptsize{24,970} & \scriptsize{1,115} & \scriptsize{668,448} & \scriptsize{914} & \scriptsize{286} \tabularnewline\midrule
\scriptsize{text} & \scriptsize{22,933} & \scriptsize{11/11/2014} & \scriptsize{46} & \scriptsize{47,847} & \scriptsize{4,155} & \scriptsize{2,073,829} & \scriptsize{1,084} & \scriptsize{322} \tabularnewline\midrule
\scriptsize{csv} & \scriptsize{4,844} & \scriptsize{25/01/2002} & \scriptsize{101} & \scriptsize{66,862} & \scriptsize{3,577} & \scriptsize{1,968,137} & \scriptsize{6,144} & \scriptsize{2,833} \tabularnewline\midrule
\scriptsize{lang} & \scriptsize{85,709} & \scriptsize{19/07/2002} & \scriptsize{66} & \scriptsize{102,072} & \scriptsize{3,891} & \scriptsize{3,885,341} & \scriptsize{7,574} & \scriptsize{959} \tabularnewline\midrule
\textbf{\scriptsize{Total}} & \scriptsize{216,489} & \scriptsize{N/A} & \textbf{\scriptsize{288}} & \textbf{\scriptsize{269,168}} & \textbf{\scriptsize{14,764}} & \textbf{\scriptsize{9,787,943}} & \textbf{\scriptsize{20,513}} & \textbf{\scriptsize{5,685}} \tabularnewline\midrule
\bottomrule
\multicolumn{9}{l}{\footnotesize \textit{ ``\# LOC'' - Lines Of Code, ``\# FOM'' - First Order Mutants, ``\# HOM'' - Higher Order Mutants, ``\# Dev. Tests'' - Developer written Tests} }\\
\end{tabular}
}
\end{table}
}
\subsection{Metrics and Measurements}\label{metrics}
\textbf{Statistical Analysis:} To answer our research questions, we performed several statistical analyses to evaluate correlations among several variables. For instance, in \RQ1, we analyzed whether the number of commit-relevant mutants correlates with the number of mutants residing on a change and whether the number of subsuming commit-relevant mutants correlates with the number of subsuming mutants.
In this study, we employ \revise{two} correlation metrics, namely \textit{Kendall rank coefficient ($\tau$) (Tau-a)}, \revise{and \textit{Spearman's rank correlation coefficient ($\rho$ - (rho)) }}, with the level of statistical significance set-up to $p-value$ 0.05. The Kendall rank coefficient ($\tau$), measures the similarity in the ordering of studied scores, while \revise{Spearman's $\rho$ (rho) measures how well the relationship between two variables can be described using a monotonic function \cite{myers2004spearman}.}
\revise{The correlation metrics calculate values between -1 to 1, where a value close to 1 or -1 indicates strong correlation, while a value close to zero indicates no correlation at all. }
Additionally, to facilitate comprehension of our figures, we employed \textit{coefficient of determination} (R$^2$ trendline) as a statistical measure that describes the proportion of the variance in the dependent variable that is predictable from the independent variable(s).
\smallskip\noindent
\textbf{Mutation-specific Measures:} We also employed mutation-specific metrics such as \textit{the commit-relevant mutation score} and \textit{subsuming commit-relevant mutation score}, to measure the effectiveness and efficiency of the selected mutants that guide the testing process.
\revise{We measure how the test suite effectiveness progresses when we analyze mutants from the different mutant sets (e.g., all mutants, relevant mutants, subsuming relevant, etc.). Similarly, we measure efficiency by counting the number of test executions involved (to identify which mutants are killed by the test suites) when test suite progresses.}
\subsection{Implementation Details} \label{subsection:implementation_details}
Our commit-relevant mutant identification approach is implemented in approximately 5 KLOC of Python code, ~600 LOC in Shell scripts and 3 KLOC of Java. It employs several external tools and libraries including \texttt{Evosuite}, \texttt{git-diff }and \texttt{PitTest}. We have also implemented additional infrastructure on \texttt{PitTest} to ensure analysis of evolving software and extract assertion information. In the following, we describe each of these tools.
\subsubsection{\texttt{EvoSuite \revise{(V1.1.0)}}}
To obtain a rich test suite for our study, we collected developer-written tests and automatically generated tests. For our mutation testing analysis, we augment developers' test suites with test cases automatically generated with \texttt{EvoSuite}~\cite{FraserZ12}. \texttt{EvoSuite} is an evolutionary testing tool that generates unit tests for Java software. In our analysis, we run \texttt{EvoSuite} against all several coverage criteria (e.g., line, branch, mutation, method, etc.); we also executed \texttt{EvoSuite} with default configurations, especially concerning running time.
\subsubsection{\texttt{Pitest (\revise{V1.5.1})} and \texttt{git-diff}}
Pitest does not have built-in functionality to satisfy the requirements of our experiment. Therefore, we extended the framework for High Order Mutants~\cite{Laurent} on top of Pitest that takes as an input the \texttt{gitdiff} output\footnote{https://git-scm.com/docs/git-diff}.
Based on the statement difference between the versions, the framework extends the mutants generation functionality by generating, i.e., mapping, mutants on the change, with the mutants around the change. Thus, creating second-order mutants for that particular commit file. Our framework is configured to generate the extended set of mutants available in Pitest, introduced by Laurent \textit{et al. } \cite{LaurentPKHTV17}. Kintis \textit{et al. } \cite{KintisPPVMT18} has also shown that this extended set of mutants is more powerful than the mutant sets produced by other mutation testing tools.
\subsubsection{\texttt{Pitest Assert}} \label{sec:pitest-assert-tool}
\texttt{Pitest (V1.5.1)} creates killing matrices and identifies whether a mutant is killed or not, based on test case oracle prediction (test fails or passes).
These matrices were not suitable for our experimental procedure.
Therefore, we built a framework on top of Pitest to extract additional information concerning each test case assertions (from tests that cover mutants).
Our framework performs bytecode instrumentation of each test executed on a specific mutant, using ASM\footnote{https://asm.ow2.io/} as an all-purpose Java bytecode manipulation and analysis framework.
By instrumenting each test case assertion, we can obtain execution information.
More precisely, each assertion has a unique test name where it locates, assertion function name, assertion line number, and assertion actual execution value.
If an assertion triggers an exception, we keep track of the stack-trace execution.
However, for this study's purpose, we disregard the assertions that trigger the exception from our relevant mutants calculation (please refer to Algorithm \ref{algo:relevant}) since we only aim at actual mutants' observable behavioral output.
Hence, the mutant-assertion matrix is a weighted matrix. For each (mutant, test-assertion) pair, the value corresponds to the actual assertion value obtained by running the test on the mutant, or the exception stack trace if an assertion throws an exception.
\revise
Concretely, we employ the \textit{JUnit4}\footnote{https://junit.org/junit4/} testing framework which contains a public class (called \texttt{Assert}) that provides
a set of assertion methods to specify test conditions.
Typically, these methods (e.g., \texttt{Assert.assertEquals(expected value, actual value)}) directly evaluate the assertion's conditions, then returns
the final assertion's output (e.g., conditions not satisfied, pass, or fail).
To obtain the value of parameters within the assert statement, in our framework, we use \texttt{Pitest Assert} to instrument each assertion method. Such that we serialize the provided input values in the assert statement before they propagate to conditional checks, i.e., before the conditional check is reached in \textit{org.junit.Assert\footnote{https://junit.org/junit4/javadoc/4.13/org/junit/Assert.html}} and the output values are fed to \textit{org.hamcrest.Matcher\footnote{http://hamcrest.org/JavaHamcrest/javadoc/1.3/org/hamcrest/Matchers.html}} for evaluation.
Specifically, we serialize both the expected and actual values after they propagate as
input parameters of the assert statement. This allows to
assess the input parameters of the \texttt{assert} statement (e.g., an expression or a method call (\texttt{assertEqual(foo(), bar())})) for concrete values.
Hence, in our setup, we compare the output values of both the expected and actual values present in each assertion. However, our experimental framework does not directly account for the
potential dependencies within assertions and test cases, we address this concern in the \textit{threats to validity} (\textit{see} \autoref{ValidityThreats}).
Our test assertion framework is built on top of \texttt{PiTest} and is publicly available.
}
\subsection{Research Protocol}\label{protocol}
\autoref{fig:Process_pipeline} highlights our experimental protocol which proceeds as follows: For each project (e.g., \textit{commons-collections}) and each mined commit (e.g., hash: \textit{03543e5f9},
we first \textit{augment the developers' test suite} with automatically generated tests using EvoSuite~\cite{FraserZ12}. Next, we \textit{obtain the commit changes} (a.k.a hunks) of the commit using the \texttt{git-diff} tool, in order to identify the changed and unchanged program statements. We then \textit{generate both first-order and second-order mutants} for the program, using Pitest Assert as our extension of Pitest Mutation Testing tool~\cite{pitest}. After mutant generation, we \textit{execute every mutant to obtain the mutant-assertion matrices}, which provides information about test assertion type, position and value. Finally, we execute our relevant mutant detection algorithm~\ref{algo:relevant} to identify commit-relevant mutants.
\begin{figure}[bt!]
\begin{center}
\includegraphics[width=0.95\textwidth,trim={0cm 8cm 0cm 7cm},clip]{figures/Process_pipeline_v3.pdf}
\caption{Research Protocol}
\label{fig:Process_pipeline}
\end{center}
\end{figure}
Our result analysis proceeds after computing mutant-assertion matrices and identifying commit-relevant mutants.
We then perform the data gathering and analysis required to answer every research question (\RQ s).
In particular, we compute subsuming mutant relations necessary to answer \RQ2, and perform the mutation testing simulation needed to answer \RQ5 and \RQ6.
\section{Related Work}\label{related_work}
In this section, we discuss closely related work in the areas of change impact analysis, regression testing, test augmentation, and commit-aware mutation testing.
\smallskip \noindent \textbf{\textit{Program slicing:}}
A related line of work regards the formulation of dynamic or observation-based slicing \cite{BinkleyH05, BinkleyHK07, BinkleyGHIKY14}. These techniques aim at identifying relevant program statements and not mutants. Though, they could be used in identifying relevant mutant locations, in which every located mutant could be declared as relevant.
\revise{For instance, \citet{guimaraes2020optimizing} proposed the use of dynamic program slicing to determine the subsumption relations among mutants, in order to detect redundant mutants and reduce the number of tested mutants. In their evaluation, the authors demonstrate that using dynamic subsumption relation among mutants reduces mutation testing time by about 53\%, on average.}
\revise{Similarly, \citet{delamaro2001interface} proposed interface mutation to reduce the mutation testing effort required during system integration. The goal of the paper is to apply interface mutants as a test adequacy criterion for system integration testing. The paper demonstrates that inter-procedural program slicing is applicable for mutation analysis, particularly for integration testing. Their approach leverages the data flow dependencies between two system units to determine the set of mutants that are relevant for the integration of both units.}
This is because many non-killable or irrelevant mutants located in dependent statements will be considered as relevant. This is evident from the previously reported results of Binkley \textit{et al. } \cite {BinkleyH05, BinkleyHK07} that showing simple changed slices occupying 35-70\% of the entire programs.
\smallskip \noindent \textbf{\textit{Change-Impact Analysis:}}
It is important to analyze and test the impact of program changes on evolving software systems. To this end, researchers have proposed several automated methods to assess the impact of program changes on the quality of the software, e.g., in terms of correctness and program failures. For instance, researchers have employed \textit{program analysis} techniques (such as program slicing) to identify relevant coverage-based test requirements, specifically,
by analyzing the impact of all control and data dependencies affected by the changed code to determine all tests that are affected by the change~\cite{RothermelH94,Binkley97}.
Unlike these works, in this paper, we focus on performing change impact using commit-aware mutation testing, in particular, we empirically evaluate the properties, distribution and prevalence of (subsuming) commit-relevant mutants.
\smallskip \noindent \textbf{\textit{Regression Testing:}}
The field of regression testing investigates how to automatically generate test cases for evolving software systems to avoid regression bugs. Researchers have proposed several approaches in this field for decades~\cite{YooH12}. The closest work to ours is in the area of regression mutation testing~\cite{ZhangMZK12} and predictive mutation testing~\cite{0050ZHH0019, MaoCZ19}. The goal of regression mutation testing is to identify change-affected mutants (i.e., mutants on execution trace affected by changes), and incrementally calculate mutation score. Meanwhile, predictive mutation testing seeks to estimate the mutation score without mutant execution using machine learning classification models trained on different static features~\cite{MaoCZ19}. These approaches are focused on speeding up test execution and mutation score computation while testing evolving software systems.
In contrast, in this paper, we focus on identifying the test requirements (mutants) relevant to the program changes. i.e., the mutants that need to be analyzed and tested by the engineer, and we provide a more refined and precise mutation testing score, specially adapted for commit-relevant mutants.
\smallskip \noindent \textbf{\textit{Test Augmentation:}}
This line of research aims to automatically generate additional test cases to improve the (fault revealing) quality of existing test suites. This is particularly important when a software system changes (often); hence, it is vital to generate new tests that exercise the program changes. Researchers have proposed several test augmentation approaches to trigger program output differences~\cite{QiRL10}, increase coverage~\cite{XuKKRC10} and increase mutation score~\cite{SmithW09JSS,SmithW09EMSE}. Some test augmentation approaches have been developed to address code coverage problems using propagation-based techniques~\cite{ApiwattanapongSCOH06,SantelicesCAOH08,SantelicesH10,SantelicesH11}. Other approaches employ symbolic execution for test augmentation by generating tests that exercise the semantic differences between program versions by incrementally searching the program path space from the changed locations and onwards, this includes approaches such as differential symbolic execution \cite{PersonDEP08}, KATCH \cite{MarinescuC13} and Shadow symbolic execution \cite{KuchtaPC18}. These techniques rely on dependency analysis and symbolic execution to decide whether changes can propagate to a user-defined distance by following the predefined propagation conditions. Hence they are considerably complex and computationally expensive, for instance, because they are limited by the state explosion problem of symbolic execution. These papers are complementary to our work since the aim is to generate additional tests to improve existing test suites. However, our work is focused on test augmentation to exercise code changes, albeit using mutation testing.
\smallskip \noindent \textbf{\textit{Commit-Aware Mutation Testing:}}
The goal of commit-aware mutation testing is to select mutants that exercise program changes in evolving software systems. The problem of commit-relevant test requirements has not been studied in depth by the mutation testing literature \cite{PapadakisK00TH19}. The closest work to ours in this area includes the formalization of commit-aware mutation testing~\cite{ma2020commit},
diff-based commit-aware mutant selection (i.e., mutants within program changes only)~\cite{petrovic2018}, and a machine learning-based mutant selection method~\cite{ma2021mudelta}. Petrovic \textit{et al. }~\cite{petrovic2018} presents a diff-based probabilistic mutation testing approach that is focused on selecting mutants within committed program changes only. Unlike this paper, this approach ignores the dependencies between program changes and the unmodified code. \texttt{Mudelta}~\cite{ma2021mudelta} presents a machine learning-based approach for selecting commit-relevant mutants. Ma \textit{et al. }~\cite{ma2020commit} defines commit-relevant mutants and evaluates their relationship with traditional mutation test criteria, emphasizing the importance of commit-aware mutation testing. Unlike these works, we conducted an in-depth empirical study to understand the characteristics of commit-relevant mutants to shed more light on their properties and provide scientific insights for future research in commit-aware mutation testing. In particular, in comparison to Ma \textit{et al. }~\cite{ma2020commit}, our work impose more generalizable and easy to fulfill requirements on the programs and test contracts. For instance, Ma \textit{et al. }~\cite{ma2020commit}, determines commit-aware mutants by comparing mutants test suites from both pre- and post-commits, under the assumption that the test contract is intact and remains the same across both program versions.
However, this critical requirement considerably limits the application and adoption of the approach of Ma \textit{et al. }~\cite{ma2020commit} in practice. Our empirical study observed that test contracts frequently change across versions in evolving software systems (>60\%).
\section{Experimental Results}\label{Results}
\smallskip\noindent
\subsection{\RQ1: Commit Relevant Mutants}
We start by studying the proportion of \emph{commit-relevant mutants} that affect the commit changes out of \emph{all mutants} by using the pipeline just introduced in Section~\ref{sec:approach}.
Thus, in this RQ,
we consider as commit-relevant mutants all mutants identified by our approach, including the set of killable mutants residing on modified statements. We distinguish commit-relevant mutants in the categories of those located on changed and unchanged code to demonstrate
This allows us to estimate the potential reduction in terms of the number of mutants requiring analysis and the number of test executions required to cover them if the tester focuses testing only on commit-relevant mutants instead of the whole set mutants, \revise{or on the mutant set consisting of all mutants residing on the modification}.
Additionally, we evaluate the properties of commit-relevant mutants that can inform their selection among all mutants. Thus, we examine the location of commit-relevant mutants, whether they are mostly located within the commit or outside the committed changes. We also assess whether there is a correlation between the number of identified commit-relevant mutants and the number of commit-relevant mutants within the committed change to determine if the number of mutants within a commit can serve as a proxy to determine the number of commit-relevant mutants.
{\Large
\begin{table}[htp]
\centering
\caption{Details of the Prevalence of Commit-relevant Mutants. }
\label{tab:prevalence-commit-relevant-mutants}
\scalebox{.9}{
\begin{tabular}{l r r r r r r r}
\toprule
\textbf{\scriptsize{Project}} & \textbf{\scriptsize{\# Commits (C)}} & \textbf{\scriptsize{\# C. All R. M.}} & \textbf{\scriptsize{\# C. No R. M.}} & \textbf{\scriptsize{\# Relevant}} & \textbf{\scriptsize{\# Not Relevant}} & \textbf{\scriptsize{Ratio}} & \textbf{\scriptsize{Reduction Ratio}} \tabularnewline
\midrule
\scriptsize{commons-collections} & \scriptsize{45} & \scriptsize{2} & \scriptsize{4} & \scriptsize{6,833} & \scriptsize{18,558} & \scriptsize{32,31\%} & \scriptsize{67,69\%} \tabularnewline\midrule
\scriptsize{commons-io} & \scriptsize{30} & \scriptsize{0} & \scriptsize{3} & \scriptsize{6,052} & \scriptsize{17,803} & \scriptsize{28,70\%} & \scriptsize{71,30\%} \tabularnewline\midrule
\scriptsize{commons-text} & \scriptsize{46} & \scriptsize{1} & \scriptsize{4} & \scriptsize{8,810} & \scriptsize{34,882} & \scriptsize{27,10\%} & \scriptsize{72,90\%} \tabularnewline\midrule
\scriptsize{commons-csv} & \scriptsize{101} & \scriptsize{4} & \scriptsize{0} & \scriptsize{27,441} & \scriptsize{35,844} & \scriptsize{47,39\%} & \scriptsize{53,61\%} \tabularnewline\midrule
\scriptsize{commons-lang} & \scriptsize{66} & \scriptsize{1} & \scriptsize{2} &\scriptsize{15,724} & \scriptsize{82,457} & \scriptsize{19,22\%} & \scriptsize{80,78\%} \tabularnewline\midrule
\textbf{\scriptsize{Total}} & \textbf{\scriptsize{288}} & \textbf{\scriptsize{8}} & \textbf{\scriptsize{13}} & \textbf{\scriptsize{64,860}} & \textbf{\scriptsize{189,544}} & \textbf{\scriptsize{N/A}} & \textbf{\scriptsize{N/A}} \tabularnewline\midrule
\textbf{\scriptsize{Average}} & \textbf{\scriptsize{58}} & \textbf{\scriptsize{N/A}}& \textbf{\scriptsize{N/A}} & \textbf{\scriptsize{225}} & \textbf{\scriptsize{658}} & \textbf{\scriptsize{29,58\%}} & \textbf{\scriptsize{70,42\%}} \\ \hline
\bottomrule
\multicolumn{8}{l}{\footnotesize \textit{ ``\# C. All R. M.'' - Number of Commits with all relevant mutants, ``\# C. No R. M.'' - Number of Commits with no relevant mutants} }\\
\end{tabular}
}
\end{table}
}
\begin{figure*}[htp]
\begin{center}
\includegraphics[width=\textwidth]{figures/Box_plot_Distribution_sort_not_relevant.pdf}
\end{center}
\caption{Distribution of mutants across all commits showing the proportion of non-relevant mutants (in \textit{ \color{blue} blue}) as well as commit-relevant mutants within committed changes (in \textit{\color{red} red}) and outside committed changes (in \textit{\color{green} green})
}
\label{fig:RQ1-Distribution_plot}
\end{figure*}
\begin{figure}[bt!]
\centering
\begin{minipage}{.45\textwidth}
\centering
\vspace{-0.4cm}
\includegraphics[width=1.0\linewidth]{figures/Ratio_of_commit-relevant_mutants_located_within_a_change_scaled_100.pdf}
\captionof{figure}{Proportion of commit-relevant mutants within the commit (18.56\%) and outside the commit (81.44\%)}
\label{fig:RQ1-ratio_ouside_methods_i}
\end{minipage}%
\hspace{0.4cm}
\begin{minipage}{.45\textwidth}
\centering
\vspace{-0.4cm}
\includegraphics[width=1.0\linewidth]{figures/Ratio_of_commit-relevant_mutants_located_within_changed_methods.pdf}
\captionof{figure}{Proportion of commit-relevant mutants within the change method (30.05\%) and the outside changed methods (69.95\%) }
\label{fig:RQ1-ratio_ouside_methods_ii}
\end{minipage}
\end{figure}
\subsubsection{\textit{\textbf{RQ1.1: }\revise{What is the proportion of commit-relevant mutants out of all mutants?}}}
\autoref{tab:prevalence-commit-relevant-mutants} and \autoref{fig:RQ1-Distribution_plot} illustrate the distribution of commit-relevant mutants among all mutants.
In our evaluation, we found that \textit{only about one in three ($\approx$30\%) mutants are commit-relevant}, on average. In particular, we observed that only about 225 mutants are relevant to a commit out of 833 mutants, on average. \revise{This implies that an \textit{effective commit-aware mutation testing technique can reduce significant mutation testing effort, both computational when executing mutants and manual when analysing mutants. }} In addition, we found some (21) outliers in our analysis of commit-aware mutants, see columns ``\# C. All R. M.'' (Number of Commits with all Relevant Mutants) and ``\# C. No R. M.'' (Number of Commits with No Relevant Mutants): In particular, we found that only 2.8\% of commits (8) had 100\% commit-relevant mutants, this portrays the importance of mutant selection for evolving software systems. On the other hand, our evaluation results show that \revise{in 4.5\% of the commits (13) we found no commit-relevant mutants outside the change}; this suggests that it is pertinent to develop commit-aware mutation testing techniques that discern relevant from non-relevant mutants.
Overall, these findings demonstrate the importance of developing commit-aware test selection for evolving software systems, in particular, in selecting relevant mutants to reduce testing effort.
\begin{result}
One in three (approximately 30\%) mutants are commit-relevant; hence, selecting commit-aware mutants can significantly reduce
\revise{mutation testing cost}.
\end{result}
\subsubsection{\textit{\textbf{RQ1.2: }\revise{Where are commit-relevant mutants located in the program, i.e., how many commit-relevant mutants are within or outside the committed changes?}}} In our evaluation, most (81\%) commit-relevant mutants are outside of developers' committed changes (\textit{see} \autoref{fig:RQ1-ratio_ouside_methods_i}). Making only about one in five (19\%) commit-relevant mutants being within the committed changes of developers. For instance, a developer that tests \textit{all commit-relevant mutants within the changed method} will test only 30\% of commit-relevant mutants, and miss almost 70\% of commit-relevant mutants (\textit{see} \autoref{fig:RQ1-ratio_ouside_methods_ii}). This result suggests that to test the impact of developer changes on the program effectively, it is important to not only test within the committed changes. It is also highly pertinent to test the interaction of committed changes with the rest of the unmodified program.
\begin{result}
Most (81\% of) commit-relevant mutants are located outside of the commit, and \\only a few (19\% of) commit-relevant mutants are within the commit.
\end{result}
\begin{figure}[htp]
\centering
\begin{minipage}{.3\textwidth}
\centering
\vspace{-0.4cm}
\includegraphics[width=1.0\linewidth]{figures/Scatter_plot_correlations_relevant_mutants-mutants_on_change_spearman_kendall.pdf}
\captionof{figure}{
\revise{
Correlation Analysis between the number \textit{mutants within a change} and the number of \textit{commit-relevant mutants}}
\label{fig:RQ1-correlation_plot_i}
\end{minipage}%
\hspace{0.2cm}
\begin{minipage}{.3\textwidth}
\centering
\vspace{-0.4cm}
\includegraphics[width=1.0\linewidth]{figures/Scatter_plot_correlations_not_relevant_mutants-mutants_on_change_spearman_kendall.pdf}
\captionof{figure}{
\revise{
Correlation Analysis between the number of \textit{mutants within a change} and the number of \textit{non-relevant mutants}}
}
\label{fig:RQ1-correlation_plot_ii}
\end{minipage}
\hspace{0.2cm}
\begin{minipage}{.3\textwidth}
\centering
\vspace{-0.4cm}
\includegraphics[width=1.0\linewidth]{figures/Scatter_plot_correlations_relevant_mutants-not_relevant_mutants_spearman_kendall.pdf}
\captionof{figure}{
\revise{
Correlation Analysis between the number of \textit{non-relevant mutants} and \textit{commit-relevant mutants}}
}
\label{fig:RQ1-correlation_plot_iii}
\end{minipage}
\end{figure}
\subsubsection{\textit{\textbf{RQ1.3: }\revise{Is there a correlation between the number of commit-relevant mutants and the number of mutants within the change?}}} Our evaluation results show that \textit{there is a weak trend between the number of commit-relevant mutants and the number of mutants within the commit}. Our statistical correlation analysis shows that there is a \textit{weak correlation} between both variables. \revise{In particular, we found a Spearman and Kendall correlation coefficients of 0.212 and 0.141, respectively. Indeed, both the Spearman and Kendall correlation coefficients are statistically significant (with p-values 0.0006 and 0.0007, respectively)}. \revise{Figures~\ref{fig:RQ1-correlation_plot_i}, \ref{fig:RQ1-correlation_plot_ii} and \ref{fig:RQ1-correlation_plot_iii} summarize the results of the different studied correlations.}
These correlation results suggest that there is a weak relationship between the number of mutants within a change and the number of commit-relevant mutants, but \textit{no robust and predictable pattern or trend between both variables}. This implies that \textit{the number of mutants within the commit can not reliably predict the number of commit-relevant mutants} (in unmodified code regions), and vice versa.
\begin{result}
There is a statistically significant weak positive correlation between the number of commit-relevant mutants and the number of mutants within the change (\revise{Spearman} and Kendall correlation coefficients of \revise{0.212} and 0.141, respectively).
\end{result}
\smallskip\noindent
\subsection{\RQ2: Subsuming Commit Relevant Mutants}
In this section, we investigate the prevalence of \emph{subsuming commit-relevant mutants} among \emph{commit-relevant mutants}. Estimating the proportion of subsuming commit-relevant mutants is important to demonstrate the further reduction (in number of mutants to analyse and test executions) achieved by ``selecting" or ``optimizing'' for effectively identifying \emph{subsuming commit-relevant mutants}, in comparison to \emph{commit-relevant mutants}, \emph{subsuming mutants} and \emph{all mutants}.
The two subsumption relations (i.e., one for the commit-relevant mutants and the other one for all mutants) are computed by following the definition introduced in Section~\ref{sec:subsuming_mutants}.
Additionally, we examine the correlation between the number of subsuming commit-relevant mutants and the number of commit-relevant mutants within a change and subsuming mutants; this is important to determine if these variables are related can predict or serve as a proxy for determining subsuming commit-relevant mutants.
\begin{figure*}[htp]
\begin{center}
\begin{minipage}{.45\textwidth}
\centering
\vspace{-0.55cm}
\includegraphics[width=1.0\linewidth]{figures/venn_diagram_proportions.pdf}
\captionof{figure}{Venn diagram showing the proportion of ``commit-relevant mutants'' (29.58\% in \textit{\color{orange} orange})) and ``subsuming commit-relevant mutants'' (6.13\% in \textit{\color{purple} purple}) among all mutants (in \textit{\color{pink} pink}).}
\label{fig:venn_3a_categories_i}
\end{minipage}%
\hspace{0.2cm}
\begin{minipage}{.45\textwidth}
\centering
\vspace{-0.4cm}
\includegraphics[width=1.0\linewidth]{figures/Venn_diagram_Categories_numbers.pdf}
\captionof{figure}{Venn diagram showing the number and intersections among ``commit-relevant mutants within commit changes'' (in \textit{\color{blue} blue}),``subsuming commit-relevant mutants'' (in \textit{\color{orange} orange}) and ``subsuming mutants'' (in \textit{\color{pink} pink}).
}
\label{fig:venn_3a_categories_ii}
\end{minipage}
\end{center}
\end{figure*}
\textit{What is the proportion of ``subsuming commit-relevant mutants'' among commit-relevant mutants,
such that a test suite that distinguishes a(ll) subsuming commit-relevant mutant(s) covers (all) other commit-relevant mutants?}
\autoref{fig:venn_3a_categories_i} illustrates the proportion of \emph{subsuming commit-relevant mutants} and their intersection with \emph{commit-relevant mutants} as well as all mutants. In our evaluation, we found that \textit{``subsuming commit-relevant mutants'' are significantly smaller than commit-relevant mutants and all mutants}. About one in 20 mutants is a \emph{subsuming commit-relevant mutant}, and about one in five (5) commit-relevant mutants is a subsuming commit-relevant mutant. Specifically, ``subsuming commit-relevant mutants'' represent 20.72\% and 6.13\% of all commit-relevant mutants and all mutants, respectively. This suggests it is worthwhile to identify and select subsuming relevant mutants from all (commit-relevant) mutants. Invariably, generating only subsuming commit-relevant mutants reduces the number of mutants to analyze by 79\% and 93\% compared to generating commit-relevant mutants and all mutants, respectively. This result implies that developing automated mutation testing methods that effectively identify, select or generate subsuming commit-relevant mutants can significantly reduce \revise{mutation testing cost}.
\begin{result}
Selecting ``subsuming commit-relevant mutants'' can reduce the number of mutants to be considered by about 79\% and 93\% in comparison to commit-relevant mutants and all mutants, respectively.
\end{result}
\textit{What is the proportion of ``subsuming commit-relevant mutants'' among ``subsuming mutants'' and ``commit-relevant mutants within a change''?}
\autoref{fig:venn_3a_categories_ii} illustrates the intersections between all three types of mutants. Notably, \textit{most (92.98\% -- 18,117 out of 19,484) subsuming commit-relevant mutants are subsuming mutants as well, and they represent 26.42\% of all subsuming mutants (68,553)}.
This implies that searching for subsuming commit-relevant mutants among subsuming mutants (instead of all mutants) is beneficial in reducing the search scope.
We also observed that all subsuming commit-relevant mutants within committed changes are subsuming mutants. Meanwhile, about one in five (19.36\% -- 3,772 out of 19,484) subsuming commit-relevant mutants are within the developers' committed changes; they represent 28.38\% (3,772 out of 13,290) of all mutants within the change. This suggests that less than one in three mutants within the change are subsuming commit-relevant mutants. Hence, it is important to search for subsuming commit-relevant mutants outside of the committed changes since most subsuming commit-relevant mutants (81\%, 15,772) are outside the committed changes.
\begin{result}
Most (92.98\% of) subsuming commit-relevant mutants are subsuming mutants, while a few (19.36\% of) subsuming commit-relevant mutants are located within committed changes.
\end{result}
\begin{figure*}[htp]
\begin{center}
\begin{minipage}{.45\textwidth}
\centering
\vspace{-0.4cm}
\includegraphics[width=1.0\linewidth]{figures/Scatter_plot_correlations_minimal_relevant_mutants-mutants_on_change_spearman_kendall.pdf}
\captionof{figure}{
\revise{
Correlation Analysis between the number mutants within a change and the number of subsuming commit-relevant mutants}
\label{fig:RQ2-correlation_plot_i}
\end{minipage}%
\hspace{0.2cm}
\begin{minipage}{.45\textwidth}
\centering
\vspace{-0.4cm}
\includegraphics[width=1.0\linewidth]{figures/Scatter_plot_correlations_minimal_relevant_mutants-minimal_mutants_spearman_kendall.pdf}
\captionof{figure}{
\revise{
Correlation Analysis between the number of subsuming mutants and the number of subsuming commit-relevant mutants}
}
\label{fig:RQ2-correlation_plot_ii}
\end{minipage}
\end{center}
\end{figure*}
\textit{Is there a correlation between the number of subsuming commit-relevant mutants and the number of mutants within a change?} Our correlation analysis shows that \textit{there is a weak positive correlation between the number of commit-relevant mutants within a change and the number of subsuming commit-relevant mutants}
(\textit{see} \autoref{fig:RQ2-correlation_plot_i}). \revise{Both Spearman and Kendall correlation coefficients report a weak positive correlation, with correlation coefficients 0.222 and 0.148, respectively, (\textit{see} \autoref{fig:RQ2-correlation_plot_i}).}
\revise{In particular,
the correlation coefficients are statistically significant with p-values less than 0.05, specifically, 0.0003 and 0.0004 for Spearman and Kendall coefficients, respectively}.
This result suggests that the number of mutants within a change can not strongly predict the number of \revise{subsuming commit-relevant mutants}; hence, it is important to identify all commit-relevant mutants that interact with the committed changes, and not only test the change itself.
\begin{result}
The number of mutants within a change can not reliably predict the number of subsuming commit-relevant mutants
since there is only a weak positive correlation between both variables.
\end{result}
\begin{figure*}[bt!]
\begin{center}
\includegraphics[width=\textwidth]{figures/Chart_14.pdf}
\caption{Distribution of the proportion of commit-relevant mutants (in \textit{\color{gray} gray}) and subsuming commit-relevant mutants (in \textit{\color{red} red}); Commits are sorted from left to right in ascending order of the proportion of subsuming relevant mutants
}
\label{fig:RQ2-Distribution_minimal_relevant}
\end{center}
\end{figure*}
\textit{What is the relationship between the number of subsuming commit-relevant mutants and the number of subsuming mutants?} \autoref{fig:RQ2-correlation_plot_ii} illustrates the distribution and correlation between the number of subsuming mutants and the number of subsuming commit-relevant mutants. In this figure, the trending line shows that \textit{there is a moderate positive correlation between both variables}. \revise{Indeed, both Spearman and Kendall correlation coefficients reports a \textit{moderate positive relationship} between both variables, with correlation coefficients 0.476 and 0.368, respectively, (\textit{see} \autoref{fig:RQ2-correlation_plot_ii}). The correlation coefficients also show that the positive relationship is statistically significant (p-value < 0.05)}. As expected, we observed that the proportion of subsuming relevant mutants per commit increases (trendline $R^2$=0.881) as the proportion of commit-relevant mutants increases (\textit{see} \autoref{fig:RQ2-Distribution_minimal_relevant}).
Overall, this result implies that these variables can serve as a proxy to each other, hence predicting one variable could help identify the other. In particular, this implies that selecting subsuming mutants significantly increases the chances of selecting subsuming commit-relevant mutants.
\begin{result}
There is a moderate positive relationship between the number of subsuming commit-relevant mutants and the number of subsuming mutants, such that one can predict the other and vice versa.
\end{result}
\smallskip\noindent
\subsection{\RQ3: Commit Size}
In this section, we investigate if there is a relationship between the number of (subsuming) commit-relevant mutants and the size of the commit, measured in terms of the number of commit hunks.
In particular, we pose the following question: \textit{Is there a relationship between the number of commit hunks and the number of (subsuming) commit-relevant mutants?}
\autoref{fig:RQ3-distribution_hunks_i} illustrates the relationship between the number of commit-relevant mutants and the number of commit hunks. For commit-relevant mutants, we found that the number of \emph{commit-relevant mutants} (moderately) \emph{increases} (trendline $R^2$=0.125) as the number of commit-hunks increases. This implies that there is \emph{positive direct relationship} between the size of the commit and the number of \emph{commit-relevant mutants}. However, \autoref{fig:RQ3-distribution_hunks_ii} shows that the number of \emph{subsuming commit-relevant mutants} (moderately) \emph{decreases} (trendline $R^2$=0.023) as the number of commit-hunks increases.
These results suggest that there is an \emph{indirect relationship} between the size of the commit and the number of \emph{subsuming commit-relevant mutants}. The size of the commit does not directly predict the number of subsuming commit-relevant mutants. Indeed, the number of \emph{subsuming commit-relevant mutants} decreases as the average size of the commit increases. Overall, this result demonstrates the effectiveness and importance of \emph{subsuming commit-relevant mutants} in reducing testing effort, even for large commit changes.
\begin{result}
The number of ``commit-relevant mutants'' increases as the size of the commit increases; however, the number of ``subsuming commit-relevant mutants'' decreases as the size of the commit increases.
\end{result}
\begin{figure*}[htp]
\begin{center}
\begin{minipage}{.45\textwidth}
\centering
\vspace{-0.4cm}
\includegraphics[width=1.0\linewidth]{figures/Chart_18.pdf}
\captionof{figure}{Average Number of Commit-relevant mutants per commit
\label{fig:RQ3-distribution_hunks_i}
\end{minipage}%
\hspace{0.2cm}
\begin{minipage}{.45\textwidth}
\centering
\vspace{-0.4cm}
\includegraphics[width=1.0\linewidth]{figures/Chart_19.pdf}
\captionof{figure}{Average Number of Subsuming Commit-relevant mutants per commit}
\label{fig:RQ3-distribution_hunks_ii}
\end{minipage}
\end{center}
\end{figure*}
\begin{figure*}[bt!]
\begin{center}
\begin{minipage}{.45\textwidth}
\centering
\vspace{-0.4cm}
\includegraphics[width=1.0\linewidth]{figures/commit_relevant_operators_distribution.pdf}
\captionof{figure}{Prevalence of Commit-relevant Mutant Types
\label{fig:RQ4-test_mutants_operators}
\end{minipage}%
\hspace{0.2cm}
\begin{minipage}{.45\textwidth}
\centering
\vspace{-0.4cm}
\includegraphics[width=1.0\linewidth]{figures/subsuming_relevant_operators_distribution.pdf}
\captionof{figure}{Prevalence of Subsuming Commit-relevant Mutant Types}
\label{fig:RQ4-test_sub_mutants_operators}
\end{minipage}
\end{center}
\end{figure*}
\begin{figure*}[bt!]
\begin{center}
\begin{minipage}{.45\textwidth}
\centering
\vspace{-0.4cm}
\includegraphics[width=1.0\linewidth]{figures/Histogram_Ratio_of_relevant_mutants_operators.pdf}
\captionof{figure}{Ratio of Commit-relevant Mutants over All Mutants per Mutant Type
\label{fig:RQ4-ratio_relevant_mutants_operators}
\end{minipage}%
\hspace{0.2cm}
\begin{minipage}{.45\textwidth}
\centering
\vspace{-0.4cm}
\includegraphics[width=1.0\linewidth]{figures/Histogram_Ratio_of_subsuming_relevant_mutants_operators.pdf}
\captionof{figure}{Ratio of Subsuming Commit-relevant Mutants over All Mutants per Mutant Type}
\label{fig:RQ4-ratio_sub_mutants_operators}
\end{minipage}
\end{center}
\end{figure*}
\smallskip\noindent
\subsection{\RQ4: Commit-relevant Mutant Types}
Let us investigate the prevalence of \emph{mutant types} among (subsuming) commit-relevant mutants, using 25 distinct mutant group types from Pitest~\cite{pitest}. This is important to determine whether the generation, selection or identification of commit-relevant mutants can be improved by focusing on specific mutant types.
\textit{What is the prevalence of mutant types among (subsuming) commit-relevant mutants?}
\autoref{fig:RQ4-test_mutants_operators} illustrates the prevalence of mutant types among commit-relevant mutants.
Our evaluation results show that some mutant types are highly prevalent, such as \textit{Unary Operator Insertion Mutators (UOIMutators)}, \textit{Relational Operator Replacement Mutator (RORMutators)} and \textit{Constant Replacement Mutator (CRCRMutators)}.
\revise{
On one hand, UOIMutators inject a unary operator (increment or decrement) on a variable, this may affect the values of local variables, arrays, fields, and parameters~\cite{pitest}, while RORMutators replace a relational operator with another one, e.g., ``$<$'' with ``$>$'' or ``$<=$'' with ``$<$''.
On the other hand, CRCRMutators mutates inline constants. For further detauls about the mutant types, the table of constants and other mutation operators can be found in the official PiTest documentation\footnote{http://pitest.org/quickstart/mutators/}.}
Specifically, 50.77\% of the commit-relevant mutants are of one of these three mutant types.
This is mainly related to the fact these three mutation operators produced the majority (54.5\%) of the mutants considered in our study.
Precisely, \autoref{fig:RQ4-ratio_relevant_mutants_operators} shows that the \emph{distribution} of commit-relevant mutants is clearly \emph{uniform} per mutant type. That is, in general, between 20\% and 30\% of the mutants for each type result to be commit-relevant. This indicates that mutants type does not increase or reduce the chances for mutants of being commit-relevant.
The outliers of \autoref{fig:RQ4-ratio_relevant_mutants_operators}, corresponding to mutant types \revise{Bitwise Operator Mutator} (\textit{OBBNMutators}) and \revise{Invert Negatives Mutator} (\textit{InvertNegsMutat}), are because of the low number of mutants for these types: 13 out of 81 (16\%) mutants are commit-relevant in the case of \textit{OBBNMutators} mutant type, while 3 out of 5 (60\%) mutants are commit-relevant for \textit{InvertNegsMutat} mutant type.
\revise
In particular, OBBNMutators mutates (i.e., reverses) bitwise ``AND'' (\&) and ``OR'' ($|$) operators,
while InvertNegsMutat operators inverts the negation of integers and floating-point numbers.}
Similarly, Figures \ref{fig:RQ4-test_sub_mutants_operators} and \ref{fig:RQ4-ratio_sub_mutants_operators} show that the ratio of subsuming commit-relevant mutants per mutant type follows a uniform distribution as well. Typically, between 5-7\% of the mutants per mutant type turn to be subsuming commit-relevant.
The outlier of \autoref{fig:RQ4-ratio_sub_mutants_operators} corresponds to \textit{InvertNegsMutat} mutant type, where none of the 3 commit-relevant mutants identified for this mutant type are subsuming (because of mutants of a different mutant type subsume them).
\begin{result}
The distribution of (subsuming) commit-relevant mutants per mutant type is uniform. Typically, between 20-30\% (5-7\%) of the mutants per mutant type are (subsuming) commit-relevant.
\end{result}
\smallskip\noindent
\subsection{\RQ5: Effectiveness of Commit-relevant Mutants Selection}
This section simulates a mutation testing scenario where the tester selects a mutant for analysis for which a test to kill it is developed.
Note that a test case that is designed to kill a mutant may collaterally kill other mutants.
Consequently, opening a space to examine the effectiveness of the test suites developed when guided by different mutant selection strategies. Accordingly, this study compares the following mutant selection strategies: “random mutants selection,” “mutants within a change,” and (subsuming) commit-relevant mutants.
We measure their effectiveness in terms of the \textit{Relevant Mutation Score} (RMS) and \textit{Minimal-Relevant Mutation Score} (RMS*), which intuitively measures the number of (subsuming) commit-relevant mutants killed by the different test suites. Specifically, we investigate the extent to which selecting and killing each aforementioned mutant types improves the test suite quality, in terms of the number of (subsuming) commit-relevant mutants killed by the test suite.
Then we pose the question: \textit{How many (subsuming) commit-relevant mutants are killed if a developer or test generator selects and kills random mutants or only mutants within a change?}
\begin{table}[]
\caption{Comparative Effectiveness of selecting and killing (subsuming) commit-relevant mutants in comparison to ``\revise{all mutants}'' and ``mutants within a change'' by observing RMS (Relevant Mutation Score) and RMS* (Subsuming Relevant Mutation Score)}
\resizebox{\textwidth}{!}{
\begin{tabular}{c|cccccccccc|cccccccccc|}
\cline{2-21}
& \multicolumn{10}{c|}{\textbf{RMS}} & \multicolumn{10}{c|}{\textbf{RMS*}} \\ \hline
\multicolumn{1}{|c|}{\textbf{Selection Strategy/Interval}} & \textit{\textbf{2}} & \textit{\textbf{4}} & \textit{\textbf{6}} & \textit{\textbf{8}} & \textit{\textbf{10}} & \textit{\textbf{12}} & \textit{\textbf{14}} & \textit{\textbf{16}} & \textit{\textbf{18}} & \textit{\textbf{20}} & \textit{\textbf{2}} & \textit{\textbf{4}} & \textit{\textbf{6}} & \textit{\textbf{8}} & \textit{\textbf{10}} & \textit{\textbf{12}} & \textit{\textbf{14}} & \textit{\textbf{16}} & \textit{\textbf{18}} & \textit{\textbf{20}} \\ \hline
\multicolumn{1}{|c|}{\textbf{Random}} & 46.67 & 70.59 & 82.42 & 88.10 & 91.95 & 95.26 & 95.74 & 96.85 & 97.50 & 98.05 & 11.11 & 35.00 & 54.17 & 66.13 & 75.00 & 81.25 & 85.71 & 88.24 & 90.89 & 92.76 \\ \cline{1-1}
\multicolumn{1}{|c|}{\textbf{Within a change}} & 46.48 & 59.52 & 65.91 & 67.48 & 68.42 & 69.47 & 69.96 & 70.18 & 70.40 & 71.09 & 11.95 & 25.00 & 28.95 & 32.31 & 33.33 & 33.67 & 34.38 & 34.88 & 35.23 & 35.29 \\\cline{1-1}
\multicolumn{1}{|c|}{\textbf{Commit-Relevant}} & 75.00 & 95.05 & 100 & 100 & 100 & 100 & 100 & 100 & 100 & 100 & 40.74 & 83.72 & 100 & 100 & 100 & 100 & 100 & 100 & 100 & 100 \\ \cline{1-1}
\multicolumn{1}{|c|}{\textbf{Subsuming Commit-Relevant}} & 80.00 & 98.51 & 100 & 100 & 100 & 100 & 100 & 100 & 100 & 100 & 65.75 & 95.35 & 100 & 100 & 100 & 100 & 100 & 100 & 100 & 100 \\
\hline
\end{tabular}
}
\label{tab:RQ5-median-developer_simulation}
\end{table}
\begin{figure*}[bt!]
\centering
\begin{subfigure}[t]{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Box_plot_Simulation_relevant_ms_v3.pdf}
\caption{Relevant Mutants Progression}
\end{subfigure}
\begin{subfigure}[t]{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/Box_plot_Simulation_minimal_relevant_ms_v3.pdf}
\caption{Subsuming Relevant Mutants Progression}
\end{subfigure}
\caption{
Comparative Effectiveness of selecting and killing (subsuming) commit-relevant mutants in comparison to ``random mutants'' and ``mutants within a change''}
\label{fig:RQ5-developer_simulation}
\end{figure*}
\autoref{tab:RQ5-median-developer_simulation} and \autoref{fig:RQ5-developer_simulation} demonstrates how the effectiveness of the developed test suites progresses when we analyze up to 20 mutants from the different mutant pools.
We observed that when the same number of mutants are selected from the different pools, better effectiveness is reached by test suites developed for killing (subsuming) commit-relevant mutants.
For instance, a test suite designed to kill six (6) selected (subsuming) commit-relevant mutants will achieve 100\% of RMS and RMS*. However, a test suite designed to kill six randomly selected mutants will achieve 82.42\% RMS and 54.17\% RMS*, while a test suite that kills six mutants within a change will achieve 65.91\% RMS and 28.95\% RMS*, respectively.
More precisely, even after selecting 20 mutants, neither random selection \revise{from all mutants} nor within a change selection achieved 100\% of RMS and RMS*.
This result demonstrates the significant advantage achieved by selecting (subsuming) commit-relevant mutants.
Moreover, we observed that random selection \revise{from all mutants} is up to 1.6 times more effective than selecting mutants within a change.
For instance, selecting 20 random mutants achieves 98.05\% RMS and 92.76\% RMS*, while selecting 20 mutants within a change only achieves 71.09\% RMS and 35.29\% RMS*.
This result demonstrates the importance of selecting mutants \emph{outside} developers' committed changes.
\begin{result}
Selecting and killing (subsuming) commit-relevant mutants led to more effective test suites. They significantly reduced the number of mutants requiring analysis compared to random mutant selection and selecting mutants within a change.
\end{result}
\smallskip\noindent
\subsection{\RQ6: Test Executions}
In this section, we study the \emph{efficiency} of the different mutant sets in terms of the number of \emph{test executions} \revise{required to run the tests resulting from the analysis of 2-20 mutants.
We thus, approximate the \revise{computational} demands involved when using all mutants, relevant mutants, (subsuming) relevant mutants and mutants located within commit changes. }
\revise{\autoref{fig:RQ6-developer_simulation} illustrates the number of test executions required by the test suites \revise{derived by the analysis of 2-20 mutants}.
We found that the analysis of commit-relevant mutants significantly reduces the number of required test executions by \revise{4.28} times on average over different intervals of analysed mutants (and \revise{16} times when using subsuming commit-relevant mutants) in comparison to \revise{test execution required when analysing all mutants}.
For instance, users will need to perform \revise{601} test executions when deriving tests based on the \revise{analysis of 2} mutants, from the set of all mutants, compared to \revise{185 or 52} test executions needed by the use of commit-relevant mutants or subsuming commit relevant mutants, respectively. }
\revise{The difference increases with the number of analysed mutants. Thus, for the 2 analysed mutants, the difference in test execution is 2.8 times. For 4 mutants, 3.65, and 6 mutants, the difference in test execution is 4 times comparing all mutants and the commit-relevant mutants.
We can also observe an increase in the difference between test executions needed by the use of subsuming commit relevant mutants and all mutants over different intervals. This difference is 11.55 times, 14.68 and 16 for analysed 2,4 and 6 mutants, respectively.
Overall, we can compare test execution needed by using commit-relevant and subsuming commit-relevant mutants and observe a 4 times difference on average, with no considerable differences between intervals. }
\begin{result}
Selecting subsuming commit-relevant mutants reduces test execution cost \revise{(i.e., the number of test executions,}) by up to \revise{16} times compared to all mutants.
\end{result}
\begin{figure*}[bt!]
\centering
\includegraphics[scale=0.3]{figures/Box_plot_Simulation_computational_effort_v5.pdf}
\caption{Efficiency, number of test executions required when deriving test suite sizes (in the range [2, 20]).}
\label{fig:RQ6-developer_simulation}
\end{figure*}
\section{Threats to Validity}\label{ValidityThreats}
Our empirical study and findings may be limited by the following validity threats.
\smallskip \noindent \textbf{\textit{External Validity:} }This refers to the generalizability of our findings. We have empirically evaluated the characteristics of commit-relevant mutants on a small set of open-source Java programs, test cases, and mutants. Hence, there is a threat that our experimental protocol and findings do not generalize to other mutants, programs, or programming languages.
\revise{Additionally, there is the threat that our findings do not generalize to other Java projects, since our subject programs are all from the Apache Commons project and may share similar characteristics in terms of architecture, implementation, coding style and contributors.}
We have mitigated these threats by conducting our experiments on five (5) matured Java programs with a varying number of tests and a considerably large number of mutants. In our experiments, we had 288 commits and 10,071,872 mutants with 25 different groups of mutant types. In addition, our subject programs have 216,489 KLOC and 17 years of maturity, on average.
Hence, \revise{we are confident that our empirical findings hold for the tested (Java) projects, programs, commits, and mutants. }
\revise{
Furthermore, we encourage other researchers to replicate this study using other (Java) programs, projects and mutation tools.
}
\revise{
In our experiments we used Pitest \cite{pitest} to perform our analysis.
However, it is likely that the use of a different mutation tool may impact our findings, since it may contain different operators than Pitest.
While this is possible, recent empirical evidence \cite{KintisPPVMT18} has shown that Pitest has one of the most complete sets of mutation operators that subsumes the operators of the most popular mutation testing tools in almost all cases. Nevertheless, we are confident on our results since Pitest includes a large sample of mutants the general results are unlikely to change with different types of simple mutations.
}
\smallskip \noindent \textbf{ \textit{Internal Validity:}} This threat refers to the \textit{incorrectness} of our implementation and analysis, especially if we have correctly implemented/deployed our experimental tools (e.g., Evosuite, Pitest and Pitest assert), performed our experiment as described and accounted for randomness in our experiments. We mitigate the threat of incorrectness by (manually) testing our implementation, tools, and experimental protocol on few programs and commits to ensure our setup works as expected. \revise{Specifically, we performed manual testing by examining five (5) representative Apache programs containing about 500 LoC per commit on average. While we inspected in total about 20 commits with over 30 LoC in patch sizes, on average.}
We also address the threat of randomness in our experiments by repeating our experiments 100 times to mitigate any random or stochastic effects.
\smallskip \noindent \textbf{\textit{Construct Validity:}} This refers to the \textit{incompleteness} of our experimental approach, in terms of \textit{identifying all commit-relevant mutants}. Despite the soundness of our approach, it only provides an approximation of commit-relevant mutants, such that the set of identified commit-relevant mutants is only a subset of the total number of all commit-relevant mutants. This is due to the finite set of test cases and mutants employed in our experiments. We have mitigated this threat by ensuring we have a reasonably large set of mutants and test cases for our experiments.
For instance, following the standards set up by previous studies \cite{KurtzAODKG16,ammann_establishing_2014,PapadakisK00TH19}, we augmented developers' written tests by automatically generating additional tests (using EvoSuite), to expand the observable input space for commit-relevant mutants.
Our experimental findings are also threatened by the potential noise introduced by \textit{equivalent mutants}. First, notice that commit-relevant mutants come either from lines within the change or outside the change. On the one hand, considering our \autoref{algo:relevant} for identifying commit-relevant mutants outside the change, you can notice that in case that mutant $X$ is equivalent, then condition $Yval \neq XYval$ in Line 9 will evaluate to \emph{false}, since mutants $Y$ and $XY$ will be equivalent as well, then mutant $X$ will not be considered as commit-relevant.
On the other hand, our approach selects by default all the mutants within the change as commit-relevant, so there is a potential threat in selecting some equivalent mutant, even though mutants within the change are a small fraction concerning the total number of mutants. To mitigate this threat, we employ standard methods in mutation testing to reduce the probability of generating equivalent mutants, for instance, by applying \texttt{Pitest} to ensure no common language frameworks are mutated.
\revise{
Furthermore, our experimental approach is limited by our measure of mutant execution effort (i.e., efficiency), as well as the granularity of our test assertion checks. Firstly, in our experiments, we have estimated the efficiency of commit-aware mutation testing using the \textit{number of mutant test executions}. This measure is limited because it assumes that all tests have similar execution time (on average).
Thus, there is a threat that our measure of efficiency may not be representative of actual execution time, especially if some mutants/tests have a longer execution time than others. Though, we argue that number of mutant test executions generalizes better than execution or CPU time because it is independent of the infrastructure, level of parallelization and test execution optimizations used.
Consider the case of a test execution optimization that avoids issues caused by infinite loops.
This optimization will result in significantly different execution times than if not employing them.
Similarly, parallelization impacts the requested execution time if different strategies are used.
Therefore, execution time measurements can be more accurate than the number of mutant executions that we use only if one uses the same infrastructure, parallelization, and mutant test execution optimizations.
In our case, we ran our experiments in our University HPC\footnote{https://hpc.uni.lu/} with a heavy parallelization scheme.
Therefore, we feel that its test execution results are hard to generalize to other environments.
We also note, that there are many test execution optimizations \cite{PapadakisK00TH19, WangLXL21} that are not implemented yet by the existing mutation testing tools, fact that may reduce the generalization of our results. }
\revise{
To determine the interactions between mutants, we employ a coarse-grained assertion check in our experiments.
Specifically, our assertion checks are at the assert parameter level.
As an example, given a first-order mutant and a second-order mutant, we directly check the equality of the parameter values (i.e., the expected and actual outcomes) for both mutants.
This raises the threat of missing more fine-grained assertion properties, especially
the effect of dependencies within assertions and test cases.
Our approach may mask such dependencies, e.g., if there is a dependency between the expected and actual value within the assertion.
Indeed, this assertion check may limit the number of observed commit-relevant mutants, as a more fine-grained approach (e.g. one that accounts for such dependencies) may reveal more commit-aware mutants.
In the future, we plan to investigate the effect of assertion granularity on (commit-aware) mutation testing.
Finally, we also encourage other researchers to investigate the effect of these issues (i.e., assertion granularity and test execution effort) on the performance of commit-aware mutation testing. }
|
2024-02-18T23:40:24.592Z
|
2021-12-31T14:49:59.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14566","language":"en","timestamp":1640962199000,"url":"https:\/\/arxiv.org\/abs\/2112.14566","yymm":"2112"}
|
proofpile-arXiv_000-10045
|
{"provenance":"002.jsonl.gz:10046"}
| null | null |
\section{Introduction}
Classical thermodynamics is of great significance to the development of physics, whose validity has been verified in different fields, including astrophysics and cosmology. In the 1970s, physicists represented by Bekenstein and Hawking, in the quest for describing black holes, discovered that the evolution equations of certain gravitational phenomena are similar to the laws and equations of thermodynamics~\cite{Bekenstein:1972tm,Bekenstein:1973ur,Hawking:1974rv,Hawking:1975vcx,Bardeen:1973gs}, which provides new inspirations for revealing the deep correlation between general relativity (GR) and quantum field theory (see Refs.~\cite{Wald:1999vt,Carlip:2014pma} for comprehensive reviews of black hole thermodynamics). Then Gibbons and Hawking developed the research in cosmology and calculated the temperature and entropy of cosmological event horizons~\cite{Gibbons:1977mu}. In the 1990s, 't Hooft and Susskind attempted to study holographic principle with the (second) laws of black hole thermodynamics, and advocated that quantum gravity theories should be holographic~\cite{tHooft:1993dmi,Susskind:1994vu,Aharony:1999ti}. Against the backdrop of the research, the connection between thermodynamic laws and the properties of gravitational systems has attracted more and more attention~\cite{Padmanabhan:2002sha,Jacobson:1995ab,Eling:2006aw,Hayward:1997jp,Mukohyama:1999sp,Cai:2005ra,Cai:2006rs,Akbar:2006kj,Padmanabhan:2003gd,Padmanabhan:2009vy,Gong:2007md}.
In fact, the research on thermodynamics in gravitational systems could be traced back to the first half of the last century~\cite{Tolman11}. Since in the 1920s, astronomical observations showed that the universe is expanding, naturally one can study thermodynamics in the context of cosmology by replacing directly the expanding volume in classical thermodynamics with the co-moving volume. In this case, the thermodynamic energy equation corresponding to the first law of thermodynamics for a closed system (such as the co-moving volume) is consistent with the energy-momentum tensor conservation equation of the matter in cosmology. In the 1980s, Prigogine et al. began to systematically study particle production and the thermodynamic energy equation in the co-moving volume. By comparing the thermodynamic energy equation of an open expanding system with the cosmological evolution equation with an irreversible matter creation process, they summarized the thermodynamic and cosmological properties of particle production~\cite{Prigoginex1,Prigoginex2}. About ten years ago, Harko et al., based on previous research, compared the thermodynamic energy equation of an open co-moving volume with the evolution equation of energy-momentum tensor in the gravity theory with non-minimal coupling between geometry and matter~\cite{Harko:2008qz,Bertolami:2008zh,Harko:2011kv}. They found that the non-conservation of the energy-momentum tensor of the matter can be explained as irreversible particle production in an open system~\cite{Harko:2014pqa,Harko:2015pma}. In general, the core of these studies is the correspondence between (thermodynamic) energy conservation and energy-momentum tensor conservation. Due to the Bianchi identities, energy-momentum tensor conservation always holds in GR. Energy conservation is a pillar in classical physics, but it encounters troubles in GR and quantum mechanics. Is energy conserved in GR? The answer to this question is controversial. The definition of localized energy density in GR is still a subtle problem~\cite{Szabados:2009eka}, which has not been completely solved even though GR has been proposed for over a hundred years. In the early research on GR, Einstein, Landau, Lifshitz and others tried to define the energy-momentum tensor of the gravitational field so that the total energy of matter fields and the gravitational field could be conserved. However, the energy-momentum tensors of the gravitational field constructed in this way are all pseudotensors which are coordinate dependent (see Refs.~\cite{Szabados:2009eka,Coller:1958tx,Arnowitt:1959ah,Arnowitt:1961zz,Maluf:1995re} and references therein for more detail). Actually, the equivalence principle implies that we can transform away gravity when we move from one frame to another, and hence there cannot be such an energy-momentum tensor of the gravitational field which is independent of coordinate. Given that, the current consensus among physicists is that the law of energy conservation in GR is not valid.
In this paper, we continue to discuss energy conservation and energy-momentum tensor conservation in cosmology based on the laws of thermodynamics. The volume of the system will be extended from the conventional co-moving volume to the volume enclosed by an arbitrary cosmological horizon~\cite{Rindler:1956yx,Rindlerb}. By defining a surface energy on the cosmological horizon, the energy conservation of the system can be ensured, so that the energy conservation equation and the energy-momentum tensor conservation equation can be consistent. In addition, by defining an effective pressure at the cosmological horizon, we may also ensure that the thermodynamic energy equation of the matter inside the horizon is consistent with the corresponding energy-momentum tensor conservation equation. Both of these correspondences can be regarded as the correspondence between energy conservation and energy-momentum tensor conservation. For these cosmological horizons, we verify whether the system satisfies the generalized second law of thermodynamics~\cite{Bekenstein:1974ax,Davies:1987ti,Wu:2007se,Wu:2008ir}. For the gravity theory with non-minimal coupling between geometry and matter (non-minimal coupling gravity), the evolution equation of the energy-momentum tensor of the matter inside the cosmological horizon could correspond to the thermodynamic energy equation of an open system with a heat source~\cite{Prigoginex1,Prigoginex2}.
Usually the energy of the (apparent) horizon refers to the energy exchanged between the inside and outside of the horizon during a certain period of time~\cite{Cai:2005ra,Cai:2006rs,Akbar:2006kj,Frolov:2002va}. Here, the surface energy on the horizon that we define could be on account of the tension on the horizon or the intrinsic property of the horizon. Moreover, when there exists particle production in the universe, we can consider the impact of particle production on the evolution of the universe as an extra pressure~\cite{Ford:1986sy,Traschen:1990sw,Abramo:1996ip,Zimdahl:1999tn}, so the effective pressure of the universe is equal to the pressure of the matter plus the extra pressure related to particle production. The effective pressure at the horizon that we define may also be caused by an unknown cosmological process or the intrinsic property of gravity.
Our research motivation is twofold. On the one hand, although the co-moving volume is the system most similar to the classical thermodynamic closed system (in which the particle number is conserved in the absence of particle production and annihilation), its boundary, which may be called co-moving horizon, has no specific physical meaning in cosmology. However, other cosmological horizons have clear physical meanings. For example, the apparent horizon is closely related to holographic principle and the thermodynamic behavior of the universe~\cite{Akbar:2006kj,Bousso:2002ju,Bak:1999hd,Faraoni:2011hf}, and the cosmic event horizon is the largest co-moving distance that a photon emitted at present can propagate in the future~\cite{Giovannini:2008zzb,Davis:2003ad}. Therefore, studying energy conservation and energy-momentum tensor conservation for different cosmological horizons may reveal more characteristics of various horizons. On the other hand, the introduction of the surface energy on the horizon can solve the energy conservation problem of the universe from a phenomenological perspective, so it may also provide an alternative to define the local energy density of the gravitational field in GR. The definition of the effective pressure at the horizon, similar to particle production pressure, may explain the expansion of the universe even without dark energy~\cite{Ford:1986sy,Traschen:1990sw,Abramo:1996ip,Zimdahl:1999tn}.
The paper is organized as follows: Sec.~\ref{sec2} is devoted to introducing energy conservation in cosmology and analyzing why it cannot be implemented by bringing in the non-minimal coupling between matter fields or defining the local energy density of the gravitational field. In Sec.~\ref{sec3}, we study how to implement energy conservation in cosmology by defining a surface energy on a cosmological horizon or an effective pressure at the horizon. With these definitions, the energy conservation equation (or the thermodynamic energy equation) of the system is consistent with the energy-momentum tensor conservation equation of the matter. In Sec.~\ref{sec4}, we calculate and analyze whether the total entropy of the system satisfies the generalized second law of thermodynamics. In Sec.~\ref{sec5}, we extend these studies to the gravity theory with non-minimal coupling between geometry and matter. The discussions and conclusions are rendered in Sec~\ref{sec6}.
\section{Energy conservation in cosmology}
\label{sec2}
In cosmology, whether the thermodynamic energy equation can correspond to the energy-momentum tensor conservation equation, is dependent on whether the energy of the system is conserved. Only when the energy of the universe is conserved, one can derive the energy-momentum tensor conservation equation ($\nabla_\nu T^{\nu\mu}=0$) from the thermodynamic energy conservation ($\text{d}E=-p\,\text{d}V$)~\cite{Tolman11}. The approach here is to default that the selected system (i.e., the co-moving volume) can do work on its surroundings, and the effective pressure at the boundary of the system is equal to the pressure of the internal substance. Such correspondence can be extended to the open system with a heat source or particle production in the non-minimal coupling gravity, i.e., $\nabla_\nu T^{\nu\mu}=\hat X^\mu$ corresponds to $\text{d}E=\text{d}Q-p\,\text{d}V$~\cite{Prigoginex1,Prigoginex2}. In this section, we briefly review energy conservation in cosmology and then analyze, from a phenomenological perspective, why energy conservation in the context of cosmology cannot be preserved by introducing the non-minimal coupling between matter fields or defining the energy density of the gravitational field~\cite{Moller1,Moller2}.
The most intuitive phenomenon about energy non-conservation in the universe is the red-shift of photons. According to the observations of the cosmic microwave background (CMB)~\cite{Penzias:1965wn,Fixsen:2009ug}, the CMB photons are red-shifted as the universe expands. Since the CMB has no interaction with other matter fields after photon decoupling (i.e., the CMB photons are free to travel), it is confusing where the energy of the CMB photons flows after photon decoupling.
First, we can interpret the energy loss of photons as the work done by the universe on its surroundings ($\text{d}E=-p\,\text{d}V$). In this case, we can ensure energy conservation (for the system and outside), while $\text{d}E=-p\,\text{d}V$ and $\nabla_\nu T_p^{\nu\mu}=0$ (where $T_p^{\nu\mu}$ is the energy-momentum tensor of the CMB photons) are consistent. However, for the photons traveling completely freely, the expansion of the universe is more similar to the free expansion in classical thermodynamics, so the energy of photons should not be lost ($\text{d}E=0$)\footnote{Physicists hold the opinion that the expansion of the universe is the intrinsic property of space-time rather than the widening of the boundary. The assumption that the boundary of the co-moving volume could do work on its surroundings is just a special way to study the thermodynamics of the universe~\cite{Prigoginex1,Prigoginex2,Harko:2014pqa,Harko:2015pma}. Regarding the intrinsic property of space-time and whether the expansion of the universe should be completely free, we will not discuss these problems further here.}. Note that the correspondence between $\text{d}E=\text{d}Q-p\,\text{d}V$ and $\nabla_\nu T^{\nu\mu}=\hat X^\mu$ in Refs.~\cite{Prigoginex1,Prigoginex2,Harko:2014pqa,Harko:2015pma} is a phenomenological result. So far we do not figure out if there are deeper reasons for the consistency. Therefore, at least from the perspective of the CMB photons, interpreting their energy loss by doing work requires us to discard the standpoint that the universe is expanding freely. In our following research, we will also use the concept of work to explain the energy transformation of the universe.
Moreover, if we can well define the energy density of the gravitational field, it is possible to suppose that there exists energy exchange between the gravitational field and the CMB photons. Therefore the energy of photons may flow into the gravitational field. However, such a definition has been proven to be impracticable or defective in GR, and so it is not likely to be feasible in cosmology. Noether's theorem tells us that energy conservation is the result of time translation symmetry. Since there is no time translation symmetry in the expanding universe, there is no energy conservation in cosmology according to classical Noether's theorem. From the perspective of GR, the ``Noether current'' related to the gravitational field (the energy-momentum tensor of the gravitational field) is a pseudo-tensor $\tilde T^{\mu\nu}$ with two indices. Although we have $\nabla_\nu \tilde T^{\mu\nu}=0$, when we try to use Gauss theorem to convert this local conservation law into a global conservation law (energy conservation), the presence of the Christoffel symbols prevent us from doing that in a valid way for all frames ($\tilde T^{\mu\nu}$ must be dependent on the coordinate system)~\cite{Szabados:2009eka,Coller:1958tx,Arnowitt:1959ah,Arnowitt:1961zz,Maluf:1995re}. Such a property is essentially different from the ``Noether current'' corresponding to the electromagnetic field. Therefore, for most cosmologists the non-conservation of energy in the universe is not ``riddle'', because it is still perfectly true that energy-momentum tensor is always conserved locally and covariantly. Regarding the issue of energy conservation in cosmology and GR, one can abandon it and think that energy is indeed not conserved, but if one insists on energy conservation from a phenomenological perspective, one might see something inspiring especially in cosmology. Next we analyze how to implement energy conservation in cosmology from the perspective of phenomenology.
Since energy non-conservation in cosmology is mainly manifested in the red-shift of the CMB photons, we start from the CMB photons and analyze whether a certain component of the universe can be coupled to photons through an unknown form, so as to explain the energy loss of photons on a phenomenological level. For our homogeneous and isotropic universe, the metric can be given as
\begin{eqnarray}
\text{d}s^2=-c^2\,\text{d}t^2+a^2(t)(\text{d}x^2+\text{d}y^2+\text{d}z^2),
\end{eqnarray}
where $a(t)$ is the scale factor and we have assumed that the curvature of the universe is zero. Therefore, Friedmann equations read
\begin{eqnarray}
3H^2=&&\!\!\kappa^2(\rho_d+\rho_r+\rho_{DM}
+\rho_{DE})\label{eq:FriedmannEq},\\
3H^2+2\dot{H}=&&\!\!-\kappa^2 (p_d+p_r+p_{DM}+p_{DE})\label{eq:PressureEq}\,,\,
\end{eqnarray}
where $\kappa^2\equiv {8\pi\, G}$, $H=\dot{a}/a$ is the Hubble rate, and the dot represents the derivative of time. The indexes $d, r, DM$, and $DE$ indicate dust, radiation (photons and neutrinos), (cold) dark matter, and dark energy, respectively. Their equations of state are given as $p_d=0$, $p_r=1/3\,\rho_r$, $p_{DM}=0$, and $p_{DE}=f(\rho_{DE})$. In the standard $\Lambda$CDM, $p_{DE}=-\rho_{DE}$.
Without loss of generality, we will study the system that is covered by a spherical shell with a time-evolving radius. The spherical shell could be referred to a cosmological horizon. Different evolution of the horizon radius represents different physical meaning. For example, if the horizon radius is proportional to the scale factor, it represents the boundary of the co-moving volume, which could be called as co-moving horizon. If the horizon is a surface that is the boundary between light rays that are directed outwards and moving outwards, and those directed outward but moving inward, it represents the apparent horizon~\cite{Wald:1991zz}. Here we define the energy inside a cosmological horizon as Misner-Sharp (MS) energy~\cite{Misner:1964je,Hayward:1993ph,Hayward:1994bu,Cai:2008gw} and suppose temporarily that the gravitational field has no energy, so the total energy of the universe inside the horizon is defined as
\begin{eqnarray}
\mathfrak{E_t}=(\rho_d+\rho_r+\rho_{DM}+\rho_{DE})V_h,
\end{eqnarray}
where $V_h$ is the volume inside the horizon. In the standard $\Lambda$CDM model, the energy-momentum tensor of photons after photon decoupling satisfies $\nabla_{\mu}T_p^{\mu\nu}=0$ and so the energy density of photons evolves as $\rho_{p}(a)=\rho_{p0}\,a^{-4}$, where $\rho_{p0}$ denotes the radiation density at present ($a=1$). Therefore, the MS energy corresponding to photons is
\begin{eqnarray}
\mathfrak{E}_p=\rho_p\,V_h=\rho_{p0}\,a^{-4}\,V_h.
\end{eqnarray}
For the co-moving volume ($V_h=\frac{4\pi}{3}a^3$) or the volume covered by the apparent horizon ($V_h=\frac{4\pi}{3}H^{-3}$ for vanishing curvature), $\mathfrak{E}_p$ is not a constant and so the energy of photons is non-conserved. Only when $V_h\sim a^{4}$, $\mathfrak{E}_p$ is conserved, but the photon number is not conserved. For a single photon, $\rho_{p}=\rho_{p0}\,a^{-4}$ means its wavelength will be elongated (shortened) in an expanding (collapsing) universe. Note that
$\rho_{p}=\rho_{p0}\,a^{-4}$ is robust since it is the direct consequence from the CMB observation. Therefore, the red-shift of a single photon in the universe is deemed to be a manifestation of the energy non-conservation in GR. If one supposes that energy non-conservation is not a sacred theorem (for example, energy conservation in quantum mechanics is not a rule that must be guarded), the red-shift of a single photon is no longer a problem. But it is still difficult to stop asking where the energy has gone.
Let us inspect whether other components of the universe have similar energy non-conservation. For dust and (cold) dark matter, since their pressure is approximate to 0, in the $\Lambda$CDM model, their total energy density can be given as $\rho_{dDM}(a)=\rho_d+\rho_{DM}=\rho_{dDM0}\,a^{-3}$, where $\rho_{dDM0}$ is the total energy density of dust and (cold) dark matter at present. Therefore, the corresponding MS energy inside the horizon is
\begin{eqnarray}
\mathfrak{E}_{dDM}=\rho_{dDM}\,V_h=\rho_{dDM0}\,a^{-3}\,V_h.
\end{eqnarray}
For the co-moving volume ($V_h=\frac{4\pi}{3}a^3$), energy conservation can be satisfied and the total number of particles is also conserved. But for other volume (such as the apparent horizon), neither total energy nor the number of particles is conserved.
As for dark energy, assuming that the energy density is given as $\rho_{DE}(a)=\rho_{DE0}\,F(a)$, where $\rho_{DE0}$ denotes the current dark energy density and $F(a)$ is a function of the scale factor satisfying $F(1)=1$, the energy inside the horizon $V_h$ is give by
\begin{eqnarray}
\mathfrak{E}_{DE}=\rho_{DE}\,V_h=\rho_{DE0}\,F(a)\,V_h.
\end{eqnarray}
Obviously, the condition for energy conservation is $V_h=F^{-1}(a)$. But whether the number of particles is conserved is unknown.
Now let us check where the energy of photons may flow. The mainstream view is that it flows into the gravitational field. However, since the energy density of the gravitational field cannot be defined well, the explanation does not satisfy many people. Some researchers believe that it may flow into other matter fields, such as dark matter or dark energy, so as to ensure the total energy conservation of the universe. The current research on photons coupling to dark matter or dark energy~\cite{Ullio:2002pj,Chen:2003gz,Galli:2009zc,Wilkinson:2013kia,Boehm:2014vja,Kumar:2018yhh,Escudero:2018thh,Opher:2004vg,Opher:2005px} has seldom paid attention to the issue of the total energy of the universe. Next, we focus on the coupling between photons and other matter fields. Our choice of the horizon is still random. If the energy loss of photons can be explained by introducing the coupling, then the energy loss (or increase) of other components of the universe is no longer a problem.
Assume that photons are coupled to an unknown substance ($T_s^{\mu\nu}$) and other matter ($T_m^{\mu\nu}$) is temporarily ignored (i.e., $\nabla_{\mu}T_m^{\mu\nu}=0$ and $\rho_m\,V_h=\text{Const.}$). The equation of state of the (unknown) substance is $p_s=G(\rho_s)$, where $G(\rho_s)$ is a reasonable function of the energy density. Combining the contracted Bianchi identities (which results in $\nabla_{\mu}T_{total}^{\mu\nu}=0$) and $\nabla_{\mu}T_m^{\mu\nu}=0$, we have $\nabla_{\mu}(T_p^{\mu\nu}+T_s^{\mu\nu})=0$. Since the CMB obeys the law of black-body radiation, so $\nabla_{\mu}T_p^{\mu\nu}=0$ and $\nabla_{\mu}T_s^{\mu\nu}=0$. The corresponding energy densities are given as
\begin{eqnarray}
\rho_p=\rho_{p0}\,a^{-4},\\
-\frac13\int\frac{\text{d}\rho_s}{G(\rho_s)+\rho_s}=\ln a.\label{rhosa}
\end{eqnarray}
In order to ensure that the total energy of the universe is conserved within the horizon $V_h$, we need the energy densities of photons and the substance to satisfy the following relationship:
\begin{eqnarray}
(\rho_{p0}\,a^{-4}+\rho_s)V_h=\text{Const.},\label{eq210}
\end{eqnarray}
where $\rho_s$ is given by Eq.~(\ref{rhosa}). For the co-moving volume $V_h=\frac{4\pi}{3}a^3$, the above equation indicates $\rho_{p0}\,a^{-4}+\rho_s=0$ or $\rho_{p0}\,a^{-4}+\rho_s\sim a^{-3}$. Note that $\rho_{p0}\,a^{-4}+\rho_s=0$ means that the energy density of the substance is negative, which is non-physical. As for the latter case, $\rho_s$ will become negative as $a\rightarrow0$, which is also non-physical. Therefore, for the co-moving volume, the disappearance of the energy of the CMB photons cannot be solved by introducing the coupling between photons and other matter. For the apparent horizon ($V_h=\frac{4\pi}{3}H^{-3}$), according to Friedmann equation $3H^2=\kappa^2(\rho_{p0}\,a^{-4}+\rho_s+\rho_m)$ and $\rho_m\,V_h=\text{Const.}$, it can also be proved that Eq.~(\ref{eq210}) is invalid.
Since gravity is also evenly distributed inside the whole horizon, even if the energy density of the gravitational field can be well defined, according to the definition of the MS energy, it is clear that the energy density of the gravitational field of the system is also proportional to the radius of the horizon to the third power. In this way, one still gets Eq.~(\ref{eq210}) with $\rho_s$ being the energy density of the gravitational field. Therefore, it seems that the definition of the energy density of the gravitational field does not help much to solve the energy loss of the CMB photons.
Reviewing Friedmann equations, it is found that as long as $V_h\sim H^{-2}$, the total energy inside the horizon will be always a conserved quantity. Therefore, choosing a special horizon ($V_h\sim H^{-2}$) as the volume of the universe may solve the energy loss of the CMB photons. However, such a special horizon has two fatal problems. First, the number of particles inside the horizon is not conserved. To achieve energy conservation inside the horizon, one has to request that there exists energy (particles) exchange between the outside and inside of the horizon. Second, we now cannot figure out what its actual physical meaning is.
In summary, by investigating the red-shift of the CMB photons in the universe, it is found that from a phenomenological point of view, energy conservation cannot be achieved through the non-minimal coupling between matter fields or the definition of the energy density of the gravitational field. So what else can we do to solve this problem? Note that the above hypothetical unknown substance always spreads all over the universe, so Eq.~(\ref{eq210}) inevitably needs to be satisfied. But if the substance interacting with photons does not spread the entire universe and even it is not a three-dimensional object, what will happen? In this case, it is easy to associate the unknown substance with the two-dimensional horizon. If there exists a kind of energy related to the area of the horizon, we may solve the problem of energy conservation in the universe. On the other hand, we know that for an expanding system in classical thermodynamics, the energy loss of the system is due to the work done by the system on its surroundings, which is related to the surface area and the pressure of the system. For the co-moving volume in cosmology, if one counts the work done by photons (if it exists) as $W=p_p\,\text{d}V$ ($p_p$ is the pressure of photons), then the red-shift of photons is naturally caused by the work, and so there is no energy non-conservation in the universe. When energy conservation could be implemented in cosmology, it will be consistent with energy-momentum tensor conservation~\cite{Prigoginex1,Prigoginex2,Harko:2014pqa,Harko:2015pma}. Next, we will discuss related issues in detail, and further extend these studies to any horizons and the non-minimal coupling gravity.
\section{The correspondence between energy conservation and energy-momentum tensor conservation with surface energy and effective pressure}
\label{sec3}
In this section, we start from the volume enclosed by any cosmological horizons in the universe. By calculating the energy of the matter inside the horizon, we can define a surface energy on the horizon to implement energy conservation of the system (the energy inside the volume plus the energy on the horizon). In this case, the matter inside the horizon could be regarded as a freely expanding object and the surface energy on the horizon is the intrinsic energy of the horizon. If the matter inside the horizon could do work on its surroundings (i.e., it is not free to expand), then we can define an effective pressure at the horizon, which could also implement energy conservation of the system. Both definitions could ensure the correspondence between energy conservation and energy-momentum tensor conservation. We first focus on two types of special horizons: the apparent horizon and the co-moving horizon. At the end of the section, we generalize our discussions to an arbitrary horizon.
For the matter contained in any horizons, during the evolution of the scale factor from $a_1$ to $a_2$, the corresponding change in the MS energy is given as
\begin{eqnarray}
\nabla\mathfrak{E}_V=\rho_t(a_2)V_h(a_2)-\rho_t(a_1)V_h(a_1),\label{aaa}
\end{eqnarray}
where $\rho_t(a)=\rho_r(a)+\rho_d(a)+\rho_{DM}(a)+\rho_{DE}(a)$.
We define the surface energy on the horizon as a function of the scale factor $\mathfrak{E}_H(a)$. Then the change in the horizon energy from $a_1$ to $a_2$ can be given as
\begin{eqnarray}
\nabla\mathfrak{E}_H=\mathfrak{E}_H(a_2)-\mathfrak{E}_H(a_1).\label{aaa1}
\end{eqnarray}
For a given horizon, if $\nabla\mathfrak{E}_V=-\nabla\mathfrak{E}_H$ is fixed for any $a_1$ and $a_2$, we can state that the total energy of the gravitational system is conserved. Therefore, the definition of the horizon energy $\mathfrak{E}_H$ is totally dependent on the volume $V_h$ and all substances inside the horizon.
Similarly, in order to guarantee energy conservation (for the system and its outside), the definition of the effective pressure $\hat P$ at the horizon needs to satisfy
\begin{eqnarray}
\nabla\mathfrak{E}_V=-\int_{r_h(a_1)}^{r_h(a_2)} \hat P\,\text{d}V_h,\label{aaa3}
\end{eqnarray}
where $r_h(a)$ is the radius of the horizon.
\subsection{Apparent horizon}
\label{sec31}
The apparent horizon is the most researched horizon in cosmological thermodynamics. Since we only consider the universe with vanishing curvature, the radius of the apparent horizon is given as $r_h(a)=H^{-1}$, and the volume inside the apparent horizon is $V_h=\frac{4\pi}{3}H^{-3}$. With Friedmann equations and $V_h=\frac{4\pi}{3}H^{-3}$, Eq.~(\ref{aaa}) can be rewritten as
\begin{eqnarray}
\nabla\mathfrak{E}_V&=&\frac{4\pi}{3}\rho_t(a_2)H^{-3}(a_2)
-\frac{4\pi}{3}\rho_t(a_1)H^{-3}(a_1)\nonumber\\
&=&\frac{4\pi}{\kappa^2}\left[H^{-1}(a_2)-H^{-1}(a_1)\right]\nonumber\\
&=&\frac{4\pi}{\kappa^2}\left[r_h(a_2)-r_h(a_1)\right].\label{aaaa2}
\end{eqnarray}
Since we expect $\nabla\mathfrak{E}_V=-\nabla\mathfrak{E}_H$ for any $a_1$ and $a_2$, then we have
\begin{eqnarray}
\mathfrak{E}_H(a)=E_0-\frac{4\pi}{\kappa^2}r_h,\label{aaaa21}
\end{eqnarray}
which is the only proper definition for the surface energy of the apparent horizon. Note that $E_0$ is an unknown constant with a dimension of energy and it must ensure that $\mathfrak{E}_H(a)$ is positive. Such a definition is just for the establishment of mathematical equations, and so it lacks a physical basis without doubt. As a two-dimensional object, the apparent horizon is actually unconvincing to say that it has energy. In this work, we put aside the physical nature of the surface energy on the horizon. Therefore, we reiterate that the assumption here is only from a phenomenological consideration. In this case, the energy density of the apparent horizon is
\begin{eqnarray}
\rho_H(a)=\frac{E_0}{4\pi\, r_h^2}-\frac{1}{\kappa^2\, r_h},\label{aaaa22222221}
\end{eqnarray}
which only evolves with the radius of the apparent horizon.
With the definition of the surface energy on the apparent horizon, the energy conservation of the entire system can be expressed as
\begin{eqnarray}
\frac{4\pi}{3}\rho_t(a)\,r_h^{3}+E_0-\frac{4\pi}{\kappa^2}\,r_h=\text{Const.}\,.\label{aaaa2111}
\end{eqnarray}
Taking the derivative of the above equation with respect to $t$, with the help of Friedmann equations (\ref{eq:FriedmannEq}) and (\ref{eq:PressureEq}), one can naturally get
\begin{eqnarray}
\dot\rho_t(a)+3H[\rho_t(a)+p_t(a)]=0,\label{aaaa2112}
\end{eqnarray}
which is exactly the energy-momentum tensor conservation equation. It is actually a very obvious result, because Eq.~(\ref{aaaa2111}) itself is an identity. The derivation process is equivalent to using Friedmann equations (\ref{eq:FriedmannEq}) and (\ref{eq:PressureEq}) to derive the energy-momentum tensor conservation equation, which is an inevitable result. Therefore, the energy conservation mentioned above is always consistent with energy-momentum tensor conservation.
Next, we try to analyze and calculate the energy change of the universe (space-time) within the apparent horizon from the perspective of work. In this case, the universe does not expand freely, which is similar to the practice used in Refs.~\cite{Prigoginex1,Prigoginex2}, except that the co-moving volume is replaced by the apparent horizon. We employ $\hat P$ to represent the effective pressure at the apparent horizon, which may be related to the internal matter, but it does not have to be equal to the total pressure of the internal matter. Since the apparent horizon is increasing during the expansion of the universe, the matter inside the apparent horizon is doing work on the outside. From $a_1$ to $a_2$, the total amount of the work can be given as
\begin{eqnarray}
\nabla W_h=\int_{r_h(a_1)}^{r_h(a_2)} \hat P\,\text{d}V_h=\int_{r_h(a_1)}^{r_h(a_2)}4\pi\, r_h^2\,\hat P\,\text{d}r_h.\label{aaaa3}
\end{eqnarray}
The requirement $\nabla\mathfrak{E}_V=-\nabla W_h$ leads to
\begin{eqnarray}
\hat P=-\frac{1}{\kappa^2\,r_h^2}=-\frac{H^2}{\kappa^2}=-\frac{1}{3}\rho_t(a).\label{aaaa3334}
\end{eqnarray}
Such a result is a little bit beyond our expectations, because $\hat P$ is only a simple function of the Hubble rate. Since $\hat P$ is negative, the effective pressure is toward inside the apparent horizon. If the total pressure of all matter in the universe satisfies $p_{t}(a)=\omega(a)\rho_{t}(a)$ with $\omega(a)$ being a function of the scale factor, the relationship between the effective pressure $\hat P$ and the total pressure of all matter is given as
\begin{eqnarray}
\hat P=-\frac{1}{3}\rho_{t}(a)=-\frac{1}{3}\frac{p_{t}(a)}{\,\omega(a)}.\label{aaaa4}
\end{eqnarray}
This result manifests that when the matter inside the apparent horizon is only radiation, i.e., $\omega(a)=1/3$, the effective pressure at the apparent horizon $\hat P$ is equal to the pressure of radiation in the opposite direction.
When the effective pressure at the apparent horizon satisfies Eq.~(\ref{aaaa3334}), the thermodynamic energy equation $\text{d}E=-\hat P\,\text{d}V$ can be expressed as
\begin{eqnarray}
\text{d}E=\frac{1}{3}\rho_t(a)\,\text{d}V_h.
\end{eqnarray}
Since we require $\text{d}E$ to be equal to the energy loss of the system, then we have
\begin{eqnarray}
-\frac{1}{3}\rho_t(a)\,{\text{d}}V_h={\text{d}}[\rho_t(a)V_h].
\end{eqnarray}
Combined with Friedmann equations (\ref{eq:FriedmannEq}) and (\ref{eq:PressureEq}), it can be proved that the above equation is also the energy-momentum tensor conservation equation (\ref{aaaa2112}).
We briefly summarize the above results. If the surface energy on the apparent horizon is determined by Eq.~(\ref{aaaa21}), during the free expansion of the universe, the energy increase of the apparent horizon is equivalent to the energy lost by the matter inside the apparent horizon. Therefore, the total energy of the system remains conserved. In this case, the total energy of the matter insider the apparent horizon (and also the surface energy on the apparent horizon) is not proportional to the area of the apparent horizon, but the radius of the apparent horizon. Moreover, we can define an effective pressure at the apparent horizon as Eq.~(\ref{aaaa3334}), so the work done by the matter inside the horizon is given as Eq.~(\ref{aaaa3}), which ensures that the work is equal to the energy loss of the matter. Note that the work is a quantity related to the thermodynamic process and the equation of state of the internal matter, but the horizon energy is a thermodynamic quantity of state, which is only related to the radius of the apparent horizon. With these two definitions, for the system composed of the apparent horizon and its internal matter, energy conservation and energy-momentum tensor conservation are consistent.
\subsection{Co-moving horizon}
\label{sec32}
In this section, we study another special horizon, i.e., the co-moving horizon, whose radius is given as $r_h(a)=C_0\,a$ with a constant $C_0$. The co-moving volume is the most common thermodynamic system studied in cosmology in previous studies. The remarkable feature of the co-moving horizon is that the number of particles inside the co-moving horizon is conserved, provided there is no particle production and annihilation. In this work, we assume that the co-moving horizon also possesses energy and thermodynamic characteristics similar to the apparent horizon. The change in the MS energy inside the co-moving horizon, from $a_1$ to $a_2$, can be written as
\begin{eqnarray}
\nabla\mathfrak{E}_V=\frac{4\pi}{3}C_0^3\left[\rho_t(a_2)
a_2^3-\rho_t(a_1)a_1^3\right].\label{aaa44}
\end{eqnarray}
Similar to the previous approach, we first define a surface energy on the co-moving horizon as $\mathfrak{E}_H(a)$. When the scale factor evolves from $a_1$ to $a_2$, the energy change of the co-moving horizon should be equal to Eq.~(\ref{aaa44}). Therefore, by defining
\begin{eqnarray}
\mathfrak{E}_H(a)=E_0-\frac{4\pi}{3}C_0^3\,\rho_t(a)a^3=E_0-4\pi \frac{H^2}{\kappa^2}r_h^3,
\end{eqnarray}
one can easily get $\nabla\mathfrak{E}_H=-\nabla\mathfrak{E}_V$. Note that $E_0$ must ensure that $\mathfrak{E}_H(a)$ is positive. The energy density of the horizon is
\begin{eqnarray}
\rho_H(a)=\frac{E_0}{4\pi \,r_h^2}-\frac{H^2}{\kappa^2}r_h,\label{aaaa223441}
\end{eqnarray}
which is dependent on the Hubble rate and the radius of the co-moving horizon.
On the other hand, from the perspective of work, the effective pressure $\hat P$ at the co-moving horizon is determined by the following equation:
\begin{eqnarray}
\nabla\mathfrak{E}_V&=&\frac{4\pi}{3}C_0^3\big\{[\rho_r(a_2)+
\rho_d(a_2)+\rho_{DM}(a_2)+\rho_{DE}(a_2)]a_2^3\nonumber\\
&&-[\rho_r(a_1)+
\rho_d(a_1)+\rho_{DM}(a_1)+\rho_{DE}(a_1)]a_1^3\big\}\nonumber\\
&=&\frac{4\pi}{3}C_0^3\big\{[\rho_r(a_2)+\rho_{DE}(a_2)]a_2^3\nonumber\\
&&-[\rho_r(a_1)+\rho_{DE}(a_1)]a_1^3\big\}\nonumber\\
&=&-\int_{r_c(a_1)}^{r_c(a_2)}4\pi\, r_c^2\,\hat P\,{\text{d}}r_c,\label{aaa5555}
\end{eqnarray}
where the second equality is based on $\rho_d(a)+\rho_{DM}(a)\sim a^{-3}$. If $\hat P=\hat P_r+\hat P_{DE}$, it can be obtained quickly that $\hat P_r=p_r=\frac{1}{3}\rho_r$ and $\hat P_{DE}=p_{DE}=-\rho_{DE}$ (for $\rho_{DE}=\text{Const.}$). It is not difficult to prove that for most matter\footnote{It needs to satisfy the conservation equation $\dot \rho+3H(p+\rho)=0$, and the equation of state is given as $p=\omega_0\rho$, where $\omega_0$ is a constant. Then the energy density can be obtained as $\rho=\rho_0\,a^{-3(\omega_0+1)}$.} in the co-moving horizon,
\begin{eqnarray}
\hat P=p=\omega\, \rho
\end{eqnarray}
is the solution to Eq.~(\ref{aaa5555}).
Comparing Eqs.~(\ref{aaa5555}) and (\ref{aaaa4}), it is found that for the apparent horizon, the relationship between $\hat P$ and $p$ relies on the equation of state $\omega(t)$, and $\omega(t)$ could evolve over time. For the co-moving horizon, when $\omega(t)$ is a constant, the relationship between $\hat P$ and $p$ is irrelevant to $\omega(t)$. But if $\omega(t)$ evolves over time, to figure out $\hat P$, we have to get the energy density of the matter with respect to the scale factor according to the specific form of $\omega(t)$. Note that for the apparent horizon, when $\omega(t)<0$, $\hat P$ and $p$ have the same sign, and when $\omega(t)>0$, their signs are opposite. The physical explanation of such a result is as follows. The energy within the apparent horizon actually increases with the radius of the apparent horizon [see Eq.~(\ref{aaaa2})]. According to the thermodynamic energy equation, the surroundings of the apparent horizon should do work on the matter inside the horizon, so the effective pressure at the horizon needs to satisfy $\hat P<0$ even though the universe is expanding. Since the energy density of all matter is always positive, $\hat P=-\frac{1}{3}\rho_t(a)$ provides a guarantee for the negative value of $\hat P$. If the universe collapses, the direction of $\hat P$ is reversed. However, for the co-moving volume, when $\omega(t)$ is a constant, $\hat P$ and $p$ always have the same sign. And since the energy density is always positive, $\hat P$ is always positive when $\omega(t)>0$, which ensures that when the energy within the co-moving horizon is reduced, the matter inside the horizon will do work on its surroundings, and vice versa. One can refer to Refs.~\cite{Tolman11,Prigoginex1,Prigoginex2,Harko:2008qz,Bertolami:2008zh,Harko:2011kv,Harko:2014pqa,Harko:2015pma} for more discussions on the thermodynamic properties of the co-moving volume. The proof that energy conservation is consistent with energy-momentum tensor conservation is straightforward, which has also been confirmed in Refs.~\cite{Tolman11,Prigoginex1,Prigoginex2,Harko:2008qz,Bertolami:2008zh,Harko:2011kv,Harko:2014pqa,Harko:2015pma}, and so we will not repeat them here again.
\subsection{Arbitrary horizon}
\label{sec33}
Now, we consider an arbitrary cosmological horizon. Suppose the radius of the horizon $R(a)$ is a positive function of the scale factor. During the time period that the scale factor evolves from $a_1$ to $a_2$, the change in the MS energy inside the horizon is also given by Eq.~(\ref{aaa}) with $V_h=\frac{4\pi}{3}R^3$. Similarly, as long as we define the surface energy on the horizon as
\begin{eqnarray}
\mathfrak{E}_H(a)=E_0-\frac{4\pi}{3}\rho_t(a)R^3,
\end{eqnarray}
energy conservation is always satisfied. The second term on the right side of the definition could directly offset the energy of the internal matter, so the total energy of the system is $E_0$. The energy density of the horizon is given as
\begin{eqnarray}\label{iopuoi}
\rho_H(a)=\frac{E_0}{4\pi\, R^2}-\frac{1}{3}\rho_t(a)R.
\end{eqnarray}
Therefore, energy conservation equation is trivial: $E_0=\text{Const.}$, which is still consistent with the energy-momentum tensor conservation equation.
Let us analyze how to achieve the energy conservation of the system from the perspective of work. We still assume that the effective pressure at the horizon could be divided into several parts according to the internal composition. Therefore, the work done by the horizon, from $a_1$ to $a_2$, is given as
\begin{eqnarray}
\nabla W_h=\int_{R(a_1)}^{R(a_2)}4\pi\, R^2\,(\hat P_d+\hat P_r+\hat P_{DM}+\hat P_{DE})\,\text{d}R.\label{ccc1}
\end{eqnarray}
For the standard $\Lambda$CDM model, since the equations of state of all matter are constants, their energy densities have a similar form: $\rho\sim a^{-3(\omega+1)}$. We find that if $R(a)$ happens to be a polynomial of the scale factor: $R(a)=Y a^n$, where $Y$ is a positive constant and $n$ could be any constants, the requirement $\nabla W_h=-\nabla\mathfrak{E}_V$ will lead to a simple correlation between the effective pressure at the horizon and the pressure of the matter inside the horizon. Taking $V_h=\frac{4\pi}{3}R^3$ and $R(a)=Y a^n$ into $\nabla W_h=-\nabla\mathfrak{E}_V$, the effective pressure is given as
\begin{eqnarray}
\hat P\!&=&\!\hat P_d+\hat P_r+\hat P_{DM}+\hat P_{DE}\nonumber\\
\!&=&\!\sum_{i=d,r,DM,DE}\frac{-n+1+\omega_i}{n\,\omega_i}p_i
\,\,\,(n\neq0)\nonumber\\
\!&=&\!\frac{1-n}{n} (\rho_d+\rho_{DM})
-\frac{3n-4}{n}p_r+p_{DE}
\,\,\,(n\neq0).\label{ccc2}
\end{eqnarray}
When $n=0$, we have $\nabla W_h=-\nabla\mathfrak{E}_V=0$ directly. For $n=1$, the above formula reverts to the case of the co-moving horizon, i.e., $\hat P_d=\hat P_{DM}=0$, $\hat P_r=p_r$, and $\hat P_{DE}=p_{DE}$.
For a general radius $R(a)$, the requirement $\nabla W_h=-\nabla\mathfrak{E}_V$ can be expressed as
\begin{eqnarray}
&&\frac{4\pi}{3}\left[\rho_{t}(a_2)R^3(a_2)-\rho_{t}(a_1)R^3(a_1)\right]=\nonumber\\
&&-4\pi\int_{a_1}^{a_2}\hat P\, R^2\, R'\,\text{d} a,\label{ccc3}
\end{eqnarray}
where $\rho_{t}(a)=\rho_d(a)+\rho_r(a)+\rho_{DM}(a)+\rho_{DE}(a)$ and the prime denotes the derivative with respect to the scale factor. Then the universal relationship between $\hat P$ and other variables is
\begin{eqnarray}
\hat P&=&-\frac{1}{3}\frac{\text{d}\rho_t(a)}{\text{d}a} \frac{\text{d}a}{\text{d}R}R-\rho_t(a)\nonumber\\
&=&H[p_t(a)+\rho_t(a)]\frac{\text{d}t}{\text{d}R}R-\rho_t(a),\label{ccc4}
\end{eqnarray}
where the second equal sign is based on Friedmann equations (\ref{eq:FriedmannEq}) and (\ref{eq:PressureEq}). When $R(a)=Y a^n$, it reverts to Eq.~(\ref{ccc2}). It is found that if $\rho_t(a)\sim a^{-3(\omega_t+1)}$ and $\omega_t$ is a constant, there is an analytical solution of $\hat P=-p_t$, which requires the radius of the horizon satisfies
\begin{eqnarray}
R(a)\sim a^{\frac{1+\omega_t}{1-\omega_t}}.\label{ccc5}
\end{eqnarray}
If $\hat P=p_t$ and $\rho_t(a)\sim a^{-3(\omega_t+1)}$ with $\omega_t$ being a constant, the radius of the horizon degrades into the situation of the co-moving horizon:
\begin{eqnarray}
R(a)\sim a.\label{ccc6}
\end{eqnarray}
With Eq.~(\ref{ccc4}), the thermodynamic energy equation is
\begin{eqnarray}
\text{d}E=&&\!\!-\left\{H[p_t(a)+\rho_t(a)]\frac{\text{d}t}{\text{d}R}R-\rho_t(a)\right\}\text{d}V_h\nonumber\\
=&&\!\!\text{d}[\rho_t(a)V_h],
\end{eqnarray}
where $V_h=\frac{4\pi}{3}R^3$. By setting $\text{d}E=-\nabla W_h=\nabla\mathfrak{E}_V$, obviously one can still get
\begin{eqnarray}
\dot\rho_t(a)+3H[\rho_t(a)+p(a)]=0,\label{ccc7}
\end{eqnarray}
which corresponds to energy-momentum tensor conservation.
\section{The generalized second law of thermodynamics}
\label{sec4}
In previous research, we have discussed some special horizons, which are endowed with a surface energy to maintain energy conservation in the context of cosmology. In this section, we analyze whether these horizons are reasonable and appropriate from the perspective of thermodynamic entropy. We assume that there exists a kind of surface entropy matching the horizon, which is similar to the surface entropy of the apparent horizon~\cite{Bousso:2002ju}. Moreover, such surface entropy is compatible with the particle entropy inside the horizon, that is, they can be directly added as an extension quantity. In this way, by inspecting the evolution of the total entropy (the horizon entropy and the internal particle entropy), we can analyze, with the generalized second law of thermodynamics~\cite{Bekenstein:1974ax,Davies:1987ti,Wu:2007se,Wu:2008ir}, the rationality of the horizons mentioned above.
We denote the area entropy of the horizon as $S_A$, which can be similarly defined as the entropy of the cosmological event horizon~\cite{Bekenstein:1972tm,Bekenstein:1973ur,Bekenstein:1974ax,Davies:1987ti,Wu:2007se,Wu:2008ir}:
\begin{eqnarray}
S_A= \frac{k_B}{4\,l_p^2}A_h,\label{ddd1}
\end{eqnarray}
where $k_B$ is the Boltzmann constant, $A_h$ is the area of the horizon, and $l_p$ is the Planck length.
For the particle entropy inside the horizon, we only consider radiation, dust, and dark matter. The radiation entropy can be directly given by the entropy formula of photon gas. For dust and dark matter, one can assign a constant specific entropy (the entropy of a single particle) for them~\cite{Prigogine:1989zz,Calvao:1991wg}, and so the corresponding entropy change can be described by calculating the change in the total number of particles inside the horizon. As for the cosmological constant or dark energy, we can regard its entropy as zero.
As a result, the total entropy of the system can be expressed as
\begin{eqnarray}
\!\!\!S_t=&&\!\!S_A+S_{dDM}+S_r\nonumber\\
=&&\!\!4\pi\, r_h^2\cdot \frac{k_B}{4\,l_p^2}\!+\!\frac{4}{3}\pi\, r_h^3\cdot\sigma_x\, n_x\!+\!\frac{4}{3}\pi\,r_h^3\cdot\frac{4\pi\, k_B^2}{45c^3\,\hbar^3} T^3,\label{ddd2}
\end{eqnarray}
where $r_h$ is the radius of the horizon. Here, $n_x$ is the total number density of dust and dark matter, $\sigma_x$ is the weighted average of the specific entropy of dust and dark matter, and $T$ is the temperature of radiation. Since normal entropy is non-negative, $\sigma_x$ is a positive constant. In the standard $\Lambda$CDM model, we have $n_x\propto a^3=n_0\,a^3$ ($n_0$ is the current number density of dust and dark matter) and $T\propto a^{-1}=T_0\, a^{-1}$ ($T_0$ is the current temperature of radiation).
If we suppose that there is no entropy exchange between all matter and the horizon, then the generalized second law of thermodynamics would be that each component (dust and dark matter have been regarded as a whole) satisfies the second law of thermodynamics, respectively:
\begin{eqnarray}
S_A'=&&\!\!8\pi\,r_h\cdot \frac{k_B}{4\,l_p^2}\cdot\frac{\text{d}r_h}{\text{d}a}>0,\label{441}\\\label{88881}
S_{dDM}'=&&\!\!4\pi\,r_h^2\cdot\sigma_x\, n_0\,a^{-3}\cdot\frac{\text{d}r_h}{\text{d}a}\nonumber\\
&&\!\!-4\pi\, r_h^3\cdot\sigma_x \,n_0\, a^{-4}>0,\\\label{8888}
S_{r}'=&&\!\!4\pi\, r_h^2\cdot\frac{4\pi\,k_B^2}{45c^3\,\hbar^3} T_0^3\,a^{-3}\cdot\frac{\text{d}r_h}{\text{d}a}\nonumber\\
&&\!\!-4\pi\, r_h^3\cdot\frac{4\pi\, k_B^2}{45c^3\,\hbar^3} T_0^3\,a^{-4}>0,\label{ddd4}
\end{eqnarray}
where the prime still indicates the derivative with respect to the scale factor. Since $a(t)$ increases with $t$, $S'>0$ is equivalent to $\text{d} S/\text{d}t>0$. For an arbitrary $r_h$, the above conditions boil down to
\begin{eqnarray}
\frac{\text{d}r_h}{\text{d}a}>\frac{r_h}{a}>0,\label{ddd5}
\end{eqnarray}
which means that as long as the entropy of particles inside the horizon satisfies the second law of thermodynamics, the entropy of the whole system will be constantly increasing. Note that if the equation of state of dark energy is set as $\omega_{DE}<-1$, it will lead to $8\pi\, r_h\cdot \frac{k_B}{4\,l_p^2}\cdot\frac{\text{d}r_h}{\text{d}a}<0$ as $a\rightarrow\infty$. Therefore, we do not consider such a case. Moreover, since the above expression of entropy only corresponds to the late stage of the universe and the time after that, we can relax the second law of thermodynamics further as: $S'>0$ as $a\rightarrow\infty$.
When $r_h=H^{-1}$, it is easy to judge the authenticity of $\frac{\text{d}H^{-1}}{\text{d}a}>\frac{H^{-1}}{a}$. With the help of Eq.~(\ref{eq:FriedmannEq}), we just need to verify $\frac{\text{d}\rho_t^{-1/2}}{\text{d}a}>\frac{\rho_t^{-1/2}}{a}$. If $\rho_t\propto a^{-3(\omega_t+1)}$ with a constant $\omega_t$, then it requires $\omega_t>-\frac{1}{3}$. It is known that the $\Lambda$CDM model predicts that the universe will eventually be dominated by the cosmological constant, and so the equation of state of the universe will approach $-1$ infinitely as $a\rightarrow\infty$. When the universe evolves to the late stage, the entropy of particles inside the apparent horizon does not satisfy the second law of thermodynamics. Note that when the effective equation of state of the universe is larger than $-1$, $\frac{\text{d}H^{-1}}{\text{d}a}>0$ is always tenable. Therefore, the entropy of the apparent horizon is always increasing.
When $r_h\sim a$, the entropy of particles inside the horizon is a constant and the horizon entropy is increasing with the expansion of the universe. The generalized second law of thermodynamics will not be violated.
If these entropy can be converted to each other, then the generalized second law of thermodynamics requires
\begin{eqnarray}
S_t'=&&\!\!8\pi\, r_h\cdot \frac{k_B}{4\,l_p^2}\cdot\frac{\text{d}r_h}{\text{d}a}+\Big(4\pi\, r_h^2\cdot\sigma_x\, n_0\, a^{-3}\cdot\frac{\text{d}r_h}{\text{d}a}\nonumber\\
-&&\!\!4\pi\, r_h^3\cdot\sigma_x\, n_0\, a^{-4}\Big)
+\Big(4\pi\, r_h^2\cdot\frac{4\pi\, k_B^2}{45c^3\,\hbar^3} T_0^3\,a^{-3}\cdot\frac{\text{d}r_h}{\text{d}a}\nonumber\\
-&&\!\!4\pi\, r_h^3\cdot\frac{4\pi\, k_B^2}{45c^3\,\hbar^3} T_0^3\,a^{-4}\Big)>0.\label{ddd3}
\end{eqnarray}
It is worth noting that some studies showed that the entropy of dust and dark matter is much smaller than the entropy of radiation in a unit volume, so the particle entropy inside the horizon is mainly dominated by radiation (see Ref.~\cite{Egan:2009yy} and references therein).
When $r_h=H^{-1}$, we already know that the entropy of particles does not satisfy the second law of thermodynamics at the late stage of the universe (i.e., $a\rightarrow\infty$). Therefore, in order to ensure that the entire system meets the second law of thermodynamics, the entropy increase of the horizon must be greater than the entropy decrease of particles. According to Eqs.~(\ref{88881}) and (\ref{8888}), the first term in $S_{dDM}'$ ($S_r'$) is positive and the second term is negative. Therefore, as $a\rightarrow\infty$, if we have
\begin{eqnarray}
S_A'-4\pi\, r_h^3\cdot\sigma_x \,n_0\, a^{-4}-4\pi\, r_h^3\cdot\frac{4\pi\, k_B^2}{45c^3\,\hbar^3} T_0^3\,a^{-4}>0,
\end{eqnarray}
the entropy of the entire system absolutely satisfies the generalized second law of thermodynamics. Considering that the second terms in $S_{dDM}'$ and $S_r'$ have a same structure, we can define
\begin{eqnarray}
Q_0=\sigma_x \,n_0+\frac{4\pi\, k_B^2}{45c^3\,\hbar^3} T_0^3=\text{Const.}\,.
\end{eqnarray}
It can be proved that as long as
\begin{eqnarray}\label{fff880}
\frac{ \frac{k_B}{\,l_p^2}}{2 Q_0}\cdot\frac{1}{\sqrt{3}}\kappa\left(\frac{3\rho_{dDM0}}
{2\sqrt{\rho_{DE0}}}\right)>1,
\end{eqnarray}
we have
\begin{eqnarray}
S_A'-4\pi\, r_h^3\cdot Q_0\, a^{-4}>0.\label{ddd6}
\end{eqnarray}
Then, $S_t'>0$ can be satisfied at the late stage of the universe (see Appendix). As $a\rightarrow\infty$, the horizon entropy tends to a constant and the particle entropy tends to zero, so the dominant term of the total entropy of the universe is indeed the horizon entropy~\cite{Egan:2009yy}.
When $r_h\propto a$, the particle entropy inside the co-moving horizon is a constant, and so the entropy change is only dependant on the horizon entropy. It is obvious that Eq.~(\ref{441}) is always tenable as $r_h\propto a$. When $r_h$ is random, it is difficult to further simplify the constraint of the generalized second law of thermodynamics on the system [see Eq.~(\ref{ddd3})]. But as long as Eq.~(\ref{ddd5}) is satisfied, Eq.~(\ref{ddd3}) definitely holds.
\section{The correspondence in the non-minimal coupling gravity}
\label{sec5}
Based on previous research, it is found that we can select a specific horizon as the boundary of the universe and assume that the energy of the system includes the particle energy inside the horizon and the surface energy on the horizon. By defining the surface energy, the total energy conservation of the system can be accomplished. Although such a definition of the surface energy lacks a physical basis, the expansion of the universe can be regarded as free expansion while the energy loss of the universe can be explained. On the other hand, we find that for any horizons in the $\Lambda$CDM model, as long as the effective pressure $\hat P$ at the horizon meets Eq.~(\ref{ccc4}), the energy change of the matter inside the horizon is equal to the work done by the matter. In both cases, the energy conservation equation and the thermodynamic energy equation of the closed system are consistent with the energy-momentum tensor conservation equation, respectively. Next, we extend the relationship to the gravity theory with a non-minimal coupling between geometry and matter, in which energy-momentum tensor conservation is generally not valid.
For the non-minimal coupling gravity, the equation for the matter field can be rewritten as
\begin{eqnarray}
\dot \rho+3H(p+\rho)=\hat X,\label{ooo7}
\end{eqnarray}
where $\hat X$ indicates the interaction between geometry and matter.
This equation can correspond to the thermodynamic equation describing the energy change of an open system, which can be given as
\begin{eqnarray}
\text{d}(\rho/n)=\text{d}q-p\,\text{d}(1/n),\label{ooo8}
\end{eqnarray}
where $\text{d}q=\text{d}Q/N$ is the average heat received by a particle and $n=N/V$ is the particle number density of the matter. Therefore, $Q$ is the heat received by the system and $N$ is the total number of the system. When the system is the co-moving volume, one can interpret $\hat X$ as an extra pressure at the horizon or an extra energy density inside the horizon. If the pressure at the horizon is modified, there will be an extra pressure in Friedmann equations~\cite{Ford:1986sy,Traschen:1990sw,Abramo:1996ip,Zimdahl:1999tn}. And if the energy density changes while the pressure does not change, there will exist particle production~\cite{Prigoginex1,Prigoginex2,Harko:2014pqa,Harko:2015pma}. The two thermodynamic explanations of Eq.~(\ref{ooo7}) are actually cannot be distinguished from the macro level.
For a system with particle production but without heat exchange, we have $\text{d}q=\text{d}Q/N=0$, so Eq.~(\ref{ooo8}) can be reexpressed as
\begin{eqnarray}
\dot \rho-(p+\rho)\frac{\dot n}{n}=0,\label{ooo9}
\end{eqnarray}
where we have used Eq.~(\ref{ooo8}) divided by $\text{d}t$. By defining a particle production rate as
\begin{eqnarray}
\Gamma=\frac{\dot N}{N}=\frac{\dot n}{n}+\frac{\dot V}{V},\label{ooo10}
\end{eqnarray}
we have
\begin{eqnarray}
\dot \rho-(p+\rho)\left(\Gamma-\frac{\dot V}{V}\right)=0.\label{ooo11}
\end{eqnarray}
For the co-moving volume without particle production ($\Gamma=0$), it degrades into Eq.~(\ref{ccc7}). If there is particle production in the co-moving volume, we have
\begin{eqnarray}
\dot \rho+3H(p+\rho)=(p+\rho)\Gamma.\label{ooo12}
\end{eqnarray}
Comparing this equation with Eq.~(\ref{ooo7}), we have
\begin{eqnarray}
\hat X=(p+\rho)\Gamma,\label{ooo13}
\end{eqnarray}
which can be explained as the phenomenon of particle production (or annihilation) due to the non-minimal coupling between geometry and matter~\cite{Harko:2014pqa,Harko:2015pma}.
For the case of the co-moving volume without particle production ($\Gamma=0$) but with heat exchange, we can write Eq.~(\ref{ooo8}) as
\begin{eqnarray}
\dot \rho+3H(p+\rho)=n\frac{\text{d}q}{\text{d}t}.\label{ooo14}
\end{eqnarray}
Comparing the above formula with Eq.~(\ref{ooo7}), one can obtain
\begin{eqnarray}
\hat X=n\frac{\text{d}q}{\text{d}t}.\label{ooo15}
\end{eqnarray}
The physical meaning is that the interaction between geometry and the matter field causes the pressure of all particles to change, which is similar to a heat source. In most of the literature, the change in particle pressure caused by this kind of the interaction is also called the production pressure, which satisfies
\begin{eqnarray}
p_c=-\frac{n}{3H}\frac{\text{d}q}{\text{d}t}.\label{ooo16}
\end{eqnarray}
Therefore, Eq.~(\ref{ooo14}) can be expressed as
\begin{eqnarray}
\dot \rho+3H(p+p_c+\rho)=0.\label{ooo17}
\end{eqnarray}
Because both $n$ and $H$ are positive, when $\frac{\text{d}q}{\text{d}t}>0$, $p_c$ is negative, which can promote the expansion of the universe~\cite{Ford:1986sy,Traschen:1990sw,Abramo:1996ip,Zimdahl:1999tn}. Combining with Eqs.~(\ref{ooo13}), (\ref{ooo15}), and (\ref{ooo16}), the thermodynamic effect brought by $\hat X$ for different physical process (with or without particle production) satisfies the following relationship:
\begin{eqnarray}
p_c=-\frac{\Gamma(p+\rho)}{3H}=-\frac{\hat X}{3H}.\label{ooo18}
\end{eqnarray}
In summary, the discussions above are the correspondence between the classical thermodynamic formula of an open co-moving volume [see Eq.~(\ref{ooo8})] and the equation of motion of the matter [see Eq.~(\ref{ooo7})] in the context of the non-minimal coupling gravity. The interaction between geometry and matter can be explained as particle production (annihilation) induced by gravity [see Eq.~(\ref{ooo12})]. The particle production rate $\Gamma$ and the interaction quantity $\hat X$ satisfy Eq.~(\ref{ooo13}). If there is no particle production, one can also explain the interaction as exciting (restraining) the kinetic energy of particles by gravity. So the pressure of particles will be changed and the extra pressure is usually called the production pressure. The corresponding thermodynamic formula (\ref{ooo8}) can be written as Eq.~(\ref{ooo17}). The production pressure $p_c$ and the interaction quantity $\hat X$ satisfies Eq.~(\ref{ooo18}).
Next, we generalize these relationships from the co-moving volume to an arbitrary volume with a horizon radius $R(a)$. First of all, the equation of motion of the matter remains unchanged. For the horizon radius $R(a)$, Eqs.~(\ref{ooo12}) and (\ref{ooo14}) become
\begin{eqnarray}
\dot \rho+3\frac{\dot R}{R}(p+\rho)=(p+\rho)\Gamma=n\frac{\text{d}q}{\text{d}t}.\label{ooo19}
\end{eqnarray}
According to the idea that energy conservation and energy-momentum tensor conservation could be equivalent in standard cosmology, Eq.~(\ref{ooo19}) can also directly correspond to Eq.~(\ref{ooo7}), which yields
\begin{eqnarray}
\hat X&=&n\frac{\text{d}q}{\text{d}t}-3(p+\rho)\left(\frac{\dot R}{R}-H\right)\nonumber\\
&=&(p+\rho)\left(\Gamma-3\frac{\dot R}{R}+3H\right).\label{ooo20}
\end{eqnarray}
When the universe expands freely, we can also define a surface energy on the horizon to keep energy conservation in the non-minimal coupling gravity. For any horizons, energy conservation demands that the surface energy $\mathfrak{E}_H(a)$ satisfies
\begin{eqnarray}
\mathfrak{E}_H(a_2)-\mathfrak{E}_H(a_1)=-\frac{4\pi}{3}\rho(a_2)R^3(a_2)\nonumber\\
+\frac{4\pi}{3}\rho(a_1)R^3(a_1).\label{ooo23}
\end{eqnarray}
Therefore, we can define $\mathfrak{E}_H(a)$ as
\begin{eqnarray}
\mathfrak{E}_H(a)&=&E_0-\frac{4\pi}{3}\rho \,R^3\nonumber\\
&=&E_0-\frac{4\pi}{3}\left(\frac{\hat X-\dot \rho}{3H}-p\right)R^3,
\end{eqnarray}
where the second equality is based on Eq.~(\ref{ooo7}). The energy density of the horizon is given by
\begin{eqnarray}\label{ooo24}
\rho_H(a)=\frac{E_0}{4\pi\, R^2}-\frac{1}{3}\left(\frac{\hat X-\dot \rho}{3H}-p\right)R.
\end{eqnarray}
If the relationship between $\hat X$ and the scale factor is confirmed, we can obtain the energy density of the horizon with respect to the scale factor. In this case, energy conservation is trivial: $E_0=\text{Const.}$, which is obviously consistent with energy-momentum tensor conservation.
Moreover, from the perspective of work, the relationship between the effective pressure $\hat P$ at the horizon surface and the radius of the horizon should be similar to Eq.~(\ref{ccc3})\footnote{Note that the energy change of the system is always equal to the value of the work done by the matter, i.e., Eq.~(\ref{ccc3}), regardless of whether there exists a heat source or particle production.}, which is given as
\begin{eqnarray}
\hat P&=&-\frac{1}{3}\frac{\text{d}\rho}{\text{d}a} \frac{\text{d}a}{\text{d}R}R-\rho\nonumber\\
&=&-\frac{1}{3}\frac{R}{\dot R}\hat X+H\frac{R}{\dot R}(p+\rho)-\rho,\label{ccc2220004}
\end{eqnarray}
where $\hat X$ is related to the particle production rate or the heat source of the system given by Eq.~(\ref{ooo20}). With this definition, we can naturally get that the thermodynamic energy equation [see Eq.~(\ref{ccc3})] is consistent with the evolution equation of energy-momentum tensor [see Eq.~(\ref{ooo7})].
For the co-moving horizon, the effective pressure at the horizon surface is equal to the internal particle pressure plus the pressure related to $\hat X$, which has been verified in Refs.~\cite{Prigoginex1,Prigoginex2,Harko:2014pqa,Harko:2015pma}. For the apparent horizon, we have
\begin{eqnarray}
\hat P=\frac{1}{3}\frac{H}{\dot H}\hat X-\frac{H^2}{\dot H}(p+\rho)-\rho.\label{ooo22}
\end{eqnarray}
When $\hat X=0$, it degrades into Eq.~(\ref{aaaa3334}).
Finally, let us calculate the net energy change of the matter when there is only a heat source in the system. Note that the system includes the matter and the heat source, whose energy change is given by $\frac{4\pi}{3}\rho(a_2)R^3(a_2)-\frac{4\pi}{3}\rho(a_1)R^3(a_1)$. Therefore, the energy change of the system is equal to the work done by the effective pressure of the matter. Since there is a heat source, during the expansion of the system, the matter absorbs energy from the heat source while consuming energy through doing work. We define the net energy change of the matter as the energy absorbed from the heat source minus the work done by the matter.
Since $\text{d}q=\text{d}Q/N$ is the average heat received by a particle, the energy that the matter absorbs from the heat source can be given as
\begin{eqnarray}
\Delta Q\!&=&\!-\int_{t_1}^{t_2}\frac{\text{d}q}{\text{d}t}N \, \text{d}t\nonumber\\
\!&=&\!-\frac{4\pi}{3}\!\!\int_{R(a_1)}^{R(a_2)}\frac{R^3 }{\dot R}\!\left[\hat X\!+\!3(p\!+\!\rho)\!\left(\frac{\dot R}{R}\!-\!H\right)\!\right]\!\, \text{d}R.
\end{eqnarray}
Therefore, the net energy change of the matter is
\begin{eqnarray}
\Delta E&=&\Delta Q-\int_{R(a_1)}^{R(a_2)}\hat P\, \text{d}V\nonumber\\
&=&-\int_{R(a_1)}^{R(a_2)}\frac{4\pi}{3}\frac{R^3 }{\dot R}\left[\hat X+3(p+\rho)\left(\frac{\dot R}{R}-H\right)\right]\text{d}R\nonumber\\
& &-\int_{R(a_1)}^{R(a_2)}\left[-\frac{1}{3}\frac{R}{\dot R}\hat X+H\frac{R}{\dot R}(p+\rho)-\rho\right]\text{d}V\nonumber\\
&=&-\int_{R(a_1)}^{R(a_2)}4\pi\, p\, R^2\,\text{d}R\nonumber\\
&=&-\int_{R(a_1)}^{R(a_2)} p\,\text{d}V.\label{uuu}
\end{eqnarray}
The above formula shows that the net energy change of the matter (not the system) is equal to the work done by the internal pressure of the matter (not effective pressure). In GR, since there are no heat sources, if the universe is not expanding freely, the net energy change of the matter inside a cosmological horizon is equal to the work done by the effective pressure. If the cosmological horizon is not the co-moving horizon, the effective pressure is generally not equal to the internal pressure of the matter. In the non-minimal coupling gravity, if the universe is not expanding freely, the total energy change of the system (which includes the matter and heat source) is also equal to the work done by the effective pressure. But the net energy change of the matter (deducting the heat source supplement) is given as Eq.~(\ref{uuu}). Therefore, as long as the pressure of the matter inside the horizon is positive (which means the net energy change of the matter is negative), all the energy provided by the heat source to the matter would be used for doing work, while the matter itself also loses energy due to the expansion of the system.
\section{Discussions and conclusions}
\label{sec6}
In cosmology, we usually choose the co-moving volume as a thermodynamic system to study the thermodynamic properties of the universe. The advantages of the co-moving volume are twofold. On the one hand, it guarantees that the number of particles in the co-moving volume is conserved when there is no particle production and annihilation. On the other hand, the boundary pressure of the co-moving volume is equal to the pressure of the internal matter (which is in accordance with the laws of classical thermodynamics). The disadvantage of the co-moving volume is that the boundary of the co-moving volume lacks clear physical meaning and so we do not know exactly where the boundary of the co-moving volume is. Given that, we study the thermodynamic properties and conservativeness of the system covered by an arbitrary horizon and generalize these studies to the non-minimal coupling gravity.
From the CMB photons, we analyzed why it is unlikely to implement energy conservation in the universe by introducing the coupling between matter fields or defining the energy density of the gravitational field. It is found that if there exists a kind of energy related to the boundary area of the system, the energy of the system could be conserved while the expansion of the universe could be completely free. By defining the surface energy density of the cosmological horizon [see Eq.~(\ref{iopuoi})], the energy of the matter inside the cosmological horizon will not disappear or be generated out of air. Then the energy conservation equation of the system is consistent with the energy-momentum tensor conservation equation of the matter. If we discard the idea that the universe is expanding freely, we can suppose an effective pressure at the cosmological horizon [see Eq.~(\ref{ccc4})] to guarantee that the energy loss of the system within the horizon is equal to the work done by the matter~\cite{Tolman11,Prigoginex1,Prigoginex2}. Then the thermodynamic energy equation of the system is still consistent with the energy-momentum tensor conservation equation of the matter.
We investigated whether the surface entropy of an arbitrary cosmological horizon and the particle entropy inside the horizon satisfy the generalized second law of thermodynamics. We found that, for the apparent horizon and the co-moving horizon, the conditions for the system satisfying the generalized second law of thermodynamics are closely related to the components of the universe and the evolution of the horizon radius. As for the general cosmological horizon, it is difficult to simplify Eq.~(\ref{ddd3}). But if Eq.~(\ref{ddd5}) is tenable, the system definitely satisfies the generalized second law of thermodynamics.
Finally, we studied the thermodynamic energy equation and the evolution equation of energy-momentum tensor for any cosmological horizons in the non-minimal coupling gravity. We can still define a surface energy density on the cosmological horizon [see Eq.~(\ref{ooo24})] or an effective pressure at the horizon [see Eq.~(\ref{ccc2220004})] to implement general energy conservation. Then the thermodynamic energy equation of the system has the characteristics of an open thermodynamic system, which can be consistent with the evolution equation of the energy-momentum tensor of the matter. We calculated the net energy change of the matter (not the system, which consists of the matter and the heat source). It is found that when the pressure of the matter is positive, all the energy that the matter obtains from the heat source is used for doing work, and the matter itself also consumes internal energy for doing work, which means that the matter cannot actually gain energy from the heat source.
The present work, from a phenomenological perspective, showed that (thermodynamic) energy conservation and energy-momentum tensor conservation always do not conflict even for the system covered by an arbitrary cosmological horizon. The key to the consistency is to implement the energy conservation of the system. However, due to the fact that the number of particles is not conserved for the system covered by the general horizon, it seems counterintuitive to regard the system covered by an arbitrary horizon as a closed thermodynamic system. Therefore, although the thermodynamic conservation equation of the system can correspond mathematically to the energy-momentum tensor conservation equation of the matter, it needs more physical explanation, which is the major drawback of this research and also is a question worthy of our follow-up study. For the non-minimal coupling gravity, there could exist particle production and annihilation, so it may alleviate the problem about the non-conservation of the number of particles for an arbitrary horizon. To calculate the particle entropy within the horizon in the non-minimal coupling gravity, one needs to make some specific assumptions~\cite{Harko:2008qz,Bertolami:2008zh,Harko:2011kv}. In the end, the surface energy may provide ideas for solving the energy conservation problem in GR, which is also a topic that we can study in the future.
\section*{Appendix}
\label{sec7}
|
2024-02-18T23:40:24.602Z
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2021-12-30T02:24:00.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14471","language":"en","timestamp":1640831040000,"url":"https:\/\/arxiv.org\/abs\/2112.14471","yymm":"2112"}
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proofpile-arXiv_000-10046
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{"provenance":"002.jsonl.gz:10047"}
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\subsection{Experimental Settings}
We train the target classification networks with batch size 32 in advance and then freeze the parameters. We train the invertible framework for 50 epochs. The hyper-parameters are set as $\alpha=0.01, \beta=1, \gamma=0.005, \epsilon=2$ and $\delta=0.01$. Adam is adopted as the optimizer, and the batch size is 8. The learning rate is $10^{-4}$. The models are all trained on NVIDIA RTX 3090 and the training finishes in a week.
\noindent\textbf{Dataset and Evaluation.}
We evaluate our proposed RAEG on three typical datasets, namely, CelebA-100, Caltech-101 and Mini-ImageNet.
We divide the images in each category or class into training set and test set at a ratio of $9:1$ and resize all the images to $256 \times 256$ pixels.
We employ peak signal to noise ratio (PSNR) and structural similarity (SSIM) in our quantitative image comparisons. We report the Top-1 accuracy during evaluation.
\noindent\textbf{Benchmark.}
There is no previous work that can simultaneously ensure the robustness of the adversary and provide invertibility. RAE and RIT embed the difference between the original and the adversarial image. However, most robust AE methods \cite{shi2021generating,zhong2021undetectable} introduces too much distortion, which exceeds the maximum capacity of state-of-the-art RDH schemes according to our experiments. Therefore, it is hard to simultaneously ensure robustness and invertibility based on RAE and RIT. We implement respectively RIT and RAE using IGSM and CW as AE and RDH-OVT \cite{zhang2012reversible} as RDH.
\section{Introduction}
Deep learning and neural network models have achieved great success in various fields including image classification and face recognition.
Such superior performance causes concerns for both individuals and companies. On the one hand, individuals concern that their data will be analyzed by unauthorized models, leading to privacy leakage. On the other hand, large-scale datasets are necessary to train a satisfying deep model. However, data collecting and labelling are time-consuming and expensive. Therefore, datasets are always seemed as digital properties. Organizations, companies and researchers may charge for their carefully labeled or annotated private datasets and only those authorized are expected to use these datasets.
However, unauthorized malicious employees may steal the dataset and send it to the company's competitors, resulting in economic losses for the company. As a result, it is vital to protect the data from both instance-level as well dataset-level.
Adversarial Examples (AE) ~\cite{szegedy2013intriguing,carlini2017towards} are specialised in confusing neural networks to make unreliable decisions. The generated examples are indistinguishable with normal ones to the human eye but function drastically different. However, since the generation of AE is gradually irreversible, it also bans the legal users from utilizing the resources. Reversible Adversarial Example (RAE) ~\cite{liu2018reversible} aims to alleviate this issue by reverting AE using reversible data hiding (RDH). The method first generates the adversarial image using existing AE methods, then embeds the adversarial perturbation into the adversarial image, and generates the stego image using RDH. Due to the characteristic of reversibility, the adversarial perturbation and the original image can be recovered.
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{figures/teaser.pdf}
\caption{A practical example of RAEG. The data owner transforms his valuable images into non-recognizable, non-trainable adversarial images, despite using defensive methods like JPEG compression or ComDefend \cite{jia2019comdefend}. Legal users can revert the protection process with trivial loss.}
\label{image_teaser}
\end{figure}
Yin \emph{et al.}~\cite{yin2019reversible} adopted Reversible Image Transform (RIT) instead of histogram shifting to improve the performance of RAE.
However, RAE and RIT are fragile to adversarial defense methods, including Gaussian filtering, JPEG compression, etc. Therefore, the malicious attackers can easily obtain an approximate version of the original images using adversarial defense.
In this paper, we propose Reversible Adversarial Example Generation (RAEG) via an invertible neural network (INN), which takes the defense operations into consideration in the process of generation. In detail, the original image is fed into a U-shaped invertible neural network to obtain the adversarial image. Through the backward process of INN, we recover the original image with trivial loss.
To enhance the robustness of RAEG against defensive operations, we mimic typical image processing operations and require that the protected images through these defenses still capable of fooling networks.
Several classifiers with typical architecture are selected as targets for RAEG to attack with. Successful data protection model should enforce both the generated protected images and the attacked images to mislead the targeted models.
Figure~\ref{image_teaser} illustrates a practical application where the images are perturbed by RAEG for copyright protection. Even though the attacker processes the images using traditional blurring process or adversarial defense methods, he cannot successfully recover the images for network training. However, the authenticated receivers can successfully revert the protection process and train their networks based on the recovered images. Therefore, the data owner is not required to launch an unnecessary extra dataset transmission which is time and network consuming.
Extensive experiments demonstrate that RAEG can better protect the data with slight distortion against image purifying operations than previous methods. The models trained upon protected images or their defensed versions are mal-functioned compared to normal ones. Besides, the proposed method can almost perfectly restore the original image, while the previous methods can only restore a blurry version.
Our main contributions are summarized as follows:
\begin{itemize}
\item We propose a reversible adversarial example generation method based on invertible neural network to protect the dataset. To the best of our knowledge, this is the first work to introduce invertible neural network for generating adversarial examples.
\item Considering defense operations, e.g., JPEG compression, we utilize the neural network to mimic these operations and force the generator to resist these operations.
\item Extensive experiments show that the quality of adversarial images is considerable, and the ability of misleading the classification model with defense operations is better than traditional RAE methods.
\end{itemize}
\begin{figure*}[t]
\centering
\includegraphics[width=1.0\textwidth]{figures/fig1_new2.pdf}
\caption{Network overview of RAEG. The generator transforms the original image into the protected image. The attack layer simulates the most commonly seen image processing attacks. We ensemble several classifiers as the joint targeted networks that the generated images should evade from. The inverse process of the generator recovers the original image.}
\label{image_framework}
\end{figure*}
\input{relatedwork}
\section{Method}
\subsection{Overview}
The sketch of our RAEG framework is presented in Figure \ref{image_framework}, which consists of a generator, an attack layer for defense simulation, several pre-trained targeted classifiers and a discriminator. The generator $\mathbf{G}$ is built on an INN, where given $\mathbf{I}$, we generate the protected version $\mathbf{I}_{\emph{prt}}$ using the forward pass. We introduce an attack layer to simulate the real-world image redistribution stage as well as enhance the robustness of the generated adversarial example.
Afterwards, we emsemble several pre-trained classifiers as the joint targeted networks. We require that both $\mathbf{I}_{\emph{atk}}$ and $\mathbf{I}_{\emph{prt}}$ should mislead the joint prediction of the emsemble models.
In the backward pass, we inversely run $\mathbf{G}$ to recover $\mathbf{I}$.
Finally, we introduce a discriminator to keep high visual quality of the adversarial image.
\subsection{Network Architecture}
\noindent\textbf{The Generator. }
As shown in Figure~\ref{image_framework}, the generator $\mathbf{G}$ is composed of three stacked downscaling
modules and three upsampling modules. Each module contains a Haar downsampling or upsampling transformation layer and four double-sided affine coupling layers.
After the Haar wavelet transformation, the input image or the feature map is decomposed into one low-frequency representation and three high-frequency representations in the vertical, horizontal, and diagonal direction. The wavelet transform is an invertible symmetric transformation, which will not affect the following invertible operations.
Afterwards, the frequency representations are fed to
double-side affine coupling (DSAC) block, which
are the crucial parts in the invertible process.
DSAC block first splits the input image or the feature map $\boldsymbol{x}^i$ into two parts, denoted as $\boldsymbol{x}_1^i$ and $\boldsymbol{x}_2^i$.
Then, we apply double-side affine transformations to both $\boldsymbol{x}_1^i$ and $\boldsymbol{x}_2^i$:
\begin{equation}
\begin{split}
\boldsymbol{x}_1^{i+1} &= \boldsymbol{x}_1^i \odot \exp(\theta_1(\boldsymbol{x}_2^i)) + \phi_1(\boldsymbol{x}_2^i) \\
\boldsymbol{x}_2^{i+1} &= \boldsymbol{x}_2^i \odot \exp(\theta_2(\boldsymbol{x}_1^{i+1})) + \phi_2(\boldsymbol{x}_1^{i+1}),
\end{split}
\end{equation}
where $\theta(\cdot)$ and $\phi(\cdot)$ produce the scale and shift coefficients. The output sub-band feature maps of each block are then concatenated into a complete output feature map, denoted as $(\boldsymbol{x}_{1}^{i+1}, \boldsymbol{x}_{2}^{i+1})$.
In the reverse process, the input feature map is first split into two sub-band feature maps $\boldsymbol{x}_{1}^{i+1}$ and $\boldsymbol{x}_{2}^{i+1}$, and then use double-side affine transformations to get two reversed output sub-band feature maps $\boldsymbol{x}_1^i$ and $\boldsymbol{x}_2^i$:
\begin{equation}
\begin{split}
\boldsymbol{x}_2^i &= (\boldsymbol{x}_2^{i+1} - \phi_2(\boldsymbol{x}_1^{i+1})) \oslash \exp(\theta_2(\boldsymbol{x}_1^{i+1})) \\
\boldsymbol{x}_1^i &= (\boldsymbol{x}_1^{i+1} - \phi_1(\boldsymbol{x}_2^i)) \oslash \exp(\theta_1(\boldsymbol{x}_2^i)),
\end{split}
\end{equation}
where $\oslash$ denotes element-wise division. Finally, we concatenate two output sub-band feature maps into the reversed feature map $\boldsymbol{x}^i$. Noted that $\theta(\cdot)$ and $\phi(\cdot)$ are not required to be invertible. We use the residual block to construct the sub-nets. Spectral Normalization (SN)~\cite{miyato2018spectral} instead of the traditional batch normalization is adopted in each block, since it helps stabilizing the training.
\noindent\textbf{The Defense Simulation.}
Similar to~\cite{zhu2018hidden}, we mimic typical image preprocessing by differentiable layers. Our defense layer contains five types of typical digital attacks, including Gaussian noise, Gaussian blurring, scaling, random cropping and JPEG compressing.
Since JPEG simulation in previous works are reported to have over-fitting problem in that the compression mode is always fixed, we propose to generated the pseudo-JPEG images $\mathbf{I}_{\emph{jpg}}$ by interpolating the results among different methods with varied quality factors.
\begin{equation}
\mathbf{I}_{\emph{jpg}}=\sum_{\mathcal{J}_{k}\in \mathcal{J}}\sum_{QF_{l}\in[10,100]}\epsilon\cdot\mathcal{J}_{k}(\mathbf{I}_{prt}, QF_{l}),
\end{equation}
where $QF$ stands for the quality factor, $\sum_{\mathcal{J}_{k}\in \mathcal{J}}\epsilon=1$ and $\mathcal{J} \in $\{JPEG-SS\cite{liu2021jpeg}, JPEG-Mask\cite{zhu2018hidden}, MBRS\cite{jia2021mbrs}\}.
\noindent\textbf{The Ensemble Targeted Networks. }
We employ four targeted classifiers with different architectures, i.e., VGG16, ResNet-50, ResNet-101 and DenseNet-121. The networks are pre-trained on clean $\mathbf{I}$.
\noindent\textbf{The Discriminator. }
Finally, we introduce a discriminator $\mathbf{D}$ to improve the quality of $\mathbf{I}_{\emph{prt}}$, which distinguishes the generated images $\mathbf{I}_{\emph{prt}}$ from the original image $\mathbf{I}$.
\begin{figure}[!t]
\centering
\includegraphics[width=0.49\textwidth]{figures/123.pdf}
\caption{Inversion test of RAEG. We perform different kinds of attacks on the protected images before reconstruction, e.g., JPEG with QF=70 on (a), Gaussian Blur on (b) and rescaling on (c). RAEG can still reverse the images despite the presence of the attacks.}
\label{image_comparison}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=0.49\textwidth]{figures/protected.pdf}
\caption{Showcase of protected images among RAE-IGSM, RIT-IGSM and RAEG. The visual quality of RAE-generated images are of good quality yet the robustness is much lower (specified in Table~\ref{table_comparison}).}
\label{image_protected}
\end{figure}
\subsection{Loss Functions}
The objective functions include the reconstruction loss $\mathcal{L}_{prt}$, the classification loss $\mathcal{L}_{\emph{loc}}$ and the adversarial loss $\mathcal{L}_{\emph{adv}}$.
In the following, $\alpha$, $\beta$, $\gamma$, $\delta$ and $\epsilon$ are hyper-parameters.
\noindent\textbf{Reconstruction Loss.}
We employ a protection loss $\mathcal{L}_{prt}$ and a recovery loss $\mathcal{L}_{\emph{rec}}$ respectively to encourage $\mathbf{I}_{\emph{prt}}$ and the recovered image $\hat{\mathbf{I}}$ to resemble $\mathbf{I}$.
For $\mathcal{L}_{\emph{rec}}$, we have \begin{equation}
\mathcal{L}_{prt}=\lVert\mathbf{I}-\mathbf{I}_{\emph{prt}}\rVert_{1}.
\end{equation}
For $\mathcal{L}_{\emph{prt}}$, we additionally employ the perceptual loss $\mathcal{L}_{\emph{per}}$ by employing a VGG16 feature extractor. We compare the feature maps of $\mathbf{I}$ and $\mathbf{I}_{prt}$ from the second max-pooling layer.
\begin{equation}
\mathcal{L}_{per} = \alpha\cdot\sum_{i=1}^N \frac{1}{H \cdot W \cdot C}|\phi_{pool_i}^{gt} - \phi_{pool_i}^{pred}|_1+\lVert\mathbf{I}-\hat{\mathbf{I}}\rVert_{1},
\end{equation}
where $H, W, C$ refer to the height, weight and channel number of features.
\noindent\textbf{Discriminator Loss.}
For the discriminator loss, we update $\mathcal{D}$ by minimizing the binary classification loss:
\begin{equation}
\mathcal{L}_{dis} = \mathbb{E}_{\mathbf{I}} \log (\mathcal{D}(\mathbf{I})) + \mathbb{E}_{\mathbf{I}_{\emph{prt}}} \log (1 - \mathbf{D}(\mathbf{I}_{\emph{prt}})).
\end{equation}
The generator aims to fool the discriminator to make wrong predictions on whether an image is an original image or a generated image. The GAN loss $\mathcal{L}_\emph{GAN}$ is therefore:
\begin{equation}
\mathcal{L}_\emph{GAN}=\mathbb{E}_{\mathbf{I}_{\emph{prt}}} \log ( \mathcal{D}(\mathbf{I}_{\emph{prt}})).
\end{equation}
\noindent\textbf{Classification Loss.}
We use the negative cross entropy (CE) loss to encourage $\mathbf{I}_{\emph{prt}}$ to be wrongly predicted by the targeted models. Considering that the $\ell_1$ distance term alone might not be powerful enough to recover the trivial details of $\mathbf{I}$, we add an additional CE loss to further encourage $\hat{\mathbf{I}}$ to be correctly classified. Therefore, the classification loss is defined as:
\begin{equation}
\mathcal{L}_\emph{cls}=\emph{CE}(\emph{f}(\hat{\mathbf{I}}),l)-\epsilon\cdot~\emph{CE}(\emph{f}(\mathbf{I}_\emph{prt}),l).
\end{equation}
\noindent\textbf{Total Loss.}
The total loss is defined as follow:
\begin{equation}
\mathcal{L} = \mathcal{L}_{prt} + \beta\cdot \mathcal{L}_{rev} + \gamma\cdot\mathcal{L}_{cls} + \delta \mathcal{L}_{\emph{GAN}}.
\end{equation}
\input{sec-experiment}
\section{Conclusion}
In this paper, we design a U-shaped invertible image transfer network to nullify the dataset by generating acceptable perturbations to alter their intrinsic properties. The reversibility of the image transfer network ensures the performance of authorized users. Extensive experiments demonstrate that the proposed method can effectively nullify the images with acceptable distortion.
\bibliographystyle{IEEEbib}
\section{Related work}
\subsection{Adversarial Attack and Defense}
The concept of the adversarial attack is first proposed by Szegedy et al.~\cite{szegedy2013intriguing}.
IGSM~\cite{kurakin2016adversarial} iteratively uses the gradients of the loss with respect to the input image to create a new image that maximizes the loss. C\&W~\cite{carlini2017towards} is an efficient optimization objective for iteratively finding the
adversarial examples with the smallest perturbation. Many other AE methods have been proposed in the past decades \cite{zhong2021undetectable}.
There exist different adversarial defense methods. Most representative methods are based on image denoising, e.g., ComDefend~\cite{jia2019comdefend} and image reconstrution, e.g., DIPDefend~\cite{dai2020dipdefend}.
Besides, traditional image processings such as JPEG compression, Gaussian filter can also serve as lightweight denoising methods. Adversarial training is another powerful technique for defense. To counter the defensive methods, there are also some robust AE methods \cite{shi2021generating}, but the perturbation is much larger and the robustness is not generalized.
\subsection{Reversible Adversarial Example}
Liu et al.~\cite{liu2018reversible} aim at to restoring the original image from the reversible adversarial image using RDH method.
In detail, the adversarial image is generated by the existing AE methods, e.g. C\&W. Then, RAE try to hide the difference into the adversarial image using RDH. Due to the limited capacity of RDH, the information of the difference is usually too large, and they alternatively downsample the images and hide the resized difference. \textit{Therefore, the original image cannot be perfect restored. Besides, the adversarial defense methods can easily invalidate the RAE.}
To perfectly restore the original image, Yin et al.~\cite{yin2019reversible} introduce reversible image transform (RIT) for generating reversible adversarial example. The essence of RIT is permuting the blocks, e.g. $2\times2$, of the original image to match the adversarial image. Then some additional information, e.g., block index, should be embedded into the permuted image using RDH to generate the final image. It can be easily inferred that the permutation causes large distortion, e.g., severe block artifacts. \textit{The adversarial ability of RIT degrades a lot resulting from the distortion. }
\section{Experiment}
\input{exp_setting}
\subsection{Qualitative and Quantitative Analysis}
\noindent\textbf{Qualitative Results.}
Figure~\ref{image_comparison} and Figure~\ref{image_protected} provides examples of RAEG inversion test and RAEG-generated protected images where $\mathbf{I}$ are acceptably modified for protection. The perturbation is generally represented by contrast change or local area modification. With $\mathbf{I}$ kept secret, the overall quality of $\mathbf{I}_{\emph{prt}}$ is decent enough for normal usage of these images. The average PSNR between $\mathbf{I}$ and $\mathbf{I}_{\emph{prt}}$ in these examples is 28.12dB. In contrast, RIT cannot generate visual-satisfactory protected images in that there are noticable distortions. Please zoom in for details. The quality of generated image by RAE is the best, but the robustness of adversary is limited according to the following tests. In Figure~\ref{image_comparison}, the protected images cannot be categorized by traditional classifiers while the reversed images can, and $\hat{\mathbf{I}}$ is nearly same as the $\mathbf{I}$.
\noindent\textbf{Quantitative Comparisons.}
Table~\ref{table-psnr-acc} presents the numerical results of our RAEG under various parameters. Their exist a trade-off between the adversary and the visual quality of the protected image. PSNR$\approx$28dB in RAEG provides a considerable
trade-off, so we adopt this experimental setting during comparison. As for the recovered image $\hat{\mathbf{I}}$, the PSNR is around 40dB and SSIM is greater than 0.97.
Table~\ref{table_comparison} compares the accuracy of the target models on images generated by different RAE methods and RAEG. our RAEG significantly outperforms RAE methods against typical attacks, including low quality JPEG compression, resize. Besides, we also report the anti-defense capability against two state-of-the-art methods, namely, ComDenfend \cite{jia2019comdefend} and DIPDefend \cite{dai2020dipdefend}. The results are also promising that the attackers cannot effectively increase the classification accuracy.
\noindent\textbf{Prevention of Training Pirated Models.}
We further explore the performance of RAEG on protecting the whole dataset. We assume that the attacker can obtain the whole dataset from collecting enough $\mathbf{I}_{\emph{prt}}$, and he trains his own ResNet-50/ResNet-101/DenseNet-121 network based on these images. As can be seen from Table~\ref{table-retrain}, the three pirated models cannot be well trained on the protected dataset, even if he pre-processes $\mathbf{I}_{\emph{prt}}$ using ComDefend. In contrast, a verified recipient can train his networks well by inverting $\mathbf{I}_{\emph{prt}}$. The results clearly point out the effectiveness of RAEG to protect the whole dataset.
\begin{table}[!t]
\footnotesize
\centering
\caption{Results of RAEG under various settings. $A$, $P$, $S$ respectively denote Accuracy, PSNR and SSIM.}
\begin{tabular}{c|c|c|c|c|c|c|c}
\hline
Dataset & $A_{ori}$ & $A_{prt}$ & $A_{rev}$ & $P_{prt}$ & $P_{rev}$ & $S_{prt}$ & $S_{rev}$ \\
\hline
\multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}Caltech\\-101\end{tabular}} & 0.953 & 0.654 & 0.944 & 30.504 & 45.499 & 0.900 & 0.989 \\
& 0.980 & 0.479 & 0.977 & 27.256 & 45.260 & 0.848 & 0.985 \\
& 0.956 & 0.166 & 0.941 & 24.167 & 41.684 & 0.728 & 0.963 \\
\hline
\multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}CelebA\\-100\end{tabular}} & 0.823 & 0.147 & 0.750 & 31.878 & 43.542 & 0.955 & 0.996 \\
& 0.820 & 0.060 & 0.823 & 28.720 & 48.042 & 0.901 & 0.998 \\
& 0.833 & 0.020 & 0.643 & 25.928 & 38.575 & 0.872 & 0.989 \\
\hline
\multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}Mini-\\ImageNet\end{tabular}} & 0.926 & 0.211 & 0.889 & 30.415 & 42.262 & 0.893 & 0.983 \\
& 0.948 & 0.068 & 0.941 & 27.366 & 39.361 & 0.871 & 0.979 \\
& 0.942 & 0.086 & 0.933 & 25.947 & 40.209 & 0.830 & 0.979 \\
\hline
\end{tabular}
\label{table-psnr-acc}
\end{table}
\begin{table}
\footnotesize
\centering
\caption{Training networks based on different datasets. The accuracies are from ResNet-50/ResNet-101/DenseNet-121.}
\begin{tabular}{c|c|c}
\hline
\multirow{2}{*}{Dataset} & \multicolumn{1}{c|}{Trained on $\mathbf{I}_{atk}$} & \multirow{2}{*}{Trained on $\hat{\mathbf{I}}$} \\
& using ComDefend & \\
\hline
Caltech-101 & 0.473 / 0.552 / 0.591 & 0.974 / 0.971 / 0.967 \\
\hline
Mini-ImageNet & 0.526 / 0.565 / 0.597 & 0.935 / 0.955 / 0.952 \\
\hline
CelebA-100 & 0.243 / 0.252 / 0.260 & 0.805 / 0.807 / 0.801 \\
\hline
\end{tabular}
\label{table-retrain}
\end{table}
\subsection{Ablation Study}
\begin{figure}[t]
\centering
\includegraphics[width=0.5\textwidth]{figures/ablation.pdf}
\caption{Ablation study on the quality of $\mathbf{I}_{\emph{prt}}$. The performances from the partial setups have noticeable defects.}
\label{image_ablation}
\end{figure}
\begin{table}[t]
\footnotesize
\centering
\caption{Ablation study on robustness, visual quality and generalization of RAEG.}
\begin{tabular}{c|c|c|c|c}
\hline
Setting & $P_{rev}$ & $A_{prt}$ & $A_{tran}$ & $A_{def}$\\
\hline
$\mathbf{G}$ using ``Enc-Dec" architecture & 31.37 & 0.479 & 0.569 & 0.803 \\
\hline
w/o discriminator & 37.53 & 0.241 & 0.265 & 0.491 \\
\hline
w/o VGG loss & 38.62 & 0.197 & 0.247 & 0.343\\
\hline
Training with one victim & $\textbf{42.61}$ & $0.124$ & $0.267$ & $0.673$ \\
\hline
Full Implementation & $40.14$ & $\textbf{0.093}$ & $\textbf{0.082}$ & $\textbf{0.122}$ \\
\hline
\end{tabular}
\label{table_ablation}
\end{table}
For space limit, we conduct ablation studies on Mini-ImageNet. The accuracies are reported based on a pretrained ResNet-50. Figure~\ref{image_ablation} and Table~\ref{table_ablation} respectively showcase and provide average results from several partial setups. For fair comparison, we let $P_{prt}\approx27$dB in each test.
\noindent\textbf{Effectiveness of INN Architecture.}
A typical alternative of INN is to model image protection and image recovery independently using Encoder-Decoder-based networks. However, invertible neural network learns stable invertible distribution mapping, where the forward and back propagation operations are in the same network. From Table~\ref{table_ablation}, we observe that INN-based RAEG provides a better performance.
\noindent\textbf{Influence of the Perceptual Loss and the Discriminator.}
As show in Table~\ref{table_ablation}, without the perceptual Loss and the discriminator, the overall performance will decline, and more visually unpleasant distortions appear. The results indicate that they do improve the visual quality.
\noindent\textbf{Investigation on the Target Model.} Training with one target model can improve the visual quality of $x_{rev}$, but it greatly degrades the robustness in the test stage, which highlights the necessity of training with emsemble targeted networks. The reason is that the attack model can easily find the weakness of a fixed classifier through iteration and utimately produces a fixed pattern. It results in poor performance generalization.
|
2024-02-18T23:40:24.606Z
|
2021-12-30T02:21:50.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14420","language":"en","timestamp":1640830910000,"url":"https:\/\/arxiv.org\/abs\/2112.14420","yymm":"2112"}
|
proofpile-arXiv_000-10047
|
{"provenance":"002.jsonl.gz:10048"}
| null | null |
\section{\bf Introduction} ~The solar, atmosphere, accelerator and reactor neutrino oscillation experiments have confirmed that neutrinos are massive and lepton flavors are mixed~\cite{Xing:2020ijf}, which provide solid evidences of new physics beyond the standard model (SM). Another evidence of new physics beyond the SM comes from astrophysical observations, which show that about $26.8\%$ of the Universe is made by dark matter (DM)~\cite{ParticleDataGroup:2016lqr}. There is no cold DM candidate in minimal SM. Various models with DM mass varying from $10^{-22}$ eV up to $10^{\rm 55}$ GeV have been proposed in past few years (for reviews, see \cite{Lin:2019uvt} and references cited therein). However, the non-observation of any signal from the DM direct and indirect experiments has shifted people's interest of DM mass from the electroweak scale to the Sub-GeV scale.
The keV scale sterile neutrino~\cite{Drewes:2016upu}, proposed to fill the ``desert" spanning six orders of magnitude between ${\cal O}(0.1)$ eV and ${\cal O}(0.1)$ MeV in the lepton mass spectrum, is a well-motivated DM candidate~\cite{Boyarsky:2018tvu}. Generally, sterile neutrinos can be produced by the neutrino oscillations in the early Universe via a tiny active-sterile neutrino mixing angle and can make up $100\%$ of the DM, which is called Dodelson-Widrow (DW) mechanism~\cite{Dodelson:1993je}. However, such a minimal mechanism is conflict with the results searching for DM decaying into monochromatic X-ray~\cite{Abazajian:2017tcc}. There are some models \cite{Shi:1998km,Nemevsek:2012cd,Shaposhnikov:2006xi,Laine:2008pg,Abazajian:2014gza,Kusenko:2006rh,Bezrukov:2009th,Merle:2013wta} appeared to avoid this constraint. A recent study shows that the X-ray constraint can be avoided by introducing the neutrino self-interactions~\cite{DeGouvea:2019wpf}, $\lambda_\phi \phi \nu_\alpha \nu_\alpha + {\rm H.c.}$, where $\phi$ is a complex scalar singlet. However, it has been pointed out in Ref.~\cite{Blinov:2019gcj} that it is difficult to construct such an interaction without violating the electroweak symmetry as the active neutrino is the neutral component of an electroweak doublet and the Yukawa coupling $\lambda_\phi$ should be proportional to the active neutrino masses~\cite{Kelly:2020aks}.
In this paper we propose a new sterile neutrino DM model, which extends the SM with a pseudo-Dirac sterile neutrino and a permutation symmetry. The X-ray constraint may be avoided in this model by setting the sterile neutrino produced from the DW mechanism as an intermediate DM state, which will eventually decay into the second sterile neutrino DM and a dark radiation in the early Universe. We study the thermodynamics of this model in the early Universe, and calculate the effective number of neutrino species, $\Delta N_{\rm eff}^{} $, which is highly correlated with the DM phenomena as the existence of the intermediate sterile neutrino state will slightly modify the evolution dynamics of the standard cosmology. We point out that the $\Delta N_{\rm eff}^{} $ predicted by this model can be tested by the CMB stage IV, which is expected to reach $\Delta N_{\rm eff}^{} \sim 0.03$.
We further investigate impact of this model on the Hubble tension problem, a discrepancy between the CMB measurement~\cite{Planck:2018vyg} of the Hubble constant and the direct measurement in the local Universe from supernovae type Ia~\cite{Riess:2018byc}. Our result shows that this tension can be relieved in this model due to the decay of the sterile neutrino, which transfers the energy into the radiation and speeds up the expansion of the Universe resulting a larger Hubble constant than the one implied by the $\Lambda$CDM. The Markov Chain Monte Carlo analysis for the Planck +BAO data and Planck + BAO+R19 data shows that $H_0=68.31_{-0.61}^{+0.54}$ and $H_0=69.2_{-0.59}^{+0.58}$, respectively.
The remaining of the paper is organized as follows: In section II we present the model. Section III is devoted to the study the thermodynamics of the early Universe. We discuss the Hubble tension in section IV. The last part is concluding remarks.
\section{Pseudo-Dirac Sterile Neutrino}
In contrast to conventional sterile neutrino model, we assume that sterile neutrino is a pseudo-Dirac particle and its Yukawa interactions satisfy the permutation symmetry, $N_L^{} \leftrightarrow N_R^C$, where $N_L^{}$ and $N_R^{}$ represent the left-handed and right-handed components of the sterile neutrino. This symmetry is similar to the $\mu\leftrightarrow \tau$ symmetry in the neutrino physics~\cite{Xing:2015fdg}. The relevant Lagrangian can be written as
\begin{eqnarray}
-{\cal L}& = & \hat m \left(\overline{\nu_L^{} } N_R^{} + \overline{\nu_L} N_L^C \right)+ {1\over 2 }\mu \left( \overline{N_R^C} N_R^{} + \overline{N_L^{} } N_L^C \right) \nonumber \\ &&+ m \overline{N_L^{}} N_R^{} + {\rm h.c.} \label{massterm}
\end{eqnarray}
where $\nu_L^{}$ is left-handed active neutrino, $\hat m$ is the Dirac mass arising from Yukawa interaction, $ m$ and $\mu$ are the Dirac mass and Majorana mass of sterile neutrinos, respectively. In the basis $(\nu_L,~ N_L,~ N_R^{} )$, the neutrino mass matrix can be written as
\begin{eqnarray}
M_\nu^{} = \left( \begin{matrix} \times & \hat m & \hat m \cr \hat m & \mu & m \cr \hat m & m & \mu \end{matrix} \right) \; ,
\end{eqnarray}
which can be diagonalized by the $(n_\nu+2)\times(n_\nu+2)$ unitary transformation with $n_\nu$ the generation of active neutrinos, resulting in the mixing between active neutrinos and sterile neutrinos. The pseudo-Dirac sterile neutrino is then decoupled into two Majorana eigenstates $\hat N_1^{} $ and $\hat N_2^{}$ with a slight mass splitting. Due to the permutation symmetry, one of the mass eigenstates, namely $\hat N_2$ with the mass $m_{N_2}^{} =\mu-m$, which comes from the unitary rotation in the $N_L-N_R^C$ plane with maximal mixing angle $45^\circ$, does not mix with the active neutrinos.
The mass of heavier sterile neutrino comes from the diagonalization of the $(n_\nu +1)\times (n_\nu+1)$ sub-mass matrix, which can be roughly written as $m_{N_1}=\mu+m$. In this paper, we assume only one generation left-handed neutrino couples to the sterile neutrino for simplification, but our conclusion does not change when extended to the three generation case. In this case the mixing angle in the $2\times 2$ unitary transformation can easily be calculated as
\begin{eqnarray}
\sin \theta \approx { \sqrt{2} \hat m \over \mu+m }
\end{eqnarray}
Apparently this mixing angle may lead to the unitarity violation in the PMNS matrix \cite{Maki:1962mu,Pontecorvo:1967fh}, and is thus constrained by the neutrino oscillation data and the meson decay results~\cite{Antusch:2006vwa}. Given the mixing angle, the neutrino flavor eigenstate can be written in terms of the mass eigenstates
\begin{eqnarray}
\nu_L^{} = \cos\theta \hat \nu_L^{} - \sin \theta\hat N_{1L}^{} \label{freezein}
\end{eqnarray}
with $\hat \nu_{}^{} $ and $\hat N_1^{} $ the mass eigenstates of active and sterile neutrinos, respectively.
Now we consider the interaction between sterile neutrinos in the mass eigenstates. Due to the permutation symmetry in Yukawa sector, interactions in terms of $\overline{N_L^{} } \phi N_R^{} $ and $ \overline{N_L^{} } \Phi N_L^C + \overline{N_R^C } \Phi N_R^{} $, where $\phi$ and $\Phi$ are new scalar singlets, lead to null interaction between $\hat N_1$ and $\hat N_2$. An exception is the gauge interaction from a local $U(1)_S$ gauge symmetry, $g \left( \overline{ N_L^{} } \gamma^\mu N_L^{} + \overline{N_R^{} } \gamma^\mu N_R^{} \right) A_\mu^{\prime} $, where $g$ is the gauge coupling and $A_\mu^{\prime} $ is the new U(1) gauge field. In the mass eigenstates, one has
\begin{eqnarray}
{\cal L}_A = -g \sin\theta \overline{\hat \nu_{}^{} } \gamma^\mu A_\mu^{\prime} P_L^{} \hat N_2^{} - g\cos\theta \overline{\hat N_1^{} } \gamma^\mu A_\mu^{\prime} P_L^{} \hat N_2^{} \; , \label{master}
\end{eqnarray}
which shows that $\hat N_2$ mainly couple to $\hat N_1$. In this case, the Majorana mass term in Eq.(\ref{massterm}) comes from the non-zero vacuum expectation value of the scalar singlet that spontaneously breaks the $U(1)_S$ symmetry, while the Dirac mass terms between active and sterile neutrinos come from dimension-5 effective operators~\cite{Chao:2010mp,Cai:2014hka}.
\section{Cosmology}
Due to the existence of pseudo-Dirac neutrino and dark radiation, the evolution dynamics of the Universe is slightly modified resulting in interesting phenomena. In this section we will study impacts of this model on the DM, effective number of neutrino species and the Hubble tension problems in turn.
\subsection{DM}
As mentioned above, $\hat N_1$ may be produced in the early Universe via neutrino oscillations, it then can decay into $\hat N_2 $ and $A^\prime$ whenever kinematically allowed. Here we take $\hat N_2$ as the DM candidate and $A^\prime $ as the dark photon which is super-light due to an extremely small gauge coupling, similar to that in fuzzy DM model~\cite{Hu:2000ke}. For the mass of $\hat N_2$, there is a lower limit $m_{\rm DM}> 2$ keV arising from phase space density derived from dwarf galaxies~\cite{Abazajian:2017tcc}. The Boltzmann equation that describes the evolution of the $\hat N_1$ is
\begin{eqnarray}
{d f_{N1}^{} \over dz } &=& {\Gamma(E,z) \sin^22\theta_{\rm eff}^{} \over 4 Hz} f_a(E, z) \Theta(E-m_{N1}^{}) \nonumber \\
&-& {f_{N1}^{} \over Hz} {m_{N1} \over E} \Gamma(\hat N_1 \to \hat N_2 + A^\prime) \label{b-DM}
\end{eqnarray}
where the first term on the right-handed side is from the neutrino oscillation\cite{Abazajian:2001nj} and the second term is from the decay of $\hat N_1$, $f_{N1}^{}$ and $f_\alpha$ are the phase space distribution function of the sterile neutrino $\hat N_1$ and active neutrino respectively, $\Theta(x)$ is the Heavy side function, $\theta_{\rm eff}^{} $ is the effective mixing angle~\cite{Abazajian:2005gj,Hansen:2017rxr}. The distribution function of active neutrinos is characterized by the temperature $T$ and the chemical potential $\mu_\nu$,
\begin{eqnarray}
f_a = {1\over 1+ \exp\left({E-\mu_\nu\over T}\right)}\; .
\end{eqnarray}
Interactions contributing to $\Gamma(E, z)$ include $\nu_a \nu_\beta \leftrightarrow \nu_a \nu_\beta$, $\nu_a \ell^\pm \leftrightarrow \nu_a \ell^\pm$, $\nu_a q \leftrightarrow \nu_a q$ and $\nu_a \nu_a \leftrightarrow \ell^\pm \ell^\mp$. The total interaction rate due to neutrino self interactions and interactions with electron positron pairs is~\cite{Abazajian:2001nj}
\begin{eqnarray}
\Gamma(E, z) \approx \left\{ \begin{matrix} 1.27 G_F^2 E T^4 & a =e \cr 0.92 G_F^2 E T^4 & a=\mu,\tau \end{matrix}\right.
\end{eqnarray}
where $G_F$ is the Fermi constant. At higher temperatures, $\mu^\pm$, $\tau^\pm$ and quarks contribute to the neutrino interaction rate, which will be included in our calculation.
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{rhof.pdf}
\caption{ The energy densities of the heavy sterile neutrino $\rho_1$, light sterile neutrino DM $\rho_2$ and the dark radiation $\rho_3$ as the function of the scale factor $a/a_0$ where $a_0$ is the present scale factor.}
\label{e_densities}
\end{figure}
The DM $\hat N_2$ is produced via the freeze-in mechanism~\cite{Hall:2009bx} with the Boltzmann equation
\begin{eqnarray}
{d f_{N2}^{} \over dz } &=&{f_{N1}^{} \over Hz} {m_{N1} \over E} \Gamma_1^{}(\hat N_1 \to \hat N_2 + A^\prime) \nonumber \\
&-& {f_{N2}^{} \over Hz} {m_{N2} \over E} \Gamma_2^{} (\hat N_2 \to \hat \nu + A^\prime) \label{true-DM}
\end{eqnarray}
Combing the Eq.(\ref{b-DM}) with Eq.~(\ref{true-DM}), one can solve the number density of DM numerically to get the final relic abundance. The decay rate of sterile neutrinos can be written as
\begin{eqnarray}
\Gamma_1^{} (\hat N_1 \to \hat N_2 + A^\prime) &=& {1 \over 4 } \alpha_X^{} m_{N_1} \cos^2 \theta (1-\xi^2 )\\
\Gamma_2^{}(\hat N_2 \to \hat \nu + A^\prime) &=& {1\over 4} \alpha_{X}^{} m_{N_2}^{} \sin^2 \theta
\end{eqnarray}
where we have assumed that masses of $\hat \nu$ and $A^\prime$ are negligible compared to those of sterile neutrinos, $\alpha_X= g^2 /4\pi$ and $\xi\equiv 2\varepsilon=1-m_{N_2}^2/m_{N_1}^2$. The decay rate of $ N_2$ is suppressed by the factor of $\tan^2 \theta$ compared with the decay rate of $N_1^{} $.
We show in the Fig.~\ref{e_densities}, energy densities of $\hat N_1$ , $\hat N_2$ and $A^\prime$, i.e. $\rho_1$, $\rho_2$, $\rho_3$, as the function of scale factor $a/a_0$, where $a_0$ is the present day scale factor, by setting $\sin^{2}\theta=7\times 10^{-11}$, $\hat m_{N_1} =7~{\rm keV}$, $\varepsilon=0.05$ and $\Gamma_{\mathrm N_1}=10^8 ~{\rm km/s/Mpc}$. One can find that $\hat N_1$ is produced relativistically via neutrino oscillation in the early Universe at about ${\cal O}(100)~{\rm MeV}$. It subsequently decays into $\hat N_2$ and $A^\prime$. Given $\rho_{2}^{}$ one can easily derive the relic density of DM today, $\Omega=\rho_2/\rho_C$ where $\rho_C= 1.05\times 10^{-5} h^{-2} ~{\rm GeV/cm^3}$ being the critical density. Alternatively, the observed value $\Omega h^2 =0.12$ may applied to constrain the parameter space of the model. Since $\rho_1$ decayed away in the early Universe and $\rho_2$ does not decay into photon, constraint from X-ray observation can be avoided.
\begin{figure}[t]
\centering
\includegraphics[width=0.49\textwidth]{Neff.pdf}
\caption{ The effective number of neutrino species as the function of the scale factor $a/a_0$.}
\label{neff}
\end{figure}
\subsection{The effective number of neutrino species}
The existence of new physics relevant to neutrino could modify the effective number of relativistic neutrino species $N_{\rm eff}$, which is an important cosmological parameter and is stringently constrained by the precision data from CMB observations. Planck 2018 gives, $N_{\rm eff}=2.99\pm0.17$~\cite{Planck:2018nkj,Planck:2018vyg}, which is consistent with the SM prediction $N_{\rm eff}^{\rm SM} =3.045$~\cite{Mangano:2005cc,Grohs:2015tfy,deSalas:2016ztq,EscuderoAbenza:2020cmq}. Recent studies show that a variety of BSM neutrino physics scenarios, including non-standard neutrino interactions, may significantly contribute to $N_{\rm eff}^{}$ \cite{Escudero:2019gzq,Kelly:2020aks}. These scenarios will be tightly constrained by the future measurement from experiments such as CMB Stage IV, which are expected to reach a precision of $\Delta N_{\rm eff}^{} =N_{\rm eff}^{} -N_{\rm eff}^{\rm SM} \sim 0.03$~\cite{CMB-S4:2016ple,Abazajian:2019eic}.
The temperature relation between neutrinos and photons after the neutrino decoupling can be derived from the entropy conservation arguments, $T_\nu=(4/11)^{1/3} T_\gamma^{} $. The radiation energy density is then the sum of energy density of photon $\rho_\gamma$, neutrino $\rho_\nu $ and dark radiation $\rho_{\rm DR}^{} $
\begin{eqnarray}
\rho_{\rm R} = \rho_\gamma + \rho_\nu + \rho_{\rm DR} = \left[ 1+ {7\over 8} \left( {4\over 11}\right)^{4/3} N_{\rm eff}\right] \rho_\gamma \; ,
\end{eqnarray}
where $N_{\rm eff}^{} $ is the effective number of neutrino species with the default value $3$ by definition, corresponding to three generation of active neutrinos. More precisely, any non-standard energy density can act as $N_{\rm eff}^{}$, which can be written as
\begin{eqnarray}
\Delta N_{\rm eff}^{} = {8\over 7} \left( {11\over 4}\right)^{4/3} {\Delta \rho \over \rho_\gamma^{} }
\end{eqnarray}
where $\Delta \rho$ is the deviation of radiation energy density from the SM prediction. In our model, $\Delta \rho $ comes from decay of the heavy sterile neutrino. $\Delta N_{\rm eff}$ is measured both at the big bang nucleosynthesis (BBN)~\cite{Cyburt:2015mya} and CMB epochs. At the BBN epoch, one has
$
\Delta N_{\rm eff}^{\rm BBN} \sim 3.046\times {\rho_R (T^{\rm BBN})/ \rho_\nu^{} (T^{\rm BBN})}
$
where the pre-factor is the SM prediction of $N_{\rm eff}^{}$. Due to a tiny gauge coupling $g$, the heavy sterile neutrino decays after the BBN epoch, so that its impact to the $\Delta N_{\rm eff}^{\rm BBN}$ is negligible and one only needs to concern $\Delta N_{\rm eff}$ at the CMB epoch.
The Boltzmann equations for the energy density of dark radiation $A^\prime$ and active neutrino are
\begin{eqnarray}
\dot{\rho}_{\rm DR}^{} + 4 H\rho_{\rm DR}^{} &=& \varepsilon \Gamma_1 \rho_{1}^{} + {1\over 2} \Gamma_{2}^{} \rho_{2}^{} \\
\dot{\rho}_\nu + 4 H \rho_\nu^{} &=& {1\over 2} \Gamma_{2}^{} \rho_{2}^{}
\end{eqnarray}
where $\rho_\nu$, $\rho_{\rm DR}$, $\rho_{1}$ and $\rho_{2}$ are energy density of active neutrino, $A^\prime$, $\hat N_1^{}$ and $\hat N_2^{}$ respectively, $H$ is the Hubble parameter, $\Gamma_{1,2}$ is the decay rates given in Eqs.(10) and (11), $\varepsilon$ being the fraction of energy transferred into dark radiation, $\varepsilon \approx 1-m_{N_2}/m_{N_1}$.
Fig.~\ref{neff} shows the $\Delta N_{\rm eff}$ as the function of the scale factor $a/a_0$. The green and blue lines correspond to $\varepsilon=0.05$ and $0.1$ respectively. The dashed horizontal line represent the reach of future CMB stage IV experiment. It shows that this model can be indirectly tested by the precision measurement of $N_{\rm eff}^{}$.
\begin{figure}[t]
\centering
\includegraphics[width=0.49\textwidth]{planck_R192.pdf}
\caption{ Marginal constraints on the fitted cosmological parameters for Planck + BAO+R19 (red) and Planck +BAO (blue). Likelihood contours show the 1$\sigma$ and $2\sigma$ confidence levels.}
\label{hubble}
\end{figure}
\subsection{The Hubble tension}
The Hubble constant, which measures the expansion rate of the Universe, is one of most important cosmological observables. The Hubble Space Telescope (HST) gives a precise estimation of $H_0$, $H_0=(73.42\pm 1.74) $ ${\rm km~s^{-1}~Mpc^{-1}}$~\cite{Riess:2019cxk,Riess:2021jrx}. Alternatively, the Planck satellite data has measured the temperature and polarization anisotropies assuming a $\Lambda$CDM model, resulting in $H_0=(67.37\pm0.54)$ ${\rm km~s^{-1}~Mpc^{-1}}$ \cite{Planck:2018vyg}. These two independent measurements of $H_0$ are in tension with each other. Attempts to relieve the tension include modifying the dark energy equation of state\cite{Poulin:2018cxd,Karwal:2016vyq,DiValentino:2017zyq} or the DM model\cite{Ko:2017uyb,DEramo:2018vss,Kumar:2019wfs,Berezhiani:2015yta,Vattis:2019efj}. For a recent review see, Ref.~\cite{DiValentino:2021izs,Shah:2021onj}. In this section, we will address the Hubble tension problem by using the decay of $\hat N_1$.
In our model, the energy conservation for DM and dark radiation fluids~\cite{Kang:1993xz,Audren:2014bca} yields
{\small
\begin{eqnarray}
\rho_{1}^{} (a)&=& \rho_{*}^{} e^{-\Gamma_1^{} [t-t_*] } \left( {a_{*}^{} \over a } \right)^3 \\
\rho_2^{} (a) &\approx & {\rho_*} (1-\varepsilon) \left( 1- e^{-\Gamma_1 (t-t_*)}\right) \left( {a_* \over a }\right)^3\\
\rho_{\rm DR}^{} (a) &=& {\varepsilon \rho_{*}^{} a_{*}^3 e^{\Gamma_1^{} t_* } \over a^4} \left[ \int_{a_*}^a e^{-\Gamma_1 t } da - \left. a e^{-\Gamma_1^{} t} \right |_{a^*}^a\right]
\end{eqnarray}
}where $a_*$ is the scale factor when $N_1$ start to decay, $\rho_*$ is the energy density of $\hat N_1$ at $a_*$, $t=t(a)$ and $t_*=t(a_*)$. In deriving the energy density of $\hat N_2$, we have neglected the impact of $\Gamma_2$, which is tens of order smaller than $\Gamma_1^{}$.
The dark radiation from the decay of $\hat N_1$ will modify the expansion of the Universe and thus may relieve the Hubble tension, which is similar to the case of decaying DM. On the other hand, the fitted cosmological parameters may put constrain on the parameter space of this model. We use the Friedmann equation for a flat geometry,
\begin{eqnarray}
H^2 (a) =\left( \dot{a} \over a \right)^2 = {8\pi G \over 3 }\hat \rho (a)
\end{eqnarray}
where $\hat \rho(a) =\rho_{1}^{} (a) +\rho_{2}^{} (a) + \rho_{\rm DR}^{} (a) + \rho_\Lambda +\rho_b + \rho_\gamma(a)+ \rho_\nu(a)$ with $\rho_\Lambda$ the energy density of dark energy and $\rho_b$ the energy density of baryons, to perform a Markov Chain Monte Carlo (MCMC) analysis.
In order to obtain the CMB constraint on cosmological parameters we use the public code CLASS \cite{Blas:2011rf} and MontePython \cite{Audren:2012wb} to run MCMC using the following set of cosmological parameters \cite{Planck:2018vyg}:
\begin{eqnarray}
\Theta = \{ \omega_{b },H_0,\ln(10^{10}A_{s }),n_{s },\tau_{reio },\sigma_8 \}
\end{eqnarray}
in addition to our model dependent parameters $\{\omega^{\rm ini}_{\mathrm N_1},\varepsilon, \Gamma_{1} \}$, where $\omega_{i} (i=b, ~{\rm N_1})$ represents $\Omega_{i}h^{2}$. We set $ \varepsilon$ and $\Gamma_{1}$ in the following range: $ \varepsilon \in[0,0.15], ~\log_{10}(\Gamma_{1}/\mathrm{Gyr}^{-1}) \in[4,6] $,
such that $\hat N_1$ decay well before the recombination and $\hat N_2$ can also be viewed as cold DM. Actually, CMB is insensitive to the clustering properties of DM~\cite{Voruz:2013vqa}.
And we further use the following datasets to analyze the model:
\begin{itemize}
\item The 2018 Planck measurements of the CMB \cite{Planck:2019nip} (via TTTEEE Plik high-l, TT and EE low-l, and lensing likelihoods).
\item Baryonic Acoustic Oscillation data (BAO) from BOSS DR12 \cite{BOSS:2016wmc}.
\item the local measurement of $H_0$ from SH0ES (R19) \cite{Riess:2019cxk}.
\end{itemize}
\begin{table}[t!]
\centering
\begin{threeparttable}
\begin{tabular}{|l|c|c|c|c|}
\hline
& $\Lambda$CDM & DNDM & DNDM\\
& Planck+BAO &Planck+BAO & Planck+BAO+R19 \\
Param & mean$\pm\sigma$ & mean$\pm\sigma$ & mean$\pm\sigma$\\ \hline
$100~\omega_{b }$ & $2.242_{-0.014}^{+0.013}$ & $2.241_{-0.016}^{+0.016}$ & $2.258_{-0.015}^{+0.015}$ \\
$\log_{10}(\Gamma_{1})$ &- & $\textgreater 5.199$ & \textgreater 5.32 \\
$H_0$ & $67.7_{-0.45}^{+0.45}$ & $68.31_{-0.61}^{+0.54}$ & $69.2_{-0.59}^{+0.58}$ \\
$\ln10^{10}A_{s }$ & $3.049_{-0.015}^{+0.014}$ & $3.056_{-0.016}^{+0.014}$ & $3.065_{-0.017}^{+0.016}$\\
$n_{s }$ & $0.9664_{-0.004}^{+0.0039}$ & $0.973_{-0.0076}^{+0.0053}$ & $0.9805_{-0.0086}^{+0.0064}$ \\
$\tau_{reio }$ & $0.05718_{-0.0076}^{+0.0072}$ & $0.05774_{-0.008}^{+0.0068}$ & $0.06155_{-0.0083}^{+0.0072}$\\
$\omega^{\mathrm{ini}}_{\mathrm{N}_{1}}$ &$0.1194_{-0.00099}^{+0.00098}$ & $0.1278_{-0.0086}^{+0.0032}$ & $0.1329_{-0.0076}^{+0.0071}$ \\
$\varepsilon$ &- & $\textless 0.06889$ & $0.09233_{-0.048}^{+0.04}$ \\
$\sigma_8$ & $0.8104_{-0.0062}^{+0.0063}$ & $0.8189_{-0.01}^{+0.0077}$ & $0.8243_{-0.012}^{+0.0096}$ \\
\hline
\end{tabular}
\caption{Table for cosmological parameters}\label{tab:tablenotes}
\end{threeparttable}
\end{table}
In fig.\ref{hubble} we show the 2D-plot of ($\omega^{\rm ini}_{\rm N_1}$, $\varepsilon$, $\log(\Gamma_{1}/\mathrm{Gyr}^{-1})$, $H_0$) by using Planck+BAO and Planck+BAO+R19. We can see that the decay of $\hat N_1$ can relieve the Hubble tension. The Hubble constant is positively related to the parameter $\varepsilon$, which controls the energy density of the dark radiation and thus $\Delta N_{\rm eff}^{} $. Enlarged $N_{\rm eff}^{} $ results in a smaller sound horizon. Thus larger Hubble constant is needed to maintain the consistency with the CMB power spectrum.
We dub our model as Decaying Neutrino DM (DNDM) and summarize in Tab.~\ref{tab:tablenotes} constraints on various cosmological parameters. The data of $\Lambda$CDM is included for comparing, in which $\omega_{\rm cdm}=\omega_{\rm N_1}$ with $\hat N_1$ the stable DM. We can see that $\epsilon < 0.069$ for Planck+BAO. For Planck+BAO+R19, constraint on $\varepsilon$ is weaken because of the large prior on $H_0$. Just like other works \cite{Abellan:2021bpx,Nygaard:2020sow,Xiao:2019ccl,Pandey:2019plg,Poulin:2016nat}, the decay DM prefers small share of dark radiation. It is natural in our model because of the Pseudo-Dirac property of the sterile neutrino. Alternatively, we can use the dark radiation density to constrain the Pseudo-Dirac property. It should be pointed out that, if the mass of $\hat N_1$ is smaller, i.e. it behaves like warm DM before recombination as in Ref.~\cite{Blinov:2020uvz}, the model will work better because of the sufficiently long free-streaming length of sterile neutrino. However it can't make up all the DM due to other cosmological bounds.
\section{Conclusion}
Sterile neutrino is a typical warm DM candidate at the keV scale. However most of its parameter space has been ruled-out by the X-ray observations. We have explored one possibility of avoiding the X-ray constraint by introducing the concept of the pseudo-Dirac sterile neutrino with permutation symmetry in the Yukawa interaction sector. Our result shows that the light component of sterile neutrino, which originates from the decay of the heavy sterile neutrino state, can be successful DM candidate in this scenario.
In contradict to traditional sterile neutrino DM, $\hat N_{\hat 1,2}$ mainly decay into dark radiation instead of photon, such that X-ray constraint can be avoided. Importantly, the effective number of neutrino species serves as an indirect detection signal of this model. Impact of the model on the cosmological parameters are explored. Typically the MCMC analysis shows that the Hubble tension problem can be relieved.
\begin{acknowledgments}
The authors thank to Dr. Andreas Nygaard and Dr. Guillermo Franco Abellan for helpful discussions. WC is supported by the National Natural Science Foundation of China under grant No. 11775025, No. 12175027, and the Fundamental Research Funds for the Central Universities under grant No. 2017NT17. ZYF is supported in part by the National Key R\&D Program of China No.~2017YFA0402204, the Key Research Program of the Chinese Academy of Sciences (CAS), Grant NO.~XDPB15, the CAS Project for Young Scientists in Basic Research YSBR-006, and
the National Natural Science Foundation of China (NSFC) No.~11825506, No.~11821505,
and No.~12047503.
\end{acknowledgments}
|
2024-02-18T23:40:24.608Z
|
2021-12-30T02:26:53.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14527","language":"en","timestamp":1640831213000,"url":"https:\/\/arxiv.org\/abs\/2112.14527","yymm":"2112"}
|
proofpile-arXiv_000-10048
|
{"provenance":"002.jsonl.gz:10049"}
| null | null |
\section{Introduction}
Coal and metals are resources necessary for industrialization.
Economic activities rely heavily on these resources, which are used not only for industries, but also for transportation and infrastructure.
Predictions regarding agglomeration theory imply that resource abundance can increase the demand for local labor in the mining sector and related industries, thereby causing an influx of population and agglomeration \citep{Rosenthal2004, Duranton2004, Lederman2007}.\footnote{Induced agglomeration can improve efficiency in the local labor markets through, for instance, better matching and learning, causing spillover effects in the non-mining sectors \citep{Black:2005uh, Bjornland:2016gz}. See \citet{Corden:1982ub, Eastwood:1982wm, Corden:1984tt, vanWijnbergen:1984wf, Neary:1984wb} for earlier theoretical studies on the impact of resource discoveries.}
However, an increase in the resource extraction industry may stagnate the development of non-resource extraction industries by reallocating resources from the non-resource tradable sector \citep{Sachs:1999ul, Sachs:2001vc}.\footnote{The wealth effect mechanism suggests that resource abundance increases the demand for commodities, thereby reallocating resources from the tradable to non-tradable sector and increasing the imports of tradables. This can reduce labor supply, leading to greater earnings in the local economy. The direct effect on relative demand of extraction firms and workers is higher prices of personal and business services. Other political-economic factors that may disrupt the growth of the non-resource sector are rent-seeking and corruption. \citet{Caselli:2009to, Caselli:2013ci} provide a good summary of these theoretical implications. See \citet{Papyrakis:2007im} for empirical evidence using US states' data.}
Another pivotal risk is pollution, which can increase the health costs in resource-abundant locations and decrease agricultural productivity \citep{Aragon:2015db, vonderGoltz:2019br}.
Here, I investigate the impact of coal mining on the regional economy and population health in industrializing Japan.
By linking the location information of mines with a set of registration- and census-based statistics from the 1930s, I estimate the impact of coal mines on the regional population, labor supply, and early-life health outcomes.
The results show that mines increase the local population, mainly via internal migration.
While the influx of workers initially disturbed the demographics of the local economy, these demographics reverted to trend when the migrants formed their families.
Importantly, this local population growth occurred with the local structural shifts from the primary to the production and service sectors.
The magnitudes depend on sex: female (male) workers' labor force participation rates decreased (did not change).
Nevertheless, adverse health effects on infants were observed in coal mining areas.
A higher risk of infant mortality was associated with the occupational hazards of female miners; however, regulatory revisions in 1933 reduced the risk of fetal and infant deaths to a certain extent.
Additional analysis using heavy metal (gold, silver, and copper; hereafter, GSC) mine samples suggests that these mines rather improved the health status of infants, suggesting the advantage of structural shifts.
This study contributes to the literature in four ways:
First, it expands our understanding of the relationship between mining and regional growth.
There are still debates on whether proximity to coal mines is necessary for regional economic growth during Europe's industrialization.
For example, some studies on economic history have argued for the importance of better access to coalfields in subsequent industrial development \citep{mathias2001, pollard1981}.
Further, a recent study by \citet{Fernihough2020} found that after the mid-eighteenth century, European cities closer to coalfields had a larger population than other cities.
By contrast, \citet{Mokyr1977} observed that the distribution of coal supplies did not contribute to industrialization.
My results support the former assertion by showing that mines lead to population growth and structural shifts in mining areas, and thus, contributed to Japan's industrialization.
Meanwhile, to the best of my knowledge, this study is the first to show the heterogeneity in the impact of mining by mineral type between coal and GSC mines.
Second, this study analyzes the mechanism behind local population growth due to resource extraction.
Although studies have analyzed the impact of resource abundance on local population growth \citep{Black:2005uh, Michaels:2010ks, Fernihough2020}, little is known about the mechanisms underlying this relationship.
I find suggestive evidence that local population growth in the mining area initially occurred due to internal migration, rather than increases in local marriage and fertility rates; however, the migrants then formed families over time.
Third, this study simultaneously investigates the effects of mines on the regional economy and health outcomes.\footnote{Development economics research has typically focused on either the regional economy or health effects of mining \citep{Aragon:2015db, BenshaulTolonen:2019kx}. Meanwhile, economic history studies have predominantly focused on the health costs of air pollution. See, for example, \citet{Heblich2017wp} and \citet{Beach2017kja} for the adverse health effects of air pollution due to coal use during historical developments in Britain. These authors found that extensive coal use during economic growth increased infant mortality rates (IMR) and decreased the average adult height. Another strand of the literature is the adverse impact of using lead pipes on mortality rates: \citet{Gronqvist2017wp} showed an association between early lead exposure and later-life socioeconomic outcomes in Sweden in the 1970s and 1980s. However, to the best of my knowledge, the potential health risks of mining during historical industrialization have not been evaluated.}
I find evidence that the occupational hazards experienced by female miners lead to mortality selection \textit{in utero}, which increases the risk of early-life deaths.\footnote{Another potential contribution is that I investigate the mechanism behind the observed adverse health effects of metal pollution exposure. While most studies did not provide empirical evidence on the mechanism behind the adverse effects of air pollution, I present a pathway for the observed damaging health effects by referring to the Trivers-Willard theory \citep{Trivers:1973fd}.
}
My findings are in line with \citet{Hanlon2019wp}, who found that coal-related pollution disturbed the growth of cities in the late nineteenth century in the UK, and with the negative aspects of rapid economic development in modern Japan \citep[c.f.,][]{Tang:2017kr}.
In contrast, my results concerning the GSC mines provide supportive evidence to \citet{Drixler2016}, who argued for the exiguous contribution of industrial pollution from copper mines to the high stillbirth rates in Japan in the late 19th century.
This finding is also consistent with \citet{BenshaulTolonen:2019kx}, who shows that in northwestern Tanzania, local industrial development by gold mining can reduce the risk of infant mortality.
Fourth, this study utilizes data on demographic, labor, and health outcomes constructed using census-based statistics.\footnote{Most studies have focused on large-scale (particularly gold) mines and utilized the location information combined with survey datasets \citep{Aragon:2013ib, Aragon:2015db, Kotsadam:2016du}.}
These data allows me to eliminate the risks caused by selection bias in the statistical inference.
Using comprehensive data, this study provides the first evidence of a past-developing country in Asia, given that research has mainly focused on European countries, the US, Latin America, and African countries.\footnote{See for example, \citet{Domenech:2008id, Fernihough2020} for European countries, \citet{Black:2005uh, Michaels:2010ks} for the US, \citet{Sachs:1999ul, Aragon:2013ib} for Latin America, and \citet{Aragon:2015db, BenshaulTolonen:2019kx} for African countries.
}
The remainder of study paper is organized as follows.
Section~\ref{sec:sec2} provides a brief overview of the historical context.
Section~\ref{sec:sec3} summarizes the data.
In Section~\ref{sec:sec4}, I describe the proposed identification strategy.
Section~\ref{sec:sec5} presents the main results, while Section~\ref{sec:sec6} summarizes the additional analyses.
Finally, Section~\ref{sec:sec7} presents the conclusion of this study.
\section{Background} \label{sec:sec2}
The Mining Code of 1892 allowed Japanese private companies to have mining concessions, which accelerated the installation of modern technologies in mines and the development of Japan's mining sector.\footnote{For instance, winding engines were installed in the coal and copper mines in the 1890s, and the flame-smelting process was adopted in gold and silver mines in the 1900s \citep[pp.220--221]{Ishii1991}.}
Before the First World War (WWI), the coal industry had developed as an export labor-intensive industry.
However, WWI increased the average wage rates of miners, which reduced the comparative advantage of the Japanese coal industry.
Figure~\ref{fig:ts_coal} illustrates the development of the coal industry and shows the negative shock in the early 1920s.
This motivated coal mining firms to reduce the number of miners as well as adopt new extraction machines to improve firm productivity.\footnote{See \citet[pp.~387--393]{sumiya1968} for details on the adoption process of these machines.}
Mining firms also reallocate resources to relatively productive mines to improve their average productivity.
Consequently, average labor productivity in the coal mining sector increased from the 1920s, particularly in the 1930s \citep{Okazaki:2021}.
Figure~\ref{fig:ts_coal} shows that total coal production increased during the interwar period.\footnote{Generally, during the interwar period, the mine and manufacturing industries experienced steady capital investments, including investments in production facilities and machines, and public investments \citep[pp.138--143]{Nakamura1971}. Consequently, the relative contribution of manufacturing and mining, which provided energy and resources to manufacturing, was the largest at 42-51\% among all sectors during this period \citep[p.90]{Minami1994}.}
In the early 1930s, coal accounted for approximately 55\% of total mineral output in Japan \citep[p.44]{mb1934}.
Thus, coal extraction was an influential economic activity within mining.
The higher wage rates of miners may have motivated peasants to move into the extraction industry, shifting the local industrial structure from the primary to the production sector.
Besides original inhabitants, mining workers also included a certain proportion of migrants.
While miners included single male workers, a large proportion of them were married couples because miners usually worked in teams of two to three miners \citep[pp.~300--308]{sumiya1968}.
This may have influenced the structure of both the marriage market and fertility in the local economy.
Importantly, married male miners worked with their wives in mines.
The male skilled miner, called \textit{saki yama}, extracted coal at the face; meanwhile, the female miner, called \textit{ato yama}, brought coal to the coal wagon at the gangway through steep pits using bamboo baskets called \textit{sura} \citep[e.g.,][p.~319]{sumiya1968}.\footnote{A few images of miners are shown in Online Appendix~\ref{sec:seca0}.}
Because coal is a bulky resource, the heavy work burden for pregnant women could have increased the risk of miscarriages.
This may have also increased infant mortality risk because the overall decline in fetal health endowments can also decrease the health status of surviving infants (Section~\ref{sec:sec53}).
Moreover, as expected, miners worked in pits with very dirty air containing particulates.
This eventually damaged their lungs and could increase the risk of respiratory diseases, such as tuberculosis \citep[pp.~410--411]{sumiya1968}.
Then, in September 1933, the Miner's Labor Assistance Regulations were revised to prohibit females and children from working in the pits.
This decreased the number of female miners, and may have improved their working conditions.
However, several female miners still engage in coal dressing and extraction of residual coal \citep{Tanaka:1977}.
Another potential risk may have been pollution.
A growing body of medical literature has revealed that several pollutants discharged from mines can increase the risk of fetal death.
For example, particulates and radionuclides from coal increase the risk of stillbirths and premature deaths owing to respiratory diseases in mothers and infants \citep{Landrigan:2017fq, Lin:2013dr}. Meanwhile, heavy metal mining releases toxic metals, such as mercury (Hg), cyanide (CN), and cadmium (Cd), which are associated with the incidence of miscarriages and stillbirths \citep{Amadi:2017fl}.
However, industrial pollution was not recognized as a common social problem among people in pre-war Japan.
Other than a few exceptions, like the Ashio Copper Mine Incident in the late 19th century \citep{Notehelfer:1975vg}, the potential health costs of mineral mining during industrialization in Japan are understudied.\footnote{Most research on mines in prewar Japan comprises business history studies that investigated the features of the management and industrial organization in coal mining companies \citep[e.g.,][220-233]{Ishii1991}. An exceptional study in the field of development economics is \citet{vonderGoltz:2019br}, who provided evidence that mining is associated with child stunting among today's developing nations.}
\section{Data}\label{sec:sec3}
This study created a unique dataset of demographic outcomes, labor supply, and early life health outcomes across mining areas in the Japanese archipelago.
Because mining is a localized economic activity, a smaller lattice dataset is preferred to identify the impact of mines better.
I collected and digitized a set of official census-based municipal-level statistics published in the 1930s.
\subsection{Mine Deposit}\label{sec:sec31}
I use official reports named \textit{Zenkoku k\=ojy\=o k\=ozan meibo} (lists of factories and mines) published by \citet[][]{ahc1932, ahc1937} (hereafter, called the AHC), which document coal mine locations measured in October 1931 and October 1936.
The AHC listed all mines with 50 miners and more, meaning that very small-scale mines with fewer than 50 mimers are out of this study's scope.
Thus, the main target of this study is the average impact of small- to large-scale coal mines and not the very small-scale collieries.
This does not discount the comprehensiveness of the AHC because it still includes information on small-scale mines, which have been neglected in the literature.
The red circles in Figure~\ref{fig:map_mines} indicate the coal mine locations from the AHC.
Most coal mines are located in three specific areas in the southernmost, central, and northeastern regions.
These are representative coalfields called Chikuh\=o, Jy\=oban, and Ishikari-Kushiro Tanden, respectively.
These agglomerations were stable over time, as shown in Figures~\ref{fig:mine_1931} and~\ref{fig:mine_1936}.
In Section~\ref{sec:sec62}, I analyze the impacts of the GSC mines to provide comparable estimates to discuss the heterogeneity in the impacts of mines by mineral type.
Unlike coal mines, GSC mines were scattered across the archipelago (Figure~\ref{fig:map_mines}).
The increase in the number of GSC mines between the two measured years is much clearer than that of coal mines (Figures~\ref{fig:mine_1931} and~\ref{fig:mine_1936}).
Next, I define the treated and control municipalities.
The treatment group includes municipalities located within 0--5 km of the centroid of a municipality with mines.\footnote{I used an official shapefile provided by the Ministry of Land, Infrastructure, Transport and Tourism for geocoding. See Online Appendix~\ref{sec:secb1} for the details.}
A 5 km threshold is considered sufficient because the mean value of the distance to the nearest neighborhood municipality is approximately 4 km.
For the control group, I set the threshold as 5--30 km from the centroid.\footnote{My definition of both groups is conservative compared with that in the literature, which uses 10--20 km and 100--200 km as the thresholds of the treatment and control groups, respectively \citep{Wilson:2012ic, Aragon:2015db, Kotsadam:2016du, BenshaulTolonen:2019kx}. This is mainly because Japan has a relatively smaller and thinner archipelago than countries studied in the literature, such as South Africa (Online Appendix Figure~\ref{fig:map_coal_mines_full}). In my flexibility analysis, I provide suggestive evidence on the heterogeneous effects of mines with respect to distance by changing the baseline thresholds of the treatment group from 0--15 km (see Section~\ref{sec:sec61}).}
Figure~\ref{fig:map_coal_mines} provides an example map for a representative coalfield called Chikuh\=o-tanden in Kyush\=u Island, Japan's southernmost region (indicated in Figure~\ref{fig:mine_1931}).\footnote{I focus on a specific coalfield for the purpose of visibility. Online Appendix Figure~\ref{fig:map_coal_mines_full} shows the spatial distribution of both treatment and control groups across the Japanese archipelago.}
As shown in Figure~\ref{fig:map_mines}, the distribution of mines was stable over time; only a few mines were opened or closed between 1931 and 1936.
This feature makes it difficult to use within-variation in the spatial distribution of coal mines for the identification strategy (Section~\ref{sec:sec63}).\footnote{I found that it was practically difficult to implement a difference-in-differences (DID) estimator under two-group and two-period setting because of the lack of information (i.e., the number of municipalities that experienced the opening/closing of the coal mines).}
Finally, I explain the creation of the analytical samples.
From $11,151$ total municipalities, I first retain the municipalities within a $30$ km radius from the centroid of the municipalities with mines.
I then exclude the intersection of the different types of mines, such as municipalities within 5 km from a GSC mine.
Panel A of Table~\ref{tab:sum} presents the summary statistics of the treatment variables.
The number of municipalities in the analytical sample based on the 1931 and 1936 AHC were $1,140$ and $1,440$, respectively.
Meanwhile, the proportion of treated municipalities were approximately 15\% and 13\%, respectively, for each sample.\footnote{The number of municipalities with coal mines in each sample is 93 and 115, respectively. The slight reduction in the share of the treatment group is due to the increase in the number of enclaves, which I explained above. A mine opening in the enclave area increases the number of controlled (i.e., surrounding) municipalities, which leads to a decrease in the relative share of the treatment group.}
\subsection{Geological Stratum}\label{sec:sec32}
The spatial distribution of the geological stratum created in the Cenozoic era, which includes some carboniferous ages, is used as an instrumental variable (IV) for the location of coal mines (Section~\ref{sec:sec42}).
Data on the geological strata are obtained from the official database of the Ministry of Land, Infrastructure, Transport and Tourism, which includes roughly nine thousand stratum-points.
I matched each municipality in my analytical sample with the nearest stratum point to identify the municipalities with the relevant stratum.
Online Appendix~\ref{sec:secb2} provides finer details of the IV's definition as well as an example of a geological columnar section of a representative coal mine to show the relevance of the IV.
Panel A of Table~\ref{tab:sum} presents the summary statistics of the indicator variables for the stratum.
\subsection{Outcomes}\label{sec:sec33}
\subsubsection*{\textit{Demographics}}\label{sec:sec331}
The main dependent variable, population, is obtained by digitizing the municipal-level statistics documented in the 1930 and 1935 Population Censuses \citep{census1930v1, census1935v1}.
As the census documents the total number of people in all municipalities in the census years,
It helps us avoid sample selection issues.
To investigate the mechanism behind the potential local population growth due to mining, I also digitized the Municipal Vital Statistics of 1930 and 1935 \citep{mvs1930, mvs1935}, which document the total number of marriages and live births in all municipalities in the survey years.
I then consider four dependent variables: crude marriage rate, crude birth rate, mean household size, and sex ratio.\footnote{The crude marriage and birth rates are defined as the number of marriages and live births per 1000 people, respectively. The mean household size is the number of people in households divided by the number of households. The sex ratio is the number of males divided by the number of females.}
These demographic variables represent the characteristics of marriage markets and family formation in the local economy.
Using these variables, I assess whether the local population growth due to mining activity depended on immigrants or changes in the family planning of local people.\footnote{For instance, if population growth is caused by increased births, both marriage and fertility rates should be high in areas around a mine. Conversely, if it is caused by immigrant laborers, the household size should be small, whereas the sex ratio must be high as migrant workers include single males (Section~\ref{sec:sec2}).}
Panel B of Table~\ref{tab:sum} presents the summary statistics of the demographic outcomes by treatment status and census year.
The mean differences are statistically significant for most variables, implying that systematic demographic changes may have occurred around the mining extraction area.
\subsubsection*{\textit{Labor Supply}}\label{sec:sec332}
I consider the labor force participation rate (the number of workers per 100 people) to investigate the impacts on local labor supplies.
To analyze structural shifts, the employment share of workers in the mining, agricultural, manufacturing, commercial, and domestic sectors are considered; these are calculated as the number of workers in each sector per 100 workers.
I use the labor force participation rates of both sexes to better understand the potential gender bias among the changes in local labor markets.
These labor statistics were obtained from the prefectural part of the 1930 Population Census.\footnote{The prefectural part comprises 47 reports (one for each prefecture), which I collected and digitized. For simplicity, I cite those reports as one document, \citet{census1930pref}. The full population census is conducted every ten years in Japan; the 1930 census surveyed all labor statistics. Thus, similar statistics were unavailable in the 1935 Population Census. The 1940 Population Census was also disorganized because of the wartime regime.}
Panel C of Table~\ref{tab:sum} presents the summary statistics of these variables by treatment status and census year.\footnote{The employment share in the mining sector has a set of censored observations because mining is a localized economic activity (Panel C of Table~\ref{tab:sum}). However, the estimates from the Tobit estimator \citep{Tobin:1958uq} confirm that the main results are not affected by censoring (not reported).}
Similar to the demographic outcomes, the mean differences are statistically significant for most variables, which implies the potential impact of mining extraction.
\subsubsection*{\textit{Early-life Health}}\label{sec:sec333}
Next, I examine the IMR to assess the impacts on early-life health.
To analyze the potential mechanisms behind the shifts in infant mortality, I also use the fetal death rate (FDR).\footnote{FDR is defined as the number of fetal deaths per $1,000$ births, while the IMR is the number of infant deaths per $1,000$ live births. The censored observations in the FDRs are less than 11\% in both census years and do not significantly change the results. The estimates from the Tobit estimator \citep{Tobin:1958uq} confirm this finding (not reported). The number of censored observations in IMRs is practically negligible (3 and 26 observations in 1933 and 1938, respectively).}
To obtain both variables, I digitized the 1933 and 1938 reports documenting municipal-level statistics on births and infant deaths published by the Aiikukai and Social Welfare Bureau.
I also consider the mortality rate of children aged 1--4 to assess the impacts of air pollution.\footnote{The child mortality rate is defined as the number of deaths of children aged 1--4 in 1933 (or 1938) per $1,000$ children aged 1--5 in 1930. Note that the number of children aged 1--5 from 1930 is used as it is only documented in the 1930 population census. Given that child deaths, especially at age 5, were rarer than infant deaths at that time, the mismatch in the age bins between numerator and denominator should not be a critical issue. The difference in the survey points does add noise to the estimation of the standard error. However, the coefficient estimate is close enough to zero, and thus, should not be a practical issue in this setting (Panel A-2 of Table~\ref{tab:r_health}).}
These statistics are based on official vital statistics.
An important advantage of using these vital statistics is that since Japan has a comprehensive registration system, the data cover almost all fetal and infant death incidences in the measured years.\footnote{The potential imprecision of birth data in prewar Japan was improved by a great degree by the 1920s \citep{Drixler2016}.}
Panel D of Table~\ref{tab:sum} shows the summary statistics of the early life health variables.
Again, the mean differences were statistically significant.
\subsection{Additional Control Variables}\label{sec:sec34}
\subsubsection*{\textit{Accessibility to Railway Stations}}\label{sec:sec341}
As coal is a bulky mineral, railways were used to transport the extracted coal.
Firms may have preferred setting extraction points closer to railway stations to reduce transportation costs \citep[pp.~440--441]{sumiya1968}.
Further, the local population size could be larger if one moves closer to stations.
Therefore, it is preferable to control for the distance between each municipality and the nearest-neighboring station to deal with this potential endogeneity issue.
I used the location information of all stations obtained from the official dataset provided by the Ministry of Land, Infrastructure, Transport and Tourism to compute the nearest-neighbor distance to stations in 1931 and 1936.
Online Appendix~\ref{sec:secb4} summarizes the details of the data, and Online Appendix Figure~\ref{fig:map_station} shows the locations of these stations across the Japanese archipelago.
Panel A of Table~\ref{tab:sum} shows the summary statistics.
\subsubsection*{\textit{Accessibility to Sea Ports}}\label{sec:sec342}
Although railways were primarily used to transport coal because coal mines were usually located inland (Online Appendix~\ref{fig:map_coal_mines_full}), marine transportation was also used for secondary logistics.
Hence, similar to railway transportation, I include the distance to the nearest neighboring seaport as a variable to control for accessibility to marine transportation (Online Appendix Figure~\ref{fig:map_station} shows seaport locations).
Online Appendix~\ref{sec:secb4} summarizes the details of the data and Panel A of Table~\ref{tab:sum} shows the summary statistics.
\section{Identification Strategy}\label{sec:sec4}
\subsection{Reduced Form Approach}\label{sec:sec41}
I leverage the random nature of mineral deposits to identify the effects of mines.
The linear regression model is as follows:
\begin{eqnarray}\label{eq1}
\footnotesize{
\begin{split}
y_{i} = \alpha + \beta \text{\textit{MineDeposit}}_{i} + \mathbf{x}_{i}' \bm{\gamma} + e_{i},
\end{split}
}
\end{eqnarray}
where $i$ indexes municipalities, $y$ is the outcome variable, $\text{\textit{MineDeposit}}$ is an indicator variable that equals one for municipalities within 5 km from a mine, $\mathbf{x}$ is a vector of control variables, and $e$ is a random error term.
Because the placement of a mine is determined by a geological anomaly, $\text{\textit{MineDeposit}}$ is random in nature \citep[p.1568]{BenshaulTolonen:2019kx}.
However, the rest of the variation may be correlated with the local variation in infrastructure.
For instance, if a mineral deposit is found at a point between a village and a city, the mining firm may choose the city as its main mining point because the city is likely to have better infrastructure than the village.
Then, $\text{\textit{MineDeposit}}$ can be positively correlated with the error term, leading to a positive (negative) omitted variable bias in the estimate of $\beta$ if the placement of the city is positively (negatively) correlated with the outcome variable.\footnote{
For example, consider a simplified projection of $y$ on $\textit{MineDeposit}$ and $\textit{City}$, $y = \alpha + \beta \text{\textit{MineDeposit}} + \gamma \textit{City} + e$.
When the municipality type ($\text{\textit{City}}$) is unobserved, the linear projection coefficient can be written as $\beta^{*}=\beta + \Xi \gamma$, where $\Xi = (E[\text{\textit{MineDeposit}}^{2}]^{-1})(E[\text{\textit{MineDeposit}}\text{\textit{City}})$.
As explained, $\text{\textit{MineDeposit}}$ and $\text{\textit{City}}$ may be positively correlated such that $\Xi > 0$.
Thus, when $y$ is the population, conditional on mine deposits, it is reasonable to suppose that the town has a greater number of people than villages ($\gamma > 0$). This result implies that $\beta^{*}=\beta + \Xi \gamma > \beta$.\label{foot:obv}}
To deal with this systematic bias in the ordinary least squares, I included two indicator variables that equal one for cities and towns (i.e., zero for villages), respectively.
As explained in Section~\ref{sec:sec34}, I also include the distances to the nearest neighboring railway station and seaport to control for accessibility to railway and marine transportation, respectively.
The differences in the estimates from both specifications, including and excluding the control variables, indicate the extent to which this omitted variable mechanism influences the results.
Thus, I can partially assess the randomness of the main exposure variable ($\text{\textit{MineDeposit}}$).
As I show later, the estimates from the simple regressions are materially similar to those from the specifications, including the control variables, thereby supporting the randomness of the exposure variable.
To be conservative, I prefer to use the specification including the control variables (equation~\ref{eq1}).
Regarding statistical inference, I use the heteroskedasticity-consistent covariance matrix (HC2) estimator proposed by \citet{Horn:1975tu} as a baseline estimator.\footnote{The results are materially similar if I use the Eicker-White type covariance matrix estimator suggested by \citet{Hinkley:1977}. To be conservative, I use the HC2 estimator because my treatment variable is relatively sparse, as explained (Section~\ref{sec:sec31}).}
For the sensitivity check, I also use the standard errors clustered at the county level based on the cluster-robust covariance matrix estimator \citep{Hansen:2007dh} to determine the influence of the potential influences of the local-scale spatial correlations.
Online Appendix~\ref{sec:app_r} summarizes the results.
Both standard errors are materially similar in most cases.
Therefore, the potential spatial correlations were negligible in this empirical setting.
\subsection{Structural Form Approach}\label{sec:sec42}
A potential threat in the identification may be measurement errors in time-dimensional assignments.
As explained in Section~\ref{sec:sec31}, mine data were surveyed in October 1931 (1936), whereas the Population Census was conducted in October 1930 (1935).
Similarly, data on fetal death and infant mortality were obtained from the 1933 (1938) Vital Statistics.
The lags in the matching allow me to consider the exposure durations.
For instance, fetus miscarriages in 1933 (1938) were \textit{in utero} conceived in 1932 (1937), whereas their mothers should have been exposed to any shocks at the time of conception in 1932 (1937).
The same argument can be applied to infant mortality: infants who passed away in 1933 (1938) were born at the beginning of 1932 (1937) at the earliest.
Therefore, they should have been \textit{in utero} in 1931 (1936).
Consequently, the mining data that list the mines in October 1931 and 1936 may be reasonably matched with the health data of 1933 and 1938, respectively.
However, one must be careful as there is still a one-year lag in matching with both the census and vital statistics datasets.
This may lead to attenuation bias due to missing assignments in the time dimension, because the number of coal mines should be increased during that year.\footnote{Note that this sort of mis-assignments (i.e., measurement error) never overstate the estimates but causes attenuation. For instance, if a few treated (untreated) municipalities were regarded as the untreated (treated) municipalities, the impacts of mines shall be discounted.}
To deal with such potential measurement error issues, I consider the IV estimator using the exogenous variation in the geological stratum.
In this IV approach, Equation ~\ref{eq1} is regarded as a structural form equation because the least-squares estimator ($\hat{\beta}$) is assumed to be attenuated by the measurement error in the exposure variable.\footnote{To understand this mechanism, consider a simplified projection of $y_{i}$ on $\textit{MineDeposit}_{i}$ as $y_{i} = \theta \textit{MineDeposit}_{i} + u_{i}$. When $\textit{MineDeposit}_{i}$ has a classical measurement error, as $\textit{MineDeposit}^{*}_{i}=\textit{MineDeposit}_{i}+u_{i}$, the observable model becomes $y_{i}=\theta^{*} \textit{MineDeposit}^{*}_{i}+\nu$. Consequently, I use $\text{cov}(\textit{MineDeposit}^{*}_{i}, \nu_{i})=E[\textit{MineDeposit}^{*}_{i}\nu] = -\theta \text{var}(u_{i}) \neq 0$ for $\theta^{*} = \theta + \text{cov}(\textit{MineDeposit}^{*}_{i}, \nu_{i})/\text{var}(\textit{MineDeposit}^{*}_{i})$.}
The reduced-form equation for $\text{\textit{MineDeposit}}$ is designed as follows:
\begin{eqnarray}\label{eq2}
\footnotesize{
\begin{split}
\textit{MineDeposit}_{i} = \kappa + \tau \textit{Stratum}_{i} + \mathbf{x}'_{i}\bm{\mu} + \epsilon_{i},
\end{split}
}
\end{eqnarray}
where $\textit{Stratum}$ is a binary IV that equals one for municipalities with sedimentary rock created during the Cenozoic era, and $\epsilon$ is a random error term.
The IV is plausibly excluded from the structural equation because the location of mines is essentially dominated by the distribution of geological stratum, which is exogenously given in nature and unobservable by people in the prewar period.
In addition, the relevance condition holds because the location of coalfields is determined by the distribution of the stratum created in the Cenozoic era, which includes the Carboniferous period when the strata containing coal were created \citep{Fernihough2020}.
Online Appendix~\ref{sec:secb2} summarizes the geological strata variable in finer detail and shows that the location of coal mines is determined by specific strata created in the Cenozoic era.
\section{Results}\label{sec:sec5}
\subsection{Population Growth}\label{sec:sec51}
Panel A-1 of Table~\ref{tab:r_pop} presents the main results for the population in 1930.
Columns (1), (2), and (3) show the results from the specifications excluding the control variables, including the city and town fixed effects, and including both the fixed effects and control variables, respectively.
Columns (4), (5), and (6) show the results for the IV estimations for the same column layouts.\footnote{The first-stage $F$-statistics reported in the same panel exceed the rule-of-thumb threshold value proposed by \citet{Staiger1997}, say 10, in all regressions; this shows that the necessary rank condition is satisfied. Online Appendix~\ref{sec:app_r} confirms that the first-stage $F$-statistics never break this rule-of-thumb threshold, even if I use a conservative variance estimator such as the cluster-robust variance covariance matrix estimator.}
First, including city and town fixed effects shrinks the estimates (column (2)).
Similarly, controlling for transportation accessibility slightly reduces the estimate (column (3)).
This indicates that the location of the coal mines may be weakly positively correlated with the development of local infrastructure and related agglomeration.
Second, the estimates based on the IV estimator are greater than those under the reduced-form assumption.
Even after conditioning on the control variables, the estimate in column (6) is substantially greater than that in column (3); $0.445$ versus $0.813$.
This implies that measurement errors may systematically attenuate the estimates.
Considering these, the baseline result from the preferred specification of column (6) of Panel A-1 indicates that coal mines increased the local population by $125$\%.\footnote{Statistically, the geometric mean of population in the treatment group with mines was approximately 125\% ($\text{exp}(0.813) = \text{exp}(\hat{\alpha}+0.813)/\text{exp}(\hat{\alpha})\approx 2.25$) greater than that of the control group. However, I use simple interpretations throughout this study to avoid redundancy.}
Panel B-1 of Table~\ref{tab:r_pop} shows the results for the population in 1935.
These results are materially similar to those in Panel A-1.
The estimate in column (6) of Panel B-1 indicates that coal mines increased the local population by $152$\%.
The magnitude of the local population changes are economically meaningful.
Thus, my results imply that municipalities with coal mines in the 1930s experienced local population growth.
A natural question may be whether it is a consequence of natural increase or labor migration.
If the population grows naturally from family planning, municipalities with mines should have had higher marriage and fertility rates, larger household sizes, and a stable sex ratio.
Conversely, if labor migrations were the main cause, municipalities with mines should have had a disproportionate (higher) sex ratio and smaller household sizes because new miners were usually single males and couples.
Panel A-2 of Table~\ref{tab:r_pop} presents the results for several demographic variables to assess this mechanism.
Columns (1)--(4) show the results for the crude marriage rate, crude birth rate, average household size, and sex ratio, respectively.
All regressions include the city and town fixed effects, and are estimated using the IV estimator.
Overall, the estimated coefficients suggest that municipalities with coal mines experienced a decline in the marriage rate, fertility rate, and household size.
In contrast, mines increased the sex ratio by $0.115$ points.
This is economically meaningful, given that it accounts for roughly one standard deviation in the sex ratio (Panel B in Table~\ref{tab:sum}).
Thus, the mechanism behind local population growth may be explained by labor migration rather than by a natural increase in the population.\footnote{This tendency is generally consistent with the fact that in prewar Japan, urban areas had a relatively higher sex ratio than localities due to the internal migration of male workers \citep[p.243--247]{Ito1990}.}
Panel B-2 of Table~\ref{tab:r_pop} presents the results for the demographic outcomes in 1935 in the same column layout as Panel A-2.
Although the signs of the estimates are unchanged from those reported in Panel A-2, the magnitudes are much smaller than those for the 1930 sample.
This result is consistent with the findings of the descriptive analysis in section ~\ref{sec:sec31}.
Although several coal mines were newly opened by 1935, most coal mines already existed in 1930.
Thus, single males (couples) entering the coal mine area around 1930 could have had their spouses (or even children) by 1935.
In fact, the estimates for marriage rate, fertility rate, and average household size are estimated to be much smaller in the absolute sense (columns (1) and (2)).
Specifically, the fertility rate estimate is close to zero and no longer statistically significant.
Correspondingly, the estimate for sex ratio decreases, meaning that the sex ratio tended to revert to its natural trend.
This may reflect the attenuation in the overall sex ratio due to the regular sex ratio of infants and children born in coal mining areas after 1930.
In summary, the coal mine leads local population growth in the following two ways.
First, internal migration to coal mining areas increases the local population, which leads to an unbalanced sex ratio and lower marriage, fertility, and household size.
This may correspond to the results for the 1930 sample (Panel A-1).
Second, as households in the coal mine areas formed their families, the local population grew naturally, which reverted the fertility rate and overall sex ratio to natural trends.
Although there may be migrants to these mining areas, the result for the 1935 sample in Panel B-1 may indeed reflect a part of the local population growth at this stage.
\subsection{Labor Supply}\label{sec:sec52}
Next, I start the analysis of labor supply by observing the results for male labor first.
Panel A-1 of Table~\ref{tab:r_labor} presents the results for the labor force participation rate of male workers.
The columns are listed in the same layout as in Panel A-1 of Table~\ref{tab:r_pop}.
The estimates in columns (1)--(3) are negative but close to zero, and statistically insignificant.
The estimates from the IV approach have much greater magnitudes in columns (4)--(6).
However, the estimated coefficients are still statistically insignificant in these columns, implying that coal mines did not influence the labor supply of male workers in the local economy.
Panel A-2 of Table~\ref{tab:r_labor} shows the results for employment share of male workers.
Column (1) indicates that coal mines increased the mining sector's employment share by approximately 35\%, which leads to a similar decline in the agricultural sector's employment share (column (2)).
This suggests that the structural shift in male workers' employment occurred mainly in the agricultural sector.
The estimate for the manufacturing sector is statistically insignificant but positive.
This may imply a moderate spillover effect of mines on the manufacturing industry in the mining areas (Column (3)).\footnote{\citet{Black:2005uh} found that the 1970s coal boom in the US increased employment and earnings with modest spillovers into the non-mining sectors.}
The estimates for the commercial and domestic sectors are moderately negative and close to zero (columns (4) and (5), respectively).
This is consistent with the fact that male workers are less likely to work in these service sectors (Panel C of Table~\ref{tab:sum}).
The results for female labor suggest different responses from those of male workers.
Panel B-1 of Table~\ref{tab:r_labor} provides evidence that the coal mines decreased the female labor force participations.
Column (1) shows a statistically significantly negative estimate, which is unchanged after considering the fixed effects in column (2) and control variables in column (3).
The results from the IV estimator provide clearer effects.
The estimate from the preferred specification suggests that coal mines decreased the female labor force participation rate by approximately 18\% (Column (6)).\footnote{The clear gender difference in the labor force participation seems to be consistent with the wealth effect (``spending effect'' in the Dutch-Disease literature) of mines: it may result in higher wage rates and lower overall non-resource GDP \citep{Caselli:2001tg}. Although it is difficult to clearly analyze the impact of mines on wages, my results are in line with the findings on structural shifts presented in this subsection.}
Panel B-2 of Table~\ref{tab:r_labor} shows the results for employment share of female workers.
Column (1) indicates that coal mines increase the mining sector's employment share by $25$\% and decrease the agricultural sector's share by the same degree (column (2)).
Interestingly, the estimated magnitude is smaller than that for male workers (Column (1) of Panel A-2).
This may be because single females were less likely to work in mines than single males (Section~\ref{sec:sec2}).
The estimates for the commercial and domestic sectors are moderately positive (columns (4) and (5), respectively), whereas the manufacturing sector's employment share decreases in the mining area (column (3)).
This is consistent with the increased relative demand for personal services from mining workers in mining areas \citep{Caselli:2001tg}.
Overall, my results suggest structural shifts from primary to other sectors.
In these shifts, the division of roles based on gender was observed: males were engaged in manual work, whereas females were more likely to work in the service industries or even be housewives instead of working in any sector.
\citet{Kotsadam:2016du} found that opening of mines in African countries decreased the workers in the agricultural sector but stimulated the non-agricultural sectors.
This increased the number of female workers in the service sector and male manual laborers in the mining sector.\footnote{\citet{Bjornland:2016gz} also provides evidence that a booming resource sector has productivity spillovers on the non-resource sectors. A contradictory view by \citet{Aragon:2015db} is that large-scale gold mining in Ghana decreased agricultural productivity. This was derived using household survey samples collected near mines. Although it is difficult to test this relationship in my setting, my results provide evidence that the employment share in the agricultural sector substantially decreased because of mines in interwar Japan.}
My findings demonstrate a similar experience as a historical case.
However, more importantly, females did work in mines in prewar Japan.
They worked with males, usually their husbands, in the pit under harsh environments (Section\ref{sec:sec2}).
This may have adversely affected not only their own health but also their children's health.
I consider this possibility in the following subsection.
\subsection{Early-life Health}\label{sec:sec53}
Panel A-1 of Table~\ref{tab:r_health} presents the results for the IMRs in 1933.
The estimates are positive and statistically significant, indicating that coal mines increased the infant mortality risk.
As expected, the estimates based on the IV estimator listed in columns (4)--(6) are substantially greater than those in columns (1)--(3).
This is consistent with the rough time-dimensional assignments in health outcomes (Section\ref{sec:sec42}).
By contrast, the inclusion of the city and town fixed effects as well as the transportation accessibility variables had little influence on the estimates.
This suggests that unobservables depending on the type of municipality are likely to be orthogonal to mine location.
This result is particularly interesting given that railways could have increased local exposure to infectious diseases \citep{Tang:2017kr}.
Thus, my result implies that the greater risk of infant mortality around coal mining areas may be explained by different pathways, such as occupational hazards and/or pollution.
Panel A-2 of Table~\ref{tab:r_health} assesses the mechanism behind the higher infant mortality risk in coal mining areas.
First, I test the biological sorting mechanism \textit{in utero} suggested by the Trivers-Willard hypothesis (TWH) \citep{Trivers:1973fd}.
If greater infant mortality risk was associated with a reduction of the fetal health endowments by coal mining, the FDRs should also be increased in the coal mining area; this leftward shift of the initial health distribution is called the ``scarring mechanism'' (see Online Appendix~\ref{sec:app_tf} for a finer theoretical illustration on the TWH).
In column (1) of Panel A-2, I run the IV regression by replacing the IMR with FDR.
The estimate is positive and weakly statistically significant, providing support for the scarring mechanism before birth.
Next, I test whether this scarring mechanism was led by the occupational hazard or air pollution due to coal consumption in the mine.
In column (2) of Panel A-2, I limited my sample to 93 municipalities with coal mines to analyze the correlation between the FDR and the number of female miners.
If heavy manual work during pregnancy affects fetal health, FDRs should be positively correlated with the number of female miners.
To control for the scale effect and provide a placebo test, I included the number of male workers at the same time in the same specification.
As shown, the estimated coefficient for female miners is significantly positive, whereas that for male miners is rather negative.
This result confirms the evidence that occupational hazards for females (not males) alone increased the risk of death before birth.
The estimate indicates that a one standard deviation increase in female miners ($=269.2$ miners) increases the number of fetal deaths by approximately $10$ ($0.036 \times 269$) per 1000 births.
This is economically meaningful because the mean difference in the FDR between the treatment and control groups was $9.4$ fetal deaths per 1000 births (Panel D of Table~\ref{tab:sum}).
Column (3) of Panel A-2 regresses the IMR on both the miner variables.
It further provides suggestive evidence that this reduction in fetal health endowment is associated with higher infant mortality risks.
Therefore, the results from columns (2) and (3) suggest that occupational hazards also increase infant mortality risk through the scarring mechanism \textit{in utero}.
If air pollution increased early-life mortality, then the estimated coefficients on the male miner should be positive in columns (2) and (3) because larger mines should emit more coal smoke.
Column (4) of Panel A-2 tests the potential influence of air pollution more directly using the mortality rate of children aged 1--4.
If air pollution mattered, the child mortality rate should be greater in the coal mining area than in surrounding areas.
However, the estimate is close to zero and is statistically insignificant.
Column (5) uses data on 93 municipalities with coal mines, and provides further evidence on this: the estimated coefficients on the number of female and male miners are very close to zero, indicating that child mortality did not depend on the scale of emissions.
The results for child mortality in 1938 are similar to those in 1933 (columns (4) and(5), respectively, in Panel B-2 of Table~\ref{tab:r_health}).
Thus, air pollution does not seem to be a channel which can explain the greater IMR in the mining area.
Panel B of Table~\ref{tab:r_health} provides the results for the 1938 sample in the same panel and column layout.
Panel B-1 shows positive, but much smaller estimates relative to those in Panel A-1.
This finding suggests that the impact of coal mining on IMRs decreased from 1933 to 1938.
Column (1) of Panel B-2 shows a positive, but much smaller and statistically insignificant estimate for FDR.
This results is in line with the results for the IMR in Panel B-1, and with the fact that the FDRs showed a secular decreasing trend at that time (Online Appendix Figure~\ref{fig:ts_imr_fdr}).
However, the results from the specifications using the variations among municipalities with coal mines in columns (2) and (3) show that the estimates were statistically significantly positive for both IMR and FDR.
This result suggests the existence of a scarring mechanism before birth.
However, both estimates in columns (2) and (3) are smaller than those listed in Panel A-2.
The estimate shows that a one standard deviation increase in female miners ($=179.3$ miners) increased the number of fetal deaths by approximately $4.5$ ($0.025 \times 179$) per 1000 births.
This magnitude was less than half of that in 1933 (i.e., $10$).
This result suggests that the mortality selection mechanism before births may have been attenuated over time.
As explained in Section~\ref{sec:sec2}, the revised Miner's Labor Assistance Regulations of September 1933 prohibited females from working in pits, which indeed decreased the number of female miners throughout the 1930s.
This may partially explain the decline in the estimated magnitude of these health outcomes between 1933 and 1938.\footnote{As explained in Section~\ref{sec:sec4}, the mortality rates measured in 1933 were mainly associated with the shocks in 1932. Thus, the estimates for the 1933 sample should capture the situation before the revision.}
However, as shown in Panel A of Table~\ref{tab:sum}, several female miners still worked in these mines.
This means that while female miners were less likely to engage in heavy manual work than before, they still worked in mines with their husbands \citep
[p.~80]{Tanaka:1977}.
The moderate adverse effects observed in 1938 may be partly associated with this change in working conditions of female miners.
Another explanation may be socioeconomic trends.
This may be associated with an improving trend in health-related risk-coping strategies, such as increments in the number of medical doctors and installation of modern water supply (Online Appendix Figure~\ref{fig:ts_imr_fdr}).
\section{Additional Analysis}\label{sec:sec6}
This section assesses the heterogeneous effects of mines by distance (Section~\ref{sec:sec61}), compares the results for heavy metal mines (Section~\ref{sec:sec62}), and discusses the feasibility of a potential alternative estimation strategy utilizing within variations of each unit rather than the cross-sectional variations used before (Section~\ref{sec:sec63}).
\subsection{Alternative Thresholds}\label{sec:sec61}
First, I provide the results from the specifications using alternative definitions of the key exposure variable (\textit{MineDeposit} in equation~\ref{eq1}).
Figure~\ref{fig:hetero_distance} shows the estimates for areas $0$, $5$, $10$, and $15$ km from the mines.
Figures~\ref{fig:hetero_lnpop_coal_distance}, \ref{fig:hetero_lpr_distance}, and \ref{fig:hetero_imr_distance} present the results for local population, labor force participation rate, and IMR, respectively.
Generally, the estimated coefficients tend to attenuate as the distance from mines increases.
While the estimates are relatively larger in the absolute sense within $5$ km, those for $10$ and $15$ km show much smaller and similar values.
This result seems to provide a basis for defining the exposure variable that uses $5$ km as the threshold (Section~\ref{sec:sec31}).
Only Figure~\ref{fig:hetero_imr_distance} shows a relatively mild reduction: the effects on infant mortality are relatively persistent, regardless of the distance.
This may be because the IMR is more likely to be sensitive to environmental factors than to the local population and labor supply.
For instance, air pollution could have influenced the health of the local population in wider areas.
Related respiratory infectious diseases can also have spatially persistent effects.
\subsection{Alternative Minerals: Heavy Metal Mines}\label{sec:sec62}
I also assess the validity of my main results on coal mines by comparing the results for heavy metal mines.
Specifically, I focus on three heavy metals: gold, silver, and copper (GSC).
This is because GSC mines comprise more than 80\% of all metal mines.
Although the AHC documented other types of metal mines, they were indeed much smaller subsets: the number of lead and zinc, pig iron and steel, and tin mines documented were only one, five, and four in 1931, respectively.
Historical evidence indicates that coal mining requires many more workers than heavy metal mining because coal is a bulky mineral (Section~\ref{sec:sec2}).
Panel A of Table~\ref{tab:sum} indicates that the scale of GSC mines was much smaller than that of coal mines.
In addition, GSC mines were less likely to agglomerate than coal mines (Section~\ref{sec:sec31}).
In fact, the ``enclave effect'' suggests that some GSC mines are less likely to interact with local economies \citep{Aragon:2015db, Aragon:2013ib}.\footnote{A potentially relevant study is \citet{Stijns:2005bz}; using a country-level dataset, the author provided suggestive evidence that the impact of resource abundance on socioeconomic outcomes may vary among resource types.}
This means that the impact of GSC mines on the regional economy would be smaller than those of coal mines.
To assess this, I compare the estimates of the reduced-form equation~\ref{eq1} for the coal and GSC mines.\footnote{Unlike coal, a fuel mineral, no specific stratum can predict the spatial distribution of GSC mines.}
Table~\ref{tab:r_gsc} lists the results.
Panel A summarizes the results for the main outcome variables.
The estimates for the local population listed in columns (1) and (3) are $0.160$ for 1930 sample and $0.232$ for 1935 sample, respectively.
These are less than half of the estimates for the coal mine sample (column (2) of Panels A-1 and B-1 in Table~\ref{tab:r_pop}).
Similarly, columns (3) and (4) list smaller estimates for the labor force participation rates of male and female workers in mining areas (column (2) of Panels A-1 and B-1 in Table~\ref{tab:r_labor}).
An interesting result was observed for early-life health outcomes.
Columns (5) and (6) indicate that the results for IMRs are rather negative and statistically significant.\footnote{The FDR does not have clear correlations in both years. Among the municipalities with GSC mines, there were no meaningful correlations between the early-life health outcomes (i.e., both the FDR and IMR) and the number of miners (not reported).}
This means that infant mortality risk is even lower in the GSC mining areas than that in the surrounding municipalities.
Investigating this difference is beyond the scope of this study; however, this result suggests that unlike in coal mines, the positive economic effect of mines may exceed the negative health effects, as found in a recent related study by \citet{BenshaulTolonen:2019kx}.
My results also provide supportive evidence for \citet{Drixler2016}, who found that industrial pollution from copper mines was not associated with the risk of high stillbirth rates in Japan in the late 19th century.\footnote{\citet{Drixler2016} illustrates higher stillbirth rates in the potentially poisoned areas in the late 19th century by effluents from Ashio, modern Japan's richest copper mine, by calculating the rate of increase in stillbirth rates in the affected districts after the toxic flood of 1896. The author also shows a relatively greater increase in the stillbirth rates in some neighboring unpolluted areas, suggesting an exiguous contribution of industrial pollution to the high stillbirth rates at that time.}
In addition, this difference may be related to the characteristics of the pollutants.
While particulates generated around coal mines may immediately deteriorate the mother's and infant's health conditions, the heavy metals from GSC mines might not immediately affected mothers.\footnote{Instantaneous effects of coal consumption on mortality rates have been widely investigated \citep[e.g.,][]{Beach2017kja}.}
In other words, my empirical setting may not have captured the accumulation effects of heavy metals on humans.
\subsection{Alternative Identification Strategy}\label{sec:sec63}
I used IV estimation to obtain the baseline estimates.
For the outcome variables that can be used as a panel dataset, an alternative estimation strategy may be the DID approach, which uses both cross-sectional and within-variation identification.
However, DID estimation could not be applied to the dataset.
To implement the DID technique, municipalities that had been treated throughout the measured years would have to be excluded, while municipalities that had newer mines during the sampled periods were left.
However, the number of municipalities with coal mines increased by only 23 between 1931 and 1936 (Online Appendix~\ref{sec:seca2} presents the raw number of mines).
Consequently, as explained in Section~\ref{sec:sec31} in detail, the proportion of treated municipalities decreased from $14.6$\% to $12.5$\% (Panel A of Table~\ref{tab:sum}).
This feature of the assignments makes it difficult to apply the DID strategy for the coal mine sample because the information that can be used for the identification is insufficient.\footnote{Similarly, the number of municipalities with GSC mines increased by only 55 mines from 57 to 112 between 1931 and 1936 (see Online Appendix~\ref{sec:seca2} for the number of mines). The proportion of treated municipalities remained almost unchanged, from $5.0$\% to $5.3$\% (Panel A of Table~\ref{tab:sum}).}
\section{Conclusion}\label{sec:sec7}
This study examines the impact of mines on the regional economy during historical industrialization in pre-war Japan.
I found that coal mines increased the local population.
This population growth was initially accomplished by internal migration rather than due to increases in local marriage and fertility rates.
However, after these migrants formed their families, these demographics started to revert to trend.
Local structural shifts from agricultural to mining, manufacturing, and service sectors also occurred in the mining areas, while a clear potential gender bias against female workers was observed in the shifts.
Coal mines also increase the risk of infant mortality.
A pivotal finding is that the magnitude of this impact varies according to the mine type.
Coal mines extracting bulky minerals caused comparatively greater changes to the local population and structures than heavy metal mines.
While the mining industry can contribute to macroeconomic growth and create a temporary boom in the local economy, it will locally experience bust in due course.
Therefore, it is important to investigate the long-run, and entire costs and benefits of mining sector in the spirit of \citet{Black:2005uh}.
\clearpage
\begin{figure}[]
\centering
\captionsetup{justification=centering,margin=1.5cm}
\includegraphics[width=0.5\textwidth]{ts_coal_prod.eps}
\caption{Coal Production in Japan from 1912 to 1940}
\label{fig:ts_coal}
\scriptsize{\begin{minipage}{400pt}
\setstretch{0.85}
Note:
The solid line shows the coal output in giga tons (Gt).
The dashed line indicates the coal price in yen per giga tons.
The unit price is defined as the total price of coal output divided by the total coal output in each year.\\
Source: Created by the author. Data on coal output are from \citet{miti1954}.
\end{minipage}}
\end{figure}
\begin{figure}[]
\centering
\captionsetup{justification=centering}
\subfloat[1931]{\label{fig:mine_1931}\includegraphics[width=0.50\textwidth]{map_mine_1931.eps}}
\subfloat[1936]{\label{fig:mine_1936}\includegraphics[width=0.50\textwidth]{map_mine_1936.eps}}
\caption{Location of Coal and GSC mines in 1931 and 1936}
\label{fig:map_mines}
\scriptsize{\begin{minipage}{450pt}
\setstretch{0.9}
Notes: The red and blue circles indicate the locations of coal and GSC (gold, silver, and copper) mines, respectively.
Chikuh\=o coalfield in Fukuoka prefecture in Kyush\=u region is indicated with an arrow in Figure~\ref{fig:mine_1931}.
Figure~\ref{fig:map_coal_mines} illustrates the coal mines around Chikuh\=o coalfield in detail.\\
Source: Created by the author.
\end{minipage}}
\end{figure}
\begin{figure}[]
\centering
\captionsetup{justification=centering}
\subfloat[1931]{\label{fig:map_coal_1931_Kyushu}\includegraphics[width=0.50\textwidth]{map_coal_1931_Kyushu.eps}}
\subfloat[1936]{\label{fig:map_coal_1936_Kyushu}\includegraphics[width=0.50\textwidth]{map_coal_1936_Kyushu.eps}}
\caption{Location of Coal Mines around Chikuh\=o Coalfield in Kyush\=u}
\label{fig:map_coal_mines}
\scriptsize{\begin{minipage}{450pt}
\setstretch{0.9}
Notes:
1.~The white circles indicate the location of municipalities with coal mines around Chikuh\=o coalfield (Figure~\ref{fig:mine_1931}) in Kyush\=u region.
Figure~\ref{fig:map_coal_1931_Kyushu} shows the coal mines in Chikuh\=o, Kasuya, and Miike coalfields in Fukuoka, Karatsu coalfield in Saga, and Miike coalfield in Kumamoto.
Amakusa coalfield in Kumamoto is added in Figure~\ref{fig:map_coal_1936_Kyushu} (on the lower left of the figure).
\=Oita prefecture does not have coal mines but is shown in the figures to explain the border of the sample.\\
2.~Treatment (control) group includes municipalities within 5 (between 5 and 30) km from a mine. The excluded municipalities are shown as empty lattices in the figures.\\
Source: Created by the author.
\end{minipage}}
\end{figure}
\begin{figure}[]
\centering
\captionsetup{justification=centering}
\subfloat[Population]{\label{fig:hetero_lnpop_coal_distance}\includegraphics[width=0.33\textwidth]{hetero_lnpop_coal_distance.eps}}
\subfloat[LFP]{\label{fig:hetero_lpr_distance}\includegraphics[width=0.33\textwidth]{hetero_lpr_distance.eps}}
\subfloat[IMR]{\label{fig:hetero_imr_distance}\includegraphics[width=0.33\textwidth]{hetero_imr_distance.eps}}
\caption{Heterogeneity in the Impact of Mines by Distance from the Mines}
\label{fig:hetero_distance}
\scriptsize{\begin{minipage}{450pt}
\setstretch{0.9}
Notes:
The dots and solid lines with caps show the estimates and their 95 percent confidence intervals, respectively, from the regressions with the specification in Equation~\ref{eq1}.
Figures~\ref{fig:hetero_lnpop_coal_distance},~\ref{fig:hetero_lpr_distance}, and~\ref{fig:hetero_imr_distance} show the results for the local population, labor force participation rate, and infant mortality rate (IMR), respectively.
The confidence intervals are calculated using the standard errors based on the heteroskedasticity-robust covariance matrix estimator.
$0$km distance from mines indicates that the municipalities with a mine are defined as the treatment group.
$5$ ($10$; $15$)km distance from mines indicates that the municipalities within 5 ($10$; $15$) km of a mine are defined as the treatment group.\\
Source: Created by the author.
\end{minipage}}
\end{figure}
\clearpage
\begin{table}[]
\def1.0{0.95}
\begin{center}
\caption{Summary Statistics}
\label{tab:sum}
\scriptsize
\scalebox{0.80}[1]{
{\setlength\doublerulesep{2pt}
\begin{tabular}{lcD{.}{.}{2}D{.}{.}{2}D{.}{.}{2}D{.}{.}{2}D{.}{.}{2}D{.}{.}{2}D{.}{.}{2}D{.}{.}{2}D{.}{.}{2}}
\toprule[1pt]\midrule[0.3pt]
\textbf{Panel A}:
&\multirow{2}{*}{Year}&&\multicolumn{3}{c}{Coal mines}&&\multicolumn{3}{c}{GSC mines}&\\
\cmidrule(rrr){4-6}\cmidrule(rrr){8-10}
Mine, Stratum, \& Controls&
&&\multicolumn{1}{c}{Mean}&\multicolumn{1}{c}{Std. Dev.}&\multicolumn{1}{c}{Obs.}
&&\multicolumn{1}{c}{Mean}&\multicolumn{1}{c}{Std. Dev.}&\multicolumn{1}{c}{Obs.}\\\hline
Mine Deposit &1931 &&0.146 &0.354 &1140 &&0.050 &0.218 &2498 \\
&1936 &&0.125 &0.330 &1429 &&0.053 &0.224 &3293 \\
Stratum &(1931) &&0.094 &0.292 &1140 &&$--$ &$--$ &$--$ \\
&(1936) &&0.082 &0.274 &1429 &&$--$ &$--$ &$--$ \\
Female Miners &1931 &&207.1 &269.20 &93 &&$--$ &$--$ &$--$ \\
&1936 &&144.73 &179.26 &116 &&40.71 &58.91 &112 \\
Male Miners &1931 &&1359.25 &1710.20 &93 &&$--$ &$--$ &$--$ \\
&1936 &&1501.91 &2035.39 &116 &&449.17 &677.79 &112 \\
Distance to station &1931 &&5.38 &5.84 &1140 &&6.12 &7.76 &2498\\
&1936 &&5.74 &6.19 &1429 &&5.56 &6.92 &3293\\
Distance to port &1931 &&26.22 &22.19 &1140 &&29.87 &22.22 &2498\\
&1936 &&25.17 &21.61 &1429 &&32.66 &24.28 &3293\\\hline
\textbf{Panel B}: &\multirow{2}{*}{Year}&\multicolumn{3}{c}{Overall}&\multicolumn{3}{c}{Treated}&\multicolumn{3}{c}{Controlled}\\
\cmidrule(rrr){3-5}\cmidrule(rrr){6-8}\cmidrule(rrr){9-11}
Demographics &&\multicolumn{1}{c}{Mean}&\multicolumn{1}{c}{Std. Dev.}&\multicolumn{1}{c}{Obs.}
&\multicolumn{1}{c}{Mean}&\multicolumn{1}{c}{Std. Dev.}&\multicolumn{1}{c}{Obs.}&\multicolumn{1}{c}{Mean}&\multicolumn{1}{c}{Std. Dev.}&\multicolumn{1}{c}{Obs.}\\\hline
Population &1930$\ddagger$ &6305.06 &14829.74 &1140 &13091.93 &25577.78 &167 &5140.20 &11692.06 &973\\
&1935$\ddagger$ &6055.41 &15256.81 &1429 &12933.10 &27931.89 &178 &5076.82 &12155.15 &1251\\
Crude marriage rate &1930$\ddagger$ &8.59 &2.28 &1140 &7.43 &2.19 &167 &8.79 &2.24 &973\\
&1935$\ddagger$ &9.03 &2.42 &1429 &7.98 &2.05 &178 &9.18 &2.43 &1251\\
Crude birth rate &1930$\ddagger$ &34.06 &5.52 &1140 &31.75 &5.62 &167 &34.46 &5.40 &973\\
&1935$\ddagger$ &34.11 &5.43 &1429 &32.36 &4.95 &178 &34.36 &5.45 &1251\\
Household size &1930$\ddagger$ &5.34 &0.47 &1140 &5.18 &0.41 &167 &5.37 &0.47 &973\\
&1935$\ddagger$ &5.31 &0.57 &1429 &5.20 &0.44 &178 &5.33 &0.59 &1251\\
Sex ratio (male/female) &1930$\ddagger$ &0.99 &0.08 &1140 &1.03 &0.10 &167 &0.98 &0.07 &973\\
&1935$\ddagger$ &0.99 &0.07 &1429 &1.03 &0.08 &178 &0.98 &0.07 &1251\\\hline
\textbf{Panel C}: &\multirow{2}{*}{Year}&\multicolumn{3}{c}{Overall}&\multicolumn{3}{c}{Treated}&\multicolumn{3}{c}{Controlled}\\
\cmidrule(rrr){3-5}\cmidrule(rrr){6-8}\cmidrule(rrr){9-11}
Labor supply &&\multicolumn{1}{c}{Mean}&\multicolumn{1}{c}{Std. Dev.}&\multicolumn{1}{c}{Obs.}
&\multicolumn{1}{c}{Mean}&\multicolumn{1}{c}{Std. Dev.}&\multicolumn{1}{c}{Obs.}&\multicolumn{1}{c}{Mean}&\multicolumn{1}{c}{Std. Dev.}&\multicolumn{1}{c}{Obs.}\\\hline
Labor force participation rate &1930$\ddagger$ &48.95 &5.96 &1140 &45.76 &6.10 &167 &49.50 &5.76 &973\\
\hspace{10pt}Male &1930~ &57.13 &3.30 &1140 &56.90 &3.61 &167 &57.17 &3.24 &973\\
\hspace{10pt}Female &1930$\ddagger$ &40.77 &10.65 &1140 &34.06 &10.99 &167 &41.93 &10.15 &973\\
Mining sector$^{\star}$ &1930$\ddagger$ &2.75 &9.12 &1140 &17.13 &17.64 &167 &0.28 &1.63 &973\\
\hspace{10pt}Male$^{\star}$ &1930$\ddagger$ &3.16 &10.09 &1140 &19.32 &19.17 &167 &0.39 &2.00 &973\\
\hspace{10pt}Female$^{\star}$ &1930$\ddagger$ &1.83 &6.92 &1140 &11.85 &14.30 &167 &0.11 &0.99 &973\\
Agricultural sector &1930$\ddagger$ &65.82 &23.27 &1140 &45.55 &25.93 &167 &69.30 &20.91 &973\\
\hspace{10pt}Male &1930$\ddagger$ &60.39 &23.62 &1140 &38.84 &25.01 &167 &64.09 &21.29 &973\\
\hspace{10pt}Female &1930$\ddagger$ &74.03 &21.90 &1140 &57.66 &25.20 &167 &76.84 &19.98 &973\\
Manufacturing sector &1930$\ddagger$ &10.85 &7.95 &1140 &12.39 &6.08 &167 &10.59 &8.21 &973\\
\hspace{10pt}Male &1930$\ddagger$ &13.91 &8.04 &1140 &16.80 &7.12 &167 &13.41 &8.08 &973\\
\hspace{10pt}Female &1930$\ddagger$ &5.93 &8.78 &1140 &4.03 &3.62 &167 &6.26 &9.35 &973\\
Commerce sector &1930$\ddagger$ &8.51 &7.13 &1140 &10.44 &6.62 &167 &8.18 &7.17 &973\\
\hspace{10pt}Male &1930$\dagger$ &7.86 &6.44 &1140 &8.97 &5.68 &167 &7.67 &6.55 &973\\
\hspace{10pt}Female &1930$\ddagger$ &10.11 &10.10 &1140 &14.28 &10.64 &167 &9.40 &9.83 &973\\
Domestic sector &1930$\ddagger$ &1.77 &1.31 &1140 &2.05 &1.05 &167 &1.72 &1.34 &973\\
\hspace{10pt}Male$^{\star}$ &1930$\dagger$ &0.37 &0.54 &1140 &0.27 &0.25 &167 &0.38 &0.57 &973\\
\hspace{10pt}Female &1930$\ddagger$ &4.21 &3.93 &1140 &5.75 &3.70 &167 &3.94 &3.91 &973\\\hline
\textbf{Panel D}:&\multirow{2}{*}{Year}&\multicolumn{3}{c}{Overall}&\multicolumn{3}{c}{Treated}&\multicolumn{3}{c}{Controlled}\\
\cmidrule(rrr){3-5}\cmidrule(rrr){6-8}\cmidrule(rrr){9-11}
Early-life health&&\multicolumn{1}{c}{Mean}&\multicolumn{1}{c}{Std. Dev.}&\multicolumn{1}{c}{Obs.}&\multicolumn{1}{c}{Mean}&\multicolumn{1}{c}{Std. Dev.}&\multicolumn{1}{c}{Obs.}&\multicolumn{1}{c}{Mean}&\multicolumn{1}{c}{Std. Dev.}&\multicolumn{1}{c}{Obs.}\\\hline
Infant mortality rate &1933$\ddagger$ &117.15 &42.66 &1140 &139.40 &44.56 &167 &113.34 &41.15 &973\\
&1938$\ddagger$ &114.08 &42.74 &1429 &131.91 &36.70 &178 &111.54 &42.95 &1251\\
Fetal death rate$^{\star}$ &1933$\ddagger$ &43.51 &27.41 &1140 &51.57 &26.25 &167 &42.13 &27.38 &973\\
&1938$\ddagger$ &37.39 &25.76 &1429 &46.79 &25.65 &178 &36.06 &25.50 &1251\\
Child mortality rate$^{\star}$ &1933$\ddagger$ &13.40 &9.58 &1140 &15.54 &7.11 &167 &13.03 &9.90 &973\\
&1938$\ddagger$ &16.59 &8.76 &1429 &19.01 &8.52 &178 &16.25 &8.74 &1251\\
\midrule[0.3pt]\bottomrule[1pt]
\end{tabular}
}
}
{\scriptsize
\begin{minipage}{445pt}
\setstretch{0.85}Notes:
$\ddagger$ and $\dagger$ in Panels B--D indicate that the mean difference between treatment and control groups is statistically significant at the 1\% and 5\% levels, respectively.
The treatment (control) group is 0--5 (5-30) km from the centroid of a municipality that has a mine.\\
\textbf{Panel A} shows the summary statistics for the indicator variable for the treatment group and geological stratum.
Geological stratum is a time constant variable: ``Year'' in parenthesis indicates the matched year.
GSC mines indicate gold, silver, and copper mines.
Distance to the nearest neighboring station/port is in kilometers.\\
\textbf{Panel B} reports the summary statistics for the demographic outcomes: population, crude marriage rates, crude divorce rates, crude birth rates, and household size.
The population is log-transformed in the regressions to deal with skewed distribution in nature.
The crude marriage rate is the number of marriages per $1,000$ people.
The crude birth rate is the number of live births per $1,000$ people.
Household size is the total number of people in households divided by the number of households.\\
\textbf{Panel C} reports the summary statistics for labor force participation rates and employment shares.
Labor force participation is the number of workers per $100$ people.
The employment share is the number of workers in each sector per $100$ workers.\\
\textbf{Panel D} reports the summary statistics for the infant mortality (IMR), fetal death rates (FDR), and child mortality (CMR).
The IMR is the number of infant deaths per $1,000$ live births.
The FDR is the number of fetal deaths per $1,000$ births.
The CMR is the number of deaths of children aged 1--4 per $1,000$ children aged 1--5.
The variables with $\star$ have a set of censored observations; the results from the Tobit models are materially similar to the main results in most cases.\\
Sources:
Data on the location of mines are from \citet[][]{ahc1932, ahc1937}.
Data on the number of people and households are from \citet{census1930v1, census1935v1}.
Data on the demographic variables are from \citet{mvs1930, mvs1935}.
Data on the labor force participation rates and employment shares are from \citet{census1930pp}.
Data on births, fetal deaths, infant deaths, and child deaths are from \citet{mvs1933aiiku}; \citet{mvs1938aiiku}.
\end{minipage}
}
\end{center}
\end{table}
\begin{table}[]
\def1.0{1.0}
\begin{center}
\captionsetup{justification=centering}
\caption{Results: Local Population Growth}
\label{tab:r_pop}
\scriptsize
\scalebox{0.88}[1]{
{\setlength\doublerulesep{2pt}
\begin{tabular}{lD{.}{.}{-2}D{.}{.}{-2}D{.}{.}{-2}D{.}{.}{-2}D{.}{.}{-2}D{.}{.}{-2}}
\toprule[1pt]\midrule[0.3pt]
\multicolumn{7}{l}{\textbf{Panel A: 1930 Census}}\\
&\multicolumn{6}{c}{Dependent Variable: ln(Population)}\\
\cmidrule(rrrrrr){2-7}
Panel A-1: Main Result
&\multicolumn{1}{c}{(1)}&\multicolumn{1}{c}{(2)}&\multicolumn{1}{c}{(3)}&\multicolumn{1}{c}{(4)}&\multicolumn{1}{c}{(5)}&\multicolumn{1}{c}{(6)}\\\hline
\textit{MineDeposit}
&0.678$***$ &0.462$***$ &0.445$***$ &1.386$***$ &0.829$***$ &0.813$***$\\
&(0.078) &(0.058) &(0.058) &(0.291) &(0.197) &(0.211)\\
City and Town FEs
&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}\\
Railway accessibility
&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{Yes}\\
Port accessibility
&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{Yes}\\
Observations
&\multicolumn{1}{c}{1,140}&\multicolumn{1}{c}{1,140}&\multicolumn{1}{c}{1,140}&\multicolumn{1}{c}{1,140}&\multicolumn{1}{c}{1,140}&\multicolumn{1}{c}{1,140}\\
Estimator
&\multicolumn{1}{c}{OLS}&\multicolumn{1}{c}{OLS}&\multicolumn{1}{c}{OLS}&\multicolumn{1}{c}{IV (Wald)}&\multicolumn{1}{c}{IV}&\multicolumn{1}{c}{IV}\\
Lower bound&$--$&0.420&0.400&$--$&$--$&$--$\\
First-stage $F$-statistic&$--$&$--$&$--$&43.39&37.70&32.23\\\hline
&&&&&&\\
&&\multicolumn{4}{c}{Dependent Variable}&\\
\cmidrule(rrrr){3-6}
Panel A-2: Mechanism
&&\multicolumn{1}{c}{(1) Marriage}&\multicolumn{1}{c}{(2) Fertility}&\multicolumn{1}{c}{(3) Size}&\multicolumn{1}{c}{(4) Sex Ratio}&\\\hline
\textit{MineDeposit}
&&-2.588$***$ &-7.019$***$ &-0.437$***$ &0.115$***$ &\\
&&(0.835) &(2.054) &(0.146) &(0.037) &\\
City and Town FEs&&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\\
Railway accessibility&&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\\
Port accessibility&\multicolumn{1}{c}{}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{}\\
Observations&&\multicolumn{1}{c}{1,140}&\multicolumn{1}{c}{1,140}&\multicolumn{1}{c}{1,140}&\multicolumn{1}{c}{1,140}&\\
Estimator&&\multicolumn{1}{c}{IV}&\multicolumn{1}{c}{IV}&\multicolumn{1}{c}{IV}&\multicolumn{1}{c}{IV}&\\\hline
&&&&&&\\
\multicolumn{7}{l}{\textbf{Panel B: 1935 Census}}\\
&\multicolumn{6}{c}{Dependent Variable: ln(Population)}\\
\cmidrule(rrrrrr){2-7}
Panel B-1: Main Result
&\multicolumn{1}{c}{(1)}&\multicolumn{1}{c}{(2)}&\multicolumn{1}{c}{(3)}&\multicolumn{1}{c}{(4)}&\multicolumn{1}{c}{(5)}&\multicolumn{1}{c}{(6)}\\\hline
\textit{MineDeposit}
&0.673$***$ &0.475$***$ &0.460$***$ &1.432$***$ &0.905$***$ &0.924$***$\\
&(0.075) &(0.057) &(0.057) &(0.275) &(0.186) &(0.195)\\
City and Town FEs
&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}\\
Railway accessibility
&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{Yes}\\
Port accessibility&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{Yes}\\
Observations
&\multicolumn{1}{c}{1,429}&\multicolumn{1}{c}{1,429}&\multicolumn{1}{c}{1,429}&\multicolumn{1}{c}{1,429}&\multicolumn{1}{c}{1,429}&\multicolumn{1}{c}{1,429}\\
Estimator
&\multicolumn{1}{c}{OLS}&\multicolumn{1}{c}{OLS}&\multicolumn{1}{c}{OLS}&\multicolumn{1}{c}{IV (Wald)}&\multicolumn{1}{c}{IV}&\multicolumn{1}{c}{IV}\\
Lower bound&$--$&0.438&0.421&$--$&$--$&$--$\\
First-stage $F$-statistic&$--$&$--$&$--$&50.31&45.69&40.85\\\hline
&&&&&&\\
&&\multicolumn{4}{c}{Dependent Variable}&\\
\cmidrule(rrrr){3-6}
Panel B-2: Mechanism
&&\multicolumn{1}{c}{(1) Marriage}&\multicolumn{1}{c}{(2) Fertility}&\multicolumn{1}{c}{(3) Size}&\multicolumn{1}{c}{(4) Sex Ratio}&\\\hline
\textit{MineDeposit}
&&-1.417$**$ &-0.899 &-0.231$*$ &0.048$**$ &\\
&&(0.675) &(1.572) &(0.125) &(0.024) &\\
City and Town FEs&&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\\
Railway accessibility&&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\\
Port accessibility&\multicolumn{1}{c}{}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{}\\
Observations&&\multicolumn{1}{c}{1,429}&\multicolumn{1}{c}{1,429}&\multicolumn{1}{c}{1,429}&\multicolumn{1}{c}{1,429}&\\
Estimator&&\multicolumn{1}{c}{IV}&\multicolumn{1}{c}{IV}&\multicolumn{1}{c}{IV}&\multicolumn{1}{c}{IV}&\\\midrule[0.3pt]\bottomrule[1pt]
\end{tabular}
}
}
{\scriptsize
\begin{minipage}{440pt}
\setstretch{0.85}
***, **, and * represent statistical significance at the 1\%, 5\%, and 10\% levels, respectively.
Standard errors based on the heteroskedasticity-robust covariance matrix estimator are reported in parentheses.\\
Notes:
Panels A and B show the results for the 1930 and 1935 samples, respectively.
Panels A-1 and B-1 show the main result for log-transformed population, whereas Panels A-2 and B-2 show the result for mechanism analysis.
``Marriage,'' ``Fertility,'' ``Size,'' and ``Sex Ratio'' indicate the crude marriage rate, crude birth rate, household size, and adult sex ratio, respectively (Panel B of Table~\ref{tab:sum}).
\end{minipage}
}
\end{center}
\end{table}
\begin{table}[]
\def1.0{1.0}
\begin{center}
\captionsetup{justification=centering}
\caption{Results: Local Labor Supply}
\label{tab:r_labor}
\scriptsize
\scalebox{0.82}[1]{
{\setlength\doublerulesep{2pt}
\begin{tabular}{lD{.}{.}{-2}D{.}{.}{-2}D{.}{.}{-2}D{.}{.}{-2}D{.}{.}{-2}D{.}{.}{-2}}
\toprule[1pt]\midrule[0.3pt]
\multicolumn{7}{l}{\textbf{Panel A: Male Workers}}\\
&\multicolumn{6}{c}{Dependent Variable: Labor Force Participation Rate (\%)}\\
\cmidrule(rrrrrr){2-7}
Panel A-1: Main Result
&\multicolumn{1}{c}{(1)}&\multicolumn{1}{c}{(2)}&\multicolumn{1}{c}{(3)}&\multicolumn{1}{c}{(4)}&\multicolumn{1}{c}{(5)}&\multicolumn{1}{c}{(6)}\\\hline
\textit{MineDeposit}
&-0.277 &-0.320 &-0.127 &-0.960 &-1.192 &-0.257\\
&(0.298) &(0.295) &(0.310) &(1.068) &(1.139) &(1.258)\\
City and Town FEs&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}\\
Railway accessibility&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{Yes}\\
Port accessibility&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{Yes}\\
Observations&\multicolumn{1}{c}{1,140}&\multicolumn{1}{c}{1,140}&\multicolumn{1}{c}{1,140}&\multicolumn{1}{c}{1,140}&\multicolumn{1}{c}{1,140}&\multicolumn{1}{c}{1,140}\\
Estimator&\multicolumn{1}{c}{OLS}&\multicolumn{1}{c}{OLS}&\multicolumn{1}{c}{OLS}&\multicolumn{1}{c}{IV (Wald)}&\multicolumn{1}{c}{IV}&\multicolumn{1}{c}{IV}\\
Lower bound&$--$&-0.327&-0.102&$--$&$--$&$--$\\
First-stage $F$-statistic&$--$&$--$&$--$&43.39&37.70&32.23\\\hline
&&&&\\
&\multicolumn{5}{c}{Dependent Variable: Employment Share (\%)}&\\
\cmidrule(rrrrr){2-6}
Panel A-2: Structural Shift
&\multicolumn{1}{c}{(1) Mining}&\multicolumn{1}{c}{(2) Agricultural}&\multicolumn{1}{c}{(3) Manufacturing}&\multicolumn{1}{c}{(4) Commerce}&\multicolumn{1}{c}{(5) Domestic}&\\\hline
\textit{MineDeposit}
&34.725$***$ &-34.649$***$ &2.278 &-1.958 &-0.450$***$&\\
&(5.221) &(6.728) &(2.235) &(1.614) &(0.144)&\\
City and Town FEs&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\\
Railway accessibility&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\\
Port accessibility&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\\
Observations&\multicolumn{1}{c}{1,140}&\multicolumn{1}{c}{1,140}&\multicolumn{1}{c}{1,140}&\multicolumn{1}{c}{1,140}&\multicolumn{1}{c}{1,140}&\\
Estimator&\multicolumn{1}{c}{IV}&\multicolumn{1}{c}{IV}&\multicolumn{1}{c}{IV}&\multicolumn{1}{c}{IV}&\multicolumn{1}{c}{IV}&\\\hline
&&&&&&\\
\multicolumn{7}{l}{\textbf{Panel B: Female Workers}}\\
&\multicolumn{6}{c}{Dependent Variable: Labor Force Participation Rate (\%)}\\
\cmidrule(rrrrrr){2-7}
Panel B-1: Main Result
&\multicolumn{1}{c}{(1)}&\multicolumn{1}{c}{(2)}&\multicolumn{1}{c}{(3)}&\multicolumn{1}{c}{(4)}&\multicolumn{1}{c}{(5)}&\multicolumn{1}{c}{(6)}\\\hline
\textit{MineDeposit}
&-7.865$***$ &-5.655$***$ &-4.910$***$ &-25.097$***$ &-20.682$***$ &-17.703$***$ \\
&(0.911) &(0.826) &(0.874) &(4.357) &(4.272) &(4.190) \\
City and Town FEs&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}\\
Railway accessibility&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{Yes}\\
Port accessibility&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{Yes}\\
Observations&\multicolumn{1}{c}{1,140}&\multicolumn{1}{c}{1,140}&\multicolumn{1}{c}{1,140}&\multicolumn{1}{c}{1,140}&\multicolumn{1}{c}{1,140}&\multicolumn{1}{c}{1,140}\\
Estimator&\multicolumn{1}{c}{OLS}&\multicolumn{1}{c}{OLS}&\multicolumn{1}{c}{OLS}&\multicolumn{1}{c}{IV (Wald)}&\multicolumn{1}{c}{IV}&\multicolumn{1}{c}{IV}\\
Lower bound&$--$&-5.182&-4.332&$--$&$--$&\\
First-stage $F$-statistic&$--$&$--$&$--$&43.39&37.70&32.23\\\hline
&&&&\\
&\multicolumn{5}{c}{Dependent Variable: Employment Share (\%)}&\\
\cmidrule(rrrrr){2-6}
Panel B-2: Structural Shift
&\multicolumn{1}{c}{(1) Mining}&\multicolumn{1}{c}{(2) Agricultural}&\multicolumn{1}{c}{(3) Manufacturing}&\multicolumn{1}{c}{(4) Commerce}&\multicolumn{1}{c}{(5) Domestic}&\\\hline
\textit{MineDeposit}
&24.777$***$ &-25.130$***$ &-8.939$**$ &2.822 &4.494$***$&\\
&(4.273) &(6.081) &(2.145) &(2.742) &(1.439)&\\
City and Town FEs&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\\
Railway accessibility&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\\
Port accessibility&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\\
Observations&\multicolumn{1}{c}{1,140}&\multicolumn{1}{c}{1,140}&\multicolumn{1}{c}{1,140}&\multicolumn{1}{c}{1,140}&\multicolumn{1}{c}{1,140}&\\
Estimator&\multicolumn{1}{c}{IV}&\multicolumn{1}{c}{IV}&\multicolumn{1}{c}{IV}&\multicolumn{1}{c}{IV}&\multicolumn{1}{c}{IV}&\\\midrule[0.3pt]\bottomrule[1pt]
\end{tabular}
}
}
{\scriptsize
\begin{minipage}{435pt}
\setstretch{0.85}
***, **, and * represent statistical significance at the 1\%, 5\%, and 10\% levels, respectively.
Standard errors based on the heteroskedasticity-robust covariance matrix estimator are reported in parentheses.\\
Notes:
Panels A and B show the results for male and female worker samples, respectively, from the 1930 Population Census (Panel C of Table~\ref{tab:sum}).
Panels A-1 and B-1 show the main result for the labor force participation rate, whereas Panels A-2 and B-2 show the result for employment share in each industrial sector (Panel C of Table~\ref{tab:sum}).
\end{minipage}
}
\end{center}
\end{table}
\begin{table}[]
\def1.0{1.0}
\begin{center}
\captionsetup{justification=centering}
\caption{Results: Early-life Health}
\label{tab:r_health}
\scriptsize
\scalebox{0.85}[1]{
{\setlength\doublerulesep{2pt}
\begin{tabular}{lD{.}{.}{-2}D{.}{.}{-2}D{.}{.}{-2}D{.}{.}{-2}D{.}{.}{-2}D{.}{.}{-2}}
\toprule[1pt]\midrule[0.3pt]
\multicolumn{7}{l}{\textbf{Panel A: 1933 Vital Statistics}}\\
&\multicolumn{6}{c}{Dependent Variable: Infant Mortality Rate}\\
\cmidrule(rrrrrr){2-7}
Panel A-1: IMR
&\multicolumn{1}{c}{(1)}&\multicolumn{1}{c}{(2)}&\multicolumn{1}{c}{(3)}&\multicolumn{1}{c}{(4)}&\multicolumn{1}{c}{(5)}&\multicolumn{1}{c}{(6)}\\\hline
\textit{MineDeposit}
&26.066$***$ &25.857$***$ &22.666$***$ &49.121$***$ &50.942$***$ &36.877$**$ \\
&(3.692) &(3.757) &(3.769) &(14.526) &(15.808) &(17.106) \\
City and Town FEs&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}\\
Railway accessibility&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{Yes}\\
Port accessibility&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{Yes}\\
Observations&\multicolumn{1}{c}{1,140}&\multicolumn{1}{c}{1,140}&\multicolumn{1}{c}{1,140}&\multicolumn{1}{c}{1,140}&\multicolumn{1}{c}{1,140}&\multicolumn{1}{c}{1,140}\\
Estimator&\multicolumn{1}{c}{OLS}&\multicolumn{1}{c}{OLS}&\multicolumn{1}{c}{OLS}&\multicolumn{1}{c}{IV (Wald)}&\multicolumn{1}{c}{IV}&\multicolumn{1}{c}{IV}\\
Lower bound&$--$&23.989&21.554&$--$&$--$&$--$\\
First-stage $F$-statistic&$--$&$--$&$--$&43.39&37.70&32.23\\\hline
&&&&\\
&\multicolumn{6}{c}{Dependent Variable}\\
\cmidrule(rrrrrr){2-7}
Panel A-2: Mechanism
&\multicolumn{1}{c}{(1) FDR}&\multicolumn{1}{c}{(2) FDR}&\multicolumn{1}{c}{(3) IMR}&\multicolumn{1}{c}{(4) CMR}&\multicolumn{1}{c}{(5) CMR}&\multicolumn{1}{c}{}\\\hline
\textit{MineDeposit}
&22.910$**$ & & &-0.115 && \\
&(10.165) & & &(2.915) && \\
\textit{Female Miner}
& &0.036$***$ &0.098$***$ & &0.004 & \\
& &(0.014) &(0.031) & &(0.004) & \\
\textit{Male Miner}
& &-0.002 &-0.008$**$ & &-0.001 & \\
& &(0.002) &(0.004) & &(0.001) & \\
City and Town FEs&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{}\\
Railway accessibility&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{}\\
Port accessibility&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{}\\
Observations&\multicolumn{1}{c}{1,140}&\multicolumn{1}{c}{93}&\multicolumn{1}{c}{93}&\multicolumn{1}{c}{1,140}&\multicolumn{1}{c}{93}\\
Estimator&\multicolumn{1}{c}{IV}&\multicolumn{1}{c}{OLS}&\multicolumn{1}{c}{OLS}&\multicolumn{1}{c}{IV}&\multicolumn{1}{c}{OLS}\\
First-stage $F$-statistic &32.23&$--$&$--$&32.23&$--$&$--$ \\\hline
&&&&&&\\
\multicolumn{7}{l}{\textbf{Panel B: 1938 Vital Statistics}}\\
&\multicolumn{6}{c}{Dependent Variable: Infant Mortality Rate}\\
\cmidrule(rrrrrr){2-7}
Panel B-1: IMR
&\multicolumn{1}{c}{(1)}&\multicolumn{1}{c}{(2)}&\multicolumn{1}{c}{(3)}&\multicolumn{1}{c}{(4)}&\multicolumn{1}{c}{(4)}&\multicolumn{1}{c}{(5)}\\\hline
\textit{MineDeposit}
&20.372$***$ &19.453$***$ &16.632$***$ &30.179$***$ &28.646$**$ &16.806\\
&(3.007) &(3.048) &(2.986) &(11.290) &(11.981) &(12.525)\\
City and Town FEs&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}\\
Railway accessibility&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{Yes}\\
Port accessibility&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{No}&\multicolumn{1}{c}{Yes}\\
Observations&\multicolumn{1}{c}{1,429}&\multicolumn{1}{c}{1,429}&\multicolumn{1}{c}{1,429}&\multicolumn{1}{c}{1,429}&\multicolumn{1}{c}{1,429}&\multicolumn{1}{c}{1,429}\\
Estimator&\multicolumn{1}{c}{OLS}&\multicolumn{1}{c}{OLS}&\multicolumn{1}{c}{OLS}&\multicolumn{1}{c}{IV (Wald)}&\multicolumn{1}{c}{IV}&\multicolumn{1}{c}{IV}\\
Lower bound &$--$ &17.803 &15.677&$--$ &$--$&$--$\\
First-stage $F$-statistic &$--$ &$--$ &$--$&50.31 &45.69&40.85\\\hline
&&&&\\
&\multicolumn{6}{c}{Dependent Variable}\\
\cmidrule(rrrrrr){2-7}
Panel B-2: Mechanism
&\multicolumn{1}{c}{(1) FDR}&\multicolumn{1}{c}{(2) FDR}&\multicolumn{1}{c}{(3) IMR}&\multicolumn{1}{c}{(4) CMR}&\multicolumn{1}{c}{(5) CMR}&\multicolumn{1}{c}{}\\\hline
\textit{MineDeposit}
&6.447 & & &1.527 & & \\
&(8.230) & & &(2.540) & & \\
\textit{Female Miner}
& &0.025$*$ &0.068$**$ & &0.005 & \\
& &(0.015) &(0.030) & &(0.004) & \\
\textit{Male Miner}
& &0.001 &0.000 & &0.000 & \\
& &(0.001) &(0.002) & &(0.000) & \\
City and Town FEs&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{}\\
Railway accessibility&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{}\\
Port accessibility&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{}\\
Observations&\multicolumn{1}{c}{1,429}&\multicolumn{1}{c}{116}&\multicolumn{1}{c}{116}&\multicolumn{1}{c}{1,429}&\multicolumn{1}{c}{116}&\multicolumn{1}{c}{}\\
Estimator&\multicolumn{1}{c}{IV}&\multicolumn{1}{c}{OLS}&\multicolumn{1}{c}{OLS}&\multicolumn{1}{c}{IV}&\multicolumn{1}{c}{OLS}&\multicolumn{1}{c}{}\\
First-stage $F$-statistic &40.85&$--$&$--$&40.85&$--$&$--$ \\\midrule[0.3pt]\bottomrule[1pt]
\end{tabular}
}
}
{\scriptsize
\begin{minipage}{420pt}
\setstretch{0.85}
***, **, and * represent statistical significance at the 1\%, 5\%, and 10\% levels, respectively.
Standard errors based on the heteroskedasticity-robust covariance matrix estimator are reported in parentheses.\\
Notes:
Panels A and B show the results for the 1933 and 1938 Vital Statistics samples, respectively (Panel D of Table~\ref{tab:sum}).
Panels A-1 and B-1 show the main results for the infant mortality rate (IMR).
Panels A-2 and B-2 show the results for the fetal death rate (FDR), IMR in the coal mining municipalities, and the child mortality rate (CMR) (Panel D of Table~\ref{tab:sum}).
\end{minipage}
}
\end{center}
\end{table}
\begin{table}[]
\def1.0{1.0}
\begin{center}
\captionsetup{justification=centering}
\caption{Additional Results: Heavy Metal Mines}
\label{tab:r_gsc}
\scriptsize
\scalebox{0.94}[1]{
{\setlength\doublerulesep{2pt}
\begin{tabular}{lD{.}{.}{-2}D{.}{.}{-2}D{.}{.}{-2}D{.}{.}{-2}D{.}{.}{-2}D{.}{.}{-2}}
\toprule[1pt]\midrule[0.3pt]
&\multicolumn{6}{c}{Dependent Variable}\\
\cmidrule(rrrrrr){2-7}
&\multicolumn{2}{c}{ln(Population)}&\multicolumn{2}{c}{LFP Rate in 1930}&\multicolumn{2}{c}{IMR}\\
\cmidrule(rr){2-3}\cmidrule(rr){4-5}\cmidrule(rr){6-7}
&\multicolumn{1}{c}{(1) 1930}&\multicolumn{1}{c}{(2) 1935}&\multicolumn{1}{c}{(3) Male}&\multicolumn{1}{c}{(4) Female}&\multicolumn{1}{c}{(5) 1933}&\multicolumn{1}{c}{(6) 1938}\\\hline
\textit{MineDeposit}
&0.160$***$ &0.232$***$ &-1.108$***$ &-3.200$***$ &-7.133$**$ &-6.414$**$ \\
&(0.047) &(0.049) &(0.311) &(0.953) &(3.102) &(3.000) \\
City and Town FEs
&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}\\
Railway accessibility
&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}\\
Port accessibility
&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}&\multicolumn{1}{c}{Yes}\\
Observations
&\multicolumn{1}{c}{2,498}&\multicolumn{1}{c}{3,293}&\multicolumn{1}{c}{2,498}&\multicolumn{1}{c}{2,498}&\multicolumn{1}{c}{2,498}&\multicolumn{1}{c}{3,293}\\
Estimator
&\multicolumn{1}{c}{OLS}&\multicolumn{1}{c}{OLS}&\multicolumn{1}{c}{OLS}&\multicolumn{1}{c}{OLS}&\multicolumn{1}{c}{OLS}&\multicolumn{1}{c}{OLS}\\
Lower bound &0.153&0.226&-1.165&-3.153&-7.364&-6.213\\\midrule[0.3pt]\bottomrule[1pt]
\end{tabular}
}
}
{\scriptsize
\begin{minipage}{440pt}
\setstretch{0.85}
***, **, and * represent statistical significance at the 1\%, 5\%, and 10\% levels, respectively.
Standard errors based on the heteroskedasticity-robust covariance matrix estimator are reported in parentheses.\\
Notes:
This table shows the results for the GSC mines sample (Panel A of Table~\ref{tab:sum}).
Panel A: Columns (1) and (2) show the results for the log-transformed population measured in the 1930 and 1935 population censuses, respectively.
Columns (3) and (4) show the results for the labor force participation rates of male and female workers, respectively.
Columns (5) and (6) show the results for the infant mortality rates (IMR) measured using the 1933 and 1938 vital statistics, respectively.\\
\end{minipage}
}
\end{center}
\end{table}
\clearpage
\bibliographystyle{plainnat
|
2024-02-18T23:40:24.611Z
|
2022-10-27T02:13:52.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14514","language":"en","timestamp":1666836832000,"url":"https:\/\/arxiv.org\/abs\/2112.14514","yymm":"2112"}
|
proofpile-arXiv_000-10049
|
{"provenance":"002.jsonl.gz:10050"}
| null | null |
\section{Introduction}
Studying the nature of black holes has led to some of the deepest and most fruitful physical insights of the last century. The realization that black holes possess an entropy proportional to their surface area \cite{PhysRevD.7.2333, hawking1975particle} was the first step toward identifying the holographic nature of spacetime \cite{tHooft:1993dmi, Susskind:1994vu, Bousso:2002ju}: information stored in any spacetime volume is related to the surface area bounding that region. The most famous concrete realization of this principle in string theory is the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence \cite{Maldacena:1997re,Witten:1998qj}, which relates a theory of gravity in negatively-curved spacetime to a highly-symmetric quantum field theory in fewer spacetime dimensions. Holographic dualities like AdS/CFT are the workhorse of modern theoretical physics, harnessing black hole physics to study a surprisingly diverse plethora of phenomena, including strongly interacting condensed matter systems \cite{Hartnoll:2016apf}, the geometrization of entanglement \cite{ryu2006holographic,ryu2006aspects,maldacena2013cool}, Hawking evaporation \cite{Penington:2019npb,Almheiri:2019psf} and quantum chaos \cite{Shenker:2013pqa,Maldacena:2015waa}.
Remarkably, holographic correspondences now exist for real, physical black holes that appear in our universe, which are maximally (or near-maximally) spinning Kerr black holes \cite{kerr1963gravitational}. The Kerr/CFT correspondence \cite{Guica:2008mu} relies on taking a near-horizon limit in the extremal Kerr metric, and showing that the isometries of the resulting metric form a copy of the conformal algebra, $\grp{SL}(2,\mathbb R)$. A version of the Kerr/CFT correspondence was also found away from the extremal limit \cite{Castro:2010fd}. The authors of \cite{Castro:2010fd} found that the price of moving away from extremality is that it is necessary to consider symmetries of the near-horizon dynamics (the wave equation), rather than only symmetries of the near-horizon metric, to uncover an underlying conformal algebra. Such symmetries of the dynamics are referred to as \textit{hidden symmetries}, to distinguish them from explicit isometries of the metric. This notion of hidden conformal symmetry associated with black hole horizons has provided new insight into interesting and potentially observable aspects of black holes, such as black hole shadows \cite{hioki2008hidden}, tidal Love numbers \cite{Charalambous:2021kcz} and near-superradiant geodesics~\cite{Porfyriadis:2016gwb}. Furthermore, the presence of hidden symmetries in a gravitational system are known to be responsible for the separability of equations of motion on that background, as well as complete integrability of geodesics \cite{Frolov:2017kze}.
A particularly useful method of examining near-horizon dynamics of a probe scalar field on a black hole background is to study the monodromy properties of solutions to the Klein-Gordon equation \cite{Castro:2013kea,Castro:2013lba,Aggarwal:2019iay,Chanson:2020hly}. The monodromy data encode information about black hole thermodynamics and the hidden conformal symmetry of \cite{Castro:2010fd}, and provide ample evidence for a two-dimensional CFT description of the thermal properties of black hole microstates. In particular, the fact that hidden conformal symmetry appears to be a feature of a large class of black holes \cite{Castro:2013kea,Sakti:2020jpo,Keeler:2021tqy} seems to indicate that a Cardy formula \cite{Cardy:1986ie} may be sensible to apply and able to reproduce the Bekenstein-Hawking entropy in scenarios beyond four- and five-dimensional black holes \cite{Strominger:1997eq,Castro:2010fd,Krishnan:2010pv}.
The goals of this work are twofold. First, since hidden conformal symmetry is only discernible through studying dynamics of a probe field, we begin this article by asking: how is hidden conformal symmetry manifest when we change the dynamics? That is, we would like to study an action such that the resulting equations of motion of our probe field are not the standard Klein-Gordon equation. In addition, we would like our chosen dynamics to potentially yield novel physical insight, while at the same time being easily comparable to the known Klein-Gordon case. To this end, we consider the higher derivative action of a massless scalar field $\Phi$:
\begin{equation}\label{Introaction}
S=-\int d^{4}x\sqrt{g}\left(\frac{1}{2}\Phi\left(\nabla^{\mu}\nabla_{\mu}\right)^n\Phi\right),
\end{equation}
with corresponding equation of motion
\begin{equation}\label{Introgenreq}
\left( \nabla^{\mu}\nabla_{\mu}\right)^n\Phi=0.
\end{equation}
When the integer $n=1$, we recover the familiar Klein-Gordon equation for a free scalar field. Higher order differential equations like this one arise in other physical settings, such as in the study of buoyant thermal convection \cite{littlefield1990frobenius}.
Although the equation of motion \eqref{Introgenreq} is a very simple extension of the standard Klein-Gordon case, these higher derivative interactions already possess interesting physical attributes. Holographic duals of logarithmic conformal field theories (logCFTs) \cite{Gurarie:1993xq} are known to involve higher derivative equations of motion \cite{Hogervorst:2016itc}. LogCFTs are interesting in their own right, with applications in percolation \cite{cardy1999logarithmic} and quenched disorder \cite{cardy1999logarithmic,Caux:1995nm,Maassarani:1996jn,Caux:1998sm,Cardy:2013rqg}. Thus our second motivating question behind looking for hidden conformal symmetry in the dynamics \eqref{Introgenreq} is: can we construct a holographic logCFT correspondence in the spirit of Kerr/CFT (in which the conformal symmetry exists at the black hole horizon) as opposed to AdS/CFT (in which the CFT is said to exist at the boundary of AdS)? Constructing a logCFT correspondence in this scenario would be particularly interesting due to their non-unitary nature: the presence of hidden conformal symmetry has prompted many authors \cite{Castro:2010fd, Castro:2013kea,Haco:2018ske,Aggarwal:2019iay} to assume the validity of a Cardy formula and show that it reproduces the Bekenstein-Hawking entropy. However, Cardy's formula is not known to hold in non-unitary settings.
This article is organized as follows. In Section \ref{sec:Hidden-symm-KG} we review how hidden conformal symmetry is found in the Klein-Gordon equation on a Kerr background. In Section \ref{sec:standardform} we briefly present a standard form for the Klein-Gordon operator that will streamline our calculations. We note that this standard form holds for a probe field of general spin, in both four and five dimensions. In Section \ref{sec:Hidden-symm-HD} we perform our analysis of hidden conformal symmetry in higher derivative dynamics. We calculate the monodromy data in Section \ref{sec:monodanal}, and move to holographic considerations in Section \ref{sec:HigherDHolography}. We discuss our findings and future work in Section \ref{sec:discussion}. In our appendix \ref{app:KGgoodform} we give more examples of the standard form Klein-Gordon operator presented in Section \ref{sec:standardform}.
\section{Hidden conformal symmetry from the Klein-Gordon equation}
\label{sec:Hidden-symm-KG}
In this section we examine the hidden conformal symmetry of the Kerr black hole, as first discovered by~\cite{Castro:2010fd}. This will provide the framework for studying how hidden conformal symmetry is manifest in theories with higher derivative interactions in Section~\ref{sec:Hidden-symm-HD}.
While most of this section will be a review, some of the discussion we present is, we believe, new.
The Kerr black hole is described by the following metric:
\begin{equation}\label{kerrMet}
ds^2=\frac{\rho^2}{\Delta}dr^2-\frac{\Delta}{\rho^2}(dt^2-a~\text{sin}^2\theta d\phi)^2+\rho^2d\theta^2+\frac{\text{sin}^2\theta}{\rho^2}((r^2+a^2)d\phi-adt)^2,
\end{equation}
where we have defined
\begin{equation}
\Delta=r^2+a^2-2 M r\,, \qquad \text{and}\qquad \rho^2=r^2+a^2\text{cos}^2\theta\,,
\end{equation} and $a\equiv\frac{J}{M}$ is the spin of the black hole of mass $M$ and angular momentum $J$. This geometry possesses an outer horizon $r_+$ and inner horizon $r_-$, the locations of which are determined by the equation $\Delta=0$.
In particular, we have
\begin{equation}
r_{\pm}= M\pm \sqrt{M^2-a^2}\,.
\end{equation}
There are two Killing vectors associated with (\ref{kerrMet}): $\partial_t$ and $\partial_{\phi}$. These generate explicit isometries of the metric (\ref{kerrMet}).
The additional hidden symmetry generators are not symmetries of the metric, but of the dynamics. Consider a massless scalar field $\Phi$ on the background (\ref{kerrMet}). Here we assume that we can treat the scalar $\Phi$ as a probe.
The Klein-Gordon equation of motion for this scalar, {\it i.e.}
\begin{equation}
{1\over \sqrt{g}}\partial_\mu(\sqrt{g} g^{\mu\nu}\partial_\nu\Phi)=0\,, \qquad \mu=0,\dots, 3\,,
\end{equation}
famously separates%
%
\footnote{Indeed, the separability of this equation is actually a direct consequence of the existence of the hidden symmetry generators that we are about to build. For a review on this point see~\cite{Frolov:2006dqt, Frolov:2017kze}, and for a more recent discussion see~\cite{Keeler:2021tqy}.}
%
under the ansatz
\begin{equation}
\Phi(t,r, \theta,\phi)=e^{i(m\phi-\omega t)}R(r)S(\theta)\,.
\end{equation}
The radial equation is
\begin{equation}\label{KleinGordon}
\begin{split}
&\left(\partial_r(\Delta\partial_r)+\frac{(\omega-\Omega_+m)^2}{4\kappa_+^2}\frac{(r_+-r_-)}{r-r_+}-\frac{(\omega-\Omega_-m)^2}{4\kappa_-^2}\frac{(r_+-r_-)}{r-r_-}\right.\\
&\left.+(r^2+2M(r+2M))\omega^2\vphantom{\frac{1}{2}}\right)R(r)=KR(r),
\end{split}
\end{equation}
where $\Omega_{\pm}$ and $\kappa_{\pm}$ are the angular velocities and surface gravities of the inner and outer horizons,
\begin{equation}\label{omegas-kappas}
\Omega_{\pm}=\frac{a}{2 M r_\pm} \,, \qquad \kappa_\pm=\frac{r_+-r_-}{4 M r_\pm}\,,
\end{equation}
and $K$ is a separation constant (which also encodes information on the spectrum of the spherical harmonic function $S(\theta)$).
\subsection{Near-region limit}
\label{sec:near-region-lim}
The authors of~\cite{Castro:2010fd, Haco:2018ske} have argued that a hidden conformal symmetry becomes manifest if we consider only soft hair modes. That is, they consider the following ``near-region'' limit of the equation \eqref{KleinGordon}:
\begin{equation}
\omega \, M \ll 1\,,\qquad \text{and} \qquad \omega \, r \ll 1\,.
\end{equation}
As emphasized in \cite{Haco:2018ske}, this limit can be thought of as a near-horizon limit taken in phase space. The resulting equation is
\begin{equation}\label{KGlimit}
\left(\partial_r(\Delta\partial_r)+\frac{(\omega-\Omega_+m)^2}{4\kappa_+^2}\frac{(r_+-r_-)}{r-r_+}-\frac{(\omega-\Omega_-m)^2}{4\kappa_-^2}\frac{(r_+-r_-)}{r-r_-}\right)R(r)=KR(r).
\end{equation}
The solutions of (\ref{KGlimit}) are hypergeometric functions. As pointed out by~\cite{Castro:2010fd}, the hypergeometric functions transform in representations of $\grp{SL}(2,\mathbb{R})$, which is the first hint of the existence of a hidden conformal symmetry. But what are the generators? To find them, we can take our inspiration from Kerr/CFT, and note that the Near-Horizon Extremal Kerr (NHEK) geometry is warped AdS$_3$, and for a particular choice of angle $\theta=\theta_0$ it is exactly the upper-half plane of AdS$_3$ (up to a conformal factor):
\begin{equation}\label{AdSPoincare}
ds^2=F(\theta_0)\left(\frac{dw^+dw^-+dy^2}{y^2}\right).
\end{equation}
Thus the existence of conformal symmetry of black horizons (hidden or otherwise) is tied to the existence of a copy of AdS$_3$ in the near-horizon limit (either in metric or in the Klein-Gordon equation).
The isometry group of AdS$_3$ is $\grp{SL}(2,\mathbb{R})\times \grp{SL}(2,\mathbb{R})$, and we already know the Killing vectors for (\ref{AdSPoincare}). They are
\begin{equation}\label{Hs}
\begin{split}
&H_1=i\partial_+, \qquad H_0=i\left(w^+\partial_++\frac{1}{2}y\partial_y\right),\qquad H_{-1}=i\left(w^{+2}\partial_++w^+y\partial_y-y^2\partial_-\right)\,,\\
&\bar{H}_1=i\partial_-, \qquad \bar{H}_0=i\left(w^-\partial_-+\frac{1}{2}y\partial_y\right),\qquad \bar{H}_{-1}=i\left(w^{-2}\partial_-+w^-y\partial_y-y^2\partial_+\right).\\
\end{split}
\end{equation}
These generators satisfy the conformal algebra
\begin{equation}\label{commutationRels}
\left[H_0,H_{\pm 1}\right]=\mp iH_{\pm}, \qquad \left[H_1,H_1\right]=-2iH_0\,,
\end{equation}
and have quadratic Casimir
\begin{equation}\label{casimir}
\begin{split}
\mathcal{H}^2&=-H_0^2+\frac{1}{2}\left(H_1H_{-1}+H_{-1}H_1\right)\\
&=\frac{1}{4}\left(y^2\partial_y^2-y\partial_y\right)+y^2\partial_+\partial_-.
\end{split}
\end{equation}
Now the only thing that remains to identify the hidden symmetry generators of (\ref{KGlimit}) is to find a suitable coordinate transformation between Boyer-Lindquist coordinates $(t,r,\phi)$ and conformal coordinates $(w^{\pm},y)$. For Kerr black holes this turns out to be
\begin{equation}\label{confcoor4d}
\begin{split}
w^+&=\left(\frac{r-r_+}{r-r_-}\right)^{1/2}e^{2\pi T_R\phi}\,,\\
w^-&=\left(\frac{r-r_+}{r-r_-}\right)^{1/2}e^{2\pi T_L\phi-\frac{t}{2M}}\,,\\
y&=\left(\frac{r_+-r_-}{r-r_-}\right)^{1/2}e^{\pi(T_L+T_R)\phi-\frac{t}{4M}}\,.
\end{split}
\end{equation}
where
\begin{equation}
T_R=\frac{r_+-r_-}{4\pi a}, \qquad T_L=\frac{r_++r_-}{4\pi a}.
\end{equation}
Much has been written about the conformal coordinates (\ref{confcoor4d}) \cite{Castro:2010fd, Haco:2018ske, Aggarwal:2019iay, Perry:2020ndy}, and we will see more directly how to build them in Subsection \ref{subsec:Monodromy} regarding monodromy analysis\footnote{For instances of conformal coordinates in other contexts, see \cite{Maldacena:1998bw,Carlip:1995qv}.}. For now, we will just note that they are of the general form
\begin{equation}\label{confcoordgen}
\begin{split}
w^{+}&=f(r) e^{t_R},
\\
w^{-}&=f(r) e^{-t_L},
\\
y&=g(r)e^{(t_R-t_L)/2},
\end{split}
\end{equation}
and we will return to $(t_L,t_R)$ later. It is important to note that plugging the coordinate transformation (\ref{confcoor4d}) into the Kerr metric (\ref{kerrMet}) does not reproduce the Poincar\'e patch metric (\ref{AdSPoincare}) exactly. Rather, near the bifurcation surface $w^{\pm}=0$, the Kerr metric becomes
\begin{equation}\label{metexpand}
ds^2=\frac{4\rho^2_+}{y^2}dw^+dw^-+\frac{16J^2\sin^2\theta_0}{y^2\rho^2_+}dy^2+\rho_+^2d\theta^2+\dots\,,
\end{equation}
where
\begin{equation}
\rho_+^2=r_+^2+a^2\cos^2\theta\,,
\end{equation}
and the terms in the ellipsis ``$\dots$'' in equation \eqref{metexpand} are at least linear order in the coordinates $w^{\pm}$. The existence of these higher order terms underscores the fact that the hidden conformal symmetry generators (\ref{Hs}) are not isometries of the Kerr metric \eqref{kerrMet}.
For clarity, expressions of the generators (\ref{Hs}) in Boyer-Lindquist coordinates are~\cite{Castro:2010fd}
\begin{equation}\label{BLgens}
\begin{split}
H_1&=ie^{-2\pi T_R\phi}\left(\Delta^{1/2}\partial_r+\frac{1}{2\pi T_R}\frac{r-M}{\Delta^{1/2}}\partial_{\phi}+\frac{2T_L}{T_R}\frac{Mr-a^2}{\Delta^{1/2}}\partial_t\right),\\
H_0&=\frac{i}{2\pi T_R}\partial_{\phi}+2iM\frac{T_L}{T_R}\partial_t,\\
H_{-1}&=ie^{2\pi T_R\phi}\left(-\Delta^{1/2}\partial_r+\frac{1}{2\pi T_R}\frac{r-M}{\Delta^{1/2}}\partial_{\phi}+\frac{2T_L}{T_R}\frac{Mr-a^2}{\Delta^{1/2}}\partial_t\right),\\
\end{split}
\end{equation}
with similar expressions for the $\bar{H}$ sector. In these coordinates, the quadratic Casimir (\ref{casimir}) becomes exactly the near-region radial Klein-Gordon operator in (\ref{KGlimit}), so that
\begin{equation}\label{HPhiKPhi}
\mathcal{H}^2\Phi=K\Phi.
\end{equation}
\subsection{Monodromy Method}
\label{subsec:Monodromy}
We will now see that it is possible to find the generators (\ref{BLgens}) without explicitly taking a ``near-region'' limit, as in~\cite{Castro:2010fd}. This subsection will largely follow~\cite{Aggarwal:2019iay}, with some new discussion.
Let's again consider the Klein-Gordon equation (\ref{KleinGordon}). This differential equation has two regular singular points at the horizons $r_{\pm}$, and one irregular singular point at infinity.%
%
\footnote{Our discussion will only require the regular singular points. For treatment of the irregular singular point see~\cite{Castro:2013kea, Castro:2013lba}.}
%
Each singular point causes a branch cut, and we are interested in studying the radial solutions $R(r)$ of (\ref{KleinGordon}) (now promoted to complex-valued functions) when we go around each of the regular singular points. In general, the solutions $R(r)$ will develop a monodromy around these singular points. To study this, we posit that $R(r)$ has a series solution of the form
\begin{equation}\label{seriessol}
R(r)=(r-r_i)^{\beta}\sum_{n=0}^{\infty}q_n(r-r_i)^n.
\end{equation}
Our immediate objective is to determine the monodromy parameter $\beta\equiv i\alpha$ using the Frobenius method. We will go through this in more detail for our more complicated higher derivative case in Section \ref{sec:Hidden-symm-HD}. For the case at hand, we just state the answer and refer the reader to~\cite{Castro:2013kea} for details. The monodromy parameters around the inner and outer horizons are
\begin{equation}\label{alphas}
\alpha_{\pm}=\frac{\omega-\Omega_{\pm}m}{2\kappa_{\pm}},
\end{equation}
where $\Omega_\pm$ and $\kappa_\pm$ where defined in \eqref{omegas-kappas}.
Next, a crucial step in obtaining the generators (\ref{BLgens}) is to implement a change of basis
\begin{equation}\label{expbasis}
e^{i(m\phi-\omega t)}=e^{-i(\omega_Lt_L+\omega_Rt_R)}.
\end{equation}
The choice employed by~\cite{Aggarwal:2019iay} is
\begin{equation}\label{omegabasis}
\omega_L=\alpha_+-\alpha_-, \qquad \omega_R=\alpha_++\alpha_-.
\end{equation}
The particular change of basis (\ref{omegabasis}) is not well-motivated in the literature, and is often taken as a purely mathematical step to match the results of~\cite{Castro:2010fd}. To determine whether this is the appropriate basis choice to use in our more complicated higher derivative analysis in Section \ref{sec:Hidden-symm-HD}, a deeper physical understanding of this choice is needed, which we outline here\footnote{Key elements of this discussion were also presented in \cite{Castro:2013kea}.}.
Let's consider how the radial solutions $R(r)$ change as we go around the singular point $r_+$. Near $r=r_+$, our radial solutions are of the form
\begin{equation}
R(r)=(r-r_+)^{\pm i\alpha_+}\left(1+\mathcal O(r-r_+)\right)\,.
\end{equation}
When we go around the singular point, $r-r_+\rightarrow e^{2\pi i}(r-r_+)$, we see that
\begin{equation}
R(r)\rightarrow R(r)e^{\mp 2\pi\alpha_+}\,.
\end{equation}
As explained in \cite{Castro:2013kea}, if we go around the singular point $r_+$ twice, i.e. $r-r_+\rightarrow e^{4\pi i}(r-r_+)$, we expect the wave equation $\Phi=e^{-i\omega t+im\phi}R(r)$ to be invariant.
The radial piece of the outgoing solution $R(r)=(r-r_+)^{i\alpha_+}$ picks up the factor $e^{-4\pi\alpha_+}$. Plugging in the value for $\alpha_+$ in (\ref{alphas}), we find that $t$ and $\phi$ must transform in such a way to cancel this factor:
\begin{equation}\label{transform}
(t,\phi)\sim(t,\phi)+\frac{2\pi i}{\kappa_+}(1,\Omega_+).
\end{equation}
We will now see that the basis choice (\ref{omegabasis}) arises from determining the appropriate conjugate variables that will lead to the more natural thermal and angular transformation properties. For example, around $r=r_+$ we would like to replace (\ref{transform}) with
\begin{equation}\label{betterTransf}
(X,Y)\sim(X,Y)+2\pi i(1,1).
\end{equation}
The authors of \cite{Castro:2013kea} presciently rename $(X, Y)$ as $(t_L, t_R)$.
%
Similar arguments for the singular point $r_-$ lead us to the transformation properties
\begin{equation}\label{betterTransf-minus}
(X,Y)\sim(X,Y)+2\pi i(-1,1).
\end{equation}
For the wavefunction $\Phi=e^{-i\omega_Lt_L-i\omega_R t_R}R(r)$, we find by using (\ref{betterTransf}) and \eqref{betterTransf-minus},
the functions $(t_L, t_R)$ that accomplish these identifications are
\begin{equation}\label{tpm}
t_R=2\pi T_R\phi\,, \qquad t_L=\frac{1}{2M}t-2\pi T_L\phi\,.
\end{equation}
With $(t_L, t_R)$ in (\ref{tpm}), we can now immediately reproduce the zero-mode generators of~\cite{Castro:2010fd}. They are
\begin{equation}
H_0=\frac{i}{2\pi T_R}\partial_{\phi}+2iM\frac{T_L}{T_R}\partial_t=i\partial_{t_R}, \qquad \bar{H}_0=-2iM\partial_t=-i\partial_{t_L}.
\end{equation}
Notice that in this discussion we avoided making the seemingly arbitrary basis choice (\ref{omegabasis}). Instead, we see that $(\omega_L,\omega_R)$ are fixed by (\ref{tpm}) and (\ref{expbasis}).
It might appear that the monodromy analysis only determines the zero-mode generators $(H_0,\bar{H}_0)$ and not $(H_{\pm 1},\bar{H}_{\pm 1})$, but we can actually go further. From equation \eqref{confcoordgen}, we see that $(t_L, t_R)$ also fix the conformal coordinates (up to a radial factors, which can be recovered from the radial behavior of the hypergeometric solutions of \eqref{KGlimit} or from the Klein-Gordon operator itself, which we argue in Section \ref{sec:Hidden-symm-HD} and Appendix \ref{app:KGgoodform}). Once we have the proper conformal coordinates, all of the $H$s are determined by \eqref{Hs}.
\section{A standard form for the Klein-Gordon operator}\label{sec:standardform}
Before we move on to our higher derivative model, it is useful for us to express the Klein-Gordon operator in a standard form. In addition to streamlining our analysis, this form highlights interesting physical structure of the operator, that persists in both higher dimensional and higher spin settings.
We begin by writing the Klein-Gordon operator on the Kerr background in the following way:
\begin{equation}
\begin{split}
\nabla^{\mu}\nabla_{\mu}&=\frac{1}{\rho^2}\left[\partial_r(\Delta\partial_r)-\frac{(r_+-r_-)}{(r-r_+)}\left(\frac{\partial_t+\Omega_+\partial_{\phi}}{2\kappa_+}\right)^2+\frac{(r_+-r_-)}{(r-r_-)}\left(\frac{\partial_t+\Omega_-\partial_{\phi}}{2\kappa_-}\right)^2\right.\\
&\left.+\frac{1}{\sin^2\theta}\partial_{\phi}^2-(a^2\cos^2\theta+4M^2)\partial_t^2-(r^2+2Mr)\partial_t^2+\frac{1}{\sin^2\theta}\partial_{\theta}\left(\sin^2\theta\partial_{\theta}\right)\right].
\end{split}
\end{equation}
This form has several useful features for our analysis. First, if we posit the standard solution $\Phi=e^{i(m\phi-\omega t)}R(r)S(\theta)$, we see that the term
\begin{equation}\label{CarterOp}
\left(\frac{1}{\sin^2\theta}\partial_{\phi}^2-(a^2\cos^2\theta+4M^2)\partial_t^2\right)\Phi=C\Phi
\end{equation}
produces a version of Carter's constant $C$ \cite{carter1968global}. This means that, if we consider a constant $\theta$ slice $\theta=\theta_0$, the only dependence on our choice of slice will be in the prefactor $\rho^{-2}$ (recall that $\rho^2=r^2+a^2\cos^2\theta$). The presence of this factor means that our higher order equation of motion $\left( \nabla^{\mu}\nabla_{\mu}\right)^n\Phi=0$ appears not to be separable. Nevertheless, we will see in Section \ref{sec:Hidden-symm-HD} that at leading order near $r=r_\pm$, all dependence on $\rho^2$ (and thus $\theta_0$) will drop out. Thus we consider a constant $\theta$ slice $\theta=\theta_0$, allowing us to study the ``radial'' operator
\begin{equation}\label{radialKG}
\begin{split}
\nabla^{\mu}\nabla_{\mu}&=\frac{1}{\rho_0^2}\left[\partial_r(\Delta\partial_r)-\frac{(r_+-r_-)}{(r-r_+)}\left(\frac{\partial_t+\Omega_+\partial_{\phi}}{2\kappa_+}\right)^2+\frac{(r_+-r_-)}{(r-r_-)}\left(\frac{\partial_t+\Omega_-\partial_{\phi}}{2\kappa_-}\right)^2\right.\\
&\left.C_{t\phi}-(r^2+2Mr)\partial_t^2\vphantom{\frac{1}{1}}\right],
\end{split}
\end{equation}
where $\rho_0^2=r^2+a^2\cos^2\theta_0$ and we have called the operator in \eqref{CarterOp} $C_{t\phi}$ for convenience.
The form of the operator \eqref{radialKG} has a further use. The terms
\begin{equation}
\frac{\partial_t+\Omega_\pm\partial_{\phi}}{2\kappa_\pm}
\end{equation} give the monodromy parameters \eqref{alphas} introduced in Section \ref{subsec:Monodromy}, that is,
\begin{equation}
\left(\frac{\partial_t+\Omega_\pm\partial_{\phi}}{2\kappa_\pm}\right)\Phi=-i\alpha_{\pm}\Phi\,.
\end{equation}
In addition, the Killing vector fields
\begin{equation}
\xi_\pm=\kappa_{\pm}\left(\partial_t+\Omega_\pm\partial_{\phi}\right)
\end{equation}
are exactly those that vanish on the inner and outer horizons $r_{\pm}$. As such, they are the same vector fields that appear in Wald's formulation \cite{Wald:1993nt} of black hole entropy as the integrated Noether charge associated with the Killing vectors vanishing on the horizons. This point was discussed in \cite{Castro:2013kea}.
Further, the radial factors $\left(\frac{r_+-r_-}{r-r_\pm}\right)$ can be directly related to the conformal coordinates as defined in \eqref{confcoordgen}, and we can finally write the Klein-Gordon operator \eqref{radialKG} acting on $\Phi$ as
\begin{equation}\label{radialKG-Phi}
\nabla^{\mu}\nabla_{\mu}\Phi=\frac{\Phi}{\rho^2R(r)} \left[\partial_r(\Delta\partial_r)+\alpha_+^2\frac{g^2(r)}{f^2(r)}-\alpha_-^2g^2(r)+(r^2+2Mr)\omega^2+C_{t\phi}\right]R(r).
\end{equation}
This form of the Klein-Gordon operator holds for higher spin fields \textit{and} in higher dimensions, even though the forms of $\alpha_{\pm}$ and $f(r)$ and $g(r)$ change. This is discussed in Appendix \ref{app:KGgoodform}. The only terms that do change for higher spin/higher dimension are the non-singular terms
\begin{equation}
(r^2+2Mr)\omega^2+C_{t\phi},
\end{equation}
which are precisely those that are dropped in the near-region limit of Section \ref{sec:near-region-lim}.
\section{Hidden conformal symmetry in higher derivative dynamics}
\label{sec:Hidden-symm-HD}
To study how hidden conformal symmetry is manifest in a theory with higher derivative dynamics, we consider the following action for a massless scalar field on a Kerr black hole background
\begin{equation}\label{action}
S=-\int d^{4}x\sqrt{g}\left(\frac{1}{2}\Phi\left(\nabla^{\mu}\nabla_{\mu}\right)^n\Phi\right),
\end{equation}
where $n$ is an integer. The equation of motion resulting from this action is
\begin{equation}\label{genreq}
\left( \nabla^{\mu}\nabla_{\mu}\right)^n\Phi=0,
\end{equation}
where $n=1$ is the Klein-Gordon equation. Our motivation for choosing this action is twofold. First, the equations of motion are simple enough as to provide a straightforward extension to previous results with $n=1$ obtained by \cite{Castro:2010fd} and \cite{Aggarwal:2019iay}. Though simple, we will see that \eqref{genreq} already provides interesting complications that give insight into whether the choice of dynamics affects hidden conformal symmetry. Our second motivation is that the action \eqref{action} is of physical interest, since known examples of holographic duals to logarithmic conformal field theories contain higher derivative equations of motion \cite{Hogervorst:2016itc}. Thus, \eqref{action} provides us with the opportunity to both study the effect of changing the dynamics on hidden conformal symmetry, while also potentially diagnosing a new instance of a logCFT correspondence.
The differential equation \eqref{genreq} can be reformulated in two ways, one of which will be of particular use to us. The aim of both approaches is to reduce the system to a series of second order equations. For example, as was discussed in \cite{Bergshoeff:2012sc}, the equation of motion \eqref{genreq} can be broken up into coupled second order equations by introducing $n-1$ auxiliary scalar fields
\begin{equation}\label{auxfields}
\begin{split}
&\nabla_\mu\nabla^\mu\Phi_1=0,\\
&\nabla_\mu\nabla^\mu\Phi_i=\Phi_{i-1}, \qquad \text{for}~ i=2, ...,n.\\
\end{split}
\end{equation}
The related and more useful alternative to this approach is to repackage the auxiliary scalar fields as higher spin objects. That is, the problem \eqref{genreq} can be expressed as
\begin{equation}\label{HigherSpinEq}
\begin{split}
& \nabla^{\mu}\nabla_{\mu}\Phi=0,\\
& \nabla^{\mu}\nabla_{\mu}\Phi_{\mu_1\mu_2}=0, \\
& \nabla^{\mu}\nabla_{\mu}\Phi_{\mu_1\mu_2\mu_3\mu_4}=0, \\
&\vdots\\
&
\nabla^{\mu}\nabla_{\mu}\Phi_{\mu_1...\mu_{2n-2}}=0.
\end{split}
\end{equation}
In the above expressions we have defined defined the higher spin fields as
\begin{equation}\label{HigherSpinDefs}
\begin{split}
&\Phi_{\mu_1\mu_2}\equiv\nabla_{\mu_1}\nabla_{\mu_2}\Phi,\\ &\Phi_{\mu_1\mu_2\mu_3\mu_4}\equiv\nabla_{\mu_1}\nabla_{\mu_2}\nabla_{\mu_3}\nabla_{\mu_4}\Phi, \\
& \vdots\\
&\Phi_{\mu_1\mu_2...\mu_{2n-2}}\equiv\nabla_{\mu_1}\nabla_{\mu_2}...\nabla_{\mu_{2n-2}}\Phi.\\
\end{split}
\end{equation}
We will return to the significance of this reformulation later in Section \ref{sec:modelinads}.
We begin this section with a monodromy analysis of higher derivative dynamics on a Kerr background in Section \ref{sec:monodanal}. As mentioned before, one initial motivation for studying higher derivative dynamics is to potentially identify a new instance of a logCFT correspondence through hidden conformal symmetry, in the spirit of the Kerr/CFT correspondence away from extremality. We study this question in Section \ref{sec:modelinads}, which we begin with a short review of how the higher derivative model \eqref{action} is used in AdS/logCFT, followed by a study of whether this is a viable model with which to build a Kerr/logCFT correspondence.
\subsection{Monodromy analysis}\label{sec:monodanal}
In this section we analyze hidden conformal symmetry in our higher derivative theories in the spirit of Section \ref{subsec:Monodromy}. We begin our analysis with $n=2$ in the equation of motion \eqref{genreq}. We will then treat $n=3$, and construct a clear pattern for the monodromy parameters $\alpha_{\pm}$ for general $n$.
We immediately encounter the would-be issue that for $n>1$ the equation of motion \eqref{genreq} appears not to be separable. Perhaps intriguingly, this turns out not to matter at leading order near $r=r_\pm$. That is, in what follows, we take a constant slice $\theta=\theta_0$, and our results for $\alpha_\pm$ do not depend on the choice of $\theta_0$.
\subsubsection{Case $n=2$}
Since we are free to take a constant $\theta$ slice, we can focus on the behavior of a radial differential equation near its singular points. In particular, we can write the radial equation in standard form near a singular point $r=r_i$:
\begin{equation}\label{fourthdiffeq}
(r-r_i)^4R^{(4)}+D(r)(r-r_i)^3R^{(3)}+C(r)(r-r_i)^2R''+B(r)(r-r_i)R'+A(r)R=0.
\end{equation}
The Frobenius method instructs us to look for series solutions of the form
\begin{equation}\label{series}
\begin{split}
R(r)&=(r-r_i)^{\beta}\sum_{k=0}^{\infty}q_k(r-r_i)^k,
\end{split}
\end{equation}
and coefficient functions expanded as
\begin{equation}\label{coefseries}
\begin{split}
D(r)&=\sum_{k=0}^{\infty}d_k(r-r_i)^k,\qquad C(r)=\sum_{k=0}^{\infty}c_k(r-r_i)^k,\\ B(r)&=\sum_{k=0}^{\infty}b_k(r-r_i)^k, \qquad A(r)=\sum_{k=0}^{\infty}a_k(r-r_i)^k.
\end{split}
\end{equation}
In order for (\ref{fourthdiffeq}) to be satisfied, the coefficient of each power of $r-r_i$ must equal zero. In particular, the coefficient of the $(r-r_i)^{\beta}$ term gives us the fourth order indicial equation
\begin{equation}
\beta(\beta-1)(\beta-2)(\beta-3)+\beta(\beta-1)(\beta-2)d_0+\beta(\beta-1)c_0+\beta b_0+a_0=0.
\end{equation}
Without loss of generality, we first study the analytic structure around $r=r_+$. The zeroth order coefficients of our series expansions (\ref{series}) are
\begin{equation}
\begin{split}
d_0=4,\qquad c_0=2(1+\alpha^2_+),\qquad b_0=0, \qquad a_0=\alpha^2_+(1+\alpha^2_+),
\end{split}
\end{equation}
where $\alpha_+$ denotes the monodromy parameter for a scalar field on a Kerr blackground near the outer horizon, as defined in \eqref{alphas}. Plugging these values back into our indicial equation, we get
\begin{equation}
(\beta^2+\alpha^2_+)((\beta-1)^2+\alpha^2_+)=0.
\end{equation}
Thus, we obtain
\begin{equation}\label{betanequals2}
\beta=\left\{\pm i\alpha_+,~1\pm i\alpha_+\right\},
\end{equation}
where $\alpha_+$ is given by the expression \eqref{alphas},
or, since $\beta\equiv i\alpha$,
\begin{equation}\label{alphapn2}
\alpha^{n=2}_+=\left\{\pm\alpha_+,~-i\pm\alpha_+\right\}.
\end{equation}
There is a similar result for $\alpha^{n=2}_-$:
\begin{equation}\label{alphaminn2}
\alpha^{n=2}_-=\left\{\pm\alpha_-,~-i\pm\alpha_-\right\}\,,
\end{equation}
where again $\alpha_-$ can be read in equation \eqref{alphas}.
At this point there are several things to point out regarding \eqref{betanequals2}-\eqref{alphaminn2}. First we can see why it is useful to reformulate our equation of motion \eqref{genreq} for $n=2$ as two coupled equations with a higher spin field, as in \eqref{HigherSpinEq}:
\begin{equation}\label{HigherSpinn2}
\begin{split}
& \nabla^{\mu}\nabla_{\mu}\Phi=0,\\
& \nabla^{\mu}\nabla_{\mu}\Phi_{\mu_1\mu_2}=0. \\
\end{split}
\end{equation}
These equations are coupled in the sense that $\Phi_{\mu_1\mu_2}$ is built from $\Phi$ as in \eqref{HigherSpinDefs}. The four monodromy parameters $\alpha^{n=2}_+$ associated with our fourth-order equation near the outer horizon $r_+$ are exactly those that were found when analyzing the second-order Klein-Gordon equation for a scalar field ($\alpha_\pm$, see equation \eqref{alphas}) and for a spin-2 field ($-i\pm\alpha_\pm$, see equation \eqref{alphaspins}).
Second, from \eqref{betanequals2} we see that two of the exponents $\beta$ differ from two others by a positive integer. The Frobenius method tells us that of the four linearly independent solutions around each singular point, two of them \textit{might be} log solutions. For example, near $r=r_+$ we have
\begin{equation}
\begin{split}
R(r)&=(r-r_+)^{1\pm i\alpha_+}\Phi_1^{\pm}(r), \\
R(r)&=a_{\pm}(r-r_+)^{1\pm i\alpha_+}\Phi_1^{\pm}(r)\log (r-r_+)+(r-r_+)^{\pm i\alpha_+}\Phi_2^{\pm}(r),
\end{split}
\end{equation}
where $a_{\pm}$ are constants which can be zero or not. This could signal that, if there is indeed a CFT description of this system, it could be a logCFT. However, it is important to note that there is a subtle difference between the log terms that appear here for Kerr and those which appear in the context of logCFTs dual to an AdS background, as discussed in Section \ref{sec:modelinads}. In the Frobenius method, when two roots are repeated (as in the AdS analysis, see equation \eqref{multiplicity}), a logarithmic part of the solution is guaranteed. In contrast, it is a theorem that when two roots differ by a positive integer (as in our case) the coefficients $a_{\pm}$ could be zero, see {\it e.g.}~\cite{Coddington1955TheoryOO}. This depends on the specific and intricate nature of the given differential equation.
Our principal goal is now to study if and how hidden conformal symmetry is manifest in our higher derivative dynamics $\left(\nabla_\mu\nabla^\mu\right)^2\Phi=0$. There are several ways to approach this problem. In the usual scenario, that is $\nabla_\mu\nabla^\mu\Phi=0$, we try to find $\grp{SL}(2,\mathbb R)$ generators $(H_0,H_{\pm 1})$ as in~\eqref{BLgens} that 1) satisfy the commutation relations \eqref{commutationRels} and 2) form a Casimir that reproduces the near-region Laplacian $\mathcal{H}^2\Phi=K\Phi$, as in \eqref{HPhiKPhi}.
Trying to find equivalent structure in the equation $\left(\nabla_\mu\nabla^\mu\right)^2\Phi=0$ directly suffers from conceptual issues, as it appears the role of a quadratic Casimir $\mathcal{H}^2$ would perhaps have to be replaced by a quartic Casimir $\mathcal{H}^4$.
However, if we take the equivalent description of our system \eqref{HigherSpinn2}, we will see that hidden conformal symmetry is still visible, and presents itself in a natural way.
Since \eqref{HigherSpinn2} is an equivalent description of our fourth order equation, we can analyze each equation in \eqref{HigherSpinn2} separately. The first equation, $\nabla^{\mu}\nabla_{\mu}\Phi=0$, is of course just the standard case that was already treated in \cite{Aggarwal:2019iay}. Now we turn to the spin-2 equation. For the reader's convenience, we reproduce this equation here from our Appendix \ref{app:KGgoodform}:
\begin{equation}
\left(\partial_r\Delta\partial_r+\alpha^2_{+, s=2}\frac{g^2(r)}{f^2(r)}-\alpha^2_{-, s=2}g^2(r)+\omega^2r^2+2(M\omega+2i)\omega r+C_{t,\phi}\right)R(r)=0,
\end{equation}
where $s$ is the spin of the auxiliary field. Again, the constant $C_{t,\phi}$ we can think of as being absorbed in a separation constant, and the terms $\omega^2r^2+2(M\omega+2i)\omega r$ can be dropped in the near-region limit. Thus the solutions to
\begin{equation}
\left(\partial_r\Delta\partial_r+\alpha^2_{+, s=2}\frac{g^2(r)}{f^2(r)}-\alpha^2_{-, s=2}g^2(r)\right)R(r)=0
\end{equation}
are also hypergeometric functions, hinting at hidden conformal symmetry.
In our review Section \ref{sec:Hidden-symm-KG} we introduced several important and interrelated quantities: monodromy exponents $\beta\equiv i\alpha$~\eqref{alphas}, the change of basis modes $(\omega_L, \omega_R)$~\eqref{omegabasis}, their conjugate variables $(t_L, t_R)$~\eqref{tpm}, the conformal coordinates $(w^\pm,y)$~\eqref{confcoor4d}, and the $\grp{SL}(2,\mathbb R)$ generators $(H_0,H_{\pm 1})$~\eqref{BLgens}. We now ask the question: which of these quantities, if any, need to be modified from their $n=1$ values so that we can still obtain the conditions for diagnosing hidden conformal symmetry? Notice that this is equivalent to finding generators that satisfy the commutation relations \eqref{commutationRels} whose Casimir reproduces the near-region Klein-Gordon operator $\mathcal{H}^2_{n=2}\Phi_{\mu_1\mu_2}=K\Phi_{\mu_1\mu_2}$, as in \eqref{HPhiKPhi}. We claim that this is accomplished by modifying only one thing: the change of basis choice $(\omega_L, \omega_R)$. Let's see how this works by discussing each of the above quantities in turn.
First, we claim that $(t_L, t_R)$ do not change from their $n=1$ values given in~\eqref{tpm}, since these were obtained by thermal considerations in our review Section \ref{sec:Hidden-symm-KG}. The conformal coordinates $(w^\pm,y)$ should also not change from \eqref{confcoor4d} and \eqref{confcoordgen}, since these are just purely geometric relations taking the Kerr background to the upper-half plane (to leading order near the bifurcation surface). Finally, the generators are built directly from the the conformal coordinates via \eqref{Hs}, so these should also remain unchanged from their $n=1$ values. This only leaves two quantities: the monodromy parameters $\alpha$, which certainly \textit{do} change, and the basis choice $(\omega_L, \omega_R)$, which must change also to account for the change in the $\alpha$s. As mentioned above and in Appendix \ref{app:KGgoodform}, the new $\alpha$s for the spin-2 equation are
\begin{equation}
\alpha^{s=2}_\pm=i\pm\alpha^{s=0}_\pm,
\end{equation}
and the basis choice to accommodate this is modified from \eqref{omegabasis} to
\begin{equation}\label{newbasischoice}
\omega_L=\alpha_+-\alpha_--2i, \qquad \omega_R=\alpha_++\alpha_-.
\end{equation}
This means that we consider the frequencies $\omega\in\mathbb{C}$.
It is not illuminating to write the full equation $\left(\nabla_\mu\nabla^\mu\right)^2\Phi=0$ in a standard form (as in Section \ref{sec:standardform}) except to point out one thing. Taking a constant $\theta$ slice, the fourth order equation of motion is of the form:
\begin{equation}\label{standardhighern}
\mathcal{D}\left[R(r)\right]+\left[\alpha_+^2\left(1+\alpha_+^2\right)\frac{g^4(r)}{f^4(r)}+\alpha_-^2\left(1+\alpha_-^2\right)g^4(r)+n.s.\right]R(r)=0,
\end{equation}
where $\mathcal{D}\left[R(r)\right]$ stands for all terms involving a derivative of $R(r)$, $n.s.$ represents nonsingular terms, and the radial functions $f$ and $g$ are defined as in \eqref{confcoor4d} and \eqref{confcoordgen}. From equation \eqref{standardhighern} we learn more about the standard form discussed in Section \ref{sec:standardform}: the coefficients of the radial functions are just the Frobenius exponents $\beta$ (see equation \eqref{betanequals2}).
\subsubsection{Case $n=3$ and higher $n$}
We now sketch the monodromy calculation for the equation of motion $\left(\nabla^\mu\nabla_\mu\right)^n\Phi=0$ with $n=3$, and establish a pattern for general $n$.
Just as in the $n=2$ case, our analysis does not depend upon our choice of constant $\theta$ slice. We are thus free to consider the sixth order radial equation in standard form
\begin{equation}
\begin{split}
&(r-r_i)^6R^{(6)}+F(r)(r-r_i)^5R^{(5)}+E(r)(r-r_i)^4R^{(4)}+\\&D(r)(r-r_i)^3R^{(3)}
+C(r)(r-r_i)^2R''+B(r)(r-r_i)R'+A(r)R=0.
\end{split}
\end{equation}
For concreteness, we again choose to study $r_i=r_+$. Upon positing a series solution for $R(r)$ and the coefficient functions (as in \eqref{series} and \eqref{coefseries}), we find the indicial equation
\begin{equation}
(\beta^2+\alpha_+^2)((\beta-1)^2+\alpha_+^2)((\beta-2)^2+\alpha_+^2)=0.
\end{equation}
So all together our indicial equations close to $r=r_+$ for $\left(\nabla_\mu\nabla^\mu\right)\Phi=0$, $\left(\nabla_\mu\nabla^\mu\right)^2\Phi=0$ and $\left(\nabla_\mu\nabla^\mu\right)^3\Phi=0$ are
\begin{equation}\label{allindicials}
\begin{split}
&(\beta^2+\alpha_+^2)=0,\\
&(\beta^2+\alpha_+^2)((\beta-1)^2+\alpha_+^2)=0,\\
&(\beta^2+\alpha_+^2)((\beta-1)^2+\alpha_+^2)((\beta-2)^2+\alpha_+^2)=0,
\end{split}
\end{equation}
respectively.
Equation \eqref{allindicials} suggests the monodromy structure of $\left(\nabla^\mu\nabla_\mu\right)^n\Phi=0$ on the Kerr background for general $n$:
\begin{equation}
\prod_{j=1}^{n}((\beta-j+1)^2+\alpha_+^2)=0.
\end{equation}
\subsection{Holographic correspondences with higher derivative dynamics}\label{sec:HigherDHolography}
\subsubsection{AdS/logCFT}\label{sec:modelinads}
Here we begin with a brief review of how higher derivative dynamics in AdS are dual to logCFTs, in order to contrast what happens when we consider these dynamics on a Kerr background in the next subsection. Holographic logCFTs have been discussed since the early days of AdS/CFT~\cite{Ghezelbash:1998rj,Kogan:1999fo}. There has been much progress in this direction, see {\it e.g.} \cite{Hogervorst:2016itc}, and here we only report the main lesson from it: logCFTs are holographically realised as higher derivative theories in AdS$_{d+1}$ spacetimes. In particular, for scalar fields with mass $\mu$ the action is
\begin{equation}\label{action-laplacian-n}
S=-\frac 12 \int \dif^{\, d+1}x \sqrt g\, \phi\left(\nabla_\mu\nabla^\mu-\mu^2\right)^n\phi,
\end{equation}
where $n\ge 2$ corresponds to the rank $n$ of the dual logCFT.
The equation of motion is then a $2n$-th order differential equation, {\it i.e.}
\begin{equation}\label{eom-laplacian-n}
\left(\nabla_\mu\nabla^\mu-\mu^2\right)^n\phi=0\,.
\end{equation}
The above action can also be formulated in terms of auxiliary fields, for example in the case $n=2$ we have
\begin{equation}\label{action-n2-aux}
S=-\frac 12 \int \dif^{\,d+1}x \sqrt g \left( g^{\mu\nu} \partial_\mu \phi_1 \partial_\nu \phi_2+ \mu^2\phi_1\phi_2+\frac 12 \phi_1^2\right)\,.
\end{equation}
The equation of motion for $\phi_1 $ and $\phi_2$ are respectively
\begin{equation}
\left(\nabla_\mu\nabla^\mu-\mu^2\right)\phi_2=\phi_1\,, \qquad
\left(\nabla_\mu\nabla^\mu-\mu^2\right)\phi_1=0\,. \\
\end{equation}
It is then clear that the $\phi_2$ has to satisfy a ``squared'' equation, that is $\left(\nabla_\mu\nabla^\mu-\mu^2\right)^2\phi_2=0$. The action \eqref{action-n2-aux} can be generalised to arbitrary rank $n$~\cite{Bergshoeff:2012bi}, and the corresponding equations of motion are given by
\begin{align}
& \left(\nabla_\mu\nabla^\mu-\mu^2\right)\phi_1=0\,,
\\ \nonumber
& \left(\nabla_\mu\nabla^\mu-\mu^2\right)\phi_{i}=\phi_{i-1}\,,\qquad i=2, \dots, n\,,
\end{align}
from which follows that the equation of motion for the $n$-th field is indeed \eqref{eom-laplacian-n}.
In terms of the auxiliary fields it is manifest a shift symmetry of the equations of motion and the on-shell action
\begin{equation}
\phi_i \to \phi_i +\sum_{p=1}^{i-1}\lambda_{p} \phi_p\,,
\end{equation}
for arbitrary constant $\lambda_{p}$.
Here we briefly illustrate the example of a higher rank wave equation for a scalar field in (Euclidean) AdS$_{d+1}$; see {\it e.g.} \cite{Hogervorst:2016itc} for a recent review. We assume the scalar field to be massless, and we use a Poincar\'e patch, that is
\begin{equation}
\label{ads-metric}
ds^2_{AdS}=\frac{\dif \zeta^2+\dif x_i \dif x^i}{\zeta^2}\,,
\end{equation}
where $i=1, \dots, d$.
We are looking at the equation \eqref{genreq} for the case $n=2$ and in the background \eqref{ads-metric}. Given the high degree of symmetry of the metric \eqref{ads-metric} the differential equation is particularly simple. We can take $d=2$ to make a direct comparison with the Kerr black hole metric \eqref{kerrMet} discussed in this work.
Then, the radial differential equation for the radial component $\psi(\zeta)$ of the scalar field is given by
\begin{equation}\label{radialODE-ads4}
\psi^{(4)}(\zeta)
+{2\over \zeta} \psi^{(3)}(\zeta)
-2\left( \mathcal{K}^2+\frac{1}{\zeta^2}\right) \psi^{(2)}(\zeta)
+\left(\frac{1}{\zeta^3}-{2 \mathcal{K}^2\over \zeta}\right)\,\psi^{(1)}(\zeta)
+\mathcal{K}^4 \psi (\zeta)=0\,,
\end{equation}
where $\mathcal K$ is a constant which depends upon the mode expansions along the directions $x^i$.
At the leading order in $\zeta$ (that is close to the AdS boundary) the radial differential equation \eqref{radialODE-ads4} becomes
\begin{equation}\label{radialODE-ads4-leading}
\psi^{(4)}(\zeta)
+\frac{2}{\zeta}\psi^{(3)}(\zeta)
-\frac{1}{\zeta^2} \psi ^{(2)}(\zeta)
+\frac{1}{\zeta^3}\,\psi^{(1)}(\zeta)=0\,.
\end{equation}
The main feature to notice here is the absence of a potential term at the leading order, in contrast with our case~\eqref{4th-kerr-leading-z0}.
Applying the Frobenius method, that is assuming $\psi(\zeta)=\zeta^\beta \sum_{n=0}^\infty c_n \zeta^n$, we obtain at the leading order the roots%
%
\footnote{These roots are nothing but the roots of the equation $\Delta(\Delta-d)=m^2$, for a scalar field with mass $m$ in AdS$_{d+1}$. Here we have set $m=0$, and use $\beta$ instead of $\Delta$ to be consistent with the notation used in this work.}
%
\begin{equation}\label{multiplicity}
\beta=0, d\,, \qquad \text{with multiplicity} ~2\,.
\end{equation}
The non-trivial multiplicity means that we obtain the following four linearly independent solutions
\begin{equation}\label{AdSlogsol}
\zeta^0,~~ \zeta^d, ~~\zeta^0 \log \zeta, ~~\zeta^d \log \zeta\,.
\end{equation}
At the next-to-leading order in small $\zeta$, the differential equation becomes
\begin{equation}\label{radialODE-ads4-nexttol}
\psi^{(4)}(\zeta)
+\frac{2}{\zeta}\psi^{(3)}(\zeta)
-2\left( \mathcal{K}^2+\frac{1}{\zeta^2}\right) \psi^{(2)} (\zeta)
+\left(\frac{1}{\zeta^3}-{2\mathcal{K}^2\over \zeta}\right)\,\psi^{(1)}(\zeta)
=0\,.
\end{equation}
The potential term is not present in the equation, but this is specific to the case $d=2$.
This equation can be solved analytically, and the four linearly independent solutions are Bessel functions of the first and second kind ($I_n, Y_n$), whose arguments depend on the dimensions of AdS and the constant $\mathcal K$, and logarithm. Again expanding these solutions around $\zeta=0$ the Bessel function $Y_n$ gives rise to another explicit logarithmic behavior close to the boundary, in agreement with the leading behaviour found in \eqref{AdSlogsol}.
With this in mind, we take inspiration from the action \eqref{action-laplacian-n} to start our investigation of higher derivative models in a Kerr black hole background and their hidden symmetries.
\subsubsection{Kerr}
We start this section by rewriting the radial Klein-Gordon equation in new coordinates, and then we examine the squared Klein-Gordon operator in this setting.
Defining
\begin{equation}\label{z-coords}
z=\frac{r-r_-}{r-r_+} \,,
\end{equation}
the radial equation \eqref{KleinGordon} becomes
\begin{equation}\label{KG-z}
\begin{aligned}
& z(1-z) f^{\prime\prime}(z)+(1-z)f^\prime(z)
\\
& -\left(\alpha_+^2-{\alpha_-^2\over z}+{K\over 1-z}-\left(\frac{(r_--r_+)^2}{(1-z)^3}+2{(r_--r_+)(M+r_+)\over (1-z)^2}+\frac{4 M^2+2 M r_++r_+^2}{1-z}\right)\omega^2\right)f(z)=0\,,
\end{aligned}
\end{equation}
where we have used the definitions \eqref{alphas} in the above equation.
The singular points of the original equation \eqref{KleinGordon}, namely $r=r_-$, $r=r_+$, and $r=\infty$ have been mapped to $z=0$, $z=\infty$ and $z=1$ respectively. In these coordinates \eqref{z-coords} the Frobenius analysis becomes more transparent, and it is clear from equation \eqref{KG-z} that, for example at the leading order, close to the regular singular point $z=0$ ($r=r_-$) the equation is simply
\begin{equation}
f^{\prime\prime}(z)+\frac 1 z f^\prime(z)+\frac{\alpha_-^2}{z^2} f(z)=0\,.
\end{equation}
The two linear independent solutions are then
\begin{equation}
z^{i\alpha_-}, \qquad z^{-i\alpha_-}\,.
\end{equation}
Before moving to higher order differential equations, it is useful to examine the second order Klein-Gordon equation \eqref{KG-generics} for generic spin $s$ in this coordinate system.
\begin{equation}\label{KG-z-generals}
\begin{aligned}
& z(1-z) f^{\prime\prime}(z)+(1-z)f^\prime(z)
\\
& -\left((\alpha^s_+)^2-{(\alpha^s_-)^2\over z}+{K+s^2\over 1-z}
+2 i s \left({ M-r_+ \over 1-z}-{r_+-r_-\over (1-z)^2}\right) \omega\right.
\\
&
-\left.\left(\frac{(r_--r_+)^2}{(1-z)^3}+2{(r_--r_+)(M+r_+)\over (1-z)^2}+\frac{4 M^2+2 M r_++r_+^2}{1-z}\right)\omega^2\right)f(z)=0\,.
\end{aligned}
\end{equation}
Notice that the terms proportional to $\omega^2$ are unaffected by the spin $s$, while now there is a linear term proportional to $\omega$ and $s$. Again, expanding at leading order for example around $z=0$ ($r=r_-$), we obtain
\begin{equation}
f^{\prime\prime}(z)+\frac 1 z f^\prime(z)+\frac{(\alpha^s_-)^2}{z^2} f(z)=0\,,
\end{equation}
where $\alpha_-^s$ is defined in equation \eqref{alphaspins}. The two independent solutions to this equation are
\begin{equation}
z^{i\alpha_-^s}\,, \qquad z^{-i\alpha_-^s}\,.
\end{equation}
Similarly, we can consider the next-to-leading order expansion of the full equation, and we have
\begin{equation}\label{eq:nexttoleading}
\begin{aligned}
&f^{\prime\prime}(z)+\frac 1 z f^\prime(z)
\\
&+\left(\frac{(\alpha^s_-)^2}{z^2} -{K+s^2+(\alpha_+^s)^2-(\alpha_-^s)^2+2 i s (M-r_-)\omega-(4 M^2+2 M r_-+r_-^2)\omega^2\over z}\right)f(z)=0\,.
\end{aligned}
\end{equation}
Notice that $$
(\alpha_++\alpha_-)(\alpha_+-\alpha_--i s)=(\alpha_+^s)^2-(\alpha_-^s)^2\,.
$$
The solutions to equation \eqref{eq:nexttoleading} are modified Bessel function of the first kind $I_n(z)$.
Continuing the expansion of equation \eqref{KG-z-generals} at the next-to-next-to-leading order, when constant terms appear in the potential, we see that the solutions are hypergeometric functions. We should stress that these equations are only valid in a neighborhood of $z=0$, and so the solutions obtained in this way are not the full solution of the original equation \eqref{KG-z-generals}.
Let us now consider equation \eqref{genreq} for the case $n=2$. Our starting point are equations \eqref{radialKG} and \eqref{radialKG-Phi} (after acting on the field $\Phi$). Applying again the Klein-Gordon operator \eqref{radialKG} to equation \eqref{radialKG-Phi}, and changing coordinate system as in \eqref{z-coords}, we obtain a rather lengthy expression. We then choose a constant $\theta$ slice, since as discussed in Section \ref{sec:Hidden-symm-HD} this does not affect the monodromy data. Again the singular points are $z=0$, $z=\infty$ and $z=1$, and by performing a series expansion around the regular singular points $z=0$ ($r=r_-$), $z=\infty$ ($r=r_+$) we obtain the roots \eqref{alphaminn2} and \eqref{alphapn2} respectively.
We find instructive to examine the fourth-order differential equation obtained in this way, close to a regular singular point at the leading order. We can focus on $z=0$ ($r=r_-$) for simplicity. Then the differential equation is given by
\begin{equation}\label{4th-kerr-leading-z0}
\begin{aligned}
f^{(4)}(z)+{4\over z} f^{(3)}(z)+\frac{2(1+\alpha_-^2)}{z^2} f^{(2)}(z
+\frac{\alpha_-^2(1+\alpha^2_-)}{z^4}f(z)=0\,,
\end{aligned}
\end{equation}
and the four independent solutions are
\begin{equation}\label{highernkerralphas}
z^{1-i\alpha_-}\,,~ z^{1+i\alpha_-}\,,~ z^{-i\alpha_-}\,,~ z^{i\alpha_-}\,.
\end{equation}
At the next-to-leading order the structure of the equation is
\begin{equation}\label{ntl-order4-z}
\begin{aligned}
&f^{(4)}(z)+\left({4\over z}+A_3(\theta_0)\right) f^{(3)}(z)
+\left(\frac{2(1+\alpha_-^2)}{z^2}+\frac{A_2(\theta_0)}{z}\right) f^{(2)}(z)
+\frac{A_1(\theta_0)}{z^2} f^{(1)}(z)
\\ \noindent
&+\left(\frac{\alpha_-^2(1+\alpha^2_-)}{z^4}
+\frac{A_0(\theta_0)}{z^3}\right)f(z)=0\,.
\end{aligned}
\end{equation}
The constants $A_0, A_1, A_2, A_3$ depend on the black hole parameters, the dynamical inputs $K, \omega, m$, as well as the choice of the constant slice $\theta_0$, as our notation underlines. We refrain to write their explicit expression here, since it is not particularly useful.
The four linear independent solutions of equation \eqref{ntl-order4-z} are hypergeometric functions and Meijer G-functions, which again depend on the monodromy data and the constant $A$s.
We now need to compare and contrast the AdS case \eqref{multiplicity} and the Kerr case \eqref{highernkerralphas} in order to determine whether an analogous formulation of a Kerr/logCFT is possible. We know from Section \ref{sec:modelinads} that the fourth order radial equation in an AdS background admits logarithmic solutions \eqref{AdSlogsol}. In fact, the logarithmic behaviour is \textit{guaranteed} at the leading order close to the boundary by the degeneracy of the indicial roots, see \eqref{radialODE-ads4-leading}-\eqref{multiplicity}. However, the Kerr case is more subtle. The crucial difference between equations \eqref{radialODE-ads4-leading} and \eqref{4th-kerr-leading-z0} is the presence of a potential term in the Kerr black hole geometry.%
%
\footnote{We should stress that we are only discussing about formal similarities. In the Kerr black hole case we zoom in a region close to the horizon, while in the AdS case, we are interested in a boundary behaviour, where the CFT lives.}
We can see from equation \eqref{highernkerralphas} that two pairs of indicial roots differ by a positive integer. This signals that a logarithmic solution \textit{may or may not} be present. In the Kerr/CFT construction, the CFT exists at the black hole horizon, and so to determine whether the logarithmic terms are there in the region that interests us, we expand the Meijer G-functions close to $z=0$.
A general expansion of this function contains terms polynomial in $z$ and also terms like $z^2 \log z$. The corresponding coefficients are very lengthy expressions, which depend on the $A$ constants.
We remind the reader that these are not solutions of the full equation \eqref{KG-z}, their validity is within the validity of the expansion of the equation \eqref{ntl-order4-z} itself, and so if the solution shows a logarithmic term beyond its perturbative regime is not meaningful in this context. We have investigated the coefficient of the logarithmic term numerically, and interestingly, we were not able to find a non-zero coefficient near the black hole horizon.
This seems to indicate that a Kerr/logCFT construction is \textit{not} possible within this framework. We discuss physical interpretations of this result in the discussion section.
It is perhaps worth stressing that a logarithmic solution can still be present outside of our regime of validity, that is at order $z^2$. Again, the presence or not of a logarithmic term in general depends on the specific dynamics, in particular on the slice $\theta_0$ which we chose. We remind the reader that the full fourth-order equation \eqref{KG-z} is not separable, and we choose a specific $\theta$-slice to perform our analysis. This is in contrast to the monodromy data, which do not depend on this choice.
\section{Discussion}\label{sec:discussion}
The goals of this article were to provide a case study of how hidden conformal symmetry is manifest when we change the dynamics on a given background, and in particular whether we could use variations of the Kerr/CFT correspondence to work toward diagnosing a new instance of a logCFT correspondence. In this section we review our results, discuss the challenges with a logCFT construction, and discuss future directions.
We found the monodromy parameters $\alpha^{(n)}$ for general number of derivatives $2n$, and show how they are related to the monodromy exponents of the regular Klein-Gordon equation of higher spin fields. We show that pairs of the indicial roots $i\alpha^{(n)}$ differ by an integer, and thus a logarithmic contribution to the radial equation could be present. However, we find that sufficiently close to the black hole horizon, potential logarithmic contributions vanish. This seems to indicate that we cannot construct a Kerr/logCFT correspondence from higher derivative theories, which have been used to construct examples of AdS/logCFT correspondences.
The difficulty in constructing a Kerr/logCFT correspondence is interesting, both from the gravitational perspective and the field theory perspective. From the field theory perspective, logCFTs have proven to be relevant in numerous areas of physics. Indeed, they can arise at critical points of various physical systems, such as those describing quantum Hall plateau transition~\cite{Flohr:1995zj, Gurarie:1997dw, Cappelli:1998ma, Ino:1998xe}, but also in models describing percolation~\cite{Saleur:1991hk}, self-avoiding walks~\cite{Duplantier:1987sh}, and systems with quenched disorder~\cite{Cardy:1999zp, Caux:1995nm, Maassarani:1996jn}.
%
These special conformal field theories are characterised by a logarithmic behaviour in correlation functions~\cite{Saleur:1991hk, Rozansky:1992td, Gurarie:1993xq}, which seems to clash with the fact that the theory is scale invariant. However, the presence of these terms is hinged on the reducible but indecomposable representations of the conformal group~\cite{Gurarie:1993xq}. The crucial point is that the conformal Hamiltonian is not diagonalizable, but rather has a Jordan cell structure (for rank $n\ge 2$), which leads to logarithmic terms in the correlation functions, and to lack of unitarity.
%
While this feature would be generally considered a red flag in quantum field theory, it does not pose any threat as a description of statistical mechanical systems, as confirmed by the examples mentioned above. We refer the reader to {\it e.g.}~\cite{Hogervorst:2016itc, Grumiller:2013at, Cardy:2013gk}, and references therein, for more recent and extensive reviews on logCFTs.
From a gravitational perspective, a natural question is whether hidden conformal symmetries are still visible (or modified) when the dynamics is encoded in higher derivative differential operators, as those in the action \eqref{action-laplacian-n}. Higher derivative theories breaks unitarity, hence we might expect that this would be reflected somehow in the hidden symmetry group. Consequently, this might hinder us in our efforts to investigate/study a Cardy-like formula in this setting. Indeed, if we expect a non-unitary CFT the partition function might not be bounded from below, we might have states with negative norm, thus it is not clear in which sense we could discuss of an entropy. Still, it might be possible to give a description of the density of states, perhaps taking into account anomalies.
There are still further challenges with trying to make a logCFT correspondence in the spirit of Kerr/CFT using the model we propose. First, there is still some work to be done regarding making a robust holographic correspondence in Kerr/CFT itself. Even though there is ample evidence that the hidden conformal symmetry found in \cite{Castro:2010fd} really is described by an underlying CFT (such as the correct computations of scattering cross-sections), many elements are still lacking, such as how to conduct an asymptotic symmetry group analysis when the symmetry generators are not all isometries, and when the conformal symmetry acts at the horizon and not the boundary. We leave this interesting problem for future work. Also, as mentioned previously, in the case of a logCFT correspondence in AdS the scalar field is guaranteed to have a logarithmic piece near the boundary, but in Kerr the logarithmic piece might vanish near the horizon.
Lastly, we illustrate once again that the monodromy method is really a powerful tool for studying hidden conformal symmetry: it allows us to study near horizon dynamics without actually taking a near-region limit in the dynamics (although sometimes we do take such a limit, purely for calculational ease). Furthermore, even though the higher order equations $\left(\nabla^\mu\nabla_\mu\right)^2\Phi=0$ are no longer separable, it does not matter for the monodromy analysis. Our results are independent of the $\theta$ slice we choose.
There are several interesting opportunities for future work. One important contribution would be to establish a non-extremal analog to the asymptotic symmetry group analysis presented for example in \cite{Guica:2008mu}. This would further strengthen the claim of a non-extremal Kerr/CFT correspondence. Another direction is that this paper is a first step towards learning what the monodromy method can tell us other equations of motion. It would be interesting, for example, to study whether hidden conformal symmetry is somehow encoded in the Dirac or geodesic equations. Further, it was recently shown in \cite{Keeler:2021tqy} that there is a difficulty in constructing conformal coordinates in six spacetime dimensions and higher that do not have branch cuts, unless an explicit near-horizon limit is taken. Thus it would be interesting to check whether the general form on the Klein-Gordon operator discussed in Section \ref{sec:standardform} and Appendix \ref{app:KGgoodform} holds in higher dimensions. We leave this for future work.
An intriguing and open question is whether a generalisation of the Cardy-formula exists for non-unitary theories, particularly those where the underlying conformal field theory is a logarithmic one ~\cite{Grumiller:2013at}. Indeed, exploring this question was one of our initial motivations for this work. The Cardy formula has played a crucial role in the AdS/CFT duality~\cite{Maldacena:1997re, Witten:1998qj, Gubser:1998bc}, and in particular in the Kerr/CFT correspondence~\cite{Guica:2008mu}. The Bekenstein-Hawking entropy of asymptotically AdS$_3$ black hole exactly reproduces (at high energy) the degeneracy of states governed by the Cardy formula~\cite{Cardy:1986ie} in two-dimensional CFTs, that is
\begin{equation}\label{S-cardy}
S_{CFT}= 2\pi \sqrt{{c_R L_0\over 6}}+2\pi \sqrt{{c_L \bar L_0\over 6}}\,,
\end{equation}
where $c_R, c_L$ are the right and left central charges, respectively, and $L_0, \bar L_0$ are the zero-th generators of the Virasoro algebra.
It was later understood that any higher-dimensional black holes with an AdS$_3$ near-horizon geometry will obey a Cardy-formula, again as a consequence of the symmetry of the given geometry.
A step further was made in \cite{Detournay:2012fk} where the Cardy formula was extended to warped AdS$_3$ geometry, that is geometries with $\grp{SL}(2,\mathbb{R})\times \grp{U}(1)$ isometries.
In particular, these are the isometries of the near-horizon geometry of the extremal Kerr black holes (and in general of extremal/near extremal black holes).
%
Two essential ingredients enter in the derivation of the Cardy formula \eqref{S-cardy}: unitarity and modular invariance, and so the presence of a Cardy formula, even with a reduced symmetry group as in the warped case, tells us that there is still a notion of modular invariance here. We leave the establishment of a Cardy-like formula in non-unitary settings for future work.
\section*{Acknowledgements}
We are indebted to Cynthia Keeler and Rahul Poddar.
We thank Prof. Norma Sanchez for inviting us to contribute with this article to the Open Access Special Issue “Women Physicists in Astrophysics, Cosmology and Particle Physics”, published in [Universe] (ISSN 2218-1997) and to M. Grana, Y. Lozano, S. Penati and M. Taylor for involving us in this special issue.
This research was supported in part by the Icelandic Research Fund under contract 195970-052 and by grants from the University of Iceland Research Fund.
\begin{appendix}
\section{Other standard form examples}\label{app:KGgoodform}
In this short appendix we would like to discuss a general form for the d'Alembertian operator $\box\equiv\nabla_\mu \nabla^\mu$ that persists when acting on fields of higher spin and in higher dimensions. We present this discussion because this form was useful to us in examining monodromies in theories with higher-order equations of motion $(\nabla_\mu\nabla^\mu)^n\Phi=0$, and because we feel that it highlights important physical structure related to monodromy parameters and conformal coordinates defined in \eqref{seriessol} and \eqref{confcoordgen} respectively.
To set the stage, let's state the results for the four-dimensional case: a scalar field $\Phi=e^{i(m\phi-\omega t)}R(r)S(\theta)$ propagating on a Kerr background \eqref{kerrMet}. The Klein-Gordon equation is
\begin{equation}\label{appKG}
\nabla^{\mu}\nabla_{\mu}\Phi=\frac{\Phi}{\rho^2R(r)} \left[\partial_r(\Delta\partial_r)+\alpha_+^2\frac{g^2(r)}{f^2(r)}-\alpha_-^2g^2(r)+(r^2+2Mr)\omega^2+C_{t,\phi}\right]R(r),
\end{equation}
where, as we discussed in Section \ref{sec:Hidden-symm-KG},
\begin{equation}\label{4dradial}
\alpha_{\pm}=\frac{\omega-\Omega_\pm m}{2\kappa_\pm}
\end{equation}
are the monodromy parameters, the functions
\begin{equation}
f(r)=\left(\frac{r-r_+}{r-r_-}\right)^{1/2}, \qquad g(r)=\left(\frac{r_+-r_-}{r-r_-}\right)^{1/2}
\end{equation}
define the radial dependence of the conformal coordinates, and $C_{t\phi}$ is a constant of motion, see equations \eqref{CarterOp}-\eqref{radialKG} and discussion below. In scenarios with higher spin fields and higher-dimensional spacetime backgrounds, the quantities $(\alpha_\pm, f(r), g(r), C_{t,\phi})$ change, but the overall form of \eqref{appKG} does not. Let's see how this works.
The hidden conformal symmetry generators for five-dimensional Myers-Perry black holes \cite{myers1986black} were first studied by \cite{Krishnan:2010pv}. The radial equation of motion for a scalar field ansatz $\Phi=e^{i(-\omega t+m_1\phi_1+m_2\phi_2)}$ can be written as
\begin{equation}\label{KG-generics}
\left[\frac{\partial}{\partial x}\left(x^2-\frac{1}{4}\right)\frac{\partial}{\partial x}+\alpha_+^2\frac{g^2(r)}{f^2(r)}-\alpha_-^2g^2(r)+\frac{x\Delta\omega^2}{4}+\tilde{C}_{t,\phi}\right]\Phi=0,
\end{equation}
where
$$x\equiv\frac{r^2-1/2(r_+^2+r_-^2)}{(r_+^2-r_-^2)}$$ is a radial coordinate and the monodromy parameters $\alpha_\pm$ were found in Appendix A of \cite{Castro:2013kea}. There are two important points here. First, the functions $f(r)$ and $g(r)$ are precisely those that define the radial behavior of the conformal coordinates in a five-dimensional setting, presented in \cite{Krishnan:2010pv}. Second, we can see a pattern emerging. The Klein-Gordon equation is expressible as a derivative piece, pieces involving the monodromy parameters (the form of which are fixed), a constant term and a non-constant $r$-dependent term that is irrelevant neither either horizon.
The monodromy analysis for higher spin $s$ perturbations on a four-dimensional Kerr background was treated in \cite{Castro:2013lba}. The equation of motion for such a perturbation can be written as
\begin{equation}
\left(\partial_r\Delta\partial_r+(\alpha^s_+)^2\frac{g^2(r)}{f^2(r)}-(\alpha^s_-)^2g^2(r)+\omega^2r^2+2(M\omega+is)\omega r+\mathcal{C}_{t,\phi}\right)R(r)=0.
\end{equation}
As expected, the functions $f(r)$ and $g(r)$ attached to the monodromy parameters $\alpha_\pm$ are the same as in \eqref{4dradial}, and the monodromy parameters themselves are\footnote{There is a typo in the $\alpha_\pm$ reported in \cite{Castro:2013lba}.}
\begin{equation}\label{alphaspins}
\alpha^s_{\pm}=\mp\frac{is}{2}+\frac{2M\omega r_{\pm}-am}{r_+-r_-}.
\end{equation}
Notice that setting $s=2$ in the expressions \eqref{alphaspins}, the monodromy parameters reduce to \eqref{alphapn2}-\eqref{alphaminn2}.
\end{appendix}
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\section{Introduction}
The study of only a few giant planets was enough to notice that presence of these planets correlates with stellar metallicity \citep{Gonzalez-97, Santos-01}. Since these pioneering works, different research groups tried to link the chemical composition of stars with the properties of planets \citep{DelgadoMena-10, Adibekyan-13, Adibekyan-15b, Suarez-Andres-17, Brewer-18, Hinkel-19, Teske-19, Adibekyan-19, Unterborn-18}.
With the increased precision of mass and radius measurements of planets, it become possible to characterize the interiors and bulk composition of low-mass exoplanets \citep{Nettelmann-21, Helled-21}. Several attempts have been made in the last years trying to link the composition of low-mass planets and their host stars. However, these attempts were either based on single planetary systems \citep{Lillo-Box-20, Mortier-20}, on a small sample of planets \citep{Santos-15, Plotnykov-20, Schulze-20}, or on a comparison of the overall properties of planets and overall properties of planet host stars in a population sense \citep{Plotnykov-20}. As a result, it was not possible to reach a firm conclusion either because of low-number statistics or because the results were not as informative (especially if the composition of the stars are not derived in a homogeneous way) as they would be if a direct star-planet comparison was performed.
On the contrary, \citet[][hereafter A21]{Adibekyan-21} adopted a different approach and looked for compositional relation between rocky exoplanets and their host stars. This approach overcomes the following potential issues:
\begin{itemize}
\item When performing a direct comparison of star-planet compositions for individual systems, large uncertainties in the compositions of planets and/or stars will naturally result in indistinguishable composition of the two.
\item Because the uncertainties in planetary compositions are typically much larger than those in host star abundances \citep[A21,][]{Schulze-20}, the necessity of precise chemical characterization of planet host stars can be overlooked \citep{Schulze-20}.
\item Finally, it is impossible to make a general conclusion about the existence of a compositional link between stars and their planets by comparing the compositions of individual planet-star systems.
\end{itemize}
The main findings of A21 can be summarized as follows. The authors selected 22 low-mass exoplanets ($M$ $<$ 10 $M_{\mathrm{\oplus}}$) with precise mass and radius measurements (uncertainty both in mass and radius below 30\%) orbiting around solar-type stars. For the sample planets they determined their normalized planet density ($\rho / \rho_{\mathrm{Earth-like}}$)\footnote{The normalization parameter $\rho_{\mathrm{Earth-like}}$, is the density of a planet with Earth-like composition \citep{Dorn17} for a given mass. The normalization is to take into account the dependence of planet density on planet mass for a given composition.} and the iron mass fraction ($f_{\mathrm{iron}}^{\mathrm{planet}}$) using the planet interior models of \cite{Dorn17} and \citet{Agol-21}. Based on the chemical abundances of the host stars and using the stoichiometric model from \cite{Santos-15,Santos-17} they also estimated the iron-to-silicate mass fraction ($f_{\mathrm{iron}}^{\mathrm{star}}$) of planetary building blocks. Based on these data the authors found: i) the normalized density and iron-mass fraction of exoplanets strongly correlate with the $f_{\mathrm{iron}}^{\mathrm{star}}$; ii) the relation between $f_{\mathrm{iron}}^{\mathrm{planet}}$ and $f_{\mathrm{iron}}^{\mathrm{star}}$ is not 1-to-1 (exoplanets have on average higher $f_{\mathrm{iron}}$ than what is expected from the host star composition), and iii) super-Earths (with $f_{\mathrm{iron}}^{\mathrm{planet}}$ $\lesssim$ 50\%) and super-Mercuries (with $f_{\mathrm{iron}}^{\mathrm{planet}}$ $\gtrsim$ 60\%) appear to be distinct populations in therm of compositions.
In this paper, we add six recently discovered exoplanets to the A21 sample and re-evaluate their claims and findings. The distribution of the exoplanets on a mass-radius diagram is shown in Fig.~\ref{m_r_diagram} where we single out the newly added planets.
\begin{figure}[ht]
\centering
\includegraphics[width=0.85\textwidth]{r_m_plot_with_errorbars_new_1.pdf}
\caption{Mass-radius diagram for RV-detected planets with masses below 10 $M_{\mathrm{\oplus}}$ and radii below 2 $R_{\mathrm{\oplus}}$ for which the uncertainty both in mass and radius is below 30\%. The blue curve shows the mass-radius relationship for Earth-like composition (32\% Fe + 68\% MgSiO$_{3}$) from \cite{Dorn17}. The six newly added planets are shown in black circles surrounded by a magenta ring. The names of these new planets are also displayed.}
\label{m_r_diagram}
\end{figure}
\section{Properties of planets and their host stars}
To determine the properties of the new planets and their host stars we closely followed the work of A21. Below we provide a brief summary of the methods.
\subsection{Exoplanet properties}
We computed the bulk density ($\rho$) and the normalized density ($\rho / \rho_{\mathrm{Earth-like}}$) of the planets from their mass and radius and the Earth-like composition model of \cite{Dorn17}. From the mass and radius of planets we also estimated their expected iron fraction $f_{\mathrm{iron}}^{\mathrm{planet}}$, which is defined as $(M_{\rm Fe, mantle}+M_{\rm core})/M_{\rm pl}$, where $M_{\rm Fe, mantle}$ and $M_{\rm core}$ are the masses of iron in mantle and core, respectively. For the planet interiors, we assume a pure iron core and a silicate mantle; We neglected possible volatile atmospheric layers.
\subsection{Host properties}
We used publicly available high-resolution spectra for Kepler-37 \citep[FIES,][]{Telting-14}, K2-36 \citep[HARPS-N,][]{Cosentino-12}, K2-199 \citep[HARPS-N,][]{Cosentino-12}, and HD\,80653 \citep[UVES,][]{Dekker-00} to determine the stellar parameters and abundances of Mg, Si and Fe. The stellar atmospheric parameters ($T_{\mathrm{eff}}$, $\log{g}$, microturbulence (Vmic), and [Fe/H]) of the stars have been determined following the methodology described in our previous works \citep{Sousa-14, Santos-13}. For the derivation of chemical abundances we closely followed the methods described in \citet{Adibekyan-12, Adibekyan-15}. The stellar parameters and abundances of Mg, Si, and Fe of HD 137496 are taken from \citet{Silva-21} and for K2-111 are taken from \cite{Mortier-20}\footnote{The adopted abundances are determined from the ESPRESSO spectrum.}.
Based on the abundances of Mg, Si, and Fe, and using the stoichiometric models of \citet{Santos-15} we estimated the iron-to-silicate mass fraction ($f_{\mathrm{iron}}^{\mathrm{star}}$) of planetary building blocks under assumption that the stellar atmospheric composition reflects the composition of the proto-stellar (proto-planetary) disk where the star and the planets are formed.
\section{Results}
Fig.~\ref{density_firon_star} shows the $\rho / \rho_{\mathrm{Earth-like}}$ as a function of $f_{\mathrm{iron}}^{\mathrm{star}}$.
The figure reveals a clear correlation between these two quantities indicating that the final planetary density is a function of the composition of the planetary building blocks. We performed an orthogonal distance regression (ODR) and $t$-statistics to quantify the relation and to assess the significance of the relation. The test suggests that the observed correlation is statistically significant with a $p$-value of $\sim$ 3x10$^{-6}$. For the same relation, the sample of A21 revealed a $p$-value of $\sim$ 7x10$^{-6}$. The slopes of the relations obtained for A21 and the extended samples agree withing one-sigma: 0.051$\pm$0.008 vs. 0.061$\pm$0.009.
\begin{figure}[ht]
\centering
\includegraphics [width=0.85\textwidth] {density_firon_2021.pdf}
\caption{Normalized density of the planets as a function of iron mass fraction of planet building blocks estimated from the host star chemistry. The super-Earths and super-Mercures are shown in blue and brown colors. The positions of K2-111\,b and TOI-561\,b - planets orbiting around metal-poor stars - are indicated. The positions of the Solar System rocky planets are indicated with their respective symbols in black. The red dashed line represent the results of the ODR fit for the super-Earths of the sample of A21. The black solid and dotted-dashed lines show the ODR results for the super-Earths with and without considering Kepler-37 d, respectively. The Solar System planets are not considered in the linear regressions. All error bars show one standard deviation.}
\label{density_firon_star}
\end{figure}
The largest deviation from the fit is observed for Kepler-37\,d. Kepler-37 is orbited by three\footnote{The presence of a forth, non-transiting planet is unlikely \citep{Rajpaul-21}.} transiting small planets. Kepler-37\,d is the largest planet of the system, the mass of which was very recently determined by \citet{Rajpaul-21} using radial velocity (RV) observations. The authors obtained a RV based mass of 5.4$\pm$1.4 $M_{\mathrm{\oplus}}$ and a dynamical mass of $\sim$ 4 $M_{\mathrm{\oplus}}$. From the low density of the planet, \citet{Rajpaul-21} concluded that either Kepler-37\,d is a water-world ($>$ 25\% H$_{2}$O) or has a gaseous envelope\footnote{Note, that the equilibrium temperature of Kepler-37\,d is about 500K which is the coldest planet in the sample.}. In either case, the planet is most probably not a rocky planet consisting of only metallic core and silicate mantle. The exclusion of Kepler-37\,d from the ODR slightly reduces the value of the slope (0.047$\pm$0.007) and makes the significance of the relation slightly higher ($p$-value of $\sim$ 1x10$^{-6}$). The results of the ODR fit without considering Kepler-37\,d is shown with a dotted-dashed line in Fig.~\ref{density_firon_star}.
We also study the relation between $f_{\mathrm{iron}}^{\mathrm{planet}}$ and $f_{\mathrm{iron}}^{\mathrm{star}}$ in Fig.~\ref{firon_planet_star}. We performed an ODR and $t$-statistics to the super-Earths and found a $p$-value of $\sim$ 6x10$^{-5}$, which is even smaller than the $p$-value (1x10$^{-4}$) obtained for the A21 super-Earths sample. The slopes of the relations obtained for the A21 and the extended samples agree withing one-sigma: 0.36$\pm$0.9 vs. 4.3$\pm$0.8.
\begin{figure}[ht]
\centering
\includegraphics [width=0.85\textwidth] {cmf_firon_2021.pdf}
\caption{Iron mass fraction of planet building blocks ($f_{\mathrm{iron}}^{\mathrm{star}}$) versus iron mass fraction of the planets ($f_{\mathrm{iron}}^{\mathrm{planet}}$). The meaning of the symbols and lines are the same as in Fig.~\ref{density_firon_star}. The error bars of $f_{\mathrm{iron}}^{\mathrm{star}}$ show one standard deviation. The error bars of $f_{\mathrm{iron}}^{\mathrm{planet}}$ cover the interval between the 16th and the 84th percentiles.}.
\label{firon_planet_star}
\end{figure}
Two planets, 55 Cnc\,e and Kepler-37\,b, show $f_{\mathrm{iron}}^{\mathrm{planet}}$ smaller than $f_{\mathrm{iron}}^{\mathrm{star}}$ of their host stars. We already discussed the possibility for Kepler-37\,b to have a significant amount of volatiles or atmosphere. 55 Cnc multi-planetary system is one of the well studied ones \citep{Bourrier-18}. Several works suggested that 55 Cnc\,e may have volatile \citep[e.g.][]{Lopez-17} and/or hydrogen \citep[e.g.][]{Hammond-17} layers which could explain the low density of the planet. In addition, it was proposed that 55 Cnc\,e can have Ca- and Al-rich interior without a significant iron core \citep{Dorn-19}.
\section{Discussion}
It is interesting to see that similarly to TOI-561\,b, the newly added planet - K2-111\,b, orbiting a metal-poor star, is also a low-density planet with low iron content. The low $f_{\mathrm{iron}}$ of rocky planets was proposed in \citet{Santos-17} where the authors studied the potential composition of planet building blocks around stars from different Galactic stellar populations. The stoichiometric model of \citet{Santos-15, Santos-17} also suggest a high water-mass-fraction for planets orbiting around metal-poor stars. It is thus possible that both TOI-561\,b and K2-111\,b have a non-negligible volatile layers, which we ignored in our analysis. In a subsequent paper, we plan to model the planet interiors considering also volatile layers and evaluate the presence of correlations between water-mass fraction of the planets and their host stars.
One of the newly added planets, HD 137496\,b, is a super-Mercury. It is intriguing to see that this planet, just like the other five super-Mercuries of A21, has a high $f_{\mathrm{iron}}^{\mathrm{star}}$. A21 suggested that the high iron content of super-Mercuries might be related to the protoplanetary disk composition, and not solely to a giant impact.
\section{Summary}
In this work we extended the sample of \citet{Adibekyan-21} by adding six recently discovered rocky exoplanets and studied the compositional link between rocky exoplanets and their host stars. The main results which confirm the recent findings of A21 are summarized below:
\begin{itemize}
\item The density ($\rho / \rho_{\mathrm{Earth-like}}$) of super-Earths correlates with the iron content ($f_{\mathrm{iron}}^{\mathrm{star}}$) of the protoplanetary disk.
\item There is a non 1-to-1 relation between $f_{\mathrm{iron}}^{\mathrm{planet}}$ and $f_{\mathrm{iron}}^{\mathrm{star}}$. $f_{\mathrm{iron}}^{\mathrm{planet}}$ of super-Earths is larger than the iron content expected from the exoplanet host stars’ composition.
\item Super-Mercuries are formed in the disks with high $f_{\mathrm{iron}}^{\mathrm{star}}$ suggesting that protoplanetary disk composition might be important for the formation/evolution of these planets.
\end{itemize}
Studying the relationship between the compositions of planets and their host stars yields a wealth of information on the processes that occur during the formation and evolution of planets. As the number of newly discovered rocky exoplanets continues to increase, we will be able to better understand the origins of these compositional links.
\section*{\small Acknowledgements}
\scriptsize{This work was supported by FCT - Funda\c{c}\~ao para a Ci\^encia e Tecnologia (FCT) through national funds and by FEDER through COMPETE2020 - Programa Operacional Competitividade e Internacionaliza\c{c}\~ao by these grants: UID/FIS/04434/2019; UIDB/04434/2020; UIDP/04434/2020; PTDC/FIS-AST/32113/2017 \& POCI-01-0145-FEDER-032113; PTDC/FIS-AST/28953/2017 \& POCI-01-0145-FEDER-028953. V.A., E.D.M, N.C.S., and S.G.S. also acknowledge the support from FCT through Investigador FCT contracts nr. IF/00650/2015/CP1273/CT0001, IF/00849/2015/CP1273/CT0003, IF/00169/2012/CP0150/CT0002, and IF/00028/2014/CP1215/CT0002, respectively, and POPH/FSE (EC) by FEDER funding through the program ``Programa Operacional de Factores de Competitividade - COMPETE''. V.A., E.D.M, N.C.S., and S.G.S. acknowledge support from FCT through Investigador FCT contracts nr. IF/00650/2015/CP1273/CT0001, IF/00849/2015/CP1273/CT0003, IF/00169/2012/CP0150/CT0002, and IF/00028/2014/CP1215/CT0002, respectively, and POPH/FSE (EC) by FEDER funding through the program ``Programa Operacional de Factores de Competitividade - COMPETE''. C.D. acknowledges support from the Swiss National Science Foundation under grant PZ00P2\_174028, and the National Center for Competence in Research PlanetS supported by the SNSF. B.B., was supported by the European Research Council (ERC Starting Grant 757448-PAMDORA). O.D.S.D. and J.P.F. are supported by contracts (DL 57/2016/CP1364/CT0004 and DL57/2016/CP1364/CT0005, respectively) funded by FCT. }
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2021-12-30T02:25:46.000Z
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proofpile-arXiv_000-10051
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\section{Introduction}
\label{sect:intro}
Learned systems, particularly learned database systems, have been a recent research hot spot~\cite{zhou2020database, han2021CEbenchmark, kraska2019sagedb}. As part of this trend, a considerable number of researchers have proposed integration of machine learning models into relational database systems in order to optimize overall system performance and cope with the ever increasing amounts of data that are being processed in modern data intensive applications.
As part of this trend, machine learning based methods for a wide range of problems have been proposed. Most efforts appear to have been spent on problems related to query planning and optimization, where machine learning models may be used e.g. for cardinality estimation~\cite{han2021CEbenchmark, wu2020bayescard, hilprecht2019deepdb, yang2019deep, zhu2020flat}, join order selection~\cite{marcus2018deep}, cost prediction~\cite{siddiqui2020cost, vu2021learned} or query planner steering through hint set proposals~\cite{marcus2020bao}.
In addition to the above, other lines of research have focused on learned data structures~\cite{kraska2018case} used for efficiently querying large indices~\cite{nathan2020learning, marcus2018deep, perera2021no, sabek2021case}, automatic database configuration tuning, accurate query run time prediction and other problems~\cite{zhou2020database}.
As is the case with many publications in the machine learning space, on paper the reported results do indeed look impressive and suggest significant performance gains, if such methods were ever deployed in the real world. Yet, as most prior research has been justified on the basis of numerical experiments, performed \textit{outside} of real systems, there has been a striking absence of work focusing on the practical deployment of ML-based methods \textit{inside} real world systems. The central goal of this work is to present a concrete proposal on how this notable gap between theory and practice may be closed. For this aim, we propose here the \textsf{Baihe}\xspace SysML framework.
\textbf{\textsf{Baihe}\xspace} is a general framework developed for integrating machine learning models into a relational database systems, meaning that its fundamental architecture should provide a blueprint for other implementation projects in the AI4DB and SysML space. To exemplify its applicability and generate some more robust evidence for the actual usefulness of prior research in AI4DB, we have developed the \textsf{Baihe}\xspace extension for \textsf{PostgreSQL}\xspace, a widely used, highly successful and extensible relational database system.
This article is now structured as follows: In Section \ref{sect:background}, we first discuss general challenges for the implementation of AI-driven databases and then present our long term vision for AI4DB. Based on this, we then describe the rationale and fundamental principles for \textsf{Baihe}\xspace's design.
Later on, in Section \ref{sect:design} we present the concrete design and features of \textsf{Baihe}\xspace in the case of \textsf{PostgreSQL}\xspace and further discuss implementation specifics. After show a typical example use case for \textsf{Baihe}\xspace on query optimizer in Section~\ref{sect:example}, we conclude this article in Section \ref{sect:conclusion}.
\section{Background and Design Rationale}
\label{sect:background}
On a fundamental level, the nature of and requirements on a database system appear to be at odds with the inherently stochastic nature of machine learning. Database developers, users, and administrators expect rock-solid, stable and most of all deterministic behavior. On the other hand, machine learning models may produce predictions with hard to estimate error bounds and their generalization ability may fail catastrophically, once the input data distribution changes. Moreover, failures may not be readily detectable, since they might only be visible through a change in some numerical value rather than a concrete and meaningful error message.
Moreover, since training of a machine learning model generally amounts to solving a mathematical optimization problem, often through the use of stochastic optimizers, model training is difficult to automate, requires close supervision by experts and may thus not be fully regarded as a well defined process. Hyperparameters, useful termination criteria and evaluation guidelines all need be carefully developed on both a per-model and per-dataset basis. All of this makes it very difficult to integrate automated training procedures into a system with strong requirements on robustness and stability.
As challenging as this may sound, we still believe that the practical combination of machine learning and databases into the next generation of database systems is not a hopeless endeavor.
To see this, we first note that some core components of a database system \textit{already} rely on \textit{statistical} estimates. Modern query optimizers in particular rely on such estimates e.g. in the context of cardinality and cost estimation, which are in turn used for join-order-selection and other automated decision making. The vision outlined by e.g. SageDB~\cite{kraska2019sagedb} has made a compelling case for using more advanced probabilistic methods - powered by machine learning - to improve existing issues arising from the relatively simple statistical estimations which are still part of the standard implementations of today's database systems.
Second, we believe that issues related to using machine learning enabled components inside a database system, should merely be regarded as \textit{additional requirements} on the engineering of such systems. Hence, we argue that one should in principle be able to satisfactorily fulfil these requirements, \textit{if} they are properly taken into account during the system design phase and possibly enhanced through system-algorithm co-design. We hope to exemplify and support this point of view through the \textsf{Baihe}\xspace framework presented in this work.
\subsection{Learned Databases: Brief Overview}
\label{sect:learned_db}
Before presenting the high-level architecture of \textsf{Baihe}\xspace and the rationale behind it, we first review related literature and describe the current stage and a future vision for learned database systems.
We categorize learned databases into four levels, corresponding roughly both with historical development, as well as the amount of ``intelligence'' added by learned components.
\begin{enumerate}
\item
\textbf{Primary utilization in DBMS:}
Many available commercial and open-source DBMS have a long tradition of using collected statistics and statistical estimators to support some of their data management functions.
Some of these methods serve as an integral parts of some components. For example, cardinality estimation (\textsf{CardEst}\xspace) consists of estimating data distribution properties used later as input for query optimization (QO). While for instance SQL Server and \textsf{PostgreSQL}\xspace build histograms and collect frequently occurring values for table attributes, MySQL and MariaDB apply a sampling strategy for \textsf{CardEst}\xspace. With respect to non-integral functions, basic statistical methods are used for advisory functions, such as index or view advisors, or knob tuning and SQL rewriters~\cite{zhou2020database}.
\item
\textbf{Individually learned components:}
With the recent increased interest in machine learning, an increased amount of work has tried to design ML-based models to replace certain components in DBMS. In this context, the most representative works address QO and indexing. For QO, a variety of supervised and unsupervised models have be proposed for \textsf{CardEst}\xspace~\cite{han2021CEbenchmark, wu2020bayescard, hilprecht2019deepdb, yang2019deep, zhu2020flat, hasan2019multi}, cost estimation~\cite{siddiqui2020cost, vu2021learned} and join order selection~\cite{marcus2018deep, sabek2021case}. Meanwhile, a number of learned data structures structures are proposed for single and multi-dimensional indexing~\cite{kraska2018case, nathan2020learning, perera2021no}. It has been shown such ML-based methods often exhibit superior performance, when evaluated in numerical experiments outside of actual systems or tightly controlled experimental setups inside of real systems (e.g. through injecting cardinality from an outside file). However, many problems relevant to real-world deployment, such as e.g. explainability vs. predictive power, robustness or fault tolerance remain largely unaddressed.
\item
\textbf{Comprehensively learned modules:}
On top of individually learned components, current work further moves forward by trying to substitute an entire functional module, e.g. the QO, executor or storage engine with a machine learning model. Some approaches combine multiple learned components together, learn to steer existing modules~\cite{marcus2020bao}, or even
attempt to "learn" an entire module in an end-to-end fashion~\cite{marcus2019neo}. Although most work of this type claims to achieve incredible performance gains, many proposed solutions appear to be impractical for actual deployment in real-world DBMS. The reasons for this are two fold.
First, learned modules of this type are highly task specific and data dependent, which may cause serious shortcomings, including but not limited to: cold start problems, lack of generalization, tuning difficulties.
Second, replacing an entire module may cause compatibility risks and may require substantial engineering efforts. Therefore, at this stage, even evaluating a learned module inside a real DBMS is a difficult task.
\item
\textbf{AI-Native databases:}
Some very recent work even proposes to redesign the whole architecture of DBMS to fully adapt AI models for data management. For example,~\cite{wu2021unified} proposes a "one model for all architecture", which learns a shared representation of data and query knowledge and then fine-tunes smaller models for each specific task.~\cite{li2019xuanyuan} proposes "AI-designed databases", where AI is integrated into the life cycle
of database design, development, evaluation, and maintenance, hoping to provide optimal performance for every scenario. Although this appears to be an intriguing vision, it is clear they are still far away from any useful implementation.
\end{enumerate}
Hence, regarding ``AI for DB'', we observe that:
1. This field is very prosperous. The research efforts range from individual components to complete modules to even the whole architectures, and include many scenarios (QO, indexing, execution, storage and etc). Enough evidence has shown that AI-based solutions could indeed improve database performance and have the potential to play significant roles in next-generation DBMS.
2. Current work mainly focuses on ``AI'' solutions but is not concerned with how to actually deploy them ``for DB''. In addition to that, more realistic tests for newly proposed methods relying on e.g. existing extension functionalities of DBMS are rarely described in the literature, possibly because it requires deep expertise beyond the design of ML models. Such expertise requires a comprehensive understanding of both AI and DB perspectives and a systematic co-design of both algorithm and systems.
Therefore, we believe that a SysML framework such as \textsf{Baihe}\xspace, which supports both evaluation and deployment of AI-driven solutions in real systems is crucial for the further development of the AI for DB field.
\subsection{High Level Architecture of \textsf{Baihe}\xspace: Fundamental Design Choices and Trade-Offs}
\label{subsect:high_level}
\begin{figure*}[h]
\centering
\includegraphics[width=0.65\linewidth]{baihe_high_level.pdf}
\caption{High Level View of \textsf{Baihe}\xspace's Architecture.}
\label{fig:arch}
\end{figure*}
We now briefly describe the high level design of \textsf{Baihe}\xspace and discuss the rationale and design philosophy behind it. We note that this design should rather be considered as a design blueprint resp. design pattern, which may then have to be adapted to a specific host database system. As an example we present \textsf{Baihe}\xspace for \textsf{PostgreSQL}\xspace in Section \ref{sect:design}.
The core part of \textsf{Baihe}\xspace is an extension which plugs directly into a database host system through an existing extension mechanism. This extension is the central component of \textsf{Baihe}\xspace, as it
\begin{enumerate}
\item intercepts the query planning and execution process, so that individual steps may be substituted with model inference calls,
\item contains a clone of the host system's planner ("shadow planner"), which can be conveniently extended and modified, so that core functionality may be overwritten without interfering with the host system itself,
\item provides convenient configuration facilities,
\item controls collection of training data.
\end{enumerate}
Most of the above functionality is encapsulated in the Integration Layer component, which is the central control unit for \textsf{Baihe}\xspace.
Model inference is decoupled from the extension itself, this means that for every model a new background worker process is used. These processes communicate with the \textsf{Baihe}\xspace extension and return results for inference requests. If the host system supports it, the control of these processes is managed using the host systems process management capabilities.
Finally, \textsf{Baihe}\xspace needs a support library. This component allows for simple access to collected training data, such as e.g. query runtime statistics or saved query plans. Furthermore, it provides functionality for convenient deployment of trained models into the host system.
The design above is based on the following high-level requirements:
\begin{itemize}
\item Separation from the core system.
\item Minimal third party dependencies.
\item Robustness, stability and fault tolerance.
\item Usability and configurability.
\end{itemize}
For the remainder of the section we now describe the impact of these requirements on our design in more details.
\newline
\textbf{Separation from the Core System.}
Both commercial and open source RDBMS have been and are being actively maintained and supported over long periods of times. Commercial license and support agreements may span years or even decades, so that customers may receive regular updates and critical patches, especially those addressing security issues. In order to impact existing processes as little as possible, all while avoiding a development of an entirely new database system, we have chosen to develop \textsf{Baihe}\xspace as an extension on top of an existing system, so that we may keep it as separate from the host system as possible. In this way, we avoid unnecessary doubling of maintenance efforts and allow for a quicker pace of development.
\textbf{Minimal Third Party Dependencies.}
Modern ML stacks are characterized by a large number of dependencies, comprising low level numerical libraries and GPU kernels, intermediate ML framework and runtime codes (typically implemented in C++), as well as high level integration code typically written in Python. To train and use what may now even be considered relatively simple deep models, a large number of packages and libraries at these three levels must be present on a host system. In practice, this typically leads to a high degree of maintenance efforts for both production, as well as test and development environments. This is particularly problematic in more traditional organizations, such as financial institutions, where - due to security or legal reasons - individual software packages may have to go through a specific vetting and approval process. We have thus designed \textsf{Baihe}\xspace such that it requires a minimal amount of external dependencies beyond those required by the core system.
\textbf{Robustness, Stability and Fault Tolerance.} For most modern applications databases are among the most fundamental and mission-critical components. It is thus imperative that additional deployment of ML-based components into such systems does not impact existing service level agreements or interfere with related operational requirements. Therefore, errors arising from e.g. model-based predictions, which might influence an individual session or the system as a whole should be detected and mitigated through fallbacks to existing core functionality.
\textbf{Usability and Configurability.} To further ease the burden of integration, it should be possible to control and configure \textsf{Baihe}\xspace through standard mechanisms offered by the host system. In the concrete case of a system such as \textsf{PostgreSQL}\xspace this means that \textsf{Baihe}\xspace should be able to be configured through the usual \textsf{PostgreSQL}\xspace configuration files, as well as provide a set of user defined functions as well as stored procedures, such that \textsf{Baihe}\xspace functionality may be configured, activated or deactivated through any authorized command session and without requiring any restarts of the system as a whole.
Based on the above four points we may now formulate more concrete design goals with respect to the machine learning aspects.
\textbf{Model Support.} As a general framework, \textsf{Baihe}\xspace should support deploying models for a range of different tasks, such as e.g. cardinality estimation, join order selection, query run time prediction. Furthermore, it should support models for learned data structures, such as e.g. learned indices.
With respect to the models themselves, \textsf{Baihe}\xspace should offer support for both neural network, resp. deep learning based, model families, as well as more traditional ones, such as e.g. probabilistic graphical models, decision trees, random forests or gradient boosted trees.
\textbf{Model Training.}
As we have discussed in the previous section - despite the recent progress in AutoML~\cite{zhou2020database} - model training still requires close expert supervision supported by suitable tooling allowing for thorough model evaluation and rapid experimentation. In \textsf{Baihe}\xspace we thus prefer to decouple model training from the rest of the system as much as possible. While \textsf{Baihe}\xspace should still provide suitable functionality for training data collection, as well as tools for model export and deployment, we believe that training itself should be set up in a separate environment under control by specialist users such as data scientists or machine learning engineers. Once training has achieved satisfactory progress, a model can then be deployed using a well-defined deployment process.
\textbf{Model Inference and Deployment}
In order to avoid expensive serialization and de-serialization steps one might want to integrate a model directly into e.g. the planner component of the host system. On the other hand, to maintain the maximal amount of flexibility with respect to software dependencies and computing resources needed for inference, one might also consider implementing model inference in a completely separate service process outside of the control of the host system. We believe that both of these extremes would clash with requirements on robustness and stability, as well as maintenability of the host system. In \textsf{Baihe}\xspace we thus choose to isolate inference in a separate process, but keep this process under management by the host system. To eliminate the need for expensive serialization steps we furthermore propose process co-location, so that existing shared memory facilities may be used as much as possible.
While current (practical) models in the SysML space, even deep ones, can still be considered relatively light-weight~\cite{han2021CEbenchmark, zhu2020flat}, we hence believe that computational resources needed by the inference process would in general not adversely affect the core databases process on the same machine. In the long term, should there be the need for computationally more expensive models to be deployed in the system, one could address this problem, at least in cloud environments, through on-demand attachable resources.
As a consequence of the co-location requirement, one could imagine having to install additional packages on the machine running the host system, which would be needed to run model inference (e.g. ML framework runtimes etc.). To address this issue \textsf{Baihe}\xspace should provide proper tools allowing for exporting models such that they can readily be used with as little extra dependencies as possible. To achieve this goal, we propose to make use of the recent advances in the context of ML model compilers~\cite{ben2019modular}, which make it possible to compile models together with custom CPU or GPU based math kernels into highly efficient binary code, that may be accessed through a C-ABI. Nevertheless, \textsf{Baihe}\xspace should support a "Development Mode", where models may be developed and tested in the host system without intermediate compilation and build steps.
\section{System Design and Implementation}
\label{sect:design}
\input{system_design}
\section{Example Use Case: Learned Query Optimizer for \textsf{PostgreSQL}\xspace}
\label{sect:example}
We describe now a typical use case of for \textsf{Baihe}\xspace: deploying a learned query optimizer into \textsf{PostgreSQL}\xspace. More concretely, we discuss here the following two variants:
\begin{enumerate}
\item QO with individual components: cardinality estimation and cost model substituted with separately trained components.
\item End-2-End QO: Here, the entire query optimizer is substituted by a trained model.
\end{enumerate}
\subsection{QO with individually learned components}
\textsf{CardEst}\xspace models and ML-based cost models are supported out-of-the box. The shadow planner in the \textsf{Baihe}\xspace integration layer intercepts all requests for a cardinality estimate for a query touching a set of tables $T$ with set of query predicates $Q$. The tuple $(T,Q)$ is obtained from internal query parse tree structures, serialized and passed into a trained model running in a background worker. The model then returns a selectivity $0 \leq s \leq 1$, which is passed on to the planner.
Training data collection for a \textsf{CardEst}\xspace model depends on whether the model is based on query-driven or data-driven \textsf{CardEst}\xspace. Specifically, data-driven \textsf{CardEst}\xspace methods build unsupervised models over the tabular data, then the cardinality of any query could be estimated over this model. For data driven \textsf{CardEst}\xspace no additional data collector is needed, since models may be trained simply using (samples of) table data and schema information provided by the user (the latter being required for models supporting multi-table \textsf{CardEst}\xspace).
Query-driven \textsf{CardEst}\xspace methods build the supervised models mapping featurized queries to the cardinality. For a query driven model, we first define a data collector, which, for every query, saves the entire query plan, together with all statistics collected during the execution (i.e. the entire output of EXPLAIN ANALYZE). Training code can then load this data and convert it to the required form of $(\text{Subquery}, \text{Cardinality})$ records needed for training.
Once a \textsf{CardEst}\xspace model has been trained it may then easily be registered as a model of "CARDEST" type using a call to the corresponding \textsf{Baihe}\xspace procedure. To make the model active, a background worker is started using another \textsf{Baihe}\xspace procedure call and the session variable "baihe\_ce\_model" is set to the model identifier. Then, all subsequent queries for this session will use learned cardinality from the deployed model.
The process works similar in the case of cost models. Out of the box, the \textsf{Baihe}\xspace will shadow planner intercepts all cost-estimation call on a node-level (e.g. sequential scan, index scan, etc.). Then, a record depending on a variable number of features (depending on node type), is built and sent to a cost estimation model running in a background worker, which then returns a predicted cost in terms of cost units.
To collect training data, we register a data collector with the same settings as used for query-driven \textsf{CardEst}\xspace. In this way, for every node in a query plan we obtain all features required for training a meaningful model, with the most important features being node type, estimated cardinality, actual cardinality and the time needed to execute a node.
\subsection{End-2-End Learned QO}
Some recent work also presents methods for learning a query plan directly. Such methods take a query as input, apply a certain featurization scheme and return an entire query plan as output. For our example we take a closer look at two major representatives of this line of work, namely NEO~\cite{marcus2019neo} and BAO~\cite{marcus2020bao} and show how they could be deployed using \textsf{Baihe}\xspace.
NEO applies tree convolution networks to extract features from structured query plans and learns a function, called value network, mapping plans to execution latency. Then, a best-first search strategy is used to find a near-optimal query plan as measured by the value network.
To deploy NEO, we first register a data collector as used for query-driven \textsf{CardEst}\xspace in the previous subsection. Then, we a background workers which implements value network inference and the best-first search strategy, respectively. During query execution, we intercept the planning process at the highest level in the \textsf{Baihe}\xspace integration layer (right after the \textsf{Baihe}\xspace extension is first called by the host system) and forward the query to the the background worker, evaluates the value network worker and returns a query plan after running the best-first search. This plan is directly sent to the \textsf{PostgreSQL}\xspace engine for execution.
BAO adapts a different strategy than NEO. It learns to steer but not the replace the QO. Specifically, it also learns a latency prediction network which maps a query plan to its execution latency. For each query, it generates several plans corresponding to different hint sets and then selects the plan with the minimum predicted latency. Hence, to deploy BAO, data collection and model deployment need to be configured in exactly the same way as BAO and NEO.
Note that the above discussion only concerns the case where both the BAO and NEO models have been trained to a certain point and then remained unchanged after deployment. However, both models have been designed to be updated in an online-manner, so that they may possibly adjust to changes in the underlying data and workload, without having to be explicitly retrained.
While online updates are not directly supported yet, we note that model code running inside background workers could easily be written in such a way that incoming inference request data may simultaneously be used to updated the model running inside a worker. However, we note that - at least for now - it is then the responsibility of each such background worker to properly manage model state, ensure that model updates don't block future inference requests and deal with errors that might occur during online updates.
\section{Open Source Release and Future Plans}
\label{sect:conclusion}
As development of \textsf{Baihe}\xspace has started only recently, it is not yet available for general use. However, we plan to release a first version of \textsf{Baihe}\xspace for \textsf{PostgreSQL}\xspace under an open source license in the beginning of 2022. This version should contain all of the essential functionality needed to build and experiment with learned query optimizers as described in the previous subsection. Later on, in the second half of 2022, we plan to release an extended version of \textsf{Baihe}\xspace which has seen first tests under real world conditions and supports production-mode deployments.
The reasons for this release schedule are as follows:
\begin{itemize}
\item First, We hope to encourage community participation in the development of \textsf{Baihe}\xspace as soon as possible.
\item Second, We wish to serve the DB research community by providing a flexible and easy to use experimental platform for for future research into AI4DB, hoping to establish a standardized and realistic test bed for future models and algorithms.
\end{itemize}
Overall, we hope to have provided convincing arguments for the soundness and practicality of \textsf{Baihe}\xspace as a design blueprint. The ongoing development of \textsf{Baihe}\xspace for \textsf{PostgreSQL}\xspace should further help refining this blueprint and serve as an implementation guide for other database systems.
Besides the ongoing development, there are many avenues for future work. For instance, the current version of \textsf{Baihe}\xspace has been designed with most applications revolving around query optimization. However, one could envision \textsf{Baihe}\xspace to be used in the context of learned indices, database configuration tuning or other advisory functions.
Another aspect that has been left out for now concerns the training process itself, as well as models which may benefit from online training. Integrating training and online updates of possibly large models directly into \textsf{Baihe}\xspace should certainly provide for many interesting system design challenges.
Finally we note that the development of production mode deployment needs a custom model compiler infrastructure, which further adds to the many engineering and research challenges that accompany this line of work. We encourage the entire community to actively participate and accept some of these challenges.
\bibliographystyle{ACM-Reference-Format}
\subsection{\textsf{Baihe}\xspace Integration Layer and IPC Module}
The \textsf{Baihe}\xspace integration layer is the central control unit of the \textsf{Baihe}\xspace extension: It is accessed from the host system by implementing some of the hooks already defined in \textsf{PostgreSQL}\xspace, where it implements \textsf{Baihe}\xspace's high level logic. All functionality for communication with background workers, as well as necessary process management for background workers is encapsulated in the IPC Module. The IPC module makes extensive use of Postgres core APIs used for management of shared memory, as well as process management.
Additionally, the \textsf{Baihe}\xspace Integration Layer defines user defined functions and session variables which are necessary for controlling model handling and data collection, e.g. starting and stopping background workers, defining which models should be used in which situations, as well as defining when and how training data should be collected.
\textbf{Shadow Planner.}
To allow for a maximum degree of flexibility with respect to models providing input for query planning, the \textsf{Baihe}\xspace extension contains a customized duplicate of the core \textsf{PostgreSQL}\xspace planner as a "Shadow Planner" component. In this way we can achieve the following: First, we may freely add new hooks into the planner without having to modify core source code\footnote{Currently, \textsf{PostgreSQL}\xspace itself offers e.g. no hooks to overwrite cardinality estimation for e.g. single table queries or join size estimates}. Second, the behavior of the planner as a whole may be adjusted and new ideas tested without having to interfere with the core source. As an additional benefit, this reduces overall compilation and build times.
Currently, in addition to the existing hooks originating from the original \textsf{PostgreSQL}\xspace code, we have equipped the shadow planner with the following hooks:
\begin{itemize}
\item Cost Model: we add an additional hook per "node" in a query plan. This allows for overwriting cost predictions for such operations as sequential scan, index scan, nested loop join, hash join etc. Furthermore, we incorporate hooks for estimating costs of query predicates making use of operations beyond comparison operators for numerical values. We plan to further support hooks for overwriting the cost estimation of user defined functions etc.
\item Cardinality Estimation: We add hooks at several levels of the cardinality estimation process, such as cardinality estimation for a query involving a single table or a join between two tables.
\end{itemize}
To improve the interplay between hooks and planner code, we furthermore design the concrete hook signatures and calling code with error handling mechanisms allowing for seamless fallback to standard planner behavior in the case of errors originating e.g. from erroneous model inference calls. We discuss how shadow planner and related hooks are specifically used for query optimization task in Section~\ref{sect:example}.
\subsection{Data Collection}
The design of \textsf{Baihe}\xspace's data collection module borrows heavily from the popular \textsf{pg\_stat\_statements} extension for \textsf{PostgreSQL}\xspace. However, since \textsf{Baihe}\xspace targets data collection for training machine learning models, it takes a more dataset-centric point of view.
More concretely, \textsf{Baihe}\xspace data collection is designed around the notion of "Data Collectors". Users may define and activate several Data collectors at the same time, where each data collector may be defined as a set of filter conditions plus a versioned data set identifier. In this way users may for each Data Collector specify the following:
\begin{itemize}
\item \textit{Filter conditions}: For which query type (SELECT, INSERT, ...) involving which tables should this data collector be applied?
\item \textit{Dataset identifier and version}: For reproducibility a debugging purposes it is essential to keep track of exactly which data was used to train a specific model version. Hence, any data set collected is identified through both a dataset identifier as well as a version number.
\item \textit{Features}: For some dataset and model combinations, only queries themselves might need to be collected, while for others it might be necessary to also collected generated query plans, together with estimated costs and actual run times both on query plan, as well as plan-node level. Users may flexibly specify, which features should be saved by a data collector
\end{itemize}
Data collection may be controlled entirely through a standard command session. After a data collector has been defined through a call to a \textsf{Baihe}\xspace stored procedure, the data collection process itself may also be started and stopped by running start and stop routines exposed from the \textsf{Baihe}\xspace extension through custom stored procedures. See Figure \ref{fig:usage_data} for an example.
While a Data Collector is active, all collected data will be stored in shared memory.In the case of very large datasets, shared memory content may be temporarily stored on disk. Once data collection is stopped, a data set with incremented version identifier will be written to disk and made available in a table specified in the Data Collector's configuration. Training data may then easily be accessed through SQL.
\subsection{Model Integration}
\begin{figure*}[h]
\centering
\begin{subfigure}{8.5cm}
{\color{darkgray} \textsf{\#\# Define a data collector}} \\
{\color{darkgray} \textsf{\#\# Filter queries by tables and query type}} \\
\textsf{CALL {\color{blue} DEFINE\_DATA\_COLLECTOR} ( ``CardEstCollector'', \\
\{ ``tbl\_users'', ``tbl\_items'', … \}, \{ ``SELECT'' \} ); } \\
{\color{darkgray} \textsf{\#\# Start data collection}} \\
\textsf{CALL {\color{blue}START\_DATA\_COLLECTOR} ( ``CardEstCollector'', \\
``Data\_Set\_1'', ``tbl\_training\_data'' ); } \\
{\color{darkgray} \textsf{\#\# Stop data collection}} \\
\textsf{CALL {\color{blue}STOP\_DATA\_COLLECTOR} ( ``Data\_Set\_1''); } \\
\caption{Configuring data collection for a single cardinality estimation model. Only data related to SELECT queries touching certain tables is collected. Data collection can be started and stopped.}
\label{fig:usage_model}
\end{subfigure}
\hfill
\begin{subfigure}{8.5cm}
{\color{darkgray} \textsf{\#\# Model Registration}} \\
\textsf{CALL {\color{blue} REGISTER\_MODEL} ( ``MyCardEstModel'', ``CARDEST'', \\
\{ ``tbl\_users'', ``tbl\_items'', … \}, ``tbl\_my\_cardest\_model\_stats'' ); } \\
{\color{darkgray} \textsf{\#\# Start Background Process for Model}} \\
\textsf{CALL {\color{blue} START\_MODEL} ( ``MyCardEstModel'' ); } \\
{\color{darkgray} \textsf{\#\# Stop Background Process for Model}} \\
\textsf{CALL {\color{blue} RESET\_MODEL} ( ``MyCardEstModel'' ); } \\
\caption{Deploying a trained model into the system: The model is used only for queries touching certain tables and maybe activated or deactivated when requested by the user.}
\label{fig:usage_data}
\end{subfigure}
\hfill
\caption{Example configuration sessions for \textsf{Baihe}\xspace}
\label{fig:usage}
\end{figure*}
Along the lines of our requirements on minimization of dependencies and model inference and deployment as described in Section \ref{sect:background}, model inference takes place in background worker processes. For every model registered in \textsf{Baihe}\xspace, a user can request \textsf{Baihe}\xspace to start a background worker process, which will
\begin{enumerate}
\item Load the saved model from disk
\item Connect to the \textsf{Baihe}\xspace shared memory space
\item Wait for incoming inference requests on a message queue.
\item Once an inference request is received, the background worker will run the request through the loaded model and return inference results (which may possible also just a flag indicating that an error has occurred).
\end{enumerate}
Once a query is submitted by a client to the corresponding \textsf{PostgreSQL}\xspace backend process, the query planning and execution process will be intercepted by the \textsf{Baihe}\xspace extension and depending on model type, a number of inference requests will be sent to the correct background workers. All communication is implemented asynchronously, so that a backend may fall back to standard functionality in case a background worker is not available.
Out of the Box \textsf{Baihe}\xspace allows for the integration of custom models for query runtime prediction, as well as cost and cardinality estimation. Models of these types may be used directly without changes to the \textsf{Baihe}\xspace extension source code. More specific types of models, requiring e.g. new hooks at certain places in the planner code, may easily be supported with slight changes to the \textsf{Baihe}\xspace extension code.
Similar to the data collection functionality described in the previous subsection, \textsf{Baihe}\xspace's model handling facilities are controlled and configured using a number of stored procedures and user defined functions implemented in the \textsf{Baihe}\xspace extension. A simple usage example is displayed in Figure \ref{fig:usage_model}: Through a standard command session users with the right permissions may request models to be registered and the corresponding background workers to be started or stopped. This allows for model updates without having to restart the entire system.
As mentioned previously in Section \ref{sect:background} \textsf{Baihe}\xspace's focus is on model inference only. This means that the training process itself, that is solving the optimization process for a certain combination of model and training data, does not run in any \textsf{Baihe}\xspace components. Instead, the usual development process can be outlined as follows: \begin{enumerate}
\item Training data is selected and downloaded using the \textsf{Baihe}\xspace support library, implemented as a Python packages.
\item A model can then be defined and trained, preferably using a framework supported by \textsf{Baihe}\xspace's production mode. Currently supported frameworks are sklearn and Tensorflow. Training is controlled entirely by an expert user, such as e.g. a data scientist or machine learning engineer.
\item Once the model has been trained and evaluated, it may be deployed using the \textsf{Baihe}\xspace support library.
\begin{itemize}
\item \textit{Development mode}: In this mode, the model is deployed as a Python model on the database servers file system, together with an automatically created environment containing all the model's dependencies. A background worker then uses an embedded Python interpreter to access the model.
\item \textit{Production mode}: In this mode, \textsf{Baihe}\xspace's support library is used to compile the model including trained parameters into a shared library that is loaded dynamically by a background worker. The shared library does not depend on any external numerical or ML framework libraries. Model code itself will be compiled together with a number of math kernels (implementing e.g. matrix multiplications, convolutions, etc.) into a self contained component with a standardized interface.
\end{itemize}
\end{enumerate}
|
2024-02-18T23:40:24.622Z
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2021-12-30T02:23:41.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14460","language":"en","timestamp":1640831021000,"url":"https:\/\/arxiv.org\/abs\/2112.14460","yymm":"2112"}
|
proofpile-arXiv_000-10052
|
{"provenance":"002.jsonl.gz:10053"}
| null | null |
\section{Introduction}
In many time series anomaly detection applications one only has access to unlabeled data. This data is usually mostly nominal but may contain some (unlabeled) anomalies.
Examples of this setting are e.g.\ the widely used anomaly detection benchmarks SMAP, MSL \citep{Hundman2018telemanon},
and SMD \citep{omnianomaly}
This ``true'' unsupervised setting with \emph{mixed} data can be contrasted with the ``nominal-only'' setting, where one assumes access to ``clean'' nominal data.
In practice, techniques that (explicitly or implicitly) assume access to nominal data can often also successfully be applied to mixed data by assuming it is nominal, as long as the proportion of anomalies is sufficiently small., they are however biased by training on some anomalous data.
While some time series anomaly detection model rely on the one class classification paradigm which does not suffer from this assumption \cite{thoc, ncad},
the vast majority of the current time series anomaly detection methods are either forecasting methods \citep{shipmon2017time, zhao2020multivariate} or reconstruction methods \citep{omnianomaly, donut, lstmvae, Zhang2019mscred}. Forecasting methods detect anomalies as deviations of observations from predictions, while reconstruction methods declare observations that deviate from the reconstruction as anomalous.
In both cases, a probabilistic model of the observed data is assumed and its parameters are learned.
However, by training the model on the observed data which contains both normal and anomalous data points, the model ultimately learns the wrong data distribution.
\citet{ehrlich2021spliced} propose an approach to make the model robust to the anomalous points, still the aim is to learn the distribution of both the normal and the anomalous points.
We propose to address this issue using a simple technique based on latent indicator variables that can readily be combined with existing probabilistic anomaly detection approaches. By using latent indicator variables to explicitly infer which observations in the training set are anomalous, we can subsequently suitably account for the anomalous observations while training the probabilistic model.
Probabilistic models that use latent (unobserved) indicator variables to explicitly distinguish between nominal and anomalous data points are well-established in the context of robust mixture models \citep[e.g.][]{fraley1998many} and classical time series models \citep[e.g.][]{wang2018robust}. However, these techniques have not yet been utilized in the context of recent advances in \emph{deep} anomaly detection and time series modeling, presumably due to the (perceived) increased complexity of the required probabilistic inference and training procedure. We show that combining latent anomaly indicators with a Monte Carlo Expectation-Maximization (EM) \citep{wei1990monte} training procedure, results in a simple yet effective technique that can be combined with (almost) all existing deep anomaly detection and time series forecasting techniques.
We demonstrate the effectiveness of our approach with a simple model for anomaly detection on the Yahoo anomaly detection dataset and on the electricity dataset for forecasting from a noisy training set.
\section{Background}
For non-time series data, one common approach of formalizing the notion of anomalies is to assume that the observed data is generated by a mixture model \citep{ruff2020unifying}: each observation $\B{x}$ is drawn from the mixture distribution $p(\B{x}) = \alpha p^{+}(\B{x}) + (1-\alpha)p^{-}(\B{x})$, where $p^{+}(\B{x})$ is the distribution of the nominal data and $p^{-}(\B{x})$ the anomalous data distribution. Typically one assumes a flexible parametrized distribution for $p^+$ and a broad, unspecific distribution for $p^{-}$ (e.g.\ a uniform distribution over the extent of the data).
This mixture distribution can equivalently be written using a binary \emph{indicator latent variable} $z$ taking value $0$ with probability $p(z=0) = \alpha$ and value 1 with probability $p(z=1)=1-\alpha$, and specifying the conditional distribution
\begin{equation}
p(\B{x}|z) = \begin{cases}
p^{+}(\B{x}) & \text{if}\ z = 0\\
p^{-}(\B{x}) & \text{if}\ z = 1,
\end{cases}
\end{equation}
so that $p(\B{x}) = \sum_z p(\B{x}|z)p(z) = \alpha p^{+}(\B{x}) + (1-\alpha)p^{-}(\B{x})$. In this setup, anomaly detection can be performed by inferring the posterior distribution $p(z|\B{x})$ (and thresholding it if a hard choice is desired). Yet another way of representing the same model is generatively: first, draw $\B{y}^+ \sim p^+(\cdot)$, $\B{y}^- \sim p^-(\cdot)$, and $z \sim \text{Bernoulli}(1-\alpha)$, and then set $\B{x} = \mathbf{I}[z=0] \, \B{y}^+ + \mathbf{I}[z=1] \, \B{y}^-$, i.e.\ the observation $\B{x}$ is equal to $\B{y}^+$ if it is nominal ($z=1$) and equal to $\B{y}^-$ otherwise. Introducing the additional latent variables $\B{y}^+$ and $\B{y}^-$ is unnecessary in the IID setting, but becomes useful in the time series setting described next.
In time series setting, where the the observations are time series $\B{x}_{1:T} = \B{x}_1, \ldots, \B{x}_T$ that exhibit temporal dependencies, and anomalies are time points or regions within these time series, we have one anomaly indicator variable $z_t$ corresponding to each time point $\B{x}_t$. Like before, the nominal data is drawn from a parametrized probabilistic model $p_\theta^+(\B{y}_{1:T})$, and the anomalies are generated from a fixed model $p^-(\B{y}_{1:T})$. For time series data, the mixture data model then amounts to drawing $\B{y}_{1:T}^+ \sim p^+(\cdot)$, $\B{y}^-_{1:T} \sim p^-(\cdot)$, and $z_{1:T} \sim p^z(z_{1:T})$, and setting $\B{x}_t = \mathbf{I}[z_t=0] \, \B{y}^+_t + \mathbf{I}[z_t=1] \, \B{y}^-_t$.
\section{Method}
Forecasting or reconstruction models are designed to learn a model of $p^+(\cdot)$ but are typically trained directly on the observed time series $\B{x}_{1:T}$.
We propose to learn the model of $p^+(\cdot)$ only from $\B{y}_{1:T}^+$ by inferring $z_{1:T} \sim p^z(z_{1:T})$ on the training set.
This way we can train the model only on the observed points that are normal, the ones that are equal to $\B{y}_{1:T}^+$. Depending on the model, the anomalous points can be treated as missing or the normal point can be inferred.
\subsection{Models}
Each of the three latent time series is modeled with a probabilistic model:
a parametrized model $p^+_\theta$ of the nominal data $\B{y}_{1:T}^+$, a fixed model $p^-$ to model the anomalous data $\B{y}_{1:T}^-$, and a model $p^z$ of the indicator time series $z_{1:T}$.
\paragraph{Nominal Data Model}
Many existing deep anomaly detection methods aim to model the nominal data (e.g.\ \citep{shipmon2017time, zhao2020multivariate, omnianomaly, donut, lstmvae, Zhang2019mscred, ehrlich2021spliced}), and any of them can be used to model $\B{y}^+$, the latent nominal time series.
Our method is agnostic to the type of model used, so that it can be combined with any probabilistic time series model, be it a deep or shallow probabilistic forecasting method, a reconstruction method, or any other type of model.
We call the model of the latent normal time series $p^+_\theta$, which is parametrised by a set of parameters $\theta$.
In our experiments we demonstrate the general setup by modeling $p^+(\B{y}^+_{1:T})$ with a simple deep probabilistic forecasting model. We decompose $p(\B{y}^+_{1:T})$ into the telescoping product $p(\B{y}^+_0) \prod_{t=0}^{T} p(\B{y}^+_{t+1} | \B{y}^+_{t:0})$ and, making an $l$-th order Markov assumption, approximate it with a network $p(\B{y}^+_{t+1} | \B{y}^+_{t:t-l} ) = \mathcal{N}(f_\theta(\bm{y}_{t:t-l}), g_\theta(\bm{y}_{t:t-l}) )$ taking as input the last $l$ time points.
\paragraph{Anomalous Data Model}
A simple model can be used to model $p^-$, it does not need to take into account the time component as there are typically few anomalous points.
It can be modeled with a mixture of Gaussian distributions for example, with the risk of over-fitting to the few anomalies of the train set.
We simply model $p^-$ with a uniform distribution over the domain of the training data, not assuming any prior on the kind of anomalies that we may expect.
\paragraph{Anomaly Indicator Model}
We model the latent anomaly indicator with a Hidden Markov Model (HMM) with two states, state ${z}_t = 0$ corresponds to the point being normal and state ${z}_t = 1$ corresponds to the point being anomalous. Any kind of time series model parameterizing a Bernoulli distribution can be used to model the latent anomaly indicators, we pick an HMM as it encodes basic time dependencies while staying a simple model.
If it is available, prior knowledge about the dataset can be used to initialise the transition matrix.
The expected length of anomalous windows can be used to initialise the transition probability $p( z_{t+1} = 1 | z_t = 1 )$.
The expected percentage of anomalous points in the dataset can be used to initialise the transition probability $p( z_{t+1} = 1 | z_t = 0 )$.
\iffalse
is the prior probability of a point being anomalous, we can either:
\begin{enumerate}
\item set it to 0.01 by default as we assume that anomalies are rare (the one thing that we can assume in anomaly detection)
\item use a prior that we have for this specific dataset
\item have a prior given by labels on the dataset : e.g. 0.9 for points that are labeled as anomalous and 0.001 for the other ones
\end{enumerate}
$p_{+}^*(\bm{y}_{t+1}) = \mathcal{N}(f_\theta(\bm{y}_{t:t-l}), g_\theta(\bm{y}_{t:t-l}) )$
We model with a forecasting model, as of now, a Gaussian parametrised by a NN :
This model takes into account the time dependencies of the points, and so knowing which points are anomalous has two advantages:
\begin{enumerate}
\item we do not train the model to predict them (not including them in the loss)
\item we can replace/mask the anomalous points in the context window and so not have the model train on anomalous input
\end{enumerate}
\fi
\subsection{Training}
Our training procedure follows Monte Carlo EM \citep{wei1990monte}.
In the E-step we infer $ p^z(z_{1:T})$.
In the M-step we sample from $p^z(z_{1:T})$, using these samples to update $p^+_\theta$ and the transition matrix of the HMM.
Algorithm \ref{alg} sketches this procedure.
\begin{algorithm}[htpb]
\SetAlgoLined
\KwIn{Observed time series $\B{x}_{1:T}$, model to be trained $p^+_\theta$
}
\For{$e \in \{1, \dots, \text{numb\_epochs}\}$}{
\tcp{E-step:}
$\rightarrow$ infer $p^z(z_{1:T})$
\tcp{M-step:}
\For{$s \in \{1, \dots, \text{numb\_samples}\}$}{
$\rightarrow$ sample indicator time series $z_s$ from $p^z(z_{1:T})$
$\rightarrow$ perform one epoch of $p^+_\theta$ on $\B{x}_{\neg z_s}$ where the points at sampled anomalous indices are replaced
}
$\rightarrow$ update the transition matrix of the HMM
}
\caption{Monte Carlo EM for Latent Anomaly Indicator}
\label{alg}
\end{algorithm}
\subsubsection{E-step}
We infer $p^z(z_{1:T})$ by using the standard forward-backward algorithm for HMMs, using the following distributions:
\begin{align}
p(\B{x}_t | z_t = 0) = p^+_\theta(\B{x}_t) \\
p(\B{x}_t | z_t = 1) = p^-(\B{x}_t)
\end{align}
and $p(z_{t+1} | z_t)$ is given by the HMM transition matrix.
\iffalse
(we abuse a bit the notation here: $p(p_- | \bm{y}_t)$ is the posterior probability that the point t comes from the anomalous data distribution)
For each point, we want to infer the probability of this point coming from the anomalous model:
$$p(p_- | \bm{y}_t) = \frac{p(\bm{y}_t | p_-) p(p_-)}{p(\bm{y}_t)} = \frac{p(\bm{y}_t | p_-) p(p_-)}{p(\bm{y}_t | p_-) + p(\bm{y}_t | p_+) } $$
First, simply using Bayes rule, then using the fact that we have: $p(\bm{y}) = p_{+}(\bm{y}) + p_{-}(\bm{y})$
are obtained with our trained model for the normal data and the model for the anomalous data
\fi
\subsubsection{M-step}
We want to train $p^+(\cdot)$ only from $\B{y}_{1:T}^+$. As most models may not allow for an analytical update using $\bm{x}_{1:T}$ and $z_{1:T}$, we propose to a Monte Carlo approximation of the expectation under $p^z(z_{1:T})$.
We draw multiple samples from $p^z(z_{1:T})$ giving us possible normal points on which $p^+_\theta$ can be trained.
Each path sampled gives us a set of observed points that can be considered as coming from the normal data distribution $p^+$.
We maximise the probability of these points under $p^+_\theta$, treating the points coming from $p^-$ points as missing.
Depending on the choice of model for $p^+_\theta$, one may not be able to simply ignore anomalous points and they would have to be imputed. For deep forecasting or reconstruction models for example the model has to be given an input for each time point.
In these cases, we propose to impute the point with the forecast or reconstruction obtained from $p^+_\theta$ at the last M-step.
This way, we use $p^+_\theta$ to infer the time points of $ \B{y}_{1:T}^+$ that were not observed.
With this method we can recover the full $\B{y}_{1:T}^+$ time series and train $p^+_\theta$ on it.
Depending on the choice of model for $p^-$, one can update it using the points that are sampled as coming from $ \B{y}_{1:T}^-$.
We can update the transition matrix of the HMM with the classical M-step. The average number of transitions from one state to the next in the samples from $p^z(z_{1:T})$ become the new transition probabilities.
\iffalse
We train the forecasting model by masking the inferred anomalous points from the input vector.
We sample independently for each time point if it comes from or not. If we sample it, we replace it by the predicted mean $ f_\theta(\bm{y}_{t:t-l})$ obtained at the last epoch.
In addition we ignore the loss for the points that we sample as coming from the anomalous distribution.
This procedure allows to use mini batch training very easily:
\begin{enumerate}
\item we sample the points to ignore for the whole training set
\item we replace them in the input time series
\item we can do minibatches using this modified input time series
\end{enumerate}
For each training epoch, we sample 20 time from the distribution.
\fi
\subsection{Inference}
At inference time, we propose to use the HMM to perform filtering on $z$ and infer if incoming points are more likely to be drawn from $p^+$ or $p^-$.
If an incoming point $\B{x}_t$ is more likely to be coming from $p^-$ it can be treated as missing or replaced with a sample from $p^+_{\theta t}$ or by its mode.
This way we ensure that the trained model is only used on points coming from $\B{y}_{1:T}^+$.
\section{Experiments}
\iffalse
\fi
We make our code available with an illustration notebook.
\footnote{ \url{https://github.com/Francois-Aubet/gluon-ts/blob/monte_carlo_em_masking_ notebook/src/gluonts/nursery/anomaly_detection/Monte-Carlo-EM- for-Time-Series-Anomaly-Detection-demo-notebook.ipynb}}
\paragraph{Model}
We evaluate our approach with a simple forecasting model on both anomaly detection and forecasting tasks. We show the performance of the model when trained in a standard way and when trained with our procedure, which we call our procedure Latent Anomaly Indicator (LAI).
We use a simple Multi-Layer Perceptron (MLP) model to parametrise the mean and the variance of a predictive Gaussian distribution. It takes as input the last 25 points.
\paragraph{Datasets}
For the anomaly detection evaluation, we use the \textbf{Yahoo} dataset, published by Yahoo labs.\footnote{\url{https://webscope.sandbox.yahoo.com/catalog.php?datatype=s&did=70}}
It consists of 367 real and synthetic time series, divided into four subsets (A1-A4) with varying level of difficulty.
The length of the series vary from 700 to 1700 observations.
Labels are available for all the series.
We use the last 50\% of the time points of each of the time series as test set, like \citep{ren2019time} did, and split the rest in 40\% training and 10\% validation set.
We evaluate the performance of the model using the adjusted F1 score proposed by \citet{donut} and subsequently used in other work.
In addition, we evaluate the method on forecasting tasks using the commonly used \textbf{electricity} dataset \cite{electricity}, composed of 370 time series of 133k points each. Given the length of the dataset, we sub-sample it by a factor 10. We select the last 50\% of the points of each time series for testing.
We scale each time series using the median and inter-quartile range on the train~set.
\subsection{Visualization on synthetic data}
Figure \ref{fig:synthetic_ts} visualizes the advantage of the method on a simple sinusoidal time series with the simple MLP for $p^+_\theta$. We generate a synthetic time series and inject outliers in it.
We observe that our approach allows to train the model $p^+_\theta$ while ignoring the outliers in the data,
whereas the outliers heavily influence the model trained conventionally.
We observe from figure \ref{fig:masking_pi} that the model is able to infer accurately which of the training points are likely to be anomalous.
\begin{figure}[t]
\begin{subfigure}{.5\textwidth}
\includegraphics[width=.99\textwidth]{figures/normal_model_fit}
\caption{The fit of a simple MLP without LAI.}
\label{fig:normal_model_fit}
\end{subfigure}
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=.99\textwidth]{figures/masking_model_fit}
\caption{The fit of a simple MLP with LAI.}
\label{fig:masking_model_fit}
\end{subfigure}
\begin{subfigure}{.5\textwidth}
\includegraphics[width=.99\textwidth]{figures/masking_pi}
\caption{The time series of latent anomaly indicator $p(z_t = 1)$.}
\label{fig:masking_pi}
\end{subfigure}
\caption{ We fit a MLP on this simple synthetic time series with anomalies. (a) shows the fit of the model trained in a conventional way, (b) shows the fit of the model trained as we propose to, (c) show the inferred $p(z_{1:T})$ distribution at the end of the training.
}
\label{fig:synthetic_ts}
\end{figure}
\subsection{Time series anomaly detection}
Table \ref{table:test_set} shows the F1 score of the model with and without LAI on the different subsets of the Yahoo dataset.
We train one MLP on each of the time series and average the F1 scores obtained on the different time series of the subset.
We observe that using our approach greatly improves the performance of the model.
\begin{table}[h]
\caption{F1 score on the different subsets of the Yahoo dataset.}
\label{table:test_set}
\centering
\begin{tabular}{l|cccccccccccc}
\toprule
Model & A1 & A2 & A3 & A4 \\
\midrule
MLP & 33.64 & 53.28 & 63.25 & 47.30 \\
MLP + LAI & 41.84 & 87.26 & 87.91 & 61.62 \\
\bottomrule
\end{tabular}
\end{table}
\iffalse
ARIMA:
Yahoo A1:
Normal average: 0.3363546742021299
LAI average: 0.4183863827943147
Yahoo A2:
Normal average: 0.5328321386575404
LAI average: 0.8726312951525773
Yahoo A3:
Normal average: 0.6324923474686052
LAI average: 0.8790611366231169
Yahoo A4:
Normal average: 0.4730399846547243
LAI average: 0.6161590733402865
LSTM:
\fi
In addition to the improved F1 score, we compare the inferred anomalous points on the training set with the actual labeled anomalous points.
Table \ref{table:train_set} shows the F1 score on the training set when using the anomaly indicator as anomaly score.
We observe that our method allows to find accurately the anomalies present in the training set.
While the training and test sets are different, we propose that the higher F1 on the train set is due to the fact that the model can use the whole training set to infer if a point is anomalous, and not only the past points.
\begin{table}[H]
\caption{F1 score on the training set the different subsets of the Yahoo dataset using the inferred $p(z_{t}=1)$ as anomaly score.}
\label{table:train_set}
\centering
\begin{tabular}{l|cccccccccccc}
\toprule
Model & A1 & A2 & A3 & A4 \\
\midrule
MLP + LAI & 59.48 & 94.02 & 81.89 & 73.77 \\
\bottomrule
\end{tabular}
\end{table}
\subsection{Forecasting using a corrupted train set}
Our method can be used more generally to train a forecasting model on a forecasting dataset containing anomalies.
We take the electricity forecasting dataset and inject point outliers in the training set so that about 0.4\% of the training point have an added or subtracted spike.
Table \ref{table:forecasting} shows the mean absolute error (MAE) on the test set in the setting where the original train set is used and in the setting where the noisy train set is used.
We see that using our method allows to reduce significantly the increase in error from the outliers in the training set, only 0.0146 increase in the mean absolute error versus 0.0542 when training the model normally.
\begin{table}[H]
\caption{MAE on electricity with and without injecting point outliers in the train set}
\label{table:forecasting}
\centering
\begin{tabular}{l|cccccccccccc}
\toprule
Model & electricity & electricity + outliers \\
\midrule
MLP & 0.1551 & 0.2092 \\
MLP + LAI & 0.1558 & 0.1704 \\
\bottomrule
\end{tabular}
\end{table}
\section{Conclusion}
We present LAI, a method that can be used to wrap any probabilistic time series model to perform anomaly detection without being impacted by unlabeled anomalies in the training set.
We present the details of the approach and propose preliminary empirical results on commonly used public benchmark datasets.
The approach seems to greatly help both for anomaly detection tasks and for training a forecasting model on a contaminated training set.
One can extend this work by wrapping other bigger models such as OmniAnomaly \citep{omnianomaly} or state-of-the-art forecasting models \citep{benidis2020neural}.
Finally, with our current method at inference time, one has to decide at each incoming point if it is to be replaced or not, one could use particles which would mimic the Monte Carlo approach of the training time.
\newpage
\
\newpage
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{"paloma_paragraphs":[]}
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{"arxiv_id":"2112.14436","language":"en","timestamp":1640830941000,"url":"https:\/\/arxiv.org\/abs\/2112.14436","yymm":"2112"}
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proofpile-arXiv_000-10053
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{"provenance":"002.jsonl.gz:10054"}
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\section{Introduction}
Let $N$ be an \emph{open} necklace
with the same number of beads belonging to each of $t$ classes,
and suppose that this number is divisible by $k$.
As was proved in \cite{Al}, it is always possible to cut $N$ in at most
$(k-1)t$ points and distribute the resulting intervals
into $k$ collections, each containing the same number
of beads of each type. This is tight for all values of the
parameters, as shown by any necklace in which the beads
of each type appear contiguously. A possible interpretation of the
result is the following. Suppose that $k$ mathematically oriented thieves
want to fairly distribute a necklace among
each other. If the necklace is comprised of the same number of beads belonging to $t$
different types (colors) and this number is a multiple of $k$, then they
can always achieve this by making at most
$(k-1)t$ cuts between beads.
\smallskip
The aim of the present paper is to study this problem for
random necklaces. The random model we consider here is a
necklace of total length $n=ktm$ consisting of exactly $km$ beads
of type $i$ for each $1 \leq i \leq t$, chosen uniformly among all
intervals of $n$ beads as above. Call a set of cuts
of such a necklace {\em fair}, if it is possible to split the resulting
intervals into $k$ collections, each containing exactly $m$ beads
of each type. For a necklace $N$, let
$X=X(N)$ be the minimum number of cuts in a fair collection. When
$N$ is chosen randomly as above, $X$ is a random variable which
we denote by $X(k,t,m)$. By the result of \cite{Al} mentioned
above, we have $X(k,t,m) \leq (k-1)t$ with probability $1$. Our
objective is to study the typical behavior of the random
variable $X=X(k,t,m)$. All results presented here are asymptotic, where at
least one of the three variables $k,t,m$ tends to infinity. As
usual, we say that
a result holds {\em with high probability} ({\em whp}, for
short), if the probability that it holds tends to $1$, when the relevant
parameter(s) tend to infinity.
The problem of determining the asymptotic behavior of $X(k,t,m)$
turns out to be connected to several seemingly unrelated topics,
including matchings in nearly regular hypergraphs and random walks
in Euclidean spaces. For some values of the parameters
$k,m$ and $t$,
we were able to determine
this behavior quite accurately,
showing that
the typical number of required cuts is sometimes significantly smaller
than the deterministic upper bound $(k-1)t$ which is always valid.
We start with the following observation.
\begin{prop}
\label{p11}
For every fixed $k$ and $t$, as $m$ tends to infinity, we have
$X=X(k,t,m) \geq\left\lceil \frac{(k-1)(t+1)}{2}\right\rceil$ whp.
\end{prop}
In our main result, we describe the asymptotic behavior of
$X=X(k,t,m)$ for two thieves ($k=2$) and any fixed number of types
$t$, as $m$ tends to infinity.
\begin{theo}\label{thm:big m}
Let $t$ be a fixed positive integer and $m\to \infty $.
\begin{enumerate}
\item
For all $1\le s < \frac{t+1}{2}$, we have that
\begin{equation}
\mathbb P \big( X(2,t,m)=s \big) =
\Theta \big( m ^{s-\frac{t+1}{2}} \big).
\end{equation}
\item
When $t$ is odd and $s=\frac{t+1}{2}$, we have
\begin{equation}
\mathbb P \big( X(2,t,m)=s \big) =
\Theta \Big( \frac{1}{\log m} \Big).
\end{equation}
\item
For all $\frac{t+1}{2} <s\le t $, we have that
\begin{equation}
\mathbb P \big( X(2,t,m)=s \big) =\Theta (1).
\end{equation}
\end{enumerate}
\end{theo}
The most surprising aspect of the last result and, in fact, of the present paper is that the distribution of the values of
$ X(2,t,m)$ is not concentrated: they are spread over the interval $(\frac{t+1}2, t]$, each value being assumed with positive probability.
We also consider the case $m=1$, in which every thief should get a
single bead of each type.
\begin{theo}
\label{t14}
For $t$ and $k/ \log t$ tending to infinity,
the random variable $X=X(k,t,1)$ is
$o(kt)$ whp.
\end{theo}
For $k=2$, $m=1$ and large $t$, we prove
\begin{theo}
\label{t15}
The random variable
$X=X(2,t,1)$ is at least $2H^{-1}(1/2)t-o(t)=0.220...t -o(t)$ whp, where
$H^{-1}(x)$ is the inverse of the binary entropy
function $H(x)=-x \log_2 x -(1-x) \log_2 (1-x)$ taking values
in the interval $[0,1/2]$.
On the other hand, $X\le 0.4 t+o(t)$ holds whp.
\end{theo}
The upper bound above was obtained jointly with Ryan Alweiss,
Colin Defant and Noah Kravitz.
We show that both the upper and the lower bounds can be
slightly improved, but the arguments leave a considerable gap
between the two estimates. We further prove that
the probability that $X(2,t,1)$ deviates from its expectation by
at least $C \sqrt t$ is at most $O(e^{-\Omega(C^2)})$.
\smallskip
The rest of this paper is organized as follows. In the next section,
we analyze the case where $k$ and $t$ are fixed. We start with the
simple proof of Proposition~\ref{p11}. Since the proof of
Theorem~\ref{thm:big m} is rather long and technical,
in Subsection~\ref{ss2.1}, we settle the special case $t=3$,
which requires similar ideas but is considerably simpler.
The argument, in its full generality,
encompasses the following three subsections.
In Section~\ref{section3}, we study the case where each thief
gets one bead of each type and the number of types, $t$, tends
to infinity and present the proofs of Theorems~\ref{t14}
and~\ref{t15}. We conclude our paper with
several remarks and open problems.
To simplify the presentation, throughout the
paper, we ignore all floor and ceiling signs, whenever they are not
crucial.
\smallskip
\section{Fixed $k$ and $t$: Proof of Theorem~\ref{thm:big m}}
Let $N$ be a random open necklace with $km$ beads in each of the $t$
types. Let $P$ be a {\em partition of the positions}
of the beads into $k$ parts.
We call
$P$ \emph{balanced} if each part contains $tm$ beads and we call it
\emph{fair} if each part contains exactly $m$ beads of each type. Thus,
only balanced partitions can be fair. Let $k=2$ and consider two balanced
partitions $P_1$ and $P_2$. For a part $H_1$ of $P_1$, we can choose a
part $H_2$ of $P_2$ with $q:=|H_1\setminus H_2|\le tm/2$. We call $q$
the \emph{distance} of the balanced partitions $P_1$ and $P_2$. (Note
that the distance is symmetric and does not depend on the choice of the
part $H_1$ of $P_1$.)
We will use the first and second moment methods. Therefore, we need to
estimate the probability that a given partition is fair and that two given
partitions are both fair. The asymptotic notation in this section refers
to behavior as $m$ goes to infinity, while parameters $k$, $t,$ and $s$
(where appropriate) remain fixed.
\begin{claim}\label{12}
\begin{itemize}
\item[(i)] The probability $P(k,t,m)$ that a fixed balanced partition is fair is
$$(1+o(1))\frac{t^{(k-1)/2}k^{(t-1)/2}}{(2\pi m)^{(k-1)(t-1)/2}}=\Theta_{k,t}(m^{-(k-1)(t-1)/2}).$$
\item[(ii)] If $k=2$ and the distinct balanced partitions $P_1$ and $P_2$ have distance $q$, then we have
$$\mathbb P(P_1, P_2\mbox{ are fair})=\Theta_t((qm)^{-(t-1)/2}).$$
\end{itemize}
\end{claim}
\noindent{\bf Proof:}\, First, we show part (i). The total number of necklaces is
$(ktm)!/((mk)!)^t$. For any fixed balanced partition, the number of necklaces making it fair is $((mt)! )^k/ (m!)^{tk}$. Therefore, we have
$$
P(k,t,m)=\frac{((mt)! )^k/ (m!)^{tk}}{(ktm)!/((mk)!)^t}=(1+o(1))\frac{t^{(k-1)/2}k^{(t-1)/2}}{(2\pi m)^{(k-1)(t-1)/2}}
$$
The claimed estimate follows by Stirling's formula.
\smallskip
For (ii), choose parts $H_1$ of $P_1$ and $H_2$ of $P_2$ with
$|H_1\setminus H_2|=q$. Assuming $P_1$ is fair, then $P_2$ is also fair
if and only if in each color, $H_1\setminus H_2$ and $H_2\setminus H_1$
contain the same number of beads. Note that the distributions of the
colors in these two intervals are independent so-called {\em multivariate
hypergeometric distributions}. In this case, we choose a uniform random
$q$-set from a base set containing $m$ beads from each of the $t$
types. The conditional probability $\mathbb P(H_2\mbox{ is fair}\mid H_1\mbox{
is fair})$ is, therefore, the same as the probability that two independent
variables from the same multivariate hypergeometric distributions
agree. This is easily seen to be $\Theta_t(q^{(1-t)/2})$. For
completeness, we sketch the proof of this statement below.
The random sample can be characterized by the numbers
$q_i$ of beads of type $i$, for every $i$. We must
have $q_i\ge0$, $\sum_{i=1}^tq_i=q$. If this holds, then the probability of this specific outcome is
$$P_{q_1,\dots,q_t}=\frac{(mt-q)!q!/
\prod_{i=1}^t((m-q_i)!q_i!)}{(mt)!/(m!)^t}.$$
To prove the upper bound $O_t(q^{(1-t)/2})$ for the probability
that two independent samples agree, it is enough to notice that
$P_{q_1,\dots,q_t}=O_t(q^{(1-t)/2})$ holds for any sequence $(q_1,\dots,q_t)$.
For the lower bound, note that each $q_i$ has a hypergeometric
distribution with mean $q/t$ and variance $O_t(q)$, so with a
constant probability the numbers $q_i$ will simultaneously satisfy
$|q_i-q/t|=O_t(\sqrt q)$ for every $i$. There are $O_t(q^{(t-1)/2})$ such
integer vectors also satisfying $\sum_{i=1}^tq_i=q$, so the collision
probability is $\Omega_t(q^{(1-t)/2})$. This proves the bound and the
claim.\hfill$\Box$
\medskip
The proof of Proposition~\ref{p11} is a simple first
moment argument based on part (i) of the previous claim.
\medskip
\noindent
{\bf Proof of Proposition \ref{p11}:}\, Let $N$ be an open necklace
consisting of $km$ beads of each of the $t$ types.
We estimate the number $n(s,k,m,t)$ of balanced partitions that can be
obtained by $s$ cuts of the open necklace. Note that $s$ cuts result in
$s+1$ intervals, and the partition can be reconstructed from the ordered
list of the lengths of theses intervals together with the information which part of
the partition contains which interval. In fact, we can save by not
specifying the length of the last interval in each part, because it can
be computed from the lengths of the other intervals in the same part, as
the partition must be balanced. The intervals have lengths at most $tm$,
so we have $$n(s,k,m,t)\le k^{s+1}(tm)^{s+1-k}=O_{s,k,t}(m^{s-k+1}).$$
From Claim~\ref{12}(i),
the expected number of fair partitions obtained by $s$ cuts is
$$
n(s,k,m,t) P(k,t,m)=O_{k,s,t}( m^{s-k+1-(t-1)(k-1)/2})
=O_{k,s,t}( m^{s-(t+1)(k-1)/2}).
$$
For any fixed $k$ and $t$, if $s< (t+1)(k-1)/2$, then the above
expectation tends to $0$ as $m$ tends to infinity. This implies the
assertion of Proposition~\ref{p11}. \hfill $\Box$
\medskip
\subsection{Proof of Theorem~\ref{thm:big m} for $t=3$}\label{ss2.1}
The proof of Theorem~\ref{thm:big m} in its full generality is rather lengthy and technical.
In the present subsection, we settle the special case $t=3$. For this special case, the argument is much simpler. However, it is based on some of the same ideas as the general proof.
One cut is sufficient to fairly distribute the random necklace $N$
(that is, $X(N)=1$) if and only if the partition of $N$ into its
first and second halves is fair, which has probability $\Theta(1/m)$
by Claim~\ref{12}(i). According to the (deterministic)
result of Alon~\cite{Al}, for every $N$, we have $X(N) \leq 3$. Thus, it remains to show that
the probability that $X(N)\le2$ is $\Theta(1/\log m)$. In order to
estimate this probability, note that two cuts suffice if and only if
there is a balanced partition of $N$ into two cyclic intervals that is
fair. There are exactly $3m$ balanced partitions into two cyclic
intervals. For $0\le i<3m$, we denote by $P_i$ the balanced partition
into an interval starting at position $i+1$ and ending at position $i+3m$,
and its complement.
Let $Y=Y(N)$ denote the random variable counting the number of fair
partitions into cyclic intervals. Clearly, $X(N) \leq 2$ if and only if $Y$
is positive. We first establish a lower bound for the probability of this
event using the second moment method. The random variable $Y$ is a sum
of $3m$ random variables $Y_i$, where $Y_i$ is the indicator variable for
$P_i$ being fair. Each of these indicator random variables has expectation
$P(2,3,m)=(1+o(1))\frac{\sqrt3}{\pi m}$ by Claim~\ref{12}(i), so
$$
\mathbb E(Y)= \sum_{i=0}^{3m-1}\mathbb E(Y_i)=3m \cdot (1+o(1)) \frac{\sqrt 3}{\pi m}=
(1+o(1)) \frac{3\sqrt 3}{\pi}=\Theta(1).
$$
The expected value of $Y^2$ is
$$
\sum_{0 \leq i, j \leq 3m-1}
\mathbb E(Y_iY_j),
$$
where the sum is taken over ordered pairs. For $i=j$, we have
$\mathbb E(Y_i^2)=\mathbb E(Y_i)=P(2,3,m)=\Theta(1/m)$ by Claim~\ref{12}(i). If $i\ne
j$ we have $\mathbb E(Y_iY_j)=\Theta(1/(mq))$ by Claim~\ref{12}(ii), where
$q$ is the distance between the corresponding partitions. Note that
$q=\min(|i-j|,3m-|i-j|)$. Therefore, for all $1\le q<3m/2$, we have $6m$
pairs $(i,j)$ with $q$ being the distance between $P_i$ and $P_j$. If
$m$ is even, we also have the special case $q=3m/2$ with half as many
terms. We can ignore this special case, as we are only interested in
the order of magnitude. Therefore,
$$
\mathbb E(Y^2) =3m\cdot\Theta\left(\frac1m\right)+ 6m \sum_{1 \leq q \leq 3m/2}
\Theta\left(\frac{1}{mq}\right)=\Theta (\log m).
$$
By the Paley-Zygmund Inequality \cite{PZ}, \cite{PZ1} or, equivalently,
by the Chung-Erd\H{o}s Inequality \cite{CE},
$$
\mathbb P(Y>0) \geq \frac{\mathbb E(Y)^2}{\mathbb E(Y^2)}=\Theta\left(\frac{1}{\log
m}\right).
$$
We next prove an upper bound for the probability that $Y$ is positive.
To this end, we define another random variable $Z=Z(N)$. We will show
that $Z$ is positive with probability $O(1/\log m)$, and the probability
that $Y$ is positive but $Z$ is not, is even lower. The crucial step in
bounding the probability that $Z$ is positive, is the analysis of the
probability that an appropriate two-dimensional random walk does not
return to the origin in a certain number of steps. For this, we apply
a slightly modified version of a classical argument of Dvoretzky and
Erd\H{o}s (\cite{DE}, see also \cite{ET}, \cite{Re}). This is the subject
of Claim~\ref{p2} below.
Let $Z=Z(N)$ denote the number of fair partitions into two cyclic
intervals such that if we shift the parts by at most, say,
$s=\left\lceil \sqrt m\,\right\rceil$ positions to the right, then the
resulting partition is no longer fair. Note, first, that if $Y$ is positive
and $Z$ is zero, then in every set of $s$ consecutive balanced partitions
into two cyclic intervals, there is at least one fair partition. However,
in this case, we have $Y\ge3m/s>\sqrt m$ and, as the expectation of $Y$
is $O(1)$, the probability of this event is $O(1/\sqrt m)$.
Next, we need to bound the probability that $Z$ is positive. For this,
we use a first moment (union bound) argument. The variable $Z$ is the
sum of $3m$ indicator variables and, by symmetry, these variables have
the same expected value. Therefore, we have
\begin{eqnarray*}
\mathbb P(Z>0)&\le&\mathbb E (Z)\\
&=&3m\cdot \mathbb P(P_0\mbox{ is fair, but no $P_i$ is fair for }1\le i\le s)\\
&=&3m\PP_1\PP_2,
\end{eqnarray*}
where $\PP_1=\mathbb P(P_0\mbox{ is fair})$ and $\PP_2=\mathbb P(\mbox{no $P_i$ is fair for }1\le i\le s\mid P_0\mbox{ is fair})$. We have $\PP_1=\Theta(1/m)$, by Claim~\ref{12}(i),\ and $\PP_2=O(1/\log m)$, by Claim~\ref{p2} below. All this yields
$$\mathbb P(Z>0)=3m\cdot\Theta(1/m)\cdot O(1/\log m)=O(1/\log m).$$
Combining this bound with our earlier estimate for the probability that $Y>0$ and $Z=0$, implies that
$$\mathbb P(Y>0)\le O(1/\log m)+O(1/\sqrt m)=O(1/\log m).$$
This completes the proof of the theorem
for $t=3$,
modulo Claim~\ref{p2}, which we still have to establish.\hfill$\Box$
\medskip
\begin{claim}\label{p2}
The conditional probability $\PP_2$ defined above satisfies $\PP_2=O(1/\log m)$.
\end{claim}
\noindent{\bf Proof:}\, To evaluate $\PP_2$, we will always assume that
$P_0$ is fair and consider $N$ to be a random necklace satisfying this
condition. For simplicity, we call the three types of beads blue, red, and green, respectively.
For $1\le i\le s$, let $b_i$ be the (signed) difference
between the number of blue beads in positions $1$ through $i$ and
the number of blue beads in positions $3m+1$ through $3m+i$. Let
$r_i$ be similarly calculated for the red beads. Clearly, $P_i$
is fair if and only if $b_i=r_i=0$. We consider the vectors $(b_i,r_i)$,
as locations of a random walk starting at $(b_0,r_0)=(0,0)$. The steps
$(b_i,r_i)-(b_{i-1},r_{i-1})$ can be calculated as the difference between
two vectors from the set $S=\{(0,0), (1,0),(0,1)\}$, one corresponding
to the color of bead at position $i+3m$, the other to the color of the
bead at position $i$.
For $1\le i\le s$, let $P(i)$ denote the probability that this walk
returns to the origin after $i$ steps, that is, $b_i=r_i=0$ (equivalently,
$P_i$ is fair), and let $Q(i)$ denote the probability that the random
walk does not return to the origin for $i$ steps. By Claim~\ref{12}(ii),
we have $P(i)=\Theta(1/i)$. We also have $\PP_2=Q(s)$.
It is simpler to analyze the random walk if it is memoryless, that is,
if the steps are independent random variables. This is not exactly the
case, but is ``almost true.'' Let the random variables $b'_i$ and $r_i'$
be calculated in the same way after we fill all the positions from
$1$ to $s$ and from $3m+1$ to $3m+s$ with beads of independently and
uniformly distributed random colors. In this case, we have a memoryless
random walk that starts at the origin, and each step can be obtained
as the difference of two vectors from $S$ selected uniformly at random.
A fixed arrangement of $b$ blue, $r$ red, and $g$ green beads in
the first $s$ positions ($b+r+g=s$) has probability exactly $1/3^s$
in the second model, while its probability in the first model is
$$\frac{(3m-s)!/((m-b)!(m-r)!(m-g)!)}{(3m)!/(m!)^3}=\left(\frac13+O\left(\frac sm\right)\right)^s=\Theta\left(\frac1{3^m}\right).$$
A similar statement is true for the colors in positions $3m+1$ through $3m+s$, and in both models, the color arrangements in the two intervals are independent. Therefore, the probabilities of the same event in the two models differ by at most a constant factor.
For $1\le i\le s$, we define $P'(i)$ as the probability that this memoryless random walk returns to the origin after $i$ steps and $Q'(i)$ as the probability that the modified random walk does not return for $i$ steps. By the above argument, we have $P'(i)=\Theta(P_i)=\Theta(1/i)$ and $\PP_2=\Theta(Q'(s))$.
In the following calculation, we split the possible walks according
to their last visit at the origin and use $P'(0)=Q'(0)=1$.
$$1 = \sum_{i=0}^s P'(i) Q'(s-i)\ge Q'(s)\sum_{i=0}^sP'(i),$$
where the inequality comes from the fact that $Q'(i)$ is a monotone
decreasing function of $i$. We have
$$\sum_{i=0}^s P'(i) =1+\sum_{i=1}^s\Theta\left(
\frac{1}{i}\right) = \Theta( \log s)=\Theta(\log m).
$$
Thus, $\PP_2=\Theta(Q'(s))=O(1/\log m)$, as needed.\hfill$\Box$
\medskip
\subsection{The second moment method}
In this subsection, we prove part (1) of Theorem~\ref{thm:big m} as well as the lower bound in part (2) of Theorem~\ref{thm:big m}.
Let $N$ be a random open necklace with $2m$ beads of each of $t$ types.
Fix an integer $s \leq t$ for the number of cuts.
Recall that in the proof of Proposition~\ref{p11}, we calculated the number of balanced partitions achievable with $s$ cuts as $n(s,2,t,m)=O_{s,t}(m^{s-1})$. However, here we can be more precise. We call a balanced partition achievable by $s$ cuts but not with fewer cuts an $s$-cut partition. In an $s$-cut partition the necklace is cut into $s+1$ non-empty segments and these segments alternate
between the two participants. Taking the odd numbered intervals we obtain an arbitrary partition of $tm$ beads to $\lceil(s+1)/2\rceil$ nonempty intervals while the even numbered intervals partition $tm$ beads to $\lfloor(s+1)/2\rfloor$ nonempty intervals. Therefore, the exact number of $s$-cut partitions is
$$
n'(s,t,m)= {{tm -1} \choose\lceil(s-1)/2\rceil} \cdot {{tm -1} \choose\lfloor(s-1)/2\rfloor}=\Theta_{s,t}(m^{s-1}).
$$
By Claim~\ref{12}(i), the probability that a fixed balanced partition
is fair is $P(2,t,m)=\Theta_t(m^{-(t-1)/2})$.
Let $Y$ be the random variable counting the
number of fair $s$-cut partitions. By linearity of
expectation and the estimates above we have
\begin{equation}\label{eq:expectation of Y}
\mathbb E (Y)=n'(s,t,m) P(2,t,m)=\Theta_{s,t}(m^{s-(t+1)/2}).
\end{equation}
It follows from Markov's inequality that
\begin{equation}\label{eq:upper bound with s cuts}
\mathbb P \big( X(2,t,m)=s \big) \le \mathbb P (Y>0 ) \le \mathbb E(Y) = O_{s,t} \big( m^{s-\frac{t+1}{2}} \big) .
\end{equation}
This finishes the proof of the upper bound in part (1) of Theorem~\ref{thm:big m}.
We note that for odd $t$ and $s=(t+1)/2$, the above expectation is
$\Theta_t(1)$ and therefore the upper bound in part (2) of
Theorem~\ref{thm:big m} does not follow from
Markov's inequality as in part (1). We will present the considerably more involved proof of that upper bound in Subsection~\ref{random walks}.
We proceed to estimate the second moment of $Y$, the number of fair $s$-cut partitions. For an $s$-cut partition $P$ let $Y_P$ denote the
indicator random variable whose value is $1$ if and only if $P$ is fair.
By Claim~\ref{12}(ii), for any two distinct balanced
partitions $P$ and $P'$, the probability that both $P$ and $P'$ are fair is
$$
\mathbb E(Y_PY_{P'})=\Theta_t((mq)^{-(t-1)/2}),
$$
where $q$ is the distance between $P$ and $P'$. For $0<q \le tm/2$,
let $n^*(q,t,m,s)$ denote the number of ordered pairs of $s$-cut
partitions $(P,P')$, such that the distance between $P$ and $P'$ is $q$.
To estimate the number $n^*(q,t,m,s)$,
consider the collection of $2s$ cuts of both partitions, where if
both contain a cut at the same point we take it twice. These cuts
partition the interval of beads into $2s+1$ pairwise disjoint intervals
(including possibly some empty intervals, when the two partitions
share a cut). Let us select a part $H$ of $P$ and a part $H'$ of $P'$ such that $|H\setminus H'|=q$. The non-empty intervals can be classified as follows: the ones belonging to $H \cap H'$, the ones belonging to $H \setminus H'$, to
$H' \setminus H$, and those not in $H \cup H'$. The total number of beads in
the intervals of the first type is $|H\cap H'|=tm-q$, and this is also the
total number of beads in the intervals of the fourth type. The
number of beads in intervals of the second type is $|H\setminus H'|=q$, and so is
the number of beads in intervals of the third type. Call the first
and fourth types {\em even} and the second and third {\em odd} (this indicates the
parity of the number of sets among $H,H'$ to which the
corresponding interval belongs). With appropriate classification of the empty intervals into one of the four types one can make sure that even and odd intervals alternate so we either have $s+1$ even and $s$ odd intervals or vice versa.
We can reconstruct both partitions $P$ and $P'$ from the ordered list of
types and lengths of all the $2s+1$ intervals. In fact, we can save by
not specifying the length of the last interval in each type as that can be
computed from the lengths of the other intervals. We clearly have at most
$4^{2s+1}$ possibilities for the sequence of types. Even intervals have
lengths between $0$ and $tm-q$ and odd intervals have lengths between $0$
and $q$. So, in total, we have at most $4^{2s+1}(tm-q+1)^{s-1}(q+1)^{s-2}$
possibilities if there are $s+1$ odd and $s$ even intervals, and at most
$4^{2s+1}(tm-q+1)^{s-2}(q+1)^{s-1}$ in the reversed case. We have
$$n^*(q,t,m,s)=O_{s,t}(m^{s-1}q^{s-2}),$$
because the estimate holds for both of
these numbers if $s$ and $t$ are fixed.
Recall that the random variable $Y$ can be written as $Y=\sum Y_P$, where
$P$ ranges over the $s$-cut partitions.
Therefore
\begin{equation}\label{eq:second moment of Y}
\begin{split}
\mathbb E(Y^2)&\leq\mathbb E(Y) + \sum_{1\le q\leq tm/2} n^*(q,t,m,s)\Theta((mq)^{-(t-1)/2})\\
&= \Theta_{s,t} (m^{s-(t+1)/2})+\sum_{1\le q\leq tm/2}
O_{s,t} (m^{s-(t+1)/2}
q^{s-1-(t+1)/2}).
\end{split}
\end{equation}
When $s<(t+1)/2$, the last inequality shows that
$\mathbb E(Y^2)=O_{s,t} \big( m ^{s-\frac{t+1}{2}} \big)$. Thus, using \eqref{eq:expectation of Y} and Paley-Zygmund Inequality, we get
\begin{equation*}
\mathbb P \big(X(2,t,m) \le s \big)\ge\mathbb P (Y>0) \ge \frac{\mathbb E(Y )^2 }{\mathbb \mathbb E(Y^2)} = \Omega _{s,t} \big( m^{s-\frac{t+1}{2}} \big) .
\end{equation*}
Combining this result with \eqref{eq:upper bound with s cuts}, we obtain for all $1\le s< \frac{t+1}{2}$,
\begin{equation*}
\mathbb P \big( X(2,t,m)=s \big)=\mathbb P \big( X(2,t,m)\le s \big)-\mathbb P \big( X(2,t,m)\le s-1 \big) = \Omega _{s,t} \big( m^{s-\frac{t+1}{2}} \big) .
\end{equation*}
This finishes the proof of part (1) of Theorem~\ref{thm:big m}.
When $t$ is odd and $s=\frac{t+1}{2}$, the inequality \eqref{eq:second moment of Y} shows that $\mathbb E (Y^2)=O_{s,t}\big( \log m \big)$. Thus, using the same arguments we get that
\begin{equation*}
\mathbb P \big( X(2,t,m)=s \big) = \Omega _{s,t} \Big( \frac{1}{\log m} \Big) .
\end{equation*}
This finishes the proof of the lower bound in part (2) of Theorem~\ref{thm:big m}.
Finally, we note that when $t$ is even and $s=t/2+1$, we get from \eqref{eq:second moment of Y} that $\mathbb E (Y^2 ) =O_{s,t}(\sqrt{m})$ and therefore by the same arguments
\begin{equation}\label{eq:bounded away for s=t/2+1}
\mathbb P \big( X(2,t,m)=s \big) =\Omega _{t} (1).
\end{equation}
\noindent
This is a special case of part (3) of Theorem~\ref{thm:big m}. The proof in the general case is similar, but requires an additional twist. We present it the next subsection. Since the computation involved is similar to the one above, we omit some of the details.
\vspace{0.2cm}
\subsection{Proof of part (3) of Theorem~\ref{thm:big m}}
We have $\mathbb P(X(2,t,m)=s)=0$ for $s>t$,
by the deterministic result, and
$\mathbb P(X(2,t,m)=s)$ goes to zero if $s\le (t+1)/2$ by parts (1) and (2) of Theorem~\ref{thm:big m}. We proved part (1) in the previous subsection and will prove the relevant direction of part (2) in the next. Our goal here is to prove
$\mathbb P(X(2,t,m)=s)=\Omega_t(1)$ in all remaining cases $t/2+1\le s\le t$. According to
\eqref{eq:bounded away for s=t/2+1},
this is true for $s=t/2+1$ and can be proved
by straightforward second moment argument. For the general case we will also use the second moment method, but with a modified distribution.
We will use the following simple claim. It holds for real intervals too, but here we use the word ``interval'' to represent any set of consecutive elements in a sequence (such as beads on a necklace).
\begin{claim}
\label{chyper}
There exists an absolute positive constant $c$ so that the
following holds.
Let $x$, $y$ and $U$ be positive and let
$X$ be a uniform random subset of $x$ points of an interval
$Y$ of length $y$.
Then the probability that there exists an interval $Z$
(of any length) in $Y$ so that $|X \cap Z|$
deviates from its expectation,
$|Z|x/y$, by at least
$U$ is at most $8e^{-cU^2/x}$.
\end{claim}
\noindent{\bf Proof:}
If there exists a interval of $Y$ with deviation above $U$, then there also exists an initial interval (starting at the left end of $Y$) with deviation larger than $U/2$, so it suffices to consider initial intervals. It is also enough to considers initial intervals of length at most $y/2$ by symmetry. Consider the
first $y/2$ elements of $Y$
one by one from left to right, exposing for each of them
in turn if it belongs to the random set $X$. If during the process we ever
reach an initial interval in which the number of elements of $X$
deviates from its expectation by more
than $U/2$, stop the process
and reveal all remaining elements of $X$.
Conditioning on having a large deviation where we stopped the process,
with probability at least, say, $1/4$ we still have deviation
of at least $U/4$ in the interval
of the $y/2$ first points of $Y$. But this probability is at most
$e^{cU^2/x}$ for some absolute positive constant $c$, by
standard estimates for large deviations of a hypergeometric
distribution (see \cite{Hoe} or \cite{JLR}, Theorem 2.10 and Theorem 2.1.)
This implies that the probability of any initial segment of $Y$ of length at most $y/2$ has deviation above $U/2$ is at most $4e^{-cU^2/x}$. The probability of an initial segment of any length existing with such a high deviation is at most twice this and if no such initial interval exists then the deviation of any interval is at most $U$.
This proves the claim. \hfill$\Box$
Returning to the proof of the theorem, recall that $s$, $t$ and $m$ are
positive integers satisfying $t/2+1\le s\le t$. We will treat $t$ and $s$
as a constants and assume in our calculation that $m$ is sufficiently
large depending on $t$. Let $D$ be the uniform distribution over necklaces
$N$ with $2m$ beads of type $i$ for every $1 \leq i \leq t$. We need to
prove that $\mathbb P_D(X(N)=s)=\Omega_t(1)$. In what follows we consider
another distribution $D'$ on some of the same possible necklaces obtained
in a two step process as follows.
We split
the necklace $N$ into $s$ intervals $I_i$, each consisting
of $2mt/s$ consecutive beads. Further we split each interval $I_i$ into three equal length subintervals $I_{i,1}$, $I_{i,2}$ and $I_{i,3}$ out which $I_{i,2}$ lies in the middle. Strictly speaking, some rounding is necessary unless $3s$ divides $2mt$, but we ignore these roundings as they do not matter in our calculations. We will choose the positive integer $L<m/2$ later. In the first step of our two step process we place $L$ random beads of type $i$ uniformly in both of the intervals $I_{i,1}$ and $I_{i,3}$, for every $i$. We call the beads so placed \emph{seeds}, so we have $2sL$ seeds in total. As the second step of our process generating the distribution $D'$ we place the remaining $2mt-2sL$ beads (the \emph{non-seeds}, $2m-2L$ of them of type $i$ for $i\le s$ and $2m$ beads of type $i$ for $i>s$) uniformly in the available slots.
For a type $i$ and an interval $J$ in the necklace we denote the number of beads of type $i$ in $J$ by $n_i(J)$. The dependence on $N$ is implicit. We call a necklace $N$ \emph{normal} if the distribution of types in every interval is close to its expectation, that is, if for every interval $J$ and every type $i$, we have $|n_i(J)-\mathbb E_{D'}(n_i(J))|<L/(4t)$.
We use the asymptotic notations $O_t(\cdot)$, $\Omega_t(\cdot)$ and $\Theta_t(\cdot)$ to hide positive multiplicative factors depending on $t$ alone. These factors are not allowed to depend on $m$ or $L$. (Dependence on $s$ is allowed as $s\le t$ can take finitely many distinct values for a fixed $t$.)
\begin{claim}\label{normal}
$$\mathbb P_{D'}(N\hbox{ is not normal})=O_t(1)e^{-\Omega_t(L^2/m)}$$
\end{claim}
\noindent{\bf Proof:}\, We can identify $2s+t$ sets placed uniformly in
the process defining the distribution $D'$. For $i\le s$ we have two
sets of seeds of type $i$ placed uniformly in the intervals $I_{i,1}$
and $I_{i,3}$, respectively. And for any type $i$ we have the set of
non-seed beads of type $i$ placed uniformly in the positions not occupied
by seeds. We apply Claim~\ref{chyper} for each of these processes. The
union bound yields the estimate stated in the lemma for the existence of
an interval in any of these processes where the number of beads placed in
the interval deviates from its expectation by more than $L/(12t)$. It
remains to prove that assuming no deviation exceeds $L/(12t)$ the
resulting necklace $N$ is normal.
To see this, let us fix an interval $J$ and a type $i$. Let $E$ stand
for the expectation of $n_i(J)$ in the distribution $D'$ and let $E'$
stand for the same expectation conditioned on the placement of the seeds
(so $E'$ is a random variable as $n_i(J)$ but $E$ is a constant). Note
that $E'$ is determined by how many of the seeds are placed inside $J$
in each of the $2s$ relevant subintervals. This is deterministic for
all but at most two of the subintervals (the ones containing the ends of
$J$). Note also that the dependence of $E'$ on these numbers is linear
with all coefficients below $1$ in absolute value. With our low deviation
assumption on the seeds this means that $|E'-E|<2L/(12t)$ as $E$ is the
expectation of $E'$. We also assumed that the number of non-seeds of
type $i$ in $J$ differs from its expectation after the seeds are placed
by at most $L/(12t)$. So we have $|n_i(J)-E'|\le L/(12t)$ and therefore
$|n_i(J)-E|<L/(4t)$ as needed.\hfill $\Box$
\begin{claim}\label{atleast}
If $N$ is normal, then $X(N)\ge s$.
\end{claim}
\noindent{\bf Proof:}\, We prove the contrapositive: If $X(N)<s$, then $N$ is not normal. So let us fix a fair partition of $N$ with fewer than $s$ cuts. Note that one of the intervals $I_i$ is not cut at all. Fix such an $i$ and note that one of the players receive no part of $I_i$. We look at the expected value of type $i$ beads (necessarily non-seeds) in the intervals he receives. For any $j$, a position of $I_{j,2}$ receives a bead of type $i$ with probability $(2m-2L)/(2mt-2sL)\le1/t$, while a position in $I_{j,1}$ or $I_{j,3}$ receives a seed with probability $3sL/(2mt)$, so it receives a non-seed of type $i$ with probability $(1-3sL/(2mt))(2m-2L)/(2mt-2sL)\le1/t-3sL/(2mt^2)$. As the partition is balanced the player receives $mt$ beads. There are only $2mt/3$ beads in the middle subintervals $I_{j,2}$, so at least $mt/3$ beads are coming from non-middle subintervals and therefore the expected number of type $i$ beads this part contains is at most $m-sL/(4t)$. Our partition is fair, so the actual number of type $i$ beads the player receives is exactly $m$. The discrepancy is coming from the at most $s$ intervals the player receives, so the actual number of type $i$ beads in one of those deviates from its expectation by at least $L/(4t)$. This proves that $N$ is not normal.\hfill $\Box$
\begin{claim}\label{atmost}
$$\mathbb P_{D'}(X(N)\le s)=\Omega_t(1)$$
\end{claim}
\noindent{\bf Proof:}\, We call a balanced partition of the necklace between the two players \emph{central} if it is obtained from $s$ cuts, one in each of the middle intervals $I_{i,2}$ by distributing the resulting $s+1$ intervals alternately between the two players. Note that the seeds are distributed equally between the players in a central partition: each players receive exactly $L$ seeds of each type. Thus, a central partition is fair if and only if both players receive an equal number on non-seeds of type $i$ for each $i$. The distribution of the non-seeds are uniform on the available slots, so both parts of Claim~\ref{12} applies: the probability under the distribution $D'$ that a central partition is fair is $\Theta_t(m^{-(t-1)/2})$ and the probability of two distinct central partitions are simultaneously fair is $\Theta_t((mq)^{-(t-1)/2})$, where $q$ is the distance between the central partitions. It should be acknowledged that the situation here differs from the situation considered in Claim~\ref{12} in that there we have $2m$ beads of each type whereas here we have $2m-2L$ non-seed beads in types $1\le i\le s$. But as $2m-2L$ is between $m$ and $2m$ the estimate still holds and the original proof of Claim~\ref{12} applies in this modified setting almost verbatim. Note that the hidden constant in our estimate does not depend on $L$.
We apply the Paley-Zygmund Inequality for the random variable $Y$ counting the fair central partitions. This argument is very similar to the one presented at the end of the previous subsection. We have $\Theta(m^{s-1})$ central partitions, so
$$\mathbb E_{D'}(Y)=\Theta_t(m^{s-1})\Theta_t(m^{-(t-1)/2})=\Theta_t(m^{s-(t+1)/2}).$$
We calculate $\mathbb E_{D'}(Y^2)$ as the sum of probabilities for ordered pairs of central partitions that they are simultaneously fair. We have $\Theta_t(s^{m-1})$ pairs of equal partitions and for $1\le q$ we have $O(m^{s-1}q^{s-2})$ pairs of distance $q$. This gives
$$\mathbb E_{D'}(Y^2)=O_t(m^{s-(t+1)/2})+\sum_{q=1}^{mt/2}O(m^{s-(t+1)/2}q^{s-(t+1)/2-1})=O(m^{2s-t-1}),$$
where we used $s>(t+1)/2$ in the last step. Using the Paley-Zygmund Inequality we obtain
$$\mathbb P_{D'}(Y>0)\ge\frac{(\mathbb E_{D'}(Y))^2}{\mathbb E_{D'}(Y^2)}=\Omega_t(1).$$
To finish the proof of the claim simply observe that $Y>0$ means that there is a fair central partition, so $X(N)\le s$ as central partitions have $s$ cuts.\hfill $\Box$
\medskip
We set $L$ such that $N$ is normal with probability at least $1-\mathbb P_{D'}(X(N)\le s)/2$. By Claims~\ref{normal} and \ref{atmost} this can be achieved by an appropriate choice also satisfying $L=O(\sqrt m)$. With this choice of $L$ we have that $N$ is normal and $X(N)\le s$ with probability $\Omega_t(1)$. In this case we actually have $X(N)=s$ by Claim~\ref{atleast}. These estimates hold for a random necklace $N$ in the distribution $D'$. To prove part (3) of Theorem~\ref{thm:big m} we need a similar estimate in the uniform distribution $D$. The following Claim finishes the proof because it establishes that if $L=O(\sqrt m)$, then $D'$-weight of a normal necklace is only constant times its $D$-weight.
\begin{claim}
We have $P'(N)/P(N)\le e^{O_t(L^2/m)}$ for any normal necklace $N$, where $P'(N)$ stands for the probability of obtaining $N$ in the distribution $D'$ and $P(N)$ is the probability of obtaining it in $D$.
\end{claim}
\noindent{\bf Proof:}\, This calculation is tedious but very elementary. The probability $P'(N)$ depends on $N$ through the number $n_i(I_{i,j})$ of beads of type $i$ in the interval $I_{i,j}$ for $1\le i\le s$ and $j=1$ or $3$. We have $\binom{n_i(I_{i,j})}L$ choices to select seeds in the interval $I_{i,j}$ consistent with $N$. Thus, the probability of selecting all seeds consistent with $N$ is
$$\frac{\prod_{i=1}^s\binom{n_i(I_{i,1})}L\binom{n_i(I_{i,3})}L}{\binom{2mt/(3s)}L^{2s}}.$$
The distribution of non-seeds is uniform, so after such a consistent choice of the seeds we obtain $N$ with probability
$$\frac{(2m-2L)!^s(2m)!^{t-s}}{(2mt-2sL)!}.$$
For a random necklace in the distribution $D'$, the interval $I_{i,j}$ ($j=1$ or 3) contains $L$ seeds of type $i$, and the expected number of non-seeds of type $i$ is $(2m-2L)(2mt/(3s)-L)/(2mt-2sL)\le2m/(3s)-L/t$. As $N$ is normal, the actual value $n_i(I_{i,j})$ deviates from its expectation by less than $L/t$, so we have $n_i(I_{i,j})<2m/(3s)+L$. Using this estimate and the calculations above, we obtain
$$P'(N)\le\frac{\binom{2m/(3s)+L}L^{2s}((2m-2L)!^s(2m)!^{t-s}}{\binom{2mt/(3s)}L^{2s}(2mt-2sL)!}.$$
As $D$ is uniform, $P(N)$ does not depend on $N$:
$$P(N)=\frac{(2m)!^t}{(2mt)!}.$$
To estimate $P'(N)/P(N)$, we use the inequalities $(a-b)^b<\binom abb!\le a^b$ and obtain
$$\frac{P'(N)}{P(N)}\le\left(\frac{2mt(2m/(3s)+L)}{(2m-2L)(2mt/(3s)-L)}\right)^{2sL}.$$
Using $s\le t$ and $L\le m/2$, we can further estimate
$$\frac{2mt(2m/(3s)+L)}{(2m-2L)(2mt/(3s)-L)}\le 1+O_t\left(\frac Lm\right),$$
so we have
$$\frac{P'(N)}{P(N)}\le\left(1+O_t\left(\frac Lm\right)\right)^{2sL}=e^{O_t(L^2/m)},$$
as claimed\hfill $\Box$
\subsection{Random walks}\label{random walks}
In this subsection, we prove the upper bound in part (2) of
Theorem~\ref{thm:big m}. The following proposition is at the heart
of the argument. In this proposition we bound the probability that a
certain equation cannot be solved in the trajectories of independent
random walks. To prove the proposition we modify and generalize a
proof of Lawler~\cite{Lawler1}. In \cite{Lawler1} and \cite{Lawler2}
Lawler studied the probability that the traces of two independent random
walks on $\mathbb Z ^4$ are disjoint. See also \cite{erdos taylor2,ET}
for works of Erd\H{o}s and Taylor on the same problem.
Throughout this subsection, we fix $t\ge 1$ odd, $s=\frac{t+1}{2}$ and let
the $O$ notations depend on $t$. Recall that an infinite two-sided random
walk $W(n)$ is a sequence of random variables in some Euclidean space such
that $W(n+1)-W(n)$ for $n\in \mathbb Z $ are independent and identically
distributed. We say that the walk has a finite range if there is a
finite set $A$ for which $\mathbb P ( W(1)-W(0)\notin A )=0$. Finally,
the walk $W$ is called centered if $\mathbb E [W(1)-W(0)]=0$.
\begin{prop}\label{prop:Lawler}
Let $\langle W_j(n) ,\ n\in \mathbb Z \rangle$ for $j\le s$ be independent and identically distributed two-sided random walks on $\mathbb Z^{2s-2}$. Suppose that $W_j(0)=0$ for all $j$ and that the walks are centered and have a finite range. Then, there exists $C>0$ depending on $s$ and the step distribution of the walks such that for all $N\ge 2$,
\begin{equation*}
\mathbb P \bigg( \forall k=(k_1,\dots ,k_{s})\in A_N, \ \sum_{j=1}^{s} W_j(k_j) \neq 0 \bigg) \le \frac{C}{\log N},
\end{equation*}
where
\begin{equation*}
A_N:=\bigg\{ k\in \mathbb Z ^{s} : \sum _{j=1}^{s} k_j =0 \text{ and } k>_{\ell } 0 \bigg\} \cap [-N,N]^{s},
\end{equation*}
and where $>_{\ell } $ is the lexicographic order on $\mathbb Z ^{s}$.
\end{prop}
We first show how to use Proposition~\ref{prop:Lawler} in order to prove the upper bound in part (2) of Theorem~\ref{thm:big m}.
\subsubsection{Proof of the upper bound in part (2) of Theorem~\ref{thm:big m}}
We parameterize a partition of the necklace with $s$ cuts by a vector of integers $i=(i_0, i_1,\dots ,i_s, i_{s+1})$ with
\begin{equation*}
0=i_0\le i_1\le \cdots \le i_s\le i_{s+1}=2mt.
\end{equation*}
This corresponds to a partition where the first thief gets beads $1$ to $i_1$, the second one gets beads $i_1+1$ to $i_2$ and so on.
Let $I$ be the set of balanced partitions. That is
\begin{equation*}
I:=\bigg\{ (i_0,i_1,\dots ,i_{s+1}) \ \Big| \ \sum _{j=0}^{s} (-1)^j \left( i_{j+1}-i_j \right) =0 \bigg\}.
\end{equation*}
For $n\le 2mt$, let $U(n) \in \mathbb N ^{t-1}$ be the random variable that, in the $j$'th coordinate, counts the number of beads of type $j$ out of the first $n$ beads. It is clear that $i=(i_0, i_1,\dots i_{s+1})\in I$ is fair if and only if
\begin{equation}\label{eq:U}
\sum _{j=0}^s (-1)^j (U(i_{j+1})-U(i_j))=0.
\end{equation}
Indeed, equation \eqref{eq:U} says that the first $t-1$ types are equally distributed between the thieves. Therefore, as each thief gets in total $mt$ beads, it follows that the last type must be equally distributed as well.
Next, let $Z$ be the number of fair partitions in $I$. It suffices to prove that
\begin{equation}\label{eq:what we need}
\mathbb P (Z>0) = O \bigg( \frac{1}{\log m} \bigg).
\end{equation}
To this end, define the sets of partitions
\begin{equation*}
I_1:=\left\{ i \in I \ \big| \ \forall 0\le j\le s, \ i_{j+1}-i_j > 2m^{\frac{1}{4}} \right\}, \quad I_2:=I \setminus I_1,
\end{equation*}
and let $Z_2$ be the number of fair partitions in $I_2$.
We define a total order on $I$. For $i,i'\in I$, we write $i'\succ i$ if
\begin{equation*}
(i_1',-i_2',i_3',-i_4',\dots ) >_{\ell } (i_1,-i_2, i_3,-i_4,\dots ),
\end{equation*}
where $<_{\ell }$ is the lexicographic order on $\mathbb Z ^s$. For a partition $i\in I_1$, define the set
\begin{equation*}
B_i: =\left\{ i'\in I \ \big| \ i'\succ i \text{ and } \forall j \le s, \ |i_j-i_j'|\le m^{\frac{1}{4}} \right\}
\end{equation*}
and the event
\begin{equation*}
\mathcal B _i := \left\{ i \text{ is fair and } \forall i'\in B_i, \ i' \text{ is not fair } \right\}.
\end{equation*}
Finally, let
\begin{equation*}
Z_1:=\sum _{i\in I_1} \mathds 1 _{\mathcal B _i}.
\end{equation*}
We claim that
\begin{equation}\label{eq:Z_1 Z_2}
\{Z>0\} \subseteq \{ Z_1>0 \} \cup \{Z_2>0\}.
\end{equation}
Indeed, suppose that $Z>0$ and let $i\in I$ be the maximal fair partition
with respect to $\succ $. If $i \in I_2$ then $Z_2>0$ and therefore we
may assume that $i \in I_1$. Since $i$ is maximal, for all $i'\in B_i$,
$i'$ is not fair. Thus, $\mathcal B _i $ holds and $Z_1>0$.
We turn to bound the probabilities of the two events on the right-hand side of \eqref{eq:Z_1 Z_2}. It is easy to check that $|I_2| =O (m^{s-\frac{7}{4}})$ and therefore by Markov's inequality and Claim~\ref{12}(i),
\begin{equation}\label{eq:bound on Z_2}
\mathbb P (Z_2>0) \le \mathbb E (Z_2) =\sum _{i \in I_2 } \mathbb P (i \text{ is fair}) \le |I_2| \cdot O \big( m^{-\frac{t-1}{2}} \big) = O\big( m^{-\frac{3}{4}} \big).
\end{equation}
Next, we bound $\mathbb P ( Z_1>0 )$. We have that
\begin{equation}\label{eq:conditional}
\mathbb P (\mathcal B _i) = O \big( m^{-\frac{t-1}{2}} \big) \cdot \mathbb P \left( \forall i'\in B_i, \ i' \text{ is not fair } | \ i \text{ is fair} \right).
\end{equation}
We bound the last probability in the following claim.
\begin{claim}\label{claim:coupling}
We have that
\begin{equation}\label{eq:conditional probability}
\mathbb P \left( \forall i'\in B_i, \ i' \text{ is not fair } | \ i \text{ is fair} \right)= O \bigg( \frac{1}{\log m} \bigg).
\end{equation}
\end{claim}
Using Claim~\ref{claim:coupling}, equation \eqref{eq:conditional} and Markov's inequality we get
\begin{equation}\label{eq:bound on Z_1}
\mathbb P (Z_1>0) \le \mathbb E (Z_1) =\sum _{i\in I_1} \mathbb P (\mathcal B _i) = O \bigg( \frac{m^{-\frac{t-1}{2}}}{\log m} \bigg) \cdot |I_1|= O \bigg( \frac{1}{\log m} \bigg).
\end{equation}
Finally, \eqref{eq:what we need} follows from \eqref{eq:bound on Z_1}, \eqref{eq:bound on Z_2} and \eqref{eq:Z_1 Z_2}. This finishes the proof of part (2) of Theorem~\ref{thm:big m}.
\medskip
It remains to prove Claim~\ref{claim:coupling}
\begin{proof}[\bf Proof of Claim~\ref{claim:coupling}:]
Throughout this proof, we consider a uniform necklace $N$ with $2m$ beads of each type such that the partition $i$ is fair. As before, we denote by $U(n)\in \mathbb N ^{t-1}$ the counting vector. In this case, by \eqref{eq:U}, $i'\in I$ is fair if and only if
\begin{equation}\label{eq:i, i'}
\sum _{j=1}^{s} (-1)^{j+1} \left( U(i'_j)-U(i_j) \right) =0.
\end{equation}
For $1 \le j \le s$ and $|n|\le m^{\frac{1}{4}}$, define
\begin{equation*}
\tilde{W}_j(n):=t(-1)^{j+1} \left( U(i_j+(-1)^{j+1}n)- U(i_j) \right) -(n,\dots ,n).
\end{equation*}
Using this notation and \eqref{eq:i, i'} we have that $i'\in B_i$ is fair if and only if
\begin{equation*}
\sum _{j=1}^s \tilde{W}_j\left( (-1)^{j+1}(i_j' -i_j) \right)=0.
\end{equation*}
Here, we also used that
\begin{equation*}
\sum _{j=1}^s (-1)^{j+1}(i_j'-i_j)=0,
\end{equation*}
which follows as $i,i'\in I$. Thus, letting
\begin{equation*}
A:= \left\{ \left( (-1)^{j+1} (i_j'-i_j) \right)_{j=1}^s \ | \ i'\in B_i \right\} \subseteq \mathbb Z ^s,
\end{equation*}
we obtain that the probability in \eqref{eq:conditional probability} is given by
\begin{equation*}
\mathbb P \bigg(\forall k\in A, \ \sum _{j=1}^s \tilde{W}_j(k_j) \neq 0 \bigg).
\end{equation*}
Moreover, from the definition of $B_i$ it is clear that
\begin{equation*}
A=\bigg\{ k\in \mathbb Z ^s : \sum _{j=1}^{s} k_j =0 \text{ and } k>_{\ell } 0 \bigg\} \cap [-m^{\frac{1}{4}},m^{\frac{1}{4}}]^s.
\end{equation*}
We cannot use Proposition~\ref{prop:Lawler} yet, because the processes $\tilde{W}_j$ are not exactly independent random walks. We
will show that it is possible to couple them with random walks. To this end, let $Y$ be a random variable in $\mathbb Z ^{t-1}$ with distribution
\begin{equation*}
\forall j \le t-1 , \ \mathbb P (Y=t e_j-(1,\dots ,1))=\frac{1}{t} \quad \text{and} \quad \mathbb P (Y=-(1,\dots ,1))=\frac{1}{t}.
\end{equation*}
Let $W_j(n)$ for $j \le s$ and $|n|\le m^{\frac{1}{4}}$ be independent two-sided random walks on $\mathbb Z ^{t-1}$ with steps distributed like $Y$ and with $W_j(0)=0$. We claim that one can couple $\{W_j \}_{j=1}^s$ and $\{ \tilde{W}_j \}_{j=1}^s$ such that
\begin{equation}\label{eq:couple W}
\mathbb P \left( \forall j, \ \tilde{W}_j=W_j \right) = 1-O\big( m^{-\frac{1}{2}} \big) .
\end{equation}
We give a sketch of proof for this fact. Consider a new necklace
$N'$ of length $l = 2t\lfloor m^{\frac{1}{4}} \rfloor$ obtained from $N$ by concatenating
intervals of beads of length $2 \lfloor m^{\frac{1}{4}} \rfloor$ around each of
the cuts. We also let $N''$ be another necklace of the same length
obtained by choosing independently and uniformly the type of each
bead. It is not hard to see that \eqref{eq:couple W} follows
from a coupling of $N'$ and $N''$ such that $\mathbb P (N'=
N'')= 1-O (m^{-\frac{1}{2}})$. The first bead of $N'$ is clearly
uniform. Conditioning on the first $m$ beads of $N'$, a simple
counting argument shows that the probability that the next bead
is of a certain type is $\frac{1}{t} +O(m^{-\frac{3}{4}})$. Thus,
the probability to get any particular sequence of beads of length
$l$ is
\begin{equation*}
\Big( \frac{1}{t} +O(m^{-\frac{3}{4}}) \Big) ^l =\frac{1}{t^l} \left( 1 +O(m^{-\frac{1}{2}}) \right) .
\end{equation*}
This shows that $N'$ and $N''$ can be coupled such that $\mathbb P (N'\neq N'') = O ( m^{-\frac{1}{2}} )$, which proves~\eqref{eq:couple W}.
By \eqref{eq:couple W} and Proposition~\ref{prop:Lawler}, we obtain that
\begin{equation*}
\mathbb P \bigg(\forall k\in A, \ \sum _{j=1}^s \tilde{W}_j(k_j) \neq 0 \bigg) = O \big( m^{-\frac{1}{2}} \big) +\mathbb P \bigg(\forall k\in A, \ \sum _{j=1}^s W_j(k_j) \neq 0 \bigg)= O \bigg( \frac{1}{\log m} \bigg).
\end{equation*}
This finishes the proof of Claim~\ref{claim:coupling}.
\end{proof}
\subsubsection{Proof of Proposition~\ref{prop:Lawler}}
The proof is somewhat similar to the proof of Claim~\ref{p2}. In the proof of Claim~\ref{p2}, we split the possible walks according to their last visit at the origin. Then, we use the Markov property to argue that the probability that the last visit is at time $k$ equals the probability that the walk returns to the origin at time $k$ times the probability that the walk avoids the origin in the remaining time.
Similarly, in the following proof, we consider the last integer vector $k$ (according to the lexicographic order) for which $\sum W_j(k_j)=0$. Then, we condition on the trajectories of all walks other then the first one and use the Markov property for the first walk. The proof becomes slightly more technical as we need to control probabilities conditioned on the last walks with high probability. Throughout the proof we let the $O$ notations depend on the step distribution of the random walk as well as on $s$.
Consider the set
\begin{equation*}
\mathcal E :=\bigg\{ \xi =(\xi _2,\dots ,\xi _{s}) \ \bigg| \ \xi _j:[-2N,2N]\cap \mathbb Z \to \mathbb Z^{2s-2} \bigg\}.
\end{equation*}
We think of $\mathcal E $ as the set of trajectories of the walks $W_2,\dots ,W_s$. Let $K=(K_1,\dots ,K_{s} )$ be the maximal element $(k_1,\dots ,k_{s})\in A_N \cup \{0\} $ with respect to $>_{\ell } $ such that
\begin{equation*}
\sum _{j=1}^{s} W_j(k_j)=0.
\end{equation*}
For $\xi \in \mathcal E$ and $k\in A_N$ define the event
\begin{equation*}
\mathcal A (\xi ,k):=\{K=k\}\cap \bigcap _{j=2}^{s} \{\forall l\in \{ -2N,\dots , 2N\}, \ W_j(k_j+l)-W_j(k_j)=\xi _j(l) \}.
\end{equation*}
It is clear that these events are disjoint and that
\begin{equation}\label{eq:supset}
\begin{split}
\mathcal A (\xi ,k) &\supseteq \bigg\{ W_1(k_1)=\sum _{j=2}^{s} \xi _j(-k_j) \bigg\} \\
&\cap \bigcap _{j=2}^{s} \Big\{ \forall l\in \{ -2N,\dots ,2N\}, \ W_j(k_j+l)-W_j(k_j)=\xi _j(l) \Big\} \\
&\cap \bigg\{ \forall n \in A_{2N} ,\ W_1(k_1+n_1)-W_1(k_1)+\sum _{j=2}^{s } \xi _j(n_j)\neq 0 \bigg\}.
\end{split}
\end{equation}
All the s+1 events whose intersection gives the right-hand side of \eqref{eq:supset} are independent (the first and last events are independent as $k_1,n_1\ge 0$) and, therefore,
\begin{equation}\label{eq:lower bound on probability}
\begin{split}
\mathbb P \left( \mathcal A (\xi ,k) \right) \ge P(\xi ) &\cdot \mathbb P \bigg( W_1(k_1)=\sum _{j=2}^{s} \xi _j(-k_j) \bigg)\\
&\cdot \mathbb P \bigg( \forall n \in A_{2N} ,\ W_1(n_1)+\sum _{j=2}^{s } \xi _j(n_j)\neq 0 \bigg),
\end{split}
\end{equation}
where
\begin{equation*}
P(\xi ):= \prod _{j=2}^{s} \mathbb P \Big( \forall l\in \{-2N,\dots ,2N\}, \ W_j(l)=\xi _j(l) \Big).
\end{equation*}
Next, for $\xi \in \mathcal E$ define the function
\begin{equation*}
G(\xi ):= \sum _{k \in A_N} \mathbb P \bigg( W_1(k_1)=\sum _{j=2}^{s} \xi _j(-k_j) \bigg).
\end{equation*}
The following lemma shows that $G$ is typically of the order of $\Omega ( \log N )$.
\begin{lemma}\label{lem:upper bound on probability}
There exists $c >0$ such that
\begin{equation*}
\mathbb P \big( G(W_2,\dots ,W_s) <c \log N \big) = O \bigg( \frac{1}{\log N} \bigg).
\end{equation*}
\end{lemma}
We postpone the proof of Lemma~\ref{lem:upper bound on probability} and proceed with the proof of Proposition~\ref{prop:Lawler}. To this end, define the set
\begin{equation*}
\mathcal G :=\Big\{ \xi \in \mathcal E : G(\xi ) \ge c \log N \Big\}.
\end{equation*}
Summing the inequality in \eqref{eq:lower bound on probability} over $\xi \in \mathcal G $ and $k \in A_N$, we get
\begin{equation*}
\begin{split}
1\ge \mathbb P &\bigg( \bigcup _{\xi \in \mathcal G } \bigcup _{k \in A_N} \mathcal A (\xi ,k) \bigg)=\sum _{\xi \in \mathcal G } \sum _{k \in A_N} \mathbb P \left( \mathcal A (\xi ,k) \right) \\
&\ge \sum _{\xi \in \mathcal G } P(\xi ) \cdot G(\xi ) \cdot \mathbb P \bigg( \forall n \in A_{2N} ,\ W_1(n_1)+\sum _{j=2}^{s } \xi _j(n_j)\neq 0 \bigg) \\
&= \Omega ( \log N ) \sum _{\xi \in \mathcal G } P(\xi ) \cdot \mathbb P \bigg( \forall n \in A_{2N} ,\ W_1(n_1)+\sum _{j=2}^{s } \xi _j(n_j)\neq 0 \bigg) \\
&=\Omega ( \log N ) \cdot \mathbb P \bigg( (W_2,\dots , W_{s}) \in \mathcal G \ \text{ and } \ \forall n \in A_{2N} , \ \sum _{j=1}^{s } W_j (n_j)\neq 0 \bigg)\\
&= \Omega ( \log N ) \cdot \mathbb P \bigg( \ \forall n \in A_{2N} , \ \sum _{j=1}^{s } W_j (n_j)\neq 0 \bigg) -O(1).
\end{split}
\end{equation*}
where the last equality follows from Lemma~\ref{lem:upper bound on probability}. Thus,
\begin{equation}
\mathbb P \bigg( \ \forall n \in A_{2N} , \ \sum _{j=1}^{s } W_j (n_j)\neq 0 \bigg) = O \bigg( \frac{1}{\log N} \bigg).
\end{equation}
This completes the proof of Proposition~\ref{prop:Lawler}.
We turn to prove Lemma~\ref{lem:upper bound on probability}. For the
proof we need the following standard claim on integer valued random walks.
\begin{claim}\label{claim:on random walk}
Let $W(n)$ be a centered, finite range random walk on $\mathbb Z ^d$ with $W(0)=0$. Suppose that the covariance matrix of $W(1)$ is $\Sigma $ and that $\Sigma $ is nonsingular. Then
\begin{enumerate}
\item
For all $n\ge 1$, $k \in \mathbb Z ^d$ and $a>0$ such that $||k|| \le a\sqrt{n}$ and $\mathbb P (W(n)=k)>0$, we have that
\begin{equation*}
\mathbb P (W(n)=k) = \Omega _a \big( n^{-\frac{d}{2}} \big).
\end{equation*}
\item
For all $n\ge 1 $ and $r>0$, we have that
\begin{equation*}
\left| \mathbb P (|| W(n) ||\le r\sqrt{n} ) - \mathbb P (||Z||\le r ) \right| = O \Big( \frac{1}{\sqrt{n}} \Big),
\end{equation*}
where $Z\sim N(0,\Sigma )$.
\end{enumerate}
\end{claim}
\begin{proof}[\bf Proof:]
The second part follows from \cite[Theorem~1.1]{Berry esseen}.
Now we prove the first part. We start by showing that without loss of generality, $W$ is irreducible. Indeed, otherwise consider the set
\begin{equation*}
\Lambda := \{ k\in \mathbb Z ^d : \exists n \ge 0, \ \mathbb P (W(n)=k)>0 \}.
\end{equation*}
For a general walk, $\Lambda $ is a semigroup. However, since $W$ is centered, it is not hard to check that $\Lambda $ is in fact a lattice. If we let $T$ be a linear transformation that maps $\Lambda $ to $\mathbb Z ^d$ then the new walk $T(W)$ is an irreducible walk. Now, we can use \cite[Theorem~3.1]{LLT} that determines the asymptotic behavior of $\mathbb P (W(n)=k)$ for an irreducible (possibly periodic) walk. See also the work of Polya on the simple random walk \cite{polya}.
\end{proof}
\begin{proof}[\bf Proof of Lemma~\ref{lem:upper bound on probability}:]
Fix $k_2,\dots ,k_{s}$ such that $-N/s \le k_2,\dots ,k_s < 0$ and let $k_1:=-k_2-\cdots -k_{s}$. It is clear that $k=(k_1,\dots ,k_{s})\in A_N$. On the event
\begin{equation*}
\mathcal B := \Big\{ \forall 2\le j \le s , \ ||W_j( -k_j )||\le \sqrt{|k_j|} \Big\},
\end{equation*}
we have that
\begin{equation*}
\bigg| \bigg| \sum _{j=2}^{s} W_j(-k_j) \bigg| \bigg| \le \sqrt{|k_2|}+\cdots +\sqrt{|k_{s}|} \le s \sqrt{k_1}
\end{equation*}
and therefore, by the first part of Claim~\ref{claim:on random walk}, on $\mathcal B$ we have
\begin{equation*}
\mathbb P \bigg( W_1(k_1) = \sum _{j=2}^{s} W_j(-k_j) \ \bigg| \ W_2,\dots W_{s}\bigg) = \Omega \Big( \frac{1}{k_1^{s-1}} \Big).
\end{equation*}
Note that in order to use Claim~\ref{claim:on random walk} we have to first verify that the last probability is positive. This is indeed the case since, with positive probability, the walk $W_1$ takes the same $-k_2$ steps as $W_2$, then the same $-k_3$ steps as $W_3$ and so on. We obtain that
\begin{equation}\label{eq:lower bound on G}
\begin{split}
G(W_2,\dots ,W_{s} ) &\ge \Omega(1) \sum _{k_2=1}^{N/s} \cdots \sum _{ k_{s} =1}^{N/s} \frac{1}{\big( k_2+\cdots +k_s\big)^{s-1}} \prod _{j=2}^{s}\mathds 1 \big\{ || W_j(k_j) ||\le \sqrt{k_j} \big\}\\
&\ge \Omega (1) \sum _{p=1} ^{\log N} 2^{-p(s-1)} \sum _{k_2=2^p}^{2^{p+1}-1} \cdots \sum _{k_s=2^p}^{2^{p+1}-1} \prod _{j=2}^{s}\mathds 1 \big\{ || W_j(k_j) ||\le \sqrt{k_j} \big\}.
\end{split}
\end{equation}
Denote by $X$ the sum on the right-hand side of \eqref{eq:lower bound on G}. Intuitively, $X$ is of order $\log N$ since each dyadic scale $p$ contributes order $1$ to the sum and far away scales are weakly correlated. We make this heuristic rigorous using the second moment method.
By the central limit theorem we have that
\begin{equation*}
\mathbb E (X) = \sum _{p=1} ^{\log N} 2^{-p(s-1)} \sum _{k_2=2^p}^{2^{p+1}-1} \cdots \sum _{k_s=2^p}^{2^{p+1}-1} \Omega(1) = \Omega ( \log N ).
\end{equation*}
We turn to bound the variance of $X$. We have that
\begin{equation*}
\begin{split}
&\var (X ) \le \sum _{p=1}^{\log N} \sum _{q=1}^{\log N} 2^{-(p+q)(s-1)} \sum _{k_2=2^p}^{2^{p+1}} \cdots \sum _{k_{s}=2^p}^{2^{p+1}} \sum _{l_{2}=2^q}^{2^{q+1}} \cdots \sum _{l_{s}=2^q}^{2^{q+1}} \prod _{j=2} ^{s} \\
&\Big( \mathbb P \left(||W_j(k_j)||\le \sqrt{k_j}, \ ||W_j(l_j)||\le \sqrt{l_j} \ \right) -\mathbb P \left( ||W_j(k_j)||\le \sqrt{k_j} \right)\mathbb P \left( ||W_j(l_j)||\le \sqrt{l_j} \right) \Big).
\end{split}
\end{equation*}
For all $j$ with $k_j\le l_j$, we have that
\begin{equation}\label{eq:corelation of walk}
\begin{split}
&\mathbb P \left(||W_j(k_j)||\le \sqrt{k_j}, \ ||W_j(l_j)||\le \sqrt{l_j} \ \right) \\
& \quad \quad \quad \quad \quad \quad \quad \le \mathbb P \left(||W_j(k_j)||\le \sqrt{k_j}, \ ||W_j(l_j)-W_j(k_j)||\le \sqrt{l_j}+\sqrt{k_j} \ \right)\\
& \quad \quad \quad \quad \quad \quad \quad = \mathbb P \left(||W_j(k_j)||\le \sqrt{k_j} \ \right) \mathbb P \left( ||W_j(l_j-k_j) ||\le \sqrt{l_j}+\sqrt{k_j} \right).
\end{split}
\end{equation}
When $l_j \ge 2 k_j$, let
\begin{equation*}
r:= \frac{\sqrt{l_j}+\sqrt{k_j}} {\sqrt{l_j-k_j} }= 1+O\bigg(\frac{\sqrt{k_j}}{\sqrt{l_j}}\bigg).
\end{equation*}
By the second part of Claim~\ref{claim:on random walk} for $l_j \ge 2 k_j$,
\begin{equation}\label{eq:use the Berry Esseen}
\begin{split}
\mathbb P \left( ||W_j(l_j-k_j) ||\le \sqrt{l_j}+\sqrt{k_j} \right)=\mathbb P &\left( ||Z||\le r \right) +O\bigg(\frac{1}{\sqrt{l_j}}\bigg)\\
=\mathbb P (||Z||\le 1) +O\bigg(\frac{\sqrt{k_j}}{\sqrt{l_j}}\bigg) &=\mathbb P \left( ||W_j(l_j) ||\le \sqrt{l_j} \right) +O\bigg(\frac{\sqrt{k_j}}{\sqrt{l_j}}\bigg).
\end{split}
\end{equation}
It is clear that the probability in the left-hand side of \eqref{eq:use the Berry Esseen} is estimated by the right-hand side of \eqref{eq:use the Berry Esseen} when $k_j \le l_j\le 2k_j$ and therefore it holds whenever $l_j\ge k_j$. Substituting this estimate into \eqref{eq:corelation of walk} and using the same arguments when $l_j \le k_j$, we get that for all $k_j,l_j$,
\begin{equation*}
\begin{split}
\mathbb P \left(||W_j(k_j)||\le \sqrt{k_j}, \ ||W_j(l_j)||\le \sqrt{l_j} \ \right) & \\
\le \mathbb P \left(||W_j(k_j)||\le \sqrt{k_j} \right) &\mathbb P \left( ||W_j(l_j)||\le \sqrt{l_j} \ \right) + O \bigg( \frac{\sqrt{\min (l_j,k_j)}}{\sqrt{\max (l_j,k_j)}} \bigg) .
\end{split}
\end{equation*}
Thus,
\begin{equation}\label{eq:variance annoying sum}
\begin{split}
\var (X ) &\le O(1) \sum _{p=1}^{\log N} \sum _{q=p}^{\log N} 2^{-(p+q)(s-1)} \sum _{k_2=2^p}^{2^{p+1}} \cdots \sum _{k_{s}=2^p}^{2^{p+1}} \sum _{l_{2}=2^q}^{2^{q+1}} \cdots \sum _{l_{s}=2^q}^{2^{q+1}} 2^{(p-q)(s-1)/2} \\
&\le O(1)\sum _{p=1}^{\log N} 2^{p(s-1)/2} \sum _{q=p}^{\log N} 2^{-q(s-1)/2} \le O(1)\sum _{p=1}^{\log N} 1 =O(\log N).
\end{split}
\end{equation}
Finally, by Chebyshev's inequality there exists $c>0$ such that
\begin{equation*}
\mathbb P (X<c \log N) = O \bigg( \frac{1}{\log N} \bigg) .
\end{equation*}
This finishes the proof of the lemma using \eqref{eq:lower bound on G}.
\end{proof}
\section{The case $m=1$, many types}\label{section3}
In this section, we prove Theorems \ref{t14} and \ref{t15}
in which the number of beads of each type is equal to the number of
thieves and each collection of intervals should contain exactly one
bead of each type. In order to prove Theorem \ref{t14}, we need a
hypergraph edge-coloring result of Pippenger and Spencer \cite{PS}.
First, we recall the terminology. A \emph{hypergraph} is a pair $H=(V,E)$, where $V$ is the set of \emph{vertices} and $E$ is a multiset consisting of subsets of vertices called the \emph{edges}. We only consider here finite hypergraphs. We say that $H$ is \emph{$C$-uniform} if every edge contains $C$ vertices. The \emph{degree} $d_H(v)$ of a vertex $v\in V$ is the number edges containing $v$. If all the vertices have the same degree $k$, we call $H$ \emph{$k$-regular}. The \emph{codegree} of two distinct vertices is the number of edges which contain both of them. The \emph {maximum degree} and \emph{maximum codegree} of the hypergraph $H$ is the maximum degree of a vertex of $H$, and the maximum codegree of two distinct vertices in $H$, respectively. A set of pairwise disjoint edges is called a \emph{matching}.
\begin{theo}[\cite{PS}]
\label{t41}
For every integer $C \geq 2$ and every $\eps>0$, there is
$\delta>0$ such that the following statement holds.
For every positive integer $k$ and any $C$-uniform hypergraph $H$ with maximum degree at most $k$ and maximum codegree at most $\delta k$, the edges of $H$ can be partitioned into at most $(1+\eps)k$ matchings.
\end{theo}
The original statement published in \cite{PS} had two extra assumptions. First, it was required that the number of vertices of the hypergraph is sufficiently large as a function of $C$ and $\eps$. Second, it was also assumed that every vertex has degree at least $(1-\delta)k$. However, these two conditions are superfluous. One can add any number of isolated vertices to a hypergraph $H$ to satisfy the former condition. Then one can use the following lemma to obtain a $k$-regular hypergraph $H'$ containing $H$ without changing the maximal codegree. Partitioning the edge set of $H'$ into few matchings also partitions the edge set of $H$ into the same number of matchings.
\begin{lemma}
\label{regular}
For any $C\ge2$ and any $C$-uniform hypergraph $H$ with at least one edge and maximum degree at most $k$, there exists a $k$-regular $C$-uniform hypergraph $H'\supseteq H$ with the same maximum codegree as $H$.
\end{lemma}
\noindent
{\bf Proof:}\,
A hypergraph is called {\em simple} if its maximum codegree is 1. It is easy to construct a (finite) $k$-regular $(C-1)$-uniform simple hypergraph $H^*=(V^*,E^*)$. For example, let $V^*=U^k$, where $U$ is a set of size $C-1$, and let $E^*$ consist of all $(C-1)$-tuples with $k-1$ fixed coordinates. Let $L$ denote the number of edges in $H^*$.
Let $H=(V,E)$ and $M=\sum_{v\in V}(k-d_H(v))$. Take $M$ isomorphic copies, $H^*_1,\dots,H^*_M,$ of $H^*$ and $L$ isomorphic copies, $H_1,\dots,H_L,$ of $H$ on pairwise disjoint vertex sets, where $H_1=H$. Construct the hypergraph $H'=(V',E')$ as follows. Let $V'$ be the disjoint union of the vertex sets of all $H_i$ and $H^*_j$. Let $E'$ consist of all the edges of the hypergraphs $H_i$ and the {\em extensions} of the edges of $H^*_j$. Here, an edge of $H^*_j$ is extended with a properly chosen vertex from a hypergraph $H_i$ to obtain a size-$C$ edge of $H'$, making $H'$ $C$-uniform, as required.
We extend different edges in the same copy $H^*_j$ of $H^*$ with vertices from different copies of $H$. This guarantees that if a pair of distinct vertices in $V'$ are contained in more than one edge of $H'$, then they belong to the same $H_i$ and their codegrees in $H'$ and in $H_i$ coincide. Hence, the maximum codegrees in $H$ and $H'$ are the same, as required.
For any $j$, we have to choose a vertex from each copy of $H_i$ to extend one of the edges of $H^*_j$. So, for a fixed copy $H_i$, we have to make $M$ such choices. By the definition of $M$, we can make these choices in such a way that every vertex $v$ in $H_i$ is chosen exactly $k-d_{H_i}(v)$ times. Thus, the degree of $v$ in $H'$ will be precisely $k$. The $H'$-degree of every vertex of $H_j^*$ is also equal to $k$, so $H'$ is $k$-regular.
It follows from the assumption $H_1=H$ that the edge set of $H$ is contained in the edge set of $H'$. This completes the proof of the lemma. \hfill $\Box$
\vspace{0.2cm}
\noindent
{\bf Proof of Theorem \ref{t14}:}\,
Let $N$ be a random necklace with exactly $k$ beads of type $i$ for
each $1 \leq i \leq t$, where $t$ and $k/\log t$ are large.
The total number of
beads is, therefore, $n=kt$. Fix a small $\eps>0$ and let
$C$ be a large integer.
Let $\delta=\delta(C,\eps)$ satisfy the assertion of Theorem
\ref{t41}. We assume without loss of generality that $\delta<\eps/C$.
Split the necklace into
disjoint intervals, each of length $C$ (with possibly one shorter
interval if $kt$ is not divisible by $C$). This requires fewer than
$n/C$ cuts.
Call an interval {\em bad} if it contains two beads of the same type (or if it is the last interval with fewer than $C$ beads); otherwise call it {\em good}.
Let $H$ be the $C$-uniform hypergraph with $t$ vertices
that represent the $t$ types, and one edge for each good interval consisting of the vertices representing the types of beads in the interval. (Occasionally we might get two good intervals containing beads of the same types, but this represents no problem as we allowed for a multiset for edges.)
The maximum degree in $H$
at most $k$. The proof proceeds by showing that, with high
probability, the maximum codegree in $H$ is at most $\delta k$ and, thus, Theorem~\ref{t41} applies. One can then take the $k$ largest matchings
and use them to partition almost all beads into $k$ collections without any further cuts,
each containing at most one bead of each type. The remaining few beads
can then be cut loose by a few additional cuts and then they can be placed in the partition classes as required. Since the number of these remaining beads is small, the
total number of cuts is $o(kt)$ as $t$ and $k/\log t$ tend to infinity.
\smallskip
In what follows, we work out the technical details.
Suppose the following two inequalities hold:
\begin{equation}
\label{e41}
t \geq \frac{2C}{\delta},
\end{equation}
\begin{equation}
\label{e42}
t^2e^{-\delta^2k/2} \leq \eps.
\end{equation}
Note that both of these inequalities hold provided that $t$
and $k/ \log t$ are sufficiently large as functions of $C$ and
$\eps$ (which determine $\delta$).
For every pair of distinct types,
$i,j \in [t]$, let $E_{i,j}$ be the event that more than
$\delta k$ edges contain both $i$ and $j$. The probability
of this event can be estimated as
follows. There are at most $k$ intervals containing a bead of type
$i$. When placing the $k$ beads of type $j$ one by one, the
conditional probability for each of them to lie in an interval
containing a bead of type $i$, given any history, is at most
$\frac{(C-1)k}{(t-2)k} <\frac{C}{t}$.
The probability
that there are at least $\delta k$ such beads is, thus, at most the
probability that a binomial random variable with parameters
$C/t$ and $k$ is at least $\delta k$. This can be estimated using (\ref{e41}) and the Chernoff bound (c.f. \cite{AS}, Theorem A.1.4) as
$\mathbb P(E_{i,j})<e^{-\delta^2k/2}$.
Let $E_i$ be the event that among the length $C$ intervals of the necklace $N$, there are at least $\delta k$ which contain at least two beads of type $i$. The probability of this event can be
estimated using the same argument as above.
Indeed, when placing the beads of type $i$ one by one,
every time the probability that it falls into an interval which already contains
a bead of type $i$, is at most $C/t$. If the event $E_i$
occurs, then this happens at least $\delta k$ times, and the
probability of this event is less than $e^{-\delta^2k/2}$.
By (\ref{e42}) and the above estimates, it follows that,
with probability at least $1-\eps$, none of the events
$E_{i,j}$ and $E_i$ holds. Let us assume that this is the case. As none of the events $E_{i,j}$ hold, the maximal codegree in $H$ is below $\delta k$. Therefore, Theorem~\ref{t41} applies and the edges of $H$ can be partitioned into at most $(1+\eps)k$ matchings. We choose the largest $k$ among these matchings (breaking ties arbitrarily) and partition the corresponding good intervals of the necklace into $k$ parts, each containing at most one bead of each
type. We cut each remaining interval (the bad ones and the good ones that do not belong to any of the largest $k$ matchings) into individual beads and distribute these beads appropriately to obtain a fair partition, where each part contains exactly one bead of each type. We used fewer than $n/C$ cuts to obtain the intervals and fewer than $\eps n$ cuts to cut up the good intervals outside the $k$ largest matchings. Finally, we used at most $C-1$ cuts for each bad interval. As none of the events $E_i$ happens, there are fewer than $\delta n+1$ bad intervals. All in all, the number of cuts was smaller than $n/C+\eps n+\delta Cn\le n/C+2\eps n$ (where we used the assumption that $\delta\le\eps/C$). As $\eps>0$
can be chosen arbitrarily small, $C$ arbitrarily
large, and the required conditions hold with probability
at least $1-\eps$ provided that $t$ and $k/\log t$ are sufficiently large
depending on $C$ and $\eps$, the theorem is true. \hfill $\Box$
\vspace{0.2cm}
\noindent
{\bf Proof of Theorem \ref{t15}:}\,
The lower bound for $X=X(2,t,1)$ is proved by a simple first
moment argument. Let $s$ be the number of cuts allowed, and let
$Y$ be the random variable counting the number of fair
balanced partitions of the necklace using $s (\leq t)$ cuts.
For each fixed
balanced partition, the probability that it is fair is
$$
(t!)^2 \cdot 2^t/(2t)!=\Theta(\frac{\sqrt t}{2^t}).
$$
The number of 2-partitions requiring exactly $s$ cuts is exactly
${{2t-1} \choose s}$. The number of balanced partitions that can be obtained by at most $s$ cuts is therefore at most $\sum_{i=0}^s{2t-1\choose i} \leq 2^{2H(s/(2t))t}$, where
$H(x)$ is the binary entropy function. Therefore, the expected
number of fair partitions with $s$ cuts is at most
$$
2^{2H(s/(2t))t} \Theta\left(\frac{\sqrt t}{2^t}\right).
$$
For any positive $\eps$, large $t>t_0(\eps)$, and
$s$ smaller than $(2H^{-1}(1/2)-\eps)t$, the last expression is smaller than $\eps$.
This implies that whp $X(2,t,1)$ is at least
$H^{-1}(1/2) 2t -o(t) =0.22...t -o(t)$.
\smallskip
The proof of the upper bound,
obtained jointly with Ryan Alweiss, Colin Defant
and Noah Kravitz, follows. Let $f(t)$ denote the expectation
of $X(2,t,1)$.
\begin{claim}
\label{c2t1}
$$
f(t+1) \leq f(t) + \frac{1}{2} -
\frac{1}{2} \frac{f(t)}{2t+1}+\frac{1}{2t+2}.
$$
\end{claim}
\noindent
{\bf Proof:}\,
Expose the random necklace with $t+1$ types and $2$
beads of each type
as follows: first expose the
last bead, without loss of generality let its type be $t+1$.
Then expose the necklace of the
$2t$ beads of
types $1, 2,\ldots ,t$ (ignoring the second bead of type $t+1$).
This is uniform random
hence in expectation there is a collection of $f(t)$ cuts of this necklace
of $t$ types that gives a fair partition.
Fix one such minimum collection of cuts.
Let their number be $f$, note that $f$ is a random variable
whose expectation is $f(t)$.
So far we have the relative order of $2t+1$ beads. Now
expose the other occurrence of the bead of type $t+1$, which we call
here the
extra bead: it clearly lies
in a uniform random place among $2t+1$ options which are the
$2t+1$ spaces before bead number $i$ for some $i \leq 2t+1$.
If this place happens to be one of the existing
$f$ cuts then no additional cut is needed, as we can append this
extra bead to an interval of each of the two thieves, as needed, without
adding any new cut. This happens with probability $f/(2t+1)$. If this
is not the case, then with probability roughly a half (computed
precisely in what follows)
the extra bead is placed in
an interval that goes to the thief who is not the one to get the last
interval. Note that in this case we do not need an extra cut either.
The probability that this occurs is
exactly $(t-f/2)/(2t+1-f)$ for even $f$ and
$(t-(f-1)/2)/(2t+1-f)$ for odd $f$. This is because the total number
of beads that this thief gets from among the $2t+1$ beads above is $t$, and
among the spaces just before them, there are $f/2$ which are spaces
among the $f$ cuts if $f$ is even, and $(f-1)/2$ if $f$ is odd.
(In the odd case, each thief gets $(f+1)/2$ intervals,
but the space before the first interval is not a cut). The above
ratio is, thus, exactly $1/2$ for odd $f$ and for even $f$ it satisfies
$$
\frac{1}{2} \frac{2t-f}{2t+1-f} = \frac{1}{2}(1- \frac{1}{2t+1-f})
\geq \frac{1}{2} (1-\frac{1}{t+1}).
$$
Here, we used the fact that $f \leq t$ always holds, by the deterministic
result.
Therefore, the probability of the event above is at least
$\frac{1}{2}-\frac{1}{2t+2}$.
In the only remaining case, which happens with probability at most
$$
(\frac{1}{2}+\frac{1}{2t+2})(1-\frac{f(t)}{2t+1})
\leq \frac{1}{2} - \frac{1}{2} \frac{f(t)}{2t+1}+\frac{1}{2t+2},
$$
we need at most one additional cut: just
before the very last bead (of type $t+1$). This proves the claim.
\hfill $\Box$
\smallskip
We next show that, by Claim~\ref{c2t1}, $\lim \sup f(t)/t \leq 0.4$.
Indeed, if for some value of $t$, $\frac{f(t+1)}{t+1}
\geq \frac{f(t)}{t}$, then by the claim
$$
\frac{f(t)}{t} \leq \frac{f(t+1)}{t+1} \leq
\frac{f(t)}{t+1} -\frac{1}{2}\frac{f(t)}{(2t+1)(t+1)} +
\frac{1}{2(t+1)}+\frac{1}{2(t+1)^2}.
$$
This gives
$$
\frac{f(t)}{t} \frac{5t+2}{2t+1} \leq 1+\frac{1}{t+1},
$$
implying that in this case $\frac{f(t)}{t} \leq 0.4+O(1/t).$
However, in this case, by the above inequality
(or by the simple fact that $f(t+1) \leq f(t)+1$), we also have
$f(t+1)/(t+1) \leq 0.4 +O(1/t).$
Hence, if there are infinitely many values of $t$
for which $f(t+1)/(t+1) > f(t)/t$ holds, then $\lim \sup f(t)/t \leq 0.4$.
Otherwise, the function $f(t)/t$ is eventually decreasing, so $x=\lim f(t)/t$ exists. In this case Claim~\ref{c2t1} implies
$f(t+1)\le f(t)+1/2-x/4+o(1)$. Summing this for all values $t<t_0$ we get $f(t_0)\le(1/2-x/4)t_0+o(t_0)$ or, equivalently,
$f(t_0)/t_0\le 1/2+x/4 +o(1)$. Taking limits, we obtain $x\le1/2-x/4$, yielding $x\le0.4$ as needed.
We have thus shown that $E(X(2,t,1))\le 0.4t+o(t)$.
We can conclude, by the Azuma-Hoeffding Inequality
(see, e.g., \cite{AS}), that $X(2,t,1)\le 0.4t+o(t)$ holds whp.
Indeed, by this inequality and the fact that the minimum
number of cuts for a fair partition can change by at most $O(1)$ when
swapping two beads, it follows that the probability that $X(2,t,1)$
deviates from its expectation by at least $C\sqrt t$ is at most
$e^{-\Omega(C^2)}$. This implies that
$X(2,t,1)\le 0.4t+o(t)$ whp, completing
the proof of Theorem
\ref{t15}. \hfill $\Box$
\vspace{0.2cm}
\noindent
{\bf Remark.}\, It is possible to slightly improve both estimates
in Theorem~\ref{t15}. The upper bound can be improved by
observing that in the argument above, if the extra bead appears
in an interval forcing us to add a cut, and it happens to appear
between two beads of types $i$ and $j$ where there is a cut just
before the type $i$ bead, then, if the second bead of type $i$ also
appears right after (or before) a cut, it is possible to shift
these two cuts and get a fair partition without increasing the total
number of cuts. There are several similar local scenarios that
can be used in a similar way and lead to small improvements in the
upper bound. Optimizing these arguments can be difficult, but they
do show that the expectation is smaller than $0.39 t +o(t)$ whp.
The lower bound can also be slightly improved as follows.
If one has a fair partition with at most $s$ cuts, then one can
choose that to be minimal: first, minimize the number of cuts, then
try to make the cuts as far to the right as possible.
Now the first beads after the cuts have all distinct types, as
otherwise two of the cuts could be shifted one position to the right (and if
they reach the next cut, then they would cancel). This gives a tiny
advantage in the probability (probability of a set of cuts being
fair
versus probability of being the minimal fair set). This gives a
lower
bound of roughly $0.227 \cdot t$,
a tiny improvement over $0.22 \cdot t$.
These considerations still leave a substantial gap between the
upper and lower bounds.
\section{Concluding remarks and open problems}\label{sectionConcl}
We have studied the minimum possible number of cuts required to
partition a random necklace with $km$ beads of each of $t$ types
fairly into $k$ collections. This
minimum is denoted by $X=X(k,t,m)$.
A better understanding of the behavior of the random variable $X$
for all admissible values of the parameters requires further study.
\begin{itemize}
\item
Some of the arguments described here
for $k=2$, fixed $t$ and large $m$ can be extended to higher
values of $k$. In particular, we note that the argument in the
proof of the upper
bound for the probability that $X(2,3,m)=O(1/\log m)$ implies that
for every fixed $k >2$ and $t=3$, the probability that
$X(k,3,m)=\frac{(k-1)(t+1)}{2}=2k-2$ is also $O(1/\log m)$.
Indeed, $2k-2$ cuts split the necklace into $2k-1$ intervals,
hence there is at least one thief who gets a single interval. This
interval has to contain exactly $m$ beads of each of the three
types. The probability of the existence of such an interval is
$O(1/\log m)$, by the argument in the proof presented in Subsection~2.1.
\item
We have seen that the
ratio $X(2,t,1)/t$ is between $0.22$ and $0.4+o(1)$ whp. Is the
ratio $c+o(1)$ whp, for some constant $c$, and if so,
what is the value of $c$?
\item
The algorithmic problem of finding a small number of cuts
yielding a fair partition efficiently for a given input random
necklace is also interesting. For the deterministic case there are
known hardness results for the problem (see \cite{FG}) and known
approximation algorithms (\cite{AG}), and it will be interesting to find
efficient algorithms that work better whp for the random case.
\item
The random variable $X(2,t,m)$ has the following interpretation in
terms of a question about folding positive random walks in $Z^t$.
Consider a random walk of $2tm$ steps in $Z^t$. Starting from the
origin, every step is one of the $t$ unit vectors $e_i$, where the
sequence of steps is a random sequence consisting of exactly $2m$
steps in each direction $e_i$. An elementary folding at $j$ of this
sequence of steps switches the signs of all steps from step number
$j$ until the end. We can apply several of these elementary foldings one after the other. The random variable
$X(2,t,m)$ is thus the minimum number of elementary foldings required to
ensure the folded sequence ends at the origin. A similar question,
avoiding parity issues,
can be considered for a sequence of $n$ random, independent,
positive steps in $Z^t$. The random variable now is the minimum
number of elementary foldings required to ensure the folded walk
ends within $\ell_{\infty}$-distance $1$ from the origin.
\end{itemize}
\noindent
{\bf Acknowledgment}.
We thank Yuval Peres for providing helpful references, and thank
Ryan Alweiss, Colin Defant and Noah Kravitz for their help in the
proof of the upper bound in Theorem \ref{t15}.
|
2024-02-18T23:40:24.628Z
|
2021-12-30T02:24:33.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14488","language":"en","timestamp":1640831073000,"url":"https:\/\/arxiv.org\/abs\/2112.14488","yymm":"2112"}
|
proofpile-arXiv_000-10054
|
{"provenance":"002.jsonl.gz:10055"}
| null | null |
\section{Introduction}
\label{sec:Introduction}
Controlling light intensity fluctuations has been demonstrated to be a topic of pivotal importance in many optical applications. In the process of imaging, the useful information can be contained not only in the light field but also in the light intensity fluctuations \cite{kolobov2007quantum,zeng2013laser,boyer2008entangled}. The role of light intensity fluctuations in improving the resolution of optical image has been proved \cite{PhysRevLett.85.3789,sprigg2016super,sroda2020sofism,DIGMAN2005L33}. The impact of light intensity fluctuations on the visibility of ghost imaging, ghost interference, and ghost diffraction were reported in \cite{ferri2005high,liu2013high,bromberg2009ghost,shapiro2008computational,gatti2004ghost}. The beam positioning was performed by the fluctuation of an actual beam oscillation \cite{treps2003quantum}. The fluctuations of the light field were widely utilized to identify diverse sources of light \cite{PhysRev.130.2529,mandel1995optical,you2020identification}. The optical rogue waves and extreme events are defined as an optical pulse whose amplitude or intensity is much higher than that of the surrounding pulses, which were closely related to the light intensity fluctuations \cite{PhysRevLett.119.223603,akhmediev2016roadmap}. The reference \cite{zhang2019superbunching} showed that the bunching property of the light field can be modified via modulation of the intensity fluctuation. In the presence of the fluctuating light pump, the multiphoton absorption \cite{lambropoulos1966coherence,jechow2013enhanced}, multiphoton ionization \cite{lecompte1975laser,mouloudakis2019revisiting}, generation of optical harmonics \cite{PhysRevLett.119.223603,lamprou2020perspective}, and modulation instability \cite{Hammani:09} were hugely enhanced. Therefore, stronger light intensity fluctuation plays an important role in many optical applications.
In general, the amplification of the light intensity fluctuations has been achieved in both linear and nonlinear methods. In the linear optical system, the increased intensity fluctuations were obtained by rotating ground glasses \cite{zhou2017superbunching} and adding electro- or acousto-optical modulator \cite{straka2018generator,zhou2019superbunching}. However, due to experimental complexity and error resulted from the increased number of linear devices, the amplification degree of the intensity fluctuations is limited in linear methods.
In the nonlinear optical system, the light intensity fluctuations were enhanced in the interaction of matters and light \cite{perina1993photon,cao2016resolution,Yu:16}, squeezing processes \cite{liu2016enhanced}, and rogue waves or extreme events \cite{PhysRevLett.119.223603,manceau2019indefinite}. However, the amplification degree of light intensity fluctuations in nonlinear processes is disproportionate and heavily dependent on particular matters and environmental factors that triggered nonlinear interaction.
In this paper, an amplification mechanism for light intensity fluctuations is investigated in an N-order nonlinear optical process both theoretically and experimentally. The Nth-order light intensity fluctuation amplifier is achieved by incident beams configures with different statistical distributions involving in an N-order nonlinear optical effects. Four-wave mixing (FWM) was chosen as verification of this amplifier, 1$\sim$N incident beams were modulated simultaneously into the pseudothermal light through the FWM process, which introduced and compared the Nth-order light intensity fluctuation amplifier. The ratio $R$ of statistical distributions and the degree of second-order coherence ${g^{(2)}}(0)$ of beams through FWM were used to evaluate the affected modulations and the increased light intensity fluctuations.
Inputting light beams configures with different statistical distributions into N-order optical nonlinear processes brings different modulations and light intensity fluctuations. In the proposed Nth-order amplifier, the amplification degree of light intensity fluctuations can be tunable by different statistical distributions configures, and the multiplex amplification can be simultaneously achieved. On the other hand, by different statistical distributions involving in the N-order nonlinear optical process, the nonlinearity changes accordingly. Thus, the amplification of light intensity fluctuations is caused by not only the fluctuating light fields of incident coupling beams, but also the fluctuating nonlinear coefficient of interaction. In principle, the amplifier can be widely used in many N-order nonlinear optical effects, including optical harmonics, electromagnetically induced transparency, and four-wave mixing. The proposal highlights the potential applications as a flexible and powerful method to amplify the intensity fluctuation of light.
\section{Theory}
\label{sec:Theory}
In the N-order nonlinear processes, the optical field in atomic systems can be described in terms of equations of motion \cite{boyd2003nonlinear}
\begin{eqnarray}
\tilde {E}(t) = {E}{e^{ - i\omega t}} + c.c.,\
\label{eq:one}
\end{eqnarray}
where ${E}$ denotes the complex amplitude of the arbitrary wave at frequency $\omega$. The nonlinear interaction response can often be described by expressing the polarization $\tilde {P}(t)$ as
\begin{eqnarray}
\tilde P(t) = {\varepsilon _0}\left[ {{\chi ^{(1)}}\tilde E(t) + {\chi ^{(2)}}{{\tilde E}^2}(t) + \ldots + {\chi ^{(N)}}{{\tilde E}^N}(t)} \right].\
\label{eq:two}
\end{eqnarray}
${\chi ^{\left( 1 \right)}}\left( \omega \right)$ is linear susceptibility and ${\chi ^{\left( N \right)}}\left( \omega \right)$ is N-order nonlinear susceptibility. As seen that, there are N incident light fields $\tilde {E}(t)$ involved in the N-order nonlinear process, and the nonlinearity results from the interaction of matters and N incident beams. Thus, the output light through N-order nonlinear processes will exhibit different modulation results affected by the nonlinearity from different incident light fields combinations.
The third-harmonic generation and FWM are the third-order nonlinear processes, we use a degenerate FWM process to explain and introduce the proposed Nth-order light intensity amplifier. In the FWM process, we use different incident light fields combinations to manipulate nonlinear interaction results. The ratio $R$ of the statistical light intensity distribution and the degree of second-order coherence ${g^{(2)}}(0)$ of beams through FWM were used to evaluate the affected modulations and the increased light intensity fluctuations. The ratio $R$ is defined as
\begin{eqnarray}
R = \frac{{{{(\Delta I)}^2}}}{{{{\bar I}^2}}},\
\label{eq:three}
\end{eqnarray}
where $(\Delta I{)^2}$ and $\bar I$ are the variance and mean value, respectively. For the degree of the second-order coherence, the normalized second-order coherence functions of beams involved in the FWM process can be calculated as \cite{mandel1995optical}
\begin{eqnarray}
{g^{(2)}}(\tau ) = \frac{{\left\langle {\bar I(t)\bar I(t + \tau )} \right\rangle }}{{{{\left\langle {\bar I(t)} \right\rangle }^2}}},\
\label{eq:four}
\end{eqnarray}
where $\tau$ denotes the delay time, $\bar I(t)$ is the average of the intensity over a cycle of oscillation and the brackets $\left\langle \ldots \right\rangle$ denote the statistical or longer-time average. The ${g^{(2)}}(\tau)$ when $\tau=0$ is defined as the degree of second-order coherence ${g^{(2)}}(0)$ \cite{Loudon1983The}.
In the degenerate FWM process, the wave vector mismatch is zero in the presence of the phase matching condition. The amplitude ${E_i}$ of any field propagating in the $+z$ direction obeys the set of coupled equations
\begin{eqnarray}
\frac{{d{E_i}}}{{dz}} = - \alpha {E_i} + \kappa E_j^*.\
\label{eq:five}
\end{eqnarray}
${E_j}$ is the amplitude of phase conjugation wave of ${E_i}$. The absorption coefficient $\alpha$ and coupling coefficient $\kappa$ are
\begin{subequations}
\label{eq:six}
\begin{equation}
\alpha = - \frac{\omega }{{2nc}}{\mathop{\rm Im}\nolimits} {\chi ^{(1)}}(\omega ),\label{subeq:1}
\end{equation}
\begin{eqnarray}
\kappa = - i\frac{{3\omega }}{{2nc}}{\chi ^{(3)}}(\omega ){E_m}{E_n}.\label{subeq:2}
\end{eqnarray}
\end{subequations}
where $n = \sqrt {1 + {\mathop{\rm Re}\nolimits} {\chi ^{(1)}}(\omega )}$ is the usual linear refractive index and $c = 3 \times {10^8}m/s$ is the speed of light in vacuum. ${E_m}$ and ${E_n}$ are the amplitudes of two pump waves. The linear susceptibility ${\chi ^{\left( 1 \right)}}\left( \omega \right)$ and the third-order nonlinear susceptibility ${\chi ^{\left( 3 \right)}}\left( \omega \right)$ are given by
\begin{subequations}
\label{eq:seven}
\begin{equation}
{\chi ^{\left( 1 \right)}}\left( \omega \right) = \frac{{N{\mu ^{\rm{2}}}\omega _{\rm{0}} \hbar }}{{{\varepsilon _{\rm{0}}}}}\frac{{{T_2}\left( { - i + \Delta {T_2}} \right)}}{{{\hbar ^2}\left( {1 + {\Delta ^2}T_2^2} \right) + 4{T_1}{T_2}{\mu ^{\rm{2}}}{E_m}{E_n}}},\label{subeq:3}
\end{equation}
\begin{eqnarray}
{\chi ^{\left( 3 \right)}}\left( \omega \right) = - \frac{{4N{\mu ^4}\omega _{\rm{0}} }}{{3{\varepsilon _{\rm{0}}}\hbar \left( {i + \Delta {T_2}} \right)}}\frac{{{T_1}T_2^2}}{{{\hbar ^2}\left( {1 + {\Delta ^2}T_2^2} \right) + 4{T_1}{T_2}{\mu ^{\rm{2}}}{E_m}{E_n}}}.\label{subeq:4}
\end{eqnarray}
\end{subequations}
In this experiment, due to definite nonlinear medium and atom level of the D2 line $(5{S_{1/2}}(F = 2) \leftrightarrow 5{P_{3/2}}(F' = 3))$ of ${}^{{\rm{87}}}Rb$ \cite{boyd2003nonlinear,steck2001rubidium}, all atomic parameters become invariant constants. The transition dipole moment $\mu {\rm{ = 1}}{\rm{.731}} \times {\rm{1}}{{\rm{0}}^{{\rm{ - 29}}}}Cm$, longitudinal and transverse relaxation time are ${T_1} = 2.62357 \times 10{}^{ - 8}s$ and ${T_2} = 5.24714 \times 10{}^{ - 8}s$. The Planck's constant and permeability of vacuum is $\hbar = 1.054 \times {10^{ - 34}}Js$ and ${\varepsilon _0} = 8.854 \times {10^{ - 12}}F/m$, respectively. For the degenerate FWM, the detuning $\Delta$ of the laser frequency from the resonant frequency equals $0$. The laser angular frequency is $\omega = {\omega _0} = 2\pi \cdot 384.230THz$. The atomic density of Rb vapor is approximately $N = 6.3 \times {10^{12}}c{m^{ - 3}}$ (cell temperature $95{}^ \circ C$) \cite{steck2001rubidium}. By combining the above constant parameters and Eq.~(\ref{eq:seven}), we reduce Eq.~(\ref{eq:six}) to
\begin{subequations}
\label{eq:eight}
\begin{equation}
\alpha = \frac{{1.14 \times {{10}^{ - 47}}}}{{1.11 \times {{10}^{ - 68}} + 1.65 \times {{10}^{ - 72}}{E_m}{E_n}}},\label{subeq:5}
\end{equation}
\begin{eqnarray}
\kappa = \frac{{1.7 \times {{10}^{ - 51}}{E_m}{E_n}}}{{1.11 \times {{10}^{ - 68}} + 1.65 \times {{10}^{ - 72}}{E_m}{E_n}}}.\label{subeq:6}
\end{eqnarray}
\end{subequations}
As seen in Eq.~(\ref{eq:eight}), if the atomic and environmental conditions of the nonlinear process are defined, the nonlinear coefficients will be dependent on the light fields of pump beams ${E_m}$ and ${E_n}$. Thus the affected modulation of the light field ${E_i}$ after nonlinear interaction is the result of combining its phase conjugation field ${E_j}$ with two pump light fields ${E_m}$ and ${E_n}$. Here the probe and pump light fields in the FWM process are considered as the light fields with different fluctuations. The output beam through nonlinear interaction is modulated by not only the fluctuating light fields of incident coupling beams but also fluctuating nonlinear coefficient. Based on the above analysis, the affected modulation and light intensity fluctuations of the output beam would be considerable and controllable.
The amplitude ${E_i}$ of any field at the exit plane of the nonlinear medium is obtained by integral operation, then the intensity ${I_i}$ of any beam involved is given by
\begin{eqnarray}
{I_i} = 2n{\varepsilon _0}c{\left| {{E_i}} \right|^2}.\
\label{eq:nine}
\end{eqnarray}
Then the ratio $R$ of the statistical light intensity distribution and the degree of second-order coherence ${g^{(2)}}(0)$ of beams through FWM were measured to evaluate the affected modulations and the increased light intensity fluctuations.
\section{Experimental setup}
\label{Experimental setup}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.85\textwidth]{01.pdf}
\caption{\label{fig:1} Schematics of the experimental setup. Placing an EOM in different color area corresponds to the Nth-order intensity fluctuations amplifier. Laser: single-mode tunable diode laser; EOM: electro-optical modulator; L$_1$-L$_2$: convex lenses; M$_1$-M$_4$: mirrors; PBS$_1$-PBS$_4$: polarization beam splitters; H$_1$-H$_4$: half-wave plates; D$_1$-D$_4$: high-speed photoelectric detectors; BS$_1$-BS$_2$: beam splitters; Rb: rubidium vapor cell.}
\end{figure}
The experimental setup of the Nth-order light intensity fluctuation amplifier is schematically depicted in Fig.~\ref{fig:1}. The laser was from a single-mode tunable diode laser with a center wavelength of $780.24 nm$ and power of $56.28 mW$. The laser beam was divided into two beams by a set of half-wave plate (H$_1$) and polarization beam splitter (PBS$_1$). The reflected beam with S-polarization component was calibrated into P-polarization component via the second set of half-wave plate (H$_2$) and polarization beam splitter (PBS$_2$), named as forward pump beam (Pump$_f$). Meantime, the transmitted beam was further divided into two parts by a set of half-wave plate (H$_3$) and polarization beam splitter (PBS$_3$). The reflected part with S-polarization served as the backward pump beam (Pump$_b$) and counter-propagated with the Pump$_f$. The transmitted part through half-wave plate (H$_4$) and polarization beam splitter (PBS$_4$) was chosen as the probe beam with P-polarization. The small angle between probe beam and the backward pump beam is ${0.7^ \circ }$. The two convex lenses (L$_1$ and L$_2$) were used to focus three incident beams at a point in the rubidium (Rb) atomic vapor cell. Since they satisfied the phase-matching condition of a degenerated-FWM process, the new FWM signal was generated from the opposite direction of the probe beam with S-polarization.
The Rb cell was set to $95^\circ C$ by a temperature control heater, which had a length of $75 mm$. The forward and backward pump beam power were $12.06 mW$ and $5.24 mW$, the probe power was $8.14 mW$. After the FWM process, the high-speed photoelectric detectors (D$_1$ and D$_3$) were employed to detect the light intensities of the Probe and Pump$_f$ via the beam splitters (BS$_1$ and BS$_2$), respectively. The light intensities of the Pump$_b$ and the FWM signal were measured by the D$_2$ and D$_4$.
In this experiment, the pseudothermal light was obtained through an electro-optical modulator (EOM) and by inputting exponential distribution signal. As shown in Fig.~\ref{fig:1}, by placing an EOM in different-colors locations, one or more incident beams were modulated simultaneously into the pseudothermal light through the FWM process, that realized the Nth-order light intensity fluctuation amplifier. In our schemes, we used three cases to introduce and compare the Nth-order light intensity fluctuation amplifier, the ratio of the statistical light intensity distribution $R$ (accurate to two decimal places) and the degree of second-order coherence ${g^{(2)}}(0)$ of beams through FWM were used to evaluate the affected modulations and the increased light intensity fluctuations.
\section{Results and discussion}
\label{Results and discussion}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.75\textwidth]{02.pdf}
\caption{\label{fig:2} The measured statistical intensity distributions of (a) FWM signal, (b) probe beam, (c) backward and (d) forward pump beams through FWM in the 1st-order case. The red and black lines represent the exponential and Gaussian distributions with the same average intensities of output beam, respectively.}
\end{figure}
In the 1st-order case, an EOM is placed in the blue area as shown in Fig.~\ref{fig:1}. The probe beam is modulated to thermal light with $R_{In}=1.00$ by EOM, the forward and backward pump beams are lasers with $R_{In}=0.00$. As seen that, two incident beams through FWM process are lasers in the 1st-order case. Figure~\ref{fig:2} shows the statistical intensity distributions of beams before and after the FWM process in the 1st-order case. The light intensity distribution of the generated FWM signal is shown in Fig.~\ref{fig:2}(a). Compared with the exponential distribution with the same average intensity (red line), the FWM signal shows a heavy-tailed distribution with $R_{Out}=0.36$ according the Ref. \cite{manceau2019indefinite}. As shown in Fig.~\ref{fig:2}(b), the original probe beam is the thermal light obeying negative exponential distribution with $R_{In}=1.00$. There is very little modulation in the output of the probe beam after the FWM process, whose statistical intensity distribution remain largely the exponential distribution with $R_{Out}=0.89$. As shown in Figs.~\ref{fig:2}(c) and \ref{fig:2}(d), the incident forward and backward pump beams are laser, whose statistical intensity distributions are Gaussian distributions with $R_{In}=0.00$. Even the intensities of backward and forward pump beams decreased after the FWM process, their statistical intensity distributions still exhibit profiles of Gaussian distributions with $R_{Out}=0.05$ and $R_{Out}=0.01$. The black lines represent the Gaussian distributions with the same average intensities. It revealed that the configurations of incident light fields in the 1st-order case had little effect on amplifying affected modulation and light intensity fluctuation.
\begin{figure}[t]
\centering
\includegraphics[width=0.75\textwidth]{03.pdf}
\caption{\label{fig:3} The measured second-order coherence functions of (a) FWM signal, (b) probe beam, (c) backward and (d) forward pump beams in the 1st-order case. Dashed green lines represent the normalized degree of second-order coherence ${g^{(2)}}(0)$ of thermal light in theory, that is boundary between bunching and superbunching effect. Solid blue lines are the measured second-order coherence functions of input beams experimentally. Solid red lines indicate the measured second-order coherence functions of output beams through FWM process.}
\end{figure}
Figure~\ref{fig:3} shows the measured second-order coherence functions of beams through the FWM process in the 1st-order case. The degree of second-order coherence ${g^{(2)}}(0)$ of the generated FWM signal reached $1.36$ in Fig.~\ref{fig:3}(a). The ${g^{(2)}}(0)$ of probe beam decreased lightly from $2.00$ to $1.89$ in Fig.~\ref{fig:3}(b). The backward and forward pump beams through FWM process basically maintain ${g^{(2)}}(0)=1.00$ in Figs.~\ref{fig:3}(c) and \ref{fig:3}(d). As seen that, the configurations of incident light fields in the 1st-order case were ineffective in improving the degree of second-order coherence of beams.
\begin{figure}[t]
\centering
\includegraphics[width=0.75\textwidth]{04.pdf}
\caption{\label{fig:4} The measured statistical intensity distributions of (a) FWM signal, (b) probe beam, (c) backward and (d) forward pump beams through FWM in the 2nd-order case. The red and black lines represent the exponential and Gaussian distributions with the same average intensities of output beam, respectively.}
\end{figure}
In the 2nd-order case, an EOM is placed in the green area as shown in Fig.~\ref{fig:1}. The probe beam and backward pump beam were modulated simultaneously to the same thermal light with $R_{In}=1.00$ by EOM, the forward pump is laser with $R_{In}=0.00$. As seen that, only one incident beam through FWM process is laser in the 2nd-order case. Figure~\ref{fig:4} shows the statistical intensity distributions of beams before and after the FWM process in the 2nd-order case. The light intensity distribution of the generated FWM signal is shown in Fig.~\ref{fig:4}(a). Compared with the exponential distribution with the same average intensity (red line), the FWM signal shows heavy-modulated statistical distribution and has strong fluctuation with $R_{Out}=2.97$. As shown in Fig.~\ref{fig:4}(b), the original probe beam is the thermal light obeying negative exponential distribution with $R_{In}=1.00$. After the FWM process, the statistical intensity distribution of the probe beam is modulated slightly with $R_{Out}=1.73$. As shown in Fig.~\ref{fig:4}(c), the forward pump beam is changed from the exponential distribution with $R_{In}=1.00$ to the heavy-modulated distribution with $R_{Out}=2.53$. The incident forward pump beam is laser obeying Gaussian distribution with $R_{In}=0.00$, yet the output basically maintains Gaussian-like distribution with $R_{Out}=0.27$ as shown in Fig.~\ref{fig:4}(d). It is concluded that the statistical intensity distributions of beams in the 2nd-order case are deeper modulated than the 1st-order case, and the light intensity fluctuations of all beams are amplified.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.75\textwidth]{05.pdf}
\caption{\label{fig:5} The measured second-order coherence functions of (a) FWM signal, (b) probe beam, (c) backward and (d) forward pump beams in the 2nd-order case. Dashed green lines represent the normalized degree of second-order coherence ${g^{(2)}}(0)$ of thermal light in theory, that is the boundary between bunching and superbunching effect. Solid blue lines are the measured second-order coherence functions of input beams experimentally. Solid red lines indicate the measured second-order coherence functions of output beams through the FWM process.}
\end{figure}
Figure~\ref{fig:5} shows the second-order coherence functions of beams through the FWM process in the 2nd-order case. The FWM signal with ${g^{(2)}}(0)=3.97$ exceeds the superbunching threshold value $2.00$ in Fig.~\ref{fig:5}(a). The ${g^{(2)}}(0)$ of probe beam slightly increased from $2.00$ to $2.73$ in Fig.~\ref{fig:5}(b). The ${g^{(2)}}(0)$ of backward pump beam substantially increased from $2.00$ to $3.53$ in Fig.~\ref{fig:5}(c). The reason for different growth of ${g^{(2)}}(0)$ is that the beam with strong power is hardly modulated deeply. The forward pump beams slightly ascended to ${g^{(2)}}(0)=1.27$ from $1.00$ in Fig.~\ref{fig:5}(d). As seen that, the FWM signal, probe beam, and backward beam through the FWM process all reached superbunching effects in the 2nd-order case \cite{boitier2011photon,iskhakov2012superbunched}. We derive a conclusion that the incident light fields configurations of the 2nd-order case had done better than the 1st-order case in improving the degree of second-order coherence of beams.
\begin{figure}[t]
\centering
\includegraphics[width=0.75\textwidth]{06.pdf}
\caption{\label{fig:6} The measured statistical intensity distributions of (a) FWM signal, (b) probe beam, (c) backward and (d) forward pump beams through FWM in the 3rd-order case. Red lines represent the exponential distribution with the same average intensity of output beam.}
\end{figure}
In the 3rd-order case, an EOM is placed in the yellow area as shown in Fig.~\ref{fig:1}. All incident beam are simultaneously modulated to the same thermal light by EOM. As seen that, there is no laser through FWM process in the 3rd-order case. Figure~\ref{fig:6} shows the statistical intensity distributions of involves beams before and after the FWM process in the 3rd-order case. The light intensity distribution of the generated FWM signal is shown in Fig.~\ref{fig:6}(a). Compared with exponential distribution with the same average intensity (red line), the FWM signal shows heavy-modulated distribution with $R_{Out}=5.16$. As shown in Fig.~\ref{fig:6}(b), \ref{fig:6}(c) and \ref{fig:6}(d), the probe, backward and forward pump beam is changed from the exponential distribution with $R_{In}=1.00$ to the heavy-modulated distribution with $R_{Out}=3.16,5.27,4.49$, respectively. It is demonstrated that the simultaneously fluctuating light configuration in the 3rd-order case brings the greatest modulation and the strongest intensity fluctuation amplification.
\begin{figure}[!htbp]
\centering
\includegraphics[width=0.75\textwidth]{07.pdf}
\caption{\label{fig:7} The measured second-order coherence functions of (a) FWM signal, (b) probe beam, (c) backward and (d) forward pump beams in the 3rd-order case. Dashed green lines represent the normalized degree of second-order coherence ${g^{(2)}}(0)$ of thermal light in theory, that is the boundary between bunching and superbunching effect. Solid blue lines are the measured second-order coherence functions of input beams experimentally. Solid red lines indicate the measured second-order coherence functions of output beams through FWM process.}
\end{figure}
The second-order coherence functions of beams in the 3rd-order case are as shown in Fig.~\ref{fig:7}. The FWM signal, probe, backward, and forward pump beams with ${g^{(2)}}(0)=6.16, 4.16, 6.27, 5.49$ all reached the superbunching effect. The degree of second-order coherence of beams in the 1$\sim$N order cases are summarized in Table~\ref{tab:table2}. As seen that, the configurations of incident light fields in the 2nd-order and the 3rd-order cases achieve great performances in amplifying light intensity fluctuation and improving the degree of second-order coherence. In conclusion, in the third-order ($N$) nonlinear process, zero and one ($k$, $0 \le k < 2/N$) incident beams are the laser or coherent light (the 2nd-order and 3rd-order case), the light intensity fluctuations of all output beams after the nonlinear interaction are enhanced.
\begin{table}[t]
\centering
\caption{\label{tab:table2} The degree of second-order coherence ${g^{(2)}}(0)$ of beams through FWM in the 1$\sim$N order cases.}
\begin{tabular}{ccccc}
\toprule
${g^{(2)}}(0)$ &\mbox{$FWM$} &\mbox{$Probe$} &\mbox{$Pum{p_b}$} &\mbox{$Pum{p_f}$}\\
\midrule
$Case 1$ & 1.36 &\mbox{1.89} &\mbox{1.05} &\mbox{1.01}\\
$Case 2$ & 3.97 &\mbox{2.73} &\mbox{3.53} &\mbox{1.27}\\
$Case 3$ & 6.16 &\mbox{4.16} &\mbox{6.27} &\mbox{5.49}\\
\bottomrule
\end{tabular}
\end{table}
In most previous references for simplicity, the pump light fields in the FWM process were generally approximated to the invariable constant. When the atomic and environmental conditions of the nonlinear process were defined, the nonlinear coefficient was considered to be constant. However in the real experiments, even the pump light fields are lasers, they obey the Gaussian statistical distributions, the nonlinear coefficient would not be a constant. Thus, when the atomic and environmental conditions of the nonlinear process were defined, even the pump light fields are laser, the statistical type of the generated FWM signal is not exactly the same as its phase conjugation light field, namely probe beam.
As seen in Eq.~(\ref{eq:eight}), we found that when the atomic and environmental conditions of the nonlinear process are defined, the nonlinear coefficients will be dependent on the light fields of pump beams ${E_m}$ and ${E_n}$. The fluctuating nonlinear coefficient of interaction results from the combination of pump light fields. By combining different statistical distributions, the fluctuation degree of the nonlinear coefficient is controllable, then the affected modulation and the increased light intensity fluctuations of the output beam would be tunable flexibly.
Furthermore, there are three same thermal light fields with synchronous fluctuations through FWM in the 3rd-order case. It is interesting to study whether the intensity fluctuations of output beam can be amplified or not when the incident beams are the same light fields without synchronous fluctuations. Therefore, to amplify intensity fluctuations of output beams in the nonlinear process, the importance of the synchronous fluctuation of incident coupling beams is worth studying.
\section{Conclusion}
\label{sec:Conclusion}
Since amplifying light intensity fluctuations plays an important role in many optical applications, we proposed an Nth-order light intensity fluctuation amplifier through N-order optical processes. In the $N$-order nonlinear processes, $k$ ($0 \le k < 2/N$) incident beams are the laser or coherent light, the light intensity fluctuations of all output beams after the nonlinear interaction will be enhanced, that realizes an Nth-order light intensity fluctuation amplifier. In addition, the most effective way to improve the light intensity fluctuations is that N incident beams are all the same light fields with synchronous fluctuations. The FWM process pumped by different statistical distributions was chosen as experimental verification of this amplifier, the ratio $R$ of the statistical light intensity distribution and the degree of second-order coherence ${g^{(2)}}(0)$ of beams are used to evaluate the affected modulation degree and the increased light intensity fluctuation. The experimental results of the FWM process are consistent with the proposed N-order amplifier.
By inputting light beams configures with different statistical distributions into N-order optical nonlinear processes, the amplification of light intensity fluctuations is caused by not only the fluctuating light fields of incident coupling beams but also the fluctuating nonlinear coefficient of interaction. The different 1~Nth-order cases exhibit different amplification degrees, the amplification of light intensity fluctuations is tunable and multiplex. The proposed Nth-order light intensity fluctuation amplifier could be widely used in many N-order nonlinear optical effects, including optical harmonics, electromagnetically induced transparency, and four-wave mixing.
\section*{Declaration of competing interest}
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
\section*{Funding}
Shaanxi Key Research and Development Project (Grant No. 2019ZDLGY09-10); Key Innovation Team of Shaanxi Province (Grant No. 2018TD-024); National Natural Science Foundation of China (Grant No. 61901353).
\bibliographystyle{ref}
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2024-02-18T23:40:24.637Z
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2021-12-30T02:23:05.000Z
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{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14443","language":"en","timestamp":1640830985000,"url":"https:\/\/arxiv.org\/abs\/2112.14443","yymm":"2112"}
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proofpile-arXiv_000-10055
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{"provenance":"002.jsonl.gz:10056"}
| null | null |
\section{}
\section{Introduction}
During the COVID-19 pandemic, we have seen a revolution of the contact tracing technology, which helped track and contain the epidemic \cite{braithwaite2020automated,kretzschmar2020impact}. Some contact tracing programs were conducted by governmental/health agencies \cite{park2020contact}, while others relied on decentralized approaches \cite{troncoso2020decentralized}. Most contact tracing approaches work by notifying people who could have received the infection from known infectious patients, i.e., they trace ``forward'' in time. However, some advocate that a ``bidirectional'' tracing, where the past history of the infection is also tracked, can be more effective \cite{bradshaw2021bidirectional,endo2020implication,kojaku2021effectiveness}. In this paper we focus on the ``backward'' direction of the problem; the task of identifying the first patient who carried the disease, also called patient zero, or the source of the epidemic. The identification of patient zero can either be limited to a smaller population cluster, in which case it can be a first step towards ``bidirectional'' tracing, or it can be more ambitious; finding the first patient who developed the mutation of a certain disease can help understanding how the mutation occurred, which can help us prevent, or better prepare for future epidemics.
Surprisingly, given the importance of the problem and the relatively large literature on the topic, we are not aware of any instance where source detection algorithms have been applied in real situations, including during the COVID-19 pandemic. Our goal in this paper is to examine the applicability of the source detection models in the literature (which we call frameworks from now on), and then propose a new framework, which improves them in several aspects. Originally, source detection was introduced in the context of rumor spreading instead of epidemics by Zaman and Shah in their pioneering Sigmetrics paper~\cite{shah2010rumors,shah2011rumors}. Translating to the language of epidemics for clarity, in the framework of \cite{shah2011rumors}, an epidemic spreads over a network of agents that is completely known to us, and we observe a \textit{snapshot} of the network, which means that every agent reveals if they are infected or not at some given time (not too early, because then the problem is trivial, nor too late, because then the problem is impossible). Shortly after \cite{shah2011rumors}, Pinto et al. proposed a different framework, in which agents (also called \textit{sensors}) reveal, in addition to their state, the time when they became infected, but where only a few of them do so and act as sensors \cite{PintoTV12}; indeed, the problem is trivial if all agents are sensors. This framework is better tailored to epidemics, as it is reasonable that obtaining any information from all the agents is much harder than asking one more question about the starting time of the symptoms of the disease to only some of them. Pinto et al. found that in their framework, if the sensors are already selected, the maximum likelihood estimator of the source has a closed form solution when the underlying network is a tree, and the time it takes for an agent to infect one of its susceptible contacts follows a Gaussian distribution. For general graphs, it is difficult to find an algorithm with any theoretical guarantees, although we note that many heuristics have been developed \cite{hu2018localization,li2019locating,paluch2018fast,paluch2020locating,shen2016locating,tang2018estimating,xu2019identifying,zhu2016locating}. The only exception is on very simple contact networks \cite{lecomte2020noisy}, or when the epidemic spreads deterministically between the agents \cite{zejnilovic2013network}, which is not a realistic assumption for epidemics, but at least the estimation algorithm is trivial, and more emphasis can be put on the question of how the sensors should be selected for good performance \cite{spinelli2017effect,spinelli2018many}, which again is studied by heuristics in the general case \cite{paluch2020optimizing}. For a recent review of source detection algorithms, see \cite{shelke2019source}.
One of the main criticisms of original framework of Pinto et al. is that, even though the contact network is fully known, it is very difficult to find the source exactly unless a large fraction (20-50\%) of the population act as sensors, which is unrealistic in the case of an epidemics, when the source is searched in a large population. An alternative recently proposed is to compute confidence sets for the source instead of finding it \cite{dawkins2021diffusion}. But if our goal is to locate the source exactly, a promising approach is to allow the sensors to be selected adaptively to previous observations \cite{zejnilovic2015sequential,zejnilovic2017sequential}, which we call \textit{adaptive sensor placement}. When the contact network is known, adaptive strategies have been studied by simulations \cite{spinelli2017general,spinelli2017back} and by theoretical analysis \cite{lecomte2020noisy}, and they show a large reduction in the number of required sensors in real networks. In this paper, we will also allow the sensors to be placed adaptively.
We believe that the most problematic assumption that is still present in source detection papers, is the full knowledge of the contact network of agents, which is unrealistic (let alone because of privacy concerns). Due to this lack of data-availability, algorithms in the source detection literature have not been tested on realistic epidemic data. Moreover, while governmental/health agencies might have access to private datasets, such as cellular location data, from which a contact network may be estimated, these networks may be very noisy, and are potentially unfit for the source detection task. We only know of a few papers that study the effect of imperfections in the network data on the source detection task \cite{mashkaria2020robustness,zejnilovic2016extending}, but these papers study epidemics that spread deterministically between the agents. Inspired by adaptive sensor placement, and by the recent implementations of contact tracing algorithms, we propose a new framework for source detection, which we call Source Detection via Contact Tracing Framework (SDCTF). In SDCTF, algorithms can have two types of queries: contact queries, which can be used to explore the network, and sensor (test) queries, after which agents reveal their symptom onset time as before. The goal of the algorithm is to find the source as accurately as possible, while minimizing the number of contact and sensor queries. The SDCTF is a way to formalize the source detection task; it determines the goal of the algorithm and how information can be gained about the epidemic, but it does not specify the underlying epidemic and mobility data models (simulated or real). In this paper, we analyse different algorithms in the SDCTF with various epidemic and mobility models.
Besides specifying the possible queries that algorithms can make, the SDCTF also determines the way the outbreak is detected, which marks the starting time of the source detection task. In sensor-based source detection, the source detection task often starts long after the outbreak, when essentially all agents in the network are infected \cite{PintoTV12}, which can be seen as a limitation of source detection frameworks. The SDCTF is also closely related to contact tracing frameworks, where it is standard to assign a probability that each node spontaneously self-reports after developing symptoms, which triggers the activation of contact tracing algorithms \cite{kretzschmar2020impact,bradshaw2021bidirectional}. In the SDCTF, we adopt the idea of self-reporting with a slight modification. We believe that the most interesting time to perform the source detection task is when a new disease (or a new mutation of the disease) appears, and therefore we tie these self-reporting events to hospitalizations, where infections are properly diagnosed by healthcare professionals.
In particular, this means that the SDCTF can only be applied to epidemic data (and models) where hospitalizations are well-defined. In this paper, we use the datasets generated by the Data-driven COVID Simulator (DCS) introduced in \cite{lorch2020quantifying}, which is one of the most realistic toolboxes that generate datasets modelling COVID-19, which we are aware of (notably, hospitalizations are part of the model). We also propose synthetic approximations for the epidemic and mobility models in the DCS; the Deterministically Developing Epidemic model and the Household Network Model, which improve the interpretability of our results since they have fewer parameters.
We propose a simple algorithm called LocalSearch (LS), which adaptively traces back the transmission path from the first hospitalized patient to the source. The LS algorithm is quite efficient at finding the source; the number of contact and sensor queries that it uses does not depend on the size of the network, but only on the local neighborhood of the source. Moreover, the LS algorithm provably finds the source with 100\% accuracy, because of our assumption that every contact and sensor query is answered without noise. However, it is well-known that data-availability is a major issue in contact tracing \cite{beidasrinad2020optimizing}, either because the agents do not comply with contact tracing efforts, or possibly (and in particular in the current COVID-19 epidemic) because they do not develop symptoms, and are unaware that they have the disease. In this paper, we model the effect of asymptomatic agents. When queried and tested, these agents do not reveal their time of infection, only whether they have or had the disease at some point. We show that the accuracy of the LS algorithm drops in the presence of asymptomatic agents, because the algorithm can get stuck while tracing back the transmission path from the first hospitalized patient to the source. Therefore, we propose an improved version of LS called LS+, which accounts for the presence of asymptomatic agents by placing more sensors. We are not aware of any previous work in the source detection literature that models the effect of asymptomatic patients, but the resulting model can be seen as a mix between the snapshot and the sensor-based models. We mention that non-complying agents or agents who provide noisy observations have been studied by \cite{altarelli2014patient,hernando2008fault,louni2015identification}. Non-complying agents could also be included in our framework by treating them as asymptomatic agents (even though in this case we have no information about whether the agent had the disease or not), without jeopardizing the correctness of our algorithms.
We benchmark the LS and LS+ algorithms in both our data-driven and our synthetic epidemic and mobility models, and we compare them to state-of-the-art adaptive \cite{spinelli2017back} and non-adaptive \cite{jiang2016rumor,lokhov2014inferring} algorithms tailored to the SDCTF, whenever possible. We find that both LS and LS+ outperform these baseline algorithms in accuracy (probability of finding the correct source).
While the LS/LS+ are designed to be simple algorithms, their theoretical analysis is quite challenging. Nevertheless, we are able to provide rigorous results about the success probability of both algorithms after a series of simplifications to the epidemic and mobility models, by extending some recent results on the theory of exponential random trees \cite{feng2018profile,mahmoud2021profile}, which have previously not been connected to the source detection literature. We present these theoretical results in Section~\ref{sec:theory}, after formally introducing the SDCTF, our models and the LS/LS+ algorithms in Section~\ref{sec:models}. By simulations, we show that our analytic results approximate the accuracy of the algorithms well, even in the most realistic setting in Section~\ref{sec:simulation}. Our analytic results provide additional insight into how the parameters of the epidemic and mobility models affect the performance of the algorithms. We discuss these insights along with some non-rigorous computations that mirror our main proof ideas in Section~\ref{sec:boe_sec}. Reading Section~\ref{sec:boe_sec} before Sections~\ref{sec:models}-\ref{sec:simulation} is useful to build intuition, but is not necessary to understand the paper.
\section{Warmup Results}
\label{sec:boe_sec}
\subsection{A Simple Network and Epidemic Model and a Simple Algorithm}
\label{sec:boe_model}
\begin{figure}
\begin{center}
\includegraphics[width=\textwidth]{images/boe_fig.pdf}
\caption{(a)-(c) shows the spread of the infection in the model considered in Section~\ref{sec:boe_model}, which is equivalent to the growth of the RERT, with $d=2$. Dark blue edges show the contacts on day~$t$, and light blue edges show contacts present on previous days (and thus subfigures). Orange (resp., red; black) nodes mark symptomatic non-hospitalized (resp, asymptomatic; symptomatic hospitalized) nodes. (d)-(f) shows the LS source detection algorithm introduced in Section~\ref{sec:boe}, which succeeds in this example because there are no asymptomatic nodes on the transmission path between the first hospitalized node and the source. Black edges show the queried edges, and black stroke marks nodes already discovered by the algorithm. A node with black X marks a negative test result, and red stroked node marks the node currently maintained as source candidate by the LS algorithm.}
\label{fig:boe}
\end{center}
\end{figure}
Let us consider a time-dependent network model, where each agent meets $d$ new agents each day in such a way that the contact network is an infinite tree (ignoring the label of the edges giving the propagation time along the edge). This network models homogeneous mixing in a very large population; we consider more realistic network models in Section~\ref{sec:models}. On this network, we consider an epidemic model that starts at $t=0$ with one infected agent, and then progresses as infected agents infect their $d$ susceptible contacts each independently with probability $p_i$ each day. Since our goal is to study the epidemic process, it is sufficient to track only the agents who are already infectious (also called \emph{internal nodes}), and the agents who are in contact with infectious agents at time~$t$ (also called \emph{external nodes}), as shown in Figure \ref{fig:boe} (a)-(c). For $d=1$, the spread of the infection is then equivalent to the growth a random tree $\mathcal{T}_t$ rooted at the source of the infection, known under the name of Random Exponential Recursive Tree (RERT) and recently introduced in \cite{mahmoud2021profile}. Because of the similarities of the models, we refer to the model with general $d$ as RERT in the remaining of this section. We point out that the standard literature on elementary branching processes such as Galton-Watson trees or random recursive trees \cite{drmota2009random} is not applicable in our scenario, because these branching processes have no notion of global time (i.e., a node in such processes becomes infectious immediately after receiving the infection), whereas nodes in diseases commonly go through an exposed, non-infectious period before becoming infectious, which is well captured by the RERT model. We mention that there is literature on more advanced branching processes that do have a notion of global time, e.g. Crump-Mode-Jagers trees \cite{jagers1984growth}, however we opt for the RERT because of its simple definition.
After a node (patient) becomes infected, the disease can take three courses (which for now do not affect $\mathcal{T}_t$): with probability $p_a$ the patient is asymptomatic, with probability $(1-p_a)p_h$ the patient is hospitalized, and with probability $(1-p_a)(1-p_h)$ the patient recovers without hospitalization. The governmental/health agency learns about the outbreak when the first hospitalization occurs (see Figure~\ref{fig:boe} (c)) and starts the source detection process right away. It can inquire about the contacts of each agent and it can test the agents. From patients that were symptomatic (at any point in time in the past), the agency learns about their symptom onset time (which, in this simple model, is always one day after the infection time), but from asymptomatic patients it only learns that they had (or have) the disease at some point when they are tested. The framework introduced in this paragraph (including both the detection of the outbreak through the first hospitalization, and the possible actions the agency can take) is a simplified version of the SDCTF (Source Detection via Contact Tracing Framework), introduced in Section~\ref{sec:SDCTF}.
The network and epidemic models introduced in this section have four parameters: $d, p_i, p_a, p_h$, and it is important to understand how each of them affects the difficulty of source detection in the SDCTF. We distinguish two important factors. First, if the outbreak is not detected rapidly enough, the length of the transmission path to the first hospitalized agent is long, and source detection becomes then difficult, because a lot of information needs to be recovered. Therefore, a low $p_i$, a low $p_h$ and/or a high $p_a$ parameter can hinder source detection (recall that the probability of hospitalization was $p_h(1-p_a)$). The second factor is related to the difficulty of recovering information about the transmission path. If $p_a$ is high, then there are a lot of nodes who are asymptotic and therefore do not reveal their symptom onset time, making source detection very difficult. Since $p_a$ affects both the length of the transmission path and the amount of collected information, it is safe to expect that, of all parameters, $p_a$ has the largest effect on the difficulty of source detection. The parameter $d$ is interesting, because a large $d$ can reduce the length of the transmission path, but it also makes the information about the transmission path less accessible as more agents need to be tested. Since in this paper we do not set a hard constraint on the total number of available tests, the advantage of a shorter path takes over the drawback of additional tests and a large $d$ increases the success probability.
To say anything quantitative about source detection in the SDCTF, we must discuss specific algorithms that solve the source detection task. In this paper we propose a simple algorithm called LocalSearch (LS), shown in Figure~\ref{fig:boe}~(d)-(f). The LS algorithm maintains one candidate node $s_c$ at each iteration (initially, the first hospitalized node), which is always symptomatic, and it updates it in a greedy way: at the time of the infection of $s_c$, all its $d$ incident edges are queried, and all its $d$ neighbors are tested. Then the agent with the lowest reported infection time will be the new candidate $s_c$. The algorithm stops when $s_c$ does not change anymore between two consecutive iterations. For simplicity, we assume that the infection does not spread any further during these iterations, however, this assumption does not affect the ability of the algorithm to find the source or not. Indeed, it is not difficult to see that on tree networks, LS succeeds if and only if there are no asymptomatic nodes on the transmission path from the source to the first hospitalized agent. This observation leads us to enhance the LS algorithm by also searching within the neighbors of asymptomatic nodes; we explore this idea in the LS+ algorithm introduced in Section~\ref{sec:localsearch}. We are not aware of this simple greedy algorithm being studied in the context of source detection, although similar ideas were implemented for non-adaptive source detection to lower the runtime of the algorithms \cite{paluch2018fast}.
\subsection{Back of the Envelope Calculation}
\label{sec:boe}
Now, we have all the tools to estimate the probability of success of the LS algorithm. First we condition on the course of the disease in the source. With probability $p_a$, the source is asymptomatic and LS can never succeed. With probability $(1-p_a)p_h$, the source itself becomes hospitalized, and LS always succeeds. Finally, with probability $(1-p_a)(1-p_h)$ the source is symptomatic but not hospitalized, which we call event $\mathcal{A}$. If event $\mathcal{A}$ happens, then LS may or may not succeed depending on whether there are any asymptomatic nodes on the transmission path. More precisely, conditioned on event $\mathcal{A}$ and on the transmission path having length $l$, the probability of success is $(1-p_a)^{l-1}$ (since there are $l-1$ nodes on the path which can be asymptomatic), which implies
\begin{equation}
\label{eq:boe_ps1}
\P(\mathrm{success}) = (1-p_a)p_h+ (1-p_a)(1-p_h) \left(\sum_{l=1}^t \P\left(\text{transmission path has length $l$} \mid \mathcal{A} \right)
(1-p_a)^{l-1} \right).
\end{equation}
The difficult part is to compute the distribution of the transmission path conditioned on event $\mathcal{A}$; indeed we already saw that all four parameters $d, p_i, p_a, p_h$ affect this distribution in a non-trivial way. Let us perform a back of the envelope computation to get more insight into the effect of these parameters. The exact structure of the infection tree will not matter for this computation, only its \textit{profile} does. It is denoted by $\mathcal{T}_t(l)$ and defined as the number of (internal) nodes at level $l$ (i.e., at distance $l$ from the source of the infection). Remember that by definition the RERT has $d \cdot \mathcal{T}_{t-1}(l-1)$ external nodes on level $l$, and that at time~$t$ each external node is promoted to be internal with probability $p_i$ to form $\mathcal{T}_t$. Consequently, the level of a node~$h$ added at time $t>0$ has the same distribution (conditioned on the tree $\mathcal{T}_{t-1}$ at the previous step) as the size (number of internal nodes) of the profile $\mathcal{T}_{t-1}(l-1)$, that is, \begin{equation}
\label{eq:back_of_envelope_def}
\P(\mathrm{level}(h)=l \mid \mathcal{T}_{t-1}=T_{t-1}) = \frac{T_{t-1}(l-1)}{|T_{t-1}|}.
\end{equation}
Working on the RERT directly can be a daunting task, therefore we propose to approximate the numerator and the denominator of equation \eqref{eq:back_of_envelope_def} by $\mathbf{E}[\mathcal{T}_{t-1}(l-1)]$ and $\mathbf{E}|[\mathcal{T}_{t-1}|]$, respectively. It can be shown by a simple inductive argument, or by generating functions as in \cite{mahmoud2021profile}, that for RERTs we have $\mathbf{E}[\mathcal{T}_{t}(l)]=\binom{t}{l} (dp_i)^l$ and $\mathbf{E}[|\mathcal{T}_{t}|]=(1+dp_i)^t$, which suggests a binomial distribution for the level of $h$. And indeed, we can approximate the distribution of the level of a node $h$ added at time $t$ as
\begin{align*}
\P(\mathrm{level}(h)=l) &\approx \frac{\mathbf{E}[\mathcal{T}_{t-1}(l-1)]}{\mathbf{E}[|\mathcal{T}_{t-1}|]} \\
&=\frac{\binom{t-1}{l-1} (dp_i)^{l-1}}{(1+dp_i)^{t-1}}\\
&=\binom{t-1}{l-1}\left(\frac{ dp_i}{1+dp_i}\right)^{l-1} \left(1-\frac{ dp_i}{1+dp_i}\right)^{t-l} \\
&= \P(\mathrm{Bin}(t-1,q) = l-1),
\end{align*}
with $q=dp_i/(1+dp_i)$.
One of the main challenges of this calculation is that we do not know the day of the first hospitalization $t$ conditioned on event $\mathcal{A}$, we only know that each node is hospitalized with probability $(1-p_a)p_h$, which means that the index of the first hospitalized node follows a geometric distribution with mean $1/((1-p_a)p_h)$. We approximate $t-1$ by the first time that the expected size of the infection tree (excluding the source since we condition on event $\mathcal{A}$) exceeds the expected index of the first hospitalized node. Therefore we solve
$$\mathbf{E}[|\mathcal{T}_{t-1}|-1]=(1+dp_i)^{t-1}-1 = \frac{1}{(1-p_a)p_h}=\mathbf{E}[\text{index of the first hospitalized node}]$$
for $t$ (relaxing the constraint that $t$ is an integer), which gives
$$t-1= \frac{\log \left(1+\frac{1}{(1-p_a)p_h} \right)}{\log(1+dp_i)}.$$
Consequently, we approximate $\P\left(\text{transmission path has length $l$} \mid \mathcal{A} \right)$ by $\P(\mathrm{Bin}(t-1,q) = l-1)$. Continuing equation \eqref{eq:boe_ps1}, and using the well-known expression of the probability generating function of the binomial distribution, we get
\begin{align}
\label{eq:two}
\P(\mathrm{success}) &\approx (1-p_a)p_h+ (1-p_a)(1-p_h) \left(\sum_{l=1}^t \P\left(\mathrm{Bin} \left(t-1, q \right)=l-1 \right) (1-p_a)^{l-1} \right) \nonumber \\
&= (1-p_a)\left(p_h + (1-p_h) \left((1-p_a)\frac{dp_i}{1+dp_i}+1-\frac{dp_i}{1+dp_i}\right) ^{\frac{\log\left(1+\frac{1}{(1-p_a)p_h}\right)}{\log(1+dp_i)}} \right).
\end{align}
One can check that this expression agrees with our qualitative intuition. However, it is not at all clear whether it is valid because of the strong approximations made in some steps of the above computation. In Section \ref{sec:theory}, we prove a rigorous upper bound on the success probability, and we also provide much more careful approximations by proving exact theorems about the simplified models that we use. Then, in Section \ref{sec:simulation} we compare our results with simulation results on synthetic data, as well as with data generated by the DCS model.
\section{Models, Methods, Algorithms}
\label{sec:models}
\subsection{Epidemic Models}
\label{sec:epidemic_models}
\subsubsection{The DCS Model}
\label{subsub:DCS}
We call DCS the model implemented by \cite{lorch2020quantifying}. The DCS model is fairly complex, and we only give a brief overview.
Each agent in the agent set $V$ can be in one of 8 states: susceptible, exposed, asymptomatic infectious, pre-symptomatic infectious, symptomatic infectious, hospitalized, recovered or dead. Transitions between different states are characterized by counting processes described by stochastic differential equations with jumps. The most important, and also most complicated of these counting processes is the exposure counting process $N_i(t)$, which is modeled by a Hawkes process for each agent~$i$. Hawkes processes are point processes with a time-dependent, self-exciting conditional intensity function $\lambda^*_i(t)$.
\begin{equation}
\label{eq:Hawkes}
\hspace*{-5pt}
\lambda^*_i(t) =
\beta
\sum_{j \in V \backslash \{i\}}
\int_{t - \delta}^{t} K_{i,j}(\tau) ~ \gamma e^{-\gamma(t-\tau)} \, d\tau
\end{equation}
where the kernel $K_{i,j}(\tau)$ indicates whether $j$ has been at time $\tau$ at the same site where $i$ is at time $t$, and whether $j$ is in the infectious state. Parameters $\gamma$ and $\delta$ are the decay of infectiousness at sites and the non-contact contamination window, respectively, and they account for the fact that $j$ can infect $i$ even if they are never at the same site, as $j$ can leave some pathogens behind (airborne for instance). Parameter $\beta$ is the transmission rate for symptomatic and asymptomatic individuals, and it comes in two versions: $\beta_c$ accounts for infections outside the household and $\beta_h$ accounts for infection in the household. Parameters $\beta_c$ and $\beta_h$ are fitted to the COVID-19 infection data of Tubingen from 12/03/2020 to 03/05/2020 using Bayesian Optimization. The model also has a parameter for the relative asymptomatic transmission rate built into the function $K_{i,j}(\tau)$, which scales down the infectiousness of asymptomatic agents (to 55\% of the infectiousness of symptomatic agents by default).
Once a susceptible agent becomes infected, the disease can take three possible courses (see Figure~\ref{fig:mobility_models}~(a)). With probability $p_a$, the agent becomes asymptomatic infectious after time $T_E$, and then recovers after time $T_I$. With probability $1-p_a$, the agent becomes pre-symptomatic infectious after time $T_E$, next symptomatic infectious after time $T_P$, and then recovers with probability $1-p_h$ after time $T_I-T_P$, or becomes hospitalized with probability $p_h$ after time $T_H$. Agents in the DCS are also assigned age values based on demographic data, and the hospitalization probability $p_h$ of each agent is determined based on its age (following COVID-19 infection data). The times $T_E, T_P, T_I$ and $T_H$ are drawn from an appropriately parametrized (using values from the COVID-19 literature) lognormal distribution as shown in Table \ref{tab:Tvalues}.
\subsubsection{The DDE Model}
\label{subsub:DDE}
We start by taking the DCS model \cite{lorch2020quantifying}, which we simplify to enable its theoretical analysis. In the Deterministically Developing Epidemic (DDE) model, continuous time (used in DCS) is replaced by discrete time-steps: we refer to one time-step in the DDE as one day. Instead of modelling the infection propagation as a Hawkes process, an infectious agent (symptomatic or asymptomatic) can infect its susceptible neighbor with probability $p_i$ each day. Thereafter, the disease progresses the same way as in the DCS, except that in the DDE model the transition times are deterministic (the infection events and the severity of the disease (i.e., the (a)symptomatic and hospitalized states) are still determined randomly), and we have a single parameter $p_h$ for the hospitalization probability (agents in this model do not have an age parameter). We discuss how we set the parameters of the DDE model in Section \ref{sec:parameters}.
\begin{figure}
\begin{center}
\includegraphics[width=\textwidth]{images/mobility_models.pdf}
\caption{(a) The flow diagram of the DCS and DDE epidemic models. (b) A possible epidemic outbreak in the Tubingen mobility model, and (c) the Household network model. The large grey circles mark households, and the purple nodes mark places, otherwise we use the same coloring as in (a). In both cases (b) and (c), the transmission paths are $(v_2, v_4, v_5, v_8)$. }
\label{fig:mobility_models}
\end{center}
\end{figure}
\subsection{Simulating Mobility}
\label{sec:mobility}
\subsubsection{Tubingen Mobility Model}
\label{sec:TU}
We briefly review the mobility model introduced in \cite{lorch2020quantifying}, and illustrated in Figure~\ref{fig:mobility_models}~(b). The population is partitioned into households of possibly varying size (usually between 1 and 5). The households are assigned a location, and we also place some external sites (shops, offices, schools, transport stations, recreating sites) on the map, which the agents may visit. The location of the households and the number of agents in them is sampled randomly based on demographic datasets. Initially, each agent is assigned a few favorite sites (randomly based on distance), and will only visit these throughout the simulation. Each agent decides to leave home after some exponentially distributed time, visits one of its (randomly chosen) favorite sites, and comes back home after another (usually much shorter) exponentially distributed time. If two agents visit the same site at the same time, or within some time $\delta$, we record them as a contact, which gives an opportunity for the infection to propagate. We denote the Tubingen mobility model as TU, and the DCS epidemic model that runs on the TU mobility model as DCS+TU.
\subsubsection{Household Network Model}
\label{sec:HNM}
The Household network model (HNM) was inspired by \cite{lorch2020quantifying}, however we note that similar models have been studied in the theoretical community by~\cite{ball2009threshold}. As in the Tubingen mobility model, in HNM $N$ nodes are assigned into households, but of constant size $d_h+1$. Every pair of nodes in the same household are connected by an edge, forming therefore cliques of size $d_h+1$. Additionally, each node is assigned $d_c$ half edges, which are paired uniformly at random with other half-edges in the beginning. Some half-edge pairings can result in self-loops or multi-edges, which are discarded. This construction defines a random graph generated by a configuration model, which shares a lot of similarities with Random Regular Graphs (RRG) \cite{wormald1999models}. In fact, if we join nodes in the same household into a single node in the HNM (which we refer to as the \textit{network of households} of the HNM), then the resulting graph is equivalent to the \textit{pairing model} of RRGs with degree $d_c(d_h+1)$. It is well-known that in the pairing model of RGGs of degree $d$, the local neighborhood (of constant radius, as the number of nodes tends to infinity) of a uniformly randomly chosen vertex is a $d$-regular tree (with probability tending to 1), which implies that locally there are asymptotically almost surely no self-loops, multi-edges or any cycles in the graph. This result has various names; in random graph theory the result is usually proved by subgraph counting \cite{wormald1999models}, in probability theory it is the basis of branching process approximations \cite{ball2009threshold}, and in graph limit theory it is called the local convergence to the infinite $d$-regular tree~\cite{benjamini2011recurrence}. In our theoretical analysis, this result motivates the approximation of the neighborhood of the source in the network of households of the HNM by an infinite $d_c(d_h+1)$-regular tree. The HNM itself is then approximated by replacing each (household) node of the infinite $d_c(d_h+1)$-regular tree of households by a $(d_h+1)$-clique, and by setting the edges so that each (individual) node has degree exactly $d_c+d_h$, while keeping the connection between cliques unchanged (see Figure~\ref{fig:mobility_models}~(c) for a visualization).
Since the HNM is a time-independent graph, we adopt the standard notations from graph theory. Formally, the HNM is given by the set of nodes and edges $G=(V,E)$. Let us denote by $H(v)$ the set of nodes that are in the same household as node $v$. The distance between two nodes $u,v \in V$ (denoted by $d(u,v)$) is defined as a number of edges of the shortest path between $u$ and $v$. We denote the DDE epidemic model that runs on the HNM network as DDE+HNM.
\subsection{The Source Detection via Contact Tracing Framework}
\label{sec:SDCTF}
We present the Source Detection via Contact Tracing Framework (SDCTF), which can be applied to both epidemic and mobility models presented so far. The framework determines how the government/health agency, which conducts the source detection task, learns about the outbreak, and how it can gather further information to locate the source. In the SDCTF, as in Section \ref{sec:boe_model}, the agency learns about the outbreak when the first hospitalization occurs, and it also learns the identity of nodes when they become hospitalized (including the identity of the first hospitalized node).
After the outbreak is detected, the agency can make three types of queries. The first type of query, the household query with parameter $v$, reveals the agents that live in the same household as $v$. The household query works the same way in both the TU and the HNM models, and we do not limit the number of times it can be called (these queries are considered as cheap in the SDCTF). The second type of query, the contact query, works differently in the TU and the HNM models. For the TU model, a contact query has two parameters: an agent $v$ and a time window $[t_1,t_2]$. As a result, all agents that have been in contact with $v$ (and therefore could have infected $v$ or could have been infected by $v$) at an external site between $t_1$ and $t_2$ are revealed. In the HNM, no time window is needed for the contact query (which we also call edge query), and all neighbors of $v$ in graph $G$ are revealed. Contact (and edge) queries are considered expensive in the SDCTF. While in this paper we do not limit the number of available queries, we track the number of contacts and edges that are revealed as the algorithm runs. Note that in the TU model if two agents $v_1$ and $v_2$ have been in contact during the time window $[t_1, t_2]$ and also during a different time window $[t_3,t_4]$, then those are counted as separate contacts, whereas in the HNM an edge between $v_1$ and $v_2$ is only counted once. Although contact queries are considered expensive, both household and contact queries are answered instantly in the SDCTF.
The third kind of query is the test query with parameter $v$, which reveals information about the course of the disease in the queried agent (see Figure \ref{fig:mobility_models}~(a)). Symptomatic patients reveal the time of their symptom onset (which exactly determines their time of infection in the DDE due to the deterministic transition times) if they are past the pre-symptomatic state (i.e., if they are either infectious or recovered). Asymptomatic and pre-symptomatic patients do not reveal any information about their infection time; they just reveal that they have the disease or had the disease at some point and have recovered. For all algorithms we assume that asymptomatic patients do not reveal whether they have the infection at the time they are queried. Finally, agents who have not been exposed, or are still in their exposed state, give a negative test result. Test queries are again considered expensive in the SDCTF, we even limit the population that can be tested on any given day to at most 1\% of the total population, due to the capacity of testing facilities. However, since in this paper we do not limit the number of days that the algorithm can use to locate the source, the limit on the number of tests does not play an important role. As opposed to household and contact queries (and the model in Section \ref{sec:boe_model}), tests results are only answered the next day in the SDCTF, which means that the algorithms must operate in ``real-time'', while the epidemic keeps propagating.
\subsection{Parameters}
\label{sec:parameters}
\begin{figure}[h]
\begin{center}
\includegraphics[width=\textwidth]{images/table1.png}
\caption{Default values for the infection parameters in the DCS+TU and the DDE+HNM models.}
\label{tab:Tvalues}
\end{center}
\end{figure}
The DCS+TU model has many parameters, most of which are fitted to COVID-19 datasets of Tubingen from 12/03/2020 to 03/05/2020 by \cite{lorch2020quantifying} (we show the most relevant parameters in Table \ref{tab:Tvalues}). We determined the parameters of the DDE+HNM model so that they fit the parameters of the DCS+TU as closely as possible (see the precise values in Table \ref{tab:Tvalues}). We determine the values of $T_E, T_P, T_I$ in the DDE+HNM by rounding the expected value of the corresponding distribution in the DCS+TU to the nearest integer. Since $p_a$ is simply a constant in both models, we keep the same numerical value in the DDE+HNM. The parameter $p_h$ is more complicated, because in the DCS+TU model there is a different hospitalization probability for each age group. We take the average hospitalization probability across the population to be $p_h$. The most complicated parameter to fit is $p_i$, because in the DCS+TU model, infections are modelled by a Hawkes process, which depends on many parameters, including whether the infectious agent is symptomatic or asymptomatic, the length of the visit, the site where the infection happens, etc (see equation \eqref{eq:Hawkes}). We empirically observe the probability of infection in every contact in several simulations, and we find that an agent has on average 15 contacts outside the household each day, and that the average probability of infection during such a contact is around 0.02. However, since we use smaller networks for the DDE+HNM ($N=400$ or $1000$, because running the baselines on larger networks is not feasible) than the DCS ($N=9054$), setting $d_c$ to be as high as 15 would violate the assumption that the network of households of the HNM can be locally approximated by a tree (see Section~\ref{sec:HNM}). Therefore we chose $d_c=3$ for the HNM and we scale $p_i$ so that $d_cp_i$ (the expected number of external infections caused by a single agent each day) is the same in the DCS+TU and the DDE+HNM models. Finally, we choose $d_h$ in the DDE+HNM by rounding the average household connections in the DCS+TU. Note that the average number of household connections is not the same as the average number of household members, because the number of connections grows quadratically in the size of the households, and thus fitting to the number of connections results in a higher $d_c$ (due to the Quadratic Mean-Arithmetic Mean inequality).
Finding the default values for the parameters is useful to create a realistic model. However, we also interested in the effect of each of the parameters on the performance of our algorithms. Therefore, in the DDE+HNM, we vary the parameters $p_a,p_h,p_i,d_h$ and $d_c$, while keeping the other ones unchanged. For the DCS+TU model, we also keep the mobility model fixed and we focus on varying the parameters $p_a,p_h$ and $p_i$. As noted above, there is no single parameter $p_h$ or $p_i$ in the DCS+TU model, therefore we change all hospitalization probabilities and all intensities of the Hawkes processes so that the hospitalization probability averaged across the population and the infection probability averaged across contacts equal the desired values.
\subsection{The LocalSearch Algorithms LS and LS+}
\label{sec:localsearch}
The LS algorithm finds patient zero by local greedy search.
It keeps track of a candidate node, which is always the node with the earliest reported symptom onset time. We denote the candidate of the algorithm at iteration $i>0$ by $s_{c,i}$. We think of $s_c$ as a list, which is updated in each iteration of the algorithm, and we use the notation $s_{c,-1}$ for the last element of the list (i.e., the current candidate). In each iteration of the algorithm, we compute a new candidate denoted by $s_c'$, and we append it at the end of the list $s_c$ at the beginning of the next iteration, unless $s_c'=s_{c,-1}$, in which case the algorithm terminates.
\begin{figure}[h]
\begin{center}
\includegraphics[width=\textwidth]{images/Alg_1.pdf}
\caption{Pseudocode and graphical explanation for the LS and LS+ algorithms. We use the same coloring as in Figure \ref{fig:mobility_models}~(a). Black edges show the queried edges, a node with black X marks a negative test result, and red stroked node marks the node currently maintained as source candidate by the LS algorithm. We denote by $t_v$ the symptom onset time of symptomatic node $v$ and by $H(v)$ the household of a node $v$ similarly to the main text. }
\label{fig:Alg1_figure}
\end{center}
\end{figure}
Since we consider the SDCTF, the outbreak is detected when the first hospitalized case is reported. At that time, $s_c'$ is initialized to be the hospitalized patient, the test queue is initialized to be empty, and the algorithm is started. In the beginning of an iteration, if the test queue is empty, the household members and the ``backward'' contacts of the current candidate $s_{c,-1}$ are queried and are added to the test queue (see Figure~\ref{fig:Alg1_figure}~(a)). We define ``backward'' contacts as the set of nodes that have been in contact with $s_{c,-1}$ in the interval $[t_{s_{c,-1}} - (T_E+T_P) - (\sigma_E +\sigma_P),t_{s_{c,-1}} - (T_E+T_P) + (\sigma_E +\sigma_P)]$, where $t_{s_{c,-1}}$ is the symptom onset time of current candidate $s_{c,-1}$. The terms $\sigma_E$ and $\sigma_P$ model the standard deviation of the transition times, and they are set to zero for the DDE and to $\sigma_E=2$ and $\sigma_P=1$ for the DCS based on Table \ref{tab:Tvalues}. We note that the notion of ``backward'' contacts is only meaningful in the case of time-dependent network models; for the HNM, all neighbors are counted as backward contacts.
After the test queue is initialized, the agents inside the queue are tested (see Figure \ref{fig:Alg1_figure}~(b)). Not all nodes can be tested on the same day because of the limitation on the number of tests available per day in the SDCTF, however, this has little effect because we do not proceed to the next iteration until the test queue becomes empty. Once the test results come back to the agency, if any of the (symptomatic) nodes $v$ reports an earlier symptom onset time than the current candidate $s_{c,-1}$, then we update our next candidate $s_c'$ to be $v$ (see Figure \ref{fig:Alg1_figure}~(c)). We note that the iteration does not stop immediately after $s_c'$ is first updated; the iteration runs until the test queue becomes empty, and until then, $s_c'$ can be updated multiple times. This is important in the theoretical results to prevent the algorithm from getting sidetracked (see Figure~\ref{fig:LSp_cases}). We also experimented with a version of the LS and LS+ algorithms where the iteration stops immediately once $s_c'$ is updated; we call these algorithms LSv2 and LS+v2.
The main drawback of the LS algorithm is that is gets stuck very easily if there is even one asymptomatic node on the transmission path. For this reason, we introduce the LS+ algorithm, in which we enter the backward contacts of the asymptomatic household members of $s_{c,-1}$, and the household members of any asymptomatic node into the testing queue (see Figure \ref{fig:Alg1_figure}~(d)-(f)). Since the symptom onset times of asymptomatic nodes $v$ are not revealed, we define backward contact in this case as any contact in the time window $[t_{s_{c,-1}} -(T_P+2T_E+T_I), t_{s_{c,-1}} - (T_P+2T_E)]$, where $t_{s_{c,-1}}$ is still the symptom onset time of the current candidate $s_{c,-1}$. Indeed, in the DDE model, since $s_{c,-1}$ was infected at $t_{s_{c,-1}} -(T_P+T_E)$, if $v$ infected $s_{c,-1}$, agent $v$ must have been infectious at that time, which implies that $v$ could not have been infected later than $t_{s_{c,-1}} -(T_P+2T_E)$ or earlier than $t_{s_{c,-1}} -(T_P+2T_E+T_I)$. In the DCS model, the terms $\sigma_E$ and $\sigma_P$ can be subtracted and added to the two ends of the queried time window to account for the randomness in the transition times.
Both algorithms stop if the testing queue becomes empty before a node with an earlier symptom onset time than $s_{c,-1}$ is discovered, and both algorithm return $s_{c,-1}$ as their inferred source. The high level pseudocode and an illustration of the LS and LS+ algorithms are given in Figure~\ref{fig:Alg1_figure}.
\section{Theoretical Results}
\label{sec:theory}
In this section we present theoretical results for the LS and LS+ algorithms described in Section~\ref{sec:localsearch}. We follow a similar approach as in the non-rigorous computation in Section~\ref{sec:boe}, which useful but not necessary for understanding this section. All the statements are rigorously established, and whenever we reach a point where the computations would become intractable, we propose a simpler approximate model to study. One of the main contributions of this paper is to identify which computations can be done on more general models, and which computations need more simplified ones (see Figure~\ref{fig:overview} for an overview of the different models used for the computations in this section).
We compute the success probability of the LS and LS+ algorithms in two steps. We first assume the length of the transmission path known in Section~\ref{sec:lsp_succes}
. This computation is then made possible by a tree approximation of the HNM, called the Red-Blue (RB) tree (defined in Section~\ref{subsub:RBtree}), and a slightly modified version of the DDE model called $\mathrm{DDE}_{\mathrm{NR}}$ (defined in Section~\ref{subsub:DDENR}). The RB tree preserves some of the household structure in the HNM, and therefore allows us gain insight into the difference between the LS and LS+ algorithms, which would be difficult to obtain if we had worked on trees without taking the household structure in account.
For the second step, we would need to compute the distribution of the transmission path on the RB tree. However, finding a closed form expression is intractable. Instead, we combine the network and epidemic models into a growing random tree model, and we consider a $d$-ary Random Exponential Tree (RET).
The $d$-ary RET model has only been studied for $d=2$ \cite{feng2018profile}; we extend the results on their expected profile for general $d$ in Section~\ref{subsub:RET}. Nevertheless, working on $d$-ary RETs still remains difficult, and therefore, in our last modeling step, we introduce a Deterministic Exponential Tree (DET) model, whose profile is close to the expected profile of the RET, and we compute the distribution of the transmission path on this model in Section~\ref{subsub:DET}.
\begin{figure}
\begin{center}
\begin{subfigure}[b]{\textwidth}
\caption{}
\includegraphics[width=\textwidth]{images/Figure1_a.pdf}
\label{fig:y equals x}
\end{subfigure}
\hfill
\begin{subfigure}[b]{\textwidth}
\caption{}
\centering
\includegraphics[width=\textwidth]{images/Figure1_b.pdf}
\label{fig:three sin x}
\end{subfigure}
\caption{The different approximation methods (a) and the distribution of the length of transmission path in the different models (b) proposed in Section~\ref{sec:theory}. Panel (b) also shows the length of the transmission path in the DCS model on the TU dynamics, to highlight the fit of our model.}
\label{fig:overview}
\end{center}
\end{figure}
To summarize all models considered in this paper, we have a data-driven and a synthetic model for simulations (DCS+TU and HNM+DDE), an analytically tractable model (RB-tree+$\mathrm{DDE}_{\mathrm{NR}}$) where we can compute the success probability if the length of the transmission path is known. In a second stage, we compute the distribution of a transmission path on a deterministic tree (DET), which has a similar profile as a random tree (RET) that approximates our analytically tractable model. We visualize these five different models in Figure~\ref{fig:overview}~(a), and we show by simulations in Figure~\ref{fig:overview}~(b) that the distribution of the transmission path is similar in all of the considered models with appropriately scaled parameters. We compare our analytic results on the success probabilities of the LS and LS+ algorithms with our simulation results in Section \ref{sec:comp_theory} in Figure~\ref{fig:theory_plots}.
\subsection{Success Probability of LS and LS+ Algorithms on the RB Tree}
\label{sec:lsp_succes}
In this section we introduce the Red-Blue (RB) tree model (which is a tree approximation to the HNM), and we calculate the exact probability that the LS and LS+ algorithms succeed, if the length of the transmission path is known.
\subsubsection{Red-Blue tree models}
\label{subsub:RBtree}
In short, a RB tree is a two-type branching process with a deterministic offspring distribution that depends on $d_h$ and $d_c$. The lack of randomness in this distribution makes us adopt the formalism of deterministic rooted trees.
\begin{definition}
\label{def:RBtree}
Let a rooted tree, denoted by $G(s)$, be a tree graph with a distinguished node root node $s$. Let $u$ and $v$ be two nodes connected by an edge in $G(s)$. If $d(u,s) < d(v,s)$, we say that $u$ is a parent of $v$, otherwise $u$ is a child of $v$. Moreover, if $d(s,v) = l$ we say that $v$ is on level $l$. An RB tree with parameters $(d_c, d_h)$ is an infinite rooted tree, such that the nodes also have an additional color property. The root is always colored red and the rest of the nodes are colored red or blue. The root has $d_c$ red and $d_h$ blue children. Every other red node has $d_c-1$ red and $d_h$ blue children, and every blue node has $d_c$ red children and no blue children. Red nodes and their $d_h$ blue children partition the nodes of the RB tree $G(s)$ into subsets of size $d_h+1$, which we call households.
\end{definition}
\begin{remark}
\label{rem:RBdef}
In the RB tree, each blue node has degree $d_c+1$, and each red node has degree $d_c+d_h$, including the root of the tree $s$ (which is
the source of the epidemic, when the RB tree is combined with an epidemic model).
\end{remark}
The RB tree can be seen as a local tree approximation of the HNM. Let $G=(V,E)$ be an HNM with parameters $(d_c,d_h)$, and let $s \in V$ be the distinguished source node. In Section \ref{sec:HNM} we noted that the HNM can be approximated locally around the source node by replacing each node of an infinite $d_c(d_h+1)$-regular tree by a $(d_h+1)$-clique, and setting the edges so that each node has degree exactly $d_c+d_h$, while keeping the connection between cliques unchanged. Let us call this infinite graph $G^*$. Although $G^*$ is not a tree, all cycles in $G^*$ must be contained entirely inside the households, which implies that in each household there exists exactly one node that has the minimal distance to the source. We will refer to these nodes with minimal distance to the source as the red nodes, and we color the rest of the nodes blue. In other words, the red nodes will be the first ones in their households to be infected. Let us now delete the edges between the blue nodes in $G^*$ to obtain graph $G'$. We claim that $G'$ is isomorphic to the RB tree $G(s)$ rooted at the source~$s$. Indeed, since the edges between blue nodes have been deleted in $G^*$ to form $G'$, each blue node has $d_c+1$ red neighbors and no blue neighbor, and since the edges incident to red nodes have been unchanged, each red node has $d_c$ red and $d_h$ blue neighbors, exactly as in the definition of RB tree above.
Note that a household in $G^*$ is completely characterized by only specifying the colors of the nodes: a household always consists of one red node and of its $d_h$ blue children. We use this characterization as a definition for households in the RB tree $G'$, because it does not depend on the edges from $G$ that are deleted in $G^*$, whereas this deletion makes the original definition of a household as a clique in $G$ unusable.
Next, we make some important observations the behaviour of the LS and the LS+ algorithms on RB trees, which we prove in Appendix \ref{sec:LS,LSp,trees_app}. We start by formalizing the notion of transmission path.
\begin{definition}
\label{def:inf_path}
Let $h$ be the first hospitalized node and $s$ be the source. We call the path $(s = v_0, v_1, ... v_l = h)$, where $v_{i}$ is the infector of $v_{i+1}$ for $0 \leq i < l$, the \emph{transmission path}. Also we call the path $(v_l, v_{l-1}, ... v_1)$ the \emph{reverse transmission path}.
\end{definition}
\begin{remark}
\label{rem:RBtree}
Note that in an RB tree, each household traversed by a transmission path shares one (the red node in the household) or two (the red node of the household and one of its $d_h$ children in the household) nodes with this path. Moreover, the red node of a household traversed by a transmission path is followed by another red node on the path (in another household) if it is the only node of that household on the transmission path, whereas it is followed by a blue node (in the same household) if two nodes of that household are on the transmission path.
\end{remark}
\begin{lemma}
\label{lem:LS,LSp,trees}
In the RB tree network, the LS algorithm succeeds if and only if all nodes on the transmission path are symptomatic, and the LS+ algorithm succeeds if among the nodes of the transmission path, there exists a symptomatic node in each household, and the source is symptomatic.
\end{lemma}
\begin{remark}
We note that the statement for LS+ in Lemma \ref{lem:LS,LSp,trees} cannot be reversed, i.e., it is possible that LS+ succeeds even if among the nodes of the transmission path, there is a household with no symptomatic node (see Figure \ref{fig:LSp_cases}~(a)). Also, the proof of Lemma \ref{lem:LS,LSp,trees} does not hold if the LS+ algorithm proceeds to the next iteration at the first time $s_c'$ is updated (see Figure \ref{fig:LSp_cases}~(b)). Finally, in the proof of Lemma \ref{lem:LS,LSp,trees}, we do not make any assumptions about asymptomatic patients having had the disease previously or not, which implies that we could treat non-complying agents as asymptomatic patients without jeopardizing the correctness of the algorithms.
\end{remark}
\subsubsection{The $\mathrm{DDE}_{\mathrm{NR}}$ Model}
\label{subsub:DDENR}
Focusing on tree networks is an important step towards making our models tractable for theoretical analysis, but it will not be enough; we will make two minor simplifications to the DDE model as well: we eliminate (i) the pre-symptomatic state and (ii) the recovered state, and we call the new model $\mathrm{DDE}_{\mathrm{NR}}$ (where NR stands for No Recovery). (i) The first assumption can be made without loss of generality, because the pre-symptomatic state does not have any effect on the disease propagation, nor on the success of the source detection algorithm. Indeed, according to Lemma \ref{lem:LS,LSp,trees}, the success of the LS and LS+ algorithms depends only on the information gained about the transmission path, and by the time of the first hospitalization, every node on the transmission path must have left the pre-symptomatic state (since we always have $T_P<T_E+T_H$), even if we include it in the model. (ii) The second assumption on the absence of recovery states amounts to take $T_I \rightarrow \infty$, which does have a small effect on the disease propagation, however, this effect is minimal
because $T_I=14$ is already quite large, and because only the very early phase of the infection is interesting for computing the success probabilities of the algorithms. Finally, this last assumption has no effect on the information gained by the algorithm since we assumed that recovered patients (who were symptomatic) can remember and reveal their symptom onset time in the same way as symptomatic infectious patients.
\subsubsection{Success Probability of LS}
Assuming that the distribution of length of the transmission path is provided for us (we give an approximation in Section \ref{sec:depth_dist}), the success probability of LS can be computed succinctly. We need a short definition before stating our result.
\begin{definition}
\label{def:p}
Let $p$ be the probability that a node is asymptomatic conditioned on the event that it is not hospitalized.
\end{definition}
A simple computation shows that
\begin{equation}
\label{C5}
p=\P ( v \text{ is asy} \mid v \text{ is not hosp})=\frac{p_a}{p_a +(1-p_a)(1-p_h)}.
\end{equation}
\begin{lemma}
\label{lem:LSsucc}
For the $\mathrm{DDE}_{\mathrm{NR}}$ epidemic model with parameters $(p_i, p_a, p_h)$ on the RB tree with parameters $(d_c, d_h)$, and with $p$ computed in equation \eqref{C5}, we have
\begin{equation}
\label{eq:LS_success_RB_tree}
\P(LS \textrm{ succeeds}) = \sum_{n=0}^{\infty} \left(1-p\right)^n \P(d(s,h) = n).
\end{equation}
\end{lemma}
\begin{proof}
Let us reveal the randomness that generates the epidemic in a slightly modified way than in the definition (Sections \ref{subsub:DDE} and \ref{subsub:DDENR}). As before, at the beginning only the source is infectious, and depending on course of the disease, the source can be symptomatic and hospitalized, symptomatic but not hospitalized, or asymptomatic with probabilities $(1-p_a)p_h, (1-p_a)(1-p_h), p_a$, respectively. In each moment, each infectious node infects each of its susceptible neighbors with probability $p_i$. If a node is infected, we reveal the information whether it will become hospitalized (which happens with the probability $(1-p_a)p_h$), but if it does not become hospitalized, we do not reveal whether the node is asymptomatic or symptomatic yet. Indeed, this information is not necessary for continuing the simulation of the epidemic since we assumed that there is no difference between the infection probabilities of symptomatic and asymptomatic nodes. Thereafter, when the first hospitalized case occurs, we reveal for each infected node $v$ on the transmission path (except the last node, which we know is hospitalised; see Definition \ref{def:inf_path}) whether it is asymptomatic or not. The only information we have about these nodes is that they are not hospitalized, which implies that the probability that a node is revealed to be asymptomatic on the transmission path is exactly the probability $p$ from Definition \ref{def:p} computed in \eqref{C5}.
By Lemma \ref{lem:LS,LSp,trees}, LS succeeds if and only if each node on the transmission path is symptomatic. Conditioning on the length of the transmission path, we can compute the probability of each node being symptomatic by equation \eqref{C5} as
\begin{equation}
\P(LS\textrm{ suceeds} | d(s,h) = n)=\left(1- \P(v \textrm{ is asy}| v\textrm{ is not hosp}) \right)^n=\left(1-p\right)^n,
\end{equation}
from which \eqref{eq:LS_success_RB_tree} follows immediately.
\end{proof}
\subsubsection{Success Probability of LS+}
Computing the success probability of the LS+ algorithm is far more challenging compared to the LS algorithm, even if the distribution of the length of the transmission path is provided to us. Indeed, since the LS+ algorithm does further testing on the contacts and household members of asymptomatic nodes, it is essential to have additional information about the number of households on the transmission path. We give our main result on the LS+ in the next theorem, which we prove in Appendix \ref{sec:thrm:LSp_suc}.
\begin{theorem}
\label{thrm:LSp_suc}
Let $p$ be as in \eqref{C5} and let $\mathcal{S}(n,\alpha,\beta)$ be the set of $k$ integer values such that $k$ and $n$ have different parity and $n+1 - 2(\alpha + \beta) \geq k \geq 2-(\alpha+\beta)$. Then, for the $\mathrm{DDE}_{\mathrm{NR}}$ epidemic model with parameters $(p_i, p_a, p_h)$ on the RB tree with parameters $(d_c, d_h)$, we have
\begin{align}
&\P(LS+ \textrm{ succeeds}) \ge \P(d(s,h) = 0) + (1-p)\P(d(s,h)=1)+ \nonumber \\
& \sum_{n = 2}^{\infty} \sum_{\substack{\alpha,\beta \in \{0,1\} \\ k \in \mathcal{S}(n,\alpha,\beta)}}
\binom{\frac{n+k-3}{2}}{k-2+\alpha+\beta}
\frac{(d_h(1-p))^{\frac{n+k-1}{2}}(d_c(1+p))^{\frac{n-k+1}{2}-\alpha-\beta}d_c(d_c-1)^{k+\alpha + \beta-2}}{\lambda_1\left( \frac{d_c-1+D}{2}\right)^n + \lambda_2\left( \frac{d_c-1-D}{2}\right)^n} \P(d(s,h)=n),
\end{align}
where
\begin{align}
D &= \sqrt{(d_c-1)^2 +4d_cd_h} \\
\lambda_1 &= \frac{(d_c+1+D)(2d_h+d_c-1+D)}{2D(d_c-1+D)} \\
\lambda_2 &= \frac{(D-d_c-1)(2d_h+d_c-1-D)}{2D(d_c-1-D)}.
\end{align}
\end{theorem}
\subsection{Approximating the Depth of the Path to the First Hospitalized Node}
\label{sec:depth_dist}
Section \ref{sec:lsp_succes} was dedicated to the success probability of the LS and LS+ algorithms, however, in these results, we are still missing the distribution of the transmission path length. In this subsection we address this problem by introducing simpler approximate models.
\subsubsection{$(d_r, d)$-ary Random Exponential Tree}
\label{subsub:RET}
When we introduced the $\mathrm{DDE}_{\mathrm{NR}}$ model in Section \ref{subsub:DDENR}, we removed both parameters $T_P$ and $T_I$ from the DDE model (by removing the presymptomatic and the recovered states, respectively), but we kept the parameter $T_E$. In this step we will rescale the time parameter to make $T_E'=1$ by changing $p_i'$ to be $1-(1-p_i)^{T_E}$. Since we had $T_E=3$ by default, using $T_E'$ and $p_i'$ instead of $T_E$ and $p_i$ means that we choose 3 days to be our time unit, and the probability of infection is scaled to be the probability that the infection is passed in at least one of three days (since the RB tree is time-independent, if two nodes are connected, the infection can spread on it every day). We drop the prime from $p_i'$ and $T_E'$ for ease of notation. As a second approximation, instead of keeping track of two types of nodes (red and blue) as it is done in the RB tree, we propose to change our network model to an infinite $d$-regular tree, where $d$ is set to be the average degree of an RB tree.
By making these two changes (tracking time at a coarser scale and simplifying the network topology to a $d$-regular tree), the growth of the epidemic becomes equivalent to a known model, the $d$-ary Random Exponential Tree ($d$-RET). Binary RETs have been introduced in \cite{feng2018profile}
. We give the definition below for completeness.
\begin{definition}
\label{def:dRET}
A $d$-ary Random Exponential Tree ($d$-RET) with parameters $d,p_i$ at time day $t$, denoted by $G_t(s)$, is a random tree rooted at node $s$. At day $0$, the tree $G_t(s)$ only has its root node~$s$. Let $\bar{G}_t(s)$ be the closure of $G_t(s)$, which is obtained by attaching external nodes to $G_t(s)$ until every internal node (a node that was already present in $G_t(s)$) has degree exactly $d$ in the graph $\bar{G}_t(s)$. Then, $G_{t+1}(s)$ is obtained from $\bar{G}_{t}(s)$ by retaining each external node with probability $p_i$, and dropping the remaining external nodes.
\end{definition}
Indeed, each node of a $d$-RET infects a new node with probability $p_i$ each day, and after a sufficiently long time, the $d$-RET becomes close to a large $d$-ary tree. Of course, we do not want to let the $d$-RET grow for a very long time, we only want it to grow until the first hospitalization occurs. So far we have not talked about the course of the disease of the nodes in the $d$-RET model because we could define the spread of the infection without it. Since we still need to do one final simplification to compute the distribution of the transmission path, we defer the discussion about hospitalizations, and how the parameters $p_a$ and $p_h$ are part of the model, to Section~\ref{subsub:DET}. Note that by considering the $d$-RET, we deviate from the idea of separating the epidemic and the network models; we only have a randomly growing tree, which is stopped at some time, when the tree is still almost surely finite.
So far we only did simplifications to the model, which resulted in further and further deviations from the original version. Now we will make a small modification that brings our model back closer to the RB tree, without complicating the computations too much. We still make almost all maximum degrees of the RET uniform $d$, but we make an exception with the root, which will have maximum degree $d_r=d_c+d_h$. This makes the maximum degree of the root the same as the degree of the root of the RB tree. We call the resulting model a $(d_r,d)$-RET with parameter $p_i$. Since the close neighborhood of the source has a high impact on the success probability, we found that this solution gives the best results while keeping the computations tractable.
In our computations, only the profile the infection tree will be important, which motivates the next definition.
\begin{definition}
\label{def:total}
In the $(d_r,d)$-RET model with parameter $p_i$, let $A_{t,l}$ be the number of nodes during day~$t$ at level~$l$, and let $a_{t,l} = \mathbb{E}[A_{t,l}]$.
Moreover, we define the random variable
\begin{align}
&A_{t} = \sum_{t = 0}^{+\infty} A_{t,l} \label{eq:def_total
\end{align}
with $A_{-1, l} = 0$ for all $l$, and its expectation $a_{t} = \mathbb{E}[A_{t}]$.
\end{definition}
As noted earlier, the $d$-RET model has only been analysed for $d=2$ to this date. We provide the expected number $a_{t,l}$ of nodes at level~$l$ in day~$t$ for the general case in the next theorem and corollary, which we prove in Appendices ~\ref{sec:theorem:a} and \ref{sec:corollary:at}.
\begin{theorem}
\label{theorem:a}
In the $(d_r, d)$-RET with parameter $p_i$, let $a_{t,l}$ be as in Definition \ref{def:total}. Then
\begin{align}
a_{t,0} &= 1 \\
a_{t,l} &= d_rp_i\sum_{m = l-1}^{t-1} \binom{m}{l-1}(1-p_i)^{m-l+1}d^{l-1}p_i^{l-1} \textrm{, for } t \geq l \geq 1 \\
a_{t,l} &= 0 \textrm{, for l > t}.
\end{align}
\end{theorem}
\begin{corollary}
\label{corollary:at}
In the RET$(p_i, d_r, d)$, let $a_{t}$ be the expectation of \eqref{eq:def_total}, as in Definition~\ref{def:total}. For $t\geq0$,
\begin{equation}
a_{t} = 1 + d_r\frac{(1-p_i+dp_i)^t - 1}{d-1}. \label{lemma:total}
\end{equation}
\end{corollary}
\subsubsection{Deterministic Exponential Tree with Parameters $p_a, p_h$ and $(c_{t,l})_{t,l \in \mathbb{N}}$}
\label{subsub:DET}
In the $(d_r,d)$-RET model it is still complicated to calculate the distribution of the depth of the first hospitalized node. For this reason, we approximate the RET model by a deterministic time-dependent tree with a prescribed profile.
\begin{definition}
Let $(c_{t,l})_{t \in \mathbb{N} \bigcup \{-1\},l \in \mathbb{N}}$ be a two-dimensional array with $c_{t,l}=0$ for $t\in\{-1,0\}$ and $l\in \mathbb{N}$, except for $c_{0,0} = 1$, and with $c_{t,l} \geq c_{t,l-1}$ for any $t$ and any $l \geq 1$. Additionally, if we define $c_t = \sum_l c_{t,l}$, then the array $(c_{t,l})$ must satisfy $c_t> c_{t-1}$ for $t\ge0$. Then, we define the Deterministic Exponential Tree (DET) with parameter $(c_{t,l})_{t \in \mathbb{N} \bigcup \{-1\},l \in \mathbb{N}}$, as a time-dependent rooted tree, that has exactly $c_{t,l}$ nodes on level $l$ at time $t$. The edges between the adjacent levels are drawn arbitrarily so that the tree structure is preserved.
\end{definition}
The formal assumptions on the array $(c_{t,l})$ are simply made to ensure that the DET starts with a single node at $t=0$, that it never shrinks on any level ($c_{t,l} \geq c_{t,l-1}$), and that it grows by at least one node in each time step ($c_t> c_{t-1}$).
We have defined the DET at any given time $t$, however, to determine the length of the transmission path, we are not interested in the DET at any given time, but only when the first hospitalization occurs.
To compute the distribution of the first hospitalized node, we would like to have an absolute order on the times when the nodes are added, which we do by randomization. We say that on day $t$, nodes are added one by one to the DET, their order given by a uniformly random permutation, and each node is hospitalized with probability $(1-p_a)p_h$ (as in the original DDE model). When the first hospitalization occurs, we stop growing the tree, and we call the resulting (now random) model a stopped DET with parameters $(c_{t,l}), p_a, p_h$. We find the transmission path length distribution on the stopped DET in the next lemma, which we prove in Appendix~\ref{sec:lem:DET}.
\begin{lemma}
\label{lem:DET}
Let us consider the stopped DET model with parameters $(c_{t,l}), p_a, p_h$, and let $h$ denote the first hospitalized node. Then
\begin{align}
\label{eq:DET_lemma}
\P(d(s,h) = l) = \sum_{t=0}^{+\infty} \frac{c_{t,l}-c_{t-1,l}}{c_t-c_{t-1}} (1-(1-p_a)p_h)^{c_{t-1}}\left(1-(1-(1-p_a)p_h)^{c_t-c_{t-1}}\right).
\end{align}
\end{lemma}
We would like to set $c_{t,l}$ so that the DET is close to the RET described in Section~\ref{subsub:RET}. For equation~\eqref{eq:DET_lemma} to make sense, we should substitute integer values for $c_{t,l}$, however, for an approximation the equation can also be evaluated for fractional values as well.
\begin{remark}
\label{rem:subs}
If we substitute $c_{t,l}=a_{t,l}$ and $c_t=a_t$ in equation \eqref{eq:DET_lemma}, where $a_{t,l}$ is given in Theorem \ref{theorem:a} and $a_{t}$ is computed in Corollary~\ref{corollary:at}, then we get the expression
\begin{align}
\label{eq:rem_approx}
d_rp_i^{l-1}d^{l-1} \sum_{t=l}^{+\infty} \frac{\binom{t-1}{l-1}(1-p_i)^{t-l}}{(1-p_i+dp_i)^{t-1}}(1-(1-p_a)p_h)^{1 + d_r \frac{(1-p_i+dp_i)^{t-1} - 1}{d-1}} \left(1-(1-(1-p_a)p_h)^{d_r(1-p_i+dp_i)^{t-1})}\right),
\end{align}
which approximates the distribution of the transmission path length in the $(d_r, d)$-ary RET stopped at the first hospitalization.
\end{remark}
\section{Simulation Results}
\label{sec:simulation}
\subsection{Baseline Algorithms}
\subsubsection{Non-adaptive Baseline: Dynamic Message Passing}
There are few sensor-based source detection algorithms that are compatible with time-varying networks in the literature \cite{huang2017source,jiang2016rumor,fan2020identifying}. The most promising one among these algorithms \cite{jiang2016rumor} has a close resemblance to the a previous work of \cite{lokhov2014inferring} on Dynamic Message Passing (DMP) algorithms. Given the initial conditions on the identity of the source node and its time of infection, the DMP algorithm approximates the marginal distribution of the outcome of an epidemic at some later time $t$. The algorithm is exact on tree networks, and it computes a good approximation when there are not too many short cycles in the network. Therefore, the DMP algorithm can be used to approximate the likelihood of the observed symptom onset times for any (source,time) pair. Due to its flexibility, we were able to adapt the DMP algorithm to the SDCTF (see Appendix \ref{sec:DMP_all} for more details).
Originally, the DMP was applied to the source detection problem by computing the likelihood values for all possible (source,time) pairs, and then choosing the source node from the most likely pair as the estimate \cite{lokhov2014inferring}. However, testing all (source,time) pairs increases the time complexity of the algorithms potentially by a factor of $N^2$, which makes the algorithm intractable in many applications.
Jiang et. al. \cite{jiang2016rumor} proposed a very similar algorithm to the DMP equations (which is unfortunately not exact even on trees), and solved the issue of intractability by a heuristic preprocessing step to the DMP algorithm. This preprocessing step, identifies a few candidate (source,time) pairs, by spreading the disease backward from the observations in a deterministic way (called reverse dissemination). Since we already approximate our data-driven model (DCS) by an epidemic model with deterministic transition times (DDE), it is natural for us to also implement the deterministic preprocessing step proposed by \cite{jiang2016rumor}. We produce 5 (source,time) pairs which are feasible for the 5 earliest symptom onset time observations (see Appendix~\ref{sec:DMP_feasible} for more details). It would have been ideal to run the algorithms for more than 5 pairs, but this was made impossible by the runtimes becoming very high. We run therefore our implementation of the DMP algorithm with the previously computed feasible (source,time) pairs as initial conditions to find the most likely source candidate.
The source estimation algorithms developed using the DMP algorithm do not specify how the sensors should be selected, and therefore place these non-adaptive sensors randomly. We refer to the resulting algorithm as random+DMP. The number of sensors is set so that it always exceeds the number that LS/LS+ would use. The simulation results are shown in Figure \ref{fig:baseline_plots} for the DDE+HNM model. Importantly, the deterministic preprocessing step of \cite{jiang2016rumor} is compatible with time-varying networks, which allows us to run the algorithm for the DCS+TU model as well (see Figure \ref{fig:DCSplots}).
\subsubsection{Adaptive Baseline: Size-Gain}
The Size-Gain (SG) algorithm was developed for epidemics which spread deterministically \cite{zejnilovic2015sequential}, and has been later extended to stochastic epidemics \cite{spinelli2017back}. It works by narrowing a candidate set based on a deterministic constraint. If $v_1, v_2$ are symptomatic observations, then $s_c$ is in the candidate set of SG if and only if
\begin{equation}
\label{eq:SG}
|(t_{v_2}-t_{v_1})- (d(v_2,s_c) - d(v_1,s_c))| < \sigma(d(v_2,s_c)+d(v_1,s_c)),
\end{equation}
where $\sigma$ is the standard deviation of the infection time of a susceptible contact. If one of the observations, say $v_2$, is negative, then SG uses a condition almost identical to equation \eqref{eq:SG}, except that the absolute value is dropped, since a negative observation at time $t_{v_2}$ is only a lower bound on the true symptom onset time of $v_2$. These deterministic conditions are checked for every symptomatic-symptomatic or symptomatic-negative pair $(v_1, v_2)$ to determine if $s_c$ can be part of the candidate set. Next, SG places the next sensor adaptively at the node which reduces the candidate set by the largest amount in expectation (assuming a uniform prior on the source and its infection time), and it terminates when the candidate set shrinks to a single node. Note that the SG algorithm can fail if at least one of the deterministic conditions in equation~\eqref{eq:SG} is violated for some $(v_1, v_2)$ because of the randomness of the epidemic.
We use the existing implementation of the SG algorithm by \cite{spinelli2017back}, and adapt it to the SDCTF. We incorporate asymptomatic-symptomatic and asymptomatic-negative observations $(v_1,v_2)$ the same way as symptomatic-negative are incorporated; we drop the absolute value sign in equation \eqref{eq:SG}, because an asymptomatic observation at time $t_{v_1}$ is only an upper bound on the true symptom onset time of $v_1$. We impose the same daily limit to the number of sensors that can be placed by the SG algorithm in a single day as for the LS/LS+ algorithm, and if the candidate set size does not shrink to one on the day when both LS and LS+ have already provided their estimates, then the SG algorithm must make a uniformly random choice from the current candidate set as its source estimate. The simulation results are shown in Figure \ref{fig:baseline_plots} for the DDE+HNM model. We do not implement the SG algorithm for the DCS+TU model, because its runtime is too high, and because it is not clear how it should be implemented for time-varying networks.
\subsection{Comparison with Baselines}
\label{sec:comp_baseline}
\begin{figure}[h]
\begin{center}
\includegraphics[width=\textwidth]{images/Figure3.pdf}
\caption{The performance of the algorithms LS, LS+, R and SG if the metric is the probability of finding the source (solid curves) or the first symptomatic patient (dashed curves). The simulations were computed on a population of $n=400$ individuals in the DDE model on the HNM, and each datapoint is the average of $4800$ independent realizations except for the SG algorithm, which was run with $192$ independent realizations. The confidence intervals for the success probabilities are computed using the Wilson score interval method, and for the tests and the queries using the Student's t-distribution.}
\label{fig:baseline_plots}
\end{center}
\end{figure}
We show our simulation results comparing the random+DMP, SG, LS and LS+ algorithms in Figure~\ref{fig:baseline_plots}. In the first row of Figure \ref{fig:baseline_plots}, we show the accuracy of the algorithms with solid curves. Since the LS/LS+ algorithms cannot detect an asymptomatic source, we also show what the accuracy would look like if the goal of the SDCTF was to detect the first symptomatic agent with dashed lines. It is clear that in both metrics and across a wide range of parameters, the LS+ algorithm performs best, followed by LS, next random+DMP, and finally SG. The only exception is for high values of $p_i$, where SG performs best. The good performance of SG for these parameters is expected, because SG was originally developed for deterministically spreading epidemics (i.e., $p_i=1$).
In the second row of Figure \ref{fig:baseline_plots}, we show the number of test/sensor queries used by the algorithms. LS uses the fewest tests, followed by LS+. The random+DMP and SG algorithms always use more tests than LS/LS+, except for large values of $p_i$. Finally, in the last row of Figure \ref{fig:baseline_plots} we show the number of contact (or in this case edge) queries used by the algorithms. Again, LS uses fewer queries than LS+, while both the random+DMP and SG algorithms query essentially the entire network.
Figure \ref{fig:baseline_plots} shows that the LS/LS+ algorithms are fairly robust to changes in the parameters of the model, except for the parameter $p_a$. Indeed, if there are many asymptomatic nodes in the network, then source detection becomes very challenging. It may be surprising that as $p_a$ grows, the number of tests that LS uses decreases, contrary to LS+. This is because as $p_a$ grows, the LS algorithm gets stuck more rapidly, while the LS+ algorithm compensates for the presence of asymptomatic nodes by using more test/sensor queries.
\subsection{Comparison of Simulations and Theoretical Results}
\label{sec:comp_theory}
\begin{figure}[h]
\begin{center}
\includegraphics[width=\textwidth]{images/Figure2.pdf}
\caption{The probability of success of the LS and LS+ algorithms (solid curves) and their theoretical estimate (dash-dotted curves) with the success probabilities computed in Lemma \ref{lem:LSsucc} and Theorem \ref{thrm:LSp_suc}, while the transmission path distribution computed in equation \eqref{eq:rem_approx}. The simulation results were generated using the DDE model on HNM networks of size $n=1000$ with $4800$ independent samples. The $95\%$ confidence intervals are computed using the Wilson score interval method.}
\label{fig:theory_plots}
\end{center}
\end{figure}
The analytic results from Section~\ref{sec:theory} are in good agreement with the simulation results in Figure \ref{fig:theory_plots}. We also experiment with changing the parameters $d_h$, $d_c$ while keeping all the parameters fixed, and with changing $d_c$ while keeping the product $d_cp_i$ fixed. We observe that LS is not affected by the parameter $d_h$, whereas LS+ performs better with a higher $d_c$, which is expected because LS+ leverages the household structure of the network to improve over LS. Somewhat surprisingly, we also observe that a higher $d_c$ also improves the performance of both algorithms. This can be explained by the fact that a larger $d_c$ implies that there are more nodes in the close neighborhood of the source, which results in shorter transmission paths, making source detection less challenging. Finally, if we increase $d_c$ but keep $d_cp_i$ fixed, the performance of the algorithms does not change as much, which confirms the intuition that it is the number of infections caused by an infectious node in a single day that matters the most (as we discussed in Section~\ref{sec:boe_sec}).
\subsection{Simulations on the DCS Model}
\label{sec:simDCS}
\begin{figure}[h]
\begin{center}
\includegraphics[width=\textwidth]{images/Figure4.pdf}
\caption{The performance of the algorithms LS, LS+ and random+DMP on the DCS model with the Tubingen dynamics if the metric is the probability of finding the source (solid curves) or the first symptomatic patient (dashed curves), together with the theoretical results (dash-dotted lines), as shown in Figure \ref{fig:theory_plots}. The simulations were computed on a population of $n=9054$ individuals, and each datapoint is the average of $2400$ independent realizations for the LS/LS+/LSv2/LS+v2 algorithms, and $48$ independent realizations for the random+DMP algorithm. The default population and infection parameters were selected to match the population and COVID-19 infection datasets of Tubingen. The confidence intervals for the success probabilities are computed using the Wilson score interval method, and for the tests and the queries using the Student's t-distribution.}
\label{fig:DCSplots}
\end{center}
\end{figure}
We show our simulation results on our most realistic DCS+TU model in Figure \ref{fig:DCSplots}. We make very similar observations on this model as the ones that we have made on the DDE+HNM model in Sections~\ref{sec:comp_baseline} and \ref{sec:comp_theory}, which shows that the LS/LS+ algorithms and our analysis of their performance is robust to changes in the epidemic and network models.
In the DCS+TU model, we used a fixed limit on the number of sensors that the random+DMP model selects, instead of setting the limit based on the LS+ algorithm. As a result, for a few parameters the LS+ algorithm used more tests than the random+DMP model. However, we note that by updating the candidate node immediately after an earlier symptom onset time is revealed (see Section \ref{sec:localsearch}), we can essentially cut the number of required tests for the LS+ algorithm by half (LSv2 and LS+v2), without sacrificing the performance of the algorithms.
\section{Discussion}
\label{sec:discussion}
We have introduced the LS and LS+ algorithms in the SDCTF, and we have used a sequence of models on which we can compute their accuracy (probability of finding the correct source) rigorously. We find that both LS and LS+ outperform baseline algorithms, even though the baselines essentially query all contacts on a transmission path between agents, while LS and LS+ query only a small neighborhood of the source. One could argue that LS and LS+ beat the baseline algorithms only because we benchmark them in our own framework, which is different from the framework for which the baseline algorithms were developed. However, we argue that the LS/LS+ algorithms are robust to changes in the framework due to their simplicity, and we support our argument by simulation results. The runtimes of the LS/LS+ algorithms are also much lower than the baselines and do not depend on the network size since they are local algorithms - as opposed to the baselines, which have quadratic or even larger dependence on the network size. The ``low-tech'' approach in the design of the LS/LS+ algorithms increases their potential to be implemented in real-world scenarios, possibly even in a decentralized way, similarly to contact tracing smart phone applications \cite{troncoso2020decentralized}, which is an interesting direction for future work.
\bibliographystyle{ACM-Reference-Format}
\section{Additional Proofs}
\begin{figure}[h]
\begin{center}
\includegraphics[width=\textwidth]{images/LSp_cases.pdf}
\caption{Illustration for Lemma \ref{lem:LS,LSp,trees} using the same coloring as Figure \ref{fig:mobility_models}~(a). (a) An example for an epidemic where among the nodes of the transmission path $(v_1, v_2, v_3, v_5)$, the middle household contains no symptomatic node (only the asymptomatic node $v_3$), but the LS+ algorithm still succeeds. Indeed, at iteration 0 we set $s_{c,0}=v_5$, after which we find that $v_3$ is asymptomatic, and next that $v_2$ is asymptomatic and $v_4$ is symptomatic, with a lower symptom onset time then $v_5$. Hence, in iteration 1 we set $s_{c,1}=v_4$, and we find that $v_3, v_2$ are asymptomatic and $v_1$ is is symptomatic, with a lower symptom onset time then $v_4$. Finally, in iteration 2 we set $s_{c,2}=v_1$, and we find $s_c'=v_1=s_{c,2}$, which implies that the algorithm stops, and returns the correct source $v_1$. (b) An example for an epidemic where the LS+ algorithm would fail if we would update the candidate before the test queue becomes empty. Similarly to subfigure (a), in iteration 0 of the algorithm first learns about asymptomatic node $v_3$ and next about asymptomatic node $v_2$ and symptomatic node $v_4$. If the algorithm updates the candidate to $v_4$ and continues further, instead of scheduling the tests of the household members of $v_2$, then it is not hard to check that $v_4$ will be the final estimate and the algorithm fails. However, if the algorithm waits until the test queue becomes empty and tests the household members of $v_2$, then $v_1$ becomes the next candidate and the algorithm succeeds.}
\label{fig:LSp_cases}
\end{center}
\end{figure}
\subsection{Proof of Lemma \ref{lem:LS,LSp,trees}}
\label{sec:LS,LSp,trees_app}
We start by restating the lemma here for convenience.
\begin{lemma}
\label{lem:LS,LSp,trees_app}
In the RB tree network, the LS algorithm succeeds if and only if all nodes on the transmission path are symptomatic, and the LS+ algorithm succeeds if among the nodes of the transmission path, there exists a symptomatic node in each household, and the source is symptomatic.
\end{lemma}
\begin{proof}
Throughout the proof we assume that there is no limitation on the number available tests. We can make this assumption because in the SDCTF there is only a daily limit on the number tests, there is no limitation on the number of days, and neither the LS nor the LS+ algorithms proceed in an iteration until the test queue becomes empty, which implies that all nodes that enter the test queue get eventually tested.
Suppose that the LS algorithm succeeds. Then the list of candidate nodes $s_c$ at different iterations forms a path that consists entirely of symptomatic nodes between the source and the first hospitalized node. In tree networks, the transmission path is the only path between the source and the first hospitalized node, which yields the "only if" part of the statement on the LS algorithm.
Next, suppose that all nodes on the transmission path are symptomatic. Then, we claim that the candidate node $s_{c,i}$ computed in the $i^{th}$ iteration of the LS algorithm is $v_{l-i}$, the $i^{th}$ node of the reverse transmission path. Our claim is definitely true for $i=0$, because $s_{c,0}$ is initialized to be the first hospitalized node $v_l$. Then, the proof proceeds by induction. By the induction hypothesis, in the $i^{th}$ step, $s_{c,i}=v_{l-i}$, and since we are on a tree, the symptom onset time of $v_{l-(i+1)}$ (which is revealed because all nodes on the transmission path are symptomatic by assumption) is the only symptom onset time among the neighbors of $s_{c,i}$ that have a lower symptom onset time than $s_{c,i}$ itself. Therefore $s_c'=v_{l-(i+1)}$, and $s_{c,i+1}$ is updated to be $v_{l-(i+1)}$ in the beginning of the next iteration, which proves that the induction hypothesis holds until the source is reached.
Finally, suppose that among the nodes of the transmission path, there exists a symptomatic node in each household, and the source is symptomatic. Let us denote by $w_i$ the $i^{th}$ symptomatic node of the \emph{reverse} transmission path. Then, we claim that the candidate list $s_{c,i}$ computed in the $i^{th}$ iteration of the LS+ algorithm equals $w_i$. Similarly to the case of the LS algorithm, the $i=0$ case holds by definition, and we proceed by induction. Suppose that $s_{c,i}=w_i$. It will also be useful to define the index of $w_i$ on the \emph{forward} transmission path (without skipping asymptomatic nodes). Let $j$ be this index, for which therefore $w_i=v_j$. Now we distinguish 3 cases: (i) $v_{j-1}=w_{i+1}$ is symptomatic, (ii) $v_{j-1}$ is asymptomatic and $v_{j-2}=w_{i+1}$ is symptomatic, and (iii)
$v_{j-1}$ and $v_{j-2}$ are asymptomatic and $v_{j-3}=w_{i+1}$ is symptomatic. We claim that there are no more cases, and that in all three cases $w_{i+1}$ is tested in the $i^{th}$ iteration of the LS+ algorithm. Case (i) is immediate because all neighbors of $s_{c,i}$ are tested. Case (ii) is only possible if either $v_{j-1} \in H(s_{c,i})$ or $v_{j-2} \in H(v_{j-1})$, otherwise $v_{j-1}$ would be a lone asymptomatic node in a household, which contradicts the assumption that there is a symptomatic node in each household. Since all the contacts of asymptomatic nodes in $H(s_{c,i})$ (see Figure \ref{fig:Alg1_figure}~(d)) and all nodes in the household of asymptomatic nodes are tested in the LS+ algorithm (see Figure \ref{fig:Alg1_figure}~(e)), $v_{j-2}$ must be tested too. Finally, case (iii) is possible only if $v_{j-1} \in H(s_{c,i})$ and $v_{j-3} \in H(v_{j-2})$ both hold, otherwise $v_{j-1}$ or $v_{j-2}$ would be a lone asymptomatic node in a household. Similarly to the previous case, $v_{j-3}$ must be tested (see Figure \ref{fig:Alg1_figure}~(f)). There are no more cases because, by Remark \ref{rem:RBtree}, on the RB tree a transmission path can only have two nodes in each household, and we assumed that there exists a symptomatic node in each household among the nodes of the transmission path.
After we proved that $w_{i+1}$ is tested in the $i^{th}$ iteration of the LS+ algorithm, we must still show that it will be the next candidate $s_{c,i+1}$ for the induction hypothesis to hold. This is true because once the symptom onset time of $w_{i+1}$ is revealed, none of its neighbors are scheduled for testing, and therefore all tested nodes have $w_{i+1}$ on their path to the source, which means that $w_{i+1}$ must have the lowest revealed symptom onset time, and therefore that it will be the next candidate~$s_{c,i+1}$.
\end{proof}
\subsection{Proof of Theorem \ref{thrm:LSp_suc}}
\label{sec:thrm:LSp_suc}
We are going to need prove a few intermediate results before proving Theorem \ref{thrm:LSp_suc}. A first step is to count all the possible paths from the source with a given length.
\begin{definition}
Let $G(s)$ be the RB tree with parameters $(d_c, d_h)$, and let $s$ be the source. A Red-Blue (RB) path of length $n$ is any path of nodes in $(s=v_0, v_1, ...v_n)$ such that $(v_i, v_{i+1}) \in E'$ for $0\leq i < n$. Let $\mathcal{C}_n$ be the set of RB paths of length $n$.
\end{definition}
\begin{lemma}
\label{lem:Cn}
In the RB tree with parameters $(d_c, d_h)$, $|C_0| = 1$, while for $n \geq 1$,
\begin{equation}
|\mathcal{C}_n| = \lambda_1\left( \frac{d_c-1+D}{2}\right)^n + \lambda_2\left( \frac{d_c-1-D}{2}\right)^n
\end{equation}
where
\begin{align}
D &= \sqrt{(d_c-1)^2 +4d_cd_h} \\
\lambda_1 &= \frac{(d_c+1+D)(2d_h+d_c-1+D)}{2D(d_c-1+D)} \\
\lambda_2 &= \frac{(D-d_c-1)(2d_h+d_c-1-D)}{2D(d_c-1-D)}.
\end{align}
\end{lemma}
\begin{proof}
Let us keep track of the number of RB paths of length $n$ depending on the color of the last node in the path. Let $r_n$ and $b_n$ be the numbers of RB paths of length $n$ such that the last node is red and blue, respectively. A RB path of length 0 consists only of the source, which implies that $r_0 = 1$ and $b_0=0$. The source has $d_c$ red and $d_h$ blue neighbours, which implies that $r_1 = d_c$ and $b_1 = d_h$.
Suppose that $P$ is an RB path of length $n \geq 2$. If the last node of $P$ is red, then the node before the last node can be both blue or red. Red nodes other than the source have $d_c-1$ red children, while blue nodes have $d_c$ red children, yielding
\begin{align}
\label{eq:rn}
r_n = (d_c-1)r_{n-1}+d_cb_{n-1}, \textrm{ for }n\geq 2.
\end{align}
If the last node of $P$ is blue, then the node before has to be red. Since every red node, including the source, has $d_h$ blue children, we have
\begin{align}
\label{eq:bn}
b_n = d_hr_{n-1}, \textrm{ for }n\geq 1 .
\end{align}
By substituting equation \eqref{eq:bn} into equation \eqref{eq:rn}, we obtain the recurrence
\begin{align}
r_n = (d_c-1)r_{n-1}+d_cd_hr_{n-2}, \textrm{ for }n\geq 2.
\end{align}
We solve this recurrence equation by calculating the characteristic equation
\begin{equation}
t^2 - (d_c-1)t - d_cd_h = 0,
\end{equation}
whose roots are
\begin{align}
t_1 &= \frac{d_c-1+\sqrt{(d_c-1)^2 +4d_cd_h}}{2} = \frac{d_c-1+D}{2} \\
t_2 &= \frac{d_c-1-\sqrt{(d_c-1)^2 +4d_cd_h}}{2} = \frac{d_c-1-D}{2}
\end{align}
yielding the the general solution
\begin{equation}
r_n = c_1t_1^n + c_2t_2^n,
\end{equation}
where $c_1, c_2$ are given by the initial conditions for $n=0,1$
\begin{align}
&c_1 + c_2 = r_0 = 1 \\
&c_1 t_1 + c_2t_2 = r_1 = d_c,
\end{align}
which are
\begin{align}
c_1 &= \frac{1}{2} + \frac{d_c+1}{2\sqrt{(d_c-1)^2+4d_cd_h}} = \frac{1}{2} + \frac{d_c+1}{2D} \\
c_2 &= \frac{1}{2} - \frac{d_c+1}{2\sqrt{(d_c-1)^2+4d_cd_h}} =\frac{1}{2} - \frac{d_c+1}{2D}.
\end{align}
From equations \eqref{eq:rn} and \eqref{eq:bn} we conclude that for $n\geq 1$,
\begin{equation}
b_n = d_h(c_1t_1^{n-1} + c_2t_2^{n-1})
\end{equation}
and therefore
\begin{align}
|C_n| = r_n + b_n = \lambda_1t_1^n + \lambda_2t_2^n,
\end{align}
where
\begin{align}
\lambda_1 &= c_1\left(1+\frac{d_h}{t_1}\right) \\
\lambda_2 &= c_2\left(1+\frac{d_h}{t_2}\right) .
\end{align}
Inserting the values for $t_1, t_2, c_1, c_2$ we obtain the desired result.
\end{proof}
Since LS+ improves on LS by making use of the household structure of the network, we need further information about the household structure of the transmission paths. Recall that by Remark~\ref{rem:RBtree}, households on transmission paths on an RB tree were characterized either by a single red node (that is followed by a red node), or a pair of consecutive red and blue nodes. The following definition and lemma refine our previous result on counting the number of RB paths by taking the household structure into account.
\begin{definition}
Let $P=\{ s=v_0,v_1,\ldots, v_n=h \}$ be a RB path of length $n$. We say that a node $v$ on the path $P$ is in a $P$-single-household if no other node from $P$ is in the same household as $v$. Otherwise, we say $v$ is in a $P$-multi-household. Given a path $P$, let $M_s: \mathcal{C}_n \rightarrow \{0,1\}$ be the indicator function that the source is in a $P$-multi-household. Similarly, let $M_l: \mathcal{C}_n \rightarrow \{0,1\}$ be the indicator function that the last node of path $P$ is in a $P$-multi-household. Finally, for $0 \leq k \leq n+1$ and $\alpha, \beta \in \{0,1\}$, let
\begin{equation}
C_{n,k,\alpha, \beta} = \{P \in \mathcal{C}_n : (\textrm{there are exactly }k \textrm{ nodes in } P-\textrm{single-households})\wedge (M_s(P)=\alpha) \wedge (M_l(P) = \beta)\}.
\end{equation}
\end{definition}
The set $C_{n,k,\alpha, \beta}$ depends on 4 parameters, but only some combinations of these parameters make it non-empty. The following definition will be useful in this regard.
\begin{condition}
\label{cond:path}
Let $\alpha, \beta \in \{0,1\}$ and $n \geq 2$. We say $k \in \mathbb{N}$ satisfies Condition~\ref{cond:path} if and only if $k$ and $n$ have different parity and $n+1 - 2(\alpha + \beta) \geq k \geq 2-(\alpha+\beta)$.
\end{condition}
\begin{lemma}
\label{lem:Cnkab}
It holds that $|C_{0,1,0,0}| = 1$, $|C_{1,0,1,1}| = d_h$ and $|C_{1,2,0,0}| = d_c$. Let $\alpha, \beta \in \{0,1\}$, let $n \geq 2$ and let $k\in \mathbb{N}$ satisfy Condition \ref{cond:path}. Then
\begin{equation}
\label{eq:Cnkab}
|\mathcal{C}_{n,k,\alpha, \beta}| = \binom{\frac{n+k-3}{2}}{k-2+\alpha+\beta}d_h^{\frac{n-k+1}{2}}d_c^{\frac{n-k+3}{2}-\beta-\alpha}(d_c-1)^{k+\alpha + \beta-2}.
\end{equation}
In all other cases $|C_{n,k,\alpha, \beta}| = 0$.
\end{lemma}
\begin{proof}
Since there are $n+1$ nodes on path $P$, with $k$ in $P$-single households and thus $n+1-k$ of them in $P$-multi-households, we must have
$$k + \frac{n+1-k}{2}=\frac{n+k+1}{2}$$
households along path~$P$ in total. Clearly, the numbers $n$ and $k$ cannot be of the same parity for any RB path $P$, which is thus assumed for the rest of the proof (this assumption is also part of Condition~\ref{cond:path}).
If $n=0$, then the source is also the first hospitalized node, and it is in a $P$-single-household, which implies that $|C_{0,1,0,0}| = 1$. If $n=1$, then there are two cases: either the source is in the same $P$-multi-household with the first hospitalized node, or both of them are in $P$-single-households. The former case is possible via $d_h$ edges from the source, which gives $|C_{1,0,1,1}| = d_h$, while the latter case is possible via $d_c$ edges, and gives $|C_{1,0,1,1}| = d_c$. Since these are the only possible RB paths of length $n\le 1$, we must have $|C_{0,k,\alpha,\beta}|=|C_{1,k,\alpha,\beta}| = 0$ for any other choice of parameters $k,\alpha$ and $\beta$.
Let us assume that $n\geq 2$. Then, the source and the first hospitalized node are not in the same household. Let us denote the household of the source by $H_s$ and the household of the first hospitalized node by $H_h$. Note that $(1-\alpha)$ and $(1-\beta)$ are the indicators of $H_s$ and $H_h$ being $P$-single-households, and therefore $k \geq (1-\alpha) + (1-\beta)$. If this inequality (which is also part of Condition \ref{cond:path}) does not hold, then clearly $|\mathcal{C}_{n,k,\alpha, \beta}|=0$. Similarly, the number of $P$-multi-households is $\frac{n-k+1}{2}$ and we must have $\frac{n-k+1}{2} \geq \alpha + \beta$ for $|\mathcal{C}_{n,k,\alpha, \beta}|>0$, which implies the inequality $n+1 - 2\alpha - 2\beta \geq k$. Therefore $\mathcal{C}_{n,k,\alpha, \beta}$ is empty if Condition \ref{cond:path} does not hold. For the rest of the proof we assume that Condition \ref{cond:path} does hold.
There are $\frac{n+k-3}{2}$ households along path~$P$, excluding $H_s$ and $H_h$. Among them, there are $k-(1-\alpha)-(1-\beta)$ $P$-single-households, which can be chosen in $\binom{\frac{n+k-3}{2}}{k-2+\alpha+\beta}$ ways. Once we know the color of each node along the path, the number of RB paths can be computed by multiplying the numbers of children with the appropriate color of each node. $P$-single-households have no blue nodes, and $P$-multi-households have exactly one, which implies that there are $\frac{n-k+1}{2}$ blue nodes. Since blue nodes are preceded by red nodes that have $d_h$ blue children, they give the multiplicative factor $d_h^{\frac{n-k+1}{2}}$. Blue nodes, except from the first hospitalized node (if it is blue), have $d_c$ red children. So far we have accounted for all of the nodes in $P$-multi-households and none of the nodes in $P$-single-households. If the source is in a $P$-single-household, then we must count its red children, whose number is $d_c$. This implies that there exist $\frac{n-k+1}{2}-\beta+(1-\alpha)$ nodes with $d_c$ red children. Finally, each $P$-single-household, except $H_s$ and/or $H_h$ in case they are $P$-single households, has $d_c-1$ red children. There are $k-(1-\alpha)-(1-\beta)$ such $P$-single-households, which gives the final term in equation \eqref{eq:Cnkab}.
\end{proof}
The sets $\mathcal{C}_{n,k,\alpha, \beta}$ define equivalence classes on the transmission paths based on their household structure. In the next lemma we show that once we know which equivalence class we are in, it is possible compute the success probability of the LS+ algorithm.
\begin{lemma}
\label{lem:LSp|Cnkab}
Let $P$ be the transmission path in the $\mathrm{DDE}_{\mathrm{NR}}$ epidemic model with parameters $(p_i, p_a, p_h)$ on the RB tree with parameters $(d_c, d_h)$, and let $p$ be as computed in \eqref{C5}. Then, it holds that
$$\P(LS+ \textrm{ succeeds} | P \in \mathcal{C}_{0,1,0,0}) = 1$$ and $$\P(LS+ \textrm{ succeeds} | P \in \mathcal{C}_{1,0,1,1}) = \P(LS+ \textrm{ succeeds} | \mathcal{C}_{1,2,0,0}) = 1-p.$$
Let $\alpha, \beta \in \{0,1\}$, let $n \geq 2$ and let $k\in \mathbb{N}$ satisfy Condition \ref{cond:path}. Then, it holds that
\begin{equation}
\P(LS+ \textrm{ succeeds} | P \in \mathcal{C}_{n,k,\alpha, \beta}) \ge (1-p)^{\frac{n+k-1}{2}}(1+p)^{\frac{n-k+1}{2}-\alpha-\beta}.
\end{equation}
In all other cases $\P(LS+ \textrm{ succeeds} | P \in \mathcal{C}_{n,k,\alpha, \beta})$ is not defined.
\end{lemma}
\begin{proof}
If $n = 0$, then $k = 1$ and $\alpha = \beta = 0$. In that case, the source is the first hospitalized node and LS+ always succeeds. If $n =1$, then the first hospitalized node is in the neighbourhood of the source, and LS+ succeeds if and only if the source is symptomatic, which happens with probability $1-p$.
Let us assume that $n \geq 2$ and that $k$ satisfies Condition \ref{cond:path} (otherwise $|\mathcal{C}_{n,k,\alpha, \beta}|=0$ and $\P(LS+ \textrm{ succeeds}| P \in \mathcal{C}_{n,k,\alpha, \beta})$ is not defined). By Lemma \ref{lem:LS,LSp,trees} the LS+ algorithm succeeds in the $\mathrm{DDE}_{\mathrm{NR}}$ model on the RB tree if, among the nodes of the transmission path, there exists a symptomatic node in each household, and the source is symptomatic, which means that we can prove a lower bound on the success probability of LS+. Let us assume that the source is indeed symptomatic. Since the first hospitalized node is symptomatic by definition, the households of the source and of the first hospitalized node cannot make the LS+ algorithm fail. Let us denote these two households by $H_s$ and $H_h$, respectively. Also, let $M$ and $S$ be the sets of all $P$-multi- and $P$-single-households, respectively, excluding $H_s$ and $H_h$. Then, LS+ succeeds if all nodes in the households of $S$ are symptomatic, and if at least one node in the households of $M$ is symptomatic, which has probability $1-p$ and $1-p^2$ for each type of household, respectively, by equation \eqref{C5}. These observations yield that
\begin{align}
\P(LS+ \textrm{ succeeds} | P \in \mathcal{C}_{n,k, \alpha, \beta}) &\ge \P(\textrm{source is sym}) (1-p)^{|S|} (1-p^2)^{|M|} \nonumber \\
&= (1-p)(1-p)^{k-2+\alpha+\beta}(1-p^2)^{\frac{n-k+1}{2}-\alpha-\beta} \nonumber \\
&= (1-p)^{k-1+\alpha+\beta}(1-p^2)^{\frac{n-k+1}{2}-\alpha-\beta}.
\end{align}
\end{proof}
Finally, we are ready to state and prove Theorem \ref{thrm:LSp_suc} on the success probability of LS+, which we restate here for convenience.
\begin{theorem}
\label{thrm:LSp_suc_app}
Let $p$ be as in \eqref{C5} and let $\mathcal{S}(n,\alpha,\beta)$ be the set of $k$ values that satisfy Condition \ref{cond:path}. Then, for the $\mathrm{DDE}_{\mathrm{NR}}$ epidemic model with parameters $(p_i, p_a, p_h)$ on the RB tree with parameters $(d_c, d_h)$ we have
\begin{align}
&\P(LS+ \textrm{ succeeds}) \ge \P(d(s,h) = 0) + (1-p)\P(d(s,h)=1)+ \nonumber \\
& \sum_{n = 2}^{\infty} \sum_{ \substack{\alpha,\beta \in \{0,1\} \\ k \in \mathcal{S}(n,\alpha,\beta)}}
\binom{\frac{n+k-3}{2}}{k-2+\alpha+\beta}
\frac{(d_h(1-p))^{\frac{n+k-1}{2}}(d_c(1+p))^{\frac{n-k+1}{2}-\alpha-\beta}d_c(d_c-1)^{k+\alpha + \beta-2}}{\lambda_1\left( \frac{d_c-1+D}{2}\right)^n + \lambda_2\left( \frac{d_c-1-D}{2}\right)^n} \P(d(s,h)=n),
\end{align}
where $D,\lambda_1$ and $\lambda_2$ are terms depending on parameters $d_c$ and $d_h$ and are computed explicitly in Lemma \ref{lem:Cn}.
\end{theorem}
\begin{proof}
Let us extend the domain of $\P(LS+ \textrm{ succeeds} | P \in C_{n,k, \alpha, \beta})$ by function $g$ defined as $g: \mathbb{N} \times \mathbb{N} \times \{0,1\} \times \{0,1\} \rightarrow [0,1] $ such that
\begin{equation}
g(n,k,\alpha, \beta) = \begin{cases}
\P(LS+ \textrm{ succeeds} | P \in C_{n,k, \alpha, \beta})& \text{ if } k \in \mathcal{S}(n,\alpha,\beta) \\
0& \text{ if } k \not\in \mathcal{S}(n,\alpha,\beta).
\end{cases}
\end{equation}
Unlike $\P(LS+ \textrm{ succeeds} | P \in C_{n,k, \alpha, \beta})$, $g$ is defined for every 4-tuple of parameters $(n,k, \alpha, \beta) \in \mathbb{N} \times \mathbb{N} \times \{0,1\} \times \{0,1\}$. By the law of total probability we expand the success probability by conditioning on the path $P$ being of length~$n$ as
\begin{align}
\label{eq:LS+expand}
\P(LS+ \textrm{ succeeds}) =& \sum_{n = 0}^{\infty} \sum_{k = 0}^{\infty} \sum_{\alpha,\beta \in \{0,1\}} g(n,k, \alpha, \beta) \P(P \in C_{n,k, \alpha, \beta}) \nonumber \\
=& \sum_{n = 0}^{\infty} \sum_{k = 0}^{\infty} \sum_{\alpha,\beta \in \{0,1\}} g(n,k,\alpha, \beta) \P(P \in C_{n,k, \alpha, \beta} | P \in C_{n}) \P(d(s,h)=n).
\end{align}
Next, we exchange the sums over $\alpha, \beta$ and $k$. This allows us to sum over only those $k$ values that satisfy Condition \ref{cond:path}, which implies that $\P(LS+ \textrm{ succeeds} | P \in C_{n,k, \alpha, \beta})$ is well-defined. As in Lemma \ref{lem:Cnkab}, we need to treat the $n=0$ and $n=1$ cases separately. Continuing equation \eqref{eq:LS+expand}, we arrive to
\begin{align}
\label{eq:LS+switch}
\P(LS+ \textrm{ succeeds}) =& \P(d(s,h) = 0) + (1-p)\P(d(s,h)=1)+ \nonumber \\
&\sum_{n = 2}^{\infty} \sum_{\alpha,\beta \in \{0,1\}}
\sum_{ k \in \mathcal{S}(n,\alpha,\beta) }
\P(LS+ \textrm{ succeeds} | P \in C_{n,k, \alpha, \beta}) \frac{ |C_{n,k, \alpha, \beta}|}{ |C_{n}|} \P(d(s,h)=n
\end{align}
Substituting in the results from Lemmas \ref{lem:Cn}, \ref{lem:Cnkab} and \ref{lem:LSp|Cnkab} into equation \eqref{eq:LS+switch} gives the desired result.
\end{proof}
\subsection{Proof of Theorem \ref{theorem:a}}
\label{sec:theorem:a}
We start by restating Theorem \ref{theorem:a} for convenience.
\begin{theorem}
\label{theorem:a_app}
In the $(d_r, d)$-RET with parameters $p_i,p_a, p_h$, let $a_{t,l}$ be as in Definition \ref{def:total}. Then
\begin{align}
a_{t,0} &= 1 \\
a_{t,l} &= d_rp_i\sum_{m = l-1}^{t-1} \binom{m}{l-1}(1-p_i)^{m-l+1}d^{l-1}p_i^{l-1} \textrm{, for } t \geq l \geq 1 \\
a_{t,l} &= 0 \textrm{, for l > t}.
\end{align}
\end{theorem}
\begin{proof}
Similarly to \cite{feng2018profile,mahmoud2021profile}, the proof relies on generating functions. We start by addressing the boundary cases. For all $t \geq 0$, it holds that $A_{t,0} = 1$, and therefore $a_{t,0} = 1$. Similarly, for all $l,t$ such that $l > t$, it holds that $A_{t,l} = 0$, and therefore $a_{t,l} = 0$. Suppose that $t\geq l = 1$. During day~$t-1$, on the first level, there are $A_{t-1,1}$ infected (internal) nodes and $d_r-A_{t-1,1}$ (external) nodes that may be infected with probability $p_i$ during day~$t$. Thus,
\begin{align}
\label{eq:At1}
A_{t,1} &= A_{t-1,1}+\mathrm{Bin}(d_r - A_{t-1, 1};p_i) .
\end{align}
Taking the expectation of both sides in equation \eqref{eq:At1} yields
\begin{align}
a_{t,1} = a_{t-1,1}(1-p_i) + d_rp_i, \textrm{ for }t\geq 1.
\end{align}
By subtracting the appropriate recurrence equations for $a_{t,1}$ and $a_{t-1,1}$ for $t\geq 2$ we obtain the homogeneous recurrence equation
\begin{align}
a_{t,1} - a_{t-1,1}(2-p_i) + (1-p_i)a_{t-2,1} = 0, \textrm{ for } t\geq 2
\end{align}
and boundary conditions $a_{0,1} = 0$ and $a_{1,1} = d_rp_i$.
We solve for $a_{t,1}$ using the same methods as in the proof of Lemma~\ref{lem:Cn} and obtain
\begin{align}
\label{eq:a_t,1}
a_{t,1} = d_r\left(1-(1-p_i)^t\right), \textrm{ for }t\geq 0.
\end{align}
Next, let us consider the general case $t \geq l > 1$. On day $t-1$, there are $A_{t-1, l-1}$ nodes on level $l-1$. Since, each node on level $l-1$ has $d$ children, there are $dA_{t-1,l-1}$ nodes on level $l$ that have an infectious parent on level $l-1$. However, $A_{t-1,l}$ of them are already infected. Therefore $dA_{t-1,l-1}-A_{t-1,l}$ nodes of level $l$ may be infected on day~$t$, each with probability $p_i$, which implies
\begin{align}
\label{eq:Arec}
A_{t,l} &= A_{t-1,l}+\mathrm{Bin}(dA_{t-1, l-1} - A_{t-1; l},p_i), \textrm{ for }t \geq l \geq 2 .
\end{align}
Taking the expectation of both sides in equation \eqref{eq:Arec} yields
\begin{align}
\label{eq:atl_rec}
a_{t,l} &= a_{t-1,l} + (da_{t-1, l-1}-a_{t-1, l})p_i \nonumber \\
&= a_{t-1,l}(1-p_i) + dp_ia_{t-1, l-1}, \textrm{ for } t\geq l \geq 2.
\end{align}
For convenience, let us introduce $\lambda = 1-p_i$ and $\mu = dp_i$, and also let
\begin{align}
\label{eq:f(x,y)}
f(x,y) = \sum_{t = 1}^{\infty} \sum_{l = 1}^{\infty} a_{t,l}x^ty^l =\sum_{t = 1}^{\infty} \sum_{l = 1}^{t} a_{t,l}x^ty^l
\end{align}
be the generating function for $a_{t,l}$ with $t,l \geq 1$. By multiplying \eqref{eq:atl_rec} by $x^ty^l$ and summing it over $t,l \geq 2$ we obtain
\begin{align}
\label{eq:a1}
\sum_{t = 2}^{\infty} \sum_{l = 2}^{t} a_{t,l}x^ty^l &= \lambda \sum_{t = 2}^{\infty} \sum_{l = 2}^{t} a_{t-1,l}x^ty^l +\mu \sum_{t = 2}^{\infty} \sum_{l = 2}^{t} a_{t-1,l-1}x^ty^l \nonumber \\
&= \lambda x\sum_{t = 1}^{\infty} \sum_{l = 2}^{t} a_{t,l}x^ty^l +\mu xy \sum_{t = 1}^{\infty} \sum_{l = 1}^{t} a_{t,l}x^ty^l.
\end{align}
Since $a_{1,l} = 0$ for $l \geq 2$,
\begin{align}
\label{eq:a2}
\sum_{t = 1}^{\infty} \sum_{l = 2}^{t} a_{t,l}x^ty^l = \sum_{t = 2}^{\infty} \sum_{l = 2}^{t} a_{t,l}x^ty^l,
\end{align}
and by inserting \eqref{eq:a2} into \eqref{eq:a1}, we obtain
\begin{align}
\label{eq:atl}
(1-\lambda x)\sum_{t = 1}^{\infty} \sum_{l = 2}^{t} a_{t,l}x^ty^l &= \mu xy \sum_{t = 1}^{\infty} \sum_{l = 1}^{t} a_{t,l}x^ty^l \stackrel{\eqref{eq:f(x,y)}}{=} \mu xy f(x,y) .
\end{align}
Now, we can also decompose the sum (\ref{eq:a2}) using geometric series as
\begin{align}
\label{eq:atl2}
\sum_{t = 1}^{\infty} \sum_{l = 2}^{t} a_{t,l}x^ty^l &= \sum_{t = 1}^{\infty} \sum_{l = 1}^{t} a_{t,l}x^ty^l - \sum_{t = 1}^{\infty} a_{t,1}x^ty \nonumber \\
&\stackrel{\eqref{eq:a_t,1}}{=} f(x,y) - d_r y \sum_{t=1}^{\infty} (1-\lambda^t)x^t \nonumber \\
&= f(x,y) - d_r xy \left( \frac{1}{1-x} - \frac{\lambda}{1-\lambda x}\right).
\end{align}
By plugging \eqref{eq:atl2} into \eqref{eq:atl}, we obtain the expression
\begin{align}
\label{eq:f_x_y_final_expression}
f(x,y) &= d_r (1-\lambda) xy\frac{1}{1-x}\frac{1}{1-\lambda x - \mu xy}.
\end{align}
Then, we expand the fractions in (\ref{eq:f_x_y_final_expression}) into a power series and we next apply the binomial theorem, we arrive to
\begin{align}
\label{eq:fnmj}
f(x,y) &= d_r(1-\lambda)xy \sum_{n=0}^{\infty}x^n \sum_{m=0}^{\infty}x^m(\lambda+\mu y)^m \nonumber \\
&= d_r(1-\lambda)xy \sum_{n=0}^{\infty}x^n \sum_{m=0}^{\infty}x^m\sum_{j = 0}^m \binom{m}{j}\lambda^{m-j} (\mu y)^{j} \nonumber \\
&= d_r(1-\lambda)\sum_{n=0}^{\infty} \sum_{m=0}^{\infty}\sum_{j = 0}^m \binom{m}{j}\lambda^{m-j} \mu ^{j} x^{1+n+m}y^{j+1}.
\end{align}
Let $t = 1+n+m$ and $l= j+1$. In order to obtain an expression for $a_{t,l}$, we must change the variables in the sums of equation \eqref{eq:fnmj} from $(n,m,k)$ to $(t,m,l)$. Changing the inner sum from variable $j$ to $l$ is simple. Changing the variables in the two outer sums is more challenging because $t,n$ and $m$ depend on each other in a nontrivial way. More precisely, since $m,n\ge 0$ we have $t\ge 1$ and also $m\le t-1$, which means that we have to set the lower limit of $t$ and the upper limit of $m$ accordingly. As for the remaining limits, variable $t$ can be arbitrary large, and $m$ can take any integer value starting from $0$ independently of $t$, which yields the expression
\begin{align}
\label{eq:ftml}
f(x,y) &= d_r(1-\lambda)\sum_{t=1}^{\infty} \sum_{m=0}^{t-1}\sum_{l = 1}^{m+1} \binom{m}{l-1}\lambda^{m-l+1} \mu ^{l-1} x^{t}y^{l} .
\end{align}
For the values of $l$ with $l \geq m+1$, the binomial coefficient $\binom{m}{l-1}$ is $0$, which implies that we can increase the upper limit of the inner sum from $m+1$ to $t$ in equation \eqref{eq:ftml}. Then,
\begin{align}
\label{eq:ftml2}
f(x,y) &= d_r(1-\lambda)\sum_{t=1}^{\infty} \sum_{m=0}^{t-1}\sum_{l = 1}^{t} \binom{m}{l-1}\lambda^{m-l+1} \mu ^{l-1} x^{t}y^{l} \nonumber \\
&= \sum_{t=1}^{\infty} \sum_{l = 1}^{t} d_r(1-\lambda) \sum_{m=0}^{t-1} \binom{m}{l-1}\lambda^{m-l+1} \mu ^{l-1} x^{t}y^{l}.
\end{align}
Finally we can read off the value of $a_{t,l}$ from equation \eqref{eq:ftml2} as
\begin{align}
a_{t,l} = d_r(1-\lambda) \sum_{m=0}^{t-1} \binom{m}{l-1} \mu^{l-1} \lambda^{m-l+1} = d_rp_i \sum_{m=0}^{t-1} \binom{m}{l-1} (dp_i)^{l-1} (1-p_i)^{m-l+1}.
\end{align}
\end{proof}
\subsection{Proof of Corollary \ref{corollary:at}}
\label{sec:corollary:at}
We start by restating Corollary \ref{corollary:at} for convenience.
\begin{corollary}
\label{corollary:at_app}
In the RET$(p_i, d_r, d)$, let $a_{t}$ be the expectation of \eqref{eq:def_total}, as in Definition~\ref{def:total}. For $t\geq0$,
\begin{equation}
a_{t} = 1 + d_r\frac{(1-p_i+dp_i)^t - 1}{d-1}. \label{lemma:total_app}
\end{equation}
\end{corollary}
\begin{proof}
By using linearity of expectation, equation \eqref{eq:def_total} and Theorem \ref{theorem:a} we obtain:
\begin{align}
\label{eq3:1}
a_{t} &= \sum_{l = 0}^{+\infty} a_{t,l} \nonumber \\
&= 1 + \sum_{l = 1}^{+\infty} a_{t,l} \nonumber \\
&= 1 + d_rp_i\sum_{l = 1}^{t} \sum_{m = l-1}^{t-1} \binom{m}{l-1}(1-p_i)^{m-l+1}d^{l-1}p_i^{l-1}
\end{align}
Before we use binomial theorem, we need to swap the sums. Boundaries from \eqref{eq3:1} are equivalent to $t-1 \geq m \geq l-1 \geq 0$, so we can rewrite this as 2 conditions: $m+1 \geq l \geq 1$ and $t \geq m \geq 0$.
\begin{align}
a_{t,l}
&= 1 + d_rp_i\sum_{m = 0}^{t-1}\sum_{l = 1}^{m+1} \binom{m}{l-1}(1-p_i)^{m-l+1}d^{l-1}p_i^{l-1} \nonumber \\
&= 1 + d_rp_i\sum_{m = 0}^{t-1}\sum_{l = 0}^{m} \binom{m}{l}(1-p_i)^{m-l}d^{l}p_i^{l}
\end{align}
Finally, by applying the binomial theorem and summing the geometric series, we obtain the desired equation:
\begin{align}
a_{t,l}
&= 1 + d_rp_i\sum_{m = 0}^{t-1} (1-p_i+dp_i)^m \nonumber \\
&= 1 + d_r\frac{(1-p_i+dp_i)^t - 1}{d-1}.
\end{align}
\end{proof}
\subsection{Proof or Lemma \ref{lem:DET}}
\label{sec:lem:DET}
We restate Lemma \ref{lem:DET} here for convenience.
\begin{lemma}
\label{lem:DET_app}
Let us consider the stopped DET model with parameters $(c_{t,l}), p_a, p_h$, and let $h$ denote the first hospitalized node. Then
\begin{align}
\label{eq:DET_lemma_app}
\P(d(s,h) = l) = \sum_{t=0}^{+\infty} \frac{c_{t,l}-c_{t-1,l}}{c_t-c_{t-1}} (1-(1-p_a)p_h)^{c_{t-1}}\left(1-(1-(1-p_a)p_h)^{c_t-c_{t-1}}\right).
\end{align}
\end{lemma}
\begin{proof}
Recall that a node added at day $t$ is uniformly distributed among the $c_{t}-c_{t-1}>0$ nodes added that day, and that the number of nodes added to level $l$ is $c_{t,l}-c_{t-1,l}$ on day $t$. If we condition on the time of the first hospitalized case, denoted by $TI_h$, then
\begin{align}
\P(d(s,h) = l) &= \sum_{t=0}^{+\infty} \P(d(s,h)=l | TI_h = t) \P(TI_h = t) \nonumber \\
& = \sum_{t=0}^{+\infty} \frac{c_{t,l}-c_{t-1,l}}{c_t-c_{t-1}} \P(\textrm{node is not hosp})^{c_{t-1}} (1-\P(\textrm{node is not hosp })^{c_t-c_{t-1}})
\nonumber \\
& = \sum_{t=0}^{+\infty} \frac{c_{t,l}-c_{t-1,l}}{c_t-c_{t-1}} (1-(1-p_a)p_h)^{c_{t-1}}\left(1-(1-(1-p_a)p_h)^{c_t-c_{t-1}}\right).
\label{eq:DET_ctl}
\end{align}
\end{proof}
\section{Dynamic Message Passing for the DDE model}
\label{sec:DMP_all}
In this section, we explain how we derived and implemented the DMP equations for the DDE+HNM model. We start by reviewing the previous work on the DMP equations for the SIR model in Appendix \ref{sec:DMP_SIR}, and then we proceed to our derivations in Appendix \ref{sec:DMP_DDE}. In Appendix \ref{sec:DMP_feasible}, we explain how we find candidate (node,time) pairs for the DMP equations, and in Appendix \ref{sec:DMP_final} we conclude by combining Appendices \ref{sec:DMP_DDE} and \ref{sec:DMP_feasible} into a source-detection algorithm.
\subsection{DMP Equations for the SIR Model}
\label{sec:DMP_SIR}
The DMP equations were first derived by \cite{lokhov2014inferring} for the SIR model in the context of source detection. Their goal is to compute the marginal probabilities that node $i$ is in a given state at time $t$ (denoted by $P_S^{i}(t), P_I^{i}(t)$ and $P_R^{i}(t)$ for the susceptible, infected and recovered states, respectively), given initial conditions $P_S^{i}(t_0), P_I^{i}(t_0)$ and $P_R^{i}(t_0)$ at some initial time $t_0$. To solve this problem in tree networks, we may consider a dynamic programming approach, where we delete a node $i$, we compute the marginal probabilities of $P_S^{j}(t-1)$ for all neighbors $j$ of $i$ in the remaining subtrees, and use this information to compute $P_S^{i}(t)$ (as the marginals are independent in each of the subtrees conditioned on the state of $i$). The DMP equations make the dynamic programming intuition explicit. Originally, the DMP equations were developed for static networks, but since the generalization to time-varying networks is straightforward, and has already been foreshadowed in a similar heuristic algorithm \cite{jiang2016rumor}, we include it in this preliminary section. For time-varying networks, we define $N_i(t)$ as the set of neighbors of node $i$ in the time-window $[t,t+1)$.
To formalize the dynamic programming approach, \cite{lokhov2014inferring} introduces some new notation. Let $\lambda$ be the probability that an infectious node infects a susceptible neighbor, and let $\mu$ be the probability that an infectious node recovers. Let $D_{i}$ be the auxiliary dynamics, where node $i$ receives infection signals, but ignores them, and thus remains in the $S$ state at all times. Let $P_{S}^{j \rightarrow i}(t)$ be the probability that node $j$ is in the state $S$ at time $t$ in the dynamics $D_{i}$, and let $\theta^{k \rightarrow i}(t)$ be the probability that the infection signal has not been passed from node $k$ to node $i$ up to time $t$ in the dynamics $D_{i}$. Finally, let $\phi^{k \rightarrow i}(t)$ be the probability that the infection signal has not been passed from node $k$ to node $i$ up to time $t$, and that node $k$ is in the state $I$ at time $t$, in the dynamics $D_{i}$. With these definitions, the dynamic programming approach is formalized by the following equations for $t\ge t_0$:
\begin{align}
P_S^{i \rightarrow j}(t+1)&=P_S^{i}(t_0)\prod_{k\in N_i(t) \backslash j}\theta^{k \rightarrow i}(t+1), \label{eq:SIRequations:Ps}
\\
\theta^{k \rightarrow i}(t+1)-\theta^{k \rightarrow i}(t) &=
-\lambda\phi^{k \rightarrow i}(t), \label{eq:SIRequations:theta}
\\
\phi^{k \rightarrow i}(t)&=(1-\lambda)(1-\mu)\phi^{k \rightarrow i}(t-1) + \left(P_S^{k \rightarrow i}(t-1)-P_S^{k \rightarrow i}(t)\right). \label{eq:SIRequations:phi}
\end{align}
The marginal probabilities that node $i$ is in a given state at time $t$ are then given by
\begin{align}
& P_S^{i }(t+1)=P_S^{i}(t_0)\prod_{k\in N_i(t)}\theta^{k \rightarrow i}(t+1)\, ,\label{eq:SIRequations:S}
\\
& P_R^{i}(t+1)=P_R^{i}(t)+\mu P_{I}^{i}(t)\, ,\label{eq:SIRequations:R}
\\
& P_I^{i}(t+1)=1-P_S^{i}(t+1)-P_R^{i}(t+1)\, .\label{eq:SIRequations:I}
\end{align}
These equations are only exact on trees, but they can also be applied to networks with cycles as a heuristic approach. The heuristic gives good approximations to the true marginals if the network is at least locally tree-like \cite{karrer2010message}.
\subsection{DMP Equations for the DDE+HNM Model}
\label{sec:DMP_DDE}
There are several differences between the SIR model on locally tree-like networks and the DDE+HNM model (see Figure \ref{fig:mobility_models} (a)). First, the DDE model has additional compartments (exposed nodes, asymptomatic nodes), which motivates the introduction of several new variables. Let $\lambda_{(a)}$ (resp., $\lambda_{(s)}$) be the probability that an asymptomatic (resp., symptomatic) node infects a susceptible node. Let $\phi^{k \rightarrow i}(t)^{(a)}$ (resp., $\phi^{k \rightarrow i}(t)^{(s)}$) be the probability that the infection signal has not been passed from node $k$ to node $i$ up to time $t$, and that node $k$ is asymptomatic (resp., symptomatic) infectious at time $t$, in the dynamics $D_{i}$.
The second important difference is that in the DDE model, the transition times between different compartments are deterministic instead of following a geometric distribution as in the standard SIR model. While deterministic transition times sound simpler at first, it turns out that they make the DMP equations more complex, because the Markovian property that each marginal probability depends only on the previous timestep is lost if the transition times are larger than $1$. Recall that the times for the transitions $E \rightarrow I$ and $I \rightarrow R$ (with their default values) are $T_E=3$ and $T_I=14$.
Let us incorporate these two differences into equations \eqref{eq:SIRequations:Ps}--\eqref{eq:SIRequations:phi} to derive the DMP equations for the DDE model. Equation \eqref{eq:DDEequations:Ps} is essentially a copy of \eqref{eq:SIRequations:Ps}. Equation \eqref{eq:DDEequations:theta} follows equation \eqref{eq:SIRequations:theta}, but we incorporate the two different variants of infected (asymptomatic and symptomatic) patients with their respective infection probabilities $\lambda_{(a)}$ and $\lambda_{(s)}$. Equation \eqref{eq:DDEequations:Pr} is a new equation, which is necessary because recovery times are no longer geometric random variables; instead we need to check the probabilities of infection $T_E+T_I$ timesteps earlier than the current time $t$. Finally, equation \eqref{eq:DDEequations:phi_a} (resp., \eqref{eq:DDEequations:phi_s}) is the asymptomatic (resp., symptomatic) version of equation \eqref{eq:SIRequations:phi}, while also incorporating the deterministic time for the transition $E \rightarrow I$. For $t\ge t_0$, this yields equations
\begin{align}
P_S^{i \rightarrow j}(t+1)&=P_S^{i}(t_0)\prod_{k\in N_i(t) \backslash j}\theta^{k \rightarrow i}(t+1)=P_S^{i}(t_0)\frac{P_S^{i}(t+1)}{\theta^{j \rightarrow i}(t+1)}, \label{eq:DDEequations:Ps}
\\
\theta^{k \rightarrow i}(t+1)-\theta^{k \rightarrow i}(t) &=
-\lambda_{(a)} \phi^{k \rightarrow i}_{(a)}(t)-\lambda_{(s)}\phi^{k \rightarrow i}_{(s)}(t),
\label{eq:DDEequations:theta}
\\
P_R^{k \rightarrow i}(t) &=P_S^{k \rightarrow i}(t-T_E-T_I-1)-P_S^{k \rightarrow i}(t-T_E-T_I) \label{eq:DDEequations:Pr}
\\
\phi^{k \rightarrow i}_{(a)}(t)&=(1-\lambda_{(a)})(1-P_R^{k \rightarrow i}(t)) \phi^{k \rightarrow i}_{(a)}(t-1) \nonumber\\
& \qquad \qquad + p_a[P_S^{k \rightarrow i}(t-T_E-1)-P_S^{k \rightarrow i}(t-T_E)]. \label{eq:DDEequations:phi_a}
\\
\phi^{k \rightarrow i}_{(s)}(t)&=(1-\lambda_{(s)})(1-P_R^{k \rightarrow i}(t)) \phi^{k \rightarrow i}_{(s)}(t-1) \nonumber\\
& \qquad \qquad + (1-p_a)[P_S^{k \rightarrow i}(t-T_E-1)-P_S^{k \rightarrow i}(t-T_E)]. \label{eq:DDEequations:phi_s}
\end{align}
We note that for early values of $t$, equations \eqref{eq:DDEequations:Pr}--\eqref{eq:DDEequations:phi_s} depend on $P_S^{k \rightarrow i}$ before $t_0$, which we initialize to be 1 (all nodes are susceptible before the first node develops the infection). The marginal probability that node $i$ is susceptible at time $t$ is still computed by equation \eqref{eq:SIRequations:S} as before. Equations \eqref{eq:SIRequations:R}--\eqref{eq:SIRequations:I} do not apply anymore; we explain it in Appendix \ref{sec:DMP_final} how to take into account observations for nodes in the infectious compartments.
The third difference between the the SIR model on locally tree-like networks and the DDE+HNM model is that the HNM model contains many short cycles inside the households. Short cycles can cause unwanted feedback loops in the DMP equations where, loosely speaking, nodes are treated as if they could reinfect themselves. We solve this issue by modifying the underlying graph to be locally tree-like (only for the computation of the DMP equations). Specifically, we introduce a new central household-node for each household, and we replace the cliques inside the households by a star graph centered at this new household-node node. Introducing such a central household-node does of course alter epidemic process, in particular it makes household infections less independent and slower (all household infections need to pass through an extra node). To mitigate this issue, we assume that central household-nodes have $T_E=1$ and that they are infected with probability 1 by any node in the same household. We tested the validity of the resulting DMP equations against simulations of the epidemic progressions and we found the results to be quite accurate, in particular, more accurate than the version without the introduction of these central household-nodes.
Note that we derived the DMP equations for the DDE+HNM model, however, since (i) the compartments are the same, (ii) the equations support temporal networks, and (iii) we have separate infection probabilities $\lambda_{(a)}$ and $\lambda_{(s)}$ for asymptomatic and symptomatic nodes, our equations can also be applied to the DCS+TU model after a discretizing (rounding) the time observations.
Finally, we touch upon the computational complexity of computing the DMP equations. In principle, we need to update $O(dN)$ equations (for each edge) over $t_{\mathrm{max}}$ timesteps, where $t_{\mathrm{max}}$ is the maximum time during which the marginals can still change, which can be as large as $O(N)$. However, since we are only interested in computing the likelihood of the 5 earliest observations, $t_{\mathrm{max}}$ is typically quite low. Moreover, since we assume to be in an early stage of the epidemic, most of the equations remain unchanged. For better computational scalability, we only compute $P_S^{i}(t)$ and $\theta^{k \rightarrow i}(t)$ for nodes $k,i$ that have $P_I^{k \rightarrow i}(t)>0.01$, i.e., we only update nodes that are at least somewhat likely to have received the infection. Otherwise, we set $P_I^{k \rightarrow i}(t)=P_I^{k \rightarrow i}(t-1)$, $\theta^{k \rightarrow i}(t)=\theta^{k \rightarrow i}(t-1)$, and in the implementation we can perform these assignments implicitly using appropriate data structures. With these adjustments, the time-complexity of the algorithm becomes independent of $N$, but remains dependent on the network parameters, the epidemic parameters and the number of sensors in a non-trivial way.
\subsection{Feasible Source-time Pairs for Source Detection}
\label{sec:DMP_feasible}
In this section we explain how we implemented the feasible source identification algorithm, which was suggested as a preprocessing step for a method very similar to the DMP equations by \cite{jiang2016rumor}. Let us define the directed graph $G_2$ on (node,infecton\_time) pairs (we use ``nodes'' for the nodes of the original graph $G$ and ``pairs'' for the nodes of $G_2$), and draw an edge between two pairs $(v_1, t_1) \rightarrow (v_2, t_2)$ if $v_1$ and $v_2$ are in contact at $t_2$, and $t_2$ is in the interval $[t_1+T_E, t_1+T_E+T_I]$. Observe that in the DDE model there is an edge $(v_1, t_1) \rightarrow (v_2, t_2)$ if and only if $v_1$ becoming infected at time $t_1$ can infect $v_2$ at time $t_2$. The definition of $G_2$ is applicable to the DCS model as well after discretization (rounding), however, since the infection times are not deterministic anymore, not all possible infections $(v_1, t_1) \rightarrow (v_2, t_2)$ have a corresponding edge in~$G_2$.
Then, we perform a breadth-first search backwards on the directed edges of $G_2$, starting from each pair $(v_i, t_i-T_E-T_P)$, where $v_i$ is a symptomatic sensor node, and $t_i$ is the symptom onset time of $v_i$ (for the DCS model, we start from integer times in the $t_i-T_E-T_P \pm (\sigma_E+\sigma_P)$ interval to account for the randomness of the transition times). To limit the time complexity of the algorithm, we only consider the $k_1$ earliest observations, which means that we start $k_1$ breadth-first searches.
With this construction, each pair $(v,t)$ discovered by a breadth-first search started from $(v_i, t_i-T_E-T_P)$ could have caused the infection in $v_i$; we say that $(v,t)$ is an explanation for observation $i$. We perform the breath-first searches until we find $k_2$ pairs that explain all of the $k_1$ earliest observations. See the pseudocode in Algorithm \ref{alg:feasible}.
\begin{claim}
In the DDE model, Algorithm \ref{alg:feasible} with $\sigma_E=\sigma_P=0$ finds the $k_2$ feasible explanations with the latest starting time of the $k_1$ earliest symptomatic nodes.
\end{claim}
\begin{proof}
By construction, a source node $v$ that becomes infectious at time $t$ can cause an observation $(v_i, t_i)$ if and only if there is a directed path from $(v,t)$ to $(v_i, t_i-T_E-T_P)$. Therefore, the breadth-first search algorithm finds all of the closest feasible sources in time.
\end{proof}
\begin{algorithm}[h]
\SetKwFunction{isOddNumber}{isOddNumber}
\KwIn{
\begin{itemize}
\item The mean exposed time $T_E$ the mean pre-infectious time $T_P$, the mean infectious time $T_I$,\\ the std of the exposed time $\sigma_E$ and the std of the pre-infectious time $\sigma_P$
\item $F(v)_{min}$ and $F(v)_{max}$ returns the minimum and maximum times when $v$ could have been exposed based on all of its (possibly asymptomatic or negative) test results
\item $S(v)$ returns the time of symptom onset for a node $v$ tested positive symtomatic.
\item $N(v, [ t_{min}, t_{max} ])$ returns the set of neighbors of node $v$ in the interval $[t_{min}, t_{max}]$
\item A lower estimate of the time the source became infectious $t_{min}$
\item Integers $k_1,k_2$
\end{itemize}}
\KwOut{A list of at most $k_2$ tuples of node and time pairs that can explain the first $k_1$ symptomatic nodes}
$l \leftarrow \{\}$; \tcp*[f]{if the list $l[t]$ contains the tuple $(v,w)$, then the infection started at $w$ at time $t$ can explain $v$}\\
$D \leftarrow \{\}$; \tcp*[f]{if the list $D[w,t]$ contains the node $v$, then the infection started at $w$ at time $t$ can explain $v$}\\
$doneList \leftarrow []$\;
\For{$v \in SortIncreasingByValues(S)[0:k_1]$} {
$t'_{min} \leftarrow S(v)-(T_E+T_P)-(\sigma_E+\sigma_P)$\;
$t'_{max} \leftarrow S(v)-(T_E+T_P)+(\sigma_E+\sigma_P)$\;
\For{$t' \leftarrow t'_{min}$ to $t'_{max}$} {
$Append((v,v), l[t'])$\;
$Append(v, D[v,t'])$\;
\uIf{$Length(D[v,t'])=k_1$} {
$Append((v,t'),doneList)$
}
}
}
$t \leftarrow SortIncreasingByValues(S)[k_1-1]$\;
$stopCondition \leftarrow False$ \;
\While{ not $stopCondition$ and $t>t_{min}$} {
\For {$v,w \in l[t]$} {
\For{$u \in N(w, [t,t-1])$} {
$t'_{min} \leftarrow \max(F(u)_{min}, t-T_E-T_I)$\;
$t'_{max} \leftarrow \min(F(u)_{max}, t-T_E)$\;
\For{$t' \leftarrow t'_{max}$ to $t'_{min}$} {
$Append((v,u), l[t'])$\;
$ Append(v,D[(u,t')])$\;
\uIf{$Length(D[u,t'])=k_1$} {
$Append((u,t'),doneList)$
}
}
}
}
$doneList \leftarrow SortBySecondElement(doneList)$\;
\uIf{$Length(doneList\ge k2$) and $t-T_E \le doneList[k2][1]$} {
$stopCondition \leftarrow True$\;
}
\Else{
$t \leftarrow t-1$\;
}
}
\KwRet{$doneList$}
\caption{Feasible source identification (reverse dissemination \cite{jiang2016rumor})}
\label{alg:feasible}
\end{algorithm}
\subsection{Source Detection via Feasible Source Identification and DMP}
\label{sec:DMP_final}
In this section we explain how to combine Algorithm~\ref{alg:feasible} with the DMP equations derived in Appendix~\ref{sec:DMP_DDE}. See the pseudocode in Algorithm~\ref{alg:Sdd}.
We start by computing the DMP equations \eqref{eq:DDEequations:Ps}-\eqref{eq:DDEequations:phi_s} and \eqref{eq:SIRequations:Ps} for the $k_2$ tuples of node and time pairs that can explain the first $k_1$ symptomatic observations returned by Algorithm \ref{alg:feasible}. Next, our goal is to use these DMP equations to compute the likelihood of each of the $k_2$ tuples using the $k_1$ observations. Similarly to \cite{lokhov2014inferring}, we make the assumption that the first $k_1$ observations are independent, and we can compute the likelihood by multiplying their respective marginals together. For symptomatic observed nodes $v$, we know the time of symptom onset, which we denote by $S(v)$. Then, the marginal probability of $v$ developing symptoms exactly at time $t$ can be computed by taking the difference of $P_S^{v}(S(v)-T_P-T_E-1)$ and $P_S^{v}(S(v)-T_P-T_E)$ and multiplying the difference by $(1-p_a)$. In Algorithm \ref{alg:Sdd} we drop the multiplicative factor $(1-p_a)$ because it is present for all of the tuples, and it does not change the final order of their scores. For asymptomatic (resp., negative) observations, we only know that at the time of testing, denoted by $A(v)$ (resp., $NE(v)$), at least a time interval of length~$T_E$ has passed (resp., $T_E$ has not passed) since the time of infection. Therefore, dropping the $p_a$ factor similarly to the symptomatic case, we compute the marginal of asymptomatic observations as $1-P_S^{v}(A(v)-T_E)$, and we compute the marginal of negative observations as $P_S^{v}(NE(v)-T_E)$. Finally, the contributions of the observations are multiplied together for each of the $k_2$ tuples returned by Algorithm \ref{alg:feasible}, and the scores approximating the likelihoods are returned.
\begin{algorithm}
\SetKwFunction{isOddNumber}{isOddNumber}
\KwIn{
\begin{itemize}
\item The mean exposed time $T_E$, the mean pre-infectious time $T_P$, the mean infectious time $T_I
\item $S(v)$ returns the time of symptom onset for a node $v$ tested positive symtomatic.
\item $A(v)$ and $NE(v)$ return the time of asymptomatic and negative test results, respectively
\end{itemize}}
\KwOut{A dictionary $L$ of $k_2$ elements, which contains a score for each $(v,t)$ pair that explains the first $k_1$ observations. Higher scores signify higher confidence of being the source.}
$L \leftarrow \{\}$\;
$doneList \leftarrow \mathrm{Algorihtm \ \ref{alg:feasible}}(k_1,k_2)$\;
\For{$v,t_0 \in doneList$} {
$P_S \leftarrow$ eq. \eqref{eq:SIRequations:Ps} based on DMP eq. \eqref{eq:DDEequations:Ps}-\eqref{eq:DDEequations:phi_s} with $P_S^v(t_0)=0$, and $P_S^w(t_0)=1$ for all $w \ne v$\;
$L[v,t_0] \leftarrow 1$\;
\For{$w \in S$} {
$L[v,t_0] \leftarrow L[v,t_0]\cdot (P_S^{v}(S(v)-T_P-T_E-1)-P_S^{v}(S(v)-T_P-T_E))$\;
}
\uIf{{$w \in A$} }{
$L[v,t_0] \leftarrow L[v,t_0]\cdot (1-P_S^{v}(A(v)-T_E))$\;
}
\For{$w\in NE$} {
$L[v,t_0] \leftarrow L[v,t_0]\cdot P_S^{v}(NE(v)-T_E)$\;
}
}
\KwRet{$L$}
\caption{Source detection via DMP}
\label{alg:Sdd}
\end{algorithm}
|
2024-02-18T23:40:24.640Z
|
2021-12-30T02:26:54.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14530","language":"en","timestamp":1640831214000,"url":"https:\/\/arxiv.org\/abs\/2112.14530","yymm":"2112"}
|
proofpile-arXiv_000-10056
|
{"provenance":"002.jsonl.gz:10057"}
| null | null |
\section{Introduction} \label{sec:intro}
It has been of crucial importance to find useful applications of quantum computers
specially since the realization of real programable quantum devices.
Considering that currently available quantum devices are prone to noise and decoherence,
it is highly desirable to find applications that can work effectively with a less number of quantum gates and qubits.
Under these conditions,
one of the promising and appealing approaches is based on variational quantum algorithms~\cite{Cerezo2021b}
because it can be applied to a wide range of applications including quantum chemistry~\cite{Peruzzo2014,Yung2014,OMalley2016,Kandala2017,Yaangchao2017,Romero2018,Evangelista2019,McArdle2020}
and quantum machine learning~\cite{Schuld2015,WittekBook,Adcock2015,Biamonte2017,SchuldBook,Perdomo2018,Mitarai2018}.
In variational quantum algorithms, a quantum circuit is composed of parametrized quantum gates
and these parameters are optimized classically on a classical computer so as to minimize or maximize a cost function
by using standard optimization techniques such as a natural gradient descent method~\cite{Amari1998,Seki2020vqe,Stokes2020}
and a sequential optimization technique~\cite{Nakanishi2020}.
In this context, there is an issue, known as barren plateau (BP) phenomena, where
the partial derivatives of the cost function vanish exponentially with increasing
the number of qubits and quantum gates~\cite{McClean2018,Cerezo2021,Wang2021,Marrero2021}.
The basic tool used to discuss the BP phenomena is the unitary $t$-design~\cite{Dankert2009}
related to the representability of a quantum circuit via the Haar measure~\cite{SimSukin2019,Hubregtsen2021}.
The theory of the BP phenomenon~\cite{McClean2018} claims that a quantum circuit
which shows unitary 2-design exhibits the BP phenomena.
Generally, one tends to increase the number of quantum gates to represent a quantum state that one intends to prepare
in the first place.
However, such a quantum circuit with enhanced capability of representation
can easily fall into the class of unitary operators belonging to the unitary 2-design~\cite{Harrow2009,Diniz2011},
thus suggesting the emergence of the BP phenomena~\cite{Holmes2021}.
Several routes have been discussed to alleviate and even avoid the BP phenomena.
A simplest way is to select a cost function appropriately because the BP phenomena is
cost function dependent~\cite{Cerezo2021}.
The importance of properly setting the initial variational parameters has also been pointed out~\cite{Mitarai2020qc}.
Another route to address the BP issue is to construct an appropriate quantum circuit.
For instance, it has been reported that a quantum circuit with a structure like
the multi-scale entanglement renormalization group ansatz (MERA)~\cite{Vidal2007,Vidal2008,Evenbly2009},
known also as quantum convolutional neural network~\cite{Cong2019}, can avoid the BP phenomena~\cite{Pesah2021}.
However, the tensor network structure of MERA is originally constructed to capture the quantum entanglement of a
particular quantum state, e.g., in a one-dimensional critical system,
and therefore it is not obvious at all and is probably not appropriate that the MERA-type quantum circuit can be
applied to general problems.
An alternative approach in this regard is a method such as adapted variational quantum eigensolver (ADAPT-VQE)~\cite{Grimsley2019,Tang2021,Liu2021,Yao2021,FengZhang2021},
where an appropriate quantum circuit is automatically generated by selecting quantum gates sequentially among a predetermined
set of quantum gates accordingly to a problem to be sloved.
Following a similar strategy of the ADAPT-VQE,
here in this paper, we propose a method that constructs an appropriate quantum circuit automatically,
named automatic quantum circuit encoding (AQCE). The AQCE algorithm proposed here is
to construct a quantum circuit $\hat{\mathcal{C}}$ that approximates a given quantum state $\vert \Psi \rangle$ such that
$\vert \Psi \rangle \approx \hat{\mathcal{C}} \vert 0 \rangle$ with controlled accuracy.
Here, $\hat{\mathcal{C}}$ is composed of a standard set of
quantum gates acting on up to two qubits and $|0\rangle$ is a trivial product state in the computational basis.
The algorithm is not based on a parametrized circuit ansatz but determines sequentially optimal two-qubit unitary operators,
including an optimal location of qubits on which each unitary operator acts, by maximizing the fidelity with
a technique inspired by the optimization algorithm in the tensor-network method~\cite{Evenbly2009}.
A standard set of quantum gates is assigned algebraically to these optimally determined unitary operators.
Therefore, the AQCE algorithm does not requires any derivatives of a cost function.
With noiseless numerical simulations,
we demonstrate the AQCE algorithm to encode a ground state of quantum many-body systems
including the spin-1/2 isotropic antiferromagnetic Heisenberg model and the spin-1/2 XY model.
We also compare the results with the quantum circuit encoding of the same quantum state onto a quantum circuit
in a given circuit structure (i.e, a quantum circuit ansatz) such as the Trotter-like~\cite{Trotter1959,Suzuki1976,Lloyd1996} and MERA-like circuit structures.
Furthermore, we apply this algorithm to encode classical data that is represented as a quantum state
via the amplitude encoding~\cite{SchuldBook}, demonstrating a potential near-term application for
a quantum state preparation of input data in quantum machine learning.
In addition, we employ a real quantum device provided by IBM Quantum~\cite{IBM}
to demonstrate experimentally that a quantum circuit generated
by the AQCE algorithm can indeed represent the original quantum state reasonably.
The rest of this paper is organized as follows.
The AQCE algorithm is first introduced in Sec.~\ref{sec:encode}.
The performance of this algorithm
is then demonstrated by numerical simulations in Sec.~\ref{sec:benchmark}.
The method is first applied to encode the ground states of the spin-1/2 Heisenberg models in Sec.~\ref{sec:res:state},
and the results are compared with those for the quantum circuit encoding of the same quantum states
onto quantum circuits with fixed circuit structures in Sec.~\ref{sec:res:comp}.
The application of the AQCE algorithm to classical data such as a classical image
represented by a quantum state via the amplitude encoding is also discussed in Sec.~\ref{sec:res:data}.
Moreover, the AQCE algorithm is partially demonstrated experimentally with a real quantum device in Sec.~\ref{sec:exp}.
Finally, the paper is concluded with a brief summary in Sec.~\ref{sec:summary}.
The details of the gate assignment of unitary operators acting on a single qubit and on two qubits are
described in Appendix~\ref{app:assign} and Appendix~\ref{app:phase}.
\section{\label{sec:encode}Quantum circuit encoding algorithm}
We first introduce the fidelity as an objective function for quantum circuit encoding in Sec.~\ref{sec:encode:obj}.
We then describe how to determine a unitary matrix of a quantum gate operation
by maximizing the objective function in Sec.~\ref{sec:encode:unitary}
and briefly explain how to assign an arbitrary unitary matrix to a standard set of single- and two-qubit
quantum gates in Sec.~\ref{sec:encode:assign}.
Based on these techniques, we introduce a prototype of the algorithm for the quantum circuit encoding in Sec.~\ref{sec:encode:algorithm}.
We also explain how to evaluate the fidelity tensor elements on a quantum computer in Sec.~\ref{sec:encode:imp}.
Although the encoding algorithm can be applied in any cases,
it might meet some difficulty when the fidelity tensor is essentially zero due to a particular symmetry reason.
We discuss this issue and introduce an alternative approach to overcome this problem in Sec.~\ref{sec:encode:init}.
This approach can be used for the initialization of the quantum circuitencoding.
Combining with these methods in Secs.~\ref{sec:encode:algorithm} and \ref{sec:encode:init},
we finally introduce an algorithm, i.e., the AQCE algorithm, to construct a quantum circuit automatically
in Sec.~\ref{sec:encode:aqce}.
\subsection{\label{sec:encode:obj}Objective for quantum circuit encoding}
We consider a quantum state defined on $L$ qubits that are enumerated as $\mathbb{L} = \{ 1,2, \Compactcdots, i, \Compactcdots, L \}$.
Let $\hat{X}_i$, $\hat{Y}_i$, and $\hat{Z}_i$
denote the $x$, $y$, and $z$ components of the Pauli operators, respectively, acting on qubit $i$.
We also introduce the notation $\hat{I}_i$ for representing the identity operator on qubit $i$.
Let $\vert \sigma_i \rangle_i=\vert 0 \rangle_i$ and $\vert 1 \rangle_i$ denote the eigenstates of the Pauli operator $\hat{Z}_i$ at
qubit $i$, i.e.,
$\hat{Z}_i \vert 0 \rangle_i = \vert 0 \rangle_i$ and $\hat{Z}_i \vert 1 \rangle = - \vert 1 \rangle_i$.
The Hilbert space $\mathbb{H}_L$ on the $L$-qubit system $\mathbb{L}$
is spanned by the basis $\{ \vert \sigma_1 \sigma_2 \Compactcdots \sigma_L \rangle \}$, where
$\vert \sigma_1 \sigma_2 \Compactcdots \sigma_L \rangle = \underset{i=1}{\overset{L}{\otimes}} \vert \sigma_i \rangle_i$.
We can label the state $\vert \sigma_1 \sigma_2 \Compactcdots \sigma_L \rangle$
by introducing the integer number
\begin{equation}
n = \sum_{i=1}^L 2^{i-1} \sigma_i \label{eq:computational:label}
\end{equation}
as $\{ \vert n \rangle = \vert \sigma_1 \sigma_2 \Compactcdots \sigma_L \rangle \}_{n=0}^{2^L-1}$
in the Hilbert space $\mathbb{H}_L = \text{span}\{ \vert \sigma_1 \sigma_2 \Compactcdots \sigma_L \rangle \}$.
Let $\vert \Psi \rangle$ be an arbitrary quantum state defined on $\mathbb{L}$ and let us assume that $\vert \Psi \rangle$ is normalized.
In addition, we can assume that $\vert \Psi \rangle$ is given generally
by a linear combination of many different quantum circuit states, i.e.,
\begin{equation}
\vert \Psi \rangle = \sum_{\gamma=1}^{\Gamma} \chi_{\gamma} \vert \psi^{(\gamma)} \rangle
\label{eq:psi:lc}
\end{equation}
with
\begin{equation}
\vert \psi^{(\gamma)} \rangle = \hat{\psi}^{(\gamma)} \vert 0 \rangle,
\label{eq:psi_lc}
\end{equation}
where $\chi_{\gamma}$ and $\hat{\psi}^{(\gamma)}$ with $\gamma=1,2,\Compactcdots,\Gamma$ are
complex-valued coefficients and quantum circuits, respectively.
Although $\langle \psi^{(\gamma)} | \psi^{(\gamma)} \rangle=1$, i.e., $\hat{\psi}^{(\gamma)}$ being unitary,
here we do not assume that the states $ \vert \psi^{(\gamma)} \rangle$ with different values of $\gamma$ are mutually orthogonal.
Note that Eq.~(\ref{eq:psi:lc}) may include the simplest and most extreme case
where the circuits $\hat{\psi}_{\gamma}$ are composed simply of
products of Pauli-X operators
\begin{equation}
\hat{\mathcal{P}}_{ \{ \sigma_i \}_{i=1}^L } \equiv \prod_{i=1}^L \hat{X}_{i}^{\sigma_i},
\end{equation}
i.e., $\hat{\mathcal{P}}_{ \{ \sigma_i \}_{i=1}^L} \vert 0 \rangle = \vert \sigma_1 \sigma_2 \Compactcdots \sigma_L \rangle$,
as in the case of a quantum state representing a classical image via the amplitude encoding discussed in Sec.~\ref{sec:res:data}.
The objective here is to represent $\vert \Psi \rangle$ by a quantum circuit $\hat{\mathcal{C}}|0\rangle$
that is a priori unknown.
The algorithm proposed here constructs a quantum circuit $\hat{\mathcal{C}}$ that approximately
represents $\vert \Psi \rangle\approx\hat{\mathcal{C}} \vert 0\rangle$ with controlled accuracy.
We should note that this can be considered
as a special case in the variational quantum state eigensolver for a density operator recently reported in Ref.~\cite{Cerezo2020a},
although the optimization method introduced here is different, as it will be described below.
For this purpose, we consider as an objective function to be maximized
the absolute value of the overlap $F$ between $\vert \Psi \rangle$ and $\hat{\mathcal{C}} \vert 0 \rangle$, i.e.,
\begin{equation}
F = \langle 0 \vert \hat{\mathcal{C}}^{\dagger} \vert \Psi \rangle.
\end{equation}
Assuming that the quantum circuit $\hat{\mathcal{C}}$ is composed of a product of unitary operators
$\hat{\mathcal{U}}_{m}$, i.e.,
\begin{equation}
\hat{\mathcal{C}}^{\dagger} = \prod_{m=1}^{M} \hat{\mathcal{U}}_{m}^{\dagger}
= \hat{\mathcal{U}}_1^{\dagger} \hat{\mathcal{U}}_{2}^{\dagger} \Compactcdots \hat{\mathcal{U}}_M^{\dagger},
\end{equation}
$F$ can be expressed as
\begin{equation}
F_m = \langle \Phi_{m-1} \vert \hat{\mathcal{U}}_m^{\dagger} \vert \Psi_{m+1} \rangle,
\end{equation}
where we have introduced the subscript $m$ explicitly for the reason clarified below
and the quantum states $\vert \Psi_m \rangle$ and
$\langle \Phi_m \vert$ defined respectively as
\begin{equation}
\vert \Psi_m \rangle = \prod_{k=m}^M \hat{\mathcal{U}}_k^{\dagger} \vert \Psi \rangle
= \hat{\mathcal{U}}_m^\dag \hat{\mathcal{U}}_{m+1}^\dag \Compactcdots \hat{\mathcal{U}}_M^\dag \vert \Psi \rangle
\label{eq:ket:m}
\end{equation}
and
\begin{equation}
\langle \Phi_m \vert = \langle 0 \vert \prod_{k=1}^{m} \hat{\mathcal{U}}_k^{\dagger}
= \langle 0 \vert
\hat{\mathcal{U}}_1^{\dagger} \hat{\mathcal{U}}_{2}^{\dagger} \Compactcdots \hat{\mathcal{U}}_m^{\dagger}.
\label{eq:bra:m}
\end{equation}
\subsection{Determination of unitary operators} \label{sec:encode:unitary}
In order to appropriately determine each unitary operator $\hat{\mathcal{U}}_m$ composing the quantum circuit $\hat{\mathcal{C}}$,
here we propose a method inspired by a tensor-network algorithm~\cite{Evenbly2009} by
introducing a fidelity tensor operator.
Let $\mathbb{I}_m = \{ i_1, i_2, \Compactcdots, i_K \}$ be a subsystem in the total qubit system $\mathbb{L}$
and assume that an unitary operator $\hat{\mathcal{U}}_m$ is defined on the subsystem $\mathbb{I}_m$.
By labeling the basis states $\{ \vert n \rangle = \vert \sigma_{i_1} \sigma_{i_2} \Compactcdots \sigma_{i_K} \rangle \}_{n=0}^{2^K-1}$
on the subsystem $\mathbb{I}_m$,
the unitary operator $\hat{\mathcal{U}}_m$ can be represented generally as
\begin{equation}
\hat{\mathcal{U}}_m
= \sum_{n=0}^{2^K-1} \sum_{n^{\prime}=0}^{2^K-1} \vert n \rangle [ {\bm U}_m ]_{nn^{\prime}} \langle n^{\prime} \vert,
\end{equation}
where ${\bm U}_m$ is a $2^K \times 2^K$ unitary matrix and $[ {\bm A} ]_{nn^{\prime}}$ denotes a matrix element in
the $n$th row and the $n^{\prime}$th column of matrix ${\bm A}$.
We shall now introduce the following fidelity tensor operator $\hat{\mathcal{F}}_m$:
\begin{equation}
\hat{\mathcal{F}}_m = {\rm Tr}_{ \bar{\mathbb{I}}_m } \left[ \vert \Psi_{m+1} \rangle \langle \Phi_{m-1} \vert \right],
\label{eq:op:fm}
\end{equation}
where $\bar{\mathbb{I}}_m$ is the complement of the subsystem $\mathbb{I}_m$ in $\mathbb{L}$
and ${\rm Tr}_{ \mathbb{A} }\hat{\mathcal{O}}$ indicates the trace of operator $\hat{\mathcal{O}}$
over the Hilbert space spanned by the basis states for subsystem $\mathbb{A} = \{ i_1 ,i_2, \Compactcdots, i_A \} \subset \mathbb{L}$, i.e.,
\begin{equation}
{\rm Tr}_{\mathbb{A}} [ \hat{\mathcal{O}} ] =
\sum_{\sigma_{i_1}=0}^1 \sum_{\sigma_{i_2}=0}^1 \Compactcdots
\sum_{\sigma_{i_A}=0}^1 \langle \sigma_{i_1} \sigma_{i_2} \Compactcdots \sigma_{i_A} \vert
\hat{\mathcal{O}}
\vert \sigma_{i_1} \sigma_{i_2} \Compactcdots \sigma_{i_A} \rangle,
\end{equation}
with $\vert \sigma_{i_1} \Compactcdots \sigma_{i_A} \rangle = \vert \sigma_{i_1} \rangle_{i_1} \vert \sigma_{i_2} \rangle_{i_2} \Compactcdots\vert \sigma_{i_A} \rangle_{i_A}$.
Since $\hat{\mathcal{F}}_m$ is an operator defined on the Hilbert space spanned by the basis states for the subsystem
$\mathbb{I}_m$, one can represent the operator $\hat{\mathcal{F}}_m$ in the matrix form as
\begin{equation}
\hat{\mathcal{F}}_m = \sum_{n=0}^{2^K-1} \sum_{n^{\prime}=0}^{2^K-1} \vert n \rangle [ {\bm F}_m ]_{nn^{\prime}} \langle n^{\prime} \vert.
\label{eq:mat:fm}
\end{equation}
We can now readily find that
\begin{equation}
{\rm Tr}_{ \mathbb{I}_m } [ \hat{\mathcal{F}}_m \hat{\mathcal{U}}_m^{\dagger} ] = \langle \Phi_{m-1} \vert \hat{\mathcal{U}}_m^{\dagger} \vert \Psi_{m+1} \rangle = F_m.
\end{equation}
We also find that
\begin{equation}
{\rm Tr}_{ \mathbb{I}_m } [ \hat{\mathcal{F}}_m \hat{\mathcal{U}}_m^{\dagger} ] = {\rm tr} [ {\bm F}_m {\bm U}_m^{\dagger} ],
\end{equation}
where ${\rm tr} {\bm A}$ indicates the trace of matrix ${\bm A}$.
Note that ${\bm F}_m$ is a $2^K\times 2^K$ matrix and is neither Hermitian nor unitary in general.
Let us now perform the singular-value decomposition (SVD) for ${\bm F}_m$ as
${\bm F}_m = {\bm X} {\bm D} {\bm Y}$, where ${\bm X}$ and ${\bm Y}$ are $2^{K} \times 2^{K}$ unitary matrices,
and ${\bm D}$ is a non-negative real diagonal matrix with its diagonal elements being the singular values $d_n$ ($n=0,1,2,\Compactcdots,2^K-1$)
of matrix ${\bm F}_m$. Note that the $m$ dependence of these matrices
${\bm X}$, ${\bm Y}$, and ${\bm D}$ is implicitly assumed.
We then find that
\begin{equation}
F_m = {\rm tr} [ {\bm X} {\bm D} {\bm Y} {\bm U}_m^{\dagger} ] = {\rm tr} [ {\bm D} {\bm Z} ]
= \sum_{n=0}^{2^K-1} d_n [ {\bm Z} ]_{nn},
\label{eq:fm_svd}
\end{equation}
where ${\bm Z} = {\bm Y} {\bm U}_m^{\dagger} {\bm X}$ is a unitary matrix.
The absolute value of $F_m$ thus satisfies that
\begin{equation}
\vert F_m \vert = \left| \sum_{n=0}^{2^K-1} d_n [ {\bm Z} ]_{nn} \right| \le \sum_{n=0}^{2^K-1} d_n \left| [ {\bm Z} ]_{nn} \right| .
\label{eq:abF}
\end{equation}
The equality in Eq.~(\ref{eq:abF}) holds if and only if $\text{arg} \left[[ {\bm Z} ]_{nn} \right]$ is the same for all $n$,
where $\text{arg}\left[ {\bm c} \right]$ denotes the phase of complex number ${\bm c}$.
Noticing also that $\sum_{n'}\left| [ {\bm Z} ]_{nn'} \right|^2 = \sum_{n'}\left| [ {\bm Z} ]_{n'n} \right|^2 =1$
because ${\bm Z}$ is unitary,
the absolute value of $F_m$ is thus maximized by choosing $[ {\bm Z} ]_{nn'} = \delta_{nn'}$~\cite{phase}, i.e.,
${\bm Z} = {\bm I}$, where ${\bm I}$ is the identity matrix.
Hence, the unitary matrix ${\bm U}_m$ that maximizes the absolute value of $F_m$ is obtained as
\begin{equation}
{\bm U}_m = {\bm X} {\bm Y}
\label{eq:optU}
\end{equation}
and therefore we can determine the optimal unitary operator $\hat{\mathcal{U}}_m$.
Three remarks are in order. First, as already stated above, a similar idea is used in the optimization of tensor
network states~\cite{Evenbly2009}.
Second, although the fidelity tensor $\hat{\mathcal{F}}_m$ can be defined for a subsystem containing many qubits,
we focus mostly on the two-qubit case in this paper.
This is simply because the assignment of elementary quantum gates for an optimal unitary operator determined in
Eq.~(\ref{eq:optU}) can be made rather simply, as described in the next section.
Third, the fidelity $F_m$ and the fidelity tensor operator $\hat{\mathcal{F}}_m$ can be more explicitly expressed for
the case when the state $|\Psi\rangle$ is given by a linear combination of several quantum states as in Eq.~(\ref{eq:psi:lc}), i.e.,
\begin{equation}
F_m = \sum_{\gamma=1}^\Gamma \chi_\gamma f_m^{(\gamma)}
\label{eq:fm_lc}
\end{equation}
and
\begin{equation}
\hat{\mathcal{F}}_m = \sum_{\gamma=1}^\Gamma \chi_\gamma \hat{f}_m^{(\gamma)},
\end{equation}
where
\begin{equation}
f_m^{(\gamma)} = {\rm Tr}_{ \mathbb{I}_m } [ \hat{f}_m^{(\gamma)} \hat{\mathcal{U}}_m^{\dagger} ]
\end{equation}
and
\begin{equation}
\hat{f}_m^{(\gamma)} = {\rm Tr}_{\bar{\mathbb{I}}_m} [ \vert \psi_{m+1}^{(\gamma)} \rangle \langle \Phi_{m-1} \vert ]
\label{eq:f:lc}
\end{equation}
with
\begin{equation}
\vert \psi_{m+1}^{(\gamma)}\rangle = \hat{\mathcal{U}}_m^\dag \hat{\mathcal{U}}_{m+1}^\dag \Compactcdots \hat{\mathcal{U}}_M^\dag
\vert \psi^{(\gamma)} \rangle.
\label{eq:phi_gamma}
\end{equation}
The optimal unitary operator $ \hat{\mathcal{U}}_m$ that maximizes the absolute value of $F_m$
is still determined by Eqs.~(\ref{eq:fm_svd})-(\ref{eq:optU}).
\subsection{\label{sec:encode:assign}Assignment of quantum gates for a general unitary operator}
Once we obtain the matrix representation ${\bm U}_m$ for the unitary operator $\hat{\mathcal{U}}_m$ in Eq.~(\ref{eq:optU}),
we have to assign a standard set of elementary quantum gates to this operator $\hat{\mathcal{U}}_m$.
It is well known that
any unitary operator can be compiled as a product of two-qubit quantum gates
by using the method that proves the universality of the quantum computation~\cite{NielsenChuangBook}.
However, this method of decomposing a unitary operator acting on $K$ qubits generates an exponentially large number of
elementary single- and two-qubit quantum gates with $K$.
Therefore, it is not practical for our purpose.
In contrast, focusing on an unitary operator acting on two qubits, there exists an optimal form decomposing it into elementary quantum gates~\cite{Kraus2001},
which can be determined from the matrix representation ${\bm U}_m$. Here we briefly outline this procedure.
As proved in Ref.~\onlinecite{Kraus2001}, any two-qubit unitary operator $\hat{\mathcal{U}}$ acting on qubits $i$ and $j$
can be decomposed into a product of elementary gate operations in the following canonical form [also see Fig.~\ref{fig:twogate}(a)]:
\begin{equation}
\hat{\mathcal{U}} = {\rm e}^{-{\rm i}\alpha_0} \hat{\mathcal{R}}_i^{\prime} \hat{\mathcal{R}}_j^{\prime}
\hat{\mathcal{D}} \hat{\mathcal{R}}_i \hat{\mathcal{R}}_j,
\label{eq:twogate:physical}
\end{equation}
where $\alpha_0$ is an overall phase factor, not relevant for the assignment,
$\hat{\mathcal{R}}_q^{\prime}$ and $\hat{\mathcal{R}}_q$ are single-qubit Euler rotations acting on qubit $q\,(=i,j)$
given by
\begin{equation}
\hat{\mathcal{R}}_q =
{\rm e}^{-{\rm i}\xi_1^q \hat{Z}_q/2}
{\rm e}^{-{\rm i}\xi_2^q \hat{Y}_q/2}
{\rm e}^{-{\rm i}\xi_3^q \hat{Z}_q/2}
\end{equation}
and
\begin{equation}
\hat{\mathcal{R}}_q^{\prime} =
{\rm e}^{-{\rm i}\zeta_1^q \hat{Z}_q/2}
{\rm e}^{-{\rm i}\zeta_2^q \hat{Y}_q/2}
{\rm e}^{-{\rm i}\zeta_3^q \hat{Z}_q/2},
\end{equation}
and $\hat{\mathcal{D}}$ is a two-qubit entangled operator
\begin{equation}
\hat{\mathcal{D}} = {\rm e}^ {-{\rm i} ( \alpha_1 \hat{X}_i \hat{X}_j
+ \alpha_2 \hat{Y}_i \hat{Y}_j
+ \alpha_3 \hat{Z}_i \hat{Z}_j )}.
\label{eq:exgate}
\end{equation}
By following the proof of Eq.~(\ref{eq:twogate:physical}) in Ref.~\onlinecite{Kraus2001},
the parameters $\xi_q^k$ ($k=1,2,3$, $q=i,j$) and $\zeta_q^k$ ($k=1,2,3$, $q=i,j$) for the Euler rotaions
and $\alpha_k$ ($k=1,2,3$) for $\hat{\mathcal{D}}$ as well as $\alpha_0$ are determined algebraically.
The details are provided in Appendix~\ref{app:assign} and Appendix~\ref{app:phase}.
The total number of parameters in the right hand side of Eq.~(\ref{eq:twogate:physical}) is 16
and is identical to the number of free real parameters in a general U(4) matrix.
\begin{figure}
\includegraphics[width=\hsize]{twogate.pdf}
\caption{General form of a two-qubit unitary gate acting on qubits $i$ and $j$.
(a) Any unitary operator $\hat{\mathcal U}$ is decomposed into four single-qubit Euler rotations and two-qubit operator
$\hat{\mathcal D}$.
(b) $\hat{\mathcal D}$ is further decomposed into a product of the most standard quantum gates, including
three controlled NOT gates.
(c) Decomposition of a unitary operator $\hat{\mathcal U}$ into a standard set of the most elementary quantum gates.
Each single-qubit operation is
defined in the text.
}
\label{fig:twogate}
\end{figure}
Next, as shown in Fig.~\ref{fig:twogate}(b),
$\hat{\mathcal{D}}$ can be represented by a product of the most standard quantum gates~\cite{Vidal2004,Mark2008}
\begin{equation}
\hat{\mathcal{D}} = \hat{w}_i \hat{w}_j^{\dagger}
\hat{\rm C}_i (\hat{X}_j) \hat{u}_i^3 \hat{v}_j^3
\hat{\rm C}_i (\hat{X}_j) \hat{u}_i^2 \hat{v}_j^2
\hat{\rm C}_i (\hat{X}_j).
\label{eq:exgate:decomposed}
\end{equation}
Here, $\hat{\rm C}_i (\hat{X}_j)$ denotes the controlled NOT gate
where the NOT operation acting on the $j$th qubit is controlled by the $i$th qubit,
and other gates are single qubit gates given by
\begin{align}
& \hat{w}_i = {\rm e}^{{\rm i}\pi \hat{X}_i/4},\
\hat{w}_j^{\dagger} = {\rm e}^{-{\rm i}\pi \hat{X}_j/4},\\
& \hat{u}_i^3 = \hat{H}_i \hat{S}_i,\
\hat{v}_j^3 = {\rm e}^{-{\rm i}\alpha_2 \hat{Z}_j}, \\
& \hat{u}_i^2 = \hat{H}_i {\rm e}^{{\rm i}\alpha_1 \hat{X}_i},\
\hat{v}_j^2 = {\rm e}^{{\rm i}\alpha_3 \hat{Z}_j}
\end{align}
with $\hat{H}_i$ and $\hat{S}_i$ being the Hadamard and shift gates, respectively.
The matrix representations ${\bm H}$ and ${\bm S}$ for these quantum gates
$\hat{H}_i$ and $\hat{S}_i$ in the computational basis are given respectively by
\begin{equation}
{\bm H} = \frac{1}{\sqrt{2}} \left(
\begin{array}{cc}
1 & 1 \\
1 & -1 \\
\end{array}
\right),\
{\bm S} = \left(
\begin{array}{cc}
1 & 0 \\
0 & {\rm i} \\
\end{array}
\right).
\end{equation}
Inserting the expression of Eq.~(\ref{eq:exgate:decomposed}) into Eq.~(\ref{eq:twogate:physical}),
we obtain that
\begin{equation}
\hat{\mathcal{U}} =
\hat{u}_i^4 \hat{v}_j^4
\hat{C}_i(\hat{X}_j)
\hat{u}_i^3 \hat{v}_j^3
\hat{C}_i(\hat{X}_j)
\hat{u}_i^2 \hat{v}_j^2
\hat{C}_i(\hat{X}_j)
\hat{u}_i^1 \hat{v}_j^1,
\end{equation}
where
\begin{align}
& \hat{u}_i^1 = \hat{\mathcal{R}}_i,\
\hat{v}_j^1 = \hat{\mathcal{R}}_j,\\
& \hat{u}_i^4 = \hat{\mathcal{R}}_i^{\prime} \hat{w}_i,\
\hat{v}_j^4 = \hat{\mathcal{R}}_j^{\prime} \hat{w}_j^{\dag}.
\end{align}
This is also schematically shown in Fig.~\ref{fig:twogate}(c).
Note that once the matrix representation for a single-qubit unitary operator is obtained,
we can reparametrize any sequential product of single-qubit operators by using an overall phase factor and
a single Euler rotation algebraically (see Appendix~\ref{app:phase}),
suggesting that all $\hat{u}_i^{k}$ ($k=1,2,3,4$) and $\hat{v}_j^{k}$ ($k=1,2,3,4$)
can be represented as single Euler rotations (apart from an overall phase factor).
We should also note that when a matrix representation ${\bm U}$ of an unitary operator $\hat{\mathcal{U}}$ happens to
be O(4), instead of U(4), the corresponding two-qubit operator for $\hat{\mathcal{U}}$ in Fig.~\ref{fig:twogate}(c) can be constructed
with two controlled NOT gates~\cite{Vatan2004}.
\subsection{Quantum circuit encoding algorithm} \label{sec:encode:algorithm}
Using the procedures described above,
we can now introduce an algorithm to construct a quantum circuit $\hat{\mathcal{C}}$ that approximately represents a given
quantum state $\vert \Psi \rangle$.
Without loss of generality, let us assume that
$\hat{\mathcal{C}}^{\dagger} = \prod_{m=1}^M \hat{\mathcal{U}}_m^{\dagger}$ is given.
For example, we set $\hat{\mathcal{U}}_m = \hat{I}$ for all $m$ as the initial condition,
where $\hat{I}$ is the identity operator of the subspace defining $\hat{\mathcal{U}}_m$.
In the algorithm, we sequentially replaces $\hat{\mathcal{U}}_m$ to a new $\hat{\mathcal{U}}_m^{\prime}$
that maximizes the absolute value of the fidelity
\begin{equation}
F_m = \langle \Phi_{m-1} \vert ( \hat{\mathcal{U}}_m^{\prime} ) ^{\dagger} \vert \Psi_{m+1} \rangle,
\end{equation}
where $\langle \Phi_{m-1} \vert$ and $\vert \Psi_{m+1} \rangle$ are given respectively by
\begin{equation}
\begin{split}
\langle \Phi_{m-1} \vert = & \langle 0 \vert \prod_{m^{\prime}=1}^{m-1} (\hat{\mathcal{U}}_{m^{\prime}}^\prime)^{\dagger} \\
= & \langle 0 \vert (\hat{\mathcal{U}}_{1}^\prime)^{\dagger} (\hat{\mathcal{U}}_{2}^\prime)^{\dagger}
\Compactcdots (\hat{\mathcal{U}}_{m-1}^\prime)^{\dagger}
\end{split}
\end{equation}
and
\begin{equation}
\begin{split}
\vert \Psi_{m+1} \rangle = & \prod_{m^{\prime}=m+1}^M ( \hat{\mathcal{U}}_{m^{\prime}} )^{\dagger} \vert \Psi \rangle \\
= & ( \hat{\mathcal{U}}_{m+1} )^{\dagger} ( \hat{\mathcal{U}}_{m+2} )^{\dagger} \Compactcdots ( \hat{\mathcal{U}}_M )^{\dagger} \vert \Psi \rangle
\end{split}
\end{equation}
with $\langle \Phi_{0} \vert = \langle 0 \vert$ and $\vert \Psi_{M+1} \rangle = \vert \Psi \rangle$.
Furthermore, we assume that the $m$th two-qubit unitary operator $\hat{\mathcal{U}}_m$
acting at $\mathbb{I}_m = \{ i_m, j_m \}$ is replaced with an unitary operator $\hat{\mathcal{U}}_m^\prime$
acting at $\mathbb{I}_k = \{ i_k, j_k \}$ that is properly selected among a set of bonds
$\mathbb{B} = \{ \mathbb{I}_1, \mathbb{I}_2, \Compactcdots, \mathbb{I}_B \}$.
However, the generalization to $K$-qubit unitary operators with $K>2$ is straightforward.
A prototype of the algorithm is then given as follows:
\begin{itemize}
\item[(1)] Set $m := 1$, $\hat{\mathcal{C}}_{0} := 1$, $\vert \Psi_{2} \rangle := \prod_{m'=2}^M ( \hat{\mathcal{U}}_{m^{\prime}})^{\dagger} \vert \Psi \rangle$, and $\langle \Phi_{0} \vert := \langle 0 \vert$.
\item[(2)] Evaluate matrix ${\bm F}_m^{(k)}$ of the fidelity tensor $\hat{\mathcal{F}}_m^{(k)} = {\rm Tr}_{\bar{\mathbb{I}}_k} [ \vert \Psi_{m+1} \rangle \langle \Phi_{m-1} \vert ]$ for all $\mathbb{I}_k \in \mathbb{B}$.
\item[(3)] Perform SVD ${\bm F}_m^{(k)} = {\bm X}_k {\bm D}_k {\bm Y}_k$ for all ${\bm F}_m^{(k)}$, and calculate $S_k = \sum_{n=0}^3 [ {\bm D}_k ]_{nn}$.
\item[(4)] Find $k=k^*$ that mximizes $S_k$.
\item[(5)] Set ${\bm U} := {\bm X}_{k^*} {\bm Y}_{k^*}$ and assign the quantum gates for the new $m$th unitary operator
$\hat{\mathcal{U}}_m^\prime$, represented by the matrix ${\bm U}$, which acts on $\mathbb{I}_m := \{ i_{k^*}, j_{k^*} \}$.
\item[(6)] Set $\hat{\mathcal{C}}_{m} := \hat{\mathcal{U}}_m^\prime \hat{\mathcal{C}}_{m-1} $,
$\vert \Psi_{m+2} \rangle := \hat{\mathcal{U}}_{m+1} \vert \Psi_{m+1} \rangle$,
and
$\langle \Phi_{m} \vert := \langle \Phi_{m-1} \vert (\hat{\mathcal{U}}_{m}^{\prime})^{\dagger}$.
\item[(7)] Set $m := m+1$ and go to (2) if $m \leq M$. Otherwise, quit and return $\hat{\mathcal{C}} := \hat{\mathcal{C}}_M$.
\end{itemize}
A schematic representation of this procedure is given in Fig.~\ref{fig:qce}(a).
Note that the procedure (2) is most time consuming and should be done on a quantum computer (see Sec.~\ref{sec:encode:imp}),
while the other procedures are performed on a classical computer.
Since the evaluation of ${\bm F}_m^{(k)}$ for different bonds $\mathbb{I}_k$ is independent,
one can trivially parallelize this part.
\begin{figure}
\includegraphics[width=\hsize]{qce_algorithm.pdf}
\caption{(a) Forward and (b) backward updates that optimize sequentially unitary operators $\hat{\mathcal U}_m$
for $m=1,2,\Compactcdots,M$.
}
\label{fig:qce}
\end{figure}
Although we have described the case for updating the unitary operators from $m=1$ to $m=M$,
it is apparent that we can reverse the order of updating the unitary operators from $m=M$ to $m=1$,
as schematically shown in Fig.~\ref{fig:qce}(b).
We shall call the algorithm for updating the unitary operators from $m=1$ to $m=M$
forward update, and the algorithm for updating the unitary operators from $m=M$ to $m=1$
backward update. Furthermore, let us refer to a single set of consecutive updates,
forward update followed by backward update, as a sweep.
\subsection{Implementation on a quantum computer} \label{sec:encode:imp}
The most demanding part computationally in the quantum circuit encoding algorithm is to evaluate
the fidelity tensor operator $\hat{\mathcal{F}}_m$ in Eq.~(\ref{eq:op:fm}).
As explained here, this part can be evaluated directly using a quantum computer
when the quantum state $\vert \Psi \rangle$ is given by a linear combination of several quantum circuits
as in Eq.~(\ref{eq:psi:lc}), which includes the extreme case where $\vert \Psi \rangle$ is given by a linear combination
of direct product states in the computational basis.
Although this procedure can be extended to the case for any number of qubits in principle,
here we consider a subsystem composed of two qubits, i.e., $\mathbb{I}_m = \{ i, j \}$, on which an unitary operator
$\hat{\mathcal U}_m$ acts.
Let us introduce the following notation
\begin{equation}
\hat{\mathcal{P}}_i^{\alpha} = \left\{
\begin{array}{cc}
\hat{I}_i & (\alpha = 0 ) \\
\hat{X}_i & (\alpha = 1 ) \\
\hat{Y}_i & (\alpha = 2 ) \\
\hat{Z}_i & (\alpha = 3 ) \\
\end{array}
\right.
\end{equation}
for the identity and Pauli operators acting on qubit $i$.
Then the fidelity tensor operator $\hat{\mathcal{F}}_m$ in the two-qubit subsystem $\mathbb{I}_m = \{ i,j \}$
can be expressed generally as
\begin{equation}
\hat{\mathcal{F}}_m = \sum_{\alpha=0}^3 \sum_{\alpha^{\prime}=0}^3 \tilde{f}_{\alpha,\alpha^{\prime}} \hat{\mathcal{P}}_i^{\alpha} \hat{\mathcal{P}}_j^{\alpha^{\prime}},
\label{eq:pauli:fm}
\end{equation}
where $\tilde{f}_{\alpha,\alpha^{\prime}}$ are complex numbers.
This is simply because the operator $\vert n \rangle \langle n^{\prime} \vert$ in Eq.~(\ref{eq:mat:fm}) for all $n,n^{\prime}=0,1,2,3$
can be expanded with a polynomial of the Pauli and identity operators.
We thus find that
\begin{equation}
{\rm Tr}_{\mathbb{I}_m} [ \hat{\mathcal{F}}_{m} \hat{\mathcal{P}}_i^{\alpha} \hat{\mathcal{P}}_j^{\alpha^{\prime}} ]
= 2^2 \tilde{f}_{\alpha,\alpha^{\prime}}
\end{equation}
because
\begin{equation}
{\rm Tr}_{\mathbb{I}_m} [ \hat{\mathcal{P}}_{i}^{\beta} \hat{\mathcal{P}}_{j}^{\beta^{\prime}}
\hat{\mathcal{P}}_i^{\alpha} \hat{\mathcal{P}}_j^{\alpha^{\prime}} ] = 2^2 \delta_{\alpha,\beta} \delta_{\alpha^{\prime},\beta^{\prime}}.
\end{equation}
On the other hand, by using the definition of the fidelity tensor operator $\hat{\mathcal{F}}_{m}$ in Eq.~(\ref{eq:op:fm}),
we find that
\begin{equation}
{\rm Tr}_{\mathbb{I}_m} [ \hat{\mathcal{F}}_m \hat{\mathcal{P}}_i^{\alpha} \hat{\mathcal{P}}_j^{\alpha^{\prime}} ]
= \langle {\Phi}_{m-1} \vert \hat{\mathcal{P}}_i^{\alpha} \hat{\mathcal{P}}_j^{\alpha^{\prime}} \vert \Psi_{m+1} \rangle.
\label{eq:tr_fm}
\end{equation}
Therefore, $\tilde{f}_{\alpha,\alpha^{\prime}}$ can be determined by
estimating the overlap between $\hat{\mathcal{P}}_i^{\alpha} \hat{\mathcal{P}}_j^{\alpha^{\prime}} \vert \Psi_{m+1} \rangle$
and $\vert \Phi_{m-1} \rangle$ for all $\alpha,\alpha^{\prime}=0,1,2,3$.
This overlap can be evaluated using a Hadamard test like circuit shown in Fig.~\ref{fig:hadamard}(a),
provided that a quantum circuit $\hat{\Psi}$ generating the quantum state $|\Psi\rangle=\hat{\Psi}|0\rangle$ is already known.
However, this is generally not the case but rather the main task of the quantum circuit encoding algorithm is to finding a quantum
circuit $\hat{\mathcal{C}}$ that approximately represents $\hat\Psi$.
Instead, here we assume that $\vert \Psi \rangle$ is given by a linear combination of quantum circuit states
as in Eqs.~(\ref{eq:psi:lc}) and (\ref{eq:psi_lc}). In this case, Eq.~(\ref{eq:tr_fm}) can be more explicitly written as
\begin{eqnarray}
{\rm Tr}_{\mathbb{I}_m} [ \hat{\mathcal{F}}_m \hat{\mathcal{P}}_i^{\alpha} \hat{\mathcal{P}}_j^{\alpha^{\prime}} ]
&=& \sum_{\gamma=1}^\Gamma \chi_\gamma
\langle {\Phi}_{m-1} \vert \hat{\mathcal{P}}_i^{\alpha} \hat{\mathcal{P}}_j^{\alpha^{\prime}} \vert \psi_{m+1}^{(\gamma)} \rangle \\
&=& \sum_{\gamma=1}^\Gamma \chi_\gamma
{\rm Tr}_{\mathbb{I}_m} [ \hat{f}_m^{(\gamma)} \hat{\mathcal{P}}_i^{\alpha} \hat{\mathcal{P}}_j^{\alpha^{\prime}} ],
\end{eqnarray}
where $\hat{f}_m^{(\gamma)} $ and $\vert \psi_{m+1}^{(\gamma)} \rangle$ are defined in Eqs.~(\ref{eq:f:lc})
and (\ref{eq:phi_gamma}), respectively.
As shown in Fig.~\ref{fig:hadamard}(b),
$f_{\alpha,\alpha^{\prime}}^{(\gamma)} = {\rm Tr}_{\mathbb{I}_m} [ \hat{f}_m^{(\gamma)} \hat{\mathcal{P}}_i^{\alpha} \hat{\mathcal{P}}_j^{\alpha^{\prime}} ]$ can now be evaluated separately for each $\gamma$ by using a Hadamard test like circuit on a quantum computer.
\begin{figure}
\includegraphics[width=\hsize]{HadamardTest.pdf}
\caption{(a) A quantum circuit to evaluate Eq.~(\ref{eq:tr_fm}).
$\hat{H}$ and $\hat{X}$ are Hadamard and Pauli-X gates, respectively.
$\hat{S}_{\theta}$ is a phase shift gate given by $\hat{S}_{\theta} \vert 0 \rangle = \vert 0 \rangle$
and $\hat{S}_{\theta} \vert 1 \rangle = {\rm e}^{-{\rm i}\theta} \vert 1 \rangle$.
The quantum circuits $\hat{\Psi}_{m+1}$ and $\hat{\Phi}_{m-1}$ are defined as $|\Psi_{m+1}\rangle =
\prod_{k=m+1}^{M} \hat{\mathcal{U}}_{k}^{\dagger} |\Psi\rangle = \hat{\Psi}_{m+1} |0\rangle $ and
$\vert \Phi_{m-1} \rangle = \prod_{k=m-1}^{1} \hat{\mathcal{U}}_{k} \vert 0 \rangle = \hat{\Phi}_{m-1} \vert 0 \rangle$, respectively.
In our protocol, we assume that $|\Psi\rangle$ is given by a linear combination of quantum circuit states $|\psi^{(\gamma)}\rangle$,
including the extreme case where it is given by a linear combination of direct product states in the computational basis.
Therefore, as indicated in the figure, $\hat{\Psi}_{m+1}$ is given by a linear combination of different quantum circuits, i.e.,
$\hat{\Psi}_{m+1} = \sum_{\gamma=1}^{\Gamma} \chi_{\gamma} \hat{\psi}_{m+1}^{(\gamma)}$, where
$\hat{\psi}^{(\gamma)}_{m+1} = \prod_{k=m+1}^{M} \hat{\mathcal{U}}_{k}^{\dagger} \hat{\psi}^{(\gamma)}$ and
$|\psi^{(\gamma)}\rangle=\hat{\psi}^{(\gamma)}|0\rangle$.
Hence, $f_{\alpha,\alpha^{\prime}}^{(\gamma)} = \langle {\Phi}_{m-1} \vert \hat{\mathcal{P}}_i^{\alpha} \hat{\mathcal{P}}_j^{\alpha^{\prime}} \vert \psi_{m+1}^{(\gamma)} \rangle$ can be evaluated separately for each $\gamma$, as shown in (b), by a
Hadamard test like circuit.
By measuring $\hat{Z}$ at the ancilla qubit, we can evaluate
${\rm Re}[ \langle \Phi_{m-1} \vert \hat{P}_i^{\alpha} \hat{P}_j^{\alpha^{\prime}} \vert \psi_{m+1}^{(\gamma)} \rangle ]$ for $\theta = 0 $
and
${\rm Im}[ \langle \Phi_{m-1} \vert \hat{P}_i^{\alpha} \hat{P}_j^{\alpha^{\prime}} \vert \psi_{m+1}^{(\gamma)} \rangle ]$ for
$\theta = \pi/2$.
A black circle in the circuits indicates a control qubit for a control gate.
}
\label{fig:hadamard}
\end{figure}
\subsection{Initialization algorithm} \label{sec:encode:init}
Although the quantum circuit encoding algorithm described above in Sec.~\ref{sec:encode:algorithm}
can be applied to general cases,
there are some exceptions for which care must be taken.
For example, when we consider a ground state $\vert \Psi \rangle$ of a quantum spin system,
the state $\vert \Psi \rangle$ is often in the spin singlet sector.
In this case, there is no overlap between $\vert \Psi \rangle$ and $\vert 0 \rangle$
because the product state $\vert 0 \rangle$ represents the fully polarized state with the maximum spin value.
Therefore, an alternative algorithm is required to construct an initial circuit $\hat{\mathcal{C}}$,
for which $\hat{\mathcal{C}}^{\dagger} \vert \Psi \rangle$ has a finite overlap with $\vert 0 \rangle$.
Let us consider the reduced density matrix $\hat{\rho}$ of a quantum state $\vert \Psi \rangle$
on the subsystem $\mathbb{I} = \{ i_1, i_2, \Compactcdots, i_K \}$ that is given by
\begin{equation}
\hat{\rho} = {\rm Tr}_{\bar{\mathbb{I}}} [ \vert \Psi \rangle \langle \Psi \vert ]
\end{equation}
with the associated eigenstates and eigenvalues being denoted as
$\vert \lambda_n \rangle$ and $\lambda_n$, respectively. Here we assume that $\lambda_n$ is in the descending order, i.e.,
$\lambda_0 \geq \lambda_1 \geq \lambda_2 \geq \Compactcdots$.
The reduced density matrix $\hat{\rho}$ is then represented as
\begin{equation}
\hat{\rho} = \sum_{n} \vert \lambda_n \rangle \lambda_n \langle \lambda_n \vert.
\end{equation}
We shall now find the unitary operator $\hat{\mathcal{V}}_1$ in the subsystem $\mathbb{I}$ such that
\begin{equation}
\underset{\hat{\mathcal{V}}_1}{\text{max}} \langle 0 \vert \hat{\mathcal{V}}^\dag_1 \hat{\rho} \hat{\mathcal{V}}_1 \vert 0 \rangle,
\label{eq:max:init}
\end{equation}
where $\vert 0 \rangle = \vert 0 \rangle_{i_1} \vert 0 \rangle_{i_2} \Compactcdots \vert 0 \rangle_{i_K}$
in the computational basis defined in the subsystem $\mathbb{I}$.
For this end, let us first expand $\hat{\mathcal{V}}_1$ in the following general form:
\begin{equation}
\hat{\mathcal{V}}_1 = \sum_{l=0}^{2^K-1} \sum_{n=0}^{2^K-1} v_{ln} \vert \lambda_l \rangle \langle n \vert,
\end{equation}
where $\{ \vert n \rangle = \vert \sigma_{i_1} \sigma_{i_2} \Compactcdots \sigma_{i_K} \rangle \}_{n=0}^{2^K-1}$ are the basis states
in the subsystem $\mathbb{I}$.
We then find that
\begin{equation}
\langle 0 \vert \hat{\mathcal{V}}^\dag_1 \hat{\rho} \hat{\mathcal{V}}_1 \vert 0 \rangle
= \sum_{l=0}^{2^K-1} \lambda_l \vert v_{l0} \vert^2.
\end{equation}
It is now easy to find that
\begin{equation}
\hat{\mathcal{V}}_1 = \sum_{n=0}^{2^K-1} \vert \lambda_n \rangle \langle n \vert
\label{eq:u:init}
\end{equation}
yields one of the solutions for Eq.~(\ref{eq:max:init}).
Once we determine the unitary operator $\hat{\mathcal{V}}_1$ in Eq.~(\ref{eq:u:init}), we can assign quantum gates
for this operator by following the prescription described in Sec.~\ref{sec:encode:assign} for the two-qubit case, if it is required.
In numerical simulations, we can determine $\hat{\mathcal{V}}_1$ in Eq.~(\ref{eq:u:init}) by directly evaluating the eigenstates
$\vert \lambda_n\rangle$ of the reduced density operator $\hat{\rho}$. $\hat{\mathcal{V}}_1$ can also be determined via a quantum computer.
To show this, let us consider the subsystem $\mathbb{I}$ composed of two qubits $\mathbb{I} = \{ i,j \}$, for simplicity.
Expanding the reduced density matrix $\hat{\rho}$ with a polynomials of the Pauli and identity operators
\begin{equation}
\hat{\rho} = \sum_{\alpha,\alpha^{\prime}} \tilde{r}_{\alpha,\alpha^{\prime}} \hat{\mathcal{P}}_i^{\alpha} \hat{\mathcal{P}}_j^{\alpha^{\prime}},
\label{eq:dopt}
\end{equation}
we find that
\begin{equation}
\langle \Psi \vert \hat{\mathcal{P}}_i^{\alpha} \hat{\mathcal{P}}_j^{\alpha^{\prime}} \vert \Psi \rangle
=
{\rm Tr}_{\mathbb{I}} \left[ \hat{\rho} \hat{\mathcal{P}}_i^{\alpha} \hat{\mathcal{P}}_j^{\alpha^{\prime}} \right]
= 2^2 \tilde{r}_{\alpha,\alpha^{\prime}}.
\label{eq:dopt2}
\end{equation}
This implies that the matrix elements $ \tilde{r}_{\alpha,\alpha^{\prime}}$ of $\hat{\rho}$ in Eq.~(\ref{eq:dopt})
can be determined
by measuring all possible pairs of products of the Pauli and identity operators
$\hat{\mathcal{P}}_i^{\alpha} \hat{\mathcal{P}}_j^{\alpha^{\prime}}$ ($\alpha,\alpha^{\prime} = 0,1,2,3$).
This can be performed on a quantum computer directly
if the state $\vert \Psi\rangle$ is given in a quantum circuit,
or by using the procedure described in Sec.~\ref{sec:encode:imp} (also see Fig.~\ref{fig:hadamard}), otherwise.
Having estimated the reduced density matrix for $\hat{\rho}$, one can determine the unitary operator $\hat{\mathcal{V}}_1$
in Eq.~(\ref{eq:u:init}) classically. The extension to the subsystem $\mathbb{I}$ composed of more than two qubits is straightforward.
We should also note that constructing a reduced density matrix by measuring a set of Pauli and identity operators
on a quantum computer is known as quantum state tomography~\cite{NielsenChuangBook},
and the technique described in Sec.~\ref{sec:encode:imp} is also along this line.
This procedure can be easily extended for further adding unitary operators $\hat{\mathcal{V}}_2, \hat{\mathcal{V}}_3, \dots$.
Let us assume that we have already determined the first unitary operator $\hat{\mathcal{V}}_1$ acting on
$\mathbb{I} = \{ i_1, i_2, \Compactcdots, i_K \}$. The location of these qubits is selected among a set of clusters of $K$ qubits
$\mathbb{C} = \{ \mathbb{I}_1, \mathbb{I}_2, \Compactcdots, \mathbb{I}_C \}$ so as to maximize Eq.~(\ref{eq:max:init}), i.e.,
\begin{equation}
\underset{\mathbb{I} \in \mathbb{C}}{\text{max}} \left[ \underset{\hat{\mathcal{V}}_1}{\text{max}} \langle 0 \vert \hat{\mathcal{V}}^\dag_1 \hat{\rho} \hat{\mathcal{V}}_1 \vert 0 \rangle \right].
\label{eq:max_v1}
\end{equation}
Let us now define a quantum state $|\tilde{\Psi}_1\rangle$ incorporating $\hat{\mathcal{V}}_1^\dag$ into the quantum state
$|\Psi\rangle$, i.e.,
\begin{equation}
|\tilde{\Psi}_1\rangle = \hat{\mathcal{V}}_1^\dag |\Psi\rangle,
\end{equation}
and consider the reduced density matrix $\hat{\rho}_1$ of $|\tilde{\Psi}_1\rangle$ on
the subsystem $\mathbb{I}^\prime = \{ i'_1, i'_2, \Compactcdots, i'_K \}$ given by
\begin{equation}
\hat{\rho}_1 = {\rm Tr}_{\bar{\mathbb{I}}^\prime} [ \vert \tilde{\Psi}_1 \rangle \langle \tilde{\Psi}_1 \vert ].
\end{equation}
$\hat{\mathcal{V}}_2$ is then determined by maximizing $ \langle 0 \vert \hat{\mathcal{V}}^\dag_2 \hat{\rho}_1 \hat{\mathcal{V}}_2 \vert 0 \rangle$, i.e.,
\begin{equation}
\underset{\mathbb{I}^\prime \in \mathbb{C}}{\text{max}} \left[ \underset{\hat{\mathcal{V}}_2}{\text{max}} \langle 0 \vert \hat{\mathcal{V}}^\dag_2 \hat{\rho}_1 \hat{\mathcal{V}}_2 \vert 0 \rangle \right].
\end{equation}
This procedure can be continued until the desired number $\delta M$ of unitary operators
$\hat{\mathcal{V}}_1, \hat{\mathcal{V}}_2, \dots, \hat{\mathcal{V}}_{\delta M}$ are added, i.e.,
$\hat{\mathcal{V}}_{\delta M}^\dag\Compactcdots\hat{\mathcal{V}}_2^\dag\hat{\mathcal{V}}_1^\dag|\Psi\rangle$.
Note that the reduced density operators $\hat{\rho}_1, \hat{\rho}_2, \dots$ can be evaluated on a quantum computer
by the quantum state tomography.
\subsection{Automatic quantum circuit encoding algorithm} \label{sec:encode:aqce}
Finally, we combine a prototype algorithm of the quantum circuit encoding described in Sec.~\ref{sec:encode:algorithm}
(also see Fig.~\ref{fig:qce}) and the initialization algorithm explained in Sec.~\ref{sec:encode:init},
in order to automatically construct an optimal quantum circuit for encoding a give quantum state.
Figure~\ref{fig:aqce} summarizes the resulting algorithm that is referred to as automatic quantum circuit encoding (AQCE) algorithm.
\begin{figure}
\includegraphics[width=0.95\hsize]{aqce_algorithm.pdf}
\caption{Automatic quantum circuit encoding (AQCE) algorithm.
(a) Inputs: a quantum states $\vert \Psi \rangle$, a quantum circuit $\hat{\mathcal{C}}$, and a set of bonds $\mathbb{B}$
(or clusters $\mathbb{C}$) of quabits
on which unitary operators act. We set $\hat{\mathcal{C}}=\hat{I}$ and thus the number $M$ of unitary operators in the
circuit is zero.
(b) Enlargement step. $\delta M$ unitary operators acting on two qubits (or $K$ qubits) are newly inserted in the circuit,
according to the optimization algorithm (precisely, the backward update) described in Sec.~\ref{sec:encode:algorithm} (also see the text).
In the initial step, $\delta M$ is set to $M_0$ and the initialization algorithm in Sec.~\ref{sec:encode:init} is employed to properly
construct the number $M_0$ of unitary operators.
(c) Optimization step. The circuit consisting of $M$ unitary operators is optimized by the forward and backward updates $N$ times,
in which each unitary operator is updated one by one, including the location of qubits that is acted on,
according to the quantum circuit encoding algorithm described in Sec.~\ref{sec:encode:algorithm}.
The overall iteration is terminated when $M$ reaches to the desired number $M_{\rm max}$ of unitary operators. Otherwise,
the algorithm goes back to the enlargement step (b) to additionally insert $\delta M$ new unitary operators.
The enlargement step (b) outputs the new quantum circuit $\hat{\mathcal{C}}_{\rm new}$ with $M+\delta M$ unitary operators
and this circuit is used as the input of the optimization step (c).
Light-green and light-red squares indicate updated unitary operators, while light-blue squares indicate input unitary operators.
Control parameters in the AQCE algorithm are $M_0$, $M_{\rm max}$, $\delta M$, and $N$.
}
\label{fig:aqce}
\end{figure}
The AQCE algorithm is composed of two steps, i.e., the enlargement step in Fig.~\ref{fig:aqce}(b) and
the optimization step in Fig.~\ref{fig:aqce}(c).
The inputs of the AQCE algorithm are a target quantum state $\vert \Psi \rangle$,
a quantum circuit $\hat{\mathcal{C}}$ set to be the identity operator $\hat{I}$,
and a set of bonds $\mathbb{B}$ of two qubits
(or a set of clusters $\mathbb{C}$ of $K$ qubits), as shown in Fig.~\ref{fig:aqce}(a).
In the first enlargement step, we employ the initialization algorithm as in Fig.~\ref{fig:aqce}(b)
to construct a quantum circuit $\hat{\mathcal{C}}$ having $M_0$ number of unitary operators
and finite overlap between $\hat{\mathcal{C}} \vert 0 \rangle$ and $\vert \Psi \rangle$.
Then, in the following optimization step, we perform the forward and backward updates of the quantum circuit encoding algorithm
to optimize these unitary operators as in Fig.~\ref{fig:aqce}(c).
The total number of sweeps for the optimization is set to $N$.
Next, we enlarge the quantum circuit by increasing the number $M$ of unitary operators by $\delta M$, i.e.,
$M = M_0+\delta M$, as in Fig.~\ref{fig:aqce}(b). This is done one by one by inserting single identity operator next to $|0\rangle$
and perform the backward update of the quantum circuit encoding algorithm to optimize the whole unitary operators.
After adding $\delta M$ new unitary operators,
we move to the optimization step and perform the forward and backward updates again to optimize each unitary operator
as in Fig.~\ref{fig:aqce}(c).
Notice that expect for the initial enlargement step,
we perform in the enlargement step the backward update of the quantum circuit encoding algorithm
described in Sec.~\ref{sec:encode:algorithm},
which is more efficient than the initialization algorithm described in Sec.~\ref{sec:encode:init}.
We repeat this whole iteration of the enlargement and optimization steps
until the quantum circuit contains the desired number $M_{\rm max}$ of unitary operators.
The control parameters in the AQCE algorithm are thus $M_0$, $M_{\rm max}$, $\delta M$, and $N$.
\section{Numerical simulation} \label{sec:benchmark}
In this section, we demonstrate by numerical simulations the AQCE algorithm for quantum many-body states and
for classical data. In particular, the latter application is potentially useful for quantum machine learning in preparing
an input quantum state that represents classical data~\cite{Schuld2019}.
For the purpose of demonstration, we consider the unitary operators $\hat{\mathcal{U}}_m$ and $\hat{\mathcal{V}}_m$
acting on two qubits. However, the AQCE algorithm can also be applied to general cases for $K$ qubits with $K>2$.
\subsection{Quantum circuit encoding of quantum many-body states} \label{sec:res:state}
Here, we show the numerical demonstration of the quantum circuit encoding for the ground states of the
one-dimensional $S=1/2$ isotropic antiferromagnetic Heisenberg model and XY model.
The Hamiltonian of these models is given as
\begin{equation}
\hat{\mathcal{H}} = \sum_{i=1}^{L} (\hat{X}_i \hat{X}_{i+1} + \hat{Y}_i \hat{Y}_{i+1} + \Delta \hat{Z}_i \hat{Z}_{i+1}),
\end{equation}
where $\hat{X}_i$, $\hat{Y}_i$, and $\hat{Z}_i$ are the $x$-, $y$-, and $z$-components of Pauli operators, respectively,
at site $i$ on a one-dimensional chain with $L$ sites under periodic boundary conditions, i.e.,
$\hat{X}_{L+1} = \hat{X}_{1}$, $\hat{Y}_{L+1} = \hat{Y}_1$,
and $\hat{Z}_{L+1} = \hat{Z}_1$.
The Hamiltonian $\hat{\mathcal{H}}$ with $\Delta =1$ and $0$ corresponds to the isotropic Heisenberg and XY models, respectively,
and the ground states of these two models are at criticality with algebraically decaying correlation functions.
The ground states $\vert \Psi \rangle$ of these models are calculated numerically
by the standard Lancozs method within the accuracy of the ground state energy $10^{-12}$.
Although the AQCE algorithm is formulated deterministically,
it turns out that the resulting structure of the quantum circuit depends on
the numerical tiny error of the quantum state $\vert \Psi \rangle$ obtained by finite precision arithmetic.
This is simply because even when the fidelity tensor $\hat{\mathcal{F}}_m$ for equivalent pairs of qubits
is exactly the same theoretically, a particular pair of qubits $\mathbb{I} = \{ i,j \}$ may be selected because of
the numerical error due to finite precision calculations.
Therefore, here we perform 100 AQCE calculations for each system size $L$, in which
the ground state $\vert \Psi \rangle$ is prepared by the Lanczos method with 100 different initial Lanczos vectors,
thus implying that the ground state $\vert \Psi \rangle$ to be encoded is slightly different among these
100 different calculations,
and select the best circuit $\hat{\mathcal{C}}$ in terms of the fidelity $|\langle 0|\hat{\mathcal{C}}^\dag|\Psi\rangle|$.
In addition, we perform 1000 sweeps to further optimize the unitary operators in the best circuit using the quantum
circuit encoding algorithm.
The parameters for the AQCE algorithm are
$(M_0,N,\delta M, M_{\rm max}) = (L, 20, L/2, L(L-5)/2)$ for the XY model
and $(M_0, N, \delta M, M_{\rm max})=(L, 20, L/2, L^2/2)$ for the isotropic Heisenberg model.
We set that $\mathbb{B}$ is composed of all pairs of two sites (i.e., qubits) $\{ i, j \}$ with $i, j\in \mathbb{L}$,
thus including pairs of distant sites.
\begin{figure}
\includegraphics[width=\hsize]{olap_xy.pdf}
\caption{
(a) Fidelity per site between the ground state $\vert \Psi \rangle$ of the one-dimensional $S=1/2$ XY model and
the quantum circuit state $\hat{\mathcal{C}} \vert 0 \rangle$
optimized by the AQCE algorithms for different system sizes $L$. (b) Semi-log plot of (a).
Vertical dotted lines with numbers in (b) indicate the number of local two-qubit unitary operators $M_{\rm e}=L^2/4$
required to represent the exact ground state by the DQAP ansatz~\cite{Shirakawa2021}.
}
\label{fig:olap:xy}
\end{figure}
Figure~\ref{fig:olap:xy} shows the fidelity between the ground state $\vert \Psi \rangle$ of the XY model and
the optimized quantum circuit state $\hat{\mathcal{C}} \vert 0 \rangle$ obtained by the AQCE algorithm.
We should first recall the previous results by the discretized quantum adiabatic process (DQAP) ansatz~\cite{Shirakawa2021},
a similar approach to digitized adiabatic quantum computing reported in Refs.~\onlinecite{Barends2015,Barends2016},
where a parametrized quantum circuit is constructed on the basis of digitized quantum adiabatic
process expressed by a product of local time-evolution unitary operators and the variational parameters
are optimized so as to minimize the expectation value of energy, as in the
variational quantum eigensolver~\cite{Peruzzo2014}.
It is found in Ref.~\onlinecite{Shirakawa2021} that the optimized DQAP ansatz gives
the exact ground state of the XY model with the minimum number $M_{\rm e} = L^2/4$
of local two-qubit unitary operators set by the Lieb-Robinson bound.
For comparison, this number $M_{\rm e}$ is also indicated for each system size $L$ in Fig.~\ref{fig:olap:xy}(b).
We find that the AQCE algorithm can generate the quantum circuit state $\hat{\mathcal{C}} \vert 0 \rangle$
that represents essentially the exact ground state $\vert \Psi \rangle$ with $M=M_{\rm e}$ for all system sizes studied
except for $L=16$, for which the convergence of the quantum circuit state $\hat{\mathcal{C}} \vert 0 \rangle$ towards
the ground state $\vert \Psi \rangle$ appears slower with increasing $M$.
However, we should note that the quantum circuit state $\hat{\mathcal{C}} \vert 0 \rangle$ with $M < M_{\rm e}$ is better
in terms of the fidelity than the DQAP ansatz composed of the same number $M$ of local time-evolution
unitary operators even for $L=16$ [see Fig.~\ref{fig:olap:compare} (a)].
\begin{figure}
\includegraphics[width=\hsize]{olap_heisen.pdf}
\caption{
Same as Fig.~\ref{fig:olap:xy} but for the ground states $\vert \Psi \rangle$ of the one-dimensional $S=1/2$
isotropic antiferromagnetic Heisenberg model.
}
\label{fig:olap:heisen}
\end{figure}
Figure~\ref{fig:olap:heisen} shows the fidelity between the ground state $\vert \Psi \rangle$ of the isotropic antiferromagnetic
Heisenberg model and
the optimized quantum circuit state $\hat{\mathcal{C}} \vert 0 \rangle$ obtained by the AQCE algorithm.
For smaller systems with $L \leq 8$, the AQCE algorithm can construct a quantum circuit state $\hat{\mathcal{C}} \vert 0 \rangle$
that represents numerically exactly the ground state $\vert \Psi \rangle$ with a less number of $M$.
For example, one of the resulting quantum circuits describing the ground state for $L=6$ is shown in Fig.~\ref{fig:qce:circuit}(a).
The number $M$ of two-qubit unitary operators contained in this particular circuit is $M=12$
and the number of independent parameters, once these unitary operators are represented by a standard set of quantum gates
(see Fig.~\ref{fig:twogate}), is
$9\times 12 + 6 \times 3 = 126$ if we combine adjacent two single-qubit Euler rotations into a single-qubit Euler rotation.
On the other hand, the dimension of the Hilbert space for the $L=6$ system is $2^L = 64$,
suggesting that there are $128-2 = 126$ independent real parameters, where two is subtracted because of the overall phase factor
and the normalization factor.
It is hence interesting to find that the number of the independent real parameters in this quantum circuit $\hat{\mathcal{C}} $
with $M=12$ coincides with that for the Hilbert space on which the quantum state $\vert \Psi \rangle$ is defined.
However, it is highly nontrivial whether the quantum circuit $\hat{\mathcal{C}}$ composed of the limited number of two-qubit
unitary operators can always represent any quantum state whenever the number of parameters
in a quantum circuit is the same as that for the Hilbert space.
\begin{figure}
\includegraphics[width=\hsize]{circuit_qce.pdf}
\caption{
Optimized quantum circuit states $\hat{\mathcal{C}} \vert 0 \rangle$ obtained by the AQCE algorithm,
which represent essentially exactly the ground states of the one-dimensional $S=1/2$ isotropic antiferromagnetic
Heisenberg model for (a) $L=6$ and (b) $L=8$, containing 12 and 24 two-qubit unitary operators $\hat{\mathcal{U}}_m$, respectively,
denoted by black squares with number $m$ in them. The location of two qubits
on which each unitary operator acts is indicated by crosses.
Each two-qubit unitary operator can be decomposed into a standard set of quantum
gates (see Fig.~\ref{fig:twogate}) with having 15 independent real parameters (apart from a single global phase). Since adjacent two
single-qubit Euler rotations are combined into a single single-qubit Euler rotation, the total number of independent real parameters
is 126 for $L=6$ in (a) and 240 for $L=8$ in (b).
}
\label{fig:qce:circuit}
\end{figure}
We should also note that the two-qubit unitary operators in the optimized quantum circuit $\hat{\mathcal{C}}$,
representing the ground state $\vert \Psi \rangle$ essentially exactly for $L=6$ and $8$ (see Figure~\ref{fig:qce:circuit}), are not
uniformly distributed, even though the ground state $\vert \Psi \rangle$ represented by the quantum circuit is translational
symmetric (apart from the finite precision numerical error).
Figure~\ref{fig:qce:circuit}(b) shows one of the resulting quantum circuits describing the ground state for $L=8$.
The circuit structure is much more complicated than that for $L=6$ shown in Fig.~\ref{fig:qce:circuit}(a).
Nonetheless, we have confirmed numerically that the resulting quantum circuit states $\hat{\mathcal{C}}|0\rangle$ for $L=6$
and $8$ are essentially translational symmetric and also spin rotation symmetric.
In contrast, for the systems with $L > 8$, we find
that the convergence of the optimized quantum circuit state $\hat{\mathcal{C}}|0\rangle$
towards the ground state $\vert \Psi \rangle$ is slower with the number $M$ of unitary operators,
although the convergence is still approximately exponential, as shown in Fig.~\ref{fig:olap:heisen}(b).
For example, the error in fidelity of the optimized quantum circuit state $\hat{\mathcal{C}}|0\rangle$ for $L=16$
is still relatively large even when $M/L = 8$.
Moreover, as observed in Fig.~\ref{fig:olap:heisen}(b), the slope of the fidelity in the semi-log plot becomes more flattered
with increasing the system size $L$.
We speculate that this is due to a difficulty of sequentially optimizing each unitary operator $\hat{\mathcal{U}}_m$, including
the location of qubits on which the unitary operator $\hat{\mathcal{U}}_m$ acts, when the system size is large.
Since much more computational resources are required for further systematic analysis with larger system sizes,
we leave this issue for a future study.
\subsection{Quantum circuit encoding with fixed Trotter- and MERA-like circuit structures} \label{sec:res:comp}
\begin{figure}
\includegraphics[width=\hsize]{circuit_fix.pdf}
\caption{(a) Trotter-like circuit and (b) MERA-like circuit for a 16-qubit system.
Two-qubit unitary operators $\hat{\mathcal{U}}_m$ are indicated by back squares and the location of two qubits
on which each unitary operator acts is indicated by crosses.
Shaded color layers are repeated $D$ times.
}
\label{fig:fix:circuit}
\end{figure}
In this section, using numerical simulations,
we shall compare the results obtained by the AQCE algorithm, which can automatically construct a quantum circuit with
a self-assembled optimal structure, and those obtained for a quantum circuit with a fixed circuit structure.
For this purpose, here we consider two particular fixed circuit structures.
One is a Trotter-like circuit structure schematically shown in Fig.~\ref{fig:fix:circuit}(a).
In this Trotter-like circuit, two-qubit unitary operators $\hat{\mathcal{U}}_m$ acting on adjoining qubits are
distributed as if a time evolution operator of the whole system is Trotter decomposed into two parts in a one dimensionally aligned
qubit ring.
The quantum circuit is composed of $D$ layers and each layer corresponds to one Trotter step, containing $L$ two-qubit unitary
operators $\hat{\mathcal{U}}_m$. Therefore, the total number $M$ of unitary operators $\hat{\mathcal{U}}_m$
in the Trotter-like circuit is $D L$.
The other circuit structure considered here is inspired by the MERA
and is shown schematically in Fig.~\ref{fig:fix:circuit}(b).
In this MERA-like circuit, each basic layer indicated by different shaded color in Fig.~\ref{fig:fix:circuit}(b) represents a different length
scale and thus two-qubit unitary operators $\hat{\mathcal{U}}_m$ in different basic layers act on two qubits that are
located in different (adjoining as well as distinct) distances.
To improve the accuracy, we also increase the number of layers in each basic layer $D$ times [see Fig.~\ref{fig:fix:circuit}(b)],
and therefore the total number $M$ of unitary operators $\hat{\mathcal{U}}_m$ in the MERA-like circuit is
$D(L+L/2+L/2^2+L/2^3+\Compactcdots+4)+(2-1)=2D(L-2)+1$, assuming that the system size $L$ is factorial of 2.
In order to optimize two-qubit unitary operators $\hat{\mathcal{U}}_m$ in the Trotter- and MERA-like circuits
for encoding a quantum state $\vert \Psi\rangle$, we perform 1000 sweeps of the forward and backward updates using the algorithm
described in Sec.~\ref{sec:encode:algorithm} (also see Fig.~\ref{fig:qce}),
i.e., the quantum circuit encoding algorithm, but with the fixed circuit structures.
\begin{figure}
\includegraphics[width=\hsize]{olap_compare.pdf}
\caption{(a) Fidelity per site between the ground state $|\Psi\rangle$ of the one-dimensional $S=1/2$ XY model and
the quantum circuit states $\hat{\mathcal{C}} \vert 0 \rangle$ obtained by the AQCE algorithm and with the Trotter-
and MERA-like circuit structures for $L=16$.
(b) Same as (a) but for the ground state $|\Psi\rangle$ of the one-dimensional $S=1/2$ isotropic
antiferromagnetic Heisenberg model.
(c) Same as (a) but for the ground state $|\Psi\rangle$ of the one-dimensional $S=1/2$ anisotropic
antiferromagnetic Heisenberg model with $\Delta=4$.
The results obtained by the AQCE algorithm in (a) and (b) are the same as those shown in Fig.~\ref{fig:olap:xy} and
Fig.~\ref{fig:olap:heisen}, respectively.
Note that the ground states in (a) and (b) are at criticality, while the ground state in (c) is away from criticality.
}
\label{fig:olap:compare}
\end{figure}
Figure~\ref{fig:olap:compare}(a) shows the fidelity between the ground state $|\Psi\rangle$
of the XY model and the optimized quantum circuit states $\hat{\mathcal{C}} \vert 0 \rangle$ obtained by the AQCE algorithm and
with the Trotter- and MERA-like circuit structures for $L=16$.
First, we find that the quantum circuit state with the Trotter-like circuit structure can represent numerically exactly the
ground state $|\Psi\rangle$ at $M=4L$ (corresponding to $L^2/4$ for $L=16)$,
which is consistent with the previous study using the DQAP ansatz~\cite{Shirakawa2021}.
This is understood simply because the Trotter-like circuit and the DQAP ansatz have the same circuit structure, although these two
approaches employ different optimization schemes to determine the optimal two-qubit unitary operators: In the DQAP ansatz,
each two-qubit unitary operator is parametrized with a single variational parameter [i.e., $\hat{\cal{D}}$ with $\alpha_1=\alpha_2=\alpha_3$ in Eq.~(\ref{eq:exgate})]
and the variational parameters are optimized
so as to minimize the expectation value of energy, while the optimal unitary operators in the Trotter-like circuit are
determined essentially deterministically by maximizing the fidelity of the ground state.
We also find in Fig.~\ref{fig:olap:compare}(a) that the quantum circuit state $\hat{\mathcal{C}}|0\rangle$ obtained by the AQCE algorithm
is much better than that with the Trotter-like circuit structure when $M < 4 L$ and
it is competitive in terms of the accuracy
with that with the MERA-like circuit structure.
Figure~\ref{fig:olap:compare}(b) shows the fidelity between the ground state $|\Psi\rangle$
of the isotropic antiferromagnetic Heisenberg model and the optimized quantum circuit states $\hat{\mathcal{C}} \vert 0 \rangle$
obtained by the AQCE algorithm and
with the Trotter- and MERA-like circuit structures for $L=16$.
Similar to the results for the XY model in Fig.~\ref{fig:olap:compare}(a),
the quantum circuit state $\hat{\mathcal{C}}|0\rangle$ obtained by the AQCE algorithm exhibits the
better accuracy than that with the Trotter-like circuit structure and it is compatible with
that with the MERA-like circuit structure when $M \leq 5 L$.
However, for $M > 5L$, all these three quantum circuit states show the similar accuracy that is
improved approximately exponentially with increasing $M$.
The ground states in these two cases are both at criticality and the MERA is known to be best suited for describing
such a quantum state~\cite{Evenbly2011}. Therefore, it is also interesting to study a case for which the ground state is
away from criticality.
Figure~\ref{fig:olap:compare}(c) shows the fidelity between the ground state $|\Psi\rangle$ of
the anisotropic antiferromagnetic Heisenberg model with $\Delta=4$ and the optimized quantum circuit states
$\hat{\mathcal{C}} \vert 0 \rangle$ obtained by the AQCE algorithm and with the Trotter- and MERA-like circuit structures for $L=16$.
In this case, the ground state is gapped and is less entangled as compared to those of the previous two models with
$\Delta=0$ and $\Delta=1$.
Therefore, one expects that the number of two-qubit unitary operators required to achieve given accuracy for $\Delta=4$
is smaller than that for $\Delta =0$ and $\Delta =1$.
We indeed find in Fig.~\ref{fig:olap:compare}(c) that the fidelity is much closer to 1 when the number of two-qubit unitary operators is small
for the optimized quantum circuit states obtained by the AQCE algorithm and with the MERA-like circuit structure,
but not for the optimized quantum circuit state with the Trotter-like circuit structure.
In the case of the Trotter-like circuit, the fidelity first remains almost constant with increasing $M$ until $M=4L$ at which
the fidelity suddenly jumps to a larger value and then again remains almost constant afterward.
It is interesting to observe in Figs.~\ref{fig:olap:compare}(b) and \ref{fig:olap:compare}(c)
that the fidelity becomes approximately independent of the quantum circuit structures employed
when $M/L$ is larger than $4$ or $5$.
A possible reason for this is due to the effect of the barren plateau phenomena.
It is known that the unitary 2-design can be realized in polynomial time for a quantum circuit
where two-qubit unitary operators are randomly distributed~\cite{Harrow2009}.
As shown in Fig.~\ref{fig:qce:circuit}(b), the distribution of two-qubit unitary operators in the quantum circuit
obtained by the AQCE algorithm for $L=8$ is quite random.
Therefore, it is naturally expected that the quantum circuit obtained
exhibits the unitary 2-design and thus might suffer from the barren plateaus phenomena.
Since the fidelities for other quantum circuits also exhibit similar values,
we expect that all of them might suffer from the barren plateau phenomena.
Indeed, we find that in all cases, the change of the quantum circuit during the optimization iteration
is very small when $M$ is large. This implies that a better quantum circuit can be generated more efficiently
when the number of two-qubit unitary operators is small enough not to exhibit the unitary 2-design.
However, we should note that the small improvement of fidelity with further increasing $M$ also simply
implies a trapping of a local minimum of the cost function.
\subsection{Quantum circuit encoding of classical data} \label{sec:res:data}
In this section, we demonstrate that the AQCE algorithm is also useful to construct a optimal
quantum circuit to represent classical data such as a classical image.
It is well known that there are several ways to encode classical data to a quantum state
(for example, see Ref.~\onlinecite{Schuld2019}). However, it is usually not obvious
how to optimally prepare such a quantum state encoding particular classical data
in a quantum circuit with a less number of quantum gates.
We show that the AQCE algorithm can be a promising approach for this purpose.
One way to express classical data in a quantum state
is the amplitude encoding~\cite{Schuld2016}, where
the classical data ${\bm x} = \{ x_0, x_1, \Compactcdots, x_n, \Compactcdots, x_{N-1} \}$
is described by using a quantum state
\begin{equation}
\vert \Psi_{\rm c} \rangle = \sum_{n=0}^{N-1} \bar{x}_n \vert n \rangle.
\label{eq:amp:encode}
\end{equation}
Here, $\vert n \rangle$ is the basis labelled by Eq.~(\ref{eq:computational:label})
with $L \geq \log_2 N$ and
\begin{equation}
\bar{x}_n = x_n / \sqrt{V_x}
\end{equation}
with $V_x$ being a volume of ${\bm x}$ given by
\begin{equation}
V_x = \sum_{n=0}^{N-1} \vert x_n \vert^2. \label{eq:x:volume}
\end{equation}
Each element $x_n$ in the classical data ${\bm x}$ is usually real number, but the amplitude encoding can also be applied to
the case of complex number.
There exist several proposals to implement the amplitude encoding~\cite{Grover2000,Shende2006,Plesch2011,Yuval2019}.
However, these are in general not best fit for a near-term application.
A variational quantum algorithm using a parametrized quantum circuit has also been proposed recently~\cite{Nakaji2021}.
In the previous sections, we have demonstrated the quantum circuit encoding of a quantum state focusing on the ground state
of a typical many-body Hamiltonian encountered in condensed matter physics and quantum statistical physics, which
is in some sense simple.
Instead, a quantum state given in Eq.~(\ref{eq:amp:encode}) representing classical data
is relatively complicated and moreover there is no prior knowledge of such a quantum state.
Therefore, the quantum circuit encoding of such a quantum state in Eq.~(\ref{eq:amp:encode}) is generally a difficult task
in any means.
Here, we employ the AQCE algorithm to demonstrate the quantum circuit encoding of a quantum state representing a classical image.
\begin{figure*}
\includegraphics[width=\hsize]{picture.pdf}
\caption{
Quantum circuit encoding of a gray scale picture known as ``{\it Peppers}" \cite{ImageDatabase}.
(a) Labeling of two-dimensional classical data (with $8\times 8$ pixels, as an example).
(b) Original picture with 256$\times$256 pixels.
(c)-(e) Pictures reconstructed by decoding the quantum circuit states $\hat{\mathcal{C}}|0\rangle$ on $L=16$ qubits
with the different number $M$ of two-qubit unitary operators, $M=32$, 112, and 520.
(f) Original picture divided into 16 pieces ($m_s=1,2,\dots,16$ indicated by yellow in the picture) with $64 \times 64$ pixels each.
(g)-(i) Pictures reconstructed by decoding each quantum circuit state $\hat{\mathcal{C}}^{(m_s)}|0\rangle$ on $L=12$ qubits
with the different number $M$ of two-qubit unitary operators, $M=24$, 42, and 450.
}
\label{fig:picture}
\end{figure*}
As an example of a classical image, we consider the gray scale picture shown in Fig.~\ref{fig:picture}(b),
which is also known as ``{\it Peppers}'' available in Ref.~\cite{ImageDatabase}.
The data size of this picture is $256 \times 256$ pixels
and each pixel in the two-dimensional array is assigned to represent each part of the picture
located at a position labeled $(i_x,i_y)$ with $i_x = 0,1,2,\Compactcdots,255$ ($=2^8-1$) in the horizontal axis from left to right and
$i_y = 0,1,2,\Compactcdots,255$ in the vertical axis from top to bottom,
as shown in Fig.~\ref{fig:picture}(a).
Therefore, the picture is fully given by a $2^{16}$ dimensional vector
${\bm x}_{\rm Pep} = \{ x_0, x_1, x_2, \Compactcdots, x_s, \Compactcdots, x_{65535} \}$ of non-negative real numbers,
where label $s= i_x + 256 i_y$.
This suggests that the data can be transformed into a quantum state $\vert \Psi_{\rm c} \rangle$ in the form given
in Eq.~(\ref{eq:amp:encode}) with $L=16$ qubits.
Using numerical simulations, we perform the AQCE algorithm to encode the quantum state $\vert \Psi_{\rm c} \rangle$
into an optimal quantum circuit state $\hat{\mathcal{C}} \vert 0 \rangle$.
For this end, we set the control parameters in the AQCE algorithm as $(M_0, N, \delta M) =(16, 100, 8)$ with varying
the total number $M$ of two-qubit unitary operators $\hat{\mathcal{U}}_m$ in the generated quantum circuit (defined as $M_{\rm max}$
in Sec.~\ref{sec:encode:aqce}).
Figures~\ref{fig:picture}(c)-\ref{fig:picture}(e) show the reconstructed pictures by decoding the quantum circuit states
$\hat{\mathcal{C}} \vert 0 \rangle$ with the different number $M$ of two-qubit unitary operators. In reconstructing the classical data
${\bm x}' = \{ x'_0, x'_1, \Compactcdots, x'_n, \Compactcdots, x'_{N-1} \}$ from the amplitude $\bar{x}_n'=\langle n|\hat{\mathcal{C}} \vert 0 \rangle$
of the quantum circuit state,
we have to rescale back the amplitude with the volume $V_x$, i.e., $x'_n=\sqrt{V'_x}\bar{x}'_n$.
It turns out that when the number $M$ of two-qubit unitary operators is extremely small,
the reconstructed picture looks more like a mosaic, as shown in Fig.~\ref{fig:picture}(c) for $M=32$. However,
as expected, the reconstructed pictures are improved with increasing $M$ [see Figs.~\ref{fig:picture}(d) and \ref{fig:picture}(e)].
To be more quantitative, we plot the fidelity between the quantum state $\vert \Psi_{\rm c} \rangle$ representing the original picture
and the quantum circuit state $\hat{\mathcal{C}} \vert 0 \rangle$ in Fig.~\ref{fig:olap:picture}.
The fidelity improves rather rapidly with increasing $M$ for $M$ up to 50, but the improvement becomes somewhat slower
for $M>100$.
\begin{figure}
\includegraphics[width=\hsize]{olap_picture.pdf}
\caption{
Fidelity per site between the quantum state $|\Psi_{\rm c}\rangle$ ($|\Psi_{\rm c}^{(m_s)}\rangle$) on $L=16$ ($L=12$) qubits
representing the original picture in Fig.~\ref{fig:picture}(b)
[the $m_s$th segment of the original picture in Fig.~\ref{fig:picture}(f)] and the quantum circuit state $\hat{\mathcal{C}}|0\rangle$
($\hat{\mathcal{C}}^{(m_s)}|0\rangle$) indicated by red triangles (green crosses).
Here, $m_s=1,2,\dots,16$ and thus 16 different results are shown for the case where the picture is divided into 16 pieces.
}
\label{fig:olap:picture}
\end{figure}
For the better performance, next we simply divide the original classical data
${\bm x}_{\rm Pep} = \{ x_0, x_1, x_2, \Compactcdots, x_{65535} \}$
into 16 pieces, each representing a $64 \times 64$ pixels picture, as shown in Fig.~\ref{fig:picture}(f).
This implies that each segment of the picture is given by a $2^{12}$ dimensional vector, i.e.,
${\bm x}_{\rm Pep}^{(m_s)} = \{ x_0^{(m_s)}, x_1^{(m_s)}, x_2^{(m_s)}, \Compactcdots, x_{4095}^{(m_s)} \}$ with $m_s=1,2,\dots, 16$.
Accordingly, a quantum state $\vert \tilde{\Psi}_{\rm c}\rangle$ for the whole picture is given by
a direct product of quantum states $|\tilde{\Psi}_{\rm c}^{(m_s)}\rangle$ representing different segments of the original picture, i.e.,
\begin{equation}
\vert \tilde{\Psi}_{\rm c}\rangle = \bigotimes_{m_s=1}^{16} |\Psi_{\rm c}^{(m_s)}\rangle,
\label{eq:amp:sum}
\end{equation}
where
\begin{equation}
\vert \Psi_{\rm c}^{(m_s)} \rangle = \sum_{n=0}^{2^{12}-1} \bar{x}_n^{(m_s)} \vert n^{(m_s)} \rangle
\label{eq:amp:encode2}
\end{equation}
with $\bar{x}_n^{(m_s)} = x_n^{(m_s)} / \sqrt{V_x^{(m_s)}}$ and
$V_x^{(m_s)} = \sum_{n=0}^{2^{12}-1} \vert x_n^{(m_s)} \vert^2$.
Note that $\vert n^{(m_s)} \rangle$ in Eq.~(\ref{eq:amp:encode2}) is the basis labelled by Eq.~(\ref{eq:computational:label})
within the $m_s$th segment. Therefore, each $\vert \Psi_{\rm c}^{(m_s)} \rangle$ is properly normalized within the segment,
i.e., $\langle \Psi_{\rm c}^{(m_s)}\vert \Psi_{\rm c}^{(m_s)} \rangle=1$.
The quantum state $\vert \Psi_{\rm c}^{(m_s)} \rangle $ in Eq.~(\ref{eq:amp:encode2}) is now expressed with a smaller number of
qubits $L=12$ and each $\vert \Psi_{\rm c}^{(m_s)} \rangle $ is encoded separately into a quantum circuit state
$\hat{\mathcal{C}}^{(m_s)}|0\rangle$ using the AQCE algorithm, which is expected to be easier than the case for $L=16$.
We should however note that the total Hilbert space defining $\vert \tilde{\Psi}_{\rm c}\rangle$ in Eq.~(\ref{eq:amp:sum}) is now
increased to $2^{12 \times 16} = 2^{192}$ from $2^{16}$ for $\vert \Psi_{\rm c}\rangle$ to represent the $2^{16}$ dimensional classical data,
suggesting that the input classical data is mapped into a higher dimensional space via a feature map $\vert \tilde{\Psi}_{\rm c}\rangle$~\cite{Stoudenmire2016,Mitarai2018}.
Although we do not perform any further explicit demonstration, this might find an interesting application of quantum machine learning
based on a kernel method~\cite{Stoudenmire2016,Mitarai2018}.
We employ the AQCE algorithm to encode separately the quantum state $\vert \Psi_{\rm c}^{(m_s)} \rangle $ representing
the $m_s$th segment of the picture
with the control parameters $(M_0,N,\delta M) = (12,100,6)$ and varying the total number $M$ of two-qubit unitary operators
in the quantum circuit $\hat{\mathcal{C}}^{(m_s)}$.
Figures~\ref{fig:picture}(g)-\ref{fig:picture}(i) show the reconstructed pictures by decoding the quantum circuit states
$\hat{\mathcal{C}}^{(m_s)}|0\rangle$ with properly rescaling back the amplitude
$\bar{x}_n^{(m_s)\prime}=\langle n^{(m_s)}|\hat{\mathcal{C}}^{(m_s)}|0\rangle$ by the $m_s$ dependent volume $\sqrt{V_x^{(m_s)}}$.
We find that the original picture is reconstructed very efficiently with a much less number $M$ of two-qubit unitary operators, as
compared with the results of encoding the whole picture without the segmentation shown
in Figs.~\ref{fig:picture}(c)-\ref{fig:picture}(e).
The fidelity between the quantum state $\vert \Psi_{\rm c}^{(m_s)} \rangle$ representing the $m_s$th segment of the picture
and the quantum circuit state $\hat{\mathcal{C}}^{(m_s)} \vert 0 \rangle$ is also shown in Fig.~\ref{fig:olap:picture}.
We observe that the fidelity can be improved much more efficiently with increasing $M$ when the whole picture is divided into many
pieces so as to decrease the dimension of the classical data that is to be encoded into a quantum circuit with a less number $L$ of
qubits.
Recall now that a $64 \times 64$ pixels picture is given by a $2^{12} = 4096$ dimensional vector.
While an SU(4) operator (i.e, a two-qubit unitary operator with ignoring a global phase) is parametrized
by 15 independent real parameters,
two consecutive single-qubit Euler rotations are redundant (see Fig.~\ref{fig:twogate}).
Removing these redundancies, the number of the independent real parameters
for a quantum circuit with $M$ number of SU(4) operators is $9M + L \times 3$.
Therefore, the number of the independent real parameters for the quantum circuit with $M = 480$ on $L=12$ qubits is
almost equal to the dimension of the segmented picture.
As shown in Fig.~\ref{fig:picture}(i), we indeed find that the reconstructed picture reproduces the original picture with
a reasonable accuracy.
How is this quantum circuit encoding of classical data potentially useful in the context of quantum machine learning?
In order to make good use of quantum computer for machine learning, classical data has to be implemented into a
quantum device in the first place. As explained above, a quantum state representing classical data via, e.g.,
the amplitude encoding is generally too complicated to be prepared in a quantum device naively.
The quantum circuit encoding proposed here can be employed for this purpose to approximately construct a quantum circuit
representing a quantum state of classical data with controlled accuracy.
This can be done on a classical computer and
the obtained quantum circuit is implemented in a quantum device for further processing of machine learning.
In the next section, we shall demonstrate experimentally some of this procedure.
\section{\label{sec:exp}Experimental demonstration using a quantum device}
Although the quantum-classical hybrid computation of the AQCE algorithm is in principle possible,
we find that the implementation using currently available quantum devices is practically difficult.
Therefore, here we instead experimentally demonstrate that
the AQCE algorithm indeed generates a quantum circuit that can be implemented on a real quantum device to produce
a desired quantum state with reasonable accuracy.
For this demonstration, we use a quantum device (ibmq\_lima) provided by IBM Quantum~\cite{IBM} and all experimental data
were collected on 15 Octorber 2021.
\subsection{Quantum states in the two-qubit space}\label{sec:twoqubits}
We first consider one of the simplest quantum states, i.e., the singlet state in the two-qubit space (one of the Bell states)
given by
\begin{equation}
\vert \Psi_{\text{2QS}} \rangle = \frac{1}{\sqrt{2}} ( \vert 0 1 \rangle - \vert 1 0 \rangle ),
\label{eq:ex:2qs}
\end{equation}
where $\vert 0 1 \rangle = |0\rangle_0\otimes|1\rangle_1$ and $\vert 1 0 \rangle = |1\rangle_0\otimes|0\rangle_1$,
following the notation introduced at the beginning of Sec.~\ref{sec:encode:obj}.
We apply the AQCE algorithm on a classical computer to encode the quantum state $\vert \Psi_{\text{2QS}}\rangle$ and
obtain within the machine precision that
\begin{equation}
\vert \Psi_{\text{2QS}} \rangle =
\hat{\mathcal{U}}_{0,1}(\mbox{\boldmath{$\theta$}}) \vert 0 \rangle,
\end{equation}
where the quantum circuit $\hat{\mathcal{U}}_{i,j}(\mbox{\boldmath{$\theta$}})$ operating on qubits $i$ and $j$
is given as
\begin{equation}
\begin{split}
\hat{\mathcal{U}}_{i,j}(\mbox{\boldmath{$\theta$}}) & =
\hat{\mathcal{R}}^z_j (\theta_{14})
\hat{\mathcal{R}}^z_j (\theta_{13})
\hat{\mathcal{R}}^z_j (\theta_{12})
\hat{\mathcal{R}}^z_i (\theta_{11})
\hat{\mathcal{R}}^z_i (\theta_{10})
\hat{\mathcal{R}}^z_i (\theta_9)
\\ & \times
\hat{\mathcal{R}}^x_j (\pi/2)
\hat{\mathcal{R}}^x_i (-\pi/2)
\hat{C}_i(\hat{X}_j)
\hat{H}_i \hat{S}_i
\\ & \times
\hat{\mathcal{R}}^z_j (2\theta_7)
\hat{C}_i(\hat{X}_j)
\hat{\mathcal{R}}^z_j (-2\theta_8)
\hat{H}_i
\hat{\mathcal{R}}^x_i (-2\theta_6)
\hat{C}_i(\hat{X}_j)
\\
&
\times
\hat{\mathcal{R}}^z_j (\theta_5)
\hat{\mathcal{R}}^y_j (\theta_4)
\hat{\mathcal{R}}^z_j (\theta_3)
\hat{\mathcal{R}}^z_i (\theta_2)
\hat{\mathcal{R}}^y_i (\theta_1)
\hat{\mathcal{R}}^z_i (\theta_0)
\end{split}
\label{eq:ex:c:qce}
\end{equation}
and the resulting set of parameters $\mbox{\boldmath{$\theta$}} = \{ \theta_0, \theta_1, \Compactcdots, \theta_{14} \}$ is displayed
in Table.~\ref{tab:2q:params}.
The explicit form of the quantum circuit $\hat{\mathcal{U}}_{i,j}(\mbox{\boldmath{$\theta$}})$ and the associated quantum gates
are shown in Fig.~\ref{fig:device:circuit}(a).
Note that the singlet state in Eq.~(\ref{eq:ex:2qs})
can also be prepared simply by
\begin{equation}
\vert \Psi_{\text{2QS}} \rangle = \hat{\mathcal{U}}_{\text{2QS}} \vert 0 \rangle
\end{equation}
with the quantum circuit
\begin{equation}
\hat{\mathcal{U}}_{\text{2QS}} = \hat{C}_0(\hat{X}_1) \hat{H}_0 \hat{X}_1 \hat{X}_0,
\label{eq:ex:c:sing:man}
\end{equation}
as shown in Fig.~\ref{fig:device:circuit}(b).
\begin{table}
\caption{
Sets of parameters $\mbox{\boldmath{$\theta$}} = \{ \theta_0, \theta_1, \Compactcdots, \theta_{14} \}$
for the quantum circuits $\hat{\mathcal{U}}_{0,1}(\mbox{\boldmath{$\theta$}})$ in Eq.~(\ref{eq:ex:c:qce})
[also see Fig.~\ref{fig:device:circuit}(a)] generated by the AQCE algorithm,
encoding the singlet state $\vert \Psi_{\rm 2QS} \rangle$
and the random state $\vert \Psi_{\rm 2QR} \rangle$ in the two-qubit space.}
\label{tab:2q:params}
\begin{tabular}{lrr}
\hline
\hline
{} & singlet state & random state \\
\hline
$\theta_0$ & 1.6823068 & 2.0216448 \\
$\theta_1$ & 3.1415927 & 1.3683389 \\
$\theta_2$ & 0 & $-$2.2863607 \\
$\theta_3$ & $-$0.9758576 & $-$2.8429004 \\
$\theta_4$ & 0 & 1.9027058 \\
$\theta_5$ & $-$1.6678105 & $-$1.8420845 \\
$\theta_6$ & 0.3926991 & 0.7086172 \\
$\theta_7$ & 3.5342917 & 1.1534484 \\
$\theta_8$ & 3.1355175 & 1.6383263 \\
$\theta_9$ & $-$2.6094912 & $-$2.6132016 \\
$\theta_{10}$ & $-$3.1415927 & $-$2.0676228 \\
$\theta_{11}$ & 3.1204519 & 2.1424122 \\
$\theta_{12}$ & $-$1.6869951 & $-$1.2293439 \\
$\theta_{13}$ & $-$3.1415926 & $-$1.8418481 \\
$\theta_{14}$ & 2.4721516 & $-$2.6729236 \\
\hline
\end{tabular}
\end{table}
\begin{figure*}
\includegraphics[width=0.9\hsize]{device_circuit.pdf}
\caption{
(a) Two-qubit unitary operator $\hat{\mathcal{U}}_{i,j}(\mbox{\boldmath{$\theta$}})$ acting on qubit $i$ and $j$ is
implemented on a quantum device by a standard set of quantum gates with 15 parameters
$\mbox{\boldmath{$\theta$}} = \{ \theta_0, \theta_1, \Compactcdots, \theta_{14} \}$
for the rotation angles of single-qubit gates.
(b) Quantum circuit $\hat{\mathcal{U}}_{\text{2QS}}$ that generates the singlet state $\vert \Psi_{\text{2QS}} \rangle $
in the two-qubit space.
(c) Quantum circuit $\hat{\mathcal{U}}_{\rm GHZ}$ that generates the GHZ state $\vert \Psi_{\rm GHZ} \rangle$ in three-qubit space.
(d), (e) Quantum circuit structures obtained by the AQCE algorithm for a quantum state in the three-qubit space,
containing two two-qubit unitary operators acting on qubits that are physically connected in the quantum device employed here.
}
\label{fig:device:circuit}
\end{figure*}
By using the quantum device,
we evaluate in Figs.~\ref{fig:ex:2q}(a) and \ref{fig:ex:2q}(b) the density matrix,
$[\bm{\rho}]_{nn'}=\langle n|\hat{\rho}|n'\rangle$,
of the singlet state generated by the quantum
circuits $\hat{\mathcal{U}}_{\text{2QS}}$ in Eq.~(\ref{eq:ex:c:sing:man})
and $\hat{\mathcal{U}}_{0,1}(\mbox{\boldmath{$\theta$}})$ in Eq.~(\ref{eq:ex:c:qce}), respectively.
Here, $|n\rangle$ and $|n'\rangle$ with $n,n'=0,1,2,3$ are the basis states of $L=2$ qubits labeled as in Eq.~(\ref{eq:computational:label}).
To evaluate the density matrix,
we perform the quantum state tomography by measuring 16 different sets of Pauli strings (including the identity operator)
with length two [see Eqs.~(\ref{eq:dopt}) and (\ref{eq:dopt2})]. Each Pauli string is measured on the quantum device
4096 times and the density matrix $[\bm{\rho}]_{nn'}$ shown in Figs.~\ref{fig:ex:2q}(a) and \ref{fig:ex:2q}(b)
is evaluated from the averaged values over these measurements.
These results are also compared with the exact values.
We find that the density matrices evaluated on the quantum device with the two different quantum circuits, one
obtained by the AQCE algorithm, are rather similar and can both reproduce the exact result with reasonable accuracy.
\begin{figure}
\begin{center}
\includegraphics[width=\hsize]{experiment_2q.pdf}
\caption{(a)
Density matrix $[\bm{\rho}]_{nn'}=\langle n|\hat{\rho}|n'\rangle$
evaluated on the quantum device (ibmq\_lima) with the
quantum circuit $\hat{\mathcal{U}}_{\rm 2QS}$ describing the singlet state $\vert \Psi_{\text{2QS}}\rangle$
in Eq.~(\ref{eq:ex:2qs}).
(b) Same as (a) but with the quantum circuit $\hat{\mathcal{U}}_{0,1}(\mbox{\boldmath{$\theta$}}) $ in Eq.~(\ref{eq:ex:c:qce})
generated by the AQCE algorithm encoding the singlet state $\vert \Psi_{\text{2QS}}\rangle$.
The parameter set $\mbox{\boldmath{$\theta$}}$ is provided in the column ``singlet state" of Table~\ref{tab:2q:params}.
Note that only the real part of density matrix is shown in (a) and (b) because the imaginary part is zero
for the singlet state $\vert \Psi_{\text{2QS}}\rangle$.
(c) Real and (d) imaginary parts of density matrix $[\bm{\rho}]_{nn'}=\langle n|\hat{\rho}|n'\rangle$ evaluated on the quantum
device (ibmq\_lima) with the quantum circuit $\hat{\mathcal{U}}_{0,1}(\mbox{\boldmath{$\theta$}}) $ in Eq.~(\ref{eq:ex:c:qce})
generated by the AQCE algorithm encoding the random state $\vert \Psi_{\text{2QR}}\rangle$ in Eq.~(\ref{eq:ex:2rnd}).
The parameter set $\mbox{\boldmath{$\theta$}}$ is provided in the column ``random state" of Table~\ref{tab:2q:params}.
For comparison, the exact results are also shown by orange bars.
The density matrix is evaluated on the quantum device by the quantum state tomography, measuring 16 different sets of Pauli
strings with length two, and the results shown here are obtained from the averaged values over 4096 measurements
of each Pauli string.
$|n\rangle$ and $|n'\rangle$ with $n,n'=0,1,2,3$ are the basis states of $L=2$ qubits labeled as in Eq.~(\ref{eq:computational:label}).
}
\label{fig:ex:2q}
\end{center}
\end{figure}
For qualitative comparison, we now introduce the following quantity:
\begin{equation}
Q = ({\rm Tr}[ \hat{\rho}_{A} \hat{\rho}_{B}])^{1/2}, \label{eq:qfid}
\end{equation}
where $\hat{\rho}_A$ and $\hat{\rho}_B$ are the density operators of two quantum states $A$ and $B$.
Here we employ this quantity as fidelity of two quantum states because some of the eigenvalues of the
density matrix evaluated from our experimental measurements are negative.
This can be justified when two quantum states $A$ and $B$ are pure states, i.e.,
$\hat{\rho}_A = \vert \Psi_A \rangle \langle \Psi_A \vert$ and $\vert \Psi_B \rangle \langle \Psi_B \vert$,
because in this case $Q = \vert \langle \Psi_A \vert \Psi_B \rangle \vert$.
Using the density matrix evaluated experimentally in Fig.~\ref{fig:ex:2q}(b),
we find that the fidelity $Q$ for the exact singlet state $\vert \Psi_{\text{2QS}} \rangle$ and the singlet state generated
by the quantum circuit $\hat{\mathcal{U}}_{0,1}(\mbox{\boldmath{$\theta$}})$ is as large as 0.9512.
This is indeed comparable to the fidelity $Q=0.9607$ for the exact singlet state $\vert \Psi_{\text{2QS}} \rangle$
and the singlet state generated by the quantum circuit $\hat{\mathcal{U}}_{\rm 2QS}$.
Next, we consider the case where a quantum state is more complex in the sense that the associated density matrix has
many nonzero elements.
To this end, we examine a random state described by the following state in the two-qubit space:
\begin{equation}
\begin{split}
\vert \Psi_{\text{2QR}} \rangle = & ( 0.36179353 + {\rm i} 0.42519915 ) \vert 0 0 \rangle \\
& + ( 0.14876111 + {\rm i} 0.33156910 ) \vert 1 0 \rangle \\
& + (-0.02356009 + {\rm i} 0.68066637 ) \vert 0 1 \rangle \\
& + ( 0.23101109 - {\rm i} 0.19752287 ) \vert 1 1 \rangle,
\end{split}
\label{eq:ex:2rnd}
\end{equation}
where $\vert 0 0 \rangle = |0\rangle_0\otimes|0\rangle_1$, etc.
The coefficients are randomly determined as follows: We first use a random generator in the c++ standard library
for the normal distribution with the mean 0 and the standard deviation 1 to determine the real and imaginary parts of each
coefficient and then normalize the resulting state~\cite{Mezzadri2006}.
We apply the AQCE algorithm on a classical computer to encode the quantum state $\vert \Psi_{\text{2QR}} \rangle$
and obtain the quantum circuit
$\hat{\mathcal{U}}_{0,1}(\mbox{\boldmath{$\theta$}})$ with the
parameter set $\mbox{\boldmath{$\theta$}} = \{ \theta_1, \theta_2, \Compactcdots, \theta_{14} \}$
given in Table~\ref{tab:2q:params}, which can represent $\vert \Psi_{\text{2QR}} \rangle$ exactly within the machine precision.
The density matrix $[\bm{\rho}]_{nn'}$
of the random state generated by the quantum
circuits $\hat{\mathcal{U}}_{0,1}(\mbox{\boldmath{$\theta$}})$ is evaluated on the quantum device in
Figs.~\ref{fig:ex:2q}(c) and \ref{fig:ex:2q}(d) by using the quantum state tomography described above.
Similar to the case of the singlet state, we find that the results evaluated on the quantum device are rather well compared
with the exact values.
Indeed, we find that the fidelity $Q$ for the exact random state $\vert \Psi_{\text{2QR}} \rangle$ and
the random state generated by the quantum circuit $\hat{\mathcal{U}}_{0,1}(\mbox{\boldmath{$\theta$}})$ is as large as 0.9592.
\subsection{Quantum states in the three-qubit space}
The demonstrations shown above are focused on quantum states in the two-qubit space.
It is also highly interesting to continue a similar demonstration for a quantum state in a larger Hilbert space.
Let us now consider the GHZ state in the three-qubit space given by
\begin{equation}
\vert \Psi_{\rm GHZ} \rangle = \frac{1}{\sqrt{2}} ( \vert 0 0 0 \rangle + \vert 1 1 1 \rangle ),
\label{eq:ghz}
\end{equation}
where $\vert 0 0 0 \rangle = |0\rangle_0\otimes|0\rangle_1\otimes|0\rangle_2$
and $\vert 1 1 1 \rangle = |1\rangle_0\otimes|1\rangle_1\otimes|1\rangle_2$,
following the notation introduced at the beginning of Sec.~\ref{sec:encode:obj}.
It is known that the GHZ state can be prepared simply by
\begin{equation}
\vert \Psi_{\rm GHZ} \rangle = \hat{\mathcal{U}}_{\rm GHZ} \vert 0 \rangle
\end{equation}
with the quantum circuit
\begin{equation}
\hat{\mathcal{U}}_{\rm GHZ} =
\hat{C}_1(\hat{X}_2)
\hat{C}_0(\hat{X}_1)
\hat{H}_0
\label{eq:ex:c:ghzs:man}
\end{equation}
acting on qubits 0, 1, and 2, as shown in Fig.~\ref{fig:device:circuit}(c).
We also perform the AQCE algorithm on a classical computer to encode the GHZ state into a quantum circuit.
Considering a set of bonds $\mathbb{B}$ in the AQCE algorithm, it is wise to include only pairs of qubits that are physically
connected in the quantum device so as to decrease the number of extra quantum gates.
In the quantum device employed in this demonstration, there are only two pairs of qubits:
$\mathbb{B} = \{ \{ 0,1 \}, \{ 1,2 \} \}$.
However, it is not obvious in advance how many two-qubit unitary operators $\hat{\mathcal{U}}_{i,j}(\mbox{\boldmath{$\theta$}})$
are necessary to encode the GHZ state.
By performing the AQCE algorithm on a classical computer, we find within the machine precision that
\begin{equation}
\vert \Psi_{\text{GHZ}} \rangle = \hat{\mathcal{U}}_{0,1,2}(\mbox{\boldmath{$\theta$}})|0\rangle
\label{eq:ex:ghz}
\end{equation}
with the quantum circuit $\hat{\mathcal{U}}_{0,1,2}(\mbox{\boldmath{$\theta$}})$ being composed of two two-qubit unitary operators,
\begin{equation}
\hat{\mathcal{U}}_{0,1,2}(\mbox{\boldmath{$\theta$}}) =
\hat{\mathcal{U}}_{1,2}(\mbox{\boldmath{$\theta$}}^1)
\hat{\mathcal{U}}_{0,1}(\mbox{\boldmath{$\theta$}}^0),
\label{eq:ex:c:qce:3}
\end{equation}
where $\mbox{\boldmath{$\theta$}} = \{ \mbox{\boldmath{$\theta$}}^{0}, \mbox{\boldmath{$\theta$}}^{1} \}$ and
the resulting sets of parameters
$\mbox{\boldmath{$\theta$}}^m = \{ \theta_0^m, \theta_1^m, \Compactcdots, \theta_{14}^m \}$ ($m=0$ and $1$)
for $\hat{\mathcal{U}}_{0,1}(\mbox{\boldmath{$\theta$}}^0)$ and
$\hat{\mathcal{U}}_{1,2}(\mbox{\boldmath{$\theta$}}^1)$ are given in Table~\ref{tab:ghz:params}.
The schematic structure of the quantum circuit is shown in Fig.~\ref{fig:device:circuit}(d).
Notice in Table~\ref{tab:ghz:params} that $\theta_0^1=\theta_1^1=\theta^1_2=0$ because these parameters correspond to the
first Euler rotation (acting on qubit 1) of the second two-qubit unitary
operator $ \hat{\mathcal{U}}_{1,2}(\mbox{\boldmath{$\theta$}}^1)$,
which can be absorbed into the last Euler rotation of the first two-qubit unitary operator
$ \hat{\mathcal{U}}_{0,1}(\mbox{\boldmath{$\theta$}}^1) $.
We should also note that since the GHZ state is translational symmetric, the quantum circuit with the structure shown
in Fig.~\ref{fig:device:circuit}(e) is topologically equivalent.
The AQCE algorithm select one of them and, in this particular demonstration,
the quantum circuit with the structure shown in Fig.~\ref{fig:device:circuit}(d) is selected.
\begin{table}
\caption{Sets of parameters $\mbox{\boldmath{$\theta$}}^m = \{ \theta_0^m, \theta_1^m, \Compactcdots, \theta_{14}^m \}$
($m=0$ and $1$)
for the quantum circuit $\hat{\mathcal{U}}_{0,1,2}(\mbox{\boldmath{$\theta$}})$ in Eq.~(\ref{eq:ex:c:qce:3})
[also see Fig.~\ref{fig:device:circuit}(d)] generated by the AQCE algorithm,
encoding the GHZ state $\vert \Psi_{\rm GHZ}\rangle$
in the three-qubit space. }\label{tab:ghz:params}
\begin{tabular}{lrr}
\hline
\hline
{} & $m=0$ & $m=1$ \\
\hline
$\theta_{0}^{m}$ & 0.70081942 & 0 \\
$\theta_{1}^{m}$ & 1.59343588 & 0 \\
$\theta_{2}^{m}$ & $-$2.99819974 & 0 \\
$\theta_{3}^{m}$ & 3.01209222 & 0.03670498 \\
$\theta_{4}^{m}$ & 1.45398911 & 1.57079633 \\
$\theta_{5}^{m}$ & $-$2.86368415 & 0 \\
$\theta_{6}^{m}$ & 0.25868788 & 0.25573854 \\
$\theta_{7}^{m}$ & 0.14505637 & 0 \\
$\theta_{8}^{m}$ & $-$0.63764672 & $-$2.35619449 \\
$\theta_{9}^{m}$ & $-$1.71718177 & 0 \\
$\theta_{10}^{m}$ & $-$2.80049545 & $-$3.14159265 \\
$\theta_{11}^{m}$ & $-$2.60125707 & 0.04066063 \\
$\theta_{12}^{m}$ & $-$2.13304918 & $-$1.57079633 \\
$\theta_{13}^{m}$ & 1.68590984 & $-$1.57079633 \\
$\theta_{14}^{m}$ & $-$2.04727870 & $-$1.11421776 \\
\hline
\end{tabular}
\end{table}
Next, using the quantum device, we evaluate in Figs.~\ref{fig:ex:3q}(a) and \ref{fig:ex:3q}(b) the density matrix,
$[\bm{\rho}]_{nn'}=\langle n|\hat{\rho}|n'\rangle$,
of the GHZ state generated by the quantum
circuits $\hat{\mathcal{U}}_{\text{GHZ}}$ in Eq.~(\ref{eq:ex:c:ghzs:man})
and $\hat{\mathcal{U}}_{0,1,2}(\mbox{\boldmath{$\theta$}})$ in Eq.~(\ref{eq:ex:c:qce:3}), respectively.
Here, $|n\rangle$ and $|n'\rangle$ with $n,n'=0,1,2,\dots,7$ are the basis states of $L=3$ qubits labeled as in Eq.~(\ref{eq:computational:label}).
Similar to the cases of $L=2$ qubits discussed in Sec.~\ref{sec:twoqubits}, we
evaluate the density matrix by performing the quantum state tomography, where 64 different sets of Pauli strings
(including the identity operator) with length three [see Eqs.~(\ref{eq:dopt}) and (\ref{eq:dopt2}) for $L=2$ qubits] are measured.
The density matrix $[\bm{\rho}]_{nn'}$ shown in Figs.~\ref{fig:ex:3q}(a) and \ref{fig:ex:3q}(b)
is evaluated from the averaged values of Pauli strings measured 4096 times each.
Although the number of quantum gates in the quantum circuit $\hat{\mathcal{U}}_{0,1,2}(\mbox{\boldmath{$\theta$}})$ is
much larger than that in the quantum circuit $\hat{\mathcal{U}}_{\text{GHZ}}$, we find that
the density matrices evaluated on the quantum device with these two different quantum circuits
are rather similar and are in reasonable agreement with the exact values.
More quantitatively, using the density matrix evaluated experimentally in Fig.~\ref{fig:ex:3q}(b),
we find that the fidelity $Q$ for the exact GHZ state $\vert \Psi_{\rm GHZ} \rangle$ and
the GHZ state generated by the quantum circuit $\hat{\mathcal{U}}_{0,1,2}(\mbox{\boldmath{$\theta$}})$
is as large as 0.8906. This is comparable to the fidelity $Q = 0.9189$ for the exact GHZ state $\vert \Psi_{\rm GHZ} \rangle$
and the GHZ state generated by the quantum circuit $\hat{\mathcal{U}}_{\rm GHZ}$.
\begin{figure}
\includegraphics[width=\hsize]{experiment_3q.pdf}
\caption{
(a) Density matrix $[\bm{\rho}]_{nn'}=\langle n|\hat{\rho}|n'\rangle$
evaluated on the quantum device (ibmq\_lima) with the
quantum circuit $\hat{\mathcal{U}}_{\rm GHZ}$ describing the GHZ state $\vert \Psi_{\text{GHZ}}\rangle$
in Eq.~(\ref{eq:ghz}).
(b) Same as (a) but with the quantum circuit $\hat{\mathcal{U}}_{0,1,2}(\mbox{\boldmath{$\theta$}}) $ in Eq.~(\ref{eq:ex:c:qce:3})
generated by the AQCE algorithm encoding the GHZ state $\vert \Psi_{\text{GHZ}}\rangle$.
The parameter set $\mbox{\boldmath{$\theta$}}$ is provided in Table~\ref{tab:ghz:params}.
Note that only the real part of density matrix is shown in (a) and (b)
because the imaginary part is zero for the GHZ state $\vert \Psi_{\rm GHZ} \rangle$.
(c) Real and (d) imaginary parts of density matrix $[\bm{\rho}]_{nn'}=\langle n|\hat{\rho}|n'\rangle$ evaluated on the quantum
device (ibmq\_lima) with the quantum circuit $\hat{\mathcal{U}}^\prime_{0,1,2}(\mbox{\boldmath{$\theta$}}) $
in Eq.~(\ref{eq:ex:c:qce:4})
generated by the AQCE algorithm encoding the random state $\vert \Psi_{\text{3QR}}\rangle$ in Eq.~(\ref{eq:3rnd}).
The parameter set $\mbox{\boldmath{$\theta$}}$ is provided in Table~\ref{tab:3rnd}.
For comparison, the exact results are also shown by orange bars.
The density matrix is evaluated on the quantum device by the quantum state tomography, measuring 64 different sets of Pauli
strings with length three, and the results shown here are obtained from the averaged values over 4096 measurements
of each Pauli string.
$|n\rangle$ and $|n'\rangle$ with $n,n'=0,1,2,\dots,7$ are the basis states of $L=3$ qubits labeled as in
Eq.~(\ref{eq:computational:label}).
}
\label{fig:ex:3q}
\end{figure}
Finally, we examine a random state in the three-qubit space:
\begin{equation}
\begin{split}
\vert \Psi_{\text{3QR}} \rangle = &
( -0.41507377 + {\rm i} 0.14526187 ) \vert 000 \rangle \\
& + ( 0.03169105 + {\rm i} 0.35848024 ) \vert 1 0 0 \rangle \\
& + ( -0.23166622 + {\rm i} 0.21332733 ) \vert 0 1 0 \rangle \\
& + ( -0.32248929 - {\rm i} 0.06104028 ) \vert 1 1 0 \rangle \\
& + ( -0.11551530 + {\rm i} 0.13972069 ) \vert 0 0 1 \rangle \\
& + ( 0.26960898 - {\rm i} 0.03973709 ) \vert 1 0 1 \rangle \\
& + ( 0.00215509 + {\rm i} 0.44364270 ) \vert 0 1 1 \rangle \\
& + ( 0.01417350 + {\rm i} 0.40747913 ) \vert 1 1 1 \rangle,
\end{split}
\label{eq:3rnd}
\end{equation}
where $\vert 0 0 0\rangle = \vert 0\rangle_0\otimes \vert 0\rangle_1 \otimes \vert 0\rangle_2$, etc. and
the coefficients are randomly determined as in the case of the random state $\vert \Psi_{\rm 2QR} \rangle$ in Eq.~(\ref{eq:ex:2rnd}).
We perform the AQCE algorithm on a classical computer and obtain within the machine precision
that
\begin{equation}
\vert \Psi_{\text{3QR}} \rangle =
\hat{\mathcal{U}}^{\prime}_{0,1,2}(\mbox{\boldmath{$\theta$}}) \vert 0 \rangle
\end{equation}
with the quantum circuit
\begin{equation}
\hat{\mathcal{U}}^{\prime}_{0,1,2}(\mbox{\boldmath{$\theta$}}) =
\hat{\mathcal{U}}_{0,1}(\mbox{\boldmath{$\theta$}}^1)
\hat{\mathcal{U}}_{1,2}(\mbox{\boldmath{$\theta$}}^0),
\label{eq:ex:c:qce:4}
\end{equation}
where $\mbox{\boldmath{$\theta$}} = \{ \mbox{\boldmath{$\theta$}}^{0}, \mbox{\boldmath{$\theta$}}^{1} \}$ and
the resulting sets of parameters
$\mbox{\boldmath{$\theta$}}^m = \{ \theta_0^m, \theta_1^m, \Compactcdots, \theta_{14}^m \}$ ($m=0$ and $1$)
for $\hat{\mathcal{U}}_{1,2}(\mbox{\boldmath{$\theta$}}^0)$ and
$\hat{\mathcal{U}}_{0,1}(\mbox{\boldmath{$\theta$}}^1)$ are given in Table~\ref{tab:3rnd}.
The schematic structure of the quantum circuit $ \hat{\mathcal{U}}^{\prime}_{0,1,2}(\mbox{\boldmath{$\theta$}})$ is
shown in Fig.~\ref{fig:device:circuit}(e).
Here, we should note that, depending of the initialization process, the AQCE algorithm also finds a quantum circuit
forming the structure shown in Fig.~\ref{fig:device:circuit}(d) with a different set of parameters $\mbox{\boldmath{$\theta$}}$,
which can encode the random state $\vert \Psi_{\text{3QR}} \rangle$ exactly within the machine precision.
This implies that two two-qubit unitary operators are enough to encode any quantum state in the three-qubit space.
\begin{table}
\caption{
Sets of parameters $\mbox{\boldmath{$\theta$}}^m = \{ \theta_0^m, \theta_1^m, \Compactcdots, \theta_{14}^m \}$
($m=0$ and $1$)
for the quantum circuit $\hat{\mathcal{U}}^\prime_{0,1,2}(\mbox{\boldmath{$\theta$}})$ in Eq.~(\ref{eq:ex:c:qce:4})
[also see Fig.~\ref{fig:device:circuit}(e)] generated by the AQCE algorithm,
encoding the random state $\vert \Psi_{\rm 3QR}\rangle$
in the three-qubit space.
}
\label{tab:3rnd}
\begin{tabular}{lrr}
\hline
\hline
{} & $m=0$ & $m=1$ \\
\hline
$\theta_{0}^{m}$ & 1.39099869 & 0.12699636 \\
$\theta_{1}^{m}$ & 1.22253363 & 1.49657252 \\
$\theta_{2}^{m}$ & $-$1.22510250 & $-$0.96112628 \\
$\theta_{3}^{m}$ & $-$1.04474694 & 0 \\
$\theta_{4}^{m}$ & 1.85347535 & 0 \\
$\theta_{5}^{m}$ & $-$2.24417198 & 0 \\
$\theta_{6}^{m}$ & $-$1.06037512 & $-$0.39222573 \\
$\theta_{7}^{m}$ & $-$0.87968547 & 0.60984155 \\
$\theta_{8}^{m}$ & $-$0.05457889 & $-$0.07696758 \\
$\theta_{9}^{m}$ & 0.03359139 & 0.48406694 \\
$\theta_{10}^{m}$ & 2.27862931 & $-$0.36703453 \\
$\theta_{11}^{m}$ & 0.49867804 & 0.19553219 \\
$\theta_{12}^{m}$ & 2.89140237 & $-$1.17312888 \\
$\theta_{13}^{m}$ & $-$0.80188802 & $-$2.29176295 \\
$\theta_{14}^{m}$ & 1.52534544 & $-$3.06220240 \\
\hline
\end{tabular}
\end{table}
We perform the quantum state tomography on the quantum device to evaluate
the density matrix $[\bm{\rho}]_{nn'}$
of the random state generated by the quantum
circuit $\hat{\mathcal{U}}^{\prime}_{0,1,2}(\mbox{\boldmath{$\theta$}})$.
The 64 different Pauli strings with length three are measured 4096 times each and the density matrix shown in
Figs.~\ref{fig:ex:3q}(c) and \ref{fig:ex:3q}(d) is obtained from the averaged values of these measurements.
We find that the fidelity $Q$ for the exact random state $\vert \Psi_{\text{3QR}} \rangle$
and the random state generated by the quantum circuit $\hat{\mathcal{U}}^{\prime}_{0,1,2}(\mbox{\boldmath{$\theta$}})$
is as large as 0.9051, suggesting good accordance with the exact result.
\section{Summary} \label{sec:summary}
We have proposed the quantum circuit encoding algorithm to encode a given quantum state $|\Psi\rangle$
onto a quantum circuit $\hat{\mathcal{C}}$ composed of $K$-qubit unitary operators $\hat{\mathcal{U}}_m$
by maximizing the absolute value of the fidelity $F=\langle 0| \hat{\mathcal{C}}^\dag |\Psi\rangle$.
The fidelity $|F|$ can be maximized deterministically by sequentially optimizing each unitary operator $\hat{\mathcal{U}}_m$
one by one via the SVD of the fidelity tensor matrix ${\bm F}_m$,
a similar scheme used for the optimization in the tensor network method.
The most demanding part of the algorithm is to construct the fidelity tensor matrix ${\bm F}_m$ and we have shown
how a quantum computer can be utilized for this task.
The AQCE algorithm proposed here determines not only the form of the individual unitary operators but also the optimal location of qubits
in the circuit on which each unitary operator acts. Therefore, it allows us to
generate an optimal quantum circuit of a given quantum state automatically.
The elementary quantum gates are algebraically assigned when the encoded quantum circuit is
composed of two-qubit unitary operators. We emphasize that the AQCE algorithm proposed here does not rely on any parametrized
quantum circuit as in variational quantum algorithms such as the variational quantum eigensolver
and thus the associated parameter optimization is not required.
Using numerical simulations, we have demonstrated the AQCE
algorithm to encode a ground state of a quantum many-body system,
such as the spin-1/2 isotropic antiferromagnetic Heisenberg model and the spin-1/2 XY model in one spatial dimension,
onto a quantum circuit composed of two-qubit unitary operators.
We have also compared the results with the quantum circuit encoding of the same quantum state onto
a quantum circuit in a given circuit structure
such as the Trotter-like and MERA-like circuit structures and found that the quantum circuit generated by the AQCE algorithm is
better than the Trotter-like circuit and is equally competitive with the MERA-like circuit.
We have also demonstrated that the AQCE algorithm can be applied to encode a quantum state
representing classical data such as a classical image. As a concrete example, we considered a gray scale picture of
$256\times256$ pixels, which can be expressed as a quantum state $|\Psi_{\rm c}\rangle$ on $16$ qubits by
using the amplitude encoding, and thus can be encoded onto a quantum circuit $\hat{\mathcal{C}}$ by employing
the AQCEalgorithm.
Although the picture reconstructed by decoding the quantum circuit state $\hat{\mathcal{C}}|0\rangle$ improves its quality
systematically with increasing the number of two-qubit unitary operators in the quantum circuit $\hat{\mathcal{C}}$,
the improvement is relatively slow if the size of the picture is large.
Therefore, we have also made a different attempt by dividing the original picture into 16 pieces, which thus allows us to
represent each segment of the picture of $64\times64$ pixels with a quantum state $|\Psi_{\rm c}^{(m_s)}\rangle$ on $12$ qubits
for $m_s=1,2,\dots,16$.
This implies that the original classical data is represented by a direct product of 16 quantum states
$|\Psi_{\rm c}^{(m_s)}\rangle$, which is thus defined in a higher dimensional space than the input classical data.
We have encoded each quantum state $|\Psi_{\rm c}^{(m_s)}\rangle$ separately onto a different quantum circuit
$\hat{\mathcal{C}}^{(m_s)}$ and found that the quality of the reconstructed picture by decoding all these quantum circuit states
$\hat{\mathcal{C}}^{(m_s)}|0\rangle$ is much improved.
This is encouraging for a near-term application because, depending on available quantum devices,
one can adjust the number of qubits by dividing classical data into multiple pieces.
In the context of quantum machine learning,
the AQCE algorithm would be potentially useful for finding an optimal quantum circuit, which can be
done classically, in order to prepare a
quantum state representing classical data that is to be processed on a quantum computer for machine learning.
Moreover, we have used the quantum device provided by IBM Quantum to demonstrate experimentally that quantum circuits
generated by the AQCE algorithm can be implemented on a real quantum device to produce a desired quantum state
with reasonable accuracy. For this purpose, we have considered the well-known quantum states, such as the singlet state and
the GHZ state, as well as random states in the two- and three-qubit spaces, and shown that the density matrix evaluated
on the quantum device for the quantum circuits obtained by the AQCE algorithm is indeed compatible with the
exact values.
As clearly demonstrated for several examples,
the AQCE algorithm can encode a given quantum state onto a quantum circuit with controlled
accuracy by varying the number $M$ of unitary operators $\hat{\mathcal{U}}_m$ in the quantum circuit. However, we have observed
that the improvement of accuracy with increasing $M$ becomes sometimes slower when the number of qubits is large.
There are two possible ways to further improve the AQCE algorithm. One is to improve the procedure of increasing the number of
unitary operators by $\delta M$ in the enlargement step of the algorithm.
The procedure adopted as a prototype algorithm in this paper is to simply insert $\delta M$ new
unitary operators at the end of the quantum circuit (see Fig.~\ref{fig:aqce}). We have found that this simple strategy
is not the most efficient.
Instead, one may as well insert these new unitary operators in any location among already existing unitary operators.
However, this is certainly more costly if a brute-force search is used.
Another way to improve the AQCE algorithm is related to how to generate and update unitary operators in the quantum circuit.
In all the demonstrations,
a quantum state is encoded directly onto a quantum circuit composed of unitary operators acting only on two qubits.
However, as described
in Secs.~\ref{sec:encode}, the space on which unitary operators act is not necessary the two-qubit space but the AQCE algorithm
can encode a quantum state more generally onto a quantum circuit composed of $K$-qubit unitary operators with $K>2$.
One possible strategy is to encode a quantum state first onto a quantum circuit composed of unitary operators acting on a large
qubit space, and these unitary operators are then encoded into unitary operators acting on a smaller qubit space.
We have found that this procedure can improve the accuracy significantly when the number of qubits is large and more details will
be reported elsewhere.
\section*{Acknowledgement}
The authors are grateful to K. Seki and Y. Otsuka for fruitful discussion.
The calculation has been performed on the RIKEN supercomputer system (HOKUSAI GreatWave) and
the supercomputer Fugaku installed in RIKEN R-CCS.
This work was supported by Grant-in-Aid for Scientific Research (A) (No.~JP21H03455),
for Young Scientists (B) (No.~JP17K14359), and
for Transformative Research Areas (A) (No.~21H05191) from MEXT, Japan,
and by JST PRESTO (No. JPMJPR1911), Japan.
This work was also supported in part by MEXT Q-LEAP (No.~JPMXS0120319794), Japan, and
by the COE research grant in computational science from
Hyogo Prefecture and Kobe City through Foundation for Computational Science.
|
2024-02-18T23:40:24.649Z
|
2021-12-30T02:26:41.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14524","language":"en","timestamp":1640831201000,"url":"https:\/\/arxiv.org\/abs\/2112.14524","yymm":"2112"}
|
proofpile-arXiv_000-10057
|
{"provenance":"002.jsonl.gz:10058"}
| null | null |
\section{Introduction}
In recent years, there have been intensive studies on the developments and applications of quantum kinetic theory (QKT) for spin transport of relativistic fermions. In order to study non-equilibrium transport phenomena beyond e.g. the renown chiral magnetic effect (CME) \cite{Vilenkin:1980fu,Nielsen:1983rb,Alekseev:1998ds,Kharzeev:2004ey,Kharzeev:2007jp,Fukushima:2008xe} and chiral vortical effect (CVE) \cite{Vilenkin:1979ui,Erdmenger:2008rm,Banerjee:2008th,Son:2009tf,Landsteiner:2011cp} in chiral matter composed of massless Weyl fermions, the chiral kinetic theory (CKT) as a QKT that captures the chiral anomaly \cite{Adler:1969gk,Bell:1969ts} and spin-orbit interaction could be constructed by introducing the Berry phase as quantum corrections from the adiabatic approximation \cite{Son:2012wh,Stephanov:2012ki,Chen:2013iga,Chen:2014cla,Chen:2015gta}. Alternatively, by employing the Wigner-function approach based on quantum field theories with the $\hbar$ expansion \cite{Gao:2012ix,Chen:2012ca,Son:2012zy}, the covariant CKT with systematic inclusions of both background electromagnetic fields and collisions involving quantum corrections has been later obtained \cite{Hidaka:2016yjf,Hidaka:2017auj}. There is a considerable number of related and follow-up work \cite{Manuel:2013zaa,Manuel:2014dza,Hidaka:2018ekt,Mueller:2017arw,Mueller:2017lzw,Huang:2018wdl,Yang:2018lew,Liu:2018xip,Lin:2019ytz,Lin:2019fqo,Shi:2020htn,Fauth:2021nwe,Chen:2021azy,Luo:2021uog,Fang:2022ttm}. In phenomenology, the CKT was broadly applied to relativistic heavy ion collisions \cite{Sun:2017xhx,Sun:2018idn,Liu:2019krs}, Weyl semimetals \cite{Gorbar:2016ygi,Gorbar:2016sey,Gorbar:2021ebc}, and neutrino transport in core-collapse supernovae \cite{Yamamoto:2020zrs,Yamamoto:2021hjs}.
On the other hand, due to the observations of spin polarization of Lambda hyperons and spin alignments of vector mesons in heavy-ion experiments at RHIC and LHC \cite{STAR:2017ckg,STAR:2019erd,ALICE:2019aid,Singha:2020qns} motivated by Refs.~\cite{Liang:2004ph,Liang:2004xn,Becattini2013a} in theory, further studies on the generalization of CKT to the QKT for massive fermions has been carried out in order to understand the dynamical spin polarization of strange quarks traveling through the quark gluon plasmas (QGP). In light of the Wigner-function approaches, such a collisionless QKT to track entangled spin and charge transport has been derived in Refs.~\cite{Gao:2019znl,Weickgenannt:2019dks,Hattori:2019ahi,Wang:2019moi}. Later, the systematic inclusion of collisions with quantum corrections was achieved by using the Kadanoff-Baym approach \cite{Yang:2020hri,Sheng:2021kfc} and the extended phase space with non-local collisions \cite{Weickgenannt:2020aaf,Weickgenannt:2021cuo}. See also Refs.~\cite{Florkowski:2018ahw,Zhang:2019xya,Kapusta:2019sad,Liu:2020flb,Li:2019qkf,Wang:2020pej,Wang:2021qnt} for related studies of QKT and spin transport. In particular, from detailed balance of effective models, the vanishing collision term in QKT yields the Wigner function with vorticity correction that gives rise to spin polarization in global equilibrium \cite{Weickgenannt:2020aaf,Wang:2020pej}. Such a result matches the one previously found from different methods \cite{Becattini2013a,Fang:2016vpj} that provides the modified Cooper-Frye formula dictated by thermal vorticity, which successfully describes the global spin polarization observed by STAR collaboration after implementing numerical simulations \cite{Karpenko:2016jyx,Li:2017slc,Xie:2017upb,Wei:2018zfb,Ryu:2021lnx}. In fact, the vanishing collision term of the CKT with Coulomb scattering yields the Wigner functions in local equilibrium, which incorporates extra corrections such as shear and chemical-potential gradient effects \cite{Hidaka:2017auj}. These corrections for massive fermions were later obtained by using the linear response theory \cite{Liu:2020dxg,Liu:2021uhn} and the statistical-field theory \cite{Becattini:2021suc} (see also Refs.\cite{Yi:2021ryh,Liu:2021nyg}). Especially, the shear correction could lead to a substantial contributions on local spin polarization \cite{Fu:2021pok,Becattini:2021iol,Yi:2021ryh,Florkowski:2021xvy,Yi:2021unq,Sun:2021nsg}, which is crucial to explain the experimental measurements \cite{STAR:2019erd}. However, it is also found that the local polarization is sensitive to the equation of state and freeze-out temperature \cite{Becattini:2021iol,Yi:2021ryh}. Therefore, further studies involving non-equilibrium corrections from collisions are still required.
Nevertheless, the role of gluons or color degrees of freedom that may potentially affect the spin transport of quarks has been relatively overlooked in previous studies. Even though the thermalized gluons could lead to the spin diffusion through classical collisional effects shown in Refs.~\cite{Li:2019qkf,Yang:2020hri}, the quantum correction as a source term triggering the spin polarization from gluons is currently unknown\footnote{There are studies for the construction of the QKT for polarized photons with possible generalization to weakly-coupled gluons \cite{Huang:2020kik,Hattori:2020gqh}, while the direct application to QGP is not yet feasible.}. On the other hand, in addition to the scattering with on-shell gluons, there exist dynamically generated chromo-electromagnetic fields originating from Weibel-type instabilities in expanding QGP \cite{Mrowczynski:1988dz,Mrowczynski:1993qm,Romatschke:2003ms} that may potentially influence the spin transport of quarks in heavy ion collisions. Such color fields could result in anomalous dissipative transport that may dominate over the collisional effects in weakly coupled quantum chromodynamics (QCD) as proposed in Refs.~\cite{Asakawa:2006tc,Asakawa:2006jn}. See also Refs.~\cite{Mrowczynski:2017kso,Carrington:2020sww} for the analogous effect upon heavy-quark transport. The possible impact upon spin transport of quarks from color fields has been recently found by using a similar analysis \cite{Muller:2021hpe}, which in particular stems from the parity-odd correlators of color fields that could be potentially generated in different phases of heavy ion collisions from distinct mechanisms \cite{Joyce:1997uy,Akamatsu:2013pjd,Mace:2016svc,Tanji:2016dka,Mace:2019cqo}. In this paper, we present the derivation of QKT for massive quarks coupled with background color fields applied in Ref.~\cite{Muller:2021hpe} via the Wigner-function approach and effective power counting. More general results and details for the derivation of source terms in color-singlet Wigner functions and kinetic equations responsible for spin polarization are shown. Furthermore, we investigate the axial Ward identity from the axial charge current led by parity-odd color-field correlators in the constant-field limit. In addition, we perform a similar analysis for massless quarks by utilizing the CKT with background color fields constructed in Ref.~\cite{Luo:2021uog}.
Since the application of the derived QKT to QGP relies on several assumptions, here we briefly summarize the validity and critical approximations of our approach. Similar to the generic QKT with $\hbar$ expansion, it is required the energy of a quasi-particle is much larger than the gradient scale, $\epsilon_{\bm p}\gg \mathcal{O}(\partial)$. For the QKT of massive quarks derived in Sec.~\ref{sec:sol_and_KE}, it is further subject to $m\gg \mathcal{O}(\partial)$ with $m$ being the quark mass due to a technical reason. Our approach is applicable to weakly coupled QCD and the collision term is assumed to be suppressed by sufficiently strong background color fields. Moreover, the other sources for spin polarization pertinent to the fluid properties such as thermal vorticity are assumed to be relatively small and thus neglected. In order to further solve for dynamical spin polarization from kinetic equations, we further assume the correlation function of color fields takes a Gaussian form depending on only the time difference in light of space-time translational invariance and propose a hierarchy for different color-field correlators according to the screening of chromo-electric fields.
This article is organized as follows: In Sec.~\ref{sec:KB_eq}, we derive the master equations from free-streaming Kadanoff-Baym equation for massive quarks under background color fields. In Sec.~\ref{sec:sol_and_KE}, we solve for the perturbative solution and kinetic equations up to $\mathcal{O}(\hbar)$ with effective power counting and further perform the color decomposition to separate the color-singlet and color-octet components. In Sec.~\ref{sec:diffusion_source}, the spin diffusion and source terms for the color-singlet kinetic equations and associated Wigner functions are obtained in weak coupling. In Sec.~\ref{sec:spin_pol_aixal_Ward}, the spin polarization, axial charge current, and axial Ward identity are investigated based on postulated color-field correlators. In Sec.~\ref{sec:massless_fermions}, similar analyses for massless quarks are shown. Finally, in Sec.~\ref{sec:summary}, we make the summary and discussions.
Throughout this paper we use
the mostly minus signature of the Minkowski metric $\eta^{\mu\nu} = {\rm diag} (1, -1,-1,-1) $
and the completely antisymmetric tensor $ \epsilon^{\mu\nu\rho\lambda} $ with $ \epsilon^{0123} = 1 $.
We use the notations $A^{(\mu}B^{\nu)}\equiv A^{\mu}B^{\nu}+A^{\nu}B^{\mu}$
and $A^{[\mu}B^{\nu]}\equiv A^{\mu}B^{\nu}-A^{\nu}B^{\mu}$. We also define $\tilde{F}^{\mu\nu}\equiv\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}/2$.
\section{Master equations from the Kadanoff-Baym equation}\label{sec:KB_eq}
To track the dynamical spin polarization for quarks in heavy ion collisions with non-Abelian color fields in quark gluon plasmas (QGP) or even pre-equilibrium phases, we generalize the so-called axial kinetic theory (AKT) constructed in Refs.~\cite{Hattori:2019ahi,Yang:2020hri}, as a quantum kinetic theory (QKT) to delineate the intertwined dynamics between charge and spin evolution of relativistic fermions, to the case with background color fields in QCD. The AKT incorporates a scalar kinetic equation (SKE) and an axial-vector kinetic equation (AKE) obtained from the Wigner-function approach with the $\hbar$ expansion generally equivalent to a gradient expansion capturing quantum corrections. In addition, we apply a slightly different approach based on Ref.~\cite{Elze:1986qd} by deriving the perturbative solution for Wigner operators and the corresponding kinetic equations in operator form first, while taking ensemble average to obtain the Wigner functions and the quantum-averaged kinetic equations in the end of calculations.
To consider a more rigorous generalization for the AKT from quantum electrodynamics (QED) to QCD, we have to introduce extended Wigner functions incorporating color degrees of freedom. According to Ref.~\cite{Elze:1986qd}, the gauge-covariant Wigner operator of quarks is defined as
\begin{eqnarray}\label{eq:WT}
\grave{S}^{<}(p,X)=\int\frac{d^4Y}{(2\pi)^4}e^{\frac{ip\cdot Y}{\hbar}}S^{<}(X,Y),
\end{eqnarray}
where
\begin{eqnarray}\label{eq:Sless}
S^{<}(X,Y)=\bar{\psi}\left(X-\frac{Y}{2}\right)U\left(-\frac{Y}{2},X\right)\otimes U\left(X,X+\frac{Y}{2}\right)\psi\left(X+\frac{Y}{2}\right)
\end{eqnarray}
as a two-point operator for Dirac fermions. The quantum expectation value of the Wigner operator, $\langle\grave{S}^{<}(p,X)\rangle$, by taking the ensemble average corresponds to the Wigner function. Here $U(x_2,x_1)$ denotes the gauge link with the path integration along a straight line between two end points, $x_2$ and $x_1$, and the tensor product $\otimes$ implies that $S^{<}(X,Y)$ is a matrix in spinor and color spaces. Also, $p_{\mu}$ represents the kinetic momentum. We will focus on the lesser propagators of quarks throughout this paper and hence add the superscript $^<$ for the Wigner operator. On the other hand, the dynamics of $\grave{S}^{<}$ is governed by the Kadanoff-Baym equation (See Ref.~\cite{Hidaka:2022dmn} for a review),
\begin{eqnarray}\label{KB_collisions}
\bigg(\gamma^{\mu}\hat{\Pi}_{\mu}-m+\frac{i\hbar}{2}\gamma^{\mu}\hat{\nabla}_{\mu}\bigg)\grave{S}^{<}=\frac{i\hbar}{2}\Big(\Sigma^{<}\star S^>-\Sigma^>\star S^<\Big),
\end{eqnarray}
where $\Sigma^{<}$ and $\Sigma^{>}$ represent the lesser and greater self energies in operator form, which characterize the collisional effects. The notation $\star$ denotes the Moyal product, whereas its explicit definition is unimportant here. The explicit expressions of the operators, $\hat{\Pi}_{\mu}$ and $\hat{\nabla}_{\mu}$, will be shown later. In the absence of background fields, they simply reduce to $\hat{\Pi}_{\mu}=p_{\mu}$ and $\hat{\nabla}_{\mu}=\partial_{\mu}$. Here we also neglect the real part of the retarded Wigner operator and of the retarded self energy. The former is dropped for the quasi-particle approximation, while the latter is omitted for simplicity\footnote{The real part of the retarded self energy leads to the modification on the dispersion relation such as the thermal mass. It may further yield a potential term in the kinetic equation when it depends on $X$. See Ref.~\cite{Hidaka:2022dmn} for detailed discussions. How such a term could possibly affect the spin transport requires further studies, which is beyond the scope of current work. }
At weak coupling, the collision term is of $\mathcal{O}(g^4\ln g)$ (e.g. \cite{Li:2019qkf,Yang:2020hri}), which could be suppressed by sufficiently strong background color fields, superficially of $\mathcal{O}(g)$ and more precisely of $\mathcal{O}(g^2)$ as will be shown in this work, giving rise to the Lorentz force and anomalous force from the $\hbar$ correction. We hence focus on the collisionless case in the present work.
Now, the free-streaming Kadanoff-Baym equation in QCD, dictating the dynamical evolution of Wigner operators, takes the form \cite{Elze:1986qd}
\begin{eqnarray}
&&\bigg(\gamma^{\mu}p_{\mu}-m+\frac{i\hbar}{2}\gamma^{\mu}D_{\mu}\bigg)\grave{S}^{<}
\\\nonumber
&&=-\frac{i\hbar}{2}\partial_{p}^{\nu}\bigg(\int^1_0ds\Big(1-\frac{s}{2}\Big)\big[e^{-\frac{i\hbar}{2}(1-s)\partial_{p}\cdot D}F_{\nu\mu}\big]\grave{S}^{<}
+\grave{S}^{<}\int^1_0ds\frac{(1-s)}{2}\big[e^{\frac{i\hbar}{2}s\partial_{p}\cdot D}F_{\nu\mu}\big]\bigg)
\\\nonumber
&&=-\frac{i\hbar}{2}\partial_{p}^{\nu}\gamma^{\mu}\bigg(F_{\nu\mu}\grave{S}^{<}
-\frac{1}{4}[F_{\nu\mu},\grave{S}^{<}]_{\rm c}-\frac{i\hbar}{6}\big(\partial_{p}\cdot DF_{\nu\mu}\big)\grave{S}^{<}-\frac{i\hbar}{24}[(\partial_{p}\cdot DF_{\nu\mu}),\grave{S}^{<}]_{\rm c}\bigg)+\mathcal{O}(\hbar^3),
\end{eqnarray}
which can be rearranged as
\begin{eqnarray}
\bigg(\gamma^{\mu}\hat{\Pi}_{\mu}-m+\frac{i\hbar}{2}\gamma^{\mu}\hat{\nabla}_{\mu}\bigg)\grave{S}^{<}=0,
\end{eqnarray}
where
\begin{eqnarray}\nonumber
\hat{\Pi}_{\mu}\grave{S}^{<}&=&p_{\mu}\grave{S}^{<}+\frac{i\hbar}{8}[F_{\nu\mu},\partial_{p}^{\nu}\grave{S}^{<}]_{\rm c}+\frac{\hbar^2}{12}\big(\partial_{p}\cdot DF_{\nu\mu}\big)\partial_{p}^{\nu}\grave{S}^{<}+\mathcal{O}(\hbar^3),
\\
\hat{\nabla}_{\mu}\grave{S}^{<}&=&D_{\mu}\grave{S}^{<}+\frac{1}{2}\{F_{\nu\mu},\partial_{p}^{\nu}\grave{S}^{<}\}_{\rm c}-\frac{i\hbar}{24}[(\partial_{p}\cdot DF_{\nu\mu}),\partial_p^{\nu}\grave{S}^{<}]_{\rm c}+\mathcal{O}(\hbar^2).
\end{eqnarray}
Here $D_{\mu}O=\partial_{\mu}O+i[A_{\mu},O]_{\rm c}$, $A_{\mu}=t^aA_{\mu}^a$, $F_{\nu\mu}=t^aF^{a}_{\nu\mu}$, and $t^a=\lambda^a/2$ with $\lambda^a$ the Gell-Mann matrices. Also, $\{\,,\,\}_{\rm c}$ and $[\,,\,]_{\rm c}$ denote the anti-commutation and commutation relations in color space. The collision term is of the higher order at weak coupling and hence omitted.
We then apply the decomposition based on the Clifford algebra \cite{Vasak:1987um} in spinor space,
\begin{eqnarray}
\grave{S}^<=\mathcal{S}+ i\mathcal{P}\gamma^5+ \mathcal{V}^{\mu}\gamma_\mu+\mathcal{A}^{\mu}\gamma^5\gamma_{\mu}+ \frac{\mathcal{S}^{\mu\nu}}{2}\Sigma_{\mu\nu},
\end{eqnarray}
where $\Sigma_{\mu\nu}=i[\gamma_{\mu},\gamma_{\nu}]/2$ and $\gamma^5=i\gamma^0\gamma^1\gamma^2\gamma^3$. In particular, the vector and axial-vector components, $\mathcal{V}^{\mu}(p,X)$ and $\mathcal{A}^{\mu}(p,X)$, directly contributes to the vector charge current and the axial charge current (also spin polarization), receptively. We hence focus on the derivation of these two components. Inserting $\grave{S}^<$ with the decomposition above into the Kadanoff-Baym equation, one should derive 10 master equations. We can further write $\mathcal{S}$, $\mathcal{P}$, and $\mathcal{S}_{\mu\nu}$ in terms of $\mathcal{V}_{\mu}$ and $\mathcal{A}_{\mu}$ via three of the master equations,
\begin{align}
\label{meq_3}
m\mathcal{S}&=\hat{\Pi}\cdot\mathcal{V},
\\\label{meq_4}
m\mathcal{P}&=- \frac{\hbar}{2}\hat{\nabla}_{\mu}\mathcal{A}^{\mu},
\\\label{meq_5}
m\mathcal{S}_{\mu\nu}&=-\epsilon_{\mu\nu\rho\sigma} \hat{\Pi}^{\rho}\mathcal{A}^{\sigma}
+\frac{\hbar}{2}\hat{\nabla}_{[\mu}\mathcal{V}_{\nu]}
.
\end{align}
We accordingly obtain the rest of master equations as
\begin{eqnarray}\label{rrr1}
&&\hat{\nabla}\cdot\mathcal{V}=0,
\\\label{rrr2}
&&(\hat{\Pi}_{\mu}\hat{\Pi}\cdot\mathcal{V}-m^2\mathcal{V}_{\mu})
=-\frac{\hbar}{2}\epsilon_{\nu\mu\rho\sigma}\hat{\nabla}^{\nu}\hat{\Pi}^{\rho}\mathcal{A}^{\sigma}+\frac{\hbar^2}{4}\hat{\nabla}^{\nu}\hat{\nabla}_{[\nu}\mathcal{V}_{\mu]},
\\\label{rrr3}
&&\hat{\Pi}\cdot\mathcal{A}=0,
\\\label{rrr4}
&&\hat{\Pi}_{\nu}\mathcal{V}_{\mu}-\hat{\Pi}_{\mu}\mathcal{V}_{\nu}
=\frac{\hbar}{2}\epsilon_{\mu\nu\rho\sigma}\hat{\nabla}^{\rho}\mathcal{A}^{\sigma},
\\\label{rrr5}
&&(\hat{\Pi}^2-m^2)\mathcal{A}^{\mu}-\hat{\Pi}_{\sigma}\hat{\Pi}^{\mu}\mathcal{A}^{\sigma}
=\frac{\hbar}{2}\epsilon^{\mu\nu\rho\sigma}\hat{\Pi}_{\sigma}
\hat{\nabla}_{\nu}\mathcal{V}_{\rho}-\frac{\hbar^2}{4}\hat{\nabla}^{\mu}\hat{\nabla}\cdot\mathcal{A}
,
\\\label{rrr6}
&&\hbar\hat{\nabla}_{\sigma}\hat{\Pi}^{\sigma}\mathcal{A}^{\mu}
-\hbar \hat{\nabla}_{\sigma}\hat{\Pi}^{\mu}\mathcal{A}^{\sigma}+\hbar\hat{\Pi}^{\mu}\hat{\nabla}_{\sigma}\mathcal{A}^{\sigma}
=\frac{\hbar^2}{4}\epsilon^{\mu\nu\rho\sigma}\hat{\nabla}_{\sigma}\hat{\nabla}_{[\nu}\mathcal{V}_{\rho]}
,
\\\label{rrr7}
&&\hbar\big(\hat{\Pi}\cdot\hat{\nabla}\mathcal{V}_{\mu}
+\hat{\nabla}_{\mu}\hat{\Pi}^{\nu}\mathcal{V}_{\nu}-\hat{\Pi}^{\nu}\hat{\nabla}_{\mu}\mathcal{V}_{\nu}\big)
=2\epsilon_{\nu\mu\rho\sigma}(\hat{\Pi}^{\nu}\hat{\Pi}^{\rho})\mathcal{A}^{\sigma}.
\end{eqnarray}
See e.g. Refs.~\cite{Hattori:2019ahi,Yang:2020hri} for an analogous derivation in QED. However, for further simplification in QCD, we may adopt the power counting in Ref.~\cite{Yang:2020hri} such that $\mathcal{V}^{\mu}\sim \mathcal{O}(\hbar^0)$ and $\mathcal{A}^{\mu}\sim \mathcal{O}(\hbar)$ due to the quantum nature of $\mathcal{A}^{\mu}$ as the spin-current density in phase space and retain only the leading-order contribution in the $\hbar$ expansion. The master equations then reduce to
\begin{eqnarray}\label{rrr1_sim}
&&\tilde{\Delta}\cdot\mathcal{V}=0,
\\\label{rrr2_sim}
&&p_{\mu}p\cdot\mathcal{V}-m^2\mathcal{V}_{\mu}
=0,
\\\label{rrr3_sim}
&&p_{[\nu}\mathcal{V}_{\mu]}=0,
\\\label{rrr4_sim}
&&p\cdot\mathcal{A}=0,
\\\label{rrr5_sim}
&&(p^2-m^2)\mathcal{A}^{\mu}
=\frac{\hbar}{2}\epsilon^{\mu\nu\rho\sigma}p_{\sigma}
\tilde{\Delta}_{\nu}\mathcal{V}_{\rho}
,
\\\label{rrr6_sim}
&&p\cdot\tilde{\Delta}\mathcal{A}^{\mu}+\frac{1}{2}\{F^{\nu\mu},\mathcal{A}_{\nu}\}_{\rm c}
=\frac{\hbar}{2}\epsilon^{\mu\nu\rho\sigma}\tilde{\Delta}_{\sigma}\tilde{\Delta}_{\nu}\mathcal{V}_{\rho},
\end{eqnarray}
where $\tilde{\Delta}_{\mu}O=D_{\mu}O+\{F_{\nu\mu}\partial_{p}^{\nu},O\}_{\rm c}/2$ and we have dropped Eq.~(\ref{rrr7}) as a redundant equation that can be obtained from the combination of other master equations. Eqs.~(\ref{rrr1_sim})-(\ref{rrr6_sim}) are same as those in QED by simply replacing the operator $\Delta_{\mu}=\partial_{\mu}+F_{\nu\mu}\partial_{p}^{\nu}$ in QED by $\tilde{\Delta}_{\mu}$ except for further corrections on Eq.~(\ref{rrr6_sim}). By using
\begin{eqnarray}\nonumber
&&D_{[\sigma}D_{\nu]}O=i[F_{\sigma\nu},O]_{\rm c},
\quad
D_{[\sigma}\{F_{\beta\nu]},\partial_{p}^{\beta},O\}_{\rm c}+\{F_{\beta[\sigma},D_{\nu]}\partial_{p}^{\beta}O\}=\{(D_{[\sigma}F_{\beta\nu]}),\partial^{\beta}_{p}O\}_{\rm c},
\\
&&\{F_{\alpha[\sigma},\{F_{\beta\nu]},\partial^{\alpha}_{p}\partial^{\beta}_{p}O\}\}_{\rm c}
=[F_{\alpha[\sigma}F_{\beta\nu]},\partial^{\alpha}_{p}\partial^{\beta}_{p}O]_{\rm c},
\end{eqnarray}
and hence
\begin{eqnarray}
\tilde{\Delta}_{[\sigma}\tilde{\Delta}_{\nu]}O=
i[F_{\sigma\nu},O]_{\rm c}+\frac{1}{2}\{(D_{[\sigma}F_{\beta\nu]}),\partial^{\beta}_{p}O\}_{\rm c}+\frac{1}{4}[F_{\alpha[\sigma}F_{\beta\nu]},\partial^{\alpha}_{p}\partial^{\beta}_{p}O]_{\rm c},
\end{eqnarray}
Eq.~(\ref{rrr6_sim}) becomes
\begin{eqnarray}\label{rrr6_sim_explicit}
p\cdot\tilde{\Delta}\mathcal{A}^{\mu}+\frac{1}{2}\{F^{\nu\mu},\mathcal{A}_{\nu}\}_{\rm c}
=\frac{\hbar}{4}\epsilon^{\mu\nu\rho\sigma}\Big(\{(D_{\nu}F_{\beta\rho}),\partial^{\beta}_{p}\mathcal{V}_{\sigma}\}_{\rm c}+i[F_{\nu\rho},\mathcal{V}_{\sigma}]_{\rm c}
+\frac{1}{2}[F_{\alpha\nu}F_{\beta\rho},\partial^{\alpha}_{p}\partial^{\beta}_{p}\mathcal{V}_{\sigma}]_{\rm c}\Big).
\end{eqnarray}
Following Ref.~\cite{Hattori:2019ahi}, we can then apply the master equations to derive a perturbative solution of Wigner operators and corresponding kinetic equations.
\section{Perturbative solutions and effective kinetic equations}\label{sec:sol_and_KE}
By using Eqs.~(\ref{rrr1_sim})-(\ref{rrr6_sim}), we will derive the leading-order solution of $\mathcal{V}^{\mu}$ and $\mathcal{A}^{\mu}$ and the corresponding SKE and AKE. Note that we will keep $\hbar$ as a parameter to specify the quantum corrections while taking the power counting for $\mathcal{V}^{\mu}\sim \mathcal{O}(\hbar^0)$ and $\mathcal{A}^{\mu}\sim \mathcal{O}(\hbar)$. Hereafter all terms in equations will be of the same order of the $\hbar$ expansion unless specified otherwise.
\subsection{Axial kinetic equation}
From Eqs.~(\ref{rrr2_sim}) and (\ref{rrr3_sim}), the vector component now takes the bookkeeping form\footnote{For brevity, we omit the sign function of energy in front of the dispersion relation, which is required to incorporate the contributions from both fermions and anti-fermions.}
\begin{eqnarray}
\mathcal{V}^{\mu}=2\pi \delta(p^2-m^2)p^{\mu}\hat{f}_V
\label{eq:V-LO}
,
\end{eqnarray}
and the SKE from Eq.~(\ref{rrr1_sim}) reads
\begin{eqnarray}\label{SKE}
\delta(p^2-m^2)p^{\mu}\Big(D_{\mu}\hat{f}_V+\frac{1}{2}\{F_{\nu\mu},\partial_{p}^{\nu}\hat{f}_V\}_{\rm c}\Big)=0,
\end{eqnarray}
which corresponds to the classical kinetic equation in Ref.~\cite{Elze:1986qd}. Here $\hat{f}_V(p,X)$ denotes the vector-charge operator, form which we have $\langle \hat{f}_V(p,X)\rangle=f_{V}(p,X)$ as the vector charge distribution function in phase space by taking the ensemble average.
In light of Refs.~\cite{Hattori:2019ahi,Yang:2020hri}, from Eqs.~(\ref{rrr4_sim}) and (\ref{rrr5_sim}), the axial charge component becomes
\begin{eqnarray}\label{axial_sol}
\mathcal{A}^{\mu}=2\pi\Big[\delta(p^2-m^2)\Big(\hat{a}^{\mu}+\hbar S^{\mu\nu}_{m(n)}\tilde{\Delta}_{\nu}\hat{f}_V\Big)
+\frac{\hbar}{2} p_{\nu}\delta'(p^2-m^2)\{\tilde{F}^{\mu\nu},\hat{f}_V\}_{\rm c}\Big],
\end{eqnarray}
where $\delta'(x)\equiv \partial\delta(x)/\partial x$ and $S^{\mu\nu}_{m(n)}=\epsilon^{\mu\nu\alpha\beta}p_{\alpha}n_{\beta}/(2(p\cdot n+m))$ represents the spin tensor and contributes to the magnetization-current term with $n^{\mu}$ being a timelike frame vector coming from a choice of the spin basis \footnote{Such a choice does not affect physical observables. See e.g. Refs~\cite{Hidaka:2016yjf,Hattori:2019ahi} for discussions.}. Here $\hat{a}^{\mu}(p,X)$ denotes the (effective) spin operator in phase space with the constraint $p\cdot \hat{a}=0$ when the quark is on-shell \footnote{Note that the number-density operator becomes a $2\times 2$ matrix in spin space, where the component proportional to an identity matrix corresponds to $\hat{f}_V$ and the other independent components can be written as $\sigma\cdot \hat{S}$. Here $\sigma^{\mu}$ is the Pauli matrix and $\hat{S}^{\mu}$ is related to $\hat{a}^{\mu}$. We could combine $p^{\mu}$ and $\hat{S}^{\mu}$ to form $\hat{a}^{\mu}$. See Appendix C of Ref.~\cite{Hattori:2019ahi} for the details of construction.}. Similarly, we have $\langle\hat{a}^{\mu}(p,X)\rangle=\tilde{a}^{\mu}(p,X)$ as the (effective) spin four vector. Note that the magnetization-current term is actually obtained by generalization from the case without background fields as in the derivation for QED \cite{Hattori:2019ahi}. Accordingly, one may insert Eq.~(\ref{axial_sol}) into Eq.~(\ref{rrr6_sim_explicit}) to derive the AKE with a general frame vector $n^{\mu}=n^{\mu}(X)$, whereas the full kinetic equation is rather complicated.
Nonetheless, to simplify the problem, we may instead work in the rest frame of massive quarks by choosing $n^{\mu}=n^{\mu}_{r}(p)=p^{\mu}/m$ \cite{Weickgenannt:2019dks} albeit the validity for $m\gg \mathcal{O}(\partial)$. In the rest frame, $S^{\mu\nu}_{m(n_r)}=0$ and hence the magnetization-current term in Eq.~(\ref{axial_sol}) vanishes. Here we compute the $\hbar$ contributions on left-hand side of Eq.~(\ref{rrr6_sim_explicit}),
\begin{eqnarray}\nonumber
&&\frac{1}{2}p\cdot\tilde{\Delta}\Big(p_{\nu}\delta'(p^2-m^2)\{\tilde{F}^{\mu\nu},\hat{f}_V\}_{\rm c}\Big)
+\frac{\delta'(p^2-m^2)}{2}\{F^{\nu\mu},\{F_{\nu\rho}p^{\rho}, \hat{f}_V\}\}_{\rm c}
\\\nonumber
&&=\frac{\delta'(p^2-m^2)}{2}\Big(p_{\nu}\{(p\cdot D\tilde{F}^{\mu\nu}),\hat{f}_V\}_{\rm c}+p_{\nu}\{\tilde{F}^{\mu\nu},p\cdot\tilde{\Delta}\hat{f}_V\}
+\frac{p^{\rho}p^{\nu}}{2}[[F_{\sigma\rho},\tilde{F}_{\mu\nu}],\partial_{p\sigma}\hat{f}_V]_{\rm c}
\\
&&\quad +\epsilon^{\mu\rho\alpha\beta}p_{\rho}\{F_{\alpha}^{\,\,\nu},\{F_{\beta\nu},\hat{f}_V\}\}_{\rm c}
\Big),
\end{eqnarray}
where we employed
\begin{eqnarray}
\frac{1}{2}\{F_{\nu\rho},\{\tilde{F}^{\mu\nu},\hat{f}_V\}\}_{\rm c}=\frac{1}{2}\{F^{\nu\mu},\{\tilde{F}_{\rho\nu},\hat{f}_V\}\}_{\rm c}+\epsilon^{\mu\rho\alpha\beta}\{F_{\alpha}^{\,\,\nu},\{F_{\beta\nu},\hat{f}_V\}\}_{\rm c}
\end{eqnarray}
by using the Schouten identity. For the right-hand side, one could show
\begin{eqnarray}
&&\frac{p_{\sigma}p^{\beta}}{2}\delta'(p^2-m^2)\epsilon^{\mu\nu\rho\sigma}\{(D_{\nu}F_{\beta\rho}),\hat{f}_V\}_{\rm c}
\\\nonumber
&&=\frac{1}{2}\delta'(p^2-m^2)\Big(p_{\nu}\{(p\cdot D)\tilde{F}^{\mu\nu},\hat{f}_V\}_{\rm c}+p_{\sigma}p^{\mu}\{D_{\nu}\tilde{F}^{\sigma\nu},\hat{f}_V\}_{\rm c}\Big)
-\frac{p^2}{2}\delta'(p^2-m^2)\{D_{\nu}\tilde{F}^{\mu\nu},\hat{f}_V\}_{\rm c}\Big).
\end{eqnarray}
We then derive a free-streaming effective AKE for massive quarks coupled with background color fields,
\begin{eqnarray}\label{AKE_nr_2}\nonumber
0&=&\delta(p^2-m^2)
\Big(p\cdot\tilde{\Delta}\hat{a}^{\mu}+\frac{1}{2}\{F^{\nu\mu},\hat{a}_{\nu}\}_{\rm c}
-\frac{i\hbar}{2}[\tilde{F}^{\mu\nu}p_{\nu},\hat{f}_V]_{\rm c}
-\frac{\hbar\epsilon^{\mu\nu\rho\sigma}p_{\sigma}}{8}[F_{\alpha\nu}F_{\beta\rho},\partial^{\alpha}_{p}\partial^{\beta}_{p}\hat{f}_V]_{\rm c}
\\\nonumber
&& -\frac{\hbar}{4}\epsilon^{\mu\nu\rho\sigma}p_{\rho}\{(D_{\sigma}F_{\beta\nu})
,\partial_{p}^{\beta}\hat{f}_{V}\}_{\rm c}\Big)
+\frac{\hbar}{2}\delta'(p^2-m^2)\Big(p_{\nu}\{\tilde{F}^{\mu\nu},p\cdot\tilde{\Delta} \hat{f}_V\}_{\rm c}
+\epsilon^{\mu\rho\alpha\beta}p_{\rho}\{F_{\alpha}^{\,\,\nu},\{F_{\beta\nu},\hat{f}_V\}\}_{\rm c}
\\
&&+\frac{p^{\rho}p^{\nu}}{2}[[F_{\sigma\rho},\tilde{F}_{\mu\nu}],\partial_{p}^{\sigma}\hat{f}_V]_{\rm c}\Big),
\end{eqnarray}
where we took the Bianchi identity $D_{\mu}\tilde{F}^{\mu\nu}=0$ for non-Abelian gauge fields.
\subsection{Color decomposition}
Now, both the SKE and AKE in Eqs.~(\ref{SKE}) and (\ref{AKE_nr_2}) are matrix equations in color space. Nonetheless, only the color-single components of $\mathcal{V}^{\mu}$ and $\mathcal{A}^{\mu}$ ($\hat{f}_V$ and $\hat{a}^{\mu}$) directly contribute to physical observables. For example, the vector and axial charge currents are defined as
\begin{eqnarray}\label{def_currents}
J^{\mu}_{V}=4\int\frac{d^4p}{(2\pi)^4}{\rm tr_c}(\langle \mathcal{V}^{\mu}\rangle), \quad J^{\mu}_{5}=4\int\frac{d^4p}{(2\pi)^4}{\rm tr_c}(\langle\mathcal{A}^{\mu}\rangle),
\end{eqnarray}
by taking the traces over color space and ensemble averages.
Therefore, we may further adopt the color decomposition for kinetic equations.
We first consider the SKE with color decomposition by taking $\hat{f}_V=\hat{f}^{\rm s}_VI+\hat{f}^a_Vt^a$, where $I$ represents an identity matrix in color space. Using
\begin{eqnarray}
[t^a,t^b]_{\rm c}=if^{abc}t^c,\quad
\{t^a,t^b\}_{\rm c}=2\bar{C}_2\delta^{ab}I+d^{abc}t^c,\quad
t^at^b=\bar{C}_2\delta^{ab}I+(d^{abc}+if^{abc})\frac{t^c}{2},
\end{eqnarray}
with $\bar{C}_2=1/(2N_c)$ and $N_c=3$ being number of colors,
Eq.~(\ref{SKE}) becomes
\begin{eqnarray}\nonumber
0&=&\delta(p^2-m^2)p^{\mu}\Big(\partial_{\mu}(\hat{f}^{\rm s}_V+\hat{f}^a_Vt^a)-gf^{bca}t^aA^b_{\mu}\hat{f}^c_V+t^aF^a_{\nu\mu}\partial_{p}^{\nu}\hat{f}^{\rm s}_V
\\
&&
+\bar{C}_2F^a_{\nu\mu}\partial_{p}^{\nu}\hat{f}^{a}_V+\frac{d^{bca}}{2}t^aF^b_{\nu\mu}\partial_{p}^{\nu}\hat{f}^{c}_V\Big),
\end{eqnarray}
which yields
\begin{eqnarray}\label{SKE_singlet}
&&\delta(p^2-m^2)\mathcal{K}_{\rm s}[\hat{f}_V]=0,
\\\label{SKE_octet}
&&\delta(p^2-m^2)\mathcal{K}_{\rm o}^a[\hat{f}_V]=0,
\end{eqnarray}
where
\begin{eqnarray}
\mathcal{K}_{\rm s}[O]&\equiv&p^{\mu}\Big(\partial_{\mu}O^{\rm s}+\bar{C}_2F^a_{\nu\mu}\partial_{p}^{\nu}O^{a}\Big),
\\
\mathcal{K}_{\rm o}^a[O]&\equiv&p^{\mu}\Big(\partial_{\mu}O^a-f^{bca}A^b_{\mu}O^c+F^a_{\nu\mu}\partial_{p}^{\nu}O^{\rm s}
+\frac{d^{bca}}{2}F^b_{\nu\mu}\partial_{p}^{\nu}O^{c}\Big),
\end{eqnarray}
for an arbitrary color object that can be decomposed into $O=IO^{\rm s}+t^aO^a$.
Eq.~(\ref{SKE_singlet}) and (\ref{SKE_octet}) correspond to the Vlasov equations for the color-singlet and color-octet components, respectively.
Although the prefactors of the force terms in Eqs.~(\ref{SKE_singlet}) and (\ref{SKE_octet}) are distinct from those in Refs.~\cite{Heinz:1983nx,Heinz:1984yq} due to different color decompositions for $\hat{f}_V$, the combination of two kinetic equations still leads to equivalent results in physics.
Analogously, we may carry out the color decomposition for the AKE. For simplicity, we focus on the AKE in the rest frame. In addition, we may apply the off-shell solution of $\hat{f}_V$ such that $p\cdot\tilde{\Delta}\hat{f}_V=0$.\footnote{In practice, one has to solve for $\hat{f}_V$ and $\hat{a}^{\mu}$ from the kinetic equations and input them into Wigner operators to evaluate physical quantities. Introducing the on-shell $\hat{f}_V$ and $\hat{a}^{\mu}$ first in kinetic equations or later in Wigner operators should not affect the final results.} It is more convenient to denote Eq.~(\ref{AKE_nr_2}) in the following form,
\begin{eqnarray}
\delta(p^2-m^2)\Big(\hat{\Pi}^{\mu\nu}\hat{a}_{\nu}+\hbar\hat{\chi}^{\mu}\hat{f}_{V}\Big)
+\delta'(p^2-m^2)\hbar\hat{\Theta}^{\mu}\hat{f}_V=0,
\end{eqnarray}
where
\begin{eqnarray}\nonumber
\hat{\Pi}^{\mu\nu}\hat{a}_{\nu}&=& p\cdot\tilde{\Delta}\hat{a}^{\mu}+\frac{1}{2}\{F^{\nu\mu},\hat{a}_{\nu}\}_{\rm c},
\\\nonumber
\hat{\chi}^{\mu}\hat{f}_{V}&=& -\frac{1}{4}\Big(2i[\tilde{F}^{\mu\nu}p_{\nu},\hat{f}_V]_{\rm c}
+\frac{\epsilon^{\mu\nu\rho\sigma}p_{\sigma}}{2}[F_{\alpha\nu}F_{\beta\rho},\partial^{\alpha}_{p}\partial^{\beta}_{p}\hat{f}_V]_{\rm c}
+\epsilon^{\mu\nu\rho\sigma}p_{\rho}\{(D_{\sigma}F_{\beta\nu})
,\partial_{p}^{\beta}\hat{f}_{V}\}_{\rm c}\Big),
\\
\hat{\Theta}^{\mu}\hat{f}_V&=&\frac{\epsilon^{\mu\rho\alpha\beta}}{2}p_{\rho}\{F_{\alpha}^{\,\,\nu},\{F_{\beta\nu},\hat{f}_V\}\}_{\rm c}
+\frac{p^{\rho}p^{\nu}}{4}[[F_{\sigma\rho},\tilde{F}_{\mu\nu}],\partial_{p}^{\sigma}\hat{f}_V]_{\rm c}.
\end{eqnarray}
These terms above without and with $\hbar$ correspond to the classical contributions and quantum corrections in the AKE \footnote{More precisely, the quantum correction in our context refers to the gradient correction. The classical part mentioned here should implicitly encode the quantum effect that generates nonzero $\hat{a}^{\mu}$, while such a term could also exist if we assume there is an external source irrelevant to the quantities of $\mathcal{O}(\partial)$.}, although they are of the same order in our power counting.
Taking $\hat{a}^{\mu}=\hat{a}^{\rm s\mu}I+\hat{a}^{a\mu }t^a$, it is found
\begin{eqnarray}\nonumber
\hat{\Pi}^{\mu\nu}\hat{a}_{\nu}&=&
\bigg[p\cdot\partial\big(\hat{a}^{\rm s\mu}+\hat{a}^{a\mu}t^a\big)-t^ap^{\nu}f^{bca}A^b_{\nu}\hat{a}^{\mu}_c
+\frac{p^{\nu}}{2}\big(2t^aF^a_{\rho\nu}\partial_{p}^{\rho}\hat{a}^{\rm s\mu}
+2\bar{C}_2F^a_{\rho\nu}\partial_{p}^{\rho}\hat{a}^{a\mu}
\\\nonumber
&&+d^{bca}t^aF^b_{\rho\nu}\partial_{p}^{\rho}\hat{a}^{\mu}_{c}\big)\bigg]
\\
&=&\mathcal{K}_{\rm s}[\hat{a}^{\mu}]+t^a\mathcal{K}_{\rm o}^a[\hat{a}^{\mu}].
\end{eqnarray}
For part of the quantum corrections, one finds
\begin{eqnarray}\nonumber
\hat{\chi}^{\mu}\hat{f}_{V}&=& \frac{1}{4}\bigg\{2t^af^{bca}\tilde{F}^{b\mu\nu}p_{\nu}\hat{f}^c_V
+\frac{\epsilon^{\mu\nu\rho\sigma}p_{\sigma}}{4}t^af^{dea}f^{bcd}F^b_{\alpha\nu}F^c_{\beta\rho}\partial^{\alpha}_{p}\partial^{\beta}_{p}\hat{f}^e_V
\\\nonumber
&&-\epsilon^{\mu\nu\rho\sigma}p_{\rho}
\Big[\big((\partial_{\sigma}F^a_{\beta\nu})-f^{bca}A^b_{\sigma}F^c_{\beta\nu}\big)t^a\partial_{p}^{\beta}\hat{f}^{\rm s}_V
+\big((\partial_{\sigma}F^d_{\beta\nu})-f^{bcd}A^b_{\sigma}F^c_{\beta\nu}\big)
\big(2\bar{C}_2\partial_{p}^{\beta}\hat{f}^d_V
\\
&&+d^{dea}t^a\partial_{p}^{\beta}\hat{f}^e_V\big)
\Big]
\bigg\},
\end{eqnarray}
which can be rearranged as
\begin{eqnarray}
\hat{\chi}^{\mu}f_{V}=\mathcal{Q}^{\mu}_{\rm s}[\hat{f}_V]+t^a\mathcal{Q}^{a\mu}_{\rm o}[\hat{f}_V],
\end{eqnarray}
where
\begin{eqnarray}
\mathcal{Q}^{\mu}_{\rm s}[O]\equiv
-\frac{1}{2}\big[\epsilon^{\mu\nu\rho\sigma}p_{\rho}
\big((\partial_{\sigma}F^a_{\beta\nu})-f^{bca}A^b_{\sigma}F^c_{\beta\nu}\big)
\bar{C}_2\partial_{p}^{\beta}O^a
\big],
\end{eqnarray}
and
\begin{eqnarray}\nonumber
\mathcal{Q}^{a\mu}_{\rm o}[O]&\equiv &\frac{1}{4}\bigg[2f^{bca}\tilde{F}^{b\mu\nu}p_{\nu}O^c
+\frac{\epsilon^{\mu\nu\rho\sigma}p_{\sigma}}{4}f^{dea}f^{bcd}F^b_{\alpha\nu}F^c_{\beta\rho}\partial^{\alpha}_{p}\partial^{\beta}_{p}O^e
\\
&&-\epsilon^{\mu\nu\rho\sigma}p_{\rho}
\bigg(2\big((\partial_{\sigma}F^a_{\beta\nu})-f^{bca}A^b_{\sigma}F^c_{\beta\nu}\big)\partial_{p}^{\beta}O_{\rm s}+\big((\partial_{\sigma}F^d_{\beta\nu})-f^{bcd}A^b_{\sigma}F^c_{\beta\nu}\big)
d^{dea}\partial_{p}^{\beta}O^e
\bigg)\bigg].
\end{eqnarray}
Finally, it is found $\hat{\Theta}^{\mu}\hat{f}_V=t^a\hat{\Theta}^{a\mu}_{\rm o}[\hat{f}_V]$, where
\begin{eqnarray}
\hat{\Theta}^{a\mu}_{\rm o}[\hat{f}_V]=\frac{\epsilon^{\mu\rho\alpha\beta}}{2}p_{\rho}\big(2\bar{C}_2F^{a\nu}_{\alpha}F_{\beta\nu}^b\hat{f}_V^b+d^{abc}d^{dec}F^{b\nu}_{\alpha}F^d_{\beta\nu}\hat{f}_V^e\big)
-\frac{p^{\rho}p^{\nu}}{4}f^{cda}f^{ebc}F^e_{\sigma\rho}F^b_{\mu\nu}\partial_{p}^{\sigma}\hat{f}^d_V.
\end{eqnarray}
Here we have used the symmetric property of $d^{abc}$ and it turns out that only part of the color-octet components of $\hat{\Theta}^{\mu}\hat{f}_V$ remains. Accordingly, the color-singlet and color-octet components of the AKE read
\begin{eqnarray}\label{AKE_singlet}
&&\delta(p^2-m^2)\big(\mathcal{K}_{\rm s}[\hat{a}^{\mu}]+\hbar\mathcal{Q}^{\mu}_{\rm s}[\hat{f}_V]\big)=0,
\\\label{AKE_octet}
&&\delta(p^2-m^2)\big(\mathcal{K}_{\rm o}^a[\hat{a}^{\mu}]+\hbar \mathcal{Q}^{a\mu}_{\rm o}[\hat{f}_V]\big)+\hbar \delta'(p^2-m^2)\hat{\Theta}^{a\mu}_{\rm o}[\hat{f}_V]=0.
\end{eqnarray}
\section{Spin diffusion and source terms}\label{sec:diffusion_source}
Given Eqs.~(\ref{SKE_singlet}), (\ref{SKE_octet}), (\ref{AKE_singlet}), and (\ref{AKE_octet}), we may follow the approach in Refs.~\cite{Asakawa:2006tc,Asakawa:2006jn} to combine the color-singlet and color-octet kinetic equations into a single equation for $\hat{f}^{\rm s}_V$ or for $\hat{a}^{\rm s\mu}$ with effective diffusion terms and quantum corrections up to $\mathcal{O}(\hbar)$ in weak coupling. The same derivation has been presented in Ref.~\cite{Muller:2021hpe}. More details for the derivation are shown below.
Considering QCD at weak coupling and assuming the color-octet distribution functions are subleading such that $O^a\sim\mathcal{O}(g)$, we may approximate
\begin{eqnarray}\label{Ko_approx}
\mathcal{K}_{\rm o}^a[O]&\approx&p^{\mu}\Big(\partial_{\mu}O^a-f^{bca}A^b_{\mu}O^c+F^a_{\nu\mu}\partial_{p}^{\nu}O^{\rm s}\Big),
\\\label{Qo_approx}
\mathcal{Q}^{a\mu}_{\rm o}[O]&\approx &-\frac{1}{2}\Big(\epsilon^{\mu\nu\rho\sigma}p_{\rho}
(\partial_{\sigma}F^a_{\beta\nu})\partial_{p}^{\beta}O^{\rm s}
\Big),
\end{eqnarray}
up to $\mathcal{O}(g)$ except for maintaining the $f^{bca}A^b_{\mu}$ term associated with the gauge link. Whereby Eq.~(\ref{SKE_octet}) gives rise to
\begin{eqnarray}\label{fV_octet_sol}
\hat{f}_{V}^a(p,X)=-i\int d^4k\int\frac{d^4X'}{(2\pi)^4}U^{ab}(X,X')\frac{e^{ik\cdot(X'-X)}}{p\cdot k+i\epsilon}p^{\mu}F^b_{\nu\mu}(X')\partial_{p}^{\nu}\hat{f}^{\rm s}_V(p,X'),
\end{eqnarray}
where
\begin{eqnarray}
U^{ac}(X,X')={\rm exp}\bigg[P\bigg(\int^{X}_{X'}f^{abc}A_{\mu}^b(s)ds^{\mu}\bigg)\bigg]
\end{eqnarray}
parallel transports the gauge field from $X'$ to $X$ with $P$ being the path ordering.
Similarly, from $\mathcal{K}_{\rm o}^a[\hat{a}^{\mu}]$ and $\mathcal{Q}^{a\mu}_{\rm o}[\hat{f}_V]$ in Eqs.~(\ref{Ko_approx}) and (\ref{Qo_approx}), Eq.~(\ref{AKE_octet}) results in
\begin{eqnarray}\nonumber\label{a_octet_sol}
\hat{a}^{a\mu}(p,X)&=&-i\int d^4k\int\frac{d^4X'}{(2\pi)^4}U^{ab}(X,X')\frac{e^{ik\cdot(X'-X)}}{p\cdot k+i\epsilon}\bigg[p^{\mu}F^b_{\nu\mu}(X')\partial_{p}^{\nu}\hat{a}^{\rm s\mu}(p,X')
\\
&&-\frac{\hbar}{2}\epsilon^{\mu\nu\rho\sigma}p_{\rho}
(\partial_{\sigma}F^b_{\beta\nu}(X'))\partial_{p}^{\beta}\hat{f}^{\rm s}_V(p,X')
\bigg]
.
\end{eqnarray}
Replacing $\hat{f}_{V}^a$ in Eq.~(\ref{SKE_singlet}) by Eq.~(\ref{fV_octet_sol})
with the relation,
\begin{eqnarray}
\frac{1}{p\cdot k+i\epsilon}= -i\pi\delta(p\cdot k)+PV(1/p\cdot k),
\end{eqnarray}
where $PV(x)$ represents the principle value of $x$, one obtains \cite{Asakawa:2006jn}
\begin{align}\label{SKE_signlet_full}
0&=&\delta(p^2-m^2)\bigg(p\cdot\partial \hat{f}_V^{\rm s}(p,X)-\bar{C}_2p^{\mu}F^a_{\nu\mu}(X)\partial_{p}^{\nu}\int^{p}_{k,X'}U^{ab}(X,X')
p^{\beta}F^b_{\alpha\beta}(X')\partial_{p}^{\alpha}\hat{f}^{\rm s}_V(p,X')\bigg),
\end{align}
where
\begin{eqnarray}
\int^{p}_{k,X'}\equiv \int d^4k\int\frac{d^4X'}{(2\pi)^4}e^{ik\cdot(X'-X)}\big(\pi\delta(p\cdot k)+iPV(1/p\cdot k)\big).
\end{eqnarray}
When the correlations of field strengths are even functions with respect to $X-X'$, only the imaginary part of $1/(p\cdot k+i\epsilon)$ contributes as considered in Ref.~\cite{Asakawa:2006jn}. However, the real part of $1/(p\cdot k+i\epsilon)$ could also gives a real contribution to $\hat{f}_{V}^a(p,X)$. See appendix A for a detailed discussion.
On the other hand, by inserting Eq.~(\ref{a_octet_sol}) into Eq.~(\ref{AKE_singlet}) and utilizing Eq.~(\ref{fV_octet_sol}), it is found
\begin{eqnarray}\nonumber\label{AKE_signlet_full}
0&=&\delta(p^2-m^2)\bigg\{p\cdot\partial\hat{a}^{\rm s\mu}(p,X)-\bar{C}_2p^{\lambda}F^a_{\kappa\lambda}(X)\partial_{p}^{\kappa}\int^{p}_{k,X'}
U^{ab}(X,X')\bigg[p^{\beta}F^b_{\alpha\beta}(X')\partial_{p}^{\alpha}\hat{a}^{\rm s\mu}(p,X')
\\\nonumber
&&-\frac{\hbar}{2}\epsilon^{\mu\nu\rho\sigma}p_{\rho}
(\partial_{\sigma}F^b_{\beta\nu}(X'))\partial_{p}^{\beta}\hat{f}^{\rm s}_V(p,X')
\bigg]
\\
&&+\frac{\hbar\bar{C}_2}{2}\epsilon^{\mu\nu\rho\sigma}p_{\rho}
(\partial_{\sigma}F^a_{\kappa\nu}(X))
\partial_{p}^{\kappa}
\int^{p}_{k,X'}U^{ab}(X,X')
p^{\beta}F^b_{\alpha\beta}(X')\partial_{p}^{\alpha}\hat{f}^{\rm s}_V(p,X')\bigg\}.
\end{eqnarray}
By taking the ensemble average, we may recast
Eq.~(\ref{AKE_signlet_full}) into
\begin{eqnarray}\label{AKE_signlet_simplify}
0=\delta(p^2-m^2)\Big(p\cdot\partial\tilde{a}^{\rm s\mu}(p,X)-\partial_{p}^{\kappa}\mathscr{D}_{\kappa}[\tilde{a}^{s\mu}]
+\hbar\partial_{p}^{\kappa}\big(\mathscr{A}^{\mu}_{\kappa}[f^{\rm s}_{V}]\big)\Big),
\end{eqnarray}
where
\begin{eqnarray}
\mathscr{D}_{\kappa}[O]=\bar{C}_2\int^{p}_{k,X'}p^{\lambda}p^{\rho}\langle F^a_{\kappa\lambda}(X)
F^a_{\alpha\rho}(X')\rangle \partial_{p}^{\alpha}O(p,X')
\end{eqnarray}
and
\begin{eqnarray}
\mathscr{A}^{\mu}_{\kappa}[O]=\frac{\bar{C}_2}{2}
\epsilon^{\mu\nu\rho\sigma}\int^{p}_{k,X'}p^{\lambda}p_{\rho}\Big(\partial_{X'\sigma}\langle F^a_{\kappa\lambda}(X)F^a_{\alpha\nu}(X')\rangle
+\partial_{X\sigma}\langle F^a_{\kappa\nu}(X)F^a_{\alpha\lambda}(X')\rangle\Big)\partial^{\alpha}_{p}O(p,X').
\end{eqnarray}
Here we have dropped the term proportional to $\epsilon^{\mu\nu\rho\sigma}\partial_{\sigma}F^a_{\rho\nu}(X)=\mathcal{O}(g^2)$ and used $\langle \hat{O} \hat{f}^{\rm s}_{V}\rangle=\langle \hat{O}\rangle \langle\hat{f}^{\rm s}_{V}\rangle$ and $\langle \hat{O} \hat{a}^{\rm s\mu}\rangle=\langle \hat{O}\rangle \langle\hat{a}^{\rm s\mu}\rangle$ with $\hat{O}$ an arbitrary operator based on the quasi-particle approximation. Also, we introduce the color-field correlator with the insertion of a gauge link,
\begin{eqnarray}\label{eq:color_field_corr}
\langle F^a_{\kappa\lambda}(X)
F^a_{\alpha\rho}(X')\rangle \equiv \langle F^a_{\kappa\lambda}(X)U^{ab}(X,X')
F^b_{\alpha\rho}(X')\rangle,
\end{eqnarray}
which comes from the ensemble average of the gauge field and color-octet distribution function.
Note that the background color fields in Eq.~(\ref{eq:color_field_corr}) in QGP originates from soft gluons emitted by stochastic sources or induced by the Weibel-type plasma instability, whereas the influence from quantum fluctuations of hard gluons characterized by on-shell gluonic Wigner functions are encoded in $\Sigma^{\lessgtr}$ in the collision term delineating the interaction with the probe strange quark with hard-momentum exchange. Such a scale separation results in the Fokker-Plank equation plus hard scattering kernel in kinetic theory \cite{Ghiglieri:2015ala,Dai:2020rlu}.
Recall that the color-singlet SKE in Eq.~(\ref{SKE_signlet_full}) with the same approximations reads \cite{Asakawa:2006jn}
\begin{eqnarray}\label{SKE_signlet_simplify}
0=\delta(p^2-m^2)\Big(p\cdot\partial f^{\rm s}_{V}(p,X)-\partial_{p}^{\kappa}\mathscr{D}_{\kappa}[f_V^{s}]\Big)=0.
\end{eqnarray}
One finds that both Eqs.~(\ref{AKE_signlet_simplify}) and (\ref{SKE_signlet_simplify}) have similar diffusion terms, $\partial_{p}^{\kappa}\mathscr{D}_{\kappa}[\tilde{a}^{s\mu}]$ and $\partial_{p}^{\kappa}\mathscr{D}_{\kappa}[f_V^{s}]$, while Eq.~(\ref{AKE_signlet_simplify}) further incorporates the quantum correction, $\hbar\partial_{p}^{\kappa}\big(\mathscr{A}^{\mu}_{\kappa}[f^{\rm s}_{V}]\big)$, as a source term for dynamical spin polarization.
In addition, from Eq.~(\ref{axial_sol}), the color-singlet component of the axial-vector Wigner function, $\langle\mathcal{A}^{\rm s\mu}\rangle={\rm tr_c}(\langle\mathcal{A}^{\mu}\rangle)/N_c$, in the rest frame reads
\begin{eqnarray}\nonumber\label{eq:Asmu_with_totalDp}
\langle\mathcal{A}^{\rm s\mu}\rangle&=&2\pi\Big[\delta(p^2-m^2)\tilde{a}^{\rm s\mu}
+\frac{\hbar\bar{C}_2}{2} \big(\partial_{p\nu}\delta(p^2-m^2)\big)\langle\tilde{F}^{a\mu\nu}\hat{f}^{a}_V\rangle\Big]
\\
&=&2\pi\bigg[\delta(p^2-m^2)\bigg(\tilde{a}^{\rm s\mu}-\frac{\hbar\bar{C}_2}{2}\langle\tilde{F}^{a\mu\nu}\partial_{p\nu}\hat{f}^{a}_V\rangle\bigg)
+\frac{\hbar\bar{C}_2}{2} \partial_{p\nu}\big(\delta(p^2-m^2)\langle \tilde{F}^{a\mu\nu}\hat{f}^{a}_V\rangle\big)\bigg]
.
\end{eqnarray}
In order to obtain the spectrum of spin polarization for on-shell fermions, we have to integrate over $p\cdot \bar{n}$ with $\bar{n}^{\mu}=(1,{\bm 0})$ being a timelike vector to define the particle energy. It turns out that
\begin{eqnarray}
\int \frac{dp_0}{2\pi}\langle\mathcal{A}^{\rm s\mu}\rangle=\frac{1}{2\epsilon_{\bm p}}\bigg(\tilde{a}^{\rm s\mu}-\frac{\hbar\bar{C}_2}{2}\langle\tilde{F}^{a\mu\nu}\partial_{p\nu}\hat{f}^{a}_V\rangle\bigg)_{p_0=\epsilon_{\bm p}}
+\frac{\hbar\bar{C}_2}{4}\partial_{p_{\perp}\nu}\left(\frac{\langle\tilde{F}^{a\mu\nu}\hat{f}^{a}_V\rangle}{\epsilon_{\bm p}}\right)_{p_0=\epsilon_{\bm p}},
\end{eqnarray}
where $p_{\perp}^{\mu}\equiv (\eta^{\mu\nu}-\bar{n}^{\mu}\bar{n}^{\nu})p_{\nu}$, $|\bm p|=\sqrt{-p_{\perp}^2}$, and $\epsilon_{\bm p}\equiv\sqrt{|\bm p|^2+m^2}$. Here we only consider the particle with positive energy. Consequently, we could introduce the color singlet of the on-shell axial charge current density in phase space,
\begin{eqnarray}\label{eq:Asmu_massive}
\mathcal{A}^{\rm s\mu}(\bm p,X)\equiv \int \frac{dp_0}{2\pi}\langle\mathcal{A}^{\rm s\mu}\rangle
=\frac{1}{2\epsilon_{\bm p}}\big(\tilde{a}^{\rm s\mu}+\hbar\bar{C}_2\mathcal{A}^{\mu}_{Q}\big)_{p_0=\epsilon_{\bm p}},
\end{eqnarray}
where $\mathcal{A}^{\mu}_{Q}=\mathcal{A}^{\mu}_{Q1}+\mathcal{A}^{\mu}_{Q2}$ and
\begin{equation}\label{AQmu_origin}
\mathcal{A}^{\mu}_{Q1}=\bigg[\frac{\partial_{p\kappa}}{2}\int^{p}_{k,X'}p^{\beta}\langle \tilde{F}^{a\mu\kappa}(X)F^a_{\alpha\beta}(X')\rangle\partial_{p}^{\alpha}f^{\rm s}_V(p,X')\bigg]_{p_0=\epsilon_{\bm p}},
\end{equation}
\begin{eqnarray}\nonumber
\mathcal{A}^{\mu}_{Q2}&=&-\frac{\epsilon_{\bm p}}{2}\partial_{p_\perp\kappa}\bigg[\int^{p}_{k,X'}\hat{p}^{\beta}\langle \tilde{F}^{a\mu\kappa}(X)F^a_{\alpha\beta}(X')\rangle\partial_{p}^{\alpha}f^{\rm s}_V(p,X')\bigg]_{p_0=\epsilon_{\bm p}}
\\
&=&\frac{1}{2\epsilon_{\bm p}^2}(p_{\perp\kappa}-\epsilon_{\bm p}^2\partial_{p_{\perp}\kappa})\bigg[\int^{p}_{k,X'}p^{\beta}\langle \tilde{F}^{a\mu\kappa}(X)F^a_{\alpha\beta}(X')\rangle \partial_{p}^{\alpha}f^{\rm s}_V(p,X')\bigg]_{p_0=\epsilon_{\bm p}},
\end{eqnarray}
with $\hat{p}_{\mu}\equiv p_{\mu}/p_0$.
One should solve for $f^{\rm s}_{V}(p,X)$ and $\tilde{a}^{\rm s\mu}$ from Eqs.~(\ref{SKE_signlet_simplify}) and (\ref{AKE_signlet_simplify}), respectively, and input the solutions to Eq.~(\ref{eq:Asmu_massive}) for evaluating the spin polarization spectrum governed by the modified Cooper-Frye formula over the freeze-out hypersurface $\Sigma_{\mu}$ \cite{Becattini2013a,Fang:2016vpj},
\begin{equation}\label{Spin_CooperFrye}
\mathcal{P}^{\mu}({\bm p})=\frac{\int d\Sigma\cdot p \hat{\mathcal{A}}^{\rm s\mu}(\bm p,X)}{2m\int d\Sigma\cdot pf_{V}^{\rm s}(\bm p,X)},
\end{equation}
where $\hat{\mathcal{A}}^{\rm s\mu}(\bm p,X)=2\epsilon_{\bm p}\mathcal{A}^{\rm s\mu}(\bm p,X)$ and we have imposed the on-shell condition with positive energy. In Ref.~\cite{Muller:2021hpe}, there is a mistake for the omission of $\mathcal{A}^{\mu}_{Q2}$.
\section{Spin polarization and axial Ward identity}\label{sec:spin_pol_aixal_Ward}
Although the genuine color-field correlators have to be obtained through the real-time simulations in heavy ion collisions, we may propose an approximated form based on the physical arguments as presented in Ref.~\cite{Muller:2021hpe}. Here we provide details for the analysis therein and further discuss the constant-field limit, which is not a realistic situation in heavy ion collisions but an ideal case for theoretical interest. We will focus on the source terms giving rise to spin polarization, while the diffusion term in the SKE has been studied in Ref.~\cite{Asakawa:2006jn} and the one in the AKE takes a similar form. Note that what we study here is a non-equilibrium (more precisely, near-equilibrium) effect on spin polarization in particular due to the involvement of color electric fields. The detailed balance for hard collisions could yield an equilibrium result for spin polarization proportional to thermal vorticity. The anomalous spin polarization from color fields should be regraded as a non-equilibrium correction on top of the equilibrium one, whereas the hydrodynamic gradient terms are assumed to be suppressed compared to the magnitude of color fields in the following analysis. One could in principle include other non-equilibrium corrections from the hard scattering kernel of QKT in e.g. Refs.~\cite{Wang:2021qnt,Fang:2022ttm} with non-QCD effective models. In practice, we focus on high-energy collisions such that the thermal vorticity and other gradient terms are expected to be negligible, which is supported by the suppressed spin polarization of $\Lambda$ hyperons in LHC.
\subsection{Spin polarization and axial charge currents}
Physically, considering (space-time) translational invariance, we may assume $\langle F^a_{\kappa\lambda}(X)
F^a_{\alpha\rho}(X')\rangle $ only depends on $X-X'$. More precisely, we may set $\langle F^a_{\kappa\lambda}(X)F^a_{\alpha\rho}(X')\rangle=\langle F^a_{\kappa\lambda}(X')
F^a_{\alpha\rho}(X)\rangle=\langle F^a_{\kappa\lambda}F^a_{\alpha\rho}\rangle\Phi(X-X')$ \footnote{Note that the Bianchi identity $D_{\mu}F^{\mu\nu}=0$ should yields $k^{\mu}\langle \tilde{F}^a_{\mu\nu}F^a_{\alpha\rho}\rangle=k^{\nu}\langle \tilde{F}^a_{\mu\nu}F^a_{\alpha\rho}\rangle=0$ as the constraint under this approximation.}. Introducing new coordinates, $\bar{X}=(X+X')/2$ and $\delta X=X-X'$, one finds $\partial_{X}=\partial_{\bar{X}}/2+\partial_{\delta X}$ and $\partial_{X'}=\partial_{\bar{X}}/2-\partial_{\delta X}$. By employing the relation in Eq.~(\ref{eq:DkintG}), the dynamical source term stemming from quantum corrections in Eq.~(\ref{AKE_signlet_simplify}) becomes
\begin{eqnarray}\nonumber
\partial_{p}^{\kappa}\big(\mathscr{A}^{\mu}_{\kappa}[f^{\rm s}_{V}]\big)&=&-\frac{\bar{C}_2}{2}
\epsilon^{\mu\nu\rho\sigma}\int^{p}_{k,\delta X}
\bigg[\partial_{\delta X\sigma}
\big(p_{\rho}\langle F^{a\lambda}_{\nu}(X)F^a_{\alpha\lambda}(X')\rangle+p^{\lambda}\langle F^{a}_{\rho\lambda}(X)F^a_{\alpha\nu}(X')\rangle\big)\partial_{p}^{\alpha}f^{\rm s}_{V}(p)
\\\nonumber
&&+p^{\lambda}p_{\rho}\partial_{\delta X\sigma}\Big(\langle F^a_{\kappa\lambda}(X)
F^a_{\alpha\nu}(X')\rangle
-\langle F^a_{\kappa\nu}(X)F^a_{\alpha\lambda}(X')\rangle
\Big)\partial_{p}^{\kappa}\partial_{p}^{\alpha}f^{\rm s}_{V}(p)\bigg]
+\chi_1+\chi_2
\\\nonumber
&=&-\frac{\bar{C}_2}{2}
\epsilon^{\mu\nu\rho\sigma}\int^{p}_{k,\delta X}
\big(p_{\rho}\langle F^{a\lambda}_{\nu}F^a_{\alpha\lambda}\rangle+p^{\lambda}\langle F^{a}_{\rho\lambda}F^a_{\alpha\nu}\rangle\big)\big(\partial_{\delta X\sigma}\Phi(\delta X )\big)\partial_{p}^{\alpha}f^{\rm s}_{V}(p)
+\chi_1+\chi_2,
\\
\end{eqnarray}
where
$\int^{p}_{k,X'}=\int^{p}_{k,\delta X}\equiv \int d^4k\int\frac{d^4\delta X}{(2\pi)^4}e^{-ik\cdot\delta X}\big(\pi\delta(p\cdot k)+iPV(1/p\cdot k)\big)$ and we have used $\langle F^a_{[\kappa\nu}F^a_{\alpha]\lambda}\rangle \partial_{p}^{\kappa}\partial_{p}^{\alpha}f^{\rm s}_{V}(p)=0$.
Here
\begin{eqnarray}
\chi_1=-\frac{\mathscr{A}^{\mu}_{0}[f^{\rm s}_{V}(p)]}{p_0}=\frac{\bar{C}_2}{2p_0}
\epsilon^{\mu\nu\rho\sigma}\int^{p}_{k,\delta X}p^{\lambda}p_{\rho}\langle F^a_{0[\lambda}F^a_{\alpha\nu]}\rangle
\big(\partial_{\delta X\sigma}\Phi(\delta X)\big)\partial^{\alpha}_{p}f^{\rm s}_{V}(p)
\end{eqnarray}
and
\begin{eqnarray}
\chi_2=-\frac{\bar{C}_2}{2p_0}
\epsilon^{\mu\nu\rho\sigma}\int^{p}_{k,\delta X}\delta X_0p^{\lambda}p_{\rho}\langle F^a_{\kappa[\lambda}F^a_{\alpha\nu]}\rangle
\big(\partial_{\delta X^{\gamma}_{\perp}}\partial_{\delta X\sigma}\Phi(\delta X)\big)\left[\partial_{p\kappa}\left(\frac{p^{\gamma}_{\perp}}{p_0}\right)\right]\partial^{\alpha}_{p}f^{\rm s}_{V}(p),
\end{eqnarray}
while their contributions will vanish in the following approximations.
We may now introduce the chromo-electric and chromo-magnetic fields explicitly via
\begin{eqnarray}
F^{a}_{\kappa\lambda}=-\epsilon_{\kappa\lambda\xi\eta}B^{a\xi}\bar{n}^{\eta}+E^a_{[\kappa}\bar{n}_{\lambda]},\quad
\tilde{F}^{a\mu\kappa}=B^{a[\mu}\bar{n}^{\kappa]}+\epsilon^{\mu\kappa\xi\eta}E^a_{\xi}\bar{n}_{\eta}.
\end{eqnarray}
When assuming the hierarchy $|\langle B^{a}_{\mu}B^{a}_{\nu}\rangle| \gg |\langle E^{a}_{\mu}B^{a}_{\nu}\rangle| \gg |\langle E^{a}_{\mu}E^{a}_{\nu}\rangle|$, stemming from the screening of chromo-electric field as opposed to the chromo-magnetic field albeit in the static case \cite{Weldon:1982aq} and amplification of the latter from plasma instability in anisotropic QGP \cite{Mrowczynski:1993qm}, the color-singlet SKE is satisfied by $f^{\rm s}_{V}(p)=f_{\rm eq}(p\cdot u)\equiv 1/(e^{\beta (p\cdot u-\mu)}+1)$ in equilibrium with $u^{\mu}$ the fluid four velocity and $\beta=1/T$ the inverse of temperature \footnote{Nonzero $\langle E^{a}_{\mu}B^{a}_{\nu}\rangle$ or $\langle E^{a}_{\mu}E^{a}_{\nu}\rangle$ can further lead to non-equilibrium corrections such that $\delta f^{\rm s}_{V}(p)= f^{\rm s}_{V}(p)-f_{\rm eq}(p\cdot u)\neq 0$, but these corrections will be at $\mathcal{O}(g^2)$ and suppressed at weak coupling.}. For the parity-odd correlation, we further assume the symmetric condition $\langle E^{a}_{\mu}B^{a}_{\nu}\rangle=\langle B^{a}_{\mu}E^{a}_{\nu}\rangle$ for simplification. One could further derive the deviation of $f^{\rm s}_{V}$ near local equilibrium to extract the anomalous transport coefficient from the diffusion term \cite{Asakawa:2006jn}. We may focus on the spin polarization when the charge distribution of quarks reaches local equilibrium with negligible corrections from spacetime gradients of fluid velocity, temperature, and chemical potential.
Assuming the absence of initial spin polarization in the non-equilibrium phase, the spin diffusion term is then relatively negligible than the source term. The suppression of the diffusion term will be explicitly shown later. Accordingly, Eq.~(\ref{AKE_signlet_simplify}) reduces to
\begin{eqnarray}\label{AKE_singlet_sourceonly}
0=\delta(p^2-m^2)\Big[p\cdot\partial\tilde{a}^{\rm s\mu}(p,X)
+\hbar\partial_{p}^{\kappa}\big(\mathscr{A}^{\mu}_{\kappa}[f^{\rm s}_{V}]\big)_{\rm eq}\Big],
\end{eqnarray}
where $\chi_1$ vanishes and we find
\begin{eqnarray}\nonumber
\partial_{p}^{\kappa}\big(\mathscr{A}^{\mu}_{\kappa}[f^{\rm s}_{V}]\big)_{\rm eq}&\approx &-\frac{\bar{C}_2}{2}
\int^{p}_{k,\delta X}\big(\partial_{p0}f_{\rm eq}(p_0)\big)\Big[\langle B^{a\mu}E^{a\nu}\rangle p_{\nu}u\cdot \partial_{\delta X}+\langle B^a\cdot E^a\rangle u^{[\mu}p^{\nu]}\partial_{\delta X\nu}
-\langle B^{a\rho}E^{a\nu}\rangle
\\
&&\times u^{\mu}p_{\nu}\partial_{\delta X\rho}
+\epsilon^{\mu\nu\rho\sigma}u_{\rho}(\langle E^a_{\nu}E^a_{\lambda}\rangle p^{\lambda}-\langle E^a\cdot E^a\rangle p_{\nu})\partial_{\delta X\sigma}
\Big]\Phi(\delta X)+\chi_2
\end{eqnarray}
by working in the fluid rest frame (not to be confused with the frame choice for the spin basis) such that $u^{\mu}\approx (1,{\bm u})$ with $|\bm u|\ll 1$.
Here we have taken $u^{\mu}\approx \bar{n}^{\mu}$
and introduced the condition $\langle B^{a\mu}(X)E^{a\beta}(X')\rangle=\langle E^{a\beta}(X)B^{a\mu}(X')\rangle$. When further imposing the spatial homogeneity of the field-strength correlators such that $\Phi(\delta X)=\Phi(\delta X_0)$ and accordingly $\chi_2=0$, it is found
\begin{eqnarray}\nonumber
\partial_{p}^{\kappa}\big(\mathscr{A}^{\mu}_{\kappa}[f^{\rm s}_{V}]\big)_{\rm eq}&=&-\frac{\bar{C}_2}{2}
\int dk_0\int\frac{d\delta X_0}{2\pi}e^{-ik_0\delta X_0}\Big[\pi\delta(p_0k_0)+i{\rm PV}\Big(\frac{1}{p_0k_0}\Big)\Big]\big(\partial_{p0}f_{\rm eq}(p_0)\big)
\\\nonumber
&&\times\big(\langle B^{a\mu}E^{a\nu}\rangle p_{\nu}-\langle B^a\cdot E^a\rangle p^{\mu}_{\perp}\big)\partial_{\delta X_0}\Phi(\delta X_0)
\\\nonumber
&=&-\frac{\bar{C}_2}{4}
\int d\delta X_0\frac{(1+{\rm sgn}(\delta X_0))}{p_0}\big(\partial_{p0}f_{\rm eq}(p_0)\big)\big(\langle B^{a\mu}E^{a\nu}\rangle p_{\nu}-\langle B^a\cdot E^a\rangle p^{\mu}_{\perp}\big)
\\
&&\times \partial_{\delta X_0}\Phi(\delta X_0).
\end{eqnarray}
Notably, when $\Phi(x)|_{x\rightarrow \pm\infty}=0$, one can carry out the integration by parts for the equation above and derive
\begin{eqnarray}
\partial_{p}^{\kappa}\big(\mathscr{A}^{\mu}_{\kappa}[f^{\rm s}_{V}]\big)_{\rm eq}=\frac{\bar{C}_2\Phi(0)}{2p_0}
\big(\partial_{p0}f_{\rm eq}(p_0)\big)\big(\langle B^{a\mu}E^{a\nu}\rangle p_{\nu}-\langle B^a\cdot E^a\rangle p^{\mu}_{\perp}\big).
\end{eqnarray}
Setting
\begin{eqnarray}
\Phi(\delta X_0)=e^{-\delta X_0^2/\tau^2_{c}}
\end{eqnarray}
as the Gaussian form with $\tau_c$ the correlation time, we further obtain
\begin{eqnarray}\label{sorce_simpl}
\partial_{p}^{\kappa}\big(\mathscr{A}^{\mu}_{\kappa}[f^{\rm s}_{V}]\big)_{\rm eq}
=\frac{\bar{C}_2}{2p_0}
\big(\partial_{p0}f_{\rm eq}(p_0)\big)\big(\langle B^{a\mu}E^{a\nu}\rangle p_{\nu}-\langle B^a\cdot E^a\rangle p^{\mu}_{\perp}\big).
\end{eqnarray}
Since now $\partial_{p}^{\kappa}\big(\mathscr{A}^{\mu}_{\kappa}[f^{\rm s}_{V}]\big)_{\rm eq}$ is independent of $X$, $\tilde{a}^{\rm s\mu}$ will be a function linear to time. More precisely, by solving Eq.~(\ref{AKE_singlet_sourceonly}) with Eq.~(\ref{sorce_simpl}), we obtain \footnote{The $\langle B^{a\mu}E^{a\nu}\rangle p_{\nu}$ term below will be in fact $\langle E^{a\mu}B^{a\nu}\rangle p_{\nu}$ without using the symmetric correlation $\langle B^{a\mu}E^{a\nu}\rangle=\langle B^{a\nu}E^{a\mu}\rangle$.}
\begin{eqnarray}\label{amu_from_source}\nonumber
\tilde{a}^{\rm s\mu}(t,p)&=&-\frac{\hbar\bar{C}_2(t-t_0)}{2p_0^2}\Phi(0)\big(\partial_{p0}f_{\rm eq}(p_0)\big)\big(\langle B^{a\mu}E^{a\nu}\rangle p_{\nu}-\langle B^a\cdot E^a\rangle p^{\mu}_{\perp}\big)
\\
&=&-\frac{\hbar\bar{C}_2(t-t_0)}{2p_0^2}
\big(\partial_{p0}f_{\rm eq}(p_0)\big)\big(\langle B^{a\mu}E^{a\nu}\rangle p_{\nu}-\langle B^a\cdot E^a\rangle p^{\mu}_{\perp}\big),
\end{eqnarray}
for $\tilde{a}^{\mu}(t_0,p)=0$. This secular solution stems from only the $\delta X$ dependence of color-field correlators. Notwithstanding the ostensible $\tau_{c}$ independence of Eq.~(\ref{sorce_simpl}), the source term should vanish when color-field correlators become constant. That is, $\partial_{p}^{\kappa}\big(\mathscr{A}^{\mu}_{\kappa}[f^{\rm s}_{V}]\big)_{\rm eq}\rightarrow 0$ when $\tau_c\rightarrow \infty$. In a finite system, there exists an upper bound for $|\delta X|$ determined by the size of the system. When $\tau_c$ is greater than the system size, $\Phi(\delta X_0)$ is peaked at $\delta X_0=0$ and the source term in the AKE is suppressed. In heavy ion collisions, one may instead assume an approximately infinite system.
In principle, at a sufficiently long time, $\tilde{a}^{s\mu}$ is no longer small and the diffusion term should start to play a role. According to the proposed hierarchy, the diffusion term in the color-singlet AKE is given by
\begin{eqnarray}\nonumber
\partial_{p}^{\kappa}\mathscr{D}_{\kappa}[\tilde{a}^{s\mu}]&\approx&\bar{C}_2\epsilon_{\alpha\beta\gamma\kappa}\epsilon_{\nu\rho\lambda\sigma}u^{\alpha}u^{\nu}\partial_{p}^{\beta}\int^p_{k,X'}\langle B^{a\kappa}(X)B^{a\sigma}(X')\rangle p^{\gamma}p^{\lambda}\partial_{p}^{\rho}\tilde{a}^{s\mu}
\\
&=&\frac{\bar{C}_2\sqrt{\pi}\tau_m}{2p_0}\epsilon_{\alpha\beta\gamma\kappa}\epsilon_{\nu\rho\lambda\sigma}u^{\alpha}u^{\nu}\langle B^{a\kappa}B^{a\sigma}\rangle\partial_{p}^{\beta}\big( p^{\gamma}p^{\lambda}\partial_{p}^{\rho}\tilde{a}^{s\mu}\big),
\end{eqnarray}
where we derive the second equality by assuming $\langle B^{a\kappa}(X)B^{a\sigma}(X')\rangle=\langle B^{a\kappa}B^{a\sigma}\rangle e^{-\delta X_0^2/\tau_{m}^2}$. Conducting a simple power counting, $\partial_{p}^{\kappa}\mathscr{D}_{\kappa}[\tilde{a}^{s\mu}]$ becomes comparable to $p\cdot \partial \tilde{a}^{s\mu}$ when
\begin{eqnarray}
(t-t_0)\tau_m|\langle B^{a\kappa}B^{a\sigma}\rangle|\sim p_0^2.
\end{eqnarray}
Based on the kinetic region we considered for the $\hbar$ expansion, $p_0^2\gg |B^{a\mu}|$, we may in general neglect the diffusion term given no other sources for spin polarization. Nevertheless, the suppression of the diffusion term is also valid because of the properties of postulated color fields, which may not be always the case in a more general condition.
Moreover, nonzero spin polarization could be engendered by the explicit $\hbar$ term in Eq.~(\ref{eq:Asmu_massive}), which could be regarded as a non-dynamical source term for spin polarization, even when the dynamical source term in the AKE is suppressed. We will analyze such a contribution from Eq.~(\ref{eq:Asmu_massive}) in the following paragraphs. When $f^{\rm s}_{V}(p)$ is in thermal equilibrium, one obtains
\begin{eqnarray}\label{AQ1_massive}
\mathcal{A}_{Q1}^{\mu} = -\frac{1}{2}\bigg[\int^{p}_{k,X'}\langle \tilde{F}^{a\mu\kappa}(X)E^a_{\beta}(X')\rangle \big(\delta^{\beta}_{\kappa}+u_{\kappa}(p^{\beta}\partial_{0}-\hat{p}^{\beta})\big)\partial_{p0}f_{\rm eq}(p_0)\bigg]_{p_0=\epsilon_{\bm p}}
+\xi_1
\end{eqnarray}
and
\begin{eqnarray}\nonumber
\mathcal{A}^{\mu}_{Q2}&=&-\frac{1}{2\epsilon_{\bm p}^2}(p_{\perp\kappa}-\epsilon_{\bm p}^2\partial_{p_{\perp}\kappa})\bigg[\int^{p}_{k,X'}p^{\beta}\langle \tilde{F}^{a\mu\kappa}(X)E^a_{\beta}(X')\rangle \partial_{p0}f_{\rm eq}(p_0)\bigg]_{p_0=\epsilon_{\bm p}}+\xi_2
\\
&=&-\frac{1}{2}\bigg[\int^{p}_{k,X'}\langle \tilde{F}^{a\mu\kappa}(X)E^a_{\beta}(X')\rangle \big(\hat{p}_{\perp\kappa}\hat{p}^{\beta}(2-p_0\partial_{p0})-\delta^{\beta}_{\kappa}\big)\partial_{p0}f_{\rm eq}(p_0)\bigg]_{p_0=\epsilon_{\bm p}}+\xi_2
\end{eqnarray}
Here
\begin{eqnarray}
\xi_1= -\frac{1}{2}\bigg[\int^{p}_{k,X'}\delta X_0\partial_{\delta X_{\perp}\rho}\langle \tilde{F}^{a\mu\kappa}(X)E^a_{\beta}(X')\rangle p^{\beta}\big(\partial_{p0}f_{\rm eq}(p_0)\big)\partial_{p\kappa}\left(\frac{p^{\rho}_{\perp}}{p_0}\right)\bigg]_{p_0=\epsilon_{\bm p}}
\end{eqnarray}
and
\begin{eqnarray}
\xi_2=\frac{1}{2}\bigg[\int^{p}_{k,X'}\delta X_0\partial_{\delta X_{\perp}\rho}\langle \tilde{F}^{a\mu\kappa}(X)E^a_{\beta}(X')\rangle p^{\beta}\partial_{p0}f_{\rm eq}(p_0)\bigg]_{p_0=\epsilon_{\bm p}}\partial_{p_{\perp}\kappa}\left(\frac{p^{\rho}_{\perp}}{\epsilon_{\bm p}}\right).
\end{eqnarray}
Further taking $\langle E^{a\mu}(X) B^{a\beta}(X')\rangle=\langle E^{a\mu}(X') B^{a\beta}(X)\rangle$, $\langle E^{a\mu}(X) E^{a\beta}(X')\rangle=\langle E^{a\mu}(X') E^{a\beta}(X)\rangle$, and $\Phi(\delta X)=\Phi(\delta X_0)$ such that $\xi_1=\xi_2=0$
and using the relation,
\begin{eqnarray}\label{EB_rel}\nonumber
\tilde{F}^{a\mu\beta}(X)F^a_{\alpha\beta}(X')&=&\delta^{\mu}_{\alpha}E^a(X)\cdot B^a(X')+\bar{n}^{\mu}\bar{n}_{\alpha}\big(B^a(X)\cdot E^a(X')-E^a(X)\cdot B^a(X')\big)
\\ &&+B^{a\mu}(X)E^a_{\alpha}(X')-E^a_{\alpha}(X)B^{a\mu}(X'),
\end{eqnarray}
with the symmetric correlation $\langle E^{a\mu}(X) B^{a\beta}(X')=\langle E^{a\beta}(X) B^{a\mu}(X')\rangle$,
it is found
\begin{eqnarray}\nonumber
\mathcal{A}_{Q1}^{\mu}(\bm p,X)&\approx&\bigg[\int d\delta X_0\frac{(1+{\rm sgn}(\delta X_0))}{4p_0}\Phi(\delta X_0)\big(\langle E^a\cdot B^a\rangle u^{\mu}-\langle B^{a\mu}E^{a\beta}\rangle (p_{\beta}\partial_{p_0}-\hat{p}_{\beta})\big)
\\
&&\times\partial_{p_0}f_{\rm eq}(p_0)\bigg]_{p_0=\epsilon_{\bm p}},
\end{eqnarray}
and
\begin{eqnarray}\nonumber
\mathcal{A}_{Q2}^{\mu}(\bm p,X)&\approx&
\bigg[\int d\delta X_0\frac{(1+{\rm sgn}(\delta X_0))}{4p_0}\Phi(\delta X_0)
\Big(\big(\hat{p}^{\alpha}\hat{p}^{\beta}\langle E^a_{\alpha}B^a_{\beta}\rangle u^{\mu}+\epsilon^{\nu\mu\rho\alpha}u_{\nu}\hat{p}_{\rho}\hat{p}^{\beta}\langle E^a_{\alpha}E^a_{\beta}\rangle\big)
\\
&&\times (2-p_0\partial_{p0})-\langle E^a\cdot B^a\rangle u^{\mu} \Big)\partial_{p_0}f_{\rm eq}(p_0)\bigg]_{p_0=\epsilon_{\bm p}},
\end{eqnarray}
which yield
\begin{eqnarray}\nonumber\label{AQ_massive}
\mathcal{A}_Q^{\mu}(\bm p,X)&=&\frac{\pi^{1/2}\tau_c}{4\epsilon_{\bm p}^3}\Big[\big(p^{\alpha}p^{\beta}\langle E^a_{\alpha}B^a_{\beta}\rangle u^{\mu}+\epsilon^{\mu\alpha\rho\nu}u_{\nu}p_{\rho}\langle E^a_{\alpha}E^a_{\beta}\rangle\big)(2p^{\beta}-\epsilon_{\bm p}^2\partial^{\beta}_{p})
\\
&&-\langle B^{a\mu}E^{a\beta}\rangle \epsilon_{\bm p}(\epsilon_{\bm p}^2\partial_{p\beta}-p_{\beta})\Big]\partial_{\epsilon_{\bm p}}f_{\rm eq}(\epsilon_{\bm p})
\end{eqnarray}
for $\Phi(\delta X_0)=e^{-\delta X_0^2/\tau^2_{c}}$, where we have used $\epsilon_{\bm p}\partial^{\beta}_{p_{\perp}}G(\epsilon_{\bm p})= p^{\beta}_{\perp}\partial_{\epsilon_{\bm p}}G(\epsilon_{\bm p})$ for an arbitrary function $G(\epsilon_{\bm p})$.
In $\mathcal{A}_{Q2}^{\mu}$, we further have nonzero contribution from $\langle E^{a}_{\nu}E^{a\beta}\rangle$ as a parity-even correlator, while it is attached to $\epsilon^{\mu\alpha\rho\nu}u_{\nu}\hat{p}_{\rho}$ and hence the overall parity-odd contribution is induced. Such a term could be understood as the combination of the Lorentz force and spin Hall effect, whereas it is subdominant according to our assumption that the average magnitude of color magnetic fields is larger than of color electric fields. This term also vanishes when assuming $\langle E^a_{\alpha}E^a_{\beta}\rangle \sim \delta_{\alpha\beta}$ for the correlation of only parallel color fields. When combining the contribution from $\tilde{a}^{\rm s\mu}$ in Eq.~(\ref{amu_from_source}), one finds
\begin{eqnarray}\nonumber\label{Amu_equil}
\hat{\mathcal{A}}^{\rm s\mu}(\bm p,X)
&=&\frac{\hbar\bar{C}_2}{4\epsilon_{\bm p}^3}\Big\{\sqrt{\pi}\tau_c\Big[(p^{\alpha}p^{\beta}\langle E^a_{\alpha}B^a_{\beta}\rangle u^{\mu}+\epsilon^{\mu\alpha\rho\nu}u_{\nu}p_{\rho}\langle E^a_{\alpha}E^a_{\beta}\rangle)(2p^{\beta}-\epsilon_{\bm p}^2\partial^{\beta}_{p})
-\langle B^{a\mu}E^{a\nu}\rangle
\\
&&\times \epsilon_{\bm p}(\epsilon_{\bm p}^2\partial_{p\nu}-p_{\nu})\Big]
+2(t-t_0)\epsilon_{\bm p}\big(\langle E^a\cdot B^a\rangle p^{\mu}_{\perp}-\langle B^{a\mu}E^{a\nu}\rangle p_{\nu}\big)
\Big\}\partial_{\epsilon_{\bm p}}f_{\rm eq}(\epsilon_{\bm p}).
\end{eqnarray}
When considering the non-relativistic limit such that $|p^{\mu}_{\perp}|\ll \epsilon_{\bm p}$, we further obtain
\begin{eqnarray}
\hat{\mathcal{A}}^{\rm s\mu}(\bm p,X)\approx -\frac{\hbar\bar{C}_2}{4\epsilon_{\bm p}}\sqrt{\pi}\tau_c\langle B^{a\mu}E^{a\nu}\rangle p_{\nu}\partial_{\epsilon_{\bm p}}f_{\rm eq}(\epsilon_{\bm p}),
\end{eqnarray}
where one component of the non-dynamical source term dominates. Such a term stems from the combination of spin polarization induced by a color magnetic field and the Lorentz force driven by a color electric field. We may further evaluate the axial charge current via Eq.~(\ref{def_currents}), where only $\mathcal{A}^{\mu}_{Q1}$ contributes. One hence obtains
\begin{eqnarray}\label{J5_massive}
J^{\mu}_5
=2\hbar \int\frac{d^4p}{(2\pi)^3}\frac{\delta(p^2-m^2)}{4p\cdot u}\sqrt{\pi}\tau_c
\langle E^a\cdot B^a\rangle u^{\mu}\partial_{p\cdot u}f_{\rm eq}(p\cdot u)
.
\end{eqnarray}
Here the contribution from $\tilde{a}^{\rm s\mu}$ in Eq.~(\ref{amu_from_source}) does not affect $J^{\mu}_5$.
\subsection{Axial Ward identity}
It is obvious that $J^{\mu}_5$ in Eq.~(\ref{J5_massive}) is independent of $X$ and thus $\partial_{\mu}J^{\mu}_5=0$.
It is however useful to investigate the vanishing axial Ward identity from Eq.~(\ref{AQ1_massive}).
When $\langle B^{a}(X)\cdot E^a(X')\rangle$ only depends on $\delta X$, it is found
\begin{eqnarray}\nonumber\label{eq:partialx_EB}
\partial_{X\mu}\int^{p}_{k,X'}
\langle B^{a}(X)\cdot E^a(X')\rangle&=&
\int^{p}_{k,\delta X}
(-ik_{\mu}+\partial_{\delta X\mu})\langle B^{a}(X)\cdot E^a(X')\rangle
\\
&=&i\int d^4k\frac{d^4\delta X}{(2\pi)^4}e^{-ik\cdot\delta X}\frac{k_{\mu}+i\partial_{\delta X\mu}}{p\cdot k+i\epsilon}\langle B^{a}(X)\cdot E^a(X')\rangle.
\end{eqnarray}
From the integration by part, $i\partial_{\delta X\mu}$ in the integrand above could be replaced by $-k_{\mu}$ and thus we find $\partial_{\mu}J^{\mu}_5(X)=0$ for finite $\tau_c$.
Nevertheless, when $\tau_c\rightarrow \infty$ such that $\langle B^{a}(X)\cdot E^a(X')\rangle\rightarrow \langle B^{a}\cdot E^a\rangle$ as the constant-field limit, $i\partial_{\delta X\mu}$ in the integrand of Eq.~(\ref{eq:partialx_EB}) no longer contributes and we find
\begin{eqnarray}
\partial_{\mu} J_5^{\mu}(X)=2\hbar \int d^4k\frac{d^4\delta X}{(2\pi)^4}e^{-ik\cdot\delta X}\langle B^{a}\cdot E^a\rangle\int\frac{d^4p}{(2\pi)^3}{{\rm sgn}(p_0)}\frac{\delta(p^2-m^2)}{2(p\cdot k+i\epsilon)}
k_0\partial_{p_0}f_{\rm eq}(p_0),
\end{eqnarray}
where we further insert ${{\rm sgn}(p_0)}$ as a sign function for energy to include the contribution from anti-quarks. In fact, the ultraviolet contribution from anti-fermions is essential to reproduce the term associated with the chiral anomaly \cite{Hidaka:2017auj}. Now, the $\delta X$ integral can be evaluated independently,
\begin{eqnarray}\label{eq:delta_k}
\int \frac{d^4\delta X}{(2\pi)^4}e^{ik\cdot\delta X}=\delta^4(k).
\end{eqnarray}
For the remaining $p$ and $k$ integrals, one may make the decomposition,
\begin{eqnarray}
\int d^4k\delta^4(k)\int\frac{d^4p}{(2\pi)^3}{{\rm sgn}(p_0)}\frac{\delta(p^2-m^2)}{2(p\cdot k+i\epsilon)}k_0\partial_{p_0}f_{\rm eq}(p_0)
=I_1+I_2,
\end{eqnarray}
where the real part of $(p\cdot k+i\epsilon)^{-1}$ contributes to \footnote{Here we first take $\lim_{\bm k\rightarrow 0}$ and then $\lim_{k_0\rightarrow 0}$. In fact, for constant color fields, we may reduce $\int^{p}_{k,X'}$ to just the one-dimensional integral as $\int^{p}_{k,X'}\equiv \int dk_0\int\frac{dX'_0}{(2\pi)}e^{ik_0(X'_0-X_0)}i(p_0k_0+i\epsilon)^{-1}$ and hence the $\bm k$ and $\bm X'$ integrals are redundant since it is natural to assume $f^a_{V}$ and $\tilde{a}^{\mu}_a$ only depend on time.}
\begin{eqnarray}
I_1=\int\frac{d^4p}{(2\pi)^3}{{\rm sgn}(p_0)}\frac{\delta(p^2-m^2)}{2p\cdot k}k_0\partial_{p_0}f_{\rm eq}(p_0)\Big|_{k_{\mu}\rightarrow 0}=\int\frac{d^3\bm p}{(2\pi)^3 4\epsilon_{\bm p}^2}\frac{d}{d\epsilon_{\bm p}}\big[f_{\rm eq}(\epsilon_{\bm p})-f_{\rm eq}(-\epsilon_{\bm p})\big]
\end{eqnarray}
with $\epsilon_{\bm p}=\sqrt{|\bm p|^2+m^2}$, and the imaginary part of $(p\cdot k+i\epsilon)^{-1}$ yields
\begin{eqnarray}
I_2&=&-i\int\frac{d^4p}{(2\pi)^3}{{\rm sgn}(p_0)}\frac{\delta(p^2-m^2)}{2}k_0\pi\delta(p\cdot k)\partial_{p_0}f_{\rm eq}(p_0)\Big|_{k_{\mu}\rightarrow 0}=0.
\end{eqnarray}
In the end, we obtain an axial Ward identity as
\begin{eqnarray}\label{divJ5_nonzero}
\partial_{\mu} J_5^{\mu}(X)=\hbar \langle B^{a}\cdot E^a\rangle
\int^{\infty}_0\frac{d|\bm p||\bm p|^2}{4\pi^2\epsilon_{\bm p}^2}\frac{d}{d\epsilon_{\bm p}}\big[f_{\rm eq}(\epsilon_{\bm p})-f_{\rm eq}(-\epsilon_{\bm p})\big].
\end{eqnarray}
As briefly discussed in Ref.~\cite{Muller:2021hpe}, the vanishing axial Ward identity at finite $\tau_c$ corresponds to a triangle diagram with two gluon legs connected and zero momentum flow from the axial vertex. However, when $\tau_c\rightarrow \infty$, two legs break apart and allow the momentum exchanges. See Fig.~\ref{fig:triangle diagrams} for a schematic description.
\begin{figure}
\begin{center}
\includegraphics[width=0.6\hsize]{triangle_diagrams}
\end{center}
\caption{The triangle diagram with finite $\tau_c$ and with $\tau_c\rightarrow \infty$ in the constant-field limit. The blobs denote the axial vertices and the wiggly line with a cross at the center represents the correlator of color fields originating from the medium.
}
\label{fig:triangle diagrams}
\end{figure}
It is found $\partial_{\mu} J^{\mu}_5(X)\neq 0$ as opposed to the conservation of a vector charge current, $\partial_{\mu} J^{\mu}_V(X)=0$, implied by Eq.~(\ref{SKE_signlet_simplify}) since the diffusion term can be recast into a total momentum derivative including the attached $\delta(p^2-m^2)$ due to $p^{\kappa}\mathscr{D}_{\kappa}[O]=0$. Now, Eq.~(\ref{divJ5_nonzero}) seems to be in connection to the chiral anomaly but it also receives the finite-mass correction. Recall that
\begin{eqnarray}
\partial_{\mu} J^{\mu}_5(X)=-\frac{E\cdot B}{2\pi^2}+2m\langle \bar{\psi}i\gamma_5\psi\rangle
\end{eqnarray}
for massive fermions, where we consider the Abelian gauge fields for simplicity yet the similar feature holds for non-Abeleian cases. Here the pseudo-scalar condensate results in further modifications when $m\neq 0$. However, it is anticipated that $\partial_{\mu} J^{\mu}_5(X)=0$ when $m\rightarrow \infty$. For example, in the vacuum with constant electromagnetic fields, it is found \cite{Copinger:2018ftr},
\begin{eqnarray}
\langle \bar{\psi}i\gamma_5\psi\rangle=\frac{E\cdot B}{4\pi^2m}\Big(1-e^{-\pi m^2/|E|}\Big)
\end{eqnarray}
and hence
\begin{eqnarray}
\partial_{\mu} J^{\mu}_5(X)=-\frac{E\cdot B}{2\pi^2}e^{-\pi m^2/|E|}.
\end{eqnarray}
As shown in Eq.~(\ref{divJ5_nonzero}), $\partial_{\mu} J^{\mu}_5(X)=0$ when $m\rightarrow \infty$. In fact, from Eq.~(\ref{divJ5_nonzero}), we can extract the pseudo-scalar condensate,
\begin{eqnarray}
\langle \bar{\psi}i\gamma_5\psi\rangle=-\frac{\hbar \langle B^{a}\cdot E^a\rangle}{8m\pi^2}
\int^{\infty}_0d|\bm p|\left(1-\frac{|\bm p|}{\epsilon_{\bm p}}\right)\frac{d}{d|\bm p|}\big[f_{\rm eq}(\epsilon_{\bm p})-f_{\rm eq}(-\epsilon_{\bm p})\big],
\end{eqnarray}
such that Eq.~(\ref{divJ5_nonzero}) is decomposed as
\begin{eqnarray}\label{divJ5_condensate}
\partial_{\mu} J_5^{\mu}(X)=-\hbar \frac{\langle B^{a}\cdot E^a\rangle}{4\pi^2}+2m\langle \bar{\psi}i\gamma_5\psi\rangle,
\end{eqnarray}
where the first term on the right-hand side above is given by
\begin{eqnarray}\nonumber
&&\frac{\hbar}{2}\langle B^{a}\cdot E^a\rangle
\int^{\infty}_0\frac{d|\bm p|}{4\pi^2}\frac{d}{d|\bm p|}\big[f_{\rm eq}(|\bm p|)-f_{\rm eq}(-|\bm p|)\big]
\\\nonumber
&&=\frac{\hbar}{4\pi^2}\langle B^{a}\cdot E^a\rangle
\big[f_{\rm eq}(|\bm p|)-f_{\rm eq}(-|\bm p|)\big]\Big|^{|\bm p|=\infty}_{|\bm p|=0}
\\
&&=-\frac{\hbar}{4\pi^2}\langle B^{a}\cdot E^a\rangle.
\end{eqnarray}
Naively, when taking $m\rightarrow 0$, it seems $m\langle \bar{\psi}i\gamma_5\psi\rangle\rightarrow 0$ and the non-Abelian axial anomaly is reproduced in the massless limit.
Nevertheless, since we work in the rest frame $n^{\mu}=n^{\mu}_{r}(p)$, the result is in principle subject to a finite-mass regime $m^2\gg |E^a|,\,|B^a|$. We may have to examine the massless limit more rigorously as will be demonstrated in the follow-up section.
\section{Massless fermions}\label{sec:massless_fermions}
In order to investigate the influence of chromo-electromagnetic fields upon light quarks in QGP. We further analyze the similar effects on spin polarization of massless quarks via the CKT with non-Abelian gauge fields to complement the previous study for massive quarks.
\subsection{Derivation of the effective AKE}
For massless fermions, it is more convenient to work in the chiral bases and the master equations reduce to
\begin{eqnarray}
&&\hat{\nabla}\cdot \mathcal{W}_{s}=0,
\\
&&\hat{\Pi}\cdot\mathcal{W}_{s}=0,
\\
&&\hat{\Pi}_{\nu}\mathcal{W}_{s\mu}-\hat{\Pi}_{\mu}\mathcal{W}_{s\nu}
=\frac{s\hbar}{2}\epsilon_{\mu\nu\rho\sigma}\hat{\nabla}^{\rho}\mathcal{W}_{s}^{\sigma},
\end{eqnarray}
where $\mathcal{W}_{s\mu}(p,X)$ denote the chiral components of Wigner operators (lesser propagators) for massless fermions with $s=1$ and $s=-1$ for right and left handed fermions (not to confuse with the superscript $\rm s$ for color singlet). Unlike the case for Dirac fermions, $\mathcal{W}_{s}$ are $2\times 2$ matrices in spinor space, which can be decomposed as $\mathcal{W}_{+}=\bar{\sigma}^{\mu}\mathcal{W}_{+\mu}$ and $\mathcal{W}_{-}=\sigma^{\mu}\mathcal{W}_{-\mu}$, where $\sigma^{\mu}=(1,{\bm \sigma})$ and $\bar{\sigma}^{\mu}=(1,-{\bm \sigma})$ with $\sigma^i$ being Pauli matrices. Based on the $\hbar$ expansion, the master equations explicit read
\begin{eqnarray}\label{Weyl_eq_1}
&&D\cdot \mathcal{W}_{s}+\frac{1}{2}\{F_{\nu\mu},\partial_{p}^{\nu}\mathcal{W}_{s}^{\mu}\}_{\rm c}-\frac{i\hbar}{24}[(\partial_{p}\cdot DF_{\nu\mu}),\partial_p^{\nu}\mathcal{W}_{s}^{\mu}]_{\rm c}=0,
\\\label{Weyl_eq_2}
&&p_{\mu}\mathcal{W}_{s}^{\mu}+\frac{i\hbar}{8}[F_{\nu\mu},\partial_{p}^{\nu}\mathcal{W}_{s}^{\mu}]_{\rm c}=0,
\\\label{Weyl_eq_3}
&&p_{[\nu}\mathcal{W}_{s\mu]}+\frac{i\hbar}{8}[F_{\rho[\nu},\partial_{p}^{\rho}\mathcal{W}_{s\mu]}]_{\rm c}
=\frac{s\hbar}{2}\epsilon_{\mu\nu\rho\sigma}\Big(D^{\rho}\mathcal{W}_{s}^{\sigma}+\frac{1}{2}\{F_{\nu\mu},\partial_{p}^{\nu}\mathcal{W}_{s}^{\sigma}\}_{\rm c}\Big),
\end{eqnarray}
up to $\mathcal{O}(\hbar)$. By contracting Eq.~(\ref{Weyl_eq_3}) with $n^{\mu}$, one can derive the Wigner operator satisfying the constrain in Eq.~(\ref{Weyl_eq_2}). It is found
\begin{eqnarray}\nonumber
\mathcal{W}_{s}^{\mu}&=&2\pi\bigg[\delta(p^2)\Big(p^{\mu}\hat{f}_s+s\hbar S^{\mu\nu}_{(n)}\tilde{\Delta}_{\nu}\hat{f}_s\Big)
+\frac{s\hbar}{2} p_{\nu}\delta'(p^2)\{\tilde{F}^{\mu\nu},\hat{f}_s\}_{\rm c}
+\frac{i\hbar}{8}\bigg(p^{\mu}\delta'(p^2)p^{\rho}[F_{\nu\rho},\partial_{p}^{\nu}\hat{f}_s]_{\rm c}
\\
&&+[F^{\rho\mu},\partial_{p\rho}\delta(p^2)\hat{f}_s]_{\rm c}-\frac{2\delta(p^2)}{p\cdot n}[F^{\mu\nu}n_{\nu},\hat{f}_s]_{\rm c}\bigg)\bigg]
\end{eqnarray}
up to $\mathcal{O}(\hbar)$, where $S^{\mu\nu}_{(n)}=S^{\mu\nu}_{m(n)}|_{m=0}$ represents the spin tensor for massless fermions. When making a conversion to the vector and axial-vector basis by $(\mathcal{V}/\mathcal{A})^{\mu}=(\mathcal{W}^{\mu}_{+}\pm \mathcal{W}^{\mu}_{-})/2$, one finds
\begin{eqnarray}\nonumber
\mathcal{A}^{\mu}&=&2\pi\bigg[\delta(p^2)\Big(p^{\mu}\hat{f}_{A}+\hbar S^{\mu\nu}_{(n)}\tilde{\Delta}_{\nu}\hat{f}_{V}\Big)
+\frac{\hbar}{2} p_{\nu}\delta'(p^2)\{\tilde{F}^{\mu\nu},\hat{f}_{V}\}_{\rm c}
+\frac{i\hbar}{8}\bigg(p^{\mu}\delta'(p^2)p^{\rho}[F_{\nu\rho},\partial_{p}^{\nu}\hat{f}_{A}]_{\rm c}
\\
&&+[F^{\rho\mu},\partial_{p\rho}\delta(p^2)\hat{f}_{A}]_{\rm c}-\frac{2\delta(p^2)}{p\cdot n}[F^{\mu\nu}n_{\nu},\hat{f}_{A}]_{\rm c}\bigg)\bigg],
\end{eqnarray}
where $\hat{f}_{V/A}=(\hat{f}_{+}\pm \hat{f}_{-})/2$. Notably, the $\hbar$ terms related to commutators in color space are all linear to $\hat{f}_A$. As a result, implementing the power counting $\hat{f}_{V}\sim \mathcal{O}(\hbar^0)$ and $\hat{f}_{A}\sim \mathcal{O}(\hbar)$, such terms involving commutators are suppressed. Similarly, the contribution from the last term on the left hand side of Eq.~(\ref{Weyl_eq_1}) in AKE is now of $\mathcal{O}(\hbar^2)$ and hence omitted. That is, the effective AKE is given by
\begin{eqnarray}\label{AKE_massless}
\tilde{\Delta}\cdot\mathcal{A}=D\cdot \mathcal{A}+\frac{1}{2}\{F_{\nu\mu},\partial_{p}^{\nu}\mathcal{A}^{\mu}\}_{\rm c}=0,
\end{eqnarray}
with
\begin{eqnarray}\label{Amu_massless}
\mathcal{A}^{\mu}=2\pi\bigg[\delta(p^2)\Big(p^{\mu}\hat{f}_{A}+\hbar S^{\mu\nu}_{(n)}\tilde{\Delta}_{\nu}\hat{f}_{V}\Big)
+\frac{\hbar}{2} p_{\nu}\delta'(p^2)\{\tilde{F}^{\mu\nu},\hat{f}_{V}\}_{\rm c}
\bigg],
\end{eqnarray}
which corresponds to the massless limit of Eq.~(\ref{axial_sol}) with $\hat{a}^{\mu}=p^{\mu}\hat{f}_{A}$ due to the spin locking by helicity.
Note that the $\mathcal{A}^{\mu}$ in Eq.~(\ref{Amu_massless}) is consistent with the result in Ref.~\cite{Luo:2021uog} \footnote{There are extra $\hbar$ terms proportional to $f_A$ in Ref.~\cite{Luo:2021uog}, while these terms are at $\mathcal{O}(\hbar^2)$ with our power counting and accordingly omitted.}. As opposed to the massive case, one is now unable to choose the particle rest frame for $n^{\mu}$. Consequently, the magnetization-current term pertinent to $S^{\mu\nu}_{(n)}$ or the so-called side-jump term \cite{Chen:2014cla,Hidaka:2016yjf} in the massless limit will be always involved. For $\mathcal{V}^{\mu}$, we simply retain the leading-order term of $\mathcal{O}(\hbar^0)$ and thus have $\mathcal{V}^{\mu}=2\pi\delta(p^2)p^{\mu}\hat{f}_V$. Also, the SKE is same as Eq.~(\ref{SKE}) with the massless on-shell condition.
Inserting Eq.~(\ref{Amu_massless}) into Eq.~(\ref{AKE_massless}), the AKE turns out to be
\begin{eqnarray}\nonumber
0&=&\delta(p^2)\Big(p\cdot\tilde{\Delta}\hat{f}_A+\hbar\big(\tilde{\Delta}_{\mu}S^{\mu\nu}_{(n)}\big)\tilde{\Delta}_{\nu}\hat{f}_V
+\hbar S^{\mu\nu}_{(n)}\tilde{\Delta}_{\mu}\tilde{\Delta}_{\nu}\hat{f}_V\Big)
+\hbar \delta'(p^2)S^{\mu\nu}_{(n)}p^{\rho}\{F_{\rho\mu},\tilde{\Delta}_{\nu}\hat{f}_V\}
\\
&&+\frac{\hbar}{2}\delta'(p^2)
p_{\nu}\tilde{\Delta}_{\mu}\{\tilde{F}^{\mu\nu},\hat{f}_{V}\}_{\rm c}+\frac{\hbar}{4}\big(\partial_{p}^{\rho}p_{\nu}\delta'(p^2)\big)\{F_{\rho\mu},\{\tilde{F}^{\mu\nu},\hat{f}_{V}\}\}_{\rm c}\Big).
\end{eqnarray}
We may now compute each term above. For the $\hbar$ corrections pertinent to $\delta(p^2)$, it is found
\begin{eqnarray}
\big(\tilde{\Delta}_{\mu}S^{\mu\nu}_{(n)}\big)\tilde{\Delta}_{\nu}\hat{f}_V=(\partial_{\mu}S^{\mu\nu}_{(n)})\tilde{\Delta}_{\nu}\hat{f}_V+\frac{1}{2p\cdot n}\Big(\{\tilde{F}^{\nu\beta}n_{\beta},\tilde{\Delta}_{\nu}\hat{f}_V\}_{\rm c}-S^{\mu\nu}_{(n)}\{n^{\rho}F_{\rho\mu},\tilde{\Delta}_{\nu}\hat{f}_V\}_{\rm c}\Big),
\end{eqnarray}
and
\begin{eqnarray}
S^{\mu\nu}_{(n)}\tilde{\Delta}_{\mu}\tilde{\Delta}_{\nu}\hat{f}_V=\frac{S^{\mu\nu}_{(n)}}{2}
\Big(i[F_{\mu\nu},\hat{f}_V]_{\rm c}+\frac{1}{2}\{(D_{[\mu}F_{\beta\nu]}),\partial^{\beta}_{p}\hat{f}_V\}_{\rm c}+\frac{1}{4}[F_{\alpha[\mu}F_{\beta\nu]},\partial^{\alpha}_{p}\partial^{\beta}_{p}\hat{f}_V]_{\rm c}\Big).
\end{eqnarray}
On the other hand, for $\delta'(p^2)$ related terms, we acquire
\begin{eqnarray}\nonumber
\delta'(p^2)S^{\mu\nu}_{(n)}p^{\rho}\{F_{\rho\mu},\tilde{\Delta}_{\nu}\hat{f}_V\}&=&-\frac{\delta'(p^2)p_{\nu}}{2}\{\tilde{F}^{\mu\nu},\tilde{\Delta}_{\mu}\hat{f}_V\}_{\rm c}
+\frac{\delta'(p^2)p_{\mu}n_{\nu}}{2p\cdot n}\{\tilde{F}^{\mu\nu},p\cdot\tilde{\Delta}\hat{f}_V\}_{\rm c}
\\
&&
-\frac{\delta(p^2)}{2p\cdot n}\{\tilde{F}^{\nu\beta}n_{\beta},\tilde{\Delta}_{\nu}\hat{f}_V\}_{\rm c}
\end{eqnarray}
by using the Schouten identity and $p^2\delta'(p^2)=-\delta(p^2)$. Also, one finds
\begin{eqnarray}
\frac{\delta'(p^2)}{2}
p_{\nu}\tilde{\Delta}_{\mu}\{\tilde{F}^{\mu\nu},\hat{f}_{V}\}_{\rm c}=\frac{\delta'(p^2)}{2}p_{\nu}\Big(\{\tilde{F}^{\mu\nu},\tilde{\Delta}_{\mu}\hat{f}_{V}\}_{\rm c}
+\frac{1}{2}[[F_{\mu\rho},\tilde{F}^{\mu\nu}],\partial^{\rho}_{p}\hat{f}_V]_{\rm c}\Big)
\end{eqnarray}
by using
\begin{eqnarray}
[A,\{B,C\}]=\{B,[A,C]\}+\{[A,B],C\},\quad
\{A,\{B,C\}\}=\{B,\{A,C\}\}+[[A,B],C].
\end{eqnarray}
Finally, the last term is given by
\begin{eqnarray}\nonumber
&&\frac{1}{4}\big(\partial_{p}^{\rho}p_{\nu}\delta'(p^2)\big)\{F_{\rho\mu},\{\tilde{F}^{\mu\nu},\hat{f}_{V}\}\}_{\rm c}
\\\nonumber
&&=\frac{1}{4}\big(\delta'(p^2)\{F_{\nu\mu},\{\tilde{F}^{\mu\nu},\hat{f}_{V}\}\}_{\rm c}+2p_{\nu}p^{\rho}\delta''(p^2)\{F_{\rho\mu},\{\tilde{F}^{\mu\nu},\hat{f}_{V}\}\}_{\rm c}\big)
\\
&&=\frac{1}{2}\delta''(p^2)p_{\nu}p^{\rho}[[\tilde{F}^{\mu\nu},F_{\rho\mu}],\hat{f}_V]_{\rm c},
\end{eqnarray}
where we have used
\begin{eqnarray}
2p_{\nu}p^{\rho}\{F_{\rho\mu},\{\tilde{F}^{\mu\nu},\hat{f}_{V}\}\}_{\rm c}
=-\frac{p^2}{2}\{F_{\mu\nu}\{\tilde{F}^{\mu\nu},\hat{f}_V\}\}_{\rm c}+2p_{\nu}p^{\rho}[[\tilde{F}^{\mu\nu},F_{\rho\mu}],\hat{f}_V]_{\rm c}
\end{eqnarray}
obtained from the Schouten identity and $p^2\delta''(p^2)=-2\delta'(p^2)$. Combining all terms together, We eventually derive a free-streaming effective AKE for massless quarks coupled with background color fields
\begin{eqnarray}\nonumber
0&=&\delta(p^2)\bigg[p\cdot\tilde{\Delta}\hat{f}_A+\hbar(\partial_{\mu}S^{\mu\nu}_{(n)})\tilde{\Delta}_{\nu}\hat{f}_V+\frac{\hbar}{2p\cdot n}S^{\mu\nu}_{(n)}\{n^{\rho}F_{\mu\rho},\tilde{\Delta}_{\nu}\hat{f}_V\}_{\rm c}
+\frac{\hbar S^{\mu\nu}_{(n)}}{2}
\Big(i[F_{\mu\nu},\hat{f}_V]_{\rm c}
\\\nonumber
&&+\frac{1}{2}\{(D_{[\mu}F_{\beta\nu]}),\partial^{\beta}_{p}\hat{f}_V\}_{\rm c}+\frac{1}{4}[F_{\alpha[\mu}F_{\beta\nu]},\partial^{\alpha}_{p}\partial^{\beta}_{p}\hat{f}_V]_{\rm c}\Big)\bigg]
+\frac{\hbar\delta'(p^2)p_{\mu}n_{\nu}}{2p\cdot n}\{\tilde{F}^{\mu\nu},p\cdot\tilde{\Delta}\hat{f}_V\}_{\rm c}
\\
&&+\frac{\hbar\delta'(p^2)}{4}p_{\nu}
[[F_{\mu\rho},\tilde{F}^{\mu\nu}],\partial^{\rho}_{p}\hat{f}_V]_{\rm c}+\frac{\hbar\delta''(p^2)}{2}p_{\nu}p^{\rho}[[F_{\mu\rho},\tilde{F}^{\mu\nu}],\hat{f}_V]_{\rm c}.
\end{eqnarray}
\subsection{Spin diffusion and source terms}
Implementing the color decomposition $\hat{f}_{V/A}=\hat{f}_{V/A}^{\rm s}I+\hat{f}_{V/A}^at^a$ and taking $p\cdot\tilde{\Delta}\hat{f}_V=0$ from the off-shell SKE, the color-singlet and color-octet components of the AKE are given by
\begin{eqnarray}\nonumber\label{AKE_singlet_massless}
0&=&\delta(p^2)\bigg[p^{\mu}\mathcal{K}_{\rm s\mu}[\hat{f}_A]+\hbar(\partial_{\mu}S^{\mu\nu}_{(n)})\mathcal{K}_{\rm s\nu}[\hat{f}_V]
+\frac{\hbar S^{\mu\nu}_{(n)}}{2p\cdot n}\bar{C}_{2}n^{\rho}F^{a}_{\mu\rho}\big(2\partial_{\nu}f_{V}^{a}+d^{abc}F^{b}_{\lambda\nu}\partial_{p}^{\lambda}\hat{f}_V^c\big)
\\
&&+\hbar S^{\mu\nu}_{(n)}
\Big(\bar{C}_{2}\big((\partial_{\mu}F^{a}_{\beta\nu})-f^{abc}A^{b}_{\mu}F^{c}_{\beta\nu}\big)\partial_{p}^{\beta}\hat{f}_{V}^{a}\bigg],
\end{eqnarray}
and
\begin{eqnarray}\nonumber\label{AKE_octet_massless}
0&=&\delta(p^2)\bigg[p^{\mu}\mathcal{K}^a_{\rm o \mu}[\hat{f}_A]+\hbar(\partial_{\mu}S^{\mu\nu}_{(n)})\mathcal{K}^a_{\rm o\mu}[\hat{f}_V]
+\frac{\hbar}{2p\cdot n}S^{\mu\nu}_{(n)}n^{\rho}\big(2F^{a}_{\mu\rho}\mathcal{K}_{\rm s\nu}[\hat{f}_V]
+d^{abc}F^{b}_{\mu\rho}\mathcal{K}^c_{\rm o\nu}[\hat{f}_V]\big)
\\\nonumber
&&+\hbar S^{\mu\nu}_{(n)}\big((\partial_{\mu}F^a_{\beta\nu})-f^{abc}A^b_{\mu}F^c_{\beta\nu}\big)\partial^{\beta}_{p}\hat{f}^{\rm s}_V
\\\nonumber
&&-\frac{\hbar S^{\mu\nu}_{(n)}}{2}
\Big(f^{abc}F^b_{\mu\nu}
-d^{abc}\big((\partial_{\mu}F^b_{\beta\nu})-f^{bef}A_{\nu}^eF^f_{\beta\nu}\big)\partial^{\beta}_{p}+\frac{f^{abc}f^{bef}}{4}F^e_{\alpha\mu}F^f_{\beta\nu}\partial^{\alpha}_{p}\partial^{\beta}_{p}\Big)\hat{f}^c_V\bigg]
\\
&&-\frac{\hbar}{4}p_{\nu}f^{abc}f^{bef}F^{e}_{\mu\rho}\tilde{F}^{f\mu\nu}\Big(\delta'(p^2-m^2)\partial^{\rho}_{p}+2p^{\rho}\delta''(p^2-m^2)\Big)\hat{f}_V^c,
\end{eqnarray}
where
\begin{eqnarray}\nonumber
\mathcal{K}_{\rm s\mu}[O]&\equiv& \partial_{\mu} O^{s}+\bar{C}_{2}F^{a}_{\nu\mu}\partial_{p}^{\nu}O^{a},
\\
\mathcal{K}^a_{\rm o\mu}[O]&\equiv&\partial_{\mu} O^{a}-f^{bca}A^{b}_{\mu}O^{c}+F^a_{\nu\mu}\partial^{\nu}_{p}O^{\rm s}+\frac{d^{bca}}{2}F^b_{\nu\mu}\partial_{p}^{\nu}O^c.
\end{eqnarray}
Given $\mathcal{O}^a\sim\mathcal{O}(g)$, Eq.~(\ref{AKE_octet_massless}) leads to
\begin{align}
0\approx p^{\mu}\big(\partial_{\mu} \hat{f}_A^a-f^{bca}A^{b}_{\mu}\hat{f}^{c}_A+F^a_{\nu\mu}\partial^{\nu}_{p}\hat{f}_V^{\rm s}\big)+\hbar(\partial_{\mu}S^{\mu\nu}_{(n)})\partial_{\nu}\hat{f}^a_V+\hbar S^{\mu\nu}_{(n)}\Big(\frac{n^{\rho}F^{a}_{\mu\rho}}{p\cdot n}\partial_{\nu}+(\partial_{\mu}F^{a}_{\beta\nu})\partial_{p}^{\beta}\Big)\hat{f}^{s}_{V}
\end{align}
and thus
\begin{eqnarray}\nonumber
\hat{f}_A^a(p,X)&=&-\int^{ab,p}_{k,X'}\Big[p^{\mu}F^b_{\nu\mu}(X')\partial^{\nu}_{p}+\hbar S^{\mu\nu}_{(n)}\Big(\frac{n^{\rho}F^{b}_{\mu\rho}(X')}{p\cdot n}\partial_{X'\nu}+\big(\partial_{X'\mu}F^{a}_{\beta\nu}(X')\big)\partial_{p}^{\beta}\Big)\Big]\hat{f}^{s}_{V}(p,X')
\\
&&+\int^{ab,p}_{k,X'}\hbar(\partial_{X'\mu}S^{\mu\nu}_{(n)})\partial_{X'\nu}\int^{bc,p}_{k',X''}\Big[p^{\alpha}F^c_{\beta\alpha}(X'')\partial^{\beta}_{X''p}\hat{f}^{\rm s}_V(p,X'')\Big]
\end{eqnarray}
up to $\mathcal{O}(g)$, where we have employed Eq.~(\ref{fV_octet_sol}) to rewrite $\hat{f}^a_{V}$ in terms of $\hat{f}^{s}_{V}$. For simplicity, we may work with a constant frame vector.
Then Eq.~(\ref{AKE_singlet_massless}) becomes
\begin{eqnarray}
0&\approx&\delta(p^2)\bigg[p\cdot\partial \hat{f}^{\rm s}_A+\bar{C}_2p^{\mu}F^a_{\nu\mu}\partial^{\nu}_{p}\hat{f}^a_A
+\frac{\hbar S^{\mu\nu}_{(n)}}{p\cdot n}\bar{C}_{2}n^{\rho}F^{a}_{\mu\rho}\partial_{\nu}\hat{f}_{V}^{a}
+\hbar S^{\mu\nu}_{(n)}
\bar{C}_{2}(\partial_{\mu}F^{a}_{\rho\nu})\partial_{p}^{\rho}\hat{f}_V^a\bigg]
\end{eqnarray}
up to $\mathcal{O}(g^2)$. By replacing $\hat{f}^a_{V/A}$ with $\hat{f}^{\rm s}_{V/A}$ and taking the ensemble averages, we eventually derive
\begin{eqnarray}\label{CKE_signlet_simplify}
0=\delta(p^2)\Big(p\cdot\partial f^{\rm s}_{A}(p,X)-\partial_{p}^{\kappa}\mathscr{D}_{\kappa}[f^{\rm s}_A]
+\hbar\partial_{p}^{\sigma}\mathscr{B}_{\sigma}[f^{\rm s}_{V}]
+\hbar S^{\mu\nu}_{(n)}\mathscr{C}_{\mu\nu}[f^{\rm s}_{V}]\Big),
\end{eqnarray}
where
\begin{align}
\mathscr{B}_{\sigma}[O]=\bar{C}_2p^{\rho}S^{\mu\nu}_{(n)}\int^p_{k,X'}\bigg[\big(\partial_{X'\mu}\langle F^a_{\sigma\rho}(X)F^a_{\nu\lambda}(X')\rangle\big) \partial_{p}^{\lambda}
-\frac{n^{\lambda}}{p\cdot n}\langle F^a_{\sigma\rho}(X)F^a_{\nu\lambda}(X')\rangle\partial_{X'\mu}
\bigg]O(p,X')
\end{align}
and
\begin{eqnarray}
\mathscr{C}_{\mu\nu}[O]=\bar{C}_2p^{\rho}\int^p_{k,X'}\bigg[\frac{n^{\lambda}}{p\cdot n}\partial_{X'\mu}\langle F^a_{\nu\lambda}(X)F^a_{\sigma\rho}(X')\rangle
+\partial_{X\mu}\langle F^a_{\nu\lambda}(X)F^a_{\sigma\rho}(X')\rangle \partial^{\lambda}_{p}\bigg]\partial^{\sigma}_{p}O(p,X').
\end{eqnarray}
Here we have utilized
\begin{eqnarray}\nonumber
\bar{C}_2p^{\mu}\langle F^a_{\nu\mu}\partial^{\nu}_{p}\hat{f}^a_A\rangle&=&\hbar\bar{C}_2p^{\rho}\partial_{p}^{\sigma}\bigg\{S^{\mu\nu}_{(n)}\int^p_{k,X'}\bigg[\big(\partial_{X'\mu}\langle F^a_{\sigma\rho}(X)F^a_{\nu\lambda}(X')\rangle\big) \partial^{\lambda}_{p}
\\
&&-\frac{n^{\lambda}}{p\cdot n}\langle F^a_{\sigma\rho}(X)F^a_{\nu\lambda}(X')\rangle\partial_{X'\mu}\bigg]
f^{\rm s}_{V}(p,X')\bigg\},
\end{eqnarray}
\begin{eqnarray}\label{mag_term_AKE}
\frac{\hbar S^{\mu\nu}_{(n)}}{p\cdot n}\bar{C}_{2}n^{\rho}\langle F^{a}_{\mu\rho}\partial_{\nu}\hat{f}_{V}^{a}\rangle=
\frac{\hbar S^{\mu\nu}_{(n)}}{p\cdot n}\bar{C}_{2}n^{\lambda}\langle F^a_{\nu\lambda}(X)\partial_{X\mu}\int^p_{k,X'}F^a_{\sigma\rho}(X')\rangle\partial^{\sigma}_{p}f^{\rm s}_{V}(p,X'),
\end{eqnarray}
\begin{eqnarray}
\hbar S^{\mu\nu}_{(n)}
\bar{C}_{2}\langle (\partial_{\mu} F^{a}_{\rho\nu})\partial_{p}^{\rho}\hat{f}_V^a\rangle=\hbar \bar{C}_2S^{\mu\nu}_{(n)}\partial^{\lambda}_{p}\int^p_{k,X'}p^{\rho}\partial_{X\mu}\langle F^a_{\nu\lambda}(X)F^a_{\sigma\rho}(X')\rangle \partial_{p}^{\sigma}f^{\rm s}_{V}(p,X'),
\end{eqnarray}
and have assumed the relation
\begin{eqnarray}\label{dX_rel}
\partial_{X\mu}\int^p_{k,X'}G(X')=-i\int^p_{k,X'}k_{\mu}G(X')
=\int^p_{k,X'}\partial_{X'\mu}G(X')
\end{eqnarray}
given $e^{ik\cdot X'}G(X')\big|_{X'^{\mu}=\pm\infty}=0$ for an arbitrary function $G(X')$ is applicable for Eq.~(\ref{mag_term_AKE}).
Although the $\hbar$ correction, corresponding to the source term for dynamical spin polarization, in the AKE for massless quarks takes a slightly different form compared to that for massive fermions, they both stem from spacetime variations of color fields. On the other hand, the color-singlet axial-vector component of the Wigner function reads
\begin{eqnarray}\nonumber\label{eq:Asmu_massless_exp}
\langle \mathcal{A}^{\rm s\mu}\rangle&=&2\pi\bigg\{\delta(p^2)\bigg[p^{\mu}f^{\rm s}_{A}+\hbar S^{\mu\nu}_{(n)}\big(\partial_{\nu}f_{V}^{\rm s}+\bar{C}_{2}\langle F^{a}_{\rho\nu}\partial_{p}^{\rho}\hat{f}_V^a\rangle\big)
-\frac{\hbar\bar{C}_2}{2}\langle\tilde{F}^{a\mu\nu}\partial_{p\nu}\hat{f}^a_{V}\rangle
\bigg]
\\
&&+\frac{\hbar\bar{C}_2}{2}\partial_{p\nu}\big(\delta(p^2)\langle\tilde{F}^{a\mu\nu}\hat{f}^a_{V}\rangle\big)\bigg\},
\end{eqnarray}
which yields the color singlet of the on-shell axial charge current density in phase space,
\begin{eqnarray}\label{eq:Asmu_massless}
\mathcal{A}^{\rm s\mu}(\bm p,X)\equiv \int \frac{dp_0}{2\pi}\langle\mathcal{A}^{\rm s\mu}\rangle
=\frac{1}{2|\bm p|}\big(p^{\mu}f_A^{\rm s}+\hbar\bar{C}_2\tilde{\mathcal{A}}^{\mu}_{Q}\big)_{p_0=|\bm p|},
\end{eqnarray}
where $\tilde{\mathcal{A}}^{\mu}_Q=\tilde{\mathcal{A}}^{\mu}_{Q_1}+\tilde{\mathcal{A}}^{\mu}_{Q2}+\tilde{\mathcal{A}}^{\mu}_{Q3}$ with $\tilde{\mathcal{A}}^{\mu}_{Q1}=\mathcal{A}^{\mu}_{Q1}|_{m=0}$, $\tilde{\mathcal{A}}^{\mu}_{Q2}=\mathcal{A}^{\mu}_{Q2}|_{m=0}$, and
\begin{eqnarray}
\tilde{\mathcal{A}}^{\mu}_{Q3}(\bm p,X)
&=&-\bigg[S^{\mu\nu}_{(n)}\partial_{p}^{\rho}\int^p_{k,X'}p^{\beta}\langle F^{a}_{\rho\nu}(X)F_{\alpha\beta}(X')\rangle\partial^{\alpha}_{p}f_V^{\rm s}(p,X')\bigg]_{p_0=|\bm p|},
\end{eqnarray}
which also contain the source terms coming from $\hbar$ corrections for spin polarization.
\subsection{Spin polarization, axial charge currents, and axial Ward identity}
We may now also investigate the spin polarization, axial charge current, and axial Ward identity of massless fermions for $f^{\rm s}_{V}$ near equilibrium with negligible gradient corrections in hydrodynamics. In such a case, it is more convenient to choose $n^{\mu}=u^{\mu}$ and work in the fluid rest frame \footnote{Similar to finding the local-equilibrium distribution functions in CKT \cite{Hidaka:2017auj}, such a frame choice is made and the distribution function in an arbitrary frame can be derived from the modified frame transformation.}. We re-emphasize that the Wigner function is however independent of the choice of a frame. Moreover, we may consider the color-field correlators only depending on $\delta X_0$ with the temporal direction now defined in the fluid rest frame. Notably, since $S^{\mu\nu}_{(n)}\partial_{\delta X\nu}\Phi(\delta X_0)=0$ for $n^{\mu}=u^{\mu}\approx(1,\bm 0)$, it turns out that the source term in Eq.~(\ref{CKE_signlet_simplify}) vanishes in our particular setup. In such a case, $f_A^{s}$ should simply diffuse to zero in equilibrium given no vortical corrections. When $f^{\rm s}_{V}$ is in equilibrium and $f^{\rm s}_A=0$, Eq.~(\ref{eq:Asmu_massless_exp}) reduces to
\begin{eqnarray}\label{Asmu_massless}
\hat{\mathcal{A}}^{\rm s\mu}(\bm p,X)=\hbar\bar{C}_2\big(\tilde{\mathcal{A}}^{\mu}_{Q1}(\bm p,X)+\tilde{\mathcal{A}}^{\mu}_{Q2}(\bm p,X)+\tilde{\mathcal{A}}^{\mu}_{Q3}(\bm p,X)\big),
\end{eqnarray}
where $\tilde{\mathcal{A}}^{\mu}_{Q1}(\bm p,X)+\tilde{\mathcal{A}}^{\mu}_{Q2}(\bm p,X)$ is nothing but Eq.~(\ref{AQ_massive}) with $m=0$, and
\begin{eqnarray}
\tilde{\mathcal{A}}^{\mu}_{Q3}(\bm p,X)
=\frac{\pi^{1/2}\tau_c}{2p_0}S^{\mu\nu}_{(n)}\langle E^{a}_{\nu}E^{a}_{\beta}\rangle \hat{p}^{\beta}\big(1-p_{0}\partial_{p0}\big)\partial_{p0}f_{\rm eq}(p_0)\Big|_{p_0=|\bm p|}
\end{eqnarray}
by similarly assuming
$\Phi(\delta X_0)=e^{-\delta X_0^2/\tau_{c}^2}$ and working in the fluid-rest frame. When dropping the subleading correlation function of two color electric fields, the non-dynamical source term for spin polarization of massless quarks matches the one of massive quarks by simply taking $m=0$.
One can analogously evaluate the axial charge current for finite $\tau_c$, which reads
\begin{eqnarray}\nonumber
J^{\mu}_5(X)&=&-2\hbar \int\frac{d^4p}{(2\pi)^3}\frac{\delta(p^2)}{2}\int^{p}_{k,X'}
\langle B^{a\rho}(X)E^a_{\rho}(X')\rangle \beta u^{\mu}f_{\rm eq}(p\cdot u)(1-f_{\rm eq}(p\cdot u))
\\
&=&2\hbar\int\frac{d^4p}{(2\pi)^3}\frac{\delta(p^2)}{4p\cdot u}\sqrt{\pi}\tau_c
\langle E^a\cdot B^a\rangle u^{\mu}\partial_{p\cdot u}f_{\rm eq}(p\cdot u)
\end{eqnarray}
and agrees with the result for massive fermions by taking $m=0$ and hence $\partial\cdot J_5=0$. In the limit for $\tau_c\rightarrow \infty$, following the previous calculations for massive fermions and taking $\langle B^{a\rho}(X)E^a_{\rho}(X')\rangle=\langle B^{a}\cdot E^a\rangle$, it turns out that
\begin{eqnarray}\label{divJ5_anomaly}
\partial_{\mu} J^{\mu}_5(X)
=-\frac{\hbar}{4\pi^2}\langle B^{a}\cdot E^a\rangle,
\end{eqnarray}
which corresponds to the non-Abelian axial anomaly as expected.
\section{Summary and discussions}\label{sec:summary}
Here we briefly summarize the primary findings. We have derived the AKE containing a classical spin diffusion term and a source term from quantum corrections that could possibly trigger spin polarization for massive quarks (with mass greater than the gradient scales) as shown in Eq.~(\ref{AKE_signlet_simplify}). A similar equation for massless quarks is also derived in Eq.~(\ref{CKE_signlet_simplify}).
On the other hand, there also exist explicit quantum corrections in Wigner functions, which severe as the additional source terms to generate spin polarization shown in Eqs.~(\ref{eq:Asmu_massive}) and (\ref{eq:Asmu_massless}) for massive and massless quarks, respectively. Given postulated color-field correlators based on spacetime translational invariance and spatial homogeneity, the parity-odd correlators of color fields, depending on their time difference can dominantly generate nonzero spin polarization as shown in Eq.~(\ref{Amu_equil}) for massive fermions near thermal equilibrium. Such a contribution further results in a nonzero axial charge current or more precisely the axial charge density in the fluid rest frame. This constant axial charge current also gives the vanishing axial Ward identity with finite correlation time of the color-field correlators. However, in the constant-field limit with infinite correlation time, the nonzero axial Ward identity associated with the pseudo-scalar condensate at finite temperature is acquired as shown in Eq.~(\ref{divJ5_condensate}). For massless fermions, a particular frame choice leads to the vanishing dynamical source term in AKE, while there exists a similar non-dynamical source term for spin polarization in the Wigner function. Note that the choice of a frame is simply for the technical reason, which does not affect the physical observables in the end. The axial Ward identity reduces to the expected axial anomaly as manifested by Eq.~(\ref{divJ5_anomaly}) in the constant-field limit as anticipated. In order to make a smooth connection between the results of the massive and massless quarks, it is necessary to further work out the AKT of massive quarks in a proper frame for arbitrary mass such as $n^{\mu}=u^{\mu}$ for $f^s_V$ near equilibrium.
In phenomenology, as proposed in Ref.~\cite{Muller:2021hpe}, such spin polarization of quarks engendered by color-field correlators could potentially affect the spin alignment of vector mesons observed in heavy ion collisions \cite{ALICE:2019aid,Singha:2020qns} as an unsettled question in theory (see Refs~.\cite{Sheng:2019kmk,Sheng:2020ghv,Xia:2020tyd,Goncalves:2021ziy} for other theoretical explanations). In order to make direct comparisons with experimental data, it is necessary to incorporate the color-field correlators from real-time simulations with prescribed initial conditions. According to the approximations adopted in Ref.~\cite{Muller:2021hpe} and this work, the parity-odd correlator between a chromo-magnetic field and a chromo-electric field is essential to generate spin polarization for massive quarks. However, due to the event-by-event fluctuations of the sign for such a correlator, the anomalous spin polarization by turbulent color fields may contribute to spin alignment of vector mesons, coming from the product of spin polarization for a quark and for an anti-quark, without affecting the spin polarization of Lambda hyperons. Further studies on simulations or more practical estimation of color-field correlators and the generalization of present QKT with source terms to arbitrary quark mass will be needed.
\acknowledgments
The author would like to thank B. M\"uller for insightful comments on the preliminary version of this work and K. Hattori for fruitful discussions. This work was supported by Ministry of Science and Technology, Taiwan under Grant No. MOST 110-2112-M-001-070-MY3.
|
2024-02-18T23:40:24.656Z
|
2022-06-01T02:10:27.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14392","language":"en","timestamp":1654049427000,"url":"https:\/\/arxiv.org\/abs\/2112.14392","yymm":"2112"}
|
proofpile-arXiv_000-10058
|
{"provenance":"002.jsonl.gz:10059"}
| null | null |
\section{Introduction}
\IEEEPARstart{I}{mage} fusion is an important task in image processing, which aims to integrate salient features of source images into a single image by using a fusion method\cite{1}.
Fusion of infrared and visible images is a chalenging task in image fusion\cite{luo2016novel}\cite{li2020nestfuse}. Due to different imaging principles of infrared and visible images, they contain different information about the same scene. Therefore, solving the problem of fusing infrared and visible images is important in many applications\cite{zheng2006nearest}\cite{sun2019effective}\cite{wang2003initial}\cite{sun2011quantum}.
Many methods have been proposed in the area of image fusion. Generally speaking, the multiscale transform (MST) method is the most common method in the field of image fusion. The main steps of MST-based methods include decomposition, fusion and reconstruction. However, the MST-based methods will loose the effective information of the source image in the process of inverse transform, thus affecting the final fusion results\cite{3}.
In recent years, image fusion methods based on representation learning have been widely developed. In sparse representation (SR), Liu et al.\cite{4} obtained global and local saliency maps of the source images based on sparse coeffcients for fusion. Liu et al.\cite{5} proposed a general framework of image fusion by combining MST and SR to overcome the inherent defects of the MST- and SR- based fusion methods. In low-rank representation (LRR), Li et al.\cite{6} obtained a better performance in both global and local structure by combining LRR and dictionary learning simultaneously.
With the development of deep learning, many image fusion methods based on deep learning were proposed. In 2017, Liu et al.\cite{7} presented a method that can generate the activity level measurement and fusion rule jointly under a convolutional neural network (CNN) model. In 2018, Li et al.\cite{2} used the pretrained VGG-19\cite{9} to extract the features. Then, they made full use of middle layer features to generate the fused image. In 2019, Li et al.\cite{10} proposed a novel deep learning architecture of image fusion, called DenseFuse. The fusion framework consists of encoding network, a fusion layer and decoding network. But they did not consider multi-scale features in their network. To solve this problem, Song et al.\cite{11} proposed a MSDNet for medical image fusion. They added a multi-scale layer at the end of the encoder of Li's\cite{10} to get multi-scale features and achieved a good fusion effect.
Therefore, from the multi-scale perspective, in this paper we embed the idea of Res2Net\cite{12} into the encoder. Res2Net is a new multi-scale backbone architecture, which will be introduced in Section\ref{Res2Net}. The main contributions of the proposed fusion method are given below:
(1) Res2Net is used for the fast time in image fusion. We applied Res2Net into our encoder, which extracts muti-scale image features at a more granular level.
(2) An interesting aspect is that we found that by using only a single natural image we can learn an effective reconstruction model, and the performance of the reconstruction model is comparable with the reconstruction model trained by 80000 images. This has greatly reduced the training time of the proposed model.
(3) A fusion strategy was developed to use the attention model to produce a weight map for the salient features of the source images.
The structure of the rest of the paper is as follows. In Section\ref{relatedworks}, we briefly introduce Res2Net and DenseFuse, respectively. In Section\ref{Methodology}, we introduce the the proposed method in detail. In Section\ref{experiment}, experiment results will be shown, and Section\ref{conclusion} is the conclusion of our paper.
\section{Related Works}
\label{relatedworks}
\subsection{Res2Net}
\label{Res2Net}
Multi-scale features play an important role in many image processing tasks. In PAMI 2020, Gao et al\cite{12} presented a novel building block for CNN, called Res2Net. The Res2Net module is shown in Fig.~\ref{Res2Net module}
\begin{figure}[!ht]
\centering
\includegraphics[width=0.8\linewidth]{Res2Net_module}
\caption{The graph of Res2Net module.}
\label{Res2Net module}
\end{figure}
In Fig.~\ref{Res2Net module}, after the $1\times1$ convolution, the feature map subsets are obtained by splitting the feature maps evenly, denoted by $x_i$, where $i\in\{1,2, \ldots,n\}$ and $n$ represents the number of subsets. Except for $x_1$, the rest of subsets have a corresponding convolution denoted as $F_i()$. The output of $F_i()$ is marked with $y_i$. Thus, $y_i$ can be calculated by Eq.\ref{equ:$y_i$}
\begin{eqnarray}\label{equ:$y_i$}
y_i=\left\{\begin{array}{ll}
x_i & \textrm{ $i=1$ } \\
F_i(x_i) & \textrm{ $i=2$ } \\
F_i(x_i+y_{i-1}) & \textrm{$2 <i\leq n$}
\end{array}\right.
\end{eqnarray}
As shown in Eq.\ref{equ:$y_i$}, when $x_i$ goes through $3\times3$ convolutional layer, the output result ($y_i$) can have a larger receptive field than $x_i$. Finally, concatenating all splits and passing them through a $1\times1$ convolution. Therefore, the idea of Res2Net is constructing hierarchical residual-like connections within a residual block. Res2Net extracts multi-scale features at the granularity level and increases the receptive field range for each network layer.
\subsection{DenseFuse}
\label{DenseFuse}
The DenseFuse architecture consists of an encoding network, fusion layer and decoding network. The training network consists of encoder and decoder, and its purpose is to reconstruct the input image. Therefore, training images are not necessarily of the same type as the test images. Hence, the training dataset is easy to get. After the training process, the trained encoder and decoder, respectively, have the ability to extract features and reconstruct the image. Both of them are used in the fusion network. The architecture of DenseFuse is described in Fig.\ref{densefuse}.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.8\linewidth]{densefuse}
\caption{The DenseFuse framework.}
\label{densefuse}
\end{figure}
In Fig.\ref{densefuse}, after encoding, two groups of features are fused in the fusion layer. Finally, the fused image is reconstructed by the decoder.
\section{Methodology}
\label{Methodology}
In this section, we will introduce in detail the proposed fusion method for grayscale images. How to process color images will be introduced in Section\ref{RGBresults}. The three subsections are: the proposed method in \ref{proposedmethod}; training network (loss function and training strategy) in \ref{Training}; fusion layer(strategy) in \ref{fusionstrategy}.
\begin{figure*}[!ht]
\centering
\includegraphics[width=0.75\linewidth]{framework}
\caption{The architecture of the proposed method for gray scale images.}
\label{framework}
\end{figure*}
\begin{figure*}[!ht]
\centering
\includegraphics[width=0.75\linewidth]{trainingnetwork}
\caption{The reconstruction model.}
\label{trainingnetwork}
\end{figure*}
\subsection{The Proposed Method}
\label{proposedmethod}
Inspired by the success of DenseFuse and Res2Net, we use the Res2Net module for the encoder because of its strong ability to extract multi-scale features.
The input images are infrared and visible images, denoted as
$I_1$ and $I_2$. We assume the input images are already registered. Our proposed architecture includes the encoder, the fusion layer and the decoder. The architecture of the proposed method is shown in Fig.~\ref{framework}.
As shown in Fig.~\ref{framework}, in the fusion model, the encoder consists of two $3\times3$ filters and a block of Res2Net. For the Res2Net block, after a $1\times1$ convolution, all features are split evenly. The operation and the specific formula for the Res2Net block are described in Section\ref{Res2Net}. At the end of the Res2Net block, all splits are concatenated and pass them through a $1\times1$ convolution. The decoder is constructed by four $3\times3$ filters. The fusion strategy in the fusion layer will be introduced later.
\subsection{Training}
\label{Training}
Regardless of color or gray scale images being used, the training network is the same. The training network has just two parts: the encoder and the decoder.
The function of the training network is to reconstruct the input image. Therefore, the training network can also be called a reconstruction model. After the training process, we add the fusion layer between the encoder and the decoder. The architecture of the training network is shown in Fig.\ref{trainingnetwork}. The parameters of the training network are outlined in Table\ref{tab:outline}.
\begin{table}[!ht]
\centering
\caption{\label{tab:outline}The outline of the training network. \textbf{Conv3} and \textbf{Conv1} denote the convolutional layer with $3\times3$ filter and $1\times1$ filter, respectively. \textbf{[Conv3]$\times2$} denotes two identical layers.}
\resizebox{3.5in}{!}{
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
& Layer & Size & Stride & \makecell[cc]{Input\\Channel} & \makecell[cc]{Output\\Channel} & Activation \\
\hline
\multirow{3}*{Encoder} &
Conv3 & 3 & 1 & 1 & 32 & ReLU \\
\cline{2-7}
~ & Conv3 & 3 & 1 & 32 & 64 & ReLU\\
\cline{2-7}
~ & Res2Net Block & -- & -- & -- & -- & --\\
\cline{2-7}
\hline
\multirow{4}*{Decoder} &
Conv3 & 3 & 1 & 64 & 64 & ReLU \\
\cline{2-7}
~ & Conv3 & 3 & 1 & 64 & 32 & ReLU \\
\cline{2-7}
~ & Conv3 & 3 & 1 & 32 & 16 & ReLU \\
\cline{2-7}
~ & Conv3 & 3 & 1 & 16 & 1 & --\\
\cline{2-7}
\hline
\multirow{6}*{\makecell[cc]{Res2Net Block}} &
Conv1 & 1 & 1 & 64 & 64 & ReLU \\
\cline{2-7}
~ & -- & -- & -- & -- & 16 & -- \\
\cline{2-7}
~ & [Conv3]$\times2$ & 3 & 1 & 16 & 16 & ReLU\\
\cline{2-7}
~ & [Conv3]$\times2$ & 3 & 1 & 16 & 16 &ReLU\\
\cline{2-7}
~ & [Conv3]$\times2$ & 3 & 1 & 16 & 16 &ReLU\\
\cline{2-7}
~ & Conv1 & 1 & 1 & 64 & 64 & ReLU \\
\hline
\end{tabular}}
\end{table}
\textbf{Reconstruction Loss}. In order to reconstruct the input image, we use the loss function\cite{10} as shown in Eq.\ref{equ:lossfunction}.
\begin{eqnarray}\label{equ:lossfunction}
L = L_{ssim} + L_{pixel}
\end{eqnarray}
The $L_{pixel}$ and the $L_{ssim}$ are respectively calculated as:
\begin{eqnarray}\label{L_pixel}
L_{pixel} = \frac{1}{BCHW}||O-I||^2_2
\end{eqnarray}
\begin{eqnarray}\label{L_ssim}
L_{ssim} = 1 - SSIM(O,I)
\end{eqnarray}
where $O$ and $I$ are output and input images respectively, $B$ represents the batch size, $C$ is the number of channels of $O$, $H$ and $W$ are height and width of $O$. In addition, SSIM($\cdot$) denotes the structural similarity\cite{14}.
\textbf{Training Strategy}. In an ICCV 2019 paper\cite{15}, the authors proved that the internal statistics of patches within a natural image own enough information for learning a powerful generative model. Authors use a single natural image at multiple scales to train a generative model, rather than many samples from a database.
Inspired by \cite{15}, at first, we tried a single natural image at different scales to train a reconstruction model. However, experiments showed that the model has the ability to reconstruct the input image, but the details of the output image are slightly blurry.
Therefore, we just tried using a whole natural image to train the reconstruction model 2000 times. Finally, the results show that the reconstruction ability of the model is comparable with DenseFuse\cite{10} and MSDNet\cite{11}, which are trained by 80000 images. Moreover, the training time is greatly reduced.
The assessment and the analysis of the training strategy will be dicussed in detail in the \ref{Reconstruction} subsection.
Hence, we just randomly choose a natural image (grayscale) from MS-COCO\cite{16} as being our training sample to train the reconstruction model.
\subsection{The Fusion Layer}
\label{fusionstrategy}
The fusion strategy plays an important role in image fusion, and this section will detail our fusion strategy.
Input images have their own distinctive features, therefore, two groups of features will be obtained after encoding. Then, we need two weight maps to fuse them, but the average strategy is too rough. Therefore, as shown in Fig.\ref{framework}, we propose to apply a spatial attention model to the fusion layer.
\textbf{1)Spatial Attention Based on $1_1$-norm. } Inspired by\cite{10}, we use an $l_1$-norm to process the features extracted by the encoder. Then, we obtain two weight maps, $w_1$ and $w_2$, which are computed by Eq.\ref{equ:weightmap1}
\begin{eqnarray}\label{equ:weightmap1}
w_i(x,y) = \frac{\sum_{j=1}^m||\phi_i^j(x,y)||_1}{\sum_{i=1}^k\sum_{j=1}^m||\phi_i^j(x,y)||_1}
\end{eqnarray}
where $\phi_i^{1:m}$ are the feature maps extracted by the encoder, $m$ and $k$, respectively, denote the number of feature maps and input images. In this paper, $k=2$.
Finally, the enhanced(fused) features denoted as $f^{1:m}$ are calculated by Eq.\ref{equ:fusedmaps1}.
\begin{eqnarray}\label{equ:fusedmaps1}
f^{1:m}=\sum_{i=1}^kw_i\times\phi_i^{1:m}
\end{eqnarray}
\textbf{2)Spatial Attention Based on Mean Operation.}
Firstly, we use mean operation to process the features
extracted by the encoder $\phi_i^{1:m}$. Then, we apply the soft-max operation to get the weight maps $w_1$ and $w_2$, as shown in Eq.\ref{equ:weightmap2}.
\begin{eqnarray}\label{equ:weightmap2}
w_i(x,y) = \frac{M(\phi_i^{1:m}(x,y))}{\sum_{i=1}^kM(\phi_i^{1:m}(x,y))}
\end{eqnarray}
where $M(\cdot)$ is the mean operation to process the pixel at position $(x,y)$ of each feature map. Then, the fused features $f^{1:m}$ are calculated by Eq.\ref{equ:fusedmaps2}.
\begin{eqnarray}\label{equ:fusedmaps2}
f^{1:m}=\sum_{i=1}^kw_i\times\phi_i^{1:m}
\end{eqnarray}
After the fusion layer, the final fused features $f^{1:m}$ are fed into the decoder and the fused image is reconstructed accordingly.
\section{Experimental Results}
\label{experiment}
This section firstly introduces the experimental settings and environment. Then, a detailed discussion on the performance of our training network that can be obtained using a single training image is done. The results of the experiments will also be presented, and, finally, we will explain how to process RGB images.
\subsection{Experimental Settings and Environment}
\label{settings}
In our experiment, there are 20 pairs of test images (infrared and visible images)\cite{10}. Several samples of test images are shown in Fig.\ref{testimages}
\begin{figure}[!ht]
\centering
\includegraphics[width=\linewidth]{testimages}
\caption{Four pairs of source images. The first row is infrared images and the second row is visible images.}
\label{testimages}
\end{figure}
Comparison methods include the discrete cosine harmonic wavelet transform method(DCHWT)\cite{19}, the joint sparse representation-based method(JSR)\cite{20}, the fusion method based on saliency detection in sparse domain(JSRSD)\cite{4}, DenseFuse\cite{10}, FusionGAN\cite{22}, and MSDNet\cite{11}.
In order to evaluate the fusion results in an objective assessment, we choose eight indices as follows: entropy(EN)\cite{23} that indicates how much information the fusion result contains; mutual information(MI)\cite{24} that measures the amount of information of the source images that the fused image contains; $Q_{abf}$\cite{25} reflects the quality of visual information; the sum of the correlations of differences(SCD)\cite{28} that indicates the amount of transferred information from each of the input images into the fused image; a new no-reference quality assessment for image fusion(MS\_SSIM)\cite{29}; $FMI_{dct}$ and $FMI_w$\cite{26}, which calculate the feature mutual information, such as discrete cosine and wavelet features; $SSIM_a$ which is calculated by Eq.\ref{ssim}.
\begin{eqnarray}\label{ssim}
&SSIM_a(f) = (SSIM(f,I_1)+SSIM(f,I_2))\times 0.5
\end{eqnarray}
where $f$ means the fused image, SSIM($\cdot$) denotes the structural similarity operation\cite{14}. The values of $SSIM_a$ measure the structural information of the source images.
In the training phase, the number of iterations is 2000 and the batch size is 1. In addition, the number of training images (grayscale) is 1 and the size of training image is $256\times256$.
Our method was implemented with NVIDIA GTX 1050Ti GPU, and Pytorch was utilized as the backend for the network framework.
\subsection{Detailed Discussion of the Training Strategy}
\label{Reconstruction}
Natural images have strong internal data repetiton\cite{shocher2018zero}. The analysis of the internal predictive-power was shown to be strong for almost any natural image\cite{glasner2009super}.
Motivated by these observations, we combined the predictive power of internal information and the generalization capabilities of deep-Learning to train our reconstruction model by a single natural image. Therefore, the training strategy consists in that we utilize a natural image to train the reconstruction model 2000 times.
Firstly, we will show the loss graph of our training strategy in Fig.\ref{loss}. As we can see, after 200 iterations, the training model begins to become stable. Then, we display the comparison results to prove the effectiveness of the trained model from the perspective of reconstruction and fusion.
\begin{figure}[!ht]
\centering
\includegraphics[width=\linewidth]{loss}
\caption{The graphs of loss. (a)$L_{pixel}$; (b)$L_{ssim}$; (c)Total loss}.
\label{loss}
\end{figure}
\begin{table*}[!ht]
\centering
\caption{\label{tab:fusionl1mean}The average values of quality metrics for 20 fused images. \textbf{\emph{$l_1$-norm}} and \textbf{\emph{mean}} means fusion strategy.}
\resizebox{\textwidth}{!}{
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
\multicolumn{2}{|c|}{Methods} & $SSIM_a$ & EN & MI & $Q_{abf}$ & $FMI_{dct}$ & $FMI_w$ & SCD & MS\_SSIM \\
\hline
\multicolumn{2}{|c|}{DenseFuse} & 0.72829 & 6.66296 & 13.32592 & 0.43454 & {\color{red}{\textbf{0.41456}}} & {\color{green}{\textbf{0.42525}}} & {\color{blue}{\textbf{1.83379}}} & {\color{red}{\textbf{0.92860}}}\\
\hline
\multirow{2}*{MSDNet}&
$l_1$-norm & {\color{blue}{\textbf{0.76365}}} & 6.40040 & 12.80079
& 0.44416 & 0.37547 & 0.41741 & 1.61825 & 0.83311 \\
\cline{2-10}
~& Mean & {\color{red}{\textbf{0.77453}}} & 6.28483 & 12.56966 & 0.39214 & 0.39046 & 0.41116 & 1.67953 & 0.87041 \\
\cline{2-10}
\hline
\multirow{2}*{Res2NetFuse\_80000}&
$l_1$-norm & 0.70881 & {\color{red}{\textbf{6.83607}}} & {\color{red}{\textbf{13.67214}}} & {\color{blue}{\textbf{0.47354}}} & 0.37620 & {\color{red}{\textbf{0.43160}}} & 1.67887 & 0.84166\\
\cline{2-10}
~& Mean & 0.72993 & {\color{green}{\textbf{6.77470}}} & {\color{green}{\textbf{13.54940}}} & 0.45175 & {\color{blue}{\textbf{0.40685}}} & 0.42417 & {\color{green}{\textbf{1.82024}}} & {\color{green}{\textbf{0.91695}}}\\
\hline
\multirow{2}*{Res2NetFuse\_1}&
$l_1$-norm & 0.72520 & {\color{blue}{\textbf{6.81785}}} & {\color{blue}{\textbf{13.63569}}} & {\color{red}{\textbf{0.48364}}} & 0.37552 & {\color{blue}{\textbf{0.42924}}} & 1.69888 & 0.85384 \\
\cline{2-10}
~& Mean & {\color{green}{\textbf{0.74206}}} & 6.76675 & 13.53350 & {\color{green}{\textbf{0.46368}}} & {\color{green}{\textbf{0.40262}}} & 0.42334 & {\color{red}{\textbf{1.83464}}} & {\color{blue}{\textbf{0.92129}}} \\
\hline
\end{tabular}}
\end{table*}
\begin{table}[!ht]
\centering
\caption{\label{tab:reconstruction}The average values of quality metrics for 20 reconstructed images. \textbf{\emph{DenseFuseRecons}} denotes the reconstruction model of DenseFuse. \textbf{\emph{MSDNetRecons}} denotes the reconstruction model of MSDNet. \textbf{\emph{Res2NetFuseRecons\_1}} denotes the reconstruction model of Res2NetFuse trained by an image. \textbf{\emph{Res2NetFuseRecons\_80000}} denotes the reconstruction model of Res2NetFuse trained by 80000 images.}
\resizebox{3.5in}{!}{
\begin{tabular}{|c|c|c|c|}
\hline
Methods & SSIM & PSNR & MSE \\
\hline
DenseFuseRecons & 0.99560 & 43.62612 & 0.00016 \\
\hline
MSDNetRecons & {\color{red}{\textbf{0.99933}}} & {\color{red}{\textbf{52.10881}}} & {\color{red}{\textbf{0.00001}}} \\
\hline
Res2NetFuseRecons\_1 & 0.99560 & 40.36431 & 0.00042 \\
\hline
Res2NetFuseRecons\_80000 & 0.99888 & 46.50366 & 0.00005 \\
\hline
\end{tabular}}
\end{table}
\subsubsection{Reconstruction Comparisons} We compare our reconstruction model with existing reconstruction models which are trained by 80000 images. The reconstruction models for comparison are from DenseFuse, MSDNet and our proposed method which is trained by 80000 images (batch size and epoch are all 4). Then, we randomly choose 20 natural images from MS-COCO in order to do reconstruction experiments. We utilize the structural similarity (SSIM), the peak signal to noise ratio (PSNR) and the mean squared error(MSE), calculated by using Eq.\ref{MSE}, in order to evaluate the reconstruction performance.
\begin{eqnarray}\label{MSE}
MSE = \frac{1}{W\times H}\sum_{x=1}^W\sum_{y=1}^H(I(x,y)-R(x,y))^2
\end{eqnarray}
where $W$ and $H$ are the width and the height of the image, $I$ is the input image and $R$ is the reconstructed image.
The reconstruction comparison results are shown in Table\ref{tab:reconstruction}.
As shown in Table\ref{tab:reconstruction}, although Res2NetFuse does not achieve the best metrics values, it still obtains comparable reconstruction ability.
Comparing with Res2NetFuseRecons\_80000, the reconstruction model trained by one image achieves acceptable reconstruction performance. This means the internal information of a single image is enough to train our lightweight network. Moreover, with this training strategy, our lightweight network needs much less training time.
\subsubsection{Fusion Comparisons} In this part, for fusion comparison purposes, we also compare Res2NetFuse\_1, trained by an image, with DenseFuse, MSDNet and Res2NetFuse\_80000 trained by 80000 images. Among the compared methods, MSDNet is also from a multi-scale perspective. Therefore, when comparing MSDNet and Res2NetFuse, we ensured that the fusion strategy is the same. The fusion comparison results are shown in Table\ref{tab:fusionl1mean}.
In Table\ref{tab:fusionl1mean}, the \textbf{\emph{red}} values mean the best values, the \textbf{\emph{blue}} values mean the second-best values, the \textbf{\emph{green}} values mean the third-best values. We can see that Res2NetFuse\_1 has comparable fusion ability with the other methods. In addition, although the MSDNet's reconstruction ability is better than that of Res2NetFuse, the fusion results of Res2NetFuse are better than that of MSDNet under the same fusion strategy. The reason is that the activity level maps of MSDNet are for different scales and, on the other hand, that of Res2NetFuse are for all scales. Therefore, the weight maps generated by the avtivity level maps of Res2NetFuse will be more stable.
Comparing with the Res2NetFuse\_80000, we find that the metrics values are similar, which means the fusion performace of Res2NetFuse\_1 is acceptable. Therefore, even the Res2Net-based fusion model is trained by a single natural image, it can obtain a good fusion performance.
The above comparisons on the performance of reconstruction and fusion proves that our training strategy is effective. Now we will summarize several reasons why the training strategy is effective:
1) The proposed method is a lightweight network, in contrast to the deep networks in other computer vision tasks. In theory, the training process could be done with less training data. And for some deep-learning-based image fusion methods, the deep network is mainly used for reconstruction, which requires less training data than a classification task.
2) In fact, natural images have strong internal data repetition and the internal statistics often provides strong predictive-power \cite{shocher2018zero}. Therefore, combining the strong generalization abilities of deep-learning, the lightweight network can be trained by a single natural image.
3) Multi-scale feature representations of Res2Net are of great importance to the proposed method, which can make it easier for the decoder to reconstruct the image.
Therefore, through a comprehensive consideration, we choose Res2NetFuse\_1 as the final fusion model. The Res2NetFuse mentioned later will be trained by a single natural image.
\subsection{Subjective \& Objective Evaluation}
\label{Assessment}
Our method is tested for 20 images, and three groups of experimental results are shown in Fig.\ref{results1}$-$Fig.\ref{results3}.
\begin{figure*}[!ht]
\centering
\includegraphics[width=0.9\linewidth]{results1}
\caption{The first group of experimental results. (a)Infrared image; (b)Visible image; (c)DCHWT; (d)JSR; (e)JSRSD; (f)DenseFuse; (g)FusionGAN; (h)MSDNet with $l_1$-norm fusion strategy; (i)MSDNet with mean fusion strategy; (j)Res2NetFuse with $l_1$ fusion strategy; (k)Res2NetFuse with mean fusion strategy.}
\label{results1}
\end{figure*}
\begin{figure*}[!ht]
\centering
\includegraphics[width=0.9\linewidth]{results2}
\caption{The second group of experimental results. (a)Infrared image; (b)Visible image; (c)DCHWT; (d)JSR; (e)JSRSD; (f)DenseFuse; (g)FusionGAN; (h)MSDNet with $l_1$-norm fusion strategy; (i)MSDNet with mean fusion strategy; (j)Res2NetFuse with $l_1$ fusion strategy; (k)Res2NetFuse with mean fusion strategy.}
\label{results2}
\end{figure*}
\begin{figure*}[!ht]
\centering
\includegraphics[width=0.9\linewidth]{results3}
\caption{The third group of experimental results. (a)Infrared image; (b)Visible image; (c)DCHWT; (d)JSR; (e)JSRSD; (f)DenseFuse; (g)FusionGAN; (h)MSDNet with $l_1$-norm fusion strategy; (i)MSDNet with mean fusion strategy; (j)Res2NetFuse with $l_1$ fusion strategy; (k)Res2NetFuse with mean fusion strategy.}
\label{results3}
\end{figure*}
Through perceptual comparison, the fused results of DCHWT have some noise and salient features are not clear. In addition, the salient features of fused results obtained by JSR and JSRSD are so sharp. Furthermore, as shown in the three figures above, we can find that the results of FusionGAN are not stable, such as shown in Fig.\ref{results2}(g).
On the other hand, the results of Res2NetFuse, MSDNet and DenseFuse are more consistent with human visual standards. But obviously, the results of Res2NetFuse are more stable and have more salient features.
In order to verify objectively the effectiveness of the proposed method, we utilize some indices to evaluate the obtained fused results. The objective evaluation is shown in Table.\ref{tab:ob}
\begin{table*}[!ht]
\centering
\caption{\label{tab:ob}The average values of 20 fused images.}
\resizebox{\textwidth}{!}{
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
\multicolumn{2}{|c|}{Methods} & $SSIM_a$ & EN & MI & $Q_{abf}$ & $FMI_{dct}$ & $FMI_w$ & SCD & MS\_SSIM \\
\hline
\multicolumn{2}{|c|}{DCHWT} & 0.73078 & 6.53455 & 13.06910 & {\color{green}{\textbf{0.45890}}} & 0.38061 & 0.39700 & 1.61007 & 0.84278 \\
\hline
\multicolumn{2}{|c|}{JSR} & 0.60912 & 6.38043 & 12.76086 & 0.36267 & 0.16738 & 0.21284 & {\color{green}{\textbf{1.75518}}} & 0.84735 \\
\hline
\multicolumn{2}{|c|}{JSRSD} & 0.54471 & {\color{green}{\textbf{6.66771}}} & {\color{green}{\textbf{13.33541}}} & 0.32914 & 0.14560 & 0.18697 & 1.59142 & 0.76548 \\
\hline
\multicolumn{2}{|c|}{DenseFuse} & 0.72829 & 6.66296 & 13.32592 & 0.43454 & {\color{red}{\textbf{0.41456}}} & {\color{blue}{\textbf{0.42525}}} & {\color{blue}{\textbf{1.83379}}} & {\color{red}{\textbf{0.92860}}}\\
\hline
\multicolumn{2}{|c|}{fusionGAN} & 0.65207 & 6.36496 & 12.72993 & 0.21853 & 0.36097 & 0.36797 & 1.45818 & 0.73233 \\
\hline
\multirow{2}*{MSDNet}&
$l_1$-norm & {\color{blue}{\textbf{0.76365}}} & 6.40040 & 12.80079
& 0.44416 & 0.37547 & 0.41741 & 1.61825 & 0.83311 \\
\cline{2-10}
~& Mean & {\color{red}{\textbf{0.77453}}} & 6.28483 & 12.56966 & 0.39214 & {\color{green}{\textbf{0.39046}}} & 0.41116 & 1.67953 & {\color{green}{\textbf{0.87041}}} \\
\cline{2-10}
\hline
\multirow{2}*{Res2NetFuse}&
$l_1$-norm & 0.72520 & {\color{red}{\textbf{6.81785}}} & {\color{red}{\textbf{13.63569}}} & {\color{red}{\textbf{0.48364}}} & 0.37552 & {\color{red}{\textbf{0.42924}}} & 1.69888 & 0.85384 \\
\cline{2-10}
~& Mean & {\color{green}{\textbf{0.74206}}} & {\color{blue}{\textbf{6.76675}}} & {\color{blue}{\textbf{13.53350}}} & {\color{blue}{\textbf{0.46368}}} & {\color{blue}{\textbf{0.40262}}} & {\color{green}{\textbf{0.42334}}} &{\color{red}{\textbf{1.83464}}} & {\color{blue}{\textbf{0.92129}}} \\
\hline
\end{tabular}}
\end{table*}
In Table.\ref{tab:ob}, the best values are in \textbf{\emph{red}}, the second-best values are marked in \textbf{\emph{blue}} and the third-best values are in \textbf{\emph{green}}. As we can see, Res2NetFuse has some advantages with respect these results of fusion evaluation.
The best values of EN, MI, $Q_{abf}$, $FMI_w$ and SCD indicate that our method can preserve more visual and salient information of the source images and contain less noise. And second-best values MS\_SSIM and $FMI_{dct}$ and third-best value $SSIM_a$ mean that our method preserve more structural information of the source images. In addition, the comparison between Res2NetFuse and MSDNet also indicates that Res2NetFuse could extract more powerful deep features, which are beneficial for image fusion tasks. Therefore, our method is an effective fusion architecture for infrared and visible images' fusion.
\begin{figure}[!ht]
\centering
\includegraphics[width=\linewidth]{rgb}
\caption{The fusion framework for color images.}
\label{rgbframework}
\end{figure}
\begin{figure*}[!ht]
\centering
\includegraphics[width=0.85\linewidth]{rgbresults}
\caption{The fused results for color images. (a)Infrared image; (b)Visible image; (c)Fused results}
\label{rgbresults}
\end{figure*}
\subsection{Additional Experiments on RGB(Visible) and Infrared Images}
\label{RGBresults}
Inspired by \cite{27}, for color images, we convert the visible image to YUV space and the infrared image to gray sale image. Y channel is the luminance component, in addition, U and V are the chrominance components. We apply our method to fuse the Y channel and the gray scale image. Then, we convert the fused image combined with the U and V channels of visible image to the RGB space. Finally, the fused color image will be obtained. The fusion framework for color images is shown in Fig.\ref{rgbframework}, and the fused color results are shown in Fig.\ref{rgbresults}.
In Fig.\ref{rgbresults}, we can see that the features of fused results are enhanced, which are consistent with human vision perception. Therefore, the fusion framework for color images is beneficial for RGB and infrared images' fusion tasks.
\section{Conclusion}
\label{conclusion}
In this paper, we presented a novel architecture for fusion of infrared and visible images based on a multi-scale backbone Res2Net. We found that a single natural image can train the network very well for image fusion. Insights on the success of the proposed method were also presented and analyzed. We proposed to apply an attention model to the fusion strategy, which can generate a more effective weight map to fuse the salient features of source images. The experimental results show that our method outperforms the state-of-the-art methods. In future work, we will apply the proposed training strategy to other tasks of image processing.
\bibliographystyle{IEEEtran}
|
2024-02-18T23:40:24.663Z
|
2022-02-01T02:15:38.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14540","language":"en","timestamp":1643681738000,"url":"https:\/\/arxiv.org\/abs\/2112.14540","yymm":"2112"}
|
proofpile-arXiv_000-10059
|
{"provenance":"002.jsonl.gz:10060"}
| null | null |
\section{Introduction}
Many optimization problems that arise in application fields such as statistics, computer science and data science can be cast into constrained optimization problems that minimize smooth functions over compact sets:
\begin{equation}\label{P00}
\begin{array}{rl}
\min\limits_{x\in{\mathbb{X}}} & f(x) \ \ \ \ {\rm s.t.} \ \ x \in C,
\end{array}
\end{equation}
where ${\mathbb{X}}$ is a finite-dimensional Euclidean space, $f:{\mathbb{X}}\rightarrow{\rm I\!R}$ is continuously differentiable and $C$ is a nonempty compact set in ${\mathbb{X}}$.
When projections onto $C$ can be efficiently computed, the classical gradient projection method \cite{Bertsekas99, Goldstein64, LevitinPolyak66} and its variants are usually the prominent choices of algorithms for solving \eqref{P00}, due to their ease of implementation and scalability.
Projections onto $C$, however, are not necessarily easy to compute; see for example \cite{FrGM17,HaJN15} for some concrete instances that arise in applications.
In this case, scalable first-order methods that do not involve projections may be employed. When the $C$ in \eqref{P00} is \emph{convex}, one popular class of such algorithms is the Frank-Wolfe (FW) method (also called the conditional gradient method) \cite{FrankWolfe56} and its variants. Unlike the
gradient projection methods which require efficient projections onto $C$, FW method, in each iteration, makes use of a linear approximation of $f$, and moves towards a minimizer of this linear function over $C$ along a straight line to generate a next iterate in $C$. In particular,
FW method uses a {\em linear-optimization oracle} in each iteration; these kind of oracles can be
much less computationally expensive than projecting onto $C$ in many applications \cite{FrGM17,GarberHazan16,Jaggi13}. Due to their low iteration costs and ease of implementations, FW method
and its variants have found applications in machine learning and have received much renewed interest in recent years \cite{FrGM17,GarberHazan16,Jaggi13,LaZh16,Lacoste15,Pedregosa20}. For example, when $f$ is also convex and satisfies certain curvature conditions, \cite{Jaggi13} established
the $O(1/k)$ complexity of FW-type methods for \eqref{P00} and presented their powerful applications for
solving sparse optimization models. In \cite{GarberHazan16,Lacoste15,Pedregosa20}, linear convergence results of some FW-type methods
were established under suitable conditions such as strong convexity of
$f$ or $C$ in \eqref{P00}.
More recently, refinements on the FW method were presented by incorporating
the so-called ``in-face" directions \cite{FrGM17}, which extended the idea of ``away-steps" proposed earlier in \cite{Guelat86,Wolfe}.
For recent development of FW-type methods, we refer the readers to \cite{BRZ21} for a recent survey.
The previous discussions were on FW-type methods for \eqref{P00} when $C$ is convex.
In the case when $C$ in \eqref{P00} is not convex, the study of FW-type methods are much more limited. Indeed, when $C$ is not convex, a notable difficulty is that one may move \emph{outside of $C$} when moving towards a minimizer of the linear-optimization oracle in an iteration of the FW method.
Despite this difficulty, some important contributions along this direction were given in \cite{RT13,BalashovPolyak20}. Specifically, in \cite{RT13}, FW-type methods have been extended to some optimization models for sparse principal component analysis, whose $C=\{x \in \mathbb{R}^n: \|x\|=1, \, \|x\|_0 \le r\}$, where $\|x\|_0$ denotes the cardinality of $x$ and $r>0$. More recently, \cite{BalashovPolyak20} further discussed how FW-type methods can be developed when $C$ is a sphere or more generally a smooth manifold. Interestingly, the feasible region $C$ considered in \cite{RT13,BalashovPolyak20} have empty interior.
In this paper, different from \cite{RT13,BalashovPolyak20}, we are interested in developing FW type methods for \eqref{P00} when $C$ is nonconvex and has (possibly) nonempty interior. Specifically, we consider the following nonconvex optimization problem:
\begin{equation}\label{P0}
\begin{array}{rl}
\min\limits_{x\in{\mathbb{X}}} & f(x) \ \ \ \ {\rm s.t.} \ \ P_1(x) - P_2(x) \leq \sigma,
\end{array}
\end{equation}
where $f:{\mathbb{X}}\rightarrow{\rm I\!R}$ is continuously differentiable,
$P_1:{\mathbb{X}}\to {\rm I\!R}$ and $P_2:{\mathbb{X}}\rightarrow{\rm I\!R}$ are convex functions, $\sigma >0$ and the feasible set $\mathcal{F} :=\{x\in {\mathbb{X}}:\; P_1(x) - P_2(x) \leq \sigma\}$ is nonempty and compact.
It is easy to see that \eqref{P0} is a special case of \eqref{P00} with $C=\{x: P_1(x)-P_2(x) \le \sigma\}$.
Notice that, the model \eqref{P0} covers, for example, sparse optimization models whose feasible region can be described as $\{x \in \mathbb{R}^n: P(x) \le r\}$, with $P$ being a weighted difference of $\ell_1$ and $\ell_2$ norms, and $r>0$. For more optimization models of this form, see Section~\ref{sec:LO} below. Our proposed method is an FW-type method in the sense that we make use of a (generalized) linear-optimization oracle (see Definition~\ref{LOdef}) where we linearize \emph{both} the objective function $f$ and the concave part, $-P_2$, of the constraint function. We invoke this linear-optimization oracle to obtain a search direction and use a line search procedure to construct the next iterate and \emph{stay feasible}. We would like to point out that, when $P_2\equiv 0$, our (generalized) linear-optimization oracle reduces to the classical linear-optimization oracle used in the classical FW method. As a result, similar to classical FW method for \eqref{P00} with convex $C$, our proposed method also does not require projections onto $C$.
On the other hand, a notable difference is that, the feasible regions of the linear-optimization oracles in our proposed method \emph{evolve} as the algorithm progresses. This is opposed to the existing FW methods in the literature where the feasible regions of the linear-optimization oracles are fixed as the feasible region of the original problem. As a result, our convergence analysis is \emph{completely different} from those in the existing literature on FW-type methods.
The organization and contribution of this paper are as follows:
\begin{itemize}
\item[{\rm (1)}] After a quick review on notation and basic mathematical tools in Section~\ref{sec2}, we formulate the (new) linear-optimization oracle of our proposed method in Section~\ref{sec:LO}. We also discuss how these linear-optimization oracle problems can be efficiently solved for several important nonconvex optimization models of the form \eqref{P0} arising from compressed sensing and matrix completion.
\item[{\rm (2)}] In Section~\ref{sec4}, we discuss optimality conditions and define a stationarity measure for \eqref{P0}, which paves the way for the convergence analysis of our new FW type methods later.
\item[{\rm (3)}] In Section~\ref{sec:convergence}, we present a new FW type method for solving \eqref{P0}. Under a suitable strict feasibility condition, we establish that the sequence generated by the proposed method is bounded and any accumulation point is a stationary point of \eqref{P0}. In the case where, additionally, the convex part $P_1$ in the constraint is strongly convex and $f$ is Lipschitz differentiable with nonvanishing gradients on ${\cal F}$, we obtain a $o(1/k)$ complexity in terms of the stationarity measure defined in Section~\ref{sec4}.
\item[{\rm (4)}] In Section~\ref{sec6}, motivated by the success of ``away steps" in accelerating FW method in the convex setting, we further introduce a new FW method with ``away steps" for \eqref{P0}, and establish its subsequential convergence under suitable conditions.
\item[{\rm (5)}] We conclude this paper with numerical experiments on our proposed FW method and its ``away-step" variant for solving some compressed sensing models in the form of \eqref{P0} in Section~\ref{sec7}.
\end{itemize}
\section{Notation and preliminaries}\label{sec2}
In this paper, we use $ {\mathbb{X}} $ to denote a finite-dimensional Euclidean space. We denote by $ \langle \cdot, \cdot\rangle $ the inner product on $ {\mathbb{X}} $ and $ \|\cdot\| $ the associated norm. Next, we let $ {\rm I\!R}^n $ denote the Euclidean space of dimension $ n $, and $ {\rm I\!R}^{m\times n} $ denote the space of all $ m\times n $ matrices. Moreover, the space of $ n\times n $ symmetric matrices will be denoted by $ S^{n} $ and the cone of $ n\times n $ positive semidefinite matrices will be denoted by $ S_+^{n} $. Finally, we let $ \mathbb N_+ $ denote the set of nonnegative integers.
For a set $ S\subseteq{\mathbb{X}} $ and an $ x\in{\mathbb{X}} $, we define the distance from $ x $ to $ S $ as $ {\rm dist}(x, S):= \inf\{\|x-y\|:\,y\in S\} $. The convex hull of $ S $ is denoted by $ {\rm conv}(S) $, and $ {{\rm bdry}\,} S $ is the boundary of $ S $. If $ S $ is a finite set, we denote by $ \lvert S\rvert $ its cardinal number. We use $ B(x, r) $ to denote the closed ball with center $ x $ and radius $ r >0 $, i.e., $ B(x, r) = \{y\in{\mathbb{X}}: \|x-y\| \le r\} $.
We say that an extended-real-valued function $ f:{\mathbb{X}}\rightarrow [-\infty, \infty] $ is proper if its effective domain $ {\rm dom}\, f:=\{x\in{\mathbb{X}}:\, f(x)<\infty \} $ is not empty and $ f(x)>-\infty $ for every $ x\in{\mathbb{X}} $. A proper function is said to be closed if it is lower semicontinuous. The limiting subdifferential of a proper closed function $ h $ at $ \bar x\in{\rm dom}\, h $ is given as
\[
\partial h(\bar x):= \left\{\upsilon\in{\mathbb{X}}:\; \exists x^k\overset{h}\rightarrow \bar x \text{ and } \upsilon^k \in \widehat{\partial} h(x^k) \text{ with } \upsilon^k \rightarrow \upsilon\right\},
\]
where $ x^k\overset{h}\rightarrow \bar x $ means $ x^k\rightarrow \bar x $ and $ h(x^k)\rightarrow h(\bar x) $; here, $\widehat{\partial} h$ is the so-called regular (or Fr\'{e}chet) subdifferential, which, for any $x\in {\rm dom\,} h$, is given by
\[
\widehat{\partial} h(x):=\left\{\upsilon\in{\mathbb{X}} :\; \liminf\limits_{z\rightarrow x,z\neq x}\frac{h(z)-h(x)-\langle \upsilon,z-x\rangle }{\|z-x\|}\ge 0 \right\}.
\]
By convention, we set $\widehat{\partial}h(x)= \partial h(x) = \emptyset$ if $ x\notin {\rm dom}\, h $. It is known that $\partial h(x) = \{\nabla h(x)\}$ if $h$ is continuously differentiable at $x$; see \cite[Exercise~8.8(b)]{RoWe98}. Moreover, when $h$ is proper closed and convex, $\partial h$ reduces to the classical subdifferential in convex analysis; see \cite[Proposition~8.12]{RoWe98}.
Next, for a locally Lipschitz continuous function $ h $, we define its Clarke subdifferential at $ \bar x$ as
\[
\partial^\circ h(\bar x):= \left\{\upsilon\in{\mathbb{X}} :\; \limsup_{x\rightarrow \bar x, t\downarrow 0} \frac{f(x+tw)-f(x)}{t}\geq \langle \upsilon, w\rangle \text{ for all } w\in{\mathbb{X}}\right\};
\]
it holds that $\widehat{\partial}h(\bar x) \subseteq \partial h(\bar x)\subseteq \partial^\circ h(\bar x)$; see \cite[Theorem~5.2.22]{BoZh04}.
We now recall a suitable constraint qualification on the constraint set of \eqref{P0} and present the notion of stationary points of \eqref{P0}.
\begin{definition}[{gMFCQ}
For \eqref{P0}, we say that the generalized Mangasarian-Fromovitz constraint qualification (gMFCQ) holds at an $x^*\in\mathcal{F}$ if the following implication holds:
\[
{\rm If }\; P_1(x^*) - P_2(x^*) = \sigma, \;{\rm then }\; 0\not\in\partial^\circ(P_1 - P_2)(x^*).
\]
\end{definition}
Note that gMFCQ reduces to the standard MFCQ when $P_1-P_2$ is smooth.
\begin{definition}[{Stationary point}]\label{Stationary}
We say that an $x^*\in{\mathbb{X}}$ is a stationary point of \eqref{P0} if there exists $\lambda_*\in{\rm I\!R}_+$ such that $(x^*,\lambda_*)$ satisfies
\begin{enumerate}[{\rm (i)}]
\item $0\in \nabla f(x^*) + \lambda_*(\partial P_1(x^*) - \partial P_2(x^*))$;
\item $\lambda_*(P_1(x^*) - P_2(x^*) - \sigma)=0$, and $P_1(x^*) - P_2(x^*) \leq \sigma$.
\end{enumerate}
\end{definition}
We next deduce that any local minimizer of \eqref{P0} is a stationary point under gMFCQ.
\begin{proposition}\label{loc-sta}
Consider \eqref{P0}. If the gMFCQ holds at every point in $\mathcal{F}$, then any local minimizer of \eqref{P0} is a stationary point of \eqref{P0}.
\end{proposition}
\begin{proof}
For any local minimizer $x^*$ of \eqref{P0}, we can deduce from \cite[Theorem~10.1]{RoWe98} that
\begin{equation}\label{subdif}
0\in\partial(f + \delta_{[P_1 - P_2\leq\sigma]})(x^*),
\end{equation}
where $ \delta_S $ is the indicator function of the set $ S $. Below, we consider two cases.
{\bf Case 1:} $P_1(x^*) - P_2(x^*) < \sigma$. As $P_1$ and $P_2$ are real-valued convex functions, they are also continuous. By the continuity of $P_1$ and $P_2$ and \eqref{subdif}, we have that $\nabla f(x^*) = 0$. Thus $x^*$ is a stationary point of \eqref{P0} (with $\lambda_* = 0$ in Definition~\ref{Stationary}).
{\bf Case 2:} $P_1(x^*) - P_2(x^*) = \sigma$. Then we have that
\[
\begin{aligned}
0&\in\partial(f + \delta_{[P_1 - P_2\leq\sigma]})(x^*) \overset{\rm (a)} = \nabla f(x^*) + \mathcal{N}_{[P_1 - P_2\leq\sigma]}(x^*)\\
&\overset{\rm (b)}\subseteq \nabla f(x^*) + \bigcup\limits_{\lambda\ge0}\lambda\partial^\circ(P_1 - P_2)(x^*)\overset{\rm (c)}\subseteq \nabla f(x^*) + \bigcup\limits_{\lambda\ge0}\lambda(\partial P_1(x^*) - \partial P_2(x^*)).
\end{aligned}
\]
where $ \mathcal{N}_S(x):= \partial \delta_S(x) $ is the limiting normal cone of the set $ S $ at $ x $, and (a) holds in view of \cite[Exercise~8.8]{RoWe98} and the smoothness of $ f $, (b) follows from \cite[Theorem~5.2.22]{BoZh04}, the first corollary to \cite[Theorem~2.4.7]{Cl90} and the fact that $0\not\in\partial^\circ(P_1 - P_2)(x^*)$ (thanks to the gMFCQ), (c) holds because of \cite[Proposition~2.3.3]{Cl90}, \cite[Proposition~2.3.1]{Cl90} and \cite[Theorem~6.2.2]{BoLe06}. Then, there exists $\lambda_*\ge0$ such that $0\in \nabla f(x^*) + \lambda_*(\partial P_1(x^*) - \partial P_2(x^*))$. Noticing that $P_1(x^*) - P_2(x^*) = \sigma$, we also have $\lambda_*(P_1(x^*) - P_2(x^*) - \sigma)=0$. Therefore, $x^*$ is a stationary point of \eqref{P0}.
\end{proof}
Before ending this section, we state two lemmas that will be used subsequently in our discussion of stationarity measure in Section~\ref{sec4} and our convergence analysis in Sections~\ref{sec:convergence} and \ref{sec6}. The first lemma is due to Robinson \cite{Rob75} and concerns error bounds for the so-called ${\cal K}-$convex functions: Given a closed convex cone ${\cal K}\subseteq {\rm I\!R}^m$, we say that $g: {\mathbb{X}}\rightarrow {\rm I\!R}^m$ is $\mathcal{K}-$convex if
\[
\lambda g(x) + (1 - \lambda)g(y)\in g(\lambda x + (1 - \lambda)y) +\mathcal{K},\ \ \forall x, y\in {\mathbb{X}}, \ \lambda\in [0,1].
\]
\begin{lemma}\label{ErrorBounded}
Let $\mathcal{K}\subseteq {\rm I\!R}^m$ be a closed convex cone and $g:{\mathbb{X}}\to {\rm I\!R}^m$
be a ${\cal K}-$convex function. Let $\Omega := \{x\in {\mathbb{X}}:\; 0 \in g(x) + {\cal K}\}$ and suppose there exist $x^s\in \Omega$ and $\delta > 0$ such that $B(0,\delta)\subseteq g(x^s) + \mathcal{K}$.
Then
\[
{\rm dist}(x,\Omega)\leq \frac{\|x - x^s\|}{\delta}{\rm dist}(0, g(x) + \mathcal{K}), \ \forall x\in {\mathbb{X}}.
\]
\end{lemma}
Our next lemma concerns the convergence of descent algorithms with line-search based on the Armijo rule, which will be used in our convergence analysis. The proof is standard and follows a similar idea as in \cite[Proposition~1.2.1]{Bertsekas99}. Here we include the proof for the ease of readers.
\begin{lemma}\label{Armijolemma}
Let $ \Gamma \subseteq {\mathbb{X}} $ be a compact set. Suppose that $ f:\; {\mathbb{X}}\rightarrow {\rm I\!R} $ is continuously differentiable on an open set containing $\Gamma$. Let $ c\in(0, 1) $, $ \eta\in(0, 1) $ and $ \{\alpha_k^{0}\} $ satisfy $ 0<\inf_k \alpha_k^0\leq \sup_k \alpha_k^0< \infty $. Let $ x^0\in\Gamma $, $\{\alpha_k\}$ be a positive sequence, and $\{x^k\}$ and $ \{d^k\} $ be bounded sequences such that the following conditions hold for each $ k\in\mathbb N_+ $:
\begin{enumerate}[{\rm (i)}]
\item $\langle \nabla f(x^k), d^k\rangle < 0$.
\item $ \alpha_k $ is computed via Armijo line search with backtracking from $\alpha^0_k$, i.e., $\alpha_k = \alpha^0_k\eta^{j_k}$ with
\begin{equation}\label{Armijols}
\!\!\!\!\!\!j_k\!:=\! \mathop{\rm arg\,min}\left\{j\!\in\! \mathbb{N}_+\!\!:\! \; f(x^k\!+\!\alpha_k^0\eta^jd^k)\!\leq\! f(x^k) \!+\! c\alpha_k^0\eta^j\langle\nabla f(x^k),d^k\rangle\right\}\!.\!\!\!\! \ \footnote{Here and throughout this paper, we use $\mathop{\rm Arg\,min}$ to denote the set of minimizers, and write $\mathop{\rm arg\,min}$ when the set of minimizers is a singleton.}
\end{equation}
\item $ x^{k+1}\in\Gamma $ and satisfies $ f(x^{k+1})\leq f(x^k+\alpha_kd^k) $.
\end{enumerate}
Then, it holds that $ \lim_{k\to\infty} \langle\nabla f(x^k),d^k\rangle =0 $.
\end{lemma}
\begin{proof}
According to \eqref{Armijols} and item (iii), we have for every $k\in\mathbb{N}_+ $,
\[
-c\alpha_k\langle \nabla f(x^k), d^k\rangle\leq f(x^{k}) - f(x^k+\alpha_kd^k) \leq f(x^k) - f(x^{k+1}).
\]
Summing from $ k=0 $ to $ \infty $ on both sides of the above display, we obtain
\[
-c\sum\limits_{k=0}^\infty \alpha_k \langle \nabla f(x^k), d^k\rangle \leq f(x^0) - \inf\limits_{x\in\Gamma} f(x) < \infty;
\]
here, $\inf_\Gamma f$ is finite because $\Gamma$ is compact and $f$ is continuously differentiable on an open set containing $\Gamma$. Using this together with item (i), we can deduce that
\begin{equation}\label{sddlim}
\lim_{k\to\infty} \alpha_k\langle \nabla f(x^k), d^k\rangle = 0.
\end{equation}
Next, notice that $\{x^{k}\}\subseteq \Gamma$ and $\Gamma$ is compact. This together with the boundedness of $\{d^k\} $ and the continuity of $ \nabla f $ implies that $ \{\langle\nabla f(x^k), d^k\rangle\} $ is bounded. Therefore, to prove the desired conclusion, it suffices to show that the limit of any convergent subsequence of $ \{\langle\nabla f(x^k), d^k\rangle\} $ is zero.
To this end, fix any convergent subsequence $ \{ \langle\nabla f(x^{k_t}), d^{k_t}\rangle \} $. Since $ \{\alpha_k\} $ is also bounded, by passing to a further subsequence if necessary, we have $\lim_{t\to\infty} (x^{k_t}, d^{k_t}, \alpha_{k_t}) = (x^*, d^*, \alpha_*)$ for some $ x^*\in\Gamma $, $ d^*\in{\mathbb{X}} $ and $ \alpha_*\geq 0 $. We consider two cases:
{\bf Case 1}: $ \alpha_*>0 $. Then $ \lim_{t\to\infty} \langle\nabla f(x^{k_t}), d^{k_t}\rangle = 0 $ follows directly from \eqref{sddlim}.
{\bf Case 2}: $\alpha_*=0$. In this case, using the fact that $ \inf \alpha_k^0 > 0 $ and the definition of $\alpha_k$ in \eqref{Armijols}, we see that backtracking must have been invoked for all large $t$, i.e., there exists $T $ such that $j_{k_t} \ge 1$ whenever $t\ge T$. Then $j = j_{k_t}-1$ violates the inequality in \eqref{Armijols}, i.e.,
\[
f(x^{k_t}+\eta^{-1}\alpha_{k_t}d^{k_t}) > f(x^{k_t}) + c\eta^{-1}\alpha_{k_t}\langle\nabla f(x^{k_t}), d^{k_t}\rangle \; \text{for every}\; t\geq T.
\]
Dividing both sides of the above inequality by $ \alpha_{k_t}/\eta $ and rearranging terms, we obtain
\[
\frac{f(x^{k_t}+\eta^{-1}\alpha_{k_t}d^{k_t}) - f(x^{k_t})}{\eta^{-1}\alpha_{k_t}} > c\langle\nabla f(x^{k_t}), d^{k_t}\rangle \; \text{for every}\; t\geq T.
\]
Passing to the limit and rearranging terms in the above display, we have
\[
(1-c)\langle\nabla f(x^*), d^*\rangle \geq 0.
\]
Since $ c\in(0, 1) $, we see further that
\[
\lim_{t\to\infty} \langle\nabla f(x^{k_t}), d^{k_t}\rangle = \langle\nabla f(x^*), d^*\rangle \geq 0.
\]
On the other hand, we have $\lim_{t\to\infty} \langle\nabla f(x^{k_t}), d^{k_t}\rangle \leq 0$
because $ \langle\nabla f(x^k), d^k\rangle < 0$ for every $ k $. Thus, we conclude that $ \lim_{t\to\infty} \langle\nabla f(x^{k_t}), d^{k_t}\rangle = 0 $.
\end{proof}
\section{Linear-optimization oracles: Examples}\label{sec:LO}
Recall that the FW method \cite{FrankWolfe56, DeRu70, Jaggi13, GarberHazan16} can be efficiently employed in some instances of \eqref{P00} when $C$ is \emph{convex} but the projections onto $C$ are difficult. Concrete examples of such $C$ include the $\ell_p$ norm ball constraint when $p\in(1, \infty)\setminus\{2\}$, the nuclear norm ball constraint that arises in matrix completion problems for recommender systems \cite{FrGM17}, and the total-variation norm ball adopted in image reconstruction tasks \cite{HaJN15}; for these examples, the so-called linear-optimization oracle can be carried out efficiently. Such oracle is used in each iteration of the FW method to generate a test point, and the next iterate of the FW method is obtained as a suitable \emph{convex combination} of the current iterate and the test point. Note that \emph{convexity} of $C$ in \eqref{P00} is crucial here so that the next iterate stays feasible. Since the constraint set of \eqref{P0} can be nonconvex in general, it appears that the classic FW method described above cannot be directly applied to solve \eqref{P0}.
As a first step towards developing FW method for \eqref{P0}, let us define a (new) notion of linear-optimization oracle for the possibly nonconvex constraint set in \eqref{P0}. We will then discuss how to solve the linear-optimization oracles that correspond to some concrete applications. Our new FW-type methods for \eqref{P0} based on this new notion of linear-optimization oracles will be presented as Algorithms~\ref{FW_nonconvex} and \ref{alg:aw-fw} in Sections~\ref{sec:convergence} and \ref{sec6}, respectively.
\begin{definition}[{Linear-optimization oracle}]\label{LOdef}
Let $P_1$, $P_2$ and $\sigma$ be defined in \eqref{P0}, $y\in \mathcal{F}$, $\xi\in\partial P_2(y)$, and define
\begin{equation}\label{Fyxi}
{\cal F}(y,\xi) := \{x\in {\mathbb{X}}:\; P_1(x) - \langle \xi, x - y\rangle - P_2(y) \leq \sigma\}.
\end{equation}
Let $a\in {\mathbb{X}}$. A linear-optimization oracle for $(a, y, \xi)$ (denoted by ${\cal LO}(a, y, \xi)$ or ${\cal LO}$ for brevity) computes a solution of the following problem
\begin{equation}\label{LO}
\begin{array}{rl}
\min\limits_{x\in{\mathbb{X}}} & \langle a, x\rangle\ \ \ \ {\rm s.t.}\ \ x\in {\cal F}(y,\xi).
\end{array}
\end{equation}
\end{definition}
\begin{remark}[{Well-definedness of ${\cal LO}$}]\label{welldefined}
Notice that problem \eqref{LO} is well-defined. Indeed, given $y\in \mathcal{F}$ and $\xi\in\partial P_2(y)$, we have
\begin{equation}\label{constrain}
\begin{aligned}
y\in {\cal F}(y,\xi) &= \{x\in {\mathbb{X}}:\; P_1(x) - P_2(y) - \langle \xi, x- y\rangle \leq \sigma\}\\
&\subseteq \{x\in {\mathbb{X}}:\; P_1(x) - P_2(x) \leq \sigma\} = {\cal F}\subseteq B(0,M),
\end{aligned}
\end{equation}
where the first set inclusion follows from the convexity of $P_2$ and the fact that $\xi\in\partial P_2(y)$, and the second set inclusion holds for some $M > 0$ due to the compactness of $\cal F$. Thus, problem \eqref{LO} is minimizing a linear objective function over a compact nonempty constraint set. Hence, its set of optimal solutions is nonempty.
\end{remark}
We now present in the following subsections some concrete examples of ${\cal LO}$ (in the sense of Definition~\ref{LOdef}) that can be carried out efficiently. Our first two examples arise from sparsity inducing problems (group sparsity) and the matrix completion problem, respectively. Our third example concerns the case when $P_1$ is strongly convex, where the ${\cal LO}$ can be shown to be related to the computation of the proximal mapping of a suitable function.
\subsection{Group sparsity}\label{gl}
In this subsection, we let $ {\mathbb{X}} ={\rm I\!R}^n $ and let $x_J$ denote the subvector of $ x \in {\rm I\!R}^n$ indexed by $ J $, where $ J\subseteq \{1,\ldots n\} $. We consider $P_1$ and $P_2$ as in the following assumption and discuss the corresponding ${\cal LO}$.
\begin{assumption}\label{assump31}
Let $ {\mathbb{X}} ={\rm I\!R}^n $ and $\mathcal{J}$ be a partition of $\{1,2,\ldots, n\}$. Let $P_1(x) = \sum_{J\in\mathcal{J}}\|x_J\|$ and $P_2$ be a norm such that $P_2\le \mu P_1$ for some $\mu\in[0,1)$.
\end{assumption}
Notice that the choice of $P_1$ and $P_2$ in Assumption~\ref{assump31} (together with $\sigma > 0$) ensures that the constraint set $\{x\in {\mathbb{X}}:\; P_1(x) - P_2(x)\le \sigma\}$ is compact and nonempty. Thus, in view of Remark~\ref{welldefined}, the corresponding ${\cal LO}$ is well-defined.
The choice of $P_1$ in Assumption~\ref{assump31} is known as the group LASSO regularizer \cite{YuLi06}. As in \cite{YinLouHeXin15}, here we consider a natural extension that subtracts a norm $P_2$ from the group LASSO regularizer. An example of $P_2$ satisfying Assumption~\ref{assump31} is $P_2(x)=\mu\|x\|$ with $\mu \in [0,1)$.
With $P_1$ and $P_2$ in \eqref{P0} chosen as in Assumption~\ref{assump31}, for any given $a\in{\rm I\!R}^n$, any $y\in {\cal F} $ and any $\xi\in\partial P_2(y)$, it holds that $P_2(y) = \langle\xi,y\rangle$ and hence the corresponding ${\cal LO}(a, y, \xi)$ solves a problem of the following form:
\begin{equation}\label{sgl}
\begin{array}{rl}
\min\limits_{x\in{\rm I\!R}^n} & \displaystyle \langle a, x\rangle \ \ \ \
{\rm s.t.} \ \ \displaystyle \sum\limits_{J\in\mathcal{J}}\|x_J\| - \langle\xi, x\rangle\leq \sigma.
\end{array}
\end{equation}
We next derive a closed form formula that describes an output of ${\cal LO}(a, y, \xi)$. To proceed with our derivation, we first establish the following lemma.
\begin{lemma}\label{gruopsub}
Let $b\in {\rm I\!R}^p\backslash \{0\}$ and $c\in {\rm I\!R}^p$ with $\|c\|<1$. Consider
\begin{equation}\label{sgleq33}
\kappa: = \min\limits_{\|x\| = 1} \frac{\langle b, x\rangle}{1 - \langle c, x\rangle}.
\end{equation}
Then $\kappa < 0$ and it holds that
\begin{equation*}
c - \frac{\langle b, c\rangle + \sqrt{\langle b, c\rangle^2 - \|b\|^2(\|c\|^2 - 1)}}{\|b\|^2}b \ \in\ \mathop{\rm Arg\,min}\limits_{\|x\| = 1} \frac{\langle b, x\rangle}{1 - \langle c, x\rangle}.
\end{equation*}
\end{lemma}
\begin{proof}
First note that \eqref{sgleq33} is equivalent to the following problem
\begin{equation}\label{sgleq34}
\min\limits_{\|x\| = 1} \frac{\langle b, x\rangle}{\|x\| - \langle c, x\rangle}.
\end{equation}
Since $\|c\| < 1$, we must have $\|x\| \pm \langle c, x\rangle > 0$ whenever $\|x\|=1$. Thus, the optimal value $\kappa$ of the above optimization problem must be negative.\footnote{Since $b\neq 0$, the objective in \eqref{sgleq34} is negative at $x = -\frac{b}{\|b\|}$.}
Since the objective of \eqref{sgleq34} is (positively) $0$-homogeneous, we see that an $x^*$ solves \eqref{sgleq34} if and only if $x^* = \frac{x_1^*}{\|x_1^*\|}$, where
\begin{equation}\label{sgleq4}
x_1^*\in \mathop{\rm Arg\,min}\limits_{x_1\in{\rm I\!R}^p\setminus\{0\}} \frac{\langle b, x_1\rangle}{\|x_1\| - \langle c, x_1\rangle}.
\end{equation}
Using the (positive) $0$-homogeneity of the objective function in \eqref{sgleq4}, one can see further that an optimal solution $x_2^*\in{\rm I\!R}^p$ of the following problem must be optimal for \eqref{sgleq4}:
\begin{equation}\label{sgleq5}
\begin{array}{rl}
\min\limits_{x_2\in{\rm I\!R}^p} & \langle b, x_2\rangle\ \ \ \ {\rm s.t.} \ \ \|x_2\| - \langle c, x_2\rangle = 1.
\end{array}
\end{equation}
We now turn to solving \eqref{sgleq5}. For any $x_2$ satisfying $\|x_2\| - \langle c, x_2\rangle = 1$, we have $x_2 \neq 0$ and $\frac{x_2}{\|x_2\|} - c \neq 0$ (thanks to $\|c\|<1$). Thus, the LICQ holds for \eqref{sgleq5}. Hence, for any optimal solution $x_2^*$ of \eqref{sgleq5},\footnote{Since $\|c\| < 1$, the feasible set of \eqref{sgleq5} is compact and nonempty. This implies that the set of optimal solutions is nonempty.} there exists $\lambda_*\in{\rm I\!R}$ such that
\[
0= b + \lambda_*\left(\frac{x_2^*}{\|x_2^*\|} - c\right)
\]
Since $b\not= 0$, we have that $\lambda_*\not= 0$ and hence $\frac{x_2^*}{\|x_2^*\|} = c - \lambda_*^{-1}b$, which further gives
\begin{equation}\label{findlambda}
\|c - \lambda^{-1}_*b\|^2 = 1.
\end{equation}
Next, since $c - \lambda^{-1}_*b$ is a positive rescaling of $x_2^*$, we conclude using the relations between \eqref{sgleq34}, \eqref{sgleq4} and \eqref{sgleq5} that $c - \lambda^{-1}_*b$ is an optimal solution of \eqref{sgleq34}. Since $\kappa < 0$, we must then have
\[
0>\langle b, c - \lambda^{-1}_*b\rangle = \langle b, c\rangle - \lambda^{-1}_*\|b\|^2 \quad\Longrightarrow\quad \lambda_*^{-1} > \frac{\langle b, c\rangle}{\|b\|^2}.
\]
Using this together with \eqref{findlambda}, we see that $\lambda^{-1}_*$ must be the larger solution of the quadratic equation $\|c - tb\|^2 = 1$. Solving this quadratic equation, we obtain that
\[
\lambda_*^{-1} = \frac{2\langle b, c\rangle + \sqrt{4\langle b, c\rangle^2 - 4\|b\|^2(\|c\|^2 - 1)}}{2\|b\|^2}.
\]
Combining this with the fact that $c - \lambda^{-1}_*b$ is an optimal solution of \eqref{sgleq34} (and hence \eqref{sgleq33}) completes the proof.
\end{proof}
We now present a closed form solution of problem~\eqref{sgl}. For notational convenience, we define for any $x\in {\rm I\!R}^p$ the following sign function,
\begin{equation}\label{sign}
{\rm Sgn}(x):= \begin{cases}
\frac{x}{\|x\|} & {\rm if}\ x\not=0,\\
\frac{e}{\|e\|} & {\rm if}\ x = 0,
\end{cases}
\end{equation}
where $e$ is the vector of all ones of dimension $p$.
\begin{theorem}[{Closed form solution for \eqref{sgl}}]
Consider \eqref{sgl} with $a\neq 0$. For each $J\in {\cal J}$, fix any vector $v_J\in {\rm I\!R}^{\lvert J\rvert}$ with $\|v_J\|=1$ and define
\begin{equation}\label{wstar}
w_J^*=
\begin{cases}
v_J & {\rm if}\ a_J = 0,\\
\xi_J - \frac{\langle a_J, \xi_J\rangle + \sqrt{\langle a_J, \xi_J\rangle^2 - \|a_J\|^2(\|\xi_J\|^2 - 1)}}{\|a_J\|^2}a_J & {\rm otherwise}.
\end{cases}
\end{equation}
Let $\mathcal{I}:= \mathop{\rm Arg\,min}_{J\in {\cal J}}\{\kappa_J\}$, where
\begin{equation}\label{sgleq3}
\kappa_J := \min\limits_{\|w_J\| = 1} \frac{\langle a_J, w_J\rangle}{1 - \langle \xi_J, w_J\rangle},
\end{equation}
and fixed any $J_0\in\mathcal{I}$.\footnote{These $\kappa_J$ are readily computable. Indeed, as we will point out later in the proof of this theorem, the $w^*_J$ in \eqref{wstar} solves the minimization problem in \eqref{sgleq3}, thanks to Lemma~\ref{gruopsub}.} Then a solution $x^*= (x^*_J)_{J\in {\cal J}}$ of \eqref{sgl} is given by
\begin{equation}\label{hahahahaha}
x^*_J = \begin{cases}
\frac{\sigma w_{J_0}^*}{1 - \langle \xi_{J_0}, w_{J_0}^*\rangle} & {\rm if}\ J = J_0,\\
0 & {\rm otherwise}.
\end{cases}
\end{equation}
\end{theorem}
\begin{proof}
The existence of optimal solutions to \eqref{sgl} follows from Remark~\ref{welldefined}.
Also, for each $J\in {\cal J}$, we have $\|\xi_J\|\leq \mu < 1$ (thanks to the fact that $P_2$ is a norm, $P_2\leq\mu P_1$ and $\mu< 1$). In addition, from \eqref{sign}, then one can check readily that $\|{\rm Sgn}(x_J)\| = 1$ and $x_J = \|x_J\|{\rm Sgn}(x_J)$. In particular, we have $1 - \langle\xi_J, {\rm Sgn}(x_J)\rangle > 0$.
For each $J\in\mathcal{J}$, let $u_J = (1 - \langle\xi_J, {\rm Sgn}(x_J)\rangle)x_J$. Then we have $\|u_J\| = \big(1 - \langle\xi_J, {\rm Sgn}(x_J)\rangle\big)\|x_J\|$ and ${\rm Sgn}(u_J) = {\rm Sgn}(x_J)$. Thus, we obtain the following equivalences:
\begin{equation*
\begin{split}
&\sum\limits_{J\in\mathcal{J}}(\|x_J\| \!-\! \langle\xi_J, x_J\rangle)\!\leq\! \sigma
\!\Longleftrightarrow\!\! \sum\limits_{J\in\mathcal{J}}\!(1 \!-\! \langle\xi_J, {\rm Sgn}(x_J)\rangle)\|x_J\|\!\leq\! \sigma
\!\Longleftrightarrow\!\! \sum\limits_{J\in\mathcal{J}}\! \|u_J\|\! \leq \sigma.
\end{split}
\end{equation*}
Hence, an $\widehat x$ solves \eqref{sgl} if and only if $\widehat x_J = \frac{\widehat u_J}{(1 - \langle\xi_J, {\rm Sgn}(\widehat u_J)\rangle)}$ for each $J\in\mathcal{J}$, where
\begin{equation}\label{sgleq1}
\widehat u\in\mathop{\rm Arg\,min}\limits_{\sum\limits_{J\in\mathcal{J}}\|u_J\| \leq \sigma} \sum\limits_{J\in\mathcal{J}}\left\langle a_J, \frac{u_J}{1 - \langle\xi_J, {\rm Sgn}(u_J)\rangle} \right\rangle
\end{equation}
We next discuss how to find such a $\widehat u$. Notice that any $u_J\in{\rm I\!R}^{\lvert J\rvert}$ can be written as $u_J = r_Jw_J$ for some $r_J\geq 0$ and $w_J\in {\rm I\!R}^{\lvert J\rvert}$ satisfying $\|w_J\| = 1$. Consequently, $\widehat u$ satisfies \eqref{sgleq1} if and only if $\widehat u_J = \widehat r_J\widehat w_J$, for each $J\in\mathcal{J}$, where
\begin{equation}\label{sgleq2}
\begin{aligned}
(\widehat r, \widehat w)\in \mathop{\rm Arg\,min}\limits_{(r, w)\in{\rm I\!R}^{\lvert\cal{J}\rvert}\times{\rm I\!R}^n}& \displaystyle\sum\limits_{J\in\mathcal{J}} \left\langle a_J, \frac{ w_J}{1 - \langle\xi_J, w_J\rangle} \right\rangle r_J\\
{\rm s.t.} & \displaystyle\sum_{J\in {\cal J}}r_J \le \sigma,\\
& \displaystyle r_J\geq 0, \ \ \|w_J\| = 1, \ \ \forall J\in\mathcal{J}.
\end{aligned}
\end{equation}
To solve \eqref{sgleq2}, we start with the minimization with respect to $w_J$. This amounts to solving the optimization problems \eqref{sgleq3} for each $J\in\mathcal{J}$. There are two cases:
\begin{itemize}
\item If $a_J = 0$, then $\kappa_J = 0$ and the minimum in \eqref{sgleq3} is achieved at any feasible $w_J$; in particular, one can take $\widehat w_J = v_J$ (with $v_J$ as in \eqref{wstar}).
\item If $a_J \not= 0$, we can apply Lemma~\ref{gruopsub} with $p := \lvert J\rvert$, $b := a_J$ and $c := \xi_J$, to deduce that $\kappa_J < 0$ and a minimizer is given as in \eqref{wstar}.
\end{itemize}
Thus, one can take $\widehat w _J = w^*_J$ in \eqref{wstar} as a minimizer (with respect to $w_J$) in \eqref{sgleq2}.
Now, to solve \eqref{sgleq2}, it remains to consider the following problem to find an $\widehat r$:
\begin{equation}\label{sgleq6}
\begin{array}{rl}
\min\limits_{r\in{\rm I\!R}^{\lvert\cal{J}\rvert}} & \displaystyle\sum\limits_{J\in\mathcal{J}} \kappa_Jr_J\ \ \ \ \ \ {\rm s.t.}\ \ \ \displaystyle\sum_{J\in {\cal J}}r_J \le \sigma,\ \ r_J \ge 0,\ \ \forall J\in {\cal J}.
\end{array}
\end{equation}
Recall that $\mathcal{I}= \mathop{\rm Arg\,min}_{J}\{\kappa_J\}$ and $J_0$ is a fixed element in $\mathcal{I}$. Since $\kappa_{J} \leq 0$ for any $J\in\mathcal{J}$, we see that $r^* = (r^*_J)_{J\in {\cal J}}$ defined below is an optimal solution of \eqref{sgleq6}:
\begin{equation}\label{rstar}
r^*_J = \begin{cases}
\sigma & {\rm if}\ J = J_0,\\
0 & {\rm otherwise}.
\end{cases}
\end{equation}
Hence, a solution to \eqref{sgleq2} is given by $\widehat r = r_J^*$ and $\widehat w = w_J^*$, for each $J$, defined in \eqref{rstar} and \eqref{wstar} respectively.
Finally, invoking the relationship between \eqref{sgleq1} and \eqref{sgleq2}, we see that $\widehat u$ with $\widehat u_J:= r^*_Jw^*_J$ for each $J$ solves \eqref{sgleq1}. Recall that \eqref{sgl} and \eqref{sgleq1} are related via $u_J = (1 - \langle\xi_J, {\rm Sgn}(x_J)\rangle)x_J$ and $x_J = \frac{1}{1 - \langle \xi_J, {\rm Sgn}(u_J)\rangle}u_J$ for all $J$. Thus, we conclude that an optimal solution $x^*= (x^*_J)_{J\in {\cal J}}$ of \eqref{sgl} is given by:
\begin{equation*}
x^*_J = \frac{1}{1 - \langle \xi_J, {\rm Sgn}(r_J^*w^*_J)\rangle}r_J^*w^*_J = \begin{cases}
\frac{\sigma w_{J_0}^*}{1 - \langle \xi_{J_0}, w_{J_0}^*\rangle} & {\rm if}\ J = J_0,\\
0 & {\rm otherwise}.
\end{cases}
\end{equation*}
This completes the proof.
\end{proof}
\begin{remark}[Element-wise sparsity]\label{cssparsity}
On passing, we discuss an interesting special case where every $J\in\mathcal{J}$ is a singleton. In this case, we have $P_1 = \|\cdot\|_1$ and $P_2:{\rm I\!R}^n\rightarrow{\rm I\!R}$ is a norm satisfying $P_2\le \mu P_1$ for some $\mu\in[0,1)$. Such a setting arises in compressed sensing \cite{LouYan18,YinLouHeXin15}, where $P_2$ can be chosen as $P_2 = \mu\|\cdot\|$ for some $\mu \in [0,1)$, resulting in the difference of $\ell_1$ and (a positive multiple of) $\ell_2$ norm regularizer. A closed form formula that describes an output of the corresponding ${\cal LO}(a, y, \xi)$ can be readily deduced from \eqref{hahahahaha} as follows, upon invoking \eqref{wstar} and \eqref{rstar}:
\begin{equation}\label{xstar}
x^*_i=\begin{cases}
\displaystyle\frac{-\sigma{\rm Sgn}(a_{i_0})}{1 + \xi_{i_0}{\rm Sgn}(a_{i_0})} & i=i_0,\\
\displaystyle 0 & i\neq i_0,
\end{cases}
\end{equation}
where $ \mathcal{I}:= \mathop{\rm Arg\,min}_{i}\left\{ \frac{-\lvert a_i\rvert }{1 + \xi_i {\rm Sgn}(a_i)} \right\} $ with $ i_0 $ being any fixed element of $\mathcal{I} $, and ${\rm Sgn}$ is defined in \eqref{sign}.
\end{remark}
\subsection{Matrix completion}\label{mc}
In this subsection, we let ${\mathbb{X}} = {\rm I\!R}^{m\times n}$. For an $ X\in{\mathbb{X}} $, we denote by $ \|X\|_* $ and $ \|X\|_F $ its nuclear norm and Frobenius norm, respectively. We consider $P_1$ and $P_2$ as in the following assumption and discuss the corresponding ${\cal LO}$.
\begin{assumption}\label{assump32}
Let ${\mathbb{X}} = {\rm I\!R}^{m\times n}$, $P_1(X) = \|X\|_*$ and $P_2$ be a norm function such that $P_2\le \mu P_1$ for some $\mu\in[0,1)$.
\end{assumption}
Observe that under the choice of $P_1$ and $P_2$ in Assumption~\ref{assump32} (together with $\sigma > 0$), the set $\{x\in {\mathbb{X}}:\; P_1(x) - P_2(x)\le \sigma\}$ is nonempty and compact. Hence, the corresponding ${\cal LO}$ is well-defined thanks to Remark~\ref{welldefined}.
An example of $P_2$ is $P_2(X)= \mu\|X\|_F$ with $\mu \in [0,1)$. In this case, the regularization $ P(X)=\|X\|_* - \mu\|X\|_F $ has been used in low rank matrix completion, which can be viewed as an extension of the $ \ell_{1-2} $ metric in compressed sensing. Exact and stable recovery condition and numerical advantages of this class of nonconvex nonsmooth low-rank metric with $ \mu=1 $ are discussed in \cite{MaLou17}.
Now, with $P_1$ and $P_2$ in \eqref{P0} chosen as in Assumption~\ref{assump32}, for any given $A\in{\rm I\!R}^{m\times n}$, any $Y\in {\cal F} $ and any $\Xi\in \partial P_2(Y)$, it holds that $P_2(Y) = \langle\Xi,Y\rangle$ and thus the corresponding ${\cal LO}(A, Y, \Xi)$ solves a problem of the form:
\begin{equation}\label{lo-mat}
\min\limits_{X\in{\rm I\!R}^{m\times n}} \langle A, X \rangle \quad \text{s.t.}\; \|X\|_* -\langle \Xi, X\rangle \leq \sigma.
\end{equation}We next present a closed form solution of \eqref{lo-mat}. For notational simplicity, we write
\begin{equation}\label{AandXi}
\widetilde{A} = \begin{bmatrix}
0 & A \\ A^T & 0
\end{bmatrix}\ \ \ {\rm and}\ \ \ \widetilde{\Xi}= \begin{bmatrix}
0 & \Xi \\ \Xi^T & 0
\end{bmatrix}.
\end{equation}
\begin{theorem}\label{cfs-mat}
Consider \eqref{lo-mat} with $A\neq 0$. Let $ z\in{\rm I\!R}^{m+n} $ be a generalized eigenvector of the smallest generalized eigenvalue of the matrix pencil $ (\widetilde A, I - \widetilde\Xi) $, and satisfy $z^T(I - \widetilde\Xi)z = 1$, where $\widetilde A$ and $\widetilde\Xi$ are given in \eqref{AandXi}. Then $ X^*=2\sigma z_1z_2^T $ is an optimal solution of \eqref{lo-mat}, where $ z=\begin{bmatrix}
z_1^T & z_2^T
\end{bmatrix}^T $ with $ z_1\in{\rm I\!R}^m $ and $ z_2\in{\rm I\!R}^n $.
\end{theorem}
\begin{proof}
For notational convenience, we write
\begin{equation}\label{Zstar}
Z^*:= 2\sigma zz^T=\begin{bmatrix}
U^* & X^* \\ (X^*)^T & V^*
\end{bmatrix}\succeq 0,
\end{equation}
where $ U^* = 2{\sigma}z_1z_1^T $ and $ V^* = 2{\sigma}z_2z_2^T $.
We first claim that $ Z^* $ is a solution of the following optimization problem:\footnote{Note that $I - \widetilde \Xi \succ 0$ because the spectral norm of $\Xi$ is at most $\mu < 1$. Thus, the feasible set of \eqref{auxp} is nonempty and compact, and hence the set of optimal solution is nonempty.}
\begin{equation}\label{auxp}
\min\limits_{Y\in{\mathcal S}^{m+n}_+}\; \langle \widetilde{A}, Y\rangle \quad \text{s.t.}\; \langle I-\widetilde{\Xi}, Y \rangle \leq 2\sigma.
\end{equation}
Indeed, by adding a slack variable $ \alpha\in{\rm I\!R}_+ $, \eqref{auxp} can be written as
\begin{equation}\label{auxp1}
\min\limits_{Y\in{\mathcal S}^{m+n}_+, \alpha \geq 0}\; \langle \widetilde{A}, Y\rangle \quad \text{s.t.}\; \langle I-\widetilde{\Xi}, Y \rangle +\alpha = 2\sigma.
\end{equation}
Note that there is only one equality constraint in the above semidefinite programming problem, and there must be a solution $ (Y^*, \alpha_*) $ that is an extreme point of the feasible set of \eqref{auxp1} (as the solution set of \eqref{auxp1} does not contain a line). According to \cite[Theorem~2.2]{Pataki98}, the rank $ r_Y $ of $ Y^* $ and the rank $ r_\alpha $ of $\alpha_* $ satisfy
\[
r_Y(r_Y+1) + r_\alpha (r_\alpha +1 ) \leq 2.
\]
Since $ A\neq 0 $, we must have $ Y^*\neq 0 $. This fact together with the above display implies that $ r_Y = 1 $ and $r_{\alpha}=0$. Therefore, we can write the rank-1 solution $ Y^* $ of \eqref{auxp1} as $ Y^* = 2\sigma y^*(y^*)^T $ for some $ y^*\in{\rm I\!R}^{m+n}$ that solves
\begin{equation*}
\min_{y\in{\rm I\!R}^{m+n}} y^T\widetilde{A}y \quad \text{s.t.}\; y^T(I-\widetilde{\Xi})y = 1.
\end{equation*}
Such a $y^*$ can be obtained as a generalized eigenvector that corresponds to the smallest generalized eigenvalue of the matrix pencil $ (\widetilde A, I-\widetilde\Xi) $ and satisfies $ y^T(I-\widetilde{\Xi})y = 1 $. Now, recalling the definitions of $ z $ and $ Z^* $, we see that $ Z^* $ is a solution of \eqref{auxp}.
We are now ready to prove that $ X^* $ is a solution of \eqref{lo-mat}. First, recall from \cite{ReFa10} that the nuclear norm of a matrix $ X\in{\rm I\!R}^{m\times n} $ can be represented as:
\begin{equation}\label{nucn}
\!\!\!\!\!\|X\|_* = \min\limits_{U, V} \left\{\frac12\left(
{\rm tr}(U)+{\rm tr}(V)\right):\; \begin{bmatrix}
U & X \\ X^T & V
\end{bmatrix}\succeq 0,\; U\in S^{m},\; V\in S^n
\right\}.\footnote{We would like to point out that the minimum is attainable according to the discussions in \cite{ReFa10}.}
\end{equation}
One can then deduce that
\begin{equation}\label{feasXstar}
\begin{split}
&\|X^*\|_*-\langle \Xi, X^* \rangle \overset{\rm (a)}\leq \frac12\left({\rm tr}(U^*)+{\rm tr}(V^*)\right) - \langle \Xi, X^*\rangle\\
& \overset{\rm (b)}=
\frac12{\rm tr}(Z^*)-\frac12 \langle \widetilde{\Xi}, Z^*\rangle = \frac12 \langle I-\widetilde{\Xi}, Z^*\rangle \leq
\sigma,
\end{split}
\end{equation}
where (a) follows from \eqref{nucn} and the definition of $U^*$ and $V^*$ in \eqref{Zstar}, and (b) uses the definitions of $ \widetilde\Xi $ and $ Z^* $. This shows that $X^*$ is feasible for \eqref{lo-mat}.
Next, for any $ \widetilde X\in{\rm I\!R}^{m\times n} $ satisfying $ \|\widetilde X\|_*-\langle \Xi, \widetilde X\rangle\leq \sigma $, we define $ (\widetilde U, \widetilde V) $ as the minimizer in \eqref{nucn} corresponding to $\|\widetilde X\|_*$.
Let
\[\widetilde Z=\begin{bmatrix}
\widetilde{U} & \widetilde{X} \\ \widetilde{X}^T & \widetilde{V}
\end{bmatrix}.
\]
We can check directly that $ \widetilde Z $ is feasible for \eqref{auxp} by using the feasibility of $ \widetilde X $ for \eqref{lo-mat} and the definitions of $ \widetilde A $ and $ \widetilde \Xi $. Then we have
\[
\langle A, X^*\rangle = \frac12 \langle \widetilde{A}, Z^*\rangle \overset{\rm (a)}\leq \frac12\langle \widetilde{A}, \widetilde{Z}\rangle \overset{\rm
(b)}= \langle A, \widetilde{X}\rangle,
\]
where (a) uses the optimality of $ Z^* $ and the feasibility of $ \widetilde Z $ for \eqref{auxp}, and (b) uses the definition of $ \widetilde{A} $. This together with \eqref{feasXstar} shows that $ X^*=2 \sigma z_1z_2^T $ solves \eqref{lo-mat}.
\end{proof}
\begin{remark}
To obtain a closed form solution of \eqref{lo-mat}, we have to compute a generalized eigenvector $ z $ as shown in Theorem~\ref{cfs-mat}. Noticing that $I - \widetilde \Xi \succ 0$ (thanks to the fact that the spectral norm of $\Xi$ is at most $\mu < 1$), such a generalized eigenvector $z$ can be found efficiently by {\sf eigifp} \cite{YeGolub02}: {\sf eigifp} is an iterative solver based on Krylov subspace methods and only requires matrix vector multiplications $ \widetilde{A}u $ and $ \widetilde{\Xi}u $ in each iteration.
\end{remark}
\subsection{The case where $P_1$ is strongly convex}\label{sec:scsubp}
In this subsection, we assume that $ P_1 $ in \eqref{P0} is strongly convex with modulus $ \rho>0 $. We will argue that the corresponding ${\cal LO}$ involves a linear-optimization problem over a strongly convex set. Also, under suitable constraint qualifications, its solution involves computation of proximal mapping.
To this end, consider any given $ a\in{\mathbb{X}}\backslash\{0\} $, any $ y \in {\cal F} $ and any $ \xi\in\partial P_2(y) $. Then the corresponding $ {\cal LO}(a, y, \xi) $ solves a problem of the following form:
\begin{equation}\label{LOsc}
\begin{array}{cl}
\min\limits_{x\in{\mathbb{X}}} & \langle a, x \rangle\ \ \ \ {\rm s.t.} \ \ \widetilde P_1(x) + \frac{\rho}{2}\|x\|^2 -\langle \xi, x\rangle \leq \widetilde{\sigma},
\end{array}
\end{equation}
where $ \widetilde P_1 := P_1 -\frac{\rho}2\|\cdot\|^2 $ and $ \widetilde\sigma = \sigma + P_2(y) - \langle\xi, y\rangle $.
Moreover, $\widetilde P_1$ is convex.
Now, suppose that Slater's condition holds for the constraint set in \eqref{LOsc}.
Let $ x^* $ be a solution of \eqref{LOsc}. Since $a\neq 0$, we see from \cite[Corollary~28.1]{Ro70} and \cite[Theorem~28.3]{Ro70} that there exists $ \lambda_* > 0 $
such that
\begin{subnumcases}{}
x^* = \mathop{\rm arg\,min}_{x\in{\mathbb{X}}} \;\left\{\langle a, x\rangle + \lambda_*\left(\widetilde P_1(x) + \frac{\rho}2\|x\|^2- \langle \xi, x\rangle - \widetilde{\sigma}\right)\right\}, \label{xlambda} \\
P_1(x^*) - \langle \xi, x^*\rangle = \widetilde{\sigma}. \label{eqconst}
\end{subnumcases}
Let $ \iota_* = \lambda_*^{-1} $. One can then deduce from \eqref{xlambda} that
\begin{equation}\label{xprox}
x^* = {\rm Prox}_{\frac 1{\rho}\widetilde P_1}\left(\frac{1}{\rho}(\xi - \iota_* a)\right),
\end{equation}
where $ {\rm Prox}_{g}(y):=\inf_{x\in {\mathbb{X}}}\left\{ g(x) + \frac{1}{2}\|x-y\|^2\right\}$ is the proximal mapping of the proper closed function $ g $ at $ y $. Substituting the above expression into \eqref{eqconst}, we obtain a one-dimensional nonlinear equation in $ \iota_* $ as follows:
\begin{equation*
P_1\left( {\rm Prox}_{\frac 1{\rho}\widetilde P_1}\left(\frac{1}{\rho}(\xi - \iota_* a)\right)\right) - \left\langle \xi, {\rm Prox}_{\frac 1{\rho}\widetilde P_1}\left(\frac{1}{\rho}(\xi - \iota_* a)\right)\right\rangle = \widetilde{\sigma}.
\end{equation*}
By standard root-finding procedures, one can solve for $ \iota_*>0 $. Then a solution $ x^* $ to \eqref{LOsc} can be obtained as \eqref{xprox}.
On passing, we would like to point out that any difference-of-convex (DC) function $ P_1-P_2 $ can be rewritten as the difference of strongly convex functions: Indeed, we trivially have for any $\rho > 0$ that $P_1 - P_2 = [P_1 + \frac\rho2\|\cdot\|^2] - [P_2 + \frac\rho2\|\cdot\|^2]$. Thus, the discussions in this section can be applied to the examples in Sections~\ref{gl} and \ref{mc} after transforming the DC functions therein to the difference of strongly convex functions. However, the ${\cal LO}$ involved will then require computing proximal mappings, which can be inefficient compared with the oracles described in Sections~\ref{gl} and \ref{mc}.
\section{Optimality condition}\label{sec4}
In this section, we discuss optimality conditions and define a stationarity measure for problem \eqref{P0} that are important for our algorithmic development later. We consider the following assumption.
\begin{assumption}\label{assumption1}
In \eqref{P0}, for any $y\in\mathcal{F}$ and $\xi\in\partial P_2(y)$, there exists $x^\odot_{(y, \xi)}\in{\mathbb{X}}$ so that the following holds:
\begin{equation}\label{assumine}
P_1(x^\odot_{(y, \xi)}) - P_2(y) - \langle\xi, x^\odot_{(y, \xi)} - y\rangle < \sigma.
\end{equation}
\end{assumption}
Note that Assumption~\ref{assumption1} holds in the examples described under Assumption~\ref{assump31} or \ref{assump32}. In fact, if we take $x^\odot_{(y, \xi)} = 0$ for any $y\in\mathcal{F}$ and $\xi\in\partial P_2(y)$, one can see that \eqref{assumine} holds for those $P_1$ and $P_2$. Moreover, it is interesting to note that Assumption~\ref{assumption1} depends on the choices of $P_1$ and $P_2$ in the DC decomposition of the constraint function. In contrast, the validity of gMFCQ is independent of the choices of $P_1$ and $P_2$.
We now study some relationships between gMFCQ and Assumption~\ref{assumption1}, and show in particular that under Assumption~\ref{assumption1} every local minimizer of \eqref{P0} is a stationary point.
\begin{proposition}\label{slaequ}
Consider \eqref{P0}. Then the following statements hold:
\begin{enumerate}[{\rm (i)}]
\item If Assumption~\ref{assumption1} holds, then the gMFCQ holds at every point in $\mathcal{F}$.
\item If gMFCQ holds and $-P_2$ is regular at every point in $\{x\in{\mathbb{X}}:\; P_1(x) - P_2(x) = \sigma\}$, then Assumption~\ref{assumption1} holds.
\end{enumerate}
\end{proposition}
\begin{remark}
From Proposition~\ref{loc-sta}, when the gMFCQ holds at every point in $\mathcal{F}$, we see that any local minimizer of \eqref{P0} is a stationary point of \eqref{P0}. Then we can deduce from Proposition~\ref{slaequ}(i) that any local minimizer of \eqref{P0} is a stationary point of \eqref{P0} when Assumption~\ref{assumption1} holds.
\end{remark}
\begin{proof}
(i): Suppose to the contrary that the gMFCQ fails at some $\bar{x}\in\mathcal{F}$, that is, there exists $\bar x$ with $P_1(\bar x) - P_2(\bar x) = \sigma$ but $0\in\partial^\circ (P_1 - P_2)(\bar x)$. This implies that $0\in\partial P_1(\bar x) - \partial P_2(\bar x)$, and hence there exists $\bar\xi\in\partial P_2(\bar x)$ satisfying $\bar\xi\in\partial P_1(\bar x)$. Moreover, by Assumption~\ref{assumption1}, there exists $x^\odot_{(\bar x, \bar\xi)}\in{\mathbb{X}}$ such that
\[
\sigma > P_1(x^\odot_{(\bar x, \bar\xi)}) - P_2(\bar x) - \langle\bar\xi, x^\odot_{(\bar x, \bar\xi)} - \bar x\rangle \overset{\rm (a)}\geq P_1(\bar x) - P_2(\bar x).
\]
where (a) follows from the convexity of $P_1$ and the fact that $\bar\xi\in\partial P_1(\bar x)$. The above display contradicts $P_1(\bar x) - P_2(\bar x) = \sigma$. Hence the gMFCQ holds at every point in $\mathcal{F}$.
(ii): Suppose to the contrary that Assumption~\ref{assumption1} fails. Then there exist $y\in {\cal F}$ and $\xi\in\partial P_2(y)$ such that for all $x\in{\mathbb{X}}$, $P_1(x) - P_2(y) - \langle\xi, x - y\rangle \geq \sigma$.
In particular, we have $P_1(y) - P_2(y) = P_1(y) - P_2(y) - \langle\xi, y - y\rangle \geq \sigma$, which together with $y\in {\cal F}$ implies
$P_1(y) - P_2(y) = \sigma$.
Since $P_1(y) - P_2(y) = \sigma$, we conclude that $y$ is a minimizer of the function $x\mapsto P_1(x) - P_2(y) - \langle\xi, x - y\rangle $. Then, we deduce from the first-order optimality condition that
\begin{equation*}
\begin{aligned}
0 &\in \partial P_1(y) - \xi\subseteq \partial P_1(y) - \partial P_2(y) \overset{\rm (a)}= \partial^\circ P_1(y) - \partial^\circ P_2(y) \\
& \overset{\rm (b)}= \partial^\circ P_1(y) + \partial^\circ(- P_2)(y) \overset{\rm (c)}= \partial^\circ(P_1 - P_2)(y),
\end{aligned}
\end{equation*}
where (a) follows from \cite[Theorem~6.2.2]{BoLe06}, (b) holds because of \cite[Proposition~2.3.1]{Cl90}, and (c) is true in view of Corollary~1 of \cite[Proposition~2.9.8]{Cl90} and the regularity properties of $-P_2$ (by assumption) and $P_1$ (thanks to convexity). The above display contradicts the gMFCQ. Thus, Assumption~\ref{assumption1} holds.
\end{proof}
Next, we present equivalent characterizations of a stationary point of \eqref{P0}.
\begin{lemma}\label{equistat}
Consider \eqref{P0} and suppose that Assumption~\ref{assumption1} holds. Let $x^*\in {\cal F}$. Then the following statements are equivalent:
\begin{enumerate}[{\rm (i)}]
\item $x^*$ is a stationary point of \eqref{P0}.
\item There exists $\xi^*\in\partial P_2(x^*)$ such that $x^*\in\mathop{\rm Arg\,min}_{x\in {\cal F}(x^*,\xi^*)}\{\langle \nabla f(x^*), x\rangle\}$.\footnote{See \eqref{Fyxi} for the definition of ${\cal F}(x^*,\xi^*)$.}
\item There exist $\xi^*\in\partial P_2(x^*)$ and $u^*\in\mathop{\rm Arg\,min}_{x\in {\cal F}(x^*,\xi^*)}\{\langle \nabla f(x^*), x\rangle\}$ such that
\[
\langle \nabla f(x^*), u^* - x^*\rangle = 0.
\]
\end{enumerate}
\end{lemma}
\begin{proof}
(i)$\Leftrightarrow$(ii): By Assumption~\ref{assumption1}, for any $x^*\in\mathcal{F}$ and for any $\xi^*\in\partial P_2(x^*)$, one
can see that the constraint set ${\cal F}(x^*,\xi^*) = \{x:\; P_1(x) - P_2(x^*) - \langle \xi^*, x - x^*\rangle \leq
\sigma\}$ contains a Slater point $x^\odot_{(x^*, \xi^*)}$. Then, in view of \cite[Corollary~28.2.1, Theorem~28.3]{Ro70}, we see that (i) is equivalent to (ii).
(ii)$\Rightarrow$(iii): Let $\xi^*$ be as in item (ii). Then $\mathop{\rm Arg\,min}_{x\in {\cal F}(x^*,\xi^*)}\{\langle \nabla f(x^*), x\rangle\}$ is nonempty thanks to \eqref{constrain}. Now, pick any $u^*\in\mathop{\rm Arg\,min}_{x\in {\cal F}(x^*,\xi^*)}\{\langle \nabla f(x^*), x\rangle\}$, we obtain $P_1(u^*) - P_2(x^*) - \langle \xi^*, u^* - x^*\rangle \leq \sigma$ and
\begin{equation*}
\langle \nabla f(x^*), u^* \rangle \leq \langle \nabla f(x^*), x^*\rangle,
\end{equation*}
because $x^*\in {\cal F}(x^*,\xi^*)$. Next, since $x^*\in\mathop{\rm Arg\,min}_{x\in {\cal F}(x^*,\xi^*)}\{\langle \nabla f(x^*), x\rangle\}$, one has
\[
\langle \nabla f(x^*), x^* \rangle \leq \langle \nabla f(x^*), u^*\rangle
\]
Combining the above two displays yields $\langle \nabla f(x^*), u^* - x^*\rangle = 0$.
(iii)$\Rightarrow$(ii): Let $\xi^*$ and $u^*$ be as in item (iii). Then
\[
\langle \nabla f(x^*), x^*\rangle = \langle \nabla f(x^*), u^*\rangle = \min\limits_{x\in {\cal F}(x^*,\xi^*)}\{\langle \nabla f(x^*), x\rangle\}.
\]
Now, since $x^*\in\mathcal{F}$ and hence $x^*\in {\cal F}(x^*,\xi^*)$, we conclude that (ii) holds.
\end{proof}
Finally, we will introduce a stationarity measure for \eqref{P0} which plays an important role in our convergence analysis later. Specifically, we define a merit function $G: \mathcal{F} \rightarrow \mathbb{R}$ by
\begin{equation}\label{defG}
G(x)=\inf_{\xi \in \partial P_2(x)} \max_{y \in \mathcal{F}(x, \xi)} \langle \nabla f(x), x-y\rangle\ \ \ \ \mbox{ for all } x \in \mathcal{F},
\end{equation}
where ${\cal F}(x,\xi)$ is defined as in \eqref{Fyxi}. Note that the maximum in the definition of $G$ is attained.\footnote{
The attainment of the maximum in the definition of $G$ follows from the fact that $\mathcal{F}(x, \xi)$ is a nonempty compact set for all $x \in \mathcal{F}$ and $\xi \in \partial P_2(x)$; see \eqref{constrain}.} In addition, notice that when $ P_2\equiv 0 $, the function $ G $ in \eqref{defG} reduces to the $g$ in \cite[Eq.~(2)]{Jaggi13}: this latter function was used as a measure for optimality in \cite{Jaggi13} where $f$ was assumed to be convex. Here, for problem \eqref{P0} with a possibly nonzero $P_2$, we argue that $G$ can be regarded as a measure for ``proximity to stationarity" for any feasible points of \eqref{P0}, under Assumption~\ref{assumption1}.
\begin{theorem}[Stationarity measure]\label{lemma:1}
Consider \eqref{P0} and suppose that Assumption~\ref{assumption1} holds. Then, the following statements hold for the $G$ in \eqref{defG}:
\begin{enumerate}[{\rm (i)}]
\item $G(x) \ge 0$ for all $x \in \mathcal{F}$.
\item Let $\{x^k\}\subseteq \mathcal{F}$. If $G(x^k) \rightarrow 0$ and $x^k \rightarrow x^*$ for some $x^*$, then $x^*\in {\cal F}$ and is a stationary point of \eqref{P0}.
\item For $x^* \in \mathcal{F}$, we have $G(x^*)=0$ if and only if $x^*$ is a stationary point of \eqref{P0}.
\end{enumerate}
\end{theorem}
\begin{proof}
Item {\rm (i)} holds in view of \eqref{constrain} and the definition of $G$.
To prove {\rm (ii)}, let $\{x^k\} \subseteq \mathcal{F}$ and $x^k \rightarrow x^*$ with $G(x^k) \rightarrow 0$. Then $x^*\in\mathcal{F}$ because ${\cal F}$ is closed.
Next, notice that
\[
0 \le G(x^k)=\inf_{\xi \in \partial P_2(x^k)} \max_{y \in \mathcal{F}(x^k, \xi)} \langle \nabla f(x^k), x^k-y\rangle \rightarrow 0.
\]
So there exist $\xi^k \in \partial P_2(x^k)$ and $y^k \in {\rm Argmax}_{y \in \mathcal{F}(x^k, \xi^k)}\langle \nabla f(x^k), x^k-y\rangle$ such that
\begin{equation}\label{eq:use1}
\langle \nabla f(x^k), x^k-y^k \rangle \rightarrow 0.
\end{equation}
Note that $\{y^k\}\subseteq {\cal F}$ in view of \eqref{constrain} and is hence bounded. Moreover, $\{x^k\}$ is bounded and hence $\{\xi^k\}$ is bounded in view of \cite[Theorem~2.6]{Tuy98} and the continuity and convexity of $P_2$.
Passing to convergent subsequences if necessary, we may assume that $y^k \rightarrow y^*$ and $\xi^k \rightarrow \xi^*$ for some
$y^* \in \mathcal{F}$ and $\xi^* \in \partial P_2(x^*)$ (thanks to the closedness of $\partial P_2$). Passing to the limit in \eqref{eq:use1} and using the continuity of $\nabla f$, we have
\begin{equation}\label{eq:use2}
\langle \nabla f(x^*), x^*-y^* \rangle =0.
\end{equation}
We next claim that
\begin{equation}\label{minsub}
x^* \in \mathop{\rm Arg\,min}\limits_{x\in{\mathbb{X}}}\{\langle\nabla f(x^*), x\rangle:\; P_1(x) - P_2(x^*) - \langle\xi^*, x -
x^*\rangle\leq\sigma \}.
\end{equation}
Granting this, in view of Lemma~\ref{equistat}(i), (ii) and the fact that $\xi^*\in \partial P_2(x^*)$, we can then conclude that $x^*$ is a stationary point of \eqref{P0}.
It now remains to establish \eqref{minsub}. To this end, we first define the following
\begin{equation}\label{Omegag}
\begin{array}{ll}
\!\!\!\!\!g_{k}(x)\!:=\! P_1(x) \!-\! P_2(x^{k}) \!-\!\langle\xi^{k}, x -x^{k}\rangle \!-\! \sigma, & \Omega_{k} \!:=\! \{x\in{\mathbb{X}}:\, g_{k}(x) \leq 0\}
\ \forall k,\!\!\!\!\!\!\\
\!\!\!\!\!g_{*}(x)\!:=\! P_1(x) \!-\! P_2(x^*) \!-\!\langle\xi^*, x -x^*\rangle \!-\! \sigma,& \Omega_{*} \!:=\! \{x\in{\mathbb{X}}:\, g_{*}(x) \leq 0\}.
\end{array}
\end{equation}
Then, because $x^{k}\to x^*$ and $\xi^{k}\to \xi^*$, we have $\lim_{k\to\infty}g_{k}(x) = g_{*}(x)$ for any $x\in{\rm I\!R}^n$.
Since $x^*\in\mathcal{F}$ and $\xi^*\in\partial P_2(x^*)$, by Assumption~\ref{assumption1}, there exists $x^\odot_{({x^*, \xi^*})}$ such that $g_{*}(x^\odot_{({x^*, \xi^*})}) < 0$, i.e. there exists $\delta_1 > 0$ such that $g_{*}(x^\odot_{({x^*, \xi^*})}) = -\delta_1 < 0$. Moreover, since $g_{k}(x^\odot_{({x^*, \xi^*})}) \to g_{*}(x^\odot_{({x^*, \xi^*})})$, there exists $N_0 > 0$, such that for any $k > N_0$ we have that $g_{k}(x^\odot_{({x^*, \xi^*})}) \leq g_*(x^\odot_{({x^*, \xi^*})}) + \frac{\delta_1}{2} = -\frac{\delta_1}{2} < 0$.
We now apply Lemma~\ref{ErrorBounded} with $\Omega := \Omega_{k}$, ${\cal K} := {\rm I\!R}_+$, $x^s = x^\odot_{({x^*, \xi^*})}$ and $\delta = \frac{\delta_1}{2}$ to obtain, for each $k > N_0$, that,
\begin{equation}\label{erro}
\begin{aligned}
{\rm dist}(z, \Omega_{k})&\leq 2\delta_1^{-1}\|z - x^\odot_{(x^*, \xi^*)}\|{\rm dist}(0, g_{k}(z) + {\rm I\!R}_+)\\
&\leq 4M\delta_1^{-1}{\rm dist}(0, g_{k}(z) + {\rm I\!R}_+) = 4M\delta_1^{-1}[g_{k}(z)]_+, \
\forall z\in B(0, M),
\end{aligned}
\end{equation}
where $M$ is defined as in \eqref{constrain}, and the second inequality follows from the fact that $\|x^\odot_{(x^*, \xi^*)}\| \le M$ (thanks to \eqref{constrain} and the definition of $x^\odot_{(x^*, \xi^*)}$).
Fix any $u\in \Omega_*$. Since $x^*\in\mathcal{F}$ and $\xi^*\in\partial P_2(x^*)$, by \eqref{constrain}, we have $u\in B(0, M)$.
Moreover, applying \eqref{erro} with $z = u$, we have, for each $k > N_0$,
\[
\|u - {\rm Proj}_{\Omega_{k}}(u)\| = {\rm dist}(u, \Omega_{k})\leq
4M\delta_1^{-1}[g_{k}(u)]_+,
\]
where ${\rm Proj}_{\Omega_{k}}$ is the projection mapping onto $\Omega_k$.
Since $P_1(u) - P_2(x^*) - \langle\xi^*, u - x^*\rangle \le \sigma $ (thanks to $u\in \Omega_*$), we have $[g_{k}(u)]_+\rightarrow 0$ since $x^{k}\rightarrow
x^*$ and $\xi^{k}\rightarrow \xi^*$. From this relation and the above display, we have shown that
\begin{equation}\label{limitproju}
\mbox{For each }u\in \Omega_*, \mbox{ it holds that } \lim_{k\to \infty}{\rm Proj}_{\Omega_{k}}(u) = u,
\end{equation}
where $\Omega_*$ and $\Omega_{k}$ are as in \eqref{Omegag}. Now, since $y^k \in {\rm Argmax}_{y \in \mathcal{F}(x^k, \xi^k)}\langle \nabla f(x^k), x^k-y\rangle$ and noting that $\Omega_k = \mathcal{F}(x^k, \xi^k)$, we see that for any $u\in \Omega_*$,
\[
\langle\nabla f(x^{k}), y^{k} - x^{k}\rangle \leq \langle\nabla f(x^{k}), {\rm Proj}_{\Omega_{k}}(u) -
x^{k}\rangle.
\]
Passing to the limits and noting \eqref{limitproju}, we have
\[
\langle\nabla f(x^*), y^* - x^*\rangle \leq \langle\nabla f(x^*), u - x^*\rangle.
\]
Moreover, in view of \eqref{eq:use2}, we have
\[
\langle\nabla f(x^*), x^* - x^*\rangle = 0 = \langle\nabla f(x^*), y^* - x^*\rangle \leq \langle\nabla f(x^*), u
- x^*\rangle,
\]
Since $u\in \Omega_*$ is chosen arbitrarily, we deduce that \eqref{minsub} holds.
To see {\rm (iii)}, applying {\rm (ii)} with $x^k\equiv x^* \in \mathcal{F}$, we see that if $G(x^*)=0$ then $x^*$ is a stationary point of \eqref{P0}. Conversely, suppose that $x^*$ is a stationary point of \eqref{P0}. Then, from Lemma \ref{equistat}, there exists $\xi^* \in \partial P_2(x^*)$ such that
\[
x^* \in \mathop{\rm Arg\,min}_{y \in \mathcal{F}(x^*, \xi^*)} \langle \nabla f(x^*), y-x^*\rangle = \mathop{\rm Arg\,max}_{y \in \mathcal{F}(x^*, \xi^*)} \langle \nabla f(x^*), x^*-y\rangle.
\]
This shows that $G(x^*) \le 0$. Note that $x^* \in \mathcal{F}$ and hence $G(x^*) \ge 0$ from item (i). Therefore, we see that $G(x^*)=0$.
\end{proof}
\section{Algorithm and convergence analysis}\label{sec:convergence}
We present our FW type algorithm for solving \eqref{P0} as Algorithm~\ref{FW_nonconvex} below, which involves an ${\cal LO}$ (defined in Definition~\ref{LOdef}) and a line-search strategy \eqref{linesearch}. Notice that the structure of the algorithm is similar to that of the classical FW algorithm \cite{LaZh16, Lacoste15,Jaggi13}. The main difference is that the constraint sets in the linear-optimization oracles involved in the classic FW algorithm are the same at each step, while the constraint sets in the ${\cal LO}$s in Algorithm~\ref{FW_nonconvex} change with iteration.
\begin{algorithm}
\caption{Frank-Wolfe type algorithm for \eqref{P0} under Assumption~\ref{assumption1}}\label{FW_nonconvex}
\begin{algorithmic
\State \vspace{-0.15 cm}
\begin{description}
\item[\bf Step 0.] Choose $x^0\in \mathcal{F}$, $c,\, \eta\in (0, 1)$, and a sequence $\{\alpha^0_k\}\subseteq(0,1]$ with $\inf_k\alpha^0_k > 0$. Set $k = 0$. \vspace{0.1 cm}
\item[\bf Step 1.] Pick $\xi^k\in\partial P_2(x^k)$ and let $u^k$ be an output of ${\cal LO}(\nabla f(x^k), x^k, \xi^k)$ (see Definition~\ref{LOdef}). Let $ d^k = d^k_{\rm fw} :=u^k-x^k $.
\noindent If $\langle\nabla f(x^k), d^k\rangle = 0$, terminate. \vspace{0.1 cm}
\item[\bf Step 2.] Find $ \alpha_k = \alpha_k^0\eta^{j_k} $ with $ j_k $ being the smallest nonnegative integer such that
\begin{equation}\label{linesearch}
f(x^k + \alpha_k d^k) \le f(x^k) + c\alpha_k\langle\nabla f(x^k), d^k \rangle.
\end{equation}
\item[\bf Step 3.] Set $\widehat x^{k+1} = x^k + \alpha_k d^k$. Choose $x^{k+1}\in\mathcal{F}$ such that $ f(x^{k+1})\leq f(\widehat x^{k+1}) $. Update $k \leftarrow k+1$ and go to Step 1.
\end{description}
\end{algorithmic}
\end{algorithm}
Before we analyze the well-definedness and other theoretical properties of Algorithm~\ref{FW_nonconvex}, we first comment on the termination condition in Algorithm~\ref{FW_nonconvex}.
\begin{remark}[Termination condition in Step 1]\label{term:step1}
Notice that $u^k$ is an output of ${\cal LO}(\nabla f(x^k), x^k, \xi^k)$. This means, in view of Definition~\ref{LOdef}, that $u^k\in\mathop{\rm Arg\,min}_{x\in {\cal F}(x^k,\xi^k)}\{\langle \nabla f(x^k), x - x^k\rangle\}$. If $\langle\nabla f(x^k), u^k - x^k\rangle = 0$, then we deduce from Lemma~\ref{equistat}(iii) that $x^k$ is a stationary point of \eqref{P0}. Moreover, if $x^k$ is not a stationary point of \eqref{P0}, we have that $\langle\nabla f(x^k), u^k - x^k\rangle < 0$ (since we always have $\langle\nabla f(x^k), u^k - x^k\rangle \leq 0$, thanks to $x^k\in {\cal F}$ and hence $x^k\in\mathcal{F}(x^k,\xi^k)$).
\end{remark}
Next we show that Algorithm~\ref{FW_nonconvex} is well-defined. By Remark~\ref{term:step1}, it suffices to show that, if $x^k\in {\cal F}$ for some $k\ge 0$ is not a stationary point of \eqref{P0} and a $ \xi^k\in\partial P_2(x^k) $ is given, then ${\cal LO}(\nabla f(x^k), x^k, \xi^k)$ has an output, the line-search step in Step 2 terminates in finitely many inner iterations, and an $x^{k+1}\in\mathcal F$ can be generated. This together with an induction argument would establish the well-definedness of Algorithm~\ref{FW_nonconvex}.
\begin{proposition}[Well-definedness of Algorithm~\ref{FW_nonconvex}]\label{Prop1}
Consider Algorithm~\ref{FW_nonconvex} for solving \eqref{P0} under Assumption~\ref{assumption1}. Suppose that an $x^k\in\mathcal{F}$ is generated at the end of the $k$-th iteration of Algorithm~\ref{FW_nonconvex} for some $k\geq 0$ and suppose that $ x^k $ is not a stationary point of \eqref{P0}. Then the following
statements hold for this $k$:
\begin{enumerate}[{\rm (i)}]
\item The ${\cal LO}(\nabla f(x^k), x^k, \xi^k)$ is well-defined, i.e. the corresponding linear-optimization problem has an optimal solution.
\item Step 2 of Algorithm~\ref{FW_nonconvex} terminates in finitely many inner iterations.
\item $\widehat{x}^{k+1}\in {\cal F}$.
\end{enumerate}
Thus, an $x^{k+1}\in {\cal F}$ can be generated at the end of the $(k+1)$th iteration of Algorithm~\ref{FW_nonconvex}.
\end{proposition}
\begin{proof}
(i): The well-definedness follows from Remark~\ref{welldefined}.
(ii): Define $\psi(\alpha) := f(x^k + \alpha(u^k - x^k))$. Since $f$ is continuously differentiable, for each $\alpha > 0$, by the mean value theorem, there exists $t_\alpha\in (0,\alpha)$ such that
\[
\begin{split}
&f(x^k + \alpha(u^k - x^k)) = \psi(\alpha) = \psi(0) + \alpha \psi'(t_\alpha) \\
&= \psi(0) + c\alpha \psi'(0) + \alpha [(1-c)\psi'(0) + \psi'(t_\alpha) - \psi'(0)]\\
& = f(x^k) + c \alpha\langle\nabla f(x^k), u^k - x^k\rangle \\
&\ \ \ \ +\alpha\left((1-c)\langle\nabla f(x^k), u^k - x^k\rangle + [\psi'(t_\alpha) - \psi'(0)]\right)
\end{split}
\]
Noting that $\langle\nabla f(x^k), u^k - x^k\rangle < 0$ (thanks to Remark~\ref{term:step1} and the fact that $x^k$ is not a stationary point of \eqref{P0}), $c\in (0,1)$ and that $\lim_{\alpha\downarrow 0}\psi'(t_\alpha) = \psi'(0)$ (thanks to the continuity of $\nabla f$), we conclude that \eqref{linesearch} is satisfied for all sufficiently small $\alpha > 0$.
(iii): Recall from \eqref{constrain} that $x^k\in {\cal F}(x^k,\xi^k)$. Moreover, since $u^k$ is an output of ${\cal LO}(\nabla f(x^k), x^k, \xi^k)$, we also have $u^k\in {\cal F}(x^k,\xi^k)$. These together with $\alpha_k\in(0,1]$ and the convexity of ${\cal F}(x^k,\xi^k)$ imply that
\[
\widehat x^{k+1} := x^k + \alpha_k(u^k - x^k) = \alpha_k u^k +(1-\alpha_k)x^k \in {\cal F}(x^k,\xi^k)\subseteq\mathcal{F}.
\]
Then an $x^{k+1}\in {\cal F}$ can be generated at the end of the $(k+1)$th iteration of Algorithm~\ref{FW_nonconvex}, since we can at least choose $x^{k+1} := \widehat x^{k+1}$.
\end{proof}
\begin{remark}[Choice of $ x^{k+1} $]
From Proposition~\ref{Prop1}, we see that one can always choose $x^{k+1} = \widehat x^{k+1}$. Here, observing that the constraint functions are all positively homogeneous (as the difference of two norms) in the examples we discussed in Sections~\ref{gl} and \ref{mc}, we introduce a boundary boosting technique to choose $ x^{k+1} $ for those examples. The main idea is to leverage the positive homogeneity of $ P_1-P_2 $. Specifically, if $ c_1 := P_1(\widehat x^{k+1}) - P_2(\widehat x^{k+1})< \sigma $ and $ c_1>0 $, we define $ \widetilde{x}^{k+1} = \frac{\sigma}{c_1}\widehat x^{k+1} $. It follows that $ P_1(\widetilde x^{k+1}) - P_2(\widetilde x^{k+1}) = \frac{\sigma}{c_1}(P_1(\widehat x^{k+1}) - P_2(\widehat x^{k+1})) = \sigma $. Then we choose
\begin{equation}\label{choosexk+1}
x^{k+1} = \begin{cases}
\widetilde x^{k+1} & \text{if}\; c_1>0 \;\text{and}\; f(\widetilde x^{k+1}) \leq f(\widehat x^{k+1}), \\
\widehat{x}^{k+1} & \text{otherwise}.
\end{cases}
\end{equation}
\end{remark}
We next show that the sequence $\{x^k\}$ generated by Algorithm~\ref{FW_nonconvex} clusters at a stationary point of \eqref{P0}. Notice from Remark~\ref{term:step1} and Proposition~\ref{Prop1} that $\{x^k\}$ is either an infinite sequence or is a finite sequence that ends at a stationary point of
\eqref{P0}. Without loss of generality, we assume that $\{x^k\}$ is an infinite sequence.
\begin{theorem}[Subsequential convergence]\label{thm-fw}
Consider \eqref{P0} and suppose that Assumption~\ref{assumption1} holds. Let $\{x^k\}$ be an infinite sequence generated by
Algorithm~\ref{FW_nonconvex} and let $G$ be defined as in \eqref{defG}. Then the sequence $\{x^k\}$ is bounded and $G(x^k) \rightarrow 0$ as $k \rightarrow \infty$. Moreover, any accumulation point of $ \{x^k\} $ is a stationary point of \eqref{P0}.
\end{theorem}
\begin{proof}
The boundedness of $\{x^k\}$ follows from the boundedness of $\mathcal{F}$ and the fact that $\{x^k\}\subseteq\mathcal{F}$. Next, we recall from the definition that $u^{k}$ is an output of ${\cal LO}(\nabla f(x^{k}), x^{k}, \xi^{k})$. This shows that
\[
0 \le G(x^{k})\le \max_{y \in \mathcal{F}(x^{k}, \xi^{k})} \langle \nabla f(x^{k}), x^{k}-y\rangle = \langle \nabla f(x^{k}), x^{k}-u^{k}\rangle,
\]
where the first inequality follows from Theorem~\ref{lemma:1}(i) and the fact that $x^{k} \in \mathcal{F}$. In addition, in view of Remark~\ref{term:step1}, we deduce from Lemma \ref{Armijolemma} (with $\Gamma:= \mathcal{F}$) that
\[
\lim_{k\to\infty}\langle \nabla f(x^{k}), u^{k}-x^{k}\rangle = 0.
\]
Thus, we have $G(x^{k}) \rightarrow 0$. Now it follows directly from Theorem~\ref{lemma:1}(ii) that any accumulation point of $ \{x^k\} $ is a stationary point of \eqref{P0}.
\end{proof}
\subsection{The case when $P_1$ is strongly convex}
We now consider the case where $P_1$ in \eqref{P0} is a strongly convex function. We consider the following additional assumption.
\begin{assumption}\label{ass:nonzerograd}
In \eqref{P0}, it holds that $\nabla f(x)\neq 0$ for all $x\in {\cal F}$ and $\nabla f$ is Lipschitz continuous on ${\cal F}$, i.e., there exists $L > 0$ such that
\[
\|\nabla f(x) - \nabla f(y)\|\le L\|x - y\|\ \ {\rm whenever}\ x,y\in {\cal F}.
\]
\end{assumption}
The Lipschitz continuity requirement on $\nabla f$ in Assumption~\ref{ass:nonzerograd} is standard when it comes to complexity analysis of first-order methods; see, for example, \cite{Ne04}.
Moreover, the condition of nonvanishing gradient in Assumption~\ref{ass:nonzerograd} eliminates the ``uninteresting" situation where a feasible point of problem~\eqref{P0} is a stationary point of the {\em unconstrained} problem $\min_{x \in {\mathbb{X}}} f(x)$.
\begin{proposition}\label{Prop5.2}
Consider \eqref{P0} with $P_1$ being strongly convex. Suppose that Assumptions~\ref{assumption1} and \ref{ass:nonzerograd} hold. Let $\{\alpha_k\}$ and $\{d^k\}$ be two infinite sequences generated by
Algorithm~\ref{FW_nonconvex}. Then
\[
\sum_{k=0}^{\infty} \|d^k\|^2 <+\infty\ \ { and}\ \ \inf_k\alpha_k > 0.
\]
\end{proposition}
\begin{proof}
Since $u^{k}$ is an output of ${\cal LO}(\nabla f(x^{k}), x^{k}, \xi^{k})$ and Assumption~\ref{assumption1} holds, by \cite[Corollary~28.2.1, Theorem~28.3]{Ro70}, there exist $\lambda_k \ge 0$ and $w^k \in \partial P_1(u^k)$ such that
\begin{equation}\label{eq:KKT0}
\begin{array}{rl}
&\nabla f(x^k)+ \lambda_k (w^k- \xi^k)=0,\\ [2 pt]
&\lambda_k (P_1(u^k)- \langle \xi^k, u^k-x^k \rangle -P_2(x^k)-\sigma)=0.
\end{array}
\end{equation}
We claim that $\underline{\lambda}:=\inf\{\lambda_k: k \in \mathbb{N}\}>0$. Suppose not, by passing to subsequences if necessary, we may assume that $\lambda_k \rightarrow 0$ and $x^k \rightarrow x^*$ for some $x^*$. Note that $x^*\in {\cal F}$. Passing to the limit in the first relation of \eqref{eq:KKT0} and noting that $\{w^k\}$ and $\{\xi^k\}$ are bounded (thanks to the boundedness of $\{x^k\}$, $\{u^k\}$, the continuity of $P_1$ and $P_2$, and \cite[Theorem~2.6]{Tuy98}), we have
\[
\nabla f(x^*)=0.
\]
This contradicts Assumption~\ref{ass:nonzerograd}. Thus, $\underline{\lambda}=\inf\{\lambda_k: k \in \mathbb{N}\}>0$.
Next, let $\rho > 0$ denote the modulus of strong convexity of $P_1$. Using \eqref{eq:KKT0}, we have
\begin{eqnarray}\label{eq:ineq}
&&\langle \nabla f(x^k), u^k-x^k\rangle = \langle -\lambda_k (w^k- \xi^k), u^k-x^k\rangle \notag \\
&& = \lambda_k \langle w^k, x^k-u^k\rangle + \lambda_k \langle \xi^k, u^k-x^k\rangle \notag \\
&& \le \lambda_k (P_1(x^k)-P_1(u^k) -\frac{\rho}{2}\|x^k-u^k\|^2) + \lambda_k (P_1(u^k)-P_2(x^k)-\sigma) \notag \\
&& = \lambda_k(P_1(x^k)-P_2(x^k)-\sigma) - \frac{\rho \lambda_k}{2}\|x^k-u^k\|^2 \le - \frac{\rho \underline{\lambda}}{2}\|x^k-u^k\|^2.
\end{eqnarray}
where the first inequality holds in view of the second relation of \eqref{eq:KKT0} and the definition of $\rho$, and the last inequality holds because $x^k \in \mathcal{F}$.
This together with the definition of $x^{k+1}$ and \eqref{linesearch} shows that
\[
f(x^{k+1}) \le f(\widehat{x}^{k+1}) \le f(x^k) + c\alpha_k\langle\nabla f(x^k), u^k - x^k\rangle \le f(x^k) - c\alpha_k\frac{\rho \underline{\lambda}}{2}\|x^k-u^k\|^2.
\]
Note that $\{x^k\} \subseteq \mathcal{F}$ (so that $\inf_k f(x^k) > -\infty$). Using this and summing the above display from $k = 0$ to $\infty$, we see further that
\begin{equation}\label{hahahaha2}
\sum_{k=0}^{\infty} \alpha_k \|d^k\|^2=\sum_{k=0}^{\infty} \alpha_k \|x^k-u^k\|^2 <+\infty.
\end{equation}
We now show that $\inf_k\alpha_k>0$. To see this, first recall from \eqref{constrain} that $x^k\in {\cal F}(x^k,\xi^k)$, and note from the definition that $u^k\in {\cal F}(x^k,\xi^k)$. Moreover, since ${\cal F}(x^k,\xi^k)$ is convex, we have $x^k+ \alpha (u^k-x^k) \in {\cal F}(x^k,\xi^k) \subseteq {\cal F}$ for all $\alpha\in [0,1]$. Then, for any $\alpha\in [0,1]$, with $L$ as in Assumption~\ref{ass:nonzerograd}, we have
\begin{align*}
&f(x^k+ \alpha (u^k-x^k)) \le f(x^k) + \alpha \langle \nabla f(x^k), u^k-x^k\rangle+ \frac{L \alpha^2}{2}\|u^k-x^k\|^2 \\
& = f(x^k) + c \alpha \langle \nabla f(x^k), u^k-x^k\rangle \\
& \ \ \ \ +\bigg[(1-c)\alpha \langle \nabla f(x^k), u^k-x^k\rangle + \frac{L \alpha^2}{2}\|u^k-x^k\|^2\bigg] \\
& \le f(x^k) + c \alpha \langle \nabla f(x^k), u^k-x^k\rangle + \alpha \|u^k-x^k\|^2 \, \bigg[- \, \frac{(1-c) \rho \underline{\lambda}}{2} + \frac{L \alpha}{2}\bigg],
\end{align*}
where the last inequality follows from \eqref{eq:ineq}.
This shows that the line search criterion \eqref{linesearch} will be satisfied as long as
$\alpha \le \frac{(1-c)\rho \underline{\lambda}}{L}$.
Consequently, we have
$\alpha_k \ge \min\{ \alpha_k^0, \frac{(1-c)\rho \underline{\lambda} \eta}{L} \}$,
and so $\inf_k\alpha_k \ge \min\{ \inf_{k}\alpha_k^0, \frac{(1-c)\rho \underline{\lambda} \eta}{L}\}>0$.
The desired conclusion now follows immediately from this and \eqref{hahahaha2}.
\end{proof}
Next, we derive a $o(1/k)$ complexity in terms of the stationarity measure $G$ defined in \eqref{defG} under the strong convexity of $P_1$.
\begin{theorem}
Consider \eqref{P0} with $P_1$ being strongly convex. Suppose that Assumptions~\ref{assumption1} and \ref{ass:nonzerograd} hold. Let $\{x^k\}$ be an infinite sequence generated by
Algorithm~\ref{FW_nonconvex} and let $G$ be defined as in \eqref{defG}. Then
\[
\sum_{k=0}^{\infty} G(x^k) <+\infty \ \mbox{ and }\ k \big[\min_{0 \le t \le k} G(x^t) \big] \rightarrow 0.
\]
\end{theorem}
\begin{proof}
Recalling that $u^k$ is an output of ${\cal LO}(\nabla f(x^{k}), x^{k}, \xi^{k})$, where $\xi^k \in \partial P_2(x^k)$, we have
\[
G(x^k)\le \max_{y \in \mathcal{F}(x^k, \xi^k)} \langle \nabla f(x^k), x^k-y\rangle = \langle \nabla f(x^k), x^k-u^k\rangle
\]
This together with the definition of $x^{k+1}$ implies that
\[
f(x^{k+1})-f(x^k) \le f(\widehat{x}^{k+1})-f(x^k) \le c \alpha^k \langle \nabla f(x^{k}), u^k-x^k\rangle \le - c \alpha^k G(x^k).
\]
Summing the above display from $k = 0$ to $\infty$ and recalling $\inf_k f(x^k) > -\infty$ (since $\{x^k\}\subseteq {\cal F}$), we see that $\sum_{k=0}^{\infty} \alpha_k G(x^k)<\infty$.
Since $\inf_k\alpha_k > 0$ from Proposition~\ref{Prop5.2} and $G(x^k)\ge 0$ (see Theorem~\ref{lemma:1}(i)), we deduce further that
$\sum_{k=0}^{\infty} G(x^k)<\infty$.
Finally, let $a_k=\min_{0 \le t \le k} G(x^t)$. Then $\{a_k\}$ is nonnnegative (see Theorem~\ref{lemma:1}(i)) and non-increasing. On the other hand, the Cauchy criterion for the convergent series gives $\lim_{k\to\infty}\sum_{t=k}^{2k} G(x^t) = 0$. This implies that
\[
(k+1) \, a_{2k} \le G(x^k)+\ldots + G(x^{2k}) = \sum_{t=k}^{2k} G(x^t) \rightarrow 0
\]
Similarly, one can show that $\lim_{k\to\infty}(k+1)a_{2k+1}= 0$.
Thus, we have $\lim_{k\to\infty}k a_{k} = 0$ as desired.
\end{proof}
\section{Away-step Frank-Wolfe type algorithm}\label{sec6}
When $f$ and $C$ in \eqref{P00} are convex and $C = {\rm conv}(\mathcal A) $ for some finite set $ \mathcal A $ (whose elements are called atoms), the so-called away-step technique was proposed in \cite{Guelat86} to accelerate the convergence of FW method; see \cite{Lacoste15, Garber16, Clarkson10} for recent developments. The key ingredient in the away-step technique is to keep track of \emph{a convex decomposition of the current iterate into atoms in $ \mathcal A $}. The away step then selects one atom from the current decomposition that ``differs" most from the gradient direction. The FW method with away step requires to maintain the set of the ``active" atoms used in the aforementioned decomposition and update this active set in each iteration (see \cite[Algorithm~1]{Lacoste15} for more details).
In our nonconvex settings, the feasible set of ${\cal LO}$ is not necessarily a convex hull of finitely many atoms, and it can vary from iteration to iteration. In particular, the active atoms used for the current iteration may be infeasible for the subproblem of the next iteration. It seems difficult to maintain the decomposition with a uniform atomic set $ \mathcal A $ and possibly meaningless to update the decomposition based on previous information. In view of this, we do not store atoms used in the decompositions of previous iterates, but generate a new set of atoms in each iteration based on the current iterate. Below we describe the details of how to decompose the iterate and construct away-step oracles for our FW method for \eqref{P0}.
We first introduce our away-step oracle, which mimics the away step used in the classical FW method.
\begin{definition}[Away-step oracles]\label{awdef}
Let $P_1$, $P_2$ and $\sigma$ be defined in \eqref{P0}, $ y\in\mathcal F $ and $ \xi\in\partial P_2(y) $.
Given $ a\in{\mathbb{X}}\setminus\{0\} $, choose a set
\begin{equation}\label{Sy}
\mathcal S(y, \xi)=\left\{v_1,\ldots, v_{m_y} \right\}\subseteq {{\rm bdry}\,} {\cal F}(y,\xi)
\end{equation}
so that $ y = \sum_{i=1}^{m_y} c_i v_i $ for some $ c_1,\ldots,c_{m_y}>0$ satisfying $ \sum_{i=1}^{m_y} c_i = 1$, where $1\le m_y\leq n+1 $ with $ n $ being the dimension of $ {\mathbb{X}} $.
We define the away-step oracle $ {\cal AWO}(a, \mathcal S(y,\xi)) $ as
\begin{equation}\label{aw-step}
\begin{array}{cl}
\max\limits_{x\in {\mathbb{X}}} & \langle a, x \rangle \ \ \ \
{\rm s.t.} \ \ x \in {\rm conv}(\mathcal{S}(y, \xi)).
\end{array}
\end{equation}
\end{definition}
The choice of the set $ \mathcal S(y, \xi) $ is essential for $ {\cal AWO}(a, \mathcal S(y,\xi)) $. Specific strategies of choosing $ \mathcal S(y,\xi) $ and how the $ {\cal AWO} $ can be carried out efficiently for some concrete examples will be discussed in Section~\ref{sec:aw3exs}. Here, we first comment on the existence of such $ \mathcal S(y,\xi) $.
\begin{remark}[Existence of $S(y,\xi)$]\label{remark1}
Since $y\in {\cal F}$ and $ \xi\in\partial P_2(y) $ in Definition~\ref{awdef}, we see from \eqref{constrain} that $y \in {\cal F}(y,\xi)$. Since ${\cal F}(y,\xi)$ is compact and convex, it is the convex hull of all its extreme points; in particular, ${\cal F}(y,\xi) = {\rm conv}\, ({{\rm bdry}\,} {\cal F}(y,\xi))$.
The existence of $v_i$ and $c_i$ in Definition~\ref{awdef} now follows from this and the Carath\'{e}odory's theorem.
\end{remark}
Next, we note that ${\cal AWO}$ is well-defined, because it is solving a maximization problem with a linear objective and a nonempty compact feasible set. We register this simple observation as our next proposition.
\begin{proposition}[Well-definedness of ${\cal AWO}$]
Consider \eqref{P0}. For any fixed $ y\in \mathcal{F} $, $ \xi\in \partial P_2(y) $ and any set $ \mathcal S(y,\xi) $ given as in \eqref{Sy}, the away-step oracle $ {\cal AWO}(\nabla f(y), \mathcal S(y,\xi)) $ in \eqref{aw-step} is well-defined.
\end{proposition}
We now present our Frank-Wolfe type algorithm with away step as Algorithm~\ref{alg:aw-fw} to enhance Algorithm~\ref{FW_nonconvex} for solving \eqref{P0} under Assumption~\ref{assumption1}.
\begin{algorithm}
\caption{Away-step Frank-Wolfe type algorithm for \eqref{P0} under Assumption~\ref{assumption1}}\label{alg:aw-fw}
\begin{algorithmic}
\State \vspace{-0.25 cm}
\begin{description}
\item[\bf Step 0.] Choose $x^0\in \mathcal{F}$, $ 0<\epsilon< \zeta < \infty$, $c,\,\eta\in (0, 1)$ and a sequence $ \{\alpha_k^0\}\subseteq (0,1] $ with $\inf_k \alpha_k^0 > 0$. Set $k = 0$.\vspace{0.1 cm}
\item[\bf Step 1.] Pick $\xi^k\in\partial P_2(x^k)$. Compute $ u^k_{\rm fw}$ by calling ${\cal LO}(\nabla f(x^k),x^k, \xi^k)$ (see Definition~\ref{LOdef}) and set $ d_{\rm fw}^k := u^k_{\rm fw} - x^k $.
\noindent If $ \langle \nabla f(x^k), u_{\rm fw}^k-x^k\rangle = 0 $, terminate. \vspace{0.1 cm}
\item[\bf Step 2.] Choose $ \mathcal S(x^k, \xi^k) $ as in \eqref{Sy} and call $ {\cal AWO}(\nabla f(x^k),\mathcal S(x^k, \xi^k)) $ (see Definition~\ref{awdef}) to compute $ u^k_{\rm aw} $.
Set $ d^k_{\rm aw} := x^k - u^k_{\rm aw} $ and choose an $ \alpha^k_{\rm aw}\le \max\left\{\alpha\geq 0:\; x^k+\alpha d^k_{\rm aw} \in {\rm conv}(\mathcal S(x^k,\xi^k)) \right\} $.\vspace{0.1 cm}
\item[\bf Step 3.] If $ \langle \nabla f(x^k), d^k_{\rm fw}\rangle > \langle \nabla f(x^k), d^k_{\rm aw}\rangle $ and $\alpha_{\rm aw}^k\in(\epsilon, \zeta] $, set $ d^k = d^k_{\rm aw} $ and $ \widetilde\alpha_k^0 = \alpha_{\rm aw}^k $; we declare that an AW step is taken.\vspace{0.1 cm}
\noindent Otherwise, set $ d^k = d^k_{\rm fw}$ and $\widetilde \alpha^0_k = \alpha^0_k$; we declare that an FW step is taken.\vspace{0.1 cm}
\item[\bf Step 4.] Find $ \alpha_k = \widetilde \alpha_k^0\eta^{j_k} $ with $ j_k $ being the smallest nonnegative integer such that
\begin{equation}\label{ls_awfw}
f(x^k + \alpha_k d^k) \le f(x^k) + c\alpha_k\langle\nabla f(x^k), d^k \rangle.
\end{equation}
\item[\bf Step 5.] Set $\widehat x^{k+1} = x^k + \alpha_k d^k$. Choose $ x^{k+1}\in\mathcal F $ such that $ f(x^{k+1})\leq f(\widehat x^{k+1}) $. Update $k \leftarrow k+1$ and go to Step 1.
\end{description}
\end{algorithmic}
\end{algorithm}
\begin{remark}[{Well-definedness of Algorithm~\ref{alg:aw-fw}}]\label{welldefined:Alg2}
Similar to Proposition~\ref{Prop1}, we can argue the well-definedness of Algorithm~\ref{alg:aw-fw} as follows: Suppose that a nonstationary $x^k\in {\cal F}$ is given for some $k \ge 0$. Note that $\langle \nabla f(x^k), u_{\rm fw}^k-x^k\rangle\le 0$ because $x^k \in {\cal F}(x^k,\xi^k)$. Since $x^k$ is not stationary, we further have $\langle \nabla f(x^k), u_{\rm fw}^k-x^k\rangle< 0$ in view of Lemma~\ref{equistat}. Then the rule of choosing $d^k$ in Step~3 of Algorithm~\ref{alg:aw-fw} yields $ \langle \nabla f(x^k), d^k\rangle < 0 $. Therefore, one can show similarly as in Proposition~\ref{Prop1} that the line-search subroutine in Step~4 of Algorithm~\ref{alg:aw-fw} can terminate in finitely many inner iterations. Furthermore, from the definition of $\alpha_k$, one can deduce that $ \widehat{x}^{k+1} \in\mathcal F(x^k,\xi^k)\subseteq {\cal F} $. Then an $ x^{k+1} \in {\cal F}$ can be generated at the end of the ($k+1$)th iteration of Algorithm~\ref{alg:aw-fw}, since we can at least choose $ x^{k+1}:=\widehat{x}^{k+1} $.
\end{remark}
We next show that the sequence $\{x^k\}$ generated by Algorithm~\ref{alg:aw-fw} clusters
at a stationary point of \eqref{P0}. From the discussion in Remarks~\ref{term:step1} and \ref{welldefined:Alg2}, we see that $\{x^k\}$ is either an infinite sequence or is a finite sequence that terminates at a stationary point of \eqref{P0}. Without loss of generality, we assume that $\{x^k\}$ is an infinite sequence.
\begin{theorem}[Subsequential convergence]\label{thm-fw-aw}
Consider \eqref{P0} and suppose that Assumption~\ref{assumption1} holds. Let $ \{x^k\} $ be an infinite sequence generated by
Algorithm~\ref{alg:aw-fw}. Then $ \{x^k\} $ is bounded and every accumulation point of $ \{x^k\} $ is a stationary point of \eqref{P0}.
\end{theorem}
\begin{proof}
Note that $\{x^k\}$ is bounded because $\{x^k\}\subseteq {\cal F}$. Similarly, $\{u^k_{\rm aw}\}$ and $\{u^k_{\rm fw}\}$ are bounded, and we also have the boundedness of $\{\xi^k\}$ in view of \cite[Theorem~2.6]{Tuy98}, the continuity of $P_2$ and the boundedness of $\{x^k\}$.
Next, observe from Step 3 of Algorithm~\ref{alg:aw-fw}, the descent property of the FW direction (i.e., $\langle \nabla f(x^k), u^k_{\rm fw} -x^k \rangle \leq \langle \nabla f(x^k), x^k -x^k\rangle = 0$ for all $k$) and the assumption that $\{x^k\}$ is an infinite sequence (so that $\langle \nabla f(x^k), u^k_{\rm fw} -x^k \rangle < 0$ according to Step 1)
that $ d^{k} $ is a descent direction for every $ k $, i.e., $ \langle \nabla f(x^k), d^k\rangle < 0 $ for every $ k $. In view of this, \eqref{ls_awfw} and Lemma~\ref{Armijolemma} (with $\Gamma:= \mathcal{F}$), we have
\begin{equation}\label{limitf}
\lim_{k\to\infty}\langle \nabla f(x^k), d^k\rangle = 0.
\end{equation}
Now, let $x^*$ be an accumulation point of $\{x^k\}$. Then, in view of the boundedness of $ \{(x^k,\xi^k,u^k_{\rm aw},u^k_{\rm fw})\} $, there exists a subsequence $ \{(x^{k_t},\xi^{k_t},u^{k_t}_{\rm aw},u^{k_t}_{\rm fw})\} $ that
converges to $ (x^*, \xi^*, u^*_{\rm aw} , u^*_{\rm fw}) $ for some $\xi^*\in\partial P_2(x^*)$ (thanks to the closedness of $\partial P_2$), $u^*_{\rm aw}\in {\cal F}$ and $u^*_{\rm fw}\in {\cal F}$. We consider two cases.
{\bf Case 1}: Suppose that $ \{x^{k_t}\} $ is followed by infinitely many FW steps and finitely many AW steps. By passing to a further subsequence, we assume without loss of generality that $ x^{k_t} $ is followed by an FW step for all $t$, i.e., $d^{k_t} = u^{k_t}_{\rm fw} - x^{k_t}$ for all $t$. Since $u^k_{\rm fw}$ is an output of ${\cal LO}(\nabla f(x^k),x^k,\xi^k)$, we have from the definition of $G$ in \eqref{defG} and \eqref{limitf} that
\[
0 \le G(x^{k_t})\le \langle \nabla f(x^{k_t}), x^{k_t}-u^{k_t}_{\rm fw}\rangle = - \langle \nabla f(x^{k_t}), d^{k_t}\rangle \to 0.
\]
This together with Theorem~\ref{lemma:1}(ii) shows that $x^*$ is a stationary point of \eqref{P0}.
{\bf Case 2}: Suppose that the AW step is invoked infinitely many times in $ \{x^{k_t}\} $.
Passing to a further subsequence, we assume without loss of generality that $ x^{k_t} $ is followed by an AW step for all $ t $, i.e., $d^{k_t} = x^{k_t} - u^{k_t}_{\rm aw}$ for all $t$. Then in view of Step 3, we have
\begin{equation}\label{dirule}
\langle \nabla f(x^{k_t}), x^{k_t}-u^{k_t}_{\rm aw}\rangle = \langle \nabla f(x^{k_t}), d^{k_t}\rangle < \langle \nabla f(x^{k_t}), u^{k_t}_{\rm fw}-x^{k_t}\rangle.
\end{equation}
Now, recalling the definition of $G$ in \eqref{defG} and the facts that $\xi^{k_t} \in \partial P_2(x^{k_t})$ and $u^{k_t}_{\rm fw}$ is an output of ${\cal LO}(\nabla f(x^k),x^k, \xi^k)$, we deduce further that
\[
\begin{aligned}
0\le G(x^{k_t})&\le \max_{y \in \mathcal{F}(x^{k_t}, \xi^{k_t})} \langle \nabla f(x^{k_t}), x^{k_t}-y\rangle = \langle \nabla f(x^{k_t}), x^{k_t}-u^{k_t}_{\rm fw}\rangle \\
& \le - \langle \nabla f(x^{k_t}), x^{k_t}-u^{k_t}_{\rm aw}\rangle = - \langle \nabla f(x^{k_t}), d^{k_t}\rangle \rightarrow 0,
\end{aligned}
\]
where the second inequality follows from \eqref{dirule} and the last relation follows from \eqref{limitf}.
This together with Theorem~\ref{lemma:1}(ii) implies that $ x^* $ is a stationary point of problem~\eqref{P0}. This completes the proof.
\end{proof}
\subsection{{$ {\cal AWO} $} for $P_1$ and $P_2$ as in Assumption~\ref{assump31} or \ref{assump32}}\label{sec:aw3exs}
In this section, we discuss how to construct the set $ \mathcal S(y,\xi) $ and obtain a solution for ${\cal AWO}$ when $ P_1 $ and $ P_2 $ are described as in Assumption~\ref{assump31} or \ref{assump32}.
Note that the $ P_1 $ in Assumption~\ref{assump31} or \ref{assump32} are ``atomic norms" that can be written as a gauge function of the form:
\begin{equation}\label{atomic}
\|x\|_\mathcal{A} = \inf\{t\geq 0:\; x\in t\;{\rm conv}(\mathcal{A}) \}
\end{equation}
for some symmetric compact atomic set $\mathcal{A}\subseteq {\mathbb{X}} $ with $0 \in {\rm conv}({\cal A})$. For more discussions on atomic norm, we refer the readers to \cite{Chandrasekaran10}. Below, we list the atomic sets for the $P_1$ in the two scenarios discussed in Assumptions \ref{assump31} and \ref{assump32}.
\begin{itemize}
\item {\bf Scenario~1.} $ \mathcal{A}_{\rm gl} = \bigcup_{J\in{\cal J}}\mathcal{A}_J$ with
\[
\mathcal A_{J} = \{x:\,\|x_{J}\| = 1,\; x_I= 0, \forall I\in \mathcal J\setminus \{J\} \}
\]
for each $ J\in\mathcal J $, where $ \mathcal J $ is a partition of $ \{1,\ldots, n\} $. This corresponds to $ P_1(x) = \sum_{J\in\mathcal J} \|x_{J}\| $ (see \cite[Corollary~2.2]{RaoRecht12}) in Assumption~\ref{assump31}.
\item {\bf Scenario~2.} $ \mathcal A_* = \{uv^T:\; u\in{\rm I\!R}^m, v\in{\rm I\!R}^n \;{\rm with}\;\|u\| = \|v\| = 1 \} $. This corresponds to $ P_1(X) = \|X\|_* $ (see \cite[Section~2.2]{Chandrasekaran10}) in Assumption~\ref{assump32}.
\end{itemize}
For the rest of this section, we will focus on \eqref{atomic} with ${\cal A} = {\cal A}_{\rm gl}$ or ${\cal A}_*$.
Based on these two atomic sets, we first construct a set that provides potential choices for the elements of $ \mathcal S(y,\xi) $ in ${\cal AWO}$ for problems in Sections~\ref{gl} and \ref{mc}.
\begin{proposition}\label{atomprop}
Consider \eqref{P0}. Let $ P_1 = \|\cdot\|_{\mathcal A} $ as in \eqref{atomic} with ${\cal A} = {\cal A}_{\rm gl}$ or ${\cal A}_*$, and let $ P_2 $ be a norm function such that $ P_2\leq \mu P_1$ for some $ \mu \in[0,1) $. Given $ y\in\mathcal F $ and $ \xi\in\partial P_2(y) $, let
\[
\mathcal V(y,\xi) = \left\{\frac{{\sigma}s}{1-\langle \xi, s\rangle}:\; s\in {\cal A} \right\}.
\]
Then the following statements hold:
\begin{enumerate}[{\rm (i)}]
\item For any $ s \in {\cal A}$, we have $ 1-\langle\xi, s\rangle >0 $;
\item For any $ v\in\mathcal{V}(y,\xi) $, it holds that
\[\|v\|_\mathcal{A} - P_2(y) - \langle \xi, v - y\rangle = \|v\|_\mathcal{A} - \langle \xi, v \rangle =\sigma.\]
\end{enumerate}
\end{proposition}
\begin{proof}
For notational simplicity, we write $ P_2(x)= \mu \gamma(x) $ for some norm function $\gamma$, and use $ \gamma^\circ $ to denote the dual norm of $ \gamma $. Since $ \xi\in\partial P_2(y) $, we see that $ \gamma^\circ(\xi)\leq \mu $. Since we have $\|s\|_{\cal A} = 1$ for all $s\in {\cal A}$ (recall that ${\cal A} = {\cal A}_{\rm gl}$ or ${\cal A}_*$), it follows that
\begin{equation*}
\begin{split}
&1-\langle\xi, s\rangle \geq \|s\|_\mathcal{A} - \gamma^\circ(\xi)\gamma(s)\geq \|s\|_\mathcal{A}-\mu \gamma(s) = P_1(s)- P_2(s) \\
&\overset{\rm (a)}\geq P_1(s)-\mu P_1(s) = 1-\mu >0,
\end{split}
\end{equation*}
where the first inequality and the last equality holds because $P_1(s) = \|s\|_\mathcal{A}=1 $, and (a) holds because $ P_2\leq \mu P_1 $. This proves (i).
We now prove (ii). Fix any $ v\in\mathcal{V}(y,\xi) $. Then we have
\begin{equation*}
\begin{split}
&\|v\|_\mathcal{A} - P_2(y)- \langle\xi, v-y\rangle= \frac{{\sigma}\left(\|s\|_\mathcal{A} -\langle \xi, s\rangle \right)}{1-\langle \xi, s\rangle} -P_2(y) +\langle \xi, y\rangle \\
&\overset{\rm (a)}= \sigma - P_2(y) +\langle\xi,y\rangle \overset{\rm (b)}= \sigma,
\end{split}
\end{equation*}
where (a) holds since $ \|s\|_\mathcal{A} = 1 $, and (b) holds as $P_2$ is a norm and $\xi\in \partial P_2(y)$.
\end{proof}
In the next two subsections, we will present a strategy of choosing $ \mathcal{S}(y,\xi) $ in {\bf Scenarios 1} and {\bf 2}, and discuss how to determine the stepsize $ \alpha_{\rm aw} $ in Step~2 in Algorithm~\ref{alg:aw-fw}. Here, we first exclude the case $ y=0 $: In this case, the away-step direction $ d_{\rm aw} $ will always be inferior to $ d_{\rm fw} $ in Step~3 if we choose $ \xi=0 $ (note that $ 0\in\partial P_2(0) $ since $P_2$ is a norm). To see this, note that $ \mathcal{F}(0,0)=\{x:\; \|x\|_\mathcal{A}\leq \sigma \} $. Therefore, we have
\[
\langle \nabla f(0), 0 - u_{\rm aw} \rangle = \langle \nabla f(0), (-u_{\rm aw}) - 0\rangle \geq \langle \nabla f(0), u_{\rm fw}-0\rangle,
\]
where we use the facts that $ -u_{\rm aw}\in\mathcal{F}(0,0) $ (thanks to the symmetry of $ \mathcal{F}(0,0) $) and $ u_{\rm fw} $ is the output of ${\cal LO}(\nabla f(0),0,0)$ for the last inequality. In other words, if $x^k = 0$ and is not stationary, and if we set $\xi^k = 0$, then $d^k = u^k_{\rm fw} - x^k$.
\subsubsection{Constructing $ \mathcal S(y, \xi) $ in ${\cal AWO}$ in Step 2 of Algorithm~\ref{alg:aw-fw}} \label{sec:syxi}
Let $ P_1 $ and $ P_2 $ be as in Assumption~\ref{assump31} or \ref{assump32}. As discussed before, these $ P_1 $ can be written as $ P_1=\|\cdot\|_\mathcal A $ with ${\cal A} = {\cal A}_{\rm gl}$ or ${\cal A}_*$.
We now discuss the choice of ${\cal S}(y,\xi)$ in \eqref{aw-step} when $y\in\mathcal{F}\setminus\{0\}$. To this end, fix any $ y\in\mathcal{F}\setminus\{0\}$ and pick $ \xi\in\partial P_2(y)$. From the definition of the atomic norm \eqref{atomic}, there exist a positive integer $ m\leq n+1 $, a set of atoms $ \{s_i\}_{i=1}^m\subset \mathcal A $ and nonnegative coefficients $ \{r_i\}_{i=1}^m $ such that
\begin{equation}\label{atomrep}
y = \|y\|_\mathcal{A} \sum\limits_{i=1}^m r_i s_i \;\text{and}\; \sum\limits_{i=1}^m r_i =1 .
\end{equation}
Specifically, for ${\cal A} = {\cal A}_{\rm gl}$ and ${\cal A}_*$ in {\bf Scenarios 1} and {\bf 2}, respectively, we can derive the representation in \eqref{atomrep} with
\begin{enumerate}[{\rm (i)}]
\item $ m = \lvert\mathcal{J}\rvert, $ $ r_i = \|y_{J_i}\|/\sum_{i=1}^m \|y_{J_i}\| $ and $ s_i = {\rm Sgn}(y_{J_i}) $ for $ i = 1,\ldots,m $, where we number the elements of ${\cal J}$ as $\{J_1,\ldots,J_m\}$, $ y=[y_{J_i}]_{i=1}^m\in{\rm I\!R}^n $ and ${\rm Sgn}$ is defined in \eqref{sign};
\item $ m = {\rm rank}(Y)$, $ r_i = \lambda_i/\|Y\|_* $ and $ s_i = u_iw_i^T $ for $ i=1, \ldots, {\rm rank}(Y) $, where $ u_i $'s (resp., $ w_i $'s) are columns of $ U $ (resp., $ W $) from the thin SVD of $ Y= U\Lambda W^T $, and $ \lambda_i $'s are the diagonal entries of $ \Lambda $.
\end{enumerate}
With respect to \eqref{atomrep}, we define $ \mathcal I_y:=\{i:\, r_i\neq 0, i=1, \ldots, m\} $ and
\begin{equation}\label{vicidef}
v_i := \frac{\sigma s_i}{1-\langle\xi, s_i\rangle}, \;{\rm and}\; c_i := \dfrac{r_i\|y\|_\mathcal{A}(1-\langle\xi, s_i\rangle)}{\sigma},\quad \forall i=1, \ldots, m.
\end{equation}
It then follows that
\begin{equation}\label{ydcomp1}
y = \sum\limits_{i=1}^m c_i v_i = \sum\limits_{i\in\mathcal I_y} c_i v_i.
\end{equation}
Now, using Proposition~\ref{atomprop} and the definition of $ c_i $ and $ v_i $ in \eqref{vicidef}, and recalling that $ y\neq 0 $ and $ r_i>0 $ for every $ i\in\mathcal{I}_y $, one can deduce that $ v_i \in {{\rm bdry}\,} \mathcal F(y,\xi) $ (defined in \eqref{Fyxi}) and $ c_i>0$ for every $ i\in\mathcal{I}_y $. Moreover, it holds that
\begin{equation}\label{sumci}
\sum\limits_{i\in\mathcal I_y} c_i = \sum\limits_{i\in\mathcal I_y} \frac{r_i\|y\|_\mathcal A - \|y\|_\mathcal A \langle\xi,r_i s_i\rangle }{\sigma} \overset{\rm (a)} = \frac{\|y\|_\mathcal A -\langle \xi, y \rangle}{\sigma} \overset{\rm (b)} \le 1,
\end{equation}
where (a) uses \eqref{atomrep} and the fact $ \sum_{i\in\mathcal I_y} r_i = \sum_{i=1}^m r_i=1 $, and (b) holds because $ y\in\mathcal F (y,\xi) $ (see \eqref{constrain}), $\xi\in \partial P_2(y)$ and $P_2$ is a norm. When $ \sum_{i\in {\mathcal{I}_y}} c_i = 1$, the decomposition \eqref{ydcomp1} of $ y $ satisfies the conditions in Definition~\ref{awdef}. Therefore, we can choose $ \mathcal{S}(y,\xi)=\{v_i:\;i\in\mathcal{I}_y\} $. Otherwise, choose any $ i_0\in\mathcal{I}_y $ and set $ \bar c=1-\sum_{i\in\mathcal{I}_y} c_i > 0$. The
\begin{equation}\label{ydcomp2}
\begin{split}
y = c_{i_0}v_{i_0} + \sum_{i\in\mathcal{I}_y\setminus\{i_0\}}c_i v_i \overset{\rm (a)}=& \sum_{i\in\mathcal{I}_y\setminus\{i_0\}}c_i v_i + \left(c_{i_0}+\frac{\bar c(1-\langle\xi, s_{i_0}\rangle )}{2}\right)v_{i_0} \\
&+ \frac{\bar c(1+\langle\xi, s_{i_0}\rangle )}{2}\cdot \frac{-\sigma s_{i_0}}{(1+\langle\xi, s_{i_0}\rangle)},
\end{split}
\end{equation}
where (a) uses the fact that $ v_{i_0}=\frac{\sigma s_{i_0}}{1-\langle\xi, s_{i_0}\rangle} $.
Note that $ \frac{-\sigma s_{i_0}}{1+\langle\xi, s_{i_0}\rangle}\in\mathcal{V}(y,\xi) $ since $-s_{i_0}\in {\cal A}$ and $ \|-s_{i_0}\|_{\cal A} = 1$. Furthermore, we can check directly that the decomposition of $ y $ in \eqref{ydcomp2} is a convex combination of elements in $ \mathcal{V}(y,\xi)\subset \mathcal F(y,\xi) $ (see Proposition~\ref{atomprop}(i) for the positivity of $1 \pm \langle\xi,s_{i_0}\rangle$). In summary, we can choose the set $ \mathcal S(y,\xi) $ in \eqref{aw-step} as
\begin{equation}\label{Syxi}
\mathcal{S}(y, \xi) = \begin{cases}
\left\{\frac{\sigma s_i}{1-\langle\xi, s_i\rangle}: i\in \mathcal I_y\right\} & {\rm if}\; \sum\limits_{i\in\mathcal{I}_y} c_i=1;\\
\left\{\frac{\sigma s_i}{1-\langle\xi, s_i\rangle}: i\in \mathcal I_y\right\}\cup \left\{\frac{-\sigma s_{i_0}}{1+\langle\xi, s_{i_0}\rangle}\right\} & {\rm otherwise,}
\end{cases}
\end{equation}
where $i_0$ is an arbitrarily chosen element in ${\mathcal{I}}_y$.
Now we give a solution $u_{\rm aw}$ of \eqref{aw-step}, which can be found by maximizing $ \langle a, x\rangle $ over the finite discrete set $ \mathcal S(y, \xi) $. In particular, a solution of \eqref{aw-step} is
\begin{equation}\label{uaw}
u_{\rm aw} =
\begin{cases}
\frac{\sigma s_{i_*}}{1-\langle\xi, s_{i_*}\rangle} & {\rm if}\; \sum\limits_{i\in\mathcal{I}_y} c_i=1 \;{\rm or}\; \langle a, \frac{-\sigma s_{i_0}}{1+\langle \xi, s_{i_0}\rangle}\rangle < \langle a, \frac{\sigma s_{i_*}}{1-\langle \xi, s_{i_*}\rangle}\rangle; \\
\frac{-\sigma s_{i_0}}{1+\langle \xi, s_{i_0}\rangle} & {\rm otherwise,}
\end{cases}
\end{equation}
where $ i_*\in\mathop{\rm Arg\,max}_{i\in\mathcal I_y} \left\langle a, \frac{\sigma s_i}{1-\langle\xi, s_i\rangle}\right\rangle $. Recall that $i_0$ can be chosen to be any element in ${\cal I}_y$. In our numerical experiments in Section~\ref{sec7} below, for simplicity, we choose $i_0 = i_*$ in \eqref{uaw}.
\subsubsection{Choosing the $ \alpha_{\rm aw} $ in Step 2 of Algorithm~\ref{alg:aw-fw}}
Let $ P_1 $ and $ P_2 $ be as in Assumption~\ref{assump31} or \ref{assump32}.
Given $ y \in\mathcal F\setminus\{0\}$ and $ \xi\in\partial P_2(y)$, with $ \mathcal S(y,\xi) $ and $ u_{\rm aw} $ determined as in \eqref{Syxi} and \eqref{uaw}, we discuss how to find an $ \alpha_{\rm aw} \le \max\{\alpha\geq 0:y+\alpha d_{\rm aw}\in {\rm conv}(\mathcal S(y,\xi))\} $ along the away-step direction $ d_{\rm aw} = y - u_{\rm aw}$. Note that the sets $ \mathcal S(y, \xi) $ in Section~\ref{sec:syxi} are all discrete sets with finite elements. For notational simplicity, we write
\[
\mathcal S(y,\xi) = \{v_1, \ldots, v_{q} \}
\]
with $ q=\lvert\mathcal{I}_y\rvert $ or $ \lvert\mathcal{I}_y\rvert+1 $, where $\mathcal{I}_y$ is defined as in Section~\ref{sec:syxi}. According to \eqref{ydcomp2} and \eqref{uaw}, we see that $ u_{\rm aw}\in\mathcal S(y,\xi) $ and
\begin{equation}\label{yrep}
y = \sum_{i=1}^{q} \widetilde c_i v_i
\end{equation}
with $\widetilde c_i> 0 $ for every $ i=1,\ldots,q $ and $ \sum_{i=1}^{q}\widetilde c_i=1 $. Suppose that $ u_{\rm aw} = v_{i_1} $ for some $ v_{i_1}\in \mathcal S(y,\xi)$. Then one can choose
\begin{equation*}
\alpha_{\rm aw} \le \max\{\alpha\geq 0:\; y+\alpha(y - v_{i_1}) \in {\rm conv} (\mathcal S(y, \xi))\}.
\end{equation*}
Define $ y(\alpha) = y+\alpha(y-v_{i_1}) $. Using \eqref{yrep}, we have
\[
\begin{split}
y(\alpha) = (1+\alpha)\sum\limits_{i\neq i_1}\widetilde c_i v_i + ((1+\alpha)\widetilde c_{i_1} -\alpha) v_{i_1}.
\end{split}
\]
To ensure that $ y(\alpha)\in {\rm conv}(S(y,\xi)) $, we impose the following conditions on $ \alpha $:
\begin{align}
&(1+\alpha)\sum_{i\neq i_1}\widetilde c_i + (1+\alpha)\widetilde c_{i_1}-\alpha = 1, \label{alpha_ineq2} \\
&(1+\alpha)\widetilde c_{i_1} -\alpha \geq 0. \label{alpha_ineq1}
\end{align}
Since $ \sum_{i=1}^{q} \widetilde c_i=1 $ (thanks to \eqref{sumci} and \eqref{ydcomp2}), we have $\sum_{i\neq i_1}\widetilde c_i =1 - \widetilde c_{i_1}$. It follows that
\[
(1+\alpha)\sum_{i\neq i_1}\widetilde c_i + (1+\alpha)\widetilde c_{i_1}-\alpha= (1+\alpha)(1-\widetilde c_{i_1})+(1+\alpha)\widetilde c_{i_1} - \alpha = 1.
\]
That is, \eqref{alpha_ineq2} holds automatically for any $ \alpha\geq 0 $. So \eqref{alpha_ineq1} is sufficient to ensure that $ y(\alpha)\in{\rm conv}(\mathcal S(y,\xi)) $. We can thus choose
\begin{equation*}
\alpha_{\rm aw} = \min\left\{\frac{\widetilde c_{i_1}}{1-\widetilde c_{i_1}},\zeta\right\},
\end{equation*}
where $\zeta$ is given in Step 0 of Algorithm~\ref{alg:aw-fw}.
Here, we would like to mention that the expression $\frac{\widetilde c_{i_1}}{1-\widetilde c_{i_1}}$ coincides with the formula of the feasible stepsize $ \gamma_{\rm max} $ given in the FW method with away-step in convex setting proposed in \cite{Lacoste15}.
\section{Numerical experiments}\label{sec7}
In this section, we conduct numerical experiments to illustrate the performances of Algorithm~\ref{FW_nonconvex} and Algorithm~\ref{alg:aw-fw}.
Our simulations are based on the compressed sensing (CS) problem \cite{CandesTao06,Donoho06,Eldar12}, which aims to recover a sparse signal $ x_0\in{\rm I\!R}^n $ from a noisy linear measurement $ b\in{\rm I\!R}^m $ given by $Ax_0 +\widehat n $ with sensing matrix $ A\in{\rm I\!R}^{m\times n} $ and noise vector $ \widehat n\in{\rm I\!R}^m $. Motivated by the model with cardinality constraint for CS in \cite{Patterson13}, here we consider a CS model that minimizes the least squares loss over a sparsity inducing DC constraint described in Remark~\ref{cssparsity} as follows:
\begin{equation}\label{prob:cs}
\begin{array}{cl}
\min\limits_{x\in{\rm I\!R}^{n}} & \frac12\|Ax-b\|^2\\
\text{s.t.} & \|x\|_1-\mu\|x\| \leq \sigma,
\end{array}
\end{equation}
where $ \mu \in (0, 1) $ and $ \sigma > 0 $. Recall from Section~\ref{sec:scsubp} that the DC constraint function in \eqref{prob:cs} can be trivially transformed to the difference of two strongly convex functions with given $ \rho>0 $. Here, we also consider the following \textit{equivalent} formulation of \eqref{prob:cs}:
\begin{equation}\label{prob:cs-sc}
\begin{array}{cl}
\min\limits_{x\in{\rm I\!R}^{n}} & \frac12\|Ax-b\|^2\\
\text{s.t.} & \left[\|x\|_1+\frac{\rho}{2}\|x\|^2\right]-\left[\mu\|x\|+\frac{\rho}{2}\|x\|^2\right] \leq \sigma.
\end{array}
\end{equation}
Observe that Assumption~\ref{assumption1} holds for \eqref{prob:cs} with $P_1 = \|\cdot\|_1$ and $P_2 = \mu \|\cdot\|$: Indeed, it suffices to take $x^\odot_{(y,\xi)} = 0$ for any $y \in {\cal F}$ and $\xi\in \mu\partial \|y\|$.
Now, using this, Proposition~\ref{slaequ}(ii) and the fact that $ -\mu\|\cdot\| - \frac{\rho}2\|\cdot\|^2 $ is regular at every $ x\in{\rm I\!R}^n\setminus\{0\} $, we deduce further that Assumption~\ref{assumption1} is also satisfied for \eqref{prob:cs-sc} if we let $ P_1=\|\cdot\|_1+\frac{\rho}{2}\|\cdot\|^2 $ and $ P_2 = \mu\|\cdot\|+\frac{\rho}{2}\|\cdot\|^2 $. Thus, we can apply Theorem~\ref{thm-fw} to conclude that the sequence generated by Algorithm~\ref{FW_nonconvex} (applied to \eqref{prob:cs} and \eqref{prob:cs-sc} with the aforementioned $P_1$ and $P_2$) clusters at a stationary point of \eqref{prob:cs}. Moreover, the convergence of Algorithm~\ref{alg:aw-fw} applied to \eqref{prob:cs} also follows from Theorem~\ref{thm-fw-aw}.
In the remainder of this section, we perform numerical experiments to study the performances of Algorithm~\ref{FW_nonconvex} and Algorithm~\ref{alg:aw-fw} for solving randomly generated instances of \eqref{prob:cs}. Specifically, we apply Algorithm~\ref{FW_nonconvex} to \eqref{prob:cs} directly (i.e., with $P_1 = \|\cdot\|_1$ and $P_2 = \mu \|\cdot\|$) as well as to \eqref{prob:cs-sc} (i.e., with $ P_1=\|\cdot\|_1+\frac{\rho}{2}\|\cdot\|^2 $ and $ P_2 = \mu\|\cdot\|+\frac{\rho}{2}\|\cdot\|^2 $) for $ \rho = 0.01$, $0.05$ and $0.1$, respectively: For ease of reference, we refer to these approaches as FW, $ {\rm FW}_{\rho_{0.01}} $, $ {\rm FW}_{\rho_{0.05}} $ and ${\rm FW}_{\rho_{0.1}}$, respectively. We also apply Algorithm~\ref{alg:aw-fw} to \eqref{prob:cs} and refer to it as AFW.\footnote{Note that the discussion in Section~\ref{sec:aw3exs} requires that $P_1$ satisfies Assumption~\ref{assump31} or \ref{assump32}, which does not cover the decomposition considered in \eqref{prob:cs-sc}. Thus, we do not consider away-step oracle for algorithms based on \eqref{prob:cs-sc}.} All numerical experiments are performed in MATLAB 2019b on a 64-bit PC with an Intel Core i7-6700 CPU (3.40GHz) and 32GB of RAM.
\paragraph{Algorithm settings.}
We initialize all algorithms at the origin, which is feasible for \eqref{prob:cs} (also for \eqref{prob:cs-sc} with any $\rho > 0$). We set $ c=10^{-4} $, $ \eta = 1/2 $ and choose $\alpha_0^0 = 1$ and, for $k\ge 1$,
\begin{equation*}
\alpha_k^0 = \begin{cases}
\max\left\{10^{-8}, \min\left\{2\alpha_{k}^{\rm fw}, 1\right\}\right\} & \;\text{if}\; d^{k-1} = d^{k-1}_{\rm fw} \;\text{and}\; j_{k-1} = 0,\\
\max\left\{10^{-8}, \min\left\{\alpha_{k}^{\rm fw}, 1\right\}\right\} & \; \text{otherwise},
\end{cases}
\end{equation*}
where $ \alpha_{k}^{\rm fw} $ is the stepsize used in the most recent FW step.\footnote{Note that by our choice of $\xi^k$ below and the discussion right before Section~\ref{sec:syxi}, we must have $d^0 = d^0_{\rm fw}$ in AFW.} In the tests of FW and AFW for \eqref{prob:cs}, we compute a closed-form solution of $ {\cal LO}$ through \eqref{xstar} with $ a = A^T(Ax^k - b) $ and $ \xi = \xi^k:= \mathop{\rm arg\,min}_{u\in \mu\partial\|x^k\|}\{\|u\|\} $ in each iteration. Moreover, when applying AFW for solving \eqref{prob:cs}, we set $ \epsilon = 10^{-5} $ and $ \zeta=10^{5} $, and we compute a solution of $ {\cal AWO}(x^k, \mathcal S(x^k, \xi^k)) $ through \eqref{uaw} with $ a = A^T(Ax^k - b) $, $ \xi = \xi^k:= \mathop{\rm arg\,min}_{u\in \mu\partial\|x^k\|}\{\|u\|\} $ and $ \mathcal I_{y}=\left\{i:\;x^k_i\neq 0\right\} $. Regarding the $ {\cal LO}$ in $ {\rm FW}_{\rho_{0.01}} $ (and, similarly, in $ {\rm FW}_{\rho_{0.05}} $ and $ {\rm FW}_{\rho_{0.1}} $) for solving \eqref{prob:cs-sc}, we compute its solution by using \eqref{xprox} with $ a= A^T(Ax^k - b)$, $ \xi = \xi^k:= \rho x^k + \mathop{\rm arg\,min}_{u\in \mu\partial\|x^k\|}\{\|u\|\}$ and $\widetilde P_1 = \|\cdot\|_1 $. Moreover, leveraging the positive homogeneity of $ \|\cdot\|_1-\mu\|\cdot\| $, we adopt the strategy in \eqref{choosexk+1} to determine $ x^{k+1}$ in all the five algorithms. Finally, we terminate these algorithms if one of the following criteria is satisfied:
\begin{itemize}
\item the number of iterations exceeds 10000;
\item the computational time exceeds a given upper bound $ {\rm T}^{\max} $;
\item $\lvert\langle A^T(Ax^k - b), d^k_{\rm fw}\rangle \rvert < 10^{-10} \cdot\max\{\frac12\|A x^k - b\|^2,1\}$.
\end{itemize}
\paragraph{Randomly generated instances.}
We first generate a matrix whose entries follow i.i.d. standard Gaussian distribution. We then normalize each column and take it as the sensing matrix $A$ in \eqref{prob:cs}. Next, we generate an $s$-sparse vector with $s$ i.i.d. standard Gaussian entries at random (uniformly chosen) positions and let it be the original signal $x_{\rm orig}\in\mathbb{R}^{n}$. Then we set $ b = Ax_{\rm orig}+0.01\varepsilon $, where $ \varepsilon\in{\rm I\!R}^{m} $ has i.i.d. standard Gaussian entries. Finally, we set $ \mu =0.5 $ and $ \sigma=\left(\|x_{\rm orig}\|_1-\mu\|x_{\rm orig}\|\right)/1.1 $.
In our tests, we consider $ (n, m, s) = (2560i, 720i, 80i) $ with $ i\in\{1, 3\} $. For each $ i = 1$ and $3$, we set $ {\rm T}^{\max} = 5$ and $30$ seconds, respectively, and generate 30 random instances as described above.
We observe that the algorithms solving \eqref{prob:cs-sc} (i.e., $ {\rm FW}_{\rho_{0.01}} $, $ {\rm FW}_{\rho_{0.05}} $ and $ {\rm FW}_{\rho_{0.1}} $) generally take more time in each iteration than the FW and AFW for \eqref{prob:cs}, but they satisfy the third termination criterion in fewer iterations if $ \rho $ is properly chosen. To better illustrate the performance of these algorithms, we follow \cite{Gillis12, YangPongChen18} and plot an evolution of the objective function value relative to the initial value and the ``best" objective value achievable. To this end, we first define the following ratio for each $ k $:
\[
e(k) = \frac{f_k - f_{\min}}{f_0 - f_{\min}},
\]
where $ f_k = \frac12\|Ax^k - b\|^2 $, $ f_0 = \frac12\|Ax^0 - b\|^2 $ and $ f_{\min} $ is the minimum of the obtained objective function values at termination among all 5 algorithms. Note that $ e(k)\in[0,1] $ for all $ k $ and it decreases with respect to $ k $ since $ f_k $ decreases with respect to $ k $ in all algorithms. Let $ \mathcal T(k) $ denote the total computational time used by the first $ k $ iterations. We next define the evolution of the (relative) objective function value with respect to the computational time as follows:
\[
E(t) = \min\limits_k\{e(k):\; \mathcal T(k)\leq t \}.
\]
Notice that $ E(t)\in[0, 1] $ for any $ t\in [0, {\rm T}^{\max}] $. For $ i=1$ and $3$, we plot the averaged $ E(t) $ for each algorithm, averaged over the 30 random instances, in Figure~\ref{fig:fig1}. From the plots, we see that $ {\rm FW}_{\rho_{0.1}} $ performs best among all algorithms, and the performance of the approaches based on solving \eqref{prob:cs-sc} decays as $\rho$ decreases. On the other hand, by comparing the performances of FW and AFW, we see that our proposed away-step technique does improve the performance of FW empirically.
\begin{figure}
\caption{Plots of the evolution of the averaged $E(t)$.}
\label{fig:fig1}
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[scale=0.45]{FvalTimeTrace_index_1-24-Dec-2021-11-26-eps-converted-to.pdf}
\end{subfigure}
\begin{subfigure}{.4\textwidth}
\centering
\includegraphics[scale=0.45]{FvalTimeTrace_index_3-24-Dec-2021-11-26-eps-converted-to.pdf}
\end{subfigure}
\end{figure}
\vspace{3mm}
{{\bf Acknowledgements.} Yongle Zhang was supported partly by the National Natural Science Foundation of China (11901414) and (11871359). Guoyin Li was partially supported by discovery projects from Australian Research Council (DP190100555 and DP210101025). Ting Kei Pong was supported in part by the Hong Kong Research Grants Council PolyU153004/18p.}
|
2024-02-18T23:40:24.667Z
|
2021-12-30T02:21:15.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14404","language":"en","timestamp":1640830875000,"url":"https:\/\/arxiv.org\/abs\/2112.14404","yymm":"2112"}
|
proofpile-arXiv_000-10060
|
{"provenance":"002.jsonl.gz:10061"}
| null | null |
\section{Introduction}
Color image steganography is one of the essential branches of information hiding. It is widely used in the military, medicine, intellectual property protection, and other fields. Through human visual characteristics, color image steganography can hide secret information (usually bitstream or text) into a color image. We call the original color image a cover image, and the modified color cover image a stego image. The stego image is almost the same as the cover image, so it is not easy to arouse the suspicion of the third party and shows good imperceptibility. In recent years, the image hiding steganography based on deep learning has emerged. Image hiding steganography can hide the image as secret information into a cover image. This kind of steganography improves the steganographic capacity and greatly expands the application scope of steganography.
Nowadays, image hiding steganography based on deep learning is a popular research direction in steganography. Image hiding steganography should have a large steganographic capacity and good imperceptibility to obtain a better application effect. However, there is a strong negative correlation between capacity and imperceptibility. The increase of steganographic capacity usually leads to a decrease in imperceptibility\cite{r1}.Therefore, it is difficult for the current image hiding steganography based on deep learning to take into account these two characteristics simultaneously\cite{r2}. As shown in Figure \ref{fig:1-1} , when the secret information is the same size image, the stego image may generate visible color distortion and artificial texture traces, which do not meet the imperceptibility requirement of image steganography. Therefore, currently controlling the steganographic traces in the invisible range of human eyes is the point of image hiding steganography. As shown in Figure \ref{fig:1-2}, some researchers tried to use the characteristics of color spaces and embedded the secret image into the channel, which does not relate to color information. This idea effectively reduces the color distortion of color stego images\cite{r4}. More than this, the idea provides researchers with a new path: to use the characteristics of color spaces to improve the imperceptibility of image steganography based on deep learning and make it fit the visual characteristics of the human eyes. Nevertheless, the influence of color spaces characteristics still needs to be verified in different deep learning steganography.
\begin{figure}[t]
\centering
\includegraphics[width=1.0\textwidth]{1-1.jpg}
\caption{\label{fig:1-1}We mark and enlarge the color distortion (a) and artificial texture traces (b) area in stego images generated by \cite{r3}.}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=1.0\textwidth]{1-2.jpg}
\caption{\label{fig:1-2}Four groups of stego images generated by the model of \cite{r4}. This model uses the characteristics of YUV color space to improve the quality of stego images. There is only a slight difference in brightness between stego images and original images.}
\end{figure}
According to the research results of traditional steganography, the embedding domain will have a significant impact on the steganographic visual effect\cite{r5}\cite{r6}\cite{r7}. According to the difference of embedding domain, image steganography can be divided into two categories: The first type is the spatial domain steganography, which directly modifies the pixels in the image space. The second type is frequency-domain steganography, which uses the frequency domain transformation method to convert images into frequency sub-bands and modify them. Contrary to image information in the spatial domain, the frequency domain sub-bands are information human eyes can not directly observe. Therefore, in the process of frequency domain steganography, the steganographic traces of stego images is invisible if the modification of frequency sub-band is controlled\cite{r8}. Unlike traditional steganography, image hiding steganography based on deep learning primarily hides secret data in the spatial domain of the cover image. The information in the spatial domain is directly visible to the human eyes. Therefore, if modified this information directly, it is easy to create a color distraction or artificial texture traces. Combined with the research results of traditional image steganography, we believe that image hiding steganography based on frequency domain will show a better imperceptibility through the learning of the deep neural network.
Based on the above inference, we try to embed a secret image into the frequency domain of a cover image to obtain better image quality. On this basis, the influence of different frequency sub-bands and color spaces on frequency domain steganography based on deep learning will be further discussed. In terms of frequency sub-bands, the characteristics of some frequency domain transform methods(such as DWT) are consistent with the characteristics of the human visual system (HVS)\cite{r9}. Hence Some traditional frequency domain steganography further improves the imperceptibility by using different frequency components and characteristics of HVS\cite{r10}. Similarly, we can further improve the imperceptibility of steganography by using the characteristics of image frequency sub-bands. In terms of the characteristics of color spaces, different color spaces will express image information in different ways. Hence they will produce different effects on steganography\cite{r11}. Moreover, each channel also has different visual characteristics in the same color space. Therefore, controlling each channel for steganography will also produce different steganography effects \cite{r12}. According to the characteristics of color spaces and the degree of image distortion, this paper selects the channel with the best steganographic effect as the embedding channel to further improve the imperceptibility of the model.
At present, image hiding steganography faces the problems of color distortion and artificial texture traces. According to the characteristics of color spaces and frequency sub-bands, a color image hiding steganography model based on frequency sub-band selection is proposed in this paper. The model is based on the deep neural network of encoder-decoder structure. Firstly, DWT is used to transform the secret image and the B channel of the cover image into the frequency domain. Then to reduce the color distortion and texture modification trace of the stego image, the high-frequency sub-band of the cover image is selected as the embedding domain. In order to keep a good balance between capacity and efficiency, we choose a gray image as the secret image and a color image as the cover image. The experimental results show that compared with other deep learning image steganography models based on spatial domain, the proposed model significantly improves image quality, reduces color distortion and artificial texture traces. The imperceptibility of the model is significantly improved. In summary, the main contributions of this paper are:
\begin{enumerate}
\item We propose a steganographic model in the frequency domain. Our model has better imperceptibility than the same category color image steganographic model in the spatial domain.
\item We analyze the influence of color spaces characteristics on the proposed model, find the most suitable embedding channel according to the characteristics of HVS and the principle of minimizing distortion.
\item Combined with the channel selection, we add the frequency sub-band selection module in the proposed model. This module selects the high-frequency sub-band as the embedding domain, which effectively improves the imperceptibility. Moreover, the relationship between frequency sub-bands characteristics and the steganographic effect is discussed.
\item To better compare the color difference between the original image and the stego image, we propose a new color image quality metric called CL-PSNR. CL-PSNR takes the color difference into account, which is more in line with the feeling of HVS to color images.
\end{enumerate}
The rest of the paper is organized as follows: In Section 2, we discuss the existing research results in related directions. In Section 3, we describe the structure of the model, the discussion of color spaces and imperceptibility, the selection of frequency sub-band, and the new color image quality metric. In Section 4, we present and analyze the experimental results. In Section 5, we conclude this work.
\section{Related Work}
Steganography is a technology of hiding information, which originated in ancient Greece and has a long history. The proposed steganography is the image hiding steganography based on deep learning, which means the steganography is based on deep learning knowledge and hiding the image with the image. Hence this section will introduce the development of image steganography based on deep learning. The representative achievements and problems of image hiding steganography will be discussed in detail. Moreover, the traditional image steganography will be used as a reference to improve imperceptibility.
\subsection{Image Steganography Based on Deep Learning}
In recent years, many image steganography methods based on deep learning have been proposed. ASDL-GAN was one of those earlier methods\cite{r13}. Although ASDL-GAN does not directly participate in the embedding and extraction process, its results inspire more scholars to improve the performance of steganography by using deep learning. Zhu et al. proposed Hidden, which first used a neural network for end-to-end training and has stronger robustness than traditional methods\cite{r14}. Hayes et al. proposed a HayesGAN to use an encoder network to hide text information in the image \cite{r15}. Tancik of Berkeley University proposed a StegaStamp framework based on the U-net structure \cite{r16}. They used a series of image enhancement methods to make the model robust in the real world. Huang et al. introduced the adversarial strategy in model training to better resist steganalysis\cite{r17}. Compared with the mature and stable traditional image steganography, the advantage of deep learning image steganography is to use the deep neural network to learn the steganographic method independently. Traditional image steganography techniques rely on artificial design, so it is difficult to improve the performance further. Therefore, the introduction of deep learning broke the deadlock. Moreover, with the development of deep learning technology, image steganography based on deep learning will have more space to develop.
\subsection{image hiding steganography}
In the early stage of deep learning image steganography, most secret information is a string or a bitstream, and the steganographic capacity is small (mostly 0.2bpp-4bpp)\cite{r2}. Fortunately, the image hiding steganography appeared. Image hiding steganography based on deep learning uses the experience of other deep learning steganography. It uses the image to hide the image, which significantly increases the capacity. The deep steganographic model proposed by Baluja is the first model using two same-size images\cite{r3}. It extracts the secret image features through a preprocessing network, then hides a secret image into a cover image by a hidden network, and finally extracts it by an extraction network. This model significantly increases capacity (24bpp). After that, Baluja tried to embed two secret images into another image and confuse the specific information of secret images\cite{r18}. This method improves capacity and enhances the security of secret information. Wu et al. proposed the StegNet. It is based on a deep neural network, and it can also hide the same size image, which only modifies 0.76\% cover image\cite{r19}. Van et al. proposed a new training scheme, which modifies the error back propagation to speed up the training of the network\cite{r20}. The above results use color images as secret information. However, gray image is also common secret information in image hiding steganography. Compared with the color image, gray image only contains semantic information, and its capacity is only one-third of the color image. The characteristics make the gray image more suitable as secret information in specific tasks. Rehman et al. proposed a steganographic model based on the coder-decoder network\cite{r21}. The model can embed a gray image into an RGB color image. Moreover, a new loss function is introduced to ensure the end-to-end joint training of the encoder-decoder network. Experiments show that the model can still obtain good image quality with a large capacity(8bpp).
The image hiding steganography based on deep learning significantly improves the steganographic capacity, but its steganographic traces(color distortion or artificial texture traces) is visible to human eyes. Zhang et al. proposed ISGAN\cite{r4} to solve the problem of color distortion, which converts the RGB cover image into YUV color space, and then selects the Y channel of the cover image as the embedding domain to avoid the modification of color information. The model effectively reduces color distortion. However, slight brightness differences still exist. Hence there is still room for improvement of imperceptibility.
Most of the image steganography methods based on deep learning are operate in the spatial domain. Unlike this kind of steganography, traditional image steganography can be carried out in both spatial and frequency domains. One of the most typical spatial domain steganography is to replace the least significant bit (LSB) of pixels with secret information\cite{r22}.On the other hand, Outguess\cite{r23} and J-UNIWARD\cite{r24} are typical frequency domain steganography. In traditional image steganography, frequency domain steganography usually obtain better robustness and can resist the attack of steganalysis more effectively. Moreover, thanks to the characteristics of frequency transformation, some frequency domain steganography use the HVS characteristics to process frequency domain information to improve the visual quality of the generated image\cite{r9}. There is almost no deep learning image steganography using frequency domain to hide the image. Therefore, relying on the powerful learning ability of deep neural networks and advantages of the frequency domain, the performance of image steganography is expected to be further improved. Although image hiding steganography based on deep learning effectively improves capacity, there are still noticeable color distortion and artificial texture traces. Inspired by traditional frequency domain steganography, this paper attempts to establish a frequency domain color image steganography model based on deep learning. We use the characteristics of color spaces and frequency sub-bands to improve imperceptibility further. Our model shows significant imperceptibility when keeping a large capacity compared with other models.
\section{Methodology}
We aim to establish a color image steganographic model using a deep neural network based on the frequency domain. We explore methods to improve imperceptibility according to characteristics of color spaces and frequency sub-bands. The proposed model mainly considers how to improve imperceptibility in the case of large capacity (8bpp). Hence the robustness is not discussed in this paper. Our model is a fully convolutional encoder-decoder network. The key to improving imperceptibility is adding the frequency sub-band selection module. The model inputs are a color cover image and a secret gray image. Encoder and Decoder networks output a color stego image and a reconstructed gray secret image. The structure of the proposed model is shown in Figure \ref{fig:3-1}.
\begin{figure}[h]
\centering
\includegraphics[width=1.0\textwidth]{3-1.png}
\caption{\label{fig:3-1}The structure of the color image steganographic model based on frequency sub-band selection. The model consists of a frequency sub-band selection module, encoder network, and decoder network. The encoder network includes a pre-processed network for feature extraction of the secret image.}
\end{figure}
The frequency sub-band selection module pre-processes the cover image. The module selects the B channel as the embedding channel and separates the diagonal high-frequency sub-band (cD). The frequency sub-bands of secret image ([sA, sH, sV, sD]) and cD are the inputs of the encoder network. The encoder network first extracts the features from [sA, sH, sV, sD] and then encodes with cD to hide the secret image. For the extraction of secret information, the decoder network receives the diagonal high-frequency sub-band (cD') as input. Finally, the decoder network outputs the reconstructed secret image through the decoding and inverse DWT.
\subsection{Why B Channel}
In order to ensure that the stego image has a quality closer to the original cover image, we hope to retain the original image information as much as possible. Therefore, the proposed model only selects one channel of cover image as the embedding channel, which means only 1/3 of the cover image is modified. The data to process is reduced, and the learning efficiency of the network is improved.
Considering the influence of color space characteristics, we need to find the optimal embedding channel to obtain the minimum image distortion and the best visual effect. Many steganographic models tried to improve the quality of the generated image by using channels independent of color information (for example, ISGAN uses the Y channel of YUV color space). Therefore, we select the embedding channel in several typical color spaces according to channel characteristics to view the image quality generated by the proposed model. The experimental results are shown Figure \ref{fig:3-2}.
\begin{figure}[t]
\centering
\includegraphics[width=1.0\textwidth]{3-2.png}
\caption{\label{fig:3-2}A group of steganographic examples based on different embedding channels. The left column is the original cover image and the secret image (the second row) The others are the stego images and reconstructed secret images generated in the corresponding color space.}
\end{figure}
We use the proposed model to hide information in different color spaces. We select the common nonlinear luminance/chroma color space (YUV, LAB), intensity/saturation/hue color space (HSV), and the original color space (RGB) and select a channel for embedding in each color space. Since modifying the brightness channel is more imperceptible, we give priority to the brightness channel for YUV, LAB, and HSV. The three channels of RGB color space represent three primary colors, so we hid the image in R, G, and B channels, respectively. As shown in Figure 4, all generated images are almost the same as the cover image. Therefore, the steganographic quality of the proposed model is hardly affected by the characteristics of color space.
It can be seen from previous studies that if the modification is visible, the sensitivity of vision to modification can be reduced by using a brightness channel as an embedding channel \cite{r4}. Vision is a subjective judgment. Hence we use the JND (the Just Notice Difference) to calculate the visual detection threshold. JND is the largest image distortion imperceptible to vision, which can judge whether the modification of the cover image reaches the threshold that human eyes can perceive. According to the calculation\cite{r25}, we can see that the JND threshold of the original cover image in Figure \ref{fig:3-2} is 7.46. However, the modification of stego images is far less than the threshold (the maximum modification is 1.30). Therefore, the modification caused by our model does not reach the range that human eyes can perceive. Moreover, in the following experimental part (4.2), we use the new metric proposed in this paper, CL-PSNR, closer to human visual perception to evaluate the quality of stego images. The results show that the CL-PSNR values of different color space experiments remain in the same range of good imperceptibility. Therefore, using other color spaces in our model can not effectively improve imperceptibility but will add unnecessary computational tasks and loss of accuracy (mostly from space transformation operations). Overall, our model uses the original RGB color space.
Although three stego images generated based on R, G, and B channels are almost the same in Figure \ref{fig:3-2}, we still select the embedding channel according to the principle of minimizing distortion. We calculate the error per pixel of each stego image, in which the image distortion from the model based on the B channel is the slightest(error per pixel is 0.66 BPP). Moreover, human vision is less sensitive to the modification of channel B than the other two color channels\cite{r26}. Hence B channel is also the best choice.
We select the B channel of RGB color space as the embedding channel based on the steganographic results in each color space. Based on the B channel, the stego image is visually close to the original image and has less distortion than that embedded in other channels.
\subsection{Selection of Frequency Sub-band}
After determining the embedding channel, the frequency sub-band selection module further selects a frequency sub-band as an embedding area.
First, the B channel information in the spatial domain will be transformed into frequency domain information. At present, DCT(discrete cosine transform) and DWT are the most commonly used methods for frequency-domain transformation. Compared with DCT, DWT does not produce a blocking effect and can restore images more accurately. Moreover, DWT can also select different wavelet functions according by actual needs to decompose the image signal at different scales\cite{r27}. Therefore, to minimize the accuracy loss caused by frequency-domain transformation, we choose the DWT method to process B channel information. After experiments, we choose the dmey wavelet, which has a better network learning effect. In the complete steganographic process, the spatial-frequency domain transformation of images needs to be carried out twice, used for the data input of Encoder and Decoder. Similarly, the frequency-spatial domain transformation also needs to be carried out twice, used for the output of Encoder and Decoder.
In order to further improve the imperceptibility, we select only part of the frequency information of the cover image as the embedding domain instead of modifying all sub-bands. The secret information is a gray image of the same size as the cover image. Hence it is not easy to hide information and maintain good invisibility in a limited embedding domain. In order to ensure that the generated stego image is as close as possible to the original image, we hope that the embedding domain has as little impact on the whole image as possible. After DWT, We will get four frequency sub-bands, [cA, cH, cV, cD]. Where cA is a low-frequency sub-band, corresponds to the region with smooth gray transformation and simple texture in the image. Besides, cH, cV, and cD are high-frequency sub-bands corresponding to the image region with intense gray transformation and complex texture. Although the four frequency sub-bands have the same dimension, most of the energy of the image will be concentrated on cA. On the contrary, cD only contains a small amount of detailed information. It means that the embedding area size provided by cD is the same as other sub-bands. However, with the same modification, its impact on the original image information is the smallest.
Figure \ref{fig:3-4} shows images after the same modification on different frequency sub-bands. We set all values in cA and cD to 0. The modification percentage of (a) (which means the percentage of pixels with value error larger than 5 bpp in the whole image) is as high as 99.39\%. In contrast, the modification percentage of (b) is only 10.97\%, and the error range of (b) is smaller than that of (a). Compared with the original image, the maximum pixel value error of (b) is 47.52, while the maximum pixel value error of (a) is as high as 255 (the maximum gray value). Incidentally, performing the same modification on cV and cH, the modification percentage is about 22\%. Although their visual results is better than (a), cD is still the best embedding area choice.
\begin{figure}[h]
\centering
\includegraphics[width=0.4\textwidth]{3-4.png}
\caption{\label{fig:3-4} (a) It is the result of setting the values of cA to 0. Almost all information in the original image is lost. (b)Tt is the result obtained by setting the value of cD to 0, which is almost the same as the original image.}
\end{figure}
In addition to minimizing the modification range of the cover image, the characteristics of HVS are also an important reason we select cD. As shown Figure \ref{fig:3-5}, cA contains almost all the semantic information of the image. Low frequency often corresponds to gently transformed gray areas in the image, such as white walls, sky, etc. Therefore, it is easier for human eyes to detect the modification on cA, which does not fit the imperceptibility requirements of image steganography. Therefore, from the characteristics of HVS, cA is not suitable as an embedding domain. Unlike cA, other sub-bands contain detailed information such as the edge contour of the image. According to HVS characteristics, the vision is relatively insensitive to the modification of areas with complex texture, high contrast, or very bright/dark areas in the image \cite{r28}. Based on this property, we separate the diagonal high-frequency sub-band (cD) as the embedding domain. The cH and cV still contain low frequency information in one direction, so they are not considered embedding domains.
\begin{figure}[h]
\centering
\includegraphics[width=0.6\textwidth]{3-5.png}
\caption{\label{fig:3-5}The process of frequency sub-band selection module. Firstly, the B channel is separated, and then DWT operation is performed on the B channel to obtain frequency sub-bands. Finally, the model selects cD as the embedded area.}
\end{figure}
Figure \ref{fig:3-5} shows the process of the frequency sub-bands selection module. According to the influence of frequency sub-bands and the characteristics of HVS, the module selects cD as the embedding domain to reduce the impact of embedding on image quality. Therefore, the generated image has fewer texture traces and color distortion.
After embedding, the output of the encoder network will combine with other frequency sub-bands and be transformed into the spatial domain through inverse DWT. According to the embedding process, we could not measure the embedding position and modification of the stego image in the spatial domain. Therefore, we use the heat map to reflect the modification of the proposed model. Figure \ref{fig:3-7} shows a cover image, a stego image, and a heat map based on the difference between the two images.
\begin{figure}[h]
\centering
\includegraphics[width=0.6\textwidth]{3-6.png}
\caption{\label{fig:3-6}The original image, the density image generated by the model, and the heat map based on the two image errors.}
\end{figure}
The heat map reflects the modification from steganography. In the heat map, red indicates that the pixel value of this position has increased, and blue indicates that the pixel value has decreased. The heat map has both red and blue areas, which means the model does not modify the cover image in only one direction. Since the color block in the heat map shows the outline of the cover image, we speculate that the network modification is determined by the data features of cD. Although almost all pixels in the heat map are colorful, the deviation of the overall pixel value is still within an acceptable range (in the example image, the maximum error is 14 bpp). As shown in Figure \ref{fig:3-6}, the stego image has a high image quality, which is almost the same as the cover image.
\subsection{Model Architecture}
After the pre-processing of the cover image, the model uses the encoder-decoder network to hide and extract the secret image. Encoder-decoder network is a common network structure in spatial domain steganography. We choose this network to explore whether it can learn the steganographic process in the frequency domain. Compared with other spatial domain steganographic models, the steganographic process is roughly the same, but the data transmitted in our model is frequency domain information.
\begin{figure}[h]
\centering
\includegraphics[width=1.0\textwidth]{3-7.png}
\caption{\label{fig:3-7}Detailed configuration of the encoder network (using 250×250 images). The blue blocks are convolution layers, which are used to extract the features of secret image and embedding.}
\end{figure}
First, the encoder network embeds the secret image into cD. Encoder network is mainly composed of three groups of convolution layers, connection layers, and the input/output layer. The encoder network receives two inputs: the input layer 1 receives the frequency sub-bands [sA, sH, sV, sD], which are obtained after DWT of the secret image. The input layer 2 receives cD from the frequency sub-band selection module. The first two groups of convolution layers pre-process [sA, sH, sV, sD] to extract the data features of the secret image. The output feature maps are connected with cD, and the last group of convolution layers is used for coding.
The encoder network outputs the diagonal high-frequency sub-band (represented by cD'), which contains the frequency information of the secret image. Therefore, the stego image still needs to be obtained through the inverse operation of the frequency sub-band selection module.
The decoder network is mainly composed of a group of convolution layers, connection layers and the input/output layer. After receiving the stego image, the decoder network separates the cD' according to the frequency sub-band selection module. Input layer 3 receives cD'. Convolution layers perform a decoding operation to extract the frequency information of secret image (represented by [sA', sH', sV', sD']). Finally, the inverse DWT is used to transform the [sA ', sH', sV', sD'] into spatial domain information, that is, to obtain the reconstructed secret image.
\begin{figure}[h]
\centering
\includegraphics[width=0.6\textwidth]{3-8.png}
\caption{\label{fig:3-8}Detailed configuration of the decoder network(using 250×250 images). The blue blocks are convolution layers, which are used to extract secret image.}
\end{figure}
In order to minimize the difference between the stego image and the original image, the secret image and the reconstructed secret image, the loss function we use is the MSE loss function calculated based on frequency domain:
\begin{equation}
\centering
\text Loss(c,c',s,s') = \Vert{c-c'}\Vert^2+\beta \Vert{s-s'}\Vert^2
\end{equation}
In equation (1), $c$ and $c'$ represent the frequency domain information of the cover image and stego image, $s$ and $s'$ represent the frequency domain information of the original secret image and reconstructed secret image. In the encoder-decoder structure, image information is transmitted in the form of frequency information. Hence the loss function is also calculated in the frequency domain. The parameter $\beta$ is used to adjust the weight of the decoder network. Although the MSE loss function has achieved a good result in the proposed model, we still expect to design a loss function based on HVS characteristics in the future, which achieves a better imperceptibility.
We call the encoder-decoder network a basic model. In early experiments, we found that the steganographic effect of the basic model is better than that of the spatial domain steganographic model with the same structure(Table 2). This not only proves that the encoder-decoder network is also suitable for the deep learning steganography in the frequency domain, but also proves that the frequency domain is more conducive to network learning. Based on the basic model, we add a frequency sub-band selection module to form the final model, which further improves the imperceptibility of steganography.
\section{Experiments}
In this section, we designed a series of ablation experiments on color spaces and frequency sub-bands, determined the final model based on performances of imperceptibility. In the deep learning steganographic model of hiding a gray image with a color image, we choose Antique's model\cite{r21} and ISGAN\cite{r4} to compare with our final model. Finally, we verified the data generalization ability of the final model.
\subsection{Implementation Details}
\subsubsection{Dataset}
We mainly used the Labeled Faces in the Wild (LFW) dataset during our experiments. This dataset provides more than 13k face images which sizes are reshaped to 250×250. Due to multi posture, illumination, expression, age, occlusion, and other factors, even the photos of the same person are very different. We randomly selected 4K images as the training set and 200 images as the testing set. The training set and testing set are equally divided into cover images and secret images. Secret images are uniformly processed into gray images. In addition, ILSVRC and PASCAL VOC 2012 are also used for model training to verify the final model has a good generalization ability.
\subsubsection{Image Quality Evaluation Metric}
Firstly, we chose the widely used image quality evaluation metrics: PSNR (Peak Signal to Noise Ratio) and SSIM (Structural Similarity) to measure the quality of generated stego image and reconstructed secret image.
During experiments, we found that PSNR sometimes can not directly reflect the perception of vision (especially when evaluating the quality of color images). In Figure 10, a and b are the stego images obtained by the proposed model in two different color spaces. For the observer, it is difficult to detect the difference between a and b, but the result difference of PSNR is more than 30. We believe that this is because PSNR is a metric designed for gray image quality evaluation. Therefore, when evaluating the quality of color image, PSNR sometimes gives results different from human eyes perception.
\begin{figure}[h]
\centering
\includegraphics[width=0.6\textwidth]{4-1.png}
\caption{\label{fig:4-1}Image a and b both have almost the same similarity with the cover image, but there is a large difference in their PSNR values. Compared with PSNR, CL-PSNR more fit with human visual perception.}
\end{figure}
For the limitations of PSNR, we introduce CIE color-difference formula\cite{r29} and propose a new image quality evaluation metric named CL-PSNR. We first calculate the color difference between generated image and the original image, then normalize it and calculate the average value to replace the MSE in the original PSNR equation, which is represented by CL-MSE.
\begin{equation}
\centering
\Delta E_{(x,y)} = \sqrt{ \left(\frac{\Delta L_{(x,y)}^{*} }{K_{L} S_{L}}\right)^2 + \left(\frac{\Delta C_{(x,y)}^{*} }{K_{C} S_{C}}\right)^2 + \left(\frac{\Delta H_{(x,y)}^{*} }{K_{H} S_{H}}\right)^2 }
\end{equation}
\begin{equation}
\centering
CL-MSE = \frac{\sum_{n=1}^{N} \frac{\Delta E_{n} }{MAX} }{N}
\end{equation}
Equation (2) calculates the CIE94 color-difference values of pixels $(L_{x}, a_{x}, b_{x})$ and $(L_{y}, a_{y}, b_{y})$, where $\Delta L_{*}$, $\Delta C_{AB}^{*}$, and $\Delta H_{AB}^{*}$ represent the lightness difference, saturation difference, and hue difference. $K_{L}$, $K_{C}$ and $K_{H}$ are parameter factors. $S_{L}$, $S_{C}$, and $S_{H}$ are correction coefficients of brightness, chroma, and hue. The default values for $K_{L}$ and $S_{L}$ are 1, and the values of $K_{C}$, $K_{H}$, $S_{C}$, and $S_{H}$ are adjusted according to different applications. In equation (3), $N$ means the total number of pixels. $MAX$ represents the maximum CIE color difference, rounded to 140.
\begin{equation}
\centering
CL-PSNR = 10 \times log_{10}\left( \frac{(2^{n}-1)^2}{CL-MSE} \right)
\end{equation}
In equation (4), we use CL-MSE to replace MSE in the original PSNR equation, so CL-PSNR will consider the influence of color difference.
Compared with PSNR, CL-PSNR is more fit with human visual perception. In order to further prove it, we select 100 pictures containing four different distortion types in TID2013 dataset\cite{r30} for verification.
\begin{figure}[t]
\centering
\includegraphics[width=0.5\textwidth]{4-2.png}
\caption{\label{fig:4-2}Selected images in TID2013. The distortion types involved JPEG compression, contrast change, color saturation change, and image color quantization with dither.}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.5\textwidth]{4-3.png}
\caption{\label{fig:4-3}CL-PSNR and MOS(Mean Opinion Score) distribution.}
\end{figure}
In Figure \ref{fig:4-3}, the vertical axis coordinate represents the Mean Opinion Score (MOS) given by the TID dataset, that is, the score for image quality obtained through subjective human eye experiment. We calculated the CL-PSNR values of sample images, represented them with points in the distribution diagram, and gave the fitting line. From the distribution trend, we can see that the MOS fraction increases with the increase of CL-PSNR, and CL-PSNR is linearly correlated with the MOS score. Therefore, CL-PSNR is suitable for the evaluation of color image distortion. We will add CL-PSNR as a new metric in experiments to evaluate the quality of the color stego image.
\subsubsection{Neural Network Training Settings}
We use Adam optimizer and train our model for 400 epochs. We set the initial learning rate to 0.001 and decreased to 0.0003 after 150 epochs. After 300 epochs, it decreases to 0.0001. The batch size is 16, and $\beta$ is 1 in the loss function. The model is implemented with TensorFlow, and an NVIDIA Tesla T4 GPU is used.
\begin{figure}[h]
\centering
\includegraphics[width=0.8\textwidth]{4-4.png}
\caption{\label{fig:4-4}Five groups of generated images by the final model. It is difficult for human eyes to distinguish the difference between original images and generated images.}
\end{figure}
Figure \ref{fig:4-4} shows the steganographic effect of the final models. The embedded secret information is a gray image of the same size as the cover image. Hence the capacity of the model is fixed at 8 bpp.
\subsection{Ablation Experiments}
We designed two ablation experiments to verify the effects of color spaces and frequency sub-bands characteristics on the proposed model. We determined the final model, which uses the cD of the B channel as the embedding domain.
\subsubsection{Color Spaces}
The following six experiments are designed to verify the superiority of the B channel as an embedding channel.
From the perspective of imperceptibility, in order to reduce the sensitivity of human eyes to steganographic traces, we select brightness channels for embedding in YUV, LAB, and HSV color spaces. The three channels of RGB color space represent three primary colors, and there is little difference in the sensitivity of human eyes to their modification. Therefore, the last three experiments are embedded in R, G, and B channels, respectively. The experimental results in Table 1 include the results of error per pixel (denoted by C\_Error per pixel and S\_Error per pixel), peak signal to noise ratio (denoted by C\_PSNR and S\_PSNR), and structural similarity (denoted by C\_SSIM and S\_SSIM) of stego images and reconstructed secret images. We also give CL-PSNR values of stego images. The steganographic performances in each color space are given in figure 4 of Section 3.1, which will not be repeated in this section.
\begin{table}[h]
\centering
\begin{tabular}{lllllll}
\hline
\begin{tabular}[c]{@{}l@{}}Color space\\ /Embedding \\ Channel\end{tabular} & \begin{tabular}[c]{@{}l@{}}C\_Error \\ per \\ pixel\end{tabular} & \begin{tabular}[c]{@{}l@{}}S\_Error\\ per\\ pixel\end{tabular} & \begin{tabular}[c]{@{}l@{}}C\_PSNR/\\ CL-PSNR\end{tabular} & S\_PSNR & C\_SSIM & S\_SSIM \\
\hline
YUV/Y & 1.1529 & 3.5101 & 46.89/40.62 & 37.22 & 0.9919 & 0.9790 \\
\hline
LAB/L & 1.3090 & 3.3554 & 45.80/34.36 & 37.61 & 0.9895 & 0.9803\\
\hline
HSV/H & 0.9815 & 3.3464 & 48.29/40.75 & 37.63 & 0.9941 & 0.9803\\
\hline
\textbf{RGB/R} & \textbf{0.7} & \textbf{3.31} & \textbf{82.12/43.37} & \textbf{37.72} & \textbf{0.9971} & \textbf{0.9999} \\
\hline
\textbf{RGB/G} & \textbf{0.69} & \textbf{3.32} & \textbf{82.31/43.67} & \textbf{37.7} & \textbf{0.997} & \textbf{0.9999} \\
\hline
\textbf{\begin{tabular}[c]{@{}l@{}}RGB/B\\ (final model)\end{tabular}} & \textbf{0.66} & \textbf{3.3586} & \textbf{82.31/44.33} & \textbf{37.75} & \textbf{0.9975} & \textbf{0.9999} \\
\hline
\end{tabular}
\caption{\label{tab:1}Comparison of steganographic results based on different color spaces or channels}
\end{table}
The experimental results show that in the above color spaces, the PSNR of stego images from each experiment is greater than 40, and the SSIM value is close to or greater than 0.99. The PSNR value of secret images is greater than 37, and the SSIM value is close to or greater than 0.98. Combined with table 1 and Figure 4, we believe that in the above color spaces, the generated images have the quality close to original images, and our model is hardly affected by the characteristics of color spaces. We believe that this is because, after frequency sub-bands selection, the model further reduces the steganographic range of the cover image. Hence the modification from embedding does not reach the JND threshold of the cover image.
From the first three experiments and the last three experiments, it can be seen that the results of error per pixel, PSNR, CL-PSNR, and SSIM of steganography model based on RGB color space are better than other models. We think this is because the rounding operation of color space transformation may bring additional loss. Based on the principle of distortion minimization, we selected the B channel as the embedding channel of the proposed model. Compared with the modification of the other two color channels, human eyes are more insensitive to the modification of channel B. Moreover, the distortion under B channel is the smallest.
\subsubsection{Frequency sub-band selection}
In this section, we verified the advantages of frequency sub-bands selection. As shown in Table \ref{tab:2}, we compared the steganographic effects of models under different embedding domains based on the B channel. The embedding domain of the four experiments is the global spatial domain, the global frequency domain(cA+cH+cV+cD), the low frequency sub-band (cA), and the diagonal high-frequency sub-band (cD).
\begin{table}[h]
\centering
\begin{tabular}{lllllll}
\hline
Embedding Area & \begin{tabular}[c]{@{}l@{}}C\_Error per \\ pixel\end{tabular} & \begin{tabular}[c]{@{}l@{}}S\_Error per \\ pixel\end{tabular} & \begin{tabular}[c]{@{}l@{}}C\_PSNR \\ /CL\_PSNR\end{tabular} & S\_PSNR & C\_SSIM & S\_SSIM \\ \hline
Spatial Domain & 5.38 & 10.69 & 76.24/27.24 & 27.54 & 0.9225 & 0.7739 \\
\hline
\begin{tabular}[c]{@{}l@{}}Frequency domain\\ (basic model)\end{tabular} & 1.73 & 3.40 & 79.52/38.03 & 37.47 & 0.9923 & 0.9785\\
\hline
\begin{tabular}[c]{@{}l@{}}Low Frequency \\ sub-band\end{tabular} & 5.51 & 11.50 & 76.17/27.29 & 26.91 & 0.9211 & 0.7964 \\
\hline
\textbf{\begin{tabular}[c]{@{}l@{}}High Frequency \\ sub-band\\ (final model)\end{tabular}} & \textbf{0.66} & \textbf{3.3586} & \textbf{82.31/44.33} & \textbf{37.75} & \textbf{0.9975} & \textbf{0.9999} \\
\hline
\end{tabular}
\caption{\label{tab:2} Results of steganography based on different embedding domains}
\end{table}
\begin{figure}[h]
\centering
\includegraphics[width=1.0\textwidth]{4-5.png}
\caption{\label{fig:4-5}The first and third columns are the original cover image and secret image, and other images are the generated images. The right side of each line shows the residual images of stego images.}
\end{figure}
According to Table \ref{tab:2} and Figure \ref{fig:4-5}, the secret image generated based on spatial domain has obvious distortion. In contrast, the distortion of the image embedded in the frequency domain is slighter and visually closer to the original image (the second line of Figure \ref{fig:4-5}). Therefore, the proposed model can obtain a better imperceptibility in the frequency domain. We believe that this is due to the different characteristics of spatial domain information and frequency domain information. Compared with spatial domain information, frequency domain information is the result of image decomposition from multiple different scales, and its characteristics are more conducive to network learning.
According to the experimental results of the last three groups in Table \ref{tab:2}, the C\_Error per pixel of the final model is only 0.66, the PSNR is as high as 82.31, and the SSIM of the stego image and the reconstructed secret image is almost saturated. In the residual images of Figure \ref{fig:4-5}, the steganographic trace generated by the final model is the smallest, so its stego image is closer to the original cover image. We attribute the advantage of the final model to the local modification of frequency domain and avoiding modifying cA, which has a large image energy concentration. When cD is selected for steganography, the dimension of its data accounts for only 1 / 4 of the overall frequency domain. And the data dimension is equal to other sub-bands, but it has less impact on the original image.
According to the above experimental results based on color spaces characteristics and frequency sub-bands characteristics, the module frequency sub-band selection of the final model selects the diagonal high-frequency sub-band of the B channel as the embedding domain.
\subsection{Comparison with Other Models}
In order to verify that the proposed model is superior to similar steganographic models in imperceptibility, we compare the steganographic effect of our model with that of Antique's model\cite{r21} and ISGAN\cite{r4}. Both of them are steganographic models based on a color image hiding a gray image, so they have the same steganographic capacity (8bpp) as the proposed model. The three models are trained with the same dataset.
\begin{table}[h]
\centering
\begin{tabular}{ccccc}
\hline
Model & C\_PSNR & S\_PSNR & C\_SSIM & S\_SSIM \\
\hline
Antique’s Model & 33.7 & 39.9 & 0.95 & 0.96 \\
\hline
ISGAN & 34.63 & 33.63 & 0.95 & 0.94 \\
\hline
Proposed Model & 82.31 & 37.75 & 0.99 & 0.99\\
\hline
\end{tabular}
\caption{\label{tab:3}Comparison of steganographic performance of Antique's model, ISGAN, and proposed model.}
\end{table}
According to Table \ref{tab:3}, except S\_PSNR, other indicators of the final model are the highest, S\_SSIM and C\_SSIM of the final model reached 0.99, which is close to saturation. In addition, it is obvious from Figure 15 that the stego image generated by the final model is closer to the original image. And the distortion of the reconstructed secret image is also really difficult to detect.
\begin{figure}[t]
\centering
\includegraphics[width=1.0\textwidth]{4-6.png}
\caption{\label{fig:4-6}The generated images and their residual images based on LFW dataset. The results show that compared with the models of ISGAN and Antique, the image generated by the final model is closer to the original image and has a good imperceptibility.}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.9\textwidth]{4-7.png}
\caption{\label{fig:4-7}(a) The color distortion between generated image and cover images; (b) The texture traces generated by steganography.}
\end{figure}
\subsubsection{Color Distortion}
Figure \ref{fig:4-6} (a) shows the color distortion of different models. It is obvious that the tone of the stego image generated by Antique's model (the second column) is yellower, the stego image of ISGAN is slightly darker, while the stego image of the final model is basically the original image, and the residual image does not show the obvious modification.
\subsubsection{Traces of Texture}
Figure \ref{fig:4-7} (b) shows the texture traces of different models. Compared with the color distortion, the texture traces of stego image are more difficult to detect by vision, but after enlarging the stego image or multiple residual processing with the original image, we can still find the texture outline of the secret image in the stego image of antique's model and ISGAN(we enhance the brightness of the residual image, so as to make the texture trace in the residual image more obvious). This means that the third party can obtain the semantic information of the secret image according to some simple image processing. Therefore, the security of the secret image is also difficult to guarantee. In contrast, the residual images of the final model still have no visible texture traces, which means our model has a better imperceptibility.
In summary, from the perspective of color distortion and texture traces, the proposed model has better performance in imperceptibility.
\subsection{Generalization Ability}
To further verify the generalization ability of the final model, we used the ILSVRC2012 and PASCAL\_VOC\_2012 dataset training model.
\begin{figure}[h]
\centering
\includegraphics[width=1.0\textwidth]{4-8.png}
\caption{\label{fig:4-8}Generated images based on different datasets. The final model can always generate images which close to original images.}
\end{figure}
\begin{table}[h]
\centering
\begin{tabular}{ccccccc}
\hline
Dataset& \begin{tabular}[c]{@{}c@{}}C\_Error \\ per \\ pixel\end{tabular} & \begin{tabular}[c]{@{}c@{}}S\_Error \\ per \\ pixel\end{tabular} & \begin{tabular}[c]{@{}c@{}}C\_PSNR\\ /CL-PSNR\end{tabular} & S\_PSNR & C\_SSIM & S\_SSIM \\
\hline
LFW& 0.66 & 3.3586 & 82.31/44.33 & 37.75 & 0.9975 & 0.9999 \\
\hline
ILSVRC2012 & 2.86 & 4.21 & 78.07/35.11 & 35.62 & 0.9914 & 0.966 \\
\hline
PASCAL\_VOC\_2012 & 2.67 & 6.02 & 78.30/35.49 & 33.05 & 0.9918 & 0.9466 \\
\hline
\end{tabular}
\caption{\label{tab:4} Training results of the final model based on different datasets}
\end{table}
The data of the ILSVRC2012 dataset comes from the ImageNet dataset. PASCAL\_VOC\_2012 dataset is commonly used for target detection, including more than 20 object classes and more than 10 action classes. Compared with the LFW dataset used in previous experiments, the data characteristics of the two datasets are more complex, which is helpful for the generalization verification of the final model.
Although the performances of training on ILSVRC2012 and Pascal\_VOC\_2012 are slightly lower than that of training on LFW, but its distortion is still within an acceptable range. In addition, all SSIM values of the three results are more than 0.99, which is close to saturation. The last three columns of images in Figure \ref{fig:4-8} are residual images of the original cover image and the stego image. When the residual is expanded to 5 times, the outline of the stego image appears, and when it is expanded to 10 times, the outline is clearer. This means that the final model can still show good imperceptibility on different datasets, and the steganographic performance remains at the same level. Therefore, the proposed model has good generalization ability.
\section{Conclusion}
In this paper, a color image steganographic model based on frequency sub-band selection is proposed. The model uses the frequency domain as the embedding domain. Moreover, we discuss and verify whether the characteristics of different color spaces and frequency sub-bands will affect the proposed model, which helps us determine the final model. Our model is an encoder-decoder structure model, so the research conclusion has reference significance for the same type of steganographic model and determined the final model. Experiments show that compared with other models, our model effectively reduces the color distortion and texture traces, which significantly improves the imperceptibility. At present, the stego image generated by our model is almost the same as the original cover image. However, there is still room for improvement in the quality of the reconstructed secret image. Moreover, our model has excellent performances in imperceptibility and capacity. Hence we will pay more attention to the improvement of its robustness in our future work.
\printbibliography{}
\end{document}
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2021-12-30T02:22:25.000Z
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{"paloma_paragraphs":[]}
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{"arxiv_id":"2112.14437","language":"en","timestamp":1640830945000,"url":"https:\/\/arxiv.org\/abs\/2112.14437","yymm":"2112"}
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proofpile-arXiv_000-10061
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{"provenance":"002.jsonl.gz:10062"}
| null | null |
\section{Introduction}
In non-central heavy ion collisions at relativistic energies, a large amount of orbital angular momentum from the two colliding nuclei are transferred to the produced quark-gluon plasma (QGP), creating thus the most vortical fluid of the order of $10^{21}-10^{22}$ s$^{-1}$ in known physical systems~\cite{STAR:2017ckg,PhysRevC.93.064907,PhysRevC.94.044910,PhysRevC.87.034906,PhysRevC.90.021904,PhysRevC.88.061901,PhysRevC.92.014906}. Due to their spin-orbit interactions, quarks in the QGP and $\Lambda$ hyperons formed after the hadronization become polarized along the direction of the total orbital angular momentum~\cite{PhysRevLett.96.039901,PhysRevC.77.044902}, as observed in experiments by the STAR Collaboration ~\cite{STAR:2017ckg,PhysRevLett.126.162301}. Because of the non-uniformity of the vorticity field, local structures in the $\Lambda$ spin polarization have also been found both in theoretical studies~\cite{PhysRevLett.117.192301,PhysRevLett.120.012302,PhysRevC.98.024905} and in experimental measurements~\cite{PhysRevLett.123.132301}. Although various theoretical models, such as those based on the hydrodynamic approach~\cite{PhysRevC.88.034905,Karpenko2017,PhysRevC.94.054907,PhysRevC.95.031901,Becattini2015,PhysRevResearch.1.033058,PhysRevC.103.024903}, the transport approach~\cite{PhysRevC.96.054908,PhysRevC.97.064902,PhysRevC.98.024905,PhysRevC.99.014905,Deng:2021miw} and the non-equilibrium chiral kinetic approach~\cite{PhysRevC.96.024906,PhysRevC.99.011903,PhysRevLett.125.062301}, have successfully described the measured $\Lambda$ global spin polarization, most of them have failed to explain the measured $\Lambda$ local spin polarizations. A plausible explanation of the latter has been provided in the chiral kinetic approach through the induced quadrupole axial charge distribution in the transverse plane of a heavy ion collision~\cite{PhysRevC.99.011903,PhysRevLett.125.062301}. Also, it has recently been pointed out that adding the thermal shear contribution to that from the thermal vorticity in the fluid dynamic approach can potentially describe the measured azimuthal angle dependence of $\Lambda$ local spin polarizations~\cite{Liu:2021uhn,Becattini:2021suc,Yi:2021ryh,Liu:2021nyg}. Indeed, the correct azimuthal angle dependence is obtained in the ``strange memory" scenario of Ref.~\cite{PhysRevLett.127.142301}, which assumes that the $\Lambda$ spin polarization is identical to its strange quark spin polarization like in the quark coalescence model for $\Lambda$ production~\cite{PhysRevC.96.024906}, after including contributions from both thermal vorticity and thermal shear. A similar result can also be achieved in the ``isothermal local equilibrium" scenario of Ref.~\cite{Becattini:2021iol}, in which the hadronization is assumed to take place at a constant temperature to eliminate the contribution from its space-time gradients in the calculation of the thermal vorticity and shear~\cite{Becattini:2021iol}.
In almost all these theoretical studies, the $\Lambda$ spin polarization in a heavy ion collision is calculated at the end of the partonic phase and compared to experimental measurements. Since the strength of the vorticity field decreases as the hadronic matter expands~\cite{PhysRevC.94.044910}, it is of interest to study how the $\Lambda$ polarization changes during the hadronic evolution of heavy ion collisions. In this work, we study the time evolution of the global and local $\Lambda$ spin polarizations during the expansion of the hadronic matter produced in non-central Au+Au collisions at $\sqrt{s_{NN}}=19.6$ GeV by using the MUSIC+UrQMD hybrid model. Because of the lack of a dynamical description of the spin degree of freedom in the hadronic transport model, we study the dependence of $\Lambda$ spin polarizations on the temperature by assuming that they are in thermal equilibrium with the vorticity field in the expanding hadronic matter.
The paper is organized as follows: In Sec. II, we discuss the relations between the spin polarization of a spin-1/2 fermion and the thermal vorticity and shear in a hadronic matter. We then present the numerical results from our study on the global and local spin polarization of $\Lambda$ hyperons at different freeze-out temperatures for the spin degree of freedom. Finally, a brief conclusion and discussion is given in Sec. IV.
\section{Spin polarization of a fermion in thermal equilibrium}
The spin polarization of a fermion in a thermal medium of temperature $T$ depends on the four-temperature vector $\beta=u/T$, where $u$ is the four flow velocity of the local medium. To the leading order in the gradient of $\beta$, the spin polarization vector $S$ of a fermion of four momentum $p$ at four space-time coordinate $x$ can be written as ~\cite{Liu:2021uhn,Becattini:2021suc,Yi:2021ryh,Liu:2021nyg}
\begin{eqnarray}
S^{\mu}(x,p)=&&-\frac{1}{8m}(1-n_F)\epsilon^{\mu\nu\rho\sigma}p_v\varpi_{\rho\sigma}(x)\nonumber
\\&&-\frac{1}{4m}(1-n_F)\epsilon^{\mu\nu\rho\sigma}p_v\frac{n_{\rho}p^{\lambda} \xi_{\lambda\sigma}(x)}{n\cdot p}.
\label{spin}
\end{eqnarray}
In the above,
\begin{eqnarray}\label{thermal}
\varpi_{\rho\sigma}=\frac{1}{2}(\partial_{\sigma}\beta_{\rho}-\partial_{\rho}\beta_{\sigma}),
\quad\xi_{\rho\sigma}=\frac{1}{2}(\partial_{\sigma}\beta_{\rho}+\partial_{\rho}\beta_{\sigma})
\end{eqnarray}
are the thermal vorticity and thermal shear of the medium, respectively, $n_F$ is the Fermi-Dirac distribution function, and $n$ is a unit four vector that specifies the frame of reference. Although different forms of $S^{\mu}$ are used in Refs.~\cite{Liu:2021uhn,Becattini:2021suc,Yi:2021ryh,Liu:2021nyg}, they all become the same if $n$ is taken to be the flow field $u$ in the fluid, which we adopt in this work. Our calculation using Eq.~(\ref{thermal}) is similar to that in Ref.~\cite{PhysRevLett.127.142301} but different from that based on the assumption of isothermal local equilibrium adopted in Ref.~\cite{Becattini:2021iol}, where both the spatial and temporal gradients of the temperature are neglected, i.e., only contributions from the kinetic vorticity and shear are included.
By decomposing the two terms in Eq.(\ref{spin}) into the following two components,
\begin{eqnarray}
&&\boldsymbol{\varpi}_T=\frac{1}{2}\left[\boldsymbol{\nabla}\left(\frac{u^0}{T}\right)+\partial_t\left(\frac{\boldsymbol{u}}{T}\right)\right]-u^0\boldsymbol{f}+\boldsymbol{u}f^0,\nonumber
\\&&\boldsymbol{\varpi}_S=\frac{1}{2}\boldsymbol{\nabla}\times\left(\frac{\boldsymbol{u}}{T}\right)+\boldsymbol{u}\times\boldsymbol{f},
\label{omega}
\end{eqnarray}
with
\begin{eqnarray}
&&u^0=\gamma,\quad \boldsymbol{u}=\gamma \boldsymbol{v},\nonumber
\\&&f^0=\frac{p^{\lambda}\xi_{0\lambda}}{u\cdot p},\quad \boldsymbol{f}=-(f_1,f_2,f_3),\quad f_i=\frac{p^{\lambda}\xi_{i\lambda}}{u\cdot p},
\end{eqnarray}
Eq. (\ref{spin}) can be rewritten as
\begin{eqnarray}
&&S^0(x,p)=\frac{1}{4m}\boldsymbol{p}\cdot\boldsymbol{\varpi}_S,\nonumber
\\&&\boldsymbol{S}(x,p)=\frac{1}{4m}(E_p\boldsymbol{\varpi}_S+\boldsymbol{p}\times\boldsymbol{\varpi}_T),
\label{spinvector}
\end{eqnarray}
where $E_p=\sqrt{m^2+{\bf p}^2}$ and ${\bf p}$ are, respectively, the time and space components of the four momentum $p$.
Boosting the spin vector $S$ in the fluid frame to the particle's rest frame and using the fact that the spin of a fermion equal to $1/2$, one finds the spin polarization ${\bf P}$ measured in experiments to be
\begin{eqnarray}
&& \boldsymbol{P}=2 \boldsymbol{S}^*=2\left[\boldsymbol{S}-\frac{ \boldsymbol{p}\times \boldsymbol{S}}{E_p(E_p+m)} \boldsymbol{p}\right].
\label{polarization}
\end{eqnarray}
\section{Numerical results}
To study the evolution of $\Lambda$ spin polarization in the hadronic phase of a heavy ion collision, we first use the MUSIC hydrodynamic model~\cite{PhysRevLett.106.042301} with the initial conditions taken from A Multi-Phase Transport (AMPT) model~\cite{PhysRevC.72.064901} to simulate the QGP phase of the collision. After converting the fluid elements on the freeze-out hypersurface into hadrons by the Cooper-Frye formula, we adopt the UrQMD model~\cite{Bass:1998ca,Bleicher:1999xi} to simulate the evolution of these hadrons due to their scatterings. This hybrid approach has successfully described various soft hadronic observables, such as the charged particle yields as well as their transverse momentum spectra and flow anisotropies in relativistic heavy ion collisions at both RHIC and LHC energies~\cite{Shen:2020jwv, Zhao:2020irc,Schenke:2020mbo}.
To determine the local temperature and flow field as well as their gradients in time and space in the hadronic matter, we use the coarse-grained method with the time step $\Delta t=0.5$ fm$/c$, and the cell size $\Delta x=\Delta y=0.5$ fm and $\Delta \eta=0.2$, where $\eta=\frac{1}{2}\ln\frac{t+z}{t-z}$ is the space-time rapidity. In this method, the temperature $T$ of a local cell is calculated from the energy density in the cell at its rest frame by using the equation of state from the lattice QCD calculations~\cite{Borsanyi:2013bia}. For the flow field $\boldsymbol{v}$ in each cell, it is calculated from the average velocity of the $N$ hadrons in the cell according to
\begin{eqnarray}
\boldsymbol{v}(t,x,y,z)=\frac{\sum_i \frac{\boldsymbol{p_i}}{E_i}}{N},
\end{eqnarray}
where ${\bf p}_i$ and $E_i$ are the momentum and energy of the $i$-th hadron in the cell.
\begin{figure}[h]
\centering
\includegraphics[width=1\linewidth]{T-Py2.eps}
\caption{(Color online) Spin freeze-out temperature dependence of $\Lambda$ hyperon spin polarization along the total angular momentum direction due to the thermal vorticity and the thermal shear in an expanding hadronic matter.}
\label{fig:T-Py}
\end{figure}
To illustrate how $\Lambda$ spin polarizations are affected by the temperature at which the spin degrees of freedom freeze out in relativistic heavy ion collisions, we consider in the present study Au+Au collisions at $\sqrt{s_{NN}}=19.6$ GeV and 20-60\% centrality. As in the experimental measurements at RHIC by the STAR Collaboration~\cite{PhysRevLett.123.132301}, we only include $\Lambda$ hyperons of $|\eta|<1$ and $p_T>$0.15 GeV$/c$ in our analysis. In Fig.~\ref{fig:T-Py}, we first show by the solid red line the global spin polarization of $\Lambda$ ($\overline{\Lambda}$) hyperons calculated from the thermal vorticity along the direction of the total angular momentum of the hadronic matter as a function of the local temperature in the hadronic matter. It is seen that if the spin degree of freedom freezes out at $T_s=160$ MeV, which is usually assumed in the literature~\cite{PhysRevC.96.054908,PhysRevC.96.024906,PhysRevC.103.024903} as mentioned in the introduction, the $\Lambda$ global polarization is 5.3$\times10^{-3}$ and is close to the value in Ref.~\cite{PhysRevC.103.024903} based on the same AMPT+MUSIC hybrid approach without the UrQMD afterburner. With decreasing temperature as the hadronic matter expands, the $\Lambda$ global spin polarization decreases to 2.5$\times10^{-3}$ at the kinetic freeze-out temperature of $T_s=110 $ MeV, when hadron momentum spectra and anisotropies stop changing. The above result thus shows that the $\Lambda$ global spin polarization has a strong temperature dependence in the expanding hadronic matter if it continues to be in thermal equilibrium .
Shown by the blue dashed line in Fig.~\ref{fig:T-Py} is the $\Lambda$ global spin polarization generated by the thermal shear in the hadronic matter, i.e., the second term in the RHS of Eq. (\ref{spin}), as a function of local temperature. It is seen that at $T_s=160$ MeV, the $\Lambda$ global polarization due to the thermal shear is about 2.9$\times10^{-3}$, which is comparable to that due to the thermal vorticity. We note that the total spin polarization of $\Lambda$ hyperons due to the thermal shear should have been zero without the cut of $p_T\ge 0.15$ GeV/$c$ and $|\eta|<$1 in the $\Lambda$ momentum. With decreasing spin freeze-out temperature, the $\Lambda$ global spin polarization due to the thermal shear decreases faster compared to that due to the thermal vorticity, and it becomes negligible at $T_s=140$ MeV.
\begin{figure}[h]
\centering
\includegraphics[width=1\linewidth]{LocalSpin2.pdf}
\caption{(Color online) The azimuthal angle dependence of the local spin polarizations $P_x, -P_y$ and $P_z$ of $\Lambda$ hyperons generated by thermal vorticity and thermal shear at the spin freeze-out temperatures $T_s=$160, 140, and 120 MeV. }
\label{fig:Localspin}
\end{figure}
As to the local spin polarization of $\Lambda$ hyperons as a function of its azimuthal angle in the transverse plane of a heavy ion collision, it has recently been extensively studied~\cite{PhysRevLett.120.012302,PhysRevC.99.011903,PhysRevLett.125.062301,PhysRevLett.123.132301}. It is found in Refs.~\cite{PhysRevLett.127.142301,Becattini:2021iol} that both thermal vorticity and thermal shear are important in determining the local spin polarization. In the left panels (a), (b) and (c) of Fig.~\ref{fig:Localspin}, we show, respectively, the spin polarization $P_x, -P_y$ and $P_z$ of $\Lambda$ hyperons generated by the thermal vorticity (red dashed lines) and the thermal shear (blue dashed-dotted lines) as functions of the azimuthal angle $\phi_p$ of the $\Lambda$ transverse momentum if the spin degree of freedom freezes out at $T_s=160$ MeV. It is seen that $-P_y$ (left-middle panel) and $P_z$ (left-lower panel) have the form of $-\cos(2\phi_p)$ and $-\sin(2\phi_p)$, respectively, due to the thermal vorticity and $\cos(2\phi_p)$ and $\sin(2\phi_p)$, respectively, due to the thermal shear, which are similar to the findings in Refs.~\cite{PhysRevLett.127.142301,Becattini:2021iol,Yi:2021ryh}. However, such an oscillatory azimuthal angle dependence essentially disappears in the total $\Lambda$ spin polarization after adding the two contributions, which disagrees with the experimental measurements~\cite{PhysRevLett.123.132301}, unless one adopts either the ``strange memory" scenario as in Ref.~\cite{PhysRevLett.127.142301} or the ``isothermal local equilibrium" scenario as in Ref.~\cite{Becattini:2021iol}. In the present work, we focus on the effect of hadronic evolution on the $\Lambda$ spin polarization and postpone the study of the above two scenarios for future study.
In the left-upper panel (a) of Fig.~\ref{fig:Localspin}, we also show for the first time the azimuthal angle dependence of $P_x$ generated by the thermal vorticity and thermal shear. It is seen that $P_x$ has the form of $-\sin(2\phi_p)$ due to the thermal vorticity and the form of $\sin(2\phi_p)$ due to the thermal shear. The total $P_x$ after adding the two oscillatory contributions is, however, negligible.
\begin{figure}[h]
\centering
\includegraphics[width=1\linewidth]{component2.eps}
\caption{(Color online) Azimuthal angle dependence of $\Lambda$ spin polarization generated by components of the thermal shear at the spin freeze-out temperature $T_s=160$ MeV.}
\label{fig:component}
\end{figure}
In Fig.~\ref{fig:component}, we further show the $\Lambda$ spin polarization generated by individual components of the thermal shear. We find that the main contribution to $-P_y$ is from the $\xi_{13}$ component and that to $P_z$ is from the $\xi_{11}$ and $\xi_{22}$ components. According to Eqs.(\ref{omega}), (\ref{spinvector}) and (\ref{polarization}), $\xi_{13}$ contributes to $-P_y$ through a term of the form $p_x^2\xi_{13}\propto (1+\cos(2\phi_p))\xi_{13}$. For $P_z$, the contribution from $\xi_{11}$ and $\xi_{22}$ is through a term of the form $p_xp_y(\xi_{11}-\xi_{22})\propto \sin(2\phi_p)(\xi_{11}-\xi_{22})$, where $(\xi_{11}-\xi_{22})$ is positive because of the positive elliptic flow of the hadronic matter in non-central Au+Au collisions at $\sqrt{s_{NN}}=19.6$ GeV.
\begin{figure}[h]
\centering
\label{fig:T-0.14}
\end{figure}
We have also studied the azimuthal angle dependence of the spin polarization of $\Lambda$ hyperons generated by the thermal vorticity and thermal shear at a lower temperature of $T_s=140$ MeV during the later stage of the hadronic evolution by assuming that the spin degree of freedom remains in thermal equilibrium. As shown by the red dashed lines in the middle panels (d), (e) and (f) of Fig.~\ref{fig:Localspin}, $P_x, -P_y$ and $P_z$ generated by the thermal vorticity at this temperature have the same azimuthal angle dependence as those at $T_s=160$ MeV. Although $P_x$ and $-P_y$ become smaller at the lower temperature, $P_z$ does not change much with temperature. For the contribution from the thermal shear at $T_s=140$ MeV, $P_x$ and $-P_y$ become negligible, while $P_z$ decreases by a factor of 3 as shown by the blue dashed-dotted lines. By adding contributions from both thermal vorticity and thermal shear, $P_x$ and $P_z$ have the form of $-\sin(2\phi_p)$ and $P_y$ has the form of $-\cos(2\phi_p)$. These results suggest that the $\Lambda$ local spin polarization depends strongly on its evolution during the hadronic phase of heavy ion collisions.
\begin{figure}[h]
\centering
\label{fig:T-0.12}
\end{figure}
Finally, we consider an even lower temperature of $T_s=120$ MeV, corresponding approximately to the kinetic freeze-out of the expanding hadronic matter, and the results are shown in the right panel{s (g), (h) and (k) of Fig.~\ref{fig:Localspin}. It is seen from the red dashed lines that the $\Lambda$ local spin polarizations $P_x, -P_y$ and $P_z$ generated by the thermal vorticity become even smaller, although they have the same $\phi_p$ dependence as that at higher temperatures. However, for the $\Lambda$ spin polarization generated by the thermal shear, its values along all $x$, $y$ or $z$ directions become negligibly small. Thus, the total $P_x$, $-P_y$ and $P_z$ have the same $\phi_p$ dependence as that at $T_s=140$ MeV, albeit somewhat smaller.
\section{Conclusion and Discussion}
Using the AMPT model initial conditions to the MUSIC hydrodynamic model for the QGP phase, which is followed by the UrQMD model for the hadronic phase in relativistic heavy ion collisions, we have studied in this paper the global and local spin polarizations of $\Lambda$ hyperons as functions of the freeze-out temperature of the spin degree of freedom in the hadronic phase. We have found that both the $\Lambda$ global and local spin polarizations due to the thermal vorticity decrease by a factor of 2 if the spin freeze-out temperature drops from 160 to 140 MeV, while those due to the thermal shear already becomes negligibly small at temperature equal to 140 MeV. This result suggests the importance of including in theoretical studies the dynamical evolution of the spin degrees of freedom and its freeze-out in the expanding hadronic matter. Such a study requires a hadronic transport model that takes into account explicitly the spin degrees of freedom of hadrons, such as in Refs.~\cite{Xu:2012hh,Xia:2014qva,Xia:2014rua,PhysRevD.104.016022,PhysRevD.104.016029} on the nucleon spin transport in heavy ion reactions at lower energies, which we plan to study in the future.
\section*{ACKNOWLEDGEMENTS}
We thank Huichao Song for helpful discussions and suggestions.
This work was supported by INFN-SIM national project and linea di intervento 2 for HQCDyn at DFA-Unict (Y.S.), the National Natural Science Foundation of China under Grant No. 11905302 (Z.Z.), the U.S. Department of Energy under Contract No. DE-SC0015266 and the Welch Foundation under Grant No. A-1358 (C.M.K.), and the U.S. National Science Foundation (NSF) under grant numbers ACI-2004571 (W.Z.).
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{"paloma_paragraphs":[]}
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{"arxiv_id":"2112.14410","language":"en","timestamp":1641176080000,"url":"https:\/\/arxiv.org\/abs\/2112.14410","yymm":"2112"}
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{"provenance":"002.jsonl.gz:10063"}
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\section*{Negative Societal Impacts}
\section{Introduction}
\input{1_Introduction/1_introduction}
\section{Related Works}
\input{2_Related_Work/2_related_work}
\section{Method}
\input{3_Method/3_0_method}
\subsection{Preliminaries}
\input{3_Method/3_1_preliminaries}
\subsection{Deformable Graph Convolution}
\input{3_Method/3_2_deformable_graph_convolution}
\input{3_Method/3_2_latent_graph_generation}
\subsection{Deformable Graph Convolutional Networks}
\input{3_Method/3_3_deformable_gcn}
\section{Experiments}
\input{4_Experiments/4_0_experiments}
\subsection{Dataset}
\input{4_Experiments/4_1_dataset}
\subsection{Baselines and Implementation Details}
\input{4_Experiments/4_2_baseline_and_experimental_settings}
\subsection{Results on Node Classification}
\input{4_Experiments/4_3_results}
\subsection{Ablation Study and Analysis}
\input{4_Experiments/4_4_ablation_study}
\input{4_Experiments/4_5_analysis}
\section{Conclusion}
\input{5_Conclusion/5_conclusion}
\section*{Acknowledgments}
This work was partly supported by MSIT (Ministry of Science and ICT), Korea, under the ICT Creative Consilience program (IITP-2021-2020-0-01819) supervised by the IITP and Samsung Research Funding \& Incubation Center of Samsung Electronics under Project Number SRFC-IT1701-51.
{
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{"paloma_paragraphs":[]}
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{"arxiv_id":"2112.14438","language":"en","timestamp":1640830947000,"url":"https:\/\/arxiv.org\/abs\/2112.14438","yymm":"2112"}
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proofpile-arXiv_000-10063
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{"provenance":"002.jsonl.gz:10064"}
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\section{Introduction}
The possible existence of weakly interacting massive particles (WIMP) is a well motivated paradigm to address
the cosmological and astrophysical observations~\cite{Planck:2018vyg, Bertone:2004pz}. On one hand this paradigm allows for an explanation of how the dark matter abundance arises during the thermal evolution of the early universe and, consequently, leads to expectation of directly detectable interactions between WIMPs and baryonic matter~\cite{Arcadi:2017kky, Roszkowski:2017nbc}. Many experiments have searched
for such interactions~\cite{Schumann:2019eaa}, with most stringent exclusion constraints~\cite{XENON:2018voc} applying to WIMPs with masses above ${\cal O}$(10 GeV).
Progress into the sub-GeV region has been demonstrated by new detector developments offering detection thresholds sensitive to WIMP-nucleus scattering at precision of single-electron excitation~\cite{Romani:2017iwi,Crisler:2018gci,DAMIC:2020cut,EDELWEISS:2020fxc}. While these technologies are currently operating with low mass targets, ${\cal O}$(g), attempts to scale up to detector masses of ${\cal O}$(kg) have been made~\cite{Mirabolfathi:2015pha,Iyer:2020nxe}. Phonon-mediated detectors with ${\cal O}$(eV) resolution are the most appropriate detectors to perform the measurements that we are proposing in this work. Phonons are among the lowest energy quantum excitations (compred to \textit{e.g.} ionization and scintillation) that can be detected after particle interactions. In addition to their excellent signal to noise, phonon-mediated detectors offer an interaction-type independent (nuclear or electron recoil) energy measurement .
Many groups have recently achieved the energy resolution that are within the required range for our proposed detection technique \cite{Strauss:2017cam, CPD:2020xvi,Verma:2022tkq}.
Recently several experiments \cite{CRESST:2019axx,CRESST:2019jnq,DAMIC:2020cut,EDELWEISS:2019vjv,EDELWEISS:2020fxc,CRESST:2017ues,NUCLEUS:2019kxv,SENSEI:2020dpa,SuperCDMS:2018mne,SuperCDMS:2020ymb} have observed a steeply rising event rate at low energies, $E_{\rm r} \lesssim 1000$ eV. The origin of these events is currently unknown, and understanding their physical character is a question of great interest for both the DM and coherent neutrino scattering experiments~\cite{Proceedings:2022hmu}. Most of the anticipated background sources, such as photons or electrons, would give rise to electron recoils. Therefore the identification of the nuclear/electron recoil character of these events would add an important piece of information towards understanding and mitigating this background.
We propose to approach this problem with detailed understanding of the response of the target material to low energy scattering events, achieved via molecular dynamics (MD) simulations~\cite{Gib60,Dia87,Nor18b,PhysRevB.78.045202,NORDLUND2006322,Sassi:2022njl}. Particularly, it was noticed in \cite{Kadribasic:2020pwx,Sassi:2022njl} that a sharp defect creation threshold in diamond leads to a prominent feature in the observed nuclear recoil spectrum: the energy readout of the phonon detector will not see the energy stored in the defect, and the affected recoil events will be shifted towards lower observed energies. If the defect creation threshold is sharp, this effect turns on abruptly at the corresponding recoil energy, resulting in a prominent peak followed by a dip in the observed event rate. On the other hand, this effect is not expected for electronic recoils, and therefore its presence or absence in the recoil spectrum allows to discriminate between these two explanations for the unknown origin of the events. Beyond particle scattering events, electrical noise and stress relaxation or microscopic fractures in the material have been suggested as possible sources of the excess events. Electrical noise would clearly not be affected by energy loss and therefore the observation of this effect would also exclude this explanation. While we do not expect the stress relaxation/fracture scenario to exhibit the same feature either, a full understanding of the expected spectrum in this case requires dedicated simulations and is beyond the scope of this work.
Phonon-mediated detectors have the advantage of fully measuring the recoil energy without making any assumption about the nature of the recoil: nuclear or electronic. Recent progress in phonon-mediated detectors allows for the low detection thresholds that are required to measure these lattice defect features~\cite{CPD:2020xvi,EDELWEISS:2019vjv,Strauss:2017cam}. In particular, gram scale diamond based detectors are expected to offer a resolution that is superior to the existing technologies~\cite{Kurinsky:2019pgb,Abdelhameed:2022skh}.
In the following, we will describe this effect in more detail in light of
recent experimental results and propose a method to assess the underlying nature of the observed low energy event rate.\\
\section{Testing the nuclear recoil origin of the excess}
To test if the excess events are due to nuclear recoils, we propose to use a diamond detector.
For a quantitative analysis, we use data from Nucleus 1g prototype \cite{NUCLEUS:2019kxv}, SuperCDMS-CPD \cite{SuperCDMS:2020aus} and Edelweiss \cite{EDELWEISS:2019vjv} shown in figure \ref{nucleusdata}, reproduced here using the data repository \cite{excess_workshop_2021}. We will parametrize this data using a three-component fit, with an exponential, a power law and a constant component, of the form
\begin{equation}
f(x) = Ae^{-\alpha x}+Bx^\beta+C,
\label{nucleusfit}
\end{equation}
where $x= E_{\rm r}/{\rm eV}$ and we have determined the best fit values for the parameters as shown in table \ref{bestfitparams}. The fit function is shown together with the data points in figure \ref{nucleusdata}. A similar fit for the SuperCDMS and Edelweiss data was used in \cite{Abbamonte:2022rfh}, where it was suggested that the exponential part is due to electronic trigger noise. We adopt this interpretation, and therefore do not apply the energy loss effect to the exponential part of the event rate in our simulations, as these counts are taken to not represent real recoil events. Furthermore the constant component $C$ is taken to represent background from mostly electron recoils, so that the power law component represents the unidentified excess events. Therefore in our simulations the energy loss is not applied to the constant component either, but we have checked that the results do not significantly differ based on this decision.
\begin{table*}[htb]
\centering
\begin{tabular}{c|c c c c c}
& $A$ & $\alpha$ & $B$ & $\beta$ & $C$ \\
\hline
Nucleus & $(9.7\pm 25.7)\times10^9$\, & $0.77\pm 0.13$\, & $(1.58\pm 0.40)\times 10^4$\, & $-1.44\pm0.05$\, & $0\pm 0.19$ \\
SuperCDMS & $(1.41\pm 0.16)\times 10^8$ & $0.61\pm 0.006$ & $(3.7\pm 4.1)\times 10^4$ & $-2.7\pm 0.3$ & $0.18\pm 0.01$\\
Edelweiss & $(1.46\pm 0.28)\times 10^5$ & $0.124\pm 0.003$ & $(1.04\pm 0.55)\times 10^5$ & $-2.6\pm 0.1$ & $0.011\pm 0.002$
\end{tabular}
\caption{Best fit values for the parametric model (\ref{nucleusfit}) for the three data sets. The parameters $A,B,C$ are in units of events/(eV g day). }
\label{bestfitparams}
\end{table*}
\begin{figure*}[htb]
\begin{center}
\includegraphics[width=0.32\linewidth]{Nucleusfit_linlog.pdf}
\includegraphics[width=0.32\linewidth]{SuperCDMSCPDfit_linlog.pdf}
\includegraphics[width=0.32\linewidth]{Edelweissfit_linlog.pdf}
\caption{The differential event rate $dR/dE_{\rm r}$ observed in the Nucleus 1g-prototype (left), SuperCDMS-CPD (center) and Edelweiss (right) measurements shown with the red points. The fit function (\ref{nucleusfit}) is shown with the gray solid line. The gray dashed line shows the exponential component, the gray dotted line the constant component and the blue solid line the power-law component of the fit function, taken here to represent the unidentified excess. }
\label{nucleusdata}
\end{center}
\end{figure*}
Based on MD simulation data presented in~\cite{Sassi:2022njl}, diamond has a very sharp threshold for defect creation, resulting in a step-like rise in the average energy loss as a function of recoil energy, shown in figure \ref{ElossAverageC}. As the energy read out from the phonon detector will not see the energy stored in the defects, this sharp onset of energy loss will result in a peak in the visible energy spectrum, if the underlying recoil spectrum is smooth. If the events are electronic recoils, no energy loss is expected and the phonon measurement should see the smooth underlying spectrum. For comparison, figure \ref{ElossAverageC} shows also the average energy loss curves for silicon in green and germanium in purple. For these semiconductor targets we can observe a threshold, but smoother than in diamond and at a lower energy. Silicon and germanium have identical crystal structure and very similar chemical properties, which explains the similar energy loss curves of these materials.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.99\linewidth]{ElossAverageCTN_Al2O3_Si_Ge.pdf}
\caption{The average (over recoil direction) energy loss in diamond (red), sapphire (blue), silicon (green) and germanium (purple) as a function of the recoil energy.}
\label{ElossAverageC}
\end{center}
\end{figure}
The blue line in figure \ref{ElossAverageC} shows the average energy loss in sapphire, based on~\cite{Sassi:2022njl} . Evidently for sapphire the energy loss is a rather smooth function of recoil energy and therefore a peak is not expected in a sapphire detector, such as the Nucleus 1g. We can therefore expect that if the low energy events observed by Nucleus are nuclear recoil events, the Nucleus data represents the true underlying recoil spectrum and a diamond detector should see a spectrum that contains a peak due to the energy loss feature. As the thresholds in germanium and silicon are modest, the same conclusion holds to a large degree also for SuperCDMS (Si) and Edelweiss (Ge).
\begin{figure*}[h!tb]
\begin{center}
\includegraphics[width=\linewidth]{dRdE_All_nonoise_combined.pdf}
\caption{Top left: The observed event rate in diamond for the underlying recoil spectrum given by the power law component of the fit function (\ref{nucleusfit}). The blue lines correspond to the best fit parameter values for Nucleus data, the green lines for SuperCDMS data and the purple lines for Edelweiss data. The solid curves show the observed rate after the energy loss, and black dotted curves if the energy loss is not simulated. The dashed curves show the exponential component of each fit. The curve in the bottom inset shows the ratio of the rate with/without the energy loss for the power law component. Top right: The same for a sapphire detector, using Nucleus fit. The bottom row shows same results for germanium (left, Edelweiss fit) and silicon (right, SuperCDMS fit).}
\label{dRdEplots}
\end{center}
\end{figure*}
To confirm this hypothesis, we have taken the power law part of the fit function (\ref{nucleusfit}) to represent the true underlying excess event rate, over which the scatter in the data is assumed to be statistical fluctuations, and simulated the resulting observed energy spectrum in diamond, sapphire, germanium and silicon detectors, assuming 1 eV energy resolution with Gaussian smearing. Due to the unknown origin of the events, we assume the same normalization for the event rate in units of events/[eV gram day] in each material, not correcting for possible differences e.g. in the nuclear scattering cross sections for different nuclei. While our sample data from the three experiments utilizing different target materials show clearly distinct event rates, we do not determine in detail to what part of the difference in overall normalization is explained e.g. by differences in the shielding or detector efficiencies of the experiments, and which part is due to underlying scattering cross sections.
The simulation is performed using 1 eV energy intervals. We sample recoil directions randomly, and for each direction increase the energy in 1 eV intervals. For each (direction,energy) combination we obtain energy loss due to defect creation by comparing the potential energy given by the MD interatomic potentials before and after the recoil event. Details of how the MD simulations are set up are provided in Refs. \cite{Sassi:2022njl,Kad17,Kadribasic:2020pwx}. This energy is then subtracted from the true recoil energy to produce the observed energy in the phonon measurement. The resulting observed recoil spectra are shown in figure \ref{dRdEplots} with the solid lines, while the black dotted lines show the observed event rate if the energy loss is not included in the simulation.
The top left panel shows the result for diamond and the top right panel for a sapphire detector. Indeed, for sapphire the solid and dotted lines are indistinguishable, confirming that we can treat the sapphire data as representing the true underlying event rate. For diamond we should instead expect to see a peak followed by a dip over the smooth power law. If the events are not due to nuclear recoils, this feature will not appear in the observed spectrum in diamond. Therefore the presence of the peak can be used as a verification of the nuclear recoil origin of the events.
In the bottom row of figure \ref{dRdEplots} we repeat this calculation for germanium, shown in the left panel, and for silicon, shown on the right panel. For these materials the defect creation threshold is in the 10--20 eV range, and therefore lies outside the selection window of our current data sample, and would in any case be masked by the exponential trigger noise. The effect of the energy loss is then just to move the expected curve slightly towards lower energy, but no prominent peak is produced such as in diamond. These considerations favor diamond as the best suited material for this task, where the threshold is both sharp and appears at high enough recoil energy, so that the peak in the observed spectrum appears over the smooth power-law component in our event rate model.
\section{Statistical analysis}
To quantify the statistical significance of the energy loss effect in a diamond detector, we generated simulated data sets from the expected event rate in diamond for the three best fit parameter sets shown in table \ref{bestfitparams}, and accounting for the energy loss as described above. We then computed the log-likelihood ratio for fitting the simulated data with the event rate containing the energy loss effect, allowing the overall normalization of the event rate vary but keeping the fit parameters $\{A,\alpha,B,\beta,C\}$ fixed to their best fit values, and with an event rate assuming no energy loss, but allowing to vary the fit parameters. This method allows to test if the effect of the energy loss could be mimicked by altering the fit parameters, which could mask the effect as the values of the parameters are not a priori known, and appear also to vary between experiments utilizing different target materials. Because we assume that the power law component of the fit function represents the excess part, we have repeated this procedure also for the fit function containing just the power law component, (i.e. omitting the exponential and constant components), and for a fit containing the power law and the constant components but omitting the exponential. The corresponding test statistic is given by
\begin{equation}
q_0 = 2\log \left( \frac{{\rm max}\, \mathcal{L}(\mu_{\rm loss})}{ {\rm max}\, \mathcal{L}(A,\alpha,B,\beta,C)} \right),
\end{equation}
where $\mathcal{L}(\mu_{\rm loss})$ and $\mathcal{L}(A,\alpha,B,\beta,C)$ are the likelihoods for drawing the data from the expected distribution of the event rate with or without the energy loss effect, respectively given by
\begin{equation}
\mathcal{L}(\{\lambda\}) = \prod\limits_{i=1}^N \frac{ e^{- n_{{\rm exp},i}(\{\lambda\})}}{n_{{\rm obs},i}!}\left( n_{{\rm exp},i}(\{\lambda\}) \right)^{n_{{\rm obs},i}}.
\end{equation}
Here $\{\lambda\} = \{ \mu_{\rm loss}\},\{ A,\alpha,B,\beta,C\}$ are the parameters that are varied to find the maximum likelihood, $n_{{\rm exp},i}(\{\lambda\})$ is the expected number of events in the energy bin $i$ for the event rate, and $n_{{\rm obs},i}$ is the 'observed' number of events in the bin $i$ in the simulated data set. For the number of bins we use $N=180$ corresponding to 1 eV bins from 20 eV to 200 eV.
For varying exposure we simulate 1000 data sets, and test if $q_0 > 9$ in at least 90\% of the iterations. If this test is successful, we conclude that the corresponding exposure is enough to identify the events as nuclear recoils at $3\sigma$ confidence level. The results of this analysis are shown in table \ref{tab:Results}, where we have used three versions of the event rate model: {\it{(i)}} the full model (\ref{nucleusfit}), {\it{(ii)}} the model without the exponential component and {\it{(iii)}} the model with only the power law component.
\begin{table*}[htb]
\centering
\begin{tabular}{c|c c | c c | c c}
& \multicolumn{2}{c}{Full fit} & \multicolumn{2}{c}{Power law + const} & \multicolumn{2}{c}{Power law only} \\
& $\mathcal{E}$ [gd] & $N_{\rm events}$ & $\mathcal{E}$ [gd] & $N_{\rm events}$ & $\mathcal{E}$ [gd] & $N_{\rm events}$ \\
\hline
Nucleus & 0.08 & 700 & 0.11 & 710 & 0.11 & 710 \\
SuperCDMS & 6.3 & 7\,900 & 17 & 2\,500 & 3.8 & 440 \\
Edelweiss & 750 & 190\,000 & 2.3 & 1\,300 & 0.75 & 440
\end{tabular}
\caption{Required exposure in gram days and the corresponding number of events for a $3\sigma$ observation of the energy loss effect in the recoil spectrum, for the three best fit parameter sets from table \ref{bestfitparams}, using the full fit function, omitting the exponential component, and omitting the exponential and constant components.}
\label{tab:Results}
\end{table*}
Looking at the first column, we notice that the larger amount of required events for SuperCDMS and especially for the Edelweiss fits is explained by the fact that the exponential part dominates the event rate up to higher energy than in the Nucleus fit, partly masking the peak feature, as shown by the dashed lines in the top left panel of figure \ref{dRdEplots}. These numbers point to the importance of suppressing the trigger noise so that it does not mask the event rate in the relevant window above $\gtrsim 20$ eV in order to make full use of the energy loss feature.
The second column shows the results for the fit function where the exponential part has been omitted; Assuming that the exponential component represents trigger noise or other measuring electronics related background, this will be device specific and not necessarily present in similar magnitude in the proposed diamond detector. Therefore it is interesting to examine how the event rate would look like in the absence of this noise component. For the Nucleus fit, we find a similar number of events required for the observation of the energy loss effect, as was the case with the exponential component present. This confirms the conclusion that if the exponential rise begins only at energies sufficiently below the defect creation threshold, as is the case for this fit, the peak feature due to energy loss will not be masked by it and the significance of the observation therefore does not depend on the presence or absence of this background. For the other two fits we instead find a clear reduction in the required number of events compared to the first column, and as expected the effect is more dramatic for the Edelweiss fit where the exponential masked a larger part of the recoil spectrum.
Finally, the third column presents the results for the pure power law model, omitting also the constant component. Since the Nucleus best fit value for the constant is zero, these numbers are identical to the second column for the Nucleus parameters. For the SuperCDMS and Edelweiss fits the power law indices are almost identical, resulting in equal numbers of required events. It appears that the steeper power law $\sim 2.7$ of SuperCDMS/Edelweiss fits compared to the \mbox{$\sim 1.4$} of the Nucleus fit makes the observation of the energy loss effect easier, with the required number of events 440 against 710 for the Nucleus fit.
\section{Conclusions}
The energy spectrum of the unidentified excess events observed at low energy can be reasonably fitted with a power law model. Assuming this smooth function as the true underlying event rate, we have shown that the nuclear recoil origin of these events can be tested with a diamond detector, due to the sharp defect creation threshold which gives rise to a prominent peak in the observed recoil spectrum. If the trigger noise can be suppressed down to $\sim\! 20$ eV this peak can be observed in a data set of $\sim\! 700$ events in the selection window [20-200]~eV with a diamond detector with 1~eV energy resolution. Assuming an event rate similar to the one observed in Nucleus, this corresponds to $\sim\!0.1$ gram days of data. In sapphire, germanium and silicon no peak is expected due to the smooth energy loss function, so that the data from existing measurements can reasonably be used as reference spectrum.
There is an ongoing research to calibrate ionization yield for very low energy nuclear recoils in Si and Ge detectors using low energy neutron beam and simultaneous scattering angle measurement\footnote{See e.g. presentation by Tarek Saab at the Excess2022 workshop, \url{https://indico.scc.kit.edu/event/2575/contributions/9684/}}. Very similar method can be used to validate and calibrate the effects of crystalline defects in nuclear recoil energy spectrum in our proposed diamond phonon mediated detectors.
\section*{Acknowledgements}
We thank Jesper Byggmästar and Antti Kuronen for their help in setting up the MD simulations, the Finnish Computing Competence Infrastructure (FCCI) for supporting this project with computational and data storage resources, and the Academy of Finland for financial support under project $\# 342777$.
|
2024-02-18T23:40:24.680Z
|
2022-10-05T02:14:01.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14495","language":"en","timestamp":1664936041000,"url":"https:\/\/arxiv.org\/abs\/2112.14495","yymm":"2112"}
|
proofpile-arXiv_000-10064
|
{"provenance":"002.jsonl.gz:10065"}
| null | null |
\section{Introduction}
Determining the allowed total angular momenta $J$ to which the individual
half-integer spins $j$ of $N$ identical particles
may couple is of primary importance in nuclear physics. Some values of $J$ are forbidden by
the Pauli exclusion principle, others occur more than once. Although that problem was investigated by many authors over the years, and despite the variety of approaches (number theory, recurrence relations, generating functions, etc.), exact analytical expressions for the number of states $P(M)$
with a given projection $M$ on the quantization axis,
the number of levels $Q(J)$ with spin $J$
or the total number of levels $Q_{\text{tot}}$
in a configuration are not known, except in very simple cases.
Zhao and Arima have shown that there are simple structures in $Q(J)$ for $j^3$ or $j^4$, and found empirical formulas \cite{Zhao2003}. In 2005, the same authors \cite{Zhao2005} showed that $Q(J)$ could be enumerated by the reduction from $SU(N+1)$ to $SO(3)$ and obtained analytical expressions of $Q(J)$ for four particles.
The same year, Talmi derived a recursion formula for $Q(J)$ \cite{Talmi2005}.
The latter quantity for $j^N$ is expressed in terms of $Q(J)$ for $(j-1)^N$, $(j-1)^{N-1}$ and $(j-1)^{N-2}$. In the same work, Talmi also proved some interesting results found empirically by Zhao and Arima \cite{Zhao2003}. Zhang \emph{et al} extended Talmi's recursion relation to boson systems and proved empirical formulas for five bosons. They also obtained the number of states with given spin for three and four bosons by using sum rules of six-$j$ and nine-$j$ symbols \cite{Zhang2008,Pain2011}. Five years later, Jiang \emph{et al} derived the analytical formulas for $Q(J)$ for three fermions in a single-$j$ shell and three bosons with spin $\ell$, by using a reduction rule from the $U(4)$ to the $O(3)$ group chain, $U(4) \supset Sp(4) \supset O(3)$ \cite{Hamermesh1962}, for $\tilde{N}$ virtual bosons which follow the $U(4)$ symmetry (i.e., spin 3/2) \cite{Jiang2013}. One has $\tilde{N}=2j-2$ for fermions and $\tilde{N}=2\ell$ for bosons. The authors were able to obtain analytical formulas of three bosons and fermions in a unified form and on a unified footing.
Let us consider a system of $N$ identical fermions in a single
$j$ (which is half-integer) shell of degeneracy $g=2j+1$, $m_i$ being the angular momentum
projection of electron state $i$ ($m_1=-j, m_2=-j+1, m_3=-j+2, \cdots, m_{g-1}=j-1, m_g=j$).
The maximum total angular momentum is
\begin{equation}J_{\text{max}}=(2j+1-N)N/2\end{equation}
and the minimum angular momentum $J_{\text{min}}$ is 0 if $N$ is even and $1/2$ if $N$ is odd.
The distribution $P(M)$ represents the number of $N$-fermion states having the total projection
(or magnetic quantum number) $M$.
The number $Q(J)$ of levels with angular momentum $J$ in a configuration can be obtained from the
distribution $P(M)$ of the $M$ values by means of the relations \cite{Bethe1936,*Landau1977}
\begin{subequations}\begin{gather}\label{eq:QvsP}
Q(J)=P\left(J\right)-P\left(J+1\right)\text{\quad if }J\le J_{\text{max}}-1,\\
Q(J_{\text{max}})=P(J_{\text{max}}).\end{gather}
\end{subequations}
In the following we use the notation $P(M;j,N)$ instead of $P(M)$ everytime it is necessary
to specify the angular momentum of the shell and the number of fermions.
The fundamental relation used in the present paper to get the number of states
$P(M;j,N)$ of $N$ fermions with spin $j$ and total magnetic quantum number $M$
has been derived by Talmi [Eq.~(1) in Ref.~\cite{Talmi2005}]
\begin{equation}\label{eq:Talmi}
P(M;j,N)=P(M;j-1,N)+P(M-j;j-1,N-1)+P(M+j;j-1,N-1)+P(M;j-1,N-2).
\end{equation}
A short alternative derivation is presented in Appendix \ref{sec:Talmirec}.
From the above relation (\ref{eq:QvsP}), one also gets easily the total number of levels
\begin{equation}\label{eq:totlev}
\sum_{J=J_\text{min}}^{J_\text{max}}Q(J;j,N)=P\left(J_{\text{min}};j,N\right)
\end{equation}
where $J_\text{min}=0$ (resp. 1/2) for $N$ even (resp. odd).
A simple expression for the total number of levels for $j^3$ was found using coefficients of fractional parentage \cite{Pain2019}. In the case of four fermions, no explicit formula could be obtained with the latter technique, only a triple summation involving nine-$j$ coefficients, or equivalently products of two six-$j$ symbols multiplied by Dunlap-Judd coefficients \cite{Dunlap1975}.
In the present work, using the recurrence relation (\ref{eq:Talmi}), we derive
explicit expressions for $P(M;j,3)$, $Q(J;j,3)$ (Section \ref{sec:PMj3}), $P(M;j,4)$,
and $Q(J;j,4)$ (Section \ref{sec:PMj4}), as well as for the total number of
$J$-levels in the case of five fermions (Section \ref{sec:Qtotj5}). This leads us
to deduce exact formulas for $Q_{\text{tot}}\left(j^3\right)$ (i.e., an alternative
derivation much simpler than the one previously published and relying on the use of
fractional parentage coefficients \cite{Pain2019}), for $Q_{\text{tot}}(j^4)$ and
for $Q_{\text{tot}}(j^5)$. To our knowledge, no expressions of the two latter
formulas were published elsewhere.
The algebraic forms of $Q(J;j,3)$ and $Q(J;j,4)$ are also likely to yield to sum
rules for six-$j$ symbols (Section \ref{sec:sum}). We also provide some additional
results, such as an alternative derivation of the Ginocchio-Haxton relation
(Section \ref{sec:sum}), cancellation properties and particular values of the
number of levels with a given angular momentum (Section \ref{sec:part}).
\section{Three-fermion systems}
\label{sec:PMj3}
\subsection{Total number of levels}\label{sec:Qtotj3}
The total number of levels will be derived from Eq.~(\ref{eq:totlev}). For three
particles, the relation (\ref{eq:Talmi}) is written as
\begin{equation}\label{eq:PN3s4t}
P\left(\frac12;j,3\right)=P\left(\frac12;j-1;3\right)+P\left(\frac12-j;j-1,2\right)
+P\left(\frac12+j;j-1,2\right)+P\left(\frac12;j-1,1\right).
\end{equation}
This provides us with a recurrence relation on $j$ for $P(1/2;j,3)$, which is
initialized by the value $P(1/2;3/2,3)$. Using the relation easily obtained by
considering the coupling of two momenta
\begin{equation}\label{eq:PMj2}
P(M;j,2)=\left\lfloor\frac{2j+1-|M|}{2}\right\rfloor
\end{equation}
where $\lfloor x\rfloor$ is the integer part of $x$, we get immediately, for $j$
half-integer,
\begin{subequations}\begin{gather}
P\left(\frac12-j;j-1,2\right)=\left\lfloor\frac{j}{2}-\frac14\right\rfloor\\
P\left(\frac12+j;j-1,2\right)=\left\lfloor\frac{j}{2}-\frac34\right\rfloor
\end{gather}\end{subequations}
and a rapid inspection of the cases $j=2n+1/2, j=2n+3/2$ shows that, since
$P(1/2;j-1,1)=1$ for $j\ge3/2$, one has
\begin{equation}
P\left(\frac12;j,3\right)=P\left(\frac12;j-1,3\right)+j-\frac12
\end{equation}
for $j\ge3/2$. Since the coupling of three angular momenta $j=1/2$ is not possible (Pauli exclusion principle), we have $P(1/2;1/2,3)=0$ and therefore
\begin{equation}\label{qtotj3}
Q_{\mathrm{tot}}\left(j^3\right)=P\left(\frac12;j,3\right)
=\sum_{i=1/2}^j(i-1/2)=\sum_{t=0}^{j-1/2}t=\frac12\left(j^2-\frac14\right)
\end{equation}
in agreement with the formula (36) of Ref.~\cite{Pain2019}.
\subsection{Determination of the \textit{M} distribution for three fermions}
\label{eq:PMj3}
\subsubsection{Case \textit{M} greater than \textit{j}}
We first determine $P(j+q;j,3)$ with $q$ positive integer ($q=1,2\dots J_\text{max}-j$). Using Talmi's formula and the explicit value (\ref{eq:PMj2})
one gets, after $p$ iterations,
\begin{subequations}\begin{align}
P(j+q;j,3) &= P(j+q;j-1,3)+P(q;j-1,2)\\
&= P(j+q;j-1,3)+\left\lfloor j-\frac{q+1}{2}\right\rfloor\\
&= P(j+q;j-2,3)+\left\lfloor j-\frac{q+1}{2}-\frac32\right\rfloor
+\left\lfloor j-\frac{q+1}{2}\right\rfloor\\
&\vdots\nonumber\\% \cdots -> \vdots 2021/12/20
&= P(j+q;j-p,3)+\sum_{t=0}^{p-1}\left\lfloor j-\frac{q+1}{2}-\frac{3t}{2}\right\rfloor,
\label{eq:PjpqjN3rec}
\end{align}
\end{subequations}
where we have used the property $P(2j+q-t;j-t,2)=0$ and $P(j+q-t;j-t,1)=0$
valid for $q>0$, and $0\le t\le p-1$.
We choose $p$ such that $P(j+q;j-p,3)$ vanishes
while $P(j+q;j-p+1,3)$ does not. This yields the conditions
\begin{equation}\label{eq:condp}
1\le j-\frac{q}{2}-\frac{3p}{2}+1,\quad
j-\frac{q}{2}-\frac{3p}{2}-\frac12 < 1,
\end{equation}
which amount to
\begin{equation}\label{eq:p_fnjmq_N3}
p=\left\lfloor\frac{2j-q}{3}\right\rfloor.
\end{equation}
Since $q$ can be even or odd, for $j$ half-integer $j-q/2$ is either integer or
half-integer. When evaluating $\lfloor(2j-q)/3\rfloor$ six cases must be
considered. One obtains for the value of the maximum index $p$
\begin{equation}\label{eq:pmaxj3}
p = \begin{cases}
2n & \quad\text{if }j-\frac{q}{2}=3n, 3n+1/2,\text{ or }3n+1\\
2n+1 & \quad\text{if }j-\frac{q}{2}=3n+3/2,3n+2,\text{ or }3n+5/2.\\
\end{cases}
\end{equation}
In the computation of the sum (\ref{eq:PjpqjN3rec}) with that value of $p$, we
note that $P(j+q;j-p,3)$ vanishes because of the conditions (\ref{eq:condp}).
We distinguish six cases, according to the maximum index (\ref{eq:pmaxj3}).
For instance if $j-q/2=3n$ the sum is, after reordering odd and even $t$ values,
\begin{subequations}\begin{align}
P(j+q;j,3) &= \lfloor1\rfloor+\lfloor5/2\rfloor+\lfloor4\rfloor+\cdots
\lfloor 3n-2\rfloor+\lfloor 3n-1/2\rfloor\\
&=1+4+\cdots+(3n-2)+2+5+\cdots+(3n-1)\\
&= \sum_{t=1}^n (3t-2)+\sum_{t=1}^n (3t-1)=3n^2
=\frac13\left(j-\frac{q}{2}\right)^2.
\end{align} \end{subequations}
\begin{table}[htb]
\centering\renewcommand*{\arraystretch}{1.75}
\begin{tabular}{ccccccc}
\hline\hline
$j-q/2$ & $3n$ & $3n+1/2$ & $3n+1$ & $3n+3/2$ & $3n+2$ & $3n+5/2$ \\
\hline
First term & $\lfloor1\rfloor$ & $\lfloor3/2\rfloor$ & $\lfloor2\rfloor$ &
$\lfloor1\rfloor$ & $\lfloor3/2\rfloor$ & $\lfloor2\rfloor$\\
Last term & $\lfloor3n-1/2\rfloor$ & $\lfloor3n\rfloor$ & $\lfloor3n+1/2\rfloor$
& $\lfloor3n+1\rfloor$ & $\lfloor3n+3/2\rfloor$ & $\lfloor3n+2\rfloor$\\
Sum & $3n^2$ & $n(3n+1)$ & $n(3n+2)$ & $3n(n+1)+1$ & $(n+1)(3n+1)$ & $(n+1)(3n+2)$\\
\hline\hline
\end{tabular}
\caption{Various cases for the computation of $P(j+q;j,3)$}\label{tab:calcPjq3}
\end{table}
The six cases are summed up in Table \ref{tab:calcPjq3}.
Expressing $n$ versus $j-q/2$, we obtain the desired formula
\begin{subequations}\begin{align}\label{eq:PjpqjN3}
P(j+q;j,3) &= \frac13\left(j-\frac{q}{2}\right)^2+\alpha(2j-q)\\
\text{with }\alpha(2j-q)&=\left(0,-\frac{1}{12},-\frac{1}{3},\frac{1}{4},
-\frac{1}{3},-\frac{1}{12}\right) \text{if }2j-q\bmod6=(0,1,2,3,4,5)
\text{ respectively.}
\end{align}\end{subequations}
For instance one can check for $q=1$
\begin{equation}\label{eq:Pj1jN3}
P(j+1;j,3) = \begin{cases}
\frac13\left(j-\frac{1}{2}\right)^2 & \text{if }j-1/2=3n\\
\frac13\left(j-\frac{3}{2}\right)\left(j+\frac{1}{2}\right)=
\frac13\left(j-\frac{1}{2}\right)^2 -\frac13 &
\text{if }j-1/2=3n+1 \text{ or }j-1/2=3n+2.
\end{cases}
\end{equation}
The formula (\ref{eq:PjpqjN3}) does not assume that $j$ is half-integer.
Instead of (\ref{eq:Pj1jN3}), we would have, for integer $j$,
\begin{equation}
P(j+1;j,3) = \begin{cases}
\frac13\left(j-\frac{1}{2}\right)^2-\frac{1}{12}= \frac13(j-1)j
& \text{if }j=3n+1 \text{ or if }j=3n\\
\frac13\left(j-\frac{1}{2}\right)^2+\frac14=\frac13(j^2-j+1)
& \text{if }j=3n+2.
\end{cases}
\end{equation}
\subsubsection{Case \textit{M} less than or equal to \textit{j}}
In this section one assumes $j$ half-integer such that $j\ge3/2$. From the basic
relation (\ref{eq:Talmi}), one writes
\begin{subequations}\label{eq:TalmiN3}\begin{gather}
P(j-q;j,3)=P(j-q;j-1,3)+S(j,q)\label{eq:recPjmq}\\
\text{where }S(j,q)=P(2j-q;j-1,2)+P(q;j-1,2)+P(j-q;j-1,1).\label{eq:defSjq}
\end{gather}\end{subequations}
The quantity $S(j,q)$ is easily transformed using the value (\ref{eq:PMj2}) and the
fact that $P(j-q;j-1,1)=1$ if $q>0$. Using this definition one easily checks that the
terms in $S(j,0)$ take the values 0, $j-1/2$, 0 respectively, so that $S(j,0)=j-1/2$.
If $q>0$ the identity (\ref{eq:PMj2}) provides the result
\begin{equation}
S(j,q)=\lfloor(q-1)/2\rfloor+\lfloor j-(q+1)/2\rfloor+1
\end{equation}
and considering the cases $q$ even or odd one easily verifies that, for $j\ge3/2$,
\begin{equation} S(j,q)=j-\frac12\end{equation}
which is also valid if $q=0$. The formula (\ref{eq:TalmiN3}) leads to a recurrence
relation
\begin{subequations}\begin{align}
P(j-q;j,3)&= P(j-q;j-1,3)+j-\frac12\\
&= P(j-q;j-2,3)+j-\frac32+j-\frac12\\
&\vdots\nonumber\\%
&= P(j-q;j-q-1,3)+\sum_{s=0}^q\left(j-\frac12-s\right)\\
&= P(j-q;j-q-1,3)+(q+1)\left(j-\frac{q+1}{2}\right).
\end{align}\end{subequations}
The initial value $ P(j-q;j-q-1,3)$ is derived from the expression (\ref{eq:Pj1jN3}).
One finds for $0\le q\le j-1/2$
\begin{equation}\label{eq:PjmqjN3}
P(j-q;j,3)=(q+1)\left(j-\frac{q+1}{2}\right)+\begin{cases}
\frac13\left(j-q-\frac32\right)^2 & \text{ if } j-q=3n+\frac32\\
\frac13\left(j-q-\frac52\right)\left(j-q-\frac12\right)
&\text{ if } j-q=3n+\frac12 \text{ or }3n+\frac52\end{cases}
\end{equation
or after simplification
\begin{subequations}\begin{align}\label{eq:PjmqjN3s}
P(j-q;j,3)&=\frac13\left(j+\frac{q}{2}\right)^2-\frac{q^2}{4}+\beta(j-q-1/2)\\
\text{with }\beta(n) &=\left(-\frac{1}{12},\frac14,-\frac{1}{12}\right),
\text{ if }n\bmod3=(0,1,2)\text{ respectively.}
\end{align}\end{subequations}
For instance, one obtains in the $q=0$ case
\begin{equation}\label{eq:PjjN3}
P(j;j,3)=\frac{j^2}3\begin{cases} +\frac14 &\text{ if } j=3n+3/2\\
-\frac{1}{12}=\frac13\left(j^2-\frac14\right)
&\text{ if } j=3n+1/2 \text{ or }3n+5/2.\end{cases}
\end{equation}
Such formulas can be generalized for $j$ integer but
the resulting expressions will be different.
The formula (\ref{eq:PjmqjN3}) was established for $q\ge0$. One can check that it
remains true for $q=-1$. Assuming (\ref{eq:PjmqjN3}) is valid for $q=-1$ we get
a piece-wise expression which is identical to (\ref{eq:Pj1jN3}).
It is worth mentioning that the relation (\ref{eq:PjmqjN3}) applies in particular
for $j-q=1/2, 3/2,\dots n+1/2$. A series of examples is provided in Appendix
\ref{sec:PlowMj3}. Finally if $j-q=n+1/2$, with $0\le n\le j-1/2$ one has
\begin{equation}
P(n+1/2;j,3)=P(n+1/2;n-1/2,3)+\frac12\left(j^2-\left(n-\frac12\right)^2\right).
\end{equation}
\subsubsection{General case}
The formulas (\ref{eq:PjpqjN3}), (\ref{eq:PjmqjN3s}) can be gathered in a single
equation, valid for any integer $q$. Using the Heaviside function $H(q)=1$ if
$q\ge0$, 0 otherwise, one has
\begin{equation}
P(j-q;j,3)=\frac13\left(j+\frac{q}{2}\right)^2-H(q)\frac{q^2}{4}
+H(q)\beta(j-q-1/2)+(1-H(q))\alpha(2j+q).
\end{equation}
Considering the various values of $2j+q\bmod6$ and $q\bmod2$, one can then
easily check that $\gamma=\alpha(2j+q)-\beta(j-q-1/2)$ is indeed a function of
$q$ and equal to $-(q\bmod2)/4$. The above equation transforms into
\begin{subequations}\begin{align}\label{eq:PMjN3gen}
P(j-q;j,3)&=\frac13\left(j+\frac{q}{2}\right)^2+\alpha(2j+q)
-H(q)\left[\frac{q^2}{4}+\gamma(q)\right]\\
\text{ with }\gamma(q)&=\left(0,-\frac14\right)\text{ for }q\bmod2=(0,1)
\end{align}\end{subequations}
if $-2j+3\le q\le j-1/2$, and $\alpha$ defined above (\ref{eq:PjpqjN3}).
\subsection{Distribution of the total angular momentum}
Using the fundamental relation (\ref{eq:QvsP}), the expression (\ref{eq:PjpqjN3})
allows us to derive the distribution of the total momentum $J$. The evaluation of
$P(j+q;j,3)-P(j+q+1;j,3)$ provides
\begin{align}\label{eq:QjpqjN3}\nonumber
Q(j+q;j,3) &= \frac{2j-q}{6}+q_{3p}\\\text{ with } q_{3p}&=
\left(0,-\frac{1}{6},-\frac{1}{3},\frac{1}{2},-\frac{2}{3},\frac{1}{6}\right)
\text{ for }2j-q\bmod6=(0,1,2,3,4,5)\text{ respectively.}
\end{align}
For instance one has $Q(j;j,3)=j/3+(-1/6,1/2,1/6)$ if $j-1/2\bmod3=0,1,2$
respectively, i.e., $Q(j;,j,3)=\lfloor(2j+1)/6\rfloor$. One also verifies that
$Q(j+1;j,3)=(2j-1)/6+(0,-1/3,-2/3)$ for $j-1/2\bmod3=0,1,2$ respectively, i.e.,
$Q(j+1,j,3)=\lfloor(2j-1)/6\rfloor$. Similarly, from (\ref{eq:PjmqjN3}), the
evaluation of $P(j-q;j,3)-P(j-q+1;j,3)$ provides the following expression
\begin{equation}\label{eq:QjmqjN3}
Q(j-q;j,3) = \frac{j-q}{3}+q_{3m}\text{ with }
q_{3m}=\left(-\frac{1}{6},\frac{1}{2},\frac{1}{6}\right)
\text{ for }j-q\bmod3=(1/2,3/2,5/2)\text{ respectively.}
\end{equation}
The expression (\ref{eq:PjmqjN3}) for $P(j-q;j,3)$ remains valid for $q=-1$,
therefore the above expression applies if $q=0$. One verifies easily that
(\ref{eq:QjpqjN3},\ref{eq:QjmqjN3}) are both correct for $q=0$.
One may also use the general expression (\ref{eq:PMjN3gen}). When computing the
difference $P(j-q;j,3)-P(j-q+1;j,3)$ some attention must be paid to the case $q=0$
for which $H(q)=1\ne H(q-1)=0$. However the term in the factor of $H(q-1)$ is
$(q-1)^2/4+\gamma(q-1)=0$ for $q=0$. The evaluation of $Q(j-q;j,3)$ is then
straightforward, defining $\overline{\alpha}(n)=\alpha(n)-\alpha(n-1)$,
$\overline{\gamma}(n)=\gamma(n)-\gamma(n-1)$. One gets
\begin{subequations}\begin{align}\label{eq:QJjN3gen}
Q(j-q;j,3)&=\frac16\left(2j+q-\frac12\right)+\overline{\alpha}(2j+q)
-H(q)\left[\frac{q}{2}-\frac14+\overline{\gamma}(q)\right]\\
\text{ with }\quad\overline{\alpha}(n)&=\left(\frac{1}{12},-\frac{1}{12},-\frac14,
\frac{7}{12},-\frac{7}{12},\frac14\right)
\text{for }n\bmod6=(0,1,2,3,4,5)\text{ respectively,}\\
\overline{\gamma}(q)&=\left(\frac14,-\frac14\right)
\text{ for }q\bmod2=(0,1) \text{ respectively,}
\end{align}\end{subequations}
with the conditions $3-2j\le q\le j-1/2$, since one must have $1/2\le j-q\le 3j-3$.
\section{Four-fermion systems}
\label{sec:PMj4}
\subsection{Determination of $P(M;j,4)$ if $M\ge2j$}
We first derive the expressions for $P(2j+p;j,4)$ which are easier to obtain than
the expressions for $P(2j-p;j,4)$. One has for any natural integer $p$
\begin{subequations}\begin{align}
P(2j+p;j,4) &= P(2j+p;j-1,4)+P(j+p;j-1,3)+P(3j+p;j-1,3)+P(2j+p;j-1,2)\\
&= P(2j+p;j-1,4)+P(j+p;j-1,3)\\
&= P(2j+p;j-2,4)+P(j+p;j-1,3)+P(j+p+1;j-2,3)\\
&= \sum_{s=1}^\sigma P(j+p+s-1;j-s,3)\label{eq:Sp2jpj4}
\end{align}\end{subequations}
where we used the properties $3j+p>J_\text{max}(j-1,3)$ and $2j+p>J_\text{max}(j-1,2)$.
The upper bound $\sigma$ in (\ref{eq:Sp2jpj4}) is chosen so that $P(j+p+s-1;j-s,3)$
is nonzero if $s=\sigma$ and zero if $s=\sigma+1$, implying that $P(2j+p;j-\sigma,4)=0$.
Explicitly
\begin{subequations}\begin{gather}
j+p+\sigma-1\le J_\text{max}(j-\sigma,3)=3j-3\sigma-3\nonumber\\
\text{and}\quad
j+p+(\sigma+1)-2> J_\text{max}(j-(\sigma+1),3)=3j-3(\sigma+1)-3\\
\sigma=\left\lfloor\frac{2j-p-2}{4}\right\rfloor\label{eq:sig}.
\end{gather}
\end{subequations}
In order that the above formulas be meaningful one must have $\sigma\ge1$ or
\begin{equation}
2j-p\ge6\quad\text{ from which }2j+p\le4j-6=J_\text{max}(j,4)\label{eq:condjp}.
\end{equation}
The sum (\ref{eq:Sp2jpj4}) will be calculated with formulas (\ref{eq:PjpqjN3}).
This lead us to define $d(s)=j'-q'/2$ with $j'=j-s, q'=p+2s-1$, or
\begin{equation}\label{eq:defds} d(s)=j-\frac{p}{2}-2s+\frac12.\end{equation}
The following analysis will be done according to the value of
$d(1)=j-p/2-3/2$. From (\ref{eq:sig})
\begin{equation}
\sigma=\left\lfloor\frac12d(1)+\frac14\right\rfloor
=\left\lfloor\frac12\left(j-\frac{p}{2}-\frac32\right)+\frac14\right\rfloor.
\end{equation}
To describe the procedure used to get $P(2j+p;j,4)$ let us consider the case
$d(1)=j-p/2-3/2=3n$, where $n$ is an integer. One has then $\sigma=\lfloor3n
+1/4\rfloor$ so that one must split the cases $n$ even and odd. If $n=2\nu$
with $\nu$ integer, then $\sigma=3\nu$. Writing $\theta(d)$ for the number on the
right of the bracket in (\ref{eq:PjpqjN3}), we have
\begin{subequations}\begin{equation}
P(2j+p;j,4) = \sum_{s=1}^\sigma\frac13\left(j-\frac{p}{2}-2s+\frac12\right)^2
+ \sum_{s=1}^\sigma \theta(d(s)).
\end{equation}
The quantity $d(s)=j-p/2-2s+1/2$ is equal to $j-p/2-3/2=3n=6\nu, 3n-2, 3n-4,
\dots 2$ for $s=1\dots\sigma$.
Since $\sigma=3\nu$, there are $\nu$ elements in the sum such that $d\bmod3=0$,
and as many such that $d\bmod3=1$ and $d\bmod3=2$. The sum of $\theta(d)$ is
according to (\ref{eq:PjpqjN3}), $(0-1/3-1/3)\nu=-2\nu/3$. The final result is
\begin{align}
P(2j+p;j,4) &= \sum_{t=1}^\sigma\frac43 t^2-\frac23\nu\\
&= \frac{2}{9}(3\nu)(3\nu+1)(6\nu+1)-\frac23\nu=6\nu^2(2\nu+1)\\
&=\frac{1}{18}\left(j-\frac{p}{2}-\frac32\right)^2\left(j-\frac{p}{2}+\frac32\right)
\text{ in the case }j-\frac{p}{2}-\frac32=6\nu.\label{eq:P2jpj4d6n0p}
\end{align}\end{subequations}
The procedure must be repeated in the cases $d(1)=3n,3n+1/2,3n+1,3n+3/2,3n+2,3n+5/2$,
with $n=2\nu$, $n=2\nu+1$ where $\nu$ is integer.
From the expression (\ref{eq:PjpqjN3}), one notes that
the sought number $P(M;j,4)$ is a sum of $P(j+t;j,3)$ that can be written as
\begin{subequations}\begin{align}\label{eq:P2jpj4s}
P(2j+p;j,4)&=\frac13\sum_{s=1}^\sigma [d(s)]^2-\sum_{s=1}^\sigma\theta(d(s))\\
&=\frac43\sum_{s=1}^\sigma \left(\frac{j}{2}-\frac{p-1}{4}-s\right)^2
-\sum_{s=1}^\sigma\theta(d(s))\\
&=\frac43\sum_{t=0}^{\sigma-1}\left(t+b\right)^2
-\sum_{s=1}^\sigma\theta(d(s))\label{eq:P2jpj4}
\end{align}\end{subequations}
where $b=j/2-(p-1)/4-\sigma$ is the smallest value of the
quantity $d(s)/2$ in the sum (\ref{eq:P2jpj4s}).
Therefore the computation of $P(2j+p;j,4)$ amounts to
obtaining the sum of the squares of numbers in arithmetical progression, which is
easy to evaluate. This sum must be corrected by the term $\sum_{s}\theta(d(s))$.
\begin{table}[htb]
\centering\renewcommand*{\arraystretch}{1.5}
\begin{tabular}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}
\hline\hline
$j-\frac{p}{2}-\frac{3}{2}$ & $6\nu$ & $6\nu+1$ & $6\nu+2$ & $6\nu+3$ & $6\nu+4$ & $6\nu+5$\\
\hline
$\sigma$ & $3\nu$ & $3\nu$ & $3\nu+1$ & $3\nu+1$ & $3\nu+2$ & $3\nu+2$ \\
$b$ & $1$ & $3/2$ & $1$ & $3/2$ & $1$ & $3/2$\\
$\sum_s\theta(d(s))$ & $-2\nu/3$ & $-2\nu/3$ & $-(2\nu+1)/3$ & $-2\nu/3$ & $-2(\nu+1)/3$ & $-(2\nu+1)/3$ \\
$P(2j+p;j,4)$& $6\nu^2(2\nu+1)$ & $3\nu(2\nu+1)^2$ & $e_2$ & $3(\nu+1)(2\nu+1)^2$ & $6(\nu+1)^2(2\nu+1)$ & $e_5$ \\
\hline\hline
\end{tabular}
\caption{Various cases for the computation of $P(2j+p;j,4)$ in the case
$p=2q$ even, i.e., $j-p/2+1/2$ integer. The number of states $P(2j+p;j,4)$ is
given by $\frac43\sum_{t=0}^{\sigma-1}(t+b)^2+S$ with $S=\sum_s\theta(d(s))$.
The number of terms in the sum defining $P(2j+p;j,4)$ is
$\sigma=\lfloor d(1)/2+1/4\rfloor$. One has $e_2=(2\nu+1)(6\nu^2+6\nu+1)$,
$e_5=(\nu+1)(12\nu^2+24\nu+11)$ for $j-p/2-3/2=6\nu+2$, $6\nu+5$ respectively.
\label{tab:calcP2jq4e}
}
\end{table}
\begin{table}[htb]
\centering\renewcommand*{\arraystretch}{1.5}
\begin{tabular}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}
\hline\hline
$j-\frac{p}{2}-\frac{3}{2}$ & $6\nu+1/2$ & $6\nu+3/2$ & $6\nu+5/2$ &
$6\nu+7/2$ & $6\nu+9/2$ & $6\nu+11/2$ \\
\hline
$\sigma$ & $3\nu$ & $3\nu+1$ & $3\nu+1$ & $3\nu+2$ & $3\nu+2$ & $3\nu+3$\\
$b$ & $5/4$ & $3/4$ & $5/4$ & $3/4$ & $5/4$ & $3/4$ \\
$\sum_s\theta(d(s))$ & $\nu/12$ & $(\nu+3)/12$ & $(\nu-1)/12$ & $(\nu+2)/12$
& $(\nu+2)/12$ & $(\nu+1)/12$ \\
$P(2j+p;j,4)$ & $\nu(12\nu^2+9\nu+2)$ & $o_1$ & $o_2$ &
$(\nu+1)(12\nu^2+15\nu+5)$& $3(\nu+1)^2(4\nu+3)$ & $3(\nu+1)^2(4\nu+5)$\\
\hline\hline
\end{tabular}
\caption{Various cases for the computation of $P(2j+p;j,4)$ in the case $p=2q+1$
odd, i.e., $j-p/2+1/2$ half-integer. One has $o_1=12\nu^3+15\nu^2+6\nu+1$ for
$j-p/2-3/2=6\nu+3/2$ and $o_2=12\nu^3+21\nu^2+12\nu+2$ for $j-p/2-3/2=6\nu+5/2$.
See Table \ref{tab:calcP2jq4e} for details.\label{tab:calcP2jq4o}
}
\end{table}
The parameters $\sigma,b,\sum_{s}\theta(d(s))$ corresponding to each case are
described in the Tables \ref{tab:calcP2jq4e} and \ref{tab:calcP2jq4o} for
$p$ even and odd respectively. The last line of these tables provides the
number of states as given by (\ref{eq:P2jpj4}). The $\nu$-dependent values
can be expressed back versus the physical quantities $j,p$. With the
additional definition
\begin{equation}X=j-\frac{p}{2}-\frac{1}{2}\label{eq:defX}\end{equation}
the expressions for $P(2j+p;j,4)$ will be even simpler.
In the case $p=2q$ even, from table \ref{tab:calcP2jq4e} results, a detailed
inspection proves that the expression of $P(2j+2q;j,4)$ versus $j,q$ is identical
for each pair of adjacent columns. Namely, columns $6\nu$ (resp. $6\nu+1$, $6\nu+2$)
and $6\nu+3$ (resp. $6\nu+4$, $6\nu+5$) provide the same result, so that, the
$P(2j+2q;j,4)$ value does not depend on $j-q-1/2\bmod6$ but on $j-q-1/2\bmod3$.
Expressing $\nu$ versus $j,q$ one has, using the definition (\ref{eq:defX}),
\begin{equation}\label{eq:P2jpj4e}
P(2j+2q;j,4)=\begin{cases}
\frac{1}{18}\left(j-q-\frac32\right)^2\left(j-q+\frac32\right)
=\frac{X^3}{18}-\frac{X}{6}+\frac19 &\quad\text{if }j-q+\frac32=3n\\
\frac{1}{18}\left(j-q-\frac52\right)\left(j-q+\frac12\right)^2
=\frac{X^3}{18}-\frac{X}{6}-\frac19 &\quad\text{if }j-q+\frac32=3n+1\\
\frac{1}{18}\left(j-q-\frac12\right)\left[\left(j-q-\frac12\right)^2-3\right]
=\frac{X^3}{18}-\frac{X}{6} &\quad\text{if }j-q+\frac32=3n+2.
\end{cases}
\end{equation}
In the case of odd $p$ one must also express $P(2j+2q+1;j,4)$ versus $j,q$ or more
precisely versus $X=j-q-1$. The six cases considered in Table \ref{tab:calcP2jq4o}
provide as many different expressions. As seen on Eq.~(\ref{eq:P2jpj4e}), the final
expressions are simpler as functions of $X$. One has
\begin{align}\label{eq:P2jpj4o}
P(2j+2q+1;j,4)&=
\frac{X^3}{18}-\frac{X}{24}+\psi\left(j-q-\frac{1}{2}\right)\nonumber\\
\text{ where }\psi\left(j-q-\frac{1}{2}\right)&=\left(-\frac{1}{72},\frac{1}{72},
-\frac{1}{8},\frac{17}{72},-\frac{17}{72},\frac{1}{8}\right)
\quad\text{if }j-q-\frac{1}{2}\bmod6=(0,1,2,3,4,5)\text{ respectively.}
\end{align}
\subsection{Determination of $P(M;j,4)$ if $M<2j$}
From Talmi's equation one has, assuming $q$ positive integer
\begin{equation}
P(2j-q;j,4)=P(2j-q;j-1,4)+P(3j-q;j-1,3)+P(j-q;j-1,3)+P(2j-q;j-1,2)
\end{equation}
which suggests implementing a recurrence on $q$. Indeed the elements $P(3j-q;j-1,3),
P(j-q;j-1,3),P(2j-q;j-1,4)$ are known. In addition
\begin{equation}P(2j-q,j-1,4)=P(2(j-1)-(q-2),j-1;4)\end{equation}
shows that the expression for $P(2j+1,j;4)$ (resp. $P(2j,j;4)$) obtained
above --- using (\ref{eq:P2jpj4o}), (\ref{eq:P2jpj4e}) respectively ---, allows us
to get $P(2j-1;j,4)$ (resp. $P(2j-2;j,4)$).
This leads us to split the discussion according to the parity of $q$.
We first define
\begin{equation} F(j,q)=P(3j-q;j-1,3)+P(2j-q;j-1,2).\label{eq:defF}\end{equation}
Using the expression (\ref{eq:PjpqjN3}), it is easy to prove that, if $j\ge q+1/2$,
\begin{equation} F(j,2q)=\begin{cases}
q^2/3 & \text{ if } q\bmod3=0\\ (q^2-1)/3 & \text{ if } q\bmod3=1, 2,
\end{cases}\label{eq:Fj2q}\end{equation}
and, assuming again $j\ge q+1/2$, that
\begin{equation} F(j,2q+1)=\begin{cases}
q(q+1)/3 & \text{ if } q\bmod3=0, 2\\ (q^2+q+1)/3 &\text{ if } q\bmod3=1.
\end{cases}\label{eq:Fj2q1}\end{equation}
\subsubsection{Computation of $P(2j-2p;j,4)$}
We first consider the case where $q$ is even.
The above formula for $F(j,2p)$ provides us with the expression for
$P(2j-2;j,4)$. Using (\ref{eq:PjmqjN3}) for $P_3=P(j-q,j-1,3)$ and
(\ref{eq:P2jpj4e})
for $P_1=P(2j-2;j-1,4)$ we get $P(2j-2;j,4)=P_1+P_3$. Writing $x=j-3/2$,
one considers three cases according to $j\bmod3$.
\begin{itemize}\begin{subequations}\label{eq:P2jm2j4}
\item[$\bullet$]If $j\bmod3=1/2$, $P_1=x^3/18-x/6-1/9,
P_3=2(j-2)+\frac13(j-7/2)^2$, so that
\begin{equation}P(2j-2;j,4)=\frac{1}{144}\left(8j^3+12j^2-16j+5\right)
=\frac{(j+1/2)^3}{18}-\frac{j+1/2}{6}+\frac19.
\end{equation}
\item[$\bullet$]If $j\bmod3=3/2$, $P_1=x^3/18-x/6,
P_3=2(j-2)+\frac13(j-9/2)(j-5/2)$, whence
\begin{equation}P(2j-2;j,4)=\frac{1}{144}\left(8j^3+12j^2-16j-27\right)
=\frac{(j+1/2)^3}{18}-\frac{j+1/2}{6}-\frac19.
\end{equation}
\item[$\bullet$]If $j\bmod3=5/2$, $P_1=x^3/18-x/6+1/9,
P_3=2(j-2)+\frac13(j-9/2)(j-5/2)$, from which
\begin{equation}P(2j-2;j,4)=\frac{1}{144}\left(8j^3+12j^2-16j-11\right)
=\frac{(j+1/2)^3}{18}-\frac{j+1/2}{6}.
\end{equation}\end{subequations}
\end{itemize}
A series of similar computations for greater values of $q$ has been performed
and leads us to propose the formula
\begin{equation}
P(2j-2q;j,4)=\frac{(j+q-1/2)^3}{18}-\frac{(j+q-1/2)}{6}
-f(q)+\theta(j+q)\label{eq:hypP2jm2q}
\end{equation}
which we will prove by recurrence on $q$. The initial computations
show that $f(0)=0, f(1)=0$, and the general expression for $f(q)$ will be
obtained below. The initial value $q=0$ (\ref{eq:P2jm2j4}) requires that
\begin{equation} \theta(j+q)=\begin{cases}
-1/9 & \text{ if } j+q+1/2\bmod3=0\\
0 & \text{ if } j+q+1/2\bmod3=1\\
+1/9 & \text{ if } j+q+1/2\bmod3=2. \end{cases}\label{eq:theta}
\end{equation}
Let us assume the recurrence (\ref{eq:hypP2jm2q}) true up to $q=p$ (e.g.,
$p=0$ or $1$), and prove it for $q=p+1$. With definition (\ref{eq:Fj2q})
\begin{equation}\label{eq:Fj2p2}
F(j,2p+2)=\frac13(p+1)^2+\varepsilon_{p+1},\text{ with }
\varepsilon_{p+1}=-\frac13\text{ if }p+1\bmod3=1 \text{ or } 2,
\text{ otherwise }0,
\end{equation}
we get from the fundamental relation (\ref{eq:Talmi})
\begin{equation}\label{eq:Pj2p}
P(2j-2p-2;j,4)=P(2(j-1)-2p;j-1,4)+P(j-2p-2;j-1,3)+F(j,2p+2).
\end{equation}
The recurrence hypothesis applies to the first term of (\ref{eq:Pj2p})
\begin{equation}
P(2(j-1)-2p;j,4)=\frac{(j+p-3/2)^3}{18}-\frac{(j+p-3/2)}{6}-f(p)
+\theta(j-1,p).
\end{equation}
The second term of (\ref{eq:Pj2p}) is obtained from (\ref{eq:PjmqjN3})
\begin{subequations}\begin{align}
P(j-2p-2;j-1,3)&=\frac{1}{3}(j+p-1/2)^2-(p+1/2)^2-\frac{1}{12}+\tau(j+p)\\
\text{ with }
\tau(j+p)&=\frac13 \text{ if }j-2p-\frac12\bmod3 = j+p-\frac12\bmod3 = 0, \\
\text{ and }\tau(j+p)&=0 \text{ if }j-2p-\frac12\bmod3 = j+p-\frac12\bmod3 = 1 \text{ or } 2.
\end{align}\end{subequations}
In order to verify the recurrence for $q=2p+2$, according to Eq.
(\ref{eq:Pj2p}) one must verify for every $j$
\begin{equation}\label{eq:cond_recpair}
\Delta-f(p+1)+\theta(j+p+1)=-f(p)+\theta(j-1+p)+\varepsilon_{p+1}+\tau(j+p)
\end{equation}
where $\Delta$ contains the terms function of $j,p$ except $f(p)$ and the
quantities defined modulo 3
\begin{align}\nonumber
\Delta=\frac{(j+p+1/2)^3}{18}-\frac{(j+p+1/2)}{6}&-\frac{(j+p-3/2)^3}{18}
+\frac{(j+p-3/2)}{6}\\&-\frac{1}{3}(j+p-1/2)^2+(p+1/2)^2+\frac{1}{12}
-\frac{(p+1)^2}{3}.
\end{align}
After some basic algebraic manipulations one obtains
\begin{equation} \Delta=\frac23p^2+\frac{p}{3}-\frac29.\end{equation}
In addition, one may verify that $\delta=\theta(j+p+1)-\theta(j-1+p)
-\tau(j+p)$ does not depend on $j$. Indeed
\begin{equation}
\delta=\left\{\begin{matrix}1/9\\-1/9\\0\end{matrix}\right.
-\left\{\begin{matrix}-1/9\\0\\1/9\end{matrix}\right.
-\left\{\begin{matrix}1/3\\0\\0\end{matrix}\right.
\text{\quad if }j+p-\frac12\bmod3=
\left\{\begin{matrix}0\\1\\2\end{matrix}\right.
\end{equation}
which leads to $\delta=-1/9$ in all cases. Equation (\ref{eq:cond_recpair})
may be rewritten, using $\varepsilon_{p+1}$ as given by (\ref{eq:Fj2p2})
\begin{subequations}\begin{align}
f(p+1)-f(p)&=\Delta+\delta-\varepsilon_{p+1}\\
&=\frac23p^2+\frac{p}{3}-\frac13
\begin{cases}+\frac13 &\text{ if }p\bmod3 = 0\text{ or }1\\
&\text{ if }p\bmod3 = 2\\
\end{cases}
\end{align}\end{subequations}
which is
\begin{equation}\label{eq:diffp}
f(p+1)-f(p)=
\begin{cases}\frac13p(2p+1) &\text{ if }p\bmod3 = 0\text{ or }1\\
\frac13(p+1)(2p-1)=\frac13p(2p+1)-\frac13 &\text{ if }p\bmod3 = 2.
\end{cases}
\end{equation}
Since $f(p+1)-f(p)$ is indeed independent of $j$ the recurrence
assumption (\ref{eq:hypP2jm2q}) is verified. The proof is completed by
the determination of $f(p)$. Applying Eq.~(\ref{eq:diffp}) for
$p,p+1,p+2$, we get, whatever $p\bmod3$,
\begin{equation}\label{eq:diff3p}
f(p+3)-f(p)=\frac{p}{3}(2p+1)+\frac{(p+1)}{3}(2p+3)
+\frac{(p+2)}{3}(2p+5)-\frac13=2p^2+5p+4.
\end{equation}
From the known initial values $f(0)=f(1)=0$ using (\ref{eq:diffp})
one gets $f(2)=1$, and more generally $f(p)$
\begin{equation}
f(3n+p_0)-f(p_0)=\sum_{t=0}^{n-1}\left[2(3t+p_0)^2+5(3t+p_0)+4\right]
\end{equation}
and considering $p_0=0,1,2$ separately we obtain
\begin{subequations}\begin{align}\label{eq:valfp}
f(p)&=f_0(p)=
\frac{m}{2}(12m^2-3m-1)=\frac29p^3-\frac{p^2}{6}-\frac{p}{6} &\text{ if }p=3m\\
f(p)&=f_1(p)= \frac{m}{2}(12m^2+9m+1)=\frac29p^3-\frac{p^2}{6}-\frac{p}{6}+\frac19
&\text{ if }p=3m+1\\
f(p)&=f_2(p)= \frac{(m+1)}{2}(12m^2+9m+2)=\frac29p^3-\frac{p^2}{6}-\frac{p}{6}+\frac29
&\text{ if }p=3m+2.
\end{align}\end{subequations}
A further generalization consists in verifying that the expression
(\ref{eq:hypP2jm2q}) may be applied \emph{even for $q$ negative}
provided one cancels the $f(q)$ term. Indeed comparing this expression
to the known values (\ref{eq:P2jpj4e}), one notes that
\begin{equation}
P(2j-2q;j,4)=\frac{(j+q-1/2)^3}{18}-\frac{(j+q-1/2)}{6}+\theta(j+p)
\quad\text{if }q<0
\end{equation}
from which one gets, whatever the sign of the integer $q$,
\begin{equation}
P(2j-2q;j,4)=\frac{(j+q-1/2)^3}{18}-\frac{(j+q-1/2)}{6}-H(q)f(q)
+\theta(j+q)
\end{equation}
$H(q)$ being the Heaviside function, $H(q)=1$ if $q\ge0$, 0 otherwise.
\subsubsection{Computation of $P(2j-2p-1;j,4)$}
As a first example, the computation of $P(2j-1;j,4)$ is detailed in
Appendix \ref{sec:P2jm1j4}. In order to discover the general formula,
we also got expressions for $P(2j-3;j,4)$ and $P(2j-5;j,4)$.
An analysis on $P(M;j,4)$ with $2j-M$ odd similar to the case $2j-M$
even leads us to propose the relation
\begin{equation}\label{eq:hypP2jm2q1}
P(2j-2q-1;j,4)=\frac{(j+q)^3}{18}-\frac{(j+q)}{24}
-g(q)+\phi(j+q)
\end{equation}
which will be demonstrated by recurrence. The direct computation in the
first two cases show that $g(0)=0$ and $g(1)=0$. From the analysis of
Appendix \ref{sec:P2jm1j4} one imposes that, if $q\ge0$,
\begin{equation} \phi(j+q)=\left(\frac{1}{72},-\frac{1}{8},
\frac{17}{72},-\frac{17}{72},\frac{1}{8},-\frac{1}{72}\right)\text{ if }
j+q-\frac12\bmod6=(0,1,2,3,4,5)\text{ respectively.}\label{eq:valphi}
\end{equation}
Assuming that (\ref{eq:hypP2jm2q1}) is true up to $q=p$, we now try to
prove it for $q=p+1$. Using the value (\ref{eq:Fj2q1})
\begin{equation}\label{eq:Fj2p3}
F(j,2p+3)=\frac13(p+1)(p+2)+\mu_{p+1},\text{ where }
\mu_{p+1}=\frac13\text{ if }(p+1)\bmod3=1, \text{ otherwise }\mu_{p+1}=0,
\end{equation}
the fundamental relation (\ref{eq:Talmi}) may be written
\begin{equation}\label{eq:Pj2p1}
P(2j-2p-3;j,4)=P(2(j-1)-2p-1;j-1,4)+P(j-2p-3;j-1,3)+F(j,2p+3).
\end{equation}
The recurrence hypothesis is again applied to the first term at the second
member of (\ref{eq:Pj2p1})
\begin{equation}
P(2(j-1)-2p-1;j,4)=\frac{(j+p-1)^3}{18}-\frac{(j+p-1)}{24}-g(p)
+\phi(j-1,p).
\end{equation}
The second term at the second member of (\ref{eq:Pj2p1}) is given by
(\ref{eq:PjmqjN3s})
\begin{subequations}\begin{align}
P(j-2p-3;j-1,3)&=\frac{1}{3}(j-1+p+1)^2-(p+1)^2-\frac{1}{12}+\upsilon(j+p)\\
\text{ with }
\upsilon(j+p)&=\frac13 \text{ if }j-2p-\frac12\bmod3 = j+p-\frac12\bmod3 = 1, \\
\text{ and }\upsilon(j+p)&=0 \text{ if }j-2p-\frac12\bmod3 = j+p-\frac12\bmod3 = 0 \text{ or } 2.
\end{align}\end{subequations}
In order to verify the recurrence for $q=2p+3$, from (\ref{eq:Pj2p1})
one must have for every $j$
\begin{equation}\label{eq:cond_recimpair}
\Delta'-g(p+1)+\phi(j+p+1)=-g(p)+\phi(j-1+p)+\mu_{p+1}+\upsilon(j+p)
\end{equation}
where $\Delta'$ contains the terms function of $j,p$ except $g(p)$ and the
terms defined modulo 3 or modulo 6
\begin{equation}
\Delta'=\frac{(j+p+1)^3}{18}-\frac{(j+p+1)}{24}-\frac{(j+p-1)^3}{18}
+\frac{(j+p-1)}{24}-\frac{1}{3}(j+p)^2+(p+1)^2+\frac{1}{12}-\frac13(p+1)(p+2).
\end{equation}
After some algebra one gets
\begin{equation} \Delta'=\frac23p^2+p+\frac49.\end{equation}
In addition one can check that $\delta'=\phi(j+p+1)-\phi(j-1+p)
-\upsilon(j+p)$ is independent of $j$. Indeed
\begin{equation} \delta'=
\left\{\begin{matrix}-1/8\\17/72\\-17/72\\1/8\\-1/72\\1/72\end{matrix}\right.
-\left\{\begin{matrix}-1/72\\1/72\\-1/8\\17/72\\-17/72\\1/8\end{matrix}\right.
-\left\{\begin{matrix}0\\1/3\\0\\0\\1/3\\0\end{matrix}\right.
\text{\quad if }j+p-\frac12\bmod3=
\left\{\begin{matrix}0\\1\\2\\3\\4\\5\end{matrix}\right.
\end{equation}
therefore $\delta'=-1/9$ in all cases. Equation (\ref{eq:cond_recimpair})
may be rewritten as
\begin{equation}
g(p+1)-g(p)=\Delta'+\delta'-\mu_{p+1}=\frac23p^2+p+\frac13-\mu_{p+1}
\end{equation}
which is, using $\mu_{p+1}$ as given by (\ref{eq:Fj2p3}),
\begin{equation}\label{eq:difgp}
g(p+1)-g(p)=
\begin{cases}
\frac13p(2p+3)=\frac13(p+1)(2p+1)-\frac13 &\text{ if }p\bmod3 = 0\\
\frac13(p+1)(2p+1) &\text{ if }p\bmod3 = 1\text{ or }2.
\end{cases}
\end{equation}
Since $g(p+1)-g(p)$ is indeed independent of $j$ the recurrence
relation (\ref{eq:hypP2jm2q1}) is proved and $g(p)$ may be computed.
One may use (\ref{eq:difgp}) in order to get $g(1)=0, g(2)=2$ from $g(0)=0$.
One has then, whatever $p\bmod3$,
\begin{equation}\begin{split}\label{eq:difg3p}
g(p+3)-g(p)=\frac{(p+1)}{3}(2p+1)+\frac{(p+2)}{3}(2p+3)
+\frac{(p+3)}{3}(2p+5)-\frac13\\=2p^2+7p+7.
\end{split}\end{equation}
From the values of $g(p)$ for $p_0=0,1,2$ one obtains the general expression
\begin{equation}
g(3n+p_0)-g(p_0)=\sum_{t=0}^{n-1}\left[2(3t+p_0)^2+7(3t+p_0)+7\right]
\end{equation}
and splitting cases $p_0=0,1,2$,
\begin{subequations}\label{eq:valgp}\begin{align}
g(p)&=g_0(p)=
\frac{n}{2}(12n^2+3n-1)=\frac29p^3+\frac{p^2}{6}-\frac{p}{6} &\text{ if }p=3n\\
g(p)&=g_1(p)= \frac{n}{2}(12n^2+15n+5)=\frac29p^3+\frac{p^2}{6}-\frac{p}{6}-\frac29
&\text{ if }p=3n+1\\
g(p)&=g_2(p)= \frac{(n+1)}{2}(12n^2+15n+4)=\frac29p^3+\frac{p^2}{6}-\frac{p}{6}-\frac{1}{9}
&\text{ if }p=3n+2.
\end{align}\end{subequations}
As for even $p$, comparing expressions (\ref{eq:P2jpj4o}) and (\ref{eq:hypP2jm2q1})
for $q$ negative or positive, one may write the general relation
\begin{equation}
P(2j-2q-1;j,4)= \frac{(j+q)^3}{18}-\frac{(j+q)}{24}-H(q)g(q)+\phi(j+q)
\end{equation}
where $H(q)$ is the Heaviside function, and $\phi(j+q)$ is given by
(\ref{eq:valphi}).
\subsubsection{General expression for the distribution of the
magnetic quantum number}
The formulas (\ref{eq:hypP2jm2q}), (\ref{eq:hypP2jm2q1}) for $P(2j-n;j,4)$ in the
cases $n$ even and odd can even be gathered in a single expression. One
notices that the first term can be simply written as $(j+(n-1)/2)^3/18$,
while the second is $-(j+(n-1)/2)/6+\pi(n)(j+(n-1)/2)/8$ where $\pi(n)$ is 0
if $n$ is even, 1 if $n$ is odd. The third term of the quoted formulas may
also be unified, noting that from the values (\ref{eq:valfp},\ref{eq:valgp}),
one has
\begin{equation}g_2(x-1/2)=f_1(x)-\frac18\end{equation}
which allows us to write $f(n/2)$ and $g((n-1)/2)$ with a single formula,
namely $f(n/2)=f_1(n/2)+\xi(n)$ with $\xi(n)=(-1/9,0,1/9)$ if $n/2\bmod3=
0,1,2$ respectively, and $g((n-1)/2)=f_1(n/2)+\xi(n)$ with
$\xi(n)=(-1/8+1/9,-1/8-1/9,-1/8)$ for $(n-1)/2\bmod3=0,1,2$ respectively.
Finally the term $\theta(j+n/2)$ and $\phi(j+(n-1)/2)$ in these formulas can
be collected in a single expression, if one considers $2j+n\bmod12$ value. If
$n$ is even, from the expression (\ref{eq:theta}) one may write this term as
$\theta(j+n/2)=\left(0,1/9,-1/9,0,1/9,-1/9\right)$ for $2j+n-1\bmod12=
(0,2,4,6,8,10)$ respectively. If $n$ is odd, from (\ref{eq:valphi}) this term
is $\phi(j+q)=\phi(j+(n-1)/2)=(1/72,-1/8,17/72,-17/72, 1/8,-1/72)$ for
$2j+n-1\bmod12=(1,3,5,7,9,11)$ respectively. One obtains the single formula
\begin{subequations}\label{eq:PMjN4gen}\begin{align}
P(2j-n;j,4)&=\frac{1}{18}\left(j+\frac{n-1}2\right)^3
-\left(\frac16-\frac{\pi(n)}{8}\right)\left(j+\frac{n-1}2\right)
-H(n)\left[f_1\left(\frac{n}2\right)+\xi(n)\right]+\omega(2j+n-1)\\
\text{with}\quad \pi(n)&=n\bmod2,\quad \label{eq:xi}
\xi(n)=\left(-\frac{1}{9},-\frac{1}{72},0,-\frac{17}{72},\frac{1}{9},
-\frac{1}{8}\right)\text{ if } n\bmod6=(0,1,2,3,4,5)\text{ respectively},\\
f_1(n)&=\frac29n^3-\frac{n^2}{6}-\frac{n}{6}+\frac19,\label{eq:f1}\\
\omega(2j+n-1)&=\nonumber
\left(0,\frac{1}{72},\frac{1}{9},-\frac{1}{8},-\frac{1}{9},\frac{17}{72},0,
-\frac{17}{72},\frac{1}{9},\frac{1}{8},-\frac{1}{9},-\frac{1}{72}\right)\\
&\quad\text{ if }2j+n-1\bmod12=(0,1,2,3,4,5,6,7,8,9,10,11)\text{ respectively}.
\label{eq:omega}\end{align}\end{subequations}
In this formula $2j-n$ must be non-negative, $n$ may be negative. Explicitly,
$n$ must be such that $-2j+6\le n\le2j$.
Though this paper is not devoted to deriving approximations, one will observe that
for $n\ge0$, one has $|\omega(2j+n-1)-\xi(n)|\le17/36$, and for $n<0$ one has
$|\omega(2j+n-1)|\le17/72$, so that the approximation
\begin{equation}\label{eq:PMjN4approx}
P(2j-n;j,4)\simeq P_\text{app}(2j-n;j,4)=\frac{1}{18}\left(j+\frac{n-1}2\right)^3
-\left(\frac16-\frac{\pi(n)}{8}\right)\left(j+\frac{n-1}2\right)
-H(n)f_1\left(\frac{n}2\right)
\end{equation}
results in an absolute error below 1/2. The relative error will be small if
conditions $(j+(n-1)/2)\gg1, n\gg 1$ are met. For really large $j$, even the
$\pi(n)$ dependent term may be omitted, but the resulting approximation is not as
good. This is illustrated by Figs.~\ref{fig:PMj7s2N4} and \ref{fig:PMj15s2N4} for
$j=7/2$ and $15/2$ respectively. As can be seen in
the approximate form above, both
approximations $\pi(n)=0$ and 1 exhibit a discontinuity of 1/9 at $n=0$ or $M=2j$
since $f_1(0)=1/9$. Though the above approximation is rough for $j=7/2$ it proves to
be fair for higher $j$, and correctly reproduces the even-odd staggering, previously
noticed in the atomic physics context \cite{Bauche1997,Poirier2021b}.
\begin{figure}[htbp]
\begin{minipage}[c]{0.49\textwidth}
\centerin
\includegraphics[width=\textwidth,angle=\anglefig,scale=\scalefig]{Approx_PMj7s2N4
\caption{(Color online) Magnetic quantum number distribution $P(M;j,4)$ for a
four-fermion system with spin $j=7/2$. The red (resp. blue) curve is the
approximation (\ref{eq:PMjN4approx}) with $\pi(n)=0$ (resp. 1). The black circles
are the exact values.\label{fig:PMj7s2N4}}
\end{minipage}\hfill
\begin{minipage}[c]{0.49\textwidth}
\centerin
\includegraphics[width=\textwidth,angle=\anglefig,scale=\scalefig]{Approx_PMj15s2N4
\caption{(Color online) Magnetic quantum number distribution $P(M;j,4)$ for a
four-fermion system with spin $j=15/2$. See Fig.~\ref{fig:PMj7s2N4} for details.
\label{fig:PMj15s2N4}}\qua
\end{minipage}
\end{figure}
\subsection{Total number of levels}
\label{subsec:Qtotj4}
A direct application of the above derived expression for $P(M;j,4)$ is the
determination of the total number of levels. From the relation (\ref{eq:QvsP}),
one verifies that the total number of levels for four fermions of spin $j$ is
given by $P(0;j,4)$, which is easily obtained with (\ref{eq:hypP2jm2q1}).
Writing $q=j-1/2$ in this equation, one gets
\begin{equation}\label{qtotj4}
P(0;j,4)= \frac{(2j-1/2)^3}{18}-\frac{(2j-1/2)}{24}-g(j-1/2)+\phi(2j-1/2).
\end{equation}
One has to consider three cases according to $j-1/2\bmod3$.
If $j-1/2=3n$, the first equation in the group (\ref{eq:valgp}) applies, and
one has $\phi(2j-1/2)=1/72$. If $j-1/2=3n+1$, the second equation in the group
(\ref{eq:valgp}) applies, and $\phi(2j-1/2)=17/72$. If $j-1/2=3n+1$, the third
equation (\ref{eq:valgp}) is relevant, and $\phi(2j-1/2)=1/8$.
One obtains the general formula
\begin{equation}
P(0;j,4)= \frac{2}{9}j^3-\frac{j^2}{6}+\frac{j}{6}
\begin{cases}
-5/72&\text{ if }j-1/2\bmod3=0,\\
+3/8&\text{ if }j-1/2\bmod3=1,\\
+11/72&\text{ if }j-1/2\bmod3=2.
\end{cases}
\end{equation}
\subsection{Distribution of the total angular momentum}
\label{subsec:Qj4}
Once again, the fundamental relation (\ref{eq:QvsP}), together with the expression
(\ref{eq:PMjN4gen}) of the $M$ distribution for a four-fermion system, allow us to
derive the distribution of the total momentum $J$. One must evaluate $Q(2j-n;j,4)=
P(2j-n;j,4)-P(2j-(n-1);j,4)$ which we will write as $Q_1+Q_2+Q_3$. The quantity
$Q_1$ consists in the contribution of first two terms of (\ref{eq:PMjN4gen}), which
is easily obtained noticing that $\pi(n-1)=1-\pi(n)$,
\begin{align}
Q_1&=\frac{X^3}{18}-\frac{X}{6}-\frac{(X-1/2)^3}{18}+\frac{X-1/2}{6}
-\frac{j+n/2-1}{8}+\frac{\pi(n)}{8}(2j+n-3/2)\quad\text{ with }X=j+\frac{n-1}{2}\\
&=\frac{1}{12}\left(j+\frac{n-3}2\right)^2-\frac{7}{72}
+\frac{\pi(n)}{8}(2j+n-3/2).
\end{align}
The quantity $Q_2$ is the difference of terms involving the Heaviside factors $H(n)$
and $H(n-1)$. These factors are equal except in the case $n=0$ which requires more
attention: one must note that the factor of $H(n-1)$ for $n=0$ is $f_1(-1/2)+
\xi(-1)$, which is zero according to the values (\ref{eq:xi},\ref{eq:f1}).
Therefore one may write
\begin{subequations}\begin{align}
Q_2&=-H(n)\left[f_1(n/2)+\xi(n)-f_1((n-1)/2)-\xi(n-1)\right]\\
&=-H(n)\left[\frac{n^2}{12}-\frac{n}{6}-\frac{1}{72}+\overline{\xi}(n)\right]
\end{align}
with
\begin{equation}\overline{\xi}(n)=\xi(n)-\xi(n-1)=\left(\frac{1}{72},\frac{7}{72},
\frac{1}{72},-\frac{17}{72},\frac{25}{72},-\frac{17}{72}\right)
\text{ for }n\bmod6=0,1,2,3,4,5 \text{ respectively.}
\end{equation}\end{subequations}
Finally the $\omega$-dependent term is simply
\begin{subequations}\begin{align}
Q_3&=\overline{\omega}(2j+n-1)=\omega(2j+n-1)-\omega(2j+n-2)\\
&=\left(\frac{1}{72},\frac{1}{72},\frac{7}{72},-\frac{17}{72},\frac{1}{72},\frac{25}{72},
-\frac{17}{72},-\frac{17}{72},\frac{25}{72},\frac{1}{72},-\frac{17}{72},\frac{7}{72}\right)
\end{align}\end{subequations}
for $2j+n-1\bmod12=0-11$ respectively. The complete formula is
\begin{align}\label{eq:Q2jmnjN4}
Q(2j-n;j,4)=&\frac{1}{12}\left(j+\frac{n-3}2\right)^2-\frac{7}{72}
+\frac{\pi(n)}{8}(2j+n-3/2)\nonumber\\
&-H(n)\left[\frac{(n-1)^2}{12}-\frac{7}{72}+\overline{\xi}(n)\right]
+\overline{\omega}(2j+n-1).
\end{align}
Similarly to the $M$-distribution study, one observes that for $n\ge0$ one has
$|\overline{\omega}(2j+n-1)-\overline{\xi}(n)|\le7/12$, while for $n<0$ one has
$|\overline{\omega}(2j+n-1)|\le25/72$, so that the congruence-free approximation
\begin{equation}\label{eq:QJjN4approx}
Q(2j-n;j,4)\simeq Q_\text{app}(2j-n;j,4)=
\frac{1}{12}\left(j+\frac{n-3}2\right)^2-\frac{7}{72}+\frac{\pi(n)}{8}(2j+n-3/2)
-H(n)\left[\frac{(n-1)^2}{12}-\frac{7}{72}\right]
\end{equation}
holds with an error less than unity. The approximation is tested in Figs.
\ref{fig:QJj7s2N4} and \ref{fig:QJj15s2N4} for $j=7/2$ and 15/2 respectively.
Since the main contribution to $Q(J;j,4)$ scales as the squares $j^2$ or $n^2$
instead of cubes in the $P(M;j,4)$ case, the above approximation is not as good
as $P_\text{app}(M;j,4)$. Nevertheless the above formula is quite simple and
efficient for moderate $j$ values. As for the above $P(M;j,4)$ analysis, one notices
a significant even-odd staggering \cite{Bauche1997,Poirier2021b} which
is correctly reproduced by the above formula. Finally one will note that
the discontinuity on the approximate values at $n=0$ is only 1/72 so that the red
and blue curves look almost continuous at $J=2j$.
\begin{figure}[htbp]
\begin{minipage}[c]{0.49\textwidth}
\centerin
\includegraphics[width=\textwidth,angle=\anglefig,scale=\scalefig]{Approx_QJj7s2N4
\caption{(Color online) Angular momentum distribution $Q(J;j,4)$ for a
four-fermion system with spin $j=7/2$. The red (resp. blue) curve is the
approximation (\ref{eq:QJjN4approx}) with $\pi(n)=0$ (resp. 1). The black circles
are the exact values.\label{fig:QJj7s2N4}}
\end{minipage}\hfill
\begin{minipage}[c]{0.49\textwidth}
\centerin
\includegraphics[width=\textwidth,angle=\anglefig,scale=\scalefig]{Approx_QJj15s2N4
\caption{(Color online) Angular momentum distribution $Q(J;j,4)$ for a
four-fermion system with spin $j=15/2$. See Fig.~\ref{fig:QJj7s2N4} for details.
\label{fig:QJj15s2N4}}\qua
\end{minipage}
\end{figure}
\section{Total number of levels in five-fermion systems}
\label{sec:Qtotj5}
The formula (\ref{eq:PMjN4gen}) allows us to get the total number of levels
for a five-fermion system, which is equal to $P(1/2;j,5)$. From (\ref{eq:Talmi}),
one may write, for $s$ from 1 to $s=j-3/2$,
\begin{equation}
P(1/2;j-s+1,5)=P(1/2;j-s,5)+P(j-s+3/2;j-s,4)+P(j-s+1/2;j-s,4)+P(1/2;j-s,3).
\end{equation}
which gives the total number of levels as a sum
\begin{subequations}\label{eq:S1S2S3}\begin{align}
P(1/2;j,5)&=S_1+S_2+S_3\\
S_1&=\sum_{s=1}^{j-5/2}P(j-s+3/2;j-s,4)\\
S_2&=\sum_{s=1}^{j-5/2}P(j-s+1/2;j-s,4)\\
S_3&=\sum_{s=1}^{j-3/2}P(1/2;j-s,3)
\end{align}\end{subequations}
knowing that for $s=j-3/2$ the elements $P(j-s+1\pm1/2;j-s,4)$ vanish. The sum
$S_3$ is easily derived from (\ref{qtotj3})
\begin{equation}
S_3=\sum_{s=1}^{j-3/2}\frac12\left[(j-s)^2-\frac14\right]
=\frac{1}{48}(2j-3)(2j-1)(2j+1)
=\frac{j^3}{6}-\frac{j^2}{4}-\frac{j}{24}+\frac{1}{16}.
\end{equation}
Using the formula for the four-fermion distribution (\ref{eq:PMjN4gen}), the sum
(\ref{eq:S1S2S3}) may be rewritten by gathering the contributions to $S_1$ and $S_2$
\begin{subequations}\begin{align}
P(1/2;j,5)&=A-\Xi+\Omega\\
A&=\sum_{u=0,1}\sum_{s=1}^{j-5/2}\left[\frac{(3j-3s-u-1/2)^3}{144}
-\frac{(3j-3s-u-1/2)}{12}-f_1\left(\frac{j-s-u-1/2}{2}\right)\right]
\nonumber\\
&\quad+\sum_{u=0,1}\sum_{s=1}^{j-5/2}\left[\pi(j-s-u-1/2)\frac{(3j-3s-u-1/2)}{16}\right]+S_3\\
\Xi&=\sum_{u=0,1}\sum_{s=1}^{j-5/2}\xi(j-s-u-1/2)\label{eq:defXi}\\
\Omega&=\sum_{u=0,1}\sum_{s=1}^{j-5/2}\omega(3j-3s-u-3/2).\label{eq:defOmega}
\end{align}\end{subequations}
In order that $P(j-s+1\pm1/2;j-s,4)$ be nonzero, on must have $j\ge7/2$. When
evaluating the last part of the sum $A$, because of the factor $\pi(j-s-u-1/2)$ one
must consider separately the cases $j-1/2$ even and odd. We define $\overline{n}=j-s-u-1/2$.
If $j-1/2=2\nu$ is even, we have $\overline{n}=2\nu-1-s$ (resp. $2\nu-s$) for $u=1$
(resp. $u=0$), and $\overline{n}$ will be odd if $s=2t, 1\le t\le(j-5/2)/2=\nu-1$
(resp. $s=2t-1, 1\le t\le (j-5/2)/2=\nu-1$). If $j-1/2=2\nu+1$ is odd, we have
$\overline{n}=2\nu-s$ (resp. $2\nu+1-s$) for $u=1,0$ respectively, and since
$\overline{n}$ must be odd, the summation index is $s=2t-1, 1\le t\le (j-3/2)/2
=\nu$ (resp. $s=2t, 1\le t\le (j-7/2)/2=\nu-1$). This allows one to compute $A$ as
a sum of first, second, and third powers of terms in arithmetic progression,
which is a simple operation. Namely we get
\begin{subequations}\begin{align}\label{eq:valA}
A&=\frac{23j^4}{288}-\frac{23j^3}{144}+\frac{49j^2}{576}-\frac{139j}{576}+\alpha\\
\alpha&=\frac{2063}{4608}\text{ if }j-1/2\text{ even},\quad
\frac{1919}{4608}\text{ if }j-1/2\text{ odd.}
\end{align}\end{subequations}
For $j\ge7/2$, specifying the contributions $u=0,1$ the sum $\Xi$ may be written
$\Xi=\Xi_1+\Xi_0, \Xi_1=\sum_{t=1}^{j-5/2}\xi(t), \Xi_0=\sum_{t=2}^{j-3/2}\xi(t)$ .
The quantities $\Xi_0,\Xi_1,\Xi$ as functions of $j$ are easily derived from the
definition (\ref{eq:xi}) of $\xi$. Since $\xi(n)$ is periodic with period 6, one
will note that $\sum_{n=0}^5\xi(n)=-3/8$, and therefore $\Xi_1(j+6)=\Xi_1(j)-3/8$,
$\Xi_0(j+6)=\Xi_0(j)-3/8$, and $\Xi(j+6)=\Xi(j)-3/4$.
We have, from the definition (\ref{eq:defXi}), defining a new table $T_\xi(n)$
equally periodic with period 6,
\begin{multline}\label{eq:valxi}
\Xi=\sum_{t=0}^{j-7/2}\left[\xi(2+t)+\xi(1+t)\right]
=-\frac34\left\lfloor\frac{j-7/2}{6}\right\rfloor+T_\xi(j-7/2)
=-\frac18\left[j-\frac72-\left(j-\frac72\bmod6\right)\right]+T_\xi(j-7/2)\\
\text{where }T_\xi(n)=\left(-\frac{1}{72},-\frac{1}{4},-\frac{3}{8},
-\frac{7}{18}-\frac{5}{8},-\frac{3}{4}\right)
\text{ if }n\bmod6=(0,1,2,3,4,5)\text{ respectively.}
\end{multline}
The sum over $\omega(n)$ is obtained in a similar way. One has, from
definition (\ref{eq:defOmega}),
\begin{equation}
\Omega=\sum_{s=1}^{j-5/2}\left[\omega(3j-3s-5/2)+\omega(3j-3s-3/2)\right]
=\sum_{t=0}^{j-7/2}\left[\omega(5+3t)+\omega(6+3t)\right]=\Omega_1+\Omega_0
\end{equation}
and using the $\omega$-value (\ref{eq:omega}) it easy to check that $\Omega_1(j+4)=
\Omega_1(j)+4/9$, $\Omega_0(j+4)=\Omega_0(j)$, $\Omega(j+4)=\Omega(j)+4/9$.
One obtains the last contribution
\begin{equation}
\Omega=\frac49\left\lfloor\frac{j-7/2}{4}\right\rfloor+U_\omega(j-7/2)
\text{ with }U_\omega(n)=\left(\frac{17}{72},\frac{17}{36},\frac{11}{24},
\frac{4}{9}\right)
\text{ if }n\bmod4=(0,1,2,3)\text{ respectively.}
\end{equation}
With $\lfloor(j-7/2)/4\rfloor=(j-7/2-(j-7/2\bmod4))/4$, one gets
\begin{equation}\label{eq:valomega}
\Omega=\frac19\left(j-\frac72-\left(j-\frac72\bmod4\right)\right)+U_\omega(j-7/2).
\end{equation}
Collecting (\ref{eq:valomega}), (\ref{eq:valxi}), one obtains
\begin{subequations}\label{eq:OmegaXi}\begin{equation}
\Omega-\Xi=\frac{17}{72}j
-\frac{119}{144}+\mathscr{T}(j-7/2\bmod6)+\mathscr{U}(j-7/2\bmod4)
\end{equation}
with, for $n=0,1,2,3,4,5$, $\mathscr{T}(n)=-T_\xi(n)-n/8$ or
\begin{equation}
\mathscr{T}(n)=
\left(\frac{1}{72},\frac18,\frac18,\frac{1}{72},\frac18,\frac18\right),
\end{equation}
and, for $n=0,1,2,3$, $\mathscr{U}(n)=U_\omega(n)-n/9$ or
\begin{equation}
\mathscr{U}(n)=
\left(\frac{17}{72},\frac{17}{36},\frac{17}{72},\frac19\right).
\end{equation}\end{subequations}
The expression for $P(1/2;j,5)=A-\Xi+\Omega$ comes from relations
(\ref{eq:valA}), (\ref{eq:OmegaXi}). We get
\begin{subequations}\begin{equation}\label{qtotj5}
P(1/2;j,5)=
\frac{23j^4}{288}-\frac{23j^3}{144}+\frac{49j^2}{576}-\frac{j}{192}+p_0
\end{equation}
with
\begin{multline}
p_0=\left(-\frac{73}{512},-\frac{737}{4608},\frac{55}{512},-\frac{25}{512},
-\frac{1169}{4608},-\frac{25}{512},\frac{55}{512},-\frac{737}{4608},
-\frac{73}{512},-\frac{25}{512},-\frac{17}{4608},-\frac{25}{512}\right)\\
\text{if }j-\frac52\bmod12=(0,1,2,3,4,5,6,7,8,9,10,11).
\end{multline}\end{subequations}
For instance, one gets $P(1/2;7/2,5)=6$.
Since a $j=7/2$ subshell has a degeneracy $g=8$, this corresponds to a three-hole
system. One expects that the total number of levels is the same for a three-fermion $j=7/2$ shell.
Using Eq. (\ref{qtotj3}) one indeed finds $P(1/2;7/2,3)=6$, in agreement with the $j^5$ number
of levels. This is a simple consistency check of Eq. (\ref{qtotj5}).
\section{Derivation of sum rules for six-\textit{j} and nine-\textit{j}
symbols}\label{sec:sum}
\subsection{Three-fermion case: Sum rules for six-$j$ symbols}
It was shown in Ref. \cite{Pain2019} that, for three-fermion systems,
\begin{equation}\label{qjj3}
Q(J,j,3)=\frac{1}{3}
\sum_{\substack{J_\text{min}\le J_1\le J_\text{max}\\J_1\text{ even}}}
\left[1+2(2J_1+1)\sixj{J_1}{j}{J}{J_1}{j}{j}\right],
\end{equation}
where $J_{\text{min}}=\left|J-j\right|$ and $J_{\text{max}}=\min(2j,j+J)$. Replacing the left-hand side of Eq. (\ref{qjj3}) by the expressions (\ref{eq:QjpqjN3}) and (\ref{eq:QjmqjN3}) of $Q(J,j,3)$ provides a new sum rule on six-$j$ coefficients
\begin{equation}
2\sum_{\substack{J_\text{min}\le J_1\le J_\text{max}\\J_1\text{ even}}}
\left[1+2(2J_1+1)\sixj{J_1}{j}{J}{J_1}{j}{j}\right]=
\begin{cases}
(3j-J)+\tilde{q}_{3p}\text{ with } \tilde{q}_{3p}=\left(0,-1,-2,3,-4,1\right)\text{ for }j\ge J\\
\text{ and }(3j-J)\bmod6=(0,1,2,3,4,5)\text{ respectively},\\
\\
2J+\tilde{q}_{3m}\text{ with } \tilde{q}_{3m}=\left(-1,3,1\right)\text{ for }\text{ for }j\le J\\
\text{ and }J\bmod3=(1/2,3/2,5/2)\text{ respectively}.
\end{cases}
\end{equation}
\begin{comment}
In addition, the expression (\ref{qtotj3}) of $Q_{\mathrm{tot}}\left(j^3\right)$ yields the following sum rule on $6j$ coefficients:
\begin{eqnarray}\label{sr31}
\sum_{J=1/2}^{3(j-1)}\sum_{J_1=J_{1,\mathrm{min}}}^{J_{1,\mathrm{max}}}\left[1+(-1)^{J_1}\right]\left[1+2(2J_1+1)\sixj{J_1}{j}{J}{J_1}{j}{j}\right]=3\left(j^2-\frac14\right).
\end{eqnarray}
\end{comment}
To our knowledge, the above sum rule is not included in reference books such as Ref. \cite{Varshalovich1988}, nor can be deduced in a simple way from elementary sum rules.
\subsection{Four-fermion case: Connection to Ginocchio-Haxton and
Rosensteel-Rowe sum rules}\label{gxrr}
The number of $J$=0 states for four fermions in a single-$j$ shell was originally solved by Ginocchio and Haxton \cite{Ginocchio1993, Zamick2005, Pain2018}. They found that
\begin{equation}\label{s1}
Q(0;j,4)=\left\lfloor\frac{2j+3}{6}\right\rfloor.
\end{equation}
Using formula (\ref{eq:Q2jmnjN4}) with $n=2j$, one gets after simple operations $Q(0;j,4)=j/3-1/12+\overline{\omega}(4j-1)-\overline{\xi}(2j)$. With the above definitions of $\overline{\xi}$ and $\overline{\omega}$, one gets $Q(0;j,4)=(j-1/2)/3$, $(j+3/2)/3$ and $(j+1/2)/3$ for $j-1/2\bmod3=0,1,2$ respectively. It is then simple to verify that such expressions are identical to $\lfloor j/3+1/2\rfloor$.
Rosensteel and Rowe showed that the number of linear constraints and algebraic expressions for conservation of seniority can be derived with the quasi-spin tensor decomposition of the two-body interaction. They proposed a matrix which can project the eigenvectors to two quasi-spin subspaces, stated that the eigenvalues of the matrix must equal to 2 or $-1$ and showed that way \cite{Rosensteel2003} that the number of $J$=0 states for four fermions is equal to
\begin{equation}\label{s2}
Q(0;j,4)=\frac{1}{3}\left(\frac{2j+1}{3}
+2\sum_{\text{even } J_0}(2J_0+1)\sixj{j}{j}{J_0}{j}{j}{J_0}\right).
\end{equation}
From Eqs. (\ref{s1}) and (\ref{s2}), Zhao pointed out that
\begin{equation}
\sum_{\text{even }J_0}(2J_0+1)\sixj{j}{j}{J_0}{j}{j}{J_0}
\end{equation}
has a modular behavior \cite{Zhao2003b} (the sum over all $J_0$ was calculated by
Schwinger for instance \cite{Schwinger1965} but none of these sums --- over all
values of $J_0$ or over even values only --- are given in the handbook by
Varshalovich \textit{et al} \cite{Varshalovich1988}). The values are $(-0.5,0.5,0)$
for $j$ values $(1/2,3/2,5/2)$, and repeat after that, i.e., are the same for $j$
values $(7/2,9/2,11/2)$, $(13/2,15/2,17/2)$, $(19/2,21/2,23/2)$, etc. The first
three values $-0.5$, $0.5$, $0$ for $j=1/2$, $3/2$ and $5/2$ respectively were
obtained by Zamick and Escuderos using recursion relations for coefficients of
fractional parentage \cite{Zamick2005c, Zamick2006}.
Noticing that the number of $J=j$ states for three fermions in equal to the number of $J=0$ states for four fermions, Zamick and Escuderos proposed an alternate derivation \cite{Zamick2005} of $Q(0;j,4)$.
In 2010, Qi \emph{et al} published an alternative proof of the Rosensteel-Rowe relation relying on a decomposition of the total angular momentum. In this work, a matrix similar to that of Ref.~\cite{Zhao2003} has been constructed from the decomposition and the eigenvalue problem was explored in a general way with symmetry properties of angular-momentum coupling coefficients \cite{Qi2010}.
All those properties (Ginocchio-Haxton and Rosensteel-Rowe relations, sum rules over six-$j$ symbols) are obtained in a straightforward way by the formulas given in the preceding sections.
\subsection{Four-fermion case: Sum rules for nine-\textit{j} symbols}
In the same paper \cite{Pain2019}, the following expression was derived for $j^4$
\begin{equation}
Q(J,j,4)=\frac{1}{6}\sum_{\substack{J_1\text{ even}\\0\le J_1\le2j}}
\sum_{\substack{J_2\text{ even}\\0\le J_2\le2j}}\Delta(J_1,J_2,J)
\left[1+(-1)^{J}\delta_{J_1,J_2}-4(2J_1+1)(2J_2+1)
\ninej{j}{j}{J_2}{j}{j}{J_1}{J_2}{J_1}{J}\right]
\end{equation}
where $\Delta(J_1,J_2,J)=1$ if $(J_1,J_2,J)$ verify the triangular conditions,
0 otherwise.
Setting $J=2j-n$ in Eq.(\ref{eq:Q2jmnjN4}), we get the sum rule
\begin{align}\label{sr40}
&4\sum_{\substack{J_1\text{ even}\\0\le J_1\le2j}}
\sum_{\substack{J_2\text{ even}\\0\le J_2\le2j}}
\Delta(J_1,J_2,J)
\left[1+(-1)^{J}\delta_{J_1,J_2}
-4(2J_1+1)(2J_2+1)\ninej{j}{j}{J_2}{j}{j}{J_1}{J_2}{J_1}{J}\right]\nonumber\\
& =2\left(2j-\frac{J+3}2\right)^2-\frac{7}{3}
-H(2j-J)\left[2(2j-J-1)^2-\frac{7}{3}+24\overline{\xi}(2j-J)\right]\nonumber\\
&\quad +24\overline{\omega}(4j-J-1) \begin{cases}
&\text{ if } (2j-J)\text{ even},\\
+3(4j-J-3/2)&\text{ if } (2j-J)\text{ odd}.\end{cases}
\end{align}
As implied by the triangular and parity conditions, the above relation is derived
assuming that $J\le4j-2$. For higher $J$, the left-hand side always vanishes while
the right-hand side does vanish if $4j-1\le J\le4j+2$, but equals 24 if $J=4j+3$.
The total number of levels in $j^4$ reads
\begin{subequations}\label{sumj4}\begin{align}
Q_{\mathrm{tot}}(j^4)&=\sum_{J=0}^{2(2j-3)}Q(J,j,4)\\
&=
\frac{1}{72}(2j+1)\left(8j^2+2j+9\right
-\frac{2}{3}\sum_{J_1,J_2\text{ even}} (2J_1+1)(2J_2+1)
\sum_{J=|J_1-J_2|}^{J_1+J_2}\ninej{j}{j}{J_2}{j}{j}{J_1}{J_2}{J_1}{J},
\end{align}\end{subequations}
and therefore expression (\ref{qtotj4}) enables one to write the sum rule
\begin{equation}\label{sr41}
4\sum_{J_1,J_2\text{ even}}
(2J_1+1)(2J_2+1)\sum_{J=|J_1-J_2|}^{J_1+J_2}\ninej{j}{j}{J_2}{j}{j}{J_1}{J_2}{J_1}{J}
=2j^2+\frac{2}{3}j
\begin{cases}+\frac{7}{6}&\quad\text{ if }\left(j-\frac12\right)\bmod3=0,\\
-\frac{3}{2}&\quad\text{ if }\left(j-\frac12\right)\bmod3=1,\\
-\frac{1}{6}&\quad\text{ if }\left(j-\frac12\right)\bmod3=2.\end{cases}
\end{equation}
Equation (\ref{sumj4}) can also be expressed using the coefficients introduced by Dunlap and Judd \cite{Dunlap1975}
\begin{equation}\label{dun}
D_{J_a,J_b;k}=\frac{1}{2k+1}\left[\frac{\left(2J_a-k\right)!\left(2J_b+k+1\right)!}{\left(2J_b-k\right)!\left(2J_a+k+1\right)!}\right]^{1/2},
\end{equation}
as
\begin{align}\label{sumdun}
Q_{\mathrm{tot}}\left(j^4\right)&=\frac{2j+1}{72}\left[2j(4j+1)+9\right]
\nonumber\\
&-\frac23\sum_{\substack{J_1,J_2\\J_1,J_2\text{ even}}}
(2J_1+1)(2J_2+1)\sum_{k=0}^{\min(2j,2J_1,2J_2)}(2k+1)(-1)^{\phi}D_{J_M,J_m;k}\sixj{j}{j}{k}{J_2}{J_2}{j}\sixj{j}{j}{k}{J_1}{J_1}{j}
\end{align}
with $\phi=J_1+J_2+k$, $J_m=\min\left(J_1,J_2\right)$, $J_M=\max(J_1,J_2)$.
Of course the sum is restricted to conditions $0\le J_1\le2j$, $0\le J_2\le2j$
imposed by the 6-$j$ symbol. The corresponding sum rule is therefore
\begin{align}\label{sr42}
&4\sum_{\substack{J_1,J_2\\J_1,J_2\text{ even}}}
(2J_1+1)(2J_2+1)
\sum_{k=0}^{\min(2j,2J_1,2J_2)}(2k+1)(-1)^{\phi}D_{J_M,J_m;k}\sixj{j}{j}{k}{J_2}{J_2}{j}\sixj{j}{j}{k}{J_1}{J_1}{j}\nonumber\\
&=2j^2+\frac{2}{3}
\begin{cases}+\frac{7}{6}&\text{ if }\left(j-\frac12\right)\bmod3=0,\\
-\frac{3}{2}&\text{ if }\left(j-\frac12\right)\bmod3=1,\\
-\frac{1}{6}&\text{ if }\left(j-\frac12\right)\bmod3=2.\end{cases}
\end{align}
To our knowledge, Eqs. (\ref{sr40}), (\ref{sr41}) and (\ref{sr42}) are not included in reference books such as Ref. \cite{Varshalovich1988}, nor can they
be deduced in a simple way from elementary sum rules.
\section{Particular values of the number of levels with a given spin \textit{J}}
\label{sec:part}
A property mentioned by Talmi is the vanishing of $Q(1/2;j,3)$. It is worth mentioning that for $j=1/2$, it is not possible to get three distinct values $m_1,m_2,m_3$ because $m_i=\pm1/2$ and therefore $Q(1/2;1/2,3)=0$. From the above relation (\ref{eq:QvsP}), one also gets
\begin{equation}\label{eq:TalmiQ}
Q(J;j,N)=Q(J;j-1,N)+Q(J-j;j-1,N-1)+Q(J+j;j-1,N-1)+Q(J;j-1,N-2).
\end{equation}
The recurrence (\ref{eq:TalmiQ}) reads, for $J=1/2, N=3$ and accounting for the formal symmetry property $Q(-J-1)=-Q(J)$,
\begin{subequations}\begin{align}\label{eq:recQj_jm1N3}
Q\left(\frac12;j,3\right)&=Q\left(\frac12;j-1,3\right)+Q\left(\frac12-j;j-1,2\right)+Q\left(\frac12+j;j-1,2\right)+Q\left(\frac12;j-1,1\right)\\
&=Q\left(\frac12;j-1,3\right)-Q\left(j-\frac32;j-1,2\right)+Q\left(j+\frac12;j-1,2\right)+Q\left(\frac12;j-1,1\right).\label{eq:Qj1s2}
\end{align}\end{subequations}
Let us note first that the fourth term of that equation is zero except if $j=3/2$. For $j=3/2$, the second term is $-Q(0;1/2,2)=-1$, the third $Q(2;1/2,2)=0$, and the fourth $Q(1/2;1/2,1)=1$ according to the elementary properties of the coupling of angular momenta $j=1/2$. The sum of the last three terms of (\ref{eq:Qj1s2}) is therefore zero. For $j=5/2$, $-Q(1;3/2,2)=Q(3;3/2,2)=0$ because the total momentum $J$ must be even, and the sum of the last three terms of (\ref{eq:Qj1s2}) cancels as well. For $j\ge7/2$, one has $j-3/2<j+1/2\le J_\text{max}=2j-3$. For $J\ge0$,
$Q(J;j-1,2)=1$ if $J$ even, 0 otherwise. One checks
\begin{equation}
-Q\left(j-\frac32;j-1,2\right)+Q\left(\frac12+j;j-1,2\right)=\begin{cases}
-1+1=0&\text{ for }j=2n-\frac12\\
0+0=0&\text{ for }j=2n+\frac12\end{cases}
\end{equation}
and therefore for each $j$ the summation of the last three terms of (\ref{eq:Qj1s2}) cancels. Such an equation implies that for $j$ half-integer
\begin{equation}
Q\left(\frac12;j,3\right)=Q\left(\frac12;j-1,3\right)=\cdots=Q\left(\frac12;\frac32,3\right)=Q\left(\frac12;\frac12,3\right)=0.
\end{equation}
In addition, it is easy to show that $Q(J_\text{max}(j,N);j,N) = 1$ and that
$Q(J_\text{max}(j,N)-1;j,N) = 0$.
Indeed, for each configuration
$j^N$, the value $J=J_\text{max}$ is realized only once. This manifests clearly if
one notes that in order to get $M=J_\text{max}$ there is only one solution except
permutations of the $m_i$, which is $m_1=j-N+1, m_2=j-N+2, \cdots, m_N=j$, yielding
$P\left(J_\text{max};j,N\right)=1$. For $M=J_{\mathrm{max}}-1$, the only possibility
is to reduce $m_1$ by one with respect to the $J_{\mathrm{max}}$ case: $m_1=j-N,
m_2=j-N+2, \cdots, m_N=j$ and one has also $P(J_\text{max}-1;j,N)=1$ and thus
$Q(J_\text{max}-1;j,N)=0$.
\section{Conclusion}
Closed-form expressions for the number of levels for three, four and five fermions
in a single-$j$ shell are obtained using recursion relations for $P(M)$, the number
of states with a given magnetic quantum number $M$. We derive exact expressions
for $P(M)$ and $Q(J)$, the number of levels with a given total angular momentum $J$,
in the cases of $j^3$ and $j^4$. The formulas involve polynomials, the coefficients
of which are defined by congruence relations. We provide supplementary results,
such as proofs of empirical formulas published by several authors over the last
years, cancellation properties and peculiar values of $Q(J)$, or new sum rules over
six-$j$ and nine-$j$ symbols.
|
2024-02-18T23:40:24.682Z
|
2021-12-30T02:25:30.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14501","language":"en","timestamp":1640831130000,"url":"https:\/\/arxiv.org\/abs\/2112.14501","yymm":"2112"}
|
proofpile-arXiv_000-10065
|
{"provenance":"002.jsonl.gz:10066"}
| null | null |
\section{Introduction}
In this note we construct an iteration scheme for finding zeros of monotone operators in Hilbert spaces. The only assumption on the operator needed is that it is defined in the whole space. The scheme produces a sequence which, independently of the initial choice, will converge strongly to a zero of the operator, if there exists any, otherwise it tends to infinity. When applied to subdifferentials we obtain a strongly convergent iteration scheme for minimising convex functionals. Recall that the (continuous version of) gradient method need not converge strongly, see Baillon [1].
We discussed iteration schemes for monotone operators in an earlier paper [3], but we had to assume that the operators were either bounded or continuous. If the operator is defined in the whole space, then it need not be bounded but it is always locally bounded [5]. In [4] Petrov and Yurgelas showed how one can in this case find the zero of a strongly monotone operator, Here we point out that their scheme can be modified to cover the general case as well.
\bigskip
\section{Iteration scheme}
Let $H$ be a real Hilbert space. A possibly multivalued operator in $H$ is said to be monotone if
\begin{equation}
(y_1-y_2, x_1-x_2) \ge 0 \ \ {\rm for \ all } \ x_i \in D(A) \ {\rm and }\ y_i \in Ax_i.
\end{equation}
Here we shall assume that $D(A)=H$, and without loss of generality, that $A$ is maximal [2], (otherwise we should call $p\in H$ a (generalised) zero if $p$ satisfies
\begin{equation}
(y, x-p) \ge 0 \ \ {\rm for \ all } \ x \in H \ {\rm and }\ y \in Ax,
\end{equation}
which is equivalent to $0 \in \tilde A p$, where $\tilde A$ denotes the maximal monotone extension of $A$).
We now give the algorithm: Choose any decreasing positive sequence $\{ \theta_\mu\}$ such that the sequence
$\{\rho_\mu\}$, where $\rho_\mu = \frac{1}{\theta_\mu+1} -\frac{1}{\theta_\mu}$, is also positive and decreasing and $\lim _{\mu\rightarrow \infty} \theta_\mu =\lim _{\mu\rightarrow \infty} \rho_\mu =0.$
\smallskip
{\bf Step 1} Choose $x_0 \in H$ and set $\mu=0$. Further choose $r_0$ such that for some $y_0 \in Ax_0$
\begin{equation}
|x_0 + \frac{1}{\theta_0} y_0| \le r_0.
\end{equation}
\smallskip
{\bf Step 2} If $r_n \le \rho_\mu$, then go to {Step 3}, otherwise take $y_n \in Ax_n$, set
\begin{equation}
\lambda_n = \min \{ 1/ 2 \theta_\mu, \ \theta_\mu r^2_n / |y_n + \theta_\mu x_n|^2\}
\end{equation}
and compute
\begin{equation}
x_{n+1} = x_n - \lambda_n (y_n + \theta_\mu x_n)
\end{equation}
and
\begin{equation}
r_{n+1}^2 = (1-2\theta_\mu \lambda_n)\ r_n^2 + \lambda_n^2 \ | y_n + \theta_\mu x_n|^2.
\end{equation}
Repeat Step 2.
\smallskip
{\bf Step 3 } Redefine $r_n=(1+|z| ) \rho_\mu$, where $z\in A0$.
Increase $\mu$ by $1$ and go back to Step 2.
\bigskip
If we denote the vectors $\theta_\mu x_n$ appearing in Step 2 by $v_n$, then we have the following
\bigskip
{\bf Theorem}
\ {\it Let $A$ be a maximal monotone operator in a Hilbert space $H$ and assume that $D(A)=H$. Suppose further that $\{x_n\}$ is generated by the iteration scheme described above. If $A^{-1} \not= \emptyset $, then $x_n$ converges strongly to $p$, where $p$ is the element in $A^{-1} 0$ with minimum norm, and if $A^{-1}=\emptyset$, then $|x_n| \rightarrow \infty$ in such a way that $v_n$ converges to $ -a^0$, where $a^0$ is the element in $\overline {R(A)}$ with minimum norm. }
\bigskip
\section{Proof of Theorem }
Since $A$ is maximal monotone there exists a unique $p_\mu \in H$ such that
\begin{equation}
\theta_\mu p_\mu + A p_\mu \ni 0.
\end{equation}
The algorithm associates to any $n$ an index $\mu= \mu(n) $ by Step 2. In the following
we denote this index simply by $\mu$. The convergence of the iterates $x_n$ follows from two facts: first, $|x_n -p_\mu|$ tends to zero as $n$ tends to infinity, and secondly, $p\mu$ tends to minimum solution of $ Ap \ni 0$, if it exists, or tends to infinity if $0\in Ap$ has no solutions.
We show first that for all $n\ge 0$
\begin{equation}
|x_n - p_\mu| \le r_n.
\end{equation}
By monotonicity of $A$ we have for all $x \in H$, $y \in Ax$
\begin{equation}
|x-p_\mu|^2 \le (x-p_\mu+ \frac { 1}{\theta_\mu}(y-q_\mu), \ x-p_\mu) = (x+\frac{1}{\theta_\mu} y, \ x-p_\mu)
\end{equation}
where $q_\mu$ is the vector in $Ap_\mu$ such that $\theta_\mu p_\mu + q-\mu =0$. But (3.3) implies
\begin{equation}
|x-p_\mu | \le |x + \frac{1}{\theta_\mu} y | , \ \ {\rm for \ all } \ \ x \in H,\ y \in Ax,
\end{equation}
and (3.2) holds for $n=0$ by (2.3).
Suppose now that (3.2) holds for the index $n$, and assume first that $r_n > \rho_\mu$.
Subtracting $p_\mu$ from both sides of (2.5), squaring both sides and using the monotonicity of $A$ yield
\begin{equation}
| x_{n+1} - p_\mu|^2 \le (1-2\theta_\mu \lambda_n) |x_n - p_\mu|^2 + \lambda_n^2 |y_n + \theta_\mu x_n|^2.
\end{equation}
But this together with the definition of $r_{n+1}$ and the assumption $|x_n -p_\mu| \le r_n$
imply $|x_{n+1} -p_\mu | \le r_{n+1}$. Suppose then that $r_n \le \rho_\mu$. We show that $|x_n -p_{\mu +1}| \le (1 + |z| ) \rho_\mu$, and hence, after we redefine $r_n = (1+|z|) \rho_\mu$, we get $|x_n - p_{\mu +1} | \le r_n$ and the previous discussion applies since $r_n >\rho_{\mu+1}$.
Since $A$ is monotone and $\theta_\mu p_\mu + q_\mu =0$, we obtain
$$
|p_\mu -p_{\mu+1}| \le |p_\mu -p_{\mu+1} + \frac{1}{\theta_{\mu+1}}(q_\mu - q_{\mu+1})|
= (\frac{\theta_\mu}{\theta_{\mu+1}} - 1) \ | p_\mu |.
$$
If $z \in A0$, then (3.4) implies $|p_\mu| \le \frac{1}{\theta_\mu} |z|$ and therefore
$$
|x_n - p_{\mu+1}| \le |x-p_\mu | + |p_\mu - p_{\mu+1}| \le (1+ |z| ) \rho_\mu
$$
because $ \rho_\mu = \frac{1}{\theta_{\mu+1}} - \frac{1}{\theta_{\mu} }$. Hence (3.2) holds for all $n \ge 0$.
From the definitions of $\lambda_n$ and $r_{n+1}$ se see that, as long as $r_n > \rho_\mu$ we have $r_{n+1} < r_n$ and therefore for all $n$ with $\mu = \mu(n) \ge 1$ we have $r_n \le (1+|z|) \rho_ {\mu-1}$. But by (3.2), $|x_n-p_\mu | \le (1+|z|) \rho_{\mu-1}$ and $ |x_n - p_\mu | $ tends to zero as $n\rightarrow \infty$ if $\mu=\mu(n)$ tends to infinity, because $ \lim_{\mu\rightarrow\infty} \rho_\mu =0$.
Assume therefore that for some $\mu$ we have $r_n >\rho_\mu $ for all large $n$. This is a contradiction to a result of Perov and Yurgelas [4], which states that the iteration (2.4), (2.5), (2.6) (with a fixed $\theta_\mu$ ) converges to the solution of $\theta_\mu p_\mu + Ap_\mu \ni 0$ and $\lim_{n\rightarrow \infty}=0$. FOr the convenience of the reader we outline the proof.
Consider the iteration (2.4), (2.5), (2.6) with $\theta_\mu = \theta$ and with $r_n > \rho_\mu$. Then $r_n$ is decreasing and $\rho=\lim r_n >0 $. From (2.6) we see that if $\lambda_n = 1/2\theta$, then $r_{n+1} ^2 \le \frac{1}{2} r_n^2$, so that for large enough $n$ we must have $\lambda_n = \theta r_n^2/ |y_n + \theta x_n|^2$, and hence for $n$ large
\begin{equation}
r_{n+1}^2 \le (1-\theta^2 r_n^2/ |y_n + \theta x_n|^2) r_n^2.
\end{equation}
Now $ \lim|y_n + \theta x_n| =\infty$ , since otherwise (3.6) would contradict $r_n \ge \rho>0$. By [5] $A$ is locally bounded, and since $|y_n + \theta x_n| \rightarrow \infty$ there exists an $R>0$ such that $|x_n - p_\mu| >R$ for all large $n$. Hence $ |x_n-p_\mu|$ is bounded both from above and below. It follows from Lemma 1 that there exists a constant $\gamma >0$ such that for large $n$
$$
(y_n + \theta x_n, x_n -p_\mu) \ge \gamma |y_n + \theta x_n|.
$$
This implies that for large $n$
$$
|x_{n+1}-p_\mu|^2 \le (1-\phi_n) |x_n -p_\mu|^2,
$$
where $\phi_n \sim C/|y_n + \theta x_n|$. Therefore $\sum 1/ |y_n + x_n| <\infty$ and
$$
\sum |x_{n+1}-x_n| = \sum \lambda_n | y_n + \theta x_n| <\infty.
$$
But if $x_n$ converges, then $ |y_n + \theta x_n$ must be bounded because $A$ is locally bounded and we arrive into a contradiction. In the discussion above we used the following
\bigskip
{\bf Lemma 1}, [4].
{\it Let $A$ be a monotone operator in a Hilbert space and assume that $B(x_0,r_0)=\{\xi \ : \
|\xi - x_0| \le r_0 \} \subset D(A)$ and $ c_0 =\sup |\eta| < \infty$ for $\eta \in A\xi, \ \xi \in B(x_0, r_0)$. If $|y| >c_0$ for $y \in Ax$, then
$$
(y, x-x_0) \ge \sigma |y| |x-x_0|,
$$
where $\sigma = \frac{r_0}{r} \{1- (\frac{c_0}{c} )^2\}^{1/2} - \frac{c_0}{c} \{1- (\frac{r_0}{r} )^2\}^{1/2},
$ and $c=|y|, \ r= |x-x_0| $. }
\bigskip
So far we have shown that $\lim_{n\rightarrow \infty} |x_n -p_{\mu(n)}| =0$ and $ \mu(n) \rightarrow \infty $ as $n \rightarrow \infty$. The conclusions of Theorem 2.1 now follow from Lemma 2, since $p_\mu = J_{1/\theta_\mu}.$ Here $J_\lambda$ denotes the resolvent: $J_\lambda = (I + \lambda A)^{-1}. $
\bigskip
{\bf Lemma 2}, [3].
{\it Let $A$ be a maximal monotone operator in a Hilbert space and $\lambda>0$. Then
$$
\lim_{\lambda\rightarrow \infty} \frac{1}{\lambda} J_\lambda = - a^0
$$
where $a^0$ is the element in $\overline {R(A)}$ with minimum norm, If $A^{-1}0 = \emptyset$ then $J_\lambda 0 \rightarrow \infty$, and if $A^{-1}0 \not= \emptyset$ then $J_\lambda 0 \rightarrow p$ as $\lambda \rightarrow \infty$, where $p $ is the element in $A^{-1}0$ with minimum norm.
}
\bigskip
Let $A_\lambda$ denote the Yosida approximation of $A$: $A_\lambda = \frac{1}{\lambda} (I-J_\lambda)$. Then Lemma 2 follows from the identity $J_\lambda 0 = (A^{-1} )_{1/\lambda} 0$ and from known properties of the resolvent and of the Yosida approximation, [2]. For a proof see [3].
\bigskip
\bigskip
{\bf References}
\bigskip
[1] J.B. Baillon: Un Exemple Concernant le Comportement Asymptotique de la Solution du Probleme $\frac {du}{dt} + partial \phi (u) \ni 0.$ J. Functional Analysis {\bf 28}, 369-376 (1978)
\smallskip
[2] H. Brezis: Operateurs maximaux monotones et semi-groups de contractions dans les espaces de Hilbert. Math.Studies {\bf 5} , North Holland, 1973
\smallskip
[3] O. Nevanlinna: Global Iteration Schemes for Monotone Operators. Nonlinear Analysis, Theory, Methods \& Applications, Vol. 3, No. 4, pp. 505-514
\smallskip
[4] A.I. Perov and V.V. Yurgelas: On the Convergence of an Iterative Process. Zh.vychil.Mat.mat.Fiz. {\bf 17}, 4, 859-870 (1977). (English translation: U.S.S.R. Comp.Maths Math.Phys. {\bf 17} , 45-56 (1978)
\smallskip
[5] R.T.Rockafellar: Local boundedness of nonlinear monotone operators. Michigan Math.J. {\bf 16}, 397-407
\bigskip
\bigskip
{\bf Postscript in December 2021}
\bigskip
The text above is as it was in the original report in November 1978, except that reference [3] was then "to appear". Here I also included the key words and MSC classifications.
I made no attempts to improve the, at places clumsy, language so as to keep it authentic. The library of Oulu University still has a copy of the original report.
The note was written while moving from Madison, Wis. to Oulu where I started in September 1978, and as it turned out, the report was never published or even submitted. I soon got a full professorship in Helsinki University of Technology and was busy running things.
At times there has been references to [3], modifications or generalisations to accretive operators in Banach spaces, in particular if the unit balls are smooth enough, The work [4] by Perov and Yurgelas has not been referenced much. As this note was partly based on their work, I find it natural to make the report available and thus also to have a pointer to [4].
\end{document}
|
2024-02-18T23:40:24.687Z
|
2021-12-30T02:25:29.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14498","language":"en","timestamp":1640831129000,"url":"https:\/\/arxiv.org\/abs\/2112.14498","yymm":"2112"}
|
proofpile-arXiv_000-10066
|
{"provenance":"002.jsonl.gz:10067"}
| null | null |
\section{Introduction}
\label{SI}
The metric of a subextremal Kerr--de~Sitter (KdS) spacetime depends on the parameters $\Lambda>0$ (cosmological constant), $\fm>0$ (mass of the black hole), and $\fa\in\mathbb{R}$ (specific angular momentum). It involves the quartic polynomial
\begin{equation}
\label{EqImu}
\mu_{\Lambda,\fm,\fa}(r) = (r^2+\fa^2)\Bigl(1-\frac{\Lambda r^2}{3}\Bigr) - 2\fm r.
\end{equation}
The spacetime, or the set of parameters $(\Lambda,\fm,\fa)$, is called \emph{subextremal} if $\mu_{\Lambda,\fm,\fa}$ has four distinct real roots
\[
r_{\Lambda,\fm,\fa}^- < r_{\Lambda,\fm,\fa}^C < r_{\Lambda,\fm,\fa}^e < r_{\Lambda,\fm,\fa}^c.
\]
For subextremal parameters, the KdS metric is given on the \emph{domain of outer communications}
\begin{equation}
\label{EqIKdSDOC}
M_{\Lambda,\fm,\fa}^{\rm DOC} = \mathbb{R}_t \times (r_{\Lambda,\fm,\fa}^e, r_{\Lambda,\fm,\fa}^c)_r \times (0,\pi)_\theta \times (0,2\pi)_\phi
\end{equation}
in Boyer--Lindquist coordinates (introduced in the special case $\Lambda=0$ in \cite{BoyerLindquistKerr}) by
\begin{align}
\label{EqIMetric}
\begin{split}
g_{\Lambda,\fm,\fa} &:= -\frac{\mu_{\Lambda,\fm,\fa}(r)}{b_{\Lambda,\fm,\fa}^2\varrho_{\Lambda,\fm,\fa}^2(r,\theta)}({\mathrm d} t-\fa\,\sin^2\theta\,{\mathrm d}\phi)^2 + \varrho_{\Lambda,\fm,\fa}^2(r,\theta)\Bigl(\frac{{\mathrm d} r^2}{\mu_{\Lambda,\fm,\fa}(r)} + \frac{{\mathrm d}\theta^2}{c_{\Lambda,\fm,\fa}(\theta)}\Bigr) \\
&\qquad + \frac{c_{\Lambda,\fm,\fa}(\theta)\sin^2\theta}{b_{\Lambda,\fm,\fa}^2\varrho_{\Lambda,\fm,\fa}^2(r,\theta)}\bigl( (r^2+\fa^2){\mathrm d}\phi - \fa\,{\mathrm d} t\bigr)^2,
\end{split} \\
\label{EqIMetric2}
&\hspace{-1em}b_{\Lambda,\fm,\fa}:=1+\frac{\Lambda\fa^2}{3},\quad
c_{\Lambda,\fm,\fa}(\theta):=1+\frac{\Lambda\fa^2}{3}\cos^2\theta,\quad
\varrho_{\Lambda,\fm,\fa}^2(r,\theta):=r^2+\fa^2\cos^2\theta.
\end{align}
Its physical relevance stems from the fact that it is a solution of the Einstein vacuum equation $\mathrm{Ric}(g_{\Lambda,\fm,\fa})-\Lambda g_{\Lambda,\fm,\fa}=0$. It was discovered by Carter \cite{CarterHamiltonJacobiEinstein}, following the earlier discovery \cite{KerrKerr} of the Kerr metric, which is obtained by formally setting $\Lambda=0$:
\[
g_{\fm,\fa} := g_{0,\fm,\fa};\qquad
\mathrm{Ric}(g_{\fm,\fa})=0\ \ \text{on}\ \ \mathbb{R}_t\times(r_{\fm,\fa}^e,\infty)_r\times(0,\pi)_\theta\times(0,2\pi)_\phi.
\]
For $\Lambda=0$, the condition for subextremality is that $r^2+\fa^2-2\fm r$ have two distinct real roots $r_{\fm,\fa}^C<r_{\fm,\fa}^e$; these roots are $\fm\mp\sqrt{\fm^2-\fa^2}$, and thus a Kerr spacetime is subextremal if and only if $|\fa/\fm|<1$. When $\Lambda\fm^2>0$ is sufficiently small, this is also a sufficient condition for the subextremality of the KdS spacetime; see Figure~\ref{FigI} below, and Lemma~\ref{LemmaKRoots} for a weaker---but sufficient for our purposes---statement.
The above expression for the metric becomes singular at $r=r_{\Lambda,\fm,\fa}^e$ and $r=r_{\Lambda,\fm,\fa}^c$. This is merely a coordinate singularity, as can be seen by passing to the coordinates
\begin{equation}
\label{EqIKerrStar}
\begin{alignedat}{2}
t_* &:= t - T_{\Lambda,\fm,\fa}(r),&\qquad T_{\Lambda,\fm,\fa}'(r)&=(r^2+\fa^2)\frac{b_{\Lambda,\fm,\fa}}{\mu_{\Lambda,\fm,\fa}(r)}F_{\Lambda,\fm,\fa}(r), \\
\phi_* &:= \phi - \Phi_{\Lambda,\fm,\fa}(r),&\qquad \Phi_{\Lambda,\fm,\fa}'(r)&=\fa \frac{b_{\Lambda,\fm,\fa}}{\mu_{\Lambda,\fm,\fa}(r)}F_{\Lambda,\fm,\fa}(r),
\end{alignedat}
\end{equation}
where $F_{\Lambda,\fm,\fa}(r)=2\frac{r-r_{\Lambda,\fm,\fa}^e}{r_{\Lambda,\fm,\fa}^c-r_{\Lambda,\fm,\fa}^e}-1$. Expressed in the coordinates $(t_*,r,\theta,\phi_*)$, the metric $g_{\Lambda,\fm,\fa}$ extends real analytically to
\begin{equation}
\label{EqIKdSExt}
\widetilde M_{\Lambda,\fm,\fa} = \mathbb{R}_{t_*} \times \widetilde X_{\Lambda,\fm,\fa},\qquad
\widetilde X_{\Lambda,\fm,\fa} := (r_{\Lambda,\fm,\fa}^C,\infty)_r \times \mathbb{S}^2_{\theta,\phi_*}.
\end{equation}
See~\cite[Equation~(5)]{PetersenVasySubextremal} for the explicit expression.\footnote{In the main part of the paper, we will make a different choice of $F_{\Lambda,\fm,\fa}(r)$ which has better properties in the limit $\Lambda\fm^2\searrow 0$; see~\S\ref{SsKL}.} The two null hypersurfaces
\[
\mc H^+_{\Lambda,\fm,\fa} = \mathbb{R}_{t_*}\times\{r_{\Lambda,\fm,\fa}^e\}\times\mathbb{S}^2_{\theta,\phi_*}, \qquad
\overline\mc H{}^+_{\Lambda,\fm,\fa} = \mathbb{R}_{t_*}\times\{r_{\Lambda,\fm,\fa}^C\}\times\mathbb{S}^2_{\theta,\phi_*}
\]
are called the \emph{(future) event horizon} and \emph{(future) cosmological horizon}, respectively.
The object of main interest in this paper is the set
\[
{\mathrm{QNM}}(\Lambda,\fm,\fa)\subset\mathbb{C}
\]
of \emph{resonances}, or \emph{quasinormal modes}, of the scalar wave operator $\Box_{g_{\Lambda,\fm,\fa}}$. Here, $\sigma\in{\mathrm{QNM}}(\Lambda,\fm,\fa)$ if and only if there exists a \emph{resonant state} $u_0(r,\theta,\phi_*)\in\cC^\infty(\widetilde X_{\Lambda,\fm,\fa})$ so that $u(t_*,r,\theta,\phi_*)=e^{-i\sigma t_*}u_0(r,\theta,\phi_*)\in\cC^\infty(M_{\Lambda,\fm,\fa})$ is a \emph{mode solution} of the wave equation $\Box_{g_{\Lambda,\fm,\fa}}u=0$. (For an equivalent definition in terms of Boyer--Lindquist coordinates, see e.g.\ \cite[Theorem~3]{DyatlovQNM} or \cite[Definition~2.4]{CasalsTeixeiradCModes}.)
\begin{thm}[Quasinormal modes of Kerr--de~Sitter black holes away from extremality: massless scalar fields]
\label{ThmI}
Fix $C>0$, and let $\epsilon>0$. Then there exists $\delta>0$ so that for\footnote{The quantities $\fa/\fm$, $\Lambda\fm^2$, and $\Lambda^{-\frac12}\sigma$ are dimensionless; see~\S\ref{SsIS}.} $|\fa/\fm|\leq 1-\epsilon$ and $\Lambda\fm^2\in(0,\delta)$, every
\[
\sigma\in{\mathrm{QNM}}(\Lambda,\fm,\fa),\qquad \Im(\Lambda^{-1/2}\sigma)>-C
\]
either satisfies $\sigma=0$ or $\sigma=-i\sqrt{\Lambda/3}(n+o(1))$ for some $n\in\mathbb{N}$ as $\Lambda\fm^2\searrow 0$. Moreover, the only mode solutions with $\sigma=0$ are constant functions. Conversely, for any $n\in\mathbb{N}$ and $\eta>0$ there exists, for sufficiently small $\Lambda\fm^2>0$ and for any $\fa/\fm\in[-1+\epsilon,1-\epsilon]$, an element $\sigma\in{\mathrm{QNM}}(\Lambda,\fm,\fa)$ with $|\sigma\sqrt{3/\Lambda}+i n|<\eta$.
\end{thm}
Thus, the set $(\Lambda/3)^{-\frac12}{\mathrm{QNM}}(\Lambda,\fm,\fa)$ converges in any half space $\Im\sigma>-C$ to the set $-i\mathbb{N}_0$ as $\Lambda\fm^2\searrow 0$ when $|\fa/\fm|$ remains bounded away from $1$. The full result, Theorem~\ref{ThmK} (together with Lemma~\ref{LemmaKdSQNM}), is more precise: we show the convergence of resonances \emph{with multiplicity}, and we also prove the convergence of (generalized) resonant states, appropriately rescaled, to (generalized) resonant states on the static patch of de~Sitter space (see~\S\ref{SsIA}). (Petersen--Vasy \cite{PetersenVasyAnalytic}, based on earlier work by Galkowski--Zworski \cite{GalkowskiZworskiHypo}, showed that resonant states are \emph{analytic}, but our analysis does not make use of this fact.) We refer the reader to \cite[\S\S 1 and 4]{HintzXieSdS} for plots and numerics in the Schwarzschild--de~Sitter case $\fa=0$, and to Figure~\ref{FigI} below for a schematic illustration of Theorem~\ref{ThmI}.
Mode stability is an immediate consequence of Theorem~\ref{ThmI}:\footnote{The KdS parameter range covered by Corollary~\ref{CorIMS} has been confirmed to constitute a ``large'' range in the sense of \cite[{\includegraphics[width=0.8em]{MichelinStar.png}\,\includegraphics[width=0.8em]{MichelinStar.png}} Conjecture~4]{ZworskiResonanceReview}.}
\begin{cor}[Mode stability of Kerr--de~Sitter black holes away from extremality]
\label{CorIMS}
For any $\epsilon>0$, there exists $\delta>0$ so that mode stability holds for the scalar wave equation on Kerr--de~Sitter black holes with parameters $\Lambda,\fm,\fa$ satisfying $|\fa/\fm|\leq 1-\epsilon$ and $\Lambda\fm^2\in(0,\delta)$. That is, no $\sigma\in\mathbb{C}$ with $\Im\sigma\geq 0$ and $\sigma\neq 0$ is a quasinormal mode; equivalently, for $\sigma\in{\mathrm{QNM}}(\Lambda,\fm,\fa)$, either $\Im\sigma<0$ or $\sigma=0$. Moreover, for $\sigma=0$, the only mode solutions are constants.
\end{cor}
In particular, when $\Lambda>0$ and the ratio $|\fa/\fm|<1$ are fixed, this implies the mode stability of KdS when the black hole mass $\fm$ is sufficiently small. Alternatively, when $\fm$ and $|\fa/\fm|<1$ are fixed, we conclude mode stability when $\Lambda>0$ is sufficiently small; this regime is of particular astrophysical interest since, according to the currently favored $\Lambda$CDM model, $\Lambda$ is indeed positive but very small.
\begin{figure}[!ht]
\centering
\includegraphics{FigI}
\caption{\textit{On the left:} illustration of Theorem~\ref{ThmI}. The wave operator on Kerr--de~Sitter spacetimes with small $\Lambda\fm^2$ has a resonance at $0$, and resonances near $-i n\sqrt{\Lambda/3}$, $n=1,2,3,\ldots$ \textit{On the right:} the dashed region is the parameter space of subextremal Kerr--de~Sitter black holes. The red region is a schematic depiction of the set of parameters to which Theorem~\ref{ThmI} applies. Mode stability is known to hold in the union of the red region (Corollary~\ref{CorIMS}) and the green region (see~\S\ref{SsIMS}).}
\label{FigI}
\end{figure}
The KdS black holes considered in Theorem~\ref{ThmI} fit into Vasy's framework \cite[\S6]{VasyMicroKerrdS}, recently extended to the full subextremal range of KdS black holes by Petersen--Vasy \cite{PetersenVasySubextremal}. This implies resonance expansions for solutions of the wave equation up to exponentially decaying remainders.\footnote{In the present context, the dynamical assumptions required by Vasy's framework follow already by combining the $r$-normal hyperbolicity for every $r$ of the trapped set of subextremal Kerr black holes, proved by Dyatlov \cite{DyatlovWaveAsymptotics}, with the structural stability of such trapped sets \cite{HirschPughShubInvariantManifolds}. In fact, however, in the course of our proof of Theorem~\ref{ThmI}, we directly prove the meromorphicity of, and high energy estimates for, the inverse of the spectral family of $\Box_{g_{\Lambda,\fm,\fa}}$ in $\Im(\Lambda^{-\frac12}\sigma)>-C$, which imply such resonance expansions.} We state this in the simplest form, and only record the terms corresponding to the quasinormal modes captured by Theorem~\ref{ThmI}:
\begin{cor}[Resonance expansions for waves]
\label{CorI}
Put $x=(r,\theta,\phi_*)$. For $C>0$ and $\epsilon>0$, let $\delta>0$ be as in Theorem~\usref{ThmI}, and suppose $|\fa/\fm|\leq 1-\epsilon$ and $\Lambda\fm^2\in(0,\delta)$. Let $X:=[r_-,r_+]\times\mathbb{S}^2_{\theta,\phi_*}$, where $r_-\in(r_{\Lambda,\fm,\fa}^C,r_{\Lambda,\fm,\fa}^e)$ and $r_+\in(r_{\Lambda,\fm,\fa}^c,\infty)$. Let $u=u(t_*,x)$ denote the solution of the initial value problem
\[
\Box_{g_{\Lambda,\fm,\fa}}u=0,\qquad (u,\partial_{t_*}u)|_{t_*=0} = (u_0,u_1) \in \cC^\infty(X)\oplus\cC^\infty(X).
\]
Then $u$ has an asymptotic expansion
\[
\biggl|u(t_*,x) - u_0 - \sum_{j=1}^N\biggl(\sum_{k=0}^{k_j} t_*^k e^{-i\sigma_j t_*}u_{j k}(x)\biggr)\biggr| \leq C_1 e^{-C\sqrt{\Lambda}\,t_*},
\]
where $u_0\in\mathbb{C}$, and where $\sigma_1,\ldots,\sigma_N\in\mathbb{C}$ (possibly with repetitions) are the quasinormal modes with $0>\Im(\Lambda^{-\frac12}\sigma_j)\geq-C$, and the $\sum_{k=0}^{k_j}t_*^k e^{-i\sigma_j t_*}u_{j k}$ are (generalized) mode solutions\footnote{We do not rule out the possibility that some of the resonances controlled by Theorem~\ref{ThmI} are not simple; hence the need to allow for $k_j\geq 1$.} of the wave equation. In particular,
\[
|u(t_*,x)-u_0| \leq C_2 \exp\biggl(-\Bigl((1+\eta)\sqrt{\frac{\Lambda}{3}}\,\Bigr)t_*\biggr),
\]
where $\eta=\eta(\Lambda\fm^2,\fa/\fm)\to 0$ as $\Lambda\fm^2\searrow 0$. Above, $C_1,C_2$ are constants depending only on $\Lambda,\fm,\fa$, and on the initial data $u_0,u_1$.
\end{cor}
See \cite[Theorem~1.5]{PetersenVasySubextremal} (based on \cite[Theorem~1.4]{VasyMicroKerrdS}) for a more precise statement which has weaker regularity requirements and allows for the presence of forcing. See moreover \cite{HintzVasyKdSStability} and \cite[Theorem~1.6]{PetersenVasySubextremal} (based on \cite{HintzVasyQuasilinearKdS}) for applications of such resonance expansions to quasilinear equations.
\begin{rmk}[Spacetime degeneration]
A uniform description of the singular limit of (waves on) KdS \emph{spacetimes} as $\fm\searrow 0$ is beyond the scope of this paper.
\end{rmk}
As an illustration of the flexibility of our method of proof, we also show:
\begin{thm}[Quasinormal modes of Kerr--de~Sitter black holes away from extremality: massive scalar fields]
\label{ThmIKG}
Let $\nu\in\mathbb{C}$. Denote by ${\mathrm{QNM}}(\nu;\Lambda,\fm,\fa)$ the set of resonances for the Klein--Gordon operator $\Box_{g_{\Lambda,\fm,\fa}}-\frac{\Lambda}{3}\nu$.\footnote{The normalization of the zeroth order term is chosen so that $\nu$ is dimensionless; see~\S\ref{SsIS}.} Put $\lambda_\pm=\frac32\pm\sqrt{\frac94-\nu}$. Let $C>0$. Then for any $\epsilon>0$, there exists $\delta>0$ so that for $|\fa/\fm|\leq 1-\epsilon$ and $\Lambda\fm^2\in(0,\delta)$, every $\sigma\in{\mathrm{QNM}}(\nu;\Lambda,\fm,\fa)$ with $\Im(\Lambda^{-\frac12}\sigma)>-C$ satisfies
\[
\sigma = -i\sqrt{\Lambda/3}(\lambda_\pm+n+o(1))
\]
for some $n\in\mathbb{N}_0$ as $\Lambda\fm^2\searrow 0$. Conversely, there does exist a quasinormal mode near each $-i\sqrt{\Lambda/3}(\lambda_\pm+n)$.
\end{thm}
Solutions of the Klein--Gordon equation admit resonance expansions in a manner analogous to Corollary~\ref{CorI}. \emph{For the remainder of this introduction, we restrict attention to the massless case (Theorem~\usref{ThmI}) unless explicitly stated otherwise.}
\subsection{Prior work on quasinormal modes and resonance expansions on de~Sitter black hole spacetimes}
\label{SsIQ}
In the special case $\fa=0$ of \emph{Schwarzschild--de~Sitter} black holes (in which case the subextremality condition becomes $0<9\Lambda\fm^2<1$), the discreteness of ${\mathrm{QNM}}(\Lambda,\fm,0)$ was shown by S\'a Barreto--Zworski \cite{SaBarretoZworskiResonances}, relying in particular on \cite{MazzeoMelroseHyp}. For fixed parameters $(\Lambda,\fm)$, they also characterized resonances in the high frequency regime $|\Re\sigma|\gg 1$, and showed that in conic sectors $\Im\sigma>-\theta|\Re\sigma|$ (with $\theta>0$ sufficiently small) they are given by
\begin{equation}
\label{EqISBZw}
\Bigl(\pm l\pm\frac12-\frac{i}{2}\Bigl(n+\frac12\Bigr)\Bigr)\frac{(1-9\Lambda\fm^2)^{\frac12}}{3\Lambda^{\frac12}\fm}\sqrt{\Lambda/3} + o(1),\qquad l\to\infty.\quad (n=0,1,2,\ldots)
\end{equation}
When $\Lambda\fm^2\searrow 0$, these approximate resonances thus leave any fixed half space $\Im(\Lambda^{-\frac12}\sigma)>-C$; in this sense, Theorem~\ref{ThmI} concerns an altogether different regime of resonances than \cite{SaBarretoZworskiResonances}. One is moreover led to conjecture that (at least away from the negative imaginary axis) Theorem~\ref{ThmI} continues to hold in the larger $\fm$-dependent range $\Im\sigma\geq-c\fm^{-1}$ for any $c<\frac{1}{12\sqrt 3}$.
Still for $\fa=0$, the author and Xie \cite{HintzXieSdS} proved a version of Theorem~\ref{ThmI} which only provides uniform control of resonances in any fixed ball $|\Lambda^{-\frac12}\sigma|\leq C$ provided they are associated with mode solutions which moreover have a \emph{fixed} angular momentum $l\in\mathbb{N}_0$ (i.e.\ their dependence on the angular variables is given by a degree $l$ spherical harmonic). The proof proceeded via uniform estimates for a degenerating family of ordinary differential equations, whereas the proof of Theorem~\ref{ThmI} requires more sophisticated tools (see~\S\ref{SsIA}).
On Schwarzschild--de~Sitter spacetimes, high energy resolvent estimates and resonance expansions similar to Corollary~\ref{CorI} were established in \cite{MelroseSaBarretoVasySdS,MelroseSaBarretoVasyResolvent} (exponential decay to constants on $\widetilde M_{\Lambda,\fm,0}$) and previously in \cite{BonyHaefnerDecay}: in the latter paper, Bony--H\"afner showed that on $M_{\Lambda,\fm,0}^{\rm DOC}$, waves are convergent sums over possibly infinitely many resonances, up to an error term which has any desired amount of exponential decay. In recent work, Mavrogiannis \cite{MavrogiannisSdSMorawetz} gives a proof of exponential decay to constants (thus exponential energy decay) using vector field (`physical space') techniques; this improves on earlier work by Dafermos--Rodnianski \cite{DafermosRodnianskiSdS} which gave superpolynomial energy decay. An alternative definition of the quasinormal mode spectrum, as the set of eigenvalues of an appropriate evolution semigroup, and a proof of some of its salient properties (such as discreteness), was given by Warnick \cite{WarnickQNMs}, and extended to asymptotically flat settings by Gajic--Warnick \cite{GajicWarnickModel}; see also \cite{GalkowskiZworskiGevrey2}.
These results were generalized to the case of slowly rotating Kerr--de~Sitter black holes in a series of papers by Dyatlov. In~\cite{DyatlovQNM}, Dyatlov defined resonances by exploiting the separability of the wave equation and proved the discreteness of the set of resonances; he moreover showed exponential decay to constants of waves first in $M_{\Lambda,\fm,\fa}^{\rm DOC}$, and then in $\widetilde M_{\Lambda,\fm,\fa}$ in~\cite{DyatlovQNMExtended} using red-shift estimates of Dafermos--Rodnianski \cite{DafermosRodnianskiRedShift} near the horizons. The paper \cite{DyatlovAsymptoticDistribution} gives a description of the high energy resonances generalizing and significantly refining~\eqref{EqISBZw}, and proves resonance expansions for solutions of the wave equation, up to error terms with any desired amount of exponential decay. As in the case $\fa=0$, the semiclassical methods of \cite{DyatlovAsymptoticDistribution} are effective only in a high frequency regime, and all resonances captured by it leave the subset of the complex frequency plane described in Theorem~\ref{ThmI} when $\Lambda\fm^2\searrow 0$.
A key ingredient in Dyatlov's works is a robust approach to the analysis of the spectral family at high frequencies near the trapped set. Wunsch--Zworski \cite{WunschZworskiNormHypResolvent,WunschZworskiNormHypResolventCorrection} showed that the trapped set of slowly rotating Kerr black holes is $k$-normally hyperbolic for every $k$ \cite{HirschPughShubInvariantManifolds}; this was later extended to the full subextremal range, and to KdS black holes with either small angular momentum or small cosmological constant by Dyatlov \cite{DyatlovWaveAsymptotics}. Moreover, \cite{WunschZworskiNormHypResolvent} provided microlocal semiclassical (i.e.\ high energy) estimates at the trapped set. Dyatlov subsequently devised a particularly elegant method \cite{DyatlovSpectralGaps} to prove semiclassical estimates at normally hyperbolic trapped sets; we will use \cite{DyatlovSpectralGaps} (rephrased as a propagation estimate as in \cite[Theorem~4.7]{HintzVasyQuasilinearKdS}) as a black box in the present paper. Dyatlov's method has since been extended to give estimates at the trapped set for waves on asymptotically Kerr(--de~Sitter) spacetimes \cite{HintzPolyTrap}.
\begin{rmk}[Further comments on trapping]
An important conceptual feature of the analysis of the trapped set in \cite{DyatlovQNM} is that it is based solely on the dynamical structure of the trapped set (which is stable under perturbations \cite{HirschPughShubInvariantManifolds}), rather than on the separability of the wave equation. Using the separability, estimates at the trapped set of rotating Kerr spacetimes can be proved using rather explicit pseudodifferential multipliers, as shown by Tataru--Tohaneanu \cite{TataruTohaneanuKerrLocalEnergy}; see also \cite{DafermosRodnianskiKerrBoundedness,DafermosRodnianskiKerrDecaySmall} and the definitive \cite{DafermosRodnianskiShlapentokhRothmanDecay} for a very explicit approach of this nature. Andersson--Blue \cite{AnderssonBlueHiddenKerr} can avoid this issue altogether by exploiting a second order `hidden' symmetry operator which is closely related to the complete integrability of the geodesic flow on Kerr spacetimes.
\end{rmk}
Vasy's influential non-elliptic Fredholm theory \cite{VasyMicroKerrdS} provides a general framework for proving the discreteness of resonance spectra and for establishing resonance expansions of waves. This framework is fully microlocal, and makes use in particular of radial point estimates (originating in \cite{MelroseEuclideanSpectralTheory}) and real principal type propagation estimates \cite{DuistermaatHormanderFIO2}, together with high energy estimates in the presence of normally hyperbolic trapping. Without having to separate variables, \cite{VasyMicroKerrdS} recovers the results on exponential decay to constants proved in \cite{BonyHaefnerDecay,MelroseSaBarretoVasySdS,DyatlovQNMExtended}. A detailed account is given by Dyatlov--Zworski \cite{DyatlovZworskiBook}.
The absence of modes for the Klein--Gordon equation in $\Im\sigma\geq 0$ can be proved directly for all $\nu>0$ (in the notation of Theorem~\ref{ThmIKG}) in the case $\fa=0$. In the case of small $\fa/\fm\neq 0$, it also follows for sufficiently small $\nu>0$ from a perturbative calculation off the massless KdS case (see \cite{DyatlovQNM} or \cite[Lemma~3.5]{HintzVasySemilinear}). We also note that Besset--H\"afner \cite{BessetHaefnerBomb} proved, by such perturbative means, the existence of exponentially growing modes for weakly charged and weakly massive scalar fields on slowly rotating Kerr--Newman--de~Sitter spacetimes.
\subsection{Prior work on mode stability}
\label{SsIMS}
We now turn to a discussion of the problem of mode stability for black hole spacetimes. Mode stability (for massless scalar waves) is a much weaker statement than Theorem~\ref{ThmI}, and by itself far from sufficient to obtain Corollary~\ref{CorI} (or even just boundedness of waves).\footnote{When combined with the Fredholm theory of \cite{PetersenVasySubextremal}, it does however imply the existence of a spectral gap, i.e.\ a small number $\alpha>0$ so that $0$ is the only resonance in $\Im(\Lambda^{-\frac12}\sigma)>-\alpha$; and this gives decay to constants, at the rate $e^{-\alpha t_*}$, of smooth linear waves.} It is, however, more amenable to direct investigations. Indeed, for $\fa=0$, mode stability can be proved via an integration by parts argument (when $\Im\sigma>0$) and a Wronskian (or boundary pairing) argument (when $\sigma\in\mathbb{R}\setminus\{0\}$); and the zero mode can be analyzed using an integration by parts argument as well. (Even for the linearized Einstein equation, an appropriate notion of mode stability for Schwarzschild--de~Sitter black holes can be proved with moderate effort, see e.g.\ \cite{KodamaIshibashiMaster} and \cite[\S7]{HintzVasyKdSStability}.) Given Vasy's general perturbation-stable framework \cite{VasyMicroKerrdS}, or using the arguments specific to the Kerr--de~Sitter metric by Dyatlov \cite[Theorem~4]{DyatlovQNM}, mode stability follows for the wave equation on KdS with parameters $(\Lambda,\fm,\fa)$ provided $|\fa/\fm|$ is sufficiently small.
For subextremal KdS black holes with $\fa\neq 0$, one can consider separated mode solutions
\[
e^{-i\sigma t}e^{i m\phi}S(\theta)R(r),\qquad m\in\mathbb{Z},
\]
where the angular function $S$ and the radial function $R$ solve decoupled ordinary differential equations (ODEs). Mode stability can then be proved for certain values $\sigma\in\mathbb{R}$ by means of Wronskian (or energy) arguments for the radial ODE. More precisely, this applies to $\sigma$ which are not superradiant (see \cite[\S1.6]{ShlapentokhRothmanModeStability} for this notion on Kerr spacetimes); the set of superradiant frequencies $\sigma\in\mathbb{R}$ is a nonempty (when $\fa\neq 0$ and $m\neq 0$) open interval centered roughly around $m\fa$. This argument also excludes resonances outside an appropriate subset of the upper half plane. (There are no superradiant modes when one restricts to axially symmetric mode solutions, i.e.\ $m=0$, so mode stability for \emph{axially symmetric} scalar perturbations holds true.) Casals--Teixeira da Costa \cite{CasalsTeixeiradCModes} exploit subtle discrete symmetries of the radial ODE, conjectured in \cite{AminovGrassiHatsudaQNM,HatsudaTeukolskyAlt}, to prove mode stability outside a smaller, but still always nonempty, subset of the closed upper half plane. Numerical evidence \cite{YoshidaUchikataFutamaseKdS,HatsudaKdSQNM} supports the conjecture that mode stability does hold in the full subextremal range.
By contrast with the Kerr--de~Sitter case, the mode stability of subextremal Kerr spacetimes is settled (and $0$ is not a resonance in this case). It was proved in $\Im\sigma>0$ by Whiting \cite{WhitingKerrModeStability} who used a carefully defined integral transform which maps the radial function $R$ to another function which satisfies an ODE for which Wronskian arguments can be applied successfully; Shlapentokh-Rothman \cite{ShlapentokhRothmanModeStability} showed that Whiting's transformation can be used to prove mode stability on the real axis.\footnote{The quantitative main result of \cite{ShlapentokhRothmanModeStability} was a key input in the proof of decay of solutions of the wave equation on subextremal Kerr spacetimes by Dafermos--Rodnianski--Shlapentokh-Rothman \cite{DafermosRodnianskiShlapentokhRothmanDecay}. The merely qualitative mode stability result is sufficient for this purpose as well if one uses it, in conjunction with strong (Fredholm and high energy) estimates for the spectral family, to exclude the presence of a nontrivial nullspace of the spectral family for $\Im\sigma\geq 0$; see \cite{HintzPrice} and also Propositions~\ref{PropKSyKerr}, \ref{PropKnfNz}, and \ref{PropKnfZ}.} Mode stability in $\Im\sigma\geq 0$ for the Teukolsky equation for other values of the spin $s\in{\tfrac{1}{2}}\mathbb{Z}$ (with $s=0$ corresponding to the scalar wave equation) was subsequently proved by Andersson--Ma--Paganini--Whiting \cite{AnderssonMaPaganiniWhitingModeStab}. A different proof of these mode stability results, based on a discrete symmetry of the relevant confluent Heun equation, was given in \cite{CasalsTeixeiradCModes}.
\begin{thm}[Mode stability of subextremal Kerr black holes \cite{WhitingKerrModeStability,ShlapentokhRothmanModeStability}]
\label{ThmIKerr}
Denote by $g_{\fm,\fa}=g_{0,\fm,\fa}$ the Kerr metric on $\mathbb{R}_{t_*}\times[r_{\fm,\fa}^e,\infty)_r\times\mathbb{S}^2_{\theta,\phi_*}$, expressed in terms of the coordinates $t_*,\phi_*$ in~\eqref{EqIKerrStar} where we take $F_{0,\fm,\fa}(r)$ to be equal to $-1$ near $r=r_{\fm,\fa}^e$ and equal to $0$ for $r>2 r_{\fm,\fa}^e$.\footnote{In these coordinates, $g_{\fm,\fa}$ extends analytically down to, and across, the future event horizon $\mc H^+_{\fm,\fa}=\mc H^+_{0,\fm,\fa}$, with the level sets of $t_*$ being transversal to $\mc H^+_{\fm,\fa}$. See~\eqref{EqKKerr} for the explicit form of this metric when the black hole mass and (specific) angular momentum are $1$ and $\hat\fa$, respectively, and the function $F_{0,\fm,\fa}$ is denoted $-\tilde\chi^e$.} Let $0\neq\sigma\in\mathbb{C}$, $\Im\sigma\geq 0$. Suppose $u(t_*,r,\theta,\phi_*)=e^{-i\sigma t_*}u_0(r,\theta,\phi_*)$ is a mode solution of $\Box_{g_{\fm,\fa}}u=0$, where $u_0$ is smooth on $[r_{\fm,\fa}^e,\infty)_r\times\mathbb{S}^2_{\theta,\phi_*}$, and so that $e^{-i\sigma r}r^{-2 i\fm\sigma}u_0(r,\theta,\phi_*)=r^{-1}v_0(r^{-1},\theta,\phi_*)$ where $v_0=v_0(\rho,\theta,\phi_*)$ is smooth on $[0,1/r_{\fm,\fa}^e)\times\mathbb{S}^2_{\theta,\phi_*}$. Then $u_0\equiv 0$ on $[r_{\fm,\fa}^e,\infty)\times\mathbb{S}^2$.
\end{thm}
In the references, this is stated for fully separated mode solutions and in Boyer--Lindquist coordinates, with the incoming condition at the event horizon being equivalent to smoothness down to $r=r_{\fm,\fa}^e$ in the $(t_*,r,\theta,\phi_*)$ coordinates. Theorem~\ref{ThmIKerr} as stated follows by applying \cite[Theorems~1.5 and 1.6]{ShlapentokhRothmanModeStability} to each individual separated component $e^{i m\phi}S(\theta)R(r)$ of $u_0$. The case of $\sigma=0$, in which the boundary condition at infinity becomes an appropriate decay requirement, is analyzed in Lemma~\ref{LemmaKz}.
Teixeira da Costa \cite{TeixeiradCModes} proved the mode stability of \emph{extremal} Kerr black holes, i.e.\ $|\fa|=\fm$, using an appropriate integral transform---which due to the different character of the radial ODE, related to the presence of a degenerate event horizon, is substantially different from that introduced by Whiting. (The exceptional values $\sigma\in (2\fm)^{-1}\mathbb{N}_0$ are not covered by this result.) See \cite[Theorem~1.2]{TeixeiradCModes}; see also Remark~\ref{RmkIExtr} for the relation between Teixeira da Costa's result and the topic of the present paper.
We remark that mode stability \emph{fails} for the Klein--Gordon equation on subextremal Kerr spacetimes for a large range of parameters, as shown by Shlapentokh-Rothman \cite{ShlapentokhRothmanBlackHoleBombs}. Moschidis \cite{MoschidisSuperradiant} proved a number of related mode instability results for deformations of the Kerr spacetime by means of potentials or metric deformations which either exhibit stable trapping or feature a non-Euclidean conic infinity. These results do not have a bearing on Theorem~\ref{ThmIKG} however, since the scalar field mass term vanishes in the appropriate Kerr limit. (In any case, depending on the value of $\nu$, Theorem~\ref{ThmIKG} implies mode stability or mode instability.)
A proof of mode stability for the scalar wave equation on Kerr--de~Sitter black holes (without restriction to axially symmetric modes), beyond the Schwarzschild--de~Sitter case and its small perturbations, has remained elusive, with all attempts so far having been based on integral transforms \cite{SuzukiTakasugiUmetsuKdS,UmetsuKdS} or discrete symmetries \cite{CasalsTeixeiradCModes}. The starting point for the present paper is the idea, substantiated in a simple special case in \cite{HintzXieSdS}, that subextremal KdS spacetimes with small $\Lambda\fm^2$ can be regarded as singular perturbations of subextremal Kerr spacetimes and of de~Sitter space, and that one can extrapolate mode stability and the approximate values of quasinormal modes from these two singular limits. We explain this in some detail in~\S\ref{SsIA}.
\begin{rmk}[KdS mode stability in the full subextremal range]
In the event that a direct proof (via an integral transform, discrete symmetries, or otherwise) of the conjectural mode stability of all subextremal KdS black holes should be found, the recent work by Petersen--Vasy \cite{PetersenVasySubextremal} would immediately imply exponential decay to constants of solutions of the wave equation. But even then, Theorem~\ref{ThmI} and Corollary~\ref{CorI} would give, in the regime in which they apply, significantly more precise information on the quasinormal mode spectrum which likely remains out of reach for any direct methods. We hope that the rather general singular perturbation perspective put forth in the present paper can be put to use in other settings involving spectral or resonance analysis in singular limits.
\end{rmk}
\subsection{Scaling}
\label{SsIS}
In order to reduce the number of parameters, we note:
\begin{lemma}[Scaling]
\label{LemmaIS}
For $s>0$, let $M_s\colon(t_*,r,\theta,\phi_*)\mapsto(s t_*,s r,\theta,\phi_*)$. Then on the extended spacetime $\widetilde M_{\Lambda s^2,\fm/s,\fa/s}$ (see~\eqref{EqIKdSExt}), we have
\begin{equation}
\label{EqISMetric}
M_s^*g_{\Lambda,\fm,\fa} = s^2 g_{\Lambda s^2,\fm/s,\fa/s}
\end{equation}
In the notation of Theorems~\usref{ThmI} and \usref{ThmIKG}, we furthermore have
\begin{equation}
\label{EqISQNM}
\begin{split}
{\mathrm{QNM}}(\Lambda,\fm,\fa) &= s^{-1}{\mathrm{QNM}}(\Lambda s^2,\fm/s,\fa/s), \\
{\mathrm{QNM}}(\nu;\Lambda,\fm,\fa) &= s^{-1}{\mathrm{QNM}}(\nu;\Lambda s^2,\fm/s,\fa/s).
\end{split}
\end{equation}
\end{lemma}
\begin{proof}
The expressions~\eqref{EqImu} and \eqref{EqIMetric2} imply that
\begin{equation}
\label{EqISPf}
\begin{alignedat}{2}
&(M_s^*\mu_{\Lambda,\fm,\fa})(r) = s^2\mu_{\Lambda s^2,\fm/s,\fa/s}(r),&\qquad
&(M_s^*\varrho^2_{\Lambda,\fm,\fa})(r,\theta)=s^2\varrho^2_{\Lambda s^2,\fm/s,\fa/s}(r,\theta), \\
&M_s^*b_{\Lambda,\fm,\fa}=b_{\Lambda,\fm,\fa}=b_{\Lambda s^2,\fm/s,\fa/s},&\qquad
&(M_s^*c_{\Lambda,\fm,\fa})(\theta)=c_{\Lambda,\fm,\fa}(\theta)=c_{\Lambda s^2,\fm/s,\fa/s}(\theta).
\end{alignedat}
\end{equation}
Therefore, $r_{\Lambda,\fm,\fa}^\bullet=s r_{\Lambda s^2,\fm/s,\fa/s}^\bullet$ for $\bullet=-,C,e,c$. Plugged into~\eqref{EqIKerrStar} (with the choice of $F_{\Lambda,\fm,\fa}$ made there), this gives
\[
(M_s^*(\partial_r T_{\Lambda,\fm,\fa}))(r)=\partial_r T_{\Lambda s^2,\fm/s,\fa/s}(r);
\]
since $M_s^*(s\partial_r)=\partial_r$, we can choose the constant of integration for $T_{\Lambda,\fm,\fa}$ so that
\[
(M_s^* T_{\Lambda,\fm,\fa})(r) = s T_{\Lambda s^2,\fm/s,\fa/s}(r).
\]
We can similarly arrange $(M_s^*\Phi_{\Lambda,\fm,\fa})(r)=s\Phi_{\Lambda s^2,\fm/s,\fa/s}(r)$. We conclude that $M_s$ takes the form $(t,r,\theta,\phi)\mapsto(s t,s r,\theta,\phi)$ in Boyer--Lindquist coordinates. The claim~\eqref{EqISMetric} then follows on $M_{\Lambda,\fm,\fa}^{\rm DOC}$ from~\eqref{EqISPf} and the explicit form~\eqref{EqIMetric} of $g_{\Lambda,\fm,\fa}$. On the extended manifold $\widetilde M_{\Lambda s^2,\fm/s,\fa/s}$, the equality~\eqref{EqISMetric} follows by analytic continuation, or directly by inspection of the explicit form~\eqref{EqKExt} of the metric in $(t_*,r,\theta,\phi_*)$ coordinates.
As a consequence of~\eqref{EqISMetric}, pulling back along $M_s^*$ or $M_{1/s}^*$ proves the equivalence
\[
\Bigl(\Box_{g_{\Lambda,\fm,\fa}}-\frac{\Lambda}{3}\nu\Bigr)\bigl(e^{-i\sigma t_*}u(r,\theta,\phi_*)\bigr)=0 \Leftrightarrow
\Bigl(\Box_{g_{\Lambda s^2,\fm/s,\fa/s}}-\frac{\Lambda s^2}{3}\nu\Bigr)\bigl(e^{-i(s \sigma)t_*}u(s r,\theta,\phi_*)\bigr)=0.
\]
Thus, $\sigma\in{\mathrm{QNM}}(\nu;\Lambda,\fm,\fa)$ if and only if $s\sigma\in{\mathrm{QNM}}(\nu;\Lambda s^2,\fm/s,\fa/s)$. This implies~\eqref{EqISQNM} and finishes the proof.
\end{proof}
It thus suffices to consider the first asymptotic regime mentioned after Corollary~\ref{CorIMS}. Concretely, we take $s=\sqrt{3/\Lambda}$ in Lemma~\ref{LemmaIS}, and henceforth work with
\[
\Lambda=3.
\]
\subsection{Singular limits and asymptotic regimes}
\label{SsIA}
We now describe a few elements of the proof of Theorem~\ref{ThmI}. Let us fix $\Lambda=3$, and fix also the ratio $\fa/\fm=\hat\fa\in(-1,1)$; thus, in this section we exclusively work with Kerr--de~Sitter metrics
\[
g_{\Lambda,\fm,\fa} = g_{3,\fm,\hat\fa\fm},
\]
and we are interested in the limit $\fm\searrow 0$. For notational simplicity, we work with Boyer--Lindquist coordinates here, and we restrict our attention to frequencies $\sigma$ which lie in a strip rather than a half space; thus, $\Im\sigma$ is bounded, but $\Re\sigma$ is unbounded.
For fixed $r>0$, the Kerr--de~Sitter metric $g_{\Lambda,\fm,\fa}=g_{3,\fm,\hat\fa,\fm}$ in~\eqref{EqIMetric} converges, as the black hole mass tends to $0$ (i.e.\ the black hole `disappears'), to the de~Sitter metric
\[
g_{\mathrm{dS}} = -(1-r^2){\mathrm d} t^2 + \frac{1}{1-r^2}{\mathrm d} r^2 + r^2\slashed{g}{},\qquad \slashed{g}{}={\mathrm d}\theta^2+\sin^2\theta\,{\mathrm d}\phi^2.
\]
This metric is singular at the cosmological horizon $r=1$, but a coordinate change similar to~\eqref{EqIKerrStar} shows that this is merely a coordinate singularity (see~\eqref{EqKdS}). Moreover, $g_{\mathrm{dS}}$ is the expression in polar coordinates $(r,\theta,\phi)$ of a metric on $\mathbb{R}_t\times B(0,1)$, where $B(0,1):=\{x\in\mathbb{R}^3\colon r=|x|<1\}$, which is smooth across $x=0$. One can then define resonances and mode solutions for $\Box_{g_{\mathrm{dS}}}$ as in the Kerr--de~Sitter setting explained before Theorem~\ref{ThmI}; the set of quasinormal modes of $\Box_{g_{\mathrm{dS}}}$ (which are known explicitly, see Lemma~\ref{LemmaKdSQNM}) is then precisely the limit of ${\mathrm{QNM}}(3,\fm,\hat\fa\fm)$ as $\fm\searrow 0$ in Theorem~\ref{ThmI}.
Now, $g_{\Lambda,\fm,\fa}$ does not converge smoothly to $g_{\mathrm{dS}}$. Rather, in rescaled coordinates
\[
\hat t = \frac{t}{\fm},\qquad
\hat r = \frac{r}{\fm},
\]
the rescaled metric $\fm^{-2}g_{\Lambda,\fm,\fa}$ converges, for fixed $\hat r>0$ and as $\fm\searrow 0$, to the metric
\begin{align*}
&\hat g = -\frac{\hat\mu(\hat r)}{\hat\varrho^2(r,\theta)}\bigl({\mathrm d}\hat t{-}\hat\fa\,\sin^2\theta\,{\mathrm d}\phi\bigr)^2 + \frac{\hat\varrho^2(r,\theta)}{\hat\mu(r)}{\mathrm d}\hat r^2 + \hat\varrho^2(\hat r,\theta){\mathrm d}\theta^2 + \frac{\sin^2\theta}{\hat\varrho^2(\hat r,\theta)}\bigl((\hat r^2{+}\hat\fa^2){\mathrm d}\phi{-}\hat\fa\,{\mathrm d}\hat t\bigr)^2, \\
&\qquad \hat\mu(\hat r):=\hat r^2-2\hat r+\hat\fa^2,\qquad
\hat\varrho^2(\hat r,\theta):=\hat r^2+\hat\fa^2\cos^2\theta,
\end{align*}
of a Kerr black hole with mass $1$ and angular momentum $\hat\fa$. Note the relationship
\begin{equation}
\label{EqIASigma}
e^{-i\sigma t}=e^{-i\tilde\sigma\hat t},\qquad \tilde\sigma=\fm\sigma,
\end{equation}
between frequencies on the KdS spacetime and frequencies for the rescaled observer on the Kerr spacetime. Thus, $\tilde\sigma$ is small compared to $\sigma$ when $\fm>0$ is small; but since $|\sigma|$ itself may be large, the rescaled frequency $\tilde\sigma$ may nonetheless be large too---or not, depending on the relative size of $|\sigma|$ and $\fm^{-1}$.
\begin{rmk}[Simple model]
\label{RmkISimple}
An operator on $(2\fm,2)_r\times\mathbb{S}^1_\theta$ that the reader may keep in mind in the subsequent discussion is
\[
P_\fm(\sigma):=\Bigl(1-\frac{2\fm}{r}-r^2\Bigr)D_r^2+r^{-2}D_\theta^2-\sigma^2,\qquad D=\frac{1}{i}\partial.
\]
(This is a poor approximation of the spectral family of the Schwarzschild--de~Sitter wave operator.) The two singular limits as $\fm\searrow 0$ are
\begin{equation}
\label{EqISimpleScale}
\begin{alignedat}{3}
P_\fm(\sigma) &\to P_0(\sigma)&&=(1-r^2)D_r^2+r^{-2}D_\theta^2-\sigma^2,&\qquad& r\simeq 1, \\
\fm^2 P_\fm(\sigma) &\to \hat P(\tilde\sigma)&&=\Bigl(1-\frac{2}{\hat r}\Bigr)D_{\hat r}^2+\hat r^{-2}D_\theta^2-\tilde\sigma^2,&\qquad& \hat r\simeq 1,\quad \tilde\sigma=\lim_{\fm\searrow 0} \fm\sigma.
\end{alignedat}
\end{equation}
(In the second line, $\sigma$ may vary with $\fm$.) Here, $P_0(\sigma)$ plays the role of the de~Sitter model, and $\hat P(\tilde\sigma)$ that of the Kerr model.
\end{rmk}
We now list the different frequency regimes for $\sigma$ and $\tilde\sigma$ as $\fm\searrow 0$, together with a brief description of the two limiting problems that one needs to study in each regime.
\begin{enumerate}
\item \textit{Bounded frequencies.} $\sigma$ remains bounded as $\fm\searrow 0$: the spectral theory for de~Sitter space for bounded frequencies enters---and thus the de~Sitter quasinormal mode spectrum---but the Kerr wave operator enters only at frequency $\tilde\sigma=0$ by~\eqref{EqIASigma}.
\item\label{ItILarge} \textit{Large frequencies.} $1\ll|\Re\sigma|\ll\fm^{-1}$, i.e.\ $\sigma$ is large but remains small compared to $\fm^{-1}$: this involves high energy (semiclassical) analysis on de~Sitter space---where there are no quasinormal modes---and low (i.e.\ near zero) frequency analysis for the Kerr wave operator. From this point onwards, we are in the high frequency regime from the perspective of the de~Sitter limit.
\item\label{ItIVLarge} \textit{Very large frequencies.} $|\Re\sigma|$ is comparable to $\fm^{-1}$: in this case, $\tilde\sigma=\fm\sigma$ is, in the limit $\fm\searrow 0$, of unit size but real. Thus, we are in a bounded real frequency regime for the Kerr wave operator. Excluding the possibility of KdS resonances in this regime thus requires as an input the absence of modes on the real axis for the Kerr wave operator (Theorem~\ref{ThmIKerr}).
\item\label{ItIELarge} \textit{Extremely large frequencies.} Finally, when $|\Re\sigma|$ is large compared to $\fm^{-1}$ as $\fm\searrow 0$, then we are in a high (real) frequency regime ($|\tilde\sigma|=|\fm\sigma|\gg 1$) also from the perspective of the Kerr model. In this case, the absence of Kerr modes follows directly using semiclassical methods.
\end{enumerate}
More concretely then, in the bounded frequency regime, the uniform analysis of the spectral family $\Box_{g_{\Lambda,\fm,\fa}}(\sigma)=e^{i t\sigma}\Box_{g_{\Lambda,\fm,\fa}}e^{-i t\sigma}$ (acting on functions of the spatial variables only) takes place on function spaces which incorporate the two different spatial limiting regimes: for $\hat r\simeq 1$, we measure regularity with respect to $\partial_{\hat r}$, $\partial_\omega$ (spherical derivatives), and for $r\simeq 1$ with respect to $\partial_r$, $\partial_\omega$; put differently, writing $x\in\mathbb{R}^3$ for spatial coordinates on de~Sitter space, we use $\partial_{\hat x}=\fm\partial_x$ (where $\hat x=\frac{x}{\fm}$) for bounded $|\hat x|$, and $\partial_x$ when $|x|\simeq 1$. (In the region $\hat r\gtrsim 1$, the vector fields $r\partial_r$, $\partial_\omega$ work in both regimes simultaneously.) This is conveniently phrased on a geometric resolution (blow-up) of the total space $[0,1]_\fm\times B(0,1)_x$ in which one introduces polar coordinates around $(\fm,x)=(0,0)$, see Figure~\ref{FigIqSingle}.
\begin{figure}[!ht]
\centering
\includegraphics{FigIqSingle}
\caption{The total space for analysis at bounded frequencies.}
\label{FigIqSingle}
\end{figure}
We call this total space the \emph{q-single space $X_{\mathrm{q}}$ of $X=B(0,1)$}, and refer to the corresponding scale of function spaces as (weighted) \emph{q-Sobolev spaces} $H_{{\mathrm{q}},\fm}^{s,l,\gamma}$: these are spaces of functions of the spatial variables, and indeed equal to $H^s$ as a set, but with norms that degenerate in a specific manner as $\fm\searrow 0$. For functions supported in $\hat r\gtrsim 1$, the $\fm$-dependent norm on $H_{{\mathrm{q}},\fm}^{s,l,\gamma}$ for integer $s$ is given by
\[
\|u\|_{H_{{\mathrm{q}},\fm}^{s,l,\gamma}}^2 = \sum_{j+|\alpha|\leq s} \Bigl\| r^{-l}\Bigl(\frac{\fm}{r}\Bigr)^{-\gamma} (r D_r)^j D_\omega^\alpha u \Bigr\|_{L^2}^2,
\]
where $L^2$ is the standard $L^2$-norm on $X$. The algebra of q-(pseudo)differential operators is described in detail in~\S\ref{Ssq}; it is a close relative of the surgery calculus of McDonald \cite{McDonaldThesis} and Mazzeo--Melrose \cite{MazzeoMelroseSurgery}, see Remark~\ref{RmkqComp}.
The proof of Theorem~\ref{ThmI} for bounded $\sigma$ uses a priori estimates on q-Sobolev spaces for $u$ in terms of $\Box_{g_{\Lambda,\fm,\fa}}(\sigma)u$, with constants that are uniform as $\fm\searrow 0$. These estimates are based on three ingredients.
\begin{enumerate}
\item \textit{Symbolic analysis: elliptic regularity, radial point estimates, microlocal propagation of regularity.} This is a direct translation to the q-calculus of the corresponding estimates introduced in the black hole setting by Vasy \cite{VasyMicroKerrdS}; by design, these q-estimates are uniform in $\fm$. They take the form
\begin{equation}
\label{EqIEst0}
\|u\|_{H_{{\mathrm{q}},\fm}^{s,l,\gamma}} \leq C \bigl( \|\Box_{g_{\Lambda,\fm,\fa}}(\sigma)u \|_{H_{{\mathrm{q}},\fm}^{s-1,l-2,\gamma}} + \|u\|_{H_{{\mathrm{q}},\fm}^{s_0,l,\gamma}} \bigr),\qquad s_0<s;
\end{equation}
that is, symbolic (or principal symbol) arguments control $u$ to leading order in the q-differentiability sense. The differential order $s-1$ on $\Box_{g_{\Lambda,\fm,\fa}}(\sigma)u$ reflects the usual loss of one derivative in radial point or hyperbolic propagation estimates. The shift of $-2$ in the weight $l-2$ reflects the scaling near the Kerr regime $\hat r\simeq 1$, cf.\ \eqref{EqISimpleScale}.
\item \textit{Estimates for the Kerr model problem.} This is a quantitative estimate for a function $v$ on $\hat X$ (i.e.\ expressed in the rescaled coordinates $\hat x$) in terms of the zero energy operator $\Box_{\hat g}(0)$ applied to $v$. Apart from involving symbolic estimates as before, such an estimate involves analysis at spatial infinity, where the operator $\Box_{\hat g}(0)$ is an elliptic element of Melrose's b-algebra \cite{MelroseTransformation,MelroseAPS}. Applying this estimate to the error term $\|u\|_{H_{{\mathrm{q}},\fm}^{s_0,l,\gamma}}$ in~\eqref{EqIEst0} (cut off to a neighborhood of $\hat X$ in Figure~\ref{FigIqSingle}) and noting that $\Box_{\hat g}(0)$ and $\fm^2\Box_{g_{\Lambda,\fm,\fa}}(\sigma)$ differ by an operator whose coefficients vanish to leading order at $\fm=0$ for bounded $\hat r$, this gives the improved estimate
\begin{equation}
\label{EqIEst1}
\|u\|_{H_{{\mathrm{q}},\fm}^{s,l,\gamma}} \leq C \bigl( \|\Box_{g_{\Lambda,\fm,\fa}}(\sigma)u \|_{H_{{\mathrm{q}},\fm}^{s-1,l-2,\gamma}} + \|u\|_{H_{{\mathrm{q}},\fm}^{s_0,l_0,\gamma}} \bigr),\qquad s_0<s,\ l_0<l.
\end{equation}
\item \textit{Estimates for the de~Sitter model problem.} This is a quantitative estimate for a function $v$ on $\dot X$ (see Figure~\ref{FigIqSingle}) in terms of $\Box_{g_{\mathrm{dS}}}(\sigma)v$ where $\Box_{g_{\mathrm{dS}}}(\sigma)$ is the spectral family of de~Sitter space. The caveat here is that the singular limit $\fm\searrow 0$ leaves a mark not just geometrically (as in Figure~\ref{FigIqSingle}) but also analytically, in that the point $x=0$ is blown up, and q-Sobolev spaces involve a choice of weight at $r=0$. Indeed, in the near-de Sitter region $\fm\lesssim r$, q-Sobolev spaces are cone Sobolev spaces (i.e.\ weighted b-Sobolev spaces) with cone point at $r=0$, and for appropriate weights one has elliptic estimates at the cone point. (This issue was already addressed in a simple setting in \cite[\S2.1]{HintzXieSdS}.) Thus, if $\sigma$ is not a de Sitter quasinormal mode, one can apply this quantitative estimate to the error term in~\eqref{EqIEst1} and thereby weaken the error term to\footnote{While not apparent from this sketch, careful accounting of the orders required to apply the two model operator estimates, and of the q-regularity of the error term, shows that the symbolic analysis is indeed necessary in order to get an error term with differential order $\leq s$ here.} $C\|u\|_{H_{{\mathrm{q}},\fm}^{s_0,l_0,\gamma_0}}$ where $\gamma_0<\gamma$. But this is bounded by $C\fm^\delta\|u\|_{H_{{\mathrm{q}},\fm}^{s,l_0+\delta,\gamma_0+\delta}}$ where $0<\delta\leq\min(l-l_0,\gamma-\gamma_0)$, and hence \emph{small} compared to $\|u\|_{H_{{\mathrm{q}},\fm}^{s,l,\gamma}}$ when $\fm$ is small. Therefore, we obtain a uniform estimate
\[
\|u\|_{H_{{\mathrm{q}},\fm}^{s,l,\gamma}} \leq C\|\Box_{g_{\Lambda,\fm,\fa}}(\sigma)u\|_{H_{{\mathrm{q}},\fm}^{s-1,l-2,\gamma}}
\]
for all sufficiently small $\fm$, and for bounded $\sigma$ which are at most at a fixed small distance away from de Sitter quasinormal modes. See Proposition~\ref{PropKBdNo}. A Grushin problem setup together with Rouch\'e's theorem takes care of KdS quasinormal modes \emph{near} de~Sitter quasinormal modes.
\end{enumerate}
\begin{rmk}[Comparison with \cite{HintzXieSdS}]
The work \cite{HintzXieSdS} demonstrated how on the spherically symmetric Schwarzschild--de~Sitter spacetime, and after separation into spherical harmonics, uniform estimates for a degenerating family of ordinary differential equations in the radial variable imply Theorem~\ref{ThmI} for bounded spectral parameters and for fixed spherical harmonic degrees. In the present paper, we adopt a point of view based fully on the analysis of \emph{partial} differential operators; the part of the proof concerned with bounded frequencies is conceptually very similar to \cite[\S3]{HintzXieSdS}, except now the uniform estimates are proved using microlocal means, as described above. The remaining three frequency regimes~\eqref{ItILarge}--\eqref{ItIELarge} are not covered by \cite{HintzXieSdS}.
\end{rmk}
The large frequency regime~\eqref{ItILarge} is the most delicate one. From the perspective of de~Sitter space, uniform analysis away from the cone point utilizes semiclassical Sobolev spaces (i.e.\ measuring regularity with respect to $h\partial_x$ for $|x|\simeq 1$ where $h=|\sigma|^{-1}$), but there is now an artificial conic point at $r=0$ through which we need to propagate semiclassical estimates (along null-bicharacteristics which hit the cone point or emanate from it). We do this by adapting the semiclassical propagation estimates which were proved in \cite{HintzConicProp} by means of the semiclassical cone calculus introduced in \cite{HintzConicPowers}: this involves radial point estimates at incoming and outgoing radial sets over the cone point, and estimates for a model operator on an exact Euclidean cone which here is the spectral family of the Laplacian at frequency $1$ (i.e.\ on the spectrum). In terms of the model of Remark~\ref{RmkISimple}, we are considering $h^2 P_0(h^{-1})=(1-r^2)(h D_r)^2+r^{-2}(h D_\theta)^2-1$, and the model operator arises by passing to $\tilde r:=r/h$ and taking the limit $h\searrow 0$ for bounded $\tilde r$, giving $D_{\tilde r}^2+\tilde r^{-2}D_\theta^2-1$. (We refer the reader to \cite{MelroseWunschConic,MelroseVasyWunschEdge,MelroseVasyWunschDiffraction,XiConeParametrix,YangDiffraction} for further results on propagation through, and diffraction by, conic singularities.)
From the perspective of the rescaled Kerr model on the other hand, the large frequency regime~\eqref{ItILarge} puts us into a regime of \emph{low} frequencies $\tilde\sigma$, and we need to prove uniform estimates for the spectral family $\Box_{\hat g}(\tilde\sigma)$ for real $\tilde\sigma$ near $0$. Uniform estimates for low energy resolvents on asymptotically flat spaces or spacetimes have a long history going back to work by Jensen--Kato \cite{JensenKatoResolvent}, with recent contributions including \cite{GuillarmouHassellResI,GuillarmouHassellResII,GuillarmouHassellSikoraResIII,BonyHaefnerResolvent,DonningerSchlagSofferPrice,DonningerSchlagSofferSchwarzschild,TataruDecayAsympFlat,VasyLowEnergy,VasyLowEnergyLag,HintzPrice,StrohmaierWatersHodge,MorganDecay,MorganWunschPrice}. Here, we use an approach that matches up exactly with the semiclassical cone analysis on the de~Sitter side: we work with function spaces (and a corresponding ps.d.o.\ algebra which we call the \emph{scattering-b-transition algebra}---see~\S\ref{SsPscbt}---which is taken directly from \cite{GuillarmouHassellResI} except for different terminology) which resolve the transition from the (elliptic) b-analysis at zero frequency to (non-elliptic) scattering theory (in the spirit of \cite{MelroseEuclideanSpectralTheory}) at nonzero frequencies. The \emph{same} model operator as above (conic Laplacian at frequency $1$) now captures the transition from zero to nonzero energies for the low energy spectral family of the Kerr wave operator. This is less precise than, but technically simpler than the very precise second microlocal approach introduced recently by Vasy \cite{VasyLowEnergyLag}. In terms of the model of Remark~\ref{RmkISimple}, we pass to $\hat\rho=\hat r^{-1}$ in order to work at spatial infinity, so $\tilde\sigma^{-2}\hat P(\tilde\sigma)=(1-2\hat\rho)\tilde\sigma^{-2}(\hat\rho^2 D_{\hat\rho})^2+\hat\rho^2\tilde\sigma^{-2}D_\theta^2-1$, then introduce $\tilde\rho=\hat\rho/\tilde\sigma$, and pass to the limit $\tilde\sigma\searrow 0$ for bounded $\tilde\rho$; this produces $(\tilde\rho^2 D_{\tilde\rho})^2+\tilde\rho^2 D_\theta^2-1$. Upon identifying $\tilde\rho=\tilde r^{-1}$, this is the same operator as the one arising from the high frequency cone point perspective above.
On the level of estimates, we combine symbolic estimates and estimates for the two model spectral families by means of an appropriate family of $(\fm,\sigma)$-dependent \emph{Q-Sobolev norms} which reduce to semiclassical cone Sobolev norms in the high energy de~Sitter regime, and to scattering-b-transition Sobolev norms in the low energy Kerr regime. Concretely, an integer order norm with these properties is
\begin{align*}
&\|u\|_{H_{{\mathrm{Q}},\fm,\sigma}^{s,(l,\gamma,l',\mathsf{r})}}^2 = \sum_{j+|\alpha|\leq s} \Bigl\| r^{-l}\Bigl(\frac{\fm}{r}\Bigr)^{-\gamma}(h+r)^{-l'+l}\Bigl(\frac{h}{h+r}\Bigr)^{-\mathsf{r}+\gamma}\Bigl(\frac{h}{h+r}r D_r\Bigr)^j\Bigl(\frac{h}{h+r}D_\omega\Bigr)^\alpha u \Bigr\|_{L^2}^2, \\
&\qquad h:=|\sigma|^{-1}\in(\fm,1],
\end{align*}
for $u$ with support in $r\gtrsim\fm$. For fixed $\fm>0$ and $\sigma$, this is equivalent to the $H^s$-norm, but it degenerates in the correct manner as $\fm\searrow 0$. (In the main part of the paper, such \emph{weighted Q-Sobolev norms} have an extra order, denoted $b$, which however does not matter outside the extremely high frequency regime. Moreover, the order $\mathsf{r}$ will be variable to accommodate incoming and outgoing radial point estimates.)
Next, in the very large frequency regime~\eqref{ItIVLarge}, we are now, from the de~Sitter perspective, fully in a semiclassical regime. The symbolic propagation through the conic singularity again follows \cite{HintzConicProp}, but the model problem at the cone point is now the spectral family of the Kerr wave operator at bounded nonzero real frequencies. Estimates for the latter are limiting absorption principle type estimates; they are proved as in \cite{MelroseEuclideanSpectralTheory} up to compact error terms, and removing these error terms precisely requires the mode stability for the Kerr spacetime \cite{ShlapentokhRothmanModeStability}. (This is reminiscent of propagation results for $3$- or $N$-body scattering \cite{VasyThreeBody,VasyManyBody}, where microlocal propagation of decay through collision planes requires the invertibility of a spectral problem for a subsystem.)
In the extremely large frequency regime~\eqref{ItIELarge} finally, we can use semiclassical methods also for the spectral family on the Kerr spacetime (and therefore the absence of extremely large frequency quasinormal modes can be proved entirely using symbolic means). Here, the full null-geodesic dynamics of the Kerr spacetime enter; this is described in detail in \cite{DyatlovWaveAsymptotics}, and we can use this and the relevant microlocal propagation results, in particular at the trapped set \cite{DyatlovSpectralGaps}, as black boxes.
While the analysis of bounded frequencies is done separately (see~\S\ref{SsKBd}), the analysis of all three high frequency regimes is phrased in terms of the single aforementioned family of weighted Q-Sobolev spaces. These capture regularity with respect to a Lie algebra of vector fields adapted to each of the regimes discussed. We adopt a fully geometric microlocal point of view and describe this underlying Lie algebra of \emph{Q-vector fields} on a suitable total space (a resolution of $\overline{\mathbb{R}_\sigma}\times[0,1]_\fm\times B(0,1)$ where $\overline\mathbb{R}=\mathbb{R}\cup\{-\infty,+\infty\}$); the full spectral family $(\sigma,\fm)\mapsto\Box_{g_{3,\fm,\hat\fa\fm}}(\sigma)$ is then (for fixed $\Im\sigma$) a \emph{single} element of a corresponding space of Q-differential operators. Its microlocal analysis is accomplished by means of an algebra of Q-pseudodifferential operators. Q-geometry and Q-analysis are developed in detail in~\S\ref{SQ}.
\begin{rmk}[Separation of variables]
It is conceivable that one can prove Theorem~\ref{ThmI} by starting with Carter's separation of variables \cite{CarterHamiltonJacobiEinstein} and extending the ODE techniques introduced in \cite{HintzXieSdS} to keep track of uniformity in half spaces $\Im\sigma>-C$ and also in the parameters $(\ell,m)$ of the spheroidal harmonics (generalizing the usual parameters $\ell\in\mathbb{N}_0$ and $m\in\mathbb{Z}\cap[-\ell,\ell]$ of spherical harmonics); we shall not pursue this possibility here. We merely note that this approach would introduce yet another large parameter ($|\ell|+|m|\to\infty$). Elements of the low frequency analysis for the Kerr model in the case $\hat\fa=0$ are developed from a separation of variables point of view in \cite{DonningerSchlagSofferPrice,DonningerSchlagSofferSchwarzschild}.
\end{rmk}
\begin{rmk}[Mode stability in the full subextremal range]
\label{RmkIExtr}
For simplicity of notation, fix the black hole mass to be $1$, and consider a sequence $(\Lambda_j,1,\fa_j)$ of subextremal KdS parameters with $\Lambda_j\searrow 0$, $|\fa_j|<1$. Then the limiting Kerr parameters $(1,\fa)$, $\fa=\lim\fa_j$, may be extremal. While the mode stability of extremal Kerr black holes is known \cite{TeixeiradCModes} (with the exceptional frequencies requiring separate treatment), there do not exist any estimates yet on the spectral family on an extremal Kerr spacetime (in any frequency regime) which could take the place of the estimates on the subextremal Kerr spectral family used above. \emph{If} such estimates were available, one could likely generalize Theorem~\ref{ThmI} to all subextremal KdS black holes (possibly even including the extremal case) when $\Lambda\fm^2$ is sufficiently small; at present, this is out of reach however.
\end{rmk}
The analytic framework introduced in this paper is very flexible. In particular, it can be generalized in a straightforward manner to degenerating families of operators acting on sections of vector bundles. In particular, for the Teukolsky equation on Kerr--de~Sitter spacetimes, we expect an analogue of Theorem~\ref{ThmI} to hold; this would be an important step towards an unconditional proof of the nonlinear stability of Kerr--de~Sitter black holes without restriction to small angular momenta. (The case of small angular momenta was treated in \cite{HintzVasyKdSStability}.) Furthermore, other singular limits with similar scaling behavior can be analyzed using the same approach. As a simple (albeit contrived) example, the operator
\[
\Box_{g_{\mathrm{dS}}}+\fm^{-2}V(x/\fm),
\]
where $V\in\cC^\infty_\cp(\mathbb{R}^3)$ (or more generally with inverse cubic decay), fits into our framework: the analogue of the de Sitter model is now simply the spectral family of $\Box_{g_{\mathrm{dS}}}$, while the analogue of the Kerr model is $\Delta_{\hat x}-\sigma^2+V(\hat x)$, i.e.\ the spectral family of the Schr\"odinger operator $\Delta+V$ on $\mathbb{R}^3_{\hat x}$. Thus, if $\Delta+V$ has no resonances in the closed upper half plane, then the resonances of $\Box_{g_{\mathrm{dS}}}+\fm^{-2}V(x/\fm)$ have the same description as in Theorem~\ref{ThmI}. (Note that separation of variables is not available at all for this operator when $V$ has no symmetries.)
On the other hand, if the Kerr model of the equation under study has zero energy resonances or bound states---as is the case for the Maxwell equations \cite{SterbenzTataruMaxwellSchwarzschild,AnderssonBlueMaxwellKerr} or the equations of linearized gravity \cite{AnderssonBackdahlBlueMaKerr,HaefnerHintzVasyKerr}---the bounded frequency analysis sketched above fails. It is an interesting open problem to analyze the limiting behavior of KdS quasinormal modes in this case.
\subsection{Outline of the paper}
\label{SsIO}
The technical heart of the paper is~\S\ref{SQ}. We first discuss in detail the geometric and analytic tools (\emph{q-analysis}) which we will use for the uniform analysis at bounded frequencies---see~\S\ref{Ssq}---before describing the appropriate large frequency generalization (\emph{Q-analysis}) in~\S\S\ref{SsQS}--\ref{SsQH}. The main result of the paper, Theorem~\ref{ThmK}, is set up in~\S\S\ref{SsKL}--\ref{SsKMain}. After placing the full spectral family of a degenerating family of Kerr--de~Sitter spacetimes into the framework of Q-analysis in~\S\ref{SsKS}, the proof of Theorem~\ref{ThmK} occupies~\S\S\ref{SsKSy}--\ref{SsKU}, with \S\ref{SsKU} describing the modifications necessary to treat resonances in a full half space (rather than merely in strips, as described in~\S\ref{SsIA}). The proof of Theorem~\ref{ThmIKG} does not require any further work, and is given in~\S\ref{SsKG}.
Appendix~\ref{SP} reviews elements of geometric singular analysis and recalls the various pseudodifferential algebras (the b-, scattering, semiclassical scattering, semiclassical cone, and scattering-b-transition algebras) that are used in the analysis of the model problems discussed in~\S\ref{SsIA}. Appendix~\ref{SQSemi} contains supplementary material for \S\ref{SsQP}; this is included for conceptual completeness, but it is not used in the proofs of the main results.
\subsection*{Acknowledgments}
I am very grateful to Simone Ferraro for an inspiring conversation during our time as Miller Research Fellows at UC Berkeley which spawned the idea for the present work (and for the earlier \cite{HintzXieSdS}). I am grateful to Andr\'as Vasy for discussions about his work \cite{PetersenVasySubextremal} with Oliver Lindblad Petersen, which prompted the writing of this paper. Thanks are also due to Dietrich H\"afner and Andr\'as Vasy who shared with me their unpublished manuscript \cite{HaefnerVasyKerrUnfinished}, as well as to Maciej Zworski for encouragement and support. This research is supported by the U.S.\ National Science Foundation under Grant No.\ DMS-1955614, and by a Sloan Research Fellowship.
\section{Geometric and analytic setup of the singular limit}
\label{SQ}
Let us fix an $n$-dimensional manifold $X$ without boundary, and fix a point $0\in X$ and local coordinates $x\in B(0,2)=\{x\in\mathbb{R}^n\colon|x|<2\}$ so that $x=0$ at the point $0$. (All constructions presented below go through whether $X$ is compact or not. The main case of interest in this paper is when $X\subset\mathbb{R}^3$ is the spatial part of the de~Sitter manifold. For compact $X$ the discussion of function spaces is slightly simplified.)
We first describe somewhat briefly the geometric and analytic setup for the degenerate limit for fixed frequencies in~\S\ref{Ssq}; we call this q-analysis. The geometric setup for uniform analysis across all frequency regimes is then discussed in detail in~\S\S\ref{SsQS}--\ref{SsQP}; we call this Q-analysis. (The letters `q' and `Q' stand for `quasinormal modes'.) We freely make use of the material in Appendix~\ref{SP}.
\subsection{q-geometry and -analysis}
\label{Ssq}
When, in the context of Theorems~\ref{ThmI} and \ref{ThmIKG}, the frequency $\sigma$ is fixed, the following space captures the geometric degeneration of the spacetime as $\fm\to 0$.
\begin{definition}[q-single space]
\label{DefqSingle}
The \emph{q-single space} of $X$ is the resolution $X_{\mathrm{q}}$ of $[0,1]_\fm\times X$ defined as the blow-up
\[
X_{\mathrm{q}} := \bigl[ [0,1]\times X; \{0\}\times\{0\} \bigr].
\]
We denote by ${\mathrm{zf}}_{\mathrm{q}}$ the front face, and by ${\mathrm{mf}}_{\mathrm{q}}$ the lift of $\{0\}\times X$. We write $\rho_{{\mathrm{zf}}_{\mathrm{q}}},\rho_{{\mathrm{mf}}_{\mathrm{q}}}\in\cC^\infty(X_{\mathrm{q}})$ for defining functions of these two boundary hypersurfaces.
\end{definition}
See Figure~\ref{FigqSingle}. Our interest will be in uniform analysis as $\fm\searrow 0$; thus, one may as well replace $[0,1]$ by any other interval $[0,\fm_0]$ with $\fm_0>0$. We work with a closed interval of values of $\fm$ since it will be convenient to keep all parameter spaces compact.
\begin{figure}[!ht]
\centering
\includegraphics{FigqqSingle}
\caption{The q-single space $X_{\mathrm{q}}$ when $\dim X=1$.}
\label{FigqSingle}
\end{figure}
\begin{rmk}[q-analysis and analytic surgery]
\label{RmkqComp}
In the case that $X$ is 1-dimensional, the set $\{0\}\subset X$ is a hypersurface, and $X_{\mathrm{q}}$ is equal to the \emph{single surgery space} defined in \cite{MazzeoMelroseSurgery}; this was first introduced by McDonald \cite{McDonaldThesis}. For higher-dimensional $X$, the single surgery space is defined via blow-up of a hypersurface of $X$, rather than a point as in the q-single space above. However, much of the discussion of the geometry, Lie algebra of vector fields, and pseudodifferential calculus carries over from \cite[\S\S3--4]{MazzeoMelroseSurgery} to the q-setting with minor changes. We shall nonetheless give a self-contained account here to fix the notation and to facilitate the subsequent generalization to the Q-calculus.
\end{rmk}
We denote by $\fm$ the lift of the first coordinate on $[0,1]\times X$ to $X_{\mathrm{q}}$; we furthermore write
\begin{gather}
\label{EqqCoord1}
x = r\omega,\qquad r\geq 0,\ \omega\in\mathbb{S}^{n-1}, \\
\label{EqqCoord2}
\hat x := \frac{x}{\fm},\qquad
\hat r := \frac{r}{\fm},\qquad
\hat\rho := \hat r^{-1} = \frac{\fm}{r}.
\end{gather}
We finally put
\begin{equation}
\label{EqqXs}
\dot X := [X;\{0\}] = [0,2)_r\times\mathbb{S}^{n-1},\qquad
\hat X := \overline{\mathbb{R}^3_{\hat x}}.
\end{equation}
Thus, $\partial\dot X=r^{-1}(0)\subset\dot X$ is the front face of $\dot X$. Moreover, $\hat X$ is the radial compactification $\overline{T_0}X$ of the tangent space $T_0 X$. We have natural diffeomorphisms
\[
{\mathrm{zf}}_{\mathrm{q}} \cong \hat X,\qquad
{\mathrm{mf}}_{\mathrm{q}} \cong \dot X,
\]
and we shall use both notations for these boundary hypersurfaces.
\begin{definition}[q-vector fields]
\label{DefqVF}
The space of \emph{q-vector fields} on $X$ is defined as
\[
\cV_\qop(X) := \{ V\in\cV_\bop(X_{\mathrm{q}}) \colon V\fm=0 \}.
\]
For $m\in\mathbb{N}_0$, we denote by $\Diff_\qop^m(X)$ the space of $m$-th order q-differential operators, consisting of locally finite sums of up to $m$-fold compositions of elements of $\cV_\qop(X)$ (a $0$-fold composition being multiplication by an element of $\cC^\infty(X_{\mathrm{q}})$). For $\alpha=(\alpha_{\mathrm{zf}},\alpha_{\mathrm{mf}})\in\mathbb{R}^2$, put
\[
\Diff_\qop^{m,\alpha}(X) = \rho_{{\mathrm{zf}}_{\mathrm{q}}}^{-\alpha_{\mathrm{zf}}}\rho_{{\mathrm{mf}}_{\mathrm{q}}}^{-\alpha_{\mathrm{mf}}}\Diff_\qop^m(X) = \bigl\{ \rho_{{\mathrm{zf}}_{\mathrm{q}}}^{-\alpha_{\mathrm{zf}}}\rho_{{\mathrm{mf}}_{\mathrm{q}}}^{-\alpha_{\mathrm{mf}}}A \colon A\in\Diff_\qop^m(X) \bigr\}.
\]
\end{definition}
Since $X_{\mathrm{q}}\cap\{\fm>0\}=(0,1]\times X$, an element $V\in\cV_\qop(X)$ is thus a smooth family $(0,1]\ni\fm\mapsto V_\fm\in\mc V(X)$ of smooth vector fields on $X$ which degenerate in a particular fashion in the limit $r\to 0$, $\fm\to 0$. Since $\cV_\bop(X_{\mathrm{q}})$ is a Lie algebra, and since $[V,W]\fm=V(W\fm)-W(V\fm)=0$ whenever $V\fm=0$ and $W\fm=0$, we conclude that also $\cV_\qop(X)$ is a Lie algebra.
\begin{rmk}[Comparison with \cite{HintzXieSdS}]
\label{RmkqXie}
The uniform ODE analysis of \cite{HintzXieSdS} was phrased in terms of horizontal b-vector fields on the subset of $[[0,1)_\fm\times[0,1)_r;\{0\}\times\{0\}]$ where $\fm\lesssim r\lesssim 1$; thus, the b-behavior at the lift of $r=0$ was excised. The q-single space and class of q-vector fields defined here, even in the ODE setting where $X$ is an open interval containing $0$, is more natural, as it does not introduce an artificial b-boundary at the lift of $r=0$.
\end{rmk}
In local coordinates $\fm\geq 0$, $\hat x\in\mathbb{R}^3$ near the interior ${\mathrm{zf}}_{\mathrm{q}}^\circ$ of ${\mathrm{zf}}_{\mathrm{q}}$, the space $\cV_\qop(X)$ is spanned by $\partial_{\hat x^j}$ ($j=1,\ldots,n$) over $\cC^\infty(X_{\mathrm{q}})$. Near the interior ${\mathrm{mf}}_{\mathrm{q}}^\circ$, $\cV_\qop(X)$ is spanned by $\partial_{x^j}$ ($j=1,\ldots,n$) or equivalently by $\partial_r$, $\partial_\omega$ (schematic notation for spherical vector fields). Near the corner ${\mathrm{zf}}_{\mathrm{q}}\cap{\mathrm{mf}}_{\mathrm{q}}$, where we have local coordinates $\hat\rho,r,\omega$, we can use $r\partial_r-\hat\rho\partial_{\hat\rho}$, $\partial_\omega$ as a spanning set. A global frame near ${\mathrm{zf}}_{\mathrm{q}}$ is given by $\sqrt{\fm^2+|x|^2}\partial_{x^j}$ ($j=1,\ldots,n$). In particular, if we regard $\mc V(X)$ as the subset of $\fm$-independent vector fields on $X_{\mathrm{q}}$, then
\begin{equation}
\label{EqqVFX}
\mc V(X)\subset\rho_{{\mathrm{zf}}_{\mathrm{q}}}^{-1}\cV_\qop(X),\qquad
\mathrm{Diff}^m(X) \subset \rho_{{\mathrm{zf}}_{\mathrm{q}}}^{-m}\Diff_\qop^m(X) = \Diff_\qop^{m,(m,0)}(X).
\end{equation}
We denote by
\[
{}^\qop T X \to X_{\mathrm{q}}
\]
the \emph{q-vector bundle} which has local frames given by the above collections of vector fields; thus there is a bundle map ${}^\qop T X\to T X_{\mathrm{q}}$ so that $\cV_\qop(X)=\cC^\infty(X,{}^\qop T X)$. From the above local coordinate descriptions, we can then also conclude that the restriction maps
\begin{equation}
\label{EqqVFNorm}
N_{{\mathrm{zf}}_{\mathrm{q}}} \colon \cV_\qop(X) \to \cV_\bop(\hat X),\qquad
N_{{\mathrm{mf}}_{\mathrm{q}}} \colon \cV_\qop(X) \to \cV_\bop(\dot X)
\end{equation}
are surjective, and their kernels are $\rho_{{\mathrm{zf}}_{\mathrm{q}}}\cV_\qop(X)$ and $\rho_{{\mathrm{mf}}_{\mathrm{q}}}\cV_\qop(X)$, respectively. These maps thus induce bundle isomorphisms
\begin{equation}
\label{EqqBundleIso}
{}^\qop T_{{\mathrm{zf}}_{\mathrm{q}}}X \cong {}^{\bop}T\hat X,\qquad
{}^\qop T_{{\mathrm{mf}}_{\mathrm{q}}}X \cong {}^{\bop}T\dot X,
\end{equation}
and corresponding isomorphisms of cotangent bundles. We can define the q-principal symbol for $V\in\cV_\qop(X)$ as ${}^\qop\upsigma^1(V)\colon{}^\qop T^*X\ni\xi\mapsto i\xi(V)$, and by linearity and multiplicativity we can define ${}^\qop\upsigma^m(A)\in P^m({}^\qop T^*X)$ for $A\in\Diff_\qop^m(X)$; the principal symbol ${}^\qop\upsigma^m(A)$ vanishes if and only if $A\in\Diff_\qop^{m-1}(X)$. We also have surjective restriction maps
\begin{equation}
\label{EqqDiffNorm}
N_{{\mathrm{zf}}_{\mathrm{q}}} \colon \Diff_\qop^m(X) \to \Diff_\bop^m(\hat X),\qquad
N_{{\mathrm{mf}}_{\mathrm{q}}} \colon \Diff_\qop^m(X) \to \Diff_\bop^m(\dot X),
\end{equation}
and ${}^\bop\upsigma^m(N_H(A))={}^\qop\upsigma^m(A)|_{{}^\qop T^*_H X}$ for $H={\mathrm{zf}}_{\mathrm{q}},{\mathrm{mf}}_{\mathrm{q}}$ under the above bundle isomorphisms. These maps can be defined completely analogously to restrictions of b-vector fields: that is, $N_{{\mathrm{zf}}_{\mathrm{q}}}(A)u=(A\tilde u)|_{{\mathrm{zf}}_{\mathrm{q}}}$ for $u\in\dot\cC^\infty(\hat X)=\cC^\infty({\mathrm{zf}}_{\mathrm{q}})$ where $\tilde u\in\cC^\infty(X_{\mathrm{q}})$ is any smooth extension of $u$; similarly for $N_{{\mathrm{mf}}_{\mathrm{q}}}$.
\begin{definition}[Weighted q-Sobolev spaces]
\label{DefqSob}
Suppose $X$ is compact, and fix a finite collection $V_1,\ldots,V_N\in\cV_\qop(X)$ of q-vector fields which at any point of $X_{\mathrm{q}}$ span the q-tangent space. Fix any weighted positive density $\nu=\rho_{{\mathrm{zf}}_{\mathrm{q}}}^{\alpha_{\mathrm{zf}}}\rho_{{\mathrm{mf}}_{\mathrm{q}}}^{\alpha_{\mathrm{mf}}}\nu_0$ where $0<\nu_0\in\cC^\infty(X_{\mathrm{q}},{}^{\qop}\Omega X)$. We then define, for $s\in\mathbb{N}_0$ and $l,\gamma\in\mathbb{R}$, the function space $H_{{\mathrm{q}},\fm}^{s,l,\gamma}(X,\nu)$ to be equal to $H^s(X)$ as a set, but equipped with the squared norm
\[
\|u\|_{H_{{\mathrm{q}},\fm}^{s,l,\gamma}(X,\nu)}^2 := \sum_{\alpha\in\mathbb{N}_0^N,\ |\alpha|\leq m} \|\rho_{{\mathrm{zf}}_{\mathrm{q}}}^{-l}\rho_{{\mathrm{mf}}_{\mathrm{q}}}^{-\gamma}V^\alpha u\|_{L^2(X,\nu_\fm)}^2,\qquad V^\alpha=\prod_{j=1}^N V_j^{\alpha_j},
\]
where we write $0<\nu_{\fm_0}\in\cC^\infty(X,\Omega X)$ for the restriction of $\nu$ to $\fm^{-1}(\fm_0)$.
\end{definition}
In particular, if $\nu=|{\mathrm d} x|$, then for $u$ supported in $|\hat x|\lesssim 1$, resp.\ $r\gtrsim 1$, the norm $\|u\|_{H_{{\mathrm{q}},\fm}^s(X)}$ is uniformly equivalent to $\fm^{n/2}\|u\|_{H_{\bop}^s(\hat X)}$ (since $|{\mathrm d} x|=\fm^n|{\mathrm d}\hat x|$), resp.\ $\|u\|_{H_{\bop}^s(\dot X)}$.
To analyze q-differential operators using microlocal techniques, we need to define a corresponding pseudodifferential algebra.
\begin{definition}[q-double space]
\label{DefqDouble}
The \emph{q-double space} of $X$ is defined as the resolution of $[0,1]_\fm\times X^2$ given by
\[
X^2_{\mathrm{q}} := \bigl[ [0,1]\times X^2; \{0\}\times\{0\}\times\{0\}; \{0\}\times\{0\}\times X, \{0\}\times X\times\{0\} \bigr].
\]
We denote the front face of $X^2_{\mathrm{q}}$ by ${\mathrm{zf}}_{{\mathrm{q}},2}$, the lift of $\{0\}\times X^2$ by ${\mathrm{mf}}_{{\mathrm{q}},2}$, and the lift of $[0,1]\times\operatorname{diag}_X$ (with $\operatorname{diag}_X\subset X^2$ denoting the diagonal) by $\operatorname{diag}_{\mathrm{q}}$. Furthermore, ${\mathrm{lb}}_{{\mathrm{q}},2}$, resp.\ ${\mathrm{rb}}_{{\mathrm{q}},2}$ denotes the lift of $\{0\}\times\{0\}\times X$, resp.\ $\{0\}\times X\times\{0\}$. See Figure~\ref{FigqDouble}.
\end{definition}
\begin{figure}[!ht]
\centering
\includegraphics{FigqDouble}
\caption{The q-double space $X^2_{\mathrm{q}}$.}
\label{FigqDouble}
\end{figure}
\begin{lemma}[b-fibrations from the q-double space]
\label{LemmaqbFib}
The left projection $[0,1]\times X\times X\ni(\fm,x,x')\mapsto(\fm,x)$ and right projection $(\fm,x,x')\mapsto(\fm,x')$ lift to b-fibrations $\pi_L,\pi_R\colon X^2_{\mathrm{q}}\to X_{\mathrm{q}}$.
\end{lemma}
\begin{proof}
We only consider the left projection. It lifts to a projection $[[0,1]\times X\times X;\{0\}\times\{0\}\times X]=X_{\mathrm{q}}\times X\to X_{\mathrm{q}}$ which is b-transversal to $\{0\}\times\{0\}\times\{0\}$, and hence lifts to a b-fibration
\begin{equation}
\label{EqqbFib}
\bigl[ [0,1]\times X\times X; \{0\}\times\{0\}\times X; \{0\}\times\{0\}\times\{0\}\bigr] \to X_{\mathrm{q}}.
\end{equation}
On the left, we can reverse the order of the two blow-ups since the second center is contained in the first. Since the map~\eqref{EqqbFib} is b-transversal to the lift of $\{0\}\times X\times\{0\}$, this lift can be blown up, and the map~\eqref{EqqbFib} lifts to the desired b-fibration.
\end{proof}
It is easy to check in local coordinates on $X^2_{\mathrm{q}}$ that the lift of $\cV_\qop(X)$ to $X_{\mathrm{q}}^2$ along $\pi_L$ is transversal to $\operatorname{diag}_{\mathrm{q}}$ (see also~\eqref{EqqQuant} below). (This can also be deduced from the analogous statement for b-double spaces by using Lemma~\ref{LemmaqBdy}, together with the analogous statement in $\fm>0$.) The resulting isomorphism ${}^\qop T X\cong N\operatorname{diag}_{\mathrm{q}}$ induces a bundle isomorphism $N^*\operatorname{diag}_{\mathrm{q}}\cong{}^\qop T^*X$.
\begin{definition}[q-pseudodifferential operators]
\label{DefqPsdo}
Let $s,l,\gamma\in\mathbb{R}$. Then $\Psi_\qop^{s,l,\gamma}(X)$ is the space of all smooth families of bounded linear operators on $\cC^\infty_\cp(X)$, parameterized by $\fm\in(0,1]$, with Schwartz kernels $\kappa\in\rho_{{\mathrm{zf}}_{{\mathrm{q}},2}}^{-l}\rho_{{\mathrm{mf}}_{{\mathrm{q}},2}}^{-\gamma}I^{m-\frac14}(X^2_{\mathrm{q}},\operatorname{diag}_{\mathrm{q}};\pi_R^*{}^{\qop}\Omega X)$ which vanish to infinite order at ${\mathrm{lb}}_{{\mathrm{q}},2}$ and ${\mathrm{rb}}_{{\mathrm{q}},2}$, and which are conormal at ${\mathrm{zf}}_{{\mathrm{q}},2}$ and ${\mathrm{mf}}_{{\mathrm{q}},2}$. When $X$ is non-compact, we furthermore demand that $\kappa$ is properly supported, i.e.\ the projection maps $\pi_L,\pi_R\colon\operatorname{supp}\kappa\to X_{\mathrm{q}}$ are proper.
\end{definition}
A typical element of $\Psi_{\mathrm{q}}^{s,l,\gamma}(X)$ is given in coordinates $\fm>0$ and $x,x'\in\mathbb{R}^n$ (the lift of coordinates on $X$ centered around $0$ to the left and right factor of $X^2$) as a quantization\footnote{In these local coordinates, we can take $\rho_{{\mathrm{zf}}_{{\mathrm{q}},2}}=\sqrt{\fm^2+|x|^2+|x'|^2}$.}
\begin{equation}
\label{EqqQuant}
(\Op_{{\mathrm{q}},\fm}(a)u)(x) = (2\pi)^{-n}\iint \exp\Bigl(i\frac{x-x'}{\rho_{{\mathrm{zf}}_{{\mathrm{q}},2}}}\xi\Bigr)\chi\Bigl(\frac{|x-x'|}{\rho_{{\mathrm{zf}}_{{\mathrm{q}},2}}}\Bigr) a(\fm,x,\xi) u(x')\,\frac{{\mathrm d} x}{\rho_{{\mathrm{zf}}_{{\mathrm{q}},2}}^n}\,{\mathrm d}\xi,
\end{equation}
where $\chi\in\cC^\infty_\cp((-{\tfrac{1}{2}},{\tfrac{1}{2}}))$ is identically $1$ near $0$, and $a$ is the local coordinate expression of an element of the symbol space $S^{s,l,\gamma}({}^\qop T^*X)$ consisting of conormal functions on $\overline{{}^\qop T^*}X$ with weights $-s$, $-l$, and $-\gamma$ at fiber infinity, over ${\mathrm{zf}}_{{\mathrm{q}},2}$, and over ${\mathrm{mf}}_{{\mathrm{q}},2}$, respectively.
\begin{lemma}[Boundary hypersurfaces of $X^2_{\mathrm{q}}$]
\label{LemmaqBdy}
In the notation of~\S\usref{SsPbsc}, We have natural diffeomorphisms
\begin{equation}
\label{EqqBdy}
{\mathrm{zf}}_{{\mathrm{q}},2} \cong \hat X^2_{\mathrm{b}},\qquad
{\mathrm{mf}}_{{\mathrm{q}},2} \cong \dot X^2_{\mathrm{b}}.
\end{equation}
\end{lemma}
\begin{proof}
The front face of $[[0,1]\times X^2;\{0\}\times\{0\}\times\{0\}]$ is the radial compactification $\overline{T_{(0,0)}}(X^2)$. The lift of $\{0\}\times\{0\}\times X$, resp.\ $\{0\}\times X\times\{0\}$ intersects this at $\{0\}\times\partial(\overline{T_0}X)$, resp.\ $\partial(\overline{T_0}X)\times\{0\}$. The first isomorphism in~\eqref{EqqBdy} is thus the same as the fact---which can be checked by direct computation---that the resolution of $\overline{\mathbb{R}^{2 n}}$ at $\{0\}\times\partial\overline{\mathbb{R}^n}$ and $\partial\overline{\mathbb{R}^n}\times\{0\}$ is naturally diffeomorphic to $(\overline{\mathbb{R}^n})^2_{\mathrm{b}}$.
For the second isomorphism in~\eqref{EqqBdy}, note that the lift of $\{0\}\times X^2$ to $[[0,1]\times X^2;\{0\}\times\{0\}\times\{0\}]$ is $[X^2;\{0\}\times\{0\}]$. In this manifold, we then further blow up the lift of $X\times\{0\}$---resulting in $[X\times\dot X;\{0\}\times\partial\dot X]$---and then we blow up the lift of $\{0\}\times\dot X$, which can in fact be done prior to blowing up $\{0\}\times\partial\dot X$ and thus results in $[X\times\dot X;\{0\}\times\dot X;\{0\}\times\partial\dot X]=[\dot X^2;(\partial\dot X)^2]=\dot X^2_{\mathrm{b}}$, as claimed.
\end{proof}
The principal symbol map ${}^\qop\upsigma^{s,l,\gamma}$ fits into the short exact sequence
\[
0 \to \Psi_\qop^{s-1,(l,\gamma)}(X) \hookrightarrow \Psi^{s,l,\gamma}(X) \xrightarrow{{}^\qop\upsigma^{s,l,\gamma}} S^{s,l,\gamma}({}^\qop T^*X)/S^{s-1,l,\gamma}({}^\qop T^*X) \to 0.
\]
Restricting to operators whose Schwartz kernels are classical (denoted by an added subscript `${\mathrm{cl}}$') at ${\mathrm{zf}}_{{\mathrm{q}},2}$ and ${\mathrm{mf}}_{{\mathrm{q}},2}$ (thus smooth when the corresponding order vanishes), we obtain from Lemma~\ref{LemmaqBdy} surjective normal operator maps
\begin{equation}
\label{EqqPsdoNorm}
N_{{\mathrm{zf}}_{\mathrm{q}}} \colon \Psi_{{\mathrm{q}},{\mathrm{cl}}}^{s,0,\gamma}(X) \to \Psi_\bop^{s,\gamma}(\hat X),\qquad
N_{{\mathrm{mf}}_{\mathrm{q}}} \colon \Psi_{{\mathrm{q}},{\mathrm{cl}}}^{s,l,0}(X) \to \Psi_\bop^{s,l}(\dot X).
\end{equation}
As in the case of q-differential operators, the principal symbols of $N_H(A)$ are related to that of $A$ by restriction using~\eqref{EqqBundleIso}. Also, the normal operators can be defined via testing, and therefore are multiplicative once we know that $\Psi_\qop(X)$ is closed under composition; we turn to this now.
Pushforward along $\pi_L$ maps the Schwartz kernel of elements of $\Psi_{{\mathrm{q}},{\mathrm{cl}}}^{s,l,\gamma}(X)$, resp.\ $\Psi_{\mathrm{q}}^{s,l,\gamma}(X)$ into $\rho_{{\mathrm{zf}}_{\mathrm{q}}}^{-l}\rho_{{\mathrm{mf}}_{\mathrm{q}}}^{-\gamma}\cC^\infty(X_{\mathrm{q}})$, resp.\ $\mc A^{(-l,-\gamma)}(X_{\mathrm{q}})$. Therefore, compositions of q-ps.d.o.s are well-defined as maps on conormal functions on $X_{\mathrm{q}}$. One can prove that the composition is again a q-ps.d.o.\ using the explicit quantization map in local coordinates above and direct estimates for the residual remainders (in $\Psi_\qop^{-\infty,l,\gamma}(X)$). A geometric proof proceeds via the construction of an appropriate triple space:
\begin{definition}[q-triple space]
\label{DefqTriple}
Define the following submanifolds of $[0,1]_\fm\times X^3$:
\begin{alignat*}{3}
&&C&=\{(0,0,0,0)\}, \\
L_F&=\{0\}\times \{0\}\times\{0\}\times X, &\quad
L_S&=\{0\}\times X\times\{0\}\times\{0\},&\quad
L_C&=\{0\}\times \{0\}\times X\times\{0\}, \\
P_F&=\{0\}\times X\times X\times\{0\},&\quad
P_S&=\{0\}\times \{0\}\times X\times X,&\quad
P_C&=\{0\}\times X\times\{0\}\times X.
\end{alignat*}
The \emph{q-triple space} of $X$ is then defined as
\[
X^3_{\mathrm{q}} := \bigl[ [0,1]\times X^3; C; L_F,L_S,L_C; P_F,P_S,P_C \bigr].
\]
We denote by ${\mathrm{zf}}_{{\mathrm{q}},3}$ and ${\mathrm{mf}}_{{\mathrm{q}},3}$ the lifts of $C$ and $\{0\}\times X^3$, respectively. For $*=F,S,C$, we denote by ${\mathrm{bf}}_{{\mathrm{q}},*}$ and ${\mathrm{mf}}_{{\mathrm{q}},*}$ the lifts of $L_*$ and $P_*$, respectively; and $\operatorname{diag}_{{\mathrm{q}},*}$ denotes the lift of $[0,1]\times(\pi^X_*)^{-1}(\operatorname{diag}_{\mathrm{q}})$ where $\pi^X_*\colon X^3\to X^2$ are the projections $\pi^X_F\colon(x,x',x'')\mapsto(x,x')$, $\pi^X_S\colon(x,x',x'')\mapsto(x',x'')$, $\pi^X_C\colon(x,x',x'')\mapsto(x,x'')$. Finally, $\operatorname{diag}_{{\mathrm{q}},3}$ is the lift of $[0,1]\times\operatorname{diag}_3$ where $\operatorname{diag}_3=\{(x,x,x)\colon x\in X\}$ is the triple diagonal.
\end{definition}
\begin{lemma}[b-fibrations from the q-triple space]
The projection map $[0,1]_\fm\times X^3\ni(\fm,x,x',x'')\mapsto(\fm,x,x')\in[0,1]\times X^2$ to the first and second factor of $X^3$ lifts to a b-fibration $\pi_F\colon X^3_{\mathrm{q}}\to X^2_{\mathrm{q}}$, similarly for the lifts $\pi_S$, $\pi_C\colon X^3_{\mathrm{q}}\to X^2_{\mathrm{q}}$ of the projections to the second and third, resp.\ first and third factor of $X^3$.
\end{lemma}
\begin{proof}
We only prove the result for $\pi_F$. Since the lifted projection $[[0,1]\times X^3;L_F]\to[[0,1]\times X^2;\{0\}\times\{0\}\times\{0\}]$ is b-transversal to the lift of $C\supset L_F$, it lifts to a b-fibration
\[
\bigl[[0,1]\times X^3;C;L_F] \to \bigl[ [0,1]\times X^2; \{0\}\times\{0\}\times\{0\} \bigr].
\]
The preimage of the lift of $\{0\}\times\{0\}\times X$, resp.\ $\{0\}\times X\times\{0\}$, is the lift of $P_S$, resp.\ $P_C$, and thus the lifted projection
\[
\bigl[ [0,1]\times X^3;C;L_F;P_S,P_C \bigr] \to X^2_{\mathrm{q}}
\]
is a b-fibration still. It is b-transversal to the lift of $L_S$, and thus lifts to a b-fibration if we blow up $L_S$ in the domain; since $L_S$ and $P_S$ are transversal, and since $L_S\subset P_C$, \cite[Proposition~5.11.2]{MelroseDiffOnMwc} implies that we can commute the blow-up of $L_S$ through that of $P_S,P_C$. Arguing similarly for $L_C$, we thus have a b-fibration
\[
\bigl[ [0,1]\times X^3; C; L_F,L_S,L_C; P_S,P_C \bigr] \to X_{\mathrm{q}}^2.
\]
This is b-transversal to the lift of $P_F$; blowing up $P_F$ in the domain thus gives the desired b-fibration $\pi_F$.
\end{proof}
For later use, we record
\begin{equation}
\label{EqqPreimages}
\begin{alignedat}{2}
\pi_F^{-1}({\mathrm{zf}}_{{\mathrm{q}},2}) &= {\mathrm{zf}}_{{\mathrm{q}},3}\cup{\mathrm{bf}}_{{\mathrm{q}},F}, &\qquad
\pi_F^{-1}({\mathrm{mf}}_{{\mathrm{q}},2}) &= {\mathrm{mf}}_{{\mathrm{q}},3}\cup{\mathrm{mf}}_{{\mathrm{q}},F}, \\
\pi_F^{-1}({\mathrm{lb}}_{{\mathrm{q}},2}) &= {\mathrm{bf}}_{{\mathrm{q}},C}\cup{\mathrm{mf}}_{{\mathrm{q}},S}, &\qquad
\pi_F^{-1}({\mathrm{rb}}_{{\mathrm{q}},2}) &= {\mathrm{bf}}_{{\mathrm{q}},S}\cup{\mathrm{mf}}_{{\mathrm{q}},C}, \\
\pi_F^{-1}(\operatorname{diag}_{\mathrm{q}}) &= \operatorname{diag}_{{\mathrm{q}},F}.
\end{alignedat}
\end{equation}
similarly for the preimages under $\pi_S$ and $\pi_C$.
\begin{prop}[Composition of q-ps.d.o.s]
\label{PropqComp}
Let $A_j\in\Psi_\qop^{s_j,l_j,\gamma_j}(X)$, $j=1,2$. Then $A_1\circ A_2\in\Psi_\qop^{s_1+s_2,l_1+l_2,\gamma_1+\gamma_2}(X)$.
\end{prop}
\begin{proof}
Since the space $\Psi_\qop^s(X)$ is invariant under conjugation by powers of $\rho_{{\mathrm{zf}}_{\mathrm{q}}}$ and $\rho_{{\mathrm{mf}}_{\mathrm{q}}}$, it suffices to prove the result for $l_1=l_2=0$ and $\gamma_1=\gamma_2=0$. Write the Schwartz kernel $\kappa$ of $A_1\circ A_2$ in terms of the Schwartz kernels $\kappa_1,\kappa_2$ of $A_1,A_2$ as
\[
\kappa = (\nu_1\nu_2)^{-1}(\pi_C)_* \bigl( \pi_F^*\kappa_1 \cdot \pi_S^*\kappa_2 \cdot \pi_C^*\nu_1\cdot\pi^*\nu_2 \bigr)
\]
where $0<\nu_1\in\cC^\infty(X_{\mathrm{q}};{}^{\qop}\Omega X)$ is an arbitrary q-density, and $\nu_2=|\frac{{\mathrm d}\fm}{\fm}|$ is a b-density on $[0,2)_\fm$ with $\pi\colon X_{\mathrm{q}}^3\to[0,1]$ denoting the lifted projection. The term in parentheses is then a bounded conormal section of $\pi_F^*{}^{\qop}\Omega X\otimes\pi_S^*{}^{\qop}\Omega X\otimes\pi_C^*{}^{\qop}\Omega X\otimes\pi^*{}^{\bop}\Omega_{[0,1]}[0,2)\cong{}^{\bop}\Omega X^3_{\mathrm{q}}$ which vanishes to infinite order at the boundary hypersurfaces of $X^3_{\mathrm{q}}$ which map to ${\mathrm{lb}}_{{\mathrm{q}},2}$ or ${\mathrm{rb}}_{{\mathrm{q}},2}$ under $\pi_C$. The conclusion then follows using pullback and pushforward results for conormal distributions, see \cite[\S4]{MelroseDiffOnMwc} and \cite{MelrosePushfwd}.
\end{proof}
A proof of the uniform (for $\fm\in(0,1]$, the point being uniformity as $\fm\searrow 0$) boundedness of elements of $\Psi_\qop^0(X)$ on $L^2(X,\nu)$ for $0<\nu\in\cC^\infty(X_{\mathrm{q}},{}^{\qop}\Omega X)$ can be reduced, using H\"ormander's square root trick (see the proof of~\cite[Theorem~2.1.1]{HormanderFIO1}), to the uniform $L^2$-boundedness of elements of $\Psi_\qop^{-\infty}(X)$. Such elements have Schwartz kernels $\kappa\in\cC^\infty(X_{\mathrm{q}}^2,\pi_R^*{}^{\qop}\Omega X)$ which vanish to infinite order at ${\mathrm{lb}}_{{\mathrm{q}},2}$ and ${\mathrm{rb}}_{{\mathrm{q}},2}$. Pushforward of $\kappa$ along $\pi_L$ thus gives an element of $\cC^\infty(X_{\mathrm{q}})$. The Schur test implies the desired $L^2$-boundedness; since $\Psi_\qop(X)$ is invariant under conjugation by weights, we deduce boundedness on $L^2(X,\nu)$ for any weighted q-density $\nu$. One can then define weighted Sobolev spaces $H_{\mathrm{q}}^{s,l,\gamma}(X)$ also for real orders $s\in\mathbb{R}$ in the usual manner (cf.\ \S\ref{SsPH}), and any $A\in\Psi_\qop^{s,l,\gamma}(X)$ defines a (uniformly as $\fm\searrow 0$) bounded map $H_{\mathrm{q}}^{\tilde s,\tilde l,\tilde\gamma}(X)\to H_{\mathrm{q}}^{\tilde s-s,\tilde l-l,\tilde\gamma-\gamma}(X)$ for all $\tilde s,\tilde l,\tilde\gamma\in\mathbb{R}$.
The normal operator maps~\eqref{EqqVFNorm}--\eqref{EqqDiffNorm} for q-differential operators imply relationships between integer order q-Sobolev spaces on $X$ and families of b-Sobolev spaces on collar neighborhoods of $\hat X$ and $\dot X$. We immediately state the version for general orders, which rests on~\eqref{EqqPsdoNorm}; for brevity, we restrict to the class of densities which we will use in~\S\ref{SK}.
\begin{prop}[Relationships between Sobolev spaces]
\label{PropqHRel}
Fix a density $\nu=\rho_{{\mathrm{zf}}_q}^{n/2}\nu_0$ where $0<\nu_0\in\cC^\infty(X,{}^{\qop}\Omega X)$.\footnote{This includes as a special case $\fm$-independent smooth positive densities on $X$.}
\begin{enumerate}
\item\label{ItqHRelzf} Consider the (change of coordinates) map $\phi_{{\mathrm{zf}}_q}\colon(0,1]_\fm\times\hat X^\circ\ni(\fm,\hat x)\mapsto(\fm,\fm\hat x)\in X_{\mathrm{q}}$, and let $\chi\in\cC^\infty(X_{\mathrm{q}})$ be identically $1$ near ${\mathrm{zf}}_{\mathrm{q}}$ and supported in a collar neighborhood of ${\mathrm{zf}}_{\mathrm{q}}\subset X_{\mathrm{q}}$. Then we have a uniform equivalence of norms
\begin{equation}
\label{EqqHRelzf}
\|\chi u\|_{H_{{\mathrm{q}},\fm}^{s,l,\gamma}(X)} \sim \fm^{\frac{n}{2}-l}\| \phi_{{\mathrm{zf}}_q}^*(\chi u)|_\fm \|_{H_{\bop}^{s,\gamma-l}(\hat X,|{\mathrm d}\hat x|)},
\end{equation}
in the sense that there exists a constant $C>1$ independent of $\fm\in(0,1]$ so that the left hand side is bounded by $C$ times the right hand side, and vice versa.
\item\label{ItqHRelmf} Consider the inclusion map $\phi_{{\mathrm{mf}}_{\mathrm{q}}}\colon(0,1]_\fm\times\dot X^\circ\hookrightarrow X_{\mathrm{q}}$, and let $\chi\in\cC^\infty(X_{\mathrm{q}})$ be identically $1$ near ${\mathrm{mf}}_{\mathrm{q}}$ and supported in a collar neighborhood of $\dot X\subset X_{\mathrm{q}}$. Then we have a uniform equivalence of norms
\begin{equation}
\label{EqqHRelmf}
\|\chi u\|_{H_{{\mathrm{q}},\fm}^{s,l,\gamma}(X)} \sim \fm^{-\gamma} \|\phi_{{\mathrm{mf}}_{\mathrm{q}}}^*(\chi u)|_\fm\|_{H_{\bop}^{s,l-\gamma}(\dot X,\nu_{\mathrm{c}})},
\end{equation}
where $\nu_{\mathrm{c}}$ is the lift of a fixed smooth positive density on $X$ to $\dot X$. (Thus, one can take $\nu_{\mathrm{c}}=|{\mathrm d} x|=r^{n-1}|{\mathrm d} r\,{\mathrm d} g_{\mathbb{S}^{n-1}}|$ near $r=0$.)
\end{enumerate}
\end{prop}
\begin{proof}
Via division by $\fm^l$, we can reduce to the case $l=0$. Moreover, $\rho_{{\mathrm{mf}}_q}:=\fm/\sqrt{|x|^2+\fm^2}$ is a defining function of ${\mathrm{mf}}_q$, and its pullback along $\phi_{{\mathrm{zf}}_q}$ is $\langle\hat r\rangle^{-1}$ which is a defining function of $\partial\hat X$; therefore, we may also reduce to the case $\gamma=0$.
For part~\eqref{ItqHRelzf}, the $L^2$-case $s=0$ now follows from the observation that $\phi_{{\mathrm{zf}}_q}^*(|{\mathrm d} x|)=\fm^n|{\mathrm d}\hat x|$. For $s>0$, fix an elliptic operator $A_0\in\Psi_\bop^s(\hat X)$ (independent of $\fm$) with Schwartz kernel $\kappa_0$, and fix also $\tilde\chi\in\cC^\infty_\cp(X_{\mathrm{q}})$ to be identically $1$ near $\operatorname{supp}\chi$ but still with support in a collar neighborhood of ${\mathrm{zf}}_{\mathrm{q}}$; define then $A\in\Psi_\qop^s(\hat X)$ via its Schwartz kernel $\kappa$ as
\begin{equation}
\label{EqqHRelzfOp}
\kappa=(\pi_L^*\tilde\chi)(\pi_R^*\tilde\chi)\cdot(\phi_{{\mathrm{zf}}_q}^{-1})^*\kappa_0
\end{equation}
where $\pi_L,\pi_R\colon X^2_{\mathrm{q}}\to X_{\mathrm{q}}$ denote the lifted left and right projections. (Thus, $\kappa$ is obtained from $\kappa_0$ via dilation-invariant extension off ${\mathrm{zf}}_{{\mathrm{q}},2}$, followed by cutting it off to a neighborhood of ${\mathrm{zf}}_{{\mathrm{q}},2}$.) In particular, $A$ is elliptic as a q-ps.d.o.\ near the q-cotangent bundle over $\operatorname{supp}\chi$. We then have a uniform equivalence of norms
\begin{align*}
\|\chi u\|_{H_{{\mathrm{q}},\fm}^s(X)} &\sim \|(\chi u)|_\fm\|_{L^2(X)} + \|A(\chi u)|_\fm\|_{L^2(X)} \\
&\sim \fm^{\frac{n}{2}} \bigl( \|\phi_{{\mathrm{zf}}_q}^*(\chi u)|_\fm\|_{L^2(\hat X,|{\mathrm d}\hat x|)} + \| A_0(\phi_{{\mathrm{zf}}_q}^*(\chi u)|_\fm)\|_{L^2(\hat X,|{\mathrm d}\hat x|)}\bigr) \\
&\sim \fm^{\frac{n}{2}} \|\phi_{{\mathrm{zf}}_q}^*(\chi u)|_\fm\|_{H_{\bop}^s(\hat X,|{\mathrm d}\hat x|)},
\end{align*}
as claimed. For $s<0$, the claim follows by duality.
The proof of part~\eqref{ItqHRelmf} is completely analogous; one now takes an elliptic operator $A_0\in\Psi_\bop^s(\dot X)$ to measure $H_{\bop}^s(\dot X)$-norms, and relates this to $H_{\mathrm{q}}^s(X)$-norms by measuring the latter using a q-ps.d.o.\ $A$ defined analogously to~\eqref{EqqHRelzfOp}.
\end{proof}
\subsection{Q-single space}
\label{SsQS}
We shall control (solutions of) the degenerating spectral family for an infinite range of spectral parameters on the following space, which is a resolution of a parameter-dependent version of the q-single space $X_{\mathrm{q}}$ from Definition~\ref{DefqSingle}:
\begin{definition}[Q-single space]
\label{DefQSingle}
The \emph{Q-single space} of $X$ is the resolution of $\overline{\mathbb{R}_\sigma}\times[0,1]_\fm\times X$ defined as the iterated blow-up
\begin{align}
\label{EqQSingleXq}
X_{\mathrm{Q}} &:= \bigl[ \overline\mathbb{R}\times X_{\mathrm{q}}; \partial\overline\mathbb{R}\times{\mathrm{zf}}_{\mathrm{q}}; \partial\overline\mathbb{R}\times{\mathrm{mf}}_{\mathrm{q}} \bigr] \\
\label{EqQSingle}
&=\bigl[ \overline\mathbb{R}\times[0,1]\times X; \overline\mathbb{R}\times\{0\}\times\{0\}; \partial\overline\mathbb{R}\times\{0\}\times\{0\}; \partial\overline\mathbb{R}\times\{0\}\times X \bigr].
\end{align}
We denote its boundary hypersurfaces as follows:
\begin{enumerate}
\item ${\mathrm{mf}}$ (the `main face') is the lift of $\overline\mathbb{R}\times\{0\}\times X$;
\item ${\mathrm{zf}}$ (the `zero energy face') is the lift of $\overline\mathbb{R}\times\{0\}\times\{0\}$;
\item ${\mathrm{nf}}$ (the `nonzero energy face') is the lift of $\partial\overline\mathbb{R}\times\{0\}\times\{0\}$;
\item ${\mathrm{if}}$ (the `intermediate semiclassical face') is the lift of $\partial\overline\mathbb{R}\times\{0\}\times X$;
\item ${\mathrm{sf}}$ (the `semiclassical face') is the lift of $\partial\overline\mathbb{R}\times[0,1]\times X$.
\end{enumerate}
The hypersurfaces ${\mathrm{nf}}$, ${\mathrm{if}}$, ${\mathrm{sf}}$ have two connected components each, denoted ${\mathrm{nf}}_\pm$, ${\mathrm{if}}_\pm$, ${\mathrm{sf}}_\pm$, corresponding to whether $\sigma=+\infty$ or $-\infty$. For $H\subset X_{\mathrm{Q}}$ equal to any one of these boundary hypersurfaces, we denote by $\rho_H\in\cC^\infty(X_{\mathrm{Q}})$ a defining function of $H$, i.e.\ $H=\rho_H^{-1}(0)$ and ${\mathrm d}\rho_H\neq 0$ on $H$. For $H={\mathrm{nf}}$, we denote by $\rho_H$ a total boundary defining function for ${\mathrm{nf}}^+\cup{\mathrm{nf}}^-$, likewise for $H={\mathrm{if}},{\mathrm{sf}}$.
\end{definition}
We introduce a variety of functions defined on (subsets of) $X_{\mathrm{Q}}$. We denote by $\sigma,\fm$ the lifts of the first two coordinates on $\mathbb{R}\times[0,1]\times X$. We furthermore write $x=r\omega$ and $\hat x=\frac{x}{\fm}$, $\hat r=\frac{r}{\fm}$, $\hat\rho=\hat r^{-1}$ as in~\eqref{EqqCoord1}--\eqref{EqqCoord2}. We also set
\begin{equation}
\label{EqQCoord}
h = |\sigma|^{-1},\qquad
\tilde r := \frac{r}{h},\qquad
\tilde\rho := \tilde r^{-1} = \frac{h}{r},\qquad
\tilde\sigma := \fm\sigma,\qquad
\tilde h := |\tilde\sigma|^{-1} = \frac{h}{\fm}.
\end{equation}
See Figure~\ref{FigQSingle}.
\begin{figure}[!ht]
\centering
\includegraphics{FigQSingle}
\caption{The Q-single space $X_{\mathrm{Q}}$ in the case $X=(-1,1)$, restricted to $\sigma>-C$, and the coordinates~\eqref{EqqCoord1}, \eqref{EqqCoord2}, and \eqref{EqQCoord}.}
\label{FigQSingle}
\end{figure}
\begin{figure}[!ht]
\centering
\includegraphics{FigQSingleLvl}
\caption{The Q-single space $X_Q$ for $X=(-1,1)$. We show here the intersections of three level sets of the (rescaled) frequency variable $\sigma$ ($\tilde\sigma$) with $\fm^{-1}(0)$: one level set $\sigma=\sigma_0$ where $|\sigma_0|\lesssim 1$ is bounded and thus the rescaled Kerr frequency $\tilde\sigma_0=\fm\sigma_0=0$ vanishes; one level set $\sigma=\sigma_1$ where $\sigma_1$ is large but $\tilde\sigma_1:=\fm\sigma_1$ still vanishes; and one level set $\tilde\sigma=\fm\sigma=\tilde\sigma_0$ where the rescaled frequency $\tilde\sigma_0$ is of order $1$.}
\label{FigQSingleLvl}
\end{figure}
\begin{prop}[Structure of boundary hypersurfaces]
\fakephantomsection
\label{PropQStruct}
\hspace{0in}\begin{enumerate}
\item\label{ItQStructzf} The restriction of $(\sigma,\hat x)$ to the interior of ${\mathrm{zf}}$ induces a diffeomorphism
\[
{\mathrm{zf}} \cong \overline{\mathbb{R}_\sigma} \times \hat X
\]
Thus, ${\mathrm{zf}}$ is the total space of the (trivial) fibration $\hat X-{\mathrm{zf}}\to\overline{\mathbb{R}_\sigma}$.
\item\label{ItQStructmf} The restriction of $(\sigma,(r,\omega))$ to the interior of ${\mathrm{mf}}$ induces a diffeomorphism
\[
{\mathrm{mf}} \cong \bigl[ \overline\mathbb{R} \times \dot X; \partial\overline\mathbb{R}\times\partial\dot X \bigr].
\]
\item\label{ItQStructnf} The restriction of $(\tilde\sigma,\hat x)$ to the interior of ${\mathrm{nf}}_\pm$ induces a diffeomorphism
\[
{\mathrm{nf}}_\pm \cong \bigl[ (\pm[0,\infty]) \times \hat X; \{0\}\times\hat X \bigr].
\]
\item\label{ItQStructif} The restriction of $(\tilde\sigma,x)$ to the interior of ${\mathrm{if}}_\pm$ induces a diffeomorphism
\[
{\mathrm{if}}_\pm \cong (\pm[0,\infty])\times\dot X.
\]
\end{enumerate}
\end{prop}
\begin{proof}
The front face of $[\overline\mathbb{R}\times[0,1]\times X;\overline\mathbb{R}\times\{0\}\times\{0\}]=\overline\mathbb{R}\times[[0,1]\times X;\{0\}\times\{0\}]$ is diffeomorphic to $\overline\mathbb{R}\times\overline{T_0}X=\overline\mathbb{R}\times\hat X$ (with coordinates $\sigma$, $\hat x$ in the interior). The boundary hypersurface ${\mathrm{zf}}$ is obtained from this front face by blowing up $\sigma=\pm\infty$ which does not change the smooth structure. (Note that the lift of the final submanifold $\partial\overline\mathbb{R}\times\{0\}\times X$ in~\eqref{EqQSingle} is disjoint from this front face.) This proves part~\eqref{ItQStructzf}.
For part~\eqref{ItQStructmf}, we note that the lift of $\overline\mathbb{R}\times\{0\}\times X$ to $X_{\mathrm{Q}}$ is given by first resolving $\overline\mathbb{R}\times X$ at $\overline\mathbb{R}\times\{0\}$ (which produces $\overline\mathbb{R}\times\dot X$) followed by the resolution of $\partial\overline\mathbb{R}\times\partial\dot X$. Within this space then, the final resolution in~\eqref{EqQSingle} only blows up the lift of $\partial\overline\mathbb{R}\times\dot X$, which does not change the smooth structure.
For part~\eqref{ItQStructnf}, we first note that the front face ${\mathrm{nf}}_\pm'$ of the blow-up of the lift of $\{\pm\infty\}\times\{0\}\times\{0\}$ to $X_{\mathrm{Q}}':=[\overline\mathbb{R}\times[0,1]\times X;\overline\mathbb{R}\times\{0\}\times\{0\}]$ is diffeomorphic to $[0,\infty]_\mu\times\hat X$ where $\mu=\frac{\nu}{h}$ with $\nu=(r^2+\fm^2)^{1/2}$ a defining function of the front face of $X_{\mathrm{Q}}'$. The final blow-up in~\eqref{EqQSingle} restricts to ${\mathrm{nf}}_\pm'$ as the blow-up of $\{\infty\}\times\partial\hat X$, that is,
\[
{\mathrm{nf}}_\pm = \bigl[ [0,\infty]_\mu\times\hat X; \{\infty\}\times\partial\hat X\bigr].
\]
Upon restriction to a compact subset $K$ of the interior $\hat X^\circ$ in the second factor (thus $r\lesssim\fm$), we can replace $\nu$ by $\fm$, and thus $\mu$ by $\frac{\fm}{h}=\pm\tilde\sigma$. (That is, $\mu/(\pm\tilde\sigma)$ is a positive smooth function on $[0,\infty]_\mu\times K$.) Near the boundary of $\hat X$ on the other hand, let us work in the collar neighborhood $[0,1)_{\hat\rho}\times\mathbb{S}^{n-1}_\omega$ of $\partial\hat X\subset\hat X$. Since there we can replace $\nu$ by $r$ and thus $\mu$ by $\tilde r$, the lift of $[0,\infty]_\mu\times[0,1)_{\hat\rho}\times\mathbb{S}^{n-1}_\omega$ to ${\mathrm{nf}}_\pm$ is
\[
\bigl[ [0,\infty]_{\tilde r} \times [0,1)_{\hat\rho}\times\mathbb{S}^{n-1}_\omega; \{\infty\}\times\{0\}\times\mathbb{S}^{n-1}_\omega \bigr] = \bigl[ [0,\infty]\times[0,1); \{\infty\}\times\{0\} \bigr] \times \mathbb{S}^{n-1}.
\]
Observe then that the map $(\tilde r,\hat\rho)\to(\tilde r\hat\rho,\hat\rho)$ induces a diffeomorphism
\begin{equation}
\label{EqQnfBlowup}
\bigl[ [0,\infty]\times[0,1); \{\infty\}\times\{0\} \bigr] \cong \bigl[ [0,\infty] \times [0,1); \{0\}\times\{0\} \bigr].
\end{equation}
Since $\tilde r\hat\rho=\pm\tilde\sigma$, this proves part~\eqref{ItQStructnf}.
Finally, for the proof of part~\eqref{ItQStructif}, we note that coordinates near the lift of $\{\infty\}\times\{0\}\times X$ to $[\overline\mathbb{R}\times[0,1]\times X;\partial\overline\mathbb{R}\times\{0\}\times\{0\}]$ are $r\geq 0$, $\omega\in\mathbb{S}^{n-1}$, $\hat\rho=\frac{\fm}{r}\geq 0$, and $\tilde\rho=\frac{h}{r}\geq 0$, with the lift of $\{\infty\}\times\{0\}\times X$ given by $\tilde\rho=\hat\rho=0$. Therefore,
\[
{\mathrm{if}} \cong [0,\infty]_{\hat\rho/\tilde\rho} \times X,
\]
and it remains to note that $\hat\rho/\tilde\rho=\fm/h=\pm\tilde\sigma$.
\end{proof}
\begin{definition}[Pieces of ${\mathrm{zf}}$, ${\mathrm{mf}}$ and ${\mathrm{nf}}_\pm$]
\label{DefQPieces}
We define
\begin{alignat*}{2}
{\mathrm{mf}}_{\pm,\hbar} &:= {\mathrm{mf}} \cap \sigma^{-1}(\pm[1,\infty]), \\
{\mathrm{nf}}_{\pm,\tilde\hbar} &:= {\mathrm{nf}}_\pm \cap \tilde\sigma^{-1}(\pm[1,\infty]), &\qquad
{\mathrm{nf}}_{\pm,{\mathrm{low}}} &:= {\mathrm{nf}}_\pm \cap \tilde\sigma^{-1}(\pm[0,1]).
\end{alignat*}
We furthermore set, for $\sigma_0\in\mathbb{R}$ and $\tilde\sigma_0\in\mathbb{R}\setminus\{0\}$,
\[
{\mathrm{zf}}_{\sigma_0} := {\mathrm{zf}} \cap \sigma^{-1}(\sigma_0),\qquad
{\mathrm{mf}}_{\sigma_0} := {\mathrm{mf}} \cap \sigma^{-1}(\sigma_0),\qquad
{\mathrm{nf}}_{\tilde\sigma_0} := {\mathrm{nf}} \cap \tilde\sigma^{-1}(\tilde\sigma_0).
\]
\end{definition}
Thus, using the notation for the single spaces for semiclassical cone, ${\mathrm{sc}\text{-}\mathrm{b}}$-transition, and semiclassical scattering analysis from~\S\S\ref{SsPch}, \ref{SsPbsc}, and \ref{SsPscbt}, respectively, Proposition~\ref{PropQStruct} provides diffeomorphisms
\begin{equation}
\label{EqQPieces}
\begin{alignedat}{2}
{\mathrm{mf}}_{\pm,\hbar} &\cong \dot X_{\mathrm{c}\hbar}&\quad&\text{(with semiclassical parameter $h=|\sigma|^{-1}\in[0,1]$)}, \\
{\mathrm{nf}}_{\pm,\tilde\hbar} &\cong \hat X_{{\mathrm{sc}},\tilde\hbar}&\quad&\text{(with semiclassical parameter $\tilde h=|\tilde\sigma|^{-1}\in[0,1]$)}, \\
{\mathrm{nf}}_{\pm,{\mathrm{low}}} &\cong \hat X_{\mathrm{sc}\text{-}\mathrm{b}}&\quad&\text{(with spectral parameter $\tilde\sigma\in\pm[0,1]$)},
\end{alignedat}
\end{equation}
as well as
\[
{\mathrm{zf}}_{\sigma_0} \cong \hat X,\qquad
{\mathrm{mf}}_{\sigma_0} \cong \dot X,\qquad
{\mathrm{nf}}_{\tilde\sigma_0} \cong \hat X.
\]
\subsection{Q-vector fields and differential operators}
\label{SsQV}
We next turn to the class of $\sigma$- and $\fm$-dependent vector fields on $X$ on which our uniform analysis will be based.
\begin{definition}[Q-vector fields]
\label{DefQVF}
The space of \emph{Q-vector fields} on $X$ is defined as
\[
\cV_\Qop(X) := \{ V\in\rho_{\mathrm{if}}\rho_{\mathrm{sf}}\cV_\bop(X_{\mathrm{Q}}) \colon V\sigma=0,\ V\fm=0 \}.
\]
\end{definition}
Since $X_{\mathrm{Q}}\cap\{\sigma\in\mathbb{R},\ \fm>0\}=\mathbb{R}_\sigma\times(0,1]_\fm\times X$, an element $V\in\cV_\Qop(X)$ is thus a smooth family $\mathbb{R}\times(0,1]\ni(\sigma,\fm)\mapsto V_{\sigma,\fm}$ of smooth vector fields on $X$ which degenerate or become singular in a particular fashion in the limits $r\to 0$, $\fm\to 0$, $|\sigma|\to\infty$, or any combination thereof.
\begin{lemma}[Properties of $\cV_\Qop(X)$]
\label{LemmaQVF}
The space $\cV_\Qop(X)$ is Lie algebra, and in fact
\begin{equation}
\label{EqQVFComm}
V,W\in\cV_\Qop(X) \implies [V,W] \in \rho_{\mathrm{sf}}\rho_{\mathrm{if}}\cV_\Qop(X).
\end{equation}
Moreover, for any weight $w=\prod_H\rho_H^{\alpha_H}$ where $H\subset X_{\mathrm{Q}}$ ranges over all boundary hypersurfaces and $\alpha_H\in\mathbb{R}$, we have $w^{-1}[V,w]\in\rho_{\mathrm{sf}}\rho_{\mathrm{if}}\cC^\infty(X_{\mathrm{Q}})$ for any $V\in\cV_\Qop(X)$.
\end{lemma}
\begin{proof}
The final claim follows from the fact that $w^{-1}[V_0,w]\in\cC^\infty(X_{\mathrm{Q}})$ for \emph{any} $V_0\in\cV_\bop(X_{\mathrm{Q}})$. In order to prove~\eqref{EqQVFComm}, we observe that $[V,W]\sigma=V W\sigma-W V\sigma=0$, likewise $[V,W]\fm=0$; moreover, we have, for $w:=\rho_{\mathrm{sf}}\rho_{\mathrm{if}}$ and $V=w V_0$, $W=w W_0\in\cV_\Qop(X)$,
\[
[V,W] = w\bigl((w^{-1}[V_0,w])W_0 - (w^{-1}[W_0,w])w V_0\bigr) \in w\cV_\bop(X_{\mathrm{Q}}),
\]
which implies the claim.
\end{proof}
We make this explicit in various local coordinate systems; we use the notation from~\eqref{EqqCoord1}, \eqref{EqqCoord2}, and \eqref{EqQCoord}.
\begin{enumerate}
\item The intersection of $X_{\mathrm{Q}}$ with $|\hat x|<C$ is $[[0,1]_\fm\times\overline{\mathbb{R}_\sigma}\times B;\{0\}\times\partial\overline\mathbb{R}\times B]$ where $B=\{\hat x\in\hat X\colon|\hat x|<C\}$ is a ball. Thus, in the coordinates $\fm,\sigma,\hat x$, a basis of Q-vector fields is given by $\partial_{\hat x^j}$ ($j=1,\ldots,n$) in the set where $\sigma$ is bounded or even where $|\sigma|\geq 1$ but $\tilde\sigma$ is bounded, and by $\tilde h\partial_{\hat x^j}$ when $|\tilde\sigma|\gtrsim 1$ (where $\tilde h$ is a defining function of ${\mathrm{sf}}$).
\item The intersection of $X_{\mathrm{Q}}$ with $r>c>0$ is $[[0,1]_\fm\times\overline{\mathbb{R}_\sigma}\times A;\{0\}\times\partial\overline\mathbb{R}\times A]$ where $A=\{x\in X\colon|x|>c\}$. A basis of Q-vector fields, in the coordinates $\fm,\sigma,x$ (or $r,\omega$ instead of $x$) is then for bounded $\sigma$ given by $\partial_{x^j}$ ($j=1,\ldots,n$) (or equivalently $\partial_r$ and spherical vector fields, which we schematically write as $\partial_\omega$), and for large $|\sigma|$ by $h\partial_{x^j}$ (or $h\partial_r$, $h\partial_\omega$).
\end{enumerate}
It remains to consider the subset of $X_{\mathrm{Q}}$ where $|\hat x|>C$ and $r<c$.
\begin{enumerate}
\setcounter{enumi}{2}
\item Near the interior of ${\mathrm{zf}}\cap{\mathrm{mf}}$, we have local coordinates $\sigma\in\mathbb{R}$, $\hat\rho\geq 0$, $r\geq 0$, $\omega\in\mathbb{S}^{n-1}$. Q-vector fields are spanned by $r\partial_r-\hat\rho\partial_{\hat\rho}$, $\partial_\omega$.
\item Near the corner ${\mathrm{zf}}\cap{\mathrm{mf}}\cap{\mathrm{nf}}_+$, local coordinates are $h\geq 0$, $\tilde r\geq 0$, $\hat\rho\geq 0$, $\omega\in\mathbb{S}^{n-1}$, and Q-vector fields are spanned by $\tilde r\partial_{\tilde r}-\hat\rho\partial_{\hat\rho}$, $\partial_\omega$.
\item Near the corner ${\mathrm{mf}}\cap{\mathrm{nf}}_+\cap{\mathrm{if}}_+$, local coordinates are $r\geq 0$, $\tilde\rho\geq 0$, $\tilde\sigma\geq 0$, $\omega$, with $\tilde\rho$ a defining function of ${\mathrm{if}}_+$. Q-vector fields are thus spanned by $\tilde\rho(r\partial_r-\tilde\rho\partial_{\tilde\rho})$, $\tilde\rho\partial_\omega$.
\item Near the corner ${\mathrm{nf}}_+\cap{\mathrm{if}}_+\cap{\mathrm{sf}}$ finally, local coordinates are $r\geq 0$, $\hat\rho\geq 0$, $\tilde h\geq 0$, $\omega$, with $\hat\rho$ and $\tilde h$ being local defining functions of ${\mathrm{if}}_+$ and ${\mathrm{sf}}$, respectively. Thus, Q-vector fields are spanned by $\hat\rho\tilde h(r\partial_r-\hat\rho\partial_{\hat\rho})$, $\hat\rho\tilde h\partial_\omega$.
\end{enumerate}
One can also give a more global description: in $|\hat x|\lesssim 1$, resp.\ $|\hat x|\gtrsim 1$, Q-vector fields are spanned by
\begin{equation}
\label{EqQSpan}
\frac{h}{h+\fm}\partial_{\hat x^j}\ (j=1,\ldots,n),\qquad \text{resp.}\qquad \frac{h}{h+r}r\partial_r,\ \frac{h}{h+r}\partial_\omega.
\end{equation}
\begin{definition}[Q-bundles]
\label{DefQBundle}
We denote by ${}^\Qop T X\to X_{\mathrm{Q}}$ the \emph{Q-vector bundle}, which is the vector bundle equipped with a smooth bundle map ${}^\Qop T X\to T X_{\mathrm{Q}}$ with the property that $\cV_\Qop(X)=\cC^\infty(X_{\mathrm{Q}},{}^\Qop T X)$. The dual bundle ${}^\Qop T^*X$ is the \emph{Q-cotangent bundle}.
\end{definition}
We next study restrictions of Q-vector fields to various boundary hypersurfaces of $X_{\mathrm{Q}}$. We use the notation from Appendix~\ref{SP}. The following result, based on~\eqref{EqQPieces} is the reason for the appearance of the various model problems in uniform singular analysis in the Q-setting:
\begin{lemma}[Restriction to boundary hypersurfaces]
\fakephantomsection
\label{LemmaQVFRes}
\hspace{0in}\begin{enumerate}
\item\label{ItQVFReszf} Restriction to ${\mathrm{zf}}$ induces a surjective map $N_{\mathrm{zf}}\colon\cV_\Qop(X)\to\cC^\infty(\overline\mathbb{R};\cV_\bop(\hat X))$ with kernel $\rho_{\mathrm{zf}}\cV_\Qop(X)$.
\item\label{ItQVFResmf} Restriction to ${\mathrm{mf}}_{\sigma_0}$ induces a surjective map $N_{{\mathrm{mf}}_{\sigma_0}}\colon\cV_\Qop(X)\to\cV_\bop(\dot X)$. Restriction to ${\mathrm{mf}}_{\pm,\hbar}$ induces a surjective map $N_{{\mathrm{mf}}_{\pm,\hbar}}\colon\cV_\Qop(X)\to\cV_\chop(\dot X)$ (see~\S\usref{SsPch}). The kernel of $\oplus_{\sigma_0\in\mathbb{R}}N_{{\mathrm{mf}}_{\sigma_0}}$ is $\rho_{\mathrm{mf}}\cV_\Qop(X)$.
\item\label{ItQVFResnf} Restriction to ${\mathrm{nf}}_{\pm,{\mathrm{low}}}$, resp.\ ${\mathrm{nf}}_{\pm,\tilde\hbar}$ induces a surjective map $N_{{\mathrm{nf}}_{\pm,{\mathrm{low}}}}\colon\cV_\Qop(X)\to\cV_\scbtop(\hat X)$ (see~\S\usref{SsPscbt}), resp.\ $N_{{\mathrm{nf}}_{\pm,\tilde\hbar}}\colon\cV_\Qop(X)\to\cV_\schop(\hat X)$ (see~\S\usref{SsPbsc}). The kernel of $(N_{{\mathrm{nf}}_{\pm,{\mathrm{low}}}},N_{{\mathrm{nf}}_{\pm,\tilde\hbar}})$ is $\rho_{{\mathrm{nf}}_\pm}\cV_\Qop(X)$.
\end{enumerate}
\end{lemma}
We could leave ${\mathrm{mf}}$ in one piece; then restriction to ${\mathrm{mf}}$ induces a map from $\cV_\Qop(X)$ onto the space of b-vector fields on $[\overline{\mathbb{R}_\sigma}\times\dot X;\partial\overline\mathbb{R}\times\partial\dot X]$ which annihilate $\sigma$ and vanish at the lift of $\partial\overline\mathbb{R}\times\dot X$. This target space consists of smooth families of b-vector fields which degenerate like semiclassical cone vector fields as $|\sigma|\to\infty$. (An analogous remark applies to ${\mathrm{nf}}_\pm$.) The reason for splitting ${\mathrm{mf}}$ (or ${\mathrm{nf}}_\pm$) is that the analysis at high energies $|\sigma|\to\infty$ (or $|\tilde\sigma|\to\infty$) will be conceptually different from the analysis at bounded frequencies $\sigma$ (or $\tilde\sigma$).
\begin{proof}[Proof of Lemma~\usref{LemmaQVFRes}]
We prove this using the coordinate systems and local spanning sets of $\cV_\Qop(X)$ listed before the statement of Lemma~\ref{LemmaQVFRes}. Thus, part~\eqref{ItQVFReszf} follows from the observation that the map $N_{\mathrm{zf}}$, in the coordinates $\fm,\sigma,\hat x$, resp.\ $\sigma,\hat\rho,r,\omega$, maps $\partial_{\hat x^j}$ to itself ($j=1,\ldots,n$), resp.\ $r\partial_r-\hat\rho\partial_{\hat\rho}$, $\partial_\omega$ to $-\hat\rho\partial_{\hat\rho}$, $\partial_\omega$, with coefficients that are smooth on $\overline{\mathbb{R}_\sigma}\times\hat X$.
For part~\eqref{ItQVFResmf}, consider first the case of bounded $\sigma$. The conclusion is then clear in $r>c>0$, whereas near ${\mathrm{mf}}\cap{\mathrm{zf}}$ and in the coordinates $\sigma,\hat\rho,r,\omega$, the map $N_{\mathrm{mf}}$ maps $r\partial_r-\hat\rho\partial_{\hat\rho}\mapsto r\partial_r$ and $\partial_\omega\mapsto\partial_\omega$, thus has range equal to smooth families (in $\sigma$) of elements of $\cV_\bop(\dot X)$. In the coordinates $h,\tilde r,\hat\rho,\omega$ near ${\mathrm{mf}}\cap{\mathrm{nf}}_\pm\cap{\mathrm{zf}}$, with $\hat\rho$ a defining function of ${\mathrm{mf}}$, the map $N_{\mathrm{mf}}$ takes $\tilde r\partial_{\tilde r}-\hat\rho\partial_{\hat\rho}\mapsto\tilde r\partial_{\tilde r}$, $\partial_\omega\mapsto\partial_\omega$, thus its range consists of ${\mathrm{c}\hbar}$-vector fields indeed. This is true also in the coordinates $r,\tilde\rho,\pm\tilde\sigma,\omega$ near ${\mathrm{mf}}\cap{\mathrm{nf}}_\pm\cap{\mathrm{if}}_\pm$ (with $\pm\tilde\sigma$ defining ${\mathrm{mf}}$), in which $N_{\mathrm{mf}}$ maps $\tilde\rho(r\partial_r-\tilde\rho\partial_{\tilde\rho})$ and $\tilde\rho\partial_\omega$ to the same expressions; since the semiclassical face of $\dot X_{\mathrm{c}\hbar}$ is defined by $\tilde\rho=0$, this proves part~\eqref{ItQVFResmf}.
For part~\eqref{ItQVFResnf}, the maps $N_{{\mathrm{nf}}_\pm,{\mathrm{low}}}$ and $N_{{\mathrm{nf}}_\pm,\tilde\hbar}$ are given by the restriction of coefficients of Q-vector fields, with respect to the bases listed in the various coordinate systems prior to the statement of Lemma~\ref{LemmaQVF}, to ${\mathrm{nf}}_\pm$. These vector fields are indeed ${\mathrm{sc}\text{-}\mathrm{b}}$-vector fields on ${\mathrm{nf}}_{\pm,{\mathrm{low}}}$ (with scattering behavior at $\tilde\rho=0$, cf.\ the coordinate system near ${\mathrm{nf}}_\pm\cap{\mathrm{mf}}\cap{\mathrm{if}}_\pm$), and semiclassical scattering vector fields on ${\mathrm{nf}}_{\pm,\tilde\hbar}$ (with $\tilde h$ the semiclassical parameter, and with scattering behavior at $\hat\rho=0$, cf.\ the coordinate system near ${\mathrm{nf}}_+\cap{\mathrm{if}}_+\cap{\mathrm{sf}}$).
\end{proof}
\begin{cor}[Bundle identifications]
\label{CorQBundle}
The restriction maps of Lemma~\usref{LemmaQVFRes} induce bundle isomorphisms
\begin{alignat*}{2}
{}^\Qop T_{\mathrm{zf}} X &\cong \overline\mathbb{R} \times {}^{\bop}T\hat X\ \text{(as bundles over ${\mathrm{zf}}=\overline\mathbb{R}\times\hat X$)},\hspace{-20em}& \\
{}^\Qop T_{{\mathrm{mf}}_{\sigma_0}}X &\cong {}^{\bop}T\dot X, &\qquad
{}^\Qop T_{{\mathrm{mf}}_{\pm,\hbar}}X &\cong {}^\chop T\dot X, \\
{}^\Qop T_{{\mathrm{nf}}_{\pm,{\mathrm{low}}}}X &\cong {}^\scbtop T\hat X, &\qquad
{}^\Qop T_{{\mathrm{nf}}_{\pm,\tilde\hbar}}X &\cong {}^{{\mathrm{sc}}\tilde\hbar}T\hat X,
\end{alignat*}
and ${}^\Qop T_{{\mathrm{nf}}_{\tilde\sigma_0}}X \cong{}^{\scop}T\hat X$, where $\sigma_0\in\mathbb{R}$ and $\tilde\sigma_0\in\mathbb{R}\setminus\{0\}$.
\end{cor}
\begin{definition}[Q-differential operators]
\label{DefQDiff}
For $m\in\mathbb{N}_0$, we denote by $\Diff_\Qop^m(X)$ the space of locally finite sums of up to $m$-fold compositions of elements of $\cV_\Qop(X)$ (a $0$-fold composition is, by definition an element of $\cC^\infty(X_{\mathrm{Q}})$). Given a collection $\alpha=(\alpha_H)$ of weights $\alpha_H\in\mathbb{R}$ for $H={\mathrm{zf}},{\mathrm{mf}},{\mathrm{nf}},{\mathrm{if}},{\mathrm{sf}}$, we denote more generally
\[
\Diff_\Qop^{m,\alpha}(X) = \biggl(\prod_H\rho_H^{-\alpha_H}\biggr)\Diff_\Qop^m(X) = \biggl\{ \biggl(\prod_H\rho_H^{-\alpha_H}\biggr) A \colon A \in \Diff_\Qop^m(X) \biggr\}.
\]
\end{definition}
Analogously to Q-vector fields, Q-differential operators $A\in\Diff_\Qop^m(X)$ are smooth families $(\fm,\sigma)\mapsto A_{\fm,\sigma}\in\mathrm{Diff}^m(X)$ of differential operators on $X$ which degenerate in a particular fashion as $\fm\to 0$, $|\sigma|\to\infty$, and/or $r\to 0$. Note that elements of $\Diff_\Qop(X)$ commute with multiplication by $\fm$ and $\sigma$, with
\begin{equation}
\label{EqQDiffEx}
\fm \in \Diff_\Qop^{0,(-1,-1,-1,-1,0)}(X),\quad
\sigma \in \Diff_\Qop^{0,(0,0,1,1,1)}(X).
\end{equation}
Thus, for instance, it suffices to restrict in Definition~\ref{DefQDiff} to the case $\alpha_{\mathrm{mf}}=\alpha_{\mathrm{nf}}=0$. We also remark that a $\sigma$-independent q-differential operator $A\in\Diff_\qop^m(X)$ defines an element $A\in\Diff_\Qop^{m,(0,0,0,m,m)}(X)$; this is a consequence of the fact that $V\in\cV_\qop(X)$, regarded as a $\sigma$-independent vector field on $X_{\mathrm{Q}}$, satisfies $V\in\rho_{\mathrm{if}}^{-1}\rho_{\mathrm{sf}}^{-1}\cV_\Qop(X)$, as follows directly from the definition. Recalling~\eqref{EqqVFX}, this implies that, regarding an operator on $X$ as an $\fm$- and $\sigma$-independent operator on $X_{\mathrm{q}}$ and $X_{\mathrm{Q}}$,
\begin{equation}
\label{EqQDiffEx2}
\mathrm{Diff}^m(X)\subset\rho_{{\mathrm{zf}}_{\mathrm{q}}}^{-m}\Diff_\qop^m(X)\subset\Diff_\Qop^{m,(m,0,m,m,m)}(X).
\end{equation}
The principal symbol ${}^\Qop\upsigma^1(V)$ of $V\in\cV_\Qop(X)$, defined as mapping $\xi\in{}^\Qop T^*X$ to $i\xi(V)$, is a fiber-linear function. The property~\eqref{EqQVFComm} implies that the principal symbol extends to a multiplicative family of maps ${}^\Qop\upsigma^m$ with the property that
\begin{equation}
\label{EqQDiffSym}
0 \to \rho_{\mathrm{sf}}\rho_{\mathrm{if}}\Diff_\Qop^{m-1}(X) \hookrightarrow \Diff_\Qop^m(X) \xrightarrow{{}^\Qop\upsigma^m} P^m({}^\Qop T^*X)/\rho_{\mathrm{sf}}\rho_{\mathrm{if}} P^{m-1}({}^\Qop T^*X) \to 0
\end{equation}
is a short exact sequence. By Lemma~\ref{LemmaQVFRes}, we get multiplicative normal operator maps
\begin{equation}
\label{EqQNormOp}
\begin{alignedat}{2}
N_{\mathrm{zf}} &\colon \Diff_\Qop^m(X) \to \cC^\infty(\overline\mathbb{R};\Diff_\bop^m(\hat X)), \\
N_{{\mathrm{mf}}_{\sigma_0}} &\colon \Diff_\Qop^m(X) \to \Diff_\bop^m(\dot X), &\qquad
N_{{\mathrm{mf}}_{\pm,\hbar}} &\colon \Diff_\Qop^m(X) \to \Diff_\chop^m(\dot X), \\
N_{{\mathrm{nf}}_{\pm,{\mathrm{low}}}} &\colon \Diff_\Qop^m(X) \to \Diff_\scbtop^m(\hat X), &\qquad
N_{{\mathrm{nf}}_{\pm,\tilde\hbar}} &\colon \Diff_\Qop^m(X) \to \mathrm{Diff}_{{\mathrm{sc}}\tilde\hbar}^m(\hat X),
\end{alignedat}
\end{equation}
as well as similar maps on spaces of weighted operators (with the weight at $H$ required to be $0$ in the definition of $N_H$). Moreover, the principal symbol of $N_{\mathrm{zf}}(P)$ is given by the restriction of ${}^\Qop\upsigma^m(P)$ to ${}^\Qop T^*_{\mathrm{zf}} X\cong\overline\mathbb{R}\times{}^{\bop}T^*\hat X$ via Corollary~\ref{CorQBundle}, similarly for the principal symbols of the other normal operators. Note also that the vanishing of $N_{\mathrm{zf}}(P)$, resp.\ $N_{{\mathrm{mf}}_{\sigma_0}}(P)$ for all $\sigma_0$, resp.\ $N_{{\mathrm{nf}}_{\pm,{\mathrm{low}}}}(P)$ and $N_{{\mathrm{nf}}_{\pm,\tilde\hbar}}(P)$ implies that $P$ vanishes to leading order the appropriate boundary hypersurface, i.e.\ $P\in\rho_{\mathrm{zf}}\Diff_\Qop^m(X)$, resp.\ $P\in\rho_{\mathrm{mf}}\Diff_\Qop^m(X)$, resp.\ $P\in\rho_{{\mathrm{nf}}_\pm}\Diff_\Qop^m(X)$. Together with ${}^\Qop\upsigma^m(P)$, these normal operators thus capture $P$ to leading order in all $6$ senses (corresponding to the $6$ orders in Definition~\ref{DefQDiff}).
Furthermore, we can restrict to level sets $\sigma^{-1}(\sigma_0)$ or $\tilde\sigma(\tilde\sigma_0)$ for $\sigma_0\in\mathbb{R}$ or $\tilde\sigma_0\in\mathbb{R}\setminus\{0\}$. This gives normal operator homomorphisms
\[
N_{\sigma_0} \colon \Diff_\Qop^m(X)\to\Diff_\qop^m(X),\qquad
N_{{\mathrm{nf}}_{\tilde\sigma_0}} \colon \Diff_\Qop^m(X)\to\Diff_\scop^m(\hat X).
\]
See~\S\ref{SsKS} for the way in which the spectral family of interest in Theorem~\ref{ThmI} fits into this framework of Q-analysis.
\subsection{Q-pseudodifferential operators}
\label{SsQP}
The microlocal analysis of Q-differential operators relies on a corresponding Q-pseudodifferential algebra, which we proceed to define; analogously to Q-differential operators, a Q-ps.d.o.\ $A$ will be a smooth family $\mathbb{R}\times(0,1]\ni(\sigma,\fm)\mapsto A_{\sigma,\fm}$ of ordinary ps.d.o.s on a manifold $X$ without boundary.
\begin{definition}[Q-double space]
\label{DefQPDouble}
Recall the q-double space $X_{\mathrm{q}}^2$ of $X$ and its submanifolds ${\mathrm{zf}}_{{\mathrm{q}},2}\cong\hat X^2_{\mathrm{b}}$, ${\mathrm{mf}}_{{\mathrm{q}},2}\cong\dot X^2_{\mathrm{b}}$, ${\mathrm{lb}}_{{\mathrm{q}},2}$, ${\mathrm{rb}}_{{\mathrm{q}},2}$, and $\operatorname{diag}_{\mathrm{q}}$ from Definition~\ref{DefqDouble}. The \emph{Q-double space} of $X$ is then defined as the resolution of $\overline{\mathbb{R}_\sigma}\times X^2_{\mathrm{q}}$ given by
\begin{equation}
\label{EqQPDouble}
\begin{split}
X^2_{\mathrm{Q}} &:= \bigl[ \overline\mathbb{R} \times X^2_{\mathrm{q}}; \partial\overline\mathbb{R}\times{\mathrm{zf}}_{{\mathrm{q}},2}; \partial\overline\mathbb{R}\times\operatorname{diag}_{\mathrm{q}}; \\
&\quad\qquad \partial\overline\mathbb{R}\times(\operatorname{diag}_{\mathrm{q}}\cap\,{\mathrm{mf}}_{{\mathrm{q}},2}), \partial\overline\mathbb{R}\times{\mathrm{lb}}_{{\mathrm{q}},2}, \partial\overline\mathbb{R}\times{\mathrm{rb}}_{{\mathrm{q}},2}; \partial\overline\mathbb{R}\times{\mathrm{mf}}_{{\mathrm{q}},2} \bigr].
\end{split}
\end{equation}
We label its boundary hypersurfaces as follows:
\begin{enumerate}
\item ${\mathrm{zf}}_2$ is the lift of $\overline\mathbb{R}\times{\mathrm{zf}}_{{\mathrm{q}},2}$;
\item ${\mathrm{mf}}_2$ is the lift of $\overline\mathbb{R}\times{\mathrm{mf}}_{{\mathrm{q}},2}$;
\item ${\mathrm{nf}}_2$ is the lift of $\partial\overline\mathbb{R}\times{\mathrm{zf}}_{{\mathrm{q}},2}$;
\item ${\mathrm{if}}_2$ is the lift of $\partial\overline\mathbb{R}\times(\operatorname{diag}_{\mathrm{q}}\cap\,{\mathrm{mf}}_{{\mathrm{q}},2})$, and ${\mathrm{if}}'_2$ is the lift of $\partial\overline\mathbb{R}\times{\mathrm{mf}}_{{\mathrm{q}},2}$;
\item ${\mathrm{sf}}_2$ is the lift of $\partial\overline\mathbb{R}\times\operatorname{diag}_{\mathrm{q}}$, and ${\mathrm{sf}}'_2$ is the lift of $\partial\overline\mathbb{R}\times X^2_{\mathrm{q}}$;
\item ${\mathrm{lb}}_2$, resp.\ ${\mathrm{rb}}_2$ is the lift of $\overline\mathbb{R}\times{\mathrm{lb}}_{{\mathrm{q}},2}$, resp.\ $\overline\mathbb{R}\times{\mathrm{rb}}_{{\mathrm{q}},2}$;
\item ${\mathrm{tlb}}_2$, resp.\ ${\mathrm{trb}}_2$ is the lift of $\partial\overline\mathbb{R}\times{\mathrm{lb}}_{{\mathrm{q}},2}$, resp.\ $\partial\overline\mathbb{R}\times{\mathrm{rb}}_{{\mathrm{q}},2}$.
\end{enumerate}
We denote by ${\mathrm{nf}}_{2,\pm}$ the connected components of ${\mathrm{nf}}_2$ corresponding to the value of $\sigma=\pm\infty$; similarly for ${\mathrm{if}}_{2,\pm}$, ${\mathrm{if}}_{2,\pm}'$, ${\mathrm{sf}}_{2,\pm}$, ${\mathrm{sf}}_{2,\pm}'$, ${\mathrm{tlb}}_{2,\pm}$, ${\mathrm{trb}}_{2,\pm}$. Furthermore, we write for $\sigma_0\in\mathbb{R}$ and $\tilde\sigma_0\in\mathbb{R}\setminus\{0\}$
\begin{alignat*}{2}
{\mathrm{mf}}_{2,\sigma_0} &:= {\mathrm{mf}}_2 \cap \sigma^{-1}(\sigma_0),&\qquad
{\mathrm{mf}}_{2,\pm,\hbar} &:= {\mathrm{mf}}_{2,\pm} \cap \sigma^{-1}(\pm[1,\infty]), \\
{\mathrm{nf}}_{2,\pm,{\mathrm{low}}} &:= {\mathrm{nf}}_{2,\pm}\cap\tilde\sigma^{-1}(\pm[0,1]),&\qquad
{\mathrm{nf}}_{2,\pm,\tilde\hbar} &:= {\mathrm{nf}}_{2,\pm}\cap\tilde\sigma^{-1}(\pm[1,\infty]),
\end{alignat*}
and ${\mathrm{nf}}_{2,\tilde\sigma_0}:={\mathrm{nf}}_2\cap\tilde\sigma^{-1}(\tilde\sigma_0)$. Finally, $\operatorname{diag}_{\mathrm{Q}}$ denotes the lift of $\overline\mathbb{R}\times\operatorname{diag}_{\mathrm{q}}$.
\end{definition}
In~\eqref{EqQPDouble}, note that $\partial\overline\mathbb{R}\times{\mathrm{lb}}_{{\mathrm{q}},2}$, $\partial\overline\mathbb{R}\times{\mathrm{rb}}_{{\mathrm{q}},2}$, and $\partial\overline\mathbb{R}\times(\operatorname{diag}_{\mathrm{q}}\cap\,{\mathrm{mf}}_{{\mathrm{q}},2})$ are disjoint, and hence they can be blown up in any order.
\begin{lemma}[b-fibrations from the Q-double space]
\label{LemmaQPbFib}
The left projection, resp.\ right projection $\mathbb{R}\times(0,1]\times X\times X\ni(\sigma,\fm,x,x')\mapsto(\sigma,\fm,x)\in\mathbb{R}\times(0,1]\times X$, resp.\ $(\sigma,\fm,x')$ lifts to a b-fibration $\pi_L$, resp.\ $\pi_R\colon X^2_{\mathrm{Q}}\to X_{\mathrm{Q}}$.
\end{lemma}
\begin{proof}
We only discuss the case of the left projection. Using Lemma~\ref{LemmaqbFib}, we start with the fact that the left projection lifts to a b-fibration $\tilde\pi_L\colon\overline\mathbb{R}\times X_{\mathrm{q}}^2\to\overline\mathbb{R}\times X_{\mathrm{q}}$; the preimages of the centers in~\eqref{EqQSingleXq} under it are
\begin{equation}
\label{EqQPbFibPre}
\begin{split}
\tilde\pi_L^{-1}(\partial\overline\mathbb{R}\times{\mathrm{zf}}_{\mathrm{q}}) &= (\partial\overline\mathbb{R}\times{\mathrm{zf}}_{{\mathrm{q}},2}) \cup (\partial\overline\mathbb{R}\times{\mathrm{lb}}_{{\mathrm{q}},2}), \\
\tilde\pi_L^{-1}(\partial\overline\mathbb{R}\times{\mathrm{mf}}_{\mathrm{q}}) &= (\partial\overline\mathbb{R}\times{\mathrm{mf}}_{{\mathrm{q}},2}) \cup (\partial\overline\mathbb{R}\times{\mathrm{rb}}_{{\mathrm{q}},2}).
\end{split}
\end{equation}
From the first line and \cite[Proposition~5.12.1]{MelroseDiffOnMwc}, we deduce that the lift of $\tilde\pi_L$ to
\[
\bigl[\overline\mathbb{R}\times X_{\mathrm{q}}^2; \partial\overline\mathbb{R}\times{\mathrm{zf}}_{{\mathrm{q}},2}; \partial\overline\mathbb{R}\times{\mathrm{lb}}_{{\mathrm{q}},2} \bigr] \to \bigl[ \overline\mathbb{R}\times X_{\mathrm{q}}; \partial\overline\mathbb{R}\times{\mathrm{zf}}_{\mathrm{q}} \bigr]
\]
is a b-fibration. Since this is b-transversal to the lift of $\partial\overline\mathbb{R}\times\operatorname{diag}_{\mathrm{q}}$ (which is mapped diffeomorphically to a copy of $X_{\mathrm{q}}$), this lifts to a b-fibration
\[
\bigl[\overline\mathbb{R}\times X_{\mathrm{q}}^2; \partial\overline\mathbb{R}\times{\mathrm{zf}}_{{\mathrm{q}},2}; \partial\overline\mathbb{R}\times\operatorname{diag}_{\mathrm{q}}, \partial\overline\mathbb{R}\times{\mathrm{lb}}_{{\mathrm{q}},2} \bigr] \to \bigl[ \overline\mathbb{R}\times X_{\mathrm{q}}; \partial\overline\mathbb{R}\times{\mathrm{zf}}_{\mathrm{q}} \bigr].
\]
By~\eqref{EqQPbFibPre}, the preimage of the lift of $\partial\overline\mathbb{R}\times{\mathrm{mf}}_{\mathrm{q}}$ under this map is the union of the lifts of $\partial\overline\mathbb{R}\times{\mathrm{mf}}_{{\mathrm{q}},2}$, $\partial\overline\mathbb{R}\times(\operatorname{diag}_{\mathrm{q}}\cap\,{\mathrm{mf}}_{{\mathrm{q}},2})$, and $\partial\overline\mathbb{R}\times{\mathrm{rb}}_{{\mathrm{q}},2}$. By \cite[Proposition~5.11.2]{MelroseDiffOnMwc}, the lift
\[
\bigl[\overline\mathbb{R}\times X_{\mathrm{q}}^2; \partial\overline\mathbb{R}\times{\mathrm{zf}}_{{\mathrm{q}},2}; \partial\overline\mathbb{R}\times\operatorname{diag}_{\mathrm{q}}, \partial\overline\mathbb{R}\times{\mathrm{lb}}_{{\mathrm{q}},2}; \partial\overline\mathbb{R}\times{\mathrm{rb}}_{{\mathrm{q}},2}; \partial\overline\mathbb{R}\times(\operatorname{diag}_{\mathrm{q}}\cap{\mathrm{mf}}_{{\mathrm{q}},2}); \partial\overline\mathbb{R}\times{\mathrm{mf}}_{{\mathrm{q}},2} \bigr] \to X_{\mathrm{Q}}
\]
is therefore a b-fibration. This finishes the proof.
\end{proof}
\begin{definition}[Q-pseudodifferential operators]
\label{DefQP}
Let $s\in\mathbb{R}$ and $\alpha=(\alpha_H)$ where $\alpha_H\in\mathbb{R}$ for $H={\mathrm{mf}},{\mathrm{zf}},{\mathrm{nf}},{\mathrm{if}},{\mathrm{sf}}$. Then $\Psi_\Qop^{s,\alpha}(X)$ consists of all smooth families $A=(A_{\fm,\sigma})_{\fm\in(0,1],\sigma\in\mathbb{R}}$ of bounded linear operators on $\cC^\infty_\cp(X)$ whose Schwartz kernels are elements of
\begin{equation}
\label{EqQP}
\rho_{{\mathrm{zf}}_2}^{-\alpha_{\mathrm{zf}}}\rho_{{\mathrm{mf}}_2}^{-\alpha_{\mathrm{mf}}}\rho_{{\mathrm{nf}}_2}^{-\alpha_{\mathrm{nf}}}\rho_{{\mathrm{if}}_2}^{-\alpha_{\mathrm{if}}}\rho_{{\mathrm{sf}}_2}^{-\alpha_{\mathrm{sf}}}I^{m-\frac12}(X^2_{\mathrm{Q}},\operatorname{diag}_{\mathrm{Q}};\pi_R^*{}^{\Qop}\Omega X)
\end{equation}
which are conormal down to all boundary hypersurfaces of $X^2_{\mathrm{Q}}$ and vanish to infinite order at all boundary hypersurfaces other than ${\mathrm{mf}}_2$, ${\mathrm{zf}}_2$, ${\mathrm{nf}}_2$, ${\mathrm{if}}_2$, ${\mathrm{sf}}_2$ (and the lift of $\fm^{-1}(1)$). The subspace of operators whose Schwartz kernels are classical conormal at ${\mathrm{zf}}_2,{\mathrm{mf}}_2,{\mathrm{nf}}_2$ is denoted $\Psi_{{\mathrm{Q}},{\mathrm{cl}}}^{s,\alpha}(X)$.
\end{definition}
\begin{rmk}[Defining functions]
\label{RmkQPDefFn}
Note that $\pi_L^{-1}({\mathrm{zf}})={\mathrm{zf}}_2\cup{\mathrm{lb}}_2$, and indeed the defining function ${\mathrm{zf}}$ lifts to $X_{\mathrm{Q}}^2$ under $\pi_L$ to a product of defining functions of ${\mathrm{zf}}_2$ and ${\mathrm{lb}}_2$. In view of the infinite order of vanishing of Schwartz kernels of Q-ps.d.o.s at ${\mathrm{lb}}_2$, we can therefore replace the weight $\rho_{{\mathrm{zf}}_2}$ in~\eqref{EqQP} by (the left lift of) $\rho_{\mathrm{zf}}$. Similarly,
\begin{alignat*}{2}
\pi_L^{-1}({\mathrm{mf}})&={\mathrm{mf}}_2\cup{\mathrm{rb}}_2,&\qquad
\pi_L^{-1}({\mathrm{nf}})&={\mathrm{nf}}_2\cup{\mathrm{tlb}}_2, \\
\pi_L^{-1}({\mathrm{if}})&={\mathrm{if}}_2\cup{\mathrm{if}}'_2\cup{\mathrm{trb}}_2,&\qquad
\pi_L^{-1}({\mathrm{sf}})&={\mathrm{sf}}_2\cup{\mathrm{sf}}'_2.
\end{alignat*}
Similar statements hold for $\pi_R$ in place of $\pi_L$. Together, they imply that $\Psi_\Qop^{s,\alpha}(X)$ is invariant under conjugation by weights $\prod\rho_H^{-\alpha_H}$ on $X_{\mathrm{Q}}$.
\end{rmk}
For local coordinate descriptions, we shall use the smooth functions on $X_{\mathrm{Q}}^2$ obtained by lifting coordinates on $X_{\mathrm{Q}}$ to the left, resp.\ right factor; the left lift will be denoted by the same symbol, and the right lift with the primed symbol. For example, $\hat x$ and $\hat x'$ denote the left and right lift of the function on $X_{\mathrm{Q}}$ denoted $\hat x$ in~\eqref{EqqCoord2}.
For bounded $\sigma$, Q-ps.d.o.s are smooth families (in $\sigma$) of q-ps.d.o.s, for which a local coordinate description was given in~\eqref{EqqQuant}. Consider next the region $|\hat x|,|\hat x'|\lesssim 1$ for $\sigma\gtrsim 1$. Near $\{\infty\}\times(\operatorname{diag}_{\mathrm{q}}\cap\,{\mathrm{zf}}_{{\mathrm{q}},2}^\circ)\subset\overline\mathbb{R}\times X_{\mathrm{q}}^2$, we can then use local coordinates $h\geq 0$, $\fm\geq 0$, $\hat x'$, and $y:=\hat x-\hat x'$, with the diagonal defined by $y=0$. Upon blowing up $h=\fm=0$, the lift of $h=0$ is defined by $\frac{h}{h+\fm}=0$; upon passing to the subsequent blow-up of the lift of $\partial\overline\mathbb{R}\times\operatorname{diag}_{\mathrm{q}}$, coordinates near the Q-diagonal are thus
\[
y_{\mathrm{Q}}:=\frac{y}{h/(h+\fm)},
\]
and therefore a typical element of $\Psi_\Qop^{s,\alpha}(X)$ is given by
\begin{equation}
\label{EqQPQuant1}
(\Op_{{\mathrm{Q}},\fm,h^{-1}}(a)u)(\hat x) = (2\pi)^{-n}\int \exp\Bigl(i\frac{\hat x-\hat x'}{h/(h+\fm)}\cdot\xi\Bigr) \chi\bigl(|\hat x-\hat x'|\bigr) a(h,\fm,\hat x,\xi)\,{\mathrm d}\xi
\end{equation}
where $a$ is a symbol, or more precisely $a$ is conormal on $X_{\mathrm{Q}}\times\overline{\mathbb{R}^n}$ with order $\alpha_H$ at $H\times\overline{\mathbb{R}^n}$ for $H={\mathrm{zf}},{\mathrm{nf}},{\mathrm{sf}}$, and order $s$ at $X_{\mathrm{q}}\times\partial\overline{\mathbb{R}^n}$; and $\chi\in\cC^\infty_\cp((-{\tfrac{1}{2}},{\tfrac{1}{2}}))$ is identically $1$ near $0$. Thus,~\eqref{EqQPQuant1} is essentially a semiclassical ps.d.o.\ with semiclassical parameter $\frac{h}{h+\fm}$. We also note that the left lift of the basis $\frac{h}{h+\fm}\partial_{\hat x^j}$ of $\cV_\Qop(X)$ in this coordinate system (see~\eqref{EqQSpan}) is given by $\partial_{y_{\mathrm{Q}}^j}$, which is transversal to $\operatorname{diag}_{\mathrm{Q}}=y_{\mathrm{Q}}^{-1}(0)$.
Working in the region $|\hat x|,|\hat x'|\gtrsim 1$ for $\sigma\gtrsim 1$, we can use as smooth coordinates near $\{\infty\}\times\operatorname{diag}_{\mathrm{q}}\subset\overline\mathbb{R}\times X^2_{\mathrm{q}}$ the functions $h\geq 0$, $\frac{\fm}{r'}\geq 0$, $r'\geq 0$, $\omega'\in\mathbb{R}^{n-1}$, $z=\frac{r-r'}{r'}$, $w=\omega-\omega'\in\mathbb{R}^{n-1}$ where we fix local coordinates on $\mathbb{S}^{n-1}$. Upon blowing up $\partial\overline\mathbb{R}\times{\mathrm{zf}}_{{\mathrm{q}},2}$ (given by $h=r'=0$), the lift of $h=0$ is given by $\frac{h}{h+r'}=0$; passing to the subsequent blow-up of the lift of $\partial\overline\mathbb{R}\times\operatorname{diag}_{\mathrm{q}}$, coordinates transversal to the lifted diagonal are thus
\[
(z_{\mathrm{Q}},w_{\mathrm{Q}}) := \frac{(z,w)}{h/(h+r')}.
\]
These coordinates remain transversal to the lift of the diagonal to the subsequent blow-ups in~\eqref{EqQPDouble}. Thus, an element of $\Psi_\Qop^{s,\alpha}(X)$ is given by
\begin{equation}
\label{EqQPQuant2}
\begin{split}
(\Op_{{\mathrm{Q}},\fm,h^{-1}}(a)u)(r,\omega) &= (2\pi)^{-n}\iint \exp\Bigl[i\Bigl(\frac{r-r'}{r'\frac{h}{h+r'}}\xi + \frac{\omega-\omega'}{h/(h+r')}\cdot\eta\Bigr)\Bigr] \\
&\quad\hspace{6em} \times \chi\Bigl(\Bigl|\frac{r-r'}{r'}\Bigr|\Bigr)\chi(|\omega-\omega'|) a(h,\fm,r,\xi,\eta)\,{\mathrm d}\xi\,{\mathrm d}\eta,
\end{split}
\end{equation}
where $a$ is conormal on $X_{\mathrm{Q}}\times\overline{\mathbb{R}^n_{(\xi,\eta)}}$ with order $\alpha_H$ at $H\times\overline\mathbb{R}^n$ for all boundary hypersurfaces $H\subset X_{\mathrm{Q}}$, and order $s$ at $X_{\mathrm{Q}}\times\partial\overline{\mathbb{R}^n_{(\xi,\eta)}}$. Since the second spanning set of Q-vector fields in~\eqref{EqQSpan} lifts to the left factor of $X_{\mathrm{Q}}^2$ as $\partial_{z_{\mathrm{Q}}}$, $\partial_{w_{\mathrm{Q}}}$, we conclude that also in this region the left lift of $\cV_\Qop(X)$ is transversal to $\operatorname{diag}_{\mathrm{Q}}$.
As a consequence of the two transversality statements, we obtain a bundle isomorphism ${}^\Qop T X\cong T_{\operatorname{diag}_{\mathrm{Q}}}X_{\mathrm{Q}}^2/T\operatorname{diag}_{\mathrm{Q}}=N\operatorname{diag}_{\mathrm{Q}}$ given by the left lift; and therefore
\begin{equation}
\label{EqQPCon}
N^*\operatorname{diag}_{\mathrm{Q}} \cong {}^\Qop T^*X.
\end{equation}
Moreover, for $m\in\mathbb{N}_0$, we conclude that $\Diff_\Qop^{m,\alpha}(X)\subset\Psi_\Qop^{m,\alpha}(X)$ consists of those operators whose Schwartz kernels are Dirac distributions at $\operatorname{diag}_{\mathrm{Q}}$. Generalizing~\eqref{EqQDiffSym}, the principal symbol map ${}^\Qop\upsigma^{s,\alpha}$ on $\Psi_\Qop^{s,\alpha}(X)$ fits into the short exact sequence
\[
0 \to \rho_{\mathrm{if}}\rho_{\mathrm{sf}}\Psi_\Qop^{s,\alpha}(X) \hookrightarrow \Psi_\Qop^{s,\alpha}(X) \xrightarrow{{}^\Qop\upsigma^{s,\alpha}} (S^{s,\alpha}/\rho_{\mathrm{if}}\rho_{\mathrm{sf}} S^{s-1,\alpha})({}^\Qop T^*X) \to 0.
\]
Finally, we conclude that pushforward along $\pi_L$ is a continuous map from $\Psi_{\mathrm{Q}}^s(X)$, resp. $\Psi_{{\mathrm{Q}},{\mathrm{cl}}}^s(X)$ into $\mc A^0(X_{\mathrm{Q}})$, resp.\ $\cC^\infty(X_{\mathrm{Q}})$; thus, Q-ps.d.o.s define bounded linear maps on $\mc A^0(X_{\mathrm{Q}})$, or on $\cC^\infty(X_{\mathrm{Q}})$ for classical ps.d.o.s.
We may allow for the orders $s$, $\alpha_{\mathrm{if}}$, $\alpha_{\mathrm{sf}}$ to be variable; in this paper we only need to consider the case that the ${\mathrm{if}}$-order is variable,
\[
\upalpha_{\mathrm{if}} \in \cC^\infty(\overline{{}^\Qop T^*_{\mathrm{if}}}X),
\]
while $s,\alpha_{\mathrm{sf}}$ are constant; for $\alpha=(\alpha_{\mathrm{zf}},\alpha_{\mathrm{mf}},\alpha_{\mathrm{nf}},\upalpha_{\mathrm{if}},\alpha_{\mathrm{sf}})$, the principal symbol map then takes values in $(S^{s,\alpha}/\rho_{\mathrm{if}}^{1-2\delta}\rho_{\mathrm{sf}} S^{s-1,\alpha})({}^\Qop T^*X)$ for any $\delta>0$.
In order to study the normal operators of Q-ps.d.o.s, we need the following result, which is the double space analogue of Lemma~\ref{LemmaQVFRes}:
\begin{prop}[Boundary hypersurfaces of $X_{\mathrm{Q}}^2$]
\label{PropQPBdy}
We have the following natural diffeomorphisms:
\begin{enumerate}
\item\label{ItQPBdyzf2} ${\mathrm{zf}}_2\cong\overline\mathbb{R}\times\hat X^2_{\mathrm{b}}$;
\item\label{ItQPBdymf2lo} ${\mathrm{mf}}_{2,\sigma_0}\cong\dot X^2_{\mathrm{b}}$ (for $\sigma_0\in\mathbb{R}$);
\item\label{ItQPBdymf2hi} ${\mathrm{mf}}_{2,\pm,\hbar}\cong\dot X^2_{\mathrm{c}\hbar}$ (see~\S\usref{SsPch}) with semiclassical parameter $h=|\sigma|^{-1}$;
\item\label{ItQPBdynf2lo} ${\mathrm{nf}}_{2,\pm,{\mathrm{low}}}\cong\hat X^2_{\mathrm{sc}\text{-}\mathrm{b}}$ (see~\S\usref{SsPscbt}) with spectral parameter $\tilde\sigma=\fm\sigma$.
\item\label{ItQPBdynf2hi} ${\mathrm{nf}}_{2,\pm,\tilde\hbar}\cong\hat X^2_{\mathrm{sc},\hbar}$ (see~\S\usref{SsPbsc}) with semiclassical parameter $\tilde h=|\tilde\sigma|^{-1}$.
\end{enumerate}
That is, the local coordinates $\sigma,\hat x,\hat x'$ restrict to a map ${\mathrm{zf}}_2^\circ\to\mathbb{R}\times\mathbb{R}^n_{\hat x}\times\mathbb{R}^n_{\hat x'}$ which extends by continuity to the diffeomorphism in part~\eqref{ItQPBdyzf2}; similarly for the other diffeomorphisms.
\end{prop}
\begin{proof}
We obtain ${\mathrm{zf}}_2$ by first blowing up $\partial\overline\mathbb{R}\times{\mathrm{zf}}_{{\mathrm{q}},2}\subset\overline\mathbb{R}\times{\mathrm{zf}}_{{\mathrm{q}},2}$, which thus does not change the smooth structure of $\overline\mathbb{R}\times{\mathrm{zf}}_{{\mathrm{q}},2}$; the lifts of the remaining submanifolds in~\eqref{EqQPDouble} to $[\overline\mathbb{R}\times X^2_{\mathrm{q}};\partial\overline\mathbb{R}\times{\mathrm{zf}}_{{\mathrm{q}},2}]$ are disjoint from the lift of $\overline\mathbb{R}\times{\mathrm{zf}}_{{\mathrm{q}},2}$. This proves part~\eqref{ItQPBdyzf2}.
Next, ${\mathrm{mf}}_2$ arises from $\overline\mathbb{R}\times{\mathrm{mf}}_{\mathrm{q}}=\overline\mathbb{R}\times\dot X^2_{\mathrm{b}}$ (see Lemma~\ref{LemmaqBdy}) by first blowing up its intersection $\partial\overline\mathbb{R}\times\mathrm{ff}_{\mathrm{b}}$ with $\partial\overline\mathbb{R}\times{\mathrm{zf}}_{{\mathrm{q}},2}$, where $\mathrm{ff}_{\mathrm{b}}$ denotes the front face of $\dot X^2_{\mathrm{b}}$; then one blows up the intersection with the lift of $\partial\overline\mathbb{R}\times\operatorname{diag}_{\mathrm{q}}$, which is equal to the intersection with the lift of $\partial\overline\mathbb{R}\times(\operatorname{diag}_{\mathrm{q}}\cap\,{\mathrm{mf}}_{{\mathrm{q}},2})$ and thus given by the lift of $\partial\overline\mathbb{R}\times\operatorname{diag}_{\mathrm{b}}$ to $[\overline\mathbb{R}\times\dot X^2_{\mathrm{b}};\partial\overline\mathbb{R}\times\mathrm{ff}_{\mathrm{b}}]$. This blow-up thus produces
\[
\bigl[ \overline\mathbb{R}\times\dot X^2_{\mathrm{b}}; \partial\overline\mathbb{R}\times\mathrm{ff}_{\mathrm{b}}; \partial\overline\mathbb{R}\times\operatorname{diag}_{\mathrm{b}} \bigr].
\]
The intersection of this space with the lift of $\partial\overline\mathbb{R}\times{\mathrm{lb}}_{{\mathrm{q}},2}$ is $\partial\overline\mathbb{R}\times{\mathrm{lb}}_{\mathrm{b}}$, similarly for the right boundary, and hence blowing up both of these lifts produces
\begin{equation}
\label{EqQPBdymf2}
\bigl[ \overline\mathbb{R}\times\dot X^2_{\mathrm{b}}; \partial\overline\mathbb{R}\times\mathrm{ff}_{\mathrm{b}}, \partial\overline\mathbb{R}\times{\mathrm{lb}}_{\mathrm{b}}, \partial\overline\mathbb{R}\times{\mathrm{rb}}_{\mathrm{b}}, \partial\overline\mathbb{R}\times\operatorname{diag}_{\mathrm{b}} \bigr].
\end{equation}
The intersections of this space with the lift of $\partial\overline\mathbb{R}\times(\operatorname{diag}_{\mathrm{q}}\cap\,{\mathrm{mf}}_{{\mathrm{q}},2})$ or with the lift of $\partial\overline\mathbb{R}\times{\mathrm{mf}}_{{\mathrm{q}},2}$ are both boundary hypersurfaces, hence their blow-up does not affect the smooth structure. Upon intersecting the space~\eqref{EqQPBdymf2} with $\sigma^{-1}(\sigma_0)$ or $\sigma^{-1}(\pm[1,\infty])$, we thus obtain the isomorphisms stated in parts~\eqref{ItQPBdymf2lo} and \eqref{ItQPBdymf2hi}.
Finally, we consider ${\mathrm{nf}}_{2,+}$. Let $[0,\epsilon)_{\rho_{{\mathrm{zf}}_{{\mathrm{q}},2}}}\times{\mathrm{zf}}_{{\mathrm{q}},2}$ be a collar neighborhood of ${\mathrm{zf}}_{{\mathrm{q}},2}\subset X^2_{\mathrm{q}}$. We take $\rho_{{\mathrm{zf}}_{{\mathrm{q}},2}}=\sqrt{\fm^2+|x|^2+|x'|^2}$ for concreteness. Then the front face of $[\overline\mathbb{R}\times X_{\mathrm{q}}^2;\{\infty\}\times{\mathrm{zf}}_{{\mathrm{q}},2}]$ is that of $[[0,1]_h\times[0,\epsilon)_{\rho_{{\mathrm{zf}}_{{\mathrm{q}},2}}}\times{\mathrm{zf}}_{{\mathrm{q}},2};\{0\}\times\{0\}\times{\mathrm{zf}}_{{\mathrm{q}},2}]$, and thus equal to
\[
{\mathrm{nf}}_{2,+}' := [0,\infty]_{\tilde h'} \times {\mathrm{zf}}_{{\mathrm{q}},2} = [0,\infty]_{\tilde h'} \times \hat X^2_{\mathrm{b}},\qquad \tilde h':=\frac{h}{\rho_{{\mathrm{zf}}_{{\mathrm{q}},2}}}.
\]
Its intersections with the lifts of
\[
\{\infty\}\times\operatorname{diag}_{\mathrm{q}},\quad
\{\infty\}\times(\operatorname{diag}_{\mathrm{q}}\cap\,{\mathrm{mf}}_{{\mathrm{q}},2}), \quad
\{\infty\}\times{\mathrm{lb}}_{{\mathrm{q}},2},\quad
\{\infty\}\times{\mathrm{rb}}_{{\mathrm{q}},2},\quad
\{\infty\}\times{\mathrm{mf}}_{{\mathrm{q}},2}
\]
with ${\mathrm{nf}}_{2,+}'$ are given by
\[
\{0\}\times\operatorname{diag}_{\mathrm{b}},\quad
\{0\}\times\partial\operatorname{diag}_{\mathrm{b}}, \quad
\{0\}\times{\mathrm{lb}}_{\mathrm{b}},\quad
\{0\}\times{\mathrm{rb}}_{\mathrm{b}},\quad
\{0\}\times\mathrm{ff}_{\mathrm{b}},
\]
respectively; we need to blow these up in the listed order. In fact, the first two blow-ups can be performed after the third and fourth (since the first/second and third/fourth submanifolds are disjoint); then, since $\partial\operatorname{diag}_{\mathrm{b}}=\operatorname{diag}_{\mathrm{b}}\cap\,\mathrm{ff}_{\mathrm{b}}$, one can blow up $\{0\}\times\operatorname{diag}_{\mathrm{b}}$, $\{0\}\times\partial\operatorname{diag}_{\mathrm{b}}$, and $\{0\}\times\mathrm{ff}_{\mathrm{b}}$ in the order $\{0\}\times\mathrm{ff}_{\mathrm{b}}$, $\{0\}\times\partial\operatorname{diag}_{\mathrm{b}}$, $\{0\}\times\operatorname{diag}_{\mathrm{b}}$. Thus,
\begin{equation}
\label{EqQPBdynf2funny}
{\mathrm{nf}}_{2,+} = \bigl[ [0,\infty]_{\tilde h'}\times\hat X_{\mathrm{b}}^2; \{0\}\times{\mathrm{lb}}_{\mathrm{b}}, \{0\}\times{\mathrm{rb}}_{\mathrm{b}}; \{0\}\times\mathrm{ff}_{\mathrm{b}}; \{0\}\times\partial\operatorname{diag}_{\mathrm{b}}; \{0\}\times\operatorname{diag}_{\mathrm{b}} \bigr].
\end{equation}
To analyze this space, we introduce $\hat\rho_\mathrm{tot}:=(1+|\hat x|^2+|\hat x'|^2)^{-\frac12}=\hat\rho_{{\mathrm{lb}}_{\mathrm{b}}}\hat\rho_{\mathrm{ff}_{\mathrm{b}}}\hat\rho_{{\mathrm{rb}}_{\mathrm{b}}}$, which is a total boundary defining function of $\hat X^2_{\mathrm{b}}$. We then claim that the change of coordinates map $(\tilde h',\hat x,\hat x')\mapsto(\tilde\sigma,\hat x,\hat x')$ with $\tilde\sigma=(1+|\hat x|^2+|\hat x'|^2)^{-\frac12}/\tilde h'=\frac{\hat\rho_\mathrm{tot}}{\tilde h'}$ induces a diffeomorphism\footnote{This is the analogue, in the double space setting, of the isomorphism~\eqref{EqQnfBlowup}.}
\begin{equation}
\label{EqQPBdynf2partial}
\begin{split}
&\bigl[ [0,\infty]_{\tilde h'}\times\hat X_{\mathrm{b}}^2; \{0\}\times{\mathrm{lb}}_{\mathrm{b}}, \{0\}\times{\mathrm{rb}}_{\mathrm{b}}; \{0\}\times\mathrm{ff}_{\mathrm{b}} \bigr] \\
&\qquad \cong \bigl[ [0,\infty]_{\tilde\sigma} \times \hat X_{\mathrm{b}}^2; \{0\}\times\mathrm{ff}_{\mathrm{b}}; \{0\}\times{\mathrm{lb}}_{\mathrm{b}}, \{0\}\times{\mathrm{rb}}_{\mathrm{b}} \bigr].
\end{split}
\end{equation}
(See Figure~\ref{FigQPBdynf2}.) This is clear over the interior $(\hat X^\circ)^2$ of $\hat X^2_{\mathrm{b}}$. We have $\mathrm{ff}_{\mathrm{b}}\cong[0,\infty]_s\times(\partial\hat X)^2$ where $s=\frac{\hat\rho}{\hat\rho'}$ with $\hat\rho=|\hat x|^{-1}=\hat\rho_{{\mathrm{lb}}_{\mathrm{b}}}\hat\rho_{\mathrm{ff}_{\mathrm{b}}}$ and $\hat\rho'=|\hat x'|^{-1}=\hat\rho_{{\mathrm{rb}}_{\mathrm{b}}}\hat\rho_{\mathrm{ff}_{\mathrm{b}}}$ for suitable defining functions $\hat\rho_{{\mathrm{lb}}_{\mathrm{b}}}$, $\hat\rho_{\mathrm{ff}_{\mathrm{b}}}$, $\hat\rho_{{\mathrm{rb}}_{\mathrm{b}}}$ of ${\mathrm{lb}}_{\mathrm{b}}$, $\mathrm{ff}_{\mathrm{b}}$, ${\mathrm{rb}}_{\mathrm{b}}\subset\hat X^2_{\mathrm{b}}$, so
\[
s=\frac{\hat\rho_{{\mathrm{lb}}_{\mathrm{b}}}}{\hat\rho_{{\mathrm{rb}}_{\mathrm{b}}}}.
\]
Thus, a collar neighborhood of $\mathrm{ff}_{\mathrm{b}}\subset\hat X^2_{\mathrm{b}}$ is given by $[0,\epsilon)_{\hat\rho_{\mathrm{ff}_{\mathrm{b}}}}\times[0,\infty]_s\times(\partial X)^2$. Upon dropping the factor $(\partial X)^2$, the claim~\eqref{EqQPBdynf2partial} thus reads
\begin{equation}
\label{EqQPBdynf2partial2}
\begin{split}
&\bigl[ [0,\infty]_{\tilde h'} \times [0,\epsilon)_{\hat\rho_{\mathrm{ff}_{\mathrm{b}}}} \times [0,\infty]_s; \{0\}\times[0,\epsilon)\times\{0\}, \{0\}\times[0,\epsilon)\times\{\infty\}; \{0\} \times \{0\} \times [0,\infty] \bigr] \\
&\quad \cong \bigl[ [0,\infty]_{\tilde\sigma} \times [0,\epsilon) \times [0,\infty]; \{0\}\times\{0\}\times[0,\infty]; \{0\}\times[0,\epsilon)\times\{0\}, \{0\}\times[0,\epsilon)\times\{\infty\} \bigr]
\end{split}
\end{equation}
via the change of coordinates map $\kappa\colon(\tilde h',\hat\rho_{\mathrm{ff}_{\mathrm{b}}},s)\mapsto(\frac{\hat\rho_{{\mathrm{lb}}_{\mathrm{b}}}\hat\rho_{\mathrm{ff}_{\mathrm{b}}}\hat\rho_{{\mathrm{rb}}_{\mathrm{b}}}}{\tilde h'},\hat\rho_{\mathrm{ff}_{\mathrm{b}}},s)$, where we put $\hat\rho_{{\mathrm{lb}}_{\mathrm{b}}}=\frac{s}{s+1}$ and $\hat\rho_{{\mathrm{rb}}_{\mathrm{b}}}=\frac{1}{s+1}$. The proof of~\eqref{EqQPBdynf2partial2} proceeds by explicit calculations in local coordinate systems, and is pictorially given in Figure~\ref{FigQPBdynf2}.
\begin{figure}[!ht]
\centering
\includegraphics{FigQPBdynf2}
\caption{Illustration of (the proof of) the diffeomorphism~\eqref{EqQPBdynf2partial2}. \textit{On the left:} the space on the left in~\eqref{EqQPBdynf2partial2}. \textit{On the right:} the space on the right in~\eqref{EqQPBdynf2partial2}. Also shown are matching local coordinate systems near the various boundary faces; in the listed coordinates systems, we have $\hat\rho_\mathrm{tot}\sim\hat\rho_{{\mathrm{lb}}_{\mathrm{b}}}\hat\rho_{\mathrm{ff}_{\mathrm{b}}}$, and we also recall that $\tilde\sigma=\hat\rho_\mathrm{tot}/\tilde h'$.}
\label{FigQPBdynf2}
\end{figure}
Using the diffeomorphism~\eqref{EqQPBdynf2partial} in~\eqref{EqQPBdynf2funny}, we then find that
\[
{\mathrm{nf}}_{2,+} = \bigl[ [0,\infty]_{\tilde\sigma}\times\hat X_{\mathrm{b}}^2; \{0\}\times\mathrm{ff}_{\mathrm{b}}; \{0\}\times{\mathrm{lb}}_{\mathrm{b}}, \{0\}\times{\mathrm{rb}}_{\mathrm{b}}; [0,\infty]\times\partial\operatorname{diag}_{\mathrm{b}}; \{\infty\}\times\operatorname{diag}_{\mathrm{b}} \bigr].
\]
This implies parts~\eqref{ItQPBdynf2lo} and \eqref{ItQPBdynf2hi}. The proof is complete.
\end{proof}
The relationship between the semiclassical, resp.\ doubly semiclassical cone algebras of \cite{HintzConicPowers} and the Q-algebra in the intermediate semiclassical regime $|\sigma|\sim\fm^{-1}$ (mentioned in the discussion of the \emph{very large frequency regime} in~\S\ref{SsIA}), resp.\ fully semiclassical regime $|\sigma|\gg\fm^{-1}$ is described in Appendix~\ref{SQSemi}.
We now switch to a less cumbersome notation for the weights, writing $l=\alpha_{\mathrm{zf}}$, $\gamma=\alpha_{\mathrm{mf}}$, $l'=\alpha_{\mathrm{nf}}$, $r=\alpha_{\mathrm{if}}$, $b=\alpha_{\mathrm{sf}}$.
\begin{cor}[Normal operators]
\label{CorQPNormal}
Restricting Schwartz kernels of classical Q-ps.d.o.s to the boundary hypersurfaces ${\mathrm{zf}}_2$, ${\mathrm{mf}}_{2,\pm,\hbar}$, ${\mathrm{nf}}_{2,\pm,{\mathrm{low}}}$, and ${\mathrm{nf}}_{2,\pm,\tilde\hbar}$ defines surjective normal operator maps
\begin{alignat*}{2}
N_{\mathrm{zf}} &\colon \Psi_{{\mathrm{Q}},{\mathrm{cl}}}^{s,(0,\gamma,l',r,b)}(X) &&\to \cC^\infty\bigl(\overline\mathbb{R};\Psi_\bop^{s,\gamma}(\hat X)\bigr), \\
N_{{\mathrm{mf}}_{\pm,\hbar}} &\colon \Psi_{{\mathrm{Q}},{\mathrm{cl}}}^{s,(l,0,l',r,b)}(X) &&\to \Psi_\chop^{s,l,l',r}(\dot X), \\
N_{{\mathrm{nf}}_{\pm,{\mathrm{low}}}} &\colon \Psi_{{\mathrm{Q}},{\mathrm{cl}}}^{s,(l,\gamma,0,r,b)}(X) &&\to \Psi_\scbtop^{s,r,\gamma,l}(\hat X), \\
N_{{\mathrm{nf}}_{\pm,\tilde\hbar}} &\colon \Psi_{{\mathrm{Q}},{\mathrm{cl}}}^{s,(l,\gamma,0,r,b)}(X) &&\to \Psi_{{\mathrm{sc}}\tilde\hbar}^{s,r,b}(\hat X).
\end{alignat*}
Moreover, for $\sigma_0\in\mathbb{R}$ and $\tilde\sigma_0\in\mathbb{R}\setminus\{0\}$, restriction to $\sigma^{-1}(\sigma_0)$, $\sigma^{-1}(\sigma_0)\cap{\mathrm{mf}}_2$, and ${\mathrm{nf}}_{2,\tilde\sigma_0}$ defines surjective maps
\begin{alignat*}{2}
N_{\sigma_0} &\colon \Psi_\Qop^{s,(l,\gamma,l',r,b)}(X) &&\to \Psi_\qop^{s,(l,\gamma)}(X), \\
N_{{\mathrm{mf}}_{\sigma_0}} &\colon \Psi_\Qop^{s,(l,0,l',r,b)}(X) &&\to \Psi_\bop^{s,l}(\dot X), \\
N_{{\mathrm{nf}}_{\tilde\sigma_0}} &\colon \Psi_\Qop^{s,(l,\gamma,0,r,b)}(X) &&\to \Psi_\scop^{s,r}(\hat X),
\end{alignat*}
respectively. All statements hold also for variable ${\mathrm{if}}$-orders $\mathsf{r}\in\cC^\infty(\overline{{}^\Qop T^*_{\mathrm{if}}}X)$.
\end{cor}
Since $\Psi_{{\mathrm{Q}},{\mathrm{cl}}}^s(X)$ acts boundedly on $\cC^\infty(X_Q)$ and is invariant under conjugation by weights, these normal operators can be defined via testing. That is, for $A\in\Psi_{{\mathrm{Q}},{\mathrm{cl}}}^{s,(0,\gamma,l',r,b)}(X)$, the operator $N_{\mathrm{zf}}(A)$ can be defined via $N_{\mathrm{zf}}(A)u:=(A\tilde u)|_{\mathrm{zf}}$ where $\tilde u\in\cC^\infty(X_Q)$ is any smooth extension of $u\in\dot\cC^\infty({\mathrm{zf}})$; likewise for the other normal operators. In particular, the above normal operator maps are homomorphisms under composition, where we compose Q-ps.d.o.s as operators between spaces of weighted smooth functions (i.e.\ classical conormal distributions) on $X_Q$.
We finally show that the spaces $\Psi_\Qop(X)$ and $\Psi_{{\mathrm{Q}},{\mathrm{cl}}}(X)$ are closed under composition. This can be done in a straightforward but tedious manner using the local coordinate descriptions~\eqref{EqQPQuant1}--\eqref{EqQPQuant2} (while residual operators, i.e.\ those with orders $s$, $\alpha_{\mathrm{if}}$, $\alpha_{\mathrm{sf}}=-\infty$ are handled directly on the level of Schwartz kernels). Keeping in line with the presentation thus far, we instead sketch the proof based on an appropriate triple space.
We use the notation for the q-triple space $X_{\mathrm{q}}^3$ from Definition~\ref{DefqTriple}, and furthermore write
\[
\partial\overline\mathbb{R}\times {\mathrm{mf}}_{{\mathrm{q}},S/C} = \{ \partial\overline\mathbb{R}\times{\mathrm{mf}}_{{\mathrm{q}},S}, \partial\overline\mathbb{R}\times{\mathrm{mf}}_{{\mathrm{q}},C} \},
\]
similarly $\partial\overline\mathbb{R}\times{\mathrm{bf}}_{{\mathrm{q}},F/S/C}$, etc.
\begin{definition}[Q-triple space]
\label{DefQPCTriple}
The \emph{Q-triple space} of $X$ is the resolution
\begin{align*}
X^3_{\mathrm{Q}} := &\bigl[ \overline\mathbb{R}\times X^3_{\mathrm{q}}; \partial\overline\mathbb{R}\times{\mathrm{zf}}_{{\mathrm{q}},3}; \partial\overline\mathbb{R}\times{\mathrm{bf}}_{{\mathrm{q}},F/S/C}; \partial\overline\mathbb{R}\times\operatorname{diag}_{{\mathrm{q}},3}; \partial\overline\mathbb{R}\times\operatorname{diag}_{{\mathrm{q}},F/S/C}; \\
&\quad \partial\overline\mathbb{R}\times(\operatorname{diag}_{{\mathrm{q}},F/S/C}\cap\,{\mathrm{mf}}_{{\mathrm{q}},F/S/C}); \partial\overline\mathbb{R}\times{\mathrm{mf}}_{{\mathrm{q}},F/S/C}; \\
&\quad \partial\overline\mathbb{R}\times(\operatorname{diag}_{{\mathrm{q}},F/S/C}\cap\,{\mathrm{mf}}_{{\mathrm{q}},3}); \partial\overline\mathbb{R}\times{\mathrm{mf}}_{{\mathrm{q}},3} \bigr].
\end{align*}
\end{definition}
\begin{lemma}[b-fibrations from the Q-triple space]
\label{LemmaQPCbFib}
The projection map $\overline{\mathbb{R}_\sigma}\times[0,1]_\fm\times X^3\to\overline\mathbb{R}\times[0,1]\times X^2$ to the first and second factor of $X$, i.e.\ $(\sigma,\fm,x,x',x'')\mapsto(\sigma,\fm,x,x')$, lifts to a b-fibration $\pi_F\colon X_{\mathrm{Q}}^3\to X_{\mathrm{Q}}^2$, similarly for the lifts $\pi_S$, $\pi_C\colon X^3_{\mathrm{Q}}\to X^2_{\mathrm{Q}}$ of the projections to the second and third, resp.\ first and third factor of $X^3$.
\end{lemma}
\begin{proof}
Denote the lifted projection from Lemma~\ref{LemmaqbFib} by $\pi_{{\mathrm{q}},F}$. We make use of the description~\eqref{EqqPreimages} of the preimages of boundary hypersurfaces of $X^2_{\mathrm{q}}$ under $\pi_{{\mathrm{q}},F}$. We start with the b-fibration $\operatorname{Id}\times\pi_{{\mathrm{q}},F}\colon\overline\mathbb{R}\times X^3_{\mathrm{q}}\to\overline\mathbb{R}\times X^2_{\mathrm{q}}$. By \cite[Proposition~5.12.1]{MelroseDiffOnMwc}, this map lifts to a b-fibration
\[
\bigl[ \overline\mathbb{R}\times X^3_{\mathrm{q}}; \partial\overline\mathbb{R}\times{\mathrm{zf}}_{{\mathrm{q}},3}; \partial\overline\mathbb{R}\times{\mathrm{bf}}_{{\mathrm{q}},F} \bigr] \to \bigl[ \overline\mathbb{R}\times X^2_{\mathrm{q}}; \partial\overline\mathbb{R}\times{\mathrm{zf}}_{{\mathrm{q}},2} \bigr].
\]
We next blow up $\partial\overline\mathbb{R}\times{\mathrm{lb}}_{{\mathrm{q}},2}$ and $\partial\overline\mathbb{R}\times{\mathrm{rb}}_{{\mathrm{q}},2}$ in the image; blowing up the preimages in the domain---see~\eqref{EqqPreimages}--- thus gives a b-fibration
\begin{align*}
&\bigl[ \overline\mathbb{R}\times X^3_{\mathrm{q}}; \partial\overline\mathbb{R}\times{\mathrm{zf}}_{{\mathrm{q}},3}; \partial\overline\mathbb{R}\times{\mathrm{bf}}_{{\mathrm{q}},F/S/C}; \partial\overline\mathbb{R}\times{\mathrm{mf}}_{{\mathrm{q}},S/C} \bigr] \\
&\hspace{6em} \to \bigl[ \overline\mathbb{R}\times X^2_{\mathrm{q}}; \partial\overline\mathbb{R}\times{\mathrm{zf}}_{{\mathrm{q}},2}; \partial\overline\mathbb{R}\times({\mathrm{lb}}_{{\mathrm{q}},2}\cup{\mathrm{rb}}_{{\mathrm{q}},2}) \bigr].
\end{align*}
We used here that $\partial\overline\mathbb{R}\times{\mathrm{mf}}_{{\mathrm{q}},S}$ and $\partial\overline\mathbb{R}\times{\mathrm{bf}}_{{\mathrm{q}},S}$ are disjoint to commute their blow-ups. Next, we blow up $\partial\overline\mathbb{R}\times\operatorname{diag}_{\mathrm{q}}$ in the range and correspondingly $\partial\overline\mathbb{R}\times\operatorname{diag}_{{\mathrm{q}},F}$ in the domain; we may subsequently also blow up $\partial\overline\mathbb{R}\times\operatorname{diag}_{{\mathrm{q}},3}$ in the domain, as the lifted projection is b-transversal to this. This produces a b-fibration
\begin{equation}
\label{EqQPCbFib0}
\begin{split}
&\bigl[ \overline\mathbb{R}\times X^3_{\mathrm{q}}; \partial\overline\mathbb{R}\times{\mathrm{zf}}_{{\mathrm{q}},3}; \partial\overline\mathbb{R}\times{\mathrm{bf}}_{{\mathrm{q}},F/S/C}; \partial\overline\mathbb{R}\times\operatorname{diag}_{{\mathrm{q}},3}; \partial\overline\mathbb{R}\times\operatorname{diag}_{{\mathrm{q}},F}; \partial\overline\mathbb{R}\times{\mathrm{mf}}_{{\mathrm{q}},S/C} \bigr] \\
&\hspace{6em} \to X^2_{{\mathrm{Q}},\flat} := \bigl[ \overline\mathbb{R}\times X^2_{\mathrm{q}}; \partial\overline\mathbb{R}\times{\mathrm{zf}}_{{\mathrm{q}},2}; \partial\overline\mathbb{R}\times\operatorname{diag}_{\mathrm{q}}, \partial\overline\mathbb{R}\times({\mathrm{lb}}_{{\mathrm{q}},2}\cup{\mathrm{rb}}_{{\mathrm{q}},2}) \bigr].
\end{split}
\end{equation}
Here we use that $\partial\overline\mathbb{R}\times\operatorname{diag}_{{\mathrm{q}},3}\subset\partial\overline\mathbb{R}\times\operatorname{diag}_{{\mathrm{q}},F}$, which implies that we can switch the order of their blow-ups; and moreover ${\mathrm{mf}}_{{\mathrm{q}},S}$ and ${\mathrm{mf}}_{{\mathrm{q}},C}$ are disjoint from $\operatorname{diag}_{{\mathrm{q}},3}$ and $\operatorname{diag}_{{\mathrm{q}},F}$, hence their blow-ups can be commuted through to the end.
In the domain, we next blow up $\partial\overline\mathbb{R}\times(\operatorname{diag}_{{\mathrm{q}},*}\cap\,{\mathrm{mf}}_{{\mathrm{q}},*})$ for $*=S,C$ (whose lifts get mapped diffeomorphically onto the lifts of $\partial\overline\mathbb{R}\times{\mathrm{lb}}_{{\mathrm{q}},2}$ and $\partial\overline\mathbb{R}\times{\mathrm{rb}}_{{\mathrm{q}},2}$); they can be commuted through the blow-ups of their supersets $\partial\overline\mathbb{R}\times{\mathrm{mf}}_{{\mathrm{q}},S/C}$. We thus obtain a b-fibration
\begin{equation}
\label{EqQPCbFib1}
\begin{split}
&\bigl[ \overline\mathbb{R}\times X^3_{\mathrm{q}}; \partial\overline\mathbb{R}\times{\mathrm{zf}}_{{\mathrm{q}},3}; \partial\overline\mathbb{R}\times{\mathrm{bf}}_{{\mathrm{q}},F/S/C}; \partial\overline\mathbb{R}\times\operatorname{diag}_{{\mathrm{q}},3}; \partial\overline\mathbb{R}\times\operatorname{diag}_{{\mathrm{q}},F}; \\
&\hspace{11em} \partial\overline\mathbb{R}\times(\operatorname{diag}_{{\mathrm{q}},S/C}\cap\,{\mathrm{mf}}_{{\mathrm{q}},S/C}); \partial\overline\mathbb{R}\times{\mathrm{mf}}_{{\mathrm{q}},S/C} \bigr] \to X^2_{{\mathrm{Q}},\flat}.
\end{split}
\end{equation}
We can then blow up $\partial\overline\mathbb{R}\times\operatorname{diag}_{{\mathrm{q}},S}$ in the domain; this blow-up can be commuted through that of $\partial\overline\mathbb{R}\times{\mathrm{mf}}_{{\mathrm{q}},*}$ for $*=S$ (since the intersection $\partial\overline\mathbb{R}\times(\operatorname{diag}_{{\mathrm{q}},S}\cap\,{\mathrm{mf}}_{{\mathrm{q}},S})$ is blown up before) and also for $*=C$ (by disjointness), and then it can be commuted further through its subset $\partial\overline\mathbb{R}\times(\operatorname{diag}_{{\mathrm{q}},S}\cap\,{\mathrm{mf}}_{{\mathrm{q}},S})$. Arguing similarly for the blow-up of $\partial\overline\mathbb{R}\times\operatorname{diag}_{{\mathrm{q}},C}$, the map~\eqref{EqQPCbFib1} thus lifts to a b-fibration
\begin{equation}
\label{EqQPCbFib2}
\begin{split}
&\bigl[ \overline\mathbb{R}\times X^3_{\mathrm{q}}; \partial\overline\mathbb{R}\times{\mathrm{zf}}_{{\mathrm{q}},3}; \partial\overline\mathbb{R}\times{\mathrm{bf}}_{{\mathrm{q}},F/S/C}; \partial\overline\mathbb{R}\times\operatorname{diag}_{{\mathrm{q}},3}; \partial\overline\mathbb{R}\times\operatorname{diag}_{{\mathrm{q}},F/S/C}; \\
&\hspace{13em} \partial\overline\mathbb{R}\times(\operatorname{diag}_{{\mathrm{q}},S/C}\cap\,{\mathrm{mf}}_{{\mathrm{q}},S/C}); \partial\overline\mathbb{R}\times{\mathrm{mf}}_{{\mathrm{q}},S/C} \bigr] \to X^2_{{\mathrm{Q}},\flat}.
\end{split}
\end{equation}
Next, blowing up $\partial\overline\mathbb{R}\times(\operatorname{diag}_{\mathrm{q}}\cap\,{\mathrm{mf}}_{{\mathrm{q}},2})$ in the range, and using~\eqref{EqqPreimages} to deduce that we need to blow up $\partial\overline\mathbb{R}\times(\operatorname{diag}_{{\mathrm{q}},F}\cap\,{\mathrm{mf}}_{{\mathrm{q}},F})$ and $\partial\overline\mathbb{R}\times(\operatorname{diag}_{{\mathrm{q}},F}\cap\,{\mathrm{mf}}_{{\mathrm{q}},3})$ in the domain, we infer that the map~\eqref{EqQPCbFib2} lifts further to a b-fibration
\begin{align*}
&\bigl[ \overline\mathbb{R}\times X^3_{\mathrm{q}}; \partial\overline\mathbb{R}\times{\mathrm{zf}}_{{\mathrm{q}},3}; \partial\overline\mathbb{R}\times{\mathrm{bf}}_{{\mathrm{q}},F/S/C}; \partial\overline\mathbb{R}\times\operatorname{diag}_{{\mathrm{q}},3}; \partial\overline\mathbb{R}\times\operatorname{diag}_{{\mathrm{q}},F/S/C}; \\
&\qquad \partial\overline\mathbb{R}\times(\operatorname{diag}_{{\mathrm{q}},F/S/C}\cap\,{\mathrm{mf}}_{{\mathrm{q}},F/S/C}); \partial\overline\mathbb{R}\times{\mathrm{mf}}_{{\mathrm{q}},S/C}; \partial\overline\mathbb{R}\times(\operatorname{diag}_{{\mathrm{q}},F}\cap\,{\mathrm{mf}}_{{\mathrm{q}},3}) \bigr] \\
&\ \to X_{{\mathrm{Q}},\sharp}^2 := \bigl[ \overline\mathbb{R}\times X^2_{\mathrm{q}}; \partial\overline\mathbb{R}\times{\mathrm{zf}}_{{\mathrm{q}},2}; \partial\overline\mathbb{R}\times\operatorname{diag}_{\mathrm{q}}, \partial\overline\mathbb{R}\times(\operatorname{diag}_{\mathrm{q}}\cap\,{\mathrm{mf}}_{{\mathrm{q}},2}), \partial\overline\mathbb{R}\times({\mathrm{lb}}_{{\mathrm{q}},2}\cup{\mathrm{rb}}_{{\mathrm{q}},2}) \bigr].
\end{align*}
For the commutation of blow-ups, we use here that $\operatorname{diag}_{{\mathrm{q}},F}$ is disjoint from ${\mathrm{mf}}_{{\mathrm{q}},S/C}$. To restore some symmetry, we then blow up $\partial\overline\mathbb{R}\times(\operatorname{diag}_{{\mathrm{q}},*}\cap\,{\mathrm{mf}}_{{\mathrm{q}},3})$ in the domain for $*=S,C$; these get mapped diffeomorphically onto the lift of $\partial\overline\mathbb{R}\times{\mathrm{mf}}_{{\mathrm{q}},2}$. Thus, we get a b-fibration
\begin{align*}
&\bigl[ \overline\mathbb{R}\times X^3_{\mathrm{q}}; \partial\overline\mathbb{R}\times{\mathrm{zf}}_{{\mathrm{q}},3}; \partial\overline\mathbb{R}\times{\mathrm{bf}}_{{\mathrm{q}},F/S/C}; \partial\overline\mathbb{R}\times\operatorname{diag}_{{\mathrm{q}},3}; \partial\overline\mathbb{R}\times\operatorname{diag}_{{\mathrm{q}},F/S/C}; \\
&\qquad \partial\overline\mathbb{R}\times(\operatorname{diag}_{{\mathrm{q}},F/S/C}\cap\,{\mathrm{mf}}_{{\mathrm{q}},F/S/C}); \partial\overline\mathbb{R}\times{\mathrm{mf}}_{{\mathrm{q}},S/C}; \partial\overline\mathbb{R}\times(\operatorname{diag}_{{\mathrm{q}},F/S/C}\cap\,{\mathrm{mf}}_{{\mathrm{q}},3}) \bigr] \to X_{{\mathrm{Q}},\sharp}^2.
\end{align*}
Finally, we again use \cite[Proposition~5.12.1]{MelroseDiffOnMwc} to lift this map to a b-fibration under the blow-up of $\partial\overline\mathbb{R}\times{\mathrm{mf}}_{{\mathrm{q}},2}$ in the range (producing $X^2_{\mathrm{Q}}$) and of the lifts of its preimages $\partial\overline\mathbb{R}\times{\mathrm{mf}}_{{\mathrm{q}},F}$ and $\partial\overline\mathbb{R}\times{\mathrm{mf}}_{{\mathrm{q}},3}$ (in this order) in the domain; the resulting domain is naturally diffeomorphic to $X^3_{\mathrm{Q}}$, since the blow-up of $\partial\overline\mathbb{R}\times{\mathrm{mf}}_{{\mathrm{q}},F}$ can be commuted through that of $\partial\overline\mathbb{R}\times(\operatorname{diag}_{{\mathrm{q}},*}\cap\,{\mathrm{mf}}_{{\mathrm{q}},3})$ for $*=F$ (since the set $\partial\overline\mathbb{R}\times(\operatorname{diag}_{{\mathrm{q}},F}\cap\,{\mathrm{mf}}_{{\mathrm{q}},F})$ containing their intersection is blown up earlier) and for $*=S,C$ (by disjointness). This finishes the proof.
\end{proof}
\begin{prop}[Composition of Q-ps.d.o.s]
\label{PropQPC}
Let $A_j\in\Psi_\Qop^{s_j,\alpha_j}(X)$, $j=1,2$. Then $A_1\circ A_2\in\Psi_\Qop^{s_1+s_2,\alpha_1+\alpha_2}(X)$. The same holds true when working with $\Psi_{{\mathrm{Q}},{\mathrm{cl}}}$ instead.
\end{prop}
\begin{proof}
The proof is similar to that of Proposition~\ref{PropqComp}. By Remark~\ref{RmkQPDefFn}, it suffices to consider the case $\alpha_1=\alpha_2=(0,0,0,0,0)$. Write the Schwartz kernel $\kappa$ of $A_1\circ A_2$ in terms of the Schwartz kernels $\kappa_1,\kappa_2$ of $A_1,A_2$ as
\[
\kappa = (\nu_1\nu_2)^{-1}(\pi_C)_* \bigl( \pi_F^*\kappa_1 \cdot \pi_S^*\kappa_2 \cdot \pi_C^*\nu_1\cdot\pi^*\nu_2 \bigr)
\]
where $0<\nu_1\in\cC^\infty(X_{\mathrm{Q}};{}^{\Qop}\Omega X)$ is an arbitrary q-density, and $\nu_2=\frac{{\mathrm d}\fm}{\fm}\frac{{\mathrm d}\sigma}{\langle\sigma\rangle}$ is a b-density on $\overline{\mathbb{R}_\sigma}\times[0,2)_\fm$ with $\pi\colon X_{\mathrm{Q}}^3\to\overline\mathbb{R}\times[0,1]$ denoting the lifted projection. One can then check that the term in parentheses is then a bounded conormal section of $\pi_F^*{}^{\Qop}\Omega X\otimes\pi_S^*{}^{\Qop}\Omega X\otimes\pi_C^*{}^{\Qop}\Omega X\otimes\pi^*{}^{\bop}\Omega_{\overline\mathbb{R}\times[0,1]}(\overline\mathbb{R}\times[0,2))$ which vanishes to infinite order at the boundary hypersurfaces of $X^3_{\mathrm{Q}}$ which map to ${\mathrm{if}}'_2$, ${\mathrm{sf}}'_2$, ${\mathrm{lb}}_2$, ${\mathrm{rb}}_2$, ${\mathrm{tlb}}_2$, or ${\mathrm{trb}}_2$ under $\pi_C$; thus, it is a bounded conormal section of ${}^{\bop}\Omega X^3_{\mathrm{Q}}$ which vanishes at the aforementioned boundary hypersurfaces. The conclusion then follows using pullback and pushforward results for conormal distributions.
\end{proof}
\subsection{Q-Sobolev spaces}
\label{SsQH}
We now assume that $X$ is compact. We can define weighted Sobolev spaces (corresponding to the Lie algebra $\cV_\Qop(X)$) of integer differential order in the usual manner, analogously to Definition~\ref{DefqSob}; we leave it to the reader to spell this out. Here, we instead immediately record the definition for general orders, allowing in particular also for variable orders at ${\mathrm{if}}$:
\begin{definition}[Weighted Q-Sobolev spaces]
\label{DefQH}
Fix any positive weighted Q-density $\nu$ on $X_{\mathrm{Q}}$, i.e.\ an element $\nu=(\prod\rho_H^{\nu_H})\nu_0$ where $0<\nu_0\in\cC^\infty(X_{\mathrm{Q}},{}^{\Qop}\Omega X)$ and $\nu_H\in\mathbb{R}$, and $H$ ranges over the boundary hypersurfaces ${\mathrm{zf}},{\mathrm{mf}},{\mathrm{nf}},{\mathrm{if}},{\mathrm{sf}}$. Thus, the restriction $\nu_{\fm,\sigma}$ is a smooth positive density on $X$ for any $\fm\in(0,1]$, $\sigma\in\mathbb{R}$. Let $s\in\mathbb{R}$ and $l,\gamma,l',r,b\in\mathbb{R}$; put $w:=\rho_{\mathrm{zf}}^l\rho_{\mathrm{mf}}^\gamma\rho_{\mathrm{nf}}^{l'}\rho_{\mathrm{if}}^r\rho_{\mathrm{sf}}^b$. Then for $s\geq 0$, and for $\fm\in(0,1]$ and $\sigma\in\mathbb{R}$, we put
\begin{subequations}
\begin{equation}
\label{EqQH}
H_{{\mathrm{Q}},\fm,\sigma}^{s,(l,\gamma,l',r,b)}(X,\nu) = H^s(X)
\end{equation}
with $(\fm,\sigma)$-dependent norm
\begin{equation}
\label{EqQHNorm}
\|u\|_{H_{{\mathrm{Q}},\fm,\sigma}^{s,(l,\gamma,l',r,b)}(X,\nu)}^2 := \|w^{-1}u\|_{L^2(X,\nu_{\fm,\sigma})}^2 + \|w^{-1}A_{\fm,\sigma} u\|_{L^2(X,\nu_{\fm,\sigma})},
\end{equation}
\end{subequations}
where $A=(A_{\fm,\sigma})\in\Psi_\Qop^s(X)$ is any fixed Q-ps.d.o.\ with elliptic principal symbol. For $s<0$, we define the space~\eqref{EqQH} as a Hilbert space by letting it be the dual space (with respect to the inner product on $L^2(X,\nu_{\fm,\sigma})$) of $H_{{\mathrm{Q}},\fm,\sigma}^{-s,(-l,-\gamma,-l',-r,-b)}(X,\nu)$.\footnote{Equivalently, fixing an elliptic operator $A\in\Psi_\Qop^{-s}(X)$, it is the space of all distributions of the form $u=u_0+A u_1$ where $u_0,u_1\in w L^2(X)$, equipped with the norm $\inf_{u=u_0+A u_1}\|w^{-1}u_0\|_{L^2(X,\nu_{\fm,\sigma})}+\|w^{-1}u_1\|_{L^2(X,\nu_{\fm,\sigma})}$; cf.\ \cite[Appendix~A]{MelroseVasyWunschEdge} for a general discussion of the underlying functional analysis.} Finally, for variable orders $\mathsf{r}\in\cC^\infty(\overline{{}^\Qop T^*_{\mathrm{if}}}X)$, we define the norm on $H_{{\mathrm{Q}},\fm,\sigma}^{s,(l,\gamma,l',\mathsf{r},b)}(X,\nu)=H^s(X)$ to be
\[
\|u\|_{H_{{\mathrm{Q}},\fm,\sigma}^{s,(l,\gamma,l',\mathsf{r},b)}(X,\nu)}^2 := \|u\|_{H_{{\mathrm{Q}},\fm,\sigma}^{s,(l,\gamma,l',r_0,b)}(X,\nu)}^2 + \|A u\|_{L^2(X,\nu_{\fm,\sigma})}^2
\]
where $r_0=\min\mathsf{r}$, and where $A\in\Psi_\Qop^{s,(l,\gamma,l',\mathsf{r},b)}(X)$ is a fixed elliptic operator.
\end{definition}
We claim that any $A\in\Psi_\Qop^0(X)$ is uniformly (for $\fm\in(0,1]$ and $\sigma\in\mathbb{R}$) bounded on $L^2(X,\nu)$ when $0<\nu\in\cC^\infty(X_{\mathrm{Q}},{}^{\Qop}\Omega X)$ is a positive Q-density. As in~\S\ref{Ssq}, the proof can be reduced, using H\"or\-man\-der's square root trick, to the case that $A\in\Psi_\Qop^{-\infty,(0,0,0,-\infty,-\infty)}(X)$; thus, the Schwartz kernel $\kappa$ of $A$ is a bounded conormal right Q-density on $X_{\mathrm{Q}}^2$ which vanishes to infinite order at all boundary hypersurfaces except ${\mathrm{zf}}_2$, ${\mathrm{mf}}_2$, and ${\mathrm{nf}}_2$. The pushforward along the projection $X_{\mathrm{Q}}^2\to X_{\mathrm{Q}}$ (see Lemma~\ref{LemmaQPbFib}) is thus bounded (on $\fm^{-1}([0,1])$) and conormal on $X_{\mathrm{Q}}$ (and vanishes to infinite order at ${\mathrm{if}}$ and ${\mathrm{sf}}$). The Schur test implies the claim. Directly from Definition~\ref{DefQH}, one can then show that for any orders $s,\tilde s\in\mathbb{R}$, $l,\tilde l,\gamma,\tilde\gamma,l',\tilde l',r,\tilde r,b,\tilde b\in\mathbb{R}$, any element $A=(A_{\fm,\sigma})\in\Psi_\Qop^{s,(l,\gamma,l',r,b)}(X)$ defines a uniformly bounded (as $\fm,\sigma$ ranges over $(0,1]\times\mathbb{R}$) family of maps
\[
A_{\fm,\sigma} \colon H_{{\mathrm{Q}},\fm,\sigma}^{\tilde s,(\tilde l,\tilde\gamma,\tilde l',\tilde r,\tilde b)}(X,\nu) \to H_{{\mathrm{Q}},\fm,\sigma}^{\tilde s-s,(\tilde l-l,\tilde\gamma-\gamma,\tilde l'-l',\tilde r-r,\tilde b-b)}(X,\nu),
\]
similarly when the ${\mathrm{if}}$-order is variable.
We next show how to relate Q-Sobolev spaces (and their norms) to b-, ${\mathrm{sc}\text{-}\mathrm{b}}$-, ${\mathrm{c}\hbar}$-, and semiclassical scattering Sobolev spaces near the various boundary hypersurfaces of $X_{\mathrm{Q}}$, see Proposition~\ref{PropQStruct}, Definition~\ref{DefQPieces}, and equation~\eqref{EqQPieces}. We restrict attention to a certain class of $\sigma$-independent densities $\nu$, which are lifts of weighted q-densities on $X_{\mathrm{q}}$ along the projection off the $\sigma$-coordinate.
\begin{prop}[Relationships between Sobolev spaces]
\label{PropQHRel}
Fix a $\sigma$-independent density $\nu$ on $X_{\mathrm{Q}}$ which is of the form $\nu=\rho_{{\mathrm{zf}}_q}^{n/2}\nu_0$, $0<\nu_0\in\cC^\infty(X,{}^{\qop}\Omega X)$, as in Proposition~\usref{PropqHRel}. Let $\mathsf{r}\in\cC^\infty(\overline{{}^\Qop T^*_{\mathrm{if}}}X)$ be an order function which in $|x|<r_0$ (for some $r_0>0$) is invariant under the lift to ${}^\Qop T^*X$ of the dilation action $(\tilde\sigma,\fm,x)\mapsto(\tilde\sigma,\lambda\fm,\lambda x)$.
\begin{enumerate}
\item\label{ItQHRelzf} Put $\phi_{\mathrm{zf}}\colon\mathbb{R}\times(0,1]\times\hat X^\circ\ni(\sigma,\fm,\hat x)\mapsto(\sigma,\fm,\fm\hat x)\in X_{\mathrm{Q}}$, and let $\chi\in\cC^\infty(X_{\mathrm{Q}})$ be identically $1$ near ${\mathrm{zf}}$ and supported in a collar neighborhood thereof. Then for $\fm\in(0,1]$ and $\sigma\in\mathbb{R}$, we have a uniform equivalence (in the same sense as in Proposition~\usref{PropqHRel})
\begin{equation}
\label{EqQHRelzf}
\|\chi u\|_{H_{{\mathrm{Q}},\fm,\sigma}^{s,(l,\gamma,l',\mathsf{r},b)}(X)} \sim \langle\sigma\rangle^{l'-l}\fm^{\frac{n}{2}-l}\|\phi_{\mathrm{zf}}^*(\chi u)|_{\sigma,\fm}\|_{H_{\bop}^{s,\gamma-l}(\hat X,|{\mathrm d}\hat x|)}.
\end{equation}
\item\label{ItQHRelmfsemi} Put $\phi_{{\mathrm{mf}},\pm,\hbar}\colon(0,1]\times(0,1]\times\dot X^\circ\ni(h,\fm,x)\mapsto(\pm h^{-1},\fm,x)\in X_{\mathrm{Q}}$, and let $\chi\in\cC^\infty(X_{\mathrm{Q}})$ be identically $1$ near ${\mathrm{mf}}$ and supported in a collar neighborhood thereof. Then, uniformly for $\fm\in(0,1]$ and $h\in(0,1]$, we have
\begin{equation}
\label{EqQHRelmfsemi}
\|\chi u\|_{H_{{\mathrm{Q}},\fm,\pm h^{-1}}^{s,(l,\gamma,l',\mathsf{r},b)}(X)} \sim \fm^{-\gamma} \| \phi_{{\mathrm{mf}},\pm,\hbar}^*(\chi u)|_{h,\fm} \|_{H_{{\mathrm{c}},h}^{s,l-\gamma,l'-\gamma,\mathsf{r}-\gamma}(\dot X,\nu_{\mathrm{c}})},
\end{equation}
where $\nu_{\mathrm{c}}$ is the lift of a smooth positive density on $X$ to $\dot X$ as in Proposition~\usref{PropqHRel}\eqref{ItqHRelmf}.
\item\label{ItQHRelnflow} Put $\phi_{{\mathrm{nf}}_{\pm,{\mathrm{low}}}}\colon\pm(0,1]\times(0,1]\times\hat X\ni(\tilde\sigma,\fm,\hat x)\mapsto(\frac{\tilde\sigma}{\fm},\fm,\fm\hat x)\in X_{\mathrm{Q}}$, and let $\chi\in\cC^\infty(X_{\mathrm{Q}})$ be identically $1$ near ${\mathrm{nf}}_\pm$ and supported in a collar neighborhood thereof. Then, uniformly for $\tilde\sigma\in(0,1]$ and $\fm\in(0,1]$,
\begin{equation}
\label{EqQHRelnflow}
\|\chi u\|_{H_{{\mathrm{Q}},\fm,\pm h^{-1}}^{s,(l,\gamma,l',\mathsf{r},b)}(X)} \sim \fm^{\frac{n}{2}-l'}\|\phi_{{\mathrm{nf}}_{\pm,{\mathrm{low}}}}^*(\chi u)|_{\tilde\sigma,\fm}\|_{H_{{\mathrm{sc}\text{-}\mathrm{b}},\tilde\sigma}^{s,\mathsf{r}-l',\gamma-l',l-l'}(\hat X,|{\mathrm d}\hat x|)}.
\end{equation}
\item\label{ItQHRelnfsemi} Put $\phi_{{\mathrm{nf}}_{\pm,\tilde\hbar}}\colon(0,1]\times(0,1]\times\hat X\ni(\tilde h,\fm,\hat x)\mapsto(\pm(\tilde h\fm)^{-1},\fm,\fm\hat x)\in X_{\mathrm{Q}}$, and let $\chi\in\cC^\infty(X_{\mathrm{Q}})$ be as in part~\eqref{ItQHRelnflow}. Then, uniformly for $\tilde h\in(0,1]$ and $\fm\in(0,1]$,
\begin{equation}
\label{EqQHRelnfsemi}
\|\chi u\|_{H_{{\mathrm{Q}},\fm,\pm(\tilde h\fm)^{-1}}^{s,(l,\gamma,l',\mathsf{r},b)}(X)} \sim \fm^{\frac{n}{2}-l'}\|\phi_{{\mathrm{nf}}_{\pm,\tilde\hbar}}^*(\chi u)|_{\tilde h,\fm}\|_{H_{{\mathrm{sc}},\tilde h}^{s,\mathsf{r}-l',b}(\hat X,|{\mathrm d}\hat x|)}.
\end{equation}
\end{enumerate}
\end{prop}
We remark that the invariance assumption on $\mathsf{r}$ is only used in parts~\eqref{ItQHRelnflow}--\eqref{ItQHRelnfsemi} and made there for simplicity; note that the assumption depends on the choice of local coordinates $x\in\mathbb{R}^n$ around $0\in X$. (Without this assumption, one gets slightly lossy two-sided estimates mirroring those in \cite[Corollary~3.7(2)]{HintzConicProp}; these would still be sufficient for our application.)
\begin{proof}[Proof of Proposition~\usref{PropQHRel}]
It suffices to consider the case that all constant weights $l,\gamma,l',b$ are equal to $0$; furthermore, one can restrict to the case $s\geq 0$ since the case of $s<0$ then follows by duality. The $L^2$-case $s=0$ follows, for all four parts, as in Proposition~\ref{PropqHRel}.
Part~\eqref{ItQHRelzf} for $s>0$ is then a parameter-dependent version of the estimate~\eqref{EqqHRelzf}, and the proof proceeds in the same manner: one extends the Schwartz kernel of an elliptic b-ps.d.o.\ $A_0\in\Psi_\bop^s(\hat X)$ via dilation-invariance (in $(\fm,x,x')$) and translation-invariance (in $\sigma$), and cuts off the resulting kernel to a collar neighborhood of ${\mathrm{zf}}_2$ to obtain a Q-ps.d.o.\ $A$ which is elliptic near ${\mathrm{zf}}$ and can thus be used to compute $H_{\mathrm{Q}}^s(X)$-norms in~\eqref{EqQHNorm}.
For part~\eqref{ItQHRelmfsemi}, we fix an operator $A_0\in\Psi_\chop^{s,0,0,\mathsf{r}}(\dot X)$ with elliptic principal symbol. By Corollary~\ref{CorQPNormal}, this is the ${\mathrm{mf}}_{\pm,\hbar}$-normal operator of some $A\in\Psi_\Qop^{s,(0,0,0,\mathsf{r},0)}(X)$, and in fact we can take the Schwartz kernel of $A$ to be given by the pushforward of the Schwartz kernel of $A_0$ (considered as a $\fm$-independent distribution) along $\phi_{{\mathrm{mf}},\pm,\hbar}$, cut off in both factors to a collar neighborhood of ${\mathrm{mf}}_2$. The uniform equivalence~\eqref{EqQHRelmfsemi} then follows by arguments completely analogous to those in the proof of Proposition~\ref{PropqHRel}.
The proof of parts~\eqref{ItQHRelnflow}--\eqref{ItQHRelnfsemi} is similar. The assumption on the order function $\mathsf{r}$ ensures that the cutoff (to a collar neighborhood of ${\mathrm{nf}}$ in both factors on the level of the Schwartz kernel) of the dilation-invariant extension off ${\mathrm{nf}}_2$ of an elliptic operator in $\Psi_\scbtop^{s,\mathsf{r},0,0}(\hat X)$ lies in $\Psi_\Qop^{s,(0,0,0,\mathsf{r},0)}(X)$.
\end{proof}
Finally, when $\Omega\subset X_{\mathrm{Q}}$ is an open set, and writing $\alpha=(l,\gamma,l',\mathsf{r},b)$, we denote by
\begin{equation}
\label{EqQHExt}
\dot H_{{\mathrm{Q}},\fm,\sigma}^{s,\alpha}(\overline\Omega) = \bigl\{ u\in H_{{\mathrm{Q}},\fm,\sigma}^{s,\alpha}(X)\colon \operatorname{supp} u\subset\overline\Omega \bigr\},\quad
\bar H_{{\mathrm{Q}},\fm,\sigma}^{s,\alpha}(\Omega) = \bigl\{ u|_\Omega \colon u\in H_{{\mathrm{Q}},\fm,\sigma}^{s,\alpha}(X) \bigr\}
\end{equation}
the spaces of supported, resp.\ extendible distribution (using H\"ormander's notation \cite[Appendix~B]{HormanderAnalysisPDE3}). The former space carries the subspace topology, and the latter space the quotient topology of $H_{{\mathrm{Q}},\fm,\sigma}^{s,\alpha}(X)/\dot H_{{\mathrm{Q}},\fm,\sigma}^{s,\alpha}(X_{\mathrm{Q}}\setminus\Omega)$. In our application, we will take, in some fixed local coordinates $x\in\mathbb{R}^n$ around $0\in X$,
\[
\Omega=X_{\mathrm{Q}}\cap\{\fm<r<2\}=X_{\mathrm{Q}}\cap\{\hat r>1,\ r<2\},
\]
and the relationships recorded in Proposition~\ref{PropQHRel} remain valid upon using extendible Q-Sobolev spaces on $\Omega$ as well as extendible (b-, ${\mathrm{sc}\text{-}\mathrm{b}}$-, and semiclassical scattering) Sobolev spaces on $\hat\Omega=\hat X\cap\{\hat r>1\}$ in~\eqref{EqQHRelzf}, \eqref{EqQHRelnflow}, \eqref{EqQHRelnfsemi}.
\section{Quasinormal modes of massless and massive scalar waves}
\label{SK}
In this section, we will prove Theorems~\ref{ThmI} and \ref{ThmIKG}. As discussed in~\S\ref{SsIS}, we may fix $\Lambda=3$. Moreover, we fix the ratio
\[
\hat\fa := \frac{\fa}{\fm} \in (-1,1).
\]
All estimates in this section will be uniform in the parameter $\hat\fa\in[-1+\epsilon,1-\epsilon]$ for any fixed $\epsilon>0$.
In~\S\ref{SsKL}, we fix some notation for the degenerating family of Kerr--de~Sitter spacetimes with parameters $(\Lambda,\fm,\fa)=(3,\fm,\hat\fa\fm)$, with $\fm\searrow 0$. In~\S\ref{SsKMain}, we recall the notions of generalized resonant states and the multiplicity of resonances; these feature in the detailed version of Theorem~\ref{ThmI}, see Theorem~\ref{ThmK}. As a preparation for the proof of Theorem~\ref{ThmK}, we show in~\S\ref{SsKS} how the spectral family of the wave operator on the degenerating Kerr--de~Sitter spacetimes fits into the framework of Q-analysis. The remaining sections, \S\S\ref{SsKSy}--\ref{SsKU}, contain the proof of Theorem~\ref{ThmK}; an outline is provided at the end of~\S\ref{SsKS}. In~\S\ref{SsKG}, we explain the minor modifications needed for the analysis of the Klein--Gordon equation, and thus prove Theorem~\ref{ThmIKG}.
\subsection{Limits of Kerr--de~Sitter metrics}
\label{SsKL}
Since the quantities involved in the definition~\eqref{EqIMetric} of the KdS metric depend only on $\fm$ via $(\Lambda,\fm,\fa)=(3,\fm,\hat\fa\fm)$, we denote them by
\begin{alignat*}{2}
\mu_\fm(r) &:= (r^2+\hat\fa^2\fm^2)(1-r^2) - 2\fm r, &\qquad
b_\fm &:= 1+\hat\fa^2\fm^2, \\
c_\fm(\theta) &:= 1+\hat\fa^2\fm^2\cos^2\theta, &\qquad
\varrho_\fm(r,\theta) &:= r^2+\hat\fa^2\fm^2\cos^2\theta.
\end{alignat*}
As we shall prove momentarily, for $\fm>0$ sufficiently small, the parameters $(3,\fm,\hat\fa\fm)$ are subextremal. We denote the roots of $\mu_\fm$ by
\[
r_\fm^-<r_\fm^C<r_\fm^e<r_\fm^c.
\]
\begin{lemma}[Roots of $\mu_\fm$]
\label{LemmaKRoots}
Define $\hat r^C:=1-\sqrt{1-\hat\fa^2}$, $\hat r^e:=1+\sqrt{1-\hat\fa^2}$. For sufficiently small $\fm_0>0$ (depending on $\hat\fa$), and writing $\cC^\infty=\cC^\infty([0,\fm_0])$, we have
\begin{alignat*}{2}
r_\fm^-&\equiv-1\bmod \fm\cC^\infty,&\quad
r_\fm^C&\equiv \fm\hat r^C \bmod \fm^2\cC^\infty, \\
r_\fm^e&\equiv \fm\hat r^e \bmod \fm^2\cC^\infty,&\quad
r_\fm^c&\equiv 1\bmod\fm\cC^\infty.
\end{alignat*}
\end{lemma}
\begin{proof}
The simple roots of $\mu_0(r)=r^2(1-r^2)$ at $r=\pm 1$ extend to real analytic functions $r_\fm^-=-1+\mc O(\fm)$ and $r_\fm^c=1+\mc O(\fm)$ for small real $\fm$. Note next that
\[
\fm^{-2}\mu_\fm(\fm\hat r) = \hat r^2-2\hat r+\hat\fa^2 - \fm^2(\hat r^2+\hat\fa^2)\hat r^2,
\]
for $\fm=0$, has two simple roots at $\hat r=\hat r^C,\hat r^e$, which extend to real analytic functions $\hat r^C_\fm,\hat r^e_\fm$ for small $\fm$, giving rise to the roots $r_\fm^C=\fm\hat r^C_\fm$, $r_\fm^e=\fm\hat r^e_\fm$, of $\mu_\fm$.
\end{proof}
We use the coordinates $(t_*,r,\theta,\phi_*)$, see~\eqref{EqIKerrStar}, which we define using
\begin{equation}
\label{EqKFm}
F_{3,\fm,\hat\fa\fm}(r) = F_\fm(r) := -\chi^e\Bigl(\frac{r-r_\fm^e}{\fm}\Bigr) + \chi^c(r-r_\fm^c),
\end{equation}
where $\chi^e\in\cC^\infty_\cp(\mathbb{R})$ and $\chi^c\in\cC^\infty_\cp((-1,\infty))$ are both equal to $1$ at $0$. (Thus, the functions $t_*,\phi_*$ defined with respect to the choice of $F_\fm$ here differ from those defined using the choice of $F_{3,\fm,\hat\fa\fm}$ in~\S\ref{SI} by the addition of smooth functions of $r$.) We fix $\chi^e,\chi^c$ in Lemma~\ref{LemmaKdSChi} below. The KdS metric $g_\fm:=g_{3,\fm,\hat\fa\fm}$ takes the form
\begin{equation}
\label{EqKExt}
\begin{split}
g_\fm &= -\frac{\mu_\fm(r)}{b_\fm^2\varrho_\fm^2(r,\theta)}\bigl({\mathrm d} t_*-\hat\fa\fm\,\sin^2\theta\,{\mathrm d}\phi_*\bigr)^2 - \frac{2 F_\fm(r)}{b_\fm}({\mathrm d} t_*-\hat\fa\fm\,\sin^2\theta\,{\mathrm d}\phi_*)\,{\mathrm d} r \\
&\quad + \varrho_\fm^2(r,\theta)\frac{1{-}F_\fm(r)^2}{\mu_\fm(r)}\,{\mathrm d} r^2 + \varrho_\fm^2(r,\theta)\frac{{\mathrm d}\theta^2}{c_\fm(\theta)} + \frac{c_\fm(\theta)\sin^2\theta}{b_\fm^2\varrho_\fm^2(r,\theta)}\bigl( (r^2+(\hat\fa\fm)^2)\,{\mathrm d}\phi_* - \hat\fa\fm\,{\mathrm d} t_*\bigr)^2.
\end{split}
\end{equation}
The dual metric is
\begin{equation}
\label{EqKExtDual}
\begin{split}
g_\fm^{-1} &= \varrho_\fm(r,\theta)^{-2}\Bigl( -\frac{b_\fm^2(1-F_\fm(r)^2)}{\mu_\fm(r)}\bigl((r^2+(\hat\fa\fm)^2)\partial_{t_*}+\hat\fa\fm\partial_{\phi_*}\bigr)^2 + \mu_\fm(r)\partial_r^2 + c_\fm(\theta)\partial_\theta^2 \\
&\quad - 2 b_\fm F_\fm(r)\bigl((r^2+(\hat\fa\fm)^2)\partial_{t_*}+\hat\fa\fm\partial_{\phi_*}\bigr)\otimes_s\partial_r + \frac{b_\fm^2}{c_\fm(\theta)\sin^2\theta}\bigl(\partial_{\phi_*}+\hat\fa\fm\,\sin^2\theta\,\partial_{t_*}\bigr)^2\Bigr).
\end{split}
\end{equation}
\begin{lemma}[Choice of time function]
\label{LemmaKdSChi}
There exist smooth functions $\chi^e\in\cC^\infty_\cp(\mathbb{R})$ and $\chi^c\in\cC^\infty_\cp((-1,\infty))$ with $\chi^e(0)=1$ and $\chi^c(0)=1$ so that ${\mathrm d} t_*$ is (past) timelike with respect to $g_\fm$ on $\mathbb{R}_{t_*}\times[\fm,2]_r\times\mathbb{S}_{\theta,\phi_*}^2$ when $\fm\in(0,\fm_0]$ with $\fm_0>0$ sufficiently small.
\end{lemma}
This can be proved directly by adapting the arguments of \cite[\S6.1]{VasyMicroKerrdS} to the present parameter-dependent setting; we postpone an alternative perturbative proof off the two limiting (de Sitter and Kerr) metrics until after the proof of Lemma~\ref{LemmaKMetric} below.
In $r>r_0$ for any $r_0>0$, the metric $g_\fm$ converges, as $\fm\searrow 0$, to the metric
\begin{equation}
\label{EqKdS}
\begin{split}
g_{\mathrm{dS}} &:= -(1-r^2){\mathrm d} t_*^2 - 2\tilde\chi^c(r){\mathrm d} t_*\,{\mathrm d} r + \frac{1-\tilde\chi^c(r)^2}{1-r^2}{\mathrm d} r^2 + r^2\slashed{g}{}, \\
g_{\mathrm{dS}}^{-1} &= -\frac{1-\tilde\chi^c(r)^2}{1-r^2}\partial_{t_*}^2 - 2\tilde\chi^c(r)\partial_{t_*}\otimes_s\partial_r + (1-r^2)\partial_r^2 + r^{-2}\slashed{g}{}^{-1},
\end{split}
\end{equation}
where $\tilde\chi^c(r):=\chi^c(r-1)$, and $\slashed{g}{}:={\mathrm d}\theta^2+\sin^2\theta\,{\mathrm d}\phi_*^2$ is the standard metric on $\mathbb{S}^2$. Thus, $g_{\mathrm{dS}}$ is the de~Sitter metric\footnote{If we change coordinates via $t=t_*+T_{\mathrm{dS}}(r)$ where $T'_{\mathrm{dS}}(r)=\frac{\tilde\chi^c(r)}{1-r^2}$, then $g_{\mathrm{dS}}=-(1-r^2){\mathrm d} t_*^2+(1-r^2)^{-1}{\mathrm d} r^2+r^2\slashed{g}{}$ is the de~Sitter metric in static coordinates.}---a nondegenerate Lorentzian metric on
\begin{equation}
\label{EqKdSSpace}
\mathbb{R}_{t_*} \times X,\qquad X := B(0,3)=\{ x\in\mathbb{R}^3\colon |x|<3 \},
\end{equation}
with $(r,\theta,\phi_*)$ denoting polar coordinates on $X$. (We stress that $g_{\mathrm{dS}}$ is in fact smooth across $x=0$, though the geometry, resp.\ analysis of the limit $\fm\searrow 0$ do see a remnant of the disappeared KdS black hole in the form of a conical singularity, resp.\ b-Sobolev spaces with weights at $r=0$.)
On the other hand, if we set $\hat t_*:=\fm t_*$ and $\hat r:=\fm r$ and express $g_\fm$ in the coordinates $(\hat t_*,\hat r,\theta,\phi_*)$, then for $\hat r$ in any closed subinterval of $(\hat r^C,\infty)$, the rescaled metric $\fm^{-2}g_\fm$ converges, as $\fm\searrow 0$, to the metric
\begin{subequations}
\begin{equation}
\label{EqKKerr}
\begin{split}
&\hat g = -\frac{\hat\mu(\hat r)}{\hat\varrho^2(r,\theta)}\bigl({\mathrm d}\hat t_*-\hat\fa\,\sin^2\theta\,{\mathrm d}\phi_*\bigr)^2 + 2\tilde\chi^e(\hat r)({\mathrm d}\hat t_*-\hat\fa\,\sin^2\theta\,{\mathrm d}\phi_*)\,{\mathrm d}\hat r \\
&\qquad\qquad + \hat\varrho^2\frac{1-\tilde\chi^e(\hat r)^2}{\hat\mu(r)}{\mathrm d}\hat r^2 + \hat\varrho^2(\hat r,\theta){\mathrm d}\theta^2 + \frac{\sin^2\theta}{\hat\varrho^2(\hat r,\theta)}\bigl((\hat r^2+\hat\fa^2){\mathrm d}\phi_*-\hat\fa\,{\mathrm d}\hat t_*\bigr)^2, \\
&\quad \hat\mu(\hat r):=\hat r^2-2\hat r+\hat\fa^2,\qquad
\hat\varrho^2(\hat r,\theta):=\hat r^2+\hat\fa^2\cos^2\theta,\qquad
\tilde\chi^e(\hat r)=\chi^e(\hat r-\hat r^e),
\end{split}
\end{equation}
of a Kerr black hole with mass $1$ and angular momentum $\hat\fa$.\footnote{A coordinate change in $\hat t_*$ and $\phi_*$ brings~\eqref{EqKKerr} into Boyer--Lindquist form.} The dual metric is
\begin{equation}
\label{EqKKerrDual}
\begin{split}
\hat g^{-1} &= \hat\varrho(\hat r,\theta)^{-2}\Bigl(-\frac{1-\tilde\chi^e(\hat r)^2}{\hat\mu(\hat r)}\bigl((\hat r^2+\hat\fa^2)\partial_{\hat t_*}+\hat\fa\partial_{\phi_*}\bigr)^2 + \hat\mu(\hat r)\partial_{\hat r}^2 + \partial_\theta^2 \\
&\qquad\qquad + 2\tilde\chi^e(\hat r)\bigl((\hat r^2+\hat\fa^2)\partial_{\hat t_*}+\hat\fa\partial_{\phi_*}\bigr)\otimes_s\partial_{\hat r} + \frac{1}{\sin^2\theta}\bigl(\partial_{\phi_*}+\hat\fa\,\sin^2\theta\,\partial_{\hat t_*}\bigr)^2\Bigr).
\end{split}
\end{equation}
This is a smooth nondegenerate Lorentzian metric on
\begin{equation}
\label{EqKKerrSpace}
\mathbb{R}_{\hat t_*}\times(\hat r^C,\infty)_{\hat r}\times\mathbb{S}^2_{\theta,\phi_*}.
\end{equation}
\end{subequations}
The wave operators associated with the metrics $g_\fm$, $g_{\mathrm{dS}}$, and $\hat g$ have as principal symbols the respective dual metric functions:
\begin{definition}[Dual metric functions]
\label{DefKDual}
Let $\widetilde X_\fm=\widetilde X_{3,\fm,\hat\fa\fm}$ (see~\eqref{EqIKdSExt}). The \emph{dual metric function} $G_\fm\in\cC^\infty(T^*(\mathbb{R}_{t_*}\times\widetilde X_\fm))$ of $g_\fm$ is defined as
\[
G_\fm(\zeta) = |\zeta|^2_{g_\fm(z)^{-1}},\qquad
z=(t_*,x)\in\mathbb{R}_{t_*}\times\widetilde X_\fm,\quad \zeta\in T^*_z(\mathbb{R}_{t_*}\times\widetilde M_\fm).
\]
The analogously defined dual metric functions of $g_{\mathrm{dS}}$ and $\hat g$ are denoted
\[
G_{\mathrm{dS}}\in\cC^\infty(T^*(\mathbb{R}_{t_*}\times X)),\quad\text{resp.}\quad\hat G\in\cC^\infty\bigl(T^*\bigl(\mathbb{R}_{\hat t_*}\times(\hat r^C,\infty)\times\mathbb{S}^2\bigr)\bigr).
\]
\end{definition}
When $\fm_0>0$ is sufficiently small, then Lemma~\ref{LemmaKRoots} implies that $r_\fm^C < \fm < r_\fm^e < r_\fm^c < 2$ for all $\fm\in(0,\fm_0]$ when $\fm_0>0$ is sufficiently small. Put
\begin{equation}
\label{EqKSpatMfd}
\Omega_\fm := (\fm,2)_r \times \mathbb{S}^2.
\end{equation}
Then, in the notation of~\eqref{EqIKdSDOC} and \eqref{EqIKdSExt}, the manifold $\mathbb{R}_{t_*}\times \overline{\Omega_\fm}\subset\widetilde M_\fm:=\widetilde M_{3,\fm,\hat\fa\fm}$ contains a neighborhood of the closure of $M_{3,\fm,\hat\fa\fm}^{\rm DOC}$.
\subsection{Resonances, multiplicity, and the main theorem}
\label{SsKMain}
We now prepare the precise statement of Theorem~\ref{ThmI}.
\begin{definition}[Spectral family]
\label{DefKFamily}
For $\sigma\in\mathbb{C}$, we define\footnote{The more usual notation would be $\widehat{\Box_{g_\fm}}(\sigma)$. We do not use a hat here, however, to avoid overloading the notation.}
\[
\Box_{g_\fm}(\sigma)\in\mathrm{Diff}^2(\overline{\Omega_\fm})
\]
to be the unique operator with $\Box_{g_\fm}(e^{-i\sigma t_*}u)=e^{-i\sigma t_*}\Box_{g_\fm}(\sigma)u$ for $u\in\cC^\infty_\cp(\Omega_\fm)$. With $\Omega_{\mathrm{dS}}:=B(0,2)\subset X=B(0,3)$, we similarly define the spectral family of $\Box_{g_{\mathrm{dS}}}$, denoted
\[
\Box_{g_{\mathrm{dS}}}(\sigma)\in\mathrm{Diff}^2(\overline{\Omega_{\mathrm{dS}}}),\qquad \sigma\in\mathbb{C}.
\]
We finally denote by
\[
\Box_{\hat g}(\tilde\sigma)\in\mathrm{Diff}^2\bigl([1,\infty)_{\hat r}\times\mathbb{S}^2\bigr),\qquad \tilde\sigma\in\mathbb{C},
\]
the spectral family of $\Box_{\hat g}$, so $\Box_{\hat g}(e^{-i\tilde\sigma\hat t_*}u)=e^{-i\tilde\sigma\hat t_*}\Box_{\hat g}(\tilde\sigma)u$ for $u\in\cC^\infty_\cp((1,\infty)\times\mathbb{S}^2)$.
\end{definition}
Informally, $\Box_{g_\fm}(\sigma)$, $\Box_{\mathrm{dS}}(\sigma)$, resp.\ $\Box_{\hat g}(\tilde\sigma)$ is obtained from $\Box_{g_\fm}$, $\Box_{g_{\mathrm{dS}}}$, resp.\ $\Box_{\hat g}$ by replacing $\partial_{t_*}$, resp.\ $\partial_{\hat t_*}$ by $-i\sigma$, resp.\ $-i\tilde\sigma$. Thus, the spectral families are polynomials (hence holomorphic) in $\sigma$, resp.\ $\tilde\sigma$.
\begin{definition}[Space of resonant states]
\label{DefKRes}
For $\sigma\in\mathbb{R}$, we define $\operatorname{Res}_\fm(\sigma)\subset\cC^\infty(\mathbb{R}_{t_*}\times \overline{\Omega_\fm})$ as the space of all \emph{generalized resonant states} $u=u(t_*,x)$ of $\Box_{g_\fm}$ at frequency $\sigma$, i.e.\ solutions $u$ of $\Box_{g_\fm}u=0$ which for some $n\in\mathbb{N}_0$ can be written as $u=\sum_{k=0}^n t_*^k e^{-i\sigma t_*}u_k(x)$ where $u_k\in\cC^\infty(\overline{\Omega_\fm})$. The \emph{multiplicity} of $\sigma$ is
\[
m_\fm(\sigma) := \dim\operatorname{Res}_\fm(\sigma).
\]
We similarly define $\operatorname{Res}_{\mathrm{dS}}(\sigma)\subset\cC^\infty(\mathbb{R}_{t_*}\times\overline{\Omega_{\mathrm{dS}}})$ and $m_{\mathrm{dS}}(\sigma)$ with respect to $\Box_{g_{\mathrm{dS}}}$.
\end{definition}
Thus, $\sigma\in{\mathrm{QNM}}(\fm):={\mathrm{QNM}}(3,\fm,\hat\fa\fm)$ if and only if $\operatorname{Res}_\fm(\sigma)\neq\{0\}$, i.e.\ $m_\fm(\sigma)\neq 0$.\footnote{Note that the existence of a smooth resonant state is independent of the choice of the function $F_\fm$ in~\eqref{EqKFm} or $F_{\Lambda,\fm,\fa}$ in~\eqref{EqIKerrStar}, as long as these functions equal $-1$, resp.\ $+1$ at the event, resp.\ cosmological horizon.} For sufficiently small $\fm$, the Fredholm theory of \cite[\S6]{VasyMicroKerrdS} can be shown to apply to $\Box_{g_\fm}(\sigma)$ (see also~\eqref{EqKBdMap} below), and thus $\Box_{g_\fm}(\sigma)^{-1}$ is a meromorphic family of operators on $\cC^\infty(\overline{\Omega_\fm})$. As shown in \cite[\S5.1.1]{HintzVasyKdSStability}, an equivalent definition of $\operatorname{Res}_\fm(\sigma)$ is then
\begin{equation}
\label{EqKResEquiv}
\operatorname{Res}_\fm(\sigma) = \biggl\{ {\mathrm{res}}_{\zeta=\sigma}\Bigl( e^{-i\zeta t_*}\Box_{g_\fm}(\zeta)^{-1}p(\zeta)\Bigr) \colon p(\zeta)\ \text{is a polynomial with values in}\ \cC^\infty(\overline{\Omega_\fm}) \biggr\},
\end{equation}
and the multiplicity can be computed via
\begin{equation}
\label{EqKResMult}
m_\fm(\sigma) = \frac{1}{2\pi i}\operatorname{tr}\oint_\sigma\Box_{g_\fm}(\zeta)^{-1}\partial_\zeta\Box_{g_\fm}(\zeta)\,{\mathrm d}\zeta,
\end{equation}
where $\oint_\sigma$ is the contour integral over a circle enclosing $\sigma$ counterclockwise which contains no resonances other than $\sigma$. (The integral is a finite rank operator on $\cC^\infty(\overline{\Omega_\fm})$, and hence its trace is well-defined.) There are analogous expressions for $\operatorname{Res}_{\mathrm{dS}}(\sigma)$ and $m_{\mathrm{dS}}(\sigma)$.
\begin{definition}[Quasinormal modes with multiplicity]
\label{DefKQNM}
For $\fm\in(0,\fm_0]$, we put
\begin{alignat*}{3}
{\mathrm{QNM}}^*(\fm) &:= \bigl\{ (\sigma,m_\fm(\sigma))\in\mathbb{C}\times\mathbb{N} \colon &&m_\fm(\sigma)\geq 1 \bigr\} &&\subset \mathbb{C}\times\mathbb{N}, \\
{\mathrm{QNM}}_{\mathrm{dS}}^* &:= \bigl\{ (\sigma,m_{\mathrm{dS}}(\sigma))\in\mathbb{C}\times\mathbb{N} \colon &&m_{\mathrm{dS}}(\sigma)\geq 1 \bigr\} &&\subset \mathbb{C}\times\mathbb{N}.
\end{alignat*}
Furthermore, ${\mathrm{QNM}}(\fm)={\mathrm{QNM}}(3,\fm,\hat\fa\fm)$ is the projection of ${\mathrm{QNM}}^*(\fm)$ to the first factor, and ${\mathrm{QNM}}_{\mathrm{dS}}$ is the projection of ${\mathrm{QNM}}^*_{\mathrm{dS}}$ to the first factor.
\end{definition}
\begin{lemma}[QNMs of de~Sitter space]
\label{LemmaKdSQNM}
We have ${\mathrm{QNM}}_{\mathrm{dS}}=-i\mathbb{N}_0$, and
\[
{\mathrm{QNM}}_{\mathrm{dS}}^* = \{ (-i\ell, m) \colon \ell\in\mathbb{N}_0,\ m=m_{\mathrm{dS}}(-i\ell) \}
\]
where
\begin{equation}
\label{EqKdSQNM}
m_{\mathrm{dS}}(-i\ell) = \begin{cases} 1, & \ell=0, \\ \ell^2+2, & \ell\geq 1. \end{cases}
\end{equation}
\end{lemma}
\begin{proof}
This follows from \cite[Proposition~2.1]{HintzXieSdS} upon setting $\nu=0$, thus $\lambda_-(\nu)=0$ and $\lambda_+(\nu)=3$ in the notation of the reference. Indeed, for $l\in\mathbb{N}_0$, the space of generalized resonant states with angular dependence given by a degree $l$ spherical harmonic is non-trivial exactly at all spectral parameters $-i\ell$ for $\ell\in(l+2\mathbb{N}_0)\cup(3+l+2\mathbb{N}_0)$, and at each such resonance has dimension $2 l+1$. This gives
\[
m_{\mathrm{dS}}(-i\ell) = \sum_{\genfrac{}{}{0pt}{}{l\in\mathbb{N}_0}{\ell-l\in(2\mathbb{N}_0)\cup(3+2\mathbb{N}_0)}} (2 l+1) = (2\ell+1) + \sum_{k=0}^{\ell-2} (2 k+1).
\]
For $\ell=0$, resp.\ $1$, this evaluates to $1$, resp.\ $3=1^2+2$. For $\ell\geq 2$ the second sum is $(\ell-1)^2=\ell^2-2\ell+1$. This gives~\eqref{EqKdSQNM}.
\end{proof}
\begin{thm}[Quasinormal modes of KdS black holes away from extremality: detailed version]
\label{ThmK}
Let $C_1>0$ be such that $\Im\sigma\neq-C_1$ for all $\sigma\in{\mathrm{QNM}}_{\mathrm{dS}}$. Let $\epsilon>0$ be such that for each $\sigma_*\in{\mathrm{QNM}}_{\mathrm{dS}}$ with $\Im\sigma_*\geq-C_1$, the only $\sigma\in{\mathrm{QNM}}_{\mathrm{dS}}$ with $|\sigma-\sigma_*|\leq 2\epsilon$ is $\sigma_*$ itself.\footnote{Thus, one can take any $\epsilon<{\tfrac{1}{2}}$. The present formulation generalizes without change to the case of the Klein--Gordon equation.} Then there exists $\fm_1>0$ so that the following statements hold.
\begin{enumerate}
\item\label{ItK1} If $\fm\in(0,\fm_1]$ and $\sigma\in{\mathrm{QNM}}(\fm)$, $\Im\sigma\geq-C_1$, then there exists $\sigma_*\in{\mathrm{QNM}}_{\mathrm{dS}}$ so that $|\sigma-\sigma_*|\leq\epsilon$.
\item\label{ItK2} The total multiplicity of QNMs near $\sigma_*\in{\mathrm{QNM}}_{\mathrm{dS}}$ with $\Im\sigma_*\geq-C_1$ is independent of $\fm$, that is,
\[
m_{\mathrm{dS}}(\sigma_*) = \sum_{\genfrac{}{}{0pt}{}{\sigma\in{\mathrm{QNM}}(\fm)}{|\sigma-\sigma_*|\leq\epsilon}} m_\fm(\sigma),\qquad \fm\in(0,\fm_1].
\]
\item\label{ItK0} The only resonance $\sigma\in{\mathrm{QNM}}(\fm)$ with $|\sigma|\leq\epsilon$ is $\sigma=0$, with $m_\fm(0)=1$, and $\operatorname{Res}_\fm(0)$ consists of all constant functions on $\mathbb{R}_{t_*}\times\overline{\Omega_\fm}$.
\item\label{ItK3} Let $K=[r_0,2]\times\mathbb{S}^2$, and let $\sigma_*\in{\mathrm{QNM}}_{\mathrm{dS}}$ with $\Im\sigma_*\geq-C_1$. Then for all sufficiently small $r_0>0$, the space
\begin{equation}
\label{EqK3}
\biggl\{ u|_{[0,1]_{t_*}\times K} \colon u \in \sum_{\genfrac{}{}{0pt}{}{\sigma\in{\mathrm{QNM}}(\fm)}{|\sigma-\sigma_*|\leq\epsilon}} \operatorname{Res}_\fm(\sigma) \biggr\}
\end{equation}
has dimension $m_{\mathrm{dS}}(\sigma_*)$ and converges to $\{u|_{[0,1]\times K}\colon u\in\operatorname{Res}_{\mathrm{dS}}(\sigma_*)\}$ in the topology of $\cC^\infty([0,1]\times K)$. (That is, there exists an $\fm$-dependent basis $u_{\fm,1},\ldots,u_{\fm,m_{\mathrm{dS}}(\sigma_*)}$ of the space~\eqref{EqK3} which converges in $\cC^\infty([0,1]\times K)$ to a basis of $\operatorname{Res}_{\mathrm{dS}}(\sigma_*)|_{[0,1]\times K}$.)
\end{enumerate}
\end{thm}
Parts~\eqref{ItK1} and \eqref{ItK2} together give precise meaning to the statement that the quasinormal modes of Kerr--de~Sitter space with parameters $(\Lambda,\fm,\fa)=(3,\fm,\hat\fa\fm)$ converge \emph{with multiplicity} to those of de~Sitter space in any half space $\Im\sigma\geq -C_1$ as $\fm\searrow 0$.
\subsection{The spectral family as a Q-differential operator}
\label{SsKS}
As the starting point for the proof of Theorem~\ref{ThmK}, we now place $\Box_{g_\fm}(\sigma)$ into the context of q- and Q-analysis. We use the terminology of~\S\ref{SQ}, with two small modifications: \begin{enumerate*} \item the mass $\fm$ will be restricted to a short interval $[0,\fm_0]$ (rather than $[0,1]$) where $\fm_0>0$ is chosen according to the requirement stated before~\eqref{EqKSpatMfd}; and \item we shall write $\sigma_0$ for the real parameter that was previously denoted $\sigma$ in~\S\S\ref{SsQP}--\ref{SsQH}.\end{enumerate*} We reserve the symbol $\sigma$ for the spectral parameter (which might be complex).
Let $X$ denote a 3-dimensional torus; we work in a local coordinate chart $B(0,3)$ near a point $0\in X$ as in~\eqref{EqKdSSpace}. (We make $X$ compact merely so that Sobolev spaces are well-defined.) At fixed (or more generally for bounded) frequencies $\sigma\in\mathbb{C}$, our analysis will take place in the domain
\begin{subequations}
\begin{equation}
\label{EqKSOmegaq}
\Omega_{\mathrm{q}} := \{ \hat r>1,\ r<2 \} \cap X_{\mathrm{q}}.
\end{equation}
Thus, $\Omega_{\mathrm{q}}$ resolves $\bigsqcup_{\fm\in(0,\fm_0]}\{\fm\}\times\Omega_\fm$ in the singular limit $\fm\searrow 0$, and we have
\begin{equation}
\label{EqKSOmegahat}
\hat\Omega := \Omega_{\mathrm{q}}\cap\,{\mathrm{zf}}_{\mathrm{q}} = \{ \hat r>1 \} \cap {\mathrm{zf}}_{\mathrm{q}},\qquad
\dot\Omega := \Omega_{\mathrm{q}}\cap\,{\mathrm{mf}}_{\mathrm{q}} = [0,2)_r\times\mathbb{S}^2.
\end{equation}
(Here, $\hat\Omega$ is a subset of the spatial manifold in~\eqref{EqKKerrSpace}; the radius $1$ is chosen for notational convenience.) We denote by $\overline{\Omega_{\mathrm{q}}}=\{\hat r\geq 1,\ r\leq 2\}\cap X_{\mathrm{q}}$ and $\overline{\hat\Omega}=\{\hat r\geq 1\}\cap\,{\mathrm{zf}}_{\mathrm{q}}$ the closures of $\Omega_{\mathrm{q}}$ and $\hat\Omega$ inside $X_{\mathrm{q}}$. See Figure~\ref{FigKSOmega}.
\begin{figure}[!ht]
\centering
\includegraphics{FigKSOmega}
\caption{The domains $\Omega_{\mathrm{q}}$, $\hat\Omega$, and $\dot\Omega\subset X_{\mathrm{q}}$ defined in~\eqref{EqKSOmegaq} and~\eqref{EqKSOmegahat}, without the factor $\mathbb{S}^2$.}
\label{FigKSOmega}
\end{figure}
On the Q-single space $X_{\mathrm{Q}}$, we shall work on the lift of $\overline{\mathbb{R}_{\sigma_0}}\times\Omega_{\mathrm{q}}$,
\begin{equation}
\label{EqKSOmegaQ}
\Omega_{\mathrm{Q}} := \{ \hat r>1,\ r<2 \} \cap X_{\mathrm{Q}}.
\end{equation}
\end{subequations}
We need to analyze also non-real frequencies $\sigma$. For now, we work in strips $\{\Im\sigma\leq C_1\}$ for arbitrary fixed $C_1>0$, and the total space of our analysis is therefore
\[
[-C_1,C_1]\times\Omega_{\mathrm{Q}} \subset [-C_1,C_1]\times X_{\mathrm{Q}}.
\]
(The modifications needed to treat all of $\{\Im\sigma\geq-C_1\}$ will be discussed in~\S\ref{SsKU}.) The total spectral family $(\fm,\sigma)\mapsto\Box_{g_\fm}(\sigma)$, where $\fm\in(0,\fm_0]$ and $\sigma=\sigma_0+i\sigma_1$ with $\sigma_0\in\mathbb{R}$, $\sigma_1\in[-C_1,C_1]$, defines an element
\[
\Box(\cdot+i\sigma_1)\in\mathrm{Diff}^2(\overline{\Omega_{\mathrm{Q}}}\cap\{\fm>0\}),
\]
with smooth dependence on $\sigma_1$. The following key result puts the total spectral family into the Q-analytic framework developed in~\S\ref{SQ}, and is indeed the motivation for the development of this framework.
\begin{prop}[Properties of the total spectral family]
\label{PropKS}
The total spectral family $\Box(\cdot+i\sigma_1)$ satisfies
\begin{equation}
\label{EqKSCat}
\Box(\cdot+i\sigma_1) \in \Diff_\Qop^{2,(2,0,2,2,2)}(\overline{\Omega_{\mathrm{Q}}}) = \rho_{\mathrm{zf}}^{-2}\rho_{\mathrm{mf}}^0\rho_{\mathrm{nf}}^{-2}\rho_{\mathrm{sf}}^{-2}\rho_{\mathrm{if}}^{-2}\Diff_\Qop^2(\overline{\Omega_{\mathrm{Q}}}),
\end{equation}
and depends smoothly on $\sigma_1\in[-C_1,C_1]$. Moreover, in the notation of Corollary~\usref{CorQPNormal}:
\begin{enumerate}
\item\label{ItKSSymb} The Q-principal symbol of $\Box(\cdot+i\sigma_1)$ is $G(\cdot+i\sigma_1,-;-,-)$, given by
\begin{equation}
\label{EqKSSymb}
G\colon(\sigma,\fm;x,\xi)\mapsto G_\fm|_x(-\sigma\,{\mathrm d} t_*+\xi),
\end{equation}
where $x\in\overline{\Omega_\fm}$, $\xi\in T^*_x\overline{\Omega_\fm}$, and $\sigma=\sigma_0+i\sigma_1$, in the sense that ${}^\Qop\upsigma^{2,(2,0,2,2,2)}(\Box(\cdot+i\sigma_1))$ is given by the equivalence class of $G(\cdot+i\sigma_1)$ in $(S^{2,(2,0,2,2,2)}/S^{1,(2,0,2,1,1)})({}^\Qop T_{\overline{\Omega_{\mathrm{Q}}}}^*X)$.
\item\label{ItKSzf} We have $N_{\mathrm{zf}}(\fm^2\Box(\cdot+i\sigma_1))=\Box_{\hat g}(0)$ (regarded as a $\sigma_0$-independent operator on $\overline{\mathbb{R}_{\sigma_0}}\times\overline{\hat\Omega}\subset{\mathrm{zf}}$, cf.\ Proposition~\usref{PropQStruct}\eqref{ItQStructzf}).
\item\label{ItKSnf} For $\tilde\sigma_0\in\mathbb{R}\setminus\{0\}$, we have $N_{{\mathrm{nf}}_{\tilde\sigma_0}}(\fm^2\Box(\cdot+i\sigma_1))=\Box_{\hat g}(\tilde\sigma_0)$.
\item\label{ItKSmf} For $\sigma_0\in\mathbb{R}$, we have $N_{{\mathrm{mf}}_{\sigma_0}}(\Box(\cdot+i\sigma_1))=\Box_{g_{\mathrm{dS}}}(\sigma)$, where $\sigma=\sigma_0+i\sigma_1$.
\end{enumerate}
\end{prop}
One can prove this by direct calculation using the form~\eqref{EqKExt} of the KdS metric. We instead give a conceptual proof, which highlights the relevant structural properties of the family of metrics $g_\fm$.\footnote{This route is longer, but it has the advantage of allowing for straightforward generalizations of Proposition~\ref{PropKS} to spectral families of other geometric operators---even if in the present paper we do not discuss such generalizations.} To begin with, we define
\[
M:=\mathbb{R}_{t_*}\times X,\qquad
\dot M:=\mathbb{R}_{t_*}\times\dot X,\qquad
\hat M:=\mathbb{R}_{\hat t_*}\times\hat X,
\]
and identify $X$ with $\{0\}\times X\subset M$, likewise $\dot X\subset\dot M$ and $\hat X\subset\hat M$. Smooth stationary metrics on $M$ can be identified with smooth sections of $S^2 T^*_X M\to X$, likewise for $\dot M$, $\hat M$.
Denote now by
\[
\pi_{\mathrm{q}}\colon X_{\mathrm{q}}\to X,\qquad
\pi_{\mathrm{Q}}\colon X_{\mathrm{Q}}\to X
\]
the lifts of the projection maps $[0,\fm_0]\times X\ni(\fm,x)\mapsto x\in X$ and $\overline\mathbb{R}\times[0,\fm_0]\times X\to X$, respectively. The pullback bundle $\pi_{\mathrm{q}}^*T_X M\to\Omega_{\mathrm{q}}$ will play two roles. Firstly, it is a bundle in (the tensor powers of) which geometric objects are valued (see Lemma~\ref{LemmaKMetric} below). Secondly, in $\fm>0$ its sections are smooth families of horizontal vector fields; in this latter regard, we note:
\begin{lemma}[Bundle isomorphisms]
\label{LemmaKBundleIso}
Let $\dot\beta\colon{\mathrm{mf}}_{\mathrm{q}}=\dot X=[X;\{0\}]\to X$ denote the blow-down map. Then the identity map $(\pi_{\mathrm{q}}^* T_X M)|_{(0,x)}=T_x M=T_x\dot M$ for $x\in X\setminus\{0\}$ extends to a bundle isomorphism
\begin{equation}
\label{EqKBundleIsomf}
(\pi_{\mathrm{q}}^*T_X M)|_{{\mathrm{mf}}_{\mathrm{q}}} = \dot\beta^* T_{\dot X}\dot M.
\end{equation}
Moreover, the map $(\pi_{\mathrm{q}}^* T X)|_{(\fm,x)}=T_x X\ni V\mapsto\fm V\in{}^\qop T_{(\fm,x)}X$ (for $\fm\in(0,\fm_0]$) extends by continuity to a smooth bundle map on $X_{\mathrm{q}}$ and then restricts to ${\mathrm{zf}}_{\mathrm{q}}=\hat X$ as an isomorphism
\begin{equation}
\label{EqKBundleIsozf}
\iota \colon (\pi_{\mathrm{q}}^*T X)|_{{\mathrm{zf}}_{\mathrm{q}}} \cong {}^{\scop}T\hat X\qquad \text{(via `multiplication by $\fm$')}.
\end{equation}
\end{lemma}
\begin{proof}
For~\eqref{EqKBundleIsomf}, simply note that both bundles have, as smooth frames, the vector fields $\partial_{t_*}$ and $\partial_{x^j}$ ($j=1,2,3$). For~\eqref{EqKBundleIsozf}, note that $\fm\partial_{x^j}=\partial_{\hat x^j}$ ($j=1,2,3$) is a frame of ${}^{\scop}T\hat X$.
\end{proof}
We shall also write $\iota$ for tensor powers of the isomorphism~\eqref{EqKBundleIsozf} or its adjoint. Writing $\ul\mathbb{R}=\hat X\times\mathbb{R}$ for the trivial bundle, we furthermore define the map
\begin{equation}
\label{EqKBundleSpacetime}
\widetilde\iota \colon (\pi_{\mathrm{q}}^* T_X M)|_{{\mathrm{zf}}_{\mathrm{q}}} \xrightarrow{\cong} T_0\mathbb{R}_{\hat t_*}\oplus{}^{\scop}T\hat X,\qquad \partial_{t_*}\mapsto\partial_{\hat t_*},\quad V\mapsto\iota(V).
\end{equation}
(This is `multiplication by $\fm$' for tangent vectors on the spacetime $M$.) Tensor powers of $\widetilde\iota$ or its adjoint are denoted by the same symbol.
\begin{lemma}[The family $g_\fm$ on the q-single space]
\label{LemmaKMetric}
For $\fm\in(0,\fm_0]$ and $x\in\overline{\Omega_\fm}$, define the symmetric 2-tensor $g(\fm,x)\in(\pi_{\mathrm{q}}^* S^2 T^*_X M)|_{(\fm,x)}=S^2 T^*_x M$ to be equal to $g_\fm|_x$. Then
\[
g \in \cC^\infty\bigl(\overline{\Omega_{\mathrm{q}}}; (\pi_{\mathrm{q}}^*S^2 T^*_X M)|_{\overline{\Omega_{\mathrm{q}}}}\bigr),\qquad
g^{-1} \in \cC^\infty\bigl(\overline{\Omega_{\mathrm{q}}}; (\pi_{\mathrm{q}}^*S^2 T_X M)|_{\overline{\Omega_{\mathrm{q}}}}\bigr).
\]
Moreover, $g|_{{\mathrm{mf}}_{\mathrm{q}}}=g_{\mathrm{dS}}$ (under the identification~\eqref{EqKBundleIsomf}), and $\widetilde\iota^{-1}(g|_{{\mathrm{zf}}_{\mathrm{q}}})=\hat g$. Furthermore, $|{\mathrm d} g|=|{\mathrm d} t_*||{\mathrm d} g_X|$ where $|{\mathrm d} g_X|\in\cC^\infty(\overline{\Omega_{\mathrm{q}}};(\pi_{\mathrm{q}}^*\Omega_X M)|_{\overline{\Omega_{\mathrm{q}}}})$. (Explicitly, we have $|{\mathrm d} g_X|=b^2\varrho^2\sin\theta\,|{\mathrm d} r\,{\mathrm d}\theta\,{\mathrm d}\phi_*|$ where $b(\fm)=b_\fm$ and $\varrho(\fm,r,\theta)=\varrho_\fm(r,\theta)$.)
\end{lemma}
\begin{proof}
On $\overline{\Omega_{\mathrm{q}}}$, the 1-forms ${\mathrm d} t_*$, ${\mathrm d} r$, $r\,{\mathrm d}\theta$, and $r\,{\mathrm d}\phi_*$ are smooth and nonzero sections of $\pi_{\mathrm{q}}^*T^*_X M$. It thus suffices to show that the coefficients of $g_\fm$ in~\eqref{EqKExt} (expressed in terms of symmetric tensor products of these 1-forms) are elements of $\cC^\infty(\overline{\Omega_{\mathrm{q}}})$. On $\overline{\Omega_{\mathrm{q}}}$, smooth coordinates are given by $\hat\rho=\frac{\fm}{r}\in[0,1]$, $r\geq 0$, and $\theta,\phi_*$, and we then note that
\[
\frac{\mu_\fm(r)}{b_\fm^2\varrho_\fm^2(r,\theta)} = \frac{(1+\hat\fa^2\hat\rho^2)(1-r^2)-2\hat\rho}{(1+\hat\fa^2\hat\rho^2 r^2)(1+\hat\fa^2\hat\rho^2\cos^2\theta)}
\]
is indeed smooth in these coordinates, similarly for the other coefficients of $g_\fm$; note in particular that $F_\fm=-\chi^e(\hat\rho^{-1})+\chi^c(r)$ is smooth. The membership of $g^{-1}$ follows similarly by inspection of the coefficients of $g_\fm^{-1}$ in~\eqref{EqKExtDual} in the basis $\partial_{t_*}$, $\partial_r$, $r^{-1}\partial_\theta$, $r^{-1}\partial_{\phi_*}$.
The computation of $g|_{{\mathrm{mf}}_{\mathrm{q}}}$ was already performed in~\eqref{EqKdS}. The computation of $\iota^{-1}(g|_{{\mathrm{zf}}_{\mathrm{q}}})$ amounts to taking the limit of $\fm^{-2}g_\fm$ as $\fm\searrow 0$ for bounded $\hat r=|\hat x|$, which was done in~\eqref{EqKKerr}.
\end{proof}
We can now give a simple proof of Lemma~\ref{LemmaKdSChi}:
\begin{proof}[Proof of Lemma~\usref{LemmaKdSChi}]
Using~\eqref{EqKdS} and writing $\tilde\chi^c(r)=1+(1-r^2)f(r)$, we compute
\[
|{\mathrm d} t_*|_{g_{\mathrm{dS}}^{-1}}^2 = -\frac{1-(1+(1-r^2)f(r))^2}{1-r^2} = 2 f(r) + (1-r^2)f(r)^2.
\]
Note that in any region $r\leq r_1<1$, this is negative for $f(r)=-\frac{1}{1-r^2}$ (in which case $\tilde\chi^c=0$). More generally, in $r<1$, resp.\ at $r=1$, we have $|{\mathrm d} t_*|_{g_{\mathrm{dS}}^{-1}}^2<0$ provided $-\frac{2}{1-r^2}<f(r)<0$, resp.\ $f(1)<0$. For $1<r\leq 3$, it is enough to ensure $f(r)<0$. We can thus use a partition of unity to construct a smooth $f$ so that ${\mathrm d} t_*$ is past timelike for $g_{\mathrm{dS}}$, and so that $\tilde\chi^c(r)=0$ for $r\leq{\tfrac{1}{2}}$.
Next, using~\eqref{EqKKerrDual} and writing $\tilde\chi^e(\hat r)=1+\hat\mu(\hat r)\hat f(\hat r)$, we compute
\begin{align*}
\hat\varrho(\hat r,\theta)^2|{\mathrm d}\hat t_*|_{\hat g^{-1}}^2 &= -\frac{1-(1+\hat\mu(\hat r)\hat f(\hat r))^2}{\hat\mu(\hat r)}(\hat r^2+\hat\fa^2)^2 + \hat\fa^2\sin^2\theta \\
&\leq \hat\mu(\hat r)(\hat r^2+\hat\fa^2)^2\hat f(\hat r)^2 + 2(\hat r^2+\hat\fa^2)^2\hat f(\hat r) + \hat\fa^2.
\end{align*}
When $\hat f(\hat r)=-\hat\mu(\hat r)^{-1}$ (so $\tilde\chi^e=0$), the right hand side evaluates to $-\frac{(\hat r^2+\hat\fa^2)^2}{\hat\mu(\hat r)}+\hat\fa^2$ which is negative when $\hat\mu(\hat r)>0$ (since upon multiplication by $\hat\mu(\hat r)$, this is $-(\hat r^2+\hat\fa^2)^2+\hat\fa^2(\hat r^2+\hat\fa^2-2\hat r)=-\hat r^4-\hat\fa^2\hat r^2-2\hat\fa^2\hat r$). At $\hat r=\hat r^e$, we require $\hat f(\hat r)<\frac{\hat\fa^2}{2(\hat r^2+\hat\fa^2)^2}$. Where $\hat\mu(\hat r)\neq 0$, the set of allowed values of $\hat f(\hat r)$ is a nonempty open interval (depending continuously on $\hat r$). We can thus find an appropriate $\hat f(\hat r)$ so that, moreover, $\tilde\chi^e(\hat r)=0$ for $\hat r\geq 3$, say.
Having fixed $\tilde\chi^c,\tilde\chi^e$ and thus $\chi^c,\chi^e$ in~\eqref{EqKFm} in this manner, the past timelike nature of ${\mathrm d} t_*$ with respect to $g_\fm$ now follows by continuity for all sufficiently small $\fm>0$ in view of Lemma~\ref{LemmaKMetric}.
\end{proof}
\begin{prop}[Spectral family of the connection of $g$]
\label{PropKSpectral}
For $\fm\in(0,\fm_0]$ and $\sigma\in\mathbb{C}$, denote by $\nabla^{g_\fm}(\sigma)\in\mathrm{Diff}^1\bigl(\overline{\Omega_\fm};T_{\overline{\Omega_\fm}}M,T_{\overline{\Omega_\fm}}^*M\otimes T_{\overline{\Omega_\fm}}M\bigr)$, $\sigma\in\mathbb{C}$, the spectral family of the Levi-Civita connection of $g_\fm$, defined analogously to Definition~\usref{DefKFamily}. Denote by $\nabla^g(\cdot+i\sigma_1)\colon(0,\fm_0]\times\mathbb{R}\ni(\fm,\sigma_0)\mapsto\nabla^{g_\fm}(\sigma_0+i\sigma_1)$ the total spectral family. Then
\begin{equation}
\label{EqKSpectralSymb}
\nabla^g(\cdot+i\sigma_1) \in \Diff_\Qop^{1,(1,0,1,1,1)}\bigl(\overline{\Omega_{\mathrm{Q}}};(\pi_{\mathrm{Q}}^*T_X M)|_{\overline{\Omega_{\mathrm{Q}}}},(\pi_{\mathrm{Q}}^*(T_X^*M\otimes T_X M))|_{\overline{\Omega_{\mathrm{Q}}}}\bigr).
\end{equation}
Its principal symbol is ${}^\Qop\upsigma^{1,(1,0,1,1,1)}(\nabla^g(\cdot+i\sigma_1))(\sigma_0,\fm,x,\xi)=(-\sigma\,{\mathrm d} t_*+\xi)\otimes(-)$,\footnote{It is irrelevant here which (rescaled) cotangent bundle $\xi$ lies in. For example, if we take $\xi\in{}^\Qop T^*X$ of unit size (with respect to any fixed positive definite fiber metric), then $\xi$ has size $(\rho_{\mathrm{zf}}\rho_{\mathrm{nf}}\rho_{\mathrm{if}}\rho_{\mathrm{sf}})^{-1}$ as an element of $\pi_{\mathrm{Q}}^*(T^*_X M)$.} where $\sigma=\sigma_0+i\sigma_1$.\footnote{The term $-\sigma\,{\mathrm d} t_*$ only contributes to the principal symbol in the high frequency regime $|\Re\sigma|=|\sigma_0|\gg 1$, where in view of the boundedness of $\sigma_1$ the contribution of $-i\sigma_1\,{\mathrm d} t_*$ is subprincipal and therefore does, in fact, not contribute to the principal symbol. When relaxing the assumption that $\Im\sigma$ be bounded, the imaginary part of $\sigma$ does matter, however; see~\S\ref{SsKU}.} Moreover,
\begin{subequations}
\begin{align}
\label{EqKSpectralzf}
N_{\mathrm{zf}}\bigl(\fm\nabla^g(\cdot+i\sigma_1)\bigr) &= \nabla^{\hat g}(0), \\
\label{EqKSpectralnf}
N_{{\mathrm{nf}}_{\tilde\sigma}}\bigl(\fm\nabla^g(\cdot+i\sigma_1)\bigr) &= \nabla^{\hat g}(\tilde\sigma), \\
\label{EqKSpectralmf}
N_{{\mathrm{mf}}_{\sigma_0}}\bigl(\nabla^g(\cdot+i\sigma_1)\bigr) &= \nabla^{g_{\mathrm{dS}}}(\sigma),\quad \sigma=\sigma_0+i\sigma_1,
\end{align}
\end{subequations}
where we use the isomorphism $\widetilde\iota$ from~\eqref{EqKBundleSpacetime} in the first two lines (to identify the bundles $(\pi_{\mathrm{Q}}^*T_X M)|_{{\mathrm{zf}}_{\sigma_0}}=(\pi_{\mathrm{Q}}^*T_X M)|_{{\mathrm{nf}}_{\tilde\sigma_0}}=(\pi_{\mathrm{q}}^*T_X M)|_{{\mathrm{zf}}_{\mathrm{q}}}$ with $T_0\mathbb{R}_{\hat t_*}\oplus{}^{\scop}T\hat X$, likewise for their duals), and the identification~\eqref{EqKBundleIsomf} in the third line.
Analogous statements hold for the spectral family of the exterior derivative ${\mathrm d}$, resp.\ the gradient $\nabla^g$ (with ${\mathrm d}(\cdot+i\sigma_1)$ a map from complex-valued functions to sections of $\pi_{\mathrm{Q}}^* T_X^* M$, resp.\ $\pi_{\mathrm{Q}}^*T_X M$, over $\overline{\Omega_{\mathrm{Q}}}$); see~\eqref{EqKSpectralD0}--\eqref{EqKSpectralD2} below for the case of ${\mathrm d}$.
\end{prop}
\begin{proof}
Consider first the exterior derivative ${\mathrm d} u=(\partial_{t_*}u){\mathrm d} t_*+{\mathrm d}_X u$, where, with $\ul\mathbb{R}=X\times\mathbb{R}$ denoting the trivial bundle, ${\mathrm d}_X\in\mathrm{Diff}^1(X;\ul\mathbb{R},T^*X)$ is the spatial exterior derivative. From~\eqref{EqQDiffEx}--\eqref{EqQDiffEx2} we then deduce that
\begin{align*}
{\mathrm d}(\sigma)=-i\sigma\,{\mathrm d} t_*+{\mathrm d}_X &\in \Diff_\Qop^{0,(0,0,1,1,1)}(X;\ul\mathbb{R},\pi_{\mathrm{Q}}^*T^*_X M) + \Diff_\Qop^{1,(1,0,1,1,1)}(X;\ul\mathbb{R},\pi_{\mathrm{Q}}^*T^*_X M) \\
&= \Diff_\Qop^{1,(1,0,1,1,1)}(X;\ul\mathbb{R},\pi_{\mathrm{Q}}^*T^*_X M),
\end{align*}
now with $\ul\mathbb{R}=X_{\mathrm{Q}}\times\mathbb{R}$. This explicit expression implies
\begin{subequations}
\begin{equation}
\label{EqKSpectralD0}
N_{{\mathrm{mf}}_{\sigma_0}}({\mathrm d}(\cdot+i\sigma_1))=\dot{\mathrm d}(\sigma)
\end{equation}
where $\dot{\mathrm d}$ is the exterior derivative operator on $\dot M=\mathbb{R}_{t_*}\times\dot X$. The principal symbol at $(\sigma_0,\fm,x,\xi)$ is $\xi$. Considering the rescaling $\fm{\mathrm d}(\sigma)=-i\fm\sigma\,{\mathrm d} t_*+\fm{\mathrm d}_X$, note that $\iota({\mathrm d}_X u)=\sum_{j=1}^3(\partial_{x^j}u){\mathrm d}\hat x^j$ and $\fm\partial_{x^j}=\partial_{\hat x^j}$, and therefore
\begin{equation}
\label{EqKSpectralD1}
N_{\mathrm{zf}}(\fm{\mathrm d}(\cdot+i\sigma_1))=\hat{\mathrm d}(0),
\end{equation}
with $\hat{\mathrm d}$ the exterior derivative on $\hat M=\mathbb{R}_{\hat t_*}\times\hat X$. When $\sigma=\tilde\sigma/\fm$, then $\fm{\mathrm d}(\sigma)=-i\tilde\sigma\,{\mathrm d} t_*+\fm{\mathrm d}_X$; thus,
\begin{equation}
\label{EqKSpectralD2}
N_{{\mathrm{nf}}_{\tilde\sigma_0}}(\fm{\mathrm d}(\cdot+i\sigma_1)) = \hat{\mathrm d}(\tilde\sigma).
\end{equation}
\end{subequations}
The analogous statements about the gradient $\nabla^g$ on functions follow from~\eqref{EqKSpectralD0}--\eqref{EqKSpectralD2} and the description of $g^{-1}$ in Lemma~\ref{LemmaKMetric}.
The analysis of the Levi-Civita connection $\nabla^{g_\fm}$ is similar. In terms of local coordinates $x=(x^1,x^2,x^3)$ on $X$ and the corresponding coordinates $(t_*,x^1,x^2,x^3)$ on $M$, the Christoffel symbols $\Gamma_{\mu\nu}^\lambda(g_\fm)$ satisfy
\[
\Gamma_{\mu\nu}^\lambda(g_\fm) = {\tfrac{1}{2}}(g_\fm^{-1})^{\lambda\kappa}\bigl(\partial_\mu(g_\fm)_{\nu\kappa} + \partial_\nu(g_\fm)_{\mu\kappa} - \partial_\kappa(g_\fm)_{\mu\nu}\bigr) \in \rho_{{\mathrm{zf}}_{\mathrm{q}}}^{-1}\cC^\infty(\overline{\Omega_{\mathrm{q}}}) \subset \rho_{\mathrm{zf}}^{-1}\rho_{\mathrm{nf}}^{-1}\cC^\infty(\overline{\Omega_{\mathrm{Q}}})
\]
in view of~\eqref{EqqVFX} and Lemma~\ref{LemmaKMetric}. Now, $\nabla^{g_\fm}(u^\mu\partial_\mu)=(\partial_\nu u^\mu){\mathrm d} x^\nu\otimes\partial_\mu+u^\mu\Gamma_{\mu\nu}^\lambda{\mathrm d} x^\nu\otimes\partial_\lambda$. Passing to the spectral family amounts to replacing $\partial_0$ by $-i\sigma$, and we therefore obtain~\eqref{EqKSpectralSymb} as in the discussion of ${\mathrm d}$ above; also~\eqref{EqKSpectralmf} follows directly by taking the limit as $\fm\searrow 0$ in $r>r_0>0$.
When analyzing the normal operators of $\fm\nabla^{g_\fm}(\sigma)$ at ${\mathrm{zf}}$ and ${\mathrm{nf}}$, one works with the coordinates $(\hat t_*,\hat x)=(t_*,x)/\fm$ and identifies the vector $u^\mu\partial_\mu$ with $u^\mu\partial_{\hat\mu}$ (where $\partial_{\hat 0}=\partial_{\hat t_*}$ and $\partial_{\hat j}=\partial_{\hat x^j}$, $j=1,2,3$); note also that the differential operator $\fm\partial_\nu$ for $\nu=1,2,3$ is equal to $\partial_{\hat\nu}$, while the spectral family of $\fm\partial_0=\fm\partial_{t_*}=\partial_{\hat t_*}$ is $-i\fm\sigma=-i\tilde\sigma$. To obtain~\eqref{EqKSpectralzf}--\eqref{EqKSpectralmf}, it then remains to note that for bounded $\hat x$, Lemma~\ref{LemmaKMetric} implies
\[
\lim_{\fm\searrow 0}\bigl(\fm\Gamma_{\mu\nu}^\lambda(g_\fm)\bigr) = \Gamma_{\hat\mu\hat\nu}^{\hat\lambda}(\hat g).\qedhere
\]
\end{proof}
\begin{proof}[Proof of Proposition~\usref{PropKS}]
Since $\Box(\cdot+i\sigma_1)=\operatorname{tr}_g(\nabla^g(\cdot+i\sigma_1)\circ\nabla^g(\cdot+i\sigma_1))$ in the notation of Lemma~\ref{LemmaKMetric}, we only need to appeal to Proposition~\ref{PropKSpectral} and use the multiplicativity of the principal symbol and normal operator maps.
\end{proof}
The plan of the remainder of this section is as follows:
\begin{itemize}
\item In~\S\ref{SsKSy}, we work exclusively with the principal (and subprincipal) symbol of $\Box(\cdot+i\sigma_1)$; this is enough to deduce the absence of extremely high energy resonances ($|\sigma|\gg\fm^{-1}$), see Remark~\ref{RmkKSyAbsence}. The same methods also prove the invertibility of the ${\mathrm{nf}}_{\pm,\tilde\hbar}$-normal operator of $\Box(\cdot+i\sigma_1)$ at high energies, see Proposition~\ref{PropKSyKerr}.
\item In~\S\ref{SsKnf}, we study the inverse of the spectral family of the wave operator on a Kerr spacetime at small and bounded (real) energies, cf.\ Proposition~\ref{PropKS}\eqref{ItKSnf}. We first prove uniform bounds on its inverse away from zero energy (Proposition~\ref{PropKnfNz})---which suffices to obtain the absence of very high energy resonances ($|\sigma|\sim\fm^{-1}$)---before proving uniform bounds down to zero energy (Lemma~\ref{LemmaKz} and Proposition~\ref{PropKnfZ}).
\item Having inverted all normal operators that are related to the singular Kerr limit, we then turn in~\S\ref{SsKmf} to the inversion of the spectral family on de~Sitter space at high energies. This then implies the absence of high energy resonances ($|\sigma|\geq h_0^{-1}\gg 1$) for all sufficiently small $\fm$, see Corollary~\ref{CorKAHigh}.
\item In~\S\ref{SsKBd}, we finally control the resonances in the compact subset of $\mathbb{C}_\sigma$ to which they have been constrained at this point.
\item In~\S\ref{SsKU}, we explain the modifications necessary to treat the singular limit $\fm\searrow 0$ not just in a strip of frequencies $\sigma$, but in a half space $\Im\sigma\geq -C_1$. This will complete the proof of Theorem~\ref{ThmK} (and thus of Theorem~\ref{ThmI}).
\item The minimal modifications necessary to treat the Klein--Gordon equation are discussed in~\S\ref{SsKG}.
\end{itemize}
Throughout, we will use the ($\fm$-dependent) spatial volume density $|{\mathrm d} g_X|$ on $X_{\mathrm{q}}$, its restriction $|{\mathrm d}(g_{\mathrm{dS}})|_X|$ to ${\mathrm{mf}}_{\mathrm{q}}$ (which is the spatial volume density for the de~Sitter metric, i.e.\ $|{\mathrm d} g_{\mathrm{dS}}|=|{\mathrm d} t_*||{\mathrm d}(g_{\mathrm{dS}})|_X|$), and the spatial volume density
\begin{equation}
\label{EqKSDensity}
0<|{\mathrm d}\hat g_{\hat X}| = \hat\varrho^2\sin\theta\,|{\mathrm d}\hat r\,{\mathrm d}\theta\,{\mathrm d}\phi_*|\in\cC^\infty\bigl(\overline{\hat\Omega};{}^{\scop}\Omega_{\overline{\hat\Omega}}\hat X\bigr)
\end{equation}
of the Kerr metric on $\hat M=\mathbb{R}_{\hat t_*}\times\hat X$, defined via $|{\mathrm d}\hat g|=|{\mathrm d}\hat t_*| |{\mathrm d}\hat g_{\hat X}|$.
\subsection{Symbolic analysis}
\label{SsKSy}
In this section, we exploit the information given by Proposition~\ref{PropKS}\eqref{ItKSSymb}. The symbolic estimates for $\Box(\cdot+i\sigma_1)$ on the Q-characteristic set are microlocal propagation estimates which are well-established in the literature.\footnote{The positive commutator arguments used for their proofs only make use of the principal symbol, and hence work in the Q-calculus as well.} Concretely, we shall use radial point estimates over the event and cosmological horizons following \cite[\S2.4]{VasyMicroKerrdS} as well as at spatial infinity for the Kerr model operators following \cite{MelroseEuclideanSpectralTheory,VasyZworskiScl}, and estimates at normally hyperbolic trapping \cite{DyatlovSpectralGaps}. The Q-algebra is furthermore related to the semiclassical cone algebra developed in \cite{HintzConicPowers,HintzConicProp}, and we use the radial point estimates established in \cite[\S4.4]{HintzConicProp} at the cone point $\partial\dot X$ in the high frequency regime (in the terminology of~\S\ref{SsIA}). There are further radial sets lying over ${\mathrm{if}}\cap{\mathrm{nf}}$ (thus in the very high frequency regime) where we will prove Q-microlocal estimates by means of standard positive commutator arguments.
We denote by
\[
\Sigma \subset \overline{{}^\Qop T^*_{\overline{\Omega_{\mathrm{Q}}}}}X
\]
the characteristic set of $\Box(\cdot+i\sigma_1)$ (which is independent of $\sigma_1$), i.e.\ the closure of the zero set of $(\rho_{\mathrm{zf}}\rho_{\mathrm{nf}}\rho_{\mathrm{sf}}\rho_{\mathrm{if}})^2 G$ in the notation of Proposition~\ref{PropKS}; more precisely, $\Sigma$ is the union of the characteristic sets of $\Box(\cdot+i\sigma_1)$ lying in
\begin{equation}
\label{EqKSySymbolicFaces}
\overline{{}^\Qop T^*_{\mathrm{sf}}}X,\quad\overline{{}^\Qop T^*_{\mathrm{if}}}X,\quad{}^\Qop S^*X,
\end{equation}
where ${}^\Qop S^*X\subset\overline{{}^\Qop T^*}X$ denotes the boundary at fiber infinity. In this section, we show:
\begin{prop}[Symbolic estimates]
\label{PropKSy}
Let $s,\gamma,l',b\in\mathbb{R}$, and let $\mathsf{r}\in\cC^\infty(\overline{{}^\Qop T^*_{\mathrm{if}}}X)$. Suppose that $s>{\tfrac{1}{2}}+C_1$, and that $\mathsf{r}-l'>-{\tfrac{1}{2}}$, resp.\ $\mathsf{r}-l'<-{\tfrac{1}{2}}$ at the incoming, resp.\ outgoing radial set over ${\mathrm{if}}\cap{\mathrm{nf}}$ (see~\eqref{EqKSyRif}, \eqref{EqKSyRifm}, \eqref{EqKSyRifp} below). Suppose moreover that $\mathsf{r}$ is non-increasing along the flow of the Hamiltonian vector field $H_G$ of the principal symbol $G$ of $\Box(\cdot+i\sigma_0)$. Then for any $s_0\in\mathbb{R}$, $\mathsf{r}_0\in\cC^\infty(\overline{{}^\Qop T^*_{\mathrm{if}}}X)$, and $b_0\in\mathbb{R}$, there exists $C>0$ so that for all $\sigma_0\in\mathbb{R}$, $\fm\in(0,\fm_0]$, and $\sigma_1\in[-C_1,C_1]$, we have
\begin{equation}
\label{EqKSy}
\| u \|_{\bar H_{{\mathrm{Q}},\sigma_0,\fm}^{s,(l,\gamma,l',\mathsf{r},b)}(\Omega_{\mathrm{Q}})} \leq C\Bigl( \| \Box_{g_\fm}(\sigma_0+i\sigma_1)u \|_{\bar H_{{\mathrm{Q}},\sigma_0,\fm}^{s-1,(l-2,\gamma,l'-2,\mathsf{r}-1,b)}(\Omega_{\mathrm{Q}})} + \|u\|_{\bar H_{{\mathrm{Q}},\sigma_0,\fm}^{s_0,(l,\gamma,l',\mathsf{r}_0,b_0)}(\Omega_{\mathrm{Q}})} \Bigr).
\end{equation}
\end{prop}
The loss of one order in the Q-differential and ${\mathrm{if}}$-decay sense ($s$ and $\mathsf{r}$) arises from real principal type or radial point propagation results, while the loss of two ${\mathrm{sf}}$-orders ($b$) arises at the trapped set.\footnote{Any loss in the ${\mathrm{sf}}$-order bigger than $1$ would be sufficient for this estimate, but we do not need a sharp estimate in the sequel.}
\begin{rmk}[Absence of very high energy resonances]
\label{RmkKSyAbsence}
For sufficiently small $\tilde h=|\tilde\sigma|^{-1}>0$, the second, error, term on the right in~\eqref{EqKSy} is smaller than ${\tfrac{1}{2}}$ times the left hand side. We conclude that any $u\in\bar H^s(\Omega_\fm)$ with $\Box_{g_\fm}(\sigma_0+i\sigma_1)u=0$ must vanish, provided $\tilde\sigma:=\fm\sigma_0$ is sufficiently large in absolute value, and $\fm>0$ is sufficiently small.
\end{rmk}
In the proof of Proposition~\ref{PropKSy}, we work our way systematically through the boundary faces~\eqref{EqKSySymbolicFaces} (over which the principal symbol is a well-defined function): first we work in $\overline{{}^\Qop T^*_{\mathrm{sf}}}X$ and $\overline{{}^\Qop T^*_{\mathrm{if}}}X$, and then at fiber infinity ${}^\Qop S^*X\subset\overline{{}^\Qop T^*}X$. Since we work over the domain $\overline{\Omega_{\mathrm{Q}}}$ where $r\geq\fm$ (which we will henceforth not state explicitly anymore), the function $r$ is a defining function of ${\mathrm{zf}}\cup{\mathrm{nf}}$. Since in $\sigma>1$, the function $h=|\sigma|^{-1}$ is a defining function of ${\mathrm{nf}}\cup{\mathrm{sf}}\cup{\mathrm{if}}$, we conclude that
\[
\frac{h}{h+r}\ \text{is a defining function of}\ {\mathrm{sf}}\cup{\mathrm{if}}.
\]
Furthermore, by the second part of~\eqref{EqQSpan}, smooth fiber-linear coordinates on ${}^\Qop T^*_{\overline{\Omega_{\mathrm{Q}}}}X$ are given in polar coordinates $(r,\omega)$ on $X$ by writing the canonical 1-form as
\begin{equation}
\label{EqKSyTQCoord}
\xi_{\mathrm{Q}} \frac{h+r}{h}\frac{{\mathrm d} r}{r} + \frac{h+r}{h}\eta_{\mathrm{Q}},\qquad \xi_{\mathrm{Q}}\in\mathbb{R},\ \eta_{\mathrm{Q}}\in T^*\mathbb{S}^2.
\end{equation}
At radial and trapped sets, the subprincipal symbol of $\Box(\cdot+i\sigma_1)$ enters.
\begin{lemma}[Imaginary part]
\label{LemmaKSyIm}
The operator
\begin{equation}
\label{EqKSyIm}
\Im\Box(\cdot+i\sigma_1) := \frac{1}{2 i}\bigl(\Box(\cdot+i\sigma_1)-\Box(\cdot+i\sigma_1)^*\bigr) \in \Diff_\Qop^{1,(2,0,1,1,1)}(X)
\end{equation}
has principal symbol $(\sigma_0,\fm;x,\xi)\mapsto 2(\Im\sigma) g_\fm^{-1}|_x(-{\mathrm d} t_*,-(\Re\sigma)\,{\mathrm d} t_*+\xi)$ where $\sigma=\sigma_0+i\sigma_1$.
\end{lemma}
\begin{proof}
Since $\Box_{g_\fm}$ is a symmetric operator on $\mathbb{R}_{t_*}\times\Omega_\fm$ with respect to the volume form $|{\mathrm d} g_\fm|$, we have $\Box_{g_\fm}(\sigma)^*=\Box_{g_\fm}(\bar\sigma)$.
For fixed $\sigma_1$, we have $\Im\Box(\cdot+i\sigma_1)\in\Diff_\Qop^{1,(2,0,2,1,1)}(X)$ since the principal symbol of $\Box(\cdot+i\sigma_1)$ is real-valued; but since the ${\mathrm{nf}}_{\tilde\sigma_0}$-normal operators are symmetric (as they involve only \emph{real} frequencies), we obtain an order of improvement at ${\mathrm{nf}}$, leading to~\eqref{EqKSyIm}.
Write $\Im\Box_{g_\fm}(\sigma_0+i\sigma_1)=\Re\int_0^{\sigma_1} \partial_\sigma\Box_{g_\fm}(\sigma_0+i\tau)\,{\mathrm d}\tau$. Now,
\[
\partial_\sigma\Box_{g_\fm}(\sigma)=\partial_\sigma(e^{i\sigma t_*}\Box_{g_\fm} e^{-i\sigma t_*})=e^{i\sigma t_*}(i t_*\Box_{g_\fm}-\Box_{g_\fm} i t_*)e^{-i\sigma t_*}
\]
is the spectral family $-(i[\Box_{g_\fm},t_*])(\sigma)$ of $-i[\Box_{g_\fm},t_*]$, the principal symbol of which at $z=(0,x)\in\Omega_\fm\subset M$ and $\zeta=-\sigma\,{\mathrm d} t_*+\xi\in T^*_z M$ (where $\xi\in T^*_x X$) is $-H_{G_\fm}t_*=\partial_\sigma G_\fm$. Note then that $\partial_\sigma G_\fm(z,\zeta)=2 g_\fm^{-1}|_z(-{\mathrm d} t_*,\zeta)$ (where $z=(t_*,x)$ and $\zeta\in T^*_X M$), and therefore the principal symbol of $-(i[\Box_{g_\fm},t_*])(\sigma)$ is $2 g_\fm^{-1}(-{\mathrm d} t_*,-\sigma\,{\mathrm d} t_*+\cdot)$. This implies the Lemma.
\end{proof}
\begin{notation}[Arbitrary orders]
\label{NotKOrder}
In the arguments below, some orders of symbols on ${}^\Qop T^*X$ will be arbitrary by virtue of the symbols being supported away from some boundary hypersurfaces; in this case, we write `$*$' instead of specifying (arbitrary) orders at those boundary hypersurfaces. As an example, the lift of a compactly supported smooth function in $\hat r$ to $X_{\mathrm{Q}}$ is an element of $S^{0,(0,*,0,*,0)}({}^\Qop T^*X)$ (i.e.\ with the orders at ${\mathrm{mf}}$ and ${\mathrm{if}}$ arbitrary). We use the same notation for Q-ps.d.o.s and Sobolev spaces.
\end{notation}
\subsubsection{Estimates near \texorpdfstring{${\mathrm{if}}$}{if}}
\label{SssKSyif}
We work at (large) positive frequencies $\sigma_0>1$ and indeed near ${\mathrm{sf}}_+\cup{\mathrm{if}}_+$; the analysis in $\sigma_0<-1$ is completely analogous. Consider the semiclassical rescaling $h=|\sigma|^{-1}$, $z=h\sigma$,
\[
G_{\hbar,z}(h,\fm,x,\xi) := |\sigma|^{-2}G(\sigma,\fm,x,\xi) = G_\fm|_x(-z\,{\mathrm d} t_*+h\xi),\qquad \xi\in T^*_x\overline{\Omega_\fm}.
\]
By Proposition~\ref{PropKS}\eqref{ItKSSymb} and the membership~\eqref{EqQDiffEx}, the symbol $G\in S^{2,(2,0,0,0,0)}$ is a quadratic form on the fibers of ${}^\Qop T^*X$ which is smooth down to ${\mathrm{sf}}_+\cup{\mathrm{if}}_+$. Since $|\Im\sigma|\leq C_1$, we have $|z-1|\leq C h$, and therefore we can replace $G_{\hbar,z}(h,\fm,x,\xi)$ by
\begin{equation}
\label{EqKSyifGsemi}
G_\hbar(h,\fm,x,\xi) := G_{\hbar,1}(h,\fm,x,\xi) = G_\fm|_x(-{\mathrm d} t_*+h\xi)
\end{equation}
without changing its principal part, i.e.\ its equivalence class modulo $S^{1,(2,0,0,-1,-1)}({}^\Qop T^*X)$.
Let us consider a neighborhood of ${\mathrm{if}}_+$. There, we have $h\lesssim r$, and thus $h/r$ is a joint defining function of ${\mathrm{if}}_+\cup{\mathrm{sf}}_+$; replacing $\frac{h}{h+r}$ by $\frac{h}{r}$ in~\eqref{EqKSyTQCoord}, we write Q-covectors as
\begin{equation}
\label{EqKSyCoordif}
h^{-1}(\xi\,{\mathrm d} r + r\eta),\quad \xi\in\mathbb{R},\ \eta\in T^*\mathbb{S}^2,
\end{equation}
with $\xi,\eta$ giving smooth fiber-linear coordinates. In terms of these, we have
\[
G_\hbar = G_\fm(-{\mathrm d} t_*+\xi\,{\mathrm d} r+r\eta).
\]
At $\fm=0$, this is the dual metric function $G_{\mathrm{dS}}$ of the de~Sitter metric, so from~\eqref{EqKdS} we find
\begin{equation}
\label{EqKSydS}
G_\hbar|_{{\mathrm{if}}_+} = G_{\mathrm{dS}} = (1-r^2)\xi^2+|\eta|_{\slashed{g}{}^{-1}}^2 + 2\tilde\chi^c(r)\xi - \frac{1-\tilde\chi^c(r)^2}{1-r^2}.
\end{equation}
The structure of the characteristic set of~\eqref{EqKSydS} (in slightly different coordinates), as well as the dynamics of the null-bicharacteristic flow, was studied in detail in~\cite[\S4]{VasyMicroKerrdS}, with the caveat that now $r=0$ is resolved, i.e.\ blown up. (Recall here that ${\mathrm{if}}_+=[0,\infty]_{\tilde\sigma}\times\dot X$ from Proposition~\ref{PropQStruct}\eqref{ItQStructif}.) We begin by noting that the Hamiltonian vector field in the coordinates~\eqref{EqKSyCoordif} takes the form
\[
r h^{-1}H_p = (\partial_\xi p)(r\partial_r-\eta\partial_\eta) - \bigl((r\partial_r-\eta\partial_\eta)p\bigr)\partial_\xi + (\partial_\eta p)\partial_\omega-(\partial_\omega p)\partial_\eta,
\]
as can be seen by changing variables from the standard variables $(\xi_0,\eta_0)$ (with covectors written as $\xi_0\,{\mathrm d} r+\eta_0$, thus $H_p=(\partial_{\xi_0}p)\partial_r-(\partial_r p)\partial_{\xi_0}+(\partial_{\eta_0}p)\partial_\omega-(\partial_\omega p)\partial_{\eta_0}$) to $(\xi,\eta)=(h\xi_0,h r^{-1}\eta_0)$. Thus, if ${\tfrac{1}{2}} h^{-1}H_{G_{\mathrm{dS}}}r=(1-r^2)\xi+\tilde\chi^c(r)=0$ on $\Sigma$, then $0=G_{\mathrm{dS}}=|\eta|_{\slashed{g}{}^{-1}}^2-\frac{1}{1-r^2}$ forces $r<1$ when $(\xi,\eta)$ is finite, and then ${\tfrac{1}{2}}(h^{-1}H_{G_{\mathrm{dS}}})^2 r=(1-r^2)h^{-1}H_{G_{\mathrm{dS}}}\xi=2 r^{-1}(1-r^2)(r^2\xi^2 + |\eta|^2_{\slashed{g}{}^{-1}}+\frac{r^2}{(1-r^2)^2})>0$. Therefore, the level sets of $r$ in $(0,1)$ are null-bicharacteristically convex.
At $r=0$ on the other hand, where $\tilde\chi^c(r)=0$, we have $G_{\mathrm{dS}}=\xi^2+|\eta|^2_{\slashed{g}{}^{-1}}-1$. The restriction of ${\tfrac{1}{2}} r h^{-1}H_{G_{\mathrm{dS}}}$, as a b-vector field on ${}^\Qop T^*X$, to the characteristic set over $r=0$ is given by $\xi(r\partial_r-\eta\partial_\eta)+|\eta|^2_{\slashed{g}{}^{-1}}\partial_\xi+\eta\cdot\partial_\omega$ (at the center of $\slashed{g}{}$-normal coordinates $\omega$), which on the characteristic set is radial (i.e.\ vanishes as a vector field) only at
\begin{equation}
\label{EqKSyRif}
\mc R_{{\mathrm{if}}_+,\pm} = \{ r=0,\ \xi=\pm 1,\ \eta=0 \} \cap {}^\Qop T^*_{{\mathrm{if}}_+}X.
\end{equation}
The linearizations of ${\tfrac{1}{2}} r h^{-1}H_{G_{\mathrm{dS}}}$ at these radial sets are
\begin{equation}
\label{EqKSyRadif}
{\pm}(r\partial_r-\eta\partial_\eta),
\end{equation}
and inside the characteristic set over $r=0$, the $r h^{-1}H_{G_{\mathrm{dS}}}$-flow flows from the source at $\xi=-1$ to the sink at $\xi=+1$. This can be translated into an estimate by means of a standard symbolic positive commutator argument at radial sets; we sketch this near $\xi=-1$. Thus, using the local defining functions
\[
\rho_{\mathrm{mf}} = \frac{\fm}{h+\fm},\qquad
\rho_{\mathrm{nf}} = r,\qquad
\rho_{\mathrm{if}} = \frac{h+\fm}{r},\qquad
\rho_{\mathrm{sf}} = \frac{h}{h+\fm},
\]
we consider a commutant (with constant orders, and recalling Notation~\ref{NotKOrder})
\begin{align*}
a &= \rho_{\mathrm{mf}}^{-2\gamma} \rho_{\mathrm{nf}}^{-2 l'+2} \rho_{\mathrm{if}}^{-2\mathsf{r}+1} \rho_{\mathrm{sf}}^{-2 b+1} \chi(\xi+1)\chi(|\eta|_{\slashed{g}{}^{-1}}^2)\chi(r)\chi(h/r)\chi(\fm/r) \\
&\qquad \in S^{*,(*,2\gamma,2 l'-2,2\mathsf{r}-1,2 b-1)}({}^\Qop T^*X)
\end{align*}
where $\chi\in\cC^\infty_\cp([0,2\delta))$ is identically $1$ on $[0,\delta]$ for some fixed small $\delta>0$, and satisfies $\chi'\leq 0$. The cutoffs localize to a neighborhood (in ${}^\Qop T^*X$) of $r=0$, $\xi=-1$, $\eta=0$ over ${\mathrm{if}}_+$. Denote by $A=A^*\in\Psi_\Qop^{*,(*,2\gamma,2 l'-2,2\mathsf{r}-1,2 b-1)}(X)$ a Q-quantization of $a$ (with Schwartz kernel supported in both factors in $\hat r>1$, $r<2$), and consider the $L^2$-pairing
\begin{equation}
\label{EqKSyCommCalc}
2\Im\langle\Box(\cdot+i\sigma_1)u,A u\rangle = \langle\mc C u,u\rangle,\qquad \mc C := i[\Box(\cdot+i\sigma_1),A]+2(\Im\Box(\cdot+i\sigma_1))A.
\end{equation}
Thus, $c={}^\Qop\upsigma(\mc C)\in S^{*,(*,2\gamma,2 l',2\mathsf{r},2 b)}$, with the second summand of $\mc C$ contributing an element of $S^{*,(*,2\gamma,2 l'-1,2\mathsf{r},2 b)}$ by Lemma~\ref{LemmaKSyIm}, which is thus of lower order at ${\mathrm{nf}}$. By~\eqref{EqKSyRadif}, the rescaled symbol $\rho_{\mathrm{mf}}^{-2\gamma}\rho_{\mathrm{nf}}^{-2 l'}\rho_{\mathrm{if}}^{-2\mathsf{r}}\rho_{\mathrm{sf}}^{-2 b}c$ is a positive multiple of $-2\mathsf{r}+2 l'-1$ at the radial set $\mc R_{{\mathrm{if}}_+,-}$; if $\mathsf{r},l'$ are such that this is negative, then differentiation of $\chi(|\eta|^2)$ along $\eta\partial_\eta$ gives a contribution of the same sign (i.e.\ non-positive, and strictly negative where $\chi'<0$), and so does differentiation of $\chi(\fm/r)$ along $-r\partial_r$ when $\delta>0$ is sufficiently small, whereas differentiation of $\chi(r)$ produces a nonnegative contribution which necessitates an a priori control assumption on $u$ on $\operatorname{supp} a\cap\operatorname{supp}\,\chi'(r)$. Therefore, in order to propagate Q-regularity from $r>0$ into the radial set, we need\footnote{This threshold condition is completely analogous to \cite[Theorem~4.10]{HintzConicProp}, where the notation $\mathsf{b},\alpha$ is used instead of $\mathsf{r},l'$.}
\begin{subequations}
\begin{equation}
\label{EqKSyRifm}
\mathsf{r} > -{\tfrac{1}{2}} + l'\quad \text{at}\ \mc R_{{\mathrm{if}}_+,-}.
\end{equation}
Under this assumption, we thus obtain a uniform (for $\sigma_0\in\mathbb{R}$, $\fm\in(0,\fm_0]$, and $\sigma_1\in[-C_1,C_1]$) estimate
\begin{equation}
\label{EqKSyEst}
\|B u\|_{H_{{\mathrm{Q}},\sigma_0,\fm}^{*,(*,\gamma,l',\mathsf{r},b)}} \leq C\Bigl( \|\Box_{g_\fm}(\sigma_0+i\sigma_1)u\|_{H_{{\mathrm{Q}},\sigma_0,\fm}^{*,(*,\gamma,l'-2,\mathsf{r}-1,b-1)}} + \|E u\|_{H_{{\mathrm{Q}},\sigma_0,\fm}^{*,(*,\gamma,l',\mathsf{r},b)}} + \|u\|_{H_{{\mathrm{Q}},\sigma_0,\fm}^{*,(*,\gamma,l',\mathsf{r}_0,b_0)}}\Bigr)
\end{equation}
for arbitrary $\mathsf{r}_0<\mathsf{r}$, $b_0<b$, for appropriate operators $B,E\in\Psi_\Qop^0$ microlocalized in a neighborhood of ${\mathrm{if}}_+$, where $B$ (quantizing a symbol arising from the elliptic leading order term of $c$ at $\mc R_{{\mathrm{if}}_+,-}$) is elliptic at $\mc R_{{\mathrm{if}}_+,-}$ and $E$ (quantizing a symbol arising from the term from differentiation of $\chi(r)$ above) can be taken to have operator wave front set contained in $r>0$.
Similarly, one can propagate regularity near $\mc R_{{\mathrm{if}}_+,+}$ from the a priori control regions $\fm/r>0$ and a punctured neighborhood of $\mc R_{{\mathrm{if}}_+,+}$ inside of $r=0$ into $\mc R_{{\mathrm{if}}_+,+}$ itself, together with a uniform estimate that takes the same form, except now $E$ controls $u$ on these changed a priori control regions; the requirement on the orders is
\begin{equation}
\label{EqKSyRifp}
\mathsf{r} < -{\tfrac{1}{2}}+l'\quad\text{at}\ \mc R_{{\mathrm{if}}_+,+}.
\end{equation}
\end{subequations}
(Thus, an ${\mathrm{if}}$-order $\mathsf{r}$ satisfying both~\eqref{EqKSyRifm} and \eqref{EqKSyRifp} must be variable. For real principal type propagation in between the two radial sets, one moreover needs $\mathsf{r}$ to be non-increasing along the direction of propagation; see e.g.\ \cite[\S4.1]{VasyMinicourse}.) We remark that if we restrict to bounded subsets of $\tilde\sigma\in[0,\infty)$, then the ${\mathrm{sf}}$-order $b$ becomes irrelevant, and thus the a priori control term in $\fm/r>0$ (where also the ${\mathrm{if}}$-order is irrelevant) is bounded by the overall error term (the last term in~\eqref{EqKSyEst}). This corresponds to the fact that the Q-calculus is not symbolic for finite Q-momenta away from ${\mathrm{if}}\cup{\mathrm{sf}}$; instead, control at ${\mathrm{nf}}_{\tilde\sigma}$ requires the inversion of a model \emph{operator}, see~\S\ref{SsKnf} below.
The (microlocal) propagation estimates near ${\mathrm{if}}_+$ but over $r>0$ are the same as those proved in \cite[\S4]{VasyMicroKerrdS}, except now the ${\mathrm{if}}$-order $\mathsf{r}$ is variable---which, under the aforementioned monotonicity assumption on $\mathsf{r}$, does not necessitate any changes in the proofs of the propagation results. We sketch the computation of the null-bicharacteristic dynamics and of the positive commutator estimates in order to determine the relevant threshold conditions. To wit, we shall work near fiber infinity of the conormal bundle of the cosmological horizon $r=1$ of de~Sitter space; we work in $\xi<0$ and with the coordinates $\rho_\infty=|\xi|^{-1}$, $\hat\eta=\xi/\eta$ near fiber infinity. We may replace $G_{\mathrm{dS}}$ by the simpler expression $G_0=(1-r^2)\xi^2+|\eta|_{\slashed{g}{}^{-1}}^2$, for which one finds
\[
\rho_\infty r h^{-1}H_{G_0} = -2(1-r^2)(r\partial_r-\hat\eta\partial_{\hat\eta}) + 2(r+r^{-1}|\hat\eta|^2)(\rho_\infty\partial_{\rho_\infty}+\hat\eta\partial_{\hat\eta}).
\]
At the radial set $w:=r-1=0$, $\rho_\infty=0$, $\hat\eta=0$, the linearization of this vector field is
\[
4 w\partial_w + 2\rho_\infty\partial_{\rho_\infty} + 2\hat\eta\partial_{\hat\eta},
\]
and therefore this radial set is a source for the rescaled Hamiltonian flow. Thus, $H_{G_0}$ is to leading order at the radial set given by $\rho_\infty^{-1}h(4 w\partial_w+2\rho_\infty\partial_{\rho_\infty}+2\hat\eta\partial_{\hat\eta})$. Since we are working near $r=1$, we can take as local defining functions
\[
\rho_{\mathrm{mf}}=\frac{\fm}{h}, \qquad
\rho_{\mathrm{if}}=h+\fm,\qquad
\rho_{\mathrm{sf}}=\frac{h}{h+\fm}.
\]
Consider the commutant
\[
a = \rho_\infty^{-2 s+1}\rho_{\mathrm{mf}}^{-2\gamma}\rho_{\mathrm{if}}^{-2\mathsf{r}+1}\rho_{\mathrm{sf}}^{-2 b+1} \chi(\rho_\infty)\chi(w^2)\chi(|\hat\eta|_{\slashed{g}{}^{-1}}^2)\chi(\fm)\chi(h),
\]
with $\chi$ as before, and let $A=A^*$ denote a Q-quantization of $a$. The rescaled symbol $\rho_\infty^{2 s}\rho_{\mathrm{mf}}^{2\gamma}\rho_{\mathrm{if}}^{2\mathsf{r}}\rho_{\mathrm{sf}}^{2 b}\,{}^\Qop\upsigma(\mc C)$ of the operator $\mc C$ in~\eqref{EqKSyCommCalc} is now a sum of three types of terms: the first type arises from differentiating the weights of $a$, giving $-2(2 s-1)$ at the radial set; the second arises from differentiating the cutoffs in $\rho_\infty$, $w^2$, $|\hat\eta|_{\slashed{g}{}^{-1}}^2$, which give non-positive terms; and the third arises from the skew-adjoint part and at the radial set contributes (using Lemma~\ref{LemmaKSyIm}) twice $2\sigma_1 g_{\mathrm{dS}}^{-1}(-{\mathrm d} t_*,-{\mathrm d} r)$, so $-4\sigma_1$. In order to propagate out of the radial set, we thus need $-2(2 s-1)-4\sigma_1 < 0$, or equivalently
\[
s > {\tfrac{1}{2}} - \sigma_1,
\]
which in view of $\sigma_1\geq-C_1$ holds provided $s>{\tfrac{1}{2}}+C_1$. Propagation out of the opposite radial set (at $r=1$ and $\xi>0$, $\xi^{-1}=0$, $\hat\eta=0$) requires the same threshold condition.
Finally, near $r=2$, say in $r\in[\frac32,2]$ for definiteness, we need to use energy estimates in order to deal with the presence of a Cauchy hypersurface at $r=2$; note that ${\mathrm d} r$ is past timelike in this region. We can thus apply the semiclassical energy estimates of \cite[\S3.3]{VasyMicroKerrdS}, extended to general orders $s$, $\mathsf{r}$ using microlocal propagation results in a manner completely analogous to \cite[\S2.1.3]{HintzVasySemilinear}, in order to estimate $u$ in $\bar H_{{\mathrm{Q}},\sigma_0,\fm}^{s,(*,\gamma,*,\mathsf{r},b)}$ near $r=2$ in terms of its norm near $r=\frac32$.
\bigskip
To summarize, we can propagate Q-Sobolev regularity from the radial sets over the cosmological horizon towards the conic point $r=0$ and into $\mc R_{{\mathrm{if}}_+,-}$. For finite $\tilde\sigma$, this can be propagated further into $\mc R_{{\mathrm{if}}_+,+}$ and then outwards into $r>0$, at which point we have microlocal control on the whole Q-phase space over ${\mathrm{if}}_+\cap\{r<2\}$; energy estimates near $r=2$ then give uniform control down to $r=2$. In order to complete the proof of the estimate~\eqref{EqKSy} for finite $\tilde\sigma$, it thus remains to control Q-regularity for bounded $\hat r$ and $r\simeq 1$, which is done in~\S\ref{SssKSyHor}.
In the semiclassical regime $\tilde\sigma\to\infty$, we cannot yet propagate into the outgoing radial set $\mc R_{{\mathrm{if}}_+,+}$ since this requires control on its unstable manifold also over ${\mathrm{nf}}^\circ\cap{\mathrm{sf}}$---which requires the analysis of the ${\mathrm{nf}}$-normal operator, i.e.\ the spectral family of the Kerr wave operator, at high energies. This is the subject of~\S\ref{SssKSyt} below. We remark that the radial point estimates at $\mc R_{{\mathrm{if}}_+,\pm}$ are, from the perspective of ${\mathrm{nf}}$, semiclassical scattering estimates in asymptotically Euclidean scattering; such estimates were first proved by Vasy--Zworski \cite{VasyZworskiScl} for high energy potential scattering on asymptotically Euclidean Riemannian manifolds.
\subsubsection{Estimates near \texorpdfstring{${\mathrm{sf}}\cap{\mathrm{nf}}$}{the intersection of sf and nf}}
\label{SssKSyt}
Since the analysis in~\S\ref{SssKSyif} covers (an open neighborhood of) the corner ${\mathrm{sf}}\cap{\mathrm{nf}}\cap{\mathrm{if}}$, we may work in a region $\hat r<\hat r_0$ for an arbitrary large $\hat r_0$; moreover, we work at large $|\tilde\sigma|=\tilde h^{-1}$, so local boundary defining functions are
\[
\rho_{\mathrm{nf}} = \fm,\quad
\rho_{\mathrm{sf}} = \tilde h = \frac{h}{\fm}.
\]
Our local coordinate system $\tilde h,\fm,\hat r,\omega$ is disjoint from the other boundary hypersurfaces of $X_{\mathrm{Q}}$. We introduce smooth fiber-linear coordinates on ${}^\Qop T^*X$ by writing the canonical 1-form as
\begin{equation}
\label{EqKSytCoord}
\tilde h^{-1}(\xi\,{\mathrm d}\hat r + \hat r\eta),\qquad \xi\in\mathbb{R},\ \eta\in T^*\mathbb{S}^2.
\end{equation}
In these coordinates, the semiclassically rescaled principal symbol $G_\hbar$ (see~\eqref{EqKSyifGsemi}) is, using the notation of Definition~\ref{DefKDual}, at $\fm=0$ given by
\begin{equation}
\label{EqKSySym}
G_\hbar = \hat G|_{\hat x}(-{\mathrm d}\hat t_*+\xi\,{\mathrm d}\hat r + \hat r\eta).
\end{equation}
Indeed, this is the limit as $\fm\searrow 0$ (for bounded $\hat x$) of $G_\fm|_{\fm\hat x}(-{\mathrm d}(\fm\hat t_*)+h\tilde h^{-1}(\xi\,{\mathrm d}\hat r+\hat r\eta))$. But~\eqref{EqKSySym} is the semiclassical principal symbol of the spectral family $\tilde h^2\Box_{\hat g}(\tilde h^{-1})$; a full description of its characteristic set and null-bicharacteristic flow in the black hole exterior $\hat r>\hat r^e$ can be found in~\cite[\S\S3.1--3.2]{DyatlovWaveAsymptotics}. We in particular note that the trapped set of $G_\fm$ lies over a fixed compact subset of radii $\hat r$ as $\fm\searrow 0$; this follows from \cite{DyatlovWaveAsymptotics}. For us, it is convenient to use the fact that the trapped set depends \emph{smoothly} on $\fm$ down to $\fm=0$.\footnote{Using this fact runs counter to our insistence that only the Kerr model needs to be analyzed explicitly, whereas the Kerr--de Sitter wave operators are exclusively treated perturbatively. One may instead use that the trapping on subextremal Kerr spacetimes is $k$-normally hyperbolic (for any fixed $k$), as proved in \cite{WunschZworskiNormHypResolvent} and \cite[\S3.2]{DyatlovWaveAsymptotics}, together with the structural stability of such trapping \cite{HirschPughShubInvariantManifolds}, and note that the microlocal estimates \cite{DyatlovSpectralGaps} at the trapped set only require some large but finite amount of regularity of the defining functions for the stable and unstable manifolds; see \cite[Remark after Theorem~2]{DyatlovSpectralGaps}. Using the structural stability, the symbols of the semiclassical ps.d.o.s involved in the proofs of these estimates typically only depend continuously on the parameter $\fm$, which is inconsequential for the standard semiclassical calculus (with continuous dependence on $\fm\in[0,\fm_0]$). The resulting uniform semiclassical estimates are then equivalent to estimates on Q-Sobolev spaces in the extremely high frequency regime under consideration here.} This is a consequence of the explicit description in \cite[Theorem~3.2]{PetersenVasySubextremal}, and by using this fact, one can apply the proof of \cite[Theorem~1]{DyatlovSpectralGaps} at once for the smooth family of trapped sets of $g_\fm$. (For a direct positive commutator proof of these trapping estimates, albeit not in a semiclassical setting, see \cite[\S3]{HintzPolyTrap}.) Near the event horizon $\hat r=\hat r^e$ on the other hand, we can follow \cite[\S\S4.6 and 6.4]{VasyMicroKerrdS}, which applies in the present subextremal Kerr context (see also \cite[Lemma~4.3]{HintzPrice} and \cite[Theorem~4.3]{HaefnerHintzVasyKerr}).
Since the spectral parameter $\tilde\sigma$ is real---thus $\Box_{\hat g}(\tilde\sigma)$ is formally symmetric---the threshold regularity at the radial set at fiber infinity of the conormal bundle of the event horizon $\hat r=\hat r^e$ is equal to ${\tfrac{1}{2}}$. For the same reason, the skew-adjoint part of $\Box_{\hat g}(\tilde\sigma)$ at the trapped set has vanishing principal symbol, and hence the estimates of \cite{DyatlovSpectralGaps} apply. (See \cite[Theorem~4.7]{HintzVasyQuasilinearKdS} for an explicit statement.)
Combining the trapping, radial point, microlocal propagation, elliptic regularity, and wave propagation (in $\hat r<\hat r^e$) results proves Proposition~\ref{PropKSy} at extremely high frequencies. We also record the following consequence of these estimates together with the radial point estimates proved in the previous section (cf.\ Proposition~\ref{PropQHRel}\eqref{ItQHRelnfsemi}):
\begin{prop}[Estimates for the Kerr spectral family at high energies]
\label{PropKSyKerr}
There exists $\tilde h_0>0$ so that the following holds. Let $\mathsf{r}\in\cC^\infty(\overline{{}^{\scop}T^*_{\partial\hat X}}\hat X)$ is a variable order function so that $\mathsf{r}>-{\tfrac{1}{2}}$, resp.\ $\mathsf{r}<-{\tfrac{1}{2}}$ at the semiclassical incoming, resp.\ outgoing radial set over $\partial\hat X$, and so that $\mathsf{r}$ is monotone along the Hamilton flow inside the characteristic set.\footnote{At positive frequencies, these radial sets are given by $\mc R_{{\mathrm{if}}_+,-}$ and $\mc R_{{\mathrm{if}}_+,+}$ under the isomorphism of Corollary~\ref{CorQBundle}. In general, in the coordinates~\eqref{EqKSytCoord}, the incoming, resp.\ outgoing radial set is located at $(\xi,\eta)=(-1,0)$, resp.\ $(\xi,\eta)=(1,0)$, over $\partial\hat X$, see e.g.\ \cite[\S4.8]{VasyMinicourse} or \cite{MelroseEuclideanSpectralTheory} for the non-semiclassical setting, further \cite{VasyZworskiScl} for a global semiclassical commutator estimate, and \cite[\S5]{VasyLAPLag} for a refined semiclassical estimate.} Suppose $s>{\tfrac{1}{2}}$. Then there exists $C>0$ so that
\[
\|u\|_{\bar H_{{\mathrm{sc}},\tilde h}^{s,\mathsf{r}}(\hat\Omega)} \leq C\tilde h^{-2}\|\tilde h^2\Box_{\hat g}(\pm\tilde h^{-1})u\|_{\bar H_{{\mathrm{sc}},\tilde h}^{s-1,\mathsf{r}+1}(\hat\Omega)},\quad 0<\tilde h\leq\tilde h_0.
\]
\end{prop}
In our application to the uniform analysis of $\Box(\cdot+i\sigma_1)$, we shall apply Proposition~\ref{PropKSyKerr} with $\mathsf{r}-l'$ in place of $\mathsf{r}$ (in particular, the threshold conditions here match those of Proposition~\ref{PropKSy}).
\subsubsection{Non-semiclassical estimates near the horizons}
\label{SssKSyHor}
Note that the only parts of the characteristic set $\Sigma$ not covered by the previous arguments are the conormal bundles over the cosmological horizon near ${\mathrm{mf}}$ and the event horizon near ${\mathrm{zf}}$, as well as their flowouts. The radial point estimates at the conormal bundles were however already discussed in the (more delicate) semiclassical setting in the previous two sections, as were the propagation estimates (including energy estimates to deal with the Cauchy hypersurfaces at $\hat r=1$ and $r=2$). This completes the proof of Proposition~\ref{PropKSy}.
\subsection{Estimates for the \texorpdfstring{${\mathrm{nf}}_\pm$-}{low/bounded energy nf-}normal operator}
\label{SsKnf}
We now turn to estimates for the various normal operators of $\Box(\cdot+i\sigma_1)$ which were computed in Proposition~\ref{PropKS}\eqref{ItKSzf}--\eqref{ItKSmf}. The symbolic estimates proved in~\S\ref{SsKSy} restrict to symbolic estimates for all model operators, in the sense that e.g.\ for positive commutator arguments the same commutants can be used (with fewer localizers, corresponding to working on a boundary hypersurface of $X_{\mathrm{Q}}$); on the level of function spaces, this relies on Proposition~\ref{PropQHRel}.
\begin{prop}[Uniform bounds on Kerr at bounded nonzero energies]
\label{PropKnfNz}
Let $c\in(0,1)$, $s>{\tfrac{1}{2}}$, and let $\mathsf{r}$ be as in Proposition~\usref{PropKSyKerr}.\footnote{For bounded nonzero $\tilde\sigma$, one can drop the rescaling of $\xi$ and $\eta$ in~\eqref{EqKSytCoord}, thus writing covectors simply as $\xi\,{\mathrm d}\hat r+\hat r\eta$; the outgoing radial set is then given by $(\xi,\eta)=(\tilde\sigma,0)$ over $\hat r=\infty$, and the incoming radial set by $(\xi,\eta)=(-\tilde\sigma,0)$.} Then there exists $C>0$ so that for all $\tilde\sigma\in\mathbb{R}$ with $|\tilde\sigma|\in[c,c^{-1}]$,
\begin{equation}
\label{EqKnfNz}
\|u\|_{\bar H_{\mathrm{sc}}^{s,\mathsf{r}}(\hat\Omega)} \leq C\|\Box_{\hat g}(\tilde\sigma)u\|_{\bar H_{\mathrm{sc}}^{s-1,\mathsf{r}}(\hat\Omega)}.
\end{equation}
\end{prop}
\begin{proof}
The same symbolic arguments as in the previous section give the estimate
\[
\|u\|_{\bar H_{\mathrm{sc}}^{s,\mathsf{r}}(\hat\Omega)} \leq C\bigl(\|\Box_{\hat g}(\tilde\sigma)u\|_{\bar H_{\mathrm{sc}}^{s-1,\mathsf{r}}(\hat\Omega)} + \|u\|_{\bar H_{\mathrm{sc}}^{-N,-N}(\hat\Omega)}\bigr).
\]
for any $N$, which we take to satisfy $-N<\min(s,\mathsf{r})$; thus, the embedding $\bar H_{\mathrm{sc}}^{s,\mathsf{r}}(\hat\Omega)\hookrightarrow\bar H_{\mathrm{sc}}^{-N,-N}(\hat\Omega)$ is compact. The estimate~\eqref{EqKnfNz} (for a different constant $C$) then follows provided we show that any $u\in\bar H_{\mathrm{sc}}^{s,\mathsf{r}}(\hat\Omega)$ with $\Box_{\hat g}(\tilde\sigma)u=0$ necessarily vanishes. We reduce this to the mode stability result of Whiting and Shlapentokh-Rothman \cite{WhitingKerrModeStability,ShlapentokhRothmanModeStability} which we recalled in Theorem~\ref{ThmIKerr}.
Radial point estimates at the conormal bundle of the event horizon, followed by propagation of regularity from there, imply that $u$ is smooth; at spatial infinity, $u$ has infinite scattering regularity since $\Box_{\hat g}(\tilde\sigma)$ is elliptic at high scattering frequencies. At the incoming radial set, $u$ has arbitrary scattering decay, and by propagating this to a punctured neighborhood of the outgoing radial set, we conclude that $u\ni\bar H_{\mathrm{sc}}^{\infty,\mathsf{r}'}(\hat\Omega)$ where $\mathsf{r}'$ is arbitrary except $\mathsf{r}'<-{\tfrac{1}{2}}$ at the outgoing radial set. This can be further improved by means of module regularity at the outgoing radial set, i.e.\ stable regularity under application of $\hat r(\partial_{\hat r}-i\tilde\sigma)$ and spherical vector fields; this goes back to \cite[\S12]{MelroseEuclideanSpectralTheory} and \cite{HassellMelroseVasySymbolicOrderZero}, and is discussed in detail in the present setting in \cite[\S2.4]{GellRedmanHassellShapiroZhangHelmholtz} (see also \cite[Proposition~4.4]{BaskinVasyWunschRadMink} and \cite{HaberVasyPropagation}). We thus conclude that $e^{-i\tilde\sigma\hat r}u\in\bar H_{\bop}^{\infty,l_0}(\hat\Omega)$ is conormal at $\hat r=\infty$ where $l_0<-{\tfrac{1}{2}}$. Taking into account the modified asymptotics of outgoing spherical waves caused by the black hole mass (here $1$), we consider
\[
u_0(\hat r,\theta,\phi_*) := e^{-i\tilde\sigma\hat r}\hat r^{-2 i\tilde\sigma}u(\hat r,\theta,\phi_*).
\]
Thus, $u_0$ is conormal at $\hat\rho=\hat r^{-1}=0$, but we need more precise information. To this end, we observe that the equation satisfied by $u_0$ in the coordinates $(\hat\rho,\omega)\in[0,1)\times\mathbb{S}^2$ takes the form
\[
\bigl(2 i\tilde\sigma\hat\rho(\hat\rho\partial_{\hat\rho}-1) + \hat\rho^2 L\bigr)u_0 = 0,
\]
where $L\in\Diff_\bop^2([0,1)_{\hat\rho}\times\mathbb{S}^2)$, see \cite[Definition~2.1, Lemma~2.7, and \S4]{HintzPrice}. Rewriting this as $(\hat\rho\partial_{\hat\rho}-1)u_0=\hat\rho L'u_0$ for a new operator $L'\in\Diff_\bop^2$, the conormality of $u_0$ at $\hat\rho=0$ can be upgraded by an iterative procedure, based on the inversion of $\hat\rho\partial_{\hat\rho}-1$, to the fact that $u_0\in\hat\rho\cC^\infty([0,1)_{\hat\rho}\times\mathbb{S}^2)$. We can now apply Theorem~\ref{ThmIKerr} to conclude that $u_0=0$ (and thus $u=0$) in $\hat r\geq \hat r^e$.
This then implies the vanishing of $u$ in $\hat r<\hat r^e$ as well: this can be shown by considering the projections of $u$ to its separated parts $e^{i m\phi_*}S(\theta)R(\hat r)$ and noting (by inspection of the dual metric~\eqref{EqKKerrDual}) that $R$ then satisfies an ODE which upon multiplication by $\hat\mu(\hat r)$ has a regular-singular point at $\hat r=\hat r^e$; hence the infinite order vanishing of $R$ at $\hat r^e$ implies $R\equiv 0$ also in $\hat r<\hat r^e$. The proof is complete.
\end{proof}
Uniform estimates near zero energy require, first of all, an estimate for the zero energy operator:
\begin{lemma}[Zero energy operator on Kerr]
\label{LemmaKz}
Let $s>{\tfrac{1}{2}}$ and $\gamma\in(-\frac32,-{\tfrac{1}{2}})$. Then
\begin{equation}
\label{EqKz}
\|u\|_{\bar H_{\bop}^{s,\gamma}(\hat\Omega)} \leq C\|\Box_{\hat g}(0)u\|_{\bar H_{\bop}^{s-1,\gamma+2}(\hat\Omega)}.
\end{equation}
\end{lemma}
Recall from Proposition~\ref{PropKS}\eqref{ItKSzf} that the ${\mathrm{zf}}$-normal operator of $\Box(\cdot+i\sigma_1)$ is independent of $\sigma_1\in[-C_1,C_1]$ and $\sigma_0\in\overline\mathbb{R}$, and equal to the Kerr zero energy operator $\Box_{\hat g}(0)$; thus, Lemma~\ref{LemmaKz} proves the invertibility of $N_{\mathrm{zf}}(\Box(\cdot+i\sigma_1))$.
\begin{proof}[Proof of Lemma~\usref{LemmaKz}]
Combining the symbolic estimates proved in~\S\ref{SsKSy}---or rather their restrictions to ${\mathrm{zf}}\cap{\mathrm{nf}}$, cf.\ Proposition~\ref{PropQHRel}---with elliptic b-theory near $\hat\rho=\hat r^{-1}=0$, we obtain the estimate
\[
\|u\|_{\bar H_{\bop}^{s,\gamma}(\hat\Omega)} \leq C\bigl(\|\Box_{\hat g}(0)u\|_{\bar H_{\bop}^{s-1,\gamma+2}(\hat\Omega)} + \|u\|_{\bar H_{\bop}^{-N,-N}(\hat\Omega)}\bigr).
\]
(The b-analysis at $\hat\rho=0$ uses that $\Box_{\hat g}(0)$ is, to leading order as a b-operator, the Euclidean Laplacian $\hat\rho^2((\hat\rho D_{\hat\rho})^2+i\hat\rho D_{\hat\rho}+\Delta_\slashed{g}{})$. Upon separation into spherical harmonics, this is a rescaling of the regular-singular ODE $(\hat\rho\partial_{\hat\rho})^2-\hat\rho\partial_{\hat\rho}-\ell(\ell+1)$, with $\ell\in\mathbb{N}_0$ labeling the degree of the spherical harmonic; the indicial solutions are $\hat\rho^{\ell+1}$ and $\hat\rho^{-\ell}$, and the choice of weight $\gamma$ ensures that the weighted $L^2$-space $\bar H_{\bop}^{0,\gamma}(\hat\Omega)$ contains, for all $\ell$, the solution $\hat\rho^{\ell+1}$ but not $\hat\rho^{-\ell}$. See also \cite[Theorem~2.1]{GuillarmouHassellResI}.) Since the inclusion $\bar H_{\bop}^{s,\gamma}(\hat\Omega)\hookrightarrow\bar H_{\bop}^{-N,-N}(\hat\Omega)$ is compact, it remains to prove the triviality of $\ker\Box_{\hat g}(0)$. This can be checked using explicit computations with special functions (as remarked in \cite{PressTeukolskyKerrII,TeukolskySeparation}), but we give a softer proof here, following \cite{HaefnerVasyKerrUnfinished}.
In view of~\eqref{EqKKerrDual} and~\eqref{EqKSDensity}, the operator $\Box_{\hat g}(0)$ is explicitly given by
\begin{align*}
\hat\varrho^2\Box_{\hat g}(0)&=D_{\hat r}\hat\mu(r) D_{\hat r} + \Delta_\slashed{g}{} - \frac{1-\chi^e(\hat r)^2}{\hat\mu(r)}(\fa D_{\phi_*})^2 + \bigl(\chi^e(\hat r)D_{\hat r}+D_{\hat r}\chi^e(\hat r)\bigr)\fa D_{\phi_*} \\
&= D_{\hat r}\hat\mu(\hat r)D_{\hat r} + \Delta_\slashed{g}{} - \frac{\fa^2}{\hat\mu(\hat r)}D_\phi^2,
\end{align*}
where in the second line we passed to $\phi=\phi_*+\Phi(\hat r)$ with $\Phi'(\hat r)=-\frac{\fa\chi^e(\hat r)}{\hat\mu(\hat r)}$; note that
\begin{equation}
\label{EqKzPhi}
\Phi(\hat r)=-\frac{\fa}{\beta}\log(\hat r-\hat r^e)+\tilde\Phi(\hat r),\qquad
\beta:=\hat\mu'(\hat r^e)=\hat r^e-\hat r^c=2\sqrt{1-\hat\fa^2},
\end{equation}
with $\tilde\Phi$ smooth down to $\hat r=\hat r^e$. We may also arrange that $\Phi(\hat r)=0$ for large $\hat r$.
Let now $u\in\ker\Box_{\hat g}(0)$. First of all, we have $u\in\bar H_{\bop}^{\infty,\gamma}(\hat\Omega)$: conormality at, and smoothness near spatial infinity follows from the ellipticity (for large $\hat r$) of $\Box_{\hat g}(0)$ as a weighted b-differential operator, whereas smoothness near the ergoregion and in the black hole interior follows by combining radial point estimates at the event horizon and propagation estimates in the ergoregion and in the black hole interior $\hat r<\hat r^e$. Sobolev embedding for $u\in\bar H_{\bop}^{\infty,\gamma+\frac32}(\hat\Omega,|\frac{{\mathrm d}\hat r}{\hat r}{\mathrm d}\slashed{g}{}|)$ implies that $|D_{\hat r}^j u|=\mc O(\hat r^{-\gamma-\frac32-j})=o(\hat r^{-j})$ for any $j\in\mathbb{N}_0$ as $\hat r\to\infty$.
Projecting $u(r,\theta,\phi_*)$ in the angular variables to a fixed spherical harmonic $Y_{\ell m}(\theta,\phi_*)=e^{i m\phi_*}S_{\ell m}(\theta)$, where $\ell\in\mathbb{N}_0$ and $m\in\mathbb{Z}\cap[-\ell,\ell]$, produces a separated solution
\begin{equation}
\label{EqKzvvstar}
v_*(\hat r)Y_{\ell m}(\theta,\phi_*)=v(\hat r)Y_{\ell m}(\theta,\phi),\qquad
v(\hat r)=e^{-i m\Phi(\hat r)}v_*(\hat r),
\end{equation}
where $v_*\in\cC^\infty([1,\infty)_{\hat r})$ satisfies $|v_*|=o(1)$ as $\hat r\to\infty$, and $v$ (which equals $v_*$ for large $\hat r$) satisfies
\begin{equation}
\label{EqKzODE}
\Bigl(D_{\hat r}\hat\mu D_{\hat r} - \frac{\fa^2 m^2}{\hat\mu} + \ell(\ell+1)\Bigr)v=0.
\end{equation}
This is a regular-singular ODE at $\hat r=\infty$, with indicial solutions $\hat r^\ell$ (which does not decay as $\hat r\to\infty$) and $\hat r^{-\ell-1}$, and therefore we have $|v|=\mc O(\hat r^{-\ell-1})$ and thus $|D_{\hat r}^j v|=\mc O(\hat r^{-\ell-1-j})$ for all $j\in\mathbb{N}_0$.
We first study the case $m\fa=0$, i.e.\ $\fa=0$ or $m=0$. Then $v$ is smooth on $[\hat r^e,\infty)$; upon multiplying~\eqref{EqKzODE} by $\bar v$ and integrating over $\hat r\in(\hat r^e,\infty)$, we may integrate by parts in view of $|v|=\mc O(\hat r^{-1})$ and $|v'|=\mc O(\hat r^{-2})$ as $\hat r\to\infty$. For $\ell=0$, we obtain $v'=0$, hence $v$ is constant and therefore must vanish since $v$ is required to decay at infinity; for $\ell\geq 1$, we obtain $v=0$ directly.
When $m,\fa\neq 0$, the rescaling of~\eqref{EqKzODE} by $\hat\mu$ is of regular-singular type at $\hat\mu=0$, and by~\eqref{EqKzPhi} and \eqref{EqKzvvstar}, we have $v(\hat r)=(\hat r-\hat r^e)^{i m\fa/\beta}w(\hat r)$ where $w(\hat r)$ is smooth down to $\hat r=\hat r^e$. The Wronskian
\[
W:=\Im\bigl(v(\hat r)\mu D_{\hat r}\bar v(\hat r)\bigr)
\]
is constant, but decays to zero as $\hat r\to\infty$, and hence $W=0$. On the other hand, by evaluating its limit as $\hat r\searrow\hat r^e$, one finds $W=m\hat\fa|w(\hat r^e)|^2$; thus, $w(\hat r^e)=0$, and since the other indicial root of~\eqref{EqKzODE} is $-i m\fa/\beta\notin i m\fa/\beta-\mathbb{N}_0$, we conclude that $w$ vanishes identically, and therefore so does $v_*$ in $\hat r\geq\hat r^e$.
Having shown that $v_*=0$ on $[\hat r^e,\infty)$, we obtain $v_*=0$ also on $[1,\hat r^e]$ since $v_*(\hat r)$ vanishes to infinite order at $\hat r=\hat r^e$ and satisfies $0=\hat\mu\hat\varrho^2\Box_{\hat g}(0)(v_* Y_{\ell m})$, which is a regular-singular ODE at $\hat r=\hat r^e$.
\end{proof}
Next, the transition between zero and nonzero frequencies is governed by a model operator on an exact cone; for purely imaginary spectral parameters, this was introduced in \cite{GuillarmouHassellResI}, while in the present context of real spectral parameters, this model operator was introduced in \cite[\S5]{GuillarmouHassellSikoraResIII}; see also \cite[Definition~2.4, \S5]{VasyLowEnergyLag}. In the following result, we work on the transition face ${\mathrm{tf}}\subset\hat X_{\mathrm{sc}\text{-}\mathrm{b}}$, which (recalling the coordinates~\eqref{EqqCoord2} and \eqref{EqQCoord}) is
\[
{\mathrm{tf}} = [0,\infty]_{\tilde r}\times\mathbb{S}^2,\qquad \tilde r=|\tilde\sigma|\hat r
\]
by Proposition~\ref{PropQStruct}\eqref{ItQStructnf}. Concretely, the ${\mathrm{tf}}$-normal operator of $\tilde\sigma^{-2}\Box_{\hat g}(\tilde\sigma)$ is
\begin{equation}
\label{EqKtfOp}
\Box_{\mathrm{tf}}(1) := N_{\mathrm{tf}}(\Box_{\hat g}(\cdot)) = \tilde\Delta + 1,\qquad \tilde\Delta=D_{\tilde r}^2-\frac{2 i}{\tilde r}D_{\tilde r} + \tilde r^{-2}\Delta_\slashed{g}{},
\end{equation}
see \cite[\S\S4.1 and 6]{VasyLowEnergy}.\footnote{This is the conjugation of the model operator in \cite[Definition~2.20]{HintzPrice} by $e^{i\tilde r}$. We remark that in \cite{HintzPrice}, which is based on \cite{VasyLowEnergyLag}, the analytic setup focuses on precise second microlocal/module regularity at the outgoing radial set, whereas in the present paper variable order estimates are sufficient.} On ${\mathrm{tf}}$, we work with the volume density $\tilde r^2|{\mathrm d}\tilde r\,{\mathrm d}\slashed{g}{}|$, and with Sobolev spaces
\[
H_{{\mathrm{sc}},{\mathrm{b}}}^{s,\mathsf{r},l}({\mathrm{tf}})
\]
which are scattering Sobolev spaces near $\tilde\rho=0$ (with variable decay order $\mathsf{r}$) and b-Sobolev spaces near $\tilde r=0$ (with decay order $l$ there). Note that
\[
\Box_{\mathrm{tf}}(1) \in \mathrm{Diff}_{{\mathrm{sc}},{\mathrm{b}}}^{2,0,2}({\mathrm{tf}}) = \Bigl(\frac{\tilde r}{\tilde r+1}\Bigr)^{-2}\mathrm{Diff}_{{\mathrm{sc}},{\mathrm{b}}}^2({\mathrm{tf}})
\]
is an unweighted scattering operator near $\tilde\rho=0$, and a weighted b-operator near $\tilde r=0$. The b-normal operator of $\tilde r^2\Box_{\mathrm{tf}}(1)$ at $\tilde r=0$ is $(\tilde r D_{\tilde r})^2-i\tilde r D_{\tilde r}+\Delta_\slashed{g}{}$, with indicial solutions $\hat r^{-\ell-1}Y_{\ell m}$ and $\hat r^\ell Y_{\ell m}$; the range $(\frac12,\frac32)$ of weights in Lemma~\ref{LemmaKtf} disallows the former, more singular, solution. The outgoing and incoming radial sets are as usual the graphs at $\tilde\rho=\tilde r^{-1}=0$ of ${\mathrm d}\tilde r$ and $-{\mathrm d}\tilde r$, respectively.
\begin{lemma}[Estimates for the ${\mathrm{tf}}$-normal operator]
\label{LemmaKtf}
Let $s\in\mathbb{R}$, $l\in(\frac12,\frac32)$, and suppose $\mathsf{r}\in\cC^\infty(\overline{{}^{{\mathrm{sc}},{\mathrm{b}}}T^*_{\hat\rho^{-1}(0)}}{\mathrm{tf}})$ is a variable order function which is monotone along the flow of the Hamiltonian vector field of the principal symbol of $\Box_{\mathrm{tf}}(1)$, and which satisfies $\mathsf{r}>-{\tfrac{1}{2}}$, resp.\ $\mathsf{r}<-{\tfrac{1}{2}}$ at the incoming, resp.\ outgoing radial set. Then there exists a constant $C>0$ so that
\begin{equation}
\label{EqKtfEst}
\|u\|_{H_{{\mathrm{sc}},{\mathrm{b}}}^{s,\mathsf{r},l}({\mathrm{tf}})} \leq C \| \Box_{\mathrm{tf}}(1)u \|_{H_{{\mathrm{sc}},{\mathrm{b}}}^{s-2,\mathsf{r}+1,l-2}({\mathrm{tf}})}.
\end{equation}
\end{lemma}
\begin{proof}
Radial point estimates at the scattering end $\tilde\rho=0$, and elliptic b-estimates at the small end $\tilde r=0$ of the cone ${\mathrm{tf}}$ give the estimate~\eqref{EqKtfEst} except for the presence of an additional, relatively compact, error term $C\|u\|_{H_{{\mathrm{sc}},{\mathrm{b}}}^{-N,-N,-N}({\mathrm{tf}})}$ on the right. The estimate~\eqref{EqKtfEst} thus follows from the nonexistence of outgoing elements in the kernel of $\Box_{\mathrm{tf}}(1)$, which is standard; it can be proved upon separation into spherical harmonics using Wronskian arguments, or by inspection of the asymptotic behavior of the explicit (Bessel function) solutions as done in \cite[\S\S3.4--3.5]{GuillarmouHassellResI} or \cite[Lemma~5.10]{HintzConicProp}.
\end{proof}
Lemmas~\ref{LemmaKz} and \ref{LemmaKtf} provide the normal operator estimates for the uniform low energy analysis of $\Box_{\hat g}(\cdot)\in\Diff_\scbtop^{2,0,2,0}(\overline{\hat\Omega})\subset\Psi_\scbtop^{2,0,2,0}(\overline{\hat\Omega})$ on the ${\mathrm{sc}\text{-}\mathrm{b}}$-transition-Sobolev spaces $\bar H_{{\mathrm{sc}\text{-}\mathrm{b}},\tilde\sigma}^{s,\mathsf{r},\gamma,l'}(\hat\Omega)$ introduced in~\S\ref{SsPH}, with $\gamma$ and $l'$ the weights at ${\mathrm{tf}}$ and ${\mathrm{zf}}$, respectively, and $\mathsf{r}\in\cC^\infty(\overline{{}^\scbtop T^*_{\mathrm{scf}}}\hat X)$ denoting a variable scattering decay order function. Near ${\mathrm{scf}}\subset\hat X_{\mathrm{sc}\text{-}\mathrm{b}}$, a defining function of ${\mathrm{scf}}$ is $\tilde\rho=\tilde r^{-1}$, and thus we can write ${\mathrm{sc}\text{-}\mathrm{b}}$-covectors (cf.\ \eqref{EqPscbtFrame}) as
\[
-\xi\frac{{\mathrm d}\tilde\rho}{\tilde\rho^2}+\frac{\eta}{\tilde\rho} = \xi\,{\mathrm d}\tilde r + \tilde r\eta = |\tilde\sigma| \bigl( \xi\,{\mathrm d}\hat r+\hat r\eta \bigr)
\]
where $\eta\in T^*\mathbb{S}^2$. For $\tilde\sigma>0$, the outgoing (incoming) radial set is then given by $\xi=1$ ($\xi=-1$), $\eta=0$, $\hat\rho=0$, and the signs are reversed when $\tilde\sigma<0$.
\begin{prop}[Uniform bounds on Kerr near zero energy]
\label{PropKnfZ}
Let $s>{\tfrac{1}{2}}$, $l,\gamma\in\mathbb{R}$, and suppose $\gamma-l\in(-\frac32,-{\tfrac{1}{2}})$. Suppose $\mathsf{r}$ is a variable order function that is monotone along the Hamiltonian flow of the principal symbol of $\Box_{\mathrm{tf}}(1)$, and which satisfies $\mathsf{r}>{\tfrac{1}{2}}$, resp.\ $\mathsf{r}<-{\tfrac{1}{2}}$ at the incoming, resp.\ outgoing radial set. Then there exists $C>0$ so that, for $\tilde\sigma\in\pm[0,1]$, we have
\begin{equation}
\label{EqKnfZ}
\|u\|_{\bar H_{{\mathrm{sc}\text{-}\mathrm{b}},\tilde\sigma}^{s,\mathsf{r},\gamma,l}(\hat\Omega)} \leq C\|\Box_{\hat g}(\tilde\sigma)u\|_{\bar H_{{\mathrm{sc}\text{-}\mathrm{b}},\tilde\sigma}^{s-1,\mathsf{r}+1,\gamma+2,l}(\hat\Omega)}.
\end{equation}
\end{prop}
These estimates are closely related to those proved by Vasy in~\cite{VasyLowEnergyLag}; but whereas Vasy uses a second microlocal algebra which allows for precise module regularity control at the outgoing radial set (roughly speaking allowing the order $\mathsf{r}$ to be constant---and thus high---except for a jump right at the outgoing radial set), we prove a less precise estimate on variable order spaces here. Thus, using the simpler ${\mathrm{sc}\text{-}\mathrm{b}}$-ps.d.o.\ algebra already introduced by Guillarmou--Hassell \cite{GuillarmouHassellResI}, we are still able to prove uniform low energy resolvent estimates.
\begin{proof}[Proof of Proposition~\usref{PropKnfZ}]
Via multiplication by $|\tilde\sigma|^l$, one may reduce to the case that $l=0$. When $|\tilde\sigma|$ is bounded away from $0$, the estimate~\eqref{EqKnfZ} is the content of Proposition~\ref{PropKnfNz}. Symbolic estimates (which at the incoming radial set only require $\mathsf{r}>-{\tfrac{1}{2}}$) give
\[
\|u\|_{\bar H_{{\mathrm{sc}\text{-}\mathrm{b}},\tilde\sigma}^{s,\mathsf{r},\gamma,0}(\hat\Omega)} \leq C\bigl( \|\Box_{\hat g}(\tilde\sigma)u\|_{\bar H_{{\mathrm{sc}\text{-}\mathrm{b}},\tilde\sigma}^{s-1,\mathsf{r}+1,\gamma+2,0}(\hat\Omega)} + \|u\|_{\bar H_{{\mathrm{sc}\text{-}\mathrm{b}},\tilde\sigma}^{s_0,\mathsf{r}_0,\gamma,0}(\hat\Omega)}\bigr)
\]
for any $s_0<s$ and $\mathsf{r}_0<\mathsf{r}$; we shall take $s_0\in({\tfrac{1}{2}},s)$, and choose $\mathsf{r}_0<\mathsf{r}-1$ with $\mathsf{r}_0>-{\tfrac{1}{2}}$ at the incoming radial set and monotone along the Hamiltonian flow. Let $\chi=\chi(\tilde\sigma/\hat\rho)=\chi(\tilde r)\in\cC^\infty_\cp([0,1))$ denote a cutoff, identically $1$ near $0$, to a neighborhood of ${\mathrm{zf}}$. Then by writing $u=\chi(\tilde\rho)u+(1-\chi(\tilde\rho))u$, with the second summand supported away from ${\mathrm{zf}}$, we have
\[
\|u\|_{\bar H_{{\mathrm{sc}\text{-}\mathrm{b}},\tilde\sigma}^{s_0,\mathsf{r}_0,\gamma,0}(\hat\Omega)} \leq \|\chi(\tilde\rho)u\|_{\bar H_{{\mathrm{sc}\text{-}\mathrm{b}},\tilde\sigma}^{s_0,\mathsf{r}_0,\gamma,0}(\hat\Omega)} + C\|u\|_{\bar H_{{\mathrm{sc}\text{-}\mathrm{b}},\tilde\sigma}^{s_0,\mathsf{r}_0,\gamma,-N}(\hat\Omega)}
\]
for any fixed $N$; we take $N=1$. Moreover, uniformly for $\tilde\sigma\in[0,1]$,
\[
\|\chi(\tilde\rho)u\|_{\bar H_{{\mathrm{sc}\text{-}\mathrm{b}},\tilde\sigma}^{s_0,\mathsf{r}_0,\gamma,0}(\hat\Omega)} \leq C\|\chi(\tilde\rho)u\|_{\bar H_{\bop}^{s_0,\gamma}(\hat\Omega)}.
\]
(In fact, the norms on both sides, in the presence of the cutoff $\chi(\tilde\rho)$, are uniformly equivalent, see~\eqref{EqPHEquivzf}.) Using Lemma~\ref{LemmaKz},
\begin{align*}
\|\chi(\tilde\rho)u\|_{\bar H_{\bop}^{s_0,\gamma}(\hat\Omega)} &\leq C\bigl( \|\chi(\tilde\rho)\Box_{\hat g}(0)u\|_{\bar H_{\bop}^{s_0-1,\gamma+2}(\hat\Omega)} + \|[\Box_{\hat g}(0),\chi(\tilde\rho)]u\|_{\bar H_{\bop}^{s_0-1,\gamma+2}(\hat\Omega)}\bigr) \\
&\leq C\bigl( \|\chi(\tilde\rho)\Box_{\hat g}(0)u\|_{\bar H_{{\mathrm{sc}\text{-}\mathrm{b}},\tilde\sigma}^{s_0-1,*,\gamma+2,0}(\hat\Omega)} + \|[\Box_{\hat g}(0),\chi(\tilde\rho)]u\|_{\bar H_{{\mathrm{sc}\text{-}\mathrm{b}},\tilde\sigma}^{s_0-1,*,\gamma+2,0}(\hat\Omega)}\bigr) \\
&\leq C\bigl( \|\Box_{\hat g}(\tilde\sigma)u\|_{\bar H_{{\mathrm{sc}\text{-}\mathrm{b}},\tilde\sigma}^{s_0-1,*,\gamma+2,0}(\hat\Omega)} + \|u\|_{\bar H_{{\mathrm{sc}\text{-}\mathrm{b}},\tilde\sigma}^{s_0,*,\gamma,-1}(\hat\Omega)}\bigr),
\end{align*}
where the `$*$' indicates that the order is arbitrary; we use here that $\chi(\tilde\rho)(\Box_{\hat g}(\tilde\sigma)-\Box_{\hat g}(0))\in\Diff_\scbtop^{1,0,2,-1}(\hat\Omega)$ and $[\Box_{\hat g}(0),\chi(\tilde\rho)]\in\Diff_\scbtop^{1,-\infty,2,-\infty}(\hat\Omega)$.
We have now obtained the improved estimate
\[
\|u\|_{\bar H_{{\mathrm{sc}\text{-}\mathrm{b}},\tilde\sigma}^{s,\mathsf{r},\gamma,0}(\hat\Omega)} \leq C\bigl( \|\Box_{\hat g}(\tilde\sigma)u\|_{\bar H_{{\mathrm{sc}\text{-}\mathrm{b}},\tilde\sigma}^{s-1,\mathsf{r}+1,\gamma+2,0}(\hat\Omega)} + \|u\|_{\bar H_{{\mathrm{sc}\text{-}\mathrm{b}},\tilde\sigma}^{s_0,\mathsf{r}_0,\gamma,-1}(\hat\Omega)}\bigr).
\]
The next step is to strengthen this further by weakening the weight of the error term at ${\mathrm{tf}}$. To this end, we fix a cutoff $\psi\in\cC^\infty(\hat X_{\mathrm{sc}\text{-}\mathrm{b}})$ which is supported in a small collar neighborhood of ${\mathrm{tf}}\subset\hat X_{\mathrm{sc}\text{-}\mathrm{b}}$ and identically $1$ near ${\mathrm{tf}}$; then for any $\delta\in(0,1]$ and $N\in\mathbb{R}$, we have
\[
\|u\|_{\bar H_{{\mathrm{sc}\text{-}\mathrm{b}},\tilde\sigma}^{s_0,\mathsf{r}_0,\gamma,\delta}(\hat\Omega)} \leq \|\psi u\|_{\bar H_{{\mathrm{sc}\text{-}\mathrm{b}},\tilde\sigma}^{s_0,\mathsf{r}_0,\gamma,\delta}(\hat\Omega)} + C\|u\|_{\bar H_{{\mathrm{sc}\text{-}\mathrm{b}},\tilde\sigma}^{s_0,\mathsf{r}_0,-N,\delta}(\hat\Omega)}.
\]
We can estimate the first term, using~\eqref{EqPHEquivtf} and Lemma~\ref{LemmaKtf}, via pullback along the coordinate change $\phi\colon(\tilde\sigma,\tilde\rho,\omega)\mapsto(\tilde\sigma,\tilde\sigma\tilde\rho,\omega)\in[0,1]\times[0,1)_{\hat\rho}\times\mathbb{S}^2_\omega$, similarly to above by
\begin{align}
&|\tilde\sigma|^{\delta+\frac32}\|\phi^*(\psi u)\|_{H_{{\mathrm{sc}},{\mathrm{b}}}^{s_0,\mathsf{r}_0,-\gamma+\delta}({\mathrm{tf}})} \nonumber\\
&\quad\leq C|\tilde\sigma|^{\delta+\frac32}\Bigl( \|\phi^*(\psi)\Box_{\mathrm{tf}}(1)(\phi^*u)\|_{H_{{\mathrm{sc}},{\mathrm{b}}}^{s_0-2,\mathsf{r}_0+1,-\gamma+\delta-2}({\mathrm{tf}})} + \|[\Box_{\mathrm{tf}}(1),\phi^*(\psi)]\phi^*u\|_{H_{{\mathrm{sc}},{\mathrm{b}}}^{s_0,\mathsf{r}_0,-\gamma+\delta}({\mathrm{tf}})}\Bigr) \nonumber\\
\label{EqKnfZalmost}
&\quad\leq C\Bigl( \|\psi\Box_{\hat g}(\tilde\sigma)u\|_{\bar H_{{\mathrm{sc}\text{-}\mathrm{b}},\tilde\sigma}^{s_0-2,\mathsf{r}_0+1,\gamma+2,\delta}(\hat\Omega)} + \|u\|_{\bar H_{{\mathrm{sc}\text{-}\mathrm{b}},\tilde\sigma}^{s_0,\mathsf{r}_0+1,\gamma-1,\delta}(\hat\Omega)}\Bigr),
\end{align}
where we fix $\delta>0$ so small that $-\gamma+\delta\in(\frac12,\frac32)$. Here, we used that $\psi(\Box_{\hat g}(\tilde\sigma)-\phi_*(\tilde\sigma^2\Box_{\mathrm{tf}}(1)))\in\Diff_\scbtop^{2,0,-3,0}(\hat\Omega)$ (which is the statement that $\tilde\sigma^2\Box_{\mathrm{tf}}(1)$ is the ${\mathrm{tf}}$-normal operator of $\Box_{\hat g}(\tilde\sigma)$).
Altogether, increasing the ${\mathrm{tf}}$-order of the final term in~\eqref{EqKnfZalmost} to $\gamma-\delta$ (thus making this term larger) for convenience, we have shown
\begin{align*}
\|u\|_{\bar H_{{\mathrm{sc}\text{-}\mathrm{b}},\tilde\sigma}^{s,\mathsf{r},\gamma,0}(\hat\Omega)} &\leq C\bigl( \|\Box_{\hat g}(\tilde\sigma)u\|_{\bar H_{{\mathrm{sc}\text{-}\mathrm{b}},\tilde\sigma}^{s-1,\mathsf{r}+1,\gamma+2,0}(\hat\Omega)} + \|u\|_{\bar H_{{\mathrm{sc}\text{-}\mathrm{b}},\tilde\sigma}^{s_0,\mathsf{r}_0+1,\gamma-\delta,\delta}(\hat\Omega)}\bigr) \\
&\leq C\|\Box_{\hat g}(\tilde\sigma)u\|_{\bar H_{{\mathrm{sc}\text{-}\mathrm{b}},\tilde\sigma}^{s-1,\mathsf{r}+1,\gamma+2,0}(\hat\Omega)} + C|\tilde\sigma|^\delta \|u\|_{\bar H_{{\mathrm{sc}\text{-}\mathrm{b}},\tilde\sigma}^{s_0,\mathsf{r}_0+1,\gamma,0}(\hat\Omega)}.
\end{align*}
Since $s_0<s$ and $\mathsf{r}_0+1<\mathsf{r}$, the second term, for sufficiently small $|\tilde\sigma|$, can be absorbed into the left hand side. The proof is complete.
\end{proof}
\subsection{Estimates for the \texorpdfstring{${\mathrm{mf}}_{\pm,\hbar}$-}{high energy mf-}normal operator}
\label{SsKmf}
Having proved estimates for all normal operators related to the Kerr model, we now turn to the de~Sitter model at ${\mathrm{mf}}$ and prove high energy estimates. Since the de~Sitter model involves, analytically and geometrically, a cone point due to the blow-up of the spatial manifold $X$ at $0\in X$, these estimates do not follow from \cite[\S4]{VasyMicroKerrdS}. Rather, they involve propagation estimates on semiclassical cone spaces; indeed one can quote \cite[Theorem~4.10]{HintzConicProp}. The details are as follows. By Proposition~\ref{PropKS}\eqref{ItKSmf}, the ${\mathrm{mf}}$-normal operator of $\Box(\cdot+i\sigma_1)$ is the operator family $\sigma_0\mapsto\Box_{g_{\mathrm{dS}}}(\sigma_0+i\sigma_1)$. In the high energy regime $h=|\sigma_0|^{-1}\leq 1$, $\pm\sigma_0>0$, we rescale this to
\begin{equation}
\label{EqKmOp}
h\mapsto h^2\Box_{g_{\mathrm{dS}}}(\pm h^{-1}+i\sigma_1).
\end{equation}
Near the lift ${\mathrm{sf}}$ of $h=0$ to $\dot X_{\mathrm{c}\hbar}\subset{\mathrm{mf}}$, we have coordinates $\tilde h=h/r$, $r$, and $\omega\in\mathbb{S}^2$, and Q-covectors can be written as $h^{-1}(\xi\,{\mathrm d} r+r\eta)$, $\xi\in\mathbb{R}$, $\eta\in T^*\mathbb{S}^2$, as in~\eqref{EqKSyCoordif}. In view of~\eqref{EqKdS}, the semiclassical cone principal symbol of~\eqref{EqKmOp} is then
\[
(1-r^2)\xi^2 + |\eta|_{\slashed{g}{}^{-1}}^2 - \frac{1}{1-r^2}.
\]
The outgoing and incoming radial sets were computed already in~\S\ref{SssKSyif}, see~\eqref{EqKSyRif}. (Indeed, in view of Corollary~\ref{CorQBundle}, we have ${}^\Qop T_{{\mathrm{mf}}_{+,\hbar}\cap\,{\mathrm{if}}_+}X\cong{}^\chop T^*_{\mathrm{sf}}\dot X$.) Furthermore, the ${\mathrm{tf}}$-model operator of~\eqref{EqKmOp} only depends on the metric $g_{\mathrm{dS}}$ at the point $0$ where it is the Minkowski metric on $\mathbb{R}_{t_*}\times X$, and therefore the model operator is
\[
\Box_{\mathrm{tf}}(1) = D_{\tilde r}^2 - \frac{2 i}{\tilde r}D_{\tilde r} + \tilde r^2\Delta_\slashed{g}{} + 1, \qquad \tilde r=\frac{r}{h}.
\]
This is of course the same operator as in~\eqref{EqKtfOp}, since it is the restriction of $h^2\Box(\cdot+i\sigma_1)$ to the boundary face ${\mathrm{mf}}\cap{\mathrm{nf}}$ (see also Figure~\ref{FigQSingle}). Notice how what here is a model problem at high energy right at the conic singularity of the spatial de~Sitter manifold blown up at $0$ is the same as a model problem at low energy at spatial infinity of the asymptotically flat spatial Kerr manifold.
\begin{prop}[High energy estimates on de~Sitter space]
\label{PropKmf}
There exists $h_0>0$ so that the following holds. Let $s>{\tfrac{1}{2}}+C_1$, $l\in(\frac12,\frac32)$, $l'\in\mathbb{R}$ and $\mathsf{r}\in\cC^\infty(\overline{{}^\chop T^*_{\mathrm{sf}}}\dot X)$, and assume that $\mathsf{r}$ is monotone along the Hamiltonian flow of the semiclassical cone principal symbol of $\Box_{g_{\mathrm{dS}}}(\sigma_0+i\sigma_1)$ (with $h=\pm\sigma_0^{-1}\geq 0$ the semiclassical parameter), and so that $\mathsf{r}-l'>{\tfrac{1}{2}}$, resp.\ $\mathsf{r}-l'<-{\tfrac{1}{2}}$ at the incoming, resp.\ outgoing radial set. Then there exists $C>0$ so that
\[
\|u\|_{\bar H_{{\mathrm{c}},h}^{s,l,l',\mathsf{r}}(\dot\Omega)} \leq C\|h^2\Box_{g_{\mathrm{dS}}}(\pm h^{-1}+i\sigma_1)u\|_{\bar H_{{\mathrm{c}},h}^{s-1,l-2,l',\mathsf{r}+1}(\dot\Omega)},\qquad 0<h\leq h_0.
\]
(Recall here the notation $\dot\Omega$ from~\eqref{EqKSOmegahat}.)
\end{prop}
\begin{proof}
Via multiplication by $h^{l'}$, we can reduce to the case $l'=0$. Using the assumptions on $s$ and $\mathsf{r}$, symbolic estimates (which control elements of semiclassical cone Sobolev spaces in the sense of regularity $s$ and semiclassical order $\mathsf{r}$) give
\begin{equation}
\label{EqKmAlmost}
\|u\|_{\bar H_{{\mathrm{c}},h}^{s,l,0,\mathsf{r}}(\dot\Omega)} \leq C\bigl(\|h^2\Box_{g_{\mathrm{dS}}}(\pm h^{-1}+i\sigma_1)u\|_{\bar H_{{\mathrm{c}},h}^{s-1,l-2,0,\mathsf{r}+1}(\dot\Omega)} + \|u\|_{\bar H_{{\mathrm{c}},h}^{-N,l,0,\mathsf{r}_0}(\dot\Omega)}\bigr)
\end{equation}
for any fixed $N$ and $\mathsf{r}_0<\mathsf{r}$, which we fix subject to $\mathsf{r}_0<\mathsf{r}-1$, and $\mathsf{r}_0>-{\tfrac{1}{2}}$ at the incoming radial set. The error term can then be estimated in terms of the ${\mathrm{tf}}$-normal operator $\Box_{\mathrm{tf}}(1)$ by using Lemma~\ref{LemmaKtf} in a manner completely analogous to the proof of Proposition~\ref{PropKnfZ}; this is where the assumption $l\in(\frac12,\frac32)$ is used. Thus, the last, error, term in~\eqref{EqKmAlmost} can be replaced by $\|u\|_{\bar H_{{\mathrm{c}},h}^{-N,l,-1,\mathsf{r}_0+1}(\dot\Omega)}\leq C h^\delta\|u\|_{\bar H_{{\mathrm{c}},h}^{-N,l,0,\mathsf{r}}(\dot\Omega)}$ if we choose $\delta>0$ small enough so that $\mathsf{r}_0+1+\delta<\mathsf{r}$ still.\footnote{The assumption $\mathsf{r}-l'>{\tfrac{1}{2}}$ can be weakened to $\mathsf{r}-l'>-{\tfrac{1}{2}}$, see the end of the proof of \cite[Theorem~4.10]{HintzConicProp}, but we do not need this precision here.} For small $h>0$, this error term can then be absorbed into the left hand side of~\eqref{EqKmAlmost}, finishing the proof.
\end{proof}
\subsection{Absence of high energy resonances}
\label{SsKA}
By combining the estimates proved in~\S\S\ref{SsKSy}--\ref{SsKmf}, we can now show:
\begin{prop}[Uniform estimates at high energies]
\label{PropKA}
Let $s,\gamma,l',b\in\mathbb{R}$, and let $\mathsf{r}\in\cC^\infty(\overline{{}^\Qop T^*_{\mathrm{if}}}X)$ be a variable order. Suppose that $s>\frac32+C_1$, $\gamma-l\in(-\frac32,-{\tfrac{1}{2}})$, and that $\mathsf{r}-l'>{\tfrac{1}{2}}$, resp.\ $\mathsf{r}-l'<-{\tfrac{1}{2}}$ at the incoming, resp.\ outgoing radial set over ${\mathrm{if}}\cap{\mathrm{nf}}$. Suppose moreover that $\mathsf{r}$ is non-increasing along the Hamiltonian flow of the principal symbol of $\Box(\cdot+i\sigma_1)$. Let $h_0>0$ be as in Proposition~\usref{PropKmf} (i.e.\ sufficiently small). Then for any fixed $s_0<s$, $l_0<l$, $\gamma_0<\gamma$, $l'_0<l'$, $\mathsf{r}_0<\mathsf{r}$, $b_0<b$, there exists a constant $C>0$ so that for $|\sigma_0|\geq h_0^{-1}$, we have the uniform (for $|\sigma_0|\geq h_0^{-1}$, $\sigma_1\in[-C_1,C_1]$, $\fm\in(0,\fm_0]$) estimate
\begin{equation}
\label{EqKAHi}
\|u\|_{\bar H_{{\mathrm{Q}},\sigma_0,\fm}^{s,(l,\gamma,l',\mathsf{r},b)}(\Omega_{\mathrm{Q}})} \leq C\bigl( \|\Box_{g_\fm}(\sigma_0+i\sigma_1)u\|_{\bar H_{{\mathrm{Q}},\sigma_0,\fm}^{s-1,(l-2,\gamma,l'-2,\mathsf{r}-1,b)}(\Omega_{\mathrm{Q}})} + \|u\|_{\bar H_{{\mathrm{Q}},\sigma_0,\fm}^{s_0,(l_0,\gamma_0,l'_0,\mathsf{r}_0,b_0)}(\Omega_{\mathrm{Q}})}\bigr)
\end{equation}
In the remaining bounded frequency range $|\sigma_0|\leq h_0^{-1}$, we have a uniform (for $\sigma_0\in[-h_0^{-1},h_0^{-1}]$, $\sigma_1\in[-C_1,C_1]$, $\fm\in(0,\fm_0]$) estimate
\begin{equation}
\label{EqKABdd}
\|u\|_{\bar H_{{\mathrm{q}},\fm}^{s,(l,\gamma)}(\Omega_{\mathrm{q}})} \leq C\bigl( \|\Box_{g_\fm}(\sigma_0+i\sigma_1)u\|_{\bar H_{{\mathrm{q}},\fm}^{s-1,(l-2,\gamma)}(\Omega_{\mathrm{q}})} + \|u\|_{\bar H_{{\mathrm{q}},\fm}^{s_0,(l_0,\gamma)}(\Omega_{\mathrm{q}})}\bigr).
\end{equation}
\end{prop}
\begin{proof}
This follows analogously to the proof of Proposition~\ref{PropKnfZ} from successive improvements of the error term of the symbolic estimate of Proposition~\ref{PropKSy} by means of the normal operator estimates proved in~\S\S\ref{SsKnf}--\ref{SsKmf}; the function spaces are related via Proposition~\ref{PropQHRel}.
Thus, we fix $s_0<s-1$, $\mathsf{r}_0<\mathsf{r}-1$, and $b_0<b-2$ subject to the conditions that $s_0>{\tfrac{1}{2}}+C_1$, and $\mathsf{r}_0-l'>-{\tfrac{1}{2}}$ at the incoming radial set, and start with the estimate~\eqref{EqKSy}. We weaken the error term in~\eqref{EqKSy} at ${\mathrm{zf}}$: let $\chi$ be a cutoff to a neighborhood of ${\mathrm{zf}}$ as in Proposition~\ref{PropQHRel}\eqref{ItQHRelzf}, then Lemma~\ref{LemmaKz} implies (omitting the coordinate change $\phi_{\mathrm{zf}}$ from the notation)
\begin{align*}
&\|\chi u\|_{\bar H_{{\mathrm{Q}},\sigma_0,\fm}^{s_0,(l,\gamma,l',*,*)}(\Omega_{\mathrm{Q}})} \\
&\quad \leq C\langle\sigma\rangle^{l'-l}\fm^{\frac32-l}\|\chi u\|_{\bar H_{\bop}^{s_0,\gamma-l}(\hat\Omega)} \\
&\quad \leq C\langle\sigma\rangle^{(l'-2)-(l-2)}\fm^{\frac32-(l-2)} \\
&\quad\qquad \times \bigl(\|\chi\fm^{-2}\Box_{\hat g}(0)u\|_{\bar H_{\bop}^{s_0-1,\gamma-l+2}(\hat\Omega)} + \|\fm^{-2}[\Box_{\hat g}(0),\chi]u\|_{\bar H_{\bop}^{s_0-1,\gamma-l+2}(\hat\Omega)}\bigr) \\
&\quad \leq C\bigl( \|\chi\Box_{g_\fm}(\sigma_0+i\sigma_1)u\|_{\bar H_{{\mathrm{Q}},\sigma_0,\fm}^{s_0-1,(l-2,\gamma,l'-2,*,*)}(\Omega_{\mathrm{Q}})} + \|u\|_{\bar H_{{\mathrm{Q}},\sigma_0,\fm}^{s_0+1,(l-1,\gamma,l',*,*)}}\bigr),
\end{align*}
where we used $\gamma-l\in(-\frac32,-{\tfrac{1}{2}})$ in the application of Lemma~\ref{LemmaKz}, and the fact that $\chi(\Box_{g_\fm}(\sigma_0+i\sigma_1)-\fm^{-2}\Box_{\hat g}(0))\in\Diff_\Qop^{2,(1,0,2,*,*)}$ (from Proposition~\ref{PropKS}\eqref{ItKSzf}); also $\fm^{-2}[\Box_{\hat g}(0),\chi]\in\Diff_\Qop^{1,(-\infty,0,2,*,*)}$ is a fortiori of this class. Since on the other hand for Q-Sobolev norms of $(1-\chi)u$ (which is supported away from ${\mathrm{zf}}$) the weight at ${\mathrm{zf}}$ is arbitrary, we can now improve the symbolic estimate~\eqref{EqKSy} (as far as the ${\mathrm{zf}}$-weight is concerned) to
\[
\| u \|_{\bar H_{{\mathrm{Q}},\sigma_0,\fm}^{s,(l,\gamma,l',\mathsf{r},b)}(\Omega_{\mathrm{Q}})} \leq C\Bigl( \| \Box_{g_\fm}(\sigma_0+i\sigma_1)u \|_{\bar H_{{\mathrm{Q}},\sigma_0,\fm}^{s-1,(l-2,\gamma,l'-2,\mathsf{r}-1,b)}(\Omega_{\mathrm{Q}})} + \|u\|_{\bar H_{{\mathrm{Q}},\sigma_0,\fm}^{s_0+1,(l-\delta,\gamma,l',\mathsf{r}_0,b_0)}(\Omega_{\mathrm{Q}})} \Bigr)
\]
for any $\delta\in(0,1]$. For any fixed compact interval of $\sigma_0$, this implies the uniform estimate~\eqref{EqKABdd}. (The weight $l-\delta$ can be reduced to any $l_0$ using an interpolation argument.) Note also that we can apply Proposition~\ref{PropKSy} to the error term here and thereby reduce its differential order back to $s_0$.
We work on the resulting error term $\|u\|_{\bar H_{{\mathrm{Q}},\sigma_0,\fm}^{s_0,(l-\delta,\gamma,l',\mathsf{r}_0,b_0)}(\Omega_{\mathrm{Q}})}$ further by inverting the ${\mathrm{nf}}$-normal operator, which is $\fm^{-2}\Box_{\hat g}(\fm\cdot)$ by Proposition~\ref{PropKS}\eqref{ItKSnf}. Thus, reusing the symbol $\chi$ to now denote a cutoff to a collar neighborhood of ${\mathrm{nf}}$ which is identically $1$ near ${\mathrm{nf}}$, we use Proposition~\ref{PropQHRel}\eqref{ItQHRelnflow} and Proposition~\ref{PropKnfZ} to estimate, for $\tilde\sigma_0=\fm\sigma_0$ with $\tilde\sigma_0\in\pm[0,1]$,
\begin{align*}
&\|\chi u\|_{\bar H_{{\mathrm{Q}},\sigma_0,\fm}^{s_0,(l-\delta,\gamma,l',\mathsf{r}_0,*)}(\Omega_{\mathrm{Q}})} \\
&\quad \leq \fm^{\frac32-l'}\|\chi u\|_{\bar H_{{\mathrm{sc}\text{-}\mathrm{b}},\tilde\sigma_0}^{s_0,\mathsf{r}_0-l',\gamma-l',l-\delta-l'}(\hat\Omega)} \\
&\quad\leq C \fm^{\frac32-(l'-2)}\bigl( \|\chi \fm^{-2}\Box_{\hat g}(\tilde\sigma_0)u\|_{\bar H_{{\mathrm{sc}\text{-}\mathrm{b}},\tilde\sigma_0}^{s_0-1,\mathsf{r}_0-l'+1,\gamma-l'+2,l-\delta-l'}(\hat\Omega)} \\
&\quad\hspace{6em} + \|\fm^{-2}[\Box_{\hat g}(\tilde\sigma_0),\chi]u\|_{\bar H_{{\mathrm{sc}\text{-}\mathrm{b}},\tilde\sigma_0}^{s_0-1,\mathsf{r}_0-l'+1,\gamma-l'+2,l-\delta-l'}(\hat\Omega)}\bigr) \\
&\quad\leq C\bigl( \bigl\| \chi\Box(\cdot+i\sigma_1)u \|_{\bar H_{{\mathrm{Q}},\sigma_0,\fm}^{s_0-1,(l-2-\delta,\gamma,l'-2,\mathsf{r}_0-1,*)}(\Omega_{\mathrm{Q}})} + \|u\|_{\bar H_{{\mathrm{Q}},\sigma_0,\fm}^{s_0+1,(l-\delta,\gamma,l'-\delta',\mathsf{r}_0,*)}(\Omega_{\mathrm{Q}})} \bigr).
\end{align*}
Here, we fix $\delta>0$ sufficiently small so that $\gamma-(l-\delta)\in(-\frac32,-{\tfrac{1}{2}})$; and $\delta'\in(0,1]$ can be chosen arbitrarily. A completely analogous argument, now using Proposition~\ref{PropQHRel}\eqref{ItQHRelnfsemi} and Propositions~\ref{PropKSyKerr} and~\ref{PropKnfNz}, gives the high energy estimate (for $\tilde\sigma_0\in\pm[1,\infty]$)
\begin{align*}
&\|\chi u\|_{\bar H_{{\mathrm{Q}},\sigma_0,\fm}^{s_0,(l-\delta,\gamma,l',\mathsf{r}_0,b_0)}(\Omega_{\mathrm{Q}})} \\
&\quad \leq C\bigl( \bigl\| \chi\Box_{g_\fm}(\sigma_0+i\sigma_1)u \|_{\bar H_{{\mathrm{Q}},\sigma_0,\fm}^{s_0-1,(l-2-\delta,\gamma,l'-2,\mathsf{r}_0-1,b_0)}(\Omega_{\mathrm{Q}})} + \|u\|_{\bar H_{{\mathrm{Q}},\sigma_0,\fm}^{s_0+1,(l-\delta,\gamma,l'-\delta',\mathsf{r}_0+1,b_0+2)}(\Omega_{\mathrm{Q}})} \bigr).
\end{align*}
(The semiclassical order $b_0+2$ of the error term arises from the fact that $\Box(\cdot+i\sigma_1)$ differs, near ${\mathrm{sf}}$, from its ${\mathrm{nf}}$-normal operator by an operator of class $\Diff_\Qop^{2,(*,*,1,2,2)}$.) On the other hand, $(1-\chi)u$ is supported away from ${\mathrm{nf}}$, and hence for its Q-Sobolev norms the order at ${\mathrm{nf}}$ is arbitrary. We have thus established the uniform estimate
\begin{equation}
\label{EqKAAlmost}
\| u \|_{\bar H_{{\mathrm{Q}},\sigma_0,\fm}^{s,(l,\gamma,l',\mathsf{r},b)}(\Omega_{\mathrm{Q}})} \leq C\Bigl( \| \Box_{g_\fm}(\sigma_0+i\sigma_1)u \|_{\bar H_{{\mathrm{Q}},\sigma_0,\fm}^{s-1,(l-2,\gamma,l'-2,\mathsf{r}-1,b)}(\Omega_{\mathrm{Q}})} + \|u\|_{\bar H_{{\mathrm{Q}},\sigma_0,\fm}^{s_0,(l-\delta,\gamma,l'-\delta',\mathsf{r}_0,b_0)}(\Omega_{\mathrm{Q}})} \Bigr),
\end{equation}
where we used the symbolic estimate~\eqref{EqKSy} again to reduce the differential and semiclassical order to $s_0$ and $b_0$ (using $b_0+2<b$).
Finally, for $|\sigma_0|^{-1}\leq h_0$, we can apply Proposition~\ref{PropKmf} to the localization of the error term in~\eqref{EqKAAlmost} to a collar neighborhood of ${\mathrm{mf}}$ and to these high frequencies; using Proposition~\ref{PropQHRel}\eqref{ItQHRelmfsemi} to pass between Q- and semiclassical cone Sobolev spaces, and using that $l-\delta-\gamma\in(\frac12,\frac32)$ and $(\mathsf{r}_0-\gamma)-(l'-\delta')>-{\tfrac{1}{2}}$, resp.\ $<-{\tfrac{1}{2}}$ at the incoming, resp.\ outgoing radial set when $\delta,\delta'>0$ are sufficiently small, an application of Proposition~\ref{PropKmf} improves~\eqref{EqKAAlmost} to the desired estimate~\eqref{EqKAHi}.
\end{proof}
\begin{cor}[Absence of high energy resonances]
\label{CorKAHigh}
There exists $\fm_1>0$ so that for all $\fm\in(0,\fm_1]$, $\sigma_0\in\mathbb{R}$ with $|\sigma_0|\geq h_0^{-1}$, and $\sigma_1\in[-C_1,C_1]$, we have $\sigma_0+i\sigma_1\notin{\mathrm{QNM}}(\fm)$.
\end{cor}
\begin{proof}
For $|\sigma_0|\geq h_0^{-1}$, the final, error, term in the estimate~\eqref{EqKAHi} is \emph{small} compared to the left hand side, since
\[
\|u\|_{\bar H_{{\mathrm{Q}},\sigma_0,\fm}^{s_0,(l_0,\gamma_0,l'_0,\mathsf{r}_0,b_0)}(\Omega_{\mathrm{Q}})} = \fm^\delta \|u\|_{\bar H_{{\mathrm{Q}},\sigma_0,\fm}^{s_0,(l_0+\delta,\gamma_0+\delta,l'_0+\delta,\mathsf{r}_0+\delta,b_0)}(\Omega_{\mathrm{Q}})},
\]
and $l_0+\delta<l$, $\gamma_0+\delta<\gamma$, $l'_0+\delta<l'$, $\mathsf{r}_0+\delta<\mathsf{r}$, and $b_0<b$ for sufficiently small $\delta>0$. Thus, when $\fm_1>0$ is sufficiently small, then for $\fm\in(0,\fm_1]$, the estimate~\eqref{EqKAHi} implies
\[
\|u\|_{\bar H_{{\mathrm{Q}},\sigma_0,\fm}^{s,(l,\gamma,l',\mathsf{r},b)}(\Omega_{\mathrm{Q}})} \leq C\|\Box_{g_\fm}(\sigma_0+i\sigma_1)u\|_{\bar H_{{\mathrm{Q}},\sigma_0,\fm}^{s-1,(l-2,\gamma,l'-2,\mathsf{r}-1,b)}(\Omega_{\mathrm{Q}})}.
\]
This implies the claim.
\end{proof}
\subsection{Uniform control of bounded frequencies}
\label{SsKBd}
Having proved that all resonances $\sigma\in{\mathrm{QNM}}(\fm)$ with $\sigma_1=\Im\sigma\in[-C_1,C_1]$ satisfy $|\Re\sigma_0|<C_0$ for $C_0=h_0^{-1}$, we may now work with the holomorphic family
\[
B := [-C_0,C_0]+i[-C_1,C_1] \ni \sigma \mapsto \bigl( (0,\fm_0]\ni\fm \mapsto \Box_{g_\fm}(\sigma) \bigr) \in \Diff_\qop^{2,2,0}(\overline{\Omega_{\mathrm{q}}})
\]
of q-differential operators. For this family, we have the uniform estimate~\eqref{EqKABdd}; for its ${\mathrm{mf}}_\sigma$-normal operator $\Box_{g_{\mathrm{dS}}}(\sigma)$, we moreover have uniform estimates
\begin{equation}
\label{EqKBddS}
\|u\|_{\bar H_{\bop}^{s,l}(\dot\Omega)} \leq C\bigl( \|\Box_{g_{\mathrm{dS}}}(\sigma)u\|_{\bar H_{\bop}^{s-1,l-2}(\dot\Omega)} + \|u\|_{\bar H_{\bop}^{s_0,l_0}(\dot\Omega)} \bigr)
\end{equation}
for any fixed $s_0<s$, $l_0<l$ when $s>{\tfrac{1}{2}}+C_1$, $l\in(\frac12,\frac32)$. (This follows under the stronger requirement $s>\frac32+C_1$ from the relationship between q- and b-Sobolev spaces, see Proposition~\ref{PropqHRel}\eqref{ItqHRelmf}; or it follows directly by combining elliptic b-theory in $r<1$, and radial point and propagation estimates in $r\geq 1$.)
For the following result, we recall that $\dot\beta\colon\dot X\to X$ is the blow-down map (used before in Lemma~\ref{LemmaKBundleIso}), and we recall $\Omega_{\mathrm{dS}}:=B(0,2)\subset X$ from Definition~\ref{DefKFamily}.
\begin{lemma}[Properties of the spectral family on de~Sitter space]
\label{LemmaKBddS}
Let $s>{\tfrac{1}{2}}+C_1$ and $l\in(\frac12,\frac32)$. Then for all $\sigma\in B$, the operator\footnote{Note that $\Box_{g_{\mathrm{dS}}}(\sigma)-\Box_{g_{\mathrm{dS}}}(0)\in r^{-2}\Diff_\bop^1(\overline{\dot\Omega})$, and therefore the domain in~\eqref{EqKBddSOp} can be defined equivalently using $\Box_{g_{\mathrm{dS}}}(\sigma)$ in place of $\Box_{g_{\mathrm{dS}}}(0)$.}
\begin{equation}
\label{EqKBddSOp}
\Box_{g_{\mathrm{dS}}}(\sigma) \colon \bigl\{ u\in\bar H_{\bop}^{s,l}(\dot\Omega) \colon \Box_{g_{\mathrm{dS}}}(0)u\in\bar H_{\bop}^{s-1,l-2}(\dot\Omega) \bigr\} \to \bar H_{\bop}^{s-1,l-2}(\dot\Omega)
\end{equation}
is Fredholm and has index $0$. Moreover, if $u$ lies in its kernel, then $u=\dot\beta^*v$ where $v\in\cC^\infty(\overline{\Omega_{\mathrm{dS}}})$.
\end{lemma}
\begin{proof}
We complement~\eqref{EqKBddS} by an analogous estimate for the adjoint operator on the dual function spaces,
\[
\|v\|_{\dot H_{\bop}^{-s+1,-l+2}\bigl(\overline{\dot\Omega}\bigr)} \leq C\bigl( \|\Box_{g_{\mathrm{dS}}}(\sigma)^*v\|_{\dot H_{\bop}^{-s,-l}\bigl(\overline{\dot\Omega}\bigr)} + \|v\|_{\dot H_{\bop}^{s_1,l_1}\bigl(\overline{\dot\Omega}\bigr)}\bigr)
\]
for any $s_1<-s+1$, $l_1<-l+2$. This is proved as in \cite[\S4]{VasyMicroKerrdS} (see also \cite{ZworskiRevisitVasy}) using radial point and propagation estimates which propagate in the opposite direction compared to the proof of~\eqref{EqKBddS}, with the caveat that at the conic singularity $r=0$, one uses elliptic b-theory and $-l+2\in({\tfrac{1}{2}},\frac32)$. Together with~\eqref{EqKBddS}, this implies that~\eqref{EqKBddSOp} is Fredholm.
The high energy estimates of Proposition~\ref{PropKmf} imply that~\eqref{EqKBddSOp} is injective when $|\Re\sigma|$ is sufficiently large. One can similarly prove adjoint versions of the high energy estimates of Proposition~\ref{PropKmf}, which imply the triviality of the kernel of the adjoint $\Box_{g_{\mathrm{dS}}}(\sigma)^*$ on $\dot H_{\bop}^{-s+1,-l+2}(\overline{\dot\Omega})$. Thus, the operator~\eqref{EqKBddSOp} is invertible for large $|\Re\sigma|$, and therefore Fredholm of index $0$ for all $\sigma\in B$ since the Fredholm index is constant.
If $u\in\ker\Box_{g_{\mathrm{dS}}}(\sigma)$, then $u\in\bar H_{\bop}^{\infty,l}(\dot\Omega)$ by elliptic regularity and radial point and propagation estimates. But interpolating between the maps $\bar H^0_{\mathrm{b}}(\dot\Omega)\hookrightarrow L^2(\Omega_{\mathrm{dS}})$ and $\bar H^{1,1}_{\mathrm{b}}(\dot\Omega)\hookrightarrow\bar H^1(\Omega_{\mathrm{dS}})$ implies that $u=\beta^*v$ where $v\in\bar H^l(\Omega_{\mathrm{dS}})\cap\cC^\infty(\overline{\Omega_{\mathrm{dS}}}\setminus\{0\})$. Therefore $\Box_{g_{\mathrm{dS}}}(\sigma)v$, as an extendible distribution on $\Omega_{\mathrm{dS}}$, has support in $\{0\}$ but Sobolev regularity $\geq l-2$ (since $\Box_{g_{\mathrm{dS}}}(\sigma)\in\mathrm{Diff}^2(\Omega_{\mathrm{dS}})$). Since $l-2>-\frac32$, we must have $\Box_{g_{\mathrm{dS}}}(\sigma)v=0$, and therefore $v$ is smooth near $0$ by elliptic regularity. (One can also prove this directly by expanding $u$ near $r=0$ into spherical harmonics and solving the resulting family of regular-singular ODEs at $r=0$.)
\end{proof}
Similarly, we can complement~\eqref{EqKABdd} by a uniform adjoint estimate
\[
\|u\|_{\dot H_{{\mathrm{q}},\fm}^{-s+1,(-l+2,-\gamma)}(\overline{\Omega_{\mathrm{q}}})} \leq C\bigl( \|\Box_{g_\fm}(\sigma_0+i\sigma_1)^*u\|_{\dot H_{{\mathrm{q}},\fm}^{-s,(-l,-\gamma)}(\overline{\Omega_{\mathrm{q}}})} + \|u\|_{\dot H_{{\mathrm{q}},\fm}^{s_0,(l_0,-\gamma)}(\overline{\Omega_{\mathrm{q}}})}
\]
for $s_0<-s+1$, $l_0<-l+2$. For any $\fm>0$, the two estimates together imply that
\begin{equation}
\label{EqKBdMap}
\Box_{g_\fm}(\sigma) \colon \mc H_\fm^s := \bigl\{ u\in\bar H^s(\overline{\Omega_\fm}) \colon \Box_{g_\fm}(0)u\in\bar H^{s-1}(\overline{\Omega_\fm}) \bigr\} \to \bar H^{s-1}(\overline{\Omega_\fm})
\end{equation}
is Fredholm; and it is invertible for $\sigma=\sigma_0+i\sigma_1$, $\sigma_1\in[-C_1,C_1]$, provided $|\Re\sigma_0|$ is sufficiently large, as follows from the absence of a kernel in this semiclassical regime (proved in Corollary~\ref{CorKAHigh}) and of a cokernel (proved by means of an adjoint version of the estimate~\eqref{EqKAHi}). Thus, the map~\eqref{EqKBdMap} has Fredholm index $0$ and a meromorphic inverse.
The following two complementary results describe KdS QNMs for small masses as perturbations of dS QNMs.
\begin{prop}[Absence of QNMs away from de~Sitter QNMs]
\label{PropKBdNo}
Suppose $\sigma_*\in B$ is such that $\cC^\infty(\overline{\Omega_{\mathrm{dS}}})\cap\ker\Box_{g_{\mathrm{dS}}}(\sigma_*)$ is trivial. Then there exists $\epsilon>0$ so that for all $\fm\in(0,\epsilon]$ and $\sigma\in R$ with $|\sigma-\sigma_*|\leq\epsilon$, we have $\sigma\notin{\mathrm{QNM}}(\fm)$.
\end{prop}
By Lemma~\ref{LemmaKdSQNM}, the assumption on $\sigma_*$ is equivalent to $\sigma_*\notin -i\mathbb{N}_0$.
\begin{proof}[Proof of Proposition~\usref{PropKBdNo}]
In view of the uniform Fredholm estimates for the spectral family of $\Box_{g_{\mathrm{dS}}}$, the assumption is satisfied for an open set of $\sigma_*$ (see \cite[\S2.7]{VasyMicroKerrdS} for the relevant functional analysis). Thus, when $\epsilon>0$ is sufficiently small, then for $\sigma\in B$ with $|\sigma-\sigma_*|\leq\epsilon$, we have
\[
\|u\|_{\bar H_{\bop}^{s,l}(\dot\Omega)} \leq C\|\Box_{g_{\mathrm{dS}}}(\sigma)u\|_{\bar H_{\bop}^{s-1,l-2}(\dot\Omega)}
\]
for any fixed $s>{\tfrac{1}{2}}+C_1$ and $l\in(\frac12,\frac32)$. Using this estimate, with ${\tfrac{1}{2}}+C_1<s_0<s-1$ and $l_0-\gamma$ in place of $s,l$, we can improve the error term in~\eqref{EqKABdd} (applied with $s>\frac32+C_1$) to $\|u\|_{\bar H_{{\mathrm{q}},\fm}^{s_0+1,(l_0,\gamma-1)}(\Omega_{\mathrm{q}})}$ (provided $l_0<l$ is sufficiently close to $l$ so that $l_0-\gamma\in(\frac12,\frac32)$ still) by exploiting the relationship between q- and b-Sobolev spaces near ${\mathrm{mf}}_{\mathrm{q}}$, see Proposition~\ref{PropqHRel}\eqref{ItqHRelmf}. But this new error term is now small when $\fm>0$ is sufficiently small, and can thus be absorbed into the left hand side of~\eqref{EqKABdd}. The resulting estimate, for $\fm\leq\epsilon$, is
\[
\|u\|_{\bar H_{{\mathrm{q}},\fm}^{s,(l,\gamma)}(\Omega_{\mathrm{q}})} \leq C \|\Box_{g_\fm}(\sigma)u\|_{\bar H_{{\mathrm{q}},\fm}^{s-1,(l-2,\gamma)}(\Omega_{\mathrm{q}})},\qquad |\sigma-\sigma_*|\leq\epsilon.
\]
This implies the triviality of $\ker\Box_{g_\fm}(\sigma)$ and finishes the proof.
\end{proof}
\begin{prop}[Kerr--de~Sitter QNMs near de~Sitter QNMs]
\label{PropKBdYes}
Let $\sigma_*\in B^\circ\cap{\mathrm{QNM}}_{\mathrm{dS}}$, and let $\epsilon_0>0$ be so small that ${\mathrm{QNM}}_{\mathrm{dS}}\cap\{|\sigma-\sigma_*|\leq 2\epsilon_0\}=\{\sigma_*\}$. Then for sufficiently small $\epsilon\in(0,\epsilon_0]$, there exists $\fm_1>0$ so that
\begin{equation}
\label{EqKBdYes}
m_{\mathrm{dS}}(\sigma_*) = \sum_{|\sigma-\sigma_*|\leq\epsilon} m_\fm(\sigma),\qquad \fm\in(0,\fm_1].
\end{equation}
Moreover, for all sufficiently small $r_0>0$ and $K:=[r_0,2]_r\times\mathbb{S}^2$, the restriction of $\sum_{|\sigma-\sigma_*|\leq\epsilon} \operatorname{Res}_\fm(\sigma)$ to $[0,1]_{t_*}\times K$ converges to $\operatorname{Res}_{\mathrm{dS}}(\sigma_*)$ in the topology of $\cC^\infty([0,1]\times K)$.
\end{prop}
\begin{proof}
The proof is an elaboration on \cite[Theorem~1.1]{HintzXieSdS}. Thus, for $s>\frac32+C_1$ and $l\in({\tfrac{1}{2}},\frac32)$, let
\[
K_0 = \ker_{\bar H_{\bop}^{s,l}(\dot\Omega)}\Box_{g_{\mathrm{dS}}}(\sigma_*),\qquad
K_0^* = \ker_{\dot H_{\bop}^{-s+1,-l-2}\bigl(\overline{\dot\Omega}\bigr)} \Box_{g_{\mathrm{dS}}}(\sigma_*)^*.
\]
(Note that $e^{-i\sigma_* t_*}K_0\subseteq\operatorname{Res}_{\mathrm{dS}}(\sigma_*)$, but equality need not hold.) By Lemma~\ref{LemmaKBddS}, the spaces $K_0$ and $K_0^*$ have equal dimension $d\geq 1$. Choose $r^\flat>0$ and functions $u_j^\flat,u_j^\sharp\in\cC^\infty_\cp(\dot\Omega\cap\{r>r^\flat\})$, $j=1,\ldots,d$, so that the maps $K_0\ni u\mapsto(\langle u,u_j^\flat\rangle)_{j=1,\ldots,d}\in\mathbb{C}^d$ and $K_0^*\ni u^*\mapsto(\langle u^*,u_j^\sharp\rangle)_{j=1,\ldots,d}\in\mathbb{C}^d$ are isomorphisms. Define the operators
\begin{alignat*}{3}
R_+ &\colon &\bar H_{\bop}^{s,l}(\dot\Omega) &\ni u && \mapsto ( \langle u,u_j^\flat\rangle )_{j=1,\ldots, d} \in \mathbb{C}^d, \\
R_- &\colon &\mathbb{C}^d &\ni (w_j)_{j=1,\ldots,d} && \mapsto \sum_{j=1}^d w_j u_j^\sharp \in \cC^\infty_\cp(\dot\Omega\setminus\partial\dot X).
\end{alignat*}
Recalling the definition of $\mc H_\fm^s$ from~\eqref{EqKBdMap}, the operator
\[
P_\fm(\sigma) := \begin{pmatrix} \Box_{g_\fm}(\sigma) & R_- \\ R_+ & 0 \end{pmatrix} \colon \mc H_\fm^s \oplus \mathbb{C}^d \to \bar H_{\bop}^{s-1,l-2}(\Omega_\fm) \oplus \mathbb{C}^d
\]
is Fredholm of index $0$.
The uniform estimate~\eqref{EqKABdd} (with $\gamma=0$) for $\Box_{g_\fm}(\sigma)$ for $\sigma\in B$ implies
\begin{equation}
\label{EqKBdYesAlmost}
\|(u,w)\|_{\bar H_{{\mathrm{q}},\fm}^{s,(l,0)}(\Omega_{\mathrm{q}}) \oplus \mathbb{C}^d} \leq C\bigl( \| P_\fm(\sigma)(u,w) \|_{\bar H_{{\mathrm{q}},\fm}^{s-1,(l-2,0)}(\Omega_{\mathrm{q}})\oplus\mathbb{C}^d} + \|(u,w)\|_{\bar H_{{\mathrm{q}},\fm}^{s_0,(l_0,0)}(\Omega_{\mathrm{q}})\oplus\mathbb{C}^d} \bigr).
\end{equation}
But now the ${\mathrm{mf}}_{\mathrm{q}}$-normal operator
\[
P_{\mathrm{dS}}(\sigma) := \begin{pmatrix} \Box_{g_{\mathrm{dS}}}(\sigma) & R_- \\ R_+ & 0 \end{pmatrix}
\]
has trivial nullspace for $\sigma=\sigma_*$ by construction, and thus for $|\sigma-\sigma_*|\leq 2\epsilon$ if we shrink $\epsilon>0$; we may assume that $2\epsilon$ is smaller than the distance from $\sigma_*$ to $\partial B$. Therefore, $P_{\mathrm{dS}}$ obeys an estimate
\[
\|(u,w)\|_{\bar H_{\bop}^{s,l}(\dot\Omega)\oplus\mathbb{C}^d} \leq C\| P_{\mathrm{dS}}(\sigma)u \|_{\bar H_{\bop}^{s-1,l-2}(\dot\Omega)\oplus\mathbb{C}^d},\qquad |\sigma-\sigma_*|\leq 2\epsilon.
\]
As in the proof of Proposition~\ref{PropKBdNo}, this can then be used to weaken the norm on the error term in~\eqref{EqKBdYesAlmost} to $\|(u,0)\|_{\bar H_{{\mathrm{q}},\fm}^{s_0+2,(l_0,-1)}(\Omega_{\mathrm{q}})\oplus\mathbb{C}^d}$; this weakened error term can be absorbed into the left hand side of~\eqref{EqKBdYesAlmost} when $\fm\in(0,\fm_1]$ for a sufficiently small $\fm_1>0$, and for all $\sigma\in\mathbb{C}$ with $|\sigma-\sigma_*|\leq 2\epsilon$. (Here, as in the proof of Proposition~\ref{PropKBdNo}, we need to use $s>\frac32+C_1$ and take ${\tfrac{1}{2}}+C_1<s_0<s-1$.) Therefore, the operator $P_\fm(\sigma)$ is injective and thus invertible for such $\fm,\sigma$; we write its inverse as
\[
P_\fm(\sigma)^{-1} = \begin{pmatrix} A_\fm(\sigma) & B_\fm(\sigma) \\ C_\fm(\sigma) & D_\fm(\sigma) \end{pmatrix},\qquad \fm\in(0,\fm_1],\quad |\sigma-\sigma_*|\leq 2\epsilon.
\]
By the Schur complement formula, $\Box_{g_\fm}(\sigma)$ is invertible on $\cC^\infty(\overline{\Omega_\fm})$ (or, equivalently, as a map~\eqref{EqKBdMap}) if and only if the $d\times d$ matrix $D_\fm(\sigma)$ is invertible; concretely, we have
\begin{align}
\label{EqKBdYesBoxInv}
\Box_{g_\fm}(\sigma)^{-1} &= A_\fm(\sigma)-B_\fm(\sigma)D_\fm(\sigma)^{-1}C_\fm(\sigma), \\
D_\fm(\sigma)^{-1} &= -R_+ \Box_{g_\fm}(\sigma)^{-1} R_-. \nonumber
\end{align}
Upon setting
\[
m'_\fm(\sigma') := \frac{1}{2\pi i}\operatorname{tr}\oint_{\sigma'} D_\fm(\sigma)^{-1}\partial_\sigma D_\fm(\sigma)\,{\mathrm d}\sigma
\]
these formulas imply $m_\fm(\sigma')\leq m'_\fm(\sigma')$ and $m'_\fm(\sigma')\leq m_\fm(\sigma)$, and therefore $m_\fm(\sigma)=m'_\fm(\sigma')$. (Since $D_\fm(\sigma)$ is an analytic family in $\sigma$ of $d\times d$ matrices, $m_\fm(\sigma')$ is the order of vanishing of $\det D_\fm(\sigma)$ at $\sigma=\sigma'$.) We similarly have
\[
P_{\mathrm{dS}}(\sigma)^{-1} = \begin{pmatrix} A_{\mathrm{dS}}(\sigma) & B_{\mathrm{dS}}(\sigma) \\ C_{\mathrm{dS}}(\sigma) & D_{\mathrm{dS}}(\sigma) \end{pmatrix},\qquad
m_{\mathrm{dS}}(\sigma')=\frac{1}{2\pi i}\oint_{\sigma'}D_{\mathrm{dS}}(\sigma)^{-1}\partial_\sigma D_{\mathrm{dS}}(\sigma)\,{\mathrm d}\sigma.
\]
Set $D_0(\sigma):=D_{\mathrm{dS}}(\sigma)$. We then claim that $D_\fm(\sigma)$ is continuous in $\fm\in[0,\fm_1]$ with values in holomorphic families (in $|\sigma-\sigma_*|\leq\frac32\epsilon$) of $d\times d$ matrices; to this end, since $D_0(\sigma)$ is holomorphic, it suffices to prove the continuity of $D_\fm(\sigma)$ in $\fm$ for any fixed $\sigma$ with $|\sigma-\sigma_*|\leq\frac32\epsilon$. Thus, let $w\in\mathbb{C}^d$ and consider
\[
(u_\fm,w_\fm) = P_\fm(\sigma)^{-1} (0,w);
\]
we need to show that $w_\fm=D_\fm(\sigma)w\to D_{\mathrm{dS}}(\sigma)w$ as $\fm\searrow 0$. But $u_\fm\in\bar H_{{\mathrm{q}},\fm}^{s,(l,0)}(\Omega_{\mathrm{q}})$ and $w_\fm\in\mathbb{C}^d$ are uniformly bounded. Fixing $\chi\in\cC^\infty_\cp([0,1))$ with $\chi=1$ on $[0,{\tfrac{1}{2}}]$, this implies in view of Proposition~\ref{PropqHRel}\eqref{ItqHRelmf} that $u'_\fm:=\chi(\fm/r)u_\fm\in\bar H_{\bop}^{s,l}(\dot\Omega)$ is uniformly bounded. Upon passing to a subsequence of black hole masses $\fm_j$ with $\fm_j\searrow 0$ as $j\to\infty$, we may assume that $u'_{\fm_j}\rightharpoonup u_0\in\bar H_{\bop}^{s,l}(\dot\Omega)$ and $w_{\fm_j}\to w_0$. When $\fm_j$ is so small that $\chi(\fm_j/r)=1$ for $r>r^\flat$, then $u'_{\fm_j}|_{r>r^\flat}=u_{\fm_j}|_{r>r^\flat}$ and therefore $R_+ u'_{\fm_j}=R_+ u_{\fm_j}=w$; thus, by taking the weak limit of
\[
P_{\fm_j}(\sigma)(u'_{\fm_j},w_{\fm_j})=\bigl([\Box_{g_{\fm_j}},\chi(\fm_j/r)]u_{\fm_j},w\bigr),
\]
as $j\to\infty$, we obtain
\[
\Box_{g_{\mathrm{dS}}}(\sigma)u_0 + R_- w_0 = 0,\qquad
R_+ u_0 = w.
\]
But since $P_{\mathrm{dS}}(\sigma)$ is invertible, we must have $(u_0,w_0)=P_{\mathrm{dS}}(\sigma)^{-1}(0,w)$, so $w_0=D_{\mathrm{dS}}(\sigma)w$. The weak subsequential limit $(u_0,w_0)=(B_{\mathrm{dS}}(\sigma)w,D_{\mathrm{dS}}(\sigma)w)$ is therefore unique, and in particular $w_\fm\to D_{\mathrm{dS}}(\sigma) w$, as claimed. For later use, we note that for any fixed $r_0>0$, this also shows that $(B_\fm(\sigma)w)|_{r>r_0}=u_\fm|_{r>r_0}=u'_\fm|_{r>r_0}\rightharpoonup u_0|_{r>r_0}=(B_{\mathrm{dS}}(\sigma)w)|_{r>r_0}$ in $\bar H^s([r_0,2]\times\mathbb{S}^2)$ as $\fm\searrow 0$ (where the second equality holds when $\fm$ is so small that $\chi(\fm/r)=1$ for $r>r_0$), and since $s$ here is arbitrary, we indeed have strong convergence
\begin{equation}
\label{EqKBdYesB}
(B_\fm(\sigma)w)|_{r>r_0} \to (B_{\mathrm{dS}}(\sigma)w)|_{r>r_0}\quad \text{in}\ \cC^\infty([r_0,2]\times\mathbb{S}^2),
\end{equation}
uniformly in $\sigma$ when $|\sigma-\sigma_*|\leq\frac32\epsilon$.
As a consequence, if $\gamma=\{|\sigma-\sigma_*|=\epsilon\}\subset B$, oriented counterclockwise, then, for $\fm_1>0$ so small that $\gamma\cap{\mathrm{QNM}}(\fm)=\emptyset$ for all $\fm\in(0,\fm_1]$ (such $\fm_1$ exists by Proposition~\ref{PropKBdNo}), we have
\begin{align*}
m_{\mathrm{dS}}(\sigma_*) = \sum_{|\sigma-\sigma_*|\leq\epsilon} m_{\mathrm{dS}}(\sigma) &= \frac{1}{2\pi i}\oint_\gamma D_{\mathrm{dS}}(\sigma)^{-1}\partial_\sigma D_{\mathrm{dS}}(\sigma)\,{\mathrm d}\sigma \\
&= \frac{1}{2\pi i}\oint_\gamma D_\fm(\sigma)^{-1}\partial_\sigma D_\fm(\sigma)\,{\mathrm d}\sigma = \sum_{|\sigma-\sigma_*|\leq\epsilon} m_\fm(\sigma),
\end{align*}
as asserted in~\eqref{EqKBdYes}.
Finally, we can choose a number $r_\flat>0$ and polynomials $p_j=p_j(\zeta)$ with values in $\cC^\infty_\cp(\dot\Omega\cap\{r>r_\flat\})$ for $j=1,\ldots,m_{\mathrm{dS}}(\sigma_*)$ so that $\operatorname{Res}_{\mathrm{dS}}(\sigma_*)$ has as a basis
\[
u_{{\mathrm{dS}},j}(t_*,x)={\mathrm{res}}_{\zeta=\sigma_*}\bigl(e^{-i t_*\zeta}\Box_{g_{\mathrm{dS}}}(\zeta)^{-1}p_j(\zeta)\bigr),\qquad j=1,\ldots,m_{\mathrm{dS}}(\sigma_*).
\]
The restrictions of $u_{{\mathrm{dS}},j}$ to $[0,1]_{t_*}\times K$ remain linearly independent for any $K=[r_0,2]\times\mathbb{S}^2$ when $r_0\in(0,2)$ is sufficiently small.\footnote{Since the $u_{{\mathrm{dS}},j}$ are analytic in an appropriate coordinate system---see \cite{HintzXiedS} for explicit formulas and \cite{GalkowskiZworskiHypo} for a general argument---the smallness requirement on $r_0$ is in fact unnecessary.} With $\gamma$ as above, we can then set
\begin{align*}
u_{\fm,j}(t_*,x) &= \frac{1}{2\pi i}\oint_\gamma e^{-i t_*\zeta} \Box_\fm(\zeta)^{-1}p_j(\zeta)\,{\mathrm d}\zeta = -\frac{1}{2\pi i}\oint_\gamma e^{-i t_*\zeta} B_\fm(\zeta)D_\fm(\zeta)^{-1}C_\fm(\zeta)p_j(\zeta)\,{\mathrm d}\zeta \\
&\in \sum_{|\sigma-\sigma_*|\leq\epsilon} \operatorname{Res}_\fm(\sigma),
\end{align*}
where we used~\eqref{EqKBdYesBoxInv} and the holomorphicity of $A_\fm(\zeta)$ in $\zeta$. Since the span of $p_j(\zeta)$, where $j$ and $\zeta$ range over $1,\ldots,m_{\mathrm{dS}}(\sigma_*)$ and $\mathbb{C}$ respectively, is a fixed finite-dimensional subspace of $\cC^\infty_\cp(\dot\Omega\cap\{r>r_\flat\})$, one can prove the uniform convergence $C_\fm(\zeta)p_j(\zeta)\to C_{\mathrm{dS}}(\zeta)p_j(\zeta)$ in $\mathbb{C}^d$ for $|\zeta-\sigma_*|\leq\frac32\epsilon$ using arguments analogous to those leading to~\eqref{EqKBdYesB}. Using the already established convergence of $D_\fm(\zeta)$ and $B_\fm(\zeta)$, we thus conclude that $u_{\fm,j}|_{[0,1]\times K}\to u_{{\mathrm{dS}},j}|_{[0,1]\times K}$ in $\cC^\infty([0,1]\times K)$. In particular, for all sufficiently small $\fm>0$, the span of $u_{\fm,1},\ldots,u_{\fm,m_{\mathrm{dS}}(\sigma_*)}$ is $m_{\mathrm{dS}}(\sigma_*)$-dimensional. But since we already proved $\dim\sum_{|\sigma-\sigma_*|\leq\epsilon}\operatorname{Res}_\fm(\sigma)=m_{\mathrm{dS}}(\sigma_*)$, the $u_{\fm,j}$, $j=1,\ldots,m_{\mathrm{dS}}(\sigma_*)$, span the full space $\sum_{|\sigma-\sigma_*|\leq\epsilon}\operatorname{Res}_\fm(\sigma)$. The proof is complete.
\end{proof}
In particular, for $\sigma_*=0$, the equation~\eqref{EqKBdYes} gives $1=m_{\mathrm{dS}}(0)=\sum_{|\sigma-\sigma_*|\leq\epsilon}m_\fm(\sigma)$, and therefore there exists a single resonance $\sigma(\fm)\in{\mathrm{QNM}}(\fm)$ with $|\sigma(\fm)|\leq\epsilon$. But since constant functions on $\mathbb{R}_{t_*}\times\overline{\Omega_\fm}$ lie in the nullspace of $\Box_{g_\fm}$, we have $0\in{\mathrm{QNM}}(\fm)$; therefore, necessarily, $\sigma(\fm)=0$, with $\operatorname{Res}_\fm(0)$ equal to the space of constant functions. This proves part~\eqref{ItK0} of Theorem~\ref{ThmK}.
In order to finish the proof of Theorem~\ref{ThmK}, it now remains to show that there exists $h_1>0$ so that for $\sigma\in{\mathrm{QNM}}(\fm)$ we have $\Im\sigma\leq h_1^{-1}$ for all sufficiently small $\fm$; that is, we need to prove uniform estimates not just in strips (as done so far) but also in the full upper half plane. We turn to this next.
\subsection{Uniform analysis in a half space}
\label{SsKU}
We now work in the complement $\{\sigma\in\mathbb{C}\colon \Im\sigma\geq 0, |\sigma|\geq 1\}$ of the unit ball in the closed upper half plane; we parameterize this set via
\[
[0,\pi]_\vartheta \times [1,\infty)_{\sigma_0} \mapsto \sigma=e^{i\vartheta}\sigma_0.
\]
We can then regard the spectral family $\Box_\fm(\sigma)$ as a smooth family
\[
[0,\pi] \ni \vartheta \mapsto \bigl( [1,\infty) \times (0,\fm_0] \ni (\sigma_0,\fm)\mapsto \Box_{g_\fm}(e^{i\vartheta}\sigma_0) \bigr).
\]
In the Q-single space $X_{\mathrm{Q}}$, we work only in $\sigma_0\geq 1$. The analogues of Proposition~\ref{PropKS} and Lemma~\ref{LemmaKSyIm} in this setting are then:
\begin{prop}[Properties of the spectral family]
\label{PropKUS}
We have
\[
\Box(e^{i\vartheta}\cdot) \in \Diff_\Qop^{2,(2,0,2,2,2)}(\overline{\Omega_{\mathrm{Q}}}),
\]
with smooth dependence on $\vartheta\in[0,\pi]$. Moreover:
\begin{enumerate}
\item the Q-principal symbol of $\Box(e^{i\vartheta}\cdot)$ is given by~\eqref{EqKSSymb} with $\sigma=e^{i\vartheta}\sigma_0$;
\item we have $N_{\mathrm{zf}}(\fm^2\Box(e^{i\vartheta}\cdot))=\Box_{\hat g}(0)$;
\item for $\tilde\sigma_0>0$, we have $N_{{\mathrm{nf}}_{\tilde\sigma_0}}(\Box(e^{i\vartheta}\cdot))=\Box_{\hat g}(e^{i\vartheta}\tilde\sigma_0)$;
\item for $\sigma_0\geq 1$, we have $N_{{\mathrm{mf}}_{\sigma_0}}(\Box(e^{i\vartheta}\cdot))=\Box_{g_{\mathrm{dS}}}(e^{i\vartheta}\sigma_0)$;
\item the principal symbol of $\Im\Box(e^{i\vartheta}\cdot)$ is
\[
(\sigma_0,\fm;x,\xi)\mapsto 2(\Im\sigma) g_\fm^{-1}|_x(-{\mathrm d} t_*,-(\Re\sigma){\mathrm d} t_*+\xi),\qquad \sigma=e^{i\vartheta}\sigma_0.
\]
\end{enumerate}
\end{prop}
Since we arranged for ${\mathrm d} t_*$ to be past timelike (see Lemma~\ref{LemmaKdSChi}), the symbolic estimates of~\S\ref{SsKSy} apply uniformly for $\vartheta\in[0,\pi]$ (thus $\Im e^{i\vartheta}=\sin\vartheta\geq 0$), cf.\ \cite[\S7]{VasyMicroKerrdS}; for $\vartheta\in(0,\pi)$, these are propagation estimates with complex absorption which permit propagation in the causal future direction along the Hamiltonian flow. In particular, at the radial points at spatial infinity from the perspective of the Kerr model problems at ${\mathrm{nf}}$, the need to obtain uniform estimates in $\Im\sigma\geq 0$ down to $\Im\sigma=0$ is what forces the choice of $\mc R_{{\mathrm{if}}_+,-}$ as the incoming and $\mc R_{{\mathrm{if}}_+,+}$ as the outgoing radial set (rather than the other way around). (This is the essence of the scattering microlocal proof of the limiting absorption principle, see \cite[\S\S 9 and 14]{MelroseEuclideanSpectralTheory}, or \cite[Proposition~4.13]{VasyMinicourse}.)
Next, the ${\mathrm{zf}}$-model problem is unchanged, with estimates for it provided by Lemma~\ref{LemmaKz}. For the ${\mathrm{nf}}$-model problem at frequencies $e^{i\vartheta}\tilde\sigma_0$ with $\tilde\sigma_0$ bounded away from $0$ and $\infty$, one similarly has uniform (in $\vartheta\in[0,\pi]$) symbolic estimates on the same function spaces as in Proposition~\ref{PropKnfNz}, and for the triviality of $\ker\Box_{\hat g}(e^{i\vartheta}\tilde\sigma_0)$ one can use Whiting's original result \cite{WhitingKerrModeStability} for $\vartheta\in(0,\pi)$ (or \cite{ShlapentokhRothmanModeStability,CasalsTeixeiradCModes} for all $\vartheta\in[0,\pi]$). For the uniform low energy estimate~\eqref{EqKnfZ} for $\tilde\sigma=e^{i\vartheta}\tilde\sigma_0$, $\tilde\sigma_0\in[0,1]$, the only additional ingredient is a uniform estimate
\[
\|u\|_{H_{{\mathrm{sc}},{\mathrm{b}}}^{s,\mathsf{r},l}({\mathrm{tf}})} \leq C\|\Box_{\mathrm{tf}}(e^{i\vartheta})u\|_{H_{{\mathrm{sc}},{\mathrm{b}}}^{s-2,\mathsf{r}+1,l-2}({\mathrm{tf}})},\qquad \vartheta\in[0,\pi],
\]
for the ${\mathrm{tf}}$-model operator $\Box_{\mathrm{tf}}(e^{i\vartheta})=\tilde\Delta+e^{i\vartheta}$ (see~\eqref{EqKtfOp}); this is again a consequence of uniform symbolic estimates together with the triviality of $\ker\Box_{\mathrm{tf}}(e^{i\vartheta})$, which for $\vartheta=0,\pi$ was proved in Lemma~\ref{LemmaKtf} and which for $\vartheta\in(0,\pi)$ follows via a direct integration by parts (since tempered elements of $\ker\Box_{\mathrm{tf}}(e^{i\vartheta})$ are then automatically rapidly decaying as $\tilde r\to\infty$). The uniform high energy estimates of Proposition~\ref{PropKmf} continue to hold for $h^2\Box_{g_{\mathrm{dS}}}(h^{-1}e^{i\vartheta})$ when $h>0$ is sufficiently small.
These symbolic and normal operator estimates can then be combined as in the proof of Proposition~\ref{PropKA} and yield, as in Corollary~\ref{CorKAHigh}, the existence of $\fm_1>0$ and $h_1>0$ so that for all $\fm\in(0,\fm_1]$ and $\sigma_0\geq h_1^{-1}$ we have $e^{i\vartheta}\sigma_0\notin{\mathrm{QNM}}(\fm)$ for all $\vartheta\in[0,\pi]$. Thus, all quasinormal modes $\sigma$ of $\Box_{g_\fm}$, $\fm\in(0,\fm_1]$, satisfy $\Im\sigma\leq h_1^{-1}$. As noted at the end of~\S\ref{SsKBd}, this completes the proof of Theorem~\ref{ThmK}.
\subsection{Quasinormal modes of massive scalar fields}
\label{SsKG}
From \cite[Proposition~2.1]{HintzXieSdS}, we recall the following analogue of Lemma~\ref{LemmaKdSQNM}:
\begin{lemma}[QNMs for massive scalar fields on de~Sitter space]
\label{LemmaKG}
Let $\nu\in\mathbb{C}$ and $\lambda_\pm=\frac32\pm\sqrt{\frac94-\nu}$ as in Theorem~\usref{ThmIKG}. Then the set ${\mathrm{QNM}}_{\mathrm{dS}}(\nu)$ of quasinormal modes of $\Box_{g_{\mathrm{dS}}}-\nu$ is equal to $\bigcup_\pm(-i\lambda_\pm-i\mathbb{N}_0)$, and the multiplicity of $\sigma\in{\mathrm{QNM}}_{\mathrm{dS}}(\nu)$ is
\begin{equation}
\label{EqKG}
m_{\mathrm{dS}}(\nu;\sigma) = \sum_{\genfrac{}{}{0pt}{}{l\in\mathbb{N}_0}{i\sigma-l\in(\lambda_-+2\mathbb{N}_0)\cup(\lambda_++2\mathbb{N}_0)}} (2 l+1).
\end{equation}
\end{lemma}
The formula~\eqref{EqKG} reduces to~\eqref{EqKdSQNM} for $\nu=0$; see the proof of Lemma~\ref{LemmaKdSQNM}. Define
\[
{\mathrm{QNM}}(\nu;\fm),\quad m_\fm(\nu;\sigma),\quad \operatorname{Res}_\fm(\nu;\sigma),
\]
and $\operatorname{Res}_{\mathrm{dS}}(\nu;\sigma)$ as in~\S\ref{SsKMain} but now using the operators $\Box_{g_\fm}-\nu$ and $\Box_{g_{\mathrm{dS}}}-\nu$. \emph{Then Theorem~\usref{ThmK}, except for part~\eqref{ItK0}, remains valid upon adding the parameter $\nu$ to the notation throughout.} (This also proves Theorem~\usref{ThmIKG}.)
The proof is \emph{the same} as that of Theorem~\ref{ThmK}; indeed, the presence of the scalar field mass term $\nu$ affects neither the principal symbol of $\Box(\cdot+i\sigma_1)-\nu$ nor any of its normal operators, with the exception of
\[
N_{{\mathrm{mf}}_{\sigma_0}}(\Box(\cdot+i\sigma_1)-\nu) = \Box_{g_{\mathrm{dS}}}(\sigma)-\nu.
\]
Thus, the invertibility properties of $\Box_{g_{\mathrm{dS}}}(\sigma)-\nu$ are what determine the limiting quasinormal mode spectrum of $\Box_{g_\fm}(\sigma)-\nu$.
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{"paloma_paragraphs":[]}
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{"arxiv_id":"2112.14431","language":"en","timestamp":1640830936000,"url":"https:\/\/arxiv.org\/abs\/2112.14431","yymm":"2112"}
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proofpile-arXiv_000-10067
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{"provenance":"002.jsonl.gz:10068"}
| null | null |
\section{Introduction}
A recent report by the World Wide Fund for Nature (WWF) confirms that biodiversity and ecosystems are deteriorating worldwide \citep{almond2020living}.
Population sizes of mammals, birds, amphibians, reptiles and fish have decreased by an average of 68\% between 1970 and 2016 across the world.
This decrease in biodiversity has several causes, such as habitat loss due to pollution, species overexploitation
or climate change.
Biodiversity
is important since it is a key indicator of overall healthy ecosystems which in their turn have important social and economic consequences for humans.
In particular, biodiversity and ecosystems influence our water quality,
air quality and climate, they secure our food production and impact the spread of infectious diseases originating from animals \citep{almond2020living,diaz2019summary}.
Machine learning (ML) can help to more efficiently measure and monitor the well-being of ecosystems and the success of biodiversity conservation efforts \citep{huynh2018annotation,joly2020lifeclef,van2014nature,park2020illuminating}.
As an example, this paper proposes a method for automatic classification of camera trap images, a type of motion triggered cameras used in biological studies to estimate animal population density and activity patterns \citep{ridout2009estimating, foster2012critique,rowcliffe2014quantifying,sollmann2018gentle,tabak2019machine, trolliet2014use}.
Since manually labeling large numbers of camera trap images is time consuming and costly \citep{kelly2008estimating},
ML could be used to automatically detect animals and the species to which they belong in images.
This work uses Convolutional Neural Networks
\citep{lecun1998gradient,lecun2015deep}
to classify camera trap images.
Training a CNN on a dataset of camera trap images is challenging, because camera trap images often only depict a part of an animal, because of high intra-class variation
due to differences in backgrounds, and because the class-distribution of camera trap datasets is typically highly imbalanced.
This imbalance is inherent to camera trap datasets since it reflects the imbalance of ecosystems \citep{trebilco2013ecosystem}, and it results in biased classifiers that perform very well for a few majority classes but poorly for many minority classes.
Classifiers that perform well on all classes would be of more value to ecologists, and moreover, rare or less observed animal species might even be of special interest to research.
Therefore, solutions are needed to mitigate this imbalance when
classifying camera trap images.
To this end, we use a two-phase training method \citep{lee2016plankton} to mitigate class imbalance, for the first time to the best of our knowledge on camera trap images. In experiments we compare it to different data-level class imbalance mitigation techniques, and show that it improves performance on minority classes, with limited loss in performance for other classes, resulting in an increase in macro F1-score.
\section{Related work}
Pioneering studies
that automatically classified camera trap images relied on manual feature extraction and smaller datasets \citep{figueroa2014fast,swinnen2014novel,yu2013automated,chen2014deep}. Better and more scalable results were later achieved with deep CNNs and larger datasets \citep{gomez2016towards,norouzzadeh2018automatically,tabak2019machine, willi2019identifying, schneider2020three}.
Generally, models trained by these scholars achieve accuracies well above 90\%, but the models are biased towards majority classes, which severely affects their class-specific performance.
Especially the
performance for rare species is poor.
Scholars dealt with this challenge by removing the rare classes from the dataset \citep{gomez2016towards,willi2019identifying}, with confidence thresholding and letting experts review the uncertain classifications \citep{willi2019identifying}, with weighted losses, oversampling and emphasis sampling \citep{norouzzadeh2018automatically} or by using a combination of additional image augmentations for rare classes and novel sampling techniques \citep{schneider2020three}.
Although \citep{norouzzadeh2018automatically} managed to greatly increase the accuracy for a few rare classes using oversampling, none of the aforementioned techniques systematically improved accuracy for most of the rare species. It can thus be concluded that dealing with class-imbalance in the context of camera trap image classification is still an unsolved issue.
Two categories of methods for mitigation of class imbalance in deep learning exist: data-level and algorithm-level techniques \citep{buda2018systematic, johnson2019survey}.
The former refers to techniques that alter the class-distribution of the data, such as random minority oversampling (ROS) and random majority undersampling (RUS),
which respectively randomly duplicate or randomly remove samples to obtain a more balanced dataset.
More advanced techniques
can also be used to synthesize new samples \citep{chawla2002smote,han2005borderline,he2008adasyn,wan2017variational,wang2017cgan,li2021plankton},
but these are computationally expensive, and they require a large number of images per class and images within a class that are sufficiently similar.
Algorithm-level techniques are techniques that work on the model itself, such as loss functions or thresholding \citep{lin2017focal,nemoto2018classification,buda2018systematic,johnson2019survey,he2013imbalanced,buda2018systematic}.
Two-phase training, a hybrid technique, was recently introduced
and shown to obtain good results
for training a CNN classifier on a highly imbalanced dataset of images of plankton \citep{lee2016plankton}, and it was later used by others for image segmentation and classification \citep{havaei2017brain,buda2018systematic}.
Because of these promising results and the broad applicability of 2-phase training, we
test 2-phase training for camera trap images.
\section{Two-phase training}
Two-phase training consists of the following steps \citep{lee2016plankton}. $\mathcal{D}_{orig}$ is the original, imbalanced dataset. Figure \ref{fig:two-phase fig} in the appendix shows an overview of two-phrase training.
\begin{enumerate}
\item \textbf{Phase 1}: a CNN $f_\theta$ is trained on a more balanced dataset $\mathcal{D}_{bal}$, obtained by any sampling method such as ROS, RUS or a combination thereof.
\item \textbf{Phase 2}: the convolutional weights\footnote{I.e. all weights except the weights of the fully connected layers that project the CNN features to the classes.} of $f_\theta$ are frozen, and the network is trained further on the full imbalanced dataset $\mathcal{D}_{org}$.
\end{enumerate}
The 1st phase trains the convolutional layers with (more) equal importance allocated to minority classes, so they learn to extract relevant features for these classes as well.
In the 2nd phase the classification layers learn to model the class imbalance present in the dataset.
\section{Dataset \& Experiments}
We used the 9th season of the publicly available Snapshot Serengeti (SS) dataset,
which is generated by a grid of 225 cameras spread over the Serengeti National Park in Tanzania \citep{swanson2015snapshot}.
The images were labeled by citizen scientists on the Zooniverse platform.
After filtering some images, the full dataset $\mathcal{D}_{orig}$ contains 194k images belonging to 52 classes.
The class-distribution of this dataset is depicted in fig. \ref{fig:season9_species fig} in the appendix, and is highly imbalanced, with the three majority classes accounting for just under 75\% of the data.
We used this smaller subset of the full SS dataset for computational tractability, and to ensure insights remain valid for ecologists with access to smaller datasets.
Appendix \ref{app:hyperparams} lists the hyperparameters\footnote{Our code is publicly available: \url{https://github.com/FarjadMalik/aigoeswild}.}.
First we trained the baseline CNN on the full dataset $\mathcal{D}_{orig}$.
Next, we trained 4 models with different instantiations of $\mathcal{D}_{bal}$ for phase 1 of two-phase training.
\begin{enumerate}
\item $\mathcal{D}_{bal}^1$: ROS (oversampling) classes with less than 5k images until 5k, see appendix fig. \ref{fig:ROS}.
\item $\mathcal{D}_{bal}^2$: RUS (undersampling) classes with more than 15k images until 15k.
\item $\mathcal{D}_{bal}^3$: ROS classes with less than 5k images until 5k as in 1., and RUS classes with more than 15k images until 15k as in 2. Shown in fig. \ref{fig:ROS&RUS} in the appendix.
\item $\mathcal{D}_{bal}^4$: ROS classes with less than 5k images until 5k as in 1., and RUS classes with more than 5k images until 5k.
\end{enumerate}
We used a lower sample ratio for classes with very few images to avoid overfitting (appendix \ref{app:Sampling}).
As evaluation metrics we used not only top-1 accuracy but also precision, recall and F1-score, since these metrics are more informative to class-imbalance.
We report their values macro-averaged over classes as well as the class specific values (in appendix tables \ref{tab:baseline1}-\ref{tab:rus_perspecies_stat}).
The results of the models after phase 1 correspond to the results that we would obtain by only using ROS, RUS or a combination of both
(and no two-phase training). These results will thus serve as a baseline.
\section{Results} \label{sec:results}
\begin{table}
\footnotesize
\centering
\begin{tabular}{lrrrr}
\toprule
\textbf{Model} & \textbf{Phase 1: Accuracy} & \textbf{Phase 2: Accuracy} & \textbf{Phase 1: F1} & \textbf{Phase 2: F1} \\ \midrule
$\mathcal{D}_{orig}$: Baseline & \textbf{0.8527} & / & {0.3944} & / \\
$\mathcal{D}_{bal}^1$: {ROS} & {0.8326} & \textbf{0.8528} & {0.3843} & {0.4012}\\
$\mathcal{D}_{bal}^2$: {RUS} & {0.8012} & {0.8491} & {0.3681} & \textbf{0.4147} \\
$\mathcal{D}_{bal}^3$: ROS\&RUS(15K) & 0.8346 & 0.8454 & \textbf{0.4179} & 0.4094\\
$\mathcal{D}_{bal}^4$: ROS\&RUS(5K) & 0.7335 & 0.8066 & 0.3620 & 0.4001\\ \bottomrule
\end{tabular}
\caption{Model Comparison - Top-1 accuracy and Macro F1-score.}
\label{tab:modelcomparison_top1_F1}
\end{table}
\paragraph{Accuracy and Macro F1.} Table \ref{tab:modelcomparison_top1_F1} shows the accuracy and F1-score of the models after the 1st and the 2nd phase\footnote{Appendix \ref{app:extra-results} contains more results and in-depth discussion.}.
Training on more balanced datasets reduces accuracy in phase 1 for all models compared to the baseline which was trained on the imbalanced dataset $\mathcal{D}_{orig}$.
However, further training the classification layers in phase 2 on the full dataset increases accuracy back to more or less the baseline level for all models (except ROS\&RUS(5K)), meaning that two-phase training lost little to no accuracy.
The phase 2 mean accuracy is substantially higher than the phase 1 mean accuracy.
The F1-scores of most models
also drop in phase 1.
Interestingly, phase 2 raises the F1-score of most models
again, and all models obtain an F1-score after phase 2 that is higher than the baseline: 3.0\% on average.
The RUS model obtains the highest F1-score after phase 2: an increase of 5.1\% compared to the baseline, while the ROS\&RUS(15K) model obtain the highest F1-score overall\footnote{We consider the F1-score of ROS\&RUS(15K) after phase 1 an anomaly which needs further analysis.}.
Most two-phase trained models outperform their counterparts which were only trained on more balanced datasets.
As for the accuracy, the mean F1-score in phase 2 is substantially higher than the mean F1-score in phase 1: 6.1\%.
These observations lead us to conclude that 1) two-phase training outperforms using only sampling techniques across most sampling regimes, and 2) two-phase training can increase the F1-score without substantial loss in accuracy, meaning it improves minority class predictions with very limited loss in majority class performance.
These findings are in line with the results of \citep{lee2016plankton}, though they report greater increases in F1-scores than us, possibly due to an even more imbalanced dataset.
They also find RUS to work best for creating $\mathcal{D}_{bal}$ for phase 1.
The F1-scores are substantially lower than the accuracies
(idem for precision and recall, appendix tables \ref{tab:modelcomparison_precision}-\ref{tab:modelcomparison_recall}).
This is because the class-specific values for these metrics are high for the majority classes, but extremely low for many minority classes, confirming that the imbalanced data creates a bias towards the majority classes.
\paragraph{Class-specific performance.}
Class-specific F1-scores increase with two-phase training for the majority of (minority) classes.
Two-phase training with RUS leads to the greatest average increase of F1-score per class: 3\% (ignoring the classes for which the F1-score remained 0.0\%).
This increase is most notable for minority classes.
RUS performing best is remarkable, since we trained the RUS model in phase 1 with only 85k images, compared to 131k--231k for the other models.
Fig. \ref{fig:f1_comparison_delta_base_rus} shows the changes in F1-score due to two-phase training with RUS.
\begin{figure}
\centering
\begin{subfigure}{0.5\textwidth}
\centering
\includegraphics[width=0.96\linewidth,keepaspectratio]{f1_comparison_delta_base_rus.png}
\caption{}
\label{fig:f1_comparison_delta_base_rus}
\end{subfigure}%
\begin{subfigure}{0.5\textwidth}
\centering
\includegraphics[width=0.96\linewidth,keepaspectratio]{f1_comparison_delta_rus_phases.png}
\caption{}
\label{fig:f1_comparison_delta_rus_phases}
\end{subfigure}
\caption{Relative difference in F1-score per species of (\subref{fig:f1_comparison_delta_base_rus}) the two-phase RUS model vs. the baseline, and (\subref{fig:f1_comparison_delta_rus_phases}) phase 2 vs. phase 1 of the RUS-model. The appendix contains larger versions: figs. \ref{fig:f1_comparison_delta_base_rus_l}, \ref{fig:f1_comparison_delta_rus_phases_l}. Species are sorted in descending order according to their occurrence frequency.}
\label{fig:f1_comparisons}
\end{figure}
\section{Conclusion}
We explored the use of two-phase training to mitigate the class imbalance issue for camera trap image classification with CNNs.
We conclude that 1) two-phase training outperforms using only sampling techniques across most sampling regimes, and 2) two-phase training improves minority class predictions with very limited loss in majority class performance, compared to training on the imbalanced dataset only.
In the future we would like to rerun our experiments with different random seeds to obtain more statistically convincing results, compare two-phase training to other algorithm-level imbalance mitigation techniques, and test it on varying dataset sizes and levels of class imbalance.
\clearpage
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{"arxiv_id":"2112.14491","language":"en","timestamp":1640831113000,"url":"https:\/\/arxiv.org\/abs\/2112.14491","yymm":"2112"}
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\section*{Supplementary Material}
See supplementary material for the electronic band structures of isolated monolayers, and the plane-averaged differential charge and electrostatic potential profiles of the MoSi$_2$N$_4$/ZnO and MoSi$_2$N$_4$/GaN heterostructures.
\section*{ACKNOWLEDGMENTS}
This work is supported by A*STAR AME IRG (A2083c0057) and Singapore University of Technology and Design Start-Up Research grant (Project No. SRG SCI 2021 163). J.Q.N. acknowledge the PhD Scholarship support from the Singapore University of Technology and Design. The calculations were carried out using the computational resources provided by the Titan supercomputing facility in SUTD and the National Supercomputing Centre (NSCC) Singapore.
\section*{Author Declarations}
\subsection*{Conflict of Interest}
\noindent The authors declare that there are no conflicts of interest.
\subsection*{Author Contributions}
\noindent J.Q.N. performed the simulations and data analysis. Q.W., L.K.A. and Y.S.A. supervised the project. All authors contributed to the writing and the revision to this work.
\section*{Data Availability}
The data that support the findings of this study are available from the corresponding author upon reasonable request.
\bibliographystyle{apsrev}
\providecommand{\noopsort}[1]{}\providecommand{\singleletter}[1]{#1}%
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{"arxiv_id":"2112.14526","language":"en","timestamp":1645755774000,"url":"https:\/\/arxiv.org\/abs\/2112.14526","yymm":"2112"}
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{"provenance":"002.jsonl.gz:10070"}
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\section{Introduction}
Non-line-of-sight (NLOS) imaging aims at recovering objects outside the direct line of sight of a sensor~\cite{2020Faccio, 2021Geng}. Most active NLOS capture systems exploit an ultra-fast pulsed laser beam that can be controlled to direct toward a relay surface (e.g., a wall). A companion time-resolved detector then collects the arrival time and number of photons that return after the first and third of three bounces: off the relay surface, off the hidden objects, and back off the relay surface. Fig.~\ref{fig:scene} illustrates a conventional setting of the NLOS capture system. The first bounce corresponds to direct reflection and enables us to recover the shape and albedo of the relay surface. By removing or gating the photons from the first bounce, photons from the third bounce are employed to reconstruct~\cite{2012Gupta, 2015Mauro, 2018LCT, 2019FK, 2019Liu, 2020ECCV, 2021Feng, 2021PNAS} and localize~\cite{2017Chan, 2018LCT, 2019FK, 2021PNAS, 2021Metzler} hidden objects in the NLOS scene. Potential applications are numerous, including autonomous driving, remote sensing, and biomedical imaging.
Substantial efforts have been made to both improve the hardware of the capture system \cite{2012Velten, 2015Mauro, 2018LCT, 2021PNAS, 2020Ceiling} and provide better NLOS reconstruction\cite{2019FK, 2019Liu, 2019Xin, 2021Ye, 2021Shen}. For the former, the efforts have been focused on using more accurate and affordable detectors and lasers. Streak cameras pioneered NLOS imaging ~\cite{2012Gupta, 2012Velten, 2014Wu, 2021Feng} by providing precise temporal resolution, e.g., down to 2 picoseconds (ps) or \SI{0.6}{mm}~\cite{2012Velten}. Such cameras, however, are overwhelmingly expensive and at the same time difficult to calibrate under nonlinear temporal-spatial transforms. Photonic mixer devices (PMDs) are compact and less expensive but can only provide low temporal resolution on the order of nanoseconds\cite{2013Heide}. In recent years, single photon avalanche diodes (SPADs) have served as an affordable and convenient alternative~\cite{2015Mauro, 2016Gariepy, 2018LCT, 2019FK, 2019Liu, 2020Ceiling, 2021PNAS}. A single-pixel SPAD with a time-correlated single photon counting (TCSPC) module produces a histogram of photon counts versus time bins of \SI{4}{ps} at a detection point. On the laser front, a picosecond or femtosecond pulsed laser is used to illuminate a spot on the relay surface, and to trigger and synchronize pulse signals with the detector so as to record the arrival times of returning photons. The properties of the laser, e.g., average power and repetition rate, affect acquisition time and measurements.
Despite all these advances, constructing a prototype system even for validation still requires elaborate calibrations to acquire accurate transients and other information such as the 3D coordinates of scanning points. In a typical NLOS camera system, a uniform grid of $N \times N$ evenly spaced points is adopted to fit most reconstruction algorithms. An emerging trend, however, is to use non-uniform sampling, in both LOS\cite{2020NeRF} and NLOS\cite{2021Shen} imaging where the capture system would require re-calibration. In real-world deployment, it is also common to adjust the position and orientation of the occluder and the hidden object for validations where re-calibration needs to be conducted.
\begin{figure}[t]
\centering
\subfloat[Conventional scenario of NLOS imaging]{
\begin{minipage}{.49\linewidth}
\centering
\includegraphics[width=.99\linewidth]{scenario.pdf}
\end{minipage}
\label{subfig:scenerio}}
\subfloat[Top-view of setting]{
\begin{minipage}{.49\linewidth}
\centering
\includegraphics[width=.99\linewidth]{scenarioTop.pdf}
\end{minipage}
\label{subfig:scenarioTop}}
\caption{A conventional setting of the capture system for NLOS imaging: (a) conventional scenario with an ultra-fast pulsed laser and a time-resolved detector; (b) top-view of the setting.}
\label{fig:scene}
\end{figure}
By far nearly all existing NLOS calibration schemes require auxiliary apparatus. For example, a checkerboard or regular point grid coupled with a visible camera\cite{2015Mauro,2019Ahn} can be used to acquire the internal and external parameters of the camera or the 3D coordinates of the grid point. Alternatives include using mirrors\cite{2020Klein} to establish correspondences of the laser point and the detection point for simultaneously estimating the mirror plane and the relay wall. When the scanning point is re-positioned, or the position of the relay surface changes relative to the data acquisition system, these methods need re-calibration. This is infeasible for onsite deployment for the process is both labor intensive and time consuming.
We present an online calibration technique that directly decouples the acquired transients at onsite scanning into the LOS and hidden components. The former, we call \textit{Gamma} map, contains information of both geometry and albedo of the relay surface and detection points. We show how to use the \textit{Gamma} map to directly (re)calibrate the system upon changes of scene/obstacle configurations, scanning regions, and scanning patterns, as well as for transient normalization. This also provides a useful preview to allow dynamic adjustment: we can scan a small number of detection points and preview the calibration results of the scanning galvanometer and the relay surface to determine if the current scanning setup is acceptable.
Furthermore, since our technique supports both uniform and non-uniform scanning patterns, we integrate existing NLOS reconstruction techniques for processing the hidden component based on spatial, frequency or learning based techniques. Many fast Fourier transform (FFT)-based algorithms, e.g., LCT\cite{2018LCT}, FK\cite{2019FK}, and PF\cite{2019Liu}, require transients in a regular grid as input whereas learning-based methods, such as neural transient fields (NeTF)~\cite{2021Shen}, allow more flexible sampling patterns but require significantly longer computational time, e.g., to train the network. We present a tailored reconstruction scheme suitable for arbitrary sampling patterns. This scheme employs the gradient descent method to iteratively reconstruct NLOS scenes. By supporting online calibration and onsite reconstruction assessment, our solution supports more feasible laboratory validations and real-world deployments and will be made open-source to the community.
\section{Related work}
NLOS imaging have used a variety of sensing technologies at the convergence of physics, optics, electronics, and signal processing. Optical detectors, including optical interferometry~\cite{2015Ioannis}, PMDs~\cite{2013Heide}, SPADs~\cite{2015Mauro, 2016Gariepy, 2018LCT}, intensified CCD cameras (iCCDs)~\cite{2014Martin}, streak cameras~\cite{2011Kirmani, 2012Gupta, 2012Velten, 2014Wu, 2021Feng}, or even non-optical devices, e.g., acoustic~\cite{2019Lindell}, thermal~\cite{2019Maeda}, and radar~\cite{2020Radar}, are exploited to collect transients of hidden objects. The temporal resolution of a detector is one of the key parameters for NLOS capture systems. Among the incoherent optical devices, PMDs and iCCDs are compact and low-cost and offer temporal resolution on the order of nanoseconds. Streak cameras and SPADs have high resolution of several picoseconds, but the former are bulky and incompatible with most capture systems in practice. For more details of time-resolved detectors, we refer readers to recent reviews~\cite{2017Jarabo, 2020Faccio, 2021Geng}. \\[-1.5ex]
\textit{SPAD-based NLOS imaging.} Buttafava et al.\cite{2015Mauro} built the first SPAD-based NLOS capture system, which is a conventional setting, using a femtosecond (fs) laser that generates \SI{250}{fs} pulses at a wavelength of \SI{515}{nm} with a \SI{55}{MHz} repetition rate and an average power of \SI{50}{mW}. SPADs are reverse-biased photodiodes in Geiger mode, which distinguishes them from linear avalanche photodiodes (APDs). In particular, a SPAD restricts one incident photon to trigger an avalanche event and remains blind for a hold-off period, and then detects the next incoming photon. Several main factors, including sensitivity, dark count rate, temporal jitter, and hold-off time, define the signal-to-noise ratio and the temporal resolution and are related to the physical configuration and the material properties of the SPAD. For instance, silicon-based SPADs cover the visible spectrum, e.g., \SI{400}-\SI{800}{nm}, with full width at half maximum (FWHM) of tens of ps~\cite{2018LCT, 2019FK, 2019Liu}, whereas InGaAs/InP-based SPADs cover the infrared spectrum with FWHM of approximately \SI{200}{ps}~\cite{2021Ye, 2017Chan, 2021PNAS}. Wu et al.\cite{2021PNAS} constructed a long-range NLOS capture system, over \SI{1.43}{km}, exploiting InGaAs/InP SPADs at \SI{1550}{nm} and a fiber laser with \SI{500}{ps} pulses and \SI{300}{mW} average power.
We opt for a silicon-based single-pixel SPAD with a fast-gating mode, which can switch off the direct light paths between the relay surface and the SPAD. A 1D or 2D SPAD array simultaneously records transients of many pixels, but is often composed of single SPADs that have a smaller active area (of $\SI{6.95}{\um} \times \SI{6.95}{\um}$) and lower temporal resolution (hundreds of ps) due to the complicated fabrication~\cite{2016Gariepy, 2017Chan, 2021Pei}. Our SPAD has an active area of $\SI{50}{\um} \times \SI{50}{\um}$ and \SI{50}{ps} temporal jitter. The single SPAD in Stanford's NLOS capture system has a $\SI{100}{\um} \times \SI{100}{\um}$ active area and the laser is \SI{670}{nm} wavelength with \SI{30}{ps} pulses, \SI{10}{MHz} repetition rate, and \SI{0.11}{mW} average power\cite{2018LCT, 2019Heide}. Their improved version uses the fast-gated SPAD and a laser at \SI{532}{nm} with \SI{35}{ps} pulses and an average power of stronger than \SI{1}{W} \cite{2019FK}. They constructed several publicly available datasets of various types of NLOS objects: retro-reflective, specular, and diffuse. Using similar hardware devices, more NLOS capture systems have been built, and have achieved a large number of returning photons from the hidden objects within short acquisition\cite{2019Liu, 2019Ahn, 2019Xin}. These NLOS capture systems, as well as ours, are all of confocal setting with a beam splitter that locates the laser and the SPAD coaxially. In contrast, our laser is lower-cost, at a \SI{670}{nm} wavelength with \SI{50}{ps} pulses, \SI{40}{MHz} repetition rate, and \SI{2}{mW} average power, and produces up to 60 photons per pulse, which is sufficient for recovering many NLOS scenes. \\[-1.5ex]
\textit{Calibration schemes.} In NLOS imaging, positions and distances of the hardware devices need to be under control to record precise transients of hidden objects. Klein et al.\cite{2020Klein} proposed a calibration scheme with mirrors as target objects placed at different positions in an NLOS scene. Since the laser is directed by a 2D galvanometer, digital cameras are useful tools to take pictures of laser spots on the relay surface and to determine the 3D coordinates of the spots using computer vision techniques\cite{2015Mauro, 2019FK, 2019Ahn}. By scanning a pre-defined uniform grid with equidistant points, the position and orientation of the relay surface can also be estimated. However, these calibration schemes require specific target objects, e.g., mirrors or a checkerboard, and the digital cameras to be known or pre-calibrated. In contrast, our calibration technique needs no additional target objects or a digital camera and can operate online the iterations of calibration procedures. In particular, most previous studies attempt to determine the coordinates of scanning points or their mapping to input voltages of the galvanometer, resulting in inflexible scanning point selection and re-calibration while adjusting the relay surface or the galvanometer. Here we describe a detailed galvanometer calibration that allows us to arbitrarily select scanning points and to determine scanning patterns by manually or automatically controlling the galvanometer. Our calibration scheme enables users to calibrate the relay surface along with the galvanometer, and to calibrate these two separately for more precise effects.
\section{NLOS imaging models}
NLOS imaging are constructed in accordance with an imaging formulation that models the physics of light traveling from a laser to a detector via LOS and NLOS objects. Fig.~\ref{fig:scene}\subref{subfig:scenarioTop} illustrates a top-view of the conventional setting. The pulsed laser beam $\mathbf{o}$ illuminates a spot $\mathbf{l}$ on a relay surface $W$. After the light scatters off the spot, some photons bounce off from a point $\mathbf{p}$ of the NLOS object $P$ and travel back onto a patch $\mathbf{s}$ of the relay surface. The detector $\mathbf{d}$ collects a number of photons from the patch at a time instant $t$. Based on the physics of light transport \cite{2006Dutre}, we define the image formation model as:
\begin{equation}
\begin{aligned}
\tau(t;\mathbf{o},\mathbf{l},\mathbf{s},\mathbf{d})
= \frac{N_{\mathbf{o}}}{c} \rdb{\frac{\rho}{\pi}}^{2} A_{\mathbf{d}} A_{\mathbf{s}} \frac{\rdb{\boldsymbol{\omega}_{\mathbf{s}\to\mathbf{d}} \cdot \mathbf{n}_{\mathbf{s}}} }{|\mathbf{s}-\mathbf{d}|^{2}}
\int_{P} \delta\rdb{|\mathbf{l}-\mathbf{o}|+|\mathbf{p}-\mathbf{l}|+|\mathbf{s}-\mathbf{p}|+|\mathbf{d}-\mathbf{s}| - ct} &\\
\cdot
\upsilon(\mathbf{p};\mathbf{l},\mathbf{s})
f(\mathbf{p};\boldsymbol{\omega}_{\mathbf{l}\to\mathbf{p}},\boldsymbol{\omega}_{\mathbf{p}\to\mathbf{s}})
\frac{\rdb{\boldsymbol{\omega}_{\mathbf{p}\to\mathbf{l}} \cdot \mathbf{n}_{\mathbf{p}}} \rdb{\boldsymbol{\omega}_{\mathbf{l}\to\mathbf{p}} \cdot \mathbf{n}_{\mathbf{l}}} \rdb{\boldsymbol{\omega}_{\mathbf{s}\to\mathbf{p}} \cdot \mathbf{n}_{\mathbf{s}}} \rdb{\boldsymbol{\omega}_{\mathbf{p}\to\mathbf{s}} \cdot \mathbf{n}_{\mathbf{p}}}}{|\mathbf{l}-\mathbf{p}|^{2}|\mathbf{p}-\mathbf{s}|^2} \mathrm{d}{A_{\mathbf{p}}} &
\end{aligned}
\label{eq:nc-model}
\end{equation}
\noindent where $N_{\mathbf{o}}$ is the number of photons emitted in one pulse of light from $\mathbf{o}$, while $\rho$ denotes the albedo of the relay surface. The coefficients $A_{\mathbf{d}}$ and $A_{\mathbf{s}}$ are the active area of the detector and the area mapped on the relay surface, respectively. The unit vector $\boldsymbol{\omega}_{\mathbf{a}\to\mathbf{b}} = \frac{\mathbf{b}-\mathbf{a}}{|\mathbf{b}-\mathbf{a}|}$ represents the direction from the input argument $\mathbf{a}$ to $\mathbf{b}$. Adopting the same notation, $\mathbf{n}_{\mathbf{a}}$ represents the surface normal vector at the point $\mathbf{a}$. The Dirac delta function $\delta(\cdot)$ relates the time $t$ to the travel distance, while $c$ is the speed of light. The geometry term $\upsilon(\mathbf{p};\mathbf{l},\mathbf{s})$ is the visibility function of a hidden point $\mathbf{p}$ from an illumination spot $\mathbf{l}$ and a sensing patch $\mathbf{s}$. The function $f(\mathbf{p};\boldsymbol{\omega}_{\mathbf{l}\to\mathbf{p}},\boldsymbol{\omega}_{\mathbf{p}\to\mathbf{s}})$ describes the bidirectional reflectance distribution function (BRDF) of a point $\mathbf{p}$ with the incident and exitant directions $\boldsymbol{\omega}_{\mathbf{l}\to\mathbf{p}}$ and $\boldsymbol{\omega}_{\mathbf{p}\to\mathbf{s}}$. The integral $\int_{P}$ represents a summation of the photons that travel back at time $t$ from a small area $A_{\mathbf{p}}$ centered at the point $\mathbf{p}$ on the hidden object $P$.
For simplicity, Equation \eqref{eq:nc-model} neglects the volatility of light, e.g., diffraction and interference. We provide its complete derivation in Supplementary Information. $\tau(t;\mathbf{o},\mathbf{l},\mathbf{s},\mathbf{d})$ considers arbitrary combinations of illumination and detection points on the relay surface and records 5D transients, each of which is a histogram of the number of photons at time bins $t$. The peaks of the histogram indicate the arrival time of photons that travel back from the relay surface and from a hidden object in the scenario. $\tau(t;\mathbf{o},\mathbf{l},\mathbf{s},\mathbf{d})$ contains two portions: direct light paths in the LOS scene between the relay surface and the detector, and the indirect light paths in the NLOS scene between the relay surface and the hidden object. As these two portions are convolutional, we separate $\tau(t;\mathbf{o},\mathbf{l},\mathbf{s},\mathbf{d})$ into portions $\tau_{\text{NLOS}}(t;\mathbf{l},\mathbf{s})$ and $\tau_{\text{LOS}}(t;\mathbf{l}, \mathbf{s}, \mathbf{d})$, resulting in:
\begin{equation}
\begin{aligned}
\tau_{\text{NLOS}}(t;\mathbf{l},\mathbf{s})
&=
\int_{P} \delta\rdb{|\mathbf{p}-\mathbf{l}|+|\mathbf{s}-\mathbf{p}| - ct}
\upsilon(\mathbf{p};\mathbf{l},\mathbf{s}) \\ &\cdot
f(\mathbf{p};\boldsymbol{\omega}_{\mathbf{l}\to\mathbf{p}},\boldsymbol{\omega}_{\mathbf{p}\to\mathbf{s}})
g(\mathbf{p}; \mathbf{l}, \mathbf{s})
\mathrm{d}{A_{\mathbf{p}}}
\end{aligned}
\label{eq:nc-model-NLOS}
\end{equation}
\noindent and
\begin{equation}
\begin{aligned}
\tau_{\text{LOS}}(t;\mathbf{l}, \mathbf{s}, \mathbf{d})
&= \Gamma(\mathbf{s}) \delta\rdb{|\mathbf{l}-\mathbf{o}|+|\mathbf{s}-\mathbf{d}| - ct}
\end{aligned}
\label{eq:nc-model-LOS}
\end{equation}
\noindent where
\begin{equation}
g(\mathbf{p}; \mathbf{l}, \mathbf{s}) =
\frac{\rdb{\boldsymbol{\omega}_{\mathbf{p}\to\mathbf{l}} \cdot \mathbf{n}_{\mathbf{p}}} \rdb{\boldsymbol{\omega}_{\mathbf{l}\to\mathbf{p}} \cdot \mathbf{n}_{\mathbf{l}}} \rdb{\boldsymbol{\omega}_{\mathbf{s}\to\mathbf{p}} \cdot \mathbf{n}_{\mathbf{s}}} \rdb{\boldsymbol{\omega}_{\mathbf{p}\to\mathbf{s}} \cdot \mathbf{n}_{\mathbf{p}}}}{|\mathbf{l}-\mathbf{p}|^{2}|\mathbf{p}-\mathbf{s}|^2}
\end{equation}
\begin{equation}
\Gamma(\mathbf{s}) = \frac{N_{\mathbf{o}}}{c} \rdb{\frac{\rho}{\pi}}^{2} A_{\mathbf{d}} A_{\mathbf{s}} \frac{\rdb{\boldsymbol{\omega}_{\mathbf{s}\to\mathbf{d}} \cdot \mathbf{n}_{\mathbf{s}}} }{|\mathbf{s}-\mathbf{d}|^{2}}
\label{eq:Gamma}
\end{equation}
The function $g(\mathbf{p}; \mathbf{l}, \mathbf{s})$ represents attenuation effects dependent on the distance and shading effects due to the surface normals of a hidden point $\mathbf{p}$, $\mathbf{l}$, and $\mathbf{s}$. $\Gamma(\mathbf{s})$ models the intensity variation of the light after scattering off the relay surface, and restricts scanning regions of the detector. For NLOS reconstruction, the LOS portion is often gated or removed to obtain the NLOS portion $\tau_{\text{NLOS}}$ by assuming a virtual light source at $\mathbf{l}$ and a virtual detector at $\mathbf{s}$. $\tau_{\text{LOS}}$ contains information, e.g., depth and reflectance, on the relay surface and enables us to calibrate the hardware devices. The calibration procedures, however, are complicated because the distances $|\mathbf{l}-\mathbf{o}|$ and $|\mathbf{s}-\mathbf{d}|$ are difficult to determine from the measurements only.
A confocal image formation model, first proposed by O'Toole et al. \cite{2018LCT}, collocates illumination and detection points, i.e., $\mathbf{l}=\mathbf{s}$ and $\mathbf{o}=\mathbf{d}$, and collects a 3D subset of the transient $\tau(t;\mathbf{o},\mathbf{l},\mathbf{s},\mathbf{d})$, as:
\begin{equation}
\begin{aligned}
\tau(t;\mathbf{o},\mathbf{s})
&= \Gamma(\mathbf{s})
\int_{P} \delta\rdb{2(|\mathbf{s}-\mathbf{o}|+|\mathbf{p}-\mathbf{s}|) - ct} \\ &\cdot \upsilon(\mathbf{p};\mathbf{s}) f(\mathbf{p};\boldsymbol{\omega}_{\mathbf{p}\to\mathbf{s}})
\frac{\rdb{\boldsymbol{\omega}_{\mathbf{p}\to\mathbf{s}} \cdot \mathbf{n}_{\mathbf{p}}}^2 \rdb{\boldsymbol{\omega}_{\mathbf{s}\to\mathbf{p}} \cdot \mathbf{n}_{\mathbf{s}}}^2}{|\mathbf{p}-\mathbf{s}|^{4}} \mathrm{d}{A_{\mathbf{p}}}
\end{aligned}
\label{eq:c-model}
\end{equation}
Similarly, we separate $\tau(t;\mathbf{o},\mathbf{s})$ into the LOS and NLOS portions of light paths as:
\begin{equation}
\begin{aligned}
\tau_{\text{NLOS}}(t;\mathbf{s})
&=
\int_{P} \delta\rdb{2|\mathbf{p}-\mathbf{s}| - ct} \upsilon(\mathbf{p};\mathbf{s}) f(\mathbf{p};\boldsymbol{\omega}_{\mathbf{p}\to\mathbf{s}})
g(\mathbf{p};\mathbf{s})
\mathrm{d}{A_{\mathbf{p}}}
\end{aligned}
\label{eq:model}
\end{equation}
\noindent and
\begin{equation}
\begin{aligned}
\tau_{\text{LOS}}(t;\mathbf{s})
&=
\Gamma(\mathbf{s}) \delta\rdb{2|\mathbf{s}-\mathbf{o}| - ct}
\end{aligned}
\label{eq:los-model}
\end{equation}
\noindent where
\begin{equation}
g(\mathbf{p}; \mathbf{s}) =
\frac{\rdb{\boldsymbol{\omega}_{\mathbf{p}\to\mathbf{s}} \cdot \mathbf{n}_{\mathbf{p}}}^2 \rdb{\boldsymbol{\omega}_{\mathbf{s}\to\mathbf{p}} \cdot \mathbf{n}_{\mathbf{s}}}^2}{|\mathbf{p}-\mathbf{s}|^{4}}
\end{equation}
The confocal NLOS imaging model is highly simplified and has advantages in terms of system calibration because the distance between $\mathbf{s}$ and $\mathbf{o}$ and the coordinates of $\mathbf{s}$ are easily determined.
\section{Adjustable NLOS imaging}
Fig.~\ref{fig:optodesign} shows the overview of our adjustable NLOS imaging with the light paths and the opto-electrical design of the hardware devices. The capture system consists of three modules: hardware module, LOS module, and NLOS module. The hardware module includes hardware devices and their controllers. The LOS module relates the hardware module and the relay surface, and corresponds to the LOS portion of transients, which is exploited to define the relay surface, scanning patterns, and a measurable bounding box where hidden objects can be situated. The NLOS module models spherical light paths between the relay surface and the hidden objects, and contains the information on the hidden objects for reconstruction. Fig.~\ref{fig:optodesign}\subref{subfig:softwareProcess} illustrates the pipeline of the procedures in software that combines the applications provided by the manufacturers of the hardware devices, e.g., the TCSPC and the galvanometer, and operates the entire procedures with minimal human interventions. A software manipulation video is shown in Supplementary Information.
\begin{figure}
\centering
\hfill
\subfloat[Light paths between the hardware devices]{
\begin{minipage}{.54\linewidth}
\centering
\includegraphics[height=24ex]{lightPath.pdf}
\end{minipage}
\label{subfig:lightPath}}
\subfloat[Opto-electrical design]{
\begin{minipage}{.36\linewidth}
\centering
\includegraphics[height=24ex]{lightCircuit.pdf}
\end{minipage}
\label{subfig:lightCircuit}}
\hspace{\fill}~\\
\subfloat[Procedural pipeline in software]{
\begin{minipage}{.93\linewidth}
\centering
\includegraphics[width=.98\linewidth]{softwareProcess.pdf}
\end{minipage}
\label{subfig:softwareProcess}}
\caption{Overview of our capture system. (a) Light paths between a pulsed laser, a single-pixel SPAD, a beam splitter, and a 2D galvanometer; emitting light from the laser in red and returning light in blue. (b) The opto-electrical design of three modules: hardware module, LOS module, and NLOS module. (c) The pipeline of procedures in software. Note that the galvanometer calibration and the temporal jitter calculation are required only once, and any of the reconstruction algorithms is available as a default.}
\label{fig:optodesign}
\end{figure}
\subsection{Gamma map}
In Equation \eqref{eq:Gamma}, we observe that in physics, $\Gamma(\mathbf{s})$ is related to the laser with $N_{\mathbf{o}}$, the detector with $A_{\mathbf{d}}$, the relay surface with $\frac{\rho}{\pi}$, and the relay surface with the detection points $\mathbf{s}$. It also relates the detector with the relay surface via the mapping from $A_{\mathbf{d}}$ to $A_{\mathbf{s}}$, and via the distance $|\mathbf{s}-\mathbf{d}|$. When the detector $\mathbf{d}$ is fixed, we can define $\Gamma(\mathbf{s})$ by determining the coordinates of detection points $\mathbf{s}$ on the relay surface. Based on the confocal NLOS imaging model, we calculate the coefficient of $\Gamma$ as the summation of the LOS portion $\tau_{\text{LOS}}(t;\mathbf{s})$ in Equation~\eqref{eq:los-model} at each detection point because the integral of $\delta$ is one. The coefficients of $\Gamma(\mathbf{s})$, or a \textit{Gamma} map, is defined for all detection points scanned on the relay surface.
Fig.\ref{fig:scanRegion}\subref{subfig:Gamma} shows three \textit{Gamma} maps. We first scanned a small number of detection points to outline the region scannable by the galvanometer and calculate the \textit{Gamma} map (upper left) to adjust where the relay surface is situated with respect to the galvanometer. The \textit{Gamma} map (lower left) is then calculated to check the scanning region where the entire setting of the capture system is occluded by an object or its parts. After selecting a scanning pattern, we can calculate a \textit{Gamma} map (upper right) to verify the scanning region defined by a scanning pattern for NLOS measurements. The \textit{Gamma} map is also employed to normalize the transients because it models intensity variation of the NLOS measurements.
The maximum intensity projection (MIP) map of LOS portions (lower right) appears similar to the corresponding \textit{Gamma} map, and can also be useful as the latter. The MIP map relies on the maximal value of the distribution function, rather than the integral, which may be less sensitive to the noise in transients measurements.
\subsection{System calibration}
The aim of calibration is to optimize our NLOS imaging by minimizing the difference between the transients that are measured from our setup and those that are theoretically computed based on the confocal NLOS imaging model. With the exception of hardware alignment, which needs human interventions to adjust light paths between hardware devices, the subsequent calibration procedures for the galvanometer, the relay surface, and the NLOS bounding box are iterative and can be operated online. The \textit{Gamma} maps are exploited to preview efficient scanning regions. \\[-1.5ex]
\noindent \textbf{Hardware alignment.} As shown in Fig.~\ref{fig:optodesign}, the laser and the SPAD are first aligned by observing the speckle of the laser beam that illuminates a surface, e.g., a white paper. We adjust the positions and angles of the laser and SPAD to maximize the intensity of the speckle. We then situate the beam splitter, the galvanometer, and the relay wall such that the speckle is clear at its focus on the wall. \\[-1.5ex]
\noindent \textbf{Galvanometer.} A dual-axis galvanometer supports optical scanning angles of about $\pm40\si{\degree}$, depending on several factors such as the laser beam diameter and the input voltage. In general, the galvanometer is coupled with a servo motor, which provides the feedback angles of the mirrors while scanning. We exploit the feedback between input voltages and angles to calibrate the galvanometer and to further determine the positions of scanning points. Fig.~\ref{fig:scanRegion}\subref{subfig:galvoSystem} illustrates the coordinate systems and scanning regions. Two Cartesian coordinate systems include $XYZ$ at the origin $\mathbf{o}$ where the laser beam is emitted into the free space toward the relay surface, and $xyz$ with the origin at the center of the detection region (in pink) on the relay surface, while $z=0$ when the relay surface is planar.
\begin{figure}
\centering
\hspace{\fill}
\subfloat[Coordinate systems and scanning regions]{
\begin{minipage}{.36\linewidth}
\centering
\includegraphics[height=28ex]{galvoSystem.pdf}
\end{minipage}
\label{subfig:galvoSystem}}
\subfloat[\textit{Gamma} and MIP maps]{
\begin{minipage}[c][28ex]{.36\linewidth}
\centering
\includegraphics[height=25ex]{Gamma.pdf}
\end{minipage}
\label{subfig:Gamma}}
\hspace{\fill}~\\\hspace{-.02\linewidth}
\subfloat[Arbitrary scanning pattern]{
\begin{minipage}{.36\linewidth}
\centering
\includegraphics[width=.75\linewidth]{validationRandom.png}
\end{minipage}
\label{subfig:validationRandom}}\hspace{-.115\linewidth}
\subfloat[Regular scanning pattern]{
\begin{minipage}{.36\linewidth}
\centering
\includegraphics[width=.75\linewidth]{validationUnifrom.png}
\end{minipage}
\label{subfig:validationUniform}}\hspace{-.105\linewidth}
\subfloat[Multi-circle scanning pattern]{
\begin{minipage}{.36\linewidth}
\centering
\includegraphics[width=.75\linewidth]{validationMulticircles.png}
\end{minipage}
\label{subfig:validationCircles}}
\caption{Scanning regions and patterns. (a) Coordinate systems and scanning regions. Two coordinate systems: XYZ and xyz. Scanning regions: blue area defined by the galvanometer, magenta area limited by $\Gamma$, and pink area determined by scanning patterns. (b) \textit{Gamma} and MIP maps. The \textit{Gamma} maps of the scannable region by the galvanometer (upper left), of the efficient scanning region with occlusions (lower left), and of a scanned region defined by a scanning pattern (upper right), and a corresponding MIP map (lower right). (c-e) Scanning patterns: arbitrary pattern, regular grid pattern, and multi-circle pattern. The scanned points are in red; the desired detection points, the regular grid, and the concentric circles in blue, which are all post-processed. }
\label{fig:scanRegion}
\end{figure}
To calibrate the galvanometer, we assume that the scanning system is linear, and formulate the relationship between the optical scanning angles $\theta_{X}$ and $\theta_{Y}$ and the input voltages $V_{X}$ and $V_{Y}$, as:
\begin{equation}
\begin{bmatrix}
\theta_{X} \\
\theta_{Y}
\end{bmatrix}
=
\begin{bmatrix}
\epsilon_{X} \\
\epsilon_{Y}
\end{bmatrix}
+
\begin{bmatrix}
\beta_{XX} & \beta_{XY} \\
\beta_{YX} & \beta_{YY}
\end{bmatrix}
\begin{bmatrix}
V_{X} \\
V_{Y}
\end{bmatrix}
\label{eq:galvanometer_model}
\end{equation}
\noindent where the initial angles $\epsilon_{X}$ and $\epsilon_{Y}$ are determined by the offset arrangement of the two mirrors prior to input voltages, and $\beta$ represents the coefficients of an angle with respect to the input voltages. The initial angles $\epsilon_{X}$ and $\epsilon_{Y}$ and the coefficients of $\beta$ may be offered by the manufacturers, whereas for precision, we collect $N$ groups of optical scanning angles $\theta_{n}$ along with input voltages $V_{n}$ in the given voltage range to calculate them. The coefficients of $\beta$ are first optimized using the multiple linear regression algorithm with a loss function $\mathcal{L}_{\text{Galvo}}$, as:
\begin{equation}
\mathcal{L}_{\text{Galvo}}(\beta) = \frac{1}{N}\sum_{n}{|\theta_{n} - \beta V_{n}|^{2}}
\label{eq:loss_galvo}
\end{equation}
We then compute $\epsilon_{X}$ and $\epsilon_{Y}$ as the average error between the calculated and measured optical scanning angles. Finally, the input voltages for each group of scanning angles are calculated to control scanning points on arbitrary surfaces and to define the scannable region (blue area in Fig.~\ref{fig:scanRegion}\subref{subfig:galvoSystem}) of the galvanometer, which is unnecessarily rectangular. \\[-1.5ex]
\noindent \textbf{Relay surface.} The relay surface plays a critical role for an NLOS capture system. The LOS portion of transients, as in Equation~\eqref{eq:los-model}, can be exploited to recover the albedo and the orientation of the relay surface. Specifically, we extract the peak of the histogram $\tau_{\text{LOS}}(t;\mathbf{s})$ at each detection point and its corresponding $t$, and calculate the depth $\ell$ of the relay surface. We then employ the optical scanning angles $\theta_{X}$ and $\theta_{Y}$ of the galvanometer and the depth $\ell$ to estimate 3D coordinates of the detection points in $XYZ$, as:
\begin{equation}
\begin{cases}
X = Z \tan{\theta_{X}} \\
Y = Z \tan{\theta_{Y}} \\
Z = \ell \rdb{1+\tan^{2}{\theta_{X}}+\tan^{2}{\theta_{Y}}}^{-1/2}
\end{cases}
\label{eq:coordinates-angledist}
\end{equation}
The relay surface $W(W_{X},W_{Y},W_{Z})$ is considered to be planar and is formulated as:
\begin{equation}
W: W_{X}X + W_{Y}X + W_{Z}Z + 1 = 0
\label{eq:wall-model}
\end{equation}
\noindent We notice that the albedo and the orientation of the relay are considered in $\Gamma(\mathbf{s})$. Our calibration technique therefore does not require any additional devices or textured targets by using the \textit{Gamma} map. We thus recover the relay when the loss function $\mathcal{L}_{W}(W_{X},W_{Y},W_{Z})$ is minimal, which is the root-mean-square error (RMSE) of distances from $N$ points to plane:
\begin{equation}
\mathcal{L}_{W}(W_{X},W_{Y},W_{Z}) = \sqrt{\frac{1}{N}\sum_{X,Y,Z}{\frac{|W_{X}X + W_{Y}X + W_{Z}Z + 1|^{2}}{W_{X}^{2}+W_{Y}^{2}+W_{Z}^{2}}}}
\label{eq:wall-loss}
\end{equation}
\noindent \textbf{NLOS bounding box.} Reconstruction accuracy depends on several factors of the capture system, including the geometric setting of the hardware module and the detection region on the relay . For efficient measurements, we define a bounding box to specify where the objects are situated in an NLOS scene.
Ahn et al.\cite{2019Ahn} have mentioned that the reconstructed shape should be within the orthogonal projection of the scanning region on the relay wall. In practice, we make a free space to allow for larger hidden objects by exploiting a relay wall whose orientation and position are adjustable. This setting is equivalent to adjusting the position of the entire setting of hardware devices. The orthogonal projection of the scanning region thus restricts a measurable bounding box of the NLOS scene with the maximal width and height of the scanning region. The minimal depth of the bounding box is approximated as:
\begin{equation}
z_{\min} = ct_{\text{delay}}
\end{equation}
\noindent where $t_{\text{delay}}$ represents the delay between the arrival times of photons that travel back from the relay and from the hidden objects. Since the distance between hidden objects and the relay plays a significant role in the attenuation of photons, we consider it to be the maximal depth $z_{\max}$ of the bounding box. Here we assume that the hidden object is a perfectly diffuse white sphere, i.e., its BRDF $f(\mathbf{p})=\frac{1}{\pi}$, and that the signal is larger than a bias $b$, as:
\begin{equation}
\Gamma(\mathbf{s}) \int_{P}{\frac{(\boldsymbol{\omega}_{\mathbf{s}\to\mathbf{p}}\cdot\mathbf{n}_{\mathbf{s}})^{2}}{\pi|\mathbf{p}-\mathbf{s}|^{4}} \mathrm{d}{A}_{\mathbf{p}}}
= \Gamma(\mathbf{s}) \int_{0}^{2\pi}\int_{0}^{\pi/2}{\frac{\cos^{2}{\theta}}{\pi z_{\max}^{4}} \mathrm{d}{\theta}\sin{\theta}\mathrm{d}{\phi}} \ge b
\end{equation}
\noindent when $\Gamma(\mathbf{s})$ is minimal. The bias $b$ includes the dark counts of the capture system and the ambient photon flux. $z_{\max}$ is then computed:
\begin{equation}
z_{\max} = \rdb{\frac{2\Gamma_{\text{min}}}{3b}}^{1/4}
\end{equation}
\noindent where $\Gamma(\mathbf{s})$ limits the scanning region and the volume of the bounding box. Fig.~\ref{fig:scanRegion}\subref{subfig:galvoSystem} shows the initial scannable region of the galvanometer (in blue), the \textit{Gamma}-restricted scanning region (in magenta), and a user-defined detection region (in pink) on the relay . The bounding box of the NLOS scene helps us to estimate the size and position of a hidden object and to roughly predict the reconstruction quality by placing a textured target, e.g., a checkerboard, at different positions.
\subsection{Scanning patterns}
The scanning process relates the hardware module to the relay and the NLOS bounding box, and can also preview the effects of the system calibration. In the literature, illumination and detection points are usually distributed in a regular grid with evenly spaced points on the relay . In contrast, our capture system addresses user-defined scanning patterns by calculating the input voltages that correspond to the coordinates of each detection point on the relay . Specifically, we first re-parameterize Equation~\eqref{eq:wall-model} with $w(w_{X},w_{Y},w_{Z})$, as:
\begin{equation}
\begin{aligned}
w &= \rdb{W_{X}^2+W_{Y}^2+W_{Z}^{2}}^{-1/2} \\
w_{X} &= w \cdot W_{X} \\
w_{Y} &= w \cdot W_{Y} \\
w_{Z} &= w \cdot W_{Z}
\label{eq:wall-parameters}
\end{aligned}
\end{equation}
In the efficient scanning region restricted by $\Gamma$, we define a scanning area (pink in Fig.~\ref{fig:scanRegion}\subref{subfig:galvoSystem}) with the origin $\mathbf{o_s}$ at its center on the relay . New sets of orthonormal basis $\hat{x},\hat{y},\hat{z}$ and $\hat{X},\hat{Y},\hat{Z}$ are constructed to normalize the surface normal of each detection point $(x,y,z)$, as:
\begin{equation}
\begin{cases}
\hat{x} = \frac{\frac{1}{w_{x}}\hat{X} - \frac{1}{w_{z}}\hat{Z}}{\sqrt{w_{x}^{-2}+w_{z}^{-2}}}\\
\hat{y} = \hat{z} \times \hat{x} \\
\hat{z} = w_{X}\hat{X} + w_{X}\hat{Y} + w_{Z}\hat{Z}
\end{cases}
\label{eq:wall-basis}
\end{equation}
\noindent where $\hat{z}$ is the unit vector of the surface normal. Note that $z=0$ for a planar relay , and the coordinates of a point $\mathbf{p}$ on the hidden object $P$ can therefore be denoted with a value of $z$ such that the two coordinate systems for a detection point are transformed as:
\begin{equation}
\begin{cases}
\mathbf{s} = \mathbf{o}_{\mathbf{s}} + x\hat{x} + y\hat{y} + 0\hat{z} & (\text{in } xyz \text{ coordinate system})\\
\mathbf{s} = \mathbf{o} + X\hat{X} + Y\hat{Y} + Z\hat{Z} & (\text{in } XYZ \text{ coordinate system})
\end{cases}
\label{eq:coordinates-trans}
\end{equation}
With Equations \eqref{eq:galvanometer_model} and \eqref{eq:wall-basis}, the input voltages of the galvanometer are determined by calculating the scanning angles from the coordinates of any detection point on the relay . To validate our scanning technique, we have tested three scanning patterns: arbitrary pattern, regular grid pattern, and multi-circle pattern.
For the arbitrary scanning pattern, we randomly raster-scan several points in different directions, and determine the coordinates of the detection points $\mathbf{s}$ and the equation of the relay . New coordinates $\mathbf{s}_{W}$ are then calculated for these points on the estimated relay and their corresponding input voltages. Using the input voltages, the galvanometer is controlled to re-scan on the relay . Fig.~\ref{fig:scanRegion}\subref{subfig:validationRandom} demonstrates the two groups of detection points: desired detection points $\mathbf{s}$ are in blue, and the re-scanned points $\mathbf{s}_{W}$ in red.
\begin{equation}
\mathbf{s}_{W} = \mathbf{s} - (\hat{z} \cdot \mathbf{s})\hat{z}
\end{equation}
We also define a regular scanning pattern with our system. $N \times N$ detection points are scanned in a region of $L \times L$, and the coordinates of the detection point $\mathbf{s}(i,j)$ are represented as:
\begin{equation}
\mathbf{s}(i,j) = \mathbf{o}_{\mathbf{s}} - \frac{L}{2}\hat{x} + \frac{L}{2}\hat{y} + \frac{j-1}{N-1}L\hat{x} - \frac{i-1}{N-1}L\hat{y}
\end{equation}
\noindent Fig.~\ref{fig:scanRegion}\subref{subfig:validationUniform} shows a regular scanning pattern with evenly spaced points (in red) on the relay . Note that the grid in blue is post-processed to identify the equidistant spaces between the scanned points.
Inspired by a circular scanning pattern in \cite{2020ECCV}, we further present a multi-circle scanning pattern. In a scanning region with the radius $R$, we define $N_{r}$ concentric circles, and $N_{\phi}$ points on each circle. The coordinates of the detection points $\mathbf{s}(i,j)$ are then determined as:
\begin{equation}
\mathbf{s}(i,j) = \mathbf{o}_{\mathbf{s}} + \frac{i}{N_{r}}R\cos\rdb{\frac{\pi}{2}-(j-1)\frac{2\pi}{N_{\phi}}}\hat{x} + \frac{i}{N_{r}}R\sin\rdb{\frac{\pi}{2}-(j-1)\frac{2\pi}{N_{\phi}}}\hat{y}
\end{equation}
The spaces between the circles and between the points on each circle may be equal or unequal. Similarly, we scan 32 sensing points on 4 circles, i.e., 8 points on each circle, and demonstrate them on additional concentric circles (in blue) as in Fig.~\ref{fig:scanRegion}\subref{subfig:validationCircles}. The results of the three scanning patterns in Fig.~\ref{fig:scanRegion}\subref{subfig:validationRandom} to \subref{subfig:validationCircles} show that the scanned points are in good agreement with the desired positions in either regular or irregular fashion, and either evenly or unevenly spaced.
\subsection{Transient enhancement}
The signal-to-noise ratio of transients that are collected from the NLOS capture system is influenced by two major factors: properties of the SPAD including photon detection efficiency, afterpulsing, and pileup; and temporal jitter of the laser and the SPAD, which models uncertainty in the time-resolving mechanism. We first re-align the histogram (or transient) of each detection point such that the arrival time of the direct reflection from the relay locates at the position of zero. Using the \textit{Gamma} map, we then normalize the transients $\tau(t;\mathbf{s})$ for higher quality.
While Hernandez et al.\cite{2017Quercus} have introduced a computational model for a SPAD, we opt for the approximation model in \cite{2017Matthew} to describe the probability of detecting individual photon events in a histogram bin as a Poisson distribution $\text{Pois}$. The bias $b$ is usually considered to be independent of time at a detection point and increases the background noise of the transient. The transients recorded with a SPAD, $\tau^{\text{SPAD}}(t;\mathbf{s})$, are formulated as:
\begin{equation}
\tau^{\text{SPAD}}(t;\mathbf{s}) = \text{Pois}\rdb{(\tau*j)(t;\mathbf{s})+b}
\label{eq:spad_model}
\end{equation}
\noindent where $j$ represents the temporal jitter of the system. Following \cite{2019Sun}, the temporal jitter typically yields a curve having two parts: a Gaussian peak and an exponential tail, as:
\begin{equation}
\begin{aligned}
j(t; \mu, \sigma, \kappa_{0}, \kappa_{1}, \gamma)
&= \text{Gaus}(t; \mu, \sigma) + \gamma \text{Exp}(t; \mu, \kappa_{0}, \kappa_{1}) \quad (t>0)
\label{eq:jitter}
\end{aligned}
\end{equation}
\noindent where
\begin{equation}
\text{Gaus}(t; \mu, \sigma) = \exp\rdb{-\frac{(t-\mu)^{2}}{2\sigma^{2}}}
\end{equation}
\begin{equation}
\text{Exp}(t; \mu, \kappa_{0}, \kappa_{1}) = \frac{1}{\sqrt{t}} \exp\rdb{-\frac{(t-\mu)^{2}}{\kappa_{0}t}}\rdb{1 + \frac{t-\mu}{\kappa_{1}t}}
\end{equation}
\noindent where $\mu, \sigma, \kappa_{0}$, and $\kappa_{1}$ are the coefficients of the temporal jitter, and $\gamma$ is the weight of the exponential term. The LOS transients $\tau_{\text{LOS}}^{\text{SPAD}}$ are exploited to calculate the temporal jitter $j$ of our capture system from the loss function $\mathcal{L}_{\text{SPAD}}$ as a cross-entropy loss:
\begin{equation}
\mathcal{L}_{\text{SPAD}}(\mu, \sigma, \kappa_{0}, \kappa_{1}, \gamma) = -\sum_{t,\mathbf{s}}{\left( \frac{\tau_{\text{LOS}}^{\text{SPAD}}(t;\mathbf{s})}{\Gamma(\mathbf{s})} \cdot \log{j(t-|\mathbf{s}|/c; \mu, \sigma, \kappa_{0}, \kappa_{1}, \gamma)} \right)}
\end{equation}
We compute the temporal jitter at several detection points and consider the average value as the temporal jitter of our capture system. In Equation~\eqref{eq:spad_model}, we notice that the temporal jitter $j(t; \sigma, \gamma)$ dominates the measurement effect with a SPAD when decomposing the Poisson distribution from $\tau^{\text{SPAD}}$. The Wiener filter can thus be applied to denoise the transients $\tau^{\text{SPAD}}$:
\begin{equation}
\tilde{\tau}(\nu;\mathbf{s}) = \tilde{k}(\nu)\tilde{\tau}^{\text{SPAD}}(\nu;\mathbf{s})
\end{equation}
\noindent where $\tilde{\tau}(\nu;\mathbf{s})$ and $\tilde{\tau}^{\text{SPAD}}$ represent the Fourier transform of $\tau(t;\mathbf{s})$ and $\tau^{\text{SPAD}}(t;\mathbf{s})$, while $\nu$ is the frequency. The kernel of the Wiener filter $\tilde{k}(\nu)$ in the frequency domain is computed from the Fourier transform of the temporal jitter, $\tilde{j}(\nu)$, and the signal-to-noise ratio $\eta$, as:
\begin{equation}
\tilde{k}(\nu) = \frac{\rdb{\tilde{j}(\nu)}^{-1}}{1+\rdb{\eta|\tilde{j}(\nu)|^{2}}^{-1}}
\end{equation}
O'Toole et al. \cite{2018LCT} embed the Wiener filter within their LCT algorithm, while Velten et al. \cite{2012Velten} use the Laplacian filter after the back-projection algorithm as a post-processing to mitigate noise. In contrast to LCT and FBP, we treat the denoising process prior to NLOS reconstruction such that we can opt for a wide variety of reconstruction algorithms. In common with the preceding studies, we ignore the effect of the Poisson distribution on denoising, while our optimization algorithm accounts for this effect on NLOS reconstruction.
\subsection{NLOS Reconstruction}
Many algorithms have been proposed to reconstruct an NLOS scene from its transients. Volume-based algorithms, e.g., LCT\cite{2018LCT}, FK\cite{2019FK}, and PF\cite{2019Liu}, assume that the NLOS scene is represented as 3D voxels of a volume, and solve the inverse problem using FFT for reconstruction. These algorithms require a regular form of transients as input. Alternatively, optimization-based and learning-based algorithms (e.g., NeTF\cite{2021Shen}) support arbitrary forms of input transients, while the latter require tens of hours for training.
We implement an optimization algorithm based on the confocal imaging model. In common with most existing methods, the optimization algorithm assumes that the hidden object is perfectly diffuse, and that there are no occlusions between the object and the relay surface, and no inter-reflection between the points on the hidden object. This implies that in Equation~\eqref{eq:model}, both $\upsilon(\mathbf{p};\mathbf{s})$ and $g(\mathbf{p}; \mathbf{s})$ are considered to be one, and the BRDF $f(\mathbf{p};\boldsymbol{\omega}_{\mathbf{p}\to\mathbf{s}})$ becomes view-independent $\mathbf{f}$. Equation~\eqref{eq:model} is thus simplified as a linear formulation:
\begin{equation}
\boldsymbol{\tau} = \Psi \mathbf{f}
\end{equation}
\noindent where $\boldsymbol{\tau}$ is the discretized measurements and $\Psi$ a linearized measurement matrix. The albedo of the hidden object, $\mathbf{f}$, can be optimized by minimizing the differences between the theoretical $\Psi \mathbf{f}$ and the measured $\boldsymbol{\tau}$. We adopt the Poisson likelihood function to evaluate the similarity of the following observation function with total variation (TV) as a regularization term\cite{2021Ye,2012Harmany}:
\begin{equation}
\mathcal{L}_{\text{NLOS}}\rdb{\mathbf{f}} = -\ln\rdb{\prod_{i}{\frac{(\mathbf{e}^{\intercal}_{i} \Psi \mathbf{f})^{\tau_{i}}}{\tau_{i}!}} \exp({-\mathbf{e}^{\intercal}_{i} \Psi \mathbf{f}})} + \lambda ||\mathbf{f}||_{\text{TV}}
\end{equation}
\noindent where $\mathbf{e}_i$ is the $i$th standard unit vector and $\tau_i$ the $i$th element of vector $\boldsymbol{\tau}$, while $\lambda$ is the weight of the TV. We adopt a gradient descent method to implement the optimization algorithm.
\section{Evaluation}
To evaluate our capture system, we use transients that are theoretically simulated and really measured and reconstruct a number of NLOS scenes with the optimization algorithm (OPT) and state-of-the-art (SOTA) methods, including LCT\cite{2018LCT}, FK\cite{2019FK}, and PF\cite{2019Liu}.
\subsection{Simulation and measurement}
\textbf{Transient simulation.} We develop an algorithm to synthesize transients based on the confocal image formation model in Equation~\eqref{eq:model}. The illumination spot collocates with the detection point $\mathbf{s}$ on the relay , and the light scatters in a spherical wavefront toward a hidden object. We assume that at the detection point, an intensity map $L(u,v;\mathbf{s})$ and a depth map $D(u,v;\mathbf{s})$ are captured, where $u,v$ represents a corresponding pixel of the pair of maps. We then extract the depth and the intensity at each pixel, and compute the transient $\tau(t;\mathbf{s})$ at the time instant $t$. This process is repeated for all detection points, as outlined in Algorithm \ref{alg:simulation}.
\begin{algorithm}
\caption{Transient simulation}
\label{alg:simulation}
\begin{algorithmic}[1]
\REQUIRE{depth maps $D(u,v;\mathbf{s})$, intensity maps $L(u,v;\mathbf{s})$, speed of light $c$}
\ENSURE{transients $\tau(t;\mathbf{s})$}
\STATE initialize $\tau(t;\mathbf{s})$
\FORALL{sensing points $\mathbf{s}$}
\FORALL{image pixels $(u,v)$}
\STATE $t = \text{round}(2*D(u,v)/c)$
\STATE $\tau(t;\mathbf{s}) = \tau(t;\mathbf{s}) + L(u,v)$
\ENDFOR
\ENDFOR
\end{algorithmic}
\end{algorithm}
We render transients $\tau(t;\mathbf{s})$ of an S-shape of \SI{0.6}{m} $\times$ \SI{0.6}{m} and a Whiteboard of \SI{0.6}{m} $\times$ \SI{0.4}{m} with \SI{4}{ps} temporal resolution and 64 $\times$ 64 spatial resolution. By considering the temporal jitter, the bias, and the Poisson distribution in Equation~ \eqref{eq:spad_model}, the transients with noise of the S-shape and the Whiteboard are also synthesized. Fig.~\ref{fig:experimentSetup}(b, lower) shows the simulated transients of S-shape with and without additional noise. \\[-1.5ex]
\noindent \textbf{Transient measurement.} Fig.~\ref{fig:experimentSetup}(a) shows a photograph of our capture system. We employ a fast-gated SPAD from Micro Photon Devices, which provides \SI{4}{ps} temporal resolution using a PicoHarp 300. An achromatic lens (Canon EF \SI{50}{mm} f/1.8) is exploited to focus the returning light onto the SPAD. A laser (PicoQuant LDH-IB-670-P) emits collimated light through a polarized beam splitter cube (Thorlabs VA5-PBS251). The light is then guided by a 2D galvanometer (Thorlabs GVS012) controlled by a data acquisition device. The SPAD records the indirect light from an NLOS scene by gating the direct light using a delayer (PicoQuant PSD-065-A-MOD). The temporal jitter of the entire system is approximately \SI{100}{ps} with a laser line filter (Thorlabs FL670-10). During the NLOS measurements, the laser power is set as \SI{1.54}{mW}, and delayer gate width is set as \SI{11}{ns}.
Our hardware devices are located \SI{1.0}{m} from the relay wall, a melamine white panel whose position and orientation can be adjusted for different sizes and shapes of NLOS scenes. We have measured the transients of a Whiteboard and an S-shape, whose settings are similar to their simulation. Fig.~\ref{fig:experimentSetup}(b, upper) shows measured transients, raw and denoised. In comparison with the simulated transients, the smaller peaks of measured transients are declined due to the properties of a SPAD, e.g., dark count rate and pileup. Other NLOS scenes include a Checkerboard, a Mannequin, and a Reso.board that is designed with stripes of different widths and lengths. All the objects measured are diffuse materials of paper, cotton or wood. Since the transients are captured automatically in our system, most experiments take \SI{30}{s} to record precise measurements at a single detection point and to transit to the next. The detailed parameters of experiments are shown in Table~\ref{tab:measurements}.
\begin{figure}
\centering
\subfloat[Photograph of our capture system]{
\begin{minipage}{.45\linewidth}
\centering
\includegraphics[height=36ex]{experimentSetup.png}
\end{minipage}
\label{subfig:experimentSetup}}
\subfloat[NLOS transients measured
and simulated
]{
\begin{minipage}{.5\linewidth}
\centering
\includegraphics[height=18ex]{measComparison.pdf}~\\
\includegraphics[height=18ex]{simComparison.pdf}
\end{minipage}
\label{subfig:dataComparison}}
\caption{Prototype and NLOS transients. (a) The prototype consists of a single-pixel SPAD, a pulsed laser, a 2D galvanometer, and a beam splitter. (b, upper) Transients measured, raw (blue) and denoised (red). (b, lower) Transients simulated with additional noise (blue) and without noise (red). Note that the transients are measured and simulated at the corresponding point of S-shape.}
\label{fig:experimentSetup}
\end{figure}
\subsection{Reconstruction results}
We have conducted experiments using the simulated and measured transients. The source codes with Matlab, including LCT, FK, and PF that are publicly available, run on a personal computer with an Intel i7-8750H CPU (2.2 GHz), 16 GB RAM, and a preliminary GPU 1050Ti. Fig.~\ref{fig:sComparison} shows the reconstruction of S-shape using the algorithms of LCT, FK, and PF. The transients of S-shape, raw or denoised, are measured in a regular grid and are simulated with and without noise. As expected, the volumes of S-shape are well reconstructed from the transients and our enhancement scheme can improve the reconstruction quality. We further carry out experiments of Checkerboard, Reso.board, and Mannequin using the normalized transients that are also measured in an equidistant grid. Fig.\ref{fig:results} demonstrates the volumes of the hidden objects reconstructed with the SOTA algorithms and OPT. The textured patterns of Checkerboard and Reso.board are shown in good agreement with the photographs or ground truth (GT), while the algorithms highly influence the reconstruction quality. The SOTA algorithms take of the order of seconds to reconstruct each NLOS scene, whereas our optimization algorithm takes several minutes for 1000 iterations.
\begin{figure}
\centering
\includegraphics[width=.98\linewidth]{sComparison.pdf}
\caption{Evaluation of simulated and measured transients of S-shape. (a) and (c) are reconstructed using the simulated transients, without and with noise. (b) and (d) are reconstructed using the measured transients, raw and denoised.}
\label{fig:sComparison}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=.98\linewidth]{results.pdf}
\caption{Evaluation of normalized transients in a regular grid. From top to bottom: Checkerboard, Reso.board, and Mannequin are reconstructed with the SOTA algorithms and OPT. The evaluation of denoised transients are shown in supplementary information}
\label{fig:results}
\end{figure}
In addition, we evaluate the reconstruction effects using our transients measured in different scanning patterns. Fig.\ref{fig:moreResults}(a) and (b) show the reconstruction results of Whiteboard with the SOTA algorithms and OPT. The transients of Whiteboard were measured in a regular grid of 16 $\times$ 16 with acquisition \SI{15}{s} and 32 $\times$ 32 with \SI{30}{s}, while the two frames in yellow were measured in a multi-circle pattern. We notice that the reconstruction quality with OPT using a smaller number of
transients remains similar while the results with the SOTA algorithms are highly degraded. The Whiteboard measured in multi-circle pattern is slightly better reconstructed than in regular grid since denser detection points are distributed at the center of the NLOS scene. Fig.\ref{fig:moreResults}(c) shows the reconstruction results with OPT in more scanning patterns. Whiteboard, S-shape, Checkerboard, Mannequin, and Reso.board1 are measured in a multi-circle scanning pattern, and their volumes are sufficiently reconstructed. Reso.board2 is measured in an arbitrary pattern, i.e., coarse detection points in the lower left area and dense detection points in the upper right area, and is reconstructed with higher quality than Reso.board1 in multi-circle pattern. More experimental results with simulated and measured transients, raw and denoised, are shown in Supplementary Information.
\begin{figure}
\centering
\includegraphics[width=.98\linewidth]{moreResults.pdf}
\caption{Evaluation of the transients measured in different scanning patterns. (a) and (b) Reconstruction results of Whiteboard with the SOTA algorithms and OPT, using transients in regular grid of 16 $\times$ 16 with \SI{15}{s} and 32 $\times$ 32 with \SI{30}{s}, while the two frames (yellow) are reconstructed with OPT using transients in a multi-circle pattern. (c) Reconstruction results with OPT of different objects. From left to right: Whiteboard, S-shape, Checkerboard, Mannequin, and Reso.board1 were measured in multi-circle pattern, while Reso.board2 was measured in an arbitrary pattern. }
\label{fig:moreResults}
\end{figure}
\begin{table}
\centering
\caption{Experimental details of transient measurements. Three scanning patterns include regular grid (grid), multi-circle (circles), and arbitrary pattern.}
\small
\begin{tabular}{cccccc}
\toprule
Experiments & Whiteboard & S-shape & Checkerboard & Reso.board & Mannequin \\
\midrule
NLOS size & \SI{0.6}{m}$\times$\SI{0.4}{m} & \SI{0.6}{m}$\times$\SI{0.6}{m} & \SI{0.8}{m}$\times$\SI{0.8}{m} & \SI{0.8}{m}$\times$\SI{0.8}{m} & \SI{0.5}{m}$\times$\SI{0.6}{m} \\
NLOS distance & \SI{0.8}{m} & \SI{0.8}{m} & \SI{0.8}{m} & \SI{0.8}{m} & \SI{0.8}{m} \\
Detection region & \SI{0.8}{m}$\times$\SI{0.8}{m} & \SI{0.8}{m}$\times$\SI{0.8}{m} & \SI{0.8}{m}$\times$\SI{0.8}{m} & \SI{0.8}{m}$\times$\SI{0.8}{m} & \SI{0.8}{m}$\times$\SI{0.8}{m} \\
Detection points & $16^{2}$/$32^{2}$ & $64^{2}$ & $64^{2}$ & $32^{2}$ & $64^{2}$ \\
Scanning patterns & grid/circles & grid/circles & grid/circles & three patterns & grid/circles \\
Exposure time & \SI{15}/\SI{30}{s} & \SI{30}{s} & \SI{30}{s} & \SI{30}{s} & \SI{30}{s} \\
\bottomrule
\end{tabular}
\label{tab:measurements}
\end{table}
\section{Conclusion}
In this work, we have presented an adjustable NLOS imaging based on the confocal imaging model. Our capture system enables us to adjust scanning regions and patterns of detection points and to calibrate online the system components such as a relay surface and a galvanometer via \textit{Gamma} maps, scanning patterns, and reconstruction. In addition, our scanning technique allows users to define proper scanning patterns for precise measurements of different NLOS scenes, and
the tailored reconstruction scheme can support arbitrary forms of input transients. Moreover, our software manipulates the procedures from system calibration to transient recording and enhancement to NLOS reconstruction except for manual adjustment of hardware devices. The experimental results for various hidden objects demonstrate that the measurements from our capture system are adequate and efficient. We believe that our work makes a significant step toward automation and convenience of NLOS acquisition and helps to facilitate NLOS imaging research.
Although our setup and techniques could be extended to support conventional NLOS imaging, estimating the coordinates of illumination and detection points may need additional devices or textured targets. The scanning patterns resemble how to align cameras and light sources in a light field imaging system and to determine rich information on the NLOS scenes in specific views. Adaptively selecting scanning points has the potential to unlock optimal numbers and positions of the illumination and detection points, resulting in minimal acquisition time and high-quality reconstruction. Our denoising scheme is preliminary and can be improved by accounting for more properties of a SPAD, e.g., Poisson distribution and pileup \cite{2019Gupta}, and by using a pulsed laser with even higher power. Further efforts and developments are thus needed for both NLOS imaging and NLOS reconstruction algorithms. \\
\noindent\textbf{Funding} This work was supported by Shanghai Science and Technology Program (21010502400) and NSFC programs (61976138, 61977047).
\noindent\textbf{Disclosures} The authors declare no conflicts of interest.
\noindent\textbf{Data availability} We will provide data underlying the results presented in this paper.
|
2024-02-18T23:40:24.710Z
|
2021-12-30T02:27:36.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14555","language":"en","timestamp":1640831256000,"url":"https:\/\/arxiv.org\/abs\/2112.14555","yymm":"2112"}
|
proofpile-arXiv_000-10070
|
{"provenance":"002.jsonl.gz:10071"}
| null | null |
\section{Introduction}
In this era of informatization and data in trajectories, nature and human activities will be recorded as a large amount of asynchronous sequential data, for instance, the record of occurrence of earthquakes and aftershocks \cite{1ogata1998space}, transaction history of financial markets \cite{2bacry2015hawkes}, electronic health record \cite{3wang2018supervised}, equipment failure history \cite{4zhang2020survival} and user behavior in social networks \cite{5zhou2013learninga,6zhao2015seismic}, such as Weibo, Twitter, Facebook, etc. all are asynchronous events sequences data.
These asynchronous sequences contain valuable knowledge and information, researchers utilize a variety of methods to better mine these knowledge and information from the data. Among a diversity of methods, point process is one of the most widely used methods in this field, and in point process models, Hawkes process \cite{7hawkes1971spectra} is one of the most commonly used model. The self-exciting progress in Hawkes process fits the interaction between events to some extent, thus the application of Hawkes process achieves certain results in sequence analysis.
For example, Zhou et al. present alternating direction method of multipliers (ADMM)-based algorithm to learn Hawkes process to discover the hidden network of social influence \cite{7hawkes1971spectra}, Xu et al. make use of non-parametric Hawkes process to reveal the Granger causality between the users activity on watching internet protocol television (IPTV) \cite{8xu2016learning}. Hansen et al. demonstrate the great expressive potential of Hawkes process in neuroscience with least absolute shrinkage and selection operator (LASSO) method \cite{9hansen2015lasso}. Reynaud-Bouret and Schbath provide a new way to detect the favored or avoided distances between genomic events along deoxyribonucleic acid (DNA) sequences \cite{10reynaud2010adaptive}. Zhang et al. \cite{4zhang2020survival} modify the traditional Hawkes process, introducing the time-dependent background intensity to Hawkes process to analyze the background probability of failure and relationship between failures in compressor station.
However, traditional Hawkes process only considers the positive superposition of effect of historical events, which severely constrains the fitting ability of this type of model. Meanwhile, the lack of nonlinear operations in traditional Hawkes process also sets an upper limit for Hawkes’ expressive ability. Thus, in recent years, due to the strong fitting ability of neural networks and especially the sequence modeling ability of RNN, the research direction in this field is transferred to the neural point process model. For instance, Du et al. \cite{11du2016recurrent} embed the information of sequence data (including time-stamp and event type) into RNN, and come up with the recurrent marked temporal point processes (RMTPP) to model the conditional intensity function by nonlinear dependency of the history. And similar to RMTPP, Mei et al. \cite{12mei2017neural} propose the continuous-time LSTM (Long Short-Term Memory) to model the conditional intensity of point process, which is called as neural Hawkes process, and in this continuous neural Hawkes process, the influence of the previous event decays with time continuously. Xiao et al. use two RNN to model the conditional intensity function, one is used to process time stamp information, and the other is used to process historical event information \cite{13xiao2017modeling}.
Inevitably, these RNN based-models also inherit the drawbacks of RNN, for instance, it may take a long time for the patient to develop symptoms due to certain sequel, which has obvious long-term characteristics, such as diabetes, cancer and other chronic diseases, while these RNN-based models are hard to reveal the long-term dependency between the distant events in the sequences \cite{14bengio1994learning}. The ideal point process model should be able to solve these problems. Moreover, in the training of RNN-based models, such problems as vanishing and exploding Gradients \cite{15pascanu2013difficulty} often occur, and then affect the performance of the model.
It’s worth noting that in traditional sequence learning problem, such as machine translation \cite{16raganato2018analysis} and speech recognition \cite{17dong2018speech}, transformer model \cite{18vaswani2017attention} based on self-attention \cite{19DBLP:journals/corr/BahdanauCB14} mechanism achieves distinct performance improvement without application of CNN and RNN, meanwhile, the transformer structure free-from recurrent modules makes the model have higher computational efficiency. These achievements give a new insight on the development of sequential data learning. On account of this fact, Zhang et al. present self-attention Hawkes process \cite{20zhang2020self}, furthermore, Zuo et al. \cite{21DBLP:conf/icml/ZuoJLZZ20} propose transformer Hawkes process based on the attention mechanism and encoder structure in transformer. This model utilizes pure transformer structure without using RNN and CNN, and achieves state-of-the-art performance, but there is still much room for improvement in the transformer models, for instance, transformer simply stack the encoder layer to learn sequence data, and foregoes the recursive bias learning in RNN, while this recursive learning might be more important than commonly believed.
Dehghani et al. point out that re-introduce recurrence calculation in transformer maybe promote the performance of transformer, which is called as universal transformer \cite{22DBLP:conf/iclr/DehghaniGVUK19}, this model combines the advantage of transformer and RNN, organically combines the self-attention and recurrence learning mechanism. In recurrence process, Dehghani et al. make use of the ACT mechanism \cite{23graves2016adaptive} to decide when the recurrence process will halt. The experimental results of universal transformer demonstrate that the effectiveness of the combination of self-attention mechanism and recurrence mechanism.
Based on the achievements of universal transformer, we tend to work out a new framework of transformer Hawkes process based on the idea of universal transformer, we name it as universal transformer Hawkes process. We introduce the recurrent structure in transformer Hawkes process, and make our model to achieve Turing completeness compared with previous transformer model. Moreover, we add a convolution module to the position-wise-feed-forward-layer, which enhance the local perception ability of universal transformer Hawkes process. We conduct the experiments on multiple dataset to compare with state-of-the-art baselines, to validate the effectiveness of our model. We also demonstrate whether the additional RNN layers will have a positive impact on fitting mutual interdependence among the events in the sequence. In addition, to demonstrate the effectiveness of the ACT mechanism, we compare the performances of universal transformer with and without the ACT mechanism, and verify that the halting mechanism of dynamic iteration will make the model perform better overall.
Our paper is organized as follows. In Section 2, we introduce the related work about Hawkes process and point process in view of the neural network. In Section 3, we are going to instruct our model in details, including structure, condition intensity function, prediction tasks and training process. At last, Section 4 lists our experimental results to illustrate the advantages of universal transformer Hawkes process and t ACT mechanism. At last, Section 5 concludes the article.
\section{Related work}
\subsection{Hawkes process}
Hawkes process has form shown as the following:
\begin{equation}
\label{eq1}
\lambda (t) = \mu (t) + \sum\limits_{i:t_i < t} {\phi (t - t_i )}
\end{equation}
where $\mu (t)$ is background intensity function, indicates the background probability of event occurrence, $\phi (t)$ is the impact function, which used to measure the historical event influence, and $\sum\limits_{i:t_i < t} {\phi (t - t_i )} $ records the impact of all historical events on the current instant. Traditional Hawkes process model in Eq. 1 assumes the positive superposition of past historical impact. Until now, there are many variants of traditional Hawkes process.
Zhao et al. firstly make use of Hawkes process to model the Twitter data to predict the final number of reshares, which only have small relative error \cite{6zhao2015seismic}. Xu et al. use the non-parametric method to represent the Hawkes process, and utilize a corresponding learning algorithm to get the better performance of the model, and then the Granger causality on of IPTV user on watching program is obtained \cite{8xu2016learning}. Kobayashi and Lamhbiotte present a time-dependent Hawkes process, whose impact functions are time-dependent. They validate the method on Twitter data and prove that there is a systematic improvement to the previous method \cite{24kobayashi2016tideh}. Yang et al. develop an online learning method of Hawkes process based on the nonparametric method \cite{25yang2017online}. In 2018, Alan reviews and summarizes the application of the Hawkes process proposed by him in the financial field \cite{26hawkes2018hawkes}. Mohler presents a novel modulated Hawkes process, to quickly identify risks and trigger appropriate public safety responses in communities to prevent a range of social harm events, such as crime, drug abuse, traffic accident and medical emergencies \cite{27mohler2018improving}.
Zhang et al. \cite{4zhang2020survival} make significant improvement to the Hawkes process model, the Weibull background intensity is used instead of constant background intensity. Weibull background intensity is a kind of time-dependent background intensity, which can better describe the trend of the base possibility of the event over time. After verifying the effectiveness of this Weibull-Hawkes process and the corresponding learning algorithm, based on this new model, Zhang et al. analyze the failure sequence of the compressor station and list the trend of background probability of failures in the compressor station over time and the Granger causality between the failures, and some suggestions are made for the production of the compressor station.
\subsection{Transformer and Universal Transformer}
In 2017, Vaswani et al. propose transformer model \cite{18vaswani2017attention}, which makes full use of self-attention \cite{19DBLP:journals/corr/BahdanauCB14} module, and discards the CNN and RNN structure, achieves a great improvement in the field of sequence learning, such as natural language processing \cite{28chowdhary2020natural}.
However, recent research shows that the recurrent learning in RNN can have a greater role beyond imagination, thus, Dehghani et al. propose universal transformer \cite{22DBLP:conf/iclr/DehghaniGVUK19} which combine the recurrent learning and self-attention mechanism, moreover, in order to better allocate model computing resources, the ACT mechanism \cite{23graves2016adaptive} is introduced into models, then, this model achieves better results than transformers.
\subsection{Neural Hawkes Process}
In general, the neural network has a stronger nonlinear fitting ability than other models, especially RNN is better at learning sequence data than other neural networks, so it has been often used in the processing of sequence data such as speech and text.
Therefore, researchers also hope to use RNN to fit asynchronous event sequence data. Du et al. present RMTPP models \cite{11du2016recurrent}, to learn the history effect via RNN, including history event type and time-stamp. For the first time, this neural model abandons the overly strong assumptions of the Hawkes process and other point process models, and achieves greater improvements. Xiao et al. make use of two RNN to model the event sequence, one of them is used to model the background intensity and the other is used to model the impact of historical events \cite{13xiao2017modeling}. This method allows a black-box treatment for modeling the conditional intensity function, and end to end training can make model event sequences easier. Mei and Eisner propose a new form of LSTM \cite{12mei2017neural}, the continuous time LSTM, whose state can decay to another with time, and based on this LSTM they come up with the neural Hawkes process to model the asynchronous event sequence, which obtains better model performance.
While the shortcomings of RNN gradually appear, researchers tend to use better neural networks to model event sequences. Zhang et al. \cite{20zhang2020self} firstly utilize self-attention mechanism to get the Hawkes process of the sequence. And based on the achievement of transformer, and enlighten by \cite{18vaswani2017attention}, Zuo et al. utilize the encoder structure in transformer, and propose the transformer Hawkes process (THP) \cite{21DBLP:conf/icml/ZuoJLZZ20}, this model encodes and converts event sequence data into hidden representations, and then maps hidden representations into continuous conditional intensity functions.
\section{Proposed Model}
\begin{table}[!htbp]
\caption{Nomenclature}
\label{tb1}
\resizebox{150mm}{93mm}{
\begin{tabular}{cccc}
\hline
Symbols & Description & Variable names in ACT mechanism & Size\\ \hline
$S_e$ &The dataset of sequences & / & / \\
$s_n$ & The $n$-th sequence &/ & / \\
$I_n$ & The length of $n$-th sequence &/ & / \\
$N$ &The total number of sequences &/ & $\mathbb{N}^ +$ \\
$C$ &The total number of type of events in sequences &/ & $\mathbb{N}^ +$ \\
$t_i$ & The time stamp of $i$-th event &/ & $\mathbb{R}$ \\
$c_i$ & The event type of $i$-th event &/ & $\mathbb{N}^ +$ \\
$D$ &The model dimension of universal transformer & /& $\mathbb{N}^ +$ \\
$\bm{K}$ & The embedding matrix of event type & /& $\mathbb{R}^ {D\times C}$ \\
max\_n & The max iteration times of ACT mechanism & /& $\mathbb{N}^ +$ \\
$\mathbb{I}( \cdot )$ &Indicator function &/& / \\
\textit{EncodingLayer} & The encoding layer in ACT mechanism &/ & / \\
$T_h$ & The threshold in ACT mechanism & threshold & $\mathbb{R}$ \\
$\bm{W}_p$ & The weight p in ACT mechanism & weight\_p & $\mathbb{R}^ {D\times 1}$ \\
$S$ & The state variable in ACT mechanism & state & $\mathbb{R}^ {I_n\times D}$ \\
$P_S$ & The previous state variable in ACT mechanism & previous\_state &
$\mathbb{R}^ {I_n \times D}$ \\
$h_p$ & The halting probability variable in ACT mechanism &halting\_probability & $\mathbb{R}^ {D}$ \\
$R_e$ &The remainders variable in ACT mechanism &remainders & $\mathbb{R}^ {D}$ \\
$n$ & The upgrade times of state in ACT mechanism & n\_updatas & $\mathbb{R}^ {D}$ \\
$S_r$ & The still running variable in ACT mechanism &still\_running & $\mathbb{R}^ {D}$ \\
$n_h$ & The new halted variable in ACT mechanism &new\_halted & $\mathbb{R}^ {D}$ \\
$W$ & The update weight of ACT mechanism &update\_weight & $\mathbb{R}^ {D}$ \\
$L$ &he number of multi-head attention & /& $\mathbb{R}$ \\
$\left\{ {{\bm{A}}_l } \right\}_{l = 1}^L $ & The number of multi-head attention &/ & $\mathbb{R}^ {D_V}$ \\
${D_K}$ &The dimension of query and key vector &/ & $\mathbb{N}^ +$ \\
${D_V}$ & The dimension of value vector &/ & $\mathbb{N}^ +$ \\
$\left\{ {{\bm{Q}}_l } \right\}_{l = 1}^L$ &The query variable of multi-head attention &/ & $\mathbb{R}^{I_n \times D_K }$ \\
$\left\{ {{\bm{K}}_l } \right\}_{l = 1}^L$ &The key variable of multi-head attention &/ & $\mathbb{R}^{I_n \times D_K }$ \\
$\left\{ {{\bm{V}}_l } \right\}_{l = 1}^L$ &The value variable of multi-head attention &/ & $\mathbb{R}^{I_n \times D_V }$ \\
$\left\{ {{\bm{W}}_Q^l } \right\}_{l = 1}^L$ &The query matrix of multi-head attention &/ &$\mathbb{R}^{D \times D_K }$ \\
$\left\{ {{\bm{W}}_K^l } \right\}_{l = 1}^L$ &The key matrix of multi-head attention &/ & $\mathbb{R}^{D \times D_K }$ \\
$\left\{ {{\bm{W}}_v^l } \right\}_{l = 1}^L$ &The value matrix of multi-head attention &/ &$\mathbb{R}^{D \times D_K }$ \\
$W_{multi}$ & The aggregation matrix of multi-head attention &/ & $\mathbb{R}^{LD_V \times D }$\\
$\bm{A}'$&Obtained from $\bm{A}$ passes through fully connected layer FC1 &/ & $\mathbb{R}^{I_n \times D }$ \\
$\bm{A}''$& Obtained from $\bm{A}'$ passes through CNN layer and fully connected layer FC2 &/ & $\mathbb{R}^{I_n \times D_H }$ \\
$D_{RNN}$& The dimension of RNN in the model &/ & $\mathbb{N}^ {+}$ \\
$S'$ & Obtained from , the output of ACT mechanism, passes through fully connected layer FC3 &/ & $\mathbb{R}^{I_n \times D_{RNN}}$ \\
$S''$ & Obtained from passes through RNN and fully connected layer FC4 &/ & $\mathbb{R}^{I_n \times D}$ \\
$\bm{H}$ &The hidden representation of sequence &/ & $\mathbb{R}^{I_n \times D}$ \\
$\bm{h}(t_i)$ & The hidden representation of \textit{i}-th event &/ & $\mathbb{R}^{ D}$ \\
$\mathcal{H}_t$ & The previous history at time \textit{t} & / & / \\
$b_c$ & The background intensity of event type \textit{c} &/ & $\mathbb{R}$ \\
${\alpha _c } $ & Thecontinuous parameter of conditional intensity function of event type &/ & $\mathbb{R}^{1\times D}$ \\
${\bm{w}}_c^T $& Historical weight parameter of conditional intensity function of event type &/ & $\mathbb{R}^{1\times D}$ \\
$\bm{W}_{time}$ & The prediction parameter of time-stamp&/ & $\mathbb{R}^{1\times D}$ \\
$\bm{W}_{type}$ &The prediction parameter of event type &/ & $\mathbb{R}^{C\times D}$ \\ \hline
\end{tabular}}
\end{table}
For sequences of asynchronous events, we need to determine what model they have. The symbols used in the paper are shown in Table 1, and in general, we formulate it as following: assume that there are $N$ sequences in the dataset of the asynchronous events, represented as $S_e = \{ s_n \} _{n = 1}^N$ , and for each sequence, note that their lengths are not the same, for the $n$-th sequence $s_n = \{ t_i ,c_i \} _{i = 1}^{I_n } $ , its length is $I_n$ , each sequence $s_n$ is composed with $I_n$ tuples, $t_i$ is the time-stamp of $i$-th event, and $c_i\in C$ is the corresponding event type. In the forward phase, for arbitrary sequence $s_n = \{ t_i ,c_i \} _{i = 1}^{I_n } $ , the entire sequence could be directly input to the universal transformer, supposing the model dimension is $D$ , then all these events in the sequences (including their time-stamp) can be represent by their corresponding $D$-dimensional vectors, then this sequence can be described by the hidden representation of event sequence ${\bm{H}} \in \mathbb{R}^{D \times I_N } $ . The continuous conditional intensity function can be calculated by ${\bm{H}} \in \mathbb{R}^{D \times I_N } $ and the equation we proposed in section 3.2.
\subsection{Universal Transformer Hawkes Process}
First, we need to map the asynchronous event sequence into the temporal encodings, which denote the occurring times of events, and event-type encodings. According to point process theory, the time-stamps of events in the event sequences are the coordinate on the temporal axis, in other words, time-stamps are temporal position of events. Analogous to the text sequence, the time-stamps in the event sequence are equivalent to the position of the words, and the event type embedding vectors are equivalent to the semantic vectors of the words. Thus, for the temporal encodings, we can leverage the position encoding approaches in nature language processing to encode the time-stamps of events, similar as [18,21], the position encoder for the occurring time-stamp of event is shown as Eq.2:
\begin{equation}
\label{eq2}
[{\bm{x}}(t_i )]_j = \left\{ {\begin{array}{*{20}c}
{\cos \left( {t_i /10000^{\frac{{j - 1}}{D}} } \right),{\rm{if}}\:j\:{\rm{is}}\:{\rm{odd}},} \\
{\sin \left( {t_i /10000^{\frac{j}{D}} } \right),{\rm{if}}\:j\:{\rm{is}}\:{\rm{even}}.} \\
\end{array}} \right.
\end{equation}
Thus, for the time-stamp $t_i$ of event, its corresponding temporal encoding is $
{\bm{x}}(t_i ) \in \mathbb{R}^D $ , where $D$ is the model dimension of universal transformer, and in order to get event-type encoding, we set embedding matrix ${\bm{K}} \in \mathbb{R}^{D \times C} $, and each kind of event is corresponding to an one-hot encoding ${\bm{c}}_i \in \mathbb{R}^C $ , then we can get the event-type encoding $
{\bm{Kc}}_i \in \mathbb{R}^D $ . We define that $\bm{X}^T$ and $({\bm{KC}}_n )^T$ are the corresponding temporal encoding and event-type encoding of the sequence, where $
{bm{X}} = \{ {bm{x}}(t_1 ),{bm{x}}(t_2 ),...,{bm{x}}(t_{I_n } )\} \in \mathbb{R}^{D \times I_n } $ and ${bm{C}}_n = [c_1 ,c_2 ,...,c_{I_n } ] \in \mathbb{R}^{C \times I_n } $ .
In order to get the hidden representation of event sequence, we need to input the above temporal and event-type encodings into the recurrent part in universal transformer, which is shown as Fig. 1 and Fig. 2.
\begin{figure}[!htbp]
\centering
\includegraphics[scale=0.7]{1.pdf}
\caption{Schematic diagram of universal transformer with pure recurrence, the model simply uses the hidden state as a recurrent variable }
\label{fig1}
\end{figure}
\begin{figure}[!htbp]
\centering
\includegraphics[scale=0.7]{2.pdf}
\caption{Schematic diagram of universal transformer based on ACT algorithm, in the model, ACT algorithm provides a measure of how well of learning of different elements in the variable.}
\label{fig1}
\end{figure}
Different with traditional transformer, for universal transformer, the temporal and event-type encodings will be alternatively and recursively refined by attention and convolutional layers until a certain stop condition is met, then the final hidden representation will be acquired. In universal transformer with pure recurrence illustrated in Fig. 1, the hidden state will iterate the predetermined number of times, and the final hidden state is set as the hidden representation of event sequence. The corresponding implementation process is shown as the Algorithm 1:
\begin{algorithm}
\renewcommand{\algorithmicrequire}{\textbf{Input:}}
\renewcommand{\algorithmicensure}{\textbf{Output:}}
\caption{Universal Transformer with pure recurrence.}
\label{alg1}
\begin{algorithmic}[2]
\REQUIRE The maximum number of iterations: max\_n, event-type encoding $({\bm{KC}}_n )^T $, temporal encoding (position encoding) $\bm{X}^T$ and \textit{EncodingLayer} in model.
\ENSURE Hidden representation of event sequence ${\bm{H}} \in \mathbb{R}^{D \times I_N } $ (State)
\STATE Initialize state $n\leftarrow0$, $ {\bm{S}} \leftarrow ({\bm{KC}}_n )^T $ .
\STATE \textbf{while} $(n<\rm{max\_n})$ \textbf{do}
\STATE ${\bm{S}} \leftarrow {\bm{S}} + {{ }}{\bm{X}}^T $
\STATE ${\bm{S}} \leftarrow output\;of\;{\rm{the}}\;i{\rm{ - th }}\;{\rm{encoding\;layer}}({\bm{S}})$
\STATE \textbf{end while}
\STATE \textbf{return} ${\bm{H}} \leftarrow {\bm{S}}$
\end{algorithmic}
\end{algorithm}
In iterative process, some symbols, such as words and phonemes, in our application scenario they are the temporal and event-type encodings, are usually more ambiguous than other elements in the sequence, the more ambiguous these symbols are, and the more refining processes are required. However, the above algorithm simply refines equally among each element in the sequence, it is easy to see that this mechanism is too rough and doesn’t meet the requirement we mentioned.
Thus, similar as \cite{22DBLP:conf/iclr/DehghaniGVUK19}, we adopt the ACT mechanism \cite{23graves2016adaptive} in our proposed universal transformer Hawkes process, which is a mechanism can dynamically adjust the number of refinedness for each symbol in the sequence. The ACT mechanism updates the symbols in the sequence according to their state parameter. Since ACT method is used in speech recognition and other scenarios and cannot be directly applied to event sequence data, we are going to make some certain modifications to the concrete implementation procedure of the ACT mechanism. The modified ACT mechanism is illustrated in Algorithm 2, and the information flow diagram of ACT mechanism in Universal transformer is shown in Fig. 3.
\begin{algorithm}
\renewcommand{\algorithmicrequire}{\textbf{Input:}}
\renewcommand{\algorithmicensure}{\textbf{Output:}}
\caption{Universal Transformer based on ACT mechanism.}
\label{alg2}
\begin{algorithmic}[2]
\REQUIRE The maximum number of iterations: max\_n, event-type encoding $({\bm{KC}}_n )^T $, temporal encoding (position encoding) $\bm{X}^T$, threshold $T_h$ and \textit{EncodingLayer} in model.\\ \textbf{Paremeter}:${\bm{W}}_p \in \mathbb{R}^{D \times 1} $
\ENSURE Hidden representation of event sequence ${\bm{H}} \in \mathbb{R}^{D \times I_N } $ (State)
\STATE Initialize state $n \leftarrow {0},R_e \leftarrow 0,h_p \leftarrow 0,P_s \leftarrow 0 $ and $ {\bm{S}} \leftarrow ({\bm{KC}}_n )^T $ .
\STATE \textbf{while} there is any element in $\left( {(h_p > T_h )\& (n < \max \_n)} \right)$ is true, \textbf{do}
\STATE ${\bm{S}} \leftarrow {\bm{S}} + {{ }}{\bm{X}}^T $
\STATE $p \leftarrow Sigmoid (S{\bm{W}}_p )$
\STATE $S_r \leftarrow \mathbb{I}(h_p < 1.0)$
\STATE $N_h \leftarrow (h_p + (p*\mathbb{I}(S_r > T_h ))*S_r$
\STATE$S_r \leftarrow (h_p + (p*\mathbb{I}(S_r \geqslant T_h ))*S_r $
\STATE$ h_p \leftarrow h_p + p*S_r $
\STATE$ R_e \leftarrow R_e + N_h *(1 - h_p ) $
\STATE$ h_p \leftarrow h_p + N_h *R_e $
\STATE$ n \leftarrow n + S_r + N_h $
\STATE$ W \leftarrow p*S_r + N_h *R_e $
\STATE$ S \leftarrow EncodingLayer(S) $
\STATE$ P_S \leftarrow S*W + P_S *(1 - W) $
\STATE \textbf{end while}
\STATE \textbf{return} ${\bm{H}} \leftarrow {{P_S}}$
\end{algorithmic}
\end{algorithm}
\begin{figure}[!htbp]
\centering
\includegraphics[scale=0.7]{3.pdf}
\caption{Diagram of universal transformer based on ACT mechanism, the figure shows the information flow and update process of different variables }
\label{fig3}
\end{figure}
We can see that universal transformer without ACT mechanism and RNN have obvious similarities, in both models, there is only state calculated recursively in the iteration, and universal transformer with ACT mechanism is more similar to LSTM, because there are multiple variables to be updated in the recurrence, and the updated weights are to some extent consistent with the forgetting gate in LSTM. These facts show that universal transformer with ACT mechanism will have a stronger learning ability than universal transformer without ACT mechanism.
\begin{figure}[!htbp]
\centering
\includegraphics[scale=0.7]{4.pdf}
\caption{The diagram of encoding layer in universal transformer, the self-attention mechanism of universal transformer is similar to the normal transformer, and we modify position-wise-feed-forward part, add a convolutional layer between the two fully connected layers to enhance the local perception ability of the model. }
\label{fig4}
\end{figure}
In both algorithms, we all are going to feed the state in Algorithm 1 and 2 into the encoding layer, whose flow-chart is illustrated in Fig. 4, encoding layer is composed with two parts, i.e., multi-head attention part and position-wise-feed-forward part. In encoding layer, we also utilize the commonly used techniques in transformer to relieve the difficulty in training, such as the residual connection \cite{29he2016deep} and layer normalization \cite{30ba2016layer}.
In multi-head attention part, we also adopt dot-product attention, which can be written as Eq.3:
\begin{equation}
{\bm{A}} = Softmax\left( {\frac{{{\bm{QK}}^T }}
{{\sqrt {D_K } }}} \right){\bm{V}}
\end{equation}
where
\begin{equation}
{\bm{Q}} = {\bm{SW}}_Q ,{\bm{K}} = {\bm{SW}}_K ,{\bm{V}} = {\bm{SW}}_V
\end{equation}
state $S$ is the input to the encoding layer, $\bm{Q}$,$\bm{K}$,and $\bm{V}$ are respectively query, key and value matrices, which are obtained by different linear transformations ${\bm{W}}_Q ,{\bm{W}}_K \in \mathbb{R}^{D \times D_K } $
, $ {\bm{W}}_V \in \mathbb{R}^{D \times D_V } $ of state $S$ . And in order to improve the diversity and expressive ability of model, we make use of multi-head self-attention to make the model perform better. If we utilize L heads self-attention, then, we can get $L$ outputs ${\bm{A}}_1 ,{\bm{A}}_2 ,...,{\bm{A}}_L
$ , which are computed by different sets of matrices $\left\{ {{\bm{W}}_Q^l ,{\bm{W}}_K^l ,{\bm{W}}_V^l } \right\}_{l = 1}^L $ . The formula for multi-head attention is shown as Eq.5:
\begin{equation}
{\bm{A}} = \left[ {{\bm{A}}_1 ,{\bm{A}}_2 ,...,{\bm{A}}_L } \right]{\bm{W}}_{multi}
\end{equation}
In Eq.5, ${\bm{W}}_{multi} \in \mathbb{R}^{LD_V \times D}$ is the aggregation matrix. It is worth noting that the events in the sequence occur in sequential order, thus, it has the characteristic called as “future invisibility”, we need to block the impact of future events on current time. We utilize the masked self-attention mechanism similar as \cite{18vaswani2017attention}. For example, we set ${\bm{E}} = {\bm{QK}}^T $ , the masked self-attention mechanism will set the element in the $i$-th row in $\bm{E}$ ranged in ${\bm{E}}(i,i + 1),{\bm{E}}(i,i + 2),..,{\bm{E}}(i,D)$
to negative infinite, which will ensure that the self-attention weight of future events obtained through softmax function is 0.
After getting the attention output ${\bm{A}}$ it will be fed into position-wise-feed-forward part, which is shown in the right part of Fig. 4. To improve model’s local perception of the sequence, we introduce the CNN module in position-wise-feed-forward part, more specifically, fully connected layer FC1 transforms attention output ${\bm{A}}$ into a higher-dimensional linear matrix ${\bm{A'}} \in \mathbb{R}^{I_n \times D_H } $ . Then it pass through the convolutional layer, nonlinear layer, and pooling layer, and fully connected layer FC2 will convert it to the original dimension ${\bm{A''}} \in \mathbb{R}^{I_n \times D}$.
After the Algorithm 1 or 2 completely stop, we will get the output $H$, and at last, will be as the input of postprocessing part, which is constituted with fully connected layer FC3, RNN layer and final fully connected layer FC4. This structure is motivated by \cite{21DBLP:conf/icml/ZuoJLZZ20},\cite{31wang2019language}, and in fact the additional RNN can make model better fit the sequential data, in a nutshell, the FC3 transforms input $S$ or $P_S$ to $S' \in \mathbb{R}^{I_n \times D_{{\rm{RNN}}} } $ and $S'$ will be recurrently iterated by the RNN, note that the types of RNN can be selected as LSTM or GRU, then the FC4 converts $S'$ into $S'' \in \mathbb{R}^{I_n \times D } $ , at last, $S''$ will be assigned to the hidden representation ${\bm{H}} \in \mathbb{R}^{I_n \times D} $ of the sequence of events. The effectiveness of postprocessing part in UTHP is decided by the characteristic of the dataset, and we will discuss this issue in subsection 4.3.2 and analyzed it in experiments.
\subsection{Conditional Intensity Function}
In general, point processes are supposed to be described by their corresponding conditional intensity functions. However, after encoding operation of universal transformer, we can only get the discrete conditional intensity established by hidden representation of event sequence. Therefore, we need to construct the continuous conditional intensity function on the foundation of hidden representation of event sequence.
For every kind of event, we denote $\lambda _c (t\left| {\mathcal{H}_t } \right.)$ as corresponding conditional intensity function of UTHP, and here $\mathcal{H}_t = (t_i ,c_i ):t_i < t$ is the history of past events at time \textit{t}. Similar as \cite{21DBLP:conf/icml/ZuoJLZZ20}, the form of conditional intensity function is shown as Eq.6:
\begin{equation}
\lambda _c (t\left| {\mathcal{H}_t } \right.) = f(b_c + \alpha _c \frac{{t - t_i }}
{{t_i }} + {\bm{w}}_c^T {\bm{h}}(t_i ))
\end{equation}
In Eq.6, $t \in [t_i ,t_{i + 1} )$, and $b_c$ is the background intensity, indicates the possibility of an event occurring without considering historical information. $\alpha _c \frac{{t - t_i }}{{t_i }}$ is continuous conditional intensity function in the interval $[t_i ,t_{i + 1} )$ , where $\alpha _c$ is predefined parameter and can be modified to trainable parameter in subsequent improvements, and ${\bm{w}}_c^T {\bm{h}}(t_i ))$ represents the history impact to the conditional intensity function.
$f( \cdot )$is the softplus function, which is a smooth improvement of the ReLU nonlinear function. Softness parameter of softplus function is $\beta$ , using this nonlinear function makes the conditional intensity function smoother and make the new model have a better expressive ability. And the overall conditional intensity function for the whole sequence is given as follow:
\begin{equation}
\lambda (t\left| {\mathcal{H}_t } \right.){\rm{ = }}\sum\limits_{c = 1}^C {\lambda _c (t\left| {\mathcal{H}_t } \right.)}
\end{equation}
\subsection{Prediction}
In point process theory, prediction is one of the most important tasks. For instance, it can be used to predict when does the patient develop which disease, and when and what events will happen in social networks. After obtaining the conditional intensity function, in view of \cite{32daley2007introduction}, we can predict the possible types and occurrence times of future events according to conditional intensity function and Eq.8:
\begin{equation}
\begin{array}{l}
p(\left. t \right|{\cal H}_t ) = \lambda (\left. t \right|{\cal H}_t )\exp t( - \int_{t_i }^t {\lambda (\left. s \right|{\cal H}_t )} ds) \\
\hat t_{i + 1} = \int_{t_i }^\infty {t\cdot p(\left. t \right|{\cal H}_t )dt} \\
\hat c_{i + 1} = \mathop {\arg \max }\limits_c \frac{{\lambda _c (\hat t_{i + 1} |{\cal H}_{i + 1} )}}{{\lambda (\hat t_{i + 1} |{\cal H}_{i + 1} )}} \\
\end{array}
\end{equation}
The existence of the integral of the conditional intensity function in Eq.8 makes the prediction become difficult, although we can use the Monte Carlo sampling method \cite{33robert2013monte} to calculate it, this is still very inefficient. Due to the powerful fitting ability of neural network, we make use of neural work to predict the future event type and time-stamps, which can be shown as Eq.9:
\begin{equation}
\begin{array}{l}
\hat t_{i + 1} = {\bm{W}}_{time} {\bm{h}}(t_i ) \\
\widehat{\bf{p}}_{i + 1} = Softmax({\bm{W}}_{type} {\bm{h}}(t_i )) \\
\hat c_{i + 1} = \mathop {\arg \max }\limits_c {\bm{\hat p}}_{i + 1} (c) \\
\end{array}
\end{equation}
According to \cite{32daley2007introduction}, given the sequence $s_n = \{ t_i ,c_i \} _{i = 1}^{I_n } $ , and conditional intensity function, we can get its log-likelihood expression:
\begin{equation}
L(s_n ) = \sum\limits_{i = 1}^{I_n } {\log } \,\lambda (\left. {t_i } \right|\mathcal{H}_i ) - \int_{t_1 }^{t_{I_n } } {\lambda (\left. t \right|\mathcal{H}_t )} dt
\end{equation}
Assuming there are $N$ sequences, then the model parameters can be solved by maximum log-likelihood principle:
\[
\max \sum\nolimits_{n = 1}^N {L(s_n )}
\]
However, there is difficulty in solving $\Lambda = \int_{t_1 }^{t_{I_n } } {\lambda \left( {\left. t \right|\mathcal{H}_t } \right)} dt$ because we use universal transformer to obtain the conditional intensity function, it is hard to obtain the closed-form of $\Lambda$ , fortunately, we have two alternative methods to solve it. The first method is the Monte Carlo sampling integration method \cite{33robert2013monte}, which is shown in Eq.11:
\begin{equation}
\label{eq11}
\hat \Lambda _{MC} = \sum\limits_{i = 2}^L {\left( {t_i - t_{i - 1} } \right)} (\frac{1}
{M}\sum\limits_{m = 1}^M {\lambda (u_m )} )
\end{equation}
where $u_m \sim U(t_{i - 1} ,t_i )$ , and $u_m$ is sampled from the uniform distribution in support set $[Ut_{i - 1} ,t_i ]$. $\hat \Lambda _{MC}$ calculated by this method is an unbiased estimate of $\Lambda$. The second method is numerical analysis method \cite{34stoer2013introduction}, for instance, based on the trapezoidal rule, we can get the estimate as shown in Eq.12:
\begin{equation}
\hat \Lambda _{NI} = \sum\limits_{i = 2}^{I_n } {\frac{{t_i - t_{i - 1} }}
{2}} (\lambda (\left. {t_i } \right|\mathcal{H}_i ) + \lambda (\left. {t_{i - 1} } \right|\mathcal{H}_{i - 1} ))
\end{equation}
In spite of the bigger estimate error than the Monte Carlo method, $\hat \Lambda _{NI}$ is still a valid approximation of $\Lambda$ . After dealing with the solution of $\Lambda$ , we are going to define the prediction loss of time-stamp and event type, for a sequence of length $I_n$ , we will make $I_n-1$ predictions of time-stamp and event type, in other words, we will not predict the first event. Therefore, the predict loss of next time-stamp and event prediction for sequence $s_n$ is shown as Eq.13 and 14:
\begin{equation}
L_{time} (s_n ) = \sum\nolimits_{i = 2}^{I_n } {(t_i - \hat t_i )^2 }
\end{equation}
\begin{equation}
L_{type} (s_n ) = \sum\nolimits_{i = 2}^{I_n } { - {\mathbf{c}}} _i^T \log (\widehat{\mathbf{p}}_i )
\end{equation}
where $\bm{c}_i$ is one-hot encoding with the ground-truth event type . In summary, given the dataset including the sequences$\{ s_n \} _{n = 1}^N$ , we need to solve:
\begin{equation}
\min \sum\limits_{n = 1}^N { - L(s_n )} + \alpha _{type} L_{type} (s_n ) + \alpha _{time} L_{time} (s_n )
\end{equation}
where $\alpha _{type}$ and $\alpha _{time}$ are hyper-parameters, which will help keep training stable. This objective function can be efficiently solved by stochastic gradient optimization algorithm, such as adaptive moment estimation (ADAM) \cite{35kingma2014adam}, and we use the default parameters of ADAM: learning rate=0.0001, betas are 0.9 and 0.999, eps = $10^{-8}$, weight\_decay = 0, for all these experiments, when we solves log-likelihood of models, we set $\alpha _{type}=0$ and $\alpha _{time}=0$ , and when performing prediction tasks, we fix $\alpha _{type}=1$ and $\alpha _{time}=0.01$ .
\section{Experiments}
We compare our model with three baselines on six events sequence datasets, we evaluate these models by per-event-loglikelihood (in nats), root mean square error (RMSE) and event prediction accuracy on held-out test sets, we first introduce the details of the data set and baselines, and then list our experimental results.
\subsection{Datasets}
In this subsection, we utilize benchmark six datasets of event sequence to conduct experiments, Table 2 introduces the characteristics of each dataset.
\begin{table}[]
\centering
\caption{Characteristics of datasets. Each row corresponds to the description of this data set, including the total type number of events, the smallest, average, and largest sequence length.}
\begin{tabular}{ccccc}
\hline
\multirow{2}{*}{Dataset} & \multirow{2}{*}{C} & \multicolumn{3}{c}{Sequence Length} \\ \cline{3-5}
& & Min & Aver. & Max \\ \hline
Synthetic & 5 & 20 & 60 & 100 \\
Retweets & 3 & 50 & 109 & 264 \\
MemeTrack & 5000 & 1 & 3 & 31 \\
MIMIC-II & 75 & 2 & 4 & 33 \\
StackOverflow & 22 & 41 & 72 & 736 \\
Financial & 2 & 829 & 2074 & 3319 \\ \hline
\end{tabular}
\end{table}
\textbf{Synthetic} \cite{12mei2017neural}: Mei generates a set of synthetic sequences based on Hawkes process with random sampling the parameters and thinning algorithm.
\textbf{Retweets} \cite{6zhao2015seismic}: This dataset contains lots of sequences of tweets, every sequence corresponds to an origin tweet (for instance, the original content some user post) and its following retweets. The time and label of user of each retweet is recorded in the sequence, and users are divided into three categories based on the number of their followers: “small”, “medium”, and “large”.
\textbf{MemeTrack} \cite{36leskovec2014snap}: This dataset contains 42000 different meme posting in 5000 websites in the duration of 10 months, each sequence represents this meme from its birth to its final disappearance, and each event in the sequence corresponds to a time stamp and a website id.
\textbf{StackOverflow} \cite{36leskovec2014snap}: StackOverflow is a famous programming question-answering website. StackOverflow rewards users with badges to encourage them to participate in community activities, meanwhile, the same badge can be given to the same user, this dataset includes the lots of users reward histories during two years, which are treated as sequences, and each event in sequences indicates the acquisition of badge.
\textbf{Electrical Medical Records} \cite{37johnson2016mimic}: MIMIC-II dataset contains patients’ visit record to a hospital’s ICU in the course of seven years. Each patient’s record is treated as a sequence, and each event contains the corresponding time-stamp and diagnosis.
\textbf{Financial Transactions} \cite{11du2016recurrent}: This financial dataset contains large number of short-term transaction records of a stock in one day. The operation of each transaction is recorded as the event, and this dataset contains several long sequences, which only have two kinds of events: “buy” and “sell”, and the unit of the time-stamp is millisecond.
\subsection{Baselines}
\textbf{RMTPP} \cite{11du2016recurrent}: Du et al. come up with a recurrent network, which can model the event types and time-stamp in the sequence by embedding history to the vector.
\textbf{NHP} \cite{12mei2017neural}: Mei and Eisner present the neural Hawkes process based on the continuous-time LSTM, which has decay property of history events impact.
\textbf{THP} \cite{21DBLP:conf/icml/ZuoJLZZ20}: On the foundation of existing achievement of transformer, Zuo et al. propose the transformer Hawkes process, which achieves state-of-the-art performance. We get the following experimental results based on the model and the hyper-parameter they provide.
For NHP model, we also use ADAM optimizer with same hyper-parameter i.e., we set learning rate=0.0001, betas are 0.9 and 0.999, eps =$10^{-8}$ , weight\_decay = 0, and the number of hidden nodes of the continuous-time LSTM is 64, for Synthetic, Retweets and MemeTrack dataset, we set batch size to 16. The hyper-parameter configurations of THP and UTHP are given in section 4.5.
\subsection{Experimental results and comparison}
In this part, we are going to compare the performance of UTHP and baselines on datasets of multiple event sequences. First, we utilize loglike (per-event-loglikelihood) as the measurement of models’ performance on Synthetic, Retweets and MemeTrack datasets, which is shown in Table 3:
\begin{table}[]
\centering
\caption{The loglike on three dataset, from left to right, each column is a different model, and each row is a different dataset}
\begin{tabular}{ccccc}
\hline
Datasets & RMTPP & NHP & THP & UTHP \\ \hline
Synthetic & \textbackslash{} & -1.33 & \textbf{0.834} & 0.796 \\
Retweets & -5.99 & -5.06 & \textbf{-4.69} & -4.75 \\
MemeTrack & -6.04 & -6.23 &\textbf{ 0.68} & 0.31 \\ \hline
\end{tabular}
\end{table}
From Table 3, we can see that transformer models achieve obvious improvement than previous one, and although our UTHP model does not reach the state-of-the-art level, it still achieves second best. After this, we compare the performance of different models on complex datasets, including StackOverflow, MIMIC-II and Financial datasets. Because of the importance of event and time prediction of point process, while we compare the performance of models on these datasets, the multiple metric criteria will be adopted, involving per-event-log-likelihood, the predict accuracy for event and RMSE of time-stamp prediction. For transformer based model, we utilize Eq. 9 to make predictions, and for RNN based model, Eq. 10 is used. Table 4, Table 5 and Table 6 summarize the experimental results on these datasets.
\begin{table}[]
\centering
\caption{Different models’ performance on StackOverflow dataset, each row is the different metric criteria, including accuracy, RMSE and loglike (per-event-loglikelihood)}
\begin{tabular}{ccccc}
\hline
StackOverflow & RMTPP & NHP & THP & UTHP \\ \hline
Accuracy & 45.9 & 46.3 & 46.8 & \textbf{46.9} \\
RMSE(d) & 9.78 & 9.83 & 4.99 & \textbf{4.42 } \\
Loglike & -2.60 & -2.55 & -0.56 & \textbf{-0.55 }\\ \hline
\end{tabular}
\end{table}
\begin{table}[]
\centering
\caption{Different models’ performance on MIMIC-II dataset.}
\begin{tabular}{ccccc}
\hline
MIMIC-II & RMTPP & NHP & THP & UTHP \\ \hline
Accuracy & 81.2 & 83.2 & 83.2 & \textbf{84.4} \\
RMSE(d) & 6.12 & 6.13 & 0.86 & \textbf{0.85} \\
Loglike & -1.35 & -1.38 & \textbf{-0.14} & -0.18 \\ \hline
\end{tabular}
\end{table}
\begin{table}[]
\centering
\caption{Different models’ performance on Financial dataset.}
\begin{tabular}{ccccc}
\hline
Financial & RMTPP & NHP & THP & UTHP \\ \hline
Accuracy & 61.95 & 62.20 & 62.23 & \textbf{62.52} \\
RMSE(s) & 1.56 & 1.56 & 0.02575 & \textbf{0.02574} \\
Loglike & -3.89 & -3.60 & -1.39 & \textbf{-1.16} \\ \hline
\end{tabular}
\end{table}
From these results, we can see that our proposed UTHP model achieves obvious improvement than the baselines among the different scenarios, different data sets have markedly different characteristics, for instance, the number of classes of MIMIC-II dataset is 75, and the average length is only 4, while the number of classes of Financial dataset is only 2, and the average length is 2074. In all of these datasets, our model improves relatively significantly. This suggests that UTHP can better model complex asynchronous event sequences and learn long-term and short-term event dependencies than existing baseline models.
\begin{figure}[!htbp]
\label{fig3}
\centering
\subfigure[MIMIC-II]{
\includegraphics[width=4.5cm]{mic_acc.pdf}
}
\quad
\subfigure[StackOverflow]{
\includegraphics[width=4.5cm]{so_acc.pdf}
}
\quad
\subfigure[Fianancial]{
\includegraphics[width=4.5cm]{fi_acc.pdf}
}
\caption{Prediction accuracies of RMTPP, NHP, THP and UTHP. Based on the five times train-dev-test partition, five experiments are performed on each dataset, the mean and standard deviations of different models are depicted.}
\end{figure}
\begin{figure}[!htbp]
\centering
\subfigure[MIMIC-II]{
\includegraphics[width=4.5cm]{mic_rsme.pdf}
}
\quad
\subfigure[StackOverflow]{
\includegraphics[width=4.5cm]{so_rmse.pdf}
}
\quad
\subfigure[Fianancial]{
\includegraphics[width=4.5cm]{fi_rmse.pdf}
}
\subfigure[Zoom in of RMSE of THP and UTHP on the datasets]{
\includegraphics[width=7cm]{rmsestd.pdf}
}
\caption{The mean and standard deviation for different models obtained from the five experiments in term of RMSE metric for RMTPP, NHP, THP and UTHP.}
\end{figure}
Fig. 5 and Fig.6 visualizes the accuracies of different models and RMSE of baselines and UHP, due to the dispersion of the data, the error bars are wide. We can find out that the results of UTHP better than the other baselines.
\subsection{Ablation study}
In this subsection, we are going to discuss the impact to the performance of three significance part in models, the postprocessing part, CNN module in the position-wise-feed-forward part and ACT mechanism, we demonstrate experimentally how the presence of these three parts affects the performance of the model.
\subsubsection{Ablation study of the postprocessing part}
As we mentioned in subsection 3.1, the introduction of the additional postprocessing part will have an impact on the transformer-based models, and the impact on the model varies with the dataset. In this subsection, we will demonstrate the impact of the introduction of RNN.
Previous experimental results prove that RNN (including LSTM and GRU) can successfully capture the short-term dependencies in the sequence, but not for long-term dependencies. Thus, we can assume that if we introduce the RNN in the additional postprocessing part to the transformer-based model, it will have better performance to capture the dependencies between the events on the dataset with short-term characteristic, and have relatively poor performance of dependencies fitting on the dataset with long-term characteristic.
To validate whether the facts are consistent with our assumptions, validation experiments are carried out on the dataset with long-term property, i.e., MIMIC-II, and dataset with short-term property, i.e., Financial dataset, and the event prediction accuracies are used to measure how well the model fits the dependencies between events. The experimental results are shown as Table 7 and Fig. 7:
\begin{table}[]
\centering
\caption{Consider that whether there is an additional RNN layer in the postprocessing part, UTHP and THP’s event prediction accuracy rates on MIMIC-II dataset and financial dataset.}
\begin{tabular}{ccccc}
\hline
& \multicolumn{2}{c}{MIMIC-II} & \multicolumn{2}{c}{Financial} \\ \hline
& With RNN & Without RNN & With RNN & Without RNN \\ \hline
\begin{tabular}[c]{@{}c@{}}Accuracy of THP\end{tabular} & 81.44 & \textbf{83.18} &\textbf{ 62.23} & 60.22 \\
\begin{tabular}[c]{@{}c@{}}Accuracy of UTHP\end{tabular} & 83.34 & \textbf{84.43 } & \textbf{62.52 } & 62.04 \\ \hline
\end{tabular}
\end{table}
From the experimental results, we can discover that for transformer-based model, addition RNN layer in the postprocessing part has a negative effect on the experimental results of MIMIC-II dataset, and has a positive impact on the financial dataset. These facts verify that our hypothesis, RNN do enhance the fitting ability to events short-term dependencies, and weaken the fitting ability to the events long-term dependencies.
\begin{figure}[!htbp]
\label{fig3}
\centering
\subfigure[MIMIC-II]{
\includegraphics[width=4.5cm]{rnnmimic.pdf}
}
\quad
\subfigure[Financial]{
\includegraphics[width=4.5cm]{rnnFinancial.pdf}
}
\quad
\caption{Visualization of ablation experimental results with and without an additional RNN layer in the postprocessing part.}
\end{figure}
In addition, UTHP model demonstrates its robustness without RNN on the financial dataset. The reason is that our model already has a recursive structure, which helps it better model short-term data. This also reflects the value of UTHP model we proposed.
\subsubsection{Ablation study of CNN module in the position-wise-feed-forward part}
In general, CNN module will enhance the local perception ability of the model, which is the reason why we add CNN module into the position-wise-feed-forward part. In order to judge whether the introduction of the CNN module is effective, we conduct the comparison experiments, compare the UTHP based on ACT algorithm with CNN and without CNN module in position-wise-feed-forward part on StackOverflow, MIMIC-II and Financial dataset. The experiment results are shown as Table 8 and Fig. 8.
\begin{table}[]
\centering
\caption{Model performance comparisons of UTHP based on ACT algorithm with CNN and without CNN module in position-wise-feed-forward part}
\begin{tabular}{cccc}
\hline
& UTHP & With CNN & Without CNN \\ \hline
\multirow{3}{*}{Accuracy} & StackOverflow & \textbf{46.87} & 46.76 \\
& MIMIC-II & \textbf{84.43} & 83.83 \\
& Financial & 62.52 & \textbf{62.52} \\ \hline
\end{tabular}
\end{table}
\begin{figure}[!htbp]
\centering
\includegraphics[scale=0.3]{cn.pdf}
\caption{Accuracy comparison of UTHP based on ACT algorithm with CNN and without CNN module in position-wise-feed-forward part.}
\label{fig3}
\end{figure}
\subsubsection{Ablation study of ACT mechanism}
From the above experimental results, we can find out that the introduction of CNN module in position-wise-feed-forward part obviously raises the event prediction accuracy. This fact indicates that CNN module enhance the ability to model connections between events of UTHP through its local perception characteristics, which indicates that the introduction of CNN module in position-wise-feed-forward part is effective.
In subsection 3.1, we introduce the reasons for adding the ACT mechanism to the model, ACT mechanism will dynamically regulate the calculation of input in the transformer, make the model use more computing resources to where it is needed more. The concept of this effect is similar to and complementary to the self-attention mechanism, and theoretically makes the model perform better.
And in order to investigate the effect of the ACT mechanism on UTHP, we conducted a series of comparative experiments, we compare pure UTHP under different recursive iteration times (the corresponding algorithm is Algorithm 1) and UTHP with ACT mechanism (the corresponding algorithm is Algorithm 2 and maximum number of iterations max\_n=2). These results are summarized in Table 9:
\begin{table}[]
\centering
\caption{Model performance comparisons of UTHP based on ACT mechanism and UTHP with pure recurrence}
\begin{tabular}{ccccccc}
\hline
& Times of iteration & ACT & 1 & 2 & 3 & 4 \\\hline
\multirow{3}{*}{StackOverflow} & Accuracy & \textbf{46.87} & 46.67 & 46.70 & 46.70 & 46.72 \\
& RMSE & \textbf{4.42} & 4.52 & 4.47 & 4.47 & 4.47 \\
& Loglike & \textbf{-0.55} & -0.60 & -0.58 & -0.58 & -0.57 \\
\multirow{3}{*}{MIMIC-II} & Accuracy & \textbf{84.43} & 82.34 & 82.63 & 83.05 & 81.95 \\
& RMSE & \textbf{0.85} & 0.90 & 0.87 & 0.89 & 0.85 \\
& Loglike & -0.18 & -0.21 & -0.19 & -0.18 & \textbf{-0.17} \\
\multirow{3}{*}{Financial} & Accuracy & \textbf{62.52} & 62.52 & 62.45 & 62.43 & 62.29 \\
& RMSE & 0.02574 & 0.02574 & 0.02572 & \textbf{0.02571} & 0.02572 \\
& Loglike & -1.16 & -1.24 & \textbf{-0.76} & -0.80 & -0.77 \\ \hline
\end{tabular}
\end{table}
For the pure recurrence of UTHP, we considered four different iteration times as the first row in Table 9, and when the number of iterations is once, it indicates that hidden representation of sequences will directly output the model, rather than recurrently iterate in the model, from the results we can find out that UTHP with ACT mechanism performs than pure recurrence UTHP overall. Especially in terms of the accuracies of event prediction, UTHP with ACT mechanism are all better than UTHP with pure recurrence.
For a more intuitive comparison, we visualize these experimental results and error bars, as shown in Fig. 9, Fig. 10, and Fig. 11.
\begin{figure}[!htbp]
\label{fig3}
\centering
\subfigure[Accuracy]{
\includegraphics[width=4.5cm]{soacc.pdf}
}
\quad
\subfigure[RMSE]{
\includegraphics[width=4.5cm]{sormse.pdf}
}
\quad
\subfigure[Loglike]{
\includegraphics[width=4.5cm]{solog.pdf}
}
\caption{Model performance curves of UTHP with ACT mechanism and pure UTHP on StackOverflow dataset.}
\end{figure}
\begin{figure}[!htbp]
\label{fig3}
\centering
\subfigure[Accuracy]{
\includegraphics[width=4.5cm]{miacc.pdf}
}
\quad
\subfigure[RMSE]{
\includegraphics[width=4.5cm]{mirmse.pdf}
}
\quad
\subfigure[Loglike]{
\includegraphics[width=4.5cm]{milog.pdf}
}
\caption{Model performance curves of UTHP with ACT mechanism and pure UTHP on MIMIC-II dataset.}
\end{figure}
\begin{figure}[!htbp]
\label{fig3}
\centering
\subfigure[Accuracy]{
\includegraphics[width=4.5cm]{bookacc.pdf}
}
\quad
\subfigure[RMSE]{
\includegraphics[width=4.5cm]{bookrmse.pdf}
}
\quad
\subfigure[Loglike]{
\includegraphics[width=4.5cm]{booklog.pdf}
}
\caption{Model performance curves of UTHP with ACT mechanism and pure UTHP on Financial dataset.}
\end{figure}
From these figures, we can see that compared to the error bar, UTHP with ACT mechanism has a clear gap with pure UTHP when it performs well, such as the accuracy, RMSE and Loglike curves on StackOverflow dataset, and the accuracy curves on MIMIC-II dataset. And comparing with the standard deviation, when pure UTHP results are better than UTHP with mechanism, the difference is not so obvious. This further verifies that the introduction of the ACT mechanism will improve the overall performance of UTHP.
In addition, with the increasing of iteration times, the Loglike on these datasets of pure UTHP are also increasing. The fact represents that the increment of iteration items enhance the modeling ability of UTHP.
\subsection{Fewer model parameters of UTHP}
Because of the recurrence structure of UTHP, there is only one encoding layer in our model, while THP utilize the stacking of multiple encoding layer, in other words, UTHP model share the parameters in the encoding layer. By this way, we find that our proposed architectures improve model performance, and also reduce model parameters.
For instance, in subsection 4.3, on each dataset, UTHP and THP adopt the similar hyper-parameters, which is shown as Table 10 and Table 11, the only difference is that UTHP has only one encoding layer, and there is a CNN module in the position-wise-feed-forward part of encoding layer.
\begin{table}[]
\centering
\caption{Hyperparameter configurations of THP.}
\begin{tabular}{ccccccc}\hline
Dataset & Synthetic & Retweets & MemeTrack & Financial & MIMIC-II & \begin{tabular}[c]{@{}c@{}}StackOverflow\end{tabular} \\ \hline
$D$ & 64 & 64 & 64 & 128 & 64 & 512 \\
$D_H$ & 256 & 256 & 256 & 2048 & 256 & 1024 \\
$D_{\rm{RNN}}$ & 128 & 128 & 128 & 128 & 0 & 128 \\
$D_Q=D_V$ & 16 & 16 & 16 & 64 & 16 & 512 \\
Heads of attention & 3 & 3 & 3 & 6 & 3 & 4 \\
Layers of transformer & 2 & 2 & 2 & 2 & 2 & 2 \\
Dropout & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 \\ \hline
\end{tabular}
\end{table}
\begin{table}[]
\centering
\caption{Hyperparameter configurations of UTHP.}
\begin{tabular}{ccccccc}\hline
Dataset & Synthetic & Retweets & MemeTrack & Financial & MIMIC-II & \begin{tabular}[c]{@{}c@{}}StackOverflow\end{tabular} \\\hline
$D$ & 64 & 64 & 64 & 128 & 64 & 512 \\
$D_H$ & 256 & 256 & 256 & 2048 & 256 & 1024 \\
$D_{\rm{RNN}}$ & 128 & 128 & 128 & 128 & 0 & 128 \\
$D_Q=D_V$ & 16 & 16 & 16 & 64 & 16 & 512 \\
Heads of attention & 3 & 3 & 3 & 6 & 3 & 4 \\
Max\_n & 2 & 2 & 2 & 2 & 2 & 2 \\
Dropout & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 \\
Size of convolution kernel & 3 & 3 & 3 & 3 & 3 & 3 \\
Stride of convolution kernel & 2 & 2 & 2 & 2 & 2 & 2 \\
Padding & 0 & 0 & 0 & 0 & 0 & 0 \\
Stride of the pooling module & 2 & 2 & 2 & 2 & 2 & 2 \\
Size of the pooling module & 2 & 2 & 2 & 2 & 2 & 2\\ \hline
\end{tabular}
\end{table}
\begin{table}[]
\centering
\caption{The numbers of parameters of the THP and UTHP on different datasets.}
\begin{tabular}{ccc}\hline
Number of parameters & THP & UTHP \\\hline
Synthetic & 245765 & 129494 \\
Retweet & 137795 & 46356 \\
MemeTrack & 1102216 & 1010777 \\
MIMIC-II & 151691 & 60252 \\
StackOverflow & 21022742 & 5281319 \\
Financial & 4343298 & 724627 \\\hline
\end{tabular}
\end{table}
As we can see from the Table 12, although we introduce a CNN module in position-wise-feed-forward part which increases the additional parameters in UTHP, it still has less parameter than THP. The parameter number of the THP’s parameter is approximately 6 to 1.1 times of the UTHP’s one. The fewer number of event types in the dataset, the fewer parameters for UTHP compared to THP. Combined with the experimental results in subsection 4.3, our model uses fewer parameters to achieve better performance, which reflects the improvement of our model.
\section{Conclusions and future works}
In this paper, we come up with UTHP, a new neural point process model to analyze the asynchronous event sequence. UTHP combines the self-attention mechanism in transformer and the recurrence mechanism in RNN, this operation allows our model to organically integrate the advantages of RNN and transformer, moreover, in order to make UTHP to adaptively determine when to stop refining processes of hidden variables, we introduce the ACT mechanism to UTHP. Experimental results verify that our model performs better than the baselines overall with fewer model parameters, and we also discuss the impact of the existence of the additional RNN layer and the ACT mechanism in the ablation study.
\section{Acknowledgment}
Thanks for Hong-yuan Mei and Si-miao Zuo for their generous help in my research state, their help greatly improved our research.
|
2024-02-18T23:40:24.714Z
|
2021-12-30T02:24:12.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14479","language":"en","timestamp":1640831052000,"url":"https:\/\/arxiv.org\/abs\/2112.14479","yymm":"2112"}
|
proofpile-arXiv_000-10071
|
{"provenance":"002.jsonl.gz:10072"}
| null | null |
\section{Introduction}
In the context of $\mathrm{CAT}(0)$ spaces, we show the strong convergence of an iterative schema which alternates between Halpern and Krasnoselskii-Mann style iterations, while also obtaining quantitative information in the form of rates of asymptotic regularity and rates of metastability.
Let $H$ be a Hilbert space and $C$ a nonempty closed convex subset. A mapping $U:C \to C$ is said to be \emph{nonexpansive} on $C$ if for all $x, y \in C$ one has $\norm{U (x) -U (y)}\leq \norm{x -y}$. A well-known method for finding fixed points of a nonexpansive mapping $U:C \to C$ is
\begin{equation}\label{e:KM}\tag{KM}
x_{n+1} = (1 - \beta_n)U(x_n) + \beta_nx_n,
\end{equation}
with $x_0 \in C$ a starting point and $(\beta_n) \subset [0,1]$. This iteration was introduced by Mann \cite{Mann(53)} and Krasnoselskii \cite{Kraso(55)}. Even though this iteration has the propitious property of being Fejér monotone with respect to the fixed point set, in general it only converges weakly to a fixed point. This motivated several modifications of \eqref{e:KM} in order to ensure strong convergence. One such version was introduced by Bo\c{t}, Csetnek, and Meier~\cite{Botetal(19)}, where strong convergence is guaranteed by introducing Tikhonov regularization terms $(\gamma_n)\subset [0,1]$
\begin{equation}\label{e:TKM}\tag{T-KM}
x_{n+1} = (1 - \beta_n)U(\gamma_nx_n) + \beta_n(\gamma_nx_n).
\end{equation}
In \cite[Theorem~3]{Botetal(19)} Bo\c{t} \emph{et al.} showed that $(x_n)$ generated by \eqref{e:TKM} converges strongly to the fixed point of $U$ with minimum norm $P_{\mathrm{Fix}\,(U)}(0)$, provided that the following conditions hold
\begin{enumerate}[$(i)$]
\item $(\gamma_n) \subset (0,1]$, \, $\lim \gamma_n =1$,\, $\sum_{n \geq 0} (1-\gamma_n)=\infty$,\, $\sum_{n \geq 0}|\gamma_{n+1}-\gamma_{n}|< \infty$,
\item $(\beta_n)\subset [0,1)$, \, $\sum_{n \geq 0}|\beta_{n+1}-\beta_{n}|< \infty$, and $\limsup \beta_n <1$.
\end{enumerate}
From this result, Bo\c{t} \emph{et al.} introduced strongly convergent versions of the forward-backward and the Douglas-Rachford algorithms \cite[Theorems~7 and 10]{Botetal(19)}, which are well-known splitting methods that weakly approximate zeros of
a sum of monotone operators.
A different iteration was introduced by Halpern in \cite{Halpern67}
\begin{equation}\label{e:Halpern}\tag{H}
x_{n+1}=(1 - \alpha_n)T(x_n) + \alpha_nu,
\end{equation}
where $T:C \to C$ is a nonexpansive map, $x_0,u \in C$ and $(\alpha_n) \subset [0,1]$. Halpern showed the strong convergence of \eqref{e:Halpern} to the
metric projection onto the set $\mathrm{Fix}\,(T)$ of the anchor point $u$, in Hilbert spaces. The conditions considered by Halpern prevented the natural choice $\alpha_n =\frac{1}{n+1}$, which was later overcome by Wittmann \cite{Wittmann(92)}. In \cite{X(02),Xu(04)}, Xu extended the strong convergence to the setting of uniformly smooth Banach spaces under conditions which are incomparable to those considered by Wittmann but which still
allowed $\alpha_n =\frac{1}{n+1}$.
As observed by Cheval and Leu\c{s}tean in \cite{CK(ta)}, the iteration \eqref{e:TKM} can be seen as alternating between a Mann style construction and a special case of the Halpern schema. Indeed, the iteration considered by Cheval and Leu\c{s}tean can be written in the following way
\begin{equation*}
\begin{cases}
x_{2n+1}&=(1-\alpha_n) x_{2n}+\alpha_nu\\
x_{2n+2}&=(1-\beta_n)U(x_{2n+1})+\beta_nx_{2n+1},
\end{cases}
\end{equation*}
which corresponds to \eqref{e:TKM} when $u=0$. They considered this iteration in the general setting of hyperbolic spaces and obtained rates of asymptotic regularity. Yet no strong convergence results were established.
Let $X$ be a $\mathrm{CAT}(0)$ space and $C$ a nonempty convex closed subset. In this paper we will consider an iteration $(x_n)$ generated by the recursive schema
\begin{equation}\label{e:MannHalpern}\tag{HM}
\begin{cases}
x_{2n+1}&=(1-\alpha_n)T(x_{2n})\oplus \alpha_n u\\
x_{2n+2}&=(1-\beta_n)U(x_{2n+1})\oplus\beta_nx_{2n+1},
\end{cases}
\end{equation}
where $x_0, u\in C$, $(\alpha_n), (\beta_n)\subset [0,1]$ are sequences of real numbers and $T,U:C \to C$ are nonexpansive mappings. The symbol ``$\oplus$'' denotes the application of the convex operator $W$ (see Section~\ref{s:preliminaries} for details), which is the usual linear convex combination in the case of normed spaces. We show the following result:
\begin{theorem}\label{t:main}
Let $X$ be a complete $\mathrm{CAT}(0)$ space and $C \subseteq X$ a nonempty convex closed subset. Consider nonexpansive mappings $T,U:C \to C$ such that $F:=\mathrm{Fix}\,(T) \cap \mathrm{Fix}\,(U) \neq \emptyset$ and $u,x_0 \in C$. Assume that $(\alpha_n) \subset [0,1]$, $(\beta_n) \subset (0,1)$ are sequences of real numbers satisfying
%
\begin{align*}
&(i)\ \lim \alpha_n =0; \qquad (ii)\ \sum_{n \geq 0} \alpha_n= \infty; \qquad (iii)\ \sum_{n \geq 0} |\alpha_{n+1} - \alpha_{n}|< \infty;\\
&(iv)\ \sum_{n \geq 0} |\beta_{n+1} - \beta_{n}|< \infty; \qquad (v)\ 0< \liminf \beta_n \leq \limsup \beta_n <1.
\end{align*}
%
Then $(x_n)$ generated by \eqref{e:MannHalpern} converges strongly to $\textup{P}_F(u)$.
\end{theorem}
Our result is a twofold generalization of \cite[Theorem~3]{Botetal(19)}. On the one hand, our iteration has \eqref{e:TKM} as a particular case. On the other hand, our strong convergence result is established in $\mathrm{CAT}(0)$ spaces -- frequently considered the non-linear generalization of Hilbert spaces. Our iteration also extends the one in \cite{CK(ta)} -- which is the particular case of \eqref{e:MannHalpern} when $T= \mathrm{Id}_C$. Even though the context of \cite{CK(ta)} is more general than that of $\mathrm{CAT}(0)$ spaces, Theorem~\ref{t:main} proves strong convergence while \cite[Theorems~4.1 and 4.2]{CK(ta)} only establish asymptotic regularity for the iteration. This greater generality is achieved at the cost of additionally assuming the mild condition $0< \liminf \beta_n$. Similarly to the strongly convergent versions of the forward-backward and the Douglas-Rachford algorithms introduced in \cite{Botetal(19)}, our generalized iteration is also used to define extended versions of these algorithms.
As an initial step into proving our main theorem we first establish quantitative results in the spirit of the proof mining program \cite{K(08),K(18)}. In this framework one looks to obtain quantitative information, such as rates of convergence and rates of metastability, guided by proof-theoretical techniques. For example, while in general it is not possible to obtain computable information on the value of $n$ in the Cauchy property (see e.g.\ \cite{N(15)}),
\begin{equation*}
\forall \varepsilon >0\,\exists n \in \mathbb{N}\, \forall i,j \geq n \left(d(x_i,x_j)\leq \varepsilon \right),
\end{equation*}
in many cases, logical results guarantee quantitative information for the equivalent finitary version
\begin{equation}\label{metastab}\tag{$\dagger$}
\forall \varepsilon >0 \,\forall f : \mathbb{N} \to \mathbb{N} \,\exists n \, \forall i,j \in [n,f(n)] \left(d(x_i,x_j)\leq \varepsilon \right),
\end{equation}
where $[n,f(n)]$ denotes the set $\{n,n+1, \cdots, f(n)\}$.
The property \eqref{metastab} was popularized by Tao \cite{T(08b),T(08a)} under the name \emph{metastability}. Quantitative information for \eqref{metastab} takes the form of a \emph{rate of metastability}, i.e.\ a computable functional $\mu: (0, +\infty) \times \mathbb{N}^\mathbb{N} \to \mathbb{N}$ satisfying
\begin{equation*}\label{metastabb}
\forall \varepsilon >0\,\forall f : \mathbb{N} \to \mathbb{N} \,\exists n \leq \mu(\varepsilon,f) \, \forall i,j \in [n,f(n)] \left(d(x_i,x_j)\leq \varepsilon \right).
\end{equation*}
Note that a rate of metastability for \eqref{metastab} does not entail computable information for the equivalent Cauchy property. Even though these two notions are equivalent, this equivalence is non-effective as the proof is by contradiction.
Still, metastability results have a far reaching scope as can be seen in several recent results, see for example \cite{K(05)i,AGT(10),GT(08),KL(09),T(08b)}. Another striking feature of proof mining is the possibility to identify precisely the required conditions needed to prove a result. This frequently allows to single out unused hypothesis and thus to obtain generalized results.
In this sense, Theorem~\ref{t:main} follows from a generalization to a non-linear setting of a corresponding quantitative result in Hilbert spaces. In Hilbert spaces the result can be established via a sequential weak compactness argument (see Section~\ref{s:final}). It is not clear how to carry out such an argument in $\mathrm{CAT}(0)$ spaces. Nevertheless, the technique developed in \cite{FFLLPP(19)} shows that, through a quantitative treatment, it is possible to bypass this argument in Hilbert spaces (as was done e.g.\ in \cite{DP(ta),DP(21)}). Extending such quantitative results to the setting of $\mathrm{CAT}(0)$ spaces allows to conclude the main theorem. These quantitative results may also be seen as a natural continuation of previous quantitative analyses (see e.g.\ \cite{DP(ta),KL(12),DP(MAR)}).
Even though our results and proofs are inspired by proof theoretical techniques, these are only used as an intermediate step and are not visible in the final product. As such we do not presuppose any particular knowledge of logical tools.
The structure of the paper is the following. Some relevant terminology and useful lemmas are recalled in Section~\ref{s:preliminaries}. The central results are obtained in Sections~\ref{s:asymptoticregularity} and \ref{s:convergence} where we show that a sequence $(x_n)$ generated by \eqref{e:MannHalpern} is asymptotically regular (Section~\ref{s:asymptoticregularity}) and has the metastability property (Section~\ref{s:convergence}), while also obtaining the corresponding quantitative information. In Section~\ref{s:Projection} we show a quantitative version of the metric projection in $\mathrm{CAT}(0)$ spaces which is used to establish the metastability of the sequence. We show how Theorem~\ref{t:main} follows from the metastability property in Section~\ref{s:Strongconvergence}. Afterwards, in Section~\ref{s:FBDR}, we study generalized versions of the forward-backward and the Douglas-Rachford algorithms (now in the context of Hilbert spaces). Some final remarks are left for Section~\ref{s:final}.
\section{Preliminaries}\label{s:preliminaries}
\subsection{Hyperbolic and $\mathrm{CAT}(0)$ spaces}
Consider a triple $(X,d,W)$ where $(X,d)$ is a metric space and $W:X \times X \times [0,1] \to X$ is a function satisfying, for all $x,y,z,w\in X$ and $\lambda, \lambda' \in [0,1]$,
\begin{enumerate}
\item[(W1)] $d(z, W(x, y,\lambda))\leq (1-\lambda)d(z, x) + \lambda d(z, y)$
\item[(W2)] $d(W(x, y, \lambda), W(x, y, \lambda'))=|\lambda-\lambda'|d(x, y)$
\item[(W3)] $W(x,y,\lambda)=W(y,x, 1-\lambda)$
\item[(W4)] $d(W(x,y,\lambda), W(z,w,\lambda))\leq (1-\lambda)d(x,z)+\lambda d(y,w)$.
\end{enumerate}
A triple in the conditions above is called a \emph{hyperbolic space}. This formulation was introduced by Kohlenbach in \cite{K(05)} under the name of $W$-hyperbolic spaces. The convexity function $W$ was first considered by Takahashi in \cite{T(70)}, where a triple $(X,d,W)$ satisfying (W1) was called a convex metric space. The notion of hyperbolic space considered here is more general than the hyperbolic spaces in the sense of Reich and Shafrir \cite{RS(90)}, and slightly more restrictive than the notion of space of hyperbolic type by Goebel and Kirk \cite{GK(83)}.
If $x,y\in X$ and $\lambda \in[0,1]$, we use the notation $(1-\lambda)x \oplus \lambda y$ to denote $W(x,y,\lambda)$. It is easy to see using (W1) that
\begin{equation}\label{e:equality}
d(x, (1-\lambda)x \oplus \lambda y)=\lambda d(x,y) \,\mbox{ and }\, d(y, (1-\lambda)x \oplus \lambda y)=(1-\lambda)d(x,y).
\end{equation}
The metric segment with endpoinds $x,y \in X$ is the set $\{(1-\lambda)x\oplus \lambda y:\lambda \in [0,1]\}$ and is denoted by $[x,y]$. A nonempty subset $C \subseteq X$ is said to be \emph{convex} if $\forall x,y \in C ([x,y] \subseteq C)$.
The class of hyperbolic spaces includes the normed spaces and their convex subsets (with $(1-\lambda)x\oplus \lambda y=(1-\lambda)x + \lambda y$, the usual convex linear combination), the Hilbert ball \cite{GR(84)} and the $\mathrm{CAT}(0)$ spaces.
The important class of $\mathrm{CAT}(0)$ spaces (introduced by Alexandrov in \cite{A(51)}, and named as such by Gromov in \cite{G(87)}; see \cite{BH(13)} for a detailed treatment) is characterized as the hyperbolic spaces that satisfy the property $\mathrm{CN}^-$ (which in the presence of the other axioms is equivalent to the Bruhat-Tits $\mathrm{CN}$-inequality \cite{BT(72)}, but contrary to the latter is purely universal):
\begin{equation}\tag{CN$^-$}
\forall x,y,z \in X\, \left( d^2\left(z, \frac{1}{2}x \oplus \frac{1}{2}y\right) \leq \frac{1}{2}d^2(z, x)+\frac{1}{2}d^2(z, y)-\frac{1}{4}d^2(x, y) \right)
\end{equation}
This inequality actually extends beyond midpoints -- see e.g. \cite[Lemma 2.5]{DP(08)}: for all $x,y,z\in X$ and $\lambda \in[0,1]$
\begin{equation}\tag{CN$^+$}\label{CN}
d^2\left(z, (1-\lambda)x \oplus \lambda y\right) \leq (1-\lambda)d^2(z, x)+\lambda d^2(z, y)-\lambda(1-\lambda)d^2(x, y).
\end{equation}
In the same way as hyperbolic spaces are considered the non-linear counterpart of normed spaces, $\mathrm{CAT}(0)$ spaces are the non-linear generalizations of Hilbert spaces.
As shown by Leu\c{s}tean \cite{L(07)}, $\mathrm{CAT}(0)$ spaces are uniformly convex with a quadratic modulus of uniform convexity:
\begin{lemma}[{\cite[Proposition 8]{L(07)}}]\label{l:CATconvexity}
Every $\mathrm{CAT}(0)$ space is a uniformly convex space and $\eta(\varepsilon)=\frac{\varepsilon^2}{8}$ is a modulus of uniform convexity, i.e. for all $\varepsilon\in(0,2]$, $r >0$, and $x,y,a\in X$
\begin{equation*}
\begin{rcases}
d(x, a)&\leq r\\
d(y,a)&\leq r\\
d(x,y) &\geq \varepsilon r\\
\end{rcases} \rightarrow d\left(\frac{1}{2}x \oplus \frac{1}{2}y, a\right)\leq (1-\eta(\varepsilon))r.
\end{equation*}
\end{lemma}
In \cite[Proposition~14]{BN(08)} it was shown that in every metric space there exists a unique function $\langle \cdot, \cdot \rangle: X^2 \times X^2 \to \mathbb{R}$
satisfying
\begin{enumerate}[$(i)$]
\item $\langle \overrightarrow{xy},\overrightarrow{xy}\rangle=d^2(x,y)$
\item $\langle \overrightarrow{xy},\overrightarrow{uv}\rangle=\langle \overrightarrow{uv},\overrightarrow{xy}\rangle$
\item $\langle \overrightarrow{yx},\overrightarrow{uv}\rangle=-\langle \overrightarrow{xy},\overrightarrow{uv}\rangle$
\item $\langle \overrightarrow{xy},\overrightarrow{uv}\rangle+ \langle \overrightarrow{xy},\overrightarrow{vw}\rangle= \langle \overrightarrow{xy},\overrightarrow{uw}\rangle$,
\end{enumerate}
where $\overrightarrow{xy}$ denotes the pair $(x,y)$.
This unique function, called the \emph{quasi-linearization function}, is defined for any $(x,y),(u,v)\in X^2$ by
\begin{equation*}
\langle \overrightarrow{xy},\overrightarrow{uv}\rangle:= \frac{1}{2}\left(d^2(x,v)+d^2(y,u)-d^2(x,u)-d^2(y,v) \right).
\end{equation*}
In $\mathrm{CAT}(0)$ spaces the quasi-linearization function satisfies the Cauchy-Schwarz inequality,
\begin{equation}\label{e:CS}
\langle \overrightarrow{xy},\overrightarrow{uv}\rangle \leq d(x,y)d(u,v).
\end{equation}
We also have the following useful inequality.
\begin{lemma}\label{l:binomial}
Let $X$ be a $\mathrm{CAT}(0)$ space. For every $x,y,z \in X$ and $t \in [0,1]$
\begin{equation*}
d^2((1-t)x\oplus t y,z)\leq (1-t)^2d^2(x,z)+2t(1-t)\langle \overrightarrow{xz},\overrightarrow{yz}\rangle+t^2 d^2(y,z).
\end{equation*}
\end{lemma}
\begin{proof}
Using \eqref{CN} we derive
\SE{d^2((1-t)x\oplus t y,z)&\leq (1-t)d^2(x,z)+td^2(y,z)-t(1-t)d^2(x,y)\\
&= (1-t)^2d^2(x,z)+t(1-t)\left(d^2(x,z)+d^2(y,z)-d^2(x,y) \right)+t^2d^2(y,z)\\
&=(1-t)^2d^2(x,z)+2t(1-t)\langle \overrightarrow{xz},\overrightarrow{yz}\rangle+t^2 d^2(y,z).\qedhere
}
\end{proof}
\subsection{Quantitative notions}
\begin{definition}\label{d:RN}
Let $(a_n)$ be a sequence of real numbers.
\begin{enumerate}[$(i)$]
\item A \emph{rate of convergence} for
$\lim a_n = 0$ is a function $\gamma:(0,+\infty)\to\mathbb{N}$ such that
\[\forall \varepsilon >0\, \forall n\geq \gamma(\varepsilon)\,\left(|a_n|\leq \varepsilon\right).\]
\item A \emph{rate of divergence} for $\lim a_n =+\infty$ is a function $\gamma:\mathbb{N}\to\mathbb{N}$ such that
\[\forall k\in\mathbb{N}\, \forall n\geq \gamma(k) \left(a_n \geq k\right).\]
\item A \emph{Cauchy rate} for $(a_n)$ is a function $\gamma:(0,+\infty)\to\mathbb{N}$ such that
\[\forall \varepsilon >0\, \forall i,j \geq \gamma(\varepsilon) \left(|a_{i}-a_{j}| \leq \varepsilon \right).\]
\end{enumerate}
\end{definition}
\begin{definition}\label{d:HS}
Let $(x_n)$ be a sequence in a metric space $(X,d)$ and $x \in X$.
\begin{enumerate}[$(i)$]
\item A \emph{rate of convergence} for $\lim x_n = x$ is a rate of convergence for $\lim d(x_n,x)= 0$.
\item A \emph{Cauchy rate} for $(x_n)$ is a function $\gamma:(0,+\infty)\to\mathbb{N}$ such that
\[\forall \varepsilon>0 \, \forall i,j \geq \gamma(\varepsilon)\left(d(x_i,x_j)\leq \varepsilon\right).\]
\item A \emph{rate of metastability} for $(x_n)$ is a functional $\Gamma:(0,+\infty) \times \mathbb{N}^{\mathbb{N}}\to\mathbb{N}$ such that
\begin{equation*}
\forall \varepsilon>0 \,\forall f : \mathbb{N} \to \mathbb{N} \,\exists n \leq \Gamma(\varepsilon,f) \, \forall i,j \in [n,f(n)] \left(d(x_i,x_j)\leq \varepsilon \right).
\end{equation*}
\end{enumerate}
\end{definition}
We say that a function $f:\mathbb{N}\to\mathbb{N}$ is \emph{monotone} if
\[
k\leq k' \to f(k)\leq f(k').
\]
Without loss of generality, we can always assume to have this property, since if needed we can replace the function $f$ with the function $f^{\max}:\mathbb{N}\to\mathbb{N}$ defined by $f^{\max}(k):=\max\{f(k')\, : \, k'\leq k\}$ for all $k\in\mathbb{N}$. We say that a function $g: (0, +\infty)\to \mathbb{N}$ is \emph{monotone} if
\[
0\leq \varepsilon \leq \varepsilon' \to g(\varepsilon) \geq g(\varepsilon').
\]
This notion follows from a proof mining treatment where one works with a function $g':\mathbb{N}\to \mathbb{N}$ satisfying
\[
g(\varepsilon)=g'(k)\, \text{ with } k=\lceil \varepsilon^{-1}\rceil.
\]
\begin{lemma}\label{l:metaCauchy}
Let $(X,d)$ be a metric space and $(x_n)$ be a sequence in $X$. Then $(x_n)$ has the metastability property, i.e.\
\begin{equation*}
\forall \varepsilon>0 \,\forall f : \mathbb{N} \to \mathbb{N} \,\exists n \in \mathbb{N} \, \forall i,j \in [n,f(n)] \left(d(x_i,x_j)\leq \varepsilon \right)
\end{equation*}
if and only if $(x_n)$ is a Cauchy sequence.
\end{lemma}
\begin{proof}
If $(x_n)$ is a Cauchy sequence then it clearly has the metastability property. Assume that $(x_n)$ is not a Cauchy sequence. Then, there exists $\varepsilon_0 >0$ such that for every $n \in \mathbb{N}$
\begin{equation*}
\exists i_n > j_n \geq n \left(d(x_{i_n},x_{j_n})>\varepsilon_0 \right).
\end{equation*}
Hence, the metastability property fails for $\varepsilon_0$ and the function $f$ defined by $f(n):=i_n$, for all $n \in \mathbb{N}$.
\end{proof}
Note that the proof that metastability implies the Cauchy property is non-effective. As a consequence, the existence of a computable rate of metastability does not entail a computable Cauchy rate. On the other hand, a function $\gamma$ is a Cauchy rate if and only if $\Gamma(\varepsilon,f):=\gamma(\varepsilon)$ is a rate of metastability (see e.g.\ \cite[Proposition~2.6]{KP(22)}).
\begin{definition}
Let $(x_n)$ be a sequence in a metric space $(X,d)$ and consider a mapping $T: X \to X$.
\begin{enumerate}[$(i)$]
\item The sequence $(x_n)$ is \emph{asymptotically regular} if $\lim d(x_{n+1},x_n)= 0$. A \emph{rate of asymptotic regularity} for $(x_n)$ is a rate of convergence for $\lim d(x_{n+1},x_n)= 0$.
\item The sequence $(x_n)$ is \emph{asymptotically regular} with respect to $T$ if $\lim d(T(x_{n}),x_n)= 0$. A \emph{rate of asymptotic regularity} for $(x_n)$ with respect to $T$ is a rate of convergence for $\lim d(T(x_{n}),x_n)= 0$.
\end{enumerate}
\end{definition}
\subsection{Useful lemmas}
We recall the following well-known result due to Xu.
\begin{lemma}[\cite{X(02)}]\label{L:Xu}
Let $(a_n) \subset (0,1)$ and $(r_n),(v_n)$ be real sequences such that
\begin{equation*}
(i) \sum a_n = \infty; \qquad (ii) \limsup r_n \leq 0; \qquad (iii)\sum v_n<\infty.
\end{equation*}
Let $(s_n)$ be a non-negative real sequence satisfying $s_{n+1}\leq (1-a_n)s_n+ a_nr_n+v_n$, for all $n \in \mathbb{N}$. Then $\lim s_n = 0$.
\end{lemma}
Xu's lemma has received several quantitative analysis. The next two results are (essentially) from \cite{LLPP(21)}.
\begin{notation}\hfill
\begin{enumerate}
\item Throughout this paper $\lceil x\rceil$ is defined as $\max\{0, \lceil x\rceil\}$
with the usual definition of $\lceil\cdot\rceil$ in the latter.
\item Consider a function $\varphi$ on tuples of variables $\bar{x}$, $\bar{y}$. If we wish to consider the variables $\bar{x}$ as parameters we write $\varphi[\bar{x}](\bar{y})$. For simplicity of notation we may then even omit the parameters and simply write $\varphi(\bar{y})$.
\end{enumerate}
\end{notation}
\begin{lemma}\label{L:xu_seq_reals_qt1}
Let $(s_n)$ be a bounded sequence of non-negative real numbers and $D\in\mathbb{N}\setminus\{0\}$ an upper bound on $(s_n)$. Consider sequences of real numbers $(a_n)\subset\,[0,1]$, $(r_n)\subset \mathbb{R}$ and $(v_n)\subset \mathbb{R}^+_0$ and monotone functions ${\rm A}:\mathbb{N} \to \mathbb{N}$ and ${\rm R}$, ${\rm V}: (0,+\infty)\to \mathbb{N}$ such that
\begin{enumerate}[$(i)$]
\item ${\rm A}$ is a rate of divergence for $\left(\sum a_n\right)$,
\item ${\rm R}$ is such that $\forall \varepsilon >0\, \forall n\geq {\rm R}(\varepsilon) \, \left( r_n \leq \varepsilon \right)$,
\item ${\rm V}$ is a Cauchy rate for $\left(\sum v_n\right)$.
\end{enumerate}
Assume that for all $n\in \mathbb{N}$, $s_{n+1}\leq (1-a_n)s_n+a_nr_n + v_n$. Then $\lim s_n =0$ with rate of convergence
\[\theta(\varepsilon):=\theta[{\rm A}, {\rm R}, {\rm V}, D](\varepsilon):={\rm A}\left(K+\left\lceil \ln\left(\frac{3D}{\varepsilon}\right)\right\rceil\right)+1,\, \text{ with }\, K:=\max\left\{ {\rm R}\left(\frac{\varepsilon}{3}\right), {\rm V}\left(\frac{\varepsilon}{3}\right)+1 \right\}.
\]
Moreover,
\begin{enumerate}[$(1)$]
\item If $r_n\equiv 0$, then the function $\theta$ is simplified to
\[
\theta(\varepsilon):=\widehat{\theta}[A,V,D](\varepsilon):={\rm A}\left({\rm V}\left(\frac{\varepsilon}{2}\right)+\left\lceil \ln\left(\frac{2D}{\varepsilon}\right)\right\rceil+1\right)+1.
\]
\item If $v_n\equiv 0$, then the function $\theta$ is simplified to
\[
\theta(\varepsilon):=\widecheck{\theta}[A,R,D](\varepsilon):={\rm A}\left({\rm R}\left(\frac{\varepsilon}{2}\right)+\left\lceil \ln\left(\frac{2D}{\varepsilon}\right)\right\rceil\right)+1.
\]
\end{enumerate}
\end{lemma}
\iffalse
In Lemma~\ref{xu_seq_reals_1}, the fact that $(s_n)$ is bounded follows trivially from the other assumptions. This translates into the easy fact that it is possible to compute a bound $D$ from the remaining data. Namely, a possible value for $D$ is $\lceil \max\{s_0, {\cal R}\} + {\cal G} \rceil$, where ${\cal R}:=\max_{n\leq {\rm R}(0)}\{1, r_n\}$ and ${\cal G}:=1+\sum_{i=0}^{{\rm G}(0)}\gamma_i$ are bounds on the sequences $(r_n)$ and $(\sum \gamma_i)$, respectively.
\fi
Instead of considering $\sum a_n = \infty$, one can work with the following equivalent condition
\begin{equation}\tag{$\star$}\label{c2}
\forall m\in \mathbb{N}\, \left(\prod_{i= m}^{\infty} (1-a_i)=0\right).
\end{equation}
Hence, it makes sense to also consider a quantitative hypothesis corresponding to \eqref{c2}:
\begin{equation}\tag{$Q{\star}$}\label{q2'}
\begin{gathered}
{\rm A'}:\mathbb{N}\times (0,+\infty) \to\mathbb{N} \text{ is a monotone function satisfying}\\
\forall \varepsilon >0 \, \forall m\in \mathbb{N}\, \left( \prod_{i=m}^{{\rm A}'(m,\varepsilon)}(1-a_i)\leq \varepsilon\right),
\end{gathered}
\end{equation}
i.e.\ ${\rm A}'(m, \cdot)$ is a rate of convergence towards zero for the sequence $\left(\prod_{i=m}^{n}(1-a_i)\right)_n$. By saying that ${\rm A}'$ is monotone we mean the following
\[
\forall \varepsilon_1, \varepsilon_2 >0 \, \forall m_1, m_2 \in \mathbb{N}\, \left( \varepsilon_1 \leq \varepsilon_2 \land m_1\leq m_2 \to {\rm A'}(m_1, \varepsilon_2)\leq {\rm A'}(m_2, \varepsilon_1)\right).
\]
Next we state a quantitative version of Lemma~\ref{L:Xu} which relies on the condition \eqref{q2'} (see also \cite[Lemma 2.4]{K(15)} and \cite{LLPP(21)}).
\begin{lemma}\label{L:xu_seq_reals_qt2}
Let $(s_n)$ be a bounded sequence of non-negative real numbers and $D\in\mathbb{N}\setminus\{0\}$ an upper bound on $(s_n)$. Consider sequences of real numbers $(a_n)\subset\, [0,1]$, $(r_n)\subset \mathbb{R}$ and $(v_n)\subset \mathbb{R}^+_0$, and monotone functions ${\rm A'}: \mathbb{N}\times (0, +\infty) \to \mathbb{N}$ and ${\rm R}$, ${\rm V}:(0,+\infty) \to \mathbb{N}$ such that
\begin{enumerate}[$(i)$]
\item ${\rm A'}$ satisfies \eqref{q2'},
\item ${\rm R}$ is such that $\forall \varepsilon>0 \, \forall n\geq {\rm R}(\varepsilon) \, \left( r_n \leq \varepsilon \right)$,
\item ${\rm V}$ is a Cauchy rate for $(\sum v_n)$.
\end{enumerate}
Assume that for all $n\in \mathbb{N}$, $s_{n+1}\leq (1-a_n)s_n+a_nr_n + v_n$. Then $ \lim s_n=0$ with rate of convergence
\[
\theta'[{\rm A'}, {\rm R}, {\rm V}, D](\varepsilon):={\rm A'}\left(K, \frac{\varepsilon}{3D}\right)+1,\, \text{ with }\, K\, \text{ as in Lemma~\ref{L:xu_seq_reals_qt1}}.
\]
Moreover,
\begin{enumerate}[$(1)$]
\item If $r_n\equiv 0$, then the function $\theta'$ is simplified to
\[
\theta'(\varepsilon):=\widehat{\theta}'[A,V,D](\varepsilon):={\rm A'}\left({\rm V}\left(\frac{\varepsilon}{2}\right)+1, \frac{\varepsilon}{2D}\right)+1.
\]
\item If $v_n\equiv 0$, then the function $\theta'$ is simplified to
\[
\theta'(\varepsilon):=\widecheck{\theta}'[A,R,D](\varepsilon):={\rm A'}\left({\rm R}\left(\frac{\varepsilon}{2}\right), \frac{\varepsilon}{3D}\right)+1.
\]
\end{enumerate}
\end{lemma}
We will also require particular instances of Lemma~\ref{L:Xu} that allow for a error term. The next two results are trivial variants of \cite[Lemmas 5.2 and 5.3]{KL(12)}. For completeness, we include the proofs nevertheless.
\begin{lemma}\label{L:xu_seq_reals_qt3}
Let $(s_n)$ be a bounded sequence of non-negative real numbers and $D\in\mathbb{N}\setminus\{0\}$ an upper bound on $(s_n)$. Consider sequences of real numbers $(a_n)\subset\, [0,1]$, $(r_n)\subset \mathbb{R}$ and assume that $\sum a_n=\infty$ with a monotone rate of divergence ${\rm A}$. Let $\varepsilon>0$, $K, P\in \mathbb{N}$ be given. If for all $n\in [K,P]$
\begin{equation*}
(i)\quad s_{n+1}\leq (1-a_n)s_n+a_nr_n+\mathcal{E}\,,\qquad\qquad (ii)\quad r_n\leq \frac{\varepsilon}{3}\,,\qquad\qquad (iii)\quad \mathcal{E}\leq \frac{\varepsilon}{3(P+1)},
\end{equation*}
then $\forall n \in [\sigma, P] \left( s_n\leq \varepsilon\right)$, where
\begin{equation*}
\sigma:=\sigma[{\rm A},D](\varepsilon,K):={\rm A}\left(K+\left\lceil\ln\left( \frac{3D}{\varepsilon}\right)\right\rceil\right)+1.
\end{equation*}
\end{lemma}
\begin{proof}
First note that if $\sigma>P$, then the result is trivial. Hence we may assume that $\sigma\leq P$, which in particular implies that $K\leq P$. Next, we may assume that $\varepsilon <D$, otherwise the result also trivially holds. Now, by induction one easily sees that for all $m\leq P-K$,
\begin{equation*}
s_{K+m+1}\leq \left( \prod_{i=K}^{K+m} (1-a_i) \right)s_K+\left( 1-\prod_{i=K}^{K+m}(1-a_i) \right)\frac{\varepsilon}{3}+(m+1)\mathcal{E}.
\end{equation*}
Hence, for all $m\leq P-K$,
\begin{equation}\label{e:Xuquant}
s_{K+m+1}\leq \left( \prod_{i=K}^{K+m} (1-a_i) \right)s_K+\frac{\varepsilon}{3}+(P+1)\mathcal{E}\leq \left( \prod_{i=K}^{K+m} (1-a_i) \right)D+\frac{2\varepsilon}{3}.
\end{equation}
Since $j+1\leq \sum_{i=0}^{A(j+1)}a_i\leq A(j+1)+1$, we have $A(j+1)\geq j$ for all $j\in\mathbb{N}$. By the assumption on $\varepsilon$, we have $3D/\varepsilon\geq 3$ and so $\lceil\ln(3D/\varepsilon)\rceil\geq 1$. Hence, by the fact that ${\rm A}$ is monotone, we have ${\rm A}(K+\lceil\ln(3D/\varepsilon)\rceil)\geq K$ and can consider the natural number $M:={\rm A}(K+\lceil\ln(3D/\varepsilon)\rceil)-K$.
For all $m\geq M$, we have
\begin{equation*}
\sum_{i=0}^{K+m}a_i\geq \sum_{i=0}^{K+M}a_i=\sum_{i=0}^{{\rm A}(K+\lceil\ln(3D/\varepsilon)\rceil)}a_i\geq K+\ln\left(\frac{3D}{\varepsilon}\right)\geq \sum_{i=0}^{K-1}a_i+ \ln\left(\frac{3D}{\varepsilon}\right).
\end{equation*}
which entails $\sum_{i=K}^{K+m}a_i\geq \ln(3D/\varepsilon)$. Using the inequality $\forall x\in\mathbb{R}^+_0 \left(1-x\leq \exp(-x)\right)$, we derive for all $m\geq M$,
\begin{equation*}
\left(\prod_{i=K}^{K+m}(1-a_i)\right)D\leq \exp\left(-\sum_{i=K}^{K+m}a_i\right)D\leq \frac{\varepsilon}{3D}D=\frac{\varepsilon}{3}.
\end{equation*}
Together with \eqref{e:Xuquant}, we have thus concluded that for $m\in[M, P-K]$,
\begin{equation*}
s_{K+m+1}\leq \left( \prod_{i=K}^{K+m} (1-a_i) \right)D+\frac{2\varepsilon}{3}\leq \varepsilon,
\end{equation*}
which entails the result.
\end{proof}
As before, instead of a rate of divergence ${\rm A}$, we can stablish an analogous of the previous result with a function ${\rm A'}$ satisfying \eqref{q2'}.
\begin{lemma}\label{L:xu_seq_reals_qt4}
Let $(s_n)$ be a bounded sequence of non-negative real numbers and $D\in\mathbb{N}\setminus\{0\}$ an upper bound on $(s_n)$. Consider sequences of real numbers $(a_n)\subset\, [0,1]$, $(r_n)\subset \mathbb{R}$ assume that $\sum a_n=\infty$ with a monotone function ${\rm A'}$ satisfying \eqref{q2'}. Let $\varepsilon>0$, $K, P\in \mathbb{N}$ be given. If for all $n\in [K,P]$
\begin{equation*}
(i)\quad s_{n+1}\leq (1-a_n)s_n+a_nr_n+\mathcal{E}\,,\qquad\qquad (ii)\quad r_n\leq \frac{\varepsilon}{3}\,,\qquad\qquad (iii)\quad \mathcal{E}\leq \frac{\varepsilon}{3(P+1)},
\end{equation*}
then $\forall n \in [\sigma', P] \left( s_n\leq \varepsilon\right)$, where
\begin{equation*}
\sigma':=\sigma'[{\rm A'},D](\varepsilon,K):={\rm A'}\left(K, \frac{\varepsilon}{3D}\right)+1.
\end{equation*}
\end{lemma}
\begin{proof}
Following the proof of Lemma~\ref{L:xu_seq_reals_qt3}, we have for all $m\leq P-K$,
\begin{equation*}
s_{K+m+1}\leq \left( \prod_{i=K}^{K+m} (1-a_i) \right)D+\frac{2\varepsilon}{3}.
\end{equation*}
Again from the assumption on $\varepsilon$, we may consider the natural number $M:={\rm A'}\left(K, \varepsilon/3D\right)-K$. For all $m\geq M$,
\begin{equation*}
\left(\prod_{i=K}^{K+m}(1-a_i)\right)D\leq \left(\prod_{i=K}^{K+M}(1-a_i)\right)D=\left(\prod_{i=K}^{{\rm A'}\left(K, \varepsilon/3D\right)}(1-a_i)\right)D\leq \frac{\varepsilon}{3},
\end{equation*}
which as before entails the result.
\end{proof}
\section{Asymptotic regularity}\label{s:asymptoticregularity}
In this section we prove that a sequence $(x_n)$ generated by \eqref{e:MannHalpern} is an asymptotically regular sequence of approximate common fixed points of $T$ and $U$.
We will assume that the conditions $(i)-(v)$ of Theorem~\ref{t:main} hold and that their corresponding quantitative information is given explicitly. Namely, given $(\alpha_n),(\beta_n)\subseteq [0,1]$, we consider the following conditions:
\begin{enumerate}[($Q_1$)]
\item\label{Q1} $\Gamma_1:(0,+\infty)\to \mathbb{N}$ is a rate of convergence for $\lim \alpha_n=0$, i.e.
\[
\forall \varepsilon >0 \, \forall n \geq \Gamma_1(\varepsilon) \left(\alpha_n \leq \varepsilon \right).
\]
\item\label{Q2} $\Gamma_2:\mathbb{N}\to\mathbb{N}$ is a rate of divergence for $\left( \sum_{n\geq 0} \alpha_n\right)$, i.e.
\[
\forall k \in \mathbb{N} \left(\sum_{i=0}^{\Gamma_2(k)}\alpha_i \geq k \right).
\]
\item\label{Q3} $\Gamma_3:(0,+\infty)\to \mathbb{N}$ is a Cauchy rate for $\left(\sum_{n\geq 0}|\alpha_{n+1}-\alpha_{n}|\right)$, i.e.
\[\forall \varepsilon >0 \, \forall n \in \mathbb{N} \left(\sum_{i=\Gamma_3(\varepsilon)+1}^{\Gamma_3(\varepsilon)+n}|\alpha_{i+1}-\alpha_{i}|\leq \varepsilon \right).\]
\item\label{Q4} $\Gamma_4:(0,+\infty)\to \mathbb{N}$ is a Cauchy rate for $\left(\sum_{n\geq 0}|\beta_{n+1}-\beta_{n}|\right)$
\item\label{Q5} $\gamma\in(0,1/2]$ is such that $\gamma \leq \beta_n \leq 1-\gamma$, for all $n \in \mathbb{N}$.
\end{enumerate}
We will assume furthermore that the functions above satisfy the natural monotonicity conditions:
\begin{enumerate}
\item if $\varepsilon \leq \varepsilon' $ then $\Gamma_i(\varepsilon)\geq \Gamma_i(\varepsilon')$, for $i=1,3,4$
\item if $k \leq k' $ then $\Gamma_2(k)\leq \Gamma_2(k')$.
\end{enumerate}
\begin{example}\label{ex:1}
Consider the sequences defined by $\alpha_n:=\frac{1}{n+1}$, $\beta_n:=\beta \in (0,1)$ for all $n \in \mathbb{N}$. Then the conditions $(Q_{\ref{Q1}})-(Q_{\ref{Q5}})$ are satisfied with
\[
\Gamma_1(\varepsilon)=\Gamma_3(\varepsilon):= \lfloor \varepsilon^{-1}\rfloor, \quad \Gamma_2(k):=\lfloor e^{k}\rfloor, \quad \Gamma_4 :\equiv 0, \quad \text{ and } \quad \gamma:=\min \{ \beta, 1- \beta\}.
\]
\end{example}
We start by showing that $(x_n)$ is bounded.
\begin{lemma}\label{l:bounded}
Consider a natural number $N \in \mathbb{N} \setminus \{0\}$ such that $N \geq \max\{d(x_0,p),d(u,p)\}$, for some $p \in F$.
We have that \begin{equation}\label{e:bounded}
d(x_{n},p) \leq N
\end{equation}
and, in particular, the sequence $(x_n)$ is bounded.
\end{lemma}
\begin{proof}
%
Since $U$ is nonexpansive and $U(p)=p$, using (W1) we have
%
\begin{equation}\label{e:evenindices}
d(x_{2n+2},p)\leq (1-\alpha_n)d(U(x_{2n+1}),p)+\alpha_nd(x_{2n+1},p)= d(x_{2n+1},p).
\end{equation}
%
By induction, we have that $\forall n \in \mathbb{N} \left(d(x_{2n},p)\leq N \right)$.
%
The base case is trivial. If the induction thesis holds for $n$, using the fact that $T$ is nonexpansive, $T(p)=p$, and again (W1), we get
%
\begin{equation}\label{e:oddindices}
d(x_{2n+1},p)\leq (1-\alpha_n)d(x_{2n},p)+\alpha_n d(u,p) \leq (1-\alpha_n)N+\alpha_n N=N.
\end{equation}
%
By \eqref{e:evenindices}, the induction step follows. Moreover, from \eqref{e:oddindices} it is clear that the bound is also valid for the odd terms, and we conclude the result.
\end{proof}
The next lemma proves that the sequence $(x_n)$ is an asymptotically regular sequence of almost fixed points for both $T$ and $U$.
\begin{lemma}\label{l:artheta}
Consider a natural number $N \in \mathbb{N} \setminus \{0\}$ such that $N \geq \max\{d(x_0,p),d(u,p)\}$, for some $p \in F$.
If conditions $(Q_{\ref{Q1}})-(Q_{\ref{Q5}})$ hold, then
\begin{enumerate}
\item[$(i)$] $\theta_1$ is a rate of convergence for $\lim d(x_{2n+2},x_{2n})=0$,
\item[$(ii)$] $\theta_2$ is a rate of convergence for $\lim d(x_{2n+3},x_{2n+1})=0$,
\item[$(iii)$] $\theta_3$ is a rate of convergence for both $\lim d(U(x_{2n+1}),x_{2n+1})=0$ and for $\lim d(x_{2n+2}, x_{2n+1})=0$,
\item[$(iv)$] $\theta_4$ is a rate of convergence for $\lim d(x_{2n+1},x_{2n})=0$.
\end{enumerate}
where
\begin{equation*}
\begin{aligned}
\theta_1(\varepsilon)&:=\widehat{\theta}[A,V,D], \, \text{ and } \widehat{\theta} \text{ is as in Lemma~\ref{L:xu_seq_reals_qt1}(1), with parameters }\\
&\bullet \, A(k):=\Gamma_2(k+1), \, \text{ for all }\, k\in \mathbb{N}\\
&\bullet\, V(\varepsilon):=\max\left\{\Gamma_3\left(\frac{\varepsilon}{4N}\right),\Gamma_4\left(\frac{\varepsilon}{4N}\right)\right\},\, \text{ for all } \, \varepsilon>0\\
&\bullet \, D:=2N - \text{which by \eqref{e:bounded} is a bound on the sequence $(d(x_{2n+2}, x_{2n}))$}\, \\
\theta_2(\varepsilon)&:=\max\left\{\theta_1\left(\frac{\varepsilon}{2}\right), \Gamma_3\left(\frac{\varepsilon}{4N}\right)+1\right\}\\
\theta_3(\varepsilon)&:=\max\left\{ \theta_1\left(\frac{(\gamma\varepsilon)^2}{8N}\right), \Gamma_1\left(\frac{(\gamma\varepsilon)^2}{2N^2}\right) \right\}\\
\theta_4(\varepsilon)&:=\max\left\{ \theta_1\left(\frac{\varepsilon}{2}\right), \theta_3\left(\frac{\varepsilon}{2}\right) \right\}.
\end{aligned}
\end{equation*}
\end{lemma}
\begin{proof}
Using (W2) and (W4) we have
%
\SE{d(x_{2n+3},x_{2n+1})& \leq d(x_{2n+3},(1-\alpha_{n+1})T(x_{2n})\oplus\alpha_{n+1}u) +d((1-\alpha_{n+1})T(x_{2n})\oplus\alpha_{n+1}u,x_{2n+1})\\
& \leq (1-\alpha_{n+1})d(T(x_{2n+2}),T(x_{2n}))+|\alpha_{n+1}-\alpha_{n}|d(T(x_{2n}),u)\\
& \leq(1-\alpha_{n+1})d(x_{2n+2},x_{2n})+|\alpha_{n+1}-\alpha_{n}|d(T(x_{2n}),u).
}
%
Hence, using \eqref{e:bounded}, for all $n \in \mathbb{N}$
%
\begin{equation}\label{e:difp}
d(x_{2n+3},x_{2n+1})\leq (1-\alpha_{n+1}) d(x_{2n+2},x_{2n})+2N|\alpha_{n+1}-\alpha_{n}|.
\end{equation}
%
Similarly,
%
\SE{d(x_{2n+4},x_{2n+2}) & \leq d(x_{2n+2},(1-\beta_{n+1})U(x_{2n+1}) \oplus \beta_{n+1}x_{2n+1})\\
& \quad +d((1-\beta_{n+1})U(x_{2n+1}) \oplus \beta_{n+1}x_{2n+1},x_{2n+2})\\
& \leq (1-\beta_{n+1})d(U(x_{2n+3}),U(x_{2n+1}))+\beta_{n+1}d(x_{2n+3},x_{2n+1})\\
& \quad +|\beta_{n+1}-\beta_{n}|d(U(x_{2n+1}),x_{2n+1})\\
& \leq d(x_{2n+3},x_{2n+1})+|\beta_{n+1}-\beta_{n}|d(U(x_{2n+1}),x_{2n+1}).
}
%
Hence, using \eqref{e:bounded}, for all $n \in \mathbb{N}$
%
\begin{equation}\label{e:difi}
d(x_{2n+4},x_{2n+2})\leq d(x_{2n+3},x_{2n+1})+2N|\beta_{n+1}-\beta_{n}|.
\end{equation}
%
From \eqref{e:difp} and \eqref{e:difi} we conclude that for all $n \in \mathbb{N}$
\SE{d(x_{2n+4},x_{2n+2})& \leq (1-\alpha_{n+1}) d(x_{2n+2},x_{2n})+2N\left(|\alpha_{n+1}-\alpha_{n}|+|\beta_{n+1}-\beta_{n}|\right).}
By $(Q_{\ref{Q2}}),(Q_{\ref{Q3}})$ and $(Q_{\ref{Q4}})$, we may apply Lemma~\ref{L:xu_seq_reals_qt1} with $s_n=d(x_{2n+2},x_{2n})$, $a_n=\alpha_{n+1}$, $r_n\equiv 0$, and $v_n=2N\left(|\alpha_{n+1}-\alpha_{n}|+|\beta_{n+1}-\beta_{n}|\right)$ to conclude that
%
\begin{equation}\label{e:difpares}
\lim d(x_{2n+2},x_{2n})=0, \text{ with rate of convergence $\theta_1$}.
\end{equation}
%
Indeed, for all $k\in \mathbb{N}$,
%
\begin{equation*}
\sum_{i=0}^{A(k)}a_i=\sum_{i=0}^{\Gamma_2(k+1)}\alpha_{i+1}\geq \sum_{i=0}^{\Gamma_2(k+1)}\alpha_{i}-\alpha_0 \geq k
\end{equation*}
%
and, for all $\varepsilon >0$ and $n \in\mathbb{N}$,
\SE{
\sum_{i=V(\varepsilon)+1}^{V(\varepsilon)+n}v_i&=2N\left(\sum_{i=V(\varepsilon)+1}^{V(\varepsilon)+n} |\alpha_{i+1}-\alpha_{i}|+\sum_{i=V(\varepsilon)+1}^{V(\varepsilon)+n} |\beta_{i+1}-\beta_i|\right)\\[2mm]
&=2N\left(\sum_{i=\Gamma_3(\frac{\varepsilon}{4N})+j+1}^{\Gamma_3(\frac{\varepsilon}{4N})+j+n} |\alpha_{i+1}-\alpha_{i}|+\sum_{i=\Gamma_4(\frac{\varepsilon}{4N})+j'+1}^{\Gamma_4(\frac{\varepsilon}{4N})+j'+n} |\beta_{i+1}-\beta_i|\right)\\[2mm]
&\leq 2N\left(\sum_{i=\Gamma_3(\frac{\varepsilon}{4N})+1}^{\Gamma_3(\frac{\varepsilon}{4N})+j+n} |\alpha_{i+1}-\alpha_{i}|+\sum_{i=\Gamma_4(\frac{\varepsilon}{4N})+1}^{\Gamma_4(\frac{\varepsilon}{4N})+j'+n} |\beta_{i+1}-\beta_i|\right)\\
&\leq 2N \left(\frac{\varepsilon}{4N}+\frac{\varepsilon}{4N}\right)= \varepsilon.
}
%
From $(Q_{\ref{Q3}})$, for any $\varepsilon >0$ and $n\geq \Gamma_3(\varepsilon/4N)+1$, we have $|\alpha_{n+1}-\alpha_n|\leq \varepsilon/4N$. Then, by \eqref{e:difp},
%
\begin{equation}\label{e:difimpares}
\lim d(x_{2n+3},x_{2n+1})=0, \, \text{ with rate of convergence }\, \theta_2.
\end{equation}
%
With $p\in F$ as in the hypothesis over $N$, using \eqref{CN} and the fact that $U$ is nonexpansive, we have
%
\SE{d^2(x_{2n+2},p)& \leq (1-\beta_{n})d^2(U(x_{2n+1}),p)+\beta_{n}d^2(x_{2n+1},p)-\beta_n(1-\beta_n)d^2(U(x_{2n+1}),x_{2n+1})\\
& \leq d^2(x_{2n+1},p)-\beta_n(1-\beta_n)d^2(U(x_{2n+1}),x_{2n+1})
}
%
and similarly, now using the fact that $T$ is nonexpansive,
%
\SE{d^2(x_{2n+1},p)& \leq (1-\alpha_{n})d^2(T(x_{2n}),p)+\alpha_{n}d^2(u,p)-\alpha_{n}(1-\alpha_{n})d^2(T(x_{2n}),u)\\
& \leq d^2(x_{2n},p)+\alpha_{n}d^2(u,p).
}
Then
%
\SE{d^2(x_{2n+2},p)&\leq d^2(x_{2n},p)+\alpha_{n}d^2(u,p)-\beta_{n}(1-\beta_{n})d^2(U(x_{2n+1}),x_{2n+1}),
}
%
which implies
%
\SE{\beta_{n}(1-\beta_{n})d^2(U(x_{2n+1}),x_{2n+1})&\leq d^2(x_{2n},p)-d^2(x_{2n+2},p)+\alpha_{n}d^2(u,p)\\
& \leq d(x_{2n+2},x_{2n})(d(x_{2n+2},x_{2n})+2d(x_{2n+2},p))+\alpha_{n}d^2(u,p).
}
%
Now, condition $(Q_{\ref{Q5}})$ entails
\begin{equation}\label{e:ineqforUimpar}
\left(\gamma \cdot d(U(x_{2n+1}),x_{2n+1})\right)^2\leq d(x_{2n+2},x_{2n})(d(x_{2n+2},x_{2n})+2d(x_{2n+2},p))+\beta_{n}d^2(u,p).
\end{equation}
For $\varepsilon >0$, using \eqref{e:bounded} and \eqref{e:difpares}, we conclude that for $n\geq \theta_1((\gamma\varepsilon)^2/8N)$,
%
\begin{equation*}
d(x_{2n+2},x_{2n})(d(x_{2n+2},x_{2n})+2d(x_{2n+2},p))\leq d(x_{2n+2},x_{2n})(2N+2N)\leq \frac{(\gamma\varepsilon)^2}{2}.
\end{equation*}
%
Moreover, from condition $(Q_{\ref{Q1}})$, we get for $n\geq \Gamma_1((\gamma\varepsilon)^2/2N^2)$
%
\begin{equation*}
\alpha_nd^2(u,p)\leq \alpha_nN^2\leq \frac{(\gamma\varepsilon)^2}{2}.
\end{equation*}
%
Hence, from \eqref{e:ineqforUimpar} we conclude that
%
\begin{equation}\label{e:Uimpar}
\lim d(U(x_{2n+1}),x_{2n+1})=0, \, \text{ with rate of convergence }\, \theta_3.
\end{equation}
%
Since $d(x_{2n+2},x_{2n+1})= (1-\beta_n)d(U(x_{2n+1}),x_{2n+1})$ by \eqref{e:equality}, from \eqref{e:Uimpar} we also get that $\lim d(x_{2n+2},x_{2n+1})=0$ with rate of convergence $\theta_3$. On the other hand, since $d(x_{2n+1},x_{2n})\leq d(x_{2n+2},x_{2n+1})+d(x_{2n+2},x_{2n})$ it follows that $\lim d(x_{2n+1},x_{2n})=0$ with rate of convergence $\theta_4$.
\end{proof}
\begin{example}
With $(\alpha_n), (\beta_n)$ as in Example~\ref{ex:1}, it is possible to verify that we have the following rates of convergence
\begin{enumerate}
\item[] $\varepsilon \mapsto \left\lfloor \exp\left(\frac{12N}{\varepsilon}+2\right)\right\rfloor +1$ for $\lim d(x_{2n+2}, x_{2n})=0$,
\item[] $\varepsilon \mapsto \left\lfloor \exp\left(\frac{24N}{\varepsilon}+2\right)\right\rfloor +1$ for $\lim d(x_{2n+3}, x_{2n+1})=0$,
\item[] $\varepsilon \mapsto \left\lfloor \exp\left(\left(\frac{14N}{\gamma\varepsilon}\right)^2+2\right)\right\rfloor +1$ for both $\lim d(U(x_{2n+1}), x_{2n+1})=0 \text{ and } \lim d(x_{2n+2},x_{2n+1})=0,$
\item[] $\varepsilon \mapsto \left\lfloor \exp\left(\left(\frac{20N}{\gamma\varepsilon}\right)^2+2\right)\right\rfloor +1$ for $\lim d(x_{2n+1},x_{2n})=0$.
\end{enumerate}
\end{example}
\begin{remark}
Lemma~\ref{l:artheta}, and Proposition~\ref{p:asymptoticregularity} below, still hold if instead of a function $\Gamma_2$ satisfying $(Q_{\ref{Q2}})$, we have a function $\Gamma'_2:\mathbb{N}\times(0,+\infty)\to \mathbb{N}$ such that for all $m\in \mathbb{N}$, $\Gamma'_2(m, \cdot)$ is a rate of convergence towards zero for the sequence $(\prod_{i=m}^{n}(1-\alpha_i))$. In this case, one defines $\theta_1$ using Lemma~\ref{L:xu_seq_reals_qt2}(1) instead of Lemma~\ref{L:xu_seq_reals_qt1}(1), namely $\theta_1=\widehat{\theta}'[A', V, 2N]$, where $A'(m,\varepsilon):=\Gamma_2'(m+1,\varepsilon)$, and the remaining functions are the same as before.
\end{remark}
Lemma~\ref{l:artheta} allows us to compute rates of convergence for the asymptotic properties of the sequence $(x_n)$.
\begin{proposition}\label{p:asymptoticregularity}
Under the conditions of Lemma~\ref{l:artheta}, we have that
\[
\lim d(x_{n+1},x_n)=\lim d(U(x_n),x_n)=\lim d(T(x_n), x_n)=0
\]
with monotone rates of convergence, respectively,
\begin{enumerate}
\item[$(i)$] $\rho_1(\varepsilon):=\max\{2\theta_3(\varepsilon)+1, 2\theta_4(\varepsilon)\}$
\item[$(ii)$] $\rho_2(\varepsilon):=2\theta_3\left(\frac{\varepsilon}{3}\right)+2$
\item[$(iii)$] $\rho_3(\varepsilon):=2\max\left\{ \theta_4\left(\frac{\varepsilon}{6}\right), \Gamma_1(\frac{\varepsilon}{4N}) \right\}+1$,
\end{enumerate}
where $\theta_1$, $\theta_2$, $\theta_3$ and $\theta_4$ are as in Lemma~\ref{l:artheta}.
\end{proposition}
\begin{proof}
Let $\varepsilon>0$ be given. For part $(i)$, take $n\geq \rho_1(\varepsilon)$. If $n=2n'+1$, then $n'\geq \theta_3(\varepsilon)$ and the result follows from Lemma~\ref{l:artheta}$(iii)$. Similarly, if $n=2n'$ we get $n'\geq \theta_4(\varepsilon)$ and the result now follows from Lemma~\ref{l:artheta}$(iv)$.
%
For part $(ii)$, using the fact that $U$ is nonexpansive we have
%
\SE{d(U(x_{2n+2}),x_{2n+2})&\leq d(U(x_{2n+2}),U(x_{2n+1}))+d(U(x_{2n+1}),x_{2n+1})+d(x_{2n+2},x_{2n+1})\\
&\leq 2d(x_{2n+2},x_{2n+1})+d(U(x_{2n+1}),x_{2n+1})\leq 3d(U(x_{2n+1}),x_{2n+1}),
}
%
which from Lemma~\ref{l:artheta}$(iii)$ implies that $\lim d(U(x_{2n+2}),x_{2n+2})=0$ with rate of convergence $\varepsilon\mapsto\theta_3(\varepsilon/3)$, and consequently
%
\begin{equation*}
\lim d(U(x_n),x_n)=0, \, \text{ with rate of convergence } \tilde{\rho}_2(\varepsilon):=\max\left\{ 2\theta_3(\varepsilon)+1, 2\theta_3\left(\frac{\varepsilon}{3}\right)+2 \right\}.
\end{equation*}
%
Note that the assumption of monotonicity on the functions $\Gamma_i$ (and the definition of the function $\theta$ in Lemma~\ref{L:xu_seq_reals_qt1}), entail that the functions $\theta_i$ (and $\rho_i$) are also monotone. In particular, the monotonicity of $\theta_3$ entails that $\theta_3(\varepsilon/3)\geq \theta_3(\varepsilon)$, and so $\tilde{\rho}_2$ coincides with $\rho_2$.
%
Finally for part $(iii)$, using (W1), the fact that $T$ is a nonexpansive map and \eqref{e:bounded}, we obtain
%
\SE{d(T(x_{2n+1}),x_{2n+1})&\leq (1-\alpha_n)d(T(x_{2n+1}), T(x_{2n})) + \alpha_n d(T(x_{2n+1}), u)\\
&\leq d(x_{2n+1},x_{2n})+2N\alpha_n
}
%
and
%
\SE{d(T(x_{2n}), x_{2n})&\leq d(T(x_{2n+1}),T(x_{2n}))+d(T(x_{2n+1}), x_{2n+1}) + d(x_{2n+1},x_{2n})\\
&\leq 2d(x_{2n+1},x_{2n})+d(T(x_{2n+1}),x_{2n+1})\\
&\leq 3d(x_{2n+1},x_{2n}) + 2N\alpha_n.
}
%
By $(Q_{\ref{Q1}})$ and Lemma~\ref{l:artheta}$(iv)$, we conclude that
\begin{equation*}
\begin{aligned}
&\varepsilon\mapsto \max\left\{ \theta_4\left(\frac{\varepsilon}{2}\right), \Gamma_1\left(\frac{\varepsilon}{4N}\right) \right\} \, \text{ is a rate of convergence for }\, \lim d(T(x_{2n+1}), x_{2n+1})=0\\
&\varepsilon\mapsto \max\left\{ \theta_4\left(\frac{\varepsilon}{6}\right), \Gamma_1\left(\frac{\varepsilon}{4N}\right) \right\}\, \text{ is a rate of convergence for }\, \lim d(T(x_{2n}), x_{2n})=0.
\end{aligned}
\end{equation*}
Using the monotonicity of $\theta_4$, we conclude part $(iii)$.
\end{proof}
\begin{example}
With $(\alpha_n), (\beta_n)$ as in Example~\ref{ex:1}, it is possible to verify that for any $\varepsilon>0$,
\begin{enumerate}
\item[] $n\geq \left\lfloor \exp\left(\left(\frac{20N}{\gamma\varepsilon}\right)^2+3\right)\right\rfloor +2 \, \to\, d(x_{n+1},x_{n})\leq \varepsilon$,
\item[] $n\geq \left\lfloor \exp\left(\left(\frac{14N}{\gamma\varepsilon}\right)^2+3\right)\right\rfloor +3 \, \to\, d(U(x_{n}),x_{n})\leq \varepsilon$,
\item[] $n\geq \left\lfloor \exp\left(\left(\frac{20N}{\gamma\varepsilon}\right)^2+3\right)\right\rfloor +3 \, \to\, d(T(x_{n}),x_{n})\leq \varepsilon$.
\end{enumerate}
\end{example}
We have the following trivial consequence of Proposition~\ref{p:asymptoticregularity}.
\begin{lemma}\label{l:rho}
For any $\varepsilon>0$ define $\rho(\varepsilon):=\max\{\rho_2(\varepsilon),\rho_3(\varepsilon) \}$, where $\rho_2,\rho_3$ are as in Proposition~\ref{p:asymptoticregularity}. We have that $\rho$ is a monotone rate of asymptotic regularity for both $U$ and $T$, i.e.\
\begin{equation*}
\forall \varepsilon >0 \, \forall n \geq \rho(\varepsilon) \left(d(x_n,T(x_n), d(x_n,U(x_n))\leq \varepsilon) \right).
\end{equation*}
\end{lemma}
\section{Metric Projection in {\rm CAT(0)} spaces}\label{s:Projection}
In this section, we show a useful quantitative version of the metric projection in $\mathrm{CAT}(0)$ spaces. The proof follows the arguments from \cite{FFLLPP(19),PP(ta)} (see also \cite{kohlenbach2011quantitative}). Let $T,U:C \to C$ be nonexpansive maps on a nonempty and convex subset of a $\mathrm{CAT}(0)$ space $X$ and assume that $F:=\mathrm{Fix}\,(T) \cap \mathrm{Fix}\,(U) \neq \emptyset$.
Given $\eta >0$ and $a \in X$ we write $$F_N(a,\eta):=\{x \in C: d(x,T(x)),d(x,U(x))\leq \eta\} \cap \overline{B}_N(a).$$
\begin{proposition}\label{P:metric}
Given $u \in C$, let $N \in \mathbb{N} \setminus \{0\}$ be such that $N \geq 2d(u,p)$, for some $p \in F$. Then, for every $\varepsilon >0$ and $\delta:(0,1] \to (0,1]$ there exists $\eta \in [\varphi(\varepsilon,\delta),1]$ and $x \in F_N(p,\delta(\eta))$ such that
\begin{equation*}
\forall y \in F_N(p,\eta) \left(d^2(u,x)\leq d^2(u,y)+\varepsilon \right),
\end{equation*}
where $\varphi(\varepsilon,\delta):=\varphi[N](\varepsilon,\delta):= \min \{\delta^{(i)}(1): i \leq r\}$ with $r:=r(N,\varepsilon):=\lceil \frac{N^2}{4\varepsilon}\rceil$.\\
Moreover, $\eta$ is witnessed by $\delta^{(i)}(1)$, for some $i \leq r$.
\end{proposition}
\begin{proof}
Suppose that for some $\varepsilon >0$ and $\delta:(0,1] \to (0,1]$ we have
\begin{equation}\label{e:contradiction}
\forall \eta \in [\varphi(\varepsilon,\delta),1] \, \forall x \in F_{N}(p,\delta(\eta))\, \exists y \in F_N(p,\eta) \left( d^2(u,y)< d^2(u,x) -\varepsilon \right).
\end{equation}
We construct a finite sequence $x_0, \dots, x_{r}$ of elements of $C$ as follows. We define $x_0:=p$. In particular, $x_0 \in F_N(p,\delta^{(r)}(1))$.
Assume that, for $j \leq r-1$ we have $x_j \in F_N(p, \delta^{(r-j)}(1))$. Then, by \eqref{e:contradiction} there exists $y_0 \in F_N(p, \delta^{(r-j-1)}(1))$ such that
\begin{equation*}
d^2(u,y_0)< d^2(u,x_j) -\varepsilon,
\end{equation*}
and define $x_{j+1}:=y_0$.%
By construction, for all $j \leq r$ we have $d^2(u,x_{j+1})<d^2(u,x_j)-\varepsilon$ and so
\begin{equation*}
d^2(u,x_{r})<d^2(u,x_0)-r\varepsilon \leq \frac{N^2}{4}- \frac{N^2}{4\varepsilon}\varepsilon=0,
\end{equation*}
which is a contradiction.
\end{proof}
The following lemmas are a generalization to $\mathrm{CAT}(0)$ spaces of Lemmas~2.3 and 2.7 of \cite{kohlenbach2011quantitative}. The proofs are similar but we include them nevertheless for completeness.
\begin{lemma}\label{L:2.3K}
Let $X$ be a $\mathrm{CAT}(0)$ space, $C \subseteq X$ a convex bounded subset and $b \in \mathbb{N} \setminus\{0\}$, a bound on the diameter of $C$. Consider $T:C \to C$ a nonexpansive map on $C$. Then
\begin{equation*}
\forall \varepsilon >0 \, \forall x_1,x_2 \in C \left(\bigwedge_{i=1}^{2} d(x_i,T(x_i))< \frac{\varepsilon^2}{12b} \to \forall t \in [0,1] \left(d(w_t,T(w_t))< \varepsilon \right)\right),
\end{equation*}
where $w_t:= (1-t)x_1 \oplus tx_2$.
\end{lemma}
\begin{proof}
Let $x_1,x_2 \in C$ be such that
\begin{equation}\label{e:wedge}
\bigwedge_{i=1}^{2} d(x_i,T(x_i))< \frac{\varepsilon^2}{12b}.
\end{equation}
For any $t \in [0,1]$ we have $K:=\max\{d(x_1,w_t),d(x_1,T(w_t))\}\leq b$. If $K < \varepsilon/2$, then $d(w_t,T(w_t))\leq d(x_1,w_t)+d(x_1,T(w_t))\leq \varepsilon$. So, we may then assume that $K\geq\frac{\varepsilon}{2}$. Since $T$ is nonexpansive, using \eqref{e:wedge} we derive
\SE{d\left(x_2,\frac{1}{2}w_t \oplus \frac{1}{2}T(w_t)\right)& \overset{\mathrm{(W1)}}{\leq} \frac{1}{2} d(x_2,w_t)+\frac{1}{2}d(x_2,T(w_t))\\
& \leq \frac{1}{2} d(x_2,w_t)+ \frac{1}{2} d(T(x_2),T(w_t))+\frac{1}{2} d(x_2,T(x_2))\\
& < d(x_2,w_t)+\frac{\varepsilon^2}{24b}\\
& \leq d(x_2,w_t)+\frac{\varepsilon^2}{24K}.
}
We have that
\begin{equation}\label{e:elsecontradiction}
d\left(x_1,\frac{1}{2}w_t \oplus \frac{1}{2}T(w_t)\right)> d(x_1,w_t)-\frac{\varepsilon^2}{24K},
\end{equation}
otherwise we would obtain the following contradiction
\SE{d(x_1,x_2)& \leq d(x_1,\frac{1}{2}w_t \oplus \frac{1}{2}T(w_t))+d\left(x_2,\frac{1}{2}w_t \oplus \frac{1}{2}T(w_t)\right)\\
&< d(x_1,w_t)-\frac{\varepsilon^2}{24K}+ d(x_2,w_t)+\frac{\varepsilon^2}{24K}\\
& = td(x_1,x_2)+(1-t)d(x_1,x_2)=d(x_1,x_2).
}
Since $d(x_1, T(w_t))< d(x_1,w_t)+\frac{\varepsilon^2}{12K}$, from \eqref{e:elsecontradiction} we obtain
\SE{d\left(x_1,\frac{1}{2}w_t \oplus \frac{1}{2}T(w_t)\right) > d(x_1,w_t)-\frac{\varepsilon^2}{24K} > K-\frac{\varepsilon^2}{12K}-\frac{\varepsilon^2}{24K}=K\left(1-\frac{(\varepsilon/K)^2}{8}\right)
}
From Lemma~\ref{l:CATconvexity} with $a=x_1$, $x=w_t$, $y=T(w_t)$, $r=K$ and $\varepsilon=\frac{\varepsilon}{K}\in (0,2]$ we obtain
\begin{equation*}
d(w_t,T(w_t))< K\frac{\varepsilon}{K}=\varepsilon.\qedhere
\end{equation*}
\end{proof}
\begin{lemma}\label{L:2.7K}
Let $X$ be a $\mathrm{CAT}(0)$ space and $x,y,u \in X, t \in [0, 1]$ and define
$w_t:= (1-t)x \oplus ty$. Then
\begin{equation*}
\forall \varepsilon \in (0,b^2] \left(\forall t \in [0,1] \left(d^2(u,x) \leq d^2(u,w_{t})+\frac{\varepsilon^2}{b^2} \right)\to \langle \overrightarrow{ux},\overrightarrow{yx}\rangle \leq \varepsilon \right),
\end{equation*}
where $b \geq d(x,y)$.
\end{lemma}
\begin{proof}
We have, for any $t \in [0,1]$
\SE{d^2(u,w_t)&\leq (1-t)d^2(u,x)+td^2(u,y)-t(1-t)d^2(x,y)\\
& =d^2(u,x)+ 2t\left(\frac{1}{2} \left(d^2(u,y)+d^2(x,x)-d^2(u,x)-d^2(x,y) \right) \right)+t^2d^2(x,y)\\
&=d^2(u,x)+2t \langle \overrightarrow{ux},\overrightarrow{xy}\rangle+t^2d^2(x,y).
}
Assume that $d^2(u,x) \leq d^2(u,w_{t})+\frac{\varepsilon^2}{b^2}$. Hence, for $t=\frac{\varepsilon}{b^2} \in [0,1]$ we have
\SE{-\frac{\varepsilon^2}{b^2}\leq 2\frac{\varepsilon}{b^2}\langle \overrightarrow{ux},\overrightarrow{xy}\rangle +\left(\frac{\varepsilon}{b^2}\right)^2d^2(x,y)\leq 2\frac{\varepsilon}{b^2}\langle \overrightarrow{ux},\overrightarrow{xy}\rangle +\left(\frac{\varepsilon}{b^2}\right)^2b^2=-2\frac{\varepsilon}{b^2}\langle \overrightarrow{ux},\overrightarrow{yx}\rangle +\frac{\varepsilon^2}{b^2}, }
from which we derive that $\langle \overrightarrow{ux},\overrightarrow{yx}\rangle \leq \varepsilon$.
\end{proof}
By combining the previous results we obtain a quantitative version of the characterization of the metric projection in $\mathrm{CAT}(0)$ spaces (cf. Lemma~\ref{l:charactProj}).
\begin{proposition}\label{p:innerproduct}
Given $u \in C$, let $N \in \mathbb{N} \setminus\{0\}$ be such that $N\geq 2 d(u,p)$ for some $p \in F$. For any
$\varepsilon >0$ and function $\delta : (0, 1] \to (0, 1]$, there exist $\eta \in [\Phi(\varepsilon,\delta),1]$ and $x \in F_N(p,\delta(\eta))$ such that
\begin{equation*}
\forall y \in F_N(p,\eta) \left(\langle \overrightarrow{ux},\overrightarrow{yx}\rangle \leq \varepsilon \right),
\end{equation*}
where $\Phi(\varepsilon,\delta):=\Phi[N](\varepsilon,\delta):=\frac{\varphi(\tilde{\varepsilon}, \tilde{\delta})^2}{24N}$, with $\tilde{\varepsilon}:=\frac{\varepsilon^2}{4N^2}$, $\tilde{\delta}(\xi):=\min \left\{\delta\left(\frac{\xi^2}{24N}\right),\frac{\xi^2}{24N}\right\}$, for any $\xi \in (0,1]$, and $\varphi:=\varphi[N]$ is as in Proposition~\ref{P:metric}.
\end{proposition}
\begin{proof}
Let $\varepsilon >0$ and $\delta : (0, 1] \to (0, 1]$ be given. We may assume that $\varepsilon \leq 4N^2$, otherwise the result follows easily using the Cauchy-Schwarz inequality. By Proposition~\ref{P:metric}, there exist $\eta_0 \in[\varphi(\tilde{\varepsilon},\tilde{\delta}),1]$ and $x \in F_N(p,\tilde{\delta}(\eta_0))$ such that
\begin{equation}\label{e:quadraticdiff}
\forall y \in F_N(p,\eta_0) \left(d^2(u,x)\leq d^2(u,y)+\tilde{\varepsilon} \right),
\end{equation}
Define $\eta_1:=\frac{\eta_0^2}{24N} \in [\Phi(\varepsilon,\delta),1]$. Since $x \in F_N(p,\eta_1)$, for any $y \in F_N(p,\eta_1)$, by Lemma~\ref{L:2.3K} (applied to $T$ and $U$ restricted to $C \cap \overline{B}_N(p)$ and $b=2N$)
\begin{equation*}
\forall t \in [0,1] \left(w_t \in F_N(p, \eta_0) \right),
\end{equation*}
with $w_t:=(1-t)x \oplus ty$. Hence, by \eqref{e:quadraticdiff}
\begin{equation*}
\forall t \in [0,1] \left(d^2(u,x)\leq d^2(u,w_t)+ \frac{\varepsilon^2}{4N^2} \right)
\end{equation*}
and the result follows from Lemma~\ref{L:2.7K} (with $b=2N$) and the fact that $d(x,T(x)),d(x,U(x))\leq \delta(\eta_1)$.
\end{proof}
We now apply, in the setting of $\mathrm{CAT}(0)$ spaces, a technique which corresponds to the quantitative removal of sequential weak compactness in Hilbert spaces -- see \cite{FFLLPP(19)}.
\begin{proposition}\label{p:removalswc}
Given $u \in C$, let $N \in \mathbb{N} \setminus\{0\}$ be such that $N\geq \max\{d(x_0,p),2 d(u,p)\}$ for some $p \in F$. Let $\rho$ be a monotone rate of asymptotic regularity for both $U$ and $T$. For any
$\varepsilon >0$ and function $\Delta : \mathbb{N} \to (0, 1]$, there exist $n\leq \Psi[N,\rho](\varepsilon,\Delta)$ and $x \in F_N(p,\Delta(n))$ such that
\begin{equation*}
\forall m \geq n \left(\langle \overrightarrow{ux},\overrightarrow{x_mx}\rangle \leq \varepsilon \right),
\end{equation*}
where $\Psi[N,\rho](\varepsilon,\Delta):=\rho(\Phi(\varepsilon,\widebar{\Delta}))$, with $\widebar{\Delta}(\eta):=\Delta(\rho (\eta))$ for all $\eta \in (0,1]$, and $\Phi:=\Phi[N]$ is as in Proposition~\ref{p:innerproduct}.
\end{proposition}
\begin{proof}
Let $\varepsilon >0$ and $\Delta : \mathbb{N} \to (0, 1]$ be given. By Proposition~\ref{p:innerproduct}, there exist $\eta_0 \in[\Phi(\varepsilon,\widebar{\Delta}),1]$ and $x \in F_N(p,\widebar{\Delta}(\eta_0))$ such that
\begin{equation}\label{e:innerprod}
\forall y \in F_N(p,\eta_0) \left(\langle \overrightarrow{ux},\overrightarrow{yx}\rangle \leq \varepsilon \right).
\end{equation}
Let $n:= \rho(\eta_0)$. Observe that $d(x,T(x)),d(x,U(x))\leq \widebar{\Delta}(\eta_0)=\Delta(n)$ and using the monotonicity of $\rho$, we have $n\leq \Psi[N,\rho](\varepsilon,\Delta)$. Since $d(x_n,p) \leq N$ by Lemma~\ref{l:bounded}, and $\forall m \geq n \left(d(x_n,T(x_n)),d(x_n,U(x_n))\leq \eta_0 \right)$, the result follows from \eqref{e:innerprod}.
\end{proof}
\section{Metastability}\label{s:convergence}
In this section we prove metastability results pertaining to the iteration $(x_n)$ and extract some consequences of that property. We also consider an iteration \eqref{e:MannHalpernerrors} which is a generalization of the iteration \eqref{e:MannHalpern} with error terms. We show that rates of metastability for a sequence generated by \eqref{e:MannHalpernerrors} follow from the metastability of $(x_n)$.
\begin{theorem}\label{t:main_meta}
Let $X$ be a $\mathrm{CAT}(0)$ space and $C \subseteq X$ a nonempty convex subset. Consider nonexpansive mappings $T,U:C \to C$ such that $F:=\mathrm{Fix}\,(T) \cap \mathrm{Fix}\,(U) \neq \emptyset$. Let $(\alpha_n),(\beta_n) \subset [0,1]$ and $u,x_0 \in C$. Assume that we have monotone functions $\Gamma_1, \Gamma_2, \Gamma_3, \Gamma_4$ and a constant $\gamma\in (0,1)$ satisfying $(Q_{\ref{Q1}})-(Q_{\ref{Q5}})$, respectively. Let $N\in\mathbb{N}\setminus\{0\}$ be a natural number such that $N\geq \max\{d(x_0,p),2 d(u,p)\}$, for some $p\in F$.
Then $(x_n)$ generated by \eqref{e:MannHalpern} is a Cauchy sequence with rate of metastability
\begin{equation*}
\mu(\varepsilon,f):=2\max\left\{\Sigma\left(\Psi\left(\frac{\tilde{\varepsilon}}{24}, \Delta_{\varepsilon,f}\right)\right), \theta_4\left(\frac{\varepsilon}{4}\right) \right\} +1
\end{equation*}
where $\mu:=\mu[N,\gamma,\Gamma_1, \Gamma_2, \Gamma_3, \Gamma_4]$ and
\begin{enumerate}
\item[] $\tilde{\varepsilon}:=\frac{\varepsilon^2}{16}$,
\item[] $\Delta_{\varepsilon,f}(n):=\min \left\{\frac{\tilde{\varepsilon}}{30N(P(n)+1)},1\right\}$,
\item[] $P(n):=f\left(2\max\left\{\Sigma(n), \theta_4\left(\frac{\varepsilon}{4}\right) \right\} +1\right)$,
\item[] $\Sigma(n):=\sigma\left[\Gamma_2, 4N^2\right]\left(\tilde{\varepsilon}, K(n)\right),\, \text{ with }\, \sigma\, \text{ as defined in Lemma~\ref{L:xu_seq_reals_qt3}}$,
\item[] $K(n):=\max\left\{n, \rho_3\left(\frac{\tilde{\varepsilon}}{36N}\right)\right\}$,
\item[] $\theta_4$ as in Lemma~\ref{l:artheta}, $\rho_3$ as in Proposition~\ref{p:asymptoticregularity},
\item[] $\Psi=\Psi[N,\rho]$ as in Proposition~\ref{p:removalswc}, with $\rho$ as in Lemma~\ref{l:rho}.
\end{enumerate}
\end{theorem}
\begin{proof}
Let $\varepsilon>0$ and a function $f:\mathbb{N}\to\mathbb{N}$ be given. By Proposition~\ref{p:removalswc} we may consider $n_0\leq \Psi(\tilde{\varepsilon}/24, \Delta_{\varepsilon,f})$ and $\tilde{x}\in F_N(p, \Delta_{\varepsilon,f}(n_0))$ such that
%
\begin{equation*}
\forall n \geq n_0 \left(\langle \overrightarrow{u\tilde{x}},\overrightarrow{x_n\tilde{x}}\rangle \leq \frac{\tilde{\varepsilon}}{24} \right).
\end{equation*}
%
Using (W1) and the nonexpansivity of $U$, we have for all $n\in \mathbb{N}$
\begin{equation*}
d(x_{2n+2}, \tilde{x})\leq (1-\beta_{n})d(U(x_{2n+1}),\tilde{x}) + \beta_{n}d(x_{2n+1},\tilde{x})\leq d(x_{2n+1},\tilde{x}) + d(U(\tilde{x}), \tilde{x}).
\end{equation*}
So, using Lemma~\ref{l:binomial}, we have
\begin{align}\label{e:dquadrado}
d^2(x_{2n+2}, \tilde{x})&\leq \left(d(x_{2n+1}, \tilde{x})+d(U(\tilde{x}), \tilde{x})\right)^2 \nonumber\\
&=d^2(x_{2n+1}, \tilde{x}) + d(U(\tilde{x}), \tilde{x})\left(d(U(\tilde{x}), \tilde{x}) + 2d(x_{2n+1}, \tilde{x})\right) \nonumber\\
&\leq (1-\alpha_n)^2d^2(T(x_{2n}), \tilde{x})+2\alpha_n(1-\alpha_n)\langle \overrightarrow{T(x_{2n})\tilde{x}},\overrightarrow{u\tilde{x}}\rangle + \alpha_n^2d^2(u, \tilde{x}) + 5N\Delta_{\varepsilon,f}(n_0) \nonumber\\
&\leq (1-\alpha_n)d^2(x_{2n}, \tilde{x})+\alpha_n\left(2\langle \overrightarrow{u\tilde{x}}, \overrightarrow{T(x_{2n})\tilde{x}}\rangle + 3N^2\alpha_n\right) + 10N\Delta_{\varepsilon,f}(n_0),
\end{align}
%
since $2d^2(u, \tilde{x})\leq (d(u,p)+d(\tilde{x},p))^2\leq (3N/2)^2\leq 3N^2$ and
%
\SE{
d^2(T(x_{2n}),\tilde{x})&\leq d^2(T(x_{2n}),T(\tilde{x}))+d(T(\tilde{x}),\tilde{x})\left(d(T(\tilde{x}),\tilde{x})+ 2d(T(x_{2n}),T(\tilde{x})) \right)\\
&\leq d^2(x_{2n},\tilde{x})+ \Delta_{\varepsilon,f}(n_0)(1+2\cdot2N) \leq d^2(x_{2n},\tilde{x})+ 5N\Delta_{\varepsilon,f}(n_0).
}
%
Thus, the inequality \eqref{e:dquadrado} can we written in the compact form
\[
s_{n+1}\leq (1-\alpha_n)s_n+ \alpha_nr_n+ \mathcal{E},
\]
where $s_n:=d^2(x_{2n}, \tilde{x})$, $r_n:=2\langle \overrightarrow{u\tilde{x}}, \overrightarrow{T(x_{2n})\tilde{x}}\rangle + 3N^2\alpha_n$ and $\mathcal{E}:=10N\Delta_{\varepsilon,f}(n_0)$. Using Cauchy-Schwarz, for $n\geq \max\{n_0, \rho_3(\frac{\tilde{\varepsilon}}{36N})\}$, we have
\begin{align*}
2\langle \overrightarrow{u\tilde{x}}, \overrightarrow{T(x_{2n})\tilde{x}}\rangle &=2\langle \overrightarrow{u\tilde{x}}, \overrightarrow{T(x_{2n})x_{2n}}\rangle +2\langle \overrightarrow{u\tilde{x}}, \overrightarrow{x_{2n}\tilde{x}}\rangle\\
&\leq 2d(u, \tilde{x})d(T(x_{2n}), x_{2n}) + 2\langle \overrightarrow{u\tilde{x}}, \overrightarrow{x_{2n}\tilde{x}}\rangle\\
&\leq 3N\frac{\tilde{\varepsilon}}{36N} + \frac{2\tilde{\varepsilon}}{24}=\frac{\tilde{\varepsilon}}{6}.
\end{align*}
By the definition of $\rho_3$, note that $\max\{n_0, \rho_3(\frac{\tilde{\varepsilon}}{36N})\}\geq \Gamma_1(\frac{\tilde{\varepsilon}}{18N^2})$ and so
\[
3N^2\alpha_n\leq \frac{\tilde{\varepsilon}}{6}.
\]
Hence, for all $n\geq K(n_0)$, $r_n\leq \frac{\tilde{\varepsilon}}{3}$. Since
\[
\mathcal{E}\leq \frac{10N\tilde{\varepsilon}}{30N(P(n_0)+1)}=\frac{\tilde{\varepsilon}}{3(P(n_0)+1)},
\]
we can apply Lemma~\ref{L:xu_seq_reals_qt3} to conclude
%
\begin{equation}\label{e:ConclusionXu}
\forall n\in [\Sigma(n_0), P(n_0)] \left( d(x_{2n}, \tilde{x})\leq \frac{\varepsilon}{4}\right).
\end{equation}
%
Since $d(x_{2n+1}, \tilde{x})\leq d(x_{2n+1}, x_{2n})+d(x_{2n}, \tilde{x})$, with $n_1:= \max\{\Sigma(n_0), \theta_4(\varepsilon/4)\}$ we obtain
\[
\forall n\in [n_1, P(n_0)] \left( d(x_{2n}, \tilde{x}),d(x_{2n+1}, \tilde{x})\leq \frac{\varepsilon}{2}\right).
\]
The latter entails that for $n\in [2n_1+1, P(n_0)]$, $d(x_n, \tilde{x})\leq \frac{\varepsilon}{2}$.
One easily sees that the function $\Sigma$ is monotone, and therefore $2n_1+1\leq \mu(\varepsilon, f)$.
Finally, noticing that $P(n_0)= f(2n_1+1)$, by the triangle inequality we conclude that $\mu$ is a rate of metastability for $(x_n)$.
\end{proof}
\begin{remark}\label{R:tomain}
\begin{enumerate}
\item If $\Gamma_2$ is replaced by a monotone function $\Gamma_2':\mathbb{N}\times (0,+\infty)\to\mathbb{N}$ satisfying
\[
\forall \varepsilon >0 \, \forall m\in \mathbb{N}\, \left( \prod_{i=m}^{\Gamma_2'(m,\varepsilon)}(1-\alpha_i)\leq \varepsilon\right),
\]
then one can compute a rate of metastability as before but applying Lemma~\ref{L:xu_seq_reals_qt4} instead of Lemma~\ref{L:xu_seq_reals_qt3}. Indeed the only difference would be in the definition of the number function $\Sigma$, which now would use $\sigma'$ instead of $\sigma$,
\[
\Sigma(n):=\sigma'\left[\Gamma_2', 4N^2\right]\left(\tilde{\varepsilon}, K(n)\right).
\]
\item The function $\rho_3$ was only required as a rate of convergence towards zero for $(d(T(x_{2n}),x_{2n}))$. It is then clear that $\rho_3$ could be replaced by $\varepsilon\mapsto \max\left\{ \theta_4\left(\frac{\varepsilon}{6}\right), \Gamma_1\left(\frac{\varepsilon}{4N}\right) \right\}$ (cf.\ the end of the proof of Proposition~\ref{p:asymptoticregularity}).
\end{enumerate}
\end{remark}
As a corollary to the proof of Theorem~\ref{t:main_meta}, we can derive quantitative information on the Halpern iteration. Previous rates of metastability were obtained by Kohlenbach and Leu\c{s}tean in \cite{KL(12)} using different methods. Their results follow from a quantitative analysis of a proof by Saejung~\cite{Saejung(10)} and rely on a technique to eliminate the use of Banach limits needed in the original proof.
\begin{corollary}\label{C:Saejungmeta}
Let $T:C \to C$ be a nonexpansive mapping, $(\alpha_n)\subset [0,1]$ and $y_0,u \in C$. Let $(y_n)$ be a sequence generated by the Halpern schema \eqref{e:Halpern}
\begin{equation*}
y_{n+1}:=(1-\alpha_n)T(y_{n}) \oplus \alpha_nu.
\end{equation*}
Assume that we have monotone functions $\Gamma_1, \Gamma_2, \Gamma_3$ satisfying $(Q_{\ref{Q1}})-(Q_{\ref{Q3}})$, respectively. Let $N\in\mathbb{N}\setminus\{0\}$ be a natural number such that $N\geq \max\{d(y_0,p),2 d(u,p)\}$, for some $p\in \mathrm{Fix}\,(T)$.
Then
\begin{enumerate}
\item[$(i)$] $(y_n)$ is asymptotically regular, and we have the following rates of asymptotic regularity
\begin{enumerate}
\item[] $\lim d(y_{n+1}, y_n)=0$ with rate of convergence $\widehat{\theta}[A,V,D]$,
\item[] $\lim d(T(y_n),y_n)=0$ with rate of convergence $\tilde{\rho}(\varepsilon):=\max\left\{\widehat{\theta}[A,V,D](\frac{\varepsilon}{2}),\Gamma_1\left(\frac{\varepsilon}{3N}\right) \right\}$,
\end{enumerate}
where $\widehat{\theta}$ is as in Lemma~\ref{L:xu_seq_reals_qt1}(1), with parameters $A(k):=\Gamma_2(k+1)$, $V(\varepsilon):=\Gamma_3\left(\frac{2\varepsilon}{3N}\right)$, and $D:=2N$.
\item[$(ii)$] $(y_n)$ is a Cauchy sequence with rate of metastability
\begin{equation*}
\zeta(\varepsilon,f):=\widetilde{\Sigma}\left(\Psi\left(\frac{\tilde{\varepsilon}}{24}, \widetilde{\Delta}_{\varepsilon,f}\right)\right),
\end{equation*}
where \begin{enumerate}
\item[] $\tilde{\varepsilon}:=\frac{\varepsilon^2}{4}$,
\item[] $\widetilde{\Delta}_{\varepsilon,f}:=\min \left\{\dfrac{\tilde{\varepsilon}}{30N\left(f\left(\widetilde{\Sigma}(n) \right)+1\right)},1\right\}$,
\item[] $\widetilde{\Sigma}(n):=\sigma\left[\Gamma_3, 4N^2\right]\left(\tilde{\varepsilon}, \widetilde{K}(n)\right),\, \text{ with }\, \sigma\, \text{ as defined in Lemma~\ref{L:xu_seq_reals_qt3}}$,
\item[] $\widetilde{K}(n):=\max\left\{n, \tilde{\rho}\left(\frac{\tilde{\varepsilon}}{36N}\right)\right\}$,
\item[] $\Psi=\Psi[N,\tilde{\rho}]\, \text{ as in Proposition~\ref{p:removalswc}}$.
\end{enumerate}
\end{enumerate}
\end{corollary}
\begin{proof}
To prove part $(i)$, we simplify the arguments in the proof of Lemma~\ref{l:artheta}. As before, one easily proves by induction that $d(y_n,p)\leq N$, for all $n\in\mathbb{N}$. Using (W2) and (W4) one shows that for all $n \in \mathbb{N}$
%
\begin{equation*}
d(y_{n+2},y_{n+1})\leq (1-\alpha_{n+1})d(y_{n+1},y_{n})+\frac{3N}{2}|\alpha_{n+1}-\alpha_n|.
\end{equation*}
%
By an application of Lemma~\ref{L:xu_seq_reals_qt1}(1) we have that $\lim d(y_{n+1},y_{n})=0$ with rate of convergence $\widehat{\theta}[A,V,D]$. Since, using (W1) and the assumption on $N$, for all $n \in \mathbb{N}$
%
\begin{equation*}
d(T(y_{n}),y_{n})\leq \frac{3N}{2}\alpha_n+d(y_{n+1},y_{n}),
\end{equation*}
we see that $\tilde{\rho}$ is a rate of convergence for $\lim d(T(y_n),y_n)=0$, and conclude the proof of part $(i)$.
%
We now argue the metastability property of $(y_n)$. First consider $(x_n)$ generated by \eqref{e:MannHalpern} with parameters $(\alpha_n, T,\beta_n, U, x_0, u):=(\alpha_n, T, 1/2, \mathrm{Id}_C, y_0, u)$. By an easy induction one sees that $y_{n}=x_{2n}$, for all $n \in \mathbb{N}$. In light of Remark~\ref{R:tomain}(2) and the fact that $\tilde{\rho}$ is a rate of convergence towards zero for $(d(T(y_n),y_n))\equiv (d(T(x_{2n}),x_{2n}))$, we can follow the proof of Theorem~\ref{t:main_meta} (with $\tilde{\varepsilon}:=\varepsilon^2/4$) until \eqref{e:ConclusionXu}, to conclude that
%
\[
\forall n\in \left[\widetilde{\Sigma}(n_0), f\left(\widetilde{\Sigma}(n_0)\right)\right] \left( d(y_{n}, \tilde{x})\leq \frac{\varepsilon}{2}\right).
\]
The result then follows using the triangular inequality.
\end{proof}
The next easy lemma is well-known (see e.g.\ \cite[Proposition~7]{LLPP(21)}). We include its short proof for the sake of completeness.
\begin{lemma}\label{l:Cauchymetric}
Consider two sequences $(w_n)$ and $(z_n)$ in a metric space. If $(w_n)$ is a Cauchy sequence with rate of metastability $\tau$ and $\lim d(w_n,z_n)=0$ with rate of convergence $\nu$, then $(z_n)$ is also a Cauchy sequence with rate of metastability
%
\begin{equation*}
\tau_{\nu}(\varepsilon,f):=\max\left\{\tau\left(\frac{\varepsilon}{3},f_{\varepsilon,\nu}\right), \nu\left(\frac{\varepsilon}{3}\right)\right\},
\end{equation*}
%
where $f_{\varepsilon,\nu}:=f(\max\{n,\nu(\varepsilon/3)\})$, for all $n \in \mathbb{N}$.
\end{lemma}
\begin{proof}
Let $\varepsilon >0$ and $f:\mathbb{N} \to \mathbb{N}$ be given. By the assumption on $\tau$, consider $n_0 \leq \tau (\varepsilon/3,f_{\varepsilon,\nu})$ such that $d(w_i,w_j)\leq \varepsilon/3$, for all $i,j \in [n_0,f_{\varepsilon,\nu}(n_0)]$. Define $n:=\max\{n_0,\nu(\varepsilon/3)\} \leq \tau_{\nu}(\varepsilon,f)$. Then, for all $i,j \in [n,f_{\varepsilon,\nu}(n_0)]=[n,f(n)]$ we have
%
\SE{
d(z_i,z_j) \leq d(z_i,w_i)+d(w_i,w_j)+d(w_j,z_j)\leq \varepsilon,
}
%
using the assumption on $\nu$ and the fact that $n \geq n_0$.
\end{proof}
The next two results take into account the presence of error terms in the inductive construction of the iteration \eqref{e:MannHalpern}.
\begin{theorem}\label{t:quantresults1}
Let $T,U:C \to C$ be nonexpansive mappings and $(\alpha_n),(\beta_n)\subset [0,1]$. Let $(x_n)$ be generated by \eqref{e:MannHalpern} and assume that $(Q_{\ref{Q2}})$ holds with rate of divergence $\Gamma_2$. Consider a sequence $(x'_n)$ satisfying $x'_0=x_0$ and
\begin{equation}\label{e:MannHalpernerrors}\tag{HM$_e$}
\begin{cases}
d(x'_{2n+1},(1-\alpha_n)T(x'_{2n})\oplus \alpha_n u)&\leq \delta_{2n}\\
d(x'_{2n+2},(1-\beta_n)U(x'_{2n+1})\oplus\beta_nx'_{2n+1})&\leq \delta_{2n+1},
\end{cases}
\end{equation}
where $(\delta_n) \subset [0,+\infty)$.
If $\sum \delta_{n} <\infty$, or $\lim \frac{\delta_{2n}+\delta_{2n+1}}{\alpha_n}=0$ (in the case $\alpha_n>0$), then $\lim d(x'_n,x_n)=0$. Moreover,
\begin{enumerate}[$(1)$]
\item If $\chi_1$ is a Cauchy rate for $\left(\sum \delta_{n} \right)$, then $\widehat{\nu}$ is a rate of convergence for $\lim d(x'_n,x_n)=0$, where
\begin{equation*}
\widehat{\nu}(\varepsilon):=2\max\left\{\theta\left(\frac{\varepsilon}{2}\right),\chi_1\left(\frac{\varepsilon}{2}\right)\right\}+3,
\end{equation*}
with $\theta:=\widehat{\theta}[\Gamma_2,\chi_1,\lceil \sum_{i=0}^{\chi_1(1)}\delta_i\rceil +1]$, where $\widehat{\theta}$ is as in Lemma~\ref{L:xu_seq_reals_qt1}(1).
\item If $\chi_2$ is a rate of convergence for $\lim \frac{\delta_{2n}+\delta_{2n+1}}{\alpha_n}=0$, then $\widecheck{\nu}$ is a rate of convergence for $\lim d(x'_n,x_n)=0$, where
\begin{equation*}
\widecheck{\nu}(\varepsilon):=2\max\left\{\theta\left(\frac{\varepsilon}{2}\right),\chi_2\left(\frac{\varepsilon}{2}\right)\right\}+1,
\end{equation*}
with $\theta:=\widecheck{\theta}\left[\Gamma_2,\chi_2,\left\lceil\max_{i \leq \chi_2(1)} \left\{\frac{\delta_{2i}+\delta_{2i+1}}{\alpha_i},1\right\}\right\rceil \right]$, where $\widecheck{\theta}$ is as in Lemma~\ref{L:xu_seq_reals_qt1}(2).
\end{enumerate}
\end{theorem}
\begin{proof}
By the nonexpansivity of $T$ and (W4), we have for all $n\in \mathbb{N}$
%
\begin{equation}\label{e:dx'toximpar}
\begin{split}
d(x'_{2n+1}, x_{2n+1})& \leq d(x'_{2n+1},(1-\alpha_{n})T(x'_{2n})\oplus \alpha_{n} u) \\
&\qquad + d((1-\alpha_{n})T(x'_{2n})\oplus \alpha_{n} u,x_{2n+1})\\
& \leq (1-\alpha_{n})d(x'_{2n},x_{2n})+\delta_{2n}.
\end{split}
\end{equation}
%
Similarly
%
\begin{equation}\label{e:dx'toxpar}
\begin{split}
d(x'_{2n+2},x_{2n+2})&\leq d(x'_{2n+2},(1-\beta_{n})U(x'_{2n+1})\oplus \beta_{n} x'_{2n+1}) \\
&\qquad + d((1-\beta_{n})U(x'_{2n+1})\oplus \beta_{n} x'_{2n+},x_{2n+2})\\
& \leq d(x'_{2n+1}, x_{2n+1})+ \delta_{2n+1}\\
& \leq (1-\alpha_{n})d(x'_{2n},x_{2n})+\delta_{2n}+\delta_{2n+1}.
\end{split}
\end{equation}
%
We then conclude that $\lim d(x'_{2n},x_{2n})=0$ by applying Lemma~\ref{L:Xu} with $r_n:=0$ and $v_n:=\delta_{2n}+\delta_{2n+1}$, if $\sum \delta_n <\infty$, or with $r_n:=\frac{\delta_{2n}+\delta_{2n+1}}{\alpha_n}$ and $v_n:=0$, if $\lim \frac{\delta_{2n}+\delta_{2n+1}}{\alpha_n}=0$. Then by \eqref{e:dx'toximpar}, since $\delta_n \to 0$, also $\lim d(x'_{2n+1},x_{2n+1})=0$ . Hence $\lim d(x'_{n},x_{n})=0$.
Let us show part (1). Clearly, \eqref{e:dx'toxpar} entails that $d(x'_{2n},x_{2n})\leq \sum_{i=0}^{2n-1}\delta_i$. With $D:= \lceil \sum_{i=0}^{\chi_1(1)}\delta_i\rceil +1$ it is easy to see that the assumption on $\chi_1$ implies that $D$ is an upper bound on $\left( \sum \delta_n \right)$, and so also on $\left(d(x'_{2n},x_{2n})\right)$.
The function $\chi_1$ is also a Cauchy rate for $v_n:=\delta_{2n}+\delta_{2n+1}$. Indeed, for all $\varepsilon >0$ and $n \in \mathbb{N}$
%
\SE{
\sum_{i=\chi_1(\varepsilon)+1}^{\chi_1(\varepsilon)+n}v_n=\sum_{i=\chi_1(\varepsilon)+1}^{\chi_1(\varepsilon)+n}\delta_{2i}+\delta_{2i+1}=\sum_{i=2\chi_1(\varepsilon)+2}^{2\chi_1(\varepsilon)+2n+1}\delta_{i}\leq \sum_{i=\chi_1(\varepsilon)+1}^{2\chi_1(\varepsilon)+2n+1}\delta_{i} \leq \varepsilon.
}
%
Now, by Lemma~\ref{L:xu_seq_reals_qt1}(1) we have that $\theta=\widehat{\theta}[\Gamma_2,\chi_1,D]$ is a rate of convergence for $\lim d(x'_{2n},x_{2n})=0$. Since $\chi_1$ is a Cauchy rate for $\left( \sum \delta_n \right)$, we have $\delta_{2n} \to 0$ with rate of convergence $\chi_1+1$. Using \eqref{e:dx'toximpar} we conclude that
%
\SE{
\forall n \geq \max\left\{\theta\left(\frac{\varepsilon}{2}\right), \chi_1\left(\frac{\varepsilon}{2}\right)+1\right\} \left(d(x'_{2n},x_{2n}), d(x'_{2n+1},x_{2n+1}) \leq \varepsilon \right),
}
%
which entails part (1).
We now turn to part (2). From the assumption on $\chi_2$, we have that $\max_{i \leq \chi_2(1)} \left\{\frac{\delta_{2i}+\delta_{2i+1}}{\alpha_i},1\right\}$ is an upper bound on the sequence $\left(\frac{\delta_{2n}+\delta_{2n+1}}{\alpha_n} \right)$. Using \eqref{e:dx'toxpar}, one shows by induction that $\left(d(x'_{2n},x_{2n})\right)$ is bounded by $D$, where $D:=\left\lceil\max_{i \leq \chi_2(1)} \left\{\frac{\delta_{2i}+\delta_{2i+1}}{\alpha_i},1\right\}\right\rceil$. Now, by Lemma~\ref{L:xu_seq_reals_qt1}(2) we have that $\theta=\widecheck{\theta}[\Gamma_2,\chi_2,D]$ is a rate of convergence for $\lim d(x'_{2n},x_{2n})=0$. Since $\chi_2$ is a rate of convergence for $\lim \frac{\delta_{2n}+\delta_{2n+1}}{\alpha_n}=0$, we have $\delta_{2n} \to 0$ with the same rate of convergence. Using \eqref{e:dx'toximpar} we conclude that
%
\SE{
\forall n \geq \max\left\{\theta\left(\frac{\varepsilon}{2}\right), \chi_2\left(\frac{\varepsilon}{2}\right)\right\} \left(d(x'_{2n},x_{2n}), d(x'_{2n+1},x_{2n+1}) \leq \varepsilon \right),
}
%
which entails the result.
\end{proof}
A rate of metastability for the generalized sequence with error terms \eqref{e:MannHalpernerrors} can now be immediately obtained from Theorem~\ref{t:main_meta} using Lemma~\ref{l:Cauchymetric} and Theorem~\ref{t:quantresults1}. Additionally, it is clear that $(x'_n)$ is asymptotically regular and rates of asymptotic regularity can be trivially computed using the rates in Proposition~\ref{p:asymptoticregularity} together with the rates of convergence in Theorem~\ref{t:quantresults1}.
\begin{corollary}
Consider sequences $(\alpha_n)$, $(\beta_n) \subset [0,1]$, a sequence of error terms $(\delta_n) \subset [0,+\infty)$ satisfying $\sum \delta_{n} < \infty$, or $\lim \frac{\delta_{2n}+\delta_{2n+1}}{\alpha_n}=0$ (if $\alpha_n >0$), and let $(x'_n)$ be generated by \eqref{e:MannHalpernerrors}. Then, under the conditions of Theorem~\ref{t:main_meta}, the sequence $(x'_n)$ is a Cauchy sequence. Moreover,
%
\begin{enumerate}[$(1)$]
\item if $\chi_1$ is a Cauchy rate for $\left(\sum \delta_{n} \right)$, then $\mu_{\widehat{\nu}}$ is a rate of metastability for $(x'_n)$;
\item if $\chi_2$ is a rate of convergence for $\lim \frac{\delta_{2n}+\delta_{2n+1}}{\alpha_n}=0$, then $\mu_{\widecheck{\nu}}$ is a rate of metastability for $(x'_n)$;
\end{enumerate}
%
where $\mu$ is as in Theorem~\ref{t:main_meta}, $\widehat{\nu},\widecheck{\nu}$ are as in Theorem~\ref{t:quantresults1} and the construction $\tau_\nu$ is as in Lemma~\ref{l:Cauchymetric}.
\end{corollary}
\section{Strong Convergence}\label{s:Strongconvergence}
We will now argue how the metastability result from the previous section actually entails the strong convergence of the iteration $(x_n)$. We begin by recalling a well-known characterization of the metric projection in terms of the quasi-linearization function.
\begin{lemma}[\cite{DehghanRooin}]\label{l:charactProj} Let $S$ be a nonempty convex closed subset of a complete $\mathrm{CAT}(0)$ space $X$. For any $u\in X$, let $P_S(u)$ denote the metric projection of $u\in X$ onto $S$. Then,
\[
\forall y\in S\, \left(\langle \overrightarrow{uP_S(u)}, \overrightarrow{yP_S(u)}\rangle \leq 0\right).
\]
\end{lemma}
We are now ready to give the proof of the main theorem.
\begin{proof}[\textbf{Proof of Theorem~\ref{t:main}}]
From Theorem~\ref{t:main_meta} and Lemma~\ref{l:metaCauchy}, $(x_n)$ is a Cauchy sequence and therefore convergent since $X$ is complete. Let $z=\lim x_n$, which is an element of $C$, because $C$ is closed. By Proposition~\ref{p:asymptoticregularity} and the continuity of $T$ and $U$, we conclude that $z\in F$. From Lemma~\ref{l:charactProj} (with $S=F$), we conclude that $\langle \overrightarrow{uP_F(u)}, \overrightarrow{zP_F(u)}\rangle \leq 0$.
Using Lemma~\ref{l:binomial}, with $s_n:=d^2(x_{2n}, P_F(u))$ we have
%
\begin{align}
s_{n+1}&\leq d^2(x_{2n+1}, P_F(u)) \nonumber\\
&\leq (1-\alpha_n)^2d^2(T(x_{2n}), P_F(u))+2\alpha_n(1-\alpha_n)\langle \overrightarrow{uP_F(u)},\overrightarrow{T(x_{2n})P_F(u)}\rangle + \alpha_n^2d^2(u, P_F(u)) \nonumber\\
&\leq (1-\alpha_n)d^2(x_{2n}, P_F(u))+\alpha_n\left(2\langle \overrightarrow{uP_F(u)}, \overrightarrow{T(x_{2n})P_F(u)}\rangle + \alpha_nd^2(u, P_F(u))\right)\nonumber\\
& = (1-\alpha_n)s_n+\alpha_n r_n,\label{ineqProofmain}
\end{align}
%
where $r_n=2\langle \overrightarrow{uP_F(u)}, \overrightarrow{T(x_{2n})P_F(u)}\rangle + \alpha_nd^2(u, P_F(u))$.
Since
%
\SE{
\langle \overrightarrow{uP_F(u)}, \overrightarrow{T(x_{2n})P_F(u)}\rangle &= \langle \overrightarrow{uP_F(u)}, \overrightarrow{T(x_{2n})x_{2n}}\rangle+\langle \overrightarrow{uP_F(u)}, \overrightarrow{x_{2n}z}\rangle+\langle \overrightarrow{uP_F(u)}, \overrightarrow{zP_F(u)}\rangle\\
&\leq d(u,P_F(u))\left(d(T(x_{2n}),x_{2n})+d(x_{2n},z)\right) \to 0,
}
%
and $\alpha_n\to 0$, we conclude that $\limsup r_n \leq 0$. Since $\sum \alpha_n=\infty$, we can apply Lemma~\ref{L:Xu} to conclude that $\lim s_n=0$. This means that $\lim x_{2n}=P_F(u)$, and so $z=P_F(u)$.
\end{proof}
As discussed in the Introduction, Theorem~\ref{t:main} is a twofold generalization of the recent strong convergence result, established in the setting of Hilbert spaces, by Bo\c{t}, Csetnek, and Meier~\cite[Theorem~3]{Botetal(19)}. On the one hand, our iteration is more general than the one considered by Bo\c{t} \emph{et al.} and, on the other hand, the strong convergence is established in $\mathrm{CAT}(0)$ spaces, which are frequently considered the non-linear generalization of Hilbert spaces. Our schema is also a generalization of the algorithm considered by Leus{\c t}ean and Cheval in \cite{CK(ta)} in the setting of hyperbolic spaces. Even though the later context is more general than that of $\mathrm{CAT}(0)$ spaces, our Theorem~\ref{t:main} succeeds in proving strong convergence while \cite[Theorems~4.1 and 4.2]{CK(ta)} only show the asymptotic regularity of the iteration. Observe that not only the restriction to $\mathrm{CAT}(0)$ spaces was required to prove strong convergence, but we also needed to consider the condition $0<\liminf \beta_n$ (which has no corresponding condition in \cite{Botetal(19)} nor in \cite{CK(ta)}) in order to obtain asymptotic regularity for the more general iteration \eqref{e:MannHalpern}.
Since the Halpern iteration follows from a particular case of \eqref{e:MannHalpern} -- cf.\ the proof of Corollary~\ref{C:Saejungmeta} --, Theorem~\ref{t:main} also allows to recover Saejung's strong convergence of the Halpern iteration in $\mathrm{CAT}(0)$ spaces from \cite[Theorem~2.3]{Saejung(10)}. We would like to point out that, contrary to Saejung's proof, we do not require the heavy machinery of Banach limits.
Finally, we point out that Theorem~\ref{t:quantresults1}, also extends the strong convergence result to the iterative schema \eqref{e:MannHalpernerrors}.
\section{Forward-backward and Douglas-Rachford algorithms}\label{s:FBDR}
In this section we work in a Hilbert space $H$ and apply our main results to obtain strongly convergent variants of the Forward-backward and Douglas-Rachford algorithms.
We recall that a multi-valued operator $\mathsf{T}:H \rightrightarrows H$ is \emph{monotone} if whenever $(x,y)$ and $(x',y')$ are elements of the graph of $\mathsf{T}$, it holds that $\langle x-x',y-y'\rangle \geq 0$. A monotone operator $\mathsf{T}$ is said to be \emph{maximal monotone} if the graph of $\mathsf{T}$ is not properly contained in the graph of any other monotone operator on $H$.
We use $J_{\mathsf{T}}$ to denote the \emph{resolvent function} of $\mathsf{T}$, i.e.\ the single-valued function defined by $J_{\mathsf{T}} = (I + \mathsf{T} )^{-1}$ and $R_{ \mathsf{T}}$ to denote the nonexpansive \emph{reflected resolvent function} defined by $R_{ \mathsf{T}}:=2J_{\mathsf{T}} - \mathrm{Id}$.
\begin{definition}
A mapping $U : H \to H$ is called \emph{firmly nonexpansive} if
$$\forall x, y \in H\left( \norm{U(x)-U(y)}^2 \leq \norm{x-y}^2-\norm{(\mathrm{Id} -U)(x)- (\mathrm{Id}-U)(y)}^2 \right).$$
\end{definition}
Clearly, if $U$ is firmly nonexpansive then it is nonexpansive.
For $c>0$, the resolvent function $J_{c\mathsf{T}}$ is firmly nonexpansive and the set of fixed points of $J_{c\mathsf{T}}$ coincides with the set of all zeros of $\mathsf{T}$.
For $\alpha \in (0,1]$, a function $U:H \to H$ is called \emph{$\alpha$-averaged}\footnote{The standard definition asks for $\alpha \in (0,1)$. With this extension, $1$-averaged is just another way of saying nonexpansive.} if there exists a nonexpansive operator $U':H \to H$ such that $U=(1-\alpha)\mathrm{Id}+\alpha U'$. The $\alpha$-averaged operators are always nonexpansive. Moreover, the $\frac{1}{2}$-averaged operators coincide with the firmly nonexpansive operators.
In the following corollaries we assume that we have functions $\Gamma_1, \Gamma_2, \Gamma_3, \Gamma_4$ satisfying conditions $(Q_{\ref{Q1}})-(Q_{\ref{Q4}})$.
\begin{corollary}\label{c:Cor1}
Let $\alpha \in (0,1]$ and $U:H \to H$ be $\alpha$-averaged. Given $(\alpha_n) \subset [0,1]$, $(\beta_n)\subset [1-\frac{1}{\alpha},1]$, and $x_0,u \in H$, consider $(x_n)$ generated by
\begin{equation}\label{e:HMaveraged}\tag{H$_{\mathrm{Id}}$M}
\begin{cases}
x_{2n+1}&=(1-\alpha_n)x_{2n}+\alpha_nu\\
x_{2n+2}&=(1-\beta_n)U(x_{2n+1})+ \beta_n x_{2n+1}.
\end{cases}
\end{equation}
Let $\sigma \in (0,1)$ be such that $\alpha\geq \sigma$.
Assume that there exist $N \in \mathbb{N} \setminus\{0\}$ such that $N \geq \max\{\norm{x_0-p},2\norm{u-p}\}$, for some $p \in \mathrm{Fix}\, U$ and $\gamma \in (0,\frac{1}{2\sigma}]$ satisfying $1-\frac{1}{\alpha}+\gamma\leq \beta_n \leq 1-\gamma$, for all $n \in \mathbb{N}$.
Then $(x_n)$ converges strongly to $P_{\mathrm{Fix}\,(U)}(u)$ and
\begin{equation*}
\mu_1[N,\sigma,\gamma,\Gamma_1, \Gamma_2, \Gamma_3, \Gamma_4]:=\mu[N,\sigma\gamma,\Gamma_1, \Gamma_2, \Gamma_3, \Gamma_4], \mbox{ with $\mu$ as in Theorem~\ref{t:main_meta},}
\end{equation*}
is a rate of metastability for $(x_n)$.
\end{corollary}
\begin{proof}
Since $U$ is $\alpha$-averaged, there exists $U'$ nonexpansive and such that $U=(1-\alpha)\mathrm{Id}+\alpha U'$. For all $n \in \mathbb{N}$
\begin{equation*}
\begin{split}
x_{2n+2}&=(1-\beta_n)U(x_{2n+1})+ \beta_n x_{2n+1}\\
&=(1-\beta_n)[(1-\alpha)\mathrm{Id}+\alpha U'](x_{2n+1})+ \beta_n x_{2n+1}\\
&=(1-\tilde{\beta}_n)U'(x_{2n+1})+ \tilde{\beta}_n x_{2n+1},
\end{split}
\end{equation*}
with $\tilde{\beta}_n:=1- \alpha+\alpha\beta_n$.
Hence $(x_n)$ is generated by \eqref{e:MannHalpern}, using the sequences $(\alpha_n),(\tilde{\beta}_n) \subset [0,1]$, $T=\mathrm{Id}$ and $U= U'$. It is easy to see that $\sigma\gamma$ satisfies condition $(Q_{\ref{Q5}})$ for the sequence $(\tilde{\beta}_n)$. Note that $\mathrm{Fix}\, U = \mathrm{Fix}\, U'=F$. Since $\alpha \in (0,1]$, for all $\varepsilon >0$ and $n \in \mathbb{N}$ we have
$$\sum_{i=\Gamma_4(\varepsilon)+1}^{\Gamma_4(\varepsilon)+n}|\tilde{\beta}_{i+1}-\tilde{\beta}_{i}|\leq \sum_{i=\Gamma_4(\varepsilon)+1}^{\Gamma_4(\varepsilon)+n}|\beta_{i+1}-\beta_{i}|\leq\varepsilon,$$
and so condition $(Q_{\ref{Q4}})$ still holds with $\Gamma_4$ for the sequence $(\tilde{\beta}_n)$.
Hence, by Theorem~\ref{t:main} the sequence $(x_n)$ converges strongly to $P_{\mathrm{Fix}\,(U)}(u)$. The rate of metastability follows from an application of Theorem~\ref{t:main_meta} with $\gamma:=\sigma\gamma$.
\end{proof}
For $\delta >0$, a function $U:H \to H$ is said to be \emph{$\delta$-cocoercive} if for all $x,y \in H$,
$$\langle x-y, U(x)-U(y)\rangle \geq \delta \norm{U(x)-U(y)}^2,$$
which is equivalent to say that $\delta U$ is firmly nonexpansive. In the following result we give a strongly convergent version of the forward-backward algorithm which extends \cite[Theorem~7]{Botetal(19)} (as well as the quantitative analysis from \cite[Corollary~2]{DP(ta)}). The algorithm \eqref{e:GFB} below can be seen as a generalization in the sense that for $\alpha_n=\beta_n\equiv 0$ the sequence of even terms $(x_{2n})$ is indeed the original weakly convergent forward-backward algorithm by Lions and Mercier \cite{LM(79)}.
\begin{corollary}\label{c:Cor2}
Let $U_1: H \rightrightarrows H$ be maximal monotone and $U_2:H \to H$ be $\delta$-cocoercive, for some $\delta >0$. Let $c \in (0,2\delta]$. Given $(\alpha_n) \subset [0,1]$, $(\beta_n) \subset [1-\frac{4\delta-c}{2\delta},1]$, and $x_0, u \in H$, consider $(x_n)$ generated by
\begin{equation}\tag{GFB}\label{e:GFB}
\begin{cases}
x_{2n+1}&=(1-\alpha_n)x_{2n}+\alpha_nu\\
x_{2n+2}&=(1-\beta_n)J_{cU_1}\left(x_{2n+1}-cU_2(x_{2n+1})\right)+ \beta_n x_{2n+1}.
\end{cases}
\end{equation}
Assume that there exist $N \in \mathbb{N} \setminus\{0\}$ such that $N \geq \max\{\norm{x_0-p},2\norm{u-p}\}$, for some $p \in zer (U_1+U_2)$ and $\gamma \in(0,1]$ satisfying $1-\frac{4\delta-c}{2\delta}+\gamma\leq \beta_n\leq 1-\gamma$, for all $n\in \mathbb{N}$. Then $(x_n)$ converges strongly to $P_{zer(U_1+U_2)}(u)$ and
$$
\mu_2:=\mu_2[N,\gamma,\Gamma_1, \Gamma_2, \Gamma_3, \Gamma_4]:=\mu[N,\gamma/2,\Gamma_1, \Gamma_2, \Gamma_3, \Gamma_4], \text{ with $\mu$ as in Theorem~\ref{t:main_meta}},
$$
is a rate of metastability for $(x_n)$.
\end{corollary}
\begin{proof}
It is straightforward to see that the iteration $(x_n)$ is generated by \eqref{e:HMaveraged} with $U:=J_{c U_1} \circ (\mathrm{Id} -c U_2)$. The resolvent function $J_{c U_1}$ is firmly nonexpansive, i.e.\ $\frac{1}{2}$-averaged. Furthermore, we have $\frac{c}{2\delta} \in (0,1]$ and so it follows from \cite[Proposition~4.39]{BC(17)} that $(\mathrm{Id} -c U_2)$ is $\frac{c}{2\delta}$-averaged. If $c < 2\delta$, then we use \cite[Theorem~3$(b)$]{OY(02)} to conclude that $U$ is $\frac{2\delta}{4\delta-c}$-averaged. If $c = 2\delta$, the fact that $U_2$ is $\delta$-cocoercive entails that $\mathrm{Id} - c U_2$ is nonexpansive. Hence $U$ is nonexpansive, or equivalently (following the extended notion of 1-averaged), $U$ is $\frac{2\delta}{4\delta-c}$-averaged since $\frac{2 \delta}{4 \delta - c}=1$.
Since $\mathrm{Fix}\, U=zer(U_1+U_2)$ \cite[Proposition~26.1$(iv)(a)$]{BC(17)}, we have $N \geq \max\{\norm{x_0-p},\norm{u-p}\}$, for some $p \in \mathrm{Fix}\, U$. From the fact that $\frac{2\delta}{4\delta-c} \geq \frac{1}{2}$, we may apply Corollary~\ref{c:Cor1} with $\sigma:=\frac{1}{2}$ to conclude the result.
\end{proof}
\begin{remark}
The fact that no quantitative information regarding `$c>0$' is present in the rate of metastability reflects the fact that this condition is only needed to make sense of the definition of the resolvent function. In fact, if in Corollary~\ref{c:Cor2} the resolvent function $J_{c U_1}$ is replaced by an arbitrary firmly nonexpansive mapping $J$, and $p$ is some point in $\mathrm{Fix}\, (J \circ (\mathrm{Id} -c U_2))$, then the result holds also for $c=0$ (immediately by Corollary~\ref{c:Cor1} with $\alpha=1/2$). In such case, the sequence $(\beta_n)$ is allowed to vary in the interval $[-1,1]$.
\end{remark}
In the following result we give a strongly convergent version of the Douglas–Rachford algorithm which extends \cite[Theorem~10]{Botetal(19)} (as well as the quantitative analysis from \cite[Corollary~3]{DP(ta)}). The algorithm \eqref{e:GDR} below can be seen as a generalization in the sense that for $\alpha_n\equiv 0$ and $\beta_n \equiv -1$ the sequence of even terms $(x_{2n})$ is indeed the original weakly convergent Douglas-Rachford algorithm \cite{DR(56)}.
\begin{corollary}\label{c:Cor3}
Let $U_1,U_2: H \rightrightarrows H$ be two maximal monotone operators and $c >0$. Given $(\alpha_n) \subset [0,1]$, $(\beta_n) \subset [-1,1]$, and $x_0, u \in H$, consider $(x_n)$ generated by
\begin{equation}\tag{GDR}\label{e:GDR}
\begin{cases}
x_{2n+1}&=(1-\alpha_n)x_{2n}+\alpha_nu\\
\quad y_n&=J_{c U_2}(x_{2n+1})\\
\quad z_n&=J_{c U_1}(2y_n-x_{2n+1})\\
x_{2n+2}&=x_{2n+1}+(1-\beta_n)(z_n-y_n).
\end{cases}
\end{equation}
Assume that there exist $N \in \mathbb{N} \setminus\{0\}$ such that $N \geq \max\{\norm{x_0-p},2\norm{u-p}\}$ for some $p \in \mathrm{Fix}\,(R_{cU_1}\circ R_{cU_2})$, and $\gamma \in(0,1]$ satisfying $-1+\gamma\leq \beta_n\leq 1-\gamma$, for all $n\in \mathbb{N}$.
Then,
\begin{enumerate}[$(i)$]
\item $(x_n)$ converges strongly to $\overline{x}=P_{\mathrm{Fix}\, (R_{c U_1}\circ R_{c U_2})}(u)$ and
$$\mu_3:=\mu_3[N,\gamma,\Gamma_1, \Gamma_2, \Gamma_3, \Gamma_4]:=\mu[N,\gamma/2,\Gamma_1, \Gamma_2, \Gamma_3, \Gamma_4], \, \text{ with $\mu$ as in Theorem~\ref{t:main_meta}},$$
is a rate of metastability for $(x_n)$;
\item $(y_n)$ and $(z_n)$ converge strongly to $J_{cU_2}(\overline{x})\in zer(U_1+U_2)$ and
\begin{equation*}
\mu_4(\varepsilon,f):=\mu_3(\varepsilon,2f+1), \qquad \mu_5(\varepsilon,f):=\mu_3(\varepsilon/3,2f+1),
\end{equation*}
are rates of metastability for $(y_n)$, $(z_n)$, respectively.
\end{enumerate}
\end{corollary}
\begin{proof}
It is easy to see that the iteration $(x_n)$ is generated by \eqref{e:HMaveraged} using the sequences $(\alpha_n), (\tilde{\beta}_n) \subset [0,1]$, where $\tilde{\beta}_n:=\frac{1+\beta_n}{2}$ for all $n\in \mathbb{N}$, and with the nonexpansive map $U:=R_{c U_1} \circ R_{c U_2}$. It is clear that we are still in the conditions of Theorem~\ref{t:main}, and thus we conclude that $(x_n)$ converges strongly to $\overline{x}=P_{\mathrm{Fix}\, (R_{c U_1}\circ R_{c U_2})}(u)$.
Since the condition (Q$_{\ref{Q4}}$) still holds for $(\tilde{\beta}_n)$ with the function $\Gamma_4$, and we have for all $n\in \mathbb{N}$, $\frac{\gamma}{2}\leq \tilde{\beta}_n\leq 1-\frac{\gamma}{2}$, we can apply Theorem~\ref{t:main_meta} to obtain the desired rate of metastability. This concludes part $(i)$.
We now prove part $(ii)$. From part $(i)$, the definition of $(y_n)$ and the continuity of $J_{cU_2}$, we conclude that $(y_n)$ converges strongly to $J_{cU_2}(\overline{x})$. By \cite[Proposition 26.1$(iii)(b)$]{BC(17)}, we have $zer(U_1+U_2)=J_{cU_2}[\mathrm{Fix}\,(U)]$, hence $J_{cU_2}(\overline{x})\in zer(U_1+U_2)$. From the recursive definition of $x_{2n+2}$, we see that
\[
\norm{z_n-y_n}=\frac{1}{1-\beta_n}\norm{x_{2n+2}-x_{2n+1}}\leq \frac{1}{\gamma}\norm{x_{2n+2}-x_{2n+1}}\to 0,
\]
which entails that $(z_n)$ also converges strongly to $J_{cU_2}(\overline{x})$. We now argue the rates of metastability for $(y_n)$ and $(z_n)$. For all $i,j\in\mathbb{N}$, we have
\begin{equation*}
\norm{y_i-y_j}= \norm{J_{cU_2}(x_{2i+1})-J_{cU_2}(x_{2j+1})} \leq \norm{x_{2i+1}-x_{2j+1}}
\end{equation*}
and
\begin{align*}
\norm{z_i-z_j}&= \norm{J_{cU_1}(2y_i-x_{2i+1})-J_{cU_1}(2y_j-x_{2j+1})}\\
& \leq \norm{2y_i-x_{2i+1}-(2y_j-x_{2j+1})}\\
& \leq 2\norm{y_i-y_j}+\norm{x_{2i+1}-x_{2j+1}}\\
& \leq 3\norm{x_{2i+1}-x_{2j+1}}.
\end{align*}
The result now follows from part $(i)$ and the fact that if $i\in [n,f(n)]$, then $2i+1\in [n,2f(n)+1]$.
\end{proof}
The algorithms considered by Bo\c{t}, Csetnek and Meier \cite{Botetal(19)} are the particular case of the algorithms considered in this section when $u=0$.
\section{Final remarks}\label{s:final}
We consider in this paper an algorithm which mixes Halpern and Krasnoselskii-Mann iterative definitions in an alternating way. Under appropriate conditions we show asymptotic regularity for any sequence $(x_n)$ generated by this algorithm, and construct effective rates of asymptotic regularity. We prove that $(x_n)$ is a Cauchy sequence and obtain a rate of metastability in the sense of Tao \cite{T(08b),T(08a)}. Using this result, we show that our iterative method strongly approximates a common fixed point of two nonexpansive mappings. This strong convergence result allowed us to define strongly convergent versions of the forward-backward and the Douglas-Rachford algorithms. Our results generalize recent work by Bo\c{t}, Csetnek and Meier \cite{Botetal(19)}, and Cheval and Leu\c{s}tean \cite{CK(ta)}.
In fact, the proof mining program enables, in certain instances, to obtain generalized versions of mathematical results. We briefly comment on three recent examples. The first example concerns the analysis, carried out in \cite{KLAN(21)}, of a proof of the ``Lion-Man'' game whose convergence crucially uses the compactness of the metric space together with its betweenness property. Proof mining allowed to weaken the compactness assumption to a boundedness condition, if \emph{betweenness} is upgraded to \emph{uniform betweenness} (which coincides with betweenness in the compact case anyway), resulting in a striking generalization since now the convergence holds in all bounded subsets of uniformly convex Banach spaces, $\mathrm{CAT}(\kappa)$-spaces ($\kappa>0$, where one anyhow always has a boundedness assumption), etc.\ The second example comes from \cite{KP(22)} where the authors analysed a proof that the strong convergence of the so-called viscosity generalizations of Browder and Halpern-type algorithms can be reduced to the convergence of the original Browder/Halpern algorithms in the setting of Banach spaces. The quantitative analysis allowed to generalize this reduction to the non-linear setting of hyperbolic spaces, which in turn allowed for several applications of previously known rates of metastability for these algorithms. Our final example can be found in \cite{S(ta)} where the author adapts the analysis of the strong convergence of the Halpern type Proximal Point Algorithm, given in \cite{K(20)}, from the setting of Banach spaces to $\mathrm{CAT}(0)$-spaces. This gave rise to new qualitative convergence results. The key observation is that instead of \emph{strong nonexpansivity} and corresponding SNE-moduli, one can also work with \emph{strong quasi-nonexpansivity} and corresponding SQNE-moduli (introduced in \cite{K(16)}).
Finally, let us explain how the proof of Theorem~\ref{t:main} was obtained. A proof of strong convergence can be carried out with the usual arguments in the setting of Hilbert spaces, as explained below. However, it is not clear if such arguments can be generalized to a non-linear setting. Nevertheless, a passage through quantitative results made it possible to overcome this problem.
Let us elaborate on this matter. A proof in Hilbert spaces is guided by the following steps:
\begin{enumerate}
\item[$(1)$] \underline{$(x_n)$ is bounded}: Follow the arguments of Lemma~\ref{l:bounded}.
\item[$(2)$] \underline{Asymptotic regularity of $(x_n)$}: Follow the arguments of Lemma~\ref{l:artheta} and Proposition~\ref{p:asymptoticregularity}.
\item[$(3)$] \underline{Projection argument}: With $\tilde{x}$ the projection point onto $F$ of $u$, we have $\forall y \in F \left(\langle u- \tilde{x}, y-\tilde{x}\rangle \leq 0\right)$.
\item[$(4)$] \underline{Sequential weak compactness and demiclosedness}: Pick a subsequence $(x_{n_j})$ of $(x_n)$ such that
\[
\limsup\, \langle u- \tilde{x}, x_n-\tilde{x}\rangle =\lim_{j \to \infty}\, \langle u- \tilde{x}, x_{n_j}-\tilde{x}\rangle,
\]
and simultaneously $(x_{n_j})$ converges weakly to some $y \in F$. Here we are using (twice) the following demiclosedness principle.
\begin{lemma}[Demiclosedness principle \cite{B(65)}]
Let $C$ be a closed convex subset of $H$ and let $f : C \to C$ be a nonexpansive mapping such that $\mathrm{Fix}(f)\neq \emptyset$. Assume that $(x_n)$ is a sequence in $C$ such that $(x_n)$ weakly converges to $x \in C$ and $((\mathrm{Id}_C - f)(x_n))$ converges strongly to $y \in H$. Then $(\mathrm{Id}_C - f)(x) = y$.
\end{lemma}
By step (3) it follows that $\limsup \,\langle u-\tilde{x},x_n-\tilde{x}\rangle \leq 0$.
\item[$(5)$] \underline{Main combinatorial part}: Following the inequalities culminating in \eqref{ineqProofmain} (in the proof of Theorem~\ref{t:main}), we are then able to apply Lemma~\ref{L:Xu} and conclude that $x_n \to \tilde{x}$.
\end{enumerate}
As shown in Section~\ref{s:asymptoticregularity}, steps (1) and (2) are valid in $\mathrm{CAT}(0)$ spaces. It is however not clear how the sequential weak compactness argument could be carried out in this more general context.
On the other hand, a quantitative analysis is indeed possible in Hilbert spaces. The only roadblocks to the analysis are the projection and the sequential weak compactness arguments. Yet, recent developments in proof mining \cite{FFLLPP(19)} provide a method to eliminate such arguments. The observation that this method also holds in the setting of $\mathrm{CAT}(0)$ spaces allowed us to extend the setting of the quantitative results. The argument in Section~\ref{s:Strongconvergence} brings us back to a ``qualitative'' statement, giving rise to our proof of Theorem~\ref{t:main}.
\section*{Acknowledgements}
The first author acknowledges the support of FCT - Funda\c{c}\~ao para a Ci\^{e}ncia e Tecnologia under the projects: UIDB/04561/2020 and UIDP/04561/2020, and the research center Centro de Matem\'{a}tica, Aplica\c{c}\~{o}es Fundamentais e Investiga\c{c}\~{a}o Operacional, Universidade de Lisboa.
\noindent The second author was supported by the German Science Foundation (DFG Project KO 1737/6-2).
\noindent This work benefited from discussions with Ulrich Kohlenbach.
|
2024-02-18T23:40:24.720Z
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2021-12-30T02:26:44.000Z
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{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14525","language":"en","timestamp":1640831204000,"url":"https:\/\/arxiv.org\/abs\/2112.14525","yymm":"2112"}
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proofpile-arXiv_000-10072
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{"provenance":"002.jsonl.gz:10073"}
| null | null |
\section{Introduction}
The breaking of the conformal symmetry in the Sachdev-Ye-Kitaev model \cite{Sachdev1, Kitaev1, Sachdev2, Sachdev3}
results in an effective Lagrangian for time reparametrization which is given by the Schwarzian. The supersymmetric versions of the SYK model up to ${\cal N}{=}2$ supersymmetry have been constructed and analyzed \cite{super1,super2,super3}.
However, the construction of ${\cal N}{=}3,4$ supersymmetric SYK models and associated super--Schwarzians is not straightforward,
especially in the case of ${\cal N}{=}4$ supersymmetry.
A new approach to the construction of Schwarzians and their supersymmetric extensions has been initiated in
\cite{Gal1} and then consistently applied to ${\cal N}{=}1,2,3,4$ supersymmetric cases in \cite{Gal2,Gal3,Gal4,Gal5}.
The cornerstone idea of this approach is based on the invariance of the bosonic Schwarzian
${\cal S}( t, \tau)$ defined as
\begin{equation}\label{SchwDer}
{\cal S}( t, \tau) = \frac{\dddot t}{\dot t} - \frac{3}{2} \left( \frac{\ddot t}{\dot t}\right)^2 , \;\; \dot t = \partial_\tau t,
\end{equation}
under $SL(2,\mathbb{R})$ M\"{o}bius transformations acting on $t[\tau]$ via
\begin{equation}\label{sl2a}
t \rightarrow \frac{a t +b}{c t + d} \,.
\end{equation}
The immediate consequence of this statement is the conclusion that the Schwarzian can be constructed in terms of
$sl(2,\mathbb{R})$ Cartan forms - which are essentially the unique geometric invariants of the conformal group $SL(2,\mathbb{R})$. This idea was realized in \cite{Gal1}. The straightforward generalization of this approach
to the supersymmetric cases means passing from one dimensional conformal group $SL(2,\mathbb{R})$ to its supersymmetric
extensions - the supergroups $OSp(1|2),SU(1,1|1), OSp(3|2), SU(1,1|2)$ and $D(1,2;\alpha)$. The relevant
super--Schwarzians must be invariant with respect to these supergroups and, therefore, should be constructed from the
corresponding Cartan forms.
While trying to construct the Cartan forms and the invariants from them, one may encounter two problems:
\begin{itemize}
\item One has to find a way to reduce the number of independent fields parametrizing the group element,
\item One has to understand how the invariant (super)space\footnote{We meant the super-partners of the time $\tau$ in \p{SchwDer}.} enters the game.
\end{itemize}
The approach initiated in \cite{Gal2} works perfectly in the cases of ${\cal N}{=}0,1$ supersymmetries, but it puts unreasonably strong conditions in the cases of higher supersymmetries. In our recent paper \cite{KK} we proposed the set of constraints which perfectly reproduced all known super--Schwarzians till ${\cal N}{=}4$ one, related to the supergroup $SU(1,1|2)$.
These constraints can be easily summarized as follows:
\begin{itemize}
\item For the supergroup containing the super Poincar\'{e} subalgebra $\left\{ Q_i, Q_j \right\} = 2 \delta_{ij} P$
the invariant superspace $\left\{\tau, \theta_i\right\}$ should be introduced as
$$ \omega_P = d\tau - {\rm i\,} d\theta^i \theta^i, \; \omega_Q^i = d \theta^i \qquad \qquad (a)$$
\item The unique additional constraint is
$$\omega_D=0 \qquad \qquad (b).$$
\end{itemize}
Here, $\omega_P, \omega_Q^i$ and $\omega_D$ are the Cartan forms for translation, super-translations and dilatation, respectively.
In this paper we are going to demonstrate that our approach works perfectly in the most complicated case - the super--Schwarzian associated with the most general ${\cal N}{=}4$ superconformal group $D(1,2;\alpha)$ \cite{Sorba}.
Despite the simplicity of the constraints $(a)$ and $(b)$, their application is not trivial. Indeed, one may quickly see that the straightforward calculations shortly become quite complicated and rather cumbersome. That is why we decided to use the Maurer--Cartan equations, which drastically simplify the analysis.
Thus, to be able to check the basic steps we put the main formulas in the body of the paper, transferring more technical things to the Appendices A, B and C.
\setcounter{equation}0
\section{Preliminary steps: superalgebra, Cartan forms and all that}
\subsection{Superalgebra $D(1,2;\alpha)$ }
The structure of the superalgebra $D(1,2;\alpha)$ is quite simple: it contains nine bosonic generators $T_1^{AB},T^{ij},J^{ab}$ spanning three commuting sub-algebras $sl(2) \times su(2) \times su(2)$. The eight fermionic
generators $G^{A,i,a}$ transform as the doublets with respect to each of these algebras. The anti-commutator of the
fermionic generators contains all bosonic generators as
\begin{equation}\label{alg1}
\left\{ G^{A,i,a}, G^{B,j,b}\right\} \sim \epsilon^{ab} \epsilon^{ij} T_1^{AB} +\alpha \epsilon^{AB} \epsilon^{ij} J^{ab} -(1+\alpha) \epsilon^{AB} \epsilon^{ab} T^{ij} .
\end{equation}
Here, all indices can take values $1$ or $2$, and $\epsilon^{ij}$, $\epsilon^{ab}$, $\epsilon^{AB}$ are antisymmetric symbols, normalized as $\epsilon^{21}=1$.
The parameter $\alpha$ measures the balance between two $su(2)$ subalgebras. For the two values of $\alpha=0,-1$ one of the $su(2)$ sub-algebras decouples and $D(1,2;\alpha)$ reduces to the $su(1,1|1)\times su(2)$ superalgebra. Another interesting case corresponds to $osp(4|2)$ algebra with $\alpha =-\frac{1}{2}$, when both $su(2)$ sub-algebras occur in the same way. In what follows, we exclude consideration of the cases with $\alpha=0,-1$ which can be found in \cite{KK}. Thus, we may easily divide any expressions by $\alpha$ and/or by $\alpha+1$.
From a physical point of view, the $sl(2)$ subalgebra $T_1^{AB}$ is the conformal algebra of one-dimensional space. Therefore, it is natural to introduce the generators of translation, dilatation and conformal boosts as \cite{ikl1}
\begin{equation}
P= T_1^{22}, D= - T_1^{12}, K = T_1^{11} .
\end{equation}
Correspondingly, the supercharges divide into ordinary $Q^{ia} $ and superconformal $S^{ia}$ ones, as
\begin{equation}
Q^{ia} = - G^{2ia}, \quad S^{ia} = G^{1ia} .
\end{equation}
The full list of the non-zero (anti)commutators can be found in the Appendix A.
\subsection{Cartan forms}
To obtain the $D(1,2,\alpha)$-invariant super--Schwarzian, we are going to use the method of nonlinear realizations, developed in \cite{coset11,coset12,coset21,coset22}. In the present case we need to construct a nonlinear realization of the superconformal group $D(1,2;\alpha)$ with the group element $g$ parameterized as
\begin{equation}\label{gD12}
g= e^{{\rm i\,} t P} e^{\xi_{ia} Q^{ia}} e^{\psi_{ia} S^{ia}} e^{{\rm i\,} z K} e^{{\rm i\,} u D}
e^{{\rm i\,} v_{ij} T^{ij}} e^{{\rm i\,} \phi_{ab}J^{ab}}.
\end{equation}
The Cartan forms $\Omega$ are defined in a standard way as
\begin{equation}\label{CF1}
\Omega = g^{-1} d g ={\rm i\,} \omega_D D + {\rm i\,} \omega_K K +{\rm i\,} \omega_P P + {\rm i\,} \big( \omega_J \big)_{ab}J^{ab} + {\rm i\,} \big( \omega_T \big)_{ij} T^{ij} + \big( \omega_Q \big)_{ia}Q^{ia} + \big( \omega_S \big)_{ia}S^{ia}.
\end{equation}
The Cartan forms for the scalar generators can be easily computed \footnote{ The $su(2)$ indices are raised and lowered as $A_i = \epsilon_{ij}A^j, \; A^j=\epsilon^{ji}A_i$, where the antisymmetric tensor $\epsilon^{ij}$ satisfies $\epsilon_{ij}\epsilon^{jk} = \delta_i^k$, $\epsilon_{12}=\epsilon^{21}=1$. }
\begin{eqnarray}\label{scalarCF}
\omega_P& = &e^{-u} \left( d t - {\rm i\,} d\xi_{ia} \xi^{ia} \right) \equiv e^{-u} \triangle t , \nonumber \\
\omega_D & = & d u - 2 z \triangle t + 2 {\rm i\,} \psi_{ia} d \xi^{ia}, \nonumber \\
\omega_K & = & e^u\left( d z + z^2 \triangle t - 2 {\rm i\,} z \psi_{ia} d\xi^{ia}
+ {\rm i\,} \psi_{ia} d\psi^{ia} -\frac{2}{3} (1+2\alpha) \psi_k^c \psi^{kb}\psi_b^j d\xi_{jc}+\frac{1}{6} \triangle t (1+2\alpha) \psi_{ia} \psi^{ib} \psi_{jb}\psi^{ja} \right).
\end{eqnarray}
The fermionic and $su(2)\times su(2)$ forms look more complicated\footnote{We define the matrix-valued functions
$\left( e^v\right)_i^j$ and $\left( e^\phi\right){}_a^b$ in a standard way: $n$-th term in Taylor series expansion
of $e^v$ is understood as $\frac{1}{n!} v_i^{k_1}v_{k_1}^{k_2} \ldots v_{k_{n-1}}^j$. In particular, this imply
$\left( e^v\right)_i^j = \cos\sqrt{\frac{v^2}{2}} \delta_i^j + \frac{\sin \sqrt{\frac{v^2}{2}}}{\sqrt{\frac{v^2}{2}}}v_i^j, \; v^2 = v^{ij}v_{ij}$, etc. }
\begin{eqnarray} \label{fermionCF}
\left(\omega_Q\right)_{ia} & = & e^{-\frac{u}{2}} \left( e^v\right)_i^j \left( e^\phi\right){}_a^b\left( d\xi_{jb}-\psi_{jb} \triangle t\right), \nonumber \\
\left(\omega_S\right)_{ia} & = & e^{\frac{u}{2}} \left( e^v\right)_i^k \left( e^\phi\right){}_a^c\left( d\psi_{kc} -\frac{{\rm i\,}}{2} \psi_{jb}d\xi^{jb} \psi_{kc}+ \frac{{\rm i\,}}{2} \alpha \left(\psi_{jb}d\xi^j_c+\psi_{jc}d\xi^j_b\right)\psi^b_k -\frac{{\rm i\,}}{2}(1+\alpha) \left( \psi_{jb}d\xi^b_k+\psi_{kb}d\xi^b_j\right)\psi^j_c -\right. \nonumber \\
&& \left. - \frac{{\rm i\,}}{3} (1+2\alpha) \triangle t \psi_k^b \psi_{jb}\psi^j_c - z ( d\xi_{kc}-\triangle t \psi_{kc})\right),
\end{eqnarray}
and
\begin{equation}\label{full_omega_T}
\left(\omega_T\right)_{km} =\epsilon_{kj}\left( e^{-v}\right)^j_{i}d \left(e^v\right){}^i_m + \left( e^v\right)^i_k \left( e^v\right)^j_m \left({\hat\omega}_T\right)_{ij}\; \mbox{and} \;
\left(\omega_J\right)_{ab} =\epsilon_{ad}\left( e^{-\phi}\right)^d_{c}d \left(e^\phi\right){}^c_b + \left( e^\phi\right){}^c_a \left( e^\phi\right){}^d_b \left({\hat\omega}_J\right)_{cd} ,
\end{equation}
where
\begin{eqnarray}\label{su2CAF}
{\rm i\,} \left({\hat\omega}_T\right)_{ij} & = & (1+\alpha) \left( - \psi_{ib} \psi^b_j \triangle t+\psi_{ia} d\xi^a_j+\psi_{ja} d \xi^a_i\right), \nonumber \\
{\rm i\,} \left({\hat\omega}_J\right)_{ab} & = & \alpha \left( \psi_{ja} \psi^j_b \triangle t - \psi_{ja} d \xi^j_b- \psi_{jb}d\xi^j_a\right).
\end{eqnarray}
\subsection{Constraints and their consequences}
As we already said in the Introduction, the constraints we have to impose on the Cartan forms to find a proper
super--Schwarzian read
\begin{eqnarray}
&& \omega_P = d\tau -{\rm i\,} d\theta_{ia} \theta^{ia} \equiv \triangle \tau, \quad \omega_Q^{ia} = d\theta^{ia}, \label{a} \\
&& \omega_D = 0 . \label{b}
\end{eqnarray}
Note that the covariant derivatives with respect to $\tau$, $\theta_{ia}$ are
\begin{equation}\label{covders}
\partial_\tau = \frac{\partial}{\partial \tau}, \;\; D^{ia} = \frac{\partial}{\partial \theta_{ia}} + {\rm i\,} \theta^{ia} \frac{\partial}{\partial \tau}, \;\; \big\{ D^{ia}, D^{jb} \big\} = 2{\rm i\,} \epsilon^{ij}\epsilon^{ab}\partial_\tau.
\end{equation}
Therefore, for any superfield ${\cal A}$ we have
\begin{equation}
d {\cal A} = \triangle \tau \; \partial_\tau {\cal A} + d\theta^{ia} D_{ia} {\cal A}.
\end{equation}
With our definitions of the Cartan forms \p{scalarCF}, \p{fermionCF}, \p{full_omega_T} the constraints \p{a} and \p{b} read
\begin{eqnarray}
&& \omega_P = e^{-u}\big( dt -{\rm i\,} d\xi_{jb}\xi^{jb} \big) = e^{-u} \triangle t = \triangle \tau , \quad
\big(\omega_Q\big)_{ia} = e^{-u/2}\big( e^v \big)_i^j \big( e^\varphi \big)_a^b \big( d\xi_{jb} - \triangle t \psi_{jb} \big) = d\theta_{ia} \label{a1} \\
&& \omega_D = d u - 2 z \triangle t + 2 {\rm i\,} \psi_{ia} d \xi^{ia} =0. \label{b1}
\end{eqnarray}
The constraints \p{a1} imply
\begin{eqnarray}
&& D^{ia}t - D^{ia}\xi_{jb}\xi^{jb} =0, \;\; \dot t - {\rm i\,} \dot\xi_{ia}\xi^{ia} = e^u, \label{a21} \\
&& D^{jb}\xi_{ia} = e^{u/2}\big( e^{-v} \big)_i^j \big( e^{-\varphi} \big)_a^b, \;\; \psi_{ia} = e^{-u} \dot\xi_{ia},
\label{a22}
\end{eqnarray}
while the constraints \p{b1} are resolved by the following relations
\begin{equation}\label{b2}
z = \frac{1}{2} e^{-u} \dot{u}, \quad D_{ia}u = 2 {\rm i\,} \psi_{jb} D_{ia} \xi^{jb} .
\end{equation}
We observe that it is possible to covariantly express some of the group parameters in terms of other parameters and their derivatives. This a manifestation of the Inverse Higgs phenomenon \cite{ih}. Note that these constraints involve both $d\tau$ and $d\theta$ projections of the forms, unlike the constraints considered in \cite{Gal5}. This allows, in particular, to express $z$ in terms of $u$ (or $\xi^{ia}$) without putting any constraint on the forms of the superconformal generators.
As a result of \p{a22} and the following identities
$$ \epsilon^{ik} \epsilon_{jl} \left( e^v\right)_k^l = - \left(e^{-v}\right)^i_j, \quad \mbox{and} \quad
\epsilon^{ac} \epsilon_{bd} \left( e^\phi\right)_c^d = - \left(e^{-\phi}\right)^a_b ,$$
$D^{ia}\xi_{jb}$ satisfies the relations
\begin{equation}\label{a3}
D_{jb}\xi^{ia} = e^{u/2} \big( e^{v} \big)_j^i \big( e^{\varphi} \big)_b^a \;\; \Rightarrow \;\; D^{ia}\xi_{kc} D_{jb}\xi^{kc} = \delta^i_j \delta^a_b e^u, \;\; D^{kc}\xi_{jb} D_{kc}\xi^{ia} = \delta^i_j \delta^a_b e^u,
\end{equation}
and, moreover,
\begin{equation}\label{a4}
D_{kc}\xi^{ia}\, D^{kb}\xi_{ld} = \frac{1}{2}\delta_l^i \, D_{kc}\xi^{ma}D^{kb}\xi_{md}, \;\; \mbox{etc.}
\end{equation}
Thus, we see that all our superfields - coordinates of the group element $g$ \p{gD12} can be expressed through the
derivatives of the superfields $\xi_{ia}$, only \footnote{The superfield $t$ can be in principle found from the equations \p{a21}.}. In principle, it should be the end of the story and the technical step is to find among the
components of the surviving Cartan forms the super--Schwarzian. Unfortunately, this technical step is too involved and the direct straightforward calculations quickly become a rather cumbersome. The simplest solution we found is to use the Maurer--Cartan equations to rewrite the Cartan forms with the constraints \p{a} and \p{b} taken into account.
\setcounter{equation}0
\section{${\cal N}{=}4$ super--Schwarzian}
\subsection{Maurer--Cartan equations}
If the Cartan form $\Omega$ is defined as in \p{CF1}
$$
\Omega(d) = g^{-1} d g = {\rm i\,} \omega_D D + {\rm i\,} \omega_K K +{\rm i\,} \omega_P P + {\rm i\,} \big( \omega_J \big)_{ab}J^{ab} + {\rm i\,} \big( \omega_T \big)_{ij} T^{ij} + \big( \omega_Q \big)_{ia}Q^{ia} + \big( \omega_S \big)_{ia}S^{ia},
$$
then by construction it obeys the Maurer--Cartan equation. We prefer to deal with this equation in the form used in \cite{OSP14}. There, two independent differentials $d_1$, $d_2$ were introduced, so that $d_1 d_2=d_2 d_1$ and differentials of bosons and fermions are bosons and fermions, respectively. Therefore, the relation
\begin{equation}\label{MaurerCartan}
d_2\Omega_1 - d_1 \Omega_2 = \big[ \Omega_1, \Omega_2 \big], \;\; \Omega_1 = \Omega(d_1), \;\; \Omega_2 = \Omega(d_2)
\end{equation}
turns into identity upon substitution $\Omega_i = g^{-1}d_i g$. At the same, one can substitute $\Omega_i$ as a general expansion in generators \p{CF1} and find the list of relations the individual forms satisfy. In the case of $D(1,2;\alpha)$ equation \p{MaurerCartan} can be expanded into following set of equations \footnote{Round brackets are used to denote symmetrization of indices, $A_{(ij)} = \frac{1}{2}\big( A_{ij} + A_{ji} \big)$}
\begin{eqnarray}\label{MCexp}
{\rm i\,} \big( d_2 \omega_{1P} - d_1 \omega_{2P} \big) &=& -{\rm i\,} \big[ \omega_{2D}\omega_{1P} - \omega_{1D}\omega_{2P} \big] +2\big(\omega_{1Q}\big)_{ia}\big(\omega_{2Q}\big)^{ia}, \nonumber \\
{\rm i\,} \big( d_2 \omega_{1K} - d_1 \omega_{2K} \big) &=& {\rm i\,} \big[\omega_{2D}\omega_{1K} - \omega_{1D}\omega_{2K} \big] +2\big(\omega_{1S}\big)_{ia}\big(\omega_{2S}\big)^{ia}, \nonumber \\
{\rm i\,} \big( d_2 \omega_{1D} - d_1 \omega_{2D} \big) &=& -2{\rm i\,} \big[ \omega_{2K}\omega_{1P} - \omega_{1K}\omega_{2P} \big] + 2\big[ \big(\omega_{1Q}\big)_{ia}\big(\omega_{2S}\big)^{ia} - \big(\omega_{2Q}\big)_{ia}\big(\omega_{1S}\big)^{ia} \big], \\
{\rm i\,} \big( d_2 \big( \omega_{1J} \big)_{ab} - d_1 \big( \omega_{2J} \big)_{ab} \big) &=& {\rm i\,} \big[ \big( \omega_{1J} \big)_{ac} \big(\omega_{2J} \big)_{b}{}^c - \big(\omega_{2J} \big)_{ac} \big(\omega_{1J} \big)_{b}{}^c \big] -2\alpha \big[ \big( \omega_{1Q} \big)_{i(a} \big( \omega_{2S} \big)_{b)}^i - \big( \omega_{2Q} \big)_{i(a} \big( \omega_{1S} \big)_{b)}^i \big], \nonumber \\
{\rm i\,} \big( d_2 \big( \omega_{1T} \big)_{ij} - d_1 \big( \omega_{2T} \big)_{ij} \big) &=& {\rm i\,} \big[ \big(\omega_{1T} \big)_{ik} \big(\omega_{2T} \big)_{j}{}^k - \big(\omega_{2T} \big)_{ik} \big(\omega_{1T} \big)_{j}{}^k \big] +2(1+\alpha) \big[ \big( \omega_{1Q} \big)_{(ia} \big( \omega_{2S} \big)_{j)}^a - \big( \omega_{2Q} \big)_{(ia} \big( \omega_{1S} \big)_{j)}^a\big], \nonumber \\
d_2 \big( \omega_{1Q} \big)_{ia} - d_1 \big( \omega_{2Q} \big)_{ia} &=& -\frac{1}{2} \big[ \omega_{2D} \big( \omega_{1Q} \big)_{ia} -\omega_{1D} \big( \omega_{2Q} \big)_{ia} \big] +\big[\big( \omega_{1J}\big)_{ab}\big( \omega_{2Q} \big)_i^b - \big( \omega_{2J}\big)_{ab}\big( \omega_{1Q} \big)_i^b \big]+\nonumber \\&& + \big[ \big( \omega_{1T}\big)_{ij}\big( \omega_{2Q} \big)_a^j - \big( \omega_{2T}\big)_{ij}\big( \omega_{1Q} \big)_a^j \big] - \big[ \omega_{1P} \big( \omega_{2S} \big)_{ia} - \omega_{2P} \big( \omega_{1S} \big)_{ia} \big], \nonumber \\
d_2 \big( \omega_{1S} \big)_{ia} - d_1 \big( \omega_{2S} \big)_{ia} &=& \frac{1}{2} \big[ \omega_{2D} \big( \omega_{1S} \big)_{ia} -\omega_{1D} \big( \omega_{2S} \big)_{ia} \big] +\big[\big( \omega_{1J}\big)_{ab}\big( \omega_{2S} \big)_i^b - \big( \omega_{2J}\big)_{ab}\big( \omega_{1S} \big)_i^b \big]+\nonumber \\&& + \big[ \big( \omega_{1T}\big)_{ij}\big( \omega_{2S} \big)_a^j - \big( \omega_{2T}\big)_{ij}\big( \omega_{1S} \big)_a^j \big] + \big[ \omega_{1K} \big( \omega_{2Q} \big)_{ia} - \omega_{2K} \big( \omega_{1Q} \big)_{ia} \big]. \nonumber
\end{eqnarray}
The forms should be subjected to the conditions
\begin{equation}\label{maincond}
\omega_P = \triangle \tau, \;\; \big(\omega_{Q}\big)_{ia} = d\theta_{ia}, \;\; \omega_D =0, \;\; \triangle\tau = d\tau - {\rm i\,} d\theta_{jb}\theta^{jb}.
\end{equation}
To analyze the consequences of these constraints let us represent other forms in most general way as
\begin{eqnarray}\label{strsf}
\big( \omega_S \big)_{ia} &=& \triangle\tau \Psi_{ia} + d\theta_{jb}A_{ia}{}^{jb}, \;\; \omega_K = \triangle\tau C + d\theta_{ia}\Gamma^{ia}, \nonumber \\
\big(\omega_J \big)_{ab} &=& \triangle\tau \big( {\cal S}_J \big)_{ab} + d\theta_{kc}\Sigma_{(ab)}{}^{kc}, \;\; \big(\omega_T \big)_{ij} = \triangle\tau \big( {\cal S}_T \big)_{ij} + d\theta_{kc}\Pi_{(ij)}{}^{kc}.
\end{eqnarray}
Here $\Psi_{ia}, A_{ia}{}^{jb}, C, \Gamma^{ia}, \big( {\cal S}_J \big)_{ab}, \Sigma_{(ab)}{}^{kc},\big( {\cal S}_T \big)_{ij}$ and $\Pi_{(ij)}{}^{kc}$ are superfields that depend on $\tau, \theta_{ia}$.
The first of equations, $d\omega_P$ in \p{MCexp}, is satisfied identically due to the condition \p{maincond}.
Indeed, the left hand side of the first equation in \p{MCexp} reads
\begin{equation}\label{lhs1}
{\rm i\,} \big( d_2 \omega_{1P} - d_1 \omega_{2P} \big)={\rm i\,}( -{\rm i\,} d_1\theta_{ia} d_2 \theta^{ia} +{\rm i\,} d_2\theta_{ia} d_1 \theta^{ia} ) =2 d_1\theta_{ia} d_2 \theta^{ia}.
\end{equation}
Clearly, \p{lhs1} coincides with
\begin{equation}
\label{rhs1}
2\big(\omega_{1Q}\big)_{ia}\big(\omega_{2Q}\big)^{ia} = 2 d_1\theta_{ia} d_2 \theta^{ia} .
\end{equation}
The analysis of other Maurer--Cartan equations in \p{MCexp} is straightforward, but it is rather involved.
These technical calculations are presented in the Appendix C. The result of these analysis can be summarized as follows:
the Cartan forms can be expressed through the fermionic superfield $\sigma^{ia}$ as
\begin{eqnarray}
&& \omega_P = \triangle \tau, \quad \omega_D=0,\quad \omega_K = \triangle \tau C + {\rm i\,} d\theta_{ia} \Psi^{ia}, \label{eq1}\\
&& \left(\omega_J\right)_{ab} = \triangle \tau \left({\cal S}_J\right)_{ab}+\frac{1}{3}\left( d\theta_{ka} \sigma^k_b+
d\theta_{kb} \sigma^k_a \right),\label{eq2} \\
&& \left(\omega_T\right)_{ij} = \triangle \tau \left({\cal S}_T\right)_{ij}-\frac{1}{3}\left( d\theta_{ic} \sigma^c_j+
d\theta_{jc} \sigma^c_i \right), \label{eq3}\\
&& \left(\omega_Q\right)_{ia} = d \theta_{ia}, \quad
\left(\omega_S\right)_{ia} = \triangle \tau \Psi_{ia} -d\theta_{ib} \left({\cal S}_J\right)^b_{a}
-d\theta_{ka} \left({\cal S}_T\right)^k_{i} , \label{eq4}
\end{eqnarray}
where the superfields $C, \Psi^{ia}, ({\cal S}_T)_{ij} ,({\cal S}_J)_{ab}$ have the form
\begin{eqnarray}\label{formssol}
\left( {\cal S}_J\right)^{ab} & = & \frac{{\rm i\,}}{1+\alpha} \left[\frac{1}{12} \left( D^{ka}\sigma_k^b+D^{kb}\sigma_k^a\right)-
\frac{1}{9} \sigma^a_m \sigma^{mb}\right], \nonumber \\
\left( {\cal S}_T\right)^{ij} & = & \frac{{\rm i\,}}{\alpha} \left[\frac{1}{12} \left( D^{ic}\sigma_c^j+D^{jc}\sigma_c^j\right)+
\frac{1}{9} \sigma^i_c \sigma^{jc}\right], \nonumber \\
\Psi_a^k & = & \frac{{\rm i\,}}{3 \alpha} \left[ D^{kb} \left({\cal S}_J\right)_{ab} +\dot{\sigma}^k_a +\frac{4}{3} \left( {\cal S}_J\right)_a^d \sigma_d^k\right], \nonumber \\
C & = & \frac{1}{4}\left[ D^{ia}\Psi_{ia} + 2 \left( {\cal S}_J\right)^2+2 \left( {\cal S}_T\right)^2 \right].
\end{eqnarray}
In addition, the superfield $\sigma^{ia}$ in virtue of the same constraints \p{maincond} has to obey the following conditions
\begin{equation}\label{sigmaconstr}
\frac{1}{2} \left[ D^{ia}, D^{jb}\right] \sigma_{jb} = 3 {\rm i\,} \dot{\sigma}^{ia}, \quad
D^{i(a} \sigma^{jb)} + D^{j(a} \sigma^{ib)} =0.
\end{equation}
Clearly, the fermionic superfield $\sigma^{ia}$ is a candidate for the super--Schwarzian. The final step is to express $\sigma^{ia}$ in terms of $\xi^{ia}$.
\subsection{The super--Schwarzian}
To find the explicit expression for the super--Schwarzian, one should calculate the $d\theta$-projections of $\omega_T$, $\omega_J$ forms, taking into account explicit consequences of conditions \p{maincond}. Expanding \p{maincond} into $\triangle\tau$ and $d\theta$ projections, one can find
\begin{eqnarray}\label{maincondcons}
\omega_P = e^{-u}\big( dt -{\rm i\,} d\xi_{jb}\xi^{jb} \big) = \triangle \tau \;\; \Rightarrow \;\; D^{ia}t - D^{ia}\xi_{jb}\xi^{jb} =0, \;\; \dot t - {\rm i\,} \dot\xi_{ia}\xi^{ia} = e^u, \nonumber \\
\big(\omega_Q\big)_{ia} = e^{-u/2}\big( e^v \big)_i^j \big( e^\varphi \big)_a^b \big( d\xi_{jb} - \triangle t \psi_{jb} \big) = d\theta_{ia}\;\; \Rightarrow \;\; D^{jb}\xi_{ia} = e^{u/2}\big( e^{-v} \big)_i^j \big( e^{-\varphi} \big)_a^b, \;\; \psi_{ia} = e^{-u} \dot\xi_{ia}.
\end{eqnarray}
As a result of \p{maincondcons}, $D^{ia}\xi_{jb}$ satisfies relations
\begin{equation}\label{Dxirels1}
D_{jb}\xi^{ia} = e^{u/2} \big( e^{v} \big)_j^i \big( e^{\varphi} \big)_b^a \;\; \Rightarrow \;\; D^{ia}\xi_{kc} D_{jb}\xi^{kc} = \delta^i_j \delta^a_b e^u, \;\; D^{kc}\xi_{jb} D_{kc}\xi^{ia} = \delta^i_j \delta^a_b e^u,
\end{equation}
and, moreover,
\begin{equation}\label{Dxirels2}
D_{kc}\xi^{ia}\, D^{kb}\xi_{ld} = \frac{1}{2}\delta_l^i \, D_{kc}\xi^{ma}D^{kb}\xi_{md}, \;\; \mbox{etc.}
\end{equation}
Using these relations, it is possible to find $D_{ld}e^u$
\begin{equation}\label{expu}
e^u = \frac{1}{4} D^{ia}\xi_{kc} D_{ia}\xi^{kc} \;\; \Rightarrow \;\; D_{ld}e^u = 2{\rm i\,} \dot\xi{}_{ia}D_{ld}\xi^{ia}.
\end{equation}
The super--Schwarzian $\sigma_{ia}$ can be obtained as a $d\theta$ -projection of either the forms $\omega_T$ or $\omega_J$. For example, $T$ part of the Cartan form reads
\begin{equation}\label{omegaTschw}
{\rm i\,} \big( \omega_T \big)_{ij} T^{ij} = {\rm i\,} \triangle\tau \big( {\cal S}_T \big)_{ij} T^{ij} -\frac{2{\rm i\,}}{3} T^{ij}d\theta_{ia}\sigma^a_j = -{\rm i\,} T_k{}^m \left( e^{-v}\right)_i^k d\left(e^v\right)_m^i + {\rm i\,} T^{km} \big( e^v \big)_k^i \big( e^v \big)_m^j \left(\hat\omega_T\right)_{ij},
\end{equation}
where $\omega_{ij}$ is given by \p{su2CAF}.
To obtain $d\theta$ -projection of $\left( e^{-v}\right)_i^k d\left(e^v\right)_m^i$, one should note that due to \p{maincondcons},
\begin{equation}\label{DDxiDxi}
D_{ld}D_{kc}\xi^{ia} D^{jb}\xi_{ia} = \frac{1}{2}\delta_k^j \delta_c^b D_{ld}e^u + e^u \delta_c^b \big( e^{-v} \big)_i^j D_{ld} \big(e^{v}\big)_k^i + e^u \delta_j^k \big( e^{-\varphi} \big)_a^b D_{ld} \big( e^\varphi \big)_c^a.
\end{equation}
Substituting this into relation
\begin{equation}\label{DDxiDxi2}
D_{ld}D_{kc}\xi^{ia} D^{jb}\xi_{ia} = \big\{ D_{kc},D_{ld} \big\}\xi^{ia} D^{jb}\xi_{ia} - D_{kc}D_{ld}\xi^{ia} D^{jb}\xi_{ia}
\end{equation}
and taking trace over $c$, $b$, one can find
\begin{eqnarray}\label{DDxiDx3}
2 e^u \big( e^{-v} \big)_i^j D_{ld} \big( e^v \big)_k^i + e^u\big( e^{-v} \big)_i^j D_{kd} \big( e^v \big)_l^i &=& -2{\rm i\,} \delta^j_k \dot\xi_{ia}D_{ld}\xi^{ia} -{\rm i\,} \delta^j_l \dot\xi_{ia}D_{kd}\xi^{ia} -\nonumber \\&&-2{\rm i\,} \epsilon_{kl} \dot\xi_{ia}D^j_{d}\xi^{ia}- \delta_l^j e^u \big( e^{-\varphi}\big)_a^b D_{kb}\big( e^\varphi \big)_d^a.
\end{eqnarray}
Therefore,
\begin{eqnarray}\label{DDxiDx4}
e^u \big( e^{-\varphi} \big)_a^b D_{kb} \big( e^\varphi \big)_d^a = - e^u \big( e^{-v} \big)_i^j D_{jd} \big( e^v \big)_k^i -3{\rm i\,} \dot\xi_{ia} D_{kd}\xi^{ia}, \nonumber \\
e^u \big( e^{-v} \big)_i^j D_{ld} \big( e^v \big)_k^i = \frac{2}{3} \delta_l^j e^u \big( e^{-v} \big)_n^m D_{md} \big( e^v \big)_k^n - \frac{1}{3} \delta_k^j e^u \big( e^{-v} \big)_n^m D_{md} \big( e^v \big)_l^n, \nonumber \\
e^u \big( e^{-v} \big)_n^m D_{md} \big( e^v \big)_k^n = \frac{1}{2}D_{jd}D_{kb}\xi^{ia}\, D^{jb}\xi_{ia} - {\rm i\,} \dot\xi^{ia}D_{kd}\xi_{ia}.
\end{eqnarray}
The rest of the form reads
\begin{eqnarray}\label{omegaTrest}
&&-d\theta^{ld}(1+\alpha)T^{km}\big( e^v \big)_k^i \big( e^v \big)_m^j \left( \psi_{ia} D_{ld} \xi^a_j + \psi_{ja} D_{ld} \xi^a_i \right) =\nonumber \\
&&= 2(1+\alpha)T^{km} d\theta_{kd} \dot\xi^a_i \big( e^{-\varphi} \big)_a^d \big( e^v \big)_k^i e^{-u/2} = 2(1+\alpha)T^{km}d\theta_{kd} e^{-u} {\dot\xi}^{ia}D^d_m \xi_{ia}.
\end{eqnarray}
These results ensure that $d\theta$-projection of the form $\omega_T$ has the structure \p{omegaJT}, with $\sigma_{kd}$ being
\begin{eqnarray}\label{sigmaT}
\sigma_{dk} = \frac{1}{4} e^{-u} \left[ D_{jd}, D_{ck}\right] \xi^{ia} D^{jc}\xi_{ia} +\frac{3}{2} {\rm i\,} \left(1+ 2\alpha\right) e^{-u} {\dot\xi}^{ia} D_{dk}\xi_{ia}.
\end{eqnarray}
Analogous study of form $\omega_J$
\begin{equation}\label{omegaTschw}
{\rm i\,} \big( \omega_J \big)_{ab} J^{ab} = {\rm i\,} \triangle\tau \big( {\cal S}_J \big)_{ab} J^{ab} +\frac{2{\rm i\,}}{3} J^{ab}d\theta_{ia}\sigma^i_b = -{\rm i\,} J_c{}^d \left( e^{-\varphi}\right)_a^c d\left(e^\varphi\right)_d^a + {\rm i\,} J^{cd} \big( e^{\varphi} \big)_c^a \big( e^\varphi \big)_d^b {\hat\omega}_{ab},
\end{equation}
leads to the same expression \p{sigmaT}.
Thus, we see that all the Cartan forms expressed through the fermionic superfield $\sigma_{ai}$ \p{sigmaT}.
We associate this field with ${\cal N}{=}4$ super--Schwarzian we are looking for:
\begin{equation}
{\cal S}(\tau, \theta)_{ia} = \frac{\left[ D_{ja}, D_{ci}\right] \xi^{kb} D^{jc}\xi_{kb}}{D^{md} \xi_{ne} D_{md}\xi^{ne}}+
6 {\rm i\,} \left(1+ 2\alpha\right) \frac{ {\dot\xi}^{dk} D_{ia}\xi_{dk}}{D^{md} \xi_{ne} D_{md}\xi^{ne}}.
\end{equation}
\section{${\cal N}=4$ Schwarzian action}
Like the previously considered cases \cite{KK}, one may ask whether the superfield Schwarzian action, which provides the $D(1,2;\alpha)$ - invariant generalization of the bosonic Schwarzian action,
\begin{equation}\label{bosact}
S = - \frac{1}{2} \int d\tau {\cal S}(t,\tau) = - \frac{1}{2}\int d\tau \left( \frac{\dddot t}{\dot t} - \frac{3}{2} \left( \frac{\ddot t}{\dot t}\right)^2 \right),
\end{equation}
could be constructed. As is shown in the Section $3$ and the Appendix C, the Maurer--Cartan equations imply that the only superfields, invariant with respect to $D(1,2;\alpha)$ group transformations, are the super--Schwarzian $\sigma^{ia}$ and its derivatives. Therefore, it would be natural to expect that the superfield action is some integral of $\sigma^{ia}$ over the part of superspace. Indeed, let us show that the expression
\begin{equation}\label{N4actcand}
S = -\frac{1}{72}\int d\tau D^{kc}D_{lc}D_{kb}\sigma^{lb} |_{\theta \rightarrow 0}
\end{equation}
is invariant with respect to ${\cal N}=4$ supersymmetry, realized on superspace coordinates $\tau$ and $\theta_{ia}$ as
\begin{equation}\label{tauthetavar}
\delta \tau = -{\rm i\,} \epsilon^{ia}\theta_{ia}, \;\; \delta\theta_{ia} = \epsilon_{ia}, \;\; \delta\triangle\tau=0, \;\; \delta d\theta_{ia} =0.
\end{equation}
The active variation of any superfield $f$ with respect to transformations \p{tauthetavar} is given by the formula
\begin{equation}\label{sfvar}
\delta^\star f = -\delta \tau \frac{\partial f}{\partial \tau} -\delta\theta_{ia} \frac{\partial f}{\partial \theta_{ia}} \equiv - \epsilon_{ia} {\widehat Q}{}^{ia} f,\;\; {\widehat Q}{}^{ia} = \frac{\partial}{\partial \theta_{ia}} - {\rm i\,} \theta^{ia} \frac{\partial}{\partial \tau}.
\end{equation}
It can be straightforwardly shown that the differential operator ${\widehat Q}{}^{ia}$ anticommutes, as expected, with the covariant derivative $D^{jb}$ and differs from it by the sign of the $\theta\partial_\tau$ -term. Therefore,
\begin{equation}\label{N4actcandvar1}
\delta^\star S = \frac{1}{72}\int d\tau D^{kc}D_{lc}D_{kb}\epsilon_{ia}{\widehat Q}{}^{ia} \sigma^{lb} |_{\theta \rightarrow 0} = \frac{1}{72}\epsilon_{ia}\int d\tau {\widehat Q}{}^{ia} D^{kc}D_{lc}D_{kb} \sigma^{lb} |_{\theta \rightarrow 0}.
\end{equation}
As after applying differential operators on $\sigma^{ia}$ we take limit $\theta \rightarrow 0$, the $\theta\partial_\tau$ -term in ${\widehat Q}{}^{ia}$ is irrelevant, and ${\widehat Q}{}^{ia}$ can be replaced with $D^{ia}$. Therefore
\begin{equation}\label{N4actcandvar2}
\delta^\star S = \frac{1}{72}\epsilon_{ia}\int d\tau D^{ia} D^{kc}D_{lc}D_{kb} \sigma^{lb} |_{\theta \rightarrow 0}= \frac{1}{72}\epsilon_{ia}\int d\tau \Big[ 2{\rm i\,} D^{ia}D^{kc}{\dot\sigma}_{kc} - {\rm i\,} D^a_l \big( D^{ib}{\dot\sigma}_b^l + D^{lb}{\dot\sigma}_b^i \big) \Big]|_{\theta \rightarrow 0}=0,
\end{equation}
where the expression for $D^{ia} D^{kc}D_{lc}D_{kb} \sigma^{lb}$ is a consequence of the constraint \p{sigmaconstr}.
The supersymmetry invariant integral over $d\tau$ can be presented as an integral over part of the superspace:
\begin{equation}\label{N4actfin}
S = \frac{1}{72}\int d\tau d\theta^{kc}d\theta_{lc}d\theta_{kb}\sigma^{lb}.
\end{equation}
One can also evaluate the component form of this action. Simplest way to do so is to observe that the $\triangle \tau$ projection of the form $\omega_K = \triangle\tau C + \ldots$ \p{formssol} contains third derivative of $\sigma^{ia}$ (this expression can be found in the Appendix C \p{C}). Comparing this with the projection which can be obtained directly from \p{scalarCF} after applying all the necessary conditions, one can obtain that
\begin{eqnarray}\label{N4actcomp}
S = -\frac{1}{72}\int d\tau D^{kc}D_{lc}D_{kb}\sigma^{lb} |_{\theta \rightarrow 0} = -\int d\tau \left[ \frac{1}{2}\alpha (1+\alpha) \left( \frac{\partial_\tau \big( \ddot t - {\rm i\,} \ddot\xi_{ia}\xi^{ia} \big)}{\dot t - {\rm i\,} \dot\xi_{jb}\xi^{jb}} - \frac{3}{2} \frac{\big( \ddot t - {\rm i\,} \ddot\xi_{ia}\xi^{ia} \big)^2 }{ \big( \dot t - {\rm i\,} \dot\xi_{jb}\xi^{jb} \big)^2}\right) \right. + \nonumber \\+\left. {\rm i\,} \alpha(1+\alpha) \frac{\dot\xi_{ia}{\ddot\xi}{}^{ia}}{\dot t - {\rm i\,} \dot\xi_{jb}\xi^{jb}} - \frac{1}{2}\alpha \partial_\tau \big( e^{-v} \big)_l^i \partial_\tau \big( e^{v} \big)_i^l + \frac{1}{2}(1+\alpha) \partial_\tau \big( e^{-\phi} \big)_d^a \partial_\tau \big( e^{\phi} \big)_a^d \right. - \nonumber \\ \left. -{\rm i\,} \alpha(1+\alpha) \frac{\big( e^{-v} \big)_k^j \partial_\tau \big( e^v \big)_j^l \dot\xi_{lb}\dot\xi{}^{kb}}{\dot t - {\rm i\,} \dot\xi_{ia}\xi^{ia} } -{\rm i\,}\alpha(1+\alpha) \frac{\big( e^{-\phi} \big)_c^b \partial_\tau \big( e^\phi \big)_b^d \dot\xi_{jd}\dot\xi{}^{jc}}{\dot t - {\rm i\,} \dot\xi_{ia}\xi^{ia} } - \frac{{\rm i\,}}{9} \dot\sigma{}^{kc}\sigma_{kc} - \frac{1}{9}F^2 \right].
\end{eqnarray}
Here, we denote the first component of each superfield with the same letter, and $4F = D^{mc}\sigma_{mc}|_{\theta\rightarrow 0}$. Note that the first component of the super--Schwarzian $\sigma^{ia}$ can be treated as an independent one, as $D_{ia}D_{jb}\xi^{kc}$ can not be expressed in terms of time derivatives of anything else. The same applies to $D^{mc}\sigma_{mc}$, too.
The action \p{N4actcomp} is invariant with respect to the whole $D(1,2;\alpha)$ group for general $\alpha$ and should contain $SU(1,1|2)$ case, which corresponds to $\alpha =0$ or $-1$, as a particular limit. However, simply setting $\alpha =0$ or $-1$ in \p{N4actcomp} would remove the most important terms in the action. To take the limit properly, we should, at first, ``renormalize'' the action by dividing it by $\alpha(1+\alpha)$, thus removing $\alpha$ dependence from the most of the terms. Secondly, one should remove $\sigma_{kc}|_{\theta\rightarrow 0}$ and $F$ by their equations of motion, $\dot\sigma{}^{ia}|_{\theta\rightarrow 0} =0$ and $F=0$. Thirdly, one should set to zero $\phi^{ab}$, if limit $\alpha\rightarrow 0$ is to be taken, or $v^{ij}$ if $\alpha\rightarrow -1$. Then the action becomes nonsingular in $\alpha$ and after taking the appropriate limit coincides with one obtained in \cite{KK}. Note that somewhat confusing difference in signs of kinetic terms of $v^{ij}$ and $\phi^{ab}$ allows to obtain proper sign of the kinetic term of the remaining field in the $SU(1,1|2)$ action for $\alpha=0$ and $\alpha=-1$.
\setcounter{equation}0
\section{Conclusion}
In this work we applied the method of nonlinear realizations to the construction of the ${\cal N}{=}4$ super--Schwarzian
associated with the $D(1,2;\alpha)$ conformal group. As compared to the previous attempt to utilize the nonlinear realizations for construction of the ${\cal N}{=}4$ super--Schwarzians \cite{Gal5} we successfully used the minimal set
of the constraints on the Cartan forms advocated in \cite{KK}:
\begin{itemize}
\item For the superalgebra containing the super Poincar\'{e} subalgebra $\left\{ Q{}^{ia}, Q{}^{jb} \right\} = -2\epsilon^{ij}\epsilon^{ab} P$ the invariant super-space $\left\{\tau, \theta_{ia}\right\}$ defined as
$$ \omega_P = d\tau - {\rm i\,} d\theta^{ia} \theta_{ia}, \; \big(\omega_Q\big){}_{ia} = d \theta{}_{ia} \qquad \qquad (a)$$
\item The unique additional constraint has to be imposed on the Cartan form for dilatation
$$\omega_D=0 \qquad \qquad (b).$$
\end{itemize}
From the general structure of the Cartan forms upon imposing the constraints (a,b), it follows that the fermionic components of the forms in \p{eq1}, \p{eq2}, \p{eq3} and \p{eq4} are quite nontrivial.
Therefore, any constraint would be imposed on these forms will result in the constraints on the super--Schwarzian $\sigma_{ia}$. That is why our minimal set of the constraints is the maximally possible one.
We also demonstrated that the Maurer--Cartan equations greatly simplified all calculations helping to express all Cartan forms in terms of the single object --- ${\cal N}{=}4$ super--Schwarzian. However, to find the expression of the ${\cal N}{=}4$ super--Schwarzian in terms of the basic superfields one has to again use all set of constraints.
We are planning to apply the proposed approach to ${\cal N}$-extended superconformal group including the variant of $OSp(n|2)$ superconformal symmetry. Another interesting problem is to obtain non-relativistic and/or Carrollian versions of the Schwarzian \cite{gomis}, as well as to the flat space analogue of the Schwarzian \cite{FlatSch}.
\section*{Acknowledgements}
The work was supported by Russian Foundation for Basic Research, grant
No~20-52-12003.
\setcounter{equation}0
\defC.\arabic{equation}{A.\arabic{equation}}
\section*{Appendix A. Superalgebra $D(1,2;\alpha)$}
The set of the generators spanning $D(1,2;\alpha)$ superalgebra includes
\begin{eqnarray}
\mbox{ Bosonic generators: } && P,D,K - \mbox{ forming } sl(2) \mbox{ algebra} \nonumber \\
&& \mbox{the $su(2) \times su(2)$ generators} \; T^{ij}=T^{ji}, J^{ab}=J^{ba}, \; i,j =1,2; \; a,b = 1,2 \nonumber\\
\mbox{ Fermionic generators: } && Q^{ia}, S^{ia} ,
\end{eqnarray}
which obey the following conjugation rules
\begin{equation}
\left(T^{ij}\right)^\dagger = T_{ij}, \; \left(J^{ab}\right)^\dagger = J_{ab},\; \left(P,D,K\right)^\dagger = (P,D,K), \qquad \left(Q^{ia}\right)^\dagger = Q_{ia}, \; \left(S^{ia}\right)^\dagger = S_{ia}.
\end{equation}
The non-zero (anti)commutators are
\begin{eqnarray}\label{D12}
&& {\rm i\,} \left[ P,K\right] = - 2 D, \; {\rm i\,} \left[ P,D\right] =-P, \;
{\rm i\,} \left[K,D\right]=K, \nonumber \\
&& {\rm i\,} \left[ T^{ij}, T^{km}\right] = \epsilon^{ik} T^{jm}+ \epsilon^{jm} T^{ik}, \quad {\rm i\,} \left[ J^{ab},J^{cd}\right] = \epsilon^{ac} J^{bd}+\epsilon^{bd}J^{ac}, \nonumber \\
&& {\rm i\,} \left[ P, S^{ia}\right] =- Q^{ia}, \; {\rm i\,} \left[ K , Q^{ia}\right] = S^{ia},\quad {\rm i\,} \left[ D , Q^{ia}\right] = \frac{1}{2} Q^{ia}, \;
{\rm i\,} \left[ D , S^{ia}\right] = -\frac{1}{2} S^{ia}, \nonumber \\
&& {\rm i\,} \left[ T^{ij}, Q^{ka}\right] = \frac{1}{2} \left( \epsilon^{ik} Q^{ja}+ \epsilon^{jk} Q^{ia}\right), \;
{\rm i\,} \left[ J^{ab}, Q^{ic}\right] = \frac{1}{2} \left( \epsilon^{ac} Q^{ib}+ \epsilon^{bc} Q^{ia}\right), \nonumber \\
&& {\rm i\,} \left[ T^{ij}, S^{ka}\right] = \frac{1}{2} \left( \epsilon^{ik} S^{ja}+ \epsilon^{jk} S^{ia}\right), \;
{\rm i\,} \left[ J^{ab}, S^{ic}\right] = \frac{1}{2} \left( \epsilon^{ac} S^{ib}+ \epsilon^{bc} S^{ia}\right), \nonumber \\
&& \left\{ Q^{ia}, Q^{jb}\right\} = -2 \epsilon^{ij} \epsilon^{ab} P, \;
\left\{ S^{ia}, S^{jb}\right\} = -2 \epsilon^{ij} \epsilon^{ab} K, \nonumber \\
&&\left\{ Q^{ia}, S^{jb}\right\} = 2 \left( -\epsilon^{ij} \epsilon^{ab} D+
\alpha \epsilon^{ij} J^{ab} - (1+\alpha) \epsilon^{ab} T^{ij}\right).
\end{eqnarray}
\setcounter{equation}0
\defC.\arabic{equation}{B.\arabic{equation}}
\section*{Appendix B. $su(2)$ rotations}
Using the commutator relations of the $D(1,2;\alpha)$ algebra \p{D12} it is not too complicated to find the effect
of the $su(2)\times su(2)$ rotations on the fermionic and $su(2)$ generators
\begin{eqnarray}
e^{-{\rm i\,} v \cdot T}\, Q^{kc}\, e^{{\rm i\,} v \cdot T}& = & \left( e^{v}\right)^k_m Q^{mc} =
\cos \sqrt{\frac{v^2}{2}}Q^k+\frac{\sin\sqrt{\frac{v^2}{2}}}{\sqrt{\frac{v^2}{2}}} v^k_m Q^{mc}, \qquad v^2 \equiv v_{ij}v^{ij}, \label{Q1}\\
e^{-{\rm i\,} \phi \cdot J}\, Q^{kc}\,e^{{\rm i\,} \phi \cdot J} & = & \left( e^{\phi}\right)^c_d Q^{kd} =
\cos \sqrt{\frac{\phi^2}{2}}Q^{kc}+\frac{\sin\sqrt{\frac{\phi^2}{2}}}{\sqrt{\frac{\phi^2}{2}}} \phi^c{}_d Q^{kd}, \qquad \phi^2 \equiv \phi_{ab}\phi^{ab}, \label{Q2}\\
e^{-{\rm i\,} v\cdot T}\, T^{km}\,e^{{\rm i\,} v\cdot T} & = & T^{km} +\frac{\sin\sqrt{2 v^2}}{\sqrt{2 v^2}}\left( v^k_n T^{nm}+v_n^m T^{nk} \right)+\frac{1-\cos\sqrt{2 v^2}}{2 v^2}\left( - v^2 T^{km}+2 v^k_i v^m_j T^{ij}\right), \label{T1}\\
e^{-{\rm i\,} \phi\cdot J}\, J^{cd}\,e^{{\rm i\,} \phi\cdot J} & = & J^{cd} +\frac{\sin\sqrt{2 \phi^2}}{\sqrt{2 \phi^2}}\left( \phi^c_b J^{bd}+\phi^d_b J^{bc} \right)+\frac{1-\cos\sqrt{2 \phi^2}}{2 \phi^2}\left( - \phi^2 J^{cd}+2 v^c_a v^d_b J^{ab}\right), \label{T2}
\end{eqnarray}
and
\begin{eqnarray}
e^{-{\rm i\,} v\cdot T} d e^{{\rm i\,} v \cdot T} & = &
{\rm i\,} d v_{km} \left[T^{km}+\frac{1-\cos\sqrt{2 v^2}}{ v^2} v^k_i T^{im}+\frac{\sqrt{2 v^2}-\sin\sqrt{2 v^2}}{2 v^2 \sqrt{ 2 v^2}}\left(- v^2 T^{km}+2 v_i^k v_j^m T^{ij}\right)\right],\label{dT1} \\
e^{-{\rm i\,} \phi\cdot J} d e^{{\rm i\,} \phi \cdot J} & = &
{\rm i\,} d \phi_{ab} \left[J^{ab}+\frac{1-\cos\sqrt{2 \phi^2}}{ \phi^2} \phi^a_d J^{db}+\frac{\sqrt{2 \phi^2}-\sin\sqrt{2 \phi^2}}{2 \phi^2 \sqrt{ 2 \phi^2}}\left(- \phi^2 J^{ab}+2 \phi_c^a \phi_d^b J^{cd}\right)\right]. \label{dT2}
\end{eqnarray}
It is less evident to note that the expressions \p{T1} and \p{T2} can be written, similarly to \p{Q1} and \p{Q2}, as
\begin{equation}
e^{-{\rm i\,} v\cdot T} T^{km} e^{{\rm i\,} v \cdot T} = \left( e^v\right)^k_i T^{ij}\left(e^v\right)_j^m \quad
\mbox{and} \quad
e^{-{\rm i\,} \phi\cdot J} J^{cd} e^{{\rm i\,} \phi \cdot J} = \left( e^\phi\right)^c_a J^{ab}\left(e^\phi\right)_b^d.
\end{equation}
Finally, the expressions \p{dT1} and \p{dT2} can be also written in a simplified way as
\begin{equation}
e^{-{\rm i\,} v\cdot T} d e^{{\rm i\,} v \cdot T} = -{\rm i\,} \left(T\right){}_k{}^m \left( e^{-v}\right)_i^k d\left(e^v\right)_m^i
\quad \mbox{and} \quad
e^{-{\rm i\,} \phi\cdot J} d e^{{\rm i\,} \phi \cdot J} = -{\rm i\,} \left(J\right){}_c{}^b \left( e^{-\phi}\right)_a^c d\left(e^\phi\right)_b^a .
\end{equation}
Note the useful identities which simplify the explicit calculations
\begin{eqnarray}
\left(e^v\right)^k_m =\cos\sqrt{\frac{v^2}{2}}\delta^k_m+\frac{\sin\sqrt{\frac{v^2}{2}}}{\sqrt{\frac{v^2}{2}}} v^k_m ,\quad \left(e^{-v}\right)^k_m =\cos\sqrt{\frac{v^2}{2}}\delta^k_m-\frac{\sin\sqrt{\frac{v^2}{2}}}{\sqrt{\frac{v^2}{2}}} v^k_m = -\epsilon^{ki}\epsilon_{mj} \left(e^v\right)^j_i.
\end{eqnarray}
\setcounter{equation}0
\defC.\arabic{equation}{C.\arabic{equation}}
\section*{Appendix C. Solution to the Maurer--Cartan equations}
As we already demonstrated the equation $d\omega_P$ \p{MCexp} is satisfied due to the condition \p{maincond}.
In contrast, $d\omega_Q$ equation is not trivial. After substitution of \p{maincond}, it separates into two equations:
\begin{eqnarray}\label{domegaQ}
\big( \triangle_1 \tau d_2 \theta_{jb} - \triangle_2 \tau d_1 \theta_{jb} \big):&& 0= -A_{ia}{}^{jb} - \delta_i^j \big( {\cal S}_J\big)_a{}^b - \delta_a^b \big( {\cal S}_T\big)_i{}^j, \nonumber \\
d_1\theta_{kc}d_2\theta_{ld}: &&0= \delta_i^l \Sigma_a{}^{d|kc} + \delta_i^k \Sigma_a{}^{c|ld} + \delta_a^d \Pi_i^{l|kc} + \delta_a^c \Pi_i{}^{k|ld}.
\end{eqnarray}
The first of these equations straightforwardly expresses $A_{ia}{}^{jb}$ in terms of $\big( {\cal S}_J\big)_a{}^b$ and $\big( {\cal S}_T\big)_i{}^j$, the second one is more elaborate. At first, multiplying it by $\delta_l^i \delta_c^a$, one can obtain
\begin{equation}\label{SigmaPi1}
\Sigma_c{}^{d|kc} + \Pi_l{}^{k|ld} =0 \;\; \Rightarrow \;\; \Sigma_c{}^{d|kc} = \sigma^{kd}, \;\; \Pi_l{}^{k|ld} =- \sigma^{kd}.
\end{equation}
Next, multiplying by just $\delta_l^i$ and taking into account \p{SigmaPi1}, one obtains
\begin{equation}\label{SigmaPi2}
2\Sigma_a{}^{d|kc} + \Sigma_{a}{}^{c|kd} - \delta_a^c \sigma^{kd} =0 \;\; \Rightarrow \;\; \Sigma_a{}^{d|kc} = \frac{2}{3}\delta_a^c \sigma^{kd} - \frac{1}{3}\delta_a^d \sigma^{kc}.
\end{equation}
Multiplying by $\delta_d^a$, one obtains
\begin{equation}\label{SigmaPi3}
2\Pi_i{}^{l|kc} + \Pi_i{}^{k|lc} + \delta_i^k \sigma^{lc}=0\;\; \Rightarrow \Pi_i{}^{k|lc} = -\frac{2}{3} \delta_i^l \sigma^{kc} + \frac{1}{3}\delta_i^k \sigma^{lc}.
\end{equation}
Substituting these relations back into \p{domegaQ}, one notes that $d\theta\times d\theta$ equation is satisfied with no further constraints on $\sigma^{kd}$, and $\omega_J$ and $\omega_T$ forms can be written as
\begin{equation}\label{omegaJT}
\big(\omega_J \big)_{ab} = \triangle\tau \big( {\cal S}_J \big)_{ab} + \frac{1}{3} \big( d\theta_{ia}\sigma^i_b + d\theta_{ib}\sigma^i_a \big), \;\; \big(\omega_T \big)_{ij} = \triangle\tau \big( {\cal S}_T \big)_{ij} - \frac{1}{3} \big( d\theta_{ia}\sigma^a_j + d\theta_{ja}\sigma^a_i \big).
\end{equation}
The fermion $\sigma_{ia}$ is an obvious candidate for the super--Schwarzian. Note that if $\alpha =0,-1$, the generators of one $SU(2)$ groups do not appear at the right hand side of commutators of supercharges. If this decoupled $SU(2)$ is dropped entirely from the coset space, equation \p{domegaQ} would not contain either $\Sigma$ or $\Pi$. This equation would, as it follows from \p{SigmaPi1}, set the remaining fermion to zero and, as was already found, the bosonic component of the automorphism form becomes the super--Schwarzian. We, therefore, assume that $\alpha\neq 0,-1$ in further considerations.
Not all the equations have been written down. The $d\omega_J$ equation also separates into two:
\begin{eqnarray}\label{domegaJ}
\triangle_1 \tau d_2\theta_{kc} - \triangle_2\tau d_1\theta_{kc} : && {\rm i\,} D^{kc}\big( {\cal S}_J \big)_{ab} - \frac{{\rm i\,}}{3} \big( \delta_a^c {\dot\sigma}_b^{k} + \delta_b^c {\dot\sigma}_a^{k} \big) = -\frac{{\rm i\,}}{3} \left( \delta_b^c \big( {\cal S}_J \big)_{a}{}^d \sigma^k_d + \delta_a^c \big( {\cal S}_J \big)_{b}{}^d \sigma^k_d \right)-\nonumber \\&& - \frac{{\rm i\,}}{3} \left( \big( {\cal S}_J \big)_{a}{}^c \sigma^k_b + \big( {\cal S}_J \big)_{b}{}^c \sigma^k_a \right) + \alpha \left( \delta_b^c \Psi_a^k + \delta_a^c \Psi_b^k \right), \\
d_1\theta_{kc} d_2\theta_{ld}: && 2\epsilon^{kl}\epsilon^{cd} \big( {\cal S}_J \big)_{ab} + \frac{{\rm i\,}}{3} \left( \delta_a^c D^{ld}\sigma_b^k + \delta_b^c D^{ld}\sigma_a^k + \delta_a^d D^{kc}\sigma_b^l + \delta_b^d D^{kc}\sigma^l_a \right) = \nonumber \\
&& =\epsilon^{kl}\epsilon^{cd} \left( -\frac{2{\rm i\,}}{9} \sigma_{ma}\sigma^m_b -2\alpha \big( {\cal S}_J \big)_{ab} \right) + \big( \delta_a^c \delta_b^d + \delta_b^c \delta_a^d \big) \left(2\alpha \big( {\cal S}_T \big)^{kl} - \frac{2{\rm i\,}}{9} \sigma^k_f \sigma^{lf} \right). \nonumber
\end{eqnarray}
Substituting $D^{ia}\sigma^{jb}$ into the second equation as most general combination of tensors of various symmetries
\begin{eqnarray}\label{DsigmaF}
D^{ia}\sigma^{jb} = \epsilon^{ij}\epsilon^{ab}F - \frac{1}{2}\epsilon^{ij}F^{(ab)}- \frac{1}{2}\epsilon^{ab}F^{(ij)} + F^{(ij)(ab)}, \nonumber \\
F = \frac{1}{4}D^{kc}\sigma_{kc}, \;\; F^{ij} = \frac{1}{2}\big( D^{ic}\sigma_c^j +D^{jc}\sigma_c^i \big), \;\; F^{ab} = \frac{1}{2}\big( D^{ka}\sigma_k^b +D^{kb}\sigma_k^a \big),
\end{eqnarray}
one can obtain that second equation \p{domegaJ} implies $F^{(ij)(ab)} =0 $, relates $F^{ij}$ and $F^{ab}$ to $\big( {\cal S}_T\big)^{ij}$ and $\big( {\cal S}_J \big)^{ab}$
\begin{equation}\label{SJST}
\big({\cal S}_J \big)^{ab} = \frac{{\rm i\,}}{1+\alpha} \left[ \frac{F^{ab}}{6} - \frac{\sigma^a_m \sigma^{mb}}{9} \right], \;\; \big({\cal S}_T \big)^{ij} = \frac{{\rm i\,}}{\alpha} \left[ \frac{F^{ij}}{6} + \frac{\sigma^i_c \sigma^{jc}}{9} \right],
\end{equation}
and places no restriction on the scalar $F$.
Therefore, $\sigma^{ia}$ satisfies the differential constraint
\begin{equation}\label{Dsigmarel}
D^{ia}\sigma^{jb} +D^{ja}\sigma^{ib} +D^{ib}\sigma^{ja} +D^{jb}\sigma^{ia} =0, \quad
D^{ia} \sigma^{jb}+D^{jb}\sigma^{ia} = \frac{1}{2} \epsilon^{ij} \epsilon^{ab} D^{kc} \sigma_{kc} .
\end{equation}
To study the first equation of \p{domegaJ}, one should find the derivative of $\big({\cal S}_J \big)^{ab}$, and, therefore, of $F$, $F^{ij}$ and $F^{ab}$. Using their definition \p{DsigmaF} and commutation relation of derivatives \p{covders}, one can obtain the relations
\begin{eqnarray}\label{DFFF}
D^{kc}F^{ab} = -\frac{1}{3}\epsilon^{ac}D^k_d F^{bd} -\frac{1}{3}\epsilon^{bc}D^k_d F^{ad}, \;\; D^{kc}F^{ij} = -\frac{1}{3}\epsilon^{ik}D^c_l F^{jl} -\frac{1}{3}\epsilon^{jk}D^c_l F^{il}, \nonumber \\
D^{kc}F = {\rm i\,} {\dot\sigma}{}^{kc}, \;\; D^i_c F^{ac} = 6{\rm i\,} {\dot\sigma}{}^{ia} - D^a_k F^{ik}.
\end{eqnarray}
We prefer to express derivatives of $F^{cd}$ in terms of derivatives of $F^{ij}$.
Substituting $\big({\cal S}_J \big)^{ab}$ \p{SJST} into first equation \p{domegaJ} and evaluating derivatives, it could be obtained that all the terms neither proportional to $\delta_a^c$ or $\delta_b^c$ vanish and the rest imply that
\begin{equation}\label{Psi}
\alpha(1+\alpha)\Psi^{ia} = - \frac{{\rm i\,}}{3} \alpha {\dot\sigma}{}^{ia} - \frac{1}{18}D^a_j F^{ij} + \frac{1}{9}F \sigma^{ia} - \frac{1}{18} F^{il}\sigma_l^a - \frac{1}{18} F^{ac}\sigma_c^i + \frac{4}{81} \sigma^i_c \sigma_m^c \sigma^{ma}.
\end{equation}
Equation $d\omega_T$ is very much similar to $d\omega_J$:
\begin{eqnarray}\label{domegaT}
\triangle_1 \tau d_2\theta_{kc} - \triangle_2\tau d_1\theta_{kc} : && {\rm i\,} D^{kc}\big( {\cal S}_T \big)_{ij} + \frac{{\rm i\,}}{3} \big( \delta_i^k {\dot\sigma}_j^{c} + \delta_j^k {\dot\sigma}_i^{c} \big) =\frac{{\rm i\,}}{3} \left( \delta_j^k \big( {\cal S}_T \big)_{i}{}^l \sigma^c_l + \delta_i^k \big( {\cal S}_T \big)_{j}{}^l \sigma^c_l \right)-\nonumber \\&& + \frac{{\rm i\,}}{3} \left( \big( {\cal S}_T \big)_{i}{}^k \sigma^c_j + \big( {\cal S}_T \big)_{j}{}^k \sigma^c_i \right) -(1+ \alpha) \left( \delta_i^k \Psi_j^c + \delta_j^k \Psi_i^c \right), \\
d_1\theta_{kc} d_2\theta_{ld}: && 2\epsilon^{kl}\epsilon^{cd} \big( {\cal S}_T \big)_{ij} - \frac{{\rm i\,}}{3} \left( \delta_i^k D^{ld}\sigma_j^c + \delta_j^k D^{ld}\sigma_i^c + \delta_i^l D^{kc}\sigma_j^d + \delta_j^l D^{kc}\sigma_i^d \right) = \nonumber \\
&& = \epsilon^{kl}\epsilon^{cd} \left( -\frac{2{\rm i\,}}{9} \sigma_{ia}\sigma^a_j + 2(1+\alpha) \big( {\cal S}_T \big)_{ij} \right) + \big( \delta_i^k \delta_j^l + \delta_j^k \delta_i^l \big) \left( -2(1+ \alpha) \big( {\cal S}_J \big)^{cd} - \frac{2{\rm i\,}}{9} \sigma^c_m \sigma^{md} \right). \nonumber
\end{eqnarray}
Substitution of relations obtained above \p{DsigmaF}, \p{Dsigmarel}, \p{SJST}, \p{Psi} guarantees that these two equations are satisfied in the same manner as $d\omega_J$ equations \p{domegaJ}.
The equation $d\omega_D$ results in the following relations:
\begin{equation}\label{omegaD}
\Gamma^{ia} = {\rm i\,} \Psi^{ia}, \quad A^{ia|jb}+ A^{jb|ia}=0.
\end{equation}
The second equation in \p{omegaD} is satisfied after substitution $A^{ia|jb} = -\epsilon^{ij} \big( {\cal S}_J \big)^{ab} - \epsilon^{ab}\big( {\cal S}_T \big)^{ij}$ \p{domegaQ}, while the first one expresses the
$\Gamma^{ia}$ through $\Psi^{ia}$.
Now, combining everything together we will have the following expressions for the Cartan forms
\begin{eqnarray}
&& \omega_P = \triangle \tau, \quad \omega_D=0,\quad \omega_K = \triangle \tau C + {\rm i\,} d\theta_{ia} \Psi^{ia}, \\
&& \left(\omega_J\right)_{ab} = \triangle \tau \left({\cal S}_J\right)_{ab}+\frac{1}{3}\left( d\theta_{ka} \sigma^k_b+
d\theta_{kb} \sigma^k_a \right), \\
&& \left(\omega_T\right)_{ij} = \triangle \tau \left({\cal S}_T\right)_{ij}-\frac{1}{3}\left( d\theta_{ic} \sigma^c_j+
d\theta_{jc} \sigma^c_i \right),\\
&& \left(\omega_Q\right)^{ia} = d \theta^{ia}, \quad
\left(\omega_S\right)^{ia} = \triangle \tau \Psi^{ia} -d\theta_{ib} \left({\cal S}_J\right)^b_{a}
-d\theta_{ka} \left({\cal S}_T\right)^k_{i} ,
\end{eqnarray}
with only one function $C$ remaining to be determined by study of $d\omega_S$ and $d\omega_K$ equations.
The $d\omega_S$ equation again separates into two
\begin{eqnarray}
D^{jb} \Psi^{ia}+\epsilon^{ij} \left( \dot{{\cal S}}_J\right){}^{ab} + \epsilon^{ab} \left( \dot{{\cal S}}_T\right){}^{ij} =
-\frac{1}{2} \epsilon^{ij}\epsilon^{ab} \left( {\cal S}_J^2 + {\cal S}_T^2 - 2 C\right) + 2 \left( {\cal S}_J\right)^{ab} \left( {\cal S}_T\right)^{ij} -\nonumber \\ -\frac{1}{3}\left( \epsilon^{ab} \big( \sigma^j_c \Psi^{ic} + \sigma^i_c \Psi^{jc}\big) -\epsilon^{ij} \big(\sigma_k^a \Psi^{kb} +\sigma_k^b \Psi^{ka} \big)\right), \label{formS1} \\
-\epsilon^{il} D^{kc} \big( {\cal S}_J \big)^{ad} -\epsilon^{ik} D^{ld} \big( {\cal S}_J \big)^{ac} - \epsilon^{ad}D^{kc}\big( {\cal S}_T \big)^{il} -\epsilon^{ac}D^{ld}\big( {\cal S}_T \big)^{ik} -2{\rm i\,} \epsilon^{kl}\epsilon^{cd}\Psi^{ia} =\nonumber \\
=-{\rm i\,} \epsilon^{il} \epsilon^{ad}\Psi^{kc} -{\rm i\,} \epsilon^{ik}\epsilon^{ac} \Psi^{ld}
+ \frac{1}{3} \big( -\epsilon^{ac} \epsilon^{il} \sigma^{kb}\big( {\cal S}_J \big)_b{}^d - \epsilon^{ad}\epsilon^{ik}\sigma^{lb}\big( {\cal S}_J \big)_b{}^c +\nonumber \\
+ \epsilon^{il}\sigma^{ka}\big( {\cal S}_J \big)^{cd} + \epsilon^{ik}\sigma^{la}\big( {\cal S}_J \big)^{cd} + \epsilon^{ik}\sigma^{lc}\big( {\cal S}_J \big)^{ad} + \epsilon^{il}\sigma^{kd}\big( {\cal S}_J \big)^{ac} + \epsilon^{kl}\epsilon^{cd} \sigma^{ib} \big( {\cal S}_J \big)_b{}^a \big) + \nonumber \\ + \frac{1}{3}\big( - \epsilon^{ac}\sigma^{kd}\big( {\cal S}_T \big)^{il} - \epsilon^{ad}\sigma^{lc}\big( {\cal S}_T \big)^{ik} -\epsilon^{ad} \sigma^{ic}\big( {\cal S}_T \big)^{kl} - \epsilon^{ac}\sigma^{id} \big( {\cal S}_T \big)^{kl} + \nonumber \\+
\epsilon^{kl}\epsilon^{cd} \sigma^a_m \big( {\cal S}_T \big)^{im} -\epsilon^{ik}\epsilon^{ad}\sigma^c_m \big( {\cal S}_T \big)^{lm} - \epsilon^{il}\epsilon^{ac}\sigma^d_m \big( {\cal S}_T \big)^{km} \big) . \label{formS2}
\end{eqnarray}
Equation \p{formS2} is satisfied identically after substitution of $\Psi^{ia}$ \p{Psi} and $\big({\cal S}_J \big)^{ab}$, $\big( {\cal S}_T \big)^{ij}$ \p{SJST}. In this calculation, one should use the formula
\begin{equation}\label{DDF}
D^{jb}D^a_k F^{ik} = 3{\rm i\,} \epsilon^{ab}{\dot F}{}^{ij} - \frac{1}{4}\epsilon^{ij}\epsilon^{ab}D^c_k D_{cl}F^{kl},
\end{equation}
which follows from \p{DFFF} and commutation relations \p{covders}. Equation \p{formS1} after substitution of $\Psi^{ia}$ \p{Psi} and $\big({\cal S}_J \big)^{ab}$, $\big( {\cal S}_T \big)^{ij}$ \p{SJST} reduces to $\epsilon^{ij}\epsilon^{ab}$-projection, which determines $C$
\begin{eqnarray}\label{C}
\alpha (1+\alpha)C &=& - \frac{{\rm i\,}}{3}\alpha {\dot F} + \frac{1}{72}D^c_k D_{lc}F^{kl} + \frac{1}{9}F^2 - \frac{1}{72\alpha}F_{kl}F^{kl} + \frac{1}{72(1+\alpha)} F_{cd}F^{cd} - \nonumber \\
&&- \frac{1}{54(1+\alpha)} F^{cd}\sigma_{mc}\sigma^m_d - \frac{1}{54\alpha} F^{kl}\sigma_{kc}\sigma^c_l + \frac{{\rm i\,}}{9}{\dot\sigma}{}^{kc}\sigma_{kc} + \frac{1}{162}\left( \frac{1}{\alpha} + \frac{1}{1+\alpha} \right) \sigma_{kc}\sigma^k_d \sigma^c_l \sigma^{ld}.
\end{eqnarray}
Finally, studying $d\omega_K$ equation we will get the following relations
\begin{eqnarray}
&& {\rm i\,} {\dot \Psi}{}^{ia}- D^{ia} C = -2{\rm i\,} \left( \Psi^i_c \left({\cal S}_J\right)^{ca}+\Psi^a_k \left({\cal S}_T\right)^{ki}\right), \label{omeKa} \\
&&2 \epsilon^{ij} \epsilon^{ab} C - D^{ia} \Psi^{jb} -D^{jb} \Psi^{ia} = \epsilon^{ij}\epsilon^{ab} \left( {\cal S}_J^2 +{\cal S}_T^2 \right) - 4 \big( {\cal S}_T\big){}^{ij} \big( {\cal S}_J \big){}^{ab} . \label{omeKb}
\end{eqnarray}
Here, ${\cal S}_J^2 \equiv ({\cal S}_J)^{ab} ({\cal S}_J)_{ab}$ and ${\cal S}_T^2 \equiv ({\cal S}_T)^{ij} ({\cal S}_T)_{ij}$.
The relation \p{omeKb} is simply a consequence of \p{formS1} and therefore, is satisfied identically. Finally, substituting $C$ \p{C}, $\Psi^{ia}$ \p{Psi} and $\big({\cal S}_J \big)^{ab}$, $\big( {\cal S}_T \big)^{ij}$ \p{SJST} into \p{omeKa} and using formula
\begin{equation}\label{DDDF}
D^{ia}D^c_k D_{lc} F^{kl} = - 4{\rm i\,} D^a_l {\dot F}{}^{il},
\end{equation}
one finds that \p{omeKa} is also identically satisfied, leaving no extra constraints on $\sigma^{ia}$.
|
2024-02-18T23:40:24.728Z
|
2022-03-17T01:29:10.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14481","language":"en","timestamp":1647480550000,"url":"https:\/\/arxiv.org\/abs\/2112.14481","yymm":"2112"}
|
proofpile-arXiv_000-10073
|
{"provenance":"002.jsonl.gz:10074"}
| null | null |
\section{Introduction}
\label{Sec1}
A variety of theories for the dark sector (dark matter and dark energy)
suppose the existence of hidden fields which couple non-universally to
the Standard Model, SM
(for reviews, see Uzan 2011; Marsh 2016; Battaglieri {et al.}\ 2017; Hui {et al.}\ 2017;
Irastorza \& Redondo 2018; Terazawa 2018; Beacham {et al.}\ 2019; Ahmed {et al.}\ 2019).
Such coupling would modulate the fermion masses in different ways
thus changing their ratios. In particular, this relates to
the fundamental constant of particle physics~--
the electron-to-proton mass ratio, $\mu = m_{\rm e}/m_{\rm p}$,~-- where
$m_{\rm e}$ is directly determined by the coupling to the Higgs-like field(s),
whereas the main input to $m_{\rm p}$ comes from the binding energy of quarks.
Hence, measurements of $\mu$ can serve as a tool to probe the coupling between the SM and dark sector.
New fields are predicted to be ultralight (Compton wavelengths $\lambda \sim 1$~kpc)
and/or to change their values depending on the environmental parameters such as
gravitational potential of baryonic matter or local baryonic mass density
(e.g., Damour \& Polyakov 1994; Khoury \& Weltman 2004; Olive \& Pospelov 2008; Brax 2018).
This makes astronomical objects preferable targets in corresponding studies.
Measurements of fractional changes in $\mu$,
\begin{equation}
\frac{\Delta\mu}{\mu} = \frac{\mu_{\rm \scriptscriptstyle obs} -
\mu_{\rm \scriptscriptstyle lab}}{\mu_{\rm \scriptscriptstyle lab}},
\label{Eq1}
\end{equation}
in astronomical objects are based on the fact that the
molecular electron-vibro-rotational transitions have specific dependences on $\mu$
(Thompson 1975) and different sensitivities to
$\mu$-variations (Varshalovich \& Levshakov 1993; Flambaum \& Kozlov 2007; Levshakov {et al.}\ 2011;
Jansen {et al.}\ 2011; Patra {et al.}\ 2018).
For a given molecular transition frequency $f$,
the dimensionless sensitivity coefficient to a possible variation of $\mu$ is defined as\footnote{To avoid
confusion, we note that if
$\mu$ is defined as the proton-to-electron mass ratio, say $\mu' = m_{\rm p}/m_{\rm e}$, then
$\Delta\mu/\mu = -\Delta\mu'/\mu'$ and $Q_i = -K_i$, where $K_i = (df_i/f_i)/(d\mu'/\mu')$. }
\begin{equation}
Q = \frac{df/f}{d\mu/\mu}\, .
\label{Eq2}
\end{equation}
The coefficients $Q$ take positive or negative signs and values ranging between
$\sim 10^{-2}$ for H$_2$ and $\sim 10$s for
CH$_3$OH and other molecules (for a review see, e.g., Kozlov \& Levshakov 2013).
The fractional changes in $\mu$ can be measured using
any pair of lines of co-spatially distributed molecular transitions ($i, j$)
with different values of $Q$ (Levshakov {et al.}\ 2011):
\begin{equation}
\frac{\Delta\mu}{\mu} = \frac{V_j - V_i}{c(Q_i - Q_j)},
\label{Eq3}
\end{equation}
where $V_j$ and $V_i$ are the LSR radial velocities
of molecular transitions with sensitivity coefficients $Q_j$ and $Q_i$,
and $c$ is the speed of light.
It is to note that spectral observations with modern facilities
provide an unprecedented accuracy in measurements of molecular transitions
and, hence, in $\Delta\mu/\mu$. Additional advantages are the relative simplicity of
interpretation of the obtained results
and a restricted number of sources of systematic
errors (cf., e.g., Touboul {et al.}\ 2020).
Presently, the most stringent limit on $\Delta\mu/\mu$,
$\Delta\mu/\mu$~$< 7\times10^{-9}$ (hereafter a $1\sigma$ confidence level is used),
was obtained from
high resolution spectral observations of Milky Way's cold
molecular cores in lines of NH$_3$, HC$_3$N, HC$_5$N, HC$_7$N, and N$_2$H$^+$
at the Effelsberg 100-m, Medicina 32-m, and Nobeyama 45-m radio telescopes
(Levshakov {et al.}\ 2010a,b; Levshakov {et al.}\ 2013, hereafter L13).
Additionally, observations of the dense dark cloud core L1498
in thermal $E$- and $A$-type methanol CH$_3$OH lines at
the IRAM 30-m telescope gave the upper limit of $\Delta\mu/\mu$~$< 2\times10^{-8}$
(Dapr\'a {et al.}\ 2017, hereafter D17).
In these studies, all molecular cores were located within a 300 pc radius from the Sun,
which is insufficient to detect the predicted gradients of hidden ultralight fields:
to probe the SM--dark sector coupling,
observations of targets spaced apart by distances of kiloparsecs are required.
Such targets are objects of the present paper.
We aim at obtaining the $\Delta\mu/\mu$\ estimate in the Milky Way disk utilizing
narrow emission lines of bright Class~I methanol (CH$_3$OH) masers
from the northern Galactic hemisphere distributed
over the galactocentric distance range of $4 \la R \la 12$ kpc.
Methanol masers are usually classified into two types: Class~I and Class~II.
The first type of sources is offset by $\sim1$ pc from star formation signposts
and mostly the result from collisional excitation
(Menten 1991a,b; Cragg {et al.}\ 2005; Leurini {et al.}\ 2016).
The sources are associated with shocks caused by molecular outflows, expansion of H\,{\sc ii} regions,
and cloud-cloud interactions (Kurtz {et al.}\ 2004; Voronkov {et al.}\ 2006, 2010, 2014).
Masers of the second type are found in the closest environment of massive young stellar
objects and are pumped by the reprocessed dust continuum radiation from these sources.
The Class~II methanol masers at 6.2 and 12.2 GHz were used
by Ellingsen {et al.}\ (2011) to limit $\mu$-variations at the level
$|\Delta\mu/\mu| < 2.7\times10^{-8}$.
The advantages of using Class~I methanol masers for the $\Delta\mu/\mu$\ measurements are the following:
\begin{itemize}
\item[$\bullet$]
They belong to a large population of
Galactic emitters which are distributed
across the Galactic plane towards both the Galactic centre and anti-centre.
This enables us to scan $\Delta\mu/\mu$\ over a large spatial range.
\item[$\bullet$]
The maser lines are strong and narrow (non-thermal) and, thus, their radial velocities
can be measured with high precision.
\item[$\bullet$]
Methanol transitions show large differences in the sensitivity coefficients
(denominator in Eq.~\ref{Eq3}). This naturally decreases
uncertainties of the $\Delta\mu/\mu$\ estimate.
\item[$\bullet$]
Class~I masers are stable, i.e., do not exhibit
flux variability at time intervals $\sim 1$~yr
in contrast to Class~II methanol masers which show greater temporal variability.
\end{itemize}
We estimate the fractional changes in $\mu$ using
the Class~I $A$-type methanol transitions at 44 and 95 GHz.
The emission in these transitions is closely associated and traces the same spots
(Val'tts {et al.}\ 2000; Voronkov {et al.}\ 2014; Leurini {et al.}\ 2016).
This minimizes the Doppler noise, which are random shifts of spectral line positions
caused by possible spatial segregation and kinematic effects.
In addition to this component, which is stochastic, the total error budget of the measured radial velocity,
$V_{\scriptscriptstyle\rm LSR}$,
of a given spectral line contains also a systematic error related to the uncertainty in the
laboratory values of rest frequencies.
The laboratory frequencies of the 44 and 95 GHz transitions are presently measured with an error of
$\sim 10$ kHz (Tsunekawa {et al.}\ 1995; M\"uller {et al.}\ 2004).
This translates into systematic errors of $V_{\scriptscriptstyle\rm LSR}$
as great as $\sim 10$ m~s$^{-1}$, restricting the limit on $\Delta\mu/\mu$\ at the level of $10^{-8}$,
i.e., the level already reached in the most accurate $\Delta\mu/\mu$\ estimates up-to-now.
In principle, it is technically possible to measure the laboratory
frequencies with accuracy $\sim 1$ kHz. However, the problem is more complicated.
Namely, the molecule CH$_3$OH possesses an underlying hyperfine structure
scaled over the same $\sim 10$ kHz
(Hougen {et al.}\ 1991; Coudert {et al.}\ 2015; Belov {et al.}\ 2016; Lankhaar {et al.}\ 2016).
The hyperfine splitting can only be partly resolved in the laboratory and
is completely convolved in astrophysical observations.
If emission is purely thermal, then the barycentre of the convolved profile
is more or less stable and its error is localized within the kHz uncertainty interval (D17).
If, however, emission is due to masing effects, then, as shown in
Lankhaar {et al.}\ (2018), population inversion may be enhanced for some hyperfine transitions,
while suppressed for others. The physical mechanisms leading to the favoured pumping of this
or that hyperfine sublevel are not yet clear enough. As a result, the barycentre of the convolved
profile will shift with an amplitude of $\sim 10$ kHz depending on the physical environment.
The investigation of this additional systematics is another purpose of the present paper.
\section{Observations and target selection}
\label{Sec2}
We use simultaneous observations of the 44 GHz ($7_0 - 6_1$~A$^+$) and 95 GHz ($8_0 - 7_1$~A$^+$)
Class I methanol masers from the so-called Red {\it MSX} Source (RMS)
catalogue\footnote{https://rms.leeds.ac.uk/cgi-bin/public/RMS\_DATABASE.cgi}
observed by Kim {et al.}\ (2018) (hereafter K18), and
from the Bolocam Galactic Plane Survey (BGPS)
sources\footnote{https://irsa.opac.caltech.edu/data/BOLOCAM\_GPS}
observed by Yang {et al.}\ (2020) (hereafter Y20).
Both surveys were performed with the Korean VLBI Network (KVN) 21-m telescopes in single-dish telescope mode.
The multifrequency receiving systems at each telescope allow us to observe 44 and 95 GHz methanol transitions
at the same time, and with velocity scales defined in the same way. These two advantages are not being
met by any other data set. Details of these observations are given in the original papers,
here we repeat only those relevant to the present study.
Both the K18 and Y20 works were surveys aimed in the first place
at the discovery of new methanol masers.
Because of that the signal-to-noise ratio (S/N) in the majority of targets
was not high enough to deduce the radial velocities with the required accuracy
of $\sim$~10s m~s$^{-1}$. As a result, only 7 objects from 229 maser sources of K18 and
11 objects from 144 sources of Y20
turned out to be suitable for our purposes; the selected targets
are listed in Table~\ref{Tb1}.
Their spatial distribution in the Galactic disk is shown in Fig.~\ref{fg1}.
The objects lie at low Galactic latitudes $-2.6^\circ < b < 2.8^\circ$ and
are located towards both the Galactic centre and anti-centre at distances
$D_a \leq 4.7$ kpc (anticentre) and $D_c \leq 12.3$ kpc (centre) from the Sun.
Thus, the maximum scale covered is approximately 16 kpc which is
comparable to the Galactic optical radius.
Both the K18 and Y20 surveys report a remarkable strong correlation between methanol
peak velocities for the 44 and 95 GHz transitions:
\begin{equation}
V_{\rm pk,95} = (0.98\pm0.05)V_{\rm pk,44} - (0.01 \pm 0.010),
\label{Kim}
\end{equation}
and
\begin{equation}
V_{\rm pk,95} = (0.9998\pm0.0010)V_{\rm pk,44} - (0.0470 \pm 0.0614),
\label{Yong}
\end{equation}
respectively.
For the mean value of ${V}_{\rm pk,44} \simeq 75$ km~s$^{-1}$\ (see Fig.~4 in Y20)
the relation (\ref{Yong}) provides for the offset $\Delta V = V_{\rm pk,44} - V_{\rm pk,95}$
an error $\sigma_{\scriptscriptstyle \Delta V} \sim 0.1$ km~s$^{-1}$, which transforms into an upper
limit on $\Delta\mu/\mu$~$< 10^{-7}$ in accord with Eq.(\ref{Eq3}) where $Q_{95} - Q_{44} = 3.3$
(see Subsec.~\ref{SSec33}). Below we show how a more stringent limit on $\Delta\mu/\mu$\ can be deduced from
the existent datasets.
In K18, the original channel widths were $\Delta_{\rm ch} = 0.053$ km~s$^{-1}$\
at 44 GHz, and 0.025 km~s$^{-1}$\ at 95 GHz, whereas in Y20, they were two times larger,
$\Delta_{\rm ch} = 0.106$ km~s$^{-1}$\ at 44 GHz, and 0.049 km~s$^{-1}$\ at 95 GHz.
The peak flux densities, $F_{44}$ and $F_{95}$, indicated in Table~\ref{Tb1},
vary between 12 Jy (95 GHz, RMS153) and 420 Jy (44 GHz, BGPS4252) providing
signal-to-noise ratios between S/N = 10 and S/N = 276.
Both the signal-to-noise ratio and channel width are crucial for our analysis since they define
the final uncertainty of the line position. Namely,
an expected statistical uncertainty of the centre of a Gaussian-like
line profile is given by (e.g., Landman {et al.}\ 1982):
\begin{equation}
\sigma_0 \simeq \frac{0.7 \Delta_{\rm ch}}{\rm S/N} \sqrt{n},
\label{Eq4}
\end{equation}
where $n = {\rm FWHM}/\Delta_{\rm ch}$ is the line's full width at half maximum (FWHM) in units of channels.
With a typical line width (FWHM) for the methanol lines in question of $\sim 0.5$ km~s$^{-1}$, one obtains
for the marginal values of S/N = 10 and 276 the following errors:
$\sigma_0 \simeq 0.014$ km~s$^{-1}$\ ($\Delta_{\rm ch} = 0.074$ km~s$^{-1}$) and
$\sigma_0 \simeq 0.0006$ km~s$^{-1}$\ ($\Delta_{\rm ch} = 0.106$ km~s$^{-1}$).
The adopted rest frequencies 44069.430 MHz ($7_0 - 6_1$~A$^+$) from Pickett {et al.}\ (1998)
and 95169.463 MHz ($8_0 - 7_1$~A$^+$) from M\"uller {et al.}\ (2004)
were utilized in the original papers of K18 and Y20.
Note that these frequencies differ from those recommended by NIST, NRAO, JPL, and CDMS
(see Table~\ref{Tb2}).
\section{Analysis}
\label{Sec3}
\subsection{Method}
\label{SSec31}
At first, we determined the baseline for each original spectrum
by choosing spectral windows without emission lines and/or noise
spikes and then calculating the mean flux densities $F_i$
along with their rms uncertainties $\sigma_i$ for each
spectral window. Using spline interpolation through this set of pairs $\{F_i,\sigma_i\}$
we calculated a baseline which was subtracted from the spectrum.
Then, the mean value of the rms uncertainties, $\sigma_{\rm rms}$, was determined
and assigned to the whole spectrum.
The radial velocities $V_{\scriptscriptstyle\rm LSR}$ in Eq.~\ref{Eq3}
are calculated as described in Levshakov {et al.}\ (2019).
$V_{\scriptscriptstyle\rm LSR}$ is attributed to the line centre which is defined as a point
where the first order derivative of the line profile is equal to zero. In order to calculate this
extremum point accurately the observed line profile, $f(x)$, is filtered.
The filtering function $y(x)$ consists of a sum of $N$ Gaussian subcomponents,
parameters of which are calculated by minimization of a $\chi^2$ function:
\begin{equation}
\chi^2_\nu = \frac{1}{\nu}\sum \{[f(x_i) - y(x_i)]/\sigma_{\rm rms}\}^2,
\label{Eq4a}
\end{equation}
where $\nu$ is the number of degrees of freedom.
The number of subcomponents, $N$, is chosen so that the $\chi^2_\nu$ function
is minimized at the level of $\chi^2_\nu \simeq 1$ to avoid under- or
over-fitting of the line profile.
The uncertainty of $V_{\scriptscriptstyle\rm LSR}$, $\sigma_v$, is determined by
three points $\{x_1,y_1; x_2,y_2; x_3,y_3\}$ with $x_1 < x_2 < x_3$
which include the flux density peak, $x_{\rm peak} \in (x_1,x_3)$:
\begin{equation}
\sigma_v = \frac{\sigma_{\rm rms}\cdot \Delta_{\rm ch}}{(y_1 - 2y_2 + y_3)^2}{\cal K},
\label{Eq5}
\end{equation}
where ${\cal K} = \sqrt{(y_3-y_2)^2 + (y_1-y_3)^2 + (y_2-y_1)^2}$, and
the channel width $\Delta_{\rm ch} = x_2-x_1 = x_3-x_2$.
\subsection{Reproducibility of radial velocities}
\label{SSec32}
The 44 GHz transition was observed by K18 in two epochs:
the first one in 2011, and the second in 2012.
The second set of observations included also the 95 GHz line.
The one year time lapse can be used to test stability
of Class~I methanol masers as was mentioned in Sect.~1.
The radial velocities $V_{\rm \scriptscriptstyle LSR}$
of the 44 GHz line measured with respect to the mean value $\bar{V}_{\rm \scriptscriptstyle LSR}$
between the two observational epochs are plotted in Fig.~\ref{fg2},
while the individual $V_{\rm\scriptscriptstyle LSR}$ values are listed in Table~\ref{Tb3}.
This table includes 10 velocity offsets between 2011 and 2012 observations towards 9 maser sources
(the 44 GHz profile in RMS3841 consists of two narrow subcomponents separated by 0.4 km~s$^{-1}$).
However, in the further analysis we used only 7 targets since the 95 GHz profiles towards
RMS2584 and RMS3841 were not good enough for precision measurements of their
$V_{\rm \scriptscriptstyle LSR}$ values.
Figure~\ref{fg2} demonstrates stability of the 44 GHz line position for all 10 pairs with
the weighted mean $\langle \Delta V_{\rm yr} \rangle_w = -1.1\pm1.6$ m~s$^{-1}$.
Taking this into account, we stack up 44 GHz spectra from both epochs
coadding them with weights inversionally proportional to their
variances, $\sigma^2_{\rm rms}$.
The measured radial velocities based on the stacked data are
listed in Table~\ref{T4}, second column.
Since for the 44 and 95 GHz data, being taken at the same time (in 2012) with the same telescopes,
the computations of velocity corrections leading to LSR velocities match each other. Thus
stacking does not introduce a statistically significant systematic error.
\subsection{Fractional changes in $\mu$ }
\label{SSec33}
As it follows from Eq.~\ref{Eq3}, the fractional changes in $\mu$ are defined by
the LSR radial velocities of a pair of methanol lines, $V_i$ and $V_j$, which have
different sensitivity coefficients, $Q_i$ and $Q_j$, to $\mu$-variations.
For the 44 and 95 GHz methanol transitions the sensitivity coefficients were calculated
in Jansen {et al.}\ (2011) and in Levshakov {et al.}\ (2011). Both groups give similar $Q$-values:
$Q_{44} = -5.2\pm0.3$,
$Q_{95} = -1.88\pm0.09$ (Jansen {et al.}), and
$Q_{44} = -5.3\pm0.6$ and
$Q_{95} = -1.9\pm0.3$ (Levshakov {et al.}).
In our analysis we use $Q_{44} = -5.2$ and $Q_{95} = -1.9$.
Their difference $\Delta Q = 3.3$ is comparable to the difference between the sensitivity
coefficients in the ammonia method where $\Delta Q = 3.46$
(Flambaum \& Kozlov 2007; Levshakov {et al.}\ 2010a).
The measured LSR radial velocities along with their uncertainties
are given in Tables~\ref{T4} and \ref{T5}, whilst the fitting procedure is
illustrated in Figs.~\ref{fg3}~--~\ref{fg5}.
In these figures, vertical panels represent the 44 GHz (upper panel) and 95 GHz (lower panel) profiles of
methanol masers whose names and peak velocities of the 44 GHz line are indicated at the top of each block.
Some spectra were smoothed in order to improve the S/N ratio
in individual channels and the used channel width is indicated in the corresponding panel.
In all calculations we used the rest frequencies adopted in the original papers of K18
and Y20 (see Table~\ref{Tb2}):
$f^{\rm\scriptscriptstyle P}_{44} = 44069.430$ GHz from Pickett {et al.}\ (1998), and
$f^{\rm\scriptscriptstyle M}_{95} = 95169.463$ GHz from M\"uller {et al.}\ (2004).
If other sets of the rest frequencies listed in Table~\ref{Tb2} would be adopted,
then the velocity offsets, $\Delta V = V_{44} - V_{95}$,
change as shown in Table~\ref{T4}, columns 4-8. These values were calculated in the following way.
The KVN telescopes adopt the radio definition of radial velocity which is given by
\begin{equation}
V_{\rm\scriptscriptstyle LSR} = c\left(1 - \frac{f_{\rm obs}}{f_{\rm lab}}\right),
\label{Eq6}
\end{equation}
where $c$ is the speed of light.
If we now consider two methanol lines with laboratory rest frequencies $f'_{0,i}$ and $f'_{0,j}$
(``old'' reference frame)
which are observed at the corresponding sky frequencies $f_i$ and $f_j$ and have the LSR radial
velocities $V'_i$ and $V'_j$, then the difference between the measured velocities of these lines
is determined for a new set of laboratory frequencies $f_{0,i}$ and $f_{0,j}$
(``new'' reference frame) as
\begin{equation}
\Delta V_{ij} = \Delta V'_{ij} + \delta V_{ij},
\label{Eq7}
\end{equation}
where $\Delta V'_{ij} = V'_i - V'_j$ is the velocity offset between the lines $i$ and $j$
in the ``old'' reference frame, and
$\delta V_{ij}$ is the Doppler correction term between the ``old'' and ``new'' reference frames:
\begin{equation}
\delta V_{ij} = c\left( \frac{f'_{0,j}}{f_{0,j}} - \frac{f'_{0,i}}{f_{0,i}} \right).
\label{Eq8}
\end{equation}
The absolute values of the weighted mean velocity offsets under different sets of laboratory frequencies
$| \langle \Delta V \rangle_w |$ range from 0.030 km~s$^{-1}$\ to 0.176 km~s$^{-1}$\ (Table~\ref{T4}).
In terms of $\Delta\mu/\mu$\ (Eq.~\ref{Eq3}) these boundaries correspond to
$\Delta\mu/\mu$~= $3.0\times10^{-8}$ and $1.8\times10^{-7}$, respectively. However, the latter clearly exceeds the
upper limits on $\Delta\mu/\mu$\ found in the Galaxy: $\Delta\mu/\mu$~$< 7\times10^{-9}$ (L13) and $\Delta\mu/\mu$~$< 2\times10^{-8}$ (D17).
This means that the uncertainties in the JPL, NIST, and CDMS catalogues
as well as those reported in Tsunekawa {et al.}\ (1995) seem to be far too small
since the published rest frequencies lead to unrealistically large estimates of $\Delta\mu/\mu$~$\sim 10^{-7}$.
Accounting for all these details, we list in Table~\ref{T5} only those
$V_{\rm\scriptscriptstyle LSR}$ and $\Delta V$\ values which
are calculated with rest frequencies taken from the original paper of Y20.
Our final sample consists of 7 sources from K18 and 11 sources from Y20
and the source BGPS6820 provides two peaks (see Fig.~\ref{fg5}),
i.e., in total we have 19 velocity offsets.
\subsection{Spatial gradient of $\Delta\mu/\mu$}
\label{SSec34}
For each of the selected targets the measured velocity offset $\Delta V$~= $V_{44} - V_{95}$
is depicted in the upper panel of Fig.~\ref{fg6} against the target's distance $R$ from the Galactic centre.
The blue squares and red dots represent the sources from Tables~\ref{T4} and \ref{T5},
respectively. The indicated numbers correspond to the numbering in Table~\ref{Tb1}.
The source BGPS6820 is represented by two red dots $15_1$ and $15_2$.
It is seen that there is a clustering of points into two groups what hints to
the possible bimodality of the underlying distribution of the velocity offsets.
This hypothesis is statistically tested (by $\chi^2$-criterium) as illustrated in the lower panel
of Fig.~\ref{fg6}.
The unimodal distribution (black curve) with the mean $\langle \Delta V \rangle = 0.024$ km~s$^{-1}$, dispersion
$\sigma = 0.012$ km~s$^{-1}$, and the number of degrees of freedom $n = 7$ gives $\chi^2 = 9.9$,
which corresponds to a significance (probability) of 20\%.
On the other hand, the bimodal distribution (red curve) with $n = 5$,
two separate means $\langle \Delta V \rangle_1 = 0.032$ km~s$^{-1}$,
$\langle \Delta V \rangle_2 = 0.010$ km~s$^{-1}$, and dispersions $\sigma_1 = 0.006$ km~s$^{-1}$,
$\sigma_2 = 0.004$ km~s$^{-1}$\ delivers $\chi^2 = 3.9$ with the significance of 60\%.
Thus, we separate the data into two subsamples with $N = 12$ and $N = 7$ points.
The corresponding sample means and their $1\sigma$ errors are
$\langle \Delta V \rangle_1 \equiv \bar{x} = 0.0316\pm0.0018$ km~s$^{-1}$, and
$\langle \Delta V \rangle_2 \equiv \bar{y} = 0.0104\pm0.0015$ km~s$^{-1}$,
(marked by the horizontal dashed and dotted lines in the upper panel of Fig.~\ref{fg6}).
The weighted means and their errors are similar:
$\langle \Delta V \rangle_{1,w} = 0.0321\pm0.0014$ km~s$^{-1}$, and
$\langle \Delta V \rangle_{2,w} = 0.0086\pm0.0009$ km~s$^{-1}$.
The significance of this difference in terms of Student's $t$-test
with $n = 17$ degrees of freedom is about $7\sigma$:
$\delta = (\bar{x}-\bar{y}) \pm \sigma(\bar{x}-\bar{y}) = 0.022 \pm 0.003$ km~s$^{-1}$.
Additional arguments in support of two groups
are that each of them contains points from both the K18 and Y20 surveys
and that there is no correlation with the galactocentric distances, $R$, and/or
the bolometric luminosities, $L_{\rm bol}$, listed in Table~\ref{Tb1}.
This minimizes the probability of possible observational selection and systematic biases.
We note that the revealed bimodality does not depend on the rest frequencies
of the 44 and 95 GHz lines
since any combination of the rest frequencies listed in Table~\ref{Tb2} would simply lead to
a parallel shift of all points in the upper panel of Fig.~\ref{fg6} along the $Y$-axis.
With our adopted set of the rest frequencies
the weighted means for $\Delta\mu/\mu$\ for each group are the following:
$(\Delta \mu/\mu)_{1,w} = (32.5\pm1.4)\times10^{-9}$
and $(\Delta \mu/\mu)_{2,w} = (8.6\pm0.9)\times10^{-9}$.
Table~\ref{Tb1} shows that the targets are distributed at galactocentric distances $4.0 \la R \la 12.3$ kpc.
In this range, the input of the baryonic gravitational potential to the circular velocity of the Galactic
rotation falls from 60-70\% at $R \sim 4$ kpc to 30-50\% at $R \sim 12$ kpc,
whereas the input attributed to the dark matter increases correspondingly
(e.g., Eilers {et al.}\ 2019; Bobylev {et al.}\ 2021; Nitschai {et al.}\ 2021).
If the putative coupling between the baryonic and dark matter exists, then one can expect
some dependence of $\Delta\mu/\mu$\ on $R$.
The linear regression analysis of the velocity offsets for both groups returns
similar constraints on the gradient of $\Delta V$:
$(k_{\scriptscriptstyle \Delta V})_1 = -0.0015 \pm 0.0020$ km~s$^{-1}$~kpc$^{-1}$ in the
range $4.0 \la R \la 12.3$ kpc, and
$(k_{\scriptscriptstyle \Delta V})_2 = -0.0002 \pm 0.0027$ km~s$^{-1}$~kpc$^{-1}$ in the
range $4.5 \la R \la 8.7$ kpc,
what gives an upper limit on the gradient $k_\mu$ of $\Delta\mu/\mu$\ in the Milky Way disk:
$k_\mu < 2\times10^{-9}$ kpc$^{-1}$.
With this gradient, we constrain the value of $\Delta\mu/\mu$\
in the range $4.0 \la R \la 12.3$ kpc as $\Delta\mu/\mu$~$< 2\times10^{-8}$.
This estimate is only slightly better than
that reported by D17 and Ellingsen {et al.}\ (2011), but it is three times less tight as in L13 due to
poor statistics of our targets.
It is interesting to compare the obtained $k_\mu$ estimate with corresponding values from other experiments.
For example, the MICROSCOPE satellite mission also aimed at measuring the effects of possible non-universal coupling,
but in a completely different way: it placed two masses of different composition (titanium
and platinum alloys) on an orbit around the Earth
and measured their circular acceleration (torsion balance). The reported upper limit on
the non-standard coupling is $< 10^{-14}$ (Touboul {et al.}\ 2020).
With our estimate of the spatial gradient and the assumption of its isotropy one
obtains for the MICROSCOPE distance from the Earth centre (7000 km) an upper limit on the non-standard
coupling of $< 4\times10^{-22}$ which is tighter by eight orders of magnitude.
This shows once more the advantages of astrophysical spectroscopic methods.
\section{Maser action in the 44 and 95 GHz hyperfine components}
\label{Sec4}
The clustering of the measured velocity offsets $\Delta V$\ into two groups separated by $\delta = 22\pm3$ m~s$^{-1}$\
raises a question about the physical mechanism behind the observed effect.
Since the total nuclear spin angular momentum, {\bf I}, of the molecule
CH$_3$OH is non-zero for the torsion-rotation states,
each state of the symmetry A participating in the maser transition
is split up by hyperfine interactions in a pattern
of $2I+1$ hyperfine states each of which is characterized by the total angular momentum
{\bf F} = {\bf J} + {\bf I}, with {\bf J} being the rotational angular momentum.
For transitions involving $A$-type levels with total nuclear spin $I = 1$ and 2,
the shift of 22 m~s$^{-1}$\ ($\sim 3$ and 7 kHz respectively, at 44 and 95 GHz) is comparable
with the hyperfine frequency shifts within the corresponding torsion-rotation pattern.
The frequency shifts, $\Delta f$, between the strongest hyperfine components with $\Delta F = \Delta J = 1$
(Lankhaar {et al.}\ 2016) are listed in Tables~\ref{T6} and \ref{T7} for, respectively,
the 44 and 95 GHz transitions and are schematically illustrated in Figs.~\ref{fg7} and \ref{fg8}.
These data show that the $\approx 22$ m~s$^{-1}$\ separation could occur if
the maser action is limited to a particular hyperfine
transition with the largest Einstein $A$-coefficient for spontaneous emission.
Namely, if the favoured transitions of the first group are
those with the smallest values of $F = J-2$,
$F = 5a \rightarrow 4a$ (44 GHz)
and $F = 6a \rightarrow 5a$ (95 GHz), whereas for the second group they are
those with the largest values of $F = J+2$,
$F = 9a \rightarrow 8a$ (44 GHz) and $F = 10a \rightarrow 9a$ (95 GHz),
then the velocity offset between the two groups would be 23 m~s$^{-1}$.
The masing process with the dominance of particular hyperfine transitions
has already been suggested by Lankhaar {et al.}\ (2018)
for interpretations of the circular-polarization observations
of Class~II methanol masers at 6.7 GHz ($5_1 \rightarrow 6_0 A^+$).
Two favoured transitions $F = 3 \rightarrow 4$ ($F = J-2$)
and $F = 7 \rightarrow 8$ ($F = J+2$)
with the largest Einstein $B$-coefficient for stimulated emission\footnote{The $B$ coefficient for
stimulated emission from an upper level 2 to a lower level 1 is defined as
$B_{21} = (\pi^2 c^3/\hbar\omega^3)A_{21}$ (e.g., Hilborn 1982).}
were used to deduce an average magnetic field strength $\langle |B| \rangle \approx 12$ mG
in protostellar disks which was in line with OH maser polarization observations.
However, including all CH$_3$OH hyperfine components would lead to a considerably larger magnetic
field strength $\langle |B| \rangle \approx 80$ mG.
Similar maser actions limited to favoured hyperfine transitions
were considered by Lankhaar {et al.}\ (2018) in their study of polarization
observations of Class~I methanol masers at 36 and 44 GHz
in the outflows of massive star-forming regions.
For the $E$-type levels $4_{-1} \rightarrow 3_0$ at 36 GHz (total nuclear spin $I = 0$ and 1),
the $F = 3 \rightarrow 2$ hyperfine line with the smallest value of $F = J-1$
was found to be dominating.
For the $A$-type levels at 44 GHz, one favoured transition was $F = 5a \rightarrow 4a$
as in our case, but for the other line there are two options with the largest
Einstein $B$-coefficients (see Table~\ref{T6}):
$F = 8a \rightarrow 7a$ and $F = 8b \rightarrow 7b$
with the intermediate values of
$F = J+1$ and the ratio $B_{8a \rightarrow 7a}/B_{8b \rightarrow 7b} = 1.02$.
The second option is shifted by only 5 m~s$^{-1}$\ from the
$F = 9a \rightarrow 8a$ transition which is a favoured transition for our case.
Probably, the observed polarization could be explained with this $F = 9a \rightarrow 8a$
hyperfine component as well.
Indeed, taking into account that masing in a spectral line occurs when population is inverted and
the absolute value of the optical depth in the line $\tau > 1$,
emission in the $F = 9a \rightarrow 8a$ component should dominate since $(i)$
the ratio of the Einstein $A$-coefficients
$A_{9a \rightarrow 8a}/A_{8 \rightarrow 7} \approx 2$, and $(ii)$
the intensity of the maser emission is exponentially dependent on $\tau$.
In any case we conclude that the dominance of different hyperfine transitions
in methanol masers should be somehow related to molecular spin alignment within hyperfine structures
when all nuclear spins are of the same sign and the total angular momentum reaches its marginal values.
The orientation of atomic and molecular spins in the interstellar medium was
considered for the first time by Varshalovich (1971) and later on in a series of publications
of different authors (e.g., Burdyuzha \& Varshalovich 1973;
Landolfi \& Landi Degl'Innoceni 1986; Matveenko {et al.}\ 1988;
Yan \& Lazarian 2007; Zhang {et al.}\ 2020).
The main physical process behind this phenomenon is, in short, the following.
Atomic and molecular species in isotropic media have randomly oriented spins.
If, however, there is a directed beam of radiation or fast particles then interaction
with the beam will compel the spins to be aligned preferentially in one direction.
In order for spins to be aligned, random collisions with the surrounding gas particles
should not be too effective to flip spins over and thus to randomize their orientation.
In a magnetized medium, each hyperfine level of the methanol molecule with an
angular momentum $F$ will split into $2F + 1$ magnetic sublevels with different energy
depending on the positive or negative Land\'e factor (Lankhaar {et al.}\ 2018).
Then a difference in populations of hyperfine levels (i.e., the dominance of some of them)
will imply that the spins of the molecules are aligned.
As the projection of the photon spin, $s$,
on its direction of motion is always fully oriented, $s = \pm1$,
whereas the photon state with $s = 0$
is absent due to the transverse character of electromagnetic waves,
the unpolarized beam contains an equal number of photon states with left- and right-hand
circular polarization.
By virtue of this, when a photon is scattered by a molecule,
the spin of the molecule either becomes
aligned anisotropically with an equal number of spins oriented towards or against
the axis of symmetry, or becomes polarized with an unequal number of spins
aligned predominantly in one direction.
And if a magnetic field is present, it will control the orientation of the
plane of polarization. It was shown that a maser with anisotropic pumping
can achieve 100\% polarization (Western \& Watson 1983, 1984; Watson 2009).
We suppose that such polarization of spins can explain the clustering of
the velocity offsets detected in the present work. The involved physical interactions
are still not clear in full detail, but such an analysis which requires extended
calculations is beyond the scope of this paper.
\section{Summary and future prospects}
\label{Sect5}
Looking for the signs of possible non-universal coupling of hypothetical hidden field(s)
to standard matter, we estimate the fractional changes in the electron-to-proton mass ratio,
$\mu = m_{\rm e}/m_{\rm p}$,
where the mass of the electron is predicted to be directly affected by the Higgs-like coupling and
the mass of the proton is determined mainly by the binding energies of quarks.
The measurements are based on observations of Class~I methanol maser transitions in
sources distributed in the Milky Way disk
over the range of the galactocentric distances $4 \la R \la 12$ kpc.
The sources were selected from two surveys performed
by Kim {et al.}\ (2018) and Yang {et al.}\ (2020) at
the Korean VLBI Network (KVN) 21-m telescopes in single-dish telescope mode.
Observed were the Class~I $A$-type methanol transitions $7_0 - 6_1$~A$^+$
at 44 GHz, and $8_0 - 7_1$~A$^+$ at 95 GHz.
The value of $\Delta\mu/\mu$\ is measured through the radial velocity offset $\Delta V = V_{44} - V_{95}$
according to Eq.~\ref{Eq3}.
Our main results are as follows.
\begin{itemize}
\item[$\bullet$]
Observations of the 44 GHz line separated by a period of one year reveal
a remarkable stability of the line position with an uncertainty of only $\pm2$ m~s$^{-1}$.
\item[$\bullet$]
The measured velocity offsets between the simultaneously measured 44 GHz and 95 GHz lines
are clustered into two groups with the mean values separated by
$\delta_{\scriptscriptstyle \Delta V} = 0.022 \pm 0.003$ km~s$^{-1}$\ ($1\sigma$ C.L.).
The presence of two distinguished groups
can be explained if the methanol maser action favors
the following hyperfine transitions:
$F = 5a \rightarrow 4a$ and $6a \rightarrow 5a$ at, respectively, 44 and 95 GHz for the first mode
with the smallest value of $F = J-2$,
and
$F = 9a \rightarrow 8a$ and $10a \rightarrow 9a$ at, respectively, 44 and 95 GHz for the second mode
with the largest value of $F = J+2$.
By this we confirm the suggestion
of Lankhaar {et al.}\ (2018) that masing involves preferred hyperfine transitions.
The revealed bimodality also confirms
that the emission from the
44 and 95 GHz transitions arises in the same environment and
is highly cospatial.
\item[$\bullet$]
The measured $\Delta V$\ values constrain the
spatial gradient $k_\mu$ of $\Delta\mu/\mu$\ in the Galactic disk
$k_\mu < 2\times10^{-9}$ kpc$^{-1}$
in the range of the galactocentric distances $4 \la R \la 12$ kpc,
while the upper limit on the changes in $\mu$ is $\Delta\mu/\mu$~$<2\times10^{-8}$.
These are the tightest constraints on the spatial $\mu$ variability at present.
\item[$\bullet$]
The rest frequencies of the 44 and 95 GHz methanol transitions
reported in the NIST, NRAO, JPL, and CDMS molecular data bases are given
with underestimated errors.
\end{itemize}
According to these results, future prospects should be the following:
\begin{itemize}
\item[$-$]
The 44 and 95 GHz methanol transitions turned out to be especially
suitable for both $\Delta\mu/\mu$\ estimations and studies of the methanol masing mechanisms.
However, any further development can be possible only if
new laboratory measurements of methanol rest frequencies
with uncertainties of $\sim 1$ kHz will be carried out.
\item[$-$]
Class~I methanol masers have numerous transitions within the millimeter-wavelength range.
Other Class~I methanol maser pairs, such as the $4_{-1}-3_0$~$E$ line
at 36 GHz and the $5_{-1}-4_0$~$E$ transition at 84 GHz
and the series of $J_2-J_1$~$E$ lines near 25 GHz, could also be suitable to estimate $\Delta\mu/\mu$.
\item[$-$]
Observations used in the present work were obtained in course of big surveys and in general were not intended for
high precision measurements of the line positions.
It is desirable to reobserve with higher S/N and better spectral resolution
at least the selected targets.
\item[$-$]
High spectral resolution polarization measurements
can be also used to obtain quantitative characteristics
of the revealed two groups of Class~I methanol masers.
\item[$-$]
Maser sources being observed with high angular resolution exhibit a complex spatial structure consisting of
multiple spots. That is why interferometric observations of such sources would be of great importance since they
make it possible to control results towards a given target using the measurements of the resolved spots.
\end{itemize}
\section*{Acknowledgements}
S.A.L. and M.G.K were supported by the Russian Science
Foundation under grant No.~19-12-00157.
\section*{Data Availability}
The data underlying this article will be shared on reasonable request to the
corresponding author.
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2024-02-18T23:40:24.732Z
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2021-12-30T02:27:59.000Z
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{"paloma_paragraphs":[]}
|
{"arxiv_id":"2112.14560","language":"en","timestamp":1640831279000,"url":"https:\/\/arxiv.org\/abs\/2112.14560","yymm":"2112"}
|
proofpile-arXiv_000-10074
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{"provenance":"002.jsonl.gz:10075"}
| null | null |
\section{Introduction}
Let $\mu:{\mathbb N}\to\{-1,0,+1\}$ denote the M\"obius function, defined multiplicatively on primes $p$ by $\mu(p)=-1$ and $\mu(p^k)=0$ for $k\ge2$. Many central results in number theory may be formulated in terms of averages of the M\"obius function. Notably, the prime number theorem is equivalent to the statement $\sum_{n\le X}\mu(n) = o(X)$, and $\sum_{n\le X}\mu(n) = O(X^\theta)$ for all $\theta>\frac{1}{2}$ is equivalent to the Riemann hypothesis.
Clearly $\mu(p)=-1$ gives $\sum_{p\le X}\mu(p) = -\pi(X)$, but less is known about the M\"obius function on shifted primes. It is a folklore conjecture that $\sum_{p\le X}\mu(p+h) \ = \ o(\pi(X))$ for any fixed shift $h>0$ (this appeared in print at least since Murty--Vatwani \cite[(1.2)]{MurtyV}, and the case $h=1$ is Problem 5.2 of \cite{Sarnk}). We answer an averaged version of this conjecture with quantitative bounds.
\begin{theorem}\label{thm:main}
If $\log H/\log_2 X \to\infty$ as $X\to\infty$, then
\begin{align}\label{eq:mainqual}
\sum_{h\le H}\bigg|\sum_{p\le X}\mu(p+h)\bigg| = o(H\pi(X)).
\end{align}
Further if $H = X^\theta$ for some $\theta\in (0,1)$, then for all $\delta>0$
\begin{align*}
\sum_{h\le H}\bigg|\sum_{p\le X}\mu(p+h)\bigg| \ \ll_\delta \ \frac{H\pi(X)}{(\log X)^{1/3-\delta}}.
\end{align*}
\end{theorem}
An immediate consequence is that
$\sum_{p<X}\mu(p+h)$ exhibits cancellation for all but $o(H)$ values of $h\le
H=(\log X)^{\psi(X)}$ provided $\psi(X)\rightarrow \infty$.
\begin{remark}
The weaker qualitative cancellation \eqref{eq:mainqual} in the longer regime $H=X^\theta$ may be obtained more directly, using a recent Fourier uniformity result of Matom\"aki-Radziwi\l\l-Tao \cite{MRTUnif}. See Theorem \ref{thm:pretend} for details.
\end{remark}
Theorem \ref{thm:main} is an illustrative example within a broader class of correlations that may be handled by the methods in this paper, see Theorem \ref{thm:mainChowla} for the full technical result. Below we highlight some further example correlations of general interest.
\subsection{Higher correlations}
The influential conjectures of Chowla \cite{Chowla} and Hardy--Littlewood \cite{HardLittl} assert that for any fixed tuple $\mathcal H = \{h_1,.., h_k\}$ of distinct integers,
\begin{align*}
\sum_{n\le X}\mu(n+h_1)\cdots\mu(n+h_k) \ &= \ o(X),\\
\sum_{n\le X}\Lambda(n+h_1)\cdots\Lambda(n+h_k) \ & = \ \frak{S}(\mathcal H)X \ + \ o(X),
\end{align*}
for the singular series $\frak{S}(\mathcal H) = \prod_{p}\frac{(1-\nu_p/p)}{(1-1/p)^k}$, where $\nu_p = \#\{h_1,..,h_k (\text{mod }p)\}$. Both conjectures remain open for any $k\ge2$.
We establish an average result for Chowla--Hardy--Littlewood correlations.
\begin{theorem}\label{thm:Chowktuple}
Suppose $\log H/\log_2 X\to\infty$ as $X\to\infty$, and define $\psi_\delta$ by
\begin{align}\label{eq:psistar}
\psi_\delta(X) = \min\Big\{\frac{\log_2 X}{\log H},(\log X)^{1/3-\delta}\Big\}.
\end{align}
Then for any $m,k\ge1$, and fixed tuple $\mathcal A = \{a_1,..,a_k\}$ of disinct integers, we have
\begin{align}\label{thm:Chowktuple}
\sum_{h_1,..,h_m\le H}\bigg|\sum_{n\le X} \prod_{j=1}^m \mu(n+h_j)\prod_{i=1}^k\Lambda(n+a_i)\bigg| \ \ll_{m,\delta,\mathcal A} \ \frac{XH^m}{\psi_\delta(X)^m}.
\end{align}
\end{theorem}
It is worth emphasizing particular aspects of this result. First, \eqref{thm:Chowktuple} holds for an arbitrary fixed prime $k$-tuple. We must average over at least $m\ge1$ copies of $\mu$ in order to obtain cancellation. Notably, the cancellation becomes quantitatively stronger for larger $m$, e.g. error savings $(\log X)^{m/3-\delta}$. For the case $m=0, k=2$, Matom\"aki-Radziwi\l\l-Tao \cite{MRTCor} handled binary correlations $\sum_{n\le X}\Lambda(n)\Lambda(n+h)$ on average with error savings $(\log X)^A$, though in the much larger regime $H \ge X^{8/33+{\varepsilon}}$.
In particular, the Chowla conjecture holds on average along the subsequence of primes.
\begin{corollary}\label{cor:Chow}
Suppose $\log H/\log_2 X\to\infty$ as $X\to\infty$. Then for any $m\ge1$,
\begin{align}
\sum_{h_1,..,h_m\le H}\bigg|\sum_{p\le X}\mu(p+h_1)\cdots\mu(p+h_m)\bigg| \ = \ o_m\big(\pi(X)H^m\big).
\end{align}
\end{corollary}
Moreover, using Markov's inequality we may obtain qualitative cancellation for almost all shifts, with arbitrary log factor savings in the exceptional set.
\begin{corollary}\label{cor:exceptmu}
Suppose $\log H/\log_2 X\to\infty$ as $X\to\infty$. Then for any $A>0$,
\begin{align*}
\sum_{p\le X}\mu(p+h_1)\cdots\mu(p+h_m) \ = \ o_m(\pi(X)),
\end{align*}
for all except $O_A(H^m(\log X)^{-A})$ shifts $(h_1,..,h_m)\in [1,H]^m$.
\end{corollary}
These results build on earlier work of Matom\"aki-Radziwi\l\l-Tao \cite{MRTChow}, who established an average form of Chowla's conjecture,
\begin{align}\label{eq:MRTchow}
\sum_{h_1,..,h_m\le H}\bigg|\sum_{n\le X}\mu(n+h_1)\cdots\mu(n+h_m)\bigg| \ = \ o_m(XH^m),
\end{align}
for any $H=H(X)\to\infty$ arbitrarily slowly. Whereas, our results require the faster growth $H=(\log X)^{\psi(X)}$ with $\psi(X)\to\infty$ arbitrarily slowly.
\subsection{Correlations with divisor functions}
Consider fixed integers $a\ge1$, $k \ge l\ge 2$. The well studied correlation of two divisor functions $d_k,d_l$ is predicted to satisfy
\begin{align*}
\sum_{n\le X}d_k(n+h)d_l(n) \ = \ C_{k,l,h} \cdot \big(X \ + \ o(X)\big)(\log X)^{k-l-2},
\end{align*}
for a certain (explicit) constant $C_{k,l,h}>0$. Recently, Matom\"aki-Radziwi\l\l-Tao \cite{MRTCor2} have shown the following averaged result, in the regime $H \ge (\log X)^{10000 k\log k}$,
\begin{align*}
\sum_{h\le H}\Big|\sum_{n\le X}d_k(n+h)d_l(n) \ - \ C_{k,l,h}\cdot X(\log X)^{k-l-2}\Big| \ &= \ o_k(HX(\log X)^{k+l-2}).
\end{align*}
For higher correlations of divisor functions with M\"obius, we obtain
\begin{theorem}\label{cor:divisorcorr}
For any $j\ge1$, $k_1,..,k_j\ge2$, let $k=\sum_{i=1}^j k_i$ and take any fixed tuple $\mathcal A = \{a_1,..,a_j\}$ of distinct integers. If $\log H/\log_2 X \to\infty$, then
\begin{align*}
\sum_{h\le H}\Big|\sum_{n\le X}\mu(n+h)\prod_{i=1}^jd_{k_i}(n+a_i)\Big| \ & = \ o_{k,\mathcal A}\big(HX(\log X)^{k-j}\big).
\end{align*}
\end{theorem}
Again, we emphasize the need to average over the shift $h$ that inputs to M\"obius $\mu(n+h)$, while $a_i$ may be fixed arbitrarily.
\begin{remark}
For simplicity, the results are stated for the M\"obius function $\mu$, but our results hold equally for its completely multiplicative counterpart, the Liouville function $\lambda$. In fact, the proof strategy is to reduce from $\mu$ to $\lambda$.
\end{remark}
The main number-theoretic input is the classical Vinogradov-Korobov zero-free region
\begin{align}\label{eq:VinKor}
\bigg\{\sigma + it \; : \; 1-\sigma < \frac{c}{\max\big\{\log q,\,\log(|t|+3)^{2/3}\log\log(|t|+3)^{1/3}\big\}} \bigg\}
\end{align}
for $L(s,\chi)$, where $\chi$ is a Dirichlet character of modulus $q\le (\log X)^A$ in the Siegel-Walfisz range, see \cite[\S8]{IK}.
\subsection{Beyond M\"obius}
We also consider general multiplicative functions $f:{\mathbb N}\to{\mathbb C}$, which do not pretend to be a character $f(n)\approx n^{it}\chi(n)$ for some $\chi$ (mod $q$). More precisely, we follow Granville and Soundararajan \cite{GranSound} and define the pretentious distance
\begin{align*}
{\mathbb D}(f,g;X) = \bigg(\sum_{p\le X}\frac{1-\Re(f(p)\overline{g(p)})}{p}\bigg)^{1/2},
\end{align*}
and the related quantity
\begin{align}
M(f;X,Q) = \inf_{\substack{|t|\le X\\\chi\,(q), \, q\le Q}} {\mathbb D}\big(f,\, n\mapsto n^{it}\chi(n); X\big)^2.
\end{align}
We may apply recent work of Matom\"aki-Radziwi\l\l-Tao \cite{MRTUnif} on Fourier uniformity, in order to more directly obtain (qualitative) cancellation for averages of non-pretentious multiplicative functions over shifted primes.
\begin{theorem}\label{thm:pretend}
Given $\theta\in(0,1)$ let $H = X^\theta$. Given a multiplicative function $f:{\mathbb N}\to{\mathbb C}$ with $|f|\le1$. There exists $\rho\in (0,\frac{1}{8})$ such that, if $M(f;X^2/H^{2-\rho},Q) \to \infty$ as $X\to \infty$ for each fixed $Q>1$, then
\begin{align*}
\sum_{h\le H}\Big|\sum_{p\le X}f(p+h)\Big| \ = \ o_{\theta,\rho}\big(H\pi(X)\big).
\end{align*}
\end{theorem}
In particular, $f=\mu$ does not pretend to be a Dirichlet character, a fact equivalent to the prime number theorem in arithmetic progressions. Indeed,
\begin{align*}
M(\mu; X,Q) \ge \inf_{\substack{|t|\le X\\\chi\,(q), \, q\le Q}} \sum_{e^{(\log X)^{2/3+{\varepsilon}}}\le p\le X}\frac{1+\Re\chi(p)p^{it}}{p} \ \ge \ \Big(\frac{1}{3}-{\varepsilon}\Big)\log\log X + O(1),
\end{align*}
where the latter inequality is well-known to follow from the zero-free region \eqref{eq:VinKor}.
\subsection{Overview of the proof of Theorem \ref{thm:main}}
We now indicate the general form of the proof. We pursue a variation on the approach of Matom\"aki-Radziwi\l\l-Tao \cite{MRTChow}. Namely, we first restrict \eqref{eq:mainqual} to `typical' terms $\mu(n)$ for $n=p+h\in\mathcal S$ that have prime factors lying in certain prescribed intervals $[P_1,Q_1], [P_2,Q_2]$. The terms with $n\notin \mathcal S$ are sparse, and thus may be shown to contribute negligibly by standard sieve estimates. (For higher correlations, one may also use sieve estimates, along with work of Henriot \cite{Hen} to handle a general class of functions with `moderate growth' that are `amenable to sieves.')
Once reduced to numbers with `typical factorization,' we decouple the short interval correlation between M\"obius and the indicator for the primes, using a Fourier identity and applying Cauchy-Schwarz (Lemma \ref{lem:Fourier}). This yields a bound of $\pi(X)\ll X/\log X$ times a Fourier-type integral for $\mu$,
\begin{align*}
\sup_\alpha \int_0^X \bigg|\sum_{\substack{x\le n\le x+H\\n\in \mathcal S}} \mu(n)e(n\alpha)\bigg|\dd{x}.
\end{align*}
This decoupling step is a gambit. It has the advantage of only needing to consider $\mu$ on its own, but loses a factor of $\log X$ from the density of the primes. To make this gambit worthwhile, we must recover over a factor of $\log X$ savings in the above Fourier integral for $\mu$. However, Matom\"aki-Radziwi\l\l-Tao \cite[Theorem 2.3]{MRTChow} bound the above integral with roughly $(\log X)^{\frac{1}{500}}$ savings (though their bound holds for any non-pretentious multiplicative function $g$.) Therefore we must refine Matom\"aki-Radziwi\l\l-Tao's argument in the special case of $g=\mu$ to win back over a full factor of $\log X$. We note this task is impossible unless $H$ is larger than a power of log (this already hints at why we must assume $\log H/\log\log X\to\infty$).
We accomplish this task in the `key Fourier estimate' (Theorem \ref{thm:mainFourier}), which bounds the above integral with $(\log X)^A$ savings for any $A>0$ (though $\mathcal S$ will implicitly depend on $A$). As with \cite{MRTChow}, this bound is proven by reducing to the analogous estimate with the completely multiplicative $\lambda$, and splitting up $\alpha\in[0,1]$ into major and minor arcs.
The main technical innovation here comes from the major arcs (Proposition \ref{prop:MRmain}), essentially saving a factor $(\log X)^A$ in the mean values of `typical' Dirichlet polynomials of the form
\begin{align*}
\sum_{\substack{X\le n\le 2X\\n\in \mathcal S}}\frac{\lambda(n)\chi(n)}{n^s}
\end{align*}
for a character $\chi$ of modulus $q\le (\log X)^A$ in the Siegel-Walfisz range. This refines the seminal work of Matom\"aki-Radziwi\l\l \ \cite{MR}, who obtained a fractional power of log savings for the corresponding mean values. However, Matom\"aki-Radziwi\l\l's results apply to the general setting of (non-pretentious) multiplicative functions and appeal to Hal\'asz's theorem, which offers small savings. By contrast, our specialization to the M\"obius function affords us the full strength of Vinogradov-Korobov estimates (Lemma \ref{lem:MR2l}).
The Matom\"aki-Radziwi\l\l \ method saves roughly a fractional power of $P_1$ in the Dirichlet mean value when $Q_1\approx H$. So in order to recover from our initial gambit, we are prompted to choose $P_1=(\log X)^C$ for some large $C>0$. Then by a standard sieve bound the size of $\overline{\mathcal S}$ is morally $O(\frac{\log P_1}{\log Q_1}) = O_C(\frac{\log\log X}{\log H})$. This highlights the need for our assumption $\log H/\log\log X\to\infty$.
We remark that the Matom\"aki-Radziwi\l\l \ method requires two intervals $[P_1,Q_1], [P_2,Q_2]$ (that define $\mathcal S$) in order to handle `typical' Dirichlet polynomials in the regime $H=(\log X)^{\psi(X)}$ for $\psi(X)\to\infty$. Note in general \cite{MR} the slower $H\to\infty$ the more intervals we require (though by a neat short argument \cite{MRshort}, only one interval is needed in the regime $H=X^\theta$ for $\theta>0$).
\section*{Notation}
The M\"obius function is defined multiplicatively from primes $p$ by $\mu(p)=-1$ and $\mu(p^k)=0$ for $k\ge2$. Similarly the Liouville function $\lambda$ is defined {\it completely} multiplicatively by $\lambda(p)=-1$.
We use standard asymptotic notation: $X\ll Y$ and $X = O(Y)$ both mean $|X|\le CY$ for some some absolute constant $C$, and $X\asymp Y$ means $X\ll Y\ll X$. If $x$ is a parameter tending to infinity, $X = o(Y)$ means that $|X| \le c(x)Y$ for some quantity $c(x)$ that tends to zero as $x\to\infty$. Let $\log_k X = \log_{k-1} (\log X)$ denote the $k$th-iterated logarithm.
Unless otherwise specified, all sums range over the integers, except for sums over the variable $p$ (or $p_1$, $p_2$,..) which are understood to be over the set of primes $\mathbb P$. Let $e(x) := e^{2\pi i x}$.
We use ${\mathbf 1}_S$ to denote the indicator of a predicate $S$, so ${\mathbf 1}_S = 1$ if $S$ is true and ${\mathbf 1}_S = 0$ if $S$ is false. When $\mathcal S$ is a set, we write ${\mathbf 1}_{\mathcal S}(n) = {\mathbf 1}_{n\in \mathcal S}$ as the indicator function of $\mathcal S$. Also let ${\mathbf 1}_\mathcal S f$ denote the function $n\mapsto{\mathbf 1}_\mathcal S(n) f(n)$.
\section{Initial reductions}
We begin with a Fourier-type bound to decouple correlations of arbitrary functions.
\begin{lemma}[Fourier bound]\label{lem:Fourier}
Given $f,g:{\mathbb N}\to{\mathbb C}$, let $F(X) := \sum_{n\le X}|f(n)|^2$. Then
\begin{align}\label{eq:mainS}
\sum_{|h|\le H}\bigg|\sum_{n\le X}f(n)\,g(n+h)\bigg|^2 \ \ll \ F(X+2H)\cdot\sup_\alpha \int_0^X \bigg|\sum_{x\le n\le x+2H}g(n)e(n\alpha)\bigg|\dd{x}.
\end{align}
\end{lemma}
\begin{proof}
First, the lefthand side of \eqref{eq:mainS} is
\begin{align}
\sum_{|h|\le H}\bigg|\sum_{n\le X}f(n)\,g(n+h)\bigg|^2 \ll \ H^{-2}\sum_{|h|\le 2H}(2H-|h|)^2\bigg|\sum_{n\le X}f(n)\,g(n+h)\bigg|^2 \ =: H^{-2}\,\Sigma \label{eq:step1}.
\end{align}
Expanding the square in $\Sigma$ and letting $h = m-n=m'-n'$, we have
\begin{align*}
\Sigma &
= \sum_{|h|\le 2H}(2H-|h|)^2 \sum_{n,n'\le X} f(n)\overline{f}(n')g(n+h)\overline{g}(n'+h)\\
& \ = \sum_{n,n'\le X}\sum_{m,m'}f(n)\overline{f}(n')g(m)\overline{g}(m') {\mathbf 1}_{m-n=m'-n'} \cdot \Big(\int_0^X {\mathbf 1}_{x\le n,m\le x+2H}\dd x\Big)\Big(\int_0^X {\mathbf 1}_{x'\le n',m'\le x'+2H}\dd x'\Big).
\end{align*}
Then orthogonality ${\mathbf 1}_{m-n=m'-n'} = \int_0^1e((m-n-m'+n')\alpha)\dd{\alpha}$ gives
\begin{align*}
\Sigma = \int_0^1 & \int_0^X \sum_{x\le n,m\le x+2H}f(n)g(m)e\big((m-n)\alpha\big)\dd{x} \cdot
\int_0^X \sum_{x'\le n',m'\le x'+2H} \overline{f}(n')\overline{g}(m') e\big((n'-m')\alpha\big)\dd{x'} \dd{\alpha} \nonumber\\
& = \int_0^1 \bigg|\int_0^X \sum_{x\le n,m\le x+2H}f(n)g(m)e\big((m-n)\alpha\big)\dd{x}\bigg|^2 \dd{\alpha}.
\end{align*}
Using Cauchy-Schwarz, we bound $\Sigma$ as
\begin{align}
\Sigma \ & \le \ \int_0^1 \int_0^X \bigg|\sum_{x\le m\le x+2H}g(m)e(m\alpha)\bigg|^2\dd{x}\cdot\int_0^X \bigg|\sum_{y\le n\le y+2H}\overline{f}(n)e(n\alpha)\bigg|^2\dd{y} \dd{\alpha} \nonumber\\
& \ \ll \ H\bigg(\sup_\alpha \int_0^X \bigg|\sum_{x\le m\le x+2H} g(m)e(m\alpha)\bigg|\dd{x}\bigg) \int_0^1\int_0^X \bigg|\sum_{y\le n\le y+2H}\overline{f}(n)e(n\alpha)\bigg|^2\dd{y}\dd{\alpha}. \label{eq:step3}
\end{align}
Using $\int_0^1 e(n\alpha)\dd{\alpha} = {\mathbf 1}_{n=0}$ again, the second integral in \eqref{eq:step3} is
\begin{align*}
\int_0^1\int_0^X\bigg|\sum_{y\le n\le y+2H} & \overline{f}(n)e(n\alpha)\bigg|^2 \dd{y}\dd{\alpha}
= \int_0^X\sum_{y\le n,n'\le y+2H} \overline{f}(n)f(n')\int_0^1 e\big((n-n')\alpha\big)\dd{\alpha}\dd{y}\\
& = \int_0^X\sum_{y\le n\le y+2H}|f(n)|^2\dd{y}
= \sum_{n\le X+2H}|f(n)|^2\int_{n-2H}^n\dd{y}
\ll HF(X+2H).
\end{align*}
Hence plugging the bound \eqref{eq:step3} for $H^{-2}\Sigma$ back into \eqref{eq:step1} gives the result.
\end{proof}
Next we consider numbers with `typical factorization.'
For $A,\delta\ge 0$, define $\psi$ via $H = (\log X)^{\psi(X)}$ so that \eqref{eq:psistar} becomes
$$\psi_\delta(X) = \min\{\psi(X),(\log X)^{1/3-\delta}\}.$$
Consider the intervals
\begin{align}\label{eq:PQj}
[P_1,Q_1] &= [(\log X)^{33A},(\log X)^{\psi(X)-4A}],\\
[P_2,Q_2] &= [\exp\big((\log X)^{2/3+\delta/2}\big),\,\exp\big((\log X)^{1-\delta/2}\big)], \nonumber
\end{align}
and define the `typical factorization' set
\begin{align}\label{eq:S}
\mathcal S \ = \ \mathcal S(X,A,\delta) := \{n\le X : \exists \text{ prime factors } p_1,p_2\mid n \text{ with }p_j\in[P_j,Q_j]\}.
\end{align}
Using the Fourier bound, we shall reduce Theorem \ref{thm:main} to the following.
\begin{theorem}[Key Fourier estimate for $\mu$] \label{thm:mainFourier}
Given any $A>5$, $\delta>0$, let $\mathcal S \ = \ \mathcal S(X,A,\delta)$ as in \eqref{eq:S}. Then if $\log H/\log_2 X \to\infty$,
\begin{align*}
\sup_\alpha \int_0^X \bigg|\sum_{\substack{x\le n\le x+H\\n\in \mathcal S}} \mu(n)e(n\alpha)\bigg|\dd{x} \ \ll_{A,\delta} \ \frac{HX}{(\log X)^{A/5}}.
\end{align*}
\end{theorem}
\begin{proof}[Proof of Theorem \ref{thm:main} from Theorem \ref{thm:mainFourier}]
By a standard sieve upper bound \cite[Theorem 7.1]{Opera}, for each $h\,\le H$, $j=1,2$ we have
\begin{align}\label{eq:sievePQ}
\#\{\,p\,\le X \; : \; q\nmid p+h\,\forall q\in [P_j,Q_j]\} \ \ll \ \pi(X)\frac{\log P_j}{\log Q_j}\frac{h}{\phi(h)}.
\end{align}
Thus, recalling the choice of $[P_j,Q_j]$ in \eqref{eq:PQj}, the terms $p+h\notin\mathcal S$ trivially contribute to \eqref{eq:mainqual}
\begin{align}\label{eq:sieveph}
\sum_{h\le H} \Big|\sum_{\substack{p\le X\\p+h\notin\mathcal S}} \mu(p+h)\Big| \ \le \sum_{1\le j\le2}\underset{q\nmid p+h\,\forall q\in [P_j,Q_j]}{\sum_{h\le H}\sum_{p\le X}}1
& \ll \ \pi(X)\Big(\frac{A}{\psi(X)} + (\log X)^{\delta-1/3}\Big)\sum_{h\le H}\frac{h}{\phi(h)} \nonumber\\
& \ll_A \ \frac{\pi(X)}{\psi_\delta(X)}\sum_{h\le H}\frac{h}{\phi(h)} \ll_A \frac{H\pi(X)}{\psi_\delta(X)}.
\end{align}
On the other hand for $p+h\in \mathcal S$, Lemma \ref{lem:Fourier} with $f(n)={\mathbf 1}_{\mathbb P}(n)$, $g(n) = {\mathbf 1}_{\mathcal S}\mu(n)$ gives
\begin{align*}
\sum_{h\le H}\bigg|\sum_{\substack{p\le X\\p+h\in\mathcal S}}\mu(p+h)\bigg|^2 \ll \pi(X+2H)\cdot\sup_\alpha \int_0^X \bigg|\sum_{\substack{x\le n\le x+2H\\n\in\mathcal S}} \mu(n)e(n\alpha)\bigg|\dd{x} \ \ll_{A,\delta} \ \frac{HX^2}{(\log X)^{A/5+1}},
\end{align*}
assuming Theorem \ref{thm:mainFourier}. Thus by Cauchy-Schwarz we obtain
\begin{align}\label{eq:inS}
\sum_{h\le H}\bigg|\sum_{\substack{p\le X\\p+h\in\mathcal S}}\mu(p+h)\bigg|
\ & \ll \ \bigg(H\sum_{h\le H}\bigg|\sum_{\substack{p\le X\\p+h\in\mathcal S}}\mu(p+h)\bigg|^2\;\bigg)^{1/2}\ \ll \ \frac{HX}{(\log X)^{A/10+1/2}}.
\end{align}
Hence \eqref{eq:sieveph} and \eqref{eq:inS} with $A = 6$ give Theorem \ref{thm:main}.
\end{proof}
Let $W = (\log X)^A$. Recall Theorem \ref{thm:mainFourier} asserts that $M_H(X) \ll XH/W^{1/5}$ for
\begin{align*}
M_{H}(X) := \sup_\alpha \int_0^X \bigg|\sum_{\substack{x\le n\le x+H\\n\in \mathcal S}} \mu(n)e(n\alpha)\bigg|\dd{x}.
\end{align*}
We first note, that, for technical convenience, it suffices to establish $M_{H_0}(X) \ll XH_0/W^{1/5}$ with $H_0:=\min\{H,\exp((\log X)^{2/3})\}$. Indeed, if $H>H_0$ then by the triangle inequality
\begin{align*}
M_{H}(X) \le \sum_{k\le \lceil H/H_0\rceil}M_{H_0}(X+kH_0) \ll \sum_{k\le \lceil H/H_0\rceil}\frac{(X+kH_0)H_0}{W^{1/5}} \ \ll \ \frac{H}{H_0}\cdot \frac{XH_0}{W^{1/5}} = \frac{XH}{W^{1/5}}
\end{align*}
as desired. Hence we may assume $H\le \exp((\log X)^{2/3})$ hereafter. This reduction is not strictly necessary, but will simplify the argument. For example, in this case $Q_1<P_2$ so the intervals $[P_j,Q_j]$ are disjoint.
Consider the `refined typical factorization' sets $\mathcal S_d=\{n/d : d\mid n\in \mathcal S\}$ for $d< P_1$, that is,
\begin{align}\label{eq:Sd}
\mathcal S_d \ = \ \mathcal S_d(X,A,\delta) = \{m\le X/d : \exists\text{ prime factors } p_1,p_2\mid m \text{ with }p_j\in[P_j,Q_j]\}.
\end{align}
So far we have reduced Theorem \ref{thm:main} to Theorem \ref{thm:mainFourier} for $\mu$. We now reduce further to the analogous estimate for its completely multiplicative counterpart $\lambda$.
\begin{proposition}[Key Fourier estimate for $\lambda$] \label{prop:compmult}
Given any $A>5$, $\delta>0$, $H=(\log X)^{\psi(X)}$ with $\psi(X)\to\infty$ and $\psi(X)\le (\log X)^{2/3}$. For $d\le W=(\log X)^A$ and $\mathcal S_d = \mathcal S_d(X,A,\delta)$ as in \eqref{eq:Sd}, we have
\begin{align*}
\sup_{\alpha}\int_0^X \bigg|\sum_{\substack{x\le nd\le x+H\\n\in \mathcal S_d}} \lambda(n)e(\alpha n)\bigg|\dd{x} \ \ll_{A,\delta} \ \frac{HX}{d^{3/4}W^{1/5}}.
\end{align*}
\end{proposition}
\begin{proof}[Proof of Theorem \ref{thm:mainFourier} from Proposition \ref{prop:compmult}.]
By M\"obius inversion, we have $\mu=\lambda\ast h$ for $h = \mu\ast (\mu\lambda)$, where $\ast$ denotes Dirichlet convolution. That is, $h(d^2) = \mu(d)$ for squarefree $d$, and zero otherwise. Thus we may write
\begin{align*}
\sum_{x\le n\le x+H}{\mathbf 1}_{\mathcal S}(n)\mu(n)e(n\alpha)
& = \sum_{d\ge 1} h(d) \sum_{x\le md\le x+H}{\mathbf 1}_{\mathcal S}(md)\lambda(m)e(md\alpha),
\end{align*}
and so the triangle inequality gives
\begin{align}
\int_0^X \bigg|\sum_{x\le n\le x+H}{\mathbf 1}_{\mathcal S}(n)\mu(n)e(n\alpha)\bigg|\dd{x} & \le \sum_{d\ge 1} |h(d)| \int_0^X \bigg|\sum_{x\le md\le x+H}{\mathbf 1}_{\mathcal S}(md)\lambda(m)e(md\alpha)\bigg|\dd{x}. \label{eq:intSfe}
\end{align}
Note, using the trivial bound and swapping the order of summation and integration, the contribution of $d> W$ to \eqref{eq:intSfe} is
\begin{align}\label{eq:moreW}
\ll \ \sum_{W<d\le X} |h(d)| \sum_{md\le X+H}H \ \ll \ \frac{HX}{W^{1/4}}\sum_{d\ge 1} \frac{|h(d)|}{d^{3/4}} \ \ll \ \frac{HX}{W^{1/4}}\sum_{d\ge 1} d^{-3/2} \ \ll \ \frac{HX}{W^{1/4}}
\end{align}
since $|h|\le1$ is supported on squares.
On the other hand the contribution of $d\le W$ to \eqref{eq:intSfe} is
\begin{align}\label{eq:lessW}
\le \sum_{d\ge 1} |h(d)| \int_0^X \bigg|\sum_{x\le md\le x+H}{\mathbf 1}_{\mathcal S_d}(m)\lambda(m)e(md\alpha)\bigg|\dd{x}
\ll \frac{HX}{W^{1/5}}\sum_{d\le W} \frac{|h(d)|}{d^{3/4}} \ \ll \ \frac{HX}{W^{1/5}}
\end{align}
assuming Proposition \ref{prop:compmult}, and noting ${\mathbf 1}_{\mathcal S}(md) = {\mathbf 1}_{\mathcal S_d}(m)$ since $d\le W<P_1$. Together \eqref{eq:moreW} and \eqref{eq:lessW} give Theorem \ref{thm:mainFourier}.
\end{proof}
\section{Key Fourier estimate}
In this section, we establish Proposition \ref{prop:compmult} by the circle method, following the argument in \cite[Proposition 2.4]{MRTChow}.
Take $\alpha\in [0,1]$. By Dirichlet's approximation theorem there exists $\frac{a}{q}\in{\mathbb Q}$ with $(a,q)=1$ and $1\le q\le Q_1$ for which
\begin{align*}
\Big|\alpha - \frac{a}{q}\Big| \ \le \ \frac{1}{qQ_1}.
\end{align*}
So we may split $[0,1]$ into major arcs $\frak M$ and minor arcs $\frak m$, according to the size of denominator $q$ compared to $W$,
\begin{align*}
\frak M = \bigcup_{q\le W}\mathfrak M(q)\quad \text{and}\quad
\frak m = [0,1] \setminus\frak M,
\end{align*}
where $\mathfrak M(q) = \bigcup_{(a,q)=1}\{\alpha : |\alpha-a/q|\le 1/qQ_1\}$.
Recall the definitions \eqref{eq:PQj}, \eqref{eq:Sd},
\begin{align*}
[P_1,Q_1] &= [(\log X)^{33A},(\log X)^{\psi(X)-4A}],\\
[P_2,Q_2] & = [\exp\big((\log X)^{2/3+\delta/2}\big),\,\exp\big((\log X)^{1-\delta/2}\big)],\\
\mathcal S_d(X,A,\delta) & = \{m\le X/d : \exists p_1,p_2\mid m \text{ with }p_j\in[P_j,Q_j]\}.
\end{align*}
We shall obtain Proposition \ref{prop:compmult} from the following results.
\begin{proposition}[Key minor arc estimate] \label{prop:minor}
Given any $A>5$, $H=(\log X)^{\psi(X)}$ with $\psi(X)\to\infty$ and $\psi(X)\le (\log X)^{2/3}$, let $d\le W=(\log X)^A$ and $\mathcal S_d = \mathcal S_d(X,A,0)$ as in \eqref{eq:Sd}. Then for any completely multiplicative $g:{\mathbb N}\to{\mathbb C}$ with $|g|\le 1$, we have
\begin{align*}
\sup_{\substack{\alpha\in\,\frak{m}}} \int_0^X \bigg|\sum_{\substack{x\le nd\le x + H\\n\in \mathcal S_d}}g(n)\,e(n\alpha)\bigg| \dd{x} \ \ll_A \ \frac{HX}{d^{3/4}W^{1/5}}.
\end{align*}
\end{proposition}
\begin{proposition}[Key major arc estimate for $\lambda$] \label{prop:major}
Given any $A>5$, $\delta>0$, $H=(\log X)^{\psi(X)}$ with $\psi(X)\to\infty$ and $\psi(X)\le (\log X)^{2/3}$, let $d\le W=(\log X)^A$ and $\mathcal S_d = \mathcal S_d(X,A,\delta)$ as in \eqref{eq:Sd}. Then we have
\begin{align*}
\sup_{\substack{\alpha\in\,\frak{M}}} \int_0^X \bigg|\sum_{\substack{x\le nd\le x + H\\n\in \mathcal S_d}}\lambda(n)\,e(n\alpha)\bigg| \dd{x} \ \ll_{A,\delta} \ \frac{HX}{dW}.
\end{align*}
\end{proposition}
We remark that the bounds in the minor arc hold for any bounded multiplicative function, whereas in the major arc the specific choice of $\lambda$ is needed.
\subsection{Minor arc}
In this subsection, we prove Proposition \ref{prop:minor}. Recall for $\alpha\in \frak{m}$ in the minor arc, $|\alpha-a/q|<W^4/qH$ with $q\in [W,H/W^4]$. It suffices to show
\begin{align}\label{eq:minor}
I_{\frak m} := \int_{{\mathbb R}} \theta(x)\sum_{x\le nd\le x + H}{\mathbf 1}_{\mathcal S_d}(n) g(n)\,e(n\alpha) \dd{x} \ \ll \ HX \big(\tfrac{\log\log X}{dW}\big)^{1/2}\psi(X),
\end{align}
uniformly for any $\alpha\in \frak m$ and measurable $\theta:[0,X]\to{\mathbb C}$ with $|\theta(x)|\le 1$. Letting $\mathcal P=\{p: P_1\le p\le Q_1\}$, by definition each $n\in \mathcal S_d$ has a prime factor in $\mathcal P$, so we use a variant of the Ramar\' e identity
\begin{align}
{\mathbf 1}_{\mathcal S_d}(n) = \sum_{\substack{p\in\mathcal P\\n=mp}} \frac{{\mathbf 1}_{\mathcal S^{(1)}_d}(mp)}{\#\{q\in \mathcal P : q\mid m\} + {\mathbf 1}_{p\nmid m}},
\end{align}
where $\mathcal S^{(1)}_d=\{m\le X/d : \,\exists p\mid m, \, p\in[P_2,Q_2]\}$. As $g$ is completely multiplicative, we obtain
\begin{align*}
I_{\frak m} = \sum_{p\in\mathcal P}\sum_{m}\frac{{\mathbf 1}_{\mathcal S^{(1)}_d}(mp) g(m)g(p)e(mp\alpha)}{\#\{q\in \mathcal P : q\mid m\} + {\mathbf 1}_{p\nmid m}} \int_{{\mathbb R}} \theta(x)& {\mathbf 1}_{x\le mpd\le x + H} \dd{x}.
\end{align*}
Next we split $\mathcal P$ into dyadic intervals $[P,2P]$. It suffices to show for each $P\in [P_1,Q_1]$,
\begin{align}\label{eq:minordyad}
\sum_{\substack{p\in\mathcal P\\P\le p\le 2P}}\sum_{m}\frac{{\mathbf 1}_{\mathcal S^{(1)}_d}(mp)g(m)g(p)e(mp\alpha)}{\#\{q\in \mathcal P : q\mid m\} + {\mathbf 1}_{p\nmid m}} \int_{{\mathbb R}} \theta(x)& {\mathbf 1}_{x\le mpd\le x + H} \dd{x} \ \ll \ \frac{HX}{\log P} \big(\tfrac{\log \log X}{dW}\big)^{1/2},
\end{align}
since then \eqref{eq:minor} will follow by \eqref{eq:PQj} and the triangle inequality, using
\begin{align*}
\sum_{\substack{P_1 \ll P\ll Q_1\\ P=2^j}} \frac{1}{\log P} \ll \sum_{\log P_1 \ll j\ll \log Q_1}\frac{1}{j} \ll \log\frac{\log Q_1}{\log P_1} = \log\frac{\psi(X)-4A}{33A} \ll_A \psi(X).
\end{align*}
Fix $P$. We may replace ${\mathbf 1}_{p\nmid m}$ with 1 in \eqref{eq:minordyad} at a cost of $O(HX/dP)$. Indeed, since the integral is $\int_{\mathbb R} \theta(x) {\mathbf 1}_{x\le mpd\le x + H} \dd{x} \ll H$, and ${\mathbf 1}_{\mathcal S^{(1)}_d}(mp)=0$ unless $m\le X/dP$, the cost of such substitution is
\begin{align*}
\ll \ \sum_{\substack{p\in\mathcal P\\P\le p\le 2P}}\sum_{\substack{m\le X/dP\\p\mid m}} H \ \ll \ P\frac{X}{dP^2} H = \frac{HX}{dP}.
\end{align*}
Now the left hand side of \eqref{eq:minordyad} becomes
\begin{align*}
\sum_{m\in \mathcal S^{(1)}_d}&\frac{g(m)}{\#\{q\in \mathcal P : q\mid m\}+1} \sum_{\substack{p\in\mathcal P\\P\le p\le 2P}}g(p)e(mp\alpha){\mathbf 1}_{mpd\le X}\int_{{\mathbb R}} \theta(x) {\mathbf 1}_{x\le mpd\le x + H} \dd{x}\\
& \ll \ \sum_{m\le X/dP}\bigg|\sum_{\substack{p\in\mathcal P\\P\le p\le 2P}}g(p)e(mp\alpha){\mathbf 1}_{mpd\le X}\int_{{\mathbb R}} \theta(x) {\mathbf 1}_{x\le mpd\le x + H} \dd{x}\bigg|\\
& \ll \ (X/dP)^{1/2}\,\bigg(\sum_{m\le X/dP}\bigg|\sum_{\substack{p\in\mathcal P\\P\le p\le 2P}}g(p)e(mp\alpha){\mathbf 1}_{mpd\le X}\int_{{\mathbb R}} \theta(x) {\mathbf 1}_{x\le mpd\le x + H}\dd{x} \bigg|^2\bigg)^{1/2},
\end{align*}
by the trivial bound and Cauchy-Schwarz. Hence for \eqref{eq:minordyad} it suffices to show
\begin{align}\label{eq:minCS}
\sum_{m\le X/dP}\bigg|\sum_{\substack{p\in\mathcal P\\P\le p\le 2P}}g(p)e(mp\alpha){\mathbf 1}_{mpd\le X}\int_{{\mathbb R}} \theta(x) {\mathbf 1}_{x\le mpd\le x + H}\dd{x} \bigg|^2 \ \ll \ \frac{H^2PX}{W}\frac{\log\log P}{(\log P)^2}.
\end{align}
We expand the left hand side of \eqref{eq:minCS} and sum the resulting geometric series on $m$,
\begin{align*}
\underset{p_1,p_2\in\mathcal P\cap [P,2P]}{\sum\sum}\int_{{\mathbb R}^2}&g(p_1)\overline{g(p_2)} \theta(x_1)\overline{\theta(x_2)}\sum_{\substack{m\le X/dp_i \,\forall i\le2\\ x_i\le mdp_i\le x_i+H}}e\big(m(p_1-p_2)\alpha\big)\dd{x_1}\dd{x_2}\\
& \ll HX \sum_{p_1, p_2 \le 2P}\min\left(\frac{H}{dP}, \frac{1}{\|(p_1-p_2)\alpha\|}\right),
\end{align*}
since for given $d,p_1,p_2$, there are $O(X)$ choices for $x_1$ and $O(H)$ subsequent choices for $x_2$ since $x_2 = x_1(p_2/p_1) + O(H)$.
Note $\|z\|$ denotes the distance of $z\in{\mathbb R}$ to the nearest integer.
Thus \eqref{eq:minCS} reduces to showing
\begin{align}\label{eq:minsieve}
\sum_{p_1, p_2 \le 2P}\min\left(\frac{H}{P}, \frac{1}{\|(p_1-p_2)\alpha\|}\right) \ \ll \ \frac{HP}{W}\frac{\log\log P}{(\log P)^2}.
\end{align}
The difference of primes is $p_1-p_2 \ll P$. Conversely, any integer $n\ll P$ may be written as $n=p_1-p_2$ for $p_1,p_2\le\, 2P$ in $\ll \frac{n}{\phi(n)}P(\log P)^{-2} \ \ll P \frac{\log\log P}{(\log P)^2}$ ways by a standard upper bound sieve, see \cite[Proposition 6.22]{Opera}. Hence for \eqref{eq:minsieve} it suffices to obtain
\begin{align*}
\sum_{1\le n \ll P}\min\left(\frac{H}{n}, \frac1{\|n\alpha\|}\right) \ \ll \ \frac{H}{W} \qquad (\alpha\in \,\frak{m}).
\end{align*}
But this follows by the standard `Vinogradov lemma' \cite[p.346]{IK}.
\begin{lemma}\label{lem:vino}
Given $H,P>1$, take $\alpha\in[0,1]$ with $|\alpha-a/q|\le 1/q^2$ for some $(a,q)=1$. Then
\begin{align*}
\sum_{1\le n \le P}\min\left(\frac{H}{n}, \frac1{\|n\alpha\|}\right) \ \ll \ \frac{H}{q} + \frac{H}{P} + (P+q)\log q.
\end{align*}
\end{lemma}
Observe $H/q + H/P + (P+q)\log q \, \ll \, H/W$ since $q\in [W,H/W^4]$, $P\in [P_1,Q_1]=[W^{24},H/W^4]$. This completes the proof in the minor arc.
\subsection{Major arc}
In this subsection, we prove the key major arc estimate assuming the following mean value result for the (twisted) Liouville function.
\begin{proposition}\label{prop:J}
Given $A>5$, $\delta>0$, let $q\le W=(\log X)^A$, $d<W^{33}$, $\chi$ \textnormal{(mod $q$)}, $h\in [H/W^5,H]$, and $\mathcal S_d=\mathcal S_d(X,A,\delta)$ as in \eqref{eq:Sd}. Then for all $Y\in [X/W^4,X]$, we have
\begin{align}
J_{d,h,q}(Y;\chi):=\int_Y^{2Y}\bigg|\frac{1}{h}\sum_{\substack{x\le m\le x + h\\m\in \mathcal S_d}} \lambda(m)\chi(m)\bigg|^2\dd{x} \ \ll_{A,\delta} \ \frac{Y}{W^{10}}.
\end{align}
\end{proposition}
\begin{proof}[Proof of Proposition \ref{prop:major} from Proposition \ref{prop:J}]
To obtain the key major arc estimate we shall prove the stronger bound,
\begin{align}\label{eq:major}
I_{\frak M} := \sup_{\alpha\in\frak M}\int_0^X \bigg|\sum_{x\le nd\le x + H}{\mathbf 1}_{\mathcal S_d}(n)\lambda(n)\,e(n\alpha)\bigg| \dd{x} \ \ll \ \frac{HX}{dW}.
\end{align}
In the major arc recall $\alpha = \frac{a}{q} + \theta$ with $q\le W$ and $|\theta| \le \frac{W^4}{qH}$. By partial summation with $a_n = {\mathbf 1}_{>x/d}(n){\mathbf 1}_{\mathcal S_d}(n) \lambda(n)\,e(na/q)$, and $A(t) = \sum_{n\le t} a_n$, we have
\begin{align*}
\sum_{x\le nd\le x + H} &{\mathbf 1}_{\mathcal S_d}(n) \lambda(n)\,e(n\alpha) = e(\tfrac{x+H}{d}\theta)A(\tfrac{x+H}{d}) - e(\tfrac{x}{d}\theta)A(\tfrac{x}{d}) - 2\pi i\theta\int_{x/d}^{(x+H)/d} e(t\theta)A(t)\dd{t} \\
\ll \ & \bigg|\sum_{x\le nd\le x + H}{\mathbf 1}_{\mathcal S_d}(n)\lambda(n)\,e(an/q)\bigg| \ + \ |\theta|\int_0^{H/d}\bigg|\sum_{x/d\le n\le x/d + h}{\mathbf 1}_{\mathcal S_d}(n)\lambda(n)\,e(na/q)\bigg|\dd{h}.
\end{align*}
Thus taking the maximizing $h$ and integrating over $x\in[0,X]$, we obtain
\begin{align}\label{eq:ImajIh}
I_{\frak M} \ \ll \ I_{H/d} \ + \ |\theta| \frac{H}{d}\sup_{h\le H/d}I_h \ \ll \ I_{H/d} \ + \ \frac{W^4}{qd}\sup_{h\le H/d}I_h,
\end{align}
where
\begin{align}\label{eq:ImajI}
I_h & :=\int_0^X \bigg|\sum_{x/d\le n\le x/d + h}{\mathbf 1}_{\mathcal S_d}(n)\lambda(n) \,e(an/q)\bigg|\dd{x}.
\end{align}
Then splitting into residues $b$ (mod $q$) gives
\begin{align*}
I_h \le \sum_{b\,(q)}|e(ab/q)| \int_0^X\bigg|\sum_{\substack{x/d\le n\le x/d + h\\n\equiv b\,(q)}}{\mathbf 1}_{\mathcal S_d}(n)\lambda(n)\bigg|\dd{x} \ = \ \sum_{b\,(q)}\int_0^X\bigg|\sum_{\substack{x/d\le n\le x/d + h\\n\equiv b\,(q)}}{\mathbf 1}_{\mathcal S_d}(n)\lambda(n)\bigg|\dd{x}.
\end{align*}
Now suppose we have the bound
\begin{align}\label{eq:Ihbound}
I_h \ \ll \ \frac{qhX}{W^5} \qquad\quad\text{for}\qquad h\in[qH/W^5,H/d].
\end{align}
Then, combining with the trivial bound $I_h\le hX$ when $h\le qH/W^5$, \eqref{eq:ImajIh} becomes
\begin{align*}
I_{\frak M} \ &\ll \ I_{H/d} \quad + \quad \frac{W^4}{qd}\Big(\sup_{qH/W^5 \le h\le H/d}I_h \ + \sup_{h\le qH/W^5}hX\Big)\\
& \ll \ \frac{qHX}{dW^5} \quad + \quad \frac{W^4}{qd}\Big(\frac{qHX}{dW^5} \ + \ \frac{qHX}{W^5}\Big) \ \ll \quad \frac{HX}{dW},
\end{align*}
for $q\le W$ in the major arc. Hence it suffices to show \eqref{eq:Ihbound}.
Now to bound $I_h$, we extract the gcd. Let $c := (b,q)$ so that $c\mid n$, and we let $b' = b/c$, $q'=q/c$, $h'=h/c$, $m=n/c$. Thus since $\lambda$ is completely multiplicative, we have
\begin{align*}
I_h \ & \le \sum_{c\mid q}|\lambda(c)|\sideset{}{^*}\sum_{b'\,(q')} \int_0^X\bigg|\sum_{\substack{x/cd\le m\le x/cd + h/c\\m\equiv b'\,(q')}}{\mathbf 1}_{\mathcal S_d}(cm)\lambda(m)\bigg|\dd{x}\\
& \le \sum_{c\mid q}cd\sideset{}{^*}\sum_{b'\,(q')} \int_0^{X/cd}\bigg|\sum_{\substack{y\le m\le y + h'\\m\equiv b'\,(q')}}{\mathbf 1}_{\mathcal S_{cd}}(m)\lambda(m)\bigg|\dd{y},
\end{align*}
using the substitution $y=x/cd$, and noting ${\mathbf 1}_{\mathcal S_d}(cm) = {\mathbf 1}_{\mathcal S_{cd}}(m)$ since $c\le q<P_1$. Then recalling orthogonality of characters $\phi(q'){\mathbf 1}_{m\equiv b'\,(q')} \ = \ \sum_{\chi\,(q')}\overline{\chi(b')}\chi(m)$, we obtain
\begin{align}
I_h \ &\le \sum_{c\mid q}cd\sideset{}{^*}\sum_{b'\,(q')}\frac{1}{\phi(q')}\sum_{\chi\,(q')}|\overline{\chi(b')}| \int_0^{X/cd}\bigg|\sum_{y\le m\le y + h'}{\mathbf 1}_{\mathcal S_{cd}}(m)\lambda(m)\chi(m)\bigg|\dd{y} \nonumber\\
&\le \sum_{c\mid q}cd \sum_{\chi\,(q')} \int_0^{X/cd}\bigg|\sum_{y\le m\le y + h'}{\mathbf 1}_{\mathcal S_{cd}}(m)\lambda(m)\chi(m)\bigg|\dd{y}. \label{eq:majII}
\end{align}
We may discard the contribution to \eqref{eq:majII} of the integral over $y\le X/dW^5$, since $h'=h/c$ and $q\le W$ imply an admissible cost
\begin{align*}
\ll \ \sum_{c\mid q}cd\phi(q') \, \frac{X}{dW^5} h'
\ \le \ \frac{hX}{W^5} \sum_{q'\mid q}\phi(q')
\ = \ \frac{qhX}{W^5}.
\end{align*}
For the remaining $y\in [\frac{X}{dW^5}, \frac{X}{cd}]$ in \eqref{eq:majII}, we split into dyadic intervals so that
\begin{align}\label{eq:ItoMR}
I_h & \ \le \sum_{c\mid q}cd \sum_{\chi\,(q')}\sum_{\substack{Y = 2^j\\\frac{X}{2dW^5}\le Y\le \frac{2X}{cd}}}\int_Y^{2Y}\bigg|\sum_{y\le m\le y + h'}{\mathbf 1}_{\mathcal S_{cd}}(m)\lambda(m)\chi(m)\bigg|\dd{y} \ + \ O\Big(\frac{qHX}{W^5}\Big).
\end{align}
By assumption, Proposition \ref{prop:J} implies $J_{cd,h',q'}(Y;\chi) \ll Y/W^{10}$, so Cauchy-Schwarz gives
\begin{align*}
\int_Y^{2Y}\bigg|\sum_{y\le m\le y + h'}{\mathbf 1}_{\mathcal S_{cd}}(m) \lambda(m)\chi(m)\bigg|\dd{y} \le h'\,\sqrt{Y\cdot J_{cd,h',q'}(Y;\chi)} \ \ll_A \frac{Yh'}{W^5}.
\end{align*}
So plugging back into \eqref{eq:ItoMR}, we obtain
\begin{align*}
I_h & \ \ll \ \sum_{c\mid q}cd \phi(q')\sum_{\substack{Y = 2^j\\\frac{X}{2W^4}\le Y\le \frac{2X}{cd}}} \frac{Yh'}{W^5} \ = \ \frac{hd}{W^5}\sum_{q'\mid q} \phi(q')\sum_{\substack{Y = 2^j\\\frac{X}{2dW^5}<Y< \frac{2X}{cd}}} Y \ \ll \ \frac{qhX}{W^5}.
\end{align*}
This gives \eqref{eq:Ihbound} as desired.
\end{proof}
\section{Preparatory lemmas}
We collect some standard lemmas on Dirichlet polynomials.
The first is the integral mean value theorem \cite[Theorem 9.1]{{IK}}.
\begin{lemma}[mean value] \label{lem:meanval}
For $D(s) = \sum_{n\le N} a_n n^{-s}$, we have
\begin{align*}
\int_{-T}^T |D(it)|^2\dd{t} = (T+O(N))\sum_{n\le N}|a_n|^2.
\end{align*}
\end{lemma}
One may discretize the mean value theorem by replacing the intergal over $[-T,T]$ with a sum over a well-spaced set $\mathcal W\subset [-T,T]$.
\begin{definition}
A set $\mathcal W\subset {\mathbb R}$ is {\it well-spaced} if $|w-w'|\ge1$ for all $w,w'\in \mathcal W$.
\end{definition}
Next is the Hal\'asz-Montgomery inequality \cite[Theorem 9.6]{{IK}}, which offers an improvement to the (discretized) mean value theorem when the well-spaced set is `sparse.'
\begin{lemma}[Hal\'asz-Montgomery]\label{lem:HalMont}
Given $D(s) = \sum_{n\le N} a_n n^{-s}$ and a well-spaced set $\mathcal W\subset [-T,T]$. Then
\begin{align*}
\sum_{t\in \mathcal W}|D(it)|^2 \ \ll \ (N + |\mathcal W|\sqrt{T})\log 2T \sum_{n\le N}|a_n|^2.
\end{align*}
\end{lemma}
We also need a bound on the size of well-spaced sets $\mathcal W$ in terms of the values of prime Dirichlet polynomials on $1+i\mathcal W$ \cite[Lemma 8]{MR}.
\begin{lemma} \label{lem:MR8}
Let $a_p\in {\mathbb C}$ be indexed by primes, with $|a_p|\le 1$, and define the prime polynomial
\begin{align*}
P(s) = \sum_{L\le p\le 2L}\frac{a_p}{p^s}.
\end{align*}
Suppose a well-spaced set $\mathcal W\subset [-T,T]$ satisfies $|P(1+it)|\ge1/U$ for all $t\in\mathcal W$. Then
\begin{align*}
|\mathcal W| \ \ll \ U^2 \,T^{2(\log U + \log\log T)/\log L}.
\end{align*}
\end{lemma}
\begin{lemma} \label{lem:MR2l}
Given $A,K>0$, $\theta > \frac{2}{3}$, and a Dirichlet character $\chi$ mod $q\le (\log X)^A$. Assume $\exp((\log X)^\theta) \le P\le Q\le X$, and let $P(s,\chi) = \sum_{P\le p\le Q}\chi(p)p^{-s}$. Then for any $|t|\le X$,
\begin{align*}
|P(1+it,\chi)| \ \ll_{A,K,\theta} \ \frac{\log X}{1+|t|} \ + \ (\log X)^{-K}.
\end{align*}
\end{lemma}
\begin{proof}
This follows as with \cite[Lemma 2]{MRshort}, except that the Vinogradov--Korobov zero-free region for $\zeta(s)$ is replaced by that of $L(s,\chi)$.
\end{proof}
We also use a Parseval-type bound. This shows that the average of a multiplicative function in almost all short intervals can be approximated by its average on a long interval, provided the mean square of the corresponding Dirichlet polynomial is small.
\begin{lemma}[Parseval bound] \label{lem:MR14}
Given $T_0 \in [(\log X)^{1/15}, X^{1/4}]$, and take a sequence $(a_m)_{m=1}^\infty$ with $|a_m|\le 1$. Assume $1\le h_1\le h_2 \le X/T_0^3$. For $x\in[X,2X]$, define
\begin{align*}
S_j(x) = \sum_{x\le m\le x+h_j}a_m, \qquad \textnormal{and}\qquad A(s) = \sum_{X\le m\le 4X}\frac{a_m}{m^s}.
\end{align*}
Then
\begin{align*}
\frac{1}{X}\int_X^{2X}\Big|\tfrac{1}{h_1}S_1(x)-\tfrac{1}{h_2}S_2(x)\Big|^2\dd{x} \ \ll \ & \frac{1}{T_0} \ + \ \int_{T_0}^{X/h_1} |A(1+it)|^2\dd{t}\\
& \ + \max_{T\ge X/h_1}\frac{X/h_1}{T}\int_T^{2T} |A(1+it)|^2\dd{t}.
\end{align*}
\end{lemma}
\begin{proof}
This follows as in \cite[Lemma 14]{MR} with $(\log X)^{1/15}$ replaced by general $T_0$.
\end{proof}
We have a general mean value of products, via the Ramar\'e identity \cite[Lemma 12]{MR}.
\begin{lemma} \label{lem:MR12}
For $V,P,Q\ge1$, denote $\mathcal P=[P,Q]\cap \mathbb P$. Let $a_m,b_m,c_p$ be bounded sequences for which $a_{mq}=b_mc_q$ when $q\nmid m$ and $q\in \mathcal P$. Let
\begin{align*}
Q_{v,V}(s) &= \sum_{\substack{q\in \mathcal P\\ e^{v/V}\le q\le e^{(v+1)/V}}}\frac{c_q}{q^s},\\
R_{v,V}(s) &= \sum_{Xe^{-v/V}\le m\le 2Xe^{-v/V}}\frac{b_m}{m^s}\cdot \frac{1}{\#\{p\mid m : p\in \mathcal P\}+1},
\end{align*}
and take a measurable set $\mathcal T\subset [-T,T]$. Then for $\mathcal I=[\lfloor V \log P\rfloor,V\log Q]\cap {\mathbb Z}$, we have
\begin{align*}
\int_{\mathcal T}\Big|\sum_{X\le n \le 2X}\frac{a_n}{n^{1+it}}\Big|^2\dd{t} \ \ll \ V\log(\tfrac{Q}{P}) & \sum_{v\in \mathcal I}\int_{\mathcal T}|Q_{v,V}(1+it)\, R_{v,V}(1+it)|^2 \dd{t}\\
& \qquad \ + \ \Big(\frac{T}{X}+1\Big)\Big(\frac{1}{V}+\frac{1}{P}+\sum_{\substack{X\le n\le 2X\\p\nmid n\forall p\in \mathcal P}}\frac{|a_n|^2}{n}\Big)
\end{align*}
\end{lemma}
In the next result we employ the Fundamental Lemma of the sieve, along with the Siegel-Walfisz theorem.
\begin{lemma}\label{lem:SxS'}
Given $A,K>0$, $q\le (\log x)^A$, Dirichlet character $\chi$ \textnormal{(mod $q$)}, and let $\mathcal D = \prod_{p\in \mathcal P}p$ for any set of primes $\mathcal P\subset (q,x^{1/\log\log x})$. Then
\begin{align*}
\sum_{\substack{m\le x\\(m,\mathcal D)=1}}\lambda(m)\chi(m) \ \ll_{A,K} \ \frac{x}{(\log x)^K}.
\end{align*}
\end{lemma}
\begin{proof}
First partition the sum on $m$ by the values of $\lambda(m),\chi(m)$,
\begin{align}\label{eq:mlamchi}
S_0 := \sum_{m\le x}{\mathbf 1}_{(m,\mathcal D)=1}\lambda(m)\chi(m) = \sum_{b\,(q), \nu\in\{\pm1\}}\nu\chi(b)\sum_{m\in \mathcal A^{(b,\nu)}}{\mathbf 1}_{(m,\mathcal D)=1}
\end{align}
for the set $\mathcal A^{(b,\nu)} = \{m\le x : m\equiv b\,(q), \lambda(m)=\nu\}$.
Now it suffices to prove
\begin{align}\label{eq:SAP}
\sum_{m\in \mathcal A^{(b,\nu)}}{\mathbf 1}_{(m,\mathcal D)=1} \ = \ \frac{x}{2 q}\prod_{p\mid \mathcal D}\Big(1-\frac{1}{p}\Big) \ + \ O_{A,K}\big(x(\log x)^{-K}\big)
\end{align}
uniformly in $b,\nu$, from which it will follow
\begin{align*}
S_0 = \frac{x}{2 qd}\prod_{p\mid \mathcal D}\Big(1-\frac{1}{p}\Big)\sum_{b\,(q), \nu\in\{\pm1\}}\nu\chi(b) \ + \ O\big(x(\log x)^{-K}\big) \ \ll \
\frac{x}{(\log x)^{K}},
\end{align*}
by pairing up terms $\nu=\pm1$. This will give the lemma.
Now to show \eqref{eq:SAP}, write $\mathcal A = \mathcal A^{(b,\nu)}$. For $d\mid \mathcal D$ the set of multiples $\mathcal A_d=\{m\in \mathcal A: d\mid m\}$ has size
\begin{align*}
|\mathcal A_d| = \sum_{\substack{m\le x\\d\mid m,\,m\equiv b\;(q)}} {\mathbf 1}_{\lambda(m)=\nu}
= \sum_{\substack{n\le x/d\\nd\equiv b\;(q)}}\frac{\nu\lambda(nd)+1}{2}
= \frac{\nu\lambda(d)}{2}\sum_{\substack{n\le x/d\\n\equiv bd^{-1}\;(q)}}\lambda(n) \ + \ \frac{x}{2qd} + O(q)
\end{align*}
noting $(q,d)=1=(q, \mathcal D)$. Moreover $\max_{c\,(q)}\big|\sum^{n\le y}_{n\equiv c\,(q)}\lambda(n)\big| \ll_{A,K} y(\log y)^{-2K}$ by Siegel-Walfisz, which is valid by the assumption $q\le (\log X)^A$. Thus
\begin{align}\label{eq:AdSW}
|\mathcal A_d| = \frac{x}{2qd} + R_d, \qquad\qquad \text{where}\quad |R_d|\ll \frac{x}{d}\log(x/d)^{-2K} + q.
\end{align}
Now for any $D>1$ the indicator ${\mathbf 1}_{(m,\mathcal D)=1}$ is bounded in between $\sum_{d<D}^{d\mid (m, \mathcal D)} \lambda^\pm_d$, for the standard linear sieve weights $\{\lambda^\pm_d\}_{d<D}$, see \cite[Lemma 6.11]{Opera}. Thus the desired sum in \eqref{eq:SAP} is bounded in between
\begin{align}
\sum_{\substack{d\mid \mathcal D\\d<D}}\lambda_d^- |\mathcal A_d| \ \le \sum_{m\in \mathcal A}{\mathbf 1}_{(m, \mathcal D)=1} \ \le \
\sum_{\substack{d\mid \mathcal D\\d<D}}\lambda_d^+ |\mathcal A_d|.
\end{align}
Note by \eqref{eq:AdSW}, the upper and lower bounds are given by
\begin{align*}
\sum_{\substack{d\mid \mathcal D\\d<D}}\lambda_d^\pm |\mathcal A_d| = \frac{x}{2 q}\sum_{\substack{d\mid \mathcal D\\d<D}}\frac{\lambda_d^\pm}{d} \ + \ \sum_{\substack{d\mid \mathcal D\\d<D}}\lambda_d^\pm R_d.
\end{align*}
Let $z=x^{1/\log_2 x}$ so that $\mathcal P\subset (q,z)$. Then choosing $D = z^s$ for $s = 2K\log_2 x/\log_3 x$, the above error is $\ll\sum_{d<D}|R_d| \ll x(\log x)^{-K}$ by \eqref{eq:AdSW}. And by the Fundamental Lemma \cite[Lemma 6.11]{Opera}, the main term is
\begin{align*}
\sum_{\substack{d\mid \mathcal D\\d<D}}\frac{\lambda_d^\pm}{d} = (1+O(s^{-s}))\prod_{p\mid \mathcal D}\Big(1-\frac{1}{p}\Big).
\end{align*}
Hence \eqref{eq:SAP} follows as claimed, noting $s^{-s}\ll (\log x)^{-K}$.
\end{proof}
\section{Mean value of multiplicative functions}
In this section we prove Proposition \ref{prop:J} based on the following mean value theorem for Dirichlet polynomials with typical factorization. This refines Matom\"aki-Radziwi\l\l \ \cite[Proposition 12]{MR} in the case of $g=\lambda\,\chi$, by leveraging Vinogradov-Korobov type bounds.
\begin{proposition}\label{prop:MRmain}
Given any $A>5$, $\delta>0$, denote $B=11A$. Write $H=(\log X)^{\psi(X)}$ with $\psi(X)\to\infty$ and $\psi(X)\le (\log X)^{2/3}$. Take $q\le (\log X)^A$, $d<(\log X)^{33A}$, a Dirichlet character $\chi$ \textnormal{(mod $q$)}, and let $\mathcal S_d=\mathcal S_d(X,A,\delta)$ as in \eqref{eq:Sd}. For any $Y\in[X^{1/2},X^2]$, define
\begin{align*}
G(s) = \sum_{\substack{Y\le n\le 2Y\\ n\in \mathcal S_d}}\frac{\lambda(n)\chi(n)}{n^s}.
\end{align*}
Then for any $T\in [Y^{1/2},Y]$, we have
\begin{align}
\int_{(\log X)^{2B}}^T |G(1+it)|^2\dd{t} \ \ll_{A,\delta} \ \Big(\frac{Q_1T}{Y}\ + \ 1\Big)(\log X)^{-B}.
\end{align}
\end{proposition}
\begin{proof}[Proof of Proposition \ref{prop:J} from Proposition \ref{prop:MRmain}]
We shall prove
\begin{align}
J:=\int_Y^{2Y}\Big|\tfrac{1}{h_1}S_{h_1}(x)\Big|^2\dd{x} \ \ll_{A,\delta} \ \frac{Y}{(\log X)^{10A}}
\end{align}
for $Y\in [X/W^4,X]$, $h_1=h\in [qH/W^5,H]$ and the sum
\begin{align*}
S_l(x) := \sum_{\substack{x\le m\le x+l\\m\in\mathcal S_d}}\lambda(m)\chi(m).
\end{align*}
First we claim $S_x(0) \ll x(\log x)^{-K}$ for all $K>0$. To this, recall from \eqref{eq:Sd} that each $m\in \mathcal S_d$ has prime factors $p_1,p_2\mid m$ with $p_j\in[P_j,Q_j]$. So by inclusion-exclusion, the indicator is ${\mathbf 1}_{\mathcal S_d}(m) = \sum_{j=0}^3 (-1)^j{\mathbf 1}_{(m,\mathcal D_j)=1}$ where $\mathcal D_j=\prod_{p\in \mathcal P_j}p$ for the sets of primes $\mathcal P_0=\emptyset$, $\mathcal P_1=[P_1,Q_1]$, $\mathcal P_3=[P_2,Q_2]$, $\mathcal P_2=\mathcal P_1\cup\mathcal P_3$. Hence applying Lemma \ref{lem:SxS'} to each $\mathcal D_j$ gives
\begin{align}
S_x(0) & = \sum_{j=0}^3 (-1)^j\sum_{m\le x}{\mathbf 1}_{(m,\mathcal D_j)=1}\lambda(m)\chi(m) \ \ll_{A,K} \ x(\log x)^{-K}.
\end{align}
In particular, letting $B = 11A$ we have
\begin{align*}
S_{h_2}(x) = S_{x+h_2}(0) - S_x(0) \ll_A \frac{x}{(\log x)^{7B}} \ll_A \frac{h_2}{(\log x)^B},
\end{align*}
where $h_2 \asymp x(\log x)^{-6B}$, and so
\begin{align}
J=\frac{1}{Y}\int_Y^{2Y}\Big|\tfrac{1}{h_1}S_{h_1}(x)\Big|^2\dd{x} \ll (\log X)^{-B} \ + \frac{1}{Y}\int_Y^{2Y}\bigg|\tfrac{1}{h_1}S_{h_1}(x) \ - \ \tfrac{1}{h_2}S_{h_2}(x)\bigg|^2\dd{x}.
\end{align}
Then Lemma \ref{lem:MR14} (Parseval) with $T_0=(\log X)^{2B}$ and $h_2 = Y/T_0^3 = Y/(\log X)^{6B}$ gives
\begin{align}\label{eq:Jparseval}
J & \ll (\log X)^{-B} + \int_{T_0}^{Y/h_1} |G(1+it)|^2\dd{t} \ + \max_{T\ge Y/h_1}\frac{Y/h_1}{T}\int_T^{2T} |G(1+it)|^2\dd{t}.
\end{align}
Now for the latter integral over $[T,2T]$, we apply the Lemma \ref{lem:meanval} (mean value) if $T\ge X/2$, and apply Proposition \ref{prop:MRmain} if $T\in[Y/h_1,X/2]$. Doing so, \eqref{eq:Jparseval} becomes
\begin{align}\label{eq:Jbounded}
J & \ll \big(\tfrac{Y/h_1}{Y/Q_1}+1\big)(\log X)^{-B}
\ + \ \max_{T\ge X/2}\frac{Y/h_1}{T}(T/X+1) \nonumber\\
& \qquad\qquad\qquad\qquad\ + \max_{Y/h_1 \le T\le X/2} \frac{Y/h_1}{T}\big(\tfrac{Q_1T}{Y}+1\big)(\log X)^{-B} \nonumber\\
& \ \ll \ (\frac{Q_1}{h_1}+1)(\log X)^{-B} + \frac{Y}{h_1 X}
\ \ll \ W(\log X)^{-B} \ = \ (\log X)^{A-B}.
\end{align}
Here we used $h_1 \ge H/W^5$, $Q_1 = H/W^4$, (and $Y\in[X/W^4,X]$). Hence recalling $B=11A$ gives Proposition \ref{prop:J} as claimed.
\end{proof}
\subsection{Mean value of Dirichlet polynomials} \label{sec:MR}
In this subsection, we prove Proposition \ref{prop:MRmain}. Recall the definitions \eqref{eq:PQj}, \eqref{eq:Sd},
\begin{align*}
[P_1,Q_1] &= [(\log X)^{33A},(\log X)^{\psi(X)-4A}],\\
[P_2,Q_2] & = [\exp\big((\log X)^{2/3+\delta/2}\big),\,\exp\big((\log X)^{1-\delta/2}\big)],\\
\mathcal S_d(X,A,\delta) & = \{m\le X/d : \exists p_1,p_2\mid m \text{ with }p_j\in[P_j,Q_j]\}.
\end{align*}
Let $B=11A$, and $\alpha = 1/5$. Let $V = P_1^{1/3}=(\log X)^B$ and define the prime polynomial
\begin{align}
Q_{v,j}(s) := \sum_{\substack{P_j\le p\le Q_j\\e^{v/V}\le p\le e^{(v+1)/V}}}\frac{\lambda(p)\chi(p)}{p^s}.
\end{align}
Note $Q_{v,j}(s)\neq0$ only if $v\in \mathcal I_j:= \{v\in{\mathbb Z}: P_j\le e^{v/V} \le Q_j\} =[\lfloor V \log P_j\rfloor, V\log Q_j]$.
We decompose $[T_0,T] = \mathcal T_1\cup \mathcal T_2$ as a disjoint union, where $\mathcal T_2=[0,1]\setminus \mathcal T_1$ and
\begin{align}
\mathcal T_1 = \{t : \ |Q_{v,1}(1+it)| \le e^{-\alpha v/V} \quad \forall v\in\mathcal I_1 \}.
\end{align}
For $j=1,2$ denote by $\mathcal S_d^{(j)}$ the integers containing a prime factor in the interval $[P_i,Q_i]$ with $i\neq j$ and possibly, but not necessarily, with $i=j$. That is,
\begin{align*}
\mathcal S_d^{(j)} & = \{m\le X/d : \exists p\mid m \text{ with }p\in[P_i,Q_i] \text{ for }i\neq j\}.
\end{align*}
Also define the polynomial
\begin{align}
R_{v,j}(s) = \sum_{\substack{Ye^{-v/V}\le m\le 2Ye^{-v/V}\\m\in\mathcal S_d^{(j)}}}\frac{\lambda(m)\chi(m)}{m^s}\cdot \frac{1}{\#\{p\mid m : P_j\le p\le Q_j\}+1}.
\end{align}
Now Lemma \ref{lem:MR12} (Ramar\'e) applies with $V = V, P=P_j, Q=Q_j$ and $a_m=\lambda\,\chi\,{\mathbf 1}_{\mathcal S}(m)$, $c_p=\lambda\,\chi(p)$, $b_m=\lambda\,\chi\,{\mathbf 1}_{\mathcal S_j}(m)$, giving
\begin{align*}
\int_{\mathcal T_j}|G(1+it)|^2\dd{t} \ \ll \ V\log Q_j \sum_{v\in \mathcal I_j} & \int_{\mathcal T_j} |Q_{v,j}(1+it)\,R_{v,j}(1+it)|^2 \dd{t}\\
& \ + \ \frac{1}{V} + \frac{1}{P_j} \ +
\sum_{\substack{Y\le n\le 2Y\\ p\nmid n\forall p\in [P_j,Q_j]}} \frac{{\mathbf 1}_{\mathcal S_d}(n)}{n}.
\end{align*}
We crucially note the latter sum vanishes since each $n\in \mathcal S_d$ has a prime factor $p\in[P_j,Q_j]$. Summing over $j=1,2$, the second and third terms above contribute
\begin{align*}
\ll \sum_{1\le j\le 2}\Big(\frac{1}{V} + \frac{1}{P_j}\Big) \ll \frac{1}{V} = (\log X)^{-B}.
\end{align*}
Hence the desired integral is
\begin{align}
\int_{T_0}^T|G(1+it)|^2\dd{t} = \int_{\mathcal T_1\cup \mathcal T_2} |G(1+it)|^2\dd{t} \ &\ll \ E_1 + E_2 + (\log X)^{-B},
\end{align}
where
\begin{align}
E_j & = V\log Q_j \sum_{v\in\mathcal I_j} \int_{\mathcal T_j}|Q_{v,j}(1+it)\,R_{v,j}(1+it)|^2 \dd{t}.
\end{align}
Hence it suffices to bound $E_1,E_2\ll (Q_1T/Y+1)(\log X)^{-B}$.
{\bf Bound for $E_1$:}
By definition of $t\in\mathcal T_1$, we have $|Q_{v,1}(1+it)| \le e^{-\alpha v/V}$ for all $v\in\mathcal I_1$, so
\begin{align*}
E_1 \ \ll \ V\log Q_1 \sum_{v\in\mathcal I_1} e^{-2\alpha v/V}\int_{\mathcal T_1}|R_{v,1}(1+it)|^2 \dd{t}
\ \ll \ V\log Q_1 \sum_{v\in\mathcal I_1} e^{-2\alpha v/V} \Big(\frac{T}{Y/e^{v/V}}+1\Big)
\end{align*}
by Lemma \ref{lem:meanval} (mean value). Summing the resulting geometric series gives
\begin{align}
E_1 \ &\ll \ V\log Q_1 \frac{P_1^{-2\alpha}}{1-e^{-2\alpha/V}} \Big(\frac{Q_1T}{Y}+1\Big) \nonumber\\
& \ll \ (\log X)^{1-6\alpha}\Big(\frac{Q_1T}{Y}+1\Big)
\ \ll \ (\log X)^{-B}\Big(\frac{Q_1T}{Y}+1\Big),
\end{align}
noting $V/(1-e^{-2\alpha/V}) = O(1)$ and $1-6B/5 < -B$.
{\bf Bound for $E_2$:}
We choose the maximizing $v\in \mathcal I_2$ for $E_2$. Thus since $|\mathcal I_2|< V\log Q_2$,
\begin{align*}
E_2 & = V\log Q_2 \sum_{v\in\mathcal I_2} \int_{\mathcal T_2} |Q_{v,2}\cdot R_{v,2}(1+it)|^2 \dd{t}
\ll (V\log Q_2)^2 \int_{\mathcal T_2}|Q_{v,2}\cdot R_{v,2}(1+it)|^2 \dd{t} \nonumber\\
& \ \le (V\log Q_2)^2 \sum_n \sup_{t_n\in [n,n+1]\cap\, \mathcal T_2}|Q_{v,2}\cdot R_{v,2}(1+it_n)|^2 \nonumber\\
\ & \ \le 2(V\log Q_2)^2\sum_{t\in \mathcal W}|Q_{v,2}\cdot R_{v,2}(1+it)|^2,
\end{align*}
for a well-spaced set $\mathcal W \subset \mathcal T_2$. For instance, one may take $\mathcal W$ as the even or odd integers in $\mathcal T_2$ (choose the parity that gives a larger contribution). We shall see $\mathcal W$ is easier to analyze than $\mathcal T_2$ itself.
Now is the critical step for the choice $g=\lambda\chi$ and $\log P_2 = (\log X)^{2/3+\delta}$: by Lemma \ref{lem:MR2l} (Vinogradov-Korobov), we have for all $t\in [T_0,T]$
\begin{align}
|Q_{v,2}(1+it)| \ \ll_{\delta,A} \ \frac{\log X}{1+T_0} + (\log X)^{-B} \ll (\log X)^{-B},
\end{align}
for $T_0=(\log X)^{2B}$ and $B=11A$.
So by Lemma \ref{lem:HalMont} (Hal\'asz-Montgomery), we have
\begin{align*}
E_2 \ & \ll \ (V\log Q_2)^2(\log X)^{2-4B} \sum_{t\in \mathcal W} |R_{v,2}(1+it)|^2 \\
& \ll (V\log Q_2)^2(\log X)^{3-4B}(Ye^{-v/V} + |\mathcal W|\sqrt{T}) \, \frac{e^{v/V}}{Y} \\
& \ \ll \ (\log X)^{-B}(1 + |\mathcal W| \frac{\sqrt{T} Q_2}{Y}),
\end{align*}
recalling $\log T\asymp \log X$, $V = (\log X)^B$, and $e^{v/V}\le Q_2$.
Thus it suffices to bound $|\mathcal W|$. We shall obtain
\begin{align}
E_2 \ \ll \ (\log X)^{-B}\Big(1+\frac{T}{Y}\Big) \ \ll_{A,\delta} \ (\log X)^{-B}
\end{align}
provided we show $|\mathcal W| \ll T^{1/2}/Q_2$. To prove this, by definition of $\mathcal T_2 \supset\mathcal W$, we first partition $\mathcal W = \bigcup_{u\in \mathcal I_1}\mathcal W^{(u)}$ where
\begin{align*}
|Q_{u,1}(1+it)|> e^{-u\alpha/V} \qquad\text{for all}\quad t\in \mathcal W^{(u)}.
\end{align*}
Hence for each $u\in\mathcal I_1$, we may apply Lemma \ref{lem:MR8} to the prime polynomial $Q_{u,1}$ with $U = e^{u\alpha/V}$ and $L = e^{u/V}$, so that
\begin{align}
|\mathcal W| & = \sum_{u\in \mathcal I_1} |\mathcal W^{(u)}|
\ll \sum_{u\in \mathcal I_1}U^2T^{2\frac{\log U+\log\log T}{\log L}} \ \ll \ |\mathcal I_1| U^2 T^{2\alpha+\frac{2\log\log T}{\log L}} \ \ll \ T^{2/5+2/5A+o(1)},
\end{align}
since $|\mathcal I_1| < V\log Q_1 \ll T^{o(1)}$, $U^2 \le Q_1^{2\alpha} \ll T^{o(1)}$, $\log L \ge \log P_1 \ge 5A\log\log T$, by recalling $[P_1,Q_1] = [(\log X)^{33A},(\log X)^{\psi(X)-4A}]$ and $T\in [X^{1/4},X^2]$.
Hence $A>5$ gives $|\mathcal W| \ll T^{1/2}/Q_2$, and completes the proof of Proposition \ref{prop:MRmain}.
\section{Average Chowla-type correlations}
In this section, we establish the results for higher correlations stated in the introduction.
We first exhibit quantitative cancellation among a broad class of correlations with a `typical' factor ${\mathbf 1}_{\mathcal S}\mu$. We use a standard `van der Corput' argument and then apply the key Fourier estimate.
\begin{lemma}\label{lem:Chowlatypical}
Given any $A>5$, $\delta>0$, let $\mathcal S \ = \ \mathcal S(X,A,\delta)$ as in \eqref{eq:S}. Assume $H<X$ satisfies $\log H/\log_2 X\to\infty$, and $G:{\mathbb N}\to{\mathbb C}$ satisfies $\sum_{n\le X}|G(n)|^2 \ll X(\log X)^{A/20}$. Then
\begin{align*}
\sum_{h\le H}\Big|\sum_{n\le X} {\mathbf 1}_{\mathcal S}\mu(n+h)G(n)\Big|
\ \ll_{A,\delta} \ \frac{HX}{(\log X)^{A/40}}.
\end{align*}
\end{lemma}
\begin{proof}
Let $g={\mathbf 1}_{\mathcal S}\mu$. By Cauchy-Schwarz it suffices to show
\begin{align*}
\frac{HX^2}{(\log X)^{A/20}} \ & \gg \ \sum_{h\le H} \bigg|\sum_{n\le X} g(n+h)G(n)\bigg|^2 \ = \ \sum_{n,n'\le X}G(n)\overline{G}(n') \sum_{h\le H} g(n+h)g(n'+h).
\end{align*}
Using Cauchy-Schwarz again, the right hand side above is bounded by
\begin{align*}
\sum_{n\le X}|G(n)|^2 \cdot\bigg(\sum_{n,n'}
\Big|\sum_{h\le H} g(n+h)g(n'+h)\Big|^2\bigg)^{\frac{1}{2}}.
\end{align*}
recalling $g$ is supported on $[1,X]$. By assumption $\sum_{n\le X}|G(n)|^2 \ll X(\log X)^{A/20}$, so it suffices to prove
\begin{align*}
\frac{H^2 X^2}{(\log X)^{A/5}} \ & \gg \ \sum_{n,n'}\Big|\sum_{h\le H} g(n+h)g(n'+h)\Big|^2
= \sum_{|h|< H}(\lfloor H\rfloor -|h|)\bigg|\sum_n g(n) g(n+h)\bigg|^2.
\end{align*}
But this indeed holds: since $g={\mathbf 1}_{\mathcal S}\mu$, we apply Lemma \ref{lem:Fourier} (Fourier bound) with $f=g={\mathbf 1}_{\mathcal S}\mu$. Thus the trivial bound $F(X)\ll X$ and Theorem \ref{thm:mainFourier} (Key Fourier estimate) give
\begin{align*}
\sum_{|h|\le H}\bigg| \sum_{n\le X} & {\mathbf 1}_{\mathcal S}\mu(n)\,{\mathbf 1}_{\mathcal S}\mu(n+h)\bigg|^2\\
\ll \ & F(X+2H)\cdot\sup_\alpha \int_0^{X} \bigg|\sum_{x\le n\le x+2H} {\mathbf 1}_{\mathcal S}\mu(n)e(n\alpha)\bigg|\dd{x} \ \ll \ \frac{HX^2}{(\log X)^{A/5}}.
\end{align*}
\end{proof}
We now prove the main technical result of the article, which handles averaged Chowla-type correlations with $m\ge1$ copies of the M\"obius function $\mu$ and with any function $G:{\mathbb N}\to{\mathbb C}$ of `moderate growth' which is `amenable to sieves.'
\begin{theorem}\label{thm:mainChowla}
Given any $A>5$, $\delta>0$, let $\mathcal S \ = \ \mathcal S(X,A,\delta)$ as in \eqref{eq:S}. Assume $H<X$ satisfies $\log H/\log_2 X\to\infty$, and $G:{\mathbb N}\to{\mathbb C}$ satisfies $\sum_{n\le X}|G(n)|^2 \ll X(\log X)^{A/20}$. Then
\begin{align}\label{eq:Chowlafixed}
\sum_{h_1,..,h_m\le H} & \bigg|\sum_{n\le X} G(n)\prod_{j=1}^m \mu(n+h_j)\bigg| \\
& \ \ll_{A,\delta} \ \sum_{h_1,..,h_m\le H} \sum_{n\le X} |G(n)|\prod_{j=1}^m {\mathbf 1}_{\overline{\mathcal S}}(n+h_j) \ + \ \frac{mXH^m}{(\log X)^{A/40}}. \nonumber
\end{align}
\end{theorem}
\begin{proof}
We observe from Lemma \ref{lem:Chowlatypical} that any correlation with a factor ${\mathbf 1}_{\mathcal S}\mu$ exhibits strong cancellation. So we split up $\mu = {\mathbf 1}_{\overline{\mathcal S}}\mu +{\mathbf 1}_{\mathcal S}\mu$ until each term has a factor ${\mathbf 1}_{\mathcal S}\mu$, except for one term with $m$ factors of ${\mathbf 1}_{\overline{\mathcal S}}\mu$. Thus the product in \eqref{eq:Chowlafixed} becomes
\begin{align*}
\prod_{j=1}^m \mu(n+h_j) = \prod_{j=1}^m{\mathbf 1}_{\overline{\mathcal S}}\mu(n+h_j) \ + \ \sum_{i=1}^m {\mathbf 1}_{\mathcal S}\mu(n+h_i)\prod_{1\le j< i} {\mathbf 1}_{\overline{\mathcal S}}\mu(n+h_j)\prod_{i<j\le m} \mu(n+h_j).
\end{align*}
Hence we bound the left hand side of \eqref{eq:Chowlafixed} by $\Sigma_1 + \Sigma_2$, where
\begin{align}
\Sigma_1 & = \ \sum_{h_1,..,h_m\le H}|G(n)|\sum_{n\le X} \prod_{j=1}^m {\mathbf 1}_{\overline{\mathcal S}}(n+h_j), \label{eq:nottypicalconv} \\
\Sigma_2 & = \sum_{i=1}^m \ \sum_{h_1,..,h_m\le H}\bigg|\sum_{n\le X}{\mathbf 1}_{\mathcal S}\mu(n+h_i) G_i(n)\bigg|, \label{eq:typicalconv}
\end{align}
where $G_i(n) = G(n)\prod_{1\le j< i} {\mathbf 1}_{\overline{\mathcal S}}\mu(n+h_j)\prod_{i<j\le m} \mu(n+h_j)$. In particular $|G_i(n)| \le |G(n)|$.
Thus Lemma \ref{lem:Chowlatypical} applies to each ${\mathbf 1}_{\mathcal S}(n+h)G_i(n)$, so that $\Sigma_2$ is bounded by
\begin{align*}
\Sigma_2 & \ \ll_A \ \frac{mH^m X}{W^{1/40}}.
\end{align*}
\end{proof}
\subsection{Deduction of results}
\begin{proof}[Proof of Theorem \ref{thm:Chowktuple}]
Let $G(n) = \prod_{j=1}^k\Lambda(n+a_j)$ for the tuple $\mathcal A=\{a_1,..,a_k\}$. Then $\sum_{n\le X}|G(n)|^2\ll X(\log X)^k$, and by a standard sieve upper bound
\begin{align*}
\sum_{n\le X}G(n)\prod_{j=1}^m {\mathbf 1}_{\overline{\mathcal S}}(n+h_j) \ \ll_{m,\mathcal A} \ X\Big(\prod_{p\in[P_1,Q_1]}+\prod_{p\in[P_2,Q_2]}\Big)\Big(1-\frac{1}{p}\Big)^m \ll_{m,\delta,\mathcal A} \frac{X}{\psi_\delta(X)^m},
\end{align*}
using Mertens' product theorem. Hence Theorem \ref{thm:mainChowla} with $A=20(m+k)$ gives
\begin{align}\label{eq:ChowlaHL}
\sum_{h_1,..,h_m\le H} & \bigg|\sum_{n\le X} \prod_{j=1}^k\Lambda(n+a_j)\prod_{j=1}^m \mu(n+h_j)\bigg|
\ \ll_{m,\delta,\mathcal A} \ \frac{XH^m}{\psi_\delta(X)^m}.
\end{align}
\end{proof}
\begin{comment}
Next we consider a class of multiplicative functions with `moderate growth.'
\begin{definition}\label{def:multclass}
For $C\ge1$ denote the set $\mathcal M(C)$ of multiplicative $f:{\mathbb N}\to{\mathbb R}_{\ge0}$ with
\begin{itemize}
\item[(i)] $f(p^\nu) \le C^\nu$ for all primes $p$, $\nu\ge1$.
\item[(ii)] for all ${\varepsilon}>0$ there exists $B=B({\varepsilon})$ such that $f(n) \le Bn^{{\varepsilon}}$ for all $n\ge1$.
\end{itemize}
\end{definition}
Now we give a special case of a general bound of Henriot \cite{Hen} for the class $\mathcal M(C)$. The result refines earlier work of Nair-Tenenbaum \cite{NairTen}, and importantly, it is uniform in the discriminant.
\begin{corollary}\label{cor:Henriot}
Given $C,k\ge1$, a tuple of distinct integers $\mathcal H=\{h_1,..,h_k\}\subset [1,X^{1/2}]$, and let $\nu_p = \#\{h_1,..,h_k\,(\textnormal{mod }p)\}$. Then for any multiplicative functions $f_j\in \mathcal M(C)$,
\begin{align}\label{eq:Henriot}
\sum_{\sqrt{X} \le n\le X}\prod_{j=1}^k f_j(n+h_j) \ \ll_{C,k} \Delta_D\,X\prod_{p\le \sqrt{X}}\Big(1-\frac{\nu_p}{p}\Big)\prod_{j=1}^k\sum_{n\le \sqrt{X}}\frac{f_j(n)}{n}
\end{align}
where $D=D(\mathcal H)=\prod_{i<j}(h_j-h_i)$ is the discriminant, and
\begin{align*}
\Delta_D = \prod_{p\mid D}\bigg(1\ + \ \frac{\#\{n\, (\textnormal{mod }p^2)\,:\, p\| n+h_j\, \forall j \}}{p^2}\prod_{j=1}^k f_j(p)\bigg)
\end{align*}
In particular, we have $\Delta_D \le \prod_{p\mid D}(1+C^k/p)$.
\end{corollary}
\begin{proof}
This is \cite[Theorem 3]{Hen} in the special case of $x=\sqrt{X}$, $y=X$, $\delta = 1/k$, linear polynomials $Q_j(n)=n+h_j$, $Q(n) = \prod_{j=1}^k Q_j(n)$, and the multiplicative product $F(n_1,..,n_k)=\prod_{j=1}^k f_j(n_j)$. Here the sum of coefficients is $\|Q\| \ll \prod_{j=1}^k h_j \ll X^{k/2}$.
\end{proof}
We remark that the bound \eqref{eq:Henriot} is of the correct order of magnitude when the functions $f_j$ are not too small, e.g. $f_j(n) \ge \eta^{\Omega(n)}$ for some $\eta>0$, see \cite[Theorem 6]{Hen}.
\end{comment}
\begin{proof}[Proof of Theorem \ref{cor:divisorcorr}]
Let $G(n) = \prod_{i=1}^jd_{k_i}(n+a_i)$ for the tuple $\mathcal A=\{a_1,..,a_j\}$ and recall $k=\sum_{i=1}^jk_i$. Using work of Henriot \cite[Theorem 3]{Hen}, we may obtain
\begin{align*}
\sum_{h\le H}\sum_{n\le X}{\mathbf 1}_{\overline{\mathcal S}}(n+h) \prod_{i=1}^j d_{k_i}(n+a_i)
\ \ll_{\mathcal A} \ \frac{HX}{(\log X)^{j+1}}
\sum_{n\le \sqrt{X}}\frac{{\mathbf 1}_{\overline{\mathcal S}}(n)}{n}\prod_{i=1}^j\sum_{n\le \sqrt{X}}\frac{d_{k_i}(n)}{n}.
\end{align*}
By the divisor bound $\sum_{n\le \sqrt{X}}d_{k_i}(n)/n\ll X(\log X)^{k_i}$, and by Mertens' product theorem
\begin{align*}
\sum_{n\le \sqrt{X}}\frac{{\mathbf 1}_{\overline{\mathcal S}}(n)}{n} \ \ll \ \log X\Big(\prod_{p\in[P_1,Q_1]}+\prod_{p\in[P_2,Q_2]}\Big)\Big(1-\frac{1}{p}\Big) \ \ll_{\delta} \ \frac{\log X}{\psi_\delta(X)}.
\end{align*}
Thus since $\sum_{n\le X}|G(n)|^2\ll X(\log X)^k$, Theorem \ref{thm:mainChowla} with $A=20k$ gives
\begin{align*}
\sum_{h\le H} & \bigg|\sum_{n\le X} \mu(n+h)\prod_{i=1}^j d_{k_i}(n+a_i)\bigg|
\ \ll_{\delta,\mathcal A} \ \frac{HX}{\psi_\delta(X)}(\log X)^{k-j}.
\end{align*}
\end{proof}
\subsection{Almost all shifts}
Corollary \ref{cor:exceptmu} follows from the following result by the triangle inequality for $g_j=\mu$.
\begin{theorem}
Suppose $\log H/\log_2 X \to\infty$ as $X\to\infty$. Let $g_1=\mu$ and take any $g_j:{\mathbb N}\to{\mathbb C}$ with $|g_j|\le 1$ for $1< j\le k$. Then for any fixed shifts $h_2,...,h_k\le H$, $K>0$ we have
\begin{align}
\sum_{p\le X}\prod_{j=1}^k g_j(p+h_j) \ = \ o(\pi(X)),
\end{align}
for all except $O_K(H(\log X)^{-K})$ shifts $h_1\le H$.
\end{theorem}
\begin{proof}
Given ${\varepsilon}>0$ and fixed shifts $h_2,..,h_k\le H$, we aim to show $|\mathcal E|\ll_{{\varepsilon}} H(\log X)^{-K}$ for the exceptional set
\begin{align}
\mathcal E = \Big\{h\le H: \Big|\sum_{p\le X}\mu(p+h)\prod_{j=2}^kg_j(p+h_j)\Big|>2{\varepsilon} \pi(X) \Big\}.
\end{align}
To this, by Markov's inequality we have
\begin{align*}
|\mathcal E|({\varepsilon} \pi(X))
& \ll \sum_{h\in\mathcal E}\bigg|\sum_{p\le X}\mu(p+h)\prod_{j=2}^kg_j(p+h_j)\bigg| \\
&\le \ \sum_{h\in \mathcal E}\bigg|\sum_{p\le X}{\mathbf 1}_{\overline{\mathcal S}}(p+h)\bigg| \ + \
\sum_{h\le H}\bigg|\sum_{p\le X}{\mathbf 1}_{\mathcal S}\mu(p+h)\prod_{j=2}^kg_j(p+h_j)\bigg| \\
& \ \ll_A \quad \frac{\pi(X)}{\psi(X)}\sum_{h\in \mathcal E}\prod_{\substack{p\mid h\\p>P_1}}(1+\tfrac{1}{p}) \ \ + \quad \frac{H\pi(X)}{(\log X)^{A/40}},
\end{align*}
using Proposition \ref{lem:Chowlatypical} when $p+h\in \mathcal S$, and a standard sieve upper bound \cite[Theorem 7.1]{Opera} when $p+h\notin \mathcal S$.
Here $\mathcal S = \mathcal S(X,A,\delta)$ as in \eqref{eq:S} with $A=80K$ and $\delta=1/10$, say.
Observe for any $h\le H=(\log X)^{\psi(X)}$ the above product is at most $\prod_{P_1<p\le z}(1+\tfrac{1}{p}) \ll \frac{\log z}{\log P_1}$ where $z = P_1+\psi(X)\log_2 X$. Recalling $P_1 = (\log X)^{33A}$ this gives
\begin{align*}
\frac{\pi(X)}{\psi(X)}\sum_{h\in \mathcal E}\prod_{\substack{p\mid h\\p>P_1}}(1+\tfrac{1}{p}) \ = \ o\big(|\mathcal E|\,\pi(X)\big).
\end{align*}
Hence we conclude $|\mathcal E| \ll \frac{1}{{\varepsilon}}H(\log X)^{-K}$ as desired.
\end{proof}
\section{Non-pretentious multiplicative functions}
In this section we prove Theorem \ref{thm:pretend}, which we restate below.
\noindent
{\bf Theorem \ref{thm:pretend}}
{\it Let $H = X^\theta$ for $\theta\in(0,1)$, and take a multiplicative function $f:{\mathbb N}\to{\mathbb C}$ with $|f|\le1$. There exists $\rho\in (0,\frac{1}{8})$ such that, if $M(f;X^2/H^{2-\rho},Q) \to \infty$ as $X\to \infty$ for each fixed $Q$, then}
\begin{align*}
\sum_{h\le H}\Big|\sum_{p\le X}f(p+h)\Big| \ = \ o_{\theta,\rho}\big(H\pi(X)\big).
\end{align*}
\begin{proof}
Consider the exponential sum $F_x(\alpha)=\sum_{x\le m\le x+H}f(m)e(m\alpha)$. The hypotheses of our theorem are made in order to satisfy \cite[Theorem 1.4]{MRTUnif}, which in this case gives
\begin{align}\label{eq:FourUnif}
\int_0^X \sup_\alpha |F_x(\alpha)| \dd x \ = \ o_{\theta,\rho}(HX).
\end{align}
We critically note the supremum is {\it inside} the integral.
Now on to the proof, it suffices to show $S_f = o(HX)$ where
\begin{align*}
S_f := \sum_{h\le H}\Big|\sum_{n\le X}\Lambda(n)f(n+h)\Big|
\ll \sum_{h\le 2H}(2H-h)\Big|\sum_{n\le X}\Lambda(n)f(n+h)\Big|.
\end{align*}
We introduce coefficients $c(h)$ to denote the phase of $\sum_{n\le X}\Lambda(n)f(n+h)$, so that
\begin{align*}
S_f & \ll \frac{1}{H}\sum_{h\le H}(H-h)c(h)\sum_{n\le X}\Lambda(n)f(n+h)\\
& = \frac{1}{H}\sum_{h\le H}c(h)\sum_{n\le X}\Lambda(n) \sum_{m\le X+H}f(m){\mathbf 1}_{m=n+h}\cdot\int_0^X {\mathbf 1}_{x\le n,m\le x+H}\dd{x}\\
& = \frac{1}{H}\int_0^X\int_0^1\sum_{h\le H}c(h)e(h\alpha) \sum_{x\le n,m\le x+H}\Lambda(n)f(m)e\big((n-m)\alpha\big)\dd{\alpha}\dd{x},
\end{align*}
by orthogonality $\int_0^1 e(n\alpha)\dd{\alpha} ={\mathbf 1}_{n=0}$. That is, we have the following triple convolution
\begin{align}\label{eq:SfFourier}
S_f \ \ll \ \frac{1}{H}\int_0^X\int_0^1 C_0(\alpha) L_x(-\alpha) F_x(\alpha)\dd{\alpha}\dd{x},
\end{align}
denoting the sums $C_0(\alpha) = \sum_{h\le H}c(h)e(h\alpha)$ and $L_x(\alpha)=\sum_{x\le n\le x+H}\Lambda(n)e(n\alpha)$.
We shall split the inner integral on $\alpha$ according to the size of $L_x$. Specifically, for each $x$ let $\mathcal T_x = \{\alpha\in[0,1] : |L_x(\alpha)| \ge \delta H \}$.
Then by Markov's inequality, $\mathcal T_x$ has measure
\begin{align}\label{eq:measT}
\int_{\mathcal T_x}\dd{\alpha} \le \frac{1}{(\delta H)^4}\int_{\mathcal T_x} |L_x(\alpha)|^4\dd{\alpha} \ll \frac{1}{\delta^4 H},
\end{align}
since the Fourier identity implies
\begin{align*}
\int_0^1 | L_x (\alpha)|^4 \dd\alpha \ & = \ \sum_{x\le n_1,n_2,n_3,n_4\le x+H}\Lambda(n_1)\Lambda(n_2)\Lambda(n_3)\Lambda(n_4) {\mathbf 1}_{n_1+n_2=n_3+n_4} \\
& \ll \ (\log X)^4\sum_{\substack{x\le p_1,p_2,p_3,p_4\le x+H\\p_1+p_2=p_3+p_4}}1 \ \ll_{\theta} \ H^3,
\end{align*}
by a standard sieve upper bound \cite[Theorem 7.1]{Opera}. Thus plugging \eqref{eq:measT} into \eqref{eq:SfFourier}, we obtain
\begin{align*}
S_f \ll \frac{1}{H}\int_0^X\int_{[0,1]\setminus \mathcal T_x} C_0(\alpha) L_x(-\alpha) F_x(\alpha)\dd{\alpha}\dd{x} \ + \ \frac{1}{\delta^4 H^2}\int_0^X \sup_{\alpha\in \mathcal T_x}|C_0(\alpha) L_x(-\alpha) F_x(\alpha)|\dd{x}.
\end{align*}
Denote the two integrals above by $I_1$ and $I_2$. Observe $I_2 \,\ll_{\theta}\, \delta^{-4}\int_0^X \sup_{\alpha}|F_x(\alpha)|\dd{x}$,
using $|C_0(\alpha)| \le H$ trivially and $|L_x(\alpha)| \ll_{\theta} H$ by the Brun--Titchmarsh theorem. Then by definition of $\mathcal T_x$, Cauchy-Schwarz implies
\begin{align*}
I_1 & \le \delta\int_0^X\int_{[0,1]\setminus \mathcal T_x} |C_0(\alpha) F_x(\alpha)|\dd{\alpha}\dd{x} \le \delta\int_0^X
\bigg(\int_0^1 |C_0(\alpha)|^2\dd{\alpha} \cdot \int_0^1 |F_x(\alpha)|^2\dd{\alpha}\bigg)^{1/2}\dd{x} \ \le \ \delta HX,
\end{align*}
by Parseval's identity applied to $C_0$ and $F_x$. Thus combining bounds for $I_1,I_2$ gives
\begin{align}
S_f \ll_{\theta} \delta HX \ + \ \delta^{-4}\int_0^X \sup_{\alpha}|F_x(\alpha)|\dd{x}.
\end{align}
Hence taking $\delta\to0$, the Fourier uniformity bound \eqref{eq:FourUnif} gives $S_f = o_{\theta,\rho}(HX)$ as claimed.
\end{proof}
\section*{Acknowledgments}
The author is grateful to Joni Ter\"av\"ainen for suggesting the problem and for many valuable discussions. The author thanks Joni Ter\"av\"ainen and James Maynard for careful readings of the manuscript and for helpful feedback. The author is supported by a Clarendon Scholarship at the University of Oxford.
\bibliographystyle{amsplain}
|
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2020-09-21T02:18:28.000Z
|
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|
{"arxiv_id":"2009.08969","language":"en","timestamp":1600654708000,"url":"https:\/\/arxiv.org\/abs\/2009.08969","yymm":"2009"}
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| null | null |
\subsection{Terminology}
In this section, we explain the different stages a vulnerability goes through in its lifecycle. Then, we define our threat levels for vulnerabilities using the different stages that a vulnerability goes through during its lifecycle.
\subsection{Vulnerability Lifecycle}
\label{sec:background2}
A software \emph{vulnerability} is a weakness that allows unauthorized actions and/or access to be performed. These actions are typically used to break through the system and violate its security policies~\cite{liu2012software,shirey2000internet}. A \emph{vulnerability threat} is a
potential danger to exploit a vulnerability in order to breach security and cause possible harm~\cite{shirey2000internet}.
As shown in Figure~\ref{vuln_timeline}, typically, and with emphasis on vulnerabilities in the Node Package Manager ({npm}\xspace ecosystem), a vulnerability goes through a number of different stages~\cite{Reporting96}.
\begin{itemize}
\item \textbf{Introduction.} This is when the software vulnerability is first introduced into the code. At this stage, no one really knows about its existence, assuming that the introduction is not malicious. Hence, the potential threat of the vulnerability is quite low.
\item \textbf{Discovery (report).} When a vulnerability is discovered, it must be reported to the npm security team. The npm team investigates to ensure that the reported vulnerability is legitimate. At this stage, only the security team and the reporter of the vulnerability know about its existence. The potential threat at this stage is still low.
\item \textbf{Notification.} Once the reported vulnerability is confirmed, the security team triages the vulnerability and notifies the vulnerable package maintainers. At this stage, only the reporter, npm team, and package maintainers know about the vulnerability, hence its potential threat to be exploited remains low.
\item \textbf{Publication without a known fix.} Once the package maintainers are notified, they have 45 days before npm publishes the vulnerability publicly.
Alongside with publishing the vulnerability, the npm team may also publish a proof-of-concept showing how the vulnerability can be exploited.
At this stage, the vulnerability is known publicly and its potential threat is high.
\item \textbf{Publication with a fix.} Another (and more common) way that a vulnerability can be published is when a fix is provided by the package maintainers. If a fix is provided (before 45 days), then npm publishes the vulnerability along with the version of the package that fixes the vulnerability. At this stage, the potential threat is not as high as when a no fix is provided, but now the onus is on the application maintainers to make sure that they pull in the latest fixes, otherwise they are risking being exploited.
\end{itemize}
Typically, the vulnerability publish date is after the report and notification dates. It is important to note that although the aforementioned stages are generally sequential, we do see cases where {it is not}. For example, in some cases we see vulnerabilities with a fix date that precedes its reporting or publication
date. The race between developers and attackers starts as soon as a vulnerability is discovered. We use the different stages of a vulnerability to examine the potential threat of software vulnerabilities next.
\subsection{Threat Levels}
\label{sec:ex}
As shown earlier, the different stages that a vulnerability goes through significantly impact its threat. Hence, our study is based on the idea that vulnerabilities should be examined while taking their threat into consideration as the \textit{vulnerability timing} makes them hard to exploit. We use the various stages to ground our argument and define three specific threat levels:
\begin{enumerate}
\item \textbf{Low threat - before discovery (report).} Since very little (or nothing at all) is known about a vulnerability before it is found, i.e., vulnerabilities are hidden in the applications, we believe that its potential threats and chances of being exploited are very low. Hence, we classify all vulnerabilities at this stage as having \emph{low threat}.
\item \textbf{Medium threat - after discovery \& before publication:} Once a vulnerability has been discovered, there is potential that others may also know about it. Moreover, since at this stage the public is still not aware of the vulnerability,
the vulnerability might be exploited by people who know about it somehow and have the capability to exploit it. Hence, we classify vulnerabilities at this stage as having \emph{medium threat}.
\item \textbf{High threat - after publication:} After publication
this is the time when the chance of exploitability is highest. Of course, if a fix is provided, then the risk is lower, however, if the application does not update then it still faces a major risk of being exploited. If a fix is not provided, then all applications are at a very high risk of being exploited, hence, we classify all vulnerabilities at this stage as having \emph{high threat}.
\end{enumerate}
\begin{figure}[tb!]
\centering
\includegraphics[width=1\linewidth]{images/vuln_timeline_new}
\caption[]{Classification of threats over the vulnerability lifecycle.}
\label{vuln_timeline}
\end{figure}
\subsection{Data Collection}
\label{subsec:data}
Our study examines vulnerable dependencies in Node.js applications. We chose to focus on Javascript due to its wide popularity amongst the development community~\cite{StackOve70:online}.
\\
\noindent\textbf{Packages vs. Applications.} The software community classifies JavaScript projects into two categories:
\textit{1) packages}, also referred to as libraries, which are included in other applications using dependency management tools to help facilitate and speed up development.
Packages are referred to as "dependencies" of an application.
\textit{2) applications} are standalone software projects, which are distinct from libraries, where they are not distributed via a package manager and are typically applications for clients and end users rather than components to build upon.
As mentioned before, the Node.js applications mainly state the packages they depend on (i.e., dependencies) in a file called package.json.
To perform our study, we leverage two datasets: (1) Node.js applications that use {npm} to manage their dependencies, and (2) Security vulnerabilities that affect npm packages. To do so, we \textbf{(i)} obtain the Node.js applications from GitHub, \textbf{(ii)} extract their dependencies, and \textbf{(iii)} obtain the security vulnerabilities for npm packages from {npm} advisories~\cite{npmadvisories}. The dataset collection took place during May and June of 2019.\\
\noindent\textbf{(i) Applications Dataset.} To analyse a large number of open source JavaScript applications that depend on npm packages and obtain insights on their security vulnerabilities, we mine the GHTorrent dataset~\cite{GHtorrent} and extract information about all Node.js applications hosted on GitHub. The GHTorrent dataset contains a total of 7,863,361 JavaScript projects hosted on GitHub, of which 2,289,130 use npm as their package management platform (i.e., these projects contain a file called package.json). Moreover, since both Node.js \textit{packages} and \textit{applications} can use GitHub as their development repository, and our applications dataset should only contain Node.js \emph{applications}, we filter out the GitHub projects
that are actually npm \emph{packages} by checking their GitHub URL on the {{npm}} registry. The main reason that we focused on applications and not packages is because packages become exploitable only when used and deployed in an application, i.e., packages do not reside on their own in production, they should be part of applications that make use of them. This filtering excludes 328,343 projects from our list of GitHub projects as they are identified as packages and not applications
As shown in previous studies~\cite{kalliamvakou2014promises,kula2018developers}, some projects on GitHub are immature, hence, to make this study more reliable we refined the dataset using additional filtering criteria to eliminate such immature projects. In particular, we gather applications that satisfy all the following criteria:
\begin{itemize}
\item Non-forked applications, as we do not want to have duplicated project history to bias our analysis.
\item Applications that depend upon more than two dependencies.
\item Applications that have at least 100 commits by more than two contributors, which indicates a minimal level of commit activity.
\item Applications that have had their creation date (first commit) before January 1st 2017.
Since vulnerabilities take on median 3 years to be discovered~\cite{decan2018impact}, applications in our dataset need to have a development history long enough to have had a chance for their vulnerabilities to be discovered.
\item Applications that have had their latest commit after January 1st 2017, as we want to analyze applications that had some level of development in the last 3 years.
\end{itemize}
After applying these refinement criteria, we end up with 6,673 Node.js applications that make use of npm packages. Table~\ref{statsApplications} shows descriptive statistics on the selected Node.js applications in our dataset. Overall, the applications in our dataset
have a rich development history (a median of 213 commits made by 4 developers and 1,657
days of development lifespan) and make ample use of external
dependencies (a median of 11 dependencies).\\
\noindent\textbf{(ii) Application Dependencies.}
After obtaining the applications dataset, we want to extract the history of dependency changes of all applications.
This is necessary to identify the exact dependency versions that would be installed by the application at any specific point-in-time.
As mentioned in Section~\ref{sec:dependency}, Node.js applications specify their dependencies in the package.json file, which contains
the dependency list, containing the dependent upon packages and their respective version constraints.
Hence, we extract all changes that touched the package.json file and associate each commit hash and commit date to their respective package.json dependency list, creating a history of dependency changes for all applications.
Note that these dependencies are not yet resolved, that is, we only have the version constraints (not the versions) for the dependencies of each application.
\\
\begin{table}[tb!]
\centering
\caption{Statistics of the 6,673 studied Node.js applications.
}
\label{statsApplications}
\begin{tabular}{l|r|r|r|r}
\toprule
\textbf{Metric} & \textbf{Min.} & \textbf{Median($\bar{x}$)} & \textbf{Mean($\mu$)} & \textbf{Max.} \\ \midrule
\textbf{Commits} & 100 & 213 & 384.60 & 53,872 \\
\textbf{Dependencies} & 3 & 11 & 14.93 & 114 \\
\textbf{Developers} & 3 & 4 & 5.33 & 52 \\
\textbf{Lifespan (in days)} & 151 & 1,657 & 1,730.07 & 3,575 \\
\bottomrule
\end{tabular}
\end{table}
\noindent\textbf{(iii) NPM Advisories Dataset.}
To identify Node.js applications that depend on vulnerable packages, we need to collect information on npm vulnerable packages. We resort to the \textit{NPM advisories} registry to obtain the required information about all npm vulnerable packages~\cite{npmadvisories}.
The npm advisories dataset is the official registry for npm vulnerability reports, which contains a number of JavaScript vulnerabilities, specific to the Node.js-platform packages.
This dataset provides several kinds of information about vulnerable packages relevant to our study.
Each report has the affected package name, the package versions affected by the vulnerability, and the versions in which the vulnerabiliy was fixed (safe versions).
The report also contains both the vulnerability discovered (reported) time and published time, which we use in our approach for identifying and classifying vulnerabilities (Section~\ref{subsec:scanning}).
Note that a vulnerable package could be affected by several vulnerabilities (i.e., a vulnerable package appears with different vulnerability reports due to different vulnerability types).
Our initial dataset contains 654 security reports that cover 601 vulnerable packages. Following the criteria filtration process applied by Decan et al.~\cite{decan2018impact}, we removed 12 vulnerable packages of the type "Malicious Package", because they do not actually introduce vulnerable code. These vulnerabilities are packages with names close to popular packages (a.k.a. typo-squatting) in an attempt to deceive users at installing harmful packages. The 12 vulnerable packages account for 12 vulnerability reports. At the end of this filtering process, we are left with 642 security vulnerabilities reports affecting 589 distinct vulnerable packages.
These packages have combined 26,462 distinct package versions of which 13,868 are affected by vulnerabilities from our report.
Table~\ref{dataset} shows the summary statistics for vulnerability reports on npm packages.
\begin{table}[tb!]
\centering
\caption{Descriptive statistics on the npm advisories dataset.}
\label{dataset}
\begin{tabular}{l|r}
\toprule
Vulnerability reports & 642 \\
Vulnerable packages & 589 \\
Versions of vulnerable packages & 26,462 \\
Affected versions by vulnerability & 13,868 \\
\bottomrule
\end{tabular}%
\vspace{-2mm}
\end{table}
\begin{figure*}[tb!]
\centering
\setlength{\abovecaptionskip}{5pt}
\includegraphics[width=0.7\linewidth]{images/approach.pdf}
\caption[]{Approach for identifying and classifying vulnerable dependencies in Node.js applications.}
\label{classifyApproach}
\end{figure*}
\subsection{Identifying and Classifying Vulnerable Dependencies in Node.js Applications}
\label{subsec:scanning}
To classify the threat level of vulnerable dependencies at a specific point in the development history of a Node.js application, which we refer to as the \textit{analyzed snapshot time}, we leverage 3-step approach. Figure~\ref{classifyApproach} provides an overview of our approach, which we detail below:
\noindent
{\textbf{Step 1. Extract dependencies and resolve versions.}}
The goal of this step is to extract applications dependencies and find the actual dependency version installed at the analyzed snapshot time.
For each application, we extract the dependency list (with the versioning constraints) at that snapshot time from the history of dependency changes.
After that, to find the actual version of each dependency at the analyzed snapshot, we utilize the \textit{semver} tool~\cite{semvernp91} that is used by npm to find the latest version that satisfies the versioning constraint, with an additional restriction that the satisfying version should have been released (in the npm registry) before the application snapshot time. For example, an application can specify a versioning constraint (``P:$>$1.0.0'') at the snapshot May 2016. Hence, the actual installed version is the latest version that is greater than 1.0.0 and also has been released in the npm registry before May 2016.
This step allows us to find the installed version of the dependency at the analyzed snapshot time.
\\
\noindent
{\textbf{Step 2. Identify vulnerable dependency versions.}}
After determining the resolved (and presumably installed) version at the analyzed snapshot time, we check whether the resolved version is vulnerably or not. To do so, we check the advisories dataset for the versions that were available at that snapshot point.
If the resolved version is covered by the advisories dataset, we label it as a vulnerable dependency version. We skip the whole next step if the dependency version has not been mentioned in any advisory, i.e., the dependency version is not vulnerable.
\\
\noindent
{\textbf{Step 3. Identify threat levels of vulnerable versions.}}
Once we identify the vulnerable dependency versions at the analyzed snapshot time, we classify each vulnerable dependency version using one of the threat levels we defined earlier (in Section~\ref{sec:ex}), i.e., we find out the threat level of each vulnerable dependency version. To do so, for each vulnerable version, we compare its vulnerability \textit{discovery (report)} and \textit{publication} time to the analyzed snapshot time. As we stated previously (in Section~\ref{sec:ex}), if the vulnerability publication time of the vulnerable dependency version is before the application's snapshot time then we mark the vulnerability as high threat vulnerability. If the vulnerability of the dependency was not published but only discovered (reported) before the application's snapshot time, then we mark it as medium. And finally, if it was neither published nor discovered (reported) before the analyzed snapshot time (i.e., no one knows about it at that snapshot time), then we mark it as low.
In cases where more than one vulnerability affects the vulnerable dependency version, we resort to a weakest link approach (i.e., we label the vulnerable dependency version with the highest threat level). For example, if we find that the vulnerable version of the dependency is affected by two vulnerabilities -one having low threat and another as high threat, we label the vulnerable dependency version as high at that snapshot time.
\subsection{Replication Package}
To facilitate verification and advancement of research in the field, a replication package comprising the data used in our study along with the analyses used in our study is publicly available\footnote{http://doi.org/10.5281/zenodo.3837397}.
\subsection{Security Migration Cost}
\label{sec:cost}
Developers of Node.js applications may use dynamic versioning
constraints if they want to install the latest version of a dependency,
allowing them to get the latest updates for security fixes of the package. In fact, npm adopts a semantic version scheme~\cite{semvernp91}, where
package maintainers are encouraged to specify the extent of their
updates in three different levels: 1) patch release, which indicates backward compatible bug fixes, 2) minor release, which indicates backwards
compatible new features and 3) major release, which informs developers of
backwards incompatible changes in the package release. While our
study (RQ$_3$) showed that 90.76\% of high-threat vulnerabilities have a safe
version available for application maintainers (at the snapshot 100\%),
we manually inspected the fixed versions and the applications version constraints and found that in 43.07\% of the cases, the fix is only
available in another major release. For instance, an application
depends on P:1.0.0, and the fix patch was only released for a major
version 2.0.0 and onwards. Hence, to benefit from a fix patch in such a case, developers are required to upgrade their dependencies at the risk of breaking their own code, since a new major release has breaking
changes compared to the version the application depends on. This
imposes significant migration costs, especially for large projects
that depend on dozens of packages. Furthermore, this shows that
using dynamic versioning at the level of patch and minor releases
(as recommended by npm) does not completely prevent high threat
level vulnerabilities for affecting Node.js applications.
\vspace{0.1in}
\subsection{Implications}
\label{sec:imp}
\textbf{Implications to researchers.}
Several studies have addressed the problem of vulnerabilities in software libraries~\cite{decan2018impact,zapata2018towards}.
Our study, however, complements previous studies by analyzing the risks of vulnerable dependencies in the Node.js applications, aggregating the vulnerability lifecycle through the threat level metric.
Researchers can use our empirical evidence to better understand the risks Node.js applications face due to their high reliance on dependencies.
Our results show that most vulnerable dependencies found in a application snapshot have a low risk of being exploited when considering the lifecycle of vulnerabilities and how applications update their dependencies. Our results also show that the time element is crucial to understanding the threat of vulnerable dependencies in applications.
Hence, a major implication of our study for researchers is that not all vulnerabilities are equal, and should not be treated and analyzed as such.
Research needs to account for more than the existence of vulnerabilities to draw more meaningful analyses regarding software security, particularly for applications in software ecosystems where the level of dependency continues to increase. Research can use our threat-level approach to provide a more refined picture when reporting the impact of vulnerabilities. Researchers can also reuse our approach to help them identify and classify vulnerable dependencies in the applications (in Section~\ref{subsec:scanning}).
Furthermore, more studies across ecosystems are necessary to get a broader perspective on the threat level of vulnerability dependencies.
npm is one of the largest ecosystems and since applications depend on an increasingly high number of packages~\cite{decan2018empirical}, Node.js applications may be subjected to higher risk of vulnerable dependencies.
Further investigation could unveil if this pattern holds in other ecosystems.
\\
\noindent
\textbf{Implications to practitioners.}
Our results revealed important takeaways for software practitioners.
First, vulnerable dependencies are common, 67.93\% of the studied Node.js applications had at least one vulnerable dependency at the last studied snapshot.
Practitioners need to be in constant alert to update their dependencies and tools that increase awareness of vulnerabilities, such as Dependabot~\cite{Dependab26:online} and \texttt{npm audit}~\cite{npmaudit59:online} are evermore crucial for the safety of software applications, especially because they warn developers as soon as the vulnerability becomes of a high threat level.
Second, practitioners also need to account for the threat level of a vulnerability to have a more correct understanding of software vulnerabilities in software ecosystems. Our method of analysis can also be used by developers to identify packages that more often raise the threat level in their applications.
Also, while vulnerabilities are widespread in open-source packages in the npm, in most cases package maintainers issue a fix patch for their vulnerability as soon as it becomes public, which is crucial to mitigate the chances of having a vulnerability exploited and cause potential harm to end-users and application maintainers.
Third, our study showed that developers are in need of more tools that go beyond simply warning them of a published vulnerability. For example, they need tools to help them understand:
1) the costs of migrating to a safer version and whether it is possible to fix a vulnerability without breaking their code,
2) the frequency in which certain dependencies have become vulnerable in the past, in order to grab the threats of depending on such packages and better plan their project maintenance,
3) history of all vulnerable dependencies of their application in order to understand the frequency and the duration in which their application became at the risk of a high threat vulnerability in the past.
Packages that do not update their code to address reported vulnerabilities incur in a high risk for applications that use them and should be avoided by critical applications.
\section{Introduction}
\label{sec:introduction}
\input{introduction}
\begin{figure*}[!t]
\centering
\end{figure*}
\section{Classifying Vulnerabilities}
\label{sec:background}
\input{background}
\section{NPM Dependency Management}
\label{sec:dependency}
\input{dependency}
\section{Case Study Design}
\label{sec:case_study_design}
\input{design}
\section{Case Study Results}
\label{sec:results}
\input{results}
\section{Discussion}
\label{sec:discussion}
\input{discussion}
\section{Related Work}
\label{sec:related_work}
\input{related_work}
\section{Threats to Validity}
\label{sec:threats}
\input{threats}
\section{Conclusion and Future Work}
\label{sec:conclusion}
\input{conclusion}
\bibliographystyle{IEEEtran}
\balance
\subsection{Terminology}
In this section, we explain the different stages a vulnerability goes through in its lifecycle. Then, we define our threat levels for vulnerabilities using the different stages that a vulnerability goes through during its lifecycle.
\subsection{Vulnerability Lifecycle}
\label{sec:background2}
A software \emph{vulnerability} is a weakness that allows unauthorized actions and/or access to be performed. These actions are typically used to break through the system and violate its security policies~\cite{liu2012software,shirey2000internet}. A \emph{vulnerability threat} is a
potential danger to exploit a vulnerability in order to breach security and cause possible harm~\cite{shirey2000internet}.
As shown in Figure~\ref{vuln_timeline}, typically, and with emphasis on vulnerabilities in the Node Package Manager ({npm}\xspace ecosystem), a vulnerability goes through a number of different stages~\cite{Reporting96}.
\begin{itemize}
\item \textbf{Introduction.} This is when the software vulnerability is first introduced into the code. At this stage, no one really knows about its existence, assuming that the introduction is not malicious. Hence, the potential threat of the vulnerability is quite low.
\item \textbf{Discovery (report).} When a vulnerability is discovered, it must be reported to the npm security team. The npm team investigates to ensure that the reported vulnerability is legitimate. At this stage, only the security team and the reporter of the vulnerability know about its existence. The potential threat at this stage is still low.
\item \textbf{Notification.} Once the reported vulnerability is confirmed, the security team triages the vulnerability and notifies the vulnerable package maintainers. At this stage, only the reporter, npm team, and package maintainers know about the vulnerability, hence its potential threat to be exploited remains low.
\item \textbf{Publication without a known fix.} Once the package maintainers are notified, they have 45 days before npm publishes the vulnerability publicly.
Alongside with publishing the vulnerability, the npm team may also publish a proof-of-concept showing how the vulnerability can be exploited.
At this stage, the vulnerability is known publicly and its potential threat is high.
\item \textbf{Publication with a fix.} Another (and more common) way that a vulnerability can be published is when a fix is provided by the package maintainers. If a fix is provided (before 45 days), then npm publishes the vulnerability along with the version of the package that fixes the vulnerability. At this stage, the potential threat is not as high as when a no fix is provided, but now the onus is on the application maintainers to make sure that they pull in the latest fixes, otherwise they are risking being exploited.
\end{itemize}
Typically, the vulnerability publish date is after the report and notification dates. It is important to note that although the aforementioned stages are generally sequential, we do see cases where {it is not}. For example, in some cases we see vulnerabilities with a fix date that precedes its reporting or publication
date. The race between developers and attackers starts as soon as a vulnerability is discovered. We use the different stages of a vulnerability to examine the potential threat of software vulnerabilities next.
\subsection{Threat Levels}
\label{sec:ex}
As shown earlier, the different stages that a vulnerability goes through significantly impact its threat. Hence, our study is based on the idea that vulnerabilities should be examined while taking their threat into consideration as the \textit{vulnerability timing} makes them hard to exploit. We use the various stages to ground our argument and define three specific threat levels:
\begin{enumerate}
\item \textbf{Low threat - before discovery (report).} Since very little (or nothing at all) is known about a vulnerability before it is found, i.e., vulnerabilities are hidden in the applications, we believe that its potential threats and chances of being exploited are very low. Hence, we classify all vulnerabilities at this stage as having \emph{low threat}.
\item \textbf{Medium threat - after discovery \& before publication:} Once a vulnerability has been discovered, there is potential that others may also know about it. Moreover, since at this stage the public is still not aware of the vulnerability,
the vulnerability might be exploited by people who know about it somehow and have the capability to exploit it. Hence, we classify vulnerabilities at this stage as having \emph{medium threat}.
\item \textbf{High threat - after publication:} After publication
this is the time when the chance of exploitability is highest. Of course, if a fix is provided, then the risk is lower, however, if the application does not update then it still faces a major risk of being exploited. If a fix is not provided, then all applications are at a very high risk of being exploited, hence, we classify all vulnerabilities at this stage as having \emph{high threat}.
\end{enumerate}
\begin{figure}[tb!]
\centering
\includegraphics[width=1\linewidth]{images/vuln_timeline_new}
\caption[]{Classification of threats over the vulnerability lifecycle.}
\label{vuln_timeline}
\end{figure}
\subsection{Security Migration Cost}
\label{sec:cost}
Developers of Node.js applications may use dynamic versioning
constraints if they want to install the latest version of a dependency,
allowing them to get the latest updates for security fixes of the package. In fact, npm adopts a semantic version scheme~\cite{semvernp91}, where
package maintainers are encouraged to specify the extent of their
updates in three different levels: 1) patch release, which indicates backward compatible bug fixes, 2) minor release, which indicates backwards
compatible new features and 3) major release, which informs developers of
backwards incompatible changes in the package release. While our
study (RQ$_3$) showed that 90.76\% of high-threat vulnerabilities have a safe
version available for application maintainers (at the snapshot 100\%),
we manually inspected the fixed versions and the applications version constraints and found that in 43.07\% of the cases, the fix is only
available in another major release. For instance, an application
depends on P:1.0.0, and the fix patch was only released for a major
version 2.0.0 and onwards. Hence, to benefit from a fix patch in such a case, developers are required to upgrade their dependencies at the risk of breaking their own code, since a new major release has breaking
changes compared to the version the application depends on. This
imposes significant migration costs, especially for large projects
that depend on dozens of packages. Furthermore, this shows that
using dynamic versioning at the level of patch and minor releases
(as recommended by npm) does not completely prevent high threat
level vulnerabilities for affecting Node.js applications.
\vspace{0.1in}
\subsection{Implications}
\label{sec:imp}
\textbf{Implications to researchers.}
Several studies have addressed the problem of vulnerabilities in software libraries~\cite{decan2018impact,zapata2018towards}.
Our study, however, complements previous studies by analyzing the risks of vulnerable dependencies in the Node.js applications, aggregating the vulnerability lifecycle through the threat level metric.
Researchers can use our empirical evidence to better understand the risks Node.js applications face due to their high reliance on dependencies.
Our results show that most vulnerable dependencies found in a application snapshot have a low risk of being exploited when considering the lifecycle of vulnerabilities and how applications update their dependencies. Our results also show that the time element is crucial to understanding the threat of vulnerable dependencies in applications.
Hence, a major implication of our study for researchers is that not all vulnerabilities are equal, and should not be treated and analyzed as such.
Research needs to account for more than the existence of vulnerabilities to draw more meaningful analyses regarding software security, particularly for applications in software ecosystems where the level of dependency continues to increase. Research can use our threat-level approach to provide a more refined picture when reporting the impact of vulnerabilities. Researchers can also reuse our approach to help them identify and classify vulnerable dependencies in the applications (in Section~\ref{subsec:scanning}).
Furthermore, more studies across ecosystems are necessary to get a broader perspective on the threat level of vulnerability dependencies.
npm is one of the largest ecosystems and since applications depend on an increasingly high number of packages~\cite{decan2018empirical}, Node.js applications may be subjected to higher risk of vulnerable dependencies.
Further investigation could unveil if this pattern holds in other ecosystems.
\\
\noindent
\textbf{Implications to practitioners.}
Our results revealed important takeaways for software practitioners.
First, vulnerable dependencies are common, 67.93\% of the studied Node.js applications had at least one vulnerable dependency at the last studied snapshot.
Practitioners need to be in constant alert to update their dependencies and tools that increase awareness of vulnerabilities, such as Dependabot~\cite{Dependab26:online} and \texttt{npm audit}~\cite{npmaudit59:online} are evermore crucial for the safety of software applications, especially because they warn developers as soon as the vulnerability becomes of a high threat level.
Second, practitioners also need to account for the threat level of a vulnerability to have a more correct understanding of software vulnerabilities in software ecosystems. Our method of analysis can also be used by developers to identify packages that more often raise the threat level in their applications.
Also, while vulnerabilities are widespread in open-source packages in the npm, in most cases package maintainers issue a fix patch for their vulnerability as soon as it becomes public, which is crucial to mitigate the chances of having a vulnerability exploited and cause potential harm to end-users and application maintainers.
Third, our study showed that developers are in need of more tools that go beyond simply warning them of a published vulnerability. For example, they need tools to help them understand:
1) the costs of migrating to a safer version and whether it is possible to fix a vulnerability without breaking their code,
2) the frequency in which certain dependencies have become vulnerable in the past, in order to grab the threats of depending on such packages and better plan their project maintenance,
3) history of all vulnerable dependencies of their application in order to understand the frequency and the duration in which their application became at the risk of a high threat vulnerability in the past.
Packages that do not update their code to address reported vulnerabilities incur in a high risk for applications that use them and should be avoided by critical applications.
\section{Introduction}
\label{sec:introduction}
\input{introduction}
\begin{figure*}[!t]
\centering
\end{figure*}
\section{Classifying Vulnerabilities}
\label{sec:background}
\input{background}
\section{NPM Dependency Management}
\label{sec:dependency}
\input{dependency}
\section{Case Study Design}
\label{sec:case_study_design}
\input{design}
\section{Case Study Results}
\label{sec:results}
\input{results}
\section{Discussion}
\label{sec:discussion}
\input{discussion}
\section{Related Work}
\label{sec:related_work}
\input{related_work}
\section{Threats to Validity}
\label{sec:threats}
\input{threats}
\section{Conclusion and Future Work}
\label{sec:conclusion}
\input{conclusion}
\bibliographystyle{IEEEtran}
\balance
\subsection*{\rqone}
\label{sec:RQ1}
{\textbf{Motivation}}:
Prior work showed that a significant amount of application code comes from third party packages, and a non-negligible amount of these packages are affected by known security vulnerabilities~\cite{williams2012unfortunate}. However, we argue that not all vulnerabilities should be treated equally. Hence, in this RQ we would like to quantify how many of our studied applications have at least one vulnerable dependency and what the threat level of these vulnerable dependencies is. Answering this question will help us understand the real risk/threat of vulnerable packages on the software applications.
\noindent
{\textbf{Approach}}:
In order to perform an unbiased analysis, we need to account for vulnerability discovery time. Prior work showed that vulnerabilities in npm take on median 3 years to be discovered and publicly announced~\cite{decan2018impact}. As a consequence, selecting snapshots of our applications in 2019 will paint an incomplete picture, as most vulnerabilities recently introduced in the package's code would remain hidden for a median of 3 years.
Since we collected the advisories dataset in May/June 2019, we chose to evaluate our applications as of May 2016 (3 years prior), which ensures that at least half the dependency vulnerabilities introduced in the code are reported in the current advisories dataset.
Then, we answer our RQ in two steps. First, we examine if the \textit{selected snapshot} of the application had at least one dependency that contains a vulnerability (irrespective of its threat level).
Then, to determine the threat level of the vulnerable dependencies in the examined applications, we focus only on the set of applications that have at least one vulnerable dependency using the methodology described in Section~\ref{subsec:scanning}.
In the second step, we quantify the number of vulnerable dependencies in the applications under each threat level.
We first check the percentage of overall vulnerable dependencies in each application and illustrate their distribution using a Boxplot. We further analyze the distribution of these vulnerable dependencies across the threat levels and plot it using three Boxplots, one for each threat level. For example, an application could have 10\% of its dependencies as vulnerable at the analysed snapshot, and such percentage (i.e., 10\%) could be distributed across the threat levels as follows: 25\% of the vulnerable dependencies are classified as low threat, 60\% of them are classified as medium, and 15\% as high.
\begin{figure}[tb!]
\centering
\includegraphics[width=1\linewidth,height=.85\linewidth]
{images/RQ1_new.pdf}
\caption{Boxplots showing the distributions of the percentage of overall vulnerable dependencies in the applications (left boxplot), and how these percentages are distributed across threat levels (right boxplot). N and M are the total number of dependencies and the total number of vulnerable dependencies, respectively.
}
\label{RQ1Perc}
\end{figure}
\noindent \textbf{Results}:
Of the 6,673 studied applications \textbf{67.93\% (4533 applications) depend on at least one vulnerable dependency}. The affected applications contains a total of 10,154 vulnerable dependencies from 149 distinct vulnerable packages. The 149 packages comprises 23.21\% of the overall vulnerable packages in the npm advisories dataset.
Figure~\ref{RQ1Perc} shows the percentage of vulnerable dependencies per application (left boxplot), and the distribution of vulnerable dependencies at different threat levels (right boxplot). It shows that, on median, 14.29\% of the dependencies in the affected application (i.e., applications with at least 1 vulnerable dependency) are vulnerable. Also, Figure~\ref{RQ1Perc} shows that such percentage of vulnerable dependencies (i.e., 14.29\%) is distributed as follows: \textbf{94.91\% of the vulnerable dependencies are classified as low threat vulnerabilities}, 2.06\% of them are classified as medium, and 3.03\% are classified as high.
\vspace{0.08in}
\begin{table}[h!]
\centering
\caption{Mann-Whitney Test (p-value) and Cliff's Delta (d) for the different threat levels.}
\label{stattest}
\begin{tabular}{l|r|r}
\toprule
\textbf{Threat Levels} & \textbf{\textit{p}-value} &\textbf{ Cliff's Delta \textit{{(d)}}} \\ \midrule
Low vs. Medium & 2.2e-16 & 0.984 (large) \\
Low vs. High & 2.2e-16 & 0.970 (large) \\
Medium vs. High & 2.2e-16 & 0.335 (medium) \\ \bottomrule
\end{tabular}%
\end{table}
\vspace{0.08in}
To statistically verify our observation, we perform a one-sided non-parametric Mann-Whitney U test~\cite{mcknight2010mann} by comparing the distributions between the different threat levels.
Table~\ref{stattest} shows the p-values and effect size values. We observe a statistically significant differences between (low and medium), (low and high), (medium and high), at p-value $<$ 0.05 for all comparisons. Furthermore, we observe, using Cliff's delta~\cite{cliff1993dominance}, a large effect size for the differences between low and medium, low and high. Also, we found a medium effect size for the difference between medium and high. This indicates that the differences between the different threat levels are statistically significant.\\
\begin{mdframed}[roundcorner=5pt,linewidth=0.5mm,
linecolor=black]
\lipsum[0]
\textbf{\textit{Our findings show that 67.93\% of the examined applications depend on at least one vulnerable package. However, the vast majority (94.91\%) of these dependencies have low threat.}
}
\end{mdframed}
\vspace{0.2in}
\subsection*{\rqtwo}
{\textbf{\\Motivation}}: Thus far, we have analyzed the vulnerability threats of a single snapshot of each application in our dataset. However, our findings may differ as the applications evolve. For example, a vulnerability with high threat on a given day could have had low threat the week before.
\begin{figure*}[tb!]
\centering
\setlength{\abovecaptionskip}{10pt}
\includegraphics[width=1\linewidth]
{images/RQ2Evol3_new.pdf}
\caption{Boxplots showing the percentage of overall vulnerable dependencies and their distribution in each threat level over the studied snapshots. N and M are the total number of dependencies and the total number of vulnerable dependencies, respectively.}
\label{RQ2Evol}
\end{figure*}
Hence, in this RQ we would like to determine whether our results generalize to different historical snapshots in the application's development lifetime. Such an evolutionary examination allows us to discover whether the trend of the threat levels changes across different stages of an application's lifetime.
\\
\noindent
{\textbf{Approach}}:
Since the different applications are of different lifespans, we want to find a measure that makes comparing them feasible. To do so, we use the number of commits as a way to divide the applications into different intervals. Since commit frequency and time between commits vary from one application to another, we normalize the applications by segmenting the lifetimes of each application into five equal intervals (each containing 20\% of an application's lifetime by {time in days}),
take one snapshot at each interval, then analyze it. Although this might seem like a straightforward task, it poses some challenges, since we have a large applications dataset and the package.json file in them is updated significantly over the application's lifetime.
For this analysis, we only consider the affected applications identified in RQ$_1$.
The last snapshot (at 100\%) is the same snapshot that we analyzed in RQ$_1$ (i.e., May 2016).
\begin{table}[tbh]
\centering
\caption{The percentage of vulnerable applications at different historical snapshots.}
\label{DisT1}
\begin{tabular}{l|r}
\toprule
\multirow{2}{*}{\textbf{Snapshot}} & \textbf{Vulnerable} \\
& \textbf{Applications} \\
\midrule
\textbf{20\%} &{ 55.31\%} \\
\textbf{40\%} & 58.17\% \\
\textbf{60\%} & 60.87\% \\
\textbf{80\%} & 63.03\% \\
\textbf{100\%} & {67.93\%} \\ \bottomrule
\end{tabular
\end{table}
\noindent
{\textbf{Results}}:
Table~\ref{DisT1} shows the percentage of applications that have at least one vulnerable dependency for the 5 analyzed snapshots across their lifetime. We observe that \textbf{the percentage of vulnerable applications steadily increases each snapshot} and varies between 55.31 - 67.93\% in the studied applications.
Figure~\ref{RQ2Evol} illustrates the distributions of the percentage of vulnerable dependencies at each threat level over the studied snapshots. The total number of dependencies (N) and the total number of vulnerable dependencies (M) in the studied applications are shown at the bottom of Figure~\ref{RQ2Evol}. The raw numbers of the dependencies show that the total number of dependencies increases over time, and so does the raw number of vulnerable dependencies.
From Figure~\ref{RQ2Evol}, we observe that the affected applications depend on vulnerable dependencies at an earlier stage (i.e., at 20\%) of their lifetime. However, we also observe that the trend observed in RQ1 remains the same, i.e., the overall percentage of vulnerable dependencies ranges between 14.29\% - 14.68\%. Also, the majority of the vulnerabilities have a low threat level, followed by high and medium threat. To sum up, our analysis shows that all trends observed in RQ$_1$
also hold at different stages of the applications, albeit the raw number of dependencies does increase.\\
\begin{mdframed}[roundcorner=5pt,linewidth=0.5mm,
linecolor=black]
\lipsum[0]
\textbf{\textit{As applications evolve, the overall number of vulnerable dependencies is increasing, however, the median percentage of vulnerable dependencies remains mostly constant. Moreover, the majority of vulnerabilities they face remain as low threat vulnerabilities, as these applications evolve.}}
\end{mdframed}
\vspace{-0.2in}
\subsection*{\rqthree}
\textbf{Motivation}:
In the previous research questions, we found that the majority of affected dependencies are impacted by low threat vulnerabilities, throughout applications development history.
However, a sizeable number of projects depend on
high threat dependencies, which are the most important. This means that those applications depend on vulnerable versions of dependencies even after the vulnerability reports have been discovered (reported)-and-published.
In such cases, the \emph{developers of the applications could know} about the presence of the vulnerability in the dependency, and hence, {should} avoid using that vulnerable version, if a fix is available.
Specifically, we want to know who is to blame - the package maintainers for not providing a version that fixes a known vulnerability - or the application maintainers for not keeping their applications up-to-date.
Answering this will help us pinpoint the causes for high threat vulnerabilities in npm applications and develop further strategies to solve this problem.
\noindent
\textbf{Approach}:
To perform our investigation and answer who is responsible for the high threat vulnerabilities in applications, we use the same method to determine high threat vulnerabilities as presented in the first two RQs.
For each high threat vulnerable dependency, we check the availability of a safe version of the package for the vulnerability at the analyzed snapshot time. Depending on such availability our analysis has one of two outcomes:
\begin{itemize}
\item \textbf{Package-to-blame:} if at the analyzed snapshot, no safe version has been provided by the package maintainers for a publicly known vulnerability. As the publication of a vulnerability comes after a period of 45 days, we consider the package maintainers the responsible for the high threat vulnerability in applications.
\item \textbf{Application-to-blame:} if there is already a released safe version of the vulnerable package but the application continues to rely on an (old) version with a publicly known vulnerability.
Application developers should monitor their dependencies and update to releases without known vulnerabilities, hence, we consider the application maintainers as responsible for the high threat vulnerability.
\end{itemize}
\begin{table}[tb!]
\centering
\caption{The percentage of vulnerabilities caused by the lack of available fix patch (Package-to-blame) vs caused by the lack of dependencies update (Application-to-blame), over the applications snapshot.}
\label{RQ2T1}
\begin{tabular}{l|r|r}
\toprule
\textbf{Snapshot} &
\textbf{Package-to-blame} & \textbf{Application-to-blame} \\
\midrule
\textbf{20\%} & {12.06\%} & {87.94\%}
\\
\textbf{40\%} & {9.52\%} & {90.48\%} \\
\textbf{60\%} & 11.91\% & {88.09\%} \\
\textbf{80\%} & {12.43\%} &{87.57\%} \\
\textbf{100\%} & \textbf{9.24\%} & \textbf{90.76\%} \\
\bottomrule
\end{tabular}%
\end{table}
\noindent
\textbf{Results}:
Table~\ref{RQ2T1} shows the percentage of high threat vulnerabilities based on our responsibility analysis. From Table~\ref{RQ2T1}, we observe that \textbf{for high threat vulnerabilities, the application is to blame in 90.76\% of the cases} at the last snapshot (i.e., 100\%).
That means that in 9 out of 10 cases the high threat vulnerability had an available fix, but the applications did not update their dependencies to receive the last fix patch.
Note that this observation holds over all snapshots, with percentages of application-to-blame cases varying from 87.94\% to 90.76\%.
Therefore, and perhaps counter-intuitively, high threat vulnerabilities do not exist because packages have unfixed vulnerabilities, rather the real cause is the fact that these applications fail to keep up or at least to inform themselves well enough about a given dependency version. Hence, a major implication of our study is that application developers need to take updates pushed from their dependencies seriously, or at least actively track their dependencies, since those can lead to very serious effects.
It is important to note that we do not argue about the severity of the vulnerabilities, but rather their likelihood threat of being exploited. Hence, a low severity vulnerability can be very dangerous if everyone knows how to exploit it (high threat level according to our classification). The inverse is also true in that a high severity vulnerability can have a very low chance of being exploited if no one knows about its existence (low threat level).\\
\begin{mdframed}[roundcorner=5pt,linewidth=0.5mm,
linecolor=black]
\sloppy
\lipsum[0]
\textbf{\textit{{Our findings show that applications not updating their dependencies, are the main cause of high threat (more than 87\%) vulnerabilities.}}
}
\end{mdframed}
\subsection{Data Collection}
\label{subsec:data}
Our study examines vulnerable dependencies in Node.js applications. We chose to focus on Javascript due to its wide popularity amongst the development community~\cite{StackOve70:online}.
\\
\noindent\textbf{Packages vs. Applications.} The software community classifies JavaScript projects into two categories:
\textit{1) packages}, also referred to as libraries, which are included in other applications using dependency management tools to help facilitate and speed up development.
Packages are referred to as "dependencies" of an application.
\textit{2) applications} are standalone software projects, which are distinct from libraries, where they are not distributed via a package manager and are typically applications for clients and end users rather than components to build upon.
As mentioned before, the Node.js applications mainly state the packages they depend on (i.e., dependencies) in a file called package.json.
To perform our study, we leverage two datasets: (1) Node.js applications that use {npm} to manage their dependencies, and (2) Security vulnerabilities that affect npm packages. To do so, we \textbf{(i)} obtain the Node.js applications from GitHub, \textbf{(ii)} extract their dependencies, and \textbf{(iii)} obtain the security vulnerabilities for npm packages from {npm} advisories~\cite{npmadvisories}. The dataset collection took place during May and June of 2019.\\
\noindent\textbf{(i) Applications Dataset.} To analyse a large number of open source JavaScript applications that depend on npm packages and obtain insights on their security vulnerabilities, we mine the GHTorrent dataset~\cite{GHtorrent} and extract information about all Node.js applications hosted on GitHub. The GHTorrent dataset contains a total of 7,863,361 JavaScript projects hosted on GitHub, of which 2,289,130 use npm as their package management platform (i.e., these projects contain a file called package.json). Moreover, since both Node.js \textit{packages} and \textit{applications} can use GitHub as their development repository, and our applications dataset should only contain Node.js \emph{applications}, we filter out the GitHub projects
that are actually npm \emph{packages} by checking their GitHub URL on the {{npm}} registry. The main reason that we focused on applications and not packages is because packages become exploitable only when used and deployed in an application, i.e., packages do not reside on their own in production, they should be part of applications that make use of them. This filtering excludes 328,343 projects from our list of GitHub projects as they are identified as packages and not applications
As shown in previous studies~\cite{kalliamvakou2014promises,kula2018developers}, some projects on GitHub are immature, hence, to make this study more reliable we refined the dataset using additional filtering criteria to eliminate such immature projects. In particular, we gather applications that satisfy all the following criteria:
\begin{itemize}
\item Non-forked applications, as we do not want to have duplicated project history to bias our analysis.
\item Applications that depend upon more than two dependencies.
\item Applications that have at least 100 commits by more than two contributors, which indicates a minimal level of commit activity.
\item Applications that have had their creation date (first commit) before January 1st 2017.
Since vulnerabilities take on median 3 years to be discovered~\cite{decan2018impact}, applications in our dataset need to have a development history long enough to have had a chance for their vulnerabilities to be discovered.
\item Applications that have had their latest commit after January 1st 2017, as we want to analyze applications that had some level of development in the last 3 years.
\end{itemize}
After applying these refinement criteria, we end up with 6,673 Node.js applications that make use of npm packages. Table~\ref{statsApplications} shows descriptive statistics on the selected Node.js applications in our dataset. Overall, the applications in our dataset
have a rich development history (a median of 213 commits made by 4 developers and 1,657
days of development lifespan) and make ample use of external
dependencies (a median of 11 dependencies).\\
\noindent\textbf{(ii) Application Dependencies.}
After obtaining the applications dataset, we want to extract the history of dependency changes of all applications.
This is necessary to identify the exact dependency versions that would be installed by the application at any specific point-in-time.
As mentioned in Section~\ref{sec:dependency}, Node.js applications specify their dependencies in the package.json file, which contains
the dependency list, containing the dependent upon packages and their respective version constraints.
Hence, we extract all changes that touched the package.json file and associate each commit hash and commit date to their respective package.json dependency list, creating a history of dependency changes for all applications.
Note that these dependencies are not yet resolved, that is, we only have the version constraints (not the versions) for the dependencies of each application.
\\
\begin{table}[tb!]
\centering
\caption{Statistics of the 6,673 studied Node.js applications.
}
\label{statsApplications}
\begin{tabular}{l|r|r|r|r}
\toprule
\textbf{Metric} & \textbf{Min.} & \textbf{Median($\bar{x}$)} & \textbf{Mean($\mu$)} & \textbf{Max.} \\ \midrule
\textbf{Commits} & 100 & 213 & 384.60 & 53,872 \\
\textbf{Dependencies} & 3 & 11 & 14.93 & 114 \\
\textbf{Developers} & 3 & 4 & 5.33 & 52 \\
\textbf{Lifespan (in days)} & 151 & 1,657 & 1,730.07 & 3,575 \\
\bottomrule
\end{tabular}
\end{table}
\noindent\textbf{(iii) NPM Advisories Dataset.}
To identify Node.js applications that depend on vulnerable packages, we need to collect information on npm vulnerable packages. We resort to the \textit{NPM advisories} registry to obtain the required information about all npm vulnerable packages~\cite{npmadvisories}.
The npm advisories dataset is the official registry for npm vulnerability reports, which contains a number of JavaScript vulnerabilities, specific to the Node.js-platform packages.
This dataset provides several kinds of information about vulnerable packages relevant to our study.
Each report has the affected package name, the package versions affected by the vulnerability, and the versions in which the vulnerabiliy was fixed (safe versions).
The report also contains both the vulnerability discovered (reported) time and published time, which we use in our approach for identifying and classifying vulnerabilities (Section~\ref{subsec:scanning}).
Note that a vulnerable package could be affected by several vulnerabilities (i.e., a vulnerable package appears with different vulnerability reports due to different vulnerability types).
Our initial dataset contains 654 security reports that cover 601 vulnerable packages. Following the criteria filtration process applied by Decan et al.~\cite{decan2018impact}, we removed 12 vulnerable packages of the type "Malicious Package", because they do not actually introduce vulnerable code. These vulnerabilities are packages with names close to popular packages (a.k.a. typo-squatting) in an attempt to deceive users at installing harmful packages. The 12 vulnerable packages account for 12 vulnerability reports. At the end of this filtering process, we are left with 642 security vulnerabilities reports affecting 589 distinct vulnerable packages.
These packages have combined 26,462 distinct package versions of which 13,868 are affected by vulnerabilities from our report.
Table~\ref{dataset} shows the summary statistics for vulnerability reports on npm packages.
\begin{table}[tb!]
\centering
\caption{Descriptive statistics on the npm advisories dataset.}
\label{dataset}
\begin{tabular}{l|r}
\toprule
Vulnerability reports & 642 \\
Vulnerable packages & 589 \\
Versions of vulnerable packages & 26,462 \\
Affected versions by vulnerability & 13,868 \\
\bottomrule
\end{tabular}%
\vspace{-2mm}
\end{table}
\begin{figure*}[tb!]
\centering
\setlength{\abovecaptionskip}{5pt}
\includegraphics[width=0.7\linewidth]{images/approach.pdf}
\caption[]{Approach for identifying and classifying vulnerable dependencies in Node.js applications.}
\label{classifyApproach}
\end{figure*}
\subsection{Identifying and Classifying Vulnerable Dependencies in Node.js Applications}
\label{subsec:scanning}
To classify the threat level of vulnerable dependencies at a specific point in the development history of a Node.js application, which we refer to as the \textit{analyzed snapshot time}, we leverage 3-step approach. Figure~\ref{classifyApproach} provides an overview of our approach, which we detail below:
\noindent
{\textbf{Step 1. Extract dependencies and resolve versions.}}
The goal of this step is to extract applications dependencies and find the actual dependency version installed at the analyzed snapshot time.
For each application, we extract the dependency list (with the versioning constraints) at that snapshot time from the history of dependency changes.
After that, to find the actual version of each dependency at the analyzed snapshot, we utilize the \textit{semver} tool~\cite{semvernp91} that is used by npm to find the latest version that satisfies the versioning constraint, with an additional restriction that the satisfying version should have been released (in the npm registry) before the application snapshot time. For example, an application can specify a versioning constraint (``P:$>$1.0.0'') at the snapshot May 2016. Hence, the actual installed version is the latest version that is greater than 1.0.0 and also has been released in the npm registry before May 2016.
This step allows us to find the installed version of the dependency at the analyzed snapshot time.
\\
\noindent
{\textbf{Step 2. Identify vulnerable dependency versions.}}
After determining the resolved (and presumably installed) version at the analyzed snapshot time, we check whether the resolved version is vulnerably or not. To do so, we check the advisories dataset for the versions that were available at that snapshot point.
If the resolved version is covered by the advisories dataset, we label it as a vulnerable dependency version. We skip the whole next step if the dependency version has not been mentioned in any advisory, i.e., the dependency version is not vulnerable.
\\
\noindent
{\textbf{Step 3. Identify threat levels of vulnerable versions.}}
Once we identify the vulnerable dependency versions at the analyzed snapshot time, we classify each vulnerable dependency version using one of the threat levels we defined earlier (in Section~\ref{sec:ex}), i.e., we find out the threat level of each vulnerable dependency version. To do so, for each vulnerable version, we compare its vulnerability \textit{discovery (report)} and \textit{publication} time to the analyzed snapshot time. As we stated previously (in Section~\ref{sec:ex}), if the vulnerability publication time of the vulnerable dependency version is before the application's snapshot time then we mark the vulnerability as high threat vulnerability. If the vulnerability of the dependency was not published but only discovered (reported) before the application's snapshot time, then we mark it as medium. And finally, if it was neither published nor discovered (reported) before the analyzed snapshot time (i.e., no one knows about it at that snapshot time), then we mark it as low.
In cases where more than one vulnerability affects the vulnerable dependency version, we resort to a weakest link approach (i.e., we label the vulnerable dependency version with the highest threat level). For example, if we find that the vulnerable version of the dependency is affected by two vulnerabilities -one having low threat and another as high threat, we label the vulnerable dependency version as high at that snapshot time.
\subsection{Replication Package}
To facilitate verification and advancement of research in the field, a replication package comprising the data used in our study along with the analyses used in our study is publicly available\footnote{http://doi.org/10.5281/zenodo.3837397}.
\subsection{Software Ecosystems}
A plethora of recent work focused on software ecosystems. Several works compare different ecosystems. For example, Decan et al.~\cite {decan2018empirical} empirically compared the evolution of 7 popular package ecosystem using different aspects, e.g., growth, changeability, resuability, and fragility. They observed that the number of packages in those ecosystems is growing over time, showing their increasing importance.
Other work focused specifically on npm~\cite{fard2017javascript,kula2017impact, wittern2016look}. For example, Fard et al.~\cite{fard2017javascript} examined the evolution of dependencies within an npm project, and showed that there is a heavily interdependence, with the average number of dependencies being 6 and growing over time. Wittern et al.~\cite{wittern2016look} investigated the evolution of npm using metrics such as dependencies between packages, download count, and usage count in JavaScript applications. They found that packages in the npm ecosystem are steadily growing. Such amounts of packages make the spread and discovery of vulnerabilities much worse, given the heavy dependence on such packages and the potential security problems in those packages.
Other studies pointed out the fragility of software ecosystems and provided insights on the challenges application developers face. For example, Bogart et al.~\cite{bogart2015breaks,bogart2016break} examined the Eclipse, CRAN, and npm ecosystems, focusing on what practices cause API breakages. They found that a main reason for breaking changes are the updates of a dependency. This finding may explain why application developers are hesitant to update and explain why we see high threat vulnerabilities impacting applications that do not update in time.
Our study differs from the prior work since we focus on the threat level of dependency vulnerabilities in Node.js applications. Moreover, we examine how this threat level changes as applications evolve and examine the reason that high threat dependency vulnerabilities exist. That said, much of the aforementioned work motivated us to study npm and focus on examining vulnerabilities in application dependencies.
\vspace{-0.05in}
\subsection{Security Vulnerabilities in Dependencies/Packages}
Several works in the literature studied vulnerabilities that come from dependencies~\cite{di2009life,pham2010detection,cox2015measuring,massacci2011after,derr2017keep}. For example, Di Penta et al.~\cite{di2009life} and Pham et al.~\cite{pham2010detection} conducted empirical studies to analyze the evolution of vulnerabilities in source code, and found that most vulnerabilities are recurring due to software code reuse or libraries (i.e., dependencies). Cox et al.~\cite{cox2015measuring} evaluated ``dependency freshness'' to understand the relationship between outdated dependencies and vulnerabilities using industry benchmarks, and found that vulnerabilities were four times as much likely to have existed in outdated systems than in updated systems. Relative studies by Massacci et al.~\cite{massacci2011after} and Derr et al.~\cite{derr2017keep} are in line with~\cite{cox2015measuring}. In general, they both reported that vulnerabilities appeared commonly in non-maintained code and old versions, and this could be fixed by just an update to a newer version. Our study complements these studies by examining the threat of these vulnerabilities in the dependent applications.
More specifically, vulnerabilities that affect packages in ecosystems have been studied broadly~\cite{kula2018developers,pashchenko2018vulnerable}. For example, Kula et al.~\cite{kula2018developers} analyzed the Maven ecosystem on more than 4,000 GitHub projects that correspond to 850,000 library migrations, and found that projects were heavily dependent on these libraries, and most projects (i.e. 81.5\%) had outdated libraries. The study also mentioned (based on interviews conducted with developers) that developers do not update dependencies, and 69\% of the interviewed developers tend to be not aware of their vulnerable dependencies. Pashchenko et al.~\cite{pashchenko2018vulnerable} studied the vulnerability impact of 200 open-source Java libraries commonly used in SAP~\cite{SAPSoftw78} organisation, and found that 20\% of the vulnerable dependencies are not deployed, and hence, they are not exploitable in practice. Moreover, they found that the majority of the vulnerable dependencies (81\%) can be fixed by a simple upgrade to a newer safe version, suggesting that software development companies have to allocate their audit tools correctly.
Other recent work focused on analyzing vulnerabilities in the npm ecosystem. For example, Hejderup’s~\cite{hejderup2015dependencies}
analysed only 19 vulnerable packages
and found that the number of vulnerabilities in them is growing over time. Similarly, Decan et al.~\cite{decan2018impact} analyzed the vulnerabilities in the npm ecosystem and found that the number of vulnerabilities is growing over time. Also, they reported that it takes a long time to discover vulnerabilities that affect npm packages. Our study complements this study by analyzing the risks of vulnerable dependencies in the Node.js applications (not addressed by the study~\cite{decan2018impact}), aggregating the vulnerability lifecycle through the threat level metric.
A recent study by Zapata et al.~\cite{zapata2018towards} assessed the danger of having vulnerabilities in dependent libraries by analyzing function calls of the vulnerable functions.
They manually analyzed 60 projects that depend on vulnerabilities, and found that 73.3\% of them were actually safe because they did not make use of the vulnerable functionality of their dependencies, showing that there is a considerable overestimation on previous reports. Our study identifies yet another source of overestimation by including a time-based analysis into a large and comprehensive set of \textit{applications} (i.e., 6,673 Node.js applications).
Zimmermann et al.~\cite{zimmermann2019small} studied the security threat of the npm ecosystem dependencies by mainly analysing the maintainers role and responsibilities for vulnerable packages. They mainly observed that a very small number of maintainers' accounts (i.e., 20 accounts) could be used to inject malicious code into thousands of npm packages, a problem that has been increasing over time. Zerouali et al.~\cite{zerouali2019impact} studied npm vulnerable packages in Docker containers, and found that they are common in the containers, suggesting that Docker containers should keep their npm dependencies updated.
To assess the impact of vulnerable dependencies in the dependent Java applications, Plate et al.~\cite{plate2015impact} proposed an approach that provides a fine-grained assessment of the vulnerabilities that affect dependencies in dependent Java applications. In particular, the approach first determines whether or not the application makes use of the library that is known to be vulnerable. Then, the approach tries to determine whether or not the application executes the fragment of the dependency where the vulnerable code is located. Furthermore, Ponta et al.~\cite{ponta2018beyond} built upon their previous approach in~\cite{plate2015impact} to generalize their vulnerability detection approach by using static and dynamic analysis to determine whether the vulnerable code in the library is reachable through the application call paths. Their proposed approach is implemented in a tool called, Vulas, which is an official software used by SAP to scan its Java code.
Our study focuses on analyzing the threat of npm vulnerabilities in dependencies, which affected applications that rely on them. In many ways, our study complements the related work since, (1) instead of studying security vulnerabilities that exist in packages, we particularly focus on the threat of such vulnerable packages by real-world open source applications; (2) we provide a threat classification for software vulnerabilities based on their lifetime, and we use our classification and perform an empirical study on Node.js applications.
\subsection{Software Ecosystems}
A plethora of recent work focused on software ecosystems. Several works compare different ecosystems. For example, Decan et al.~\cite {decan2018empirical} empirically compared the evolution of 7 popular package ecosystem using different aspects, e.g., growth, changeability, resuability, and fragility. They observed that the number of packages in those ecosystems is growing over time, showing their increasing importance.
Other work focused specifically on npm~\cite{fard2017javascript,kula2017impact, wittern2016look}. For example, Fard et al.~\cite{fard2017javascript} examined the evolution of dependencies within an npm project, and showed that there is a heavily interdependence, with the average number of dependencies being 6 and growing over time. Wittern et al.~\cite{wittern2016look} investigated the evolution of npm using metrics such as dependencies between packages, download count, and usage count in JavaScript applications. They found that packages in the npm ecosystem are steadily growing. Such amounts of packages make the spread and discovery of vulnerabilities much worse, given the heavy dependence on such packages and the potential security problems in those packages.
Other studies pointed out the fragility of software ecosystems and provided insights on the challenges application developers face. For example, Bogart et al.~\cite{bogart2015breaks,bogart2016break} examined the Eclipse, CRAN, and npm ecosystems, focusing on what practices cause API breakages. They found that a main reason for breaking changes are the updates of a dependency. This finding may explain why application developers are hesitant to update and explain why we see high threat vulnerabilities impacting applications that do not update in time.
Our study differs from the prior work since we focus on the threat level of dependency vulnerabilities in Node.js applications. Moreover, we examine how this threat level changes as applications evolve and examine the reason that high threat dependency vulnerabilities exist. That said, much of the aforementioned work motivated us to study npm and focus on examining vulnerabilities in application dependencies.
\vspace{-0.05in}
\subsection{Security Vulnerabilities in Dependencies/Packages}
Several works in the literature studied vulnerabilities that come from dependencies~\cite{di2009life,pham2010detection,cox2015measuring,massacci2011after,derr2017keep}. For example, Di Penta et al.~\cite{di2009life} and Pham et al.~\cite{pham2010detection} conducted empirical studies to analyze the evolution of vulnerabilities in source code, and found that most vulnerabilities are recurring due to software code reuse or libraries (i.e., dependencies). Cox et al.~\cite{cox2015measuring} evaluated ``dependency freshness'' to understand the relationship between outdated dependencies and vulnerabilities using industry benchmarks, and found that vulnerabilities were four times as much likely to have existed in outdated systems than in updated systems. Relative studies by Massacci et al.~\cite{massacci2011after} and Derr et al.~\cite{derr2017keep} are in line with~\cite{cox2015measuring}. In general, they both reported that vulnerabilities appeared commonly in non-maintained code and old versions, and this could be fixed by just an update to a newer version. Our study complements these studies by examining the threat of these vulnerabilities in the dependent applications.
More specifically, vulnerabilities that affect packages in ecosystems have been studied broadly~\cite{kula2018developers,pashchenko2018vulnerable}. For example, Kula et al.~\cite{kula2018developers} analyzed the Maven ecosystem on more than 4,000 GitHub projects that correspond to 850,000 library migrations, and found that projects were heavily dependent on these libraries, and most projects (i.e. 81.5\%) had outdated libraries. The study also mentioned (based on interviews conducted with developers) that developers do not update dependencies, and 69\% of the interviewed developers tend to be not aware of their vulnerable dependencies. Pashchenko et al.~\cite{pashchenko2018vulnerable} studied the vulnerability impact of 200 open-source Java libraries commonly used in SAP~\cite{SAPSoftw78} organisation, and found that 20\% of the vulnerable dependencies are not deployed, and hence, they are not exploitable in practice. Moreover, they found that the majority of the vulnerable dependencies (81\%) can be fixed by a simple upgrade to a newer safe version, suggesting that software development companies have to allocate their audit tools correctly.
Other recent work focused on analyzing vulnerabilities in the npm ecosystem. For example, Hejderup’s~\cite{hejderup2015dependencies}
analysed only 19 vulnerable packages
and found that the number of vulnerabilities in them is growing over time. Similarly, Decan et al.~\cite{decan2018impact} analyzed the vulnerabilities in the npm ecosystem and found that the number of vulnerabilities is growing over time. Also, they reported that it takes a long time to discover vulnerabilities that affect npm packages. Our study complements this study by analyzing the risks of vulnerable dependencies in the Node.js applications (not addressed by the study~\cite{decan2018impact}), aggregating the vulnerability lifecycle through the threat level metric.
A recent study by Zapata et al.~\cite{zapata2018towards} assessed the danger of having vulnerabilities in dependent libraries by analyzing function calls of the vulnerable functions.
They manually analyzed 60 projects that depend on vulnerabilities, and found that 73.3\% of them were actually safe because they did not make use of the vulnerable functionality of their dependencies, showing that there is a considerable overestimation on previous reports. Our study identifies yet another source of overestimation by including a time-based analysis into a large and comprehensive set of \textit{applications} (i.e., 6,673 Node.js applications).
Zimmermann et al.~\cite{zimmermann2019small} studied the security threat of the npm ecosystem dependencies by mainly analysing the maintainers role and responsibilities for vulnerable packages. They mainly observed that a very small number of maintainers' accounts (i.e., 20 accounts) could be used to inject malicious code into thousands of npm packages, a problem that has been increasing over time. Zerouali et al.~\cite{zerouali2019impact} studied npm vulnerable packages in Docker containers, and found that they are common in the containers, suggesting that Docker containers should keep their npm dependencies updated.
To assess the impact of vulnerable dependencies in the dependent Java applications, Plate et al.~\cite{plate2015impact} proposed an approach that provides a fine-grained assessment of the vulnerabilities that affect dependencies in dependent Java applications. In particular, the approach first determines whether or not the application makes use of the library that is known to be vulnerable. Then, the approach tries to determine whether or not the application executes the fragment of the dependency where the vulnerable code is located. Furthermore, Ponta et al.~\cite{ponta2018beyond} built upon their previous approach in~\cite{plate2015impact} to generalize their vulnerability detection approach by using static and dynamic analysis to determine whether the vulnerable code in the library is reachable through the application call paths. Their proposed approach is implemented in a tool called, Vulas, which is an official software used by SAP to scan its Java code.
Our study focuses on analyzing the threat of npm vulnerabilities in dependencies, which affected applications that rely on them. In many ways, our study complements the related work since, (1) instead of studying security vulnerabilities that exist in packages, we particularly focus on the threat of such vulnerable packages by real-world open source applications; (2) we provide a threat classification for software vulnerabilities based on their lifetime, and we use our classification and perform an empirical study on Node.js applications.
\subsection*{\rqone}
\label{sec:RQ1}
{\textbf{Motivation}}:
Prior work showed that a significant amount of application code comes from third party packages, and a non-negligible amount of these packages are affected by known security vulnerabilities~\cite{williams2012unfortunate}. However, we argue that not all vulnerabilities should be treated equally. Hence, in this RQ we would like to quantify how many of our studied applications have at least one vulnerable dependency and what the threat level of these vulnerable dependencies is. Answering this question will help us understand the real risk/threat of vulnerable packages on the software applications.
\noindent
{\textbf{Approach}}:
In order to perform an unbiased analysis, we need to account for vulnerability discovery time. Prior work showed that vulnerabilities in npm take on median 3 years to be discovered and publicly announced~\cite{decan2018impact}. As a consequence, selecting snapshots of our applications in 2019 will paint an incomplete picture, as most vulnerabilities recently introduced in the package's code would remain hidden for a median of 3 years.
Since we collected the advisories dataset in May/June 2019, we chose to evaluate our applications as of May 2016 (3 years prior), which ensures that at least half the dependency vulnerabilities introduced in the code are reported in the current advisories dataset.
Then, we answer our RQ in two steps. First, we examine if the \textit{selected snapshot} of the application had at least one dependency that contains a vulnerability (irrespective of its threat level).
Then, to determine the threat level of the vulnerable dependencies in the examined applications, we focus only on the set of applications that have at least one vulnerable dependency using the methodology described in Section~\ref{subsec:scanning}.
In the second step, we quantify the number of vulnerable dependencies in the applications under each threat level.
We first check the percentage of overall vulnerable dependencies in each application and illustrate their distribution using a Boxplot. We further analyze the distribution of these vulnerable dependencies across the threat levels and plot it using three Boxplots, one for each threat level. For example, an application could have 10\% of its dependencies as vulnerable at the analysed snapshot, and such percentage (i.e., 10\%) could be distributed across the threat levels as follows: 25\% of the vulnerable dependencies are classified as low threat, 60\% of them are classified as medium, and 15\% as high.
\begin{figure}[tb!]
\centering
\includegraphics[width=1\linewidth,height=.85\linewidth]
{images/RQ1_new.pdf}
\caption{Boxplots showing the distributions of the percentage of overall vulnerable dependencies in the applications (left boxplot), and how these percentages are distributed across threat levels (right boxplot). N and M are the total number of dependencies and the total number of vulnerable dependencies, respectively.
}
\label{RQ1Perc}
\end{figure}
\noindent \textbf{Results}:
Of the 6,673 studied applications \textbf{67.93\% (4533 applications) depend on at least one vulnerable dependency}. The affected applications contains a total of 10,154 vulnerable dependencies from 149 distinct vulnerable packages. The 149 packages comprises 23.21\% of the overall vulnerable packages in the npm advisories dataset.
Figure~\ref{RQ1Perc} shows the percentage of vulnerable dependencies per application (left boxplot), and the distribution of vulnerable dependencies at different threat levels (right boxplot). It shows that, on median, 14.29\% of the dependencies in the affected application (i.e., applications with at least 1 vulnerable dependency) are vulnerable. Also, Figure~\ref{RQ1Perc} shows that such percentage of vulnerable dependencies (i.e., 14.29\%) is distributed as follows: \textbf{94.91\% of the vulnerable dependencies are classified as low threat vulnerabilities}, 2.06\% of them are classified as medium, and 3.03\% are classified as high.
\vspace{0.08in}
\begin{table}[h!]
\centering
\caption{Mann-Whitney Test (p-value) and Cliff's Delta (d) for the different threat levels.}
\label{stattest}
\begin{tabular}{l|r|r}
\toprule
\textbf{Threat Levels} & \textbf{\textit{p}-value} &\textbf{ Cliff's Delta \textit{{(d)}}} \\ \midrule
Low vs. Medium & 2.2e-16 & 0.984 (large) \\
Low vs. High & 2.2e-16 & 0.970 (large) \\
Medium vs. High & 2.2e-16 & 0.335 (medium) \\ \bottomrule
\end{tabular}%
\end{table}
\vspace{0.08in}
To statistically verify our observation, we perform a one-sided non-parametric Mann-Whitney U test~\cite{mcknight2010mann} by comparing the distributions between the different threat levels.
Table~\ref{stattest} shows the p-values and effect size values. We observe a statistically significant differences between (low and medium), (low and high), (medium and high), at p-value $<$ 0.05 for all comparisons. Furthermore, we observe, using Cliff's delta~\cite{cliff1993dominance}, a large effect size for the differences between low and medium, low and high. Also, we found a medium effect size for the difference between medium and high. This indicates that the differences between the different threat levels are statistically significant.\\
\begin{mdframed}[roundcorner=5pt,linewidth=0.5mm,
linecolor=black]
\lipsum[0]
\textbf{\textit{Our findings show that 67.93\% of the examined applications depend on at least one vulnerable package. However, the vast majority (94.91\%) of these dependencies have low threat.}
}
\end{mdframed}
\vspace{0.2in}
\subsection*{\rqtwo}
{\textbf{\\Motivation}}: Thus far, we have analyzed the vulnerability threats of a single snapshot of each application in our dataset. However, our findings may differ as the applications evolve. For example, a vulnerability with high threat on a given day could have had low threat the week before.
\begin{figure*}[tb!]
\centering
\setlength{\abovecaptionskip}{10pt}
\includegraphics[width=1\linewidth]
{images/RQ2Evol3_new.pdf}
\caption{Boxplots showing the percentage of overall vulnerable dependencies and their distribution in each threat level over the studied snapshots. N and M are the total number of dependencies and the total number of vulnerable dependencies, respectively.}
\label{RQ2Evol}
\end{figure*}
Hence, in this RQ we would like to determine whether our results generalize to different historical snapshots in the application's development lifetime. Such an evolutionary examination allows us to discover whether the trend of the threat levels changes across different stages of an application's lifetime.
\\
\noindent
{\textbf{Approach}}:
Since the different applications are of different lifespans, we want to find a measure that makes comparing them feasible. To do so, we use the number of commits as a way to divide the applications into different intervals. Since commit frequency and time between commits vary from one application to another, we normalize the applications by segmenting the lifetimes of each application into five equal intervals (each containing 20\% of an application's lifetime by {time in days}),
take one snapshot at each interval, then analyze it. Although this might seem like a straightforward task, it poses some challenges, since we have a large applications dataset and the package.json file in them is updated significantly over the application's lifetime.
For this analysis, we only consider the affected applications identified in RQ$_1$.
The last snapshot (at 100\%) is the same snapshot that we analyzed in RQ$_1$ (i.e., May 2016).
\begin{table}[tbh]
\centering
\caption{The percentage of vulnerable applications at different historical snapshots.}
\label{DisT1}
\begin{tabular}{l|r}
\toprule
\multirow{2}{*}{\textbf{Snapshot}} & \textbf{Vulnerable} \\
& \textbf{Applications} \\
\midrule
\textbf{20\%} &{ 55.31\%} \\
\textbf{40\%} & 58.17\% \\
\textbf{60\%} & 60.87\% \\
\textbf{80\%} & 63.03\% \\
\textbf{100\%} & {67.93\%} \\ \bottomrule
\end{tabular
\end{table}
\noindent
{\textbf{Results}}:
Table~\ref{DisT1} shows the percentage of applications that have at least one vulnerable dependency for the 5 analyzed snapshots across their lifetime. We observe that \textbf{the percentage of vulnerable applications steadily increases each snapshot} and varies between 55.31 - 67.93\% in the studied applications.
Figure~\ref{RQ2Evol} illustrates the distributions of the percentage of vulnerable dependencies at each threat level over the studied snapshots. The total number of dependencies (N) and the total number of vulnerable dependencies (M) in the studied applications are shown at the bottom of Figure~\ref{RQ2Evol}. The raw numbers of the dependencies show that the total number of dependencies increases over time, and so does the raw number of vulnerable dependencies.
From Figure~\ref{RQ2Evol}, we observe that the affected applications depend on vulnerable dependencies at an earlier stage (i.e., at 20\%) of their lifetime. However, we also observe that the trend observed in RQ1 remains the same, i.e., the overall percentage of vulnerable dependencies ranges between 14.29\% - 14.68\%. Also, the majority of the vulnerabilities have a low threat level, followed by high and medium threat. To sum up, our analysis shows that all trends observed in RQ$_1$
also hold at different stages of the applications, albeit the raw number of dependencies does increase.\\
\begin{mdframed}[roundcorner=5pt,linewidth=0.5mm,
linecolor=black]
\lipsum[0]
\textbf{\textit{As applications evolve, the overall number of vulnerable dependencies is increasing, however, the median percentage of vulnerable dependencies remains mostly constant. Moreover, the majority of vulnerabilities they face remain as low threat vulnerabilities, as these applications evolve.}}
\end{mdframed}
\vspace{-0.2in}
\subsection*{\rqthree}
\textbf{Motivation}:
In the previous research questions, we found that the majority of affected dependencies are impacted by low threat vulnerabilities, throughout applications development history.
However, a sizeable number of projects depend on
high threat dependencies, which are the most important. This means that those applications depend on vulnerable versions of dependencies even after the vulnerability reports have been discovered (reported)-and-published.
In such cases, the \emph{developers of the applications could know} about the presence of the vulnerability in the dependency, and hence, {should} avoid using that vulnerable version, if a fix is available.
Specifically, we want to know who is to blame - the package maintainers for not providing a version that fixes a known vulnerability - or the application maintainers for not keeping their applications up-to-date.
Answering this will help us pinpoint the causes for high threat vulnerabilities in npm applications and develop further strategies to solve this problem.
\noindent
\textbf{Approach}:
To perform our investigation and answer who is responsible for the high threat vulnerabilities in applications, we use the same method to determine high threat vulnerabilities as presented in the first two RQs.
For each high threat vulnerable dependency, we check the availability of a safe version of the package for the vulnerability at the analyzed snapshot time. Depending on such availability our analysis has one of two outcomes:
\begin{itemize}
\item \textbf{Package-to-blame:} if at the analyzed snapshot, no safe version has been provided by the package maintainers for a publicly known vulnerability. As the publication of a vulnerability comes after a period of 45 days, we consider the package maintainers the responsible for the high threat vulnerability in applications.
\item \textbf{Application-to-blame:} if there is already a released safe version of the vulnerable package but the application continues to rely on an (old) version with a publicly known vulnerability.
Application developers should monitor their dependencies and update to releases without known vulnerabilities, hence, we consider the application maintainers as responsible for the high threat vulnerability.
\end{itemize}
\begin{table}[tb!]
\centering
\caption{The percentage of vulnerabilities caused by the lack of available fix patch (Package-to-blame) vs caused by the lack of dependencies update (Application-to-blame), over the applications snapshot.}
\label{RQ2T1}
\begin{tabular}{l|r|r}
\toprule
\textbf{Snapshot} &
\textbf{Package-to-blame} & \textbf{Application-to-blame} \\
\midrule
\textbf{20\%} & {12.06\%} & {87.94\%}
\\
\textbf{40\%} & {9.52\%} & {90.48\%} \\
\textbf{60\%} & 11.91\% & {88.09\%} \\
\textbf{80\%} & {12.43\%} &{87.57\%} \\
\textbf{100\%} & \textbf{9.24\%} & \textbf{90.76\%} \\
\bottomrule
\end{tabular}%
\end{table}
\noindent
\textbf{Results}:
Table~\ref{RQ2T1} shows the percentage of high threat vulnerabilities based on our responsibility analysis. From Table~\ref{RQ2T1}, we observe that \textbf{for high threat vulnerabilities, the application is to blame in 90.76\% of the cases} at the last snapshot (i.e., 100\%).
That means that in 9 out of 10 cases the high threat vulnerability had an available fix, but the applications did not update their dependencies to receive the last fix patch.
Note that this observation holds over all snapshots, with percentages of application-to-blame cases varying from 87.94\% to 90.76\%.
Therefore, and perhaps counter-intuitively, high threat vulnerabilities do not exist because packages have unfixed vulnerabilities, rather the real cause is the fact that these applications fail to keep up or at least to inform themselves well enough about a given dependency version. Hence, a major implication of our study is that application developers need to take updates pushed from their dependencies seriously, or at least actively track their dependencies, since those can lead to very serious effects.
It is important to note that we do not argue about the severity of the vulnerabilities, but rather their likelihood threat of being exploited. Hence, a low severity vulnerability can be very dangerous if everyone knows how to exploit it (high threat level according to our classification). The inverse is also true in that a high severity vulnerability can have a very low chance of being exploited if no one knows about its existence (low threat level).\\
\begin{mdframed}[roundcorner=5pt,linewidth=0.5mm,
linecolor=black]
\sloppy
\lipsum[0]
\textbf{\textit{{Our findings show that applications not updating their dependencies, are the main cause of high threat (more than 87\%) vulnerabilities.}}
}
\end{mdframed}
|
2024-02-18T23:40:24.741Z
|
2020-09-22T02:01:21.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2009.09019","language":"en","timestamp":1600740081000,"url":"https:\/\/arxiv.org\/abs\/2009.09019","yymm":"2009"}
|
proofpile-arXiv_000-10076
|
{"provenance":"002.jsonl.gz:10077"}
| null | null |
\section{Post- vs.\ pre-update synchronization}
\nk{Say a few words about post versus pre-update synchronization for AISTATS. Perhaps omitting the naive one in this comparison would make sense.}
\section{Conclusion}
\label{sec:conclusion}
We have shown that naive deployment of LinUCB in two-stage recommender systems may result in suboptimal performance, and close to linear regret in extreme cases.
The suboptimality is due to the mismatch between both the embeddings used by the ranker and the nominators, and the gap between the amount of training data the ranker has seen compared to the nominators.
Both of these issues are inherent to the setting in which two-stage recommenders are typically deployed in industry, and thus pose important barriers to achieving a good exploration-exploitation trade-off.
We have proposed a simple modification of the LinUCB algorithm based on communication of inferred statistics between the ranker and the nominators.
Our algorithm can be implemented with minimal computational overhead, and achieves superior empirical results compared to the naive two-stage LinUCB implementation.
While focusing solely on LinUCB, we suspect the `deadlock' problem identified in \Cref{sect:motivating_example} is pertinent to any exploration algorithm which in part selects its actions based on the level of uncertainty about them (e.g., all `optimism in the face of uncertainty' type algorithms, \citealp{lattimore2020bandit}).
Since the principles motivating \Cref{eq:kl_sync} do not hinge on LinUCB in particular, we hope these issues could also be addressed by communication of statistics from the ranker.
This preprint is based on a workshop version of our work.
We plan to publish an extended version with additional experiments on real-world data and extended discussion in the near future.
\section{Introduction}
\label{sec:intro}
Contemporary recommender systems are tasked with finding a small number of relevant items among millions or billions of candidates, personalized for each of hundreds of thousands or millions of users and their always changing needs, all of which has to happen in order of milliseconds so as not to negatively impact webpage loading speeds.
One of the most widely used solutions to the problem are \emph{two-stage recommender systems} \citep{borisyuk2016casmos,covington2016deep,eksombatchai2018pixie} in which (i)~a set of \emph{computationally efficient} nominators narrows down the search from millions to only hundreds of items, and (ii)~the \emph{slower but more accurate} ranker selects and reorders a few items which are eventually served to the user.
For example, a nominator can use a two-tower architecture \citep{yi2019sampling} and focus on a narrower set of features, whereas the ranker would rely on a more powerful model and consider additional features extracted from, e.g., ratings, specialized user and item attributes, or the number and type of past interactions with given user \citep{covington2016deep,ma2020off}.
Importantly, the nominators are often heterogeneous both in terms of the size and type of the items from which they select the candidate items, and the algorithm used to select candidates ranging from simple associative rules to recurrent neural networks \citep{chen2019top}.\jh{I'm pretty sure I'm missing citations throughout; please do not hold back from adding as many as you can.}
We will focus on nominators which utilize statistical learning methods in the two-stage setup as in the paper most relevant to our work \citep{ma2020off}.
\citet{ma2020off} study off-policy learning for two-stage recommender systems where the goal is learning a good recommendation policy from the typically abundant logged data.
The main proposal of \citeauthor{ma2020off}\ is to modify the nominator training objective by adding importance weights based on the ranker's probability of recommending each item.
With adjustments facilitating gradient descent optimization, the authors show significant empirical improvements not only compared to a system trained without importance weighting, but also relative to nominators importance weighted only based on the past \emph{nominator} policy (ignoring the presence of the ranker).
These results thus demonstrate that local optima of individual components do not translate to optimality of the system as a whole.
Naturally, we can ask whether there are other aspects of the recommendation problem where optimal solutions for a single-stage system result in suboptimal performance when deployed in a two-stage system.
We answer this question affirmatively in the case of \emph{exploration} by which we mean the task of learning an optimal recommendation policy under uncertainty in a sample efficient way.
An effective exploration strategy then needs to balance greedy actions based on past interactions with exploratory recommendations targetting items about which there is little or no information.
While many strategies have been proposed in the literature, ranging from more conservative ones like Boltzmann exploration \citep{daw2006cortical,chen2019top} to more optimistic contextual bandit algorithms \citep{lattimore2020bandit}, we will restrict our discussion to the popular LinUCB algorithm \citep{auer2002using,dani2008stochastic,li2010contextual} and the associated contextual bandit recommendation setup.
A similar setup has been explored by \citep{ma2020off} as it is rich enough to exhibit many of the salient properties encountered in the real world, while abstracting away some of the complexities involved in deployment of large scale recommendation systems.
\textbf{Contributions:} We
(i)~show that a mismatch of feature mappings (architecture) and the amount of data seen (regularization) between LinUCB ranker and nominators can result in large, even linear regret;
(ii)~demonstrate said effects and their dependence on the level of mismatch empirically on a toy dataset; and
(iii)~propose a simple algorithm based on synchronization of inferred statistics between the ranker and the nominators which addresses the issue. We demonstrate the efficacy of our algorithm on simulated data.
\section{Experimental results}
\label{sec:experiments}
\def0.24{0.24}
\begin{figure*}
\centering
\textbf{Noise level 0.1:} $\hat{\theta}_0 \sim \mathcal{N} (\theta_\star, 10^{-2})$\\
\hrulespace{1mm}
\hspace{2cm}$\gamma = 1$ \hfill $\gamma = 10$ \hfill $\gamma = 25$ \hfill $\gamma = 50$ \hspace{1.3cm}\phantom{}
\\
\includegraphics[width=0.24\textwidth]
{klv2-pseudo_1-noise_1e-01-post}
\hfill
\includegraphics[width=0.24\textwidth]
{klv2-pseudo_10-noise_1e-01-post}
\hfill
\includegraphics[width=0.24\textwidth]
{klv2-pseudo_25-noise_1e-01-post}
\hfill
\includegraphics[width=0.24\textwidth]
{klv2-pseudo_50-noise_1e-01-post}
\\
\textbf{Noise level 0.2:} $\hat{\theta}_0 \sim \mathcal{N} (\theta_\star, 5^{-2})$\\
\hrulespace{1mm}
\hspace{2cm}$\gamma = 1$ \hfill $\gamma = 10$ \hfill $\gamma = 25$ \hfill $\gamma = 50$ \hspace{1.3cm}\phantom{}
\\
\includegraphics[width=0.24\textwidth]
{klv2-pseudo_1-noise_2e-01-post}
\hfill
\includegraphics[width=0.24\textwidth]
{klv2-pseudo_10-noise_2e-01-post}
\hfill
\includegraphics[width=0.24\textwidth]
{klv2-pseudo_25-noise_2e-01-post}
\hfill
\includegraphics[width=0.24\textwidth]
{klv2-pseudo_50-noise_2e-01-post}
\caption{Naive vs.\ synced 2-stage recommendation.}\label{fig:linUCB_toy}
\end{figure*}
\section{Setup}
\label{sec:setup}
\begin{figure}
\centering
\includegraphics[width=0.7\columnwidth]{setup.png}
\caption{The two-stage recommendation setup.}
\label{fig:setup}
\end{figure}
We consider a scenario in which a single item is to be recommended in each of the $T$ rounds.
The recommendation problem is modeled in a contextual bandit setup where the individual items correspond to arms $a \in A$ (we thus also refer to items as arms or actions).
For the first stage, we assume a fixed number of $N \in \mathbb{N}$ \emph{nominators}, each of which has access only to a fixed non-empty subset of arms $A_n \subseteq A$.
At every round $t \in [T]$, each nominator observes contexts $x_{t, a}$ for all $a \in A_n$, and selects a \emph{single} action $a_{t, n}$.
The ranker then chooses a \emph{single} final recommendation among the ones nominated in the first stage $a_t \in \{ a_{t, 1}, \ldots a_{t, N} \}$ \emph{based on the corresponding contexts}.\jh{Maybe cite Singla et al.\ (LEARNING WITH LEARNING EXPERTS) and Aggarwall et al. (CORALL) and explain that this is the main distinction of our setup from theirs?!}
Both the ranker and the nominators can ultimately be updated using the reward $r_t$ obtained by pulling arm $a_t$ as well as all the revealed contexts $x_{t, a}$.
This two-stage recommendation process is illustrated in \Cref{fig:setup}.
We restrict our attention to the stochastic linear bandit setting \citep{abe1999associative,lattimore2020bandit}, and the LinUCB algorithm \emph{with ellipsoidal confidence sets} for both the ranker and the nominators \citep{auer2002using,dani2008stochastic}.
The linear contextual bandit setting assumes existence of a fixed embedding for each context $\phi(x_{t , a})$ such that
\begin{equation}
\mathbb{E} [ r_t \, | \, x_{t, a_t} , a_t ] = \langle \phi(x_{t, a_t}) , \theta_\star \rangle
\end{equation}
for all $x_{t, a}$, $a_t$, and a fixed $\theta_\star \in \mathbb{R}^{d}$.
Since $\theta_\star$ is assumed unknown, a \emph{single-stage} LinUCB estimates it by ridge regression with regularization parameter $\lambda > 0$
\begin{align}
\begin{split}\label{eq:blr_posterior_params}
\Sigma_t
&\coloneqq
\Bigl[ \lambda\, I_{d} + \sum_{i = 1}^t \phi(x_{i, a_i}) \phi(x_{i, a_i})^\top\Bigr]^{-1}
\, ,
\\
\hat{\theta}_t
&\coloneqq
\Sigma_t \sum_{i = 1}^t r_i\, \phi(x_{i, a_i})
\, .
\end{split}
\end{align}
The actions are then selected according to
\begin{align}\label{eq:action_selection}
a_{t+1} &\in \argmax_{a \in A}\, \text{UCB}_{t+1} (a)
\, ,
\end{align}
where as in \citep[p.~239--241]{lattimore2020bandit}
\begin{align*}
\text{UCB}_{t+1} (a)
&\coloneqq
\langle \phi(x_{t, a}), \hat{\theta}_t \rangle + \sqrt{\beta_t}\, \| \phi(x_{t, a}) \|_{\Sigma_t}
\, ,
\\
\sqrt{\beta_t}
&\coloneqq
\sqrt{\lambda}
+
\sqrt{
2 \log t
+
d
\log \left(
\frac{d \lambda + t}{d \lambda}
\right)
}
\, .
\end{align*}
LinUCB with such $\beta_t$ achieves near optimal regret
\begin{align}\label{eq:regret_definition}
\begin{aligned}[c]
R_T
&\coloneqq
\mathbb{E} \left[
\sum_{t=1}^T
r_{t,\star} - r_t
\right]
\\
&=
\sum_{t=1}^T
\langle
\theta_\star ,
\phi(x_{t, a_{t, \star}})
-
\mathbb{E} [\phi(x_{t, a_t})]
\rangle
\, ,
\end{aligned}
\end{align}
when the reward noise is sub-Gaussian \citep{dani2008stochastic,lattimore2020bandit},
where
\begin{equation}
a_{t, \star} \coloneqq \argmax_{a \in A} \langle \theta_\star , \phi (x_{t, a}) \rangle \, ,
\end{equation}
$r_{t, \star}$ is the reward obtained by choosing $a_{t, \star}$, and the expectation is taken with respect to the randomness of the rewards and the policy (uniform tie breaking).
Importantly for our later development, LinUCB can be interpreted in Bayesian terms in the following sense: the current mean and covariance estimates $\theta \sim \mathcal{N}(\hat{\theta}_t , \Sigma_t)$ correspond to the posterior distribution obtained by combining the prior $\mathcal{N}(0, \lambda^{-1} I_{d} )$ with the likelihood $r_{t , a} \sim \mathcal{N} (\langle \phi(x_{t, a}), \theta \rangle , 1)$.
Since
\begin{equation}
\langle \phi(x_{t, a}), \theta \rangle \sim \mathcal{N} \bigl(\langle \phi(x_{t, a}), \hat{\theta}_{t-1} \rangle, \| \phi(x_{t, a}) \|_{\Sigma_{t-1}}\bigr)\, ,
\end{equation}
the selection rule employed by LinUCB can be viewed as selecting the action with the highest posterior $\Phi (\sqrt{\beta_t})$-th quantile, where $\Phi$ is the cumulative distribution function (CDF) of the standard normal distribution.
In other words, the maintained estimates define a \emph{confidence set} for the true parameter $\theta$, and LinUCB chooses the best action compatible with this set.
As mentioned, we will assume that each of the nominators and the ranker use the LinUCB algorithm also in the two-stage setup.\footnote{While not in line with the usual heterogeneity of the nominator algorithms, we believe this setting captures much that is salient to the interaction between general simultaneously learning ranker and nominators. It also goes one step beyond the setup in \citep{ma2020off} by considering the existence of more than one nominator.}
Since the nominators often need to rapidly sift through millions of items, we assume \emph{only} the ranker has access to the true but expensive to compute embeddings $\phi$, while each nominator $n \in [N]$ uses computationally cheaper embeddings $\phi_n(x) \in \mathbb{R}^{d_n}$.
In the rest of this document, we use the term \emph{naive} two-stage LinUCB to refer to the algorithm where each nominator \emph{independently} maintains its own estimates $\hat{\theta}_{n,t}$, $\Sigma_{n,t}$ defined as in \Cref{eq:blr_posterior_params} with $\phi$ replaced by $\phi_n$, nominate actions analogously to \Cref{eq:action_selection}, and update their posterior \emph{only} with $r_t$ and $\phi_n (x_{t, a_t})$, where we recall $a_t$ need not equal $a_{t, n}$.
Moreover, the ranker independently maintains its own estimates $\hat{\theta}$, $\Sigma_{t}$ used to select the item ultimately served to the user.
In the next section, we will first show on a simple example that such independently maintained uncertainty estimates lead to suboptimal global performance.
We then propose a solution based on synchronization of the upper bound estimates $\text{UCB}_t (a)$ used in \Cref{eq:action_selection} between the ranker and the nominators.
\section{Coordinated exploration}
\label{sec:algo}
To understand when the \emph{naive} two-stage LinUCB implementation does not work, it is useful to know when it does.
In particular, consider the case when all the nominators are allowed to use the \emph{same features} as the ranker $\phi_n = \phi$, and employ the \emph{same prior} $\mathcal{N} (0, \lambda_n^{-1} I_d)$ with $\lambda_n = \lambda$.
It is not hard to see that in this case $\hat{\theta}_{n, t} = \hat{\theta}_t$ and $\Sigma_{n, t} = \Sigma_{t}$ for all $t \in [T]$, and thus each nominator selects the same action as would be selected by the ranker constrained to the same action pool $A_n$.
Since $\max \{ c_1, \ldots, c_k \} = \max \{ \max \{ c_1 , \ldots , c_{k_1} \} , \ldots , \max \{ c_{k_{N - 1} + 1} , \ldots , c_k \} \}$ for any partition of $c_1, \ldots , c_k \in \mathbb{R}$, this then implies that the naive two-stage system behaves \emph{exactly} as a single-stage LinUCB with access to all actions would.
Because we know single-stage LinUCB is close to optimal, the above implies that any potential increase in regret must come from either the already discussed mismatch of the embeddings inherent to two-stage systems, or mismatch of the prior.
The latter is then most often caused by the ranker being deployed for much longer time and thus better trained.
Such a scenario is common in many contemporary industrial practices \citep{covington2016deep,ma2020off}, and can be modelled in our setup by using a ranker prior with lower initial uncertainty then the nominators.
As demonstrated next, naive two-stage LinUCB is poorly equipped to handle such discrepancies.
\def0.24{0.24}
\begin{figure*}
\centering
\textbf{Noise level $\sigma_{\theta_\star} = 0.1$:} $\hat{\theta}_0 \sim \mathcal{N} (\theta_\star, 10^{-2})$\\
\hrulespace{0.5mm}
\vspace{1mm}
\hspace{2cm}$\gamma = 1$ \hfill $\gamma = 10$ \hfill $\gamma = 25$ \hfill $\gamma = 50$ \hspace{1.3cm}\phantom{}
\\
\includegraphics[width=0.24\textwidth]
{klv2-pseudo_1-noise_1e-01-post}
\hfill
\includegraphics[width=0.24\textwidth]
{klv2-pseudo_10-noise_1e-01-post}
\hfill
\includegraphics[width=0.24\textwidth]
{klv2-pseudo_25-noise_1e-01-post}
\hfill
\includegraphics[width=0.24\textwidth]
{klv2-pseudo_50-noise_1e-01-post}
\\
\textbf{Noise level $\sigma_{\theta_\star} = 0.2$:} $\hat{\theta}_0 \sim \mathcal{N} (\theta_\star, 5^{-2})$\\
\hrulespace{0.5mm}
\vspace{1mm}
\hspace{2cm}$\gamma = 1$ \hfill $\gamma = 10$ \hfill $\gamma = 25$ \hfill $\gamma = 50$ \hspace{1.3cm}\phantom{}
\\
\includegraphics[width=0.24\textwidth]
{klv2-pseudo_1-noise_2e-01-post}
\hfill
\includegraphics[width=0.24\textwidth]
{klv2-pseudo_10-noise_2e-01-post}
\hfill
\includegraphics[width=0.24\textwidth]
{klv2-pseudo_25-noise_2e-01-post}
\hfill
\includegraphics[width=0.24\textwidth]
{klv2-pseudo_50-noise_2e-01-post}
\caption{Naive vs.\ synchronized 2-stage recommendation. Setup described in \Cref{sect:motivating_example}. Expected regret and its 2-sigma confidence intervals were estimated over 400 runs. The level of pretraining of the ranker $\gamma$ has outsized effect on the naive but not the synchronized two-stage LinUCB, overcoming the `deadlock' effect.}\label{fig:linUCB_toy}
\end{figure*}
\begin{algorithm}[tbp]
\caption{\label{alg:sync_linUCB}Two-stage {\color{SteelBlue}synchronized} LinUCB. Here $\phi_t \coloneqq \phi(a_t)$ and $\phi_{n, t} \coloneqq \phi(a_{n, t})$ for all $n, t$.}
\footnotesize
\textbf{Inputs:} $\hat{\theta}_0 \, , \Sigma_0 \, , (\beta_{t})_{t} \, ;\; \forall n \colon \hat{\theta}_{n , 0} \, , \Sigma_{n, 0} , (\beta_{n , t})_t$ \\
\For{$t=1,2,\ldots, T$}{
$\forall n \colon a_{n, t} \gets \argmax_{a \in A_n} \text{UCB}_{n, t} (a)$ \\
$a_t \gets \argmax_{a \in \{ a_{1, t}, \ldots , a_{N, t} \}} \text{UCB}_{t} (a)$ \\
$\Sigma_t^{-1} \gets \Sigma_{t-1}^{-1} + \phi_t \phi_t^\top$ \\
$\hat{\theta}_t \gets \Sigma_t \left(\Sigma_{t-1}^{-1} \hat{\theta}_{t-1} + r_t \phi_t\right)$ \\
\For{$n=1, 2, \ldots, N$} {
$\Sigma_{n, t}^{-1} \gets \Sigma_{n, t-1}^{-1} + \phi_{n,t} \phi_{n, t}^\top$ \\
$\hat{\theta}_{n, t} \gets \Sigma_{n, t} \left(\Sigma_{n, t-1}^{-1} \hat{\theta}_{n, t-1} + r_t \phi_{n, t}\right)$ \\
{\color{SteelBlue}
\uIf{$\| \phi_n (a_{n, t}) \|_{\Sigma_{n, t}} > \| \phi (a_{n, t}) \|_{\Sigma_{t}}$}{
$\hat{\theta}_{n, t} \gets \hat{\theta}_{n, t} + \frac{\langle\hat{\theta}_{t}, \phi_{t}\rangle - \langle \hat{\theta}_{n, t}, \phi_{n, t}\rangle}{\|\phi_{n, t}\|_{\Sigma_{n, t}}^2}\Sigma_{n, t}\phi_{n, t}$ \\
$\Sigma_{n, t}^{-1} \gets \Sigma_{n,t}^{-1} + \bigl( \frac{1}{\|\phi_t\|_{\Sigma_{t}}^{2}} - \frac{1}{\|\phi_{n, t}\|_{\Sigma_{n, t}}^{2}} \bigr)
\phi_{n, t}\phi_{n, t}^{\top}$
}
}
}
}
\end{algorithm}
\subsection{Motivating example}\label{sect:motivating_example}
Consider a setting with \emph{only one} context, two nominators, three actions split between them as $A_1 = \{ a_1 \}$, $A_2 = \{ a_2, a_3 \}$, $\phi$ returning \emph{one-hot encodings} of the actions, and $\phi_n = \phi$ for all nominators.
The expected rewards from $a_1$ to $a_3$ are $[1/2, 1/4, 3/4] = \theta_*$ (one-hot action encodings),
and observed rewards are generated by adding i.i.d.\ Gaussian noise $\mathcal{N} (\theta_\star, 10^{-2} I_3)$.
The ranker's parameters are initialized to $\hat{\theta}_0 \sim \mathcal{N} (\theta_\star, \sigma_{\theta_\star}^2 I_3)$, and $\Sigma_{0} = (\lambda + \gamma)^{-1} I_{3}$ where $\gamma$ represents how many more samples per action the ranker has seen at $t = 1$ compared to the nominators.
For both nominators $n \in [2]$, we take $\hat{\theta} = 0$, and $\Sigma_{n, 0} = \lambda_n^{-1} I_{3}$ with regularization parameter (prior precision) $\lambda_n = \lambda = 10^{-3}$.
To see what can go wrong in this scenario, consider the extreme case $\sigma_{\theta_\star}^2 = 0$ and $\gamma \gg 0$, i.e., the ranker has seen enough data to essentially recover the true parameter.
Since $A_1 = \{ a_1 \}$, such a ranker always picks $a_3$ (the best action) when $a_{2, t} = a_3$, and $a_1$ otherwise.
In the naive implementation, this results in a \emph{`deadlock'} where the
second nominator's uncertainty about $a_2$ never decreases, leading it to mostly nominate $a_2$ over $a_3$, entailing \emph{linear} regret.
While the extreme case may be rare in practice, \Cref{fig:linUCB_toy} shows the effect remains significant even when the ranker is not fully trained, i.e., $\sigma_{\theta_\star} > 0$, and $\gamma > 1$ but not overly large.
\subsection{Synchronized two-stage LinUCB}
The deadlock observed in the previous section is due to a lack of communication of uncertainty between the ranker and the nominators.
Since the computational constraints inherent to the two-stage setup entail distinct embedding functions, the issue cannot be addressed by simply setting nominator parameters to those of the ranker.
However, because action selection only depends on the estimated marginal quantiles of the \emph{rewards}, we can update the nominator's estimates based on the reward statistics computed by the ranker.
We propose to \emph{synchronize} each nominator $n$ in the rounds where $\| \phi_n (a_{n, t}) \|_{\Sigma_{n, t}} > \| \phi (a_{n, t}) \|_{\Sigma_{t}}$, i.e., when the nominator is more uncertain about its selected action than the ranker (the cause of the `deadlock' in \Cref{sect:motivating_example}).
In particular, we want to minimally adjust the nominator posterior so that it matches the ranker's mean and variance which fully determine $\text{UCB}_{n, t}(a_{n, t})$ (see \Cref{sec:setup}).
Defining the minimality in terms of Kullback-Leibler (KL) divergence, this can be achieved by solving the constrained optimization problem:
\begin{align}\label{eq:kl_sync}
\min_{m, S}
\quad
&
\text{KL}\left(\mathcal{N}\left(m, S\right)\;\middle \|\;\mathcal{N}\left(\hat{\theta}_{n, {t}}, \Sigma_{n, {t}}\right)\right)
\\
\text{subject to}
\quad
&\langle m, \phi_{n} (a_{n, t}) \rangle = \langle \hat{\theta}_{t}, \phi (a_{n, t}) \rangle
\, , \qquad\,\,
{\color{gray}\text{\small(mean)}}
\nonumber
\\
&\|\phi_n (a_{n, t})\|_{S} = \|\phi (a_{n, t})\|_{\Sigma_{t}}
\, , \quad
{\color{gray}\text{\small(covariance)}}
\nonumber
\end{align}
When synchronization is performed after the usual update, the solution to \Cref{eq:kl_sync} gives us \Cref{alg:sync_linUCB}.
Note that the selected KL divergence penalizes overdispersion compared to the previous distribution, meaning the resulting replacement for $\mathcal{N}(\hat{\theta}_{n, {t}}, \Sigma_{n, {t}} )$ should not have more uncertainty.
Furthermore, if we modify \Cref{alg:sync_linUCB} to perform the synchronization before the usual update---i.e., swap the black and blue lines within the inner-most for-loop and replace the if-condition by $\| \phi_n (a_{n, t}) \|_{\Sigma_{n, t-1}} > \| \phi (a_{n, t}) \|_{\Sigma_{t-1}}$---we arrive at an algorithm that only uses quantities already computed during selection of $a_{n, t}$ and $a_t$, minimizing the additional computation required;
in experiments, both versions of the algorithm performed essentially the same, which we show in an example setting in \Cref{fig:post_vs_pre}.
\nk{For a future version we should probably only show the compared lines (not naive) and make them somewhat transparent. Or perhaps even show absolute difference on log scale?}
\begin{figure}
\centering
\includegraphics[width=0.8\columnwidth]{klv2-pseudo_50-noise_2e-01-full}
\caption{Comparison of post- vs.\ pre-update synchronization for a noise level of $0.2$ and $\gamma = 50$.
In this setting, the line for pre-update synchronization is barely visible, because it is covered by the line for post-update synchronization.}
\label{fig:post_vs_pre}
\end{figure}
Finally, let us consider our synchronized two-stage LinUCB algorithm in the context of the motivating example from \Cref{sect:motivating_example}.
Because we assumed $\phi = \phi_n$ are one-hot encodings of the actions, we see that the ranker variance for an action $j \in [3]$ at time $t$ is $\| \phi(a_j) \|_{\Sigma_t}^2 = (\lambda + \gamma + n_{tj})^{-1}$ where $n_{tj}$ is the number of times the ranker selected $a_j$.
The \emph{first} time a nominator select $a_j$, its variance before the round update will be $\| \phi_n(a_j) \|_{\Sigma_{n, t}}^2 = (\lambda + n_{tj})^{-1}$.\footnote{Since we allow the pools $A_n$ to be overlapping, $n_{tj}$ could generally be greater than zero here. This is not the case for the example from \Cref{sect:motivating_example} though.}
Inspecting the synchronization update for $\Sigma_{n, t}^{-1}$ in blue (\Cref{alg:sync_linUCB}), the new value for the $(jj)$\textsuperscript{th} entry of $\Sigma_{n, t}^{-1}$ amounts to
\begin{align*}
\underbrace{\lambda + n_{tj}}_{\Sigma_{n, t}^{-1}}
+
\underbrace{\lambda + \gamma + n_{tj}}_{
\| \phi(a_j) \|_{\Sigma_t}^{-2}
}
-
\underbrace{(\lambda + n_{tj})}_{
\| \phi_n(a_j) \|_{\Sigma_{n, t}}^{-2}
}
=
\| \phi(a_j) \|_{\Sigma_t}^2
\, ,
\end{align*}
while the other variances will remain as in $\Sigma_{n, t}$ at time $t$.
Since an analogous claim holds for the mean $\hat{\theta}_{n, t}$, we conclude the synchronization update ensures the posteriors of the ranker and the nominators match after each nominator \emph{selected} (but not necessarily seen \emph{recommended}) each $a \in A_n$ exactly once.
Because the posteriors never diverge after they are fully matched, our algorithm starts behaving like single-stage LinUCB from thereon, which we know is near optimal for the task.
This is confirmed in \Cref{fig:linUCB_toy} where the more pretrained the ranker is (higher $\gamma$), the better the synchronized and the worse the naive LinUCB do.
\subsubsection*{\bibname}}
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\setitemize{leftmargin=*}
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\begin{document}
\runningauthor{Hron, Krauth, Jordan, Kilbertus}
\twocolumn[
\aistatstitle{Exploration in two-stage recommender systems}
\aistatsauthor{Jiri Hron${}^{1,*}$, Karl Krauth${}^{2,*}$, Michael I.~Jordan${}^2$, Niki Kilbertus${}^{1,3}$}
\aistatsaddress{${}^1$University of Cambridge, ${}^2$UC Berkeley, ${}^3$Max Planck Institute for Intelligent Systems
}
]
\begin{abstract}
\input{content/00abstract}
\end{abstract}
\input{content/01intro}
\input{content/02setup}
\input{content/03algo}
\input{content/05conclusion}
|
2024-02-18T23:40:24.747Z
|
2020-09-21T02:18:13.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2009.08956","language":"en","timestamp":1600654693000,"url":"https:\/\/arxiv.org\/abs\/2009.08956","yymm":"2009"}
|
proofpile-arXiv_000-10077
|
{"provenance":"002.jsonl.gz:10078"}
| null | null |
\section{Introduction}
\if2 1
\vspace{-10pt}
\else
\fi
\subsection{Industrial Wireless Control}
Wireless \gls{iiot} in the next generation of industrial control systems require communications with sub-ms, extreme low latency and ``cable-like" ultra-high reliability. For a large-scale network of sensor and actuator devices in factory automation applications, different wireless transmission schemes have recently been proposed to exploit spatial and multi-user diversity gain in the network. The challenge in this new paradigm of wireless communication is that the design requires guaranteed service to all, including the \emph{weakest} user, as opposed to the classic paradigm of network design that targets average performance.
Traditionally, such industrial automation requirements are realized on the factory floor through wired communications e.g., using fieldbus and Ethernet based solutions. The wired solutions, however, are considered to be cumbersome and expensive in many applications. Moreover, the future industrial automation targets a highly flexible and dynamic environment of production stations that support robotic mobility to be able to seamlessly re-arrange according to production requirements \cite{Cena:2008,Bennis:2018,Harish:6G2020}. As a result, there is an increased desire to replace wired communication systems for factory automation with wireless alternatives to reduce bulk as well as installation and maintenance costs \cite{Gungor:2009}. This calls for innovative solutions in constrast to existing wireless technologies that are designed for delay tolerant consumer solutions, making them unsuited for industrial automation \cite{weiner2014design}. In the era beyond the \gls{5g} mobile networks, \gls{urllc} is promised to deliver such demanding requirements using the advanced physical layer technologies, including communications in \gls{mmwave} and \gls{uwb} spectrum access \cite{Gilberto:2018,mazgula2020ultra}, accompanied by the improvements in network architecture in bringing the cloud close to the edge to reduce latency, and using machine learning for a fast and reliable prediction of channels and traffic~\cite{Bennis:2020arxiv}.
\if2 1
\vspace{-10pt}
\else
\fi
\subsection{Prior Work}
\if2 1
\vspace{-5pt}
\else
\fi
To fulfill the requirements of ultra-high reliability within a stringent latency constraint, different diversity techniques are suggested in the literature. Time and frequency diversity techniques \cite{zand2012wireless} as well as spatial diversity and multi-user cooperation \cite{swamy2015cow} can reduce the required reliability-achieving \gls{snr}. For example, the importance of multi-antenna receive diversity in improving reliability and coverage was pointed out in \cite{brahmi2015gc}. In low-latency industrial automation applications, the cycle time is shorter than the fading channel coherence time, which rules out viability of \gls{arq}-based time diversity techniques~\cite{Khosravirad:2017}. In \cite{swamy2015cow}, it is shown that relying solely on frequency diversity to achieve $10^{-9}$ error rate requires impractically high \gls{snr} values in realistic channel conditions. It is further shown that multi-user diversity, even in low or moderate \gls{snr} regime, can achieve ultra-reliability. Incidentally, increasing transmit diversity by engaging multiple transmitting \glspl{ap}, similar to \gls{comp} technology \cite{randa:2012}, is also not a straightforward path to reliability. In fact, in the absence of \gls{csi} at the transmitter, transmit diversity falls dramatically short of achieving high reliability as discussed in \cite{Rebal:2018}. The study in \cite{Rebal:2018} further shows that by using \gls{csi} to adapt transmission rate at the transmitter, the inherent multi-user diversity gain of a large size network can be exploited to achieve high reliability. To that point, the cooperative transmission in \cite{swamy2015cow} attempts to exploit the full potential of wireless network by enabling cooperative device-to-device relaying to improve reliability. The focus of \cite{swamy2015cow} is to devise transmission schemes that do not require transmitter \gls{csi}. Such \gls{csi}-agnostic schemes are not able to take the differences in the instantaneous channel conditions into account, resulting in a sub-optimal and conservative choice of transmission rate determined by the worst user's conditions, and loss of spectral efficiency. On the other hand, in a multi-user wireless network, the overhead for acquiring \gls{csi} can grow large as the size of the network grows. The works in \cite{8259329} and \cite{Rebal:2018} study the impact of such overhead. Particularly, \cite{8259329} considers the overhead of \gls{csi} acquisition when communicating with small packets, thus characterizing the error performance under the finite block length regime.
System-level simulations for multi-user networks under \gls{urllc} requirements is highly time-consuming and complex. We acknowledge that several works in the literature have addressed and dealt with those complexities, including \cite{Klaus:2018system,7247339}, and have provided insightful conclusions for system design of cellular networks with extreme reliability requirements.
\if2 1
\vspace{-10pt}
\else
\fi
\subsection{Exploiting Diversity ``On-demand''}
\if2 1
\vspace{-5pt}
\else
\fi
Spatial diversity transmission, as the most dependable source of achieving high reliability when communicating over fading channels, is a viable solution for reliability that may be achieved by using multiple transmission points or antennas. In low cost deployments, however, it is desirable to have a small number of spatially distributed, simple \glspl{ap} with limited number of antennas. Spatial diversity transmission, however, needs to also exploit multi-user diversity stemming from the fact that several users with different channel conditions are part of the communication system. On the other hand, cooperative relaying among users (as proposed in \cite{swamy2015cow}) essentially also achieves spatial diversity through multi-user diversity.
The core question this paper tries to answer is \emph{how to use channel awareness at the transmitter to efficiently allocate radio and cooperation resources, and to exploit diversity according to the instantaneous needs of the users}. We introduce a transmission protocol that is capable of identifying users' channel strength and allows for exploiting different sources of diversity, \emph{on demand}. We propose to adapt the transmission rate for users with strong channel from the \glspl{ap} according to their channel, while exploiting cooperative diversity for the remaining users with weak channel. This introduces a robust way of deploying the \emph{emergency} resources of network cooperation, only for the devices that absolutely need them. We study the improvements offered by this protocol on the operating spectral efficiency and the minimum required \gls{snr} for reliability. In high \gls{snr}, this impacts the slope of the outage probability curve and improves diversity order by deploying network-device cooperation for the poor links. Meanwhile, it brings better multiplexing gain by smartly exploiting the difference in channel conditions.
\if2 1
\vspace{-15pt}
\else
\fi
\subsection{Channel Estimation in Wireless Networks}
\if2 1
\vspace{-5pt}
\else
\fi
We assume a general transmission framework where data is accompanied with proper amount of pilot signal in all transmissions \cite{hassibi03}. The estimated channel at a receiving node can then be reported back to the transmitting node, e.g. in form of \gls{cqi}. With an adequate frequency of \gls{cqi} updates, transmitter can improve resource utilization efficiency by adapting the transmission attributes to the channel. In fact, such an approach is widely adopted in multi-user cellular technologies such as the \gls{lte} and \gls{nr}. More interestingly, by assuming channel reciprocity in \gls{tdd} transmission mode for the industrial wireless control problem of our interest with isochronous traffic pattern in the \gls{dl} and the \gls{ul} directions, \gls{cqi} acquisition requires no feedback exchange between the nodes. Instead, \gls{cqi} can be estimated when the node performs channel estimation while in receiving mode and can be used when the node switches to transmit mode. In this paper, we adopt this assumption and present a new transmission protocol that utilizes channel state information to best exploit sources of diversity in the wireless network.
\if2 1
\vspace{-15pt}
\else
\fi
\subsection{Contributions}
\if2 1
\vspace{-5pt}
\else
\fi
The objective of this paper is to design an ultra-reliable transmission scheme for extreme low-latency applications which uses minimal control signaling for scheduling. To this end, we aim to exploit full spatial and multi-user diversity potential of the network in favor of system reliability and spectral efficiency. Our contributions are summarized as follows.
\if2 1
\vspace{-5pt}
\else
\fi
\begin{itemize}
\item We identify and analyze different sources of diversity gain for ultra-reliable wireless communications in an industrial wireless control network. A network with multiple fully-connected \glspl{ap} is assumed where the \glspl{ap} coordinate their transmissions similar to \gls{comp}. We formulate the achievable multi-user and multi-antenna diversity gain in the low-latency regime, and propose a new scheme for exploiting those in favor of reliability and efficiency.
\item A new ultra-reliable transmission scheme dubbed \gls{andcoop} that exploits different sources of diversity in the network is introduced. The proposed scheme uses the approximate knowledge of \gls{csi} to categorizes the devices into two groups, namely, group of devices with \emph{strong} instantaneous channel, and the group of devices with \emph{weak} channel. The two groups are then scheduled in separate scheduling phases: first, each of the strong devices receives its \gls{dl} message with a unique transmission rate that is adapted to its instantaneous channel state; next, the second group of devices are scheduled with a fixed rate through two-hop cooperative transmission where all the devices in both groups can potentially contribute in as \gls{df} relays.
\item Reliability performance of the proposed transmission scheme is analytically formulated, in order to characterize the system outage probability. The analysis is then extended to diversity-multiplexing trade-off, where closed-form formulations for the achievable diversity order are derived. We further formulate the optimization problem of allotting time between the two scheduling phases and provide numerical solutions to the optimization problem.
\item Comprehensive and detailed system-level simulations are reported to identify guidelines for optimal system design. The proposed protocol is compared against the existing transmission protocols in the literature. The numerical analysis demonstrates significant concurrent improvement in spectral efficiency (approximately 0.5 \gls{bpcu} per \gls{ap} antenna) and reliability. Alternatively, under fixed spectral efficiency setup, the proposed algorithm acheives the desired reliability at significantly smaller transmit power (around 15 dB improvement compared to the existing schemes), while utilizing around $40\%$ less relay nodes' energy, which in turn reduces the interference footprint. Moreover, the impact of \gls{csi} estimation error is carefully studied, suggesting that the proposed \gls{andcoop} transmission scheme consistently reduces the impact of such error on system reliability, thanks to the strategy of grouping devices according to channel quality. We identify significant potential in cost reduction for the future private industrial wireless control network, thanks to the improved operation efficiency using the proposed \gls{andcoop} scheme.
\end{itemize}
\if2 1
\vspace{-15pt}
\else
\fi
\subsection{Organization of the Paper}
\if2 1
\vspace{-5pt}
\else
\fi
The sequence of this paper is as follows: in \secref{Sec:ProblemSetup} we present the problem description and the assumed network setup; further, we provide motivations for designing a new ultra-reliable transmission scheme; in \secref{Sec:Model} the proposed channel-aware \gls{urllc} solution is presented and analyzed for outage probability and diversity order; \secref{Sec:Results} presents and discusses the numerical analysis of the proposed scheme; and finally, \secref{Sec:Conclusion} covers the concluding remarks.
\section{Problem Setup}
\label{Sec:ProblemSetup}
\begin{figure}[t]
\begin{center}
\psfrag{u}[c][c][0.59]{wireless link}
\psfrag{v}[c][c][0.59]{wired link}
\psfrag{x}[c][c][0.59]{wireless connected}
\psfrag{w}[c][c][0.59]{devices}
\psfrag{y}[c][c][0.59]{access points}
\psfrag{z}[c][c][0.59]{controller}
\includegraphics[width=0.8\columnwidth,keepaspectratio]{topology.eps}
\caption{Wireless network model for industrial wireless control. Wireless devices with \emph{weak} channel conditions (highlighted) are identified for a two-hop communication.}
\label{Fig:Model}
\end{center}
\if2 1
\vspace{-35pt}
\else
\fi
\end{figure}
In this section, we first describe the communications system model of interest and highlight the main system assumptions we use in our analysis. Then, we discuss exploiting diversity gain a multi-user wireless network under the paradigm of ultra-reliable communications to motivate our design target in exploiting full diversity potential of the network for industrial wireless control.
\if2 1
\vspace{-10pt}
\else
\fi
\subsection{System Model}
\paragraph*{Network} $N$ devices are scattered on a factory floor and are wirelessly connected with the controller \glspl{ap}. \figref{Fig:Model} illustrates the considered network where a controller is wired to $M$ fully synchronized \glspl{ap}. This paper considers a \gls{comp} setting where all \glspl{ap} are synchronized and they coordinate their transmission attributes for transmission to every device. All the communicating nodes have a single antenna for transmission and reception. Every device expects an independent $B$ bytes of data to be delivered every $T$ seconds over a bandwidth of $W$ Hertz. We use $\eta = \frac{N B}{T W}$, measured in \gls{bpcu}, to denote the overall spectral efficiency of the system.
Let $\mathscr{A} = \set{1,2,\dots,M}$ denote the set of \glspl{ap}, where $M = \card{\mathscr{A}}$ is the number of \glspl{ap}. Similarly, let $\mathscr{D} = \set{1,2,\dots,N}$ be the set of device IDs where $N = \card{\mathscr{D}}$. Throughout the paper, we reserve the letter $R$ to denote transmit rate measured in \gls{bps}.
\paragraph*{Channel dynamics} Wireless channels linking every \gls{ap}-device and device-device pair are assumed to undergo independent frequency-flat Rayleigh fading. We note that this assumption is adopted for analytical tractability, although measurement campaigns for industrial environments show frequency-selectivity over wide bandwidth \cite{nist1951,rapp91}. We assume a setting where each time-cycle experiences a constant channel which fades independently from one cycle to the next. Let $\ha{i}{j}$ and $\ra{i}{j}$ denote the channel fade random variable and the average received \gls{snr} (which includes the effect of path loss and is averaged with respect to fading distribution) of the transmission from \gls{ap} $i$ to device $j$, where $i \in \mathscr{A}$ and $j \in \mathscr{D}$. We use $\ga{i}{j} = \ra{i}{j}|\ha{i}{j}|^2$ to denote the instantaneous received \gls{snr}. Similarly, let $\hd{k}{j}$, $\rd{k}{j}$ and $\gd{k}{j}$ denote the same variables for the link from device $k$ to device $j$, where $k, j \in \mathscr{D}$. Note that $\ra{i}{j} = P_a / (W \cdot \sigma_0)$ and $\rd{k}{j} = P_d / (W \cdot \sigma_0)$, where $P_a$ and $P_d$ denote the transmit power of an access point and a device, respectively, and $\sigma_{\tr{o}}$ denotes \gls{psd} of the \gls{awgn}.
\paragraph*{Outage model}
A device is said to be in outage if the transmission rate $R$ exceeds the instantaneous channel capacity, and is considered successful otherwise. We assume distributed space-time coding that collects spatial diversity through summation of the received signal powers from all transmitters. Therefore, with $\mathscr{C}_j$ denoting the set of nodes cooperatively transmitting with rate $R$ to node $j$ over bandwidth $W$, the transmission fails if
\begin{align}\label{Eq:linkoutage}
W \log \left( 1 + \sum_{i \in \mathscr{A} \cap \mathscr{C}_j}\ga{i}{j} + \sum_{k \in \mathscr{D} \cap \mathscr{C}_j}\gd{k}{j} \right) < R.
\end{align}
The expression in \eqref{Eq:linkoutage} implicitly assumes long block-length transmission. While admittedly, the transmission of small packets in line with what is typically expected in \gls{urllc} scenarios and also suitable for the block-fading model challenges the assumption that the packets are long enough for \eqref{Eq:linkoutage}, we note that the impact of such assumptions can be further evaluated by adopting the finite block-length regime outage models \cite{PolyanskiyFBL:2010}. More importantly, the recent findings in \cite{6802432} suggest that in a fading channel, the effect of outage dominates the effect of short block-length, so the outage capacity is in fact a fair substitute for the finite block-length fundamental limits. For this reason, the rest of this paper focuses on the outage model in \eqref{Eq:linkoutage}.
Similar to the previous works in \cite{Rebal:2018TCOM,swamy2015cow,Khosravirad:vtc2019}, we analyze \emph{system outage probability} as the key performance metric, denoted by $\mathbb{P}_\tr{\tiny{out}}(.)$ and defined as the probability that at least one device fails to decode its own message at the end of time-cycle $T$. This is a more appropriate measure for reliability of communication in an industrial wireless control setup compared to e.g., average outage probability across devices. The argument is that the industrial wireless control system may only continue its operation when all devices follow the controller instructions, and the system fails if at least one devices fails. Note that $\mathbb{P}_\tr{\tiny{out}}(.)$ is a function of the channel random variables, as well as the parameters of the system. Moreover, such definition complies with the joint definition of reliability and latency requirements in the context of \gls{urllc}. In essence, a \gls{urllc} system satisfies its requirements only if it can guarantee the desired reliability level \emph{within} the desired latency budget \cite{3gpp38913}. Therefore, in this work, instead of the statistics of the experience delay, we are interested in the outage probability within a constrained latency of $T$ seconds.
\paragraph*{Diversity-multiplexing}
It is widely accepted that the end goal of \gls{urllc} systems is to increase reliability, and therefore, the system outage probability curve is the natural benchmark for performance evaluation. However, the true performance of such system can only be evaluated if data rate is monitored alongside the reliability. Thanks to the choice of \emph{system outage probability} (described above) to represent error rate in the system model, the diversity gain can be captured as the slope at which the error rate decays in the high \gls{snr} regime. Moreover, we define the multiplexing gain $r$ as the ratio at which the payload size per device $B$ increases with transmit power $P_t$ in log scale, i.e., $B \propto r \log P_t$. Thus, the dual benefits can be captured by the diversity-multiplexing tradeoff in the high \gls{snr} regime, where similar to \cite{Zhao:2007diversity,Tse2003divmux}, we say that a diversity gain $d(r)$ is achieved at multiplexing gain $r$, if $\eta = r \log P_t$ and,
\begin{align}\label{Eq:dive_order_def}
d(r) = - \lim_{P_t \rightarrow \infty} \frac{\log \mathbb{P}_\tr{\tiny{out}}(r \log P_t)}{\log P_t},
\end{align}
thus capturing the tradeoff between data rate increase (i.e., $r$) and diversity order, in high \gls{snr}.
\input{table.tex}
\paragraph*{Channel estimation} We assume instantaneous \gls{csi} of the \gls{ap}-device pairs are present at the controller in the form of link strength $\ga{i}{j}$'s, which doesn't require the knowledge of channel phase. This can be for instance provided by frequent transmission of uplink pilot sequences by the devices similar to \gls{srs} in \gls{lte}.
Each \gls{ap} estimates its channel from all the devices, using those pilot sequences. The \gls{csi} will be used to identify groups of devices with \emph{strong} and \emph{weak} channel conditions and to adapt the transmission rate for the former group. The variance of channel estimation error can be arbitrarily minimized by increasing the number of pilot sequences and transmit power of the pilots \cite{hassibi03,yoo06,Rebal:2018}.
Assuming that the channel \gls{snr}, $\rho$, is known, we use $\hat{h}$ and $\hat{g}$ to denote the estimated channel fade and estimated \gls{snr}, respectively. We further use $L$ to denote the length of the pilot training sequence for each device, with duration of $T_P = L \cdot T_S$ seconds over orthogonal time-frequency resources, where $T_S = 1/W$ is the symbol period. The total overhead cost of pilot transmission then equals to $N \cdot T_P$, and is paid out of the time budget $T$, leaving $T_D = T - N\cdot T_P$ seconds for data transmission. Using recursive \gls{mmse} channel estimation \cite{hassibi03,yoo06}, the true Rayleigh fade $h$ can be written as $h=\hat{h}+\epsilon$, where $\epsilon\sim\mathcal{CN}(0,\sigma_e(L))$, $\hat{h}\sim\mathcal{CN}(0,1-\sigma_e(L))$, and
\begin{align}\label{eq_mmse}
\sigma_e(L)=\frac{1}{1+L\cdot\rho}.
\end{align}
With respect to channel estimation, and for completeness of the investigation, we adopt two scenarios in this paper; namely, genie-aided \gls{pcsi}, i.e., where fade is perfectly estimated as $\hat{h}=h$, at the cost of zero pilot overhead $L = 0$, leaving $T_D = T$; and the case of \gls{icsi}, where channel estimation error is a function of the pilot training sequence length $L$ based on \eqref{eq_mmse}, resulting in $T_D = T - N \cdot L \cdot T_S$.
\paragraph*{Notations} Throughout the paper, we use the notations listed in \tabref{table:notation}.
\if2 1
\vspace{-10pt}
\else
\fi
\subsection{On the Role of Multi-User Diversity in Low-Latency Regime}
In a \emph{large} network with multiple users, each fading independently, there is likely to be a user whose channel is near its peak, at any time. This can be utilized to maximize the long term total throughput by use of \gls{csi} feedback and always serving the user with the strongest channel \cite{Knopp:1995,Grossglauser:2002} hence, exploiting \emph{multi-user diversity} gain.
Similarly, for a given spectral efficiency, the per-user reliability of transmission can be maximized by choosing the user with strongest channel at any time. Therefore, with loose latency requirement, multi-user diversity gain is a natural source of reliability and efficiency. However, in low-latency regime, where tolerated latency is smaller or equal to the channel coherence time, it is likely to have one or few users whose channels are poor, at any time. It is therefore challenging to exploit multi-user diversity while guaranteeing timely reliability to multiple users with asymmetric channel statistics.
To further analyze the diversity gain in low-latency regime, let's assume the network setup described earlier in this section with $M = 1$, where all the channel gain $\ga{i}{j}$'s are perfectly known and thus, the controller can precisely determine the achievable rate $\aR_{j}$ for device $j$, and the \gls{ap} targets an average spectral efficiency of $\frac{N B}{T W}$ in each time cycle $T$, with equal packet size for every scheduled device. Let's consider the case where the scheduler has the complete freedom to choose any nonempty subset of the $N$ devices in each time cycle. We model the average latency based on $K$, the number of users that are scheduled within a given time cycle. The average experienced latency by a device to be scheduled can be shown to be $\frac{(N+K)}{2K}T$\footnote{Derived by averaging across all devices, knowing that the first $K$ devices experience latency of $T$, while the last $K$ devices to be scheduled in a round experience latency of $NT/K$.}. For example under round-robin scheduling, by scheduling all devices in every time cycle $T$, i.e., $K = N$, the average latency is $T$, and by scheduling only one device in each time cycle the average latency is increased to $\frac{(N+1)}{2}T$. The scheduler is thus rewarded with reduced average latency, for scheduling every additional device out of the $N$. The cost of scheduling an additional device is going to be a loss in the collected multi-user diversity order. The following proposition addresses the trade-off between average latency and the collected multi-user diversity gain, assuming \gls{iid} Rayleigh fading on every link.
\if2 1
\vspace{-5pt}
\else
\fi
\begin{proposition}
\label{Prop:motiv1}
The maximum diversity order exploited at zero multiplexing point by the scheduler described above for scheduling exactly $K$ users in each time cycle is $N - K + 1$ for the case of $M=1$, and $M(N-K+1)$ for general $M$.
\end{proposition}
\if2 1
\vspace{-10pt}
\else
\fi
\begin{proof}
See \appref{App:motiv}.
\end{proof}
In other words, with strict latency requirements, i.e., all the $N$ devices must be scheduled over one coherence time, which is the case for an industrial wireless control network as described earlier in this section, then the system experiences diversity order of $1$ (for the case of M=1), meaning that no multi-user diversity gain is exploited. On the other hand, the maximum multi-user diversity order of $N$, is only exploited when the latency requirement is maximally loose and the controller gets to schedule the best device in each time cycle. Midway, the controller can trade off latency with diversity order by transmitting to $K \leq N$ devices over each coherence time, thus gaining the diversity order of $N - K + 1$. In fact, as we see in the \secref{Sec:Model}, the transmission protocol proposed in this paper benefits from such trade-off in exploiting the multi-user diversity gain by scheduling a subset of the devices with the highest channel strengths. For the remaining devices, the protocol seeks for the gain of cooperation among nodes, which is the topic of the following discussion.
\if2 1
\begin{figure*}[t]
\begin{center}
\begin{subfigure}{.25\textwidth}
\begin{center}
\begin{psfrags}
\psfrag{xlabel}[c][c][0.6]{X (m)}
\psfrag{ylabel}[c][c][0.6]{Y (m)}
\psfrag{title}[c][c][0.6]{\gls{snr} (dB)}
\psfrag{xlabel2}[c][c][0.6]{X (m)}
\psfrag{ylabel2}[c][c][0.6]{Y (m)}
\psfrag{title2}[c][c][0.6]{coverage map}
\includegraphics[width=0.6\columnwidth,keepaspectratio]{snr_map_1hop.eps}
\end{psfrags}
\caption{Coverage of single-hop transmission.}
\label{fig:coverage_1h}
\end{center}
\end{subfigure}%
\begin{subfigure}{.55\textwidth}
\begin{subfigure}{.45\textwidth}
\begin{center}
\begin{psfrags}
\psfrag{xlabel}[c][c][0.6]{X (m)}
\psfrag{ylabel}[c][c][0.6]{Y (m)}
\psfrag{title}[c][c][0.6]{\gls{snr} (dB)}
\psfrag{xlabel2}[c][c][0.6]{X (m)}
\psfrag{ylabel2}[c][c][0.6]{Y (m)}
\psfrag{title2}[c][c][0.6]{coverage map}
\psfrag{xlabel3}[c][c][0.6]{X (m)}
\psfrag{ylabel3}[c][c][0.6]{Y (m)}
\psfrag{title3}[c][c][0.6]{\gls{snr} (dB)}
\psfrag{xlabel4}[c][c][0.6]{X (m)}
\psfrag{ylabel4}[c][c][0.6]{Y (m)}
\psfrag{title4}[c][c][0.6]{coverage map}
\includegraphics[width=.6\columnwidth,keepaspectratio]{snr_map_2hop_2.eps}
\end{psfrags}
\end{center}
\end{subfigure}%
\begin{subfigure}{.45\textwidth}
\begin{center}
\begin{psfrags}
\psfrag{xlabel1}[c][c][0.6]{X (m)}
\psfrag{ylabel1}[c][c][0.6]{Y (m)}
\psfrag{title1}[c][c][0.6]{\gls{snr} (dB)}
\psfrag{xlabel2}[c][c][0.6]{X (m)}
\psfrag{ylabel2}[c][c][0.6]{Y (m)}
\psfrag{title2}[c][c][0.6]{\gls{snr} (dB)}
\psfrag{xlabel3}[c][c][0.6]{X (m)}
\psfrag{ylabel3}[c][c][0.6]{Y (m)}
\psfrag{title3}[c][c][0.6]{\gls{snr} (dB)}
\psfrag{xlabel4}[c][c][0.6]{X (m)}
\psfrag{ylabel4}[c][c][0.6]{Y (m)}
\psfrag{title4}[c][c][0.6]{coverage map}
\includegraphics[width=.6\columnwidth,keepaspectratio]{snr_map_2hop_1.eps}
\end{psfrags}
\end{center}
\end{subfigure}%
\caption{Coverage of two-hop transmission with cooperative relaying.}
\label{fig:coverage_2h}
\end{subfigure}%
\end{center}
\vspace{-15pt}
\caption{Map of \gls{snr} and coverage of $10^{-9}$ outage probability for spectral efficiency of 1 \gls{bpcu}, in an area of $100\times100$ $m^2$, in presence of a blockage (cyan color wall); the bright point in the center locates the \gls{ap} while the rest of the bright points locate the relay devices; (a) \gls{ap} with 4 antennas in the center provides $95.5\%$ when transmitting over the full time cycle $T$, i.e., transmission rate of 1 \gls{bpcu}; (b) on the left, $87.95\%$ coverage from an \gls{ap} with 4 antennas at transmission rate of 2 \gls{bpcu}, over first $T/2$ duration; on the right, $72.1\%$ coverage using three single antenna relay devices at transmission rate of 2 \gls{bpcu}, over the second $T/2$ duration; $100\%$ combined coverage of the two phases.}
\label{fig:coverage_map}
\if2 1
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\else
\vspace{-10pt}
\fi
\end{figure*}
\else
\begin{figure*}[t]
\begin{center}
\begin{subfigure}{.35\textwidth}
\begin{center}
\begin{psfrags}
\psfrag{xlabel}[c][c][0.6]{X (m)}
\psfrag{ylabel}[c][c][0.6]{Y (m)}
\psfrag{title}[c][c][0.6]{\gls{snr} (dB)}
\psfrag{xlabel2}[c][c][0.6]{X (m)}
\psfrag{ylabel2}[c][c][0.6]{Y (m)}
\psfrag{title2}[c][c][0.6]{coverage map}
\includegraphics[width=0.6\columnwidth,keepaspectratio]{snr_map_1hop.eps}
\end{psfrags}
\caption{Coverage of single-hop transmission.}
\label{fig:coverage_1h}
\end{center}
\end{subfigure}%
\begin{subfigure}{.6\textwidth}
\begin{subfigure}{.5\textwidth}
\begin{center}
\begin{psfrags}
\psfrag{xlabel}[c][c][0.6]{X (m)}
\psfrag{ylabel}[c][c][0.6]{Y (m)}
\psfrag{title}[c][c][0.6]{\gls{snr} (dB)}
\psfrag{xlabel2}[c][c][0.6]{X (m)}
\psfrag{ylabel2}[c][c][0.6]{Y (m)}
\psfrag{title2}[c][c][0.6]{coverage map}
\psfrag{xlabel3}[c][c][0.6]{X (m)}
\psfrag{ylabel3}[c][c][0.6]{Y (m)}
\psfrag{title3}[c][c][0.6]{\gls{snr} (dB)}
\psfrag{xlabel4}[c][c][0.6]{X (m)}
\psfrag{ylabel4}[c][c][0.6]{Y (m)}
\psfrag{title4}[c][c][0.6]{coverage map}
\includegraphics[width=.7\columnwidth,keepaspectratio]{snr_map_2hop_2.eps}
\end{psfrags}
\end{center}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
\begin{center}
\begin{psfrags}
\psfrag{xlabel1}[c][c][0.6]{X (m)}
\psfrag{ylabel1}[c][c][0.6]{Y (m)}
\psfrag{title1}[c][c][0.6]{\gls{snr} (dB)}
\psfrag{xlabel2}[c][c][0.6]{X (m)}
\psfrag{ylabel2}[c][c][0.6]{Y (m)}
\psfrag{title2}[c][c][0.6]{\gls{snr} (dB)}
\psfrag{xlabel3}[c][c][0.6]{X (m)}
\psfrag{ylabel3}[c][c][0.6]{Y (m)}
\psfrag{title3}[c][c][0.6]{\gls{snr} (dB)}
\psfrag{xlabel4}[c][c][0.6]{X (m)}
\psfrag{ylabel4}[c][c][0.6]{Y (m)}
\psfrag{title4}[c][c][0.6]{coverage map}
\includegraphics[width=.7\columnwidth,keepaspectratio]{snr_map_2hop_1.eps}
\end{psfrags}
\end{center}
\end{subfigure}%
\caption{Coverage of two-hop transmission with cooperative relaying.}
\label{fig:coverage_2h}
\end{subfigure}%
\end{center}
\caption{Map of \gls{snr} and coverage of $10^{-9}$ outage probability for spectral efficiency of 1 \gls{bpcu}, in an area of $100\times100$ $m^2$, in presence of a blockage (cyan color wall); the bright point in the center locates the \gls{ap} while the rest of the bright points locate the relay devices; (a) \gls{ap} with 4 antennas in the center provides $95.5\%$ when transmitting over the full time cycle $T$, i.e., transmission rate of 1 \gls{bpcu}; (b) on the left, $87.95\%$ coverage from an \gls{ap} with 4 antennas at transmission rate of 2 \gls{bpcu}, over first $T/2$ duration; on the right, $72.1\%$ coverage using three single antenna relay devices at transmission rate of 2 \gls{bpcu}, over the second $T/2$ duration; $100\%$ combined coverage of the two phases.}
\label{fig:coverage_map}
\if2 1
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\else
\vspace{-10pt}
\fi
\end{figure*}
\fi
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\else
\fi
\subsection{Motivation for Multi-Hop Transmission}
Cooperative relaying has been studied recently in several works as an enabler of \gls{urllc}, e.g., see \cite{swamy2015cow,swamy:2018relsel,Arvin:2019,Kaiming:2019ratespliting}, leveraging on the spatial diversity gain from engaging multiple relay devices, which increases robustness to fading variations. The focus of the design in cooperative relaying scheme in \cite{swamy2015cow} has been to mitigate the effect of small-scale fading. Such approach is highly beneficial in absence of \gls{mimo} techniques. However, with increasing deployment of massive \gls{mimo}, in practice, those benefits can be largely undermined. Particularly, cellular communication technologies typically rely on channel hardening effect of \gls{mimo} to mitigate the effect of small-scale fading \cite{Hochwald2004channelhardening}.
Nevertheless, multi-hop relaying has historically been considered as a means of extending coverage in various wireless technologies, such as \gls{wimax}, \gls{lte}, and recenlty in \gls{5g} \gls{nr}, e.g., see \cite{kim2008relaycoverage,Gui2018relaycoverage,Lang2009relaycoverage,DawidKoziol:2017,3gpp22866}. The coverage problem caused by static or mobile blockages is in fact a challenging issue in industrial environments. Blockage can generally impede the link \gls{snr} by obstructing the \gls{los}. More significantly, when blockage size is several times larger than the electromagnetic wavelength, the diffraction around the obstacle becomes weaker, making the impact of blockage stronger. Consequently, blockages becomes a more severe challenge in higher frequencies, or the so called \gls{mmwave} \cite{5262296}. Moreover, the dynamic nature of factory floor with large number of static and moving objects makes it difficult to provide every time/everywhere wireless link availability. This further motivates the use of cooperative relaying to deal with temporary and/or zonal loss of coverage.
To this point, the example in \figref{fig:coverage_map} illustrates the \gls{snr} and coverage of $10^{-9}$ outage probability for spectral efficiency of 1 \gls{bpcu} over an area of $100\times100$ $m^2$. A simple static blockage is positioned on the right side of the area. First, in \figref{fig:coverage_1h}, it is shown that using the total time budget $T$ (i.e., transmitting at 1 \gls{bpcu}), direct transmission by the \gls{ap} with 4 antennas provides $95.5\%$ coverage. This means that only around $95.5\%$ of the points over the area can achieve the required target outage probability of $10^{-9}$ with a single-hop transmission. Then, in \figref{fig:coverage_2h}, the time budget $T$ is divided by two, and the transmission is done twice at the doubled rate of 2 \gls{bpcu}. Therefore, the coverage of the direct transmission from access point reduces to $87.95\%$ (left hand side of \figref{fig:coverage_2h}), due to the increase in transmission rate. However, in the second $T/2$ portion of the time, three devices randomly positioned around the blockage, and each having a single transmit antenna, relay the transmission from the \gls{ap}. The relaying phase at 2 \gls{bpcu}, also provides a complementary coverage of $72.1\%$, mostly around the area affected by the blockage. But more interestingly, the overall coverage of the two phases reaches the desirable $100\%$.
Knowing that the blockage affects the coverage around the right hand side of the square area, such coverage enhancement from two-hop relaying can in practice be directed towards devices in the same area. This in fact increases the efficiency of ultra-reliable communications, by deploying the cooperative relaying in an \emph{on-demand} fashion, only for the devices with coverage issues.
Improving coverage for \gls{iiot} applications can alternatively be done by densification of the \glspl{ap}. However, over-provisioning is not efficient in terms of the cost of the network deployment. It should be noted that the intention of the example in \figref{fig:coverage_map} is not to claim that two-hop cooperative relaying is always better than single-hop transmission. In fact, as we will discuss in the following sections, with a smart and dynamic algorithm to use the cooperative relaying gain in an \emph{on-demand} manner, the overall spectral efficiency can be improved compared to the case where all the devices are served with two-hop transmission.
\if2 1
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\else
\fi
\section{Proposed Channel-Aware Transmission Protocol}
\label{Sec:Model}
In this section we first introduce the proposed transmission protocol. Then outage and diversity order analysis of the protocol are presented.
\if2 1
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\else
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\subsection{Transmission Protocol}
\label{Sec:transmissionprotocol}
The total \gls{dl} transmission time $T_D$ is divided into two parts, using a partitioning factor $0 \leq \beta \leq 1$, where $\Toh = \beta T_D$ is used for rate-adaptive single-hop transmission of independent messages to devices with strong instantaneous channel to the controller \glspl{ap} in a \gls{tdma} fashion. The remaining $\Tth = (1-\beta)T_D$ is used for two-hop cooperative transmission phase to the rest of the devices, where the messages of all remaining devices are aggregated and transmitted in two hops. In addition, the controller acquires the knowledge of $\gha{i}{j}$'s for the \gls{ap}-device pairs via channel estimation using pilots of length $L$. Based on this, devices are put in an order according to the instantaneous transmission rate that they can receive at from the \glspl{ap}. The achievable transmission rate for device $j$, $\aR_{j}$, can be computed using \eqref{Eq:linkoutage}. The controller estimates the achievable rate, using
\begin{align}\label{Eq:achievablerate}
\ahR_{j} = W \log \left( 1 + \sum_{i \in \mathscr{A}}\gha{i}{j} \right).
\end{align}
Note that, for \gls{icsi}, i.e., $\sigma_{e} > 0$, $\prob{\ahR_{j} > \aR_{j}} = 0.5$. In other words, regardless of the channel estimation precision, the estimated transmission rate is above the achievable rate $50\%$ of the time. To curb the impact of channel estimation error in case of \gls{icsi}, we use a rate back-off parameter $0 \leq \theta \leq 1$ to adjust the transmission rate to $\theta \cdot \ahR$. The following definition denotes the largest subset of the devices that the controller can accommodate with single-hop transmission over time $\tau$.
\begin{align}\label{Eq:strongdevices}
\pmb{S}(\tau, \{\gha{i}{j} \}) = \; \underset{\mathscr{S}}{\arg} \; \max_{\mathscr{S} \subset \mathscr{D}} \; \; & \card{\mathscr{S}} \\ \nonumber
\text{subject to} & \sum_{j \in \mathscr{S}} \frac{1}{\theta \cdot \ahR_{j}} \leq \frac{\tau}{B}, \\
& \ahR_{k} \leq \ahR_{j}, \; \forall k \in \mathscr{D} \setminus \mathscr{S}, \forall j \in \mathscr{S} \nonumber
\end{align}
Let $\mathscr{D}_{1\text{h}}$, where $\mathscr{D}_{1\text{h}} \subset \mathscr{D}$, and $\mathscr{D}_{2\text{h}} = \mathscr{D} \setminus \mathscr{D}_{1\text{h}}$ denote the random set of the \emph{strong} and \emph{weak} devices, respectively. Let $K_{1\text{h}} = \card{\mathscr{D}_{1\text{h}}}$ and $K_{2\text{h}} = \card{\mathscr{D}_{2\text{h}}}$ be the discrete random variables of size of those sets, which follows $K_{1\text{h}} + K_{2\text{h}} = N$. Let $\Roh{j} = \theta \cdot \ahR_{j}$ denote the transmission rate for device $j \in \mathscr{D}_{1\text{h}}$. We assume that $\beta$ is known by all the nodes in the network. Therefore, upon generating the set $\mathscr{D}_{1\text{h}}$ for a given realization of the channels, the controller sends the set of indexes in $\mathscr{D}_{1\text{h}}$ and the transmission rates $\Roh{j}$, over a control channel to the strong devices. Such information is necessary for the devices to be able to follow the scheduling order of transmission in the single-hop phase. The devices in $\mathscr{D}_{2\text{h}}$ are then scheduled over a two-hop transmission over the remaining $\Tth$, where their messages are aggregated, and $\alpha \cdot \Tth$ and $(1-\alpha)\cdot \Tth$ are used for broadcasting and relaying the aggregated messages respectively, with $0 \leq \alpha \leq 1$.
In \figref{fig:scheduling} the time scheduling of the proposed \gls{andcoop} transmission protocol is illustrated\footnote{Note that the time dimension is chosen in here as an example. In practice the division of the time-frequency resources between the single-hop and two-hop phases of the protocol can be in either the time domain, frequency domain, or both.}. In the following the proposed \gls{andcoop} transmission protocol is summarized. We assume that the controller has the knowledge of the appropriate $\beta$ and $\alpha$ design parameters, which are acquired off-line and are shared with the devices (we discuss the optimization of those parameters in the following subsection). A summary of the proposed algorithm in each time cycle is as follows:
\begin{enumerate}
\item Using the knowledge of \gls{ap}-device \gls{csi}, the controller finds the set of devices, $\mathscr{D}_{1\text{h}}$, that will be scheduled over single-hop rate-adaptive transmission. This is done according to $\mathscr{D}_{1\text{h}} = \pmb{S}(\Toh, \{\gha{i}{j} \})$ in \eqref{Eq:strongdevices}, while $\Toh = \beta \cdot T$.
\item The controller adapts transmission rate for each device in $\mathscr{D}_{1\text{h}}$ according to their instantaneous channel by setting $\Roh{j} = \theta \cdot \ahR$, thus allocating $B/\Roh{j}$ seconds of the total time for transmission to device $j$. The controller \glspl{ap} will then perform \gls{comp} transmission of the message for each node $j \in \mathscr{D}_{1\text{h}}$ with the adapted rate in a \gls{tdma} fashion.
\item \label{step-b} All the $B$-bit messages intended for devices in $\mathscr{D}_{2\text{h}} = \mathscr{D} \setminus \mathscr{D}_{1\text{h}}$ are aggregated together. The controller \glspl{ap} jointly broadcast the aggregated message at rate $\Rth{b} = \frac{B K_{2\text{h}}}{\alpha \Tth}$, over the first $\alpha$ portion of $\Tth$ time\footnote{In practice, the messages can be concatenated before encoding which will potentially increase coding gain and reduce number of decoding attempts for relay devices.}.
\item \label{step-receive} All devices in $\mathscr{D}$ attempt decoding the broadcast message from previous step. The successful devices to decode will act as relays.
\item The \glspl{ap} broadcast the message from step~\ref{step-b} at rate $\Rth{r} = \frac{B K_{2\text{h}}}{(1-\alpha) \Tth}$. The relay devices from step~\ref{step-receive} re-encode with the same code rate, and cooperate as simultaneous relays.
\end{enumerate}
\begin{figure}[t]
\begin{center}
\psfrag{a}[c][c][0.59]{$\Toh = \beta \cdot T$}
\psfrag{b}[c][c][0.59]{$\alpha \cdot \Tth$}
\psfrag{c}[c][c][0.59]{$(1-\alpha)\cdot \Tth$}
\psfrag{t}[c][c][0.59]{$T$}
\psfrag{u}[c][c][0.59]{$\ldots$}
\psfrag{v}[c][c][0.59]{$\Rth{r} = \frac{B K_{2\text{h}}}{(1-\alpha) \Tth}$}
\psfrag{w}[c][c][0.59]{$\Rth{b} = \frac{B K_{2\text{h}}}{\alpha \Tth}$}
\psfrag{x}[c][c][0.59]{$\Roh{j_{1}}$}
\psfrag{s}[c][c][0.59]{$\Roh{j_{2}}$}
\psfrag{z}[c][c][0.59]{$\Roh{j_{K_{1\text{h}}}}$}
\includegraphics[width=0.8\columnwidth,keepaspectratio]{scheduling.eps}
\caption{Time scheduling illustrated for the proposed scheme. The transmission rate for each scheduled time slot is identified, assuming $j_1, j_2, \ldots, j_{K_{1\text{h}}} \in \mathscr{D}_{1\text{h}}$. For the devices in $\mathscr{D}_{2\text{h}}$, the aggregated message is first broadcast with rate $\Rth{b}$ and then relayed with rate $\Rth{r}$.}
\label{fig:scheduling}
\end{center}
\if2 1
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\else
\fi
\end{figure}
\if2 1
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\else
\fi
\subsection{A Note on Optimization of the Design}
The proposed transmission protocol can be optimized based on the design parameters $L$, $\beta$ and $\theta$, to achieve the minimum $\mathbb{P}_\tr{\tiny{out}}$. We perform numerical optimization in two scenarios of \gls{pcsi} and \gls{icsi}. Note that in \gls{pcsi} scenario, we fix $L = 0$ and $\theta = 1$, only optimizing with respect to $\beta$, for given transmit power and spectral efficiency. To this end, we fix the value of the time division parameter $\beta$ for all realizations of the channel. Therefore, we are able to numerically optimize $\beta$ for a given setup by running the simulation for all values of $\beta$ from a finite set of real numbers uniformly chosen from the $[0, 1]$ interval to derive $\hat{\beta} = \underset{\beta}{\arg \min} \mathbb{P}_\tr{\tiny{out}}(\beta)$.
Optimization of the \gls{icsi} scenario with exhaustive search across $L$, $\beta$ and $\theta$, is computationally costly. However, as we will show in the next section, parameter $L$ can be fixed with marginal effect on the outage, which can significantly reduces the search for the optimal $L$ , $\beta$ and $\theta$.
Parameter $\alpha$ is used to partition the time $\Tth$ between broadcast and relaying hops of the two-hop cooperative transmission. Increasing $\alpha$ results in smaller $\Rth{b}$ which increases chances of decoding for all nodes and results in larger expected number of relay nodes for the second hop. In turn, it will increase $\Rth{r}$ and decrease chance of decoding in the second hop.
We note that the optimization of $\alpha$ is not a trivial problem however, following the observation in \cite{swamy2015cow}, the average reliability gain from optimizing $\alpha$ with respect to $\alpha = 0.5$ can be marginal, therefore, in the numerical analysis that will follow, we fix $\alpha = 0.5$.
\if2 1
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\else
\fi
\subsection{Outage Analysis}
We analyze \emph{system outage probability} of the proposed system, denoted by $\mathbb{P}_\tr{\tiny{out}}$ and defined as the average probability that at least one device fails to decode its own message at the end of time-cycle $T$. Let $P_{1h}(\mathscr{D}_{1\text{h}})$ and $P_{2h}(\mathscr{D}_{2\text{h}})$ denote the probability of outage for at least one device in, respectively, the adaptive-rate single-hop phase and the two-hop cooperative phase of the transmission protocol.
\paragraph*{Single-hop rate-adaptive transmission} Conditioned on \gls{csi}, we have
\begin{align}\label{Eq:outage-1hop-icsi}
P_{1h}(\mathscr{D}_{1\text{h}} | \{ \hha{i}{j} \}) =\prob{ \exists j \in \mathscr{D}_{1\text{h}}, \theta \cdot \ahR > \aR },
\end{align}
where expectation is over channel gains.
For the case of \gls{icsi} with channel estimation error, the outage probability in \eqref{Eq:outage-1hop-icsi} is non-zero as also discussed in \cite[Sec. III-A]{Rebal:2018TCOM}. Characterization of $P_{1h}$ is not analytically tractable. However, when $\beta = 1$, i.e., all the devices are scheduled with single-hop transmission, outage probability is equivalent to the probability of \emph{time overflow}, i.e., the chance that $\Toh$ is too short given the channel gains to successfully accommodate all the packets. In this case, analytical bounds on the outage probability can be found in \cite{Rebal:2018TCOM}, where it is shown that in practical ranges of \gls{snr}, as cell load $N \times B$ grows, the chance of time overflow increases above \gls{urllc} target outage probability and dominates system outage regardless of the precision of channel estimation.
For the case of \gls{pcsi}, when $\beta < 1$, all the devices in $\mathscr{D}_{1\text{h}}$ pass the success condition in \eqref{Eq:linkoutage}, resulting in $P_{1h}(\mathscr{D}_{1\text{h}} | \{ \ha{i}{j} \}) = 0$. Therefore, the following is valid for \gls{pcsi} scenario.
\begin{align}\label{Eq:outage-1hop}
P_{1h}(\mathscr{D}_{1\text{h}} | \{ \ha{i}{j} \}) = \left\{\begin{matrix}
0 & 0 \leq \beta <1 \\
\prob{\sum_{j \in \mathscr{D}} \frac{1}{\aR_j} > \frac{ \Toh}{B}} & \beta = 1.
\end{matrix}\right.
\end{align}
\paragraph*{Two-hop cooperative transmission}
Conditioned on \gls{csi}, the outage probability of the two-hop phase is given by
\begin{align}
P_{2h}(\mathscr{D}_{2\text{h}} \mid \{ \hha{i}{j} \}) = \prob{ \exists j \in \mathscr{D}_{2\text{h}} \setminus \mathscr{R} : \sum_{k \in \mathscr{R}}\gd{k}{j} < \zeta },
\nonumbe
\end{align}
where $\zeta = 2^{\Rth{r}/W} -\sum_{i \in \mathscr{A}}\ga{i}{j} -1$, and $\mathscr{R}$ is the set of relay devices, defined as
\begin{align}
\mathscr{R} = \{j \in \mathscr{D}: \Rth{b} \leq \aR_{j} \}.
\end{align}
\if2 2
\begin{figure*}[b]
\normalsize
\hrulefill
\begin{align}\label{Eq:Ptwohop}
P_{2h}\left(\mathscr{D} \right) = \sum_{\substack{ \forall \mathscr{S} \subset \mathscr{D} }} & \prob{\forall j \in \mathscr{S} : \; W \log \bigg( 1 + \sum_{i \in \mathscr{A}}\ga{i}{j} \bigg) \geq \Rth{b} } \cdot \prob{\forall j \in \mathscr{D} \setminus \mathscr{S} : \; W \log \bigg( 1 + \sum_{i \in \mathscr{A}}\ga{i}{j} \bigg) < \Rth{b} } \\
& \cdot \left( 1 - \prob{\forall j \in \mathscr{D} \setminus \mathscr{S} : \; W \log \bigg( 1 + \sum_{i \in \mathscr{A}}\ga{i}{j} + \sum_{k \in \mathscr{S}}\gd{k}{j} \bigg) \geq \Rth{r} \; \middle| \; \forall j \in \mathscr{D} \setminus \mathscr{S} : \; W \log \bigg( 1 + \sum_{i \in \mathscr{A}}\ga{i}{j} \bigg) < \Rth{b} } \right)
\nonumber
\end{align}
\end{figure*}
Although, averaging $P_{2h}(\mathscr{D}_{2\text{h}} \mid \{ \ga{i}{j} \})$ over channel gains is not analytically tractable, for the purpose of analyzing the diversity order gain of the transmission protocol, we are interested in the outage probability when all devices are served in the two-hop cooperative phase.
An expression for $P_{2h}$, conditioned on $K_{2\text{h}} = N$ can be written as follows in \eqref{Eq:Ptwohop} at the bottom of the page.
\else
Although, averaging $P_{2h}(\mathscr{D}_{2\text{h}} \mid \{ \ga{i}{j} \})$ over channel gains is not analytically tractable, for the purpose of analyzing the diversity order gain of the transmission protocol, we are interested in the outage probability when all devices are served in the two-hop cooperative phase.
An expression for $P_{2h}$, conditioned on $K_{2\text{h}} = N$ can be written as follows in \eqref{Eq:Ptwohop}.
\fontsize{9}{11}\selectfont
\begin{align}\label{Eq:Ptwohop}
P_{2h}\left(\mathscr{D} \right) = \sum_{\substack{ \forall \mathscr{S} \subset \mathscr{D} }} & \prob{\forall j \in \mathscr{S} : \; W \log \bigg( 1 + \sum_{i \in \mathscr{A}}\ga{i}{j} \bigg) \geq \Rth{b} } \cdot \prob{\forall j \in \mathscr{D} \setminus \mathscr{S} : \; W \log \bigg( 1 + \sum_{i \in \mathscr{A}}\ga{i}{j} \bigg) < \Rth{b} } \\
& \cdot \left( 1 - \prob{\forall j \in \mathscr{D} \setminus \mathscr{S} : \; W \log \bigg( 1 + \sum_{i \in \mathscr{A}}\ga{i}{j} + \sum_{k \in \mathscr{S}}\gd{k}{j} \bigg) \geq \Rth{r} \; \middle| \; \forall j \in \mathscr{D} \setminus \mathscr{S} : \; W \log \bigg( 1 + \sum_{i \in \mathscr{A}}\ga{i}{j} \bigg) < \Rth{b} } \right)
\nonumber
\end{align}
\normalsize
\fi
For a practical wireless network in which the channel fading distribution depends on path-loss and shadowing, the evaluation of \eqref{Eq:Ptwohop} is challenging. Simplification of \eqref{Eq:Ptwohop} can be obtained in the following special case. Let's assume a setup with fixed nominal \gls{snr} $\rho$ on all \gls{ap}-device and device-device links with \gls{iid} small-scale fading $h$.
In such a scenario, using \eqref{Eq:linkoutage}, the probability of decoding failure with $m$ cooperating transmitters at rate $R$ is as follows.
\begin{align}\label{Eq:pfail}
\Pm{m,R} = \prob{W \log\Big( 1 + \rho \sum_{l = 1}^{m}|h_l|^2 \Big) < R},
\end{align}
where $h_l \sim CN(0,1)$, are \gls{iid} random variables. Therefore, $\sum_{l = 1}^{m}|h_l|^2$ has the Erlang distribution and $p^m$ can be computed as \cite{Laneman:2003}
\begin{align}
\Pm{m,R} = \gamma\left( m,\frac{\omega}{P_t} \right),
\end{align}
where $\gamma(x,y) = \int_{0}^{y} t^{x-1} e^{-t} dt$ is the incomplete Gamma function. Moreover, $\omega = W \cdot \sigma_0 \cdot (2^{R/W} - 1)$ and $\sigma_0$ denotes \gls{psd} of the \gls{awgn} and $P_t$ denotes the transmit power at each transmitting node.
In such simplified scenario, as similarly suggested in \cite{swamy2015cow}, \eqref{Eq:Ptwohop} can be reformulated as follows.
\if2 1
\begin{align}\label{Eq:Ptwohop-4}
P_{2h}\left(\mathscr{D} \right) = \sum_{n = 0}^{N-1}\left( q_{b}^{M}\right)^{(N-n)} \left(1-q_{b}^M \right)^{n} \binom{N}{n} \left(1 - \left( 1 - {q_{r}^{(M+n)}} \right)^{(N-n)} \right),
\end{align}
\else
\begin{align}\label{Eq:Ptwohop-4}
&P_{2h}\left(\mathscr{D} \right) = \\
&\sum_{n = 0}^{N-1}\left( q_{b}^{M}\right)^{(N-n)} \left(1-q_{b}^M \right)^{n} \binom{N}{n} \left(1 - \left( 1 - {q_{r}^{(M+n)}} \right)^{(N-n)} \right),
\nonumber
\end{align}
\fi
where $q_{b}^M = \Pm{M,\Rth{b}}$ and
\begin{align}\label{Eq:qr}
q_{r}^{(M+n)} = \min \{1, \Pm{M+n,\Rth{r}} / \Pm{M,\Rth{b}} \},
\end{align}
is the conditional failure probability of a device in relaying hop given
\if2 1
\else
that
\fi
it failed in broadcast hop.
\if2 1
\vspace{-10pt}
\else
\fi
\subsection{Diversity-Multiplexing Tradeoff Analysis}
\label{Sec:DMT}
In this section, we analyze the diversity-multiplexing trade-off of the proposed scheme for the case of \gls{pcsi}, assuming independent Rayleigh distributed fading for all links, using the definition in \eqref{Eq:dive_order_def}. First, note that when transmit power goes to $\infty$, the achievable transmission rate from \eqref{Eq:achievablerate} also approaches $\infty$, resulting in $ \mathscr{D}_{1\text{h}} = \mathscr{D}$, when $0 < \beta \leq 1$. Therefore, unless $\beta = 0$, all the devices are scheduled in the single-hop rate-adaptive phase. For that reason, we analyze the diversity-multiplexing tradeoff for two extreme cases of $\beta = 1$ and $\beta = 0$, respectively, when all the devices are scheduled in single-hop rate-adaptive phase and, when all the devices are scheduled in two-hop cooperative phase. This also provides lower bounds for the diversity-multiplexing tradeoff of the proposed \gls{andcoop} scheme.
\begin{proposition}
\label{Prop:1h}
When all the devices are scheduled in single-hop rate-adaptive phase, the diversity order of the considered system at zero multiplexing is given by
\begin{align}
\label{Eq:div_order_1hop}
d_\textit{single-hop}(0) = M.
\end{align}
\end{proposition}
\begin{proof}
See \appref{App:A}.
\end{proof}
Moreover, following the proof in \appref{App:A}, upper and lower bounds of the diversity-multiplexing tradeoff are readily derived as follows for the case where all the devices are scheduled over single-hop transmission.
\begin{align}\label{Eq:divmux_1hop}
M (1-r) \leq d_\textit{single-hop}(r) \leq M (1-\frac{r}{N}).
\end{align}
For the case when all the devices are scheduled in two-hop cooperative phase (i.e., the OccupyCoW protocol in \cite{swamy2015cow}), the following proposition presents the diversity-multiplexing tradeoff for the general case of $0 < \alpha <1$, when $M>1$.
\if2 1
\vspace{-10pt}
\else
\fi
\begin{proposition}
\label{Prop:2h}
When all the devices are scheduled in two-hop cooperative phase, the diversity-multiplexing tradeoff is given by
\begin{align}\label{Eq:DMT-2hop}
d_\textit{two-hop}(r) = (M+N-1)\left( 1- \frac{r}{1-\alpha} \right),
\end{align}
where at zero multiplexing gain it yields
\begin{align}
\label{Eq:div_orde_2hop}
d_\textit{two-hop}(0) = M+N-1.
\end{align}
\end{proposition}
\begin{proof}
See \appref{Proof_prop_2h}.
\end{proof}
Intuitively, diversity order $M + N -1$, i.e., the slope of outage probability curve in high \gls{snr}, corresponds to the case when all the $M$ \glspl{ap} cooperate with $N-1$ strongest devices to transmit to the weakest device. It is an interesting observation from \eqref{Eq:DMT-2hop} that the tradeoff between diversity order $d$ and multiplexing gain $r$ in high \gls{snr} is controlled by the inverse of $1- \alpha$. In other words, the duration of the relaying phase, which directly affects the outage probability for the weakest devices, controls the diversity multiplexing tradeoff in high \gls{snr}. Therefore, for a given multiplexing gain $r$, the maximum diversity is achieved when $\alpha$ is the minimum allowed value for \eqref{Eq:DMT-2hop} to be non-negative, i.e., $0 < \alpha \leq 1-r$. Since $\alpha$ cannot be zero in \eqref{Eq:DMT-2hop}, the following upper bound is valid for the two-hop transmission for $r > 0$.
\begin{align}
\label{Eq:div_orde_2hop_up}
d_\textit{two-hop}(r) < (M+N-1) ( 1- r ).
\end{align}
This upper bound is depicted by the dashed green line in \figref{fig:div_mux_tradeoff}. For the case of $\alpha = \frac{1}{2}$, which is the exercised design in \cite{Arvin:2019,swamy2015cow,Kaiming:2019ratespliting}, the diversity-multiplexing tradeoff in \eqref{Eq:DMT-2hop} becomes
\begin{align}\label{Eq:2hop_divmux_half}
d_\textit{two-hop}(r)|_{\alpha=1/2} = (M+N-1) ( 1- 0.5 \cdot r )
\end{align}
As also depicted in \figref{fig:div_mux_tradeoff}, in such case the maximum achievable multiplexing gain is $r = 0.5$, intuitively corresponding to the fact that each message is transmitted twice with the same rate, once in broadcast phase and once more in the relaying phase.
It is clear that the two-hop operation in \eqref{Eq:2hop_divmux_half} is spectrally inefficient, due to the fact that at very low system outage probability the gain from cooperative relaying is small, thus only a small percentage of the devices will benefit from relaying. On the other hand, the single-hop transmission from \propref{Prop:1h} can achieve higher multiplexing gain than \eqref{Eq:2hop_divmux_half}. This further suggests that it is best to capture the relaying benefit only for that small percentage of the devices that experience a \emph{weak} channel to the \glspl{ap}, while enjoying the high rate single-hop transmission to the devices with \emph{strong} channel, which in turn increases the overall spectral efficiency. Moreover, note that since by design the outage probability of the proposed \gls{andcoop} scheme is upper bounded by that of the two extremes studied in \propref{Prop:1h} and \propref{Prop:2h}, then from \eqref{Eq:dive_order_def} we readily have $d_\textit{single-hop}(r) \leq d_\textit{\gls{andcoop}}(r)$ and $d_\textit{two-hop}(r) \leq d_\textit{\gls{andcoop}}(r)$ for given $\alpha$ and $r$.
\begin{figure}[t]
\begin{center}
\psfrag{xlabel}[c][c][0.9]{multiplexing gain, $r$}
\psfrag{ylabel}[c][c][0.9]{diversity order, $d(r)$}
\psfrag{x1}[lc][lc][0.59]{\propref{Prop:andcoop}}
\psfrag{XXXXXXXXXx1}[lc][lc][0.59]{two-hop; up}
\psfrag{x2}[lc][lc][0.59]{two-hop; $\alpha = \frac{1}{2}$}
\psfrag{x3}[lc][lc][0.59]{single-hop; low}
\psfrag{x4}[lc][lc][0.59]{single-hop; up}
\psfrag{y1}[rc][rc][0.59]{$M$}
\psfrag{ye}[rc][rc][0.59]{$M+N-1$}
\psfrag{xe}[rc][rc][0.59]{$\frac{N}{N+1}$}
\if2 1
\includegraphics[width=.32\columnwidth,keepaspectratio]{div_mux_tradeoff.eps}
\else
\includegraphics[width=0.82\columnwidth,keepaspectratio]{div_mux_tradeoff.eps}
\fi
\caption{Diversity-multiplexing tradeoff.}
\label{fig:div_mux_tradeoff}
\end{center}
\if2 1
\vspace{-30pt}
\else
\fi
\end{figure}
\section{Numerical Results}
\label{Sec:Results}
This section presents numerical results from simulating \gls{dl} operation of the network described in \secref{Sec:ProblemSetup}, where $M$ single-antenna \glspl{ap} and $N$ single-antenna devices are randomly distributed across the factory area. To this end, we start by analyzing the performance with the assumption of \gls{pcsi}, to focus on system parameter designs. Then, we extend the analysis to the case of \gls{icsi}, and study the impact of \gls{csi} estimation error on the performance indicators.
The plots presented throughout this section are generated using a system-level simulation, where we adopt the parameters summarized in \tabref{Tab:Parameters} (except in cases where stated otherwise). Path loss exponents are determined using the factory and open-plan building channel model in \cite{rapp91}. The probability that a link is \gls{los} is derived based on the distance of the communicating nodes, $\nu$, using the following function.
\begin{align}
p_{\text{L}}(\nu) = a + \indic{\nu \leq b}\,\frac{1-a}{b^2}\,(\nu-b)^2,
\end{align}
where $a$ is a fixed probability mass, and $b$ is the cutoff above which the probability of a link being \gls{los} becomes fixed to $a$. The link is therefore \gls{nlos} with probability of $1-p_{\text{L}}(\nu)$. The cycle duration, number of devices, data per device and bandwidth are given values similar to those in \cite{swamy2015cow} and \cite{weiner2014design}.
With respect to the optimization of $\beta$, the following schemes are studied in this section.
\begin{itemize}
\item $\beta = \hat{\beta}$: This represents our proposed \gls{andcoop} scheme, where an optimal $\hat{\beta}$ portion of the total time is allocated for rate-adaptive single-hop transmission to a subset of the devices with the highest instantaneous channel quality. The remaining $1 - \hat{\beta}$ portion of the time is then used for two-hop transmission to the remaining devices.
\item $\beta = 1$, ideal rate adaptation: This transmission scheme mimics the typical transmission in cellular technologies, such as \gls{lte}, where, the transmission rate for each user is adapted to the instantaneous channel quality.
\item $\beta = 0$, two-hop transmission: This is a special case of our proposed scheme, where the total time resources is allocated to the two-hop transmission towards all the users. The scheme was originally proposed in \cite{swamy2015cow} and is known as the OccupyCoW protocol.
\end{itemize}
\input{table2.tex}
\if2 1
\vspace{-10pt}
\else
\fi
\subsection{Performance Analysis with Perfect \gls{csi}}
Let's start by assuming that the controller has perfect knowledge of the channel gains for all the \gls{ap}-device links. We emphasize that in the present work, the \gls{csi} is solely used for the purpose of transmission rate adaptation, meaning that the channel coefficient phase is not collected nor utilized. This differentiates our work from \gls{csi}-based distributed multi-antenna systems, such as in \cite{alonzo2020urllc}, which rely on coherent joint transmission and beam-forming at the transmitter. This further relaxes the assumption of tight synchronization among antennas for distributed cooperative transmission.
\if2 1
\begin{figure*}[t]
\centering
\begin{minipage}[b]{.47\textwidth}
\begin{center}
\psfrag{xlabel}[c][c][0.9]{spectral efficiency, $\eta$ (\gls{bpcu})}
\psfrag{ylabel}[c][c][0.9]{probability of system outage, $\mathbb{P}_\tr{\tiny{out}}$}
\psfrag{XXX1}[lc][lc][0.59]{$\beta = \hat{\beta}$}
\psfrag{x2}[lc][lc][0.59]{$\beta = 0$}
\psfrag{x3}[lc][lc][0.59]{$\beta = 1$}
\psfrag{XXX4}[lc][lc][0.59]{$M = 1$}
\psfrag{x5}[lc][lc][0.59]{$M = 2$}
\psfrag{x6}[lc][lc][0.59]{$M = 3$}
\includegraphics[width=0.95\columnwidth,keepaspectratio]{comp_id_rate.eps}
\caption{System outage probability against spectral efficiency at $P_a = P_d = 23$ dBm.}
\label{fig:comp_id_rate}
\end{center}
\end{minipage}\qquad
\begin{minipage}[b]{.47\textwidth}
\begin{center}
\psfrag{xlabel}[c][c][0.9]{number of weak devices, $K_{2\text{h}}$}
\psfrag{ylabel}[c][c][0.9]{cumulative distribution function}
\psfrag{XXX1}[lc][lc][0.59]{$M = 1$}
\psfrag{x2}[lc][lc][0.59]{$M = 2$}
\psfrag{x3}[lc][lc][0.59]{$M = 3$}
\includegraphics[width=0.95\columnwidth,keepaspectratio]{weak_node_stats_id.eps}
\caption{Statistics of $K_{2\text{h}}$
with $N = 50$, $P_a = P_d = 23$ dBm at $\mathbb{P}_\tr{\tiny{out}} = 10^{-5}$ points of \figref{fig:comp_id_rate}.}\label{fig:weak_node_stats_id}
\end{center}
\end{minipage}
\if2 1
\vspace{-35pt}
\else
\fi
\end{figure*}
\fi
\if2 2
\begin{figure}[t]
\begin{center}
\psfrag{xlabel}[c][c][0.9]{spectral efficiency, $\eta$ (\gls{bpcu})}
\psfrag{ylabel}[c][c][0.9]{probability of system outage, $\mathbb{P}_\tr{\tiny{out}}$}
\psfrag{XXX1}[lc][lc][0.59]{$\beta = \hat{\beta}$}
\psfrag{x2}[lc][lc][0.59]{$\beta = 0$}
\psfrag{x3}[lc][lc][0.59]{$\beta = 1$}
\psfrag{XXX4}[lc][lc][0.59]{$M = 1$}
\psfrag{x5}[lc][lc][0.59]{$M = 2$}
\psfrag{x6}[lc][lc][0.59]{$M = 3$}
\includegraphics[width=0.95\columnwidth,keepaspectratio]{comp_id_rate.eps}
\caption{System outage probability against spectral efficiency at $P_a = P_d = 23$ dBm.}
\label{fig:comp_id_rate}
\end{center}
\if2 1
\vspace{-35pt}
\else
\fi
\end{figure}
\fi
\paragraph*{Proposed scheme improves spectral efficiency} In \figref{fig:comp_id_rate}, system outage probability is shown against spectral efficiency $\eta$, derived as $\eta = \frac{N B}{T W}$. The proposed \gls{andcoop} scheme with optimized time division of $\beta = \hat{\beta}$, improves spectral efficiency by at least 0.5 \gls{bpcu} when $M = 1$ \gls{ap} is deployed. The gain is higher when a larger number of \glspl{ap} are deployed. Namely, with $M = 2$ and $M = 3$, more than 1 and 1.5 \gls{bpcu} increase in spectral efficiency is achieved with respect to the case of $\beta = 0$, i.e., when only two-hop cooperative transmission is deployed.
\if2 2
\begin{figure}[t]
\begin{center}
\psfrag{xlabel}[c][c][0.9]{number of weak devices, $K_{2\text{h}}$}
\psfrag{ylabel}[c][c][0.9]{cumulative distribution function}
\psfrag{XXX1}[lc][lc][0.59]{$M = 1$}
\psfrag{x2}[lc][lc][0.59]{$M = 2$}
\psfrag{x3}[lc][lc][0.59]{$M = 3$}
\includegraphics[width=0.95\columnwidth,keepaspectratio]{weak_node_stats_id.eps}
\caption{Statistics of $K_{2\text{h}}$
with $N = 50$, $P_a = P_d = 23$ dBm at $\mathbb{P}_\tr{\tiny{out}} = 10^{-5}$ points of \figref{fig:comp_id_rate}.}
\label{fig:weak_node_stats_id}
\end{center}
\if2 1
\vspace{-35pt}
\else
\fi
\end{figure}
\fi
The gain in spectral efficiency is thanks to transmitting packets to \emph{strong} devices with high rate in the single-hop phase, allowing the two-hop phase to accommodate the \emph{weak} devices reliably at even large packet sizes. This way, the robustness of rate-adaptation to increase in load is combined together with the robustness of OccupyCoW to fading, providing an improved reliability at a higher spectral efficiency. \figref{fig:weak_node_stats_id} shows the \gls{cdf} of the number of weak devices scheduled with two-hop transmission by the proposed \gls{andcoop} scheme with optimized $\beta$. The statistics are collected for the points in \figref{fig:comp_id_rate} where the proposed \gls{andcoop} achieves $\mathbb{P}_\tr{\tiny{out}} = 10^{-5}$. Interestingly, the average number of users scheduled with two-hop transmission is just below 13, 7 and 3 respectively for the case of 1, 2 and 3 \glspl{ap}.
\if2 2
\begin{figure}[t]
\begin{center}
\psfrag{xlabel}[c][c][0.9]{transmit power (dBm)}
\psfrag{ylabel}[c][c][0.9]{probability of system outage, $\mathbb{P}_\tr{\tiny{out}}$}
\psfrag{XXX1}[lc][lc][0.59]{$\beta = \hat{\beta}$}
\psfrag{x2}[lc][lc][0.59]{$\beta = 0$}
\psfrag{x3}[lc][lc][0.59]{$\beta = 1$}
\psfrag{XXX4}[lc][lc][0.59]{$M = 1$}
\psfrag{x5}[lc][lc][0.59]{$M = 2$}
\psfrag{x6}[lc][lc][0.59]{$M = 3$}
\includegraphics[width=0.95\columnwidth,keepaspectratio]{comp_id_power.eps}
\caption{System outage probability against transmit power, $P_a = P_d$, for $B = 50$ bytes per device.}
\label{fig:comp_id_power}
\end{center}
\if2 1
\vspace{-35pt}
\else
\fi
\end{figure}
\fi
\paragraph*{Ultra-reliability at lower transmit power} By fully exploiting the diversity gain in the network, the proposed \gls{andcoop} scheme relaxes the need for high \gls{snr} to achieve ultra-reliability. As it is shown in \figref{fig:comp_id_power}, system outage probability of $\mathbb{P}_\tr{\tiny{out}} = 10^{-5}$, the required transmit power of the proposed scheme reduces by a few dB compared to the OccupyCoW protocol. Such transmit power gap, when compared against the case of single-hop with ideal rate adaptation, can grow to tens of dB. Aside from improving the overall energy efficiency of the system, operating at a lower transmit power can also reduce the interference generated by the cell towards neighbouring cells, in case of a multi-cell operation, as also studied in \cite{Arvin:2019}. Moreover, the proposed \gls{andcoop} can naturally reduce the average relaying time per relay device, by reducing the overall duration of the relaying phase. The combined effect is a significant reduction in the average consumed energy in the relaying phase across devices, as depicted in \figref{fig:relay_power_consumption_stats_id_jules}. The curves suggest $30\%$ to $40\%$ reduction in average relaying energy consumption in all cases.
\if2 2
\begin{figure}[t]
\begin{center}
\psfrag{xlabel}[c][c][0.9]{average transmit energy per device ($\mu$J)}
\psfrag{ylabel}[c][c][0.9]{cumulative distribution function}
\psfrag{XXX1}[lc][lc][0.59]{$\beta = \hat{\beta}$}
\psfrag{x3}[lc][lc][0.59]{$\beta = 0$}
\psfrag{XXX4}[lc][lc][0.59]{$M = 1$}
\psfrag{x5}[lc][lc][0.59]{$M = 2$}
\psfrag{x6}[lc][lc][0.59]{$M = 3$}
\includegraphics[width=0.95\columnwidth,keepaspectratio]{relay_power_consumption_stats_id_jules.eps}
\caption{Statistics of average consumed energy per device for relaying with $N = 50$, $B = 50$ bytes at $\mathbb{P}_\tr{\tiny{out}} = 10^{-5}$ points of \figref{fig:comp_id_power}.}
\label{fig:relay_power_consumption_stats_id_jules}
\end{center}
\if2 1
\vspace{-35pt}
\else
\fi
\end{figure}
\fi
\if2 1
\begin{figure*}
\centering
\begin{minipage}[b]{.47\textwidth}
\begin{center}
\psfrag{xlabel}[c][c][0.9]{transmit power (dBm)}
\psfrag{ylabel}[c][c][0.9]{probability of system outage, $\mathbb{P}_\tr{\tiny{out}}$}
\psfrag{XXX1}[lc][lc][0.59]{$\beta = \hat{\beta}$}
\psfrag{x2}[lc][lc][0.59]{$\beta = 0$}
\psfrag{x3}[lc][lc][0.59]{$\beta = 1$}
\psfrag{XXX4}[lc][lc][0.59]{$M = 1$}
\psfrag{x5}[lc][lc][0.59]{$M = 2$}
\psfrag{x6}[lc][lc][0.59]{$M = 3$}
\includegraphics[width=0.95\columnwidth,keepaspectratio]{comp_id_power.eps}
\caption{System outage probability against transmit power, $P_a = P_d$, for $B = 50$ bytes per device.}
\label{fig:comp_id_power}
\end{center}
\end{minipage}\qquad
\begin{minipage}[b]{.47\textwidth}
\begin{center}
\psfrag{xlabel}[c][c][0.9]{average transmit energy per device ($\mu$J)}
\psfrag{ylabel}[c][c][0.9]{cumulative distribution function}
\psfrag{XXX1}[lc][lc][0.59]{$\beta = \hat{\beta}$}
\psfrag{x3}[lc][lc][0.59]{$\beta = 0$}
\psfrag{XXX4}[lc][lc][0.59]{$M = 1$}
\psfrag{x5}[lc][lc][0.59]{$M = 2$}
\psfrag{x6}[lc][lc][0.59]{$M = 3$}
\includegraphics[width=0.95\columnwidth,keepaspectratio]{relay_power_consumption_stats_id_jules.eps}
\caption{Statistics of average consumed energy per device for relaying with $N = 50$, $B = 50$ bytes at $\mathbb{P}_\tr{\tiny{out}} = 10^{-5}$ points of \figref{fig:comp_id_power}.}
\label{fig:relay_power_consumption_stats_id_jules}
\end{center}
\end{minipage}
\if2 1
\vspace{-35pt}
\else
\fi
\end{figure*}
\fi
\if2 2
\begin{figure}[t]
\begin{center}
\psfrag{xlabel}[c][c][0.9]{probability of system outage, $\mathbb{P}_\tr{\tiny{out}}$}
\psfrag{ylabel}[c][c][0.9]{empirical outage exponent}
\psfrag{XXX1}[lc][lc][0.59]{$\beta = \hat{\beta}$}
\psfrag{x2}[lc][lc][0.59]{$\beta = 0$}
\psfrag{x3}[lc][lc][0.59]{$\beta = 1$}
\psfrag{XXX4}[lc][lc][0.59]{$M = 1$}
\psfrag{x5}[lc][lc][0.59]{$M = 2$}
\psfrag{x6}[lc][lc][0.59]{$M = 3$}
\includegraphics[width=0.95\columnwidth,keepaspectratio]{div_order_snr.eps}
\caption{Empirical diversity order in finite \gls{snr} against system outage probability for $B = 50$ bytes per device.}
\label{fig:div_order}
\end{center}
\if2 1
\vspace{-35pt}
\else
\fi
\end{figure}
\fi
\paragraph*{Quick reach to the maximum diversity gain} We study the empirical outage exponent of the proposed transmission scheme in \figref{fig:div_order}. For that purpose, we adopt a simulation setup where all links exhibit a single nominal average \gls{snr} value, i.e., removing the effect of large-scale fading in channel gain. The system outage probability is then simulated across a finite range of link \gls{snr}. As depicted in \figref{fig:div_order}, with optimal $\beta = \hat{\beta}$, the proposed protocol reaches quickly to the maximum achievable diversity order of $M+N-1$ at $M = 3$, confirming the derivation in \eqref{Eq:div_orde_2hop}. This is thanks to collecting the multi-user diversity gain at its best, by rate-adaptive scheduling of the devices with strong channel condition, while exploiting the spatial diversity gain from cooperative relaying towards the devices with poor channel condition. Interestingly, with $\beta = 1$, the gain from multi-user diversity can initially increase the outage exponent. By increasing \gls{snr}, where all devices will naturally be scheduled with the same transmission rate, the diversity order approaches to $M$, as was also suggested by the derivation in \eqref{Eq:div_order_1hop}.
\if2 2
\begin{figure}[t]
\begin{center}
\psfrag{ylabel}[c][c][0.9]{probability of system outage, $\mathbb{P}_\tr{\tiny{out}}$}
\psfrag{xlabel}[c][c][0.9]{number of devices, $N$}
\psfrag{XXX1}[lc][lc][0.59]{$\beta = \hat{\beta}$}
\psfrag{x2}[lc][lc][0.59]{$\beta = 0$}
\psfrag{x3}[lc][lc][0.59]{$\beta = 1$}
\psfrag{XXX4}[lc][lc][0.59]{$M = 1$}
\psfrag{x5}[lc][lc][0.59]{$M = 3$}
\includegraphics[width=0.95\columnwidth,keepaspectratio]{comp_id_outage_population_pwr_14.eps}
\caption{System outage probability against number of devices, $N$, for $P_a = P_d = 14$ dBm and $B = 50$ bytes per device.}
\label{fig:comp_id_outage_population_pwr_14}
\end{center}
\if2 1
\vspace{-35pt}
\else
\fi
\end{figure}
\fi
\if2 1
\begin{figure*}
\centering
\begin{minipage}[b]{.47\textwidth}
\begin{center}
\psfrag{xlabel}[c][c][0.9]{probability of system outage, $\mathbb{P}_\tr{\tiny{out}}$}
\psfrag{ylabel}[c][c][0.9]{empirical outage exponent}
\psfrag{XXX1}[lc][lc][0.59]{$\beta = \hat{\beta}$}
\psfrag{x2}[lc][lc][0.59]{$\beta = 0$}
\psfrag{x3}[lc][lc][0.59]{$\beta = 1$}
\psfrag{XXX4}[lc][lc][0.59]{$M = 1$}
\psfrag{x5}[lc][lc][0.59]{$M = 2$}
\psfrag{x6}[lc][lc][0.59]{$M = 3$}
\includegraphics[width=0.95\columnwidth,keepaspectratio]{div_order_snr.eps}
\caption{Empirical diversity order in finite \gls{snr} against system outage probability for $B = 50$ bytes per device.}
\label{fig:div_order}
\end{center}
\end{minipage}\qquad
\begin{minipage}[b]{.47\textwidth}
\begin{center}
\psfrag{ylabel}[c][c][0.9]{probability of system outage, $\mathbb{P}_\tr{\tiny{out}}$}
\psfrag{xlabel}[c][c][0.9]{number of devices, $N$}
\psfrag{XXX1}[lc][lc][0.59]{$\beta = \hat{\beta}$}
\psfrag{x2}[lc][lc][0.59]{$\beta = 0$}
\psfrag{x3}[lc][lc][0.59]{$\beta = 1$}
\psfrag{XXX4}[lc][lc][0.59]{$M = 1$}
\psfrag{x5}[lc][lc][0.59]{$M = 3$}
\includegraphics[width=0.95\columnwidth,keepaspectratio]{comp_id_outage_population_pwr_14.eps}
\caption{System outage probability against number of devices, $N$, for $P_a = P_d = 14$ dBm and $B = 50$ bytes per device.}
\label{fig:comp_id_outage_population_pwr_14}
\end{center}
\end{minipage}
\if2 1
\vspace{-35pt}
\else
\fi
\end{figure*}
\fi
\paragraph*{Improved scalability with network size} In practical industrial networks, the number of devices connected to the same controller can become largely dynamic. Therefore, it is crucial for the transmission protocol to be able to scale with the network size, up or down, without depriving it of reliability. To that end, in \figref{fig:comp_id_outage_population_pwr_14} the system outage probability of the three transmission schemes are tested against a range of network sizes, i.e., the number of devices in the network, $N$. We fixed the packet size for each device to $B = 50$ bytes to imitate the realistic conditions.
The system outage probability for the case of single-hop transmission with ideal rate adaptation increases, almost linearly, by increasing the network size. This is an expected outcome since the system is unable to gain from the increase in number of devices (maximum diversity order or $M$, as proposed in \eqref{Eq:div_order_1hop}). Thus, increasing the number of devices, merely translates into a higher likelihood of scheduling time-overflow.
On the contrary, the proposed \gls{andcoop} protocol and the OccupyCoW protocol can benefit from the increase in network device. In fact, by increasing the network size, the potential cooperative diversity gain also increases, which in turn reduces the system outage probability. Meanwhile, by increasing the network size, the average transmission rate increases too, which has an opposite effect on outage probability. Therefore, for those two schemes, we observe a turning point for system outage probability. Overall, the proposed \gls{andcoop} can guarantee $\mathbb{P}_\tr{\tiny{out}} \leq 10^{-5}$ for $N \leq 180$ with $M = 3$ \glspl{ap} in this example. For the OccupyCoW scheme this reduces to only $2<N\leq 120$. It should be noted that at $N = 1$, the proposed scheme is equivalent to single-hop transmission since the total resources are allotted to the single device. Moreover, the overall scheduling overhead also increases with the number of devices increasing. However, since a fixed packet size per device is assumed in this analysis, by introducing a fixed scheduling overhead per device, the trend of the curves in \figref{fig:comp_id_outage_population_pwr_14} and the above conclusions will remain intact.
\if2 1
\vspace{-10pt}
\else
\fi
\subsection{Effect of Imperfect \gls{csi}}
As discussed earlier in \secref{Sec:transmissionprotocol}, due to the inevitable \gls{csi} estimation error, the transmitter must use a back-off parameter $0 < \theta \leq 1$ to adjust the transmission rate that is adapted to the \gls{icsi}.
Moreover, the effect of \gls{icsi} is only on the single-hop rate-adaptive transmission phase. The two-hop phase, thanks to the fixed-rate transmission, encounters no impact from the \gls{icsi}.
Numerical optimization in presence of \gls{csi} error is a demanding task which requires exhaustive search for the optimal operating point across three parameters $L$, $\beta$ and $\theta$. Therefore, it is crucial to understand the impact that each of those parameters have on the performance in order to reduce the optimization complexity.
\if2 1
\begin{figure*}
\centering
\begin{minipage}[b]{.47\textwidth}
\begin{center}
\psfrag{xlabel}[c][c][0.9]{number of pilot symbols per device, $L$}
\psfrag{ylabel}[c][c][0.9]{system outage probability, $\mathbb{P}_\tr{\tiny{out}}$}
\psfrag{XXXXx1}[lc][lc][0.59]{best case}
\includegraphics[width=0.95\columnwidth,keepaspectratio]{T1_fixed_tradeoff.eps}
\caption{System outage probability of the proposed \gls{andcoop} for different $\theta$ values. We fixed $P_a = P_d = 14$ dBm, $M = 1$, $N = 50$, $B = 50$ bytes and $\beta = 0.1$.}
\label{fig:T1_tradeoff}
\end{center}
\end{minipage}\qquad
\begin{minipage}[b]{.47\textwidth}
\begin{center}
\psfrag{xlabel}[c][c][0.9]{number of pilot symbols per device, $L$}
\psfrag{ylabel}[c][c][0.9]{system outage probability, $\mathbb{P}_\tr{\tiny{out}}$}
\psfrag{XXXXx1}[lc][lc][0.59]{best case}
\includegraphics[width=0.95\columnwidth,keepaspectratio]{Rbo_fixed_tradeoff.eps}
\caption{System outage probability of the proposed \gls{andcoop} for different $\beta$ values. We fixed $P_a = P_d = 14$ dBm, $M = 1$, $N = 50$, $B = 50$ bytes and $\theta = 0.6$.}
\label{fig:Rbo_tradeoff}
\end{center}
\end{minipage}
\if2 1
\vspace{-35pt}
\else
\fi
\end{figure*}
\fi
\if2 2
\begin{figure}[t]
\begin{center}
\psfrag{xlabel}[c][c][0.9]{number of pilot symbols per device, $L$}
\psfrag{ylabel}[c][c][0.9]{system outage probability, $\mathbb{P}_\tr{\tiny{out}}$}
\psfrag{XXXXx1}[lc][lc][0.59]{best case}
\includegraphics[width=0.95\columnwidth,keepaspectratio]{T1_fixed_tradeoff.eps}
\caption{System outage probability of the proposed \gls{andcoop} for different $\theta$ values. We fixed $P_a = P_d = 14$ dBm, $M = 1$, $N = 50$, $B = 50$ bytes and $\beta = 0.1$.}
\label{fig:T1_tradeoff}
\end{center}
\if2 1
\vspace{-35pt}
\else
\fi
\end{figure}
\begin{figure}[t]
\begin{center}
\psfrag{xlabel}[c][c][0.9]{number of pilot symbols per device, $L$}
\psfrag{ylabel}[c][c][0.9]{system outage probability, $\mathbb{P}_\tr{\tiny{out}}$}
\psfrag{XXXXx1}[lc][lc][0.59]{best case}
\includegraphics[width=0.95\columnwidth,keepaspectratio]{Rbo_fixed_tradeoff.eps}
\caption{System outage probability of the proposed \gls{andcoop} for different $\beta$ values. We fixed $P_a = Pd = 14$ dBm, $M = 1$, $N = 50$, $B = 50$ bytes and $\theta = 0.6$.}
\label{fig:Rbo_tradeoff}
\end{center}
\if2 1
\vspace{-35pt}
\else
\fi
\end{figure}
\fi
\paragraph*{Fixing a small number of pilot symbols per device} To reduce the complexity of the exhaustive numerical optimization we restrict each of the three parameters $L$, $\beta$ and $\theta$, to a finite set of values. Then, for each triple, we simulate the system outage probability. The curves depicted against parameter $L$ in \figref{fig:T1_tradeoff}, show the course of system outage probability across different values of $\theta$ when we fixed $\beta = 0.1$. Similarly, in \figref{fig:Rbo_tradeoff}, system outage probability is depicted for different $\beta$ values while fixing $\theta = 0.6$. The best case system outage probability in both those figures, represents the minimum outage probability that is attainable at a given $L$ while optimizing against $\beta$ and $\theta$. The following observations are given.
\begin{itemize}
\item By increasing $L$, channel estimation error decreases, which in turn lowers the impact of $\theta$ on system outage probability for fixed $\beta$.
\item By optimization across different values of $\beta$ and $\theta$, it becomes evident that system outage probability is within a small margin of the optimal value, across a wide range of $L$. In this example, choosing $2 \leq L \leq 30$, system outage probability remains roughly unchanged.
\end{itemize}
From those observations, to simplify the optimization process we propose to fix $L = 10$ and optimize only across $\beta$ and $\theta$.
\if2 2
\begin{figure}[t]
\begin{center}
\psfrag{ylabel}[c][c][0.9]{probability of system outage, $\mathbb{P}_\tr{\tiny{out}}$}
\psfrag{xlabel}[c][c][0.9]{transmit power (dBm)}
\psfrag{XXX1}[lc][lc][0.59]{$\beta = \hat{\beta}$}
\psfrag{x2}[lc][lc][0.59]{$\beta = 0$}
\psfrag{x3}[lc][lc][0.59]{\gls{icsi}}
\psfrag{XXX4}[lc][lc][0.59]{\gls{pcsi}}
\psfrag{XXX5}[lc][lc][0.59]{$M = 1$}
\psfrag{x6}[lc][lc][0.59]{$M = 3$}
\includegraphics[width=0.95\columnwidth,keepaspectratio]{comp_power_outage_reVSid.eps}
\caption{Comparison of outage probability between \gls{icsi} and \gls{pcsi} for $N = 50$, $B = 50$ bytes per device and $L = 10$.}
\label{fig:comp_power_outage_reVSid}
\end{center}
\if2 1
\vspace{-35pt}
\else
\fi
\end{figure}
\fi
\if2 1
\begin{figure*}
\centering
\begin{minipage}[b]{.47\textwidth}
\begin{center}
\psfrag{ylabel}[c][c][0.9]{probability of system outage, $\mathbb{P}_\tr{\tiny{out}}$}
\psfrag{xlabel}[c][c][0.9]{transmit power (dBm)}
\psfrag{XXX1}[lc][lc][0.59]{$\beta = \hat{\beta}$}
\psfrag{x2}[lc][lc][0.59]{$\beta = 0$}
\psfrag{x3}[lc][lc][0.59]{\gls{icsi}}
\psfrag{XXX4}[lc][lc][0.59]{\gls{pcsi}}
\psfrag{XXX5}[lc][lc][0.59]{$M = 1$}
\psfrag{x6}[lc][lc][0.59]{$M = 3$}
\includegraphics[width=0.95\columnwidth,keepaspectratio]{comp_power_outage_reVSid.eps}
\caption{Comparison of outage probability between \gls{icsi} and \gls{pcsi} for $N = 50$, $B = 50$ bytes per device and $L = 10$.}
\label{fig:comp_power_outage_reVSid}
\end{center}
\end{minipage}\qquad
\begin{minipage}[b]{.47\textwidth}
\begin{center}
\psfrag{xlabel}[c][c][0.9]{system outage probability, $\mathbb{P}_\tr{\tiny{out}}$}
\psfrag{ylabel}[c][c][0.9]{}
\psfrag{X1}[lc][lc][0.59]{$\hat{\beta}$}
\psfrag{x2}[lc][lc][0.59]{$\hat{\theta}$}
\psfrag{XXX4}[lc][lc][0.59]{$M = 1$}
\psfrag{x6}[lc][lc][0.59]{$M = 3$}
\includegraphics[width=0.95\columnwidth,keepaspectratio]{opt_beta_rbo_re.eps}
\caption{The optimal values $\hat{\beta}$ and $\hat{\theta}$ for \gls{icsi}, where $N = 50$, $B = 50$ bytes per device and $L = 10$.}
\label{fig:opt_beta_rbo_re}
\end{center}
\end{minipage}
\if2 1
\vspace{-35pt}
\else
\fi
\end{figure*}
\fi
\paragraph*{Impact of channel estimation error is marginal} For fixed number of pilot symbols per device $L = 10$, we examine the performance degradation of the proposed \gls{andcoop} from \gls{icsi} with respect to the case of \gls{pcsi} under similar setup as in \figref{fig:comp_id_power}. As shown, for both cases of $M = 1$ and $M = 3$, in presence of \gls{icsi} the proposed scheme with $\beta = \hat{\beta}$ operates within a small 1-2 dB gap from the case of \gls{pcsi}. On the other hand, the gap for the case of single-hop rate-adaptive transmission can become very large, e.g., $~15$ dB for the case of $M = 3$. Such large \gls{snr} gap owes to high channel estimation error of the devices with poor channel quality during rate adaptation for single-hop transmission. Our proposed \gls{andcoop} circumvents that by identifying those devices and scheduling them over fixed-rate two-hop transmission.
\if2 2
\begin{figure}[h]
\begin{center}
\psfrag{xlabel}[c][c][0.9]{system outage probability, $\mathbb{P}_\tr{\tiny{out}}$}
\psfrag{ylabel}[c][c][0.9]{}
\psfrag{X1}[lc][lc][0.59]{$\hat{\beta}$}
\psfrag{x2}[lc][lc][0.59]{$\hat{\theta}$}
\psfrag{XXX4}[lc][lc][0.59]{$M = 1$}
\psfrag{x6}[lc][lc][0.59]{$M = 3$}
\includegraphics[width=0.95\columnwidth,keepaspectratio]{opt_beta_rbo_re.eps}
\caption{The optimal values $\hat{\beta}$ and $\hat{\theta}$ for \gls{icsi}, where $N = 50$, $B = 50$ bytes per device and $L = 10$.}
\label{fig:opt_beta_rbo_re}
\end{center}
\if2 1
\vspace{-35pt}
\else
\fi
\end{figure}
\fi
\paragraph*{Optimal value for $\beta$ and $\theta$ for fixed $L$} In \figref{fig:opt_beta_rbo_re}, the optimal values $\hat{\beta}$ and $\hat{\theta}$ are reported for fixed $L = 10$. Those values were used in \figref{fig:comp_power_outage_reVSid} for the proposed \gls{andcoop} in case of \gls{icsi}. It is evident that with a larger number of \gls{ap} antennas, on average a larger number of devices are scheduled with single-hop transmission (i.e., larger $\hat{\beta}$). Moreover, the optimal rate adjustment factor $\hat{\theta}$ is smaller, when the number of \gls{ap} antennas is smaller. As could be expected, both $\hat{\beta}$ and $\hat{\theta}$ have a non-increasing trend with system outage probability decreasing. This parallels our design analogy for ultra-reliable communication, where only the devices with strong channel conditions should undergo rate-adaptive transmission (i.e., smaller $\hat{\beta}$), and for those, a more conservative transmission rate adjustment factor is necessary (i.e., smaller $\hat{\theta}$).
\section{Conclusion}
\label{Sec:Conclusion}
We proposed a channel-aware \gls{urllc} transmission protocol for industrial wireless control where a controller communicates with several devices in the \gls{dl}. The proposed transmission protocol uses the knowledge of instantaneous channel conditions of \gls{ap}-device links to identify devices with \emph{strong} and \emph{weak} channel conditions. The strong devices are served with a single-hop rate-adaptive transmission where the rate is adapted to the instantaneous channel. With this approach, multi-user diversity gain is exploited and frequency resources are efficiently utilized. Meanwhile, the weak devices enjoy a two-hop cooperative communication in which transmission rate is fixed and all the nodes in the network cooperate in relaying. We analyzed the system outage probability and the diversity order under the proposed transmission protocol. Through numerical analysis we derived optimal time division between the two sets of \emph{strong} and \emph{weak} devices and showed that such optimization can improve spectral efficiency by more than 0.5, 1 and 1.5 \gls{bpcu}, when the controller is equipped with 1, 2 and 3 \gls{ap} antennas, respectively. Moreover, we observed that the proposed \gls{andcoop} transmission protocol can effectively reduce the required transmit power for reliable industrial wireless control, improve the scalability with respect to network size, and marginalize the impact of channel estimation error on outage probability. The improvements are thanks to the instantaneous awareness to channel conditions that allows to exploit different sources of diversity gain in the network. As the continuation of this work in future, we will look into evaluating the performance of the proposed transmission protocol considering spatio-temporal correlation of shadowing caused by blockages. Further, we will study the use of dedicated full-duplex relay nodes in the proposed transmission protocol.
\input{app.tex}
\bibliographystyle{IEEEtran}
\section{}
\subsection{Proof of Proposition 1}
\label{App:motiv}
The achievable rate $\aR_{j}$ for device $j$ is given as
\if2 1
\fontsize{9}{11}\selectfont
\begin{align}
\aR_{j} = W \log \left( 1 + \sum_{i \in \mathscr{A}} \ga{i}{j} \right),
\end{align}
\normalsize
\else
\begin{align}
\aR_{j} = W \log \left( 1 + \sum_{i \in \mathscr{A}} \ga{i}{j} \right),
\end{align}
\fi
measured in \gls{bps}. Then, the random time duration $\tau_j$, in seconds, required for successful transmission to device $j$ is equal to $\tau_j = \frac{B'}{\aR_{j}}$, where $B' = \frac{N}{K}B$ is the adjusted packet size per scheduled device for a given $K$. Note that $\aR_{j}$'s are \gls{iid} random variables which result in \gls{iid} $\tau_j$'s. The transmission from source to $K$ arbitrarily chosen devices is then \emph{successful} if the sum of $\tau_j$'s for those $K$ devices is not larger than $T$. Without loss of generality, let's assume $\tau_1 \leq \tau_2 \leq \ldots \leq \tau_N$, meaning that $\aR_1 \geq \aR_2 \geq \ldots \geq \aR_N$, where $\aR_j$'s form \emph{order statistics} drawn from \gls{cdf} $\cdf_{\aR}$.
For the sake of better reliability (i.e., maximum diversity gain), the scheduler chooses the $K$ devices with best channels to transmit to, where $K \in \{1, 2, \ldots, N\}$.
Thus, the probability of transmission error is equivalent to the probability of \emph{time overflow} given as
\if2 1
\fontsize{9}{11}\selectfont
\begin{align}
\mathbb{P}_\tr{\tiny{out}} = \prob{ \sum_{j = 1}^{K} \tau_j \geq T} = \prob{ \sum_{j = 1}^{K} \frac{1}{\aR_{j}} \geq \frac{K T}{N B}}.
\label{Eq:Pout-k-best}
\end{align}
\normalsize
\else
\begin{align}\nonumber
\mathbb{P}_\tr{\tiny{out}} & = \prob{ \sum_{j = 1}^{K} \tau_j \geq T} \\
& = \prob{ \sum_{j = 1}^{K} \frac{1}{\aR_{j}} \geq \frac{K T}{N B}}.
\label{Eq:Pout-k-best}
\end{align}
\fi
The diversity gain $d$ from \eqref{Eq:dive_order_def} can be derived for the case of \gls{iid} Rayleigh fading for all links and fixed average \gls{snr} of $\rho$ over all links, as follows. First, note that
\if2 1
\fontsize{9}{11}\selectfont
\begin{align}
\prob{ \frac{1}{\aR_{j}} \geq \frac{1}{R}} \overset{(a)}{=} \sum_{k = N - j + 1}^{N} \binom{N}{k} \cdf_{\aR}(R)^{k} \left(1 - \cdf_{\aR}(R) \right)^{N-k}
\overset{(b)}{=} \sum_{k = N - j + 1}^{N} \binom{N}{k} \Pm{M,R}^{k} \left(1 - \Pm{M,R} \right)^{N-k},
\end{align}
\normalsize
\else
\begin{align}
\prob{ \frac{1}{\aR_{j}} \geq \frac{1}{R}} & \overset{(a)}{=} \sum_{k = N - j + 1}^{N} \binom{N}{k} \cdf_{\aR}(R)^{k} \left(1 - \cdf_{\aR}(R) \right)^{N-k} \\ \nonumber
& \overset{(b)}{=} \sum_{k = N - j + 1}^{N} \binom{N}{k} \Pm{M,R}^{k} \left(1 - \Pm{M,R} \right)^{N-k},
\end{align}
\fi
where $(a)$ follows from the \gls{cdf} of the $N - j + 1$th order statistics \cite[Chapter~6]{Mittelhammer99} and $(b)$ follows from \eqref{Eq:pfail}. Using the approximations $\Pm{m,R} \approx (\frac{\omega}{P_t})^m$ and $\left( 1 - \Pm{m,R} \right) \approx 1$ for $\frac{\omega}{P_t} \rightarrow 0$ \cite{Laneman:2003}, the following holds for any bounded real value $R$.
\if2 1
\fontsize{9}{11}\selectfont
\begin{align}
d_j = - \lim_{P_t \rightarrow \infty} \frac{\log \prob{ \frac{1}{\aR_{j}} \geq \frac{1}{R}}}{\log P_t} = - \lim_{P_t \rightarrow \infty} \frac{\log \sum_{k = N - j + 1}^{N} \binom{N}{k} (\frac{\omega}{P_t})^{k \cdot M} } {\log P_t} = M (N - j + 1)
\label{Eq:div-best-j}
\end{align}
\normalsize
\else
\begin{align}\nonumber
d_j & = - \lim_{P_t \rightarrow \infty} \frac{\log \prob{ \frac{1}{\aR_{j}} \geq \frac{1}{R}}}{\log P_t} \\ \nonumber
& = - \lim_{P_t \rightarrow \infty} \frac{\log \sum_{k = N - j + 1}^{N} \binom{N}{k} (\frac{\omega}{P_t})^{k \cdot M} } {\log P_t} \\
& = M (N - j + 1)
\label{Eq:div-best-j}
\end{align}
\fi
For the case where $K = 1$ device with the best channels is transmitted to, the diversity gain follows from \eqref{Eq:div-best-j} as
\begin{align}
d = M N.
\end{align}
For $K > 1$, the maximum diversity gain follows from the choice of $K$ devices with best channels where the probability of outage $\mathbb{P}_\tr{\tiny{out}}$ from \eqref{Eq:Pout-k-best} is bounded as follows
\if2 1
\fontsize{9}{11}\selectfont
\begin{align}\nonumber
\prob{ \frac{1}{\aR_{K}} \geq \frac{K T}{N B}} \leq \prob{ \sum_{j = 1}^{K} \frac{1}{\aR_{j}} \geq \frac{K T}{N B}} \leq \prob{ \frac{1}{\aR_{K}} \geq \frac{T}{N B}}.
\end{align}
\normalsize
\else
\begin{align}\nonumber
\prob{ \frac{1}{\aR_{K}} \geq \frac{K T}{N B}} \leq \prob{ \sum_{j = 1}^{K} \frac{1}{\aR_{j}} \geq \frac{K T}{N B}} \leq \prob{ \frac{1}{\aR_{K}} \geq \frac{T}{N B}}.
\end{align}
\fi
Therefore, the diversity order of $\mathbb{P}_\tr{\tiny{out}}$ is bounded on both sides by $M(N - K + 1)$ according to \eqref{Eq:div-best-j}, which concludes
\if2 1
\fontsize{9}{11}\selectfont
\begin{align}
d = M (N - K + 1).
\end{align}
\normalsize
\else
\begin{align}
d = M (N - K + 1).
\end{align}
\fi
\if2 1
\vspace{-15pt}
\fi
\if2 1
\def \scaletemp {0.55}
\else
\def \scaletemp {1}
\fi
\subsection{Proof of Proposition 2}
\label{App:A}
Although a closed form of outage probability in \eqref{Eq:outage-1hop} is not available, the outage probability $\mathbb{P}_\tr{\tiny{out}} = P_{1h}(\Toh)$ is bounded as follows.
\if2 1
\fontsize{9}{11}\selectfont
\begin{align}\label{Eq:bounds}
\resizebox{\scaletemp\columnwidth}{!}{$1 - \left(1 - \Pm{M,\frac{B}{\Toh}}\right)^N \leq \mathbb{P}_\tr{\tiny{out}} \leq 1 - \left(1 - \Pm{M,\frac{N B}{\Toh}}\right)^N$}
\end{align}
\normalsize
\else
\begin{align}\label{Eq:bounds}
\resizebox{\scaletemp\columnwidth}{!}{$1 - \left(1 - \Pm{M,\frac{B}{\Toh}}\right)^N \leq \mathbb{P}_\tr{\tiny{out}} \leq 1 - \left(1 - \Pm{M,\frac{N B}{\Toh}}\right)^N$}
\end{align}
\fi
The upper bound in \eqref{Eq:bounds} is realized by dividing the time $\Toh$ equally among the devices, while the lower bound is realized by allotting the time $\Toh$ to every device. Therefore, denoting the diversity order of the upper and the lower bounds in \eqref{Eq:bounds}, respectively, by $d_\text{\tiny{upper}}$ and $d_\text{\tiny{lower}}$, it can be concluded from \eqref{Eq:dive_order_def} that
\begin{align}\label{Eq:bounds_d}
d_\text{\tiny{upper}} \leq d \leq d_\text{\tiny{lower}}.
\end{align}
Using the approximation $\Pm{m,R} \approx (\frac{\omega}{P_t})^m$ for $\frac{\omega}{P_t} \rightarrow 0$ \cite{Laneman:2003}, the following holds for any bounded real value $R$.
\if2 1
\fontsize{9}{11}\selectfont
\begin{align}\nonumber
- \lim_{P_t \rightarrow \infty} \frac{\log 1 - \left(1 - \Pm{M,R} \right)^N}{\log P_t} = - \lim_{P_t \rightarrow \infty} \frac{\log N (\frac{\omega}{P_t})^M}{\log P_t} = M
\end{align}
\normalsize
\else
\begin{align}\nonumber
- \lim_{P_t \rightarrow \infty} \frac{\log 1 - \left(1 - \Pm{M,R} \right)^N}{\log P_t} = - \lim_{P_t \rightarrow \infty} \frac{\log N (\frac{\omega}{P_t})^M}{\log P_t} = M
\end{align}
\fi
Therefore, we have $d_\text{\tiny{upper}} = d_\text{\tiny{lower}} = M$, which according to \eqref{Eq:bounds_d} yields \eqref{Eq:div_order_1hop}.
\subsection{Proof of Proposition 3}
\label{Proof_prop_2h}
We start from the probability of outage $\mathbb{P}_\tr{\tiny{out}} = P_{2h}(\mathscr{D})$ in \eqref{Eq:Ptwohop-4}. First note that for $P_t \rightarrow \infty$, where $n>0$ we have
\if2 1
\fontsize{9}{11}\selectfont
\begin{align}\label{Eq:qr-prop}
\frac{\Pm{M+n,\Rth{r}} }{ \Pm{M,\Rth{b}} } = \frac{ \left( \frac{\omega_{2\text{h},r}}{P_t} \right)^{M+n}}{\left( \frac{\omega_{2\text{h},b}}{P_t} \right)^M },
\end{align}
\normalsize
\else
\begin{align}\label{Eq:qr-prop}
\frac{\Pm{M+n,\Rth{r}} }{ \Pm{M,\Rth{b}} } = \frac{ \left( \frac{\omega_{2\text{h},r}}{P_t} \right)^{M+n}}{\left( \frac{\omega_{2\text{h},b}}{P_t} \right)^M },
\end{align}
\fi
where $\omega_{2\text{h},b} = W \cdot \sigma_0 \cdot (2^{\Rth{b}/W} - 1)$, and $\omega_{2\text{h},r} = W \cdot \sigma_0 \cdot (2^{\Rth{r}/W} - 1)$, and
\if2 1
\fontsize{9}{11}\selectfont
\begin{align}
\Rth{b} & = \frac{NB}{\alpha T} = \frac{W r \log P_t}{\alpha}, \\
\Rth{r} & = \frac{NB} { (1-\alpha) T} = \frac{W r \log P_t}{1- \alpha}.
\end{align}
\normalsize
\else
\begin{align}
\Rth{b} & = \frac{NB}{\alpha T} = \frac{W r \log P_t}{\alpha}, \\
\Rth{r} & = \frac{NB} { (1-\alpha) T} = \frac{W r \log P_t}{1- \alpha}.
\end{align}
\fi
For the case where $n = 0$ and $\alpha \geq 0.5$, we have $q_{r}^{(M+n)} = 1$. Otherwise, $q_{r}^{(M+n)}$ is derived using \eqref{Eq:qr-prop}. Therefore, in the limit of $P_t \rightarrow \infty$, we use the following approximation,
\if2 1
\fontsize{9}{11}\selectfont
\begin{align}\label{Eq:a-qr-approx}
1 - \left( 1 - {q_{r}^{(M+n)}} \right)^{(N-n)} \approx
\left\{\begin{matrix}
1 & n = 0 \; \& \; \alpha \geq \frac{1}{2} \\
(N-n) q_{r}^{(M+n)} & \tr{otherwise}
\end{matrix}\right.
\end{align}
\normalsize
\else
\begin{align}\label{Eq:a-qr-approx}
1 - \left( 1 - {q_{r}^{(M+n)}} \right)^{(N-n)} \approx
\left\{\begin{matrix}
1 & n = 0 \; \& \; \alpha \geq \frac{1}{2} \\
(N-n) q_{r}^{(M+n)} & \tr{otherwise}
\end{matrix}\right.
\end{align}
\fi
We further use the approximation $\left(1-q_{b}^M \right)^{n} \approx 1$ in high \gls{snr}. Thus, we obtain
\if2 1
\fontsize{9}{11}\selectfont
\begin{align}\label{Eq:two-hop-approx-outage-div}
\mathbb{P}_\tr{\tiny{out}} & \approx \sum_{n = 0}^{N-1}\left( q_{b}^{M}\right)^{(N-n)} \binom{N}{n} \left( 1 - \left( 1 - {q_{r}^{(M+n)}} \right)^{(N-n)} \right).
\end{align}
\normalsize
\else
\begin{align}\label{Eq:two-hop-approx-outage-div}
\mathbb{P}_\tr{\tiny{out}} & \approx \sum_{n = 0}^{N-1}\left( q_{b}^{M}\right)^{(N-n)} \binom{N}{n} \left( 1 - \left( 1 - {q_{r}^{(M+n)}} \right)^{(N-n)} \right).
\end{align}
\fi
Using the approximation in \eqref{Eq:a-qr-approx} it easily follows that for $M > 1$, the slowest term in \eqref{Eq:two-hop-approx-outage-div} approaching zero as $P_t \rightarrow \infty$ is the term of $n = N - 1$. Therefore, we use this term to approximate $\mathbb{P}_\tr{\tiny{out}}$, where substituting in \eqref{Eq:dive_order_def} yields \eqref{Eq:DMT-2hop}.
\end{appendix}
|
2024-02-18T23:40:24.749Z
|
2020-09-22T02:00:07.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2009.08975","language":"en","timestamp":1600740007000,"url":"https:\/\/arxiv.org\/abs\/2009.08975","yymm":"2009"}
|
proofpile-arXiv_000-10078
|
{"provenance":"002.jsonl.gz:10079"}
| null | null |
\section{Introduction}
The quality of automatic speech recognition and machine translation of texts is constantly increasing. It leads to an opportunity to connect these two components and use them for spoken language translation (SLT).
The output of the SLT system can be delivered to users either as speech or text. In simultaneous SLT, where the output has to be delivered during the speech with as low delay as possible, there is a trade-off between latency and quality. With textual output, it is possible to present users with early, partial translation hypotheses in low latency, and correct them later by final, more accurate updates, after the system receives more context for disambiguation, or after a secondary big model produces its translation. Rewriting brings another challenge, the stability of output. If the updates are too frequent, the user is unable to read the text. The problem of unstable output could be solved by using big space for showing subtitles. The unstable, flickering output would appear only at the end, allowing the user to easily ignore the flickering part and read only the stabilized part of the output. However, in many situations, the space for subtitles is restricted. For example, if the users have to follow the speaker and slides at the same time, they lack mental capacity for searching for the stabilized part of translations. It is, therefore, necessary to put the subtitles and slides on the same screen, restricting the subtitling area to a small window.
In this paper, we propose an algorithm for presenting SLT subtitles in limited space, a way for estimating the overall usability of simultaneous SLT subtitling in a limited area, and an improved translation latency measure for SLT comparison.
\hidelong{\cref{pipeline} describes the properties of SLT for use with our subtitler. \cref{subtitler} details the main new component for presenting a text stream as readable subtitles. \cref{empirical} proposes the estimation of the usability of the subtitling of multiple realistic SLT systems.
We conclude the paper in \cref{conclusion}.
}
\section{Re-Translating Spoken Language Translation}
\label{pipeline}
Our subtitler solves the problem of presentation of SLT output with a re-translating early hypothesis, similarly to \perscite{niehues2016inter,arivazhagan2019retranslation,Arivazhagan2020RetranslationVS}. Although it can also present the subtitles from the automatic speech recognition (ASR) that re-estimates the early hypothesis, or generally any audio-to-text processor, we limit ourselves only SLT in this paper for brevity.
\hidelong{\subsection{Stable and Unstable Segments}}
\hidelong{
SLT systems output a potentially infinite stream of segments containing the beginning and final timestamps of an interval from the source audio, and the translated text in the interval. We assume that the segments can be marked as \emph{stable} and \emph{unstable},
depending on whether the system has the possibility to change them or not.
This is a realistic assumption because the ASR and SLT systems usually process a limited window of the source audio. Whenever a part of source audio exceeds this window, the corresponding output becomes stable.
}
\begin{figure*}[t]
\centering
\includegraphics[width=\textwidth]{brief-subtitler.pdf}
\caption{Illustration of speech translation subtitling in two subsequent inputs from SLT. The input arrives as a sequence of quadruples: segment beginning time, segment end time, stable/unstable flag, text. The rectangles indicate the content of the subtitling area of one line.}
\label{fig:illustration}
\end{figure*}
\begin{figure}
\centering
\includegraphics[width=0.4\textwidth]{diagram.pdf}
\caption{Subtitler processing of the inputs in \cref{fig:illustration} with different timings. In the left one, Input 2 changes the word ``Es'', which has been read by the user and scrolled away and causes a reset of a window start. In the right one, the word ``Es'' is changed in the window on the current display.}
\label{fig:diagrams}
\end{figure}
\hideXXX{
\subsection{Sentence-Level MT for On-line Translation}
\XXX{R1: You go into a lengthy explanation and then say that actual implementation is part of future work.}
With all the messages aligned to sentence boundaries, we can use off-the-shelf
MT systems. Most of these systems expect to translate individual sentences, so it does not cause any further harm if we indeed send them the input
sentence by sentence. However, it is essential to know that this can be only a
temporary solution.
Many sentences crucially depend on the context of the previous ones, and ignoring
this context will lead to translation errors. An ideal MT system for our use-case
would still process individual sentences, but it would be trained with
the context from previous sentences. The API of this system should expect not
only the new input sentence but also a representation of the previous sentences,
e.g., a
context vector. The new sentence would be translated taking this context vector
into account, and an updated context vector would be returned along with the
translation. \XXX{R2: unproven hypothesis} The actual implementation of such an approach is our
planned future work.
Another essential feature of an ideal system would be the \emph{stability} of
translation candidates for partial sentences. As mentioned above, we feed not
only complete sentences to the system but also the \emph{incoming} ones. If some new words appended to a previous version of the incoming sentence lead MT to word reordering, the presentation of such translation would ``flicker'', making the output impossible to follow. While there are already strategies to reducing this problem, e.g., wait-$k$ policy \parcite{Ma2018STACLST}, an ultimate solution may not be possible for some language pairs without an extended delay to receive all critical elements of the sentence.
In this paper, we experiment with open-source MT solution Marian \parcite{mariannmt}, which still does not have this improved stability. \XXX{R2: unproven hypothesis} Empirical results of our solution presented in \cref{empirical} can only get better when adopting a stable MT system.
}
\section{Subtitler}
\label{subtitler}
This section presents the design and algorithm of ``subtitler''.
The subtitler is a cache on a stream of input messages aiming
to satisfy the following conflicting needs:
\begin{itemize}[nosep]
\item The output should be presented with the lowest possible delay to
achieve the effect of simultaneous translation as much as possible.
\item The flickering of the partial outputs is partially desired because it
highlights the simultaneity of the translation and comforts the user in knowing
that the system is not stuck.
\item The flickering should be minimized. If some output was presented at a
position of the screen, it should keep the position until it is outdated.
\item The user must have enough time to read the message.
\item Only a predefined space of $w$ (width) characters and $h$ (height) lines
are available.
\end{itemize}
Given an input stream of stable and unstable segments as described
above, the subtitler emits a stream of ``subtitle windows''. On every update, the former window is replaced by a new one.
The basic operation of subtitler is depicted in \cref{fig:illustration,fig:diagrams}. The elements of subtitler are a
buffer of input segments, a presentation window, and two independent
processing threads.
The buffer is an ordered list of segments. The
presentation window is organized as a list of text lines of the required width and
count. The count corresponds to the height of subtitling window plus one, to allow scrolling-up the top line after displaying it for minimum reading time. This line view is regenerated whenever needed from the current starting position of
the window in the buffer, wrapping words into lines.
The \emph{input thread} receives the input stream and updates the buffer.
It replaces outdated segments with their new versions, extends the buffer, and
removes old unnecessary segments. If an update happens within
or before the current position of the presentation window, the \emph{output
thread} is notified for a forced update.
Independently, the \emph{output thread} updates the position
of the presentation window in the buffer, obeying the following timeouts and triggers:
\begin{itemize}[nosep]
\item On forced updates, the output thread detects if any content changed
before the beginning of the already presented window, which would cause
a \emph{reset}.
In that case, the window position on the window buffer has to be
moved back, and the content for the user can no longer be presented
incrementally. Instead, the beginning of the first line in the window shows
a newer version of an old sentence that has already scrolled away.
\item If the first line of the presentation window has not been changed for a
minimum reading time and if there is any input to present in the extra line of the
window, the window is ``scrolled'' by one line, i.e., the first line is discarded,
the window starting position within the buffer is updated, and the extra line is shown as the last line of the window.
\item If the whole presentation window has not been changed for a long time, e.g., 5 or 20 seconds, it is blanked by emitting empty lines.
\end{itemize}
\subsection{Timing Parameters}
The subtitler requires two timing parameters. A line of subtitles is displayed to a user for a ``minimum reading time'' before it can be scrolled away. If no input arrives for a ``blank time'', the subtitling window blanks to indicate it and to prevent the user from reading the last message unnecessarily.
We suggest adopting the minimum reading time parameter from the standards for subtitling films and videos (e.g., \inparcite{fotios}), before standards for simultaneous SLT subtitling will be established.
\perscite{Szarkowska2018ViewersCK} claim that 15 characters per second is a standard reading time in English interlingual subtitling of films and videos for deaf and hard hearing. The standards in other European regions are close to 15 characters per second. We use this value for the evaluation in this work.
\section{Estimating Usability}
\label{empirical}
The challenges in simultaneous SLT are quality, latency, and stability \cite{niehues2016inter,arivazhagan2019retranslation}. All of these properties are critical for the overall usability of the SLT system. The quality of translation is a property of the SLT system. The subtitler has no impact on it.
The minimum reading time ensures the minimum level of stability, ensuring that every stable content is readable, and may increase the latency if the original speech is faster than reading time.
The size of the subtitling window and timing parameters affect overall latency and stability.
The bigger the window, the longer updates of translations fit into it without a reset. The \hideXXX{myslim, ze tu ma byt minimum reading time, vsude, az do konce odstavce: DM: nema. Blanking time ovlivnuje resety. Mozna ma byt jak minimum reading, tak blanking... OB: tak tedy oba, protoze minimum reading urcite. Kdyz uz radku prectes (tj. uplyne minumum reading), tak se odroluje a hrozi reset. Blanking typicky *nezazijes*, protoze je az po 20 vterinach bez vstupu.
OB: napisme proste "timing constants"? timing parameters OK,
DM: Ale nekdy jo. Spocital jsem resety pro 3 radky a pro ASR, kde se vsechny maji vejit, to nekde udelalo reset kvuli tomu. OK, chapu. Myslim, ze ted je to dobre. Pozor, dal jsou dalsi veci.}
timing parameters determine how long the content stays unchanged in the window before scrolling. A small subtitling window or a short reading or blanking time may cause a reset. Every reset increases latency because it returns to the already displayed content. On the other hand, the significant latency may improve stability by skipping the early unstable hypotheses and present only the stable ones.
We provide three automatic measures for assessing the practical usability of simultaneous SLT subtitling on the test set. The automatic evaluation may serve for a rough estimation of the usefulness, or for selection of the best candidate setups. We do not provide a strict way to judge which SLT system and subtitling setup are useful and which are not. The final decision should ideally consider the particular display conditions, expectations, and needs of the users, and should be based on a significant human evaluation.
\subsection{Evaluation Measures}
For quality, we report an automatic machine translation measure BLEU computed by sacrebleu \cite{post-2018-call} after automatic sentence alignment using mwerSegmenter \cite{segmentation:matusov:2005:iwslt}. BLEU is considered as correlating with human quality judgement. The higher BLEU, the higher translation quality.
To explain the measure of latency and stability, let us use the terminology of \perscite{arivazhagan2019retranslation}. The \emph{EventLog} is an ordered list of \emph{events}. The $i^{th}$ \emph{event} is a triple $s_i, o_i, t_i$, where $s_i$ is the source text recognized so far, $o_i$ is the current SLT output, and $t_i$ is the time when this event was produced. Source and output, $s_i$ and $o_i$, are sequences of tokens. Let us denote $c(o_i)$ a transformation of a token sequence into a sequence of characters, including spaces and punctuation. Let $I$ be the number of all events, with an update either in source or output, and $T$ the number of events with an update in translation.
\subsubsection{Character Erasure}
To evaluate how many updates fit into the subtitling window, we define \emph{character erasure} (cE). It is the number of characters that must be deleted from the tail of the current translation hypothesis to update it to a new one. If a new translation only appends words to the end, the erasure is zero. The \emph{character erasure} is
cE$(i) = |c(o_{i-1})| - |LCP(c(o_i),c(o_{i-1}))|$, where the $LCP$ stands for the longest common prefix. The average character erasure is AcE = $1/T \sum_{i=1}^I $cE$(i)$. It is inspired by the normalized erasure (NE) by \perscite{arivazhagan2019retranslation}, but we do not divide it by the output length in the final event, but only by the number of translation events.
\subsubsection{Translation Latency with Sentence-Alignment Catch-up}
The translation latency may be measured with the use of a \emph{finalization event} of the $j$-th word in output. It is $f(o,j) = min_i$ such that $o_{i',j'} = o_{I,j'}$ $\forall i'\ge i$ and $\forall j'\le j$. In other words, the word $j$ is finalized in the first event $i$, for which the word $j$ and all the preceding words $j'$ remain unchanged in all subsequent events $i'$.
The translation latency of output word $j$ is the time difference of the finalization event of the word $j$ in the output and its \emph{corresponding} word $j^*$ in the source. \perscite{arivazhagan2019retranslation} estimate the source word simply as $j^* = (j/|o_I|) |s_I|$. This is problematic if the output is substantially shorter than input, because then it may incorrectly base the latency on a word which has not been uttered yet, leading to a negative time difference.
A proper word alignment would provide the most reliable correspondence. However, we propose a simpler and appropriately reliable solution. The following improved measure is our novel contribution. We use it to compare the SLT systems.
We utilize the fact that our ASR produces punctuated text, where the sentence boundaries can be detected. The sentences coming from SLT and ASR in their sequential order are parallel. They can be simply aligned because our SLT systems translate the individual sentences and keep the sentence boundaries.
If the SLT does not produce individual sentences, then we use a rule-based sentence segmenter, e.g. from \perscite{moses}, and must be aware of the potential inaccuracy
We use the sentence alignment for a catch-up, and the simple temporal correspondence of \perscite{arivazhagan2019retranslation} only within the last sentence.
To express it formally, let us assume that the EventLog has also a function $S(o,j)$, returning the index of the sentence containing the word $j$ in $o$, and $L(o,k)$, the length of the sentence $k$ in $o$. Let $x(j) = j-\sum_{i=1}^{S(o,j)-1} L(o,i)$ be the index of an output word $j$ in its sentence. Then we define our caught-up correspondence as
\DeclarePairedDelimiter\floor{\lfloor}{\rfloor}
$j^{**} = \sum_{i=1}^{S(o,j)-1} L(s,i) + x(j) \floor*{\frac{L(s,S(o,j))}{L(o,S(o,j))}}$
\label{form:sdfsd}
\defTL$^{*}${TL$^{*}$}
Finally, our \emph{translation latency with sentence-alignment catch-up} is TL$^{*}(o,j) = t_{f(o,j)} - t_{f(s,j^{**})}$.
This is then averaged for all output words in the document: TL$^{*}$ $= \frac{1}{|o_I|} \sum_{j=1}^{|o_I]} TL^*(o,j)$.
\hidelong{\footnote{For a set of documents $D$, the TL$^{*}$ $= \frac{\sum_{o,I\in D}\sum_{j=1}^{|o_I]} TL^*(o,j)}{\sum_{o,I\in D}|o_I|}$.}}
\subsection{SLT Evaluation}
We use one ASR system for English and nine SLT systems from English into Czech (three different models differing in the data and training parameters), German (2 different systems), French (2 different systems), Spanish and Russian. All the SLT systems are cascades of an ASR, a punctuator, which inserts punctuation and capitalization to unsegmented ASR output, and a neural machine translation (NMT) from the text.
\hidelong{The systems and their quality measures are in \cref{tab:scores}.}
DE A, ES, and FR B are NMT adapted for spoken translation as in \perscite{Niehues_2018}. The others are basic sentence-level Transformer NMT connected to ASR.
The ASR is a hybrid DNN-HMM by \perscite{janus-iwslt}.
We evaluate the systems on IWSLT tst2015 dataset.
\hidelong{We downloaded the referential translation from the TED website as \perscite{arivazhagan2019retranslation}, and removed the single words in parentheses because they were not verbatim translations of the speech, but marked sounds such as applause, laughter, or music.
}
\hidelong{
\begin{table}[t]
\centering
\caption{Quality measure of the English ASR and SLT systems from English into the target language in the left-most column, on IWSLT tst2015. The letters A, B, C denote different variants of SLT systems with the same target. Translation lag (TL$^{*}$) is in seconds. AcE is average character erasure, NE is normalized erasure.}
\label{tab:scores}
\small
\begin{tabular}{l|rrrrrr}
\bf SLT & \bf BLEU & \bf TL$^{*}$ & \bf AcE & \bf NE\\
\hline
EN (ASR) &58.4747 & & 29.22 & 5.88 \\
\hline
CZ A & 17.5441 & 2.226 & 24.20 & 7.05 \\
CZ B &12.2914 & 2.622 & 29.48 & 5.30 \\
CZ C &18.1505 & 2.933 & 27.90 & 3.93 \\
\hline
DE A &15.2678 & 3.506 & 47.32 & 1.39 \\
DE B &15.9672 & 1.845 & 38.12 & 5.46 \\
\hline
ES &21.8516 &5.429 & 43.30 & 1.49 \\
\hline
FR A & 25.8964 & 1.269 & 31.97 & 3.32 \\
FR B &20.5367 & 5.425 & 47.92 & 1.46 \\
\hline
RU &11.6279 & 3.168 & 31.78 & 4.05 \\
\end{tabular}
\end{table}
}
\subsection{Reset Rate}
\begin{figure}
\centering
\includegraphics[width=0.7\columnwidth]{plot.pdf}
\caption{The percentage of translation updates in the validation set with the character erasure less than or equal to the value on the $x$-axis, for all our ASR and SLT systems.%
\hideXXX{Nesla by osa x nazvat jednoduseji? Nebo aspon napsat: DM: jde to pripsat.}
The $x$-axis corresponds with the size of the subtitling window.}
\label{fig:cdf}
\end{figure}
The average character erasure does not reflect the frequency and size of the individual erasures. Therefore, in \cref{fig:cdf}, we display the cumulative density function of character erasure in the dataset. The vertical axis is the percentage of all translation updates, in which the character erasure was shorter or equal than the value on the horizontal axis. E.g., for the subtitler window with a total size of 140 characters, 99.03 \% of SLT updates of the SLT CZ A fit into this area.
\cref{tab:cdf} displays the same for selected sizes, which fit into 1, 2, and 3 lines of subtitler window of size 70, and also the percentage of updates without any erasure ($x=0$).
The values approximate the expected number of resets. However, the resets are also affected by the blanking time, so the real number of resets may be higher if the speech contains long pauses. The percentage in \cref{fig:cdf} serves as a lower bound.
\hidelong{
\begin{table}[]
\small
\centering
\caption{Percentage of character erasures in all translation updates, which are shorter or equal than $x$ characters, for selected values of $x$.}
\label{tab:cdf}
\begin{tabular}{l|ccccc}
SLT & $x = 0$ & $x = 70$ & $x = 140$ & $x = 210$ \\
\hline
EN (ASR) & 20.76 & 84.23 & 99.96 & 100.00 \\
CZ A & 41.37 & 91.98 & 99.03 & 99.76 & \\
CZ B & 28.61 & 89.78 & 98.63 & 99.77 & \\
CZ C & 30.93 & 88.31 & 98.53 & 99.72 & \\
FR A & 31.65 & 84.47 & 98.14 & 99.51 & \\
RU & 35.42 & 85.17 & 97.82 & 99.38 & \\
ES & 29.01 & 71.71 & 97.08 & 99.43 & \\
DE B & 27.89 & 80.90 & 97.05 & 99.38 & \\
DE A & 30.85 & 67.65 & 95.83 & 99.13 & \\
FR A & 30.39 & 66.15 & 95.67 & 99.39 & \\
\end{tabular}
\end{table}
}
\subsection{Subtitling Latency}
\hidelong{
\begin{figure}[t]
\centering
\includegraphics[width=0.7\columnwidth]{subtitler-lag-only-1.pdf}
\caption{Subtitling latency (y-axis) over time (x-axis) for tst2015.en.talkid1922 translated by CZ A.
The subtitling window has the width 70 and height 1, 2 and 3 lines. The minimum reading time is 15 characters per second (one line per 4.7s).}
\label{fig:subtitler-lag}
\end{figure}
}
The \emph{subtitling latency} is the difference of the finalization time of a word in subtitler and in the SLT.
\hidelong{We count it similarly as the translation latency, but the word correspondence is the identity function because the language in SLT and subtitler is the same.}
We computed the latency caused by the subtitler with 1, 2, and 3 lines of width 70 for one talk and SLT systems, see \cref{fig:subtitler-lag}.
Generally, the bigger the translation window, the lower latency.
\hideXXX{
The subtitling is useful only when the latency is continuously low. We can observe that in the case of the talk tst2015.en.talkid1922 translated by CZ A (\cref{fig:subtitler-lag}, top), the latency is around zero for 3-line window and below 10 seconds for 2-line window only at the first 400 seconds of the talk. After that, the latency is increasing up to 40 seconds, so the subtitling is desynchronized with the original and probably useless. In the second example (bottom plot), the subtitling is useless from the point where the talk starts.
}
\hideXXX{The long latency may be induced by the low stability of SLT, by the length of incoming segments, by the talking speed, or by other reasons. Our primary motivation in this work is to propose how to estimate the usefulness of the simultaneous subtitling of SLT in a limited area. We leave the analysis of subtitling latency for future work.}
\subsection{User Evaluation}
\begin{table}[]
\centering
\caption{Results of user evaluation with three subtitling windows of different heights (h). Quality level 4 is the highest, 1 is the lowest. The right-most column is the percentage of erasures fitting into the subtitling window.}
\label{tab:rating}
\small
\begin{tabular}{r|r@{~}r@{~}r@{~}r|r}
& \multicolumn{4}{c|}{Percentage of quality levels} \\
height & level=1 & level=2 & level=3 & level=4 & cE $<70\cdot h$\ \\
\midrule
$h = 1$ & 35.27 \% & 28.79 \% & 14.95 \% & 20.99 \% & 88.59 \% \\
$h = 2$ & 11.08 \% & 29.94 \% & 35.73 \% & 23.24 \% & 98.73 \% \\
$h = 3$ & 16.33 \% & 19.90 \% & 33.67 \% & 30.11 \% & 99.64 \% \\
\end{tabular}
\end{table}
We asked one user to rate the overall fluency and stability of subtitling for
the first 7-minute part of tst2015.en.talkid1922 translated by CZ A. We presented the user with the subtitles three times, in a window of width 70 and heights 1, 2 and 3. The minimum reading time parameter was 15 characters per second. The user was asked to express his subjective quality assessment by pressing one of five buttons: undecided (0), horrible (1), usable with problems (2), minor flaws, but usable (3), and perfect (4). The user was asked to press them simultaneously with reading subtitles, whenever the assessment changes. The source audio or video was not presented, so this setup is comparable to situations where the user does not understand the source language at all.
The user is a native speaker of Czech.
\cref{tab:rating} summarizes the percentage of the assessed duration and the quality levels. The user has not used the level undecided (0). The main problem that the user reported was limited readability due to resets and unstable translations. The flaws in usable parts of subtitling were subtle changes of subtitles which did not distract from reading the new input, or disfluent formulations.
In the right-most column of \cref{tab:rating} we show the percentage of erasures in the part of the evaluated document which fit into the subtitling window.
We hypothesize that the automatic measure of character erasure may be used to estimate the user assessment of readability.
\section{Conclusion}
\label{conclusion}
We proposed an algorithm for presenting automatic speech translation simultaneously in the limited space of subtitles. The algorithm is independent of the SLT system. It ensures the minimum level of stability and allows simultaneity.
Furthermore, we propose a way of estimating the reader's comfort and overall usability of the SLT with subtitling in limited space, and observe correspondence with user rating. Last but not least, we suggested a catch-up based on sentence-alignment in ASR and SLT to measure the translation latency simply and realistically.
\section*{Acknowledgments}
The research was partially supported by
the grant CZ.07.1.02/0.0/0.0/16\_023/0000108 (Operational Programme -- Growth Pole of the
Czech Republic),
H2020-ICT-2018-2-825460 (ELITR) of the EU,
398120 of the Grant Agency of Charles University,
and by SVV project number 260 575.
\small
|
2024-02-18T23:40:24.755Z
|
2020-09-22T02:01:10.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2009.09016","language":"en","timestamp":1600740070000,"url":"https:\/\/arxiv.org\/abs\/2009.09016","yymm":"2009"}
|
proofpile-arXiv_000-10079
|
{"provenance":"002.jsonl.gz:10080"}
| null | null |
\section*{Acknowledgements}
The work of EM was supported in part by NSF Grant Nos. NSF CCF-1907591, DMS-1800446.
FAK acknowledges the support of the National Science Foundation under grants CMMI-1759823 and DMS1759824.
\bibliographystyle{plain}
\section{Introduction}
Topological data analysis (TDA) is a field consisting of tools aimed at extracting shape in data.
Persistent homology, one of the most commonly used tools from TDA, has proven useful in the field of time series analysis.
Specifically, persistent homology has been shown to quantify features of a time series such as periodic and quasiperiodic behavior \cite{Perea2015,Sanderson2017,Tempelman2019,Maletic2016,Xu2018} or chaotic and periodic behavior \cite{Myers2019,Khasawneh2015}.
Existing applications in time series analysis include studying machining dynamics \cite{Khasawneh2017,Khasawneh2018a,Yesilli2019b,Khasawneh2015,Yesilli2019a,Khasawneh2018b,Khasawneh2014a}, gene expression \cite{Perea2015, Berwald2014}, financial data \cite{Gidea}, video data \cite{tralie2018quasi,tralie2016high}, and sleep-wake states \cite{Chung2019, Tymochko2020}.
These applications typically involve summarizing the underlying topological shape of each time series in a persistence diagram then using additional methods to analyze the resulting collection of persistence diagrams.
While these applications have been successful, the task of analyzing a collection of persistence diagrams can still be difficult.
Many methods have been created to convert persistence diagrams into a form amenable for machine learning \cite{Bubenik2015,Adams2017,Reininghaus2015,Perea2019}.
However, so many methods have been developed, it can be difficult to choose one appropriate for the task.
Additionally, the task of computing numerous persistence diagrams is computationally expensive.
Our method aims to circumvent these issues using zigzag persistence, a generalization of persistent homology that is capable of summarizing information from a sequence of point clouds in a single persistence diagram.
While less popular than standard persistent homology, zigzag persistence has been used in applications, including studying optical flow in computer generated videos \cite{Adams2019,Adams2020}, analyzing stacks of neuronal images \cite{Mata2015} and comparing different subsamples of the a dataset \cite{Tausz2011}.
However, to the best of our knowledge, it has not been used in the context of dynamical systems or time series analysis.
In this paper, we present Bifurcations using ZigZag (BuZZ), a one-step method to analyze bifurcations in dynamical systems using zigzag persistence.
\section{Materials and Methods}
Here we will present tools needed to build our method, including the time delay embedding, and an overview of the necessary topological tools.
Specifically, we briefly introduce homology, persistent homology and zigzag persistent homology.
However, we will not go into detail and instead direct the interested reader to \cite{Hatcher,Edelsbrunner2010,Carlsson2010} for more detail on homology, persistent homology, and zigzag homology, respectively.
\subsection{Homology and persistent homology}
\label{ssec:homology}
Homology is a tool from the field of algebraic topology that encodes information about shape in various dimensions.
Zero-dimensional homology studies connected components, 1-dimensional homology studies loops, and 2-dimensional homology studies voids.
Persistent homology is a method from TDA which studies the homology of a parameterized space.
For the purposes of this paper, we will focus on persistent homology applied to point cloud data.
Here, we need only assume a point cloud is a collection of points with a notion of distance, however in practice, this distance often arises from a point cloud in Euclidean space inheriting the ambient metric.
Given a collection of points, we will build connections between points based on their distance.
Specifically, we will build simplicial complexes, which are spaces built from different dimensional simplices.
A 0-simplex is a vertex, a 1-simplex is an edge, a 2-simplex is a triangle, and in general, a $p$-simplex is the convex hull of $p+1$ affinely independent vertices.
Abstractly, a $p$-simplex can be represented by the set of $p+1$ vertices it is built from.
So a simplicial complex, $\mathcal{K}$, is a family of sets that is closed under taking subsets.
That is, given a $p$-simplex, $\sigma$, in $\mathcal{K}$, then any simplex consisting of a subset of the vertices of size $0<k\leq p$, called a $k$-dimensional face of $\sigma$, is also in $\mathcal{K}$.
To create a simplicial complex from a point cloud, we use the Vietoris-Rips complex (sometimes just called the Rips complex).
Given a point cloud, $X$, and a distance value, $r$, the Vietoris-Rips complex, $\mathcal{K}_r$, consists of all simplices whose vertices have maximum pairwise distance at most $r$.
Taking a range of distance values, $r_0\leq r_1\leq r_2\leq \cdots r_n$ gives a set of simplicial complexes, $\{\mathcal{K}_{r_i}\}$.
Since the distance values are strictly increasing, we have a nested sequence of simplicial complexes
\begin{equation}\label{eqn:filtration}
\mathcal{K}_{r_0} \subseteq \mathcal{K}_{r_1} \subseteq \cdots \subseteq \mathcal{K}_{r_n}
\end{equation}
called a filtration.
Computing $p$-dimensional homology, $H_p(\mathcal{K})$, for each complex in the filtration gives a sequence of vector spaces and linear maps,
\begin{equation}
H_p(\mathcal{K}_{r_0}) \to H_p(\mathcal{K}_{r_1}) \to \cdots \to H_p(\mathcal{K}_{r_n}).
\end{equation}
Persistent homology tracks homological features such as connected components and loops as you move through the filtration.
Specifically, it records at what distance value a feature first appears, and when a feature disappears or connects with another feature.
These distance values are called the ``birth'' and ``death'' times respectively.
These birth and death times are represented as a persistence diagram, which is a multiset of the birth death pairs $\{(b,d)\}$.
\subsection{Time delay embedding}
\label{ssec:timedelay}
One way to reconstruct the underlying dynamics given only a time series is through a time delay embedding.
Given a time series, $[x_1,\ldots,x_n]$, a choice of dimension $d$ and lag $\tau$, the delay embedding is the point cloud $\mathbf{X} = \{\mathbf{x}_i:= (x_i,x_{i+\tau}, \ldots, x_{i+(d-1)\tau}) \} \subset \mathbb{R}^d$.
Takens' theorem \cite{Takens1981} shows that given most choices of parameters, the embedding retains the same topological structure as the state space of the dynamical system and that this is in fact a true embedding in the mathematical sense.
In practice, not all parameter choices are optimal, so heuristics for making reasonable parameter choices have been developed \cite{Fraser1986,Kennel,Pecora2007,Chelidze2017,Pan2020,Kim1999}.
In existing methods combing TDA with time series analysis, most works analyze a collection of time series by embedding each one into a point cloud using the time delay embedding.
However, this can be computationally expensive and requires an analysis of the collection of computed persistence diagrams.
Instead, we will employ a generalized version of persistent homology to avoid these additional steps.
\subsection{Zigzag persistent homology}
\label{ssec:zigzag}
Zigzag persistence is a generalization of persistent homology that can study a collection of point clouds simultaneously.
In persistent homology, you have a nested collection of simplicial complexes as in Eqn.~\ref{eqn:filtration}.
However, for zigzag persistence, you can have a collection of simplicial complexes where the inclusions go in different directions.
Specifically, the input to zigzag persistence is a sequence of simplicial complexes with maps,
\begin{equation}
\mathcal{K}_0 \leftrightarrow \mathcal{K}_1 \leftrightarrow \cdots \leftrightarrow \mathcal{K}_n
\end{equation}
where $\leftrightarrow$ is either an inclusion to the left or to the right.
While in general these inclusions can go in any direction in any order, for this paper we will focus on a specific setup for the zigzag based on a collection of point clouds.
Given an ordered collection of point clouds, $X_0,X_1,\ldots, X_n$, we can define a set of inclusions,
\begin{equation} \label{eqn:zz_pc}
\begin{tikzcd}[column sep=tiny]
X_0 \arrow[hookrightarrow,dr] &
&
X_1 \arrow[hookrightarrow,dl]\arrow[hookrightarrow,dr] &
&
\arrow[hookrightarrow,dl] X_2 &
\cdots & X_{n-1}
\arrow[hookrightarrow,dr] &
&
\arrow[hookrightarrow,dl] X_n \\
&
X_0 \cup X_1 &
&
X_1 \cup X_2 &
& & &
X_{n-1}\cup X_n. &
\end{tikzcd}
%
\end{equation}
However, these are all still point clouds, which have uninteresting homology.
Thus, we can compute the Vietoris-Rips complex of each point cloud for a fixed radius, $r$.
This results in the diagram of inclusions of simplicial complexes
\begin{equation} \label{eqn:zz_rips_fixed}
\begin{tikzcd}[column sep=-1.5em]
R(X_0,r) \arrow[hookrightarrow,dr]
&
& R(X_1,r) \arrow[hookrightarrow,dl]\arrow[hookrightarrow,dr]
&
& R(X_2,r) \arrow[hookrightarrow,dl]
& ~~~~~\cdots~~~~~
& R(X_{n-1},r) \arrow[hookrightarrow,dr]
&
& R(X_n,r) \arrow[hookrightarrow,dl] \\
& R(X_0 \cup X_1,r)
&
& R(X_1 \cup X_2,r)
&
&
&
& R(X_{n-1}\cup X_n,r).
&
\end{tikzcd}
%
\end{equation}
Computing the 1-dimensional homology of each complex in Eqn.~\ref{eqn:zz_rips_fixed} will result in a zigzag diagram of vector spaces and induced linear maps,
\begin{equation} \label{eqn:zz_rips_hom}
%
\begin{tikzcd}[column sep=-1.5em]
H_1( R(X_0,r) ) \arrow[dr]
&
& H_1( R(X_1,r) ) \arrow[dl]
& ~~~~~\cdots~~~~~
& H_1( R(X_{n-1},r) ) \arrow[dr]
&
& H_1( R(X_n,r) ) \arrow[dl] \\
& H_1( R(X_0 \cup X_1,r) )
&
&
&
& H_1( R(X_{n-1}\cup X_n,r) ).
&
\end{tikzcd}
%
\end{equation}
Zigzag persistence tracks features that are homologically equivalent through this zigzag.
This means it records the range of the zigzag filtration where the same feature appears.
The zigzag persistence diagram records ``birth'' and ``death'' relating to location in the zigzag.
If a feature appears in $R(X_i,r),$ it is assigned birth time $i$, and if it appears at $R(X_i\cup X_{i+1},r),$ it is assigned birth time $i+0.5$.
Similarly, if a feature last appears in $R(X_j,r)$, it is assigned a death time $j+0.5$, while if it last appears in $R(X_j \cup X_{j+1},r)$, it is assigned a death time of $j+1$.
A small example is shown in Fig.~\ref{fig:zz_example}.
In this example, there is a 1-dimensional feature that appears in $R(X_0,r)$ and disappears going into $R(X_0 \cup X_1,r)$, thus it appears in the zigzag persistence diagram as the point $(0,0.5)$.
Additionally, there is one connected component that exists through the zigzag, corresponding to the 0-dimensional persistence point $(0,2)$.
By default, we assume all features die at the end of the zigzag, rather than having infinite points as in standard persistence.
There is one additional connected component that first appears in $R(X_0\cup X_1,r)$, corresponding to the 0-dimensional persistence point $(0.5,2).$
\begin{figure}
\centering
\includegraphics[width=0.75\textwidth]{Figures/ZZ_Example.png}
\caption{Small example of zigzag filtration with corresponding zigzag persistence diagram.}
\label{fig:zz_example}
\end{figure}
Note that we can easily generalize this idea to use a different radius for each Rips complex, $R(X_i,r_i)$.
For the unions we choose the maximum radius between the two individual point clouds, $R(X_i\cup X_{i+1}, \max\{r_i,r_{i+1}\})$, to ensure the inclusions hold.
\subsection{Bifurcations using ZigZag (BuZZ)}
\label{ssec:ourmethod}
\begin{figure}
\centering
\includegraphics[width=\textwidth]{Figures/Pipeline_Horizontal.png}
\caption{
Outline of BuZZ method. The input time series is converted to an embedded point cloud via the time delay embedding. The Rips complexes are constructed for either a fixed $r$ or a choice of $r_i$ for each point cloud. Then, the zigzag persistence diagram is computed for the collection. }
\label{fig:pipeline}
\end{figure}
We can now present our method, Bifurcations using ZigZag (BuZZ) for combining the above tools to detect changes in circular features in dynamical systems.
We will focus on Hopf bifurcations \cite{Guckenheimer1983}, which are seen when a fixed point loses stability and a limit cycle is introduced.
These types of bifurcations are particularly topological in nature, as the state space changes from a small cluster, to a circular structure, and sometimes reduces back to a cluster.
The necessary data for our method is a collection of time series for a varying input parameter value, as shown in Fig.~\ref{fig:pipeline}(a).
This particular example is a collection of time series given by $\{a\sin(t)\}$ for $a=0.5,1.0,1.5,2.0$ (going from top to bottom).
Each time series is then embedded using the time delay embedding (shown in Fig.~\ref{fig:pipeline}(b) using $d=2$ and $\tau=3$).
While in general, the delay could be varied for each time series, the embedding dimension needs to be fixed so each time series is embedded in the same space.
For the sake of interpretability and visualization, we will use a dimension of $d=2$ throughout this paper.
Sorting the resulting point clouds based on the input parameter value, the zigzag filtration can be formed from the collection of point clouds, as shown in Fig.~\ref{fig:pipeline}(c).
Lastly, computing zigzag persistence gives a persistence diagram, as shown in Fig.~\ref{fig:pipeline}(d), encoding information about the structural changes moving through the zigzag.
With the right choices of parameters, the 1-dimensional persistence point with the longest lifetime in the zigzag persistence diagram will have birth and death time corresponding to the indices in the zigzag where the Hopf bifurcation appears and disappears.
Lastly, mapping the birth and death times back to the parameter values used to create the corresponding point clouds will give the range of parameter values where the Hopf bifurcation occurs.
Note that there are several parameter choices that need to be selected during the course of the BuZZ method.
First, the dimension $d$ and delay $\tau$ for converting each time series into a point cloud.
Fortunately, there is a vast literature from the time series analysis literature for this, which leads to standard heuristics.
The second and more difficult parameter is the choice of radius (or radii) for the Rips complexes.
In this paper, the given examples are simple enough that the choice of radii in the BuZZ method can be tuned by the user.
However, in future work, we would like to create new methods and heuristics for choosing these radii.
\subsection{Algorithms}
\label{ssec:algorithm}
While zigzag persistence has been in the literature for a decade, it has not often been used in application, and thus the software that computes it is not well developed.
A C++ package with python wrappers, Dionysus 2\footnote{ \url{https://www.mrzv.org/software/dionysus2/} }, has implemented zigzag persistence; however, it requires significant preprocessing to create the inputs.
We have developed a python package \footnote{\url{https://github.com/sarahtymochko/BuZZ}}
that, provided the collection of point clouds and radii, will perform all the necessary preprocessing to set up the zigzag diagram as shown in Eqn.~\ref{eqn:zz_rips_fixed} to pass as inputs to Dionysus.
Dionysus requires two inputs, a list of simplices, \texttt{simplex\_list}, and a list of lists, \texttt{times\_list}, where the \texttt{times\_list[i]} consists of a list of indices in the zigzag where the simplex, \texttt{simplex\_list[i]}, is added and removed.
A small example is shown in Fig.~\ref{fig:dio_input_fixed}.
Looking at that example, the two vertices and one edge in $R(X_0)$ appear at time 0, and disappear at time 1.
There are two edges and a triangle in $R(X_0\cup X_1)$ that appear there at time 0.5 (recalling that $R(X_i\cup X_{i+1})$ is time $i+0.5$) and disappear at time 1.
Lastly, the one vertex in $R(X_1)$ appears at time 0.5, and never disappears in the zigzag sequence, so by default we set death time to be $2$, which is the next index beyond the end of the zigzag sequence.
This is done to avoid infinite lifetime points, as our zigzag sequences are always finite and an infinite point has no additional meaning.
Note there are other special cases that can occur.
If a simplex is added and removed multiple times, then the corresponding entry in \texttt{times\_list} has more than two entries, where the zero and even entries in the list correspond to when it appears, and the odd entries correspond to when it disappears.
An example with this special case is shown in Fig.~\ref{fig:dio_input_changing} and will be described in more detail later.
If we are using a fixed radius across the whole zigzag, these inputs can be computed rather easily.
In this setting, we only need to compute the Rips complex of the unions, $R(X_i\cup X_{i+1},r)$, which can be done using the Dionysus package, and the list of simplices can be created by combining lists of simplices for all $i$, removing duplicates.
Next, we will outline how to construct the times list.
Starting with the set of simplices in $R(X_i\cup X_{i+1},r)$, we can split them into three groups: (a) simplices for which all 0-dimensional faces are in $X_i$, (b) simplices for which all 0-dimensional faces are in $X_{i+1}$, or (c) simplices for which some 0-dimensional faces are in $X_i$ and some are in $X_{i+1}$.
Because of the construction of the zigzag, all simplices in group (a) appear at time $i-0.5$, since $R(X_i,r)$ also includes backwards into $R(X_{i-1}\cup X_i,r)$, and disappear at time $i+1$, since the union $R(X_i \cup X_{i+1},r)$ is the last time simplices in $X_{i}$ are included.
Similarly, all simplices in group (b) appear at time $i+0.5$, since this is the first time simplices $X_{i+1}$ are included, and disappear at time $i+2$, since $R(X_{i+1},r)$ also includes forward into $R(X_{i+1}\cup X_{i+2},r)$.
Lastly, all simplices in group (c) exist only at $R(X_i\cup X_{i+1},r)$, so they appear at time $i+0.5$ and disappear at $i+1$.
Note that the first case needs to be treated separately, since in $R(X_0\cup X_1,r)$, all vertices in group (a) will appear at $0$.
\begin{figure}
\centering
\includegraphics[width=0.6\textwidth]{Figures/Dio_Input_Example.pdf}
\caption{Example zigzag using fixed radius with computed inputs for Dionysus.}
\label{fig:dio_input_fixed}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.99\textwidth]{Figures/Dio_Input_Example_Changing.pdf}
\caption{Example zigzag using a changing radius with computed inputs for Dionysus. In this example $r_0>r_1$ and $r_2>r_1$.}
\label{fig:dio_input_changing}
\end{figure}
Using a varied radius, as described in Sec.~\ref{ssec:zigzag}, complicates the above procedure.
Using the same radius, we are guaranteed all simplices in group (a) are in both $R(X_i,r)$ and $R(X_i\cup X_{i+1},r)$, and similarly for group (b), thus we only need to compute $R(X_i\cup X_{i+1}, r)$.
However, with a changing radius this is no longer true.
In the example shown in Fig.~\ref{fig:dio_input_changing}, the edge $e_{1,2}$ appears in both $R(X_0\cup X_1, r_0)$ and $R(X_1\cup X_2, r_2)$ since $r_2>r_1$ and $r_0>r_1$, but it is not in $R(X_1,r_1)$.
Thus, its corresponding list in \texttt{times\_list} is $[0.5,1,1.5,2]$.
Thus the inputs to Dionysus can be computed using the same method as above, except the rips complex needs to be computed for each point cloud, not just the unions, and additional checks need to be done to make sure a simplex being added did not already appear and disappear once before.
If it did, the entry in \texttt{times\_list} needs to be extended to account for the newest appearance and disappearance.
Because of the additional Rips complex computations, and the checks for the special case, the case of a changing radius is significantly more computationally expensive than the case of a fixed radius.
In both cases, there is the computational cost of the zigzag persistence computation as well.
The computational complexity of zigzag persistence is $O(nm^2)$ where $n$ is the number of simplices in the entire filtration and $m$ is the number of simplices in the largest single complex \cite{Carlsson2009}.
Thus, the largest barrier to computation is the zigzag itself, so choosing a radius that is as small as possible without breaking the topology is the goal.
\section{Results}
We will test the BuZZ method on three different examples.
The first example is not based on time series data, but is instead a simple proof-of-concept example to test our methods ability to detect changing circular behavior.
The second example is based on synthetic time series data generated from noisy sine waves of varying amplitude.
This lets us fully utilize the BuZZ method, including the time delay embedding, as well as test resiliency to noise.
The last example is detecting a Hopf bifurcation in the Sel'kov model of glycolysis \cite{Selkov1968}.
\subsection{Synthetic Point Cloud Example}
\begin{figure}[]
\centering
\includegraphics[width=\textwidth]{Figures/Circle_Example_2.png}
\includegraphics[width=\textwidth]{Figures/Circle_Example.png}
\includegraphics[width=0.25\textwidth]{Figures/Circle_Example_PD.png}
\caption{
Top: Example zigzag of point clouds with unions considered in Sec.~\ref{ssec:SyntheticExample}.
Middle: Zigzag filtration applied to point clouds using the Rips complex with specified radii. Note that 2-simplices are not shown in the complexes.
Bottom: The resulting zigzag persistence diagram.}
\label{fig:circles}
\end{figure}
To start, we will consider a small, synthetic example generating point cloud circles of varying size as shown in Fig.~\ref{fig:circles}.
Note, because we are starting with point clouds, we skip the time delay embedding step for this example.
While each point cloud is sampled from a circle, the first and last point clouds consist of relatively small circles.
So the strongest circular structure we can see visually starts with $X_1$ and ends with $X_3$.
This is the range we would like to detect using zigzag persistence.
For this example, we will use the generalized version of the zigzag filtration in (\ref{eqn:zz_rips_fixed}) using a changing radii.
Computing the zigzag persistence gives the persistence diagram shown in Fig.~\ref{fig:circles}.
Recall that birth and death times are assigned based on the location in the zigzag that a feature appears and disappears.
Thus, the one-dimensional point $(1,3.5)$ in the persistence diagram corresponds to a feature that first appears at $R(X_1)$ and last appears in $R(X_3)$
Thus, using the persistence diagram we can detect the appearance and disappearance of the circular feature.
This is clearly an overly simplified situation as each point cloud is sampled from a perfect circle.
Next, we will look at a more realistic example.
\subsection{Synthetic Time Series Example}
\label{ssec:SyntheticExample}
\begin{figure}
\centering
\includegraphics[width=\textwidth]{Figures/Sine-TS-0p1.png}
\includegraphics[width=\textwidth]{Figures/Sine-TDE-0p1.png}
\includegraphics[width=0.69\textwidth]{Figures/Sine-Rips-0p1.png}
\includegraphics[width=0.3\textwidth]{Figures/Sine-PD-0p1.png}
\caption{First and second rows: Generated time series data and corresponding time delay embeddings.
Bottom left: The zigzag filtration using Rips complex with fixed radius of 0.72. Note that 2-simplices are not shown in the complexes.
Bottom right:the corresponding zigzag persistence diagram.
}
\label{fig:sine_ts}
\end{figure}
For the second example, we generate synthetic time series data and apply the full method described in Sec.~\ref{ssec:ourmethod}.
We start by generating sine waves of varying amplitudes and add noise drawn from uniformly from $[-0.1,0.1]$.
The time series are then each embedded using the time delay embedding with dimension $d=2$ and delay $\tau=4$.
The time series and corresponding time delay embeddings are shown in Fig.~\ref{fig:sine_ts}.
Looking at the time series, in the first and last time series any signal is mostly obscured by noise, resulting in a small clustered time delay embedding.
However, for the other time series, the time delay embedding is still circular, picking up the periodic behavior even with the noise.
Next we compute zigzag persistence, resulting in the zigzag of rips complexes and zigzag persistence diagram shown in Fig.~\ref{fig:sine_ts}.
The zigzag persistence diagram has a one-dimensional point with coordinates $(1,7.5)$, indicating the circular feature appears in $R(X_1)$, and disappears going into $R(X_8)$.
This is the region we would expect to see a circular feature.
\subsection{Sel'kov Model}
\begin{figure}[]
\centering
\includegraphics[width=0.8\textwidth]{Figures/Selkov_Model_FULL.png}
\includegraphics[width=0.69\textwidth]{Figures/Selkov_Rips_TDE.png}
\includegraphics[width=0.3\textwidth]{Figures/Selkov_PD.png}
\caption{
Top: Examples of samplings of the state space of the Sel'kov model for varying parameter value $b$.
Bottom left: zigzag filtration using Rips complex with fixed radius of 0.25. Note that 2-simplices are not shown in the complexes.
Bottom right: resulting zigzag persistence diagram.}
\label{fig:selkov}
\end{figure}
Our last experiment is trying to detect a bifurcation in the Sel'kov model \cite{Selkov1968}, a model for glycolysis which is a process of breaking down sugar for energy.
This model is defined by the system of differential equations,
\begin{align*}
\dot{x} & = -x+ay + x^2y \\
\dot{y} & = b - ay - x^2y
\end{align*}
where the overdot denotes a derivative with respect to time.
In this system, $x$ and $y$ represent the concentration of ADP (adenosine diphosphate) and F6P (fructose-6-phosphate), respectively.
This system has a Hopf bifurcation for select choices of parameters $a$ and $b$.
This limit cycle behavior corresponds to the oscillatory rise and fall of the chemical compounds through the glycolysis process.
For our experiments, we will fix $a=0.1$ and vary the parameter $b$.
We generate 500 time points of the data ranging between 0 and 500 using \texttt{odeint} in python, with initial conditions $(0,0)$.
We also remove the first 50 points to remove transients at the beginning of the model (this is sometimes referred to as a ``burn-in period'').
This data is constructed using full knowledge of the model, however, in practice, you typically only have one measurement function and then the time-delay embedding is used to reconstruct the underlying system.
To mimic this setup, we will only use the time series corresponding to the $x$-coordinates from the model and use the delay embedding.
These time series are then embedded using the time delay embedding with dimension $d=2$ and delay $\tau=3$.
The next step would be to compute zigzag persistence as described in Sec.~\ref{ssec:zigzag}, however due to the large number of points in the time delay embeddings, this becomes computationally expensive.
In order to reduce the computation time, we subsample these point clouds using the furthest point sampling method (also called a greedy permutation) \cite{Cavanna2015}.
We subsample down to only 20 points in each point cloud, compute the Rips complex zigzag for a fixed radius value of $0.25$, and then compute the zigzag persistence.
Figure~\ref{fig:selkov} shows the zigzag filtration of Rips complexes along with the resulting zigzag persistence diagram.
In the zigzag persistence diagram, the point with the longest lifetime has coordinates $(2,8.5)$.
Again, since these coordinates correspond to the index in the zigzag sequence, this point corresponds to a feature appearing at $R(X_2)$ and disappearing at $R(X_8 \cup X_9)$.
Looking back at which values of $b$ were used to generate these point clouds, we see this corresponds to a feature appearing at $b=0.45$ and disappearing at $b=0.8$.
For the fixed parameter value of $a=0.1$, the Sel'kov model has a limit cycle approximately between the parameter values $0.4 \leq b \leq 0.8$ \cite{Strogatz2014}.
Our method is picking up approximately that same range.
These results use the $x$-coordinates of the model, however the same results can be obtained using the $y$-coordinates and a slightly larger radius value.
\section{Discussion}
Here we have introduced a method of detecting Hopf bifurcations in dynamical systems using zigzag persistent homology called BuZZ.
This method was shown to work on two synthetic examples as well as a more realistic example using the Sel'kov model.
Our method is able to detect the range of the zigzag filtration where circular features appear and disappear.
Thus, this method could be applied to any application with an ordered set of point clouds and a changing topological structure.
While this method has shown success, it also has its limitations.
The method is computationally expensive due to numerous Rips complex computations in addition to the zigzag persistence computation itself.
This issue can be alleviated using subsampling, as shown with the Sel'kov model, but this may not be feasible depending on the application.
Future extensions of this project could include improvements of the algorithms described in Sec.~\ref{ssec:algorithm}.
Additionally, while the method works well in practice, it lacks theoretical guarantees.
Given the method requires parameter choices for the radii of the Vietoris-Rips complexes, we would like some heuristics to be used in practice to choose these radii more easily.
Because our examples in this paper are small, selecting parameters by hand is reasonable.
However, in the future when applied to larger, experimental data, these sorts of heuristics will be necessary.
|
2024-02-18T23:40:24.757Z
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2020-09-21T02:18:32.000Z
|
{"paloma_paragraphs":[]}
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{"arxiv_id":"2009.08972","language":"en","timestamp":1600654712000,"url":"https:\/\/arxiv.org\/abs\/2009.08972","yymm":"2009"}
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proofpile-arXiv_000-10080
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{"provenance":"002.jsonl.gz:10081"}
| null | null |
\section{Introduction}
In the AdS/CFT correspondence, the entanglement structure of the boundary CFT encodes the geometry and topology of the bulk AdS space \cite{vanRaams, Maldacena_2003, Balasubramanian_2014, Marolf_2015, doi:10.1002/prop.201300020}, albeit in a complicated and nonlocal way. We consider the dual of a state in the $\mathcal{N} = 4$ $SU(2N)$ Super Yang-Mills (SYM) theory in a Coulomb phase where the infrared modes are thermally entangled. We argue that this system should correspond to an asymptotically AdS$_5\times S^5$ geometry with a single boundary and a long-lived interior wormhole. The purpose of this paper is to construct this wormhole and describe its properties.
The simplest connection between wormholes and entanglement involves two copies of a CFT entangled in the ``thermofield double" (TFD) state, a two-party purification of the thermal state on each factor:
\begin{align}
|\text{TFD}\rangle = \sum_i e^{-\beta E_i / 2} |i\rangle \otimes |i^{\ast}\rangle \, .
\end{align}
where $|i^{\ast}\rangle$ indicates the CPT conjugate of $|i\rangle$. This system is dual to the eternal black hole, i.e., a wormhole between two asymptotically AdS universes \cite{Maldacena_2003}. To construct a wormhole between distant regions of a \emph{single} universe, we will study a state in the Coulomb branch of the $\mathcal{N}=4$ SYM theory \cite{KLEBANOV199989, PhysRevD.62.086003}, where the $SU(2N)$ gauge symmetry has been partially broken down to $S(U(N) \times U(N))$. In type IIB supergravity, the low energy effective theory dual to $\mathcal{N}=4$ SYM, this configuration corresponds to a multicenter solution sourced by two stacks of $N$ D3-branes \cite{CoulombBranch, PhysRevD.60.127902}. The procedure for constructing this geometry is illustrated in Fig.~\ref{fig:wormhole1}. We will begin with a two-centered BPS \cite{DUFF1991409} harmonic function solution as originally found in \cite{HOROWITZ1991197}, corresponding to two stacks of $N$ extremal D3-branes in (9+1)D Minkowski space and controlled by a parameter $L$. These stacks of D3-branes are separated by a distance $\Lambda$, corresponding to the Higgs scale. In the limit that $L\gg \Lambda$, there will be an AdS$_5 \times S^5$ geometry outside the region containing the two stacks of branes, which splits into two smaller AdS$_5 \times S^5$ regions as one nears either stack. Taking the limit $\alpha^{\prime} \to 0$ with the ratios of the five-sphere coordinates to $\alpha^{\prime}$ held fixed decouples the AdS regions from the asymptotically flat space, leaving a geometry which is asymptotically AdS \cite{largeN}.
Now we heat up the solution by entangling the degrees of freedom living on each brane (in each $SU(N)$ sector of the Higgsed SYM) up to the Higgs scale $\Lambda$ in an approximate thermofield double state. This has the effect that in the IR of the field theory, i.e. the deep bulk, the approximate thermofield double state will be dual to the two-sided planar AdS-Schwarzschild black brane \cite{Maldacena_2003} plus corrections due to the multicenter nature of the exterior solution. No known solution exists for the multicenter black brane geometry at nonzero temperature, so we solve for these corrections in perturbation theory. Matching these corrections in different coordinate patches glues together the wormhole solution. We will find that this gluing must introduce a global monodromy that inverts some spatial directions between the two throats in order to respect flux conservation. However, the full ten-dimensional spacetime remains globally orientable. The complete solution is unstable, as finite temperature breaks the supersymmetry of the BPS solution and turns on an effective potential for the scalar fields that break the $SU(2N)$ symmetry \cite{Kraus_1999, NAYEK2017192}. Supergravity wormhole solutions have been previously studied e.g. in \cite{BERGMAN2009300, Maldacena_2004, susywormhole}, but only in the case of connecting two different asymptotic spaces.
\begin{center}
\begin{figure}[hbtp!]
\includegraphics[width=\textwidth]{Wormhole1KleinUpdated.pdf}
\caption{Perturbative construction of an AdS wormhole solution in a single asymptotic space by taking the decoupling limit of two stacks of D3-branes and correcting for the finite temperature caused by the thermofield double entanglement structure below the scale $\Lambda$. Figure inspired by the near-horizon limit as depicted in \cite{costa, Michelson_1999}.\label{fig:wormhole1}}
\end{figure}
\end{center}
In Fig.~\ref{fig:wormhole2} we have labeled the different regimes in which a different coordinate patch or limit will be used to describe the solution. In region I, the solution is approximately the two-sided non-extremal black brane. Far from the horizon, in regions II and III, the effects of non-extremality are small and the solution is close to vacuum AdS$_5 \times S^5$. In this region, the perturbative corrections from the nonzero temperature and from the two throats can simultaneously be treated as linear corrections to the vacuum AdS$_5 \times S^5$ background, and therefore they linearly superpose. The leading corrections from the left throat are monopole corrections and do not break spherical symmetry around the right throat, which defines region II. However, we can include multipole effects from the left throat as linearized corrections, and these will be dominant over nonlinearities up to fourth order in the multipole expansion. This captures effects of the left throat breaking the spherical symmetry around the right throat, which defines region III. We will also be able to present solutions which are valid in regions I, II and parts of III simultaneously. These will be linearized perturbations of the finite temperature black brane geometry. These solutions show that leading multipole effects remain small near the causal horizons, but we find that they grow in the interior towards the singularity. Therefore the singularity in this wormhole is not of the AdS-Schwarzschild type. Finally, region IV is where both throats have non-perturbative effects, but their non-extremality is negligible and the solution is approximately the multicenter BPS solution, which in region V approaches that of pure AdS$_5 \times S^5$ with a larger AdS radius.
\begin{figure}[hbtp!]
\begin{center}
\includegraphics[width=.4\textwidth]{Wormhole2Klein.pdf}
\end{center}
\caption{Region I: Perturbatively corrected black brane glued with inversion to the rest of the geometry. Region II: Linearized perturbations to vacuum AdS. Region III: Perturbatively corrected throat solution. Region IV: The (extremal) two-throat solution. Region V: Far from both throats, vacuum AdS with larger radius.\label{fig:wormhole2}}
\end{figure}
Entanglement between disconnected non-interacting boundary theories gives rise to wormholes where the boundaries are separated by causal horizons \cite{GALLOWAY2001255}. A large body of recent work has also been directed towards finding mechanisms that can create and send signals through traversable wormholes in the context of AdS/CFT. In general, supporting a traversable wormhole requires that one violate the averaged null energy condition (ANEC) \cite{Morris:1988tu, PhysRevLett.90.201102, PhysRevLett.61.1446, PhysRevLett.81.746, GALLOWAY2001255}, meaning that there exists an infinite null geodesic with tangent $k^{\mu}$ and affine parameter $\lambda$ such that \cite{GaoJafferisWall}\footnote{The authors of \cite{susywormhole} argued that this condition could be avoided for supersymmetric traversable wormholes connecting two asymptotic AdS universes in the context of pure gauged $\mathcal{N} = 2$ supergravity in four dimensions.}
\begin{align}
\int_{-\infty}^{\infty} k^{\mu} k^{\nu} T_{\mu \nu} d\lambda < 0 \,.
\end{align}
Consequently, in order to build a traversable wormhole, there must be a negative source of stress-energy in the bulk. Several suggestions for introducing this negative stress-energy include inserting explicit double-trace couplings between the boundaries of the wormhole \cite{GaoJafferisWall, GJWBounds}, incorporating the perturbative gravitational back-reaction of bulk quantum fields \cite{Fu_2019, Fu_2019b}, including the Casimir energy of bulk fields running in non-contractible cycles \cite{SM_wormhole, selfsupp}, and nucleating and supporting wormholes via cosmic strings \cite{Horowitz_2019, Fu_2019a}. A particularly productive setting has been the correspondence between the SYK model and AdS$_2$ Jackiw-Teitelboim gravity \cite{doi:10.1002/prop.201700034, SYKwormhole, GaoJafferisSYK, SYKwormhole2}, which, while not an exact duality, has provided further support that explicit boundary couplings may render the bulk geometry traversable and provided an experimental setting by which probing wormhole traversability may be possible in the lab \cite{GJWprobes, wormhole_signaling}.
These constructions use the fact that the eternal AdS-Schwarzschild wormhole is marginally non-traversable in the sense that the null energy vanishes along the causal horizons, so arbitrarily small negative energy perturbations render the wormhole traversable. We will find that the leading classical corrections coming from the global structure of our single-boundary geometry preserve this marginal non-traversability.\footnote{We thank Simon Ross for discussions regarding this point.} This motivates us to describe a mechanism by which our single-boundary wormhole may become traversable by the presence of a natural ``double-trace" type operator in the IR of $\mathcal{N} = 4$ SYM generated by the Wilsonian RG flow \cite{connectivity}, although the presence of bulk fermions in the supergravity spectrum implies that other mechanisms mentioned above may also be a possibility depending, e.g., on the final sign of cancellations between Casimir energies.
The rest of this paper is organized as follows. In Sec.~\ref{sec:fieldtheory} we explain the pattern of symmetry breaking in the field theory and describe a particular entangled state in the IR. In Sec.~\ref{sec:wormholeGeomSoln} we solve for the metric and five-form of the wormhole solution in perturbation theory in type IIB supergravity and describe its global structure. In Sec.~\ref{sec:stability} we use the DBI action to estimate the instability timescale of the wormhole, and show that it is controlled by the same ratio of scales that governs the thermal effective potential in the SYM theory. We also show that it is unlikely that the wormhole could be stabilized by adding rotation. In Sec.~\ref{sec:traverse} we discuss a mechanism for rendering our wormhole traversable, and in Sec.~\ref{sec:doublewormhole} we explain how to use our results to construct a double wormhole between two asymptotic universes. We conclude in the Discussion with comments and remarks for future directions of study.
\noindent\textbf{Conventions} \\
We work in ``mostly-plus" signature for Lorentzian metrics. The convention for five-form components is that
$F = \frac{1}{5!}F_{\mu \alpha \beta\gamma \delta} dx^{\mu} \wedge \ldots \wedge dx^{\delta} = F_{t123r} dt \wedge \ldots \wedge dr + F_{\theta_1 \ldots \theta_5} d\theta_1 \wedge \ldots \wedge d\theta_5$ (all other components will be zero throughout this paper). The notation and combinatorial factors used in symmetrization of indices are for example $A_{(\mu}B_{\nu)} = \frac{1}{2!} (A_{\mu} B_{\nu} + A_{\nu} B_{\mu})$. The indices of all perturbative geometric quantities are raised with the background metric. In general this means index raising and lowering does not commute with perturbative variation. The action of the Hodge star on the components of $p$-forms in $d$ spacetime dimensions is $(\ast F)_{\nu_1 \ldots \nu_{d-p}} = \frac{1}{p!} \sqrt{-g} \epsilon_{\nu_1 \ldots \nu_{d-p} \sigma_1 \ldots \sigma_{p}} g^{\mu_1 \sigma_1} \ldots g^{\mu_p \sigma_p} F_{\mu_1 \ldots \mu_p}$ where $\epsilon_{\mu_1 \ldots \mu_n}$ is the Levi-Civita symbol and $\epsilon_{01\ldots (d-1)} = 1$.
\section{Description in Super-Yang Mills}\label{sec:fieldtheory}
The Lagrangian of $\mathcal{N} = 4$ SYM in terms of component fields is \cite{BRINK197777}
\begin{align}
\mathcal{L}_0 &= \text{tr} \biggl(-\frac{1}{2g_{YM}^2} F_{\mu \nu} F^{\mu \nu} + \frac{\theta}{16\pi^2} F_{\mu \nu} \tilde{F}^{\mu \nu} - i\bar{\lambda}^a \bar{\sigma}^{\mu} D_{\mu} \lambda_a - \sum_i D_{\mu} \phi^i D^{\mu} \phi^i \nonumber \\
&+g_{YM} \sum_{a,b,i} C^{ab}_i \lambda_a [\phi^i, \lambda_b] + g_{YM} \sum_{a,b,i} \bar{C}_{iab} \bar{\lambda}^a [\phi^i, \bar{\lambda}^b] + \frac{g_{YM}^2}{2} \sum_{i,j} [\phi^i, \phi^j]^2\biggr)\, ,
\end{align}
where $F_{\mu \nu} = \partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu} + i[A_{\mu}, A_{\nu}]$ is an $SU(2N)$ gauge field, $D_{\mu} \cdot = \partial_{\mu} \cdot + i[A_{\mu}, \cdot]$ is the covariant derivative on fields in the adjoint representation, $\lambda_a$ are four adjoint Weyl fermions, $\phi^i$ are six adjoint real scalars, and the $C^{ab}_i$ are the Clebsch-Gordon coefficients that couple two $\mathbf{4}$ representations of the $SU(4)_R$ symmetry to the $\mathbf{6}$ antisymmetric representation. The diagonal elements of the vacuum expectation value (vev) of the adjoint scalars $\phi^i$ in the AdS/CFT correspondence map to the positions of D3-branes in ten-dimensional flat space, while the off-diagonal elements are excitations of open strings stretching between the branes \cite{DouglasTaylor}.
At zero temperature, any diagonal configuration of $\phi^i$ gives rise to a vanishing commutator in the potential for the scalars, so there is a large moduli space of stable vacua. In the dual gravity theory, this is equivalent to the statement that an arbitrary number of D3-branes can be superposed at any location in space without any force between them. We choose a vev that will correspond simply to separating two stacks of $N$ D3-branes in a single transverse coordinate by distance $\Lambda$, by expanding
\begin{equation}
\phi^1 \to \psi + \varphi^1
\end{equation}
with the background $\psi$ given by
\begin{align}
\psi = \frac{1}{2\pi \alpha'}\text{diag}(\Lambda \sqrt{N} , \Lambda \sqrt{N} , \ldots, 0, 0, \ldots) \, , \label{eq:scalarvev}
\end{align}
where the eigenvalue $\Lambda \sqrt{N} $ is repeated $N$ times and the vevs of all other $\phi^i$ are zero.
We introduce an explicit factor of $\sqrt{N}$ to appropriately normalize the gauge-invariant classical observable $\text{tr} \, \psi^2$. In detail, we would like the classical value of the observable $\text{tr}\, \phi_1^2$, after proper normalization, to have a finite $\mathcal{O}(1)$ expectation in the large $N$ limit so that there is a well-defined classical gravitational dual. Next we observe that $\text{tr}\, \varphi_1^2$ scales as $\mathcal{O}(N)$, since the components of $\varphi_1$ are $\mathcal{O}(1)$, so its connected two-point function scales as $\langle \text{tr}\, \varphi_1^2 \text{tr}\, \varphi_1^2 \rangle \sim \mathcal{O}(N^2)$. But we know from large-N index counting that if the expectation value of a classical observable is taken to be $O(1)$, then the connected component of the two-point function of its quantum fluctuations should be $O(1/N^2)$. Therefore, the observable $\text{tr}\, \phi_1^2$ requires an overall normalization proportional to $N^{-2}$. Consequently, to make the properly normalized value $N^{-2} \text{tr}\, \psi^2$ of the classical observable $\mathcal{O}(1)$ in the large $N$ limit, a factor of $\sqrt{N}$ should be included in $\psi$.
The factor of $\alpha' = \ell_s^2$ is required by dimensional analysis since the field $\phi^1$ has mass dimension one in four spacetime dimensions. This should be understood as a scale coming from open string theory, since it provides the energy cutoff such that the massless excitations of the open string endpoints moving on the D-brane world-volumes are described by $\mathcal{N}=4$ SYM\footnote{The $\mathcal{N}=4$ SYM does not include $\alpha'$ as a parameter. We introduce $\alpha'$ in order to set the dimensions of $\psi$ correctly as a fiducial scale where we expect the SYM description to break down, anticipating the correspondence with the supergravity description to be discussed later.}. The effective Lagrangian for fluctuations about this background is, rescaling the gauge field $A_{\mu} \to g_{YM} A_{\mu}$ to canonically normalize its kinetic term,
\begin{align}
\mathcal{L} = \mathcal{L}_0\, + \,\text{tr} \biggl(&2 i g_{YM} [\psi, A_{\mu}] \partial^{\mu} \varphi^1 + g_{YM}^2 ([\psi, A_{\mu}]^2 + 2 [\psi, A_{\mu}][\varphi^1, A^{\mu}]) +g_{YM} \sum_{a,b} C_1^{ab} \lambda_a [\psi, \lambda_b] \nonumber\\
&+ g_{YM} \sum_{a,b} \bar{C}_{1ab} \bar{\lambda}^a [\psi, \bar{\lambda}^b]+g^2_{YM} \sum_i ([\psi, \phi^i]^2 + 2 [\psi, \phi^i][\varphi^1, \phi^i])\biggr)\, ,
\end{align}
where $\psi$ should be treated as a classical source.
To understand this effective Lagrangian, it is instructive to expand the commutator of the vev $\psi$ with an arbitrary Hermitian matrix $M$ in the adjoint of $SU(2N)$, which may be written in block form as $M = \begin{pmatrix} M_A & M_B \\ M_B^{\dagger} & M_C \end{pmatrix}$ where each block is $N \times N$:
\begin{align}
[\psi, M] &= \frac{\Lambda \sqrt{N} }{2\pi \alpha^{\prime} }\left(\begin{array}{cc} 0 & \:\:M_B \\ -M_B^{\dagger} & \:\:0\end{array}\right) = \frac{\Lambda \sqrt{N}}{2\pi \alpha^{\prime} } M_{AO} \, .
\end{align}
We have labeled the final matrix $M_{AO}$ for the ``antihermiticized off-diagonal" piece of $M$. Note that $\text{tr}\, M_{AO}^2$ is strictly negative, which is required to give the correct signs below. Armed with this knowledge we further rewrite the commutators in the effective Lagrangian to make the dependence on the coupling $\Lambda$ clear, defining the 't Hooft coupling $\lambda = g_{YM}^2 N$ \cite{HOOFT1974461}:
\begin{align}
\mathcal{L} = \mathcal{L}_0 \, + \, & \frac{1}{2\pi \alpha^{\prime}}\text{tr}\biggl( \Lambda\sqrt{\lambda} \, A_{\mu, AO} \partial^{\mu} \varphi^1 + \frac{\lambda\Lambda^2 }{2\pi \alpha' } \,A_{\mu, AO}^2+ 2 \frac{\lambda \Lambda}{\sqrt{N}} \,A_{\mu, AO} [\varphi^1, A^{\mu}]+ \Lambda \sqrt{\lambda}\sum_{a,b} C_1^{ab}\, \lambda_a \lambda_{b,AO} \nonumber \\
&+ \Lambda \sqrt{\lambda} \sum_{a,b} \bar{C}_{1ab}\, \bar{\lambda}^a \bar{\lambda}^b_{AO} + \frac{\lambda\Lambda^2 }{2\pi \alpha' } \sum_i (\phi^{i}_{AO})^2 + 2 \frac{\lambda \Lambda}{\sqrt{N}} \sum_i \phi^i_{AO}[\varphi^1, \phi^i] \biggr) \, . \label{eq:efflagrang}
\end{align}
The off-diagonal pieces of the gauge field, scalars, and fermions have acquired a mass \\ $ \Lambda \sqrt{\lambda} / (2\pi \alpha^{\prime} ) = \Lambda \sqrt{\lambda} / (2\pi \alpha^{\prime}) $.\footnote{This tree-level mass defines the Higgs scale at weak coupling. At strong coupling, the dependence on $\lambda$ may be different and we comment on this at the end of this section.} The diagonal blocks of the adjoint fields remain massless, so the background $\psi$ has Higgsed the theory $SU(2N) \to S(U(N) \times U(N))$. The extra terms remaining in \eqref{eq:efflagrang} coupling the gauge field to the scalar $\varphi^1$ are typical of those that appear in spontaneously broken non-Abelian gauge theories; we expect that there is a gauge choice which is an analog of the unitary gauge \cite{PhysRevLett.27.1688} where these terms vanish.
In the symmetry-broken phase that we have chosen, low-lying excitations above the vacuum are local to only one of the $SU(N)$ factors of the gauge group. This is because, as we have shown above, the off-diagonal degrees of freedom can be made very heavy by choosing a large Higgs scale $\Lambda\sqrt{\lambda} / (2\pi \alpha')$. Therefore, at energies below the Higgs scale, the Hilbert space of the theory approximately factorizes into that of two separate $SU(N)$ gauge theories, each of which is dual to an AdS throat in the 10D supergravity. Following the ER=EPR conjecture, a state in SYM that possesses the appropriate entanglement between the $\mathcal{O} (N^2)$ light degrees of freedom in each $SU(N)$ factor should be dual to two AdS throats connected by a wormhole in the bulk \cite{doi:10.1002/prop.201300020}. Specifically, we build the approximate thermofield double state\footnote{In the UV it is not possible to factorize the Hilbert spaces due to the $SU(2N)$ being gauged, but it is approximately possible below the Higgs scale.} coupling the energy eigenstates of the Hamiltonian for the effective IR fields in each $SU(N)$ sector
\begin{align}
|\text{TFD}_{\Lambda}\rangle = \sum_{i=1}^{E_i < E_c} e^{-\beta E_i / 2} |i\rangle_L \otimes |i^*\rangle_R \, ,\label{eq:tfdapprox}
\end{align}
where the sum runs over eigenstates of energy less than a cutoff energy scale $E_c$ set by the Higgs scale $\Lambda \sqrt{\lambda} / (2\pi \alpha')$, and the subscripts $L$ and $R$ refer to each of the two $SU(N)$ factors in the symmetry-broken theory. When the thermal energy density is much smaller than the cutoff energy density, we expect this state to be very close to the exact thermofield double state. In the deep bulk, this state is approximately dual to the planar two-sided AdS-Schwarzschild black brane which at fixed times describes a spacelike wormhole or ``Einstein-Rosen bridge" between two asymptotically AdS regions. However, in the ultraviolet of the field theory, the state \eqref{eq:tfdapprox} is embedded in a single $SU(2N)$ SYM theory, so in fact the wormhole begins and ends in the same asymptotic region.
Let us be a bit more precise about how close \eqref{eq:tfdapprox} is to the thermofield double. The dominant contribution to \eqref{eq:tfdapprox} comes at energies where the Boltzman factor offsets the growth coming from the number of states. At low enough temperatures we can think of the individual $SU(N)$ theories as being conformal and therefore their canonical energy density and entropy are fixed by scale invariance and dimensional analysis\footnote{These relations are derived from $\log Z = c V\beta^{-3}$.}
\begin{equation}
\langle E \rangle_\beta/V = 3c\beta^{-4}, \quad S_{th}/V = 4c \beta^{-3}\, ,
\end{equation}
where $c$ is a constant proportional to the central charge and $V$ is the spatial volume. The contribution of a canonical window of states at the cutoff energy $E_c/V=3c \beta_c^{-4}$ is then estimated to be
\begin{equation}
\label{eq:corrections}
e^{S_c-\beta E_c}\approx e^{V c \beta_c^{-4}(4 \beta_c-3 \beta)}\, ,
\end{equation}
i.e. we have an exponential suppression of these contributions if $\beta>4 \beta_c/3$. We take the cutoff temperature to be set by the Higgs scale $\beta_c^{-1} \lesssim \Lambda \sqrt{\lambda}/(2\pi \alpha')$.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=.5\textwidth]{Ueffplot.pdf}
\end{center}
\caption[The shape of the effective potential at finite temperature (orange) vs. zero temperature (blue). At zero temperature, the potential is flat and the components of the scalar background vev $\psi^i$ are free, while at finite temperature, the only stable configuration has all $\psi^i = 0$.]{The shape of the effective potential at finite temperature (orange) vs. zero temperature (blue). At zero temperature, the potential is flat and the components of the scalar background vev $\psi^i$ are free, while at finite temperature, the only stable configuration has all $\psi^i = 0$.\footnote[7]{\mbox{}}\label{fig:effpotential}}
\end{figure}
Reducing $|\text{TFD}_\Lambda\rangle$ onto the Hilbert space of either $SU(N)$ factor yields a state which looks approximately thermal in the infrared (up to corrections of order \eqref{eq:corrections}), with a temperature $\beta \sim r_0^{-1}$ that corresponds to a black brane of horizon radius $r_0$ in the gravity dual. In finite temperature field theory the supersymmetry of the SYM Lagrangian is broken and in particular the effective potential for the scalars $\phi^i$ is modified so that the only stable vacuum configuration is the one where the vevs of all the $\phi^i$ sit at the origin in moduli space. See Fig.~\ref{fig:effpotential} for a schematic depiction of the effective potential at finite temperature. Consequently, an initial configuration of the form \eqref{eq:scalarvev} is unstable and the vev will roll down the potential towards the origin. In the bulk dual, this has the well-known effect that nonextremal D-branes exert a nonzero attractive force on each other. Parametrically, at weak coupling $\lambda$ the thermal effective potential is controlled by the perturbatively small ratio of the thermal and Higgs scales, $\frac{2\pi \alpha'}{\beta \Lambda \sqrt{\lambda}}= \epsilon$. The weak coupling effective potential cannot be directly compared with the dual semiclassical gravity, since the latter is only valid when the field theory coupling is strong. Nonetheless, in Sec.~\ref{sec:stability} we will estimate the timescale of the instability from the gravity dual using the DBI action of the underlying branes. In terms of bulk quantities in the gravity dual, the perturbative parameter controlling the thermal effective potential at small 't Hooft coupling is
\begin{align}
\epsilon = \frac{\sqrt{2}}{\lambda} \frac{ r_0}{\Lambda }\, .
\end{align}
\setcounter{footnote}{7} \footnotetext{It may be surprising that the effective potential at zero temperature for the scalars is flat since the SYM theory is dual to the asymptotically AdS geometry that remains after the decoupling limit, and radially separated branes in asymptotically AdS space are subject to a potential barrier at infinity. However, the geometry sourced by two stacks of branes is a full ten-dimensional geometry that only approximately fibers into an AdS$_5$ and an $S^5$ close to each stack and near infinity. From the perspective of the asymptotic $S^5$, the branes are located at opposite poles and are not radially separated in the AdS space, so the AdS potential barrier does not apply.
}
At \emph{strong} coupling in SYM, where the bulk dual admits a semiclassical description in supergravity, quantities computed at weak field theory coupling are often rescaled by functions of $\lambda$ (see \cite{doi:10.1063/1.1372177} for a concrete example). Therefore, away from weak coupling we expect that the perturbative parameter that will control the bulk geometry in supergravity (and therefore parameterize the instability timescale of the wormhole) will take the form
\begin{align}
\epsilon = f(\lambda) \frac{\sqrt{2}}{\lambda} \frac{ r_0}{\Lambda } \, ,
\end{align}
for some function $f(\lambda)$. In the limit of large coupling $\lambda$ we will see that $f(\lambda) \sim \lambda$, so that the perturbative description of the classical geometry is naturally controlled by $\epsilon \sim r_0 / \Lambda$, which is independent of the string scale. It would be interesting to see if the function $f(\lambda)$ can be determined as an exact function of the coupling $\lambda$ using integrability techniques.
\section{Wormhole Geometry in Supergravity} \label{sec:wormholeGeomSoln}
In subsequent sections, we will write down the detailed solution to the equations of motion in each region. Our starting point for the construction, following \cite{pol95}, is the action of type IIB supergravity in string frame, restricted to the metric, dilaton, and five-form\footnote{The reviews \cite{DUFF1995213, peet} and textbooks \cite{kiritsis, johnson, amerd} provide compact and relevant introductions to D-brane solutions to type IIB supergravity that may be useful for subsequent sections.}:
\begin{align}
S_{\text{IIB}} = \frac{1}{2\kappa^2_{10}} \int d^{10} x \sqrt{-g} \left( e^{-2\phi} (R + 4\partial_{\mu} \phi \partial^{\mu} \phi) - \frac{1}{4\cdot 5! } F_{\mu \nu \rho \sigma \tau} F^{\mu \nu \rho \sigma \tau}\right) \, ,\label{eq:IIBaction}
\end{align}
where $2\kappa_{10}^2 = (2\pi)^7 \alpha'^{4} g_s^2$. The asymptotic value of the dilaton has already been scaled out so that $e^{\phi} = 1$ at infinity. We work in the strongly coupled limit of the field theory, $g_s N \to \infty$, such that classical supergravity is valid. The background values of the various fermions of type IIB are taken to be zero self-consistently.
Taking as an ansatz that the dilaton will be constant everywhere so that we can drop terms involving its gradient, the classical equations of motion are
\begin{align}
e^{-2\phi} (R_{\mu \nu} - \frac12 R g_{\mu \nu}) &= -\frac{1}{8\cdot 5!} g_{\mu \nu} F_{\alpha \beta \gamma \delta \epsilon} F^{ \alpha \beta \gamma \delta \epsilon} + \frac{1}{4\cdot 4!} F_{\mu \alpha \beta \gamma \delta} F^{ \:\:\alpha \beta\gamma \delta}_{\nu} \label{eq:geomEOM1} \\
\partial_{\mu} (\sqrt{-g} F^{ \mu \nu \rho \sigma\tau}) &= 0 \, , \label{eq:maxwellbgd}
\end{align}
to be supplemented by the self-duality constraint $F = \ast F$, and with $e^{\phi}= 1$ everywhere. Due to the self-duality constraint, $ F_{\alpha \beta \gamma \delta \epsilon} F^{ \alpha \beta \gamma \delta \epsilon} \sim F \wedge \ast F = F \wedge F = 0$ since the wedge product is antisymmetric on five-forms. A straightforward computation by taking traces and using this identity shows that $R = 0$. Consequently, \eqref{eq:geomEOM1} simplifies to
\begin{align}
R_{\mu \nu} &= \frac{1}{4\cdot 4!} F_{\mu \alpha \beta \gamma \delta} F^{ \:\:\alpha \beta\gamma \delta}_{\nu} \, . \label{eq:geomEOM2}
\end{align}
In subsequent sections we will write down perturbative corrections to solutions to the background equations of motion. We add these perturbative corrections at first order to the metric and five-form, $g_{\mu \nu} \to \bar{g}_{\mu \nu} + h_{\mu \nu}$ and $F \to \bar{F} + \delta F$, where the bar indicates quantities at background order, i.e. that solve \eqref{eq:maxwellbgd} and \eqref{eq:geomEOM2}. The perturbative equations of motion are
\begin{align}
\nabla_{\lambda} \nabla_{(\mu} h^{\lambda}_{\:\:\nu)} - \frac12 \nabla_{\mu}\partial_{\nu} h - \frac12 \nabla_{\lambda} \nabla^{\lambda} h_{\mu \nu} &= \frac{1}{4 \cdot 4!} (\delta F_{\mu \alpha \beta \gamma \delta} \bar{F}_{\nu}^{\:\:\alpha \beta \gamma \delta} + \bar{F}_{\mu \alpha \beta \gamma \delta} \delta F_{\nu}^{\:\: \alpha \beta \gamma \delta}) \label{eq:perturbativeEOMs1}\\
\partial_{\mu} \bigl[\sqrt{-\bar{g}} \bigl(\frac{h}{2} \bar{F}^{\mu \nu \rho \sigma \tau} + \delta F^{\mu \nu \rho \sigma \tau} \bigr) \bigr]&= 0 \, ,
\label{eq:perturbativeEOMs2}
\end{align}
where $h = \bar{g}^{\mu \nu} h_{\mu \nu}$ is the trace of the metric perturbation and $\nabla_{\mu}$ is the covariant derivative with respect to $\bar{g}$. These equations must be supplemented with the self-duality constraint at all orders, such that $\bar{F} + \delta F= \ast (\bar{F} + \delta F)$, a nontrivial constraint since the Hodge dual involves the metric perturbations. The derivation of \eqref{eq:perturbativeEOMs1} and \eqref{eq:perturbativeEOMs2} can be found in Appendix~\ref{sec:perturbEOM}. In the following sections, we will exhibit solutions to the background equations of motion \eqref{eq:maxwellbgd} and \eqref{eq:geomEOM2} and to their first-order variation in \eqref{eq:perturbativeEOMs1} and \eqref{eq:perturbativeEOMs2}. Although we have labeled the regions of the geometry I - V in order of the flow from the IR to the UV in the field theory, we will describe the solutions below in a different order that will be more convenient for intuition.
\subsection{Region IV: Two-Center Harmonic Solution} \label{sec:twocenter}
The general two-center solution at nonzero temperature is not known even in perturbation theory, so we will first write down the background solution to \eqref{eq:maxwellbgd} and \eqref{eq:geomEOM2} without perturbations in this region. We begin with the solution corresponding to two stacks of $N$ extremal D3-branes placed at a separation $\Lambda$ in 10D Minkowski spacetime. The coordinates $t, x^1, \ldots, x^3$ extend parallel to the brane worldvolumes; we label the other six directions transverse to the branes as $r^1$ through $r^6$. Without loss of generality let the two stacks of branes be displaced in the $r^1$ direction. The solution in asymptotically flat space is BPS and the metric and five-form are given by \cite{DUFF1991409, HOROWITZ1991197}
\begin{align}
\label{eq:harmonicmulticenter}
ds^2 &= H^{-1/2} (- dt^2 + d\vec{x}^2) + H^{1/2} \delta_{ij} dr^i dr^j \\
F&= (1+\ast) dt \wedge dx^1 \wedge dx^2 \wedge dx^3 \wedge dH^{-1} \, ,
\end{align}
where $H$ is a two-center harmonic function:
\begin{align}
H= 1 + \frac{L^4}{r^4} + \frac{L^4}{|\vec{r} \pm\vec{\Lambda}|^4} \, .
\end{align}
Here $\vec{\Lambda} = (\Lambda,0,0,0,0,0)$ and $r^2 = \sum_i (r^i)^2$. The choice of sign fixes the direction of displacement of the stacks of branes.
The flux through the five-sphere jumps discontinuously when the radius of the five-sphere around one stack of branes crosses through the other stack. When the radius is smaller than $\Lambda$, the charge of a single stack is, by Stokes' theorem,
\begin{align}
Q = \frac{1}{2\kappa^2}\int_{S^5} \ast F = \frac{L^4}{2g_s^2 (2\pi)^4 (\alpha')^4} \, .
\end{align}
The normalization comes from the normalization of the kinetic term for the five-form in \eqref{eq:IIBaction} \cite{pol95}. By the BPS condition, the charge is equal to the number of branes $N$ times the tension of a single extremal brane, $\tau =(2\pi)^{-3} (\alpha')^{-2} g_s^{-1}$, so
\begin{align*}
Q = N\tau \implies L^4 = 4\pi g_s N(\alpha')^2 \, .
\end{align*}
We now take the decoupling limit $\alpha' \to 0$ keeping fixed $\Lambda / \alpha'$ and $r^i / \alpha'$ \cite{costa}. Fixing $\Lambda/\alpha'$ amounts to fixing the Higgs scale of \eqref{eq:scalarvev} in the dual theory. To write a non-singular metric, we rescale $L^2 \to \alpha' L^2$, $r^i \to \alpha' r^i$, and $\Lambda \to \alpha' \Lambda$. The harmonic function becomes
\begin{align}
H = \alpha'^{-2} \left(\frac{L^4}{r^4} + \frac{L^4}{|\vec{r} \pm\vec{\Lambda}|^4} \right) \, .
\end{align}
Lastly, we nondimensionalize coordinates by the rescaling
\begin{align}
\frac{r_0 t}{L^2} = \tilde{t}, \qquad \frac{r}{r_0} = \tilde{r}, \qquad \frac{r_0 x^i}{L^2} = \tilde{x}^i \, . \label{eq:nondim}
\end{align}
Here $r_0$ is the wormhole horizon radius, to be introduced in subsequent sections. This nondimensionalization will be convenient in other regions where it removes the length scale $r_0$. We introduce the parameter $\epsilon = \frac{r_0}{ \Lambda}$. The full wormhole solution will only be valid in the limit $\epsilon \ll 1$ where the horizons of the two stacks of branes are well-separated. Given these definitions, the full solution in region IV is
\begin{align}
\frac{1}{\alpha'}ds^2 &= L^2 \biggl[\biggl(\frac{1}{\tilde{r}^4} + \frac{1}{|\vec{\tilde{r}} \pm \vec{\epsilon}^{-1}|^4} \biggr)^{-1/2} (- d\tilde{t}^2 + d\vec{\tilde{x}}^2) + \biggl(\frac{1}{\tilde{r}^4} + \frac{1}{|\vec{\tilde{r}} \pm \vec{\epsilon}^{-1}|^4} \biggr)^{1/2} \delta_{ij} d\tilde{r}^i d\tilde{r}^j\biggr] \\
\frac{1}{\alpha'^2} F&= L^4 \biggl[ (1+\ast) d\tilde{t} \wedge d\tilde{x}^1 \wedge d\tilde{x}^2 \wedge d\tilde{x}^3 \wedge d \biggl(\frac{1}{\tilde{r}^4} + \frac{1}{|\vec{\tilde{r}} \pm \vec{\epsilon}^{-1}|^4} \biggr)^{-1}\biggr] \, ,
\end{align}
where $\vec{\epsilon}^{-1} = (\epsilon^{-1},0,0,0,0,0)$.
It is convenient for subsequent sections to expand this solution close to the stack of branes at the origin, $r^i \ll \Lambda$. In this limit the (nondimensionalized) harmonic function becomes simply
\begin{align}
H = \frac{1}{\tilde{r}^4} + \epsilon^4 \, .\label{eq:Hlimit}
\end{align}
That is, all dependence on $r^1$ is subleading, so spherical symmetry about the stack of branes is valid in this limit. Furthermore, this expansion is valid around either stack provided the radial coordinate is defined appropriately. Defining the hyperspherical coordinates
\begin{align}
\begin{aligned}
r_1 &= r\cos \theta_1 \\
r_2 &=r\cos \theta_2\sin \theta_1 \\
r_3 &= r\cos \theta_3\sin \theta_2\sin \theta_1\\
r_4 &= r\cos \theta_4\sin \theta_3\sin \theta_2\sin \theta_1\\
r_5 &= r\cos \theta_5\sin \theta_4\sin \theta_3\sin \theta_2\sin \theta_1\\
r_6 &= r\sin \theta_5\sin \theta_4\sin \theta_3\sin \theta_2\sin \theta_1 \, ,
\end{aligned}
\end{align}
and series expanding $H^{\pm 1/2}$ yields
\begin{align}
\frac{1}{\alpha'} ds^2
&= L^2\biggl[ \tilde{r}^2 \left(1- \frac12 (\epsilon \tilde{r})^4\right)(-d\tilde{t}^2+d\vec{\tilde{x}}^2) + \left(1+\frac12 (\epsilon \tilde{r})^4 \right)(\frac{d\tilde{r}^2}{\tilde{r}^2} + d\Omega_5^2) \biggr] \label{eq:throatcorrect1}\\
\frac{1}{\alpha'^2} F &= 4L^4 \biggl[\tilde{r}^3 \left(1 - 2(\epsilon \tilde{r})^4\right) d\tilde{t} \wedge d\tilde{x}^1 \wedge d\tilde{x}^2 \wedge d\tilde{x}^3 \wedge d\tilde{r} \nonumber\\
&\qquad\qquad\qquad+ \sin^4 \theta_1 \sin^3 \theta_2 \sin^2 \theta_3 \sin \theta_4 d\theta_1 \wedge d\theta_2 \wedge d\theta_3 \wedge d\theta_4 \wedge d\theta_5\biggr] \, .\label{eq:throatcorrect2}
\end{align}
In this form, the linearized corrections to vacuum AdS$_5 \times S^5$ deep within a single throat are apparent. These corrections will be useful in subsequent sections.
\subsection{Region V: Asymptotics}
In the region far from both stacks of branes, we take the limit $r^i \gg \Lambda$, by which the harmonic function $H$ simplifies to
\begin{align}
H = \frac{2L^4}{r^4}
\end{align}
Writing $L_{\infty}^4 = 2L^4$, taking the decoupling limit, rescaling, and nondimensionalizing, the solution in region V is
\begin{align}
\frac{1}{\alpha'} ds^2 &= L_{\infty}^2\biggl[ \tilde{r}^2 (-d\tilde{t}^2+d\vec{\tilde{x}}^2) + \frac{d\tilde{r}^2}{\tilde{r}^2} + d\Omega_5^2 \biggr] \\
\frac{1}{\alpha'^2} F &= 4L_{\infty}^4 \biggl[\tilde{r}^3 d\tilde{t} \wedge d\tilde{x}^1 \wedge d\tilde{x}^2 \wedge d\tilde{x}^3 \wedge d\tilde{r} + \sin^4 \theta_1 \sin^3 \theta_2 \sin^2 \theta_3 \sin \theta_4 d\theta_1 \wedge d\theta_2 \wedge d\theta_3 \wedge d\theta_4 \wedge d\theta_5\biggr] \, .
\end{align}
This solution is vacuum AdS$_5 \times S^5$ with AdS length $L_{\infty}^4 = 2L^4$. The charge is
\begin{align}
Q = \frac{L_{\infty}^4}{2g_s^2 (2\pi)^4 (\alpha')^4} \, ,
\end{align}
leading to
\begin{align}
L_{\infty}^4 = 4\pi g_s (2N)(\alpha')^2
\end{align}
from the flux quantization condition. At infinity, the flux sees both stacks of branes as if they are at the origin, as expected. This region corresponds to the UV in the field theory where the $SU(2N)$ symmetry is unbroken.
\subsection{Region I: Black Brane}
In region I, the solution is the geometry of the two-sided black brane with perturbative corrections coming from the second throat in the full geometry. In this region, the solution will describe the geometry close to one of the two stacks with the origin of coordinates placed at the location of the stack, that is, in the limit $r^i \ll \Lambda$ of Sec.~\ref{sec:twocenter}. The solution preserves the $SO(3,1) \times SO(6)$ isometries induced by the brane locations. Following the conventions of \cite{peet}, the metric and five-form of the asymptotically flat solution are \cite{HOROWITZ1991197, DUFF1991409, DUFF1995213}
\begin{align}
ds^2 &= H(r)^{-1/2} (-f(r) dt^2 + d\vec{x}^2) + H(r)^{1/2} (dr^2 / f(r) + r^2 d\Omega_5^2) \\
F&= \sqrt{1+\frac{r_0^4}{L^4}}(1+\ast) dt \wedge dx^1 \wedge dx^2 \wedge dx^3 \wedge dH^{-1} \, ,
\label{eq:flatnonextremal}
\end{align}
with $H(r) = 1+\frac{L^4}{r^4}$ and $f(r) = 1- \frac{r_0^4}{r^4}$. As the horizon radius $r_0 \to 0$ one approaches the extremal limit of the brane solution. The five-form can be written explicitly in coordinates as
\begin{align}
F&= \sqrt{1+\frac{r_0^4}{L^4}} \biggl[\frac{4L^4}{ r^5 H(r)^2} dt \wedge dx^1 \wedge dx^2 \wedge dx^3 \wedge dr \nonumber \\
& \qquad\qquad\qquad\qquad + 4L^4 \sin^4 \theta_1 \sin^3 \theta_2 \sin^2 \theta_3 \sin \theta_4 d\theta_1 \wedge d\theta_2 \wedge d\theta_3 \wedge d\theta_4 \wedge d\theta_5\biggr] \, .
\end{align}
Consequently, the charge is
\begin{align}
Q = \frac{1}{2\kappa^2} \int_{S^5} \ast F = \frac{L^4}{2g_s^2 (2\pi)^4 (\alpha')^4 }\sqrt{1+\frac{r_0^4}{L^4}} \, .
\end{align}
The charge remains equal to the number of branes $N$ times the tension $\tau$ of an extremal brane, so
\begin{align}
Q = N\tau \implies L^4 = -\frac12 r_0^4 + \sqrt{(4\pi g_s N(\alpha')^2 )^2 + \frac14 r_0^8} \, .
\end{align}
We now take the decoupling limit $\alpha' \to 0$ keeping fixed $r^i / \alpha'$ and $r_0 / \alpha'$, rescaling $L^2 \to \alpha' L^2$, $r_0\rightarrow \alpha' r_0$ and $r \to \alpha' r $. The resulting solution has the same form as \eqref{eq:flatnonextremal} with $H=L^4/r^4$, no overall scaling on the five-form, and $f(r)$ unchanged. Nondimensionalizing following \eqref{eq:nondim} it can be written as
\begin{align}
\frac{1}{\alpha'} ds^2 &= L^2 \biggl[ -\tilde{r}^2(1-\frac{1}{\tilde{r}^4}) d\tilde{t}^2 + \tilde{r}^2 d\vec{\tilde{x}}^2 + \frac{d\tilde{r}^2}{\tilde{r}^2(1-\frac{1}{\tilde{r}^4})} + d\Omega_5^2 \biggr] \label{eq:blackbranebgdgeom} \\
\frac{1}{\alpha'^{2}} F &= 4L^4 \biggl[\tilde{r}^3 d\tilde{t} \wedge d\tilde{x}^1 \wedge d\tilde{x}^2 \wedge d\tilde{x}^3 \wedge d\tilde{r} + \sin^4 \theta_1 \sin^3 \theta_2 \sin^2 \theta_3 \sin \theta_4 d\theta_1 \wedge d\theta_2 \wedge d\theta_3 \wedge d\theta_4 \wedge d\theta_5 \biggr] \, . \label{eq:blackbranebgdform}
\end{align}
This is the background solution in region I. Note that the finite temperature factor has dropped out of the five-form after the decoupling limit. Consequently, the charge in the decoupling limit is simply
\begin{align}
Q = \frac{L^4}{2g_s^2 (2\pi)^4 (\alpha')^4 } \, .
\end{align}
This leads to the extremal quantization condition for the black brane in AdS,
\begin{align}
L^4 = 4\pi g_s N(\alpha')^2 \, .
\end{align}
In Sec.~\ref{sec:monopole} we will describe the perturbative corrections to \eqref{eq:blackbranebgdgeom}, \eqref{eq:blackbranebgdform} coming from the second throat, though we first describe the general structure of the multipole expansion that gives rise to these perturbative corrections in Sec.~\ref{sec:linearized}.
\subsection{Regions II-III: Linearized Regime} \label{sec:linearized}
Regions II and III are the intermediate regimes far from the horizon and sufficiently deep within a single throat such that the corrections to vacuum AdS$_5 \times S^5$ both from the throat and from the wormhole can be linearized. These regions are defined by $\tilde{r} \sim \mathcal{O}(\epsilon^{-1/2})$, where the background is empty AdS and corrections to this coming both from the harmonic function and the blackening factor are $\mathcal{O}(\epsilon^2)$. Since both of these corrections can be treated as linearized and the equations of motion are linear in the perturbations, the full solution can be written simply as the linear superposition of the two,
\begin{align}
\frac{1}{\alpha'}ds^2 &=H_0(r)^{-1/2} \left[ 1-\frac{1}{2}\frac{\delta H(r,\theta_1)}{H_0(r)}\right] \left(-\left[1-\frac{r_0^4}{r^4} \right] dt^2 + d\vec{x}^2 \right) \nonumber \\ &+ H_0(r)^{1/2} \left[ 1+\frac{1}{2}\frac{\delta H(r,\theta_1)}{H_0(r)}\right] \left(dr^2 \left[1+\frac{r_0^4}{r^4} \right] + r^2 d\Omega_5^2\right) \\
\frac{1}{{\alpha'}^2}F&= (1+\ast) dt \wedge dx^1 \wedge dx^2 \wedge dx^3 \wedge d\left[ H_0(r)^{-1}\left( 1-\frac{\delta H(r,\theta_1)}{H_0(r)}\right)\right] \, ,
\end{align}
where $H_0(r)=\frac{L^4}{r^4}$ and
\begin{equation}
\label{eq:harmonicmultipole}
\delta H(r,\theta_1)=\frac{L^4}{\Lambda ^4}+\frac{4 L^4 r \cos (\theta_1 )}{\Lambda ^5} +\frac{2 L^4 r^2 (3 \cos (2 \theta_1 )+2)}{\Lambda ^6}+\frac{4 L^4 r^3 (3 \cos (\theta_1 )+2 \cos (3 \theta_1 ))}{\Lambda ^7}+ \mathcal{O}(\Lambda^{-8}) \, ,
\end{equation}
and the Hodge star must be applied so that in the result we linearize both in $\delta H$ and $r_0^4$. This solves the linearized equations of motions simply because it is the sum of two linear perturbations of AdS$_5\times S^5$, one defined by expanding \eqref{eq:harmonicmulticenter} in $\delta H$ with $H=H_0+\delta H$, and the other by expanding \eqref{eq:blackbranebgdgeom} in $r_0^4/r^4$. Note that the leading nonlinearity from the two centered harmonic function comes at $(\delta H)^2 \sim \Lambda^{-8}$ and therefore we can keep the multipole expansion \eqref{eq:harmonicmultipole} up to $\mathcal{O}(\Lambda^{-7})$ in the linearized regime.\footnote{After nondimensionalizing, the multipole expansion is controlled by powers of $\epsilon$, so the leading nonlinearity is at $\mathcal{O}(\epsilon^8)$.}
We define region II as the patch where spherical symmetry around the throat is approximately unbroken and hence we can stop in the multipole expansion of $\delta H (r, \theta_1)$ at monopole order. This means keeping only the $L^4/\Lambda^4$ term in \eqref{eq:harmonicmultipole}. In this case the solution explicitly reads in nondimensionalized coordinates
\begin{align}
\frac{1}{\alpha'} ds^2 &= L^2 \biggl[-\tilde{r}^2\left(1- \frac{(\epsilon \tilde{r})^4}{2} -\frac{1}{\tilde{r}^4} \right) d\tilde{t}^2 +\tilde{r}^2\left(1- \frac{(\epsilon \tilde{r})^4}{2}\right) d\vec{\tilde{x}}^2 \nonumber\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad + \frac{1}{\tilde{r}^2}\left(1+\frac{(\epsilon \tilde{r})^4}{2} + \frac{1}{\tilde{r}^4}\right)d\tilde{r}^2 + \left(1+\frac{(\epsilon \tilde{r})^4}{2} \right) d\Omega_5^2 \biggr] \label{eq:linear1}\\
\frac{1}{\alpha^{'2}} F &= 4L^4 \biggl[\tilde{r}^3 \left(1 - 2(\epsilon \tilde{r})^4 \right) d\tilde{t} \wedge d\tilde{x}^1 \wedge d\tilde{x}^2 \wedge d\tilde{x}^3 \wedge d\tilde{r} \nonumber \\&\qquad\qquad\qquad\qquad\qquad+ \sin^4 \theta_1 \sin^3 \theta_2 \sin^2 \theta_3 \sin \theta_4 d\theta_1 \wedge d\theta_2 \wedge d\theta_3 \wedge d\theta_4 \wedge d\theta_5 \biggr] \, .\label{eq:linear2}
\end{align}
In this regime, the equations of motion can be solved by hand; see Appendix~\ref{sec:linearizedApp}. The procedure involves several undetermined constants and an undetermined function as a consequence of a residual diffeomorphism freedom.
\subsection{Joint solution in regions I-II: Monopole contribution}
\label{sec:monopole}
Now, we solve \eqref{eq:perturbativeEOMs1} and \eqref{eq:perturbativeEOMs2} for the perturbations to the metric and the five-form in the backgrounds of \eqref{eq:blackbranebgdgeom} and \eqref{eq:blackbranebgdform}, i.e. the non-extremal black brane. In this subsection we deal with the case when spherical symmetry is intact, that is, we solve for the monopole contribution of the far throat down the near throat. We begin with an ansatz for the perturbations consistent with the $SO(3,1) \times SO(6)$ symmetry
\begin{align}
\frac{1}{\alpha'}ds^2 &= L^2 \left[-\tilde{r}^2(1-\frac{1}{\tilde{r}^4})(1+\delta g_{\tilde{t}\tilde{t}}) d\tilde{t}^2 + \tilde{r}^2 (1+ \delta g_{\tilde{i}\tilde{i}}) d\vec{\tilde{x}}^2 + \frac{d\tilde{r}^2}{\tilde{r}^2(1-\frac{1}{\tilde{r}^4})} (1+\delta g_{\tilde{r}\tilde{r}}) + (1+\delta g_{\Omega \Omega}) d\Omega_5^2 \right] \\
\frac{1}{\alpha^{'2}} F &=4L^4\biggl[\tilde{r}^3 (1+a(\tilde{r})) d\tilde{t} \wedge d\tilde{x}^1 \wedge d\tilde{x}^2 \wedge d\tilde{x}^3 \wedge d\tilde{r} \nonumber \\
&\qquad + (1+b(\tilde{r}))\sin^4 \theta_1 \sin^3 \theta_2 \sin^2 \theta_3 \sin \theta_4 d\theta_1 \wedge d\theta_2 \wedge d\theta_3 \wedge d\theta_4 \wedge d\theta_5\biggr] \, ,
\end{align}
where the metric perturbations are all functions only of the radial coordinate $r$. In terms of the perturbations, the Maxwell equations and self-duality constraint reduce simply to
\begin{align}
2a-2b-3\delta g_{\tilde{i}\tilde{i}} - \delta g_{\tilde{t}\tilde{t}}-\delta g_{\tilde{r}\tilde{r}}+5\delta g_{\Omega \Omega} &= 0 \label{eq:selfdualBB} \\
2a'-3\delta g_{\tilde{i}\tilde{i}}' - \delta g_{\tilde{t}\tilde{t}}'-\delta g_{\tilde{r}\tilde{r}}'+5\delta g_{\Omega \Omega}' &= 0 \, .\label{eq:maxwellBB}
\end{align}
The geometric equations of motion are:
\begin{align}
\begin{aligned}
&\left(1-5 \tilde{r}^4\right) \delta g_{\Omega \Omega }'- \tilde{r} \left(\tilde{r}^4-1\right) \delta g_{\Omega \Omega }''-16\tilde{r}^3 (b-2 \delta g_{\Omega \Omega }) = 0 \\
&-16\tilde{r}^3 a + 24 \tilde{r}^3 \delta g_{\tilde{i}\tilde{i}} + 8\tilde{r}^3 \delta g_{\tilde{t}\tilde{t}} + 3(\tilde{r}^4 + 1)\delta g_{\tilde{i}\tilde{i}}' + 6\tilde{r}^4 \delta g_{\tilde{t}\tilde{t}}' \\
&\qquad\qquad\qquad- (\tilde{r}^4 + 1)\delta g_{\tilde{r}\tilde{r}}' + 5(\tilde{r}^4 + 1) \delta g_{\Omega \Omega}' + (\tilde{r}^5 - \tilde{r} ) \delta g_{\tilde{t}\tilde{t}}'' = 0 \\
&-16\tilde{r}^3 a + 24 \tilde{r}^3 \delta g_{\tilde{i}\tilde{i}} + 8\tilde{r}^3 \delta g_{\tilde{t}\tilde{t}} -4(2\tilde{r}^4-1) \delta g_{\tilde{i}\tilde{i}}'- (\tilde{r}^4-1)\delta g_{\tilde{t}\tilde{t}}' \\&\qquad\qquad\qquad+(\tilde{r}^4-1)\delta g_{\tilde{r}\tilde{r}}' - 5(\tilde{r}^4-1)\delta g_{\Omega \Omega}' - (\tilde{r}^5 - \tilde{r}) \delta g_{\tilde{i}\tilde{i}}'' = 0 \\
&-16\tilde{r}^3 a + 24 \tilde{r}^3 \delta g_{\tilde{i}\tilde{i}} + 8\tilde{r}^3 \delta g_{\tilde{t}\tilde{t}} + 3 (3\tilde{r}^4-1)\delta g_{\tilde{i}\tilde{i}}' + 3(\tilde{r}^4+1)\delta g_{\tilde{t}\tilde{t}}'-2(2\tilde{r}^4-1) \delta g_{\tilde{r}\tilde{r}}' \\
&\qquad\qquad\qquad+5(\tilde{r}^4+1)\delta g_{\Omega \Omega}' + 3(\tilde{r}^5-\tilde{r})\delta g_{\tilde{i}\tilde{i}}'' + (\tilde{r}^5-\tilde{r})\delta g_{\tilde{t}\tilde{t}}''+5(\tilde{r}^5-\tilde{r}) \delta g_{\Omega \Omega}'' = 0 \, .
\end{aligned}
\end{align}
Note that we are in the region $\tilde{r}>1$ outside the horizon at all times. We solve the equations of motion by the ansatz that the perturbations will be equal to the near-horizon corrections of the two-throat solution given in \eqref{eq:throatcorrect1} and \eqref{eq:throatcorrect2}. This ansatz turns out to be correct provided that we include an additional contribution to the five-form perturbations $a(\tilde{r})$ and $b(\tilde{r})$. The full perturbative solution in region I is
\begin{align}
\label{eq:monopole}
\frac{1}{\alpha'}ds^2 &= L^2 \biggl[-\tilde{r}^2(1-\frac{1}{\tilde{r}^4})(1-\frac12 (\epsilon \tilde{r})^4) d\tilde{t}^2 + \tilde{r}^2 (1-\frac12 (\epsilon \tilde{r})^4 ) d\vec{\tilde{x}}^2 \nonumber\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad + \frac{d\tilde{r}^2}{\tilde{r}^2(1-\frac{1}{\tilde{r}^4})} (1+\frac12 (\epsilon \tilde{r})^4) + (1+\frac12 (\epsilon \tilde{r})^4 ) d\Omega_5^2 \biggr] \\
\frac{1}{\alpha^{'2}} F &=4L^4\biggl[\tilde{r}^3 \left(1-2(\epsilon \tilde{r})^4 + \frac{\epsilon^4}{2}\right) d\tilde{t} \wedge d\tilde{x}^1 \wedge d\tilde{x}^2 \wedge d\tilde{x}^3 \wedge d\tilde{r} \nonumber \\
&\qquad\qquad\qquad\qquad\qquad +(1+\frac{\epsilon^4}{2})\sin^4 \theta_1 \sin^3 \theta_2 \sin^2 \theta_3 \sin \theta_4 d\theta_1 \wedge d\theta_2 \wedge d\theta_3 \wedge d\theta_4 \wedge d\theta_5\biggr] \, .
\end{align}
In this regime, we are close to the horizon, so $\tilde{r} \sim \mathcal{O}(1)$ and the perturbative corrections are $\mathcal{O}(\epsilon^4)$. When $\tilde{r}$ gets large, this solution matches onto \eqref{eq:linear1} in the linearized regime, region II.
The physical interpretation of the leading monopole contribution from the presence of the other throat is to create a small region of flat space around the black brane. This can be seen by noting that \eqref{eq:monopole} can be obtained by linearizing in $1/\Lambda^4$ the non-perturbative solution (in dimensionful coordinates):
\begin{align}
\frac{1}{\alpha'}ds^2 &=\left(\frac{L^4}{r^4}+\frac{L^4}{\Lambda^4} \right)^{-1/2} \left(-\left[1-\frac{r_0^4}{r^4} \right] dt^2 + d\vec{x}^2 \right) + \left(\frac{L^4}{r^4}+\frac{L^4}{\Lambda^4} \right)^{1/2} \left(dr^2 \left[1-\frac{r_0^4}{r^4} \right]^{-1} + r^2 d\Omega_5^2\right) \label{eq:metricnonpert}\\
\frac{1}{{\alpha'}^2}F&= \sqrt{1+\frac{r_0^4}{\Lambda^4}} (1+\ast) dt \wedge dx^1 \wedge dx^2 \wedge dx^3 \wedge d\left(\frac{L^4}{r^4}+\frac{L^4}{\Lambda^4} \right)^{-1} \label{eq:fiveformnonpert}\, ,
\end{align}
which is obtained by truncating the multipole expansion at monopole order but keeping the nonlinear dependence on the harmonic functions, as well as the blackening factor. It is easy to see that this solution is just a single non-extremal black brane in asymptotically flat space, in rescaled coordinates $t'=\frac{\Lambda}{L}t$, $\vec{x}'=\frac{\Lambda}{L}\vec{x}$, $r'=\frac{L}{\Lambda}r$. This rescaling puts the solution \eqref{eq:metricnonpert} in the form \eqref{eq:flatnonextremal} but with a rescaled horizon radius $r_0'= \frac{L}{\Lambda}r_0$.
\subsection{Joint solution in regions I-II-III: Dipole contribution}
\label{sec:dipole}
It is interesting to ask if we can capture the leading effect of spherical symmetry breaking on the wormhole. The above solutions contain the monopole contribution from the presence of the other throat. At next order, there is a dipole contribution from the harmonic function
\begin{equation}
H=L^4\left(\frac{1}{r^4} + \frac{1}{\Lambda^4} + 4 \frac{ r \cos \theta_1}{\Lambda^5} + \cdots \right)\, , \label{eq:dipoleharmonic}
\end{equation}
We look for a solution including dipole effects in all three regions I-III, that is, we keep the blackening factor exact. We take a general ansatz where the harmonic functions $H_g$ in the metric and $H_F$ in the five-form are allowed to be different,
\begin{align}
\label{eq:dipole}
\frac{1}{\alpha'}ds^2 &= H_g(r,\theta_1)^{-1/2} (-f(r) dt^2 + d\vec{x}^2) + H_g(r,\theta_1)^{1/2} (dr^2 / f(r) + r^2 d\Omega_5^2) \\
\frac{1}{{\alpha'}^2}F&= B(1+\ast) dt \wedge dx^1 \wedge dx^2 \wedge dx^3 \wedge dH_F^{-1} \, ,
\end{align}
with $f(r)=1-r_0^4/r^4$ and $B=1+\frac{r_0^4}{2 \Lambda^4}$. Similarly, we take a general ansatz for $H_g$ and $H_F$ whereby both must be asymptotically equal to \eqref{eq:dipoleharmonic} as $r\to\infty$,
\begin{align}
H_g &= L^4\left(\frac{1}{r^4} + \frac{1}{\Lambda^4} + 4 \frac{ h_g(r) \cos \theta_1}{\Lambda^5} + \cdots \right) \\
H_F &= L^4\left(\frac{1}{r^4} + \frac{1}{\Lambda^4} + 4 \frac{ h_F(r) \cos \theta_1}{\Lambda^5} + \cdots \right)\, ,
\end{align}
that is, $h_g\sim h_F \sim r$ as $r\rightarrow \infty$. Requiring the $r\theta_1$ component of the curvature equation of motion \eqref{eq:geomEOM2} to vanish at order $1/\Lambda^5$ gives
\begin{equation}
h_F(r)=\frac{2r^4-r_0^4}{2r^4}h_g(r) \, .
\end{equation}
Imposing this, it turns out all the remaining components of \eqref{eq:geomEOM2}, as well as the only non-vanishing component of Maxwell's equation, $(d F)_{r \theta_1...\theta_5}=0$ are proportional to the equation
\begin{equation}
-5 r^3 h_g+(r^4-r_0^4)(5 h_g'+r h_g'')=0 \, .
\end{equation}
This is a second order equation with two initial conditions. One is fixed by $h_g(r\rightarrow \infty ) \rightarrow r$. The other is fixed by requiring the solution to stay real in the interior of the wormhole, $r<r_0$. It turns out that the latter condition translates into $h_g(r_0)=0$, so that the location of the horizon is not affected by the perturbation. The solution is then
\begin{align}
h_g(r)&=r_0 Q\left( \frac{r_0^4}{r^4}\right), \\
Q(x)&=\frac{\, _2F_1\left(-\frac{5}{4},-\frac{1}{4};-\frac{1}{2};x\right)+\frac{8 x^{3/2} \Gamma \left(\frac{5}{4}\right) \Gamma \left(\frac{9}{4}\right) }{3 \Gamma \left(-\frac{1}{4}\right) \Gamma \left(\frac{3}{4}\right)}\, _2F_1\left(\frac{1}{4},\frac{5}{4};\frac{5}{2};x\right)}{x^{1/4}}\, ,
\end{align}
and is analytic at $r=r_0$ ($x=1$) due to the cancellation of the branch cuts starting at $x=1$ that are separately present in the two hypergeometric functions.
It would be interesting to further analyze this solution. It seems like it is not possible to have a perturbation that decays towards the singularity $r\rightarrow 0$. Instead, the perturbation decays towards the horizon, i.e. it is decaying in tortoise coordinates. So the presence of the other throat appears to have a significant effect on the interior, where the perturbation becomes large again as we approach the singularity, since $Q(x\rightarrow \infty) \sim \frac{2 \sqrt{2 \pi } x}{\Gamma \left(-\frac{1}{4}\right) \Gamma \left(\frac{3}{4}\right)}$. The singularity inside this wormhole is therefore not of the AdS-Schwarzschild type. The $S^5$ does not factorize, so the geometry is really a full ten-dimensional wormhole.
\subsection{Global Structure and Flux Conservation}
\label{sec:global}
Here we discuss how regions I-II-III (the wormhole) should be glued to regions IV-V (the two throats in a single spacetime) so that the five form flux is conserved.\footnote{We thank Juan Maldacena for raising this point.} The gluing procedure leads to some interesting global properties of the wormhole. We will show that the spacetime has a moduli space coming from the freedom to add a certain amount of twisting during gluing.
In the geometry that we have described, both AdS throats have a positive net five-form flux towards infinity, so that there are $2N$ units of flux near infinity and $N$ near each throat. This presents a puzzle: if there are no sources in the wormhole, flux conservation demands that the flux should thread through the wormhole and close in the outside, giving zero net flux far away from the throats rather than $2N$. Our setup is analogous to the circuital law for a magnetic field in two dimensions, where the closed line integral must be conserved if there are no sources for the curl. This is illustrated for a 2D wormhole in Fig.~\ref{fig:wormholeflux}, where again, the wormhole without sources has zero line integral for the magnetic field on a loop enclosing both throats. However, one may support a non-vanishing line integral on such a loop purely by modification of the geometry, without adding sources. This is achieved by cutting open the wormhole, and gluing it back to the ambient space while twisting to invert the angular coordinate. This results in a sourceless ``Klein-bottle" wormhole, which is a non-orientable surface that supports a nonzero circuital flux at infinity. This is shown on Fig.~\ref{fig:kleinflux}.
\begin{figure}[hbtp!]
\begin{center}
\subfloat[\label{fig:wormholeflux}]{\includegraphics[width =.5\textwidth]{WormholeFlux.pdf}}
\subfloat[\label{fig:kleinflux}]{\includegraphics[width =.5\textwidth]{KleinFig.pdf}}
\end{center}
\caption{Spacetime wormholes with orientable (a) and non-orientable (b) Cauchy slices. In the orientable wormhole, conservation of circuital flux demands that the flux reverses direction at the second throat relative to the first throat, while in the non-orientable wormhole the flux at both throats points the same direction.\label{fig:fluxgeom}}
\end{figure}
The way that flux conservation works in our supergravity wormhole is very similar, although there are some technical differences because the flux comes from a five-form and lives in ten dimensions. In particular, the Cauchy slices of the wormhole will remain orientable. We illustrate on a spacetime diagram in Fig.~\ref{fig:penroseglueing} the two throats and the wormhole before we glue them together, and the orientation of the coordinate differentials. The left and right throats share a time coordinate $t_g$ and three spatial coordinates $x^i_g$ which are globally defined with the same orientation in the ambient space outside the throats. However, the natural radial and angular coordinates $r_L$ and $\theta_L^i$ at the left throat do not coincide with the corresponding coordinates $r_R$, $\theta_R^i$ at the right throat. This is because the geometry only fibers into AdS$_5 \times S^5$ near each throat, so the two five-spheres are centered at different points.
\begin{figure}[hbtp!]
\begin{center}
\includegraphics[width=.7\textwidth]{penroseglueing2.pdf}
\end{center}
\caption{Spacetime diagram of gluing the wormhole to the throat regions. In the middle we have the Penrose diagram of the eternal black brane, and the sides represent the throat regions. We show the orientation of the coordinate differentials $dr$, $dt$, and $d\theta^1$. The $d\theta^1$ differential points out from the plane of the figure in the left throat and in the wormhole, but it points inwards in the right throat. The three differentials must always form the same right handed system.\label{fig:penroseglueing} }
\end{figure}
For the following discussion, by the \emph{electric part} of the five-form we refer to the term proportional to $dt\wedge dx^1\wedge dx^2 \wedge dx^3 \wedge dr$ and by the \emph{magnetic part} we refer to the term proportional to $d\text{Vol}_{S^5}$. Now, both electric and magnetic parts of the five-form are oriented in the same direction in the ambient spacetime, so deep in each throat, both have the same expression in local coordinates: $dt_g \wedge dx^1_g \wedge dx^2_g \wedge dx^3_g \wedge dr_{(L/R)}$ for the electric part and $d\text{Vol}_{S^5 (L/R)}$ for the magnetic part. At the left side of the wormhole, we choose the exterior Schwarzschild coordinates in the left wedge of the Penrose diagram to match the direction of local coordinates of the left throat: $dt = dt_g$, $dx^i = dx^i_g$, $dr = dr_L$, $d\theta_i = d\theta^i_L$ \footnote{The equalities that describe the ``gluing" between the left/right throats and left/right exterior wedges should be understood to be specifying the transition functions on the wormhole manifold in the coordinate patches where they are defined.}. In the right wedge, the radial coordinate points outwards towards the right throat, and the Schwarzschild time coordinate runs downward, in the opposite direction as the left wedge. However, we would like to glue the throats to the wormhole so that time points up on both sides. This is what we expect from the field theory, since after Higgsing the SYM Hamiltonian looks like $H_L + H_R$ in the IR, which generates upwards time evolution on both sides. Therefore, in the right wedge, we must take $-dt = dt_g$ and $dr = dr_R$.
The gluing of the rest of the coordinate directions at the right interface is determined by requiring the five form to be continuous. Consider starting with the five-form in the left throat and continuing into the left exterior wedge and across the wormhole to the right exterior wedge, where we must glue the geometry back to the right throat. On Fig.~\ref{fig:penroseglueing} we show the orientation of the coordinate differentials that are changing during this process. The rest of the coordinate differentials, $dx^i$, $i=1,2,3$ and $d\theta^i$, $i=2,\ldots ,5$ are oriented the same way throughout the figure. In the Schwarzschild coordinates, the five-form has the same solution in both the left and right wedges. But note that in the right exterior wedge of the wormhole, both $dr$ and $dt$ are flipped in Schwarzschild coordinates relative to the left exterior wedge. Since both of these are flipped, the electric part of the five-form, $F_5 \sim dt\wedge dx^1\wedge dx^2 \wedge dx^3 \wedge dr$ actually keeps its orientation throughout the wormhole region. The same applies for the magnetic part, since the $S^5$ approximately factorizes in the wormhole.
On the other hand, in the right throat, the basis differentials $dr_R$ and $d\theta^1_R$ are flipped relative to the left throat. However, the solution for the five-form looks the same in terms of these coordinate differentials in both throats. So the orientation of \textit{both} the electric part $dt_R\wedge dx^1\wedge dx^2 \wedge dx^3 \wedge dr_R$ \textit{and} the magnetic part $d\theta^1_R \wedge d\theta^2\wedge d\theta^3 \wedge d\theta^4\wedge d\theta^5$ of the five form appear reversed compared to the right wedge of the wormhole for the purpose of gluing them.\footnote{What we mean here is that tracking the global five-form from the left throat to the right throat on the outside results in five-forms that point in opposite directions in the left and right throats if we draw them as in Fig.~\ref{fig:penroseglueing}.} So a direct gluing would lead to a discontinuous five-form. However, we can follow the idea from Fig. \ref{fig:kleinflux} and perform the gluing by twisting the $x^i$ coordinates (parallel to the brane) and the $\theta^1$ coordinate by an inversion at the gluing surface.
In terms of transition functions between the right wedge and the right throat, this works as follows. First we align the basis of coordinate differentials on the two sides by introducing new coordinates $t'=-t$ and $(\theta^1)'=-\theta^1$ in the right Schwarzschild wedge. In this properly aligned basis, there is an explicit sign difference in both the electric and magnetic components of the five form compared to the right throat. Then in order to make the five form components continuous, we glue with the transition functions $t_R\equiv t_g=t'$, $x_R^i\equiv x^i_g=-x^i$ and $\theta^1_R=-(\theta^1)'$. This way, in the electric part we make up for the sign by inverting $x^i$ in the gluing function, while in the magnetic part we invert the $\theta^1$ direction. This makes the complete five form continuous. Note that since the total determinant of this twist is positive, the Cauchy slice remains orientable, as opposed to the 2d example of Fig. \ref{fig:kleinflux}.\footnote{The total spacetime is also orientable since there exists a globally defined ``upwards" time, which in the wormhole region is just Kruskal time.}
Note that this twisted gluing results in a smooth geometry, because the $O(3)$ symmetry of the $x^i$ subspace is unbroken by the configuration of two throats, so nothing will depend on these coordinates even in the fully nonlinear time-dependent solution that we have not written down. Similarly, there is an unbroken $O(5)$ subgroup of the $O(6)$ acting on the $r_i$ coordinates, where the $O(5)$ fixes $r_1$, the direction in which the throats are separated. The $\theta^1$ twisting is an inversion of the $r_2,\ldots ,r_6$ coordinates, which is an element $R\in O(5)\subset O(6)$ of this unbroken symmetry with $\det R=-1$.
Now we briefly discuss the moduli space of solutions. Note that the only restrictions on the spatial twisting at the right gluing surface are that (i) it is from the subgroup which remains a symmetry of the solution, (ii) it reverses the orientation of both $dx^1\wedge dx^2 \wedge dx^3$ and $d\text{Vol}_{S^5}$. This gives a freedom in picking the group element with which we twist the gluing, resulting in a moduli space. In addition to the spatial twisting, as pointed out in \cite{Verlinde2020}, one may introduce a constant time shift in the identification of Schwarzschild time with the global time in the throat, which gives an extra real parameter (the difference between constant time shifts between left and right). Therefore, the total moduli space of single boundary wormholes is
\begin{equation}
\mathbb{R} \times ISO(3) \times SO(5).
\end{equation}
Here, $ISO(3)$ denotes the group of (orientation-preserving) rotations and translations of the $x^i$ coordinates.
Finally, let us comment on the field theory interpretation of the gluing twisted by inversions. The inversion of Schwarzschild time and the parallel coordinates $x^i$ correspond to time reversal and parity (TP) in the right IR $\mathcal{N}=4$ SYM factor. The inversion on the $S^5$ corresponds to inverting the $R$ charges in the field theory, so it is natural to think about it as the action of charge conjugation $C$. Therefore, from the field theory point of view, the right IR field theory factor is ``glued back" to the UV field theory by an action of CPT. This is natural for the following reason. As discussed before, the state in the IR looks like the thermofield double state. The TFD state is defined from the square root of the thermal density matrix $\rho^{1/2}$, which is an element of $\mathcal{H}\otimes \mathcal{H}^*$, where $\mathcal{H}^*$ denotes the dual Hilbert space, where bra vectors live. In order to define the thermofield double, which lives on a doubled Hilbert space $\mathcal{H}\otimes \mathcal{H}$, one needs to turn the bra vectors into ket vectors with an anti-unitary symmetry. There is one such anti-unitary transformation that is a symmetry in any quantum field theory, which is CPT.
\section{Instability of the Solution}\label{sec:stability}
\subsection{Instability Timescale}
The wormhole solution we have described is not an extremal (BPS) solution of type IIB: its mass is larger than its charge. Consequently, it suffers from an instability: the attractive gravitational (NS-NS) force is larger than the repulsive five-form (R-R) force between the underlying branes, so that at late times the wormhole disappears as the two stacks of $N$ branes collide and form a single stack of $2N$ branes at nonzero temperature. We can compute the time scale of this instability by examining the tree-level effective action governing the dynamics of one stack of branes in the background of the other stack. The full action is the Dirac-Born-Infeld (DBI) action describing the geometric dynamics of the branes and their coupling to open strings, plus the coupling of the branes to the five-form \cite{GIBBONS1998603}:
\begin{align}
S = -T \int d^{p+1} \xi \:\sqrt{-\text{det}\,G_{ab}} + \mu\int d^{p+1} \xi \: C_4 \, . \label{eq:dbiaction}
\end{align}
Here the $\xi^i$ are coordinates on the world-volume of a brane, $T$ is the tension of the stack of branes, $\mu$ is the charge density coupling to the five-form, $G_{ab}$ is the pullback of the background metric $g_{\mu \nu}$ to the brane, and $C_4$ is the pullback of the potential for the five-form. Evaluating the DBI action using the classical metric and five-form gives the effective action at tree level where we have taken the backgrounds for the antisymmetric two-form and the gauge field on the brane to be zero.
In order to study the dynamics of one stack of branes as a probe, the backreaction of the probe on the background geometry should be negligible. However, this is not the case in the full two-center geometry, as each stack of branes sources its own independent AdS throat. Moreover, each of the stacks have a field that is the size of the AdS radius, and since they are separated in the asymptotic $S^5$ directions which have comparable size, we cannot treat the two stacks as point-like objects interacting via weak fields. Nevertheless, we may obtain a lower bound on the timescale of the instability by considering the motion of an extremal probe brane located halfway between the two stacks of branes, where we heat up one stack slightly and leave the other extremal. We may think about the extremal probe as being separated from the extremal stack. Such a brane experiences a higher acceleration than a brane located deep in the AdS throat of the extremal stack, where it is further from the thermal stack. We will show that the temperature and separation of the branes can be chosen so as to make this lower bound on the instability timescale arbitrarily high, i.e. the wormhole is long-lived.
The extremal probe brane starts at a point in the geometry which can be approximated by the flat-space region of \eqref{eq:metricnonpert}, \eqref{eq:fiveformnonpert}, far from both the horizon of the thermal branes and the AdS throat of the extremal branes. To compute the pullback of the metric and four-potential, we use spacetime Lorentz transformations and world-volume reparameterizations to work in ``static gauge" in which the world-volume coordinates are parallel to the spacetime coordinates.
\begin{align}
\xi^0 = t, \qquad \xi^i = x^i \, ,
\end{align}
where $i = 1,2,3$. The pullback of the potential to a brane sitting at distance $r$ from the thermal stack is then
\begin{align}
C_4 = \sqrt{1+\frac{r_0^4}{\Lambda^4}} \left(\frac{L^4}{r^4} + \frac{L^4}{\Lambda^4} \right)^{-1} dt \wedge dx^1 \wedge dx^2 \wedge dx^3 \, .
\end{align}
In spherical coordinates, the probe brane moves only in the radial direction, so the pullback of the metric is given by
\begin{align}
G_{00} = g_{00} + \dot{r}^2 g_{rr}, \qquad G_{ii} = g_{ii} \, ,
\end{align}
and all other components are zero. Consequently, the effective action experienced by the probe brane is, defining $M = TV$ and $Q = \mu V$ as the effective mass and charge of the probe brane where $V = \int d^3\xi^i$ is the (regularized) brane world-volume,
\begin{align}
S = \frac{\Lambda^2}{L^4}\int dt\, \frac{r^4}{r^4+\Lambda^4}\biggl( -M \Lambda^2 \sqrt{1-\frac{r_0^4}{r^4}- \dot{r}^2 \frac{L^4(r^4+\Lambda^4)}{\Lambda^4(r^4-r_0^4)}} + Q\sqrt{r_0^4+\Lambda^4}\biggr) \, .
\end{align}
This expression should be expanded at large radius compared to the horizon $r_0$, but keeping $r/\Lambda$ fixed since $r \sim \mathcal{O}(\Lambda)$ at the scale of the dynamics. Therefore, we introduce the dimensionless radial coordinate $\hat{r} = r/\Lambda$ that we imagine to be order one, and dimensionless time $\hat{t} = \frac{r_0 t}{L^2}$ as before. Since we take the probe to be extremal, we set $Q=M$, and all dimensionful quantities then scale out in front of the action,
\begin{align}
S = \frac{M \Lambda^4}{r_0 L^2} \int d\hat{t} \,\frac{\hat{r}^4}{\hat{r}^4+1}\biggl(\sqrt{1+\epsilon^4} - \sqrt{1-\frac{\epsilon^4}{\hat{r}^4}-\epsilon^2 \left(\frac{d\hat{r}}{d\hat{t}} \right)^2 \frac{\hat{r}^4+1}{\hat{r}^4-\epsilon^4}} \biggr) \, ,
\end{align}
where $\epsilon=r_0/\Lambda$ as before. We may now expand in $\epsilon$ and take $d\hat{r} / d\hat{t}\ll 1$, the Newtonian slow-moving approximation for the probe\footnote{The speed $d\hat{r} / d\hat{t}$ is $\mathcal{O}(\epsilon^3)$ in this expansion, coming from balancing the orders of the leading potential and kinetic terms and/or from the equation of motion.}, to find at lowest order
\begin{align}
S = \frac{M\Lambda^2 r_0}{ L^2} \int d\hat{t} \left(\frac12 \left(\frac{d\hat{r}}{d\hat{t}}\right)^2 + \frac{\epsilon^2}{2} + \frac{\epsilon^6}{8} \left(\frac{1}{\hat{r}^4} - 1\right) + \ldots \right) \, .
\end{align}
This is motion in a flat space attractive Coulomb potential which scales as $\mathcal{O}(\hat{r}^{-4})$ as expected for a charged object of codimension six in ten spacetime dimensions, in agreement with what would have been found from the tree-level closed string exchange. The resulting dynamics are simply
\begin{align}
\frac{d^2 \hat{r}}{d\hat{t}^2} = -\frac12 \frac{\epsilon^6}{\hat{r}^5} \, ,
\end{align}
and so the acceleration can be made small by making $\epsilon=r_0/\Lambda$ small. In terms of the original time coordinate, the instability timescale is $t \sim \frac{L^2 \Lambda^3}{r_0^4}\sim \beta \epsilon^{-3}$, where $\beta=\frac{\pi L^2}{r_0}$ is the inverse temperature of the black brane. That is, taking the thermal branes to be very cold or the stacks of branes to be widely separated, the wormhole solution can be made arbitrarily long-lived.
\subsection{Stabilizing with Rotation}
\label{sec:rotation}
One may wonder whether our wormhole can be stabilized by making the throats spin around each other in the transverse $r_i$ directions. We will not attempt to perturbatively construct such a spinning solution in the present work. On the other hand, we can repeat the DBI analysis above for the case where the extremal probe brane rotates around the non-extremal black branes. We parameterize the brane trajectory in a circular orbit around the equator $\theta_1=\ldots=\theta_4=\pi/2$ by $r(t)$, $\theta_5(t)$. In this case, the $G_{00}$ component of the pullback of the metric is
\begin{align}
G_{00} = g_{00} + \dot{r}^2 g_{rr} + \dot{\theta}_5^2 g_{\theta_5 \theta_5} \, .
\end{align}
Using the same coordinates as the previous section, the DBI action for the extremal probe constrained to the equator is
\begin{align}
S = \frac{M\Lambda^4}{r_0 L^4} \int d\hat{t} \frac{\hat{r}^4}{\hat{r}^4+1} \left(\sqrt{1+\epsilon^4} - \sqrt{1-\frac{\epsilon^4}{\hat{r}^4}-\epsilon^2 (\hat{r}^4+1) \left[\frac{1}{\hat{r}^4-\epsilon^4} \left(\frac{d\hat{r}}{d\hat{t}} \right)^2+\frac{1}{\hat{r}^2} \left(\frac{d\theta_5}{d\hat{t}} \right)^2 \right]} \right) \, . \label{eq:dbirotate}
\end{align}
Expanding in $\epsilon$ and taking the slow-moving approximation yields
\begin{align}
S = \frac{M\Lambda^2 r_0}{ L^2} \int d\hat{t} \left(\frac12 \left(\frac{d\hat{r}}{d\hat{t}}\right)^2 + \frac12 {\hat r}^2 \left(\frac{d\theta_5}{d\hat{t}} \right)^2+ \frac{\epsilon^2}{2} + \frac{\epsilon^6}{8} \left(\frac{1}{\hat{r}^4} - 1\right) + \ldots \right) \, ,
\end{align}
the same result as previously with the Newtonian rotational kinetic energy added. The radial equation of motion, assuming the existence of a solution with a constant rotational velocity $\frac{d\theta_5}{d\hat{t}} = \hat{\omega}$, is
\begin{align}
\frac{d^2 \hat{r}}{d\hat{t}^2} = \hat{r} \omega^2 - \frac{\epsilon^6}{2\hat{r}^5} \, ,
\end{align}
and therefore we can obtain circular orbits of radius $\hat{r}_c$ when the rotation speed is
\begin{align}
\hat{\omega} = \frac{\epsilon^3}{\sqrt{2} \hat{r}_c^3} \, .
\end{align}
Therefore, the angular speed needed to obtain circular orbits is $\omega \sim \epsilon^3/\beta$, the inverse of the instability time scale. In dimensionful coordinates, this speed is
\begin{align}
\omega = \frac{r_0^4}{\sqrt{2} L^2 r_c^3} \, .
\end{align}
We can check if the circular orbit radius $\hat{r}_c = \frac{\epsilon}{(\sqrt{2} \hat{\omega})^{1/3}}$ leads to stable or unstable orbits. For this, we examine the effective potential written in terms of conserved angular momentum $\ell = \frac{{\hat r}^2}{\epsilon^3} \frac{d \theta_5}{d\hat t}$,
\begin{equation}
V(r)=\epsilon^6 \left( \frac{1}{4}-\frac{1}{4 \hat r^4}+\frac{\ell^2}{\hat r^2} \right) \, .
\end{equation}
We see that the circular orbit corresponds to a maximum, i.e. it is unstable. The reason this happens is that the centrifugal piece in the effective potential dies off slower than the attractive force, which is the opposite of the situation in normal 4D Kepler motion. Based on this analysis, it is unlikely that the wormhole solution can be stabilized by rotation, unless nonlinear effects conspire to stabilize a circular orbit.
We can also try to solve \eqref{eq:dbirotate} for circular orbit frequencies directly without series expanding by taking the circular orbit as an ansatz. In that case the equation of motion reduces to the algebraic equation
\begin{align}
\hat{r}^{10} \omega^2 \epsilon^8+4 \hat{r}^6 \omega^2 \epsilon^8+3 \hat{r}^2 \omega^2 \epsilon^8+\hat{r}^4 \left(4 \sqrt{-\frac{\left(\epsilon^4+1\right) \left(\hat{r}^6 \omega^2 \epsilon^8-\hat{r}^4+r^2 \omega^2 \epsilon^8+\epsilon^4\right)}{\hat{r}^4}}-2 \epsilon^4-4\right)+2 \epsilon^4 = 0 \, .
\end{align}
We find an additional solution in this case with angular speed at leading order in $\epsilon$ given by
\begin{align}
\hat{\omega} = \frac{2\sqrt{2}\hat{r}_c}{(\hat{r}_c^4+3) \epsilon} \, ,
\end{align}
or $\omega = \frac{\Lambda^4}{L^2} \frac{2\sqrt{2} r_c }{(r_c^4 + \Lambda^4)}$ in dimensionful coordinates. This solution did not appear previously from perturbing the action around small $\epsilon$ simply because it is inversely proportional to $\epsilon$ and therefore not perturbatively slow-moving\footnote{One might worry that $\omega$ exceeds light-speed, even if it does not diverge. One can check that the maximum value of $\omega$ is $\frac{3^{3/4} \Lambda}{\sqrt{2}L^2}$ which is certainly small as $L \gg \Lambda$, and that this occurs at the reasonable radius $r_c = \Lambda / 3^{1/4}$.}. This value of $\hat{\omega}$ supports two different possible radii,
\begin{align}
\hat{r}_c = \frac{3\hat{\omega} \epsilon}{2\sqrt{2}} \qquad \text{and} \qquad \hat{r}_c = \frac{\sqrt{2}}{(\hat{\omega} \epsilon)^{1/3}} \, ,
\end{align}
or in dimensionful coordinates,
\begin{align}
r_c = \frac{3L^2 \omega}{2\sqrt{2}} \qquad \text{and} \qquad r_c= \sqrt{2} \left(\frac{\Lambda^4}{\omega L^2} \right)^{1/3} \, .
\end{align}
We can study the stability of these circular orbits by linearizing around the solution. One finds that the (dimensionless) frequency-squared of the radial oscillation, to leading order in $\epsilon$ is
\begin{equation}
\Omega^2 = \frac{4 \left(5 \hat{r}_c^8+12 \hat{r}_c^4-9\right) \hat{r}_c^2 \left| \hat{r}_c^4-1\right| +4 \left(\hat{r}_c^{12}-5 \hat{r}_c^8-33 \hat{r}_c^4+5\right) \hat{r}_c^2}{\epsilon^2 \left({\hat r}_c^4+1\right)^3 \left(\hat{r}_c^4+3\right)^2} \, ,
\end{equation}
that is, the orbit is stable when ${\hat r}_c\gtrsim 1.281$ and unstable when ${\hat r}_c\lesssim 1.281$. In the extremal limit, these circular orbits rotate with a finite angular velocity, as $\omega$ does not depend on $r_0$. These orbits are not directly relevant for stabilizing our wormhole, which is a perturbation to a non-rotating solution. This is because for self-consistency we would want the rotation in the circular orbits to be perturbatively small in $\epsilon=r_0/\Lambda$, while we saw that the rotation persists even in the extremal limit. One may therefore wonder if there exists a rotating version of the extremal two-center solution that is perturbatively stable. An exact solution is likely not possible due to gravitational and five form radiation, but we really just want a long-lived rotating binary black hole. This would provide a starting point for a wormhole solution stabilized by rotation.
\section{Traversing the Wormhole} \label{sec:traverse}
The two throats in our wormhole are separated by causal horizons, so it is not possible to traverse through it. Near the horizons, the wormhole looks like a perturbation of the planar AdS-Schwarzschild black brane, which is a marginally non-traversable solution in the sense that it can be made traversable by a small negative energy perturbation \cite{GaoJafferisWall}. Here we wish to analyse if the perturbation of the geometry near the horizon spoils this property. In two-sided null Kruskal coordinates $U,V$ (which exist for both the eternal black brane and for our wormhole geometry) the requirement to violate the ANEC is written $\int dU \, T_{UU} < 0$ along $V = 0$. In the absence of any stress-energy, $T_{UU} = 0$, null rays along $V = 0$ pass through the bifurcation surface and asymptote to infinity in either direction. Consequently, any negative perturbation will pull back the horizons and create traversability.
We will now evaluate the ANEC for the monopole- and dipole-corrected Einstein-Rosen bridges of Sec.~\ref{sec:monopole} and Sec.~\ref{sec:dipole}. The monopole corrections \eqref{eq:monopole} do not affect the marginal traversability since $T_{UU}$ vanishes along the horizon. This follows because as noted at the end of Sec.~\ref{sec:monopole}, this correction can be obtained by linearizing \eqref{eq:metricnonpert}, which is the asymptotically flat black brane in rescaled coordinates. The dipole contribution \eqref{eq:dipole} is more complicated to analyze because the $t-r$ plane is no longer decoupled from the $\theta_1$ angle, so the near-horizon geometry is effectively three-dimensional. Regardless, the location of the horizon stays at $r=r_0$ since the location of the zero of the blackening factor is not affected. Moreover, the null geodesics comprising the horizon remain on the $t-r$ plane at fixed $\theta_1$. This can be seen by examining the $\theta_1$ component of the geodesic equation,
\begin{equation}
\frac{d}{d\lambda} (r^2\dot{\theta_1}) = -\frac{r^2 \sin \left(\theta _1\right) h_g(r) \left(\dot{r}^2 L^4 r^4+\dot{t}^2 \left(r^4-r_0^4\right){}^2\right)}{L^2 \Lambda^5 \left(r^4-r_0^4\right)} \, ,
\end{equation}
where dot indicates derivative with respect to affine parameter $\lambda$. We have $h_g(r)=\frac{20 \pi ^{3/2} (r-r_0)}{\Gamma \left(-\frac{1}{4}\right)^2} + \cdots$ around $r=r_0$, so in order to have $\ddot{\theta_1}=0$ at the horizon, we need that $\dot{r}$ vanishes at $r=r_0$. Examining the condition $g_{mn}\dot{x^a}\dot{x^b}=0$ around $r=r_0$ one finds that $\dot{r} \propto \sqrt{r-r_0}\dot{\theta_1}$. Therefore, $\dot{r}=\dot{\theta_1}=0$ and $\dot{t}= \text{const}$ is a null geodesic at $r=r_0$ for any fixed $\theta_1$ (and the rest of the seven coordinates fixed as well). Therefore, $t$ also affinely parameterizes the null worldlines, so the ANEC quantity can be written $\int dt\, T_{tt}$. We identify the stress-energy tensor from the right-hand side of \eqref{eq:geomEOM2} as
\begin{align}
T_{\mu \nu} = \frac{1}{4\cdot 4!} F_{\mu \alpha \beta \gamma \delta} F_{\nu}^{\:\:\alpha \beta \gamma \delta} \, .
\end{align}
Applying this to the solution \eqref{eq:dipole} we find that $T_{tt}=0$ at $r=r_0$ up to $O(1/\Lambda^6)$ corrections. Therefore, at the order $\mathcal{O}(\Lambda^{-5})$ of the dipole corrections, spherical symmetry breaking does not affect the marginal non-traversability of the single-boundary wormhole.
As in \cite{GaoJafferisWall, SYKwormhole}, one mechanism to generate negative contributions to the ANEC that allow traversability is to introduce a nonlocal coupling between the two throats of the wormhole by adding a double-trace type interaction in the field theory. In fact, the field theory symmetry breaking $SU(2N) \to S(U(N) \times U(N))$ that we have described already generates couplings between the two $SU(N)$ effective subfactors in the IR from the Wilsonian RG flow \cite{INTRILIGATOR200099, connectivity}. At leading order, these include single-trace interactions of the form $g_I V_I$, where $V_I$ is proportional to
\begin{align}
V_I \propto \text{tr} \left(F_{\mu \nu} F^{\nu \rho} F_{\rho \sigma} F^{\sigma \mu} - \frac14 (F_{\mu \nu} F^{\mu \nu})^2 \right) \, ,
\end{align}
and $I=1,2$ are the two $U(N)$ factors. The couplings $g_I$ are dynamically determined by abelian singleton degrees of freedom in the other CFT factors (i.e., the Goldstone modes associated to the moduli of branes in the other stack(s)). Of more interest to us with respect to traversability are the double-trace interactions that are generated. These directly couple the IR factors in the CFT:
\begin{align}
V_{IJ} \propto \text{tr}_I \left(F_{\mu \nu} F^{\mu \nu} \right) \text{tr}_J \left(F_{\mu \nu} F^{\mu \nu} \right) \, .\label{eq:symdoubletrace}
\end{align}
When the full UV CFT is genuinely a product of $n$ individual subfactors $\text{CFT} = \prod_{i=1}^n \text{CFT}_i$, the dual bulk geometry generally consists of $n$ different asymptotic universes. In this case, the single-trace terms $\text{tr}_I F^2$ that comprise the operator \eqref{eq:symdoubletrace} are dual to the bulk dilaton in component $I$ \cite{OGwitten, PhysRevD.59.104021}. In our setting, the components $\text{tr}_1 F^2$ and $\text{tr}_2 F^2$ are dual to the bulk dilaton in the vicinity of the first and second throats, as these deep bulk regions correspond to the IR of the CFT where the approximate factorization into two $SU(N)$ gauge theories holds. Therefore, the double-trace interactions $V_{12}$ are structurally of the form $h_{12} \phi_1 \phi_2$ required to generate negative contributions to the ANEC as shown in \cite{GaoJafferisWall}. This indicates that the natural operators that arise from the Wilsonian RG flow in the IR of the symmetry-broken theory are of the correct form to generate traversability, albeit possibly weak traversability. However, in the Gao-Jafferis-Wall protocol, only one sign results in a traversable wormhole, while the opposite sign lengthens the wormhole. It would therefore be interesting to determine the sign of the coefficient of \eqref{eq:symdoubletrace} as generated by the Wilsonian RG, at least in perturbation theory. In fact, in our setting, there are various other double-trace operators that can be generated by the supersymmetry transformations of \eqref{eq:symdoubletrace}. A full analysis should understand the net effect of all such RG-generated double-trace operators on the sign of the null stress-energy.
In \cite{GaoJafferisWall}, the double-trace interactions are taken to be relevant deformations of the Hamiltonian so that they are renormalizable and there is no backreaction at the AdS boundary. The term $V_{12}$ generated by the RG flow is an irrelevant deformation; nonetheless, this is not a concern as we know that the theory is UV-complete, since above the Higgs scale it flows to the $SU(2N)$ $\mathcal{N}=4$ SYM theory. Furthermore, \cite{GaoJafferisWall} take the deformation to be a quench, turned on after some time $t_0$. Since our solution is perturbatively unstable, we also expect the coupling strength to be time-dependent, although we have not analyzed this in detail. Lastly, \cite{GaoJafferisWall} takes the boundaries to be connected with the same time orientation by taking the deformation to be structurally $h(t) \phi_1 (t, \vec{x}) \phi_2 (-t, \vec{x})$. This is because the asymptotic time on one boundary of the eternal black hole runs in the opposite direction on the other boundary. In our setting, the wormhole resides in a single universe and we have taken time to run upwards on both sides, so there is a unique asymptotic time $t$ and we need not flip the time orientation between the two throats.
In addition to the terms that are naturally generated by RG-flow, we can try, like \cite{GaoJafferisWall}, to add by hand some deformation that generates traversability in the IR wormhole. This should be a relevant operator in order for it not to destroy the UV $SU(2N)$ $\mathcal{N}=4$ SYM theory. The lightest single trace operators in a single factor of $SU(N)$ $\mathcal{N}=4$ SYM are the $\Delta=2$ scalars in the $\textbf{20}$ of the $SO(6)$ $R$-symmetry. They are of the form $\mathcal{O}^{ij}=\text{Tr} \phi^{(i}\phi^{j)}$. The possible deformations $\mathcal{O}^{ij}_L \mathcal{O}^{kl}_R$ therefore furnish $\textbf{20}\times \textbf{20}$. These are marginal to leading order in $1/N$ due to large $N$ factorization. In order to work out the effects of deforming by these operators (with either sign of the coefficient) we would need to understand their RG flow and the $1/N$ corrections to their dimension. The corresponding single-sided double-trace operators (i.e. an operator in one of the low energy $SU(N)$ factors) $\mathcal{O}^{ij}_L \mathcal{O}^{kl}_L$ are well understood in the strong coupling regime \cite{Arutyunov:2000ku}, and they all have either vanishing or negative anomalous dimensions. The negative anomalous dimensions are intuitively understood as binding energies coming from the attractive nature of the bulk interaction between two particles. The same intuitive reasoning applies to the two-sided operator (i.e. an operator connecting the two low energy $SU(N)$ factors) $\mathcal{O}^{ij}_L \mathcal{O}^{kl}_R$, which suggests that these operators should be marginally relevant at strong coupling, and one should be able to use them to make our single boundary wormhole traversable.
\section{A Double Wormhole Between Universes} \label{sec:doublewormhole}
The solutions we discussed in Sec.~\ref{sec:wormholeGeomSoln} capture certain effects of two non-extremal throats living in a single asymptotically AdS spacetime, and in Sec.~\ref{sec:global} we explained how to join these throats so that we end up with a wormhole in a single universe. There are also other ways to join the solutions of Sec.~\ref{sec:wormholeGeomSoln} to get interesting new wormhole configurations. For example, one could duplicate the spacetime with two throats and join them in a way shown in Fig.~\ref{fig:twouniverse}. In this case, the global time runs in opposite way in the two asymptotic regions and no twisting is required to enforce flux conservation (see Fig.~\ref{fig:twouniverse}). This spacetime is patch-wise described by the same solutions that we have discussed in Sec.~\ref{sec:wormholeGeomSoln}, but the patches are glued together differently.
\begin{figure}[hbtp!]
\begin{center}
\includegraphics[width=.45\textwidth]{twouniverse.pdf}
\end{center}
\caption{A double wormhole between two asymptotically AdS universes. This geometry, which can be constructed from the solutions in the text, is dual to a pair of Higgsed Yang-Mills theories, with IR factors entangled pairwise between them.
\label{fig:twouniverse}}
\end{figure}
In the dual field theory we now start with two copies of $\mathcal{N}=4$ $SU(2N)$ SYM, and we Higgs each copy. Let us label the two theories $A$ and $B$, while the low energy factors are called $L$ and $R$. Then, the low energy Hilbert space is
\begin{equation}
\mathcal{H}_{A,L}\otimes \mathcal{H}_{A,R} \otimes \mathcal{H}_{B,L}\otimes \mathcal{H}_{B,R} ,
\end{equation}
and we expect a wormhole configuration like Fig. \ref{fig:twouniverse} to be approximately dual in the IR to a tensor product of two thermofield double states
\begin{equation}
|\text{TFD}\rangle_{A,L;B,L}\otimes |\text{TFD}\rangle_{A,R;B,R}.
\end{equation}
We can embed this state in the UV Hilbert space $\mathcal{H}_A\otimes \mathcal{H}_B$ as explained in Sec.~\ref{sec:fieldtheory}, that is, we must take the temperatures of the thermofield doubles to be much smaller than the Higgs scale.
\section{Discussion}
In this paper, we constructed an asymptotically AdS$_5 \times S^5$ single boundary wormhole solution by matching a two-center extremal black brane solution to a two-sided AdS black brane in perturbation theory. Preserving continuity of the five-form in the solution required a global monodromy in some of the coordinates, although the total geometry remains orientable. The small parameter in the problem is the horizon radius compared to the separation of the throats, $r_0/\Lambda$. We argued that the solution is dual to an approximate thermofield double state in a single copy of $\mathcal{N}=4$ SYM, where the gauge group is Higgsed into two copies of $SU(N)$, which are entangled. In the field theory the small parameter is the ratio of the thermal scale to the Higgs scale.
\subsubsection*{Thermal effective potential and $R$ charge}
Our wormhole is non-extremal, and consequently has to be unstable. This instability is dual in $\mathcal{N}=4$ SYM to the scalar vevs developing an effective potential at finite temperature, as illustrated in Fig. \ref{fig:effpotential}. We have argued that the wormhole can be made parametrically long lived by making $r_0/\Lambda$ small.
Another possibility is to stabilize the wormhole by making the throats rotate around each other. In Sec.~\ref{sec:rotation} we found that an extremal probe brane can be put on a stable circular orbit around a non-extremal black brane. This is surprising since planetary orbits are unstable in more than four dimensions, and is possible here due to the five-form interaction. The stable orbit we find has finite angular velocity in the extremal limit, so it is not possible to add this effect perturbatively to our solution. Nevertheless, this finding suggests that in the dual theory one can create a local minimum in the effective potential of the scalar vevs away from the origin by adding $R$ charge. This should lead to long-lived states with finite temperature symmetry breaking. The states are only long-lived, since from the supergravity picture, we expect them to decay due to gravitational and five form radiation. This is consistent with the expectation that all symmetries must be restored at sufficiently high temperatures: see \cite{Chai:2020zgq} for a recent discussion in the case of global symmetries. It would be interesting to understand this effect better.
\subsubsection*{Global monodromy and moduli space}
As emphasized in \cite{Verlinde2020}, gluing the two sides of a wormhole to a single asymptotic region breaks the two-sided boost-like Killing symmetry of the eternal black brane geometry and correspondingly, there is a one-parameter family of wormholes labeled by the ``monodromy" of Schwarzschild time as one goes between the two throats on the outside. In addition to this, we have found that there is a freedom of introducing a global monodromy consisting of rotating and translating the parallel spatial directions to the brane, and also rotating by the unbroken $SO(5)$ subgroup of the $SO(6)$ symmetry of the $S^5$.
Therefore, there is a moduli space $\mathbb{R}\times ISO(3)\times SO(5)$ of locally equivalent but globally different solutions. It would be interesting to understand the interpretation of this in the dual $\mathcal{N}=4$ $SU(2N)$ SYM theory. It is tempting to speculate that it is related to some ambiguity in embedding the IR state \eqref{eq:tfdapprox} into the UV theory, which possibly includes an ambiguity in the implementation of the energy cutoff in the state \eqref{eq:tfdapprox}.
\subsubsection*{Making the wormhole traversable}
We have showed that corrections coming from the two throats being in the same spacetime in the first few orders in perturbation theory do not spoil the marginal traversability of the wormhole, in the sense that the ANEC quantity $\int dU T_{UU}$ remains zero along the causal horizons. It would thus be interesting to see if the wormhole can be made traversable using the ideas in \cite{GaoJafferisWall}. This requires a double trace coupling between the two $SU(N)$ factors in the Higgsed $\mathcal{N}=4$ SYM theory.
We have pointed out that such couplings are naturally generated in RG due to the fact that in the UV the two $SU(N)$ factors are part of the total $SU(2N)$. It would require a careful analysis to account for the net effect of all these double trace interactions and to see if the resulting sign makes the wormhole traversable. This is beyond the scope of the present paper but is certainly an interesting problem.
One may also try to make the wormhole traversable by adding a double trace coupling by hand. This would have to be a relevant double trace operator, otherwise the theory will no longer flow to a single $SU(2N)$ $\mathcal{N}=4$ SYM in the UV (or to a wormhole in a single spacetime). We have argued that such relevant double traces can be formed from the $\Delta=2$ scalar operators of the theory, though it would also be useful to check that the two-sided operators $\mathcal{O}_L^{ij} \mathcal{O}_R^{kl}$ have negative anomalous dimensions.
In \cite{SM_wormhole} negative contributions to the ANEC were generated by negative Casimir-like vacuum energies coming from the lowest Landau levels of the bulk fermion running in a cycle threading their wormhole solution. In our solution, there are various fermions in the spectrum of type IIB supergravity which have vacuum fluctuations, though we have set their classical backgrounds to vanish. These fermions, and the bulk bosonic fields, should similarly provide Casimir-like vacuum energies in our setup. The sign of the total Casimir energy is important, as before; so it is important to check which contributions ultimately win out. There is potentially the possibility that the underlying supersymmetry enforces a vanishing total Casimir energy. In any case, the vacuum energies provide another potential mechanism for traversability in competition or collusion with the other effects that we have discussed.
\subsubsection*{Probing the monodromy through the wormhole}
As we discussed, continuity of the five-form requires a twisted gluing of the interior AdS-Schwarzschild geometry to the two-center ambient spacetime, although the complete spacetime remains orientable. An interesting way of probing the resulting monodromy is to send a giant graviton through the wormhole. Giant gravitons are spherical D3-branes localized on the $S^5$ in the geometry, and are supported by their angular momentum and by interactions with the five-form flux \cite{Myers:1999ps,McGreevy:2000cw}. These brane states are created by determinant and subdeterminant operators in the field theory \cite{Balasubramanian:2001nh,Corley:2001zk,Balasubramanian:2004nb}. To use these branes to probe the wormhole in the field theory, we would want to construct such operators in the light infrared factors after Higgsing. On the gravitational side, we could explicitly test what happens to the corresponding giant gravitons as they are moved through the wormhole, expecting them to emerge with inverted $\theta^1$.
\subsection*{Acknowledgements}
We are grateful to Alexandre Belin, Vishnu Jejjala, Arjun Kar, Guram Kartvelishvili, Lampros Lamprou, Juan Maldacena, Onkar Parrikar, Simon Ross, and Tomonori Ugajin for useful conversations. MD is supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE-1845298. The research of VB, MD, and GS was supported in part by the Simons Foundation through the It From Qubit Collaboration (Grant No.~38559), and by the Department of Energy through grants DE-SC0013528, and QuantISED DE-SC0020360. VB also thanks the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611, for hospitality while this work was in progress.
\begin{appendices}
\section{Perturbative Equations of Motion}\label{sec:perturbEOM}
In this appendix, we derive the perturbative equations of motion \eqref{eq:perturbativeEOMs1} and \eqref{eq:perturbativeEOMs2}. In general, bars will indicate background-order quantities. We begin by variation of the full geometric equation of motion \eqref{eq:geomEOM1}. Since we impose self-duality at all orders,
\begin{align}
0 = (F + \delta F)\wedge \ast(F+\delta F) \sim F_{\alpha \beta \gamma \delta \epsilon} F^{\alpha \beta \gamma \delta \epsilon} \, ,
\end{align}
where the last expression is to all orders. Consequently, the variation of \eqref{eq:geomEOM1} is
\begin{align}
\delta R_{\mu \nu} -\frac12 \delta R \bar{g}_{\mu \nu} &= \frac{1}{4\cdot 4!} (\delta F_{\mu \alpha \beta \gamma \delta} \bar{F}_{\nu}^{\:\:\alpha \beta\gamma\delta} + \bar{F}_{\mu \alpha \beta \gamma \delta} \delta F_{\nu}^{\:\:\alpha \beta \gamma \delta} ) \, .
\end{align}
$\bar{R} = 0$ at background order, but it is not obvious that $\delta R = 0$ perturbatively, so we have retained this term for now. Now we can constrain $\delta R$ by tracing both sides, noting that $\bar{g}^{\mu \nu} \delta R_{\mu \nu} = \delta R - \bar{R}_{\mu \nu} \delta g^{\mu \nu}$,
\begin{align}
-4\delta R - \bar{R}_{\mu \nu} \delta g^{\mu \nu}
= \frac{1}{4\cdot 4!} \bar{g}^{\mu \nu} (\delta F_{\mu \alpha \beta \gamma \delta} \bar{F}_{\nu}^{\:\:\alpha \beta\gamma\delta} + \bar{F}_{\mu \alpha \beta \gamma \delta} \delta F_{\nu}^{\:\:\alpha \beta \gamma \delta} ) \, .
\end{align}
Substituting $\bar{R}_{\mu \nu}$ with its background equation of motion and rearranging for $\delta R$ one finds:
\begin{align}
\delta R = \frac{1}{16\cdot 4!} \delta \left(g^{\mu \nu} F_{\mu \alpha \beta \gamma \delta} F_{\nu}^{\:\:\alpha\beta\gamma\delta} \right) = 0 \, .
\end{align}
That is, $\delta R$ vanishes to all orders as a consequence of the self-duality constraint. The perturbative equation of motion for the metric is therefore simply
\begin{align}
\delta R_{\mu \nu}&= \frac{1}{4\cdot 4!} (\delta F_{\mu \alpha \beta \gamma \delta} \bar{F}_{\nu}^{\:\:\alpha \beta\gamma\delta} + \bar{F}_{\mu \alpha \beta \gamma \delta} \delta F_{\nu}^{\:\:\alpha \beta \gamma \delta} ) \, .
\end{align}
We now make use of the formula for $\delta R_{\mu \nu}$ in terms of the metric perturbation $\delta g_{\mu \nu} = h_{\mu \nu}$, to first order in the perturbation, to arrive at \eqref{eq:perturbativeEOMs1}:
\begin{align}
\delta R_{\mu \nu} = \nabla_{\lambda} \nabla_{(\mu}h_{\nu)}^{\lambda} - \frac12 \nabla_{\mu} \partial_{\nu} h - \frac12 \nabla_{\lambda} \nabla^{\lambda} h_{\mu \nu} \, ,
\end{align}
where the covariant derivative is taken with respect to $\bar{g}$. We review the derivation of this formula below. The Ricci tensor is
\begin{align}
R_{bd} = R^a_{bad} = \partial_a \Gamma^a_{bd} - \partial_d \Gamma^a_{ba} + \Gamma^s_{bd} \Gamma^a_{sa} - \Gamma^s_{ba} \Gamma^a_{sd} \, .
\end{align}
Relabeling indices and varying each term independently gives a formula in terms of the variation $\delta \Gamma$
\begin{align}
\begin{aligned}
\delta R_{ab} &= \partial_c \delta \Gamma^c_{ab} - \partial_b \delta \Gamma^c_{ac} + \delta(\Gamma^s_{ab} \Gamma^c_{sc}) - \delta(\Gamma^s_{ac} \Gamma^c_{sb} ) \\
&= \partial_c \delta \Gamma^c_{ab} - \partial_b \delta \Gamma^c_{ac} + \delta\Gamma^s_{ab} \Gamma^c_{sc} +\Gamma^s_{ab}\delta \Gamma^c_{sc} - \delta\Gamma^s_{ac} \Gamma^c_{sb} -\Gamma^s_{ac}\delta \Gamma^c_{sb} \\
&= \nabla_c (\delta \Gamma^c_{ab}) - \nabla_b (\delta \Gamma^c_{ac} ) \, .
\end{aligned}
\end{align}
To compute the variation of the Christoffel symbols, expand the covariant derivative of the metric perturbations
\begin{align}
\begin{aligned}
\nabla_a h_{bc} &= \nabla_a (\delta g_{bc}) = \partial_a (\delta g_{bc}) - \Gamma^s_{ab} \delta g_{sc} - \Gamma^s_{ac} \delta g_{bs} \\
&= \delta (\partial_a g_{bc}) - \delta(\Gamma^s_{ab}g_{sc}) + \delta \Gamma^s_{ab}g_{sc} - \delta ( \Gamma^s_{ac} g_{bs} )+ \delta\Gamma^s_{ac} g_{bs} \\
&= \delta (\partial_a g_{bc} - \Gamma^s_{ab}g_{sc}- \Gamma^s_{ac} g_{bs} )+ \delta \Gamma^s_{ab}g_{sc} + \delta\Gamma^s_{ac} g_{bs} \\
&= \delta (\nabla_a g_{bc})+ \delta \Gamma^s_{ab}g_{sc} + \delta\Gamma^s_{ac} g_{bs} \\
&= \delta \Gamma^s_{ab}g_{sc} + \delta\Gamma^s_{ac} g_{bs} \, ,
\end{aligned}
\end{align}
using metric compatibility. Now cyclically permuting and adding a convenient sign gives
\begin{align}
\begin{aligned}
\nabla_a h_{bc} + \nabla_b h_{ca} - \nabla_c h_{ab} &= \delta \Gamma^s_{ab}g_{sc} + \delta\Gamma^s_{ac} g_{bs} + \delta \Gamma^s_{bc}g_{sa} + \delta\Gamma^s_{ba} g_{cs} - \delta \Gamma^s_{ca}g_{sb} - \delta\Gamma^s_{cb} g_{as} \\
&= 2\delta \Gamma^s_{ab}g_{sc} \, .
\end{aligned}
\end{align}
Rearranging and permuting the indices gives
\begin{align}
\delta \Gamma^a_{bc} = \frac12 (\nabla_b h_{c}^a + \nabla_c h_{b}^a - \nabla^a h_{bc}) \, .
\end{align}
Expanding the variation $\delta R_{ab}$ with this formula, one finds
\begin{align}
\begin{aligned}
\delta R_{ab} &= \nabla_c (\delta \Gamma^c_{ab}) - \nabla_b (\delta \Gamma^c_{ac} ) \\
&= \frac12 \nabla_c (\nabla_a h^c_b + \nabla_b h^c_a - \nabla^c h_{ab})- \frac12 \nabla_b (\nabla_a h + \nabla_c h^c_a - \nabla^c h_{ac}) \\
&= \nabla_c \nabla_{(a} h_{b)}^c - \frac12 \nabla^2 h_{ab} -\frac12 \nabla_a \partial_b h \, ,
\end{aligned}
\end{align}
which was the claimed formula for the variation of the Ricci tensor.
Now we must consider the variation of Maxwell's equations:
\begin{align}
\delta \left( \partial_{\mu} (\sqrt{-g} F^{ \mu \nu \rho \sigma\tau})\right) &= \partial_{\mu} (\delta \sqrt{-g} \bar{F}^{\mu \nu \rho \sigma\tau} + \sqrt{-\bar{g}} \delta F^{ \mu \nu \rho \sigma\tau}) \, .
\end{align}
Recall the variation of $\sqrt{-g}$, from Sylvester's formula:
\begin{align}
\delta \sqrt{-g} = -\frac12 \sqrt{-\bar{g}} \bar{g}_{\mu \nu} \delta g^{\mu \nu} = \sqrt{-\bar{g}} \frac{h}{2} \, .
\end{align}
Therefore, we find \eqref{eq:perturbativeEOMs2}:
\begin{align}
\partial_{\mu} (\sqrt{-\bar{g}} (\frac{h}{2} \bar{F}^{ \mu \nu \rho \sigma\tau} + \delta F^{ \mu \nu \rho \sigma\tau})) = 0 \, .
\end{align}
Lastly, the perturbation to the five-form must leave it to be self-dual. However, one must be careful because the Hodge dual involves factors of the metric that also contribute perturbatively. Let us assume the metric is diagonal and that the only nonzero independent components of the five-form are $F_{t123r}$ and $F_{\theta_1 \ldots \theta_5}$. In terms of the components of the metric and five-form the constraint can be written explicitly as
\begin{align}
F_{t123r} + \delta F_{t123r} &= \sqrt{-g} g^{\theta_1 \theta_1} \ldots g^{\theta_5 \theta_5} (F_{\theta_1 \ldots \theta_5} + \delta F_{\theta_1 \ldots \theta_5}) \\
F_{\theta_1 \ldots \theta_5} + \delta F_{\theta_1 \ldots \theta_5} &= -\sqrt{-g} g^{tt} g^{11} g^{22} g^{33} g^{rr} (F_{t123r} + \delta F_{t123r}) \, .
\end{align}
Now removing the background-order equations and expanding perturbatively, we find
\begin{align}
\delta F_{t123r} &= \sqrt{-\bar{g}} \bar{g}^{\theta_1 \theta_1} \ldots \bar{g}^{\theta_5 \theta_5} \delta F_{\theta_1 \ldots \theta_5} - \sqrt{-\bar{g}} (h^{\theta_1 \theta_1} \ldots \bar{g}^{\theta_5 \theta_5} + \ldots + \bar{g}^{\theta_1 \theta_1} \ldots h^{\theta_5 \theta_5} ) \bar{F}_{\theta_1 \ldots \theta_5} + \frac{h}{2} \bar{F}_{t123r} \label{eq:selfdual1}\\
\delta F_{\theta_1 \ldots \theta_5} &= -\sqrt{-\bar{g}} \bar{g}^{tt} \bar{g}^{11} \bar{g}^{22} \bar{g}^{33} \bar{g}^{rr} \delta F_{t123r} + \sqrt{-\bar{g}} (h^{tt} \bar{g}^{11} \bar{g}^{22} \bar{g}^{33} \bar{g}^{rr} + \ldots +\bar{g}^{tt} \bar{g}^{11} \bar{g}^{22} \bar{g}^{33} h^{rr} ) F_{t123r} + \frac{h}{2} \bar{F}_{\theta_1 \ldots \theta_5} \, .\label{eq:selfdual2}
\end{align}
To proceed further, one requires more details about the background metric and five-form of interest.
\section{Solving the Linearized Equations} \label{sec:linearizedApp}
In this appendix, we demonstrate the procedure to solve the perturbative equations of motion \eqref{eq:perturbativeEOMs1} and \eqref{eq:perturbativeEOMs2} by hand in the linearized regime, region II. In this regime the ansatz for the metric and five-form perturbations takes the form
\begin{align}
\frac{1}{\alpha'} ds^2 &= L^2 \biggl[-\tilde{r}^2\left(1 + \delta g_{\tilde{t}\tilde{t}} \right) d\tilde{t}^2 +\tilde{r}^2\left(1 + \delta g_{\tilde{i}\tilde{i}}\right) d\vec{\tilde{x}}^2 + \frac{1}{\tilde{r}^2}\left(1+\delta g_{\tilde{r}\tilde{r}}\right)d\tilde{r}^2 + (1+\delta g_{\Omega \Omega}) d\Omega_5^2 \biggr] \\
\frac{1}{\alpha^{'2}} F &= 4L^4 \biggl[\tilde{r}^3 \left(1 +a(\tilde{r}) \right) d\tilde{t} \wedge d\tilde{x}^1 \wedge d\tilde{x}^2 \wedge d\tilde{x}^3 \wedge d\tilde{r} \nonumber \\&\qquad\qquad\qquad\qquad\qquad+ (1+b(\tilde{r})) \sin^4 \theta_1 \sin^3 \theta_2 \sin^2 \theta_3 \sin \theta_4 d\theta_1 \wedge d\theta_2 \wedge d\theta_3 \wedge d\theta_4 \wedge d\theta_5 \biggr] \, ,
\end{align}
where all metric perturbations are functions only of $\tilde{r}$. To begin, we evaluate the self-duality constraint equations \eqref{eq:selfdual1} and \eqref{eq:selfdual2} on the background metric of vacuum AdS$_5 \times S_5$. They reduce to only one independent equation,
\begin{align}
2 a-3 \delta g_{\tilde{i}\tilde{i}}- \delta g_{\tilde{t}\tilde{t}}+5 \delta g_{\Omega \Omega }- \delta g_{\tilde{r}\tilde{r}} &= 2b \, ,
\end{align}
identical to \eqref{eq:selfdualBB}. Plugging into Maxwell's equations we also find only one independent equation
\begin{align}
2 a'-3 \delta g_{\tilde{i}\tilde{i}}'- \delta g_{\tilde{t}\tilde{t}}'+5 \delta g_{\Omega \Omega}'- \delta g_{\tilde{r}\tilde{r}}' &= 0 \, ,
\end{align}
which is identical to \eqref{eq:maxwellBB}. Combining Maxwell's equations and self-duality gives $b' = 0$, meaning that the five-form charge is conserved. The independent geometric equations of motion are
\begin{align}
-8 (b -
2\delta g_{\Omega\Omega } ) - \frac {1} {2} \tilde{r}\left (\tilde{r} g_{\Omega\Omega }'' +
5g_{\Omega\Omega }' \right) &= 0 \label{eq:geom1}\\
-16 a+\tilde{r}^2 \delta g_{\tilde{t}\tilde{t}} ''+3 \tilde{r} \delta g_{\tilde{i}\tilde{i}} '+6 \tilde{r} \delta g_{\tilde{t}\tilde{t}}'+5 \tilde{r} \delta g_{\Omega \Omega}'- \tilde{r} \delta g_{\tilde{r}\tilde{r}}'+24 \delta g_{\tilde{i}\tilde{i}} +8 \delta g_{\tilde{t}\tilde{t}} &= 0\label{eq:geom2}\\
-16 a+\tilde{r}^2 \delta g_{\tilde{i}\tilde{i}}''+8 \tilde{r}\delta g_{\tilde{i}\tilde{i}}'+\tilde{r} \delta g_{\tilde{t}\tilde{t}}'+5 \tilde{r} \delta g_{\Omega \Omega }'- \tilde{r} \delta g_{\tilde{r}\tilde{r}}'+24 \delta g_{\tilde{i}\tilde{i}}+8 \delta g_{\tilde{t}\tilde{t}} &= 0 \label{eq:geom3}\\
-16 a +3 \tilde{r}^2 \delta g_{\tilde{i}\tilde{i}}''+\tilde{r}^2 \delta g_{\tilde{t}\tilde{t}}''+5 \tilde{r}^2 \delta g_{\Omega \Omega}''+9 \tilde{r}\delta g_{\tilde{i}\tilde{i}}'
+3 \tilde{r}\delta g_{\tilde{t}\tilde{t}}'\qquad\qquad\qquad\qquad\qquad&\nonumber \\+5 \tilde{r} \delta g_{\Omega \Omega }'-4 \tilde{r} \delta g_{\tilde{r}\tilde{r}}'+24 \delta g_{\tilde{i}\tilde{i}}+8 \delta g_{\tilde{t}\tilde{t}} &= 0 \, .\label{eq:geom4}
\end{align}
Notice that the equation of motion \eqref{eq:geom1} for $\delta g_{\Omega \Omega}$ is independent of the others and may be solved directly, yielding:
\begin{align}
\delta g_{\Omega \Omega} = \frac{b}{2} + a_1 \tilde{r}^4 + a_2/\tilde{r}^8 \, ,
\end{align}
for constants $a_1, a_2$. Now examine the three remaining geometric equations of motion. By taking the linear combination of \eqref{eq:geom2} + 3\eqref{eq:geom3} - \eqref{eq:geom4}, we find an equation determining $\delta g_{\tilde{r}\tilde{r}}$ in terms of the others:
\begin{align}
4\delta g_{\tilde{r}\tilde{r}} = 2b + 3\tilde{r}\delta g^{'}_{\tilde{i}\tilde{i}}+ \tilde{r}\delta g^{'}_{\tilde{t}\tilde{t}} +20 (a_1 \tilde{r}^4 -3 a_2 / \tilde{r}^8) \, . \label{eq:rreqn}
\end{align}
Plugging this back into all three equations we find that all three are solved as long as:
\begin{align}
5(\delta g_{\tilde{i}\tilde{i}}^{'} - \delta g_{\tilde{t}\tilde{t}}^{'}) + r(\delta g_{\tilde{i}\tilde{i}}^{''} - \delta g_{\tilde{t}\tilde{t}}^{''}) = 0 \, .
\end{align}
This is a differential equation in $f (\tilde{r}) = \delta g_{\tilde{i}\tilde{i}} - \delta g_{\tilde{t}\tilde{t}}$ which is solved by $f(\tilde{r}) = c_2 - \frac{c_1}{4\tilde{r}^4}$ where $c_1, c_2$ are constants. Therefore we can relate $\delta g_{\tilde{i}\tilde{i}}$ to $\delta g_{\tilde{t}\tilde{t}}$ via
\begin{align}
\delta g_{\tilde{i}\tilde{i}} = c_2 - \frac{c_1}{4\tilde{r}^4}+\delta g_{\tilde{t}\tilde{t}} \, . \label{eq:iieqn}
\end{align}
Plugging \eqref{eq:iieqn} into \eqref{eq:rreqn} reduces it to
\begin{align}
\delta g_{\tilde{r}\tilde{r}} = \frac{b}{2} + \tilde{r}\delta g_{\tilde{t}\tilde{t}}^{'} + \frac34\frac{c_1}{\tilde{r}^4} - 15 \frac{a_2}{\tilde{r}^8} + 5a_1 \tilde{r}^4 \, .
\end{align}
We have consequently fixed the general perturbative solution in the linearized regime in terms of one arbitrary function $\delta g_{\tilde{t}\tilde{t}}$ and five constants $a_1, a_2, c_1, c_2, b$:
\begin{align}
\frac{1}{\alpha'} ds^2 &= L^2 \biggl[-\tilde{r}^2\left(1 + \delta g_{\tilde{t}\tilde{t}} \right) d\tilde{t}^2 +\tilde{r}^2\left(1 + c_2 - \frac{c_1}{4\tilde{r}^4}+\delta g_{\tilde{t}\tilde{t}}\right) d\vec{\tilde{x}}^2 \nonumber\\
&+ \frac{1}{\tilde{r}^2}\left(1+\frac{b}{2} + \tilde{r}\delta g'_{\tilde{t}\tilde{t}} + \frac34\frac{c_1}{\tilde{r}^4} - 15 \frac{a_2}{\tilde{r}^8} + 5a_1 \tilde{r}^4\right)d\tilde{r}^2 + (1+ \frac{b}{2} + a_1 \tilde{r}^4 + \frac{a_2}{\tilde{r}^8}) d\Omega_5^2 \biggr] \\
\frac{1}{\alpha^{'2}} F &= 4L^4 \biggl[\tilde{r}^3 \left(1 +\frac32 c_2 - 10 \frac{a_2}{\tilde{r}^8} + 2 \delta g_{\tilde{t}\tilde{t}} + \frac{\tilde{r}}{2} \delta g'_{\tilde{t}\tilde{t}}\right) d\tilde{t} \wedge d\tilde{x}^1 \wedge d\tilde{x}^2 \wedge d\tilde{x}^3 \wedge d\tilde{r} \nonumber \\&\qquad\qquad\qquad\qquad\qquad+ (1+b) \sin^4 \theta_1 \sin^3 \theta_2 \sin^2 \theta_3 \sin \theta_4 d\theta_1 \wedge d\theta_2 \wedge d\theta_3 \wedge d\theta_4 \wedge d\theta_5 \biggr] \, .
\end{align}
To fix $\delta g_{\tilde{t}\tilde{t}}$ and the five constants, we compare the solution to the linearized expansion of the solution \eqref{eq:metricnonpert}-\eqref{eq:fiveformnonpert}. A consistent solution is found by taking $a_1 = \epsilon^4 / 2$, $a_2 = c_2 = 0$, $c_1 = -4$, $b=0$, and
\begin{align}
\delta g_{\tilde{t}\tilde{t}} = -\frac{1}{\tilde{r}^4} - \frac12 (\epsilon \tilde{r})^4 \, ,
\end{align}
leading to the solution \eqref{eq:linear1}-\eqref{eq:linear2}.
\end{appendices}
\bibliographystyle{JHEP}
\section{Introduction}
In the AdS/CFT correspondence, the entanglement structure of the boundary CFT encodes the geometry and topology of the bulk AdS space \cite{vanRaams, Maldacena_2003, Balasubramanian_2014, Marolf_2015, doi:10.1002/prop.201300020}, albeit in a complicated and nonlocal way. We consider the dual of a state in the $\mathcal{N} = 4$ $SU(2N)$ Super Yang-Mills (SYM) theory in a Coulomb phase where the infrared modes are thermally entangled. We argue that this system should correspond to an asymptotically AdS$_5\times S^5$ geometry with a single boundary and a long-lived interior wormhole. The purpose of this paper is to construct this wormhole and describe its properties.
The simplest connection between wormholes and entanglement involves two copies of a CFT entangled in the ``thermofield double" (TFD) state, a two-party purification of the thermal state on each factor:
\begin{align}
|\text{TFD}\rangle = \sum_i e^{-\beta E_i / 2} |i\rangle \otimes |i^{\ast}\rangle \, .
\end{align}
where $|i^{\ast}\rangle$ indicates the CPT conjugate of $|i\rangle$. This system is dual to the eternal black hole, i.e., a wormhole between two asymptotically AdS universes \cite{Maldacena_2003}. To construct a wormhole between distant regions of a \emph{single} universe, we will study a state in the Coulomb branch of the $\mathcal{N}=4$ SYM theory \cite{KLEBANOV199989, PhysRevD.62.086003}, where the $SU(2N)$ gauge symmetry has been partially broken down to $S(U(N) \times U(N))$. In type IIB supergravity, the low energy effective theory dual to $\mathcal{N}=4$ SYM, this configuration corresponds to a multicenter solution sourced by two stacks of $N$ D3-branes \cite{CoulombBranch, PhysRevD.60.127902}. The procedure for constructing this geometry is illustrated in Fig.~\ref{fig:wormhole1}. We will begin with a two-centered BPS \cite{DUFF1991409} harmonic function solution as originally found in \cite{HOROWITZ1991197}, corresponding to two stacks of $N$ extremal D3-branes in (9+1)D Minkowski space and controlled by a parameter $L$. These stacks of D3-branes are separated by a distance $\Lambda$, corresponding to the Higgs scale. In the limit that $L\gg \Lambda$, there will be an AdS$_5 \times S^5$ geometry outside the region containing the two stacks of branes, which splits into two smaller AdS$_5 \times S^5$ regions as one nears either stack. Taking the limit $\alpha^{\prime} \to 0$ with the ratios of the five-sphere coordinates to $\alpha^{\prime}$ held fixed decouples the AdS regions from the asymptotically flat space, leaving a geometry which is asymptotically AdS \cite{largeN}.
Now we heat up the solution by entangling the degrees of freedom living on each brane (in each $SU(N)$ sector of the Higgsed SYM) up to the Higgs scale $\Lambda$ in an approximate thermofield double state. This has the effect that in the IR of the field theory, i.e. the deep bulk, the approximate thermofield double state will be dual to the two-sided planar AdS-Schwarzschild black brane \cite{Maldacena_2003} plus corrections due to the multicenter nature of the exterior solution. No known solution exists for the multicenter black brane geometry at nonzero temperature, so we solve for these corrections in perturbation theory. Matching these corrections in different coordinate patches glues together the wormhole solution. We will find that this gluing must introduce a global monodromy that inverts some spatial directions between the two throats in order to respect flux conservation. However, the full ten-dimensional spacetime remains globally orientable. The complete solution is unstable, as finite temperature breaks the supersymmetry of the BPS solution and turns on an effective potential for the scalar fields that break the $SU(2N)$ symmetry \cite{Kraus_1999, NAYEK2017192}. Supergravity wormhole solutions have been previously studied e.g. in \cite{BERGMAN2009300, Maldacena_2004, susywormhole}, but only in the case of connecting two different asymptotic spaces.
\begin{center}
\begin{figure}[hbtp!]
\includegraphics[width=\textwidth]{Wormhole1KleinUpdated.pdf}
\caption{Perturbative construction of an AdS wormhole solution in a single asymptotic space by taking the decoupling limit of two stacks of D3-branes and correcting for the finite temperature caused by the thermofield double entanglement structure below the scale $\Lambda$. Figure inspired by the near-horizon limit as depicted in \cite{costa, Michelson_1999}.\label{fig:wormhole1}}
\end{figure}
\end{center}
In Fig.~\ref{fig:wormhole2} we have labeled the different regimes in which a different coordinate patch or limit will be used to describe the solution. In region I, the solution is approximately the two-sided non-extremal black brane. Far from the horizon, in regions II and III, the effects of non-extremality are small and the solution is close to vacuum AdS$_5 \times S^5$. In this region, the perturbative corrections from the nonzero temperature and from the two throats can simultaneously be treated as linear corrections to the vacuum AdS$_5 \times S^5$ background, and therefore they linearly superpose. The leading corrections from the left throat are monopole corrections and do not break spherical symmetry around the right throat, which defines region II. However, we can include multipole effects from the left throat as linearized corrections, and these will be dominant over nonlinearities up to fourth order in the multipole expansion. This captures effects of the left throat breaking the spherical symmetry around the right throat, which defines region III. We will also be able to present solutions which are valid in regions I, II and parts of III simultaneously. These will be linearized perturbations of the finite temperature black brane geometry. These solutions show that leading multipole effects remain small near the causal horizons, but we find that they grow in the interior towards the singularity. Therefore the singularity in this wormhole is not of the AdS-Schwarzschild type. Finally, region IV is where both throats have non-perturbative effects, but their non-extremality is negligible and the solution is approximately the multicenter BPS solution, which in region V approaches that of pure AdS$_5 \times S^5$ with a larger AdS radius.
\begin{figure}[hbtp!]
\begin{center}
\includegraphics[width=.4\textwidth]{Wormhole2Klein.pdf}
\end{center}
\caption{Region I: Perturbatively corrected black brane glued with inversion to the rest of the geometry. Region II: Linearized perturbations to vacuum AdS. Region III: Perturbatively corrected throat solution. Region IV: The (extremal) two-throat solution. Region V: Far from both throats, vacuum AdS with larger radius.\label{fig:wormhole2}}
\end{figure}
Entanglement between disconnected non-interacting boundary theories gives rise to wormholes where the boundaries are separated by causal horizons \cite{GALLOWAY2001255}. A large body of recent work has also been directed towards finding mechanisms that can create and send signals through traversable wormholes in the context of AdS/CFT. In general, supporting a traversable wormhole requires that one violate the averaged null energy condition (ANEC) \cite{Morris:1988tu, PhysRevLett.90.201102, PhysRevLett.61.1446, PhysRevLett.81.746, GALLOWAY2001255}, meaning that there exists an infinite null geodesic with tangent $k^{\mu}$ and affine parameter $\lambda$ such that \cite{GaoJafferisWall}\footnote{The authors of \cite{susywormhole} argued that this condition could be avoided for supersymmetric traversable wormholes connecting two asymptotic AdS universes in the context of pure gauged $\mathcal{N} = 2$ supergravity in four dimensions.}
\begin{align}
\int_{-\infty}^{\infty} k^{\mu} k^{\nu} T_{\mu \nu} d\lambda < 0 \,.
\end{align}
Consequently, in order to build a traversable wormhole, there must be a negative source of stress-energy in the bulk. Several suggestions for introducing this negative stress-energy include inserting explicit double-trace couplings between the boundaries of the wormhole \cite{GaoJafferisWall, GJWBounds}, incorporating the perturbative gravitational back-reaction of bulk quantum fields \cite{Fu_2019, Fu_2019b}, including the Casimir energy of bulk fields running in non-contractible cycles \cite{SM_wormhole, selfsupp}, and nucleating and supporting wormholes via cosmic strings \cite{Horowitz_2019, Fu_2019a}. A particularly productive setting has been the correspondence between the SYK model and AdS$_2$ Jackiw-Teitelboim gravity \cite{doi:10.1002/prop.201700034, SYKwormhole, GaoJafferisSYK, SYKwormhole2}, which, while not an exact duality, has provided further support that explicit boundary couplings may render the bulk geometry traversable and provided an experimental setting by which probing wormhole traversability may be possible in the lab \cite{GJWprobes, wormhole_signaling}.
These constructions use the fact that the eternal AdS-Schwarzschild wormhole is marginally non-traversable in the sense that the null energy vanishes along the causal horizons, so arbitrarily small negative energy perturbations render the wormhole traversable. We will find that the leading classical corrections coming from the global structure of our single-boundary geometry preserve this marginal non-traversability.\footnote{We thank Simon Ross for discussions regarding this point.} This motivates us to describe a mechanism by which our single-boundary wormhole may become traversable by the presence of a natural ``double-trace" type operator in the IR of $\mathcal{N} = 4$ SYM generated by the Wilsonian RG flow \cite{connectivity}, although the presence of bulk fermions in the supergravity spectrum implies that other mechanisms mentioned above may also be a possibility depending, e.g., on the final sign of cancellations between Casimir energies.
The rest of this paper is organized as follows. In Sec.~\ref{sec:fieldtheory} we explain the pattern of symmetry breaking in the field theory and describe a particular entangled state in the IR. In Sec.~\ref{sec:wormholeGeomSoln} we solve for the metric and five-form of the wormhole solution in perturbation theory in type IIB supergravity and describe its global structure. In Sec.~\ref{sec:stability} we use the DBI action to estimate the instability timescale of the wormhole, and show that it is controlled by the same ratio of scales that governs the thermal effective potential in the SYM theory. We also show that it is unlikely that the wormhole could be stabilized by adding rotation. In Sec.~\ref{sec:traverse} we discuss a mechanism for rendering our wormhole traversable, and in Sec.~\ref{sec:doublewormhole} we explain how to use our results to construct a double wormhole between two asymptotic universes. We conclude in the Discussion with comments and remarks for future directions of study.
\noindent\textbf{Conventions} \\
We work in ``mostly-plus" signature for Lorentzian metrics. The convention for five-form components is that
$F = \frac{1}{5!}F_{\mu \alpha \beta\gamma \delta} dx^{\mu} \wedge \ldots \wedge dx^{\delta} = F_{t123r} dt \wedge \ldots \wedge dr + F_{\theta_1 \ldots \theta_5} d\theta_1 \wedge \ldots \wedge d\theta_5$ (all other components will be zero throughout this paper). The notation and combinatorial factors used in symmetrization of indices are for example $A_{(\mu}B_{\nu)} = \frac{1}{2!} (A_{\mu} B_{\nu} + A_{\nu} B_{\mu})$. The indices of all perturbative geometric quantities are raised with the background metric. In general this means index raising and lowering does not commute with perturbative variation. The action of the Hodge star on the components of $p$-forms in $d$ spacetime dimensions is $(\ast F)_{\nu_1 \ldots \nu_{d-p}} = \frac{1}{p!} \sqrt{-g} \epsilon_{\nu_1 \ldots \nu_{d-p} \sigma_1 \ldots \sigma_{p}} g^{\mu_1 \sigma_1} \ldots g^{\mu_p \sigma_p} F_{\mu_1 \ldots \mu_p}$ where $\epsilon_{\mu_1 \ldots \mu_n}$ is the Levi-Civita symbol and $\epsilon_{01\ldots (d-1)} = 1$.
\section{Description in Super-Yang Mills}\label{sec:fieldtheory}
The Lagrangian of $\mathcal{N} = 4$ SYM in terms of component fields is \cite{BRINK197777}
\begin{align}
\mathcal{L}_0 &= \text{tr} \biggl(-\frac{1}{2g_{YM}^2} F_{\mu \nu} F^{\mu \nu} + \frac{\theta}{16\pi^2} F_{\mu \nu} \tilde{F}^{\mu \nu} - i\bar{\lambda}^a \bar{\sigma}^{\mu} D_{\mu} \lambda_a - \sum_i D_{\mu} \phi^i D^{\mu} \phi^i \nonumber \\
&+g_{YM} \sum_{a,b,i} C^{ab}_i \lambda_a [\phi^i, \lambda_b] + g_{YM} \sum_{a,b,i} \bar{C}_{iab} \bar{\lambda}^a [\phi^i, \bar{\lambda}^b] + \frac{g_{YM}^2}{2} \sum_{i,j} [\phi^i, \phi^j]^2\biggr)\, ,
\end{align}
where $F_{\mu \nu} = \partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu} + i[A_{\mu}, A_{\nu}]$ is an $SU(2N)$ gauge field, $D_{\mu} \cdot = \partial_{\mu} \cdot + i[A_{\mu}, \cdot]$ is the covariant derivative on fields in the adjoint representation, $\lambda_a$ are four adjoint Weyl fermions, $\phi^i$ are six adjoint real scalars, and the $C^{ab}_i$ are the Clebsch-Gordon coefficients that couple two $\mathbf{4}$ representations of the $SU(4)_R$ symmetry to the $\mathbf{6}$ antisymmetric representation. The diagonal elements of the vacuum expectation value (vev) of the adjoint scalars $\phi^i$ in the AdS/CFT correspondence map to the positions of D3-branes in ten-dimensional flat space, while the off-diagonal elements are excitations of open strings stretching between the branes \cite{DouglasTaylor}.
At zero temperature, any diagonal configuration of $\phi^i$ gives rise to a vanishing commutator in the potential for the scalars, so there is a large moduli space of stable vacua. In the dual gravity theory, this is equivalent to the statement that an arbitrary number of D3-branes can be superposed at any location in space without any force between them. We choose a vev that will correspond simply to separating two stacks of $N$ D3-branes in a single transverse coordinate by distance $\Lambda$, by expanding
\begin{equation}
\phi^1 \to \psi + \varphi^1
\end{equation}
with the background $\psi$ given by
\begin{align}
\psi = \frac{1}{2\pi \alpha'}\text{diag}(\Lambda \sqrt{N} , \Lambda \sqrt{N} , \ldots, 0, 0, \ldots) \, , \label{eq:scalarvev}
\end{align}
where the eigenvalue $\Lambda \sqrt{N} $ is repeated $N$ times and the vevs of all other $\phi^i$ are zero.
We introduce an explicit factor of $\sqrt{N}$ to appropriately normalize the gauge-invariant classical observable $\text{tr} \, \psi^2$. In detail, we would like the classical value of the observable $\text{tr}\, \phi_1^2$, after proper normalization, to have a finite $\mathcal{O}(1)$ expectation in the large $N$ limit so that there is a well-defined classical gravitational dual. Next we observe that $\text{tr}\, \varphi_1^2$ scales as $\mathcal{O}(N)$, since the components of $\varphi_1$ are $\mathcal{O}(1)$, so its connected two-point function scales as $\langle \text{tr}\, \varphi_1^2 \text{tr}\, \varphi_1^2 \rangle \sim \mathcal{O}(N^2)$. But we know from large-N index counting that if the expectation value of a classical observable is taken to be $O(1)$, then the connected component of the two-point function of its quantum fluctuations should be $O(1/N^2)$. Therefore, the observable $\text{tr}\, \phi_1^2$ requires an overall normalization proportional to $N^{-2}$. Consequently, to make the properly normalized value $N^{-2} \text{tr}\, \psi^2$ of the classical observable $\mathcal{O}(1)$ in the large $N$ limit, a factor of $\sqrt{N}$ should be included in $\psi$.
The factor of $\alpha' = \ell_s^2$ is required by dimensional analysis since the field $\phi^1$ has mass dimension one in four spacetime dimensions. This should be understood as a scale coming from open string theory, since it provides the energy cutoff such that the massless excitations of the open string endpoints moving on the D-brane world-volumes are described by $\mathcal{N}=4$ SYM\footnote{The $\mathcal{N}=4$ SYM does not include $\alpha'$ as a parameter. We introduce $\alpha'$ in order to set the dimensions of $\psi$ correctly as a fiducial scale where we expect the SYM description to break down, anticipating the correspondence with the supergravity description to be discussed later.}. The effective Lagrangian for fluctuations about this background is, rescaling the gauge field $A_{\mu} \to g_{YM} A_{\mu}$ to canonically normalize its kinetic term,
\begin{align}
\mathcal{L} = \mathcal{L}_0\, + \,\text{tr} \biggl(&2 i g_{YM} [\psi, A_{\mu}] \partial^{\mu} \varphi^1 + g_{YM}^2 ([\psi, A_{\mu}]^2 + 2 [\psi, A_{\mu}][\varphi^1, A^{\mu}]) +g_{YM} \sum_{a,b} C_1^{ab} \lambda_a [\psi, \lambda_b] \nonumber\\
&+ g_{YM} \sum_{a,b} \bar{C}_{1ab} \bar{\lambda}^a [\psi, \bar{\lambda}^b]+g^2_{YM} \sum_i ([\psi, \phi^i]^2 + 2 [\psi, \phi^i][\varphi^1, \phi^i])\biggr)\, ,
\end{align}
where $\psi$ should be treated as a classical source.
To understand this effective Lagrangian, it is instructive to expand the commutator of the vev $\psi$ with an arbitrary Hermitian matrix $M$ in the adjoint of $SU(2N)$, which may be written in block form as $M = \begin{pmatrix} M_A & M_B \\ M_B^{\dagger} & M_C \end{pmatrix}$ where each block is $N \times N$:
\begin{align}
[\psi, M] &= \frac{\Lambda \sqrt{N} }{2\pi \alpha^{\prime} }\left(\begin{array}{cc} 0 & \:\:M_B \\ -M_B^{\dagger} & \:\:0\end{array}\right) = \frac{\Lambda \sqrt{N}}{2\pi \alpha^{\prime} } M_{AO} \, .
\end{align}
We have labeled the final matrix $M_{AO}$ for the ``antihermiticized off-diagonal" piece of $M$. Note that $\text{tr}\, M_{AO}^2$ is strictly negative, which is required to give the correct signs below. Armed with this knowledge we further rewrite the commutators in the effective Lagrangian to make the dependence on the coupling $\Lambda$ clear, defining the 't Hooft coupling $\lambda = g_{YM}^2 N$ \cite{HOOFT1974461}:
\begin{align}
\mathcal{L} = \mathcal{L}_0 \, + \, & \frac{1}{2\pi \alpha^{\prime}}\text{tr}\biggl( \Lambda\sqrt{\lambda} \, A_{\mu, AO} \partial^{\mu} \varphi^1 + \frac{\lambda\Lambda^2 }{2\pi \alpha' } \,A_{\mu, AO}^2+ 2 \frac{\lambda \Lambda}{\sqrt{N}} \,A_{\mu, AO} [\varphi^1, A^{\mu}]+ \Lambda \sqrt{\lambda}\sum_{a,b} C_1^{ab}\, \lambda_a \lambda_{b,AO} \nonumber \\
&+ \Lambda \sqrt{\lambda} \sum_{a,b} \bar{C}_{1ab}\, \bar{\lambda}^a \bar{\lambda}^b_{AO} + \frac{\lambda\Lambda^2 }{2\pi \alpha' } \sum_i (\phi^{i}_{AO})^2 + 2 \frac{\lambda \Lambda}{\sqrt{N}} \sum_i \phi^i_{AO}[\varphi^1, \phi^i] \biggr) \, . \label{eq:efflagrang}
\end{align}
The off-diagonal pieces of the gauge field, scalars, and fermions have acquired a mass \\ $ \Lambda \sqrt{\lambda} / (2\pi \alpha^{\prime} ) = \Lambda \sqrt{\lambda} / (2\pi \alpha^{\prime}) $.\footnote{This tree-level mass defines the Higgs scale at weak coupling. At strong coupling, the dependence on $\lambda$ may be different and we comment on this at the end of this section.} The diagonal blocks of the adjoint fields remain massless, so the background $\psi$ has Higgsed the theory $SU(2N) \to S(U(N) \times U(N))$. The extra terms remaining in \eqref{eq:efflagrang} coupling the gauge field to the scalar $\varphi^1$ are typical of those that appear in spontaneously broken non-Abelian gauge theories; we expect that there is a gauge choice which is an analog of the unitary gauge \cite{PhysRevLett.27.1688} where these terms vanish.
In the symmetry-broken phase that we have chosen, low-lying excitations above the vacuum are local to only one of the $SU(N)$ factors of the gauge group. This is because, as we have shown above, the off-diagonal degrees of freedom can be made very heavy by choosing a large Higgs scale $\Lambda\sqrt{\lambda} / (2\pi \alpha')$. Therefore, at energies below the Higgs scale, the Hilbert space of the theory approximately factorizes into that of two separate $SU(N)$ gauge theories, each of which is dual to an AdS throat in the 10D supergravity. Following the ER=EPR conjecture, a state in SYM that possesses the appropriate entanglement between the $\mathcal{O} (N^2)$ light degrees of freedom in each $SU(N)$ factor should be dual to two AdS throats connected by a wormhole in the bulk \cite{doi:10.1002/prop.201300020}. Specifically, we build the approximate thermofield double state\footnote{In the UV it is not possible to factorize the Hilbert spaces due to the $SU(2N)$ being gauged, but it is approximately possible below the Higgs scale.} coupling the energy eigenstates of the Hamiltonian for the effective IR fields in each $SU(N)$ sector
\begin{align}
|\text{TFD}_{\Lambda}\rangle = \sum_{i=1}^{E_i < E_c} e^{-\beta E_i / 2} |i\rangle_L \otimes |i^*\rangle_R \, ,\label{eq:tfdapprox}
\end{align}
where the sum runs over eigenstates of energy less than a cutoff energy scale $E_c$ set by the Higgs scale $\Lambda \sqrt{\lambda} / (2\pi \alpha')$, and the subscripts $L$ and $R$ refer to each of the two $SU(N)$ factors in the symmetry-broken theory. When the thermal energy density is much smaller than the cutoff energy density, we expect this state to be very close to the exact thermofield double state. In the deep bulk, this state is approximately dual to the planar two-sided AdS-Schwarzschild black brane which at fixed times describes a spacelike wormhole or ``Einstein-Rosen bridge" between two asymptotically AdS regions. However, in the ultraviolet of the field theory, the state \eqref{eq:tfdapprox} is embedded in a single $SU(2N)$ SYM theory, so in fact the wormhole begins and ends in the same asymptotic region.
Let us be a bit more precise about how close \eqref{eq:tfdapprox} is to the thermofield double. The dominant contribution to \eqref{eq:tfdapprox} comes at energies where the Boltzman factor offsets the growth coming from the number of states. At low enough temperatures we can think of the individual $SU(N)$ theories as being conformal and therefore their canonical energy density and entropy are fixed by scale invariance and dimensional analysis\footnote{These relations are derived from $\log Z = c V\beta^{-3}$.}
\begin{equation}
\langle E \rangle_\beta/V = 3c\beta^{-4}, \quad S_{th}/V = 4c \beta^{-3}\, ,
\end{equation}
where $c$ is a constant proportional to the central charge and $V$ is the spatial volume. The contribution of a canonical window of states at the cutoff energy $E_c/V=3c \beta_c^{-4}$ is then estimated to be
\begin{equation}
\label{eq:corrections}
e^{S_c-\beta E_c}\approx e^{V c \beta_c^{-4}(4 \beta_c-3 \beta)}\, ,
\end{equation}
i.e. we have an exponential suppression of these contributions if $\beta>4 \beta_c/3$. We take the cutoff temperature to be set by the Higgs scale $\beta_c^{-1} \lesssim \Lambda \sqrt{\lambda}/(2\pi \alpha')$.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=.5\textwidth]{Ueffplot.pdf}
\end{center}
\caption[The shape of the effective potential at finite temperature (orange) vs. zero temperature (blue). At zero temperature, the potential is flat and the components of the scalar background vev $\psi^i$ are free, while at finite temperature, the only stable configuration has all $\psi^i = 0$.]{The shape of the effective potential at finite temperature (orange) vs. zero temperature (blue). At zero temperature, the potential is flat and the components of the scalar background vev $\psi^i$ are free, while at finite temperature, the only stable configuration has all $\psi^i = 0$.\footnote[7]{\mbox{}}\label{fig:effpotential}}
\end{figure}
Reducing $|\text{TFD}_\Lambda\rangle$ onto the Hilbert space of either $SU(N)$ factor yields a state which looks approximately thermal in the infrared (up to corrections of order \eqref{eq:corrections}), with a temperature $\beta \sim r_0^{-1}$ that corresponds to a black brane of horizon radius $r_0$ in the gravity dual. In finite temperature field theory the supersymmetry of the SYM Lagrangian is broken and in particular the effective potential for the scalars $\phi^i$ is modified so that the only stable vacuum configuration is the one where the vevs of all the $\phi^i$ sit at the origin in moduli space. See Fig.~\ref{fig:effpotential} for a schematic depiction of the effective potential at finite temperature. Consequently, an initial configuration of the form \eqref{eq:scalarvev} is unstable and the vev will roll down the potential towards the origin. In the bulk dual, this has the well-known effect that nonextremal D-branes exert a nonzero attractive force on each other. Parametrically, at weak coupling $\lambda$ the thermal effective potential is controlled by the perturbatively small ratio of the thermal and Higgs scales, $\frac{2\pi \alpha'}{\beta \Lambda \sqrt{\lambda}}= \epsilon$. The weak coupling effective potential cannot be directly compared with the dual semiclassical gravity, since the latter is only valid when the field theory coupling is strong. Nonetheless, in Sec.~\ref{sec:stability} we will estimate the timescale of the instability from the gravity dual using the DBI action of the underlying branes. In terms of bulk quantities in the gravity dual, the perturbative parameter controlling the thermal effective potential at small 't Hooft coupling is
\begin{align}
\epsilon = \frac{\sqrt{2}}{\lambda} \frac{ r_0}{\Lambda }\, .
\end{align}
\setcounter{footnote}{7} \footnotetext{It may be surprising that the effective potential at zero temperature for the scalars is flat since the SYM theory is dual to the asymptotically AdS geometry that remains after the decoupling limit, and radially separated branes in asymptotically AdS space are subject to a potential barrier at infinity. However, the geometry sourced by two stacks of branes is a full ten-dimensional geometry that only approximately fibers into an AdS$_5$ and an $S^5$ close to each stack and near infinity. From the perspective of the asymptotic $S^5$, the branes are located at opposite poles and are not radially separated in the AdS space, so the AdS potential barrier does not apply.
}
At \emph{strong} coupling in SYM, where the bulk dual admits a semiclassical description in supergravity, quantities computed at weak field theory coupling are often rescaled by functions of $\lambda$ (see \cite{doi:10.1063/1.1372177} for a concrete example). Therefore, away from weak coupling we expect that the perturbative parameter that will control the bulk geometry in supergravity (and therefore parameterize the instability timescale of the wormhole) will take the form
\begin{align}
\epsilon = f(\lambda) \frac{\sqrt{2}}{\lambda} \frac{ r_0}{\Lambda } \, ,
\end{align}
for some function $f(\lambda)$. In the limit of large coupling $\lambda$ we will see that $f(\lambda) \sim \lambda$, so that the perturbative description of the classical geometry is naturally controlled by $\epsilon \sim r_0 / \Lambda$, which is independent of the string scale. It would be interesting to see if the function $f(\lambda)$ can be determined as an exact function of the coupling $\lambda$ using integrability techniques.
\section{Wormhole Geometry in Supergravity} \label{sec:wormholeGeomSoln}
In subsequent sections, we will write down the detailed solution to the equations of motion in each region. Our starting point for the construction, following \cite{pol95}, is the action of type IIB supergravity in string frame, restricted to the metric, dilaton, and five-form\footnote{The reviews \cite{DUFF1995213, peet} and textbooks \cite{kiritsis, johnson, amerd} provide compact and relevant introductions to D-brane solutions to type IIB supergravity that may be useful for subsequent sections.}:
\begin{align}
S_{\text{IIB}} = \frac{1}{2\kappa^2_{10}} \int d^{10} x \sqrt{-g} \left( e^{-2\phi} (R + 4\partial_{\mu} \phi \partial^{\mu} \phi) - \frac{1}{4\cdot 5! } F_{\mu \nu \rho \sigma \tau} F^{\mu \nu \rho \sigma \tau}\right) \, ,\label{eq:IIBaction}
\end{align}
where $2\kappa_{10}^2 = (2\pi)^7 \alpha'^{4} g_s^2$. The asymptotic value of the dilaton has already been scaled out so that $e^{\phi} = 1$ at infinity. We work in the strongly coupled limit of the field theory, $g_s N \to \infty$, such that classical supergravity is valid. The background values of the various fermions of type IIB are taken to be zero self-consistently.
Taking as an ansatz that the dilaton will be constant everywhere so that we can drop terms involving its gradient, the classical equations of motion are
\begin{align}
e^{-2\phi} (R_{\mu \nu} - \frac12 R g_{\mu \nu}) &= -\frac{1}{8\cdot 5!} g_{\mu \nu} F_{\alpha \beta \gamma \delta \epsilon} F^{ \alpha \beta \gamma \delta \epsilon} + \frac{1}{4\cdot 4!} F_{\mu \alpha \beta \gamma \delta} F^{ \:\:\alpha \beta\gamma \delta}_{\nu} \label{eq:geomEOM1} \\
\partial_{\mu} (\sqrt{-g} F^{ \mu \nu \rho \sigma\tau}) &= 0 \, , \label{eq:maxwellbgd}
\end{align}
to be supplemented by the self-duality constraint $F = \ast F$, and with $e^{\phi}= 1$ everywhere. Due to the self-duality constraint, $ F_{\alpha \beta \gamma \delta \epsilon} F^{ \alpha \beta \gamma \delta \epsilon} \sim F \wedge \ast F = F \wedge F = 0$ since the wedge product is antisymmetric on five-forms. A straightforward computation by taking traces and using this identity shows that $R = 0$. Consequently, \eqref{eq:geomEOM1} simplifies to
\begin{align}
R_{\mu \nu} &= \frac{1}{4\cdot 4!} F_{\mu \alpha \beta \gamma \delta} F^{ \:\:\alpha \beta\gamma \delta}_{\nu} \, . \label{eq:geomEOM2}
\end{align}
In subsequent sections we will write down perturbative corrections to solutions to the background equations of motion. We add these perturbative corrections at first order to the metric and five-form, $g_{\mu \nu} \to \bar{g}_{\mu \nu} + h_{\mu \nu}$ and $F \to \bar{F} + \delta F$, where the bar indicates quantities at background order, i.e. that solve \eqref{eq:maxwellbgd} and \eqref{eq:geomEOM2}. The perturbative equations of motion are
\begin{align}
\nabla_{\lambda} \nabla_{(\mu} h^{\lambda}_{\:\:\nu)} - \frac12 \nabla_{\mu}\partial_{\nu} h - \frac12 \nabla_{\lambda} \nabla^{\lambda} h_{\mu \nu} &= \frac{1}{4 \cdot 4!} (\delta F_{\mu \alpha \beta \gamma \delta} \bar{F}_{\nu}^{\:\:\alpha \beta \gamma \delta} + \bar{F}_{\mu \alpha \beta \gamma \delta} \delta F_{\nu}^{\:\: \alpha \beta \gamma \delta}) \label{eq:perturbativeEOMs1}\\
\partial_{\mu} \bigl[\sqrt{-\bar{g}} \bigl(\frac{h}{2} \bar{F}^{\mu \nu \rho \sigma \tau} + \delta F^{\mu \nu \rho \sigma \tau} \bigr) \bigr]&= 0 \, ,
\label{eq:perturbativeEOMs2}
\end{align}
where $h = \bar{g}^{\mu \nu} h_{\mu \nu}$ is the trace of the metric perturbation and $\nabla_{\mu}$ is the covariant derivative with respect to $\bar{g}$. These equations must be supplemented with the self-duality constraint at all orders, such that $\bar{F} + \delta F= \ast (\bar{F} + \delta F)$, a nontrivial constraint since the Hodge dual involves the metric perturbations. The derivation of \eqref{eq:perturbativeEOMs1} and \eqref{eq:perturbativeEOMs2} can be found in Appendix~\ref{sec:perturbEOM}. In the following sections, we will exhibit solutions to the background equations of motion \eqref{eq:maxwellbgd} and \eqref{eq:geomEOM2} and to their first-order variation in \eqref{eq:perturbativeEOMs1} and \eqref{eq:perturbativeEOMs2}. Although we have labeled the regions of the geometry I - V in order of the flow from the IR to the UV in the field theory, we will describe the solutions below in a different order that will be more convenient for intuition.
\subsection{Region IV: Two-Center Harmonic Solution} \label{sec:twocenter}
The general two-center solution at nonzero temperature is not known even in perturbation theory, so we will first write down the background solution to \eqref{eq:maxwellbgd} and \eqref{eq:geomEOM2} without perturbations in this region. We begin with the solution corresponding to two stacks of $N$ extremal D3-branes placed at a separation $\Lambda$ in 10D Minkowski spacetime. The coordinates $t, x^1, \ldots, x^3$ extend parallel to the brane worldvolumes; we label the other six directions transverse to the branes as $r^1$ through $r^6$. Without loss of generality let the two stacks of branes be displaced in the $r^1$ direction. The solution in asymptotically flat space is BPS and the metric and five-form are given by \cite{DUFF1991409, HOROWITZ1991197}
\begin{align}
\label{eq:harmonicmulticenter}
ds^2 &= H^{-1/2} (- dt^2 + d\vec{x}^2) + H^{1/2} \delta_{ij} dr^i dr^j \\
F&= (1+\ast) dt \wedge dx^1 \wedge dx^2 \wedge dx^3 \wedge dH^{-1} \, ,
\end{align}
where $H$ is a two-center harmonic function:
\begin{align}
H= 1 + \frac{L^4}{r^4} + \frac{L^4}{|\vec{r} \pm\vec{\Lambda}|^4} \, .
\end{align}
Here $\vec{\Lambda} = (\Lambda,0,0,0,0,0)$ and $r^2 = \sum_i (r^i)^2$. The choice of sign fixes the direction of displacement of the stacks of branes.
The flux through the five-sphere jumps discontinuously when the radius of the five-sphere around one stack of branes crosses through the other stack. When the radius is smaller than $\Lambda$, the charge of a single stack is, by Stokes' theorem,
\begin{align}
Q = \frac{1}{2\kappa^2}\int_{S^5} \ast F = \frac{L^4}{2g_s^2 (2\pi)^4 (\alpha')^4} \, .
\end{align}
The normalization comes from the normalization of the kinetic term for the five-form in \eqref{eq:IIBaction} \cite{pol95}. By the BPS condition, the charge is equal to the number of branes $N$ times the tension of a single extremal brane, $\tau =(2\pi)^{-3} (\alpha')^{-2} g_s^{-1}$, so
\begin{align*}
Q = N\tau \implies L^4 = 4\pi g_s N(\alpha')^2 \, .
\end{align*}
We now take the decoupling limit $\alpha' \to 0$ keeping fixed $\Lambda / \alpha'$ and $r^i / \alpha'$ \cite{costa}. Fixing $\Lambda/\alpha'$ amounts to fixing the Higgs scale of \eqref{eq:scalarvev} in the dual theory. To write a non-singular metric, we rescale $L^2 \to \alpha' L^2$, $r^i \to \alpha' r^i$, and $\Lambda \to \alpha' \Lambda$. The harmonic function becomes
\begin{align}
H = \alpha'^{-2} \left(\frac{L^4}{r^4} + \frac{L^4}{|\vec{r} \pm\vec{\Lambda}|^4} \right) \, .
\end{align}
Lastly, we nondimensionalize coordinates by the rescaling
\begin{align}
\frac{r_0 t}{L^2} = \tilde{t}, \qquad \frac{r}{r_0} = \tilde{r}, \qquad \frac{r_0 x^i}{L^2} = \tilde{x}^i \, . \label{eq:nondim}
\end{align}
Here $r_0$ is the wormhole horizon radius, to be introduced in subsequent sections. This nondimensionalization will be convenient in other regions where it removes the length scale $r_0$. We introduce the parameter $\epsilon = \frac{r_0}{ \Lambda}$. The full wormhole solution will only be valid in the limit $\epsilon \ll 1$ where the horizons of the two stacks of branes are well-separated. Given these definitions, the full solution in region IV is
\begin{align}
\frac{1}{\alpha'}ds^2 &= L^2 \biggl[\biggl(\frac{1}{\tilde{r}^4} + \frac{1}{|\vec{\tilde{r}} \pm \vec{\epsilon}^{-1}|^4} \biggr)^{-1/2} (- d\tilde{t}^2 + d\vec{\tilde{x}}^2) + \biggl(\frac{1}{\tilde{r}^4} + \frac{1}{|\vec{\tilde{r}} \pm \vec{\epsilon}^{-1}|^4} \biggr)^{1/2} \delta_{ij} d\tilde{r}^i d\tilde{r}^j\biggr] \\
\frac{1}{\alpha'^2} F&= L^4 \biggl[ (1+\ast) d\tilde{t} \wedge d\tilde{x}^1 \wedge d\tilde{x}^2 \wedge d\tilde{x}^3 \wedge d \biggl(\frac{1}{\tilde{r}^4} + \frac{1}{|\vec{\tilde{r}} \pm \vec{\epsilon}^{-1}|^4} \biggr)^{-1}\biggr] \, ,
\end{align}
where $\vec{\epsilon}^{-1} = (\epsilon^{-1},0,0,0,0,0)$.
It is convenient for subsequent sections to expand this solution close to the stack of branes at the origin, $r^i \ll \Lambda$. In this limit the (nondimensionalized) harmonic function becomes simply
\begin{align}
H = \frac{1}{\tilde{r}^4} + \epsilon^4 \, .\label{eq:Hlimit}
\end{align}
That is, all dependence on $r^1$ is subleading, so spherical symmetry about the stack of branes is valid in this limit. Furthermore, this expansion is valid around either stack provided the radial coordinate is defined appropriately. Defining the hyperspherical coordinates
\begin{align}
\begin{aligned}
r_1 &= r\cos \theta_1 \\
r_2 &=r\cos \theta_2\sin \theta_1 \\
r_3 &= r\cos \theta_3\sin \theta_2\sin \theta_1\\
r_4 &= r\cos \theta_4\sin \theta_3\sin \theta_2\sin \theta_1\\
r_5 &= r\cos \theta_5\sin \theta_4\sin \theta_3\sin \theta_2\sin \theta_1\\
r_6 &= r\sin \theta_5\sin \theta_4\sin \theta_3\sin \theta_2\sin \theta_1 \, ,
\end{aligned}
\end{align}
and series expanding $H^{\pm 1/2}$ yields
\begin{align}
\frac{1}{\alpha'} ds^2
&= L^2\biggl[ \tilde{r}^2 \left(1- \frac12 (\epsilon \tilde{r})^4\right)(-d\tilde{t}^2+d\vec{\tilde{x}}^2) + \left(1+\frac12 (\epsilon \tilde{r})^4 \right)(\frac{d\tilde{r}^2}{\tilde{r}^2} + d\Omega_5^2) \biggr] \label{eq:throatcorrect1}\\
\frac{1}{\alpha'^2} F &= 4L^4 \biggl[\tilde{r}^3 \left(1 - 2(\epsilon \tilde{r})^4\right) d\tilde{t} \wedge d\tilde{x}^1 \wedge d\tilde{x}^2 \wedge d\tilde{x}^3 \wedge d\tilde{r} \nonumber\\
&\qquad\qquad\qquad+ \sin^4 \theta_1 \sin^3 \theta_2 \sin^2 \theta_3 \sin \theta_4 d\theta_1 \wedge d\theta_2 \wedge d\theta_3 \wedge d\theta_4 \wedge d\theta_5\biggr] \, .\label{eq:throatcorrect2}
\end{align}
In this form, the linearized corrections to vacuum AdS$_5 \times S^5$ deep within a single throat are apparent. These corrections will be useful in subsequent sections.
\subsection{Region V: Asymptotics}
In the region far from both stacks of branes, we take the limit $r^i \gg \Lambda$, by which the harmonic function $H$ simplifies to
\begin{align}
H = \frac{2L^4}{r^4}
\end{align}
Writing $L_{\infty}^4 = 2L^4$, taking the decoupling limit, rescaling, and nondimensionalizing, the solution in region V is
\begin{align}
\frac{1}{\alpha'} ds^2 &= L_{\infty}^2\biggl[ \tilde{r}^2 (-d\tilde{t}^2+d\vec{\tilde{x}}^2) + \frac{d\tilde{r}^2}{\tilde{r}^2} + d\Omega_5^2 \biggr] \\
\frac{1}{\alpha'^2} F &= 4L_{\infty}^4 \biggl[\tilde{r}^3 d\tilde{t} \wedge d\tilde{x}^1 \wedge d\tilde{x}^2 \wedge d\tilde{x}^3 \wedge d\tilde{r} + \sin^4 \theta_1 \sin^3 \theta_2 \sin^2 \theta_3 \sin \theta_4 d\theta_1 \wedge d\theta_2 \wedge d\theta_3 \wedge d\theta_4 \wedge d\theta_5\biggr] \, .
\end{align}
This solution is vacuum AdS$_5 \times S^5$ with AdS length $L_{\infty}^4 = 2L^4$. The charge is
\begin{align}
Q = \frac{L_{\infty}^4}{2g_s^2 (2\pi)^4 (\alpha')^4} \, ,
\end{align}
leading to
\begin{align}
L_{\infty}^4 = 4\pi g_s (2N)(\alpha')^2
\end{align}
from the flux quantization condition. At infinity, the flux sees both stacks of branes as if they are at the origin, as expected. This region corresponds to the UV in the field theory where the $SU(2N)$ symmetry is unbroken.
\subsection{Region I: Black Brane}
In region I, the solution is the geometry of the two-sided black brane with perturbative corrections coming from the second throat in the full geometry. In this region, the solution will describe the geometry close to one of the two stacks with the origin of coordinates placed at the location of the stack, that is, in the limit $r^i \ll \Lambda$ of Sec.~\ref{sec:twocenter}. The solution preserves the $SO(3,1) \times SO(6)$ isometries induced by the brane locations. Following the conventions of \cite{peet}, the metric and five-form of the asymptotically flat solution are \cite{HOROWITZ1991197, DUFF1991409, DUFF1995213}
\begin{align}
ds^2 &= H(r)^{-1/2} (-f(r) dt^2 + d\vec{x}^2) + H(r)^{1/2} (dr^2 / f(r) + r^2 d\Omega_5^2) \\
F&= \sqrt{1+\frac{r_0^4}{L^4}}(1+\ast) dt \wedge dx^1 \wedge dx^2 \wedge dx^3 \wedge dH^{-1} \, ,
\label{eq:flatnonextremal}
\end{align}
with $H(r) = 1+\frac{L^4}{r^4}$ and $f(r) = 1- \frac{r_0^4}{r^4}$. As the horizon radius $r_0 \to 0$ one approaches the extremal limit of the brane solution. The five-form can be written explicitly in coordinates as
\begin{align}
F&= \sqrt{1+\frac{r_0^4}{L^4}} \biggl[\frac{4L^4}{ r^5 H(r)^2} dt \wedge dx^1 \wedge dx^2 \wedge dx^3 \wedge dr \nonumber \\
& \qquad\qquad\qquad\qquad + 4L^4 \sin^4 \theta_1 \sin^3 \theta_2 \sin^2 \theta_3 \sin \theta_4 d\theta_1 \wedge d\theta_2 \wedge d\theta_3 \wedge d\theta_4 \wedge d\theta_5\biggr] \, .
\end{align}
Consequently, the charge is
\begin{align}
Q = \frac{1}{2\kappa^2} \int_{S^5} \ast F = \frac{L^4}{2g_s^2 (2\pi)^4 (\alpha')^4 }\sqrt{1+\frac{r_0^4}{L^4}} \, .
\end{align}
The charge remains equal to the number of branes $N$ times the tension $\tau$ of an extremal brane, so
\begin{align}
Q = N\tau \implies L^4 = -\frac12 r_0^4 + \sqrt{(4\pi g_s N(\alpha')^2 )^2 + \frac14 r_0^8} \, .
\end{align}
We now take the decoupling limit $\alpha' \to 0$ keeping fixed $r^i / \alpha'$ and $r_0 / \alpha'$, rescaling $L^2 \to \alpha' L^2$, $r_0\rightarrow \alpha' r_0$ and $r \to \alpha' r $. The resulting solution has the same form as \eqref{eq:flatnonextremal} with $H=L^4/r^4$, no overall scaling on the five-form, and $f(r)$ unchanged. Nondimensionalizing following \eqref{eq:nondim} it can be written as
\begin{align}
\frac{1}{\alpha'} ds^2 &= L^2 \biggl[ -\tilde{r}^2(1-\frac{1}{\tilde{r}^4}) d\tilde{t}^2 + \tilde{r}^2 d\vec{\tilde{x}}^2 + \frac{d\tilde{r}^2}{\tilde{r}^2(1-\frac{1}{\tilde{r}^4})} + d\Omega_5^2 \biggr] \label{eq:blackbranebgdgeom} \\
\frac{1}{\alpha'^{2}} F &= 4L^4 \biggl[\tilde{r}^3 d\tilde{t} \wedge d\tilde{x}^1 \wedge d\tilde{x}^2 \wedge d\tilde{x}^3 \wedge d\tilde{r} + \sin^4 \theta_1 \sin^3 \theta_2 \sin^2 \theta_3 \sin \theta_4 d\theta_1 \wedge d\theta_2 \wedge d\theta_3 \wedge d\theta_4 \wedge d\theta_5 \biggr] \, . \label{eq:blackbranebgdform}
\end{align}
This is the background solution in region I. Note that the finite temperature factor has dropped out of the five-form after the decoupling limit. Consequently, the charge in the decoupling limit is simply
\begin{align}
Q = \frac{L^4}{2g_s^2 (2\pi)^4 (\alpha')^4 } \, .
\end{align}
This leads to the extremal quantization condition for the black brane in AdS,
\begin{align}
L^4 = 4\pi g_s N(\alpha')^2 \, .
\end{align}
In Sec.~\ref{sec:monopole} we will describe the perturbative corrections to \eqref{eq:blackbranebgdgeom}, \eqref{eq:blackbranebgdform} coming from the second throat, though we first describe the general structure of the multipole expansion that gives rise to these perturbative corrections in Sec.~\ref{sec:linearized}.
\subsection{Regions II-III: Linearized Regime} \label{sec:linearized}
Regions II and III are the intermediate regimes far from the horizon and sufficiently deep within a single throat such that the corrections to vacuum AdS$_5 \times S^5$ both from the throat and from the wormhole can be linearized. These regions are defined by $\tilde{r} \sim \mathcal{O}(\epsilon^{-1/2})$, where the background is empty AdS and corrections to this coming both from the harmonic function and the blackening factor are $\mathcal{O}(\epsilon^2)$. Since both of these corrections can be treated as linearized and the equations of motion are linear in the perturbations, the full solution can be written simply as the linear superposition of the two,
\begin{align}
\frac{1}{\alpha'}ds^2 &=H_0(r)^{-1/2} \left[ 1-\frac{1}{2}\frac{\delta H(r,\theta_1)}{H_0(r)}\right] \left(-\left[1-\frac{r_0^4}{r^4} \right] dt^2 + d\vec{x}^2 \right) \nonumber \\ &+ H_0(r)^{1/2} \left[ 1+\frac{1}{2}\frac{\delta H(r,\theta_1)}{H_0(r)}\right] \left(dr^2 \left[1+\frac{r_0^4}{r^4} \right] + r^2 d\Omega_5^2\right) \\
\frac{1}{{\alpha'}^2}F&= (1+\ast) dt \wedge dx^1 \wedge dx^2 \wedge dx^3 \wedge d\left[ H_0(r)^{-1}\left( 1-\frac{\delta H(r,\theta_1)}{H_0(r)}\right)\right] \, ,
\end{align}
where $H_0(r)=\frac{L^4}{r^4}$ and
\begin{equation}
\label{eq:harmonicmultipole}
\delta H(r,\theta_1)=\frac{L^4}{\Lambda ^4}+\frac{4 L^4 r \cos (\theta_1 )}{\Lambda ^5} +\frac{2 L^4 r^2 (3 \cos (2 \theta_1 )+2)}{\Lambda ^6}+\frac{4 L^4 r^3 (3 \cos (\theta_1 )+2 \cos (3 \theta_1 ))}{\Lambda ^7}+ \mathcal{O}(\Lambda^{-8}) \, ,
\end{equation}
and the Hodge star must be applied so that in the result we linearize both in $\delta H$ and $r_0^4$. This solves the linearized equations of motions simply because it is the sum of two linear perturbations of AdS$_5\times S^5$, one defined by expanding \eqref{eq:harmonicmulticenter} in $\delta H$ with $H=H_0+\delta H$, and the other by expanding \eqref{eq:blackbranebgdgeom} in $r_0^4/r^4$. Note that the leading nonlinearity from the two centered harmonic function comes at $(\delta H)^2 \sim \Lambda^{-8}$ and therefore we can keep the multipole expansion \eqref{eq:harmonicmultipole} up to $\mathcal{O}(\Lambda^{-7})$ in the linearized regime.\footnote{After nondimensionalizing, the multipole expansion is controlled by powers of $\epsilon$, so the leading nonlinearity is at $\mathcal{O}(\epsilon^8)$.}
We define region II as the patch where spherical symmetry around the throat is approximately unbroken and hence we can stop in the multipole expansion of $\delta H (r, \theta_1)$ at monopole order. This means keeping only the $L^4/\Lambda^4$ term in \eqref{eq:harmonicmultipole}. In this case the solution explicitly reads in nondimensionalized coordinates
\begin{align}
\frac{1}{\alpha'} ds^2 &= L^2 \biggl[-\tilde{r}^2\left(1- \frac{(\epsilon \tilde{r})^4}{2} -\frac{1}{\tilde{r}^4} \right) d\tilde{t}^2 +\tilde{r}^2\left(1- \frac{(\epsilon \tilde{r})^4}{2}\right) d\vec{\tilde{x}}^2 \nonumber\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad + \frac{1}{\tilde{r}^2}\left(1+\frac{(\epsilon \tilde{r})^4}{2} + \frac{1}{\tilde{r}^4}\right)d\tilde{r}^2 + \left(1+\frac{(\epsilon \tilde{r})^4}{2} \right) d\Omega_5^2 \biggr] \label{eq:linear1}\\
\frac{1}{\alpha^{'2}} F &= 4L^4 \biggl[\tilde{r}^3 \left(1 - 2(\epsilon \tilde{r})^4 \right) d\tilde{t} \wedge d\tilde{x}^1 \wedge d\tilde{x}^2 \wedge d\tilde{x}^3 \wedge d\tilde{r} \nonumber \\&\qquad\qquad\qquad\qquad\qquad+ \sin^4 \theta_1 \sin^3 \theta_2 \sin^2 \theta_3 \sin \theta_4 d\theta_1 \wedge d\theta_2 \wedge d\theta_3 \wedge d\theta_4 \wedge d\theta_5 \biggr] \, .\label{eq:linear2}
\end{align}
In this regime, the equations of motion can be solved by hand; see Appendix~\ref{sec:linearizedApp}. The procedure involves several undetermined constants and an undetermined function as a consequence of a residual diffeomorphism freedom.
\subsection{Joint solution in regions I-II: Monopole contribution}
\label{sec:monopole}
Now, we solve \eqref{eq:perturbativeEOMs1} and \eqref{eq:perturbativeEOMs2} for the perturbations to the metric and the five-form in the backgrounds of \eqref{eq:blackbranebgdgeom} and \eqref{eq:blackbranebgdform}, i.e. the non-extremal black brane. In this subsection we deal with the case when spherical symmetry is intact, that is, we solve for the monopole contribution of the far throat down the near throat. We begin with an ansatz for the perturbations consistent with the $SO(3,1) \times SO(6)$ symmetry
\begin{align}
\frac{1}{\alpha'}ds^2 &= L^2 \left[-\tilde{r}^2(1-\frac{1}{\tilde{r}^4})(1+\delta g_{\tilde{t}\tilde{t}}) d\tilde{t}^2 + \tilde{r}^2 (1+ \delta g_{\tilde{i}\tilde{i}}) d\vec{\tilde{x}}^2 + \frac{d\tilde{r}^2}{\tilde{r}^2(1-\frac{1}{\tilde{r}^4})} (1+\delta g_{\tilde{r}\tilde{r}}) + (1+\delta g_{\Omega \Omega}) d\Omega_5^2 \right] \\
\frac{1}{\alpha^{'2}} F &=4L^4\biggl[\tilde{r}^3 (1+a(\tilde{r})) d\tilde{t} \wedge d\tilde{x}^1 \wedge d\tilde{x}^2 \wedge d\tilde{x}^3 \wedge d\tilde{r} \nonumber \\
&\qquad + (1+b(\tilde{r}))\sin^4 \theta_1 \sin^3 \theta_2 \sin^2 \theta_3 \sin \theta_4 d\theta_1 \wedge d\theta_2 \wedge d\theta_3 \wedge d\theta_4 \wedge d\theta_5\biggr] \, ,
\end{align}
where the metric perturbations are all functions only of the radial coordinate $r$. In terms of the perturbations, the Maxwell equations and self-duality constraint reduce simply to
\begin{align}
2a-2b-3\delta g_{\tilde{i}\tilde{i}} - \delta g_{\tilde{t}\tilde{t}}-\delta g_{\tilde{r}\tilde{r}}+5\delta g_{\Omega \Omega} &= 0 \label{eq:selfdualBB} \\
2a'-3\delta g_{\tilde{i}\tilde{i}}' - \delta g_{\tilde{t}\tilde{t}}'-\delta g_{\tilde{r}\tilde{r}}'+5\delta g_{\Omega \Omega}' &= 0 \, .\label{eq:maxwellBB}
\end{align}
The geometric equations of motion are:
\begin{align}
\begin{aligned}
&\left(1-5 \tilde{r}^4\right) \delta g_{\Omega \Omega }'- \tilde{r} \left(\tilde{r}^4-1\right) \delta g_{\Omega \Omega }''-16\tilde{r}^3 (b-2 \delta g_{\Omega \Omega }) = 0 \\
&-16\tilde{r}^3 a + 24 \tilde{r}^3 \delta g_{\tilde{i}\tilde{i}} + 8\tilde{r}^3 \delta g_{\tilde{t}\tilde{t}} + 3(\tilde{r}^4 + 1)\delta g_{\tilde{i}\tilde{i}}' + 6\tilde{r}^4 \delta g_{\tilde{t}\tilde{t}}' \\
&\qquad\qquad\qquad- (\tilde{r}^4 + 1)\delta g_{\tilde{r}\tilde{r}}' + 5(\tilde{r}^4 + 1) \delta g_{\Omega \Omega}' + (\tilde{r}^5 - \tilde{r} ) \delta g_{\tilde{t}\tilde{t}}'' = 0 \\
&-16\tilde{r}^3 a + 24 \tilde{r}^3 \delta g_{\tilde{i}\tilde{i}} + 8\tilde{r}^3 \delta g_{\tilde{t}\tilde{t}} -4(2\tilde{r}^4-1) \delta g_{\tilde{i}\tilde{i}}'- (\tilde{r}^4-1)\delta g_{\tilde{t}\tilde{t}}' \\&\qquad\qquad\qquad+(\tilde{r}^4-1)\delta g_{\tilde{r}\tilde{r}}' - 5(\tilde{r}^4-1)\delta g_{\Omega \Omega}' - (\tilde{r}^5 - \tilde{r}) \delta g_{\tilde{i}\tilde{i}}'' = 0 \\
&-16\tilde{r}^3 a + 24 \tilde{r}^3 \delta g_{\tilde{i}\tilde{i}} + 8\tilde{r}^3 \delta g_{\tilde{t}\tilde{t}} + 3 (3\tilde{r}^4-1)\delta g_{\tilde{i}\tilde{i}}' + 3(\tilde{r}^4+1)\delta g_{\tilde{t}\tilde{t}}'-2(2\tilde{r}^4-1) \delta g_{\tilde{r}\tilde{r}}' \\
&\qquad\qquad\qquad+5(\tilde{r}^4+1)\delta g_{\Omega \Omega}' + 3(\tilde{r}^5-\tilde{r})\delta g_{\tilde{i}\tilde{i}}'' + (\tilde{r}^5-\tilde{r})\delta g_{\tilde{t}\tilde{t}}''+5(\tilde{r}^5-\tilde{r}) \delta g_{\Omega \Omega}'' = 0 \, .
\end{aligned}
\end{align}
Note that we are in the region $\tilde{r}>1$ outside the horizon at all times. We solve the equations of motion by the ansatz that the perturbations will be equal to the near-horizon corrections of the two-throat solution given in \eqref{eq:throatcorrect1} and \eqref{eq:throatcorrect2}. This ansatz turns out to be correct provided that we include an additional contribution to the five-form perturbations $a(\tilde{r})$ and $b(\tilde{r})$. The full perturbative solution in region I is
\begin{align}
\label{eq:monopole}
\frac{1}{\alpha'}ds^2 &= L^2 \biggl[-\tilde{r}^2(1-\frac{1}{\tilde{r}^4})(1-\frac12 (\epsilon \tilde{r})^4) d\tilde{t}^2 + \tilde{r}^2 (1-\frac12 (\epsilon \tilde{r})^4 ) d\vec{\tilde{x}}^2 \nonumber\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad + \frac{d\tilde{r}^2}{\tilde{r}^2(1-\frac{1}{\tilde{r}^4})} (1+\frac12 (\epsilon \tilde{r})^4) + (1+\frac12 (\epsilon \tilde{r})^4 ) d\Omega_5^2 \biggr] \\
\frac{1}{\alpha^{'2}} F &=4L^4\biggl[\tilde{r}^3 \left(1-2(\epsilon \tilde{r})^4 + \frac{\epsilon^4}{2}\right) d\tilde{t} \wedge d\tilde{x}^1 \wedge d\tilde{x}^2 \wedge d\tilde{x}^3 \wedge d\tilde{r} \nonumber \\
&\qquad\qquad\qquad\qquad\qquad +(1+\frac{\epsilon^4}{2})\sin^4 \theta_1 \sin^3 \theta_2 \sin^2 \theta_3 \sin \theta_4 d\theta_1 \wedge d\theta_2 \wedge d\theta_3 \wedge d\theta_4 \wedge d\theta_5\biggr] \, .
\end{align}
In this regime, we are close to the horizon, so $\tilde{r} \sim \mathcal{O}(1)$ and the perturbative corrections are $\mathcal{O}(\epsilon^4)$. When $\tilde{r}$ gets large, this solution matches onto \eqref{eq:linear1} in the linearized regime, region II.
The physical interpretation of the leading monopole contribution from the presence of the other throat is to create a small region of flat space around the black brane. This can be seen by noting that \eqref{eq:monopole} can be obtained by linearizing in $1/\Lambda^4$ the non-perturbative solution (in dimensionful coordinates):
\begin{align}
\frac{1}{\alpha'}ds^2 &=\left(\frac{L^4}{r^4}+\frac{L^4}{\Lambda^4} \right)^{-1/2} \left(-\left[1-\frac{r_0^4}{r^4} \right] dt^2 + d\vec{x}^2 \right) + \left(\frac{L^4}{r^4}+\frac{L^4}{\Lambda^4} \right)^{1/2} \left(dr^2 \left[1-\frac{r_0^4}{r^4} \right]^{-1} + r^2 d\Omega_5^2\right) \label{eq:metricnonpert}\\
\frac{1}{{\alpha'}^2}F&= \sqrt{1+\frac{r_0^4}{\Lambda^4}} (1+\ast) dt \wedge dx^1 \wedge dx^2 \wedge dx^3 \wedge d\left(\frac{L^4}{r^4}+\frac{L^4}{\Lambda^4} \right)^{-1} \label{eq:fiveformnonpert}\, ,
\end{align}
which is obtained by truncating the multipole expansion at monopole order but keeping the nonlinear dependence on the harmonic functions, as well as the blackening factor. It is easy to see that this solution is just a single non-extremal black brane in asymptotically flat space, in rescaled coordinates $t'=\frac{\Lambda}{L}t$, $\vec{x}'=\frac{\Lambda}{L}\vec{x}$, $r'=\frac{L}{\Lambda}r$. This rescaling puts the solution \eqref{eq:metricnonpert} in the form \eqref{eq:flatnonextremal} but with a rescaled horizon radius $r_0'= \frac{L}{\Lambda}r_0$.
\subsection{Joint solution in regions I-II-III: Dipole contribution}
\label{sec:dipole}
It is interesting to ask if we can capture the leading effect of spherical symmetry breaking on the wormhole. The above solutions contain the monopole contribution from the presence of the other throat. At next order, there is a dipole contribution from the harmonic function
\begin{equation}
H=L^4\left(\frac{1}{r^4} + \frac{1}{\Lambda^4} + 4 \frac{ r \cos \theta_1}{\Lambda^5} + \cdots \right)\, , \label{eq:dipoleharmonic}
\end{equation}
We look for a solution including dipole effects in all three regions I-III, that is, we keep the blackening factor exact. We take a general ansatz where the harmonic functions $H_g$ in the metric and $H_F$ in the five-form are allowed to be different,
\begin{align}
\label{eq:dipole}
\frac{1}{\alpha'}ds^2 &= H_g(r,\theta_1)^{-1/2} (-f(r) dt^2 + d\vec{x}^2) + H_g(r,\theta_1)^{1/2} (dr^2 / f(r) + r^2 d\Omega_5^2) \\
\frac{1}{{\alpha'}^2}F&= B(1+\ast) dt \wedge dx^1 \wedge dx^2 \wedge dx^3 \wedge dH_F^{-1} \, ,
\end{align}
with $f(r)=1-r_0^4/r^4$ and $B=1+\frac{r_0^4}{2 \Lambda^4}$. Similarly, we take a general ansatz for $H_g$ and $H_F$ whereby both must be asymptotically equal to \eqref{eq:dipoleharmonic} as $r\to\infty$,
\begin{align}
H_g &= L^4\left(\frac{1}{r^4} + \frac{1}{\Lambda^4} + 4 \frac{ h_g(r) \cos \theta_1}{\Lambda^5} + \cdots \right) \\
H_F &= L^4\left(\frac{1}{r^4} + \frac{1}{\Lambda^4} + 4 \frac{ h_F(r) \cos \theta_1}{\Lambda^5} + \cdots \right)\, ,
\end{align}
that is, $h_g\sim h_F \sim r$ as $r\rightarrow \infty$. Requiring the $r\theta_1$ component of the curvature equation of motion \eqref{eq:geomEOM2} to vanish at order $1/\Lambda^5$ gives
\begin{equation}
h_F(r)=\frac{2r^4-r_0^4}{2r^4}h_g(r) \, .
\end{equation}
Imposing this, it turns out all the remaining components of \eqref{eq:geomEOM2}, as well as the only non-vanishing component of Maxwell's equation, $(d F)_{r \theta_1...\theta_5}=0$ are proportional to the equation
\begin{equation}
-5 r^3 h_g+(r^4-r_0^4)(5 h_g'+r h_g'')=0 \, .
\end{equation}
This is a second order equation with two initial conditions. One is fixed by $h_g(r\rightarrow \infty ) \rightarrow r$. The other is fixed by requiring the solution to stay real in the interior of the wormhole, $r<r_0$. It turns out that the latter condition translates into $h_g(r_0)=0$, so that the location of the horizon is not affected by the perturbation. The solution is then
\begin{align}
h_g(r)&=r_0 Q\left( \frac{r_0^4}{r^4}\right), \\
Q(x)&=\frac{\, _2F_1\left(-\frac{5}{4},-\frac{1}{4};-\frac{1}{2};x\right)+\frac{8 x^{3/2} \Gamma \left(\frac{5}{4}\right) \Gamma \left(\frac{9}{4}\right) }{3 \Gamma \left(-\frac{1}{4}\right) \Gamma \left(\frac{3}{4}\right)}\, _2F_1\left(\frac{1}{4},\frac{5}{4};\frac{5}{2};x\right)}{x^{1/4}}\, ,
\end{align}
and is analytic at $r=r_0$ ($x=1$) due to the cancellation of the branch cuts starting at $x=1$ that are separately present in the two hypergeometric functions.
It would be interesting to further analyze this solution. It seems like it is not possible to have a perturbation that decays towards the singularity $r\rightarrow 0$. Instead, the perturbation decays towards the horizon, i.e. it is decaying in tortoise coordinates. So the presence of the other throat appears to have a significant effect on the interior, where the perturbation becomes large again as we approach the singularity, since $Q(x\rightarrow \infty) \sim \frac{2 \sqrt{2 \pi } x}{\Gamma \left(-\frac{1}{4}\right) \Gamma \left(\frac{3}{4}\right)}$. The singularity inside this wormhole is therefore not of the AdS-Schwarzschild type. The $S^5$ does not factorize, so the geometry is really a full ten-dimensional wormhole.
\subsection{Global Structure and Flux Conservation}
\label{sec:global}
Here we discuss how regions I-II-III (the wormhole) should be glued to regions IV-V (the two throats in a single spacetime) so that the five form flux is conserved.\footnote{We thank Juan Maldacena for raising this point.} The gluing procedure leads to some interesting global properties of the wormhole. We will show that the spacetime has a moduli space coming from the freedom to add a certain amount of twisting during gluing.
In the geometry that we have described, both AdS throats have a positive net five-form flux towards infinity, so that there are $2N$ units of flux near infinity and $N$ near each throat. This presents a puzzle: if there are no sources in the wormhole, flux conservation demands that the flux should thread through the wormhole and close in the outside, giving zero net flux far away from the throats rather than $2N$. Our setup is analogous to the circuital law for a magnetic field in two dimensions, where the closed line integral must be conserved if there are no sources for the curl. This is illustrated for a 2D wormhole in Fig.~\ref{fig:wormholeflux}, where again, the wormhole without sources has zero line integral for the magnetic field on a loop enclosing both throats. However, one may support a non-vanishing line integral on such a loop purely by modification of the geometry, without adding sources. This is achieved by cutting open the wormhole, and gluing it back to the ambient space while twisting to invert the angular coordinate. This results in a sourceless ``Klein-bottle" wormhole, which is a non-orientable surface that supports a nonzero circuital flux at infinity. This is shown on Fig.~\ref{fig:kleinflux}.
\begin{figure}[hbtp!]
\begin{center}
\subfloat[\label{fig:wormholeflux}]{\includegraphics[width =.5\textwidth]{WormholeFlux.pdf}}
\subfloat[\label{fig:kleinflux}]{\includegraphics[width =.5\textwidth]{KleinFig.pdf}}
\end{center}
\caption{Spacetime wormholes with orientable (a) and non-orientable (b) Cauchy slices. In the orientable wormhole, conservation of circuital flux demands that the flux reverses direction at the second throat relative to the first throat, while in the non-orientable wormhole the flux at both throats points the same direction.\label{fig:fluxgeom}}
\end{figure}
The way that flux conservation works in our supergravity wormhole is very similar, although there are some technical differences because the flux comes from a five-form and lives in ten dimensions. In particular, the Cauchy slices of the wormhole will remain orientable. We illustrate on a spacetime diagram in Fig.~\ref{fig:penroseglueing} the two throats and the wormhole before we glue them together, and the orientation of the coordinate differentials. The left and right throats share a time coordinate $t_g$ and three spatial coordinates $x^i_g$ which are globally defined with the same orientation in the ambient space outside the throats. However, the natural radial and angular coordinates $r_L$ and $\theta_L^i$ at the left throat do not coincide with the corresponding coordinates $r_R$, $\theta_R^i$ at the right throat. This is because the geometry only fibers into AdS$_5 \times S^5$ near each throat, so the two five-spheres are centered at different points.
\begin{figure}[hbtp!]
\begin{center}
\includegraphics[width=.7\textwidth]{penroseglueing2.pdf}
\end{center}
\caption{Spacetime diagram of gluing the wormhole to the throat regions. In the middle we have the Penrose diagram of the eternal black brane, and the sides represent the throat regions. We show the orientation of the coordinate differentials $dr$, $dt$, and $d\theta^1$. The $d\theta^1$ differential points out from the plane of the figure in the left throat and in the wormhole, but it points inwards in the right throat. The three differentials must always form the same right handed system.\label{fig:penroseglueing} }
\end{figure}
For the following discussion, by the \emph{electric part} of the five-form we refer to the term proportional to $dt\wedge dx^1\wedge dx^2 \wedge dx^3 \wedge dr$ and by the \emph{magnetic part} we refer to the term proportional to $d\text{Vol}_{S^5}$. Now, both electric and magnetic parts of the five-form are oriented in the same direction in the ambient spacetime, so deep in each throat, both have the same expression in local coordinates: $dt_g \wedge dx^1_g \wedge dx^2_g \wedge dx^3_g \wedge dr_{(L/R)}$ for the electric part and $d\text{Vol}_{S^5 (L/R)}$ for the magnetic part. At the left side of the wormhole, we choose the exterior Schwarzschild coordinates in the left wedge of the Penrose diagram to match the direction of local coordinates of the left throat: $dt = dt_g$, $dx^i = dx^i_g$, $dr = dr_L$, $d\theta_i = d\theta^i_L$ \footnote{The equalities that describe the ``gluing" between the left/right throats and left/right exterior wedges should be understood to be specifying the transition functions on the wormhole manifold in the coordinate patches where they are defined.}. In the right wedge, the radial coordinate points outwards towards the right throat, and the Schwarzschild time coordinate runs downward, in the opposite direction as the left wedge. However, we would like to glue the throats to the wormhole so that time points up on both sides. This is what we expect from the field theory, since after Higgsing the SYM Hamiltonian looks like $H_L + H_R$ in the IR, which generates upwards time evolution on both sides. Therefore, in the right wedge, we must take $-dt = dt_g$ and $dr = dr_R$.
The gluing of the rest of the coordinate directions at the right interface is determined by requiring the five form to be continuous. Consider starting with the five-form in the left throat and continuing into the left exterior wedge and across the wormhole to the right exterior wedge, where we must glue the geometry back to the right throat. On Fig.~\ref{fig:penroseglueing} we show the orientation of the coordinate differentials that are changing during this process. The rest of the coordinate differentials, $dx^i$, $i=1,2,3$ and $d\theta^i$, $i=2,\ldots ,5$ are oriented the same way throughout the figure. In the Schwarzschild coordinates, the five-form has the same solution in both the left and right wedges. But note that in the right exterior wedge of the wormhole, both $dr$ and $dt$ are flipped in Schwarzschild coordinates relative to the left exterior wedge. Since both of these are flipped, the electric part of the five-form, $F_5 \sim dt\wedge dx^1\wedge dx^2 \wedge dx^3 \wedge dr$ actually keeps its orientation throughout the wormhole region. The same applies for the magnetic part, since the $S^5$ approximately factorizes in the wormhole.
On the other hand, in the right throat, the basis differentials $dr_R$ and $d\theta^1_R$ are flipped relative to the left throat. However, the solution for the five-form looks the same in terms of these coordinate differentials in both throats. So the orientation of \textit{both} the electric part $dt_R\wedge dx^1\wedge dx^2 \wedge dx^3 \wedge dr_R$ \textit{and} the magnetic part $d\theta^1_R \wedge d\theta^2\wedge d\theta^3 \wedge d\theta^4\wedge d\theta^5$ of the five form appear reversed compared to the right wedge of the wormhole for the purpose of gluing them.\footnote{What we mean here is that tracking the global five-form from the left throat to the right throat on the outside results in five-forms that point in opposite directions in the left and right throats if we draw them as in Fig.~\ref{fig:penroseglueing}.} So a direct gluing would lead to a discontinuous five-form. However, we can follow the idea from Fig. \ref{fig:kleinflux} and perform the gluing by twisting the $x^i$ coordinates (parallel to the brane) and the $\theta^1$ coordinate by an inversion at the gluing surface.
In terms of transition functions between the right wedge and the right throat, this works as follows. First we align the basis of coordinate differentials on the two sides by introducing new coordinates $t'=-t$ and $(\theta^1)'=-\theta^1$ in the right Schwarzschild wedge. In this properly aligned basis, there is an explicit sign difference in both the electric and magnetic components of the five form compared to the right throat. Then in order to make the five form components continuous, we glue with the transition functions $t_R\equiv t_g=t'$, $x_R^i\equiv x^i_g=-x^i$ and $\theta^1_R=-(\theta^1)'$. This way, in the electric part we make up for the sign by inverting $x^i$ in the gluing function, while in the magnetic part we invert the $\theta^1$ direction. This makes the complete five form continuous. Note that since the total determinant of this twist is positive, the Cauchy slice remains orientable, as opposed to the 2d example of Fig. \ref{fig:kleinflux}.\footnote{The total spacetime is also orientable since there exists a globally defined ``upwards" time, which in the wormhole region is just Kruskal time.}
Note that this twisted gluing results in a smooth geometry, because the $O(3)$ symmetry of the $x^i$ subspace is unbroken by the configuration of two throats, so nothing will depend on these coordinates even in the fully nonlinear time-dependent solution that we have not written down. Similarly, there is an unbroken $O(5)$ subgroup of the $O(6)$ acting on the $r_i$ coordinates, where the $O(5)$ fixes $r_1$, the direction in which the throats are separated. The $\theta^1$ twisting is an inversion of the $r_2,\ldots ,r_6$ coordinates, which is an element $R\in O(5)\subset O(6)$ of this unbroken symmetry with $\det R=-1$.
Now we briefly discuss the moduli space of solutions. Note that the only restrictions on the spatial twisting at the right gluing surface are that (i) it is from the subgroup which remains a symmetry of the solution, (ii) it reverses the orientation of both $dx^1\wedge dx^2 \wedge dx^3$ and $d\text{Vol}_{S^5}$. This gives a freedom in picking the group element with which we twist the gluing, resulting in a moduli space. In addition to the spatial twisting, as pointed out in \cite{Verlinde2020}, one may introduce a constant time shift in the identification of Schwarzschild time with the global time in the throat, which gives an extra real parameter (the difference between constant time shifts between left and right). Therefore, the total moduli space of single boundary wormholes is
\begin{equation}
\mathbb{R} \times ISO(3) \times SO(5).
\end{equation}
Here, $ISO(3)$ denotes the group of (orientation-preserving) rotations and translations of the $x^i$ coordinates.
Finally, let us comment on the field theory interpretation of the gluing twisted by inversions. The inversion of Schwarzschild time and the parallel coordinates $x^i$ correspond to time reversal and parity (TP) in the right IR $\mathcal{N}=4$ SYM factor. The inversion on the $S^5$ corresponds to inverting the $R$ charges in the field theory, so it is natural to think about it as the action of charge conjugation $C$. Therefore, from the field theory point of view, the right IR field theory factor is ``glued back" to the UV field theory by an action of CPT. This is natural for the following reason. As discussed before, the state in the IR looks like the thermofield double state. The TFD state is defined from the square root of the thermal density matrix $\rho^{1/2}$, which is an element of $\mathcal{H}\otimes \mathcal{H}^*$, where $\mathcal{H}^*$ denotes the dual Hilbert space, where bra vectors live. In order to define the thermofield double, which lives on a doubled Hilbert space $\mathcal{H}\otimes \mathcal{H}$, one needs to turn the bra vectors into ket vectors with an anti-unitary symmetry. There is one such anti-unitary transformation that is a symmetry in any quantum field theory, which is CPT.
\section{Instability of the Solution}\label{sec:stability}
\subsection{Instability Timescale}
The wormhole solution we have described is not an extremal (BPS) solution of type IIB: its mass is larger than its charge. Consequently, it suffers from an instability: the attractive gravitational (NS-NS) force is larger than the repulsive five-form (R-R) force between the underlying branes, so that at late times the wormhole disappears as the two stacks of $N$ branes collide and form a single stack of $2N$ branes at nonzero temperature. We can compute the time scale of this instability by examining the tree-level effective action governing the dynamics of one stack of branes in the background of the other stack. The full action is the Dirac-Born-Infeld (DBI) action describing the geometric dynamics of the branes and their coupling to open strings, plus the coupling of the branes to the five-form \cite{GIBBONS1998603}:
\begin{align}
S = -T \int d^{p+1} \xi \:\sqrt{-\text{det}\,G_{ab}} + \mu\int d^{p+1} \xi \: C_4 \, . \label{eq:dbiaction}
\end{align}
Here the $\xi^i$ are coordinates on the world-volume of a brane, $T$ is the tension of the stack of branes, $\mu$ is the charge density coupling to the five-form, $G_{ab}$ is the pullback of the background metric $g_{\mu \nu}$ to the brane, and $C_4$ is the pullback of the potential for the five-form. Evaluating the DBI action using the classical metric and five-form gives the effective action at tree level where we have taken the backgrounds for the antisymmetric two-form and the gauge field on the brane to be zero.
In order to study the dynamics of one stack of branes as a probe, the backreaction of the probe on the background geometry should be negligible. However, this is not the case in the full two-center geometry, as each stack of branes sources its own independent AdS throat. Moreover, each of the stacks have a field that is the size of the AdS radius, and since they are separated in the asymptotic $S^5$ directions which have comparable size, we cannot treat the two stacks as point-like objects interacting via weak fields. Nevertheless, we may obtain a lower bound on the timescale of the instability by considering the motion of an extremal probe brane located halfway between the two stacks of branes, where we heat up one stack slightly and leave the other extremal. We may think about the extremal probe as being separated from the extremal stack. Such a brane experiences a higher acceleration than a brane located deep in the AdS throat of the extremal stack, where it is further from the thermal stack. We will show that the temperature and separation of the branes can be chosen so as to make this lower bound on the instability timescale arbitrarily high, i.e. the wormhole is long-lived.
The extremal probe brane starts at a point in the geometry which can be approximated by the flat-space region of \eqref{eq:metricnonpert}, \eqref{eq:fiveformnonpert}, far from both the horizon of the thermal branes and the AdS throat of the extremal branes. To compute the pullback of the metric and four-potential, we use spacetime Lorentz transformations and world-volume reparameterizations to work in ``static gauge" in which the world-volume coordinates are parallel to the spacetime coordinates.
\begin{align}
\xi^0 = t, \qquad \xi^i = x^i \, ,
\end{align}
where $i = 1,2,3$. The pullback of the potential to a brane sitting at distance $r$ from the thermal stack is then
\begin{align}
C_4 = \sqrt{1+\frac{r_0^4}{\Lambda^4}} \left(\frac{L^4}{r^4} + \frac{L^4}{\Lambda^4} \right)^{-1} dt \wedge dx^1 \wedge dx^2 \wedge dx^3 \, .
\end{align}
In spherical coordinates, the probe brane moves only in the radial direction, so the pullback of the metric is given by
\begin{align}
G_{00} = g_{00} + \dot{r}^2 g_{rr}, \qquad G_{ii} = g_{ii} \, ,
\end{align}
and all other components are zero. Consequently, the effective action experienced by the probe brane is, defining $M = TV$ and $Q = \mu V$ as the effective mass and charge of the probe brane where $V = \int d^3\xi^i$ is the (regularized) brane world-volume,
\begin{align}
S = \frac{\Lambda^2}{L^4}\int dt\, \frac{r^4}{r^4+\Lambda^4}\biggl( -M \Lambda^2 \sqrt{1-\frac{r_0^4}{r^4}- \dot{r}^2 \frac{L^4(r^4+\Lambda^4)}{\Lambda^4(r^4-r_0^4)}} + Q\sqrt{r_0^4+\Lambda^4}\biggr) \, .
\end{align}
This expression should be expanded at large radius compared to the horizon $r_0$, but keeping $r/\Lambda$ fixed since $r \sim \mathcal{O}(\Lambda)$ at the scale of the dynamics. Therefore, we introduce the dimensionless radial coordinate $\hat{r} = r/\Lambda$ that we imagine to be order one, and dimensionless time $\hat{t} = \frac{r_0 t}{L^2}$ as before. Since we take the probe to be extremal, we set $Q=M$, and all dimensionful quantities then scale out in front of the action,
\begin{align}
S = \frac{M \Lambda^4}{r_0 L^2} \int d\hat{t} \,\frac{\hat{r}^4}{\hat{r}^4+1}\biggl(\sqrt{1+\epsilon^4} - \sqrt{1-\frac{\epsilon^4}{\hat{r}^4}-\epsilon^2 \left(\frac{d\hat{r}}{d\hat{t}} \right)^2 \frac{\hat{r}^4+1}{\hat{r}^4-\epsilon^4}} \biggr) \, ,
\end{align}
where $\epsilon=r_0/\Lambda$ as before. We may now expand in $\epsilon$ and take $d\hat{r} / d\hat{t}\ll 1$, the Newtonian slow-moving approximation for the probe\footnote{The speed $d\hat{r} / d\hat{t}$ is $\mathcal{O}(\epsilon^3)$ in this expansion, coming from balancing the orders of the leading potential and kinetic terms and/or from the equation of motion.}, to find at lowest order
\begin{align}
S = \frac{M\Lambda^2 r_0}{ L^2} \int d\hat{t} \left(\frac12 \left(\frac{d\hat{r}}{d\hat{t}}\right)^2 + \frac{\epsilon^2}{2} + \frac{\epsilon^6}{8} \left(\frac{1}{\hat{r}^4} - 1\right) + \ldots \right) \, .
\end{align}
This is motion in a flat space attractive Coulomb potential which scales as $\mathcal{O}(\hat{r}^{-4})$ as expected for a charged object of codimension six in ten spacetime dimensions, in agreement with what would have been found from the tree-level closed string exchange. The resulting dynamics are simply
\begin{align}
\frac{d^2 \hat{r}}{d\hat{t}^2} = -\frac12 \frac{\epsilon^6}{\hat{r}^5} \, ,
\end{align}
and so the acceleration can be made small by making $\epsilon=r_0/\Lambda$ small. In terms of the original time coordinate, the instability timescale is $t \sim \frac{L^2 \Lambda^3}{r_0^4}\sim \beta \epsilon^{-3}$, where $\beta=\frac{\pi L^2}{r_0}$ is the inverse temperature of the black brane. That is, taking the thermal branes to be very cold or the stacks of branes to be widely separated, the wormhole solution can be made arbitrarily long-lived.
\subsection{Stabilizing with Rotation}
\label{sec:rotation}
One may wonder whether our wormhole can be stabilized by making the throats spin around each other in the transverse $r_i$ directions. We will not attempt to perturbatively construct such a spinning solution in the present work. On the other hand, we can repeat the DBI analysis above for the case where the extremal probe brane rotates around the non-extremal black branes. We parameterize the brane trajectory in a circular orbit around the equator $\theta_1=\ldots=\theta_4=\pi/2$ by $r(t)$, $\theta_5(t)$. In this case, the $G_{00}$ component of the pullback of the metric is
\begin{align}
G_{00} = g_{00} + \dot{r}^2 g_{rr} + \dot{\theta}_5^2 g_{\theta_5 \theta_5} \, .
\end{align}
Using the same coordinates as the previous section, the DBI action for the extremal probe constrained to the equator is
\begin{align}
S = \frac{M\Lambda^4}{r_0 L^4} \int d\hat{t} \frac{\hat{r}^4}{\hat{r}^4+1} \left(\sqrt{1+\epsilon^4} - \sqrt{1-\frac{\epsilon^4}{\hat{r}^4}-\epsilon^2 (\hat{r}^4+1) \left[\frac{1}{\hat{r}^4-\epsilon^4} \left(\frac{d\hat{r}}{d\hat{t}} \right)^2+\frac{1}{\hat{r}^2} \left(\frac{d\theta_5}{d\hat{t}} \right)^2 \right]} \right) \, . \label{eq:dbirotate}
\end{align}
Expanding in $\epsilon$ and taking the slow-moving approximation yields
\begin{align}
S = \frac{M\Lambda^2 r_0}{ L^2} \int d\hat{t} \left(\frac12 \left(\frac{d\hat{r}}{d\hat{t}}\right)^2 + \frac12 {\hat r}^2 \left(\frac{d\theta_5}{d\hat{t}} \right)^2+ \frac{\epsilon^2}{2} + \frac{\epsilon^6}{8} \left(\frac{1}{\hat{r}^4} - 1\right) + \ldots \right) \, ,
\end{align}
the same result as previously with the Newtonian rotational kinetic energy added. The radial equation of motion, assuming the existence of a solution with a constant rotational velocity $\frac{d\theta_5}{d\hat{t}} = \hat{\omega}$, is
\begin{align}
\frac{d^2 \hat{r}}{d\hat{t}^2} = \hat{r} \omega^2 - \frac{\epsilon^6}{2\hat{r}^5} \, ,
\end{align}
and therefore we can obtain circular orbits of radius $\hat{r}_c$ when the rotation speed is
\begin{align}
\hat{\omega} = \frac{\epsilon^3}{\sqrt{2} \hat{r}_c^3} \, .
\end{align}
Therefore, the angular speed needed to obtain circular orbits is $\omega \sim \epsilon^3/\beta$, the inverse of the instability time scale. In dimensionful coordinates, this speed is
\begin{align}
\omega = \frac{r_0^4}{\sqrt{2} L^2 r_c^3} \, .
\end{align}
We can check if the circular orbit radius $\hat{r}_c = \frac{\epsilon}{(\sqrt{2} \hat{\omega})^{1/3}}$ leads to stable or unstable orbits. For this, we examine the effective potential written in terms of conserved angular momentum $\ell = \frac{{\hat r}^2}{\epsilon^3} \frac{d \theta_5}{d\hat t}$,
\begin{equation}
V(r)=\epsilon^6 \left( \frac{1}{4}-\frac{1}{4 \hat r^4}+\frac{\ell^2}{\hat r^2} \right) \, .
\end{equation}
We see that the circular orbit corresponds to a maximum, i.e. it is unstable. The reason this happens is that the centrifugal piece in the effective potential dies off slower than the attractive force, which is the opposite of the situation in normal 4D Kepler motion. Based on this analysis, it is unlikely that the wormhole solution can be stabilized by rotation, unless nonlinear effects conspire to stabilize a circular orbit.
We can also try to solve \eqref{eq:dbirotate} for circular orbit frequencies directly without series expanding by taking the circular orbit as an ansatz. In that case the equation of motion reduces to the algebraic equation
\begin{align}
\hat{r}^{10} \omega^2 \epsilon^8+4 \hat{r}^6 \omega^2 \epsilon^8+3 \hat{r}^2 \omega^2 \epsilon^8+\hat{r}^4 \left(4 \sqrt{-\frac{\left(\epsilon^4+1\right) \left(\hat{r}^6 \omega^2 \epsilon^8-\hat{r}^4+r^2 \omega^2 \epsilon^8+\epsilon^4\right)}{\hat{r}^4}}-2 \epsilon^4-4\right)+2 \epsilon^4 = 0 \, .
\end{align}
We find an additional solution in this case with angular speed at leading order in $\epsilon$ given by
\begin{align}
\hat{\omega} = \frac{2\sqrt{2}\hat{r}_c}{(\hat{r}_c^4+3) \epsilon} \, ,
\end{align}
or $\omega = \frac{\Lambda^4}{L^2} \frac{2\sqrt{2} r_c }{(r_c^4 + \Lambda^4)}$ in dimensionful coordinates. This solution did not appear previously from perturbing the action around small $\epsilon$ simply because it is inversely proportional to $\epsilon$ and therefore not perturbatively slow-moving\footnote{One might worry that $\omega$ exceeds light-speed, even if it does not diverge. One can check that the maximum value of $\omega$ is $\frac{3^{3/4} \Lambda}{\sqrt{2}L^2}$ which is certainly small as $L \gg \Lambda$, and that this occurs at the reasonable radius $r_c = \Lambda / 3^{1/4}$.}. This value of $\hat{\omega}$ supports two different possible radii,
\begin{align}
\hat{r}_c = \frac{3\hat{\omega} \epsilon}{2\sqrt{2}} \qquad \text{and} \qquad \hat{r}_c = \frac{\sqrt{2}}{(\hat{\omega} \epsilon)^{1/3}} \, ,
\end{align}
or in dimensionful coordinates,
\begin{align}
r_c = \frac{3L^2 \omega}{2\sqrt{2}} \qquad \text{and} \qquad r_c= \sqrt{2} \left(\frac{\Lambda^4}{\omega L^2} \right)^{1/3} \, .
\end{align}
We can study the stability of these circular orbits by linearizing around the solution. One finds that the (dimensionless) frequency-squared of the radial oscillation, to leading order in $\epsilon$ is
\begin{equation}
\Omega^2 = \frac{4 \left(5 \hat{r}_c^8+12 \hat{r}_c^4-9\right) \hat{r}_c^2 \left| \hat{r}_c^4-1\right| +4 \left(\hat{r}_c^{12}-5 \hat{r}_c^8-33 \hat{r}_c^4+5\right) \hat{r}_c^2}{\epsilon^2 \left({\hat r}_c^4+1\right)^3 \left(\hat{r}_c^4+3\right)^2} \, ,
\end{equation}
that is, the orbit is stable when ${\hat r}_c\gtrsim 1.281$ and unstable when ${\hat r}_c\lesssim 1.281$. In the extremal limit, these circular orbits rotate with a finite angular velocity, as $\omega$ does not depend on $r_0$. These orbits are not directly relevant for stabilizing our wormhole, which is a perturbation to a non-rotating solution. This is because for self-consistency we would want the rotation in the circular orbits to be perturbatively small in $\epsilon=r_0/\Lambda$, while we saw that the rotation persists even in the extremal limit. One may therefore wonder if there exists a rotating version of the extremal two-center solution that is perturbatively stable. An exact solution is likely not possible due to gravitational and five form radiation, but we really just want a long-lived rotating binary black hole. This would provide a starting point for a wormhole solution stabilized by rotation.
\section{Traversing the Wormhole} \label{sec:traverse}
The two throats in our wormhole are separated by causal horizons, so it is not possible to traverse through it. Near the horizons, the wormhole looks like a perturbation of the planar AdS-Schwarzschild black brane, which is a marginally non-traversable solution in the sense that it can be made traversable by a small negative energy perturbation \cite{GaoJafferisWall}. Here we wish to analyse if the perturbation of the geometry near the horizon spoils this property. In two-sided null Kruskal coordinates $U,V$ (which exist for both the eternal black brane and for our wormhole geometry) the requirement to violate the ANEC is written $\int dU \, T_{UU} < 0$ along $V = 0$. In the absence of any stress-energy, $T_{UU} = 0$, null rays along $V = 0$ pass through the bifurcation surface and asymptote to infinity in either direction. Consequently, any negative perturbation will pull back the horizons and create traversability.
We will now evaluate the ANEC for the monopole- and dipole-corrected Einstein-Rosen bridges of Sec.~\ref{sec:monopole} and Sec.~\ref{sec:dipole}. The monopole corrections \eqref{eq:monopole} do not affect the marginal traversability since $T_{UU}$ vanishes along the horizon. This follows because as noted at the end of Sec.~\ref{sec:monopole}, this correction can be obtained by linearizing \eqref{eq:metricnonpert}, which is the asymptotically flat black brane in rescaled coordinates. The dipole contribution \eqref{eq:dipole} is more complicated to analyze because the $t-r$ plane is no longer decoupled from the $\theta_1$ angle, so the near-horizon geometry is effectively three-dimensional. Regardless, the location of the horizon stays at $r=r_0$ since the location of the zero of the blackening factor is not affected. Moreover, the null geodesics comprising the horizon remain on the $t-r$ plane at fixed $\theta_1$. This can be seen by examining the $\theta_1$ component of the geodesic equation,
\begin{equation}
\frac{d}{d\lambda} (r^2\dot{\theta_1}) = -\frac{r^2 \sin \left(\theta _1\right) h_g(r) \left(\dot{r}^2 L^4 r^4+\dot{t}^2 \left(r^4-r_0^4\right){}^2\right)}{L^2 \Lambda^5 \left(r^4-r_0^4\right)} \, ,
\end{equation}
where dot indicates derivative with respect to affine parameter $\lambda$. We have $h_g(r)=\frac{20 \pi ^{3/2} (r-r_0)}{\Gamma \left(-\frac{1}{4}\right)^2} + \cdots$ around $r=r_0$, so in order to have $\ddot{\theta_1}=0$ at the horizon, we need that $\dot{r}$ vanishes at $r=r_0$. Examining the condition $g_{mn}\dot{x^a}\dot{x^b}=0$ around $r=r_0$ one finds that $\dot{r} \propto \sqrt{r-r_0}\dot{\theta_1}$. Therefore, $\dot{r}=\dot{\theta_1}=0$ and $\dot{t}= \text{const}$ is a null geodesic at $r=r_0$ for any fixed $\theta_1$ (and the rest of the seven coordinates fixed as well). Therefore, $t$ also affinely parameterizes the null worldlines, so the ANEC quantity can be written $\int dt\, T_{tt}$. We identify the stress-energy tensor from the right-hand side of \eqref{eq:geomEOM2} as
\begin{align}
T_{\mu \nu} = \frac{1}{4\cdot 4!} F_{\mu \alpha \beta \gamma \delta} F_{\nu}^{\:\:\alpha \beta \gamma \delta} \, .
\end{align}
Applying this to the solution \eqref{eq:dipole} we find that $T_{tt}=0$ at $r=r_0$ up to $O(1/\Lambda^6)$ corrections. Therefore, at the order $\mathcal{O}(\Lambda^{-5})$ of the dipole corrections, spherical symmetry breaking does not affect the marginal non-traversability of the single-boundary wormhole.
As in \cite{GaoJafferisWall, SYKwormhole}, one mechanism to generate negative contributions to the ANEC that allow traversability is to introduce a nonlocal coupling between the two throats of the wormhole by adding a double-trace type interaction in the field theory. In fact, the field theory symmetry breaking $SU(2N) \to S(U(N) \times U(N))$ that we have described already generates couplings between the two $SU(N)$ effective subfactors in the IR from the Wilsonian RG flow \cite{INTRILIGATOR200099, connectivity}. At leading order, these include single-trace interactions of the form $g_I V_I$, where $V_I$ is proportional to
\begin{align}
V_I \propto \text{tr} \left(F_{\mu \nu} F^{\nu \rho} F_{\rho \sigma} F^{\sigma \mu} - \frac14 (F_{\mu \nu} F^{\mu \nu})^2 \right) \, ,
\end{align}
and $I=1,2$ are the two $U(N)$ factors. The couplings $g_I$ are dynamically determined by abelian singleton degrees of freedom in the other CFT factors (i.e., the Goldstone modes associated to the moduli of branes in the other stack(s)). Of more interest to us with respect to traversability are the double-trace interactions that are generated. These directly couple the IR factors in the CFT:
\begin{align}
V_{IJ} \propto \text{tr}_I \left(F_{\mu \nu} F^{\mu \nu} \right) \text{tr}_J \left(F_{\mu \nu} F^{\mu \nu} \right) \, .\label{eq:symdoubletrace}
\end{align}
When the full UV CFT is genuinely a product of $n$ individual subfactors $\text{CFT} = \prod_{i=1}^n \text{CFT}_i$, the dual bulk geometry generally consists of $n$ different asymptotic universes. In this case, the single-trace terms $\text{tr}_I F^2$ that comprise the operator \eqref{eq:symdoubletrace} are dual to the bulk dilaton in component $I$ \cite{OGwitten, PhysRevD.59.104021}. In our setting, the components $\text{tr}_1 F^2$ and $\text{tr}_2 F^2$ are dual to the bulk dilaton in the vicinity of the first and second throats, as these deep bulk regions correspond to the IR of the CFT where the approximate factorization into two $SU(N)$ gauge theories holds. Therefore, the double-trace interactions $V_{12}$ are structurally of the form $h_{12} \phi_1 \phi_2$ required to generate negative contributions to the ANEC as shown in \cite{GaoJafferisWall}. This indicates that the natural operators that arise from the Wilsonian RG flow in the IR of the symmetry-broken theory are of the correct form to generate traversability, albeit possibly weak traversability. However, in the Gao-Jafferis-Wall protocol, only one sign results in a traversable wormhole, while the opposite sign lengthens the wormhole. It would therefore be interesting to determine the sign of the coefficient of \eqref{eq:symdoubletrace} as generated by the Wilsonian RG, at least in perturbation theory. In fact, in our setting, there are various other double-trace operators that can be generated by the supersymmetry transformations of \eqref{eq:symdoubletrace}. A full analysis should understand the net effect of all such RG-generated double-trace operators on the sign of the null stress-energy.
In \cite{GaoJafferisWall}, the double-trace interactions are taken to be relevant deformations of the Hamiltonian so that they are renormalizable and there is no backreaction at the AdS boundary. The term $V_{12}$ generated by the RG flow is an irrelevant deformation; nonetheless, this is not a concern as we know that the theory is UV-complete, since above the Higgs scale it flows to the $SU(2N)$ $\mathcal{N}=4$ SYM theory. Furthermore, \cite{GaoJafferisWall} take the deformation to be a quench, turned on after some time $t_0$. Since our solution is perturbatively unstable, we also expect the coupling strength to be time-dependent, although we have not analyzed this in detail. Lastly, \cite{GaoJafferisWall} takes the boundaries to be connected with the same time orientation by taking the deformation to be structurally $h(t) \phi_1 (t, \vec{x}) \phi_2 (-t, \vec{x})$. This is because the asymptotic time on one boundary of the eternal black hole runs in the opposite direction on the other boundary. In our setting, the wormhole resides in a single universe and we have taken time to run upwards on both sides, so there is a unique asymptotic time $t$ and we need not flip the time orientation between the two throats.
In addition to the terms that are naturally generated by RG-flow, we can try, like \cite{GaoJafferisWall}, to add by hand some deformation that generates traversability in the IR wormhole. This should be a relevant operator in order for it not to destroy the UV $SU(2N)$ $\mathcal{N}=4$ SYM theory. The lightest single trace operators in a single factor of $SU(N)$ $\mathcal{N}=4$ SYM are the $\Delta=2$ scalars in the $\textbf{20}$ of the $SO(6)$ $R$-symmetry. They are of the form $\mathcal{O}^{ij}=\text{Tr} \phi^{(i}\phi^{j)}$. The possible deformations $\mathcal{O}^{ij}_L \mathcal{O}^{kl}_R$ therefore furnish $\textbf{20}\times \textbf{20}$. These are marginal to leading order in $1/N$ due to large $N$ factorization. In order to work out the effects of deforming by these operators (with either sign of the coefficient) we would need to understand their RG flow and the $1/N$ corrections to their dimension. The corresponding single-sided double-trace operators (i.e. an operator in one of the low energy $SU(N)$ factors) $\mathcal{O}^{ij}_L \mathcal{O}^{kl}_L$ are well understood in the strong coupling regime \cite{Arutyunov:2000ku}, and they all have either vanishing or negative anomalous dimensions. The negative anomalous dimensions are intuitively understood as binding energies coming from the attractive nature of the bulk interaction between two particles. The same intuitive reasoning applies to the two-sided operator (i.e. an operator connecting the two low energy $SU(N)$ factors) $\mathcal{O}^{ij}_L \mathcal{O}^{kl}_R$, which suggests that these operators should be marginally relevant at strong coupling, and one should be able to use them to make our single boundary wormhole traversable.
\section{A Double Wormhole Between Universes} \label{sec:doublewormhole}
The solutions we discussed in Sec.~\ref{sec:wormholeGeomSoln} capture certain effects of two non-extremal throats living in a single asymptotically AdS spacetime, and in Sec.~\ref{sec:global} we explained how to join these throats so that we end up with a wormhole in a single universe. There are also other ways to join the solutions of Sec.~\ref{sec:wormholeGeomSoln} to get interesting new wormhole configurations. For example, one could duplicate the spacetime with two throats and join them in a way shown in Fig.~\ref{fig:twouniverse}. In this case, the global time runs in opposite way in the two asymptotic regions and no twisting is required to enforce flux conservation (see Fig.~\ref{fig:twouniverse}). This spacetime is patch-wise described by the same solutions that we have discussed in Sec.~\ref{sec:wormholeGeomSoln}, but the patches are glued together differently.
\begin{figure}[hbtp!]
\begin{center}
\includegraphics[width=.45\textwidth]{twouniverse.pdf}
\end{center}
\caption{A double wormhole between two asymptotically AdS universes. This geometry, which can be constructed from the solutions in the text, is dual to a pair of Higgsed Yang-Mills theories, with IR factors entangled pairwise between them.
\label{fig:twouniverse}}
\end{figure}
In the dual field theory we now start with two copies of $\mathcal{N}=4$ $SU(2N)$ SYM, and we Higgs each copy. Let us label the two theories $A$ and $B$, while the low energy factors are called $L$ and $R$. Then, the low energy Hilbert space is
\begin{equation}
\mathcal{H}_{A,L}\otimes \mathcal{H}_{A,R} \otimes \mathcal{H}_{B,L}\otimes \mathcal{H}_{B,R} ,
\end{equation}
and we expect a wormhole configuration like Fig. \ref{fig:twouniverse} to be approximately dual in the IR to a tensor product of two thermofield double states
\begin{equation}
|\text{TFD}\rangle_{A,L;B,L}\otimes |\text{TFD}\rangle_{A,R;B,R}.
\end{equation}
We can embed this state in the UV Hilbert space $\mathcal{H}_A\otimes \mathcal{H}_B$ as explained in Sec.~\ref{sec:fieldtheory}, that is, we must take the temperatures of the thermofield doubles to be much smaller than the Higgs scale.
\section{Discussion}
In this paper, we constructed an asymptotically AdS$_5 \times S^5$ single boundary wormhole solution by matching a two-center extremal black brane solution to a two-sided AdS black brane in perturbation theory. Preserving continuity of the five-form in the solution required a global monodromy in some of the coordinates, although the total geometry remains orientable. The small parameter in the problem is the horizon radius compared to the separation of the throats, $r_0/\Lambda$. We argued that the solution is dual to an approximate thermofield double state in a single copy of $\mathcal{N}=4$ SYM, where the gauge group is Higgsed into two copies of $SU(N)$, which are entangled. In the field theory the small parameter is the ratio of the thermal scale to the Higgs scale.
\subsubsection*{Thermal effective potential and $R$ charge}
Our wormhole is non-extremal, and consequently has to be unstable. This instability is dual in $\mathcal{N}=4$ SYM to the scalar vevs developing an effective potential at finite temperature, as illustrated in Fig. \ref{fig:effpotential}. We have argued that the wormhole can be made parametrically long lived by making $r_0/\Lambda$ small.
Another possibility is to stabilize the wormhole by making the throats rotate around each other. In Sec.~\ref{sec:rotation} we found that an extremal probe brane can be put on a stable circular orbit around a non-extremal black brane. This is surprising since planetary orbits are unstable in more than four dimensions, and is possible here due to the five-form interaction. The stable orbit we find has finite angular velocity in the extremal limit, so it is not possible to add this effect perturbatively to our solution. Nevertheless, this finding suggests that in the dual theory one can create a local minimum in the effective potential of the scalar vevs away from the origin by adding $R$ charge. This should lead to long-lived states with finite temperature symmetry breaking. The states are only long-lived, since from the supergravity picture, we expect them to decay due to gravitational and five form radiation. This is consistent with the expectation that all symmetries must be restored at sufficiently high temperatures: see \cite{Chai:2020zgq} for a recent discussion in the case of global symmetries. It would be interesting to understand this effect better.
\subsubsection*{Global monodromy and moduli space}
As emphasized in \cite{Verlinde2020}, gluing the two sides of a wormhole to a single asymptotic region breaks the two-sided boost-like Killing symmetry of the eternal black brane geometry and correspondingly, there is a one-parameter family of wormholes labeled by the ``monodromy" of Schwarzschild time as one goes between the two throats on the outside. In addition to this, we have found that there is a freedom of introducing a global monodromy consisting of rotating and translating the parallel spatial directions to the brane, and also rotating by the unbroken $SO(5)$ subgroup of the $SO(6)$ symmetry of the $S^5$.
Therefore, there is a moduli space $\mathbb{R}\times ISO(3)\times SO(5)$ of locally equivalent but globally different solutions. It would be interesting to understand the interpretation of this in the dual $\mathcal{N}=4$ $SU(2N)$ SYM theory. It is tempting to speculate that it is related to some ambiguity in embedding the IR state \eqref{eq:tfdapprox} into the UV theory, which possibly includes an ambiguity in the implementation of the energy cutoff in the state \eqref{eq:tfdapprox}.
\subsubsection*{Making the wormhole traversable}
We have showed that corrections coming from the two throats being in the same spacetime in the first few orders in perturbation theory do not spoil the marginal traversability of the wormhole, in the sense that the ANEC quantity $\int dU T_{UU}$ remains zero along the causal horizons. It would thus be interesting to see if the wormhole can be made traversable using the ideas in \cite{GaoJafferisWall}. This requires a double trace coupling between the two $SU(N)$ factors in the Higgsed $\mathcal{N}=4$ SYM theory.
We have pointed out that such couplings are naturally generated in RG due to the fact that in the UV the two $SU(N)$ factors are part of the total $SU(2N)$. It would require a careful analysis to account for the net effect of all these double trace interactions and to see if the resulting sign makes the wormhole traversable. This is beyond the scope of the present paper but is certainly an interesting problem.
One may also try to make the wormhole traversable by adding a double trace coupling by hand. This would have to be a relevant double trace operator, otherwise the theory will no longer flow to a single $SU(2N)$ $\mathcal{N}=4$ SYM in the UV (or to a wormhole in a single spacetime). We have argued that such relevant double traces can be formed from the $\Delta=2$ scalar operators of the theory, though it would also be useful to check that the two-sided operators $\mathcal{O}_L^{ij} \mathcal{O}_R^{kl}$ have negative anomalous dimensions.
In \cite{SM_wormhole} negative contributions to the ANEC were generated by negative Casimir-like vacuum energies coming from the lowest Landau levels of the bulk fermion running in a cycle threading their wormhole solution. In our solution, there are various fermions in the spectrum of type IIB supergravity which have vacuum fluctuations, though we have set their classical backgrounds to vanish. These fermions, and the bulk bosonic fields, should similarly provide Casimir-like vacuum energies in our setup. The sign of the total Casimir energy is important, as before; so it is important to check which contributions ultimately win out. There is potentially the possibility that the underlying supersymmetry enforces a vanishing total Casimir energy. In any case, the vacuum energies provide another potential mechanism for traversability in competition or collusion with the other effects that we have discussed.
\subsubsection*{Probing the monodromy through the wormhole}
As we discussed, continuity of the five-form requires a twisted gluing of the interior AdS-Schwarzschild geometry to the two-center ambient spacetime, although the complete spacetime remains orientable. An interesting way of probing the resulting monodromy is to send a giant graviton through the wormhole. Giant gravitons are spherical D3-branes localized on the $S^5$ in the geometry, and are supported by their angular momentum and by interactions with the five-form flux \cite{Myers:1999ps,McGreevy:2000cw}. These brane states are created by determinant and subdeterminant operators in the field theory \cite{Balasubramanian:2001nh,Corley:2001zk,Balasubramanian:2004nb}. To use these branes to probe the wormhole in the field theory, we would want to construct such operators in the light infrared factors after Higgsing. On the gravitational side, we could explicitly test what happens to the corresponding giant gravitons as they are moved through the wormhole, expecting them to emerge with inverted $\theta^1$.
\subsection*{Acknowledgements}
We are grateful to Alexandre Belin, Vishnu Jejjala, Arjun Kar, Guram Kartvelishvili, Lampros Lamprou, Juan Maldacena, Onkar Parrikar, Simon Ross, and Tomonori Ugajin for useful conversations. MD is supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE-1845298. The research of VB, MD, and GS was supported in part by the Simons Foundation through the It From Qubit Collaboration (Grant No.~38559), and by the Department of Energy through grants DE-SC0013528, and QuantISED DE-SC0020360. VB also thanks the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611, for hospitality while this work was in progress.
\begin{appendices}
\section{Perturbative Equations of Motion}\label{sec:perturbEOM}
In this appendix, we derive the perturbative equations of motion \eqref{eq:perturbativeEOMs1} and \eqref{eq:perturbativeEOMs2}. In general, bars will indicate background-order quantities. We begin by variation of the full geometric equation of motion \eqref{eq:geomEOM1}. Since we impose self-duality at all orders,
\begin{align}
0 = (F + \delta F)\wedge \ast(F+\delta F) \sim F_{\alpha \beta \gamma \delta \epsilon} F^{\alpha \beta \gamma \delta \epsilon} \, ,
\end{align}
where the last expression is to all orders. Consequently, the variation of \eqref{eq:geomEOM1} is
\begin{align}
\delta R_{\mu \nu} -\frac12 \delta R \bar{g}_{\mu \nu} &= \frac{1}{4\cdot 4!} (\delta F_{\mu \alpha \beta \gamma \delta} \bar{F}_{\nu}^{\:\:\alpha \beta\gamma\delta} + \bar{F}_{\mu \alpha \beta \gamma \delta} \delta F_{\nu}^{\:\:\alpha \beta \gamma \delta} ) \, .
\end{align}
$\bar{R} = 0$ at background order, but it is not obvious that $\delta R = 0$ perturbatively, so we have retained this term for now. Now we can constrain $\delta R$ by tracing both sides, noting that $\bar{g}^{\mu \nu} \delta R_{\mu \nu} = \delta R - \bar{R}_{\mu \nu} \delta g^{\mu \nu}$,
\begin{align}
-4\delta R - \bar{R}_{\mu \nu} \delta g^{\mu \nu}
= \frac{1}{4\cdot 4!} \bar{g}^{\mu \nu} (\delta F_{\mu \alpha \beta \gamma \delta} \bar{F}_{\nu}^{\:\:\alpha \beta\gamma\delta} + \bar{F}_{\mu \alpha \beta \gamma \delta} \delta F_{\nu}^{\:\:\alpha \beta \gamma \delta} ) \, .
\end{align}
Substituting $\bar{R}_{\mu \nu}$ with its background equation of motion and rearranging for $\delta R$ one finds:
\begin{align}
\delta R = \frac{1}{16\cdot 4!} \delta \left(g^{\mu \nu} F_{\mu \alpha \beta \gamma \delta} F_{\nu}^{\:\:\alpha\beta\gamma\delta} \right) = 0 \, .
\end{align}
That is, $\delta R$ vanishes to all orders as a consequence of the self-duality constraint. The perturbative equation of motion for the metric is therefore simply
\begin{align}
\delta R_{\mu \nu}&= \frac{1}{4\cdot 4!} (\delta F_{\mu \alpha \beta \gamma \delta} \bar{F}_{\nu}^{\:\:\alpha \beta\gamma\delta} + \bar{F}_{\mu \alpha \beta \gamma \delta} \delta F_{\nu}^{\:\:\alpha \beta \gamma \delta} ) \, .
\end{align}
We now make use of the formula for $\delta R_{\mu \nu}$ in terms of the metric perturbation $\delta g_{\mu \nu} = h_{\mu \nu}$, to first order in the perturbation, to arrive at \eqref{eq:perturbativeEOMs1}:
\begin{align}
\delta R_{\mu \nu} = \nabla_{\lambda} \nabla_{(\mu}h_{\nu)}^{\lambda} - \frac12 \nabla_{\mu} \partial_{\nu} h - \frac12 \nabla_{\lambda} \nabla^{\lambda} h_{\mu \nu} \, ,
\end{align}
where the covariant derivative is taken with respect to $\bar{g}$. We review the derivation of this formula below. The Ricci tensor is
\begin{align}
R_{bd} = R^a_{bad} = \partial_a \Gamma^a_{bd} - \partial_d \Gamma^a_{ba} + \Gamma^s_{bd} \Gamma^a_{sa} - \Gamma^s_{ba} \Gamma^a_{sd} \, .
\end{align}
Relabeling indices and varying each term independently gives a formula in terms of the variation $\delta \Gamma$
\begin{align}
\begin{aligned}
\delta R_{ab} &= \partial_c \delta \Gamma^c_{ab} - \partial_b \delta \Gamma^c_{ac} + \delta(\Gamma^s_{ab} \Gamma^c_{sc}) - \delta(\Gamma^s_{ac} \Gamma^c_{sb} ) \\
&= \partial_c \delta \Gamma^c_{ab} - \partial_b \delta \Gamma^c_{ac} + \delta\Gamma^s_{ab} \Gamma^c_{sc} +\Gamma^s_{ab}\delta \Gamma^c_{sc} - \delta\Gamma^s_{ac} \Gamma^c_{sb} -\Gamma^s_{ac}\delta \Gamma^c_{sb} \\
&= \nabla_c (\delta \Gamma^c_{ab}) - \nabla_b (\delta \Gamma^c_{ac} ) \, .
\end{aligned}
\end{align}
To compute the variation of the Christoffel symbols, expand the covariant derivative of the metric perturbations
\begin{align}
\begin{aligned}
\nabla_a h_{bc} &= \nabla_a (\delta g_{bc}) = \partial_a (\delta g_{bc}) - \Gamma^s_{ab} \delta g_{sc} - \Gamma^s_{ac} \delta g_{bs} \\
&= \delta (\partial_a g_{bc}) - \delta(\Gamma^s_{ab}g_{sc}) + \delta \Gamma^s_{ab}g_{sc} - \delta ( \Gamma^s_{ac} g_{bs} )+ \delta\Gamma^s_{ac} g_{bs} \\
&= \delta (\partial_a g_{bc} - \Gamma^s_{ab}g_{sc}- \Gamma^s_{ac} g_{bs} )+ \delta \Gamma^s_{ab}g_{sc} + \delta\Gamma^s_{ac} g_{bs} \\
&= \delta (\nabla_a g_{bc})+ \delta \Gamma^s_{ab}g_{sc} + \delta\Gamma^s_{ac} g_{bs} \\
&= \delta \Gamma^s_{ab}g_{sc} + \delta\Gamma^s_{ac} g_{bs} \, ,
\end{aligned}
\end{align}
using metric compatibility. Now cyclically permuting and adding a convenient sign gives
\begin{align}
\begin{aligned}
\nabla_a h_{bc} + \nabla_b h_{ca} - \nabla_c h_{ab} &= \delta \Gamma^s_{ab}g_{sc} + \delta\Gamma^s_{ac} g_{bs} + \delta \Gamma^s_{bc}g_{sa} + \delta\Gamma^s_{ba} g_{cs} - \delta \Gamma^s_{ca}g_{sb} - \delta\Gamma^s_{cb} g_{as} \\
&= 2\delta \Gamma^s_{ab}g_{sc} \, .
\end{aligned}
\end{align}
Rearranging and permuting the indices gives
\begin{align}
\delta \Gamma^a_{bc} = \frac12 (\nabla_b h_{c}^a + \nabla_c h_{b}^a - \nabla^a h_{bc}) \, .
\end{align}
Expanding the variation $\delta R_{ab}$ with this formula, one finds
\begin{align}
\begin{aligned}
\delta R_{ab} &= \nabla_c (\delta \Gamma^c_{ab}) - \nabla_b (\delta \Gamma^c_{ac} ) \\
&= \frac12 \nabla_c (\nabla_a h^c_b + \nabla_b h^c_a - \nabla^c h_{ab})- \frac12 \nabla_b (\nabla_a h + \nabla_c h^c_a - \nabla^c h_{ac}) \\
&= \nabla_c \nabla_{(a} h_{b)}^c - \frac12 \nabla^2 h_{ab} -\frac12 \nabla_a \partial_b h \, ,
\end{aligned}
\end{align}
which was the claimed formula for the variation of the Ricci tensor.
Now we must consider the variation of Maxwell's equations:
\begin{align}
\delta \left( \partial_{\mu} (\sqrt{-g} F^{ \mu \nu \rho \sigma\tau})\right) &= \partial_{\mu} (\delta \sqrt{-g} \bar{F}^{\mu \nu \rho \sigma\tau} + \sqrt{-\bar{g}} \delta F^{ \mu \nu \rho \sigma\tau}) \, .
\end{align}
Recall the variation of $\sqrt{-g}$, from Sylvester's formula:
\begin{align}
\delta \sqrt{-g} = -\frac12 \sqrt{-\bar{g}} \bar{g}_{\mu \nu} \delta g^{\mu \nu} = \sqrt{-\bar{g}} \frac{h}{2} \, .
\end{align}
Therefore, we find \eqref{eq:perturbativeEOMs2}:
\begin{align}
\partial_{\mu} (\sqrt{-\bar{g}} (\frac{h}{2} \bar{F}^{ \mu \nu \rho \sigma\tau} + \delta F^{ \mu \nu \rho \sigma\tau})) = 0 \, .
\end{align}
Lastly, the perturbation to the five-form must leave it to be self-dual. However, one must be careful because the Hodge dual involves factors of the metric that also contribute perturbatively. Let us assume the metric is diagonal and that the only nonzero independent components of the five-form are $F_{t123r}$ and $F_{\theta_1 \ldots \theta_5}$. In terms of the components of the metric and five-form the constraint can be written explicitly as
\begin{align}
F_{t123r} + \delta F_{t123r} &= \sqrt{-g} g^{\theta_1 \theta_1} \ldots g^{\theta_5 \theta_5} (F_{\theta_1 \ldots \theta_5} + \delta F_{\theta_1 \ldots \theta_5}) \\
F_{\theta_1 \ldots \theta_5} + \delta F_{\theta_1 \ldots \theta_5} &= -\sqrt{-g} g^{tt} g^{11} g^{22} g^{33} g^{rr} (F_{t123r} + \delta F_{t123r}) \, .
\end{align}
Now removing the background-order equations and expanding perturbatively, we find
\begin{align}
\delta F_{t123r} &= \sqrt{-\bar{g}} \bar{g}^{\theta_1 \theta_1} \ldots \bar{g}^{\theta_5 \theta_5} \delta F_{\theta_1 \ldots \theta_5} - \sqrt{-\bar{g}} (h^{\theta_1 \theta_1} \ldots \bar{g}^{\theta_5 \theta_5} + \ldots + \bar{g}^{\theta_1 \theta_1} \ldots h^{\theta_5 \theta_5} ) \bar{F}_{\theta_1 \ldots \theta_5} + \frac{h}{2} \bar{F}_{t123r} \label{eq:selfdual1}\\
\delta F_{\theta_1 \ldots \theta_5} &= -\sqrt{-\bar{g}} \bar{g}^{tt} \bar{g}^{11} \bar{g}^{22} \bar{g}^{33} \bar{g}^{rr} \delta F_{t123r} + \sqrt{-\bar{g}} (h^{tt} \bar{g}^{11} \bar{g}^{22} \bar{g}^{33} \bar{g}^{rr} + \ldots +\bar{g}^{tt} \bar{g}^{11} \bar{g}^{22} \bar{g}^{33} h^{rr} ) F_{t123r} + \frac{h}{2} \bar{F}_{\theta_1 \ldots \theta_5} \, .\label{eq:selfdual2}
\end{align}
To proceed further, one requires more details about the background metric and five-form of interest.
\section{Solving the Linearized Equations} \label{sec:linearizedApp}
In this appendix, we demonstrate the procedure to solve the perturbative equations of motion \eqref{eq:perturbativeEOMs1} and \eqref{eq:perturbativeEOMs2} by hand in the linearized regime, region II. In this regime the ansatz for the metric and five-form perturbations takes the form
\begin{align}
\frac{1}{\alpha'} ds^2 &= L^2 \biggl[-\tilde{r}^2\left(1 + \delta g_{\tilde{t}\tilde{t}} \right) d\tilde{t}^2 +\tilde{r}^2\left(1 + \delta g_{\tilde{i}\tilde{i}}\right) d\vec{\tilde{x}}^2 + \frac{1}{\tilde{r}^2}\left(1+\delta g_{\tilde{r}\tilde{r}}\right)d\tilde{r}^2 + (1+\delta g_{\Omega \Omega}) d\Omega_5^2 \biggr] \\
\frac{1}{\alpha^{'2}} F &= 4L^4 \biggl[\tilde{r}^3 \left(1 +a(\tilde{r}) \right) d\tilde{t} \wedge d\tilde{x}^1 \wedge d\tilde{x}^2 \wedge d\tilde{x}^3 \wedge d\tilde{r} \nonumber \\&\qquad\qquad\qquad\qquad\qquad+ (1+b(\tilde{r})) \sin^4 \theta_1 \sin^3 \theta_2 \sin^2 \theta_3 \sin \theta_4 d\theta_1 \wedge d\theta_2 \wedge d\theta_3 \wedge d\theta_4 \wedge d\theta_5 \biggr] \, ,
\end{align}
where all metric perturbations are functions only of $\tilde{r}$. To begin, we evaluate the self-duality constraint equations \eqref{eq:selfdual1} and \eqref{eq:selfdual2} on the background metric of vacuum AdS$_5 \times S_5$. They reduce to only one independent equation,
\begin{align}
2 a-3 \delta g_{\tilde{i}\tilde{i}}- \delta g_{\tilde{t}\tilde{t}}+5 \delta g_{\Omega \Omega }- \delta g_{\tilde{r}\tilde{r}} &= 2b \, ,
\end{align}
identical to \eqref{eq:selfdualBB}. Plugging into Maxwell's equations we also find only one independent equation
\begin{align}
2 a'-3 \delta g_{\tilde{i}\tilde{i}}'- \delta g_{\tilde{t}\tilde{t}}'+5 \delta g_{\Omega \Omega}'- \delta g_{\tilde{r}\tilde{r}}' &= 0 \, ,
\end{align}
which is identical to \eqref{eq:maxwellBB}. Combining Maxwell's equations and self-duality gives $b' = 0$, meaning that the five-form charge is conserved. The independent geometric equations of motion are
\begin{align}
-8 (b -
2\delta g_{\Omega\Omega } ) - \frac {1} {2} \tilde{r}\left (\tilde{r} g_{\Omega\Omega }'' +
5g_{\Omega\Omega }' \right) &= 0 \label{eq:geom1}\\
-16 a+\tilde{r}^2 \delta g_{\tilde{t}\tilde{t}} ''+3 \tilde{r} \delta g_{\tilde{i}\tilde{i}} '+6 \tilde{r} \delta g_{\tilde{t}\tilde{t}}'+5 \tilde{r} \delta g_{\Omega \Omega}'- \tilde{r} \delta g_{\tilde{r}\tilde{r}}'+24 \delta g_{\tilde{i}\tilde{i}} +8 \delta g_{\tilde{t}\tilde{t}} &= 0\label{eq:geom2}\\
-16 a+\tilde{r}^2 \delta g_{\tilde{i}\tilde{i}}''+8 \tilde{r}\delta g_{\tilde{i}\tilde{i}}'+\tilde{r} \delta g_{\tilde{t}\tilde{t}}'+5 \tilde{r} \delta g_{\Omega \Omega }'- \tilde{r} \delta g_{\tilde{r}\tilde{r}}'+24 \delta g_{\tilde{i}\tilde{i}}+8 \delta g_{\tilde{t}\tilde{t}} &= 0 \label{eq:geom3}\\
-16 a +3 \tilde{r}^2 \delta g_{\tilde{i}\tilde{i}}''+\tilde{r}^2 \delta g_{\tilde{t}\tilde{t}}''+5 \tilde{r}^2 \delta g_{\Omega \Omega}''+9 \tilde{r}\delta g_{\tilde{i}\tilde{i}}'
+3 \tilde{r}\delta g_{\tilde{t}\tilde{t}}'\qquad\qquad\qquad\qquad\qquad&\nonumber \\+5 \tilde{r} \delta g_{\Omega \Omega }'-4 \tilde{r} \delta g_{\tilde{r}\tilde{r}}'+24 \delta g_{\tilde{i}\tilde{i}}+8 \delta g_{\tilde{t}\tilde{t}} &= 0 \, .\label{eq:geom4}
\end{align}
Notice that the equation of motion \eqref{eq:geom1} for $\delta g_{\Omega \Omega}$ is independent of the others and may be solved directly, yielding:
\begin{align}
\delta g_{\Omega \Omega} = \frac{b}{2} + a_1 \tilde{r}^4 + a_2/\tilde{r}^8 \, ,
\end{align}
for constants $a_1, a_2$. Now examine the three remaining geometric equations of motion. By taking the linear combination of \eqref{eq:geom2} + 3\eqref{eq:geom3} - \eqref{eq:geom4}, we find an equation determining $\delta g_{\tilde{r}\tilde{r}}$ in terms of the others:
\begin{align}
4\delta g_{\tilde{r}\tilde{r}} = 2b + 3\tilde{r}\delta g^{'}_{\tilde{i}\tilde{i}}+ \tilde{r}\delta g^{'}_{\tilde{t}\tilde{t}} +20 (a_1 \tilde{r}^4 -3 a_2 / \tilde{r}^8) \, . \label{eq:rreqn}
\end{align}
Plugging this back into all three equations we find that all three are solved as long as:
\begin{align}
5(\delta g_{\tilde{i}\tilde{i}}^{'} - \delta g_{\tilde{t}\tilde{t}}^{'}) + r(\delta g_{\tilde{i}\tilde{i}}^{''} - \delta g_{\tilde{t}\tilde{t}}^{''}) = 0 \, .
\end{align}
This is a differential equation in $f (\tilde{r}) = \delta g_{\tilde{i}\tilde{i}} - \delta g_{\tilde{t}\tilde{t}}$ which is solved by $f(\tilde{r}) = c_2 - \frac{c_1}{4\tilde{r}^4}$ where $c_1, c_2$ are constants. Therefore we can relate $\delta g_{\tilde{i}\tilde{i}}$ to $\delta g_{\tilde{t}\tilde{t}}$ via
\begin{align}
\delta g_{\tilde{i}\tilde{i}} = c_2 - \frac{c_1}{4\tilde{r}^4}+\delta g_{\tilde{t}\tilde{t}} \, . \label{eq:iieqn}
\end{align}
Plugging \eqref{eq:iieqn} into \eqref{eq:rreqn} reduces it to
\begin{align}
\delta g_{\tilde{r}\tilde{r}} = \frac{b}{2} + \tilde{r}\delta g_{\tilde{t}\tilde{t}}^{'} + \frac34\frac{c_1}{\tilde{r}^4} - 15 \frac{a_2}{\tilde{r}^8} + 5a_1 \tilde{r}^4 \, .
\end{align}
We have consequently fixed the general perturbative solution in the linearized regime in terms of one arbitrary function $\delta g_{\tilde{t}\tilde{t}}$ and five constants $a_1, a_2, c_1, c_2, b$:
\begin{align}
\frac{1}{\alpha'} ds^2 &= L^2 \biggl[-\tilde{r}^2\left(1 + \delta g_{\tilde{t}\tilde{t}} \right) d\tilde{t}^2 +\tilde{r}^2\left(1 + c_2 - \frac{c_1}{4\tilde{r}^4}+\delta g_{\tilde{t}\tilde{t}}\right) d\vec{\tilde{x}}^2 \nonumber\\
&+ \frac{1}{\tilde{r}^2}\left(1+\frac{b}{2} + \tilde{r}\delta g'_{\tilde{t}\tilde{t}} + \frac34\frac{c_1}{\tilde{r}^4} - 15 \frac{a_2}{\tilde{r}^8} + 5a_1 \tilde{r}^4\right)d\tilde{r}^2 + (1+ \frac{b}{2} + a_1 \tilde{r}^4 + \frac{a_2}{\tilde{r}^8}) d\Omega_5^2 \biggr] \\
\frac{1}{\alpha^{'2}} F &= 4L^4 \biggl[\tilde{r}^3 \left(1 +\frac32 c_2 - 10 \frac{a_2}{\tilde{r}^8} + 2 \delta g_{\tilde{t}\tilde{t}} + \frac{\tilde{r}}{2} \delta g'_{\tilde{t}\tilde{t}}\right) d\tilde{t} \wedge d\tilde{x}^1 \wedge d\tilde{x}^2 \wedge d\tilde{x}^3 \wedge d\tilde{r} \nonumber \\&\qquad\qquad\qquad\qquad\qquad+ (1+b) \sin^4 \theta_1 \sin^3 \theta_2 \sin^2 \theta_3 \sin \theta_4 d\theta_1 \wedge d\theta_2 \wedge d\theta_3 \wedge d\theta_4 \wedge d\theta_5 \biggr] \, .
\end{align}
To fix $\delta g_{\tilde{t}\tilde{t}}$ and the five constants, we compare the solution to the linearized expansion of the solution \eqref{eq:metricnonpert}-\eqref{eq:fiveformnonpert}. A consistent solution is found by taking $a_1 = \epsilon^4 / 2$, $a_2 = c_2 = 0$, $c_1 = -4$, $b=0$, and
\begin{align}
\delta g_{\tilde{t}\tilde{t}} = -\frac{1}{\tilde{r}^4} - \frac12 (\epsilon \tilde{r})^4 \, ,
\end{align}
leading to the solution \eqref{eq:linear1}-\eqref{eq:linear2}.
\end{appendices}
\bibliographystyle{JHEP}
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2024-02-18T23:40:24.761Z
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2020-10-01T02:21:37.000Z
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{"paloma_paragraphs":[]}
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{"arxiv_id":"2009.08980","language":"en","timestamp":1601518897000,"url":"https:\/\/arxiv.org\/abs\/2009.08980","yymm":"2009"}
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proofpile-arXiv_000-10081
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{"provenance":"002.jsonl.gz:10082"}
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\subsection{Immersive Sketching and Modeling}
\label{sec:relatedDrawing}
Immersive creation has a long history in computer graphics. Immersive 3D sketching was pioneered by the HoloSketch system \cite{deering1995holosketch}, which used a 6-DoF wand as the input device for creating polyline sketches, 3D tubes, and primitives. In a similar vein, various subsequent systems have explored the creation of freeform 3D curves and swept surfaces \cite{schkolne2001surface, keefe2001cavepainting, google2020tilt}.
While directly turning 3D input to creative output is acceptable for ideation, the inherent imprecision of 3D sketching is quickly apparent when more structured creation is desired.
The perceptual and ergonomic challenges in precise control of 3D input is well-known~\cite{keefe2007drawing, wiese2010investigating, arora2017experimental, machuca2019effect, machuca2018multi}, resulting in various methods for correcting 3D input.
Input 3D curves have been algorithmically regularized to snap onto existing geometry, as with the FreeDrawer~\shortcite{wesche2001freedrawer} system, or constrained physically to 2D input with additional techniques for ``lifting'' these curves into 3D \cite{jackson2016lift, arora2018symbiosis, kwan2019mobi, paczkowski2011insitu}. Haptic rendering devices \cite{keefe2007drawing, kamuro20113d} and tools utilizing passive physical feedback \cite{grossman2002creating} are an alternate approach to tackling the imprecision of 3D inputs. We are motivated by similar considerations.
Arora et al.~\shortcite{arora2017experimental} demonstrated the difficulty of creating curves that lie exactly on virtual surfaces in VR, even when the virtual surface is a plane. This observation directly motivates our exploration of techniques for projecting 3D strokes onto surfaces, instead of coercing users to awkwardly draw exactly on a virtual surface.
\subsection{Drawing Curves on, near, and around Surfaces}
\label{sec:relatedProjecting}
Curve creation and editing on or near the surface of 3D virtual objects is fundamental for a variety of artistic and functional shape modeling tasks.
Functionally, curves on 3D surfaces are used to model or annotate structural features \cite{iwires, neobarok}, define trims and holes \cite{schmidt2010meshmixer}, and to provide handles for shape deformation \cite{wires, kara2007sketch,nealen2007fibermesh}, registration \cite{gehre2018interactive} and remeshing \cite{krishlevoy,takayama2013sketch}.
Artistically, curves on surfaces are used in painterly rendering \cite{goochbook}, decal creation \cite{schmidt2006discrete}, texture painting \cite{adobe2020substance}, and even texture synthesis \cite{fisher2007design}. Curve on surface creation in this body of research typically uses the established view-centric WYSIWYG projection of on-screen sketched 2D strokes. While the sketch view-point in these interfaces is interactively set by the user, there has been some effort in automatic camera control for drawing \cite{ortega2014direct}, auto-rotation of the sketching view for 3D planar curves \cite{flatfab}, and user assistance in selecting the most \emph{sketchable} viewpoints \cite{bae08ilovesketch}. Immersive 3D drawing enables direct, view-point independent 3D curve sketching, and is thus an appealing alternative to these 2D interfaces.
Our work is also related to drawing curves \emph{around} surfaces. Such techniques are important for a variety of applications: modeling string and wire that wrap around objects \cite{coleman2006cords}; curves that loosely conform to virtual objects or define collision-free paths around objects \cite{krs2017Skippy}; curve patterns for clothing design on a 3D mannequin model \cite{turquin2007sketch}; curves for layered modeling of shells and armour \cite{depauli2015secondskin}; and curves for the design and grooming of hair and fur \cite{fu2007sketching,schmidt2011overcoat, xing2019hairbrush}. Some approaches such as SecondSkin~\shortcite{ depauli2015secondskin} and Skippy~\shortcite{krs2017Skippy} use insights into spatial relationship between a 2D stroke and the 3D object, to infer a 3D curve that lies on and around the surface of the object. Other techniques like Cords~\shortcite{coleman2006cords} or hair and clothing design \cite{xing2019hairbrush} are closer to our work, in that they drape 3D curve input on and around 3D objects using geometric collisions or physical simulation.
In contrast, this paper is application agnostic, and remains focused on the general problem of projecting a drawn 3D stroke to a real-time inked curve on the surface of a virtual 3D object. While we do not address curve creation with specific geometric relationships to the object surface (like distance-offset curve), our techniques can be extended to incorporate geometry-specific terms (\S~\ref{sec:conclusion}).
\subsection{Sketch-based 3D Modeling}
\label{sec:relatedModeling}
Sketch-based 3D modeling is a rich ongoing area of research (see survey by Olsen et al.~\shortcite{olsen2009sketch}). Typically, these systems interpret 2D sketch inputs for various shape modeling tasks. One could categorize these modeling approaches as single-view (akin to traditional pen on paper) \cite{schmidt2009analytic,andre11single,chen20133sweep,xu2014true2form} or multi-view (akin to 3D modeling with frequent view manipulation) \cite{igarashi1999teddy,fan2004sketch,nealen2007fibermesh,bae08ilovesketch,fan2013modeling}. Single-view techniques use perceptual insights and geometric properties of the 2D sketch to infer its depth in 3D, while multi-view techniques explicitly use view manipulation to specify 3D curve attributes from different views. While our work utilizes mid-air 3D stroke input, the ambiguity of projection onto surfaces connects it to the interpretative algorithms designed for sketch-based 3D modeling.
We aim to take advantage of the immersive interaction space by allowing view manipulation as and when desired, independent of geometry creation.
\subsection{Context-Free Projection Techniques}
\label{sec:historyFree}
Context-free techniques project points independent of each other, simply based on the spatial relationships between the controller, HMD, and 3D object ( Figure~\ref{fig:historyFree}).
We can further categorize techniques as raycast or proximity based.
\subsubsection{Raycast Projections}
\label{sec:raycasting}
View-centric projection in 2D interfaces project points from the screen along a ray from the eye through the screen point, to where the ray first intersects the 3D object. In an immersive setting, raycast approaches similarly use a ray emanating from the 3D stroke point to intersect 3D objects. This ray $(\mathbf{o}, \mathbf{d})$ with origin $\mathbf{o}$ and direction $\mathbf{d}$ can be defined in a number of ways. Similar to pointing behavior, {\it Occlude} defines this ray from the eye through the controller origin (also the stroke point, Figure~\ref{fig:historyFree}a) $\left(\mathbf{c}_i, (\mathbf{c}_i - \mathbf{h}_i) / \|\mathbf{c}_i - \mathbf{h}_i\|\right)$. If the ray intersects $\mathcal{M}$, then the closest intersection to $\mathbf{p}_i$ defines $\mathbf{q}_i$. In case of no intersection, $\mathbf{p}_i$ is ignored in defining the projected curve, i.e., $\mathbf{q}_i$ is marked undefined and the projected curve connects $\mathbf{q}_{i-1}$ to $\mathbf{q}_{i+1}$ (or the proximal index points on either side of $i$ for which projections are defined).
The {\it Spraycan} approach treats the controller like a spraycan, defining the ray like a nozzle direction in the local space of the controller (Figure~\ref{fig:historyFree}b). For example the ray could be defined as $(\mathbf{c}_i, \mathbf{f}_i)$, where the nozzle $\mathbf{f}_i = \mathsf{c}_i\cdot [0, 0, 1]^T$ is the controller's local z-axis (or \emph{forward} direction).
Alternately, {\it Head-centric} projection can define the ray using the HMD's view direction as $(\mathbf{h}_i, \mathsf{h}_i\cdot [0, 0, 1]^T)$ (Figure~\ref{fig:historyFree}c).
{\bf Pros and Cons:} The strengths of raycasting are: a predictable visual/proprioceptive sense of ray direction; a spatially continuous mapping between user input and projection rays; and AR/VR scenarios where it is difficult or undesirable to reach and draw close to the virtual object.
The biggest limitation of raycast projection stems from the controller/HMD-based ray direction being completely agnostic of the shape or location of the 3D object. Projected curves can consequently be very different in shape and size from drawn strokes, and ill-defined for stroke points with no ray-object intersection.
\subsubsection{Proximity-Based Projections}
\label{sec:proximity}
In 2D interfaces, the on-screen 2D strokes are typically distant to the viewed 3D scene, necessitating some form of raycast projection onto the visible surface of 3D objects. In AR/VR, however, users are able to reach out in 3D and directly draw the desired curve on the 3D object.
While precise mid-air drawing on a virtual surface is very difficult in practice (Figure~\ref{fig:imprecise3D}), projection methods based on proximity between the mid-air stroke and the 3D object are certainly worth investigation.
The simplest proximity-based projection technique {\it Snap}, projects a stroke point $\mathbf{p}_i$ to its closest-point in $\mathcal{M}$ (Figure~\ref{fig:historyFree}d).
\begin{equation}
\mathbf{q}_i = \pi_{snap}(\mathbf{p}_i) = \argmin_{\mathbf{x}\in\mathcal{M}}\ d(\mathbf{p}_i, \mathbf{x}),
\end{equation}
\begin{wrapfigure}[10]{r}{0.3\columnwidth}
\begin{center}
\vspace{-0.7cm}
\hspace{-0.7cm}
\includegraphics[width=0.3\columnwidth]{cp_scp}
\end{center}
\end{wrapfigure}
where $d(\cdot, \cdot)$ is the Euclidean distance between two points.
Unfortunately, for triangle meshes, closest-point projection tends to snap to the edges of the mesh (\textcolor{matlabblue}{\textbf{blue}} curve inset), resulting in unexpectedly jaggy projected curves, even for smooth 3D input strokes (\textbf{black} curve inset)~\cite{panozzo2013weighted}.
These discontinuities are due to the discrete nature of the mesh representation, as well spatial singularities in closest point computation even for smooth 3D objects. We mitigate this problem by formulating an extension of Panozzo et al.'s \emph{Phong} projection~\shortcite{panozzo2013weighted} in \S~\ref{sec:phong}, that simulates projection of points onto an imaginary smooth surface approximated by the triangle mesh.
We denote this {\it smooth-closest-point} projection as $\pi_{SCP}$ (\textcolor{matlabred}{\textbf{red}} curve inset).
{\bf Pros and Cons:} The biggest strength of proximity-based projection is it exploits the immersive concept of drawing directly on or near an object, using the spatial relationship between a 3D stroke point and the 3D object to determine projection. The main limitation is that since users rarely draw precisely on the surface, discontinuities and local extrema persist when projecting distantly drawn stoke points, even when using {\it smooth-closest-point}. In \S~\ref{sec:anchoredProximity}, we address this problem using stroke {\it mimicry} to anchor distant stroke points close to the object to be finally projected using {\it smooth-closest-point}.
\subsection{Smooth-Closest-Point Projection}
\label{sec:phong}
\tikzcdset{column sep/huge=3.5cm}
\begin{figure}
\centering
\begin{subfigure}{\linewidth}
\begin{tikzcd}
\{\mathbf{x}^d\}\subset\mathcal{M}^d \arrow[r, "\text{Def.}"] & \mathbf{y}^d=\sum w_i\mathbf{x}_i^d \arrow[r, "\mathcal{P}_{Phong}"] & \mathbf{z}^d\in\mathcal{M}^d \arrow[d, "\bary(\mathcal{M})"] \\
\{\mathbf{x}^d\}\subset\mathcal{M}^3 \arrow[u, "{\bary(\mathcal{M}),\ \mathbf{e}^d(\mathcal{M})}"] \arrow[r, "\text{Def.}"] & \mathbf{y}^3=\sum w_i\mathbf{x}_i^3 & \mathbf{z}^3\in\mathcal{M}^3
\end{tikzcd}
\caption{Computing weighted averages in Panozzo et al.~\shortcite{panozzo2013weighted}}
\end{subfigure}
\begin{subfigure}{\linewidth}
\begin{tikzcd}[column sep = huge]
\mathbf{y}^d\in\mathcal{T}_\mathcal{M}^d \arrow[r, "\mathcal{P}_{Phong}"] & \mathbf{z}^d\in\mathcal{M}^d \arrow[d, "\bary(\mathcal{M})"] \\
\mathbf{y}^3\in\mathcal{T}_\mathcal{M}^3 \arrow[u, "{\bary(\mathcal{T}_\mathcal{M}),\ \mathbf{e}^d(\mathcal{T}_\mathcal{M})}"] & \mathbf{z}^3\in\mathcal{M}^3
\end{tikzcd}
\caption{Computing smooth-closest-point projection.}
\end{subfigure}
\begin{subfigure}{\linewidth}
\centering
\includegraphics[width=.9\linewidth]{higher_dim_embedding}
\caption{Computing a $d$-dimensional embedding for $\mathcal{M}$ (a) and $\mathcal{T}_\mathcal{M}$ (b).}
\end{subfigure}
\caption{Panozzo et al.~\shortcite{panozzo2013weighted} compute weighted averages on surfaces (a), while we want to compute a smooth closest-point projection for an arbitrary point \emph{near} the mesh in $\mathbb{R}^3$ (b). We therefore embed $\mathcal{T}_\mathcal{M}$---the region \emph{around} the mesh---in higher-dimensional space $\mathbb{R}^d$, instead of just $\mathcal{M}$ (c).}
\label{fig:phongAdapt}
\end{figure}
Our goal with \emph{smooth-closest-point} projection is to define a mapping from a 3D point to a point on $\mathcal{M}$ that approximates the closest point projection but tends to be functionally smooth, at least for points near the 3D object. We note that computing the closest point to a Laplacian-smoothed mesh proxy, for example, will also provide a smoother mapping than $\pi_{snap}$, but a potentially poor closest-point approximation to the original mesh.
\emph{Phong projection}, introduced by Panozzo et al.~\shortcite{panozzo2013weighted}, addresses these goals for points expressible as weighted-averages of points on $\mathcal{M}$, but we extend their technique to define a smooth-closest-point projection for points in the neighbourhood of the mesh. For completeness, we first present a brief overview of their technique.
Phong projection is a two-step approach to map a point $\mathbf{y}^3 \in \mathbb{R}^3$ to a manifold triangle mesh $\mathcal{M}$ embedded in $\mathbb{R}^3$, emulating closest-point projection on a smooth surface approximated by the triangle mesh.
First, $\mathcal{M}$ is embedded in a higher dimensional Euclidean space $\mathbb{R}^d$ such that Euclidean distance (between points on the mesh) in $\mathbb{R}^d$ better approximates geodesic distances in $\mathbb{R}^3$.
Second, analogous to vertex normal interpolation in Phong shading, a smooth surface is approximated by blending tangent planes across edges. Barycentric coordinates at a point within a triangle are used to blend the tangent planes corresponding to the three edges incident to the triangle.
We extend the first step to a higher dimensional embedding of not just the triangle mesh $\mathcal{M}$, but a tetrahedral representation of an offset volume around the mesh $\mathcal{M}$ (Figure~\ref{fig:phongAdapt}).
The second step remains the same, and we refer the reader to Panozzo et al.~\shortcite{panozzo2013weighted} for details.
For clarity, we refer to $\mathcal{M}$ embedded in $\mathbb{R}^3$ as $\mathcal{M}^3$, and the embedding in $\mathbb{R}^d$ as $\mathcal{M}^d$.
Panozzo et al. compute $\mathcal{M}^d$ by first embedding a subset of the vertices in $\mathbb{R}^D$ using metric multi-dimensional scaling (MDS)~\cite{cox2008multidimensional}, aiming to preserve the geodesic distance between the vertices.
This subset consists of the high-curvature vertices of $\mathcal{M}$.
The embedding of the remaining vertices is then computed using LS-meshes~\cite{sorkine2004LSmeshes}.
For the problem of computing weighted averages on surfaces, one only needs to project 3D points of the form $\mathbf{y}^3=\sum w_i\mathbf{x}_i^3$, where all $\mathbf{x}_i^3\in\mathcal{M}^3$.
The point $\mathbf{y}^3$ is \emph{lifted} into $\mathbb{R}^d$ by simply defining $\mathbf{y}^d = \sum w_i\mathbf{x}_i^d$, where $\mathbf{x}_i^d$ is defined as the point on $\mathcal{M}^d$ with the same implicit coordinates (triangle and barycentric coordinates) as $\mathbf{x}_i^3$ does on $\mathcal{M}^3$.
Therefore, their approach only embeds $\mathcal{M}$ into $\mathbb{R}^d$ (Figure~\ref{fig:phongAdapt}a,c).
In contrast, we want to project arbitrary points \emph{near} $\mathcal{M}^3$ onto it using the Phong projection.
Therefore, we compute the offset surfaces at signed-distance $\pm\mu$ from $\mathcal{M}$.
We then compute a tetrahedral mesh $\mathcal{T}_\mathcal{M}^3$ of the space between these two surfaces in $\mathbb{R}^3$.
In the final step, we embed the vertices of $\mathcal{T}_\mathcal{M}$ in $\mathbb{R}^d$ using MDS and LS-Meshes as described above.
Note that all of the above steps are realized in a precomputation.
Now, given a 3D point $\mathbf{y}^3$ within a distance $\mu$ from $\mathcal{M}^3$, we situate it within $\mathcal{T}_\mathcal{M}^3$, use tetrahedral Barycentric coordinates to infer its location in $\mathbb{R}^d$, and then compute its Phong projection (Figure~\ref{fig:phongAdapt}b,c).
We fallback to closest-point projection for points outside $\mathcal{T}_\mathcal{M}^3$, since Phong projection converges to closest-point projection when far from $\mathcal{M}$. Furthermore, we set $\mu$ large enough to easily handle our smooth-closest-point queries in \S~\ref{sec:anchoredProximity}.
\subsection{Analysis of Context-Free Projection}
\label{sec:historyFreeProblems}
We implemented the four different context-free projection approaches in Figure~\ref{fig:historyFree}, and had 4 users informally test each, drawing a variety of curves on the various 3D models seen in this paper.
Qualitatively, we made a number of observations:
\begin{itemize}[label=--, leftmargin=1em]
\item {\it Head-centric} and {\it Occlude} projections become unpredictable if the user is inadvertently changing their viewpoint while drawing. These projections are also only effective when drawing frontally on an object, like with a 2D interface. Neither as a result exploits the potential gains of mid-air drawing in AR/VR.
\item {\it Spraycan} projection was clearly the most effective context-free technique. Commonly used for graffiti and airbrushing, usually on fairly flat surfaces, we noted however, that consciously reorienting the controller while drawing on or around complex objects was both cognitively and physically tiring.
\item {\it Snap} projection was quite sensitive to changes in the distance of the stroke from the object surface, and in general produced the most undulating projections due to closest-point singularities.
\item All projections converge to the mid-air user stroke when it precisely conforms to the surface of the 3D object. But as the distance between the object and points on the mid-air stroke increases, their behavior diverges quickly.
\item While users did draw in the vicinity and mostly above the object surface, they rarely drew precisely on the object. The average distance of stroke points from the target object was observed to be 4.8 cm in a subsequent user study (\S~\ref{sec:study}).
\item The most valuable insight however, was that the user stroke in mid-air often tended to {\bf mimic} the expected projected curve.
\end{itemize}
Context-free approaches, by design, are unable to capture this mimicry, i.e., the notion that the change between projected point as we draw a stroke is commensurate with the change in the 3D points along the stroke. This inability due to a lack of curve history or context,
materializes as problems in different forms.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{history_free_problems}
\caption{Context-free projection problems: projection discontinuities (a), undesirable snapping (b) large depth disparity (c) and unexpected jumps (d).}
\label{fig:historyFreeProblems}
\end{figure}
\subsubsection{Projection Discontinuities}
\label{sec:discontinuity}
Proximal projection (including \emph{smooth-closest-point}) can be highly discontinuous with increasing distance from the 3D object, particularly in concave regions (Figure~\ref{fig:historyFreeProblems}a).
Mid-air drawing along valleys without staying in precise contact with virtual object is thus
extremely difficult.
Raycast projections can similarly suffer large discontinuous jumps across occluded regions (in the ray direction) of the object (Figure~\ref{fig:historyFreeProblems}d).
While this problem theoretically exists in 2D interfaces as well, it is less observed in practice for two reasons: 2D drawing on a constraining physical surface is significantly more precise than mid-air drawing in AR/VR~\cite{arora2017experimental}; and artists minimize such discontinuities by carefully choosing appropriate views (raycast directions) before drawing each curve. Automatic diretion control of view or controller, while effective in 2D \cite{ortega2014direct}), is detrimental to a sense of agency and presence in AR/VR.
\subsubsection{Undesirable Snapping}
\label{sec:undesirableSnapping}
Proximity-based methods also tend to get stuck on sharp (or high curvature) convex features of the object (Figure~\ref{fig:historyFreeProblems}b).
While this can be useful to trace along a ridge feature, it is particularly problematic for general curve-on-surface drawing.
\subsubsection{Projection depth disparity}
\label{sec:geometryIgnorance}
The relative orientation between the 3D object surface and raycast direction can cause large depth disparities between parts of user strokes and curves projected by raycasting (Figure~\ref{fig:historyFreeProblems}c). Such irregular bunching or spreading of points on the projected curve also goes against our observation of stroke mimicry. Users can arguably reduce this disparity by continually orienting the view/controller to keep the projection ray well aligned with object surface normal. Such re-orientation however can be tiring, ergonomically awkward, and deviates from 2D experience, where pen/brush tilt only impacts curve aesthetic, and not shape.
We noted that the {\it Occlude} and {\it Spraycan} techniques were complementary: drawing with {\it Occlude} on parts of an object frontal to the view provided good comfort and control, which degraded when drawing closer to the object silhouette, and observed the opposite when drawing with {\it Spraycan}. We thus implemented a hybrid projection, where the ray direction was interpolated between {\it Occlude} and {\it Spraycan} based on alignment with the most recently projected smooth surface normal. Unfortunately, the difference between {\it Occlude} and {\it Spraycan} ray directions was often large enough to make even smooth ray transitions abrupt and hard to control.
All these problems point to the projection function ignoring the shape of the mid-air stroke $\mathbf{P}$ and the projected curve $\mathbf{Q}$, and can be addressed using projection functions that explicitly incorporate both. We call these functions \emph{anchored}.
\subsection{Mimicry Projection}
\label{sec:anchoredProximity}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{anchored_proximity}
\caption{Anchored smooth-closest-point (a), and refinements: using a locally-fit plane (b), and anchor point constrained to an offset (c) or parallel surface (d). $\mathbf{q}_i$, is obtained by projecting $\mathbf{r}_i$ (a), $\mathbf{r}_i'$ (c), or $\mathbf{r}_i''$ (d) onto $\mathcal{M}$ via smooth-closest-point; or closest-point to $\mathbf{r}_i$ in $\mathcal{M}\cap N_i$ (b). }
\label{fig:historyProximity}
\end{figure}
Controller sampling rate in current AR/VR systems is 50Hz or more, meaning that even during ballistic movements, the distance $\|\Delta\mathbf{p}_i\|$ for any stroke sample $i$ is of the order of a few millimetres.
Consequently, the anchored stroke point $\mathbf{r}_i$ is typically much closer to $\mathcal{M}$, than the stroke point $\mathbf{p}_i$, making closest-point {\it snap} projection a compelling candidate for projecting $\mathbf{r}_i$.
Such an \emph{anchored closest-point} projection explicitly minimizes $\|\Delta\mathbf{p}_i - \Delta\mathbf{q}_i\|$, but precise minimization is less important than avoiding projection discontinuities and undesirably snapping, even for points close to the mesh. Our formulation of a \emph{smooth-closest-point} projection $\pi_{SCP}$ in \S~\ref{sec:phong} addresses these goals precisely. Also note that the maximum observed $\|\Delta\mathbf{p}\|$ for the controller readily defines the offset distance $\mu$ for our pre-computed tet mesh $\mathcal{T}_\mathcal{M}^3$. We define \emph{mimicry} projection as
\begin{equation}
\label{eq:acp}
\Pi_{mimicry}(\mathbf{p}_i) =
\begin{cases}
\pi_{spray}(\mathbf{p}_i) & \text{if } i=0, \\
\pi_{SCP}(\mathbf{r}_i) & \text{otherwise.}
\end{cases}
\end{equation}
\subsection{Refinements to Mimicry Projection}
We further explore refinements to \emph{mimicry} projection, that might improve curve projection in certain scenarios.
{\bf Planar curves} are very common in design and visualization \cite{mccrae2011slices}. We can locally encourage planarity in mimicry projection by constructing a plane $N_i$ with normal $\Delta\mathbf{p}_i \times \Delta\mathbf{p}_{i-1}$ (i.e. the local plane of the mid-air stroke) and passing through the anchor point $\mathbf{r}_i$ (Figure~\ref{fig:historyProximity}b).
We then intersect $N_i$ with $\mathcal{M}$. $\mathbf{q}_i$ is defined as the closest-point to $\mathbf{r}_i$ on the intersection curve that contains $\mathbf{q}_{i-1}$.
Note, we use $\pi_{spray}(\mathbf{p}_i)$ for $i<2$, and we retain the most recently defined normal direction ($N_{i-1}$ or prior) when $N_i= \Delta\mathbf{p}_i \times \Delta\mathbf{p}_{i-1}$ is undefined.
We find this method works well for near-planar curves, but the plane is sensitive to noise in the mid-air stroke (Figure~\ref{fig:anchoredProblems}f), and can feel {\it sticky} or less responsive for non-planar curves.
{\bf Offset and Parallel surface drawing} captures the observation that
users tend to draw an intended curve as a corresponding mid-air stroke on an imaginary offset or parallel surface of the object $\mathcal{M}$. While we do not expect users to draw precisely on such a surface, we note that is unlikely a user would intentionally draw orthogonal to such a surface along the gradient of the 3D object.
In scenarios when a user is sub-consciously drawing on a offset surface of $\mathcal{M}$ (an isosurface of its signed-distance function $d_\mathcal{M}(\cdot)$), we can remove the component of a user stroke segment that lies along the gradient $\nabla d_\mathcal{M}$, when computing the desired anchor point $\mathbf{r}_i$ in Equation~\ref{eq:offsetSurface} as (Figure~\ref{fig:historyProximity}c):
\begin{equation}
\label{eq:offsetSurface}
\mathbf{r}_i' = \mathbf{q}_{i-1} + \Delta\mathbf{p}_i - \Big(\Delta\mathbf{p}_i \cdot \nabla d_\mathcal{M}(\mathbf{p}_i) \Big)\nabla d_\mathcal{M}(\mathbf{p}_i)
\end{equation}
We can similarly locally constrain user strokes to a parallel surface of $\mathcal{M}$ in Equation~\ref{eq:parallelSurface} as:
\begin{equation}
\label{eq:parallelSurface}
\mathbf{r}_i'' = \mathbf{q}_{i-1} + \Delta\mathbf{p}_i - \Big(\Delta\mathbf{p}_i \cdot \nabla d_\mathcal{M}(\mathbf{r}_i) \Big)\nabla d_\mathcal{M}(\mathbf{r}_i)\text{.}
\end{equation}
Note that the difference from Eq.~\ref{eq:offsetSurface} is the position where $\nabla d_\mathcal{M}$ is computed, as shown in Figure~\ref{fig:historyProximity}d.
A parallel surface better matched user expectation than an offset surface in our pilot testing, but both techniques produced poor results when user drawing deviated from these imaginary surfaces (Figure~\ref{fig:anchoredProblems}g--l).
\subsection{Anchored Raycast Projection}
\label{sec:anchoredRaycast}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{anchored_raycast}
\caption{Anchored raycast techniques: ray direction defined orthogonal to $\Delta\mathbf{p}_i$ in a local plane (a); parallel transport of ray direction along the user stroke (b). The cast rays (forward/backward) are shown in blue.}
\label{fig:historyRaycast}
\end{figure}
For completeness, we also investigated raycast alternatives to projection of the anchored stroke point $\mathbf{r}_i$. We used similar priors of local planarity and offset or parallel surface transport as with mimicry refinement, to define ray directions. Figure~\ref{fig:historyRaycast} shows two such options.
In Figure~\ref{fig:historyRaycast}a, we cast a ray in the local plane of motion, orthogonal to the user stroke, given by $\Delta\mathbf{p}_i$.
We construct the local plane containing $\mathbf{r}_i$ spanned by $\Delta\mathbf{p}_i$ and $\mathbf{p}_{i-1} - \mathbf{q}_{i-1}$, and then define the direction orthogonal to $\Delta\mathbf{p}_i$ in this plane. Since $\mathbf{r}_i$ may be inside $\mathcal{M}$, we cast two rays bi-directionally $(\mathbf{r}_i, \pm \Delta\mathbf{p}_i^\bot)$, where
\[\Delta\mathbf{p}_i^\bot =
\Delta\mathbf{p}_i \times
\big(
\Delta\mathbf{p}_i \times
\left( \mathbf{p}_{i-1} - \mathbf{q}_{i-1} \right)
\big)
\]
If both rays successfully intersect $\mathcal{M}$, we choose $\mathbf{q}_i$ to be the point closer to $\mathbf{r}_i$, a heuristic that works well in practice.
As with locally planar mimicry projection, this technique suffered from instability in the local plane.
Motivated by mimicry, in Figure~\ref{fig:historyRaycast}b, we also explored parallel transport of the projection ray direction along the user stroke.
For $i>0$, we parallel transport the previous projection direction $\mathbf{q}_{i-1} - \mathbf{p}_{i-1}$ along the mid-air curve by rotating it with the rotation that aligns $\Delta\mathbf{p}_{i-1}$ with $\Delta\mathbf{p}_{i}$. Once again bi-directional rays are cast from $\mathbf{r}_i$, and $\mathbf{q}_i$ is set to the closer intersection with $\mathcal{M}$.
In general, we found that all raycast projections, even when anchored, suffered from
unpredictability over long strokes and stroke discontinuities when there are no ray-object intersections (Figure~\ref{fig:anchoredProblems}n,o).
\begin{figure*}[htb]
\centering
\includegraphics[width=\linewidth]{problems_anchored}
\caption{{\it Mimicry} vs. other anchored stroke projections: Mid-air strokes are shown in \textbf{black} and {\it mimicry} curves in \textcolor{matlabred}{\textbf{red}}. Anchored closest-point (\textcolor{matlabblue}{\textbf{blue}}), is similar to {\it mimicry} on smooth, low-curvature meshes (a,b) but degrades with mesh detail/noise (c,d). Locally planar projection (\textcolor{matlabblue}{\textbf{blue}}) is susceptible local plane instability (e,f). Parallel (\textcolor{matlabpurple}{\textbf{purple}} h,k) or offset (\textcolor{matlabblue}{\textbf{blue}} i,l) surface based projection fail in (h,l) when the user stroke deviates from said surface, while {\it mimicry} remains reasonable (g, j). Compared to {\it mimicry} (m), anchored raycasting based on a local plane (\textcolor{matlabpurple}{\textbf{purple}} n), or ray transport (\textcolor{matlabblue}{\textbf{blue}} o) can be discontinuous.}
\label{fig:anchoredProblems}
\end{figure*}
\subsection{Final Analysis and Implementation Details}
\label{sec:implementation}
In summary, extensive pilot testing of the anchored techniques revealed that they seemed generally better than context-free approaches, specially when users drew further away from the 3D object. Among the anchored techniques, stroke mimicry captured as an \emph{anchored-smooth-closest-point} projection, proved to be theoretically elegant, and practically the most resilient to ambiguities of user intent and differences of drawing style among users.
\emph{Anchored closest-point} can be a reasonable proxy to \emph{anchored smooth-closest-point} when pre-processing the 3D virtual objects is undesirable.
Our techniques are implemented in C\#, with interaction, rendering, and VR support provided by the Unity Engine.
For the smooth closest-point operation, we modified Panozzo et al.'s~\shortcite{panozzo2013weighted} reference implementation, which includes pre-processing code written in MATLAB and C++, and real-time code in C++.
The real-time projection implementation is exposed to our C\# application via a compiled dynamic library.
In their implementation, as well as ours, $d=8$; that is, we embed $\mathcal{M}$ in $\mathbb{R}^8$ for computing the Phong projection.
We use $\mu=20\text{cm}$, and compute the offset surfaces using \texttt{libigl}~\cite{libigl}.
We then improve the surface quality using TetWild~\cite{hu2018tetwild}, before computing the tetrahedral mesh $\mathcal{T}_\mathcal{M}$ between the two surfaces using TetGen~\cite{si2015tetgen}.
We support fast closest-point queries, using an AABB tree implemented in \texttt{geometry3Sharp}~\cite{geometry3sharp}.
Signed-distance is also computed using the AABB tree and fast winding number~\cite{barill2018fast}, and gradient $\nabla d_\mathcal{M}$ computed using central finite differences.
To ease replication of our various techniques and aid future work, we will open-source our implementation.
We now formally compare our most promising projection \emph{mimicry}, to the best state-of-the-art context-free projection {\it spraycan}.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{all_shapes}
\caption{The six shapes utilized in the user study. The \emph{torus} shape was used for tutorials, while the rest were used for the recorded experimental tasks.}
\label{fig:studyShapes}
\end{figure}
\subsection{Sampling the Target Curves}
\label{sec:sampling}
We wanted to design target curves that could be executed using a single smooth motion. Since users typically draw sharp corners using multiple strokes \cite{bae08ilovesketch}, we constrain our target curves to be smooth, created using cardinal cubic B-splines on the meshes, computed using Panozzo et al.~\shortcite{panozzo2013weighted}.
We also control the length and curvature complexity of the curves, as pilot testing showed that very simple and short curves can be reasonably executed by almost any projection technique.
Curve length and complexity is modeled by placing spline control points at mesh vertices, and specifying the desired geodesic distance and Gau{\ss} map distance between consecutive control points on the curve.
We represent a target curve using four parameters $\langle n, i_0, k_G, k_N \rangle$, where $n$ is the number of spline control points, $i_0$ the vertex index of the first control point, and $k_G, k_N$ constants that control the geodesic and normal map distance between consecutive control points.
We define the desired geodesic distance between consecutive control points as,
$D_G = k_G\times\|\mathtt{BBox}(\mathcal{M})\|$, where $\|\mathtt{BBox}(\mathcal{M})\|$ is the length of the bounding box diagonal of $\mathcal{M}$. The desired Gau{\ss} map distance (angle between the unit vertex normals) between consecutive control points is simply $k_N$.
A target curve $\mathbf{C}_0, \ldots, \mathbf{C}_{n-1}$ starting at vertex $\mathbf{v}_{i_0}$ of the mesh is generated incrementally for $i>0$ as:
\begin{equation}
\mathbf{C}_i = \argmin_{\mathbf{v}\in V'} \ \big(d_G(\mathbf{C}_{i-1}, \mathbf{v}) - D_G\big)^2 + \big(d_N(\mathbf{C}_{i-1}, \mathbf{v}) - k_N\big)^2,
\end{equation}
where $d_G$ and $d_N$ compute the geodesic and normal distance between two points on $\mathcal{M}$, and $V'\subset V$ contains only those vertices of $\mathcal{M}$ whose geodesic distance from $\mathbf{C}_0, \ldots, \mathbf{C}_{i-1}$ is at least $D_G/2$.
The restricted subset of vertices conveniently helps prevent (but doesn't fully avoid) self-intersecting or nearly self-intersecting curves.
Curves with complex self-intersections are less important practically, and can be particularly confusing for the curve re-creation task.
All our target curve samples were generated using $k_G\in[0.05, 0.25]$, $k_N\in[\pi/6, 5\pi/12]$, $n=6$, and a randomly chosen $i_0$.
The curves were manually inspected for self-intersections, and infringing curves rejected.
We then defined keypoints on the target curves as follows: curve endpoints were chosen as keypoints;
followed by greedily picking extrema of geodesic curvature, while ensuring that the arclength distance between any two consecutive keypoints was at least 3cm;
and concluding the procedure when the maximum arclength distance between any consecutive keypoints was below 15cm. Our target curves had between 4--9 keypoints (including endpoints).
\subsection{Experiment Design}
\label{ref:expDesign}
The main variable studied in the experiment was \emph{Projection method}---\emph{spraycan} vs. \emph{mimicry}---realized as a within-subjects variable.
The order of methods was counterbalanced between participants.
For each method, participants were exposed to all the six objects.
Object order was fixed as torus, cube, trebol, bunny, hand, and fertility, based on our personal judgment of drawing difficulty.
The torus was used as a tutorial, where participants had access to additional instructions visible in the scene and their strokes were not utilized for analysis.
For each object, the order of the 10 target strokes was randomized.
The first five were used for the tracing curves task, while the remaining five were used for re-creating curves.
The target curve for the first tracing task was repeated after the five unique curves, to gauge user consistency and learning effects. A similar repetition
was used for curve re-creation.
Participants thus performed 12 curve drawing tasks per object, leading to a total of $12 \times 5$ (objects) $\times\ 2$ (projections) $= 120$ strokes per participant.
Owing to the COVID-19 physical distancing guidelines, the study was conducted in the wild, on participants' personal VR equipment at their homes. A 15-minute instruction video introduced the study tasks and the two projection methods.
Participants then filled out a consent form and a questionnaire to collect demographic information.
This was followed by them testing the first projection method and filling out a questionnaire to express their subjective opinions of the method.
They then tested the second method, followed by a similar questionnaire, and questions involving subjective comparisons between the two methods.
Participants were required to take a break after testing the first method, and were also encouraged to take breaks after drawing on the first three shapes for each method. The study took approximately an hour, including the questionnaires.
\subsection{Participants}
\label{participants}
Twenty participants (5 female) aged 21--47 from five countries participated in the study.
All but one were right-handed.
Participants self-reported a diverse range of artistic abilities (min. 1, max. 5, median 3 on a 1--5 scale), and had varying degrees of VR experience, ranging from below 1 year to over 5 years.
Thirteen participants had a technical computer graphics or HCI background, while ten had experience with creative tools in VR, with one reporting professional usage.
Participants were paid $\approx22$ USD as a gift card.
\subsection{Apparatus}
\label{sec:apparatus}
As the study was conducted on personal VR setups, a variety of commercial VR devices were utilized---Oculus Rift, Rift S, and Quest using Link cable, HTC Vive and Vive Pro, Valve Index, and Samsung Odyssey using Windows Mixed Reality.
All but one participant used a standing setup allowing them to freely move around.
\subsection{Procedure}
\label{sec:procedure}
Before each trial, participants could use the ``grab'' button on their controller (in the dominant hand) to grab the mesh to position and orient it as desired.
The trial started as soon as the participant started to draw by pressing the ``main trigger'' on their dominant hand controller.
This action disabled the grabbing interaction---participants could not draw and move the object simultaneously.
As noted earlier, for curve re-creation tasks, this had the additional effect of hiding the target curve, but leaving keypoints visible.
\subsection{Data Processing and Filtering}
\label{sec:dataProcess}
We formulated three criteria to filter out meaningless user strokes:\\
{\it Short Curves:} we ignore projected curves $\mathcal{Q}$ that are too short as compared to the length of the target curves $\mathcal{X}$ (conservatively curves less than half as long as the target curve). While it is possible that the user stopped drawing mid-way out of frustration, we found it was more likely that they prematurely released the controller trigger by accident.
Both curve lengths are computed in $\mathbb{R}^3$ for efficiency.\\
{\it Stroke Noise:} we ignore strokes for which the mid-air stroke is too noisy.
Specifically, mid-air strokes with distant consecutive points
($\exists\ i\ \text{s.t.}\ \|\mathbf{p}_i - \mathbf{p}_{i-1}\| > 5\text{cm}$) are rejected.\\
{\it Inverted Strokes:} while we labelled keypoints with numbers and marked start and end points in green and red (Figure~\ref{fig:studyTasks}), some users occasionally drew the target curve in reverse.
The motion to draw a curve in reverse is not symmetric, and such curves are thus rejected. We detect inverted strokes by look at the indices $i_0, i_1, \ldots, i_l$ of the points in $\mathcal{Q}$ which are closest to the keypoints $\mathbf{x}_{k_0}, \mathbf{x}_{k_1}, \ldots, \mathbf{x}_{k_l}$ of $\mathcal{X}$. Ideally, the sequence $i_0,\ldots, i_l$ should have no inversions, i.e., $\forall\ 0\le j<k\le l,\ i_j \le i_k$; and maximum $l(l+1)/2$ inversions, if $\mathcal{Q}$ is aligned in reverse with $\mathcal{X}$.
We consider curves $\mathcal{Q}$ with more than $l(l+1)/4$ (half the maximum) inversions, to be inadvertently inverted and reject them.
We compute distances to the keypoints in $\mathbb{R}^3$ for efficiency.
Despite conducting our experiment remotely without supervision, we found that 95.6\% of the strokes satisfied our criteria and could be utilized for analysis.
For comparisons between $\pi_{spray}$ and $\pi_{mimicry}$, we reject curve pairs where either curve did not satisfy the quality criteria.
Out of 1200 curve pairs (2400 total strokes), 1103 (91.9\%) satisfied the quality criteria and were used for analysis, including 564 pairs for the curve re-creation task and 539 for the tracing task.
\setlength{\tabcolsep}{3pt}
\begin{table}[tbh]
\centering
\caption{Quantitative results (mean $\pm$ std-dev.) of the comparisons between \emph{mimicry} and \emph{spraycan} projection.
All measures are analyzed using Wilcoxon signed-rank tests, lower values are better, and significantly better values ($p<.05$) are shown in \textbf{boldface}.
Accuracy, aesthetic, and physical effort measures are shown with \textcolor{tablegreen}{green}, \textcolor{tablered}{red}, and \textcolor{tableblue}{blue} backgrounds, respectively.
}
\label{tbl:quantitativeResults}
\begin{tabular}{@{}lccrr@{}}
\midrule
\multicolumn{5}{@{}c@{}}{\large{\textsc{Tracing}}}\\
\midrule
\textbf{Measure\hspace{-1ex}} & \textbf{Spraycan} & \textbf{Mimicry} & \textbf{$p$-value} & \textbf{$z$-stat} \\
\rowcolor{tablegreen!30}
$D_{ep}$ & 2.31 $\pm$ 2.64 mm & \textbf{1.13 $\pm$ 1.11 mm} & <.001 & 8.36 \\
\rowcolor{tablegreen!15}
$D_{sym}$ & 0.64 $\pm$ 0.66 mm & 0.56 $\pm$ 0.44 mm & >.05 & -0.09 \\
\rowcolor{tablered!30}
$K_E$ & 280 $\pm$ 262 rad/m & \textbf{174 $\pm$ 162 rad/m} & <.001 & 15.59 \\
\rowcolor{tablered!15}
$K_g$ & 249 $\pm$ 245 rad/m & \textbf{152 $\pm$ 157 rad/m} & <.001 & 15.42 \\
\rowcolor{tablered!30}
$F_g$ & 394 $\pm$ 413 rad/m & \textbf{248 $\pm$ 285 rad/m} & <.001 & 14.82 \\
\rowcolor{tableblue!15}
$T_h$ & 0.81 $\pm$ 0.70 & \textbf{0.58 $\pm$ 0.40 } & <.001 & 7.93 \\
\rowcolor{tableblue!30}
$R_h$ & 1.63 $\pm$ 2.18 rad/m & \textbf{1.18 $\pm$ 1.63 rad/m} & <.001 & 4.82 \\
\rowcolor{tableblue!15}
$T_c$ & \textbf{1.05 $\pm$ 0.36 } & 1.10 $\pm$ 0.29 & <.001 & -3.36 \\
\rowcolor{tableblue!30}
$R_c$ & 5.12 $\pm$ 5.88 rad/m & \textbf{3.79 $\pm$ 4.84 rad/m} & <.001 & 5.51 \\
\rowcolor{tableblue!15}
$\tau$ & \textbf{4.69 $\pm$ 1.85 s} & 5.29 $\pm$ 2.17 s & <.001 & -7.32 \\
\midrule
\multicolumn{5}{@{}c@{}}{\large{\textsc{Memory}}}\\
\midrule
\textbf{Measure\hspace{-1ex}} & \textbf{Spraycan} & \textbf{Mimicry} & \textbf{$p$-value} & \textbf{$z$-stat} \\
\rowcolor{tablegreen!30}
$D_{ep}$ & 2.34 $\pm$ 2.49 mm & \textbf{2.24 $\pm$ 23.32 mm} & <.001 & 8.63 \\
\rowcolor{tablegreen!15}
$D_{sym}$ & 0.75 $\pm$ 0.65 mm & 1.12 $\pm$ 11.51 mm & >.05 & 0.55 \\
\rowcolor{tablered!30}
$K_E$ & 254 $\pm$ 236 rad/m & \textbf{155 $\pm$ 127 rad/m} & <.001 & 14.70 \\
\rowcolor{tablered!15}
$K_g$ & 223 $\pm$ 219 rad/m & \textbf{132 $\pm$ 123 rad/m} & <.001 & 14.95 \\
\rowcolor{tablered!30}
$F_g$ & 348 $\pm$ 371 rad/m & \textbf{215 $\pm$ 227 rad/m} & <.001 & 14.11 \\
\rowcolor{tableblue!15}
$T_h$ & 0.72 $\pm$ 0.54 & \textbf{0.54 $\pm$ 0.35 } & <.001 & 6.78 \\
\rowcolor{tableblue!30}
$R_h$ & 1.50 $\pm$ 2.19 rad/m & \textbf{1.32 $\pm$ 1.99 rad/m} & .002 & 3.07 \\
\rowcolor{tableblue!15}
$T_c$ & \textbf{1.05 $\pm$ 0.37 } & 1.11 $\pm$ 0.23 & <.001 & -5.94 \\
\rowcolor{tableblue!30}
$R_c$ & 5.23 $\pm$ 6.36 rad/m & \textbf{3.63 $\pm$ 5.13 rad/m} & <.001 & 4.00 \\
\rowcolor{tableblue!15}
$\tau$ & \textbf{4.33 $\pm$ 1.57 s} & 4.92 $\pm$ 1.89 s & <.001 & -7.12 \\
\end{tabular}
\end{table}
\setlength{\tabcolsep}{6pt}
\subsection{Quantitative Analysis}
\label{sec:quantitative}
We define 10 different statistical measures (Table~\ref{tbl:quantitativeResults}) to compare $\pi_{spray}$ and $\pi_{mimicry}$ curves in terms of their accuracy, aesthetic, and effort in curve creation.
We consistently use the non-parametric Wilcoxon signed rank test for all quantitative measures instead of a parametric test such as the paired $t$-test,
since the recorded data for none of our measures was normally distributed (normality hypothesis rejected via the Kolmogorov-Smirnov test, $p<.005$).
\subsubsection{Curve Accuracy}
\label{sec:strokeAccuracy}
Accuracy is computed using two measures of distance between points on the projected curve $\mathcal{Q}$ and target curve $\mathcal{X}$. Both curves are densely re-sampled using $m=101$ sample points equi-spaced by arc-length.
Given $\mathcal{Q} = {\mathbf{q}_0, \ldots, \mathbf{q}_{m-1}}$ and $\mathcal{X} = {\mathbf{x}_0, \ldots, \mathbf{x}_{m-1}}$,
we compute the \emph{average equi-parameter distance} $D_{ep}$ as
\begin{equation}
\label{eq:distanceEquiparameter}
D_{ep}(\mathcal{Q}) = \frac1m \sum_{i=0}^{m-1} d_E\left( \mathbf{q}_i, \mathbf{x}_i \right) \text{,}
\end{equation}
where $d_E$ computes the Euclidean distance between two points in $\mathbb{R}^3$.
We also compute the \emph{average symmetric distance} $D_{sym}$ as
\begin{equation*}
\label{eq:distanceSymmetric}
D_{sym}(\mathcal{Q}) = \frac1{2m} \sum_{i=0}^{m-1}\left( \min_{\mathbf{x}\in X} d_E\left( \mathbf{q}_i, \mathbf{x} \right) \right) +
\frac1{2m} \sum_{i=0}^{m-1}\left( \min_{\mathbf{q}\in Q} d_E\left( \mathbf{q}, \mathbf{x}_i \right) \right)
\end{equation*}
In other words, $D_{ep}$ computes the distance between corresponding points on the two curves and $D_{sym}$ computes the average minimum distance from each point on one curve to the other curve.
For both tracing and re-creation tasks, $D_{ep}$ indicated that \emph{mimicry} produced significantly better results than \emph{spraycan} (see Table~\ref{tbl:quantitativeResults}, Figure~\ref{fig:teaser}c,~\ref{fig:smoothnessResult}).
The $D_{sym}$ difference was not statistically significant, evidenced by users correcting their strokes to stay close to the intended target curve (at the expense of curve aesthetic).
\subsubsection{Curve Aesthetic}
\label{sec:strokeQuality}
\begin{figure}
\centering
\begin{subfigure}{\linewidth}
\centering
\includegraphics[width=0.44\linewidth]{quant_gkmean_tracing}
\includegraphics[width=0.44\linewidth]{quant_gkmean_memory}
\caption{Normalized geodesic curvature $K_g$.}
\end{subfigure}
\begin{subfigure}{\linewidth}
\centering
\includegraphics[width=0.44\linewidth]{quant_gfair_tracing}
\includegraphics[width=0.44\linewidth]{quant_gfair_memory}
\caption{Normalized fairness deficiency $F_g$.}
\end{subfigure}
\begin{subfigure}{\linewidth}
\centering
\includegraphics[width=0.88\linewidth]{example_strokes}
\caption{Example strokes, \textcolor{matlaborange}{orange} points in (a, b) above.}
\end{subfigure}
\caption{Curvature measures (a,b) indicate that \emph{mimicry} produces significantly smoother and fairer curves than \emph{spraycan} for both tracing (left) and re-creating tasks (right). Pairwise comparison plots between \emph{mimicry} (y-axis) and \emph{spraycan} (x-axis), favour \emph{mimicry} for the vast majority of points (points below the $y=x$ line). A linear regression fit (on the log plots) is shown as a dashed line.
Example curve pairs (\textcolor{matlaborange}{orange} points) for curve tracing (left) and re-creating (right) are also shown with the target curve $\mathcal{X}$ shown in gray (c).}
\label{fig:smoothnessResult}
\end{figure}
For most design applications, jagged projected curves, even if geometrically quite accurate, are aesthetically undesirable~\cite{mccrae2008sketching}.
Curvature-based measures are typically used to measure fairness of curves. We report three such measures of curve aesthetic for the projected curve $\mathcal{Q}$. We note that the smoothness quality of the user stroke $\mathcal{P}$, was similar to $\mathcal{Q}$ and significantly poorer than the target curve $\mathcal{X}$. This is expected since drawing in mid-air smoothly and with precision is difficult, and such strokes are usually neatened post-hoc \cite{arora2018symbiosis}. We therefore avoid comparisons to the target curve and simply report three aesthetic measures for a projected curve $\mathcal{Q}=\mathbf{q}_0,\ldots, \mathbf{q}_{n-1}$. We first refine $\mathcal{Q}$ by computing the exact geodesic on $\mathcal{M}$ between consecutive points of $\mathcal{Q}$ ~\cite{surazhsky2005fast}, to create $\widehat{\mathcal{Q}}$ with points $\widehat{\mathbf{q}}_0, \ldots, \widehat{\mathbf{q}}_{k-1}$, $k\ge n$. We choose to normalize our curvature measures using $L_{\mathcal{X}}$, the length of the corresponding target stroke $\mathcal{X}$.
The \emph{normalized Euclidean curvature} for $\mathcal{Q}$ is defined as
\begin{equation}
\label{eq:euclideanCurvature}
K_E(\mathcal{Q}) = \frac1{L_{\mathcal{X}}} \sum_{i=1}^{k-1} \theta_i
\end{equation}
where $\theta_i$ is the angle between the two segments of $\widehat{\mathcal{Q}}$ incident on $\widehat{\mathbf{q}}_i$.
Thus, $K_E$ is the total discrete curvature of $\widehat{\mathcal{Q}}$, normalized by the target curve length.
Since $\widehat{\mathcal{Q}}$ is embedded in $\mathcal{M}$, we can also compute discrete \emph{geodesic} curvature, computed as the deviation from the straightest geodesic for a curve on surface. Using a signed $\theta^g_i$ defined at each point $\widehat{\mathbf{q}}_i$ via Polthier and Schmies's definition~\shortcite{polthier2006straightest}, we compute \emph{normalized geodesic curvature} as
\begin{equation}
\label{eq:geodesicCurvature}
K_g(\mathcal{Q}) = \frac1{L_{\mathcal{X}}} \sum_{i=1}^{k-1} |\theta_i^g|\text{.}
\end{equation}
Finally, we define fairness~\cite{arora2017experimental, mccrae2008sketching} as a first-order variation in geodesic curvature, thus defining the \emph{normalized fairness deficiency} as
\begin{equation}
\label{eq:fairness}
F_g(\mathcal{Q}) = \frac1{L_{\mathcal{X}}} \sum_{i=2}^{m-1} |\theta_i^g - \theta_{i-1}^g|\text{,}
\end{equation}
For all three measures, a lower value indicates a smoother, pleasing, curve.
Wilcoxon signed-rank tests on all three measures indicated that {\it mimicry} produced significantly smoother and better curves than {\it spraycan} (Table~\ref{tbl:quantitativeResults}).
\subsubsection{Physical Effort}
\label{sec:effortQuantitative}
Quantitatively, we use the amount of head (HMD) and hand (controller) movement, and stroke \emph{execution time} $\tau$, as proxies for physical effort.
For head and hand translation, we first filter the position data with a Gaussian-weighted moving average filter with $\sigma=20\text{ms}$.
We then define \emph{normalized head/controller translation} $T_h$ and $T_c$ as the length of the poly-line defined by the filtered head/controller positions normalized by the length of the target curve $L_\mathcal{X}$.
An important ergonomic measure is the amount of head/hand rotation required to draw the mid-air stroke.
We first de-noise or filter the forward and up vectors of the head/controller frame, using the same filter as for positional data.
We then re-orthogonalize the frames and compute the length of the curve defined by the filtered orientations in $\mathrm{SO(3)}$, using the angle between consecutive orientation data-points.
We define \emph{normalized head/controller rotation} $R_h$ and $R_c$ as its orientation curve length, normalized by $L_\mathcal{X}$.
Table~\ref{tbl:quantitativeResults} summarizes the physical effort measures.
We observe lower controller translation (effect size $\approx 5\%$) and execution time (effect size $\approx 12\%$) in favour of {\it spraycan}; lower head translation and orientation (effect sizes $\approx 36\%, 26\%$) in favour of {\it mimicry}. Noteworthy, is the significantly reduced controller rotation using {\it mimicry}, with {\it spraycan} unsurprisingly requiring $35\%$ (tracing) and 44\% (re-creating) more hand rotation from the user.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{qual_difficulty}
\caption{Perceived difficulty of drawing for the six 3D shapes in the study.}
\label{fig:difficulty}
\end{figure}
\begin{figure}
\centering
\begin{subfigure}{\linewidth}
\centering
\includegraphics[width=\linewidth]{qual_accuracy}
\caption{Perceived accuracy.}
\end{subfigure}
\begin{subfigure}{\linewidth}
\centering
\includegraphics[width=\linewidth]{qual_smoothness}
\caption{Perceived smoothness.}
\end{subfigure}
\begin{subfigure}{\linewidth}
\centering
\includegraphics[width=\linewidth]{qual_effort}
\caption{Physical and mental effort ratings.}
\end{subfigure}
\caption{Participants perceived \emph{mimicry} to be better than \emph{spraycan} in terms of accuracy (a), curve aesthetic (b) and user effort (c).}
\label{fig:qualitativeComparison}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{preference_understand}
\caption{Participants stated understanding \emph{spraycan} projection better (left); 17/20 users stated an overall preference for \emph{mimicry} over \emph{spraycan} (right). }
\label{fig:prefenceUnderstanding}
\end{figure}
\subsubsection{Quantifying Users' Tendency to Mimic}
\label{Sec:mimicriness}
The study also provided an opportunity to test if the users actually tended to mimic their intended curve $\mathcal{X}$ in the mid-air stroke $\mathcal{P}$. To quantify the ``mimcriness'' of a stroke, we subsample $\mathcal{P}$ and $\mathcal{X}$ into $m$ points as in \S~\ref{sec:strokeAccuracy}, use the correspondence as in Eq.~\ref{eq:distanceEquiparameter} and look at the variation in the distance (distance between the closest pair of corresponding points subtracted from that of the farthest pair) as a percentage of the target length $L_\mathcal{X}$. We call this measure the \emph{mimicry violation} of a stroke. Intuitively, the lower the \emph{mimicry violation}, the closer the stroke $\mathcal{P}$ is to being a perfect mimicry of $\mathcal{X}$, going to zero if it is a precise translation of $\mathcal{X}$. Notably, users depicted very similar trends to mimic for both the techniques---with 86\% (\emph{mimicry}), 80\% (\emph{spraycan}) strokes exhibiting \emph{mimicry violation} below 25\% of $L_\mathcal{X}$, and 71\%, 66\% below 20\% of $L_\mathcal{X}$---suggesting that mimicry is indeed a natural tendency.
\subsubsection{Consistency across Repeated Strokes}
\label{sec:repeat}
Recall that users repeated 2 of the 10 strokes per shape for both the techniques. To analyze consistency across the repeated strokes, we compared the values of the stroke accuracy measure $D_{eq}$ and the aesthetic measure $F_g$ between the original stroke and the corresponding repeated stroke. Specifically, we measured the relative change $|f(i) - f(i')|/f(i)$, where $(i, i')$ is a pair of original and repeated strokes, and $f(\cdot)$ is either $D_{eq}$ or $F_g$. Users were fairly consistent across both the techniques, with the average consistency for $D_{eq}$ being 35.4\% for \emph{mimicry} and 36.8\% for \emph{spraycan}, while for $F_g$, it was 36.5\% and 34.1\%, respectively. Note that the averages were computed after removing extreme outliers outside the $5\sigma$ threshold.
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{complex_curves}
\caption{Gallery of free-form curves in \textcolor{matlabred}{\textbf{red}}, drawn using \emph{mimicry}. From left to right, tracing geometric features on the bunny, smooth maze-like curves on the cube, maze-like curve with sharp corners and a spiral on the trebol, and artistic tattoo motifs on the hand. Some mid-air strokes (\textbf{black}) are hidden for clarity.}
\label{fig:curveGallery}
\end{figure*}
\subsection{Qualitative Analysis}
\label{sec:qualitative}
The mid- and post-study questionnaires elicited qualitative responses from participants on their perceived difficulty of drawing, curve accuracy and smoothness, mental and physical effort, understanding of the projection methods, and overall method of preference.
Participants rated their perceived difficulty in drawing on the 6 study objects (Figure~\ref{fig:difficulty}), validating our ordering of shapes in the experiment based on expected drawing difficulty.
Accuracy, smoothness, physical/mental effort responses were collected via 5-point Likert scales. We consistently order the choices from 1 (worst) to 5 (best) in terms of user experience in Figure~\ref{fig:qualitativeComparison}, and reported median ($M$) scores here.
{\it Mimicry} was perceived to be a more accurate projection method (tracing, re-creating $M=3,3.5$) compared to {\it spraycan} ($M=2,2$), with 9 participants perceiving their traced curves to be either \emph{very accurate} or \emph{somewhat accurate} with {\it mimicry} (compared to 2 for {\it spraycan}) (Figure~\ref{fig:qualitativeComparison}a).
User perception of stroke smoothness was also consistent with quantitative results, with {\it mimicry} (tracing, re-creating $M=4,4$) clearly outperforming {\it spraycan} (tracing, re-creating $M=1,2$) (Figure~\ref{fig:qualitativeComparison}b).
Lastly, with no need for controller rotation, {\it mimicry} ($M=3$) was perceived as less physically demanding than {\it spraycan} ($M=2$), as expected (Figure~\ref{fig:qualitativeComparison}c).
The response to understanding and mental effort was more complex. {\it Spraycan}, with its physical analogy and mathematically precise definition was clearly understood by all 20 participants (17 very well, 3 somewhat)
(Figure~\ref{fig:prefenceUnderstanding}a). {\it Mimicry}, conveyed as ``drawing a mid-air stroke on or near the object as similar in shape as possible to the intended projection'', was less clear to users (7 very well, 11 somewhat, 3 not at all). Despite not understanding the method, the 3 participants were able to create curves that were both accurate and smooth. Further, users perceived {\it mimicry} ($M=2.5$) as less cognitively taxing than {\it spraycan} ($M=2$) (Figure~\ref{fig:qualitativeComparison}c). We believe this may be because users were less prone to consciously controlling their stroke direction and rather focused on drawing. The tendency to mimic may have thus manifested sub-consciously, as we had observed in pilot testing.
The most important qualitative question was user preference (Figure~\ref{fig:prefenceUnderstanding}b).
$85\%$ of the 20 participants preferred {\it mimicry} (10 highly preferred, 7 somewhat preferred). The remaining users were neutral (1/20) or somewhat preferred {\it spraycan} (2/20).
\begin{figure}[b!]
\centering
\includegraphics[width=\linewidth]{texture}
\caption{{\it Mimicry} used to interactively paint textures on 3D objects.
}
\label{fig:texturing}
\end{figure}
\subsection{Participant Feedback}
We also asked participants to elaborate on their stated preferences and ratings.
Participants (\emph{P4,8,16,17}) noted discontinuous \pquote{jumps} caused by {\it spraycan}, and felt the continuity guarantee of {\it mimicry}: \pquote[P6]{seemed to deal with the types of jitter and inaccuracy VR setups are prone to better}; \pquote[P9]{could stabilize my drawing}.
\emph{P9,15} felt that {\it mimicry} projection was smoothing their strokes (no smoothing was employed): we believe this may be the effect of noise and inadvertent controller rotation, which {\it mimicry} ignores, but can cause
large variations with {\it spraycan}, perceived as curve smoothing.
Some participants (\emph{P4,17}) felt that rotating the hand smoothly while drawing was difficult, while others missed the {\it spraycan} ability to simply use hand rotation to sweep out long projected curves from a distance (\emph{P2,7}).
Participants commented on physical effort: \pquote[P4]{Mimicry method seemed to required [sic] much less head movement, hand rotation and mental planning}.
Participants appreciated the anchored control of {\it mimicry} in high-curvature regions (\emph{P1,2,4,8}) also noting that with {\it spraycan}, \pquote[P1]{the curvature of the surface could completely mess up my stroke}.
Some participants did feel that {\it spraycan} could be preferable when drawing on near-flat regions of the mesh (\emph{P3,14,19,20}).
Finally, participants who preferred spraycan felt that mimicry required more thinking: \pquote[P3]{with mimicry, there was extra mental effort needed to predict where the line would go on each movement}, or because mimicry felt \pquote[P7]{unintuitive} due to their prior experience using a {\it spraycan} technique. Some who preferred {\it mimicry} found it difficult to use initially, but felt it got easier over the course of the experiment (\emph{P4,17}).
\subsection{Texture Painting}
\emph{Texture Painting:} Figures~\ref{fig:teaser}e, \ref{fig:texturing} show examples of textures painted in VR using {\it mimicry}.
The long, smooth, wraparound curves on the torus, are especially hard to draw with 2D interfaces.
Our implementation uses Discrete Exponential Maps (DEM)~\cite{schmidt2006discrete} to compute a dynamic local parametrization around each projected point $\mathbf{q}_i$, to create brush strokes or geometric stamps on the object.
\emph{Mesh Segmentation: }
Figures~\ref{fig:teaser}e and \ref{fig:segmentation} show {\it mimicry} used for interactive segmentation in VR.
In our implementation users draw an almost-closed curve $\mathcal{Q} = \{\mathbf{q}_0,\ldots,\mathbf{q}_{n-1}\}$ on the object using {\it mimcry}. We snap points $\mathbf{q}_i$ to their nearest mesh vertex, and use Dijkstra's shortest path to connect consecutive vertices, and to close the cycle of vertices. A mesh region is selected or segmented using mesh faces partitioned by these cycles that are easy to draw in AR/VR, but often require view changes and multiple strokes in 2D.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{segmentation}
\caption{Interactive segmentation by drawing curves onto torus and bunny meshes. Each segmented portion is shown with a unique colour.}
\label{fig:segmentation}
\end{figure}
\emph{Vector Field Design: }
Vector fields on meshes are commonly used for
texture synthesis~\cite{turk2001texture}, guiding fluid simulations~\cite{stam2003flows}, and non-photorealistic rendering~\cite{hertzmann2003illustrating}.
We use {\it mimicry} curves as soft constraints to guide the vector field generation of Fisher et al.~\shortcite{fisher2007design}.
Figure~\ref{fig:vectorFieldDesign} shows example vector fields, visualized using Line Integral Convolutions~\cite{cabral1993imaging} in the texture domain.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{vector_field_design}
\caption{Smooth {\it mimicry} curves (\textcolor{matlabred}{\textbf{red}}) provide constraints for vector field design~\cite{fisher2007design}, which we visualize via Line Integral Convolutions~\cite{cabral1993imaging}.}
\label{fig:vectorFieldDesign}
\end{figure}
\subsection{Mimicry Limitations}
\paragraph{Mimicry Limitations}
Our lack of a concise mathematical definition of observed stroke mimicry, makes it harder to precisely communicate it to users. While a precise mathematical formulation may exist, conveying it to non-technical users can still be a challenging task.
{\it Mimicry} ignores controller orientation, producing smoother strokes with less effort, but can give participants a reduced sense of sketch control (\emph{P2,3,6}). We hypothesize that the reduced sense of control is in part due to the tendency for anchored smooth-closest-point to shorten the user stroke upon projection, sometimes creating a feeling of lag. {\it Spraycan} like techniques in contrast, have a sense of amplified immediacy, and the explicit ability to make lagging curves catch-up by rotating a controller in place.
\paragraph{Future work}
Our goal was to develop a general real-time inked projection with minimal stroke context via anchoring.
Optimizing the method to account for the entire partially projected stroke may improve the projection quality. Relaxing the restriction of real-time inking would allow techniques such as spline fitting and global optimization that can account for the entire user stroke and geometric features of the target object. Local parametrizations such as DEM (\S~\ref{sec:applications}) can be used to incrementally grow or shrink the projected curve, so it does not lag the user stroke. Hybrid projections leveraging both proximity and raycasting are also subject to future work.
On the interactive side, we did experiment with feedback to encourage users to draw closer to a 3D object. For example, we tried varying the appearance of the line connecting the controller to the projected point based on line length; or providing aural/haptic feedback if the controller got further than a certain distance from the object. While these techniques can help users in specific drawing or tracing tasks, we found them to be distracting and harmful to stroke quality for general stroke projection. Bimanual interaction in VR, such as rotating the shape with one hand while drawing on it with the other (suggested by \emph{P3,19}), can also be explored.
Perhaps the most exciting area of future work is employing data-driven techniques to infer the user-intended projection, perhaps customized to the drawing style of individual users. Our study code and data will be made publicly available to aid in such endeavours.
In summary, this paper presents early research on processing and projection of mid-air strokes drawn on and around 3D objects, that we hope will inspire further work and applications in AR/VR.
\begin{acks}
We are thankful to Michelle Lei for developing the initial implementation of the context-free techniques, and to Jiannan Li and Debanjana Kundu for helping pilot our methods.
We also thank various 3D model creators and repositories for the models we utilized: Stanford bunny model courtesy of the Stanford 3D Scanning Repository, trebol model provided by Shao et al.\shortcite{shao2012crossshade}, fertility model courtesy the Aim@Shape repository, hand model provided by \texttt{Jeffo89} on \url{turbosquid.com}, and cup model (Figure~\ref{fig:2DProblems}) provided by Daniel Noree on \url{thingiverse.com} under a CC BY 4.0 license.
\end{acks}
\section{Introduction}
\label{sec:intro}
\input{1_intro}
\section{Related Work}
\label{sec:related}
\input{2_related}
\section{Projecting Strokes on 3D Objects}
\label{sec:drawing}
\input{3_drawing}
\section{Anchored Stroke Projection}
\label{sec:historical}
\input{4_historical}
\section{User Study}
\label{sec:study}
\input{5_study}
\section{Study Results and Discussion}
\label{sec:results}
\input{6_results}
\section{Applications}
\label{sec:applications}
\input{7_applications}
\section{Conclusion}
\label{sec:conclusion}
\input{8_conclude}
\bibliographystyle{ACM-Reference-Format}
|
2024-02-18T23:40:24.778Z
|
2020-09-22T02:01:44.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2009.09029","language":"en","timestamp":1600740104000,"url":"https:\/\/arxiv.org\/abs\/2009.09029","yymm":"2009"}
|
proofpile-arXiv_000-10082
|
{"provenance":"002.jsonl.gz:10083"}
| null | null |
\section{Introduction}
\label{sec:intro}
\noindent
One of the major challenges faced by an advanced-technology node IC designer is
the overhead of large run-times of analysis tools. Fast and accurate analysis tools that aid quick design
turn-around are particularly important for two critical, time-consuming simulations that are performed
several times during the design cycle:
\begin{itemize}
\item {\em Thermal analysis}, which checks the feasibility of a
placement/floorplan solution by computing on-chip temperature distributions in
order to check for temperature hot spots.
\item {\em IR drop analysis} in power distribution networks (PDNs), which
diagnoses the goodness of the PDN by determining voltage (IR) drops from the
power pads to the gates.
\end{itemize}
The underlying computational engines that form the crux of both analyses are
similar: both simulate networks of conductances and current/voltage
sources by solving a large system of equations of the form $G {\bf V} = {\bf J}$~\cite{Zhan08,Zhong05} with millions to billions of
variables. In modern
industry designs, a single full-chip temperature or IR drop simulation can take hours to several
hours.
Accelerating these analyses opens the door to optimizations in the design cycle that
iteratively invoke these engines under the hood.
The advent of machine learning (ML) has presented fast and fairly accurate solutions to these
problems~\cite{zhang18, juan12,
tan19, Lin18,incpird, powernet} which can successfully be used in early design cycle optimizations, operating within larger allowable error margins at these stages.
To the best of our knowledge, no published
work addresses full-chip ML-based thermal analysis: the existing literature
focuses on coarser-level thermal modeling at the system level~\cite{zhang18,
juan12, tan19}. For PDN analysis, the works in~\cite{Lin18, incpird} address
incremental analysis, and are not intended for full-chip
estimation. The work in~\cite{powernet} proposes
a convolutional neural network (CNN)-based implementation for full-chip IR drop
prediction, using cell-level power maps as features. However, it assumes
similar resistance from each cell to the power pads, which may not be valid for practical power
grids with irregular grid density. The analysis divides the chip into regions ({\it
tiles}), and the CNN operates on each tile and its near
neighbors. Selecting an appropriate tile and window size is nontrivial -- small windows could violate the principle of
locality~\cite{Chiprout04}, causing inaccuracies, while large windows could
result in large models with significant runtimes for training and inference.
Our approach bypasses
window size selection by providing the entire power map
as a feature, allowing ML to learn the window size for accurate estimation.
We translate static analysis problems to an image-to-image translation task and
dynamic analysis problems to video-to-video translation, where the inputs are
the power/current distributions and the required outputs are the temperature or
IR drop contours. For static analysis, we employ fully convolutional (FC) EDGe
networks for rapid and accurate thermal and IR drop analysis. FC EDGe networks
have proven to be very successful with image-related problems with 2-D
spatially distributed data~\cite{fcn, unet, mao16, segnet} when compared to
other networks that operate without spatial correlation
awareness. For transient analysis, we use long-short-term-memory (LSTM)
based EDGe networks that maintain memory of analyses at prior time steps.
\begin{figure}[tb]
\centering
\includegraphics[width=0.43\textwidth]{figs/enc-dec-arch.pdf}
\caption{Image-to-image translation using EDGe network.}
\label{fig:enc-dec-arch}
\vspace{-1.5em}
\end{figure}
Based on these concepts, this work proposes two novel ML-based analyzers: {\bf
ThermEDGe} for both \textit{full-chip} static and transient thermal analysis, and {\bf IREDGe} for
\textit{full-chip} static IR drop estimation. The fast inference times of ThermEDGe and IREDGe enable {\bf full-chip} thermal and IR drop analysis in milliseconds, as opposed to runtimes of several hours using
commercial tools. We obtain average error of
0.6\% and 0.008\% for ThermEDGe and IREDGe, respectively, over a range of testcases. {\em We will open-source our software.}
Fig.~\ref{fig:enc-dec-arch} shows a general top-level structure of an EDGe
network. It consists of two parts: (i) the encoder/downsampling path, which
captures global features of the 2-D distributions of power dissipation, and
produces a low-dimensional state space and (ii) the decoder/upsampling path,
which transforms the state space into the required detailed outputs
(temperature or IR drop contours).
The EDGe network is well-suited for PDN/thermal
analyses because: \\
{\em (a)} The convolutional nature of the encoder
{\em captures the dependence of both problems on the spatial distributions of power}.
Unlike CNNs, EDGe networks contain a decoder which acts as
a generator to convert the extracted power and PDN density features into
accurate high-dimensional temperature and IR drop contours across the chip. \\
{\em (b)} The trained EDGe network model for static analysis is
{\em chip-area-independent}: it only stores the weights of the convolutional kernel,
and the same filter can be applied to a chip of any size. The selection of the
network topology (convolution filter size, number of convolution layers) is
related to the expected sizes of the hotspots rather than the size of the chip:
these sizes are generally similar for a given application domain, technology,
and packaging choice. \\
{\em (c)} Unlike prior methods~\cite{powernet} that operate tile-by-tile, where finding the right tile and window size for accurate analysis
is challenging, {\em the choice of window size is treated as an ML
hyperparameter tuning problem} to decide the necessary amount of input spatial information.
\section{EDGe Network for PDN and Thermal Analysis}
\subsection{Problem formulations and data representation}
\label{sec:problem-def}
\noindent
This section presents the ML-based framework for ThermEDGe and IREDGe. The
first step is to extract an appropriate set of features from a standard
design-flow environment. The layout database provides the
locations of each instance and block in the layout, as
outlined in Fig.~\ref{fig:data-rep}(a). This may be combined with information
from a power analysis tool such as~\cite{voltus} (Fig.~\ref{fig:data-rep}(b))
that is used to build a 2-D spatial power map over the die area.
\begin{figure}[tb]
\centering
\includegraphics[width=8.8cm]{figs/data-rep.pdf}
\caption{Data representation: Mapping PDN and thermal analysis problems into
image-to-image translations tasks.}
\label{fig:data-rep}
\vspace{-1em}
\end{figure}
For thermal analysis using ThermEDGe, both the inputs and outputs are images
for the static case, and a sequence of images for the transient case. Each
input image shows a 2-D die power distribution (static) image, and each output
image is a temperature map across the die (Fig.~\ref{fig:data-rep}). For static PDN analysis, the output is an IR drop map across the full chip. However, in addition to the 2-D power distributions, IREDGe has two other inputs:
\noindent
(i)~\underline{\it A PDN density map}: This feature is generated by extracting the average PDN
pitch in each region of the chip. For example, when used in conjunction with the PDN styles in~\cite{jsingh,OpeNPDN}, where the chip uses regionwise uniform PDNs, the
average PDN density in each region, across all metal layers, is provided as an input (Fig.~\ref{fig:data-rep}(e)).
\noindent
(ii)~\underline{\it An effective distance to power pad}: This feature represents the equivalent distance from an instance to all power pads in the package. We compute the effective distance of each instance, $d_e$, to $N$ power pads on the chip as the harmonic sum of the distances to the pads:
\begin{equation}
d_e^{-1} = d_1^{-1} + d_2^{-1} + ...+d_N^{-1}
\label{eq:deffective}
\end{equation}
where $d_i$ is the distance of the $i^{th}$ power pad from the instance. Intuitively, the effective distance metric and the PDN density map together, represent the equivalent resistance between the instance and the pad. The equivalent resistance is a parallel combination of each path from the instance to the pad. We use distance to each pad as a proxy for the resistance in Eq.~\eqref{eq:deffective}. Fig.~\ref{fig:data-rep}(f) shows a typical ``checkerboard'' power pad layout for flip-chip packages~\cite{checkerboard1, checkerboard2}.
Temperature depends on
the ability of the package and system to conduct heat to the ambient, and IR drop depends on off-chip (e.g., package) parasitics. In this work, our focus is strictly on-chip, and
both ThermEDGe and IR-EDGe are trained for fixed models of a given technology, package, and system.
Next, we map these
problems to standard ML networks:
\begin{itemize}
\item
For static analysis, the problem formulations require a translation from an
input power image to an output image, both corresponding to contour maps over the
same die area, and we employ a {\em U-Net-based EDGe network}~\cite{unet}.
\item
The dynamic analysis problem requires the conversion of a sequence of input
power images, to a sequence of output images of temperature contours, and this problem is
addressed using an {\em LSTM-based EDGe network}~\cite{seq2seq}.
\end{itemize}
We describe these networks in the rest of this section.
\begin{figure}[htb]
\centering
\includegraphics[width=8.8cm]{figs/thermedge-arch.pdf}
\caption{U-Net-based EDGe network for static thermal and PDN analysis.}
\label{fig:thermedge}
\vspace{-1.5em}
\end{figure}
\subsection{U-Nets for static thermal and PDN analysis}
\label{sec:unet}
\subsubsection{Overview of U-Nets}
CNNs are successful in extracting 2-D spatial information for image
classification and image labeling tasks, which have low-dimensional outputs (class or label). For PDN and thermal analysis tasks, the required outputs
are high-dimensional distributions of IR drop and temperature
contour, where the dimensionality corresponds to the number of
pixels of the image and the number of pixels
is proportional to the size of the chip. This calls for a generator network
that can translate the extracted low-dimensional power and PDN features from a
CNN-like encoder back into high-dimensional representing the required output data.
Fig.~\ref{fig:thermedge} shows the
structure of the EDGe network used for static PDN and thermal analysis.
At the top level, it consists of two networks:
\noindent
(a)~\underline{\em Encoder/downsampling network} Like a CNN, the network utilizes a sequence of 2-D
convolution and max pooling layer pairs that extract key features from the
high-dimensional input feature set. The convolution operation performs a
weighted sum on a sliding window across the image~\cite{conv-deconv}, and the
max pooling layer reduces the dimension of the input data by extracting the
maximum value from a sliding window across the input image. In
Fig.~\ref{fig:thermedge}, the feature dimension is halved at each stage by each
layer pair, and after several such operations, an encoded, low-dimensional,
compressed representation of the input data is obtained. For this reason, the
encoder is also called the downsampling path: intuitively, downsampling
helps understand the {\em ``what"} (e.g., ``Does the image contain power or IR
hotspots?'') in the input image but tends to be imprecise with the {\em ``where"}
information (e.g., the precise locations of the hotspots). The latter is
recovered by the decoder stages.
\noindent
(b)~\underline{\em Decoder/upsampling network} Intuitively, the generative decoder is
responsible for retrieving the {\em ``where"} information that was lost during
downsampling, This distinguishes an EDGe network from its CNN counterpart. The
decoder is implemented using the transpose convolution~\cite{conv-deconv} and
upsampling layers. Upsampling layers are functionally the opposite of a
pooling layer, and increase the dimension of the input data matrix by replicating the rows and columns.
\subsubsection{Use of skip connections}
Static IR drop and temperature are strongly correlated to the input power --
a region with high power on the chip could potentially have an IR or temperature
hotspot in its vicinity. U-Nets~\cite{unet} utilize {\it skip} connections
between the downsampling and upsampling paths, as shown in
Fig.~\ref{fig:thermedge}. These connections take information from one layer
and incorporate it using a {\it concatenation} layer at a deeper stage skipping intermediate layers, and appends it to the embedding
along the z-dimension.
For IR analysis, skip connections combine the local power, PDN information, and power pad locations from the
downsampling path with the global power information from the upsampling path,
allowing the underlying input features to
and directly shuttle to the layers closer to the output, and are similarly helpful for thermal analysis. This helps
recover the fine-grained ({\em ``where"}) details that are lost in the encoding part
of the network (as stated before) during upsampling in the decoder for detailed temperature and IR
drop contours.
\subsubsection{Receptive fields in the encoder and decoder networks}
The characteristic of PDN and thermal analyses problems is that the IR drop and
temperature at each location depend on both the local and global power
information.
During convolution, by sliding averaging windows of an appropriate size
across the input power image, the network captures local spatially correlated
distributions. For capturing the larger global impact of power on temperature
and IR drop, max pooling layers are used after each convolution to
appropriately increase the size of the {\it receptive field} at each stage of
the network. The {\it receptive field} is defined as the region in the input
2-D space that affects a particular pixel, and it determines the impact
of the local, neighboring, and global features on PDN and thermal
analysis.
In a deep network, the value of each pixel feature is
affected by all of the other pixels in the receptive field at the previous convolution stage, with the largest contributions coming from pixels near the center
of the receptive field. Thus, each feature not only captures its receptive field in the input image, but also gives an exponentially higher weight to the middle of that region~\cite{receptive-field}. This matches
with our applications, where both thermal and IR maps for a pixel are most
affected by the features in the same pixel, and partially by features in
nearby pixels, with decreasing importance for those that are farther away.
The size of the receptive field at each stage in the network is determined by the convolutional filter size, number of convolutional
layers, max pooling filter sizes, and number of max pooling layers.
On both the encoder and decoder sides in Fig.~\ref{fig:thermedge}, we use three
stacked convolution layers, each followed by 2$\times$2 max-pooling to
extract the features from the power and PDN density images. The number of layers and filter sizes are determined based on the magnitude of the hotspot
size encountered during design iterations.
\subsection{LSTM-based EDGe network for transient thermal analysis}
Long short term memory (LSTM) based EDGe networks are a special kind of recurrent neural
network (RNN) that are known to be capable of learning long term dependencies
in data sequences, i.e., they have a memory component and are capable of learning
from past information in the sequence.
\begin{figure}[h]
\centering
\includegraphics[width=0.47\textwidth]{figs/lstm-edge.pdf}
\caption{LSTM-based EDGe network for transient analysis in ThermEDGe.}
\label{fig:lstm-edge}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=8.5cm]{figs/conv-lstm.pdf}
\caption{A fully connected LSTM cell (left) and a ConvLSTM cell (right).}
\label{fig:conv-lstm}
\end{figure}
For transient thermal analysis, the structure of ThermEDGe is shown in Fig.~\ref{fig:lstm-edge}. The core architecture
is an EDGe network, similar to the static analysis problem described in Section~\ref{sec:unet}, except that the
network uses additional LSTM cells to account for the time-varying component. The figure demonstrates the time-unrolled LSTM where input power
frames are passed to the network one frame at a time. The LSTM cell accounts
for the history of the power maps to generate the output temperature frames for all time steps. The network is used for sequence-to-sequence translation in transient thermal analysis, where the input is a set of time-varying power maps and the output is a set of time-varying temperature maps (Section~\ref{sec:problem-def}).
Similar to the static ThermEDGe network (Fig.~\ref{fig:thermedge}), the encoder consists of convolution and max pooling layers to downsample and extract critical local and global spatial information and the decoder consists of upsampling and transpose convolution layers to upsample the encoded output.
However, in addition, transient ThermEDGe has LSTM layers in both the encoding and decoding paths.
A standard LSTM cell is shown in Fig.~\ref{fig:conv-lstm} (left).
While the basic LSTM cell uses fully connected layers within each gate, our application
uses a variation of an LSTM cell called a convolutional LSTM
(ConvLSTM)~\cite{conv-lstm}, shown in Fig.~\ref{fig:conv-lstm} (right). In
this cell, the fully connected layers in each gate are replaced by convolution
layers that capture spatial information. Thus, the LSTM-based EDGe network
obtains a spatiotemporal view that enables accurate inference.
\section{ThermEDGe and IREDGe Model Training}
\noindent
We train the models that go into ThermEDGe and IREDGe to learn the temperature
and IR contours from the ``golden" commercial tool-generated or ground truth
data. We train ThermEdge using the full physics-based thermal simulations from
the Ansys-Icepak~\cite{icepak} simulator, incorporating off-chip thermal
dynamics from package and system thermal characteristics. IREDGe is trained
using static IR drop distribution from a PDN analyzer~\cite{pdnsim, voltus} for
various power, PDN density, and power pad distributions.
\subsection{Generating training data}
\label{sec:train}
\noindent
{\bf Static ThermEDGe and IREDGe}
A challenge we faced to evaluate our experiments is the dearth of public domain benchmarks that fit these applications. The IBM benchmarks~\cite{ibm}, are potential candidates for our applications, but they assume constant currents per region and represent an older technology node. Therefore, we generate our
dataset which comprises of 50 industry-relevant testcases, where each testcase represents industry-standard workloads for commercial designs implemented in a FinFET technology. The power images of size
34$\times$32 pixels, with each pixel representing the power/temperature a
250$\mu$m$\times$250$\mu$m tile on an 8.5mm$\times$8mm chip.\footnote{Note that although the temperature and power map work at this resolution, the actual simulation consists of millions of nodes; using fewer node (e.g., one node per pixel) is grossly insufficient for accuracy.} Our training is
specific to the resolution: for another image resolution, the model must be
retrained. We reiterate that although the training is performed on chips of fixed size, as
we show (Section~\ref{sec:results}), inference can be performed on a chip of any size as long as the
resolution remains the same.
For static ThermEDGe our training data is based on static Ansys-Icepak~\cite{icepak}
simulations of these 50 testcases. For IREDGe, we synthesize irregular PDNs of
varying densities for each dataset element using {\it PDN templates}, as
defined by OpeNPDN~\cite{OpeNPDN}. These templates are a set of PDN building
blocks, spanning multiple metal layers in a 14nm commercial FinFET technology, which vary in their metal
utilization. For our testcases, we use three templates (high, medium, and low
density) and divide the chip into nine regions. As outlined in Section~\ref{sec:problem-def}), we use a checkerboard pattern of power pads that vary in the bump pitch and offsets across the dataset.
The synthesized full-chip PDN, power pad locations, and power distributions are taken as inputs into
the IR analyzer~\cite{pdnsim} to obtain training data for IREDGe. For
each of the 50 testcases, we synthesize 10 patterns of PDN densities, and for each combination of combination of power and PDN distribution we synthesize 10 patterns of power pad distributions, creating a dataset with 5000 points.
\noindent
{\bf Transient ThermEDGe}
For the transient analysis problem, our training data is based on transient
Ansys-Icepak~\cite{icepak} simulations. The size of the chip is the same as that of the static ThermEDGe testcases.
For each testcase, we generate 45 time-step simulations that range from 0 to
3000s, with irregular time intervals from the thermal simulator. Each
simulation is expensive in terms of the time and memory
resources: one simulation of a 3000s time interval with 45
time-steps can take 4 hours with 2 million nodes.
Transient ThermEDGe is trained using constant time steps of 15s which enables
easy integration with existing LSTM architectures which have an implicit
assumption of uniformly distributed time steps, without requiring additional
features to account for the time.
The model is trained on 150 testcases with time-varying workloads as features, and their
time-varying temperature from Ansys-Icepak as labels.
\vspace{-1.0em}
\subsection{Model training}
\label{sec:training}
\noindent
For the static analysis problem, ThermEDGe and IREDGe use a
static power map as input and PDN density map (for IR analysis only) to predict
the corresponding temperature and IR drop contours. For the transient thermal
analysis problem, the input is a sequence of 200 power maps and the output is a
sequence of 200 temperature contours maps at a 15s time interval. The ML model
and training hyperparameters used for these models are listed in
Table~\ref{tbl:parameters}.
\begin{table}[h]
\centering
\caption{ThermEDGe and IREDGe ML hyperparameters}
\label{tbl:parameters}
\resizebox{\linewidth}{!}{%
\begin{tabular}{||l||l|l||l|l|l||}
\hhline{|t:===:t:===:t|}
\multicolumn{3}{||l||}{ML hyperparameters} & \begin{tabular}[c]{@{}l@{}}Static\\ThermEDGe \end{tabular} & IREDGe & \begin{tabular}[c]{@{}l@{}}Transient\\ThermEDGe\end{tabular} \\
\hhline{|:=:t:==::===:|}
\multirow{9}{*}{\begin{tabular}[c]{@{}l@{}}Model layer \\parameters \end{tabular}} & \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}2D\_conv1 \\2D\_conv\_trans1 \end{tabular}} & filter size & 5x5 & 3x3 & 5x5 \\
\cline{3-6}
& & \# filters & 64 & 64 & 64 \\
\cline{2-6}
& \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}2D\_conv2 \\2D\_conv\_trans2 \end{tabular}} & filter size & 3x3 & 3x3 & 3x3 \\
\cline{3-6}
& & \# filters & 32 & 32 & 32 \\
\cline{2-6}
& \multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}2D\_conv3 \\2D\_conv\_trans3 \end{tabular}} & filter size & 3x3 & 3x3 & -- \\
\cline{3-6}
& & \# filters & 16 & 16 & -- \\
\cline{2-6}
& Max pool layers & filter size & 2x2 & 2x2 & 2x2 \\
\cline{2-6}
& \multirow{2}{*}{ConvLSTM} & filter size & -- & -- & 7x7 \\
\cline{3-6}
& & \# filters & -- & -- & 16 \\
\hhline{|:=::==::===:|}
\multirow{8}{*}{\begin{tabular}[c]{@{}l@{}}Training \\parameters \end{tabular}} & \multicolumn{2}{l||}{Epochs} & \multicolumn{3}{l||}{500} \\
\cline{2-6}
& \multicolumn{2}{l||}{Optimizer} & \multicolumn{3}{l||}{ADAM} \\
\cline{2-6}
& \multicolumn{2}{l||}{Loss function} & \multicolumn{3}{l||}{Pixelwise MSE} \\
\cline{2-6}
& \multicolumn{2}{l||}{Decay rate} & \multicolumn{3}{l||}{0.98} \\
\cline{2-6}
& \multicolumn{2}{l||}{Decap steps} & \multicolumn{3}{l||}{1000} \\
\cline{2-6}
& \multicolumn{2}{l||}{Regularizer} & \multicolumn{3}{l||}{L2} \\
\cline{2-6}
& \multicolumn{2}{l||}{Regularization rate} & \multicolumn{3}{l||}{1.00E-05} \\
\cline{2-6}
& \multicolumn{2}{l||}{Learning rate} & \multicolumn{3}{l||}{1.00E-03} \\
\hhline{|b:=:b:==:b:===:b|}
\end{tabular}
}
\end{table}
We split the data in each set, using 80\% of the data points for
training, 10\% for test, and 10\% for validation. The training dataset is
normalized by subtracting the mean and dividing by the standard deviation. The
normalized golden dataset is used to train the network using an ADAM
optimizer~\cite{adam} where the loss function is a pixel-wise mean square error
(MSE). The convolutional operation in the encoder and the transpose
convolution in the decoder are each followed by ReLU activation to add
non-linearity and L2 regularization to prevent over fitting.
The model is trained in Tensorflow 2.1 on an NVIDIA GeForce RTX2080Ti GPU.
Training run-times are: 30m each for static ThermEDGe and IREDGe, and
6.5h for transient ThermEDGe.
We reiterate that this is
a one-time cost for a given technology node
and package, and this cost is amortized over
repeated use over many design iterations for multiple chips.
\section{Temperature/IR drop Results using ThermEDGe/IREDGe}
\section{Results and Analysis using TherEDGe/IREDGe}
\label{sec:results}
\subsection{Experimental setup and metric definitions}
\noindent
ThermEDGe and IREDGe are implemented using Python3.7 within a Tensorflow 2.1
framework. We test the performance of our models on the 10\% of datapoints
reserved for the testset (Section~\ref{sec:training}) which are labeled T1--T21. As mentioned earlier in Section~\ref{sec:train}, due the unavailability of new, public domain benchmarks to evaluate our experiments,
we use benchmarks that
represent commercial industry-standard design workloads.
\noindent
{\bf Error metrics} As a measure of goodness of ThermEDGe and IREDGe predictions, we define a discretized regionwise error, $T_{err}~=~\left | T_{true} - T_{pred} \right | $, where $T_{true}$ is ground truth image, generated by commercial tools, and $T_{pred}$ the predicted image, generated by ThermEDGe. $IR_{err}$ is computed in a similar way. We report the average and maximum values of $T_{err}$ and $IR_{err}$ for each testcase. In addition, the percentage mean and maximum error are listed as a fraction of a temperature corner, i.e., 105$^\circ$C for thermal analysis and as a fraction of VDD$=0.7$V for IR drop analysis.
\subsection{Performance of ThermEDGe and IREDGe: Accuracy and speed}
\noindent
\noindent
{\bf Static ThermEDGe results}
A comparison between the commercial tool-generated temperature and the
ThermEDGe-generated temperature map for T1--T5 are listed in
Table~\ref{tbl:thermal-results}. The runtime of static ThermEDGe for
each the five testcases which are of size 34$\times$32 is approximately 1.1ms in
our environment. On average across the five testcases (five rows of the table), ThermEDGe has an average $T_{err}$ of 0.63$^\circ$C and a maximum $T_{err}$ of 2.93$^\circ$C.\footnote{Achieving this accuracy requires much finer discretization in Icepak.} These numbers are a small fraction when compared to the maximum ground truth temperature of these testcases (85 -- 150$^\circ$C).
The fast runtimes imply that our method can be used in the
inner loop of a thermal optimizer, e.g., to evaluate various chip configurations
under the same packaging solution (typically chosen early in the
design process). For such applications, this level of error is very
acceptable.
\begin{table}[h]
\centering
\caption{Summary of ThermEDGe results for static and transient analysis across 10 testcases.}
\label{tbl:thermal-results}
\resizebox{\linewidth}{!}{%
\begin{tabular}{||l|l|l||l|l|l||}
\hhline{|t:===:t:===:t|}
\multicolumn{3}{||c||}{{\bf Static ThermEDGe
}} & \multicolumn{3}{|c||}{{\bf Transient ThermEDGe}} \\
\hhline{|:===::===:|}
\textbf{\#Testcase} & \textbf{Avg. $\bf T_{err}$} & \textbf{Max $\bf T_{err}$} & \textbf{\#Testcase} & \textbf{Avg. $\bf T_{err}$} & \textbf{Max $\bf T_{err}$} \\
\hline \hline
T1 & 0.64C (0.61\%) & 2.76C (2.63\%) & T6 & 0.51C (0.49\%) & 5.59C (5.32\%) \\
\hline
T2 & 0.63C (0.60\%) & 2.67C (2.54\%) & T7 & 0.58C (0.55\%) & 6.17C (5.88\%) \\
\hline
T3 & 0.65C (0.62\%) & 2.93C (2.79\%) & T8 & 0.57C (0.54\%) & 5.83C (5.55\%) \\
\hline
T4 & 0.48C (0.46\%) & 2.22C (2.11\%) & T9 & 0.52C (0.50\%) & 6.32C (6.02\%) \\
\hline
T5 & 0.75C (0.71\%) & 2.86C (2.72\%) & T10 & 0.56C (0.53\%) & 7.14C (6.80\%) \\
\hhline{|b:===:b:===:b|}
\end{tabular}
}
\end{table}
\begin{figure}[h]
\centering
\includegraphics[width=6.5cm]{figs/static-thermal-results.pdf}
\caption{ThermEDGe static temperature estimation on T1: (a) input normalized power distribution, (b) histogram of $T_{err}$ where maximum error is 2.76$^\circ$C which is very small compared to the maximum temperature of 85$^\circ$C, (c) ground truth temperature map, and (d) predicted temperature map.}
\label{fig:static-thermal-results}
\end{figure}
A graphical view of the predicted map for T1 is
depicted in Fig.~\ref{fig:static-thermal-results}. For a given input power
distribution in Fig.~\ref{fig:static-thermal-results}(a), ThermEDGe generates
the temperature contour plots, as shown in
Fig.~\ref{fig:static-thermal-results}(d). We compare the predicted value against the true value
(Fig.~\ref{fig:static-thermal-results}(c)). The discrepancy is visually seen to
be small. Numerically, the histogram in Fig.~\ref{fig:static-thermal-results}(b) shows the distribution of \%$T_{err}$ across regions (Fig.~\ref{fig:static-thermal-results}(b). The average $T_{err}$ 0.64$^\circ$C and the maximum $T_{err}$ is 2.93$^\circ$C. This corresponds an average error of 0.52\% and worst-case error of 2.79\% as shown in the figure.
\noindent
{\bf Transient ThermEDGe results}
The transient thermal analysis problem is a sequence-to-sequence prediction
task where each datapoint in the testset
has 200 frames of power maps at a 15s interval. Trained transient ThermEDGe
predicts the output temperature sequence for the input power sequence.
We summarize the results in Table~\ref{tbl:thermal-results}. The
inference run-times of T6--10 to generate a sequence 200 frames of
temperature contours is approximately 10ms in our setup.
Across the five testcases, the prediction has an average $T_{err}$ of 0.52\%
and a maximum $T_{err}$ of 6.80\% as shown. The maximum $T_{err}$ in our testcases occur during transients which do not have long-last effects (e.g., on IC reliability). These errors are reduced to the average $T_{err}$ values at sustained peak temperatures.
Fig.~\ref{fig:video} (left) shows an animated video of the time-varying
power map for T6, where each frame (time-step) is after a 15s
time interval. As before, the corresponding ground truth and predicted
temperature contours are depicted in center and right,
respectively, of the figure.
\begin{figure}[ht]
\centering
\includegraphics[width=1.05\linewidth]{figs/video-frames/frame120.png}
\caption{{\em [For an animated version, visit the GitHub repository: https://github.com/asp-dac/asp-dac-1323.git to view the video.]} Video comparing the prediction of transient ThermEDGe against commercial tool-generated temperature contours for T6: (i) left video shows the time-varying power map, (ii) center video shows the commercial tool-generated temperature maps, and (iii) right video shows ThermEDGe-generated temperature maps}
\label{fig:video}
\end{figure}
\noindent
{\bf IREDGe results}
We compare IREDGe-generated contours against the contours generated by ~\cite{pdnsim} across 500 different testcases (10\% of the data, orthogonal to the training set) with varying PDN densities and power distributions. Across the five testcases in Table~\ref{tbl:iredge-qcomm-results}, IREDGe has an average $IR_{err}$ of 0.053mV and a worstcase max $IR_{err}$ of 0.34mV which corresponds to 0.008\% and 0.048\% of VDD respectively. Given that static
IR drop constraints are 1--2.5\% of VDD, a worstcase error of 0.34mV is acceptable in light of the rapid runtimes. We list the results of five representative testcases in
Table~\ref{tbl:iredge-qcomm-results} where the percentage errors in $IR_{err}$ are listed as fraction of VDD$=0.7$V.
\begin{table}[htb]
\caption{Summary of results from IREDGe for 10 different testcases. T16-T20 are testcases which have a chip size that was not in the training set. }
\centering
\label{tbl:iredge-qcomm-results}
\resizebox{\linewidth}{!}{%
\begin{tabular}{||l|l|l||l|l|l||}
\hhline{|t:===:t:===:t|}
\multicolumn{3}{||c||}{{\bf Chip size: 34x32
}} & \multicolumn{3}{|c||}{{\bf Chip size: 68x32}} \\
\hhline{|:===::===:|}
\textbf{\#Testcase} & \textbf{Avg. $\bf IR_{err}$} & \textbf{Max $\bf IR_{err}$} & \textbf{\#Testcase} & \textbf{Avg. $\bf IR_{err}$} & \textbf{Max $\bf IR_{err}$} \\
\hline \hline
T11 & 0.052mV (0.007\%) & 0.26mV (0.03\%) & T16 & 0.035mV (0.005\%) & 0.16mV (0.02\%)\\ \hline
T12 & 0.074mV (0.011\%) & 0.34mV (0.05\%) & T17 & 0.054mV (0.008\%) & 0.42mV (0.06\%)\\ \hline
T13 & 0.036mV (0.005\%) & 0.21mV (0.03\%) & T18 & 0.035mV (0.005\%) & 0.35mV (0.05\%)\\ \hline
T14 & 0.053mV (0.008\%) & 0.24mV (0.03\%) & T19 & 0.068mV (0.010\%) & 0.22mV (0.03\%)\\ \hline
T15 & 0.051mV (0.007\%) & 0.23mV (0.03\%) & T20 & 0.061mV (0.009\%) & 0.38mV (0.05\%)\\
\hhline{|b:===:b:===:b|}
\end{tabular}
}
\end{table}
\begin{figure}[htb]
\centering
\includegraphics[width=9cm]{figs/ir-results.pdf}
\vspace{-3em}
\caption{IREDGe static IR drop estimation on T11: (a) input power map, (b) input PDN density map, (c) effective distance to power pad map (d) ground truth IR drop map, (e) predicted IR drop map, and (f) histogram of $IR_{err}$ showing a worstcase error of 0.16mV.}
\label{fig:ir-result}
\end{figure}
\begin{figure}[htb]
\centering
\vspace{-1.0em}
\includegraphics[width=8.5cm]{figs/size-independent.pdf}
\caption{Size independent nature of IREDGe: Comparison between (a) Actual IR drop contours and (b) IREDGe-predicted contours for a power map (T16) with size 68$\times$32 using a model that
was trained on images of size 34$\times$32.}
\label{fig:size-independent}
\vspace{-0.5em}
\end{figure}
A detailed view of T11 is shown in Fig.~\ref{fig:ir-result}.
It compares the IREDGe-generated IR drop contour plots
against contour plot generated by~\cite{pdnsim}.
The input power maps, PDN density maps, and effective distance to power pad maps are shown
in Fig.~\ref{fig:ir-result}(a), (b), and (c) respectively.
Fig.~\ref{fig:ir-result}(d) and (e) shows the comparison between ground truth and predicted value for the corresponding inputs. It is evident that the plots are similar; numerically, the histogram in Fig.~\ref{fig:ir-result}(f)
shows the \%$IR_{err}$ where the worst \%$IR_{err}$ is less than 0.02\% of VDD.
\noindent
{\bf Size-independence}
One of the primary advantages of using IREDGe for static IR
estimation is that its fully-convolutional nature enables the use of input
images of any size, and the size of the hotspot determines the model rather
than the size of the chip. Since the trained model comprises only of the
trained weights of the kernel, the same kernel can be used to predict the
temperature contours of chip of any size as long as resolution of the represented image remains the same. We test static IREDGe on chips of a different size (T16 -- T20), using a
power distribution of size $68\times32$ as input.
Fig.~\ref{fig:size-independent}(a) compares the actual IR drop of T16 (Fig.~\ref{fig:size-independent}(a)) and the IREDGe-predicted (Fig.~\ref{fig:size-independent}(b)) solution of T16 using a model which was trained on $34\times32$ power maps.
We summarize the results for the rest of the testcases in Table~\ref{tbl:iredge-qcomm-results}.
\noindent
{\bf Runtime analysis} A summary of the runtime comparison of our ML-based EDGe network approach against the temperature and IR drop golden solvers is listed in Table~\ref{tab:runtime}. The runtimes are reported on a NVIDIA GeForce RTX 2080Ti GPU. With the millisecond inference times, and the transferable nature of our trained models, the one-time cost of training the EDGe networks is easily amortized over multiple uses within a design cycle, and over multiple designs.
\begin{table}[ht]
\centering
\caption{Runtime comparison between EDGe networks and golden thermal analysis and IR drop analysis tools}
\label{tab:runtime}
\resizebox{0.4\textwidth}{!}{%
\begin{tabular}{|l|l|l|l|l|}
\hline
\textbf{Analysis type} & \textbf{\# Nodes} & \textbf{\begin{tabular}[c]{@{}l@{}}Design \\ Area \\ (mm$^2$)\end{tabular}} & \textbf{\begin{tabular}[c]{@{}l@{}}Icepak/\\ PDNSim \\ (minutes)\end{tabular}} & \textbf{\begin{tabular}[c]{@{}l@{}}ThermEDGe/\\ IREDGe\\ (milli seconds)\end{tabular}} \\ \hline
Static thermal & 2.0 million & 64 & 30 mins & 1.1 ms \\ \hline
Transient thermal & 2.0 million & 64 & 210 mins & 10 ms \\ \hline
Static IR drop & 5.2 million & 0.16 & 310 mins & 1.1 ms \\ \hline
\end{tabular}%
}
\end{table}
\begin{figure}[ht]
\centering
\vspace{-0.5em}
\includegraphics[width=8.5cm]{figs/power-net-comparison.pdf}
\caption{IR drop comparisons on T21: (a) ground truth, (b) from IREDGe, and (c) from our implementation of PowerNet.}
\vspace{-1.5em}
\label{fig:powernet-comp}
\end{figure}
\subsection{IREDGe compared with PowerNet}
\noindent
We compare the performance of IREDGe against our implementation of
PowerNet, based on its description in~\cite{powernet}.
The layout is divided into tiles, and the CNN features are the
2-D power distributions (toggle rate-scaled switching and internal power, total
power, and leakage power) within each tile and in a fixed window of surrounding tiles.
The trained CNN is used to predict the IR drop on a tile-by-tile basis by
sliding a window across all tiles on the chip. The work uses a tile size of
5$\mu$m$\times$5$\mu$m and takes into consideration a 31$\times$31 tiled
neighborhood (window) power information as features.
For a fair comparison, we train IREDGe under a fixed PDN density and fixed power pad locations that is used to train PowerNet. Qualitatively, IREDGe is superior on three aspects: \\
(1) {\em Tile and window size selection:} It is stated in~\cite{powernet} that
when the size of the tile is increased from
1$\mu$m$\times$1$\mu$m to 5$\mu$m$\times$5$\mu$m and the size of the resulting window is increased
to represent 31$\times$31 window of 25$\mu$m$^2$ tiles instead of 1$\mu$m$^2$ tiles, the
accuracy of the PowerNet model improves.
In general, this is the
expected behavior with an IR analysis problem where the accuracy increases as
more global information is available, until a certain radius after which the
principle of locality holds~\cite{Chiprout04}. IREDGe bypasses
this tile-size selection problem entirely by providing the entire power map
as input to IREDGe and allowing the network to learn the window size that is
needed for accurate IR estimation. \\
(2) {\em Run times:} Unlike PowerNet, which trains and infers IR drop on a sliding tile-by-tile
basis, IREDGe has faster training and
inference. IREDGe requires a {\em single} inference, irrespective of the size of the chip while PowerNet performs an inference for every tile in the chip. For this setup and data, it takes 75 minutes to train and implementation of PowerNet, as against 30 minutes for IREDGe. For inference, PowerNet takes 3.2ms while IREDGe takes 1.1ms for a 34$\times$32 chip size. For a chip of $68\times32$ IREDGe takes 1.3ms to generate IR drop contours while PowerNet takes 6.2ms.
(3) {\em Model accuracy:} Since PowerNet uses a CNN to predict IR drop on a
region-by-region basis, where each region is 5$\mu$m by 5$\mu$m,
the resulting IR drop image is pixelated, and the predicted region prediction value does not correlate well with the neighboring regions.
We compare IREDGe against our implementation of PowerNet on five different testcases T21--25.
These testcases have the same power distribution in T11--15 except that all
the five testscases have identical uniform PDNs, and identical power pad
distributions, as required by PowerNet; IREDGe does not require this.
Fig.~\ref{fig:powernet-comp} shows a comparison between the IR drop solutions
from a golden solver (Fig.~\ref{fig:powernet-comp}(a)), IREDGe
(Fig.~\ref{fig:powernet-comp}(b), and our implementation of
PowerNet (Fig.~\ref{fig:powernet-comp}(c)) for T21 (a representative testcase).
On average, across
T21--25 IREDGe has an average $IR_{err}$ of 0.028mV and a maximum $IR_{err}$ of 0.14mV as against 0.042mV and 0.17mV respectively for PowerNet.
\section{Conclusion}
\noindent
This paper addresses the compute-intensive tasks of thermal and IR analysis by proposing the use EDGe networks as apt ML-based solutions. Our EDGe-based solution not only improves runtimes but overcomes the window size-selection challenge (amount of neighborhood information required for accurate thermal and IR analysis), that is faced by other ML-based techniques, by allowing ML to learn the window size.
We successfully evaluate EDGe networks for these applications by developing two ML software solutions (i) ThermEDGe and (ii) IREDGe for rapid on-chip (static and dynamic) thermal and (static) IR analysis respectively. In principle, our methodology is applicable to dynamic IR as well, but is not shown due to the unavailability of public-domain benchmarks.
|
2024-02-18T23:40:24.783Z
|
2020-09-22T02:01:02.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2009.09009","language":"en","timestamp":1600740062000,"url":"https:\/\/arxiv.org\/abs\/2009.09009","yymm":"2009"}
|
proofpile-arXiv_000-10083
|
{"provenance":"002.jsonl.gz:10084"}
| null | null |
\section{Introduction}
The cross-entropy (CE) method is a probabilistic optimization approach that attempts to iteratively fit a distribution to elite samples from an initial input distribution \cite{rubinstein2004cross,rubinstein1999cross}.
The goal is to estimate a rare-event probability by minimizing the \textit{cross-entropy} between the two distributions \cite{de2005tutorial}.
The CE-method has gained popularity in part due to its simplicity in implementation and straightforward derivation.
The technique uses \textit{importance sampling} which introduces a proposal distribution over the rare-events to sample from then re-weights the posterior likelihood by the \textit{likelihood ratio} of the true distribution over the proposal distribution.
There are a few key assumptions that make the CE-method work effectively.
Through random sampling, the CE-method assumes that there are enough objective function evaluations to accurately represent the objective.
This may not be a problem for simple applications, but can be an issue for computationally expensive objective functions.
Another assumption is that the initial parameters of the input distribution are wide enough to cover the design space of interest. For the case with a multivariate Gaussian distribution, this corresponds to an appropriate mean and wide covariance.
In rare-event simulations with many local minima, the CE-method can fail to find a global minima especially with sparse objective function evaluations.
This work aims to address the key assumptions of the CE-method.
We introduce variants of the CE-method that use surrogate modeling to approximate the objective function, thus updating the belief of the underlying objective through estimation.
As part of this approach, we introduce evaluation scheduling techniques to reallocate true objective function calls earlier in the optimization when we know the covariance will be large.
The evaluation schedules can be based on a distribution (e.g., the Geometric distribution) or can be prescribed manually depending on the problem.
We also use a Gaussian mixture model representation of the prior distribution as a method to explore competing local optima.
While the use of Gaussian mixture models in the CE-method is not novel, we connect the use of mixture models and surrogate modeling in the CE-method.
This connection uses each elite sample as the mean of a component distribution in the mixture, optimized through a subroutine call to the standard CE-method using the learned surrogate model.
To test our approach, we introduce a parameterized test objective function called \textit{sierra}.
The sierra function is built from a multivariate Gaussian mixture model with many local minima and a single global minimum.
Parameters for the sierra function allow control over both the spread and distinction of the minima.
Lastly, we provide an analysis of the weak areas of the CE-method compared to our proposed variants.
\section{Related Work} \label{sec:related_work}
The cross-entropy method is popular in the fields of operations research, machine learning, and optimization \cite{kochenderfer2015decision,Kochenderfer2019}.
The combination of the cross-entropy method, surrogate modeling, and mixture models has been explored in other work \cite{bardenet2010surrogating}.
The work in \cite{bardenet2010surrogating} proposed an adaptive grid approach to accelerate Gaussian-process-based surrogate modeling using mixture models as the prior in the cross-entropy method. They showed that a mixture model performs better than a single Gaussian when the objective function is multimodal.
Our work differs in that we augment the ``elite'' samples both by an approximate surrogate model and by a subroutine call to the CE-method using the learned surrogate model.
Other related work use Gaussian processes and a modified cross-entropy method for receding-horizon trajectory optimization \cite{tan2018gaussian}.
Their cross-entropy method variant also incorporates the notion of exploration in the context of path finding applications.
An approach based on \textit{relative entropy}, described in \cref{sec:background_ce}, proposed a model-based stochastic search that seeks to minimize the relative entropy \cite{NIPS2015_5672}. They also explore the use of a simple quadratic surrogate model to approximate the objective function.
Prior work that relate cross-entropy-based adaptive importance sampling with Gaussian mixture models show that a mixture model require less objective function calls than a na\"ive Monte Carlo or standard unimodal cross-entropy-based importance sampling method \cite{kurtz2013cross,wang2016cross}.
\section{Background} \label{sec:background}
This section provides necessary background on techniques used in this work. We provide introductions to cross-entropy and the cross-entropy method, surrogate modeling using Gaussian processes, and multivariate Gaussian mixture models.
\subsection{Cross-Entropy} \label{sec:background_ce}
Before understanding the cross-entropy method, we first must understand the notion of \textit{cross-entropy}.
Cross-entropy is a metric used to measure the distance between two probability distributions, where the distance may not be symmetric \cite{de2005tutorial}.
The distance used to define cross-entropy is called the \textit{Kullback-Leibler (KL) distance} or \textit{KL divergence}.
The KL distance is also called the \textit{relative entropy}, and we can use this to derive the cross-entropy.
Formally, for a random variable $\mat{X} = (X_1, \ldots, X_n)$ with a support of $\mathcal{X}$, the KL distance between two continuous probability density functions $f$ and $g$ is defined to be:
\begin{align*}
\mathcal{D}(f, g) &= \mathbb{E}_f\left[\log \frac{f(\vec{X})}{g(\vec{X})} \right]\\
&= \int\limits_{\vec{x} \in \mathcal{X}} f(\vec{x}) \log f(\vec{x}) d\vec{x} - \int\limits_{\vec{x} \in \mathcal{X}} f(\vec{x}) \log g(\vec{x}) d\vec{x}
\end{align*}
We denote the expectation of some function with respect to a distribution $f$ as $\mathbb{E}_f$.
Minimizing the KL distance $\mathcal{D}$ between our true distribution $f$ and our proposal distribution $g$ parameterized by $\vec{\theta}$, is equivalent to choosing $\vec\theta$ that minimizes the following, called the \textit{cross-entropy}:
\begin{align*}
H(f,g) &= H(f) + \mathcal{D}(f,g)\\
&= -\mathbb{E}_f[\log g(\vec{X})] \tag{using KL distance}\\
&= - \int\limits_{\vec{x} \in \mathcal{X}} f(\vec{x}) \log g(\vec{x} \mid \vec{\theta}) d\vec{x}
\end{align*}
where $H(f)$ denotes the entropy of the distribution $f$ (where we conflate entropy and continuous entropy for convenience).
This assumes that $f$ and $g$ share the support $\mathcal{X}$ and are continuous with respect to $\vec{x}$.
The minimization problem then becomes:
\begin{equation} \label{eq:min}
\begin{aligned}
\operatornamewithlimits{minimize}_{\vec{\theta}} & & - \int\limits_{\vec{x} \in \mathcal{X}} f(\vec{x}) \log g(\vec{x} \mid \vec{\theta}) d\vec{x}
\end{aligned}
\end{equation}
Efficiently finding this minimum is the goal of the cross-entropy method algorithm.
\subsection{Cross-Entropy Method} \label{sec:background_cem}
Using the definition of cross-entropy, intuitively the \textit{cross-entropy method} (CEM or CE-method) aims to minimize the cross-entropy between the unknown true distribution $f$ and a proposal distribution $g$ parameterized by $\vec\theta$.
This technique reformulates the minimization problem as a probability estimation problem, and uses adaptive importance sampling to estimate the unknown expectation \cite{de2005tutorial}.
The cross-entropy method has been applied in the context of both discrete and continuous optimization problems \cite{rubinstein1999cross,kroese2006cross}.
The initial goal is to estimate the probability
\begin{align*}
\ell = P_{\vec{\theta}}(S(\vec{x}) \ge \gamma)
\end{align*}
where $S$ can the thought of as an objective function of $\vec{x}$, and $\vec{x}$ follows a distribution defined by $g(\vec{x} \mid \vec{\theta})$.
We want to find events where our objective function $S$ is above some threshold $\gamma$.
We can express this unknown probability as the expectation
\begin{align} \label{eq:expect}
\ell = \mathbb{E}_{\vec{\theta}}[\mathbbm{1}_{(S(\vec{x}) \ge \gamma)}]
\end{align}
where $\mathbbm{1}$ denotes the indicator function.
A straightforward way to estimate \cref{eq:expect} can be done through Monte Carlo sampling.
But for rare-event simulations where the probability of a target event occurring is relatively small, this estimate becomes inadequate.
The challenge of the minimization in \cref{eq:min} then becomes choosing the density function for the true distribution $f(\vec{x})$.
Importance sampling tells us that the optimal importance sampling density can be reduced to
\begin{align*}
f^*(\vec{x}) = \frac{\mathbbm{1}_{(S(\vec{x}) \ge \gamma)}g(\vec{x} \mid \vec{\theta})}{\ell}
\end{align*}
thus resulting in the optimization problem:
\begin{align*}
\vec{\theta}_g^* &= \operatornamewithlimits{arg\,min}_{\vec{\theta}_g} - \int\limits_{\vec{x} \in \mathcal{X}} f^*(\vec{x})\log g(\vec{x} \mid \vec{\theta}_g) d\vec{x}\\
&= \operatornamewithlimits{arg\,min}_{\vec{\theta}_g} - \int\limits_{\vec{x} \in \mathcal{X}} \frac{\mathbbm{1}_{(S(\vec{x}) \ge \gamma)}g(\vec{x} \mid \vec{\theta})}{\ell}\log g(\vec{x} \mid \vec{\theta}_g) d\vec{x}
\end{align*}
Note that since we assume $f$ and $g$ belong to the same family of distributions, we get that $f(\vec{x}) = g(\vec{x} \mid \vec{\theta}_g)$.
Now notice that $\ell$ is independent of $\vec{\theta}_g$, thus we can drop $\ell$ and get the final optimization problem of:
\begin{align} \label{eq:opt}
\vec{\theta}_g^* &= \operatornamewithlimits{arg\,min}_{\vec{\theta}_g} - \int\limits_{\vec{x} \in \mathcal{X}} \mathbbm{1}_{(S(\vec{x}) \ge \gamma)}g(\vec{x} \mid \vec{\theta}) \log g(\vec{x} \mid \vec{\theta}_g) d\vec{x}\\\nonumber
&= \operatornamewithlimits{arg\,min}_{\vec{\theta}_g} - \mathbb{E}_{\vec{\theta}}[ \mathbbm{1}_{(S(\vec{x}) \ge \gamma)} \log g(\vec{x} \mid \vec{\theta}_g)]
\end{align}
The CE-method uses a multi-level algorithm to estimate $\vec{\theta}_g^*$ iteratively.
The parameter $\vec{\theta}_k$ at iteration $k$ is used to find new parameters $\vec{\theta}_{k^\prime}$ at the next iteration $k^\prime$.
The threshold $\gamma_k$ becomes smaller that its initial value, thus artificially making events \textit{less rare} under $\vec{X} \sim g(\vec{x} \mid \vec{\theta}_k)$.
In practice, the CE-method algorithm requires the user to specify a number of \textit{elite} samples $m_\text{elite}$ which are used when fitting the new parameters for iteration $k^\prime$.
Conveniently, if our distribution $g$ belongs to the \textit{natural exponential family} then the optimal parameters can be found analytically \cite{Kochenderfer2019}. For a multivariate Gaussian distribution parameterized by $\vec{\mu}$ and $\mat{\Sigma}$, the optimal parameters for the next iteration $k^\prime$ correspond to the maximum likelihood estimate (MLE):
\begin{align*}
\vec{\mu}_{k^\prime} &= \frac{1}{m_\text{elite}} \sum_{i=1}^{m_\text{elite}} \vec{x}_i\\
\vec{\Sigma}_{k^\prime} &= \frac{1}{m_\text{elite}} \sum_{i=1}^{m_\text{elite}} (\vec{x}_i - \vec{\mu}_{k^\prime})(\vec{x}_i - \vec{\mu}_{k^\prime})^\top
\end{align*}
The cross-entropy method algorithm is shown in \cref{alg:cem}.
For an objective function $S$ and input distribution $g$, the CE-method algorithm will run for $k_\text{max}$ iterations.
At each iteration, $m$ inputs are sampled from $g$ and evaluated using the objective function $S$.
The sampled inputs are denoted by $\mat{X}$ and the evaluated values are denoted by $\mat{Y}$.
Next, the top $m_\text{elite}$ samples are stored in the elite set $\mathbf{e}$, and the distribution $g$ is fit to the elites.
This process is repeated for $k_\text{max}$ iterations and the resulting parameters $\vec{\theta}_{k_\text{max}}$ are returned.
Note that a variety of input distributions for $g$ are supported, but we focus on the multivariate Gaussian distribution and the Gaussian mixture model in this work.
\begin{algorithm}[ht]
\begin{algorithmic}
\Function{CrossEntropyMethod}{}($S, g, m, m_\text{elite}, k_\text{max}$)
\For {$k \in [1,\ldots,k_\text{max}]$}
\State $\mat{X} \sim g(\;\cdot \mid \vec{\theta}_k)$ where $\mat{X} \in \mathbb{R}^m$
\State $\mat{Y} \leftarrow S(\vec{x})$ for $\vec{x} \in \mat{X}$
\State $\mathbf{e} \leftarrow$ store top $m_\text{elite}$ from $\mat{Y}$
\State $\vec{\theta}_{k^\prime} \leftarrow \textproc{Fit}(g(\;\cdot \mid \vec{\theta}_k), \mathbf{e})$
\EndFor
\State \Return $g(\;\cdot \mid \vec{\theta}_{k_\text{max}})$
\EndFunction
\end{algorithmic}
\caption{\label{alg:cem} Cross-entropy method.}
\end{algorithm}
\subsection{Mixture Models}
A standard Gaussian distribution is \textit{unimodal} and can have trouble generalizing over data that is \textit{multimodal}.
A \textit{mixture model} is a weighted mixture of component distributions used to represent continuous multimodal distributions \cite{kochenderfer2015decision}.
Formally, a Gaussian mixture model (GMM) is defined by its parameters $\vec{\mu}$ and $\mat{\Sigma}$ and associated weights $\mathbf{w}$ where $\sum_{i=1}^n w_i = 1$. We denote that a random variable $\mat{X}$ is distributed according to a mixture model as $\mat{X} \sim \operatorname{Mixture}(\vec{\mu}, \vec{\Sigma}, \vec{w})$.
The probability density of the GMM then becomes:
\begin{gather*}
P( \mat{X} = \vec{x} \mid \vec{\mu}, \mat{\Sigma}, \vec{w}) = \sum_{i=1}^n w_i \Normal(\vec{x} \mid \vec{\mu}_i, \mat{\Sigma}_i)
\end{gather*}
To fit the parameters of a Gaussian mixture model, it is well known that the \textit{expectation-maximization (EM)} algorithm can be used \cite{dempster1977maximum,aitkin1980mixture}.
The EM algorithm seeks to find the maximum likelihood estimate of the hidden variable $H$ using the observed data defined by $E$.
Intuitively, the algorithm alternates between an expectation step (E-step) and a maximization step (M-step) to guarantee convergence to a local minima.
A simplified EM algorithm is provide in \cref{alg:em} for reference and we refer to \cite{dempster1977maximum,aitkin1980mixture} for further reading.
\begin{algorithm}[ht]
\begin{algorithmic}
\Function{ExpectationMaximization}{$H, E, \vec{\theta}$}
\For{\textbf{E-step}}
\State Compute $Q(h) = P(H=h \mid E=e, \vec{\theta})$ for each $h$
\State Create weighted points: $(h,e)$ with weight $Q(h)$
\EndFor
\For{\textbf{M-step}}
\State Compute $\mathbf{\hat{\vec{\theta}}}_{\text{MLE}}$
\EndFor
\State Repeat until convergence.
\State \Return $\mathbf{\hat{\vec{\theta}}}_{\text{MLE}}$
\EndFunction
\end{algorithmic}
\caption{\label{alg:em} Expectation-maximization.}
\end{algorithm}
\subsection{Surrogate Models}
In the context of optimization, a surrogate model $\hat{S}$ is used to estimate the true objective function and provide less expensive evaluations.
Surrogate models are a popular approach and have been used to evaluate rare-event probabilities in computationally expensive systems \cite{li2010evaluation,li2011efficient}.
The simplest example of a surrogate model is linear regression.
In this work, we focus on the \textit{Gaussian process} surrogate model.
A Gaussian process (GP) is a distribution over functions that predicts the underlying objective function $S$ and captures the uncertainty of the prediction using a probability distribution \cite{Kochenderfer2019}.
This means a GP can be sampled to generate random functions, which can then be fit to our given data $\mat{X}$.
A Gaussian process is parameterized by a mean function $\mathbf{m}(\mat{X})$ and kernel function $\mat{K}(\mat{X},\mat{X})$, which captures the relationship between data points as covariance values.
We denote a Gaussian process that produces estimates $\hat{\vec{y}}$ as:
\begin{align*}
\hat{\vec{y}} &\sim\mathcal{N}\left(\vec{m}(\mat{X}),\vec{K}(\mat{X},\mat{X})\right)\\
&= \begin{bmatrix}
\hat{S}(\vec{x}_1), \ldots, \hat{S}(\vec{x}_n)
\end{bmatrix}
\end{align*}
where
\begin{gather*}
\vec{m}(\mat{X}) = \begin{bmatrix} m(\vec{x}_1), \ldots, m(\vec{x}_n) \end{bmatrix}\\
\vec{K}(\mat{X}, \mat{X}) = \begin{bmatrix}
k(\vec{x}_1, \vec{x}_1) & \cdots & k(\vec{x}_1, \vec{x}_n)\\
\vdots & \ddots & \vdots\\
k(\vec{x}_n, \vec{x}_1) & \cdots & k(\vec{x}_n, \vec{x}_n)
\end{bmatrix}
\end{gather*}
We use the commonly used zero-mean function $m(\vec{x}_i) = \vec{0}$.
For the kernel function $k(\vec{x}_i, \vec{x}_i)$, we use the squared exponential kernel with variance $\sigma^2$ and characteristic scale-length $\ell$, where larger $\ell$ values increase the correlation between successive data points, thus smoothing out the generated functions. The squared exponential kernel is defined as:
\begin{align*}
k(\vec{x},\vec{x}^\prime) = \sigma^2\exp\left(- \frac{(\vec{x} - \vec{x}^\prime)^\top(\vec{x} - \vec{x}^\prime)}{2\ell^2}\right)
\end{align*}
We refer to \cite{Kochenderfer2019} for a detailed overview of Gaussian processes and different kernel functions.
\section{Algorithms} \label{sec:algorithms}
We can now describe the cross-entropy method variants introduced in this work.
This section will first cover the main algorithm introduced, the cross-entropy surrogate method (CE-surrogate).
Then we introduce a modification to the CE-surrogate method, namely the cross-entropy mixture method (CE-mixture).
Lastly, we describe various evaluation schedules for redistributing objective function calls over the iterations.
\subsection{Cross-Entropy Surrogate Method} \label{sec:alg_ce_surrogate}
The main CE-method variant we introduce is the cross-entropy surrogate method (CE-surrogate).
The CE-surrogate method is a superset of the CE-method, where the differences lie in the evaluation scheduling and modeling of the elite set using a surrogate model.
The goal of the CE-surrogate algorithm is to address the shortcomings of the CE-method when the number of objective function calls is sparse and the underlying objective function $S$ has multiple local minima.
The CE-surrogate algorithm is shown in \cref{alg:ce_surrogate}.
It takes as input the objective function $S$, the distribution $\mathbf{M}$ parameterized by $\vec{\theta}$, the number of samples $m$, the number of elite samples $m_\text{elite}$, and the maximum iterations $k_\text{max}$.
For each iteration $k$, the number of samples $m$ are redistributed through a call to \smallcaps{EvaluationSchedule}, where $m$ controls the number of true objective function evaluations of $S$.
Then, the algorithm samples from $\mathbf{M}$ parameterized by the current $\vec{\theta}_k$ given the adjusted number of samples $m$.
For each sample in $\mat{X}$, the objective function $S$ is evaluated and the results are stored in $\mat{Y}$.
The top $m_\text{elite}$ evaluations from $\mat{Y}$ are stored in $\mathbf{e}$.
Using all of the current function evaluations $\mat{Y}$ from sampled inputs $\mat{X}$, a modeled elite set $\mathbf{E}$ is created to augment the sparse information provided by a low number of true objective function evaluations.
Finally, the distribution $\mathbf{M}$ is fit to the elite set $\mathbf{E}$ and the distribution with the final parameters $\vec{\theta}_{k_\text{max}}$ is returned.
\begin{algorithm}[ht]
\begin{algorithmic}
\Function{CE-Surrogate}{$S$, $\mathbf{M}$, $m$, $m_\text{elite}$, $k_\text{max}$}
\For {$k \in [1,\ldots,k_\text{max}]$}
\State $m, m_\text{elite} \leftarrow \textproc{EvaluationSchedule}(k, k_\text{max})$
\State $\mat{X} \sim \mathbf{M}(\;\cdot \mid \vec{\theta}_k)$ where $\mat{X} \in \mathbb{R}^m$
\State $\mat{Y} \leftarrow S(\vec{x})$ for $\vec{x} \in \mat{X}$
\State $\mathbf{e} \leftarrow$ store top $m_\text{elite}$ from $\mat{Y}$
\State $\mathbf{E} \leftarrow \textproc{ModelEliteSet}(\mat{X}, \mat{Y}, \mathbf{M}, \mathbf{e}, m, m_\text{elite})$
\State $\vec{\theta}_{k^\prime} \leftarrow \textproc{Fit}(\mathbf{M}(\;\cdot \mid \vec{\theta}_k), \mathbf{E})$
\EndFor
\State \Return $\mathbf{M}(\;\cdot \mid \vec{\theta}_{k_\text{max}})$
\EndFunction
\end{algorithmic}
\caption{\label{alg:ce_surrogate} Cross-entropy surrogate method.}
\end{algorithm}
The main difference between the standard CE-method and the CE-surrogate variant lies in the call to \smallcaps{ModelEliteSet}.
The motivation is to use \textit{all} of the already evaluated objective function values $\mat{Y}$ from a set of sampled inputs $\mat{X}$.
This way the expensive function evaluations---otherwise discarded---can be used to build a surrogate model of the underlying objective function.
First, a surrogate model $\surrogate$ is constructed from the samples $\mat{X}$ and true objective function values $\mat{Y}$.
We used a Gaussian process with a specified kernel and optimizer, but other surrogate modeling techniques such as regression with basis functions can be used.
We chose a Gaussian process because it incorporates probabilistic uncertainty in the predictions, which may more accurately represent our objective function, or at least be sensitive to over-fitting to sparse data.
Now we have an approximated objective function $\surrogate$ that we can inexpensively call.
We sample $10m$ values from the distribution $\mathbf{M}$ and evaluate them using the surrogate model.
We then store the top $10m_\text{elite}$ values from the estimates $\mathbf{\hat{\mat{Y}}}_\text{m}$.
We call these estimated elite values $\mathbf{e}_\text{model}$ the \textit{model-elites}.
The surrogate model is then passed to \smallcaps{SubEliteSet}, which returns more estimates for elite values.
Finally, the elite set $\mathbf{E}$ is built from the true-elites $\mathbf{e}$, the model-elites $\mathbf{e}_\text{model}$, and the subcomponent-elites $\mathbf{e}_\text{sub}$.
The resulting concatenated elite set $\mathbf{E}$ is returned.
\begin{algorithm}[ht]
\begin{algorithmic}
\Function{ModelEliteSet}{$\mat{X}, \mat{Y}, \mathbf{M}, \mathbf{e}, m, m_\text{elite}$}
\State $\surrogate \leftarrow \textproc{GaussianProcess}(\mat{X}, \mat{Y}, \text{kernel}, \text{optimizer})$
\State $\mat{X}_\text{m} \sim \mathbf{M}(\;\cdot \mid \vec{\theta}_k)$ where $\mat{X}_\text{m} \in \mathbb{R}^{10m}$
\State $\mathbf{\hat{\mat{Y}}}_\text{m} \leftarrow \surrogate(\vec{x}_\text{m})$ for $\vec{x}_\text{m} \in \mat{X}_\text{m}$
\State $\mathbf{e}_\text{model} \leftarrow$ store top $10m_\text{elite}$ from $\mathbf{\hat{\mat{Y}}}_\text{m}$
\State $\mathbf{e}_\text{sub} \leftarrow \textproc{SubEliteSet}(\surrogate, \mathbf{M}, \mathbf{e})$
\State $\mathbf{E} \leftarrow \{ \mathbf{e} \} \cup \{ \mathbf{e}_\text{model} \} \cup \{ \mathbf{e}_\text{sub} \}$ \algorithmiccomment{elite set}
\State \Return $\mathbf{E}$
\EndFunction
\end{algorithmic}
\caption{\label{alg:model_elite_set} Modeling elite set.}
\end{algorithm}
To encourage exploration of promising areas of the design space, the algorithm \smallcaps{SubEliteSet} focuses on the already marked true-elites $\mathbf{e}$.
Each elite $e_x \in \mathbf{e}$ is used as the mean of a new multivariate Gaussian distribution with covariance inherited from the distribution $\mathbf{M}$.
The collection of subcomponent distributions is stored in $\mathbf{m}$.
The idea is to use the information given to us by the true-elites to emphasize areas of the design space that look promising.
For each distribution $\mathbf{m}_i \in \mathbf{m}$ we run a subroutine call to the standard CE-method to fit the distribution $\mathbf{m}_i$ using the surrogate model $\surrogate$.
Then the best objective function value is added to the subcomponent-elite set $\mathbf{e}_\text{sub}$, and after iterating the full set is returned.
Note that we use $\theta_\text{CE}$ to denote the parameters for the CE-method algorithm.
In our case, we recommend using a small $k_\text{max}$ of around $2$ so the subcomponent-elites do not over-fit to the surrogate model but have enough CE-method iterations to tend towards optimal.
\begin{algorithm}[ht]
\begin{algorithmic}
\Function{SubEliteSet}{$\surrogate, \mathbf{M}, \mathbf{e}$}
\State $\mathbf{e}_\text{sub} \leftarrow \emptyset$
\State $\mathbf{m} \leftarrow \{ e_x \in \mathbf{e} \mid \Normal(e_x, \mathbf{M}.\Sigma) \}$
\For {$\mathbf{m}_i \in \mathbf{m}$}
\State $\mathbf{m}_i \leftarrow \textproc{CrossEntropyMethod}(\surrogate, \mathbf{m}_i \mid \theta_{\text{CE}})$
\State $\mathbf{e}_\text{sub} \leftarrow \{\mathbf{e}_\text{sub}\} \cup \{\textproc{Best}(\mathbf{m}_i)\}$
\EndFor
\State \Return $\mathbf{e}_\text{sub}$
\EndFunction
\end{algorithmic}
\caption{\label{alg:sub_elite_set} Subcomponent elite set.}
\end{algorithm}
\subsection{Cross-Entropy Mixture Method} \label{sec:alg_ce_mixture}
We refer to the variant of our CE-surrogate method that takes an input \textit{mixture model} $\mathbf{M}$ as the cross-entropy mixture method (CE-mixture).
The CE-mixture algorithm is identical to the CE-surrogate algorithm, but calls a custom \smallcaps{Fit} function to fit a mixture model to the elite set $\mathbf{E}$.
The input distribution $\mathbf{M}$ is cast to a mixture model using the subcomponent distributions $\mathbf{m}$ as the components of the mixture.
We use the default uniform weighting for each mixture component.
The mixture model $\mathbf{M}$ is then fit using the expectation-maximization algorithm shown in \cref{alg:em}, and the resulting distribution is returned.
The idea is to use the distributions in $\mathbf{m}$ that are centered around each true-elite as the components of the casted mixture model.
Therefore, we would expect better performance of the CE-mixture method when the objective function has many competing local minima.
Results in \cref{sec:results} aim to show this behavior.
\begin{algorithm}[ht]
\begin{algorithmic}
\Function{Fit}{$\mathbf{M}, \mathbf{m}, \mathbf{E}$}
\State $\mathbf{M} \leftarrow \operatorname{Mixture}( \mathbf{m} )$
\State $\mathbf{\hat{\vec{\theta}}} \leftarrow \textproc{ExpectationMaximization}(\mathbf{M}, \mathbf{E})$
\State \Return $\mathbf{M}(\;\cdot \mid \mathbf{\hat{\vec{\theta}}})$
\EndFunction
\end{algorithmic}
\caption{\label{alg:ce_mixture_fit} Fitting mixture models (used by CE-mixture).}
\end{algorithm}
\subsection{Evaluation Scheduling} \label{sec:alg_eval_schedule}
Given the nature of the CE-method, we expect the covariance to shrink over time, thus resulting in a solution with higher confidence.
Yet if each iteration is given the same number of objective function evaluations $m$, there is the potential for elite samples from early iterations dominating the convergence.
Therefore, we would like to redistribute the objective function evaluations throughout the iterations to use more truth information early in the process.
We call these heuristics \textit{evaluation schedules}.
One way to achieve this is to reallocate the evaluations according to a Geometric distribution.
Evaluation schedules can also be ad-hoc and manually prescribed based on the current iteration.
We provide the evaluation schedule we use that follows a Geometric distribution with parameter $p$ in \cref{alg:evaluation_schedule}.
We denote $G \sim \Geo(p)$ to be a random variable that follows a truncated Geometric distribution with the probability mass function $p_G(k) = p(1 - p)^k$ for $k \in \{0, 1, 2, \ldots, k_\text{max}\}$.
Note the use of the integer rounding function (e.g., $\round{x}$), which we later have to compensate for towards the final iterations.
Results in \cref{sec:results} compare values of $p$ that control the redistribution of evaluations.
\begin{algorithm}[ht]
\begin{algorithmic}
\Function{EvaluationSchedule}{$k, k_\text{max}$}
\State $G \sim \Geo(p)$
\State $N_\text{max} \leftarrow k_\text{max} \cdot m$
\State $m \leftarrow \round{N_\text{max} \cdot p_G(k)}$
\If{$k = k_\text{max}$}
\State $s \leftarrow \displaystyle\sum_{i=1}^{k_\text{max}-1} \round{N_\text{max} \cdot p_G(i)}$
\State $m \leftarrow \min(N_\text{max} - s, N_\text{max} - m)$
\EndIf
\State $m_\text{elite} \leftarrow \min(m_\text{elite}, m)$
\State \Return ($m, m_\text{elite}$)
\EndFunction
\end{algorithmic}
\caption{\label{alg:evaluation_schedule} Evaluation schedule using a Geometric distr.}
\end{algorithm}
\section{Experiments} \label{sec:experiments}
In this section, we detail the experiments we ran to compare the CE-method variants and evaluation schedules.
We first introduce a test objective function we created to stress the issue of converging to local minima.
We then describe the experimental setup for each of our experiments and provide an analysis and results.
\subsection{Test Objective Function Generation}
\begin{figure*}[!t]
\centering
\resizebox{0.8\textwidth}{!}{\input{sierra_group.tex}}
\caption{
\label{fig:sierra}
Example test objective functions generated using the sierra function.
}
\end{figure*}
To stress the cross-entropy method and its variants, we created a test objective function called \textit{sierra} that is generated from a mixture model comprised of $49$ multivariate Gaussian distributions.
We chose this construction so that we can use the negative peeks of the component distributions as local minima and can force a global minimum centered at our desired $\mathbf{\tilde{\vec{\mu}}}$.
The construction of the sierra test function can be controlled by parameters that define the spread of the local minima.
We first start with the center defined by a mean vector $\mathbf{\tilde{\vec{\mu}}}$ and we use a common covariance $\mathbf{\tilde{\mat{\Sigma}}}$:
\begin{align*}
\mathbf{\tilde{\vec{\mu}}} &= [\mu_1, \mu_2], \quad \mathbf{\tilde{\mat{\Sigma}}} = \begin{bmatrix}\sigma & 0\\ 0 & \sigma \end{bmatrix}
\end{align*}
Next, we use the parameter $\delta$ that controls the clustered distance between symmetric points:
\begin{align*}
\mat{G} &= \left\{[+\delta, +\delta], [+\delta, -\delta], [-\delta, +\delta], [-\delta, -\delta]\right\}
\end{align*}
We chose points $\mat{P}$ to fan out the clustered minima relative to the center defined by $\mathbf{\tilde{\vec{\mu}}}$:
\begin{align*}
\mat{P} &= \left\{[0, 0], [1, 1], [2, 0], [3, 1], [0, 2], [1, 3]\right\}
\end{align*}
The vector $\vec{s}$ is used to control the $\pm$ distance to create an `s' shape comprised of minima, using the standard deviation $\sigma$:
$\vec{s} = \begin{bmatrix}+\sigma, -\sigma \end{bmatrix}$.
We set the following default parameters: standard deviation $\sigma=3$, spread rate $\eta=6$, and cluster distance $\delta=2$.
We can also control if the local minima clusters ``decay'', thus making those local minima less distinct (where $\text{decay} \in \{0, 1\})$.
The parameters that define the sierra function are collected into $\vec{\theta} = \langle \mathbf{\tilde{\vec{\mu}}}, \mathbf{\tilde{\mat{\Sigma}}}, \mat{G}, \mat{P}, \vec{s} \rangle$.
Using these parameters, we can define the mixture model used by the sierra function as:
\begin{gather*}
\mathbf{M}_\mathcal{S} \sim \operatorname{Mixture}\left(\left\{ \vec{\theta} ~\Big|~ \Normal\left(\vec{g} + s\vec{p}_i + \mathbf{\tilde{\vec{\mu}}},\; \mathbf{\tilde{\mat{\Sigma}}} \cdot i^{\text{decay}}/\eta \right) \right\} \right)\\
\text{for } (\vec{g}, \vec{p}_i, s) \in (\mat{G}, \mat{P}, \vec{s})
\end{gather*}
We add a final component to be our global minimum centered at $\mathbf{\tilde{\vec{\mu}}}$ and with a covariance scaled by $\sigma\eta$. Namely, the global minimum is $\vec{x}^* = \mathbb{E}[\Normal(\mathbf{\tilde{\vec{\mu}}}, \mathbf{\tilde{\mat{\Sigma}}}/(\sigma\eta))] = \mathbf{\tilde{\vec{\mu}}}$.
We can now use this constant mixture model with $49$ components and define the sierra objective function $\mathcal{S}(\vec{x})$ to be the negative probability density of the mixture at input $\vec{x}$ with uniform weights:
\begin{align*}
\mathcal{S}(\vec{x}) &= -P(\mathbf{M}_\mathcal{S} = \vec{x}) = -\frac{1}{|\mathbf{M}_\mathcal{S}|}\sum_{j=1}^{n}\Normal(\vec{x} \mid \vec{\mu}_j, \mat{\Sigma}_j)
\end{align*}
An example of six different objective functions generated using the sierra function are shown in \cref{fig:sierra}, sweeping over the spread rate $\eta$, with and without decay.
\subsection{Experimental Setup} \label{sec:experiment_setup}
Experiments were run to stress a variety of behaviors of each CE-method variant.
The experiments are split into two categories: algorithmic and scheduling.
The algorithmic category aims to compare features of each CE-method variant while holding common parameters constant (for a better comparison).
While the scheduling category experiments with evaluation scheduling heuristics.
\begin{figure*}[!t]
\centering
\subfloat[The cross-entropy method.]
\resizebox{0.3\textwidth}{!}{\input{k5_ce_method.pgf}}
}
\subfloat[The cross-entropy surrogate method.]
\resizebox{0.3\textwidth}{!}{\input{k5_ce_surrogate.pgf}}
}
\subfloat[The cross-entropy mixture method.]
\resizebox{0.3\textwidth}{!}{\input{k5_ce_mixture.pgf}}
}
\caption{
\label{fig:k5} Iteration $k=5$ illustrated for each algorithm. The covariance is shown by the contours.
}
\end{figure*}
Because the algorithms are stochastic, we run each experiment with 50 different random number generator seed values.
To evaluate the performance of the algorithms in their respective experiments, we define three metrics.
First, we define the average ``optimal'' value $\bar{b}_v$ to be the average of the best so-far objective function value (termed ``optimal'' in the context of each algorithm). Again, we emphasize that we average over the 50 seed values to gather meaningful statistics.
Another metric we monitor is the average distance to the true global optimal $\bar{b}_d = \norm{\vec{b}_{\vec{x}} - \vec{x}^*}$, where $\vec{b}_{\vec{x}}$ denotes the $\vec{x}$-value associated with the ``optimal''.
We make the distinction between these metrics to show both ``closeness'' in \textit{value} to the global minimum and ``closeness'' in the \textit{design space} to the global minimum.
Our final metric looks at the average runtime of each algorithm, noting that our goal is to off-load computationally expensive objective function calls to the surrogate model.
For all of the experiments, we use a common setting of the following parameters for the sierra test function (shown in the top-right plot in \cref{fig:sierra}):
\begin{equation*}
(\mathbf{\tilde{\vec{\mu}}} =[0,0],\; \sigma=3,\; \delta=2,\; \eta=6,\; \text{decay} = 1)
\end{equation*}
\subsubsection{Algorithmic Experiments} \label{sec:alg_experiments}
We run three separate algorithmic experiments, each to test a specific feature.
For our first algorithmic experiment (1A), we want to test each algorithm when the user-defined mean is centered at the global minimum and the covariance is arbitrarily wide enough to cover the design space.
Let $\mathbf{M}$ be a distribution parameterized by $\vec{\theta} = (\vec{\mu}, \mat{\Sigma})$, and for experiment (1A) we set the following:
\begin{equation*}
\vec\mu^{(\text{1A})} = [0, 0] \qquad
\mat\Sigma^{(\text{1A})} = \begin{bmatrix}
200 & 0\\
0 & 200
\end{bmatrix}
\end{equation*}
For our second algorithmic experiment (1B), we test a mean that is far off-centered with a wider covariance:
\begin{equation*}
\vec\mu^{(\text{1B})} = [-50, -50] \qquad
\mat\Sigma^{(\text{1B})} = \begin{bmatrix}
2000 & 0\\
0 & 2000
\end{bmatrix}
\end{equation*}
This experiment is used to test the ``exploration'' of the CE-method variants introduced in this work.
In experiments (1A) and (1B), we set the following common parameters across each CE-method variant:
\begin{equation*}
(k_\text{max} = 10,\; m=10,\; m_\text{elite}=5)^{(\text{1A,1B})}
\end{equation*}
This results in $m\cdot k_\text{max} = 100$ objective function evaluations, which we define to be \textit{relatively} low.
For our third algorithmic experiment (1C), we want to test how each variant responds to an extremely low number of function evaluations.
This sparse experiment sets the common CE-method parameters to:
\begin{equation*}
(k_\text{max} = 10,\; m=5,\; m_\text{elite}=3)^{(\text{1C})}
\end{equation*}
This results in $m\cdot k_\text{max} = 50$ objective function evaluations, which we defined to be \textit{extremely} low.
We use the same mean and covariance defined for experiment (1A):
\begin{equation*}
\vec\mu^{(\text{1C})} = [0, 0] \qquad
\mat\Sigma^{(\text{1C})} = \begin{bmatrix}
200 & 0\\
0 & 200
\end{bmatrix}
\end{equation*}
\subsubsection{Scheduling Experiments} \label{sec:schedule_experiments}
In our final experiment (2), we test the evaluation scheduling heuristics which are based on the Geometric distribution.
We sweep over the parameter $p$ that determines the Geometric distribution which controls the redistribution of objective function evaluations.
In this experiment, we compare the CE-surrogate methods using the same setup as experiment (1B), namely the far off-centered mean.
We chose this setup to analyze exploration schemes when given very little information about the true objective function.
\subsection{Results and Analysis} \label{sec:results}
\begin{figure}[!hb]
\resizebox{0.9\columnwidth}{!}{\input{experiment1a.tex}}
\caption{
\label{fig:experiment_1a}
Average optimal value for experiment (1A) when the initial mean is centered at the global minimum and the covariance sufficiently covers the design space.
}
\end{figure}
\Cref{fig:experiment_1a} shows the average value of the current optimal $\bar{b}_v$ for the three algorithms for experiment (1A).
One standard deviation is plotted in the shaded region.
Notice that the standard CE-method converges to a local minima before $k_\text{max}$ is reached.
Both CE-surrogate method and CE-mixture stay below the standard CE-method curve, highlighting the mitigation of convergence to local minima.
Minor differences can be seen between CE-surrogate and CE-mixture, differing slightly towards the tail in favor of CE-surrogate.
The average runtime of the algorithms along with the performance metrics are shown together for each experiment in \cref{tab:results}.
\begin{table}[!ht]
\centering
\caption{\label{tab:results} Experimental results.}
\begin{tabular}{cllll}
\toprule
\textbf{Exper.} & \textbf{Algorithm} & \textbf{Runtime} & $\bar{b}_v$ & $\bar{b}_d$\\
\midrule
\multirow{3}{*}{1A} & CE-method & \textbf{0.029 $\operatorname{s}$} & $-$0.0134 & 23.48\\
&CE-surrogate & 1.47 $\operatorname{s}$ & \textbf{\boldmath$-$0.0179} & \textbf{12.23}\\
&CE-mixture & 9.17 $\operatorname{s}$ & $-$0.0169 & 16.87\\
\midrule
\multirow{3}{*}{1B} & CE-method & \textbf{0.046 $\operatorname{s}$} & $-$0.0032 & 138.87\\
&CE-surrogate & 11.82 $\operatorname{s}$ & \textbf{\boldmath$-$0.0156} & \textbf{18.24}\\
&CE-mixture & 28.10 $\operatorname{s}$ & $-$0.0146 & 33.30\\
\midrule
\multirow{3}{*}{1C} & CE-method & \textbf{0.052 $\operatorname{s}$} & $-$0.0065 & 43.14\\
&CE-surrogate & 0.474 $\operatorname{s}$ & \textbf{\boldmath$-$0.0156} & \textbf{17.23}\\
&CE-mixture & 2.57 $\operatorname{s}$ & $-$0.0146 & 22.17\\
\midrule
\multirow{3}{*}{2} & CE-surrogate, $\operatorname{Uniform}$ & --- & \textbf{\boldmath$-$0.0193} & \textbf{8.53}\\
&CE-surrogate, $\Geo(0.1)$ & {\color{gray}---} & $-$0.0115 & 25.35\\
&CE-surrogate, $\Geo(0.2)$ & {\color{gray}---} & $-$0.0099 & 27.59\\
&CE-surrogate, $\Geo(0.3)$ & {\color{gray}---} & $-$0.0089 & 30.88\\
\bottomrule
& & & \multicolumn{2}{l}{$-\text{0.0220} \approx\vec{x}^*$}\\
\end{tabular}
\end{table}
An apparent benefit of the standard CE-method is in its simplicity and speed.
As shown in \cref{tab:results}, the CE-method is the fastest approach by about 2-3 orders of magnitude compared to CE-surrogate and CE-mixture.
The CE-mixture method is notably the slowest approach.
Although the runtime is also based on the objective function being tested, recall that we are using the same number of true objective function calls in each algorithm, and the metrics we are concerned with in optimization are to minimize $\bar{b}_v$ and $\bar{b}_d$.
We can see that the CE-surrogate method consistently out performs the other methods.
Surprisingly, a uniform evaluation schedule performs the best even in the sparse scenario where the initial mean is far away from the global optimal.
\begin{figure}[!ht]
\resizebox{0.9\columnwidth}{!}{\input{experiment1b.tex}}
\caption{
\label{fig:experiment_1b}
Average optimal value for experiment (1B) when the initial mean is far from the global minimum with a wide covariance.
}
\end{figure}
When the initial mean of the input distribution is placed far away from the global optimal, the CE-method tends to converge prematurely as shown in \cref{fig:experiment_1b}.
This scenario is illustrated in \cref{fig:example_1b}.
We can see that both CE-surrogate and CE-mixture perform well in this case.
\begin{figure}[!h]
\centering
\resizebox{0.7\columnwidth}{!}{\input{example1b.pgf}}
\caption{
\label{fig:example_1b}
First iteration of the scenario in experiment (1B) where the initial distribution is far away form the global optimal. The red dots indicate the true-elites, the black dots with white outlines indicate the ``non-elites'' evaluated from the true objective function, and the white dots with black outlines indicate the samples evaluated using the surrogate model.
}
\end{figure}
\begin{figure}[!ht]
\resizebox{0.9\columnwidth}{!}{\input{experiment1c.tex}}
\caption{
\label{fig:experiment_1c}
Average optimal value for experiment (1C) when we restrict the number of objective function calls.
}
\end{figure}
Given the same centered mean as before, when we restrict the number of objective function calls even further to just 50 we see interesting behavior.
Notice that the results of experiment (1C) shown in \cref{fig:experiment_1c} follow a curve closer to the far away mean from experiment (1B) than from the same setup as experiment (1A). Also notice that the CE-surrogate results cap out at iteration 9 due to the evaluation schedule front-loading the objective function calls, thus leaving none for the final iteration (while still maintaining the same total number of evaluations of 50).
\section{Conclusion} \label{sec:conclusion}
We presented variants of the popular cross-entropy method for optimization of objective functions with multiple local minima.
Using a Gaussian processes-based surrogate model, we can use the same number of true objective function evaluations and achieve better performance than the standard CE-method on average.
We also explored the use of a Gaussian mixture model to help find global minimum in multimodal objective functions.
We introduce a parameterized test objective function with a controllable global minimum and spread of local minima.
Using this test function, we showed that the CE-surrogate algorithm achieves the best performance relative to the standard CE-method, each using the same number of true objective function evaluations.
\printbibliography
\end{document}
|
2024-02-18T23:40:24.786Z
|
2020-09-22T02:02:21.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2009.09043","language":"en","timestamp":1600740141000,"url":"https:\/\/arxiv.org\/abs\/2009.09043","yymm":"2009"}
|
proofpile-arXiv_000-10084
|
{"provenance":"002.jsonl.gz:10085"}
| null | null |
\section{Introduction}\label{sec:introduction}}
With today’s ever-increasing ease of access to the internet and information, we have reached a point of information overload. Users often find themselves in a dilemma where they spend more time sorting through the clutter of information rather than absorbing the information itself. Over the last decade, there has been a push towards creating accurate and efficient recommendation systems to help users sort through this clutter and improve the experience of obtaining information. However, with the recent increase in research in Machine Learning and Deep Learning, researchers have been experimenting with using deep learning techniques to provide recommendations rather than relying on traditional recommendation techniques like probabilistic matrix factorization \cite{mnih2008probabilistic} and Naive Bayes classifier \cite{aiguzhinov2010similarity}.
The recent emergence of information overload has caused decreased productivity, increased frustration, and an overall more negative user experience \cite{zou2014information}. For example, users now often spend more time trying to find a movie on Netflix to watch rather than actually watching movies \cite{gomez2016netflix}.
As with any industry, technology companies ultimately focus on their revenues and profits as key metrics for optimization. Recommendation systems play a significant role in increasing revenues, retaining customers and users, and gaining competitive advantages over competitors \cite{recsys}. By using recommendation systems, companies can encourage users to stay engaged with their application/website/service, which allows the companies to show more advertisements, attract new clients, and retain existing clients \cite{vaidya2017recommender}. From an e-commerce perspective, recommendation systems can entice customers to buy more. Carefully selected recommendations can make customers more satisfied, leading them to buy more items than they previously anticipated.
Companies store large amounts of data in databases containing transactional, accounting, inventory, and customer contact information. These data can be analyzed to understand customer purchasing behavior, which allows companies to target customers through product recommendations based on past purchases or activities leading to increased revenue. Companies such as Amazon and Netflix are already using algorithms to give recommendations, and other companies are looking to implement similar recommendation system algorithms. The development of e-commerce solutions pioneered by Amazon not only demonstrated the importance of incorporating web data to meet customers’ shopping needs but also leveraging customer data to enhance sales using machine learning algorithms. In another example, Uber Eats food recommendations are based on many factors that take into account real-time information such as user queries, location, time of day/week and historical information about the user’s purchases, restaurants and delivery partners used \cite{uber}.
A large parts supply company (the Company) has gathered a large amount of sales and transactional information and has an extensive product database. However, the Company performed minimal analysis of their databases from the perspective of product recommendations to customers. Furthermore, the Company’s databases are large, containing more than 500,000 stock keeping units (SKUs), 20 million rows of transactional data as well as information related to their 200,000 customers. The two primary databases are Invoiced Orders (Invoicedorders.csv) and Current Items (ITEMCURRENT.csv). The \say{Invoiced Orders} database contains the customers’ transaction data. The \say{Current Items} database contains detailed information about the products offered by the Company. The Company wants to use the available data to provide a list of 12 products the customers may be interested in purchasing and to improve the user experience of the Company's website.
Collaborative Filtering (CF) was used to solve this problem, it relies on the intuitive idea that similar users tend to like different items similarly. CF methods use an a-priori available set of user-item ratings to learn the interdependencies among users and items, and then predict a user’s rating of an item either via the neighboring items’ ratings (neighbor-based \cite{das2007google}, \cite{sarwar2001item} ) or by inferring latent factors that find similar users and items in a low-dimensional embedding (latent factor-based \cite{hu2008collaborative}, \cite{he2017neural}, \cite{rendle2009bpr}, \cite{moussawi2018towards} ). The feedback can be explicit, i.e., the user either likes an item or dislikes an item. For example, the user either clicks “thumbs up” or “thumbs down” on a youtube video or rates a movie on a scale of one to five stars. Alternatively, the feedback can be implicit, i.e., the feedback derived from the browsing behavior of the user e.g., the user clicked on/purchased a product, checked into a venue, or viewed an article.
The objective of this paper is to build a recommendation system for the Company’s online customers using machine learning techniques, which will allow the Company to predict their customers’ needs better and provide product recommendations tailored to the customers’ purchasing behaviors. The contributions associated with this work are listed below:
\begin{itemize
\item The evaluation and testing of the hypothesis that Collaborative Filtering with latent variable models can capture the historical, transactional and sparse customer purchasing behaviors to provide personalized product recommendations. This paper work continues related efforts in \cite{hu2008collaborative}, \cite{he2017neural}, \cite{rendle2009bpr}, \cite{moussawi2018towards} to understand the hidden factors in customers’ purchasing behaviors.
\item The construction and evaluation of ranking metrics for four different models based on Matrix Factorization using Alternative Least Squares, Bayesian Personalized Ranking, Neural Collaborative Filtering and Autoencdoer Collaborative Filtering.
\end{itemize}
\section{Related Work}
A recommendation system, often also called a \say{recommender system}, is used to estimate user preferences of items or objects they have not seen yet. Recommendation systems commonly use inputs such as user preferences, item/object features, histories of users-items’ past interactions, temporal data, and spatial data. Generally, there are three types of recommendation systems: Collaborative Filtering systems \cite{ekstrand2011collaborative}, \cite{zhao2010user}, \cite{sarwar2001item} , Content-Based Recommendation systems \cite{zenebe2009representation}, \cite{lops2011content}, and Hybrid systems\cite{burke2002hybrid}, \cite{burke2007hybrid}. Latent-factor or matrix factorization (MF) methods are popular in recommendation systems \cite{koren2009matrix} both for implicit \cite{rendle2009bpr},\cite{hu2008collaborative} and explicit feedback \cite{mnih2008probabilistic}. The latent factor model tries to predict the users' rating on an item by optimizing an objective function and by reconstructing the rating matrix into low-rank dimensional latent factors. Bayesian Personalized Ranking (BPR) \cite{rendle2009bpr} has emerged as one of the best Top-K recommendation models for implicit data. Popular deep learning-based recommendation techniques that utilize multi-layer perceptrons are Neural Collaborative Filtering (NCF) \cite{he2017neural} and deep factorization machine \cite{guo2017deepfm}. In MLP techniques, recommendations are considered as two-way interactions between users’ preferences and the features of an item/object. NCF uses the binary cross-entropy loss function for implicit feedback and the weighted square loss for explicit feedback. Further extension of the work using pairwise ranking loss are proposed by the authors in \cite{niu2018neural}, \cite{song2018neural} to improve the results. The NCF model is extended to cross-domain recommendations like \cite{lian2017cccfnet}, \cite{wang2016collaborative}. DeepFM \cite{guo2017deepfm} is an end-to-end model, able to integrate factorization machine and MLP to find lower and higher-order feature representation. DeepFM is similar to the wide and deep model, a two network deep learning architecture gives recommendations \cite{cheng2016wide}; however, it does not require sophisticated feature engineering. The authors in \cite{lian2018xdeepfm} proposed eXtreme deep factorization machine, which models implicit and explicit features together to improve the performance over DeepFM. Covington et al. proposed an MLP based YouTube recommendation model \cite{covington2016deep}, and the predictor generates a top-n list of videos based on the nearest neighbor scores from the several hundred videos. The authors in \cite{alashkar2017examples} explored and applied MLP in makeup recommendations. This work uses two identical MLPs to model labeled examples and expert rules, respectively, which provides highly precise recommendations.
Autoencoders are also widely used in building recommendation systems. There are two general techniques for using autoencoders as recommendation systems. First, the autoencoder can be used to learn a lower-dimensional feature representation at the bottleneck layer, which means the employement of autoencoder as a dimensionality reduction tool, which is used sequentially with other deep learning techniques for recommendations \cite{zhang2019deep}. On the other hand, one can use autoencoders to fill in the blank values of the rating matrix directly in the reconstruction or decoder layer \cite{zhang2019deep}. Nearly all of the autoencoder alternatives, such as denoising autoencoder, variational autoencoder, contactive autoencoder, and marginalized autoencoder, can be employed to the recommendation task \cite{zhang2019deep}. The AutoRec \cite{sedhain2015autorec} system is a specific implementation of the technique to use the autoencoder as a generative tool for recommendations. The Collaborative Filtering Neural Network \cite{strub2015collaborative}, \cite{strub2016hybrid} is an extension of the AutoRec System. It uses denoising techniques to make the recommendation system more robust. It additionally uses side information (user profile information / item descriptions) to reduce data sparsity issues and the cold start problem, which in return, increases the training speed and robustness, and improves the prediction accuracy. Unlike other autoencoders, which are designed to output rating predictions, Collaborative Denoising Autoencoders (CDAE) models \cite{wu2016collaborative} are used to output ranking predictions and are prone to overfitting, because CDAE provides multiple predictions at once; for example, “What are the top 10 best movies for User A”. The authors in \cite{liang2018variational} proposed a modification of the variational autoencoder called Multi-VAE and Multi-DAE for recommendation tasks using implicit data. The proposed alternative showed better performance than CDAE.
Matrix Factorization with Alternating Least Square (ALS) \cite{hu2008collaborative}, Bayesian Personalized Ranking (BPR) \cite{rendle2009bpr}, Neural Collaborative Filtering (NCF) \cite{he2017neural} and Autoencoder for Collaborative Filtering (ACF) \cite{moussawi2018towards} are the four different latent variable methods that were selected to analyze large customer purchasing behavior patterns because of the implicit feedback between customers and products. These modelling approaches were selected to examine different aspects of providing product recommendations.
\section{Methodology}
The specific details of the dataset cannot be directly summarized in this paper since the data are proprietary. The two primary databases are Invoiced Orders (Invoiced\_orders.csv) and Current Items (ITEM\_CURRENT.csv).
The \say{Invoiced Orders} database contains 20 million historical transactional data from August 2016 to June 2019. It has more than 100 fields such as geographic location, selling location, regional order number, rebate value,
stock quantity, etc. The \say{Current Items} database contains 500,000 unique products and has detailed information about the items offered by the Company, including dimensions, quantity, English, and French text descriptions, etc.
The focus of the analysis was to recommend products/items based on past transaction history; therefore, most of the information is filtered out like location, manufacturer, order\_date, etc. The dataset is filtered using a structured query language (SQL) query where the final dataset has the following information: customer ID, product ID, English and French descriptions of the products, and the number of times a customer purchased a product (ratings).
\subsection{Matrix Factorization with Alternating Least Squares (ALS)}
Latent-factor or Matrix Factorization (MF) \cite{hu2008collaborative} methods are popular in recommendation systems, MF algorithms work by decomposing the user-item interaction matrix into the product of two lower dimensionality rectangular matrices. The latent factors (lower dimensions) otherwise called features, can be found using Singular Value Decomposition (SVD). Let $M$ and $N$ denote the number of users and items, repectively and $k$ denotes the dimensions of latent space. Let $p \in \mathbb{R}^{k \times M} $ be the latent factor matrix for the users, where the $u^{th}$ column $p_u \in \mathbb{R}^k$ is the latent factor for user $u$. Similarly, let $q \in \mathbb{R}^{k \times N} $ be the latent factor for the items, where the $i^{th}$ column $q_i \in \mathbb{R}^k$ is the latent factor for item $i$. The ratings ($\hat{r}_{u, i}$) for user $u$ and item $i$ are given by $\hat{r}_{u, i} = q_i^T p_u$. For implicit rating, the MF can be formulated \cite{hu2008collaborative} as
\begin{equation}
min \sum c_{(u, i)} ( p_{(u, i)} - q_i^T p_u)^2 + \lambda ( (\|q_i\| )^2 + (\|p_u\| )^2 )
\end{equation}
where $c_{(u,i)} = 1 + \alpha r_{(u, i)}$ and $p_{(u, i)} = 1$ if $r_{(u, i)}>0$ and $p_{(u, i)} = 0$ if $r_{(u, i)} = 0$. $r_{(u, i)}$ is a numerical representation of users' preferences (e.g., number of purchases, number of clicks etc.) and $\lambda$ is a regularization parameter. Owing to the term of $q_i^T p_u$, the loss function is non-convex. The gradient descent method can be applied, but this will incur expensive computations. An Alternating Least Square (ALS) algorithm was therefore developed to overcome this issue \cite{hastie2015matrix}. The basic idea of ALS is to learn one of $q$ or $p$ at the time of optimization while keeping the other as constant. ALS makes the objective at each iteration convex and solvable. The alternating between $q$ and $p$ stops when the convergence is optimal. It is worth noting that the ALS iterative computation can be parallelized and/or distributed, which makes the algorithm desirable \cite{yu2014parallel} for use cases where the dataset is large and thus the user-item rating matrix is highly sparse (as typical in recommendation system scenarios).
\subsection{Bayesian Personalized Ranking (BPR)}
BPR \cite{rendle2009bpr} has emerged as one of the best Top-K recommendation models for implicit data. BPR is also a strong baseline, which makes it difficult to beat. BPR falls under the category of one-class collaborative filtering (for example 0 or 1) and pairwise comparison. It considers the recommendation task as a ranking problem and assumes that the user prefers items that they have already observed/interacted with, rather than unobserved/not interacted with items. To learn the relative ranking of items for each user, BPR needs to model negative feedback. BPR uses the pairwise interpretation of positive-only feedback by creating triplets (user, observed item, unobserved item). The positive-only feedback is then transformed into positive and negative feedback in pairs of item $(i,j)$. $D_s$ represents triplets $(u,i,j)$ such that a user $u$ prefers item $i$ over item $j$. The triplets ($D_s$) are sampled \cite{rendle2009bpr} from
\begin{equation}
D_s := (u,i,j)|i \in I_u^+ \land j \in I\backslash I_u^+
\end{equation}
where we let $U$ be the set of all users and $I$ the set of all items, with implicit feedback $S$ defined as $ S \subseteq U \times I$; $I_u^+$ is defined as the items $I_u^+ := { i \in I : (u , i) \in S } $ where user $u$ gives positive feedback; $i$ is a positive item taken from $I_u^+$ and $j$ is a negative item randomly sampled from unobserved/not interacted with items. The author derives an optimization criterion called BPR-OPT \cite{rendle2009bpr}, which is an optimization framework. To provide a personalized ranking of items or recommendations, we need to train a separate model using BPR-OPT. Compared to standard MF or kNN methods, BPR ensures not only the rating predictions but also optimizes the item rankings.
\subsection{Neural Collaborative Filtering (NCF)}
\begin{figure}
\centering
\includegraphics[width=0.49\textwidth]{Images/chap3/ncf.png}
\caption{Neural Collaborative Filtering model architecture}
\label{fig:ncf}
\end{figure}
NCF \cite{he2017neural} is a new neural matrix factorization model that combines both Generalized Matrix Factorization (GMF) and Multi-Layer Perceptron (MLP) to combine the strengths of the linearity and non-linearity given by MF and MLP, respectively, for modeling user-item latent features. The input layer consists of latent vectors of items and users. Figure \ref{fig:ncf} shows the architecture of NCF. The User (u) and Item (i) are used to create low-dimensional embeddings for the user and item. Generalized Matrix Factorization (GMF) combines the two embeddings using the dot product. Multi-layer perceptron (MLP) also produces embeddings for the user and items. However, instead of taking a dot product to obtain the rating, the embeddings are concatenated to create a feature vector that can be passed on to the further layers. The outputs from the final layers of the MLP and GMF are concatenated, called a NeuMF layer to obtain the prediction score. The final layer output of GMF ($\hat{r}_{u,i}$) can be formulated as follows:
\begin{equation}
\hat{r}_{u,i} = a_{out} ( h^T (q_i \odot p_u ) )
\end{equation}
where we let the user latent vector $p_u$ be $P^T v_u^U$ and item latent vector $q_i$ be $Q^T v_i^I$, where $ \odot $ is an element-wise product of vector terms. Additionally, $a_{out}$ and $h$ represent the activation function and weights of the output layer, respectively. The MLP model under the NCF framework is defined as follows. In the input layer, we concatenate the user latent vector $p_u$ and item latent vector $q_i$ as follows:
\begin{equation}
z_1 = \phi_1 (p_u, q_i) = \begin{bmatrix} p_u \\ q_i\end{bmatrix}
\end{equation}
where $z_1$ represents the concatenation of $p_u$ and $q_i$ at the input layer. The hidden layers are formulated as:
\begin{equation}
\phi_l(z_l) = a_{out} (W_l^T z_l + b_l) , (l = 2, 3, ..., L-1)
\end{equation}
where $W_l$, $b_l$, and $a_{out}$ denote the weight matrix, bias vector, and activation function for the $l$-th layer's perceptron, respectively and the output layer is formulated as:
\begin{equation}
\hat{r}_{u,i} = \sigma ( h^T \phi(z_{L-1}))
\end{equation}
where $\hat{r}_{u,i}$ is the predicted rating, $h$ denotes the edge weights and the activation function (sigmoid) of the output layer is defined as $\sigma (x) = \frac{1}{1 + \mathrm{e}^{-x} }$ to restrict the predicted score to be in (0,1). To have more flexibility in the fused model, we use GMF and MLP to learn the embeddings separately and then combine these two models by concatenating their last hidden layer \cite{he2017neural}. We get $\phi^{GMF}$ from GMF and obtain $\phi^{MLP}$ from MLP:
\begin{equation}
\phi^{GMF}_{u, i} = p^{GMF}_u \odot q^{GMF}_i
\end{equation}
\begin{equation}
\phi^{MLP}_{u,i} = a_{out} \big( W_L^T \big( a_{out} \big( .. a_{out} \big( W_2^T \begin{bmatrix} p_u^{MLP} \\ q_i^{MLP}\end{bmatrix} + b_2 \big) ..\big) \big) + b_L \big)
\end{equation}
Lastly, we fuse the output from GMF and MLP:
\begin{equation}
\hat{r}_{u,i} = \sigma \bigg( h^T \begin{bmatrix} \phi^{GMF} \\ \phi^{MLP}\end{bmatrix} \bigg)
\end{equation}
where $\hat{r}_{u,i}$, $\sigma$, $h$, $\phi^{GMF}$ and $\phi^{MLP}$ denote the predicted ratings, sigmoid activation function, edge weights of the output layer, last hidden layer of GMF and last hidden layer of MLP, respectively. By taking a negative log likelihood, we obtain the objective function to minimize for the NCF method:
\begin{equation}
L = - \sum_{ (u,i) \in \mathbb{O} \cup \mathbb{O}^{-}} r_{u,i} log \hat{r}_{u,i} + ( 1 - r_{u,i} ) log ( 1 - \hat{r}_{u,i} )
\end{equation}
where $\mathbb{O}$ denotes the set of observed interactions, $\mathbb{O}^{-}$ denotes the set of negative instances (unobserved interactions), $r_{u,i}$ denotes the actual ratings and $\hat{r}_{u,i}$ denotes the predicted ratings.
\subsection{Autoencoder for Collaborative Filtering (ACF)}
Recently, there has been a significant focus in research on using autoencoders for recommendation systems \cite{sedhain2015autorec}, \cite{strub2015collaborative}. Many different techniques have been proposed, such as denoising architecture \cite{wu2016collaborative}, dropout to increase efficiency, etc. Here, we use a particular type of autoencoder (ACF) \cite{moussawi2018towards} whose model is described in this section. Consider the user-item interaction matrix is represented as $ X \in \{0,1 \}^{|U| \times |I|} $ where $U$ and $I$ are the set of users and items, respectively. If there is an interaction between user $u$ and item $i$ then $X_{u,i} = 1$; otherwise, $X_{u,i} = 0$. Given user $u$ and item $i$, $I_u$ represents a set of items, $u$ has interacted with, and $U_i$ represents a set of users who have interacted with $i$. The autoencoder learns a model $p(x_u | z_u , \theta) = h(g_\theta(z_u))$, where $x_u$ is the user $u$ vector of interactions, $z_u$ represents the user latent factor, $g_\theta$ is an autoencoder parameterized by $\theta$ and $h$ is an activation function that maps the output of $g_\theta$ to probabilities based on the logistic likelihood distribution used to model $p(x_u | z_u , \theta)$. $z_u$ can be computed as a function of $f_\lambda(x_u)$, where $f_\lambda$ is an autoencoder parameterized by $\lambda$. The negative log-likelihood loss function of our model to be minimized is then:
\begin{eqnarray}
- \sum_i \log p(x_u | z_u, \theta)_i = - x_u \cdot \log (g_\theta ( z_u)) \nonumber\\
- ( 1 - x_u) \cdot \log (1 - g_\theta ( z_u))
\end{eqnarray}
For regularization, dropouts are applied at the input layers, and also applied L2 weight decay on $\theta$ and $\lambda$.
\section{Evaluation and Model Specifications}
\subsection{Evaluation Protocols}
To evaluate the efficiency of a product recommendation, we followed the leave-one-out evaluation \cite{he2017neural}, which has been widely used in the literature \cite{bayer2017generic}, \cite{he2016fast}, \cite{rendle2012bpr}. In leave-one-out evaluation, for each user, we keep their latest interaction with the item as a test set, and we use the remaining as a training set for the model. During evaluation, it is too tedious and time-consuming to rank all the items for each user. To overcome this problem, we adopted a common strategy used by the practitioners where we sample 100 random items that are not interacted with by the user as a test set and then rank those 100 items with respect to that user. Leave-one-out evaluation with negative sampling is implemented in this work. The efficiency of the ranked list is measured by a widely used metric in the learning-to-rank problem called Normalized Discounted Cumulative Gain (NDCG) \cite{yilmaz2008simple}. NDCG gives more weight to relevant predictions at the starting of the ranked list of items and discounts relevant predictions that occur farther from the beginning of the ranked list. One-product Hit Ratio is a metric we devised, which is relevant for the business. One-product Hit Ratio intuitively measures whether the test item is present on the ranked list. For example, if the test item present on the ranked list, then the score is 1; otherwise 0.
\subsection{Model Specifications}
The model specifications include the model training procedures and selection of hyperparameters for the Collaborative Filtering models. These models are prepared using Tensorflow \cite{abadi2016tensorflow} and PyTorch \cite{paszke2017automatic} frameworks with the Python programming language. These models are trained on a server with 32 cores of CPU and 256 GB of RAM. The dataset is split into a train, validation and a test set using a customized function, where the split is stratified so that the same set of customers and products will appear in all train, validation and test sets. Hyperparameter tuning is an essential part in machine learning; grid search, random search and bayesian optimization are the three standard methods for hyperparameter tuning. The Bayesian Optimization approach has been proven to outperform other state-of-the-art hyperparameter tuning approaches \cite{snoek2012practical}. Bayesian approach uses past choices made to make a smart choice of hyperparameters for the next set of values to evaluate, through which it reduces the cost of searching for parameters. In this paper, we use Bayesian Optimization for hyperparameter tuning.
For the ALS methodology, we closely followed the implementation of \cite{hu2008collaborative} as mentioned in Section 3.1. We find that training the model in this fashion takes more time, so we experimented with different optimizers and used the Adam optimizer \cite{kingma2014adam} to reduce the training time. To make the training even faster, we converted the python code to Cython \cite{behnel2011cython}, where Cython converts the python code to C code to boost performance. We used Tensorflow's embedding layer to create user-item latent vectors. We used Bayesian Optimization to find the optimal hyperparameters. Table \ref{tab:MF_ALS} displays the hyperparameters of the ALS model. The hyperparameters include user-item latent dimensions, regularization, iterations and scaling factors. We find that the user-item latent dimensions have the most significant impact on performance. To create latent dimension vectors, we used embedding layers whose values were uniform and randomly initialized, and then the values were learned during the model training process. L2 regularization was used. The scaling factor was set to 15. During hyperparameter tuning, the number of iterations was found to be optimal at 50.
\begin{comment}
\begin{figure*}[!t]
\centering
\subfloat[Case I]{\includegraphics[width=2.5in]{Images/chap4/ALS_TL_k_200_reg_le4.png}
\label{fig_first_case}}
\hfil
\subfloat[Case II]{\includegraphics[width=2.5in]{Images/chap4/BPR_TL_k_200_reg_le4.png}
\label{fig_second_case}}
\caption{Simulation results for the network.}
\label{fig_sim}
\end{figure*}
\begin{figure}[!t]
\centering
\includegraphics[width=2.5in]{Images/chap4/ALS_TL_k_200_reg_le4.png}
\caption{MF-ALS Training Loss Graph}
\label{fig_sim}
\end{figure}
\end{comment}
\begin{table}[ht]
\caption{Hyperparameters for MF with ALS models}
\begin{center}
\begin{tabular}{ |c|c|c|c| }
\hline
\textbf{Hyperparameter} & \textbf{Value} \\
\hline
user-item Latent Dimension (k) & 200 \\
Regularization & 0.0001 \\
Learning Rate & 0.01 \\
Iterations (Epochs) & 30 \\
Batch Size & 512 \\
Scaling Factor ($\alpha$) & 15 \\
\hline
\end{tabular}
\end{center}
\label{tab:MF_ALS}
\end{table}
In MF with the ALS method described above, we focused on pointwise loss minimization, which captured the positive or negative user-item interactions separately. However, there may be some hidden information available in the negative user-item interactions for positive user-item interactions. The idea was formulated into pairwise loss minimization by \cite{rendle2009bpr}. We train the triplets using the pairwise loss function, as discussed in Section 3.5. The BPR latent factors are constructed using TensorFlow’s embedding layer, and we find that BPR has the same training characteristics as MF with the ALS method. However, BPR takes more iterations or epochs to converge to local minima. This problem arises because of the pairwise loss function, which needs to compute more gradients during the training procedure. The hyperparameter for this methodology consists of user-item latent dimensions (factors), the number of epochs, the learning rate for the optimizer, and regularization to prevent overfitting. By using Bayesian Optimization, we find the best hyperparameters for the BPR approach. Table \ref{tab:BPR} displays the hyperparameters of the BPR model.
\begin{table}[ht]
\caption{Hyperparameters for BPR model}
\begin{center}
\begin{tabular}{ |c|c|c|c| }
\hline
\textbf{Hyperparameter} & \textbf{Value} \\
\hline
user-item Latent Dimension (k) & 200 \\
Learning Rate & 0.01 \\
Regularization & 0.0001 \\
Iterations (Epochs) & 200 \\
Batch Size & 512\\
\hline
\end{tabular}
\end{center}
\label{tab:BPR}
\end{table}
In ALS and BPR methods, we tried to find the linear relationships that exist between the user and the item. However, in NCF, we try to find both linear and non-linear relationships in user-item interactions by utilizing the power of neural networks. As discussed in Section 3.3, we create GMF, MLP, and Fusion of GMF \& MLP (NeuMF) models. The goal of the NCF model is to train and minimize the binary cross-entropy loss function as defined in equation (10). The hyperparameters for the NCF methodology consist of n\_factors, layer\_size, n\_epochs, learning\_rate, and batch\_size, where n\_factors represents the dimensions of the latent space; layer\_size represents the sizes of the input and hidden layers of the MLP; n\_epochs is the number of iterations to run the training. In general, we find that increasing n\_factors increases the quality of predictions. The user/item labels are mapped to real-valued latent vectors with Tensorflow's embedding layers. Table \ref{tab:NCF} describes the optimal hyperparameters for the NCF models using Bayesian Optimization. We found that, in training a NeuMF model, using pre-trained model weights of GMF and MLP is far better in reducing cross-entropy loss than using gaussian normal sampled initialized weights.
\begin{table}[ht]
\caption{Hyperparameters for NCF model}
\begin{center}
\begin{tabular}{ |c|c|c|c| }
\hline
\textbf{Hyperparameter} & \textbf{Value} \\
\hline
n\_factors & 16 \\
layer\_size & [64,32,16] \\
n\_epochs & 50 \\
learning\_rate & 0.001 \\
batch\_size & 256 \\
\hline
\end{tabular}
\end{center}
\label{tab:NCF}
\end{table}
The ACF approach is entirely different from previous approaches. Here, we use the Autoencoder both as a tool for dimensionality reduction as well as a learning algorithm to discover the hidden user-item latent features in the dataset. We use the PyTorch framework to build the ACF model. The objective of the ACF model is to minimize the loss function which was defined in equation (11). Typically, an Autoencoder has two parts: an encoder and a decoder. PyTorch’s nn.embedding layers are used to build the encoder, whereas the same nn.embedding layers are used to build the decoder, but we reverse the encoder’s architecture. User/item labels are mapped to the latent space using an encoder. As the name suggests, the decoder is used to decode the encoder to get the original user/item labels from the latent space. Instead of randomly assigning weights to the embedding layer, Xavier’s initialization is used. An Adam optimizer, as well as the ReLU activation function, are used throughout the process. The hyperparameter for the ACF model consists of hidden\_layer, noise\_prob (dropout probability at the input layer), dropout\_prob (dropout probability at the bottleneck layer), lr (learning rate), weight\_decay, batch\_size, and num\_epochs. Like ALS, BPR and NCF methodologies, we use Bayesian Optimization to find the best hyperparameters. Table \ref{tab:ACF} shows the hyperparameters for the ACF model.
\begin{table}[ht]
\caption{Hyperparameters for ACF model}
\begin{center}
\begin{tabular}{ |c|c|c|c| }
\hline
\textbf{Hyperparameter} & \textbf{Value} \\
\hline
hidden\_layer & 7 \\
noise\_prob & 0.3 \\
dropout\_prob & 0.2 \\
lr (learning rate) & 0.001 \\
weight\_decay & 2e-5 \\
batch\_size & 256 \\
num\_epochs & 30 \\
\hline
\end{tabular}
\end{center}
\label{tab:ACF}
\end{table}
\section{Results and Discussion}
\subsection{Results}
The average performance of the CF model predictions for the test set using leave-one-out evaluation are displayed in Table \ref{tab:Results}. The NDCG metric was used to calculate the performance of the model, according to the business objective, we choose twelve products to recommend; hence, we used NDCG@12. NDCG@12 gives us the twelve most relevant ranked products based on customer purchasing behavior. The results in the Table \ref{tab:Results} demonstrate that the NCF model achieved the best average performance for NDCG@12 over the leave-one-out evaluation test interval. One-Product Hit Ratio is 1 for ALS, BPR, NCF, and ACF; which means the algorithms at least predict one product that the customer intend to buy.
\begin{table}[ht]
\caption{CF average performance metrics}
\begin{center}
\begin{tabular}{ |c|c|c|c| }
\hline
\textbf{Average Predictions} & \textbf{NDCG@12} & \textbf{One-Product Hit Ratio} \\
\hline
ALS & 0.577 $\pm$ 0.055 & 1 \\
\hline
BPR & 0.636 $\pm$ 0.048 & 1\\
\hline
NCF & 0.724 $\pm$ 0.049 & 1\\
\hline
ACF & 0.604 $\pm$ 0.069 & 1\\
\hline
\end{tabular}
\end{center}
\label{tab:Results}
\end{table}
We consider ALS as the baseline since it tries to find the most direct relationship between the user and item using the matrix factorization technique. From the results in the Table \ref{tab:Results}, the performance of ALS is good over the test period. BPR performed better than ALS, which was expected, because it also implemented matrix factorization with the same user/item latent factors to discover the hidden features in the dataset. BPR varies from ALS in that we implemented pairwise training between positive and negative interactions, which was described in Section 3.2. NCF performed better than all other models, which was expected since it combines the linearity and non-linearity of matrix factorization and neural networks, respectively, which allows NCF to find more relevant hidden features over the dataset. The ACF performed better than ALS but not better than BPR and NCF, which is not always the case. The efficiency of an autoencoder depends on the data; it learns to capture as much information as possible rather than relevant information \cite{ghasemi2018neural}. This means that if we have less user-item interactions, the autoencoder may lose this information.
\subsection{Discussion}
For the specific dataset investigated in this paper, CF with NCF produced the best test performance. MF with ALS is considered as a baseline in this paper since it focused on direct linear relationships between a user and an item. From the results shown in Table 5.1, we can see that all the models performed relatively well compared with the baseline model. NCF has a significant 14.7\% increase in NDCG over the baseline and BPR has a 5.9\% increase in NDCG. Compared with BPR and NCF, ACF performance is relatively less, with a 2.7\% increase in NDCG over the baseline. The BPR matrix factorization model achieved better NDCG performance than the MF with the ALS model (the baseline), which indicates that the pairwise training in BPR was effective in exploring the parameter space as opposed to the pointwise training used in ALS, leading to the improved performance of NDCG, though both BPR and ALS used matrix factorization with the same user/item latent factor vector dimensions. The ACF did not perform as well as the BPR approach in terms of NDCG metrics. The huge sparsity in the dataset investigated likely limited the usefulness of the autoencoder-based deep learning approaches. As discussed before, the autoencoder captures more irrelevant information as opposed to relevant information; since the dataset is sparse in nature, it is likely to capture non-relevant information on the user-item interactions. Although the autoencoder can be used as a dimensionality reduction tool, the sparsity issue does not allow us to use the autoencoder to its full potential. The NCF model combines matrix factorization and neural networks at the final layer, which allows the model to create a more effective and useful hidden representation of the sparse dataset that allows the model to perform better than other models.
\begin{figure}
\centering
\begin{subfigure}[b]{0.24\textwidth}
\centering
\includegraphics[width=\textwidth]{Images/chap5/BPR_k_500_lr_01_reg_01.png}
\caption{k = 500}
\label{fig:BPR500}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.24\textwidth}
\centering
\includegraphics[width=\textwidth]{Images/chap5/BPR_k_200_lr_01_reg_01.png}
\caption{k = 200}
\label{fig:BPR200}
\end{subfigure}
\caption{BPR Training Loss Graph}
\label{fig:BPRK}
\end{figure}
Since the factors determined by the models are hidden, it is difficult to understand which factors of the user/item interaction are driving the results. We know that the hyperparameters of the model greatly affect performance. For the BPR model, the major hyperparameters that drive the results are user-item latent dimensional factors (k) and regularization. From Figure \ref{fig:BPR500}, we can see that for higher k values, the loss is relatively higher in the first few epochs compared with lower k values (Figure \ref{fig:BPR200}) by keeping other hyperparameters the same. Also, the model takes a significantly higher time to train when the k value is large. Another important hyperparameter is regularization; this is a technique used for tuning the loss function by adding an additional penalty term. From Figure \ref{fig:BPRKR1}, we can see that the pairwise loss is starting around 0.90 for a regularization value of 0.001. From Figure \ref{fig:BPRKR2}, it can be seen that the pairwise loss starts at 0.68 for a regularization value of 0.0001. Therefore, it is apparent that regularization is a driving factor in reducing the loss.
\begin{figure}
\centering
\begin{subfigure}[b]{0.24\textwidth}
\centering
\includegraphics[width=\textwidth]{Images/chap5/BPR_k_200_reg_001.png}
\caption{k = 200, reg = 0.001}
\label{fig:BPRKR1}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.24\textwidth}
\centering
\includegraphics[width=\textwidth]{Images/chap5/BPR_k_200_reg_0001.png}
\caption{k = 200, reg = 0.0001}
\label{fig:BPRKR2}
\end{subfigure}
\caption{BPR Training Loss Graph}
\label{fig:BPRKR}
\end{figure}
For the NCF model, pre-training GMF and MLP was also an essential hyperparameter in reducing the loss function and achieving better NDCG performance. As described in Section 3.3, the NCF model was constructed by concatenating the last hidden layer of GMF and MLP to learn the hidden features of user-item interactions. The concatenated layer is called the NeuMF layer. The training loss for the GMF and MLP are shown in Figure \ref{fig:GMF} and \ref{fig:MLP}. From Figure \ref{fig:GMF}, we can see that the binary cross-entropy loss for GMF starts at approximately 0.50 and drops down to 0.10 and from Figure \ref{fig:MLP}, it can be seen that the binary cross-entropy loss for MLP begins around at 0.35 and ends at 0.07. We trained the NCF model without pre-training weights and observed the training loss. From Figure \ref{fig:WPT}, we can see that the binary cross-entropy loss is very high, which starts at approximately 0.35. \cite{he2017neural} argued that the pre-trained weights reduced the training loss; we trained the GMF and MLP model separately with an Adam optimizer. The trained GMF and MLP model weights are used to train the NeuMF layer. As a result, the binary cross-entropy training loss reduced drastically and started at 0.064, which is shown in Figure \ref{fig:PT}, whereas the non-pre-trained model started at 0.35. The pre-trained model achieved 0.724 on the NDCG metric, while the non-pre-trained model achieved 0.698, a 2.6\% drop in NDCG performance, showing that pre-training has a significant impact on the training loss. For the ACF model, we implemented different loss functions like mean squared loss and binary cross entropy loss to see any improvement in the overall performance of the model. We found that changing the loss function did not improve the NDCG performance; we believe this is likely due to the sparse nature of the dataset.
\begin{figure}
\centering
\begin{subfigure}[b]{0.24\textwidth}
\centering
\includegraphics[width=\textwidth]{Images/chap5/NCF-GMF_loss.png}
\caption{GMF}
\label{fig:GMF}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.24\textwidth}
\centering
\includegraphics[width=\textwidth]{Images/chap5/NCF-MLP_loss.png}
\caption{MLP}
\label{fig:MLP}
\end{subfigure}
\caption{Training Loss Graph}
\label{fig:NCF1}
\end{figure}
\begin{figure}
\centering
\begin{subfigure}[b]{0.24\textwidth}
\centering
\includegraphics[width=\textwidth]{Images/chap5/NCF-NeuMF_loss.png}
\caption{Without Pre-training}
\label{fig:WPT}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.24\textwidth}
\centering
\includegraphics[width=\textwidth]{Images/chap5/NCF-NeuMF-Pretrained_loss.png}
\caption{With Pre-training}
\label{fig:PT}
\end{subfigure}
\caption{NCF Training Loss Graph}
\label{fig:NCF2}
\end{figure}
Hyperparameter tuning by Bayesian Optimization (BO) sometimes results in overfitting, the knowledge of the validation data split(s) increases as the number of iterations increases on optimizing hyperparameters. k-fold cross-validation is the standard solution to the problem of overfitting in hyperparameter optimization. The authors \cite{wainer2017empirical} conducted an experimental study to find the number of validation folds to be used for cross-validation evaluation, and they found that two or three validation folds are adequate for finding optimal hyperparameters. In this paper, we used NDCG as a target function and applied Bayesian Optimization to find the optimal combination of parameters that maximized the NDCG. We selected the iterations (number of validation folds) to be three, and the hyperparameters include the number of epochs, learning rate, batch size, weight decay, etc. Table \ref{tab:BO} shows the Bayesian Optimization results for the NCF model with two hyperparameters, namely learning rate and n\_factors. From Table \ref{tab:BO} it can be seen that learning rate 0.001 and n\_factors 16 gives the maximum NDCG (0.708). This discussion motivates further work on how to best represent sparse datasets to predict product recommendations efficiently. The CF with NCF predictions demonstrated in this paper discovers the hidden features of user-item interactions that lead to a good generalization, though a hybrid approach of combining item-based and user-based recommenders for predictions on a more complex representation of user-item interactions may improve the NDCG results presented in this paper. The extensions to matrix factorization and the usage of click data for future work are discussed in the next section.
\begin{table}[ht]
\caption{Bayesian Optimization for NCF}
\begin{center}
\begin{tabular}{ |c|c|c|c| }
\hline
\textbf{Iterations} & \textbf{NDCG@12} & \textbf{Learning Rate} & \textbf{n\_factors} \\
\hline
1 & 0.698 & 0.003 & 12\\
\hline
2 & 0.703 & 0.09 & 14\\
\hline
3 & 0.708 & 0.001 & 16\\
\hline
\end{tabular}
\end{center}
\label{tab:BO}
\end{table}
\section{Future Work and Conclusions}
\subsection{Future Work}
Research into collaborative filtering with matrix factorization has demonstrated algorithmic, logical, and practical insights that explain the NDCG performance observed in this paper and has provided future directions of research in predicting sparse user-item interactions with implicit feedback by creating a similarity metric over the latent variable model \cite{borgs2017thy}.
The cold start problem is an active area of research that also applies to the work in this paper. The cold start problem can be relevant to both users and items that have no reviews or history; in simple terms, what to recommend to users/items that are not part of a training dataset. Different procedures have been proposed to address the cold start problem. A user/item deep learning content-based recommender was proposed by \cite{volkovs2017dropoutnet} gives a probability of user/item interaction in the absence of particular user/item interaction in the training dataset.
Similar to ACF, which is implemented in this paper, VAEs also learn hidden non-linear representations of the data using a probabilistic framework \cite{kingma2013auto}. VAEs that implemented CF demonstrated state-of-the-art results for movie recommendations on Netflix dataset \cite{liang2018variational}.
Another critical area of future work is to utilize click-data. In this work, we did not have click-data. However, we suggested the Company to consider implementing click-data so that it can be incorporated in future versions of the Company’s recommender system. Clickthrough Rate
(CTR) is a ratio that indicates how often the people who
see the recommendation ends up clicking it. In
future, after accumulating click-data, the Deep Interest Network \cite{zhou2018deep} model could
be implemented with the current use case and investigated
further.
\subsection{Conclusions}
In this paper, the problem of predicting sparse product recommendations at a specific customer level was studied on a proprietary dataset from August 2016 to June 2019. It was hypothesized that CF with latent variable models could discover hidden features in the broad range of products by exploiting the similarity of customers’ past purchasing behaviors, which could be generalized to future purchases. CF with NCF using latent hidden factors for customers and products presented the highest average performance metrics over the period of the dataset studied. The BPR model is shown to be nearly as efficient as CF with NCF. Deep Learning strategies for CF with latent variables like ACF were investigated and revealed to not be as powerful as NCF. A benefit of the NCF approach is that it has discovered that non-linear hidden relationships exist between customers and products, information that was very valuable in the product recommendations. A limitation of NCF is that the learned hidden representation of user/item matrices is not interpretable, which makes it complicated to comprehend why NCF recommends these products to a customer. Recent work illustrated that matrix factorization with latent variables not limited to low-rank approximation shows excellent generalization properties, enduring sparsity and predicting CTR using Deep Interest Network are promising for further application development in predicting product recommendations where data may be sparse and underlying factors may be complicated, noisy and challenging to apprehend explicitly.
\ifCLASSOPTIONcompsoc
\section*{Acknowledgments}
\else
\section*{Acknowledgment}
\fi
The authors would like to thank the Center for Management of Technology and Entrepreneurship (CMTE) and their sponsors for providing funding to our project.
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\bibliographystyle{IEEEtran}
|
2024-02-18T23:40:24.789Z
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2020-09-21T02:18:09.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2009.08950","language":"en","timestamp":1600654689000,"url":"https:\/\/arxiv.org\/abs\/2009.08950","yymm":"2009"}
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proofpile-arXiv_000-10085
|
{"provenance":"002.jsonl.gz:10086"}
| null | null |
\section{Introduction}
Understanding the symplectic fillings of contact manifolds is a problem with a long history in contact and symplectic geometry. In this article, we focus on symplectic filling of lens spaces. The lens space $L(p,q)$ is the manifold obtained by $-p/q$-surgery on the unknot where $p > q > 0$ are relatively prime integers.
Eliashberg~\cite{Eliashberg:1990-1} proved that every symplectic filling for the standard tight contact structure on $S^3$ is obtained by a blowup of the standard symplectic $B^4$. Moreover, McDuff~\cite{McDuff:1990-1} classified the symplectic fillings for the standard tight contact structures on the lens spaces $L(p, 1)$.
Lisca~\cite{Lisca:2008-1} extended these results by classifing the symplectic fillings of the standard tight contact structures on every lens space. Moreover, recently Etnyre-Roy~\cite{Etnyre-Roy:2020-1} and Christian-Li \cite{Christian-Li:2020-1} classified the symplectic fillings of all the tight contact structures on lens spaces. A symplectic filling $X$ is said to be minimal if it does not admit any symplectic blow-downs, that is it doesn't contain any embedded symplectic 2-spheres with self-intersection $-1$. Since every tight contact structure on a lens space is planar~\cite{Schonenberger}, every minimal symplectic filling is symplectically deformation equivalent to a Stein filling \cite{Wendl-Niederkruger}. So in this setting the study of Stein fillings and minimal symplectic fillings are effectively equivalent. In this paper, we find several bounds on the topology of minimal symplectic fillings of lens spaces. These results unify several preexisting results in the literature and the general theme is that a large fundamental group implies a small second Betti number.
\begin{thm}\label{thm:main2}
If $X$ is a minimal symplectic filling of $(L(p,q), \xi)$ with $|\pi_1(X)| =d$, then $d^2$ divides $p$ and
\begin{equation}\label{eq:b2-bound}
b_2(X)\leq \frac{p}{d^2} -1.
\end{equation}
Moreover, if $\xi$ is virtually overtwisted then this inequality is strict.
\end{thm}
This extends an observation of Fossati, who showed that $d$ divides $p$ and that every symplectic filling of $L(p,q)$ is simply connected if $p$ is prime \cite[Corollary 14]{Fossati:2019-1} (see also \cite[Theorem~1.16]{Etnyre-Roy:2020-1}). Whereas Theorem~\ref{thm:main2} implies that every symplectic filling is simply connected if $p$ is square-free. Finally, Theorem~\ref{thm:main2} also recovers the fact that a virtually overtwisted contact structure on a lens space cannot be filled with a rational homology ball \cite[Proposition~A.1]{Golla-Starkston}, \cite[Lemma 1.5]{Etnyre-Roy:2020-1} (see also \cite[Theorem~4]{Fossati:2019-1}).
Furthermore the bound \eqref{eq:b2-bound} is sharp and one can characterize those lens spaces attaining equality.
\begin{prop}\label{prop:main2_optimality}
A lens space $(L(p,q),\xi)$ admits a minimal symplectic filling $X$ with
\[
|\pi_1(X)|= d \quad\text{and}\quad b_2(X) =\frac{p}{d^2} -1
\]
if and only if $L(p,q)$ is homeomorphic to $L(nd^2,ndc-1)$ where $n=\frac{p}{d^2}$; $c$ is an integer satisfying $\gcd(c,d)=1$ and $1\leq c \leq d$; and $\xi$ is universally tight.
\end{prop}
Note that setting $n=1$ in Proposition~\ref{prop:main2_optimality} recovers the classification of lens spaces which bound symplectic rational balls.
\iffalse
The class of lens spaces appearing in Proposition \ref{prop:main2_optimality} plays a prominent role in the study of cyclic quotient, and more generally rational, surface singularities and their smoothings (see \cite{Looijenga-Wahl1986}, \cite{Kollar}, and \cite{Nemethi}). For a given $L(p,q)$, let $(X_{p,q},0)$ be the corresponding cyclic quotient singularity. It is shown in \cite[Proposition 5.9]{Looijenga-Wahl1986} that $(X_{p,q},0)$ has a smoothing which is a quotient of a Gorenstein smoothing if and only if the pair $(p,q)$ is of the form described in Proposition \ref{prop:main2_optimality}. In fact, one can verify that the Milnor fiber corresponding to the smoothing obtained in \cite{Looijenga-Wahl1986} is a symplectic filling with fundamental group and second Betti number as in Proposition \ref{prop:main2_optimality}.\fi
\iffalse
\textcolor{blue}{Moreover, by using \cite[Theorem~1.1]{Lisca:2008-1} it can be seen that $(L(p,q), \xi)$ has a unique minimal symplectic filling with the fundamental group and the second Betti number as in Proposition~\ref{prop:main2_optimality} up to diffeomorphism (see Remark~\ref{rem:univ_tight}).} It is also interesting to point out that the lens spaces appearing in Proposition~\ref{prop:main2_optimality} play a prominent role in singularity theory (see \cite{Looijenga-Wahl1986}, \cite{Kollar}, and \cite{Nemethi}). Every $L(p,q)$ arises as the link of $(X_{p,q},0)$ a cyclic quotient singularity. It is shown in \cite[Proposition 5.9]{Looijenga-Wahl1986} that $(X_{p,q},0)$ has a smoothing which is a quotient of a Gorenstein smoothing if and only if the pair $(p,q)$ is of the form described in Proposition~\ref{prop:main2_optimality}. Moreover, one can verify that the Milnor fibers corresponding to these smoothings are, in fact, diffeomorphic to the symplectic fillings arising in Proposition~\ref{prop:main2_optimality}.\fi
Moreover, it turns out that a symplectic filling $X$ of $L(p,q)$ with $|\pi_1(X)|= d$ and $b_2(X) =\frac{p}{d^2} -1$, as in Proposition~\ref{prop:main2_optimality}, is unique up to diffeomorphism (see Remark~\ref{rem:uniqueness}) and plays an interesting role in singularity theory. Every $L(p,q)$ arises as the link of $(X_{p,q},0)$ a cyclic quotient singularity (See \cite{Nemethi} for a nice introduction to these matters). It is shown in \cite[Proposition 5.9]{Looijenga-Wahl1986} that $(X_{p,q},0)$ has a smoothing which is a quotient of a Gorenstein smoothing if and only if the pair $(p,q)$ is of the form described in Proposition~\ref{prop:main2_optimality}. In fact, one can verify that fillings under consideration are precisely those arising as the Milnor fibers of smoothings obtained as quotients of Gorenstein smoothings (again, see Remark~\ref{rem:uniqueness}).
To state our second bound we need to establish some notation. If $p > q > 0$ are relatively prime integers, then the rational number $p/q$ admits a unique Hirzebruch-Jung continued fraction expansion $$p/q=[a_1,a_2,\dots, a_k]^-$$ where $a_i$ are integers with $a_i \geq 2$. We use $\len(p/q)=k$ to denote the length of this continued fraction. If $X$ is a minimal symplectic filling of $L(p,q)$ then the following inequality holds \cite[Theorem~1]{Fossati:2019-1}, \cite[Theorem~1.4]{Etnyre-Roy:2020-1}:
$$b_2(X)\leq \len(p/q).$$
Moreover, Fossati showed that this inequality is strict if $\pi_1 (X)$ is non-trivial. We show that in fact the gap between $b_2(X)$ and $\len(p/q)$ grows with the size of the fundamental group $\pi_1 (X)$. Let $F_i$ denote the Fibonacci sequence indexed so that $F_1=F_2=1$.
\begin{thm}\label{thm:main}
If $X$ is a minimal symplectic filling of $(L(p,q), \xi)$ with $|\pi_1(X)| \geq F_{\ell+2}$, then
\begin{equation}\label{eq:b2_lengthbound}
b_2(X)\leq \len(p/q) - \ell.
\end{equation}\end{thm}
In fact, since the Fibonacci numbers are approximately exponential in the golden ratio $\varphi$, there is an explicit upper bound for $b_2(X)$ of the form:
$$b_2(X) < \len(p/q) -\log_\varphi |\pi_1(X)| +1.$$
We can also exhibit examples showing that the bound \eqref{eq:b2_lengthbound} is sharp.
\begin{prop}\label{prop:main_optimality}
A lens space $(L(p,q),\xi)$ admits a non-simply connected minimal symplectic filling $X$ with
\[
|\pi_1(X)|= F_{\ell+2} \quad\text{and}\quad b_2(X)= \len(p/q) -\ell
\]
if and only if $L(p,q)$ is homeomorphic to $L(nF_{\ell+2}^2,nF_{\ell}F_{\ell+2}-1)$ for some integer $n\geq 1$, and $\xi$ is universally tight.
\end{prop}
Lisca gave an explicit construction which yields all symplectic fillings of the standard contact structures on lens spaces \cite{Lisca:2008-1} and Etnyre-Roy \cite{Etnyre-Roy:2020-1} and Christian-Li \cite{Christian-Li:2020-1} showed that in fact any minimal symplectic filling of a lens space is diffeomorphic to one of the fillings constructed by Lisca. Thus we prove our results by analyzing the topology of Lisca's fillings, which have a somewhat intricate description in terms continued fraction expansions and admissable tuples.
\subsection*{Acknowledgments}
The authors would like to thank John Etnyre and Edoardo Fossati for enlightening correspondence.
\section{Continued fractions}\label{sec:propcont}
Consider Hirzebruch-Jung continued fraction expansions:
\[
[a_1,\dots, a_k]^- := a_1 - \cfrac{1}{a_2 - \cfrac{1}{\ddots - \cfrac{1}{a_k}}}.
\]
If $p>q>0$ are relatively prime integers, then the rational number $p/q$ admits a unique expression as a continued fraction in the form
\[
p/q=[a_1, \dots, a_k]^-,
\]
where the $a_i$ are integers with $a_i \geq 2$. Using this expansion we can define several notions of `size' for rational numbers. We define $\len,U,V:\mathbb{Q}_{>1} \rightarrow \mathbb{Z}$ as follows. We define
\[\len(p/q):=k\]
to be the length of the expansion and we define
\[
U(p/q):=\sum_{i=1}^k (a_i-2)
\quad\text{and}\quad
V(p/q):=\sum_{i=1}^k(a_i -1).
\]
Notice that these satisfy
\[
V(p/q)=\len(p/q)+U(p/q).
\]
We will make use of the following properties of continued fractions. These facts are all well-known and are straightforward to prove by induction. For example, proofs of \eqref{it:frac0} and \eqref{it:frac1} can be found in \cite[Proposition~10.2.2]{Cox-Little-Schenck}. Proofs of \eqref{it:reverse_order} and \eqref{it:frac2} can be obtained similarly.
\begin{lem}\label{lem:cont_frac_properties}
Let $\{a_i\}_{i\geq 1}$ be a sequence of integers. Let $p_0=1$, $q_0=0$, $p_1=a_1$, $q_1=1$ and for $k\geq 2$ define $p_k$ and $q_k$ recursively by
\begin{equation}\label{eq:define_convergents}
p_k=a_k p_{k-1} - p_{k-2} \quad\text{and}\quad q_k=a_k q_{k-1} - q_{k-2}.
\end{equation}
Then these satisfy the following properties:
\begin{enumerate}[font=\upshape]
\item\label{it:frac0} $p_k/q_k=[a_1,\dots, a_k]^-$ whenever $q_k\neq 0$,
\item\label{it:reverse_order} $p_k/p_{k-1}=[a_k,\dots, a_1]^-$ whenever $p_{k-1}\neq 0$,
\item\label{it:frac1} $q_k p_{k-1} - p_k q_{k-1}=1$ for all $k\geq 1$, and
\item\label{it:frac2} $[a_1,\dots, a_k,x]^-=\frac{x p_k -p_{k-1}}{x q_k -q_{k-1}}$ for any $x\in \mathbb{R}$ with $x\neq \frac{q_{k-1}}{q_k}$.\qed
\end{enumerate}
\end{lem}
\subsection{The Fibonacci numbers}\label{subsec:fib}
It will be convenient to define the following operations $S,T : \mathbb{Q}_{>1} \rightarrow \mathbb{Q}_{>1}$ on the rationals by setting
\[S(p/q)=\frac{p+q}{q} \quad\text{and}\quad T(p/q)=\frac{2p-q}{p}.\]
These are chosen so that if $a_i$ are integers with $a_i \geq 2$, then
\[
S([a_1, \dots, a_k]^-)=[a_1+1, \dots, a_k]^-
\quad\text{and}\quad
T([a_1, \dots, a_k]^-)=[2,a_1, \dots, a_k]^-.
\]
Since every rational $p/q>1$ has a unique continued fraction expansion with integer coefficients greater than 1, we see that every $p/q>1$ is obtained from $2/1$ by a unique sequence of applications of the $S$ and $T$ operators. Furthermore, note that $p/q>1$ is in the image of the $T$ operator if and only if $p/q<2$ and $p/q$ is in the image of the $S$ operator if and only if $p/q>2$.
Under these definitions we can easily see the following:
\begin{enumerate}[label=(\roman*)]
\item $V(T(p/q))=V(S(p/q))=V(p/q)+1$,
\item $U(S(p/q))=U(p/q)+1 \quad\text{and}\quad U(T(p/q))=U(p/q)$, and
\item $\len(S(p/q))=\len(p/q) \quad\text{and}\quad \len(T(p/q))=\len(p/q)+1.$
\end{enumerate}
So we see that $V(p/q)-1$ is counting the number of applications of $S$ and $T$ required to obtain $p/q$ from $2/1$; that $U(p/q)$ is counting the number of applications of $S$ required to obtain $p/q$ from $2/1$ and that $\len(p/q)-1$ is counting the number of applications of $T$ required to obtain $p/q$ from $2/1$.
\begin{lem}\label{lem:ULidentities}
If $p>q>0$ are relatively prime integers, then we have that
\begin{enumerate}[font=\upshape]
\item\label{it:1ULidentities} $V(p/q)= V(p/(p-q))$,
\item $\len(p/q)=U(p/(p-q))+1$,
\item $\len(p/q)+ \len(p/(p-q)) =V(p/q)+1$, and
\item\label{eq:UUL} $U(p/q)+ U(p/(p-q)) =V(p/q)-1.$
\end{enumerate}
\end{lem}
\begin{proof} We only sketch the proof and the details are left to the reader. We use induction. The base case is $p/q=2/1$ for which the identities are evident. Now suppose that
\[p/q=S(p'/q')= \frac{p'+q'}{q'}.\]
for some relatively prime integers $p'>q'>0$, then
\[T\left(\frac{p'}{p'-q'}\right) = \frac{p'+q'}{p'}=\frac{p}{p-q}.\]
Similarly (by the symmetry of the situation), if
\[p/q=T(p'/q') = \frac{2p'-q'}{p'} \]
for some relatively prime integers $p'>q'>0$, then
\[
S\left(\frac{p'}{p'-q'}\right) = \frac{2p'-q'}{p'-q'} = \frac{p}{p-q}.
\]
This allows us to prove all the identities by observing how $U$, $V$ and $\len$ are changed by applications of the $S$ and $T$ operators.
\end{proof}
Let $F_n$ denote the Fibonacci numbers indexed so that $F_1=F_2=1$.
\begin{lem}\label{lem:fibonacci}
Let $p>q>0$ be relatively prime integers. If $V(p/q)=L$, then
\[p\leq F_{L+2}\]
with equality if and only if $p/q$ takes the form
\[
p/q=F_{L+2}/F_L \quad \text{or} \quad p/q=F_{L+2}/F_{L+1}.
\]
\end{lem}
\begin{proof}
Suppose that we have some $p/q>1$ which maximizes $p$ for a given $L$. Suppose that we can write this as $p/q =G\circ T (r/s)$ where $G$ is some composition of the $S$ and $T$ operators and $r>s>0$ are relatively prime integers. Thus we see that $p/q$ can be written in the form
\[
p/q=[b_1, \dots, b_k, r/s]^-
\]
for some integers $b_i$ with $b_i \geq 2$. Set
\[
p'/q'=\left[b_1, \dots, b_k, \frac{r}{r-s}\right]^-,
\]
where $p'>q'>0$ are relatively prime integers. By Lemma~\ref{lem:cont_frac_properties} \eqref{it:frac2} there are integers $n$ and $m$, such that $p=nr-ms$ and $p'=nr - m(r-s)$. The maximality of $p$ we have that $p'\leq p$ (note that by Lemma~\ref{lem:ULidentities} \eqref{it:1ULidentities} we have that $r/s$ and $r/(r-s)$ have the same $V$ value). This implies that $r/s\geq 2$, which implies that either $r/s=2/1$ or $r/s$ is in the image of the $S$ operator.
Likewise suppose that
$p/q =G\circ S (r/s)$ where $G$ is some composition of the $S$ and $T$ operators.
Thus we see that $p/q$ can be written in the form
\[
p/q=\left[b_1, \dots, b_k, t + \frac{r}{s}\right]^-
\]
for some integers $b_i$ and $t$ with $b_i\geq 2$ and $t\geq 1$. Set
\[
p'/q'=\left[b_1, \dots, b_k, t+ \frac{r}{r-s}\right]^-.
\]
Similarly, there are integers $n>m$ such that
\[p=n(r+ts)-ms \quad\text{and}\quad p'=n(r+t(r-s)) - m(r-s).\]
The maximality of $p$ implies that $p'\leq p$. This implies that $r/s\leq 2$, which implies that either $r/s=2/1$ or $r/s$ is in the image of the $T$ operator. Thus we have shown that if $p$ is maximal then $p/q= G (2/1)$ where $G$ is an alternating product of the $S$ and $T$ operators. This implies that a $p/q$ with $p$ maximal takes one of the forms
\[
p/q=(ST)^n(2/1) \quad \text{or}\quad p/q=(TS)^n(2/1)
\]
if $L=2n+1$ is odd or
\[
p/q=T(ST)^n(2/1) \quad \text{or}\quad p/q=S(TS)^n(2/1)
\]
if $L=2n+2$ is even. In terms of continued fractions these give
\[
p/q=[3,\dots, 3]^- \quad \text{or}\quad p/q=[2,3,\dots, 3,2]^-
\]
or
\[
p/q=[3,\dots, 3,2]^- \quad \text{or}\quad p/q=[2,3,\dots, 3]^-
\]
depending on whether $L$ is odd or even. It easy to check that these are continued fraction expansions for
\[
p/q=F_{L+2}/F_L \quad \text{and} \quad p/q=F_{L+2}/F_{L+1},
\]
as required.
\end{proof}
\subsection{Matrix identities}
Given a tuple of real numbers $\mathbf x=(x_1, \dots, x_k)$ we will use $M(\mathbf x)$ to denote the matrix
\[
M(\mathbf x)= \begin{pmatrix}
x_1 & -1 & 0 & 0 \\
-1 & x_2 & -1 & 0 \\
0 &-1 & \ddots & -1 \\
0 & 0 & -1 &x_k
\end{pmatrix}.
\]
If one attempts to calculate the determinant of $M(\mathbf x)$ by expanding the first row or column one quickly arises at the following recursive formula:
\begin{equation}\label{eq:det_recursion}
\det(M(x_1, \dots, x_k))=x_1 \det (M(x_2, \dots, x_k))- \det (M(x_3,\dots,x_k)),
\end{equation}
valid when $k\geq 3$. By comparing \eqref{eq:det_recursion} to \eqref{eq:define_convergents} in Lemma~\ref{lem:cont_frac_properties} one can easily prove the following lemma by induction. We will leave the proof to the reader.
\begin{lem}\label{lem:cont_frac_matrices}
Let $\{a_i\}_{i\geq 1}$ be a sequence of positive integers. If $k \geq 2$, then the convergents of the continued fraction
\[
p_k/q_k=[a_1, \dots, a_k]^-
\]
where $p_k$ and $q_k$ are positive relatively prime integers, can be computed as
\[\pushQED{\qed}
p_k=\det M(a_1, \dots, a_k)
\quad\text{and}\quad
q_k=\det M(a_2, \dots, a_k).
\qedhere \]
\end{lem}
The following identity, which is an easy application of the multilinearity of the determinant, will also be useful.
\begin{lem}\label{lem:det_multilinear} Let $\{a_i\}_{i\geq 1}$ be a sequence of positive integers. If $1 < i < k$, then
\begin{align*}\pushQED{\qed}
\det M(a_1, \dots, a_{i-1}, a_i +m, a_{i+1}, \dots, a_k)&=
\det M(a_1, \dots, a_k)\\ &+m \det M(a_1, \dots, a_{i-1})\det M(a_{i+1}, \dots, a_k).\qedhere
\end{align*}
\end{lem}
\subsection{Admissable tuples}
Let $\mathbf n=(n_1, \dots, n_k)$ be a tuple of non-negative integers. For each $1\leq i \leq k$, let
\[
\alpha_i/ \beta_i= [n_1, \dots, n_i]^-.
\]
We say that $\mathbf n$ is an \emph{admissable tuple} if the matrix $M(\mathbf n)$ is positive semi-definite with rank at least $k-1$.
It follows that $\mathbf n$ is admissable then for $1\leq i \leq j \leq k$ we have
\begin{equation*}\label{eq:det_positivity}
\det (M(n_i, \dots, n_j))\geq 0
\end{equation*}
with equality only if $i=1$ and $j=k$.
Let $\mathbf n=(n_1, \dots, n_k)$ be a tuple with $n_j=1$ for some $1 \leq j \leq k$. We define an operation called \emph{blow down} of $\mathbf n$ at $j$ as follows:
\[
\mathbf n' =
\begin{cases}
(n_1, \dots, n_{j-1}-1, n_{j+1}-1, \dots, n_k) &\text{if $1<j<k$,}\\
(n_1, \dots, n_{k-2}, n_{k-1}-1) &\text{if $j=k$,}\\
(n_2-1, n_3, \dots, n_{k}) &\text{if $j=1$.}\\
\end{cases}
\]
It readily follows from the definition of admissibility that $\mathbf n$ is admissable if and only if $\mathbf n'$ is admissable.\footnote{The point is that one can perform a change of basis that converts the matrix $M(\mathbf n)$ into $\begin{pmatrix}
M(\mathbf n') & 0\\
0& 1
\end{pmatrix}$.}
\begin{lem}\label{lem:U_bound}
Suppose that $\mathbf n=(n_1, \dots, n_k)$ is an admissable tuple with
\[
\alpha/ \beta =[n_1, \dots, n_k]^-,
\]
where $\alpha$ and $\beta$ are positive relatively prime integers. Then
\begin{equation}\label{eq:U_bound}
\sum_{i=1}^k (n_i -2)\geq U(\alpha/\beta^*)
\end{equation}
where $1\leq \beta^*<\alpha$ and $\beta^* \equiv \beta \bmod{\alpha}$. Moreover, we have equality only if $n_i\geq 2$ for all $1< i<k$.
\end{lem}
\begin{proof}
Suppose that we $n_j=1$ for some $j$ and let
\[\mathbf n'=(n_1', \dots, n_{k-1}')\]
be the admissable tuple obtained by blowing down $\mathbf n$ at $j$.
\begin{equation*}
[n_1', \dots, n_{k-1}']^- = \begin{cases}
\frac{\alpha}{\beta} &\text{if $j>1$}\\
\frac{\alpha}{\beta-\alpha} &\text{if $j=1$}
\end{cases}.
\end{equation*}
However we find that
\begin{equation}\label{eq:blowdown_U}
\sum_{i=1}^{k-1} (n_i'-2) =
\begin{cases}
-1+\sum_{i=1}^{k} (n_i-2) & \text{if $j<k$,}\\
\sum_{i=1}^{k} (n_i-2) &\text{if $j=1$ or $j=k$.}
\end{cases}
\end{equation}
That is blow-downs cannot increase the sum in the left hand side of \eqref{eq:U_bound}.
Since the operation of blowing down decreases the length of the tuple it cannot be repeated indefinitely. Thus after some finite number of blowdowns the tuple $\mathbf n$ will be converted to the tuple
$(c_1, \dots, c_{k'})$ where $c_i\geq 2$ for all $i$ and
\[\alpha/\beta^* =[c_1, \dots, c_{k'}]^-.\]
We have
\[
\sum_{i=1}^{k} (n_i-2) \geq \sum_{i=1}^{k'} (c_i-2) = U(\alpha/\beta^*).
\]
The statement about equality comes from observing that if $n_j=1$ for some $1<j<k$, then \eqref{eq:blowdown_U} shows that blowing down strictly decreases the quantity $\sum_{i=1}^{k} (n_i-2)$.
\end{proof}
The following facts concerning admissable tuples will also be useful.
\begin{lem}\label{lem:admissable_facts}
Let $\mathbf n=(n_1, \dots, n_k)$ be an admissable tuple such that
\[
[n_1, \dots, n_k]^-=0
\]
and for each $1\leq i \leq k$ let
\[
\alpha_i/\beta_i = [n_1, \dots, n_i]^-,
\]
where $\alpha_i$ and $\beta_i$ are positive relatively prime integers. Then we have the following:
\begin{enumerate}[font=\upshape]
\item\label{admissable1} $[n_i, \dots, n_k]^-=\alpha_{i-2}/\alpha_{i-1}$ for any $1<i \leq k$,
\item\label{admissable2} $\det M(n_1, \dots, n_{i-1})=\det M(n_{i+1}, \dots, n_k)$ for any $1<i<k$, and
\item\label{admissable3}
if $n_i=1$ for some $1<i<k$, then
\[
[n_k, \dots, n_{i+1} ]^- = \frac{\alpha_{i-1}}{m\alpha_{i-1}-\beta_{i-1}}.
\]
for some integer $m\geq 1$.
\end{enumerate}
\end{lem}
\begin{proof}
For~\eqref{admissable1}, suppose that $r/s=[n_i, \dots, n_k]^-$, then by Lemma~\ref{lem:cont_frac_properties} \eqref{it:frac2} we have that
\[
0=[n_1, \dots, n_{i-1}, r/s]^-=\frac{ \alpha_{i-1}(r/s)-\alpha_{i-2}}{\beta_{i-1}(r/s)-\beta_{i-2}}.
\]
This implies that $r/s = \alpha_{i-2}/\alpha_{i-1}$ as required.
For~\eqref{admissable2}, we observe that Lemma~\ref{lem:cont_frac_matrices} combined with \eqref{admissable1} implies that $\alpha_{i-1}$ is computed as a determinant in two ways:
\[
\det M(n_1, \dots, n_{i-1})=\det M(n_{i+1}, \dots, n_k)=\alpha_{i-1}.
\]
For~\eqref{admissable3}, suppose that $n_i=1$. This implies that
\[
\alpha_{i-2}/\alpha_{i-1}= [1,n_{i+1}, \dots, n_k]^- = 1-\frac{1}{[n_{i+1}, \dots, n_{k} ]^-},
\]
and hence that
\[
[n_{i+1}, \dots, n_{k} ]^-=\frac{\alpha_{i-1}}{\alpha_{i-1}-\alpha_{i-2}}.
\]
Now suppose that
\[
[n_{k}, \dots, n_{i+1} ]^-=c/d,
\]
where $c,d>0$ are relatively prime integers. By Lemma~\ref{lem:cont_frac_properties}~\eqref{it:reverse_order} we have that $c=\alpha_{i-1}$. Furthermore, by applying Lemma~\ref{lem:cont_frac_properties}~\eqref{it:frac1} we see that $d$ satisfies
\[
(\alpha_{i-1}-\alpha_{i-2})d \equiv 1 \bmod{\alpha_{i-1}}.
\]
This implies that $d\equiv -\beta_{i-1}\bmod{\alpha_{i-1}}$, since Lemma~\ref{lem:cont_frac_properties}~\eqref{it:frac1} shows that $\beta_{i-1}\alpha_{i-2}\equiv 1 \bmod{\alpha_{i-1}}$.\end{proof}
\section{Symplectic fillings}\label{sec:symplectic}
We first recall Lisca's~\cite{Lisca:2008-1} construction of symplectic fillings for lens spaces. Given a lens space $L(p,q)$, consider continued fraction expansion
\[
\frac{p}{p-q}=[b_1, \dots, b_k]^-,
\]
where $b_i$ are integers with $b_i \geq 2$. Let $\mathbf n=(n_1, \dots, n_k)$ be an admissible tuple with
\begin{equation}\label{eq:admissable}
[n_1,\dots, n_k]^-=0 \quad\text{and}\quad b_i\geq n_i \text{ for $i=1, \ldots, k$}.
\end{equation}
Note that this condition implies that the following surgery diagram describes $S^1 \times S^2$.
\begin{figure}[h]
\includegraphics[width=.9\textwidth]{figures/s1s2}
\caption{A surgery description of $S^1 \times S^2$ corresponding to $\mathbf n$}\label{fig:s1s2}
\end{figure}
\noindent Moreover, we construct a 4-manifold $W_{p,q}(\mathbf n)$ by attaching $2$-handles to $S^1 \times D^3$ as in Figure~\ref{fig:wn}.
\begin{figure}[h]
\includegraphics[width=.9\textwidth]{figures/wn}
\caption{The manifold $W_{p,q}(\mathbf n)$.}\label{fig:wn}
\end{figure}
\noindent By~\cite[Theorem 1]{Lisca:2008-1}, every minimal symplectic filling of a universally tight contact structures on $L(p,q)$ is diffeomorphic to $W_{p,q}(\mathbf n)$ for some $\mathbf n$ satisfying \eqref{eq:admissable}. It is straight forward to calculate some invariants of $W_{p,q}(\mathbf n)$. First note that by Lemma~\ref{lem:ULidentities}, we have
\[
\len(p/q)=U\left(\frac{p}{p-q}\right)+1 = \sum_{i=1}^k (b_i-2)+1.
\]
Also, by construction of $W_{p,q}(\mathbf n)$ we have that
\begin{equation}\label{eq:b2_formula}
b_2(W_{p,q}(\mathbf n))=\sum_{i=1}^k (b_i-n_i)-1.
\end{equation}
Thus we see that
\begin{equation}\label{eq:b2_len_difference}
\len(p/q)- b_2(W_{p,q}(\mathbf n))= \sum_{i=1}^k (n_i-2)+2.
\end{equation}
Now we compute the fundamental group. Given an admissable tuble $\mathbf n=(n_1, \dots, n_k)$ satisfying \eqref{eq:admissable} and an integer $i$ with $1\leq i \leq k$, let
\[
\alpha_i/ \beta_i= [n_1, \dots, n_i]^-,
\]
where $\alpha_i$ and $\beta_i$ are positive relatively prime integers. Also, we set $\alpha_0 = 1$ and $\beta_0 = 0$.
\begin{lem}\label{lem:pi1_calc}
For each integer $i$ with $1\leq i \leq k$, let $d_i=\alpha_{i-1}$ if $b_i>n_i$ and $d_i=0$ otherwise. If $d=\gcd (d_1, \ldots, d_k)$, then
\[\pi_1(W_{p,q}(\mathbf n)) \cong \mathbb{Z} /d\mathbb{Z}.\]
\end{lem}
\begin{proof}
Since $W_{p,q}(\mathbf n)$ is obtained by attaching 2-handles to a single 1-handle, it's fundamental group is cyclic and hence abelian. Thus it suffices to compute $H_1(W_{p,q}(\mathbf n))$. Let $\mu_i$ denote the homology class of the meridian of the $i$th unknot component of the link in Figure~\ref{fig:s1s2}. Notice that these satisfy the relation
\[
-\mu_{i-1}+n_i \mu_i - \mu_{i+1} =0.
\]
for $1\leq i \leq k$ (with the convention that $\mu_0=\mu_{k+1}=0$). Thus notice that if $\mu_i=\gamma_i \mu_1$ and $3 \leq i \leq k$, then these satisfy the recursion relation
\[
\gamma_i= n_{i-1} \gamma_{i-1} - \gamma_{i-2}
\]
where $\gamma_1=1$ and $\gamma_2=n_1$. However we see that the sequence $\alpha_{i-1}$ satisfies the same recursion relation by \eqref{eq:define_convergents} in Lemma~\ref{lem:cont_frac_properties} and we have $\alpha_{0}=1$ and $\alpha_{1}=n_1$. Hence we conclude that $\gamma_i=\alpha_{i-1}$ for each $1\leq i \leq k$. In particular, we have that $\gamma_k=\alpha_{k-1}=1$ since $[n_1, \dots, n_k]^-=0$. Thus we conclude that $H_1(W_{p,q}(\mathbf n))$ is generated by $\mu_1=\mu_k$ with a relation $\alpha_{i-1} \mu_1=0$ for each 2-handle attached along $\mu_i$ (i.e.\ when $b_i < n_i$).
\end{proof}
We will later make use of the following consequence.
\begin{cor}\label{cor:divide}
If $b_i>n_i$ for some $1<i<k$, then $|\pi_1(W_{p,q}(\mathbf n))|$ divides
\[
\det(M(n_1, \dots, n_{i-1}))=\det(M(n_{i+1}, \dots, n_{k})).
\]
\end{cor}
\begin{proof} By Lemma~\ref{lem:cont_frac_matrices} and Lemma~\ref{lem:admissable_facts} \eqref{admissable2}, we have \[
\alpha_{i-1}=\det(M(n_1, \dots, n_{i-1}))=\det(M(n_{i+1}, \dots, n_{k})).
\] The proof is complete by applying Lemma~\ref{lem:pi1_calc}.
\end{proof}
\begin{lem}\label{lem:extremal_example}
Suppose $W_{p,q}(\mathbf n)$ is a non-simply connected symplectic filling of $L(p,q)$ where there is a unique $j$ such that $b_j>n_j$.
Then $j>1$ and there are coprime integers $c,d$ with $0< c < d$ and $d/c=[n_1, \dots, n_{j-1}]^-$, such that
\begin{enumerate}[font=\upshape]
\item $L(p,q)$ is diffeomorphic to $L(nd^2, ncd-1)$,
\item $|\pi_1(W_{p,q}(\mathbf n))| = d$, and
\item $\len(p/q)-b_2(W_{p,q}(\mathbf n)) = V(d/c)$,
\end{enumerate}
where $b_2(W_{p,q}(\mathbf n))= n-1$.\end{lem}
\begin{proof}
Since we are assuming that $W_{p,q}(\mathbf n)$ is not simply connected, $\mathbf n$ takes the form $[n_1, \dots, n_k]^-=0$, where $k>1$. The tuple $\mathbf n$ must have some $n_j=1$. Since $b_j\geq 2$, this implies that this must be the unique index with $n_j=1$. Moreover thus, $\mathbf n$ takes the form
\[
\mathbf n=(n_1, \dots, n_{j-1}, 1, n_{j+1} , \dots, n_{k}).
\]
Moreover, as $W_{p,q}(\mathbf n)$ is not simply connected, we have that $1<j<k$. Also, the assumption $b_2(W_{p,q}(\mathbf n))=n-1$ and equation \eqref{eq:b2_formula} implies that
\[
\frac{p}{p-q}=[n_1, \dots, n_{j-1}, 1+n, n_{j+1} , \dots, n_{k}]^-.
\]
Since $n_1, \dots, n_{j-1}\geq 2$, then there are relatively prime integers $c,d$ such that $d>c>0$ and
\[
\frac{d}{c}=[n_1, \dots, n_{j-1}]^-
\]
By Lemma~\ref{lem:pi1_calc} we have $\pi_1 (W_{p,q}(\mathbf n)) = \mathbb{Z}/d\mathbb{Z}$.
Lemma~\ref{lem:admissable_facts} \eqref{admissable3} combined with the fact that $d>c>0$ shows that
\[
\frac{d}{d-c}=[n_{k} , \dots, n_{i+1}]^-.
\]
Thus by equation~\eqref{eq:b2_len_difference} and Lemma~\ref{lem:ULidentities} \eqref{eq:UUL}, we see that
\[
\len(p/q)- b_2(W_{p,q}(\mathbf n))= 2 + \sum_{i=1}^k (n_i-2)=1+U\left(\frac{d}{c}\right)+U\left(\frac{d}{d-c}\right)=V\left(\frac{d}{c}\right).
\]
It is an exercise in continued fractions to calculate $p/(p-q)$ as
\[
\frac{p}{p-q}= \frac{nd^2}{ndc+1}
\]
Thus, $p/q$ takes the form:
\[\frac{p}{q}= \frac{nd^2}{nd^2- ndc -1}.\]
However since $(nd^2+ ndc -1)(ndc-1)\equiv 1 \bmod nd^2$, we see that $L(p,q)$ is homeomorphic to $L(nd^2, ndc-1)$, as required.
\end{proof}
\begin{lem}\label{lem:univ_tight}
Let $X$ be a minimal symplectic filling of $(L(nd^2, q), \xi)$ where $q\equiv -1 \bmod nd$ and $|\pi_1 (X) |=d$. Then $\xi$ is universally tight.\end{lem}
\begin{proof}
By deforming the symplectic structure on $X$, we can assume that it is a Stein filling of $(L(p,q),\xi)$ \cite{Wendl-Niederkruger}. We have a surjection $\pi_1 (L(p,q)) \rightarrow \pi_1 (X)$ induced by inclusion. Thus taking the universal cover of $X$ yields a Stein filling $\widetilde{X}$ of the lens space $(L(p',q'), \widetilde{\xi})$, where $L(p',q')\cong L(nd, q') \cong L(nd, nd-1)$. Since $\widetilde{X}$ is Stein, the contact structure $\widetilde{\xi}$ is tight \cite{Gromov1985}. The classification of tight contact structures on lens space shows that every tight contact structure on $L(nd, nd-1)$ is universally tight \cite{Honda2000}. Thus we have universally tight $\widetilde{\xi}$ covering $\xi$. This implies that $\xi$ is itself universally tight.\end{proof}
\subsection{Proof of Theorem~\ref{thm:main} and Proposition~\ref{prop:main_optimality}}
\begin{proof}[Proof of Theorem~\ref{thm:main} and Proposition~\ref{prop:main_optimality}]
By \cite[Theorem 1.1]{Etnyre-Roy:2020-1} and \cite[Theorem 1.4]{Christian-Li:2020-1} any minimal symplectic filling of a lens space is diffeomorphic to one constructed by Lisca~\cite{Lisca:2008-1} as described at the begining of Section~\ref{sec:symplectic}. So suppose that $X\cong W_{p,q}(\mathbf n)$ is a filling of $L(p,q)$ corresponding to an admissable tuple
\[
\mathbf n = (n_1, \dots, n_k),
\]
satisfying \eqref{eq:admissable}.
Note that if $X$ simply connected, the theorem is equivalent to the assertion that $\len(p/q) - b_2(X)\geq 0$. This was established Fossati~\cite[Theorem 1]{Fossati:2019-1}. Alternatively the enthusiastic reader can deduce it from \eqref{eq:b2_len_difference} and properties of admissable tuples\footnote{\textbf{Hint:} The admissable tuple $\mathbf n$ can be reduced to the admissable tuple $(0)$ by a sequence of blow-downs.}.
In any case, we may assume that $X$ is not simply connected. In particular, by Lemma~\ref{lem:pi1_calc} we can assume that $n_1,n_k\geq 2$. Now the admissable tuple must have $n_j=1$ for some $1<j<k$. By picking $j$ minimal with this property we can assume that the admissible tuple takes the form
\[
\mathbf n=(n_1, \dots, n_{j-1}, 1, n_{j+1}, \dots, n_k),
\]
where we can assume that $n_1,\dots, n_{j-1}\geq 2$.
Suppose that
\[
[n_{1}, \dots, n_{j-1}]^-=\frac{\alpha}{\beta}>1,
\]
where $\alpha$ and $\beta$ are positive relatively prime integers. Then by Lemma~\ref{lem:admissable_facts}~\eqref{admissable3}, we have that
\[
[n_{k}, \dots, n_{j+1}]^-=\frac{\alpha}{m\alpha-\beta}.
\]
for some integer $m\geq 1$. Since $\alpha>\beta$, Lemma~\ref{lem:U_bound} implies that
\begin{equation}\label{eq:proof_main_sum}
\sum_{i=j+1}^k (n_i-2) \geq U\left(\frac{\alpha}{\alpha-\beta}\right)
\end{equation}
Also, Corollary~\ref{cor:divide} implies that
\begin{equation}\label{eq:pi1leqalpha}
|\pi_1(X)|\leq \alpha.
\end{equation}
Moreover, by equation~\eqref{eq:b2_len_difference}, Lemma~\ref{lem:ULidentities} \eqref{eq:UUL}, and equation~\eqref{eq:proof_main_sum}, we have
\begin{equation}\label{eq:proof_main1}
\len(p/q)-b_2(X)\geq U\left(\frac{\alpha}{\beta}\right)+U\left(\frac{\alpha}{\alpha-\beta}\right)+1 = V\left(\frac{\alpha}{\beta}\right).
\end{equation}
Thus if we write $L=V(\alpha/\beta)$, then equation~\eqref{eq:proof_main1} along with equation~\eqref{eq:pi1leqalpha} and Lemma~\ref{lem:fibonacci} show that
\[
b_2(X)\leq \len(p/q)- L \quad\text{and}\quad | \pi_1(X)|\leq F_{L+2}.
\]
Thus if $| \pi_1(X)|\geq F_{\ell+2}$ for some integer $\ell \geq 0$, we have that $L\geq \ell$ and the bound \eqref{eq:b2_lengthbound} follows.
Next we deduce Proposition~\ref{prop:main_optimality}. We continue to work with $X$ non-simply connected and the same notation as before. Suppose that we have
\[
|\pi_1(X)|=F_{\ell+2} \quad\text{and}\quad b_2(X)= \len(p/q)- \ell
\]
for some $\ell\geq 1$. This implies that $|\pi_1(X)|=\alpha= F_{\ell+2}$ and $\ell=V(\alpha/\beta)$. By Lemma~\ref{lem:fibonacci} this implies that $\alpha/\beta =F_{L+2}/F_L$ or $\alpha/\beta= F_{L+2}/F_{L+1}$. Lemma~\ref{lem:U_bound} implies that if
\[
\len(p/q)-b_2(X) =2+\sum_{i=1}^k (n_i-2) = V(\alpha/\beta)
\]
then $n_{j+1}, \dots, n_k\geq 2$. Thus $\mathbf n$ has the unique index $j$ with $n_j=1$. Moreover, since $n_1, \dots, n_{j-1}\geq 2$, the sequence of denominators $\alpha_i$ is increasing for $i<j$ and so for $i<j$ we have $\alpha_{i-1}<\alpha_{j-1}$. Thus if $b_i>n_i$ for some $i< j$, we would have $|\pi_1 (X)|<\alpha$. By reversing the order of the admissable tuple we can show similarly that if $i>j$ satisfies $b_i>n_i$ then we would have $|\pi_1 (X)|<\alpha$. Thus we see that $j$ is the unique index with $b_j>n_j$. Thus it follows from Lemma~\ref{lem:extremal_example} and Lemma~\ref{lem:univ_tight} that $X$ is a filling of $L(n F_{\ell +2}^2, n F_{\ell+2} F_\ell -1)$ or $L(n F_{\ell +2}^2, n F_{\ell+2} F_{\ell+1} -1)$ with a universally tight contact structure. However since $F_{\ell+2} =F_\ell + F_{\ell +1}$ these two lens spaces are diffeomorphic. This proves the only if statement. Conversely, guided by Lemma~\ref{lem:extremal_example}, it is easy to produce an admissable tuple $\mathbf n$ for which $W_{p,q}(\mathbf n)$ has all the necessary properties.\end{proof}
\subsection{Proof of Theorem~\ref{thm:main2} and Proposition~\ref{prop:main2_optimality}}
\begin{proof}[Proof of Theorem~\ref{thm:main2} and Proposition~\ref{prop:main2_optimality}]
First we verify Theorem~\ref{thm:main2} and Proposition~\ref{prop:main2_optimality} in the case that $d=1$. Let $X$ be a minimal simply connected filling of $(L(p,q), \xi)$. Since $b_2(X)\leq \len(p/q)\leq q \leq p-1$, the bound \eqref{eq:b2-bound} is satisfied in this case. Moreover, we have equality implies that $q=p-1$. Lemma~\ref{lem:univ_tight} shows that if $q=p-1$, then $\xi$ must be universally tight. This proves the moreover statement in Theorem~\ref{thm:main2} and establishes the only if direction of Proposition~\ref{prop:main2_optimality}. To complete the proof of Proposition~\ref{prop:main2_optimality}, note that a standard plumbing fills $L(p,p-1)$ and has all the necessary properties.
Thus from now now on we can assume that $X$ is a non-simply connected minimal symplectic filling of $(L(p,q),\xi)$. Then \cite[Theorem 1.1]{Etnyre-Roy:2020-1} and \cite[Theorem 1.4]{Christian-Li:2020-1} implies $X$ is diffeomorphic to some $W_{p,q}(\mathbf n)$ as constructed by Lisca~\cite{Lisca:2008-1}. So suppose that $X\cong W_{p,q}(\mathbf n)$ is a filling of $L(p,q)$ corresponding to an admissable tuple
\[
\mathbf n = (n_1, \dots, n_k),
\]
satisfying \eqref{eq:admissable}, and
\[\frac{p}{p-q}=[b_1, \dots, b_k]^-.\]
By Lemma~\ref{lem:cont_frac_matrices} We have that
\[
p=\det M(b_1, \dots, b_k).
\]
Let $S\subseteq \{1,\dots, k\}$ be the set of $i$ with $b_i>n_i$. As we are assuming that $X$ is not simply-connected and Lemma~\ref{lem:pi1_calc} shows that $1$ and $k$ are not in $S$. By using multilinearity of the determinant as in Lemma~\ref{lem:det_multilinear} and inducting on the size of the set $S$, we see that $p$ can be expressed here as
\begin{align*}
p&= \sum_{i\in S} (b_i-n_i) \det M(n_1, \dots, n_{i-1}) \det M(n_{i+1}, \dots, n_{k})\\
&+ \sum_{\substack{i,j\in S \\ i<j}} K_{i,j} \det M(n_1, \dots, n_{i-1}) \det M(n_{j+1}, \dots, n_{k}),
\end{align*}
where $K_{i,j}$ is the integer
\[
K_{i,j}= (b_i-n_i)(b_j-n_j) \det M(b_{i+1}, \dots, b_{j-1}).
\]
In any case we see that for any $i\in S$ we have that $d$ divides both
$\det M(n_{1}\dots , n_{i-1})$ and $\det M(n_{i+1}\dots , n_{k})$. So the above expression implies that $d^2$ divides $p$. Morever the property that the tuple is admissable implies that the coefficients satisfy $K_{i,j}\geq 1$. Thus by ignoring terms that correspond to subsets of $S$ with more than one element we obtain the bound
\begin{equation}\label{eq:d2_bound}
p\geq d^2\sum_{i\in S} (b_i-n_i)= d^2(b_2(X)-1),
\end{equation}
with equality only if $S$ contains a single element. Rearranging the above inequality yields \eqref{eq:b2-bound}.
Now suppose that $X$ is a filling with $b_2(X)=p/d^2-1$ and $|\pi_1(X)|=d$. Since equality in \eqref{eq:d2_bound} implies that the admissible tuple $\mathbf n$ has a unique index $j$ with $b_j>n_j$, we see that $X$ is a filling of the form studied in Lemma~\ref{lem:extremal_example}. This shows that $L(p,q)$ is diffeomorphic to $L(nd^2, ndc-1)$, where $n=b_2(X)+1$. Moreover, Lemma~\ref{lem:univ_tight} shows that the contact structure $\xi$ is universally tight. This shows that the filling is in the form required by Proposition~\ref{prop:main2_optimality} and establishes the moreover statement in Theorem~\ref{thm:main2}.
Guided by Lemma~\ref{lem:extremal_example}, one can easily find admissable tuples necessary to construct fillings completing the proof of Proposition~\ref{prop:main2_optimality}.
\end{proof}
\begin{rem}\label{rem:uniqueness}
\iffalse\textcolor{blue}{In fact, Lemma~\ref{lem:extremal_example} and the proof of Theorem~\ref{thm:main2} and Proposition~\ref{prop:main2_optimality} imply that there is a unique $\mathbf n$ such that $W_{p,q}(\mathbf n)$ has the fundamental group and the second Betti number as in the hypothesis of Proposition~\ref{prop:main2_optimality}. Hence, by \cite[Theorem~1.1]{Lisca:2008-1} we conclude that there is a unique minimal symplectic filling up to diffeomorphism with the topological constraint as above.}\fi
Lemma~\ref{lem:extremal_example} and the proof of Proposition~\ref{prop:main2_optimality} imply that $L(p,q)$ has a unique minimal symplectic filling $X$ with with $|\pi_1(X)|= d$ and $b_2(X) =\frac{p}{d^2} -1$. It is established in the proof Proposition~\ref{prop:main2_optimality} that this $X$ is diffeomorphic to $W_{p,q}(\mathbf n)$, where there is exactly one index $j$ such that $b_j>n_j$. It follows from Lemma~\ref{lem:extremal_example} the admissable tuple $\mathbf n$ is determined by the fraction $p/q$ and the integer $d$. Hence, by \cite[Theorem~1.1]{Lisca:2008-1} we conclude that the filling $X$ is unique up to diffeomorphism.
In particular, since $X$ is unique it must coincide with the filling obtained as the Milnor fiber of a quotient of a Gorenstein smoothing (cf. \cite[Proposition 5.9]{Looijenga-Wahl1986})
\end{rem}
\bibliographystyle{alpha}
\def\MR#1{}
|
2024-02-18T23:40:24.792Z
|
2020-09-21T02:18:19.000Z
|
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proofpile-arXiv_000-10086
|
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\section{Introduction}
Understanding the global star formation process in galaxies is of key importance in the comprehension of galaxy formation and evolution. One of the biggest challenges faced by numerical models of galaxy formation derived directly from cosmological models is to explain why the stellar masses of galaxies are consistently lower than those expected from the simulations \citep{2012RAA....12..917S}. This difference has been bridged by invoking internal mechanisms capable of regulating the star formation rate. Two regimes have been generally used: for massive galaxies their nuclear activity is found to be a mechanism which acts in this way \citep{2018Natur.553..307M}. But for low mass galaxies the star formation itself, through feedback, appears to offer a satisfactory mechanism to reduce the star formation rate, making star formation an inefficient process when comparing the stars which are formed with the availability of gas to form them
\citep{2008AJ....136.2846B,2014MNRAS.445..581H,2019Natur.569..519K}.
Star formation self-regulates by expelling gas, and the amount of gas that flows out of any system is considered to depend on the mass of stars formed.
The models used to explain the consistently low mean star formation rate (SFR) efficiency
use sub-grid physics parametrised by a mass loading factor, $\eta$, relating the mass outflow rate $\dot{M}_{\rm{out}}$ and the SFR by $\dot{M}_{\rm{out}}=\eta {\rm{SFR}}$ \citep{2010MNRAS.402.1536S,2013MNRAS.436.3031V,2015ARA&A..53...51S,2018MNRAS.480..800H}.
This factor can be predicted by modelling the feedback process \citep{2013MNRAS.429.1922C,2015MNRAS.454.2691M,2017ApJ...841..101L}, or inferred from observations \citep{2019MNRAS.490.4368S,2019ApJ...886...74M,2019Natur.569..519K,2020MNRAS.493.3081R}.
However feedback modelling has many uncertainties, and the required observations are scarce and also subject to uncertainty. The present article marks a significant step in making up for the observational deficiencies.
In order to see whether the star formation at different epochs is correlated and to quantify it by estimating the mass-loading factor we apply an empirical method based on stellar population synthesis and the self-regulator model of star formation, which has been presented previously \citep{2019MNRAS.487L..61Z}.
The star formation self-regulator model
\citep{2010ApJ...718.1001B,2013ApJ...772..119L,2014MNRAS.444.2071D,2014MNRAS.443..168F,2015MNRAS.448.2126A}
assumes mass conservation for a galaxy,
which implies that the change per unit time of the gas mass,
$\dot{M}_{\rm{gas}}$, equals the
inflow rate into the galaxy, $\dot{M}_{\rm{in}}$, minus the gas that goes into star formation, SFR,
and the gas which flows out of the galaxy, $\dot{M}_{\rm{out}}$:
\begin{equation}
\dot{M}_{\rm{gas}}=\dot{M}_{\rm{in}}-\rm{SFR}(1-R+\eta)\rm{,}
\end{equation}
\noindent where $R$ is the fraction of the mass which is returned to the interstellar medium from the stellar population.
The spatially resolved star formation self-regulator model applies to segments of a galaxy \citep{2019MNRAS.487L..61Z}, where by segment we mean any spatially resolved region of a galaxy.
In these resolved regions we also assume conservation of mass:
the time change of the gas mass surface density in a segment, $\dot{\Sigma}_{\rm{gas}}$,
is equal to the surface density of the net gas flow rate, $\dot{\Sigma}_{\rm{net\thinspace flow}}$,
minus the surface density of gas that goes into new stars through star formation, ${\Sigma_{\rm{SFR}}}$, and
minus the surface density of gas that is
expelled from the segment by stellar processes, $\dot{\Sigma}_{\rm{out}}$:
\begin{equation}
\dot{\Sigma}_{\rm{gas}}={\dot{\Sigma}}_{\rm{net \thinspace flow}}-{\Sigma_{\rm{SFR}}}(1-R+\eta)
\label{eq_bath_res}
\end{equation}
\noindent where $R$ is the fraction of the mass that is returned to the interstellar medium,
and
\begin{equation}
{\dot{\Sigma}}_{\rm{out}}=\eta{\Sigma_{\rm{SFR}}}.
\label{eq_loading_res}
\end{equation}
This model allows us to relate the star formation rate surface density in a segment, $\Sigma_{\rm{SFR}}$, to the change in gas mass in that segment.
The complex processes of stellar feedback are parameterized
by the mass-loading factor:
$\dot{\Sigma}_{\rm{out}}=\eta {\Sigma_{\rm{SFR}}}$.
We present the galaxy sample and the data in section \S2. In section \S3 we give the stellar population synthesis fits and also fit the observables to the star formation self-regulator model. In section \S4 we show the results obtained, and the variation of $\eta$, while in section \S5 we convert local values of $\eta$ into global ones. We discuss our results
in section \S6 and present our conclusions in section \S7.
\section{Galaxy Sample and data}
\subsection{Galaxy sample}
\begin{table*}
\centering
\caption{Galaxy sample.}
\label{tab_sample}
\begin{tabular}{rrrrrr}
\hline
Galaxy identifier & PGC identifier $^a$ & $D$ $^b$ & z $^c$ & Type $^d$ & i $^e$ \\ & & Mpc & & & $^{\circ}$ \\
\hline
pgc33816 & PGC33816 & 23.6 & 0.005187 & 7.0 & 19.9 \\
eso184-g082 & PGC63387 & 35.2 & 0.00867 & 4.1 & 32.6 \\
eso467-062 & PGC68883 & 57.5 & 0.013526 & 8.6 & 49.9 \\
ugc272 & PGC1713 & 55.6 & 0.012993 & 6.5 & 70.7 \\
ngc5584 & PGC51344 & 23.1 & 0.005464 & 5.9 & 42.4 \\
eso319-g015 & PGC34856 & 37.5 & 0.009159 & 8.6 & 54.2 \\
ugc11214 & PGC61802 & 38.0 & 0.008903 & 5.9 & 16.5 \\
ngc6118 & PGC57924 & 20.5 & 0.005247 & 6.0 & 68.7 \\
ic1158 & PGC56723 & 24.5 & 0.006428 & 5.1 & 62.2 \\
ngc5468 & PGC50323 & 30.0 & 0.00948 & 6.0 & 21.1 \\
eso325-g045 & PGC50052 & 75.9 & 0.017842 & 7.0 & 40.2 \\
ngc1954 & PGC17422 & 38.0 & 0.010441 & 4.4 & 61.5 \\
ic5332 & PGC71775 & 9.9 & 0.002338 & 6.8 & 18.6 \\
ugc04729 & PGC25309 & 57.0 & 0.013009 & 6.0 & 35.2 \\
ngc2104 & PGC17822 & 16.4 & 0.003873 & 8.5 & 83.6 \\
eso316-g7 & PGC28744 & 47.5 & 0.01166 & 3.3 & 70.0 \\
eso298-g28 & PGC8871 & 70.1 & 0.016895 & 3.8 & 64.4 \\
mcg-01-57-021 & PGC69448 & 30.6 & 0.009907 & 4.0 & 52.2 \\
pgc128348 & PGC128348 & 61.1 & 0.014827 & 5.0 & 36.7 \\
pgc1167400 & PGC1167400 & 60.0 & 0.01334 & 4.0 & 30.5 \\
ngc2835 & PGC26259 & 10.1 & 0.002955 & 5.0 & 56.2 \\
ic2151 & PGC18040 & 30.6 & 0.010377 & 3.9 & 61.5 \\
ngc988 & PGC9843 & 17.3 & 0.005037 & 5.9 & 69.1 \\
ngc1483 & PGC14022 & 16.8 & 0.003833 & 4.0 & 37.3 \\
ngc7421 & PGC70083 & 24.2 & 0.005979 & 3.7 & 36.2 \\
fcc290 & PGC13687 & 19.0 & 0.004627 & 2.1 & 48.1 \\
ic344 & PGC13568 & 75.6 & 0.018146 & 4.0 & 60.7 \\
ngc3389 & PGC32306 & 21.4 & 0.004364 & 5.3 & 66.2 \\
eso246-g21 & PGC9544 & 76.6 & 0.018513 & 3.0 & 52.4 \\
pgc170248 & PGC170248 & 85.1 & 0.019163 & 4.7 & 76.4 \\
ngc7329 & PGC69453 & 45.7 & 0.010847 & 3.6 & 42.7 \\
ugc12859 & PGC72995 & 78.3 & 0.018029 & 4.0 & 72.8 \\
ugc1395 & PGC7164 & 74.1 & 0.017405 & 3.1 & 55.1 \\
ngc5339 & PGC49388 & 27.0 & 0.009126 & 1.3 & 37.5 \\
ngc1591 & PGC15276 & 55.8 & 0.013719 & 2.0 & 56.8 \\
pgc98793 & PGC98793 & 55.2 & 0.01292 & 5.0 & 0.0 \\
ugc5378 & PGC28949 & 56.5 & 0.01388 & 3.1 & 64.1 \\
ngc4806 & PGC44116 & 29.0 & 0.008032 & 4.9 & 32.9 \\
ngc1087 & PGC10496 & 14.4 & 0.00506 & 5.2 & 54.1 \\
ngc4980 & PGC45596 & 16.9 & 0.004767 & 1.1 & 71.5 \\
ngc6902 & PGC64632 & 46.6 & 0.009326 & 2.3 & 40.2 \\
ugc11001 & PGC60957 & 63.3 & 0.01406 & 8.1 & 78.7 \\
ic217 & PGC8673 & 27.0 & 0.006304 & 5.8 & 82.6 \\
eso506-g004 & PGC39991 & 57.5 & 0.013416 & 2.6 & 67.2 \\
ic2160 & PGC18092 & 64.7 & 0.015809 & 4.6 & 62.7 \\
ngc1385 & PGC13368 & 22.7 & 0.005 & 5.9 & 52.3 \\
mcg-01-33-034 & PGC43690 & 32.0 & 0.008526 & 2.1 & 56.6 \\
\end{tabular}
\\
\hrulefill
\\
{\raggedright
$^a$ Principal General Catalog of Galaxies identifier from Hyperleda database \citep{2003A&A...412...45P}.
$^b$ Distance from the z=0 Multi-wavelength Galaxy Synthesis (z0MGS from \cite{2019ApJS..244...24L}, when available) and HyperLeda database best homogenized distances \citep{2014A&A...570A..13M}.
$^c$ Redshift, from Nasa Ned.
$^d$ Numerical morphologycal type, from the HyperLeda database.
$^e$ Inclination from the HyperLeda database. } \\
\hrulefill
\end{table*}
\begin{table*}
\centering
\contcaption{}
\begin{tabular}{rrrrrr}
\hline
Galaxy identifier & PGC identifier $^a$ & $D$ $^b$ & z $^c$ & Type $^d$ & i $^e$ \\ & & Mpc & & & $^{\circ}$ \\
\hline
ngc4603 & PGC42510 & 33.1 & 0.008647 & 5.0 & 44.8 \\
ngc4535 & PGC41812 & 15.8 & 0.006551 & 5.0 & 23.8 \\
ngc1762 & PGC16654 & 76.5 & 0.015854 & 5.1 & 51.5 \\
ngc3451 & PGC32754 & 26.1 & 0.00445 & 6.5 & 62.7 \\
ngc4790 & PGC43972 & 15.3 & 0.004483 & 4.8 & 58.8 \\
ngc3244 & PGC30594 & 42.7 & 0.009211 & 5.6 & 49.3 \\
ngc628 & PGC5974 & 9.8 & 0.002192 & 5.2 & 19.8 \\
pgc30591 & PGC30591 & 35.5 & 0.006765 & 6.8 & 86.6 \\
ngc5643 & PGC51969 & 11.8 & 0.003999 & 5.0 & 29.6 \\
ngc1309 & PGC12626 & 24.1 & 0.007125 & 3.9 & 21.2 \\
ngc1084 & PGC10464 & 17.3 & 0.004693 & 4.8 & 49.9 \\
ngc7580 & PGC70962 & 65.3 & 0.01479 & 3.0 & 36.5 \\
ngc692 & PGC6642 & 87.9 & 0.021181 & 4.1 & 45.2 \\
eso462-g009 & PGC64537 & 83.2 & 0.019277 & 1.1 & 58.8 \\
ic5273 & PGC70184 & 14.7 & 0.004312 & 5.6 & 50.8 \\
pgc3140 & PGC3140 & 81.3 & 0.019029 & 1.4 & 62.7 \\
ic1553 & PGC1977 & 35.0 & 0.00979 & 7.0 & 78.6 \\
ugc11289 & PGC62097 & 59.7 & 0.013333 & 4.5 & 53.7 \\
ic4582 & PGC55967 & 37.3 & 0.007155 & 3.8 & 83.1 \\
ngc2466 & PGC21714 & 73.1 & 0.017722 & 5.0 & 16.0 \\
eso443-21 & PGC44663 & 41.9 & 0.009404 & 5.7 & 79.0 \\
ic4452 & PGC51951 & 65.3 & 0.014337 & 1.3 & 20.6 \\
eso498-g5 & PGC26671 & 40.7 & 0.008049 & 4.3 & 41.8 \\
eso552-g40 & PGC16465 & 95.5 & 0.022649 & 2.1 & 54.4 \\
eso163-g11 & PGC21453 & 33.0 & 0.009413 & 3.0 & 70.9 \\
ngc7582 & PGC71001 & 18.7 & 0.005254 & 2.1 & 68.0 \\
ngc1620 & PGC15638 & 39.6 & 0.011715 & 4.5 & 81.2 \\
ic1320 & PGC64685 & 73.6 & 0.016548 & 2.9 & 58.1 \\
ngc3393 & PGC32300 & 52.8 & 0.012509 & 1.2 & 30.9 \\
ngc2370 & PGC20955 & 79.8 & 0.018346 & 3.4 & 56.8 \\
ngc4981 & PGC45574 & 21.0 & 0.005604 & 4.0 & 44.7 \\
ngc3783 & PGC36101 & 25.1 & 0.00973 & 1.4 & 26.6 \\
ngc1285 & PGC12259 & 74.1 & 0.017475 & 3.4 & 59.3 \\
ngc5806 & PGC53578 & 26.2 & 0.004533 & 3.2 & 60.4 \\
eso018-g018 & PGC26840 & 71.1 & 0.017572 & 4.2 & 38.9 \\
ngc6754 & PGC62871 & 38.4 & 0.010864 & 3.2 & 61.0 \\
ic2560 & PGC29993 & 32.5 & 0.009757 & 3.4 & 65.6 \\
ngc7140 & PGC67532 & 36.0 & 0.009947 & 3.8 & 49.6 \\
ngc3464 & PGC833131 & 52.8 & 0.012462 & 4.9 & 50.8 \\
mcg-02-13-38 & PGC16605 & 55.2 & 0.013293 & 1.2 & 73.6 \\
ngc1590 & PGC15368 & 55.2 & 0.012999 & 5.0 & 27.9 \\
pgc8822 & PGC8822 & 74.1 & 0.017555 & 5.0 & 58.2 \\
ngc7721 & PGC72001 & 21.2 & 0.006721 & 4.9 & 81.4 \\
pgc28308 & PGC28308 & 43.0 & 0.00907 & 6.7 & 85.5 \\
ngc1137 & PGC10942 & 42.8 & 0.010147 & 3.0 & 59.5 \\
eso478-g006 & PGC8223 & 74.8 & 0.017786 & 4.2 & 57.7 \\
ngc1448 & PGC13727 & 16.8 & 0.003896 & 6.0 & 86.4 \\
ngc3278 & PGC31068 & 42.7 & 0.009877 & 5.1 & 41.0 \\
ngc4030 & PGC37845 & 19.0 & 0.004887 & 4.0 & 47.0 \\
ngc3363 & PGC32089 & 85.1 & 0.019233 & 3.5 & 45.3 \\
ngc7780 & PGC72775 & 76.2 & 0.017195 & 2.0 & 61.2 \\
ic1438 & PGC68469 & 42.5 & 0.008659 & 1.2 & 23.8 \\
ngc4666 & PGC42975 & 15.7 & 0.005101 & 5.0 & 69.6 \\
ngc7396 & PGC69889 & 71.8 & 0.016561 & 1.0 & 59.5 \\
ngc716 & PGC6982 & 65.9 & 0.015204 & 1.1 & 75.9 \\
\hline
\end{tabular}
\end{table*}
A significant number of galaxies have been observed with the MUSE instrument on the VLT in different surveys \citep{2017ApJ...844...48P,2018A&A...609A.119S,2019ApJ...887...80K,2019MNRAS.484.5009E,2020AJ....159..167L}.
To use these observations we build our sample using the Hyperleda database and looking for them in the MUSE archive.
To be able to apply the method for a given galaxy,
we need to resolve the galaxy at a specific spatial scale.
Based on results of NGC 628 \citep{2019MNRAS.487L..61Z}, we choose the 500pc scale to study
the star formation self-regulation so we are limited to galaxies
closer than 100Mpc to resolve 500pc at $1\thinspace\rm{arcsec}$ resolution. We also need enough ($\sim16$)
resolution elements, so very nearby galaxies with low number of 500pc resolution elements are not useful. We will divide the MUSE field of view in squares, so we will need at least $4\times4$ 500pc squares per galaxy, limiting us to galaxies further away than 7Mpc.
We need galaxies with recent star formation to study star formation self-regulation. To ensure that we will detect recent star formation, we just consider Sa or later types morphology (Hubble type $T\geq1.0$ in Hyperleda). We discard edge-on galaxies ($i=90\rm{\rm{\deg}}$), galaxies classified as multiple, Irregulars (Hubble type $T\geq9$ in Hyperleda), and LIRGs (in NASA Ned). We just select galaxies with declination lower than $45\rm{\deg}$N to be observable from Paranal Observatory.
The SQL (Structured Query Language) search through Hyperleda \footnote{\url{http://leda.univ-lyon1.fr/fullsql.html}} selects 13636 galaxies, of which 164 have been observed with MUSE on the VLT and have publicly available data with an exposure time at least of 1600 seconds. We also removed galaxies in Arp \citep{1966ApJS...14....1A}, Vorontsov-Velyaminov \citep{1959VV....C......0V}, and Hickson Compact Group \citep{1982ApJ...255..382H} catalogs, to get rid out of strong external effects on the star formation history (SFH) and gas flows due to interactions. We have a total of 148 galaxies satisfying these conditions in the public MUSE archive. Of these, 9 galaxies did not pass our requirements in a spectral inspection by eye, because of clear spectral artifacts, or not having enough H$\alpha$ emission in the pointing (MUSE has a square $1\rm{arcmin}\times1\rm{arcmin}$ FOV). We initially analysed the single stellar populations (SSP's) of the remaining 139 galaxies, to apply the method described in this article. Since the method requires enough regions to be include in the analysis, we set this limit to 16 (4x4). However each of the 16 regions sampled per galaxy needs sufficient current SFR, sufficient signal to noise, and that can be properly reproduced with stellar population synthesis models. Finally, only 102 galaxies satisfied all the conditions allowing us to estimate $\eta$. We present their parameters in Table \ref{tab_sample}.
\subsection{Muse spectral data}
\begin{figure}
\includegraphics[width=0.42\textwidth]{ugc11001_color.pdf}
\caption{Colour composite RGB image recovered from MUSE data of one of the studied galaxies, UGC 11001. The red, green and blue images used to create RGB are obtained by integrating MUSE spectra in R, V and B filters, respectively.
500pc wide regions where spectra was extracted are overplotted as green
and the regions identified as those on the envelope (defined above in section \S4) are marked as red squares.}
\label{fig_colormuse}
\end{figure}
We use the MUSE \citep{2010SPIE.7735E..08B} reduced publicly available data for the galaxies
listed in Table \ref{tab_sample}, from the ESO
archive\footnote{\url{http://archive.eso.org/wdb/wdb/adp/phase3_spectral/form?collection_name=MUSE}}.
We first made a visual inspection to remove galaxies with no H$\alpha$ emission in the MUSE pointing, and thus to select MUSE fields where H$\alpha$ was observed, to be able to estimate recent star formation. After delimiting the regions with recent star formation, we divide each field into an integer number of observing squares, giving us squares with the closest (and larger than) size value to 500 pc.
We show an example in Fig. \ref{fig_colormuse}, we do not use the squares outside the MUSE pointing.
We choose 500 pc because it gives us a scale on which, from previous work, we expect to observe the self-regulation of star formation \citep{2019MNRAS.487L..61Z}, and it allows us to include galaxies at distances of up to 100 Mpc where 500 pc corresponds to 1 arcsec.
We also need the foreground stars to be masked. We extract the spectrum for each defined region, correct it for Galactic extinction, and associate each with a redshift estimate, using the H$\alpha$
or [NII] at $6583.4\rm{\AA}$ if the later has a stronger peak than the former. We next estimate the [NII]/H$\alpha$ and the [OIII]/H$\beta$ flux ratios, and remove the regions which are classified as Seyfert-LINER in the BPT diagram \citep{2006MNRAS.372..961K}.
\section{Stellar population synthesis and model fits}
\subsection{Stellar population synthesis}
\begin{figure}
\begin{center}
\includegraphics[width=0.5\textwidth]{sinopsis_fits_ngc716_34_786.pdf}
\end{center}
\caption{Two characteristic spectra for the galaxy NGC 716. Observed spectra are shown as blue lines. Top: the total star formation is dominated by the past star formation. Bottom: the total star formation is dominated by the recent star formation. The model spectrum which best fits the observed spectrum is shown as a red line. The continuum bands used to fit the observed spectrum to the combination of SSPs are shown as black lines.
}
\label{fig_specfit}
\end{figure}
We use SINOPSIS code \footnote{\url{https://www.irya.unam.mx/gente/j.fritz/JFhp/SINOPSIS.html}}
\citep{2007A&A...470..137F,2017ApJ...848..132F} to fit combinations of SSPs to the
observed spectra. SINOPSIS fits equivalent widths of emission and absorption lines, as well as
defined continuum bands. In this work, we use the H$\alpha$ and H$\beta$ equivalent widths, and
the 9 continuum bands shown in Fig. \ref{fig_specfit}, where we show two observed spectra of the galaxy NGC 716 and the resulted fits as an example.
We use the updated version of the Bruzual \& Charlot models \citep{2019MNRAS.483.2382W}. We used SSPs of 3 metallicities ($Z=0.004$, $Z=0.02$, and $Z=0.04$)
in 12 age bins (2 Myr, 4 Myr, 7 Myr, 20 Myr, 57 Myr, 200 Myr,
570 Myr, 1 Gyr, 3 Gyr, 5.75 Gyr, 10 Gyr, and 14 Gyr). We assume a free form of SFH, the Calzetti dust attenuation law \citep{2000ApJ...533..682C}, and the Chabrier 2003 IMF \citep{2003PASP..115..763C} for stellar masses between $0.1\rm{M_{\odot}}$
and $100\rm{M_{\odot}}$. The emission lines for the SSPs younger than 20 Myr are computed
using the photoionisation code
{\sc{Cloudy}} \citep{1993hbic.book.....F,1998PASP..110..761F,2013ApJ...767..123F},
assuming case B recombination \citep{1989agna.book.....O},
an electron temperature of $10^4 \rm{K}$, an electron density of $100\rm{cm}^{-3}$,
and a gas cloud with an inner radius of $10^{-2} \rm{pc}$
\citep{2017ApJ...848..132F}.
SINOPSIS uses the degeneracies
between age, metallicity, and dust attenuation, to compute the uncertainties in the
derived parameters \citep{2007A&A...470..137F}.
We rebin the different age bins into 4 bins: at $20\rm{Myr}$, $570\rm{Myr}$, $5.7\rm{Gyr}$,
and $14\rm{Gyr}$.
Simulated and observed spectra have been used to prove the validity of using SINOPSIS to recover these
4 age bins \citep{2007A&A...470..137F,2011A&A...526A..45F}. Additionally, SINOPSIS and similar codes
have shown the reliability of recovering the SFH using synthesis of SSPs in, at least, 4 age bins \citep{2005MNRAS.358..363C,2007A&A...470..137F,2011A&A...526A..45F,2016RMxAA..52...21S}. However, since we are interested in the recent star formation variations, we consider only the two most recent age bins, $20\rm{Myr}$, $570\rm{Myr}$, and call them recent, and past age bins, respectively. In this way we recover the recent, and the past star formation rate surface densities, $\Sigma_{\rm{SFR\thinspace recent}}$ and $\Sigma_{\rm{SFR\thinspace past}}$,
which improves the confidence in the results presented in this work, since the two most recent age bins are better constrained than the oldest ones.
In order to use regions with a meaningful
result, we take into account only regions with a signal to noise ratio (SNR) larger than 20 over the $[5350-5420]\thinspace \rm{\AA}$ range, and $\chi^2<3$. Due to IMF sampling effects, we also consider only regions where the recent SFR is larger than $10^{-3}\rm{M_{\odot}/yr}$ and the past SFR is larger than $10^{-5}\rm{M_{\odot}/yr}$.
Because we are limited to galaxies from Hubble Type Sa to Sdm, also excluding interacting galaxies and (U)LIRGs, the galaxy sample, by construction, is defined by galaxies that are probably on the star formation galaxy main sequence, and probably evolved via secular evolution in the studied age range (last 570 Myr), where
by secular evolution we mean evolution dominated by slow processes (slower than many galaxy rotation periods \citet{2004ARA&A..42..603K}).
The galaxies probably evolved through more violent episodes in the past, but we are not affected by them in the studied age range.
Nevertheless, individual zones such as the centres of the galaxies, might have evolved via rapid evolution due to high gas flows even in the studied age range. Because of this, we removed regions whose centres are at a distance of 500 pc or less from the centre of the galaxy, as well as regions
having a very high recent SFR compared to the rest of the galaxy, specifically, we removed regions having $\Sigma_{\rm{SFR\thinspace recent}}$ larger than $\overline{\Sigma}_{\rm{SFR\thinspace recent}}+3\sigma_{\Sigma_{\rm{SFR\thinspace recent}}}$ for each galaxy. We will discuss how affects the results the removal of very high recent SFR regions in the discussion section (\S6.4).
\subsection{Fitting data to self-regulator the model}
We have made the same assumptions made in \citet{2019MNRAS.487L..61Z} in order to fit
our observables to the self-regulator model. For completeness, we briefly describe them here.
The self-regulator model (Eq. \ref{eq_bath_res}) is valid for a star or a group of co-rotating stars in the galaxy such as a massive star cluster (>500$\rm{M_{\odot}}$ \citet{2003ARA&A..41...57L}). Assuming $\eta$ constant, Eq. \ref{eq_bath_res} is linear, so we can add up regions obeying that equation, and still obey the equation. In this context, the mass-loading factor would be representative of massive star clusters scales ($\sim$pc \citet{2003ARA&A..41...57L}). Although we find below that $\eta$ varies (Eq. \ref{eq_eta_den}), the variation is smooth enough to consider it approximately constant here.
Therefore, the group of stars which are massive enough to produce bound clusters can be considered as a whole, while the less massive ones are splitted into individual stars. Feedback between different regions is then not considered here. We assume that our 500 pc wide regions are made of individual smaller regions obeying Eq \ref{eq_bath_res}, so we can rewrite Eq. \ref{eq_bath_res} to be valid for our larger regions as the average of individual regions:
\begin{equation}
\dot{\overline{{\Sigma}}}_{\rm{gas}}=\dot{\overline{{\Sigma}}}_{\rm{net \thinspace flow}}-\overline{\Sigma}_{\rm{SFR}}(1-R+\eta).
\label{eq_bath_res_sum}
\end{equation}
We already showed in \cite{2019MNRAS.487L..61Z} that the resulted mass-loading factor was independent of the chosen scale (from 87pc to 1kpc) in NGC 628. Hence, regions can be added up while Eq. \ref{eq_bath_res_sum} is still valid.
The value of the $\overline{\Sigma}_{\rm{SFR\thinspace past}}$ we are able to measure is a time average over 550Myr. Since Eq. \ref{eq_bath_res_sum} is linear, we can substitute the time differentials by time average values over our age bin, and we will not be affected by possible bursts of the star formation, as long as the variation of $\eta$ is small enough (as we do find below).
The net gas flow rate surface density, $\dot{\overline{{\Sigma}}}_{\rm{net \thinspace flow}}$, is the change in gas density due to gas flows (independently of star formation), which can be negative, although in that case, the star formation is quenched \citep{2019MNRAS.487L..61Z}. This term, $\dot{\overline{{\Sigma}}}_{\rm{net \thinspace flow}}$, also includes the possibility of gas return from different regions and the same region at a later epoque, an effect known as galactic fountains \citep{2017ASSL..430..323F}.
The observables are $\overline{\Sigma}_{\rm{SFR\thinspace recent}}$ and $\overline{\Sigma}_{\rm{SFR\thinspace past}}$. Let us assume that we can estimate
$\dot{\overline{{\Sigma}}}_{\rm{gas}}$ from the star formation change considering the KS law, $\overline{\Sigma}_{\rm{SFR}}=A{\overline{{\Sigma}}}_{\rm{gas}}^N$, and rewrite Eq. \ref{eq_bath_res}:
{\small
\begin{equation}
\begin{split}
& \overline{\Sigma}_{\rm{SFR\thinspace recent}}= \\
& A\left\{ \left[ \dot{\overline{\Sigma}}_{\rm{net \thinspace flow}}-\overline{\Sigma}_{\rm{SFR\thinspace past}}\left( 1-R+\eta \right) \right]\Delta t + \left[\frac{\overline{\Sigma}_{\rm{SFR\thinspace past}}}{A} \right]^{\frac{1}{N}} \right\}^N.
\end{split}
\label{eq_etafit}
\end{equation}
}
\noindent where there is a relation between our two observables ($\overline{\Sigma}_{\rm{SFR\thinspace recent}}$ and $\overline{\Sigma}_{\rm{SFR\thinspace past}}$), $\dot{\overline{{\Sigma}}}_{\rm{net \thinspace flow}}$, and $\eta$. We will use a simplistic approximation to estimate $\dot{\overline{{\Sigma}}}_{\rm{net \thinspace flow}}$, since we do not observe it. As explained in \cite{2019MNRAS.487L..61Z}, we assume that several regions have an approximate value close to the maximum value of $\dot{\overline{{\Sigma}}}_{\rm{net \thinspace flow}}$, for a given galaxy.
In the case of the estimation of $\eta$, although $\eta$ could vary between regions, we will find that the variation is smooth enough
(Eq. \ref{eq_eta_den}) to
consider the existence of a representative value for specific regions.
In the following, for simplicity since we are only dealing with one type of regions, the 500 pc wide ones, we will be using the analysed terms (e.g. $\overline{\Sigma}_{\rm{SFR\thinspace recent}}$, $\overline{\Sigma}_{\rm{SFR\thinspace past}}$, $\dot{\overline{{\Sigma}}}_{\rm{net \thinspace flow}}$) without the need of using the average symbols ($\Sigma_{\rm{SFR\thinspace recent}}$, $\Sigma_{\rm{SFR\thinspace past}}$, $\dot{{\Sigma}}_{\rm{net \thinspace flow}}$). Therefore, when we present an average, the average will be for several 500pc wide regions.
Assuming the instantaneous recycling approximation \citep{2014ARA&A..52..415M}
for stars more massive than $3\rm{M_{\odot}}$ ($\tau_{\rm{MS}}\sim 0.6\rm{Gyr}$, where $\tau_{\rm{MS}}$ is the main sequence lifetime), and a Chabrier IMF
\citep{2003PASP..115..763C},
we obtain a value of $R=0.27$. We use the values $A=10^{-4.32}\rm{M_{\odot}/kpc^2/yr}$, and $N=1.56$ for the KS law\cite{2007ApJ...671..333K}.
\section{Results}
\begin{figure}
\includegraphics[width=0.5\textwidth]{ugc11001_500_sfr1_sfr2.pdf}
\caption{Recent star formation rate surface density, $\Sigma_{\rm{SFR\thinspace recent}}$, versus the past
star formation rate surface density, $\Sigma_{\rm{SFR\thinspace past}}$, for the UGC 11001 galaxy. The red dots are the regions identified as those on the envelope. We plot the fit of Eq. \ref{eq_etafit} to the regions on the envelope as well as the result of the fit and the 1-$\sigma$ uncertainty range of the fit as shaded region.
}
\label{fig_diagram_ex}
\end{figure}
As an example, we plot the $\Sigma_{\rm{SFR\thinspace recent}}$ versus $\Sigma_{\rm{SFR\thinspace past}}$ diagram for one of the galaxies, UGC 11001, in Fig. \ref{fig_diagram_ex}. We plot the $\Sigma_{\rm{SFR\thinspace recent}}$ versus $\Sigma_{\rm{SFR\thinspace past}}$ diagrams for all of the galaxies in Fig. \ref{sfr_diagrams}.
Each of the points in these plots can be seen as the relation between the $\Sigma_{\rm{SFR\thinspace recent}}$ and the $\Sigma_{\rm{SFR\thinspace past}}$
which depends on the value of $\dot{\Sigma}_{\rm{net\thinspace flow}}$, and $\eta$ (Eq. \ref{eq_etafit}).
We identify those regions having the maximum $\Sigma_{\rm{SFR\thinspace recent}}$, per bin of $\Sigma_{\rm{SFR\thinspace past}}$, as the regions on the envelope.
We see the regions on the envelope as red dots
in Fig. \ref{fig_diagram_ex}, and as red squares in the MUSE recovered false color image of UGC 11001 in Fig. \ref{fig_colormuse}.
\begin{table}
\centering
\caption{Estimated mass-loading factors, $\eta$, maximum flow gas surface density term, $\dot{\Sigma}_{\rm{flow\thinspace max}}$, and the associated average stellar mass surface density for the regions on the envelope, $\Sigma_{*}$.}
\label{tab_results}
\begin{tabular}{rrrr}
\hline
Galaxy identifier & $\eta$ $^a$ & $\dot{\Sigma}_{\rm{flow\thinspace max}}$ $^b$ & $\Sigma_{*}$ $^c$ \\ & & $10^{-8}\rm{M_{\odot}\thinspace yr^{-1}\thinspace kpc^{-2}}$ & $10^{6}\rm{M_{\odot}\thinspace kpc^{-2}}$ \\
\hline
pgc33816 & $4.8 \pm 0.9$ & $4.9\pm 0.9$ & $27\pm 15 $ \\
eso184-g082 & $5.0 \pm 1.0$ & $8.7\pm 0.6$ & $49\pm 20 $ \\
eso467-062 & $8.0 \pm 2.0$ & $14.0\pm 1.0$ & $51\pm 39 $ \\
ugc272 & $3.4 \pm 0.2$ & $6.9\pm 0.4$ & $57\pm 44 $ \\
ngc5584 & $2.2 \pm 0.4$ & $4.0\pm 1.0$ & $60\pm 31 $ \\
eso319-g015 & $5.0 \pm 2.0$ & $11.0\pm 3.0$ & $66\pm 65 $ \\
ugc11214 & $2.6 \pm 0.6$ & $9.0\pm 2.0$ & $84\pm 33 $ \\
ngc6118 & $2.19 \pm 0.09$ & $2.8\pm 0.3$ & $90\pm 51 $ \\
ic1158 & $6.8 \pm 0.4$ & $16.3\pm 0.3$ & $109\pm 28 $ \\
ngc5468 & $2.2 \pm 0.8$ & $23.0\pm 2.0$ & $113\pm 66 $ \\
eso325-g045 & $1.7 \pm 0.2$ & $12.0\pm 1.0$ & $121\pm 57 $ \\
ngc1954 & $3.3 \pm 0.5$ & $23.8\pm 0.8$ & $121\pm 31 $ \\
ic5332 & $3.0 \pm 1.0$ & $12.0\pm 2.0$ & $120\pm 100 $ \\
ugc04729 & $2.8 \pm 0.6$ & $7.0\pm 2.0$ & $126\pm 57 $ \\
ngc2104 & $1.7 \pm 0.6$ & $4.0\pm 2.0$ & $132\pm 62 $ \\
eso316-g7 & $2.0 \pm 1.0$ & $12.0\pm 6.0$ & $136\pm 39 $ \\
eso298-g28 & $6.0 \pm 0.7$ & $47.0\pm 2.0$ & $136\pm 46 $ \\
mcg-01-57-021 & $7.0 \pm 1.0$ & $17.0\pm 2.0$ & $137\pm 24 $ \\
pgc128348 & $2.9 \pm 0.1$ & $11.6\pm 0.7$ & $140\pm 92 $ \\
pgc1167400 & $2.3 \pm 0.3$ & $4.1\pm 0.8$ & $141\pm 78 $ \\
ngc2835 & $2.2 \pm 0.7$ & $7.0\pm 3.0$ & $144\pm 40 $ \\
ic2151 & $2.0 \pm 0.5$ & $5.0\pm 3.0$ & $146\pm 61 $ \\
ngc988 & $1.2 \pm 0.4$ & $1.0\pm 2.0$ & $158\pm 63 $ \\
ngc1483 & $3.0 \pm 2.0$ & $13.0\pm 5.0$ & $158\pm 40 $ \\
ngc7421 & $1.1 \pm 0.3$ & $1.0\pm 0.9$ & $167\pm 78 $ \\
fcc290 & $2.0 \pm 0.4$ & $3.0\pm 1.0$ & $169\pm 42 $ \\
ic344 & $2.5 \pm 0.2$ & $11.0\pm 1.0$ & $171\pm 78 $ \\
ngc3389 & $4.4 \pm 0.7$ & $32.4\pm 0.7$ & $190\pm 130 $ \\
eso246-g21 & $2.9 \pm 0.7$ & $7.0\pm 2.0$ & $188\pm 67 $ \\
pgc170248 & $4.9 \pm 0.7$ & $16.0\pm 1.0$ & $192\pm 77 $ \\
ngc7329 & $4.1 \pm 0.2$ & $7.8\pm 0.4$ & $200\pm 120 $ \\
ugc12859 & $2.8 \pm 0.3$ & $5.1\pm 0.9$ & $202\pm 90 $ \\
ugc1395 & $2.7 \pm 0.3$ & $8.0\pm 1.0$ & $200\pm 160 $ \\
ngc5339 & $2.9 \pm 0.4$ & $5.0\pm 1.0$ & $210\pm 150 $ \\
ngc1591 & $2.3 \pm 0.7$ & $19.0\pm 4.0$ & $212\pm 93 $ \\
pgc98793 & $1.7 \pm 0.1$ & $6.6\pm 0.8$ & $214\pm 95 $ \\
ugc5378 & $2.7 \pm 0.4$ & $10.0\pm 1.0$ & $223\pm 93 $ \\
ngc4806 & $1.9 \pm 0.2$ & $10.3\pm 0.9$ & $230\pm 170 $ \\
ngc1087 & $1.3 \pm 0.3$ & $14.0\pm 2.0$ & $230\pm 170 $ \\
ngc4980 & $1.8 \pm 0.3$ & $6.6\pm 0.9$ & $240\pm 170 $ \\
ngc6902 & $2.4 \pm 0.3$ & $8.0\pm 1.0$ & $240\pm 13 $ \\
ugc11001 & $2.5 \pm 0.1$ & $21.8\pm 0.5$ & $250\pm 140 $ \\
ic217 & $1.2 \pm 0.4$ & $3.0\pm 2.0$ & $266\pm 88 $ \\
eso506-g004 & $3.4 \pm 0.2$ & $13.9\pm 0.4$ & $270\pm 170 $ \\
ic2160 & $2.4 \pm 0.4$ & $12.0\pm 2.0$ & $270\pm 260 $ \\
ngc1385 & $0.9 \pm 0.1$ & $9.7\pm 0.9$ & $272\pm 44 $ \\
mcg-01-33-034 & $1.0 \pm 0.09$ & $7.9\pm 0.5$ & $270\pm 150 $ \\
ngc4603 & $0.9 \pm 0.1$ & $3.0\pm 1.0$ & $276\pm 85 $ \\
ngc4535 & $4.1 \pm 0.7$ & $15.0\pm 2.0$ & $280\pm 200 $ \\
ngc1762 & $2.3 \pm 0.2$ & $8.2\pm 0.7$ & $290\pm 140 $ \\
ngc3451 & $3.7 \pm 0.4$ & $11.0\pm 1.0$ & $300\pm 180 $ \\
ngc4790 & $1.2 \pm 0.3$ & $9.0\pm 2.0$ & $334\pm 63 $ \\
ngc3244 & $1.8 \pm 0.2$ & $8.0\pm 1.0$ & $340\pm 270 $ \\
ngc628 & $1.8 \pm 0.1$ & $19.0\pm 1.0$ & $360\pm 170 $ \\
pgc30591 & $0.7 \pm 0.4$ & $0.0\pm 2.0$ & $360\pm 160 $ \\
ngc5643 & $1.1 \pm 0.3$ & $3.0\pm 2.0$ & $410\pm 160 $ \\
ngc1309 & $1.3 \pm 0.2$ & $17.0\pm 2.0$ & $420\pm 220 $ \\
\end{tabular}
\\
\hrulefill
\\
{\raggedright
$^a$ Mass-loading factor derived in this work.\
$^b$ Maximum flow gas surface density term derived in this work.\
$^c$ Stellar mass surface density for the regions on the envelope obtained in this work. }
\\
\hrulefill
\end{table}
\begin{table}
\centering
\contcaption{}
\begin{tabular}{rrrr}
\hline
Galaxy identifier & $\eta$ $^a$ & $\dot{\Sigma}_{\rm{flow\thinspace max}}$ $^b$ & $\Sigma_{*}$ $^c$ \\ & & $10^{-8}\rm{M_{\odot}\thinspace yr^{-1}\thinspace kpc^{-2}}$ & $10^{6}\rm{M_{\odot}\thinspace kpc^{-2}}$ \\
\hline
ngc1084 & $0.6 \pm 0.2$ & $13.0\pm 6.0$ & $420\pm 170 $ \\
ngc7580 & $1.6 \pm 0.2$ & $18.0\pm 1.0$ & $420\pm 110 $ \\
ngc692 & $2.7 \pm 0.2$ & $14.0\pm 1.0$ & $420\pm 220 $ \\
eso462-g009 & $4.0 \pm 1.0$ & $8.0\pm 2.0$ & $440\pm 310 $ \\
ic5273 & $1.5 \pm 0.7$ & $9.0\pm 4.0$ & $450\pm 220 $ \\
pgc3140 & $1.0 \pm 0.1$ & $4.0\pm 1.0$ & $460\pm 210 $ \\
ic1553 & $0.6 \pm 0.3$ & $5.0\pm 2.0$ & $460\pm 290 $ \\
ugc11289 & $1.8 \pm 0.3$ & $8.0\pm 3.0$ & $472\pm 46 $ \\
ic4582 & $2.0 \pm 0.1$ & $9.1\pm 0.5$ & $480\pm 280 $ \\
ngc2466 & $1.5 \pm 0.2$ & $18.0\pm 2.0$ & $480\pm 470 $ \\
eso443-21 & $1.9 \pm 0.3$ & $94.0\pm 4.0$ & $490\pm 180 $ \\
ic4452 & $0.37 \pm 0.05$ & $5.2\pm 0.6$ & $500\pm 320 $ \\
eso498-g5 & $1.5 \pm 0.3$ & $20.0\pm 2.0$ & $510\pm 320 $ \\
eso552-g40 & $1.7 \pm 0.4$ & $12.0\pm 3.0$ & $540\pm 280 $ \\
eso163-g11 & $0.48 \pm 0.06$ & $1.0\pm 2.0$ & $570\pm 300 $ \\
ngc7582 & $0.6 \pm 0.1$ & $0.0\pm 2.0$ & $570\pm 280 $ \\
ngc1620 & $1.4 \pm 0.3$ & $8.0\pm 2.0$ & $580\pm 400 $ \\
ic1320 & $1.4 \pm 0.1$ & $4.6\pm 0.9$ & $590\pm 270 $ \\
ngc3393 & $1.9 \pm 0.3$ & $11.0\pm 1.0$ & $590\pm 270 $ \\
ngc2370 & $0.87 \pm 0.08$ & $4.0\pm 2.0$ & $600\pm 310 $ \\
ngc4981 & $0.85 \pm 0.08$ & $7.0\pm 1.0$ & $630\pm 580 $ \\
ngc3783 & $1.3 \pm 0.3$ & $12.0\pm 3.0$ & $680\pm 170 $ \\
ngc1285 & $0.8 \pm 0.1$ & $7.0\pm 2.0$ & $700\pm 460 $ \\
ngc5806 & $1.7 \pm 0.3$ & $17.0\pm 3.0$ & $720\pm 530 $ \\
eso018-g018 & $1.0 \pm 0.3$ & $13.0\pm 5.0$ & $720\pm 420 $ \\
ngc6754 & $0.44 \pm 0.06$ & $1.0\pm 1.0$ & $750\pm 480 $ \\
ic2560 & $1.2 \pm 0.2$ & $11.0\pm 1.0$ & $760\pm 370 $ \\
ngc7140 & $2.4 \pm 0.7$ & $13.0\pm 6.0$ & $770\pm 210 $ \\
ngc3464 & $1.9 \pm 0.3$ & $9.0\pm 3.0$ & $780\pm 380 $ \\
mcg-02-13-38 & $1.3 \pm 0.1$ & $20.0\pm 1.0$ & $790\pm 450 $ \\
ngc1590 & $0.83 \pm 0.08$ & $25.0\pm 3.0$ & $810\pm 460 $ \\
pgc8822 & $1.15 \pm 0.09$ & $4.0\pm 1.0$ & $970\pm 430 $ \\
ngc7721 & $1.3 \pm 0.5$ & $10\pm 10$ & $990\pm 540 $ \\
pgc28308 & $0.2 \pm 0.1$ & $0.0\pm 8.0$ & $1030\pm 450 $ \\
ngc1137 & $0.8 \pm 0.2$ & $1.0\pm 4.0$ & $1030\pm 830 $ \\
eso478-g006 & $1.1 \pm 0.2$ & $10.0\pm 4.0$ & $1050\pm 650 $ \\
ngc1448 & $1.05 \pm 0.09$ & $15.0\pm 2.0$ & $1130\pm 480 $ \\
ngc3278 & $1.3 \pm 0.2$ & $31.0\pm 3.0$ & $1170\pm 710 $ \\
ngc4030 & $0.6 \pm 0.1$ & $14.0\pm 3.0$ & $1310\pm 550 $ \\
ngc3363 & $0.41 \pm 0.08$ & $0.0\pm 0.9$ & $1330\pm 590 $ \\
ngc7780 & $1.2 \pm 0.2$ & $8.0\pm 3.0$ & $1450\pm 590 $ \\
ic1438 & $1.5 \pm 0.3$ & $11.0\pm 4.0$ & $1520\pm 250 $ \\
ngc4666 & $0.3 \pm 0.1$ & $10.0\pm 3.0$ & $2900\pm 1300 $ \\
ngc7396 & $0.5 \pm 0.2$ & $0.0\pm 4.0$ & $4100\pm 1900 $ \\
ngc716 & $0.5 \pm 0.2$ & $70\pm 10$ & $5200\pm 1900 $ \\
\hline
\end{tabular}
\end{table}
We can see in Eq. \ref{eq_etafit} that we need the value of $\dot{\Sigma}_{\rm{net \thinspace flow}}$ in combination with the $\Sigma_{\rm{SFR\thinspace recent}}$ and $\Sigma_{\rm{SFR\thinspace past}}$ values, in order to quantify $\eta$.
For a given galaxy, there should be a maximum value for the net flow gas surface density term, $\dot{\Sigma}_{\rm{flow\thinspace max}}$. Although in principle $\dot{\Sigma}_{\rm{flow\thinspace max}}$ is unknown for us, if we assume that there are several segments where $\dot{\Sigma}_{\rm{net\thinspace flow}}\sim\dot{\Sigma}_{\rm{flow\thinspace max}}$, then these regions are those on the $\Sigma_{\rm{SFR\thinspace recent}}$ versus $\Sigma_{\rm{SFR\thinspace past}}$ diagram envelope.
Assuming $\dot{\Sigma}_{\rm{net\thinspace flow}}$ constant, we fit Eq. \ref{eq_etafit} to the regions on the envelope and estimate the mass-loading factor, representative of those specific regions.
We have selected a galaxy sample mainly composed by galaxies on the main sequence of star formation, and remove segments having very high $\Sigma_{\rm{SFR\thinspace recent}}$ compared to the rest of the segments of a galaxy.
Therefore, the galaxy sample, as well as the segments, have been chosen to assure the $\dot{\Sigma}_{\rm{net\thinspace flow}}\sim\dot{\Sigma}_{\rm{flow\thinspace max}}$ hypothesis for several regions.
The regions below the envelope are due, to a greater extent, to regions having a smaller value
of $\dot{\Sigma}_{\rm{net\thinspace flow}}<\dot{\Sigma}_{\rm{flow\thinspace max}}$, and to a much lesser
extent, to regions having different $\eta$ values.
We have fitted Eq. \ref{eq_etafit} to the envelopes of the 102 galaxy listed in Table \ref{tab_sample} and show the results in Fig. \ref{fig_diagram_ex} for UGC 11001, and in Fig. \ref{sfr_diagrams} for the rest of the galaxies, as a red line.
\subsection{ Variations of $\eta$ }
\begin{figure}
\begin{center}
\includegraphics[width=0.49\textwidth]{eta_density_sinop.pdf}
\end{center}
\caption{Local mass-loading factor, $\eta$, versus the average stellar mass surface density
of the regions on the envelope for each galaxy, $\Sigma_{*}$, as black stars. We plot the linear fit to the empirically derived quantities and the 1-$\sigma$ uncertainty range of the fit as shaded regions.
The blue line shows the $\eta$ reported for supernova explosions hydrodynamical simulations \citep{2013MNRAS.429.1922C} with cut-off cooling modeling, and as blue line the $\eta$ reported for supernova explosions hydrodynamical
simulations \citep{2017ApJ...841..101L} with gas cold-phase modeling included.
}
\label{fig_eta_den}
\end{figure}
Although we have an $\eta$ value for each galaxy, $\eta$ is an average value representative only of the regions on the envelope, instead of the whole galaxy. We associate the estimated $\eta$ for a given envelope with the average surface stellar mass density, $\Sigma_{*}$, of those regions on the envelope where we estimate $\eta$. Therefore, $\eta$ is a local average value, representative only of the regions on the envelope, and their mean value of $\Sigma_{*}$.
Although $\eta$ value might vary through the regions on the envelope, we assume that the variation is smooth enough and associate the average stellar mass
density, $\Sigma_{*}$ and the standard deviation, to each envelope. The correlation found between $\eta$ and $\Sigma_{*}$ (Eq. \ref{eq_eta_den} and Fig. \ref{fig_eta_den}) is in fact smooth enough to make the association between $\eta$ and $\Sigma_{*}$ for the regions on the envelope.
We report $\eta$, $\dot{\Sigma}_{\rm{flow\thinspace max}}$, and $\Sigma_{*}$ values in Table \ref{tab_results}.
We plot in Fig. \ref{fig_eta_den}, $\eta$ versus $\Sigma_{*}$, and find that the mass-loading factors strongly correlate with the local $\Sigma_{*}$ measured on the envelope regions:
\begin{equation}
\log(\eta+1)=(-0.32\pm0.03)\log(\Sigma_{*})+(3.2\pm0.3)
\label{eq_eta_den}
\end{equation}
This correlation means that the denser the region, the lower is the mass-loading factor, which means that the amount of outflowing gas mass per unit star formation rate depends inversely on the stellar mass surface density. This is so because the denser the region, the larger is the local gravitational pull, making it harder for the gas to be expelled. The local chemical enrichment of galaxies \citep{2018ApJ...852...74B} also favours the gas regulator model and finds that the mass-loading factor depends on the local escape velocity.
This new empirical $\eta$-$\Sigma_{*}$ relation, can be used to check if stellar feedback implementations in numerical simulations \citep{2014MNRAS.445..581H,2017ApJ...841..101L,2013MNRAS.429.1922C} are realistic. In particular, we can
compare our $\eta$-$\Sigma_{*}$ relation with the $\eta$-$\Sigma_{\rm{gas}}$ relation from supernova explosion feedback simulations \citep{2017ApJ...841..101L,2013MNRAS.429.1922C},
using the observed $\Sigma_{*}$-$\Sigma_{\rm{gas}}$ relation \citep{2020MNRAS.492.2651B}. The inverse correlation between $\eta$ and $\Sigma_{*}$ is clear from observations and simulations. However, our empirical result supports that the radiative cooling below $10^{4}\rm{K}$ is important. This is because when ignoring gas cooling below $10^4\rm{K}$, there is an excess of warm gas compared with the models including a multiphase cold/warm gas \citep{2017ApJ...841..101L}.
Thus, in the warm gas excess scenario, there is a layer of gas with higher ISM pressure which prevents the gas to be expelled from the supernova explosions.
\section{Local to global mass-loading factors}
The mass-loading factor derived here
is representative of local scales. However, other observational and theoretical studies report global
mass-loading factors \citep{2015MNRAS.454.2691M,2016MNRAS.455.2592R,2017MNRAS.465.1682H,2019MNRAS.490.4368S,2019ApJ...886...74M}. We estimate global mass-loading factors, $\eta_{\rm{G}}$, from the empirical
$\eta$-$\Sigma_{*}$ relation reported here (Eq. \ref{eq_eta_den}), integrating over observed stellar mass density profiles.
To convert $\eta$ to $\eta_{\rm{G}}$, we assume that we can estimate the total outflow due to stellar feedback, $\dot{M}_{\rm{out}}$, by adding up $\dot{\Sigma}_{\rm{out}}$ over each individual segment where stellar feedback acts. Since we are interested in galaxy discs, we assume a radial characterization for the properties of interest, i.e., $\eta$,
$\dot{\Sigma}_{\rm{out}}$, $\Sigma_{\rm{SFR}}$, and $\Sigma_{\rm{SFR}}$ depend on $R$, the radial distance to the centre of the disc:
\begin{equation}
\eta_{\rm{G}}=\frac{\dot{M}_{\rm{out}}}{\rm{SFR}}=\frac{\int_{0}^{\infty}\dot{\Sigma}_{\rm{out}}(R)R\thinspace{\rm{d}}R}{\int_{0}^{\infty} \Sigma_{\rm{SFR}}(R)R\thinspace{\rm{d}}R}=\frac{\int_{0}^{\infty}\eta(R) \Sigma_{\rm{SFR}}(R)R\thinspace{\rm{d}}R}{\int_{0}^{\infty} \Sigma_{\rm{SFR}}(R)R\thinspace{\rm{d}}R}.
\label{eta_global}
\end{equation}
The conversion between $\eta$ to $\eta_{\rm{G}}$ is a $\Sigma_{\rm{SFR}}$ weighted average of $\eta$. Stellar mass density profiles, $\Sigma_{*}(R)$, are better constrained than $\Sigma_{\rm{SFR}}(R)$ profiles, so we decide to use the empirical relation between
$\Sigma_{\rm{SFR}}\propto \Sigma_{*}(R)^n$. For simplicity, we assume $n=1$, which is consistent with the latest results of this relation for galaxy discs \citep{2019MNRAS.488.3929C}, although we found that different $n$ values close to 1 do not change the results significantly. Using our empirical relation (Eq. \ref{eq_eta_den}) we rewrite Eq. \ref{eta_global}:
\begin{equation}
\eta_{\rm{G}}=\frac{\int_{0}^{\infty}\eta(R) \Sigma_{*}(R)R\thinspace{\rm{d}}R}{\int_{0}^{\infty} \Sigma_{*}(R)R\thinspace{\rm{d}}R}=\frac{10^{3.2}\int_{0}^{\infty}{\Sigma_{*}(R)}^{0.68} R\thinspace{\rm{d}}R}{\int_{0}^{\infty} \Sigma_{*}(R)R\thinspace{\rm{d}}R}.
\label{eta_global2}
\end{equation}
The stellar mass surface density profile, $\Sigma_{*}(R)$, is therefore all we need to compute the global mass-loading factor. We use the deepest stellar mass surface density profiles from the Spitzer Survey of Stellar Structure in Galaxies (S4G) \citep{2016A&A...596A..84D} to compute the global mass-loading factor as a function of stellar mass. S4G stellar mass surface density profiles are divided into 5 mass bins: $[10^{8.5}$-$10^{9}]$, $[10^{9}$-$10^{9.5}]$, $[10^{9.5}$-$10^{10}]$, $[10^{10}$-$10^{10.5}]$, and $[10^{10.5}$-$10^{11}]\thinspace\rm{M_{\odot}}$. We used these 5 mass bins to compute the $\eta_{\rm{G}}$ shown in Fig. \ref{eta_mstar}.
\subsection{Variations of $\eta_{\rm{G}}$}
\begin{figure}
\begin{center}
\includegraphics[width=0.49\textwidth]{eta_mstar500.pdf}
\end{center}
\caption{Global mass-loading factor, $\eta_{\rm{G}}$, versus the stellar mass of the galaxy, $M_*$.
The black lines show the empirical quantities which have been derived combining the distribution of $\eta$ as a function of stellar mass surface density (Fig. \ref{fig_eta_den}) with the stellar mass surface density profiles from S4G results \citep{2016A&A...596A..84D}, where the shaded region is the corresponding uncertainty propagated from the 1-$\sigma$ uncertainty found in the $\eta$-$\Sigma_{*}$ relation (Eq. \ref{eq_eta_den}). The discontinuity is due to the division in different parametrizations of the stellar mass surface density profiles in 5 mass bins reported by \citet{2016A&A...596A..84D}. The orange line shows the $\eta_{\rm{G}}$ reported for cosmological zoom-in galaxy hydrodynamical simulations \citep{2015MNRAS.454.2691M}, the green line the shows an analytical feedback model \citep{2017MNRAS.465.1682H}, and the blue line shows the $\eta_{\rm{G}}$ reported using the N-body Bolshoi-Planck simulation \citep{2016MNRAS.455.2592R}.
}
\label{eta_mstar}
\end{figure}
We present our
empirically derived global mass-loading factors as a function of stellar mass in Fig. \ref{eta_mstar} as black lines. The discontinuity is due to the division in different parametrizations of the stellar mass surface density profiles in 5 mass bins reported by \citet{2016A&A...596A..84D} .
The most important feature we find in Fig. \ref{eta_mstar} is that the smaller the stellar mass of the galaxy, the larger the mass-loading factor, as required to reconcile the ratio of halo to stellar mass in low-mass galaxies \citep{2010ApJ...717..379B,2016MNRAS.455.2592R}. We compare our empirical estimates with predictions based on N-body \citep{2016MNRAS.455.2592R} and hydrodynamical simulations \citep{2015MNRAS.454.2691M}, as well as with an analytical model \citep{2017MNRAS.465.1682H}.
Some simulations, as well as the analytic model we used to compare with, define the mass-loading factor using the outflowing mass that escapes the galaxy forever \citep{2016MNRAS.455.2592R,2017MNRAS.465.1682H}. Therefore, they
do not consider the outflowing gas which returns to the galaxy at a later time which, depending on the outflow velocity, will result in smaller mass-loading factors. We define the mass-loading factor as the total mass ejected
independently of velocity, and the mass can come back at a later time (outside the 550Myr time range). There is a remarkable agreement between our
empirical result and that from hydrodynamical simulations that define the mass-loading factor as independent of velocity \citep{2015MNRAS.454.2691M}. In fact, for lower masses ($M_* < 10^{9.5}\rm{M_{\odot}}$), where the
stellar feedback is thought to be more important regulating galaxy stellar mass growth, and the escape velocity lower, the agreement with the analytic model is also remarkably good.
\section{Discussion}
\subsection{Comparison with other studies}
The results reported here are slightly different when we compare them with
recent observational studies reporting local \citep{2019Natur.569..519K,2020MNRAS.493.3081R} and global \citep{2019MNRAS.490.4368S,2019ApJ...886...74M} mass-loading factors.
The reported local mass-loading factors using the spatial de-correlation between star formation and molecular gas \citep{2019Natur.569..519K} differ by less than $3\sigma$ from our reported values, and it might be due to the smaller time scale of $\sim 1.5\rm{Myr}$ for which they report efficient gas dispersal, while our reported time scale is $\sim550\rm{Myr}$.
The previously reported local mass-loading factors using the Na {\sc{D}} absorption \citep{2020MNRAS.493.3081R} are consistent with ours within $1\sigma$.
The use of Mg {\sc{II}} absorption of the circum-galactic medium to derive global mass-loading factors gives no
clear dependence on the total mass of the galaxy \citep{2019MNRAS.490.4368S}. However, the Mg {\sc{II}} absorption method gives $\eta_{\rm{G}}$ with very
high uncertainties, mainly due to the uncertainty when deriving the HI
column density from the Mg {\sc{II}} equivalent width \citep{2015ApJ...804...83S}. Due to these large uncertainties, their results are apparently consistent
with our results within 1$\sigma$ for most of their reported
$\eta_{\rm{G}}$'s.
The mass-loading factor estimates using deep H$\alpha$ imaging give smaller mass-loading factors compared to those reported here and give no correlation with the stellar mass of the galaxy \citep{2019ApJ...886...74M}. Nevertheless, the method using H$\alpha$ imaging derives the amount of outflowing gas from the H$\alpha$ surface brightness background, so it neglects H$\alpha$ emission stronger than this background emission.
The estimated outflowing mass could be inferior to the real one since we already know that H$\alpha$ emission has a component due to expansive bubbles \citep{2005A&A...430..911R,2015MNRAS.447.3840C}.
\subsection{Discussion on envelope's shapes}
Eq. \ref{eq_etafit} depends on $\dot{\Sigma}_{\rm{flow \thinspace max}}$, $\eta$, $\Sigma_{\rm{SFR\thinspace recent}}$, and $\Sigma_{\rm{SFR\thinspace past}}$ values. The case shown in Fig. \ref{fig_diagram_ex}, where there is no high recent star formation rate for those regions where the past star formation rate was the highest, is a common case, but not the only one. For instance, there are cases where $\Sigma_{\rm{SFR\thinspace recent}}$ values are approximately constant through the regions on the envelope or even increase as
$\Sigma_{\rm{SFR\thinspace past}}$ increase (e.g. NGC 988, NGC 7421, IC 217, IC 4452, PGC 28308 in Fig. \ref{sfr_diagrams}).
Essentially, there are two terms depending on $\Sigma_{\rm{SFR\thinspace past}}$ in Eq. \ref{eq_etafit},
one with a direct proportionality and the other with an inverse one.
The latter dominates for larger $\eta$ values meaning that the larger the mass-loading factor, the larger is the effect in reducing the amount of gas to form stars, as expected. However, for small enough $\eta$ and $\Sigma_{\rm{SFR\thinspace past}}$ values, the directly dependent term can dominate producing a direct relation between $\Sigma_{\rm{SFR\thinspace recent}}$ and $\Sigma_{\rm{SFR\thinspace past}}$, as we see in some galaxies in Fig. \ref{sfr_diagrams} (e.g. IC 4452).
The extrapolation of Eq. \ref{eq_etafit} to large enough values of $\Sigma_{\rm{SFR\thinspace past}}$ would give always a decrease in $\Sigma_{\rm{SFR\thinspace recent}}$, as long as $\eta>0$, and that is the reason to observe some inverted U-shape envelopes (e.g. NGC 1084, PGC 3140).
Finally, for large enough $\dot{\Sigma}_{\rm{flow \thinspace max}}$ values, the directly $\Sigma_{\rm{SFR\thinspace past}}$ dependent term is almost negligible for small values of $\Sigma_{\rm{SFR\thinspace past}}$, and then the recent star formation depends on $\dot{\Sigma}_{\rm{flow \thinspace max}}$ for the low $\Sigma_{\rm{SFR\thinspace past}}$, while decreases proportionally with $\Sigma_{\rm{SFR\thinspace past}}$ depending on the mass-loading factor (e.g UGC 5378, UGC 11001).
Therefore, the combination of $\dot{\Sigma}_{\rm{flow \thinspace max}}$ and $\eta$ variations, as well as the range covered by the past and recent star formation rate surface densities, is what gives the different envelope's shapes.
The uncertainties obtained when fitting Eq. \ref{eq_etafit} show reliable estimates, except for one case, the PGC 30591 galaxy, which is the one having the highest inclination of our sample.
\subsection{High-inclination galaxies}
Although we have removed edge-on ($i=90^{\circ}$) galaxies from our sample, high-inclination ($i>70^{\circ}$) galaxies might not be good candidates to apply the method used in this study, as in the case of the PGC 30591 galaxy. In the case of edge-on and very high-inclination galaxies, two main effects can affect the pertinence of applying the method. Firstly, one can have in the line of sight a large combination of different regions of the galaxy. Secondly, the higher the inclination, the smaller is the number of resolved regions, while the method relies on a large enough number of resolved regions.
Nevertheless, except for PGC 30591, we found reliable $\dot{\Sigma}_{\rm{flow \thinspace max}}$ and $\eta$ estimates, even for
high-inclination galaxies. Including or excluding high-inclination galaxies does not make any changes to the reported $\eta$-$\Sigma_{*}$ relation presented here. We show in Fig. \ref{eta_highincl} the $\eta$-$\Sigma_{*}$ observed and fitted relations for high-inclination galaxies, as star symbols and black line, respectively. When we compare the high-inclination galaxies results with the fit obtained using the full sample (Eq. \ref{eq_eta_den}), shown as a red line, both samples are compatible within $1\sigma$.
\begin{figure}
\begin{center}
\includegraphics[width=0.49\textwidth]{eta_density_sinop_highincl.pdf}
\end{center}
\caption{Local mass-loading factor, $\eta$, versus the average stellar mass surface density
of the regions on the envelope for each galaxy, $\Sigma_{*}$, as black stars, for high-inclination galaxies. We plot, as a black line, the linear fit to the empirically derived quantities.
We plot as a red line, the fit for the full sample (Eq. \ref{eq_eta_den}), and the 1-$\sigma$ uncertainty range of the fits as shaded regions.
}
\label{eta_highincl}
\end{figure}
\subsection{Effects of very high recent SFR regions removal}
\begin{figure}
\begin{center}
\includegraphics[width=0.49\textwidth]{eta_density_sinop_rem.pdf}
\end{center}
\caption{ Local mass-loading factor, $\eta$, versus the average stellar mass surface density
of the regions on the envelope for each galaxy, $\Sigma_{*}$, as black stars. Same plot as in Fig. \ref{fig_eta_den}, but this version of the plot has been done without removing any region based on its $\Sigma_{\rm{SFR\thinspace recent}}$ value. The 4 galaxies where we find a significant difference in the derived $\eta$ value are marked as orange circles.
We plot the linear fit to the empirically derived quantities and the 1-$\sigma$ uncertainty range of the fit as shaded regions.
The blue line shows the $\eta$ reported for supernova explosions hydrodynamical simulations \citep{2013MNRAS.429.1922C} with cut-off cooling modeling, and as blue line the $\eta$ reported for supernova explosions hydrodynamical
simulations \citep{2017ApJ...841..101L} with gas cold-phase modeling included.
}
\label{fig_eta_den_rem}
\end{figure}
We think is important to remove regions having very high values of $\Sigma_{\rm{SFR\thinspace recent}}$ compared to the rest of the galaxy, since these regions probably have a very high $\dot{\Sigma}_{\rm{net\thinspace flow}}$ value compared to the rest of the regions identified to be on the envelope. The effect of considering these high $\Sigma_{\rm{SFR\thinspace recent}}$ regions affects our assumption about the approximate equal value of $\dot{\Sigma}_{\rm{net\thinspace flow}}$ for the regions on the envelope.
However, in order to explore the effect of this removal in our results, we have performed the same analysis without removing any region based on its $\Sigma_{\rm{SFR\thinspace recent}}$ value. We plot the derived $\eta$ versus the local $\Sigma_{*}$ of the regions on the envelope for the whole sample of galaxies in Fig. \ref{fig_eta_den_rem}. We also plot the resulted $\eta$-$\Sigma_{*}$ fit to the observed data. We find a very similar correlation between $\eta$ and the local $\Sigma_{*}$:
\begin{equation}
\log(\eta+1)=(-0.32\pm0.05)\log(\Sigma_{*})+(3.2\pm0.4)
\label{eq_eta_den_rem}.
\end{equation}
There are only 4 galaxies where the $\eta$ value significantly changes: IC 1158, NGC 3389, ESO 298-G28, and MCG 01-57-021. We marked these galaxies as orange circles in Fig. \ref{fig_eta_den_rem}. However, the changes are not significant enough to change the resulted $\eta$-$\Sigma_{*}$ relation, but just a slightly increase in the obtained uncertainties. Therefore, although the removal could be important for some specific individual galaxies, when compared with the full sample of galaxies we still find a consistent $\eta$-$\Sigma_{*}$ relation.
\section{Conclusions}
We have used MUSE observations of a sample of 102 galaxy discs. We extracted
the spectra of 500 pc wide regions and apply them stellar population synthesis
using SINOPSIS code. We obtained the star formation histories of those regions
and we analysed the recent and past star formation rate densities. We compared
the $\Sigma_{\rm{SFR\thinspace recent}}$ with the $\Sigma_{\rm{SFR\thinspace past}}$ and
found that, for each galaxy, there is an envelope of regions formed by those regions having the maximum $\Sigma_{\rm{SFR\thinspace recent}}$, per bin of $\Sigma_{\rm{SFR\thinspace past}}$.
We fitted the resolved star formation self-regulator model (Eq. \ref{eq_etafit}) to those regions on the envelope
and quantify the
mass-loading factor, $\eta$.
We find correlations locally between $\eta$ and the stellar mass surface density, $\Sigma_{*}$, and globally between the averaged value of $\eta$ for a galaxy, $\eta_{\rm{G}}$, and the stellar mass of the galaxy, $M_{*}$, which are strong indications of how stellar feedback {\bf locally} regulates the mass growth of galaxies, especially those of lower masses.
The comparison between our empirical local $\eta$-$\Sigma_{*}$ relation with that from hydrodynamical simulations of supernova explosions \citep{2017ApJ...841..101L} is remarkably in agreement. In the case of our empirical global $\eta_{\rm{G}}$-$M_{*}$ relation, the
comparison with hydrodynamical cosmological zoom-in galaxy simulations \citep{2015MNRAS.454.2691M} is also in excellent agreement.
We note that the value of $\eta$ depends on the time scale over which the feedback is analysed, and can be defined either including or excluding posterior gas return, so comparison with other observations and with theory must be done with care.
These empirical relations offer excellent tools to confront with stellar feedback models which are crucial for understanding galaxy formation and evolution.
\section*{Acknowledgements}
The authors thank the anonymous referee whose comments have led to significant improvements in the paper. The authors also thank Aldo Rodr\'iguez-Puebla for sharing their stellar halo accretion rate coevolution models from \citet{2016MNRAS.455.2592R}.
JZC and IA's work is funded by a CONACYT grant through project FDC-2018-1848.
DRG acknowledges financial support through CONACYT project A1-S-22784.
GB acknowledges financial support through PAPIIT project IG100319 from DGAPA-UNAM.
This research has made use of the services of the ESO Science Archive Facility,
Astropy,\footnote{\url{http://www.astropy.org}} a
community-developed core Python package for Astronomy \citep{2013A&A...558A..33A,2018AJ....156..123A}, APLpy,
an open-source plotting package for Python \citep{2012ascl.soft08017R}, Astroquery, a package that contains a collection of tools to access online Astronomical data \citep{2019AJ....157...98G}, and pyregion ({\url{https://github.com/astropy/pyregion}), a python module to parse ds9 region files.
We acknowledge the usage of the HyperLeda database ({\url{http://leda.univ-lyon1.fr}}).
Based on observations collected at the European Southern Observatory
under ESO programmes 095.D-0172(A), 196.B-0578(B), 096.D-0263(A), 097.D-0408(A), 095.B-0532(A), 096.D-0296(A), 60.A-9319(A), 1100.B-0651(B), 0102.B-0794(A), 098.B-0551(A), 0101.D-0748(A), 099.B-0397(A), 1100.B-0651(A), 0100.D-0341(A), 096.B-0309(A), 296.B-5054(A), 095.D-0091(B), 0101.D-0748(B), 095.D-0091(A), 097.B-0640(A), 096.B-0054(A), 1100.B-0651(C), 098.D-0115(A), 097.B-0165(A), 098.C-0484(A), 094.C-0623(A), 60.A-9339(A), 099.B-0242(A), 0102.A-0135(A), 097.B-0041(A), 095.B-0934(A), 099.D-0022(A), and 60.A-9329(A).
\bibliographystyle{mnras}
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2024-02-18T23:40:24.795Z
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2020-09-22T02:00:23.000Z
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{"paloma_paragraphs":[]}
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{"arxiv_id":"2009.08992","language":"en","timestamp":1600740023000,"url":"https:\/\/arxiv.org\/abs\/2009.08992","yymm":"2009"}
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proofpile-arXiv_000-10087
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{"provenance":"002.jsonl.gz:10088"}
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\section{Introduction} \label{sec:intro}
Protoplanetary disks (PPDs) are produced by the gravitational collapse and angular momentum mediated flattening of dense rotating cores in molecular clouds. Their initial formation and subsequent evolution will be strongly impacted by the presence or absence of a magnetic field \cite[e.g.,][and references therein]{li2014}.
Observations of (sub)millimeter continuum dust polarization in cloud complexes reveal suggestive (e.g., ``hour-glass") linear polarization patterns on $\lesssim$1000 AU scales in both low- and high- mass regimes \citep{girart2006,beltran2019}. This structure is commonly interpreted as evidence of magnetic field structure in these environments, with the polarization thought to arise from alignment, through ``radiative torques," of dust grains orthogonal to the local magnetic field \citep{lazarian2007}.
Indeed, a magnetic field with this morphology (pinched toward the center of the collapsing core) is consistent with standard theoretical models for magnetized star formation \citep{galli1993,fielder1993}.
The magnetism of interstellar clouds has also been probed by Zeeman splitting measurements
(e.g., of CN, OH and HI), and studies to this end \citep{falgarone2008,troland2008,heiles2004} reveal that cores are moderately magnetized, with mean line-of-sight $\boldsymbol{B}$-field strengths up to $\approx30$ $\mu$G.
\citet{crutcher2010} concluded through Bayesian analysis of a large sample of dense cores that the most strongly magnetized cores have approximately critical mass-to-flux ratios, suggesting a dynamically important magnetic field regulating the star formation infall process.
Since PPDs form in molecular cloud core environments, it would not be surprising if they inherit some seed magnetization as well, which could be amplified by sheering effects within the disk.
It is difficult to determine the magnetic field morphology of a protoplanetary disk based on core-scale constraints, however, because a large amount of physical evolution and dynamical processing occurs as the disk forms \citep{li2014}.
For example, as gas flows onto the proto-stellar disk and local densities increase, the ionization level drops sufficiently low that non-ideal MHD effects, such as ambipolar diffusion, the Hall effect, and Ohmic dissipation, become important \citep[for review, see e.g.,][]{armitage2019}.
Simulation work that incorporates these physics has been successful in informing how PPDs evolve dynamically under these conditions \citep{turner2014}, but there remains significant ambiguity in determining what constitutes a reasonable initial set-up. We do not have firm answers to some basic questions. How strong should the magnetic field be? How should it be configured?
These questions are of critical importance, as $\boldsymbol{B}-$fields remain central to the study of PPDs and are thought to play a key role in
gas dynamics, which in turn controls the concentration and growth of dust grains that are crucial to the formation of planetesimals and eventually planets \citep{armitage2019}.
In particular, magnetic fields can cause magneto-rotational instability \citep[MRI;][]{balbus1991}, which is widely believed to be a
dominant driver of gas accretion in disk systems.
This interpretation remains uncertain in light of observations that suggest
ionization rates that are too low for the MRI to operate efficiently \citep{cleeves2015}, which is consistent with the low levels of turbulence inferred in some disks \citep[e.g.,][]{flaherty2015}.
Poloidal field components may also launch jets and winds perpendicular to the
disk plane \citep[e.g.,][]{blandford1982, simon2013} that
mediate gas accretion.
These flows have been proposed to trigger the formation of rings and gaps \citep{suriano2017}, and field-dependent mechanisms \citep[e.g., ``zonal flows,"][]{johansen2009,bai2013} can lead to planetesimal formation as well.
Since there is a wealth of disk physics that depends on the magnetic field strength and orientation, observational constraints are important.
To date there has never been an independently confirmed direct measurement of a magnetic field in a protoplanetary disk.
This is largely because linear polarization, the historically available technique for inferring magnetic information, has yielded results on the disk-scale that are difficult to reconcile with any clear $\boldsymbol{B}-$field interpretation.
Though magnetic alignment is expected \citep{cho2007,bertrang2017},
recent work has demonstrated that a variety of other mechanisms may also produce millimeter linear polarization in disks, including self-scattering of thermal dust emission \citep{kataoka2015,yang2016}, radiation field (e..g ``k-RAT") alignment \citep{kataoka2017,tazaki2017}, and gas flow alignment \citep{kataoka2019}, none of which depend explicitly on the magnetic field geometry.
Fortunately, circular polarization is now possible with the Atacama Large Millimeter/submillimeter Array (ALMA), providing us with the opportunity to carry out ``Zeeman effect" observations as a more definitive technique for accessing line-of-sight magnetic information in PPDs \citep[e.g.,][]{vlemmings2019}.
With more observations on the horizon, this paper aims to elucidate physical interpretation of disk-scale circular polarization and address the main difficulties associated with inferences of magnetic structure in PPDs.
We perform full radiative transfer simulations of Zeeman observations of the CN $J = 1 - 0$ transition for several different disk set-ups (in terms of CN distribution and magnetic field configuration), then interpret the emission and assess its detectability under a variety of conditions.
Finally, we address the importance of beam size, which presents challenges that are unique to circular polarization observations.
\section{Zeeman Effect Primer} \label{sec:zeemanreview}
For a parcel of gas threaded by a magnetic field, Zeeman-sensitive species' line emission is split into two circularly polarized components:
\begin{itemize}[topsep=8pt,itemsep=4pt,partopsep=4pt, parsep=4pt]
\item[] $\sigma_+(\nu)$: line center at $\nu = \nu_0 - \Delta \nu_z$,
\item[] $\sigma_-(\nu)$: line center at $\nu = \nu_0 + \Delta \nu_z$
\end{itemize}
where $\Delta \nu_z = z_B B$.
The value of $z_B$ (the so-called ``Zeeman-factor") is calculated as
\begin{equation}
z_B = \frac{\mu_B}{h}(g'M'-g''M'')\,,
\end{equation}
where $\mu_B$ is the Bohr magneton, $g$ is the Land\'e factor, and $M$ is the projection of the total angular momentum quantum number onto the magnetic field.
Single and double primes represent the upper and lower levels of a given line transition, respectively.
The $\sigma_+$ and $\sigma_-$ line profiles have the same intrinsic width, $\Delta \nu_{\rm line}$, as determined by the typical environmental processes (e.g., thermal, pressure, natural broadening),
and the Stokes $I$ and Stokes $V$\footnote{The choice $V = \sigma_+ - \sigma_-$ (instead of $V = \sigma_- - \sigma_+$) is purely a matter of convention.} of the emission are as follows:
\begin{align}
I(\nu) &= \sigma_+(\nu) + \sigma_-(\nu) \\
V(\nu) &= \sigma_+(\nu) - \sigma_-(\nu) \,.
\end{align}
If the magnetic field is uniform along the line-of-sight and sufficiently weak such that $\Delta \nu_z \lesssim \Delta \nu_{\rm line}$ (i.e., unresolved splitting, which is expected for both molecular cloud-like and PPD environments), the Stokes $V$ can be related approximately to the Stokes $I$ as
\begin{align}
V = \frac{dI}{d\nu} \Delta \nu_z \cos{\theta_B}\,. \label{eq:VB}
\end{align}
Here, $\theta_B$ is the inclination of the magnetic field relative to the line-of-sight \citep{crutcher1993}.
In Figure \ref{fig:splitDemo}, we demonstrate the $I$ and $V$ profiles obtained from Doppler broadened lines for a variety of choices of $\Delta \nu_z / \Delta \nu_{\rm line}$, showing that Equation \ref{eq:VB} is an increasingly good approximation in the $\Delta \nu_z \rightarrow 0$ limit.
It is worth stressing that $dI/d\nu$ mimicks the shape of the $V$ profile only for \textit{uniform} magnetic fields.
Magnetic field configurations with significant sub-structure (e.g., toroidal or radial components) can cause the relationship to break down.
\begin{figure*}
\includegraphics[width=1.05\textwidth]{realMulti.png}
\caption{Stokes $I$ (top) and Stokes $V$ (bottom) profiles for a variety of choices of $\Delta \nu_z / \Delta \nu_{\rm line}$ in the case of Doppler (i.e., Gaussian) broadened lines. In this demonstration we set $T = 20$ K, $\nu_0 = 113.144$ GHz (the frequency of the CN $J = 1 - 0$ transiton), and calculate $\Delta \nu_d = \frac{\nu_0}{c}\sqrt{\frac{2kT}{m_p}}$. We then vary $B$ to calculate each $\Delta \nu_z$. In the unresolved limit, the magnitude of $V$ scales linearly with the magnetic field strength. Protoplanetary disks fall in this regime, since their field strengths are expected to be relatively weak. Note that in each plot on the bottom row, the $\frac{dI}{d\nu}$ curve has been scaled down to match the magnitude of the $V$ curve for figure clarity.}
\label{fig:splitDemo}
\end{figure*}
\section{Parametric Modeling}\label{sec:model}
We explore a fully parametric disk model for use in our radiative transfer simulations, to allow us to probe a variety of disk chemical and physical set-ups.
We first produce ``simple'' models with purely vertical and purely toroidal field configurations as morphological case studies.
We then invoke a more complex magnetic field geometry and vary parameters related to the distribution of our emitting molecule (CN) and the magnetic field strength.
A description and list of chosen values for our parameter exploration is given in Table \ref{tab:parVary}.
Our fiducial disk structure is inspired by AS 209, a nearby ($d \approx 126$ pc), approximately solar-mass star with a minimally extincted \citep[$A_{\rm v}$ = 0.8;][]{avenhaus2018}, moderately inclined ($i = 38^{\circ}$) disk that has been observed to have CN $J = 2-1$ emission to a radial extent of $\sim 200$ AU \citep{oberg2011}.
These favorable observational characteristics have made AS 209 a common choice for pilot circular polarization studies with ALMA (e.g., 2018.1.01030.S, PI: R. Harrision; 2018.1.00298.S, PI: L. Cleeves).
It should be noted that though we use gas and dust distributions specifically fitted to AS 209 (see Section \ref{ssec:phys}), the bulk structure is not dissimilar from a variety of other disks \citep{andrews2009}.
In addition, recent sub-millimeter observations \citep[e.g., DSHARP;][]{andrews2018} also show that dust sub-structure is common in PPDs.
Therefore, the model presented in this work is intended to serve as an example of a ``typical" disk, and we expect the general trends found here to be broadly applicable.
\begin{deluxetable*}{llll}
\tablecaption{Selected values for our parameter exploration. For each parameter we run a batch of line emission simulations of the 113.144 GHz CN $J = 1-0$ transition over the specified range, with all other parameters set to their fiducial values. \label{tab:parVary}}
\setlength{\tabcolsep}{5pt}
\renewcommand{\arraystretch}{1}
\tablehead{Parameter & Fiducial Value & Range & Description}
\startdata
$\rm X_{\rm CN}$ & $\mathbf{10^{-8}}$ & $5 \times 10^{-10}$ - $5 \times 10^{-7}$ & CN abundance in slab (relative to $H_2$) \\
$R_{\rm in, CN}$ (AU) & \textbf{30} & 1 - 60 & Inner radius of CN slab \\
$R_{\rm out, CN}$ (AU) & \textbf{150} & 90 - 200 & Outer radius of CN slab \\
$N_{\rm min, CN}$ ($\times 10^{21}$ cm$^{-2}$) & \textbf{0.5} & 0.05 - 3 & Minimum column density of CN slab \\
$N_{\rm max, CN}$ ($\times 10^{21}$ cm$^{-2}$) & \textbf{10} & 5 - 200 & Maximum column density of CN slab \\
$B_{\rm sum,0}$ (mG) & \textbf{40} & 5 - 100 & Sum\tablenotemark{a} of magnetic field components at $R = 1$ AU \\
$\beta_{B_{\rm r}}$ & \textbf{-0.75} & -0.3 to -1.3& Power law index for radial falloff in magnetic field strength \\
$f_1$ & \textbf{0.3} & - & $B_{\rm vert,0}$/$B_{\rm sum,0}$ \\
$f_2$ & \textbf{0.36} & - & $B_{\rm rad,0}$/($B_{\rm sum,0} (1-f_1)$) \\
$i$ ($^{\circ}$) & - & 0, 40, 90 & Disk inclination ($0^{\circ} =$ face-on, $90^{\circ} =$ edge-on) \\
$f_{\rm lg}$ & \textbf{0.85} & - & Fraction of $M_{\rm dust}$ put into the large dust population \\
\enddata
\tablenotetext{a}{$B_{\rm sum,0}$ = $B_{\rm rad,0}$ + $B_{\rm tor,0}$ + $B_{\rm vert,0}$, the sum of the radial, toroidal, and vertical magnetic field components, respectively.}
\end{deluxetable*}
\subsection{Density Structure}\label{ssec:phys}
Our gas density distribution is based on the best-fit self-similar accretion disk solution obtained through multi-wavelength fitting of AS 209 by \citet{tazzari2016}.
Their reported gas surface density profile has a power-law falloff and exponential taper
\begin{align}
\Sigma_{\rm g}(R) = \Sigma^0_{\rm g} \big(\frac{R}{R_0}\big)^{\gamma_0} e^{-\big(\frac{R}{R_{\rm c}}\big)^{2+\gamma_0}}
\end{align}
with parameter choices $R_0 = 40$ AU, critical radius $R_{\rm c} = 78$ AU, and $\gamma_0 = -0.91$.
Dust plays an important role in radiative transfer and should be modeled as accurately as possible to produce a reasonable calculation of the disk's temperature.
We include two dust density distributions to simultanously account for a puffed-up, hydrostatically supported layer of small grains and a midplane-settled population of large grains.
Both are set to have MRN \citep{mathis1977} power-law size distributions, with the small population ranging from 0.005-1 $\mu$m and the large population ranging from 0.005-2000 $\mu$m.
We take the small dust to be spatially co-located with the gas, and set the large dust distribution based on the best-fit surface density profile from ALMA 1.3 mm observations \citep{fedele2018},
\begin{align}
\Sigma_{\rm d,lg}(R) = \Sigma^0_{\rm d,lg} \delta(R) \big(\frac{R}{R_{\rm c}}\big)^{\gamma_1} e^{-\big(\frac{R}{R_{\rm c}}\big)^{\gamma_2}} \,,
\end{align}
with $\gamma_1 = 0.3$ and $\gamma_2 = 2.0$. The scaling parameter $\delta(R)$ models the observed ring/gap sub-structure in AS 209 and is written as
\begin{align}
\delta(R)=
\begin{cases}
\delta_{\rm G1} = 0.1 & \text{$R$ $\in$ [$R_{\rm G1} - hw_{\rm G1}, R_{\rm G1} + hw_{\rm G1}$]}\\
\delta_{\rm R1} = 0.75 & \text{$R$ $\in$ [$R_{\rm G1} + hw_{\rm G1}, R_{\rm G2} - hw_{\rm G2}$]}\\
\delta_{\rm G2} = 0.01 & \text{$R$ $\in$ [$R_{\rm G2} - hw_{\rm G2}, R_{\rm G2} + hw_{\rm G2}$]}\\
\delta_{\rm R2} = 4.5 & \text{$R$ $\in$ [$R_{\rm G2} + hw_{\rm G2}, R_{\rm R2, out} $] } \\
\delta_{\rm out} = 1.5 & \text{$R$ $\geq R_{\rm R2, out}$}\\
1 & \text{otherwise}\,,
\end{cases} \label{eq:delta}
\end{align}
with the gaps parameterized by best-fit radii ($R_{\rm G1} = 62$ AU, $R_{\rm G2} = 103$ AU) and half-widths ($hw_{\rm G1} = 8$ AU, $hw_{\rm G2} = 16$ AU). The outer ring has an outer radius of $R_{\rm R2,out} = 140$ AU. Outside of disk radius $R_{\rm out} = 200$ AU, we set both the gas and dust surface densities to zero.
Our fiducual disk model does not include gas deficits.
However, there is observational evidence from near-infrared scattered light \citep{avenhaus2018} and CO line transition data \citep{favre2019} that gas gaps may be present in AS 209 and similar disks, perhaps co-located with the dust gaps.
We explore their impact on Zeeman observations in Section \ref{ssec:gas}.
The 2.5-dimensional distributions used in our simulations are constructed from the above detailed 1-dimensional surface density profiles using the general conversion
\begin{align}
\rho_i(R,z) = \Sigma_i(R) \frac{e^{-\frac{1}{2}(\frac{\theta_z}{h_i})^2}}{\sqrt{2\pi}R h_i}\,,
\end{align}
where $\theta_z = \arctan{(|z|/R)}$.
The scale height $h_i$ for each distribution allows for flaring and is parameterized as
\begin{align}
h_i = \chi_i h_{\rm c} \big(\frac{R}{R_{\rm c}}\big)^{\psi}\,,
\end{align}
where $h_{\rm c}$ is a dimensionless critical scale height (normalized to radius), $R_{\rm c}$ is the critical radius of the disk, and $\psi$ is the disk flaring parameter. For consistency with the \citet{fedele2018} results, we choose $\psi = 0.1$, $h_{\rm c} = 0.133$, $\chi_{\rm g} = \chi_{\rm d,sm} = 1$, and $\chi_{\rm d,lg} = 0.2$, where subscripts correspond to gas, small dust, and large dust, respectively.
We also set the total disk dust mass as $M_{dust} = 3.5 \times 10^{-4}$ $M_{\odot}$. To determine the normalizations for our density distributions, we assume a gas-to-dust mass ratio of 100 and set the fraction of dust mass in the large grain distribution by parameter $f_{\rm lg}$. Numerical integration then easily yields appropriate values for $\Sigma^0_{\rm g}$, $\Sigma^0_{\rm d,sm}$, and $\Sigma^0_{\rm d,lg}$. In Figure \ref{fig:densities} we show edge-on midplane cuts of $\rho_{\rm g}$, $\rho_{\rm d,sm}$, and $\rho_{\rm d,lg}$ for our ``AS 209"-like density structure.
\begin{figure*}
\includegraphics[width=\textwidth]{lol2_new.png}
\caption{Density structure of our AS 209 based model. \textbf{Panel (a)}: Gas density, overlaid with vertically computed H$_2$ column density contours relative to ($N_{H_2}/10^{21}\text{ cm}^{-2}$).
$N_{H_2}$ bounds the allowed limits for CN to reside in our simulations. The small dust is co-located with the gas.
\textbf{Panel (b)}: A large dust density slice. The gaps at 62 AU and 103 AU are based on previous modeling of AS 209's sub-mm dust continuum observations and aim to make our model more realistic due to the observed prevalence of substructure in disks \citep{andrews2018}. \textbf{Panel (c)}: Midplane number densities as a function of radius for our gas, large dust, and small dust distributions.}
\label{fig:densities}
\end{figure*}
\subsection{Velocity Field}
The bulk gas motions are assumed to be Keplerian, i.e.,
\begin{align}
\boldsymbol{v}(R,z) = \sqrt{\frac{GM_*}{R}} \boldsymbol{\hat{\phi}}\,,
\end{align}
where $\boldsymbol{\hat{\phi}}$ is the azimuthal unit vector in cylindrical coordinates and $M_*=0.9$ $M_{\odot}$ \citep{andrews2009}.
In addition, the line emission simulations include thermal broadening and an additional turbulent component specified by the user, where we choose $v_{\rm turb} = 0.1$ km/s \citep{pietu2007,chapillon2012}.
\subsection{Magnetic Field}
We adopt a parametric description of the disk magnetic field. The magnetic field strength is set to obey a radial power-law
\begin{align}
B_{\rm sum}(R) = B_{\rm sum,0} \big(\frac{R}{\rm 1\, AU}\big)^{\beta_{B_{\rm r}}}\,,
\end{align}
with $B_{\rm sum}$ assumed to be constant as a function of $z$, approximately consistent with the results of magnetized simulations from \citet{suriano2017} that include a disk-wind. A routinely used method for deriving reasonable values for $\beta_{B_{\rm r}}$ is to invoke self-similarity between the radial gas density and magnetic field strength profiles. Taking $P \propto \rho^{\Gamma}$, it
is straightforward to show \citep{zanni2007} that $\beta_{B_{\rm r}}$ is a function of the radial gas density power law, $\beta_{\rho}$:
\begin{align}
\beta_{B_{\rm r}} = \frac{\Gamma \beta_{\rho}}{2} \,.
\end{align}
Adopting $\Gamma = 5/3$ and setting $\beta_{\rho} = \gamma_0 = -0.91$ from the AS 209 gas density distribution modeled in Section \ref{ssec:phys}, we obtain $\beta_{B_{\rm r}} = -0.758$.
We use this calculation as a guide for our fiducial value.
At each radial location in the disk, we divide the magnetic field strength into independent toroidal, radial, and vertical components as
\begin{multline}
\boldsymbol{B}(R,z) = (1 - f_1) f_2 B_{\rm sum} \hat{\boldsymbol{r}}^{\prime}+ \\ (1 - f_1) (1-f_2) B_{\rm sum} \hat{\boldsymbol{\phi}}^{\prime} + f_1 B_{\rm sum} \hat{\boldsymbol{z}}\,,
\end{multline}
where $f_1 \leq 1$ and $f_2 \leq 1$.
Also, we prescribe
\begin{align}
\hat{\boldsymbol{r}}^{\prime} =
\begin{cases}
\hat{\boldsymbol{r}} & \text{if } z > 0\\
-\hat{\boldsymbol{r}} & \text{otherwise} \,,
\end{cases} \label{eq:cond1}
\end{align}
and
\begin{align}
\hat{\boldsymbol{\phi}}^{\prime} =
\begin{cases}
\hat{\boldsymbol{\phi}} & \text{if } z > 0\\
-\hat{\boldsymbol{\phi}} & \text{otherwise} \,.
\end{cases} \label{eq:cond2}
\end{align}
Equation \ref{eq:cond1} is included to model the ``wind-up" that occurs in the toroidal $\boldsymbol{B}$-field component due to disk rotation \citep[per simulations, e.g.][]{romanova2012}, and Equation \ref{eq:cond2} accounts for the reversal of the radial component that occurs due to inward dragging in accretion disks.
Our fiducial choices (see Table \ref{tab:parVary}) for $f_1$, $f_2$, and $B_{\rm sum,0}$ are guided by the results of disk wind simulations \citep{suriano2017} after 1000 orbits.
A few other values are also explored to examine a diverse variety of potential magnetic field configurations.
\subsection{CN Distribution}
Chemical modeling of PPDs with many different physical structures by \citet{cazzoletti2018} suggests that it is ubiquitous for CN to reside in a relatively thin layer in the upper and outer regions of the disk.
This structure arises because CN abundance is mainly governed by the balance between ionizing far ultraviolet photons (which produce overwhelming photodissociation and photoionization at $N_{\text{H}_2} \lesssim 10^{20}$ cm$^{-2}$) and freeze-out onto grains deep in the disk at low temperatures, $\lesssim32$~K.
Chemical models also find CN abundances are approximately constant (to within a factor of $\approx$2) within this intermediate layer irrespective of radius, modulo an inner deficit of CN.
Given these constraints, we set the distribution of CN in our simulations to be a constant abundance slab.
The slab is defined to have inner and outer radii, $R_{\rm in,CN}$ and $R_{\rm out,CN}$, and the vertical extent is set by upper and lower H$_2$ column densities, $N_{\rm min,CN}$ and $N_{\rm max,CN}$.
Expected values for $N_{\rm in,CN}$, $N_{\rm out,CN}$, $R_{\rm in,CN}$, and $R_{\rm out,CN}$ are not precisely constrained, so we vary each over a few different reasonable possibilities in Section \ref{sssec::parsli}.
\section{Simulation Methods}\label{sec:simmeth}
We perform our simulations using the POLARIS 3D radiative transfer code \citep{reissl2016,brauer2017b}.
Radiative transfer in POLARIS is solved using Mol3D \citep{ober2015}, and spectral line Zeeman splitting and polarization is based on the Stokes formalism implementation by \citet{larsson2014}.
We specify physical quantities in an octree format,
with grid sub-division set using a variable refinement scheme based on local gas density.
The densest regions have $\sim 0.2$ AU resolution, with reduced resolution approximately linearly down to $\sim 8$ AU in the most diffuse parts of the disk, such as the upper atmosphere above the CN emitting region.
Each simulation involves two computations: first a temperature calculation based on the dust density structure, then the CN line emission. Each step is detailed further in the following sections.
\subsection{Temperature Calculation}\label{ssec:rt}
The disk is heated by irradiation from a central point source, set to have luminosity consistent with a blackbody that has AS 209 stellar parameters \citep[$R = 2.3$ R$_{\odot}$, $T = 4250$~K; ][]{tazzari2016}.
We use $10^7$ photons in this calculation to ensure good coverage in all regions of the disk.
After each photon is generated (with characteristic wavelength, energy per unit time, and randomly chosen direction), it is allowed to scatter on dust grains according to an isotropic phase function. Dust heating is handled with continuous absorption \citep{lucy1999} and immediate re-emission \citep{bjorkman2001} methods.
After all photons from the central star have been propagated, $T_{\rm dust}$ at each location in the disk is determined based on the temperature of local grains.
We then set $T_{\rm gas} = T_{\rm dust}$ for simplicity in our parametric model; however, we note that the disk gas in the atmosphere is likely warmer than the dust temperature, due to additional UV heating from the star. This could result in generally brighter CN emission than what is predicted here.
\subsection{Emission from CN Spectral Lines}\label{ssec:zeeman}
The $J = 1-0$ transition of CN presents nine hyperfine Zeeman components, seven of which are strong enough to be of potential astronomical relevance. In Table \ref{tab:CNLines} we give the rest frequency ($\nu_0$), relative intensity (RI), and Zeeman factor ($z_B$) for each of these lines, as originally tabulated by \citet{falgarone2008}.
For our main set of models we only consider the 113.144 GHz transition, since it is a good representative case with high relative sensitivity to $B_{\rm LOS}$ and a large $z_B$.
In Section \ref{sssec:transitions}, we simulate (and stack) the emission from all seven lines for our fiducial disk.
\begin{deluxetable}{cccc}
\tablecaption{The seven strong hyperfine lines for the CN $J = 1-0$ transition. RI$\times$$z_b$ quantifies relative sensitivity to $B_{\rm LOS}$. \label{tab:CNLines}}
\setlength{\tabcolsep}{5pt}
\renewcommand{\arraystretch}{1}
\tablehead{$\nu_0$ (GHz) & RI & $z_B$ (Hz/$\mu$G) & RI$\times z_B$}
\startdata
113.144 & 8 & 2.18 & 17.4 \\
113.171 & 8 & -0.31 & 2.5 \\
113.191 & 10 & 0.62 & 6.2 \\
113.488 & 10 & 2.18 & 21.8 \\
113.491 & 27 & 0.56 & 15.1 \\
113.500 & 8 & 0.62 & 5.0 \\
113.509 & 8 & 1.62 & 13.0 \\
\enddata
\end{deluxetable}
Zeeman-splitting line emission in POLARIS is computed using the ZRAD extension \citep{brauer2017b}.
ZRAD makes use of energy level and transition data from the Leiden Atomic and Molecular DAtabase \citep[LAMDA;][]{schoier2005} and the JPL spectral line catalog \citep{pickett1998}.
This work uses the CN hyperfine data set, with rates from \citet{kalugina2015}.
Natural, collisional, and Doppler broadening, as well as the magneto-optic effect \citep{larsson2014}, are all considered in determining the line shape, and the final profile is calculated with a Faddeeva function solver\footnote{http://ab-initio.mit.edu/wiki/index.php/Faddeeva\_Package, Copyright \copyright 2012 Massachusetts Institute of Technology}.
For the turbulent component we choose $v_{\rm turb} = 0.1$ km/s \citep{pietu2007,chapillon2012}, or about 30\% of the sound speed.
We initialize our line radiative transfer simulations with $10^5$ unpolarized background photons and assume local thermodynamic equilibrium (LTE) for all level population calculations.
Photons are ray traced to a $256\times 256$ pixel detector, where the Stokes $I$ and $V$ of the emission are recorded.
We set the detector to observe in 181 velocity channels in range $[v_0-6\text{ km/s},v_0+6\text{ km/s}]$, producing 0.067 km~s$^{-1}$ resolution data. The source velocity is set to $v_0 = 0$ km~s$^{-1}$.
\section{Results}\label{sec:results}
Our POLARIS simulations yield 3D data cubes with spatially resolved $I$, $V$, and optical depth ($\tau$) information for each pixel in each of the 181 channels.
We then bin the data to 0.4 km/s wide frequency bins
and convolve the data with a Gaussian kernel to simulate a $1''$ beam.
From these processed data, we produce channel maps and spatially integrated line profiles.
\subsection{Vertical and Toroidal Magnetic Field Case Studies}\label{ssec:magCases}
Presented here are the results of simulations with either vertical or toroidal magnetic field configurations. All the parameters from Table \ref{tab:parVary} (except $f_1$ and $f_2$) are set to their fiducial values for these models, except for the maximum column density of the CN slab which we set to $N_{\rm max, CN} = 20 \times 10^{21}$ cm$^{-2}$ here. While this choice is arbitrary, it ensures that the CN is not too optically thick such that the Stokes $V$ is dominated by magnetic effects rather than opacity.
Opacity varies due to the geometry of the CN emitting gas and sight line effects, but aside from some regions in the vertical magnetic field case when viewed face-on,
$\tau < 1$ at all locations in observer space across all frequencies for these runs.
Therefore, these models are reasonable approximations of the ``optically-thin" limit.
\subsubsection{Vertical Magnetic Field}\label{sssec:vertCase}
The top two panels of Figure \ref{fig:compChan} show our results for face-on and intermediate inclination views of our purely vertical, $f_1 = 1$ and $f_2 = 0$, simulation.
In the face-on case, the Keplerian rotation of the disk is in the plane-of-the-sky, so its contribution to the line-of-sight velocity field is zero everywhere.
The emission is therefore spread in frequency space only due to line broadening, distributed primarily among the central three channels.
Since $v_{\rm LOS, Kep} = 0$ km~s$^{-1}$ and the line-of-sight magnetic field is pointed entirely toward the observer at all locations, the $\boldsymbol{B}$-field configuration and viewing angle combination produces Stokes $I$ and $V$ profiles that are morphologically similar to the $\Delta \nu_z < \Delta \nu_{\rm line}$ case for the simple model (uniform magnetic field threading a uniform, non-moving parcel of gas) illustrated in Figure \ref{fig:splitDemo}.
Notably, in the central (zero velocity) channel the Stokes $V$ is zero due to $\sigma_+$ and $\sigma_-$ cancellation.
Unlike the face-on case, the intermediate ($i = 40^{\circ}$) inclination case produces line-of-sight velocity contributions.
For $\Delta x_{\rm obs} < 0$, $v_{\rm LOS, Kep} > 0$, and for $\Delta x_{\rm obs} > 0$, $v_{\rm LOS, Kep} < 0$, resulting in a double-peaked Stokes $I$ line profile.
Since the magnetic field here is again pointed in the same direction across all space (the inclination simply results in a $\cos{\theta_B}$ reduction of its line-of-sight strength), the shape of the Stokes $V$ profile is well mimicked by $dI/d\nu$.
Each channel in the Stokes $V$ map has positive and negative regions.
This pattern arises due to the varying amounts of red and blue shifted emission, and
can be understood most clearly by considering the central ($v_{\rm LOS} = 0$) channel.
In this channel, all the positive $V$ is located at $\Delta x_{\rm obs} < 0$ (where $v_{\rm LOS, Kep} > 0$) and all the negative $V$ is located at $\Delta x_{\rm obs} > 0$ (where $v_{\rm LOS, Kep} < 0$).
This flip occurs because, as demonstrated in Figure \ref{fig:splitDemo}, for a parcel of gas with line-of-sight velocity $v_0$, the peaks of the Stokes $V$ profile occur at $v_0 \pm \sim 0.4$ km/s (the precise value depends on the temperature and turbulence of the gas, which sets the slope of the Stokes $I$ over frequency).
As a result, the positive Stokes $V$ emission we observe in the zero velocity channel is dominated by red-shifted regions in the disk, and the negative Stokes $V$ arises in the blue-shifted regions.
In general, for a channel centered at $v = v_{\rm channel}$ the crossover ``line" from positive $V$ to negative $V$ occurs where $v_{\rm LOS} = v_{\rm channel}$.
\begin{figure*}
\makebox[\textwidth][c]{\includegraphics[width=0.99\textwidth]{chanCompMaps_300.png}}
\caption{Model Stokes $I$ and $V$ channel maps of the 113.144 GHz CN $J = 1 - 0$ transition. The top two panels show vertical magnetic field simulations at the labelled inclinations. The bottom two panels show toroidal magnetic field simulations at edge on and $40^{\circ}$ inclinations. Stokes $V$ contours are drawn at $\pm$0.1 and $\pm$1 mJy~beam$^{-1}$, and optical depth contours (over-plotted on the Stokes $I$ maps) are drawn at $\tau$ of 0.5, 0.75, and 1. In the bottom three panels, the major axis of the disk lies along the $\Delta x_{\rm obs}$-axis. Below each set of channel maps we include disk-integrated spectra. As described in the text, morphological differences between the vertical and toroidal field cases are readily apparent. Also of note, the edge-on toroidal case shows bright Stokes $V$ emission in the channel maps (with some regions producing $>10$ mJy/beam), but roughly zero signal in the integrated profile (due to spatial cancellation). This demonstrates the importance of leveraging spatial information when observing sources with sub-structured magnetic field configurations.}
\label{fig:compChan}
\end{figure*}
\subsubsection{Toroidal Magnetic Field}\label{sssec:torCase}
In the bottom two panels of Figure \ref{fig:compChan}, we plot the results for our toroidal-only model ($f_1 = 0, f_2 = 0$).
Viewed edge-on ($i = 90^{\circ}$), we see the Stokes $V$ image is clearly split into four distinct regions in most channels.
The divide across the midplane (at $\Delta y = 0$) reflects the crossover from the magnetic field being oriented parallel to the Keplerian rotation to it being anti-parallel. Recall this feature aims to simulate $\boldsymbol{B}$-field ``wind-up" due to disk rotation.
Meanwhile, the vertical divide occurs because of the Keplerian rotation itself and is similar to the effect observed in the vertical magnetic field viewed at $i = 40^{\circ}$ case.
This divide is absent in the center-most channels, due to the co-locality of the velocity field and the magnetic field sign flips.
Together, these effects make it such that gas with slightly negative or slightly positive line-of-sight velocity components will both produce the same handedness of circular polarization at $v_{\rm LOS} = 0$.
Note also that the shape of the spatially integrated Stokes $V$ is no longer mimicked by $\frac{dI}{d\nu}$ due to the non-uniform magnetic field geometry.
We discuss this break down in more detail in Section \ref{sssec:fourB}.
Viewed at intermediate inclination, the emission from the toroidal $\boldsymbol{B}$-field is still split into four distinct sub-regions in most velocity channels.
This morphology arises because the CN slab traces out the disk surface, with each line of sight piercing the upper and lower surface at different radial positions, mirrored over
the major axis of our axisymmetric disk.
When rotated to $i = 40^{\circ}$,
this arrangement gives four regions of coherent emission in the central velocity channel, because the magnetic field sign flips and velocity field sign flips are co-local (similar to the scenario for the \textit{two} regions for the edge-on case described above).
A few channels, e.g $\pm0.7$ km/s, express additional substructure.
This is due to the combined effects of the emitting layer height, the viewing geometry, and the Keplerian rotation.
\begin{figure*}
\includegraphics[width=\textwidth]{inc_tor.png}
\caption{Illustration of the geometry of a disk with a toroidal magnetic field viewed at $i = 40^{\circ}$. \textit{Top:} Side view of the disk geometry. The four closed regions denote the locations where CN is placed in our model, with each one schematically color-coded to assist with interpretation of the ``LOS position" plot below. \textit{Bottom:} 3D Line-of-sight (LOS) velocity, LOS position, and LOS magnetic field maps. These visualizations are scatter plots, created by selecting 10,000 random locations in the disk, then color-coding the points accordingly and projecting them into the observer plane (notated as the $xy$-plane here). In the ``LOS position" plot, $z$ denotes LOS deviation from the center of the disk model space. The four CN slabs are clearly discernible, and this is why emission for the $i = 40^{\circ}$, toroidal case in Figure \ref{fig:compChan} is distributed into four distinct clumps (especially evident in the centermost channels). The clumps alternate between positive and negative $V$ because the magnetic field sign flips across the midplane, as illustrated in the ``LOS magnetic field" panel here.}
\label{fig:torInter}
\end{figure*}
\subsection{Parameter Space Exploration}\label{ssec:parExp}
We now assess the observational impact of varying the parametric set-up of our model disk.
This analysis is performed in two parts.
First, we explore factors related to CN configuration and magnetic field strength (the first seven parameters listed in Table \ref{tab:parVary}).
Starting from our fiducial model (plotted in the top panel of Figure \ref{fig:fidModel}), which has a magnetic field component ratio of $B_{\rm vert}:B_{\rm tor}:B_{\rm rad} = 30\%:45\%:25\%$, we independently vary each parameter with the other parameters held fixed to examine \textit{par}ameter \textit{sli}ces (hereafter referred to as our \textit{parsli} analysis) through the model space.
This produces an easily digestible set of data to consider (as opposed to a full $n$-dimensional parameter space, it is instead $n$ 1-dimensional cuts).
In the subsequent section, we revert back to our fiducial model for those parameters and examine some different magnetic field geometries by varying $f_1$ and $f_2$.
\subsubsection{parsli}\label{sssec::parsli}
We vary the following parameters within the ranges specified in Table \ref{tab:parVary}: X$_{\rm CN}$, $R_{\rm in,CN}$, $R_{\rm out,CN}$, $N_{\rm min,CN}$, $N_{\rm max,CN}$, $B_{\rm sum, 0}$, and $\beta_{B_r}$.
After binning the simulation data to 0.4 km/s resolution and producing channel maps, we calculate the maximum flux (Stokes $I$ and $V$) and optical depth obtained for each model.
The results of these computations are provided in Figure \ref{fig:parsli}.
Among the parameters related to the distribution of CN in the disk, X$_{\rm CN}$, $R_{\rm out,CN}$, and $N_{\rm max,CN}$ are the most important.
Sensibly, flux scales linearly with CN abundance until there are enough molecules to produce $\tau \sim 1$, at which point optical depth effects become important and some of the emission is suppressed.
Extending the outer radius of the slab has a large effect due to the increase in emitting area.
Extending the CN slab deeper into the disk to higher H$_2$ column densities (i.e., increasing $N_{\rm max,CN}$) incorporates more high density gas and thus also significantly boosts CN $J=1-0$ emission.
\begin{figure*}
\makebox[\textwidth][c]{\includegraphics[width=1.2\textwidth]{Vplots_paper_rb_v4.png}}
\caption{Parameter space cuts for several variables, plotting the peak Stokes $I$, $V$, and $V$/$I$ for a 1'' beam as a function of parameter values. Peak flux is defined as the maximum value obtained for a given Zeeman simulation spatially and spectrally. The orange, blue, and pink curves correspond to $0^{\circ}$, $40^{\circ}$, and $90^{\circ}$ inclinations, respectively. In the top panel the black points indicate a transition to optically thick Stokes $I$. This only occurs for large X$_{\rm CN}$ or $N_{\rm max, CN}$.}
\label{fig:parsli}
\end{figure*}
Meanwhile, there is very little dependence on the inner radius or the minimum hydrogen column density limits of the CN slab (i.e., the upper CN slab surface).
This is because there is a relatively small volume of gas at small $R$ (between the prescribed column density limits) and relatively low emissive material in the disk upper atmosphere.
Ultimately, for all these scaling relationships the operative quantity being modified is the total number of emitting CN molecules added or subtracted, so expansion of the CN slab into high density regions (or by a large volume) is what produces the largest increases in $I$ and, for a fixed magnetic field strength, $V$. Furthermore, we find that the magnetic quantities ($B_{\rm sum, 0}$ and $\beta_{B_r}$) scale with Stokes $V$ proportionally as expected from Eq.~\ref{eq:VB}.
\subsubsection{Extracting Magnetic Field Strengths}\label{sssec:fourB}
Given the complications of the disk magnetic structure, in this section we explore how the ``true'' value of the magnetic field put into the simulation compares to what one would extract using conventional line fitting techniques like Eq.~\ref{eq:VB}.
Included in this analysis are a subset\footnote{Some might say a sprig.} of the {\em parsli} simulations, including our purely toroidal (\texttt{tor}), purely vertical (\texttt{vert}), and fiducial (\texttt{fid}) models, as well as a ``fiducial-like" model with a boosted toroidal component (\texttt{fidtc}).
All four configurations (summarized in Table \ref{tab:magConfigs}) have the same scaling for the magnetic field strength $B_{\rm sum,0} = 0.4$ Gauss and power law dependence $\beta_{B_r} = -0.75$, and therefore have the same mean (mass weighted) magnetic field strength of $B_{\rm avg} = 1.4$ mG within the CN emitting region.
Nevertheless, these models give different amounts of Stokes $V$ emission since the magnitude of the line-of-sight component of the magnetic field naturally changes.
In Table \ref{tab:magConfData} we list the mean line-of-sight magnetic field strength for each case, where
\begin{align}
B_{\rm LOS, avg} =
\frac{\int \frac{B_z + B_y \tan i}{\sqrt{1+\tan^2 i}}\rho(\boldsymbol{r}) d\boldsymbol{r}}{\int \rho(\boldsymbol{r}) d\boldsymbol{r}} \,,
\end{align}
integrated over the CN emitting region.
Due to symmetry, toroidal field components always produce $B_{\rm LOS, avg} = 0$.
Though this usefully expresses the importance of cancellation, most cancellation is due to spatial confusion rather than line-of-sight effects.
To get a sense of the magnitude of all the Zeeman-relevant emission, irrespective of whether $\boldsymbol{B}$ is directed toward or away from the observer, we also report the absolute value of the line-of-sight magnetic field strength as well, again integrated over the whole CN slab.
We also give the maximum Stokes $V$ found anywhere in the observation, with corresponding profiles (Peak $V$ vs. frequency) plotted in Figure \ref{fig:magConfigs}.
We find that the strongest Stokes $V$ emission in a given simulation is a strong function of the geometry of the underlying magnetic field.
Due to the lack of spatial cancellation within a given beam, vertical $\boldsymbol{B}$-field components produce by far the largest $B_{\rm LOS, avg}$ and peak $V$.
A face-on disk with a purely vertical magnetic yields a peak signal that is about a factor of seven larger than an edge-on disk with a purely toroidal magnetic field, even though $|B_{\rm LOS}|_{\rm, avg}$ is only $\approx$1.5 times larger.
In terms of producing a detection, intermediate inclination and edge-on viewing geometries are only preferable for field configurations that are almost entirely toroidal.
For our \texttt{fid} and \texttt{fidtc} cases, face-on observations result in emission that peaks 2.2 and 1.8 times higher than $i = 40^{\circ}$ observations, respectively.
Of course, due to Keplerian rotation, high inclination sources will have their emission distributed across a larger chunk of frequency space.
This geometry can be advantageous for some analysis goals, like localizing the emission along a given column of gas based on an assumed velocity profile \citep[e.g.,][]{teague2019}. However it can also have some disadvantages, like decreasing the line peak, thereby making detection more challenging.
As described previously, fitting the derivative of the Stokes $I$ to the Stokes $V$ profile is a conventional technique for inferring line-of-sight magnetic field strengths from Zeeman observations (Eq~ \ref{eq:VB}).
This methodology may be applied to disk-scale observations, but we must be aware that the obtained $B_{\rm LOS}$ value may be significantly reduced due to field sub-structure in these environments.
In Figure \ref{fig:fitB} we plot the spatially integrated Stokes $V$ profiles for each of the magnetic field configurations, viewed at both face-on and intermediate inclinations. Note, the edge-on case produces $V \approx 0$~mJy for all four magnetic field geometries. This includes the toroidal field case due to the sign flip cancellation across the midplane.
In the same figure, we overplot the $dI/d\nu$ curve scaled to represent the $V$ inferred by setting $B = 1.4$ mG (the density-weighted average field strength for these runs).
In the face-on case, the shape of $dI/d\nu$ mimics the $V$ curves well because this view picks out the vertical field component, which is not subject to any cancellation.
The \texttt{fid} and \texttt{fidtc} curves are reduced in magnitude because they have a small fraction of their $\boldsymbol{B}$-field strength put into the vertical component.
At $i = 40^{\circ}$, the shape of $dI/d\nu$ still reasonably matches the \texttt{vert}, \texttt{fid}, \texttt{fidtc} Stokes $V$ profiles.
This highlights the dominance of the vertical field component, even when it is down to a factor 3.5 weaker than the toroidal component (as in the \texttt{fidtc} model).
However, in the fully toroidal model the profile is both substantially reduced and has a different morphology, owing to the sign flips in the magnetic field geometry.
If this magnetic sub-structure is not taken into account, fitting these curves using the conventional method results in considerable underestimates of the magnetic field strength.
\begin{figure*}
\centering
\includegraphics[width=0.9\textwidth]{magConfigs_V.pdf}
\caption{Spatially integrated Stokes $V$ profiles for the different magnetic field geometries we considered, as viewed at $i = 0^{\circ}$ and $i = 40^{\circ}$. Also plotted is the derivative of the Stokes $I$, scaled to fit the $V$ curve for a uniform magnetic field with a strength consistent with that put into our simulations. Magnetic field geometries with substructure produce significantly reduced Stokes $V$ magnitudes. If the toroidal field component is large enough, it can yield a profile that is different in shape from $dI/d\nu$.}
\label{fig:fitB}
\end{figure*}
\begin{deluxetable}{cccc}
\tablecaption{The magnetic field configurations we consider in Section \ref{sssec:fourB} and Section \ref{sssec:beam}. Percentages represent the fraction of the total magnetic field strength allocated to each of the components. \label{tab:magConfigs}}
\setlength{\tabcolsep}{5pt}
\renewcommand{\arraystretch}{.9}
\tablehead{\colhead{Name} & \colhead{Vertical (\%)} & \colhead{Toroidal (\%)} & \colhead{Radial (\%)}}
\startdata
\texttt{vert} & 100 & 0 & 0 \\
\texttt{tor} & 0 & 100 & 0 \\
\texttt{fid} & 30 & 45 & 25 \\
\texttt{fidtc} & 20 & 70 & 10 \\
\enddata
\end{deluxetable}
\begin{figure*}
\centering
\includegraphics[width=0.9\textwidth]{magConfigs.pdf}
\caption{Peak $|V|$ as a function of frequency for each of the magnetic field configurations from Table \ref{tab:magConfigs}. The maximum value for each of these curves (i.e., the peak $|V|$ across all frequencies) is listed in Table \ref{tab:magConfData}. These data are binned to the same resolution (0.4 km/s) as the channel maps.}
\label{fig:magConfigs}
\end{figure*}
\begin{deluxetable}{ccccc}
\tablecaption{Mean line-of-sight magnetic field strength ($B_{\rm LOS, avg}$), mean absolute value of the line-of-sight magnetic field strength ($|B_{\rm LOS}|_{\rm, avg}$), and peak $|V|$ obtained from the channel maps for each of the magnetic field configurations we simulated. Values are reported for emission from the 113.144 GHz component only. A vertical field viewed face-on yields a peak $V$ flux that is a factor of $\sim7$ larger than a toroidal field viewed edge-on, even though $|B_{\rm LOS}|_{\rm, avg}$ is only a factor of $\sim1.6$ larger. This highlights the importance of cancellation for sub-structured (e.g., toroidal) magnetic field configurations. \label{tab:magConfData}}
\setlength{\tabcolsep}{5pt}
\renewcommand{\arraystretch}{1}
\tablehead{ & & $i = 0^{\circ}$ & $i = 40^{\circ}$ & $i = 90^{\circ}$}
\startdata
& $B_{\rm LOS, avg}$ (mG) & 1.40 & 1.07 & 0 \\
\texttt{vert} & $|B_{\rm LOS}|_{\rm, avg}$ (mG) & 1.40 & 1.07 & 0 \\
& Peak $V$ (mJy/beam) & 0.78 & 0.29 & 0 \\ \midrule
& $B_{\rm LOS, avg}$ (mG) & 0 & 0 & 0 \\
\texttt{tor} & $|B_{\rm LOS}|_{\rm, avg}$ (mG) & 0 & 0.57 & 0.89 \\
& Peak $V$ (mJy/beam) & 0 & 0.09 & 0.11 \\ \midrule
& $B_{\rm LOS, avg}$ (mG) & 0.42 & 0.32 & 0 \\
\texttt{fid} & $|B_{\rm LOS}|_{\rm, avg}$ (mG) & 0.42 & 0.37 & 0.46 \\
& Peak $V$ (mJy/beam) & 0.23 & 0.10 & 0.05 \\ \midrule
& $B_{\rm LOS, avg}$ (mG) & 0.28 & 0.21 & 0 \\
\texttt{fidtc} & $|B_{\rm LOS}|_{\rm, avg}$ (mG) & 0.28 & 0.43 & 0.63 \\
& Peak $V$ (mJy/beam) & 0.15 & 0.08 & 0.07 \\ \bottomrule
\enddata
\end{deluxetable}
\begin{figure*}
\makebox[\textwidth][c]{\includegraphics[width=1.1\textwidth]{fid_fidtc_ch.png}}
\caption{\textit{Top panel:} Same as Figure \ref{fig:compChan}, now for our ``fiducial" case, viewed at $i = 40^{\circ}$. This model has its magnetic field strength divided such that $B_{\rm vert}:B_{\rm tor}:B_{\rm rad} = 30\%:45\%:25\%$. \textit{Bottom panel:} ``Toroidally boosted" version of the fiducial model, with $B_{\rm vert}:B_{\rm tor}:B_{\rm rad} = 20\%:70\%:10\%$.}
\label{fig:fidModel}
\end{figure*}
\section{Discussion}\label{sec:discussion}
\subsection{Evidence of Magnetic Complexity in Stokes $V$ Channel Maps}\label{ssec:compDisc}
One of the principle results of this work is that channel map information from spatially resolved observations can be used to distinguish vertical and toroidal magnetic field geometries in intermediate inclination disks.
The features of the emission produced in these respective case studies are individually discussed in detail in Sections \ref{sssec:torCase} and \ref{sssec:vertCase}, but we can also use them to make a broader point about identifying magnetic sub-structure in general.
For the purely vertical $\boldsymbol{B}$-field geometry, each channel has exactly one region of positive $V$ and one region of negative $V$.
The spatial span of these regions changes for different channels (due to the Keplerian rotation of the disk), but at all velocities they are both continuous and symmetric about the major axis of the disk.
We can think of this as the ``unsubstructured" baseline --- that is, a rotating, axisymmetric disk with a uniform magnetic field threading through it will always produce Stokes $V$ channel map emission that looks like this.
Therefore, any deviation from this picture is suggestive of magnetic sub-structure.
The purely toroidal channel map is an extreme example of such deviation.
We see well-defined, interlaced regions of positive and negative $V$ emission, and the placement of these regions are not symmetric about the disk's major axis (due to the combined effects of CN positioning and viewing geometry, as illustrated in Figure \ref{fig:torInter}).
A key point here is that the morphology of the emission in the vertical $\boldsymbol{B}$-field case essentially only reflects the impact of Keplerian rotation (since the magnetic field is uniform), whereas the toroidal $\boldsymbol{B}$-field case is sensitive to the inherent nearside/farside asymmetries that arise in an inclined disk (since, unlike a uniform field, a sub-structured magnetic field is itself affected by the asymmetry).
Interestingly, this is also the reason why the toroidal $\boldsymbol{B}$-field case (at $i = 40^{\circ}$) does not have zero spatially-integrated $V$ emission.
Even though the disk's mean line-of-sight magnetic field strength is zero (see Table \ref{tab:magConfData}), the asymmetry results in non-zero emission for many velocity channels.
Our fiducial disk has a complicated magnetic field geometry ($B_{\rm vert}:B_{\rm tor}:B_{\rm rad} = 30\%:45\%:25\%$) and is intended to model a ``realistic" situation.
In the context of the discussion above, we can use it to make an important qualitative point about general interpretation of Zeeman observations in disk environments.
Looking at the channel map for the fiducial model (given in the top panel of Figure \ref{fig:fidModel}, as viewed at intermediate inclination), it is obvious that its morphology much more closely resembles the purely vertical case than the purely toroidal case.
This tells us that the observed Stokes $V$ will be dominated by any vertical field component, if present.
As a result, the shape of the integrated $V$ profile is almost identical to that of the purely vertical model.
However, as we know from the model set-up, the disk's intrinsic $\boldsymbol{B}$-field is \textit{not} primarily vertical --- only 30\% of the field strength is in the vertical component.
The only clear evidence of the other (sub-structured) components is the slight asymmetry in the Stokes $V$ emission across the disk's major axis.
This asymmetry is of course more pronounced if the toroidal component is boosted (as in the bottom panel of Figure \ref{fig:fidModel}), but even in that case the integrated $V$ profile shows virtually no evidence of the non-vertical magnetic field.
The channel map information therefore provides crucial context for interpreting $\boldsymbol{B}$-field orientation and strength.
It is important to be aware that even small asymmetries in the emission can represent a relatively high degree of complexity (and therefore cancellation) in the disk's intrinsic magnetic field.
\subsection{Detectability Analysis}\label{ssec:detect}
Apart from the characteristics of the source itself, there are a few observational effects that can play a role in governing the level of detectability for our emission of interest.
We first evaluate the importance of beam size, then discuss the potential efficacy of velocity-based stacking of the hyperfine transitions (listed in Table \ref{tab:CNLines}) to boost the total Stokes $V$ flux.
\subsubsection{Beam Size}\label{sssec:beam}
In the case of observations for which the total emission is the quantity of interest, there is a direct proportionality between the size of the beam, $\theta_{\rm beam}$, and the maximum flux observed per beam.
This relationship is not necessarily true for observations of the Stokes $V$, because the positive and negative components of the emission become more prone to cancellation when integrated over more area.
Therefore, larger beams are liable to wash out signals of opposite polarity.
In Figure \ref{fig:beamComp}, we choose a representative velocity channel (0.4 km/s wide, centered at 1 km/s) and for each of the magnetic field configurations discussed in Section \ref{sssec:fourB} show $V$ emission maps using $\theta_{\rm beam} = 0.5,1,1.5$ and $2''$, viewed at $i = 40^{\circ}$.
We also plot peak $V$ (mJy/beam) vs. $\theta_{\rm beam}$.
In the 100\% vertical magnetic field simulation, the emission scales approximately linearly with the size of the beam.
This scaling occurs because the magnetic field has uniform direction in this case, and as such there is no sub-structure to produce cancellation.
In the other models we introduce toroidal (and radial) $\boldsymbol{B}$-field components, and the impact this has in suppressing signal is clear.
The most striking example is the 100\% toroidal case, for which we observe a turnover in peak $V$ at $\theta_{\rm beam} \approx 0.8''$.
The signal becomes almost completely washed out for very large beams.
For the more complicated magnetic field geometries, the $V$ vs. $\theta_{\rm beam}$ plots for those cases exhibit a knee at $\approx 0.8''$, the scale at which toroidal field cancellation becomes important.
For larger $\theta_{\rm beam}$, the rate of increase of the $V$ emission tapers considerably.
Since simulations generally predict substantial toroidal $\boldsymbol{B}$-field components, these results suggest that $\theta_{\rm beam} \approx 0.8''$ is the most reasonable choice for observations to maximize signal and preserve good spatial resolution when little is known about the true magnetic field geometry.
\begin{figure*}
\includegraphics[width=\textwidth]{washOut.png}
\caption{Comparison plots of a 0.4 km/s wide channel (centered at 1 km/s) for several choices beam size, viewed at intermediate ($40^{\circ}$) inclination. Each row reflects a different magnetic field geometry. The left panel shows how the maximum observable intensity (e.g., flux coming from the brightest pixel) changes as a function of beam size. Note that in the fully toroidal case, there is a turnover in Peak $V$ at $\theta_{\rm beam} = 0.8$ arcsec. This demonstrates the importance of spatial cancellation in poorly resolved observations of sources with toroidally dominated magnetic fields.}
\label{fig:beamComp}
\end{figure*}
\subsubsection{Hyperfine Component Stacking}\label{sssec:transitions}
For the CN $J = 1-0$ transition, there are 7 observable hyperfine components.
So far we have only considered the 113.144 GHz line (as a representative case), but it is in principle possible to leverage the flux from multiple lines to produce a stronger detection.
In Figure \ref{fig:trans} we plot the (spatially integrated) line flux results of simulations for all the transitions, performed for our fiducial model at $i = 40^{\circ}$.
Since the lines are entirely non-blended, stacking is possible.
The stacked line profile has a peak flux that is a factor of $\sim$5 larger than that produced solely by the 113.144 GHz component.
\begin{figure*}
\includegraphics[width=\textwidth]{V_vs_nu_transitions_40_png.png}
\caption{\textbf{Panels (a) - (g):} Stokes $V$ profiles for each of the CN $J = 1 - 0$ hyperfine transitions. Also included in each panel are optical depth profiles, plotting the peak $\tau$ (across all space) found at each frequency. \textbf{Panel (h):} A plot of where the lines lie in frequency space with respect to each other. They are mostly well separated. In the sub-panel we show that the 113.488 GHz and 113.491 GHz, which are relatively nearby, are still completely non-blended. \textbf{Panel (i):} Stacked profile of all 7 lines. Note that because the 113.171 GHz transition has negative $z_B$, its profile should be negated before stacking.}
\label{fig:trans}
\end{figure*}
\subsection{Sub-structured Gas Distribution}\label{ssec:gas}
Our fiducial disk includes rings in the large dust population.
As part of our modeling work we also tested disk scenarios with smooth (re-normalized to the same mass) large dust distributions, and found that the presence or absence of dust sub-structure has a negligible effect on the line emission results.
However, it is possible that this sub-structure may exist in the gas as well.
Observations of C$^{18}$O ($ J = 1-0$) emission in AS 209 by \citet{favre2019} show evidence of gas deficits that are spatially coincident with the dust gaps.
To model this scenario, we ran additional versions of our fiducial simulations with gas density gaps carved out according to the $\delta(R)$ prescription given in Equation \ref{eq:delta}.
The density distribution is renormalized such that the total gas mass is kept the same as it was in the original runs.
In Figure \ref{fig:gasGaps} we compare the emission profiles from these sub-structured runs with the original smooth ones.
For intermediate inclination and edge-on models, the Stokes $I$ is redistributed in velocity space when sub-structure is introduced, yielding more ``peaky" profiles since more of the CN gas is constrained to specific radii.
The opacity in these regions is slightly higher, exceeding $\tau = 1$ only near the peaks (this results in a $\sim20$\% lower maximum in $I$ than the original).
For most frequencies the emission remains optically thin, but there are still differences in the profile morphology as a result of the added gas sub-structure.
This is an important point to consider --- in the case of sub-structured disks, it is possible that some of the features in the $V$ profile are \textit{not} the result of magnetic complexity.
Observers should be cautious of this when searching for signatures of $\boldsymbol{B}$-field morphology in their data.
For the face-on model, the opacity increases dramatically with the addition of gaps.
This is because the emission, already distributed over a relatively narrow range in frequency space (since $v_{\rm LOS} = 0$ everywhere), is now pushed to smaller regions in observer space.
As a result of these optical depth effects, the $I$ and $V$ emission are both reduced significantly (by a factor of $\sim2$).
\begin{figure*}
\includegraphics[width=\textwidth]{gasGaps.pdf}
\caption{Comparison plots of the spatially integrated Stokes $I$ and $V$ profiles for a model with smooth gas distributions (e.g., \texttt{fid}) and one with gas gaps added. Due to increased opacity in the ring regions, the face-on view yields significantly reduced emission when gas sub-structure is introduced. This effect is present in the intermediate and edge-on cases as well, but to a smaller extent since the emission is spread over a larger range of velocity space. At $i = 40^{\circ}$, the gas gaps affect the morphology of the $V$ profile as well.}
\label{fig:gasGaps}
\end{figure*}
\subsection{Comparison to ALMA Percentage Polarization Limits}\label{ssec:alma}
ALMA's current circular polarization instrumentation is nominally stated to have a 1.8\% percentage polarization limit.
In the bottom row of Figure \ref{fig:parsli}, we give percentage polarization for the models in our main \textit{parsli} grid.
Since values for $I$ and $V$ vary across the observer plane, we report peak values for each run.
Our fiducial model yields percentages of 0.1\%, 0.3\%, and 0.4\% for 90$^{\circ}$ (edge-on), 40$^{\circ}$, and 0$^{\circ}$ (face-on) viewing angles, respectively.
Increasing the CN abundance or the depth of the CN slab (to larger $N_{\rm max,CN}$) increases $V/I$ in the face-on case, and extending the maximum radius of the slab leads to larger $V/I$ in the intermediate inclination and edge-on cases.
Peak percentage polarization also scales with $B_{\rm sum,0}$, of course.
Increasing the values of these parameters in various combinations produces a parameter space of optimistic disk scenarios that reach the nominal ALMA limit of 1.8\%.
For instance, if we set $B_{\rm sum,0}$ to 1.0 G (corresponding to $B_{\rm avg} = 3.5$ mG), we could produce 1.8\% polarization by also increasing the CN abundance to $\approx 3 \times 10^{-7}$ (per H$_{2}$) or increasing $N_{\rm max,CN}$ to about $10^{23}$ cm$^{-2}$.
It should be noted that at these high values of CN abundance and $N_{\rm max,CN}$, opacity effects will start to come into play as some regions of the disk reach $\tau > 1$.
Based on their circular polarization (non-detection) observations of TW Hydra, \citet{vlemmings2019} suggest ALMA may be capable of substantially better polarization performance, inferring a $<$0.8\% detection level.
For our face-on fiducial model, 0.8\% polarization can be reached if we set $B_{\rm sum,0} = 0.8$ G, which corresponds to a mean magnetic field in the CN emitting region of $B_{\rm avg} = 2.8$ mG.
This agrees reasonably well with the 2.6 mG limit \citet{vlemmings2019} report.
We note however that, as discussed above, there are also factors related to the disk set-up that can affect percentage polarization --- namely the abundance of the emitting molecule and the depth of the molecular layer.
\section{Conclusions}\label{sec:conclusions}
We simulated the Stokes $I$ and $V$ CN $J = 1-0$ emission arising from a ringed disk (modelled after the AS 209 disk system) with the POLARIS radiative transfer code. We produced synthetic observations viewed at face-on, intermediate ($i = 40^{\circ}$), and edge-on inclinations.
We varied several parameters in our model to probe how the emission changes as a function of the magnetic field configuration and the properties of the CN emitting region.
Our main conclusions are as follows:
\begin{enumerate}
\item Vertical and toroidal magnetic field configurations produce substantially different Stokes $V$ emission, and it is possible to distinguish them based on channel map morphology. At intermediate inclination, vertical $\boldsymbol{B}$-field components produce blotches of positive and negative $V$ emission that are symmetric about the major axis of the disk. Asymmetries to this end are a telltale sign of magnetic complexity, and even small ones can signify a relatively strong toroidal magnetic field component.
For sources with both vertical and toroidal components, the toroidal component must be much stronger than the vertical component for it to contribute significantly to the spatially integrated Stokes $V$ emission, unless the disk is viewed close to edge-on.
\item For our fiducial disk model, which has ``realistic" distributions of magnetic field strength and CN, the maximum Stokes $V$ signal obtained from our synthetic observations (at 0.4 km/s velocity resolution, with a $1''$ beam) is 0.6, 0.2, and 0.04 mJy/beam for face-on, $i=40^{\circ}$, and edge-on observations, respectively. Note that these values are for the 113.144 GHz transition only --- considering the other hyperfine components can fruitfully improve the signal (see item 6 below).
\item The Stokes $V$ scales with the strength of the magnetic field, and both the Stokes $I$ and Stokes $V$ scale with the total number of CN molecules. For our fiducial model the line emission is optically thin, but if CN exists deep enough into the disk (at column densities $\gtrsim 3 \times 10^{22}$ cm$^{-2}$) or if it is abundant enough ($\gtrsim 4 \times 10^{-8}$ CN molecules per H$_2$), the emission can transition to optically thick in some regions.
\item The traditional method for inferring magnetic field strength from Zeeman observations (i.e., fitting with Equation \ref{eq:VB}) must be approached with caution in disk environments, because PPDs are expected to have significant magnetic sub-structure. If the magnetic field has a strong vertical component, this component will be picked out effectively for face-on or intermediate inclination observations. However, its magnitude will imply a magnetic field strength that may be significantly reduced from the true value, depending on how much of the field is distributed into the other components. For close to edge-on sources or disks with dominant toroidal fields, the spatially integrated Stokes $V$ profile will be greatly diminished due to cancellation, and its shape will not be matched by $dI/d\nu$ due to the non-uniformity of the magnetic field. In this case, leveraging spatial information becomes crucial.
\item Choice of beam size can play an important role in the detectability of the Stokes $V$ emission in sources with magnetic sub-structure. If the magnetic field is toroidally dominated, there is a turnover in flux per beam at $\theta_{\rm beam} \approx 0.8''$ in our model. This beam size corresponds to a physical size of $\sim100$ au. Larger beams wash out the signal due to cancellation.
\item The 7 observable hyperfine components in the CN $J = 1 - 0$ suite are well-resolved in frequency space.
Due to optical depth effects and differing critical densities, the profiles of these components are not all identical. Nonetheless, they are similar enough that stacking is feasible. We demonstrate that stacking can increase the total signal by a factor of $\sim$5 over just using the strongest 113.144 GHz line.
\item The presence of gas sub-structure in the disk can have important effects on the Stokes $V$ emission, both in terms of magnitude and morphology. Face-on disks with gaps have substantially elevated optical depth (in the rings) compared to equal mass gap-less counterparts. If some regions (i.e., the rings) reach $\tau > 1$, this is liable to produce reduced emission in the spatially integrated profile. Intermediate inclination disks are also susceptible to this effect, but to a lesser extent since the emission is spread over a wider breadth of frequency space. As our $i = 40^{\circ}$ simulation shows, gas gaps in intermediate inclination sources also produce perturbations in the Stokes $V$ profile, which could in principle be interpreted (incorrectly) as evidence of magnetic sub-structure. Observers should be cautioned of this when inferring magnetic field information from Zeeman observations.
\end{enumerate}
In this work we considered one disk structure and only performed line emission simulations. Natural future extensions could include testing different density distributions (in both gas and dust) and simulating the continuum emission.
Namely, one potentially important factor we have not accounted for here is that some sources may have thick dust midplanes that could block up to half of the disk, depending on the viewing geometry.
This could of course reduce total emission, but also may eliminate some of the cancellation that occurs in the Stokes $V$ emission of sub-structured magnetic field configurations, which could have interesting effects on both the morphology and detectability of the signal.
In the simulations we performed for this work the midplane was optically thin at 113 GHz, so dust did not play a role in the radiative transfer beyond factoring into the calculation of the dust and gas temperature. However, future simulations of Zeeman at higher $J$ rotational transitions should take the possibility of optically thick continuum emission into account when simulating line observations.
\section*{Acknowledgements}
RRM is supported in part by the National Radio Astronomy Observatory (NRAO) through a Student Observing Support (SOS) award and by NASA 80NSSC18K0481. RRM acknowledges additional support from the Virginia Space Grant Consortium (VSGC) Graduate Fellowship Award, as well as useful communication with Crystal Brogan, Scott Suriano, Daniel Lin, Robert Brauer, and Stefan Reissl. LIC gratefully acknowledges support from the David and Lucille Packard Foundation and the VSGC New Investigators Award and NASA 80NSSC20K0529. ZYL is supported in part by NASA 80NSSC20K0533 and NSF AST-1815784.
\bibliographystyle{aasjournal}
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{"arxiv_id":"2009.08996","language":"en","timestamp":1600740030000,"url":"https:\/\/arxiv.org\/abs\/2009.08996","yymm":"2009"}
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\section{Introduction}
\textbf{The significance of recommender systems.}
In an increasingly digital world, institutions must accurately anticipate the next movie, song, product, or job that a user will interact with. From YouTube to Netflix, Spotify to Amazon, Facebook to Linkedin, recommender systems are a staple of our daily routines. They influence how we perceive our environment, from media content to human relationships.
\textbf{Standard evaluation does not reflect real-life use-cases.}
Traditional methods of evaluation that entail random sampling over a long period of time are perfect if the system is designed to remain unchanged for an equally long and predefined period.
However, if the system is to be used in a dynamic setting, the way it is evaluated must reflect that.
Inadequate evaluation techniques can lead to a false confidence, which is especially detrimental in commercial settings.
Evaluating a recommender system can be done \emph{online} or \emph{offline}. \emph{Online} evaluation entails deployment of the recommender in a commercial setting. While this may be the best way to measure the real-life impact of a system, it is also costly and therefore rarely used in research and benchmarking.
\emph{Offline} evaluation is far more common in recommender systems research. Here, the model is evaluated on historical data, by selecting some portion of the data to train on, while some other subset is used for performance testing.
\textbf{Many existing recommenders ignore temporal information.}
Most recommender systems fall into one of the two main categories: content-based and collaborative filtering. Content-based filtering relies on recommended items having similar attributes to those that the user has previously interacted with. Collaborative filtering methods base the recommendation on items bought by similar users.
However, most models ignore temporal information, except a subtype of recommenders that focuses on the order of interactions. Time-aware recommender systems
-- called sequence-aware recommenders --
introduce additional information to the interactions: the time at which the users' decisions were made.
Consequently, there is a need to incorporate the temporal context into traditional recommenders.
\textbf{In this work,} we first focus on the importance of temporal dynamics in recommender system creation, training, and offline evaluation. While much effort is directed towards establishing the importance of proper evaluation design, it is generally focused on implementing relevant metrics to avoid under- or over-estimating real-world performance \cite{aggarwal2016recommender}.
We draw attention to the lack of standardization in this domain, and the differences between research settings and the systems' ultimate applications.
Then, we propose two temporal evaluation protocols and show how they attain a closer approximation of the real-life conditions in which recommender systems are deployed.
Second, we present a multi-objective approach to time-unaware recommender systems to incorporate the temporal context without any change in the model architecture.
We demonstrate the advantages of such systems. Then, we introduce recency as an objective and as a means to include temporal dynamics in typically time-independent recommender systems. We also provide a measure of recency in the form of a performance metric.
Experiments on three real-world publicly available datasets show both improvements in recency and relevance. Finally, we demonstrate that the Pareto Fronts obtained with the added objective dominate those produced by state-of-the-art models.
To the best of our knowledge, this is the first study quantifying the difference in recommender system performance when evaluated using methods that model real-world environments, as opposed to traditional techniques. After demonstrating the impact of recency, we show that a recommender system can be optimized for both the relevance and recency objectives simultaneously.
To summarize, the main contributions of this paper are as follows:
\begin{itemize}
\item We demonstrate how commonly used evaluation protocols do not provide adequate modeling of real-world deployment settings. To combat this, we propose two evaluation techniques to facilitate offline modeling of online production environments that inherently incorporate temporal dynamics;
\item We introduce a recency function that can be utilized to create a recency objective. We show that optimizing for both recency and relevance \cite{milojkovic2019multi} leads to solutions that dominate those optimized just for relevance in both dimensions.
\end{itemize}
\section{Related Work}
\subsection{Evaluating Recommender Systems}
\subsubsection{Traditional Recommender Systems.} Inputs and outputs share similarities with classification and regression modeling: a class variable is predicted from a set of given features. Therefore, given that recommendation tasks can be seen as a generalization of these, some evaluation techniques used for classification are transferrable to recommender systems.
In collaborative filtering research, recommenders are generally evaluated either through \emph{strong} or \emph{weak generalization}, characterized by \cite{marlin2004collaborative}.
In both approaches, models are trained on observed interactions and validated or tested on those that are held-out.
\emph{Weak generalization} is introduced in \cite{10.5555/2074094.2074100}, where the held-out set is created through random sampling of the available interactions.
\emph{Strong generalization} differs by taking disjoint sets of users for the training, validation, and testing sets. Following this, some interactions are held-out from the validation and test sets and then approximated using the recommender.
Methods that encode user representation cannot apply \emph{strong generalization}, as they cannot generate outputs for previously unseen users.
An example of the \emph{strong generalization} approach can be seen in \cite{liang2018variational}, whereas \cite{ning2011slim, wu2016collaborative, rendle2012bpr} all use \emph{weak generalization}.
Several of these works emphasize that the application of their recommender system would be in predicting future user actions, yet all validation and testing is done with randomly selected interactions. This can break the time linearity as the knowledge of future interactions can help predict an anterior interaction.
\subsubsection{Temporal Recommender Systems.} They denote time-aware models (TARS), and incorporate time explicitly or implicitly. Temporal recommender systems include, but are not limited to, sequence-aware recommender systems (SARS).
As previously stated, SARS can be evaluated similarly to TARS. \cite{campos2014time} provide an extensive overview of possible evaluation techniques, which served as an inspiration and point of reference for this work.
While traditional evaluation protocols may be used on temporal recommenders, it is more representative to preserve the temporal ordering between interactions since this is something that the recommender aims to learn. By extension, train, validation, and test splits should also be~ordered.
\cite{quadrana2018sequence} state that they were unable to find a consensus among evaluation protocols used in recent sequence-aware recommender work, which is mirrored in our findings.
Yet we did determine that most recent SARS focus only on next item prediction, meaning they output one recommendation. They also typically employ certain target item conditions to decrease computational cost \cite{campos2014time}.
The target item conditions determine the (sub)set of items for which a recommender should produce predictions and are specific for top-N recommendation evaluation.
The reduction of the computational costs is generally done through conditions that rank one ground truth item against a set of other items false items.
Examples can be found in \cite{sun2019bert4rec, kang2018self, hidasi2018recurrent}. We return to the problem of subsampling in Section \ref{sec:proposed}.
\subsection{Temporal Context in Recommender Systems}
In this paper, we introduce the concept of recency.
An important note is that there are multiple definitions of recency in recommender systems literature.
In fact, this lack of consensus has persisted for years.
\cite{ding2006recency, vinagre2015collaborative} treat the recency of an item as an attribute that is user-dependent. The value is determined by the last time the user interacted with a given item. \cite{chakraborty2017optimizing, gabriel2019contextual} also claim to incorporate recency into their research: when recommending news articles, they measure recency as the age of the item on the platform. Our analysis will follow the latter definition.
\section{Proposed Evaluation Protocols}
\label{sec:proposed}
We propose that the temporal dimension should be considered when evaluating the performance of any recommender.
While random sampling may be an appropriate target selection technique for some classification or regression tasks, we argue that this is not the case when it comes to predicting a user's subsequent move.
Unlike the vast majority of evaluation methods applied to traditional recommenders, temporal recommender systems literature does model the passage of time.
However, as stated above, the performance is often computed over a subset of the itemset and the user's true chosen item.
The argument is that subsampling is necessary due to the complexity of the ranking task. While this has some validity, itemsets of around 10,000 datapoints can be ranked highly efficiently, especially when taking into consideration recent advancements in machine learning libraries and GPU programming. Therefore, we do not utilize subsampling in our work.
The adoption of a recommender system in real scenarios has two major phases. The first, called the development phase, is purely offline and theoretical. In this part, three separate sets of data must be created: a training set that the model will use to learn item and user representations, a validation set for hyperparameter tuning, and a test set to evaluate how well the model performs.
The second, called the deployment-ready phase, include interactions with end-users. The maximum amount of data is leveraged to train a model with as much information as possible, evaluate its performance, and then deploy it into production. In this case, only two sets are needed: a training and a validation set.
One downside of collaborative filtering methods is that most models are incapable of incorporating new items without retraining. While ways to alleviate this problem have been explored \cite{luo2012incremental}, the issue remains widespread and worthy of more study, but lies outside the scope of this paper.
Therefore, we assume an industry-like environment: the recommender system will be retrained regularly and will be exposed to clients for a relatively short period, ranging from a couple of days to a maximum of a few months.
We postulate that the performance of the recommender on the last portion of historically available data is most indicative of how it will behave when deployed.
\begin{figure}[t]
\centering
\includegraphics[width=.85\linewidth]{Figures/proportional.pdf}
\caption{Proportional Temporal Selection.}
\label{fig:prop}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=.85\linewidth]{Figures/cutoff.pdf}
\caption{Strict Temporal Cutoff.}
\label{fig:cutoff}
\end{figure}
Our protocols focus on set creation. When selecting the target values in a validation set, we take two possible approaches.
The first, \emph{proportional selection}, depicted in Figure \ref{fig:prop}, selects the final \(X\%\) of each user's interactions and uses these to create target items.
Here we preserve the time ordering of the input and target interactions, maintaining similarity with the real-life use-case.
However, there is no strict time cutoff, as is the case when we train a system on data available to a certain point and then deploy it.
The second approach, shown in Figure \ref{fig:cutoff}, is precisely based on a \emph{strict time cutoff} to select the target items of the validation set. This method is even closer to the real-world use case. However, it does suffer from certain drawbacks as user interactions are not necessarily evenly distributed through time, leading to some users being more represented than others in the target set.
While these are similar to the suggestions developed in \cite{campos2014time}, we underline that these approaches should not be limited to evaluating TARS.
It is crucial to approximate with maximum precision the performance of a model when developing a novel system, before it is released into production.
The second approach directly models the real-world context and contains user-item interaction sequences created after a specific strict time cutoff.
\section{Recency to Improve Recommendation}
The main task of a recommender system is to anticipate users' future desires and suggest content that they would find relevant.
However, just recommending the most relevant items does not always satisfy all the concerns of those building the system. The \emph{relevance} objective is the one that is most commonly found in recommender systems literature and accounts for the accuracy or correctness. It actively focuses the recommender on selecting the item(s) with which the users will most likely interact.
However, relevance is not the only objective used in practice. We distinguish two types of objectives: \emph{correlated} and \emph{uncorrelated} to relevance.
The former ones correspond to those whose optimization is linked to the relevance objective. Examples are novelty \cite{vargas2011rank}, serendipity \cite{ge2010beyond}, and utility-based objectives, such as revenue.
The latter, \emph{not correlated to relevance}, can be diversity and fairness.
We introduce such a utility-based objective used to inject temporal information alongside the relevance objective. While the exploration of uncorrelated objectives is essential for the future of recommender systems, we leave it for future~work.
\subsection{Adding Temporal Context}
Based on the our experience with real-life use-cases, we discovered that users seem to gravitate towards purchasing content that had more recently been added to a given platform.
Building on these findings, and works such as \cite{chakraborty2017optimizing} and \cite{gabriel2019contextual}, we decided to explore the effects of incorporating recency as an objective during the learning phase. Given an item \(x\) with a timestamp \(t_x\), we further define the recency function \(f\) as:
{\small
\begin{equation}
f(x) =
\left\{
\begin{array}{ll}
1 & \frac{t_{x} - t_{min}}{t_{max} - t_{min}} \geq 0.8 \\
0.3^{(0.8 - \frac{t_{x} - t_{min}}{t_{max} - t_{min}})\times \frac{10}{3}}& otherwise \\
\end{array}
\right.
\end{equation}
}
where \(t_{max}\) and \(t_{min}\) are the maximum (most recent) and minimum (oldest) timestamps over the itemset.
In \(f\), we first scale all timestamps to \([0, 1]\) using the min-max scaler, and then apply a transformation inspired by \cite{huang2013incorporating}. A plot of the function is shown in Figure \ref{fig:recency}.
\begin{figure}[t]
\centering
\includegraphics[width=.7\linewidth]{Figures/recency_function.pdf}
\caption{Our proposed recency function (Equation 1).}
\label{fig:recency}
\end{figure}
The recency objective is formulated as a loss that stimulates the recommendation of recent items.
Each item in the itemset is assigned a recency weight, based on the recency function. The vector is then used to weigh item importance when calculating the loss. Adding weights into a traditional loss does not affect the differentiability of the function.
To illustrate how our temporal objective can be easily integrated into a traditionally time-unaware recommender system, we take as a use-case the state-of-the-art variational autoencoder Mult-VAE\textsuperscript{PR} of \cite{liang2018variational}. For the sake of brevity, we refer the reader to \cite{liang2018variational} for more details about the model.
We thus propose an extension of Mult-VAE\textsuperscript{PR}, where the loss function for user $u$ is modified to:
\begin{align*}
\mathcal{L}_{\beta}(x_u;\theta, \phi) = \mathbb{E}_{q_\phi(z_{u}|x_{u})}\left[ \log p_\theta(f(x_u) \ast x_{u}|z_{u})\right] \\
- \beta \cdot \mathrm{KL}(q_\phi(z_{u}|x_{u}) || p(z_{u}))
\end{align*}
where the expected negative log-likelihood is modified to include the element-wise multiplication of input vector $x_u$ by $f(x_u)$, which corresponds to the recency scores of the given items in $x_u$. $\beta$ controls how much importance is given to the KL term, \(z_u\) is a variational parameter of the variational distribution $\theta$ and $\phi$ are model parameters.
\begin{algorithm}[t]
\scriptsize
\caption{SMSGDA with Gradient Normalization. }\label{alg:multiobj}
\label{vanila_alg}
\begin{algorithmic}[1]
\State $initialize()$
\For{$i \in {1,...,n}$}
\State $empirical\_loss_i = \mathcal{L}_i(w)$
\EndFor
\For{$epoch \in {1,...,M}$}
\For{$batch \in {1,...,B}$}
\State $do\_forward\_pass()$
\For{$i \in {1,...,n}$}
\State $calculate \, loss \;\; \mathcal{L}_i(w)$
\State $calculate \, gradient \;\; \nabla_{w} \mathcal{L}_i(w)$
\State $normalize \, gradient \;\; \hat{\nabla_{w}\mathcal{L}_i(w)} = \frac{\nabla_{w} \mathcal{L}_i(w)}{empirical\_loss_i} $
\EndFor
\State $\alpha_{1}, \ldots, \alpha_{n} = QCOPSolver\left(\hat{\nabla_{w}\mathcal{L}_1(w)}, \ldots, \hat{\nabla_{w}\mathcal{L}_n(w)}\right)$
\State $\nabla_{w} \mathcal{L}(w)=\sum_{i=1}^{n} \alpha_{i} \hat{\nabla_{w}\mathcal{L}_i(w)}$
\State $w = w - \eta \nabla_{w} \mathcal{L}(w)$
\EndFor
\State $evaluate\_model()$
\State $update\_pareto\_set()$
\EndFor
\end{algorithmic}
\end{algorithm}
\subsection{Multi-Objective Optimization}
Optimizing a recommender on multiple objectives is non-trivial. Thanks to the recent work of \cite{milojkovic2019multi}, we employ the proposed multi-gradient descent algorithm for multiple objectives to train our recommenders. Additionally, the authors show that the algorithm is efficient and does not impact on training time, as it can be seen in Algorithm~\ref{alg:multiobj}, where $n$ is the number of objectives, $M$ is the number of epochs, and $B$ the number of batches.
After a standard forward pass (Line 7), the loss and gradient are computed for each objective (Line 8-10). Then, weights of the gradients are computing as a Quadratic Constrained Optimization Problem \cite{desideri2012multiple}, which can be solved analytically for two objectives, or solved as a constrained optimization problem as proposed in \cite{sener2018multi} for more than two objectives. Solving it allows us to obtain the common descent vector and update the parameters (Line 14-15).
This training procedure enables us to incorporate both our temporal context and the relevance objectives to retrieve time-aware recommendations. The algorithm adapts~the weight repartition between the two objectives in an advanced manner to optimize both during~training.
\section{Experiments}
\label{ExperimentsSection}
\subsection{Datasets}
\begin{table}[t]
\centering
\begin{threeparttable}
\begin{tabular}{@{}lccc@{}}
\textbf{ } & \textbf{ML-20M} & \textbf{Steam} & \textbf{Netflix$\geq$4} \\
\toprule
\# of users & 46,295 & 257,775 & 471,457 \\
\# of items & 9479 & 13,018 & 13,995\\
\# of interactions & 3.76M & 3.14M & 38.87M \\
\% of interactions & 0.86\% & 0.09\% & 0.59\% \\
\bottomrule
\end{tabular}
\end{threeparttable}
\caption{Statistics of the datasets, after preprocessing.}
\label{table:datasets}
\end{table}
We study the performance of various models on three real-world publicly available datasets. The characteristics of the preprocessed datasets are summarized in Table \ref{table:datasets}.
\textbf{MovieLens-20M}
contains about 20 million ratings\footnote{\url{https://grouplens.org/datasets/movielens/20m/}.}, with values between 1 and 5.
To transform it into implicit feedback, we binarize the user-item interaction matrix, keeping ratings of~4 and above as positive feedback. We filter out all users with less than five ratings, and all movies rated by less than five users.
Since this dataset contains entries from 1999 up to 2015, we chose to focus on the last ten years of available data.
\textbf{Steam}
has review information from the gaming platform Steam\footnote{\url{https://cseweb.ucsd.edu/~jmcauley/datasets.html}.}. We converted user-item interactions into a positive feedback signals.
The dataset contains reviews from 2010 to 2018; however, the platform only sees an uptick in review activity after 2014, which is why we select the last four years available for further analysis.
\textbf{Netflix}
is the well-known Netflix Prize Competition dataset\footnote{\url{https://www.kaggle.com/netflix-inc/netflix-prize-data}.}. It consists of over 100 million ratings. The ratings are on a scale from 1 to 5 and were collected between 1998 and 2005.
We filter these ratings in the same way as the MovieLens ratings, and take the last two years of activity. Because of low performance on certain baselines, we denote two variants for the implicit feedback: one with threshold of~$4$ and above (Netflix$\geq$4), the other one with a threshold of~$5$ (Netflix$\geq$5).
\subsection{Recommendation Techniques}
For all models, we ensured that the items that the user had previously interacted with were removed from the output before the top-k results were selected for metric calculation. All models were trained with the Adam optimizer, with a learning rate of 0.001.
\textbf{Mult-VAE\textsuperscript{PR}: }
All experiments with the Mult-VAE\textsuperscript{PR} \cite{liang2018variational} were conducted using the implementation from the MAMO framework\footnote{\url{https://github.com/swisscom/ai-research-mamo-framework}.}.
We used the same setup as in the original paper.
\textbf{SVD: }
We utilize the PyTorch implementation\footnote{\url{https://pytorch.org/docs/stable/generated/torch.svd.html}.} of the Singular Value Decomposition \cite{sarwar2000application}, taking only the top 100 dimensions.
\textbf{NCF: }
For Neural Collaborative Filtering \cite{he2017neural}, we take the implementation from \footnote{\url{https://github.com/guoyang9/NCF}.}, sample four negative instances for every existing user-item interaction, set the predictive factor of 64, and the number of hidden layers for the multilayer perceptron (MLP) to three.
We do not present results obtained using pre-trained NeuMF, as they exhibited the same patterns as generalized matrix factorizaion (GMF) and MLP, but did not give a significant improvement.
To resolve the difficulties to obtain good results with the \emph{Netflix$\geq$4} dataset for GMF and MLP models, we used instead the \emph{Netflix$\geq$5} dataset. The main difference being that only ratings of five and above are considered as positive.
\textbf{BERT4Rec: }
This sequence-aware recommender system was introduced in \cite{sun2019bert4rec}. We implemented it in PyTorch and integrated it with the MAMO framework.
Most of the hyperparameters used were taken from the original paper.
The number of transformer layers is set to 2, the head number is 4, head dimensionality is 64, and the dropout is 0.1.
We use a sequence length of 100, while the proportion of masked inputs is 0.2.
The model is trained using the Adam optimizer with a learning rate of 1e-4.
\textbf{BERT4Rec Extension: }
We propose an extension to BERT4Rec.
BERT4Rec consumes sequences of items. The positions in the sequence that the user wishes to predict are filled with a special mask identifier. When predicting the next item(s) in a sequence, \cite{sun2019bert4rec} place a mask on the last position in the sequence, and then take the top-$k$ items from the probability distribution of this position, as generated by the model.
Instead of selecting the top-\(k\) items from the last (masked) position in the sequence, we suggest to select them from the last-\(p\) positions, all of which are masked in the input sequence. From each position in \(p\) we select the top-\(\lfloor \frac{k}{p} \rfloor\) items, making sure that there are no repeated items.
If \(k\) is not divisible by \(p\), the leftover elements are selected from the first masked position.
\begin{table*}[t]
\centering
\begin{threeparttable}
\begin{tabular}{@{}llcccc@{}}
\textbf{Dataset}&\textbf{Model}& \textbf{Val\textsuperscript{trad}} & \textbf{Val\textsuperscript{prop}} & \textbf{Val\textsuperscript{cutoff}}& \textbf{Test\textsuperscript{temp}} \\
\toprule
\multirow{4}{*}{ML-20M} & Mult-VAE\textsuperscript{PR} & 0.32 / 0.18 & 0.26 / 0.13 & 0.11 / 0.06 & 0.11 / 0.07\\
& SVD & 0.25 / 0.22 & 0.14 / 0.11 & 0.07 / 0.03 & 0.11 / 0.07\\
& GMF & 0.25 / 0.22 & 0.11/ 0.10 & 0.08 / 0.03 & 0.10 / 0.07\\
& MLP & 0.25 / 0.23 & 0.12 / 0.10 & 0.07 / 0.03 & 0.11 / 0.07\\
\midrule
\multirow{2}{*}{Steam} & Mult-VAE\textsuperscript{PR} & 0.20 / 0.02 & 0.14 / 0.02 & 0.11 / 0.01 & 0.13 / 0.01\\
& SVD & 0.10 / 0.02 & 0.10 / 0.02 & 0.09 / 0.01 & 0.08 / 0.01\\
\midrule
\multirow{2}{*}{Netflix$\geq$4} & Mult-VAE\textsuperscript{PR} & 0.35 / 0.18 & 0.22 / 0.10 & 0.12 / 0.05 & 0.10 / 0.05\\
& SVD & 0.23 / 0.16 & 0.23 / 0.16 & 0.09 / 0.05 & 0.07 / 0.04\\
\midrule
\multirow{3}{*}{Netflix$\geq$5} & SVD & 0.23 / 0.10 & 0.23 / 0.11 & 0.12 / 0.05 & 0.09 / 0.03\\
& GMF & 0.31 / 0.14 & 0.30 / 0.14 & 0.14 / 0.05 & 0.12 / 0.04\\
& MLP & 0.31 / 0.14 & 0.30 / 0.14 & 0.14 / 0.05 & 0.12 / 0.04\\
\bottomrule
\end{tabular}
\end{threeparttable}
\caption{\label{table:combinedResults}Results of the Mult-VAE\textsuperscript{PR}, SVD, GMF, and MLP evaluated on a traditional, proportionally selected temporal, and strict cutoff validation set, as well as on a temporally shifted test set. We report Recall / Precision at $k=20$.}
\end{table*}
\subsection{Experimental Setup}
The experiments conducted show how an inadequate manner of creating the validation sets in the \emph{deployment-ready} phase leads to false confidence in the performance of the evaluated model.
In the \emph{deployment-ready} phase, what we call the validation set is not necessarily used for hyperparameter tuning, but to assess the performance of the model before it is deployed.
There are minor differences in the datasets used for the models with and without user representation. Models without user representation require some input interactions to be able to predict targets, while those without simply need to be passed a user identifier.
We divide our experiments into three sets, corresponding to the type of evaluation.
\subsubsection{Traditional Evaluation.}
Similarly to \cite{liang2018variational}, we divide the data into a train set with 80\% of users, validation set with 10\% and test set with 10\%.
The target user-item interactions are selected by randomly sampling 20\% of the user-item interactions in the validation and test sets.
We show that if a model is evaluated on and later applied to a task that entails predicting randomly held-out interactions, the performance achieved on both validation and test sets is comparable.
This traditional approach is typically used to report model performance.
We then contrast performance on randomly held-out interactions in the validation set against temporally held-out interactions in the test set.
We take 5\% of the users from the train set to create the validation set and hold-out 20\% of their interactions.
The test set contains the interactions and users from the train and validation sets as inputs, and the temporally held-out interactions are targets.
\subsubsection{Temporal Evaluation.}
We show that when evaluated with either a proportional or hard temporal cutoff, the model's performance is closer to what would be observed in a real-life setting.
However, it is important to note the ideal evaluation technique is heavily domain dependent.
First, we hold out the last 20\% of user-item interactions from each user in the validation set. In the second approach, we hold out the last couple of months of activity and evaluate the model's ability to predict these interactions.
We create the validation and test sets as before.
\subsubsection{Temporal Evaluation with Added Temporal Context.}
We introduce temporal context into the traditionally time-independent Mult-VAE\textsuperscript{PR} by using the work from \cite{milojkovic2019multi} to optimize the model for accuracy and recency.
To calculate the recency score we must determine a timestamp for every item in the itemset. We take the timestamp of the moment that the item first became available, or the first recorded instance of any user interacting with the given item.
The strict temporal cutoff validation set is utilized, as well as the temporal test set described previously.
\subsection{Evaluation Metrics}
We evaluate models using three ranking metrics, as recommenders can often only show a predefined number of recommendations.
\begin{itemize}
\item \textbf{Precision@K: }calculates how many of the recommended items are relevant to the user;
\item \textbf{Recall@K: }quantifies the proportion of relevant items in the top-k recommended items by calculating how many of the desirable items are are suggested to the end-user. We take our definition from \cite{liang2018variational};
\item \textbf{Recency@K: }assigns a recency score to each item, calculating the rating of the top-k recommended and relevant items. For user \(u\) with relevant items \(I_{u}\) we define \(\omega(k)\) as the item at rank \(k\), where \(\mathbb{I}\) is the indicator function: \begin{equation}
Recency@K(u, \omega, f) = \sum_{k = 1}^{K}{\mathbb{I}
[\omega(k) \in I_{u}]\times f(\omega(k))}
\end{equation}
\end{itemize}
\section{Results}
\subsubsection{Traditional Evaluation.}
This experiment aims to show that the traditional way of evaluation recommender systems, shown in Table \ref{table:vaeTraditional}, is not a faithful representation of the environments in which they are actually deployed. The good performance achieved by evaluating in this way can provide a false sense of security.
\begin{table}[t]
\centering
\begin{threeparttable}
\begin{tabular}{@{}lcc@{}}
\textbf{Dataset} & \textbf{Val\textsuperscript{trad}} & \textbf{Test\textsuperscript{trad}} \\
\toprule
ML-20M & 0.31 / 0.17 & 0.31 / 0.17 \\
Steam & 0.20 / 0.02 & 0.20 / 0.02\\
Netflix$\geq$4 & 0.35 / 0.19 & 0.35 / 0.19 \\
\bottomrule
\end{tabular}
\end{threeparttable}
\caption{Results of initial Mult-VAE\textsuperscript{PR} experiments, evaluated on a traditional evaluation protocol. We report Recall / Precision at $k=20$.}
\label{table:vaeTraditional}
\end{table}
Our claim is supported by the values highlighted by Table \ref{table:combinedResults}. Even though the validation sets are not identical to the ones before, the performance observed is very similar. However, it degrades on the time delayed test set, or to be more precise, when it simulates what would happen in a production setting. Drops in performance of -65.63\%, -35.00\%, and -71.43\% can be observed, on the Recall@20 values.
We postulate that this discrepancy leads to significant dissonance between the results of certain recommenders as reported in literature, and those observed in their real-life application.
\subsubsection{Temporal Evaluation.}
The results shown in Table \ref{table:combinedResults} depict what happens when using traditional validation as apposed to our proposed evaluation sets.
The table illustrates how the strict cutoff validation set approximates the \emph{deployment} behavior.
For all datasets, this approach seems to be a closer estimation of the ``real-life" performance.
For example, the drop in performance is reduced from -71.43\% to -16.67\% on the Netflix$\geq$4 dataset.
The proportionally selected validation sets seems to work well for the Steam dataset, and we know from industry experience that it can be good on others. However, it seems to be highly dataset specific and is something that should be kept in mind.
Table \ref{table:combinedResults} also shows that this phenomenon is not isolated to the Mult-VAE\textsuperscript{PR}, but can be repeated with the SVD, GMF, and MLP models. As mentioned before, we were unable to conduct experiments on Netflix$\geq$4 with the GMF and MLP models; therefore we report their results on Netflix$\geq$5.
The most severe drop in performance is in the case of traditional evaluation on the GMF and MLP on the Netflix$\geq$5 dataset, where the Recall@20 decreases by -61.29\%.
It is important to note that simpler methods, especially those based on matrix factorization, do not deal well with the Steam dataset.
This is the sparsest dataset that we work with, as shown in Table \ref{table:datasets}, and this seems to make it difficult to learn anything meaningful.
Following this conclusion, we exclude the Steam dataset results for GMF and MLP.
However, we keep the results for SVD.
\begin{figure*}[t]
\centering
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[width=\linewidth]{Figures/VAE_ml20m_strict_last_single_vs_multi_150.pdf}
\caption{ML20m dataset.}
\label{fig:sub1}
\end{subfigure}
\hfill
\begin{subfigure}[b]{.335\textwidth}
\centering
\includegraphics[width=\linewidth]{Figures/VAE_steam_strict_last_single_vs_multi_100.pdf}
\caption{Steam dataset.}
\label{fig:sub2}
\end{subfigure}
\hfill
\begin{subfigure}[b]{.32\textwidth}
\centering
\includegraphics[width=\linewidth]{Figures/VAE_netflix_strict_last_single_vs_multi_75.pdf}
\caption{Netflix$\geq$4 dataset.}
\label{fig:sub3}
\end{subfigure}
\caption[Comparison of Pareto Fronts obtained through optimizing on one objective and two objectives]{Pareto Fronts obtained through optimizing on one objective (accuracy), and two objectives (accuracy and recency).}
\label{fig:paretofronts}
\end{figure*}
We strongly recommend that these evaluation methods be taken into account when presenting novel achievements in the field.
When feasible, we recommend to apply both evaluation protocols.
\subsubsection{Temporal Evaluation and Temporal Models.} The results presented so far were achieved using traditional recommender architectures, with no way of learning temporal dynamics. Our next contribution is to integrate the temporal dynamic in the training process. Table \ref{table:bertpropcomp} shows the results obtained with BERT4Rec and BERT4Rec\textsuperscript{(5)}. We apply this extension in the testing phase only. The results show that while it boosts the predictive power of the model in some cases, the benefits vary from case to case.
\begin{table}[t]
\centering
\begin{threeparttable}
\begin{tabular}{@{}llcc@{}}
\textbf{Dataset} & \textbf{Model} & \textbf{Val\textsuperscript{cutoff}} & \textbf{Test\textsuperscript{temp}}\\
\toprule
\multirow{2}{*}{ML-20M} & BERT4Rec & 0.20 / 0.09 & 0.15 / 0.08 \\
& BERT4Rec\textsuperscript{(5)} & - & 0.15 / 0.08 \\
\midrule
\multirow{2}{*}{Steam} & BERT4Rec & 0.21 / 0.02 & 0.17 / 0.02\\
& BERT4Rec\textsuperscript{(5)} & - & 0.18 / 0.02 \\
\midrule
\multirow{2}{*}{Netflix$\geq$4} & BERT4Rec & 0.24 / 0.13 & 0.20 / 0.05\\
& BERT4Rec\textsuperscript{(5)} & - & 0.21 / 0.06 \\
\bottomrule
\end{tabular}
\end{threeparttable}
\caption{Results of BERT4Rec and BERT4Rec\textsuperscript{(5)} evaluated on a strict cutoff validation set and a time delayed test set.
BERT4Rec\textsuperscript{(5)} is only applied on the test set. We report Recall / Precision at $k=20$.}
\label{table:bertpropcomp}
\end{table}
By comparing Table~\ref{table:bertpropcomp} and Table~\ref{table:combinedResults} (that contain results achieved with traditional, time-independent recommenders), BERT4Rec achieves the best performance on the test set.
This confirms our hypothesis that temporal dynamics should be accounted for in both evaluation design and model architecture in order to attain the best possible recommenders.
\subsubsection{Temporal Evaluation with Added Temporal Context.} To further integrate the temporal context, our following contribution has the recency included as an objective influencing the optimization.
We refer to the multi-objective Mult-VAE\textsuperscript{PR} as the Multi-Objective Recency Enriched mult-VAE\textsuperscript{PR}(MOREVAE).
\begin{table}[t]
\centering
\begin{threeparttable}
\begin{tabular}{@{}llccc@{}}
\textbf{Dataset} & \textbf{Model} & \textbf{R} & \textbf{P} & \textbf{Re}\\
\toprule
\multirow{2}{*}{ML-20M} & Mult-VAE\textsuperscript{PR} & 0.11 & 0.07 &0.23 \\
& MOREVAE & \textbf{0.13} & \textbf{0.08} & \textbf{0.47}\\
\midrule
\multirow{2}{*}{Steam} & Mult-VAE\textsuperscript{PR} & \textbf{0.13} & \textbf{0.01}& 0.15\\
& MOREVAE & \textbf{0.13} & \textbf{0.01} & \textbf{0.18}\\
\midrule
\multirow{2}{*}{Netflix$\geq$4} & Mult-VAE\textsuperscript{PR} & 0.10 & 0.04 &0.34\\
& MOREVAE & \textbf{0.12} & \textbf{0.05} & \textbf{0.66} \\
\bottomrule
\end{tabular}
\end{threeparttable}
\caption{\label{table:vaeRecncyAdded}Comparison of Mult-VAE\textsuperscript{PR} and MOREVAE results obtained on temporally shifted test sets. We report \textbf{R}ecall, \textbf{P}recision, and \textbf{Re}cency at $k=20$.}
\end{table}
We present both the Pareto Fronts obtained during training and the results of the best models on the test sets. Those results were obtained through more intense training than those shown in the previous sections.
The Pareto Fronts were generated by evaluating on the strict cutoff validation sets during training, and the best models were chosen by selecting those with the highest Recall@20 and applying them to the time delayed test sets.
Figure \ref{fig:paretofronts} shows that the multi-objective approach not only dominates the single objective one in terms of recency, but that optimizing for recency also increases the relevance of the recommendations, validating our initial intuition.
The results of the best models over the test sets are shown in Table \ref{table:vaeRecncyAdded}.
The improvements obtained are 18.18\%, 0.00\%, and 20\% for Recall@20; 14.29\%, 0.00\%, and 25.00\% for Precision@20.
The improvements seen in Recency@20 are 104.35\%, 20.00\%, and 94.12\%.
\section{Conclusion}
Following standard offline recommendation evaluations during development, based on random sampling of user-item interactions as held-out data, leads to false confidence when deploying models in real-life scenarios. Previous research generally focused on developing better metrics to reflect real-world performance, but still omitted temporal context. We highlighted the lack of standardization and proposed two temporal evaluation protocols that empirically better approximate real-life conditions.
Our second contribution is a novel recency objective, that can be used to integrate temporal information in existing time-unaware recommenders. We propose to leverage~a multi-objective approach and train models on relevance and recency simultaneously. Experiments on three real-world publicly available datasets showed that our method produced solutions that strictly dominate those obtained with a model trained on a single-objective optimization.
We explored datasets that are frequently used in recommender systems research, all related to digital media content. Digital media content is consumed frequently and generally without much repetition. The importance of recency and capturing transient behavioral trends may not be equivalent in other recommender systems applications, such as grocery or clothes shopping. The influence of temporal dynamics on these sectors is an exciting topic, and we leave it to future academic and commercial research.
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2024-02-18T23:40:24.805Z
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2020-09-22T02:00:08.000Z
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{"paloma_paragraphs":[]}
|
{"arxiv_id":"2009.08978","language":"en","timestamp":1600740008000,"url":"https:\/\/arxiv.org\/abs\/2009.08978","yymm":"2009"}
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proofpile-arXiv_000-10089
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{"provenance":"002.jsonl.gz:10090"}
| null | null |
\section{Introduction}
Over the past few decades, the quasi normal modes (QNMs) of various black holes and exotic compact objects (ECOs) have been studied extensively \cite{Kokkotas:1999bd,Berti:2009kk}. The ringdown in the postmerger phase of a binary coalescence can be described in terms of a superposition of the QNM. This is sensitive to any modification to the near-horizon physics of the compact object that can modify/remove the horizon, as the boundary condition imposed at the horizon for obtaining the QNM spectrum, depends on the nature/presence of the horizon. Precise detection of the QNM by analysing the ringdown spectrum can give us a fair idea about the exact nature of the black hole (or ECO) horizon and test the accuracy of general relativity in the strong gravity regime.
From the ringdown signal, it is possible to probe some of the quantum aspects of a black hole and analyse key signatures of quantum gravity leading to near-horizon modification of the black hole geometry. Such near-horizon modifications of the black hole are usually realised in models (Firewalls \cite{Almheiri:2012rt}, fuzzball geometry \cite{Mathur:2005zp}, gravastars \cite{Mazur:2001fv}, 2-2 holes \cite{Holdom:2016nek}) resolving the information loss paradox or stems from the modified dynamics in the strong gravity regime addressing the dark energy problem \cite{PrescodWeinstein:2009mp}.
Echoes in the ringdown part of the gravitational wave spectrum \cite{Cardoso:2016rao,Cardoso:2016oxy,Abedi:2016hgu,Abedi:2020ujo,Oshita:2020dox} are considered as a signature of quantum modifications to the classical black hole horizon and reports of tentative detection of these echoes, \cite{abedi2018echoes,Abedi:2016hgu,Holdom:2019bdv} makes the subject even more interesting. Attempts to provide a microscopic description of the spectrum of quantum black holes include a quantum multilevel system \cite{Wang:2019rcf,Oshita:2019sat}, quantum corrected black holes in the braneworld scenario \cite{Dey:2020lhq}, interpreting the echo time in terms of the scrambling time \cite{Saraswat:2019npa}, black hole area quantization \cite{Cardoso:2019apo,Coates:2019bun}, or 2-2 holes in asymptotically free Quadratic Quantum Gravity \cite{Holdom:2016nek,Conklin:2017lwb}. Removal of the black hole horizon due quantum effects in the near-horizon region (e.g., through introducing a partially reflective membrane in front of the would-be horizon) would modify the quasi normal modes and the QNM spectrum of these horizonless ECOs are different from that of the classical black holes as a purely ingoing boundary condition at the horizon cannot be imposed for these horizonless compact objects. Since the motivation for near-horizon modifications to the black hole geometry comes from quantum gravitational effects, in principle it must be possible to explain the removal/modification of the event horizon from the near-horizon degrees of freedom within a theory of quantum gravity.
The Kerr/CFT correspondence conjectures that quantum gravity in the near-horizon region of the Kerr black hole is dual to a two-dimensional thermal CFT
\cite{Guica:2008mu,Compere:2012jk}. The correspondence \cite{Guica:2008mu} was originally shown to exist for an extremal Kerr black holes and it relied heavily on the decoupling of the near-horizon region (NHEK) with an exact $SL(2,R)\times U(1)$ symmetry. The $SL(2,R)$ symmetry governed the behaviour of the near-horizon scattering cross-section and it was explicitly shown that the dual two-dimensional chiral CFT gives the same results as the gravity computation.
For non-extremal Kerr black holes, the realisation of the correspondence was less trivial but it was shown that there is an underlying hidden $SL(2,R)\times SL(2,R)$ present in the dynamics of a probe scalar field close to the horizon (in the low frequency limit)\cite{Castro:2010fd}.
Based on this ``hidden conformal symmetry'', and the dual two-dimensional CFT living in the near-horizon region of the Kerr black hole, the black hole entropy as well as the absorption cross-section of the black hole \cite{Castro:2010fd,Haco:2018ske} was computed. It was further shown that the probe scalar field equation can be written as the $SL(2,R)\times SL(2,R)$ Casimir, establishing the conjectured correspondence even further.
In this case the local conformal symmetry in the solution space of the wave equation for propagating fields in the Kerr background was the sufficient condition for obtaining the correct scattering cross-section. In \cite{Haco:2018ske,Haco:2019ggi,Chen:2020nyh} the central charge of the dual CFT was computed using the covariant phase space formalism, making the correspondence for non-extremal Kerr black holes even stronger. In this paper, we look for a plausible way to interpret the near-horizon quantum modifications to a Kerr black hole within this holographic setup.
For a Kerr-like ECO with a partially reflective membrane placed in front of the horizon, a probe scalar field would inherit the same hidden conformal symmetry in the near-horizon region as shown in \cite{Castro:2010fd}.
In this paper, we study the appropriate holographic CFT description that is dual to the microscopic degrees of freedom in the near-horizon region of such an horizonless ECO.
We argue that the near-horizon modifications due to quantum effects can originate as finite size effects in the dual field theory, living on a circle of length $L$ (or an Euclidean torus having periodicities of $L$ and $1/T$). For Kerr-like ECOs the reflectivity of the membrane is
not well understood and it is assumed that an exact quantum gravity computation is required to determine the reflectivity accurately. In this paper, our dual CFT analysis suggests that the reflectivity can be interpreted in terms of a Boltzmann factor which matches with the reflectivity of a quantum horizon as derived in \cite{Oshita:2020dox}.
In the context of AdS/CFT correspondence, attempts were made to understand the bulk geometry dual to a boundary field theory in the strongly coupled regime having a finite volume \cite{Birmingham:2002ph,Solodukhin:2004rv,Solodukhin:2005qy}. Usually, finite-$N$ effects in the boundary theory would correspond to a modification of the semi-classical description of the black hole in the bulk and in particular the black hole horizon is removed (e.g., replaced by a wormhole or a``fractal brick-wall'' \cite{Solodukhin:2005qy}) due to breakdown of the semi-classical physics close to the horizon at finite Planck length\cite{Kabat:2014kfa}. The brick-wall scenario \cite{tHooft:1984kcu} would be a classic example of such non-perturbative near-horizon modifications where a ``brick-wall'' (or Dirichlet wall) is placed few Planck length away from the horizon to regularize and interpret the black hole entropy as the entropy of particles forming a thermal atmosphere outside the black hole horizon. One must consider the finite-$N$ corrections of the boundary theory to match the spectrum of the bulk having such a brick-wall within the context of AdS/CFT \cite{Iizuka:2013kma,Kay:2011np,Solodukhin:2005qy}. Since the exact dual field theory is not known at a microscopic level for the Kerr/CFT correspondence, we do our computation in terms of a generic thermal CFT.
In this context, perturbation of the black hole corresponds to perturbation of the dual CFT state with some relevant operator. Thus, to understand the perturbation on the gravity side with a modified boundary condition at the horizon, we must study the thermal two-point function of certain conformal operators.
Usually for a classical black hole, the dual thermal field theory lives on a toroidal two-manifold and is considered in the high temperature limit so the effective spatial length of the torus becomes infinite. For the computation of the two-point function in the high temperature limit, a cylindrical approximation of the torus is used which in terms of the spatial cycle ($L$) and temporal cycle ($1/T$) of the torus can be written as $L\gg 1/T$ . However, one must note that the cylindrical limit makes the discrete spectrum of the theory continuous as one might expect for an infinite volume theory. As we will see, the QNM spectrum of the Kerr-like ECO is discrete, and hence we must consider the dual field theory on a circle of finite length $L$ to establish the duality between the two descriptions.
We study the finite-size effects of the covering space (for a dual CFT living on an Euclidean torus having periodicities $L$ and $1/T$) to the thermal two-point function in order to understand the spectrum of the horizonless ECO. As given in \cite{Birmingham:2002ph,Solodukhin:2005qy},
such finite-size effects of the dual CFT can be contrasted with the behaviour of finite $N$ field theories in the strong coupling regime dual to supergravity on $AdS_3$.
Following the original derivation for BTZ black holes \cite{Birmingham:2001pj}, we show that the poles of the CFT two-point correlation function on a finite torus match the QNM spectrum of the Kerr-like ECO. The absorption cross-section is also correctly predicted by the dual CFT when the length of the circle on which the dual theory lives, is related to the distance of the reflective membrane from the (would-be) horizon, or equivalently the ``echo time-delay"\cite{Wang:2019rcf} for the Kerr black hole/ECO.
The paper is organized as follows: In section \ref{ECO} we review some of the basic aspects of a Kerr-like ECO and demonstrate how the absorption cross-section and the quasi normal modes differ from the Kerr black hole with a classical event horizon.
In section \ref{hidden}, we discuss the hidden conformal symmetry associated with a probe scalar field in Kerr spacetime and establish the conformal coordinates based on which the Kerr/CFT correspondence is conjectured. We demonstrate how the dual field theory living on a finite circle captures the necessary near-horizon modification of a Kerr-like ECO, finally predicting the QNM spectrum and absorption cross-section from the dual field theory which matches with the direct bulk computations. In section \ref{observation} we comment on the observational aspects of our findings. Finally, Section \ref{conclusion} summarizes our results and provides some open questions for future study.
\section{Kerr-like Exotic compact object (ECO)}
\label{ECO}
We consider a model of exotic compact object (ECO) where the exterior spacetime is described by the Kerr metric but the near-horizon geometry is modified due to the presence of quantum structures originating from quantum gravitational effects \cite{Maggio:2017ivp,Maggio:2018ivz}. Usually such ECOs do not have an event horizon but a partially reflective membrane is placed slightly outside the usual position of the event horizon for the stability of these compact objects. For a rotating Kerr-like compact object with mass $M$ and angular momentum $J=aM$, the metric in the Boyer-Lindquist coordinate can be written as
\begin{eqnarray} \label{kerr}
ds^2 =-\big(1-{2Mr \over \rho^2}\big)dt^2+\bigg(r^2+a^2+{2a^2Mr \sin^2\theta \over \rho^2}\bigg)\sin^2\theta d\phi^2
-{4aMr \sin^2\theta \over \rho^2}d\phi dt
+{\rho^2 \over \Delta}dr^2+\rho^2 d\theta^2,
\end{eqnarray}
where we defined \begin{align}
~~~~~~\rho^2=r^2+a^2\cos^2\theta, ~~~~~~\Delta=r^2+a^2-2Mr=(r-r_+)(r-r_-).
\end{align}
The position of the classical black hole horizons and the angular velocity of the horizon is given as
\begin{align}
r_{\pm}=M\pm \sqrt{M^2-a^2},\qquad \Omega_H={a\over 2Mr_+}.
\end{align}In this section, we study the QNM spectrum and the absorption cross-section of the Kerr-like ECO with a modified boundary condition at the horizon. Later we will reproduce these results from a dual CFT analysis by establishing a correspondence between quantum gravity in the near-horizon region and a thermal two-dimensional CFT. We study a massless field in the Kerr background and the Klein-Gordon equation for the scalar field is given as:
\begin{align} \label{scalareq}
\frac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g}g^{\mu \nu }\partial_{\nu}\Phi)=0
\end{align}
We will assume excitation wavelengths larger than the black hole radius i.e. $\omega M\ll 1$ to simplify the wave equation, which can be solved analytically by using the method of asymptotic solution matching, where the background spacetime is divided into a ``near-region"($\omega r\ll 1$), a ``far-region"($r\gg M$) and a intermediate matching region($M\ll r\ll 1/\omega$). As shown in Appendix \ref{app_a}, we solve the wave equation in near/far-region and match the solutions in the intermediate region($M\ll r_{match}\ll 1/\omega$) at some arbitrary position( given as ``r$_{match}$"). %
\subsection{Scattering }
\label{flux}
We use the approximate wave function obtained in Appendix \ref{app_a} to compute the absorption cross-section of a Kerr-like ECO and analyze the effects due to the presence of the near-horizon quantum structure. We assume, at $(r_{h}+\epsilon)$ there is a reflective membrane with reflectivity $\mathcal{R}$, where $\epsilon$ is usually assumed to be of the order of the Planck length. This reflective membrane can be seen as an effective description of the quantum corrections at the horizon such as a firewall or as seen in the fuzzball scenario. The scattering cross-section of the ECO is sensitive to the near-horizon geometry as presence of such reflective membranes would modify the boundary condition at the horizon.
\begin{figure}[h]
\includegraphics[width=11cm]{ECO_potential.png}
\caption[] {{The difference in the near-horizon region between a black hole and ECO is shown in the above schematic diagram. Due to the presence of a quantum structure at $r^*_0$ the near-horizon region would have an additional wave reflected by the membrane placed in front of the would-be horizon. This wave would eventually tunnel through the potential barrier and reach the asymptotic region after a time delay.}}\label{eco}
\end{figure}
For a scalar perturbation in the Kerr background the asymptotic behaviour of the waveform would be given as
\begin{align}\label{asymp_soln}
R\sim
\begin{cases}
C {e^{-i\omega r^*}\over r}+D{e^{i\omega r^*}\over r }& \text{for}\ \quad r^*\to \infty \\
Ae^{-i\tilde\omega r^*}+Be^{i\tilde\omega r^*} & \text{for} \quad r^*\to -\infty,
\end{cases}
\end{align}
where $\tilde \omega$ is the horizon-frame frequency and the asymptotic amplitudes ($A,B,C,D$) are shown in Fig (\ref{eco}). The reflectivity of the membrane can be defined in terms of these asymptotic amplitudes as
\begin{align} \label{ref}
\mathcal{R} e^{i\pi\delta}\equiv {B\over A}z_0^{2i\sigma},
\end{align}
where $\delta$ is a phase determined by the quantum properties of the ECO, $\sigma={\omega_H-m\Omega\over4\pi T_H} $ and $z_0$ corresponds to a particular point outside the horizon, where the reflective membrane is placed, in terms of the coordinate defined as $z={r-r_+\over r-r_-}$. The conserved flux associated to the radial wave equation \eqref{wave_eq_hd} for the radial wavefunction $R$ is given as
\begin{align}
F=-i2\pi\big( R^*\Delta\partial_rR-R\Delta \partial_rR^*\big) .
\end{align}
Using this expression we can calculate the ingoing/outgoing flux at the horizon ($F^{in}_{r\to r_+}, F^{out}_{r\to r_+} $) and the ingoing flux from infinity ($F^{in}_\infty$),
to obtain the absorption cross-section as
\begin{align} \label{eco_abs}
\sigma_{abs}={F^{in}_{r\to r_+}+F^{out}_{r\to r_+}\over F^{in}_{r\to \infty}}=4\pi\sigma(r_+-r_-){1-\mathcal{|R|}^2\over |\alpha/A|^2+|\beta/A|^2}.
\end{align}
To arrive at the last expression we have used the reflectivity of the ECO \eqref{ref} to write the asymptotic amplitude $B$ in terms of $A$ and $\alpha, \beta$ can be expressed as some linear combination of $C,D$ as given in Appendix \ref{app_a}. As we are performing the calculation in the low frequency limit, $|\beta/A|$ is suppressed compared to $|\alpha/A|$ due to the positive power of $\omega$ in the former. Using $|\alpha/A|$ as given in \eqref{amplitude_rel} we can write the absorption cross-section, \eqref{eco_abs}, as
\begin{align} \label{kerrabs}
&\sigma_{abs}\sim
\nonumber\\
&\omega^{2l+1}\sinh\bigg({4\pi Mr_+\over r_+-r_-}(\omega-m\Omega_H)\bigg)
\left|\Gamma\bigg(1+l-i2M\omega\bigg)\Gamma\bigg(1+l-i{4M^2\over r_+-r_-}\omega+i{2a\over r_+-r_-} m\bigg)\right|^2
{1-|\mathcal{R}|^2\over \left|1-\mathcal{R}e^{-2ir_0^*(\omega_H-m\Omega)+i\delta}\right|^2}.
\end{align}
As an example, Figure (\ref{fig1}a) shows $\sigma_{abs}$ for an ECO with different constant values of ${\cal R}$ (with ${\cal R} =0$, corresponding to a classical black hole).
However, like any physical system, the reflectivity can change with frequency, depending on its microscopic structure and energy levels. The Boltzmann frequency-dependent reflectivity for a quantum black hole was first derived in \cite{Oshita:2019sat,Wang:2019rcf} and is given by
\begin{align} \label{ref_B}
\mathcal{R}\sim e^{-|\omega-m\Omega_H|/2T_{QH}},
\end{align}
where $T_{QH}$ is defined as the ``Quantum Horizon temperature'' \cite{Oshita:2020dox} and as shown, it is expected to be comparable to the Hawking temperature:
\begin{align}
T_H={1\over 8\pi}{r_+-r_-\over Mr_+}.
\end{align}Thus we can write $T_{QH}=\alpha T_H$ , where $\alpha$ is the proportionality constant that depends on the dispersion and dissipation effects in graviton propagation \cite{Oshita:2018fqu}. In order to avoid instability of the ergoregion (for all BH spins) one must satisfy the condition, $\alpha <1.86 $ \cite{Oshita:2020dox}.
In Fig.(\ref{fig1})b we have plotted the absorption cross-section for different positions of the reflective membrane (using the reflectivity defined in Equation \ref{ref_B}) in front of the horizon.
\begin{figure}[h]
\begin{minipage}{14cm}
\includegraphics[width=14cm]{abs.png}
\end{minipage}
%
\begin{minipage}{14cm}
\includegraphics[width=14cm]{abs_wall_B.png}
\end{minipage}
\caption{ The figure on the top shows the absorption cross-section for different values of the reflectivity keeping the position of the membrane fixed at $r_0^*=50$. For the bottom figure we used a frequency dependent reflectivity (Boltzmann reflectivity \cite{Oshita:2019sat}) and plotted the absorption cross-section for different positions of the wall. The dotted line in the bottom figure corresponds to the classical black hole absorption cross-section with zero reflectivity. For this plot we assumed $\alpha=1$. For both the plots we used $a=0.67, m=2, l=2.$ }\label{fig1}
\end{figure}
For a Kerr black hole, the low frequency absorption cross-section is negative due to superradiance \cite{Brito:2015oca} $(\omega<m\Omega_H)$ but the novel feature of the absorption cross-section of the ECO's are the oscillatory features, superposed on top of the classical cross-section in Fig. (\ref{fig1}). These oscillations corresponds to resonances at the ECO quasi-normal frequencies.
\subsection{Quasi normal modes}
To obtain the quasi normal modes of a classical black hole, usually an ingoing boundary condition is imposed at the horizon and the boundary condition at asymptotic infinity would be set by the fact that there is no incoming wave from infinity (in Fig\eqref{eco} this would correspond to setting $C=0$). There are various approaches used in the literature to obtain the QNM spectrum of a black hole/ECO. Analytically it is difficult to obtain the QNM of ECOs without making assumptions as the wave equation governing the perturbations are difficult to solve, depending on the model of the ECO. Numerical techniques are used to solve the wave equation or one can obtain the QNM spectrum from the poles of the Green's function expressed in terms of a transfer function\cite{Mark:2017dnq}.
As demonstrated in Appendix \ref{app_a} we use the low frequency approximation to solve the wave function and obtain the QNM spectrum.
Using the frequency dependent Boltzmann reflectivity \eqref{ref_B}, in \eqref{qnf} the QNM spectrum is given as\cite{Wang:2019rcf}
\begin{align} \label{qnf_B}
\omega_{n}-m\Omega_H\sim{\pi (2n+1+\delta)\over 2 r^*_0}\left[1-{i\times \text{sgn}(2n+1+\delta)\over 4r^*_0\alpha T_H}\right].
\end{align}
This matches exactly with the quasi normal frequencies obtained in \cite{Oshita:2019sat}, assuming the Boltzmann reflectivity for the near-horizon membrane in Kerr spacetime.
One must note that one major difference between the QNM spectra obtained for the ECOs and the black holes is that the quality factor, $Q \equiv \Re\omega_n/\Im\omega_n $ for ECO QNM is parametrically enhanced, and approaches infinity in the continuum limit $r^*_0 T_H \to \infty$. In contrast, $Q\lesssim 1$ for classical black hole QNMs.
\section{Towards a Holographic Interpretation of Kerr-like ECOs}
\label{hidden}
\subsection{Hidden conformal symmetry of classical Kerr spacetime}
Let us start by reviewing the current progress in the Kerr/CFT correspondence for non-extremal Kerr black holes.
Castro, Maloney, and Strominger have found that Kerr black holes with generic mass and spin inherit a hidden local conformal symmetry acting on the low frequency modes \cite{Castro:2010fd}. When solving for the wave equation using the method of the asymptotic solution matching, we saw that the final solution does not depend on $r_{match}$ and this indicates that the solution in the near region must inherit some special symmetry keeping it invariant under transformations of $r_{match}$.
The symmetry becomes evident in the {\it near-horizon} region of the phase space defined as
$
\omega(r-r_+) \ll 1,
$
where $r$ is the Boyer-Lindquist radial coordinate and $r_+$ is the radius of the outer horizon. The near-horizon wave function for a scalar field in Kerr background is given by hypergeometric functions \eqref{sol_near}, one can understand the emergent conformal symmetry from the fact that these hypergeometric functions fall into the $SL(2,R)$ representation. The conformal symmetry is called hidden in this case as it acts locally on the solution space and it is globally broken to $U(1) \times U(1)$ by the periodic identification of the azimuthal angle $\phi$.
To understand the hidden conformal symmetry explicitly, one can introduce a set of
conformal coordinates \cite{Castro:2010fd,Haco:2018ske,Chen:2020nyh} (similar to the coordinate transformation that turns Poincare $AdS_3$ into BTZ black hole \cite{KeskiVakkuri:1998nw,Maldacena:1998bw}):
\begin{align}
\label{cftcoordinate}
& w^+= \sqrt{r-r_+ \over r-r_-}e^{{2\pi T_R}\phi},
\\
&w^-= \sqrt{r-r_+ \over r-r_-}e^{{2\pi T_L}\phi-{t \over 2M}},
\\
&y= \sqrt{r_+ -r_-\over r-r_-}e^{{\pi(T_R+T_L)}\phi-{t \over 4M}},
\end{align}
where we defined a right and left temperature as
\begin{align}
T_R={r_+-r_-\over4\pi a},~~~~~~T_L={r_++r_-\over4\pi a}.
\end{align}
In terms of the conformal coordinates, the past horizon and the future horizons are at $w^\pm=0$, respectively.
The inverse transformation is given by
\begin{align} \phi&={1 \over 4\pi T_R}\ln{w^+(w^+w^-+y^2) \over w^-},
\\
r&=r_++4\pi a T_R{w^+w^-\over y^2},
\\
t&={M(T_R+T_L) \over T_R}\ln{w^+\over w^-}+{M(T_L-T_R)\over T_R}\ln (w^+w^-+y^2).
\end{align}
To leading order around the bifurcation surface, it is possible to obtain a metric in terms of the conformal coordinates from \eqref{kerr} as \cite{Haco:2018ske}
\begin{align}\label{conformalkerr}
ds^2={4\rho_+^2\over y^2}dw_+dw_-+{16J^2sin^2\theta\over y^2 \rho_+^2}dy^2+\rho_+^2d\theta^2+...
\end{align}
Using these coordinate transformations one can express the scalar wave equation in the {\it near-horizon} region as the $SL(2,R)$ Casimir with conformal weights \cite{Castro:2010fd, Compere:2012jk}
\begin{align}
(h_L,h_R)=(l,l).
\end{align}
As mentioned earlier the $SL(2,R)_L\times SL(2,R)_R$ symmetry is spontaneously broken by the periodic identification of the angular coordinate
\begin{align}\label{2pi}
\phi \to \phi+2\pi.
\end{align}
For the given periodicity, the conformal coordinates would be identified as
\begin{align}
\label{idn} w^+ \sim e^{4\pi^2 T_R}w^+, ~~ w^- \sim e^{4\pi^2 T_L}w^-, ~~
y\sim e^{2\pi^2 (T_R+T_L)}y.\end{align}
From \eqref{cftcoordinate}, at fixed radial distance $r$ we can write the relation between the conformal coordinates and Boyer-Lindquist coordinate as
\begin{align} \label{cftrindler}
w^{\pm}=e^{\pm t_{R,L}}.
\end{align}
This relation looks analogous to the relation that identifies the Minkowski coordinates $(w^{\pm})$ to the Rindler coordinates$(t_{R,L})$ where we defined
\begin{align} \label{monocoordinate}
&t_R=2\pi T_R\phi,
\nonumber\\
&t_L={t\over 2M}-2\pi T_L\phi.
\end{align}
The periodic identification of the angular coordinate $\phi$ in \eqref{monocoordinate} gives the periodicity of these coordinates as
\begin{align} \label{cor_periodicity}
(t_L,t_R)\sim (t_L,t_R)+4 \pi^2(-T_L,T_R).
\end{align}
The frequencies $(\omega_L,\omega_R)$ associated with the Killing vectors $(i\partial_{t_L},i\partial _{t_R})$ are conjugate to $(t_L,t_R)$. We can write the relation between $(\omega,m)$, which are eigenvalues of the operators $(i\partial _t,-i\partial _{\phi})$, and $(\omega_L,\omega_R)$ through \cite{Castro:2013kea}
\begin{align}
e^{-i \omega t+im\phi}=e^{-i \omega_Lt_L-i\omega_Rt_R}.
\end{align}
This relation between the frequencies along with \eqref{monocoordinate} would explicitly give the left/right frequencies as
\begin{align} \label{qnf_matching0}
&\omega_L=2M\omega,
\\
& \omega_R={2M^2\over \sqrt{M^2-a^2}}\omega-{a\over \sqrt{M^2-a^2}}m.
\end{align}
As given in \cite{Castro:2010fd}, another way of writing the relation between the left, right frequencies ($\omega_L,\omega_R$) and $(\omega,m)$ of Kerr spacetime having an entropy $S_{BH}$, Hawking temperature $T_H $, is by taking a thermodynamic route using the first law of thermodynamics, $T_H\delta S_{BH}=\delta M-\Omega \delta J $. Identifying $\omega=\delta M$ and $m=\delta J$, we need to consider the conjugate charges ($\delta E_R, \delta E_L$), following the relation
\begin{align}
\delta S_{BH}={\delta E_R \over T_R}+{\delta E_L \over T_L},
\end{align}to show the relation between $(\omega,m)$ and $(\delta E_R, \delta E_L)$ as\footnote{One must note that $\delta E_{L,R}=\tilde \omega_{L,R}$ is conjugate to the re-scaled coordinate $\tilde t_{L,R}=t_{L,R}/2\pi T_{L,R}$ (in terms of which we will later define the torus coordinates \eqref{torus_coordinate} on which the CFT lives), thus drawing the equivalence between this identification and \eqref{qnf_matching0}.}
\begin{align} \label{qnf_matching}
&\delta E_L=\tilde\omega_L= \omega_L 2\pi T_L ={2M^3\over J}\omega,
\nonumber
\\
&\delta E_R=\tilde\omega_R= \omega_R 2\pi T_R ={2M^3\over J}\omega-m.
\end{align}
In the context of Kerr/CFT correspondence this identification between the frequencies were used to show the consistency between the absorption cross-section of the dual CFT (calculated from the thermal two-point function) and the calculation performed in Kerr spacetime \cite{Castro:2010fd,Chen:2010xu}.
\subsection{Dual CFT of the Kerr-like ECO}
Now that we have reviewed the current understanding of the classical Kerr/CFT correspondence, we shall derive a CFT dual to a Kerr-like ECO.
As we have discussed in Section \ref{ECO},
for a Kerr-like ECO the horizon is modified due to quantum gravitational effects in the near-horizon region of a black hole, while the exterior spacetime is well approximated by the Kerr metric. The equations governing the evolution of the perturbation in the ECO background is exactly the same as in classical Kerr spacetime; the only difference is in the boundary condition at the horizon. Hence, in the case of the ECO, the solution of the near-horizon scalar perturbation is given in terms of hypergeometric functions as well (hinting at the hidden conformal symmetry). One can define a set of vector fields
(similar to the classical Kerr case \cite{Castro:2010fd}) falling in the $SL(2,R)$ representation and the Casimir would be the same as the near-horizon scalar perturbation equation, allowing us to identify the conformal dimension of the CFT.
Usually, quantum gravity modifications to the bulk geometry can be understood as non-perturbative effects in the dual field theory, finite $N$ and/or finite size effects in the usual AdS/CFT terminology \cite{Kabat:2014kfa,Birmingham:2002ph,Solodukhin:2005qy}. We can then use the fact that the dual CFT lives in a space of finite volume to reproduce the discrete QNM spectrum of the Kerr-like ECO. We can see similar finite $N$/finite size effects in the holographic interpretation of the brick wall \cite{Iizuka:2013kma,Solodukhin:2005qy}; the near-horizon cut-off can be understood in terms of finite $N$ effects in the dual CFT in order to make the free energy and the entropy finite (on either side of the duality).
In order to compute the QNM spectrum and the absorption cross-section from the dual CFT, we start with the thermal two-point function of a CFT living on an Euclidean torus having a spatial cycle of length $L$ and an temporal cycle of length $1/T$. Unlike in the limit of, $L\gg 1/T$ or $L\ll 1/T$ where the torus decompactifies to a cylinder, it is usually difficult to determine the two-point function on a torus keeping both the lengths finite, as conformal symmetry is not enough to determine the universal form of the correlator. As shown by \cite{Maldacena:2001kr,Kleban:2004rx,Birmingham:2002ph,Solodukhin:2005qy}, some progress can be made in special cases such as the supersymmetric CFT in the strong coupling regime dual to string theory in $AdS_3$ which describes the low energy excitation of D1-D5 branes \cite{Aharony:1999ti}.
In the usual description of Kerr/CFT correspondence, the cylinder approximation of the torus is considered based on the assumption $L\gg 1/T$ and thus the spatial direction effectively decompactifies \cite{Castro:2010fd,Maldacena:1997ih}. In the context of $AdS_3/CFT_2$, we have seen that a similar approximation is done for the BTZ black holes whereas, for its T-dual, thermal $AdS_3$ one can assume $1/T\gg L$. However, to include the finite size/$N$ effects, we can no longer use the $L\gg1/T$ approximation. Therefore, for computation of physical quantities such as the absorption cross-section, the precise doubly-periodic two-point function on the torus is needed. The latter is computed in Appendix \ref{2point}.
One way to put the dual CFT on a circle of finite length, in the context of Kerr CFT, is by considering a rescaling of the temperatures$(T_R,T_L)$, that appear in the conformal coordinates \eqref{cftcoordinate}, by some constant.
One can verify the consistency of the Cardy entropy on rescaling of the temperatures in the conformal coordinates from the recent arguments given in \cite{Chen:2020nyh}. We modify the size of the covering space, where the CFT lives, by introducing a factor of $L$ ($L$ is an integer) in the periodicity of the azimuthal angle that appears in the torus coordinates. As we will see, $L$ will correspond to the length of the circle on which the CFT lives, thus allowing us to study the finite size/N effects. We must warn the reader that the gravity/CFT duality is not well defined in the finite size/N limit and hence, to a certain extent, we have to guess how the finite size/N effects of the holographic CFT reflects on the gravity side. Our objective is to verify if the QNM spectrum (calculated from the poles of two-point function of a CFT living on a torus) matches with the spectrum obtained on the gravity side, once the black hole horizon is removed and replaced by a partially reflective membrane in front of the would-be horizon.
For the ECO the usual periodic identification of the azimuthal coordinate is there, as given in \eqref{2pi}. We can combine this azimuthal periodicity (times an integer multiple of $L$) with thermal periodicity of imaginary time to get
\begin{align}\label{eco_period}
&\phi \to \phi + 2L \pi +i\Omega_H/T_H,
\\
&t=t+i/T_H.
\end{align}
Now, under the periodic identifications \eqref{eco_period} for the coordinates of the torus \eqref{monocoordinate} we get
\begin{align}\label{cor_periodicity_new}
( t_L, t_R)\sim ( t_L, t_R)+(-4 \pi^2T_LL-i(2\pi {T}_L\Omega_H/T_H-1/2MT_H),4\pi^2{T}_RL+i2\pi {T}_R\Omega_H/T_H).
\end{align}
\subsection{Quasi normal modes spectrum from the dual CFT}
The postmerger ringdown phase after a binary black hole coalescence is described in terms of QNM and can carry information about the near-horizon quantum structures of an ECO. In the context of AdS/CFT, it was shown in \cite{Birmingham:2001pj} that the black hole QNM spectrum can be obtained from the poles of the retarded CFT correlation function. Assuming that quantum gravity in the near-horizon region of a Kerr black hole is dual to a two-dimensional thermal CFT, as conjectured by the Kerr/CFT correspondence, the consistency between the gravity results and CFT computations of the QNM was shown in \cite{Chen:2010xu,Chen:2010ni}.
Here, we conjecture that quantum gravity living in the near-horizon region of a Kerr-like ECO is dual to a two-dimensional CFT that lives on a circle of finite length $L$, where the CFT coordinates are defined by the relation \eqref{monocoordinate} having periodicities as given in \eqref{cor_periodicity_new}. To obtain the QNM spectrum from the poles of the retarded CFT correlator, we need to perform the Fourier transform of the retarded correlation function to momentum space. It is complicated to perform this computation due to the presence of the $\Theta-$function in the retarded two-point correlation function. Instead, we perform the Fourier transform of the two-point function given in
\eqref{2pointtorus} and look at the poles lying in the lower half-plane as these would match with the poles of the retarded correlation function. The Fourier transform of the two-point function is given as \footnote{ The infinite sum appears in front of the two-point function as we are implementing the method of images. To describe an ECO/black hole in the dual CFT one must take into account the correct periodic identification as given in \eqref{cor_periodicity_new} and hence we are shifting the periodicity by an integer multiples and summing over the images }
\begin{align}
&\bar G(\omega_L,\omega_R)=\int d t_R \,d t_L\,e^{-i\omega_R t_R}e^{-i\omega_L t_L}\langle\mathcal{O}( t_R, t_L)\mathcal{O}(0,0)\rangle_{torus}
\nonumber\\
&=\int d t_R \,d t_L \,e^{-i\omega_R t_R}e^{-i\omega_L t_L} \sum_{p\in \mathbb{Z}}{(\pi T_R)^{2h_R}(\pi T_L)^{2h_L} \over[\sinh({ t_R\over 2}+p(2\pi ^2LT_R+i\pi T_R{\Omega_H\over T_H}))]^{2h_R}[\sinh({ t_L\over 2}+p(2\pi^2 L T_L-i\pi T_L{\Omega_H\over T_H}+{i\over 4MT_H}))]^{2h_L}}
\nonumber\\
&=\int d t_R \,d t_L \,e^{-i\omega_R t_R}e^{-i\omega_L t_L} \sum_{p\in \mathbb{Z}}{(\pi T_R)^{2h_R}(\pi T_L)^{2h_L} \over[\sinh[\pi T_R({ t_R\over 2\pi T_R}+p(2 \pi L +i/T_R))]]^{2h_R}[\sinh[\pi T_L({ t_L\over 2\pi T_L}+p(2 \pi L+i/T_L ))]^{2h_L}}. \label{fourier2pointtorus2}
\end{align}
The CFT two-point function derived here based on the torus coordinate along with their periodicities \eqref{cor_periodicity_new} matches with the generic two-point function of a CFT living on a Euclidean torus as given in \eqref{2pointtorus} if we choose the group parameters as $(\mathbf{a}=1,\mathbf{b}=1,\mathbf{c}=-1,\mathbf{d}=0)$. However,the CFT two-point function can be written in a more general way by keeping the value of the group parameter $\mathbf{a}$ arbitrary.
Without fixing the value of $\mathbf{a}$, the two-point function \eqref{fourier2pointtorus2} is now modified as
\begin{align} \label{fourier2pointtorusA}
&\bar G(\omega_L,\omega_R)=
\nonumber\\
&\int d t_R \,d t_L \,e^{-i\omega_R t_R}e^{-i\omega_L t_L} \sum_{p\in \mathbb{Z}}{(\pi T_R)^{2h_R}(\pi T_L)^{2h_L} \over[\sinh[\pi T_R({ t_R\over 2\pi T_R}+p(2 \pi L +i\mathbf{a}/T_R))]]^{2h_R}[\sinh[\pi T_L({ t_L\over 2\pi T_L}+p(2 \pi L+i\mathbf{a}/T_L ))]^{2h_L}}.
\end{align}
In order to do the Fourier transform in a convenient way we perform a coordinate transformation, defining the torus coordinates $(\tilde t_{R,L})$ as
\begin{align}\label{torus_coordinate}
&\tilde t_{R}= t_{R}/2\pi T_R+p(2\pi L+i\mathbf{a} /T_R).
\nonumber\\
&\tilde t_{L}= t_{L}/2\pi T_L+p(2\pi L +i\mathbf{a} /T_L).
\end{align}
These newly defined coordinates along with \eqref{fourier2pointtorusA} would give us
\begin{align} \label{2pointtorusk}
&\bar G(\tilde \omega_L,\tilde \omega_R)\sim
\nonumber\\
&\sum_{p\in \mathbb{Z}}
e^{ip (2 \pi \tilde\omega_LL+2 \pi \tilde\omega_RL+i\mathbf{a} {\tilde\omega_R\over T_R}+i\mathbf{a}{\tilde\omega_L\over T_L})}
\int d\tilde t_{R} \,d \tilde t_{L} \,e^{-i\tilde\omega_R \tilde t_{R}}e^{-i\tilde\omega_L \tilde t_{L}} {(\pi T_R)^{2h_R}(\pi T_L)^{2h_L} \over[\sinh(\pi T_R\tilde t_{R})]^{2h_R}[\sinh(\pi T_L\tilde t_{L})]^{2h_L}}
\nonumber\\
&\propto
T_{L}^{2h_L-1}T_{R}^{2h_R-1}e^{-{\tilde\omega_L\over 2T_L}-{\tilde\omega_R\over 2T_R}}\left|\Gamma\bigg(h_R+i{\tilde\omega_R\over2\pi T_R}\bigg)\Gamma\bigg(h_L+i{\tilde\omega_L\over2\pi T_L}\bigg)\right|^2\nonumber\\&
\hspace{5cm}\times\bigg[
{1\over 1-e^{i2\pi L(\tilde\omega_R+\tilde\omega_L)-\mathbf{a}\left|{\tilde \omega_R\over T_R}+{\tilde \omega_L\over T_L}\right|}}
-{1 \over 1-e^{i2\pi L(\tilde\omega_R+\tilde\omega_L)+\mathbf{a}\left|{\tilde \omega_R\over T_R}+{\tilde \omega_L\over T_L}\right|}}\bigg]
.
\end{align}
The CFT two-point function in the momentum space has two set of poles coming from the exponential part of \eqref{2pointtorusk} lying in the upper and lower half of the $\omega$ plane. As we discussed above, the QNM spectrum is given by the poles of the retarded correlation function, the poles of \eqref{2pointtorusk} in the lower half plane are the ones relevant for us and these are given as\footnote{There are other poles coming from the singularities of the Gamma function and it can be shown that these poles match with the usual Kerr black hole QNM spectrum obtained by imposing an ingoing boundary condition at the horizon.}
\begin{align}\label{torusqnf}
2\pi L(\tilde \omega_L+\tilde\omega_R)+i\mathbf{a}\bigg|{\tilde \omega_R\over T_R}+{\tilde \omega_L\over T_L}\bigg|=2\pi n.
\end{align}
Using the relation between the CFT frequencies and the ECO frequencies as given in \eqref{qnf_matching} we get
\begin{align} \label{QNM_CFT}
&{4M^2\over a}\omega-m+i\mathbf{a}\bigg({1\over 2\pi L T_H}|\omega-m\Omega_H|\bigg)={\pi\over 2\pi L}2n,
\nonumber\\
& \implies \omega-m\Omega_H\simeq{a\over 8M^2 L}(2n+1+\delta)\bigg[1-i\mathbf{a}{a\times ~\text{sgn}[2n+1+\delta]\over 8M^2\pi LT_H }\bigg].
\end{align}
where we have defined the quantity $\delta$ as
\begin{align}\label{phase}
\delta=-2mLr_-/ r_+-1.
\end{align}
For the Kerr-like ECO the QNM spectrum is calculated in the low frequency approximation as given in \eqref{qnf_B} considering a Boltzmann reflectivity of the near-horizon membrane \cite{Wang:2019rcf}.
We notice that {\it both} the real and imaginary parts of the quasi normal frequency for Boltzmann ECOs \eqref{qnf_B} would match with the CFT result if we take, $\mathbf{a}=1/2\alpha$ and the length of the torus as
\begin{align} \label{echotimeL}
L={a|r^*_0|\over 4\pi M^2}.
\end{align}
Comparing \eqref{QNM_CFT} with \eqref{qnf} we can explicitly read off the reflectivity of the membrane as
\begin{align}
\mathcal{R}= e^{-|\omega-m\Omega_H|\mathbf{a}/T_H},
\end{align}
which is identical to the generalized Boltzmann reflectivity introduced in \cite{Oshita:2020dox}, reducing to the standard Boltzmann for ${\bf a} =1/2$ \cite{Oshita:2019sat}.
Furthermore, from \eqref{phase} we get the phase of reflection as defined in \eqref{ref}. The phase can depend on the frequency but as we are working in the low frequency limit such frequency dependence is not evident in our calculation and we determine $\delta$ as a constant.
\subsection{CFT interpretation of Absorption probability}
In the previous section, we saw that the identification \eqref{echotimeL} makes the QNM of the ECO consistent with the dual CFT computation. The other important check about the accuracy of this identification is to compute the absorption cross-section from the CFT two-point function and see if it can determine the near-horizon contribution to the scattering cross-section correctly, as predicted by the gravity result in \eqref{kerrabs} (Fig (\ref{fig1})). The low frequency absorption cross-section for a massless scalar in Kerr spacetime is a well known result \cite{Maldacena:1997ih}. In \cite{Castro:2010fd,Maldacena:1997ih,Bredberg:2009pv} the absorption cross-section was computed for a scalar field in Kerr background with an ingoing boundary condition at the horizon and its equivalence with the CFT computation was shown.
In this section we will compute the absorption cross-section from the CFT two-point function given in \eqref{2pointtorus}. We again consider the dual field theory on a circle of length $L$ and study the finite size effects of the boundary theory on the absorption cross-section.
Given the two-point function of a two-dimensional CFT, $G( \tilde t_R, \tilde t_L)=\langle\mathcal{O}( \tilde t_R, \tilde t_L)\mathcal{O}(0,0)\rangle$, having right and left moving coordinates $( \tilde t_R, \tilde t_L)$, the absorption cross-section can be defined using Fermi's golden rule (to leading order in perturbation theory) as
\begin{align} \label{cft_abs}
\sigma_{abs} \sim
\int d t_R ~d t_L e^{-i \omega_R t_R-i\omega_L t_L}[G( t_R-i\epsilon, t_L-i\epsilon)-G( t_R+i\epsilon, t_L+i\epsilon)].
\end{align}
We perform the integral in the same way as we obtained the momentum space two-point function in \eqref{2pointtorusk}. In this case the $\pm i\epsilon$ prescription determined which poles contribute while performing the integral.
The absorption cross-section is given as:\footnote{Following \cite{Cvetic:2009jn} we can fix the prefactor on the right side of \eqref{momabs} based on the fact that the emission of any quanta in the dual CFT takes place due to an interaction, between the bulk modes and the CFT operator $\mathcal{O}^{(h_R,h_L)}$, given as
$
\phi_{bulk}\mathcal{O}^{(h_R,h_L)}.
$ It was argued that due to the presence of derivatives acting on the outgoing wave function in the general form of the coupling and the normalization of the outgoing modes, the prefactor would be $(\omega)^{2l-1}$.}
\begin{align} \label{momabs}
&\sigma_{abs} \sim
\nonumber\\
&T_L^{2h_L-1} T_R^{2h_R-1}(\omega)^{2l-1}\text{sinh}\bigg({\tilde\omega_R \over 2T_R} +{\tilde\omega_L \over 2T_L}\bigg)
\left|\Gamma\bigg(h_R+i{\tilde\omega_R\over2\pi T_R}\bigg)\right|^2\left|\Gamma\bigg(h_L+i{\tilde\omega_L\over2\pi T_L}\bigg)\right|^2
{1-e^{-2\mathbf{a}\left|{\tilde\omega_R \over T_R} +{\tilde\omega_L \over T_L}\right|}\over \left|1-e^{i2\pi L(\tilde\omega_R+\tilde\omega_L)-\mathbf{a}\left|{\tilde \omega_R\over T_R}+{\tilde \omega_L\over T_L}\right|}\right|^2}
\end{align}
In section \ref{flux}, we obtained the low frequency scattering cross-section by using a matching procedure where we divided the Kerr spacetime into a near-horizon region $(\omega r\ll 1)$, far region $(r\gg M)$ and an overlapping matching region $(1/\omega\gg r\gg M)$. Using the relation between the CFT frequencies $(\tilde \omega_R,\tilde \omega_L)$ and $(\omega,m)$, as given in \eqref{qnf_matching}, along with \eqref{echotimeL} and \eqref{phase} we get the absorption cross-section as given in \eqref{kerrabs}. Plotting \eqref{momabs} for the value of $L$ given by the condition \eqref{echotimeL}, corresponding to a particular position of the membrane, we can see the correspondence with the gravity result in a better way.
\begin{figure}[h]
\includegraphics[width=15cm]{abs_cft_compare_B.png}
\caption[] {{Comparison of the CFT absorption cross-section corresponding to $r^*_0=50$ for the position of the reflective membrane in front of the ECO. For the gravity computation we choose $\alpha=1$ so that $T_{QH}\sim T_{H}$. For the CFT cross-section we see that $\mathbf{a}=1/2$ exactly matches with the gravity computation as we expected. For $\mathbf{a}=1$ we can infer that the reflectivity is suppressed. Once again we assumed $a=0.67, m=2, l=2$ for the plots} }\label{figure:abs_cft}
\end{figure}
In \eqref{phase} we found the phase change due to the reflection at the near-horizon membrane from the dual CFT analysis.
In Fig (\ref{phase_plot}) we have plotted the absorption cross-section taking $\delta=0$ and the specific value that we found in \eqref{phase}. It is interesting to note that the behavior of the absorption cross-section is different at the superradiant frequency ($\omega=m\Omega_H$) for the two values of $\delta$ that we have chosen for the plot. For $\delta=0$ there is a discontinuity at the superradiant frequency as the Boltzmann reflectivity becomes unity and is non-analytic at $\omega=m\Omega_H$. On the contrary when $\delta=-2mLr_-/ r_+-1$ the discontinuity disappears.
\begin{figure}[h]
\includegraphics[width=15cm]{phase.png}
\caption[] {{The absorption cross-section for two different values of the phase ($\delta$), placing the reflective membrane at $r_0^*$=50, is shown. The magnified part shows the behaviour of the absorption cross-section at the superradiant frequency, $\omega=m\Omega_H$. }}
\label{phase_plot}
\end{figure}
\section{Observable signatures }
\label{observation}
It is well-established that possible near-horizon modifications to the Kerr spacetime could have signatures in terms of echoes in the ringdown phase of the detected gravitational wave signal \cite{Cardoso:2016rao,Oshita:2018fqu,Abedi:2020ujo}. In the presence of the partially reflective membrane in front of the horizon, we saw that the ringdown of the black hole is modified, leading to a new tower of QNMs for the ECO \eqref{qnf_B}. These QNMs will manifest as repeating echoes, with amplitudes decaying as a power-law \cite{Wang_2018,Wang:2019rcf}, in contrast to the fast exponential decay in case of the classical black hole. The time delay between two consecutive echoes would give the echo time-delay and can be written in terms of, $r^*_0$, the distance between the photon sphere and the near-horizon membrane in tortoise coordinates:
\begin{align}\label{techo}
\Delta t_{echo}=2|r^*_0|.
\end{align}
In geometric optics approximations, one can understand this echo time-delay as the time it would take for the classical QNM, generated at the photon sphere due to the perturbation of the ECO, to travel to the membrane placed in front of the horizon and then reflected back to the photon sphere. This argument, as well as the expression \eqref{techo} makes it evident that $\Delta t_{echo}$ is sensitive to the position of the membrane. Converting the tortoise coordinate to the proper distance of the membrane from the would-be horizon $d_{wall}$, Eq. (\ref{techo}) becomes:
\begin{align}\label{echotime}
\Delta t_{echo}\simeq {4 Mr_+\over r_+-r_-}\ln\left[{8M^2r_+\over (r_+-r_-)d_{wall}^2}\right]={1\over 2\pi T_H}\ln\left[{M\over \pi T_H d_{wall}^2}\right].
\end{align}
In terms of the echo time-delay \eqref{echotime} we can write the length of the circle as
\begin{align}
L=\Delta t_{echo}{a\over 8\pi M^2}=
{a\over 16\pi^2 M^2 T_H}\ln\left[{M\over \pi T_H d_{wall}^2}\right] \in \mathbb{N}.
\end{align}
From this relation the position of the wall ($d_{wall}$) can be written as
\begin{align} \label{dwall}
d_{wall}=\sqrt{{M\over \pi T_H}}e^{-L16\pi^2M^2T_H/a}.
\end{align}
Further, $a,M$ can be written in terms of $S_{BH},T_H$ as
\begin{align}
&M=T_HS_{BH}\left[-1+\left(1+{1\over 2\pi T_H^2 S_{BH}}\right)^{1/2}\right],
\\
&a=T_H S_{BH}\left[\left(-1+\left(1+{1\over 2\pi T_H^2 S_{BH}}\right)^{1/2}\right)^2-4\right]^{1/2}.
\end{align}
From \eqref{dwall} we can see that the position of the membrane ($d_{wall}$) in front of the horizon is determined by the size of the covering space $L$, on which the dual field theory lives. If the membrane is pushed all the way to the horizon, i.e. $d_{wall}\to 0$, we get $L\to \infty$, which is the well-defined classical limit on both sides of the duality. Also we note that $L$ is an integer and this puts a constraint on the relation between the echo time-delay and BH properties. In Fig (\ref{echoplot}) we show how the ringdown of the ECO is sensitive to the value of $d_{wall}$. We see that for a larger $d_{wall}$, the distance between the angular momentum barrier and the partially reflective membrane in front of the would-be horizon decreases, leading to shorter echo time-delays.
\begin{figure}[h]
\includegraphics[width=14cm]{echoplot.png}
\caption[] {{ Echoes in the ringdown for different positions of the reflective membrane placed in front of the would-be horizon. It is often assumed that the membrane is few Planck lengths, $l_p$, away from the horizon. }}
\label{echoplot}
\end{figure}
\section{Discussion}
\label{conclusion}
In this paper, we analysed the holographic CFT dual to a Kerr spacetime whose near-horizon region is modified due to quantum gravitational effects. One motivation for such near-horizon modification is to address the information loss paradox and we have seen models (such as the firewall or fuzzball) where the horizon is removed or replaced by a partially reflective membrane a few Planckian distance away from the position of the would-be horizon. According to holographic renormalization group (RG) flow program, any UV (near-horizon) modifications to the bulk geometry gets mapped onto an IR modification in the dual CFT \cite{Ishikawa:2013wf,Faulkner:2010jy,Cai:2007zw}. To be more precise, for the Kerr-like ECO, removal of the horizon can be interpreted as a UV cutoff on the gravity side. In this context, we proposed that the dual field theory lives on a finite \emph{toroidal} two-manifold, where the limit $LT \to \infty$ for the periodicity of the torus in the spatial and temporal direction is not taken (i.e the spatial circle has a finite length given as $L$). In this picture, the spatial length of the torus, $L$, acts as an IR regulator on the field theory side, and as we do not know the microscopic realization of the Kerr/CFT correspondence, we kept the arguments simply in terms of the basic features of a thermal CFT. One might interpret this as a non-perturbative or finite $N$ effect in the dual field theory which captures the near-horizon modifications of the Kerr black hole as expected.
We showed the consistency between the quasi normal modes and the absorption cross-section computed from the gravity side and similar quantities computed from the dual CFT. On the gravity side, this leads to echoes in the ringdown which repeat with a period of $\Delta t_{echo}$ (which in turn depends on the position of the reflective membrane), and decay as a power law. We proposed a relation between the length of the circle, $L \in \mathbb{N}$ on which the CFT lives and the echo time-delay, so that there is an exact matching of the CFT results with the gravity computation.
This relation between the echo time-delay and $L$ determines the position of the membrane in front of the horizon in terms of the length of the covering space on which the CFT lives.
For an ECO, the reflectively of the membrane depends on the quantum properties of the near-horizon membrane. Since a rigorous derivation of the reflectivity from a quantum theory of gravity is not available one needs to make a guess about this feature of the membrane. The reflectivity is usually assumed to be a constant, or as proposed recently based on a semi-classical analysis, the reflectivity is given in terms of the frequency dependent Boltzmann factor \cite{Oshita:2019sat}. From the dual CFT computation, we could determine what the reflectivity would be and surprisingly it does take the form of the Boltzmann factor. We believe the derivation of the reflectivity of the membrane based on the holographic CFT opens up the possibility of its interpretation in terms the quantum theory of gravity in the near-horizon region of a black hole/ECO.
The ingoing modes would also undergo a phase change upon reflection by the near-horizon membrane. This phase depends on the intrinsic quantum properties of the membrane, which also follows from our CFT computation. Furthermore, we showed the absorption cross-section has a discontinuity at the superradiance frequency when the phase change at the near-horizon membrane is taken to be zero. However, if we use the the phase predicted by the CFT computation, the absorption cross-section is well-behaved at superradiant frequency.
The Kerr/CFT correspondence was originally established for the NHEK geometry in \cite{Guica:2008mu} and later a microscopic description of the Kerr/CFT for the extremal case was established in \cite{Guica:2010ej,Compere:2010uk} by embedding the NHEK geometry in string theory. For the near-horizon quantum gravity modifications to the Kerr geometry, we saw that in the corresponding dual CFT one needs to incorporate finite size effects and we can conjecture that such effects would come from finite N corrections in the usual CFT terminology. Understanding the quantum corrections to the ``near-NHEK'' geometry from the CFT side (e.g., \cite{Bredberg:2009pv}) as a result of which the horizon may be removed (e.g., \cite{Solodukhin:2005qy,Bueno:2017hyj}), would make it possible for a more rigorous analysis in terms of the microscopic theory.
\section*{Acknowledgment }
We would like to thank Steve Carlip, Alejandra Castro, Geoffrey Compère, Lin-Qing Chen, Wan Zhen Chua, Malcolm Perry, and Andy Strominger for invaluable discussions.
Research at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Colleges and Universities.
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2024-02-18T23:40:24.807Z
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2020-09-22T02:01:41.000Z
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{"paloma_paragraphs":[]}
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{"arxiv_id":"2009.09027","language":"en","timestamp":1600740101000,"url":"https:\/\/arxiv.org\/abs\/2009.09027","yymm":"2009"}
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proofpile-arXiv_000-10090
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{"provenance":"002.jsonl.gz:10091"}
| null | null |
\section{\textbf{Introduction}}
In the last decades, the interest in studying two-dimensional theories has been increased since the experimental realization of graphene \cite{grapexp,review} and other related materials, such as the transition metal dichalcogenide monolayers \cite{TMDs}. The main goal is to derive either new quantum phases of matter \cite{Haldane,Bernevig,QSHE,prx} or calculate renormalized parameters that change the electronic properties of these materials \cite{Voz1,foster,Gui,popovici,geim, PQEDGap}, opening possibilities for future technological applications (in particular spintronics \cite{spintronicsnature}, valleytronics \cite{valleytronics}, and electric-field tunning of energy bands \cite{WSe2gap,MoS2gap}). Before this context, in the realm of high-energy physics, several works discussed the possibility of dynamical mass generation for massless Dirac particles, yielding a phase transition to a new quantum state of matter in which the chiral symmetry is broken \cite{maris,robert,curtis,appelquist}. This is generated due to the electronic interactions in the plane and it may occur even at finite temperatures. Because electrons in these materials obey a Dirac-like equation, therefore, some authors have also considered the realization of dynamical mass generation in these two-dimensional systems \cite{Herbut,VLWJF,ChunXu,teber1}.
The possibility of chiral symmetry breaking, i.e, dynamical mass generation in quantum electrodynamics (QED) has been discussed in several Refs. \cite{maris,bashir1,bashir2,bloch,robert,curtis,appelquist} for both (2+1)D and (3+1)D cases, just to cite a few. Furthermore, this symmetry-broken phase also has been shown to occur due to four-fermion interactions (such as Gross-Neveu and Thirring interactions) in Refs.~\cite{akram, gomes1, GNT=0, semmenoff}. In this case, the fermionic field exhibits a massive phase due to the spontaneous symmetry breaking, described by the so-called gap equation. This is usually calculated at the large-$N$ expansion, where $N$ is the number of fermion fields and the gap equation is calculated at order of $1/N$ \cite{akram,gomes1}. The dynamical mass generation is a typical effect of the nonperturbative regime, hence, it is common the application of the Schwinger-Dyson equations (SDEs) for calculating the critical parameters that describe the phase transition.
The SDEs are an infinite set of coupled-integral equations, relating all of the interacting Green functions of the model \cite{curtis, robert, fayzullaev}. Fortunately, within some approximative scheme, it is possible to find a truncated set of equations for calculating the desired Green functions, in particular, the electron-self energy that provides the chiral symmetry breaking. For pseudo-quantum electrodynamic (PQED) \cite{marino,VLWJF}, also called reduced quantum electrodynamics \cite{teber0}, the dynamical mass generation has been studied both at finite temperatures \cite{PQEDT} and at zero temperature with the presence of the Gross-Neveu interaction \cite{GNT=0}. Nevertheless, the effect of the Gross-Neveu interaction in PQED at finite temperatures has not been considered until now.
In this work, we describe the dynamical mass generation in PQED coupled to a Gross-Neveu interaction at finite temperature. We use the Matsubara formalism in order to include the effect of the thermal bath into the Schwinger-Dyson equation for the electron. This is dependent on both the gauge-field propagator and the auxiliary-field propagator, obtained after we use a Hubbard-Stratonovich transformation into the four-fermion interaction. These two bosonic propagators are calculated in order of $1/N$, which is consistent with our assumption of strong-coupling limit. Thereafter, we use this result into the Schwinger-Dyson equation for the electron self-energy and calculate the full electron propagator in the dominant order of $1/N$ in the nonperturbative limit. From this result, we conclude that a mass function is dynamically generated whether the number of fermions is less than a temperature-dependent critical parameter $N_c(T)$. At large temperatures, we find that $N_c(T)\ll 1$, hence, the dynamically generated mass vanishes and the system is in the gapless phase.
This paper is organized as follow. In Sec. II. we show our model and perform the large-$N$ approximation. In Sec. III. we write the truncated set of Schwinger-Dyson equations within the unquenched-rainbow approximation. In Sec. IV. we calculate the mass function in the static regime and obtain the critical parameters for the phase transition. In Sec.V. we use the zero-mode approximation in the dynamical regime for calculating the mass function. In Sec.VI. we summarize and discuss our main results. We also include three appendices, where we give details about the angular integral, kernel expansion, and the numerical results for the mass function.
\section{\textbf{Pseudo-quantum electrodynamics with Gross-Neveu interaction}}
We consider $N$ fermion fields constrained to the plane, whose interaction is described by Pseudo-quantum electrodynamics (PQED) \cite{marino}. Furthermore, we assume that these particles involve a contact interaction, given by the Gross-Neveu (GN) action \cite{gross}. Therefore, in the Euclidean space-time, the action of the model reads
\begin{equation}
\begin{split}
{\cal L}&=\frac{1}{2} \frac{F_{\mu \nu}F^{\mu\nu}}{\sqrt{-\Box}} +i \bar\psi_a\partial\!\!\!/ \psi_a + e \bar{\psi}_a\gamma^\mu \psi_a A_{\mu} + \\ & -\frac{G_0}{2}(\bar\psi_a \psi_a)^2 - \frac{\xi}{2}A_{\mu}\frac{\partial^{\mu}\partial^{\nu}}{\sqrt{-\Box}}A_{\nu},
\label{action}
\end{split}
\end{equation}
where $F_{\mu \nu}$ is the field intensity tensor of $A_{\mu}$, which is the gauge field, $\xi$ is the gauge-fixing parameter, $\psi_a$ is a four-component Dirac field, with the flavor index $a = 1, \ldots, N$, the dimensionless coupling constant $e$ is the electric charge, $\gamma_{\mu}$ are the Dirac matrices in the four-rank representation, whose algebra is defined by $\left\lbrace \gamma_{\mu}, \gamma_{\nu} \right\rbrace = -2 \delta_{\mu \nu}$, the coupling constant $G_0$ describes the strength of the GN interaction and has unit of mass in the natural system of units $(\hbar=c =1)$.
In the perturbative regime at zero temperature, the model in Eq.~(\ref{action}) for massive fermions has been used to describe the renormalization of the bandgap for WSe$_2$ and MoS$_2$ \cite{PQEDGap}. This result, nevertheless, requires a bare-mass term such as $m_0 \bar\psi_a \psi_a$ which is renormalized at one-loop and provides a beta function for the mass in terms of the RG scale. From this result, one concludes that the renormalized mass is dependent on the electronic density and a good agreement with experimental data has been found in Ref.~\cite{PQEDGap}
In the nonpertubative limit, the mass term is generated due to interactions even if we start with $m_0\rightarrow 0$ \cite{GNT=0}. Here, we generalize this result by including a thermal bath of temperature $T$.
From Eq.~(\ref{action}), we find the gauge-field propagator, namely,
\begin{equation}
\Delta_{0,\,\, \mu\nu}(p)=\frac{1}{2\sqrt{p^2}} \left[\delta_{\mu\nu} - \left(1-\frac{1}{\xi}\right) \frac{p_{\mu}p_{\nu}}{p^2} \right] \label{photonbare}
\end{equation}
and the fermion propagator
\begin{equation}
S_{0,\,F}(p)=-\frac{1}{\gamma^\mu p_\mu}.
\label{fermionfree}
\end{equation}
Before we discuss the vertex interactions, let us apply the Hubbard-Stratonovich transformation, which converts the four-fermion interaction into a Yukawa-type interaction by including an auxiliary field $\varphi$ and the coupling constant $g=G_0N$. In this case, we replace the four-fermion interaction by the following scheme
\begin{equation}
{\cal L}_{\rm{GN}}\rightarrow {\cal L}_{\rm{GN}}+\frac{1}{2g}\left(\varphi-\frac{g}{\sqrt{N}}\bar\psi_a\psi_a\right)^2, \label{transf}
\end{equation}
where ${\cal L}_{\rm{GN}}=-G_0 (\bar\psi_a \psi_a)^2/2$. Therefore, after that we find
\begin{equation}
{\cal L}_{\rm{GN}}= -\frac{\varphi}{\sqrt{N}} \bar\psi_a\psi_a+\frac{\varphi^2}{2g}.
\label{action2}
\end{equation}
Equation \eqref{action2} must be supplemented by the motion equation of $\varphi$, given by
\begin{equation}
\varphi=\frac{g}{\sqrt{N}}\bar\psi_a\psi_a,
\end{equation}
which proves that the transformation does not change the dynamics of the model at classical level. Furthermore, the bare auxiliary-field propagator is
\begin{equation}
\Delta_{0, \,\varphi}=\frac{1}{1/g},
\end{equation}
and Eq. \eqref{action} reads
\begin{equation}
\begin{split}
{\cal L}&= \frac{1}{2} \frac{F_{\mu \nu}F^{\mu\nu}}{\sqrt{-\Box}} + i\bar\psi_a\partial\!\!\!/ \psi_a+e\bar{\psi}_a \gamma^{\mu}\psi_a A_{\mu} \\
&-\frac{\varphi}{\sqrt{N}} \bar\psi_a\psi_a+\frac{\varphi^2}{2g}- \frac{\xi}{2}A_{\mu}\frac{\partial^{\mu}\partial^{\nu}}{\sqrt{-\Box}}A_{\nu}.
\label{actionend}
\end{split}
\end{equation}
The Yukawa-type vertex function is easily obtained from Eq. \eqref{actionend} and is given by $-1/\sqrt{N}$. On the other hand, for summing the self-energies in the large-$N$ expansion for PQED, we shall replace $e^2\rightarrow \lambda/N$, where $\lambda$ is taken fixed at large $N$. This allow us to sum over all of the diagrammatic contributions in order of $1/N$, which is an infinite sum, unlike the standard perturbation in $e$. Therefore, the PQED vertex reads $\sqrt{\lambda/N} \gamma_\mu$, describing the electromagnetic interaction.
\section{ \textbf{Truncated Schwinger-Dyson equation at finite temperatures}}
In this section we present the Schwinger-Dyson equations, obtained from Eq. \eqref{actionend}, that describes the quantum corrections for the two-point functions. In principle, this is a very complicated set of integral equations for all of the full Green functions of the model. From now on, we assume the ladder approximation, also called rainbow approach \cite{rainbow}, which consists of neglecting quantum corrections to the vertex functions. It is worth to note that this may be corrected by the Ball-Chiu vertex \cite{ball} when one wishes to preserve the Ward-Takahashi identity.
Because we are interested in dynamical mass generation for the electrons, we need to find a closed solution for both gauge-field and auxiliary-field propagators. These shall be given by the large-$N$ approximation, thus the bosonic-field propagators are calculated in the unquenched approximation \cite{kondo}.
\subsection{ \textbf{Auxiliary field}}
In Fig. \ref{Fig1}, we show the diagrammatic representation of the Schwinger-Dyson equation for the auxiliary field.
\begin{figure}[H]
\centering
\includegraphics[scale=0.8]{SDEauxiliaryfield.eps}
\caption{The Schwinger-Dyson equation for the auxiliary field. The left-hand side is the inverse of the full propagator of the auxiliary field, the first term in the right-hand side corresponds to the bare auxiliary-field propagator, and the second term $\Pi(p)$ is the quantum correction. The function $\Gamma[p,k]\rightarrow \mathbbm{1}$ corresponds to the approximated vertex.} \label{Fig1}
\end{figure}
Its analytical expression is given by \cite{GNT=0}
\begin{equation}
\Delta^{-1}_\varphi(p)=\Delta^{-1}_{0, \,\varphi}-\Pi(p),
\end{equation}
where $\Delta^{-1}_\varphi$ is the full propagator of $\varphi$ and $\Pi(p)$ is given by the fermionic loop, hence
\begin{equation}
\Pi(p)=-\frac{1}{N}\textrm{Tr} \int\frac{d^3k}{(2\pi)^3}\mathbbm{1} S_F(p-k)\Gamma(p,k) S_F(k),
\end{equation}
where $S_F(p)$ is the full fermion propagator. In the lowest order of $1/N$, we find that $\Pi(p)=-\sqrt{p^2} g_0$, with $g_0=1/4$ \cite{appelquist}. Therefore, the full propagator of the auxiliary field is given by
\begin{equation}
\Delta_\varphi(p)=\frac{1}{g_0}\frac{1}{(gg_0)^{-1}+\sqrt{p^2}}. \label{aux}
\end{equation}
\subsection{ \textbf{Gauge field}}
In Fig.~(\ref{Fig2}), we show the diagrammatic representation of the Schwinger-Dyson equation for the gauge-field propagator \cite{GNT=0}.
\begin{figure}[H]
\centering
\includegraphics[scale=0.8]{SDEphotonfield.eps}
\caption{The Schwinger-Dyson equation for the gauge-field propagator. The left-hand side is the inverse of the full propagator of the gauge field, the first in the right-hand side corresponds to the bare gauge-field propagator, and the second term $\Pi_{\mu\nu}(p)$ is the exact vacuum polarization tensor. The function $\Gamma^{\mu}[p,k]\rightarrow \gamma^{\mu}$ corresponds to the approximated vertex.} \label{Fig2}
\end{figure}
Its analytical expression is given by
\begin{equation}
\Delta^{-1}_{\mu\nu}(p)=\Delta^{-1}_{0,\,\mu\nu}(p)-\Pi_{\mu\nu}(p),
\label{SDphoton}
\end{equation}
where $\Delta^{-1}_{\mu\nu}$ is the full propagator and $\Pi_{\mu \nu}$ is the polarization tensor due to the electromagnetic interaction. This is given by
\begin{equation}
\Pi_{\mu \nu}(p) = \!- \frac{\lambda}{N} \textrm{Tr}\!\! \int\!\! \frac{d^3 k}{(2 \pi)^3} \gamma_{\mu} S_F(p+k)\Gamma_{\nu}(p,k)S_F(k).
\label{polgauge}
\end{equation}
Similarly to the previous case, in the lowest order of $1/N$, we find $\Pi_{\mu\nu}= \lambda \sqrt{p^2}/8 P_{\mu \nu}$, where $P_{\mu \nu}= \delta_{\mu \nu}-p_{\mu}p_{\nu}/p^2$. Therefore, the full propagator reads
\begin{equation}
\Delta_{\mu\nu}=\Delta_{0, \,\mu\alpha}[\delta^{\alpha}_\nu-\Pi^{\alpha\beta}\Delta_{0,\,\beta\nu}]^{-1}. \label{solphoton}
\end{equation}
Using Eqs. \eqref{photonbare} and \eqref{polgauge} in Eq. \eqref{solphoton}, we find
\begin{equation}
\Delta_{\mu\nu}(p)=\frac{P_{\mu\nu}}{\sqrt{p^2}\left(2+\frac{\lambda}{8}\right)}
\label{gaugefull}
\end{equation}
in the Landau gauge, i.e, with $\xi=\infty$.
\subsection{ \textbf{Matter field}}
In Fig.~(\ref{Fig3}), we show the diagrammatic representation of the Schwinger-Dyson equation for the fermions field \cite{GNT=0}.
\begin{figure}[H]
\centering
\includegraphics[scale=0.8]{SDEfermionfield.eps}
\caption{The Schwinger-Dyson equation for the electron. The left-hand side is the inverse of the full propagator of the fermion field, the first term in the right-hand side corresponds to the bare Dirac propagator, and the other terms in the right-hand side are the electron self-energy $\Xi(p)$.} \label{Fig3}
\end{figure}
Its analytical expression is given by
\begin{equation}
S_F^{-1}(p)=S_{0,\,F}^{-1}(p)-\Xi(p),
\label{sdefermion}
\end{equation}
where $S_F$ is the full fermion propagator and $\Xi$ is the electron self-energy, given by
\begin{equation}
\Xi(p) = \Xi^{\lambda}(p) + \Xi^{g}(p),
\end{equation}
where $\Xi^{\lambda}$ is the PQED contribution and $\Xi^{g}$ is the GN contribution, namely,
\begin{equation} \label{Xi1}
\Xi^{\lambda}(p)= \frac{\lambda}{N} \int \frac{d^3 k}{(2 \pi)^3} \gamma^{\mu}S_F(k)\Gamma^{\nu}(k;p)\Delta_{\mu \nu}(p-k),
\end{equation}
\begin{equation}\label{Xi2}
\begin{split}
\Xi^g(p)&=\frac{1}{N} \int \frac{d^3 k}{(2 \pi)^3}\mathbbm{1}S_F(k)\Gamma(k;p)\Delta_{\varphi}(p-k)+\\
&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!-\frac{1}{N} \int \frac{d^3 k}{(2 \pi)^3} \mathbbm{1} \textrm{Tr} \left[S_F(k)\Gamma(k;q=0) \Delta_{\varphi}(q=0) \right].
\end{split}
\end{equation}
In order to find the full fermion propagator, it is convenient to decompose this Green function into its irreducible parts. Hence, we adopt the following ansatz
\begin{equation}
S_F^{-1}(p)= -\gamma^{\mu}p_{\mu}A(p)+\Sigma(p),
\label{fullfermion}
\end{equation}
where $A(p)$ yields the wavefunction renormalization and $\Sigma(p)$ is called mass function. Note that $\Sigma(p)\neq 0$ implies a dynamical mass generation \cite{appelquist}. In order to obtain the mass function, we take the trace operation in both sides of Eq. \eqref{sdefermion}. Thereafter, we substitute Eqs. \eqref{fermionfree}, \eqref{Xi1}, \eqref{Xi2} and \eqref{fullfermion}, within the rainbow approximation, $\Gamma^{\nu}(p;k) \rightarrow \gamma^{\nu}$ e $\Gamma(p;k) \rightarrow \mathbbm{1}$, we obtain
\begin{equation}
\begin{split}
\Sigma(p)&= \frac{\lambda}{N}\int\frac{d^3 k}{(2 \pi)^3} \frac{\delta^{\mu \nu} \Sigma(k)}{k^2 A^2(k)+\Sigma^2(k)}\Delta_{\mu \nu}(p-k)\\
&+ \frac{1}{N} \int \frac{d^3 k}{(2 \pi)^3} \frac{\Sigma(k)}{k^2 A^2(k)+ \Sigma^2(k)}\Delta_{\varphi}(p-k)\\
& - \Delta_{\varphi}(0)\int \frac{d^3 k}{(2 \pi)^3} \frac{\Sigma(k)}{k^2 A^2(k)+ \Sigma^2(k)}.
\label{massfunction}
\end{split}
\end{equation}
On the other hand, for calculating the wavefuntion renormalization, we multiply Eq. \eqref{sdefermion} by $\slashed{p}$ and, after calculating the trace operation, we obtain
\begin{equation}
\begin{split}
A(p)&=\!1\! -\!\frac{\lambda}{N p^2}\!\!\int\!\! \frac{d^3 k}{(2 \pi)^3} \left(\delta^{\beta \mu}\delta^{\alpha\nu}-\delta^{\beta \alpha}\delta^{\mu \nu} \right. \\ & \left. +\delta^{\beta \nu}\delta^{\mu \alpha}\right) \frac{p_{\beta}k_{\alpha}A(k)}{k^2A^2(k)+\Sigma^2(k)}\Delta_{\mu \nu}(p-k) \\& +\! \frac{1}{N p^2}\!\!\int\!\! \frac{d^3 k}{(2\pi)^3} \frac{\delta^{\beta \alpha}p_{\beta}k_{\alpha}A(k)}{k^2 A^2(k)+\Sigma^2(k)}\Delta_{\varphi}(p-k).
\label{wavefunction}
\end{split}
\end{equation}
It is clear that Eqs. \eqref{massfunction} and \eqref{wavefunction} are a coupled set of equations for $\Sigma(p)$ and $A(p)$. However, in the dominant order $1/N$, we may replace $A(p)\rightarrow 1$ in Eq. \eqref{massfunction}. This is also in agreement with the rainbow approximation.
Next, we use the Matsubara formalism for introducing temperature effects into Eq. \eqref{massfunction}. In the imaginary-time formalism, the main step is to change the time-component integrals into a sum over Matsubara frequencies \cite{matsubara}, i.e, we must perform
\begin{equation}
\int \frac{d k_0}{2\pi} f(k_0,\textbf{k},\beta) \rightarrow \frac{1}{\beta} \sum_{n=-\infty}^{\infty} f(n, \textbf{k},\beta),
\end{equation}
with
\begin{equation}
k_{\mu}=(k_0,\textbf{k}) \rightarrow (\omega_n,\textbf{k}),
\end{equation}
and
\begin{eqnarray*}
\omega_n = \begin{cases}
\frac{(2n+1)\pi}{\beta} \,\, \textrm{for fermions},\\
\frac{2n\pi}{\beta} \hspace{0.7cm} \textrm{for bosons},
\end{cases}
\end{eqnarray*}
where $\omega_{n}$ are the Matsubara frequencies, $T=\beta^{-1}$ is the temperature of the thermal bath, and $(m, n)$ are the vibration modes related to each momentum component. Note that we are considering the Boltzmann constant as $k_{B}= 1$, in the natural system of units. After including the thermal bath in Eq.~(\ref{massfunction}), we have
\begin{equation}
\begin{split}
\!\!\!\Sigma_m(\textbf{p},T)&= C_1 \sum \!\!\!\!\!\!\!\! \int \frac{\Sigma_n(\textbf{\textbf{k}, T})}{(2n+1)^2 \pi^2 T^2 + \textbf{k}^2 +\Sigma^2_{n}(\textbf{k},T)} \\ & \!\!\!\! \times \frac{1}{\sqrt{4 (n-m)^2 \pi^2 T^2 +(\textbf{p}-\textbf{k})^2}} \,\,+ \\
&+C_2 \sum \!\!\!\!\!\!\!\! \int \frac{\Sigma_n(\textbf{k},T)}{(2n+1)^2 \pi^2 T^2 + \textbf{k}^2 +\Sigma^2_{n}(\textbf{k},T)} \\
& \!\!\!\! \times \frac{1}{(g g_0)^{-1}+\sqrt{4 (n-m)^2 \pi^2 T^2 +(\textbf{p}-\textbf{k})^2}},
\label{masstemperature}
\end{split}
\end{equation}
where
\begin{equation}
\sum \!\!\!\!\!\!\!\! \int \rightarrow T\sum_{n = -\infty}^{\infty} \int \frac{d^2 k}{(2 \pi)^2},
\end{equation}
\begin{equation}
C_1 = \frac{2 \lambda }{\left(2+\frac{\lambda}{8}\right)N},
\end{equation}
\begin{equation}
C_2 = \frac{1}{g_0 N}.
\end{equation}
From now on, we assume that $\Sigma_m(\textbf{p},T)=\Sigma(\textbf{p},T)$ and, therefore, the mass function only depends on the dominant vibrational mode $m=0$. The standard procedure for calculating the mass function is to convert the integral equation into a differential equation for $\Sigma(\textbf{p},T)$. It turns out that there exist two main operations, namely, the sum over $n$ and the angular integral that must be performed before finding this result. We shall explore some approximations for these operations.
Before performing further approximations, let us show that the sum over the vibrational modes $n$ is convergent. The convergence of this sum for PQED at finite temperature has been shown in \cite{PQEDT}. In Eq.~(\ref{masstemperature}), the sum over $n$ may be written as
\begin{equation}
U=\sum_{n=-\infty}^{\infty}\frac{1}{(2n+1)^2+A^2}\frac{1}{B+\sqrt{n^2+C^2}},
\label{GNconv}
\end{equation}
where we conclude that $(A^2,C^2)>0$ and, because we consider $g>0$ \cite{gross,van}, we also find that $B>0$.
Next, we apply the Cauchy integral test \cite{arfken}, where the sum is written as
\begin{equation}
U=2 \sum_{n=1}^{\infty}u^{(+)}_n + u_0,
\label{GN2}
\end{equation}
with $u^{(+)}_n = (u_n+u_{-n})/2$. The test imposes that whether $u(x)=u_x$ is positive, continuous, and decreasing in the interval $[1, \infty]$, hence, the integral over $u(x)$ is finite and, therefore, convergence is derived. This is exactly our case and one may easily conclude that
\begin{equation}
u(x)>0, \,\,\,\,\, \forall x \mid x\,\in [1,\infty],
\end{equation}
\begin{equation}
u'(x)<0,
\end{equation}
\begin{equation}
\lim_{x\rightarrow a} u(x) = u(a), \,\, \forall a \mid \,\, a \in [1,\infty].
\end{equation}
Therefore, Eq.~\eqref{GN2} is a convergent series for all $n$.
\section{\textbf{STATIC REGIME} $p_0 =0$}
The static regime, also called instantaneous-exchange approximation \cite{exchange}, consists of neglecting all of the time-components of the bosonic-field propagator in Eq. \eqref{massfunction}, with the consideration that the interaction vertex being only $\gamma^0$ in Eq. \eqref{Xi1}. This is usually realistic in cases where the electron velocity is much less than the light velocity. In these systems, it is possible to show that charged particles interacts through the Coulomb potential. This is given by the Fourier transform of the gauge-field propagator in Eq.~(\ref{photonbare}) after we perform $p_\mu=(p_0,\textbf{p}) \rightarrow (0,\textbf{p})$, where we will use the following notation $(0,\textbf{p})\equiv (0,P)$. Next, let us consider both zero and finite temperature cases for calculating $\Sigma(P,T)$.
\subsection{ \textbf{Zero Temperature Case}}
After considering the static regime with $T\rightarrow 0$ in Eq. \eqref{masstemperature}, we find
\begin{equation}
\begin{split}
\Sigma(P)&=\frac{C_1}{2} \int \frac{d^3k}{(2\pi)^3} \frac{\Sigma(K)}{k_0^2+K^2+\Sigma^2(K)}\Delta(P-K) \\
&+C_2\!\! \int \frac{d^3k}{(2\pi)^3} \frac{\Sigma(K)}{k_0^2+K^2+\Sigma^2(K)}\Delta_\varphi(P-K), \label{SP0}
\end{split}
\end{equation}
where $d^3k=dk_0 d^2K$,
\begin{equation}
\Delta_\varphi(P-K)=\frac{1}{(gg_0)^{-1}+\sqrt{(P-K)^2}},
\end{equation}
and
\begin{equation}
\Delta(P-K)=\frac{1}{\sqrt{(P-K)^2}}.
\end{equation}
The integral over $k_0$ in Eq. \eqref{SP0} is easily solved and the angular integral may be solved by using the identity
\begin{equation}
\int_0^{2\pi} d\theta \Delta_\varphi(P-K)=\frac{4K(x_0)}{\sqrt{(P-K)^2}}+O[(gg_0)^{-2}], \label{angT}
\end{equation}
where $K(x_0)$ is the complete elliptic integral of the first kind with $x_0\equiv -4 PK/(P-K)^2$. Note that the angular integral for the gauge-field term is very similar and that we will neglect the terms in order of $O[(gg_0)^{-2}]$. Replacing Eq.~(\ref{angT}) in Eq.~(\ref{SP0}), we find
\begin{equation}
\Sigma(P)=4\left(\frac{C_1}{2}+C_2\right) \int_0^\Lambda \frac{dK}{(2\pi)^2} \frac{K \,\Sigma(K) {\cal K}(K,P)}{2\sqrt{K^2+\Sigma^2(K)}}, \label{SigP}
\end{equation}
where the kernel ${\cal K}(K,P)$ reads
\begin{equation}
{\cal K}(K,P)=\frac{K(x_0)}{\sqrt{(P-K)^2}}.
\end{equation}
Next, we approximate the kernel for its lowest order terms, namely,
\begin{equation}
{\cal K}(K,P) \approx \frac{\pi}{2P} \Theta(P-K)+\frac{\pi}{2K} \Theta(K-P). \label{kernelApp}
\end{equation}
Note that $\Theta(P-K)$ is the standard step function. Using Eq.~(\ref{kernelApp}) in Eq.~(\ref{SigP}), we have
\begin{equation}
\begin{split}
\Sigma(P)&=\frac{(C_1/2+C_2)}{4\pi} \left[\int_0^P K dK \frac{\Sigma(K)}{P\sqrt{K^2+\Sigma^2(K)}} \right. \\
& \left. +\int_P^\Lambda dK \frac{\Sigma(K)}{\sqrt{K^2+\Sigma^2(K)}}\right].
\label{SigP2}
\end{split}
\end{equation}
On the other hand, by taking derivatives in respect to $P$ in both sides of Eq.~(\ref{SigP2}), we obtain a differential equation for the mass function, given by
\begin{equation}
P^2 \Sigma''(P)+2P \Sigma'(P)+\frac{N_c}{4N} \frac{P \Sigma(P)}{\sqrt{P^2+\Sigma^2(P)}}=0, \label{EulerT0}
\end{equation}
where the critical number of fermions is given by
\begin{equation}
N_c=\frac{1}{\pi}\left(\frac{C_1}{2}+C_2\right)=\frac{ \lambda }{\pi\left(2+\frac{\lambda}{8}\right)}+\frac{4}{\pi}. \label{Nc0}
\end{equation}
Eq. \eqref{EulerT0} is also supplemented by two equations, namely,
\begin{equation}
\lim_{P\rightarrow 0} P^2 \Sigma'(P)=0,
\end{equation}
which provides the infrared behavior and
\begin{equation}
\lim_{P\rightarrow \Lambda} \left[ P \Sigma'(P)+\Sigma(P) \right]=0,
\end{equation}
which shows that the mass function is expected to vanish at large $P$.
Using the approximation $P/\sqrt{P^2+\Sigma^2(P)} \approx 1$ in Eq. \eqref{EulerT0}, which holds for large-external momentum, we find the analytical solution
\begin{equation}
\Sigma(P)=A_1 P^\gamma+A_2 P^{\gamma^*}, \label{SolEuler}
\end{equation}
where $\gamma=-1/2+i/2 \sqrt{N_c/N-1}$ and $(A_1,A_2)$ are arbitrary constants. In the dynamical regime, it has been show that the dynamical mass generation only occurs for $N<N_c$ \cite{GNT=0}. The same conclusion form Eq. \eqref{Nc0} is also obtained from the static regime. Using Eq. \eqref{Nc0} in $\gamma$, $\lambda = 4 \pi N \alpha_c$, and solving $N_c(\alpha_c)/N=1$ for $\alpha_c$, we find the critical fine-structure constant, given by
\begin{equation}
\alpha_c = \frac{ \frac{16}{N \pi} - 4}{N \pi - 12 },
\end{equation}
in terms of fine-structure constant the mass generation condition is $\alpha > \alpha_c$.
\subsection{ \textbf{Finite Temperature}}
In order to describe temperature effects, we include the Matsubara frequencies in Eq. \eqref{SP0}. Thereafter, we solve the angular integral using Eq.~(\ref{angT}) while for the Matsubara sum we use the identity
\begin{equation}
T\sum_n \frac{1}{\omega^2_n+\epsilon^2_k}=\frac{1}{2\epsilon_k}\left[1-2n_F(\epsilon_k)\right],
\end{equation}
where $\epsilon_K \equiv \sqrt{K^2+\Sigma^2(K)}$ and
\begin{equation}
n_F(\epsilon_K)=\frac{1}{e^{\beta \epsilon_K}+1}
\end{equation}
is the Fermi-Dirac distribution. After using these properties, we find
\begin{equation}
\!\!\!\!\!\Sigma(P,T)\!=\!4 \left(\!\frac{C_1}{2}\!+\!C_2\!\right) \int_0^\Lambda \!\!\!\!dK \frac{K}{(2\pi)^2} \frac{ \Sigma(K){\cal K}_\beta (K,P)}{2\sqrt{K^2+\Sigma^2(K)}}, \label{IEST}
\end{equation}
where the temperature-dependent kernel ${\cal K}_\beta(K,P)$ reads
\begin{equation}
{\cal K}_\beta(K,P)=\frac{K(x_0)}{\sqrt{(P-K)^2}}\left[1-2n_F(\epsilon_K)\right].
\end{equation}
Next, following the same steps as in the previous section, we find the differential equation for the mass function, namely,
\begin{equation}
\!\!P^2 \Sigma''(P)\!+\!2P \Sigma'(P)\!+\!\frac{N_c h_\beta(P)}{4N}\frac{P \Sigma(P)}{\sqrt{P^2+\Sigma^2(P)}}=0, \label{EulerT}
\end{equation}
where
\begin{equation}
h_\beta(P)=\left[1-2n_F(\epsilon_P)\right].
\end{equation}
The linearized version of Eq.~(\ref{EulerT}) may be obtained at large-external momentum and by doing $P\rightarrow \Lambda$ in $h_\beta(P)$. This allow us to obtain a temperature-dependent critical number $N_c(T)$, given by
\begin{equation}
N_c(T)=N_c(0) \left[1-\frac{2}{e^{\Lambda/T}+1}\right], \label{NcT}
\end{equation}
where $N_c(0)\equiv N_c=(C_1/2+C_2)/\pi$ is given by Eq. \eqref{Nc0} in terms of $\lambda$. On the other hand, for large-$T$, we obtain $N_c(T)\rightarrow 0$, which implies that there exist no dynamical mass generation for any $\lambda$.
In the large-external momentum, the solution of Eq. \eqref{EulerT} is given by Eq. \eqref{SolEuler}, after we replace $N_c \rightarrow N_c(T)$. Hence, we obtain a temperature-dependent exponent $\gamma(T)=-1/2+ i/2\sqrt{N_c(T)/N-1}$. Therefore, we find
\begin{equation}
\Sigma(P,T)=A_1(T) P^{\gamma(T)}+A_2(T) P^{\gamma^*(T)}, \label{SolEulerT}
\end{equation}
where the arbitrary constants $(A_1,A_2)$ may be dependent on the temperature. At the critical point $N=N_c(T)$, we have $\gamma(T)=-1/2$. This critical point also may be described in terms of $\alpha$, using Eq.~(\ref{Nc0}) and that $e^2=4\pi\alpha= \lambda N$. Indeed, after a simple calculation, we obtain a temperature-dependent critical coupling constant $\alpha_c(T)$, namely,
\begin{equation}
\alpha_c(T)\!=-\frac{4 \left[4\left(1-e^{\Lambda/T}\right) + N\pi\left(1+e^{\frac{\Lambda}{T}}\right)\right]}{ N \pi \left[12\left(1-e^{\Lambda/T}\right) +N \pi\left(1+e^{\frac{\Lambda}{T}}\right)\right]}. \label{alphacT}
\end{equation}
Because the ratio $N_c(T)/N$ is a monotonically increasing function of $\alpha$, it follows that the mass generation only occurs for $\alpha>\alpha_c(T)$. In Fig.~\ref{Fig4}, we plot both $N_c(T)$ and $\alpha_c(T)$ and show that whether the temperature increases, hence, the system quickly goes to a gapless phase.
\begin{figure}[H]
\centering
\includegraphics[scale=0.9]{NcStatic.eps}
\caption{The critical parameters for the dynamical mass generation. We plot both Eq.~(\ref{NcT}) (the common line) and Eq.~(\ref{alphacT}) (the inset) with $\Lambda=10$ (units of energy). For the common line we also use $\lambda=3.0$ while for the inset we use $N=2$. Note that for small enough temperatures, the critical coupling constants remain almost unchanged.} \label{Fig4}
\end{figure}
We also may find numerical results (see App.~A for more details) of the integral equation for $\Sigma(P,T)$ in Eq.~(\ref{IEST}). In Fig.~\ref{Fig5}, we plot these results and compare this with our analytical solution given by Eq.~(\ref{SolEulerT}).
\begin{figure}[H]
\includegraphics[scale=0.9]
{PQEDGNstaticwithfiniteTNf1.eps}
\caption{The analytical and numerical solutions for $\Sigma(P,T)$. The dashed line is the numerical solution of the integral equation given by Eq. \eqref{IEST} with $N=1$. The common line is the analytical solution given by Eq. \eqref{SolEulerT} with $N=1$ and $A_1(T)=A_2(T)^*=0.13+0.05i$. We have used $\Lambda=10$ (units of energy), $\lambda=3.0$ and $T=0.1$ (units of $\Lambda$) for all of the curves. The analytical solution is in good agreement with the numerical results.} \label{Fig5}
\centering
\end{figure}
\section{\textbf{DYNAMICAL REGIME} $p_0 \neq 0$}
In this section we recover the retardation effects by assuming $p_0 \neq 0$ in Eq. \eqref{massfunction}. For $T=0$, this regime has been discussed in several contexts.
After a simple changing of variables, the angular integral of the PQED sector (see details in App. B) reads
\begin{equation}
\mathcal{I}_n^{PQED}(p,k;T)=\frac{4}{w_{p,k}^n}K\left[x_n\right],
\end{equation}
where,
\begin{equation}
w_{p,k}^n = \sqrt{4n^2\pi^2T^2 +(p-k)^2} \label{omega},
\end{equation}
and
\begin{equation}
x_n = -\frac{4pk}{4n^2\pi^2T^2 +(p-k)^2} \label{x}.
\end{equation}
Similarly, for the GN sector, we find
\begin{equation}
\mathcal{I}_n^{GN}(p,k,T)\! =\!\frac{4}{w_{p,k}^n}\left[K(x_n)-\frac{\Pi(y_n,x_n)}{1-g^2 g_0^{2}\left(w_{p,k}^n\right)^2}\right]
\end{equation}
with
\begin{equation}
y_n=-\frac{4pk}{4n^2\pi^2T^2 + (p-k)^2 -\frac{1}{g^2 g_0^{2}}},
\label{y}
\end{equation}
where $K(x_n)$ and $\Pi(y_n,x_n)$ are the elliptic integral of the first and third kind, respectively. In particular, note that for $g=0$, we have that $\Pi(0,x_n) = K(x_n)$, hence, the kernel of the GN sector vanishes, as expected. The integral representation for these functions are
\begin{align}
K(x_n)= \int_0^{\pi/2} \!\!\!\! d\theta\frac{1}{\sqrt{1-x_n \sin^2(\theta)}}
\label{ellipticK}
\end{align}
and
\begin{equation}
\Pi(y_n,x_n)=\!\! \int_0^{\pi/2} \!\!\!\! d\theta\frac{1}{[1-y_n \sin^2(\theta)]\sqrt{1-x_n \sin^2(\theta)}}.
\label{ellipticPi}
\end{equation}
The resulting kernel, after using the angular integral, is a very complicated function of $n$. This preclude us of solving the whole Matsubara sum, as we have did in the static regime. Therefore, we shall investigate the lowest order contribution, provided by the zero-mode $n=0$ approximation.
\section{\textbf{Zero-mode approximation}}
In this section we consider the fundamental vibrational mode $n=0$ in Eq. \eqref{masstemperature}. Using Eqs.~\eqref{ellipticK} and \eqref{ellipticPi}, we find
\begin{equation}
\begin{split}
\Sigma(p,T)&=\frac{C_1}{\pi^2}\!\! \int_0^{\infty} \!\!\!\!\!\!dk \frac
{k K(x_0)}{\|p-k \|} \frac{T \Sigma(k,T)}{\pi^2 T^2 + k^2 +\Sigma^2(k,T)}\\
& + \frac{C_2}{\pi^2}\!\!\int_0^{\infty}\!\!\!\!\!\!dk \frac{k}{\|p-k \|} \frac{T\Sigma(k,T)}{\pi^2 T^2 + k^2 + \Sigma^2(k,T)} \\
& \times \left[K(x_0)-\frac{\Pi(y_0,x_0)}{1-g^2 g_0^{2}(p-k)^2}\right]
\label{zeromode}
\end{split}
\end{equation}
For calculating an analytical solution of Eq.~(\ref{zeromode}), we write the elliptic integral as the hypergeometrics Appell $F_1$. Because $F_1$ has an expansion in its parameters, we may find a simplified version of the kernel (see App. C for more details). Hence, Eq. \eqref{zeromode} yields
\begin{equation}
\begin{split}
\Sigma(p)&=\frac{C_1 T}{2 \pi}\int_0^{\Lambda} \!\!dk f(k) \left[\frac{\Theta(p-k)}{p}+\frac{\Theta(k-p)}{k}\right] \\ & + \frac{C_2 T}{2 \pi}\int_0^{\Lambda} \!\!\! dk f(k) \left[\frac{\Theta(p-k)}{p}+\frac{\Theta(k-p) k}{k^2 -\frac{1}{(g g_0)^2}}\right], \label{auxST}
\end{split}
\end{equation}
where $f(k)\equiv k \Sigma(k)/\left[(\pi T)^2+k^2+\Sigma^2(k)\right]$. By taking derivatives in respect to $p$ in both sides of Eq.~(\ref{auxST}), we find
\begin{equation}
\begin{split}
\Sigma'(p)+\frac{b \Sigma(p)}{(\pi T)^2+p^2+\Sigma^2(p)}= -\frac{a}{p^2}\int_0^{p}dk f(k), \label{Sigmaprime}
\end{split}
\end{equation}
with $a=(C_1+C_2)T/2 \pi$ and $b=C_2 T/2 \pi$, both constants with dimension of mass. Next, we neglect the nonlinear terms by using $(\pi T)^2+p^2+\Sigma^2(p) \approx (\pi T)^2+p^2$, which is expected to be reasonable at large $p$ \cite{cutinfrared, PQEDT}. After deriving Eq.~(\ref{Sigmaprime}) in respect to $p$, we obtain the differential equation
\begin{equation}
\begin{split}
p^3 & \Sigma''(p) +\left[2p^2+\frac{p^3 b}{(\pi T)^2+p^2}\right]\Sigma'(p) \\
& +\left[\frac{2 p^2 b}{(\pi T)^2+p^2}-\frac{2 p^4 b}{\left((\pi T)^2+p^2\right)^2}+a\right]\Sigma(p)=0,
\label{diff}
\end{split}
\end{equation}
which is supplemented by two conditions for the infrared and ultraviolet regimes. These are given by
\begin{equation}
\lim_{p \rightarrow \Lambda} \Sigma(p)=0
\end{equation}
and
\begin{equation}
\lim_{p \rightarrow 0} p^2 \left[\frac{d \Sigma(p)}{d p} +\frac{b \Sigma(p)}{(\pi T)^2+p^2}\right] = 0. \label{IR}
\end{equation}
Unfortunately, Eq.~(\ref{diff}) has not an analytical solution for arbitrary values of the set $(p,T)$ of variables. Nevertheless, we may consider a possible linearized version for large external momentum.
\subsection{\textbf{Large-external-momentum approximation}}
In this section we investigate the linearized version of Eq.~(\ref{diff}) by assuming $p\gg T,\Sigma(p)$. In this case, the behavior of the generated mass is given by
\begin{equation}
p^3 \Sigma''(p) +\left[2p^2+pb\right]\Sigma'(p) +a\Sigma(p)=0,
\label{diffp}
\end{equation}
which yields
\begin{align}
\!\!\Sigma(p)\!=\!A_1 \frac{b \,_1F_1\left[1-\frac{a}{b},2,\frac{b}{p}\right]}{p}\! + A_2 G^{2\, 0}_{1\,2} \left[ -\frac{b}{p} \big{|}
\begin{array}{c l}
\frac{a+b}{b}\\
0,1
\end{array} \right], \label{anazero}
\end{align}
where $_1F_1$ is the confluent hypergeometric function and G is the Meijer G-function \cite{handbook}. The sign of this Meijer G-function changes for different values of the external momentum. Because we have not observed such behavior in the numerical results of Eq.~(\ref{zeromode}), from now on, we shall take $A_2=0$ for the sake of agreement with the integral equation. In Fig.~(\ref{Fig6}), we plot our analytical solution in Eq.~(\ref{anazero}) and compare this with the numerical results of the integral equation in Eq.~(\ref{zeromode}).
\begin{figure}[H]
\includegraphics[scale=0.6]{compPQEDGNdynamic.eps}
\centering
\caption{The analytical and numerical solutions for $\Sigma(p,T)$ in the zero-mode approximation. The dashed line is the numerical solution of the integral equation given by Eq.~(\ref{zeromode}) with $N=2$. The common line is the analytical solution given by Eq.~(\ref{anazero}) with $N=2$ and $A_1(T)=0.6$ (the best fit for large-$p$). We have used $\Lambda=10$ (units of energy), $\lambda=3.0$ and $T=0.1$ (units of $\Lambda$) for all of the curves. Note that the analytical solution is in good agreement with the numerical results only at large-$p$ limit, as expected.} \label{Fig6}
\end{figure}
\vspace{-1.1cm}
\subsection{\textbf{Zero-external-momentum approximation}}
In this section, we consider the limit $\Sigma(p=0,T)=m(T)$ in Eq. \eqref{masstemperature}, using the zero-frequency mode. In this case, the angular integral provides a constant factor of $2 \pi$. Hence, we find
\begin{equation}
\begin{split}
1&= \frac{1}{4 \pi}\int_{0}^{\Lambda}\!\!\!\! dk \,\, \frac{1}{\pi^2 T^2 + k^2 +m^2(T)} \\ & \times\left\lbrace C_1 T + \frac{C_2 T k}{(g g_0)^{-1}+k}\right\rbrace.
\end{split}
\end{equation}
After calculating the integral over $k$, we have
\begin{widetext}
\begin{equation}
\begin{split}
1&= \frac{C_1 T}{2 \pi} \frac{\arctan \left(\frac{\Lambda}{\sqrt{\pi^2 T^2 + m^2(T)}}\right)}{ \sqrt{m^2(T)+ \pi^2T^2}} + \frac{C_2 T}{2 \pi} \left\lbrace \frac{\sqrt{\pi^2 T^2 + m^2(T)}\arctan \left(\frac{\Lambda}{\sqrt{\pi^2 T^2 + m^2(T)}}\right)}{\left[\pi^2 T^2 + m^2(T) + (\frac{1}{g g_0})^2\right]} \right. \\ & \left. + \frac{g g_0 \ln\left[\frac{\pi^2 T^2 + m^2(T)+\Lambda^2}{(\pi^2 T^2 + m^2(T))(1 +g g_0 \Lambda)^2}\right]}{2\left[(g g_0)^2(\pi^2 T^2 +m^2(T))+1\right]}\right\rbrace.
\label{m(T)}
\end{split}
\end{equation}
\end{widetext}
Equation \eqref{m(T)} is our gap equation for the dynamical mass as a function of temperature and the coupling constants . Similarly to the BCS model, we define the critical temperature as the point $T=T_c$ in which $m$ vanishes. Here, nevertheless, we define that $C_1 T$ and $C_ 2 T$ are fixed in order to be in agreement with the fact that $m\neq 0$ for $T=0$.
\begin{figure}[H]
\centering
\includegraphics[width=8cm, height=4cm]{mg1.eps}
\caption{The numerical result of the mass function given by Eq. \eqref{m(T)}. We use $\Lambda =10$, $N=2$ and $\lambda=3$ in the blue points with $g = 0$, and in the square point $g=1$. Where, we see the aument of the critical temperature due at the GN-coupling.}
\label{mT}
\end{figure}
In Fig. \ref{mT}, we plot the numerical solutions of $m(T)$ for both $g=0$ and $g=1.0$, which proves that the GN interaction increases the gapped phase. Furthermore, in Fig. \ref{T(g)}, we show that the GN interaction increases the critical temperature, as expected.
\begin{figure}[H]
\centering
\includegraphics[width=8cm, height=5cm]{TCversusg.eps}
\caption{Behaviour of critical temperature for different values of the GN coupling constant given by Eq. \eqref{m(T)} numerically treated. Where we use $\Lambda=10$, $N=2$ and $\lambda=3$.}
\label{T(g)}
\end{figure}
\section{\textbf{Summary and Outlook}}
In this work, we investigate the influence of temperature on the dynamical mass generation for fermions in PQED with a Gross-Neveu interaction. In order to do so, we used the Schwinger-Dyson equation for the electron in the dominant order $1/N$, negleting quantum corrections to the vertices at the finite temperature. In the static regime, we were able to solve the whole Matsubara sum. Thereafter, we calculated the critical coupling constant $\alpha_c$ and the critical numbers of fermions $N_c$ as a function of $T$ and the cutoff $\Lambda$. From these results, we concluded that the system goes to a gapless phase whether we increase the ratio $T/\Lambda$. Furthermore, we obtained an analytical solution for the mass function $\Sigma(p)$, which is in good agreement with our numerical results.
In the dynamical regime, we used the zero-mode approximation in the sum over the Matsubara frequencies, which allowed us to calculate the analytical solution $\Sigma(p)$. This function agrees with our numerical results for large-exteral momentum, while for small-external momentum some deviations have been found. Moreover, we found a gap equation that provides the value of the mass function in the zero-external momentum approximation, i.e, $\Sigma(p=0,T) \rightarrow m(T)$. In this case, the numerical tests show that the critical temperature increases as we increase the strength of the fermionic self-interaction.
Several generalizations of this work may be perfomed. For instance, one would investigate the main effects of vertices quantum corrections into the critical parameters for the mass generation. In particular, it would be relevant to find a way to increase the gapped phase for finite temperature. In this phase, the competition between $\alpha$ and $g$ yields a tunneable mass which may be relevant for applications in two-dimensional materials \cite{PQEDGap}.
\section*{\textbf{ACKNOWLEDGMENTS}}
L. F. is partially supported by Coordena\c{c}\~{a}o de Aperfei\c{c}oamento de Pessoal de N\'{i}vel Superior Brasil (CAPES), finance code 001. V. S. A. and L. O. N. are partially supported by Conselho Nacional de Desenvolvimento Cient\'{i}fico e Tecnol\'{o}gico (CNPq) and by CAPES/NUFFIC, finance code 0112. F. P. acknowledge the financial support from DIUFRO Grant DI20-0005 of the Direcci\'{o}n de Investigaci\'{o}n y Desarrollo, Universidad de La Frontera.
|
2024-02-18T23:40:24.811Z
|
2020-09-22T02:01:11.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2009.09017","language":"en","timestamp":1600740071000,"url":"https:\/\/arxiv.org\/abs\/2009.09017","yymm":"2009"}
|
proofpile-arXiv_000-10091
|
{"provenance":"002.jsonl.gz:10092"}
| null | null |
\section{Introduction}
Graph embeddings aim to learn a low-dimensional representation for each node in a graph accurately capturing relationships among the nodes. This has wide applicability in many graph analysis tasks including node classification \cite{bhagat2011node}, clustering \cite{ding2001min}, recommendation \cite{liben2007link}, and visualization \cite{maaten2008visualizing}. Various embedding methods have been proposed, including classical spectral embedding algorithms (\cite{balasubramanian2002isomap}, \cite{de2002locally}, \cite{belkin2002laplacian}), factorization (\cite{ahmed2013distributed}), neural embedding (\cite{he2017neural}, \cite{hamilton2017inductive}).
The above approaches aim to provide point estimators of embedding features, and a significant shortcoming is their inability to express variation, especially the dynamic variation when data are sequentially collected. Being able to accurately represent variation is critical. In sequence-aware recommender systems (RS), where data are collected from sessions or transactions, besides the intrinsic features, user preferences for products varies over time. Item popularity also changes with time. We use the user-movie rating in the benchmark dataset MovieLens (\cite{harper2016movielens}) as an illustration example. Figure \ref{fig:movielens} (a) displays that for a particular movie, the number of ratings varies with time; Figure \ref{fig:movielens} (b) shows that for a certain user, the number of ratings also varies with time. This example reveals the users' behavior and item popularity may dynamically change based on factors such as the available time, interest shift, and environment changes. These dynamic changes suggest not relying too much on exploitation of past behavior in favor of exploration. However, research focusing on this aspect is still lacking; see Section \ref{sec:related_ref} for detailed reference review.
\begin{figure}[h]
\centering
\begin{tabular}{cc}
\includegraphics[scale=0.4]{img/movies0_bar_plot.png} & \includegraphics[scale=0.4]{img/users0_bar_plot.png}\\
~~~~~$(a)$ & ~~~~~ $(b)$\\
%
%
\end{tabular}
\caption{ $(a)$: the barplot shows the number of ratings versus time for movie ``The Perfect Storm'' which is released on June, 2000. $(b)$: the barplot shows the number of ratings versus time for the user with user\_id 423.}\label{fig:movielens}
\end{figure}
In this paper, we propose a novel dynamic variational embedding (DVE) method for sequence-aware data to address the aforementioned gap in quantifying variation. We assume that the embedding feature for each node consists of two parts: one is the intrinsic nature and one is the variational feature that captures temporal changes sequentially.
Recurrent neural networks (RNN) have been shown to be successful at capturing the nuances of nodes' interactions in the short and long term (\cite{sutskever2014sequence}, \cite{lecun2015deep}). Therefore, we develop an RNN architecture to characterize the dynamic variance. We design an embedding feature that can characterize both short and long range dependence. A distinguishing feature of our method is that the individualized dynamic variance can be explicitly included in the model, providing a strong guidance in exploration.
Applying DVE to sequence-aware recommender systems (RS), we further develop an end-to-end deep neural network (DNN) to study link prediction. Given the explosive growth of information available on the web, RS have been widely adopted by many online services, including e-commerce, and social media sites. Personalized RS is an essential demand for facilitating a better user experience. One of the most popular RS approaches is collaborative filtering (CF) (\cite{sarwar2001item}, \cite{schafer2007collaborative}) that aims to model the users' preference on items based on their previous behavior (e.g., ratings, clicks, buy).
Among the various CF techniques, a mainstream is measuring the interactions between users and items through products of their latent features (\cite{koren2008factorization}). However, it has been shown in \cite{he2017neural} that such inner product-based models may not be sufficient to capture the complex structure of user interaction data.
DNN is flourishing in recent years (\cite{sutskever2014sequence}, \cite{he2017neural} etc). It endows the model with a large level of flexibility and non-linearity to learn the interactions between the embedding features of users and items. In this paper, we built a sequence-aware RS by fully utilizing a neural collaborative filtering framework based on DVE. The main contributions of our work are summarized below.
\begin{itemize}
\item We propose a novel dynamic variational embedding (DVE) approach to learn nodes' intrinsic and variational features simultaneously. The dynamic variational feature is achieved by introducing a recurrent neural network (RNN) into the neural embedding architecture. This is crucial for facilitating exploration.
\item We consider a sequence-aware recommender system, and show that handling temporal information plays a vital role in
improving the accuracy of the RS.
\item Based on DVE, we develop an end-to-end deep neural architecture for our sequence-aware recommender system, where user's and item's embedding features exhibit temporal dependencies, to study the link prediction. The whole neural architecture is constructed in two parts: one is the embedding layers for DVE, and one is the neural collaborative filtering layers to explore the non-linear interaction between users and items.
\end{itemize}
\vspace{-5pt}
\section{Related Works}\label{sec:related_ref}
One classical direction of embedding is factorization-based; examples include spectral embedding algorithms like IsoMap, LLE, Laplacian eigenmap in \cite{balasubramanian2002isomap}, \cite{de2002locally}, \cite{belkin2002laplacian}, and matrix factorization in (\cite{ahmed2013distributed}).
Neural networks are also used in graph embedding in recent years.
Deep neural networks have proven successful due in part to their ability to model complicated non-linear data representations. The neural collaborative filtering (NCF) proposed by \cite{he2017neural} fuses matrix factorization and one-hot embedding, and feeds them into a deep neural network framework, showing significant gains in accuracy in prediction. Recently, Graphsage \cite{hamilton2017inductive} proposes an inductive learning approach for node features based on graph convolutions, having wide applicability in massive graph problems. However, the above methods aim to provide single estimator, without characterizing variation. Latent space embedding is one approach to learn features with variation, i.e., using latent representations to characterize features of each node with Bayesian probabilistic models, including latent space models \cite{hoff2002latent}.
A variety of sequence-aware recommender systems have been proposed in the literature. \cite{koren2009collaborative} developed a time-aware factor model to address the temporal changes in collaborative filtering. \cite{ying2018sequential} developed attention-based RS based on DNN. Again, these methods are lacking variation quantification compared with our DVE based recommender architecture.
Variational autoencoders (VAEs), combining the deep latent variable model and variational learning technique, are popular in the application of recommender systems recently. \cite{li2017collaborative} proposes collaborative variational autoencoder (CVAE) approach to learn the item-based embedding in an unsupervised manner. \cite{liang2018variational} constructs a generative model with multinomial likelihood for each user’s preference on all items by assigning a low dimensional latent vector for the user's preference.
The above approaches are either item-based or user-based unsupervised learning; while our approach is supervised learning, and can learn the variational features of the users and items simultaneously. Another crucial limitation of current VAE-based recommender learnings is their insufficiency in exploration, since the key idea of VAE (e.g.,\cite{liang2018variational}) is minimizing the KL distance between the input behavior and its latent representer, which only focuses on the exploitation of the previous behaviors. However, our DVE-based approach considers both the long-term feature and the dynamic variation for each user and item, thus enabling exploration. Furthermore, the computational and storage bottleneck of the above user-based VAEs become critical for RS with millions of users and items, since the input and the training target include the whole preference for each user. In contrast, the input of our approach includes only the individualized records at time $t$, such as (user\_id, item\_id, click (or score)) at $t$, i.e., the nonzero entries in the sparse preference matrix. Thus we enjoy relatively high computational efficiency.
Our work is related to the study of NCF in \cite{he2017neural}, but
distinguished from \cite{he2017neural} by the following aspects: (1) our embedding feature is random and dynamic by considering the temporal information; (2) we consider sequence-aware RS instead of static RS.
\section{Method}
The key idea behind our dynamic variational embedding (DVE) approach is that we assume the embedding features have both an intrinsic and variational nature. In the following, we first introduce neural variational embedding algorithm in which the embedding is learned in two parts: the (intrinsic) mean and its variance. Based on such structure, we further illustrate the construction of DVE, incorporating temporal changes into the variance by employing recurrent neural networks (RNN).
\subsection{Neural Variational Embedding}\label{subsec:ve}
Suppose we have $n$ nodes. We first express each node as a binarized sparse vector with one-hot encoding. Denote the input feature of the $i$th node as $u_i$. For simplicity, we only use the identity of the node as the input feature, i.e. $u_i$ is a binary vector with $i$th entry being $1$ and other entries being $0$. Note that $u_i$ can be easily extended to content-based or neighbor-based features.
Denote the embedding feature for each node as $w_i$. Suppose $w_i$ follows the regression function as
\begin{equation}\label{eq:embed_decom}
w_i = W_1 u_i + z_{u_i},
\end{equation}
where $W_1 \in \mathbb{R}^{R\times n}$ is the mean embedding matrix, $z_{u_i}\sim N(0, \sigma_{u_i}^2 I_R)$ with $I_R$ as an $R\times R$ identity matrix, and $R$ is the embedding dimension. Define $\mu_{u_i}= W_1 u_i$, i.e., the $i$th column of $W_1$. E.q. (\ref{eq:embed_decom}) says that our embedding feature consists of two parts: the mean $\mu_{u_i}$ and the random variation $z_i$ induced by $\sigma_{u_i}$. As shown in Figure \ref{fig:VE}, the mean embedding vector $\mu_{u_i}$ can be achieved via learning $W_1$; and we learn the variance $\sigma_{u_i}^2$ from the variance embedding vector through fully connected layers. That is,
$$
\sigma_{u_i}^2 = g(W_3W_2u_i),
$$
where $W_2$ and $W_3$ are weight matrices. To guarantee $\sigma_{u_i}^2\geq 0$, the activation function $g$ for the output layer can be chosen from the following candidates based on the performance.
\begin{equation}\label{eq:active_function}
g(x)=
\begin{cases}
\max\{0,x\}, \\ %
|x|, & \\ %
x^2. & %
\end{cases}
\end{equation}
Then following (\ref{eq:embed_decom}), the embedding feature $w_i$ can be achieved by combining $\mu_{u_i}$
and $z_{u_i}$ generated from $N(0, \sigma_{u_i}^2I_R)$.
\begin{figure}[h]
\centering
\includegraphics[width=0.60\textwidth]{img/VE.pdf}
\caption{Variational embedding architecture.}
\label{fig:VE}
\end{figure}
\subsection{Dynamic Variational Embedding}
In sequence-aware data, the variation changes dynamically, which further leads to the dynamic embedding for each node. Denoting $w_i^{(t)}$ as the embedding feature of node $i$ at time $t$, we update (\ref{eq:embed_decom}) to the following temporal model
\begin{equation}\label{eq:embed_dve}
w_i^{(t)} = W_1 u_i^{(t)} + z_{u_i}^{(t)},
\end{equation}
where $u_i^{(t)}$ is the input feature of the $i$th node at time $t$, and $z_{u_i}^{(t)}$, representing the variational part, is generated from $N(0, \sigma_{u_i}^{2(t)}I_R)$. We incorporate recurrent neural networks (RNN) to learn $\sigma_{u_i}^{2(t)}$, which is different from the variational learning in Section \ref{subsec:ve}.
RNNs are powerful sequence models that take as their input not just the current input example they see, but also what they have perceived previously in time. However, it is well-known that vanilla RNNs suffer from the vanishing gradient problem. Long short-term memory units (LSTM) are a special kind of RNNs that retains similar structure to the vanilla RNN, but can solve the problem of vanishing and exploding gradients faced while training vanilla RNNs. In this part, we utilize the LSTM to train the dynamic variation of each node.
As shown in Figure \ref{fig:DVE}, the variance embedding vector $W_2u_i^{(t)}$ is fed into a recurrent neural architecture. The output dense vector depends on the current history $h_{u_i}^{(t-1)}$ by means of a recurrent layer $h_{u_i}^{(t)}$:
$$
h_{u_i}^{(t)} = RNN(h_{u_i}^{(t-1)}, W_2u_i^{(t)}).
$$
The $h_{u_i}^{(t)}$ is then fed into the fully connected layers via the weight matrix $W_3$, and finally outputs the dynamic variance $\sigma_{u_i}^{2(t)}$ based on the activation function $g$ specified in (\ref{eq:active_function}).
\begin{figure}[h]
\centering
\includegraphics[width=0.60\textwidth]{img/DVE.pdf}
\caption{Dynamic variational embedding architecture.}
\label{fig:DVE}
\end{figure}
After obtaining $\sigma_{u_i}^{2(t)}$, we generate $z_{u_i}^{(t)}$ from $N(0, \sigma_{u_i}^{2(t)} I_R)$, where $I_R$ is the identity matrix with dimension $R$. The final DVE of the $i$th node can be achieved by combining $\mu_{u_i}$ and $z_{u_i}^{(t)}$ by e.q. (\ref{eq:embed_dve}).
\section{DVE-based Neural Collaborative Filtering }
In this section, we apply the DVE to sequence-aware recommender systems, and construct a neural collaborative filtering architecture to learn the model parameters and the user-item interaction. We first provide a brief introduction of the graph notation in recommender systems.
\subsection{Notations in Recommender Systems}
Denote $G= (U,V,Y)$, where $U$ consists of $n$ users, $V$ consists of $m$ items, $Y= (Y^{(1)}, \cdots, Y^{(T)})$, and each $Y^{(t)}$ is an $n\times m$ incidence matrix, with each entry $y^{(t)}_{ij}$ denoting the value of the interaction between user $i$ and item $j$ at time $t$, where $i=1, \cdots, n$, $j=1, \cdots, m$, $0<t \leq T$. For example, in e-commercial recommender systems, $y^{(t)}_{ij} =0,1,2,3$ represents that user $i$ has no access/no response, click, add to cart and buy actions on item $j$ at time $t$, respectively. In recommender rating systems, $y^{(t)}_{ij}=0,1,\cdots, 5$ denotes the possible ratings of user $i$ on item $j$ at time $t$: $0$ denotes no access, $1$ denotes a poor rating and $5$ is the maximum value allowed. Denote $W^{(t)}= (w^{(t)}_1, \cdots, w^{(t)}_n)\in \mathbb{R}^{R\times n}$ as the embedding matrix of $n$ users at time t, with $w^{(t)}_i\in \mathbb{R}^{R\times 1}$ as the embedding feature of the $i$th user at time $t$. Similarly, define $Q^{(t)} = (q^{(t)}_1, \cdots, q^{(t)}_m)\in \mathbb{R}^{R\times m}$ as the embedding matrix of $m$ items at time $t$, with $q^{(t)}_j\in \mathbb{R}^{R\times 1}$ as the embedding feature of the $j$th item at time $t$.
Define $u_i^{(t)}$ and $v_j^{(t)}$ as the input feature of user $i$ and item $j$ at time $t$, respectively. In fact, for user $i$, $u_i^{(1)} = \cdots = u_i^{(T)}= u_i$; for item $j$, $v_j^{(1)} = \cdots = v_j^{(T)}= v_j$, with $u_i$, $v_j$ encoded following Section \ref{subsec:ve}. The purpose of introducing the index ${(t)}$ here is to activate the current history in the RNN layer when learning DVE. Notations are summarized in Table \ref{table:notation}.
\begin{table}[h]
\small
\centering
%
\begin{tabular}{c|p{0.3\textwidth}}
\hline\hline
{\bf{Notation}} & {\bf{Description}}\\
\hline
U & the set of $n$ users\\
V & the set of $m$ items\\
$Y^{(t)}$ & the $n\times m$ incidence matrix at time $t$ \\
$W^{(t)}$ & the embedding matrix of users at $t$\\
$w_i^{(t)}$ & the embedding feature of user $i$ at $t$\\
$Q^{(t)}$ & the embedding matrix of items at $t$\\
$q_j^{(t)}$ & the embedding feature of item $j$ at $t$\\
$u_i^{(t)}$ & the input feature of user $i$ at $t$\\
$v_j^{(t)}$ & the input feature of item $j$ at $t$\\
\hline\hline
\end{tabular}
\caption{Notations.}\label{table:notation}
\end{table}
\subsection{Dynamic Neural collaborative filtering}
Collaborative filtering predicts what items a user will prefer by discovering and exploiting the similarity patterns across users and items. Here we use the DVE layers to learn user/item embedding features. Inspired by \cite{he2017neural}, we construct the neural collaborative filtering (NCF) layers, and combine them with the DVE layers to learn the model parameters and the user-item interaction. Figure \ref{fig:NCF} illustrates the dynamic NCF architecture. As shown in Figure \ref{fig:NCF}, we fed the user/item embeddings based on DVE into a multi-layer neural architecture, and finally output the predicted score $\hat{y}_{ij}^{(t)}$. The training is performed by minimizing the loss function as specified in the following part.
Given the embedding feature matrix $W^{(t)}$ and $Q^{(t)}$, the predicted score between user $i$ and item $j$ at time $t$ can be expressed as
\begin{equation}\label{eq:est_y}
\hat{y}_{ij}^{(t)} = f(W^{(t)}u_i^{(t)}, Q^{(t)}v_j^{(t)} | W^{(t)}, Q^{(t)}),
\end{equation}
where $f(\cdot)$ is the interaction function defined as
\begin{equation*}
f(W^{(t)}u_i^{(t)}, Q^{(t)}v_j^{(t)}) = \varphi_{out}\Big(\varphi_M\big(\dots \varphi_1(W^{(t)}u_i^{(t)}, Q^{(t)}u_j^{(t)})\big)\Big),
\end{equation*}
where $\varphi_{out}$ and $\varphi_M$, respectively, denote the mapping function for the output layer and the $M$-th neural collaborative filtering layer, and there are $M$ NCF layers in total. Therefore, $M$ determines the model's learning capacity. Note that $w_i^{(t)} = W^{(t)}u_i^{(t)}$ is user $i$'s embedding feature obtained via DVE, i.e., model (\ref{eq:embed_dve}). Similarly, $q_j^{(t)}=Q^{(t)}u_j^{(t)}$ is the item $j$'s embedding feature obtained via DVE with
$
q_j^{(t)} = Q_1 v_j^{(t)} + z_{v_j}^{(t)}.
$
\begin{figure}[h]
\centering
\includegraphics[width=0.70\textwidth]{img/NCF.pdf}
\caption{Neural collaborative filtering architecture with DVE. }
\label{fig:NCF}
\end{figure}
Commonly used feedbacks in RS include two categories: explicit (e.g., ratings, votes) and implicit (e.g., clicks, purchases). Explicit feedback data are often in the form of numeric ratings from users to express their preferences regarding specific items. In this case, we can view $y_{ij}^{(t)}$ as a continuous variable. We use squared loss to learn model parameters, and the loss function is defined as
\begin{equation}
\mathcal{L}_{sq} = \sum_{t=1}^T \sum_{(i,j)\in (U\cup V)^{(t)}} \big(y_{ij}^{(t)} - \hat{y}_{ij}^{(t)}\big)^2,
\end{equation}
where $(U\cup V)^{(t)}$ refers to the observed interaction between user and item nodes at time $t$.
Implicit feedback data are easier to collect, it is also called one-class RS in which only positive implicit feedback can be observed. The target value $y_{ij}^{(t)}$ is $1$ if user $i$ and item $j$ have interaction at time $t$, and $0$ otherwise. For the binary response case, in order to guarantee $\hat{y}_{ij}^{(t)}\in \{0,1\}$, we impose a logistic model on the activation function for the output layer $\varphi_{out}$, i.e.,
$\varphi_{out}(x) =\frac{e^x}{1+e^x}$. Denote $\mathcal{Y}^{(t)}$ as the set of observed interactions in $Y$ at time $t$, and $\mathcal{Y}^{(t)-}$ as the set of negative instances, which can be no interactions or unobserved interactions. Define $\mathcal{Y} = (\mathcal{Y}^{(1)}, \cdots, \mathcal{Y}^{(T)})$, $\mathcal{Y}^{-} = (\mathcal{Y}^{(1)-}, \cdots, \mathcal{Y}^{(T)-})$, $W= (W^{(1)}, \cdots, W^{(T)})$, $Q = (Q^{(1)}, \cdots, Q^{(T)})$, also denote $\Theta$ as the model parameters in the neural architecture. Then the likelihood function can be written as
$$
P(\mathcal{Y}, \mathcal{Y}^{-}|W, Q, \Theta) = \prod_{t=1}^T \prod_{(i,j)\in \mathcal{Y}^{(t)}}\hat{y}_{ij}^{(t)} \prod_{(i,j)\in \mathcal{Y}^{(t)-}} (1-\hat{y}_{ij}^{(t)}),
$$
where $\hat{y}_{ij}^{(t)}$ is estimated by e.q.(\ref{eq:est_y}). The log-likelihood loss function can be written as
$$
\mathcal{L} = \sum_{t=1}^T \sum_{(i,j) \in \mathcal{Y}^{(t)}\cap \mathcal{Y}^{(t)-}} \big(y_{ij}^{(t)} \log \hat{y}_{ij}^{(t)} + (1-y_{ij}^{(t)})\log(1-\hat{y}_{ij}^{(t)})\big).
$$
In practice, when the interactions are sparse, we uniformly sample from the negative sets $\mathcal{Y}^{(t)-}$ at each time $t$, and control the sampling ratio to the range between $1:3$ and $1:5$.
\section{Experimental Results}
In this section, we implement our proposed DVE method on sequence-aware recommender systems, and compare with existing methods using one of the most popular public data sets: Movielens. We first compare our proposed method with several competitors which are designed for explicit data. We further exam its performance on implicit data, by transforming the ratings into 0 or 1 based on whether the user has rated the item or not.
The numerical studies are run on a computing workstation with two Titan-V GPU processors and 64GB RAM.
\subsection{Movielens data description}
The Movielens-1M data set is collected by GroupLens Research and is downloaded from \url{http://grouplens.org/datasets/movielens}. It contains $1,000,209$ ratings of $3883$ movies by $6040$ users, and the rating scores range from 1 to 5. The data are collected from April 2000 to February 2003. The timestamps are recorded to show when a user rates a movie.
We observe that the variation of the popularity of the movie and the preference of the user dramatically changes over time. For example, the number of viewers might be large in the first few months following its release date, then decrease after that. We plot the number of ratings versus time for the movie titled as ``The Perfect Storm'' in Figure \ref{fig:movielens}(a). It shows that the number of ratings is increasing from May to December of $2000$ and dropped to less than $20$ in $2001$. The strong temporal pattern motivates us to model the variance dynamically. Movies having features with large variance will tend to be recommended to a broader range of users, while lower feature variance narrows the recommendation range.
\subsection{MovieLens data with explicit feedback}
We directly use the rating as explicit feedback. The proposed method is compared with the following competitors designed for explicit feedback recently.
\begin{itemize}
\item \citet{agarwal2009regression} proposes a regression-based latent factor model.
\item \citet{mazumder2010spectral} provides a soft-impute algorithm to replace the missing elements with those obtained from a soft-thresholded SVD.
\item \citet{zhu2016personalized} proposes a likelihood method to seek a sparse latent factorization, from a class of overcomplete factorizations, possibly with a high percentage of missing values.
\item \citet{he2017neural} establishes the general NCF framework based on one-hot embedding layer for latent features of each user and item.
\item \citet{bi2017} proposes a group-specific method to use dependency information from users and items which share similar characteristics under the singular value decomposition framework.
\end{itemize}
We first order all the ratings based on their timestamps. Then we set the first $75\%$ as the training data set and set aside the last $25\%$ of ratings as the testing data set.
The root mean square error (RMSE) of the testing set is reported in Table \ref{tab:1}.
\begin{table}[h!]
\centering
\begin{tabular}{l|c}
\hline
\multicolumn{2}{c}{Dataset: Movielens-1m}\\
\hline
Method & RMSE\\
\hline
DVE (our proposed) & \textbf{0.891} \\
\citet{agarwal2009regression} & 1.197 \\
\citet{mazumder2010spectral} & 1.073 \\
\citet{zhu2016personalized}& 1.063\\
\citet{bi2017} & 0.964 \\
\citet{he2017neural} & 0.933 \\
\hline
\end{tabular}
\caption{Experimental results of Movielens-1M dataset with explicit feedback. }\label{tab:1}
\end{table}
Table \ref{tab:1} provides the prediction results on the testing set, and shows that our proposed method outperforms other methods significantly. The RMSE of the proposed method is 25.6\% less than \citet{agarwal2009regression}, 17.0\% less than \citet{mazumder2010spectral}, 16.2\% less than \citet{zhu2016personalized}, 7.6\% less than \cite{bi2017},
and 4.5\% less than \citet{he2017neural}.
\subsection{Movielens data with implicit feedback}
In this setting, we code the user-movie interaction as a binary variable in which $1$ indicates that the user rates the movie and $0$ indicates that the rating is missing. %
We use leave-one-out evaluation to evaluate the performance of item recommendation; see \cite{he2017neural,bayer2017generic,he2016fast} etc. For each user, we hold-out his/her latest interaction as the test item and utilize the remaining data for training. To increase the computational efficiency, we randomly sample $100$ items that are not interacted with by the user, and rank the test item among the $100$ items.
The evaluation is done on top-$k$ recommendation. The performance of a ranked list is judged by the overall {\it{top-$k$ Hit Ratio}} (HR@k)
and overall {\it{top-$k$ Normalized Discounted Cumulative Gain}} (NDCG@k). The HR@k measures whether the test item is included in the top-$k$ list.
NDCG@k gives more weight to the relevant items on top of the recommender list, and is defined as
\begin{equation}
NDCG@k = \frac{\sum_{i=1}^k r_i/\log_2(i+1)}{\sum_{i=1}^{k}1/\log_2(i+1)},
\end{equation}
The performance of a ranked list is judged by the averaged HR@10 and NDCG@10 for all users. We compare with \cite{he2017neural} for their three methods: the generalized matrix factorization method (GM), the pure neural network framework with one-hot embeddings (MPL), and a fusion of the two (Neural-GM). We implement their algorithm using the docker image provided in \url{https://github.com/hexiangnan/neural_collaborative_filtering}.
\begin{table}[h!]
\centering
\begin{tabular}{l|c|c}
\hline
\multicolumn{3}{c}{Movielens-1M with implicit feedback}\\
\hline
Method & NDCG@10 & HR@10 \\
\hline
Our proposed & \textbf{0.4211} & \textbf{0.6924}\\
\citet{he2017neural}-GM & 0.3676 & 0.6358 \\
\citet{he2017neural}-MPL & 0.3942 & 0.6737 \\
\citet{he2017neural}-Neural-GM & 0.4073 & 0.6790 \\
\citet{liang2018variational}-VAE& 0.0416 & 0.4481\\
\hline
\end{tabular}
\caption{Experimental results for Movielens-1M data set with implicit feedback.}\label{tab:s2}
\end{table}
Table \ref{tab:s2} shows that our proposed method outperforms the state-of-the-art methods \citet{he2017neural}-GM, \citet{he2017neural}-MPL and \citet{he2017neural}-Neural-GM by a large
3\% -15\% and 2\%-9\%, respectively. We also compare our proposed method with the VAE-based collaborative filtering studied in \cite{liang2018variational}. As shown in Table \ref{tab:s2}, the NDCG@10 is lower than $0.1$, and HR@10 is lower than $0.5$. In \cite{liang2018variational}, they use the evaluation strategy by holding out several users as the testing set. However, in this study, for each user, we hold out the latest interaction as the testing set for evaluation. The VAE-based approach has limitation in exploration under such evaluation strategy, since its input treats all the unobserved movies as $0$ and the VAE is designed to learn the representation of the input. Instead, our proposed method uses negative sampling to sample a small portion of the negative set, thus it turns out to have better performance in exploration for existing users.
\section{Discussion}
We propose a dynamic variational embedding framework and implement it for collaborative filtering with temporal information. Our method is simple and generic; it is not limited to the applications presented in this paper, but is designed to any embedding task. This work complements the mainstream embedding models by incorporating variation and dynamic changes, opening up a new avenue of research possibilities for wide range of embedding models. In the future, we will study the knowledge-based embedding to model auxiliary information, such as user reviews, user geographical information, and movie reviews. Individualized or itemized information could help us to better understand the uncertainty and dynamic pattern of the embeddings.
\bibliographystyle{plainnat}
|
2024-02-18T23:40:24.814Z
|
2020-09-21T02:18:18.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2009.08962","language":"en","timestamp":1600654698000,"url":"https:\/\/arxiv.org\/abs\/2009.08962","yymm":"2009"}
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proofpile-arXiv_000-10092
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{"provenance":"002.jsonl.gz:10093"}
| null | null |
\section{Conclusion}
\par Our work studies how adversarially perturbing feature statistics simulates domain shift in image data. We find that adversarial fine-tuning on features perturbed in this way improves robustness to data stylization and corruption without ever training on auxiliary data. Training with Adversarial Batch Normalization (AdvBN) is computationally cheap and can quickly make pre-trained models less brittle. We fine-tune a ResNet-50 with our algorithm and surpass the performance of state-of-the-art methods on both ImageNet-Instagram and Stylized-ImageNet. Adversarial feature statistics are a promising direction for creating models that generalize well to a variety of domains.
\section{Adversarial Batch Normalization}
\label{sec:AdvBN}
We propose {\em Adversarial Batch Normalization} (AdvBN), a module that adversarially perturbs deep feature distributions such that the features confuse CNN classifiers. We iteratively compute adversarial directions in feature space by taking PGD steps on batch statistics.
In the next section, we will train on these perturbed feature distributions in order to produce models robust to domain shifts.
Consider a pre-trained classification network, \(g\), with $L$ layers. We divide \(g\) into two parts, $g^{1,l}$ and $g^{l+1, L}$, where $g^{m ,n}$ denotes layers $m$ through $n$. Now, consider a batch of data, $x$, with corresponding labels, $y$. Formally, the AdvBN module is defined by
\begin{equation}
\begin{array}{cccc}
\text{BN}_{\text{adv}}(x;g,l, y) =
\delta_{\sigma}'\cdot(f-\mu(f))+\delta_{\mu}' \cdot \mu(f)
\text{, where
f = g^{1,l}(x),\\
&\\
(\delta_{\mu}', \delta_{\sigma}') = \argmax_{(\delta_\mu, \delta_\sigma)} \mathcal{L}\bigg[g^{l+1, L}\bigg(\delta_\sigma \cdot (f-\mu(f))+\delta_{\mu} \cdot \mu(f)\bigg), y\bigg],\\
&\\
\text{subject to } \Vert \delta_{\mu}-1\Vert_{\infty} \leq \epsilon, \Vert \delta_{\sigma}-1 \Vert_{\infty}\leq \epsilon,
\end{array}
\end{equation}
where the maximization problem is solved with projected gradient descent. Simply put, the AdvBN module is a PGD attack on batch norm statistics which can be inserted inside a network. Note that $\delta_\mu, \delta_\sigma$ are vectors with length equal to the number of channels in the output of layer $l$, and we multiply by them entry-wise, one scalar entry per channel, similarly to Batch Normalization. Additionally, note that this module acts on a per-batch basis so that features corresponding to an individual image may be perturbed differently depending on the batch the image is in.
We formulate the perturbation by first subtracting out the mean so that $\delta_\mu\cdot\mu(f)$ is the new mean of the adversarial features, and $\delta_\mu$ directly controls the new mean. We choose $\delta_\mu\cdot \mu(f)$, rather than simply $\delta_\mu$, to represent the new mean of the perturbed features so that $\ell_\infty$ bounds and steps size do not need to depend on the mean of $f$. Intuitively, we permit the mean of adversarial features to vary more when the clean features have a mean of higher magnitude.
\begin{figure}[b!]
\centering
\resizebox{\linewidth}{!}{
\includegraphics{figs/demo.pdf}
}
\caption{\textbf{Examples of perturbed ImageNet images generated by AdvBN with a decoder}. The first row contains the original versions from the ImageNet validation set.}
\label{fig:demo1}
\end{figure}
\paragraph{Visualizing feature shifts}
To verify our assumption that adversarially perturbing feature statistics corresponds to transforming the distribution in image space, we visualize the effects of AdvBN.
We adopt the VGG-19 based autoencoder from \citet{huang2017arbitrary}. At the bottleneck of the autoencoder, we plug in an AdvBN module followed by a VGG-19 classifier \citep{vgg}. We visualize perturbations by feeding the AdvBN outputs to the decoder.
In Figure~\ref{fig:demo1}, images crafted through this procedure are visibly different from the originals; semantic content in the original images is preserved, but the new images exhibit differences in color, texture, and edges. We draw two major conclusions from these visualizations which highlight the adversarial properties of these domains. The first one concerns textures: according to \citet{geirhos2018imagenet}, CNNs rely heavily on image textures for classification. Images from the adversarial domain, on the other hand, have inconsistent textures across samples. For example, the furry texture of a dog is smoothed in column 2, and the stripes disappear from a zebra in column 4, whereas visible textures appear in columns 6 and 8. The second conclusion pertains to color. Results in \citet{zhang2016colorful} suggest that colors serve as important information for CNNs. In the adversarial domain, we find suppressed colors (columns 1, 3) and unnatural hue (columns 5, 7). See Appendix A.3 for additional example images generated by this procedure. Figure \ref{fig:advsteps1} illustrates how the appearance of reconstructed images shifts as adversarial perturbations to feature statistics become larger.
We use this visualization technique to process the entire ImageNet validation set and denote it as ImageNet-AdvBN in Figure~\ref{fig:datasets}. By evaluating different methods on this dataset, we observe that accuracy on ImageNet-AdvBN is consistently lower than that on the original ImageNet dataset. This degradation in performance validates the adversarial property of features generated by AdvBN. Experiments concerning performance on ImageNet-AdvBN are listed in Appendix A.2.
\begin{figure}[h!]
\centering
\resizebox{\linewidth}{!}{
\includegraphics{figs/advstrength.pdf}
}
\caption{\textbf{The effect of adversarial strength on visualized examples}. $\epsilon = 0$ corresponds to images reconstructed by our autoencoder without AdvBN.}
\label{fig:advsteps1}
\end{figure}
\section{Training with Adversarial Batch Normalization}\label{training}
In this section, we use the proposed AdvBN module to train networks on the perturbed features. The goal of this adversarial training routine is to produce networks that both perform well on the data distribution on which they are trained and also generalize well to many new domains, all without having to obtain auxiliary data from new domains.
We start with a pre-trained model, \(g = g^{l+1, L}\circ g^{1,l}\), and we fine-tune the subnetwork, \(g^{l+1, L}\), on clean and adversarial features simultaneously. To this end, we solve the following min-max problem,
\begin{equation}
\min_{\theta} \mathbb{E}_{(x,y)\sim \mathcal{D}} \bigg[ \mathcal{L}(g_{\theta}^{l+1,L}\circ \text{BN}_\text{adv}\circ g^{1,l} (x),y) + \mathcal{L}(g_{\theta}^{l+1,L}\circ g^{1,l} (x),y)\bigg],
\end{equation}
where $\mathcal{L}$ denotes cross-entropy loss, and $\mathcal{D}$ is the distribution of batches of size $n$. This optimization problem contains a maximization problem inside the $\text{BN}_\text{adv}$ layer. In order to maintain the network's performance on natural images, we adopt a similar approach to \citet{xie2019adversarial} by using auxiliary batch normalization in \(g^{l+1, L}\) for adversarial features; we use the original BNs when propagating clean features, and we use auxiliary ones for adversarial features. See Algorithm \ref{algo:adv1} for a detailed description of our method for fine-tuning the parameters of \(g^{l+1, L}\).
\par Since we start with pre-trained models, we only need to fine-tune for $20$ epochs, yielding improved robustness with little additional compute. Moreover, we only modify the parameters of later layers, so we do not need to backpropagate through the first half of the network. See Appendix B for an analysis of training times using our method.
In the following section, we measure the performance, on several datasets, of our model fine-tuned using adversarial training with AdvBN.
\begin{algorithm}[ht]
\KwIn{Training data, pretrained network \(g = g_{\theta}^{l+1, L}\circ g^{1,l}\), PGD bound \(\epsilon\), and PGD step size \(\tau\) }
\KwResult{Updated network parameters, \(\theta\), of subnetwork \(g_{\theta}^{l+1, L}\) }
\For{epoch = 1, \dots, N}{
Sample mini-batch \(x\) with label \(y\)\;
Obtain feature map \(f = g^{1,l}(x)\)\;
Initialize perturbation: \(\delta = (\delta_{\mu}, \delta_{\sigma}\))\;
Let \(f_{adv} = f\)\;
\For{adversarial step = 1, \dots, m}{
$f_{adv} \leftarrow \delta_{\sigma}\cdot (f - \mu(f)) + \delta_{\mu} \cdot \mu(f)$\;
Update $\delta$:
$\delta \leftarrow \delta + \tau \cdot sign(\nabla_{\delta} \mathcal{L}(g_\theta^{l+1,L}(f_{adv}), y))$\;
$\delta \leftarrow$ clip $(\delta, 1-\epsilon, 1+\epsilon)$\;
}
$f_{adv} \leftarrow \delta_{\sigma}\cdot (f - \mu(f)) + \delta_{\mu} \cdot \mu(f)$\;
Minimize the total loss w.r.t. network parameter:
$\theta \leftarrow \argmin\limits_{\theta}\mathcal{L}(g_\theta^{l+1,L}(f_{adv}), y) + \mathcal{L}(g_{\theta}^{l+1,L}(f), y)$\;
}
\KwRet{$\theta$}
\caption{Training with Adversarial Batch Normalization}
\label{algo:adv1}
\end{algorithm}
\section{ImageNet-AdvBN Experiments}
\subsection{Creation of the ImageNet-AdvBN dataset}
We process the entire ImageNet validation set using the visualization technique introduced in Section~\ref{sec:AdvBN}. We consider two encoder architectures: one is the VGG-19 encoder we use for visualization, another consists of layers of a ResNet-50 up to \texttt{conv2\_3}. Both encoders are paired with the same decoder architecture from \citet{huang2017arbitrary}.The resulting datasets, denoted by ImageNet-AdvBN-VGG and ImageNet-AdvBN-ResNet respectively, contain 50000 images each. The data we synthesize for testing other models is generated using these autoencoders that contain the AdvBN module but on ImageNet validation data. AdvBN is conducted with 6 steps, stepsize $= 0.20$, $\epsilon = 1.1$, and a batchsize of $32$. We do not shuffle the ImageNet validation data when generating these batches.
\subsection{Classification on ImageNet-AdvBN}
\input{tables/AdvBN}
Table~\ref{a:advbn} shows the classification performance of various models on the two ImageNet-AdvBN variants, denoted as Im-Adv-VGG and Im-Adv-ResNet respectively.
We also test these models on ImageNet images that are reconstructed using our autoencoders, denoted as VGG Reconstructed and ResNet Reconstructed, for each autoencoder. The performance gap between ImageNet-AdvBN and Reconstructed ImageNet indicates that the degradation on ImageNet-AdvBN is not solely caused by the reconstruction loss due to the autoencoders we use.
\subsection{Additional Example Images}
We include more images from ImageNet-AdvBN-VGG in this section. Example images in Figure~\ref{fig:examples} are randomly chosen. We do not include the ImageNet-AdvBN-ResNet, because the resulting images are mostly in extreme contrast with small textures that are hard to observe. It is possible that features output from ResNet based encoders are more sensitive to AdvBN perturbations; another explanation is that the features we extract from ResNet-50 are relatively shallow features compared to their VGG counterparts.
\begin{figure}[!ht]
\centering
\resizebox{\linewidth}{!}{
\includegraphics{figs/examples.pdf}
}
\caption{More example images. For each pair of adjacent columns, original versions are on the left, ImageNet-AdvBN-VGG is on the right.}
\label{fig:examples}
\end{figure}
\section{Runtime Analysis}
\subsection{Runtime of training with AdvBN}
We evaluate the training time of our method on a workstation with 4 GeForce RTX 2080 Ti GPUs. We use the default settings for AdvBN on ResNet-50: an AdvBN module after the \texttt{conv2\_3} layer, a fixed AutoAugment policy, and 20 epochs of fine-tuning with 6-step PGD inside the AdvBN module. Fine-tuning is conducted on the ImageNet training set, containing 1.3 million images. Training in this setting takes approximately 40 hours with batchsize set to 256 .
\subsection{Comparison with other methods concerning training budget}
We use the same infrastructure above to evaluate the training time of other ResNet-50 models that appear in Table~\ref{crossDomain}. Training code for all other methods are obtained from official repository. For all methods, we use a batch size of $128$, because augmix\citep{hendrycks2019augmix} cannot run with a batch size of $256$ on our workstation due to limited GPU memory. The number of processes in the dataloader is set to be 16. The speed values in Table~\ref{speed} are averaged over 100 iterations. The estimated training duration is calculated by multiplying the speed and corresponding total number of iterations. Time spent on evaluation after each epoch is not considered in this estimation.
\input{tables/speed}
Training with AdvBN takes a long time per iteration because each iteration contains 6 PGD steps. The SIN~\citep{geirhos2018imagenet} runtime speed is estimated based on standard training, since the model architectures and training procedures of these two methods are the same, except that SIN is trained for half the epochs but twice the data and thus twice the number of iterations per epoch. Training SIN requires additional access to Stylized-ImageNet as training data, which takes 134GB disk space; the time for generating the Stylized-ImageNet dataset and the extra storage cost are not considered in Table~\ref{speed}.
\section{Inference using models trained with AdvBN}
Models containing Batchnorm layers will have two set of BN statistics in deeper layers that have been fine-tuned by AdvBN because we use auxiliary BNs introduced by \citep{xie2019adversarial} for propagating adversarial features crafted by the AdvBN module.
During evaluation, we can choose either of the BN statistics to normalize features. The results we report in previous sections with regard to ImageNet, ImageNet-C and ImageNet-Instagram are obtained by using BN statistics corresponding to original features. We only use auxiliary BNs, which keep the batch statistics of adversarial features, to test performance on Stylized-ImageNet in Table~\ref{crossDomain}. We also use auxiliary BNs for evaluating performances on ImageNet-AdvBN and Reconstructed ImageNet in Table~\ref{a:advbn}.
\section{Background}
\subsection{Feature normalization}
Feature normalization is an important component of modern neural networks that stabilizes training and improves model generalization. Let $f \in \mathbb{R}^{N \times C \times H \times W}$ denote feature maps output by a layer, where $N$ is the batch size, $C$ is the number of channels, and $H$ and $W$ represent the height and width of the feature maps, respectively. Different normalization methods compute the mean, $\mu$, and standard deviation, $\sigma$, over different dimensions of the feature maps. They use the derived feature statistics, often along with learned multiplicative and additive parameters, to produce normalized features, $f'$:
\begin{equation}
f' = \gamma\cdot\frac{f-\mu(f)}{\sigma(f)}+\beta,
\end{equation}
where $\gamma$ and $\beta$ are learnable parameters which re-scale and shift normalized features. For example, Batch Normalization (BN)~\citep{ioffe2015batch} estimates feature statistics along the $N, H, W$ dimensions.
On the other hand, Instance Normalization (IN)~\citep{ulyanov2016instance} computes $\mu$ and $\sigma$ for each individual sample in the batch and only normalizes across the $H$ and $W$ dimensions.
Although feature normalization was originally proposed to accelerate the training process~\citep{bjorck2018understanding}, previous work~\citep{huang2017arbitrary, li2016revisiting} has shown that feature statistics effectively capture information concerning the appearance of images. Motivated by this observation, we impose uncertainty on these statistics during training in order to obtain models that are less sensitive to non-semantic characteristics, thus generalizing to images with different appearances.
\subsection{Adversarial training}
Untargeted adversarial examples are generated by maximizing classification loss with respect to the input. One popular method, projected gradient descent (PGD), involves performing gradient ascent in the signed gradient direction and projecting the perturbation in order to enforce an $\ell_{\infty}$-norm constraint \citep{madry2017towards}. Adversarial training aims to solve the saddlepoint optimization problem,
\begin{align} \min_{\theta}\mathbb{E}_{(X,y)\sim\mathcal{D}}\left[ \max_{\|\mathbf{\delta}\|_{p}<\epsilon}\mathcal{L}(g_{\theta}(X+\mathbf{\delta}),y) \right],\end{align}
where $g_\theta$ is a model with parameter vector $\theta$, $X,y$ is a clean input and the corresponding label drawn from distribution $\mathcal{D}$, and $\mathcal{L}$ denotes cross-entropy loss. Adversarial training solves this problem by iteratively sampling a batch of data, perturbing the batch adversarially, and performing a parameter update on the new adversarial batch \citep{madry2017towards}. We harness adversarial training in order to create models robust to distributional shifts rather than the standard pixel-wise adversarial attacks.
\iffalse
\begin{algorithm}[h]
\KwIn{Input \(x\), label \(y\), network \(g \), \(\ell_{\infty}\) bound \(\epsilon\), and step size \(\tau\);\\}
\KwResult{Adversarial example \(x_adv\);\\ }
Initialize \(\delta \sim \mathcal{U}(-\epsilon,\epsilon)\);\\
\For{adversarial step = 1, \dots, m}{
\(\delta \leftarrow \delta + \tau \cdot sign(\nabla_{\delta} \mathcal{L}(g(x+\delta), y))\);\\
\(\delta \leftarrow \text{clip}(\delta, -\epsilon, \epsilon)\);\\
}
\KwRet{\(x_adv = x+\delta\)}
\caption{PGD attack}
\label{algo:PGD}
\end{algorithm}
\fi
\section{Experiments}
\subsection{Implementation}
Our method begins with a torchvision ImageNet pre-trained ResNet-50~\citep{resnet}. We insert the AdvBN module at the end of the 2\textsuperscript{nd} convolutional stage. The model is fine-tuned following Algorithm~\ref{algo:adv1} for 20 epochs. The learning rate starts at 0.001 and decreases by a factor of 10 after 10 epochs with a batch size of 256. We use SGD with momentum 0.9 and weight decay coefficient $10^{-4}$. We augment inputs with a fixed AutoAugment~\citep{cubuk2019autoaugment} policy. Adversarial parameters are \(\tau=0.2\) and \(\epsilon=1.1\) with 6 repeats.
\subsection{Generalization to ImageNet Variants}
\paragraph{Datasets.} We compare our method to other methods designed to produce classification networks which generalize better. The datasets we consider are ImageNet \citep{imagenet_cvpr09} and its variants:
\begin{itemize}
\item \textbf{ImageNet-C}~\citep{hendrycks2019benchmarking} contains distorted images with 15 categories of common image corruption applied, each with 5 levels of severity. Performance on this dataset is measured by mean Corruption Error (mCE), the average classification error over all 75 combinations of corruption type and severity level, weighted by their difficulty.
\item \textbf{ImageNet-Instagram}~\citep{instagram} is composed of ImageNet images filtered with a total of 20 different Instagram filters. This dataset contains 20 versions of each ImageNet image, each with a different filter applied.
\item \textbf{Stylized-ImageNet}~\citep{geirhos2018imagenet} consists of images from the ImageNet dataset, each stylized using AdaIN~\citep{huang2017arbitrary} with a randomly selected painting. Textures and colors of images in this dataset differ heavily from the originals.
\end{itemize}
\paragraph{Models.} Our baseline model is the publicly available torchvision ResNet-50 pre-trained on ImageNet, denoted as ``Standard'' in Table~\ref{crossDomain}. All models we compare to, aside from SIN \citep{geirhos2018imagenet}, are not trained on any of the ImageNet variants that are used for evaluation. The PGD model is adversarially trained with the PGD attack on inputs and is provided by \citet{madrylab}. MoEx~\citep{li2020feature} changes feature moments inside networks but not in an adversarial manner. IBN-Net~\citep{ibnnet} improves the generalization of networks by combining batch normalization and instance normalization. AugMix~\citep{hendrycks2019augmix} is a data augmentation method that solves the distributional mismatch between training and testing data and increases classification robustness. SIN is a network trained on both Stylized ImageNet and ImageNet. We do not measure the accuracy of SIN on Stylized-ImageNet since it acquires knowledge of the target domain during training. Note that all models we use in our comparisons are the original versions released by the authors of the original work.
\input{tables/crossDomain}
\paragraph{Results.} As shown in Table~\ref{crossDomain}, the performance of baseline model significantly degrades on all three ImageNet variants, highlighting the brittleness of this high performance classification model when tested on novel distributions. Fine-tuning with AdvBN, on the other hand, substantially improves the performance of the standard ResNet-50 model. In particular, we achieve an 8.1\% accuracy gain on ImageNet-C through fine-tuning with AdvBN. On Stylized-ImageNet and ImageNet-Instagram, our model not only vastly improves upon the baseline model but also achieves the best performance among all methods with which we compare.
The consistent performance boost across all three benchmarks demonstrates that adversarial training with AdvBN can effectively enhance robustness against various distributional shifts. Note that this AdvBN model has additional auxiliary BN layers. See Appendix C for details concerning inference.
\subsection{Ablation Study}
\paragraph{Where should the AdvBN module be placed within a network?} The proposed AdvBN module can be inserted after any layer in a deep network.
In this part, we try inserting the AdvBN at deeper layers, \emph{i.e.}\@\xspace, $ \texttt{conv3\_4}$ and $\texttt{conv4\_6}$.
From the results in Table~\ref{abaltion}, we observe that $\texttt{conv4\_6}$ yields the worst performance among all three ImageNet variants, indicating that using AdvBN at deeper layers is not as helpful as at shallower layers.
We hypothesize two possible explanations for this phenomenon: (1) there are fewer trainable parameters when only very deep layers are fine-tuned; (2) features are more abstract in deeper layers, and perturbing the mean and standard deviation of high-level features can lead to extremely chaotic feature representations that are harmful for classification.
\input{tables/ablation}
\paragraph{Adversarial strength.} The strength of the adversarial attack in the adversarial training framework has a major impact on model performance \citep{madry2017towards}.
We test a range of PGD parameters to demonstrate how the strength of AdvBN affects model performance.
We measure strength by the perturbation bound $\epsilon$, where we fix $\tau$ to be 0.2 for all settings, and change the number of repeats for different bounds. The number of repeats $n$ for each $\epsilon$ is set to be $n = [\epsilon / 0.2] +1$.
Results concerning the impact of adversarial strength are listed in Table~\ref{abaltion}.
\paragraph{Multiplicative vs. additive perturbations.}
We originally chose a multiplicative noise so that a single perturbation bound can be applied to various layers and architectures regardless of the range in which feature values lie. To adjust the perturbation bound for an additive noise for each batch of feature, we compute maximum mean and standard deviation values across channels, $\mu_{max}$ and $\sigma_{max}$. Then, in the projection step of the PGD attack, we project perturbations to the mean and standard deviation into the ranges $(-\epsilon \cdot \mu_{max}, \epsilon \cdot \mu_{max})$ and $(-\epsilon \cdot \sigma_{max}, \epsilon \cdot \sigma_{max})$, respectively.
The model is denoted ``additive $\delta$'' in Table~\ref{abaltion}. This variant results in slightly degraded accuracy on each dataset.
\paragraph{Data augmentation.} AdvBN performing in feature space can easily be combined with input space data augmentation. To determine what portion of the improvements we observed can be credited to AdvBN, we fine-tune a pre-trained ResNet-50 following the same routine as before but without the AdvBN module, adopting the same fixed AutoAugment policy along with all other hyperparameters. This method is denoted by AutoAugment$^{\ast}$ in Table~\ref{abaltion}. We see that fine-tuning with AutoAugment alone does not result in nearly as much improvement as the combined method on all datasets we consider; even performance on the original ImageNet dataset benefits from the AdvBN module.
\subsection{Feature divergence analysis}
We compare the features extracted by our network to those of a standard ResNet-50 trained on ImageNet.
Following \citet{ibnnet}, we model features from each channel using a normal distribution with the same mean and standard deviation, and we compute the symmetric KL divergence between the corresponding distributions on the two datasets ($A$ and $B$). For two sets of deep features, $F_A$ and $F_B$, each with $C$ channels, the divergence $D(F_A || F_B)$ is computed using the formula,
\begin{equation}
\begin{array}{cc}
D(F_A || F_B) = \frac{1}{C}\sum^{C}_{i=1}(KL(F^i_A||F^i_B) + KL(F^i_B || F^i_A)),\\
&\\
KL(F^i_A || F^i_B) = \log\frac{\sigma^i_B}{\sigma^i_A} + \frac{\sigma_A^{i^2} + (\mu^i_A - \mu^i_B)^2}{2\sigma_B^{i^2}} - \frac{1}{2},
\end{array}
\end{equation}
where $F^i$ denotes the features of $i$-th channel with mean $\mu^i$ and standard deviation $\sigma^i$.
In Figure~\ref{fig:kldiv}, we compare the baseline model with our own on two pairs of datasets in the fine-tuned layers. Since ImageNet-Instagram contains 20 filter versions, we use the ``Toaster'' filter found in \citep{instagram} to cause the sharpest drop in classification performance.
We find that the feature divergence in our network trained with AdvBN is substantially smaller on all layers in the fine-tuned subnetwork. In other words, the distribution of deep features corresponding to shifted domains is very similar to the distribution of deep features corresponding to standard ImageNet data. The small divergence between feature representations explains the effectiveness of AdvBN from a different angle and explains why our model generalizes well across datasets.
\begin{figure}[!ht]
\centering
\resizebox{\linewidth}{!}{
\includegraphics{figs/feature_divergence.pdf}
}
\caption{\textbf{Feature divergence between pairs of datasets using features extracted by AdvBN and a standard ResNet50}.}
\label{fig:kldiv}
\end{figure}
\subsection{Generalization on Semantic Segmentation}
We now evaluate AdvBN in the context of semantic segmentation. We train a ResNet-50 based network with dilated convolutions~\citep{dilatedconv} on Cityscapes~\citep{Cordts2016Cityscapes} as our baseline model following the training protocol in~\citet{ibnnet}. Cityscapes comprises urban street scenes from various cities and contains 2975 images with fine annotations for training and 500 images for validation. To evaluate generalizability, we test the performance of the model on another dataset, GTA5~\citep{GTA5}. The GTA5 dataset consists of synthetic images extracted from computer games, which have similar street scene content as Cityscapes and compatible label categories. The GTA5 validation set we use contains 6382 densely annotated images.
\input{tables/cityscapes}
To apply AdvBN, we fine-tune the baseline model with AdvBN plugged in after layer \texttt{conv2\_3} on Cityscapes for 20 epochs with adversarial training parameters $\tau = 0.15$, $\epsilon=0.4$, and 4 repeats.
In table~\ref{seg-cityscapes}, we observe a performance gain of 8.2\% in mean IoU on the GTA5 dataset compared to the baseline model. The pixel accuracy also improves by 11.3\%. Numbers on the left side of arrows denote performance on Cityscapes, and numbers on the right side of arrows denote performance on the GTA5 dataset.
\section{Introduction}
Robust optimization for neural networks has been a major focus of recent research. A mainstream approach to reducing the brittleness of classifiers is {\em adversarial training}, which solves a min-max optimization problem in which an adversary makes perturbations to images to degrade network performance, while the network adapts its parameters to resist degradation \citep{goodfellow2014explaining, kurakin2016adversarial, madry2017towards}. The result is a hardened network that is no longer brittle to small perturbations to input pixels. While adversarial training makes networks robust to adversarial perturbations, it does not address other forms of brittleness that plague vision systems. For example, shifts in image style, lighting, color mapping, and domain shifts can still severely degrade the performance of neural networks \citep{hendrycks2019benchmarking}.
We propose adapting adversarial training to make neural networks robust to changes in image style and appearance, rather than small perturbations at the pixel level. We formulate a min-max game in which an adversary chooses {\em adversarial feature statistics}, and network parameters are then updated to resist these changes in feature space that correspond to appearance differences of input images. This game is played until the network is robust to a variety of domain shifts in image space including texture, color, brightness, \emph{etc}\@\xspace.
The idea of adversarial feature statistics is inspired by the observation that the mean and variance of features maps encode style information, and thus, they enable the transfer of style information from a source image to a target image through normalization \citep{huang2017arbitrary,ulyanov2016instance}. Unlike standard approaches that rely on feature statistics from auxiliary images to define an image style, we use adversarial optimization of feature statistics to prepare classifiers for the worst-case style that they might encounter.
We propose training with {\em Adversarial Batch Normalization} (AdvBN) layers. Before each gradient update, the AdvBN layers perform an adversarial feature shift by re-normalizing with the most damaging mean and variance.
By using these layers in a robust optimization framework, we create networks which are resistant to any domain shift caused by feature statistics shift. An advantage of this method is that it does not require additional auxiliary data from new domains. We show that robust training with AdvBN layers hardens classifiers against changes in image appearance and style using a range of vision tasks including Stylized-ImageNet~\citep{geirhos2018imagenet} and ImageNet-Instagram~\citep{instagram}.
\section{Related work}
\paragraph{Adversarial training} Adversarial training and its variants \citep{goldblum2019adversarially2, madry2017towards, shafahi2019adversarial} have been widely studied for producing models that are robust to adversarial examples \citep{moosavi2016deepfool, intriguing}. Recent work considers adversarial training as data augmentation \citep{odds2019accuracy}. \citet{xie2019adversarial} finds that deep features corresponding to adversarial examples have different mean and standard deviation than those corresponding to natural images. This work takes advantage of the distributional discrepancy to improve performance on non-adversarial data. Our work also adopts the adversarial training framework to make models robust against other kinds of perturbations, but
instead of crafting adversarial examples in image space, we craft adversarial feature distributions by perturbing feature statistics.
\paragraph{Robustness to distributional shifts}
While extensive effort has been made to improve the robustness of classifiers to adversarial examples, there are other kinds of robustness that deep neural networks must address in order for them to be reliable. Corrupted images and new domains pose major challenges to networks with standard training~\citep{geirhos2018imagenet, hendrycks2019benchmarking, instagram}. Performance degradation on these images can be attributed to shifts in data distributions \citep{distribution}. In order to produce networks which generalize well, one common practice is to perform data augmentation \citep{cubuk2019autoaugment, hendrycks2019augmix, yun2019cutmix}. However, the benefits of data augmentation are largely limited by the types of augmentations used during training \citep{geirhos2018generalisation}. Feature space augmentation \citep{li2020feature} replaces feature statistics corresponding to one sample with ones corresponding to another sample. Our work can be also considered as feature space augmentation, we instead consider a worst-case scenario in the context of feature space distributional shifts by adopting the adversarial training framework.
|
2024-02-18T23:40:24.816Z
|
2020-09-23T02:02:04.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2009.08965","language":"en","timestamp":1600826524000,"url":"https:\/\/arxiv.org\/abs\/2009.08965","yymm":"2009"}
|
proofpile-arXiv_000-10093
|
{"provenance":"002.jsonl.gz:10094"}
| null | null |
\section{Introduction}
As neural networks are being adopted to solve real-world problems, while some parts of the network may be easy to develop, other unknown aspects such as hyperparameters, have no clear method of derivation. Ongoing research focuses on developing new network architectures and training methods. When developing neural networks, the question at hand is how to set the hyperparameter values to maximize results and set the training configuration. For network architecture design, important hyperparameters include the type of network, the number of layers, the number of units per layer, and unit type. For training configurations, important hyperparameters include learning algorithm, learning rate, and dropout ratio. All these hyperparameters interact with each other and affect the performance of neural networks. This interaction between hyperparameters can be referred to as epistasis. Thus they need to be tuned simultaneously to get optimum results.\\
The motivation behind this research is to replace tedious manual tuning of hyperparameters with an automatic method performed by computers. Current methods of optimization are limited to trivial methods like Grid search. Grid search is a simple method for hyperparameter optimization. However, as the number of hyperparameters increases, Grid search becomes time consuming and computationally taxing. This is because the number of lattice points increases in an exponential way with an increase in the number of hyperparameters \cite{qin2017evolution}. For example, if there are ten hyperparameters to be tuned and we only try five values for each parameter, and this alone requires more than 9 Million evaluations: \begin{math}5^{10} = 9765625\end{math}. For this reason, the grid search is not feasible for certain applications. To solve this, we look to a GA for a higher-performing and less computationally taxing solution. The use of a GA for neural network hyperparameter optimization has been explored previously in \cite{suganuma2017genetic, moriya2018evolution}. \\
We present an empirical study of GAs for neural network models in machine translation of natural language specifically Japanese to English. We describe the experiment setup in Section 2, our GA method in Section 3, and results in Section 4. The preliminary findings suggest that a simple GA encoding has the potential to find optimum network architectures compared to a random search baseline.
\section{Experimental Setup}
Genetic Algorithms (GA) are a class of optimization methods where each individual, a neural network, represents a solution to the optimization problem and the population is evolved in hopes of generating good solutions. In our case, each individual represents the hyperparameters of a neural machine translation system, and our goal is to find hyperparameters that will lead to good systems. The defining factor when measuring an individual's fitness is its BLEU score, a measurement of the individual's translation quality which is dependent on the individuals hyperparameters. The data set used for experimentation was limited to 150 individuals each consisting of 6 hyperparameters. We use a benchmark data set provided by \cite{zhang2020benchmarks} where a grid of hyperparameter settings and the resulting model BLEU scores are pre-computed for the purpose of reproducible and efficient hyperparameter optimization experiments. In particular, we use the Japanese-to-English data set which consists of various kinds of Transformer models (described later) trained on the WMT2019 Ro-bust Task \cite{li-EtAl:2019:WMT1}, with BLEU scores ranging from 9 to 16.\vspace{6pt}
Every test was done over three trials each consisting of 1000 optimization iterations. One iteration involves a process, elaborated below, to arrive at the target BLEU score of 16. The higher the BLEU score the better. While testing, both the GA and the Baseline (random search) received the same initial population of 5, 10, 15, 20, or 25 individuals. Throughout experimentation, the goal was to arrive at one individual with a BLEU score, fitness, of 16 or higher. The fitness is a measure of the likelihood of the individual remaining in the population.\vspace{6pt}
A practical limitation of the data set includes a possibility that a combination of hyperparameters found by the GA may not be represented in the data set. If the combination is non-existent, we assign a fitness value of 0 to that respective individual. This is imperative as we do not want a non-existant individual remaining in the population. By assigning the individual a fitness value of 0, it is guaranteed be replaced in the next breeding cycle. Furthermore, the possibility of two individuals with a fitness 0 is impossible as the population starts with individuals with representation in the data set. This is a benefit of an individual based measurement rather than a generational measure. Additionally, during experimentation, individuals were not added to the population if they have already been in the population, or if their genes exist in the current population. In traditional metrics, the entire current population is replaced by a new generation consisting of new offspring. In our implementation, only one offspring is generated, and that replaces the weakest individual. The performance of the GA and baseline algorithm is represented with a value based on the total amount of individuals added. This value increases every time an individual is added to the population. The performance of both algorithms is determined by averaging the performance value measured from all the iterations. An iteration consists of one optimization cycle. The performance value measured in individuals per iteration is displayed in the tables in section 4.
\section{Method}
The GA system used in this experiment is based on the natural selection process theorized by Darwin. This theory states that the stronger/fitter individuals survive while the weaker individuals do not. Thus over time, when the surviving stronger individuals reproduce, you get a population that as a whole carries genes that make them more resilient. However, Darwin only theorized this process in the natural world. The idea for using a GA in optimization problems and machine learning was thought of by Goldberg and Holland and expressed in \cite{goldberg1988genetic}. Throughout this experiment, we simulate their logic for machine translation neural network optimization. We begin with an initial population that "reproduces" until we get a group of individuals that collectively hold hyperparameters that perform better and one individual that is the "fittest" and meets the BLEU score target. The parts of the algorithm are represented through 3 objects: Individuals, Populations, and the Genetic Algorithm itself. The GA and Baseline algorithm were developed in the Python programming language. The GA and Baseline algorithm (random), elaborated later, are compared via an evaluator. This hierarchy, explained in \cite{whitley1994genetic}, and is represented visually below:\vspace{6pt}
\begin{figure}[ht]
\begin{center}
\begin{forest}
[Evaluator
[Genetic Algorithm
[Population
[Individuals]
]
]
[Random Selection
[Population
[Individuals]
]
]
]
\end{forest}
\caption*{Figure 1: Hierarchy of the Objects in Experimentation}
\end{center}
\end{figure}
\subsection{Individual}
Every individual consists of two defining characteristics: 1) the "chromosome", consisting of 6 hyperparameters, and 2) the fitness score, which is the BLEU value. An individual's chromosome can look like [10000.0, 4.0, 512.0, 1024.0, 8.0, 0.0006]. The fitness of the machine translation neural networks (a.k.a individual) is defined as the BLEU score or how well the neural network can translate from Japanese to English. BLEU score is measured by an algorithm for evaluating the quality of translation performed by the machine to that performed by a human. The BLEU score is calculated via a simple proportion. The two translations, the human translation, referred to as reference translation, and the translation done by the machine, the candidate translation. To compute, one counts up the number of candidate translation words, unigrams, that occur in any reference translation and the total number of words found in the reference translation \cite{papineni2002bleu}. Once we get the two values, we divide them to get a precision component of the BLEU score.\vspace{6pt}
However, simple proportions are not the only thing used during calculation of BLEU score. Another property of BLEU score worth noting is the use of a brevity penalty, a penalty based on length. Since BLEU is a precision based method, the brevity penalty assures that a system does not only translate fragments of the test set of which it is confident, resulting in high precision \cite{koehn2004statistical}. Is has become common practice to include a word penalty component dependant on length of the phrase for translation. This is especially relevant for the BLEU score that harshly penalizes translation output that is too short. Finally, in this study, we use alternate notation for readability. A BLEU score of 0.289 is reported as a percent 28.9\%.\\\\\\\\
\begin{figure}[ht]
\centering
\begin{tabular}{ |l | c |}
\hline
\textbf{\# BPE subword units} (1k) & 10, 30, 50\\
\textbf{\# encoder/decoder layers} & 2, 4\\
\textbf{\# word embedding dimensions} & 256, 512, 1024 \\
\textbf{\# hidden units} & 1024, 2048 \\ \textbf{\# attention heads} & 8, 16\\ \textbf{initial learning rate} ($10^{-4}$) &3, 6, 10\\
\hline
\end{tabular}
\caption*{Table 1: Hyperparameter search space for the Tranformer NMT systems}
\label{tab:datasets_hyps}
\end{figure}
Amongst the 150 individuals there are 7 target BLEU scores above the 16 goal: 16.04, 16.09, 16.02, 16.41, 16.03, 16.13, and 16.21. Ranges of hyperparameter values, referred to as Genes, are as follows: Fitness(9.86 - 16.41), Gene 1 (10,000, 30,000, or 50,000), Gene 2 (2.0 or 4.0), Gene 3 (256.0, 512.0, or 1024.0), Gene 4 (1024.0 or 2048.0), Gene 5 (8.0 or 16.0), and Gene 6 (0.001, 0.0003, or 0.0006). This information is shown above.\\
Additionally, a convolutional or recurrent model was not implemented for these translation networks. The networks in this study implemented transformers. On top of higher translation quality, the transformers requires less computation to train and are a much better fit for modern machine learning hardware, speeding up the training process immensely. In regards to the less computational power needed, the ease of training of transformers can be accredited to its lack of growth based on amount of words \cite{dehghani2018universal}. Specifically, a transformer is not recurrent meaning it does not need the translation of the previous word to translate the next. For example lets take the example a translation for German to English. Let's use the German sentence, Das Haus ist groß, meaning the house is big. In a Recurrent Neural Network (RNN), the network identifies das and the, and then uses that as a reference for the next word to translate Haus to house, etc. However, in a Transformer, we can treat each word as a separate object and for translation rather than the translation of the previous word, it uses the embedding value of all of the other words. Because they are treated independently, we can have the translation operation occur in parallel. So in this example every word das, Haus, ist, groß are all vectored and use the other words' embeddings. Additionally, a notable characteristic of a transformer is its use of an attention mechanism. The Attention mechanism allows for the network to direct its focus, and it pays greater attention to certain factors when processing the data resulting in a higher performing network. For these three main reasons, the lack of recurrence, the use of an attention mechanism, and the ability for parallel computation, transformers are a preferred choice as a network architecture in Machine Translation. The hyperparameters searched for in our Transformer models are shown in Figure 2.
\subsection{Population}
The population of individuals consist of three instance variables: the population itself, consisting of an initial x individuals implemented through an array, and two individuals that resemble the fittest and second fittest individuals in the population. For a population of 5 individuals it will look something like [Individual 1, Individual 2, Individual 3, Individual 4, Individual 5]. Examples of the fittest and second fittest individual follow: Fittest: Individual 1 can be represented as the fitness with BLEU score of 16.41, and second fittest: Individual 2 can be represented as second fitness with BLEU score of 16.04. For experimentation, however the algorithm stopped when the goal of 16 or above is reached, so the situation above would not occur.
\subsection{Genetic Algorithm (GA)}
In a broad sense, a genetic algorithm is any population-based model that uses various operators to generate new sample points. The GA system used during experimentation abides to most conventional characteristics of GA: a population, individuals, selection/mutation/crossover operations, etc. Our GA is comprised of three objects: the population, a list of all individuals, and an individual that acts as the child, referred as place holder. The population holds the individuals, the list of all individuals allows us to make sure that the child is not a repeated individual, and the individual allows for an object to store information on the offspring. The following describes the structure of the GA: First, an array of individuals with only the current population. The current population is defined as the population before the selection process, elaborated below. Second, it will contain a List of all the individuals that have been introduced to the population. We iterate through this list every time before adding an individual to make sure that the new individual, or placeholder, has not been introduced before. Place holder is an individual that is initially set to have values of 0: [0.0, 0.0, 0.0, 0.0, 0.0, 0.0]. During the crossover process, elaborated later, all the genes are changed to that of the offspring. Our implementation, however, differs from convention in two main ways: our implementation includes an Integer Representation rather than bit value, and an individual based measure for optimization rather than a generational measure. At the end of the process, a value that represents the total number of individuals added to the population added to the initial population size is returned.\vspace{6pt}
\begin{center}
The process the Genetic Algorithm goes through is depicted below:
\begin{tikzpicture}[node distance=1.5cm]
\node (io) [rectangle] {Genetic Algorithm};
\node (pro1) [rectangle, below of=io] {Selection};
\node (dec1) [rectangle, below of=pro1] {Crossover};
\node (pro2) [rectangle, below of=dec1] {Mutation};
\node (dec2) [rectangle, below of=pro2] {Check Validity of Individual};
\node (pro3) [rectangle, below of=dec2] {Add Individual};
\node (dec3) [rectangle, right of=dec1, xshift=2cm,yshift=-.75cm] {Not Valid};
\node (dec4) [rectangle, below of=pro3] {Check if target is reached};
\node (pro4) [rectangle, below of=dec4] {Optimization Complete};
\node (dec5) [rectangle, left of=pro2, xshift=-3cm,yshift=-.75cm] {Target not Reached};
\draw [->] (io) -- (pro1);
\draw [->] (pro1) -- (dec1);
\draw [->] (dec1) -- (pro2);
\draw [->] (pro2) -- (dec2);
\draw [->] (dec2) -- node {Valid} (pro3);
\draw [->] (dec4) -- node {Target Reached} (pro4);
\draw [->] (dec3) |- (pro1);
\draw [->] (dec5) |- (pro1);
\draw [->] (dec3) |- (pro1);
\draw [-] (dec4) -| (dec5);
\draw [->] (pro3) -- (dec4);
\draw [-] (dec2) -| (dec3);
\end{tikzpicture}
\\The processes used during the GA, selection, crossover, and mutation, and their functionalities are elaborated on below.
\end{center}
\subsubsection{Selection}
The selection process allowed for the GA to select two parents to mate to form a new offspring. The two parents are selected by weighted probability, proportional to their fitness. For example, lets take an initial population of 5 individuals with fitness values 10, 25, 15, 5, 45. The probability of selecting an individual is calculated by finding the sum of the fitness values, in this example 100, divided by the individuals fitness. Individual 1 has a 10\% of being selected, 10/100, Individual 2 has a 25\% chance, 25/100, etc. After repeating this process we get a list of percentages 10\%, 25\% , 15\% , 5\% , 45\%. As you can see, the percentages will always add up to 100. These values are then used to get an array with range values for each individual. In the aforementioned example, a list will look like [10, 35, 50, 55, 100]. These values are determined by adding the percentage value of an individual to that of all the previous individuals. From here, we select a random integer value from 0 to 100. This value is then correlated to an individual. For the above example with a population size of 5 Individual 1 is selected if the random value is less than 10, Individual 2 is selected if the value is greater than 10 and less than 35, Individual 3 if the value is greater than 50 and less than 55, etc. This approach is optimal as its adaptable based of size of population, and gives an accurate weighted representation.
\subsubsection{Crossover}
The crossover process simulates the breeding part of the natural selection process. Following selection, the selected two individuals are used to create an offspring. Initially, a random gene in the chromosome is selected as the "crossover point". Up to that point genes of the fittest, Parent 1 in Figure 2, are selected, and from that point to the end, genes of the second fittest, Parent 2 in Figure 2, are added. For a cross over point of 2, the example below depicts the cross over process.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=1\textwidth]{Crossover_Image.png}
\caption*{Figure 2: Visual depicting the Crossover process}
\end{center}
\end{figure}
\\ From here we now need to assign a fitness a value to the individual. Later we explain how we derived the 7 lists, for now just understand that we have 7 lists, one list with all potential values for each hyperparameter respectively, and another list for every fitness a value. These lists are ordered by individuals. For example the first index in all 7 lists correspond to individual one, the second index to individual 2, etc. We initially iterate through the entire first list, an array containing all values for the 150 individuals ordered, until we arrive at a match. In the example above this can be seen when we first arrive at the value of 10000. Then we store the index of the 10000 and see if that same index in list 2,3,4 etc also match the individuals gene. If all 6 genes correlate to values at one index, we assign a fitness value from the fitness list at the same index. Though an inefficient operator for assigning fitness, this approach makes the mutation operator easier.
\subsubsection{Mutation}
During the experiment, a mutation rate of 1/8, 12.5\%, was selected. When mutation occurs, an individual in the population is randomly selected, and one gene of that individual is also randomly selected. That gene is then given a new random value from the data set. Mutation process excludes the weakest Individual from being selected as that Individual would be replaced. If the weakest individual were not excluded, it would be replaced in during the addition of the offspring making the mutation irrelevant. Below is a mutation example with the mutation occurring on Gene 5. As shown, the mutation is resulting in an increased fitness.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=1\textwidth]{Mutation_Image.png}
\caption*{Figure 3: Visual depicting the Mutation process}
\end{center}
\end{figure}
\subsection{Evaluator}
All of the below results are measured using an average of individuals needed to be added over 1000 iterations. Each iteration consists of a process of going through optimization until a target score of 16 is reached. As previously stated, both algorithms return a value that represents the number of individuals that were added to reach the goal. However, before returning the values, we add the initial population size to account for individuals in the initial population. These values are then stored in two lists one for each algorithm, one list for the GA and another for the Baseline algorithm. At the very of the end of the evaluator the average of both lists is used to arrive at performance measure of the algorithms.
\subsection{Baseline (Comparison Algorithm)}
As a measure of the quality of the GA, a baseline algorithm that optimizes by randomly selecting hyperparameters was used to compare. The baseline algorithm selected a random index value from 0 to one less than the size of the data set. From this, an individual is made with values from that index in every list from the initialization of data points, elaborated below. At the end of the process, a value that represents the total number of individuals added to the population added to the initial population size is returned.
\begin{center}
The process the Baseline Algorithm goes through is depicted below:
\begin{tikzpicture}[node distance=1.5cm]
\node (io) [rectangle] {Baseline Algorithm};
\node (pro1) [rectangle, below of=io] {Select Random Value};
\node (dec4) [rectangle, below of=pro1] {Get Individual that corresponds to the Random Value};
\node (dec1) [rectangle, below of=dec4] {Check Validity of Individual};
\node (pro2) [rectangle, below of=dec1] {Add Individual};
\node (dec2) [rectangle, right of=dec4, xshift=4.5cm] {Not Valid};
\node (pro3) [rectangle, below of=pro2] {Check if target is reached};
\node (dec3) [rectangle, below of=pro3] {Optimization Complete};
\node (pro4) [rectangle, left of=dec1, xshift=-5cm] {Target not Reached};
\draw [->] (io) -- (pro1);
\draw [->] (pro1) -- (dec4);
\draw [->] (dec4) -- (dec1);
\draw [->] (dec1) -- node {Valid} (pro2);
\draw [->] (pro2) -- (pro3);
\draw [->] (pro3) -- node {Target Reached} (dec3);
\draw [-] (dec1) -| (dec2);
\draw [->] (dec2) |- (pro1);
\draw [->] (pro3) -| (pro4);
\draw [->] (pro4) |- (pro1);
\end{tikzpicture}
\end{center}
\subsection{Initialization of Data Points}
The initialization uses the python pandas library to create seven arrays that are representative of all possible Individual combinations. For example, all potential Gene 1 values are stored in an array. Similarly, there are arrays for the other genes and the fitness values. Additionally, these arrays store the fitnesses that are in order of fitness. Index 0 of all the lists correlate to the characteristics of individual 1, index 1 for individual 2, etc. These arrays are implemented to assign fitness values when a new off-springs are created, generate the initial population, and during the mutation operator.
\section{Results}
Every trial in the tables below consists of 1000 iterations. An iteration entails the repetition of the GA and Baseline processes, elaborated above, until the target goal is reached. Thus the average is calculated over an accumulative 3000 (per the tables below) iterations. We measure how long it takes for an algorithm to find the a good solution (i.e. $\ge 16$ BLEU), so the lower the better.
\begin{center}
\end{center}
\begin{figure}[ht]
\begin{center}
\begin{tabular}{ |p{2.5cm}||p{2.5cm}|p{2.5cm}|p{2.5cm}| p{2.5cm}| }
\hline
\multicolumn{5}{|c|}{Genetic Algorithm's Results} \\
\hline
Initial Population Size & Trial 1 (Average Number of Individuals Added)& Trial 2 (Average Number of Individuals Added)&Trial 3 (Average Number of Individuals Added) & Average\\
\hline
5&20.056&19.91&20.247&20.071\\
10&21.011&20.739&20.392&20.714\\
15&23.91&23.933&23.54&23.794\\
20&27.555&28.019&27.267&27.614\\
25&31.475&30.816&32.099&31.463\\
\hline
\end{tabular}
\end{center}
\caption*{Table 2: Genetic Algorithm Results}
\label{fig:1}
\end{figure}
\begin{figure}[ht]
\begin{center}
\begin{tabular}{ |p{2.5cm}||p{2.5cm}|p{2.5cm}|p{2.5cm}| p{2.5cm}| }
\hline
\multicolumn{5}{|c|}{Baseline Algorithm Results} \\
\hline
Initial Population Size & Trial 1 (Average Number of Individuals Added)& Trial 2 (Average Number of Individuals Added)&Trial 3 (Average Number of Individuals Added)&Average\\
\hline
5&20.435&21.949&21.891&21.425\\
10&23.513&22.545&23.29&23.116\\
15&26.056&25.611&25.345&25.671\\
20&28.332&28.292&28.365&28.330\\
25&31.857&30.819&32.117&31.598\\
\hline
\end{tabular}
\end{center}
\caption*{Table 3: Baseline Algorithm Results}
\label{fig:2}
\end{figure}
\begin{figure}[h]
\begin{center}
\vspace{1em}
\vspace{1em}
\begin{tabular}{|p{4cm}||p{4cm}||p{4cm}|}
\hline
\multicolumn{3}{|c|}{Performance Difference} \\
\hline
Initial Population Size & Winner & Difference in Performance\\
\hline
5 & Genetic Algorithm & 1.354\\
10 & Genetic Algorithm & 2.402\\
15 & Genetic Algorithm & 1.877\\
20 & Genetic Algorithm & .716\\
25 & Genetic Algorithm & .135\\
\hline
\end{tabular}
\end{center}
\caption*{Table 4: Difference in Performance between GA and Baseline Algorithm:}
\label{fig:3}
\end{figure}
\vspace{144pt}Using the average values, shown below, we can get a numerical representation of how much better a GA is. The average of the values showing difference in the performance shows that the GA can reach the desired goal with an average of 1.27 individuals fewer. The values in the table above were found by taking the average from the Baseline and subtracting the correlating GA value to find how many more individuals the Baseline needs on average. The value 1.27 was found by averaging all the values in the table below. Although saving 1.27 iterations is not large in the grander scheme of things, it is promising to see that GA gives consistent gains, implying that there are patterns to be exploited in the hyperparameter optimization process.\\\\\\\\\\
\section{Future Work}
The study proves validity of a GA being implemented for hyperparameter optimization. Future work falls into three main categories: general, structural (changes how the system works), and behavioral changes (variations in individual methods implementations which can cause varying results). Additionally, there is the possibility for variations to be implemented for the Baseline algorithm. Lastly, we will explore results of the generational measure rather than the individual based. We realized that a plateua occurs as the population increases. This is due to the chance of picking individuals with lower fitness values increasing as the population increases for the GA. Therefore, as the initial population size increases, the GA's performance approaches that of the Baseline algorithm.
\subsection{General Changes}
When reflecting on the experiment as a whole, there are many key aspects that should be changed to further test the validity of a GA. One such example includes the range of Individuals represented in the data set. For example, in the data set above, many combinations of hyperparameters where not pre-trained so a fitness value was not assigned. This results in the GA iterating extra times due to non-existant individuals. Additionally, as previously mentioned, one goal of a GA is to outperform the grid search, a more primitive type of optimization. To further test the performance, a larger data set will need to be used to test the efficiency compared to grid search, a higher performing solution than random. Also, additional evaluation metrics can be used such as the populations average fitness and run time of the algorithm. Another potential change can be discounting individuals that are not represented. While impractical to have all combinations represented, the Baseline algorithm could not select a combination that was not represented while the GA could. For further comparison, not adding an individual to the population would decrease the individual count and enlarge the performance gap between the GA and Baseline Algorithm.
\subsection{Structural Changes}
An example of a structural change includes: Having mutation occur before selection. The mutation before selection can result in different weights of individuals during the selection process. This can result in varying fitness values which can ultimately change the entire optimization process.
\subsection{Behavioral Changes}
A glaring example of a behavioral change can be seen in how the individual that is being replaced is selected. Unlike Darwin's theory, the replaced individual was selected definitively during the experiment. The individual with the lowest fitness was replaced. Changing this to weighted probability similar to selecting the fittest and second fittest can affect the tuning process as hyperparameters work simultaneously. Two individuals with low fitness values can have a child that has a high fitness value due to the lack of direct correlation between individual hyperparameter values and BLEU score. Another example includes a two-point crossover operator rather than the one-point crossover currently implemented. A two-point cross over has the individuals exchange the genes that fall between these two points.
\section{Conclusion}
This work introduces an advanced GA for hyperparameter optimization and applies it to machine translation optimization. We demonstrate that optimization of hyperparameters via a GA can outperform a random selection of hyperparameters. Specifically, outperform is defined by the ability of the algorithm to arrive at the goal with less individuals added. Finally, we propose future research directions which are expected to provide additional gains in the efficacy of GAs.
\section{Acknowledgements}
I would like to thank and acknowledge Dr. Kevin Duh (JHU) for giving me the opportunity to pursue research with him. Thank you for the continuous support, patience, and motivation throughout the entire process. Additional thanks for the feedback and comments on this paper, and for the invaluable guidance regarding programming, resources for help, and much more.\\
Also, I am very appreciative of and grateful to my family and friends for their love, patience and support, without which this project would not have been possible.
|
2024-02-18T23:40:24.818Z
|
2020-09-21T02:17:35.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2009.08928","language":"en","timestamp":1600654655000,"url":"https:\/\/arxiv.org\/abs\/2009.08928","yymm":"2009"}
|
proofpile-arXiv_000-10094
|
{"provenance":"002.jsonl.gz:10095"}
| null | null |
\section{Introduction}
\label{sec:intro}
Quickest change detection (QCD) is the problem of sequentially detecting a change in the statistical properties of a signal. Given a sequence of independent and identically distributed (i.i.d.) observations $\{x_t:t\in\mathbb{N}\}$ with distribution $f$ up to an unknown change point $\nu$ and distribution $g\neq f$ after, the goal is to detect this change as quickly as possible, subject to false alarm constraints. Traditionally, applications of QCD can be found in manufacturing, in areas such as quality control\cite{lai1995sequential} where any change in the quality of products must be quickly detected. With the proliferation of low-cost sensors, QCD methods have also found applications in other areas such as fraud detection\cite{bolton02}, cognitive radio\cite{lai2008quickest} and power system line outage detection\cite{banerjee2014power}.
As sensor and computing technology become increasingly ubiquitous and powerful, it becomes easier for an adversary to infer sensitive information, such as lifestyle preferences and location information, from available data. In many practical QCD applications, rather than having one distribution, the distribution that generates the signal in the post-change regime belongs to a finite set $G=\{g_1,\ldots,g_{|G|}\}$. In some applications, the distribution of some sensitive information $U$ may depend on the post-change distribution. Thus, knowing which distribution $g\in G$ generates the signal in the post-change regime may reveal some information about $U$. One example of such an application is occupancy detection. In an Internet of Things (IoT) based occupancy detection system, the occupancy sensor continuously measures attributes such as infrared radiation, temperature, humidity and carbon dioxide levels, to quickly detect when a room becomes occupied so that certain functions like turning on the air conditioning system can be automated. A baseline distribution can be used to model the fluctuation of these attributes with time when the room is empty. However, fluctuations in these attributes when the room is occupied can reveal the number of people and the activity conducted in the room, leading to privacy leakage if an adversary has access to the raw attribute signals. The goal of privacy-aware QCD is to sanitize the attribute signals so that the change from a vacant to an occupied room can still be detected, while preserving the privacy of the occupants in the room. In practice, we are unable to know the adversary's intent and hence would like to sanitize the signal so that the largest improvement an adversary can achieve, over all possible queries, is controlled.
In other applications, $G$ may be partitioned into two sets, a private set $I_1$ and a public set $I_2$. We would like to sanitize the signal so that it is difficult for an adversary to deduce the distribution when a post-change distribution from the private set $I_1$ is generating the signal. On the other hand, we would like to retain the ability to deduce the distribution when a post-change distribution from the public set $I_2$ is generating the signal. One example of such an application is activity monitoring using wearables. In this application, we would like to quickly detect any change from a resting state to an active state. There are many possible active states such as walking, running, typing on a computer, and using a mobile phone. We would like to protect the privacy of some of these active states, like typing on a computer or using a mobile phone, while still being able to accurately track the other active states. Hence, the goal of privacy-aware QCD is to perform QCD while protecting the privacy of some active states and maintaining some distinguishability for the other active states.
In this paper, we address the QCD problem with multiple post-change distributions while maintaining a privacy constraint for two different privacy metrics. For each of the privacy metrics, we propose a signal sanitization algorithm and a stopping time that is able to identify the critical change quickly while preserving a pre-determined level of privacy.
\subsection{Related Work}
In the QCD problem with a single post-change distribution, when the pre- and post-change distributions are fully specified and the change point $\nu$ is unknown but deterministic, the Cumulative Sum (CuSum) test, developed by Page \cite{page54}, is optimal as the false alarm rate goes to zero. For the case where the post-change distribution is not fully specified, the GLR CuSum test is asymptotically optimal for the case of finite multiple post-change distributions. For a comprehensive overview of the QCD problem, we refer the reader to \cite{tartakovsky2014sequential,poor2009quickest,LauTayVee:J19,LauTay:J19} and the references therein.
In many applications, observations are obtained through measurements taken from several wireless sensors in the network and a fusion center decides if a change has taken place based on the information received from these sensors. Due to power and bandwidth constraints, the sensors are constrained to send messages belonging to a finite alphabet to the fusion center. This is an example in which the information for decision-making is \emph{decentralized}. The QCD problem with a decentralized framework was first introduced in \cite{veeravalli2001decentralized} and further studied in \cite{mei2005information,tartakovsky2008asymptotically,hadjiliadis2009one,banerjee2013decentralized} under various settings. All the aforementioned works on QCD do not consider any privacy constraints.
Existing work on protecting or quantifying privacy can be divided into two main categories: \emph{data} privacy and \emph{inference} privacy\cite{WangTIT2016,SunTay:C17,SunTay:J20b}. Data privacy refers to the protection of the sensors' raw information from being obtained by the fusion center. In contrast, inference privacy refers to the protection from an adversary's attempt to deduce properties of an underlying distribution. Privacy metrics proposed to quantify data privacy include local differential privacy \cite{Dwork2006,xiong2016randomized,duchi2013local}, $k$-anonymity\cite{GupRao:J17} and homomorphic encryption \cite{Boneh2005}. On the other hand, privacy metrics proposed to quantify inference privacy include average information leakage\cite{salamatian2013hide,khouzani2019generalized}, mutual information privacy \cite{WangTIT2016,song2017tutorial}, information privacy\cite{CalFaw:C12,SunTay:C16,SunTayHe:J18,HeTayHua:J19,SunTay:J20a}, maximal leakage privacy\cite{issa2016operational,issa2017operational}, local differential privacy, hypothesis testing adversary privacy\cite{LiOechtering:J19}, and compressive privacy\cite{KunSPM2017,SonWanTay:J20}. We refer the reader to \cite{SunTay:J20b,WangTIT2016} for a comprehensive discussion on the relationship between data and inference privacy. In this paper, we use maximal leakage privacy, and sequential hypothesis testing adversary privacy, which is the sequential analog to the privacy metric proposed in \cite{LiOechtering:J19}, to quantify the gain which the adversary obtains through the observation of the signal.
There are several works in the literature that present privacy-preserving frameworks for different signal processing tasks\cite{SunTay:J20a,rassouli2020,liao2017hypothesis,cummings2018differentially}. In \cite{cummings2018differentially}, the authors developed differentially private algorithms, which assume that the adversary knows all entries of the database except one, for the purpose of change-point detection. Unlike \cite{cummings2018differentially}, we consider a weaker form of privacy as this assumption may be too strong for some applications. Furthermore, we provide theoretical guarantees on the average run length to false alarm and the worst-case detection delay, which are more relevant to the QCD task as compared to guarantees on the accuracy of the estimated change-point provided by \cite{cummings2018differentially}, which are more suited for the change-point detection task. In \cite{rassouli2020}, the sanitization channel is designed for general signal processing tasks while preserving privacy, where the utility of the sanitized signal is measured using general information theoretic quantities such as mutual information, minimum mean-square error (MMSE), and probability of error. In \cite{liao2017hypothesis,SunTay:J20a}, the authors consider a fixed sample size problem of hypothesis testing while preserving privacy, where the utility of the sanitized signal is measured by the Bayes error or the Type II error of the test.
\subsection{Our Contributions}
In this paper, we consider the problem of optimizing QCD performance while preserving privacy. Furthermore, unlike the papers mentioned above, we consider sequential signals. It is thus possible for the adversary to obtain an arbitrarily large number of samples to improve his guess. Our main contributions are summarized as follows:
\begin{itemize}
\item We formulate the QCD problem with two privacy constraints, the maximal leakage privacy metric and the sequential hypothesis testing privacy metric.
\item We show that the
GLR CuSum stopping time together with a properly designed sanitization
channel is asymptotically optimal.
\item We propose relaxations and algorithms for both the centralized and decentralized versions of the QCD problem with privacy constraints.
\end{itemize}
A preliminary version of this paper was presented in \cite{lau2020quickest}.
The rest of this paper is organized as follows. In \cref{sec:problem}, we present our signal model and problem formulation. We derive the asymptotic optimality of the GLR CuSum stopping time and formulate an optimization problem to design the optimal sanitization channel in \cref{sec:asym_opt}. We propose relaxations to the channel design problem in \cref{sec:relaxation} and present methods to solve the relaxed channel design problem in \cref{sec:centralized_algorithms}. We present the signal and sanitization model for the decentralized privacy-aware QCD problem in \cref{sec:iid} and propose methods to solve the corresponding relaxed channel design problem in \cref{sec:decentralized_algorithms}. Results from numerical experiments are presented in \cref{sec:numerical}. We conclude in \cref{sec:conclusion}.
\section{Problem formulation}\label{sec:problem}
Let $\mathcal{X}$ be a measurable space, where $\mathcal{X}$ is a finite alphabet. We consider a sequence of random variables $X_1, X_2, \ldots$ taking values in $\mathcal{X}$ and \gls{iid} according to different distributions before and after an unknown change point. Let $f$ be the pre-change distribution and $G=\{g_1,g_2,\ldots,g_{|G|}\}$ be the set of possible post-change distributions on $\mathcal{X}$ such that $f\neq g_i$ for all $i\in\{1,2,\ldots,|G|\}$. Let $I$ be a random variable on the indices of $G$ with distribution $p_I$.We assume that the sequence of random variables $X_1,X_2,\ldots$ satisfy the following:
\begin{align}\label{eqn:signalmodel}
\begin{cases}
X_{t} \sim f \quad \text{i.i.d.\ for all $t< \nu$},\\
X_{t} \sim g_{i} \quad \text{i.i.d.\ for all $t\geq \nu$},\\
\end{cases}
\end{align}
where $\nu\geq 0$ is an unknown but deterministic change point, $i$ is the realization of the random variable $I$ which remains fixed for all $t\geq\nu$. We further assume that an adversary is interested in obtaining information about a random variable $U$, unknown to the data curator, which takes on values in a finite set $\mathcal{U}$, and that $U$ can be expressed as a randomized function of $I$, i.e., the identity of the post-change distribution informs us about $U$.
We restrict our analysis to memoryless privacy mechanisms. A privacy mapping or sanitization channel $q$ maps an observation $X \in \mathcal{X}$ to a random variable $Y \in \mathcal{Y}$, where $\mathcal{Y}$ is a discrete alphabet such that $|\mathcal{Y}| \leq |\mathcal{X}|$. The sanitization channel $q$ can be represented by the conditional probability $\P(Y=y \ | X=x)$. Let $T_q$ be a column-stochastic matrix with $[T_q]_{y,x}=\P(Y=y){X=x}$ where $ [T]_{y,x}$ denotes the $(y,x)$ entry of a matrix $T$. Likewise, we represent a distribution $h$ on $\mathcal{X}$ as a column vector with $[h]_x=h(x)$ and similarly for a distribution on $\mathcal{Y}$.
At each time $t$, we apply a sanitization channel $q$ to obtain $Y_t=q(X_t)$. The sanitized signal $Y_t$ is generated i.i.d.\ by the distribution $\widetilde{f}=T_{q}f$ in the pre-change regime and by the distribution $T_{q}g_i$ in the post-change regime, for some $1\leq i\leq |G|$. For a fixed $q$, we let
\begin{align*}
\widetilde{G}=\{T_{q}g_i\ :\ \text{$1\leq i\leq|G|$}\}
\end{align*}
to be the set of possible post-change distributions. Since the distributions $T_{q}g_i$ may not be distinct, we have $|\widetilde{G}|\leq |G|$.
In this paper, we study the QCD problem with privacy constraints using two different privacy metrics. We assume that, at each time $t$, the adversary knows the pre-change distribution $f$, the set $G$ of post-change distributions, change-point $\nu$, the sanitization channel $q$, the current and all previous observations $Y^{1:t}=\{Y_1,Y_2,\ldots,Y_t\}$.
The first privacy metric we consider is maximal leakage, first proposed in \cite{issa2016operational}, to quantify the amount of information leakage an adversary is able to gain from observing the signal $\{Y_t\ :\ t\in\mathbb{N}\}$, where $\mathbb{N}$ is the set of positive integers. Given two random variables $A \in\mathcal{A}$ and $B\in\mathcal{B}$, the maximal leakage from $A$ to $B$ is defined as
\begin{align}\label{eqn:maxleak}
\mathcal{L}_{\text{max}}(A\to B)=\sup_{U-A-B-\widehat{U}}\log\frac{\P(\widehat{U}=U)}{\max_{u\in\mathcal{U}}\P(U=u)},
\end{align}
where $U - A - B - \hat{U}$ denotes a Markov chain and the supremum is taken over all such Markov chains. The quantity $\mathcal{L}_{\text{max}}(A\to B)$ can be interpreted as the maximum gain in bits (if $\log$ is base 2) an adversary can achieve in guessing $U$ by observing $B$, where $U$ is a randomized function of $A$. The expression in \eqref{eqn:maxleak} is equivalent\cite{issa2016operational} to
\begin{align}\label{eqn:maximal_leakage_closed_form}
\mathcal{L}_{\text{max}}(A\to B)=\log\left(\sum_{a\in \mathcal{A}}\max_{b\in\mathcal{B}}\P(B=b){A=a}\right).
\end{align}
Thus, at each time $t$, the maximum gain in bits an adversary can achieve in guessing $U$ is $\mathcal{L}_{\text{max}}(I\to Y^{\nu:t})$.
The second privacy metric we consider is the \emph{sequential hypothesis testing privacy} metric. It quantifies the amount of gain an adversary is able to achieve by performing a sequential hypothesis test. Using this privacy metric, we are able to protect a subset of the post-change hypotheses from the inference of an adversary while ensuring the distinguishability of the rest of the post-change hypotheses. Given a sanitization channel $q$ and a partition $I_1\cup I_2$ of the index set of $G$ with $|I_1|>1$, such that $I_1$ is the set of indices of the post-change hypotheses to be protected, we define
\begin{align*}
\mathcal{K}_1(T_q)=\max_{i\in I_1} \min_{j\in I_1}\KLD{ T_qg_{i}}{T_qg_{j}},\\
\mathcal{K}_2(T_q)=\min_{i\in I_2} \min_{j\in I_1\cup I_2}\KLD{T_qg_{i}}{T_qg_{j}},
\end{align*}
where $\KLD{\cdot}{\cdot}$ is the Kullback-Leibler (KL) divergence.
Using standard results from sequential hypothesis testing\cite[Theorem 4.3.1]{tartakovsky2014sequential}, assuming that the adversary is willing to accept a misclassification rate of $\eta$, the expected number of samples required to identify a distribution with index in $I_1$ is \emph{at least} $|\log \eta|/\mathcal{K}_1(T_q)$ asymptotically as $\eta\to 0$. Similarly, the expected number of samples required to identify a distribution with index in $I_2$ is \emph{at most} $|\log \eta|/\mathcal{K}_2(T_q)$ asymptotically as $\eta\to 0$.
For a fixed sanitization channel $q$, the QCD problem is to detect a change in distribution as quickly as possible by observing the sanitized signal $Y_1=q(X_1),Y_2=q(X_2),\ldots$, while keeping the false alarm rate low. In a typical sequential change detection procedure, at each time $t$, a test statistic $S(t)$ is computed based on the observations $Y_1,\ldots,Y_t$ up to time $t$, and the observer decides that a change has occurred at a stopping time $\tau=\inf\{t:S(t)>b\}$, which is the first $t$ such that $S(t)$ exceeds a pre-determined threshold $b$. The QCD performance of a stopping time $\tau$ can be quantified using the trade-off between two quantities, the average run length to false alarm, $\text{ARL}(\tau)$, and the expected worst-case average detection delay, $\text{EWADD}(\tau)$, defined as
\begin{align*}
\text{ARL}(\tau)&=\E{\infty}[\tau],\\
\text{EWADD}(\tau)&=\E[\text{WADD}_I(\tau)], \\
\text{WADD}_i(\tau)&=\sup_{\nu\geq 1}\esssup \E{\nu,i}[(\tau-\nu_c+1)^+|Y_1^{\nu_c-1}],
\end{align*}
where $\esssup$ is the essential supremum operator, $\E_{\nu,i}$ is the expectation operator assuming the change-point is at $\nu$ with post-change distribution $g_i$, and $\E_{\infty}$ is the expectation operator assuming the change does not occur. We should note that typically, there are several sanitization channels $q$ that satisfy a privacy constraint. As the pre and post-change distributions, $\{\widetilde{f}\}\cup\widetilde{G}$, vary with the sanitization channel $q$, we expect that the QCD performance varies with $q$ as well. It is then important for us to select the sanitization channel $q$ that provides the best QCD performance while satisfying the privacy constraint.
Our privacy-aware QCD problem can be formulated as an optimization problem as follows: given a privacy admissible set $\mathcal{Q}$ and an average run length requirement $\gamma$, we seek a stopping time $\tau$, and a sanitization channel $q$ such that they are optimal solutions to the following problem:
\begin{equation}\label{eqn:optimize_formulation}
\begin{aligned}
& \underset{\tau,q}{\text{minimize}}
& & \text{EWADD}(\tau) \\
& \text{subject to}
& & \text{ARL}(\tau)\geq\gamma,\\
& &&q\in \mathcal{Q}
\end{aligned}
\end{equation}
where under the first privacy metric, the privacy admissible set $\mathcal{Q}$ is defined as
\begin{align*}
\mathcal{Q}=\{q\ :\ \sup_{0\leq\nu\leq t}\mathcal{L}_{\text{max}}(I\to Y^{\nu:t})\leq\epsilon\}
\end{align*}
for some given privacy budget $\epsilon>0$, and under the second privacy metric,
\begin{align*}
\mathcal{Q}=\{q\ :\ \mathcal{K}_1(T_q)\leq \epsilon_1,\ \mathcal{K}_2(T_q)\geq \epsilon_2\}
\end{align*}
for some given privacy budget $\epsilon_1$ and distinguishability level $\epsilon_2>0$.
\section{Asymptotic optimality}\label{sec:asym_opt}
In this section, we present the GLR CuSum stopping time for the privacy-aware QCD problem and study its asymptotic properties as $\gamma\to\infty$. First, we note that the minimization over the sanitization channel $q$ and stopping time $\tau$ can be decoupled in Problem~\cref{eqn:optimize_formulation}. For a fixed $q\in\mathcal{Q}$, we define the GLR CuSum stopping time $\omega_{q}$ and the GLR CuSum test statistic $S(t)$ as follows
\begin{align*}
&\ \ \ \omega_{q}=\inf\left\{t\ :\ S(t)\geq b\right\},\\
&\ \ \ S(t)=\max_{1\leq j\leq |\widetilde{G}|}S_j(t),\\
&\begin{cases}
S_j(t)&=\max\left(S_j(t-1)+\log\frac{\widetilde{g}_j(y_t)}{\widetilde{f}(y_t)},0\right)\\
S_j(0)&=0,
\end{cases}\text{for $1\leq j\leq |\widetilde{G}|$. }
\end{align*}
When the signal $\{\mathbf{X}_t\ :\ t\in\mathbb{N}\}$ is sanitized using the channel $q$,
the GLR CuSum stopping time $\omega_{q}$ is asymptotically optimal\cite{lorden71} for the following problem:
\begin{equation}\label{eqn:optimize_formulation_lorden}
\begin{aligned}
& \underset{\tau}{\text{minimize}}
& & \text{EWADD}(\tau)\\
& \text{subject to} &&\text{ARL}(\tau)\geq\gamma,
\end{aligned}
\end{equation}
with the asymptotic $\text{ARL}$-$\text{EWADD}$ trade-off given as
\begin{align*}
\text{EWADD}(\omega_{q})=\E[\frac{\log \gamma}{\KLD{T_{q}g_I}{T_{q}f}}](1+o(1))
\end{align*}
as $\gamma\to\infty$, where the expectation is taken with respect to $I$.
Let $q^*$ be an optimal solution to the following problem:
\begin{equation}\label{eqn:optimize_formulation_asymptotic}
\begin{aligned}
& \underset{q}{\text{minimize}}
& & \E[\frac{1}{\KLD{T_{q}g_I}{T_{q}f}}], \\
&\text{subject to} && q\in\mathcal{Q}.
\end{aligned}
\end{equation}
Using similar arguments from \cite{tartakovsky2014sequential,lau2017optimal}, it can be shown that $\omega_{q^*}$ is asymptotically optimal for Problem~\cref{eqn:optimize_formulation} as $\gamma\to \infty$.
We call Problem~\cref{eqn:optimize_formulation_asymptotic} the channel design problem and note that it is challenging to solve for several reasons. First, the objective function is neither concave nor convex. Second, it is difficult to obtain a closed form expression for the maximal leakage privacy constraint since the alphabet size of $Y^{\nu:t}$ increases quickly as $t\to\infty$. The sequential hypothesis testing privacy constraints are also neither concave nor convex. In the next section, we present relaxations of the constraint and objective function of Problem~\cref{eqn:optimize_formulation_asymptotic} to improve its computational tractability.
\section{Relaxation of the Channel Design Problem}\label{sec:relaxation}
\subsection{Relaxation of the Objective Function}
In this subsection, we provide a relaxation of the objective function in Problem~\cref{eqn:optimize_formulation_asymptotic}. We propose to relax the objective function using Jensen's inequality:
\begin{align*}
\E[\frac{1}{\KLD{T_{q}g_I}{T_{q}f}}] \geq \frac{1}{\E[\KLD{T_{q}g_I}{T_{q}f}]} .
\end{align*}
By replacing the objective function with its lower bound, we obtain the following relaxed problem:
\begin{equation}\label{eqn:optimize_formulation_asymptotic_relaxed_objective}
\begin{aligned}
& \underset{q}{\text{maximize}}
& & \E[\KLD{T_{q}g_I}{T_{q}f}] \\
& \text{subject to} &&q\in\mathcal{Q}.
\end{aligned}
\end{equation}
For a fixed post-change distribution $g_i$, with $1\leq i\leq N$, the expected rate of growth of $S(t)$ in the post-change regime is given as $\KLD{T_q g_i}{T_qf}$. Thus, we can interpret the new objective function as the expected rate of growth of $S(t)$ averaged over the different post-change distributions. Since the stopping time $\omega_{q}$ declares that a change has taken place when the test statistic $S(t)$ exceeds a pre-defined threshold $b$, this means that, heuristically, a larger expected rate of growth of $S(t)$ gives a smaller $\text{EWADD}$. This intuition agrees with the relaxed problem \cref{eqn:optimize_formulation_asymptotic_relaxed_objective} obtained by replacing the objective function with its lower bound ${1}/{\E[\KLD{T_{q}g_I}{T_{q}f}]}$ in Problem~\cref{eqn:optimize_formulation_asymptotic}.
\subsection{Relaxation of Privacy Constraints}
\subsubsection{Maximal Leakage privacy}\label{subsec:ml_constraint}
In this subsection, we focus on the relaxation of the constraint $q\in\mathcal{Q}$ when the privacy metric is the maximal leakage privacy. Under the maximal leakage privacy metric, Problem~\cref{eqn:optimize_formulation_asymptotic_relaxed_objective} becomes:
\begin{equation}\label{eqn:optimize_formulation_asymptotic_relaxed_objective_ml}
\begin{aligned}
& \underset{q}{\text{maximize}}
& & \E[\KLD{T_{q}g_I}{T_{q}f}] \\
& \text{subject to} &&\sup_{0\leq\nu\leq t}\mathcal{L}_{\text{max}}(I\to Y^{\nu:t})\leq\epsilon .
\end{aligned}
\end{equation}
As it is difficult to obtain a closed form expression for the maximal leakage $\mathcal{L}_{\max}(I\to Y^{\nu:t})$, we approximate it using an upper bound which is easily computable. Let $J$ be a random variable on the indices of $\widetilde{G}=\{\widetilde{g}_1,\widetilde{g}_2,\ldots,\widetilde{g}_{|\widetilde{G}|}\}$, such that
\begin{align*}
\P(J=j){I=i}&=\begin{cases}
1 \quad\text{if $\widetilde{g}_j= T_{q}g_i$,}\\
0\quad\text{otherwise.}
\end{cases}
\end{align*}
Let $U$ be a randomized function of $I$. According to our signal model, we have the following factorization,
\begin{align*}
P_{J,I,Y^{\nu:t},U}&=P_{J,I,U}\ P_{Y^{\nu:t}|I,J,U}\\
&=P_{I,J,U}\ P_{Y^{\nu:t}| J}\\
&=P_I\ P_{J | I}\ P_{U |I}\ P_{Y^{\nu:t} | J},
\end{align*}
where we use $P_X$ to denote the probability mass function (pmf) of $X$ and $P_{X|Y}$ to denote the conditional pmf of $X$ given $Y$.
The following proposition provides the motivation to relax the privacy constraint in Problem~\eqref{eqn:optimize_formulation_asymptotic_relaxed_objective_ml} to $\mathcal{L}_{\text{max}}(I\to J)\leq \epsilon$.
\begin{Proposition}\label{prop:upperbound_max_leakage}
For any $t\in\mathbb{N}$, we have
$\mathcal{L}_{\text{max}}(I\to Y^{\nu:t})\leq\mathcal{L}_{\text{max}}(I\to J).$ Hence, we have
\begin{align*}
\sup_{0\leq \nu\leq t}\mathcal{L}_{\text{max}}(I\to Y^{\nu:t})\leq\mathcal{L}_{\text{max}}(I\to J).
\end{align*}
\end{Proposition}
\begin{IEEEproof}
See \cref{sec:AppProp1}.
\end{IEEEproof}
Replacing the privacy constraint $ \sup_{0\leq \nu\leq t}\mathcal{L}_{\text{max}}(I\to Y^{\nu:t})\leq \epsilon$ in \cref{eqn:optimize_formulation_asymptotic_relaxed_objective_ml} with $\mathcal{L}_{\text{max}}(I\to J)\leq \epsilon$, we obtain the relaxed channel design problem:
\begin{equation}\label{eqn:optimize_relaxed_formulation}
\begin{aligned}
& \underset{q}{\text{maximize}}
& & \E[\KLD{T_{q}g_I}{T_{q}f}]\\
& \text{subject to}
& & \mathcal{L}_{\text{max}}(I\to J)\leq\epsilon.
\end{aligned}
\end{equation}
\cref{prop:upperbound_max_leakage} guarantees that any solution to Problem~\cref{eqn:optimize_relaxed_formulation} satisfies the original privacy constraint $\sup_{0\leq \nu\leq t}\mathcal{L}_{\text{max}}(I\to Y^{\nu:t})\leq\epsilon$.
\subsubsection{Relaxation of the sequential hypothesis testing privacy constraint}\label{subsec:sht_constraint}
In this subsection, we focus on the relaxation of the constraint $q\in\mathcal{Q}$ when the privacy metric is the sequential hypothesis testing privacy metric. Under the sequential hypothesis testing privacy metric, Problem~\cref{eqn:optimize_formulation_asymptotic_relaxed_objective} becomes:
\begin{equation}\label{eqn:optimize_formulation_asymptotic_relaxed_objective_sht}
\begin{aligned}
& \underset{q}{\text{maximize}}
& & \E[\KLD{T_{q}g_I}{T_{q}f}] \\
& \text{subject to} &&\mathcal{K}_1(T_q)\leq \epsilon_1,\ \mathcal{K}_2(T_q)\geq \epsilon_2.
\end{aligned}
\end{equation}
Problem~\cref{eqn:optimize_formulation_asymptotic_relaxed_objective_sht} is non-convex as the constraints $\mathcal{K}_1(T_q)\leq \epsilon_1$ and $\mathcal{K}_2(T_q)\geq \epsilon_2$ are non-convex. This makes it difficult to have any theoretical guarantees of the global optimality of solutions found for Problem~\cref{eqn:optimize_formulation_asymptotic_relaxed_objective_sht}. We focus on a restricted sanitization model in order to improve the computational tractability of Problem~\cref{eqn:optimize_formulation_asymptotic_relaxed_objective_sht}. Let $\mathcal{C}=\{q_1,q_2,\ldots,q_n\}$ be a finite set of sanitization channels with column-stochastic matrices $T_1,\ldots,T_n$. At each time instance $t$, we assume that the observer obtains an observation $Y_t=(Z_t,A_t)$ where $Z_t=q_{A_t}(X_t)$ is a randomized function of the random variable $X_t$ under the sanitization channel $q_{A_t}$. We further assume that $\{A_t\}_{t\in\mathbb{N}}$ are i.i.d. generated with distribution $\phi$ on $\{1,2,\ldots,n\}$. Under the restricted sanitization model, rather than designing the sanitization channel $q$, we design the distribution $\phi$ that samples a sanitization channel $q_{A_t}$ from $\mathcal{C}$ at each time instance $t$ and use $q_{A_t}$ to sanitize the signal.
For a fixed set of sanitization channels $\mathcal{C}=\{q_1,q_2,\ldots,q_n\}$ and a fixed distribution $\phi$, the asymptotic $\text{ARL}$-$\text{EWADD}$ trade-off of the GLR CuSum stopping time under the restricted sanitization model can be derived, using similar arguments from \cite{lau2017optimal}, to be
\begin{align*}
\text{EWADD}(\omega_{q})=\E[\frac{\log \gamma}{\sum_{c=1}^n\phi(c)\KLD{T_c g_I}{T_c f}}](1+o(1)),
\end{align*}
as $\gamma\to\infty$. Furthermore, we have
\begin{align*}
\E[\KLD{T_{q}g_I}{T_{q}f}]=\sum_{c=1}^n \phi(c)\E[\KLD{T_{c}g_I}{T_{c}f}].
\end{align*} Thus, Problem~\cref{eqn:optimize_formulation_asymptotic_relaxed_objective_sht} becomes
\begin{equation}\label{eqn:optimize_formulation_asymptotic_relaxed_objective_sht_linearized}
\begin{aligned}
& \underset{\phi}{\text{maximize}}
& & \sum_{c=1}^n \phi(c)\E[\KLD{T_{c}g_I}{T_{c}f}] \\
& \text{subject to} &&\max_{i\in I_1} \min_{j\in I_1}\sum_{c=1}^n\phi(c)\KLD{T_cg_i}{T_cg_j}\leq \epsilon_1,\\
& & &\sum_{c=1}^n \phi(c)=1,\\
& & &\phi(c)\geq 0, \quad\text{$c\in\{1,\ldots,n\}$,}\\
& & &\sum_{c=1}^n\phi(c)\KLD{T_cg_i}{T_cg_j}\geq \epsilon_2\\
& & &\text{for $i\in I_2$ and $j\in I_1\cup I_2$}.
\end{aligned}
\end{equation}
Since $\KLD{T_{c}g_I}{T_{c}f},\KLD{T_{c}g_i}{T_{c}g_j}$ can be pre-computed for all $c\in\{1,\ldots,n\}$ and $i,j\in \{1,\ldots,|G|\}$, Problem~\cref{eqn:optimize_formulation_asymptotic_relaxed_objective_sht_linearized} without the constraint
\begin{align*}
\max_{i\in I_1} \min_{j\in I_1}\sum_{c=1}^n\phi(c)\KLD{T_cg_i}{T_cg_j}\leq \epsilon_1
\end{align*} is a linear program. In the next section, we show that Problem~\cref{eqn:optimize_formulation_asymptotic_relaxed_objective_sht_linearized} can be expressed as a mixed-integer linear program (MILP).
\section{Algorithms for Privacy-Aware QCD}\label{sec:centralized_algorithms}
In this section, methods that provide globally and locally optimal solutions for Problems~\cref{eqn:optimize_relaxed_formulation,eqn:optimize_formulation_asymptotic_relaxed_objective_sht_linearized} are presented.
\subsection{Maximal Leakage Privacy}\label{subsec:algorithms_ml}
\subsubsection{Exact Method}
From \eqref{eqn:maximal_leakage_closed_form}, $2^{\mathcal{L}_{\text{max}}(I\to J)}$ is an integer since the conditional probability $\P(J=j){I=i}\in\{0,1\}$ is an integer. Hence, the constraint $\mathcal{L}_{\text{max}}(I\to J)\leq \epsilon$ in Problem~\cref{eqn:optimize_relaxed_formulation} is equivalent to
\begin{align}\label{eqn:L(IJ)=m}
\sum_j\max_i\P(J=j){I=i}\leq m,
\end{align}
where $m=\lfloor2^\epsilon\rfloor$. There are at most $\stirling{|G|}{m}\leq m^{|G|}$ different conditional pmfs $P_{J|I}$ satisfying \eqref{eqn:L(IJ)=m}, where the Stirling number of the second kind $\stirling{a}{b}$ counts the number of ways to partition a set of $a$ labeled objects into $b$ nonempty unlabeled subsets\cite{riordan2012introduction}. For each conditional pmf $P_{J|I}$, we solve the following problem:
\begin{equation}\label{eqn:N=1,epsilon>0,cases}
\begin{aligned}
& \underset{q}{\text{maximize}}
& & \E[\KLD{T_{q}g_i}{T_{q}f}] \\
& \text{subject to}
& & T_{q}g_i=\widetilde{g}_j \\
& & &\text{for all $i,j$ such that $\P(J=j){I=i}=1$}.\\
\end{aligned}
\end{equation}
For a fixed conditional pmf $P_{J|I}$, Problem~\eqref{eqn:N=1,epsilon>0,cases} is maximizing a convex function over a convex bounded polytope. Therefore, an extreme point achieves the maximum value, and we are able to solve Problem~\eqref{eqn:N=1,epsilon>0,cases} by enumerating over the finite number of extreme points of the convex bounded polytope defined by the linear constraints of \eqref{eqn:N=1,epsilon>0,cases}. A globally optimal solution for Problem~\cref{eqn:optimize_relaxed_formulation} can be obtained by enumerating over all conditional pmfs $P_{J|I}$ represented by a zero-one matrix satisfying \eqref{eqn:L(IJ)=m}, and solving Problem~\eqref{eqn:N=1,epsilon>0,cases} for each of these conditional pmfs.
However, we still need to solve at least exponentially many convex maximization problems with respect to the number of post-change distributions $|G|$, since $\stirling{|G|}{m}\sim\frac{m^{|G|}}{m!}$ as $|G|\to \infty$. This may be computationally undesirable when $|G|$ is large.
\subsubsection{Augmented Lagrangian Method}
We further relax the channel design problem by relaxing the discrete constraint \eqref{eqn:L(IJ)=m}. This relaxation allows the application of the augmented Lagrangian method for cases where the exact method is computationally undesirable.
The constraint \eqref{eqn:L(IJ)=m} is equivalent to
$
|\{T_qg_i \ :\ 1\leq i\leq |G|\}|\leq m.
$
In order to count the number of distinct elements in the set $\{T_qg_i \ :\ 1\leq i\leq |G|\}$, we can use the following
continuous approximation,\begin{align*}
&|\{T_qg_i \ :\ 1\leq i\leq |G|\}|\\
&=1+\sum_{i=2}^{|G|}\prod_{j=1}^{i-1} \mathbf{1}_{T_qg_i\neq T_qg_j}\\
&\approx 1+\sum_{i=2}^{|G|}\prod_{j=1}^{i-1} \left(\frac{1}{2}+ \frac{1}{\pi}\arctan (k\|T_qg_i-T_qg_j\|_1)\right),
\end{align*}
where $k$ is a chosen to be large and $\|\cdot\|_1$ refers to the $L_1$ norm. Putting this back into Problem~\cref{eqn:optimize_relaxed_formulation}, we obtain the following continuous optimization problem:
\begin{equation}\label{eqn:optimize_continuous_relaxed_formulation}
\begin{aligned}
&\underset{q}{\text{maximize}}\ \E[\KLD{T_{q}g_n}{T_{q}f}] \\
& \text{subject to} \\
& \sum_{i=2}^{|G|}\prod_{j=1}^{i-1} \left(\frac{1}{2}+ \frac{1}{\pi}\arctan (k\|T_qg_i-T_qg_j\|_1)\right)\leq m-1,
\end{aligned}
\end{equation}
for which an augmented Lagrangian Solver\cite{pyopt-paper,conn2013lancelot} can be used to obtain locally optimal solutions\cite{fernandez2012local}.
\subsection{Sequential Hypothesis Testing Privacy}\label{subsec:algorithms_sht}
In this subsection, we show that Problem~\cref{eqn:optimize_formulation_asymptotic_relaxed_objective_sht_linearized} is equivalent to a MILP. First, we require the following proposition.
\begin{Proposition}\label{prop:equiv_conditions}
For distribution $\phi$ on $\{1,\ldots,n\}$, $\phi$ satisfies
\begin{align}\label{eqn:equiv_prop_1}
\max_{i\in I_1} \min_{j\in I_1}\sum_{c=1}^n\phi(c)\KLD{T_cg_i}{T_cg_j}\leq \epsilon_1
\end{align}
if and only if there exist functions
\begin{align*}
\xi&:I_1\to \mathbb{R},\\
\delta&:I_1\times I_1 \to\{0,1\}
\end{align*}
such that
\begin{align}
&\xi(i)\leq \epsilon_1,\label{eqn:eqn:equiv_prop_2_1}\\
&\sum_{j\in I_1}\delta(j,i)=1,\label{eqn:eqn:equiv_prop_2_2}\\
&\sum_{c=1}^n\phi(c)\KLD{T_cg_i}{T_cg_j}\geq \xi(i),\label{eqn:eqn:equiv_prop_2_3}\\
&\sum_{c=1}^n\phi(c)\KLD{T_cg_a}{T_cg_j}\leq \xi(i)+(1-\delta(j,i))M,\label{eqn:eqn:equiv_prop_2_4}
\end{align}
where
$M=\max_{c\in\{1,\ldots,n\}}\max_{i,j\in I_1}\KLD{T_cg_i}{T_cg_j},$ and $i,j\in I_1$.
\end{Proposition}
\begin{IEEEproof}
See \cref{sec:AppProp2}.
\end{IEEEproof}
By \cref{prop:equiv_conditions}, Problem~\cref{eqn:optimize_formulation_asymptotic_relaxed_objective_sht_linearized} is equivalent to the following MILP:
\begin{align}\label{eqn:optimize_formulation_asymptotic_relaxed_objective_sht_MILP}
&\underset{\phi}{\text{maximize}} \ \sum_{c=1}^n \phi(c)\E[\KLD{T_{c}g_I}{T_{c}f}] \nonumber\\
&\text{subject to} \nonumber\\
& \sum_{c=1}^n\phi(c)\KLD{T_cg_i}{T_cg_j}\geq \epsilon_2 \nonumber\\
& \quad\quad\text{for $i\in I_2$ and $j\in\{1,\ldots,|G|\}$},\nonumber\\
& \xi(i)\leq \epsilon_1\quad\text{ for $i\in I_1$},\nonumber\\
& \delta(j,i)\in\{0,1\}\text{ for $i,j\in I_1$},\\
& \sum_{c=1}^n \phi(c)=1\nonumber\\
& \phi(c)\geq 0 \quad\text{$c\in\{1,\ldots,n\}$,}\nonumber\\
& \sum_{j\in I_1}\delta(j,i)=1\text{ for $i\in I_1$},\nonumber\\
& \sum_{c=1}^n\phi(c)\KLD{T_cg_i}{T_cg_j}\geq \xi(i) \ \text{for $i,j\in I_1$},\nonumber\\
& \sum_{c=1}^n\phi(c)\KLD{T_cg_i}{T_cg_j}\nonumber\\
& \quad\quad\leq \xi(i)+(1-\delta(j,i))M\quad\text{for $i,j\in I_1$}.\nonumber
\end{align}
A global optimal solution to Problem~\cref{eqn:optimize_formulation_asymptotic_relaxed_objective_sht_MILP} can be obtain using branch-and-bound methods on a linear program solver\cite{cvx}.
\section{Decentralized QCD with Independent Sensor observations}\label{sec:iid}
We assume that the sequence of random variables $X_1,X_2,\ldots$ satisfy the observations obtained at each sensor at any time instance are independent before the change point, and conditionally independent given $I$ after the change point. The observation obtained by the $k$-th sensor at time $t$, $X_{k,t}$, taking values in $\mathcal{X}$, satisfy the following:
\begin{align}\label{eqn:iidsignalmodel}
\begin{cases}
X_{k,t} \sim f_k \quad \text{i.i.d. for all $t< \nu$},\\
X_{k,t} \sim g_{k,i} \quad \text{i.i.d. for all $t\geq \nu$},\\
\end{cases}
\end{align}
for $k\in\{1,\ldots,K\}$. The observations $X_{1,t},\ldots,X_{K,t}$ are mutually independent, $\nu\geq 0$ is an unknown but deterministic change point and $i$ is the realization of the random variable $I$ which remains fixed for all $t\geq\nu$. We denote the marginal distribution of $X_{k,t}$ under $f$ and $g_i$ as $f_k$ and $g_{k,i}$ respectively.
For the task of privacy-aware decentralized QCD, we apply a memoryless privacy mechanism locally at each sensor $k$ for $k\in\{1,\ldots,K\}$. For each sensor $k$, a local privacy mapping or sanitization channel $q_k$ maps the observation $X \in \mathcal{X}$ obtained at sensor $k$ to a random variable $Y \in \mathcal{Y}$, where $\mathcal{Y}$ is a discrete alphabet such that $|\mathcal{Y}| \leq |\mathcal{X}|$. The local privacy mechanism $q_k$ can be represented by a conditional probability $\P(Y=y \ | X=x)$. Let $T_{q_k}$ be a column-stochastic matrix with $[T_{q_k}]_{y,x}=\P(Y=y){X=x}$ where $ [T_{q_k}]_{y,x}$ denotes the $(y,x)$ entry of the matrix $T_{q_k}$. Likewise, we represent a distribution $h$ on $\mathcal{X}$ as a column vector with $[h]_{x}=h(x)$ and similarly for a distribution on $\mathcal{Y}$.
At each time $t$ and sensor $k$, we apply the local sanitization channel $q_k$ to obtain $Y_{k,t}=q_k(X_{k,t})$. The sanitized signal $Y_{k,t}$ is generated i.i.d.\ by the distribution $T_{q_k} f_k$ in the pre-change regime and by the distribution $T_{q_k} g_{k,i}$ in the post-change regime, for some $1\leq i\leq |G|$. For a fixed set of sanitization channels $\{q_1,\ldots,q_K\}$, we let
$\widetilde{G}_{\{q_1,\ldots,q_K\}}=\{\widetilde{g}_1,\ldots,\widetilde{g}_{|G|}\}$
to be the set of possible post-change distributions where $\widetilde{g}_j$ is the post-change distribution generating the sanitized signal by applying the set of sanitization channels $\{q_1,\ldots, q_K\}$ to the observations generated by $g_i$ for some $i\in\{1,\ldots,|G|\}$.
\subsection{Algorithms for Decentralized Privacy-Aware QCD}\label{sec:decentralized_algorithms}
For the task of decentralized privacy-aware QCD , the sanitization channels are only allowed to use local observations to achieve sanitization of the signal. This introduces additional constraints on the structure of the sanitization channel $q$. In this section, we present algorithms for solving Problems~\cref{eqn:optimize_relaxed_formulation,eqn:optimize_formulation_asymptotic_relaxed_objective_sht_linearized} for the task of decentralized privacy-aware QCD.
\subsection{Maximal Leakage Privacy}\label{subsec:decen_algorithms_ml}
In this subsection, we present the Local Exact Method which solves Problem~\cref{eqn:optimize_relaxed_formulation} exactly and has computational complexity that scales linearly with respect to the number of sensors $K$.
First, for each conditional pmf $P_{J|I}$ satisfying
$
\P(J=j){I=i}\in\{0,1\}
$ for all $1\leq i\leq |G|$, $1\leq j \leq |\widetilde{G}|$ and $\mathcal{L}(I\to J)\leq \epsilon$, we solve the following problem:
\begin{equation}\label{eqn:cases}
\begin{aligned}
& \underset{q_k}{\text{maximize}}
& & \E[\KLD{T_{q_k} g_{k,I}}{T_{q_k} f_k}] \\
& \text{subject to}
& & T_{q_k} g_{k,i}=\widetilde{g}_j \quad \text{for all $i,j$},\\
& & & \text{such that $\P(J=j){I=i}=1$},
\end{aligned}
\end{equation}
for $k\in\{1,\ldots,K\}$. Similar to Problem~\eqref{eqn:N=1,epsilon>0,cases}, Problem~\eqref{eqn:cases} is maximizing a convex function over a convex bounded polytope. Therefore, we are able to solve Problem~\eqref{eqn:cases} by enumerating over the finite number of extreme points on the convex bounded polytope.
Next, for each $k\in\{1,\ldots,K\}$, we let $q_k^*(P_{J|I})$ be an optimal solution to Problem~\eqref{eqn:cases} corresponding to the conditional probability distribution $P_{J|I}$ and solve the following problem:
\begin{equation}\label{eqn:cases_2}
\begin{aligned}
P_{J|I}^*=&\underset{P_{J|I}}{\text{argmax}}
& &\sum_{k=1}^K \E[\KLD{T_{q_k^*(P_{J|I})} g_{k,I}}{T_{q_k^*(P_{J|I})} f_k}] \\
& \text{subject to}
& & \P(J=j){I=i}\in\{0,1\}\quad\text{for all $i,j$},\\
& & & \mathcal{L}(I\to J)\leq \epsilon.
\end{aligned}
\end{equation}There is a maximum of $\stirling{|G|}{m}$ conditional distributions $P_{J|I}$ that satisfy $\mathcal{L}(I\to J)\leq \epsilon$ where $m=\lfloor2^\epsilon\rfloor$ and $\stirling{a}{b}$ is the Stirling number of the second kind\cite{riordan2012introduction}. Thus, a solution for Problem~\eqref{eqn:cases_2} can be obtained by enumerating over the finite set of conditional probabilities $P_{J|I}$.
In the next proposition, we show that the set of sanitization channels obtained by the Local Exact method above is an optimal solution to Problem~\cref{eqn:optimize_relaxed_formulation} under the decentralized QCD setting.
\begin{Proposition}\label{prop:optimality}
Suppose the observations follow the signal model described in \eqref{eqn:iidsignalmodel}. Under the decentralized QCD setting, $\{q_1^*(P_{J|I}^*),\ldots,q_k^*(P_{J|I}^*)\}$ is an optimal solution to Problem~\cref{eqn:optimize_relaxed_formulation}. In particular, if $g_{1,i}=g_{2,i}=\ldots=g_{K,i}$ for $i\in\{1,\ldots,|G|\}$ then $\{q_1^*(P_{J|I}^*),\ldots,q_1^*(P_{J|I}^*)\}$ is an optimal solution to Problem~\cref{eqn:optimize_relaxed_formulation}.
\end{Proposition}
\begin{IEEEproof}
See \cref{sec:AppProp3}.
\end{IEEEproof}
\subsection{Sequential Hypothesis Testing Privacy}\label{subsec:decen_algorithms_sht}
Similar to the general case, Problem~\cref{eqn:optimize_formulation_asymptotic_relaxed_objective_sht} for the decentralized privacy-aware QCD problem is non-convex as the constraints $\mathcal{K}_1(T_q)\leq \epsilon_1$ and $\mathcal{K}_2(T_q)\geq \epsilon_2$ are non-convex. We use a restricted sanitization model to improve the computational tractability of Problem~\cref{eqn:optimize_formulation_asymptotic_relaxed_objective_sht}. However, in the decentralized version of the problem, each sensor is allowed to select its sanitization channel independent of the rest of the sensors. Let $\mathcal{C}_k=\{q_{k,1},q_{k,2},\ldots,q_{k,n}\}$ be a finite set of local sanitization channels at sensor $k$ with column-stochastic matrices $T_{k,1},\ldots,T_{k,n}$. At each time instance $t$, we assume that sensor $k$ obtains an observation $Y_{k,t}=(Z_{k,t},A_{k,t})$ where $Z_{k,t}=q_{A_{k,t}}(X_{k,t})$ is a randomized function of the random variable $X_{k,t}$ under the sanitization channel $q_{A_{k,t}}$. We further assume that $\{A_{k,t}\}_{t\in\mathbb{N}}$ are i.i.d. generated with distribution $\phi_k$. Under the restricted sanitization model, rather than designing the sanitization channel $q$, we design the distribution $\phi_k$ that samples the sanitization channel $q_{A_{k,t}}$ from $\mathcal{C}_k$ at each sensor $k$ and time instance $t$. We then apply $q_{A_{k,t}}$ to $X_{k,t}$ to locally sanitize the signal.
Using similar arguments from \cite{lau2017optimal}, the asymptotic $\text{ARL}$-$\text{EWADD}$ trade-off of the GLR CuSum stopping time under the restricted sanitization model is given as
\begin{align*}
&\text{EWADD}(\omega_{q})\\
&=\E[\frac{\log \gamma}{\sum_{k=1}^K\sum_{c=1}^n\phi_k(c)\KLD{T_{q_{k,c}} g_{k,I}}{T_{q_{k,c}} f_k}}](1+o(1)),
\end{align*}
as $\gamma\to\infty$. We also have
\begin{align*}
&\E[\KLD{T_{q}g_I}{T_{q}f}]\\
&=\E[\sum_{k=1}^K\sum_{c=1}^n\phi_k(c)\KLD{T_{q_{k,c}} g_{k,I}}{T_{q_{k,c}} f_k}].
\end{align*}
Thus, Problem~\cref{eqn:optimize_formulation_asymptotic_relaxed_objective_sht} becomes
\begin{equation}\label{eqn:optimize_formulation_asymptotic_relaxed_objective_sht_linearized_decent}
\begin{aligned}
& \underset{\phi}{\text{maximize}}
& & \sum_{k=1}^K\sum_{c=1}^n \phi_k(c)\E[\KLD{T_{q_{k,c}} g_{k,I}}{T_{q_{k,c}} f_k}] \\
& \text{subject to} &&\max_{i\in I_1} \min_{j\in I_1}\sum_{k=1}^K\sum_{c=1}^n\phi_k(c)\KLD{T_{q_{k,c}}g_{k,i}}{T_{q_{k,c}}g_{k,j}}\leq \epsilon_1,\\
& & &\sum_{k=1}^K\sum_{c=1}^n \phi_k(c)=1\\
& & &\phi_k(c)\geq 0, \quad\text{$c\in\{1,\ldots,n\}$ and $k\in\{1,\ldots,K\}$}\\
& & &\sum_{k=1}^K\sum_{c=1}^n\phi_k(c)\KLD{T_{q_{k,c}}g_{k,i}}{T_{q_{k,c}}g_{k,j}}\geq \epsilon_2\\
& & & \quad \text{for $i\in I_2$ and $j\in \{1,\ldots,|G|\}$}.
\end{aligned}
\end{equation}
Using similar arguments from \cref{prop:equiv_conditions}, Problem~\cref{eqn:optimize_formulation_asymptotic_relaxed_objective_sht_linearized_decent} is equivalent to the following MILP,
\begin{align}\label{eqn:optimize_formulation_asymptotic_relaxed_objective_sht_MILP_decent}
& \underset{\phi}{\text{maximize}}\
\sum_{k=1}^K\sum_{c=1}^n \phi_k(c)\E[\KLD{T_{q_{k,c}} g_{k,I}}{T_{q_{k,c}} f_k}] \\
& \text{subject to} \nonumber\\
& \xi(i)\leq \epsilon_1\quad\text{ for $i\in I_1$},\nonumber\\
& \delta(j,i)\in\{0,1\}\text{ for $i,j\in I_1$},\nonumber\\
& \sum_{k=1}^K\sum_{c=1}^n \phi_k(c)=1\nonumber\\
& \phi_k(c)\geq 0 \quad\text{$c\in\{1,\ldots,n\}$ and $k\in\{1,\ldots,K\}$} \nonumber\\
& \sum_{j\in I_1}\delta(j,i)=1\text{ for $i\in I_1$},\nonumber\\
& \sum_{k=1}^K\sum_{c=1}^n\phi_k(c)\KLD{T_{q_{k,c}}g_{k,i}}{T_{q_{k,c}}g_{k,j}}\nonumber\\
& \quad\quad\leq \xi(i)+(1-\delta(j,i))M\quad \text{for $i,j\in I_1$}.\nonumber\\
& \sum_{k=1}^K\sum_{c=1}^n\phi_k(c)\KLD{T_{q_{k,c}}g_{k,i}}{T_{q_{k,c}}g_{k,j}}\geq \epsilon_2\nonumber\\
& \quad \text{for $i\in I_2$ and $j\in \{1,\ldots,|G|\}$},\nonumber\\
& \sum_{k=1}^K\sum_{c=1}^n\phi_k(c)\KLD{T_{q_{k,c}}g_{k,i}}{T_{q_{k,c}}g_{k,j}}\geq \xi(i) \nonumber\\
& \quad\quad \text{for $i,j\in I_1$}.\nonumber
\end{align}
A global optimal solution to Problem~\cref{eqn:optimize_formulation_asymptotic_relaxed_objective_sht_MILP_decent} can be obtained using branch-and-bound methods on a linear program solver\cite{cvx}.
\section{Numerical Experiments}\label{sec:numerical}
In this section, we present numerical results of experiments of the various methods introduced under the different signal models. The simulations were performed using MATLAB R2017a on a laptop with Intel(R) Core(TM) i7-6500U [email protected] and 16.0GB RAM.
\subsection{Privacy-Aware QCD}
\subsubsection{Maximal Leakage Privacy}
\begin{figure}[!htbp]
\centering
\includegraphics[width=8cm]{compare_exact_al_method.pdf}
\caption{Trade-off between the privacy budget $\epsilon$ and the expected KL divergence $\E[\KLD{T_{q}g_n}{T_{q}f}]$ for the Exact and augmented Lagrangian method
}
\label{fig:compare_exact_al_method}
\end{figure}
In this subsection, we present numerical results for the privacy-aware QCD task under the maximal leakage privacy metric. We consider the signal model with pre-change distribution $f$, set of possible post-change distribution $G=\{g_1,g_2,\ldots,g_5\}$ where $f$ and $G$ are randomly generated, uniform prior $p_I$ on the post-change distributions and $\mathcal{X}=\mathcal{Y}=\{1,2,\ldots,7\}$.
Using algorithms described in \cref{subsec:algorithms_ml}, we solve the relaxed channel design problem~\cref{eqn:optimize_relaxed_formulation} exactly and problem \cref{eqn:optimize_continuous_relaxed_formulation} using the augmented Lagrangian method. First, we present results to illustrate the trade-off between the privacy budget $\epsilon$ and the expected KL divergence. In \cref{fig:sanitized5,fig:sanitized1,fig:sanitized3}, we plot the pmf of the original distributions, sanitized distributions when $\epsilon=\log_2 1$ and sanitized distributions when $\epsilon=\log_2 3$, respectively. In \figref{fig:compare_exact_al_method}, the expected KL divergence obtained by each of the methods is plotted against the privacy budget $\epsilon$. When the privacy budget $\epsilon=\log_2 5\approx 2.3$, the privacy constraint $\mathcal{L}_{\text{max}}(I\to J)\leq \epsilon$ becomes redundant as it is trivially satisfied. Problem~\eqref{eqn:optimize_formulation} then reduces to a standard QCD problem without any sanitization of the observations. It should be noted that two graphs intersect the y-axis at a positive value rather than at zero. This is because when our privacy budget $\epsilon$ is zero, we only remove all information that allows us to identify the post-change distributions. Thus, it is possible to distinguish the post-change distributions from the pre-change distribution in some cases. As the augmented Lagrangian method only guarantees local optimality, the channel obtained by solving Problem \cref{eqn:optimize_continuous_relaxed_formulation} achieves a lower value of the objective function as compared to the channel obtained by solving Problem~\cref{eqn:optimize_relaxed_formulation}. Next, we plot the compute time required for each of the methods against the privacy budget $\epsilon$ in \figref{fig:compare_exact_al_method_time}. The results indicate that the augmented Lagrangian method requires significantly lesser compute time compared to the Exact method. It should be noted that the compute time required by the Exact Method is directly proportional to $\stirling{|G|}{m}$. The peak observed in \figref{fig:compare_exact_al_method_time} corresponds to $\stirling{|G|}{m}$ achieving its maximum at $m=3$ when $|G|=5$.
\begin{figure}[!htbp]
\centering
\includegraphics[width=8cm]{pmf_sanitied_5.pdf}
\caption{Pmf of pre- and post-change distributions.}
\label{fig:sanitized5}
\end{figure}
\begin{figure}[!htbp]
\centering
\includegraphics[width=8cm]{pmf_sanitied_1}
\caption{Pmf of pre- and post-change sanitized distributions for $\epsilon=\log_2 1$.}
\label{fig:sanitized1}
\end{figure}
\begin{figure}[!htbp]
\centering
\includegraphics[width=8cm]{pmf_sanitied_3}
\caption{Pmf of pre- and post-change sanitized distributions for $\epsilon=\log_2 3$.}
\label{fig:sanitized3}
\end{figure}
\begin{figure}[!htbp]
\centering
\includegraphics[width=8cm]{compare_exact_al_method_time.pdf}
\caption{Comparison of the time taken to solve the relaxed channel design problem for the Exact and augmented Lagrangian method.}
\label{fig:compare_exact_al_method_time}
\end{figure}
\subsubsection{Sequential Hypothesis Testing Privacy}
In this subsection, we present numerical results for the centralized QCD task under the sequential hypothesis testing privacy metric. We consider the signal model with pre-change distribution $f$, set of possible post-change distribution $G=\{g_1,g_2,\ldots,g_6\}$ with $I_1=\{1,2,3\}$ and $I_2=\{4,5,6\}$, a uniform prior $p_I$ on the post-change distributions, $\mathcal{X}=\mathcal{Y}=\{1,2,3,4\}$ and the set of deterministic functions from $\mathcal{X}$ to $\mathcal{Y}$ as the finite set of sanitization channel $\mathcal{C}$. We generate $f$ and $G$ randomly.
Using algorithms described in \cref{subsec:algorithms_sht}, we solve the relaxed channel design problem~\cref{eqn:optimize_formulation_asymptotic_relaxed_objective_sht_linearized} by solving the MILP in Problem~\cref{eqn:optimize_formulation_asymptotic_relaxed_objective_sht_MILP}. By the data processing inequality $\mathcal{K}_1(T_q)\leq \max_{i\in I_1}\min_{j\in I_1}\KLD{g_i}{g_j}=0.9387$ and $\mathcal{K}_2(T_q)\leq \min_{i\in I_2}\min_{j\in I_1\cup I_2}\KLD{g_i}{g_j}=0.0855$, thus we focus our attention to the region where $0\leq \epsilon_1\leq 0.9387 $ and $0\leq\epsilon_2\leq 0.0855$. In \cref{fig:original_seq,fig:sanitized_seq}, we plot the pmf of the original distributions, sanitized distributions when $\epsilon_1=0.0012, \epsilon_2=0.0024$, respectively. We compare the trade-off between the privacy $\epsilon_1$ of $I_1$ in $G$ and the expected KL divergence in \figref{fig:compare_privacy_vs_kl} and the trade-off between the distinguishability $\epsilon_2$ of $I_1$ in $G$ and the expected KL divergence in \figref{fig:compare_distinguishability_vs_kl}. In both \figref{fig:compare_privacy_vs_kl} and \figref{fig:compare_distinguishability_vs_kl}, we observe that the average KL divergence increases as $\epsilon_1$ increases and $\epsilon_2$ decreases which is consistent with the behaviour expected of the optimal value of Problem~\cref{eqn:optimize_formulation_asymptotic_relaxed_objective_sht_linearized}.
\begin{figure}[!htbp]
\centering
\includegraphics[width=8cm]{pmf_seq_testing_original.pdf}
\caption{Pmf of pre- and post-change original distributions.}
\label{fig:original_seq}
\end{figure}
\begin{figure}[!htbp]
\centering
\includegraphics[width=8cm]{pmf_seq_testing_sanitized.pdf}
\caption{Pmf of pre- and post-change sanitized distributions for $\epsilon_1=0.0012$ and $\epsilon_2=0.0024$.}
\label{fig:sanitized_seq}
\end{figure}
\begin{figure}[!htbp]
\centering
\includegraphics[width=8cm]{privacy_vs_kl}
\caption{Trade-off between the privacy budget $\epsilon_1$ and the expected KL divergence for different distinguishability levels $\epsilon_2$.}
\label{fig:compare_privacy_vs_kl}
\end{figure}
\begin{figure}[!htbp]
\centering
\includegraphics[width=8cm]{distinguishability_vs_kl}
\caption{Comparison of trade-off between the distinguishability $\epsilon_2$ of $I_2$ in $G$ and the expected KL divergence for different privacy $\epsilon_1$ of $I_1$ in $G$.}
\label{fig:compare_distinguishability_vs_kl}
\end{figure}
\subsection{Decentralized Privacy-Aware QCD}
\subsubsection{Maximal Leakage Privacy}
In this subsection, we present numerical results for the decentralized QCD task under the maximal leakage privacy metric. We consider the signal model described in \eqref{eqn:iidsignalmodel}. We generate $f_k$ and $g_{k,i}$ randomly such that $g_{k_1,i}=g_{k_2,i}$ for all $k_1,k_2\in\{1,\ldots,K\}$ and $i\in\{1,\ldots,5\}$, use a uniform prior $p_I$ on the post-change distributions and let $\mathcal{X}=\mathcal{Y}=\{1,2,\ldots,5\}$.
\begin{figure}[!htbp]
\centering
\includegraphics[width=8cm]{privacy_util_tradeoff_max_leak_decent}
\caption{The trade-off between the expected KL divergence and the privacy budget $\epsilon$ for different number of sensors.}
\label{fig:privacy_util_tradeoff_decent}
\end{figure}
\begin{figure}[!htbp]
\centering
\includegraphics[width=8cm]{privacy_util_tradeoff_max_leak_decent_vs_sensors.pdf}
\caption{The expected KL divergence achieved by the solved sanitization channel for different number of sensors.}
\label{fig:privacy_util_tradeoff_decent_linear}
\end{figure}
We solve the relaxed channel design problem~\cref{eqn:optimize_relaxed_formulation} using the Local Exact method described in \cref{subsec:decen_algorithms_ml} for different number of sensors $K$. In \figref{fig:privacy_util_tradeoff_decent}, we plot the trade-off between the expected KL divergence and the privacy budget $\epsilon$ for different number of sensors $K$. In \figref{fig:privacy_util_tradeoff_decent_linear}, we plot the expected KL divergence for the privacy budget $\epsilon=0,1,\log_2 5$ as the number of sensors vary between $1$ and $150$. The results from the experiments indicate that for the case where the signal observed at each of sensors are identically distributed, the expected KL divergence grows linearly with respect the number of sensors $K$. Next, we compare the compute time taken to solve Problem~\eqref{eqn:optimize_relaxed_formulation} using the Exact method and the Local exact method. We solve the relaxed channel design problem~\cref{eqn:optimize_relaxed_formulation} with a privacy budget of $\epsilon=1$ using the Exact method described in \cref{subsec:algorithms_ml} and the Local Exact method described in \cref{sec:iid}. \begin{figure}[!htbp]
\centering
\includegraphics[width=8cm]{time_vs_sensors}
\caption{Comparison of compute time for the Exact and Local Exact method.}
\label{fig:compare_exact_method}
\end{figure}In \figref{fig:compare_exact_method}, we plot the compute time required to solve Problem~\eqref{eqn:optimize_relaxed_formulation} against the number of sensors $K$ for the Exact method and the Local exact method respective. The simulation result indication that the compute time required by Exact method increases exponentially with respect to the number of sensors $K$ while the compute time required by the Local Exact method remains reasonably low.
\subsubsection{Sequential Hypothesis Testing Privacy}
In this subsection, we present numerical results for the decentralized QCD task under the sequential hypothesis testing privacy metric. We consider the signal model described in \eqref{eqn:iidsignalmodel}. We generate $f_k$ and $g_{k,i}$ randomly such that $g_{k_1,i}=g_{k_2,i}$ for all $k_1,k_2\in\{1,\ldots,K\}$ and $i\in\{1,\ldots,6\}$, use a uniform prior $p_I$ on the post-change distributions, partition the index set of $G$ into $I_1=\{1,2,3,4\}$ and $I_2=\{5,6\}$, and let $\mathcal{X}=\mathcal{Y}=\{1,2,3\}$.
\begin{figure}[!htbp]
\centering
\includegraphics[width=8cm]{privacy_util_tradeoff_sequential_hypo_decent_vs_sensors.pdf}
\caption{The trade-off between the expected KL divergence and the privacy budget $\epsilon$ for different number of sensors.}
\label{fig:privacy_util_tradeoff_decent_sht}
\end{figure}
We solve the relaxed channel design problem~\cref{eqn:optimize_relaxed_formulation} using by solving the MILP described in \cref{subsec:decen_algorithms_sht} for the privacy budget $\epsilon_1=0.025,\epsilon_2=0.05$. In \figref{fig:privacy_util_tradeoff_decent_sht}, we present the simulation results to illustrate the relationship between the number of sensors and the expected KL divergence under the decentralized signal model. As the number of sensor increases, the privacy constraint $\mathcal{K}_1(T_q)\leq\epsilon_1$ becomes more difficult to satisfy and the distinguishably constraint $\mathcal{K}_2(T_q)\geq \epsilon_2$ becomes easier to satisfy. Thus, we do not expect that the expected KL divergence to grow linearly with the number of sensors. The simulation indicates that the expected KL divegerence does not increase linearly with the number of sensors $K$.
Next, we present the relationship between the average compute time required to solve Problem~\eqref{eqn:optimize_relaxed_formulation} and the number of sensors $K$. We randomly generate 500 sets of distributions satisfying $g_{k_1,i}=g_{k_2,i}$ for all $k_1,k_2\in\{1,\ldots,K\}$ and $i\in\{1,\ldots,6\}$. For each set of distribution, we randomly sample $\epsilon_1$ with uniform probability in the interval $[0,\max_{i\in I_1}\min_{j\in I_1}\KLD{g_i}{g_j}]$ and $\epsilon_2$ with uniform probability in the interval $[0,\min_{i\in I_2}\min_{j\in I_1\cup I_2}\KLD{g_i}{g_j}]$. In the event when the sampled $\epsilon_1,\epsilon_2$ makes Problem~\eqref{eqn:optimize_relaxed_formulation} infeasible, we resample $\epsilon_1,\epsilon_2$. The compute time taken to solve the MILP for each set of distributions is recorded and the relationship between the average compute time and the number of sensors is present in \figref{fig:compare_seq_hypo_time}. The simulation results suggests that the average compute time increases with respest to the number of sensors $K$ in a super-linear manner. This may be a potential challenge for applications when the number of sensors is large.
\begin{figure}[!htbp]
\centering
\includegraphics[width=8cm]{compare_decent_seq_hypo_time}
\caption{Relationship between average compute time and the number of sensors $K$ .}
\label{fig:compare_seq_hypo_time}
\end{figure}
\section{Conclusion and future work}\label{sec:conclusion}
In this paper, we have proposed a framework for privacy-aware QCD using two different privacy metrics for both centralized and decentralized QCD tasks. We also proposed optimization problems for which the solution provides a sanitization channel where the GLR CuSum stopping time is asymptotically optimal under each of the scenarios. We derived relaxations to the channel design problem and provided algorithms to obtain exact solutions to the relaxed channel design problem. For the maximal leakage privacy metric, a continuous relaxation for the channel design problem is proposed so that locally optimal solutions can be obtained when the exact solutions are computationally intractable. An algorithm that scales linearly with the number of sensors is also proposed for the decentralized QCD tasks when the signal recieved at the sensors are mutually independent. One drawback of the proposed signal and sanitization model for privacy-aware QCD under maximal leakage privacy metric is the discreteness of the privacy constraint $\mathcal{L}_{\text{max}}(I\to J)$. Using a randomization of multiple possible sanitization channels, the privacy constraint $\mathcal{L}_{\text{max}}(I\to J)$ is able to achieve a continuous interval of values. The optimization problem related to this new channel design problem is, however, much harder to solve and is a potential direction for future work. For the sequential hypothesis testing privacy metric, a modified sanitization model is proposed to relax the channel design problem into a MILP for both centralized and decentralized QCD tasks. One interesting direction of future work can be done to determine the pairs $(\epsilon_1,\epsilon_2)$ where Problem~\cref{eqn:optimize_formulation_asymptotic_relaxed_objective_sht_linearized} is feasible. It will also be interesting to design an algorithm with computation complexity that grows sub-linearly with the number of sensors for the decentralized QCD task with the sequential hypothesis testing privacy metric.
\appendices
\section{Proof of \cref{prop:upperbound_max_leakage}}\label[appendix]{sec:AppProp1}
We define the following matrices
$\Delta\in\mathbb{R}^{G\times\mathcal{U}}$, $\Theta\in\mathbb{R}^{\mathcal{Y}^{t-\nu+1}\times\widetilde{G}}$, $\Lambda\in\mathbb{R}^{\widetilde{G}\times G}$, $\Phi_0\in\mathbb{R}^{U\times\mathcal{Y}^{t-\nu+1}}$ and $\Phi_1\in\mathbb{R}^{U\times\widetilde{G}}$ such that
\begin{align}\label{eqn:matrices_1}
&[\Delta]_{i,u}= P_{U|I}(u\ |\ i)P_I(i),\nonumber\\
&[\Theta]_{y^{\nu:t},j}=P_{Y^{\nu:t}|J}(y^{\nu:t}\ |\ j),\\
&[\Lambda]_{j,i}==P_{J|I}(j\ |\ i),\nonumber
\end{align}
and
\begin{align}\label{eqn:matrices_2}
[\Phi_0]_{u,y^{\nu:t}}&=\begin{cases}
1\quad\text{if $u=\argmax_{u\in\mathcal{U}}P_{\mathcal{U},Y^{\nu:t}}(u,y^{\nu:t})$,}\\
0\quad\text{otherwise.}
\end{cases}\\
[\Phi_1]_{u,j}&=\begin{cases}
1\quad\text{if $u=\argmax_{u\in\mathcal{U}}P_{\mathcal{U},J}(u,j)$,}\\
0\quad\text{otherwise.}
\end{cases}\nonumber
\end{align}
For a fixed random variable $U$ that is a randomized function of $I$, we have
\begin{align}\label{eqn:demoninator_ml_1}
&\sum_{y^{\nu:t}}\max_{u\in\mathcal{U}}P_{\mathcal{U},Y^{\nu:t}}(u,y^{\nu:t})\nonumber\\
&=\sum_{y^{\nu:t}}\max_{u\in\mathcal{U}}\sum_{j=1}^{|\widetilde{G}|}\sum_{i=1}^{|G|}P_{J,I,Y^{\nu:t},\mathcal{U}}(j,i,y^{\nu:t},u)\nonumber\\
&=\sum_{y^{\nu:t}}\max_{u\in\mathcal{U}}\sum_{i=1}^{|G|}P_{\mathcal{U}|I}(u\ |\ i)P_I(i)\nonumber\\ &\quad\quad\quad\left(\sum_{j=1}^{|\widetilde{G}|}P_{Y^{\nu:t}|J}(y^{\nu:t}\ |\ j)P_{J|I}(j\ |\ i)\right).
\end{align}
Similarly, we have
\begin{align}\label{eqn:demoninator_ml_2}
\sum_{j=1}^{|\widetilde{G}|}\max_{u\in\mathcal{U}}P_{\mathcal{U},J}(u,j)&=\sum_{j=1}^{|\widetilde{G}|}\max_{u\in\mathcal{U}}\sum_{y^{\nu:t}}\sum_{i=1}^{|G|}P_{J,I,Y^{\nu:t},\mathcal{U}}(j,i,y^{\nu:t},u)\nonumber\\
&=\sum_{j=1}^{|\widetilde{G}|}\max_{u\in\mathcal{U}}\sum_{i=1}^{|G|}P_{\mathcal{U}|I}(u\ |\ i)P_{I}(i) P_{J|I}(j\ |\ i).
\end{align}
From \cref{eqn:matrices_1,eqn:matrices_2}, we can see that \cref{eqn:demoninator_ml_1,eqn:demoninator_ml_2} can be expressed using matrix operations
\begin{align*}
&\sum_{j=1}^{|\widetilde{G}|}\max_{u\in\mathcal{U}}P_{\mathcal{U},J}(u,j)=\text{Trace}\left(\Lambda\Delta\Phi_1\right),\\
&\sum_{y^{\nu:t}}\max_{u\in\mathcal{U}}P_{\mathcal{U},Y^{\nu:t}}(u,y^{\nu:t})=\text{Trace}\left(\Theta\Lambda\Delta\Phi_0\right),
\end{align*}
Furthermore, $\Phi_1$ is a solution to the following problem,
\begin{align*}
\Phi_1=\argmax_{\Gamma}\text{Trace}(\Lambda\Delta\Gamma),
\end{align*}
where the maximization is taken over all column stochastic matrices $\Gamma$. Since $\Phi_0,\Theta$ are column stochastic matrices, we have
\begin{align*}
\text{Trace}\left(\Lambda\Delta\Phi_0\Theta\right)\leq \text{Trace}\left(\Lambda\Delta\Phi_1\right),
\end{align*}
and thus \begin{align*}
\sum_{y^{\nu:t}}\max_{u\in\mathcal{U}}P_{\mathcal{U},Y^{\nu:t}}(u,y^{\nu:t})\leq\sum_{j=1}^{|\widetilde{G}|}\max_{u\in\mathcal{U}}P_{\mathcal{U},J}(u,j).
\end{align*}
Hence, for a fixed random variable $U$ that is a randomized function of $I$, we have
\begin{align*}
\frac{\sum_{y^{\nu:t}}\max_{u\in\mathcal{U}}P_{\mathcal{U},Y^{\nu:t}}(u,y^{\nu:t})}{\max_{u\in\mathcal{U}}P_{\mathcal{U}}(u)}\leq\frac{\sum_{j=1}^{|\widetilde{G}|}\max_{u\in\mathcal{U}}P_{\mathcal{U},J}(u,j)}{\max_{u\in\mathcal{U}}P_{\mathcal{U}}(u)}.
\end{align*}
Using the following equations\cite{issa2016operational},
\begin{align*}
\mathcal{L}_{\text{max}}(I\to Y^{\nu:t})&=\sup_U \frac{\sum_{y^{\nu:t}}\max_{u\in\mathcal{U}}P_{\mathcal{U},Y^{\nu:t}}(u,y^{\nu:t})}{\max_{u\in\mathcal{U}}P_{\mathcal{U}}(u)},\\
\mathcal{L}_{\text{max}}(I\to J)&=\sup_U \frac{\sum_{j}\max_{u\in\mathcal{U}}P_{\mathcal{U},J}(u,j)}{\max_{u\in\mathcal{U}}P_{\mathcal{U}}(u)}.
\end{align*}
to compute maximal leakage, where the supremum is taken over all randomized functions $U$ of $I$, we have
\begin{align*}
\mathcal{L}_{\text{max}}(I\to Y^{\nu:t})\leq\mathcal{L}_{\text{max}}(I\to J).
\end{align*}
The proof is now complete.
\section{Proof of \cref{prop:equiv_conditions}}\label[appendix]{sec:AppProp2}
$(\Rightarrow)$ Suppose the distribution $\phi$ on $\{1,\ldots,n\}$ satisfies \cref{eqn:equiv_prop_1}. Define $\xi:I_1\to \mathbb{R}$ and $\delta:I_1\times I_1 \to\{0,1\}$ such that
\begin{align*}
\xi(i)&=\min_{j\in I_1}\sum_{c=1}^n\phi(c)\KLD{T_cg_i}{T_cg_j}\quad\text{for $j\in I_1$},\\
\delta(j,i)&=\begin{cases}
1\quad\text{if $j=\argmin \sum_{c=1}^n\phi(c)\KLD{T_cg_i}{T_cg_j}$},\\
0\quad\text{otherwise.}\\
\end{cases}
\end{align*}
We can check that $\xi$ and $\delta$ as defined satisfy \cref{eqn:eqn:equiv_prop_2_1,eqn:eqn:equiv_prop_2_2,eqn:eqn:equiv_prop_2_3,eqn:eqn:equiv_prop_2_4}.
$(\Leftarrow)$Now suppose that there exist functions $\xi:I_1\to \mathbb{R}$ and $\delta:I_1\times I_1 \to\{0,1\}$ such that the distribution $\phi$ on $\{1,\ldots,n\}$ satisfies \cref{eqn:eqn:equiv_prop_2_1,eqn:eqn:equiv_prop_2_2,eqn:eqn:equiv_prop_2_3,eqn:eqn:equiv_prop_2_4}. Fix $i\in I_1$. From \eqref{eqn:eqn:equiv_prop_2_3}, we have
\begin{align*}
\sum_{c=1}^n\phi(c)\KLD{T_cg_i}{T_cg_j}\geq \xi(i),
\end{align*} for each $j\in I_1$. Taking minimum over $j\in I_1$, we obtain
\begin{align*}
\min_{j\in I_1}\sum_{c=1}^n\phi(c)\KLD{T_cg_i}{T_cg_j}\geq \xi(i).
\end{align*} From \eqref{eqn:eqn:equiv_prop_2_2}, we have
\begin{align*}
\sum_{j\in I_1}\delta(j,i)=1.
\end{align*} Since $\delta(j,i)\in\{0,1\}$, there exists a unique $j'$ such that $\delta(j',i)=1$. Putting this together with \cref{eqn:eqn:equiv_prop_2_4}, we obtain \begin{align*}
\min_{j\in I_1}\sum_{c=1}^n\phi(c)\KLD{T_cg_i}{T_cg_j}=\xi(i).
\end{align*}
By \cref{eqn:eqn:equiv_prop_2_1}, we obtain\begin{align*}
\max_{i\in I_1} \min_{j\in I_1}\sum_{c=1}^n\phi(c)\KLD{T_cg_i}{T_cg_j}\leq \epsilon_1.
\end{align*}
The proposition is proved.
\section{Proof of \cref{prop:optimality}}\label[appendix]{sec:AppProp3}
Let $q_1^\dagger,\ldots,q_K^\dagger$ be an optimal solution to Problem~\eqref{eqn:optimize_relaxed_formulation} under the decentralized QCD setting, $P_{J|I}^\dagger$ be the corresponding conditional probability distribution of $J$ given $I$, and $\mu^\dagger$ be the corresponding optimal value.
Since the observations obtained by the sensors are mutually independent,
we have
\begin{align*}
\E[\KLD{\widetilde{g}_I}{\widetilde{f}}]=\sum_{k=1}^K\E[\KLD{T_{q_k}g_{k,I}}{T_{q_k}f_{k}}].
\end{align*}
By our construction of $q^*_k(P_{J|I}^\dagger)$, we have
\begin{align}
\label{eqn:prop2_1_1}
\begin{split}
&\E[\KLD{T_{q_k^*\left(P_{J|I}^\dagger\right)} g_{k,I}}{T_{q_k^*\left(P_{J|I}^\dagger\right)} f_k}]
\\
&\quad\geq \E[\KLD{T_{q_k^\dagger} g_{k,I}}{T_{q_k^\dagger} f_k}],
\end{split}
\end{align}
for each $k\in\{1,\ldots,K\}$. Let $\mu^*$ be the value achieved by the cost function in Problem~\eqref{eqn:optimize_relaxed_formulation} corresponding to $\{q_1^*(P_{J|I}^*),\ldots,q_k^*(P_{J|I}^*)\}$ and we obtain
\begin{align}
&\mu^*=\sum_{k=1}^K\E[\KLD{T_{q_k^*\left(P_{J|I}^*\right)} g_{k,I}}{T_{q_k^*\left(P_{J|I}^*\right)} f_k}]\label{eqn:prop2_1}\\
&\quad\geq \sum_{k=1}^K\E[\KLD{T_{q_k^*\left(P_{J|I}^\dagger\right)} g_{k,I}}{T_{q_k^*\left(P_{J|I}^\dagger\right)} f_k}]\label{eqn:prop2_2}\\
&\quad\geq\sum_{k=1}^K \E[\KLD{T_{q_k^\dagger} g_{k,I}}{T_{q_k^\dagger} f_k}]=\mu^\dagger\label{eqn:prop2_3}
\end{align}
where the inequality from \eqref{eqn:prop2_1} to \eqref{eqn:prop2_2} is a consequence of our choice of $P_{J|I}^*$ in \eqref{eqn:cases_2} and the inequality from \eqref{eqn:prop2_2} to \eqref{eqn:prop2_3} is obtained by summing \eqref{eqn:prop2_1_1} over $k\in\{1,\ldots,K\}$. Since $P_{J|I}^*$ also satisfies \begin{align*}\mathcal{L}(I\to J)\leq\epsilon,\end{align*} $\{q_k^*(P_{J|I}^*),\ldots,q_k^*(P_{J|I}^*)\}$ is also an optimal solution to Problem~\eqref{eqn:optimize_relaxed_formulation}. The proof is now complete.
\bibliographystyle{IEEETran}
|
2024-02-18T23:40:24.821Z
|
2020-09-21T02:18:18.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2009.08963","language":"en","timestamp":1600654698000,"url":"https:\/\/arxiv.org\/abs\/2009.08963","yymm":"2009"}
|
proofpile-arXiv_000-10095
|
{"provenance":"002.jsonl.gz:10096"}
| null | null |
\section{Introduction}
We consider the following
nonlinear eigenvalue problem with eigenvector-dependency (NEPv):
find an orthonormal matrix $V\in \mathbb{C}^{n\times k}$, i.e.,
$V^{\HH}V = I_k$, and
a square matrix $\Lambda\in\mathbb{C}^{k\times k}$ satisfying
\begin{equation}\label{eq:nepv}
H(V)V=V\Lambda,
\end{equation}
where $H\colon \mathbb{C}^{n\times k}\to \mathbb{C}^{n\times n}$ is a continuous
Hermitian matrix-valued function of $V$.
Necessarily, $\Lambda = V^{\HH} H(V) V$ and the eigenvalues
of $\Lambda$ are $k$ eigenvalues of $H(V)$, often
either the $k$ smallest or largest ones. Our later analysis
will focus on $\Lambda$
associated with the $k$ smallest eigenvalues of $H(V)$,
but it works equally well for the case when
$\Lambda$ is associated with the $k$ largest ones.
We assume throughout this paper
that $H(V)$ is right-unitarily invariant in $V$, i.e.,
\begin{equation}\label{eq:univar}
H(VQ)=H(V) \quad \mbox{for any unitary $Q\in\mathbb{U}^{k\times k}$},
\end{equation}
where $\mathbb{U}^{k\times k}$ is the set of all $k\times k$ unitary matrix.
This property~\eqref{eq:univar} essentially says that
NEPv~\eqref{eq:nepv} is eigenspace-dependent,
to be more precise.
However, we will adopt the notion of nonlinear eigenvalue problem
with eigenvector-dependency commonly used in literature.
Furthermore, the assumption~\eqref{eq:univar} implies that
if $(V,\Lambda)$ is a solution of NEPv~\eqref{eq:nepv},
then so is $(VQ,Q^{\HH}\Lambda Q)$ for any unitary $Q$.
We therefore view $V$ and $\widetilde V$
as an identical solution, if the two share
a common range ${\cal R}(V)={\cal R}(\widetilde V)$.
NEPv in the form of~\eqref{eq:nepv} arises frequently in
a number of areas of computational science and engineering.
They are the discrete representations of
the Kohn-Sham equation of the density functional theory in electronic structure
calculations~\cite{Martin:2004,Szabo:2012},
and the Gross-Pitaevskii equation in modeling the ground state wave function
in a Bose-Einstein condensate~\cite{Bao:2004,Jarlebring:2014}.
In particular, $H(V) = \Phi(P)$, where $\phi$
is a Hermitian matrix-valued function of $P=VV^{\HH}$, known as
the density matrix in the density functional
theory~\cite{Martin:2004,Szabo:2012}.
NEPv have also long played important roles in the classical
methods for data analysis, such as multidimensional scaling \cite{Meyer:1997}.
It has become increasingly popular recently in the fields of
machine learning and network science, such as
the trace ratio maximizations for dimensional
reduction~\cite{Ngo:2012,Zhang:2014},
balanced graph cut~\cite{Jost:2014},
robust Rayleigh quotient maximization for
handling data uncertainty~\cite{Bai:2018},
core-periphy detection in networks~\cite{tuhi:2019},
and orthogonal canonical correlation analysis \cite{zhwb:2020}.
The unitary invariance~\eqref{eq:univar} holds in all those
practical NEPv except few.
The Self-Consistent Fields (SCF) iteration
is the most general and widely-used method to
solve NEPv~\eqref{eq:nepv}.
SCF, first introduced in molecular quantum mechanics back
to 1950s~\cite{Roothaan:1951}, serves as an entrance to
all other approaches.
Starting with an orthonormal matrix $V_0\in\mathbb{U}^{n\times k}$,
SCF computes iteratively $V_{i+1}$ and $\Lambda_{i+1}$ satisfying
\begin{equation}\label{eq:pscf}
H(V_i) V_{i+1} = V_{i+1}\Lambda_{i+1},
\quad\text{for}\quad i = 0,1,2,\dots,
\end{equation}
where $V_{i+1}\in\mathbb{C}^{n\times k}$ is orthonormal and
$\Lambda_{i+1}$ is a diagonal matrix consisting of the $k$ smallest
eigenvalues of $H(V_i)$.
Since unit eigenvectors associated with simple eigenvalues can
differ by scalar factors of unimodular complex numbers
and those associated with multiple eigenvalues have even more freedom,
the iteration matrix $V_{i+1}$ cannot be uniquely defined.
But thanks to the property~\eqref{eq:univar}, the computed subspaces
${\cal R}(V_1),{\cal R}(V_2),\dots $ are always the same,
provided the $k$th and $(k+1)$st eigenvalues of $H(V_i)$ are
distinct at the $i$th
iteration. Because of this, SCF can be interpreted as an iteration
of subspaces of dimension $k$, i.e., elements in the Grassmann
manifold $\mathbf{Gr}(k,\mathbb{C}^n)$ of all $k$ dimensional subspaces of $\mathbb{C}^n$.
The procedure in~\eqref{eq:pscf} is an SCF in its simplest form,
also known as the plain SCF iteration.
In practice, such a procedure is prone to slow convergence
and sometimes may not converge~\cite{Koutecky:1971}.
Therefore it has been a fundamental problem of
intensive research for decades to understand when and how the plain SCF
would converge, so as to develop remedies to stabilize and
accelerate the SCF iteration.
For the applications of solving the Kohn-Sham equation in physics and quantum
chemistry, the solution of the associated NEPv corresponds to the minimizer
of an energy function.
In such context, optimization techniques can be employed to establish
convergence results of SCF.
A number of convergence conditions have been
investigated~\cite{Cances:2000,Liu:2014,Liu:2015,Yang:2009}.
For solving general NEPv, one may view the plain SCF~\eqref{eq:pscf}
as a simple fixed-point iteration.
Sufficient conditions for the fixed-point map being a contraction,
in terms of the sines of the canonical angles between subspaces,
has been studied in~\cite{Cai:2018}, where the authors revealed
a convergence rate for SCF based on the Davis-Kahan Sin$\Theta$
theorem for eigenspace perturbation~\cite{Davis:1970}.
Another approach for the fixed-point analysis is to look at the
spectral radius of
the Jacobian supermatrix of the fixed-point map, be it differentiable.
When $H(V)$ is a smooth function explicitly in the density matrix $P=VV^{\HH}$,
a closed-form expression of the Jacobian has been obtained in a
recent work~\cite{Upadhyaya:2018}. Similar analysis also appeared
in an earlier work \cite{Stanton:1981} by focusing on
the Hartree--Fock equation.
What is often different in the existing convergence analysis
is the way of measuring the approximation error.
Since SCF is a subspace iteration,
how to assess the distance between two subspaces ${\cal R}(V)$ and ${\cal R}(V_*)$
is the key to the convergence analysis.
Various distance measures have been applied in the literature,
leading to different approaches of analysis and different types of convergence
results. In particular,
the difference in density matrices in 2-norm is
used as a measure of distance in~\cite{Yang:2009};
A chordal 2-norm is used in~\cite{Liu:2014};
More recent work \cite{Cai:2018} turned to the sines
of the canonical angles between subspaces;
The work~\cite{Cances:2000} as well as \cite{Upadhyaya:2018} though
not explicitly specified, used the difference of density matrices
in the Frobenius norm. We believe that those distance measures
may not necessarily be the best to capture
the intrinsic feature of the SCF iteration.
The results presented in this paper is a refinement and extension
of the previous ones in~\cite{Cai:2018,Liu:2014,Upadhyaya:2018,Yang:2009}.
We aim to provide a comprehensive and unified local convergence analysis
of SCF.
Rather than resorting to a specific distance measure, our development is based
on the tangent-angle matrix, associated with the tangents of canonical
angles of subspaces. Such matrices can precisely capture the error
recurrence of SCF when close to convergence,
and they can act as intermediate measurements,
by which various distance measures can be evaluated as needed.
Despite less popular than sines, the tangents of canonical angles
have also been used to assess the distance between subspaces,
and can lead to tighter bounds when
applicable, see~\cite{Davis:1970,Zhu:2013}, and references therein.
The use of tangent-angle matrix allows us to take a closer
examination at the local error recursion of SCF, leading
to the following new contributions presented in this paper:
\begin{enumerate}[(a)]
\item
A precise characterization for the local contraction
factor of SCF for both continuous and differentiable $H(V)$.
This improves over the previous work~\cite{Cai:2018,Liu:2014,Yang:2009},
where only upper bounds of such a quantity were obtained.
\item
A closed-form formula for the local asymptotic average contraction factor
of SCF in terms of the spectral radius of an underlying linear operator
when $H(V)$ is differentiable.
The formula is optimal for providing
a sufficient and almost necessary local convergence
condition of SCF.
It extends the previous work in~\cite{Stanton:1981,Upadhyaya:2018}
to general $H(V)$ functions, and has a compact
expression that is convenient to work with in both theory and computation.
\item
A new justifications for a commonly used
level-shifting scheme for the stabilization and acceleration
of SCF~\cite{Cances:2000}. A closed-form lower
bound on the shifting parameter to guarantee local convergence
is obtained.
\end{enumerate}
The rest of the paper is organized as follows.
\Cref{sec:prelim} presents some preliminaries to set up basic definitions
and assumptions.
\Cref{sec:tangent} introduces the tangent-angle matrix and
establishes the recurrence relation of such matrices in consecutive SCF
iteration.
\Cref{sec:conv} is devoted to the local convergence theory
of the plain SCF iteration.
\Cref{sec:ls} deals with the level-shifted SCF and its convergence.
Numerical illustrations are in~\Cref{sec:examples},
followed by conclusions in~\Cref{sec:conclusion}.
We follow the notation convention in matrix analysis:
$\mathbb{R}^{m\times n}$ and $\mathbb{C}^{m\times n}$ are the sets of
$m\times n$ real and complex matrices, respectively,
and $\mathbb{R}^n=\mathbb{R}^{n\times 1}$ and $\mathbb{C}^n=\mathbb{C}^{n\times 1}$.
$\mathbb{U}^{m\times n}\subset \mathbb{C}^{m\times n}$ denotes the set of
$m\times n$ complex orthonormal matrices.
$A$, $A^{\mbox{\scriptsize \rm T}}$ and $A^{\HH}$ are the transpose and conjugate
transpose of a matrix or a vector $A$, respectively, and $\overline{A}$ takes entrywise conjugate.
$H_1\succeq H_2$ means that $H_1$ and $H_2$ are Hermitian
matrices and $H_1-H_2$ is positive semi-definite.
For a matrix $H\in\mathbb{C}^{n\times n}$ known to have real eigenvalues only,
$\lambda_i(H)$ is the $i$th eigenvalue of $H$ in the
ascending order, i.e., $\lambda_1(H)\leq \lambda_2(H)\leq\cdots\leq \lambda_n(H)$, and
$\lambda_{\min}(H)=\lambda_1(H)$ and
$\lambda_{\max}(H)=\lambda_n(H)$.
$\Diag(x)$ is a diagonal matrix formed by the vector $x$,
$\diag (X)$ is a vector consisting of the diagonal elements of a matrix $X$;
${\cal R}(X)$ is the range of $X$;
$\sigma(X)$ is the collection of all singular values of $X$.
$\Re(\cdot)$ and $\Im(\cdot)$ extract the real and imaginary parts of
a complex number and, when applied to a matrix/vector,
they are understood in the elementwise sense.
Standard big-O and little-o notations in mathematical analysis are used:
for functions $f(x), g(x)\to 0$ as $x\to 0$,
write $f(x) = \mathcal{O}(g(x))$ if $|f(x)|\leq c|g(x)|$
for some constant $c$ as $x\to 0$,
and write $f(x) = \mathbf{o}(g(x))$ if $|f(x)|/|g(x)| \to 0$ as $x\to 0$.
Other notations will be explained at their first appearance.
\section{Preliminaries}\label{sec:prelim}
Throughout this paper, we denote by $V_*\in\mathbb{U}^{n\times k}$
a solution of NEPv~\eqref{eq:nepv}.
The eigen-decomposition of $H(V_*)$ is given by
\begin{equation}\label{eq:eigdec}
H(V_*)\,[V_*,V_{*\bot}] = [V_*,V_{*\bot}]\,\twobytwo{\Lambda_*}{}{}{\Lambda_{*\bot}},
\end{equation}
where $ [V_*,V_{*\bot}]\in \mathbb{U}^{n\times n}$ is unitary,
\[
\text{$\Lambda_*=\diag(\lambda_1,\dots,\lambda_k)$
and
$\Lambda_{*\bot}=\diag(\lambda_{k+1}, \dots, \lambda_n)$}
\]
are diagonal matrices containing the eigenvalues
of $H(V_*)$ in the ascending order, i.e., $\lambda_i=\lambda_i(H(V_*))$.
We make the following assumption for the solution $V_*$ of
NEPv~\eqref{eq:nepv} under consideration.
\begin{assumption}\label{ass:gap}
There is a positive eigenvalue gap:
\begin{equation}\label{eq:gap}
\delta_*:=\lambda_{k+1}(H(V_*))-\lambda_k(H(V_*)) > 0.
\end{equation}
\end{assumption}
Such an assumption,
which is commonly applied in the convergence analysis of
SCF guarantees the uniqueness of the eigenspace corresponding
to the $k$ smallest eigenvalues of
$H(V_*)$~\cite{Cai:2018,Cances:2000,Liu:2014,Upadhyaya:2018,Yang:2009}.
\paragraph{Sylvester equation}
The following Sylvester equation in $X\in\mathbb{C}^{n\times k}$ will be needed in
our analysis
\begin{equation} \label{eq:sylvester}
\Lambda_{*\bot} X - X\Lambda_*=V_{*\bot}^{\HH}[H(V_*)-H(V)]V_*.
\end{equation}
Under~\Cref{ass:gap}, this equation has a unique solution $X\equiv \SL(V)$
for each $V\in\mathbb{U}^{n\times k}$, given by
\begin{equation}\label{eq:xv}
\SL(V) = D(V_*) \odot \left(V_{*\bot}^{\HH}[H(V_*) - H(V)]V_*\right),
\end{equation}
where
\begin{equation}\label{eq:dvs}
D(V_*)\in\mathbb{R}^{(n-k)\times k}
\quad \text{with $D(V_*)_{ij} = (\lambda_{k+i}(H(V_*))- \lambda_{j}(H(V_*)))^{-1}$},
\end{equation}
and $\odot$ denotes the Hadamard product, i.e. elementwise multiplication.
\paragraph{Unitarily invariant norm}
We denote by $\|\cdot\|_{\UI}$ a unitarily invariant norm,
which, besides being a matrix norm, also satisfies
the following two additional conditions:
\begin{enumerate}[(1)]
\item
$\|XAY\|_{\UI} = \|A\|_{\UI}$ for any unitary matrices $X$ and $Y$;
\item
$\|A\|_{\UI} = \|A\|_2$ whenever $A$ is rank-1, where $\|\cdot\|_2$ is the
spectral norm.
\end{enumerate}
It is well-known that $\|A\|_{\UI}$ is dependent
only on the singular values of $A$.
In this paper, we assume any $\|\cdot \|_{\UI}$ we use
is applicable to matrices of all sizes in
a compatible way, i.e., $\|A\|_{\UI} =\|B\|_{\UI}$ for $A$, $B$ sharing
a same set of non-zero singular values
(see, e.g., \cite[Thm 3.6, pp 78]{Stewart:1990}).
The spectral norm $\|\cdot\|_2$ and Frobenius norm $\|\cdot \|_{\F}$
are two particular examples of such unitarily invariant norms.
\paragraph{Canonical angles between subspaces}
Let $X, Y\in\mathbb{U}^{n\times k}$.
The $k$ canonical angles between the range spaces of $\mathcal X={\cal R}(X)$ and
$\mathcal Y = {\cal R}(Y)$ are defined as
\begin{equation}\label{eq:indv-angles-XY}
0\le\theta_j(\mathcal X,\mathcal Y):=\arccos\sigma_j\le\frac {\pi}2\quad\mbox{for $1\le j\le k$},
\end{equation}
where $\sigma_1\ge\cdots\ge\sigma_k$ are singular values of
the matrix $Y^{\HH}X$ (see, e.g.,~\cite[Sec 4.2.1]{Stewart:1990}).
Put $k$ canonical angles all together to define
\begin{equation}\label{eq:mat-angles-XY}
\Theta(\mathcal X,\mathcal Y)=\diag(\theta_1(\mathcal X,\mathcal
Y),\ldots,\theta_k(\mathcal X,\mathcal Y)).
\end{equation}
Since the canonical angles so defined are independent of the basis matrices $X$ and
$Y$, for convenience, we use the notation
$\Theta({X},{Y})$ interchangeably with $\Theta(\mathcal X,\mathcal Y)$.
Canonical angles provide a natural distance measure for subspaces.
For any unitarily invariant norm $\|\cdot\|_{\UI}$,
it holds that both $\|\Theta({X},{Y})\|_{\UI}$ and $\|\sin\Theta({ X},{ Y})\|_{\UI}$ are
unitarily invariant metrics on the Grassmann manifold $\mathbf {Gr}(k,\mathbb{C}^n)$
(see e.g.,~\cite[Thm. 4.10, pp 93]{Stewart:1990} and~\cite{Qiu:2005}).
In our analysis, the tangents of canonical angles will play an important role.
By trigonometric function analysis,
tangents provide good approximation to the canonical angles as
$\Theta(X,Y)\to 0$:
\begin{equation}\label{eq:ts}
\tan\Theta({X},{Y}) = \Theta({X},{Y}) + \mathcal{O}(\|\Theta({X},{Y})\|_{\UI}^3).
\end{equation}
\paragraph{$\mathbb{R}$-linear mapping}
A mapping $\mathscr{L}} %{\mathcal{L}\colon \mathbb{C}^{n\times k}\to \mathbb{C}^{p\times q}$ is called
$\mathbb{R}$-linear, if it satisfies
\begin{equation}\label{eq:rlinear}
\mathscr{L}} %{\mathcal{L}(X+Y) = \mathscr{L}} %{\mathcal{L}(X) + \mathscr{L}} %{\mathcal{L}(Y)
\quad\text{and}\quad
\mathscr{L}} %{\mathcal{L}(\alpha\, X) = \alpha\, \mathscr{L}} %{\mathcal{L}(X)
\end{equation}
for all $X, Y\in\mathbb C^{n\times k}$ and $\alpha\in\mathbb R$.
When we talk about an $\mathbb R$-linear mapping,
the complex matrix space $\mathbb{C}^{m\times n}$ is viewed as a vector space over
the field $\mathbb{R}$ of real numbers, denoted by $\mathbb{C}^{m\times n}(\mathbb{R})$.
By elementary linear algebra, $\mathbb{C}^{m\times n}(\mathbb{R})$ is a $(2mn)$-dimensional
inner product space, equipped with
the inner product $\langle X,Y\rangle:= \Re \tr(X^{\HH}Y)$
and the induced norm $\|X\|_{\F} = \left(\Re \tr(X^{\HH}X)\right)^{1/2}$.
We can see that
$\mathscr{L}} %{\mathcal{L}\colon \mathbb{C}^{n\times k}(\mathbb{R})\to \mathbb{C}^{p\times q}(\mathbb{R})$
is a linear mapping (over $\mathbb{R}$).
For convenience, we use $\mathbb C^{n\times k}$ and $\mathbb C^{n\times
k}(\mathbb{R})$ interchangeably in future discussions when referring
to an $\mathbb{R}$-linear mapping.
The \emph{spectral radius} of
an $\mathbb{R}$-linear operator $\mathscr{L}} %{\mathcal{L}: \mathbb{C}^{n\times k} \to \mathbb{C}^{n\times k}$
is defined as the largest eigenvalue in magnitude of a
matrix representation
$\mathbf L\in\mathbb{R}^{(2nk)\times (2nk)}$ of $\mathscr{L}} %{\mathcal{L}$:
\begin{equation}\label{eq:sprad}
\rho(\mathscr{L}} %{\mathcal{L}) := \max \left\{\ |\lambda|\colon \mathbf L \, \mathbf x =
\lambda \, \mathbf x,\, \mathbf x\in\mathbb{C}^{2nk}\ \right\}.
\end{equation}
Notice that $\mathbf x$ is allowed to be a complex vector,
because a real matrix can have complex eigenvalues.
Here we do not make any assumption on the basis used to obtain $\mathbf L$,
the choice of the basis does not affect the spectrum of $\mathbf L$,
therefore $\rho(\mathscr{L}} %{\mathcal{L})$.
\paragraph{Derivative operator}
Let $V=V_r+\imath V_i\in \mathbb{C}^{n\times k}$ with
$V_r,V_i\in\mathbb{R}^{n\times k}$ being the real and
imaginary parts of $V$, respectively.
A Hermitian matrix-valued function $H(V)$ is called differentiable, if each
element $h_{ij}(V)$ is a smooth function in the real and imaginary parts
$(V_r,V_i)$ of $V$.
Such differentiability is different from the one in the
holomorphic sense, which generally cannot hold for $H(V)$ with real
diagonal elements.
For $H(V)$ differentiable at $V_*$,
we can define a derivative operator
\begin{equation}\label{eq:dh}
\text{ $\mbox{\bf D}H(V_*)[\cdot ]\colon \mathbb{C}^{n\times k} \to
\mathbb{C}^{n\times n}$ \quad with\quad }
\mbox{\bf D}H(V_*)[X] = \left[\frac{d}{dt} H(V_*+t
X)\right]_{t=0},
\end{equation}
where $t\in\mathbb{R}$.
$\mbox{\bf D}H(V_*)[X]$ represents the
derivative of $H(V)$ at $V_*$,
in the direction of $X\in\mathbb C^{n\times k}$.
A direct verification shows that
$\mbox{\bf D}H(V_*)[\cdot ]$
is an $\mathbb{R}$-linear mapping satisfying \eqref{eq:rlinear}.
By Taylor's expansion of $H(V_*+tX)$ at $t=0$,
as $V$ close to $V_*$ (in the Euclidean sense),
it holds
\begin{equation}\label{eq:taylor}
H(V) = H(V_*) + \mbox{\bf D}H(V_*)[V - V_*] + o (\|V-V_*\|_2).
\end{equation}
Therefore, $\mbox{\bf D}H(V_*)[\cdot ]$
is the Fr\'echet derivative of
$H: \mathbb{C}^{n\times k}(\mathbb{R})\to \mathbb{C}^{n\times n}(\mathbb{R})$.
Note that the expansion~\eqref{eq:taylor} does not take into account
the unitary invariance \eqref{eq:univar} of $H(V)$, and that is
why the remainder term is in the Euclidean difference $V-V_*$.
\section{Tangent-angle matrix}\label{sec:tangent}
Let $V\in \mathbb{U}^{n\times k}$ be an approximation to the solution $V_*$ of
NEPv~\eqref{eq:nepv}. Each $V$ represents an orthonormal basis matrix
of a subspace.
As far as a solution of NEPv~\eqref{eq:nepv} is concerned, it is the subspaces
that matter.
To assess the distance of $V$ to the solution $V_*$ in
terms of the subspaces their columns span,
we define the {\em tangent-angle matrix} from $V$ to $V_*$ as
\begin{equation} \label{eq:tv}
T(V) := (V_{*\bot}^{\HH}V)(V_*^{\HH} V)^{-1}\in \mathbb{C}^{(n-k)\times k},
\end{equation}
provided $V_*^{\HH} V$ is invertible.
By definition, $T(V)$ can be viewed as a function of
$\mathbb{U}^{n\times k} \to \mathbb{C}^{(n-k)\times k}$.
The name of `tangent-angle matrix' comes from
the fact that
\begin{equation}\label{eq:tmat}
\|\tan\Theta(V,V_*)\|_{\UI} = \|(V_{*\bot}^{\HH}V)(V_*^{\HH} V)^{-1}\|_{\UI}
= \|T(V)\|_{\UI},
\end{equation}
for all unitarily invariant norms.
Recall that
the unitarily invariant norm
$\|A\|_{\UI}$ is defined by the singular values of $A$,
equation~\eqref{eq:tmat} is a direct consequence of the
identity of singular values
$
\sigma(\tan \Theta(V,V_*)) = \sigma\left((V_\perp^{\HH} V_*)(V^{\HH}
V_*)^{-1}\right)$,
which follows from the definition of canonical angles in~\eqref{eq:mat-angles-XY}
(see, e.g., \cite[Thm. 2.2, 2.4, Chap 4]{Stewart:2001} and~\cite{Zhu:2013}).
The tangents of canonical angles have long been used in numerical matrix
analysis,
and we refer to~\cite{Zhu:2013} and references therein.
By definition~\eqref{eq:indv-angles-XY},
the singular values of $V_*^{\HH}V$ consist of
those of the matrix
$\cos\Theta(V,V_*) = I + \mathcal{O}(\|\Theta(V,V_*)\|_{\UI}^2)$.
Therefore, it can be seen from~\eqref{eq:tmat} that $T(V)$
is well defined for sufficiently small canonical angles $\Theta(V,V_*)$.
Meanwhile, $\Theta(V,V_*)\to 0$ iff $T(V)\to 0$.
By the unitary invariance \eqref{eq:univar} and
the continuity of $H(V)$,
we have $H(V)\to H(V_*)$ as the tangent-angle matrix $T(V)\to 0$.
This is more precisely described in the following lemma.
\begin{lemma}\label{lem:hv}
Let $V\in\mathbb{U}^{n\times k}$. Then as $T(V)\to 0$, it holds that
\begin{equation}\label{eq:uninvar}
H(V) = H\big( V_* + V_{*\bot}T(V) + \mathcal{O}(\|T(V)\|_{\UI}^2)\big).
\end{equation}
If $H(V)$ is also differentiable, then
\begin{equation}\label{eq:diff}
H(V) = H(V_*) + {\bf D}H(V_*)[V_{*\bot}T(V)] + \mathbf{o}(\|T(V)\|_{\UI}).
\end{equation}
\end{lemma}
\begin{proof}
The singular values of $V_*^{\HH}V$ consist of
$\cos\Theta(V,V_*) = I + \mathcal{O}(\|\Theta(V,V_*)\|_{\UI}^2) $.
So we have
$V_*^{\HH}V = W + \mathcal{O}(\|\Theta(V,V_*)\|_{\UI}^2)$ for some
unitary $W \in\mathbb{U}^{k\times k}$.
It follows that
\begin{equation} \label{eq:vw}
VW^{-1} = V(V_*^{\HH}V)^{-1} + \mathcal{O}(\|\Theta(V,V_*)\|_{\UI}^2)
= V_* + V_{*\bot}\, T(V) + \mathcal{O}(\|T(V)\|_{\UI}^2),
\end{equation}
where we used $V= V_*(V_*^{\HH}V)+V_{*\bot}(V_{*\bot}^{\HH}V)$
and $T(V) = \mathcal{O}(\|\Theta(V,V_*)\|_{\UI})$ in the last equation.
The unitary invariance property $H(V) = H(VW^{-1})$ leads to~\eqref{eq:uninvar}.
Combining \eqref{eq:vw} with~\eqref{eq:taylor},
we obtain~\eqref{eq:diff}.
\end{proof}
The following lemma, which is the key to establishing our local convergence
results, describes the relation between the tangent-angle matrices of
two consecutive SCF iterations.
\begin{lemma}\label{lem:onestep}
Suppose Assumption~\ref{ass:gap} holds.
Let $\widetilde V$ be an orthonormal basis matrix associated with
the $k$ smallest eigenvalues of $H(V)$,
and let $\SL(V)$ be the unique solution of the Sylvester equation
defined in~\eqref{eq:xv}.
Then
\begin{enumerate}[{\rm (a)}]
\item \label{item:tx}
$\SL(V)\to 0$ as $T(V)\to 0$;
\item \label{item:tv}
the tangent-angle matrix $T(\widetilde V)$ of $\widetilde V$ satisfies
\begin{equation}\label{eq:onestep}
T(\widetilde V) = \SL(V) + \mathbf{o}( \|\SL(V)\|_{\UI});
\end{equation}
\item \label{item:ls}
if $H(V)$ is differentiable at $V_*$, then
\begin{equation}\label{eq:onestepdiff}
T(\widetilde V) = \mathscr{L}} %{\mathcal{L}\left(T(V)\right) + \mathbf{o}(\|T(V)\|_{\UI}),
\end{equation}
where $\mathscr{L}} %{\mathcal{L}:\mathbb C^{(n-k)\times k}\to \mathbb C^{(n-k)\times k}$ defined by
\begin{equation}\label{eq:ls}
\mathscr{L}} %{\mathcal{L}(Z) = D(V_*)\odot (V_{*\bot}^{\HH} \, \mbox{\bf D}H(V_*)[V_{*\bot} Z]\, V_*)
\end{equation}
is an $\mathbb R$-linear operator, called the {\emlocal $\bbR$-linear operator} of the plain SCF.
\end{enumerate}
\end{lemma}
\begin{proof}
For item~\ref{item:tx},
by~\eqref{eq:uninvar} and the continuity of $H$,
it holds $H(V) \to H(V_*)$ as $T(V)\to 0$.
Hence, $\SL(V)\to 0$ by the definition of $\SL(V)$.
For item~\ref{item:tv},
we begin with the eigen-decomposition of $H(V)$:
\[
H(V) \begin{bmatrix} \widetilde V & \widetilde V_{\bot} \end{bmatrix}
=
\begin{bmatrix} \widetilde V& \widetilde V_{\bot} \end{bmatrix}
\begin{bmatrix} \widetilde \Lambda & \\ &\widetilde \Lambda_{\bot}
\end{bmatrix},
\]
where $ \begin{bmatrix} \widetilde V & \widetilde V_{\bot}
\end{bmatrix}\in\mathbb{U}^{n\times n}$ is unitary,
$\widetilde \Lambda=\diag(\widetilde \lambda_1,\dots, \widetilde \lambda_k)$ and
$\widetilde \Lambda_{\bot}=\diag(\widetilde \lambda_{k+1},\dots, \widetilde \lambda_n)$ with
$\widetilde \lambda_i=\lambda_i(H(V))$.
Due to \Cref{ass:gap}, as $H(V) \to H(V_*)$,
we can apply the standard perturbation analysis of eigenspaces~\cite[Sec.~V.2]{Stewart:1990} to obtain
\begin{equation}\label{eq:paras:VtV}
\begin{bmatrix} \widetilde V & \widetilde V_{\bot} \end{bmatrix}
=
\begin{bmatrix} V_* & V_{*\bot} \end{bmatrix}
\begin{bmatrix} I_k & - Z^{\HH} \\ Z & I_{n-k}\end{bmatrix}
\begin{bmatrix} (I_k+Z^{\HH}Z)^{-1/2} & \\ & (I_{n-k}+ZZ^{\HH})^{-1/2}
\end{bmatrix}
\begin{bmatrix}
Q & \\ & P
\end{bmatrix},
\end{equation}
where $Z\in\mathbb{R}^{(n-k)\times k}$,
$Q\in\mathbb{U}^{k\times k}$, and $P\in\mathbb{U}^{(n-k)\times (n-k)}$
are parameter matrices, and
\begin{equation}\label{eq:zt0}
\text{ $Z\to 0$ \quad as\quad $H(V)\to H(V_*)$}.
\end{equation}
The parameterization from \eqref{eq:paras:VtV} can be equivalently put as
\begin{align*}
\widetilde V &= (V_* + V_{*\bot} Z)\, (I_k+Z^{\HH}Z)^{-1/2}Q,\\
\widetilde V_{\bot} &= (-V_*Z^{\HH} + V_{*\bot})\, (I_{n-k}+ZZ^{\HH})^{-1/2} P.
\end{align*}
By the first equation,
$Z$ is identical to the tangent-angle matrix from $\widetilde V$ to $V_*$:
\begin{equation}\label{eq:z}
T(\widetilde V) =
( V_{*\bot}^{\HH} \widetilde V) (V_{*}^{\HH} \widetilde V)^{-1} =Z,
\end{equation}
where we have used $V_{*}^{\HH} \widetilde V = (I_k+Z^{\HH}Z)^{-1/2}Q$ and $V_{*\bot}^{\HH}
\widetilde V = Z(I_k+Z^{\HH}Z)^{-1/2}Q$.
Next, we establish an equation to characterize $Z$.
From $\widetilde V_{\bot}^{\HH} H(V) \widetilde V= \widetilde V_{\bot}^{\HH}
\widetilde V\widetilde \Lambda = 0$,
we get
\begin{align*}
0&=\begin{bmatrix} -Z & I_{n-k} \end{bmatrix}[V_*, V_{*\bot}]^{\HH}
H(V) [V_*, V_{*\bot}]\begin{bmatrix}I_k \\ Z\end{bmatrix} \\
&=\begin{bmatrix} -Z & I_{n-k} \end{bmatrix}[V_*, V_{*\bot}]^{\HH}
[H(V_*) + (H(V) - H(V_*))] [V_*, V_{*\bot}]\begin{bmatrix}I_k \\
Z\end{bmatrix}\\
&=\Lambda_{*\bot}Z-Z\Lambda_*+(-ZV_*^{\HH}+V_{*\bot}^{\HH})[H(V) - H(V_*)](V_*+V_{*\bot}Z).
\end{align*}
Therefore, $Z$ satisfies the Sylvester equation (view the right hand side as fixed)
\[ \Lambda_{*\bot}Z-Z\Lambda_* =
(ZV_*^{\HH}-V_{*\bot}^{\HH})[H(V) - H(V_*)](V_*+V_{*\bot}Z).
\]
By \Cref{ass:gap}, we can solve the Sylvester equation to obtain
\begin{align}\label{eq:zxphi}
Z= \SL(V) + \Phi(Z),
\end{align}
where
\[
\Phi(Z)=D(V_*)\odot\left(ZV_*^{\HH}[H(V) - H(V_*)](V_*+V_{*\bot}Z) -
V_{*\bot}^{\HH}[H(V) - H(V_*)]V_{*\bot}Z\right),
\]
and $D(V_*)$ is defined as in~\eqref{eq:dvs}.
A quick calculation shows that
\begin{equation}\label{eq:phiz}
\|\Phi(Z)\|_{\F}\le \delta_*^{-1} \, \|H(V) - H(V_*)\|_{\F}(2\|Z\|_2+\|Z\|_2^2)
=\mathbf{o}( \|Z\|_{\UI}),
\end{equation}
where the last equation is due to $H(V)\to H(V_*)$ and $Z\to 0$, as $T(V)\to 0$,
and the equivalency of matrix norms.
Recall $T(\widetilde V) = Z$. \Cref{eq:zxphi,eq:phiz} lead directly to~\eqref{eq:onestep}.
For item~\ref{item:ls}, we derive from the definition of $\SL(V)$ and
the expansion~\eqref{eq:diff} that
\begin{align*}
\SL(V)
& = D(V_*) \odot \left(V_{*\bot}^{\HH}\, {\bf D}H(V_*)[V_{*\bot}T(V)]\, V_*\right) +
\mathbf{o}(\|T(V)\|_{\UI}).
\end{align*}
Plugging it into~\eqref{eq:onestep}, and
exploiting $\|\mathscr{L}} %{\mathcal{L}(T(V))\|_{\UI} = \mathcal{O}(\|T(V)\|_{\UI})$
since $\mathscr{L}} %{\mathcal{L}$ is an $\mathbb{R}$-linear operator of finite dimension
(which is bounded), we complete the proof.
\end{proof}
We should mention that the tangent-angle matrix in the form of~\eqref{eq:tv}
appeared in the so-called
McWeeny transformation~\cite{Mcweeny:1960,Stanton:1981b,Stanton:1981}
in the density matrix theory for electronic structure calculations, where
the matrix was treated as an independent parameter that is not connected
with canonical angles of subspaces. This lack of geometric interpretation
makes it difficult to proceed a comprehensive convergence analysis
as developed in the following sections, and extend to
the treatment of a continuous $H(V)$.
\section{Convergence analysis}\label{sec:conv}
Because of the invariance property \eqref{eq:univar},
the plain SCF iteration~\eqref{eq:pscf} should be inherently understood as a
subspace iterative scheme and the convergence of the basis matrices
$\{V_i\}_{i=0}^\infty$ to a solution $V_*$ should be measured by a metric on the
Grassmann manifold $\mathbf{Gr}(k,\mathbb{C}^n)$. Let $d(\cdot,\cdot)$ be a metric
on $\mathbf{Gr}(k,\mathbb{C}^n)$. Without causing any ambiguity, in what follows we will not
distinguish an element ${\cal R}(V)\in\mathbf{Gr}(k,\mathbb{C}^n)$ from its representation $V\in\mathbb{U}^{n\times k}$.
The following notions are straightforward extensions of the existing ones:
\begin{enumerate}[(i)]
\item
SCF~\eqref{eq:pscf} is {\em locally convergent} to $V_*$, if
$d(V_i, V_*)\to 0$ as $i\to \infty$ for any initial $V_0$ that is sufficiently close to $V_*$ in
the metric, i.e., $d(V_0,V_*)$ is sufficiently small.
\item SCF~\eqref{eq:pscf} is {\em locally divergent} from
$V_*$, if for all $\varepsilon> 0$
there exists $V_0$ with $d(V_0,V_*)\leq \varepsilon$ such that
$d(V_i,V_*)$ doesn't converge to $0$, i.e.,
either $d(V_i,V_*)$ doesn't converge at all or converges to something not $0$.
\end{enumerate}
\subsection{Contraction factors}\label{sec:local}
There are two fundamental quantities
that provide convergence measures of SCF on $\mathbf{Gr}(k,\mathbb{C}^n)$:
{\em local contraction factor\/} and
{\em local asymptotic average contraction factor}.
The former, which is a quantity to assess local convergence,
accounts for the worst case error reduction of SCF per iterative
step. The latter captures the asymptotic average convergence rate of SCF,
and provides a sufficient and almost necessary condition for
the local convergence.
Since SCF is a fixed-point iteration on the
Grassmann manifold $\mathbf{Gr}(k,\mathbb{C}^n)$,
the {\em local contraction factor of SCF} is defined as
\begin{equation}\label{eq:bestlip}
\eta_{\sup}
:= \limsup_{V_0\in \mathbb{U}^{n\times k} \atop d(V_0,V_*)\to 0}
\frac{ d(V_1,V_*)}{d(V_0,V_*)}
\end{equation}
Such a constant can be viewed as the (best) local Lipschitz constant for
the fixed-point mapping of SCF.
We observe that the condition $\eta_{\sup}<1$, which implies SCF is locally error
reductive, is sufficient for local convergence.
In the convergent case, it follows from the definition~\eqref{eq:bestlip} that
\[
\limsup_{k\to \infty} \frac{d(V_{k+1},V_*)}{d(V_{k},V_*)} \leq \eta_{\sup},
\]
namely, the {\em (asymptotic) convergence rate} of SCF is bounded by
$\eta_{\sup}$.
To take into the account of oscillation
and to obtain tighter convergence bounds,
the one-step contraction factor \eqref{eq:bestlip}
can be generalized to multiple iterative steps.
Let $m$ be a given positive integer, and define
\begin{equation}\label{eq:mstepconv}
\eta_{\sup,m} :=
\limsup_{V_0\in \mathbb{U}^{n\times k} \atop d(V_0,V_*)\to 0}
\left(\frac{d(V_{m},V_*)}{d(V_0,V_*)}\right)^{1/m}.
\end{equation}
Then $\eta_{\sup,m}$ is an average contraction factor
per $m$ consecutive iterative steps of SCF~\eqref{eq:pscf}.
The limit of the average contraction factor,
as $m\to \infty$,
\begin{equation}\label{eq:assfac}
\eta_{\sup,\infty} := \limsup_{m\to\infty} \ \eta_{\sup,m}
\end{equation}
defines a \emph{local asymptotic average contraction factor} of SCF.
By definition, the number $\eta_{\sup,\infty}$ measures
the average convergence rate of SCF.
The average convergence rate is a conventional tool to study matrix iterative
methods~\cite{Varga:1999} and typically leads to tight convergence rates
in practice.
It follows from item~\ref{i:lem:2sup:b} of the lemma below
that $\eta_{\sup,\infty}$
is the optimal local convergence rate and
thereby the optimal contraction factor of SCF.
We caution the reader that $\eta_{\sup}$, $\eta_{\sup,m}$
and $\eta_{\sup,\infty}$
depend on the metric $d(\cdot,\cdot)$ and the dependency is
suppressed for notational clarity.
\begin{lemma}\label{lem:2sup}
Suppose~\Cref{ass:gap} and $\eta_{\sup}<\infty$.
\begin{enumerate}[{\rm (a)}]
\item\label{i:lem:2sup:a}
It holds that for any $m>1$
\begin{equation}\label{eq:eee}
\eta_{\sup,\infty}\leq \eta_{\sup,m} \leq \eta_{\sup}.
\end{equation}
\item\label{i:lem:2sup:b}
If $\eta_{\sup,\infty} < 1$, then SCF is locally convergent to $V_*$, with its asymptotic average convergence rate bounded by $\eta_{\sup,\infty}$. If $\eta_{\sup,\infty} > 1$, then SCF is locally divergent from $V_*$.
\end{enumerate}
\end{lemma}
\begin{proof}
For item~\ref{i:lem:2sup:a},
first from definition~\eqref{eq:bestlip} and $\eta_{\sup}<\infty$,
we conclude that ${d(V_p,V_*)}\to 0$ for $p=0,1,\dots, m-1$ as $d(V_0,V_*)\to 0$.
Therefore,
\begin{align*}
\limsup_{V_0\in \mathbb{U}^{n\times k} \atop d(V_0,V_*)\to 0}
\left(\frac{d(V_{m},V_*)}{d(V_0,V_*)}\right)^{1/m} & =
\limsup_{V_0\in \mathbb{U}^{n\times k} \atop d(V_0,V_*)\to 0}
\left(\prod_{p=0}^{m-1}\frac{d(V_{p+1},V_*)}{d(V_p,V_*)}\right)^{1/m} \\
& \leq
\left(\prod_{p=0}^{m-1}
\limsup_{V_p\in \mathbb{U}^{n\times k} \atop d(V_p,V_*)\to 0}
\frac{d(V_{p+1},V_*)}{d(V_p,V_*)}
\right)^{1/m},
\end{align*}
and $\eta_{\sup,m}\leq \eta_{\sup}$ follows.
Now fix $m$. Any integer $m'>m$ can be expressed as $m' = sm + p$,
for some $s\ge 0$ and $0\leq p\leq m-1$.
Using the same arguments as from above, and noticing that
\begin{align*}
\left(\frac{d(V_{m'},V_*)}{d(V_0,V_*)}\right)^{1/m'}
& =
\left( \frac{d(V_{m'},V_*)}{d(V_p,V_*)} \frac{d(V_{p},V_*)}{d(V_0,V_*)} \right)^{1/m'} \\
& =
\left( \prod_{\ell = 0}^{s-1} \frac{d(V_{m(\ell+1)+p},V_*)}{d(V_{m\ell
+p},V_*)} \cdot \frac{d(V_{p},V_*)}{d(V_0,V_*)} \right)^{1/m'},
\end{align*}
we obtain by taking $\limsup$ that
\[
\eta_{\sup,m'} \leq
(\eta_{\sup,m})^{sm/m'}\cdot (\eta_{\sup,p})^{p/m'}
=
\eta_{\sup,m}\cdot
\left(\eta_{\sup,p}\,/\,\eta_{\sup,m}\right)^{p/m'}.
\]
We can always assume $\eta_{\sup,m}\neq 0$, otherwise SCF converges
in $m$ iterations and $\eta_{\sup,\infty}=0$.
Letting $m'\to \infty$ and noticing that $\eta_{\sup,p}\leq \eta_{\sup}$ is bounded, we get
$\eta_{\sup,\infty}=\limsup_{m'\to\infty}\eta_{\sup,m'}\leq \eta_{\sup,m}$.
For item~\ref{i:lem:2sup:b},
consider first $\eta_{\sup,\infty} < 1$. Pick
a constant $c$ such that $\eta_{\sup,\infty} <c< 1$.
Because of how $\eta_{\sup,m}$ is defined in \eqref{eq:assfac}, we see that
$\eta_{\sup,m} \leq c$ for $m$ sufficiently large and for
all $V_0$ sufficiently close to $V_*$ in the metric $d(\cdot,\cdot)$.
Equivalently,
there exist $\delta_1 >0$ and $m_0>0$ such that
\begin{equation}\label{eq:2sup:pf-1}
d(V_{m},V_*) \leq c^{m} \, d(V_0,V_*)
\end{equation}
for all $V_0$ with $d(V_0,V_*) < \delta_1$ and for all $m\ge m_0$.
Recall that $\eta_{\sup}<\infty$ and pick a finite constant
$c_2>\max\{1,\eta_{\sup}\}\ge 1$.
By \eqref{eq:bestlip},
there exists $\delta_2\in(0,\delta_1)$ such that
\begin{equation}\label{eq:2sup:pf-2}
d(V_1,V_*) \leq c_2 \, d(V_0,V_*)
\end{equation}
for all $V_0$ with $d(V_0,V_*) < \delta_2$.
Let $\delta_3= c_2^{-(m_0-1)}\times\delta_2<\delta_2<\delta_1$.
For any $V_0$ with $d(V_0,V_*) < \delta_3$, we have by \eqref{eq:2sup:pf-2}
\begin{subequations}\label{eq:2sup:pf-3}
\begin{align}
d(V_1,V_*)&\leq c_2 \, d(V_0,V_*)<c_2\delta_3\le\delta_2, \label{eq:2sup:pf-3a}\\
d(V_2,V_*)&\le c_2 d(V_1,V_*)\le c_2^2d(V_0,V_*)<c_2^2\delta_3\le\delta_2, \label{eq:2sup:pf-3b}\\
&\vdots \nonumber\\
d(V_{m_0-1},V_*)&\le c_2^{m_0-1}d(V_0,V_*)<c_2^{m_0-1}\delta_3\le\delta_2. \label{eq:2sup:pf-3c}
\end{align}
\end{subequations}
For any $m>m_0$, we can write $m=sm_0+p$ for some $0\le p\le m_0-1$.
We have by \eqref{eq:2sup:pf-1} and \eqref{eq:2sup:pf-3} that for $m>m_0$ and for any $V_0$ with $d(V_0,V_*) < \delta_3$
\[
d(V_{m},V_*)
\leq c^{sm_0} \cdot d(V_p,V_*)
=c^{m} \cdot \frac{ d(V_p,V_*)}{c^{p} }
\leq c^{m} \cdot \frac{ \delta_2}{c^{m_0-1}}.
\]
Letting $m\to \infty$ yields $d(V_{m},V_*) \to 0$, as expected.
On the other hand, if $\eta_{\sup,\infty} > 1$,
then there exist $c > 1$ and a subsequence $\{m_i\}_{i=0}^\infty$ of positive integers such that
$\eta_{\sup,m_i} \geq c$ as $i\to\infty$.
Let $\delta>0$ be a constant satisfying $c-\delta >1$.
It follows from the definition of $\eta_{\sup,m}$ that
for all $\varepsilon> 0$ there exists $V_0$, with
$d(V_0,V_*) \leq \varepsilon$, s.t.,
$d(V_{m_i},V_*)/ d(V_0,V_*) \geq (c-\delta )^{m_i}$,
which is arbitrarily large as $m_i\to\infty$.
Hence the iteration is locally divergent.
\end{proof}
\subsection{Characterization of contraction factors}\label{sec:sprd}
The definitions of $\eta_{\sup}$ in~\eqref{eq:bestlip}
and $\eta_{\sup,\infty}$ in~\eqref{eq:assfac} are generic.
A meaningful characterization
of $\eta_{\sup}$ and $\eta_{\sup,\infty}$ will have to involve
the specific choice of the metric $d(\cdot,\cdot)$ and
the detail of $H(V)$.
\Cref{thm:lnorm} below contains the main contributions of this paper.
It reveals for a class of metrics a direct characterization of $\eta_{\sup}$ by $H(V)$,
as compared to the previous works~\cite{Cai:2018,Liu:2014,Yang:2009}
on the upper bounds of $\eta_{\sup}$.
Furthermore, for differentiable $H(V)$,
it provides closed-form expressions for
$\eta_{\sup}$ and the optimal contraction factor
$\eta_{\sup,\infty}$.
\begin{theorem}\label{thm:lnorm}
Suppose~\Cref{ass:gap} and let $d(\cdot,\cdot):=\|\Theta(\cdot,\cdot)\|_{\UI}$.
\begin{enumerate}[{\rm (a)}]
\item \label{i:thm:lnorm:a1}
If $H(V)$ is Lipschitz continuous at $V_*$,
then
\begin{equation} \label{eq:etasup2}
\eta_{\sup} =
\limsup_{V\in \mathbb{U}^{n\times k} \atop \|\tan\Theta(V,V_*)\|_{\UI}\rightarrow 0}
\frac {\| \SL(V)\|_{\UI}} { \|\tan\Theta(V,V_*)\|_{\UI}}
\quad <\ \infty,
\end{equation}
where $\SL(V)$ is the unique solution of the Sylvester equation
defined in~\eqref{eq:xv}.
\item \label{i:thm:lnorm:a2}
If $H(V)$ is differentiable at $V_*$. Then
\begin{equation}\label{eq:etainfty}
\eta_{\sup} = \vvvert {\mathscr{L}} %{\mathcal{L}}\vvvert_{\UI}\ge \eta_{\sup,\infty}=\rho({\mathscr{L}} %{\mathcal{L}}),
\end{equation}
where
$\mathscr{L}} %{\mathcal{L}$ is the {local $\bbR$-linear operator} of the plain SCF defined in~\eqref{eq:ls},
$\vvvert\mathscr{L}} %{\mathcal{L} \vvvert_{\UI}$ is the operator norm of $\mathscr{L}} %{\mathcal{L}$
induced by the unitarily invariant norm $\|\cdot\|_{\UI}$, i.e.,
$\vvvert\mathscr{L}} %{\mathcal{L}\vvvert_{\UI} := \sup_{Z\neq 0}
\frac{\|\mathscr{L}} %{\mathcal{L}(Z)\|_{\UI}}{\|Z\|_{\UI}}.$
Consequently, the plain SCF \eqref{eq:pscf} is locally convergent to
$V_*$ with its asymptotic average convergence rate bounded by $\rho(\mathscr{L}} %{\mathcal{L})$ if $\rho(\mathscr{L}} %{\mathcal{L}) < 1$, and
locally divergent at $V_*$ if $\rho(\mathscr{L}} %{\mathcal{L}) > 1$.
\end{enumerate}
\end{theorem}
\begin{proof}
For item~\ref{i:thm:lnorm:a1},
by definition~\eqref{eq:bestlip} with $d(\cdot,\cdot):=\|\Theta(\cdot,\cdot)\|_{\UI}$,
we obtain
\begin{equation}\label{eq:etaui}
\eta_{\sup} =
\limsup_{V_0\in \mathbb{U}^{n\times k} \atop \|\Theta(V_0,V_*)\|_{\UI}\to 0}
\frac{ \|\Theta(V_1,V_*)\|_{\UI}}{\|\Theta(V_0,V_*)\|_{\UI}} =
\limsup_{V_0\in \mathbb{U}^{n\times k} \atop \|\tan\Theta(V_0,V_*)\|_{\UI}\to 0}
\frac{ \|\tan\Theta(V_1,V_*)\|_{\UI}}{\|\tan\Theta(V_0,V_*)\|_{\UI}},
\end{equation}
where the second equality is a consequence of~\eqref{eq:ts},
together with $\Theta(V_1,V_*)\to 0$ as $\Theta(V_0,V_*)\to 0$ due to~\eqref{eq:onestep}.
Then, a direct application of~\eqref{eq:onestep} leads
to~\eqref{eq:etasup2}.
For the boundedness of $\eta_{\sup} < \infty$,
by taking norms on the Sylvester solution~\eqref{eq:xv}
and exploiting the $2$-norm consistency property $\|AB\|_{\UI}\leq
\|A\|_2\|B\|_{\UI}$ of unitarily invariant norms, we have
\begin{equation}\label{eq:xvv}
\|\SL(V)\|_{\UI}\leq
\delta_*^{-1}\|V_{*\bot}^{\HH}[H(V_*)-H(V)]V_*\|_{\UI}.
\end{equation}
On the other hand,
it follows from the Lipschitz continuity of $H(V)$ and~\eqref{eq:uninvar} that
\[
\|H(V)-H(V_*)\|_{2}
\leq \alpha \, \left(\| \tan\Theta(V,V_*) \|_{2} + \mathcal{O}(\|
\tan\Theta(V,V_*) \|_{2}^2)\right),
\]
for some constant $\alpha<\infty $.
Combining this with~\eqref{eq:xvv}
and~\eqref{eq:etasup2}, we conclude $\eta_{\sup} < \infty $.
For item~\ref{i:thm:lnorm:a2},
the inequality in \eqref{eq:etainfty} has already been established
in~\eqref{eq:eee},
and the formula of $\eta_{\sup}$ follows directly from~\eqref{eq:etasup2}
and the expansion~\eqref{eq:onestepdiff}.
It remains to find the expressions for $\eta_{\sup,\infty}$.
Denote by $T_m=(V_{*\bot}^{\HH}V_m)(V_*^{\HH} V_m)^{-1}$ for
$m=0,1,\dots$.
It follows from~\Cref{lem:onestep} that
\[
T_m = \mathscr{L}} %{\mathcal{L}^m(T_0) + \mathbf{o}(c_m\|T_0\|_{\UI}),
\]
where $\mathscr{L}} %{\mathcal{L}^m =\mathscr{L}} %{\mathcal{L}\circ\dots\circ\mathscr{L}} %{\mathcal{L}$
represents the composition of the linear operator $\mathscr{L}} %{\mathcal{L}$ for $m$ times,
and $c_m$ is a constant independent of $T_0$.
Hence for any given $m$
\begin{align*}
\eta_{\sup,m} & =
\limsup_{\|\Theta(V_0,V_*)\|_{\UI}\to 0}
\left(\frac{\|\Theta(V_m,V_*)\|_{\UI}}{\|\Theta(V_0,V_*)\|_{\UI}}\right)^{1/m}
=
\limsup_{\|T_0\|_{\UI}\to 0}
\left(\frac{\|T_m\|_{\UI}}{\|T_0\|_{\UI}}\right)^{1/m} \\
& =
\limsup_{T_0\to 0}
\left(\frac{\|{\mathscr{L}} %{\mathcal{L}^m(T_0)}\|_{\UI}}{\|T_0\|_{\UI}}\right)^{1/m},
\end{align*}
where the second equation is due to~\eqref{eq:ts}, together with the continuity $T_m\to 0$ as $T_0\to 0$,
implied by~\eqref{eq:onestep}.
Since $\mathscr{L}} %{\mathcal{L}$ is a finite dimensional linear operator,
we have that $\eta_{\sup,m} = (\vvvert \mathscr{L}} %{\mathcal{L}^m \vvvert_{\UI})^{1/m}$.
The expression for $\eta_{\sup,\infty}$ in~\eqref{eq:etainfty}
is a consequence of
Gelfand's formula, which says
$\lim_{m\to \infty} \vvvert{\mathscr{L}} %{\mathcal{L}}^m\vvvert^{1/m}= \rho({\mathscr{L}} %{\mathcal{L}})$ for
any operator norm $\vvvert\cdot\vvvert$
in a finite dimensional vector space (see, e.g.,~\cite[Thm 17.4]{Lax:2002}).
\end{proof}
In recent years,
a series of works, e.g.,~\cite{Cai:2018,Liu:2014,Yang:2009},
have been published to improve the upper bounds of the local contraction factor
$\eta_{\sup}$.
Those bounds were typically established for particular choices
of the metric $d(\cdot,\cdot)$ between subspaces and for a class of $H(V)$.
Let us revisit particularly the following convergence factor
of the plain SCF iteration presented recently in~\cite{Cai:2018}:
\begin{align}\label{eq:czbl}
\eta_{\czbl}:=
\limsup_{V\in \mathbb{U}^{n\times k} \atop
\|\sin\Theta(V,V_*)\|_{\UI}\rightarrow 0}
\frac {\delta_*^{-1}\|V_{*\bot}^{\HH}[H(V_*)-H(V)]V_*\|_{\UI}} {\|
\sin\Theta(V,V_*)\|_{\UI}}.
\end{align}
For a differentiable $H(V)$ with the expansion~\eqref{eq:diff},
$\eta_{\czbl}$ can be simplified as
\begin{equation}\label{eq:etac}
\eta_{\czbl} = \delta_*^{-1} \cdot \vvvert\mathscr{L}} %{\mathcal{L}_{\czbl} \vvvert_{\UI},
\end{equation}
where $\mathscr{L}} %{\mathcal{L}_{\czbl}:\mathbb{C}^{(n-k)\times k} \to \mathbb{C}^{(n-k)\times k}$
is an $\mathbb{R}$-linear operator:
\begin{equation}\label{eq:OP-czbl}
\mathscr{L}} %{\mathcal{L}_{\czbl}(Z) = V_{*\bot}^{\HH}\, \mathbf D H(V_*)[V_{*\bot} Z]\, V_*.
\end{equation}
The convergence factor $\eta_{\czbl}$ in~\eqref{eq:czbl} has significantly
improved several previously established results
in \cite{Liu:2014,Yang:2009}.
However, it follows from the characterization of $\eta_{\sup}$ in~\eqref{eq:etasup2}
and the bound of $\SL(V)$ in~\eqref{eq:xvv} that
\begin{align}\label{eq:ub-czbl}
\eta_{\sup} \leq \eta_{\czbl}.
\end{align}
Therefore, the quantity $\eta_{\czbl}$ is an upper bound of
$\eta_{\sup}$, and could
substantially underestimate the convergence rate of SCF in practice,
see numerical examples in~\Cref{sec:examples}.
We have already seen from~\Cref{lem:2sup} that $\eta_{\sup,\infty}$ is
an optimal convergence factor for SCF and $\eta_{\sup,\infty}\leq \eta_{\sup}$.
To see how large the gap between
$\eta_{\sup,\infty}$ and $\eta_{\sup}$ in~\eqref{eq:etainfty} might get, we consider in particular the
local contraction factor $\eta_{\sup}$ in the commonly used Frobenius norm:
\begin{align}
\vvvert\mathscr{L}} %{\mathcal{L}\vvvert_{\F}
&:=\sup_{Z\neq 0} \frac{\|\mathscr{L}} %{\mathcal{L}(Z)\|_{\F}}{\|Z\|_{\F}}
= \sup_{Z\neq 0 } \frac{\langle \mathscr{L}} %{\mathcal{L}(Z),\mathscr{L}} %{\mathcal{L}(Z)\rangle^{1/2} }{\langle Z,Z\rangle^{1/2} } \nonumber\\
&= \sup_{Z\neq 0 } \frac{\langle Z,\mathscr{L}} %{\mathcal{L}^*\circ\mathscr{L}} %{\mathcal{L}(Z)\rangle^{1/2} }{\langle Z,Z\rangle^{1/2} }
= \left(\lambda_{\max} (\mathscr{L}} %{\mathcal{L}^*\circ\mathscr{L}} %{\mathcal{L})\right)^{1/2},\label{eq:lam}
\end{align}
where $\langle X,Y\rangle = \Re (\tr(X^{\HH}Y))$ denotes the inner product
for $\mathbb{C}^{(n-k)\times k}(\mathbb{R})$,
and $\mathscr{L}} %{\mathcal{L}^{*}$ is the adjoint of $\mathscr{L}} %{\mathcal{L}$.
It follows from~\eqref{eq:etainfty} that
\begin{equation}\label{eq:eta-sup-F}
\eta_{\sup}= |\lambda_{\max} (\mathscr{L}} %{\mathcal{L}^*\circ\mathscr{L}} %{\mathcal{L})|^{1/2}
\geq |\rho(\mathscr{L}} %{\mathcal{L})| = \eta_{\sup,\infty}.
\end{equation}
By the standard matrix analysis, the equality in \eqref{eq:eta-sup-F}
holds if $\mathscr{L}} %{\mathcal{L}$ is a {\em normal} linear operator on $\mathbb{C}^N(\mathbb{R})$,
and the gap between the two numbers can be arbitrarily large when $\mathscr{L}} %{\mathcal{L}$ is far from normal.
For practical NEPv, such as the ones in Section~\ref{sec:examples},
we have observed that $\mathscr{L}} %{\mathcal{L}$ is usually
a slightly non-normal operator, causing a small gap between
the two contraction factors.
Recall that $\eta_{\sup}$ is dependent of the metrics $d$.
Another possibility for the equality in~\eqref{eq:eee} to hold is through
a particular choice of metric.
Unfortunately, the optimal metric for $\eta_{\sup}$ is generally hard to know.
Finally, we comment on another recent work~\cite{Upadhyaya:2018}
on the local convergence analysis of SCF using the spectral radius.
In~\cite{Upadhyaya:2018},
SCF is viewed as a fixed-point iteration $P_{k+1} = \psi(P_k)$
in the density matrix $P_k = V_kV_k^{\HH}\in\mathbb{C}^{n\times n}$,
rather than in $V_k$ directly.
The authors showed that the fixed-point mapping $\psi(P)$ has a closed-form
Jacobian supermatrix $J$, assuming $H(V)$ is a linear function in
$P=VV^{\HH}$. So the spectral radius of $J$ also provides
a convergence criterion.
Since $P$ has $p=(n+1)n/2$ free variables, the corresponding
supermatrix $J$ is
of size $p$-by-$p$.
This is in contrast to the $\mathbb{R}$-linear operator
$\mathscr{L}} %{\mathcal{L}$~\eqref{eq:ls} in tangent-angle matrices, which is only of
size $q$-by-$q$ with $q=2(n-k)k = \mathcal{O} ({p}^{1/2})$.
In addition to the reduced size, the use of linear operator, rather than
a supermatrix, allows for more convenient computation of the spectral radius in
practice, as will be discussed in~\Cref{sec:setup}.
Furthermore, $\mathscr{L}} %{\mathcal{L}$ is also easier to work with theoretically and numerically,
thanks to its simplicity in formulation and more explicit
dependencies on key variables, such as derivatives and eigenvalue gaps.
In the next section, we will show how to apply the spectral radius
$\rho(\mathscr{L}} %{\mathcal{L})$ to analyze the so-called {\em level-shifting scheme\/}
for stabilizing and accelerating the plain SCF iteration.
\section{Level-shifted SCF iteration}\label{sec:ls}
In the previous section, we have discussed that
if the spectral radius $\rho(\mathscr{L}} %{\mathcal{L}) >1$ (or more generally $\eta_{\sup,\infty}>1$ in the case when $H(V)$ is simply just continuous),
then the plain SCF~\eqref{eq:pscf} is locally divergent at $V_*$.
However, even if $\rho(\mathscr{L}} %{\mathcal{L}) <1$, the process is prone
to slow convergence or oscillation before reaching local convergence.
To address those issues, the plain SCF may be applied in practice
with some stabilizing schemes to help with convergence.
Among the most popular choices is the level-shifting strategy
initially developed
in computation chemistry~\cite{Saunders:1973,Thogersen:2004,Yang:2007}.
In this section, we discuss why such a scheme can work through the lens of
spectral radius when $H(V)$ is differentiable.
\subsection{Level-shifted SCF iteration}
The level-shifting scheme
modifies the plain SCF~\eqref{eq:pscf} with a parameter $\sigma$
as follows:
\begin{equation}\label{eq:levelshift}
[H(V_i)-\sigma \, V_iV_i^{\HH}] V_{i+1} = V_{i+1}\Lambda_{i+1},
\quad\text{for}\quad i = 0,1,2,\dots,
\end{equation}
where $V_{i+1}$ is an orthonormal basis matrix of the invariant subspace associated with the $k$ smallest eigenvalues of
the matrix $H(V_i)-\sigma\, V_iV_i^{\HH}$.
It can be viewed simply as the plain SCF~\eqref{eq:pscf}
applied to the level-shifted NEPv
\begin{equation}\label{eq:shifted}
H_{\sigma}(V) V = V \Lambda
\quad\text{with\quad $H_{\sigma}(V):= H(V)-\sigma VV^{\HH}$}.
\end{equation}
Note that $H_{\sigma}(V)$ is again unitarily invariant as in~\eqref{eq:univar}.
The level-shifting transformation does not alter the solutions
of the original NEPv~\eqref{eq:nepv}, but shifts related eigenvalues of $H(V)$ by $\sigma$:
\[
H(V) V = V \Lambda
\qquad\Longleftrightarrow\qquad
H_{\sigma} (V) V = V (\Lambda-\sigma I_k).
\]
Hence if $(V_*,\Lambda_*)$ is a solution of
the original NEPv~\eqref{eq:nepv},
then $(V_*,\Lambda_* -\sigma I_k)$ will solve
the level-shifted NEPv
\begin{equation} \label{eq:nepvls}
H_{\sigma}(V)V = V\Lambda.
\end{equation}
In the following discussion, we assume the parameter $\sigma$ is a constant
for convenience. In practice, it can change iteration-by-iteration.
One direct consequence of the level-shifting transformation
is that it enlarges the eigenvalue gap at the solution
$V_*$. By the eigen-decomposition~\eqref{eq:eigdec},
we obtain
\begin{equation}\label{eq:eighsigma}
H_{\sigma} (V_*) \, [V_*,V_{*\bot}]
= [V_*,V_{*\bot}]\,
\begin{bmatrix}
\Lambda_*- \sigma I_k & \\
& \Lambda_{*\bot}
\end{bmatrix}.
\end{equation}
Recall that $\Lambda_*=\diag (\lambda_1,\dots,\lambda_k)$
and $\Lambda_{*\bot}=\diag (\lambda_{k+1},\dots,\lambda_n)$
consist of the ordered eigenvalues of $H(V_*)$ as in~\eqref{eq:eigdec}.
Therefore, the gap between the $k$th and $(k+1)$st eigenvalue
of $H_{\sigma}(V_*)$ becomes
\begin{equation}
\delta_{\sigma*}: = \lambda_{k+1} - (\lambda_k-\sigma) = \delta_{*} +
\sigma,
\end{equation}
where $\delta_*$ denotes the eigenvalue gap~\eqref{eq:gap} of
the original NEPv~\eqref{eq:nepv} at $V_*$.
So the level-shifted NEPv~\eqref{eq:nepvls} always has a larger
eigenvalue gap $\delta_{\sigma*}$ if $\sigma > 0$.
It is well-known that
the larger the eigenvalue gap between the desired eigenvalues
and the rest ones, the easier and more robust it will become to
compute the desired eigenvalues and the associated eigenspace~\cite{Davis:1970,Parlett:1998,Stewart:1990}.
Therefore, it is desirable to have
a large eigenvalue gap $\delta_{\sigma *}$
for the sequence of matrix eigenvalue problems
in the SCF iteration~\eqref{eq:levelshift}, but on the other hand too large a $\sigma$ negatively affects
the local convergence rate of SCF as numerical evidences suggest.
Presently, there are heuristic schemes to choose the level-shift
parameter $\sigma$ in practice, e.g., see~\cite{Yang:2007}.
However, those heuristics cannot explain how
the level-shifting parameter $\sigma$ is
directly affecting the convergence behavior of SCF~\eqref{eq:levelshift}
for NEPv~\eqref{eq:shifted}.
We should mention that the conventional restriction of $\sigma>0$
for the level-shift parameter~\cite{Saunders:1973, Thogersen:2004,Yang:2007}
is not necessary.
We can see from the eigen-decomposition~\eqref{eq:eighsigma} that,
provided $\sigma \in (-\delta_*,+\infty)$, the eigenvectors $V_*$
always correspond to the $k$ simallest eigenvalues of $H_{\sigma}(V_*)$.
\subsection{Spectral radius for level-shifted {local $\bbR$-linear operator}}
In what follows, we investigate the local convergence behavior
of the level-shifting scheme by examining
the spectral radius $\rho(\mathscr{L}} %{\mathcal{L}_{\sigma})$
for the {local $\bbR$-linear operator} $\mathscr{L}} %{\mathcal{L}_{\sigma}$ of the level-shifted SCF~\eqref{eq:levelshift}.
We will focus on a class of NEPv where certain conditions on the
derivatives of $H(V)$ will apply.
Those conditions hold for NEPv arising in optimization
problems with orthogonality constraints,
as is usually the case for most practical NEPv.
\subsubsection{NEPv from optimization with orthogonality constraints}
~Let $H(V)$ be differentiable.
Define the $\mathbb{R}$-linear operator
$\mathscr{Q}\colon \mathbb{C}^{(n-k)\times k} \to \mathbb{C}^{(n-k)\times k}$
by
\begin{equation}\label{eq:qs}
\mathscr{Q}(Z) :=
V_{*\bot}^{\HH} \, \mbox{\bf D} H(V_*)[V_{*\bot} Z]\, V_* + \Lambda_{*\bot} Z - Z \Lambda_*.
\end{equation}
We call $\mathscr{Q}$ a {\em restricted derivative operator} of
NEPv~\eqref{eq:nepv}, and make the following assumption.
\begin{assumption}\label{ass:qs}
The linear operator $\mathscr{Q}$ is self-adjoint and positive definite
with respect to the standard inner product on $\mathbb{C}^{(n-k)\times k}$,
i.e.,
$$
\Re(\tr (Z^{\HH}\mathscr{Q}(Z))) = \Re (\tr ([\mathscr{Q}(Z)]^{\HH}Z))
\quad \mbox{and} \quad
\mbox{$\Re (\tr (Z^{\HH}\mathscr{Q}(Z))) > 0$ for all $Z\neq 0$}.
$$
\end{assumption}
To justify~\Cref{ass:qs}, let us take a quick review of a
class of NEPv arising from the following optimization problems
with orthogonality constraints
\begin{equation}\label{eq:manopt}
\min_{V\in\mathbb{C}^{n\times k}} E(V) \quad \text{s.t.}\quad V^{\HH}V = I_k,
\end{equation}
where $E$ is some energy function satisfying $\nabla E(V) = H(V)V$
(see, e.g.,~\cite{Bao:2013,Yang:2007,Zhang:2014}).
We will make no assumption on the specific form of $E(\cdot)$ to be used.
For the constrained optimization problem~\eqref{eq:manopt}, the associated
Lagrangian function is given by
\[
L(V):= E(V) +
\frac{1}{2}\tr \left(\Lambda^{\HH} (V^{\HH}V - I_k)\right),
\]
where $\Lambda=\Lambda^{\HH}$ is the $k$-by-$k$ matrix
of Lagrange multipliers.
We have suppressed $L$'s dependency on $\Lambda$
for notation simplicity.
The first order optimization condition
$\nabla_{V} L(V) = H(V)V - V\Lambda=0$
leads immediately to NEPv~\eqref{eq:nepv}.
Because the target solution $V_*$ of interest is also
a minimizer of~\eqref{eq:manopt}, it needs to
satisfy certain second order condition as well.
Assuming $E(V)$ is also second order differentiable,
by straightforward derivation,
the Hessian operator of $L(V)$ is given by
\[
\nabla^2_V L(V_*)[X] = H(V_*)X +
\left(\mathbf D H(V_*)[X]\right) V_* - X\Lambda_*,
\]
where $X$ denotes the direction for the evaluation,
and $\mathbf D H(V_*)[\cdot]$ denotes the directional derivative of $H$
as defined in~\eqref{eq:dh}.
By the standard second-order optimization condition~\cite{Nocedal:2006},
this operator needs to be at least positive semi-definite
when restricted to $X=V_{*\bot}Z$ for all $Z\in\mathbb{C}^{(n-k)\times k}$,
namely, within the tangent space of the feasible set $V^{\HH}V=I_k$ at $V_*$.
Such a condition is included in Assumption~\ref{ass:qs},
where we further assume the positive definiteness of $\mathscr{Q}(\cdot)$.
\subsubsection{Spectral radius of level-shifted {local $\bbR$-linear operator}}
We can immediately draw from \Cref{lem:onestep} and \Cref{thm:lnorm}
a conclusion that
the local convergence behavior of
the level-shifted SCF~\eqref{eq:levelshift} is characterized by
the {local $\bbR$-linear operator} corresponding to the level-shifted
NEPv~\eqref{eq:shifted}. To show the dependency on $\sigma$, we denote
this {local $\bbR$-linear operator} as
\begin{equation}\label{eq:lsigma}
\mathscr{L}} %{\mathcal{L}_{\sigma}(Z) = D_{\sigma}(V_*)\odot (V_{*\bot}^{\HH} \,
\mbox{\bf D}H_{\sigma}(V_*)[V_{*\bot} Z]\, V_*),
\end{equation}
where $D_{\sigma}(V_*)\in\mathbb{R}^{(n-k)\times k}$ has elements
$D_{\sigma}(V_*)_{(i,j)} = (\lambda_{k+i}(H(V_*)) - \lambda_j(H(V_*)) +
\sigma)^{-1}$.
A representation of $\mathscr{L}} %{\mathcal{L}_{\sigma}$ in terms of {restricted derivative operator} $\mathscr{Q}$ and
a bound of the spectral radius of $\mathscr{L}} %{\mathcal{L}_{\sigma}$ are given
in the following theorem.
\begin{theorem}\label{thm:rhosigma}
Suppose~\Cref{ass:gap,ass:qs}, and $\sigma \in(-\delta_*,+\infty)$.
The {local $\bbR$-linear operator} $\mathscr{L}} %{\mathcal{L}_{\sigma}(\cdot)$ of the level-shifted SCF~\eqref{eq:levelshift}
for the level-shifted NEPv~\eqref{eq:nepvls} is given by
\begin{equation}\label{eq:lsigma2}
\mathscr{L}} %{\mathcal{L}_{\sigma}(\cdot) = D_{\sigma} (V_*)\odot \mathscr{Q}(\cdot ) - I_{\rm id},
\end{equation}
where $\mathscr{Q}$ is the {restricted derivative operator} defined in~\eqref{eq:qs}
and $I_{\rm id}$ denotes the identity operator on the vector space $\mathbb{C}^{(n-k)\times k}(\mathbb{R})$.
Moreover, the spectral radius of $\mathscr{L}} %{\mathcal{L}_{\sigma}$ is bounded:
\begin{equation}\label{ineq:rhosigma}
\rho(\mathscr{L}} %{\mathcal{L}_{\sigma}) \le
\max\left\{ \left|\frac{\mu_{\max}}{\sigma + \delta_*} -1\right|,
\left|\frac{\mu_{\min}}{\sigma + s_*} -1\right| \right\},
\end{equation}
where $\mu_{\max}\geq \mu_{\min}>0$ denote
the largest and smallest eigenvalues of the $\mathbb{R}$-linear operator $\mathscr{Q}$,
$\delta_*$ and $s_*$ are the spectral gap and span, respectively, i.e.,
$$
\delta_* = \lambda_{k+1}(H(V_*)) - \lambda_k(H(V_*))
\quad \mbox{and} \quad
s_* = \lambda_{n}(H(V_*)) - \lambda_1(H(V_*)).
$$
\end{theorem}
\begin{proof}
By the definition of $H_{\sigma}(V)$ in~\eqref{eq:shifted} and the derivative
operator~\eqref{eq:dh}, it holds that
\[
\mbox{\bf D} H_{\sigma}(V_*)[X] =
\mbox{\bf D} H(V_*)[X] - \sigma \mbox{\bf D} (V_*V_*^{\HH})[X]
= \mbox{\bf D} H(V_*)[X] - \sigma (V_*X^{\HH} + XV_*^{\HH}).
\]
Hence
\begin{align}
V_{*\bot}^{\HH} \,
\mbox{\bf D} H_{\sigma}(V_*)[V_{*\bot} Z]\, V_*
&= V_{*\bot}^{\HH} \, \mbox{\bf D} H(V_*)[V_{*\bot} Z]\, V_*
-\sigma\, Z \nonumber \\
&= \mathscr{Q}(Z) + Z (\Lambda_* -\sigma I_k) - \Lambda_{*\bot} Z
= \mathscr{Q}(Z) - Z\oslash D_{\sigma}(V_*), \label{eq:vs}
\end{align}
where the second equation is by~\eqref{eq:qs},
and `$\oslash$' denotes the elementwise division.
Plug \eqref{eq:vs} into~\eqref{eq:lsigma} to obtain
\begin{align*}
{\mathscr{L}} %{\mathcal{L}}_{\sigma}(Z)
= D_{\sigma} (V_*)\odot [\mathscr{Q}(Z) - Z\oslash D_{\sigma}(V_*) ]
= D_{\sigma} (V_*)\odot \mathscr{Q}(Z) - Z.
\end{align*}
This proves~\eqref{eq:lsigma2}.
The vector space $\mathbb{C}^{(n-k)\times k}(\mathbb{R})$ has a natural basis
$\mathcal B := \{E_{ij},\, \imath E_{ij} \colon i=1,\dots,n-k,\ j = 1,\dots,
k\}$, where the entries of $E_{ij}\in\mathbb{R}^{(n-k)\times k}$ are all zeros but 1 as its
$(i,j)$th entry.
Let $\mathbf L_{\sigma},\mathbf D_{\sigma}, \mathbf Q\in\mathbb{R}^{2N\times 2N}$
be the matrix representations of the operators $\mathscr{L}} %{\mathcal{L}_{\sigma}(\cdot)$,
$D_{\sigma}(V_*)\odot (\cdot)$, and $\mathscr{Q}(\cdot)$ with respect to the basis
$\mathcal B$, respectively, where $N=(n-k)\times k$.
It follows from~\eqref{eq:lsigma2} that
\[
\mathbf L_{\sigma} = \mathbf D_{\sigma} \mathbf Q - I_{2N}.
\]
Observe that $\mathbf D_{\sigma}$ is a diagonal matrix consisting of elements
of $D_{\sigma}$,
and $\mathbf Q$ is symmetric positive definite due to~\Cref{ass:qs}.
Hence the eigenvalues of $\mathbf D_{\sigma} \mathbf Q $ are all positive, and
\begin{equation}\label{eq:rhosigma:pf-1}
\rho(\mathscr{L}} %{\mathcal{L}_{\sigma})=\max\{|\lambda_{\max}(\mathbf D_{\sigma} \mathbf Q)-1|,|\lambda_{\min}(\mathbf D_{\sigma} \mathbf Q)-1|\}.
\end{equation}
Since the eigenvalues of $\mathbf D_{\sigma} \mathbf Q$ are the same as those of
$\mathbf Q^{1/2}\mathbf D_{\sigma} \mathbf Q^{1/2}$ and
$$
\lambda_{\max}(\mathbf D_{\sigma}) \mathbf Q
\succeq \mathbf Q^{1/2}\mathbf D_{\sigma} \mathbf Q^{1/2}\succeq \lambda_{\min}(\mathbf D_{\sigma}) \mathbf Q,
$$
we have $\lambda_{\max}(\mathbf D_{\sigma} \mathbf Q)\le \mu_{\max} /(\sigma+\delta_*)$
and $\lambda_{\min}(\mathbf D_{\sigma} \mathbf Q)\ge\mu_{\min}/(\sigma+s_*)$.
Inequality \eqref{ineq:rhosigma} is now a simple consequence of \eqref{eq:rhosigma:pf-1}.
\end{proof}
It follows immediately from~\Cref{thm:rhosigma} that
\begin{equation*}
\rho(\mathscr{L}} %{\mathcal{L}_{\sigma}) < 1 \quad
\mbox{if}\quad
0 <
\frac{\mu_{\min}}{\sigma + s_*}\leq
\frac{\mu_{\max}}{\sigma + \delta_*}\le 2,
\end{equation*}
or equivalently,
\begin{equation}\label{eq:lsbnd}
\rho(\mathscr{L}} %{\mathcal{L}_{\sigma}) < 1 \quad
\mbox{if}\quad \sigma\geq \frac{\mu_{\max}}{2} -\delta_*.
\end{equation}
Hence for a sufficiently large $\sigma$,
the level-shifted SCF is locally convergent!
On the other hand,
it also reveals that
$\rho_{\sigma}(\mathscr{L}} %{\mathcal{L})\to 1$ as $\sigma\to +\infty$, implying the slow convergence of
the level-shifted SCF.
Further, if
good estimates to $\mu_{\min}$, $\mu_{\max}$, $\delta_*$, and $s_*$
are available,
we may find a decent $\sigma$ by minimizing the upper bound
in~\eqref{ineq:rhosigma} as follows:
the minimizer is achieved when the two terms
in the right-hand side of~\eqref{ineq:rhosigma}
coincide, which can happen only if
$$
\frac{\mu_{\max}}{\sigma + \delta_*} -1=1-\frac{\mu_{\min}}{\sigma + s_*},
$$
due to $\sigma\in ( -\delta_*, +\infty)$.
This equation has a unique solution $\sigma_*\in (-\delta_*,+\infty)$.
Hence the operator $\mathscr{L}} %{\mathcal{L}_{\sigma}$ and
its spectral radius provide us the understanding of level-shifting strategy
and an approach to seek an optimal choice of the level-shifting
parameter $\sigma$, see numerical examples in Section~\ref{sec:examples}.
To end this section, we note that
the results in this section is consistent with,
and also complements, the convergence analysis
of the level-shifted methods applied to Hatree-Fock
equations~\cite{Cances:2000}.
Using optimization approaches, the authors
showed that a sufficiently large shift $\sigma$
can lead to global convergence.
The condition~\eqref{eq:lsbnd}, on the other hand, provided
a closed-form lower bound on the size of $\sigma$ needed to achieve
local convergence. The bound of~\eqref{eq:lsbnd} involves the
exact solution $V_*$ and is mostly of theoretical interest.
For particular applications, it may be possible to have an
{\em a-priori} estimate of $V_*$, as demonstrated in the examples
in the next section.
\section{Numerical examples}\label{sec:examples}
In this section, we provide numerical examples to demonstrate
the sharpness and optimality
of the convergence rate estimates presented in the
previous sections.
Specifically, the purpose of the examples
is two-fold: Firstly, to illustrate how these convergence results are
manifest in practice, where various convergence rate estimates are compared
and their sharpness in estimating the actual convergence rate is demonstrated;
Secondly, to investigate and gain insight into the influence of the
level-shifting parameter $\sigma$ on the convergence rate
of SCF \eqref{eq:levelshift}.
\subsection{Experiment setup}\label{sec:setup}
We will perform two case studies, one is a discrete Kohn-Sham equation
with real coefficient matrices $H(V)$, and the other from a discrete
Gross-Pitaevskii equation with complex matrices.
All our experiments are implemented and conducted in MATLAB 2019.
In each simulation, the ``exact'' solution $V_*$ is computed by
the plain SCF \eqref{eq:pscf}, when it is convergent,
to achieve a residual tolerance $\|H(V_*)V_* -V_*\Lambda_*\|_2\leq 10^{-14}$.
When the plain SCF failed to converge, $V_*$
is computed by the level-shifted SCF \eqref{eq:levelshift}
with a properly chosen shift $\sigma$.
The convergence rate estimates to be investigated include:
\begin{enumerate}[i)]
\item $\eta_{\czbl}$ by~\cite{Cai:2018}, computed
as~\eqref{eq:etac} in the Frobenius norm,
\item $\eta_{\sup} = \vvvert \mathscr{L}} %{\mathcal{L}\vvvert_{\F}$ in~\eqref{eq:etainfty}
in the Frobenius norm, and
\item $\eta_{\sup,\infty}=\rho(\mathscr{L}} %{\mathcal{L})$ in~\eqref{eq:etainfty}.
\end{enumerate}
These convergence rate estimates will be compared with the {\em observed convergence rate} of SCF,
estimated from the convergence history of the SCF iteration
by the least squares approximation on the last few iterations.
\paragraph{Evaluation of $\eta_{\sup,\infty} (=\rho(\mathscr{L}} %{\mathcal{L}))$}
Despite a matrix representation $\mathbf L$ is involved in the
definition~\eqref{eq:sprad},
its explicit formulation is not needed for computing
$\rho(\mathscr{L}} %{\mathcal{L})$.
Recall that
$\mathscr{L}} %{\mathcal{L}\colon \mathbb{C}^{p\times k}\to \mathbb{C}^{p\times k}$ is an
$\mathbb{R}$-linear operator.
By viewing a complex matrix $X = X_r + \imath X_i\in\mathbb{C}^{p\times k}$
as a pair of real matrices $(X_r,X_i)$ consisting of the real
and imaginary parts,
we express $\mathscr{L}} %{\mathcal{L}$
as a linear operator
$\widehat{\mathscr{L}} %{\mathcal{L}}\colon
\mathbb{R}^{p\times k}\times\mathbb{R}^{p\times k} \to \mathbb{R}^{p\times
k}\times\mathbb{R}^{p\times k}$,
\begin{equation}
\widehat{\mathscr{L}} %{\mathcal{L}}(X_r,X_i) = \left(\Re(\mathscr{L}} %{\mathcal{L}(X)), \Im(\mathscr{L}} %{\mathcal{L}(X))\right).
\end{equation}
The input (as well as the output) matrix pair $(X_r,X_i)$
can be regarded as a real ``vector'' of length-$2N$.
The largest eigenvalue in magnitude of the linear
operator $\widehat{\mathscr{L}} %{\mathcal{L}}$ can be computed conveniently by
MATLAB \texttt{eigs} function as follows:
\begin{verbatim}
v2m = @(x) reshape(x(1:N)+1i*x(N+1:end), p, []);
m2v = @(X) [real(X(:)); imag(X(:))];
hatL = @(x) m2v(L(v2m(x))));
lam_max = eigs(hatL, 2*N, 1);
\end{verbatim}
\paragraph{Evaluation of $\eta_{\sup}$ and $\eta_{\czbl}$}
The induced norm $\vvvert\mathscr{L}} %{\mathcal{L}\vvvert_{\F}$ in~\eqref{eq:lam} is defined as the square root of the
largest eigenvalue of $\mathscr{L}} %{\mathcal{L}^* \circ \mathscr{L}} %{\mathcal{L}$, which is also an $\mathbb{R}$-linear
operator.
We can use exactly the same approach above to obtain
$\lambda_{\max}(\mathscr{L}} %{\mathcal{L}^* \circ \mathscr{L}} %{\mathcal{L})$.
Since the operator $\mathscr{L}} %{\mathcal{L}^* \circ \mathscr{L}} %{\mathcal{L}$ is self-adjoint,
the largest eigenvalue is always a real number.
In analogy, for $\eta_{\czbl}$ in~\eqref{eq:etac}, $\vvvert \mathscr{L}} %{\mathcal{L}_{\czbl}\vvvert_{\F}$
can be computed as
the square root of ${\lambda_{\max}(\mathscr{L}} %{\mathcal{L}_{\czbl}^* \circ \mathscr{L}} %{\mathcal{L}_{\czbl})}$.
\subsection{Single particle Hamiltonian}
Let us consider an NEPv~\eqref{eq:nepv} with a real coefficient matrix-valued function
\begin{equation}\label{eq:hamilton}
H(V) = L + \alpha \, \Diag (L^{-1} \diag (VV^{\mbox{\scriptsize \rm T}})),
\end{equation}
where tridiagonal matrix $L=\tridiag (-1,2,-1)\in\mathbb{R}^{n\times n}$ is a discrete 1D
Laplacian, $\alpha>0$ is a given parameter,
and $V\in\mathbb{O}^{n\times k}:=\{X\in\mathbb{R}^{n\times k}\,:\, X^{\mbox{\scriptsize \rm T}}X=I_k\}$.
$H(V)$ is known as the single-particle Hamiltonian
arising from discretizing an 1D Kohn-Sham equation in electronic structure
calculations, and has become a standard testing problem for investigating
the convergence of SCF due to its simplicity,
see, e.g.,~\cite{Cai:2018,Liu:2014,Yang:2009,Zhao:2015}.
$H(V)$ is differentiable.
By a straightforward calculation, the directional derivative operator $\mbox{\bf D}H(V_*)$ defined in~\eqref{eq:dh}
is given by
\[
\mathbf D H(V)[X] = 2\alpha\, \Diag (L^{-1} \diag (XV^{\mbox{\scriptsize \rm T}})),
\]
which is linear in $X$.
The {local $\bbR$-linear operator} $\mathscr{L}} %{\mathcal{L}$ in~\eqref{eq:ls}
of the plain SCF \eqref{eq:pscf} is given by
\begin{equation}\label{eq:lreal}
\mathscr{L}} %{\mathcal{L}(Z) = 2\alpha \, D(V_*)\odot \bigg(V_{*\bot}^{\mbox{\scriptsize \rm T}}\,
\Diag (L^{-1}\diag (V_{*\bot} Z V_*^{\mbox{\scriptsize \rm T}}))\, V_* \bigg).
\end{equation}
The adjoint operator $\mathscr{L}} %{\mathcal{L}^*$
is given by
\begin{equation}\label{eq:lsreal}
\mathscr{L}} %{\mathcal{L}^*(Y) = 2\alpha \, V_{*\bot}^{\mbox{\scriptsize \rm T}}\,
\Diag \left(L^{-\mbox{\scriptsize \rm T}}\diag \big(V_{*\bot}( D(V_*)\odot
Y)V_*^{\mbox{\scriptsize \rm T}}\big)\right)\, V_* ,
\end{equation}
see Appendix~\ref{app:adj} for the derivation.
The {local $\bbR$-linear operator}
$\mathscr{L}} %{\mathcal{L}_{\sigma}$ \eqref{eq:lsigma}
of the level-shifted SCF \eqref{eq:levelshift}
is given by
\begin{equation}
\mathscr{L}} %{\mathcal{L}_{\sigma}(Z)
= D_{\sigma} (V_*) \odot \mathscr{Q}(Z) - I_{\rm id},
\end{equation}
where $\mathscr{Q}$ is the {restricted derivative operator} $\mathscr{Q}$ defined in~\eqref{eq:qs} is
given by
\begin{equation}
\mathscr{Q}(Z) = 2\alpha\, V_{*\bot}^{\mbox{\scriptsize \rm T}}\, \Diag (L^{-1}\diag (V_{*\bot} Z
V_*^{\mbox{\scriptsize \rm T}}))\, V_* + (\Lambda_{*\bot} Z- Z\Lambda_*).
\end{equation}
The largest eigenvalue $\mu_{\max}$ of $\mathscr{Q}$ can be bounded
as follows: let $Z\in\mathbb{R}^{(n-k)\times k}$ be the
corresponding eigenvector of
$\mu_{\max}$, then
\begin{align}
\mu_{\max}
= \frac{\|\mathscr{Q}(Z)\|_{\F}}{\|Z\|_{\F}}
&\leq 2\alpha\, \frac{\|\Diag (L^{-1}\diag (V_{*\bot} Z
V_*^{\mbox{\scriptsize \rm T}}))\|_{\F}}{\|Z\|_{\F}} + s_*\notag \\
& \leq 2\alpha \, \|L^{-1}\|_2 + s_*
\leq 3\alpha \, \|L^{-1}\|_2 + 4,\notag
\end{align}
where $s_*$ is the spectral span of $H(V_*)$,
and for the last inequality we have used
the inequalities $s_* \leq \lambda_n(H(V)) \leq
\|L\|_2 + \alpha \|L^{-1}\|_2$ due to~\eqref{eq:hamilton},
and $\|L\|_2 \leq 4$.
Recalling the lower bound in \eqref{eq:lsbnd} for the level-shifting parameter
$\sigma$, we find
\begin{equation}\label{eq:sbnd}
\sigma \geq \frac{3}{2}\, \alpha\, \|L^{-1}\|_2 + 2
\geq \frac{\mu_{\max}}{2} - \delta_*
\end{equation}
is sufficient to ensure local convergence of SCF \eqref{eq:levelshift}.
The first inequality provides an {\em a-priori} lower bound on the shift
$\sigma$. In practice, this crude bound is a bit pessimistic though.
But it does reveal two key contributing factors
--- the parameter $\alpha$ and size $n$ of the
problem due to the fact that
$\|L^{-1}\|_2 = 2^{-1}(1-\cos(\frac{\pi}{n+1}))^{-1} = \mathcal{O}(n^2) $
for the 1D Laplacian \cite[Lemma~6.1]{Demmel:1997}
--- that tend to negatively affect the size of shift.
\begin{example}\label{ex:single}
In this example, we compare the sharpness
of the three convergence rate estimates of the plain SCF.
We take $n=10$ and $k=2$, and use different $\alpha$
ranging from $0$ to $1$ in the Hamiltonian~\eqref{eq:hamilton}.
For each run of SCF, the starting vectors are set to be
the basis of the $k$ smallest
eigenvalues of $L$. The results are
shown in Figure~\ref{fig:ex1}.
A few observations are summarized as follows:
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.49\textwidth]{Figs/convhist10}
\includegraphics[width=0.49\textwidth]{Figs/confac10}
\end{center}
\caption{
Example~\ref{ex:single}:
convergence history
of residual norm $\|H(V_i)V_i-V_i\Lambda_i\|_2$ by the plain SCF \eqref{eq:pscf}
for selected $\alpha$ (left); and
convergence rate estimates as $\alpha$ varies (right).
}\label{fig:ex1}
\end{figure}
\begin{enumerate}[(a)]
\item
For $\alpha=0$, the NEPv reduces to a standard eigenvalue problem
$LV=V\Lambda$, for which SCF converges in one iteration.
As $\alpha$ increases, SCF faces increasing challenges to converge.
In particular, for $\alpha$ larger than $0.85$, the plain SCF becomes divergent.
For those $\alpha$, the ``exact'' solutions $V_*$ used to calculate convergence factors are
computed by the level-shifted SCF.
\item
The asymptotic average contraction factor $\eta_{\sup,\infty} (= \rho(\mathscr{L}} %{\mathcal{L}))$
successfully
predicts the convergence of SCF in all cases, and perfectly captures the
convergence rate.
The factor $\eta_{\sup,\infty}$ yields excellent estimation after only
a small number of iterative steps, although
strictly speaking, it is conclusive only as the iteration number
approaches infinity.
\item
The contraction factor estimate $\eta_{\sup}$ is an
overestimate and usually provides a good prediction of local convergence.
It failed slightly at $\alpha=0.85$, where up to 10 digits:
\[
\begin{array}{rclrrcl}
\mbox{observed} & = & 0.9913931781, \,\, & \, \, \eta_{\sup,\infty} & = & 0.9913931591, \\
\eta_{\sup} & = & 1.028434776, \,\, & \, \, \eta_{\czbl} & = & 1.430511920.
\end{array}
\]
The gap between $\eta_{\sup,\infty}$ and
$\eta_{\sup}$ implies $\mathscr{L}} %{\mathcal{L}$ is a non-normal operator as discussed
in~\Cref{sec:sprd}.
\item
In comparison, the estimate
$\eta_{\czbl}$ by~\cite{Cai:2018} is less precise.
In particular, it fails to correctly indicate
the convergence of the plain SCF starting at $\alpha=0.55$,
which is in contrast to $\eta_{\sup}$ starting at $0.85$.
We mention that, for this same experiment,
it was illustrated in~\cite{Cai:2018} that
$\eta_{\czbl}<1$ for all $\alpha\leq 0.6$
(marked as dashed vertical line).
\end{enumerate}
\end{example}
\begin{example}\label{ex:single:ls}
In this example, we examine
{the convergence of the level-shifted SCF~\eqref{eq:levelshift}
with respect to the shift $\sigma$}.
The testing problem is the same as~\Cref{ex:single}
but with a fixed $\alpha = 1$,
for which the plain SCF \eqref{eq:pscf} is divergent.
We apply the level-shifted SCF with various choices of $\sigma$ for the solution.
The convergence history and
the corresponding spectral radius of the operator $\mathscr{L}} %{\mathcal{L}_{\sigma}$
in~\eqref{eq:lsigma} is depicted in Figure~\ref{fig:ex1rhos}.
From the spectral radius plot on the right side of
Figure~\ref{fig:ex1rhos}, we observe that
$\rho(\mathscr{L}} %{\mathcal{L}_{\sigma})$ dropped quickly below $1$.
The minimal value $\rho(\mathscr{L}} %{\mathcal{L}_{\sigma})\approx 0.33$
at $\sigma\approx 0.36$ and leads to rapid convergence of SCF
as shown in the left plot.
As $\sigma$ grows, $\rho(\mathscr{L}} %{\mathcal{L}_{\sigma})$ monotonically increases towards 1.
Such a behavior of $\rho(\mathscr{L}} %{\mathcal{L}_{\sigma})$ is consistent with the
bound obtained in~\Cref{thm:rhosigma}, governed by rational functions
in the form of $|1-a/(\sigma+b)|$ with $a,b>0$.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.49\textwidth]{Figs/rhos_conv}
\includegraphics[width=0.49\textwidth]{Figs/rhos}
\end{center}
\caption{Example~\ref{ex:single:ls}:
convergence history of residual norm $\|H(V_i)V_i-V_i\Lambda_i\|_2$
by the level-shifted SCF \eqref{eq:levelshift} with selected $\sigma$ (left);
spectral radius of $\rho(\mathscr{L}} %{\mathcal{L}_{\sigma})$ as
shift $\sigma$ varies (right), where the first vertical dash line
is $\sigma=\frac{\mu_{\max}}{2} -\delta_*$
suggested by~\eqref{eq:lsbnd} and
the second is {\em a-priori}
$\sigma = \frac{3}{2}\, \alpha\, \|L^{-1}\|_2 + 2$
suggested by~\eqref{eq:sbnd},
and the optimal shift is $\sigma\approx 0.36$.
The $H(V)$ is given by~\eqref{eq:hamilton} with $\alpha=1$.
}\label{fig:ex1rhos}
\end{figure}
The sharp turning of the curve of $\rho(\mathscr{L}} %{\mathcal{L}_{\sigma})$ reveals the
challenge in finding the optimal $\sigma$.
The values of spectral radius grows quickly as $\alpha$ moves away
from the optimal shift.
We note that both the theoretic lower bound in~\eqref{eq:lsbnd}
and {\em a-priori} estimate~\eqref{eq:sbnd} fall correctly
into the convergence region.
The {\em a-priori} bound provided a pessimistic estimate of $\sigma$, that
leads to a less satisfactory convergence rate of
the level-shifted SCF~\eqref{eq:levelshift}.
\end{example}
\subsection{Gross--Pitaevskii equation
In this experiment, we consider NEPv with complex
coefficient matrices $H(V)$ given by
\begin{equation}\label{eq:gpe}
H(V) = A_f + \beta\, \Diag (|V|)^2,
\end{equation}
where $A_f\in\mathbb{C}^{n\times n}$ is a Hermitian matrix
and positive definite,
$\beta>0$ is a parameter, $V\in\mathbb{C}^n$ is a complex vector,
and $|\cdot|$ takes elementwise absolute value.
Such an NEPv arises from discretizing the Gross-Pitaevskii equation (GPE)
for modeling the physical phenomenon of
Bose--Einstein condensation~\cite{Bao:2004,Jarlebring:2014,Jia:2016,Li:2020}.
The matrix $A_f$ in~\eqref{eq:gpe} is dependent of a potential
function $f$. For illustration, we will discuss
a model 2D GPE studied in~\cite{Jarlebring:2014},
where for a given potential function $f(x,y)$ over a two dimension domain
$[-\ell,\ell]\times[-\ell,\ell]$,
the corresponding matrix
\begin{equation}\label{eq:af}
A_f = \Diag(\widetilde f)-\frac{1}{2}M -\imath \omega M_{\phi},
\end{equation}
where
\[
\text{$\widetilde f = h^2\,
\left[f(x_1,y_1),\dots,f(x_N,y_1),f(x_1,y_2),\dots,f(x_N,y_2),\dots,f(x_N,y_N)\right]^{\mbox{\scriptsize \rm T}}\in\mathbb{R}^{N^2}$}
\]
with $\{x_i\}_{i=1}^N$ and $\{y_i\}_{i=1}^N$ being interior points of the interval $[-\ell ,\ell ]$ from
the $N+2$ equidistant discretization
with spacing $h=\frac{2\ell}{N+1}$.
The matrices $M$, $M_{\phi}$ are given by
\[
M = D_{2,N}\otimes I + I\otimes D_{2,N},\,\,
M_{\phi} = h\, \Diag(y_1,\dots,y_N) \otimes D_N - D_N \otimes
\left(h\, \Diag(x_1,\dots,x_N)\right),
\]
with $N \times N$ tridiagonal matrices
$D_{N}=\tridiag (-\frac{1}{2},0,\frac{1}{2})$
and $D_{2,N}= \tridiag (1,-2,1)$.
Since $V$ is a vector,
by definition~\eqref{eq:dh}
the directional derivative operator of $H(V)$ is given by
\[
\mathbf D H(V)[X] = 2\beta\, \Diag (\Re (\overline{V}\odot X)).
\]
The {local $\bbR$-linear operator} of the plain SCF $\mathscr{L}} %{\mathcal{L}:\mathbb{C}^{n-1}\to \mathbb{C}^n$ in~\eqref{eq:ls} is
\begin{equation}\label{eq:lcmpx}
\mathscr{L}} %{\mathcal{L}(Z) = 2\beta \, D(V_*)\odot (V_{*\bot}^{\HH}\,
\Diag (
\Re (\overline{V}_*\odot (V_{*\bot} Z)))\, V_*),
\end{equation}
and its adjoint operator $\mathscr{L}} %{\mathcal{L}^*$, with respect to the standard inner product in $\mathbb{C}^{(n-k)\times k}$ ($k=1$), i.e.,
$\langle \mathscr{L}} %{\mathcal{L}(Z), Y\rangle\equiv \Re (\tr(Y^{\HH}\mathscr{L}} %{\mathcal{L}(Z)))=\langle Z, \mathscr{L}} %{\mathcal{L}^*(Y)\rangle\equiv \Re (\tr([\mathscr{L}} %{\mathcal{L}^*(Y)]^{\HH} Z))$ for any $Y,\,Z\in\mathbb{C}^{(n-k)\times k}$,
is given by
\begin{equation}\label{eq:lscmpx}
\mathscr{L}} %{\mathcal{L}^*(Y) = 2\beta \, V_{*\bot}^{\HH}\,
\left(\Re \left(\diag (V_{*\bot}( D(V_*)\odot
Y)V_*^{\HH})\right)\odot V_*\right),
\end{equation}
see Appendix~\ref{app:adj} for the derivation.
For the level-shifted SCF, the {local $\bbR$-linear operator}
$\mathscr{L}} %{\mathcal{L}_{\sigma}$ in \eqref{eq:lsigma} is given
by
\begin{equation}
\mathscr{L}} %{\mathcal{L}_{\sigma}(Z)
= D_{\sigma} (V_*) \odot \mathscr{Q}(Z) - I_{\rm id},
\end{equation}
where the {restricted derivative operator} $\mathscr{Q}(Z)$
is given by
\begin{equation}
\mathscr{Q}(Z) =
2\beta\, V_{*\bot}^{\HH}\, \Diag (
\Re (\overline{V_*}\odot (V_{*\bot}Z))\, V_*
+ (\Lambda_{*\bot} Z- Z\Lambda_*).
\end{equation}
The largest eigenvalue $\mu_{\max}$ of $\mathscr{Q}$ can be bounded
as follows. Let $Z\in\mathbb{C}^{n-1}$ be the eigenvector associated with $\mu_{\max}$. Then
\begin{align}
\mu_{\max}
= \frac{\|\mathscr{Q}(Z)\|_{\F}}{\|Z\|_{\F}}
&\leq 2\beta\, \frac{\|
\Diag ( \Re (\overline{V}\odot (V_{*\bot}Z))\|_{\F}}{\|Z\|_{\F}} + s_*\notag \\
& \leq 2\beta + s_*
\leq 3\beta + \|A_f\|_2,\notag
\end{align}
where $s_* = \lambda_n(H(V_*)) - \lambda_1(H(V_*))$ is the spectral
span, and for the last inequality we have used
the inequalities $s_* \leq \lambda_n(H(V_*)) \leq \beta + \|A_f\|_2 $
due to $H(V)$ in \eqref{eq:gpe} being positive definite.
Consequently, the lower bound on $\sigma$ in~\eqref{eq:lsbnd} yields
\begin{equation}\label{eq:sigmagp}
\sigma \geq \frac{1}{2}(3\beta+\|A_f\|_2)
\end{equation}
to ensure the local convergence of the level-shifted SCF.
\begin{example}\label{ex:gp}
In this example, we select the parameters
$\ell=1$, $\omega = 0.85$, and $N=10$ (hence $n=100$).
We use a radial harmonic potential
$f(x,y) = (x^2+y^2)/2$.
Various values of $\beta$ ranging from
$0.5$ to $5$ have been tried. The simulation results
are shown in~Figure~\ref{fig:ex:gp}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.49\textwidth]{Figs/convhist_gp}
\includegraphics[width=0.49\textwidth]{Figs/confac_gp}
\end{center}
\caption{Example~\ref{ex:gp}: (Left) Convergence history
of residual norm $\|H(V_i)V_i-V_i\Lambda_i\|_2$ by the plain SCF \eqref{eq:pscf}
for selected $\beta$; (Right)
Convergence rate estimates as $\beta$ varies.
}\label{fig:ex:gp}
\end{figure}
It is observed that
the plain SCF becomes slower and slower and eventually divergent
as $\beta$ increases.
Again, the spectral radius $\rho(\mathscr{L}} %{\mathcal{L}_{\sigma})$ and $\eta_{\sup}$ can
well capture true convergence behavior.
In particular, at $\beta = 3.5$, we find that up to 7 digits,
\[
\mbox{observed} = 0.9136140, \quad
\eta_{\sup,\infty} = 0.9136173, \quad
\eta_{\sup} = 1.019727, \quad
\eta_{\czbl} = 2.342686
\]
Again, we see the sharpness of the estimate $\eta_{\sup,\infty}$.
The performance of the level-shifted SCF with respect to different
shifts $\sigma$ is shown in Figure~\ref{fig:ex:gp1:rhos},
where we observe a similar convergence behavior to
Figure~\ref{fig:ex1rhos} of Example~\ref{ex:single:ls}
on the impact of the choice of shift $\sigma$.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.49\textwidth]{Figs/rhos_conv_gp1}
\includegraphics[width=0.49\textwidth]{Figs/rhos_gp1}
\end{center}
\caption{Example~\ref{ex:gp}:
convergence history of residual norm $\|H(V_i)V_i-V_i\Lambda_i\|_2$
by the level-shifted SCF \eqref{eq:levelshift} with selected $\sigma$ (left);
spectral radius of $\rho(\mathscr{L}} %{\mathcal{L}_{\sigma})$ as
shift $\sigma$ varies (right), where the first vertical dash line
is $\sigma=\frac{\mu_{\max}}{2} -\delta_*$
suggested by~\eqref{eq:lsbnd} and
the second is {\em a-priori} $\sigma =
\frac{1}{2}(3\beta+\|A_f\|_2)$ suggested by~\eqref{eq:sigmagp},
and the optimal shift is $\sigma\approx 0.08$.
The $H(V)$ is given by~\eqref{eq:gpe} with $\beta=5$.
}\label{fig:ex:gp1:rhos}
\end{figure}
\end{example}
\begin{example}\label{ex:gp2}
This is a repeat of Example~\ref{ex:gp}, except using
a non-radical harmonic potential function $f(x,y) = (x^2+100y^2)/2$.
The plots in Figure~\ref{fig:ex:gp:2} show
a slightly different performance of the plain SCF \eqref{eq:pscf}
compared to the radical harmonic case of Example~\ref{ex:gp}.
The sharpness of the estimate $\eta_{\sup,\infty}$
on the local convergence rate can be seen at $\beta = 2.2$, where
up to 7 digits:
\[
\mbox{observed} = 0.9652599, \quad
\eta_{\sup,\infty} = 0.9652614, \quad
\eta_{\sup} = 1.073434, \quad
\eta_{\czbl} = 2.043247
\]
The performance of the level-shifted SCF is depicted
in Figure~\ref{fig:ex:gp2:rhos}.
Again we observe a similar convergence behavior
to Example~\ref{ex:gp} with repect to the choice of shift $\sigma$.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.49\textwidth]{Figs/convhist_gp_ub}
\includegraphics[width=0.49\textwidth]{Figs/confac_gp_ub}
\end{center}
\caption{Example~\ref{ex:gp2}: convergence history
of residual norm $\|H(V_i)V_i-V_i\Lambda_i\|_2$ by
the plain SCF \eqref{eq:pscf} for selected $\beta$ (left);
convergence rate estimates as $\beta$ varies (right).
}\label{fig:ex:gp:2}
\end{figure}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.49\textwidth]{Figs/rhos_conv_gp2}
\includegraphics[width=0.49\textwidth]{Figs/rhos_gp2}
\end{center}
\caption{Example~\ref{ex:gp2}:
convergence history of residual norm $\|H(V_i)V_i-V_i\Lambda_i\|_2$
by the level-shifted SCF \eqref{eq:levelshift} with selected $\sigma$ (left);
spectral radius of $\rho(\mathscr{L}} %{\mathcal{L}_{\sigma})$ as
shift $\sigma$ varies (right), where the first vertical dash line is
$\sigma=\frac{\mu_{\max}}{2} -\delta_*$
suggested by~\eqref{eq:lsbnd} and the second is {\em a-priori}
$\sigma =
\frac{1}{2}(3\beta+\|A_f\|_2)$ suggested by~\eqref{eq:sigmagp},
and the optimal shift is $\sigma\approx 0.08$.
The $H(V)$ is given by~\eqref{eq:gpe} with $\beta=3$.
}\label{fig:ex:gp2:rhos}
\end{figure}
\end{example}
\section{Concluding remarks}\label{sec:conclusion}
We have presented a comprehensive local convergence analysis
of the plain SCF iteration and its level-shifted variant
for solving NEPv. The optimal convergence rate and
estimates are established.
Our analysis is in terms of the tangent-angle matrix to measure the
approximation error between consecutive SCF iterates
and the intended target.
We first established a
relation between the tangent-angle matrices associated
with any two consecutive SCF approximates, and with it
we developed new formulas for the local error contraction factor and
the asymptotic average contraction factor of SCF.
The new formulas are sharper and
complement previously established local convergence results.
With the help of new convergence rate estimates, we
derive an explicit lower-bound on the shifting parameter
to guarantee local convergence of the level-shifted SCF.
These results are numerically confirmed by examples from
applications in computational physics and chemistry.
Our analysis
does not cover more sophisticated variants of SCF
such as the damped SCF~\cite{Cances:2000b}
and the Direct Inversion of
Iterative Subspace (DIIS)~\cite{Pulay:1980,Pulay:1982}.
It is conceivable that by the tangent-angle matrix
and the eigenspace perturbation theory, one can pursue
the local convergence analysis of those variants.
Finally, we note that we focused on NEPv \eqref{eq:nepv}
satisfying the invariant property \eqref{eq:univar}.
While this property is formulated as a result of some practically
important applications,
there are recent emerging NEPv \eqref{eq:nepv} that do not have
this property, such as the one in \cite{zhwb:2020}, and
yet similar SCF iterations can be used. It would be interesting
to find out what now determines the optimal
local convergence rate. This will be a future project to pursue.
|
2024-02-18T23:40:24.825Z
|
2020-09-22T02:01:35.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2009.09022","language":"en","timestamp":1600740095000,"url":"https:\/\/arxiv.org\/abs\/2009.09022","yymm":"2009"}
|
proofpile-arXiv_000-10096
|
{"provenance":"002.jsonl.gz:10097"}
| null | null |
\section{Introduction}
The \textit{Hershcel Space Observatory$^\ast$} is an observatory class mission of the European Space Agency (ESA) \citep{Pilbratt10} that completed four years of observations exploring the far-infrared (FIR) and submillimeter (sub-mm) Universe in April 2013 with the depletion of its liquid cryogens \citep{herschelEnd}. The Spectral and Photometric Imaging REceiver (SPIRE) was one of three focal plane instruments on board \textit{Herschel}, consisting of both an imaging photometric camera and an imaging Fourier Transform Spectrometer (FTS) \citep{Griffin10}. The SPIRE FTS has two detector arrays, the Spectrometer Long Wavelength (SLW) and the Spectrometer Short Wavelength (SSW), that simultaneously cover a frequency band of 447--1546 GHz (SLW: 447--990 GHz, \mbox{SSW: 958--1546 GHz}). SPIRE FTS observations provide a wealth of molecular and atomic fine-structure spectral lines including the $[$\ion{N}{II}$]$ $^3$P$_1$--$^3$P$_0$, $[$\ion{C}{I}$]$ $^3$P$_2$--$^3$P$_1$, and $[$\ion{C}{I}$]$ $^3$P$_1$--$^3$P$_0$ transitions. During \textit{Herschel's} mission, the SPIRE FTS instrument made three high resolution ($\Delta \nu \sim 1.2$ GHz) spectral observations of the spiral galaxy NGC~891 (observation IDs 1342224765, 1342224766, and 1342213376) that are publicly available through the \mbox{\textit{Herschel Science Archive}} (HSA)\footnote{\url{http://archives.esac.esa.int/hsa/whsa/}} \citep{spire_handbook}. The $[$\ion{N}{II}$]$ $^3$P$_1$--$^3$P$_0$ line has been measured with exceptionally high signal-to-noise ratios (SNRs) in these observations which also contain lower SNR CO and $[$\ion{C}{I}$]$ features.
Recently the SPIRE FTS observations have become more accessible through the SPIRE Spectral Feature Finder Catalogue\footnote{\url{https://www.cosmos.esa.int/web/herschel/spire-spectral-feature-catalogue}}, which includes a collection of significant spectral features extracted from all publicly available high resolution (HR) single-pointing and mapping observations by the automated SPIRE Feature Finder (FF) routine \citep{FFtech,FFredshift,FFlineID,FFncc}. Due to repeated referencing of \mbox{\citet{FFtech}, \citet{FFredshift}}, \mbox{\citet{FFlineID}}, and \mbox{\citet{FFncc}}, the abbreviations \citetalias{FFtech}, \citetalias{FFredshift}, \citetalias{FFlineID}, and \citetalias{FFncc} are used in further text of this paper. The full SPIRE Automated Feature Extraction Catalogue (SAFECAT), contains the central frequency and SNR of $\sim165\,000$ features at SNRs greater than 5 (some lower SNR features may be found by the line identification routine, see \citetalias{FFtech, FFlineID}) from all publicly available SPIRE FTS observations in the HSA. From the spectral content of each observation and through literature cross references, radial velocity estimates for each observation are provided \citepalias{FFredshift}. Initial line identification estimates are also catalogued \citepalias{FFlineID} and the FF also employs a subroutine to better detect $[$\ion{C}{I}$]$ features that can be difficult to detect in automated line fitting routines of SPIRE spectra \citepalias{FFncc}.
To demonstrate the utility of the FF and its catalogue, we have used the central line frequencies catalogued by the FF from the three HR SPIRE FTS observations of NGC~891 to measure the rotational kinematics of ionized nitrogen and neutral carbon near the plane of the galaxy at FIR wavelengths. This data is in complement to \mbox{CO(1-2)} and $[$\ion{H}{I}$]$\,21\,cm measurements presented in \citet{Garcia1992,Scoville1993,Sofue1996}, and \citet{Kijeong2011} in order to associate different energy regimes of the interstellar medium (ISM) in NGC~891.
\vspace{-18pt}
\section{Observational Information}
NGC~891 is a nearby ($D\sim9.5$ Mpc for $H_0 = 75$ km\,$s^{-1}$\,Mpc$^{-1}$) edge-on spiral galaxy (inclination angle $\gtrsim 89^\circ$) and is similar in structure to the Milky Way \citep{vanDerKuit1981}. CO observations of the galaxy have shown a 4\,kpc molecular ring and a high velocity disk surrounding the nucleus \citep{Garcia1992, Scoville1993, Sofue1996}. $[$\ion{H}{I}$]$ measurements of this galaxy are more extend ($R>10$\,kpc) and both CO and $[$\ion{H}{I}$]$ measurements show some asymmetries with the south end of the galaxy being more extended than the north end \citep{Sancisi1979, Sofue1996, Kijeong2011}.
Two of the SPIRE FTS observations of NGC~891 are sparsely sampled single-pointing observations (1342224766 and 1342224765, hereafter referred to as \textbf{N}orth and \textbf{S}outh, respectively). These observations were taken on 2011-07-26 (operational day 804) and both have an integration time corresponding to 55 FTS HR scans. The FF has catalogued lines from both the extended and point source calibrations of the SPIRE FTS for these observation and the extended source calibration minimizes the gap in the SLW--SSW overlap region (\mbox{see \citealt{spire_handbook,Wu2013}}). The third observation of NGC~891 is an intermediately sampled mapping observation (1342213376, hereafter referred to as \textbf{M}) corresponding to four jiggle positions of the SPIRE FTS Beam Steering Mirror (BSM). Mapping observations of the SPIRE FTS are projected onto hyper-spectral cubes and the FF extracts lines from the spectra from each pixel separately (see \citetalias{FFtech}). This observation was taken on 2011-01-28 (operational day 625) with an integration time at each jiggle position corresponding to 32 FTS HR scans. The FF only extracts lines from an extended source calibration for mapping observations \citepalias{FFtech}. All three observations have been reduced by the SPIRE calibration tree \texttt{spire\_cal\_14\_3} and the emission from NGC~891 does not fill the entirety of the SPIRE beam for each detector \citep{Hughes2015}. Since emission from the galaxy is also not uniformly point-like for each detector, a proper determination of line-flux values requires semi-extended considerations outlined in \citet{Wu2013}.
\begin{figure}
\includegraphics[trim = 1mm 0mm 0mm 0mm, clip, width=\columnwidth]{Figures/Observational_foots.pdf}
\vspace{-22pt}
\caption{The SSW array of the FTS for each sparse observation (N and S) is shown with the footprint of the SSW cube from the mapping observation (M). These are imposed on a SPIRE photometer short wavelength (PSW) map of NGC~891 centered at 250$\mu$m. Each circle shows the full width at half maximum (FWHM) of the SPIRE FTS beam \protect\citep{Makiwa2013} and detectors that have observed the $[$\protect\ion{N}{II}$]$ feature have a thicker outline with colours assigned to each unique detector. The central rectangle indicates the extent of observation M with the small inset rectangle and circle showing the on-sky size of a pixel in the associated hyperspectral cube and SPIRE SSW beam.}
\label{fig:ObsFootPrint}
\vspace{-12pt}
\end{figure}
\begin{figure*}
\includegraphics[trim = 0mm 0mm 0mm 0mm, clip, width=\textwidth]{Figures/1342213376_postcard_comb_2x3.png}
\caption{A mapping postcard of observation M as provided in the main FF catalogue \protect\citepalias{FFtech}. These postcards are more fully described in \protect\citetalias{FFtech}. The first column shows the intensity of each pixel integrated across each spectrometer band (SLW top, SSW bottom). The central column displays a few spectra of interest including the brightest and dimmest pixels in the cube as well as the pixel with the most lines extracted by the FF (SLW top, SSW bottom). The third column shows the number of lines discovered by the FF in each pixel (top) as well as the velocity estimate obtained for each pixel (bottom, see \protect\citetalias{FFredshift}). In the velocity map, pixels that do not have reliable velocity estimates are coloured grey and those that have lines extracted by the FF are outlined in green. The dark green number on each pixel indicates the number of CO features found by the velocity estimate routine, an `N' character indicates when the $[$\protect\ion{N}{II}$]$ $^3$P$_1$--$^3$P$_0$ feature is used for a velocity estimate in the FF. A full list of lines extracted from every pixel is provided in the accompanying appendix material for this letter.}
\label{fig:mappingPost}
\end{figure*}
The footprint of each SSW detector in observations N and S are shown with that of the SSW cube from observation M in Fig.\,\ref{fig:ObsFootPrint}. It is important to note that the full extent of observation M is not perfectly rectangular (see Fig.\,\ref{fig:mappingPost}) and that intermediately sampled mapping observations only provide 16 arcsecond ($\sim$ 1 beamwidth) sampling rather than Nyquist resolved spatial sampling of the observed region \citep{spire_handbook}. Sparse observations have a 32 arcsecond sampling determined by the detector spacing of the FTS \citep{spire_handbook}. In Fig.\,\ref{fig:ObsFootPrint} the spatial resolution of each observation is shown by the FWHM of the SSW beam for each detector in observations N and S. The pixel size of the hyperspectral cube in observation M is shown by the inset rectangle in the lower left of its footprint. Each pixel is only spatially sampled by the SPIRE beam once.
In all three observations, the prominent $[$\ion{N}{II}$]$ $^3$P$_1$--$^3$P$_0$ feature is readily detected by the SSW array. SLW detectors measure a number of neutral carbon fine-structure lines and the occasional CO rotational feature albeit at a much lower SNR. The full results from the FF line fitting are provided in the accompanying appendix material (Tables \ref{tab:fullLineRes} and \ref{tab:fullLineResMap}) along with their molecular and atomic transitions determined by template matching in the FF's line identification routine (see \citealt{FFlineID}). Some lines extracted by the FF remain unidentified by the FF routine, often due to highly uncertain velocity estimates for their respective detector/pixel (see \citetalias{FFredshift}) or the lines being absent from the identification template (see \citetalias{FFlineID}). A few of these have been manually identified in this letter.
Neutral carbon (see \citetalias{FFncc}) and ionized nitrogen provide complementary measurements to CO and $[$\ion{H}{I}$]$ measurements of NGC~891 that probe different energy regimes within the galaxy. Neutral carbon will ionize at 11.26\,eV \citep{ionizationEnergies} but will be readily bonded into CO at energies of 11.1\,eV \mbox{\citep{dysonISM, bondDis}} thus $[$\ion{C}{I}$]$ features originate from a very narrow energy range separating the ionized and molecular phases of the ISM. The first ionization energy of nitrogen is 14.52\,eV \citep{ionizationEnergies} thus the $[$\ion{N}{II}$]$ $^3$P$_1$--$^3$P$_0$ feature provides a method to trace ionization regions \citep{ReviewHaffner}.
\vspace{-18pt}
\section{Results and Discussion}
The $[$\ion{N}{II}$]$ $^3$P$_1$--$^3$P$_0$ is detected at exceptionally high SNR in most spectra (detected at an average FF calculated SNR of 30, \mbox{see \citetalias{FFtech,FFlineID}}) making it an excellent tool to measure the rotation of ionized gas in the galaxy at FIR frequencies. Fig.\,\ref{fig:NIIspecs} shows the $[$\ion{N}{II}$]$ feature from each detector that observed it in observations N and S. The continuum has been extracted using the fitted continuum parameters provided by the FF. This figure also demonstrates the sinc-like line profile of the SPIRE FTS (see \citealt{spire_handbook}, \citealt{Naylor:16}). Due to this line shape, it can often prove difficult to extract low SNR lines via visual inspection which is one of the motivations for the FF project.
Fig.\,\ref{fig:Velcurve} shows the heliocentric radial velocity of the galaxy along its major axis from all three observations (N,S,M). Each point represents a radial velocity measurement from a $[$\ion{N}{II}$]$ $^3$P$_1$--$^3$P$_0$, $[$\ion{C}{I}$]$ $^3$P$_1$--$^3$P$_0$, or $[$\ion{C}{I}$]$ $^3$P$_2$--$^3$P$_1$ feature. Measurements corresponding to each observation are shown by the insets in the lower panel. The SPIRE resolution is shown in position-velocity space at the rest frequency of the $[$\ion{N}{II}$]$ $^3$P$_1$--$^3$P$_0$ feature determined by the FWHM of the SPIRE SSW beam \citep{Makiwa2013} and the 1.2\,GHz frequency resolution of the spectrometer \citep{spire_handbook}. In reality, the frequency calibration of the SPIRE FTS has been shown to allow the accurate measurement of line centers up to a factor of 1/50 the resolution of the FTS \citep{Swinyard2014, Spencer:15}; however the spectral resolution of the SPIRE FTS does not allow for the study of the internal structure of detected $[$\ion{N}{II}$]$ lines. Velocity error is determined by the error in the central line frequency determined by the line fitting of the FF. $[$\ion{C}{I}$]$ features are detected by the SLW array which has a beam that is 14--25 arcseconds greater than the SSW array. The axis of the galaxy is defined by a linear fit of all points in the SPIRE PSW map (see Fig.\,\ref{fig:ObsFootPrint}) that are at least 2.5\% of the peak intensity. This cutoff was chosen based upon visual inspection of the photometer data. Error in these positional measurements based upon this axis was estimated using a Monte-Carlo simulation varying the 2.5\% intensity cutoff 5000 times with a standard deviation of 0.3\% and again with a cutoff centered at 20\% of the peak intensity with a standard deviation of 1\%. Both simulations provided statistically equivalent results. We have defined the FIR center of the galaxy by taking the median coordinates of all pixels in the SPIRE PSW map that have $\geq 70$\% the flux of the pixel with the greatest flux (the median coordinates of pixels containing the bulge). This places the center at coordinates $\alpha=2\text{h}22\text{m}33.189\text{s}$ $\delta = 42^\circ20' 52.583"$, $\sim 8$ arcseconds from the radio center reported by \citet{Kijeong2011} (see the top panel of Fig.\,\ref{fig:Velcurve}).
\begin{figure}
\includegraphics[trim = 0mm 0mm 0mm 0mm, clip, width=\columnwidth]{Figures/1342224766_NIISpecs_norm.pdf}
\includegraphics[trim = 0mm 0mm 0mm 0mm, clip, width=\columnwidth]{Figures/1342224765_NIISpecs_norm.pdf}
\caption{A section from the continuum subtracted spectra from each SSW detector that observed the $[$\ion{N}{II}$]$ $^3$P$_1$--$^3$P$_0$ feature in observations N (top) and S (bottom). Each detector matches the assigned colour in Fig.\,\protect\ref{fig:ObsFootPrint} and an artificial vertical offset is applied to each spectrum. Spectra have been normalized to the peak amplitude of their respective $[$\protect\ion{N}{II}$]$ feature. The rest-frame frequency of the $[$\protect\ion{N}{II}$]$ feature in the local standard of rest for NGC~891 is marked by the dashed vertical line. $[$\protect\ion{N}{II}$]$ lines extracted by the FF are marked by circles at their peak intensity while other FF lines are marked by a vertical black dash.}
\label{fig:NIIspecs}
\end{figure}
The halo of NGC~891 is known to contain slower rotating \ion{H}{I} gas and diffuse ionized gas up to altitudes greater than 4\,kpc from the disk \citep{Swaters1997, Rand1990} and thus there is potential for slower rotating $[$\ion{N}{II}$]$ emission that can contaminate SPIRE observations of the disk. \citet{Swaters1997} have shown a significant decrease in the rotational speeds of \ion{H}{I} gas at $30<|z|<60$ arcseconds. With the 16.6 arcsecond FWHM SPIRE beam at the rest frequency of the $[$\ion{N}{II}$]$ $^3$P$_1$--$^3$P$_0$ line, any observations with an elevation ($|z|$) greater than $\sim 20$ arcseconds is likely subject to a significant amount of contamination from slow-moving halo gas. Based upon 16 SPIRE $[$\ion{N}{II}$]$ measurements that are within 20 arcseconds of the galaxy's disk and are less then a beam-width apart we have found that their radial velocities may disagree by 22--64\,km\,s$^{-1}$. Fig.\,\ref{fig:Velcurve} demonstrates that a single spectral resolution element of the FTS is large in velocity space; this coupled with the spatial resolution of the instrument may result in the FF reported line centers being subject to line-blending from $[$\ion{N}{II}$]$ features at multiple velocities. \citet{Keppel1991} have studied the kinematics of ionized hydrogen perpendicular to the major axis of NGC~891 and have shown that radial velocities measured by H$\alpha$ emission may decrease by as much as $\sim$55\,km\,s$^{-1}$ within perpendicular distances as wide as the SPIRE SSW beam.
\begin{figure}
\includegraphics[trim = 0mm 0mm 0mm 0mm, clip, width=\columnwidth]{Figures/VelPlots_stack_radCent.pdf}
\caption{The top panel shows a section of the SPIRE PSW photometer map focused on NGC~891 from Fig.\,\protect\ref{fig:ObsFootPrint} in negative color. Contour levels are 10, 21.5, 46.4, 100, 215.4, 464.2, and 1000 Mj\,sr$^{-1}$ and the radio center of the galaxy reported by \protect\citet{Kijeong2011} is marked by the pink `x'. The bottom panel displays velocity measurements from FF catalogued $[$\protect\ion{N}{II}$]$ and $[$\ion{C}{I}$]$ lines as a position-velocity diagram along the axis of the galaxy. Points are coloured by their orthogonal distance from the axis of the galaxy, $z$. The dashed cyan line shows the Keplerian curve fit to the center of the rotation curve in Fig.\,\protect\ref{fig:rotCurve}. Insets in the bottom panel show the observation each set of points correspond to. Both panels share a common axis position.}
\label{fig:Velcurve}
\end{figure}
Our measurements of the galactic rotation from the \mbox{$[$\ion{N}{II}$]$ $^3$P$_1$--$^3$P$_0$} feature follow a similar trend to the higher spatially resolved $[$\ion{H}{I}$]$ measurements presented in \citet{Sancisi1979} and \citet{Kijeong2011} showing no major deviations from these within SPIRE's limited spatial resolution. Keplerian rotation is seen within the extent of SPIRE observations with the furthest points beginning to show the flattening of the rotation curve due to dark matter. The limited extent of the SPIRE FTS observations ($R<11.5$\,kpc) does not allow us to determine if the $[$\ion{N}{II}$]$ emission experiences the same asymmetry as $[$\ion{H}{I}$]$ emission that occurs at $R>14$\,kpc for the south end of the galaxy \citep{Sancisi1979}.
Measurements of $[$\ion{C}{I}$]$ features near the center of the galaxy do not show the characteristic profile of the rapidly rotating molecular disk near the galactic nucleus \citep{Garcia1992, Sofue1996} and are instead more in agreement with $[$\ion{H}{I}$]$ measurements. This provides further evidence that this central disk is completely molecular. It should be noted that SPIRE measurements of $[$\ion{C}{I}$]$ features in the galaxy tend to be of much lower SNR than measurements of the $[$\ion{N}{II}$]$ $^3$P$_1$--$^3$P$_0$ feature and thus are subject to a much higher degree of uncertainty. These lines are also detected in the SLW band and their spatial resolution is significantly worse than measurements of the $[$\ion{N}{II}$]$ $^3$P$_1$--$^3$P$_0$ feature.
The rotation curve in Fig.\,\ref{fig:rotCurve} is calculated from spectra containing the $[$\ion{N}{II}$]$ $^3$P$_1$--$^3$P$_0$ feature that are within a half beam-width of the galaxy's major axis. Radial velocity has been corrected for the slight inclination of the galaxy. Within the extent of near-disk SPIRE observations, the motion is Keplerian and in agreement with the rotation curve for neutral hydrogen measured by \citet{Sancisi1979}. The large uncertainty in velocity values is indicative of the maximum velocity discrepancy (64\,km/s) between complementary measurement of $[$\ion{N}{II}$]$ emission within one beam-width of each other.
\begin{figure}
\includegraphics[trim = 0mm 0mm 0mm 0mm, clip, width=\columnwidth]{Figures/rotCurve_Full.pdf}
\caption{The rotation curve of NGC~891 measured by $[$\ion{N}{II}$]$ $^3$P$_1$--$^3$P$_0$ fine structure lines. The dashed cyan line shows a Keplerian curve fit to the central region of the rotation curve. The pink error bars show the systematic velocity at the radio center reported by \protect\citet{Sancisi1979} and \protect\citet{Kijeong2011} while the thin black error bars show the systematic velocity 547$\pm$24\,km\,$s^{-1}$ determined from the rotation curve measured by SPIRE detected $[$\ion{N}{II}$]$ features.}
\label{fig:rotCurve}
\end{figure}
The agreement between our measurements of $[$\ion{N}{II}$]$ and $[$\ion{C}{I}$]$ with measurements of $[$\ion{H}{I}$]$ suggest a correlation between the Warm Neutral Medium (WNM) associated with $[$\ion{H}{I}$]$ 21\,cm emission and the Warm Ionized Medium (WIM) probed by the $[$\ion{N}{II}$]$ feature (see \citealt{ReviewHaffner}). Correlations between these two phases of the ISM have been shown qualitatively at high altitudes from the galactic plane in spiral galaxies (see \citealt{ReviewHaffner,HartmanBook1997}) and for intermediate to high velocity clouds that are not co-rotating with the disk of the galaxy \citep{Tufte1998,Haffner2001}. Our results suggest that such a correlation exists within the disk of NGC~891. This result is consistent with a clumpy model of the ISM in which neutral condensations exist within a $[$\ion{H}{II}$]$ region with complex interfaces between neutral and ionized regions \citep{ReviewHollenbach,ReviewHaffner}.
\section{Conclusion}
The Herschel SPIRE FF includes provision of line frequencies, signal-to-noise ratios, velocity information, and an initial line identification estimate for three HR spectral observations of the edge-on galaxy NGC~891. Using these catalogued measurements we have made rotational measurements of the ionized gas in NGC~891 with $[$\ion{N}{II}$]$ $^3$P$_1$--$^3$P$_0$ fine-structure lines that are detected at high SNR in these observations. We have also included measurements of the narrow neutral carbon energy regime with measurements of the $[$\ion{C}{I}$]$ $^3$P$_1$--$^3$P$_0$ and $[$\ion{C}{I}$]$ $^3$P$_2$--$^3$P$_1$ fine-structure lines.
We present our results in complement to radio measurements of CO and $[$\ion{H}{I}$]$ presented in \citet{Garcia1992,Scoville1993,Sofue1996} and \citet{Kijeong2011}. In spite of the limited spatial and velocity resolution of the SPIRE FTS We have found that the position-velocity profile of $[$\ion{N}{II}$]$ and $[$\ion{C}{I}$]$ lines closely match that of atomic hydrogen in NGC~891. This result evidences the formation of ionization interfaces on the exterior of neutral clumps in ionization regions within the disk of NGC~891.
This letter demonstrates through this simple example that the information collected by the FF and made available through SAFECAT greatly aids the exploitation of observations made with the Herschel SPIRE FTS and enhances the legacy value of the instrument and associated data archive. The wealth of FIR data contained in SAFECAT aids in interpreting a large portion of the highest resolution and most sensitive observations of the FIR universe to date.
\section*{Acknowledgements}
\emph{Herschel} is an European Space Agency (ESA) space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from the National Aeronautics and Space Administration (NASA). This research acknowledges support from ESA, the Canadian Space Agency (CSA), the Canada Research Council (CRC), CMC Microsystems, and the Natural Sciences and Engineering Research Council of Canada (NSERC).
This research has made use of the NASA/IPAC Infrared Science Archive, which is funded by the National Aeronautics and Space Administration and operated by the California Institute of Technology.
This research has made use of the \textsc{SciPy} (\url{www.scipy.org}) and \textsc{Astropy} (\url{www.astropy.org}) Python packages.
\nocite{2020SciPy-NMeth, astropy:2013, astropy:2018}\\
\section*{Data Availability}
The \textit{Herschel} SPIRE Spectral Feature Catalogue has been assigned an ESA Digital Object Identifier (DOI) and is available at: \href{https://doi.org/10.5270/esa-lysf2yi}{doi.org/10.5270/esa-lysf2yi}. The FF code and all FF products are publicly available via the \textit{Herschel} Science Archive.
\nocite{chrisThesis}
\bibliographystyle{mnras}
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2020-09-25T02:00:51.000Z
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{"arxiv_id":"2009.09014","language":"en","timestamp":1600999251000,"url":"https:\/\/arxiv.org\/abs\/2009.09014","yymm":"2009"}
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proofpile-arXiv_000-10097
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{"provenance":"002.jsonl.gz:10098"}
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\section{Introduction}
By a theorem of Lazar and Lindenstrauss, any Choquet simplex can be expressed as a projective limit of finite dimensional simplices (see \cite{lazar-lindenstrauss}, \cite{EHS}). This has implications for C*-algebras: given a Choquet simplex $K$, there exists a simple, unital, approximately finite dimensional (AF) C*-algebra whose set of tracial states is isomorphic to $K$ (\cite{blackadar, EHS}). In the investigations on the structure of a C*-algebra, another kind of trace is also of interest, namely, the lower semicontinuous traces with values in $[0,\infty]$. These traces form a non-cancellative topological cone. (By cone we understand an abelian monoid endowed with a scalar multiplication by positive scalars.) Our goal here is to characterize through intrinsic properties the topological cones arising as the lower semicontinuous $[0,\infty]$-valued traces on an AF C*-algebra. These are also the projective limits of cones of the form $[0,\infty]^n$, with $n\in\mathbb{N}$, and also, the cones arising as the $[0,\infty]$-valued monoid morphisms on the positive elements of a dimension group.
Let $A$ be C*-algebra. Denote its cone of positive elements by $A_+$. A map $\tau\colon A_+\to [0,\infty]$
is called a trace if it is linear (additive, homogeneous, mapping 0 to 0) and satisfies that $\tau(x^*x)=\tau(xx^*)$
for all $x\in A$. We are interested in the lower semicontinuous traces. Let $T(A)$ denote the cone of $[0,\infty]$-valued lower semicontinuous traces on $A_+$. By the results of \cite{ERS}, $T(A)$ is a complete lattice when endowed with the algebraic order, and addition in $T(A)$ is distributive with respect to the lattice operations. Further, one can endow $T(A)$ with a topology that is locally convex, compact and Hausdorff. We call an abstract topological cone with these properties an \emph{extended Choquet cone} (see Section \ref{sectionECCs}).
By an AF C*-algebra we understand an inductive limit, over a possibly uncountable index set, of finite dimensional C*-algebras. Not every extended Choquet cone arises as the cone of lower semicontinuous traces on an AF C*-algebra. The requisite additional properties are sorted out in the theorem below.
An element $w$ in a cone is called idempotent if $2w=w$. Given a cone $C$, we denote by $\mathrm{Idem}(C)$ the set of idempotent elements of $C$.
\begin{theorem}\label{mainchar}
Let $C$ be an extended Choquet cone (see Definition \ref{defExtCho}). The following are equivalent:
\begin{enumerate}[\rm (i)]
\item
$C$ is isomorphic to $T(A)$ for some AF C*-algebra $A$.
\item
$C$ is isomorphic to $\mathrm{Hom}(G_+,[0,\infty])$ for some dimension group $(G,G_+)$. (Here
$\mathrm{Hom}(G_+,[0,\infty])$ denotes the set of monoid morphisms from $G_+$ to $[0,\infty]$.)
\item
$C$ is a projective limit of cones of the form $[0,\infty]^n$, $n\in \mathbb{N}$.
\item
$C$ has the following properties:
\begin{enumerate}
\item $\mathrm{Idem}(C)$ is an algebraic lattice under the opposite algebraic order,
\item for each $w\in \mathrm{Idem}(C)$, the set $\{x\in C:x\leq w\}$ is connected.
\end{enumerate}
\end{enumerate}
Moreover, if $C$ is metrizable and satisfies (iv), then the C*-algebra in (i) may be chosen separable, the group $G$ in (ii) may be be chosen countable, and the projective limit in (iii) may be chosen over a countable index set.
\end{theorem}
We refer to property (a) in part (iv) as ``having an abundance of compact idempotents". The fact that the primitive spectrum of an AF C*-algebra has a basis of compact open sets makes this condition necessary.
We call property (b) ``strong connectedness". The existence of a non-trivial trace on every simple ideal-quotient of an AF C*-algebra makes this condition necessary. In general, if a C*-algebra $A$ is such that
its primitive spectrum has a basis of compact open sets, and every simple quotient $I/J$, where $J\subsetneq I$ are ideals of $A$, has a non-zero densely finite trace, then $T(A)$ has an abundance of compact idempotents and is strongly connected, i.e., properties (a) and (b) above hold. For example, if $A$ has real rank zero, stable rank one, and is exact, then these conditions are met. Theorem \ref{mainchar} then asserts the existence of an AF C*-algebra $B$ such that $T(A)\cong T(B)$.
The crucial implication in Theorem \ref{mainchar} is (iv) implies (iii).
A reasonable approach to proving it is to first prove that (iv) implies (ii) by directly constructing a dimension group $G$ from the cone $C$, very much in the spirit of the proof of the Lazar-Lindenstrauss theorem obtained by Effros, Handelmann, and Shen in \cite{EHS} (which, unlike the proof in \cite{lazar-lindenstrauss}, also deals with non-metrizable Choquet simpleces). If the cone $C$ is assumed to be finitely generated, then we indeed obtain a direct construction of an ordered vector space with the Riesz property $(V,V^+)$ such that $\mathrm{Hom}(V^+,[0,\infty])$ is isomorphic to $C$. This is done in the last section of the paper. In the general case, however, such an approach has eluded us.
To prove Theorem \ref{mainchar} we first establish a duality between extended Choquet cones with an abundance of compact idempotents and certain abstract Cuntz semigroups. Briefly stated, this duality works as follows:
\[
C\mapsto \mathrm{Lsc}_\sigma(C)\hbox{ and }S\mapsto F(S).
\]
That is, to an extended Choquet cone $C$ with an abundance of compact idempotents one assigns the Cu-cone $\mathrm{Lsc}_\sigma(C)$ of lower semicontinuous linear functions $f\colon C\to [0,\infty]$ with ``$\sigma$-compact support". In the other direction, to a Cu-cone $S$ with an abundance of compact ideals one assigns the cone of functionals $F(S)$; see Section \ref{duality} and Theorem \ref{dualitythm}. In the context of this duality, strong connectedness in $C$ translates into the property of weak cancellation in
$\mathrm{Lsc}_\sigma(C)$. We then use this arrow reversing duality to turn the question of finding a projective limit representation for a cone into one of finding an inductive limit representation for a Cu-cone. To achieve the latter, we follow the strategy of proof of the Effros-Handelmann-Shen theorem, adapted to the category at hand. The main technical complication here is the non-cancellative nature of Cu-cones, but this is adequately compensated by the above mentioned property of ``weak cancellation" (dual to strong connectedness).
A question that is closely related to the one addressed by Theorem \ref{mainchar} asks for a characterization of the lattices arising as (closed two-sided) ideal lattices of AF C*-algebras. For separable AF C*-algebras, this problem was solved by Bratteli and Elliott in \cite{bratteli-elliott}, and independently by Bergman in unpublished work: Any complete algebraic lattice with a countable set of compact elements is the lattice of closed two-sided ideals of a separable AF C*-algebra. A thorough discussion of this result is given by Goodearl and Wehrung in \cite{goodearl-wehrung}. The cardinality restriction on the set of compact elements is necessary, as demonstrated by examples of R\r{u}\v{z}i\v{c}ka and Wehrung (\cite{ruzicka}, \cite{wehrungexample}). Now, the lattice of closed two-sided ideals of a C*-algebra $A$ is in order reversing bijection with the lattice of idempotents of $T(A)$ via the assignment $I\mapsto \tau_I$, where $\tau_I$ is the $\{0,\infty\}$-valued trace vanishing on $I_+$. Thus, the realization of a cone $C$ in the form $T(A)$ entails the realization of $(\mathrm{Idem}(C),\leq^{\mathrm{op}})$ as the ideal lattice of $A$. Curiously, no cardinality restriction is needed in Theorem \ref{mainchar} above. This demonstrates that the examples of R\r{u}\v{z}i\v{c}ka and Wehrung are also examples of algebraic lattices that cannot be realized as the lattice of idempotents of a cone $C$ satisfying any of the equivalent conditions of Theorem \ref{mainchar}.
This paper is organized as follows: In Section \ref{sectionECCs} we define extended Choquet cones and prove a number of background results on their structure. In Section \ref{sec:conesfromfunctionals} we go over three constructions---starting from a C*-algebra, a
dimension group, and a Cu-semigroup---yielding extended Choquet cones that are strongly connected and have an abundance of compact idempotents. Sections \ref{functionspaces} and \ref{duality} delve into spaces of linear functions on extended Choquet cones with an abundance of compact idempotents. In Theorem \ref{dualitythm} we establish the above mentioned duality assigning to a cone $C$ the Cu-cone $\mathrm{Lsc}_\sigma(C)$, and conversely to a Cu-cone $S$ its cone of functionals $F(S)$. In Section \ref{proofofmainchar} we prove Theorem \ref{mainchar}. In Section \ref{fingen} we assume that the cone $C$ is finitely generated. In this case we give a direct construction of an ordered vector space with the Riesz property $(V,V^+)$ such that $C\cong \mathrm{Hom}(V^+,[0,\infty])$. The vector space $V$ is described as $\mathbb{R}$-valued functions on a certain spectrum of the cone $C$.
\textbf{Acknowledgement}: The second author thanks Hannes Thiel for fruitful discussions on the topic of topological cones and for sharing his unpublished notes \cite{thiel}.
\section{Extended Choquet Cones}\label{sectionECCs}
\subsection{Algebraically ordered compact cones}
We call cone an abelian monoid $(C,+)$ endowed with a scalar multiplication by positive real numbers $(0,\infty)\times C\to C$ such that
\begin{enumerate}[\rm (i)]
\item
the map $(t,x)\mapsto tx$ is additive on both variables,
\item
$s(tx)=(st)x$ for all $s,t\in (0,\infty)$ and $x\in C$,
\item
$1\cdot x=x$ for all $x\in C$.
\end{enumerate}
We do not assume that the addition operation on $C$ is cancellative. In fact, the primary example of the cones that we investigate below is $[0,\infty]$ endowed with the obvious operations.
The algebraic pre-order on $C$ is defined as follows: $x\leq y$ if there exists $z\in C$ such that $x+z=y$. We say that $C$ is algebraically ordered if this pre-order is an order.
We call $C$ a topological cone if it is endowed with a topology for which the operations of addition and multiplication by positive scalars are jointly continuous.
\begin{definition}\label{defExtCho}
An algebraically ordered topological cone $C$ is called an extended Choquet cone if
\begin{enumerate}[\rm (i)]
\item
$C$ is a lattice under the algebraic order, and the addition operation is distributive over both $\wedge$ and $\vee$:
\begin{align*}
x + (y\wedge z) &=(x+y)\wedge (x+z),\\
x + (y\vee z) &=(x+y)\vee (x+z),
\end{align*}
for all $x,y,z\in C$,
\item
the topology on $C$ is compact, Hausdorff, and locally convex, i.e., it has a basis of open convex sets.
\end{enumerate}
\end{definition}
\begin{remark}
It is a standard result that in a compact algebraically ordered monoid both upward and downward directed sets converge to their supremum and infimum, respectively (\cite[Proposition 3.1]{edwards}, \cite[Proposition VI-1.3, p441]{GHK}). We shall make frequent use of this fact applied to extended Choquet cones. It readily follows from this and the existence of finite suprema and infima that extended Choquet cones are complete lattices.
\end{remark}
\begin{remark}
By Wehrung's \cite[Theorem 3.11]{wehrung}, the algebraic and order theoretic properties of an extended Choquet cone may be summarized as saying that it is an injective object in the category of positively ordered monoids.
\end{remark}
\begin{example}
The set $[0,\infty]$ is an extended Choquet cone when endowed with the standard operations of addition and scalar multiplication and the standard topology. More generally, the powers $[0,\infty]$, endowed with coordinatewise operations and the product topology are extended Choquet cones.
\end{example}
Let $C$ and $D$ be extended Choquet cones. A map $\phi\colon C\to D$ is a morphism in the extended Choquet cones category if $\phi$ is linear (additive, homogeneous with respect to scalar multiplication, and mapping 0 to 0) and continuous.
\begin{theorem}
The category of extended Choquet cones has projective limits.
\end{theorem}
\begin{proof}
Let $\{C_i:i\in I\}$, $\{\varphi_{i,j}\colon C_i\to C_j:i,j\in I \hbox{ with }j\leq i\}$, be a projective system of extended Choquet cones, where $I$ is an upward directed set. Define
\[
C=\bigg\{(x_i)_{i}\in \prod_{i\in I}
C_i: x_{j}=\varphi_{i,j}(x_i)\hbox{ for all }i,j\in I\hbox{ with }j\leq i\bigg\}.
\]
Endow the product $\prod_{i\in I} C_i$ with coordinatewise operations, coordinatewise order, and with the product topology; endow $C$ with the topological cone structure induced by inclusion. Let $\pi_i\colon C\to C_i$, $i\in I$, denote the projection maps. It follows from well known arguments that $\{C,\pi_i|_C:i\in I\}$ is the projective limit of the system $\{C_i,\phi_{i,j}:i,j\in I\}$ as compact Hausdorff topological cones (cf. \cite[Theorem 13]{davies}). Since for each $i$ the topology of $C_i$ has a basis of open convex sets, the product topology on $\prod_{i} C_i$ also has a basis of open convex sets. Further, since $C$ is a convex subset of $\prod_i C_i$, the induced topology on $C$ is locally convex as well.
Let us now prove that $C$ is a lattice. The proof runs along the same lines as the one in \cite[Theorem 13]{davies} for projective limits of Choquet simplices. We show that $C$ is closed under finite suprema; the argument for finite infima is similar. Let $x=(x_i)_i$ and $y=(y_i)_i$ be in $C$. Their coordinatewise supremum exists in $\prod_{i} C_i$, but does not necessarily belong to $C$. For each $k\in I$ define $z^{(k)}\in \prod_i C_i$ by
\[
(z^{(k)})_i=\begin{cases}
\phi_{k,i}(x_k\vee y_k)& \hbox{if }i\leq k,\\
x_i\vee y_i &\hbox{otherwise}.
\end{cases}
\]
If $k'\geq k$, then
\[
\phi_{k',k}(x_{k'}\vee y_{k'})\geq \phi_{k',k}(x_{k'})=x_k,
\]
and similarly $\phi_{k',k}(x_{k'}\vee y_{k'})\geq y_k$, whence $\phi(x_{k'}\vee y_{k'})\geq x_k\vee y_k$. It follows that
\[
(z^{(k')})_i=\phi_{k',i}(x_{k'}\vee y_{k'})\geq \phi_{k,i}(x_k\vee y_k)=(z^{(k)})_i,
\]
for $i\leq k$, while
\[
(z^{(k')})_i\geq x_i\vee y_i=(z^{(k)})_i
\]
for $i\nleq k$. Thus, $(z^{(k)})_{k\in I}$ is an upward directed net. Set $x\vee y:=\lim_k z^{(k)}$, which is readily shown to belong to $C$. Then $x\vee y\geq z^{(k)}\geq x,y$ for all $k$. Suppose that $w=(w_i)_i\in C$ is such that $w\ge x,y$. Then $w_i\ge x_i\vee y_i$ for all $i$, and further
\[
w_i=\varphi_{k,i}(w_k)\ge\varphi_{k,i}(x_k\vee y_{k}).
\]
Hence, $w\ge z^{(k)}$ for all $k$, and so $w\geq x\vee y$. This proves that $x\vee y$ is in fact the supremum of $x$ and $y$ in $C$.
Let us prove distributivity of addition over $\vee$. Let $x,y,v\in C$. Fix an index $i$.
Then
\begin{align*}
((x+v)\vee (y+v))_i &=\lim_k \phi_{k,i}((x_k+v_k)\vee (y_k+v_k))\\
&=\lim_{k} \phi_{k,i}((x_k\vee y_k)+v_k)\\
&=\lim_{k}\phi_{k,i}(x_k\vee y_k)+v_i\\
&=(x\vee y+v)_i,
\end{align*}
where we have used the distributivity of addition over $\vee$ on each coordinate and the construction of joins
in $C$ obtained above. Thus, $(x+v)\vee (y+v)=(x\vee y)+v$. Distributivity over $\wedge$ is handled similarly.
\end{proof}
\subsection{Lattice of idempotents}
Throughout this subsection $C$ denotes an extended Choquet cone.
An element $w\in C$ is called idempotent if $2w=w$. It follows, using that $C$ is algebraically ordered, that $t w=w$ for all $t\in (0,\infty]$. We denote the set of idempotents of $C$ by $\mathrm{Idem}(C)$. The set $\mathrm{Idem}(C)$ is a sub-lattice of $C$: if $w_1$ and $w_2$ are idempotents then
\[
2(w_1\vee w_2)=(2w_1\vee 2w_2)=w_1\vee w_2,
\]
where we have used that multiplication by $2$ is an order isomorphism. Hence, $w_1\vee w_2$ is an idempotent. Similarly, $w_1\wedge w_2$ is shown to be an idempotent. Moreover, $w_1\vee w_2=w_1+w_2$, a fact easily established.
In the lattice $\mathrm{Idem}(C)$, we use the symbol $\gg$ to denote the way-below relation under the opposite order.
That is, $w_1\gg w_2$ if whenever $\inf_i v_i\leq w_2$ for a decreasing net $(v_i)_i$ in $\mathrm{Idem}(C)$, we have $ v_{i_0}\leq w_1$ for some $i_0$. We call $w\in \mathrm{Idem}(C)$ a compact idempotent if $w\gg w$. More explicitly, $w$ is compact if whenever $\inf_i v_i\leq w$ for a decreasing net $(v_i)_i$ in $\mathrm{Idem}(C)$, we have $v_{i_0}\leq w$ for some $i_0$. Note: we only use the notion of compact element in $\mathrm{Idem}(C)$ in the sense just defined, i.e., applied to the \emph{opposite order}.
A complete lattice is called algebraic if each of its elements is a supremum of compact elements
(\cite[Definition I-4.2]{GHK}).
\begin{definition} We say that an extended Choquet cone $C$ has an abundance of compact idempotents if $(\mathrm{Idem}(C), \leq^{\mathrm{op}})$ is an algebraic lattice, i.e., every idempotent in $C$ is an infimum of compact idempotents.
\end{definition}
Let $x\in C$. Consider the set $\{z\in C:x+z=x\}$. This set is closed under addition and also closed in the topology of $C$. It follows that it has a maximum element $\epsilon(x)$. Since $2\cdot \epsilon(x)$ is also absorbed additively by $x$, we have $\epsilon(x)=2\epsilon(x)$, i.e., $\epsilon(x)$ is an idempotent. We call $\epsilon(x)$ the support idempotent of $x$.
\begin{lemma}\label{supportlemma}(Cf. \cite[Lemma 3.2]{edwards})
Let $x,y,z\in C$.
\begin{enumerate}[\rm (i)]
\item
$\epsilon(x) = \lim_n\frac{1}{n}x$.
\item
If $x+z\leq y+z$ then $x+\epsilon(z)\leq y+\epsilon(z)$.
\end{enumerate}
\end{lemma}
\begin{proof}
(i) Observe that $w:=\lim_n\frac{1}{n}x$ exists, since the infimum of a decreasing sequence is also its limit. It is also clear that $2w=w$, and that $x+w=x$. Let $z\in C$ be such that $x+z=x$. Then $x+nz=x$, i.e.,
$\frac1n x+ z=\frac1nx$, for all $n\in \mathbb{N}$. Letting $n\to \infty$, we get that
$w+z=w$, and in particular, $w\leq z$. Thus, $w$ is the largest element absorbed by $x$, i.e. $w=\epsilon(x)$.
(ii) This is \cite[Lemma 3.2]{edwards}. Here is the argument: We deduce, by induction, that $nx+z\leq ny+z$ for all $n\in \mathbb{N}$. Hence, $x+\frac1nz\leq y+\frac1nz$. Letting $n\to \infty$ and using (i), we get $x+\epsilon(z)\leq y+\epsilon(z)$.
\end{proof}
\begin{lemma}\label{supportmax}
Let $K\subseteq C$ be closed and convex. Then the map $x\mapsto \epsilon(x)$ attains a maximum on $K$.
\end{lemma}
\begin{proof}
Let $W=\{\epsilon(x):x\in K\}$. Let $x_1,x_2\in K$, with $\epsilon(x_1)=w_1$ and $\epsilon(x_2)=w_2$. Since $K$ is convex,
$(x_1+x_2)/2\in K$. Since
\[
\epsilon\Big(\frac{x_1+x_2}{2}\Big)=\lim_n \frac{1}{2n}x_1+\frac{1}{2n}x_2=\epsilon(x_1)+\epsilon(x_2),
\]
the set $W$ is closed under addition. For each $w\in W$, let us choose $x_w\in K$ with $\epsilon(x_w)=w$.
By compactness of $K$, the net $(x_w)_{w\in W}$ has a convergent subnet. Say $x_{h(\lambda)}\to x\in K$, where $h\colon \Lambda\to W$ is increasing and with cofinal range. For each $\lambda$ we have $x_{h(\lambda')}+h(\lambda)=x_{h(\lambda')}$ for all $\lambda'\geq \lambda$. Passing to the limit in $\lambda'$ we get $x+h(\lambda)=x$. Since $h(\lambda)$ ranges through a cofinal set in $W$, $x+w=x$ for all $w\in W$. Thus, $\epsilon(\cdot)$
attains its maximum on $W$ at $x$.
\end{proof}
\begin{lemma}\label{opensetsupport}
For each idempotent $w\in C$ the set $\{x\in C:w\gg \epsilon(x)\}$ is open. (Recall that $\gg$ is the way below relation
in the lattice $(\mathrm{Idem}(C),\leq^{\mathrm{op}})$.)
\end{lemma}
\begin{proof}
Let $x\in C$ be such that $w\gg \epsilon(x)$. By Lemma \ref{supportmax}, for each closed convex neighborhood $K$ of $x$, there exists $x_K\in K$ at which $\epsilon(\cdot)$ attains its maximum. By the local convexity of $C$, the system of closed convex neighborhoods of $x$ is downward directed. It follows that $(\epsilon(x_K))_K$ is downward directed. Moreover, $x_K\to x$, since the topology is Hausdorff. We claim that $\epsilon(x)=\inf_K \epsilon(x_K)$, where $K$ ranges through all the closed convex neighborhoods of $x$. Proof: Set $y=\inf_K \epsilon(x_K)$. We have $y\leq \epsilon(x_K)\leq \frac{x_K}{n}$ for all $K$ and $n\in \mathbb{N}$. Passing to the limit, first in $K$ and then in $n$, we get that $y\leq \epsilon(x)$. On the other hand, $\epsilon(x)\leq \epsilon(x_K)$ for all $K$ (since $x\in K$ and $\epsilon$ attains its maximum on $K$ at $x_K$). Thus, $\epsilon(x)\leq y$, proving our claim.
We have $w\gg \epsilon(x)=\inf_K \epsilon(x_K)$. Hence, there is $K$ such that $w\gg \epsilon(x_K)$. So, there is a neighborhood of $x$ all whose members belong to $\{z\in C:w\gg \epsilon(z)\}$. This shows that $\{z\in C:w\gg \epsilon(z)\}$ is open.
\end{proof}
\subsection{Cancellative subcones}
Fix an idempotent $w\in C$. Let
\[
C_w=\{x\in C:\epsilon(x)=w\}.
\]
Then $C_w$ is closed under sums, scalar multiplication by positive scalars, finite infima, and finite suprema.
By Lemma \ref{supportlemma} (ii), $C_w$ is also cancellative: $x+z\leq y+z$ implies that $x\leq y$ for all $x,y,z\in C_w$. It follows that $C_w$ embeds in a vector space; namely, the abelian group of formal differences $x-y$, with $x,y\in C_w$ endowed with the unique scalar multiplication extending the scalar multiplication on $C_w$. Let $V_w$ denote the vector space of differences $x-y$, with $x,y\in C_w$. Let $\eta\colon C_w\times C_w\to V_w$ be defined by $\eta(x,y)=x-y$. We endow $C_w$ with the topology that it receives as a subset of $C$. We endow $V_w$ with the quotient topology coming from the map $\eta$.
\begin{theorem}\label{compactbase}
Let $w\in \mathrm{Idem}(C)$ be a compact idempotent. Then $V_w$ is a locally convex topological vector space whose topology restriced to $C_w$ agrees with the topology on $C_w$. Moreover, either $C_w=\{w\}$ or $C_w$ has a compact base.
\end{theorem}
Note: A subset $B$ of a cone $T$ is called a base if for each nonzero $x\in T$ the intersection of $(0,\infty)\cdot x$ with $B$ is a singleton set.
\begin{proof}
Let us first show that the topology on $C_w$ is locally compact. Since $w$ is compact, the set $\{x\in C:w\geq \epsilon(x)\}$ is open by Lemma \ref{opensetsupport}. We then have that $C_w$ is the intersection of the closed set $\{x\in C:w\leq x\}$ and the open set $\{x:w\geq \epsilon(x)\}$. Hence, $C_w$ is locally compact in the induced topology.
We can now apply \cite[Theorem 5.3]{lawson}, which asserts that if $C_w$ is a locally compact cancellative cone, then indeed $V_w$ is a locally convex topological vector space whose topology extends that of $C_w$. Finally, by \cite[Theorem II.2.6]{alfsen}, a locally compact nontrivial cone in a locally convex topological space has a compact base.
\end{proof}
\subsection{Strong connectedness}
Let $v,w\in \mathrm{Idem}(C)$ be such that $v\leq w$. Let's say that $v$ is compact relative to $w$ if $v$ is a compact idempotent in the extended Choquet cone $\{x\in C:x\leq w\}$. Put differently, if a downward directed net $(v_i)_i$ in $C$ satisfies that $\inf_i v_i\leq v$, then $v_i\wedge w\leq v$ for some $i$.
\begin{theorem}\label{kernelpropertythm}
Let $C$ be an extended Choquet cone. The following are equivalent:
\begin{enumerate}[\rm (i)]
\item
For any two $w_1,w_2\in \mathrm{Idem}(C)$ such that $w_1 \leq w_2$, $w_1\neq w_2$, and $w_1$ is compact relative to $w_2$, there exists $x\in C$ such that $w_1\leq x\leq w_2$ and $x$ is not an idempotent.
\item
The set $\{x\in C:x\leq w\}$ is connected for all $w\in \mathrm{Idem}(C)$.
\end{enumerate}
Moreover, if the above hold then the element $x$ in (i) may always be chosen such that $\epsilon(x)=w_1$.
\end{theorem}
\begin{proof}
We show that the negations of (i) and (ii) are equivalent.
Not (ii)$\Rightarrow$ not (i): Suppose that $\{x\in C:x\leq w\}$ is disconnected for some idempotent $w$. Working in the cone $\{x\in C:x\leq w\}$ as the starting extended Choquet cone, we may assume without loss of generality that $w=\infty$ (the largest element of $C$). Let $U$ and $V$ be open disjoint sets whose union is $C$. Assume that $\infty\notin U$. Observe that totally ordered subsets of $U$ have an upper bound:
if $(x_i)_i$ is a chain then $x_i\to \sup_i x_i$, and since $U$ is closed, $\sup x_i\in U$. By Zorn's lemma, $U$ contains a maximal element $v$. Since $2v$ is connected to $v$ by the path $t\mapsto tv$ with $t\in [1,2]$, we must have that $2v=v$, i.e., $v$ is an idempotent. Let's show that $v$ is compact: Let $(v_i)_i$ be a decreasing net of idempotents with infimum $v$. Suppose, for the sake of contradiction, that $v_i\neq v$ for all $i$. Then $v_i\in U^c$ for all $i$. Since $U^c$ is closed and $v_i\to v$, $v\in U^c$, which is a contradiction. Thus, $v$ is compact. Let $x\in C$
be such that $v\leq x\leq \infty$. If $\epsilon(x)=\infty$, then $x=\infty$. Suppose that $\epsilon(x)=v$. Since $x$ is connected to $v$ by the path $t\mapsto tx$, $t\in(0,1]$, we have $x\in U$. But $v$ is maximal in $U$. Thus, $x=v$. This proves not (i).
Not (i)$\Rightarrow$ not (ii): Suppose that there exist $w_1,w_2\in \mathrm{Idem}(C)$ such that $w_1<w_2$, $w_1$ is relatively compact in $w_2$, and there is no non-idempotent $x\in C$ such that $w_1\leq x\leq w_2$. By Zorn's lemma, we can choose $w_2$ minimal among the idempotents such that $w_1\leq w_2$ and $w_1\neq w_2$. Then
\begin{equation}\label{gap}
w_1\leq x\leq w_2\Rightarrow x\in \{w_1,w_2\}\hbox{ for all }x\in C.
\end{equation}
Let us show that $\{x\in C:x\leq w_2\}$ is disconnected. Let $U_{1}=\{x\in C:x\leq w_1\}$ and $U_2=\{x\in C:x\nleq w_1\}$. These sets are clearly disjoint, non-empty ($w_1\in U_1$ and $w_2\in U_2$), and cover $\{x\in C:x\leq w_2\}$. It is also clear that $U_2$ is open in $C$. Let's consider $U_1$. By \eqref{gap}, $x\in U_1$ if and only if $\epsilon(x)\leq w_1$ and $x\leq w_2$. Further, since $w_1$ is a compact idempotent in the extended Choquet cone $\{z\in C:z\leq w_2\}$, the set $U_1$ may be described as all $x$ in the cone $\{z\in C:z\leq w_2\}$ such that $w_1\gg \epsilon(x)$, where the relation $\gg$ is taken in the idempotent lattice of the cone $\{z\in C:z\leq w_2\}$. Thus, by Lemma \ref{opensetsupport} applied in the extended Choquet cone $\{z\in C:z\leq w_2\}$, the set $U_1$ is (relatively) open in $\{x\in C:x\leq w_2\}$.
Finally, let us argue that $x$ in (i) may be chosen such that $\epsilon(x)=w_1$: Starting from $w_1\leq w_2$, with $w_1$ relatively compact in $w_2$, choose $w_2'$ minimal element in $\{w\in \mathrm{Idem}(C):w_1\leq w\leq w_2,\, w\neq w_1\}$, which exists by Zorn's lemma. Let $x\in C$ be a non-idempotent such that $w_1\leq x\leq w_2'$. Then $\epsilon(x)\in \{w_1,w_2'\}$, but we cannot have $\epsilon(x)=w_2'$, since this entails that $x=w_2'$. So $\epsilon(x)=w_1$.
\end{proof}
\begin{definition} Let $C$ be an extended Choquet simplex. Let us say that $C$ is strongly connected if it satisfies either one of the equivalent properties listed in Theorem \ref{kernelpropertythm}.
\end{definition}
\begin{proposition}\label{coneslimits}
If $C$ is a projective limit of extended Choquet cones of the form $[0,\infty]^n$, then $C$ is strongly connected and has an abundance of compact idempotents.
\end{proposition}
\begin{proof}
Suppose that $C=\varprojlim \{C_i,\phi_{i,j}:i,j\in I\}$, where $C_i\cong [0,\infty]^{n_i}$ for all $i\in I$.
A projective limit of continua (compact Hausdorff connected spaces) is again a continuum. Since each $C_i$ is a continuum, so is $C$. In particular, $C$ is connected. If $w\in \mathrm{Idem}(C)$, with $w=(w_i)_i\in \prod_i C_i$, then
\[
\{x\in C:x\leq w\}=\varprojlim \{x\in C_i:x\leq w_i\}.
\]
Thus, the same argument shows that $\{x\in C:x\leq w\}$ is connected.
The lattice of idempotent elements of $C_i$ is finite, whence algebraic under the opposite order, for all $i$. Further, by additivity and continuity, the maps $\phi_{i,j}$ preserve directed infima and arbitrary suprema (i.e., directed suprema and arbitrary infima under the opposite order). That $\mathrm{Idem}(C)$ is algebraic under the opposite order can then be deduced from the fact that a projective limit of algebraic lattices is again an algebraic lattice, where the morphisms preserve directed suprema and arbitrary infima. Let us give a direct argument instead: Let $w\in \mathrm{Idem}(C)$, with $w=(w_i)_i\in \prod_i C_i$. For each index $k\in I$ define $w^{(k)} \in \prod_i C_i$ as the unique element in
$C$ such that
\[
(w^{(k)})_i=
\sup\{z\in C_i:\phi_{i,k}(z)=w_k\}\hbox{ for all }i\geq k.
\]
It is not hard to show that $(w^{(k)})_{k\in I}$ is a decreasing net in $\mathrm{Idem}(C)$ with infimum $w$. Moreover, from the compactness of $w_k\in \mathrm{Idem}(C_k)$ we deduce that $w^{(k)}\in \mathrm{Idem}(C)$ is compact for all $k\in I$. Thus, $\mathrm{Idem}(C)$
is an algebraic lattice under the opposite order.
\end{proof}
\section{Cones of traces and functionals}\label{sec:conesfromfunctionals}
Here we review various constructions giving rise to extended Choquet cones.
Let $A$ be a C*-algebra. Let $A_+$ denote the cone of positive elements of $A$. A map $\tau\colon A_+\to [0,\infty]$
is called a trace if it maps 0 to 0, it is additive, homogeneous with respect to scalar multiplication, and satisfies $\tau(x^*x)=\tau(xx^*)$ for all $x\in A$. The set of all lower semicontinuous traces on $A$ is denoted by $T(A)$. It is endowed with the pointwise operations of addition and scalar multiplication. $T(A)$ is endowed with the topology such that a net $(\tau_i)_i$ in $T(A)$ converges to $\tau\in T(A)$ if
\[
\limsup \tau_i((a-\epsilon)_+) \leq \tau(a)\leq \liminf \tau_i(a)
\]
for all $a\in A_+$ and $\epsilon>0$. By \cite[Theorems 3.3 and 3.7]{ERS}, $T(A)$ is an extended Choquet cone.
\begin{proposition}\label{cstarECC}
Let $A$ be a C*-algebra.
\begin{enumerate}[\rm (i)]
\item
If the primitive spectrum of $A$ has a basis of compact open sets, then $T(A)$ has an abundance of compact idempotents. In particular, this holds if $A$ has real rank zero.
\item
Suppose that for all $J\subsetneq I\subseteq A$, closed two-sided ideals of $A$ such that $I/J$ has compact primitive spectrum, there exists a non-zero lower semicontinuous densely finite trace on $I/J$. Then $T(A)$ is strongly connected. In particular, this holds if $A$ has stable rank one and is exact.
\end{enumerate}
\end{proposition}
\begin{proof}
(i) The lattice of closed two-sided ideals of $A$ is in order reversing bijection with the lattice of idempotents of $T(A)$ via the assignment $I\mapsto \tau_I$, where
\[
\tau_I(a):=\begin{cases}
0&\hbox{ for }a\in I_+,\\
\infty&\hbox{otherwise.}
\end{cases}
\]
On the other hand, the lattice of closed two-sided ideals of $A$ is isomorphic to the lattice of open sets of the primitive spectrum of $A$ (\cite[Theorem 4.1.3]{pedersen}). Thus, the lattice of idempotents of $T(A)$ is algebraic (under the opposite order) if and only if the lattice of open sets of the primitive spectrum is algebraic. The latter is equivalent to the existence of a basis of compact open sets for the topology.
(ii) Let us check that $T(A)$ satisfies condition (i) of Theorem \ref{kernelpropertythm}. Recall that idempotents in $T(A)$ have the form $\tau_I$, where $I$ is a closed two-sided ideal. Let $I$ and $J$ be (closed, two-sided) ideals of $A$, with $J\subseteq I$, so that $\tau_I\leq \tau_J$. The property that $\tau_I$ is compact relative to $\tau_J$ means that if $(I_i)_i$ is an upward directed net of ideals such that $J\subseteq I_i\subseteq I$ for all $i$ and $I=\bigcup I_i$, then $I=I_{i_0}$ for some $i_0$. This, in turn, is equivalent to $I/J$ having compact primitive spectrum. By assumption, there exists $\tau\in T(I/J)$ that is densely finite and non-zero. Pre-composed with the quotient map $\pi\colon I\to I/J$ (which maps $I_+$ onto $(I/J)_+$), $\tau$ gives rise to a trace $\tau\circ\pi\in T(I)$. Let $\tilde\tau$ be the extension of $\tau\circ\pi$ to $A_+$ such that $\tilde\tau(a)=\infty$ for all $a\in A_+\backslash I_+$. Then $\tau_I\leq \tilde \tau\leq \tau_J$ and $\tilde\tau$ is not an idempotent, as it attains values other than $\{0,\infty\}$. This proves that $T(A)$ is strongly connected.
Suppose now that $A$ has stable rank one and is exact. By the arguments from the previous paragraph, it suffices to show that if $I/J$ is a non-trivial ideal-quotient with compact primitive spectrum, then there is a nontrivial lower semicontinuous densely finite trace on $I/J$. Observe that $I/J$ has stable rank one and is exact, since both properties pass to ideals and quotients. An exact C*-algebra of stable rank one with compact primitive spectrum always has a nonzero, densely finite lower semicontinuous trace; see \cite[Theorem 2.15]{rordam-cones}.
\end{proof}
Let $(G,G_+)$ be a dimension group, i.e., an ordered abelian group that is unperforated and has the Riesz refinement property. Let $\mathrm{Hom}(G_+,[0,\infty])$ denote the set of all $[0,\infty]$-valued monoid morphisms on $G_+$ (i.e., $\lambda\colon G_+\to [0,\infty]$ additive and mapping 0 to 0). Endow $\mathrm{Hom}(G_+,[0,\infty])$ with pointwise cone operations and with the topology of pointwise convergence.
\begin{proposition}\label{HomG}
Let $G$ be a dimension group. Then $\mathrm{Hom}(G_+,[0,\infty])$ is an extended Choquet cone that is strongly connected and has an abundance of compact idempotents.
\end{proposition}
\begin{proof}
By \cite[Theorem 2.33]{wehrung}, $\mathrm{Hom}(G_+,[0,\infty])$ is a complete positively ordered monoid, which entails that
it is a complete lattice and that addition distributes over $\wedge$ and $\vee$. The topology on $\mathrm{Hom}(G_+,[0,\infty])$ is that induced by its inclusion in $[0,\infty]^{G_+}$. Since the latter is compact and Hausdorff, so is $\mathrm{Hom}(G_+,[0,\infty])$. Further, since $\mathrm{Hom}(G_+,[0,\infty])$ is a convex subset of $[0,\infty]^{G_+}$, the induced topology is locally convex. Thus, $\mathrm{Hom}(G_+,[0,\infty])$ is an extended Choquet cone. To see that it is strongly connected and has an abundance of compact idempotents, we can first express $(G,G_+)$ as an inductive limit of $(\mathbb{Z}^n,\mathbb{Z}_+^n)$ using the Effros-Handelmann-Shen theorem (\cite[Theorem 2.2]{EHS}), apply the functor $\mathrm{Hom}(\cdot,[0,\infty])$ to this limit, and then apply Proposition \ref{coneslimits}. We give a direct argument in the paragraphs below.
A subgroup $I\subseteq G$ is an order ideal if $I_+ :=G_+\cap I$ is a hereditary set and $I=I_+-I_+$. Idempotent elements of $\mathrm{Hom}(G_+,[0,\infty])$ have the form $\lambda_I(g)=0$ if $g\in I_+$ and $\lambda_I(g)=\infty$ if $g\in G_+\backslash I_+$, for some ideal $I$. Moreover, the map $I\mapsto \lambda_I$ is an order reversing bijection between the two lattices. It is well known that the lattice of ideals of an ordered group is algebraic. Thus, $\mathrm{Hom}(G_+,[0,\infty])$ has abundance of compact idempotents.
Let us now prove strong connectedness. Let $I,J\subseteq G$ be order ideals such that $J\subsetneq I$ and $\lambda_I$ is compact relative to $\lambda_J$. In this case, this means that $I/J$ is finitely (thus, singly) generated. Thus, it has a finite nonzero functional $\lambda\colon (I/J)_+\to [0,\infty)$ (e.g., by \cite[Theorem 1.4]{EHS}). As in the proof of Proposition \ref{cstarECC} (ii), we define a functional on all of $G_+$ by pre-composing $\lambda$ with the quotient map $I\mapsto I/J$ and setting it equal to $\infty$ on $G_+\backslash I_+$. This produces a functional $\tilde \lambda\in \mathrm{Hom}(G_+,[0,\infty])$ such that $\lambda_I\leq \tilde\lambda\leq \lambda_J$ and $\tilde\lambda$ is not an idempotent.
\end{proof}
Yet another construction yielding an extended Choquet cone is the dual of a Cu-semigroup. Let us first briefly recall the definition of a Cu-semigroup. Let $S$ be a positively ordered monoid. Given $x,y\in S$, let us write $x\ll y$ (read ``$x$ is way below $y$'') if whenever $(y_n)_{n=1}^\infty$ is an increasing sequence in $S$ such that $y\leq \sup_n y_n$, there exists $n_0$ such that $x\leq y_{n_0}$.
We call $S$ a Cu-semigroup if it satisfies the following axioms:
\begin{enumerate}
\item[O1.] For every increasing sequence $(x_n)_n$ in $S$, the supremum $\sup_{n}x_n$ exists.
\item[O2.] For every $x\in S$ there exists a sequence $(x_n)_n$ in $S$ such that $x_n\ll x_{n+1}$ for all $n\in\mathbb{N}$ and $x=\sup_n x_n$.
\item[O3.] If $(x_n)_n$ and $(y_n)_n$ are increasing sequences in $S$, then $\sup_n(x_n+y_n)=\sup_nx_n+\sup_n y_n$.
\item[O4.] If $x_i\ll y_i$ for $i=1,2$, then $x_1+x_2\ll y_1+y_2$.
\end{enumerate}
Observe that in our definition of the way-below relation above we only consider increasing sequences $(y_n)_n$, rather than increasing nets. In the context of Cu-semigroups we always use the symbol $\ll$ to indicate this sequential version of the way below relation.
Two additional conditions that we often impose on Cu-semigroups are the following:
\begin{enumerate}
\item[O5.] If $x'\ll x\le y$ then there exists $z$ such that $x'+z\le y\le x+z$.
\item[O6.] If $x,y,z\in S$ are such that $x\le y+z$, then for every $x'\ll x$ there are elements $y',z'\in S$ such that $x'\le y'+z'$, $y'\le x,y$ and $z'\le x,z$.
\end{enumerate}
An ordered monoid map $\lambda\colon S\to [0,\infty]$ is called a functional on $S$ if it preserves the suprema of increasing sequences. The collection of all functionals on $S$, denoted by $F(S)$, is a cone, with the cone operations defined pointwise. $F(S)$ is endowed with the topology such that a net $(\lambda_i)_{i\in I}$ in $F(S)$ converges to a functional $\lambda$ if
\[
\limsup_i \lambda_i(s')\leq \lambda(s) \leq \liminf_i \lambda_i(s)
\]
for all $s'\ll s$, in $S$. By \cite[Theorem 4.8]{ERS} and \cite[Theorem 4.1.2]{robertFS}, if $S$ is a Cu-semigroup satisfying O5 and O6, then $F(S)$ is an extended Choquet cone. In Section \ref{duality} we address the problem of what conditions on $S$ guarantee that $F(S)$ has an abundance of compact idempotents and is strongly connected.
\section{Functions on an extended Choquet cone}\label{functionspaces}
Throughout this section we let $C$ denote an extended Choquet cone with an abundance of compact idempotents, i.e., such that the lattice $(\mathrm{Idem}(C),\leq^{\mathrm{op}})$ is algebraic.
\subsection{The spaces $\mathrm{Lsc}(C)$ and $\mathrm{A}(C)$}
Let us denote by $\mathrm{Lsc}(C)$ the set of all functions $f\colon C\to [0,\infty]$ that are linear (additive, homogeneous with respect to scalar multiplication, and mapping 0 to 0) and lower semicontinuous ($f^{-1}((a,\infty])$ is open for any $a\in [0,\infty)$). The linearity of the functions in $\mathrm{Lsc}(C)$ implies that they are also order preserving, for if $x\leq y$ in $C$, then $y=x+z$ for some $z$, and so $f(y)=f(x)+f(z)\geq f(x)$. We endow $\mathrm{Lsc}(C)$ with the operations of pointwise addition and scalar multiplication, and with the pointwise order. $\mathrm{Lsc}(C)$ is thus an ordered cone.
Further, the pointwise supremum of functions in $\mathrm{Lsc}(C)$ is again in $\mathrm{Lsc}(C)$; thus, $\mathrm{Lsc}(C)$ is a directed complete
ordered set (dcpo).
Let us denote by $\mathrm{Lsc}_\sigma(C)$ the subset of $\mathrm{Lsc}(C)$ of functions $f\colon C\to [0,\infty]$ for which the set $f^{-1}((a,\infty])$ is $\sigma$-compact---in addition to being open---for all $a\in [0,\infty)$ (equivalently, for $a=1$, by linearity.) We denote by $\mathrm{A}(C)$ the functions in $\mathrm{Lsc}(C)$ that are continuous. Notice that $\mathrm{A}(C)\subseteq \mathrm{Lsc}_\sigma(C)$, since
\[
f^{-1}((a,\infty])=\bigcup_n f^{-1}([a+\frac1n,\infty]),
\]
and the right side is a union of closed (hence, compact) subsets of $C$.
Our goal is to show that every function in $\mathrm{Lsc}(C)$ ($\mathrm{Lsc}_\sigma(C)$) is the supremum of an increasing net (sequence) of functions in $\mathrm{A}(C)$. We achieve this in Theorem \ref{ACsuprema} after a number of preparatory results.
Given $f\in \mathrm{Lsc}(C)$, define its support $\mathrm{supp}(f)\in C$ as
\[
\mathrm{supp}(f)=\sup\{x\in C:f(x)=0\}.
\]
Since $f(x)=0\Rightarrow f(2x)=0$, it follows easily that $\mathrm{supp}(f)$ is an idempotent of $C$.
For each $w\in \mathrm{Idem}(C)$, let
\[
\chi_w(x)=
\begin{cases}
0 & \hbox{ if }x\leq w,\\
\infty & \hbox{otherwise.}
\end{cases}
\]
This is a function in $\mathrm{Lsc}(C)$.
\begin{lemma}\label{inftyfsupp}
We have $\infty\cdot f=\chi_{\mathrm{supp}(f)}$, for all $f\in \mathrm{Lsc}(C)$. (Here
$\infty \cdot f:=\sup_{n\in \mathbb{N}} nf$.)
\end{lemma}
\begin{proof}
The set $\{x\in C:f(x)=0\}$ is upward directed and converges to its supremum, i.e., to $\mathrm{supp}(f)$. It follows, by the lower semicontinuity of $f$, that $f(\mathrm{supp}(f))=0$.
If $x\leq \mathrm{supp}(f)$, then $f(x)\leq f(\mathrm{supp}(f))=0$. Hence, $(\infty \cdot f)(x)=0$.
If on the other hand $x\nleq \mathrm{supp}(f)$, then $f(x)\neq 0$, which implies that $(\infty \cdot f)(x)=\infty$.
We have thus shown that $\infty\cdot f=\chi_{\mathrm{supp}(f)}$.
\end{proof}
Let $w\in C$ be an idempotent. Define $\mathrm{A}_w(C)=\{f\in \mathrm{A}(C): \mathrm{supp}(f)=w\}$ and
\[
\mathrm{A}_{+}(C_w)=\{f\colon C_w\to [0,\infty):\hbox{ $f$ is continuous, linear, and $f(x)=0\Leftrightarrow x=w$}\}.
\]
(Recall that we have defined $C_w=\{x\in C:\epsilon(x)=w\}$.)
\begin{theorem}\label{ACwbijection}
If $f\in \mathrm{A}(C)$ then $\mathrm{supp}(f)$ is a compact idempotent. Further, given a compact idempotent $w\in \mathrm{Idem}(C)$, the restriction map $f\mapsto f|_{C_w}$ is an ordered cone isomorphism from $\mathrm{A}_w(C)$ to $\mathrm{A}_{+}(C_w)$.
\end{theorem}
\begin{proof}
Let $f\in \mathrm{A}(C)$. We have already seen that $\mathrm{supp}(f)$ is an idempotent. To prove its compactness, let
$(w_i)_{i\in I}$ be a downward directed family of idempotents with infimum $\mathrm{supp}(f)$. By the continuity of $f$, we have $\lim_i f(w_i)=f(\mathrm{supp}(f))=0$. But $f(w_i)\in \{0,\infty\}$ for all $i$. Therefore, there exists $i_0$ such that $f(w_i)=0$ for all $i\geq i_0$. But $\mathrm{supp}(f)$ is the largest element on which $f$ vanishes. Hence, $w_i=\mathrm{supp}(f)$ for all $i\geq i_0$. Thus, $\mathrm{supp}(f)$ is a compact idempotent.
Now, fix a compact idempotent $w$. Let $f\in \mathrm{A}_w(C)$. Clearly, $f$ is continuous and linear on $C_w$, and $f(w)=0$. Let $x\in C_w$. If $f(x)=0$, then $x\leq w$, which implies that $x=w$. Thus, $f(x)>0$ for all $x\in C_w\backslash\{w\}$. Suppose that $f(x)=\infty$. Then $f(w)=\lim_n f(\frac 1n x)=\infty$, contradicting that $w=\mathrm{supp}(f)$. Thus, $f(x)<\infty$ for all $x\in C_w$. We have thus shown that $f|_{C_w}\in \mathrm{A}_{+}(C_w)$.
It is clear that the restriction map $\mathrm{A}_w(C)\ni f\mapsto f|_{C_w}\in \mathrm{A}_+(C_w)$ is additive and order preserving. Let us show that it is an order embedding. Let $f,g\in \mathrm{A}_w(C)$ be such that $f|_{C_w}\leq g|_{C_w}$. Let $x\in C$. Suppose that $x+w\in C_w$. Then
\[
f(x)=f(x+w)\leq g(x+w)=g(x).
\]
If, on the other hand, $x+w\notin C_w$, then $\epsilon(x+w)>w$. Hence,
\[
f(x)=f(x+w)\geq f(\epsilon(x+w))=\infty.
\]
We argue similarly that $g(x)=\infty$. Thus, $f(x)=g(x)$.
Let us finally prove surjectivity. Suppose first that $C_w=\{w\}$. Then $\mathrm{A}_{+}(C_w)$ consists of the zero function only. Clearly then, $\chi_w|_{C_w}=0$ and $\mathrm{supp}(\chi_w)=w$. It remains to show that $\chi_w$ is continuous. The set $\chi^{-1}_w(\{\infty\})=\{x\in C:x\nleq w\}$ is open. On the other hand, $\chi_w^{-1}(\{0\})=\{x\in C:x\leq w\}$ agrees with $\{x\in C:\epsilon(x)\leq w\}$ (since we have assumed that $C_w=\{w\}$). The set $\{x\in C:\epsilon(x)\leq w\}$ is open by the compactness of $w$ (Lemma \ref{opensetsupport}). Thus, $\chi_w$ is continuous.
Suppose now that $C_w\neq \{w\}$. Let $\tilde f\in \mathrm{A}_{+}(C_w)$. Define $f\colon C\to [0,\infty]$ by
\[
f(x)=\begin{cases}
\tilde{f}(x+w) &\hbox{ if }x+w\in C_{w},\\
\infty&\hbox{otherwise.}
\end{cases}
\]
Observe that $f|_{C_w}=\tilde f$. Let us show that $f\in \mathrm{A}_w(C)$. To show that $\mathrm{supp}(f)=w$, note that
\[
f(x)=0\Leftrightarrow \tilde f(x+w)=0\Leftrightarrow x+w=w\Leftrightarrow x\leq w.
\]
Thus, $w$ is the largest element on which $f$ vanishes, i.e., $w=\mathrm{supp}(f)$. We leave the not difficult verification that $f$ is linear to the reader. Let us show that $f$ is continuous. Let $(x_i)_i$
be a net in $C$ with $x_i\to x$. Suppose first that $x+w\in C_w$, i.e, $\epsilon(x)\leq w$. Since the set
$\{y\in C:\epsilon(y)\leq w\}$ is open (Lemma \ref{opensetsupport}), $\epsilon(x_i)\leq w$ for large enough $i$. Therefore,
\[
\lim_i f(x_i)=\lim_i \tilde f(x_i+w)=\tilde f(x+w)=f(x).
\]
Now suppose that $x+w\notin C_w$, in which case $f(x)=\infty$. To show that $\lim_i f(x_i)=\infty$,
we may assume that $x_i\in C_w$ for all $i$ (otherwise $f(x_i)=\infty$ by definition). Observe also that $x_i\neq w$ for large enough $i$. Let us thus assume that $x_i\in C_w\backslash \{w\}$ for all $i$. Since $w$ is a compact idempotent, $C_{\omega}$ has a compact base $K\subseteq C_{w}\setminus\{w\}$ (Theorem \ref{compactbase}). Write $x_i=t_i\tilde{x}_i$ with $\tilde{x}_i\in K$ and $t_i>0$ for all $i$. Passing to a convergent subnet and relabelling, assume that $\tilde{x}_i\to y\in K$ and $t_i\to t\in [0,\infty]$. If $t<\infty$, then $x=\lim_it_i\tilde{x}_i=ty\in C_{w}$, contradicting our assumption that $x+w\notin C_{w}$. Hence $t=\infty$. Let $\delta>0$ be the minimum value of $\tilde{f}$ on the compact set $K$. Then
\[
f(x_i)=\tilde{f}(x_i)=t_if(\tilde{x}_i)\ge t_i\delta.
\]
Hence $f(x_i)\to\infty$, thus showing the continuity of $f$ at $x$.
\end{proof}
We will need the following theorem from \cite{edwards}:
\begin{theorem}[{\cite[Theorem 3.5]{edwards}}]\label{fromedwards}
Given $f,g\in \mathrm{Lsc}(C)$ there exists $f\wedge g$ and further,
\[
f\wedge \sup_i f_i=\sup_i(f\wedge f_i),
\]
for any upward directed set $(f_i)_{i\in I}$ in $\mathrm{Lsc}(C)$.
\end{theorem}
Recall that throughout this section $C$ denotes an extended Choquet cone with an abundance of compact idempotents.
\begin{theorem}\label{ACsuprema}
Every function in $\mathrm{Lsc}(C)$ is the supremum of an upward directed family of functions in $\mathrm{A}(C)$, and
every function in $\mathrm{Lsc}_\sigma(C)$ is the supremum of an increasing sequence in $\mathrm{A}(C)$.
\end{theorem}
\begin{proof}
Let $f\in \mathrm{Lsc}(C)$ and set $w=\mathrm{supp}(f)$. We first consider the case that $w$ is compact and then deal with the general case.
Assume that $w$ is compact. If $C_w=\{w\}$, then $f=\chi_w$. Further, $\chi_w$ is continuous, as shown in the proof of Theorem \ref{ACwbijection}. Suppose that $C_w\neq \{w\}$. Consider the restriction of $f$ to $C_w$. By \cite[Corollary I.1.4]{alfsen}, $f|_{C_w}$ is the supremum of an increasing net $(\tilde h_i)_{i}$ of linear continuous functions $\tilde h_i\colon C_w\to \mathbb{R}$. Since $f|_{C_w}$ is strictly positive on $C_{w}\backslash\{w\}$, it is separated from 0 on any compact base of $C_w$. It follows that the functions $\tilde h_i$ are eventually strictly positive on $C_{w}\backslash\{w\}$. Indeed, the sets $U_{i,\delta}=\tilde h_i^{-1}((\delta,\infty])\cap C_w$, where $i\in I$ and $\delta>0$, form an upward directed open cover of $C_w\backslash \{w\}$. Thus, for some $\delta>0$ and $i_0\in I$, $\tilde h_{i}$ is greater than $\delta$ on a (fixed) compact base of $C_w$ for all $i\geq i_0$. Let us thus assume that $\tilde h_i\in \mathrm{A}_{+}(C_w)$ for all $i$. By Theorem \ref{ACwbijection}, each $\tilde h_i$ has a unique continuous extension to an $h_i\in \mathrm{A}_w(C)$. Further, $(h_i)_i$ is also an increasing net. We claim that $f=\sup_i h_i$. Let us first show that $h_i\leq f$ for all $i$. Let $x\in C$ be such that $f(x)<\infty$. Then
\[
0=\lim_{n}\frac 1nf(x)=f(\epsilon(x))=0.
\]
Hence, $\epsilon(x)\leq w$, i.e., $x+w\in C_w$. We thus have that
\[
h_i(x)=\tilde h_i(x+w)\leq f(x+w)=f(x).
\]
Hence, $h_i\leq f$ for all $i$. Set $h=\sup_i h_i$. Clearly $h\leq f$. If $\epsilon(x)\leq w$ then
\[
h(x)=h(x+w)=\sup_i h_i(x+w)=f(x).
\]
If, on the other hand, $\epsilon(x)\nleq w$, then $h_i(x)=\infty$ for all $i$ and $h(x)=\infty=f(x)$. Thus, $h=f$.
Let us now consider the case when $w$ is not compact. Define
\[
H=\{h\in \mathrm{A}(C):h\leq (1-\epsilon)f\hbox{ for some }\epsilon>0\}.
\]
Let us show that $H$ is upward directed and has pointwise supremum $f$. Let $h_1,h_2\in H$. Set $v_1=\mathrm{supp}(h_1)$ and $v_2=\mathrm{supp}(h_2)$, which are compact idempotents, by Theorem \ref{ACwbijection}, and satisfy that $w\leq v_1,v_2$. Set $v=v_1\wedge v_2$, which is also compact and such that $w\leq v$. Set $g=f\wedge \chi_v$, which exists by Theorem \ref{fromedwards}. Since scalar multiplication by a non-negative scalar is an order isomorphism on $C$, we have $tg=(tf)\wedge \chi_v$. Letting $t\to \infty$ and using Theorem \ref{fromedwards}, we get $\infty\cdot g=\chi_w\wedge \chi_v=\chi_v$. Thus, $\mathrm{supp}(g)=v$ (Lemma \ref{inftyfsupp}). Let $\epsilon>0$ be such that $h_1,h_2\leq (1-\epsilon)f$. Then, $h_1,h_2\leq (1-\epsilon)g$. Since we have already established the case of compact support idempotent, there exists an increasing net $(g_i)_i$ in $\mathrm{A}_v(C)$ such that $g=\sup_i g_i$. By \cite[Proposition 5.1]{ERS}, $h_1,h_2\ll (1-\epsilon/2)g$ in the directed complete ordered set $\mathrm{Lsc}(C)$ (see also the definition of the relation $\lhd$ in the next section). Thus, there exists $i_0$ such that $h_1,h_2\leq (1-\epsilon/2)g_{i_0}$. Now $h=(1-\epsilon/2)g_{i_0}$ belongs to $H$ and satisfies that $h_1,h_2\leq h$. This shows that $H$ is upward directed.
Let us show that $f$ is the pointwise supremum of the functions in $H$. It suffices to show that $f$ is the supremum of functions in $\mathrm{A}(C)$, as we can then easily arrange for the $1-\epsilon$ separation. Choose a decreasing net
of compact idempotents $(v_i)_i$ with $w=\inf v_i$ (recall that $C$ has an abundance of compact idempotents). For each fixed $i$, $f\wedge \chi_{v_i}$ has support idempotent $v_i$, which is compact. Thus, as demonstrated above, $f\wedge \chi_{v_i}$ is the supremum of an increasing net in $\mathrm{A}(C)$. But $f=\sup_i f\wedge \chi_{v_i}$ (Theorem \ref{fromedwards}). It follows that $f$ is the pointwise supremum of functions in $\mathrm{A}(C)$.
Finally, suppose that $f\in \mathrm{Lsc}_\sigma(C)$, and let us show that there is a countable set in $H$ with pointwise supremum $f$. For each $h\in H$, let $U_{h}=h^{-1}((1,\infty])$. The sets $(U_{h})_{h\in H}$ form an open cover of $f^{-1}((1,\infty])$. Since the latter is $\sigma$-compact, we can choose a countable set $H'\subseteq H$ such that $(U_{h})_{h\in H'}$ is also a cover of $f^{-1}((1,\infty])$. Observe that for each $x\in C$, $f(x)>1$ if and only if $h(x)>1$ for some $h\in H'$. It follows, by the homogeneity with respect to scalar multiplication of these functions, that $\sup_{h\in H'}h(x)=f(x)$ for all $x\in C$. Now using that $H$ is upward directed we can construct an increasing sequence with supremum $f$.
\end{proof}
\begin{theorem}\label{metrizableC}
Let $C$ be a metrizable extended Choquet cone with an abundance of compact idempotents. Then
there exists a countable subset of $\mathrm{A}(C)$ such that every function in $\mathrm{Lsc}(C)$ is the supremum
of an increasing sequence of functions in this set.
\end{theorem}
\begin{proof}
Let us first argue that the set of compact idempotents is countable. Let $(U_{i})_{i=1}^\infty$ be a countable basis for the topology of $C$. Let $w\in \mathrm{Idem}_c(C)$ be a compact
idempotent. Since $\{x\in C:w\leq x\}$ is an open set, by Lemma \ref{opensetsupport}, there exists $U_i$ such that $w\in U_i\subseteq \{x\in C:w\leq x\}$. Clearly then $w=\inf U_i$. Thus, the set of compact idempotents embeds in the countable set $\{\inf U_i:i=1,2,\ldots\}$.
Now fix a compact idempotent $w$. Recall that $\mathrm{A}_w(C)$ is isomorphic to the cone $\mathrm{A}_+(C_w)$ of positive linear functions on the cone $C_w$. Suppose that $C_w\neq \{w\}$. Let $K$ denote a compact base of $C_w$, which exists by Theorem \ref{compactbase}, and is metrizable since $C$ is metrizable by assumption. Then $\mathrm{A}_+(C_w)$ is separable in the metric induced by the uniform norm on $K$, since it embeds in $C(K)$, which is separable. Let $\tilde B_w\subseteq \mathrm{A}_+(C_w)$ be a countable dense subset. It is not hard now to express any function in $\mathrm{A}_+(C_w)$ as the supremum of an increasing sequence in $\tilde B_w$. Indeed, it suffices to show that for any $\epsilon>0$ and $f\in \mathrm{A}_{+}(C_w)$, there exists $g\in \tilde B_w$ such that $(1-\epsilon)f\leq g\leq f$. Keeping in mind that $f$ is separated from 0 on $K$, we can choose $g\in\tilde B_w$ such that
\[
\Big\|(1-\frac\epsilon{2})f|_K-g|_K\Big \|_{\infty}<\frac\epsilon{2} \min_{x\in K}|f(x)|.
\]
Then $g$ is as desired. Let $B_w\subseteq \mathrm{A}_w(C)$ be the set mapping bijectively onto $\tilde B_w\subseteq \mathrm{A}_+(C_w)$ via the restriction map. By Theorem \ref{ACwbijection}, every function in $\mathrm{A}_w(C)$ is the supremum of an increasing sequence in $B_w$. If, on the other hand, $C_w=\{w\}$, then
$\mathrm{A}_w(C)=\{\chi_w\}$. In this case we set $B_w=\{\chi_w\}$.
Let $B=\bigcup_w B_w$, where $w$ ranges through the set of compact idempotents, and $B_w$ is as in the previous paragraph. Observe that $B$ is countable. Let us show that every function in $f\in \mathrm{Lsc}(C)$ is the supremum of an increasing sequence in $B$. Observe that $\mathrm{Lsc}(C)=\mathrm{Lsc}_\sigma(C)$, since all open subsets of a compact metric space are $\sigma$-compact. Thus, $f=\sup_n h_n$, where $(h_n)_{n=1}^\infty$ is an increasing sequence in $\mathrm{A}(C)$. The sequence $h_n'=(1-\frac 1n)h_n$ is also increasing, with supremum $f$, and $h_{n}'\ll h_{n+1}'$
in the directed complete ordered set $\mathrm{Lsc}(C)$ (see \cite[Proposition 5.1]{ERS} and also the definition of the relation $\lhd$ in the next section). Say $h_{n+1}'\in \mathrm{A}_{w_n}(C)$ for some compact idempotent $w_n$. Since $h_{n+1}'$
is the supremum of a sequence in $B_{w_n}$, we can choose $g_n\in B_{w_n}$ such that $h_n'\leq g_n\leq h_{n+1}'$. Then $(g_n)_{n=1}^\infty$ is an increasing sequence in $B$ with supremum $f$.
\end{proof}
\section{Duality with Cu-cones}\label{duality}
By a Cu-cone we understand a Cu-semigroup $S$ that is also a cone, i.e., it is endowed with a scalar multiplication by $(0,\infty)$ compatible with the monoid structure of $S$; see Section \ref{sectionECCs}. Further we ask that
\begin{enumerate}
\item
$t_1\leq t_2$ and $s_1\leq s_2$ imply $t_1s_1\leq t_2s_2$ for all $t_1,t_2\in (0,\infty)$ and $s_1,s_2\in S$,
\item
$\sup_n t_ns_n=(\sup_n t_n)(\sup_n s_n)$ where $(t_n)_{n=1}^\infty$ and $(s_n)_{n=1}^\infty$ are increasing sequences in $(0,\infty)$ and $S$, respectively.
\end{enumerate}
Cu-cones are called Cu-semigroups with real multiplication in \cite{robertFS}. They are also Cu-semimodules over the Cu-semiring $[0,\infty]$, in the sense of \cite{tensorthiel}.
In this section we prove a duality between extended Choquet cones with an abundance of compact idempotents and certain Cu-cones. Throughout this section, $S$ denotes a Cu-cone satisfying the axioms O5 and O6, so that $F(S)$ is an extended Choquet cone.
Let us recall the relation $\lhd$ in $\mathrm{Lsc}(C)$ defined in \cite{ERS}: Given $f,g\in \mathrm{Lsc}(C)$, we write $f\lhd g$ if $f\le (1-\varepsilon)g$ for some $\varepsilon>0$ and $f$ is continuous at each $x\in C$ such that $g(x)<\infty$.
By \cite[Proposition 5.1]{ERS}, $f\lhd g$ implies that $f$ is way below $g$ in the dcpo $\mathrm{Lsc}(C)$, meaning that for any upward directed net $(g_i)_i$ such that $g\leq \sup g_i$, there exists $i_0$ such that $f\leq g_{i_0}$.
\begin{lemma}({Cf. \cite[Lemma 3.3.2]{robertFS}})\label{lhdll}
Let $f,g\in \mathrm{Lsc}(C)$ be such that $f\lhd g$. Then here exists $h\in \mathrm{Lsc}(C)$ such that $f+h=g$ and $h\geq \epsilon g$
for some $\epsilon>0$. Moreover, if $f,g\in \mathrm{Lsc}_\sigma(C)$, then $h$ may be chosen in $\mathrm{Lsc}_\sigma(C)$, and if $f,g\in \mathrm{A}(C)$, then $h$ may be chosen in $\mathrm{A}(C)$.
\end{lemma}
\begin{proof}
Define $h\colon C\to [0,\infty]$ by
\[
h(x) = \begin{cases}
g(x) - f(x) & \hbox{ if }g(x)<\infty,\\
\infty&\hbox{ otherwise}.
\end{cases}
\]
Then $f+h=g$. The linearity of $h$ follows from a straightforward analysis. Since $f\lhd g$, there exists $\epsilon>0$ such that $f\leq (1-\epsilon)g$. Then $g(y)-f(y)\geq \epsilon g(y)$ whenever $g(y)<\infty$, while if $g(y)=\infty$ then $g(y)=\infty=h(y)$. This establishes that $h\geq \epsilon g$.
The proof of \cite[Lemma 3.3.2]{robertFS} establishes the lower semicontinuity of $h$. Let us recall it here: Let $(x_i)_i$ be a net in $C$ such that $x_i\to x$. Suppose first that $g(x)<\infty$. Then $f(x)<\infty$, and by the continuity of $f$ at $x$, $f(x_i)<\infty$ for large enough $i$. Then,
\[
\liminf_i h(x_i)\geq \liminf_i g(x_i)-f(x_i)\geq g(x)-f(x)=h(x).
\]
Suppose now that $g(x)=\infty$, so that $h(x)=\infty$. Since $h\geq \epsilon g$,
\[
\liminf_i h(x_i)\geq \epsilon \liminf_i g(x_i)\geq \epsilon g(x)=\infty,
\]
thus showing lower semicontinuity at $x$.
Assume now that $f,g\in \mathrm{Lsc}_\sigma(C)$. It is not difficult to show that $h(x)>1$ if and only if $g(x)>1/\epsilon$ or $g(x)>1+r$ and $f(x)\leq r$ for some $r\in \mathbb{Q}$. Thus,
\[
h^{-1}((1,\infty])=g^{-1}((1/\epsilon,\infty])\cup \bigcup_{r\in \mathbb{Q}}g^{-1}((1+r,\infty])\cap f^{-1}([0,r]).
\]
The right side is $\sigma$-compact. Hence, $h\in \mathrm{Lsc}_\sigma(C)$.
Assume now that $f,g\in \mathrm{A}(C)$. Continuity at $x\in C$ such that $h(x)=\infty$ follows automatically from lower semicontinuity. Let $x\in C$ be such that $h(x)<\infty$, i.e., $g(x)<\infty$. If $x_i\to x$ then $g(x_i)<\infty$
and $f(x_i)<\infty$ for large enough $i$. Then
\[
h(x_i)=g(x_i)-f(x_i)\to g(x)-f(x)=h(x),\]
where we used the continuity of $g$ and $f$. Thus, $h$ is continuous at $x$.
\end{proof}
By an ideal of a Cu-cone we understand a subcone that is closed under the suprema of increasing sequence. There is an order reversing bijection between the ideals of $S$ and the idempotents of $F(S)$:
\[
I\mapsto \lambda_I(x):=\begin{cases}
0&\hbox{if }x\in I\\
\infty&\hbox{otherwise,}
\end{cases}
\]
where $I$ ranges through the ideals of $S$.
Let us say that a Cu-cone $S$ has an abundance of compact ideals if the lattice of ideals of $S$ is algebraic, i.e., every ideal of $S$ is a supremum of compact ideals.
\begin{theorem}\label{dualitythm}
Let $S$ be a Cu-cone satisfying O5 and O6 and having an abundance of compact ideals. Then $F(S)$ is an extended Choquet cone with an abundance of compact idempotents. Moreover, $S\cong \mathrm{Lsc}_\sigma(F(S))$ via the assignment
\[
S\ni s\mapsto \hat s\in \mathrm{Lsc}_\sigma(F(S)),
\]
where $\hat{s}(\lambda):=\lambda(s)$ for all $\lambda\in F(S)$.
Let $C$ be an extended Choquet cone with an abundance of compact idempotents. Then $\mathrm{Lsc}_\sigma(C)$ is a Cu-cone satisfying O5 and O6 and having an abundance of compact ideals. Moreover, $C\cong F(\mathrm{Lsc}_\sigma(C))$ via the assignment
\[
C\ni x\mapsto \hat x\in F(\mathrm{Lsc}_\sigma(C)),
\]
where $\hat x(f):=f(x)$ for all $f\in \mathrm{Lsc}_\sigma(C)$.
\end{theorem}
\begin{proof}
As recalled in Section \ref{sec:conesfromfunctionals}, by the results of \cite{robertFS}, $F(S)$ is an extended Choquet cone. The bijection between the ideals of $S$ and the idempotents of $F(S)$ translates the abundance of compact ideals of $S$ directly into the abundance of compact idempotents of $F(S)$. By \cite[Theorem 3.2.1]{robertFS}, the mapping
\[
S\ni s\mapsto \hat s\in \mathrm{Lsc}(F(S))
\]
is an isomorphism of the Cu-cone $S$ onto the space of functions $f\in \mathrm{Lsc}(F(S))$ expressible as the pointwise supremum of an increasing sequence $(h_n)_{n=1}^\infty$ in $\mathrm{Lsc}(F(S))$ such that $h_n\lhd h_{n+1}$ for all $n$. The set of all such functions is denoted by $L(F(S))$ in \cite{robertFS}. Let us show that, under our present assumptions, $L(F(S))=\mathrm{Lsc}_\sigma(F(S))$. Let $f\in \mathrm{Lsc}(F(S))$ be such that $f=\sup h_n$, where $h_n\lhd h_{n+1}$ for all $n$. We have $\overline{h_n^{-1}((1,\infty])}\subseteq f^{-1}((1,\infty])$ for all $n$ (\cite[Proposition 5.1]{ERS}). Hence,
\[
f^{-1}((1,\infty])=\bigcup_n \overline{h_n^{-1}((1,\infty])}.
\]
Thus, $f\in\mathrm{Lsc}_\sigma(F(S))$. Suppose, on the other hand, that $f\in \mathrm{Lsc}_\sigma(F(S))$.
Then, by Theorem \ref{ACsuprema}, there exists an increasing sequence $(h_n)_{n=1}^\infty$ in $\mathrm{A}(F(S))$ with supremum $f$. Clearly, $h_n'=(1-\frac1n)h_n$ is also increasing, has supremum $f$, and $h_n'\lhd h_{n+1}'$ for all $n$. Hence, $f\in L(F(S))$.
Let's turn now to the second part of the theorem. Let $C$ be an extended Choquet cone with an abundance of compact idempotents. Let us show that $\mathrm{Lsc}_\sigma(C)$ satisfies all axioms O1-O6 (Section \ref{sec:conesfromfunctionals}). Let us show first that $\mathrm{Lsc}_\sigma(C)$ is closed under the suprema of increasing sequences: Let $f=\sup_n f_n$, with $(f_n)_{n=1}^\infty$ an increasing sequence in $\mathrm{Lsc}_\sigma(C)$. Then $f^{-1}((1,\infty])=\bigcup_{n=1}^\infty f_n^{-1}((1,\infty])$. Since the sets on the right side are $\sigma$-compact, so is the left side. Thus, $f\in \mathrm{Lsc}_\sigma(C)$.
Let $f\in \mathrm{Lsc}_\sigma(C)$, and let $(h_n)_{n=1}^\infty$ be an increasing in $\mathrm{A}(C)$ with supremum $f$. Then $h_n'=(1-\frac1n)h_n$ has supremum $f$ and $h_n'\ll h_{n+1}'$ for all $n$ (since $h_n'\lhd h_{n+1}'$). This proves O2. Axiom O3 follows at once from the fact that suprema in $\mathrm{Lsc}_\sigma(C)$ are taken pointwise. Suppose that $f_1\ll g_1$ and $f_2\ll g_2$. Choose $h_1,h_2\in \mathrm{A}(C)$ such that $f_i\leq h_i\lhd g_i$ for $i=1,2$. Then $f_1+f_2\leq h_1+h_2\lhd g_1+g_2$, from which we deduce O4.
Let's prove O5: Suppose that $f',f,g\in \mathrm{Lsc}_\sigma(C)$ are such that $f'\ll f\leq g$. Choose $h\in \mathrm{A}(C)$ such that $f'\leq h\lhd f$. By Lemma \ref{lhdll}, there exists $h'\in \mathrm{Lsc}_\sigma(C)$ such that $h+h'=g$. Then, $f'+h'\leq g\leq f+h'$, proving O5.
Let us prove O6. We prove the stronger property that $\mathrm{Lsc}_\sigma(C)$ is inf-semilattice ordered, i.e., pairwise infima exist and addition distributes over infima. Recall that, by the results of \cite{edwards}, $\mathrm{Lsc}(C)$ is inf-semilattice ordered (see Theorem \ref{fromedwards}). Let us show that if $f,g\in \mathrm{Lsc}_\sigma(C)$, then $f\wedge g$ is also in $\mathrm{Lsc}_\sigma(C)$. By \cite[Lemma 3.4]{edwards}, for every $x\in C$ there exist $x_1,x_2\in C$, with $x_1+x_2=x$, such that $(f\wedge g)(x)=f(x_1)+g(x_2)$. It is then clear that
\[
(f\wedge g)^{-1}((1,\infty])=\bigcup_{\substack{a_1,a_2\in \mathbb{Q}, \\ a_1+a_2>1} }
f^{-1}((a_1,\infty])\cup g^{-1}((a_2,\infty]).
\]
Since the right side is a $\sigma$-compact set, $f\wedge g\in \mathrm{Lsc}_\sigma(C)$. To verify O6, suppose that $f\leq g_1+g_2$, with $f,g_1,g_2\in \mathrm{Lsc}_\sigma(C)$. Then, using the distributivity of addition over $\wedge$, $f\leq g_1+g_2\wedge f$, which proves O6.
Finally, let us prove that $C\ni x\mapsto \hat x\in F(\mathrm{Lsc}_\sigma(C))$ is an isomorphism of extended Choquet cones.
We consider injectivity first: Let $x,y\in C$ be such that $f(x)=f(y)$ for all $f\in \mathrm{Lsc}_\sigma(C)$. Choose $f\in \mathrm{A}(C)$. Passing to the limit as $n\to \infty$ in $f(\frac1n x)=f(\frac 1ny)$ we deduce that $f(\epsilon(x))=f(\epsilon(y))$ for all $f\in \mathrm{A}(C)$.
Since every function in $\mathrm{Lsc}(C)$ is the supremum of a directed net of functions in $\mathrm{A}(C)$, we have that $f(\epsilon(x))=f(\epsilon(y))$ for all $f\in \mathrm{Lsc}(C)$. Now choosing $f=\chi_w$, for $w\in \mathrm{Idem}(C)$, we conclude that $\epsilon(x)=\epsilon(y)$, i.e., $x$ and $y$ have the same support idempotent. Set $w=\epsilon(x)=\epsilon(y)$. Choose a compact idempotent $v$ such that $w\leq v$. Then $x+v,y+v\in C_v$, and $f(x+y)=f(y+v)$ for all $f\in \mathrm{A}(C)$. By Theorem \ref{ACwbijection},
$f(x+v)=f(y+v)$ for all $f\in \mathrm{A}_+(C_v)$. Recall that $C_v$ has a compact base and embeds in a locally convex Hausdorff vector space $V_v$ (Theorem \ref{compactbase}). We have $f(x+v)=f(y+v)$ for all $f\in \mathrm{A}_+(C_v)-\mathrm{A}_+(C_v)$. But $\mathrm{A}_+(C_v)-\mathrm{A}_+(C_v)$ consists of all the affine functions on $C_v$ that vanish at the origin. Thus, $f(x+v)=f(y+v)$
for all such functions, and in particular, for all continuous functionals on $V_v$. Since the weak topology on $V_v$
is Hausforff, $x+v=y+v$. Passing to the infimum over all compact idempotents $v$ such that $w\leq v$,
and using that $C$ has an abundance of compact idempotents, we conclude that $x=x+w=y+w=y$. Thus, the map $x\mapsto \hat x$ is injective.
Let us prove continuity of the map $x\mapsto \hat x$. Let $(x_i)_i$ be a net in $C$ with $x_i\to x$. Let $f',f\in \mathrm{Lsc}_\sigma(C)$, with $f'\ll f$. By the lower semicontinuity of $f$, we have
\[
\hat x(f)=f(x)\leq \liminf_i f(x_i)=\liminf_i \hat x_i(f).
\]
Choose $h\in \mathrm{A}(C)$ such that $f'\leq h\leq f$, which is possible since $f$ is supremum of an increasing sequence in $\mathrm{A}(C)$. Then
\[
\limsup_i \hat x_i(f')\leq \limsup_i \hat x_i(h)=\limsup_i h(x_i)=h(x)\leq f(x)=\hat x(f).
\]
This shows that $\hat x_i\to \hat x$ in the topology of $F(\mathrm{Lsc}_\sigma(C))$.
Let us prove surjectivity of the map $x\mapsto \hat x$. (Linearity is straightforward; continuity of the inverse is automatic from the fact that the cones are compact and Hausdorff.) The range of the map $x\mapsto \hat x$ is a compact subcone of $F(\mathrm{Lsc}_\sigma(C))$ that separates elements of $\mathrm{Lsc}_\sigma(C)$ and contains 0. By the separation theorem \cite[Corollary 4.6]{ultraCu}, it must be all of $F(\mathrm{Lsc}_\sigma(C))$.
\end{proof}
Let $S$ be a Cu-cone. We say that $S$ has weak cancellation if $x+z\ll y+z$ implies $x\ll y$ for all $x,y,z\in S$.
\begin{lemma}
Let $C$ be an extended Choquet cone. Let $h,h',g\in \mathrm{Lsc}(C)$ be such that $h\lhd g+h'$ and $h'\lhd h$.
Then $\mathrm{supp}(g+h')$ is relatively compact in $\mathrm{supp}(g)$.
\end{lemma}
\begin{proof}
Set $w_1=\mathrm{supp}(g+h')$ and $w_2=\mathrm{supp}(g)$. Let $(v_i)_i$ be a downward directed net of idempotents with $\bigwedge_i v_i\leq w_1$. Then the functions $(\chi_{v_i})_i$ form an upward directed net such that
$g+h'\leq \chi_{w_1}\leq \sup_i \chi_{v_i}$. Since $h\lhd g+h'$, there exists $i_0$ such that $h\leq \chi_{v_{i_0}}$. We have that
\[
g+h'\leq g+h\leq \chi_{w_2}+\chi_{v_{i_0}}=\chi_{w_2\wedge v_{i_0}}.
\]
Hence, $w_2\wedge v_{i_0}\leq w_1$, which proves the lemma.
\end{proof}
\begin{theorem}\label{weakcancellation}
Let $C$ be an extended Choquet cone with an abundance of compact idempotents. Then $C$ is strongly connected if and only if $\mathrm{Lsc}_\sigma(C)$ has weak cancellation.
\end{theorem}
\begin{proof}
Suppose first that $C$ is strongly connected. Let $f,g,h\in \mathrm{Lsc}_\sigma(C)$ be such that $f+h\ll g+h$.
Choose $\lhd$-increasing sequences $(g_n)_{n=1}^\infty$ and $(h_n)_{n=1}^\infty$ in $\mathrm{A}(C)$ such that $g=\sup_n g_n$ and $h=\sup_n h_n$. Then $f+h\ll g_m+h_m$ for some $m$. We will be done once we have shown that $f\leq g_m$.
Let $x\in C$. If $g_m(x)=\infty$, then indeed $f(x)\leq \infty=g_m(x)$. Suppose that $g_m(x)<\infty$. If $h_m(x)<\infty$, then we can cancel $h_m(x)$ in $f(x)+h_m(x)\leq g_m(x)+h_m(x)$ to obtain the desired $f(x)\leq g_m(x)$. It thus suffices to show that $g_m(x)<\infty$ implies $h_m(x)<\infty$, i.e., that $\mathrm{supp}(g_m)\le \mathrm{supp}(h_m)$. Let $w_1=\mathrm{supp}(g_m+h_m)$ and $w_2=\mathrm{supp}(g_m)$. Then $w_1\le w_2$ and $w_1$ is relatively compact in $w_2$, by the previous lemma. Suppose for the sake of contradiction that $w_1\ne w_2$. By strong connectedness, there exists $x\in C$ such that $w_1\le x\le w_2$, with $\epsilon(x)=w_1$ and $x\ne w_1$. Then,
\begin{align*}
h(x) &\leq g_m(x)+h_m(x)\\
&=h_m(x)\le (1-\delta)h(x),
\end{align*}
for some $\delta>0$. Hence, $h(x)\in\{0,\infty\}$. If $h(x)=0$, then $h_m(x)=g_m(x)=0$, while if $h(x)=\infty$, then
$g_m(x)+h_m(x)\geq h(x)=\infty$. In either case, we get a contradiction with $0<(g_m+h_m)(x)<\infty$, which holds by Theorem \ref{ACwbijection}. Hence, $w_1=w_2$. We thus have that $\mathrm{supp}(g_m)=\mathrm{supp}(g_m+h_m)\le \mathrm{supp}(h_m)$.
Suppose conversely that $\mathrm{Lsc}_\sigma(C)$ has weak cancellation. Let $w_1\leq w_2$ be idempotents in $C$, with $w_1$ relatively compact in $w_2$, and $w_1\neq w_2$. Further, using Zorn's lemma, choose $w_2$ minimal such that $w_1\neq w_2$ and $w_1$ is relatively compact in $w_2$. Suppose for the sake of contradiction that $w_1\leq x\leq w_2$ implies $x\in \{w_1,w_2\}$. Let $D=\{x\in C:x\leq w_2\}$. Then $D$ is an extended Choquet cone and $w_1$ is a compact idempotent in $D$. Further, $D_{w_1}=\{w_1\}$. So, as shown in the course of the proof of Theorem \ref{ACwbijection}, $\chi_{w_1}|_D$ is continuous on $D$. Let $(h_i)_i\in \mathrm{A}(C)$ be an upward directed net with supremum $\chi_{w_1}$. Since $\chi_{w_1}|_D\lhd \chi_{w_1}|_D$, there exists $i$
such that $\chi_{w_1}|_D\leq h_i|_D$. It follows that $\chi_{w_1}\leq h_i+\chi_{w_2}$ (as functions on $C$).
Fix an index $j\geq i$. Then
\[
3h_j\lhd \chi_{w_1}\leq h_i+\chi_{w_2}.
\]
Now let $(l_k)_k$ be an upward directed net in $\mathrm{A}(C)$ with supremum $\chi_{w_2}$. Then there exists an index $k$ such that $3h_j\leq h_i+l_k$. Observe that $h_i\lhd 2h_k$. By weak cancellation in $\mathrm{Lsc}_\sigma(C)$, we conclude that $h_j\leq l_k$. (Note: we have used weak cancellation in the form $f+h\leq g+h'$ and $h'\ll h$ imply $f\leq g$.) Thus, $h_j\leq \chi_{w_2}$ for all $j\geq i$, implying that $\chi_{w_1}\leq \chi_{w_2}$. This contradicts that $w_1\neq w_2$.
\end{proof}
In the following section we will make use of the following form of Riesz decomposition:
\begin{theorem}\label{ACRiesz}
Let $C$ be an extended Choquet cone that is strongly connected and has an abundance of compact idempotents. Let $f,g_1,g_2\in \mathrm{A}(C)$ be such that $f\lhd g_1+g_2$. Then there exist $f_1,f_2\in \mathrm{A}(C)$ such that $f=f_1+f_2$, $f_1\lhd g_1$, and $f_2\lhd g_2$.
\end{theorem}
\begin{proof}
Let $\epsilon>0$ be such that $f\le (1-\varepsilon)g_1+(1-\varepsilon)g_2$. Then, using the distributivity of addition over $\wedge$,
\begin{align*}
f\le f\wedge ((1-\varepsilon)g_1)+(1-\varepsilon)g_2=(1-\varepsilon)( (f\wedge g_1)+g_2).
\end{align*}
Thus, $f\lhd (f\wedge g_1)+g_2$ (recall that $f$ is continuous). By Theorem \ref{ACsuprema}, $f\wedge g_1$ is the supremum of a net of functions in $\mathrm{A}(C)$. Thus, there exists $h\in \mathrm{A}(C)$ such that
$f\lhd h+g_2$ and $h\lhd (f\wedge g_1)$. By Lemma \ref{lhdll}, we can find $l\in \mathrm{A}(C)$ such that $f=h+l$.
Then $h+l\lhd h+g_2$. By weak cancellation in $\mathrm{Lsc}_\sigma(C)$ (Theorem \ref{weakcancellation}), we have that $l\lhd g_2$. Setting $f_1=h$ and $f_2=l$ yields the desired result.
\end{proof}
\section{Proof of Theorem \ref{mainchar}}\label{proofofmainchar}
Throughout this section $C$ denotes an extended Choquet cone that is strongly connected and has an abundance of compact idempotents.
\subsection{The triangle lemma}
To prove Theorem \ref{mainchar} we follow a strategy similar to the proof of the Effros-Handelman-Shen theorem (\cite{EHS}). The key step in this proof is establishing a ``triangle lemma'', Theorem \ref{triangletheorem} below.
\begin{lemma}\label{Cugenerators}
A linear map $\phi\colon [0,\infty]\to \mathrm{Lsc}_\sigma(C)$ is a Cu-morphism if and only if $\phi(\infty)=\infty \cdot \phi(1)$ and $\phi(1)\in \mathrm{A}(C)$.
\end{lemma}
\begin{proof}
Suppose that $\phi$ is a Cu-morphism. That $\phi(\infty)=\infty \cdot \phi(1)$ follows at once from $\phi$ being
supremum preserving and additive. Set $f=\phi(1)$. To prove the continuity of $f$, it suffices to show that it is upper semicontinuous, since it is already lower semicontinuous by assumption. Fix $\epsilon>0$. Since $1-\epsilon \ll 1$ in $[0,\infty]$,
we have $(1-\epsilon)f \ll f$ in $\mathrm{Lsc}_\sigma(C)$. Choose $g\in \mathrm{A}(C)$ such that $(1-\epsilon)f\leq g\leq f$. Let $x_i\to x$ be a convergent net in $C$. Then,
\[
(1-\epsilon)\limsup_i f(x_i)\leq \limsup g(x_i)=g(x)\leq f(x).
\]
Letting $\epsilon\to 0$, we get that $\limsup f(x_i)\leq f(x)$. Thus, $f$ is upper semicontinuous.
Conversely, suppose that $\phi(1)\in \mathrm{A}(C)$ and $\phi(\infty)=\infty \cdot \phi(1)$. Observe that if $f\in \mathrm{A}(C)$ then
$\alpha f\lhd \beta f$ for all scalars $0\leq \alpha<\beta\leq \infty$. Hence, $\phi(\alpha)\ll \phi(\beta)$ in $\mathrm{Lsc}_\sigma(C)$ whenever $\alpha\ll \beta$ in $[0,\infty]$, i.e., $\phi$ preserves the way below relation. The rest of the properties of $\phi$ are readily verified.
\end{proof}
The core of the proof of Theorem \ref{triangletheorem} (the ``triangle lemma") is contained in the following lemma:
\begin{lemma}\label{coretrianglelemma}
Let $\phi\colon [0,\infty]^n\to \mathrm{Lsc}_\sigma(C)$ be a Cu-morphism. Let $x,y\in [0,\infty)^n\cap \mathbb{Z}^n$ be such that $\phi(x)\ll \phi(y)$. Then there exist $N\in \mathbb{N}$ and Cu-morphisms
\[
[0,\infty]^n \stackrel{Q}{\longrightarrow} [0,\infty]^N\stackrel{\psi}{\longrightarrow} \mathrm{Lsc}_\sigma(C),
\]
such that $\psi Q=\phi$ and $Qx\leq Qy$. Moreover, $Q$ maps $[0,\infty)^n\cap \mathbb{Z}^n$ to $[0,\infty)^N\cap \mathbb{Z}^N$
\end{lemma}
\begin{proof}
Let $x=(x_1,\ldots,x_n)$, $y=(y_1,\ldots,y_n)$, and $\phi$ be as in the statement of the lemma. Let $(E_i)_{i=1}^n$ denote the canonical basis of $[0,\infty]^n$. Set $f_i=\phi(E_i)$ for $i=1,\ldots,n$, which belong to $\mathrm{A}(C)$ by Lemma \ref{Cugenerators}. Let
\[
M=\max_i |x_i-y_i|, \, n_1=\#\{i:x_i-y_i=M\}, \, n_2=\#\{i:y_i-x_i=M\}.
\]
Let us define the degree of the triple $(\phi,x,y)$, denoted $\deg (\phi,x,y)$, as the vector $(M,n_1, n_2,n)$. We order the degrees lexicographically. We will prove the lemma by induction on the degree of the triple $(\phi, x, y)$. Let us first deal with the case $n=1$, i.e., the domain of $\phi$ is $[0,\infty]$. Since $[0,\infty]$ is totally ordered, either $x\leq y$ or $y<x$. In the first case, setting $Q$ the identity and $\phi=\psi$ gives the result. If $y<x$, then $\phi(y)\ll \phi(x)$, which, together with $\phi(x)\ll \phi(y)$, implies that $\phi(x)=\phi(y)$ is a compact element in $\mathrm{A}(C)$. The only compact element in $\mathrm{A}(C)$ is 0, for if $f\ll f$, then $f\ll (1-\epsilon)f$ for some $\epsilon>0$, and so $f=0$ by weak cancellation (Theorem \ref{weakcancellation}). Thus, $\phi(x)=0$, which in turn implies that $\phi=0$. We can then choose $Q$ and $\psi$ to be the 0 maps.
Suppose now that $\phi$, $x$, $y$ are as in the lemma, and that the lemma holds for all triples $(\phi', x', y')$ with smaller degree. If $x\leq y$, then we can choose $Q$ the identity map, $\phi=\psi$, and we are done. Let us thus assume that $x\nleq y$. If $x_{i_0}=y_{i_0}$ for some index $i_0$, then we can write $x=x_{i_0}E_{i_0}+\tilde x$ and $y=x_{i_0}E_{i_0}+\tilde y$, where $\tilde x,\tilde y$ belong to $S:=\mathrm{span}(E_i)_{i\neq i_0}\cong [0,\infty]^{n-1}$. By weak cancellation, $\phi(x)\ll \phi(y)$ implies that $\phi(\tilde x)\ll \phi(\tilde y)$. Since $\tilde x,\tilde y$ belong to a space of smaller dimension, the degree of $(\phi|_S, \tilde x,\tilde y)$ is smaller than that of $(\phi,x,y)$ ($M,n_1,n_2$ have not increased, while $n$ has decreased). By the induction hypothesis, there exist maps $\tilde Q\colon S\to [0,\infty]^N$ and $\tilde\psi\colon [0,\infty]^N\to \mathrm{Lsc}_\sigma(C)$ such that $\tilde Q\tilde x\leq \tilde Q\tilde y$
and $\phi|_S=\tilde\psi\tilde Q$. Define $Q\colon [0,\infty]^n\to [0,\infty]^{N+1}$ as the extension of $\tilde Q$ such that $QE_{i_0} = E_{N+1}$. Extend $\tilde \psi$ to $[0,\infty]^{N+1}$ setting $\psi(E_{N+1})=f_{i_0}$. Then $\phi=\psi Q$ and
\[
Qx=\tilde Q\tilde x+x_{i_0}E_{N+1}\leq \tilde Q\tilde y+y_{i_0}E_{N+1}=Qy,
\]
thus again completing the induction step.
We assume in the sequel that $x_i\neq y_i$, i.e., either $x_i<y_i$ or $x_i>y_i$, for all $i=1,\ldots,n$.
Let $I=\{i:x_i>y_i\}$ and $J=\{j:y_j>x_j\}$. Let $M_1=\max_{i\in I} x_i-y_i$ and $M_2=\max_{j\in J} y_j-x_j$. Then $M=\max (M_1,M_2)$. We break-up the rest of the proof into two cases.
\emph{Case $M_1\geq M_2$}. Using weak cancellation in
\[
\sum_{i=1}^n x_if_i = \phi(x) \ll \phi(y)=\sum_{i=1}^n y_if_i
\]
we get
\[
\sum_{i\in I} (x_i-y_i)f_i \ll \sum_{j\in J}(y_j-x_j)f_j.
\]
Let $i_1\in I$ be such that $x_{i_1}-y_{i_1}=M_1$. From the last inequality we deduce that
\[
M_1 f_{i_1}\ll \sum_{j\in J} M_2f_j,
\]
and since $M_2\leq M_1$, we get $f_{i_1}\ll \sum_{j\in J} f_j$. By the Riesz decomposition property in $\mathrm{A}(C)$ (Theorem \ref{ACRiesz}), there exist $g_j,h_j\in \mathrm{A}(C)$, with $j\in J$, such that
\[
f_{i_1}=\sum_{j\in J} g_j\hbox{ and }f_j=g_j+h_j\hbox{ for all }j\in J.
\]
Let $N_1=n+|J|-1$, and let us label the canonical generators of $[0,\infty]^{N_1}$ with the set $\{E_i:i=1,\ldots,n, \,i\neq i_1\}\cup\{G_j:j\in J\}$. Define $Q_1\colon [0,\infty]^{n}\to [0,\infty]^{N_1}$ as follows:
\begin{align*}
Q_1E_i &=E_i\hbox{ if }i\in I\backslash \{i_1\},\\
Q_1E_{i_1} &= \sum_{j\in J} G_j,\\
Q_1E_{j} &= E_j+G_j\hbox{ if }j\in J,
\end{align*}
and extend $Q_1$ to a Cu-cone morphism on $[0,\infty]^n$. Next, define a Cu-cone morphism $\psi_1\colon [0,\infty]^{N_1}\to \mathrm{Lsc}_\sigma(C)$ on the same generators as follows:
\begin{align*}
\psi_1(E_i) &= f_i,\hbox{ if }i\in I\backslash\{i_1\},\\
\psi_1(E_j) &= h_j, \hbox{ if }j\in J,\\
\psi_1(G_j) &= g_j, \hbox{ if }j\in J.
\end{align*}
It is easily checked that $\psi_1Q_1=\phi$ and that $Q_1$ maps $[0,\infty]^n\cap \mathbb{Z}^n$ to $[0,\infty]^{N_1}\cap [0,\infty]^{N_1}$. Also,
\begin{align*}
Q_1x &=\sum_{i\in I\backslash\{i_1\}}x_iE_i + \sum_{j\in J}x_{i_1}G_j+\sum_{j\in J}x_j(E_j+G_j)\\
&=\sum_{i\neq i_1} x_iE_i + \sum_{j\in J}(x_{i_1}+x_j)G_j.
\end{align*}
Similarly,
\[
Q_1y =\sum_{i\neq i_1} y_iE_i + \sum_{j\in J}(y_{i_1}+y_j)G_j.
\]
We claim that $\deg (\psi_1,Q_1x,Q_1y)< \deg (\phi,x,y)$. Indeed, the maximum of the differences of the coordinates ($M$ above) has not gotten larger. Moreover, the number of times that $M_1$ is attained ($n_1$ above) is smaller, since we have removed the coordinate $i_1$ and added new coordinates for which
\[
(x_{i_1}+x_j)-(y_{i_1}+y_j)=M_1 +x_j-y_j\in [0,M_1-1].
\]
By induction, the lemma holds for $(\psi_1,Q_1x,Q_1y)$. Thus, there exist Cu-morphisms
$Q_2\colon [0,\infty]^{N_1}\to [0,\infty]^{N_2}$ and $\psi_2\colon [0,\infty]^{N_2}\to \mathrm{Lsc}_\sigma(C)$ such that $Q_2Q_1x\leq Q_2Q_1y$ and $\psi_1=\psi_2Q_2$. Setting $Q=Q_1Q_2$ and $\psi=\psi_2$, we get the desired result.
\emph{Case $M_2>M_1$}. This case is handled similarly to the previous case, though with a few added complications. Observe first that $M_2\geq 2$ (since $M_1\geq 1$; otherwise $x\leq y$). Choose $\epsilon>0$ such that $\phi(x)\ll (1-\epsilon)\phi(y)$. If necessary, make $\epsilon$ smaller, so that we also have
\[
\epsilon<\min\{\frac 1{4x_i},\frac{1}{4y_j}:x_i\neq 0, y_j\neq 0\}.
\]
Notice that this implies that
\begin{equation}
\begin{aligned}\label{ineqsxiyi}
x_i>(1-2\epsilon)y_i&\Leftrightarrow x_i>y_i, \hbox{ for }i=1,2,\ldots,n,\\
x_i<(1-2\epsilon)y_i&\Leftrightarrow x_i<y_i, \hbox{ for }i=1,2,\ldots,n.
\end{aligned}
\end{equation}
Let $h\in \mathrm{A}(C)$ be such that $h+\phi(x)=(1-\epsilon)\phi(y)$, which exists by Lemma \ref{lhdll}.
Enlarge the domain of $\phi$ to $[0,\infty]^{n+1}$, labelling the new generator by $H$ ($=(0,\ldots,0,1)$),
and setting $\phi(H)=h$. We then have $(1-2\epsilon)\phi(y)\ll h+\phi(x)$, i.e.,
\[
\sum_{i=1}^n (1-2\epsilon)y_if_i \ll h+\sum_{i=1}^n x_if_i.
\]
Using weak cancellation and the inequalities \eqref{ineqsxiyi} we can move terms around to get
\[
\sum_{j\in J} ((1-2\epsilon)y_j-x_j)f_j \ll h+\sum_{i\in I}(x_i-(1-2\epsilon)y_i)f_i.
\]
Let $j_1\in J$ be such that $y_{j_1}-x_{j_1}=M_2$. Then
\[
((1-2\epsilon)y_{j_1}-x_{j_1})f_{j_1} \ll h +\sum_{i\in I} (x_i-(1-2\epsilon)y_i)f_i.
\]
By our choice of $\epsilon$, we have the inequalities
\[
(1-2\epsilon)y_{j_1}-x_{j_1}\geq M_2-\frac12\hbox{ and }x_{i}-(1-2\epsilon)y_i\leq M_1+\frac12\hbox{ for all }i.
\]
Hence,
\[
(M_2-\frac12)f_{j_1}\ll h+\sum_{i\in I} (M_1+\frac12)f_i.
\]
Further, $M_1+\frac12\leq M_2-\frac{1}{2}$ (since $M_2>M_1$) and $M_2-\frac12>1$ (since $M_2\geq 2$). So
\[
f_{j_1}\ll h+\sum_{i\in I} f_i.
\]
By the Riesz decomposition property in $\mathrm{A}(C)$ (Theorem \ref{ACRiesz}), $f_{j_1}=h'+\sum_{i\in I}g_i$ for some $h'\ll h$ and $g_i\ll f_i$, with $i\in I$. Let us choose $h'',h_i\in \mathrm{A}(C)$ such that $h=h'+h''$ and $f_i=g_i+h_i$ for all $i\in I$ (Lemma \ref{lhdll}). Label the canonical generators of the Cu-cone $[0,\infty]^{N_1}$, where $N_1=n+|I|+1$, with the set
\[
\{E_{j}:j=1,\dots,n,\, j\neq j_1\}\cup \{G_i:i\in I\}\cup \{H,H'\}.
\]
Define a Cu-cone morphism $Q_1\colon [0,\infty]^{n+1}\to [0,\infty]^{N_1}$ as follows:
\begin{align*}
Q_1E_j &=E_j\hbox{ for }j\in J\backslash \{j_1\},\\
Q_1E_{j_1} &= H'+\sum_{i\in I}G_i,\\
Q_1E_i &= E_i+G_i \hbox{ for }i\in I,\\
Q_1H &=H+H',
\end{align*}
Next, define a Cu-cone map $\psi_1\colon [0,\infty]^{N_1}\to \mathrm{Lsc}_{\sigma}(C)$ by
\begin{align*}
\psi_1 E_j &= f_j\hbox{ for }j\in J\backslash \{j_1\}\\
\psi_1 E_i &=h_i, \hbox{ for }i\in I,\\
\psi_1 G_i &= g_i,\hbox{ for }i\in I,\\
\psi_1 H &=h''\hbox{ and }\psi_1 H'=h'.
\end{align*}
Now $\psi_1 Q_1E_j=f_j$ for $j\in J\backslash\{j_1\}$, and
\[\psi_1 Q_1E_{j_1}=\psi_1\left(H'+\sum_{i\in I}G_i\right)=h'+\sum_{i\in I} g_i=f_{j_1}.\]
Also,
\[
\psi_1 Q_1E_i=\psi_1(E_i+G_i)=h_i+g_i=f_i,\hbox{ for }i\in I.
\]
Finally, $\psi_1 Q_1H=h'+h''=h$. Thus, we have checked that $\psi_1 Q_1=\phi$. Clearly, $Q_1$ maps
integer valued vectors to integer valued vectors.
Let us examine the degree of $(\psi_1, Q_1(x+H),Q_1y)$. We have that
\begin{align*}
Q_1(x+H)&=\sum_{j\in J\backslash\{j_1\}}x_jE_j+\sum_{i\in I}x_{j_1}G_i+x_{j_1}H'+\sum_{i\in I}x_i(E_i+G_i)+(H+H')\\
&=\sum_{j\ne j_1}x_jE_j+\sum_{i\in I}(x_{j_1}+x_i)G_i+H+(x_{j_1}+1)H'.
\end{align*}
Similarly, we compute that
\[
Q_1y=\sum_{j\ne j_1}y_jE_j+\sum_{i\in I}(y_{j_1}+y_i)G_i+H+y_{j_1}H'.
\]
We claim that the $\deg(\psi_1,Q_1(x+H),Q_1y)<\deg(\phi,x,y)$. To show this we check that for the pair $(Q_1(x+H),Q_1y)$ we have that:
\begin{enumerate}
\item
the maximum coordinates difference for the indices $i$ such that $x_i>y_i$ (number $M_1$ above) is strictly less than $M_2$,
\item
the maximum coordinates difference for the indices where $y_j>x_j$ is at most $M_2$,
\item
the number of indices for which $M_2$ is attained (number $n_2$ above) has decreased relative to the pair $(x,y)$.
\end{enumerate}
The first two points are straightforward to check. The last point follows from the fact that we have removed the coordinate $j_1$, and that for the new coordinates that we have added we have
\begin{align*}
(y_{j_1}+y_i)-(x_{j_1}+x_i)&=M_2+(y_i-x_i)\in [0,M_2-1],\\
y_{j_1}-(x_{j_1}+1)&=M_2-1<M_2.
\end{align*}
Observe that
\[
(\psi_1Q_1)(x+H)=h+\phi(x)=(1-\epsilon)\phi(y)\ll \phi(y)=\psi_1Q_1y.
\]
Hence, by the induction hypothesis, there exist $Q_2$ and $\psi_2$ such that $\psi_1=\psi_2Q_2$ and $Q_2Q_1(x+H)\leq Q_2Q_1y$. Then $Q=Q_2Q_1$ and $\psi=\psi_2$ are as desired, thus completing the step of the induction.
\end{proof}
\begin{theorem}\label{triangletheorem}
Let $\phi\colon [0,\infty]^n\to \mathrm{Lsc}_\sigma(C)$ be a Cu-morphism. Let $F\subset [0,\infty)^n$
be a finite set. Then there exist $N\in \mathbb{N}$ and Cu-morphisms
\[
[0,\infty]^n \stackrel{Q}{\longrightarrow} [0,\infty]^N\stackrel{\psi}{\longrightarrow} \mathrm{Lsc}_\sigma(C),
\]
such that $\psi Q=\phi$,
\[
\phi x\ll \phi y\implies Qx\ll Qy\hbox{ for all }x,y\in F,
\]
and $Q$ maps $[0,\infty]^n\cap \mathbb{Z}^n$ to $[0,\infty]^N\cap \mathbb{Z}^{N}$.
\end{theorem}
\begin{proof}
We start by noting that given elements $x=(x_i)_{i=1}^n$ and $y=(y_i)_{i=1}^n$ in $[0,\infty]^n$, we have $x\ll y$ if and only if
$x_i<y_i$ or $x_i=y_i=0$ for all $i=1,\dots,n$.
Suppose first that $F=\{x,y\}\subseteq [0,\infty)^n$ and that $\phi(x)\ll\phi(y)$. Choose $\varepsilon>0$ such that $(1+\varepsilon)\phi(x)\ll (1-\varepsilon)\phi(y)$. Choose $x',y'\in [0,\infty)^n\cap\mathbb{Q}^n$ such that $x\ll x'\le (1+\varepsilon)x$ and $(1-\varepsilon)y\le y'\ll y$. Then $\phi(x')\ll \phi(y')$. Let $m\in \mathbb{N}$ be such that $mx',my'\in [0,\infty)^n\cap\mathbb{Z}^n$. By Lemma \ref{coretrianglelemma}, there exist $Q,\psi$ such that $\phi=\psi Q$ and $Q(mx')\le Q(my')$, i.e., $Qx'\le Qy'$. Then
\[
Qx\ll Qx'\le Qy'\ll Qy.
\]
Lemma \ref{coretrianglelemma} also guarantees that $Q$ maps integer valued vectors to integer valued vectors. Thus, $Q$ and $\psi$ are as desired.
To deal with an arbitrary finite set $F\subseteq [0,\infty)^n$, choose $x,y\in F$ such that $\phi(x)\ll \phi(y)$ and obtain $Q_1,\psi_1$ such that $\phi=\psi_1Q_1$ and $Q_1x\ll Q_1y$. Set $F_1=Q_1F$ and apply the same argument to a new pair $x',y'\in F_1$ to obtain maps $Q_2,\psi_2$. Continue inductively until all pairs have been exhausted. Set $Q=Q_k\cdots Q_1$ and $\psi=\psi_k$.
\end{proof}
\subsection{Building the limit}
\begin{theorem}\label{inductiveCucones}
Let $C$ be an extended Choquet cone that is strongly connected and has an abundance of compact idempotents. Then $\mathrm{Lsc}_\sigma(C)$ is an inductive limit in the Cu-category of an inductive system of Cu-cones of the form $[0,\infty]^n$, $n\in \mathbb{N}$, and with Cu-morphisms that map integer valued vectors to integer valued vectors. Moreover, if $C$ is metrizable, then this inductive system can be chosen over a countable index set.
\end{theorem}
\begin{proof}
For each $n=1,2,\ldots$, choose an increasing sequence $(A_k^{(n)})_{k=1}^\infty$ of finite subsets of $[0,\infty)^n$ with dense union in $[0,\infty]^n$.
We will construct an inductive system of Cu-cones $\{S_F,\phi_{G,F}\}$, where $F,G$ range through the finite subsets of $\mathrm{A}(C)$, such that $S_F\cong [0,\infty]^{n_F}$ for all $F$. We also construct Cu-morphisms $\psi_{F}\colon S_F\to \mathrm{Lsc}_\sigma(C)$ for all $F$, finite subset of $\mathrm{A}(C)$, making the overall diagram commutative. We follow closely the presentation of the proof of the Effros-Shen-Handelmann theorem given in \cite{goodearl-wehrung}, adapted to the category of Cu-cones.
For each $f\in \mathrm{A}(C)$, define $S_{\{f\}}=[0,\infty]$ and $\psi_{\{f\}}\colon [0,\infty]\to \mathrm{Lsc}_\sigma(C)$ as the Cu-morphism such that $\psi_{\{f\}}(1)=f$. Fix a finite set $F\subseteq \mathrm{A}(C)$. Suppose that we have defined $S_G$ and $\psi_G$ for all proper subsets $G$ of $F$. Set $S^F:=\prod_G S_G$, where $G$ ranges though all proper subsets of $F$.
Define $\phi^F\colon S^F\to \mathrm{Lsc}_\sigma(C)$ as
\[
\phi^F((s_G)_G)=\sum_G \psi_G(s^G).
\]
Next, we construct $Q\colon S^F\to S_F$ and $\psi\colon S_F\to \mathrm{Lsc}_\sigma(C)$ using Theorem \ref{triangletheorem}. Here is how: For each $G$, proper subset of $F$, let $n_G$ be such that $S_G\cong [0,\infty]^{n_G}$. Let $A=\prod_G A^{(n_G)}_{k}$, where $k=|F|$ and where $G$ ranges through all proper subsets of $F$. Then $A$ is a finite subset of $S^F$. Let us apply Theorem \ref{triangletheorem} to $\phi^F$ and the set $A$, in order to obtain maps $Q\colon S^F\to S_F\cong [0,\infty]^{n_F}$ and $\psi\colon S_F\to \mathrm{Lsc}_\sigma(C)$ such that $\phi^F=\psi Q$ and
\[
\phi^F(x)\ll \phi^F(y)\Rightarrow Qx\ll Qy\hbox{ for all }x,y\in A.
\]
Set $\psi_{F}=\psi$, and for each proper subset $G$ of $F$, define $\phi_{G,F}\colon S_G\to S_F$ as the composition of the embedding of $S_G$ in $S^F$ with the map $Q$:
\[
S_G\hookrightarrow S^F\stackrel{Q}{\to} S_F.
\]
Observe that $\phi_{G,F}$ maps $[0,\infty]^{n_G}\cap \mathbb{Z}^{n_G}$ to $[0,\infty]^{n_F}\cap \mathbb{Z}^{n_F}$,
as both $Q$ and $S_G\hookrightarrow S^F$ map integer valued vectors to integer valued vectors.
Continuing in this way we obtain an inductive system $\{S_F,\phi_{G,F}\}$, indexed by the finite subsets of $\mathrm{A}(C)$, and maps $\psi_F\colon S_F\to \mathrm{Lsc}_{\sigma}(C)$ for all $F$. By construction, the overall diagram is commutative. To show that $\mathrm{Lsc}_\sigma(C)$ is the inductive limit in the Cu-category of this inductive system, we must check that
\begin{enumerate}
\item
every element in $\mathrm{Lsc}_\sigma(C)$ is supremum of an increasing sequence contained in the union of the ranges of the maps $\psi_F$,
\item
for each finite set $F$ (index of the system) and elements $x',x,y\in S_F$ such that $x'\ll x$ and $\psi_{G}(x)\leq\psi_G(y)$ in $\mathrm{Lsc}_\sigma(C)$, there exists $F'\supset F$ such that $\phi_{F,F'}(x')\ll \phi_{F,F'}(y)$.
\end{enumerate}
Let's check the first property. By construction, if $F=\{f\}$ then $f$ is contained in the range of $\psi_F$. Examining the construction of $\psi_F$ for arbitrary $F$, it becomes clear that $F$ is contained in the range of $\psi_F$. Thus, as $F$ ranges through all finite subsets of $\mathrm{A}(C)$, the union of the ranges of the maps $\psi_F$ contains $\mathrm{A}(C)$. Moreover, by Theorem \ref{ACsuprema}, every function in $\mathrm{Lsc}_\sigma(C)$ is the supremum of an increasing sequence in $\mathrm{A}(C)$.
Suppose that $x',x,y\in S_F$ are such that $\psi_F(x)\leq \psi_F(y)$ and $x'\ll x$.
Then $x'\in [0,\infty)^{n_F}$ and $\psi_F(x')\ll \psi_F(y)$. Choose $y'\ll y$ and $x'\ll x''\ll x$ such that $\psi_F(x'')\ll \psi_F(y')$. Next, choose $v,w\in A_k^{(n_F)}$ for some $k$, such that $x'\ll u \ll x''$ and $y'\ll v \ll y$. Observe then that $\psi_F(u)\ll \psi_F(v)$. Let $F'\subset \mathrm{A}(C)$ be a finite set such that $F\subset F'$ and $|F'|\geq k$. Then, by our construction of the inductive system, we have that $\phi_{F,F'}(u)\ll \phi_{F,F'}(v)$. This implies that $\phi_{F,F'}(x')\ll \phi_{F,F'}(y)$, thus proving the second property of an inductive limit.
Let us address the second part of the theorem. Suppose that $C$ is metrizable. By Theorem \ref{metrizableC}, there exists a countable set $B\subseteq \mathrm{A}(C)$ such that every function in $\mathrm{Lsc}_\sigma(C)$ is the supremum of an increasing sequence
in $B$. The construction of the inductive limit for $\mathrm{Lsc}_\sigma(C)$ in the preceding paragraphs can be repeated mutatis mutandis, letting the index set of the inductive limit be the set of finite subsets of $B$, rather than the finite subsets of $\mathrm{A}(C)$. The resulting inductive limit is thus indexed by a countable set.
\end{proof}
We are now ready to proof Theorem \ref{mainchar} from the introduction.
\begin{proof}[Proof of Theorem \ref{mainchar}]
(i)$\Rightarrow$(iv): An AF C*-algebra has real rank zero, stable rank one, and is exact (these properties hold for finite dimensional C*-algebras and are passed on to their inductive limits). Thus, (i) implies (iv) by Proposition \ref{cstarECC}.
(iv)$\Rightarrow$(iii): Suppose that we have (iv). By Theorem \ref{inductiveCucones}, $\mathrm{Lsc}_{\sigma}(C)$ is an inductive limit in the Cu-category of Cu-cones of the form $[0,\infty]^n$, with $n\in \mathbb{N}$. We have $F([0,\infty]^n)\cong [0,\infty]^n$
via the map
\[
F([0,\infty]^n)\ni \lambda\mapsto (\lambda(E_1),\ldots,\lambda(E_n))\in [0,\infty]^n,
\]
where $E_1,\ldots,E_n$ are the canonical generators of $[0,\infty]^n$.
Applying the functor $F(\cdot)$ to the inductive system with limit $\mathrm{Lsc}_\sigma(C)$ we obtain a projective system in the category of extended Choquet cones where each cone is isomorphic to $[0,\infty]^n$ for some $n$. By the continuity of the functor $F(\cdot)$ (\cite[Theorem 4.8]{ERS}), and the fact that $F(\mathrm{Lsc}_\sigma(C))\cong C$ (Theorem \ref{dualitythm}), we get (iii).
(iii)$\Rightarrow$(ii): Suppose that we have (iii). Say $C=\varprojlim_{i\in I} ([0,\infty]^{n_i}, \alpha_{i,j})$. Observe that $\alpha_{i,j}$ maps $[0,\infty)^{n_i}$ to $[0,\infty)^{n_j}$. Indeed, the support idempotent of an element in $[0,\infty)^{n_i}$ is 0. By continuity of $\alpha_{i,j}$, the same holds for the image of these elements; thus, they belong to $[0,\infty)^{n_j}$. It follows then that $\alpha_{i,j}$ is given by multiplication by a matrix $M_{i,j}$ with non-negative finite entries: $\alpha_{i,j}(v)=M_{i,j}v$ for all $v\in [0,\infty]^{n_i}$ (in $M_{i,j}v$ we regard $v$ as a column vector and use the rule $0\cdot \infty =0$). The transpose matrix $M_{i,j}^t$ can then be regarded as a map from $\mathbb{R}^{n_j}$ to $\mathbb{R}^{n_i}$. Let us form an inductive system of dimension groups whose objects are $\mathbb{R}^{n_i}$, endowed with the coordinatewise order, with $i\in I$, and with maps $M_{i,j}^t\colon \mathbb{R}^{n_j}\to \mathbb{R}^{n_i}$. This inductive system of dimension groups gives rise to the original system after applying the functor $\mathrm{Hom}(\,\cdot\,,[0,\infty])$ to it, and making the isomorphism identifications $\mathrm{Hom}(\mathbb{R}_+^{n_i},[0,\infty])\cong [0,\infty]^{n_i}$. Let $G$ be its limit in the category of dimension groups ($G$ is in fact a vector space). By the continuity of the functor $\mathrm{Hom}(\,\cdot\,,[0,\infty])$, we have $\mathrm{Hom}(G_+,[0,\infty])\cong C$. Thus, (iii) implies (ii).
(ii)$\Rightarrow$(i): By Elliott's theorem, there exists an AF C*-algebra $A$ whose Murray-von Neumann monoid of projections $V(A)$ is isomorphic to $G_+$. The result now follows from the fact, well known to experts, that $T(A)\cong \mathrm{Hom}(V(A),[0,\infty])$ for an AF $A$ (where $\mathrm{Hom}(V(A),[0,\infty])$ denotes the cone of monoid morphisms). Let us sketch a proof of this fact here:
Since AF C*-algebras are exact, we have by Haagerup's theorem that 2-quasitraces on $A$, and on the ideals of $A$, are traces. We apply here the version due to Blanchard and Kirchberg that includes densely finite lower semicontinuous 2-quasitraces; see \cite[Remark 2.29 (i)]{blanchard-kirchberg}. Thus, $T(A)=QT(A)$, where $QT(A)$ denotes the cone of lower semicontinuous $[0,\infty]$-valued 2-quasitraces on $A$. Further, by \cite[Theorem 4.4]{ERS}, $QT(A)\cong F(\mathrm{Cu}(A))$ for any C*-algebra $A$. Thus, we must show that $F(\mathrm{Cu}(A))\cong \mathrm{Hom}(V(A),[0,\infty])$ when $A$ is an AF C*-algebra. Let $\mathrm{Cu}_c(A)$ denote the submonoid of $\mathrm{Cu}(A)$ of compact elements, i.e., of elements $e\in \mathrm{Cu}(A)$ such that $e\ll e$. By \cite[Theorem 3.5]{brown-ciuperca} of Brown and Ciuperca, for stably finite $A$ the map from $V(A)$ to $\mathrm{Cu}(A)$ assigning to a Murray-von Neumann class $[p]_{\mathrm{MvN}}$ the Cuntz class $[p]_{\mathrm{Cu}}\in \mathrm{Cu}(A)$ is a monoid isomorphism with $\mathrm{Cu}_c(A)$. This holds in particular for $A$ AF. Thus, we must show that $F(\mathrm{Cu}(A))\cong \mathrm{Hom}(\mathrm{Cu}_c(A),[0,\infty])$. This isomorphism is given by the restriction map. Indeed, since $A$ has real rank zero and stable rank one, every element of $\mathrm{Cu}(A)$ is supremum of an increasing sequence of compact elements (\cite[Corollary 5]{CEI}). This shows that $\lambda\mapsto \lambda|_{\mathrm{Cu}_c(A)}$ is injective. To prove surjectivity, suppose that we have a monoid morphism $\tau\colon \mathrm{Cu}_c(A)\to [0,\infty]$. Define
\[
\lambda(x)=\sup \{\tau(e):e\leq x,\, e\in \mathrm{Cu}_c(A)\}.
\]
Then $\lambda$ is readily shown to be a functional on $\mathrm{Cu}(A)$ that extends $\tau$. Finally, from the definition of the topology on $F(\mathrm{Cu}(A))$ it is evident that a convergent net $(\lambda_i)_i$ in $F(\mathrm{Cu}(A))$ converges pointwise on compact elements of $\mathrm{Cu}(A)$. This shows that the map $\lambda\mapsto \lambda|_{\mathrm{Cu}_c(A)}$ is continuous. Since it is a bijection between compact Hausdorff spaces, its inverse is also continuous. In summary, we have the following chain of extended Choquet cones isomorphisms when $A$ is AF:
\[
T(A)=QT(A)\cong F(\mathrm{Cu}(A))\cong \mathrm{Hom}(\mathrm{Cu}_c(A),[0,\infty])\cong \mathrm{Hom}(V(A),[0,\infty]).
\]
Finally, suppose that $C$ is metrizable and satisfies (iv). Then, in the proof of (iv)$\Rightarrow$(iii) above,
Theorem \ref{inductiveCucones} allows us to start with an inductive limit for $\mathrm{Lsc}_\sigma(C)$ over a countable index set. Applying the functor $F(\cdot)$, we get a projective limit for $C$ over a countable index set. Moreover, the Cu-morphisms in the inductive system of Theorem \ref{inductiveCucones}
map integer valued vectors to integer valued vectors. Thus, the matrices $M_{i,j}$ implementing these morphisms have nonnegative integer entries. Thus,
in the proof of (iii)$\Rightarrow$(ii) we start with $C=\varprojlim_{i\in I} ([0,\infty]^{n_i}, \alpha_{i,j})$, where $\alpha_{i,j}$ is implemented by a matrix with nonnegative integer entries. We can thus construct an inductive system $(\mathbb{Z}^{n_i}, M_{i,j})_{i,j\in I}$, in the category of dimension groups, whose limit is a countable dimension group $G$ such that $\mathrm{Hom}(G_+,[0,\infty])\cong C$, as desired.
\end{proof}
\section{Finitely generated cones}\label{fingen}
A cone $C$ is called finitely generated if there exists a finite set $X\subseteq C$
such that for every $x\in C$ we have $x=\sum_{i=1}^n \alpha_ix_i$ for some $\alpha_i\in (0,\infty)$ and $x_i\in X$.
In this section we give a direct construction of an ordered vector space (over $\mathbb{R}$) $(V,V^+)$ with the Riesz property
and such that $\mathrm{Hom}(V^+,[0,\infty])$ is isomorphic to a given finitely generated, strongly connected, extended Choquet cone $C$. Here $\mathrm{Hom}(V^+,[0,\infty])$ denotes the monoid morphisms from $V^+$ to $[0,\infty]$. These maps are automatically homogeneous with respect to scalar multiplication; thus, they are also cone morphisms.
\begin{lemma}\label{fingenCw}
Let $C$ be a finitely generated extended Choquet cone. Then $\mathrm{Idem}(C)$ is finite and for each $w\in \mathrm{Idem}(C)$ the sub-cone $C_w$ is either isomorphic to $\{0\}$ or to $[0,\infty)^{n_w}$ for some $n_w\in \mathbb{N}$. (Recall that we have defined $C_w=\{x\in C:\epsilon(x)=w\}$.)
\end{lemma}
\begin{proof}
Let $Z$ be a finite set that generates $C$. Let $w\in C$ be an idempotent, and write $w=\sum_{i=1}^n \alpha_ix_i$, with $x_i\in Z$ and $\alpha_i\in (0,\infty)$. Multiplying both sides by a scalar $\delta>0$ and passing to the limit as $\delta\to 0$, we get that $w$ is the sum of support idempotents of elements in $Z$. It follows that $\mathrm{Idem}(C)$ is finite.
Next, let $w\in \mathrm{Idem}(C)$. Define $Z_w=\{x+w:x\in Z\hbox{ and }\epsilon(x)\leq w\}$, which is a finite subset of $C_w$. We claim that $Z_w$ generates $C_w$ as a cone. Indeed, let $x\in C_w$ and write $x=\sum_{i=1}^n \alpha_i x_i$, with $x_i\in Z$ and $\alpha_i\in (0,\infty)$. Adding $w$ on both sides we get $x=\sum_{i=1}^n \alpha_i(x_i+w)$. Since $\epsilon(x_i)\leq \epsilon(x)=w$, the elements $x_i+w$ are in $Z_w$. If $Z_w=\{w\}$ then $C_w$ is isomorphic to $\{0\}$. Suppose that $Z_w\neq \{w\}$. Since $w$ is a compact idempotent, $C_w$ has a compact base $K$ which is a Choquet simplex (Theorem \ref{compactbase}). Further, $K$ is finitely generated (by the set $(0,\infty)\cdot Z_w\cap K$). Hence, $K$ has finitely many extreme points, which in turn implies that $C_w\cong [0,\infty)^{n_w}$ for some $n_w\in \mathbb{N}$.
\end{proof}
\emph{For the remainder of this section we assume that $C$ is a finitely generated, strongly connected, extended Choquet cone.} Thus, each idempotent $w\in \mathrm{Idem}(C)$ is compact and, by strong connectedness, $C_w\neq \{w\}$ for all $w\neq \infty$ (here $\infty$ denotes the largest element in $C$).
Let $w\in \mathrm{Idem}(C)$ and $x\in C_w$. If $z\in C$ is such that $z+w=x$, we call $z$ and extension of $x$. The set of extensions of $x$ is downward directed: if $z_1$ and $z_2$ are extensions of $x$, then so is $z_1\wedge z_2$. Consider the element $\tilde x=\inf\{z\in C:z+w=x\}$. By the continuity of addition, $\tilde x$ is also an extension of $x$, which we call the minimum extension.
\begin{lemma}\label{irreducibles}
Let $w \in \mathrm{Idem}(C)$. Let $x\in C_w\backslash\{w\}$ be an element generating an extreme ray in $C_w$, and let $\tilde x$ denote the minimum extension of $x$.
\begin{enumerate}[\rm (i)]
\item
$\tilde x$ generates an extreme ray in $C_{\epsilon(\tilde x)}$.
\item
If $y,z\in C$ are such that $y+z=\tilde x$, then either $y\leq z$ or $z\leq y$.
\end{enumerate}
\end{lemma}
\begin{proof}
Set $v=\epsilon(\tilde x)$.
(i) Let $y,z\in C_v$ be such that $y+z=\tilde x$. Adding $w$ on both sides we get $(y+w)+(z+w)=x$. Since
$y+w,z+w\in C_w$, and $x$ generates an extreme ray in $C_w$, both $y+w$ and $z+w$ are either positive scalar multiples of $x$ or equal to $w$. Assume that $y+w=w$ and $z+w=x$. The latter says that $z$ is an extension of $x$. Hence $y+z=\tilde x\leq z$ in $C_v$. By cancellation in $C_v$ (Lemma \ref{supportlemma}), we get $y=v$ and $z=\tilde x$. Suppose on the other hand that $y+w=\alpha x$ and $z+w=\beta x$ for positive scalars $\alpha,\beta$ such that $\alpha+\beta=1$. Then $y/\alpha$ and $z/\beta$ are extensions of $x$. We deduce that $\alpha \tilde x\leq y$ and $\beta \tilde x\leq z$. Hence,
\[
\alpha \tilde x + z \leq y+z=\tilde x=\alpha \tilde x+\beta \tilde x.
\]
By cancellation in $C_v$, $z\leq \beta\tilde x$, and so $z=\beta\tilde x$. Similarly, $y=\alpha\tilde x$. Thus, $\tilde x$ generates an extreme ray in $C_v$.
(ii) The argument is similar to the one used in (ii). After arriving at $y+w=\alpha x$ and $z+w=\beta x$, we assume without loss of generality that $\alpha\leq \frac 12\leq \beta$. Using again that $\tilde x$ is the minimum extension of $x$, we get $z\geq \tilde x/2\geq y/2+z/2$, and applying Lemma \ref{supportlemma} (ii), we arrive at $z/2\geq y/2$.
\end{proof}
\begin{remark}
The property of $\tilde x$ in Lemma \ref{irreducibles} (ii) says that $\tilde x$ is an irreducible element of the cone $C$ in the
sense defined by Thiel in \cite{thiel}.
\end{remark}
Next, we construct a suitable set of generators of $C$. For each $w\in \mathrm{Idem}(C)$, let $X_w$ denote the set of minimal extensions of all elements $x\in C_w\backslash\{w\}$ that generate an extreme ray in $C_w$. Consider the set $\bigcup_{w\in \mathrm{Idem}(C)} X_w$, which is closed under scalar multiplication. We form a set $X$ by picking a representative from each ray in $\bigcup_{w\in \mathrm{Idem}(C)} X_w$.
\begin{proposition}\label{Xrepresentation}
Let $X\subseteq C$ be as described in the paragraph above. Each $y\in C$ has a unique representation of the form
\[
y=\sum_{i=1}^n \alpha_ix_i + w,
\]
where $x_i\in X$ and $\alpha_i\in (0,\infty)$ for all $i$, and $w\in \mathrm{Idem}(C)$ is such that
$\epsilon(x_i)\leq w$ but $x_i\nleq w$ for all $i$.
\end{proposition}
\begin{proof}
Let $y\in C$, and set $w=\epsilon(y)$. If $y=w$ then its representation is simply $y=w$. Suppose that $y\neq w$. In $C_w$, express $y$ as a sum of elements that lie in extreme rays (Lemma \ref{fingenCw}). By the construction of $X$, these elements have the form $\alpha_i(x_i+w)$, with $x_i\in X$ and $\alpha_i\in (0,\infty)$. We thus have that
\[
y=\sum_{i=1}^n\alpha_i(x_i+w) =\sum_{i=1}^n \alpha_i x_i + w.
\]
We have $x_i+w\in C_w\backslash\{w\}$ for all $i$; equivalently, $\epsilon(x_i)\leq w$ and $x_i\nleq w$ for all $i$. Thus, this is the desired representation.
To prove uniqueness of the representation, suppose that
\[
y=\sum_{i\in I} \alpha_i x_i + w=\sum_{j\in J} \beta_j x_j + w'.
\]
Since $\epsilon(x_i)\leq w$ for all $i$, the support of $y$ is $w$. Thus, $w=w'$. We can now rewrite the equation above as
\[
y=\sum_{i\in I} \alpha_i (x_i + w)=\sum_{j\in J} \beta_j (x_j + w).
\]
This equation occurs in $C_w\cong [0,\infty)^{n_w}$. Further, $x_i+w$ and $x_j+w$ generate extreme rays of $C_w$ for all $i,j$. It follows that $I=J$ and that the two representations are the same up to relabeling of the terms.
\end{proof}
\subsection{Constructing the vector space}
We continue to denote by $X$ the subset of $C$ defined in the previous subsection. For each $w\in \mathrm{Idem}(C)$, define
\[
O_w=\{x\in X: x\nleq w\}.
\]
\begin{lemma}\label{Otopology}
Let $w_1,w_2\in \mathrm{Idem}(C)$. Then
\begin{enumerate}[\rm (i)]
\item
$O_{w_1}\cup O_{w_2}=O_{w_1\wedge w_2}$.
\item
$O_{w_1}\cap O_{w_2}=O_{w_1+w_2}$.
\item
$O_{w_1}\subseteq O_{w_2}$ if and only if $w_1\geq w_2$.
\end{enumerate}
\end{lemma}
\begin{proof}
(i) It is more straightforward to work with the complements of the sets: $x\notin O_{w_1\wedge w_2}$ if and only if $x\leq w_1\wedge w_2$, if and only if $x\leq w_1$ and $x\leq w_2$, i.e., $x\notin O_{w_1}$ and $x\notin O_{w_2}$.
(ii) Again, we work with complements. Let's show that $O_{w_1+w_2}^c\subseteq O_{w_1}^c\cup O_{w_2}^c$ (the opposite inclusion is clear). Let $x\in O_{w_1+w_2}^c$, i.e., $x\leq w_1+w_2$. Choose $z$ such that $x\wedge w_1+z=x$. Recall that the elements of $X$ are minimal extensions of non-idempotent elements that generate an extreme ray. Thus,
by Lemma \ref{irreducibles} (ii), either $x\wedge w_1\leq z$ or $z\leq x\wedge w_1$. If $z\leq x\wedge w_1$,
then
\[
x=x\wedge w_1+z\leq 2(x\wedge w_1)\leq w_1.
\]
Hence $x\in O_{w_1}^c$, and we are done. Suppose that $x\wedge w_1\leq z$. It follows that $2(x\wedge w_1)\leq x$. Now repeat the same argument with $x$ and $w_2$. We are done unless we also have that $2(x\wedge w_2)\leq x$. In this case, adding the inequalities we get
$2(x\wedge w_1)+ 2(x\wedge w_2)\leq 2x$, i.e., $x\wedge w_1 + x\wedge w_2 \leq x$.
But $x\leq x\wedge w_1+x\wedge w_2$ (since $x\leq w_1+w_2$). Hence, $x=x\wedge w_1+x\wedge w_2$. Applying Lemma \ref{irreducibles} (ii) again we get that either $x\leq 2(x\wedge w_1)\leq w_1$ or $x\leq 2(x\wedge w_2)\leq w_2$. Hence, $x\in O_{w_1}^c\cup O_{w_2}^c$, as desired.
(iii) Suppose that $O_{w_1}\subseteq O_{w_2}$. By (i), $O_{w_1\wedge w_2}=O_{w_1}\cup O_{w_2}=O_{w_2}$. Assume, for the sake of contradiction, that $w_1\wedge w_2\neq w_2$. Since $C$ is strongly connected, there exists $x\in C_{w_1\wedge w_2}\setminus\{w_1\wedge w_2\}$ such that $x \leq w_2$. We can choose $x$ in an extreme ray of $C_{w_1\wedge w_2}$, since the set of all $x\in C_{w_1\wedge w_2}$ such that $x\leq w_2$ is a face. Consider the minimum extension $\tilde x$ of $x$. Adjusting $x$ by a scalar multiple, we may assume that $\tilde x\in X$. Now $\tilde x\leq w_2$, i.e, $\tilde x\notin O_{w_2}$. But we cannot have $\tilde x\leq w_1\wedge w_2$, since this would imply that
\[
x=\tilde x+w_1\wedge w_2 = w_1\wedge w_2.
\]
Thus, $x\in O_{w_1\wedge w_2}$. This contradicts that $O_{w_1\wedge w_2}=O_{w_2}$.
\end{proof}
Let $w\in \mathrm{Idem}(C)$. Define
\begin{align*}
P_w &=\{x\in O_w:\epsilon(x)\leq w\},\\
\widetilde P_{w} &=P_w\cup O_w^c=\{x\in X:\epsilon(x)\leq w\}.
\end{align*}
Observe that if $y\in C$, and $y=\sum_{i=1}^{n}\alpha_ix_i+w$ is the representation of $y$ described in Proposition \ref{Xrepresentation}, then $x_i\in P_{w}$ for all $i, 1\le i\le n$.
\begin{lemma}\label{MTconditions}
Let $w_1,w_2\in \mathrm{Idem}(C)$. The following statements hold:
\begin{enumerate}[\rm (i)]
\item
$\widetilde P_{w_1\wedge w_2}=\widetilde P_{w_1}\cap \tilde P_{w_2}$.
\item
If $w_1\ngeq w_2$ then $P_{w_1}\backslash O_{w_2}\neq \varnothing$.
\end{enumerate}
\end{lemma}
\begin{proof}
(i) This is straightforward: $\epsilon(x)\le w_1$ and $\epsilon(x)\leq w_2$ if and only if $\epsilon(x)\le w_1\wedge w_2$.
(ii) Suppose that $w_1\not\ge w_2$. Let $w_3=w_1+w_2$. By Lemma \ref{Otopology} (ii), $O_{w_1}\cap O_{w_2}=O_{w_3}$. Also $w_1\le w_3$ and $w_1\ne w_3$. Since $C$ is strongly connected, there exists $y\in C_{w_1}\setminus\{w_1\}$
such that $w_1\le y\le w_3$. Choose $y$ on an extreme ray (always possible, since the set of all $y\in C_{w_1}$ such that $y\le w_3$ is a face) and adjust it by a scalar so that its minimum extension $\tilde y$ belongs to $X$. Since $\tilde y+w_1\in C_{w_1}\backslash\{w_1\}$, we have that $\tilde y\nleq w_1$ and $\epsilon(\tilde y)\leq w_1$.
That is, $\tilde y\in P_{w_1}$. Since $\tilde y\le w_3$, we also have that $\tilde y\in O_{w_3}^c\subseteq O_{w_2}^c$. We have thus obtained an element $\tilde y\in P_{w_1}\setminus O_{w_2}$.
\end{proof}
Let us say that a function $f\colon X\to \mathbb{R}$ is positive provided that there exists $w\in \mathrm{Idem}(C)$ such that $f(x)=0$ for $x\notin O_w$ and $f(x)>0$ for $x\in P_w$. We call $w$ the support of $f$ and denote it by $\mathrm{supp}(f)$.
\begin{lemma}\label{supportV}
The support of a positive function is unique. Further, if $f,g\colon X\to \mathbb{R}$ are positive then
$\mathrm{supp}(f+g)=\mathrm{supp}(f)\wedge \mathrm{supp}(g)$.
\end{lemma}
\begin{proof}
Let $w_1,w_2\in \mathrm{Idem}(C)$ be both supports of $f$. Suppose that $w_1\ne w_2$, and without loss of generality, that $w_1\not\ge w_2$. Then there exists $x\in P_{w_1}\cap O_{w_2}^c$ (by Lemma \ref{MTconditions}). On one hand, $x\in P_{w_1}$ implies that $f(x)>0$. On the other hand, $x\in O_{w_2}^c$ implies that $f(x)=0$, a contradiction. Thus $w_1=w_2$, whereby proving the first part of the lemma.
To prove the second part, assume that $f$ and $g$ are positive functions on $X$, and set $v= \mathrm{supp}(f)$ and $w= \mathrm{supp}(g)$. Clearly $f+g$ vanishes on $ O_{v}^c\cap O_{w}^c= O_{v\wedge w}^c$. Let $x\in P_{v\wedge w}$. Then, by Lemma \ref{MTconditions} (i), $x\in \tilde{P}_{v}\cap\tilde{P}_{w}$. Thus, $x$ is in one of the following sets: $P_{v}\cap P_{w}$, $P_{\nu}\cap O_{w}^c$, or $P_{w}\cap O_{\nu}^c$. In all cases we see that $(f+g)(x)>0$. Indeed, if $x\in P_{\nu}\cap P_{w}$ then $f(x),g(x)>0$; if $x\in P_{\nu}\cap O_{w}^c$ then $f(x)>0$ and $g(x)=0$; if $x\in P_{w}\cap O_{\nu}^c$ then $f(x)=0$ and $g(x)>0$. Therefore $ \mathrm{supp}(f+g)=v\wedge w$.
\end{proof}
Let us denote by $V_C$ the vector space of $\mathbb{R}$-valued functions on $X$ and by $V_C^+$ the set of positive functions in $V_C$.
\begin{theorem}
The pair $(V_C,V_C^+)$ is an ordered vector space having the Riesz interpolation property.
\end{theorem}
\begin{proof}
By the previous lemma, $V_C^+$ is closed under addition. Clearly, $V_C^+$ is closed under multiplication by positive scalars. Since the pointwise strictly positive functions belong to $V_C^+$ and span $V_C$, we have $V_C^+-V_C^+=V_C$. Also, $V_C^+\cap -V_C^+=\{0\}$, for if $f$ and $-f$ are positive then, by the previous lemma,
\[
\mathrm{supp}(f)\geq \mathrm{supp}(f+-f)=\mathrm{supp}(0)=\infty,
\]
which implies that $f=0$. Thus, $(V_C,V_C^+)$ is an ordered vector space.
In \cite{maloney-tikuisis}, Maloney and Tikuisis obtained conditions guaranteeing that the Riesz interpolation property holds in a finite dimensional ordered vector space. The properties of the sets $P_w$ obtained in Lemma \ref{MTconditions} (i) and (ii) are precisely those properties in \cite[Corollary 5.1]{maloney-tikuisis} shown to guarantee that the Riesz interpolation property holds in $(V_C,V_C^+)$.
\end{proof}
Let us define a pairing $(\cdot,\cdot)\colon C\times V_C^+\to [0,\infty]$ as follows: for each $y\in C$ and $f\in V_C^+$, write $y=\sum_{i=1}^n \alpha_i x_i + w$, the representation of $y$ described in Proposition \ref{Xrepresentation}, and then set
\[
(y,f)=
\begin{cases}
\sum\limits_{i=1}^{n}\alpha_if(x_i) & \hbox{if }w\le \mathrm{supp}(f),\\
\infty &\hbox{otherwise}.
\end{cases}
\]
\begin{theorem}
The pairing defined above is bilinear. Moreover, the map $x\mapsto (x,\cdot)$, from $C$
to $\mathrm{Hom}(V_C^+,[0,\infty])$, is an isomorphism of extended Choquet cones.
\end{theorem}
\begin{proof}
Let $x,y\in C$ and $f\in V_C^+$. Write
\begin{align*}
x &= \sum_{i=1}^m \alpha_ix_i+v,\\
y &= \sum_{j=1}^n \beta_j y_j+w,
\end{align*}
with $v,w\in \mathrm{Idem}(C)$ and $x_i,y_j\in X$ as in Proposition \ref{Xrepresentation}. Then
\[
x+y=\sum_{i=1}^m \alpha_ix_i+\sum_{j=1}^n \beta_j y_j+v+w.
\]
Observe that $\epsilon(x_i),\epsilon(y_j)\leq v+w$ and that $\alpha_i,\beta_j\in (0,\infty)$ for all $i,j$.
Thus, the sum on the right side is the representation of $x+y$ described in
Proposition \ref{Xrepresentation}, except for the possible repetition of elements of $X$ appearing both among the $x_i$s and the $y_j$s. If $v+w\leq \mathrm{supp}(f)$, then $v\leq \mathrm{supp}(f)$ and $w\leq \mathrm{supp}(f)$, and
so
\[
(x,f)+(y,f)=\sum_{i=1}^m \alpha_i f(x_i) + \sum_{j=1}^n \beta_jf(y_j)=(x+y,f).
\]
If, on the other hand, $v+w\nleq \mathrm{supp}(f)$, then either $v\nleq \mathrm{supp}(f)$ or $w\nleq \mathrm{supp}(f)$,
and in either case $(x,f)+(y,f)=\infty=(x+y,f)$. This proves additivity on the first coordinate.
Homogeneity with respect to scalar multiplication follows automatically from additivity.
Let $f,g\in V^+_C$ and $w\in \mathrm{Idem}(C)$. Then $w\le\mathrm{supp}(f+g)$ if and only if $w\le\mathrm{supp}(f)$ and
$w\le\mathrm{supp}(g)$ (Lemma \ref{supportV}). This readily shows linearity on the second coordinate.
For each $x\in C$, let $\Lambda_x\in \mathrm{Hom}(V_C^+,[0,\infty])$ be defined by the pairing above: $\Lambda_x(f)=(x,f)$ for all $f\in V_C^+$. Let $\Lambda\colon C\to \mathrm{Hom}(V_C^+,[0,\infty])$ be the map given by $y\mapsto\Lambda_y$ for all $y\in C$. To prove that $\Lambda$ is injective, suppose that $y,z\in C$ are such that $\Lambda_y=\Lambda_z$. Choose any $f\in V^+_C$ such that $\mathrm{supp}(f)=\epsilon(y)$. If $\epsilon(y)\not\le \epsilon(z)$ then $\Lambda_y(f)$ is finite, while $\Lambda_z(f)=\infty$. This contradicts that $\Lambda_y=\Lambda_z$. Hence $\epsilon(y)\le \epsilon(z)$. By a similar argument $\epsilon(z)\le \epsilon(y)$, and so we get equality. Set $w=\epsilon(y)=\epsilon(z)$. Then we can write
\begin{align*}
y &=\sum_{i=1}^m\alpha_iy_i+w,\\
z &=\sum_{i=1}^n\beta_iz_i+w
\end{align*}
with $y_i,z_i\in P_{w}$ for all $i$. Let $f\in V_C$ be such that $\mathrm{supp}(f)=w$. Then
$f(y_i),f(z_i)>0$ and
\begin{equation}\label{Lambdayz}
\sum_{i=1}^{m}\alpha_if(y_i)=\Lambda_y(f)=\Lambda_z(f)=\sum_{i=1}^{n}\beta_if(z_i).
\end{equation}
Let $V_{w}^+= \{f\in\ V^+_C\colon \mathrm{supp}(f)=w\}$, i.e., $f\in V_w^+$
if $f$ is positive on $P_w$ and zero outside $O_w$. It is clear that $V_w^+-V_w^+$ consists of all the functions
on $X$ that vanish outside $O_w$. It then follows from \eqref{Lambdayz} that $n=m$ and that, up to relabelling, $y_i=z_i$ for all $1\le i\le n$. Consequently $y=z$.
Let us show that $\Lambda$ is surjective. Let $\lambda\in \mathrm{Hom}(V_C^+,[0,\infty])$. By Lemma \ref{supportV}, the set
\[
\{w\in \mathrm{Idem}(C): w=\mathrm{supp}(f)\hbox{ for some }f\in V_C^+\hbox{ such that }\lambda(f)<\infty\}
\]
is closed under infima. Since this set is also finite, it has a minimum element $w$. We claim that for each $f\in V_C^+$ we have
\[
\lambda(f)<\infty\Leftrightarrow w\leq \mathrm{supp}(f).
\]
Indeed, from the definition of $w$ it is clear that if $\lambda(f)<\infty$ then $w\leq \mathrm{supp}(f)$. Suppose on the other hand that $f\in V_C^+$ is such that $w\leq \mathrm{supp}(f)$. Let $f_0\in V_w^+$ be such that $\lambda(f_0)<\infty$. Then $\alpha f_0 - f$ is positive (with support $w$) for a sufficiently large scalar $\alpha\in (0,\infty)$. Thus, $\lambda(f)\leq \alpha\lambda(f_0)<\infty$.
Let us extend $\lambda$ by linearity to the vector subspace $V_w:=V_w^+-V_w^+$. As remarked above, $V_{w}$ consists of all the functions $f\colon X\to \mathbb{R}$ vanishing on the complement of $O_w$. That is, $V_{w}=\mathrm{span}(\{\mathbbm{1}_x\colon x\in O_w\})$, where $\mathbbm{1}_x$ denotes the characteristic function of $\{x\}$.
If $x\in P_w$, then $\mathbbm{1}_x+\epsilon\mathbbm{1}_{P_w}\in V_w^+$ for all $\epsilon>0$; here $\mathbbm{1}_{P_w}$ denotes the characteristic function of $P_w$. It follows that $\lambda(\mathbbm{1}_x+\epsilon\mathbbm{1}_{P_w})\geq 0$, and letting $\epsilon\to 0$, that $\lambda(\mathbbm{1}_x)\geq 0$ for all $x\in P_w$. If $x\in O_w\backslash P_w$, then $\lambda(\mathbbm{1}_{P_w}-\alpha\mathbbm{1}_x)\geq 0$ for all $\alpha\in \mathbb{R}$. It follows that $\lambda(\mathbbm{1}_x)=0$ for all $x\in O_w\setminus P_w$. Thus
\[
\lambda(f)=\sum_{x\in P_{w}}\lambda(\mathbbm{1}_x)f(x)
\]
for all $f\in V_w$. Since $V_v^+\subseteq V_w$ for any idempotent $v$ such that $w\leq v$, the formula above holds for all $f\in V_C^+$ such that $w\leq \mathrm{supp}(f)$.
Define
\[
y=\sum_{x\in P_{w}}\lambda(\mathbbm{1}_x)x+w.
\]
By the previous arguments, $\lambda(f)=\Lambda_y(f)$ for all $f$ such that $w\leq \mathrm{supp}(f)$. On the other hand, $\lambda(f)=\infty=\Lambda_y(f)$ for all $f$ such that $w\nleq \mathrm{supp}(f)$. Hence, $\lambda=\Lambda_y$.
\end{proof}
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\section{Introduction}
The spectral continuity is a relevant subject in Banach theory and operator theory. Several authors have studied this topic using different types of convergence. Regarding norm convergence, Newburgh in \cite{MR0051441} was one of the first to do systematically investigated in general Banach algebras. More completely results in the case of the algebra $B(H)$ for $H$ a Hilbert space are given in the series of papers of Conway and Morrel, see \cite{MR543882,MR634170,MR701023}. Other articles on this topic are \cite{MR1212728,MR1457127,MR1814478,MR1793814,MR2894894,sanchez4}. Another type of convergence for the study of spectral continuity is the $\nu$-convergence given by Ahues in \cite{MR1886113}. Also developed by S\'anchez et al. \cite{MR3388799, MR3626681,MR4043866} and Ammar et al. \cite{MR3624565, MR4075017}.
In this paper we study the $\nu$-continuity of the spectrum function. In section 2, we review this concept on arbitrary unital Banach algebras. We show that in an unital abelian Banach algebra the spectrum function is $\nu$-continuous. In section 3, we study the $\nu$-continuity of the spectrum and the approximate point spectrum on the algebra $B(X)$, for a Banach space $X$. We show that for the particular case of a Hilbert space $H$, the $\nu$-continuity of spectrum is equivalent under certain conditions to the usual continuity of the spectrum. Also, we show that the approximate point spectrum is upper semi-$\nu$-continuous at every Fredholm operator, then we give sufficient conditions to guarantee the $\nu$-continuity of this function. In section 4, we show that the restriction of the Weyl spectrum on the class of essentially $G_1$ operators is $\nu$-continuous. Finally, we investigate the $\nu$-continuity of the spectrum on the class of $p$-hyponormal operators.
The next concepts are part of classical point set topology. Let $(E_n)$ be a sequence of arbitrary subsets of $\mathbb{C}$ and define the limits inferior and superior of
$(E_n)$ as follows:
\begin{itemize}
\item[] $\liminf E_n=\{\lambda\in \mathbb{C}\,|\,$ for every $\epsilon>0$, there exists $N\in\mathbb{N}$ such that $B(\lambda, \epsilon)\cap E_n\neq \emptyset$ for all $n\geq N \}$.
\item[] $\limsup E_n=\{\lambda\in \mathbb{C}\,|\,$ for every $\epsilon>0$, there exists $J\subseteq \mathbb{N}$ infinite such that $B(\lambda, \epsilon)\cap E_n\neq \emptyset$ for all $n\in J \}$.
\end{itemize}
\begin{rem}\label{rem2}
Let $(E_n)$ be a sequence of non-empty subsets of $\mathbb{C}$. The following properties are hold:
\begin{enumerate}[(a)]
\item $\liminf E_n$ and $\limsup E_n$ are closed subsets of $\mathbb{C}$.
\item $\lambda\in\limsup E_n$ if and only if there exists an increasing sequence of natural numbers $n_1<n_2<n_3<\cdots$\, and points $\lambda_{n_k}\in E_{n_k}$, for all $k\in\mathbb{N}$, such that $\lim \lambda_{n_k}=\lambda$.
\item $\lambda\in \liminf E_n$ if and only if there exists a sequence $\{\lambda_n\}$ such that $\lambda_n\in E_n$ for all $n\in\mathbb{N}$, and $\lim \lambda_n = \lambda$.
\end{enumerate}
\end{rem}
\begin{prop}\label{prop0}
Let $K,E,E_n$ be non-empty compact subsets of $\mathbb{C}$ such that $E_n\subseteq K$, for all $n\in\mathbb{N}$. Then $E_n\to E$ in the Hausdorff metric if and only if $\limsup E_n\subseteq E$ and
$E\subseteq\liminf E_n$.
\end{prop}
\section{Spectral continuity on complex Banach algebras}
Let $\mathcal{A}$ be a complex Banach algebra with identity $1_\mathcal{A}$. For $x\in\mathcal{A}$ the resolvent of $x$ is defined by $\rho(x)=\{\lambda\in\mathbb{C}: x-\lambda1_\mathcal{A} \textrm{ is invertible in }\mathcal{A}\}$ and the spectrum of $x$ is given by $\sigma(x)=\mathbb{C}\setminus\rho(x)$. The spectral radius $r(x)$ of $x$ is the number $r(x)=\max\{|\lambda|: \lambda\in\sigma(x)\}$. It is well known that $r(x)=\underset{n\to\infty}{\lim}\|x^n\|^\frac{1}{n}=\underset{n}{\inf}\|x^n\|^\frac{1}{n}$.
A sequence $(x_n)$ in $\mathcal{A}$ is said to be norm convergent to $x$ (in notation $x_n\overset{n}{\to} x$), if $\|x_n-x\|\to 0$. M. Ahues in \cite{MR1886113} introduces a new mode of convergence on $B(X)$, named $\nu-$convergence. This type of convergence can be generalized in the same way to complex unital Banach algebras: a sequence $(x_n)$ in $\mathcal{A}$ is said to be $\nu-$convergent to $x$, denoted by $x_n\overset{\nu}{\to}x$, if $(\|x_n\|)$ is bounded, $\|(x_n-x)x\|\to 0$ and $\|(x_n-x)x_n\|\to0$. The $\nu$-convergence is a pseudo-convergence in the sense that it is possible to have $x_n\overset{\nu}{\to}x$ and $x_n\overset{\nu}{\to}y$ but $x\neq y$, see for instance \cite[Example 1]{MR1886113}. There is a connection between norm convergence and $\nu$-convergence as follows: if $x_n\overset{n}{\to} x$ then $x_n\overset{\nu}{\to}x$, also, if $x_n\overset{\nu}{\to}x$ and $x$ is right invertible then $x_n\overset{n}{\to} x$.
A function $\tau$ defined on $\mathcal{A}$ whose values are non-empty compact sets in $\mathbb{C}$ is said to be continuous ($\nu$-continuous) at $x$, if $\tau(x_n)\to\tau(x)$ with respect to the Hausdorff metric, for all sequence $(x_n)$ in $\mathcal{A}$ such that $x_n\overset{n}{\to}x$ ($x_n\overset{\nu}{\to}x$). It is clear that if $\tau$ is $\nu$-continuous at $x$, then $\tau$ is continuous at $x$. $\tau$ is said to be upper semi-continuous (upper semi-$\nu$-continuous) at $x$, if $\limsup\tau(x_n)\subseteq \tau(x)$ for all sequence $(x_n)$ in $\mathcal{A}$ such that $x_n\overset{n}{\to}x$ ($x_n\overset{\nu}{\to}x$). Also, $\tau$ is said to be lower semi-continuous (lower semi-$\nu$-continuous) at $x$, if $\tau(x)\subseteq\liminf\tau(x_n)$ for all $(x_n)$ in $\mathcal{A}$ such that $x_n\overset{n}{\to}x$ ($x_n\overset{\nu}{\to}x$). From Proposition \ref{prop0} we have next remark.
\begin{rem}\label{rem0}
Let $\tau$ be a function defined on $\mathcal{A}$ whose values are non-empty compact sets in $\mathbb{C}$ such that $\tau(y)\subseteq\sigma(y)$ for all $y\in\mathcal{A}$. Then
\begin{enumerate}
\item $\tau$ is continuous at $x\in\mathcal{A}$ if and only if $\tau$ is upper and lower semi-continuous at $x$.
\item $\tau$ is $\nu$-continuous at $x\in\mathcal{A}$ if and only if $\tau$ is upper and lower semi-$\nu$-continuous at $x$.
\end{enumerate}
\end{rem}
Next theorem is proved in \cite[Corollary 2.7]{MR1886113}.
\begin{thm}\label{prop2a}
For each $x\in\mathcal{A}$, $\sigma$ is upper semi-$\nu$-continuous at $x$.
\end{thm}
In the following proposition we will use the notation $\sigma_\mathcal{B}(x)$ for the spectrum of $x\in\mathcal{B}$ with respect to a given subalgebra $\mathcal{B}$ of $\mathcal{A}$. A character on an abelian Banach algebra $\mathcal{B}$ is a non-zero homomorphism $\varphi:\mathcal{B}\to\mathbb{C}$. The set of all characters on $\mathcal{B}$ is denoted by $\mathcal{M}(\mathcal{B})$. It is well known that if $\mathcal{B}$ is an unital abelian Banach algebra then
\begin{enumerate}
\item $\varphi(x)\in\sigma_\mathcal{B}(x)$ for all $x\in\mathcal{B}$ and $\varphi\in\mathcal{M}(\mathcal{B})$;
\item for every $\lambda\in\sigma_\mathcal{B}(x)$, there exists $\varphi\in \mathcal{M}(\mathcal{B})$ such that $\varphi(x)=\lambda$;
\item for each $\varphi\in\mathcal{M}(\mathcal{B})$, $\|\varphi\|=1$.
\end{enumerate}
A very well known result in spectral continuity is that if the elements of a sequence $(x_n)$ in a Banach algebra $\mathcal{A}$ commute with their limit, i.e. if $x_n\overset{n}{\to}x$ and $x_nx=xx_n$, then $\sigma(x_n)\to\sigma(x)$. We generalize this result for the $\nu$-convergence, as shown in the following proposition.
\begin{prop}
Let $a\in\mathcal{A}$ and $(a_n)$ be a sequence in $\mathcal{A}$ such that $a_na=aa_n$ and $a_na_m=a_ma_n$ for all $n,m\in\mathbb{N}$. If $a_n\overset{\nu}{\to}a$ and $0\in\textup{acc\,}\sigma(a)$, then $\sigma(a_n)\to\sigma(a)$.
\end{prop}
\begin{proof}
By Theorem \ref{prop2a}, $\sigma$ is upper semi-$\nu$-continuous at $a$, thus $\limsup\sigma_\mathcal{A}(a_n)\subseteq\sigma_\mathcal{A}(a)$. Hence by Proposition \ref{prop0}, we only need to prove that $\sigma_\mathcal{A}(a)\subseteq\liminf\sigma_\mathcal{A}(a_n)$.
Consider $\mathcal{B}_0$ the subalgebra of $\mathcal{A}$ which consists of all linear combinations of finite products of elements in $\{a_n:n\in\mathbb{N}\}\cup\{a,1_\mathcal{A}\}$. From hypothesis, $\mathcal{B}_0$ is commutative. Thus by Zorn's lemma, there exists $\mathcal{B}$ the maximal abelian subalgebra of $\mathcal{A}$ such that $\mathcal{B}_0\subseteq\mathcal{B}$. Therefore by \cite[Exercise 8, p. 8]{MR1074574}, $\sigma_\mathcal{A}(a)=\sigma_\mathcal{B}(a)$ and $\sigma_\mathcal{A}(a_n)=\sigma_\mathcal{B}(a_n)$ for all $n\in\mathbb{N}$. Let $\lambda\in\sigma_\mathcal{A}(a)$ with $\lambda\neq 0$. Then there exists $\varphi\in \mathcal{M}(\mathcal{B})$ such that $\varphi(a)=\lambda$. Observe that $$\|(\varphi(a_n)-\varphi(a))\varphi(a)\|=\|\varphi((a_n-a)a)\|\leq\|\varphi\|\|(a_n-a)a\|\to0.$$
Thus $\|(\varphi(a_n)-\varphi(a))\lambda\|\to0$ which implies that $\varphi(a_n)\to\lambda$. Now, since $\varphi(a_n)\in\sigma_\mathcal{B}(a_n)(=\sigma_\mathcal{A}(a_n))$ for all $n\in\mathbb{N}$, it follows from Remark \ref{rem2} that $\lambda\in\liminf\sigma_\mathcal{A}(a_n)$. Consequently, since $0\in\textup{acc\,}\sigma_\mathcal{A}(a)$, $$\sigma_\mathcal{A}(a)=\overline{\sigma_\mathcal{A}(a)\setminus\{0\}}\subseteq\liminf\sigma_\mathcal{A}(a_n).$$
\end{proof}
An elementary Cauchy domain is a bounded open connected subset of $\mathbb{C}$ whose boundary is the union of a finite number of nonintersecting Jordan curves. A finite union of elementary Cauchy domains having disjoint closures is called a Cauchy domain. Let $D$ be a Cauchy domain, if each curve involved in the boundary of $D$ is oriented in such a way that points in $D$ lie to the left as the curve is traced out, then the oriented boundary $C$ of $D$ is called a Cauchy contour. The interior of the Cauchy contour $C$ is defined as $\textrm{int}(C)=D$ and the exterior of $C$ is defined by $\textrm{ext}(C)=\mathbb{C}\setminus(D\cup C)$.
A set $\Lambda\subseteq\sigma(a)$ is a spectral set for $a$ if $\Lambda$ is closed as well as open in $\sigma(a)$. We set $\mathcal{C}(a,\Lambda)$ the set of all Cauchy contours $C$ separating $\Lambda$ from $\sigma(a)\setminus\Lambda$, i.e. $\Lambda\subseteq \textrm{int}(C)$ and $\sigma(a)\setminus\Lambda\subseteq \textrm{ext}(C)$. It is clear that if $C\in \mathcal{C}(a,\Lambda)$ then $C\subseteq \rho(a)$. For a spectral set $\Lambda$ for $a$ and $C\in \mathcal{C}(a,\Lambda)$, define $$p(a,\Lambda)=-\frac{1}{2\pi i}\int_C(a-z)^{-1}dz.$$ The element $p(a,\Lambda)$ does not depend on the choice of $C\in \mathcal{C}(a,\Lambda)$.
\begin{rem}\label{remc}
Let $a\in \mathcal{A}$, $\Lambda$ a spectral set for $a$ and $C\in \mathcal{C}(a,\Lambda)$. If $p=p(a,\Lambda)$, then
\begin{enumerate}
\item $p^2=p$ and $pa=ap$
\item $\Lambda=\emptyset$ if and only if $p(a,\Lambda)=0$.
\end{enumerate}
\end{rem}
\begin{prop}\label{prop1}
Let $p,q\in\mathcal{A}$ be such that $p^2=p$. If $p\neq 0$ and $r(p-q)<1$ then $q\neq 0$.
\end{prop}
\begin{proof}
Since $r(p-q)<1$ it follows that $(p-q)-1_\mathcal{A}$ is invertible. Suppose that $q=0$, then $p-1_\mathcal{A}$ is invertible. Thus there exists $z\in\mathcal{A}$ such that $(p-1_\mathcal{A})z=1_\mathcal{A}$. This implies that $p(p-1_\mathcal{A})z=p1_\mathcal{A}$ and so $(p^2-p)z=p$ i.e. $0=(p-p)z=p$, which is a contradiction.
\end{proof}
\begin{thm}\cite[Proposition 2.9]{MR1886113}\label{thm1}
Let $a\in\mathcal{A}$, $\Lambda$ be a spectral set for $a$, $C\in\mathcal{C}(a,\Lambda)$ and $(a_n)$ be a sequence in $\mathcal{A}$ such that $a_n\overset{\nu}{\to}a$. Then
\begin{enumerate}
\item There exists $n_0\in\mathbb{N}$ such that for every $n\geq n_0$, $C$ lies in $\rho(a_n)$.
\item If $\Lambda_n:=\sigma(a_n)\cap \textup{int}(C)$ for all $n\geq n_0$, then $\Lambda_n$ is a spectral set for $a_n$ and $C\in \mathcal{C}(a_n,\Lambda_n)$. Further, if $0\in \textup{ext}(C)$ then $p(a_n,\Lambda_n)\overset{\nu}{\to}p(a,\Lambda)$.
\end{enumerate}
\end{thm}
Next lemma is a generalization of \cite[Lemma 1.5]{MR543882}.
\begin{lem}\label{lemmma1}
Let $a\in \mathcal{A}$ and $(a_n)$ be a sequence in $\mathcal{A}$ such that $a_n\overset{\nu}{\to} a$. If $U$ is an open set for which $0\not\in U$ and $U$ contains a component of $\sigma(a)$, then there exits $n_0\in\mathbb{N}$ such that $U$ contains a component of $\sigma(a_n)$ for all $n\geq n_0$.
\end{lem}
\begin{proof}
Let $\Omega$ be a component of $\sigma(a)$ and $U$ be an open set of $\mathbb{C}$ such that $0\not\in U$ and $\Omega\subseteq U$. Since $\Omega\cap [\sigma(a)\setminus U]=\emptyset$, $\sigma(a)\setminus U$ is closed and $\sigma(a)$ is compact, it follows that there exists $\Lambda\neq\emptyset$ open and closed set in $\sigma(a)$ such that $\Omega\subseteq \Lambda$ and $\Lambda\cap [\sigma(a)\setminus U]=\emptyset$. This implies that $\Lambda\subseteq U$. From \cite[Theorem 1.21]{MR1886113}, there exists a Cauchy domain $D$ such that
\begin{equation}\label{equation1}
\Lambda\subset D\hspace{.2cm}\textrm{ and }\hspace{.2cm} \overline{D}\subset U\cap [\mathbb{C}\setminus(\sigma(a)\setminus \Lambda)].
\end{equation}
Let $C$ be the boundary of $D$ oriented in way that $C$ be a Cauchy contour. It is clear by (\ref{equation1}) that $C\in\mathcal{C}(a,\Lambda)$. Then from Theorem \ref{thm1}, there exits $n_1\in\Bbb{N}$ such that for every $n\geq n_1$, $C$ lies in $\rho(a_n)$. Further, if $\Lambda_n:=\sigma(a_n)\cap D$ for all $n\geq n_1$ then since $0\not\in U$ we have that
\begin{equation*}
[r(p-p_n)]^2\leq \|(p-p_n)^2\|\leq \|(p_n-p)p_n\|+\|(p_n-p)p\|\to 0,
\end{equation*}
where $p_n=p(a_n,\Lambda_n)$ and $p=p(a,\Lambda)$. Thus there exists $n_0\in\mathbb{N}$ with $n_0\geq n_1$ such that $r(p-p_n)<1$ for all $n\geq n_0$. Since $\Lambda\neq\emptyset$ we have from Remark \ref{remc} that $p\neq 0$. Therefore, by Proposition \ref{prop1}, $p_n\neq 0$ for all $n\geq n_0$. Thus by Remark \ref{remc}, $\Lambda_n\neq \emptyset$ for all $n\geq n_0$. This implies, since $\Lambda_n$ is both open and closed in $\sigma(a_n)$, that there exists $\Omega_n$ a component of $\sigma(a_n)$ such that $\Omega_n\subseteq\Lambda_n$. Observe that $\Lambda_n\subseteq D \subseteq U$. Thus $\Omega_n\subseteq U$. Therefore $U$ contains a component of $\sigma(a_n)$ for all $n\geq n_0$.
\end{proof}
\begin{rem}
From Lemma \ref{lemmma1} we have that if $\lambda\in\textup{iso\,}\sigma(a)$ with $\lambda\neq 0$, then $\lambda\in\liminf\sigma(a_n)$ for all sequence $(a_n)$ in $\mathcal{A}$ such that $a_n\overset{\nu}\to a$.
\end{rem}
\section{Spectral continuity on the algebra $B(X)$}
Let $X$ be a Banach space and let $B(X)$ be the algebra of all bounded linear operators defined on $X$. For $T\in B(X)$, let $N(T)$ and $R(T)$ be denote the null space and the range of the mapping $T$. Let $\alpha(T) = \dim N(T)$ and $\beta(T) = \dim X/R(T)$, if these spaces are finite dimensional, otherwise let $\alpha(T)=\infty$ and $\beta(T)=\infty$. If the range $R(T)$ of $T\in B(X)$ is closed and $\alpha(T)<\infty$ then $T$ is said to be an {\it upper semi-Fredholm} operator ($T\in \Phi_+(X)$). Similarly, if $\beta(T)<\infty$ then $T$ is said to be a {\it lower semi-Fredholm} operator ($T\in\Phi_-(X)$). If $T\in \Phi_-(X)\cup\Phi_+(X)$ then $T$ is called a
{\it semi-Fredholm} operator ($T\in \Phi_{\pm}(X)$) and for $T\in \Phi_-(X)\cap\Phi_+(X)$ we say that $T$ is a {\it Fredholm} operator ($T\in \Phi(X)$). For $T\in \Phi_{\pm}(X)$, the {\it index} of $T$ is defined by $i(T)=\alpha(T)-\beta(T).$ It is well known that the index is a continuous function on the set of semi-Fredholm operators. This property also holds for the $\nu$-convergence in the following sense:
\begin{thm}\cite[Theorem 3.4]{MR3388799}\label{thmindex}
Let $T\in \Phi(X)$ and $(T_n)$ be a sequence in $B(X)$ such that $(T_n-T)T\overset{n}{\to}0$. If $\lambda,\lambda_n$ are complex numbers such that $\lambda_n\to\lambda$ and $$T-\lambda,T_n-\lambda_n\in \Phi_\pm(X)$$ for all $n\in\mathbb{N}$, then $i(T_n-\lambda_n)\to i(T-\lambda)$.
\end{thm}
For $T\in B(X)$ and $k\in\mathbb{N}\cup\{-\infty,\infty\}$, let $\rho_{sf}^k(T)$ be denote the set of $\lambda\in\mathbb{C}$ for which $ T-\lambda\in\Phi_{\pm}(X)$ and $i(T-\lambda )=k$. Put
\begin{align*}
\rho^{-}_{sf}(T)=\underset{-\infty\leq k\leq -1}{\cup}\rho_{sf}^k(T),\hspace{.3cm}\rho^{+}_{sf}(T)=\underset{1\leq k\leq\infty}{\cup}\rho_{sf}^k(T),\hspace{.3cm}\rho_{sf}^\pm(T)=\rho_{sf}^-(T)\cup\rho_{sf}^+(T).
\end{align*}
All these sets are open and are contained in $\sigma(T)$. Let $\sigma_{sf}(T)$ be denote the set of all $\lambda\in \mathbb{C}$ such that $T-\lambda\not\in\Phi_\pm(X)$.
Let $K(X)$ be denote the set of all compact linear operators in $B(X$). If $\pi : B(X) \to B(X)/K(X)$ is the canonical homomorphism, then the essential spectrum of an operator $T \in B(X)$, $\sigma_e(T)$, is the spectrum of $\pi(T)$ in the Calkin algebra $B(X)/K(X)$. Also, the left essential spectrum $\sigma_{le}(T)$ (the right essential spectrum $\sigma_{re}(T)$) is the left spectrum (right spectrum) of $\pi(T)$. We set $\sigma_{lre}(T)=\sigma_{le}(T)\cap\sigma_{re}(T)$. It is clear that
\begin{equation}\label{eq0}
\sigma_{sf}(T)\subseteq\sigma_{lre}(T)
\end{equation}
but the opposite inclusion is not always satisfied in general Banach spaces. These classes of sets coincide in the case of Hilbert spaces.
The approximate point spectrum, the surjective spectrum, the point spectrum, the Weyl spectrum and the set Riesz points of $T\in B(X)$ are defined by $\sigma_{ap}(T)=\{\lambda\in\mathbb{C}:T-\lambda \textrm{ is not bounded below}\}$, $\sigma_s(T)=\{\lambda\in\mathbb{C}:T-\lambda \textrm{ is not surjective}\}$, $\sigma_p(T)=\{\lambda\in\mathbb{C}:\lambda$ is an eigenvalue of $T\}$, $\sigma_w(T)=\{\lambda\in\mathbb{C}:T-\lambda$ is not a Fredholm operator of index zero$\}$ and $\pi_0(T)=\{\lambda\in\mathbb{C}: \lambda$ is an isolated eigenvalue of $T$ of finite algebraic multiplicity$\}$.
\begin{rem}\label{rem1}
Let $T\in B(X)$ and $(T_n)$ be a sequence in $B(X)$ such that $T_n\overset{\nu}{\to} T$. The following inclusions are hold:
\begin{enumerate}
\item $\pi_0(T)\subseteq\liminf\pi_0(T_n)$. See, \cite[Corollary 2.13]{MR1886113}.
\item $\Delta\subseteq\liminf\sigma(T_n)$, where $\Delta=\big\{\lambda\in\sigma(T)\setminus(\overline{\rho_{sf}^\pm(T)}\cup\{0\}): $ for every $\epsilon>0$, the ball $B(\lambda,\epsilon)$ contains a component of $\sigma_{sf}(T)\cup\pi_0(T)\big\}$ and $T$ satisfies Browder's theorem. Really, take $\lambda\in \Delta$ then $\lambda\in\sigma(T)\setminus\overline{\rho_{sf}^\pm(T)}$ and $\lambda\neq 0$, so there exists $r>0$ such that $B(\lambda,r)\cap \overline{\rho_{sf}^\pm(T)}=\emptyset$ and $0\not\in B(\lambda,r)$. Let $\epsilon>0$ with $\epsilon<r$, since $\lambda\in\Delta$, the ball $B(\lambda,\epsilon)$ contains a component $\Omega$ of $\sigma_{sf}(T)\cup\pi_0(T)$. By \cite[Lemma 3.6]{sanchez4}, $\Omega$ is a component of $\sigma(T)$. Therefore by Lemma \ref{lemmma1}, there exists $n_0\in\mathbb{N}$ such that $B(\lambda,\epsilon)$ contains a component $\Omega_n$ of $\sigma(T_n)$ for all $n\geq n_0$. Thus $B(\lambda,\epsilon)\cap \sigma(T_n)\neq \emptyset$ for all $n\geq n_0$. This implies that $\lambda\in \liminf\sigma(T_n)$.
\item $[\textup{iso}\,\sigma(T)]\setminus\{0\}\subseteq\liminf\sigma(T_n)$.
\suspend{enumerate}
Moreover, if $T\in \Phi(X)$ then
\resume{enumerate}
\item $\overline{\rho_{sf}^+(T)}\subseteq\liminf\sigma_a(T_n)$. See, \cite[Theorem 3.6]{MR3388799}.
\item $\overline{\rho_{sf}^-(T)}\subseteq\liminf\sigma_s(T_n)$.
\item $\overline{\rho_{sf}^\pm(T)}\subseteq\liminf\sigma_w(T_n)$.
\item $\Gamma\subseteq\liminf\sigma_{sf}(T_n)$, where $\Gamma=\big\{\lambda\in\sigma(T):$ for every $\epsilon>0$, there exist points $\mu_1,\mu_2\in B(\lambda,\epsilon)$ such that $T-\mu_1,T-\mu_2\in \Phi_\pm(X)$ and $i(T-\mu_1)\neq i(T-\mu_2)\big\}$. This result was established in \cite[Proposition 2.1]{MR634170} for the norm convergence. The proof for the $\nu$-convergence is in a similar way.
\end{enumerate}
\end{rem}
\begin{thm}\label{thm3.2}
Let $T\in \Phi(X)$ be such that $0\in\textup{acc}\sigma(T)$. If for each $\lambda\in\sigma(T)\setminus\overline{\rho_{sf}^\pm(T)}$ and $\epsilon>0$, the ball $B(\lambda,\epsilon)$ contains a component of $\sigma_{sf}(T)\cup\pi_0(T)$, then $\sigma$ is $\nu$-continuous at $T$.
\end{thm}
\begin{proof}
First observe that $\sigma(T)\setminus\sigma_w(T)=\pi_0(T)\cup\textrm{int}[\sigma(T)\setminus\sigma_w(T)]$, see for example \cite[Proposition 2.1]{MR2894894}. By hypothesis we have that $\textrm{int}[\sigma(T)\setminus\sigma_w(T)]=\emptyset$, thus $T$ satisfies Browder's theorem.
From Remark \ref{rem0} and Theorem \ref{prop2a} it is sufficient to prove that $\sigma$ is lower semi-$\nu$-continuous at $T$. Let $(T_n)$ be a sequence in $B(X)$ such that $T_n\overset{\nu}{\to}T$. By Remark \ref{rem1} (6), $\overline{\rho_{sf}^\pm(T)}\subseteq\liminf\sigma(T_n)$. Now, from hypothesis, $\sigma(T)\setminus(\overline{\rho_{sf}^\pm(T)}\cup\{0\})\subseteq\Delta$, therefore by Remark \ref{rem1} (2), $\sigma(T)\setminus(\overline{\rho_{sf}^\pm(T)}\cup\{0\})\subseteq\liminf\sigma(T_n)$. Consequently, $\sigma(T)=\overline{\sigma(T)\setminus\{0\}}\subseteq\liminf\sigma(T_n)$. Thus $\sigma$ is lower semi-$\nu$-continuous at $T$.
\end{proof}
\begin{cor}
Let $H$ be a Hilbert space and $T\in \Phi(H)$ be such that $0\in\textup{acc}\sigma(T)$. Then, $\sigma$ is continuous at $T$ if and only if $\sigma$ is $\nu$-continuous at $T$.
\end{cor}
\begin{proof}
It is clear that the $\nu$-continuity of $\sigma$ at $T$ implies the continuity of $\sigma$ at $T$. Now, if $\sigma$ is continuous at $T$, then by \cite[Theorem 3.1]{MR543882}, for each $\lambda\in\sigma(T)\setminus\overline{\rho_{sf}^\pm(T)}$ and $\epsilon>0$, the ball $B(\lambda,\epsilon)$ contains a component of $\sigma_{sf}(T)\cup \pi_0(T)$. Therefore, by Theorem \ref{thm3.2}, $\sigma$ is $\nu$-continuous at $T$.
\end{proof}
\begin{thm}\label{limsupap}
Let $T\in \Phi(X)$. If $(T_n)$ is a sequence in $B(X)$ such that $T_n\overset{\nu}{\to}T$ then $$[\lim\sup \sigma_{ap}(T_n)]\setminus\{0\}\subseteq\sigma_{ap}(T).$$
\end{thm}
\begin{proof}
Let $D,E$ be closed subspaces of $X$ with $\dim E<\infty$ such that
\begin{equation}\label{sum}
X=N(T)\oplus D \hspace{.2cm}\textrm{ and }\hspace{.2cm} X=R(T)\oplus E.
\end{equation}
Consider $(T_n)$ a sequence in $B(X)$ which is $\nu-$convergent to $T$. Let $\lambda\in \limsup\sigma_{ap}(T_n)$ with $\lambda\neq 0$. By Remark \ref{rem2}, there exist an increasing sequence of natural numbers $(n_k)$ and points $\lambda_{n_k}\in\sigma_{ap}(T_{n_k})$ such that $\lambda_{n_k}\to\lambda$. Suppose that $\lambda\not \in \sigma_{ap}(T)$. Then $T-\lambda\in\Phi_+(X)$ and $\alpha(T-\lambda)=0$. By (\ref{sum}), $R(T|_D)=R(T)$ and so $T|_D$ is bounded below, therefore by \cite[Theorem 5.26]{MR1861991}, $(T-\lambda)T|_D\in \Phi_+(D,X)$ and $\alpha((T-\lambda)T|_D)=0$.
On the other hand, observe that
\begin{align*}
(T_{n_k}-\lambda_{n_k})T|_D&=(T-\lambda)T|_D+(T_{n_k}-T)T|_D+(\lambda-\lambda_{n_k})T|_D.
\end{align*}
From $\|(T_n-T)T\|\to 0$ we have that $(T_{n_k}-\lambda_{n_k})T|_D$ converges in norm to $(T-\lambda)T|_D$. Consequently by \cite[Theorem 5.23]{MR1861991}, there exists $k_0\in\mathbb{N}$ such that every $k\geq k_0$, $(T_{n_k}-\lambda_{n_k})T|_D\in \Phi_+(D,X)$ and $\alpha((T_{n_k}-\lambda_{n_k})T|_D)=0$.
Suppose that for each $k\geq k_0$, $$N(T_{n_k}-\lambda_{n_k})\cap E\neq\{0\}.$$ Take $v_k\in N(T_{n_k}-\lambda_{n_k})\cap E$ with $\|v_k\|= 1$ for all $k\geq k_0$. Since $\dim E<\infty$ it follows that $F:=\{e\in E:\|e\|=1\}$ is a compact set. Therefore we may assume without loss of generality that there exists $v\in F$ such that $v_{k}\to v$. Observe that
\begin{align*}
\|(T_{n_k}-T)T_{n_k}\|&\geq \|(T_{n_k}-T)T_{n_k}v_{k}\|\\
&=\|(T_{n_k}-T)\lambda_{n_k}v_{k}\|\\
&=|\lambda_{n_k}|\|\lambda_{n_k}v_k-Tv_k\|
\end{align*}
for all $k\geq k_0$, and $ |\lambda_{n_k}|\|\lambda_{n_k}v_k-Tv_k\|\to |\lambda|\|\lambda v-Tv\|$. This implies that $$|\lambda|\|\lambda v-Tv\| = \lim|\lambda_{n_k}|\|\lambda_{n_k}v_k-Tv_k\|\leq \lim \|(T_{n_k}-T)T_{n_k}\|=0,$$ and so $\|\lambda v-Tv\|=0$, i.e. $Tv=\lambda v$. Consequently, $\lambda\in\sigma_p(T)(\subseteq\sigma_{ap}(T))$, which is a contradiction. Therefore there exists $k'\geq k_0$ such that $N(T_{n_{k'}}-\lambda_{n_{k'}})\cap E=\{0\}$. Thus
\begin{equation}\label{suma2}
X=R(T)\oplus N(T_{n_{k'}}-\lambda_{n_{k'}})\oplus E.
\end{equation}
Then by (\ref{sum}) and (\ref{suma2}),
\begin{align*}
\dim E=\dim X/R(T)& =\dim[N(T_{n_{k'}}-\lambda_{n_{k'}})\oplus E]\\
&=\dim N(T_{n_{k'}}-\lambda_{n_{k'}})+\dim E.
\end{align*}
Hence $\dim N(T_{n_{k'}}-\lambda_{n_{k'}})=0$ and so $N(T_{n_{k'}}-\lambda_{n_{k'}})=\{0\}$. From (\ref{suma2}), $$R(T_{n_{k'}}-\lambda_{n_{k'}})=(T_{n_{k'}}-\lambda_{n_{k'}})T(D)+(T_{n_{k'}}-\lambda_{n_{k'}})(E),$$ which implies that $R(T_{n_{k'}}-\lambda_{n_{k'}})$ is closed. Therefore $\lambda_{n_{k'}}\not\in\sigma_{ap}(T_{n_{k'}})$, a contradiction. Consequently, $\lambda\in \sigma_{ap}(T)$.
\end{proof}
Let $\phi_+(T)$ be the set of $\lambda\in\rho^{+}_{sf}(T)$ such that $N(T-\lambda)$ is complemented in $X$ and $\phi_-(T)$ the set of $\lambda\in \rho^-_{sf}(T)$ such that $R(T-\lambda)$ is complemented in $X$. We set $\phi_\pm(T)=\phi_+(T)\cup\phi_-(T)$, $\phi_{+\infty}(T)=\{\lambda\in\phi_+(T)\ |\ i(T-\lambda)=\infty\}$, $\phi_{-\infty}(T)=\{\lambda\in\phi_-(T)\ |\ i(T-\lambda)=-\infty\}$ and $\phi_{\pm\infty}(T)=\phi_{+\infty}(T)\cup\phi_{-\infty}(T)$. Al these sets are open and it is clear that
\begin{equation}
\sigma_e(T)=\sigma_{lre}(T)\cup\phi_{\pm\infty}(T).
\end{equation}
With this equality, \cite[Proposition 1.3]{MR543882} can be extended to general Banach spaces:
\begin{prop}\label{prop2}
If $C$ is a component of $\sigma_{lre}(T)$ and $C\cap\overline{\phi_\pm(T)}=\emptyset$ then $C$ is a component of $\sigma_e(T)$.
\end{prop}
\begin{thm}
If the following conditions hold
\begin{enumerate}[i.]
\item $0\in\textup{acc}\sigma_{ap}(T)$
\item $T\in \Phi(X)$
\item $T$ satisfies Browdwer's theorem
\item $\sigma_p(T)\cap\phi_{-}(T)=\emptyset$
\item for every $\epsilon>0$ and $\lambda\in \sigma_{lre}(T)\setminus\overline{\phi_\pm(T)}$, the ball $B(\lambda,\epsilon)$ contains a component of $\sigma_{lre}(T)$,
\end{enumerate}
then $\sigma_{ap}(T_n)\to\sigma_{ap}(T)$ for all $T_n\overset{\nu}{\to}T$.
\end{thm}
\begin{proof}
From conditions (i) and (ii) we have by Theorem \ref{limsupap} that $$\limsup\sigma_{ap}(T_n)\subseteq\sigma_{ap}(T).$$
Now, we show that $\sigma_{ap}(T)\subseteq\liminf\sigma_{ap}(T_n)$. Take $\lambda\in\sigma_{ap}(T)$ with $\lambda\neq 0$. Suppose that $\lambda\in\sigma_{lre}(T)\setminus\overline{\phi_\pm(T)}$. Let $\epsilon>0$, there exists $0<\epsilon_1<\epsilon$ such that $B(\lambda,\epsilon_1)\cap\overline{\phi_\pm(T)}=\emptyset$ and $0\not\in B(\lambda,\epsilon_1)$. By hypothesis (v), $B(\lambda,\epsilon_1)$ contains a component $\Omega$ of $\sigma_{lre}(T)$. Then by Proposition \ref{prop2}, $\Omega$ is a component of $\sigma_e(T)$. Since $T_n\overset{\nu}{\to}T$ we have that $\pi(T_n)\overset{\nu}{\to}\pi(T)$ in the Calkin algebra $B(X)/K(X)$. Thus by Lemma \ref{lemmma1}, there exists $n_0\in\mathbb{N}$ such that $B(\lambda,\epsilon_1)$ contains a component $\Omega_n$ of $\sigma_e(T_n)$ for all $n\geq n_0$. Consequently $$\emptyset\neq \partial \Omega_n\subseteq\partial \sigma_e(T_n)\cap B(\lambda,\epsilon)\subseteq \sigma_{sf}(T_n)\cap B(\lambda,\epsilon)$$
for all $n\geq n_0$. Therefore $\lambda\in\liminf\sigma_{sf}(T_n)(\subseteq\liminf\sigma_{ap}(T_n))$.
If $\lambda\not\in\sigma_{lre}(T)\setminus\overline{\phi_\pm(T)}$, then $\lambda\in\phi_\pm(T)$ or $i(T-\lambda)=0$. From condition (iv), we have that $\lambda\not\in\phi_-(T)$. Thus from condition (iii), $\lambda\in \phi_+(T)\cup\pi_0(T)$. Therefore, by Remark \ref{rem1}, $\lambda\in\liminf\sigma_{ap}(T_n)$.
\end{proof}
\section{On certain class of operators}
We say that an operator $T\in B(H)$ is essentially $G_1$ if $\|(\pi(T)-z)^{-1}\|=\frac{1}{d(z,\sigma_e(T))}$ for all $z\not\in \sigma_e(T)$. In \cite[Theorem 6]{MR433257} it is shown that the restriction of the Wey spectrum on the class of essentially $G_1$ operators is continuous. This is also true for $\nu$-continuity, as the following theorem states.
\begin{thm}\label{thmB}
Let $T\in \Phi(H)$ be such that $0\not\in \sigma_w(T)$ or $0\in\textrm{acc}\sigma_w(T)$. If $(T_n)$ is a sequence of essentially $G_1$ operators such that $T_n\overset{\nu}{\to}T$ then $\sigma_w(T_n)\to\sigma_w(T)$.
\end{thm}
\begin{proof}
From \cite[Theorem 3.3]{MR3624565} we have that $\limsup\sigma_w(T_n)\subseteq \sigma_w(T)$. We show that $\sigma_w(T)\subseteq\liminf\sigma_w(T_n)$. Let $\lambda\in\sigma_w(T)\setminus\{0\}$ and suppose that $\lambda\not\in\liminf\sigma_w(T_n)$. Then, there exist $\epsilon>0$ such that $B_\epsilon(\lambda)\cap\sigma_w(T_{n}) = \emptyset$ for infinite number of $n$'s. Without loss of generality assume that this holds for all $n$. This implies that $d(\lambda,\sigma_e(T_{n}))\geq d(\lambda,\sigma_w(T_{n}))\geq \epsilon>0$. Thus $\lambda\not\in\sigma_e(T_{n})$, now from the fact that $T_{n}$ is essentially $G_1$, we have that $\|(\pi(T_{n})-\lambda)^{-1}\|=\frac{1}{d(\lambda,\sigma_e(T_{n}))}\leq\frac{1}{\epsilon}$. This implies that
$\Big\|\Big[\big(\pi(T_n)-\pi(T)\big) (\pi(T_{n})-\lambda)^{-1}\Big]^2\Big\| \leq$
\begin{align}\label{eqvn0}
&\leq \frac{1}{|\lambda|}\Big[\|\big(\pi(T_n)-\pi(T)\big)\pi(T_{n})\|\|(\pi(T_{n})-\lambda)^{-1}\|\|\pi(T_n)-\pi(T)\|\nonumber\\
&\ \ \ \ \ +\ \|\big(\pi(T_n)-\pi(T)\big)\pi(T)\|+\|\big(\pi(T_n)-\pi(T)\big)\pi(T_n)\|\Big]\|(\pi(T_{n})-\lambda)^{-1}\|\nonumber\\
&\leq\frac{1}{|\lambda|}\Big[\|(T-T_{n})T_{n}\|\|T_n-T\|\frac{1}{\epsilon}+\|(T-T_{n})T\|+\|(T-T_{n})T_{n}\|\Big]\frac{1}{\epsilon}.
\end{align}
Now, since $T_n\overset{\nu}{\to}T$, it follows that $\|(T_n-T)T\|\to 0$, $\|(T_n-T)T_{n}\|\to 0$ and $(\|T_{n}\|)$ is bounded. Therefore the expression (\ref{eqvn0}) tends to 0. Thus, there exists $n^*\in\mathbb{N}$ such that $$\Big\|\Big[\big(\pi(T_{n^*})-\pi(T)\big) (\pi(T_{n^*})-\lambda)^{-1}\Big]^2\Big\|<1.$$
Then $r\Big(\big(\pi(T_{n^*})-\pi(T)\big) (\pi(T_{n^*})-\lambda)^{-1}\Big)<1,$ which implies that $\big(\pi(T_{n^*})-\pi(T)\big) (\pi(T_{n^*})-\lambda)^{-1}-1$ is invertible and so
\begin{align*}
\pi(T)-\lambda&=(\pi(T_{n^*})-\lambda)-(\pi(T_{n^*})-\pi(T))\\
&=-\Big[\big(\pi(T_{n^*})-\pi(T)\big) (\pi(T_{n^*})-\lambda)^{-1}-1\Big](\pi(T_{n^*})-\lambda)
\end{align*}
is invertible. Consequently, $\lambda\not\in\sigma_e(T)$, i.e. $T-\lambda$ is a Fredholm operator. Finally, since $\lambda\not\in\sigma_w(T_n)$ for all $n\in\mathbb{N}$, it follows by Theorem \ref{thmindex} that $i(T_n-\lambda)\to i(T-\lambda)$, and so $i(T-\lambda)=0$, which is a contradiction.
\end{proof}
\begin{thm}
Let $T\in \Phi(H)$ be such that satisfies Browder's theorem. If $0\in\textrm{acc}\sigma_w(T)$ and $(T_n)$ is a sequence of essentially $G_1$ operators such that $T_n\overset{\nu}{\to}T$, then $\sigma(T_n)\to\sigma(T)$.
\end{thm}
An operator $A\in B(H)$ is called $p$-hyponormal if $(A^*A)^p-(AA^*)^p\geq 0$. For the case $p=1$ the operator $A$ is called hyponormal. It is well known that the restriction of the spectrum on the class of $p$-hyponormal operators is continuous. See, \cite{MR1814478} and \cite{MR1785076}. In the following theorem we use the idea of \cite{MR1814478} and adapt it for the case of $\nu$-convergence. Note first that if $A\in B(H)$ is a $p$-hyponormal operator and $0\in \sigma_p(A)$ then $0\in \sigma_{jp}(A)$ and so $N(A)=N({A}^*)$ which implies that $N(A)$ is invariant for both $A$ and ${A}^*$. Therefore
\begin{equation}\label{eqbc}
A=\begin{bmatrix}
0&0\\
0&B
\end{bmatrix}
\end{equation}
on $N(A)\oplus N(A)^\perp$, where $B=A|_{N(A)^\perp}$ and $0\not\in\sigma_p(B)$. From \cite[Lemma 4]{MR1670405}, $B$ is also $p$-hyponormal. We claim that $0\not\in\sigma(|B|)$, indeed if $0\in\sigma(|B|)(=\sigma_{ap}(|B|))$ then there exists a sequence $(x_m)$ of unit vectors such that $|B|x_m\to0$. This implies that $Bx_m\to0$, thus $0\in\sigma_{ap}(B)$, but since $R(A)$ is closed i.e. $R(B)$ is closed, it follows that $0\in \sigma_p(B)$, which is a contradiction.
\begin{thm}\label{thmd}
If $T_n,T$ are operators in $B(H)$ such that
\begin{enumerate}
\item $T_{n}\overset{\nu}{\to}T$,
\item $T\in \Phi(H)$ and $T_n$ is $p$-hyponormal for all $n\in\mathbb{N}$,
\item $0\in\sigma_p(T_n)$ for all $n\in\mathbb{N}$, and the sequence $(\||B_n|^{-1}\|)$ is bounded, where the operators $|B_n|$ are as in (\ref{eqbc}),
\end{enumerate}
then $\sigma(T_{n})\to\sigma(T)$.
\end{thm}
\begin{proof}
First observe that $\|T_nT-T^2\|=\|(T_n-T)T\|\to0$, thus $T_nT\overset{n}{\to}T^2$, which implies that $T_nT\in \Phi(H)$ for all $n$ large. Thus we may suppose that $R(T_n)$ is closed for all $n\in\mathbb{N}$. We show that there exist a sequence $(S_n)$ of hyponormal operators and a sequence $(X_n)$ of invertible operators such that $S_n=X_nT_nX^{-1}_n$ for all $n\in\mathbb{N}$, and $(\|X_{n}\|)$, $(\|X_{n}^{-1}\|)$ are bounded.
From condition (2), $0\in\sigma_{p}(T_{n})$ for all $n\in\mathbb{N}$. Then $T_{n}=\begin{bmatrix}
0&0\\
0&B_{n}
\end{bmatrix}$ on $N(T_{n})\oplus N(T_{n})^{\perp}$, $0\not\in\sigma_p(B_n)$, $B_n$ is $p$-hyponormal and $0\not\in\sigma(|B_n|)$. Consider the polar decomposition $B_n=U_n|B_n|$ and define $\widehat{B}_n=|B_n|^{1/2}U_n|B_n|^{1/2}$. Observe that if $x\in N(\widehat{B}_n)$ then $|B_n|^{1/2}U_n|B_n|^{1/2}x=0$ and so $B_n|B_n|^{-1}|B_n|^{1/2}x=0$ which implies that $|B_n|^{-1}|B_n|^{1/2}x=0$ because $0\not\in\sigma_p(B_m)$, hence $x=0$. Thus $N(\widehat{B}_n)=\{0\}$ i.e. $0\not\in\sigma_p(\widehat{B}_n)$. This implies that $0\not \in\sigma(|\widehat{B}_n|)$. Let $\widehat{B}_n$ have the polar decomposition $\widehat{B}_n=V_n|\widehat{B}_n|$, then by \cite[Corollary 3]{MR1047771}, the operator $\widetilde{B}_n$, defined by $\widetilde{B}_n=|\widehat{B}_n|^{1/2}V_n|\widehat{B}_n|^{1/2}$, is hyponormal. Define
$$X_n=\begin{bmatrix}
1&0\\
0&|\widehat{B}_n|^{1/2}|B_n|^{1/2}
\end{bmatrix} \hspace{.3cm}\textrm{and}\hspace{.3cm} S_n=\begin{bmatrix}
0&0\\
0&\widetilde{B}_n
\end{bmatrix}.$$ Then $S_n$ is hyponormal, $X_n$ is invertible and $S_n=X_nT_nX_n^{-1}$. From condition (2) we have that $(\|X_n^{-1}\|)$ is bounded. Also it is clear that $(\|X_n\|)$ is bounded.
We show that $\sigma(T)\subseteq\liminf\sigma(T_n)$. Take $\lambda\in\sigma(T)\setminus\{0\}$ and suppose that $\lambda\not\in\liminf\sigma(T_n)$. Then we may assume that there exists $\epsilon>0$ such that $B(\lambda,\epsilon)\cap \sigma(T_n)=\emptyset$ for all $n\in\mathbb{N}$. This implies that $T_n-\lambda$ is invertible. In a similar way to proof of Theorem \ref{thmB}, we have that
\begin{align}\label{eqvn}
\Big\|\Big[\big(T_n-T) (T_{n}-\lambda)^{-1}\Big]^2\Big\| &\leq \frac{1}{|\lambda|}\Big[\|(T_n-T)T_{n}\|\|(T_{n}-\lambda)^{-1}\|\|T_n-T\|\nonumber\\
&\ \ \ +\|(T_n-T)T\|+\|(T_n-T)T_n\|\Big]\|(T_{n}-\lambda)^{-1}\|.
\end{align}
Since $T_n$ and $S_n$ are similar, it follows that $\sigma(T_n)=\sigma(S_n)$. Therefore, $d(\lambda,\sigma(S_n))$ $\geq \epsilon$ and $S_n-\lambda$ is invertible. Note that $(T_n-\lambda)^{-1}=X_n^{-1}(S_n-\lambda)^{-1}X_n$. Moreover, since $S_n$ is hyponormal it follows that $\|(S_n-\lambda)^{-1}\|=\frac{1}{d(\lambda,\sigma(S_n))}\leq\frac{1}{\epsilon}$. Thus the right term of (\ref{eqvn}) is bounded by
\begin{equation}\label{eq13}
\frac{1}{|\lambda|}\Big[\|(T_n-T)T_{n}\|\frac{M_1M_2}{\epsilon}\|T_n-T\|+\|(T_n-T)T\|+\|(T_n-T)T_n\|\Big]\frac{M_1M_2}{\epsilon},
\end{equation}
where $M_1,M_2$ are constants such that $\|X_n\|\leq M_1$ and $\|X_n^{-1}|\leq M_2$ for all $n\in\mathbb{N}$. Since $T_n\overset{\nu}{\to}T$ it follows that the expresion in (\ref{eq13}) tends to zero. Proceeding similarly to the final part of the proof of Theorem \ref{thmB}, we obtain that $T-\lambda$ is invertible, which is a contradiction.
\end{proof}
\begin{rem}
The conclusion of Theorem \ref{thmd} holds if we replace the hypothesis by the following conditions:
\begin{enumerate}
\item $T_{n}\overset{\nu}{\to}T$, $T^*(T_n-T)\overset{n}{\to}0$ and $T_n^*(T_n-T)\overset{n}{\to}0$;
\item $T\in \Phi(H)$ and $\{0\}\neq N(T)\subseteq N(T_{n})$ for all $n\in\mathbb{N}$;
\item $T,T_n$ are $p$-hyponormal operators.
\end{enumerate}
\end{rem}
Indeed, from condition (1), $|T_n|^2-|T|^2=T_n^*T_n-T^*T=T_n^*(T_n-T)+[T^*(T_n-T)]^*\overset{n}{\to}0$, thus $|T_n|^{1/2}\overset{n}{\to}|T|^{1/2}$. Since $0\in \sigma_p(T)$, it follows that $T=0\oplus B$ on $N(T)\oplus N(T)^{\perp}$ with $0\not\in\sigma(|B|)$. Then there exists $\alpha>0$ such that $\alpha\|y\|\leq \||B|^{1/2} y\|$ for all $y\in N(T)^\perp$. This implies by condition (2) that for $0<\epsilon<\alpha$, there exists $N\in\mathbb{N}$ such that for every $n\geq N$, $(\alpha-\epsilon)\|y\|\leq \||B_n|^{1/2}y\|$ for all $y\in N(T_n)^\perp$. Consequently, $\|(|B_n|^{1/2})^{-1}\|\leq \frac{1}{\alpha-\epsilon}$ for all $n\geq N$.
\medskip
Berberian shows that for every Hilbert space $H$, there exists a Hilbert space $K\supset H$ and a faithful $*-$representation $T\to T^\circ$ from $B(H)$ to $B(K)$: $(S+T)^\circ=S^\circ+T^\circ$, $(\lambda T)^\circ=\lambda T^\circ$, $(ST)^\circ = S^\circ T^\circ$, $(T^*)^\circ=(T^\circ)^*$, $(I_H)^\circ=I_K$ and $\|T^\circ\|=\|T\|$ such that
\begin{enumerate}
\item $T\geq0$ if and only if $T^\circ\geq 0$,
\item $\sigma_p(T^\circ)=\sigma_a(T^\circ)=\sigma_a(T)$.
\end{enumerate}
\begin{rem}
Observe that in the previous theorem, $\sigma_p(B_n)=\sigma_{ap}(B_n)$ due to $R(T_n)$ is closed. In \cite{MR1814478} the authors use the Berberian extension $T_n^\circ$ of a $p$-hyponormal operator $T_n$ and state that if $0\in\sigma_p(T_n^\circ)$, then
\begin{equation}\label{equation14}
\sigma_p(B_n)=\sigma_{ap}(B_n),
\end{equation}
where $T_n^\circ=0\oplus B_n$ on $N(T_n)\oplus N(T_n)^\perp$ and $0\not\in \sigma_p(B_n)$, without the need for $R(T_n^\circ)$ to be closed. This fact was also established in \cite{MR2651679}, page 586, line 20. The authors claim that
\begin{equation}\label{equation15}
\sigma_{ap}(B_\lambda)=\sigma_p(B_\lambda)
\end{equation}
for all non-zero $\lambda\in\sigma_p(A^\circ),$ where $A\in \mathcal{C}(i)$, and this collection is defined as the set of all operators $T\in B(H_i)$ for which $\sigma(T)=\{0\}$ implies $T$ is nilpotent and $T^\circ$ satisfies the property:
\begin{equation*}
T^\circ =\begin{bmatrix}
\lambda&X_\lambda\\
0&B_\lambda
\end{bmatrix} \hspace{.2cm}\textrm{ on }\hspace{.2cm} N(T^\circ-\lambda)\oplus N(T^\circ-\lambda)^\perp
\end{equation*}
at every non-zero $\lambda\in\sigma_p(T^\circ)$ for some operators $X_\lambda$ and $B_\lambda$ such that $\lambda\not\in\sigma_p(B_\lambda)$ and $\sigma(T^\circ)=\{\lambda\}\cup\sigma(B_\lambda)$. However, equalities (\ref{equation14}) and (\ref{equation15}) are not necessary hold. We prove only that (\ref{equation15}) is false. It is clear that $\sigma_a(B_\lambda)\setminus\{\lambda\}=\sigma_p(B_\lambda)$ and $\alpha(B_\lambda-\lambda)=0$, but $R(B_\lambda-\lambda)$ is not necessarily closed. Indeed, consider a normal operator $A\in B(H_i)$ such that $\sigma(A)=[0,1]$ (for example, the multiplication operator $A:L^2([0,1])\to L^2([0,1])$ defined by $A(f)(x)=xf(x)$). Then $A\in\mathcal{C}(i)$. We show that for every $\lambda\in \sigma_p(A^\circ)$, $R(B_\lambda-\lambda)$ is not closed. By contradiction, suppose that there exists $\lambda\in\sigma_p(A^\circ)$ such that $R(B_\lambda-\lambda)$ is closed. Then $B_\lambda-\lambda$ is a semi-Fredholm operator such that $\alpha(B_\lambda-\lambda)=0$. By \cite[Theorem 4.2.1]{MR0415345}, there exists $\epsilon>0$ such that if $|\gamma-\lambda|<\epsilon$ then $B_\lambda-\gamma\in \Phi_+(N(A^\circ-\lambda)^\perp)$ and $\alpha(B_\lambda-\gamma)=0$. This implies that $R(A^\circ-\gamma)=(\lambda-\gamma)N(A^\circ-\lambda)\oplus R(B_\lambda-\gamma)$ is closed and $\alpha(A^\circ-\gamma)=\alpha((\lambda-\gamma)I)+\alpha(B_\lambda-\gamma)=0$ for all $\gamma\in B(\lambda,\epsilon)$ with $\gamma\neq \lambda$. Therefore, $\lambda\in\textrm{iso}\,\sigma_a(A^\circ)$. On the other hand, since $A$ is a normal operator it follows that $\sigma(A)=\sigma_a(A)=\sigma_a(A^\circ)$. Thus, $\lambda\in\textrm{iso}\,\sigma(A)$, which is a contradiction, because $\sigma(A)=[0,1]$. Consequently, the equality (\ref{equation15}) is not true. This affects the proof of the main result in the paper \cite{MR2651679}.
\end{rem}
\bibliographystyle{abbrv}
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2024-02-18T23:40:24.841Z
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2020-09-22T02:00:08.000Z
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{"paloma_paragraphs":[]}
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{"arxiv_id":"2009.08977","language":"en","timestamp":1600740008000,"url":"https:\/\/arxiv.org\/abs\/2009.08977","yymm":"2009"}
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proofpile-arXiv_000-10099
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{"provenance":"002.jsonl.gz:10100"}
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\section{Introduction}
Data taken at the Large Hadron Collider (LHC) has been essential to confirm Standard Model (SM) predictions and perform precision measurements \cite{2012PhLB..716...30C, 2012PhLB..716....1A}.
Extensive efforts have been made to search for phenomena beyond the Standard Model (BSM) as well, such as supersymmetry, dark matter, extra dimensions, and many others.
Thus far, no BSM physics has been observed.
%
The lack of evidence for new physics suggests that the simplest scenarios are not good models of nature.
%
However, one should worry that more complicated signals might have evaded current LHC search strategies.
%
A large class of unusual BSM signals arise within the ``hidden-valley" (HV) scenario, in which the SM is weakly coupled to a hidden sector of self-interacting particles neutral under the SM gauge groups, at least one of which decays visibly \cite{Strassler:2006im, Strassler:2006ri, Strassler:2006qa, Han:2007ae}.
Similarly unusual signals arise in theories of ``quirks'' \cite{Okun:1980kw, Okun:1980mu,Strassler:2006im, Kang:2008ea}, and also in the context of unparticles \cite{Georgi:2007ek, Grinstein:2008qk, Randall:1999vf} which in the presence of a mass gap may produce hidden-valley phenomenology \cite{Strassler:2008bv}.
%
Since hidden sectors can be arbitrarily complex, many models of this type are poorly constrained by LHC searches, and since indirect limits on hidden sectors are often very weak, there are few other constraints.
%
But it is impossible to search through the full space of HV models, or of models in other general scenarios.
%
Instead, the practical approach to discovering a new signal is to carry out searches for distinct and parametrizable {\it signatures}, and to make this possible, models that produce these signatures must be developed. Among the unusual signatures identified in HV models to date are soft, unclustered energy patterns (SUEPs) \cite{Strassler:2008bv, Harnik:2008ax, Knapen:2016hky}; lepton jets \cite{ArkaniHamed:2008qp, Baumgart:2009tn}; emerging jets \cite{Strassler:2006im,Schwaller:2015gea}; semi-visible jets \cite{Strassler:2006im, Strassler:2008fv, Cohen:2015toa, Cohen:2017pzm}; and dark jets with unusual substructure \cite{Cohen:2020afv}.
%
General, flexible searches for new physics at the LHC require us not only to widen the range of models and signatures, but also to expand the tools available for data analysis, both offline and at the trigger stage (where fewer than 1\% of LHC collisions are recorded). Search strategies for general hidden sectors are confounded by the many free parameters, including the masses, couplings, and lifetimes of new particles. The overall event shape of different scenarios can also take many forms. This motivates the development of diagnostic tools to characterize anomalous event shapes which are unlikely to arise from Standard Model processes.
In this paper we will introduce a class of simplified models that produce a range of new signatures, and our goal will be to characterize them using event shape observables. It would be premature to consider how to search for these signals at the LHC, as we should first understand the signatures themselves. For this reason we focus our attention on an idealized situation: a pure signal at an $e^+e^-$ collider of the future with no background.
There exist several well-known observables that characterize the shapes of events at $e^+e^-$ colliders. A commonly used infrared and collinear (IRC) safe event shape observable
is thrust, defined as \cite{Brandt:1964sa,Farhi:1977sg,DeRujula:1978vmq}
\begin{equation}
T = \text{max}_{\hat{n}} \frac{\sum_i|\hat{n}\cdot \vec{p}_i|}{\sum_i |\vec{p}_i|}.
\label{eq:thrust}
\end{equation}
It has a range $T \in [0.5,1]$, where $T=1$ corresponds to two back-to-back particles, and $T=0.5$ is an isotropic radiation pattern. While thrust has provided essential insight on the perturbative nature of QCD, it is most sensitive to event shape deviations from the two-particle dijet configuration and has less sensitivity in the quasi-spherical regime (see Fig. 8 of \cite{Cesarotti:2020hwb}). To complement such standard observables, we also make use of a recently proposed event shape observable, the event isotropy \cite{Cesarotti:2020hwb}.
Event isotropy is defined using the energy mover's distance (EMD) \cite{Komiske:2019fks,Komiske:2020qhg}.
Given two radiation patterns of massless particles $P$, $Q$, the EMD is the minimum work necessary to rearrange $P$ into $Q$.
A radiation pattern is defined as a set of particles, each of which are specified by their position on the unit sphere and fraction of the total energy.
To reorganize the pattern $P$ into $Q$, we construct a transport map $f_{ij}$ that tracks the total energy fraction moved from position $i$ to position $j$.
The total work done in the rearrangement can be written as a sum over the distance $d_{ij}$ from $i$ to $j$ weighed by the fraction of energy moved $f_{ij}$.
The EMD is the minimum work for all possible rearrangements of $P$ to $Q$:
\begin{equation}
\text{EMD}\left(P,Q\right) = \min\limits_{\{f_{ij}\}} \sum_{ij} f_{ij}d_{ij}.
\end{equation}
The distance measure we use in this paper is
\begin{equation}
d_{ij} \equiv \frac{3}{2}\sqrt{1-\hat{n}_i \cdot \hat{n}_j} = \frac{3}{2}\sqrt{1-\cos\theta_{ij}}
\label{eq:dijmeasure}
\end{equation}
for $\hat{n}_i$ the unit vector proportional to the three-momentum of the element $p_i$, etc.
Note this differs from \cite{Cesarotti:2020hwb}, where the distance measure was proportional to $d_{ij} \sim 1-\cos\theta_{ij}$.
See Appendix B of that paper for a discussion of the different distance measures.
The ideal event isotropy of an event would be its EMD to a uniform radiation pattern $\mathcal{U}$ of equal total energy:
\begin{equation}
\mathcal{I}(\mathcal{E}) = \text{EMD} \left( \mathcal{E}, \mathcal{U}\right).
\end{equation}
which we would take as a uniform spherical distribution for $e^+ e^-$ colliders.
However
to make computation times practical we calculate the EMD to a uniformly tiled, high multiplicity sphere generated with HEALPix \cite{2005ApJ...622..759G}. Specifically, we will use multiplicity 192 for the tiling, and we denote the event isotropy variable by $\iso{sph}{192}$.
\begin{figure}[!h]
\centering
\includegraphics[width=0.55\textwidth]{figures/SMbenchmark}
\caption{Event isotropy distributions for $e^+ e^- \to q {\bar q}$ and $e^+ e^- \to t {\bar t}$ (hadronic decays only) at a center-of-mass energy just above the $t\bar t$ threshold. The distance measure \Eq{eq:dijmeasure} gives higher $\iso{sph}{192}$ values than the corresponding plot in Ref.~\cite{Cesarotti:2020hwb}.}
\label{fig:SMbenchmark}
\end{figure}
Note that event isotropy is defined as a {\em distance to an isotropic event}. Thus, highly isotropic events have {\em low} values of $\iso{sph}{}$, despite the name. In particular, $\iso{sph}{}$ is 0 for a spherical event and 1 for a pencil dijet event with two back-to-back momenta. More generally, for an event with $k$ isotropically distributed particles of equal energy, a sphere tiled with multiplicity $N$, and the distance measure \Eq{eq:dijmeasure}, the event isotropy is\footnote{We may imagine spreading each particle's energy over a region of area $4\text{\greektext p}/k$, which requires moving the energy a distance of order $1/\sqrt{k}$. The normalization is obtained by requiring that for $k=2$, the extreme dijet limit, $\iso{sph}{192}= 1$. Any non-uniformity in the distribution generally increases the isotropy when averaged over many orientations. For the distance measure used in \cite{Cesarotti:2020hwb}, there is no square root.
See Appendix A of \cite{Cesarotti:2020hwb} for more details.}
\begin{equation}
\iso{sph}{N}
\gtrsim \sqrt{\frac{2}{\text{min} (k,N)}}.
\label{eq:isoest}
\end{equation}
The minimum value of $\iso{sph}{192} \approx \sqrt{\frac{2}{192}}\approx 0.1$ represents a slight loss of dynamic range, a price one pays for much faster computation.
To illustrate the behavior of this variable, we used \texttt{Pythia 8.243} \cite{Sjostrand:2014zea} to generate, at a center-of-mass energy of 350 GeV, the processes $e^+ e^-\to q{\bar q}$ and $e^+ e^- \to t {\bar t}$.
The resulting event isotropy distributions are shown in Fig.~\ref{fig:SMbenchmark}; the values of $\iso{sph}{192}$ are higher than the corresponding plot in Ref.~\cite{Cesarotti:2020hwb} because of our choice of distance measure \Eq{eq:dijmeasure}.
QCD radiation and radiative-return to the $Z$ boson both reduce the $q\bar q$ isotropy from 1 to approximately 0.8.
Meanwhile, since the top quarks are produced near threshold, their six jets are distributed quasi-isotropically, and the distribution peaks near the value $\sqrt{1/3}\sim 0.58$ that \Eq{eq:isoest} would suggest.
%
\par HV extensions to the SM often take the form of confining hidden sectors. However, confinement is compatible with a wide range of event shapes. The 't Hooft coupling $\lambda = \alpha_s N_c$ plays a major role in determining the shape of events. In a QCD-like, asymptotically free theory, $\lambda \gg 1$ near the confinement scale but runs to be $\ll 1$ at energies well above $\Lambda_\text{QCD}$. Gluon radiation in this regime is characterized by perturbative showering, in which a hard quark or gluon is dressed with moderate amounts of collinear radiation, leading to a classic QCD jet. However, a near-conformal field theory may maintain $\lambda\gg 1$ over a wide range of energies above the confinement scale. In this regime collinear radiation is extremely rapid, and all hard partons lose their energy; only soft physics survives. Such a theory will produce events which, because of greatly enhanced radiation \cite{Polchinski:2002jw}, are spherically symmetric in the extreme $\lambda\to\infty$ limit \cite{Strassler:2008bv,Hofman:2008ar,Hatta:2008tx}.
Although one can do reliable computations for $\lambda\ll 1$ (where field-theory perturbation theory is valid) and sometimes for $\lambda\gg 1$ (where gauge/string duality furnishes us with an alternative perturbation expansion), BSM physics could fall in the intermediate regime. This regime is poorly understood, as there are no methods for detailed calculation at intermediate $\lambda$.
Because of this obstacle, we pursue a more pragmatic approach. In this paper we seek to develop a procedure for generating events that is physically reasonable and has parameters that allow it to interpolate between jetty and spherical. We hope that a flexible method for producing events with intermediate values of event shape observables may allow the design of analysis and trigger strategies that are sensitive to a broader class of new physics models, even ones whose signals cannot currently be calculated.
One widely-adopted strategy in the search for new physics at colliders is the use of simplified models. These models can abstract away many details of a theory while preserving key elements of the collider phenomenology, and have been designed for a broad range of signals. However, they are intrinsically `simple': the models include a small number of new light particles and interactions \cite{Meade:2006dw, ArkaniHamed:2007fw, Alwall:2008ag}.
While it is beneficial to have fewer parameters, we would also like to consider more complex theories with many interactions and heavier particles, as in Hidden Valleys.
There have been studies of simplified models with relatively high final-state multiplicities \cite{Strassler:2008fv, Evans:2013jna, Fan:2015mxp, Cohen:2016nzv}, as well as studies of specific theories that give signals made of many soft particles \cite{Kang:2008ea, Harnik:2008ax, Knapen:2016hky}. However, there is a gap which can be filled by a straightforward approach to formulating simplified models that are flexible enough to span a wide range of event shapes.
In this paper, we show that a simplified model within an extra dimension, with a small number of parameters, allows for the generation of a wide range of event shapes. Specifically, we consider a warped extra dimension, in the form of a slice of a 5d AdS space \cite{RandallSundrum:1999}. This is motivated by the AdS/CFT or ``gauge/string'' correspondence, which relates gauge theories to string theory \cite{Maldacena:1997re,Gubser:1998bc,Witten:1998qj}. The correspondence suggests that a set of interacting fields (including gravity) on such a finite-size warped extra dimension can be interpreted as the dual of a (large $N$) confining gauge theory. (This relation has been made precise for a few purely 4d gauge theories, as in \cite{Polchinski:2000uf, Klebanov:2000hb}.) The infinite tower of massive Kaluza-Klein modes (KK modes) in the extra dimension is equivalent to a tower of hadrons of the confining gauge theory. Rather than choosing some ansatz for the masses and couplings of hadrons in a strongly-coupled gauge theory where we cannot do calculations, we choose a small set of masses and couplings in 5d which determine the entire infinite set of masses and couplings in 4d. This reduces the number of arbitrary choices to make, while still allowing enough flexibility to generate a wide range of collider event shapes. This use of AdS as a simplified dual to a Hidden Valley is in the spirit of previous uses of AdS to model low energy QCD \cite{Csaki:1998qr, Polchinski:2001tt, Polchinski:2001ju, Polchinski:2002jw, Csaki:2003zu, erlich2005qcd,DaRold:2005mxj}. The fact that KK-mode cascades in a warped extra dimension can produce approximately spherical events was previously examined in \cite{csaki2009ads}. We build on this by constructing a wider range of models, which lead to a wider range of collider events. We also make use of the new tool of event isotropy to obtain an improved characterization of these events.
We emphasize that what we present here is a physical model (morally dual to a field theory at large $\lambda$) which can interpolate {\it phenomenologically} between the jetty regime (which arises at small $\lambda$ as it does in QCD) and the quasi-spherical regime (which appears at very high energy at large $\lambda$.)
Because multiple, qualitatively different interpolations between the two regimes likely exist, our model may have little to do with what one would observe in a real confining gauge theory whose ultraviolet value of $\lambda$ is varied from small to large. Nevertheless, our approach widens the space of sensible targets for experimenters, and one may hope any search strategies that it inspires may be sensitive to a variety of models, not just the one proposed here, that sit between the jetty and spherical extremes.
We proceed with a brief introduction of simplified models in extra dimensions in \Sec{sec:xdimsimple}.
In \Sec{sec:simResults}, we present results from simulations of cascades generated with our models, for a variety of parameter values and interaction terms.
We show that these models can accommodate varied distributions of event isotropy. The features of the model are driven by basic properties of the couplings among various KK modes.
In simple scenarios with one self-coupled bulk field, near-threshold decays are often preferred, while decays with greater available phase space are suppressed. This leads to low-momentum daughter particles with no preferred boost axis, and thus to nearly isotropic events. The degree of isotropy depends on further details, such as the extent to which there is an approximately conserved KK number, and the number of stable KK modes at the bottom of the spectrum. In scenarios with multiple bulk fields, we find that there are ``plateaus'' in phase space with relatively high decay rates, far from threshold. These lead to less isotropic events. Finally, cases with boundary-localized couplings can have branching ratios determined mostly by phase space, and lead to much less isotropic events. In each case, we explain how the pattern of branching ratios is reflected in properties of the event: thrust, particle multiplicity, the energy distribution of daughter particles, and the new event isotropy observable.
The structures that we find in the patterns of couplings among various modes are determined by overlap integrals involving products of three Bessel functions.
In \Sec{sec:analytic}, we give an analytic understanding of these integrals. In particular, we show that the overlap integrals can be separated into two terms, one of which can be computed approximately and one of which can be computed exactly. The latter term often dominates, and allows us to obtain a clear analytic understanding of both the regime in which near-threshold decays are preferred and the regime with plateaus of enhanced decays away from threshold. All of the important qualitative features determining the event spectra can thus be extracted from the analytic results.
In \Sec{sec:conclusions}, we conclude and summarize both forthcoming work and open questions for the future.
A preliminary version of some of our results was reported in \S7.3 of a recent white paper on long-lived particles at the LHC \cite{Alimena:2019zri}. This also included a comparison to a parton shower algorithm pushed to strong coupling (work of Marat Freytsis), which may be of interest to some readers.
In a companion paper \cite{paper2}, we will provide a more detailed understanding of the event shape observables, and establish that event isotropy captures features of events that are not easily extracted from traditional variables (thrust, eigenvalues of the sphericity tensor, and jet multiplicities).
\section{Extra dimensional simplified models: a brief introduction}
\label{sec:xdimsimple}
One approach to robust searches for new physics at colliders is the use of simplified models. An extensive summary can be found in \cite{Alves:2011wf}. Because these models are characterized by effective Lagrangians with only a few new particles, they are not representative of the rich spectra of new particles and decay chains that can arise within hidden sectors. It is unreasonable to select, by hand, the masses and couplings of large numbers of particles. Various approaches to this problem have been chosen in the literature, which we will briefly discuss below in \Sec{subsec:comparisons}. For our purposes, an efficient approach to generating simplified models with a small number of free parameters is to consider theories with an extra dimension containing a small number of bulk fields and interactions, which then produce many modes and couplings in the four-dimensional reduction. This choice has the advantage that gauge/string duality furnishes us with an interpretation of the extra-dimensional simplified model in terms of a toy model of a confining hidden sector at large 't Hooft coupling.
Readers familiar with RS models \cite{RandallSundrum:1999} can skim this section: the main message is that we consider scalars with trilinear bulk interactions as a simplified model for the spectrum and essential
interactions in the hidden sector.
\subsection{Spectrum of masses}
We will imagine coupling the SM to a hidden sector which consists of states that propagate in at least five dimensions. Such a hidden sector might in principle be dual to a gauge theory via the gauge/string correspondence, and we will often use the language of this correspondence in describing it.
Specifically, let us begin by considering a slice of (4+1)d AdS space (RS1). We will denote the extra spatial coordinate as $z$. This spacetime geometry is specified by the curvature radius $R$ of the 5d geometry, with metric
\begin{equation}
{\rm d}s^2 = \frac{R^2}{z^2}(\eta_{\mu\nu}{\rm d}x^\mu {\rm d}x^\nu -{\rm d}z^2)
\label{eq:AdSmetric}
\end{equation}
where $z_\text{UV}<z< z_\text{IR}$. In this paper we take $z_\text{UV}=0$, in order to focus purely on modeling the hidden sector; coupling the sector to the SM may require reintroducing $z_\text{UV}$ depending on the nature of the interaction between the two sectors.\footnote{SM fields do not propagate in the bulk, because they are not composite states of the hidden sector. One possibility would be to couple to them to the bulk fields by UV-brane localized interactions, but we will not pursue the details here.}
A theory of fields propagating on AdS$_5$ for $z<z_\text{IR}$ is often called the ``hard-wall'' model and has been extensively studied as a model for QCD \cite{Polchinski:2001tt, Polchinski:2001ju, Polchinski:2002jw, erlich2005qcd,DaRold:2005mxj}.
The dimensionful parameter $z_\text{IR}$ plays the role of the confinement length scale in pure Yang-Mills theory. Indeed, this type of model is a cartoon of sorts, representing more realistic string constructions that are dual to quasi-conformal field theories which vaguely resemble QCD. More precisely, these field theories are asymptotically conformal at high energy (corresponding to small $z$) with a continuous coupling constant, and their conformal invariance is broken at a scale $\Lambda$ that corresponds to $z\sim z_\text{IR}$. In some cases the breaking of conformal invariance is due to confinement. Simple versions of the hard-wall model, like pure Yang-Mills theory, have a mass gap and towers of states, the details depending on the 5d fields that the model contains. We will consider theories of this type below.
For simplicity only, we will consider interacting scalars propagating in the bulk. These could be a subset of fields in a more realistic theory, or could serve as warm-ups for gauge and/or gravity fields. The scalars will satisfy a 5d Klein-Gordon equation with mass $M$, and can be written as an infinite sum of scalar modes that propagate in the 4d bulk modified by wavefunctions in the fifth dimension:
\begin{equation}
\Phi\left(x^\mu, z \right) = \sum_{n=1}^\infty \phi_n \left( x^\mu \right) \psi_n \left(z\right).
\end{equation}
The 5d wavefunctions have the form of Bessel functions: $\psi_n(z) \propto z^2 J_\nu(m_n z)$, where
\begin{equation}
\nu \equiv \sqrt{4+M^2R^2}.
\label{eq:nuFromMass}
\end{equation}
The tower of massive Kaluza-Klein (KK) modes of the scalars can be interpreted, through gauge/string duality, as a tower of hadrons of the quasi-conformal confining 4d theory dual to the bulk description. These hadrons are sourced by a field-theory operator ${\cal O}$ of scaling dimension $d_{\cal O} \equiv \nu + 2$. The smallest 5d mass-squared is set by the Breitenlohner-Freedman bound \cite{Breitenlohner:1982jf}
\begin{equation}
M^2R^2 \geq -4.
\end{equation}
This corresponds to values of $\nu$
\begin{equation}
\nu \equiv d_{\cal O} - 2 \geq 0.
\end{equation}
The range $1 \leq d_{\cal O} < 2$ is allowed by unitarity but requires an alternative boundary condition at $z_\text{UV}$ \cite{Klebanov:1999tb}, and will not be considered in this paper.
\begin{figure}[t]
\centering
\includegraphics[width=0.55\textwidth]{figures/kkTowers}
\caption{The mass spectra of KK towers for $\nu = 0$, $1$, $5$, and $10$ respectively, in units of $1/z_\mathrm{IR}$ starting at the lowest mode. We assume Dirichlet boundary conditions on the IR brane. We highlight two important trends. First, the lowest mass in the tower increases as $\nu$ increases. Second, the mass splittings sufficiently high in each tower are approximately equal and independent of $\nu$.}
\label{fig:towers}
\end{figure}
One can estimate the masses of the Kaluza-Klein modes by using the asymptotic expansion of the Bessel function for large (positive real) arguments,
\begin{equation}
\J{\nu}{x} \approx \sqrt{\frac{2}{\text{\greektext p} x}} \left[\cos\left(x - \frac{\text{\greektext p}}{2} \nu - \frac{\text{\greektext p}}{4}\right) - \frac{\nu^2 - \frac{1}{4}}{2 x} \sin\left(x - \frac{\text{\greektext p}}{2} \nu - \frac{\text{\greektext p}}{4}\right) + {\cal O}(1/x^2)\right].
\label{eq:besselasymptotic}
\end{equation}
In particular, the $n^\text{th}$ KK mode mass is approximately given by
\begin{equation}
m_n^{(\nu)} \approx \frac{\text{\greektext p}}{2} (\nu + C + 2n) z_\mathrm{IR}^{-1},
\label{eq:KKmassestimate}
\end{equation}
where $C$ is an ${\cal O}(1)$ number that depends on the choice of boundary condition on the IR brane. For Dirichlet boundary conditions ($\psi_n|_{z_\text{IR}} =0$), we have $C = -1/2$, while for Neumann boundary conditions ($\partial_z\psi_n|_{z_\text{IR}} =0$), we have $C = -3/2$.
The details of the mass spectrum impose constraints on particle decays; examples of spectra for different $\nu$ are given in \Fig{fig:towers}. For the Dirichlet case, $C=-1/2$ implies that for $\nu<1/2$, a decay of KK mode $n_1$ to KK modes $n_2$ and $n_3$ of the same field, with $n_1=n_2+n_3$, is always kinematically allowed. In particular, barring some additional constraints, the only stable mode is $n=1$. Conversely, for $\nu>1/2$, some of these decays are always disallowed, and (to a very good approximation) $1+\floor*{(\nu+\frac{3}{2})/2}$ modes are stable against decay. In particular, the mass spectrum for $\nu=\frac{1}{2}$ is exactly that of a 5d-massless field in a flat extra dimension, $m_n=\text{\greektext p} n$; as $\nu\to\frac{1}{2}$ from below, the phase space for the decay $2\to 1+1$ closes off and the second KK mode becomes kinematically stable. Note also that for all such spectra, a decay $n_1\to n_2+n_3$ is always forbidden for modes of the same field if $n_1<n_2+n_3$.
The wavefunction of the $n$th mode is
\begin{equation}
\psi^{(\nu)}_n(z) = N^{(\nu)}_n \,z^2 J_{\nu}(m_n^{(\nu)} z), \quad
N^{(\nu)}_n \equiv \left({\int_0^{z_\mathrm{IR}} {\rm d}z\,(R/z)^3\, \left[z^2 J_{\nu}(m^{(\nu)}_n z)\right]^2}\right)^{-\frac12} \ .
\label{eq:wavefunctionestimate}
\end{equation}
The coefficient $N^{(\nu)}_n$ is determined by requiring that the 4d field $\phi_n(x)$ be canonically normalized at tree level.
The Dirichlet case has a simple normalization:
\begin{equation}
N_n^{(\nu)} = \frac{1}{z_\text{IR}R^{3/2}}\frac{\sqrt{2}}{|J_{\nu+1}(m_n^{(\nu)} z_\text{IR})|}\approx
\sqrt{\frac{\text{\greektext p} m_n^{(\nu)}}{R^3 z_\text{IR}}}.\qquad \text{ (Dirichlet)}
\label{eq:dirNorm}
\end{equation}
For the Neumann case the closed form is more complicated, but the final approximate expression in \Eq{eq:dirNorm} remains true at large $n$.
In the case of Neumann boundary conditions, the value of the wavefunction on the IR boundary can be approximated by:
\begin{equation}
\psi^{(\nu)}_n(z_\mathrm{IR}) \approx \frac{z_\mathrm{IR}}{R^{3/2}} \left[(-1)^{n+1} \sqrt{2} + O(1/n)\right].
\label{eq:neumannIRvalue}
\end{equation}
This implies that a $\phi^3$ interaction localized on the IR boundary will give rise to couplings of approximately equal magnitude between any three KK modes, a fact that we will make use of in \Sec{subsec:boundary}.
\subsection{Interaction terms}
Each scalar field in 5d provides a tower of massive particles. To induce the decays among these particles that will create a range of signals, we next turn on interactions among the scalars. In many models, the dominant decays are all two-body, and in such cases, the only important interactions are cubic. We therefore consider cubic couplings of scalar fields in the 5d bulk,
\begin{equation}
\int \sqrt{g} \ {\rm d}^4x \ {\rm d}z \ \mathcal{L}_\text{int} = - \int \sqrt{g}\ {\rm d}^4x \ {\rm d}z \ c\, \Phi_{1}\Phi_{2}\Phi_{3}
\label{eq:interactions}
\end{equation}
where $\Phi_{1,2,3}$ are potentially different fields with corresponding bulk mass parameters $\nu_{1,2,3}$, and $c$ is a coupling constant.\footnote{Throughout the paper, we only use $c$ to denote this coupling constant; it should not be confused with the central charge $c$ that is often discussed in the context of AdS/CFT.} We implicitly assume that the unboundedness of this Lagrangian is corrected by higher-order terms which make the theory well-behaved but do not affect decays.
In our studies below, we will focus only on the ``single field case'', where all three fields are the same, and the ``two field case'' where $\Phi_2=\Phi_3$. The single-field case captures some features of self-interacting bulk fields such as a dilaton, non-abelian vectors or scalars, or the gravitational field. The two-field case captures features of situations in which a scalar, gauge field, or gravity couples to a second field that carries charge under either a ${\bf Z}_2$ symmetry, or perhaps $\mathsf{CP}$, forbidding modes of $\Phi_2$ from being created singly. It also is similar to cases in which the second field is complex and carries a $U(1)$ charge, since the spectrum and decay modes of $\Phi_2$ and $\Phi_2^*$ are the same in such a case. The field $\Phi_1$ may have its own cubic interaction, but we will assume here for simplicity that its coupling is relatively small compared to the $\Phi_1\Phi_2^2$ coupling, and so plays an insignificant role in decay chains. The cubic interaction could also be zero, as for an abelian gauge field.
We will denote the $i^\mathrm{th}$ KK mode of the field $\Phi_n$ by $\phi_{n,i}(x)$. In the 4d effective theory, the 5d interaction translates into an infinite set of couplings among the 4d modes:
\begin{equation}
\mathcal{L}_\mathrm{4d} \supset \sum_{i,j,k}^\infty c_{ijk} \phi_{1,i}(x) \phi_{2,j}(x) \phi_{3,k}(x),
\end{equation}
where the effective couplings of the 4d scalars are determined by the overlaps of the wavefunctions in the extra dimension,
\begin{equation}
c_{ijk} = c N^{(\nu_1)}_i N^{(\nu_2)}_j N^{(\nu_3)}_k \int_{z_\text{UV}}^{z_\text{IR}} \left(\frac{R}{z}\right)^5 {\rm d}z \ \big[z^2 J_{\nu_1}(m^{(\nu_1)}_i z)\big] \big[z^2 J_{\nu_2}(m^{(\nu_2)}_j z)\big]\big[z^2 J_{\nu_3}(m^{(\nu_3)}_k z)\big] \ .
\label{eq:4dcoup}
\end{equation}
Here we substituted $\det g = (R/z)^{10}$ for the metric \Eq{eq:AdSmetric} into \Eq{eq:interactions}.
These 4d coupling constants depend on various dimensionful 5d quantities: $c$, $R$, $M_n$, and $z_\mathrm{IR}$. However, it turns out that if we write $c_{ijk}$ in terms of the $\nu_n$ (which depend only on the product $M_n R$) and the dimensionless coupling $c_0 \equiv c \sqrt{R}$, then all remaining dependence on $R$ drops out of the equation. Hence, we never need to specify the 5d length scale $R$ to do a calculation. Furthermore, we are primarily interested in branching ratios (rather than total widths), for which the value of $c_0$ cancels out as well. Consequently, we can set both $R$ and $c_0$ to 1 for convenience. Meanwhile the scale $z_\mathrm{IR}$, corresponding to the confinement scale of a dual field theory, is the only dimensionful quantity that appears in physical measurements. We may express masses and widths of the KK modes, and other dimensionful measurements, in units of $z_\mathrm{IR}$, and so we can also set this quantity to 1 if we choose. The only non-trivial parameters left, then, are the 5 dimensional masses, which we express using the dimensionless parameters $\nu_i$, which in the context of a gauge/string duality are related to 4d operator dimensions.
As we will see, depending on the choices of $\nu_i$ and the boundary conditions, the $c_{ijk}$ will often (but not always) respect an approximate KK-number symmetry. Were the extra dimension flat, it would have a conserved KK-number with Neumann boundary conditions (or with periodic boundary conditions) and an approximately conserved KK-number with Dirichlet boundary conditions. More precisely, in the latter case, $c_{ijk}$ vanishes if $i+j+k$ is even, and falls off as $\sim 1/\Delta_\text{KK}$ when odd, where (if $i>j+k$) the violation of KK-number is $\Delta_\text{KK}= i-j-k$. Once we replace the flat space with a slice of AdS, however, additional effects from the bulk will break the KK symmetry, sometimes leaving it approximately conserved as in the flat Dirichlet example, but sometimes not.
The degree and pattern of KK-number violation has an intricate structure. Much of it emerges from the $c_{ijk}$, through the very interesting properties of the triple Bessel function integrals in \Eq{eq:4dcoup}. Specifically, it is convenient to rewrite the integral of interest as a difference of two easier integrals:
\begin{align}
I(\nu_i, m_i) \equiv \int_0^1 {\rm d}z\, z\, \prod_{k=1}^3 \J{\nu_k}{m_k z}
&= \left(\int_0^\infty - \int_1^\infty\right) {\rm d}z\, z\, \prod_{k=1}^3 \J{\nu_k}{m_k z}
\nonumber \\
&\equiv I_+(\nu_i, m_i) - I_-(\nu_i, m_i)
\label{def:IplusIminus}
\end{align}
Except right at threshold, detailed understanding of the $I_+$ and $I_-$ integrals can be obtained using approximation methods described in \Sec{sec:analytic}. We will use specific cases in the studies of our model presented in \Sec{sec:simResults}.
Additional KK-number violation can arise from kinematic constraints. As already noted (see \Eq{eq:KKmassestimate} and following), certain decays are kinematically forbidden in the single field case for $\nu>1/2$. These constraints can be more complex in a two field case.
In addition, KK-number might be further violated by interactions at the IR boundary of the space at $z = z_\mathrm{IR}$. Specifically, in addition to or as an alternative to the bulk coupling \eqref{eq:interactions}, we could add an interaction term:
\begin{equation}
\int \sqrt{g} \ {\rm d}^4x \ {\rm d}z \ \mathcal{L}_\text{bdry} = - \int \sqrt{g}\ {\rm d}^4x \ {\rm d}z \ z_\mathrm{IR} \delta(z - z_\mathrm{IR}) \ {\tilde c}\, \Phi_{1}\Phi_{2}\Phi_{3}.
\label{eq:bdryinteractions}
\end{equation}
This leads to nontrivial interactions if the fields satisfy Neumann boundary conditions at $z = z_\mathrm{IR}$. The boundary-localized interaction leads to approximately equal couplings among all the modes, due to \eqref{eq:neumannIRvalue}. This contrasts with bulk couplings for small $\nu_i$, where the $c_{ijk}$ generally have more structure.
A mode's decay width is determined by the available phase space for its potential decays.
Recall that the decay width of a scalar $\phi_i \to \phi_j + \phi_k$ through a constant matrix element $c_{ijk}$ is
\begin{equation}
\Gamma = \frac{c_{ijk}^2}{16\text{\greektext p} m_i^3} \lambda_{\rm PS}^{1/2}(m_i^2, m_j^2, m_k^2),
\label{eq:width}
\end{equation}
where the phase-space function $\lambda_{\rm PS}$ is defined as
\begin{equation}
\lambda_{\rm PS}(m_1^2, m_2^2, m_3^2) \equiv (m_1 + m_2 + m_3)(m_1 - m_2 + m_3)(m_1 + m_2 - m_3)(m_1 - m_2 - m_3).
\label{eq:lambdaPS}
\end{equation}
We will see cases where near-threshold decays are favored, because approximate KK-number conservation in the $c_{ijk}$ overcompensates the phase space suppression. This situation leads to near-spherical distributions in decay chains. In other cases this is not so, and decay chains lead to much less spherical events.
\subsection{Comparison to other high-multiplicity models in the literature}
\label{subsec:comparisons}
The use of a warped extra dimension to provide a model for a dark or hidden sector is natural following \cite{Randall:1999vf,Verlinde:1999fy}, and is well-established in the literature \cite{Stephanov:2007ry, Strassler:2008bv, Falkowski:2008fz, Gherghetta:2010cq, Bunk:2010gb, McDonald:2010iq, McDonald:2010fe}. In this subsection, we will briefly comment on some related or alternative approaches to modeling high-multiplicity hidden sectors.
Recently, cascade decays in warped dark sectors have been discussed in a series of papers \cite{Fichet:2019hkg, Brax:2019koq, Costantino:2020msc}. While the basic RS framework of these papers is similar to ours, they have focused especially on the regime in which individual KK modes become sufficiently broad that they should be described as a continuum, rather than as narrow resonances. On the other hand, all of our calculations will be done in the regime in which each KK mode is narrow and we can model a cascade decay as a sequence of $1 \to 2$ decays. This can be done consistently provided that our bulk couplings are sufficiently small, while at the same time not {\em so} small that resonances acquire a long lifetime and alter the collider phenomenology. (The long-lived regime may be of independent interest, but its additional complications are beyond the scope of this paper.) In every case that we consider, the couplings can be chosen so that this consistency condition is met. In particular, in every case the width-to-mass ratio of the $n^\text{th}$ KK mode grows more slowly than it would in a model with decays determined by pure phase space. With pure phase space decays, the $n^\text{th}$ KK mode has of order $n^2$ decay modes available to it, with comparable widths. On the other hand, the partial width of a given decay mode scales as in \eqref{eq:width}, i.e., roughly as $1/m_n$. As a result, the {\em total} width of the KK mode scales linearly with the mode number, and the width-to-mass ratio is constant. If we take the bulk couplings to be small but not too small, the width-to-mass ratio of each mode will be small but every unstable KK mode will still decay promptly on collider time scales. The regime of small bulk couplings is where gauge/string duality is well-understood. If the bulk couplings are too small, there is a potential ``empty universe problem'' in cosmology related to the slow first-order confining phase transition in the dark sector \cite{Creminelli:2001th,Randall:2006py,Kaplan:2006yi}. However, the details of such a phase transition can be model-dependent (see, e.g., \cite{Agashe:2019lhy}). We assume that potential cosmological problems can be addressed without qualitatively altering the collider phenomenology.
Other recent work \cite{Dienes:2019krh, Dienes:2020bmn} has studied dark sector particles with simple ans\"atze for the masses and couplings, e.g., $m_n = m_0 + n^\delta \Delta m$ for some positive exponent $\delta$, and couplings depending on factors $(m_\ell - m_i -m_j)^r$ and $(1 + |m_i - m_j|/(\Delta m))^{-s}$ favoring decays to lighter daughter particles or nearby daughter particles, respectively. The model of \cite{Dienes:2019krh} involves neutral particles $\chi_n$ decaying to ${\bar q} q' \chi_l$, and leads to high-multiplicity event shapes with missing momentum, which are qualitatively similar to events we will discuss. The fully dark decay chains discussed in \cite{Dienes:2020bmn} are even more similar to ours, but are discussed in the context of cosmology rather than collider physics. It is no accident that these decay chains have similar properties, as the ans\"atze used have been motivated, in part, by models of extra dimensions \cite{Dienes:2011ja,Dienes:2011sa,Dienes:2012jb,Buyukdag:2019lhh}.
Further models for high-multiplicity events have been proposed based on a variety of ideas. SUEP events have been modeled using thermal spectra \cite{Knapen:2016hky}. Models of black hole production at colliders predict similar spectra \cite{Giddings:2001bu, Dimopoulos:2001hw}, as do ``string balls,'' lower-mass precursors of black holes \cite{Dimopoulos:2001qe, Gingrich:2008di}. Another model with interesting phenomenology, albeit without a well-motivated UV completion, achieves a wide range of couplings within a large ensemble of particles through a random mass matrix \cite{DAgnolo:2019cio} (see also \cite{Dienes:2016kgc, Craig:2017ppp, Tropper:2020yew}). It would be interesting, in the future, to apply the event isotropy variable to more of these models.
\section{Simulation Results}
\label{sec:simResults}
In this section we study event shapes of our toy model for different parameters, by simulating decay cascades of a heavy KK mode with $n=n_p\gg 1$, in its rest frame, to light and stable KK modes.
(We will often refer to the KK modes as ``hadrons,'' using the dual viewpoint, but one should keep in mind that these are {\it hidden-sector} hadrons, not SM hadrons.)
We further decay the hidden-sector stable hadrons (HSH) into two massless particles each, to mimic decays back into the SM. Then we calculate observables from the collection of massless, final-state momenta.
In this simplified model, mass and KK-number are closely related,
as we have seen in \Eq{eq:KKmassestimate}. It follows that the degree of violation of KK-number is directly tied to kinematics. Roughly, if KK-number is conserved or lightly violated in a two-body decay, the final state particles tend to be slow in the initial particle's rest frame, while if KK-number is strongly violated, the final state particles are produced with a substantial boost. It is not surprising then that KK-number violation correlates closely with event-shape variables, as we will show in this section.
In extreme limits, it is clear how this should work.
Were KK-number precisely conserved in all decays, then every decay would occur at or near threshold, and the final state of the hidden sector cascade would be a collection of slow HSHs. When the HSHs themselves decay to the visible sector, they would produce an array of roughly back-to-back massless particles, produced at random angles. The expected distribution of such particles is roughly spherical. Note, however, that even with dozens of particles, random fluctuations are large and observed events are far from spherical, both to the eye and to event shape variables.
Conversely, in cases with large KK-number violation, the first decay in the cascade produces two relatively light KK states at high boost. Once this occurs, all ensuing decays of the lighter daughters will be highly collimated, and so, independent of the details, two hard jets result.\footnote{These jets, neither pencil-like nor QCD-like, will have opening angles and subjets that depend on the kinematics of the initial steps in the cascade, especially on the boost of the initial decay's daughters.} Thus KK-number violating decays early in the cascade leads to a highly non-spherical pattern.
The simulations that we describe in this chapter interpolate between these extremes. We will demonstrate this using thrust and event isotropy, both of which are sensitive to the features of the events.
These variables are somewhat correlated with a third, namely particle multiplicity. The possible maximum particle multiplicity in our simulations is 2$n_p$. This occurs when KK-number is conserved in every decay, and all hidden hadrons can decay except the $n=1$ state, the unique HSH. Then the decay cascade leads to $n_p$ HSH's, and to $2n_p$ massless particles once these decay. Violations of KK-number in the cascade, and the existence of multiple HSH's with $n>1$, will decrease this number. This tends to increase the event isotropy, since, as noted in \Eq{eq:isoest}, for a multiplicity $k<192$, $\iso{sph}{192} \gtrsim \sqrt{2/k}$. (The average isotropy tends to be higher than this estimate, which holds for maximally symmetric events, because of random fluctuations in the angles.) Despite this we will see that isotropy and particle multiplicity are not redundant.
We now present our results, progressing from the most isotropic scenario to the least.
For each choice of bulk masses and couplings, we generate $10^4$ events starting at KK mode $n_p = 100$ and allow it to cascade into stable hadrons, each of which then decays to a pair of massless particles. In each case, we will see that the degree of KK-number conservation, as reflected in the couplings $c_{ijk}$ and the resulting branching fractions, determines the qualitative properties of the event shapes.
Note that we mainly limit ourselves to small values of $\nu$. This is because scalars with large 5d mass correspond to 4d operators with large scaling dimension ($d_{\cal O}=\nu+2$), and it is difficult to imagine coupling the SM to them.
\subsection{Spherical and Near-Spherical Cases}
\label{sec:sphereNearsphere}
To set a baseline, we begin with the most spherical case in our AdS-based simplified model, a single-field model with $\nu=0$. This case corresponds to a five-dimensional scalar with mass-squared $-2$, at the Breitenlohner-Freedman bound, and thus to a dual CFT operator of dimension 2 with a non-zero three-point function. We will compare it a pair of toy models in which KK-number is exactly conserved and spherical events are to be expected. The event shapes for $\nu=0$ are virtually the same as for the toy models, despite the former's mild KK-number violation and semi-relativistic velocities.
The $\nu=0$ single field model has a spectrum approximately given by $m_n \approx \text{\greektext p} \left(n-\frac{1}{4}\right)$. The only HSH is the mode with $n=1$; all decays $\phi_i \to \phi_j + \phi_k$ are open if $i\geq j + k$.
For any single field with a cubic self-interaction and even integer $\nu$, the couplings of its modes satisfy a simple approximate formula.
As noted in \Eq{def:IplusIminus}, the triple Bessel integral \Eq{eq:4dcoup} can be conveniently written as a difference of integrals. For even $\nu\geq 0$, $I_+$ vanishes due to a factor of $1/\Gamma(-\nu/2)$ which can be seen in \Eq{eq:GandR}. Meanwhile $I_-$ is approximately given by \Eq{eq:IminusDirichletSmallPS}. Using \Eq{eq:dirNorm}, we find
\begin{equation}
c_{ijk} \approx \frac{(-1)^{n_1+n_2+n_3+1}}{\sqrt{2}} \frac{8 m_i m_j m_k}{\lambda_\text{PS}(m_i^2,m_j^2,m_k^2)}
\label{eq:cijk_nu_even}
\end{equation}
While this is accurate only away from threshold, for $\nu=0$ the approximation works to within 2\% percent except for $n_2+n_3=n_1-1$, where the real coupling is smaller in magnitude by up to $5\%$, and $n_2+n_3=n_1$, where the difference reaches nearly $30\%$.
From this formula it follows that partial widths behave as $\lambda_\text{PS}^{-3/2}$, and so decays tend to occur at or very near threshold. This implies that the leading decays for each hadron conserve KK-number. (For instance, the particle with $n=n_p=100$ has a 77\% branching fraction to conserve KK-number, and this varies slowly with $n_p$.) Even those decays that violate KK-number do so by small amounts, and in the end the average HSH multiplicity at the end of the cascade is reduced only to 93 from its maximum of 100; this is shown later in \Fig{fig:singleFieldDist}. The decays of the HSHs produce nearly 200 massless particles with an energy distribution, shown in \Fig{fig:toyModels}(c); note that $m_1\approx 2.40\approx m_{100}/130$, and the distribution peaks at about $m_1/2$, with a tail up to $\sim 2m_1$. Thus velocities of the HSHs tend to be only semi-relativistic, and the angular distribution of their massless daughters is largely random.
\begin{figure}[t!]
\centering
\subfloat[]{
\includegraphics[width=0.45\textwidth]{figures/emdNuSpecToy}
}
\hfill
\subfloat[]{
\includegraphics[width=0.45\textwidth]{figures/thrustNuSpecToy}
}
\hfill
\subfloat[]{
\includegraphics[width=0.45\textwidth]{figures/energySpecNu0KKConsGraphComp}
}
\caption{The (a) event isotropy, (b) thrust, and (c) energy spectra of the final state for cascades generated with $\nu=0$ with and without KK-number conservation, and the flat case.
The multiplicity is not shown as both the KK-number conserving $\nu=0$ and the flat case distributions are delta functions at 200 particles.
The energy spectrum of the flat case is not plotted as it is a delta function at $\text{\greektext p}/2$; the other two spectra peak at approximately half the mass of the HSH.}
\label{fig:toyModels}
\end{figure}
Our first toy model has the same masses and KK-number-conserving couplings as the $\nu=0$ case just described, but we set all KK-number violating couplings to zero by hand:
\begin{equation}
c_{ijk} = 0 \ \ (i\neq j+k) \ .
\end{equation}
We will call this the KK-conserving (KKC) $\nu=0$ model. Its final state consists of exactly 200 massless particles, with an energy distribution slightly narrower than the full $\nu=0$ model, as the latter has KK-number-violating decays with more kinetic energy. We expect it to have slightly more spherical events.
The second toy model (the ``flat case'') is a single field on Minkowski space times an interval, $M_4\times S^1/Z_2$, with Neumann boundary conditions on the field. As noted earlier, the spectrum has $m_n\propto n$ exactly, and KK-number is conserved. Strictly speaking all particles are marginally stable, but we imagine deforming the model infinitesimally so that all decays can occur. The final state from an initial heavy hadron with quantum number $n_p$ consists of $n_p$ HSHs with $n=1$, all at rest. The decay to SM massless particles then produces events with exactly 200 massless particles in back-to-back pairs, distributed randomly in angle. Each particle has energy exactly $m_1/2 = \frac{\text{\greektext p}}{2}$.
Now we compare the event shapes for these three cases.
In \Fig{fig:toyModels} we show the distributions in energy, event isotropy, and thrust for the particles in the final state.
For consistency in range, we plot the scaled thrust $\tilde{T}$
\begin{equation}
\tilde{T} \equiv 2 (T-1)
\label{eq:ScaleThrust}
\end{equation}
such that all variables have a range of $[0,1]$ with $0$ being the most isotropic and $1$ being the least.
All three of these examples are very similar as seen by event-shape variables. The most notable differences are percent-level shifts in event shapes, in tails that arise from small numbers of somewhat less isotropic events. We may therefore treat any one of them as a benchmark against which to compare other cases.
All three cases have event isotropy that peaks in the range 0.15-0.20.
From \Eq{eq:isoest}, maximally isotropic events with $192$ particles $\mathcal{U}^\text{sph}_{192}$ would have
\begin{equation}
\iso{sph}{}\left(\mathcal{U}^\text{sph}_{192}\right) \approx \sqrt{\frac{2}{192}} \approx 0.10
\end{equation}
Naively we might have expected the flat case, with 200 particles of equal energy and random angles, to approximate this value.
However, the random fluctuations in angle (but not in energy, which remains $m_1/2$ for each particle) lead to a significantly higher event isotropy, closer to $0.16$; we do not know a method to compute this number without simulation.\footnote{For KK-number conserving scenarios, events become more spherical at high $n_p$, as shown in App.~\ref{app:npdependence}, and the theoretical limit is reached at large $n_p$.} Despite the wider energy distribution of the $\nu=0$ and KKC cases, their event isotropies are quite similar to the flat case. We will explore the causes of this in \cite{paper2}.
\subsection{Single field, general $\nu$}
\label{subsec:singlefield}
Next, still studying a single scalar field with Dirichlet boundary conditions, we consider other values of $\nu$. Each has a different degree of KK-number violation.
Although the amount of KK-number violation is still relatively small, and the decays are still mostly close to threshold, the effects are large enough to observably shift event shapes relative to the $\nu=0$ benchmark.
As noted in \Eq{def:IplusIminus}, the integral \Eq{eq:4dcoup} can be written in terms of a difference of two integrals $I_+$ and $I_-$. Substituting $\nu_i=\nu$ into \Eq{eq:Iplus2F1} and using the Euler reflection formula, one finds
\begin{equation}
I_+ = \sin\left(\frac{\nu}{2}\text{\greektext p}\right) \left(\frac{m_2m_3}{m_1^2}\right)^\nu
{\lambda_\text{PS}^{-1/2}} H(\nu,m_i)
\label{eq:Iplussingle}
\end{equation}
where $H$ is a function whose dependence on $\nu$ and the $m_i$ is subleading compared to the terms shown explicitly.\footnote{The full integral $I$ in \Eq{def:IplusIminus} is elementary at $\nu=\frac{1}{2}$, and the ${\lambda_\text{PS}^{-1/2}}$ factor is easily seen there.}
When $\nu$ is an even integer, $I_+$ vanishes, so the couplings are determined entirely by $I_-$. Since, from \Eq{eq:IminusDirichletSmallPS}, $I_-\sim \lambda_\text{PS}^{-1}$ near but below threshold, the branching fractions decrease as $\sim {\lambda_\text{PS}^{-3/2}}$, strongly suppressing KK-number violation. For $\nu=0$, where there are no other sources of KK-number violation, this gives quasi-spherical events. For general $\nu$, however, $I_+\sim {\lambda_\text{PS}^{-1/2}}$ falls off more slowly away from threshold than $I_-$. Although for $\nu$ very large this does not matter, because the power of the mass ratio in \Eq{eq:Iplussingle} decreases rapidly with $\nu$ and tends to disfavor decays to light hadrons, for $\nu\sim 1$ one finds $|I_+|\sim |I_-|$, and so KK-number violation in the couplings is considerably larger than for $\nu\approx 0,2,4,\dots$.
The dominant decays remain KK-number conserving for $\nu\lesssim 0.1$. For the $\nu=0.15$ case below, KK-number violation in the couplings is already significant, and reduces the multiplicity of HSHs well below the $\nu=0$ benchmark, as we will see shortly.
For $\nu\geq 0.5$, a new effect reduces multiplicity further: $m_n < m_{n'} + m_{n-n'}$, so KK-number conserving decays all become kinematically forbidden, and the minimal amount of KK-number-violation per decay is $> 0$, even for even integer $\nu$. This reduces the number of typical decays in the cascade and the number of hadrons at the end of the cascade. However, even though KK-number conservation is forbidden, decays with large KK-number violation are still somewhat suppressed, and so the leading decays are {\it minimally KK-number-violating} --- that is, they have the smallest amount of violation consistent with kinematic constraints. These decays are generally the ones closest to threshold.
When decays can typically only violate KK-number by a small amount (per decay), they remain near kinematic threshold, so boosted hadrons and ensuing jetty structures in the events do not arise. Nevertheless, $\iso{sph}{192}$ increases. In part, this is due to a decrease in particle multiplicity. The total amount of KK-number $\Delta_{\text{KK},\text{tot}}$ lost in the cascade (equal to the sum over decays of the KK violation $\Delta_{\text{KK}, s}$ in each decay $s$) is given by the parent mode number $n_p$ at the beginning of the cascade minus the sum over KK-numbers $n_i$ of the $N_{\text{HSH}}$ HSHs in the final state,
\begin{equation}
\Delta_{\text{KK},\text{tot}}=
\sum\limits_s^\text{S} \Delta_{\text{KK}, s}=
n_p - \sum\limits_{i}^{N_\text{HSH}} n_i \ ,
\end{equation}
where
$S$ is the total number of decays in the cascade. The reduced final-state hadron multiplicity $N_{\text{HSH}}$ increases the minimum value of $\iso{sph}{192}$ by a factor of order $\sqrt{100/N_{\text{HSH}}}$, from \Eq{eq:isoest}, even if the HSHs are rarely boosted and their final decay products are quasi-isotropically distributed.
If KK-number is exactly conserved, $S=n_p-1$ and $N_{\text{HSH}}=n_p$. More generally $N_{\text{HSH}}\leq n_p-
\Delta_{\text{KK},\text{tot}}
$, with the equality holding only if the only HSH has $n=1$. This is true for $\nu<\frac{1}{2}$, and in particular for the case $\nu=0.15$ that we show below. For larger $\nu$, there are HSH's with $n>1$, so $N_{\text{HSH}}$ is even smaller; and on top of this, decays with $\Delta_{\text{KK}, s}=0$ are forbidden, so
$\Delta_{\text{KK},\text{tot}}$ is of the same order as $S$ and $n_p$, leading to a substantial reduction in $N_{\text{HSH}}$. For the $\nu=0.75$ case we show below, $S\sim n_p/2$ and $N_{\text{HSH}}<n_p/2$, leading to a reduction of the multiplicity by more than half compared to the $\nu=0$ case, and a corresponding substantial increase in $\iso{sph}{192}$ .
\begin{figure}[t!]
\centering
\subfloat[]{
\includegraphics[width=0.45\textwidth]{figures/BRWidthnu0nModes100PlotTot}
%
}
%
\hfill
%
\subfloat[]{
\includegraphics[width=0.45\textwidth]{figures/BRWidthnu0p15nModes100PlotTot}
}
\hfill
\subfloat[]{
\includegraphics[width=0.45\textwidth]{figures/BRWidthnu0p75nModes100PlotTot}
%
}
\hfill
\subfloat[]{
\includegraphics[width=0.45\textwidth]{figures/br_singleField_1dplot}
}
%
\caption{(a--c) Branching ratios, for $\nu=\{0, 0.15, 0.75\}$, of the $100$th KK mode into all kinematically allowed two-body final states, as a function of the daughter KK-numbers $n_1 \geq n_2$.
Dominant decays occur at or near kinematic threshold, along the line of minimum KK violation.
The projection of the branching ratios for all values of $\nu$ on the $n_1=n_2$ line is shown in (d), where the relative suppression of KK violation is made obvious especially for the $\nu=0$ case.}
\label{fig:brratios}
\end{figure}
The branching ratios for the $n_p=100$ mode into daughter modes $n_1$, $n_2$ with $\nu = \{0, 0.15, 0.75\}$ are shown in \Fig{fig:brratios}. For convenience we will refer to these plots throughout the paper as ``branching fraction triangles." For $\nu=0$, the only HSH is the $n=1$ state at the bottom of the tower, and conservation of KK-number is both kinematically allowed for all decays in the cascade and dominant; as noted earlier, about 77\% of decays are KK-number conserving.
For $\nu=0.15$, KK-number conservation is kinematically allowed, but the probability of conservation in the decay of heavy modes is only $\sim 30\%$.
Decays with KK-number violation $\Delta_\text{KK} = 1$ occur with comparable probability ($\sim 30\%$). Technically, this is due to a cancellation in the integral for $c_{ijk}$; the integrals $I_+$ and $I_-$ are similar in magnitude for $\nu \sim 0.1-1.9$ and interfere destructively (constructively) when $i+j+k$ is even (odd).
The checkerboard pattern in \Fig{fig:brratios} (b,c) arises from this effect.
The pattern of branching ratios for $\nu = 0.75$ is similar to that of $\nu=0.15$. However, $\Delta_\text{KK}=0$ decays are kinematically forbidden, as can be seen by careful examination of the upper right edge of the triangle.
While \Fig{fig:brratios} applies for $n_p=100$, it illustrates qualitative features that apply, at fixed $\nu$, for smaller values of $n$, and thus for the whole cascade. This is because the integrals $I_+$ and $I_-$ have relatively simple behavior under changes of $n$; see App.~\ref{app:npdependence} for some discussion.
\begin{figure}[t!]
\centering
\subfloat[]{
\includegraphics[width=0.45\textwidth]{figures/emdNuSpecFull}
}
\hfill
\subfloat[]{
\includegraphics[width=0.45\textwidth]{figures/thrustNuSpecFull}
}
\hfill
\subfloat[]{
\includegraphics[width=0.45\textwidth]{figures/multNuSpecFull}
}
\hfill
\subfloat[]{
\includegraphics[width=0.45\textwidth]{figures/energySpecNu0GraphComp}
}
\caption{The (a) event isotropy, (b) thrust, (c) multiplicity, and (d) energy spectra of the final state of $\nu=\{0, 0.15, 0.75\}$ cascades.
The samples comprise $10^4$ events with $n_p = 100$, and all final state observables are computed after splitting the HSHs into two massless particles.
}
\label{fig:singleFieldDist}
\end{figure}
We present results for the event shapes of the cascades in \Fig{fig:singleFieldDist}, where we show distributions of the multiplicity and energy of the massless particles in the sample, along with the sample's event isotropy and thrust distributions.
KK-number violation reduces the multiplicity for $\nu=0.15$ relative to the $\nu=0$ benchmark. The $\nu=0.75$ case has even smaller multiplicity and a wider energy distribution, due to additional sources of KK-number violation and the fact that the $n=2$ state is also an HSH.
Meanwhile the event isotropy increases (the events become less spherical) for $\nu=0.15$ and even more so for $\nu=0.75$. Interestingly, this pattern does not apply for thrust; the thrust distributions for $\nu=0.15$ and $0.75$ are very similar.
This suggests that event isotropy and thrust are sensitive to different event shape characteristics and are not redundant variables.
Further study of this issue will be presented in \cite{paper2}.
Comparing \Fig{fig:singleFieldDist} (a) and (c), one might wonder if event isotropy and particle multiplicity are redundant variables, especially considering the natural correlation between them that we described earlier. Although the correlation seems particularly strong in these examples, other simulations shown below clearly demonstrate these two variables are independent.
\begin{figure}[!h]
\centering
\subfloat[]{
\includegraphics[width=0.3\textwidth]{figures/nu0event}
}
\hfill
\subfloat[]{
\includegraphics[width=0.3\textwidth]{figures/nu0p15event}
}
\hfill
\subfloat[]{
\includegraphics[width=0.3\textwidth]{figures/nu0p75event}
}
\caption{Visualizations of a characteristic final state for (a) $\nu = 0$, (b) $\nu = 0.15$, and (c) $\nu = 0.75$.
The chosen events have $\iso{sph}{192}$ equal to the mean in each distribution: (a) $\langle \iso{sph}{192} \rangle = 0.20$, (b) $\langle \iso{sph}{192} \rangle = 0.26$, (c) $\langle \iso{sph}{192} \rangle = 0.29$.
For ease of viewing, all momenta with magnitude above the average in the event are plotted, with the length proportional to the magnitude.
}
\label{fig:singleFieldVis}
\end{figure}
We illustrate some characteristic events for each of these samples in \Fig{fig:singleFieldVis}.
An event of average isotropy in the most isotropic sample ($\nu=0$) does not show any clear boost axis, whereas the samples with greater amounts of KK violation ($\nu =\{0.15, 0.75\}$) begin to show collimated prongs of energy.
We end our discussion of the single field case with a few comments on large values of $\nu$. With increasing $\nu$ but fixed $n_p$, $I_+$ is suppressed by a factor of $\left(m_2m_3/m_1^{2}\right)^\nu$, so $I_-$ dominates in most decay channels. When this happens, the couplings conserve KK-number similarly to \Eq{eq:cijk_nu_even} even when $\nu$ is not an even integer. However, this fact is irrelevant for $n_p=100$ because the kinematic constraints imposed by the mass spectrum require large violation of KK-number in each decay, which grows with $\nu$. We noted earlier that KK-number conservation becomes kinematically disallowed for $\nu>1/2$. More generally, decays with KK-number violation $\Delta_\text{KK}$ begin to be kinematically constrained when $\nu\gtrsim 2\Delta_\text{KK}+1/2$, starting with the most symmetric decays, and once $\nu$ reaches the values $0.5, 2.92, 5.56, 8.30, 11.1, \dots$, all decays with $\Delta_\text{KK} =0,1,2,3,4,\dots$ are forbidden. Moreover, there are roughly $1+\floor*{\frac{1}{2}(\nu+\frac{3}{2})}$ HSHs in the spectrum. All of these effects drastically reduce multiplicity and widen energy distributions, further increasing the event isotropy and thrust, so we do not expect highly spherical distributions to be common in this regime.
\subsection{Two field ($\nu_1 \neq \nu_2 = \nu_3$)}
\label{subsec:twofield}
Now we consider the two field scenario, where \Eq{eq:interactions} includes two distinct 5d scalar fields.
Here the cascade will be populated with $\phi_{1,i} \rightarrow \phi_{2,j} \phi_{2,k}$ and $\phi_{2,i} \rightarrow \phi_{1,j} \phi_{2,k}$ decays.
Because there are two towers of hadrons with different mass spectra, the correlation between KK-number and mass is not as simple as in the single-field case.
However, these complications are relatively unimportant compared to the dramatic change in the pattern of couplings $c_{ijk}$.
In the single field case, we have seen that couplings near threshold, with zero or minimal KK-number violation, are strongly enhanced, because $I_-\sim \lambda_\text{PS}^{-1}$ (away from threshold) and $I_+\sim \lambda_\text{PS}^{-1/2}$.
This leads to events that, to a greater or lesser extent depending on $\nu$, tend to be quasi-spherical; jetty events are rare.
But this behavior does not extend to general $\nu_i$.
For the two field case, once $\nu$ for a decaying particle is significantly larger than the sum of the $\nu_i$ for its daughters, KK-number violation becomes large.
We can see this by examining $I_+$ (which typically is much greater than $I_-$ in this regime) using \Eq{eq:GandR} and the paragraph following it; see also \Sec{sec:Cheby}.
If, without loss of generality, we take the decaying particle to be from field $\Phi_1$ and its daughters from fields $\Phi_2,\Phi_3$, and set $\Delta\nu\equiv \nu_1-\nu_2-\nu_3$, then when $\Delta\nu=2k$, where $k$ is any positive integer, $I_+$ goes to a constant at threshold.
Consequently branching fractions are {\it suppressed} near threshold by $\lambda_\text{PS}^{+1/2}$, and instead peak elsewhere in the branching fraction triangle.
Moreover, $I_+$ has $k-1$ lines of zeroes, and so the branching fraction triangle has $k$ plateaus separated by valleys.
Some of these plateaus have large KK-number violation.
When $\Delta\nu\neq 2k$ these plateaus survive, but are supplemented by a return of the $\lambda_\text{PS}^{-1/2}$ behavior near threshold.
Although this near-threshold enhancement favors KK-number-conserving decays, the large KK-number violating decays in the more distant plateaus often remain dominant, making jetty events common.\footnote{In this discussion we have neglected $I_-$. It can be seen from the formulas of \Sec{sec:analytic} that $|I_+|\gg |I_-|$ for most decays, but there is a subtlety near threshold, where our approximation $I_-\sim \lambda_{\text{PS}}^{-1}$ seems to blow up faster than $I_+$ does. This effect is merely due to our approximation \Eq{eq:IminusDirichletSmallPS}, however, which is not valid at threshold. Instead, when $m_1-m_2-m_3\to 0$ for some decay, which requires tuning of the $\nu_i$, both $I_+$ and $I_-\sim \lambda_{\text{PS}}^{-1/2}$ extremely close to threshold, as noted in \Eq{eq:IminusDirichletVerySmallPS}, and in fact the original integral in \Eq{eq:4dcoup} is always finite there. (See also App.~\ref{app:npdependence}.) In practice, then, our discussion here of $I_+$ captures all the important features of the branching fraction triangles, except for the precise details in the near-threshold region.}
Focusing now on $\nu_2=\nu_3=0$, we will illustrate the behavior just described in the cases $\nu_1=2,3,4$, whose branching fractions for $n_p=100$ are shown in \Fig{fig:brTwoField}.
For $\nu_1=2$, using \Eq{eq:dirNorm}, \Eq{eq:4dcoup}, \Eq{eq:GandR}, and the remark below \Eq{eq:GandR} that $\mathrm{F}_4 \to 1$ in this case, we find (to a very good approximation)
\begin{equation}
c_{ijk} \propto \sqrt{m_2 m_3} \ .
\label{eq:cijknu2}
\end{equation}
The branching fractions are then proportional to $m_2m_3\lambda_\text{PS}^{1/2}$, so they vanish at all boundaries of the branching fraction triangle and are broadly distributed, as seen in \Fig{fig:brTwoField}(a), peaking $\sim10\%$ below threshold.
For $\nu_1=4$, \Eq{eq:F4k2} gives
\begin{equation}
c_{ijk} \propto \sqrt{m_2 m_3} \left(1-3\frac{m_2^2 + m_3^2}{m_1^2}\right),
\label{eq:cijknu4}
\end{equation}
which creates a zero between the threshold region at upper right and the $m_2,m_3\to 0$ corner at lower left.
The branching fraction triangle then has two plateaus, one far from threshold and one nearby, as seen in \Fig{fig:brTwoField}(c).
The probability for a particle to decay via either plateau is comparable.
%
Finally, for $\nu_1=3$, the zero seen for $\nu_1=4$ is still present, closer to threshold, but in addition there is enhancement right near threshold.
%
Despite this enhancement, there are so many decay paths in the plateau that the total probability to decay at or near threshold is only $\sim 1/4$, and so large KK-number violation is the norm.
A cascade in the two field case involves both $\phi_{1,i} \rightarrow \phi_{2,j} \phi_{2,k}$ and $\phi_{2,i} \rightarrow \phi_{1,j} \phi_{2,k}$ decays.
To compute the branching fractions for the latter, we need to exchange $\nu_1$ and $\nu_2$ (but not $\nu_3$) in our analytic formulas.
For reasons similar to those leading to \Eq{eq:Iplussingle}, $I_+$ vanishes for $\nu_1=2,4$, giving the nearly KK-number-conserving result \Eq{eq:cijk_nu_even}, but is important for $\nu=3$, leading to slightly higher KK-number violation.
Thus it is no accident that the branching fraction triangles for these decays, shown in \Fig{fig:brTwoField2}, resemble those of the single field case, \Fig{fig:brratios}.
However, the KK-number violation in $\phi_{2,i}$ decays is subleading compared to the much larger KK-number violation that can occur in $\phi_{1,i}$ decays, and its details do not much impact the results.
As was true also for \Fig{fig:brratios}, the qualitative features of the branching fraction triangles are present also for smaller values of $n_p$, and thus apply for the whole cascade. See App.~\ref{app:npdependence} for further discussion of the $n_p$ dependence.
Particle multiplicities are affected not only by these KK-number-violating processes but also by the increasing number of HSHs. For $\nu_2=\nu_3=0$, and $\nu_1\gtrsim 1.75$, the decay $\phi_{1,1}\to \phi_{2,1}+\phi_{2,1}$ is always open, so only states of $\Phi_2$ are stable against decays within the hidden sector. The mode $\phi_{2,1}$, as the lightest mode in the hidden sector, is of course an HSH, while $\phi_{2,2}$ is stable for $\nu_1\geq 0.5$, $\phi_{2,3}$ is stable for $\nu_1\gtrsim2.90$, and $\phi_{2,4}$ is stable for $\nu_1\gtrsim5.53$. But the effect on event shapes of reduced multiplicity is subleading compared to jet creation through boosts of daughter particles, as we will now see.
\begin{figure}[t!]
\centering
\subfloat[]{
\includegraphics[width=0.45\textwidth]{figures/brNu2Nu0Nu0}
}
\hfill
\subfloat[]{
\includegraphics[width=0.45\textwidth]{figures/brNu3Nu0Nu0}
}
\hfill
\subfloat[]{
\includegraphics[width=0.45\textwidth]{figures/brNu4Nu0Nu0}
}
\hfill
\subfloat[]{
\includegraphics[width=0.45\textwidth]{figures/br_twoField_1dplot}
}
\caption{The branching ratios of the $n_p=100$ mode for decays of the field corresponding to (a) $\nu_1=2$, (b) $\nu_1=3$, and (c) $\nu_1=4$ into two $\nu_2 = 0$ fields with KK modes $n_1$, $n_2$. %
The centralization of the plateau is a function of all $\nu_i$, but generally the plateaus are more strongly centralized for larger values of $\Delta \nu \equiv \nu_1 -\nu_2-\nu_3$.
The 1d projection of the branching ratios for all values of $\nu_1$ along the $n_1 = n_2 + 1$ line is given in (d).
}
\label{fig:brTwoField}
\end{figure}
\begin{figure}[t!]
\centering
\subfloat[]{
\includegraphics[width=0.45\textwidth]{figures/brNu0Nu2Nu0}
}
\hfill
\subfloat[]{
\includegraphics[width=0.45\textwidth]{figures/brNu0Nu3Nu0}
}
\hfill
\subfloat[]{
\includegraphics[width=0.45\textwidth]{figures/brNu0Nu4Nu0}
}
\hfill
\subfloat[]{
\includegraphics[width=0.45\textwidth]{figures/br_twoField_other_1dplot}
}
\caption{The branching fraction triangle of the $n_p=100$ mode for the field corresponding to $\nu_2 =0$ into another $\nu_2$ field and a (a) $\nu_1=2$, (b) $\nu_1=3$, or (c) $\nu_1=4$ field. %
These decays do not exhibit the same plateau structure as in \Fig{fig:brTwoField}.
The 1d projection of the branching ratios for all values of $\nu_1$ along the $n_1 = n_2$ line is given in (d).
}
\label{fig:brTwoField2}
\end{figure}
\begin{figure}[t!]
\centering
\subfloat[]{
\includegraphics[width=0.45\textwidth]{figures/emdNuSpecFull2F}
}
\hfill
\subfloat[]{
\includegraphics[width=0.45\textwidth]{figures/thrustNuSpecFull2F}
}
\hfill
\subfloat[]{
\includegraphics[width=0.45\textwidth]{figures/multNuSpecFull2F}
}
\hfill
\subfloat[]{
\includegraphics[width=0.45\textwidth]{figures/energySpec2FGraphComp}
}
\caption{Same as \Fig{fig:singleFieldDist} but for the two field scenario with $\nu_2 = \nu_3 = 0$, and $\nu_1 =2$ (blue), $\nu_1=3$ (green), and $\nu_1=4$ (red).
Note that average multiplicity is lower and event isotropy is higher than the single field cases in \Fig{fig:singleFieldDist}.
The appearance of plateaus in the branching ratios can greatly increase the amount of KK violation per event.
}
\label{fig:emdMultTwoField}
\end{figure}
Our results are shown in \Fig{fig:emdMultTwoField}, based again on cascades starting at the 100$^\text{th}$ KK mode of the field $\Phi_1$, with $10^4$ events per sample.
Just as the branching fractions are much more widely distributed in \Fig{fig:brTwoField} than in \Fig{fig:brratios}, all the distributions in event isotropy and thrust are much wider than for the single field case in \Fig{fig:singleFieldDist}.
As is evident by comparing these figures with \Fig{fig:brTwoField}, the plateaus and valleys in the branching fraction triangles of the $\nu_1$ field lead directly to structure in the event shape variables.
It is easy to see why this is so.
The early stages in the cascade are most important, because decays far from threshold early in the cascade create boosted particles with a substantial fraction of the event energy, and their collimated decay products lead to hard jets, while reducing the overall multiplicity of hadrons.
Such events will differ strongly from the near-spherical events we saw in the single-field examples.
If instead the initial decays nearly conserve KK-number and create several slow heavy hadrons, KK-number violation in their decays will still lead to jets, but these will have a much smaller fraction of the event's energy.
Thus, depending on the details of a particular cascade decay, an event may present a small number of hard jets at one extreme, or a small number of soft jets on top of a quasi-spherical background at the other.
As the probability for far-from-threshold decays increases, so will the fraction of events with larger thrust and event isotropy.
This is seen most dramatically for $\nu_1=4$, where the bimodal feature in the branching fractions, \Fig{fig:brTwoField}(c), with one plateau that weakly violates KK-number and another that violates KK-number significantly, leads to bimodal event shapes at small and large event isotropy and thrust respectively.
These correspond to two classes of events, one relatively spherical and the other relatively jetty. Note that the lobe at larger event isotropy is comparable to the event isotropy distribution for threshold $t\bar t$ events, shown in \Fig{fig:SMbenchmark}, and actually peaks at higher $\iso{sph}{192}$, though it is not as jetty as a $q\bar q$ sample.
Comparing $\nu_1=2$ and $\nu_1=3$, one sees the latter's ensemble of events is less spherical on average.
Despite the near-threshold enhancement for $\nu_1=3$, the majority of decays occur in the plateau far from threshold.
This plateau is located further from threshold than that of $\nu_1=2$, so the most common decays for $\nu_1=3$ have larger boost on average than those for $\nu_1=2$, and this leads to harder jets, moving the isotropy and thrust distributions to larger values.
The near-threshold decays partly compensate for this effect, making the event-shape variable distributions particularly broad.
The multiplicities of massless particles are of order 80 for $\nu_1=2$ and (partly because of the additional HSH) only 50 for $\nu_1=3,4$.
Although this was not evident in the single-field cases, here one can see clearly that the event shape variables are not perfectly correlated with multiplicity.
For instance the multiplicity distribution for $\nu_1=4$ does not show the obvious two-lobe structure seen in the event shape variables.
We will return to this issue briefly in \Sec{subsec:evolmult}.
It is also noteworthy that the isotropy distribution is narrower than the thrust distribution, a hint of imperfect correlation which we will study further in \cite{paper2}.
Turning our attention to the energy distributions of \Fig{fig:emdMultTwoField}, it is interesting to note that all the spectra (including the reference case $\nu=0$) are peaked at the same value.
The peak is at half the lightest mass in the cascade: $m^{\nu=0}_1/2 = 1.2$.
This tells us that for all cascades there are many soft particles produced in the decay of nonrelativistic $n=1$, $\nu=0$ particles.
The width of the distribution is much wider for $\nu_1 = 2$, $\nu_2 =0$ than for $\nu=0$, and even wider for $\nu_2 = 3, 4$.
This tail is due to the fact that modes heavier than the lightest mode are stable in the two field cases, whereas for $\nu=0$, only the lightest mode is stable.
Note that these observations also apply to all the single field cases shown in \Fig{fig:singleFieldDist}.
Visualizations of characteristic events from each sample are shown in \Fig{fig:twoFieldVis}.
The effects of the unsuppressed KK violating decays are evident.
\begin{figure}[!h]
\centering
\subfloat[]{
\includegraphics[width=0.23\textwidth]{figures/nu2nu0event.pdf}
}
\hfill
\subfloat[]{
\includegraphics[width=0.23\textwidth]{figures/nu3nu0event.pdf}
}
\hfill
\subfloat[]{
\includegraphics[width=0.23\textwidth]{figures/nu4nu0Isoevent.pdf}
}
\hfill
\subfloat[]{
\includegraphics[width=0.23\textwidth]{figures/nu4nu0Jetevent.pdf}
}
\caption{ Visualizations of a characteristic final state radiation pattern for the (a) $\nu_1 = 2$, (b) $\nu_1 = 3$, (c,d) $\nu_1 = 4$ and $\nu_2 = \nu_3 = 0$ samples.
The chosen events for $\nu_1 = \{2,3\}$ have event isotropy that are equal to the mean value of the distributions: (a) $\langle \iso{sph}{192} \rangle = 0.44$ and (b) $\langle \iso{sph}{192} \rangle = 0.54$.
We show two events for the $\nu_1=4$ sample, selected from the peaks of the bimodal distribution in event isotropy:
(c) $\langle \iso{sph}{192} \rangle = 0.31$ and (d)$\langle \iso{sph}{192} \rangle = 0.69$.
All momenta in the event are plotted, with the length proportional to the magnitude.
}
\label{fig:twoFieldVis}
\end{figure}
\subsection{Cascades with boundary couplings}
\label{subsec:boundary}
In the scenarios discussed above, we have assumed that fields interact in the bulk of the extra dimension, as in \eqref{eq:interactions}, and we have taken the boundary conditions to be Dirichlet. A very different phenomenology arises if we assume that the fields have Neumann boundary conditions, and add the interaction term \eqref{eq:bdryinteractions} localized on the IR boundary of the space. As indicated in \eqref{eq:neumannIRvalue}, the boundary values of the different KK modes are approximately equal, and so decays are governed approximately by the phase space of the final state. The branching ratios in this case are illustrated in Fig.~\ref{fig:BRnu0p3bdry}, which also shows the (similar) branching ratios that would result from pure phase space factors. Of course, one can also consider a model containing both the bulk coupling $c$ \eqref{eq:interactions} and the boundary coupling $\tilde c$ \eqref{eq:bdryinteractions}, obtaining physics that interpolates between the two. Below we will show some results from such a case with ${\tilde c} = 0.015 c$, which turns out to produce event-shape distributions roughly midway between those of the pure bulk and pure boundary cases.
\begin{figure}[!h]
\centering
\subfloat[]{\includegraphics[width=0.47\textwidth]{figures/BRWidthnu0p3nModes100NeumannPlotTot}}\qquad
\subfloat[]{\includegraphics[width=0.47\textwidth]{figures/BRWidthnu0p3nModes100BdyPlotTot}}\\\subfloat[]{\includegraphics[width=0.47\textwidth]{figures/BRWidthnu0p3nModes100PSPlotTot}}\qquad\subfloat[]{\includegraphics[width=0.41\textwidth]{figures/BRbulkvsbdry1d}}
\caption{The branching ratios for $\nu=0.3$ of the $100$th KK mode into all kinematically allowed two-body final states, in scenarios with Neumann boundary conditions. The cases are (a) bulk couplings; (b) boundary-localized coupling; and (c) pure phase space. In (d), we show a 1d plot of the branching ratios along the $n_1 = n_2$ line. For the boundary-localized case (b), in contrast to Fig.~\ref{fig:brratios} and the bulk case, the decays significantly populate the full triangle, and favor daughter particles with substantial momentum. The distribution is similar to that of pure phase space (c).
}
\label{fig:BRnu0p3bdry}
\end{figure}
\begin{figure}[t!]
\centering
\subfloat[]{\includegraphics[width=0.47\textwidth]{figures/eviso_1d_bulkvbdry}}
\quad \subfloat[]{\includegraphics[width=0.47\textwidth]{figures/thrust_1d_bulkvbdry}}\\
\subfloat[]{\includegraphics[width=0.47\textwidth]{figures/mult_1d_bulkvbdry}}
\quad
\subfloat[]{\includegraphics[width=0.47\textwidth]{figures/energies_1d_bulkvbdry}}
\caption{The distributions of (a) event isotropy $\iso{sph}{192}$; (b) rescaled thrust $\tilde T$; (c) particle multiplicity; and (d) energy of final-state particles for three different scenarios, all with $\nu = 0.3$, Neumann boundary conditions, and a decay cascade beginning with the $100$th KK mode. The couplings are: in blue, pure bulk ($\tilde c = 0$); in orange, pure boundary ($c = 0$); in green, a mixed case ($\tilde c = 0.015 c$). For ease of comparison with other figures, we also show the $\nu = 0$ Dirichlet case (originally displayed in Fig.~\ref{fig:toyModels}). We see that the cascade with a boundary coupling produces less isotropic events (larger $\iso{sph}{192}$). Turning on both couplings interpolates between the bulk and boundary case, with a broad distribution of $\iso{sph}{192}$.
}
\label{fig:evisobulkvbdry}
\end{figure}
In Fig.~\ref{fig:evisobulkvbdry}, we illustrate how boundary couplings affect the event shape. The blue curve shows the case of pure bulk couplings with Neumann boundary conditions, for a single self-interacting field with $\nu=0.3$; as in Fig.~\ref{fig:singleFieldDist}, this leads to approximately spherical events with small values of $\iso{sph}{192}$ and thrust.\footnote{In the Neumann case, $I_-$ is suppressed and $I_+$ dominates; consequently the checkerboard pattern seen in \Fig{fig:brratios} is absent.} The orange curve shows events with boundary decays, which are substantially less isotropic. They also have a broader distribution of both event isotropy and thrust, although the event isotropy distribution is more peaked than the thrust distribution. Comparing to Fig.~\ref{fig:SMbenchmark}, we see that the typical event with boundary decays is more isotropic than a QCD dijet event, but has similar isotropy to a near-threshold $t \bar{t}$ event. Finally, we show the case with a mixed bulk/boundary coupling, $\tilde c = 0.015 c$, in green. As expected, this interpolates between the bulk and boundary cases. It does so by broadening the distribution, rather than producing a narrow peak in between the two cases. Visualizations of typical events are shown in Fig.~\ref{fig:bdryVis}.
\begin{figure}[!h]
\centering
\subfloat[]{
\includegraphics[width=0.31\textwidth]{figures/bulkAvgEvent}
}
\hfill
\subfloat[]{
\includegraphics[width=0.31\textwidth]{figures/boundaryAvgEvent}
}
\hfill
\subfloat[]{
\includegraphics[width=0.31\textwidth]{figures/mixedAvgEvent}
}
\caption{ Visualizations of a characteristic final state radiation pattern for the (a) bulk coupling, (b) boundary coupling, (c) mixed ($\tilde c = 0.015 c$) samples. In each case, $\nu = 0.3$ and we assume Neumann boundary conditions.
Each event has event isotropy equal to the mean value for the corresponding event sample: (a) $\langle \iso{sph}{192} \rangle = 0.33$; (b) $\langle \iso{sph}{192} \rangle = 0.67$; and (c) $\langle \iso{sph}{192} \rangle = 0.55$.
The plots show that events with boundary couplings are visibly less isotropic than the sample with only a bulk coupling.
}
\label{fig:bdryVis}
\end{figure}
Fig.~\ref{fig:evisobulkvbdry} also shows the typical multiplicity and energy of the individual final-state massless particles in the events. We see that the boundary cascades have much lower multiplicity, because the decays more often go directly to lighter daughters, so it takes fewer steps to reach the HSHs at the bottom of the cascade. Consequently, the individual particles also have more energy than they would for bulk couplings. This figure suggests that the event isotropy may be highly correlated with the particle multiplicity. Although this is true when we start all cascades with the same initial KK-number, it is not the case in general: boundary cascades remain much less isotropic even if they begin with a much larger choice of $n_p$. Thus event isotropy captures {\em different} information from particle multiplicity, or even from pairs of observables like particle multiplicity and thrust. We will discuss this more in the companion paper \cite{paper2}.
\subsection{Evolution of Multiplicity, Event Isotropy in Cascade}
\label{subsec:evolmult}
\begin{figure}[t!]
\centering
\includegraphics[width=0.46\textwidth]{figures/nuAllFullcascade}
\caption{The average multiplicity vs. average event isotropy at each step in the cascade decays for the single field samples and two field samples. }
\label{fig:multEMD}
\end{figure}
To investigate further the correlations between multiplicity and isotropy, it is interesting to see how these evolve through the decay cascade.
We do the following exercise.
At each step in the cascade, we take all the hadrons present at that step (independent of whether they are stable against hadronic decays in following steps) and artificially force them to decay to massless particles.
At the initial step of the cascade, with just the $n_p$ mode at rest, every event has a pencil-dijet with $\iso{sph}{192}=\tilde T=1$, while at the end of the cascade we obtain the samples studied above.
In between, the multiplicity in each event gradually increases and $\iso{sph}{192}$ decreases.
In \Fig{fig:multEMD}, we plot the average multiplicity and the average isotropy (averaged over $10^4$ events) at each step in the cascade.
We also show, as a dashed line, the theoretical lower limit on $\iso{sph}{192}$ for the corresponding multiplicity; see \Eq{eq:isoest}.
If multiplicity and event isotropy were perfectly correlated, then all of the cascades would lie on the same curve, even though at any given step, and at the end of the cascade, they would sit at different values.
Instead, we see that the cascades for different choices of $\nu_i$ can give different curves.
This is additional evidence that isotropy measures more than multiplicity.
%
We will explore the independence and correlation of multiplicity, isotropy, thrust and other event shapes in \cite{paper2}.
\section{Analytical Estimates for Couplings}
\label{sec:analytic}
In \Sec{sec:simResults}, we saw that different parameter choices lead to qualitatively different patterns of couplings $c_{ijk}$ and branching fractions, and from there to qualitatively different event shapes.
In this section, we provide analytic calculations of the overlap integrals that determine the couplings, in order to substantiate and explain the results presented above. In particular, an {\em inverse} dependence on the phase-space function $\lambda_\text{PS}$ will be manifest in our results, explaining the cases in which we have observed a preference for near-threshold decays. We will also understand the oscillatory behavior that gives rise to the observed plateaus in the two-field case.
\subsection{Basic ingredients and general strategy}
Our goal is to gain an analytic understanding of integrals of the form
\begin{equation}
I(\nu_i, m_i) \equiv \int_0^1 {\rm d}z\, z\, \J{\nu_1}{m_1 z} \, \J{\nu_2}{m_2 z} \, \J{\nu_3}{m_3 z}.
\label{def:Iintegral}
\end{equation}
We choose a convention where particle \#1 is the heaviest particle, i.e., we assume without loss of generality that $m_1 \geq m_2 + m_3$. We can separate our integral into two pieces,
\begin{align}
I(\nu_i, m_i) &= I_+(\nu_i, m_i) - I_-(\nu_i, m_i), \quad \text{where:} \nonumber \\
I_+(\nu_i, m_i) &\equiv \int_0^{\infty} {\rm d}z\, z\, \J{\nu_1}{m_1 z} \, \J{\nu_2}{m_2 z} \, \J{\nu_3}{m_3 z}, \nonumber \\
I_-(\nu_i, m_i) &\equiv \int_1^{\infty} {\rm d}z\, z\, \J{\nu_1}{m_1 z} \, \J{\nu_2}{m_2 z} \, \J{\nu_3}{m_3 z}. \label{eq:IpImdef}
\end{align}
The reason for doing this is that when all of the masses (in units of the IR brane scale) are sufficiently large, i.e., when $m_i \gg \nu_i^2$, we can approximate the Bessel functions in the integrand of $I_-(\nu_i, m_i)$ by their large-argument asymptotic expansions. This makes approximating $I_-(\nu_i, m_i)$ into an analytically tractable problem. On the other hand, the integral $I_+(\nu_i, m_i)$ over the whole positive real axis is known analytically. By combining the exact analytic answer for $I_+$ and the approximate answer for $I_-$, we obtain an analytic approximation to the $I(\nu_i, m_i)$ and hence to the couplings among Kaluza-Klein modes.
\begin{figure}[!h]
\centering
\includegraphics[width=1.0\textwidth]{figures/IpmdecompRevised}
\caption{Examples of the decomposition of the integral $I$ (solid) into a sum of $I_+$ (dashed) and $-I_-$ (dotted), in the single-field case where $\nu_1 = \nu_2 = \nu_3 \equiv \nu = 0.75$. In both plots, the inset triangle shows the slice of the branching ratio triangle (as in Fig.~\ref{fig:brratios}) along which we have done the calculation. At left, we take near threshold decays (right-hand edge of the triangle); at right, we take decays along a slice through the middle of the triangle. For near-threshold decays, we see that $I_+$ and $I_-$ are comparable. Away from threshold, $I_+$ dominates, with $I_-$ contributing a small oscillatory pattern.}
\label{fig:Ipmdecomp}
\end{figure}
In Fig.~\ref{fig:Ipmdecomp}, we show examples of how $I$ breaks down into contributions from $I_+$ and $-I_-$, along two slices of the branching ratio triangles in the single-field case with $\nu = 0.75$. Typically, $I_+$ dominates, and the integral is suppressed for decays to light modes. Near threshold, $I_-$ gives a contribution comparable to that of $I_+$. As we will see below, in certain special cases (like $\nu = 0$), the integral $I_+$ is identically zero, but the behavior seen in the plot is representative of more general $\nu$ values.
Our goal in the remainder of this section is to provide some analytic insight into the behavior of the integrals $I_+$ and $I_-$. We will first discuss an analytic approximation to $I_-(\nu_i, m_i)$, and compare it to numerical results. Then we will present an exact analytic formula for $I_+(\nu_i, m_i)$, and comment on a special case to elucidate the ``plateau'' structure observed in Fig.~\ref{fig:brTwoField}.
\subsection{Approximating the integral $I_-(\nu_i, m_i)$}
We apply the large-argument asymptotic approximation of the Bessel function, \eqref{eq:besselasymptotic}, to estimate $I_-(\nu_i, m_i)$:
\begin{equation}
I_-(\nu_i, m_i) \approx I^{(0)}_-(\nu_i, m_i) \equiv \int_1^\infty {\rm d}z\, z\prod_{i=1}^3\sqrt{\frac{2}{\text{\greektext p} m_i z}} \cos\left(m_i z - \frac{1}{2} \text{\greektext p} \nu_i - \frac{1}{4} \text{\greektext p}\right).
\label{eq:Iminus0def}
\end{equation}
We can rewrite the product of three cosines as a sum of four cosines by repeated use of the identity $2\cos a\cos b = \cos(a+b)+\cos(a-b)$. Given signs $\sigma, \sigma' \in \{+1,-1\}$, we define
\begin{equation}
m_{\sigma \sigma'} \equiv m_1 + \sigma m_2 + \sigma' m_3, \quad \nu_{\sigma \sigma'} \equiv \nu_1 + \sigma \nu_2 + \sigma' \nu_3.
\end{equation}
In labels, we will suppress the ``1'' and write the $\sigma$'s as $+$ or $-$; for example, we denote $m_1 + m_2 - m_3$ by $m_{+-}$. Then \eqref{eq:Iminus0def} is equivalent to
\begin{equation}
I^{(0)}_-(\nu_i, m_i) = \frac{1}{\text{\greektext p}^{3/2} \sqrt{2m_1 m_2 m_3}} \int_1^\infty \frac{{\rm d}z}{\sqrt{z}} \sum_{\sigma, \sigma' \in \{+,-\}} \cos\left(m_{\sigma \sigma'}z - \frac{\text{\greektext p}}{2} \nu_{\sigma \sigma'} - \frac{\text{\greektext p}}{4}(1 + \sigma + \sigma')\right).
\label{eq:Iminusfourterms}
\end{equation}
This integral can be performed analytically in terms of the Fresnel cosine and sine integrals. Our conventions for these are specified in Appendix \ref{sec:fresnel}. We obtain:
\begin{align}
I^{(0)}_-(\nu_i, m_i) = \frac{1}{\text{\greektext p} \sqrt{m_1 m_2 m_3}} \sum_{\sigma, \sigma' \in \{+,-\}} \sqrt{\frac{1}{m_{\sigma \sigma'}}} \Bigg[ & \cos\left(\frac{\text{\greektext p}}{2} \nu_{\sigma\sigma'} + \frac{\text{\greektext p}}{4}(1+\sigma+\sigma')\right)\left(\frac{1}{2} - \FC{\sqrt{\frac{2m_{\sigma \sigma'}}{\text{\greektext p}}}}\right) + \nonumber \\
& \sin\left(\frac{\text{\greektext p}}{2} \nu_{\sigma\sigma'} + \frac{\text{\greektext p}}{4}(1+\sigma+\sigma')\right)\left(\frac{1}{2} - \FS{\sqrt{\frac{2m_{\sigma \sigma'}}{\text{\greektext p}}}}\right)\Bigg].
\label{eq:IminusFresnel}
\end{align}
This is already a useful approximation to $I_-(\nu_i, m_i)$. We can go further by noting that, provided the masses are all large and that we do not consider decays too close to threshold, we can exploit the large-argument asymptotics of the Fresnel integrals, \eqref{eq:Fresnelasymptotics}. In this case, we find that the answer depends on $\sin$ and $\cos$ of $m_{\sigma \sigma'}$, so we make use of the mass eigenvalue estimates in \eqref{eq:KKmassestimate}. We will provide the estimate in the case of Dirichlet boundary conditions.
\subsubsection{$I_-(\nu_i,m_i)$ estimate for Dirichlet boundary conditions}
Keeping the first subleading, ${\cal O}(1/z)$, term in the Fresnel integral asymptotics, the expression in brackets in \eqref{eq:IminusFresnel} becomes
\begin{align}
\frac{1}{\sqrt{2\text{\greektext p} \,m_{\sigma \sigma'}}} \Bigg[ &- \cos\left(\frac{\text{\greektext p}}{2} \nu_{\sigma\sigma'} + \frac{\text{\greektext p}}{4}(1+\sigma+\sigma')\right) \sin(m_{\sigma \sigma'}) + \sin\left(\frac{\text{\greektext p}}{2} \nu_{\sigma\sigma'} + \frac{\text{\greektext p}}{4}(1+\sigma+\sigma')\right) \cos(m_{\sigma \sigma'})\Bigg],
\end{align}
Using the sine sum-of-angles identity, then using \eqref{eq:KKmassestimate} to replace the masses in the argument of the sine by an approximate expression in terms of the KK mode numbers $n_i$, we obtain a simple approximate formula for the Dirichlet case:
\begin{align}
I^{(1)}_-(\nu_i, m_i) &\equiv -\frac{1}{\text{\greektext p}^{3/2} \sqrt{2m_1 m_2 m_3}} \sum_{\sigma, \sigma' \in \{+,-\}} \frac{1}{m_{\sigma \sigma'}} \sin\left(\text{\greektext p} (n_1 + \sigma n_2 + \sigma' n_3) - \frac{\text{\greektext p}}{2} (1 + \sigma + \sigma')\right) \nonumber \\
&= \frac{(-1)^{n_1 + n_2 + n_3 + 1}}{\text{\greektext p}^{3/2} \sqrt{2m_1 m_2 m_3}} \sum_{\sigma, \sigma' \in \{+,-\}} \frac{\sigma \sigma'}{m_{\sigma \sigma'}},
\label{eq:IminusDirichlet}
\end{align}
where $n_i$ is the integer mode number of the corresponding KK mode. In particular, there is a term that scales as $1/(m_1 - m_2 - m_3)$ that dominates when the phase space is relatively small (but not so small that our approximations break down). We can write this term using the phase space factor \eqref{eq:lambdaPS}
\begin{equation}
I^{(1)}_-(\nu_i, m_i) \approx \frac{(-1)^{n_1 + n_2 + n_3 + 1} 4 \sqrt{2}}{\text{\greektext p}^{3/2}} \frac{\sqrt{m_1 m_2 m_3}}{\lambda_\text{\rm PS}(m_1^2,m_2^2,m_3^2)},
\label{eq:IminusDirichletSmallPS}
\end{equation}
which we used in \eqref{eq:cijk_nu_even}.
The large-argument expansion of the Fresnel functions breaks down at $m_{\sigma \sigma'} \lesssim 1$, very close to threshold. This case generally only arises if we tune the values of $\nu$ so that the offsets in the Bessel function zeros align to allow for $m_3 \approx m_1 + m_2$, so we do not expect that it is generally relevant, but we discuss it for completeness. In this case, the sum is dominated by the {\em small}-argument expansion of the Fresnel functions for the $\sigma, \sigma' = -1$ case,
\begin{align}
I^{(\mathrm{threshold})}_-(\nu_i, m_i) &\approx \frac{1}{\text{\greektext p} \sqrt{2 m_1 m_2 m_3}} \frac{1}{\sqrt{m_{--}}} \sin\left(\frac{\text{\greektext p}}{2} (\nu_1 - \nu_2 - \nu_3)\right) \nonumber \\
&\approx \frac{2}{\text{\greektext p}} \frac{ \sin\left[\frac{\text{\greektext p}}{2} (\nu_1 - \nu_2 - \nu_3)\right] }{\lambda_\mathrm{PS}^{1/2}(m_1^2,m_2^2,m_3^2)}.
\label{eq:IminusDirichletVerySmallPS}
\end{align}
Thus, very close to threshold, we expect that the divergence is ameliorated to $\lambda_\mathrm{PS}^{-1/2}$, except in cases where $\nu_1 - \nu_2 - \nu_3$ is an even integer, when this term has coefficient zero and subleading terms dominate. As we will see below, this is also the near-threshold behavior of the integral $I_+$, so we predict that $I_-$ does not parametrically dominate over $I_+$ in the small phase-space region. In fact the original integral is finite at threshold, so the singular behavior of $I_+$ and $I_-$ must cancel there.
Unlike the Dirichlet case, note that the large-argument approximation for the Fresnel function would have given zero for the Neumann case, because the different constant term in \eqref{eq:KKmassestimate} removes the $\frac{\text{\greektext p}}{2} (1 + \sigma + \sigma')$ term in the argument of the sine and leads to zero. To obtain a similar approximation in the Neumann case, we must keep subleading terms in the various approximations we have made. We will not do so here.
\begin{figure}[!h]
\centering
\includegraphics[width=0.6\textwidth]{figures/IminusanalyticRevised}
\caption{Comparison of a numerical computation of the integral $I_-$ (solid curves) with the two analytic approximations $I_-^{(0)}$, \eqref{eq:IminusFresnel} (dashed curves) and $I_-^{(1)}$, \eqref{eq:IminusDirichlet} (dotted curves). The dashed curves always fall so close to the solid curve that they are indistinguishable. The simpler analytic formulation, $I_-^{(1)}$, works very well away from threshold but deviates close to the threshold, leading to the visible dotted curves. As in Fig.~\ref{fig:Ipmdecomp}, the inset triangles illustrate the slice through the branching ratio triangle that is plotted. We show the case $\nu = 0.3$ because it shows a larger discrepancy, and thus a more visible dotted curve, than the case $\nu = 0.75$.}
\label{fig:Iminusanalytic}
\end{figure}
We compare a numerical computation of the integral $I_-$ with the two approximations $I_-^{(0)}$ \eqref{eq:IminusFresnel} and $I_-^{(1)}$ \eqref{eq:IminusDirichlet} in Fig.~\ref{fig:Iminusanalytic}. The first approximation, based on the large-argument expansion of the Bessel functions, works extremely well. The subsequent approximations made in the Dirichlet case lead to an imprecise estimate, but one which is useful since the formula \eqref{eq:IminusDirichlet} makes the enhancement of the integral in the small phase-space region obvious.
\subsection{Computing the integral $I_+(\nu_i, m_i)$}
\subsubsection{General formula}
In the case of $I_-$, we used the asymptotic expansion of the Bessel function to obtain an analytic approximation. We can do better with $I_+$: the integral is analytically known (eq.~(7.1) of \cite{bailey1936some}, in the special case $\lambda = 2$) to be:
\begin{align}
I_+(\nu_i, m_i) &= \frac{2 m_2^{\nu_2} m_3^{\nu_3} m_1^{-\nu_2 - \nu_3 - 2} \ \Gamma\left(\frac{\nu_1 + \nu_2 + \nu_3}{2}+1\right)}{\Gamma \left( \nu_2 + 1 \right) \Gamma \left(\nu_3 + 1 \right) \Gamma \left( \frac{\nu_1 - \nu_2 - \nu_3}{2}\right)} \nonumber \\
&\times \mathrm{F}_4 \left( 1+\frac{\nu_2+\nu_3-\nu_1}{2},1+\frac{\nu_1+\nu_2+\nu_3}{2}; \nu_2+1, \nu_3 +1; \frac{m_2^2}{m_1^2}, \frac{m_3^2}{m_1^2}\right).
\label{eq:GandR}
\end{align}
The function $\mathrm{F}_4$ is known as an Appell function; it is a two-variable generalization of a hypergeometric function. This integral has previously appeared in the physics literature on 3-point correlators in momentum space in conformal field theories \cite{Coriano:2013jba, Bzowski:2013sza, Gillioz:2019lgs} and de Sitter space \cite{Antoniadis:2011ib, Sleight:2019mgd}. For convenience, we include its definition in Appendix~\ref{sec:appell}.
The behavior of the Appell function leads to one of the important qualitative features we have observed in our numerical results: the existence of plateaus of large branching fractions separated by valleys of suppressed branching fractions. It is manifest from the series definition of the Appell function that it becomes a polynomial when its first argument is a non-positive integer. When $\nu_1=\nu_2+\nu_3+2k$, for $k$ a positive integer, the polynomial has degree $k-1$, and has $k-1$ curves of zeroes. Numerical results, supported by incomplete analytic arguments, indicate that these zeros always lie in the physical region $m_2+m_3<m_1$.\footnote{Along the line $m_3 = 0$, this follows from the fact that $\mathrm{F}_4$ is simply ${}_2\mathrm{F}_1$, discussed in Appendix~\ref{sec:appell}, together with Theorem 3.2.i of Ref.~\cite{driver2008zeros}. In the special case $\nu_2 = \nu_3$, it can also be proven along the line $m_2 = m_3$, using the same theorem together with \Eq{eq:Iplus2F1} below. However, we lack a completely general proof.} Two special cases of interest to our two field case are $k=1$, for which the Appell function is 1, and $k=2,\nu_2=\nu_3$, for which it is
\begin{equation}
\mathrm{F}_4 \left( -1,2 \nu_2+3; \nu_2+1, \nu_2 +1; \frac{m_2^2}{m_1^2}, \frac{m_3^2}{m_1^2}\right) = 1 - \frac{2\nu_2+3}{\nu_2+1}\left(\frac{m_2^2+m_3^2}{m_1^2}\right).
\label{eq:F4k2}
\end{equation}
The zeros of the Appell functions form the valleys between plateaus observed in Fig.~\ref{fig:brTwoField}. Even in cases where the Appell function is not a polynomial, we expect that $\floor*{\frac{\nu_1 - \nu_2 - \nu_3}{2}}$, when positive, approximately counts the number of plateaus.
The expression \Eq{eq:GandR} can always be reduced to an expression in terms of ordinary hypergeometric functions \cite{bailey1933reducible, gervois1985integrals, gervois1986some, GervoisNaveletUnpublished}
{\small
\begin{align}
I_+(\nu_i, m_i) &= \left(\frac{m_2}{m_1}\right)^{\nu_2}\left(\frac{m_3}{m_1}\right)^{\nu_3} \frac{2}{\lambda_{\rm PS}^{1/2}(m_1^2,m_2^2,m_3^2)} \frac{\Gamma\left(\frac{\nu_1 + \nu_2 + \nu_3}{2}+1\right)}{\Gamma \left( \nu_3 + 1 \right) \Gamma \left(\nu_2 + 1 \right) \Gamma \left( \frac{\nu_1 - \nu_2 - \nu_3}{2}\right)} \nonumber \\
\times &\Bigg[\frac{\nu_1 + \nu_3 - \nu_2}{2\nu_1} {}_2\mathrm{F}_1\left(1 + \frac{\nu_2+\nu_3 - \nu_1}{2},\frac{\sum_i \nu_i}{2},\nu_2 + 1; X\right) {}_2\mathrm{F}_1\left(\frac{\nu_2+\nu_3 - \nu_1}{2},1+\frac{\sum_i \nu_i}{2},\nu_3 + 1; Y\right) \nonumber \\
+ & ~ \frac{\nu_1 + \nu_2 - \nu_3}{2\nu_1} {}_2\mathrm{F}_1\left(1 + \frac{\nu_2+\nu_3 - \nu_1}{2},\frac{\sum_i \nu_i}{2},\nu_3 + 1; Y\right) {}_2\mathrm{F}_1\left(\frac{\nu_2+\nu_3 - \nu_1}{2},1+\frac{\sum_i \nu_i}{2},\nu_2 + 1; X\right) \Bigg], \nonumber \\
\text{where } X &\equiv \frac{2m_2^2}{m_1^2+m_2^2 - m_3^2 + \lambda_{\rm PS}^{1/2}(m_1^2,m_2^2,m_3^2)}, ~Y \equiv \frac{2m_3^2}{m_1^2-m_2^2 + m_3^2 + \lambda_{\rm PS}^{1/2}(m_1^2,m_2^2,m_3^2)}.
\label{eq:Iplus2F1}
\end{align}
}
The variables $X$ and $Y$ are defined so that $(m_2/m_1)^2 = X(1-Y)$ and $(m_3/m_1)^2 = Y(1-X)$.
We comment on some details of the reduction from $\mathrm{F}_4$ to ${}_2\mathrm{F}_1$ in Appendix \ref{sec:hypergeom}.
We can extract from \eqref{eq:Iplus2F1} a few simple, general points:
\begin{itemize}
\item The integral $I_+$ has a phase-space factor $\lambda_{\rm PS}^{1/2}(m_1^2,m_2^2,m_3^2)$ in the {\em denominator}. Thus, precisely when the general decay width formula \eqref{eq:width} would lead one to naively expect a decay to be rare, the coupling squared enhances it via an {\em inverse} dependence of the decay width on the phase space. This accounts for the general tendency of the 5d models to feature many near-threshold decays.
\item The hypergeometric functions have the property $\lim_{z \to 0} {}_2\mathrm{F}_1(a,b,c; z) = 1$, so they are unsuppressed when $X$ or $Y$ is small. However, the prefactors $(m_2/m_1)^{\nu_2} (m_3/m_1)^{\nu_3}$ indicate that, in general, the decay rates to light daughters are suppressed. Even in the special case $\nu_2 = \nu_3 = 0$ when this suppression is absent, the normalization factor $N^{(\nu)}_n$ {\em is} smaller for light daughters than heavy ones, as indicated in \eqref{eq:dirNorm}. Thus, in general, we expect that decays to light daughters are rare, as confirmed by the numerical results in Fig.~\ref{fig:brTwoField}, for example.
\item Recall that $\Gamma(x)$ has poles whenever $x$ is a nonpositive integer. As a result, whenever $\nu_2 + \nu_3 - \nu_1$ is a positive even integer, the integral $I_+(\nu_i, m_i)$ vanishes. In the special case $\nu_1 = 0$, there are also zeros in the denominator, but taking the limit from nonzero $\nu_1$'s shows that $I_+$ vanishes in this case as well (as we saw in Fig.~\ref{fig:Ipmdecomp}, where all $\nu$'s were zero).
\item We have $X, Y \in (0,1)$, so that the hypergeometric functions are nonsingular in the physical region except perhaps near threshold. In the near threshold region $m_2 + m_3 \to m_1$, we have $X \to m_2/m_1$ and $Y \to m_3/m_1$. The hypergeometric functions may be singular when $X \to 1$ or $Y \to 1$. For concreteness, consider the case $X \to 1$. This requires that $m_3 \to 0, m_2 \to m_1$. In this case, the ${}_2\mathrm{F}_1$ functions of $X$ diverge as $(1-X)^{-\nu_3}$. However, this is compensated by the prefactor $(m_3/m_1)^{\nu_3}$. As a result, the prefactor of $\lambda_{\rm PS}^{-1/2}$ is the only source of singularities at the boundary of the physical region.
\end{itemize}
\subsubsection{Plateau structure in the special case $\nu_2 = \nu_3 = 1/2, \nu_1 \in \mathbb{Z}$}
\label{sec:Cheby}
We already noted that ${\rm F}_4$ becomes a polynomial when $\nu_1 - \nu_2 - \nu_3$ is a positive even integer; in this case, the hypergeometric functions in \eqref{eq:Iplus2F1} are also simply polynomials in $X$ and $Y$. We will now present a special case for which we can give a straightforward derivation of the integral $I_+$ in terms of Chebyshev polynomials. This is one of the simplest cases in which the existence of plateaus and valleys can be deduced.
Spherical Bessel functions have simple expressions in terms of trigonometric functions. In particular, for $\nu = 1/2$, we have the identity
\begin{equation}
\J{1/2}{x} = \sqrt{\frac{2}{\text{\greektext p} x}} \sin x.
\end{equation}
In the special case $\nu_2 = \nu_3 = 1/2$, this reduces our overlap integral to
\begin{equation}
\frac{2}{\text{\greektext p}} \int_0^{z_{\rm IR}} {\rm d}z\, \J{\nu_1}{m_1 z} \sin(m_2 z) \sin(m_3 z) = \frac{1}{\text{\greektext p}} \int_0^{z_{\rm IR}} {\rm d}z\, \J{\nu_1}{m_1 z} \left[\cos\left((m_2 - m_3) z\right) - \cos\left((m_2 + m_3) z \right)\right].
\end{equation}
The infinite integral $\int_0^\infty {\rm d}t\, \J{\nu}{a t}\cos(b t)$ is known \cite[\S 13.42]{WatsonTreatise}. Here, we will provide a clear derivation in the special case when $\nu$ is an integer. The result is 0 if $\nu$ is odd, whereas if $\nu$ is even and $b < a$, it is given by $(-1)^{\nu/2} T_{\nu}(b/a) / \sqrt{a^2 - b^2}$ \cite{bateman1954tables,gradshteyn2014table}.
The link between Chebyshev polynomials, defined by $T_n(\cos \phi) = \cos(n \phi)$, and Bessel functions of integer index arises from the decomposition of plane waves in cylindrical coordinates ($\rho, z, \phi$):
\begin{equation}
{\mathrm e}^{{\mathrm i} x} = {\mathrm e}^{{\mathrm i} \rho \cos \phi} = \sum_{n = -\infty}^\infty {\mathrm i}^n \J{n}{\rho} {\mathrm e}^{{\mathrm i} n \phi}. \label{eq:besselgenerating}
\end{equation}
One could take this to be a {\em definition} of the integer-index Bessel functions via a generating function. Sending ${\mathrm i} \mapsto -{\mathrm i}$, we learn that $\J{-m}{\rho} = (-1)^m \J{m}{\rho}$. By integrating both sides of \eqref{eq:besselgenerating} against $\cos(n \phi)$ from $-\text{\greektext p}$ to $\text{\greektext p}$ and exploiting orthogonality, we obtain the integral representation
\begin{equation}
\J{n}{\rho} = \frac{{\mathrm i}^{-n}}{\text{\greektext p}} \int_0^\text{\greektext p} {\rm d}\phi\, {\mathrm e}^{{\mathrm i} \rho \cos \phi} \cos(n \phi), \label{eq:Jnintegral}
\end{equation}
where we have used that the cosine is even to rewrite the integral from $0$ to $\text{\greektext p}$ instead of $-\text{\greektext p}$ to $\text{\greektext p}$. In this expression we see that taking a derivative with respect to $\rho$ brings down a factor of ${\mathrm i} \cos \phi$. Using the fact that $\cos(n\phi) = T_n(\cos \phi)$, we then obtain the identity
\begin{equation}
\J{n}{\rho} = {\mathrm i}^{-n} T_n\left(-{\mathrm i} \frac{\rm d}{{\rm d}\rho}\right) \J{0}{\rho}. \label{eq:besselchebyshev}
\end{equation}
This will allow us to easily compute the integral $\int_0^\infty {\rm d}t\, \J{n}{a t}\cos(b t)$ for arbitrary integer $n$ once we know the integral for the special case $n = 0$.
For $n = 0$, we directly use the representation \eqref{eq:Jnintegral} of $J_0$:
\begin{equation}
\int_0^\infty {\rm d}t\, \J{0}{a t}\cos(b t) = \frac{1}{\text{\greektext p}} \int_0^\text{\greektext p} {\rm d}\phi\, \int_0^\infty {\rm d}t\, {\mathrm e}^{{\mathrm i} a t \cos \phi} \cos(b t) = \begin{cases}
\frac{1}{\sqrt{a^2 - b^2}}, & 0 < b < a \\
0, & 0 < a < b
\end{cases} .
\end{equation}
One way to see this is to view the integral over $\phi$ as a contour integral, which picks up poles where $\cos \phi = \pm b/a$ which are in the domain of integration ($-1 \leq \cos \phi \leq 1$) when $b < a$ but not otherwise.
Next, we can obtain the result for general $J_n$ by using \eqref{eq:besselchebyshev} and then integrating by parts to move the derivatives onto the $\cos(b t)$ factor. If $n$ is even, $T_n(x)$ contains only even terms and the derivatives produce a $\cos(b t)$ factor; if $n$ is odd, a similar argument leads to a $\sin(b t)$ factor. Every two derivatives acting on a $\cos(bt)$ factor will multiply by $-b^2 = ({\mathrm i} b)^2$, effectively absorbing an extra factor of ${\mathrm i} b$ into the argument of the polynomial. Hence:
\begin{equation}
\begin{aligned}
\int_0^\infty {\rm d}t\, \J{2k}{a t}\cos(b t) &= {\mathrm i}^{-2k} \int_0^\infty {\rm d}t\, \cos(b t) T_{2k}\left(-{\mathrm i} \frac{1}{a} \frac{{\rm d}}{{\rm d}t}\right) \J{0}{a t} \\
&= (-1)^k \int_0^\infty {\rm d}t\, \J{0}{a t} T_{2k}\left(-{\mathrm i} \frac{1}{a} \frac{{\rm d}}{{\rm d}t}\right) \cos(b t) \\
&= (-1)^k \,T_{2k}(b/a) \int_0^\infty {\rm d}t\, \J{0}{a t} \cos(b t) = \frac{(-1)^k \,T_{2k}(b/a)}{\sqrt{a^2 - b^2}},
\end{aligned}
\end{equation}
agreeing with the results in the literature \cite{bateman1954tables,gradshteyn2014table}.
The polynomial behavior of $T_{2k}(b/a)$ leads to several zeros as a function of $b/a$, and hence to ``plateau'' structure in the Bessel overlap integrals like that we have previously observed (for different choices of $\nu$) in Fig.~\ref{fig:brTwoField}. More generally, the plateau structure arises due to similar oscillatory behavior in the hypergeometric functions in \eqref{eq:Iplus2F1}.
\section{Conclusions}
\label{sec:conclusions}
One of the challenges facing LHC studies is that new physics with unusual signatures might be able to escape trigger strategies or hide in the large data sets. It is important therefore to consider these unusual signatures carefully, especially those that rarely appear in classic BSM models and are difficult or impossible to calculate with confidence. Among these signatures are complex high-multiplicity final states. Events with a small number of QCD-like jets are well-studied, and various approximately spherical high-multiplicity signals have also been considered, but little is known about signatures that, in some sense, lie between these extremes.
It is well-known that large 't Hooft coupling gauge theories can produce spherical events \cite{Strassler:2008bv,Hofman:2008ar,Hatta:2008tx}, and that RS-like 5d models with cascade decays of Kaluza-Klein modes can produce {\em approximately} spherical events \cite{csaki2009ads}. Here we have shown that, in fact, 5d simplified models with tunable parameters (including a small number of bulk fields with bulk or boundary interactions) can produce a wide range of event shapes in cascade decays of their heavy states. These 5d simplified models are well-suited to serve as templates when designing collider searches for unusual events that might be hiding in samples of events with high jet multiplicity.
Specifically, we saw that a key determinant in the cascade decays was the degree to which KK-number is violated, which in turn determines how close to threshold are the majority of decays. With some choices of bulk parameters, KK-number is approximately conserved, leading to quasi-spherical event samples; see \Fig{fig:singleFieldDist}. For other choices of bulk parameters, or with the addition of boundary interactions, KK-number can be strongly violated, making events with a few hard jets commonplace. In the latter cases, one can find samples as jetty as threshold $t\bar t$ events, with many individual events as jetty as SM $q\bar q$ events; compare \Fig{fig:emdMultTwoField} or \Fig{fig:evisobulkvbdry} to \Fig{fig:SMbenchmark}. In demonstrating this, we have relied not only on thrust, a classic event shape variable, but also on event isotropy, a newly-introduced variable which appears well-suited to this purpose.
One of the main results of this paper is an approximate analytic understanding of the Bessel function overlap integrals \Eq{def:Iintegral} that determine the couplings among different KK modes in the 5d simplified model, which are the most important source of the KK-number violation. We are not aware of any previous detailed studies of this definite integral. In particular, we have seen that, aside from decays very near threshold, the integral over the finite fifth dimension is generically well approximated by the integral $I_+$ over an infinite interval; see \Eq{eq:IpImdef}. This integral has an analytic expression, \eqref{eq:Iplus2F1}, in terms of hypergeometric functions. This integral has a factor of the phase space function $\lambda_{\rm PS}^{1/2}$, defined in \eqref{eq:lambdaPS}, in the {\em denominator}. As a result, the naive expectation that decays far from threshold are favored due to the larger available phase space is precisely inverted within the context of these extra-dimensional models. Decays with small phase space are often favored. Similar results hold for the complementary integral $I_-$, which dominates in special cases, and for which we have provided an approximate analytic understanding. However, there are also cases where $I_+$ exhibits quasi-polynomial behavior with multiple zeroes, and is consequently enhanced far from the threshold region and/or suppressed at the threshold region. Then decays near threshold are no longer dominant and the cascades are more likely to produce kinematic jets.
In a companion paper \cite{paper2}, we will explore the event shapes that arise from our 5d simplified models in more depth. In particular, we will illustrate, using both simulation and analytic estimates, that event isotropy provides an important complementary probe, capturing aspects of event shapes that are distinct from those captured by thrust, the eigenvalues of the sphericity tensor, or jet multiplicities.
A remaining task is to connect our 5d simplified models with the Standard Model. In this paper, we have started our events with a single heavy KK mode which then cascades into many daughter particles. In order to use event shape observables based on massless momenta, we have assumed that all of the final-state daughter particles decay into two massless particles. However, we have stopped short of a full model for the interaction with the SM, which would allow for the production of many different modes and for a more general set of decays.
More complete models could be an interesting topic of further investigation, and could allow our 5d simplified models to be used in full event generators for experimental studies. In this regard, it would be interesting to determine the effect on event shape variables of replacing our massless particles, which stand in for SM particles in the current study, with QCD jets from quarks and gluons, or with relatively soft photons. In the LHC context, one would need to consider carefully the impact of initial state radiation, the underlying event and pileup. It is also not clear what event-shape variables would be most effective in reducing backgrounds at a hadronic collider. All of these issues must be addressed before optimized searches for phenomena of this type can be designed.
\section*{Acknowledgments}
We thank Marat Freytsis, Gavin Salam, Jesse Thaler, David Pinner, and Andr{\'e} Frankenthal for useful discussions. MR and CC are supported in part by the DOE Grant DE-SC0013607. CC is supported in part by an NSF Graduate Research Fellowship Grant DGE1745303. We have made use of the Python Optimal Transport \cite{flamary2017pot} package when computing event isotropy.
|
2024-02-18T23:40:24.845Z
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2020-09-22T02:00:12.000Z
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{"paloma_paragraphs":[]}
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{"arxiv_id":"2009.08981","language":"en","timestamp":1600740012000,"url":"https:\/\/arxiv.org\/abs\/2009.08981","yymm":"2009"}
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proofpile-arXiv_000-10100
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{"provenance":"002.jsonl.gz:10101"}
| null | null |
\section{Introduction}
Since the beginning of this century, the simultaneous consideration of families of prior distributions instead of a single probability measure has become of fundamental importance for the risk assessment of financial positions. In this context, one often speaks of \textit{model uncertainty} or \textit{ambiguity}, where the uncertainty is modeled by a set $\mathfrak P$ of probability measures. Especially after the subprime mortgage crisis, the desire for mathematical models based on \textit{nondominated} families of priors arose: no single reference probability measure can be chosen which determines whether an event is deemed certain or negligible. To date, the most prominent example for a model consisting of a nondominated set of probability distributions is a Brownian motion with uncertain volatility, the so-called $G$-Brownian motion, cf.\ Peng~\cite{PengG, MR2474349}.\footnote{~More precisely, the resulting measures are mutually singular.} The latter is intimately related to the theory of second order backward stochastic differential equations, cf. Cheridito et al.~\cite{MR2319056}, and an extensive strand of literature has formed around this model. Another seminal contribution to nondominated sets of priors is Bouchard \& Nutz~\cite{BN}.
On another note, there has been renewed interest in the role of \textit{Orlicz spaces} in mathematical finance in the past few years. They have, for instance, appeared as canonical model spaces for risk measures, premium principles, and utility maximisation problems; see Bellini \& Rosazza Gianin~\cite{OrliczQuant}, Biagini \& \v{C}ern\'y~\cite{Cerny}, Cheridito \& Li~\cite{CheriditoLi}, Gao et al.\ \cite{FatouRep,Namioka}, and many others.
The aim of the present manuscript is to investigate robust Orlicz spaces in a setting of potentially nondominated sets $\mathfrak P$ of probability priors instead of a single reference probability measure $\P$. \textit{A priori}, such a construction faces a choice among two paths, though, which often lead to the same result if a probability space $(\Omega,\mathcal F,\P)$ is underlying. As an illustrative example, consider the case of a classical $L^p(\P)$-space, for $p\in[1,\infty)$. It can be obtained either by a \textit{top-down} approach, considering the maximal set of
all equivalence classes of real-valued measurable functions on which the norm $\|\cdot\|_{L^p(\P)}:=\mathbb{E}_\P[|\, \cdot\, |^p]^{1/p}$ is finite. Or, equivalently, by proceeding in a \textit{bottom-up} manner, closing a smaller test space of, say, bounded random variables w.r.t.\ the norm $\|\cdot\|_{L^p(\P)}$. If the underlying state space is topological, one has even more degrees of freedom and may select suitable spaces of \textit{continuous functions} as a test space. Both approaches lead to a \textit{Banach lattice}, which naturally carries the $\P$-almost-sure order, and this turns out to be $L^p(\P)$ in both cases. Morally speaking, the reason for this equivalence is that $\|\cdot\|_{L^p(\P)}$ is not very robust and rather insensitive to the tail behaviour of a given random variable. It may therefore come as no surprise that the two paths tend to diverge substantially for \textit{robust Orlicz spaces}, and this note may be understood as a comparison of the two approaches.
As top-down approach, we suggest to consider a fixed measurable space $(\Omega,\mathcal F)$, a nonempty set of probability measures $\mathfrak P$ on $(\Omega,\mathcal F)$, and a family $\Phi=(\phi_\P)_{\P\in\mathfrak P}$ of Orlicz functions. As usual, we consider the quotient space $L^0(\mathfrak P)$ of all real-valued measurable functions on $(\Omega,\mathcal F)$ up to $\mathfrak P$-quasi-sure ($\mathfrak P$-q.s.) equality. On $L^0(\mathfrak P)$, we define the robust Luxemburg norm
\begin{equation}\label{eq:introduction}
\|X\|_{L^\Phi(\mathfrak P)}:=\sup_{\P\in \mathfrak P}\|X\|_{L^{\phi_\P}(\P)}\in [0,\infty],\quad \text{for }X\in L^0(\mathfrak P),
\end{equation}
where $\|\cdot\|_{L^{\phi_\P}(\P)}$ is the Luxemburg seminorm for $\phi_\P$ under the probability measure $\P\in \mathfrak P$. In line with Pitcher \cite{MR182874} and Roy \& Chakraborty \cite{MR856304,MR1038172}, the robust Orlicz space $L^\Phi(\mathfrak P)$ is then defined to be the space of all $X\in L^0(\mathfrak P)$ with $\|X\|_{L^\Phi(\mathfrak P)}<\infty$. Notice that these spaces arise naturally in the context of \textit{variational preferences} as axiomatised by Maccheroni et al.\ \cite{Variational}. These encompass prominent classes of preferences, such as multiple prior preferences introduced by Gilboa \& Schmeidler \cite{GilboaSchmeidler} and multiplier preferences introduced by Hansen \& Sargent \cite{Hansen} -- see also \cite[Section 4.2.1]{Variational}. More precisely, one considers a cloud of agents operating on, say, bounded random variables $X$, endowed with variational preferences represented by expressions of the form
\[\inf_{\mathbb Q\in\mathfrak Q}\mathbb{E}_\mathbb Q[u(X)]+c(\mathbb Q),\]
where $u$ is a utility function, $\mathfrak Q$ is a set of probability priors, and $c$ is a prior-dependent cost function. Then, robust Orlicz spaces arise as a canonical maximal model space to which all individual preferences can be extended continuously. For the details, we refer to Section~\ref{sec:utility}.
Special cases of robust Orlicz spaces have also been studied in the $G$-Framework by, e.g., Nutz \& Soner \cite{MR2974730} and Soner et al. \cite{MR2925572}, and for general measurable spaces by Gao \& Munari~\cite{surplus}, Kupper \& Svindland~\cite{KupSvi}, Liebrich \& Svindland~\cite{Liebrich}, Maggis et al.~\cite{Maggisetal}, Owari \cite{Owari}, and Svindland~\cite{Subgradients}. Note that they are Banach lattices when endowed with the $\mathfrak P$-q.s.\ order which additionally have the desirable order completeness property of \emph{Dedekind $\sigma$-completeness}, the existence of \textit{suprema} for bounded countable families of contingent claims. We point out that $\sigma$-order properties are of fundamental interest in many financial applications, for example, in the context of \textit{weak free lunch with vanishing risk}. We refer to Ob\l \'{o}j \& Wiesel \cite{wiesel} for an overview on no-arbitrage conditions. Moreover, the set of all regular pricing rules on $L^\Phi(\mathfrak P)$ reflects the full uncertainty given by $\mathfrak P$; cf.\ Section~\ref{sec:regular}.
An alternative top-down approach to robust Orlicz spaces present in the literature, cf.\ Nutz \cite{Nutz} and Soner et al.\ \cite{MR2842089}, is given by the space
\[\mathfrak L^\Phi(\mathfrak P):=\big\{X\in L^0(\mathfrak P)\,\big|\, \forall\,\P\in\mathfrak P\,\exists\,\alpha>0:~\mathbb{E}_\P[\phi_\P(\alpha|X|)]<\infty\big\},\]
the ``intersection'' of the individual Orlicz spaces. In the situation of a cloud of variational preference agents above, this space collects minimal agreement among all agents under consideration, that is, they all can attach a well-defined utility to each of the objects in $\mathfrak L^\Phi(\mathfrak P)$. One may therefore be tempted to prefer $\mathfrak L^\Phi(\mathfrak P)$ over $L^\Phi(\mathfrak P)$ and drop the modelling choice of the worst-case approach represented by the supremum over all priors in $\mathfrak P$ in~\eqref{eq:introduction}.
In Proposition~\ref{prop:equality} and Theorem~\ref{thm:equality}, we show that, in many situations,
\[\mathfrak L^\Phi(\mathfrak P)=L^\Phi(\mathfrak P),\]
a uniform-boundedness type result, which proves the equivalence of both constructions in terms of the extension of the resulting spaces in $L^0(\mathfrak P)$.
In order to maintain a certain degree of analytic tractability while still allowing for uncertainty in terms of nondominated priors, an huge strand of literature has preferred the bottom-up construction of robust Orlicz spaces (at least for special cases of Orlicz functions). First, Owari~\cite{Owari} shows that, on a probability space $(\Omega,\mathcal F,\P)$, the largest ideal of $L^1(\P)$ on which a convex finite function on $L^\infty(\P)$ has a finite extension with the Lebesgue property is the closure of $L^\infty(\P)$ w.r.t.\ a multiplicatively weighted $L^1$-norm (see also Lemma~\ref{lem.Linfty} below). Second, closures $\mathcal C^p$ of the space $C_b$ of bounded continuous functions under robust $L^p$-norms $\|\cdot\|_{L^p(\mathfrak P)}$, for a nonempty set of priors $\mathfrak P$ -- which we shall cover in Section~\ref{sec:continuous} -- have become a frequent choice for commodity spaces or spaces of contingent claims in the context of a $G$-Brownian motion, see, for instance, Beissner \& Denis \cite{Beissner}, Beissner \& Riedel \cite{MR3816550}, Bion-Nadal \& Kervarec \cite{BionNadal}, Denis et al.\ \cite{Peng}, or Denis \& Kervarec \cite{MR3048198}. One reason for this choice is certainly that, roughly speaking, all ``nice'' analytic properties of $C_b$ carry over to the $\|\cdot\|_{L^p(\mathfrak P)}$-closure. As a consequence, in the past decades, the analytic properties of these spaces have been studied extensively, see, e.g., Beissner \& Denis \cite{Beissner} or Denis et al.\ \cite{Peng}, and a complete stochastic calculus has been developed based on these spaces, cf.\ Peng \cite{peng2019nonlinear}. However, very little is known about their properties as Banach lattices. Third, the bottom-up approach appears naturally in the study of option spanning under uncertainty in the spirit of Ross~\cite{Ross}, cf.\ Section~\ref{sec:options}.
The guiding question in their studies is the following: Which properties of the larger robust Orlicz space $L^\Phi(\mathfrak P)$ do these smaller (Banach) sublattices share, and which properties do not carry over? The second set of results of the present manuscript addresses this question from an order-theoretic perspective and with a particular view towards a possibly nondominated underlying set of priors. We sometimes reduce our attention to Banach lattices which arise from closures w.r.t.~a robust Luxemburg norm introduced in equation~\eqref{eq:introduction} below of certain classes of \textit{bounded} functions, e.g.~bounded continuous functions.
We would like to draw attention to two types of results.\\
Our first main result in this direction, Theorem \ref{thm.main}, states that every separable subspace of a robust Orlicz space is order-isomorphic to a subspace of a classical $L^1$-space without changing the measurable space. As a consequence, its elements are still dominated by a single probability measure, and the $\mathfrak P$-q.s.\ order collapses to an almost sure order -- even for nondominated sets $\mathfrak P$ of priors.\\
In particular, Theorem \ref{thm.main} applies to the situation considered in the $G$-framework. We show in Proposition~\ref{prop:density} that, under very mild and typically satisfied conditions, robust closures of $C_b$ are \textit{separable}. Therefore, the special robust closure $\mathcal C^p$ not only inherits all nice analytic properties from $C_b$, but also its dominatedness, a result that, in this special case, has already been obtained by Bion-Nadal \& Kervarec \cite{BionNadal}. Theorem \ref{thm.main} and Proposition~\ref{prop:density} therefore provide decisive generalisations of results by Bion-Nadal \& Kervarec \cite{BionNadal}, showing that the very general condition of separability is what causes a collapse of the $\mathfrak P$-q.s.\ order.
In a similar spirit, our second main result, Theorem \ref{thm.super}, concerns Dedekind $\sigma$-completeness of sublattices of robust Orlicz spaces. Theorem \ref{thm.super} states that, in typical situations, there exists at most one separable Dedekind $\sigma$-complete Banach sublattice that generates the $\sigma$-algebra, and if it exists the family of priors $\mathfrak P$ is already dominated with uniformly integrable densities. We thereby qualify that what \textit{prevents} nondominated models from being dominated is the lack of all order completeness properties for separable Banach sublattices that generate the $\sigma$-algebra. In the case of the robust closure $\mathcal C^p$ of $C_b$, one concludes that this space is too similar to the original space $C_b$ in terms of its order completeness properties.
In conclusion, our results highlight both the advantages and the cost of taking a top-down or a bottom-up approach to robust Orlicz spaces, respectively. Whereas the former may lack good dual behaviour, it has reasonable order completeness properties and reflects the full nondominated nature of the underlying uncertainty structure. The latter may be handy analytically, but either ignores the nondominated uncertainty structure \textit{a posteriori}, or tends to lead to a complete breakdown of almost all lattice properties.\\
{\bf Structure of the Paper:} In Section \ref{sec.orlicz}, we start with the announced top-down construction of robust Orlicz spaces and discuss its basic properties. We derive equivalent conditions for a robust Orlicz space to coincide with a robust multiplicatively penalised $L^1$-space, cf.\ Theorem \ref{thm:L1}, and show that every separable subspace of $L^\Phi(\mathfrak P)$ is order-isomorphic to a sublattice of $L^1(\P^*)$ for a suitable probability measure $\P^*$ (Theorem \ref{thm.main}). We further provide sufficient conditions for the equality $L^\Phi(\mathfrak P)=\mathfrak L^\Phi(\mathfrak P)$ to be valid, see Proposition \ref{prop:equality} and Theorem \ref{thm:equality}. In Section \ref{sec.completion}, we consider sublattices of $L^\Phi(\mathfrak P)$. Theorem \ref{thm.super}, Proposition \ref{prop:equivalences0}, and Proposition \ref{prop:equivalences} provide a set of equivalent conditions for the Dedekind $\sigma$-completeness of sublattices of $L^\Phi(\mathfrak P)$. In particular, we prove that separability together with Dedekind $\sigma$-completeness for \textit{any} generating sublattice already implies the dominatedness of the set of priors $\mathfrak P$. We further give, in special yet relevant cases, an explicit description of the dual space of sublattices of $L^\Phi(\mathfrak P)$, see Proposition~\ref{lem:representation0}. In Section \ref{sec.applications}, we discuss applications and implications of the results obtained in Section \ref{sec.orlicz} and Section \ref{sec.completion} to the field of robust finance. In particular, we discuss the relation to aggregating variational preferences, regular pricing rules, and implications for option spanning, arbitrage theory, and the $G$-Framework. The proofs of Section \ref{sec.orlicz} can be found in the Appendix \ref{app.A}, the proofs of Section \ref{sec.completion} are collected in the Appendix \ref{app.B}, and the proofs of Section \ref{sec.applications} are collected in the Appendix \ref{app.C}.\\
{\bf Notation:} For a set $S\neq\emptyset$ and a function $f\colon S\to(-\infty,\infty]$, the \emph{effective domain} of $f$ will be denoted by $\textnormal{dom}(f):=\{s\in S\mid f(s)<\infty\}$. For a normed vector space $\mathcal H$ we denote by $\mathcal H^*$ its dual space and by $\|\cdot\|_{\mathcal H^*}$ the operator norm.
Throughout, we consider a measurable space $(\Omega,\mathcal F)$ and a nonempty set $\mathfrak P$ of probability measures $\P$ on $(\Omega,\mathcal F)$.
The latter give rise to an equivalence relation on the real vector space $\mathcal L^0(\Omega,\mathcal F)$ of all real-valued random variables on $(\Omega,\mathcal F)$:
\[f\sim g\quad :\iff\quad \forall\,\P\in\mathfrak P:~\P(f=g)=1.\]
The quotient space $L^0(\mathfrak P):=\mathcal L^0(\Omega,\mathcal F)/\sim$ is the space of all real-valued random variables on $(\Omega,\mathcal F)$ up to $\mathfrak P$-quasi-sure ($\mathfrak P$-q.s.) equality. The elements $f\colon\Omega\to\R$ in the equivalence class $X\in L^0(\mathfrak P)$ are called \emph{representatives}, and are denoted by $f\in X$. Conversely, for $f\in\mathcal L^0(\Omega,\mathcal F)$, $[f]$ denotes the equivalence class in $L^0(\mathfrak P)$ generated by $f$.
For $X$ and $Y$ in $L^0(\mathfrak P)$,
$$X\preceq Y\quad:\iff\quad\forall\, f\in X\,\forall \,g\in Y\,\forall\,\P\in\mathfrak P:~\P(f\leq g)=1,$$
defines a vector space order on $L^0(\mathfrak P)$, the \emph{$\mathfrak P$-q.s.\ order} on $L^0(\mathfrak P)$, and $(L^0(\mathfrak P),\preceq)$ is a vector lattice. In fact, for $X,Y\in L^0(\mathfrak P)$ and representatives $f\in X, g\in Y$, the formulae
\[X\wedge Y=[f\wedge g]\quad\text{and}\quad X\vee Y=[f\vee g]\]
hold for the minimum and the maximum, respectively. We denote the vector sublattice of all bounded real-valued random variables up to $\mathfrak P$-q.s.\ equality by $L^{\infty}(\mathfrak P)$. The latter is a Banach lattice, when endowed with the norm
\[\|X\|_{L^\infty(\mathfrak P)}:=\inf\big\{m>0\, \big|\, X\preceq m\mathbf{1}_\Omega\big\},\quad X\inL^{\infty}(\mathfrak P).\]
As usual, $\mathbf{ca}$ denotes the space of all signed measures with finite total variation. We denote by $\mathbf{ca}_+$ or $\mathbf{ca}_+^1$ the subset of all finite measures or probability measures, respectively. For $\mu\in \mathbf{ca}$, let $|\mu|$ denote the total variation measure of $\mu$. Given two nonempty sets $\mathfrak Q, \mathfrak R\subset \mathbf{ca}$, we write $\mathfrak Q\ll\mathfrak R$ if $\sup_{\mu\in \mathfrak Q}|\mu|(N)=0$ for all events $N\in \mathcal F$ with $\sup_{\nu\in \mathfrak R}|\nu|(N)=0$. We write, $\mathfrak Q \approx \mathfrak R$ if $\mathfrak Q \ll \mathfrak R$ and $\mathfrak R \ll \mathfrak Q$. For singletons $\mathfrak Q=\{\mu\}$, we use the notation $\mu \ll \mathfrak R$, $\mathfrak R\ll \mu$, and $\mathfrak R \approx \mu$. Finally, $\mathbf{ca}(\mathfrak P):=\{\mu\in \mathbf{ca}\, |\, \mu\ll \mathfrak P\}$ denotes the space of all countably additive signed measures, which are absolutely continuous with respect to $\mathfrak P$. The subsets $\mathbf{ca}_+(\mathfrak P)$ and $\mathbf{ca}_+^1(\mathfrak P)$ are defined in an analogous way. For all $\mu\in \mathbf{ca}$, $X\in L^0(\mathfrak P)_+$, and $f,g\in X$, $\int f\,d\mu$ and $\int g\,d\mu$ are well-defined and satisfy
\[\int f\,d\mu=\int g\,d\mu.\]
We shall therefore henceforth write
\[\mu X:= \int X\,d\mu:=\int f\,d\mu,\quad \text{for }f\in X,\]
if $X\in L^0(\mathfrak P)_+$ or it has $|\mu|$-integrable representatives.
\section{Robust Orlicz spaces: definition and first properties}\label{sec.orlicz}
In this section, we introduce the main objects of this manuscript, robust versions of Orlicz spaces, and investigate their basic properties. For the theory of classical Orlicz spaces, we refer to \cite[Chapter 2]{Edgar}. An \emph{Orlicz function} is a function $\phi\colon[0,\infty)\to[0,\infty]$ with the following three properties:
\begin{enumerate}[(i)]
\item $\phi$ is lower semicontinuous, nondecreasing, and convex.
\item $\phi(0)=0$.
\item there are $x_0,x_1>0$ with $\phi(x_0)\in [0,\infty)$ and $\phi(x_1)\in (0,\infty]$.\footnote{~This definition precludes triviality of $\phi$, i.e.\ the cases $\phi\equiv 0$ and $\phi=\infty \cdot 1_{(0,\infty)}$.}
\end{enumerate}
Throughout this section, we consider a general measurable space $(\Omega,\mathcal F)$, a nonempty set of probability measures $\mathfrak P$, a family $\Phi=(\phi_\P)_{\P\in\mathfrak P}$ of Orlicz functions, and define
\begin{equation}\label{eq.phimax}
\phi_{\rm Max}(x):=\sup_{\P\in \mathfrak P} \phi_\P(x),\quad \text{for all }x\in [0,\infty).
\end{equation}
Notice that, by definition, $\phi_{\rm Max}$ is a lower semicontinuous, nondecreasing, and convex function $[0,\infty)\to [0,\infty]$ with $\phi_{\rm Max}(0)=0$. However, in general, $\phi_{\rm Max}$ is not an Orlicz function, since $\phi_{\rm Max}(x_0)\in [0,\infty)$ for some $x_0\in (0,\infty)$ cannot be guaranteed.
\subsection{Robust Orlicz spaces and penalised versions of robust $L^p$-spaces}
\begin{definition}\label{def:robustOrlicz}\
For $X\in L^0(\mathfrak P)$, the \emph{($\Phi$-)Luxemburg norm} is defined via
\[
\|X\|_{L^\Phi(\mathfrak P)}:=\inf\big\{ \lambda>0\, \big|\,\sup_{\P\in \mathfrak P} \mathbb{E}_\P\left[\phi_\P(\lambda^{-1}|X|)\right]\leq 1\big\}=\sup_{\P\in \mathfrak P}\|X\|_{L^{\phi_\P}(\P)}\in [0,\infty].\footnote{~Defining $\|\cdot\|_{L^{\phi_\P}(\P)}$ in the usual way, we obtain a seminorm on $L^\Phi(\mathfrak P)$, not a norm as on the classical Orlicz space $L^{\phi_\P}(\P)$.}
\]
We define by $L^\Phi(\mathfrak P):=\textnormal{dom}(\|\cdot\|_{L^\Phi(\mathfrak P)})$ the \emph{($\Phi$-)robust Orlicz space.}
\end{definition}
\begin{example}\label{ex.1}
Let $(\Omega,\mathcal F)$ be a measurable space, $\mathfrak P$ a nonempty set of probability priors, and $\phi\colon [0,\infty)\to [0,\infty]$ be an Orlicz function.
\begin{enumerate}
\item[(1)] For an arbitrary function $\gamma\colon \mathfrak P\to [0,\infty)$, consider
\[
\phi_\P(x):=\frac{\phi(x)}{1+\gamma(\P)},\quad \text{for }x\geq0.
\]
This leads to an \emph{additively penalised robust Orlicz space} with Luxemburg norm
\[
\|X\|_{L^{\Phi}(\mathfrak P)}=\inf\big\{ \lambda>0\, \big|\,\sup_{\P\in \mathfrak P} \mathbb{E}_\P\left[\phi(\lambda^{-1}|X|)\right]-\gamma(\P) \leq 1\big\},\quad \text{for }X\in L^0(\mathfrak P).
\]
For $\phi:=\infty\cdot \mathbf{1}_{(1,\infty)}$, the Luxemburg norm is, independently of $\gamma$, given by
\[
\|X\|_{L^\Phi(\mathfrak P)}=\sup_{\P\in \mathfrak P}\|X\|_{L^\infty(\P)}=\|X\|_{L^\infty(\mathfrak P)},\quad \text{for }X\in L^0(\mathfrak P).
\]
Introducing the, up to a sign, convex monetary risk measure
\[
\rho(X):=\sup_{\P\in \mathfrak P} \mathbb{E}_\P[X]-\gamma(\P)\in [0,\infty],\quad \text{for }X\in L^0(\mathfrak P)_+,
\]
the robust Luxemburg norm can be expressed as
\[
\|X\|_{L^\Phi(\mathfrak P)}=\inf\left\{ \lambda>0\, \left|\,\rho\left(\phi(\lambda^{-1}|X|)\right)\leq 1\right.\right\},\quad \text{for }X\in L^0(\mathfrak P).
\]
\item[(2)] For $\theta\colon \mathfrak P\to (0,\infty)$ with $\sup_{\P\in \mathfrak P}\theta(\P)<\infty$, we consider
\[
\phi_\P(x):=\phi\big(\theta(\P)x\big),\quad\text{for }\P\in \mathfrak P\text{ and } x\geq 0.
\]
This leads to a \emph{multiplicatively penalised robust Orlicz space} with Luxemburg norm
\[
\|X\|_{L^\Phi(\mathfrak P)}=\sup_{\P\in \mathfrak P}\theta(\P)\|X\|_{L^\phi(\P)},\quad \text{for }X\in L^0(\mathfrak P).
\]
For $p\in [1,\infty)$ and $\phi(x)=x^p$, $x\ge 0$, we obtain the weighted robust $L^p$-norm
\[
\|X\|_{L^\phi(\mathfrak P)}=\sup_{\P\in \mathfrak P}\theta(\P)\|X\|_{L^p(\P)},\quad \text{for }X\in L^0(\mathfrak P),
\]
and, for $\phi(x)=\infty\cdot 1_{(1,\infty)}$, the Luxemburg norm is given by
\[
\|X\|_{L^\Phi(\mathfrak P)}=\sup_{\P\in \mathfrak P}\theta(\P)\|X\|_{L^\infty(\P)},\quad \text{for }X\in L^0(\mathfrak P).
\]
The resulting spaces will be referred to as \emph{weighted robust $L^p$-spaces}, for $1\leq p\leq \infty$.
\end{enumerate}
\end{example}
Before we prove that, as in the classical case, robust Orlicz spaces are Banach lattices, we introduce $\mathbf{ca}\big(L^\Phi(\mathfrak P)\big)$ as the set of all signed measures $\mu\in \mathbf{ca}(\mathfrak P)$ for which each $X\in L^\Phi(\mathfrak P)$ is $|\mu|$-integrable and the map
\begin{equation}\label{eq.measfunct}
L^\Phi(\mathfrak P)\to \R, \quad X\mapsto |\mu|X
\end{equation}
is continuous. Moreover, we set $\mathbf{ca}_+\big(L^\Phi(\mathfrak P)\big):=\mathbf{ca}\big(L^\Phi(\mathfrak P)\big)\cap \mathbf{ca}_+$ and $\mathbf{ca}_+^1\big(L^\Phi(\mathfrak P)\big):=\mathbf{ca}\big(L^\Phi(\mathfrak P)\big)\cap \mathbf{ca}_+^1$.
\begin{proposition}\label{prop:Banach}
The space $\big(L^\Phi(\mathfrak P),\preceq, \|\cdot\|_{L^\Phi(\mathfrak P)}\big)$ is a Dedekind $\sigma$-complete Banach lattice and $L^\Phi(\mathfrak P)\subset L^0(\mathfrak P)$ is an ideal. Moreover, for all $\P\in \mathfrak P$, $a_\P>0$, and $b_\P\leq 0$ with $a_\P x+b_\P\le \phi_\P(x)$ for all $x\ge 0$,
\begin{equation}\label{eq:normP}\|\P\|_{L^\Phi(\mathfrak P)^*}\le \frac{1-b_\P}{a_\P}.\end{equation}
In particular, $\mathfrak P\subset \mathbf{ca}_+^1\big(L^\Phi(\mathfrak P)\big)$.
\end{proposition}
\begin{remark}\label{rem:measures}
The lattice norm property of $\|\cdot\|_{L^\Phi(\mathfrak P)}$ which is proved in the preceding proposition admits two conclusions: (i) For each $\mu\in\mathbf{ca}(L^\Phi(\mathfrak P))$, the functional $L^\Phi(\mathfrak P)\ni X\mapsto\mu X$ is continuous. This is due to the fact that the Radon-Nikodym derivative $\frac{{\rm d}\mu}{{\rm d}|\mu|}$ takes values in $[-1,1]$ $|\mu|$-almost everywhere. (ii) $\mathbf{ca}(L^\Phi(\mathfrak P))$ is a vector sublattice of $\mathbf{ca}(\mathfrak P)$.
\end{remark}
\begin{example}
Suppose $\mathcal H\subset L^0(\mathfrak P)$ is an ideal which is a Banach lattice when endowed with a norm $\|\cdot\|_\mathcal H$. Furthermore assume the norm is completely determined by $\sigma$-finite measures, i.e., there is a set $\mathfrak D\ll \mathfrak P$ of $\sigma$-finite measures such that, for all $X\in \mathcal H$,
\[
\|X\|_\mathcal H=\sup_{\mu\in \mathfrak D} \mu |X|.
\]
Then $\mathcal H$ is a robust Orlicz space after a potential modification of $\mathfrak P$.
\end{example}
A robust Orlicz space may be reduced to a weighted robust $L^1$-space if and only if it contains all bounded random variables.
\begin{theorem}\label{thm:L1}
The following statements are equivalent:
\begin{enumerate}[(\textnormal{1})]
\item[\textnormal{(1)}] $L^{\infty}(\mathfrak P)\subset L^\Phi(\mathfrak P)$,
\item[\textnormal{(2)}] $\phi_{\rm Max}$ is an Orlicz function, i.e., there exists some $x_0\in (0,\infty)$ with $\phi_{\rm Max}(x_0)\in [0,\infty)$,
\item[\textnormal{(3)}] There exists a nonempty set $\mathfrak Q\subset \mathbf{ca}_+^1\big(L^\Phi(\mathfrak P)\big)$ of probability measures with $\mathfrak P\subset\mathfrak Q$ and a weight function $\theta\colon \mathfrak Q\to (0,\infty)$ with $\sup_{\mathbb Q\in \mathfrak Q}\theta(\mathbb Q)<\infty$ such that $\|\cdot\|_{L^\Phi(\mathfrak P)}=\sup_{\mathbb Q\in \mathfrak Q}\theta(\mathbb Q)\|\cdot\|_{L^1(\mathbb Q)}$.
\end{enumerate}
In this case, $L^\Phi(\mathfrak P)$ is a weighted robust $L^1$-space, and there is a constant $\kappa>0$ such that
\[
\|X\|_{L^\Phi(\mathfrak P)}\leq \kappa\|X\|_{L^\infty(\mathfrak P)},\quad\text{for } X\in L^{\infty}(\mathfrak P).
\]
\end{theorem}
For $\P\in \mathbf{ca}_+^1(L^\Phi(\mathfrak P))$, we define the \emph{canonical projection} $J_\P\colon L^\Phi(\mathfrak P)\to L^1(\P)$ via
\[
J_\P(X):=\big\{g\in \mathcal L^0(\Omega,\mathcal F)\,\big|\, \exists\,f\in X\colon \P(f\neq g)=0\big\},\quad \text{for }X\in L^\Phi(\mathfrak P).
\]
Since $\P\in \mathbf{ca}_+^1\big(L^\Phi(\mathfrak P)\big)$, $J_\P$ is well-defined, linear, continuous, and a lattice homomorphism, i.e., it is order-preserving in that
\begin{center}$J_\P(X\wedge Y)=J_\P(X)\wedge J_\P(Y)$,\quad for $X,Y\in L^\Phi(\mathfrak P)$.\end{center}
However, in general, it fails to be a lattice isomorphism onto its image, i.e., it is not injective. Still, the following surprising result holds.
\begin{theorem}\label{thm.main}
Suppose $\mathcal H$ is a separable subspace of $L^\Phi(\mathfrak P)$. Then, there is a probability measure $\P^*\in \mathbf{ca}_+^1\big(L^\Phi(\mathfrak P)\big)$ such that $\mathcal H$ is isomorphic to a subspace of $L^1(\P^*)$ via the canonical projection $J_{\P^*}$.\ In particular, the following assertions hold:
\begin{itemize}
\item[(1)]$\P^*$ defines a strictly positive linear functional on $\mathcal H$.
\item[(2)]The $\mathfrak P$-q.s.\ order and the $\P^*$-a.s.\ order coincide on $\mathcal H$.
\item[(3)]If $\mathfrak P$ is countably convex, $\P^*$ can be chosen as an element of $\mathfrak P$.
\end{itemize}
\end{theorem}
We could rephrase the previous theorem as the fact that, on separable subspaces of $L^\Phi(\mathfrak P)$, the $\mathfrak P$-q.s.~order collapses to a $\P^*$-a.s.~order for some $\P^*\in \mathbf{ca}_+^1(L^\Phi(\mathfrak P))$.
\begin{corollary}\label{cor.main}
Assume that one of the three equivalent conditions of Theorem \ref{thm:L1} is satisfied. Then, for every separable subspace $\mathcal H$ of $L^\Phi(\mathfrak P)$, there exists a countable set $\mathfrak Q_\mathcal H\subset \mathfrak Q$ such that
$$\|X\|_{L^\Phi(\mathfrak P)}=\sup_{\mathbb Q\in \mathfrak Q_\mathcal H}\theta(\mathbb Q)\|X\|_{L^1(\mathbb Q)},\quad\text{for all }X\in \mathcal H.$$
\end{corollary}
\begin{example}\label{ex.doubpen}\
\begin{enumerate}[(1)]
\item We consider the setup of Example \ref{ex.1}. Let $\theta\colon\mathfrak P\to(0,\infty)$ with $c:=\sup_{\P\in \mathfrak P}\theta(\P)<\infty$, $\gamma\colon\mathfrak P\to[0,\infty)$, and $\phi$ be a joint Orlicz function. Let
\[
\phi_\P(x):=\frac{\phi\big(\theta(\P)x\big)}{1+\gamma(\P)}, \quad \text{for }x\geq 0,
\]
corresponding to the case of a \emph{doubly penalised robust Orlicz space}. Then, for $x_0\in (0,\infty)$ with $c x_0\in\textnormal{dom}(\phi)$,
\[\phi_{\rm Max} (x_0)=\sup_{\P\in\mathfrak P}\phi_\P(x_0)=\sup_{\P\in\mathfrak P}\frac{\phi\big(\theta(\P)x_0\big)}{1+\gamma(\P)}\le \phi(c x_0)<\infty.\]
By Proposition \ref{thm:L1}, we obtain that $L^\Phi(\mathfrak P)$ is a weighted robust $L^1$-space.
\item For each fixed probability measure $\P^*$ on $(\Omega,\mathcal F)$, Proposition~\ref{thm:L1} shows that the classical space $L^\infty(\P^*)$ is a robust $L^1$-space, although this result could, of course, also be obtained in a more direct manner. Let $\mathfrak P$ be the set of all probability measures $\P$ on $(\Omega,\mathcal F)$ that are absolutely continuous with respect to $\P^*$. Consider $\phi_\P(x)=x$ for all $x\geq 0$ and $\P\in \mathfrak P$, leading a robust $L^1$-space over $\mathfrak P$. Then,
\[
\|X\|_{L^\Phi(\mathfrak P)}=\|X\|_{L^\infty(\P^*)},\quad \text{for }X\in L^0(\mathfrak P)=L^0(\P^*).
\]
\item Let $\P^*$ be a probability measure on $(\Omega,\mathcal F)$, and consider a convex monetary risk measure $\rho\colon L^\infty(\P^*)\to\R$, which enjoys the Fatou property, and satisfies $\rho(0)=0$. The dual representation, up to a sign,
\[\rho(X)=\sup_{Z\in\textnormal{dom}(\rho^*)\cap L^1(\P^*)}\mathbb{E}[ZX]-\rho^*(Z),\quad \text{for }X\in L^\infty(\P^*),\]
is a well-known consequence, where $\rho^*$ is the convex conjugate of $\rho$.
In the situation of Example \ref{ex.1} (1), set
\begin{align*}\mathfrak P&:=\big\{Z{\rm d}\P^*\, \big|\, Z\in\textnormal{dom}(\rho^*)\cap L^1(\P^*)\big\},\\
\gamma\big(Z{\rm d}\P^*\big)&:=\rho^*(Z),\quad \text{for }Z\in \textnormal{dom}(\rho^*)\cap L^1(\P^*),\\
\phi_\P(x)&:=x,\quad \text{for }x\geq 0\text{ and }\P\in\mathfrak P.
\end{align*}
Then, $L^\Phi(\mathfrak P)$ contains $L^{\infty}(\mathfrak P)$ as a sublattice. In general, we have $\mathfrak P\ll\P^*$, but $\mathfrak P\approx\P^*$ may fail without further conditions on $\rho$. We can always define the ``projection''
\[
\widehat\rho(Y):=\rho\big(J^{-1}(Y)\big), \quad\text{for }Y\inL^{\infty}(\mathfrak P),
\]
though, where $J\colon L^\infty(\P^*)\toL^{\infty}(\mathfrak P)$ is the natural projection. In that case, $L^\Phi(\mathfrak P)$ serves as the maximal sensible domain of definition of $\widehat\rho$. Various aspects of such spaces have been studied in~\cite{KupSvi,Liebrich,Owari,Pichler,Subgradients}.
\end{enumerate}
\end{example}
\subsection{An alternative path to robust Orlicz spaces}
In this section, we focus on a way to translate the concept of Orlicz spaces to a robust setting \textit{without} using a robust Luxemburg norm and the worst-case approach represented by the supremum over all models $\P\in\mathfrak P$. One may indeed wonder if this modelling assumption is actually necessary to produce the largest commodity space on which the analytic behaviour of utility can be captured well with respect to any model considered in the uncertainty profile. An alternative would be provided by the space
\begin{equation}\label{eq:curlyL}\mathfrak L^\Phi(\mathfrak P):=\big\{X\in L^0(\mathfrak P)\,\big|\, \forall\,\P\in\mathfrak P\,\exists\,\alpha_\P>0:~\mathbb{E}_\P[\phi_\P(\alpha_\P|X|)]<\infty\big\}.\end{equation}
A special case of this space has, e.g., been studied in \cite{Nutz} and \cite{MR2842089}.
One can show that $\mathfrak L^\Phi(\mathfrak P)$ is a vector sublattice of $L^0(\mathfrak P)$. Moreover, independent of $\Phi$, $L^\Phi(\mathfrak P)\subset\mathfrak L^\Phi(\mathfrak P)$ holds \textit{a priori}, and the inclusion can be strict as the following example demonstrates.
\begin{example}\label{ex.eqorlicz}\
Fix two constants $0<c<1<C$ and consider the case where $\Omega=\R$ is endowed with the Borel $\sigma$-algebra $\mathcal F$, and $\mathfrak P$ is given by the set of all probability measures $\P$, which are equivalent to $\P^*:=\mathcal N(0,1)$ with bounded density $c\le\frac{{\rm d}\P}{{\rm d}\P^*}\le C$. Moreover, fix a partition $(\mathfrak P_n)_{n\in\mathbb{N}}$ of $\mathfrak P$ into nonempty sets. We set
\[
\phi_\P(x):=x^n,\quad \text{for } x\geq 0,\; n\in \mathbb{N},\text{ and }\P\in \mathfrak P_n.
\]
Then,
\[
\mathfrak L^\Phi(\mathfrak P)=\big\{X\in L^0(\P^*)\, \big|\, \forall\,n\in \mathbb{N}:~\mathbb{E}_{\P^*}[|X|^n]<\infty\big\},
\]
and thus $U\in\mathfrak L^\Phi(\mathfrak P)$ if $U\colon \Omega\to \R$ is the identity, i.e., $U\sim\mathcal N(0,1)$ under $\P^*$. However, Stirling's formula implies that, for all $\alpha>0$,
\[\sup_{\P\in\mathfrak P}\mathbb{E}_\P [\phi_\P(\alpha|U|)]\ge c\sup_{n\in\mathbb{N}}\mathbb{E}_{\P^*}[\alpha^n|U|^n]=\infty,\]
and $U\notin L^\Phi(\mathfrak P)$ follows. It is easy to see that $\mathfrak L^\Phi(\mathfrak P)$ is a Fr\'echet space, but not a Banach space.
\end{example}
The next proposition shows that $L^\Phi(\mathfrak P)$ can always be seen as a space of type \eqref{eq:curlyL} if $L^{\infty}(\mathfrak P)\subset L^\Phi(\mathfrak P)$, and more can be said if $\mathfrak P$ is countably convex.
\begin{proposition}\label{prop:equality}
The following are equivalent:
\begin{itemize}
\item[(1)]$L^{\infty}(\mathfrak P)\subset L^\Phi(\mathfrak P)$.
\item[(2)]There is a set of probability measures $\mathfrak R\subset\mathbf{ca}^1_+(L^\Phi(\mathfrak P))$ and a family $\Psi=(\psi_\mathbb Q)_{\mathbb Q\in\mathfrak R}$ of Orlicz functions such that $\mathfrak R\approx\mathfrak P$ and
\[L^\Phi(\mathfrak P)=\mathfrak L^\Psi(\mathfrak R).\]
\end{itemize}
In particular, if $\mathfrak P$ is countably convex and there exist constants $(c_\P)_{\P\in \mathfrak P}\subset (0,\infty)$ such that
\begin{equation}\label{eq.thm:equality}
\phi_{\rm Max}(x)\leq \phi_\P(c_\P x),\quad \text{for all }x\geq 0 \text{ and }\P\in \mathfrak P,
\end{equation}
then \textnormal{(1)} and \textnormal{(2)} hold and one can choose $\mathfrak R=\mathfrak P$ as well as $\Psi=\Phi$ or $\Psi=(\phi_{\rm Max})_{\P\in\mathfrak P}$.
\end{proposition}
The next theorem, which generalises \cite[Proposition 4.2(ii)]{Liebrich} and \cite[Theorem 4.4]{MR1038172}, shows that, for doubly penalised Orlicz spaces, cf.~Example \ref{ex.doubpen}(1), the assumption of countable convexity of the set $\mathfrak P$ can be further relaxed.
\begin{theorem}\label{thm:equality}
Suppose that $\mathfrak P$ is convex. Assume that $\Phi$ is doubly penalised with joint Orlicz function $\phi$, multiplicative penalisation $\theta$, and convex additive penalty function $\gamma\colon\mathfrak P\to[0,\infty)$ with countably convex lower level sets.
Then,
\[\mathfrak L^\Phi(\mathfrak P)=L^\Phi(\mathfrak P).\]
\end{theorem}
\begin{example}
An example for an additive penalty function as demanded in Theorem~\ref{thm:equality} is given by the set $\mathfrak P$ of all probability measures in $\textnormal{dom}(\rho^*)$ for a convex monetary risk measure $\rho:\mathcal L^\infty(\Omega,\mathcal F)\to\R$ with $\textnormal{dom}(\rho^*)\cap \mathbf{ca}_+^1\neq\emptyset$. However, this set is typically not countably convex, as the choice $\Omega=\mathbb{N}$, $\mathcal F=2^\mathbb{N}$, and
\[\rho(f)=\sup_{n\in\mathbb{N}}f(n)-2^{2n},\quad\text{for }f\in\mathcal L^\infty(\Omega,\mathcal F),\]
demonstrates. Indeed, the Dirac measure $\delta_n$ lies in $\textnormal{dom}(\rho^*)\cap \mathbf{ca}^1_+$, $n\in\mathbb{N}$, but $\rho^*(\sum_{n=1}^\infty2^{-n}\delta_n)=\infty$.
\end{example}
\begin{remark}
Assume that, in the situation of Theorem~\ref{thm:equality}, the multiplicative penalty is $\theta\equiv 1$. Then, there are two equally consistent ways to translate convergence in $L^\phi(\P)$ to a robust setting given by the set $\mathfrak P$ of priors. One could either declare a net $(X_\alpha)_{\alpha\in I}$ to be convergent if it (i) converges with respect to each seminorm $\|\cdot\|_{L^\phi(\P)}$, for $\P\in\mathfrak P$, at equal or comparable speed to the same limit, or (ii) converges to the same limit with respect to each seminorm $\|\cdot\|_{L^\phi(\P)}$, for $\P\in\mathfrak P$. Convergence (i) is reflected by the norm $\|\cdot\|_{L^\Phi(\mathfrak P)}$, and the equality of speeds may be relaxed by the additive penalty, whereas (ii) would be the natural choice of a topology on $\mathfrak L^\Phi(\mathfrak P)$. Even though $\mathfrak L^\Phi(\mathfrak P)=L^\Phi(\mathfrak P)$ holds, convergence (ii) might not be normable or even sequential. However, having both options at hand provides a degree of freedom to reflect different economic phenomena on an applied level.
\end{remark}
\section{Generating sublattices of $\Phi$-robust Orlicz spaces}\label{sec.completion}
By construction, $\Phi$-robust Orlicz spaces are ideals in $L^0(\mathfrak P)$ with respect to the $\mathfrak P$-q.s.\ order, and therefore particularly well-behaved with respect to order properties. Each $\Phi$-robust Orlicz space is Dedekind $\sigma$-complete. Moreover, using arguments as in \cite[Lemma 8]{surplus}, (super) Dedekind completeness of $L^0(\mathfrak P)$ implies (super) Dedekind completeness of $L^\Phi(\mathfrak P)$, and the converse implications hold in the situation of Theorem~\ref{thm:L1}.\footnote{~For the definition of these notions, we refer to \cite{AliBurk2}.}
In conclusion, $\Phi$-robust Orlicz spaces not only have the desirable Banach space property, but also behave reasonably well as vector lattices.
In contrast to the top-down construction of $\Phi$-robust Orlicz spaces, one could also build a robust space bottom-up, a path taken in, e.g., \cite{Beissner,BionNadal,Peng}. Starting with a space of \textit{test random variables}, one could consider closing the test space in the larger ambient space $L^\Phi(\mathfrak P)$ with respect to the risk-uncertainty structure as given by $\|\cdot\|_{L^\Phi(\mathfrak P)}$. Such a procedure leads to smaller spaces in general. The existing literature typically discusses (special cases of) these spaces as \textit{Banach spaces} without further going into detail on their order-theoretic properties. This section therefore fills this gap, and explores their properties as Banach lattices.
We shall observe that they tend to be not very tractable as vector lattices.
If they behave well with respect to order properties, this usually has strong consequences.
\begin{assumption}\label{assumption}
Throughout this section, we assume that there exists some $x_0\in (0,\infty)$ with $\phi_{\rm Max}(x_0)\in [0,\infty)$, or, equivalently, that $L^\Phi(\mathfrak P)$ contains the equivalence class of the constant function $\mathbf{1}_\Omega$.
\end{assumption}
In the following, we consider a sublattice $\mathcal H$ of $L^\Phi(\mathfrak P)$ containing the equivalence class of the constant function $\mathbf{1}_\Omega$. We assume that $\mathcal H$ \textit{generates} $\mathcal F$ in that the $\sigma$-algebra $\sigma(\mathcal L)$ generated by the lattice $\mathcal L:=\{f\in\mathcal L^0(\Omega,\mathcal F)\mid [f]\in\mathcal H\}$ equals $\mathcal F$.
Note that the latter assumption does not restrict generality and merely simplifies the exposition of our results. They transfer to smaller $\sigma$-algebras otherwise.
By $\mathcal C$ we denote the $\|\cdot\|_{L^\Phi(\mathfrak P)}$-closure of $\mathcal H$ in $L^\Phi(\mathfrak P)$, i.e.
$$\mathcal C={\rm cl}(\mathcal H).$$
We define the subspaces $\mathbf{ca}(\mathcal H)$ and $\mathbf{ca}(\mathcal C)$ of $\mathbf{ca}(\mathfrak P)$ in complete analogy with $\mathbf{ca}(L^\Phi(\mathfrak P))$ (see equation \eqref{eq.measfunct}). Using Remark~\ref{rem:measures}, one can show that, for each $\mu\in\mathbf{ca}(\mathcal H)$, the functional $\mathcal H\ni X\mapsto\mu X$ is continuous.
\begin{definition}\label{def:sigmaorder}
Let $(\mathcal X,\preceq)$ be a vector lattice. A possibly nonlinear functional $\ell\colon \mathcal X\to\R$ is \emph{$\sigma$-order continuous} if the following two properties hold: (i) for all $x,y\in\mathcal X$, the set $\{\ell(z)\mid x\preceq z\preceq y\}\subset\R$ is bounded, (ii) whenever for a sequence $(x_n)_{n\in\mathbb{N}}\in\mathcal X$ there is another sequence $(y_n)_{n\in\mathbb{N}}\subset\mathcal X_+$ such that $|x_n|\preceq y_n$ for all $n\in\mathbb{N}$, $y_n\downarrow$, i.e., $y_{n+1}\preceq y_n$ for all $n\in \mathbb{N}$, and $\inf_{n\in\mathbb{N}}y_n=0$ in $\mathcal X$, then $\lim_{n\to\infty}|\ell(x_n)|=0$.
\end{definition}
In a first step, we characterise \textit{$\sigma$-order continuous functionals} on $\mathcal C$ and $\mathcal H$.
\begin{lemma}\label{lem:sigmaorder}
For each $\sigma$-order continuous functional $\ell\colon\mathcal H\to\R$ there is a unique signed measure $\mu\in\mathbf{ca}(\mathfrak P)$ such that, for all $X\in\mathcal H$, all representatives of $X$ are $|\mu|$-integrable and
\[\ell(X)=\mu X.\]
In particular, $\ell$ satisfies $\ell(X)\ge 0$ for all $X\in\mathcal H_+$ if and only if the associated measure $\mu$ lies in $\mathbf{ca}_+(\mathfrak P)$.
\end{lemma}
This motivates to denote the space of all $\sigma$-order continuous linear functionals on $\mathcal H$ by $\mathbf{ca}^\sigma(\mathcal H)$ and the cone of positive $\sigma$-order continuous linear functionals by $\mathbf{ca}_+^\sigma(\mathcal H):=\mathbf{ca}^\sigma(\mathcal H)\cap \mathbf{ca}_+$.
\begin{lemma}\label{lem:representation00}
Assume that $\|\cdot\|_{L^\Phi(\mathfrak P)}$ is $\sigma$-order continuous on $\mathcal H$. Then, $$\mathcal H^*=\mathbf{ca}(\mathcal H)=\mathbf{ca}^\sigma(\mathcal H)\cap \mathcal H^*.$$
\end{lemma}
\begin{proposition}\label{lem:representation0}
The space $\big(\mathcal C,\preceq,\|\cdot\|_{L^\Phi(\mathfrak P)}\big)$ is a Banach lattice and $\mathbf{ca}(\mathcal C)=\mathbf{ca}(\mathcal H)$. If $\mathcal H\subset L^\infty(\mathfrak P)$ and $\|\cdot\|_{L^\Phi(\mathfrak P)}$ is $\sigma$-order continuous on $\mathcal H$, then
$$\mathcal C^*=\mathbf{ca}(\mathcal C)=\mathbf{ca}(\mathcal H)=\mathbf{ca}^\sigma(\mathcal H)\cap \mathcal C^*.$$
\end{proposition}
Throughout the remainder of this section, the closure $\textnormal{cl}\big(L^\infty(\mathfrak P)\big)$ of $L^{\infty}(\mathfrak P)$ plays a fundamental role. The following lemma is a slight generalisation of \cite[Proposition 18]{Peng} and provides an explicit description of the closure of $L^{\infty}(\mathfrak P)$ in our setup.
\begin{lemma}\label{lem.Linfty}
Let $X\in L^\Phi(\mathfrak P)$. Then, $X\in \textnormal{cl}\big(L^\infty(\mathfrak P)\big)$ if and only if, for all $\alpha>0$,
$$ \lim_{n\to \infty}\sup_{\P\in \mathfrak P}\mathbb{E}_\P\big[\phi_\P(\alpha |X|)\mathbf 1_{\{|X|> n\}}\big]= 0.$$
That is,
\[
\textnormal{cl}\big(L^\infty(\mathfrak P)\big)=\big\{X\in L^\Phi(\mathfrak P)\, \big| \, \lim_{n\to \infty}\big\|X\mathbf{1}_{\{|X|> n\}}\big\|_{L^\Phi(\mathfrak P)}=0\big\}.
\]
\end{lemma}
For the remaining results of this section, we emphasise that, if we view $\mathcal H$ or $\mathcal C$ as spaces of \textit{measurable functions}, two properties should not be far-fetched: (i) Dedekind $\sigma$-completeness, (ii) many positive functionals which are integrals with respect to a measure are $\sigma$-order continuous.\footnote{~Tellingly, the early literature on vector lattices refers to $\sigma$-order continuous linear functionals as ``integrals".}\\
The following theorem shows that, if $\mathfrak P$ is nondominated, the Banach lattice $\mathcal C$ cannot be separable \textit{and} simultaneously have the mild order completeness property of Dedekind $\sigma$-completeness.
\begin{theorem}\label{thm.super}
Suppose that the Banach lattice $\big(\mathcal C,\preceq,\|\cdot\|_{L^\Phi(\mathfrak P)}\big)$ is separable, and let $\P^*$ be a probability measure as in Theorem~\ref{thm.main}. Then, the following are equivalent:
\begin{itemize}
\item[(1)] $\mathcal C$ is Dedekind $\sigma$-complete.
\item[(2)] $\mathcal C$ is super Dedekind complete.
\item[(3)] $\mathcal C=\textnormal{cl}\big(L^\infty(\mathfrak P)\big)$.
\item[(4)] $\mathcal C$ is an ideal in $L^\Phi(\mathfrak P)$.
\item[(5)]$\mathcal C^*=\mathbf{ca}(\mathcal C)=\mathbf{ca}(L^\Phi(\mathfrak P))\approx\P^*$ and the unit ball therein is weakly compact in $L^1(\P^*)$.
\end{itemize}
Moreover, they imply both of the following assertions:
\begin{itemize}
\item[(6)]$\mathfrak P\approx \P^*$.
\item[(7)]If, additionally,
\begin{equation}\label{eq:growth}\inf_{\P\in \mathfrak P}\phi_\P(x_0)\in (0,\infty]\quad\text{for some }x_0\in (0,\infty),\end{equation}
the set $\big\{\frac{{\rm d}\P}{{\rm d}\P^*}\, \big|\, \P\in \mathfrak P\big\}$ of densities of priors in $\mathfrak P$ is uniformly $\P^*$-integrable.
\end{itemize}
\end{theorem}
We thus see that, in typical situations encountered in the literature, all order completeness properties agree, and their validity usually implies dominatedness of the underlying set of priors in a particularly strong form.
Although separability is a desirable property from an analytic point of view, we have hereby shown that it has very strong implications for uncertainty robust spaces. One may wonder what happens if one drops this assumption.
We start with the following version of the Monotone Class Theorem.
\begin{lemma}\label{lem:monclass}
Assume that $\mathcal H$ is Dedekind $\sigma$-complete and $\mathfrak P\approx \mathbf{ca}_+^\sigma(\mathcal H)$. Then, $L^\infty(\mathfrak P)\subset \mathcal H$.
\end{lemma}
The next proposition now shows that the only (generating) sublattice of $L^{\infty}(\mathfrak P)$ satisfying the requirements (i) and (ii) above is $L^{\infty}(\mathfrak P)$ itself.
\begin{proposition}\label{prop:equivalences0}
The following statements are equivalent:
\begin{itemize}
\item[(1)] $\mathcal H$ is Dedekind $\sigma$-complete and $\mathfrak P\approx \mathbf{ca}_+^\sigma(\mathcal H)$,
\item[(2)] $\mathcal H$ is Dedekind $\sigma$-complete and $\mathbf{ca}(\mathcal H)=\mathbf{ca}^\sigma(\mathcal H)\cap \mathcal H^*$,
\item[(3)] $\mathcal H$ is an ideal in $L^\Phi(\mathfrak P)$.
\end{itemize}
If $\mathcal H\subset L^\infty(\mathfrak P)$, \textnormal{(1)--(3)} are furthermore equivalent to
\begin{itemize}
\item[(4)] $\mathcal H=L^\infty(\mathfrak P)$.
\end{itemize}
\end{proposition}
Considering $\mathcal C$ instead of $\mathcal H$ does not change the picture, since the closure of any (generating) sublattice of $L^{\infty}(\mathfrak P)$ satisfying (i) and (ii) leads to the same Banach lattice, the closure of $L^{\infty}(\mathfrak P)$.
\begin{proposition}\label{prop:equivalences}
The following statements are equivalent:
\begin{itemize}
\item[(1)] $\mathcal C$ is Dedekind $\sigma$-complete and $\mathfrak P\approx \mathbf{ca}_+^\sigma(\mathcal C)$,
\item[(2)] $\mathcal C$ is Dedekind $\sigma$-complete and $\mathbf{ca}(\mathcal C)=\mathbf{ca}^\sigma(\mathcal C)$,
\item[(3)] $\mathcal C$ is an ideal in $L^\Phi(\mathfrak P)$.
\end{itemize}
If $\mathcal H\subset L^\infty(\mathfrak P)$, \textnormal{(1)--(3)} are furthermore equivalent to
\begin{itemize}
\item[(4)]$\mathcal C=\textnormal{cl}\big(L^\infty(\mathfrak P)\big)$.
\end{itemize}
\end{proposition}
\section{Applications}\label{sec.applications}
This section is devoted to the discussion of the financial and economic implications of our theoretical results.
\subsection{Closures of continuous functions}\label{sec:continuous}
Prominent sublattices of $L^\Phi(\mathfrak P)$ appearing in the literature -- at least for special cases of $\Phi$ -- are $\|\cdot\|_{L^\Phi(\mathfrak P)}$-closures of sets of continuous functions on a separable metrisable space $\Omega$, such as in \cite{BionNadal}, where a general lattice of bounded continuous functions generating the Borel-$\sigma$-algebra and containing $\mathbf{1}_\Omega$ is considered. Other examples include bounded Lipschitz functions, or bounded cylindrical Lipschitz functions, respectively, cf. \cite{Peng,HuWang}. The usual minimal assumption on $\mathfrak P$ is tightness, sometimes one imposes that $\mathfrak P$ is convex and weakly compact, cf.\ \cite{Beissner}. In that case, $\mathfrak P$ has the stronger property of being countably convex.
Throughout this subsection, we assume that $\Omega$ is a separable and metrisable topological space, endowed with the Borel $\sigma$-algebra $\mathcal F$. Let $C_b$ be the space of bounded continuous functions on $\Omega$, and let $\mathcal H$ be a lattice of bounded continuous functions containing $\mathbf{1}_\Omega$.\ We shall again impose Assumption~\ref{assumption}, which yields that
\[\iota\colon C_b\to L^\Phi(\mathfrak P),\quad f\mapsto [f].\]
is a well-defined, continuous, and injective lattice homomorphism.\ We shall abuse notation and also refer to $\iota(C_b)$ as $C_b$, to the equivalence classes by capital letters though. As before, let
\[\mathcal C:=\textnormal{cl}\big(\mathcal H\big),\]
endowed with $\|\cdot\|_{L^\Phi(\mathfrak P)}$ and the $\mathfrak P$-q.s.~order.
Our first main observation is that the results in \cite{Beissner,BionNadal} are based on separability of the primal space, which holds under a comparatively mild tightness condition. The following result is a decisive generalisation of \cite[Proposition 2.6]{BionNadal}.
\begin{proposition}\label{prop:density}
Suppose that, for every $\varepsilon>0$, there exists a compact set $K\subset \Omega$ with
\begin{equation}\label{eq:compact}
\|\mathbf{1}_{\Omega\setminus K}\|_{L^\Phi(\mathfrak P)}<\varepsilon.\end{equation}
Then, $\mathcal C$ is separable.
\end{proposition}
\begin{lemma}\label{lem:density}
Condition \eqref{eq:compact} is met in any of the following situations:
\begin{itemize}
\item[(1)] $\Omega$ is compact.
\item[(2)] $\textnormal{dom}(\phi_{{\rm Max}})=[0,\infty)$ and, for all $t>0$, the set $\mathfrak P_t:=\{\P\in\mathfrak P\mid \phi_\P(t)>1\}$ is tight.
\item[(3)] $\textnormal{dom}(\phi_{{\rm Max}})=[0,\infty)$ and $\mathfrak P$ is tight.
\end{itemize}
If $\Phi$ satisfies \eqref{eq:growth}, the validity of \eqref{eq:compact} implies that $\mathfrak P$ is tight.
\end{lemma}
We emphasise that (3) is the typical minimal assumption in the literature. It is, in particular, satisfied in the $G$-framework, see \cite[Theorem 2.5]{peng2019nonlinear}.
\begin{example}\
\begin{enumerate}[(1)]
\item Let $p\in[1,\infty)$, and consider the case, where $\phi_\P(x)=x^p$ for all $x\ge 0$ and $\P\in\mathfrak P$. Then, Lemma~\ref{lem:density} implies that \eqref{eq:compact} holds if and only if $\mathfrak P$ is tight.
\item Consider the case of a doubly penalised robust Orlicz space as in Example~\ref{ex.doubpen}(1) with bounded multiplicative penalty $\theta\colon\mathfrak P\to(0,\infty)$ and additive penalty $\gamma\colon\mathfrak P\to[0,\infty)$. Then, $\textnormal{dom}(\phi_{\rm Max})=[0,\infty)$ if and only if the joint Orlicz function $\phi$ satisfies $\textnormal{dom}(\phi)=[0,\infty)$. Moreover, condition (2) in Lemma~\ref{lem:density} is met if the lower level sets of $\gamma$ are tight. Notice that, in this case, the validity of \eqref{eq:growth} implies the boundedness of $\gamma$, and thus, naturally, the tightness of $\mathfrak P$.
\end{enumerate}
\end{example}
\begin{remark}\
\begin{enumerate}[(1)]
\item Strictly speaking, Bion-Nadal \& Kervarec \cite{BionNadal} work with the \textit{Lebesgue prolongation} of a capacity $\mathfrak c$ defined on a generating lattice of continuous functions. In most of their results, they assume that $\mathfrak c$ is a \textit{Prokhorov capacity} on a separable metrisable space. As $\mathbf{1}_{\Omega\setminus K}$ is l.s.c.\ for every compact $K\subset\Omega$, one thus obtains, for each $\varepsilon>0$, the existence of a compact $K\subset\Omega$ such that
\[\mathfrak c(\mathbf{1}_{\Omega\setminus K})\le\varepsilon.\]
This counterpart of \eqref{eq:compact} admits to perform our proof of Proposition~\ref{prop:density} in their framework, and the result transfers.
\item We comment here on the role of Proposition~\ref{prop:equivalences} and Theorem~\ref{thm.super} in the present setting. As is remarked after \cite[Corollary 5.6]{deJeu}, $C_b$ over a Polish space does not admit any nontrivial $\sigma$-order continuous linear functional when endowed with the pointwise order. One could therefore interpret Proposition~\ref{prop:equivalences} as a \textit{dichotomy}: either the closure $\mathcal C$ of $C_b$ in $L^\Phi(\mathfrak P)$ behaves very much like the space of continuous functions, or it is an ideal of $L^\Phi(\mathfrak P)$, which could be obtained more directly as the closure of $L^{\infty}(\mathfrak P)$ and to which in most typical cases Theorem~\ref{thm.super} applies. \\
As an illustrative example, consider $\Omega=[0,1]$ endowed with its $\sigma$-algebra $\mathcal F$ of Borel sets and set $\mathfrak P$ to be the set of all atomless probability measures. Consider the robust weighted $L^1$-space, where $\theta\equiv 1$. One shows that each $X\in\mathcal C$ has a unique continuous representative $f$ and satisfies $\|X\|_{L^\phi(\mathfrak P)}=\|f\|_\infty$. In this setting, the inclusions
\[\{0\}=\mathbf{ca}^\sigma(\mathcal C)\subsetneq\mathbf{ca}(\mathcal C)\subsetneq\mathcal C^*\cap\mathbf{ca}\]
hold. For the first equality, note that $\mathcal C$ is lattice-isometric to $C_b$, and the existence of a nontrivial $\sigma$-order continuous linear functional would contradict the result cited above. For the second strict inclusion, consider the linear bounded functional $\ell(X):=f(1)$, $X\in\mathcal C$, where $f\in X$ is a continuous representative. Although it corresponds to the Dirac measure concentrated at 1, it cannot be identified with a measure absolutely continuous with respect to $\mathfrak P$.
\end{enumerate}
\end{remark}
We conclude with a Riesz representation result for the dual of $\mathcal C$, which follows directly from the more general observations in Section~\ref{sec.completion} and extends~\cite[Proposition 4]{Beissner} to our setting.
\begin{corollary}\label{cor:representation}
Assume that $\mathcal H=C_b$, $\mathfrak P$ is weakly compact, and that $\textnormal{dom}(\phi_{\rm Max})=[0,\infty)$. Then,
$$\mathcal C^*=\mathbf{ca}\big(\mathcal C\big).$$
\end{corollary}
\subsection{Option spanning under uncertainty}\label{sec:options}
A rich strand of literature deals with the power of options to complete a market, at least in an approximate sense. These studies date back to \cite{GreenJarrow,Ross} for finite and arbitrary numbers of future states of the economy, respectively. They have since been extended to a multitude of model spaces, such as the space of continuous functions over a compact Hausdorff space, $L^p$-spaces, or even more general ideals of $L^0$ over a probability space. We refer to \cite{Options} and the references therein. In the present example we study option spanning under potentially nondominated uncertainty.
Fix a \textit{limited liability claim} $X\in L^\Phi(\mathfrak P)$, i.e., $X\succeq 0$ holds. Its \textit{option space}
\[\mathcal H_X:=\textnormal{span}\left(\{\mathbf{1}_\Omega\}\cup\{(X-k\mathbf{1}_\Omega)^+\mid k\in\R\}\right)\]
is the collection of all portfolios of call and put options written on $X$. In line with the simplifying assumption in Section~\ref{sec.completion}, we will assume w.l.o.g.\ that
\[\mathcal F=\sigma(\{f\mid f\in X\}),\]
a condition studied in detail in the existing literature on option spanning. We also introduce the norm closure
\[\mathcal C_X:=\textnormal{cl}(\mathcal H_X),\]
the space of all contingent claims, which can be approximated by linear combinations of call and put options.
\begin{proposition}\label{prop:option}
$\big(\mathcal H_X,\preceq,\|\cdot\|_{L^\Phi(\mathfrak P)}\big)$ is a separable normed sublattice of $L^\Phi(\mathfrak P)$, and $\big(\mathcal \mathcal C_X,\preceq,\|\cdot\|_{L^\Phi(\mathfrak P)}\big)$
is a separable Banach sublattice of $L^\Phi(\mathfrak P)$. Thus, Theorem \ref{thm.main} and Theorem \ref{thm.super} apply.
\end{proposition}
We may thus draw two interesting financial conclusions from our results.
First, by Theorem~\ref{thm.main}, \textit{nondominated} uncertainty collapses both over the option space $\mathcal H_X$ and its closure $\mathcal C_X$. In fact, the same reference probability measure $\P^*$ can be chosen for both spaces (Corollary~\ref{cor.main}). $\P^*$ can be interpreted as intrinsic to and $J_{\P^*}(\mathcal H_X)\subset L^1(\P^*)$ as a copy of the original option space $\mathcal H_X$. This motivates the following corollary.
\begin{corollary}\label{cor:ae}
Let $\P^*\in\mathbf{ca}^1_+(L^\Phi(\mathfrak P))$ be a dominating probability measure for $\mathcal H_X$ as constructed in Theorem~\ref{thm.main}. Then, for each $Y\in L^\Phi(\mathfrak P)$, there is a sequence $(Y_n)_{n\in\mathbb{N}}\subset\mathcal H_X$ such that $Y_n\to Y$ $\P^*$-a.s.\ as $n\to\infty$.
\end{corollary}\label{cor:spanningpower}
The second conclusion concerns the \textit{topological spanning power of $X$} and follows directly from Theorem~\ref{thm.super}.
\begin{corollary}Suppose $X$ has topological spanning power in that
\begin{equation}\label{eq:options}\mathcal C_X\text{ is an ideal of }L^\Phi(\mathfrak P).\end{equation}
Let $\P^*$ be a probability measure as in Theorem~\ref{thm.main}. Then, the following assertions hold:
\begin{itemize}
\item[(1)]$\mathcal C_X$ is super Dedekind complete.
\item[(2)]$\mathbf{ca}(\mathcal C_X)\approx\mathfrak P\approx \P^*$ and the unit ball of $\mathbf{ca}(\mathcal C_X)$ is weakly compact in $L^1(\P^*)$.
\item[(3)]$\mathcal C_X$ is lattice-isomorphic to an ideal of $L^1(\P^*)$.
\item[(4)]$X\in\textnormal{cl}(L^{\infty}(\mathfrak P))=\mathcal C_X$.
\end{itemize}
In particular, the topological spanning power of limited liability claims is always weaker than \eqref{eq:options} unless $\mathfrak P$ is dominated.
\end{corollary}
Moreover, under the mild growth condition \eqref{eq:growth} on $\Phi$, which does not depend on the concrete choice of the limited liability claim $X$ whatsoever, \eqref{eq:options} implies that all densities of priors in $\mathfrak P$ w.r.t.\ $\P^*$ are uniformly $\P^*$-integrable. This can be seen as a converse to the spanning power results on classical $L^p$-spaces, $1\le p<\infty$. In conclusion, the topological spanning power of options under nondominated uncertainty is \textit{always} weaker than \eqref{eq:options}, whereas $X$ always has full spanning power with respect to the reference measure $\P^*$ by Corollary~\ref{cor:ae}.
\subsection{Regular pricing rules and the Fatou property}\label{sec:regular}
\textit{Positive linear functionals} on a space of contingent claims are commonly interpreted as linear pricing rules. Applying this to a generating lattice $\mathcal H\subset L^\Phi(\mathfrak P)$ as studied in Section~\ref{sec.completion} and a positive linear functional $\ell\colon\mathcal H\to\R$, $\sigma$-order continuity of $\ell$ now has the following economic interpretation. Whenever a sequence of contingent claims $(X_n)_{n\in\mathbb{N}}$ satisfies $X_n\downarrow$ and $\inf_{n\in\mathbb{N}}X_n=0$, that is, the payoffs $X_n$ become arbitrarily invaluable in the objective $\mathfrak P$-q.s.\ order, their prices $\ell(X_n)$ under $\ell$ vanish:
\[\lim_{n\to\infty}\ell(X_n)=0.\]
Pricing with such functionals does not exaggerate the value of (objectively) increasingly invaluable contingent claims.
We have seen in Lemma~\ref{lem:sigmaorder} that such functionals correspond to measures. The condition
\[\mathfrak P\approx\mathbf{ca}^\sigma_+(\mathcal H)\]
encountered in Lemma~\ref{lem:monclass} and Propositions~\ref{prop:equivalences0} and \ref{prop:equivalences} means that the set of all regular pricing rules (in the sense described above) holds the same information about certainty and impossibility of events as the set $\mathfrak P$ of ``physical priors". In the case $\mathcal H=L^\Phi(\mathfrak P)$, Proposition~\ref{prop:equivalences} shows $\mathbf{ca}^\sigma(L^\Phi(\mathfrak P))=\mathbf{ca}(L^\Phi(\mathfrak P))$. \textit{A fortiori}, $\mathbf{ca}^\sigma_+(L^\Phi(\mathfrak P))\approx\mathfrak P$ holds.
For smaller generating lattices, our results thus describe a dichotomy in the case of full information: either the lattice does not generally admit aggregation even of countable order bounded families of contingent claims, or it is an ideal. In the case where bounded contingent claims are dense in $\mathcal H$, the latter further specialises to $\mathcal H=L^{\infty}(\mathfrak P)$, or, if $\mathcal H$ is closed in $L^\Phi(\mathfrak P)$, $\mathcal H=\textnormal{cl}(L^{\infty}(\mathfrak P))$.
We would like to make another comment on the conjunction of $\mathfrak P\approx\mathbf{ca}^\sigma_+(\mathcal H)$ and Dedekind $\sigma$-completeness of $\mathcal H$ concerning the \emph{Fatou property}. The latter is one of the most prominent phenomena studied in theoretical mathematical finance and relates order closedness properties of convex sets to dual representations of these in terms of measures. In a dominated framework, say, $L^\infty(\P)$ for a single reference measure $\P$, a subset $\mathcal B\subset L^\infty(\P)$ is called \textit{Fatou closed} if, for each sequence $(X_n)_{n\in\mathbb{N}}\subset\mathcal B$ which converges $\P$-a.s.\ to some $X\inL^{\infty}(\mathfrak P)$ and whose moduli are dominated by some $Y\inL^{\infty}(\mathfrak P)$, the limit satisfies $X\in\mathcal B$. Due to the super Dedekind completeness of $L^{\infty}(\mathfrak P)$, Fatou closedness is equivalent to the so-called order closedness of $\mathcal B$. It is well known that $\mathcal B\subset L^\infty(\P)$ is Fatou closed if and only if it has a representation of shape
\[\mathcal B=\{X\in L^\infty\mid \forall\,\mu\in\mathfrak D:~\mu X\le h(\mu)\}\]
for a suitable set $\mathfrak D\subset\mathbf{ca}(\P)$ and a function $h:\mathfrak D\to\R$.
This observation does not directly transfer to nondominated frameworks. In our setting, consider the family of seminorms
\[\rho_\mu:\mathcal H\to[0,\infty),\quad X\mapsto |\mu|(|X|),\quad\text{for }\mu\in\mathbf{ca}^\sigma_+(\mathcal H),\]
and let $\tau$ be the locally convex topology on $\mathcal H$ generated by them. In the literature, $\tau$ is often referred to as the \emph{absolute weak topology} $|\sigma|(\mathcal H,\mathbf{ca}^\sigma_+(\mathcal H))$.
\begin{observation}
If $\mathfrak P\approx\mathbf{ca}^\sigma_+(\mathcal H)$, $\tau$ is a locally convex Hausdorff topology and -- by Kaplan's Theorem~\cite[Theorem 3.50]{AliBurk2} and \cite[Theorem 1.57]{AliBurk2} -- the dual of $(\mathcal H,\tau)^*$ is $\mathbf{ca}^\sigma(\mathcal H)$. In particular, $\tau$ admits the application of separating hyperplane theorems.
\end{observation}
Let $\mathcal B\subset\mathcal H$ be a nonempty convex set.
\begin{observation}
If $\mathcal H$ is Dedekind $\sigma$-complete, the following are equivalent:
\begin{enumerate}[(1)]
\item $\mathcal B$ is sequentially order closed.
\item For all sequences $(X_n)_{n\in\mathbb{N}}$ whose moduli $(|X_n|)_{n\in\mathbb{N}}$ admit some upper bound $Y\in\mathcal H$ and which converge $\mathfrak P$-q.s.\ to some $X\in\mathcal H$, this limit $X$ lies in $\mathcal B$.
\end{enumerate}
\end{observation}
Note that (2) is the direct counterpart of Fatou closedness as formulated above.
Hence, in the outlined situation, there is a locally convex Hausdorff topology $\tau$ on $\mathcal H$ such that each $\tau$-closed convex set $\mathcal B\subset\mathcal H$ is sequentially order closed. Although the power of the Fatou property lies in the \emph{converse} implication, the assumptions in the preceding two observations appear to be analytic minimal requirements for a fruitful study. Accepting this, Proposition~\ref{prop:equivalences0} states that studying the (sequential) Fatou property only makes sense on \textit{ideals} of robust Orlicz spaces.
However, in robust frameworks, there is a caveat concerning the Fatou property: sequences are usually not sufficient to unfold its full analytic power, and intuitive reasoning learned in dominated frameworks usually fails. We refer to \cite{Maggisetal} for a detailed discussion on this issue.
\subsection{Utility theory for multiple agents}\label{sec:utility}
As anticipated in the introduction, robust Orlicz spaces are canonical model spaces for aggregating \emph{variational preferences} of a cloud of agents. Variational preferences encompass other prominent classes of preferences, such as multiple prior preferences introduced by \cite{GilboaSchmeidler} and multiplier preferences introduced by \cite{Hansen} -- see also \cite[Section 4.2.1]{Variational}. One of the most appealing qualities of variational preference relations is the handy separation of risk attitudes (measured by the prior-wise expected utility approach) and ambiguity or uncertainty attitudes (as expressed by the choice of the underlying set of priors and their additive penalisation). Aggregating expert opinions or the preferences of a cloud of variational preference agents, however, requires to consider more than one utility function.
Consider a nonempty set ${\mathcal I}$ of agents. Each agent $i\in{\mathcal I}$ has preferences over, say, $\mathcal L^\infty(\Omega,\mathcal F)$, the space of bounded real-valued random variables on $(\Omega,\mathcal F)$, given by
\[f\preceq_i g\quad:\iff\quad \inf_{\P\in\mathfrak P_i}\mathbb{E}_\P[u_i(f)]+c_i(\P)\le \inf_{\P\in\mathfrak P_i}\mathbb{E}_\P[u_i(g)]+c_i(\P).\]
We shall assume that $\mathfrak P_i$ is a nonempty set of probability measures on $(\Omega,\mathcal F)$ equivalent to a reference probability measure $\P_i^*$, $c_i\colon \mathfrak P_i\to[0,\infty)$ satisfies $\inf_{\P\in\mathfrak P_i}c_i(\P)=0$, and that the utility function $u_i\colon \R\to\R$ is concave, upper semicontinuous, nondecreasing, and satisfies $u_i(0)=0$. For the details of this axiomatisation, we refer to \cite{Variational}. For the sake of brevity, we set
\[\mathfrak U_i\colon\mathcal L^\infty(\Omega,\mathcal F)\to\R,\quad f\mapsto \inf_{\P\in\mathfrak P_i}\mathbb{E}_\P[u_i(f)]+c_i(\P).\]
By an affine transformation, we can w.l.o.g.\ assume that
\[\mathfrak U_i(-\mathbf{1}_\Omega)=\inf_{\P\in\mathfrak P_i}u_i(-1)+c_i(\P)=-1,\quad\text{for all }i\in{\mathcal I}.\]
Aggregating the preferences of all agents in ${\mathcal I}$ leads to the \emph{unanimous preference relation}
\[f\trianglelefteq g\quad:\iff\quad \forall\,i\in{\mathcal I}:~f\preceq_ig.\]
For its closer study, we first observe that, setting $\mathfrak P:=\bigcup_{i\in{\mathcal I}}\mathfrak P_i$, $f=g$ $\mathfrak P$-q.s.\ implies $f\trianglelefteq g$ and $g\trianglelefteq f$, i.e., all agents are indifferent between $f$ and $g$. Hence, we may consider the preference relations $\preceq_i$ on the space $L^\infty(\mathfrak P)$ instead without losing any information. The definition of $\mathfrak U_i$ on $L^\infty(\mathfrak P)$ is immediate.
Next, for $\P\in\mathfrak P$, consider
\[\phi_\P(x):=\sup_{i\in{\mathcal I}:\,\P\in\mathfrak P_i}\frac{-u_i(-x)}{1+c_i(\P)},\quad\text{for }x\in[0,\infty).\]
$\phi_\P$ is convex, lower semicontinuous, nondecreasing, and satisfies $\phi_\P(0)=0$. We shall impose the condition that $\phi_\P$ is an Orlicz function. Moreover, one easily obtains
\[\phi_\P(1)\le 1,\quad\text{for all }\P\in\mathfrak P.\]
Set $\Phi:=(\phi_\P)_{\P\in\mathfrak P}$ and consider the associated robust Orlicz space which satisfies $L^{\infty}(\mathfrak P)\subset L^\Phi(\mathfrak P)$.
We claim that $L^\Phi(\mathfrak P)$ is a canonical maximal model space to study all individual preferences $\preceq_i$, $i\in{\mathcal I}$, simultaneously. This is due to the observation that each preference relation $\preceq_i$ canonically extends to a \textit{continuous} preference relation on $L^\Phi(\mathfrak P)$. Indeed, note that by concave duality, for each $i\in{\mathcal I}$, there is a set $\mathbf M_i\ll\mathfrak P$ of finite measures and a non-negative function $h_i:\mathbf M_i\to[0,\infty)$ such that
\[\mathfrak U_i(X)=\inf_{\mu\in\mathbf M_i}\mu X+h_i(\mu),\quad\text{for all }X\inL^{\infty}(\mathfrak P).\]
Moreover, one can show that, for each $i\in{\mathcal I}$, all $X\inL^{\infty}(\mathfrak P)$, and all $\mu\in\mathbf M_i$,
\[-1\le \mathfrak U_i\left(-\|X\|_{L^\Phi(\mathfrak P)}^{-1}|X|\right)\le\mu\big(-\|X\|_{L^\Phi(\mathfrak P)}^{-1}|X|\big)+h_i(\mu).\]
From this, we infer
\[\mu |X|\le (1+h_i(\mu))\|X\|_{L^\Phi(\mathfrak P)},\quad\text{for all }X\inL^{\infty}(\mathfrak P),\,i\in{\mathcal I},\text{ and }\mu\in\mathbf M_i.\]
By monotone convergence, the same estimate holds for all $X\in L^\Phi(\mathfrak P)$. Hence, setting
\[\mathfrak U_i^\sharp(X):=\inf_{\mu\in\mathbf M_i}\mu X+h_i(\mu),\quad\text{for all }X\in L^\Phi(\mathfrak P),\]
and
\[X\preceq_i Y\quad:\iff\quad\mathfrak U_i^\sharp(X)\le\mathfrak U_i^\sharp(Y),\quad\text{for }X,Y\in L^\Phi(\mathfrak P),\]
we have extended the initial preference relations to $L^\Phi(\mathfrak P)$ in a continuous manner.
In case that all agents have the same attitude towards \textit{risk}, i.e., the utility function $u_i$ does not depend on $i$, it is straightforward to construct examples where Theorem~\ref{thm:equality} is applicable and we have the identity
\[L^\Phi(\mathfrak P)=\mathfrak L^\Phi(\mathfrak P),\]
i.e., $L^\Phi(\mathfrak P)$ is the model space for the minimal agreement among all agents under consideration on which well-defined utility can be attached to all objects.
\begin{appendix}
\section{Proofs of Section \ref{sec.orlicz}}\label{app.A}
\begin{proof}[Proof of Proposition~\ref{prop:Banach}]
The fact that $L^\Phi(\mathfrak P)$ is an ideal of $L^0(\mathfrak P)$ follows directly from the fact that each $\phi_\P$ is nondecreasing and convex and the fact that the supremum is subadditive. Hence, it is a Dedekind $\sigma$-complete vector lattice with respect to the $\mathfrak P$-q.s.\ order because $L^0(\mathfrak P)$ is Dedekind $\sigma$-complete. In a similar way, it follows that $\|\cdot\|_{L^\Phi(\mathfrak P)}$ defines a norm on $L^\Phi(\mathfrak P)$. Let $(X_n)_{n\in\mathbb{N}}$ be a Cauchy sequence. Notice that, since $\phi_\P$ is convex and nontrivial for all $\P\in \mathfrak P$, there exist $a_\P> 0$ and $b_\P\in\R$ such that
\begin{equation}\label{eq.convlin}
\phi_\P(x)\geq(a_\P x+b_\P)^+,\quad \text{for all }x\in \R.
\end{equation}
By possibly passing to a subsequence, we may assume that
\[\|X_n-X_{n+1}\|_{L^\Phi(\mathfrak P)}<4^{-n},\quad \text{for all }n\in\mathbb{N}.\]
For all $n\in \mathbb{N}$, let $\lambda_n>0$ with $\|X_n-X_{n+1}\|_{L^\Phi(\mathfrak P)}< \lambda_n\leq 4^{-n}$. In particular, $\lambda_n^{-1}2^{-n}\ge 2^n$, i.e.\ we can fix $n_\P\in\mathbb{N}$ such that $a_\P\lambda_n^{-1}2^{-n}+b_\P>0$ holds for all $n\ge n_\P$.
Markov's inequality together with equation \eqref{eq.convlin} shows, for all $\P\in\mathfrak P$,
\begin{align*}
\sum_{n=n_\P}^\infty\P\big(|X_n-X_{n+1}|\ge 2^{-n}\big)&\le\sum_{n=n_\P}^\infty\P\left(\big(a_\P(\lambda_n^{-1}|X_n-X_{n+1}|)+b_\P\big)^+\ge \big(a_\P\lambda_n^{-1} 2^{-n}+b_\P\big)^+\right)\\
&\leq \sum_{n=n_\P}^\infty\left(a_\P2^n+b_\P\right)^{-1}\mathbb{E}_\P\left[\phi_\P\left(\lambda_n^{-1}|X_n-X_{n+1}|\right)\right]\\
&\leq \sum_{n=n_\P}^\infty\frac{1}{a_\P2^n+b_\P}<\infty.
\end{align*}
Applying the Borel-Cantelli Lemma yields that
\[\inf_{\P\in\mathfrak P}\P\big(|X_n-X_{n+1}|\leq 2^{-n}\textnormal{ eventually}\big)=1.\]
Hence, the event $\Omega^*:=\{\lim_{n\to \infty} X_n\textnormal{ exists in }\R\}\in\mathcal F$ satisfies $\P(\Omega^*)=1$ for all $\P\in\mathfrak P$. We set $X$ to be (the equivalence class in $L^0(\mathfrak P)$ induced by) $\limsup_{n\to\infty}X_n$. Now, let $\P\in \mathfrak P$ and $\alpha>0$ be arbitrary. Choose $k\in\mathbb{N}$ such that $\sum_{i\ge k}\lambda_i\alpha\le 1$.
For $l>k$, we can estimate
\begin{align*}
\phi_\P(\alpha|X_{n_k}-X_{n_l}|)&\le\phi_\P\left(\sum_{i=k}^{l-1}\alpha|X_{n_{i+1}}-X_{n_i}|\right)\le\sum_{i=k}^{l-1}\lambda_i\alpha\phi_\P\left(\lambda_i^{-1}|X_{n_{i+1}}-X_{n_i}|\right)\le\sum_{i=k}^\infty\lambda_i\alpha.
\end{align*}
Notice that the last bound is uniform in $l$ and $\P$. Letting $l\to\infty$ and using lower semicontinuity of $\phi_\P$,
\begin{align*}
\phi_\P(\alpha|X_{n_k}-X|)&\le\sum_{i=k}^\infty\lambda_i\alpha.
\end{align*}
This implies
\[\limsup_{k\to\infty}\sup_{\P\in\mathfrak P}\mathbb{E}_\P[\phi_\P\left(\alpha|X_{n_k}-X|\right)]\le \lim_{k\to\infty}\sum_{i=k}^\infty\lambda_i\alpha=0.\]
As $\alpha>0$ was arbitrary, $X\in L^\Phi(\mathfrak P)$ and $\lim_{k\to\infty}\|X_k-X\|_{L^\Phi(\mathfrak P)}=0$ follow.
At last, let $X\in L^\Phi(\mathfrak P)$. By lower semicontinuity of $\phi_\P$ and Fatou's Lemma, for all $\P\in \mathfrak P$,
\[
\mathbb{E}_\P[a_\P\|X\|_{L^\Phi(\mathfrak P)}^{-1} |X|+b_\P]\leq \mathbb{E}_\P\big[\phi_\P(\|X\|_{L^\Phi(\mathfrak P)}^{-1} |X|)\big]\leq 1,
\]
showing that $\mathbb{E}_\P[|X|]\leq \frac{1-b_\P}{a_\P}\|X\|_{L^\Phi(\mathfrak P)}$, that is, \eqref{eq:normP}.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:L1}]
Suppose $L^{\infty}(\mathfrak P)\subset L^\Phi(\mathfrak P)$. Then, we can find some $\alpha>0$ such that
\[\sup_{\P\in\mathfrak P}\phi_\P(\alpha)= \sup_{\P\in\mathfrak P}\mathbb{E}_\P[\phi_\P(\alpha\mathbf{1}_\Omega)]\leq 1.\]
Now let $\alpha>0$ with $\phi_{\rm Max}(\alpha)=\sup_{\P\in \mathfrak P}\phi_\P(\alpha)< \infty$. Since $\phi_{\rm Max}$ is convex, we may w.l.o.g.\ assume that $\phi_{\rm Max}(\alpha)\le 1$. For $\P\in \mathfrak P$ and a finite measure $\mu\ll\P$, let
\[
\|\mu\|_{\P}':=\sup\big\{\mu|X|\,\big|\, \|X\|_{L^{\phi_\P}(\P)} =1\big\}.
\]
Then, by \cite[Theorem 2.6.9 \& Corollary 2.6.6]{MeyNie},\footnote{~The cases $L^{\phi_\P}(\P)\in \{L^1(\P),L^\infty(\P)\}$ are not treated in this reference, but equation~\eqref{eq:koethe} is well known for them.}
\begin{equation}\label{eq:koethe}
\|X\|_{L^{\phi_\P}(\P)}=\sup\big\{\mu|X|\, \big| \, \mu\in\mathbf{ca}_+(\P),\, \|\mu\|_{\P}' = 1\big\},\quad \text{for all }\P\in \mathfrak P\text{ and }X\in L^{\phi_\P}(\P).
\end{equation}
Since $\sup_{\P\in \mathfrak P}\phi_\P(\alpha)\leq 1$, $\|\mathbf{1}_\Omega\|_{L^{\phi_\P}(\P)}\le\alpha^{-1}$. Hence, for all $\mu\in\mathbf{ca}_+(\P)$ with $\|\mu\|_\P'=1$,
\begin{equation}\label{eq:bound1}
\mu(\Omega)=\|\mathbf{1}_\Omega\|_{L^{\phi_\P}(\P)}\mu(\|\mathbf{1}_\Omega\|_{L^{\phi_\P}(\P)})^{-1}\mathbf{1}_\Omega)\le\frac{1}\alpha.
\end{equation}
For $\P\in \mathfrak P$, let
\[\mathfrak Q_\P:=\{\tfrac{1}{\mu(\Omega)}\mu\mid\mu\in\mathbf{ca}_+(\P),\,\|\mu\|_\P'=1\}.\]
By \eqref{eq:normP}, $\P\in\mathfrak Q_\P$ holds for all $\P\in\mathfrak P$.
We also define
\[\mathfrak Q:=\{\mathbb{Q}\in \mathbf{ca}_+^1(\mathfrak P)\mid \exists\,\P\in\mathfrak P:~\mathbb{Q}\in\mathfrak Q_\P\}\supset\mathfrak P.\]
Fix $\mathbb Q\in\mathfrak Q$, let $\P\in\mathfrak P$ such that $\mathbb Q\in\mathfrak Q_\P$, and let $\mu\in\mathbf{ca}_+(\P)$ such that $\mathbb Q=\mu(\Omega)^{-1}\mu$. Then, \eqref{eq:bound1} implies that
\[\|\mathbb Q\|_\P'=\mu(\Omega)^{-1}\ge\alpha.\]
The function
\[\theta(\mathbb{Q}):=\frac{1}{\inf_{\P\in\mathfrak P\colon \mathbb{Q}\in\mathfrak Q_\P}\|\mathbb Q\|_\P'},\quad \text{for }\mathbb Q\in\mathfrak Q,\]
is thus bounded and takes positive values. Moreover, for $X\in L^0(\mathfrak P)$,
\begin{align*}
\|X\|_{L^\Phi(\mathfrak P)}&=\sup_{\P\in\mathfrak P}\|X\|_{L^{\phi_\P}(\P)}=\sup_{\P\in\mathfrak P}\sup_{\mathbb Q\in\mathfrak Q_\P}\frac{1}{\|\mathbb Q\|_\P'}\mathbb{E}_\mathbb Q[|X|]\\
&=\frac{1}{\inf_{\P\in\mathfrak P\colon \mathbb{Q}\in\mathfrak Q_\P}\|\mathbb Q\|_\P'}\sup_{\mathbb Q\in\mathfrak Q}\mathbb{E}_\mathbb Q[|X|]=\sup_{\mathbb Q\in\mathfrak Q}\theta(\mathbb Q)\mathbb{E}_\mathbb Q[|X|].
\end{align*}
This is (3).\\
At last, suppose that $L^\Phi(\mathfrak P)$ reduces to a weighted robust $L^1$-space as in the assertion. From $\mathfrak Q\approx\mathfrak P$, we infer that the latter space contains $L^\infty(\mathfrak Q)=L^{\infty}(\mathfrak P)$.\\
For the last statement, choose $\kappa:=\sup_{\mathbb Q\in \mathfrak Q} \theta(\mathbb Q)$ or, equivalently, $\kappa:=\|\mathbf{1}_\Omega\|_{L^\Phi(\mathfrak P)}$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm.main}]
The separability of $\mathcal H$ implies that the unit ball $\big\{\ell\in\mathcal H^*\, \big|\, \|\ell\|_{\mathcal H^*}\le 1\big\}$ endowed with the weak* topology is compact, metrisable, and thus separable, cf.~\cite[Theorem 3.16]{MR1157815}. Hence, the set
\[
\bigg\{ \frac{\P}{\|\P\|_{\mathcal H^*}}\, \bigg|\, \P\in \mathfrak P\bigg\}\subset \big\{\ell\in\mathcal H^*\, \big|\, \|\ell\|_{\mathcal H^*}\le 1\big\}
\]
is separable, and there exists a countable family $(\P_n)_{n\in \mathbb{N}}$ such that, for all $X\in \mathcal H$, $\sup_{n\in \mathbb{N}} \mathbb{E}_{\P_n}[|X|]>0$ holds if and only if $X\neq 0$.
Consider the measure
\[\mu^*:=\sum_{n\in \mathbb{N}}2^{-n}\min\{1,\|\P_n\|_{\mathcal H^*}^{-1}\}\P_n\in\mathcal H^*,\]
which satisfies $\mu^*(\Omega)\le 1$. For $s>0$ appropriately chosen, the probability measure $\P^*:=s\mu^*\in\mathcal H^*$ is a countable convex combination of $(\P_n)_{n\in\mathbb{N}}$, and the functional $\P^*$ is strictly positive by construction. Hence, for $X,Y\in \mathcal H$, $X\preceq Y$ if and only if $\mathbb{E}_{\P^*}[(Y-X)^-]=0$, which immediately proves that the canonical projection $J_{\P^*}\colon \mathcal H\to L^1(\P^*)$ is injective. By construction, we see that $\P^*\in \mathfrak P$ if $\mathfrak P$ is countably convex.
\end{proof}
\begin{proof}[Proof of Corollary \ref{cor.main}]
As in the proof of the previous theorem, we see that the set
\[
\big\{ \theta (\mathbb Q)\cdot \mathbb Q \, \big|\, \mathbb Q\in \mathfrak Q\big\}\subset \big\{\ell\in\mathcal H^*\, \big|\, \|\ell\|_{\mathcal H^*}\le 1\big\}
\]
is separable with respect to the relative weak* topology. Hence, there exists a countable family $(\mathbb Q_n)_{n\in \mathbb{N}}\subset \mathfrak Q$ such that, for all $X\in \mathcal H$,
\[
\sup_{n\in \mathbb{N}} \theta (\mathbb Q_n)\|X\|_{L^1(\mathbb Q_n)}=\sup_{\mathbb Q\in \mathfrak Q} \theta (\mathbb Q)\|X\|_{L^1(\mathbb Q)}=\|X\|_{L^\Phi(\mathfrak P)}.
\]
\end{proof}
\begin{proof}[Proof of Proposition \ref{prop:equality}]
(2) clearly implies (1). Now suppose that (1) holds. By Theorem~\ref{thm:L1}, we have
\[\|X\|_{L^\Phi(\mathfrak P)}=\sup_{\mu\in\mathfrak D}\mu|X|,\]
where $\mathfrak D:=\{\mu\in\mathbf{ca}_+(L^\Phi(\mathfrak P))\mid \|\mu\|_{L^\Phi(\mathfrak P)^*}\le 1\}$ satisfies $\sup_{\mu\in\mathfrak D}\mu(\Omega)<\infty$ because of the assumption $L^{\infty}(\mathfrak P)\subset L^\Phi(\mathfrak P)$. Set
\begin{align*}&\mathfrak R:=\{\mu(\Omega)^{-1}\mu\mid \mu\in\mathfrak D\}\subset\mathbf{ca}^1_+(L^\Phi(\mathfrak P)),\\
&\psi_\mathbb Q(x):=\mu(\Omega)x,~x\ge 0,~\textnormal{for }\mathbb Q=\mu(\Omega)^{-1}\mu\in \mathfrak R,\\
&\Psi=(\psi_\mathbb Q)_{\mathbb Q\in\mathfrak R}.\end{align*}
Then, $L^\Phi(\mathfrak P)\subset\mathfrak L^\Psi(\mathfrak R)$ holds by construction. Suppose now that $X\in L^0(\mathfrak P)\setminus L^\Phi(\mathfrak P)$. Then, we must be able to find a sequence $(\mu_n)_{n\in\mathbb{N}}\subset\mathfrak D$ such that $\mu_n|X|\ge 2^{n}$, $n\in\mathbb{N}$. By the Banach space property of $L^\Phi(\mathfrak P)^*$, $\mu^*:=\sum_{n=1}^\infty2^{-n}\mu_n\in\mathfrak D$, and we observe
\[\mu^*|X|=\sum_{n=1}^\infty2^{-n}\mu_n|X|\ge\sum_{n=1}^\infty 1=\infty.\]
This completes the proof of the identity $L^\Phi(\mathfrak P)=\mathfrak L^\Psi(\mathfrak R)$. \\
Consider now the special case of $\mathfrak P$ being countably convex and \eqref{eq.thm:equality} being satisfied.
Observe that, for all $\alpha>0$, $\P\in\mathfrak P$, and all $X\in L^0(\mathfrak P)$,
\[\mathbb{E}_\P[\phi_\P(\alpha|X|)]\le\mathbb{E}_\P[\phi_{\rm Max}(\alpha|X|)]\le\mathbb{E}_\P[\phi_\P(\alpha c_\P|X|)].\]
If we set $\Psi=(\phi_{\rm Max})_{\P\in\mathfrak P}$, this is sufficient to prove the following chain of inclusions:
\[L^\Psi(\mathfrak P)\subset L^\Phi(\mathfrak P)\subset\mathfrak L^\Phi(\mathfrak P)=\mathfrak L^\Psi(\mathfrak P).\]
The proof is complete if we can show $\mathfrak L^\Psi(\mathfrak P)\subset L^\Psi(\mathfrak P)$. To this end,
let $X\in L^0(\mathfrak P)\setminus L^\Phi(\mathfrak P)$. Then, there exists a sequence $(\P_n)_{n\in \mathbb{N}}\subset \mathfrak P$ with
\[
\|X\|_{L^{\phi_{{\rm Max}}}(\P_n)}>2^nn\quad \text{for all }n\in \mathbb{N}.
\]
Define $\P:=\sum_{n\in \mathbb{N}}2^{-n}\P_n\in\mathfrak P$ (because $\mathfrak P$ is countably convex), and let $s>0$ be arbitrary. Then,
\[
\mathbb{E}_\P[\phi_{\rm Max}(s|X|)]=\sum_{n=1}^\infty 2^{-n}\mathbb{E}_{\P_n}\left[\phi_{{\rm Max}}(s|X|)\right]\ge\sum_{n=1}^\infty \mathbb{E}_{\P_n}\left[\phi_{{\rm Max}}(2^{-n}s|X|)\right]=\infty,
\]
which proves that $X\notin\mathfrak L^\Psi(\mathfrak P)$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:equality}]
Let $X\in L^0(\mathfrak P)\setminus L^\Phi(\mathfrak P)$.
Then, there is a sequence
$(\P_n)_{n\in\mathbb{N}}\subset\mathfrak P$ such that, for all $n\in\mathbb{N}$, $\|X\|_{L^{\phi_{\P_n}}(\P_n)}>2^{2n}$, which in particular entails
\[\mathbb{E}_{\P_n}[\phi(\theta(\P_n)2^{-n}|X|)]>2^n(1+\gamma(\P_n)).\]
Fix $\P^*\in\mathfrak P$ and consider the measure
\[\mathbb Q:=\sum_{n=1}^\infty2^{-n}\big(\tfrac{\gamma(\P_n)}{1+\gamma(\P_n)}\P^*+\tfrac{1}{1+\gamma(\P_n)}\P_n\big).\]
By convexity of $\gamma$ and the countable convexity of its lower level sets, $\gamma(\mathbb Q)\le\gamma(\P^*)+1$, $n\in\mathbb{N}$. For $\alpha>0$ arbitrary, set $I:=\{n\in\mathbb{N}\mid \theta(\mathbb Q)\alpha\ge \theta(\P_n)2^{-n}\}$, an infinite set. Then,
\begin{align*}\mathbb{E}_{\mathbb Q}[\phi(\theta(\mathbb Q_n)\alpha|X|)]&\ge \sum_{n\in I}\tfrac{1}{2^n(1+\gamma(\P_n)}\mathbb{E}_{\P_n}[\phi(\theta(\P_n)2^{-n}|X|)]=\infty.
\end{align*}
This proves $\|X\|_{L^{\phi_{\mathbb Q}}(\mathbb Q)}=\infty$, which means $X\notin\mathfrak L^\Phi(\mathfrak P)$.
\end{proof}
\section{Proofs of Section \ref{sec.completion}}\label{app.B}
\begin{proof}[Proof of Lemma~\ref{lem:sigmaorder}]
Let $\mathcal Y$ denote the real vector space of all $\sigma$-order continuous linear functionals on $\mathcal H$. As $\mathcal H$ is a vector lattice, ${\mathcal Y}$ is a vector lattice itself when endowed with the order
\[\ell\preceq^*\ell'\quad:\iff\quad \forall\,X\in\mathcal H:~\ell(X)\le \ell'(X);\]
cf.\ \cite[Theorem 1.57]{AliBurk2}. As such, for each $\ell\in{\mathcal Y}$ there are unique $\ell^+,\ell^-\succeq^* 0$ such that $\ell=\ell^+-\ell^-$. We may hence assume for the moment that $\ell\succeq^* 0$. \\
Then, for each sequence $(X_n)_{n\in\mathbb{N}}\subset\mathcal H$ possessing representatives $(f_n)_{n\in\mathbb{N}}$ such that $f_n\downarrow 0$ holds pointwise, $\inf_{n\in\mathbb{N}}X_n=0$ holds in $\mathcal H$. Consider the vector lattice
\[\mathcal L:=\big\{f\in\mathcal L^0(\Omega,\mathcal F)\, \big|\, [f]\in\mathcal H\big\}.\]
and the linear map $J\colon \mathcal L\to\mathcal H$ defined by $J(f)=[f]$, $f\in\mathcal L$.
The linear functional $\ell_0:=\ell\circ J$
satisfies $\ell_0(f_n)\downarrow 0$ for all sequences $(f_n)_{n\in\mathbb{N}}\subset\mathcal L$ such that $f_n\downarrow 0$ pointwise.
Since, by our assumption on $\mathcal H$, $\mathcal F=\sigma(\mathcal L)$, \cite[Theorem 7.8.1]{Bogachev} provides a unique finite measure $\mu$ on $(\Omega,\mathcal F)$ such that
\[\ell_0(f)=\int f\,{\rm d}\mu\quad\text{for all } f\in \mathcal L.\]
As $|f|\in\mathcal L$ for all $f\in\mathcal L$, each $f\in\mathcal L$ is $\mu$-integrable. Moreover, for all $X\in\mathcal H$ and $f,g\in X$,
\[\int f\,{\rm d}\mu=\ell_0(f)=\ell(X)=\ell_0(g)=\int g\,{\rm d}\mu.\]
In particular, considering that $\mathbf{1}_N\in\mathcal L$ for all $N\in\mathcal F$ satisfying $\sup_{\P\in\mathfrak P}\P(N)=0$, $\mu\in\mathbf{ca}_+(\mathfrak P)$ follows.\\
Finally, for a general $\ell\in{\mathcal Y}$, let $\nu,\eta\in\mathbf{ca}_+(\mathfrak P)$ be the finite measures corresponding to $\ell^+$ and $\ell^-$, respectively. Setting $\mu:=\nu-\eta$, we obtain for all $X\in\mathcal H$ that
\[\ell(X)=\ell^+(X)-\ell^-(X)=\nu X-\eta X=\mu X.\]
Moreover, the total variation measure $|\mu|$ satisfies $\int |f|\,{\rm d}|\mu|\le \int |f|\,{\rm d}(\nu+\eta)<\infty$, $f\in\mathcal L$. \\
At last, suppose that the representing signed measure of $\ell\in{\mathcal Y}$ is a measure. Then, $\ell\succeq^* 0$ holds automatically, and the proof is complete.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lem:representation00}]
Let $\ell\in\mathcal H^*$. In order to verify $\ell\in\mathbf{ca}^\sigma(\mathcal H)$, let $X,Y\in\mathcal H$. Then, all $Z\in\mathcal H$ with the property $X\preceq Z\preceq Y$ satisfy $|Z|\le X^-+Y^+$. We obtain
\[\sup\{\ell(Z)\mid X\preceq Z\preceq Y\}\le\|\ell\|_{\mathcal H^*}\sup\{\|Z\|_{L^\Phi(\mathfrak P)}\mid X\preceq Z\preceq Y\}\le\|\ell\|_{\mathcal H^*}\big(\|X^-\|_{L^\Phi(\mathfrak P)}+\|Y^+\|_{L^\Phi(\mathfrak P)}\big)<\infty.\]
This gives condition (i) in Definition~\ref{def:sigmaorder}. The validity of condition (ii) is a direct consequence of the lattice norm property and $\sigma$-order continuity of $\|\cdot\|_{L^\Phi(\mathfrak P)}$ on $\mathcal H$. The inclusion $\mathcal H^*\subset\mathbf{ca}^\sigma(\mathcal H)$ together with Lemma~\ref{lem:sigmaorder} implies $\mathcal H^*\subset\mathbf{ca}(\mathfrak P)$.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{lem:representation0}]
By \cite[Proposition 1.2.3(ii)]{MeyNie}, the closure $(\mathcal C,\preceq)$ of the sublattice $(\mathcal H,\preceq)$ of $L^\Phi(\mathfrak P)$ is a sublattice as well. As $\|\cdot\|_{L^\Phi(\mathfrak P)}$ is a lattice norm on $\mathcal C$, $\big(\mathcal C,\preceq,\|\cdot\|_{L^\Phi(\mathfrak P)}\big)$ is a Banach lattice by construction.\\
The inclusion $\mathbf{ca}(\mathcal C)\subset\mathbf{ca}(\mathcal H)$ is trivial. For the converse inclusion, let $\mu \in \mathbf{ca}(\mathcal H)$, i.e., $|\mu|\in\mathcal H^*$.
Since $\mathcal H$ is dense in $\mathcal C$, there exists a unique $\ell\in \mathcal C^*$ with
$$\ell(X)=|\mu|X,\quad \text{for all }X\in \mathcal H.$$
Let $X\in \mathcal C\cap L^{\infty}(\mathfrak P)$. Then, by Proposition \ref{prop:Banach}, there exists a sequence $(X_n)_{n\in \mathbb{N}}\subset \mathcal H\cap L^{\infty}(\mathfrak P)$ with $\sup_{n\in \mathbb{N}}\|X_n\|_{L^{\infty}(\mathfrak P)}<\infty$, $\|X-X_n\|_{L^\Phi(\mathfrak P)}\to 0$, and $X_n\to X$ $\mathfrak P$-q.s.~as $n\to \infty$. Since $|\mu|\ll \mathfrak P$, dominated convergence implies
\[
|\mu|X=\lim_{n\to \infty}|\mu|X_n=\lim_{n\to \infty}\ell (X_n)=\ell (X).
\]
Now, let $X\in \mathcal C$ arbitrary. Then,
\begin{align}\begin{split}\label{eq:approximation}|\mu|(|X|) &\leq \sup_{n\in \mathbb{N}}|\mu| \big(|X|\wedge n\mathbf{1}_\Omega\big)=\sup_{n\in \mathbb{N}}\ell\big(|X|\wedge n\mathbf{1}_\Omega\big)\\
&\leq \sup_{n\in \mathbb{N}}\|\ell\|_{\mathcal C^*}\big\||X|\wedge n\mathbf{1}_\Omega\big\|_{L^\Phi(\mathfrak P)}\leq \|\ell\|_{\mathcal C^*}\|X\|_{L^\Phi(\mathfrak P)}.
\end{split} \end{align}
From this observation, the equality $|\mu|=\ell$ follows, which is sufficient to prove that $\mu\in \mathbf{ca}(\mathcal C)$. \\
The remaining assertions easily follow with $\mathcal C^*=\mathcal H^*$ and Lemma~\ref{lem:representation00}.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem.Linfty}]
Let $X\in L^\Phi(\mathfrak P)$. First, notice that
\[
\sup_{\P\in \mathfrak P}\mathbb{E}_\P\big[\phi_\P(\alpha |X|)\mathbf 1_{\{|X|> n\}}\big]\to 0,\quad \text{as } n\to \infty
\]
for all $\alpha>0$ is equivalent to $\|X\mathbf 1_{\{|X|> n\}}\|_{L^\Phi(\mathfrak P)}\to 0$ as $n\to \infty$.
If $\|X\mathbf 1_{\{|X|> n\}}\|_{L^\Phi(\mathfrak P)}\to 0$ as $n\to \infty$, it follows that $\|X-X_n\|_{L^\Phi(\mathfrak P)}\to 0$ as $n\to \infty$ with $X_n:=X\mathbf 1_{\{|X|\leq n\}}\in L^{\infty}(\mathfrak P)$ for all $n\in \mathbb{N}$.
Now, assume that $X\in \textnormal{cl}\big(L^{\infty}(\mathfrak P)\big)$. Let $(Y_n)_{n\in \mathbb{N}}\subset L^{\infty}(\mathfrak P)$ with $\|X-Y_n\|_{L^\Phi(\mathfrak P)}\to 0$ as $n\to \infty$. Let $X_m:=(X\wedge m\mathbf{1}_\Omega)\vee (-m\mathbf{1}_\Omega)$ for all $m\in \mathbb{N}$. Then, for all $m,n\in \mathbb{N}$ with $m\geq\|Y_n\|_{L^\infty(\mathfrak P)}$, it follows that
\[
|X-X_m|\leq |X-Y_n|,
\]
which implies that $\|X-X_m\|_{L^\Phi(\mathfrak P)}\to 0$ as $m\to \infty$. Finally notice that
\begin{align*}
|X|\mbf1_{\{|X|>2m\}}&=(|X|-m\mathbf{1}_\Omega)\mathbf 1_{\{|X|>2m\}}+m \mathbf 1_{\{|X|>2m\}}\leq 2\big(|X|-m\mathbf{1}_\Omega\big)\mathbf 1_{\{|X|>2m\}}\\
&\leq 2\big(|X|-m\mathbf{1}_\Omega\big)\mathbf 1_{\left\{|X|>m\right\}}= 2|X-X_m|,
\end{align*}
which shows that $\|X\mbf1_{\{|X|>m\}}\|_{L^\Phi(\mathfrak P)}\to 0$ as $m\to \infty$.
\end{proof}
For the sake of clarity, we give the proofs of Lemma~\ref{lem:monclass}, Proposition~\ref{prop:equivalences0}, and Proposition~\ref{prop:equivalences} in advance of Theorem~\ref{thm.super}.
\begin{proof}[Proof of Lemma~\ref{lem:monclass}]
Let $X\in\mathcal H$ and $c\in\R$. Consider $Y_k:=k(X-c\mathbf{1}_\Omega)^+\wedge\mathbf{1}_\Omega\in\mathcal H$, $k\in\mathbb{N}$. The sequence $(Y_k)_{k\in\mathbb{N}}$ is nondecreasing and satisfies $0\preceq Y_k\preceq \mathbf{1}_\Omega$. By monotone convergence,
\begin{equation}\label{eq:1}\mu(\{X>c\})=\sup_{k\in\mathbb{N}}\mu Y_k=\lim_{k\to\infty}\mu Y_k\end{equation}
holds for all $\mu\in\mathbf{ca}_+(\mathfrak P)$. Moreover, by Dedekind $\sigma$-completeness of $\mathcal H$, $U:=\sup_{k\in\mathbb{N}}Y_k$ exists and lies in $\mathcal H_+$. \textit{A priori}, $\mathbf{1}_{\{X>c\}}\preceq U$ has to hold. Moreover, one can show that $U=(nU)\wedge\mathbf{1}_\Omega$ holds for all $n\in\mathbb{N}$. Hence, there is an event $B\in\mathcal F$ such that $\mathbf{1}_B=U$ in $\mathcal H$. For each $\mu\in\mathbf{ca}_+^\sigma(\mathcal H)$,
\begin{equation}\label{eq:4}\lim_{k\to\infty}\mu Y_k=\mu(B).\end{equation}
Equations \eqref{eq:1} and \eqref{eq:4} together with $\mathfrak P\approx \mathbf{ca}_+^\sigma(\mathcal H)$ now imply that $\mathbf{1}_{\{X>c\}}=U\in\mathcal H$, that is, for every $X\in\mathcal H$, $f\in X$, and $c\in\R$, the equivalence class generated by $\mathbf{1}_{\{f>c\}}$ lies in $\mathcal H$.
At last, consider the $\pi$-system $\Pi:=\{\{f>c\}\, |\, X\in\mathcal H,\,f\in X,\,c\in\R\}$, which generates $\mathcal F$ and is a subset of
\[\Lambda:=\{A\in\mathcal F\, |\, \mathbf{1}_A\in\mathcal H\}.\]
Since $\mathcal H$ is Dedekind $\sigma$-complete and $\mathfrak P\approx \mathbf{ca}_+^\sigma(\mathcal H)$, the latter can be shown to be a $\lambda$-system. By Dynkin's Lemma, it follows that $\Lambda=\mathcal F$. We have thus shown that $\mathcal H$ contains all representatives of $\mathcal F$-measurable simple functions. Each $X\inL^{\infty}(\mathfrak P)$ is the supremum of a countable family of simple functions in $L^\Phi(\mathfrak P)$. As $\mathcal H$ is Dedekind $\sigma$-complete and $\mathfrak P\approx \mathbf{ca}_+^\sigma(\mathcal H)$, we conclude that $L^{\infty}(\mathfrak P)\subset \mathcal H$.
\end{proof}
\begin{proof}[Proof of Proposition \ref{prop:equivalences0}]
(2) clearly implies (1). In order to see that (1) implies (3), note first that $L^{\infty}(\mathfrak P)\subset\mathcal H$ holds by Lemma~\ref{lem:monclass}. Now let $X\in L^\Phi(\mathfrak P)$, $Y\in\mathcal H$, and assume $0\preceq X\preceq Y$ holds. The set $\{X_n:=X\wedge n\mathbf{1}_\Omega\mid n\in\mathbb{N}\}\subset\mathcal H$ is order bounded above by $Y$ in $\mathcal H$. By Dedekind $\sigma$-completeness, $X^*:=\sup_{n\in\mathbb{N}}X_n$ exists in $\mathcal H$ and satisfies $X\preceq X^*$ \textit{a priori}. Arguing as in Lemma~\ref{lem:monclass}, one verifies $X=X^*\in\mathcal H$. \\
In order to see that (3) implies (2), we first show that $\mathcal H$ is Dedekind $\sigma$-complete. Let $\mathcal D\subset\mathcal H$ be order bounded from above and countable. Since $L^\Phi(\mathfrak P)$ is Dedekind $\sigma$-complete, $U:=\sup\mathcal D$ exists in $L^\Phi(\mathfrak P)$. Let $Y\in\mathcal H$ be any upper bound of $\mathcal D$ and $X\in\mathcal D$. Then, $X\preceq U\preceq Y$. As $\mathcal H$ is an ideal in $L^\Phi(\mathfrak P)$, $U\in \mathcal H$ has to hold and we have proved that $\mathcal H$ is Dedekind $\sigma$-complete.\\
Now we prove that each $\mu \in \mathbf{ca}(\mathcal H)$ is $\sigma$-order continuous. For condition (i) in Definition~\ref{def:sigmaorder}, we can argue as in the proof of Lemma~\ref{lem:representation00}. For condition (ii), let $(X_n)_{n\in \mathbb{N}}\in \mathcal H$ be a sequence with $X_{n+1}\preceq X_n$ for all $n\in \mathbb{N}$ and $\inf_{n\in \mathbb{N}} X_n=0$ in $\mathcal H$. By \cite[Theorem 1.35]{AliBurk2}, $\inf_{n\in\mathbb{N}}X_n=0$ holds in $L^\Phi(\mathfrak P)$, which is equivalent to $\inf_{\P\in\mathfrak P}\P(X_n\downarrow 0)=1$.
Moreover, by definition of $\mathbf{ca}(\mathcal H)$, $|\mu|X_1<\infty$. Dominated convergence yields
$$\lim_{n\to\infty}\mu X_n=\lim_{n\to\infty}|\mu|\big(\tfrac{{\rm d}\mu}{{\rm d}|\mu|}X_n\big)=0.$$
Now assume that, additionally, $\mathcal H\subsetL^{\infty}(\mathfrak P)$. If (3) holds, $\mathcal H$ is an ideal containing the equivalence class of $\mathbf{1}_\Omega$ and must therefore also be a superset of $L^{\infty}(\mathfrak P)$. Trivially, (4) implies (3), and the proof is complete.
\end{proof}
\begin{proof}[Proof of Proposition \ref{prop:equivalences}]
The equivalence of (1)--(3) follows directly from Proposition~\ref{prop:equivalences0} up to two additional observations: $\mathcal C$ is a Banach lattice by Proposition~\ref{lem:representation0}, and therefore each element of $\mathbf{ca}^\sigma(\mathcal C)$ is a continuous linear functional by \cite[Proposition 1.3.7]{MeyNie}.
If $\mathcal H\subsetL^{\infty}(\mathfrak P)$, $\mathcal C\subset\textnormal{cl}(L^{\infty}(\mathfrak P))$ must hold, and the converse inclusion is a direct consequence under (3). (4) implies (3) because $L^{\infty}(\mathfrak P)$ is an ideal and norm closures of ideals in Banach lattices remain ideals (\cite[Proposition 1.2.3(iii)]{MeyNie}).
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm.super}]
(1) is equivalent to (2): Theorem~\ref{thm.main} provides a strictly positive linear functional in the present situation. Hence, the equivalence of (1) and (2) follows with \cite[Lemma A.3]{nend20}.\\
(1) implies (3): Under assumption (1), $\mathcal C$ is thus a separable and Dedekind $\sigma$-complete Banach lattice. From \cite[Corollary 4.52]{AliBurk2}, we deduce that $\|\cdot\|_{L^\Phi(\mathfrak P)}$ is $\sigma$-order continuous on $\mathcal C$. Now, in view of Lemma \ref{lem:sigmaorder} and \cite[Proposition 1.3.7]{MeyNie}, $\sigma$-order continuity of the norm on $\mathcal C$ shows
$$\mathcal C^*=\mathbf{ca}(\mathcal C)=\mathbf{ca}^\sigma(\mathcal C).$$
In particular, each $\P\in\mathfrak P$ satisfies $\P\in\mathbf{ca}_+^\sigma(\mathcal C)$.
Lemma~\ref{lem:monclass} implies $L^{\infty}(\mathfrak P)\subset\mathcal C$, which entails that, for all $X\in\mathcal C$, $|X|\mathbf{1}_{\{|X|\le n\}}\uparrow|X|$ as $n\to\infty$, both in order and in norm. This proves $\mathcal C=\textnormal{cl}(L^{\infty}(\mathfrak P))$, which is (3).
(3) always implies (4).
(4) implies (1): This has been demonstrated already in the proof of Proposition~\ref{prop:equivalences}.
(1)--(4) implies (5): Note that the equivalent assertions (1)--(4) have already been demonstrated to imply $\mathcal C^*=\mathbf{ca}(\mathcal C)\supset\mathbf{ca}(L^\Phi(\mathfrak P)$. For the converse inclusion $\mathbf{ca}(\mathcal C)\subset\mathbf{ca}(L^\Phi(\mathfrak P))$, note that each $|\mu|\in\mathbf{ca}(\mathcal C)$ extends to a continuous linear functional $\ell\in L^\Phi(\mathfrak P)^*$. Arguing as in \eqref{eq:approximation}, we obtain
\[\ell(X)=|\mu|X,\quad \text{for all }X\in L^\Phi(\mathfrak P),\]
which means that $|\mu|$ (or equivalently, $\mu$) lies in $\mathbf{ca}(L^\Phi(\mathfrak P))$. Finally, let $\P^*\in \mathbf{ca}\big(L^\Phi(\mathfrak P))$ as in Theorem \ref{thm.main} and let $A\in\mathcal F$. $\mathbf{1}_A\in \mathcal C$ is implied by (3), and it follows that $\mathbf{ca}(L^\Phi(\mathfrak P))\approx \mathfrak P\approx \P^*$.\\
In order to see that the densities of measures in the unit ball of $\mathbf{ca}(\mathcal C)$ form a weakly compact subset of $L^1(\P^*)$, note that \eqref{eq:approximation} admits the representation
\[\{\mu\in\mathbf{ca}(\mathcal C)\mid \|\mu\|_{\mathcal C^*}\le 1\}=\{\mu\in\mathbf{ca}(\mathfrak P)\mid\forall\,X\in L^\infty(\P^*):~|\mu X|\le \|X\|_{L^\Phi(\mathfrak P)}\}.\]
The right-hand side is clearly weakly closed in $L^1(\P^*)$.\\
Now we consider a sequence $(A_n)_{n\in\mathbb{N}}$ such that
\[\P^*(A_n)\le 2^{-n}\text{ and }\|\mathbf{1}_{A_n}\|_{L^\Phi(\mathfrak P)}\ge \tfrac 1 2 \sup\{\|\mathbf{1}_B\|_{L^\Phi(\mathfrak P)}\mid B\in\mathcal F,\,\P^*(B)\le 2^{-n}\}.\]
Then, $\mathbf{1}_{A_n}$ is a sequence in $\mathcal C$ converging to 0 in order, and $\lim_{n\to\infty}\|\mathbf{1}_{A_n}\|_{L^\Phi(\mathfrak P)}=0$.\footnote{~More precisely, set $B_n:=\bigcup_{k\ge n}A_k$, a decreasing sequence of events. As $\P^*(B_n)\downarrow 0$, $\mathbf{1}_{B_n}\downarrow 0$ holds w.r.t.\ the $\mathfrak P$-q.s.\ order in $\mathcal C$. It remains to note that $\mathbf{1}_{A_n}\preceq\mathbf{1}_{B_n}$, $n\in\mathbb{N}$.} Set $\mathfrak B$ to be the set of all $\mu\in\mathbf{ca}_+(\mathcal C)$ with $\|\mu\|_{\mathcal C^*}\le 1$. We obtain
\[\sup\{\mu(B)\mid \mu\in\mathfrak B,\,B\in\mathcal F,\,\P^*(B)\le 2^{-n}\}=\sup\{\|\mathbf{1}_B\|_{L^\Phi(\mathfrak P)}\mid B\in\mathcal F,\,\P^*(B)\le 2^{-n}\}\to 0,\quad n\to\infty.\]
This shows that, for all $\varepsilon>0$, there is $\delta>0$ such that $\P^*(B)\le \delta$ implies $\mu(B)\le\varepsilon$, no matter the choice of $\mu\in\mathfrak B$. Moreover, $\mathfrak B$ is bounded in total variation. By \cite[Theorem 4.7.25]{Bogachev1}, $\mathfrak B$ and thus also the unit ball of $\mathbf{ca}(\mathcal C)^*$ is weakly compact in $L^1(\P^*)$. This completes the verification of (5).
(5) implies (3): Under assumption (5), the unit ball of $\mathbf{ca}\big(L^\Phi(\mathfrak P)\big)$, which is sufficient to determine $\|\cdot\|_{L^\Phi(\mathfrak P)}$ on all of $L^\Phi(\mathfrak P)$, can be identified with a weakly compact subset $\mathcal Z\subset L^1(\P^*)$. Each $X\inL^{\infty}(\mathfrak P)=L^\infty(\P^*)$ can be identified with a (linear) continuous function on $\mathcal Z$, and if $L^{\infty}(\mathfrak P) \ni X_n\downarrow 0$ $\mathfrak P$-q.s.\ (or $\P^*$-a.s.), the associated sequence of functions converges pointwise to $0$ on $\mathcal Z$. As this pointwise convergence must be uniform, $\sigma$-order continuity of $\|\cdot\|_{L^\Phi(\mathfrak P)}$ on $L^{\infty}(\mathfrak P)$ follows.
We now observe
\begin{itemize}
\item For each $X\in\mathcal C$ and each $c\in\R$, the sequence $Y_k:=k(X-c\mathbf{1}_\Omega)^+\wedge\mathbf{1}_\Omega\in\mathcal C\capL^{\infty}(\mathfrak P)$, $k\in\mathbb{N}$, satisfies $Y_k\uparrow \mathbf{1}_{\{X>c\}}$ in $L^{\infty}(\mathfrak P)$.
\item For each increasing sequence $(A_n)_{n\in\mathbb{N}}$ of events in $\Lambda:=\{A\in\mathcal F\mid \mathbf{1}_A\in\mathcal C\}$, $\mathbf{1}_{A_n}\uparrow \mathbf{1}_{\bigcup_{k\in\mathbb{N}}A_k}$ holds in $L^{\infty}(\mathfrak P)$.
\end{itemize}
Arguing as in Lemma~\ref{lem:monclass} and using $\sigma$-order continuity of the norm as well as closedness of $\mathcal C$ shows $\mathcal C\capL^{\infty}(\mathfrak P)=L^{\infty}(\mathfrak P)$, i.e., $\mathcal M:=\textnormal{cl}(L^{\infty}(\mathfrak P))\subset\mathcal C$.
Towards a contradiction, assume that we can find $X\in\mathcal C\setminus \mathcal M$. Then, there is a measure $0\neq\mu\in\mathcal C^*=\mathbf{ca}(L^\Phi(\mathfrak P))$ such that
\[\mu|_{\mathcal M}\equiv 0\quad\text{and}\quad \mu X\neq 0.\]
This however would mean $\mu|_{L^{\infty}(\mathfrak P)}\equiv 0$, which is impossible. $\mathcal C\subset\mathcal M$ follows.
We have already proved (6) above.
For (7), assume that condition \eqref{eq:growth} holds. Then, there exist $a>0$ and $b\leq 0$ such that $\phi_\P(x)\geq ax+b$ for all $\P\in \mathfrak P$ and $x\in [0,\infty)$. By~\eqref{eq:normP},
\begin{equation}\label{eq:supnormP}
\sup_{\P\in\mathfrak P}\|a(1-b)^{-1}\P\|_{L^\Phi(\mathfrak P)^*}\le 1.
\end{equation}
The assertion follows with (5), and the proof is complete.
\end{proof}
\section{Proofs of Section \ref{sec.applications}}\label{app.C}
\begin{proof}[Proof of Proposition~\ref{prop:density}]
As each subset of a separable normed space is separable itself, we can w.l.o.g.\ consider the maximal case $\mathcal H=C_b$.
By Theorem \ref{thm:L1}, there exists some constant $\kappa>0$ such that $$\|X\|_{L^\Phi(\mathfrak P)}\leq \kappa \|X\|_{L^\infty(\mathfrak P)},\quad\text{for all }X\in C_b.$$ Let $d$ be a metric consistent with the topology on $\Omega$, and $(\omega_n)_{n\in \mathbb{N}}$ dense in $\Omega$. For $m,n\in \mathbb{N}$ and $\omega\in \Omega$, let $X_{m,n}(\omega):=d(\omega,\omega_n)\wedge m$. The algebra $\mathcal A\subset C_b$ generated by $\{\mathbf{1}_\Omega\}\cup\{(X_{m,n})\mid m,n\in \mathbb{N}\}$ is separable and separates the points of each compact set $K\subset \Omega$. We show that the separable set
\[
\mathcal M:=\big\{(X_0\wedge m\mathbf{1}_\Omega)\vee(-m\mathbf{1}_\Omega) \,\big|\, X_0\in \mathcal A, \, m\ge 0\big\}
\]
is dense in $L^\Phi(\mathfrak P)$. To this end, let $X\in C_b$, $\varepsilon>0$, and $K\subset \Omega$ compact with $$\|\mathbf{1}_{\Omega\setminus K}\|_{L^\Phi(\mathfrak P)}<\frac{\varepsilon}{2(1+2\|X\|_\infty)}.$$
By the Stone-Weierstrass Theorem, there exists some $X_0\in \mathcal M$ with $\|X_0\|_{L^\infty(\mathfrak P)}\leq 1+\|X\|_{L^\infty(\mathfrak P)}$ and
\[
\big\|(X-X_0)\mathbf{1}_K\big\|_{L^\infty(\mathfrak P)}<\frac{\varepsilon}{2\kappa}.
\]
Hence,
\begin{align*}
\|X-X_0\|_{L^\Phi(\mathfrak P)}&\leq \big\|(X-X_0)\mathbf{1}_K\big\|_{L^\Phi(\mathfrak P)}+\big\|(X-X_0)\mathbf{1}_{\Omega\setminus K}\big\|_{L^\Phi(\mathfrak P)}\\
&\leq \kappa \big\|(X-X_0)\mathbf{1}_K\big\|_{L^\infty(\mathfrak P)} + \big(\|X\|_{L^{\infty}(\mathfrak P)}+\|X_0\|_{L^{\infty}(\mathfrak P)}\big) \|\mathbf{1}_{\Omega\setminus K}\|_{L^\Phi(\mathfrak P)}< \varepsilon.
\end{align*}
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lem:density}]
(1) trivially implies \eqref{eq:compact}.
Under condition (2), let $\varepsilon>0$ and set $t:=\varepsilon^{-1}$. Choose $K\subset\Omega$ compact with $$\phi_{\rm Max}(t)\sup_{\P\in\mathfrak P_t}\P(\Omega\setminus K)\le 1.$$ Then,
\[\sup_{\P\in\mathfrak P_t}\mathbb{E}_\P[\phi_\P(t\mathbf{1}_{\Omega\setminus K})]\le\sup_{\P\in\mathfrak P_t}\mathbb{E}_\P[\phi_{\rm Max}(t\mathbf{1}_{\Omega\setminus K})]=\phi_{\rm Max}(t)\sup_{\P\in\mathfrak P_t}\P(\Omega\setminus K)\le 1.\]
Moreover,
\[\sup_{\P\in\mathfrak P\setminus\mathfrak P_t}\mathbb{E}_\P[\phi_\P(t\mathbf{1}_{\Omega\setminus K})]\le 1.\]
This entails $\|\mathbf{1}_{\Omega\setminus K}\|_{L^\Phi(\mathfrak P)}\le \varepsilon$.
Condition (3) is a special case of condition (2).
Suppose now that \eqref{eq:compact} holds and let $x_0>0$ be as in \eqref{eq:growth}.
By~\eqref{eq:supnormP}, $\mathfrak P$ is a bounded subset of $\mathcal C^*$, and we have
\[\sup_{\P\in\mathfrak P}\mathbb{E}_\P[|X|]\le \frac{1-b}{a}\|X\|_{L^\Phi(\mathfrak P)}\quad\text{for all }X\in L^\Phi(\mathfrak P),\]
where $a>0$ and $b\le 0$ are suitably chosen.
Replacing $X$ by $\mathbf{1}_{\Omega\setminus K}$ for suitable $K\subset\Omega$ compact immediately yields tightness of $\mathfrak P$.
\end{proof}
\begin{proof}[Proof of Corollary~\ref{cor:representation}]
Let $(X_n)_{n\in \mathbb{N}}\subset C_b$ with $X_n\downarrow 0$ as $n\to \infty$ and $\alpha>0$. Then, $\phi_{\rm Max}(\alpha X_n)\in C_b$ for all $n\in \mathbb{N}$, and $\phi_{\rm Max}(\alpha X_n)\downarrow 0$ as $n\to \infty$. Since $\mathfrak P$ is weakly compact and the functions $\mathfrak P\ni\P\mapsto\mathbb{E}_\P[\Phi_{\rm Max}(\alpha X_n)]$ are weakly continuous, Dini's Theorem implies
\[
\lim_{n\to\infty}\sup_{\P\in \mathfrak P}\mathbb{E}_\P[\phi_\P(\alpha |X_n|)]=0.
\]
This suffices to conclude $\lim_{n\to\infty}\|X_n\|_{L^\Phi(\mathfrak P)}=0$. The assertion now follows from Proposition \ref{lem:representation0}.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:option}]
We first prove that $\mathcal H_X$ is a sublattice of $L^\Phi(\mathfrak P)$. The latter space is Dedekind $\sigma$-complete and thus also uniformly complete (\cite[Proposition 1.1.8]{MeyNie}). As such, we may replicate the argument in the proof of \cite[Theorem 3.1]{Options}.
$\big(\mathcal C_X,\preceq,\|\cdot\|_{L^\Phi(\mathfrak P)}\big)$ is a Banach lattice by Proposition~\ref{lem:representation0}.
Now, the span of the countable set $\{\mathbf{1}_\Omega\}\cup\{(X-k\mathbf{1}_\Omega)^+\mid k\textnormal{ rational}\}$ over the rational numbers lies dense in $\mathcal H_X$, whence separability of $\mathcal H_X$ and its norm closure $\mathcal C_X$ follow.
\end{proof}
\begin{proof}[Proof of Corollary~\ref{cor:ae}]
Note that $J_{\P^*}(L^\Phi(\mathfrak P))\subset L^1(\P^*)$ is an ideal on which $\P^*$ acts as a strictly positive bounded linear functional. The assertion thus follows directly from \cite[Corollary 3.2(b)]{Options}.
\end{proof}
\end{appendix}
\bibliographystyle{abbrv}
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\section{Entanglement Hamiltonian Tomography} \label{methods:EHT}
In EHT we construct an estimator of the state $\rho_A$ of the form $\rho_{A}(\tilde{\boldsymbol{g}})=\exp[-\tilde{H}_{A}(\tilde{\boldsymbol{g}})] / Z(\tilde{\boldsymbol{g}})$, with $Z(\tilde{\boldsymbol{g}}) = \mathrm{Tr}\left(\exp[-\tilde{H}_{A}(\tilde{\boldsymbol{g}})]\right)$ a constant to ensure unit trace. The ansatz for the entanglement Hamiltonian $\tilde{H}_{A}(\tilde{\boldsymbol{g}})$ is constructed from quasi-local few-body operators, as detailed in the main text. The (polynomially many) real-valued coefficients $\tilde{\boldsymbol{g}}$ of each of the operators are inferred from quantum measurements on $\rho_A$.
In particular, we consider measurements of bitstrings $s$ in the computational basis after randomized local rotations ${U}=\bigotimes_{i\in A}{u}_{i}$, where ${u}_{i}$ is a local basis rotation at site $i$, and is sampled from a unitary 2-design \cite{PhysRevA.80.012304}. Denoting by $P_U(\mathbf{s})$ the frequency of having observed a particular bitstring $\mathbf{s}$ in the experiment, one can define a cost function
\begin{align}
\chi^2 = \sum_U \sum_\mathbf{s} \left[ P_U(\mathbf{s}) - \mathrm{Tr}\left( \rho_A{(\boldsymbol{\tilde g}}) U\ket{\mathbf{s}}\bra{\mathbf{s}}U^\dagger \right) \right] ^2, \label{eq:method}
\end{align}
which is to be minimized over the parameters $\tilde{\boldsymbol{g}}$ of the estimator. Our protocol thus learns the quasi-local EH from data, in contrast to the learning of quasi-local system Hamiltonians from pure (eigen-) states \cite{Qi2019,Bairey2019,Wang2017}. We note that choices of cost-functions and random unitary ensembles \cite{elben2018renyi} other than in Eq.~\eqref{eq:method} are possible, and devote their detailed investigation to future work.
Extracting the EH and ES from experimental data, as described in context of Fig.~(\ref{fig:Exp}), requires the EHT protocol outlined above to be adapted to account for decoherence and imperfections. First, the experimental time evolution will be weakly coupled to an environment, which in the present case is well modeled by a dephasing master equation \cite{Brydges2019,ElbenXPlatform}.
Second, experimental randomized measurements suffer from coherent rotation errors on the percent level, as the dominant error source in the experiment, and possibly spin imperfect readout \cite{ElbenXPlatform}. In \cite{SM} we argue, supported by detailed numerical simulations and previous work, that decoherence and imperfections are well accounted for by a depolarizing channel, i.e.~we modify our ansatz for EH in Eq.~(\ref{eq:method}) to
\begin{equation}\label{eq:depol}
\rho_A (\tilde{\boldsymbol{g}},p) \equiv (1-p)\exp [-\tilde{H}_{A}(\tilde{\boldsymbol{g}})] /Z(\tilde{\boldsymbol{g}}) + p \mathbb{1}/{\mathcal{D}_A},
\end{equation}
introducing a depolarization parameter $0\le p\le 1$, and with $\mathcal{D}_A = 2^{N_A}$ the dimension of $A$. This allows us to extract both the EH (first term on the RHS) and $p$ from experimental data, isolating `coherent' and `incoherent' contributions to $\rho_A$.
While EHT is efficient in the number of measurements required, presently (\ref{eq:method}) is implemented as classical postprocessing of probabilities $P_U(\mathbf{s})$ measured on the quantum device, with corresponding requirements on classical computing, i.e.~feasible for subsystem sizes $N_A$ not much larger than $12$. Recent ideas \cite{Huang20, Bairey2019} allow, in principle, the determination of $\tilde{H}_A(\tilde{\boldsymbol{g}})$ in a scalable way, up to an unknown scaling factor for the coefficients $\tilde{\boldsymbol{g}}$. Determining the scaling factor, corresponding to the inverse temperature, opens interesting perspectives for measuring entanglement properties for larger subsystems. Variational quantum algorithms are potential candidates for extracting entanglement properties on a larger scale \cite{vqsd_coles, PhysRevA.101.062310}. In unpublished work we have developed hybrid classical-quantum algorithms where classical optimization of variational parameters is preceded by \textit{in situ} quantum postprocessing operations on $\rho_A$, with spins representing $A$ acting as quantum memory.
\section{Verification \& Fidelity Estimation\label{Methods:XP}}
We determine a (mixed-state) fidelity between the experimental quantum state under study, described by the density matrix $\rho_A \equiv \rho_1$, and the reconstructed density matrix from EHT, $\rho_A(\tilde{\boldsymbol{g}}) \equiv \rho_2$. To this end, we consider the fidelity \cite{Liang2019}
\begin{align}
\mathcal{F}_{\textrm{max}}(\rho_{1},\rho_{2})=\frac{\mathrm{Tr}(\rho_{1}\rho_{2})}{\max\{\mathrm{Tr}(\rho_{1}^{2}),\mathrm{Tr}(\rho_{2}^{2})\}},
\label{eq:Fmax}
\end{align}
which measures the overlap between $\rho_{1}$ and $\rho_{2}$, respectively, normalized by their purities.
As shown in Ref.~\cite{ElbenXPlatform}, $\mathcal{F}_{\textrm{max}}(\rho_{1},\rho_{2})$, i.e.\ terms of the form $\mathrm{Tr}(\rho_i\rho_j)$ for $i,j=1,2$, can be evaluated from second-order \emph{cross-correlations} between the outcomes of randomised measurements. We set $P^{(1)}_U(\mathbf{s})$ the frequency of having observed a particular bitstring $\mathbf{s}$ in the experiment (where $\rho_A$ is realized) and $P^{(2)}_U(\mathbf{s})= \mathrm{Tr}\left( \rho_A{(\boldsymbol{\tilde g}}) U\ket{\mathbf{s}}\bra{\mathbf{s}}U^\dagger \right)$. Then, we obtain the overlap $\mathrm{Tr}(\rho_i\rho_j)$ for $i=1,j=2$ and purities $\mathrm{Tr}(\rho_i\rho_j)$ for $i=j=1,2$ via \cite{ElbenXPlatform}
\begin{align}
\mathrm{Tr}(\rho_i\rho_j)= \label{eq:ovl}
\frac{2^{N_{A}}}{N_U}\sum_U \sum_{\mathbf{s},\mathbf{s}'}(-2)^{-\mathcal{D}[\mathbf{s},\mathbf{s}']}{P_{U}^{(i)}(\mathbf{s})P_{U}^{(j)}(\mathbf{s}')},
\end{align}
where the Hamming distance $\mathcal{D}[\mathbf{s},\mathbf{s}']$ between two strings $\mathbf{s}$ and $\mathbf{s}'$ is defined as the number of local constituents where $s_{k}\neq{s}'_{k}$, i.e.\ $\mathcal{D}[\mathbf{s},{\mathbf{s}}']\equiv\#\left\{ k\in \{1,\dots, N_A\}\,|\,s_{k}\neq{s}'_{k}\right\} $.
Eq. (\ref{eq:ovl}) provides a direct experimental verification of the fidelity of the reconstructed density matrix, requiring no further theory input such as simulations, and can be evaluated from the same type of randomised measurements employed for EHT. Importantly, the measurements used for fidelity estimation should be independent from those used in EHT, to avoid false correlations and biasing. We note also that in principle more advanced measurement schemes exist, performing importance sampling on (\ref{eq:ovl}), thereby dramatically reducing the number of measurements required \cite{Flammia2011,da_Silva_2011}.
\section{ Operator content of the EHT ansatz for quench experiments}\label{Methods:K}
In order to achieve good fidelities for EHT on quench experiments (see Fig.~\ref{fig:Exp}), the EHT ansatz $\tilde{H}_A(\boldsymbol{g})$ needs to be amended with additional operators $\mathcal{K}_A$ whose coefficients provide additional free fit parameters. CFT suggest lattice momenta, obtained from commutators of various terms of $H_A$, of the form
\begin{align}
\mathcal{K}_A^{(1)} &= \sum_{k<l \in A} \tilde{J}^{XY}_{kl}(\sigma^x_k\sigma^y_l - \sigma^y_k\sigma^x_l), \\
\mathcal{K}^{(2)}_A &= \sum_{k<l}\sum_{m \neq k,l} \tilde{J}^{XYZ}_{klm}(\sigma^x_k\sigma^y_l\sigma^z_m - \sigma^y_k\sigma^x_l\sigma^z_m).
\end{align}
Further improvements of the fidelity are obtained by including magnetization conserving operators of the form
\begin{align}
\mathcal{K}_A^{(3)} &= \sum_{k<l\in A} \tilde{J}_{kl}^{ZZ} \sigma^z_k\sigma^z_l + \! \! \sum_{k<l<m\in A}\! \! \! \tilde{J}^{ZZZ}_{klm} \sigma^z_k\sigma^z_l\sigma^z_m, \\
\mathcal{K}^{(4)}_A &= \sum_{k<l}\sum_{m \neq k,l} \tilde{J}^{XXZ}_{klm}(\sigma^x_k\sigma^x_l\sigma^z_m + \sigma^y_k\sigma^y_l\sigma^z_m).
\end{align}
\section{ Quantum State Tomography}
\label{Methods:QST}
For comparison with EHT, we consider in the main text two further well-known tomographical methods, Low Rank Least Squares (LRLS) and Projected Least Squares (PLS). Similar to EHT, these methods attempt to construct a density matrix estimator, $ {\rho}(X)$, depending on parameters $X$, by matching predictions of the estimator in a least squares sense with experimental observations from \textit{local} measurements performed on independent single copies of the density matrix. Here, the same cost function defined in Eq.~\ref{eq:method} is minimized, replacing $\tilde{\boldsymbol{g}}$ with $X$, with the only difference between LRLS and PLS being the form of the ansatz $\rho(X)$.
\textit{Low-Rank Least Squares (LRLS)} \cite{Riofrio2017} takes as an ansatz $ \rho(X) = X^\dagger X$, i.e. matrices that are by construction positive semidefinite. Since the true density matrix is in practice often not full rank, $X$ can be a rectangular (complex valued) $r \times d$ matrix, with $r$ the rank of the density matrix estimator. The total measurement effort needed to obtain a fixed precision scales proportional to the number of unknown variables, i.e. the $r \times d$ entries of $X$. This method comes with a substantial computational overhead associated with determining the $r \times d$ unknown variables.
\textit{Projected Least Squares (PLS)} \cite{Sugiyama_2013, Guta20} consists of first finding the Hermitian, but not necessarily positive semidefinite, matrix that would produce the observations exactly. This can be done analytically, thereby eliminating the computational overhead of LRLS. For a given set of measurement results in the computational basis, obtained after applying basis transformations U, the resulting matrix is given by
\begin{align}\label{EqRhoRT}
{\rho}_{RT} = \sum_{\mathbf{s}, \mathbf{s}'} \sum_{ {U}}P_U(\mathbf{s}) (-2)^{-\mathcal{D}[\mathbf{s}, \mathbf{s}']} {U} \ket{\mathbf{s}'} \bra{\mathbf{s}'} {U},
\end{align}
with $\mathcal{D}[\mathbf{s},\mathbf{s}']$ the Hamming distance defined in Eq.~(\ref{eq:ovl}).
For the randomised local unitary transformations $ {U}$ considered here, the estimator $ {\rho}_{RT}$ is the one obtained in randomised tomography \cite{Elben19PRA} and shadow tomography \cite{Huang20}.
The matrix (\ref{EqRhoRT}) is generally not positive semi-definite, and hence does not represent a physical density matrix. However, the matrix can be projected onto the space of positive semi-definite matrices via a simple procedure, by rescaling the eigenvalues and truncating them to positive values \cite{Smolin12}.
Again, the scaling of the number of measurements needed to obtain a certain fidelity is exponential in the system size and proportional to $r \times d$, where $r$ is a measure of the effective rank of the reduced density matrix.
\section{Conformal Field Theory \& Entanglement Hamiltonian} \label{Methods:CFT}
Conformal field theory (CFT) in $(1+1)$ dimensions provides us with explicit expression for entanglement (modular) Hamiltonians \cite{Cardy2016,Wen2018}. For connected subsystems $A$, it has been shown that for a broad range of CFT states, including ground states, thermal states and states generated via quench dynamics, the entanglement Hamiltonian density $H_E(t,x)$ can be written in terms of the energy momentum tensor $T(x,t)$ multiplied with a local weight factors \cite{Cardy2016,Wen2018}. This suggests ansätze for the EH also for the lattice models considered in this work, which, complemented with additional few body terms, provide an accurate description of the EH, even beyond the regime of applicability of a CFT description (see e.g.~Fig.~1).
In the following, we briefly summarize CFT predictions for entanglement Hamiltonians. We take $A=[0,l]$ to be a partition at the end of a system $S=[0,L]$, with $L \gg l$ much larger than any length scale in the system, such that the right boundary $L$ can be neglected. We consider a quantum quench from an initial state $|\psi_0\rangle \sim e^{-(\beta_0/4) H_{\text{CFT}}}|b\rangle$ where $b\rangle$ is a conformally invariant boundary state and $\beta_0 > 0$ introduces a finite correlation length $\ell \sim \beta_0$ \cite{Calabrese2016,Cardy2016,Wen2018}. It has been shown that in this setting, $|\psi_0\rangle$ represents generic ground states of Hamiltonians $H_0$ with inverse mass gap $m_0^{-1}\sim \beta_0$ \cite{Calabrese2016}.
For this initial state, the entanglement Hamiltonian
\begin{align}
e^{-H_E(t)} = \mathrm{Tr}_A\left[ e^{-i H_{\textrm{CFT}}t} \ket{\psi_0}\bra{\psi_0 }e^{i H_{\textrm{CFT}}t} \right]
\end{align}
can be calculated for all times $t$ exactly via a conformal mapping to an annulus. While we refer for the general result to Refs.~\cite{Cardy2016,Wen2018}, we consider here illustrative, limiting cases.
\paragraph{Ground state with long-range correlations}
For $\beta_0 \gg l$, $|\psi_0\rangle$ is locally indistinguishable from the ground state of the CFT $ \ket{ \psi_{\textrm{GS}}}$. The partition size $l$ is the only remaining length-scale and
the entanglement Hamiltonian is given by the Hamiltonian density $\mathcal{H}(x)$ modified with a parabolic weight factor \cite{hislop1982,Cardy2016,Wen2018,casini2011}
\begin{align}
\left. H_E(0) \right|_{\beta_0 \gg l} \simeq \int_0^l \! \text{d}x \, \frac{l^2-x^2}{2l} \mathcal{H}(x) \; .
\end{align}
We note that this is a direct CFT generalization of the Bisognano-Wichmann theorem to describe the entanglement Hamiltonian in a finite interval $A=[0,l]$ embedded in a semi-infinite system \cite{Cardy2016,Wen2018}. As expected, close to the entanglement cut at $x=l$, the local weight factor increases linearly $\sim (l-x)$.
\paragraph{Ground state with short-range correlations}
For $\beta_0 \ll l$, $|\psi_0\rangle$ represents the groundstate of a generic many-body Hamiltonian with (short) correlation length $\beta_0$ \cite{Calabrese2016,Cardy2016,Wen2018}. The entanglement Hamiltonian is given by \cite{Cardy2016,Wen2018}
\begin{align}
\left. H_E(0)\right|_{\beta_0 \ll l } \simeq {\beta_0} \int_0^l \sinh\left[ \frac{2\pi}{\beta_0}(l-x)\right] \mathcal{H}(x) \;.
\end{align}
For a short range entangled state, the dominant contribution to the Schmidt spectrum (i.e.\ to the entanglement between $A$ and the remainder of the system) is expected to arise from regions close to the entanglement cut $(l-x) \ll \beta_0$. Here, the hyperbolic weight factor can be expanded to linear order $\sim (l-x)$, as expected from the Bisognano-Wichmann theorem. Contributions from far from the boundary, where the entanglement Hamiltonian density is large, are exponentially suppressed.
\paragraph{Quantum quench}
For general $t>0$, the entanglement Hamiltonian receives contributions from energy $T_{00}(x)=\mathcal{H}(x)$ and momentum $T_{01}(x)$ density \cite{Cardy2016,Wen2018}. The latter can be interpreted as emerging quasi-particle currents spreading entanglement through the system \cite{Zhu2020}.
At long times $t \gg l \gg \beta_0 $, when the system approaches thermal equilibrium, the expressions considerably simplify. The contributions from the momentum density vanish and the entanglement Hamiltonian $
H_E(t\rightarrow \infty) \simeq \int_0^l \! \text{d}x \, \beta(x) \, \mathcal{H}(x)
$ is determined by \cite{Cardy2016,Wen2018}
\begin{align}
\beta(x) = 2 \beta_0 \, \frac{\sinh\left[ \pi (l-x) / \beta_0 \right]\sinh\left[ \pi (l+x) / \beta_0 \right]}{\sinh\left[ 2 \pi l / \beta_0 \right]}
\end{align}
Remarkably, $ H_E(t\rightarrow \infty) $ equals \textit{exactly} to the entanglement Hamiltonian in the thermal state \cite{Cardy2016,Wen2018}
\begin{align}\exp[-H_E(t\rightarrow \infty)]= \mathrm{Tr}_A\left[ \exp(-\beta_0 H_{\text{CFT}}) \right] .
\end{align}
Thus, CFT provides an explicit demonstration of the eigenstate thermalization hypothesis \cite{Deutsch1991,Srednicki1994,Rigol2008,Garrison2018}. The weight factor $\beta(x)$ can be interpreted as a spatially varying (inverse) temperature. Close to the edge $x\sim l$, $\beta(x)^{-1}$
is largest as entropy is generated at the entanglement cut. With the distance from the cut, the temperature decreases, and saturates to the thermal value $\beta(0)^{-1}=\beta_0^{-1}$.
\bibliographystyle{apsrev4-1}
\input{main.bbl}
\end{document}
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2024-02-18T23:40:24.860Z
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2020-09-22T02:00:39.000Z
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{"paloma_paragraphs":[]}
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{"arxiv_id":"2009.09000","language":"en","timestamp":1600740039000,"url":"https:\/\/arxiv.org\/abs\/2009.09000","yymm":"2009"}
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proofpile-arXiv_000-10102
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{"provenance":"002.jsonl.gz:10103"}
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\section{Introduction}
The Loop Current System, namely, the Loop Current itself and the
anticyclonic (counterclockwise) mesoscale (100--200-km radius)
vortices, so-called rings, shed from it, strongly influences the
circulation, thermodynamics, and biogeochemistry of the Gulf of
Mexico (GoM).\citep{Sturges-05} As important long-range carriers,
westward-propagating Loop Current rings (LCRs) provide a potential
mechanism for the remote connectivity between the GoM's western
basin and the Caribbean Sea.\citep{Lewis-Kirwan-87, Forristall-etal-92,
Kuznetsov-etal-02, Lipphardt-etal-08, Andrade-etal-13, Donohue-etal-16,
Beron-etal-18} In particular, bringing warm Caribbean Sea water
within, the heat content of the LCRs is believed to be as significant
as for LCRs to promote the intensification of tropical cyclones
(hurricanes).\citep{Shay-etal-00} On the other hand, as regions of
strong flow shear, LCRs may be capable of producing structural
damage on offshore oil drilling rigs.\citep{Kantha-14} For all these
reasons, LCRs are routinely monitored.\citep{Sturges-Leben-00}
LCRs leave footprints in satellite altimetric sea-surface height
(SSH) maps,\citep{Leben-05} so sharp that the routine detection of
LCRs consists in the identification of regions filled with closed
streamlines of the SSH field assuming a geostrophic balance, a
practice widely followed in oceanography.\citep{Chelton-etal-11a}
However, this eddy detection approach uses instantaneous Eulerian
information to reach long-term conclusions about fluid (i.e.,
Lagrangian) transport, which are invariably surrounded by uncertainty
due to the unsteady nature of the underlying flow. At the heart
of the issue with Eulerian eddy diagnostics of this type is their
dependence on the observer viewpoint:\citep{Peacock-etal-15} they
give different results for observers that rotate and translate
relative to one another. The issue is most easily grasped by
bringing up, one more time as we believe is central yet widely
overlooked, the example first discussed by \citet{Haller-05} and
thereafter by others.\citep{Beron-etal-13, Haller-Beron-13,
Haller-etal-16, Haller-15} Consider the exact solution to the
Navier--Stokes equation in two spatial dimensions: $v(x,t) =
(x^1\sin4t + x^2(2+\cos4t), x^1(\cos4t-2) - x^2\sin4t)$, where $x
= (x^1,x^2)\in \mathbb R^2$ denotes position and $t\in\mathbb R$
is time. The flow streamlines are closed at all times suggesting
an elliptic structure (i.e., a vortex). However, this flow actually
hides a rotating saddle (pure deformation), as it follows by making
$(x^1,x^2)\mapsto (\bar x^1,\bar x^2) = (x^1\cos2t - x^2\sin2t,
x^1\sin2t + x^2\cos2t)$, under which $v(x,t) \mapsto (\bar x^2,\bar
x^1) \equiv \bar v(\bar x)$. In other words, the de-facto oceanographic
eddy detection diagnostic \citep{Chelton-etal-11a} misclassifies
the flow as vortex-like. The $\bar x$-frame is special inasmuch the
flow in this frame is steady, and thus flow streamlines and fluid
trajectories coincide. Hence short-term exposition pictures of the
velocity field by the observer in the $\bar x$-frame determine the
long-term fate of fluid particles. The only additional observation
to have in mind to fully determine the Lagrangian motion is that
the observer in the $\bar x$-frame rotates (at angular speed $2$).
This tells us that the flow under consideration is not actually
unsteady as there is a frame ($\bar x$) in which it is steady. In
a truly unsteady flow there is no such distinguished observer for
whom the flow is steady.\citep{Lugt-79} Thus one can never be sure
which observer gives the right answer when the de-facto oceanographic
eddy detection diagnostic \citep{Leben-05, Chelton-etal-11a} is
applied. As a consequence, neither false positives nor false
negatives can be ruled out,\citep{Beron-etal-13} and thus the
significance of life expectancy estimates is unclear.
Our goal here is to carry out objective (i.e, observer-independent)
assessments of the life cycle of LCRs. This will be done in line
with recent but growing work that makes systematic use of geometric
tools from nonlinear dynamics to frame vortices objectively.
\citep{Haller-Beron-12, Beron-etal-13, Haller-Beron-13, Haller-Beron-14,
Karrasch-etal-14, Haller-etal-16, Beron-etal-19-PNAS, Haller-15,
Haller-16, Serra-Haller-16, Hadjighasem-etal-17, Haller-etal-18,
Haller-etal-19} We will specifically apply three methods which
define \emph{coherent Lagrangian vortex} boundaries as material
loops that (i) defy stretching,\citep{Haller-Beron-13, Haller-Beron-14}
(ii) resist diffusion,\citep{Haller-etal-18} and (iii) whose elements
rotate evenly,\citep{Haller-etal-16} respectively. The fluid enclosed by such
loops can be transported for long distances without noticeable
dispersion.\citep{Wang-etal-15, Wang-etal-16}
The rest of the paper is organized as follows. In the next section
we briefly review the formal definition of each of the above coherent
Lagrangian vortex notions. In Sec.\ \ref{sec:gen} we present a
technology that enables framing vortex genesis and apocalypse
robustly and with precision. Section \ref{sec:dat} presents the
data (satellite altimetry) on which our assessments of the life
cycle of LCRs are applied. It also presents numerical details of
the implementation of the vortex detection methods, and introduces
the databases which deliver Eulerian assessments of the ``birth''
and ``decease'' dates of LCRs, which are used for reference. The
results of our study are presented in Section \ref{sec:res}. Finally,
concluding remarks are offered in Section \ref{sec:con}.
\section{Coherent Lagrangian ring detection}\label{sec:met}
Consider
\begin{equation}
F_{t_0}^t : x_0 \mapsto x(t;x_0,t_0),
\end{equation}
the flow map resulting from integrating a two-dimensional incompressible
velocity field, namely, $v(x,t) = \nabla^\perp\psi(x,t)$ ($x\in\mathbb
R^2$ and $t\in\mathbb R$ as stated above), where $\psi$ denotes
sea-surface height. If the pressure gradient force is exclusively
due to changes in the SSH field, the latter is given by $g^{-1}f\psi(x,t)$,
where $g$ is gravity and $f$ is the Coriolis parameter, assuming a
quasigeostrophic balance.
\subsection{Null-geodesic (NG) rings}
Following \citet{Haller-Beron-13, Haller-Beron-14} we aim to identify
fluid regions enclosed by exceptional material loops that \emph{defy
the typical exponential stretching experienced by generic material
loops in turbulent flows}. This is achieved by detecting loops with
small annular neighborhoods exhibiting no leading-order variation
in averaged material stretching.
These considerations lead to a variational problem whose solutions
are loops such that any of their subsets are stretched by the same
factor $\lambda>0$ under advection by the flow from $t_0$ to $t_0
+ T$ for some $T$. The time-$t_0$ positions of such uniformly
$\lambda$-stretching material loops turn out to be limit cycles of
one of the following two bidirectional vector or \emph{line} fields:
\begin{widetext}
\begin{equation}
\eta_\lambda^\pm(x_0) \coloneqq
\sqrt{
\frac
{\lambda_2(x_0) - \lambda^2}
{\lambda_2(x_0) - \lambda_1(x_0)}
}
\,\xi_1(x_0)
\pm
\sqrt{
\frac
{\lambda^2 - \lambda_1(x_0)}
{\lambda_2(x_0) - \lambda_1(x_0)}
}
\,\xi_2(x_0),
\label{eq:eta}
\end{equation}
\end{widetext}
where $\lambda_1(x_0) < \lambda^2 < \lambda_2(x_0)$. Here,
$\{\lambda_i(x_0)\}$ and $\{\xi_i(x_0)\}$ satisfying
\begin{equation}
0 < \lambda_1(x_0) \equiv
\frac{1}{\lambda_2(x_0)} < 1,\quad
\langle\xi_i(x_0),\xi_j(x_0)\rangle = \delta_{ij},
\label{eq:eig}
\end{equation}
$i,j = 1,2$, are eigenvalues and (orientationless) normalized
eigenvectors, respectively, of the Cauchy--Green (strain) tensor,
\begin{equation}
\C_{t_0}^{t_0+T}(x_0) \coloneqq \D{F}_{t_0}^{t_0+T}(x_0)^\top
\D{F}_{t_0}^{t_0+T}(x_0).
\label{eq:C}
\end{equation}
The tensor field $\C_{t_0}^{t_0+T}(x_0)$ objectively measures
material deformation over the time interval $[t_0,t_0+T]$. Limit
cycles of \eqref{eq:eta} or \emph{$\lambda$-loops} either grow or
shrink under changes in $\lambda$, forming smooth annular regions
of non-intersecting loops. The outermost member of such a band of
material loops is observed physically as the boundary of a
\emph{coherent Lagrangian ring}. The $\lambda$-loops can also be
interpreted as so-called null-geodesics of the indefinite tensor
field $\C_{t_0}^{t_0+T}(x_0)-\lambda\Id$, which is why we also refer
to them as \emph{null-geodesic} (or \emph{NG}) \emph{rings}.
\subsection{Diffusion-barrier (DB) rings}
Another recent approach to coherent vortices in geophysical flows
has been put forward in \citet{Haller-etal-18}. In this case one
aims at identifying fluid regions that \emph{defy diffusive transport
across their boundaries}. Note that by flow invariance, any fluid
region has vanishing advective transport across its boundary. In
turbulent flows, however, a generic fluid region has massive
\emph{diffusive} leakage through its boundary, which correlates
with the typical exponential stretching of the latter.
A technical challenge is that the diffusive flux of a virtual
diffusive tracer through a material surface over a finite time
interval $[t_0,t_0+T]$ depends on the concrete evolution of the
scalar under the advection--diffusion equation. In the limit of
vanishing diffusion, however, \citet{Haller-etal-18} show that the
diffusive flux through a material surface can be determined by the
gradient of the tracer at the initial time instance and a tensor
field $\T$ that can be interpreted as the time average of the
diffusion tensor field along a fluid trajectory. In the case of
isotropic diffusion, this reduces to the average of inverse
Cauchy--Green tensors,
\begin{equation}\label{eq:transporttensor}
\T(x_0) \coloneqq \frac{1}{T}\int_{t_0}^{t_0+T}
\left(\C_{t_0}^{t_0+t}(x_0)\right)^{-1}\d{t}.
\end{equation}
Searching for material loops with small annular neighborhoods
exhibiting no leading-order variation in the vanishing-diffusivity
approximation of diffusive transport leads to a variational problem
whose solutions are limit cycles of \eqref{eq:eta}, where now
$\lambda_i$ and $\xi_i$ are, respectively, eigenvalues and eigenvectors
of the time-averaged Cauchy--Green tensor
\begin{equation}\label{eq:avCG}
\bar{\C}_{t_0}^{t_0+T}(x_0) \coloneqq
\frac{1}{T}\int_{t_0}^{t_0+T} \C_{t_0}^{t_0+t}(x_0)\d{t}.
\end{equation}
This simple tensor structure assumes isotropic diffusion and an
incompressible fluid flow. We refer to vortices obtained by this
methodology as \emph{diffusion-barrier} (or \emph{DB}) \emph{rings}.
Due to the mathematical similarity to the geodesic ring approach,
we may use the same computational method as for NG rings, simply
by replacing $\C_{t_0}^{t_0+T}$ by $\bar{\C}_{t_0}^{t_0+T}$.
\subsection{Rotationally-coherent (RC) rings}
In our analysis, we also employ a third methodology, which was
developed by \citet{Haller-etal-16}. It puts less emphasis on
specific properties of the boundary (like stretching or diffusive
flux) of coherent vortices, but highlights that coherent vortices
are often associated with \emph{concentrated regions of high
vorticity}. Defining vortices in terms of vorticity has a long
tradition,\citep{Okubo-70, Weiss-91} but in unsteady fluid flows
it comes with a number of drawbacks, one of which is the lack of
objectivity.\citep{Haller-15} In \citet{Haller-etal-16}, the authors
overcome these challenges by showing that the \emph{Lagrangian
averaged vorticity deviation} (or \emph{LAVD}) field
\begin{equation}
\mathrm{LAVD}_{t_0}^{t_0+T}(x_0) \coloneqq \int_{t_0}^{t_0+T}\left\lvert
\omega\big(F_{t_0}^t(x_0),t\big)-\bar{\omega}(t)\right\rvert\d{t},
\end{equation}
is an objective scalar field. Here, $\omega(x,t)$ is the vorticity
of the fluid velocity at position $x$ and time $t$, and $\bar{\omega}(t)$
is the vorticity at time $t$ averaged over the tracked fluid bulk.
In this framework, vortex centers are identified as maxima of the
LAVD field, and vortex boundaries as outermost convex LAVD-level
curves surrounding LAVD maxima. Because loops are composed of fluid
elements that complete the same total material rotation relative
to the mean material rotation of the whole fluid mass, we will refer
to the vortices as \emph{rotationally-coherent} (or \emph{RC})
\emph{rings}. In practice, the convexity requirement is relaxed,
using a ``tolerable'' convexity deficiency.\citep{Haller-etal-16}
In contrast to the two previously described methods, the LAVD
approach therefore does not address vortex boundaries directly (say,
via a variational approach), but deduces them as level-set features
of the objective LAVD field.
\section{Genesis and apocalypse}\label{sec:gen}
Our main goal is to study genesis, evolution, and apocalypse of
LCRs from an objective, Lagrangian point of view. Since there is
no generally agreed definition of the concept of a coherent vortex,
we need to employ several proposed methods to rule out the possibility
that the results are biased by the specific choice of method.
To determine the ``birth'' or the ``decease'' of a coherent Lagrangian
vortex in a robust fashion, we need to eliminate a couple of
potentially biasing issues. First, as stated above, we include
several Lagrangian methodologies in our study. Second, we want to
avoid potential sensitivities due to implementation details (such
as algorithm or parameter choices). Recall that Lagrangian approaches
choose not only an initial time instance $t_0$, but also a flow
horizon $T$. A naive approach to the determination of the decease
of a coherent vortex would be to simply take the maximum of $t_0+T$
for which a Lagrangian method detects a coherent vortex, where $t_0$
and $T$ are taken from a range of reasonable values. While this
approach yields a definite answer, it may be totally inconsistent
with other computations run for different choices of $t_0$ and $T$.
For instance, if a Lagrangian computation detects a coherent vortex
over the time interval $[t_0, t_0+T]$, it should also detect a
vortex over the time interval $[t_0+\delta t, (t_0+\delta t) +
(T-\delta t)] = [t_0+\delta t, t_0+T]$ for small $|\delta t|$, if
$t_0+T$ was really the date of breakdown.
In order to make our predictions statistically more robust and prove
internal consistency, we employ the following approach. First, we
run Lagrangian simulations on a temporal double grid as follows.
We roll the initial time instance $t_0$ over a time window roughly
covering the time interval of vortex existence, which we seek to
determine. For each $t_0$, we progress $T$ in 30-day steps as long
as the Lagrangian method successfully detects a coherent vortex.
Thus, we obtain for each $t_0$ a \emph{life expectancy} $T_{\max}(t_0)$,
which is the maximum $T$ for which a Lagrangian simulation starting
at $t_0$ successfully detected a coherent vortex.
Ideally, we would like to see the following $T_{\max}(t_0)$ pattern.
Assume a coherent Lagrangian vortex breaks down on day 200, counted
from day 0. Then for $t_0 = 0$ the longest successful vortex detection
should yield a $T_{\max}(0) = 180$ d. Similarly, for $t_0 = 5, 10,
15, 20$ we should get a $T_{\max}=180$ d. From $t_0=25$ on, however,
we should start seeing $T_{\max}$ dropping down to $150$ days,
because for $T=180$ days, the Lagrangian flow horizon reaches beyond
the vortex breakdown. As a consequence, we would like to see a
wedge-shaped $T_{\max}(t_0)$ distribution, which would indicate
that all Lagrangian coherence assessments predict the breakdown
consistently, though slightly smeared out regarding the exact date.
If encountered, such a consistent prediction of breakdown would
arguably remove the possibility of degenerate results. To summarize,
in an ideal case, a Lagrangian simulation of the lifespan of a
coherent vortex would therefore start with a large $T_{\max}$-value,
which consistently decreases as $t_0$ progresses forward in time.
It turns out that in many cases such wedge-shaped $T_{\max}(t_0)$-patterns
can be indeed observed, sometimes with astonishing clarity,
given the finitely resolved velocity fields and the complexity of
the Lagrangian calculation and vortex detection algorithms.
\section{Data and numerical implementation}\label{sec:dat}
The SSH field from which the flow is derived is given daily on a
0.25$^{\circ}$-resolution longitude--latitude grid. This represents
an absolute dynamic topography, i.e., the sum of a (steady) mean
dynamic topography and the (transient) altimetric SSH anomaly. The
mean dynamic topography is constructed from satellite altimetry
data, in-situ measurements, and a geoid model.\citep{Rio-Hernandez-04}
The SSH anomaly is referenced to a 20-yr (1993--2012) mean, obtained
from the combined processing of data collected by altimeters on the
constellation of available satellites.\citep{LeTraon-etal-98}
Computationally, we detect NG and DB rings from the altimetry-derived
flow by the method devised in \citet{Karrasch-etal-14} and recently
extended for large-scale computations in Karrasch and
Schilling,\citep{Karrasch-Schilling-20} as implemented in the package
\texttt{CoherentStructures.jl}. It is written in the modern
programming language \texttt{Julia}, and is freely available from
\texttt{https://\allowbreak github.com/\allowbreak
CoherentStructures/\allowbreak CoherentStructures.jl}. In turn, RC
ring detection, computationally much more straightforward, was
implemented in \texttt{MATLAB}\textsuperscript{\ttfamily\textregistered}
as described in \citet{Beron-etal-19-PNAS} (a software tool, not
employed here, is freely distributed from \texttt{https://\allowbreak
github.com/\allowbreak LCSETH/\allowbreak Lagrangian-\allowbreak
Averaged-\allowbreak Vorticity-\allowbreak Deviation-\allowbreak
LAVD}). The spacing of the grid of initial trajectory positions
in all cases is set to 0.1 km as in earlier Lagrangian coherence
analyses involving altimetry data. \citep{Beron-etal-13,
Olascoaga-etal-13, Beron-etal-18} Trajectory integration is
carried out using adaptive time-stepping schemes and involves
cubic interpolation of the velocity field data. NG and DB rings
are sought with stretching parameter ($\lambda$) ranging over the interval
$\lambda \in [1 \pm 0.5]$. Recall that $\lambda = 1$ NG-vortices
reassume their arc length at $t_0+T$.\citep{Haller-Beron-13} When
the flow is incompressible (as is the case of the altimetry-derived
flow) such $\lambda = 1$ vortices stand out as the most coherent
of all as their boundaries resist stretching while preserving the
area they enclose. Following \citet{Haller-etal-16} the convexity
deficiency is set to $10^{-3}$ for the RC ring extractions.
As our interest is in LCRs, we concentrate on the time intervals
on which these were identified by Horizon Marine, Inc.\ as part of the \texttt{EddyWatch}\textsuperscript{\ttfamily\textregistered}\, program.
This program identifies LCRs as regions instantaneously filled with
altimetric SSH streamlines.\citep{Leben-05} The \texttt{EddyWatch}\textsuperscript{\ttfamily\textregistered}\, program has
been naming LCRs and reporting their birth and decease dates since
1984. Our analysis is restricted to the period 2001--2013, long
enough to robustly test theoretical expectations and for the results
to be useful in applications such as ocean circulation model
validation.
Alternative assessments of the genesis and apocalypse of LCRs are
obtained from the \texttt{AVISO+}\, Mesoscale Eddy Trajectory Atlas Product,
which is also computed from the Eulerian footprints left by the
eddies on the global altimetric SSH field.\citep{Chelton-etal-11a}
\section{Results}\label{sec:res}
We begin by testing our expectation that Lagrangian life expectancy
($T_{\max}$) should decrease with increasing screening time ($t_0$),
exhibiting a wedge shape. We do this by focusing on LCR \emph{Kraken},
so named by \texttt{EddyWatch}\textsuperscript{\ttfamily\textregistered}\, and recently subjected to a Lagrangian coherence
study.\citep{Beron-etal-18} In that study the authors characterized
\emph{Kraken} as an NG ring using altimetry data. Furthermore,
they presented support for their characterization by analyzing
independent data, namely, satellite-derived color (Chl concentration)
and trajectories from satellite-tracked drifting buoys. This rules
out the possibility that LCR \emph{Kraken} is an artifact of the
satellite altimetric dataset, thereby constituting a solid benchmark
for testing our expectation. The authors of the aforementioned
study estimated a Lagrangian lifetime for \emph{Kraken} of about
200 d, but framing the genesis and apocalypse of the ring with
precision was beyond the scope of their work.
\begin{figure*}[t!]
\centering%
\includegraphics[width=\textwidth]{lcrgen-fig01} \caption{(top
panel) For Gulf of Mexico's Loop Current ring (LCR) \emph{Kraken},
life expectancy as a function of screening time according
null-geodesic (NG), diffusion-barrier (DB), and rotationally-coherent
(RC) Lagrangian vortex assessments. Indicated are the birth date
of the ring (b), and three decease date estimates (d1, d2, and
d3); cf.\ text for details. Birth and decease dates according
to \texttt{EddyWatch}\textsuperscript{\ttfamily\textregistered}\, and \texttt{AVISO+}\, Eulerian vortex assessments are indicated with
open and filled triangles, respectively. (bottom panel) Based on
the NG assessment, LCR \emph{Kraken} on birth date and the three
decease date estimates.}
\label{fig:kra-fwd}%
\end{figure*}
The top panel of Fig.\ \ref{fig:kra-fwd} shows $T_{\max}(t_0)$ for
\emph{Kraken} based on NG (red), DB (solid black), and RC (dashed
black) coherence assessments. First note that the NG and DB
assessments are largely consistent, producing a wide-base $T_{\max}(t_0)$
wedge with height decreasing with increasing $t_0$ in addition to
a short, less well-defined wedge prior to it. This becomes very
evident when compared to the RC assessment, which produces intermittent
wedge-like $T_{\max}(t_0)$ on various short $t_0$-intervals. We
have observed that this intermittency is typical, rather than
exceptional, for the RC assessment. Thus we consider NG coherent
Lagrangian vortex detection, which in general produces nearly
identical results as DB vortex detection, in the genesis and
apocalypse assessments that follow.
Indicated in the top panel of Fig.\ \ref{fig:kra-fwd} (with a
vertical dashed line) is our estimate of the birth date of LCR
\emph{Kraken}, $t_0=$ 18/May/2013, and three estimates of its decease
date, to wit, $t_0=$ 12/Jan/2014, 21/Feb/2014, and 21/Apr/2014. The
birth date corresponds to the $t_0$ marking the leftmost end of the
$T_{\max}(t_0)$ wedge with the longest base (highlighted). Our
first decease date estimate (d1) is given by the birth date plus its
life expectancy, set by the height of the wedge or 12/Jan/2014 $-$
18/May/2013 $=239$ d. The second decease date estimate (d2) is given
by the $t_0$ marking the rightmost end of the wedge, which is 40-d
longer than its life expectancy. Our third decease date estimate (d3)
is given by the second decease date estimate plus the height of the
wedge at its rightmost end, namely, 21/Apr/2014 $-$ 21/Feb/2014
$=59$ d.
The bottom panel of the Fig.\ \ref{fig:kra-fwd} shows, in orange,
LCR \emph{Kraken} on its estimated birth date (b), and its advected
image under the altimetry-derived flow on the first (d1), second
(d2), and third (d3) decease date estimates. Overlaid on the later
on the second and third decease date estimated are (shown in cyan)
the NG vortex extracted on the second decease date estimate and its
advected image on the third decease date estimate. Note that on
the first and second decease date estimates \emph{Kraken} does not
show any noticeable signs of outward filamentation. On the third
decease date estimate most of the original fluid mass enclosed by
the ring boundary exhibits a coherent aspect. Evidently, the first
and second decease date estimates are too conservative, so it is
reasonable to take the third one as the most meaningful decease
date estimate of the three. We will refer to it as \emph{the}
decease date.
Indicated by open and filled triangles in the abscissa of the
$T_{\max}(t_0)$ plot in the top panel of the Fig.\ \ref{fig:kra-fwd}
are the Eulerian assessment of birth and decease dates of \emph{Kraken}
by \texttt{EddyWatch}\textsuperscript{\ttfamily\textregistered}, and \texttt{AVISO+}, respectively. \texttt{EddyWatch}\textsuperscript{\ttfamily\textregistered}\, overestimates the decease date
by about 180 d, while \texttt{AVISO+}, underestimates it somewhat, by 19 d. To
evaluate the performance of Eulerian vortex detection in assessing
the birth date of \emph{Kraken} an additional analysis is needed.
\begin{figure*}[t!]
\centering%
\includegraphics[width=\textwidth]{lcrgen-fig02} \caption{As in
Fig.\ \ref{fig:kra-fwd}, but for assessments made in backward
time.}
\label{fig:kra-bwd}%
\end{figure*}
The results from such an analysis are presented in Fig.\
\ref{fig:kra-bwd}, which shows the same as in Fig.\ \ref{fig:kra-bwd}
but as obtained from applying all the Lagrangian vortex detection
methods \emph{backward} in time, i.e., with $T<0$, around the
\emph{Kraken}'s decease date. The top panel of the figure shows
(now) $|T_{\max}|$ as a function of screening time $t_0$. Note
that the NG and DB coherence assessments produce single wide-base
$|T_{\max}|(t_0)$ wedges with height decreasing with decreasing
$t_0$, nearly indistinguishable from one another. As in the
forward-time analysis, the RC assessment shows intermittent wedge-like
$|T_{\max}|(t_0)$ on various short $t_0$-intervals, suggesting a
much shorter life expectancy than observed in reality. Thus we turn
our attention to the NG (or DB) assessment. This produces a
backward-time birth date estimate on $t_0=$ 02/Apr/2014, and three
backward-time decease date estimates on 06/Apr/203, 08/Apr/2013,
and 10/Mar/2013. In forward time, 02/Apr/2014 represents a decease
date estimate, which is only 19-d earlier than the decease date
obtained above from forward-time computation. The largest discrepancy
between forward- and backward-time assessments are seen for the
birth date. Following the forward-time computation reasoning above,
the backward-time computations sets it 296 d earlier, on $t_0=$
10/Mar/2013. This lies 21 and about 30 d later and earlier than to
the \texttt{AVISO+}\, and \texttt{EddyWatch}\textsuperscript{\ttfamily\textregistered}, assessments, respectively, which are instantaneous,
i.e., they do not depend on the time direction on which they are
made.
The backward-time estimate of \emph{Kraken}'s decease date can be
taken to represent a forward-time \emph{conception} date estimate
for the ring. This is quite evident from the inspection of the
bottom panel of Fig.\ \ref{fig:kra-bwd}, which shows (in orange)
LCR \emph{Kraken} as extracted from backward-time computation on
the backward-time birth date estimate, and images thereof under the
backward-time flow on the three backward-time decease date estimates.
On the last two decease date estimates, these are shown overlaid
on the ring extracted from backward-time computation on the second
backward-time decease date estimate and its backward-advected image
on the third backward-time decease date estimate, which represents,
as noted above, a conception date for \emph{Kraken}.
\begin{figure}[t!]
\centering%
\includegraphics[width=\columnwidth]{lcrgen-fig03}%
\caption{(left panel) LCR \emph{Kraken} on birth date overlaid
in orange on the forward-advected image of the ring extracted
from backward-time computation on 08/Apr/2013 (third backward-time
decease estimate). (right panel) Backward-advected image of the
fluid region indicated in cyan in the left panel.}
\label{fig:kra-gen}%
\end{figure}
Indeed, the fluid region indicated in cyan contains at all times
the fluid region indicated in orange. Thus the orange fluid is
composed of the same fluid as the cyan fluid. Furthermore, the
cyan fluid, which can be traced back into the Caribbean Sea, ends
up forming the fluid that forms \emph{Kraken} on its (forward-time)
birth date. This is illustrated in left panel of Fig.\ \ref{fig:kra-gen},
which shows \emph{Kraken} (in orange) as obtained from forward-time
computation on its birth date overlaid on the forward-advected image
of the cyan fluid. In the right panel we show a backward-advected
image of the cyan fluid that reveals its origin in the Caribbean
Sea. The supplementary material includes an animation (Mov.\ 1)
illustrating the full life cycle of LCR \emph{Kraken}.
\begin{figure*}[t!]
\centering%
\includegraphics[width=\textwidth]{lcrgen-fig04}%
\caption{As in Fig.\ \ref{fig:kra-fwd}, but for LCR \emph{Yankee}.}
\label{fig:yan}%
\end{figure*}
We note that the need of introducing the conception date estimate
could have been anticipated from the inspection of the forward-time
assessment. Note the short wedge-like $T(t_0)$ before the long-base
wedge in Fig.\ \ref{fig:kra-fwd} employed in assessing genesis and
apocalypse. In a way the presence of that short wedge-like $T(t_0)$
was already insinuating that coherence was building sometime before
the ring was declared born. Similar disconnected wedge-like $T(t_0)$
patterns may be observed past the main wedge, as can be seen in
Fig.\ \ref{fig:yan}, which shows the same as Fig.\ \ref{fig:kra-fwd}
but for LCR \emph{Yankee}. These wedge-like patterns, however, are
not signs of the ring's ``resurrection,'' but actually correspond
to vortex structures in general unrelated or only partly related
to the ring in question.
\begin{figure*}[t!]
\centering%
\includegraphics[width=\textwidth]{lcrgen-fig05} \caption{As in
Fig.\ \ref{fig:yan}, with a focus on the highlighted piece of the
$T_{\max}(t_0)$ plot.}
\label{fig:cri}%
\end{figure*}
We illustrate the above in Fig.\ \ref{fig:cri}. Note the appearance
of two short wedge-like patterns past the main $T(t_0)$ wedge. Let
us concentrate attention on the earliest of the two short wedges.
We infer a forward-time birth date is 30/Jun/2007, and two forward-time
decease dates on 18/Sep/2017 and 17/Oct/2017. The bottom panels
of the figure show how these characterize the life cycle of a vortex,
newly formed and composed only in part of LCR \emph{Yankee}'s fluid.
This is evident by comparing the position of the vortex on birth
date and first decease date estimates and their forward-advected
images with those of \emph{Yankee} as revealed on 13/Sep/2006. The
\texttt{EddyWatch}\textsuperscript{\ttfamily\textregistered}\, and \texttt{AVISO+}\, nonobjective Eulerian vortex assessments fail to frame
this, largely overestimating the decease date of \emph{Yankee}.
We compile in Table \ref{tab:cen} the objective Lagrangian estimates
of conception, birth, and decease dates for all LCRs during the
altimetry era. An entry of table left blank means that the ring
could not be classified as coherent. The objective estimates are
compared with nonobjective Eulerian estimates by \texttt{EddyWatch}\textsuperscript{\ttfamily\textregistered}\, and \texttt{AVISO+}, with
the former only providing the month of the year when birth and
decease take place. Note the tendency of the Eulerian assessments
to overestimate the birth and decease dates of the rings. Indeed,
the Eulerian assessments cannot distinguish between conception and
birth. They typically keep track of vortex-like structures past the
decease date of the rings, which, present around that date, are not
formed by the fluid mass contained by the rings. Moreover, the
Eulerian assessments classify as coherent, and even name, rings
that turn out not to be so. The supplementary material includes
two animations supporting these conclusions for features classified
as LCRs \emph{Quick} (Mov.\ 2) and \emph{Sargassum} (Mov.\ 3) by
the \texttt{EddyWatch}\textsuperscript{\ttfamily\textregistered}\, and \texttt{AVISO+}\, nonobjective Eulerian assessments.
\renewcommand{\arraystretch}{1.25}
\begin{table*}[t!]
\linespread{1}\selectfont{}
\centering
\begin{tabular}{ccccccccccc}%
\hline\hline%
&& && \multicolumn{3}{c}{Birth date} && \multicolumn{3}{c}{Decease date}\\
\cline{5-7}\cline{9-11}
\multicolumn{1}{l}{Ring}&&\multicolumn{1}{c}{Conception date}&&Objective&\texttt{EddyWatch}\textsuperscript{\ttfamily\textregistered}&\texttt{AVISO+}&&Objective&\texttt{EddyWatch}\textsuperscript{\ttfamily\textregistered}&\texttt{AVISO+}\\
\hline
\multicolumn{1}{l}{\emph{Nansen}}&&27/02/01&&13/03/01&04/01&18/03/00&&31/07/01&12/01&04/01/02\\
\multicolumn{1}{l}{\emph{Odesa}}&&02/07/01&&31/07/01&09/01&23/03/01&&09/10/01&12/01&05/11/01\\
\multicolumn{1}{l}{\emph{Pelagic}}&&&&&12/01&07/09/01&&&05/02&17/02/02\\
\multicolumn{1}{l}{\emph{Quick}}&&&&&03/02&19/02/02&&&04/03&05/05/03\\
\multicolumn{1}{l}{\emph{Sargasum}}&&&&&05/03&30/03/02&&&12/03&16/01/04\\
\multicolumn{1}{l}{\emph{Titanic}}&&10/11/03&&09/12/03&10/03&01/08/03&&21/06/04&10/04&13/11/04\\
\multicolumn{1}{l}{\emph{Ulises}}&&22/11/04&&11/12/04&05/04&07/12/03&&15/05/05&09/05&07/10/05\\
\multicolumn{1}{l}{\emph{Extreeme}}&&&&&03/06&13/01/06&&&09/06&13/11/06\\
\multicolumn{1}{l}{\emph{Yankee}}&&04/09/06&&13/09/06&07/06&26/04/06&&31/05/07&01/08&03/01/08\\
\multicolumn{1}{l}{\emph{Zorro}}&&10/03/07&&24/03/07&04/07&26/08/06&&16/08/07&08/07&17/08/07\\
\multicolumn{1}{l}{\emph{Albert}}&&02/11/07&&01/12/07&11/07&21/03/07&&05/03/08&05/08&23/04/08\\
\multicolumn{1}{l}{\emph{Cameron}}&&20/06/08&&20/06/08&07/08&15/06/08&&10/02/09&05/09&18/06/09\\
\multicolumn{1}{l}{\emph{Darwin}}&&29/01/09&&02/02/09&12/08&13/02/08&&25/10/09&11/09&10/11/09\\
\multicolumn{1}{l}{\emph{Ekman}}&&11/04/09&&09/07/09&07/09&04/02/09&&20/05/10&03/11&28/08/10\\
\multicolumn{1}{l}{\emph{Hadal}}&&22/06/11&&11/07/11&08/11&23/11/10&&23/12/11&03/12&26/12/11\\
\multicolumn{1}{l}{\emph{Icarus}}&&08/10/11&&22/10/11&11/11&19/07/11&&11/09/12&02/13&04/10/12\\
\multicolumn{1}{l}{\emph{Jumbo}}&&28/04/12&&28/04/12&06/12&18/04/12&&21/08/12&02/13&10/11/12\\
\multicolumn{1}{l}{\emph{Kraken}}&&10/03/13&&08/04/13&04/13&17/02/13&&02/04/14&10/14&10/04/14\\
\hline%
\end{tabular}%
\caption{Objective Lagrangian estimates of conception, birth, and
decease dates of Loop Current rings in the Gulf of Mexico identified
from satellite altimetry over 2001--2013 along with nonobjective
Eulerian estimates of birth and decease dates.} \label{tab:cen}
\end{table*}\renewcommand{\arraystretch}{1}
\begin{figure}[t!]
\centering%
\includegraphics[width=\columnwidth]{lcrgen-fig06}%
\caption{As a function of time, difference (in d) between objective
Lagrangian and nonobjective Eulerian estimates of birth (left
panel) and death (right panel) dates. Dots (resp., circles) involve
\texttt{EddyWatch}\textsuperscript{\ttfamily\textregistered}, (resp., \texttt{AVISO+}) assessments.} \label{fig:diff}
\end{figure}
We conclude by highlighting the disparities between the objective
Lagrangian and nonobjective Eulerian assessments of the genesis and
apocalypse of LCR rings in Fig.\ \ref{fig:diff}. The figure presents,
as a function of time over 2001--2011, the difference (in d) between
NG and E\, (dots), and NG and A\, (circles) assessments of birth
(left) and decease (right) dates. The differences can
be quite large (up to 1 yr!) with Eulerian assessments, which
in general underestimate the birth dates of the rings and overestimate
their decease dates.
\section{Conclusions}\label{sec:con}
We have carried out an objective (i.e., observer-independent)
Lagrangian assessment of the life cycle of the Loop Current rings
(LCRs) in Gulf of Mexico detected from satellite altimetry. Three
objective methods of coherent Lagrangian vortex detection were
considered here. These reveal material vortices with boundaries
that defy stretching or diffusion, and whose elements rotate evenly.
A modest technology advance was performed which enabled framing
vortex genesis and apocalypse with robustness and precision. We
found that the stretching- and diffusion-defying assessments produce
consistent results. These in general showed large discrepancies
with Eulerian assessments which identify vortices with regions
instantaneously filled with streamlines of the SSH field. The
Eulerian assessments were found incapable to distinguish conception
from birth of the rings. They also tended to track past their
decease dates vortex-like features unrelated to the rings in question.
The even-rotation assessment, which is vorticity-based, was found
to be quite unstable, suggesting life expectancies much shorter
than those produced by all other assessments. The inconsistency
found adds to the list of known issues of LAVD-based vortex statistics,
\cite{Tarshish-etal-18} including high sensitivity with respect to
the choice of computational parameter values. Our results can find
value in drawing unambiguous evaluations of material transport and
should represent a solid metric for ocean circulation model
benchmarking.
\section*{Supplementary material}
The supplementary material contains three animations. Movie 1
illustrates the complete life cycle of LCR \emph{Kraken} as assessed
objectively using NG-ring detection. Movies 2 and 3 show sequences
of advected images of the features classified as LCRs \emph{Quick}
and \emph{Sargassum}, respectively, by the \texttt{EddyWatch}\textsuperscript{\ttfamily\textregistered}\, and \texttt{AVISO+}\, nonobjective
Eulerian assessments.
\section*{Author's contributions}
All authors contributed equally to this work.
\begin{acknowledgments}
This work was initiated during the ``Escuela interdisciplinaria
de transporte en fluidos geofísicos: de los remolinos oce\'anicos
a los agujeros negros,'' Facultad de Ciencias Exactas y Naturales,
Universidad de Buenos Aires, 5--16/Dec/2016. Support from Centro
Latinoamericano de Formaci\'on Interdisciplinaria is sincerely
appreciated. This work was supported by CONACyT--SENER (Mexico)
under Grant No.\ 201441 (FAC, FJBV) as part of the Consorcio de
Investigaci\'on del Golfo de M\'exico (CIGoM). FAC thanks CICESE
(Mexico) for allowing him to use their computer facilities
throughout the CIGoM project.
\end{acknowledgments}
\section*{AIP Publishing data sharing policy}
The gridded multimission altimeter products were produced by
SSALTO/DUACS and distributed by \texttt{AVISO+}\, (\texttt{https://\allowbreak
www.aviso.altimetry.fr/}), with support from CNES. The Mesoscale
Eddy Trajectory Atlas Product was produced by SSALTO/DUACS and
distributed by \texttt{AVISO+}\, (\texttt{https://\allowbreak www.aviso.altimetry.fr/})
with support from CNES, in collaboration with Oregon State University
with support from NASA. The \texttt{EddyWatch}\textsuperscript{\ttfamily\textregistered}\, data are available from Horizon Marine, Inc.'s
website at \texttt{https://\allowbreak www.horizonmarine.com/}.
|
2024-02-18T23:40:24.862Z
|
2020-09-22T02:02:31.000Z
|
{"paloma_paragraphs":[]}
|
{"arxiv_id":"2009.09050","language":"en","timestamp":1600740151000,"url":"https:\/\/arxiv.org\/abs\/2009.09050","yymm":"2009"}
|
proofpile-arXiv_000-10103
|
{"provenance":"002.jsonl.gz:10104"}
| null | null |
\section{Introduction and scope of the study \label{intro}}
To model the propagation of large water waves in the ocean or the coastal zone and their interaction with offshore structures fast and accurate numerical models are needed by scientists and engineers. Questions related to an accurate estimation of wave loads and structure dynamics in moderate to severe sea state conditions are of utmost importance for optimizing the design of such structures at sea and, for instance, for assessing their behavior in extreme storm conditions and/or for fatigue analysis in the life-cycle of the structure. A wide range of models have been developed in order to predict wave fields and hydrodynamic loads at any scale, from the simple linear potential boundary element method to complex Computational Fluid Dynamics (CFD) codes directly solving the Navier-Stokes equations.
In the last decade, the use of CFD code has become increasingly popular in the scientific community, see for instance the applications of industrial or research codes like OpenFOAM\textregistered\
\citep{jacobsenFuhrmanFredsoe2012,hu2016numerical,higuera2013realistic,windt2019assessment},
STAR-CCM+
\citep{oggiano2017reproductio
,jiao2020cfd},
ANSYS-FLUENT
\citep{kim2016numerical,
feng2019generation
, and REEF3D \citep{bihs2016new
, just to mention few of them.
Although these CFD models are well adapted to solve the wave-structure interaction at small scale, in particular when complex physical processes such as wave breaking, formation of jets, air entrapment, etc. occur, their use remains quite limited due to the computational cost of such models when targeting applications on ocean domains whose extent is larger than, say, about 10 typical wavelengths. Limitations associated with the employed numerical methods (\emph{e.g.}~ numerical diffusion and difficulty to resolve the dynamics of the free surface) are other reasons which still hinder the applicability of such models to large scale wave propagation problems. Thus, models based on a potential flow approach (\emph{i.e.}~ neglecting viscous effects and assuming irrotational flow) are widely used to describe the dynamics of wave-structure interaction flows \citep[see \emph{e.g.}~ the review by][]{tanizawa2000state
.
Among potential models, simplified linear versions are often used in the engineering field due to their low computational cost, allowing to capture the main effects on a wide range of parameters at a contained CPU cost. WAMIT \citep{lee1995wamit
, ANSYS-AQWA \citep{ansys2013aqwa}
or Nemoh \citep{babarit2015theoretical} are examples of such widely used linear models. However, the linear assumption is often used outside its prescribed validity domain, for example in extreme cases where the allegedly small parameters, usually the wave steepness, becomes large. In those conditions many aspects of the dynamics of both the incident waves and the wave-body interaction are not properly modeled.
Efforts have been done to extend these potential linear models to weakly nonlinear conditions at second order, mostly by adding terms like the Froude-Krylov forces or the complete Quadratic Transfer Functions, see \emph{e.g.}~ \citet{pinkster1980low} or more recently \citet{philippe:hal-01198807}.
However in some cases, mostly extreme wave conditions, even third order effects can have a significant effect on the wave load \citep[see \emph{e.g.}~,][]{fedele2017sinking}. Both the nonlinear wave dynamics and the nonlinear wave structure interaction need to be taken into account. In those cases, alternative approaches capturing higher-order or fully nonlinear effects are needed. Several developments have been made to compute such nonlinear effect exactly or at high orders in the case of periodic regular waves in uniform water depth. For example, models based on analytical theories such as the Stokes wave theory \citep[see \emph{e.g.}~ ][]{sobey1989variations}
or the so-called stream function method \citep{dean1965stream, fenton1988numerical} can only be applied with constant or simple geometry of the sea floor. A review of several methods to describe wave propagation in a potential flow framework is given by \citet{fenton1999numerical}.
High order spectral (HOS) methods are also fast and accurate to compute the flow behavior even up to large wave steepness. Such methods, though very efficient in computing the wave elevation even for large domains, are difficult to apply for complex geometry as well as for wide ranges of parameters. They are mostly applied to for wave maker modelling \citet{ducrozet2012modified}, or to compute the incident wave field within a more complex method to resolve the wave structures interactions. For example, the SWENSE method \citep[see \emph{e.g.}~][]{luquest2007simulation} on which the diffracted field is computed separately with a Navier-Stokes based solver.
In order to develop a versatile numerical wave tank, with the possibility to include body or sea bed with complex geometries, a time domain resolution involving a mesh in the spatial coordinates appears to be practical at the cost of an increase in the computational cost.
The commonly used Boundary Element Method (BEM), in which the Laplace equation is projected onto the spatially discretized boundaries with the Green's identities, has been proven to be effective in both 2D and 3D cases. For example, \citet{grilli1997numerical} used an high order BEM method to generate and absorb waves in 2D. For an overview of work on the BEM methods up to the end of the 20\textsuperscript{th} century, the reader is referred to \citet{kim1999recent}. More recently \citet{guerber2012fully} presented in great detail and implemented a complete NWT. Note that under the BEM potential scheme, a special attention must be given to the treatment of the sharp corners \citep{hague_multiple_2009}.
Another type of time domain wave simulators is volume field solvers
With the cost of increasing the number of unknowns, the resulting matrix is mostly sparse allowing the use of efficient solvers.
Navier-Stokes based solvers fall in this category. A notable work on a NWT based on the finite difference method (FDM) is given by \citet{tavassoli2001interactions}. The Finite element method (FEM) was also successfully applied to both the potential problem \citep[\emph{e.g.}~][]{ma_quasi_2006, yan_numerical_2007} and the Navier-Stokes equations \citep[\emph{e.g.}~][]{wu_simulation_2013}.
A different potential model that solves for the volume field was recently proposed by \citet{HPC:shao2012towards,HPC:shao2014fully,HPC:shao2014harmonic} and tested against several methods including the BEM, FDM and FEM. This innovative technique, called the "harmonic polynomial cell" (HPC) method, was proven to be promising both in 2D and 3D \citep{HPC:shao2014fully,HPC:hanssen2015harmonic,HPC:hanssen2017free}. Although relatively new, this method was used to study a relatively important range of flows and phenomena: from a closed flexible fish cage \citep{strand2019linear} to hydrodynamics lifting problems \citep{liang2015application}. It was also extended to solve the Poisson equation by \citet{HPC:bardazzi2015generalized}. The numerical aspects of the method were studied in details by \citet{HPC:ma2017local} and applied as a 2D NWT by \citet{hpc:zhu2017improved}.
The treatment of the free surface condition and body boundary condition can be done in several ways. A classical and straightforward approach is to use a grid that conforms the boundary shape. With this method, the boundary nodes values are explicitly enforced in the linear system. The drawback of this method is that the grid needs to be deformed at each time step so as to match the free surface. A lot of solutions have been successfully applied to tackle this issue. For instance, \citet{ma_quasi_2006} used a Quasi Arbitrary Lagrangian-Eulerian (ALE) method combined with the FEM spatial discretization to prevent the mesh to have to be regenerated at each time step. \citet{yan_numerical_2007} extended the mesh conformation technique to include a freely floating body. With this same FEM discretization
\citet{wu_simulation_2013} used in addition an hybrid Cartesian immersed boundary method for the body boundary condition.
In this work, a NWT based on the HPC method is developed, its convergence assessed, and tested against both numerical and experimental data. Relaxations zones are used to generate and absorb the waves in a similar manner as in the OpenFOAM\textregistered \ toolbox waveFoam \citep{jacobsenFuhrmanFredsoe2012}: the values and positions of the free surface nodes are imposed from a stream-function theory over a given distance. The free surface is tracked in a semi-Lagrangian way following \citet{HPC:hanssen2017free} whereas for the solid bodies, an additional grid fitted to the boundary is defined using the advances presented in \citet{HPC:ma2017local}. In order for the body to pierce the free surface, an additional free surface marker list is defined in the body fitted grid. "External" and "internal" curves also overlap each other and communicate through relaxation zones. In order to tackle the singular nodes, a null flow flux method is imposed on the sharp corners which lay on a Neumann body boundary condition.
The remainder of this article is organized as follows. In \S~\ref{sec:math_pb} the mathematical formulation of the problem in a potential flow framework is recalled. The numerical methods based on the HPC approach are presented in \S~\ref{sec:HPC}, with particular attention devoted to the treatment of the free surface dynamics. In \S~\ref{sec:standing_w}, a convergence study is performed on a freely evolving standing wave case compared to a highly accurate numerical solution from the stream function theory. Then, a fitted mesh overlapping grid method is described and implemented in \S~\ref{sec:theory_fixedBody}. This double mesh strategy is tested on two selected cases in \S~\ref{sec:valid}. The first one is an horizontal fixed circular cylinder, completely immersed although close from the free surface, and the second one involves a free surface piercing body. Results of the present NWT on the two test cases are compared to experimental data while the first one is also compared to numerical results from the literature. The data on the immersed cylinder are extracted from \citet{chaplin1984nonlinear}, whereas the free surface piercing rectangular barge data are obtained during this work. In \S~\ref{sec:conclusions}, the main findings from this study are summarized and perspectives for further research are discussed.
\section{Fully nonlinear potential flow modeling approach} \label{sec:math_pb}
Three main assumptions are used in the potential model: i. we consider a fluid of constant and homogeneous density $\rho$ (incompressible flow); ii. the flow is assumed to be irrotational, implying that $\nabla\times{\tI{v}}=0$, where $\tI{v}(x,y,z,t)$ denotes the velocity field; and iii. viscous effects are neglected (ideal fluid). Here, $(x,y)$ denote horizontal coordinates, $z$ the vertical coordinate on a vertical axis pointing upwards, and $t$ the time. The gradient operator is defined as $\nabla f \equiv (f_x,f_y,f_z)^T$, where subscripts denote partial derivatives (\emph{e.g.}~ $f_x=\frac{\partial f}{\partial x}$).
The complete description of the velocity field $\tI{v}$ can thus be reduced to the knowledge of the potential scalar field $\phi(x,y,z,t)$, such that:
$\tI{v}=\nabla \phi$.
Due to the incompressibility of the flow, the potential $\phi$ satisfies the Laplace equation inside the fluid domain:
\begin{equation}
\nabla^2 \phi = 0 , \hspace{2cm} -h(x,y) \leq z \leq \eta(x,y,t) \label{eq:Laplace}
\end{equation}
where $\eta(x,y,t)$ is the free surface elevation and $h(x,y)$ the water depth relative to the still water level (SWL).
In order to solve this equation, boundary conditions need to be considered. On the time varying free surface $z=\eta(x,y,t)$, the Kinematic Free Surface Boundary Condition (KFSBC) and the Dynamic Free Surface Boundary Condition (DFSBC) apply:
\begin{align}
\label{eq:KFSBC} \eta_{t}+\nabla_H \eta \cdot \nabla_H \phi-\phi_z &=0 \quad \text{on } z = \eta(x,y,t),\\
\label{eq:DFSBC} \phi_{t}+\frac{1}{2}\left(\nabla_H \phi\right)^2+g\eta &=0 \quad \text{on } z = \eta(x,y,t),
\end{align}
where $\nabla_H f \equiv (f_x,f_y)^T$ denotes the horizontal gradient operator and $g$ the acceleration due to gravity. At the bottom (impermeable and fixed in time), the Bottom Boundary Condition (BBC) reads:
\begin{equation}
\nabla_H h \cdot \nabla_H \phi+\phi_z =0 \quad \text{on } z = -h(x,y). \label{eq:BBC}
\end{equation}
On the body surface, the slip boundary condition expresses that the velocity component of the flow normal to the body face equals the normal component of the body velocity. Here, we restrict our attention to fixed bodies, thus, denoting $\tI{n}$ unit vector normal to the body boundary, this condition reduces to:
\begin{equation}
\frac{\partial \phi}{\partial \tI{n}} = \nabla \phi \cdot \tI{n} =0 \quad \text{on the body}. \label{eq:bodyBC}
\end{equation}
Note that, using the free surface velocity potential and the vertical component of the velocity at the free surface, defined respectively as:
\begin{eqnarray}
\tilde{\phi}(x,y,t) & = & \phi(x,y,\eta(x,y,t),t) \label{eq:fspot}\\
\tilde{w}(x,y,t) & = & \frac{\partial \phi}{\partial z}(x,y,\eta(x,y,t),t) \label{eq:fsw}
\end{eqnarray}
the KFSBC and the DFSBC can be reformulated following~\citet{zakharov1968} as:
\begin{align}
\eta_{t} & = -\nabla_H \eta \cdot \nabla_H \tilde{\phi}+\tilde{w}(1+(\nabla_H \eta)^2) \label{eq:zakharov1}\\
\tilde{\phi}_t & = -g\eta-\frac{1}{2}(\nabla_H \tilde{\phi})^2+\frac{1}{2}\tilde{w}^2(1+(\nabla_H \eta)^2) \label{eq:zakharov2}
\end{align}
It can be noted that the Laplace equation~(\ref{eq:Laplace}), the BBC~(\ref{eq:BBC}) and the body BC~(\ref{eq:bodyBC}) are all linear equations. Thus, the nonlinearity of the problem originates uniquely from the free surface boundary conditions (\ref{eq:KFSBC}-\ref{eq:DFSBC}) or (\ref{eq:zakharov1}-\ref{eq:zakharov2}).
At that point, the commonly used linear wave theory assumes small amplitude surface waves, so that these boundary conditions can be linearized and applied at the SWL (\emph{i.e.}~ at $z=0$). Here, we intend to retain full nonlinearity of wave motion by considering the complete conditions~(\ref{eq:zakharov1} and~\ref{eq:zakharov2}). These two equations are used to compute the time evolution of the free surface elevation $\eta$ and the free surface potential $\tilde{\phi}$. This requires obtaining $\tilde{w}$ from $(\eta,\tilde{\phi})$, a problem usually referred to as Dirichlet-to-Neumann (DtN) problem.
For given values of $(\eta,\tilde{\phi})$, the DtN problem is here solved by solving a BVP problem in the fluid domain on the wave potential $\phi(x,y,z,t)$, composed of the Laplace equation~(\ref{eq:Laplace}), the BBC~(\ref{eq:BBC}), the body BC~(\ref{eq:bodyBC}), the imposed value $\phi(z=\eta)=\tilde \phi$ (Dirichlet condition) on the free surface $z=\eta$, supplemented with boundary conditions on lateral boundaries of the domain (of \emph{e.g.}~ Dirichlet, Neumann, etc. type). The numerical methods to solve the BVP are presented in the next section.
\section{The Harmonic Polynomial Cell method (HPC) with immersed free surface} \label{sec:HPC}
\subsection{General principle of the HPC method} \label{ssec:HPC_0}
In order to solve the above mentioned BVP at a given time, the HPC method introduced by~\citet{HPC:shao2012towards} is used. It is briefly described here, and more details can be found in~\citet{HPC:shao2014harmonic},~\citet{HPC:hanssen2015harmonic,HPC:hanssen2017free},~\citet{HPC:hanssen2019non} and~\citet{HPC:ma2017local}. In this work, the HPC approach is implemented and tested in 2 spatial dimensions, \emph{i.e.}~ in the vertical plane $(x,z)$, for a wide range of parameters.
The fluid domain is discretized with overlapping macro-cells which are composed of 9 nodes in 2 dimensions. Those macro-cells are obtained by assembling four adjacent quadrilateral cells on an underlying quadrangular mesh. The four cells of a macro-cell share a same vertex node, called the "central node" or "center" of the macro-cell. A typical macro-cell is schematically shown in figure~(\ref{sch:HPC_cell}), with the corresponding local index numbers of the 9 nodes. With this convention, any node with global index $n$ has the local index "9" in the considered macro-cell and is considered as an interior fluid point, whereas for example node with local index "4" can either be a fluid point or a point lying an a boundary. If node "4" is also inside the fluid domain, it also defines another macro-cell whose the center right point (with local index "5") will then correspond to the node with global index $n$.
\begin{figure}[htbp!]
\begin{center}
\includegraphics{standalone/fluidpoint}
\caption{Definition sketch of 9-node macro-cell used for the HPC method, with local numbering of the nodes}
\label{sch:HPC_cell}
\end{center}
\end{figure}
In each macro-cell, the velocity potential is approximated as a weighted sum of the 8 first harmonic polynomials (HP), the later being fundamental polynomial solutions of the Laplace equation~(\ref{eq:Laplace}).
A discussion about which of the HP are to be chosen is given in \citet{HPC:ma2017local}. Here, we follow \citet{HPC:shao2012towards}, and select all polynomials of order 0 to 3 plus one fourth-order polynomial, namely: $f_1(\tI{x})=1$, $f_2(\tI{x})=x$, $f_3(\tI{x})=z$, $f_4(\tI{x})=xz$, $f_5(\tI{x})=x^2-z^2$, $f_6(\tI{x})=x^3-3xz^2$, $f_7(\tI{x})=-z^3+3x^2z$ and $f_8(\tI{x})=x^4-6x^2z^2 +z^4$. Here, $\tI{x}=(x,z)$ represents the spatial coordinates. Thereafter, we define $\tI{\bar{x}} = \tI{x}-\tI{x}_9$ the same spatial coordinate in the local reference frame of the macro-cell, with $\tI{x}_9$ being the center node of the macro cell.
From a given macro-cell, the potential can be approximated at a location $\tI{x}$ as:
\begin{equation}
\phi(\tI x)=\sum_{j=1}^{8}b_j f_j(\tI{\bar{x}}) \label{eq:interp}
\end{equation}
As every HP is a solution of the Laplace equation~(\ref{eq:Laplace}) which is linear, any linear combination of them is also solution of this equation. Thus, the goal now becomes to match the local expressions (LE) given by equation~(\ref{eq:interp}) such that every local expression is verified at each participating node. For that reason, macro-cells overlap each other.
Note that this study deals with 2D problems, but the method can be extended to 3D cases as shown by~\citet{HPC:shao2014fully} considering cubic-like macro-cells with 27 nodes.
The first objective is to determine the vector of coefficients $b_j, j=1,...,8$ for the selected macro-cell. Recalling that equation~(\ref{eq:interp}) should be verified at the location of each point of the macro-cell (with local index running from $1$ to $9$), this equation applied at the 8 neighboring nodes $(1-8)$ of the center yields:
\begin{equation}
\phi_i=\phi(\tI{x}_i) = \sum_{j=1}^{8} b_j f_j(\tI{\bar{x}}_i) \quad \text{for} \ i=1,...,8
\end{equation}
which represents, in vector notation, a relation between the vector of size 8 of the values of the potential $\phi_i$ at the outer nodes with the vector of size 8 of the $b_j$ coefficients.
The 8x8 local matrix linking this two vectors is denoted $\tII{C}$, and defined by $C_{ij}=f_j(\tI{x}_i)$.
Note that $\tII{C}$ is defined geometrically, thus it only depends on the position of the outer nodes $i$ relatively to the position of the central node.
$\tII{C}$ can be inverted and its inverse is denoted $\tII{C}^{-1}$. The $b_j$ coefficients are then obtained for the given macro-cell as a function of the potentials at the 8 neighboring nodes of the central node of that macro-cell:
\begin{equation}
b_j = \sum_{i=1}^{8} C^{-1}_{ji}\phi_i \quad \text{for} \ j=1,...,8.
\end{equation}
Injecting this result into the interpolation equation~(\ref{eq:interp}), a relation is found providing an approximation for the potential at any point located inside the macro-cell using the values of the potential of the eight surrounding nodes of the central node:
\begin{equation}
\phi(\tI x)= \sum_{i=1}^{8} \left(\sum_{j=1}^{8} C^{-1}_{ji} f_j(\tI{\bar{x}}) \right) \phi_i \label{eq:mainwithcij}
\end{equation}
This equation will be referred to as local expression (LE) of the potential. It will be used to derive the boundary conditions equations and the fluid node equations that need to be solved in the BVP. Also note that this LE provides a really good interpolation function that can be used for every additional computation once the nodal values of the potential are known (\emph{i.e.}~ potential derivatives at the free surface or close to the body to compute the pressure field).
Note that the accuracy of LE depends only on the geometry: coordinates at which this equation is applied, shape of the macro-cell, etc. Those dependencies are investigated in details by~\citet{HPC:ma2017local}.
\subsection{Treatment of nodes inside the fluid domain} \label{ssec:HPC_1}
We first consider the general case of macro-cells whose central node is an interior node of the fluid domain. Applying the LE~(\ref{eq:mainwithcij}) at the central node yields a linear relation between the values of the potential at the nine nodes of this macro-cell:
\begin{equation}
\phi_9=\phi(\tI{x}_9)= \sum_{i=1}^{8} \left(\sum_{j=1}^{8} C^{-1}_{ji} f_j(\tI{ \bar{x}}_9) \right) \phi_i
\label{eq:mainatx9}
\end{equation}
We may further simplify this equation by noting that, as $\tI{\bar{x}}_9=(0,0)$ in local coordinates, all $f_j(\tI{\bar{x}}_9)$ vanish, except $f_1(\tI{\bar{x}}_9)$ which is constant and equal to $1$. Equation~(\ref{eq:mainatx9}) then simplifies to:
\begin{equation}
\phi_9= \sum_{i=1}^{8} C^{-1}_{1i} \phi_i
\label{eq:mainatx9b}
\end{equation}
meaning that only the first row of the matrix $\tII{C}^{-1}$ is needed here.
In order to solve the global potential problem, \emph{i.e.}~ to find the value of the nodal values of the potential at all grid points (whose total number is denoted $N$), a global linear system of equations is formed, with general form $\tII{A}.\tI{\phi}=\tI{B}$, or:
\begin{equation}
\sum_{l=1}^{N} A_{kl}\phi_l=B_k \quad \text{for} \ k=1,...,N.
\end{equation}
where $k$ and $l$ are global indexes of the nodes.
For each interior node in the fluid domain, with global index $k$ and associated macro-cell, an equation of the form~(\ref{eq:mainatx9b}) allows to fill a row of the global matrix $\tII{A}$. This row $k$ of the matrix involves only the considered node and its 8 neighboring nodes, making the matrix $\tII{A}$ very sparse (at most 9 non-zero elements out of $N$ terms). Moreover, the corresponding right-hand-side (RHS) term $B_k$ is null. Note that all the 8 neighboring nodes of the macro-cell associated with center $k$ should also have a dedicated equation in the global matrix in order to close the system.
\subsection{Nodes where a Dirichlet or Neumann boundary condition is imposed
\label{sec:sub:dirandneucond}}
If a Dirichlet boundary condition with value $\phi_D$ of the potential has to be imposed at the node of global index $k$, the corresponding equation is simply $\phi_k=\phi_D$, so that only the diagonal element of the global matrix is non-null and equal to 1 for the corresponding row $k$: $A_{kl}=\delta_{kl} \ \ \forall l \in [1,N]$. The corresponding term on the RHS is set to $B_k=\phi_D$.
If a Neumann condition has to imposed at a given node of global index $k$, the relation set in the global matrix is found trough the spatial derivation along the imposed normal $\tI{n}$ of the LE~(\ref{eq:mainwithcij}) of any macro-cell on which $k$ appears. In practice, the macro-cell whose center is the closest from the node $k$ is chosen, and we then use:
\begin{equation}
\nabla \phi(\tI{ x}_k) \cdot \tI{n} = \sum_{i=1}^{8} \left(\sum_{j=1}^{8} C^{-1}_{ji} \nabla f_j(\tI{ \bar{x}}_k) \cdot \tI{n} \right) \phi_i \label{eq:mainderivated}
\end{equation}
Thus, a relation is set in the row $k$ of the global matrix to enforce the value of $B_k=\nabla \phi(\tI{x}_k) \cdot \tI{n}$ at position $\tI{x}_k$. In that case, a maximum of 8 non-zero values appear in this row on the global matrix as the potential of the central node of the macro-cell does not intervene here.
\subsection{Treatment of the free surface}
As already mentioned, in order to solve the BVP at a given time-step, the system of equations needs to be closed, meaning that each neighbor of a node in the fluid domain should have a dedicated equation. We consider now the case of nodes lying on or in the vicinity of the (time varying) free surface. The free surface potential should be involved here, either directly at a node fitted to the free surface through a Dirichlet condition described in the previous sub-section, or through alternative techniques.
For instance, an Immersed Boundary Method (IBM) was first suggested in the HPC framework by \citet{HPC:hanssen2015harmonic} to tackle body boundary conditions.
More recently,~\citet{HPC:ma2017local} compared a modified version of the IBM with two different multi-grid (MG) approaches (fitted or combined with an IBM) for both body and free surface boundary conditions. \citet{HPC:hanssen2017free} and \citet{HPC:hanssen2017free} also made in-depth comparisons of the MG and IB approaches, focusing on the free surface tracking. Both methods showed promising results. \citet{hpc:zhu2017improved} introduced a similar yet slightly different IB approach with one or two ghost node layers, then realized a comparison between this IB approach and the original fitted mesh approach.
In the present work, the IBM was chosen for the treatment of the free surface, though the fitting mesh method is shortly described thereafter.
\subsubsection{Fitted mesh approach for the free surface \label{sec:sub:sub:oldfittedMesh}}
The first possibility is to fit the mesh to the actual free surface position at any time when the BVP has to solved. The mesh is deformed so that the upper node at any abscissa always lies on the free surface.
That way, the computational domain is completely closed and the free surface potential is simply enforced as a Dirichlet boundary condition at the correct position $z=\eta$ as explained in \S~\ref{sec:sub:dirandneucond}.
With this approach, the algorithm, given the boundary values at the considered time, can be summarized as:
\begin{itemize}
\item Deform the mesh to fit the current free surface elevation,
\item Build and then invert the local geometric matrices $\tII{C}$,
\item Fill the global matrix $\tII{A}$ and RHS $\tI{B}$, using the corresponding Dirichlet conditions at nodes lying on the free surface,
\item Invert the global problem to obtain the potential everywhere
\end{itemize}
Recently, \citet{HPC:ma2017local} pointed out that the HPC method efficiency (in terms of accuracy and convergence rate) is greatly improved when a fixed mesh of perfectly-squared cells is used. In this work, the negative effects of a deforming mesh outlined in the previous subsection were also encountered. Especially, for some particular cell shapes, a high increase of the local condition number was observed, leading to difficulty of matrix inversion and important errors on the approximated potential.
As a consequence, results were highly dependent on the mesh deformation method employed, especially in the vicinity of a fixed fully-immersed body.
\subsubsection{Immersed free surface approach \label{sec:sub:ifs}}
In order to work with regular fixed grids, an IBM technique was developed and implemented to describe the free surface dynamics. \citet{HPC:hanssen2015harmonic} introduced a first version of this method applied on the boundaries of a moving body. This method was recently extended to the free surface and compared to a fitted MG method by \citet{HPC:ma2017local} and \citet{HPC:hanssen2017free}. In the current work, a semi-Lagrangian IB method introduced by \citet{HPC:hanssen2017free} is chosen.
In this method, the free surface is discretized with markers, evenly spaced and positioned at each vertical intersection with the background fixed grid, as shown in figure~(\ref{sch:ifreesurface}). Those markers are semi-Lagrangian in such a way that they are only allowed to move vertically, following equations~(\ref{eq:zakharov1}-\ref{eq:zakharov2}).
\begin{figure}[htbp!]
\begin{center}
\includegraphics{standalone/ifreesurface}
\caption{Schematic representation of the immersed free surface in a fixed grid}
\label{sch:ifreesurface}
\end{center}
\end{figure}
At a given time, every node located below the free surface (\emph{i.e.}~ below a marker) is considered as a node in the fluid domain ("fluid" node), and defines a macro-cell with its 8 neighbors.
The global matrix is classically filled with the local expression~(\ref{eq:mainwithcij}) at those nodes.
As a consequence, in order to close the system, each neighbor of a node just above the free surface must also have a dedicated equation in the global matrix. These neighbors, represented with grey circles on figure~(\ref{sch:ifreesurface}), are denoted as "ghost" nodes.
The chosen equation to close the system at a node of this type is the local expression~(\ref{eq:mainwithcij}) applied at the marker position in a given macro-cell:
\begin{equation}
\phi_m=\sum_{i=1}^8 \left(\sum_{j=1}^8 C_{ji}^{-1}f_j(\tI{\bar {x}}_m)\right) \phi_i
\label{eq:markerinterp}
\end{equation}
where $\bar{\tI{x}}_m = (x_m,\eta(x_m))- \tI{x}_c$ is the position of the marker in the macro-cell's reference frame ($\tI{x}_c$ is the global position of the center node of the chosen macro-cell) and $\phi_m$ its potential (known at this stage).
This ensures that the potential at free surface point is equal to the potential at the position of the marker from the interpolation equation. In other words, if one wants to interpolate the computed field $\phi$ at the particular location of the marker $\tI{x}_m$, the results should be consistent and yield the potential $\phi_m$.
Note that this equation~(\ref{eq:markerinterp}) is cell dependent (through $C_{ji}^{-1}$, the involved $\phi_i$ and the position of the
center node $\tI{x}_c$)
, but also depends on the chosen marker (trough $\tI{x}_m$ and $\phi_m$).
The only mathematical restriction on the choice the macro-cell to consider is that the ghost point potential should intervene as one of the $\phi_i$ in order to impose the needed constraint at this point.
An important note is that the later equation~(\ref{eq:markerinterp}) is not dependent on the ghost point in any fashion.
This implies that if the same couple (marker, macro-cell) is chosen to close the system at two different ghost points, the global matrix will have to strictly identical rows. Its inversion would thus not be possible
Particularly, two vertically aligned ghost points cannot use the same macro-cell equation at the same marker position.
Here stands the differences between the IB method of \citet{HPC:hanssen2017wave}, \citet{HPC:ma2017local} and the one chosen by \citet{hpc:zhu2017improved}.
\citet{hpc:zhu2017improved} decided to only impose the marker potential once in the first layer (or two first layers) and to constraint the upper potentials to an arbitrary value (in practice if the point is not used directly, the potential is set to the first point below which potential is used).
The method used during this work is closer to the one by \citet{HPC:hanssen2017wave} and \citet{HPC:ma2017local}: if a node needs a constraint but does not have a maker directly underneath (case of two ghost points vertically aligned), the ghost point on the top should invoke the local expression of the macro-cell centered on the closest fluid point instead of the cell centered on the vertically aligned fluid point (case indicated by an arrow in figure~\ref{sch:ifreesurface}).
With that method, in such a situation, the potential of vertically aligned marker is imposed twice in two different adjacent macro-cells.
A comparison between those two methods had not been conducted and would be of great interest.
Whatever IB the method, the main goal is achieved: it is not needed to deform the mesh in time.
As a consequence, the computation and inversion of the local (geometric) matrices is only done once, at the beginning of the computation.
However, a step of identification of the type of each node, which was proven to be time consuming, is needed instead. Note that this identification algorithm could be greatly improved and is relatively slow in its current implementation.
The general algorithm at one time step becomes:
\begin{itemize}
\item Identify nodes inside the fluid domain,
\item Identify ghost nodes needed to close the system, associated markers and macro-cells,
\item Fill global matrix $\tII{A}$ and RHS $\tI(B)$,
\item Invert global problem to obtain the potential everywhere,
\end{itemize}
\subsection{Linear solver and advance in time} \label{ssec:linsol_time}
To solve the global linear sparse system of equations, an iterative GMRES solver, based on Arnoldi inversion, was used for all computations. The base solver was developed by \citet{gmres:saad2003iterative} for sparse matrix (SPARSEKIT library), and includes an incomplete LU factorization preconditionner. During this work, a modified version was implemented with the improvement proposed by \citet{baker_simple_2009}. Except for staling during the study of a standing wave at very long time, this solver was proven to be robust.
Improvements of the construction step of the global matrix could further made in order to increase the efficiency of its inversion. Also, the initial guess in the GMRES solver could also be improved taking advantages of the already computed potential values.
The number of inner iterations of the GMRES algorithm was chosen as $m \in [30, 60]$ and the iterative solution is considered converged when the residual is lower that $5.10^{-9}$.
Marching in time thanks to equations~(\ref{eq:zakharov1}-\ref{eq:zakharov2}) yields the free surface elevation and the free surface potential at the next time step.
Note that the steps of computing the RHS terms of these equations are straightforward for most terms directly from the local expression~(\ref{eq:mainwithcij}) of the closest macro-cell.
In addition, the spatial derivative of $\eta$ is computed with a finite difference method. A centered scheme of order 4 is chosen for this work with the objective to maintain the theoretical order 4 of spatial convergence provided by the HPC method.
In order to integrate equations~(\ref{eq:zakharov1}-\ref{eq:zakharov2}), the classical four-step explicit Runge-Kutta method of order 4 (RK4) was selected as time-marching algorithm. During a given simulation, the time step $(\delta t)$ was chosen to remain constant. Its value is made nondimensional by considering the Courant-Friedrichs-Lewy (CFL) number $C_o$ based on the phase velocity $C=\lambda/T$, where $\lambda$ is the wavelength and $T$ the wave period:
\begin{equation}
C_o=\frac{C \delta t}{\delta x}=\frac{\lambda/\delta x}{T/\delta t}
\label{eq:CFL}
\end{equation}
The CFL number thus corresponds to the ratio of the number of spatial grid-steps per wavelength $(N_x=\lambda/\delta x)$ divided by the number of time-steps per wave period $(N_t=T/\delta t)$, \emph{i.e.}~ $C_o=N_x/N_t$.
\subsection{Computation of the time derivative of the potential}
The pressure inside the fluid domain is obtained from the Bernoulli equation:
\begin{equation}
p(x,z,t)=-\rho \left( \dfrac{\partial \phi}{\partial t} + \dfrac{1}{2}(\nabla \phi)^2 + gz \right) \label{eq:Bernoulli}
\end{equation}
Any potential based NWT needs to solve this equation as the loads applied on the tested bodies are most of the time of prime importance.
In a first attempt, the time derivative of the potential was estimated using a backward finite difference scheme. However, this method is not well suited when important variations of the potential are at play. Moreover, in the case of the IB method, it is not possible to obtain the value of the pressure at a point that was previously above the free surface, and thus for which a time derivative of the potential cannot be computed by the finite difference scheme.
A fairly accurate method is to introduce the (Eulerian) time derivative of the potential as a new variable $\phi_t=\dfrac{\partial \phi}{\partial t}$ and to solve a similar BVP as described previously on this newly defined variable, noting that $\phi_t$ has to satisfy the same Laplace equation as $\phi$ in the fluid domain. This method has been used by \emph{e.g.}~ \citet{guerber2011modelisation} in the BEM framework or by \citet{HPC:ma2017local} in the HPC method.
Note that the local macro-cell matrices and coefficients, which are only geometrically dependent, do not change.
In the different expressions presented above that are used to fill the global matrix, the coefficients linking the different potentials are not time dependent. Thus, the matrix to invert is exactly the same for the $\phi_t$ field and the $\phi$ field.
However, the constant boundary conditions (and thus the RHS) may change. This is the case only when the RHS is different from zero, as for example, for free surface related closure points.
Remember that the equations at a (non-moving) Neumann condition and at a point inside the fluid domain yield a zero value in the RHS, and thus the equations at those points are exactly the same for the potential variable and for its derivative.
At a (non-moving) Dirichlet boundary condition, one would simply impose $\phi_t=0$ instead of $\phi =\phi_D$. At the IB ghost points, the $\phi_t$ is imposed to match the derivative of the potential with respect to time, known at the marker positions from equations~(\ref{eq:zakharov1}-\ref{eq:zakharov2}).
Even tough the global matrices are exactly the same, the RHS being different and the chosen resolution method being iterative (GMRES solver), the easiest way is just to solve twice the almost same problem. A more clever way could maybe be investigated by taking advantage of the previous inversion, but this is left for future work.
\section{Validation and convergence study on a nonlinear standing wave \label{sec:standing_w}}
\subsection{Presentation of the test-case \label{ssec:standing_w_intro}}
The first case consists in simulating a nonlinear standing wave in a domain of uniform water depth $h$ whose extent is equal to one wavelength $\lambda$. This case is actually challenging as the wave height $H$ (difference between the maximum and minimum values of free surface elevation at anti-node locations) is fixed by choosing a large value of wave steepness $H/\lambda=10\%$ (or $kH/2=\pi/10\approx 0.314$). We also choose to work in deep water conditions by selecting $h=\lambda=64$~m (or $kh=2\pi\approx 6.28$). The water domain at rest has thus a square shape in the $(x,z)$ plane, as illustrated in figure~(\ref{sch:standingwave}).
Initial elevations of free surface $\eta(x,t=0)$ are computed from the numerical method proposed by \citet{tsai1994numerical}. The initial phase is chosen such that the imposed potential field is null at $t=0$ at any point in the water domain. This initial state corresponds to a maximum wave elevation at the beginning and the end of the domain ($x/\lambda=0$ and $1$), and a minimum wave elevation at the center point of the domain ($x/\lambda=0.5$), these three locations being anti-nodes of the standing wave.
\begin{figure}[htbp]
\centering
\includegraphics{standalone/standingwave}
\caption{Schematic representation of the nonlinear standing wave with steepness $H/\lambda=10\%$ at $t=0$.}
\label{sch:standingwave}
\end{figure}
The wave is freely evolving under the effect of gravity: in theory one should observe a fully periodic motion without any damping as the viscosity is neglected. At each time step, a spatial $L_2(\eta)$ error on $\eta$ is computed relative to the theoretical solution of \citet{tsai1994numerical} (denoted $\eta^{th}$ hereafter), and normalized with the wave height:
\begin{equation}
L_{2}(\eta,t)=\dfrac{1}{H}\sqrt{\frac{1}{n_p}\sum_{i=1}^{n_p}{\left(\eta(x_i,t)-\eta^{th}(x_i,t)\right)^2}}
\end{equation}
where $i$ represents the index of a point on the free surface and $n_p$ the total number of points on the free surface.
\subsection{Evolution of \texorpdfstring{$L_2(\eta)$}{L2(n)} error with space and time discretizations \label{ssec:standing_w_error}}
The result of this $L_2(\eta)$-error is represented as a color map at four different times $t/T =$ 1, 10, 50 and 100 in figure~(\ref{fig:ErrorTotStanding}) as a function of the number of nodes per wavelength ($N_x=\lambda/\delta x$, where $\delta x$ is the spatial step-size) and the CFL number $C_o$ (defined in \S~\ref{ssec:linsol_time}). Wide ranges of the two discretization parameters are explored, namely $N_x\in[10, 90]$ and $C_o$ $\in[0.05, 4.0]$. Simulations that ran till the end of the requested duration of $100T$ are represented with coloured squares. A circle is chosen as a marker when the computation breaks down before the end of that duration. Nonetheless the markers are colored if the computation did not yet diverged at the time instant shown on the corresponding panel.
\begin{figure}[htbp!]
\centering
\includegraphics{ErrorTotStanding}
\caption{$L_2$ error on $\eta$ on the nonlinear standing wave case at four time instants ($t/T=1$, $10$, $50$ and $100$) as a function of the spatial and temporal discretizations. The color scale indicates the $L_2(\eta)$ error respective to the theoretical solution by \citet{tsai1994numerical}. See text for explanations on the significance of the markers shapes.}
\label{fig:ErrorTotStanding}
\end{figure}
As seen on figure~(\ref{fig:ErrorTotStanding}), a large number of simulations were completed over this rather long physical time of $100T$. Note that no filter were used along this work, so some of the numerical simulations tend to be unstable for extreme values of the discretization parameters.
For instance, when $C_o \le 1$, the computation is mostly unstable and breaks down: before 50$T$ when $N_x$ is small (\emph{i.e.}~ below 40) and between 50$T$ and 100$T$ when $N_x$ is larger. Note that $C_o=1$ corresponds to a time step ranging from $N_t = T/\delta t=10$ to $90$, for $N_x=10$ and $90$ respectively. This value of $C_o=1$, and associated time step, is the lower stable limit exhibited by these simulations.
On the other hand, when the $C_o$ is too high (\emph{i.e.}~ larger than 3.5) instabilities also occur almost at the beginning of the simulation ($t/T<10$), particularly when $N_x$ is small.
For very small $N_x$ (in the range 10-15) and whatever the $C_o$, the computation tends to be unstable. This is probably due to the discretization of the immersed free surface being the same as the discretization of the background mesh.
A coarse discretization of the free surface leads to an inaccurate computation of the spatial derivative of $\eta$: instabilities may then occur.
A suitable range of parameters is thus determined to avoid instabilities: $1.5 \le C_o \le 3.5$ and $40 \le N_x \le 90$. This zone is represented in figure~(\ref{fig:ErrorTotStanding}) as a rectangular box with a dashed contour. In that zone, all the computations ran with the requested time step over a duration of $100T$. Note that the CPU cost scales with $N_x^2 N_t \sim N_x^3 /C_{o}$ and thus the most expensive computations in this stable zone are approximately 25 times slower than the least expensive ones in the same zone.
Also note that the lowest error is almost systematically reached in this zone. The $L_2(\eta)$ error is as small as $2.10^{-6}$ after $1T$. After $100T$, the lowest error is approximately $10^{-4}$. Moreover, the evolution of the value of the error is qualitatively consistent with the mesh refinement and time refinement.
The stability was not assessed for finer mesh than $N_x=90$ points per wave length, due to increasing computation cost on one hand, and the fact that finer resolutions would lie out of the range of discretizations targeted for real-case applications. In addition, at long time, a discretization of $N_x= 90$ already exhibits behavior that does not match the expected convergence rate, as will be discussed hereafter in greater detail.
\subsection{Convergence with time discretization \label{ssec:standing_w_conv_t}}
In order to study the convergence of the method in a more quantitative manner, the $L_2(\eta)$ error is shown as a function of the $C_o$ number for different spatial discretizations $N_x$ at $t/T=1$ in figure~(\ref{fig:sub:convt1}) and at $t/T=100$ in figure~(\ref{fig:sub:convt100}). A $C_o^\alpha$ regression line is computed and fitted on the linear convergence range of the log-log of the error. That will be called "linear range" for simplicity, though it correspond to an algebraic rate of convergence of the error. Note that this linear range corresponds exactly to the zone in which the computations remain stable (with the exception of one particular point at $t/T=100$ and $N_x=90$ excluded from the determination of the convergence rate). At $t/T=1$, the minimum error is, as expected, obtained for small $C_o$ numbers and large $N_x$: $L_2(\eta) \sim 10^{-5}$ in the linear range and the minimal error reached is $2.10^{-6}$ for the finer discretization $N_x = 90$. The algebraic order of convergence is close to 4. This was expected as the temporal scheme is the RK4 method at order 4. Moreover, at this early stage of the simulation the error decreases with a power 4 law only when $C_o$ $\gtrsim 1.0$. The lowest errors are achieved at $C_o$ $\approx 0.75$. Below that $C_o$ number, a threshold is met: the error remains constant when the time step (and $C_o$ number) is further decreased; it is then controlled by the spatial discretization.
It is also possible to note that the CFL number $C_o$ seems to be a relevant metric when testing the convergence of the method: the range of $C_o$ in which the results converge is the same across the 4 considered spatial discretizations.
\begin{figure}[htbp!]
\centering
\subfloat{\label{fig:sub:convt1}\includegraphics{convtStanding1}}
\subfloat{\label{fig:sub:convt100}\includegraphics{convtStanding100}}
\caption{Convergence of the $L_2$ error on $\eta$ (crosses) with respect to the temporal discretization at two different physical times: $t/T=1$ (left panel) and $t/T=100$ (right panel). The spatial discretization is fixed for a given line. Solid lines represent power regression of the error in the "linear range", the computed power is reported in the legend of the fitted straight lines.}
\label{fig:convergencett1t100}
\end{figure}
\begin{rmk}
Significant differences in terms of $C_o$ with the work of \citet{HPC:hanssen2017free} have to be stressed. In their simulations the chosen numbers of points per wavelength were similar to the ones used here ($N_x\in[15,90]$), but the time step was constant and fixed at a small value of $\delta t/ T=1/N_t=1/250$. This value yields a $C_o$ between 0.06 and 0.36. This range of $C_o$ was shown to be out of the domain of convergence in time in our case. For the same $C_o$ (and apparently the same RK4 time scheme), the computation is indeed converged with respect to the time discretization and yields low error during the first periods, but instabilities then occur when the wave are freely evolving on a longer time scale. \citet{HPC:hanssen2017free} also encountered instabilities with this IB method. To counteract these instabilities, they used a 12\textsuperscript{th} order Savitzky-Golay filter in order to suppress, or at least, attenuate them. No filtering nor smoothing was used in our simulations. This may explain the differences of behavior with \citet{HPC:hanssen2017free} in terms of $C_o$ number. Similarly, in \citet{hpc:zhu2017improved}, also with the RK4 time scheme, the time-step is chosen as $\delta t/T=1/200$ for a spatial discretization of $\delta x = h/10$. Converted to our numerical case, this would correspond to $N_x=100$, and so a $C_o$ fixed at $0.5$.
On the contrary, the time-step chosen by \citet{HPC:ma2017local} to compute the potential flow around a rigid body in infinite fluid domain, $\delta t/T=1/40$, is closer to the current range of time-steps. During their investigations on periodic wave propagation, their spatial discretizations ranged from $N_x=16$ to $N_x=128$. The equivalent $C_o$ number is thus comprised between $0.4$ and $3.2$.
\end{rmk}
\subparagraph{}
At long time $t/T=100$ (see figure~\ref{fig:sub:convt100}), the error behaves differently. First, the error is approximately one order of magnitude higher compared to the time $t/T=1$, but the convergence rate is also slightly different, actually higher.
As a matter of fact, at $t/T=100$, an order 4 of convergence is still found on the wave period and on the amplitude of the computed wave: Figure~(\ref{fig:tempx32}) shows the evolution of the error on wave period and amplitude at the center of the domain $x/\lambda=0.5$ at $t/T=100$ for a fixed $N_x=30$ as a function of the CFL number $C_o$. The reference case used here is the one with $N_x=100$ at $C_o=0.05$ computed on one period. On a given case, the period is computed trough the mean time separating two successive maximums, then a sliding Fast Fourier Transformation (FFT) is performed to obtained an accurate estimation of the amplitude of the free surface elevation.
\begin{figure}[!htbp]
\centering
\includegraphics{tempx32}
\caption{Convergence of the error on wave period and amplitude of the wave elevation at $x/\lambda=0.5$ for $N_x=30$. The $L_2(\eta)$ error -combination of both- is also added.}
\label{fig:tempx32}
\end{figure}
However, the behavior of the $L_2(\eta)$ error results from a combined effect of both the error on the wave period and the error on the amplitude. The relative effect of those errors on the total error is analyzed in detail in \ref{apd:convergencelongt}, and a brief summary is given here.
Let $e$ be the relative error between two cosine functions. The first is the target function and the second one tends to the first one in amplitude as $\epsilon_a=f_A d^4$ and in period as $\epsilon_t=f_Td^4$. Here $d$ is a discretization variable -either $C_o$ or $1/N_x$ in our case-, which drives the convergence. $f_A$ and $f_T$ are constants with respect to $d$. A Taylor expansion of $e$ in the vicinity of a time $t/T$ corresponding to a whole number of periods gives:
\begin{equation}
e = f_A d^4+ 2\pi^2\dfrac{t^2}{T^2} f_T^2 d^8 + (2\pi \dfrac{t}{T}f_T^3 - 2 f_A \pi^2\dfrac{t^2}{T^2} f_T^2 ) d^{12} +O(d^{16})
\label{eq:errordev}
\end{equation}
Note that $f_A$ depends on $t/T$ because the error on amplitude increases with time (in practice a linear dependence was observed at long time, \emph{i.e.}~ $f_A=\bar{f_A} t/T$). However they should not depend on the convergence parameter $d$.
The order 8 of convergence should disappear for small enough $d$ whatever $t/T$. In that case, the error on period is negligible compared to the error amplitude: this results from the presence of the cosine, which elevates the error to the power 2.
However, if the error in amplitude $f_A$ increases in time slower than $t^2f_T^2$, there exists a time after which the error on the wave period will play an important role (order 8 will be predominant). This effect is thought to explain the seemingly high order of convergence of the $L_2(\eta)$ error in figure~(\ref{fig:sub:convt100}). \ref{apd:convergencelongt} shows detailed comparisons at $t/T=100$ with values of $f_A$ and $f_T$ extracted from our results.
\subsection{Convergence with spatial discretization \label{ssec:standing_w_conv_x}}
The convergence with spatial refinement (\emph{i.e.}~ as a function of $N_x=\lambda/\delta x$) is analyzed in the same way and shown in figure~(\ref{fig:convergencext1t100}). The order of convergence in space is again 4.
Due to the choice of the set of HP including polynomials up to order 4, and the fact that finite difference scheme of order 4 are used to compute the derivatives of free surface variables, this order 4 was the expected order of convergence.
\begin{figure}[htbp!]
\centering
\subfloat{\label{fig:sub:convx1}\includegraphics{convxStanding1}}
\subfloat{\label{fig:sub:convx100}\includegraphics{convxStanding100}}
\caption{Convergence of the $L_2$ error on $\eta$ in mesh refinement, with temporal discretization fixed. Solid lines correspond to the power regression of the error.}
\label{fig:convergencext1t100}
\end{figure}
At long time the convergence rate exhibits the same behavior as shown in the convergence with time resolution. The latter comments concerning the long time evolution of the error still holds (with, here, $d\equiv \delta x$), and is still thought to explain the increasing order of convergence of the total error $L_2(\eta)$ with time.
Of course the values of the corresponding constants $f_A$ and $f_T$ are different.
Another effect also occurs: when the $C_o$ number increases, so does the order of convergence. This small yet clear effect at time $t/T=100$ is not completely understood. It would mean that the value of $f_T$ increases faster with the $C_o$ number than the value of $f_A$ does.
\subsection{Summary of numerical convergence study \label{ssec:standing_w_sum}}
After a comprehensive numerical study of the HPC method on a challenging nonlinear standing wave case in deep water conditions, the efficiency and accuracy of the Immersed Boundary modeling of the free surface applied on a fixed underlying spatial mesh was demonstrated (again without using filtering nor smoothing of the free surface). Optimal ranges of spatial and temporal discretization parameters were determined:
\begin{itemize}
\item the spatial discretization $\delta x$ should be chosen to have $N_x=\lambda / \delta x$ between $40$ and $90$ nodes per wavelength, which is a reasonable range of values for practical applications.
\item the temporal discretization $\delta t$ should be best selected to have a CFL number $1.5 \le C_o \le 3.5$, meaning that the number of time steps per period $N_t=T/\delta t$, also given by $N_x/C_o$, is then comprised between $N_x/3.5$ and $N_x/1.5$. This is highly beneficial as it authorizes rather large time-steps for practical applications.
\item the present implementation of the HPC shows an algebraic convergence rate with the spatial resolution of order greater than 4.
\item it also shows an algebraic convergence rate with the temporal resolution of order comprised between 4 and 5 (for long time simulations).
\end{itemize}
\section{Introduction of a fixed body in the NWT \label{sec:theory_fixedBody}}
\subsection{A double mesh strategy to adapt the resolution in the vicinity of a body \label{sec:sub:twoMeshIntro}}
After having validated the method with nonlinear waves in the previous section, and particularly the immersed free-surface strategy, the next objective is to include a body in the fluid domain, either fully submerged or floating. Obviously, a desirable solution would retain cells of square shape and constant geometry as much as possible, even in the case of a moving body (not treated here however).
Once again, different strategies are possible. \citet{HPC:hanssen2015harmonic} first introduced an IB method for bodies in waves in the HPC framework. \citet{HPC:ma2017local} compared this method with an immersed overlapping grid fitted to the boundaries (corresponding to the body in our case). This newly introduced grid will often be referred to as the "fitted mesh" for simplicity.
In the current study, the later strategy is chosen. Two main reasons led to this choice: first, an oscillatory behavior was exhibited by \citet[Figs. 24,25]{HPC:ma2017local} when studying the spatial convergence with an IBM. This oscillatory behavior is also present when the body is moving, with a large magnitude. This is mainly due to an incremental change in the chosen ghost nodes which can turn to be favorable at some time steps (\emph{i.e.}~ for certain grid configurations) and unfavorable at some other ones. Moreover, this oscillatory behavior does not come with a reduction of the error, both for the fixed and oscillating body.
The second reason is that adding a new fitted mesh allows to decouple the discretization of the wave propagation part (usually defined with respect to the wavelength, \emph{e.g.}~ approximately $N_x \in [40-90]$ as shown in the previous section) from the discretization appropriate for the resolution of the potential close to the body (usually defined with respect to the body characteristic dimension, denoted $D$). Thus, by using two different grid, a suitable discretization for both the wavelength and the computation of the loads would be possible. For instance, including a small body relative to the incoming wavelength would be challenging with the IBM: too many nodes would be required so as to correctly solve the BVP in the vicinity of the body whereas less nodes would be needed further away. From a quantitative point of view, the case inspired from \citet{chaplin1984nonlinear} and treated in \S~\ref{ssec:chaplin} hereafter involves an important ratio $\lambda/D\approx 15$. Thus, if the far-field discretization is set as $N_x=90$ to correctly capture the propagation of the waves, then $D$ is discretized with only $6$ nodes. That high ratio $\lambda/D$, often encountered in practical engineering applications, is more easily taken into account with a second grid fitting the body than with a choice of higher order cell or with a local refinement of the grid. Note that the solution combining both a secondary grid and a solid immersed boundary has not yet been tested and would be of great interest. On this subject \citet[\S~4.2.2]{HPC:ma2017local} applied this combination to model the free surface and obtained a important reduction of the resulting error.
\subsection{The two way-communication inside the fluid domain \label{sec:sub:twoMeshComm}}
Thus, a boundary fitted grid (BFG) is added locally around the body, overlapping the background grid (BGG). These grids and the points associated to the method are represented on figure~(\ref{sch:twoMeshesCom}). The Laplace problem is solved on both these grids simultaneously, \emph{i.e.}~ both domains are solved in the same global matrix problem. The global matrix size is increased by the number of nodes of the BFG and decreased by the number of nodes inactivated in the BGG. One can note that the global matrix is thus almost defined by block, each corresponding to a grid.
\begin{figure}[htbp!]
\begin{center}
\includegraphics{standalone/twomeshcomm}
\caption{Schematic representation of the immersed free surface and body below the free surface. A circle means an "interpolation" node (described in the text) while the colors are used to identify particular nodes on the two grids, as indicated in the legend.}
\label{sch:twoMeshesCom}
\end{center}
\end{figure}
Thus, the boundary nodes of the new BFG also need an dedicated equation in order for the system to be closed.
For a node laying on the body boundary, a simple Neumann boundary condition is set and enforced in the global matrix, as described in \S~\ref{sec:sub:dirandneucond}.
For an "interpolation node" $P_f$ (green circle markers on figure~\ref{sch:twoMeshesCom}) located on the outer contour of the BFG, the imposed equation in the global matrix is the interpolation equation from the closest macro-cell in the BGG (equation~\ref{eq:mainwithcij}).
On figure~(\ref{sch:twoMeshesCom}), a double arrow gives a representative example of the link between $P_f$ and the center of the closest macro-cell in the BGG.
This ensures - in a implicit manner - that the potential at the location $\tI{x}_{P_f}$ is the same in both meshes:
\begin{equation}
\phi^{(f)}_{P_f} =\phi^{(bg)}(\tI{x}_{P_f})
\label{eq:communitationInterpolationFromBg}
\end{equation}
where $\phi^{(f)}_{P_f}$ is the potential of the particular node $P_f$ (directly an unknown of our system of equations), and $\phi^{(bg)}(\tI{x}_{P_f})$ represents the value of the interpolation equation from the background potential field at the given coordinate $\tI{x}_{P_f}$. Further developing equation (\ref{eq:communitationInterpolationFromBg}) and using the closest macro-cell local expression (\ref{eq:mainwithcij}) yield an implicit interpolation equation:
\begin{equation}
\phi^{(f)}_{P_f} =\left. \sum_{i=1}^8 \left(\sum_{j=1}^8 C_{ji}^{-1} f_j(\tI{\bar{x}}_{P_f})\right) \phi_{P_i}\right|_{(bgc)}
\end{equation}
Where the notation $\left.\right|_{(bgc)}$ emphasis that the local expression is applied on the closest background macro-cell. $\phi_{P_i}$ are the potential of the bounding nodes of this cell.
Only for those interpolation nodes, the part of the matrix corresponding to the BFG is not defined by block: the potential of a node $P_f$ in the fitted grid is implicitly linked with the potentials of the neighboring background nodes.
Points belonging to the BGG situated inside the body are inactivated (black triangles on figure~(\ref{sch:twoMeshesCom}).
Thus, equations are needed for the points of the BGG surrounding those inactive points (red circle markers on figure~\ref{sch:twoMeshesCom}).
The same method is used here: the interpolation equation~\ref{eq:mainwithcij} is enforced such that the interpolation of the fitted grid potential matches the node potential. In other words, the interpolation is effective from the BFG to the BGG, using the LE of the closest cell of the fitted mesh. Thus, denoting this point $P_{bg}$ and its coordinates $\tI{x}_{bg}$:
\begin{equation}
\phi^{(bg)}_{P_{bg}}=\phi^{(f)}(\tI{x}_{P_{bg}})
\label{eq:communitationInterpolationFromFitted}
\end{equation}
Here again, this relation is represented for one particular node on figure~(\ref{sch:twoMeshesCom}) by a double arrow.
So, by considering the various type of nodes discussed above, the proposed method ensures a consistent implicit two-way communication between the two meshes of interest, as the BVP problems (on $\phi$ and $\phi_t$) are solved on both grids simultaneously.
\subsection{Free surface piercing body \label{sec:sub:Fscom}}
At this stage, a two way communication is ensured between the fitted grid (BFG) and the background grid (BGG).
The problem is closed in the sense that every node involved in the global matrix has its own dedicated equation.
Still, a difficulty arises when the fitted mesh pierces the free surface.
Indeed, it is not possible to interpolate outer points of the fitted mesh where no solution is computed (above the free surface):
This issue is solved by introducing a new free surface, evolving in the BFG. This method is a variation of the presented technique in \citet{tong_numerical_2019} applied on a piston-type wave maker.
This allows to solve and advance the free surface locally at the scale of the body. In this work, having a dedicated discretization in the vicinity of the body was proven to be necessary, when for example the reflection on the body resulted in waves of short wavelength and large steepness.
The free surface evolving in the background grid is truncated such that no marker is defined inside the body (\emph{i.e.}~ markers are only present in the fluid domain) as can be seen in figure~(\ref{sch:twoMeshesFs})
\begin{figure}[htbp!]
\begin{center}
\includegraphics{standalone/twomeshfs}
\caption{Schematic representation of the background free surface and the fitted free surface. A matching is enforced between the two free surfaces.
The difference between the two free surfaces is here exaggerated for clarity.}
\label{sch:twoMeshesFs}
\end{center}
\end{figure}
For simplification purposes, the new free surface evolving in the BFG will be called "fitted free surface" even tough this free surface also uses the IBM described in \S~\ref{sec:sub:ifs}.
Thus, we obtain two free surfaces, with different resolutions in space, following their respective grid discretizations, that overlap each other in the vicinity of the body.
To ensure the communication between both free surface curves, the outer nodes positions and values of variables $\phi$ and $\phi_t$ of one free surface are interpolated and enforced trough a 1D B-spline interpolation from the other free surface. Referring to the schematic representation (\ref{sch:twoMeshesFs}), the position and values of the outer right node of the background free surface is enforced so as to match the fitted free surface. Reciprocally, the position and values of the outer left node of the fitted free surface is enforced so as to match the background free surface.
However, if this enforcement affects only one marker at the extremity, instabilities may occur. For example a stencil of two points on each side is the minimal length to maintain a 4\textsuperscript{th} order of spatial convergence with a 1D centered finite difference scheme.
To prevent this from having an important impact, relaxation zones are set to incrementally match the free surfaces at their extremities. Relaxation formulas and weights are thus needed for every marker:
\begin{equation}
\gamma_e = (1-\alpha) \gamma_i + \alpha \gamma_t
\end{equation}
where $\gamma_e$ represents the value to enforce, $\gamma_i$ the initial marker value, $\gamma_t$ the target value (interpolated value from the other free surface) and $\alpha$ an arbitrary weight function of the marker position that evolves between $0$ and $1$. Note that for this application, $\gamma$ stands for either $\phi$, $\phi_t$ or $\eta$.
Many different functions for $\alpha$ were tested and implemented without significant impact. In practice free surfaces are completely matched over a given length (\emph{i.e.}~ $\alpha=1$ if the marker distance to the free surface extremity is lower than a certain threshold, $\alpha=0$ otherwise). The figure~(\ref{sch:twoMeshesFs}) emphasizes the effect of such relaxation functions:
it shows the free surfaces before (dotted lines) and after the matching (solid lines) using this method.
Note that if the body is not at the extremity of the computational domain, a second fitted free surface is needed on the other side of the body. This will be used in \S~\ref{ssec:barge_ECM}.
From a numerical point of view, the fitted mesh is considered as an unstructured grid. At a price of additional coding efforts and an increase of CPU time when identifying points (as well as an increase in the memory usage), a gain is made on the simplicity of inclusion of complex bodies of arbitrary shape. However, as already stated, the HPC method implemented requires square cells to be most effective \citep{HPC:ma2017local}.
Taking advantages of the fact that the BGG will remain a structured mono block, it would thus be possible to modify the methods on this grid to reduce both the RAM requirement and the necessary CPU time associated with identification of node types and interpolations between the resolutions of the BVP themselves.
\section{Validation of wave-body interaction against two flume experiments \label{sec:valid}}
In order to validate the method presented above, two experimental test cases were selected. The first one is chosen so as to verify the boundary-fitted overlapping grid method selected to include a fully immersed body: a fully submerged horizontal cylinder of circular cross-section. The second one is a free surface piercing case: a rectangular barge.
\subsection{Fixed horizontal submerged cylinder \label{ssec:chaplin}}
\citet{chaplin1984nonlinear} studied in detail a fixed horizontal cylinder, with a low submergence below the SWL, in regular waves of period $T=1$~s. Accurate experimental results about the nonlinearities of wave loads on the cylinder were given and are often used in order to validate NWTs \citep[\emph{e.g.}~][]{guerber2011modelisation}. The total water depth is $d=0.85$~m which, together with the period, imposes a wavelength of approximately $\lambda=1.56$~m (slightly varying with the wave height).
The cylinder of diameter $D=0.102$~m is immersed with its center located at $z_c=-D=-0.102$~m below the SWL.
This problem is numerically difficult to solve for volume field methods as the cylinder is close to the free surface, such that the fluid domain right above the cylinder is reduced to a small water gap of height $D/2 \approx \lambda/30$ (when the water is at rest). This water gap needs to be meshed and resolved with the HPC method. Thus, a spatial discretization of $\lambda/\delta x\in [40,90]$ would yield a discretization of this gap with only $\sim 2-3$ nodes.
The nonlinear regular incident waves are generated using the so-called stream function theory \citep{fenton1988numerical}.
To avoid reflection on both the inlet wave maker side and the outlet Neumann wall and to impose the target incident wave field at the inlet, relaxations zones are introduced to enforce the requested values over a distance chosen as $L_{relax}=\lambda$ trough a commonly used exponential weighting function \citep[\emph{e.g.}~][]{jacobsenFuhrmanFredsoe2012}. Note that other techniques of waves generation and absorption are possible. For example \citet{clamond_efficient_2005} introduced a damping term in the Bernoulli equation in order to modify the DFSBC: The wave elevation is smoothly driven to the SWL. Figure~(\ref{sch:cylinder}) shows at scale the computational domain used for the numerical simulations including the two relaxation zones.
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=0.95\textwidth]{standalone/schcylindercase}
\caption{Schematic representation of the numerical set-up inspired from \citet{chaplin1984nonlinear}. Note that the mesh fitted to the cylinder is not represented in this figure.}
\label{sch:cylinder}
\end{center}
\end{figure}
We focus our attention on the vertical force exerted on the cylinder once the periodic wave motion is established in the NWT. A Fourier analysis is applied to the computed times-series of vertical force. The normalized amplitudes of the harmonics of the vertical force and the mean vertical force (drift force) are plotted in Figure~(\ref{fig:Error2DCyl}) as a function of the Keulegan-Carpenter number. On this figure, the results from the present NWT are compared with the experimental values from \citet{chaplin1984nonlinear} and the numerical results from \citet{guerber2011modelisation}. The linear theory results from \citet{ogilvie_1963} are added for comparison of the amplitude of the first harmonic. The Keulegan-Carpenter ($KC$) is defined as:
\begin{equation}
KC= \pi\frac{H}{D}\exp(k z_c)
\label{eq:KC}
\end{equation}
and is thus directly proportional to the incident wave height.
\begin{figure}[!htbp]
\centering
\includegraphics{cylinderharmonics}
\caption{Amplitudes of the various harmonic components of the vertical load on the horizontal circular cylinder. Current results (crosses) compared to numerical simulations from~\citet{guerber2011modelisation} (lines with diamonds) and experiments from~\citet{chaplin1984nonlinear} (empty circles). The amplitude of the first order harmonic based on linear prediction is added as well as a third order model line, to compare with the evolution of the amplitude of the third order harmonic.}
\label{fig:Error2DCyl}
\end{figure}
All harmonics amplitudes up to third order are in relative good agreement with the BEM simulations from \citet{guerber2011modelisation}, although some discrepancies can be denoted.
The mean value and first order are difficult to distinguish between the two sets of numerical results, whereas the amplitude of the second order harmonics is closer from the experimental data with the current HPC method. For the third harmonic, which is of very small relative amplitude, is is difficult to identify the most accurate method.
Moreover, the behavior of the different harmonics amplitudes seems to agree with the expected theoretical results: an increase as $KC^1$ of the first amplitude of the harmonic (\emph{i.e.}~ $\hat{f_{z1}}/KC$ is constant), as $KC^2$ of the drift force and second harmonic amplitude, and as $KC^3$ of the third harmonic amplitude. regarding the latter, the HPC method reproduces more closely an order 3 in $KC$ than both \citet{guerber2011modelisation} and the experimental results.
Of course, limitations in the comparison with the experiments can be observed, in particular for larger wave heights. This is mainly due to the viscous effects (not considered here, nor in the simulations of \citet{guerber2011modelisation}) as this stalling is retrieved in the viscous computations of \citet{tavassoli2001interactions}.
However, a difficulty arises with the HPC method when $KC$ increases (\emph{i.e.}~ for larger incident wave heights). The Laplace equation is solved in the different cells inside the fluid volume. Thus, there should always be at least a fluid point in the volume above the cylinder top. In waves of large amplitude, the cylinder is very close to the free surface, in particular when a wave trough passes over the cylinder. In this situation, it is not possible to keep the number of point per wave length $N_x$ in the previously selected range $[40,90]$ and keep square cells. Saw-tooth instabilities appear as $KC$ exceeds $0.80$ approximately. No filter were used in this work, as the main objective is to emphasis the limits of the method itself. Above a wave steepness of $H/\lambda = 2.6\%$, (\emph{i.e.}~ $KC=0.86$), no computation could remain stable after two or three wave periods.
As a conclusion, even if a limitation in wave height is met, those results, correct up to third order, give us confidence on a case which is particularly challenging for volume field methods.
\subsection{Rectangular barge, experiments and numerical comparison \label{ssec:barge_ECM}}
In order to validate the HPC method with a (fixed) free surface piercing body and sharp corners, dedicated experiments were conducted in a wave flume at Centrale Marseille with a body of rectangular cross-section.
\subsubsection{Experimental setup}
The wave flume is $17$~m long and $0.65$~m wide. The water depth was set to $d=0.509$~m. The body is a rectangular barge of draft $0.10$~m for a length of $0.30$~m (in the longitudinal direction of the flume) mounted on a 6-axis load cell measuring device. The width of the barge spans the width of the flume minus a small water gap of about 2 mm on both sides between the barge and the flume walls. A perforated metallic beach is placed at the end of the wave flume to dissipate the energy of the transmitted waves, and so to avoid reflection. The waves are generated with a flap type wave maker.
The body is placed such that its front face is located at $x_b=11.52$~m from the wave maker. 13 wave gauges are installed all along the wave flume. Unfortunately, the wave gauge dedicated to measure the run-up on the front face of the barge was found \emph{a posteriori} to be defective. For some cases, a video of the experiment was recorded in the vicinity of the body, allowing to extract free surface profile and run-up at the front and rear faces of the barge.
\begin{figure}[!htbp]
\centering
\includegraphics[width=0.60\textwidth]{Experimentbarge}
\caption{Photograph of the experimental setup of the rectangular barge in the wave flume close to the body.}
\label{fig:Experimentbarge}
\end{figure}
Approximately 40 cases were tested in regular wave conditions, with varying wave period and height. A focus is made here on two periods: $T=1.1$~s and $T=1.5$~s. For the latter one, however, high wave steepness yielded high wave heights, resulting in dewetting and breaking, with a lot of turbulent effects, recirculations, and air entrainment. Thus, for this period, few relevant comparisons can be made with the potential model (which neglects all those effects).
\begin{figure}[!htbp]
\centering
\includegraphics{figuresbarge/tabexperiments}
\caption{Overview of the conducted experiments in the $(T, H/\lambda)$ plane.}
\label{fig:tabexperiment}
\end{figure}
\subsubsection{Numerical setup}
The computational domain is automatically generated and mesh by the NWT depending on the case: its length is chosen as 8$\lambda$ and relaxation zones are set in the inlet and outlet, with an exponential function over a length of $2\lambda$. A second mesh is fitted to the body in order to ensure a precise computation of the flow dynamics in its vicinity. The mesh fitting the body is of breadth $0.30$~m on each side (\emph{i.e.}~ a total extent of $0.90$~m). Both meshes communicate trough previously described interpolation boundaries. Their free surface curves communicate trough relaxation zones of length $0.14$~m and constant function of unitary weight. In order to interpolate the free surface from the other one, a 1D B-spline interpolation is used. Different discretizations of the fitted mesh were tested, without significant difference.
\subsubsection{Numerical results and comparisons with experiments}
For the wave period $T=1.1$~s, the steepest cases (31-33) were found to numerically break down. This is due to an important run down which leads to a dewetting at the bottom left corner of the barge. During the experiments, recirculations and turbulent effects were clearly visible on those steeper cases.
\begin{figure}[!htbp]
\centering
\includegraphics{figuresbargeRes/ComparisonSondesVideoHPC30Nx2_125}
\caption{Comparisons of the free surface elevations recorded at different positions: HPC (dashed lines), experimental wave gauges (solid lines) and extracted elevation from the experimental video (circle markers). Waves gauges are divided in three groups (corresponding to subfigures): Incident waves upstream (subfig 1), Front and rear run-up (subfig 2), transmitted waves (downstream waves, subfig 3). Case 30: $T=1.1$~s, $H/\lambda=4.8\%$.}
\label{fig:SondesCase30}
\end{figure}
\begin{figure}[!htbp]
\centering
\includegraphics{figuresbargeRes/ComparisonSondesVideoHPC21Nx2_125}
\caption{Comparisons of the free surface elevations recorded at different positions: HPC (dashed lines), experimental wave gauges (solid lines) and extracted elevation from the experimental video (circle markers). Waves gauges are divided in three groups (corresponding to subfigures): Incident waves upstream (subfig 1), Front and rear run-up (subfig 2), transmitted waves (downstream waves, subfig 3). Case 21: $T=1.1$~s, $H/\lambda=1.2\%$.}
\label{fig:SondesCase21}
\end{figure}
On figures~(\ref{fig:SondesCase30} and \ref{fig:SondesCase21}), wave gauge measurements are compared to the free surface elevation time series from the HPC method for cases 30 and 21 respectively. Only the waves gauges appearing in both domains are shown (the computational domain is shorter than the experimental one). Note the time synchronisation between measurements and simulations was done on the first gauge only, and the determined phase shift was then applied to all the remaining gauges.
This means that relative phases of the wave elevations in the numerical simulations are represented adequately in these figures. More generally, a very good agreement is found on both wave amplitudes and phases, at the various locations along the wave flume. However, some discrepancies of the run-up at the front and rear faces of the barge as well as concerning the downstream wave elevations can be observed. Those discrepancies become more marked when the steepness increases. It can be observed that the HPC potential model tends to overestimate larger run-up events, and transmitted wave heights.
It is thought that the main cause of those discrepancies is related to the mathematical model itself. It is well known that non-dissipative models cannot correctly describe the flow in the vicinity of sharp angles even for linear incident waves. Thus, the present potential model is not perfectly appropriate to model the behavior of this type of flow. In practice, this effect is often counteracted by introducing a numerical lid in the vicinity of the body to numerically dissipate some energy (so trying to mimic dissipation due to viscosity). With such a method, the length and strength of the lid need to be tuned to match the expected result. This option was not tested here.
Obviously, those differences are expected to impact the loads exerted on the barge, mainly through the difference of run-up on the front and rear faces, impacting for example the elevation at which the pressure is set to the atmospheric pressure (dynamic boundary condition at the free surface).
\begin{figure}[!htbp]
\centering
\includegraphics{figuresbargeRes/compareFreeSurface30}
\caption{Case 30 $T=1.1$~s, $H/\lambda=4.8\%$. Comparison of the computed free surface elevation in front of the rectangular barge with the experiment at six different time instants. Incident waves come from the left.}
\label{fig:compareFreeSurface30}
\end{figure}
Figure~(\ref{fig:compareFreeSurface30}) shows at different time instants the free surface elevation in front of the body obtained from the HPC computation superimposed on snapshots from the experiments. On these pictures, we clearly denote aspects which cannot be taken into account in the potential model: during the rise of the water level, air entrainment and wave breaking take place, leading to important complex turbulent effects. At this stage, it is expected that the viscous effect play an important role. Although those dissipative effects are neglected in the HPC model, a relative fair agreement can be seen on the figure. The run-up is approximately correctly captured, as well as the reflected wave emerging during the elevation of the run-up. Moreover, the numerically computed reflected wave seems to be slightly faster than the experimental one. The difference can again be attributed to viscous effects, delaying the apparition of the reflected wave.
\begin{figure}[!htbp]
\centering
\subfloat[Case 21]{\label{fig:sub:f21}\includegraphics{figuresbargeRes/temporalLoads21Nx2_125.pdf}}
\subfloat[Case 30]{\label{fig:sub:f30}\includegraphics{figuresbargeRes/temporalLoads30Nx2_125.pdf}}
\caption{Case 21 and 30 $T=1.1s$, $H/\lambda=1.2\%$ and $H/\lambda=4.8\%$. Time series of the hydrodynamic loads on the barge: experiments, linear results and HPC simulations.}
\label{fig:temporalLoads21}
\end{figure}
As the objective is to compare loads applied to the barge, time series of horizontal ($F_x$) and vertical ($F_z$) force components on the body are depicted on figure~(\ref{fig:temporalLoads21}) for the cases 21 and 30, respectively the most linear and nonlinear cases with $T=1.1$~s. The horizontal orange lines (symmetric with respect to the zero-force line) represent the amplitude of linear predictions of these loads.
When the incoming waves are close to be linear (case 21), a good agreement is found between the linear and HPC models: the amplitudes are almost equal though the mean value computed with the HPC model slightly moves the vertical load extrema from the symmetric horizontal lines of the linear model result.
Note that the effective (local) steepness, which determines the degree of nonlinearity, close to the body is approximately twice the incident steepness $H/\lambda$: only a small part of the energy is transmitted and thus the reflected wave adds to the incident wave locally. For a quantitative analysis, the amplitudes of incident and transmitted waves elevation can be compared in figure~(\ref{fig:SondesCase21}).
When compared with the experimental data, the computed loads are of approximately $15\%$ higher magnitude.
Note that, from an engineering point of view, the model is conservative in this case, and as expected, the differences in terms of run-up on the front and rear sides of the body lead to a slight over-prediction of both the horizontal and vertical loads. This over-prediction is seen for all cases, and consistently with approximately the same relative value.
For steep incoming waves, the discrepancies between the linear and nonlinear models increase, as one could expect. In particular, the mean value (drift force) is no longer null. For the case 30, one can note that, although the loads are over predicted in magnitude, the shape of the time series of the loads (and thus the magnitude of the higher order harmonics of the loads) seems to be in good agreement with the experiment.
\begin{figure}[!htbp]
\centering
\includegraphics{figuresbargeRes/harmonicsT11Nx2_250}
\caption{$T=1.1$~s, Amplitudes of harmonics of the vertical and horizontal loads on the barge. HPC models (dashed lines), experimental results (dots) and linear transfer functions (solid orange line). A power function of order 3 is added to compare the increase rate (solid purple line).}
\label{fig:harmonicsT11}
\end{figure}
In order to compare the computed loads with the experimental ones more precisely, a Fourier decomposition of the time series is performed. The Fourier coefficients of this decomposition are shown for all the cases performed at $T=1.1$~s on figure~(\ref{fig:harmonicsT11}).
As expected the amplitudes of the first harmonic exhibit an over-estimation compared to the experimental data, whereas the mean values ($\hat{f}_{xz}^0)$ are in correct agreement.
However, the evolution of all the harmonics with the wave height is found to be exactly the same as measured during the experiments (experimental and computed lines are parallel).
For the vertical loads, all harmonics are roughly consistent with the measurements.
An important difference can be noticed for the third order harmonic of the vertical load for small steepness. First, it can be denoted that this harmonic is 2 orders of magnitude lower than the first order harmonic, and even 3 orders magnitude lower than the hydrostatic force. Thus, an accurate computation as well as an accurate measurement of this small contribution is difficult to achieve. Moreover the behavior of the potential values seems to be more easily fitted with an order 3 model in steepness, which, together with the fact that numerical and experimental third order harmonics match for higher steepness, tends to validate the numerical results. It is thought that either measurements is not accurate enough to capture such a small contribution, or that a physical effect giving energy to the third harmonic plays a role not properly taken into account by the model.
For the horizontal loads, more discrepancies are visible. First order and drift force show an overestimation of about 15\%. The amplitude of the third order harmonic seems to be well captured, but the amplitude of the second order one exhibits a larger over-estimation by about 70\%. This effect is still under investigation, but the impact on the free surface of the viscous effects seems to be significant in this case. Note that the hydrostatic contribution originating from instantaneous wet free surface is one of the main contributions to the second order component. In the present setup, this term will play a role only on the horizontal load as the wet free surface changes only on the vertical walls of the barge. Thus, an important difference between the experiments and the numerical model is expected on this particular second order harmonic of the horizontal load: it was shown previously that our computations do not capture exactly the effective run-up elevations, probably due to the viscous effects not taken into account in the HPC model.
\section{Summary of conclusions and outlook} \label{sec:conclusions}
In this work, an implementation of the Harmonic Polynomial Cell (HPC) method for solving the fully nonlinear potential flow problem in combination with an Immersed Boundary Method (IBM) to track the free surface was achieved and validated. The newly developed Numerical Wave Tank (NWT) allows to simulate a wide range of situations in ocean and coastal engineering applications, involving nonlinear waves in uniform or variable water depth as well as wave-body interaction. In this version, the bodies in interaction with waves can be immersed or surface piercing, but have to be fixed. The numerical techniques of this NWT have been described, in particular the implications of choosing to work with non-deforming grids which has lead to the development of an accurate and robust IBM variant.
After investigating the convergence properties of the numerical methods at long time with respect to a refinement in space and time on a freely evolving standing wave, it was shown that the NWT exhibits a high level of accuracy and other interesting features. First, a large range of time steps are shown to lead to stable computations over the target duration of 100 wave periods, although for small CFL $C_o$ numbers the results tend to become unstable. This difference can however mostly be explained by the fact that no filtering nor smoothing were used during this work. Most importantly, the recommended range of CFL for this HPC implementation was shown to be $C_o \in [1.5, 3.5]$, which is of great benefit as these large values permit to use large time steps, and so to reduce the computational burden. The refinement in space discretization should lie in the range $[40,90]$ for the number of nodes per wavelength. On this case, results are found to be accurate and converging with an order of convergence comprised between 4 and 5, when refining either the spatial or temporal discretization.
A second Boundary Value Problem is introduced and solved on the time derivative of the potential in order to obtain an accurate estimation of the pressure in the fluid domain through the Bernoulli equation.
This method ensures a precise computation of loads on bodies, which will further be needed to compute the movements of a freely floating body.
An multi overlapping-mesh method was also developed as a way to compute the flow precisely in the vicinity of an object subjected to incoming regular waves with a finer local grid. In case the body pierces the free surface, a second free surface is added close to the body and evolves in the body fitted mesh. Specific strategies to couple the two (here fixed) grids and associated (time varying) free surface curves were proposed. This double mesh technique ensures a precise computation of complex flow patterns which can arise due to the presence of the body.
Coupled with relaxation zones to both generate and absorb waves at extremities of the NWT, the method is shown to be accurate for several 2D cases. In particular, the double mesh approach is shown to give accurate results on a fully submerged horizontal cylinder located very close to the free surface \citep[flume experiments by][]{chaplin1984nonlinear}. We have shown that the mean (drift) vertical force on the cylinder and the amplitudes of the three first harmonic of this force were properly estimated by the NWT for a range of Keulegan-Carpenter number. The nonlinear capabilities of the model are thus confirmed.
As a final validation test, the NWT is used to simulate a series of dedicated experiments performed in the wave flume at Centrale Marseille with a surface piercing rectangular barge. As the present model does not consider viscous effects, this last case proved to be really challenging, in particular in high wave conditions and due to the presence of sharp corners at the bottom of the barge. In these conditions, indeed, dissipative terms cannot be neglected if one wants an accurate prediction of the physical values.
Nevertheless, the NWT showed very good results for cases with low steepness incident waves (loads of the barge, run-up on the barge's vertical sides, transmitted and reflected waves). As the wave steepness increases, the simulations still reproduce the time evolution of free surface elevation, loads and run-up signals, but the differences in amplitude become more marked. In these conditions, errors versus the experiments can be up to 15\% on the amplitude of the first harmonic of the loads. Higher harmonics tend also to be overestimated compared to the experimental results. We however like to insist of the fact that some of the cases simulated here were extreme in the sense that waves started to break on the barge and we almost reached the dewetting of the front face of the barge (as shown in the pictures of the experiment on figure~\ref{fig:compareFreeSurface30}). This being considered, the ability of the NWT to run on such cases and to deliver a correct (though slightly overestimated) order of magnitude of loads and run-up elevations is regarded as a valuable outcome of this study.
Future developments of this work will encompass four main aspects, briefly outlined hereafter.
First, the HPC model needs to be extended to 3D in order to simulate more realistic cases.
Second, we plan to implement the case of moving bodies in waves. In this direction the code has already been developed so as to allow for a relative motion between the body fitted mesh and the background mesh. In fact, first tests were conducted but some developments are still required, particularly the complete computation of the derivative of the potential $\phi_t$ at the moving nodes where a Neumann condition is imposed.
Third, implementing a fully Lagrangian tracking method for the free surface nodes would be desirable. This is also of particular importance for nodes attached to body boundaries. The current semi-Lagrangian method can only describe a vertically moving marker.
Fourth, we aim at coupling this potential NWT with local viscous models in the vicinity of bodies. Current developments focus on the latter axis, with the aim to improve the simulation of cases where viscous effects are significant, as was shown here for the case of the rectangular barge in large wave conditions.
\section*{Acknowledgments}
The participation and help provided by Dr. Olivier Kimmoun during the experiments conducted in the wave flume at Centrale Marseille and presented in \S~\ref{ssec:barge_ECM} are gratefully acknowledged. The authors also express their gratitude to Bernard Molin for his comments on a preliminary version of this paper.
\paragraph{Funding}
This work was supported by the Ecole Normale Sup\'erieure (ENS) de Cachan in the form of a \emph{Ph.D.} grant attributed to Fabien Robaux. This research did not receive any other specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
\pagebreak
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2024-02-18T23:40:24.866Z
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2020-09-21T02:17:41.000Z
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{"paloma_paragraphs":[]}
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{"arxiv_id":"2009.08937","language":"en","timestamp":1600654661000,"url":"https:\/\/arxiv.org\/abs\/2009.08937","yymm":"2009"}
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proofpile-arXiv_000-10104
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{"provenance":"002.jsonl.gz:10105"}
| null | null |
\section{Introduction \label{sec:intro}}
The nuclei of active galaxies are remarkable in possessing observed bolometric luminosities up to $10^{48}-10^{49}\,\rm erg/s$, surpassing the energy output of their host galaxies, while being unresolved down to the smallest physical scales observable. The stringent requirements of extreme energy outputs in restricted volumes, together with rapid variability and the presence of powerful relativistic jets, led early on to modeling active galactic nuclei (AGN) as accreting supermassive black holes of masses up to $10^{9}~\rm M_{\odot}$ and accretion rates exceeding $1~\rm M_{\odot}/year$~\citep{1963MNRAS.125..169H,1999qagn.book.....K}. The geometry of an inner accretion disk and an outer dusty torus surrounding the black hole allows for a qualitative view of the different types of AGN in terms of the orientation of the line-of-sight relative to the disk and torus axes aligned with the black hole rotation axis~\citep{1993ARA&A..31..473A}. Anisotropic emission causes over-estimates of the AGN luminosity for privileged lines-of sight. In standard AGN scenarios, jets can be powered either by the inner regions of the radiation dominated accretion disk~\citep{1982MNRAS.199..883B,2015SSRv..191..441H}, or by the rapid rotation of the black hole~\citep{1977MNRAS.179..433B,2011MNRAS.418L..79T}.
AGN constitute the most common type of GeV $\gamma$-ray source in the sky. Most of the objects detected with the Large Area Telescope (LAT) on board {\em Fermi} are extragalactic, and the vast majority of them are blazars, either BL Lac objects or flat-spectrum radio quasars~\citep{2015ApJS..218...23A,2017ApJS..232...18A}. Relativistic models considering the alignment of AGN jets with our line of sight can account, in most cases, for the high energetics and rapid variability observed in $\gamma$-ray emission up to TeV energies~\citep{1993ApJ...416..458D,2018Galax...6...68L}. The detection of radiogalaxies and Seyfert galaxies with the {\em Fermi}-LAT and Imaging Atmospheric Cherenkov Telescopes (IACTs) provides further insight into the jet powered view of AGN under a unified scheme, as probes of off-axis $\gamma$-ray emission~\citep{2018Galax...6..116R}.
For over three decades AGN have been suspected to be the sources of ultra-high-energy cosmic rays: acceleration may occur either in the central engine, the relativistic jet, or in distant radio lobes~\citep{1984ARA&A..22..425H,2011ARA&A..49..119K}. The coincidence of the 290~TeV neutrino event IceCube-170922A with the $\gamma$-ray emitting BL Lac TXS~0506+056 provided fresh observational support~\citep{2018Sci...361..147I}. Observations at the highest photon energies are important to characterize the extreme energetics of AGN. While space borne instruments conduct deep and wide field-of-view observations leading to all-sky surveys in the high-energy range (HE; 0.1-100~GeV), most of our knowledge in the very-high energy (VHE; $\gtrsim 100\,\rm GeV$) regime comes from pointed observations with IACTs, nowadays able to detect individual sources with fluxes greater than 1/1000 that of the Crab Nebula. IACT observatories rely on performing deep, but sparse, follow-up observations of active objects, adequate selections of targets and surveying regions of particular interest. They are limited by their relatively small fields-of-view, with the most extensive survey performed with these instruments to date, the dedicated H.E.S.S. Galactic Plane Survey, covering $\sim$0.3~steradians in the course of a decade~\citep{2018A&A...612A...1H}.
Compensating their lower instantaneous sensitivity with steradian fields-of-view, high altitude air shower arrays are now able to perform unbiased continuous monitoring of known AGN, in particular Markarian~421 and Markarian~501~\citep{2017ApJ...841..100A}. This preludes their potential to conduct large surveys with sufficient depth to detect extragalactic TeV sources. The main hurdle for reaching the extragalactic sky is the access to the lowest photon energies, as our view of the TeV sky is impaired by extragalactic background light (EBL). Photon-photon pair production of TeV $\gamma$ rays with EBL photons sets a physical limit on how far in distance and in spectral range extragalactic source can be observed~\citep{1998ApJ...493..547S}. Pair production proceeds efficiently just above its kinematic threshold, making TeV $\gamma$ rays prone to interact with infrared radiation.
This paper presents an AGN follow-up survey performed with four and a half years of full-operations data from the High Altitude Water Cherenkov (HAWC) Gamma-Ray Observatory, building on the preliminary release of this work with a somewhat smaller HAWC dataset~\citep{2019arXiv190806831C}. The HAWC time-integrated data cover $\sim 60\%$ of the sky, extending the {\em Fermi}-LAT all-sky survey to a search for persistent TeV $\gamma$-ray emission. The HAWC survey encompasses all AGN in the Third Catalog of Hard {\em Fermi}-LAT sources (3FHL) accesible from the HAWC site with a redshift $z\leq 0.3$, of which about 20\% have been previously reported in the VHE range through IACT pointed observations in different states of activity.
The paper is organized as follows:
\S\ref{sec:hawcgro} describes the HAWC observatory, its data and standard analysis;
\S\ref{sec:ebl} provides a brief review of the EBL and its effect on TeV spectra;
\S\ref{sec:sample} presents the sample of AGN drawn from the 3FHL catalog, with due considerations of ground-based IACT observations;
\S\ref{sec:3fhl-follow-up} discusses the dedicated follow-up study of the sample, leading to the summary and conclusions in \S\ref{sec:summary}.
\section{The HAWC Gamma-Ray Observatory \label{sec:hawcgro}}
HAWC is a wide field-of-view TeV $\gamma$-ray observatory optimized for surveying cosmic high-energy sources. It is located inside the Parque Nacional Pico de Orizaba, in the Mexican state of Puebla. The HAWC array occupies a relatively flat area of the Volc\'an Sierra Negra mountain, at an altitude of 4100~meters, centered at geographical latitude $18.995^\circ\rm N$ and longitude $97.308^\circ\rm W$. HAWC has achieved a $\gtrsim 95\%$ duty cycle, allowing it to survey two thirds of the sky every sidereal day with sufficient depth to detect the Crab Nebula at the $5\sigma$ level~\citep{2017ApJ...843...39A}. This study improves on the 2HWC survey performed with the first year and a half of data, that allowed the detection of 39 sources of TeV $\gamma$ rays~\citep{2017ApJ...843...40A}. The most recent HAWC all-sky survey, 3HWC, uses the same data as in here. It contains 65~TeV $\gamma$-ray sources, most of them along the Galactic Plane, including the $189\sigma$ detection of the Crab Nebula~\citep{2020arXiv200708582A}.
\subsection{The HAWC detector}
HAWC is an extensive air shower (EAS) array sampling in detail secondary particles produced by primary cosmic rays in the upper atmosphere. HAWC data analysis can distinguish between hadronic and $\gamma$-ray induced cascades through their different charge distributions at the ground. The observatory consists of a dense array of 300 large water Cherenkov detectors (WCDs) covering collectively a physical area larger than 22,000~m$^2$. Each WCD is a cylindrical tank of 7.3~m diameter and 5~m height, filled with 180~m$^3$ of water and instrumented with four upward-facing photomultiplier tubes (PMTs) at its base. The signals from the 1200 PMT channels are brought to the data acquisition system located near the center of the array to be processed in real time. HAWC has been in full-operations since its inauguration on March 20, 2015, after two years of gathering data with a partial array configuration~\citep{2016ApJ...817....3A}. Further details about the observatory can be found in \citet{2017ApJ...843...39A}.
\subsection{HAWC data and standard analysis \label{subsec:analysis}}
The HAWC array records about 25,000 events per second, the vast majority caused by hadronic cosmic rays. Each data record contains the particle arrival timing and deposited charge on each of the 1200 PMTs which are used to locate the event in the sky and to perform photon/hadron discrimination. The analysis presented here follows the validation observation of the Crab Nebula, both in the $\gamma$-hadron cuts used and in partitioning the data in nine bins according to the fraction of channels hit, as indicated in Table~2 of \citet{2017ApJ...843...39A}. The bin number provides a coarse measure of the primary energy, with an important overlap in the energy distributions of different bins due to the fluctuations inherent in the development of particle cascades. Lower bins relate to lower energies, and the spatial resolution improves with increasing bin number, with the detailed detector response depending on the spectrum of the source and its declination. {The lowest bin used in this analysis, ${\cal B}=1$, has peak sensitivity around $0.5~\rm TeV$ for a source with a power-law spectrum of index 2.63 culminating at the zenith, as the Crab Nebula.}
The HAWC sky surveys have a mean photon energy of about 7~TeV and one-year sensitivity between 50 and 100~mCrab~\citep{2017ApJ...843...40A}. HAWC data analysis is based on computing the likelihood ratio of a source+background to a background only models, given by the test statistic,
\begin{equation}
{TS} = 2 \ln\left\{ {\cal L}(S+B) \over {\cal L}(B) \right\} , \label{ts}
\end{equation}
where ${\cal L}(A)$ is the likelihood of model $A$, given by the product of the probability density function computed at each point of the region of interest. Given a $TS$ value, its statistical significance can be approximated by $s=\pm\sqrt{TS}$, with the sign indicating {an excess or deficit} relative to the background. For all-sky surveys, like 2HWC, the analysis is performed by optimizing $TS$ on every pixel of a $N_{\rm side}=1024$ HEALPix grid model of the observable sky~\citep{2005ApJ...622..759G}. The source model generally consists of either a point source or an extended source hypothesis following a simple power-law spectrum of fixed spectral index, with free normalization. Joint normalization and spectral index optimizations are then performed to further characterize detected sources. The analysis presented here is performed similarly to that of the 2HWC, although on pre-defined sky locations and including the attenuation of TeV photons caused by their interaction with extragalactic background light. We use 1523~days of live data acquired between November 26, 2014, and June 3, 2019. The live data comprises 92.3\% of the total time span. The data deficit is due mostly to quality cuts and run losses during bad weather.
The comparison of HAWC and {\em Fermi}-LAT data requires the consideration of the respective systematic uncertainties of each experiment. HAWC fluxes presented here have an estimated 15\% systematic uncertainty (\S\ref{subsec:analysis}); those in the 3FHL catalog are quoted to have uncertainties of 9\% in the 150-500~GeV band, and 15\% in 0.5-2.0~TeV~\citep{2017ApJS..232...18A}.
\section{Photon-photon attenuation by extragalactic background light} \label{sec:ebl}
The astrophysical relevance of the $\gamma\gamma\to {\rm e}^{+}{\rm e}^{-}$ process as an absorption mechanism for distant sources was pointed out by~\citet{1966PhRvL..16..252G,1967PhRv..155.1404G,1967PhRv..155.1408G}, first in consideration of the cosmic microwave background, and later for more generic backgrounds. Photon-photon pair production is described by the cross section $\sigma_{\gamma\gamma}=\pi r_e^{2}\,\psi(\omega)$, with $r_e$ the classical electron radius, and $\psi$ an analytical function of \mbox{$\omega=\sqrt{E_1 E_2 (1-\mu)/2}$},~the energy of each photon in the center of momentum frame, a relativistic invariant given by the product of the energies of the two photons in an arbitrary frame, $E_{1}=E_{\gamma},\, E_{2}=h\nu$, and $\mu=\cos\theta$, where $\theta$ is their interaction angle. Pair-production requires $\omega\geq m_ec^{2}$, with the cross section maximized at $\omega\approx 1.4~m_ec^{2}\,\Rightarrow\, E_\gamma h\nu\approx 2 (m_e c^{2})^{2} \approx 0.5~\rm TeV\cdot eV$. Hence, 1~TeV $\gamma$ rays are prone to interact with near-infrared photons of 0.5~eV ($\lambda\simeq 2.5\,\mu{\rm m}$), while a 100~TeV photon is to interact with far-infrared extragalactic light, $\lambda\simeq 250\,\mu{\rm m}$.
The absorption of high-energy photons from a distant source of redshift $z$ traversing through intervening radiation is governed by the optical depth,
\begin{equation}
\tau(E_\gamma, z) = {1\over 2} \int_{0}^{d(z)} \int_{0}^{\infty} \int_{-1}^{+1} \sigma_{\gamma\gamma}\left(\omega\right)\, n_\nu(\ell) \, d\mu \, d\nu\, d\ell , \label{ebl-tau}
\end{equation}
where the photon density $n_\nu$ may describe local and/or cosmic intervening radiation fields\footnote{eq.~\ref{ebl-tau} assumes an isotropic $n_\nu$; this may not describe a local radiation field.}. In the case of an evolving cosmic field, a dependence on redshift $n_\nu=n_\nu(z)$ may be introduced. The photon path is integrated using the cosmological light-travel distance $d\ell = c\,dz/(1+z)H(z)$. The probability that the $\gamma$ ray survives the journey is $\exp(-\tau)$. Given the usual functional form of $\tau(E_\gamma,z)$, the survival probability behaves close to a cutoff once $\tau=1$ is reached. In principle the opacity of the Universe to VHE $\gamma$ rays is calculated given $n_\nu(z)$. In practice measurements of the light backgrounds are difficult to perform, particularly in the infrared and far-infrared, and observations of distant $\gamma$-ray sources become relevant for constraining the spectral shape of the EBL, both in the local Universe and as a function of redshift~\citep{2019MNRAS.486.4233A}. In here we use the EBL model of~\citet{2011MNRAS.410.2556D}, which fits well observations by {\em Fermi}-LAT and IACTs.
We note that two of the sources in our sample NGC~1068 and M87, are located below the lower bound of the redshift range of most EBL models, which start at $z=0.01$. For these sources we assume that the photon density has not changed between now and then, to approximate,
\begin{equation}
\tau(E_\gamma, z) \approx \tau(E_\gamma, 0.01) \left(z/0.01\right) \, .
\end{equation}
The analysis in here assumes intrinsic power-law spectra for the sources. We can describe the overall effect of the EBL estimating the observed integrated photon flux ($N_{obs}$; photons cm$^{-2}$s$^{-1}$) for an intrinsic spectrum of index $\alpha$, introducing the relation,
\begin{equation}
N_{obs}(\geq E_0) = \int_{E_0}^{\infty} \left(dN\over dE\right)_{intr} e^{-\tau(E,z)} dE \quad = e^{-z/z_h} N_{intr}(\geq E_0) \, , \label{nobs-ebl}
\end{equation}
where $N_{intr}(\geq E_0)\propto E_0^{-\alpha+1}/(\alpha-1)$ is the integral of the differential intrinsic spectrum. The resulting horizon scale $z_h$ depends strongly on $E_0$ and weakly on the power-law index $\alpha$. Using the EBL model of~\citet{2011MNRAS.410.2556D}, we get $z_h=0.106$ for $\alpha=2.5$ and $E_0=0.5\,\rm TeV$, justifying the bound $z\leq 0.3$ considered for this study. The value of $z_h$ at 0.5~TeV ranges from 0.096 for $\alpha=2.0$, to 0.113 for $\alpha=3.0$; on the other hand, the dependence of $z_h$ with $E_0$ is exponential, going from $z_h=0.728$ at 0.1~TeV, to $z_h=0.068$ at $E_0=1.0\,\rm TeV$, for $\alpha=2.5$, as presented in \citet{2019arXiv190806831C}.
{We compared three main EBL models for the case $\alpha=2.5,\; E_0=0.5\,\rm TeV$. When considering the upper and lower uncertainties of \citet{2011MNRAS.410.2556D}, we get that $z_h$ is in the interval $\left(0.098,\, 0.121\right)$, which is consistent to $z_h=0.102$ obtained with \citet{2012MNRAS.422.3189G} and $z_h=0.099$ with \citet{2017A&A...603A..34F}. These three models coincide within a $\sim 10\%$ systematic uncertainty.}
\section{Active galaxies above 10 GeV: 3FHL and TeV pointed observations \label{sec:sample}}
\subsection{A Sample of Active Galaxies from the 3FHL Catalog \label{subsec:sample}}
The 3FHL catalog contains 1556 objects detected at photon energies between 10~GeV and 2~TeV in the first seven years of {\em Fermi} operations, from August 4, 2008, to August 2, 2015~\citep{2017ApJS..232...18A}. Of the 3FHL entries, 79\% are identified or associated with extragalactic objects, mostly BL Lac objects and flat-spectrum radio quasars (48\% and 11\% of the 3FHL, respectively). As defined in the different LAT catalogs, an association refers to the positional coincidence of the HE $\gamma$-ray source with an object having suitable properties; while an identification requires measuring correlated variability between the $\gamma$-ray source and its associated counterpart. These criteria result in 9\% of the sources in 3FHL listed as identified and 78\% as associated, with the remaining 13\% are unassociated or unclassified. It is customary in {\em Fermi} catalogs to distinguish between identifications using upper case letters, as RDG for identified radio-galaxies, and associations using lower case letters, as rdg for an association with a radio-galaxy.
\begin{deluxetable}{lrc}
\tablecaption{Classes of AGN selected. \label{3fhl-selection}}
\tablehead{\colhead{Source class} & \colhead{Id + As} & \colhead{Total} }
\startdata
\hline
BL Lac objects (BLL + bll) & 6 + 111 & 117 \\
Blazars candidates of uncertain type (bcu) & 8 & 8 \\
Radiogalaxies (RDG + rdg) & 2 + ~~~4 & 6 \\
Flat spectrum radio quasars (FSRQ + fsrq) & 1 + ~~~5 & 6 \\
Starburst galaxies (SBG + sbg) & 1 & 1 \\
\hline
{\bf Total in sample} & 9 + 129 & {\bf 138} \\
\hline
\enddata
\tablecomments{Id = identified sources; As = associated sources}
\end{deluxetable}
The 3FHL catalog partitions its nominal wide 10~GeV-2~TeV spectral interval into five bands. The majority of AGN are detected at $TS>10$ in the two lower energy bands, 10-20~GeV and 20-50~GeV. Spectral cutoffs at energies $\lesssim 50\,\rm GeV$ are common, as it can be noticed in our sample: of the 138 AGN, 50 (14) are detected above $5\sigma$ in the LAT 50-150~GeV (150-500~GeV) intermediate band(s). Furthermore, Mkn~421 and Mkn~501 are the only two AGN studied here with $TS>25$ in the 0.5-2.0~TeV LAT band, the spectral intersection with HAWC.
The 3FHL catalog assigns a flag to sources as follows: TeV=P, when reported at VHE energies; TeV=C for candidates for TeV detection; and TeV=N for non-reported and not candidates. TeV candidates are by definition sources undetected with VHE ground based instruments whose LAT data satisfy three conditions: (i) significance $s>3$ above 50~GeV; (ii) spectral index $< 3$; (iii) integrated photon flux $N(>50\,{\rm GeV})> 10^{-11}\,{\rm cm^{-2}s^{-1}}$. In addition, the 3FHL catalog assesses source variability through the $V_{\rm bayes}$ parameter, the number of Bayes blocks needed to model the light curve. A source with $V_{\rm bayes}=1$ is consistent with a constant flux~\citep{2017ApJS..232...18A}.
We select 3FHL catalog sources identified or associated with AGN with redshifts $z\leq 0.3$ that culminate within $40^\circ$ of the zenith as viewed from the HAWC site.
For the selection of the follow-up sample we used the current version of the 3FHL catalog available at the Fermi Science Support Center\footnote{fits file gll$\_$psch$\_$v13.fits, dated July 2017 at https://fermi.gsfc.nasa.gov/ssc/.}. The sample of 138 objects is summarized in Table~\ref{3fhl-selection}. It contains 32 objects flagged TeV=P (positive VHE detections), and 32 TeV=C (candidates). The sample is grouped in five source classes, defined in the 3FHL, of distinct properties (Table~\ref{3fhl-selection}):
\begin{itemize}
\item {\bf Starburst galaxies} are the nearest and apparently less luminous AGNs in {\em Fermi}-LAT. While prone to host active nuclei, the prevailing $\gamma$-ray emission is dominated by cosmic rays produced in star formation processes. NGC~1068 is the only 3FHL starburst inside the declination range of our selection, and the lowest redshift AGN in our sample~(Figure~\ref{muestra-z-lumi}). Intriguingly, it has recently been associated to one of the hot spots in the neutrino sky, as observed by the IceCube observatory~\citep{2020PhRvL.124e1103A}.
\item {\bf Radiogalaxies (RDG)} are the nearest extragalactic GeV sources dominated by an active nucleus. They appear up to three orders of magnitude more luminous than starbursts~(Figure~\ref{muestra-z-lumi}). Radiogalaxies are attractive targets for HAWC as the relatively close distance translates in a reduced photon-photon attenuation, potentially allowing to sample the far-infrared portion of the EBL through observations above $\sim 10-30~\rm TeV$. Six 3FHL catalog radiogalaxies transit through the field-of-view of HAWC, with redshifts between $z=0.0042$, for M87, and $z=0.029$, for NGC~1218. Four of them have been claimed as VHE sources by IACT collaborations\footnote{3C~264 has a TeV=N flag, as its VHE detection occurred after the publication of the 3FHL catalog.}~\citep{2018Galax...6..116R}.
\item {\bf BL Lacertae objects (BLL)} constitute the majority of known GeV and VHE $\gamma$-ray emitters. They dominate the 3FHL catalog, in particular for redshifts $z\lesssim 0.7$. As expected, BL Lac objects completely dominate our sample with 6 identifications and 111 associations. They span most of the $z\leq 0.3$ range, led by Mkn~421 at $z=0.031$, the nearest and brightest 3FHL AGN.
\item {\bf Flat Spectrum Radio Quasars (FSRQ)} are the most distant and seemingly luminous blazars. Our sample includes six such sources, with redshifts ranging between $z=0.158$ to $z=0.222$. Of these, only PKS~0736+017 has been reported as a TeV source, the nearest FSRQ claimed in the VHE range so far~\citep{2017ICRC...35..627C}.
\item {\bf Blazars Candidates of Uncertain type (bcu)} are AGN poorly characterized across the electromagnetic spectrum. The 3FHL catalog contains 290 bcu's, a good fraction of them in the Southern sky, all with radio-loud associations, and 90\% of them lacking redshift measurements. Only eight such objects satisfy our selection criteria. They are relatively near objects, between $z=0.036$ and $z=0.128$~(Figure~\ref{muestra-z-lumi}). None is a VHE source, but three of them are flagged as TeV candidates.
\end{itemize}
Figure~\ref{muestra-z-lumi} shows the luminosity per solid angle unit, $d_L(z) f_e$, with $f_e$ being the (10~GeV-1~TeV) energy flux from the 3FHL catalog, and $d_L(z)$ the luminosity distance. The actual luminosity of each source depends on the unknown solid angle of emission.
{Prior to the analysis, we identified five objects in our sample close in the sky to bright 2HWC sources that could affect their analysis. We set a conservative distance threshold of 5$^\circ$, equivalent for five times the 68\% containment radius of ${\cal B}=1$, the bin where contamination by a bright source nearby is more likely. As our sample excludes by construction low Galactic latitudes, with all selected sources located at $|b|>5^\circ$, the only concerns were for:
\begin{itemize}
\item 3FHL~J0521.7+2112 located at 3.07$^{\circ}$ from the Crab Nebula;
\item 3FHL~J1041.7+3900, 3FHL~J1100.3+4020, and 3FHL~J1105.8+3944 located respectively at $4.51^\circ$, $2.28^\circ$ and $1.57^\circ$ from Markarian~421;
\item 3FHL~J1652.7+4024 located at $0.68^\circ$ from Markarian~501.
\end{itemize}
Only 3FHL~J0521.7+2112 is associated to a well-known VHE source, the BL Lac object VER~J0521+211. The other four sources are flagged as TeV=N. These five objects were analyzed and tested later for contamination from the bright neighbor source, as detailed for VER~J0521+211 in \S\ref{ver_j0521+211}. From the respective tests we decided to exclude 3FHL~J1105.8+3944 and 3FHL~J1652.7+4024 from our sample.}
We also revised the redshift measurements listed in the 3FHL catalog. Our sample is dominated by BL Lac objects, often with questionable redshifts. Redshifts from the 3FHL catalog were systematically collated with the SIMBAD and NED databases, with the SDSS database consulted in some particular cases. We used as references the redshift surveys of \citet{2013ApJ...764..135S} and \citet{2008ApJS..175...97H}, and in particular the dedicated survey of TeV sources performed with the 10.4m Gran Telescopio Canarias (GTC) by \citet{2017ApJ...837..144P}. While we decided to systematically use the redshifts listed in the 3FHL catalog, we list in Table~\ref{revision-redshift} sources with disputed values, for future reference.
\begin{deluxetable*}{llcccc}
\tablecaption{Selected sources with questioned redshift.\label{revision-redshift}}
\tablehead{
\colhead{3FHL name} & \colhead{Association} & \colhead{$z_{3FHL}$} & \colhead{$z_{rev}$} & \colhead{Notes} & \colhead{References}}
\startdata
3FHL~J0112.1+2245 & S2~0109+22 & 0.265 & [0.35, 0.67] & Detected up to 200~GeV & (1,2,3) \\
3FHL~J0521.7+2112 & TXS 0518+211 & 0.108 & $>0.18$ &Emission up $\gtrsim 1\,\rm TeV$. & (4,5,6) \\
3FHL~J0650.7+2503 & 1ES~0647+250 & 0.208 & $>0.29$ & $z\simeq 0.41$ from host properties. & (4,5,7) \\
3FHL~J1230.2+2517 & ON~246 & 0.135 & $>0.10$ & -- & (5,8) \\
3FHL~J2323.8+4210 & BZB~J2323+4210 & 0.059 & $\geq 0.267$ & TeV candidate & (5) \\
\enddata
\tablerefs{(1)\citet{2008ApJS..175...97H}; (2) \citet{2016MNRAS.458.2836P}; (3) \citet{2018MNRAS.480..879M}; (4) \citet{2013ApJ...764..135S}; (5) \citet{2017ApJ...837..144P}; (6) \citet{2015ICRC...34..864P};
(7) Kotilainen et al. (2011); (8) Nass et al. (1996). }
\end{deluxetable*}
\begin{figure}
\centerline{\includegraphics[width=\hsize]{agnsurvey_hawc_f1.pdf}}
\caption{Luminosity as a function of redshift for the 3FHL sample studied with HAWC. Note the clear separation in redshift and luminosities between the single starburst (yellow point, the nearest object), radio-galaxies (red), intermediate distance bcu's (in blue) and the more distant group of FSRQs (orange). BL Lac objects, in grey, span most of the redshift interval and have the highest observed luminosities.
\label{muestra-z-lumi}}
\end{figure}
\subsection{VHE Ground-Based Observations Related to Our Sample}
AGN have been extensively studied with IACTs, with evidence of VHE $\gamma$-ray emission in almost a hundred of them~\citep{2008ICRC....3.1341W,2016ARA&A..54..725M}. These observations have shown that TeV flaring is an intrinsic characteristic of blazars and radiogalaxies. IACT deep observations, able to reach down a few percent of the flux of the Crab Nebula in a single run, have identified high, medium and low states of activity in several sources. However, continuous long-term coverage of the AGN population cannot be performed with IACTs. The definition of the base level of AGN VHE emission is a pending task. Despite their lower instantaneous sensitivity, EAS arrays with efficient $\gamma$/hadron discrimination can quantify better time-averaged fluxes, integrated over long periods of time, while monitoring for flaring activity, as they continuously drift over large portions of the sky.
The first significant AGN detections at TeV energies were those of Mkn~421 and Mkn~501~\citep{1992Natur.358..477P,1996ApJ...456L..83Q}. These two Markarian galaxies have been extensively studied in the TeV range for over two decades. They are highly variable but remain bright enough over long periods of time to have been detected by EAS arrays like the Tibet Air Shower Array and MILAGRO, both providing first unbiased views of their TeV emission on timescales of years~\citep{2011ApJ...734..110B,2014ApJ...782..110A}. In the last five years, HAWC has been performing an increasingly deeper monitoring of these two BL Lac objects~\citep{2017ApJ...841..100A}. Measurements of their long-term averaged emission are presented in this paper.
IACTs have been able to go deeper and beyond these two well-known blazars by implementing sophisticated technologies to achieve lower energy thresholds, around or below 100~GeV, with improved sensitivities. This has permitted to peer through the EBL horizon up to redshifts $z\lesssim 0.9$~\citep{2015ApJ...815L..22A,2015ApJ...815L..23A}, while sampling different types of extragalactic sources, like:
\begin{itemize}
\item the starbursts galaxies M82 and NGC~253, detected by VERITAS and H.E.S.S. respectively, with fluxes below 1\% of the Crab~\citep{2016CRPhy..17..585O}.
\item four Faranoff-Riley type I {radiogalaxies}: Centaurus~A (too Southern for HAWC); NGC~1275, the most prominent galaxy of the massive Perseus cluster; 3C~264, newly reported as a VHE source; and M87, the massive central elliptical galaxy in the nearby Virgo cluster. We note the ambiguous classification of IC~310 and PKS~0625--35, referred as RDG in the 3FHL, and as unknown type of AGN in TeVCat~\citep{2008ICRC....3.1341W} and by~\citet{2018Galax...6..116R}.
\item at least sixty {BL Lacertae objects} as VHE $\gamma$-ray sources~\citep{2008ICRC....3.1341W,2017ApJS..232...18A}, mostly high-frequency peaked BL Lac objects (HBL; 50 sources), plus a few intermediate-frequency peaked BLL (IBL; 8 sources, including BL~Lac, W~Comae, and VER~J0521+211, in our sample); and only two Low-frequency peak BL Lacs (LBL). While most of the IACT observations have been reported with sub-TeV thresholds, we note the observations of H~1426+428 by VERITAS and HEGRA at energies between 0.25 and 2.5~TeV, reporting fluxes from 3 to 10\% of the Crab~\citep{2002ApJ...580..104P,2002A&A...384L..23A}.
\item seven {FSRQs}, of which PKS~0736+017 is the only FSRQ detected at VHE with $z<0.3$ and entering the field-of-view of HAWC. This object was found in a flaring state at a flux level of 100~mCrab between 100 and 300~GeV, and showing a steep spectrum~\citep{2017ICRC...35..627C}.
\end{itemize}
\subsection{Photon Flux Extrapolations\label{extrapol}}
{We calculated the expected integral photon flux above 0.5~TeV by extrapolating the spectral models and parameters listed in the 3FHL.} While in HAWC a distinction is made between the {\em intrinsic} spectrum of a source (as emitted) and the related {\em observable} parameters (attenuated by the EBL), {\em Fermi}-LAT spectral models do not need to make this distinction. The difference is minor in most of the LAT energy regime, with only a moderate increase in spectral indices for sources with $z\gtrsim 1$~\citep{2017ApJS..232...18A}. Therefore, we added the attenuation effect of the EBL to the LAT spectral models. The extrapolated fluxes as function of redshift are shown in Figure~\ref{3fhl-xtrapol}. The horizontal red line corresponds to a photon flux of 3\% of the Crab Nebula, indicative of potential HAWC detectability within the current data.
Mkn~421 and Mkn~501 stand clearly at about an order of magnitude above the red line, with a few other target candidates above the 30~mCrab reference: M87, the nearest radiogalaxy in our sample; IC~310, already reported as a VHE source; 1ES~2344+514, TXS~0210+515, and I~Zw~187 (1ES1727+502), three of the nearest BL Lac objects known; and the mildly distant B3~2247+381, clearly detected by {\em Fermi}-LAT up to 500~GeV. Only TXS~0210+515 is undetected in the VHE range.
\begin{figure}
\includegraphics[width=\hsize]{agnsurvey_hawc_f2.pdf}
\caption{Photon flux extrapolations to $E>0.5\,\rm TeV$ using the spectral models of the 3FHL catalog and $\gamma\gamma$ attenuation by the EBL. Colors in the dots indicate the TeV flag assigned in the 3FHL catalog. The horizontal red dotted line represents a flux equivalent 30~mCrab, which we take as a benchmark of potential HAWC detectability. We identify the VHE sources VER~J0521+211 and 1ES~1215+303 with respective green rings below the red line.
\label{3fhl-xtrapol}}
\end{figure}
We also computed photon flux extrapolations of VHE sources within TeVCat, using simple power laws and approximating the EBL attenuation by a direct exponential in redshift (\S\ref{sec:ebl}). The predictions from this second set are more uncertain, as IACTs observations are by nature short and sparse, reflect different activity states, and spectra are not always fitted above 1~TeV. Five candidates stood out with extrapolated fluxes $N(>0.5\,{\rm TeV})\geq 20~{\rm mCrab}$, four of them in common with the 3FHL extrapolation: M87, Mkn~421, Mkn~501, H~1426+428, and 1ES~2344+514. Results for the sources named in this section are shown and discussed in the following section~\S\ref{sec:3fhl-follow-up}.
\section{HAWC follow-up survey of 3FHL AGN \label{sec:3fhl-follow-up}}
The HAWC follow-up survey consists of a systematic search for TeV $\gamma$-ray emission from each of the 138 AGN selected from the {\em Fermi}-LAT 3FHL catalog. We performed a maximum likelihood test assuming a point source at the location of the presumed 3FHL counterpart with an intrinsic power-law differential flux spectra of index $\alpha$ attenuated by the EBL,
\begin{equation}
\left(dN\over dE\right)_{\rm obs} = K \left(E/E_0\right)^{-\alpha}\, e^{-\tau(E,z)} \, . \label{hawc-fit}
\end{equation}
A first analysis run was performed with a fixed spectral index, $\alpha=2.5$, fitting only the normalization $K$ for a pivot energy $E_0=1\,\rm TeV$. From this we obtained:
\begin{itemize}
\item The compilation of $TS$ values and 1~TeV normalizations $K$ for the revised sample of 136 AGN\footnote{3FHL~J1105.8+3944 and 3FHL~J1652.7+4024 excluded.};
\item Statistics of significances ($s\equiv \pm \sqrt{TS}$) for source classes and TeV flags. Following Wilk's theorem, under the null hypothesis the behavior of $s$ tends to a Gaussian distribution, of mean $\mu(s)\to 0$ and standard deviation $\sigma(s)\to 1/\sqrt{N}$, for $N$ points;
\item The corresponding list of 2$\sigma$ ($\sim 95\%$ confidence level) upper limits on the normalization, $K_{2\sigma}$, computed under the~\citet{1998PhRvD..57.3873F} approach, that allows only non-negative fluxes;
\item The comparison of flux values and upper limits with the extrapolations performed in \S\ref{extrapol};
\item The characterization of upper limits as function of declination and redshift.
\end{itemize}
For those sources with test-statistic $TS>9$ we then computed optimized spectral fits, allowing both $K$ and $\alpha$ to vary. The discussion of the five sources complying with this criterion is presented in \S\ref{subsec:optimized}. Finally, we also computed quasi differential limits for all the sources in the sample in three energy intervals, (0.5-2.0)~TeV, (2.0-8.0)~TeV, and (8.0-32)~TeV, following the procedure implemented in~\citet{2017A&A...607A.115I}. The first energy interval, the HAWC overlap with 3FHL, is where we expect AGN emission to be brighter, while the second interval includes the peak of HAWC sensitivity. The (8.0-32.0)~TeV range put bounds where the EBL attenuation is more severe. These quasi-differential limits are computed assuming $\alpha=2.0$ {\em without} consideration of EBL attenuation, as detailed in \S4.3.3 of~\citet{2017A&A...607A.115I}.
\subsection{The $\alpha=2.5$ AGN search \label{subsec:alpha2.5}}
{The default spectral model for the AGN search was an index $\alpha=2.5$, as used in the 3HWC~\citep{2020arXiv200708582A}. However note that in here the index refers to the intrinsic spectrum and that the observed spectra will be softer. While we know from previous analyses that this index can represent fairly well the data from both Mkn~421 and Mkn~501~\citep{2019ICRC...36..654C}, photon indices in extragalactic 3FHL sources peak in the interval between 2.0 and 2.5~\citep{2017ApJS..232...18A}.}
\subsubsection{Overall results}
The results for the complete AGN sample are shown in Table~\ref{results-2.5}. The left-hand-side of Figure~\ref{signif-all} shows the histogram of significances for the sample, also displayed as the first entry of Table~\ref{stats-groups}. Mkn~421 is detected with a significance $\sqrt{TS}=+64.6$, and Mkn~501 with $\sqrt{TS}=+16.6$. These two high-significance detections drive up the statistics of the complete sample to an overall $9\sigma$ deviation from the null hypothesis. When removing Mkn~421 and Mkn~501, the joint significance drops to a p-value of 2.2\% (Figure~\ref{signif-all}, right-hand side).
{\startlongtable
\begin{deluxetable*}{lllchrrcc}
\tablecaption{HAWC power-law fits and significances for the revised sample of 136 AGN from the 3FHL catalog. \label{results-2.5}}
\tablehead{\colhead{3FHL source} & \colhead{Counterpart} & \colhead{Class} & \colhead{$z$} & \nocolhead{~$(\alpha_{2000}; \delta_{2000})$} & \colhead{$TS$} &
\colhead{$\pm\sqrt{TS}$} & \colhead{$K \pm \Delta K$} & \colhead{$K_{2\sigma}$} }
\startdata
\input{table3data}
\enddata
\tablecomments{
Test-statistics ($TS$) and significances ($\pm\sqrt{TS}$) are estimated allowing fluxes to be positive or negative.
$K$ is the power-law normalization a 1~TeV in units of $10^{-12}\,\rm TeV^{-1}\, cm^{-2}\, s^{-1}$, and $K_{2\sigma}$ its corresponding 2$\sigma$ upper limit in the same units, computed using the method of \citet{1998PhRvD..57.3873F}.}
\end{deluxetable*}}
\begin{figure}
\includegraphics[width=0.495\hsize]{agnsurvey_hawc_f3a.pdf} \hfill
\includegraphics[width=0.495\hsize]{agnsurvey_hawc_f3b.pdf}
\caption{Histogram of significances for the sample whole ({\em left}) and for the clean sample, excluding Mkn~421 and Mkn~501 ({\em right}). The point with $\sqrt{TS}=65$ is Mkn~421, while that at 16 is Mkn~501. The three bins at $s>3$ on the right hand-side histogram contain VER~J0521+211, 1ES~1215+303 and M87, each in one bin.
\label{signif-all}}
\end{figure}
{Three more objects showed test-statistic $TS>9$: M87 ($TS=12.9$); 1ES~1215+303 ($TS=11.4$); and VER~J0521+211 ($TS=9.5$). We note that the random probability of having three $TS>9$ values in 134 trials, once the two Markarians are excluded, is $8.5\times 10^{-4}$. These three fainter sources have favorable declinations for HAWC, $\left |\delta -19^\circ \right | \lesssim 10^\circ$, allowing for transits of over 6~hours. Spectral fits for the five $TS>9$ sources are presented in~\S\ref{subsec:optimized}}.
\subsubsection{Upper limits and sensitivity}
\begin{figure}
\includegraphics[width=\hsize]{agnsurvey_hawc_f4.pdf}
\caption{Fit of $K_{2\sigma}$, the upper-limit normalizations at 1~TeV, as function of declination and redshift for the ``clean'' sample. The curved dashed lines are the fits given by eq.~\ref{sensitivity} to the upper-limits at the redshifts indicated in the plot. Blue arrows are upper limits for AGN with $z\leq 0.1$, green for $0.1<z\leq 0.2$ and red for $0.2<z\leq 0.3$. Measured values of $K$ for the $TS>9$ sources are indicated by stars. They are at declinations $12^\circ$ (M87, in blue), $21^\circ$ (VER~J0521+211, in green), $30^\circ$ (1ES~1215+303, in green), and close to $40^\circ$ (Mkn~421 as the blue star above the red curve and Mkn~501 the blue star intersecting the green curve). Statistical errors bars are shown, thought smaller than the markers for the Markarian sources.
\label{ulims-fit}}
\end{figure}
The $K_{2\sigma}$ upper limits span a large range of values: from $2.4\times 10^{-13}\,\rm TeV^{-1}cm^{-2}s^{-1}$, (3FHL~J1543.6+0452 at $z=0.040$), to $10^{-10}\,\rm TeV^{-1}cm^{-2}s^{-1}$ (3FHL~J0325.6--1646 at $z=0.291$). Upper limits are sensitive to both the declination and redshift of the source, as shown in Figure~\ref{ulims-fit}. {We performed a fit to the set of $K_{2\sigma}$ values that assumes a Gaussian dependence in declination and exponential in redshift. Three parameters were computed: the normalization limit at the reference point $\delta=19^\circ$ and $z=0$, the angular width in declination, and the redshift exponential scale. The fit obtained is,}
\begin{equation}
\log_{10}K_{2\sigma} = -12.33 + {1\over 2\ln 10} \left(\delta-19^\circ \over 18.93^{\circ} \right)^{2}+ {1\over \ln 10} \left(z\over 0.089 \right) \, . \label{sensitivity}
\end{equation}
{This fit excludes the five $TS>9$ sources. Its correlation coefficient is $r=+0.870$ and the dispersion with the data 0.241~dex. The value for the fit of $K_{2\sigma}$ at $z=0,\,\delta =19^\circ$ corresponds to 14~mCrab, a factor of two lower than the predefined depth of the survey. This would apply to near and optimally located sources only. The sensitivity of the survey degrades with a Gaussian angle of $18.9^\circ$, leading to a 11\% response at $40^\circ$ from zenith. The dependence of the upper limit normalizations with redshift gives $z_h=0.089$, matching that of an observed power-law spectrum integrated from $E_0 = 0.63 \,\rm TeV$, close to the initial assumption $E_0 = 0.5 \,\rm TeV$ (\S\ref{sec:ebl}). }
\subsubsection{Comparison with LAT extrapolations \label{sec:extra-compa}}
In Figure~\ref{extrapol-ulims} we compare photon fluxes, $N(>0.5\,\rm TeV)$, computed from the power-law fit (eq.~\ref{hawc-fit}) with the extrapolations of LAT spectra. HAWC photon fluxes relate to the normalizations $K$ through the integration of the differential spectra~(eqs.~\ref{nobs-ebl} and~\ref{hawc-fit}),
\begin{equation}
N_{obs}(>0.5~{\rm TeV}) = {4\sqrt{2}\over 3}\,K \cdot 1\, {\rm TeV}\cdot\,e^{-z/z_h}\, . \label{phi1}
\end{equation}
The (green, TeV=P) dots shown in Figure~\ref{extrapol-ulims} are for the objects with $TS>9$, while the rest are shown with the respective HAWC upper limit, using $K_{2\sigma}$ in expression~(\ref{phi1}). Thirteen objects have HAWC limits below, or close to, the LAT extrapolations. The VHE sources undetected by HAWC with limits below the LAT extrapolation are: IC~310, I~Zw~187 (1ES~1727+502), B3~2247+381, and 1ES~2344+514 (Figure~\ref{3fhl-xtrapol}). These are discussed in \S\ref{sec:undetected}. The HAWC limits for the TeV candidates PMN~J0816--1311, SBS~0812+578, and 1ES~2321+419 are also below the LAT extrapolations.
\begin{figure}
\includegraphics[width=\hsize]{agnsurvey_hawc_f5.pdf}
\caption{Flux extrapolations of 3FHL spectra versus HAWC measurements. Color code indicates the TeV flag: green TeV=P; orange TeV=C; red TeV=N. The green points are the measured fluxes of the five objects with $TS>9$, the rest of the sample indicated by upper limits. Errors shown are statistical only.
\label{extrapol-ulims}}
\end{figure}
\subsubsection{Source classes}
\begin{deluxetable}{lcrrc}
\tablecaption{Average significances for different groups of sources. \label{stats-groups}}
\tablehead{
\colhead{Source} & \colhead{Number} & \multicolumn{2}{c}{Significances} & \colhead{p-value} \\
\colhead{group} & \colhead{of objects} & \colhead{Mean} & \colhead{Std. dev} & \colhead{\empty} \\
\colhead{\empty} & \colhead{$N$} & \colhead{$\mu(s)$} & \colhead{$\sigma_{s}$} & \colhead{$P\left[\mu(s)>x\right]$} }
\startdata
All & 136 & +0.769 & 5.762 & $5.58\times 10^{-19}$ \\
All--Mkn & 134 & +0.175 & 1.209 & 0.022 \\
\hline
Starburst & 1 & +1.21 & \nodata & 0.113 \\
Radiogalaxies & 6 & +1.082 & 1.403 & $4.03\times 10^{-3}$ \\
BL Lacs--Mkn & 113 & +0.220 & 1.179 & $9.75\times 10^{-3}$ \\
FSRQ + fsrq & 6 & $-0.872$ & 0.609 & 0.984 \\
bcu & 8 & $-0.487$ & 1.028 & 0.916 \\
\hline
TeV = P (clean) & 30 & +0.768 & 1.228 & $1.29\times 10^{-5}$ \\
TeV = C (clean) & 32 & +0.124 & 1.152 & 0.241 \\
TeV = N (clean) & 72 & $-0.050$ & 1.142 & 0.665 \\
\enddata
\end{deluxetable}
Table~\ref{stats-groups} presents average significances for distinct groups of sources. While GeV emission is known to occur in these sources, 80\% of the objects in our sample have not been detected in the VHE range. Furthermore, HAWC is testing long-term average emission, in contrast to the relatively brief IACTs observations. The group statistics, reinforced by the HAWC measurements as function of redshift for different source classes (Figures~\ref{nobs-1} and~\ref{nobs-2}), are briefly summarized as follows: \begin{itemize}
\item Except for the one starburst, radiogalaxies constitute the nearest class of sources in our sample (Figure~\ref{nobs-1}). All six are nearer than $z=0.03$, and happen to be at favorable declinations, from $\delta=+4^\circ$ (NGC 1278) to $\delta=+41^\circ$ (IC~310 and NGC~1275). Four of them are known VHE sources, including 3C~264, reported after the publication of the 3FHL, that shows a $+1.9\sigma$ excess in the HAWC data. As shown in Figure~\ref{nobs-1}, their bounds are at the $N(>0.5\,{\rm TeV})\sim 10^{-12}\,\rm cm^{-2}s^{-1}$ level. The mean significance for this group has a p-value of 0.4\%.
\item The blazars candidates of uncertain type studied here are at redshifts intermediate between those of radiogalaxies and FSRQs. None of the bcu's studied here has been reported in the VHE regime. The HAWC upper limits shown in Figure~\ref{nobs-1} are not particularly constraining, owing to the unfavorable declinations of these sources, seven out of eight outside $+0^\circ \leq \delta\leq +50^\circ$.
\item Flat-spectrum radio quasars constitute the most distant class of source in our sample. Five out of six have not been detected as VHE sources, and do not comply with the LAT requirement for the TeV=C flag. They appear mostly as underfluctuations in the HAWC data. Figure~\ref{nobs-1} shows the upper limits on photon fluxes, most of them below the 30~mCrab level.
\item BL Lacertae objects constitute the clear majority of our sample, and of VHE sources. Still, 89 of the 117 BL Lac objects studied here have not been reported in the VHE regime. Figure~\ref{nobs-2} shows the HAWC fluxes for Mkn~421, Mkn~501, VER J0521+211 and 1ES~1215+303, at about 2 to $3\times 10^{-12}\,\rm cm^{-2}s^{-1}$ for the two weaker cases.
\end{itemize}
The TeV=P flagged sources appear with a p-value at the $10^{-5}$ level, {excluding Mkn~421 and Mkn~501}, as expected for a sub-threshold persistent TeV emission in these known sub-TeV emitters. The TeV=C candidates and TeV=N groups do not provide any collective hint of emission. The p-values quoted do not account for the number of trials used in testing different groups; they are quoted as indicative of the potential presence of persistent TeV emission at levels $\lesssim 10^{-12}\,\rm cm^{-2}s^{-1}$.
\begin{figure}
\includegraphics[width=\hsize]{agnsurvey_hawc_f6.pdf}
\caption{HAWC photon flux upper limits for radiogalaxies (RDG), blazars candidates of uncertain type (BCU), and flat-spectrum radio quasars (FSRQ), naturally segregated in redshift integrals. The green point at the lowest redshift correspond to M87. Only three RDGs and one FSRQ have a TeV=P flag. 3C~264 was reported as a relatively faint VHE source after the release of the 3FHL catalog. The red dotted line is the 30~mCrab reference, while the vertical dotted lines separate the types of sources. \label{nobs-1}}
\end{figure}
\begin{figure}
\includegraphics[width=\hsize]{agnsurvey_hawc_f7.pdf}
\caption{HAWC photon flux upper limits for BL Lac objects and the starburst galaxy NGC~1068. Only BL Lac objects with LAT extrapolation above $2\times 10^{-13}\,\rm cm^{-2}s^{-1}$ are shown, as in Figure~\ref{3fhl-xtrapol}. The redshift interval is restricted to $z<0.23$, as in Figure~\ref{nobs-1}. No BL Lac is extrapolated above that flux beyond that distance. Four flux points are shown: Mkn~421 and Mkn~501 above $10^{-11}\,{\rm cm^{-2}s^{-1}}$; and VER~J0521+211 and 1ES~1215+303 slightly below and above the red line at redshifts 0.11 and 0.13, respectively. \label{nobs-2}}
\end{figure}
\subsection{Spectral fits for the most significant sources\label{subsec:optimized}}
We computed optimized spectra for the $TS>9$ sources, fitting together normalizations and spectral indices. The fits are summarized in Figure~\ref{spectral-fits}, together with the systematic uncertainties. These have been quantified as 15\% in $K$, the 1~TeV normalization, and 5\% in $\alpha$, the spectral index. A short discussion on each of these five sources follows.
\begin{figure}
\includegraphics[width=\hsize]{agnsurvey_hawc_f8.pdf}
\caption{Optimized spectral fits for five selected sources. Mkn~421 and Mkn~501, the two highest points, present the best statistics. VER~J5021+211 and 1ES~1215+303 have the extreme spectral indices, although with a larger uncertainty. M87 appears with the smallest normalization and a spectral index consistent with $\alpha=2.5$. The black and grey oval on the upper left represents systematic uncertainties of 15\% in the 1~TeV normalization ($K$) and 5\% on spectral index ($\alpha$).
\label{spectral-fits}}
\end{figure}
\subsubsection{Markarian 421}
\begin{figure}
\includegraphics[width=\hsize]{agnsurvey_hawc_f9.pdf}
\caption{{\bf Spectra} of Mkn~421 measured by LAT, including EBL attenuation (black and grey, on the left) and by HAWC (blue, on the right). The full-dotted line represents the intrinsic spectrum, with statistical errors, and the full line the fit with EBL attenuation. The two attenuated fits match well at their (0.5-2.0)~TeV intersection.
\label{mkn421-spectrum}}
\end{figure}
Markarian~421 is the brightest persistent extragalactic object in the TeV sky. For the default $\alpha=2.5$ search we computed ${TS}=4167$ for $K=(29.5\pm 0.5_{\rm stat} \pm 4.4_{\rm syst})\times 10^{-12}\,\rm TeV^{-1}cm^{-2}s^{-1}$. The optimized power-law fit for the intrinsic spectrum resulted in,
\begin{equation}
{dN\over dE} = (33.0\pm 0.6_{\rm stat} \pm 4.9_{\rm syst})\times 10^{-12}\, \left(E \over 1\,{\rm TeV}\right)^{-2.63\pm 0.02_{\rm stat} \pm 0.13_{\rm syst}} {\rm TeV^{-1}cm^{-2}s^{-1}} \, ,
\end{equation}
with $TS=4193$, i.e. an increase in the test-statistic $\Delta TS=26$ relative to $\alpha$ fixed at 2.5. The integration to the observed photon flux, using $z_h=0.116$ for $\alpha=2.63$ in eq.~\ref{nobs-ebl}, gives $N_{\rm obs}=(48\pm 8)\times 10^{-12}\,\rm cm^{-2}s^{-1}$, somewhat larger than the $\alpha=2.5$ estimate shown in Figure~\ref{extrapol-ulims}. The {\em intrinsic} energy flux, $f_E=(132\pm 22)\times 10^{-12}\,\rm erg\, cm^{-2}s^{-1}$, translates into a luminosity per solid angle of $f_E d_L(z)^{2}= L(>0.5\,{\rm TeV})/\Delta\Omega = (2.1\pm 0.4)\times 10^{43} \,\rm erg~s^{-1} sr^{-1}$, about one third (32\%) of the (10~GeV-1~TeV) luminosity per steradian inferred from the {\em Fermi}-LAT measurements (Figure~\ref{muestra-z-lumi}).The HAWC spectrum is consistent with IACT observations made by MAGIC between 2007 and 2009 \citep{2016A&A...593A..91A}. Using an energy threshold of 400~GeV, these authors found photon fluxes varying from $N_{\rm min}(>0.5~{\rm TeV}) = 9.33\times 10^{-12}\,\rm cm^{-2}s^{-1}$, to $N_{\rm max}(>0.5~{\rm TeV}) = 2.22\times 10^{-10}\,\rm cm^{-2}s^{-1}$, weakly dependent on their assumed differential index of 2.5.
{The 3FHL catalog and HAWC spectra for Mkn~421 are shown together in Figure}~\ref{mkn421-spectrum}. There is a fairly good match between the two fits, with the curves intersecting at about 0.66~TeV. The local LAT spectral index at 1~TeV, $2.5\pm 0.2$, is consistent within uncertainties with the HAWC spectral index. IACT observations between 100~GeV and 5~TeV, performed around 2005 with MAGIC, resulted in a spectral index of $(2.20\pm 0.08)$, with indications of a cut-off in the intrinsic spectrum~\citep{2007ApJ...663..125A}. A detailed spectral analysis of Mkn~421, and Mkn~501, with HAWC data will be presented in a separate publication. Preliminary results can be found in~\citet{2019ICRC...36..654C}.
\subsubsection{Markarian 501}
\begin{figure}
\includegraphics[width=\hsize]{agnsurvey_hawc_f10.pdf}
\caption{Spectrum of Mkn~501 measured by LAT and EBL attenuated (black and grey) and by HAWC (blue). The full-dotted line represents the HAWC intrinsic spectrum, with statistical errors, and the full line the fit with EBL attenuation.
\label{mkn501-spectrum}}
\end{figure}
The TeV emission from Markarian~501 observed by HAWC is not as steady as that of Mkn~421~\citep{2017ApJ...841..100A}. In fact, the statistical significance of the time-averaged TeV emission of Mkn~501 has {\em decreased} with increased HAWC exposure. The $\alpha=2.5$ search resulted in a test-statistic ${TS}=276.97$ for $K=(7.74\pm 0.49_{\rm stat}\pm 1.16_{\rm syst})\times 10^{-12}\,\rm TeV^{-1}cm^{-2}s^{-1}$. The optimized power-law fit is,
\begin{equation}
{dN\over dE} = (6.21\pm 0.69_{\rm stat} \pm 0.93_{\rm syst})\times 10^{-12}\, \left(E \over 1\,{\rm TeV}\right)^{-2.31\pm 0.08_{\rm stat} \pm 0.12_{\rm syst}} {\rm TeV^{-1}cm^{-2}s^{-1}} \, , \label{mkn501-fit}
\end{equation}
with $TS=280.28$, representing a moderate increase in the test-statistic $\Delta TS=3.33$. The integrated observed photon flux is $N_{\rm obs}=(8.5\pm 2.6)\times 10^{-12}\,\rm cm^{-2}s^{-1}$, accounting for EBL attenuation. The {\em intrinsic} energy flux, $f_E=(40\pm 16)\times 10^{-12}\,\rm erg\, cm^{-2}s^{-1}$, translates into a luminosity per solid angle of $d_L(z)^{2}f_E = L(>0.5\,{\rm TeV})/\Delta\Omega = (7.3\pm 2.8)\times 10^{42} \,\rm erg~s^{-1} sr^{-1}$, which is about 25\% of the (10~GeV-1~TeV) luminosity per steradian measured by {\em Fermi}-LAT (Figure~\ref{muestra-z-lumi}). This fraction is similar to that observed for Mkn~421.
The 3FHL catalog and HAWC spectra for Mkn~501 are shown together in Figure~\ref{mkn501-spectrum}. The agreement is not as good as for Mkn~421: the Mkn~501 spectrum is harder and lies below the measurement by the LAT. The local LAT spectral index at 1~TeV is $2.58\pm 0.35$, just consistent with HAWC when accounting for the propagation of the uncertainty in the curvature parameter $\beta$ of the 3FHL fit. We note that Mkn~501 has a variability index $V_{\rm bayes}=4$ in the 3FHL catalog.
The literature on the VHE characteristics of Mkn~501 is extensive. Its high TeV variability was noted shortly after its 1996 discovery, when the HEGRA group reported flux variations of an order of magnitude observed in mid-1997~\citep{1996ApJ...456L..83Q,1997A&A...327L...5A}. Contemporaneous 10~m Whipple data confirmed the high state, adding indications of a curved spectrum favored over a simple power-law~\citep{1998ApJ...501L..17S}. Further monitoring showed strong variations in flux, although with stable spectra best-fitted by a power-law plus an intrinsic cut-off at around 6~TeV~\citep{1999A&A...349...11A}. Observations in 1998-1999 showed lower activity, with evidence for spectral curvature and steepening~\citep{2001ApJ...546..898A}. Mkn~501 observations renewed in 2005, when the MAGIC telescope measured strong and very fast variability, with spectral indices ranging from $\sim 2.0$ to $2.7$~\citep{2007ApJ...669..862A}. Data taken the following year (2006) by MAGIC permitted to characterize a low activity state, with fluxes similar to those reported by VERITAS and MAGIC from data taken in the 2009 joint observations with the {\em Fermi}-LAT~\citep{2009ApJ...705.1624A,2011ApJ...729....2A}. The contemporaneous LAT data showed spectral variability also present in the GeV range, with index variations $\Delta\alpha\sim 1$ during the first 480~days of {\em Fermi} observations~\citep{2011ApJ...727..129A}.
In the last decade, EAS arrays have been able to reach the sensitivities needed to perform long term monitoring of Mkn~501: the ARGO collaboration reported variations of a factor of six in flux observed between October 2011 and April 2012~\citep{2012ApJ...758....2B}. The HAWC collaboration presented light curves for the Crab, Mkn~421 and Mkn~501 from its first 17~months of observations~\citep{2017ApJ...841..100A}. The HAWC light curve of Mkn~501 showed a low flux baseline with a handful of very short strong flares. The simple power-law fit (of index $2.84$) was disfavored against a power-law with exponential cut-off.
The variability of Mkn~501 has prevented a baseline characterization of this object. In \citet{2019ICRC...36..654C} we presented the HAWC spectrum of Mkn~501 using data acquired between June 2015 and December 2017. Even though spectra with exponential cuts were tested, the Mkn~501 data proved consistent with a pure power-law of index~$2.40\pm 0.06$. While there might be a slight decrease in the spectral index ($2.31\pm 0.08 \pm 0.20$), the flux reported here is about half that reported in~\citet{2019ICRC...36..654C}. The four year average TeV flux observed by HAWC is a factor of two above the lowest activity observed so far~\citep{2009ApJ...705.1624A,2011ApJ...729....2A}.
\subsubsection{M87}
\begin{figure}
\includegraphics[width=\hsize]{agnsurvey_hawc_f11.pdf}
\caption{High energy spectra of M87 from the LAT Catalog and HAWC observations. The data suggest a steepening of the spectrum at energies $\sim 1~\rm TeV$. The HAWC optimized power-law fit has $TS=13.2$. The quasi-differential computation for the (2.0-8.0)~TeV interval is at $TS=13.1$.
\label{m87-espectro}}
\end{figure}
M87 is the central galaxy of the Virgo cluster, a giant elliptical at a distance of just $(16.4\pm 0.5)~\rm Mpc$, as measured independently of its redshift, $z=0.0042$~\citep{2010A&A...524A..71B}. With an optical magnitude $V=8.6$ and an angular diameter of about 8~arcmin, M87 has been imaged in detail for over a century, showing a distinct bright active nucleus and a single optical jet~\citep{1918PLicO..13....9C,1998ApJ...493L..83T}. The mass of the central black hole has been measured to be $(6.5\pm 0.7)\times 10^{9}\,M_\odot$, through its imaging with the Event Horizon Telescope~\citep{2019ApJ...875L...1E}.
Also known as Virgo~A, it is a bright object all throughout the electromagnetic spectrum. As 3FHL~J1230.8+1223, it is a $12.1\sigma$ detection above 10~GeV as reported in the 3FHL catalog, with $4.8\sigma$ in 150-500~GeV, and a $2\sigma$ upper limit in the top (0.5-2.0)~TeV band. M87 has been observed frequently in the VHE regime since its 2003 discovery by the HEGRA collaboration~\citep{2003A&A...403L...1A}. The temporal behavior of M87 in the high-energy and VHE bands has been reviewed by~\citet{2019A&A...623A...2A}. Fluxes can vary by a factor of ten between low, mid and high states, with indications of the spectral index varying from $2.6$ in low states, to $2.2$ in high states. In addition to the low and high states, time variability on single-day timescales during high states has been reported by the H.E.S.S. and MAGIC collaborations~\citep{2007AIPC..921..147B,2008ApJ...685L..23A}.
The default HAWC search at the M87 location gave a test-statistic of $TS=12.93$ for a normalization $K=(0.56\pm 0.16_{\rm stat}\pm 0.08_{\rm syst})\times 10^{-12}\,\rm TeV^{-1}cm^{-2}s^{-1}$. The optimized power-law fit for the intrinsic spectrum is,
\begin{equation}
{dN\over dE} = (0.69\pm 0.22_{\rm stat} \pm 0.10_{\rm syst})\times 10^{-12}\, \left(E \over 1\,{\rm TeV}\right)^{-2.63\pm 0.18_{\rm stat} \pm 0.13_{\rm syst}} {\rm TeV^{-1}cm^{-2}s^{-1}} \, ,
\end{equation}
with $TS=13.19$, i.e. a non-significant increase in the test-statistic, $\Delta TS=0.26$, showing $\alpha=2.5$ to be an acceptable solution within the statistics of the optimized fit.
The integrated photon flux is $N_{\rm obs}(>0.5~{\rm TeV})=(1.3\pm 0.9)\times 10^{-12}\,\rm cm^{-2}s^{-1}$, with less than 4\% of attenuation by the EBL. The energy flux, $f_E=(2.7\pm 2.4)\times 10^{-12}\,\rm erg\,cm^{-2}s^{-1}$, translates into a luminosity per solid angle of $L(>0.5~{\rm TeV})/\Delta\Omega = (6.9\pm 6.3)\times 10^{39}\,\rm erg\,s^{-1}sr^{-1}$, which is about 25\% of that in the 10~GeV-1~TeV range, from the respective 3FHL parameters. The lower apparent power with respect to the two nearest BL Lacs is attributed to the off-axis viewing of the jet. Still, the relative power when compared to the LAT regime appears to be similar.
From \citet{2019A&A...623A...2A}, we computed photon fluxes from M87 observations by IACTs, in different activity states:
\begin{itemize}
\item High state: $N(>0.5\,{\rm TeV}) = (5.74^{+1.14}_{-1.47})\times 10^{-12}\,\rm cm^{-2}s^{-1}$,
\item Mid state: $N(>0.5\,{\rm TeV}) = (1.39^{+0.47}_{-0.43})\times 10^{-12}\,\rm cm^{-2}s^{-1}$,
\item Low state: $N(>0.5\,{\rm TeV}) = (2.85^{+2.07}_{-1.53})\times 10^{-13}\,\rm cm^{-2}s^{-1}$.
\end{itemize}
The 4.5~year averaged emission, as indicated by the HAWC data, matches the mid state, with a relatively steep spectral index. The comparison with the 3FHL data, shown in Figure~\ref{m87-espectro}, points to a steepening in the spectrum. While the LAT and HAWC data are not contemporaneous, with a variability index $V_{\rm bayes}=1$ M87 does not stand as a variable in the 3FHL catalog. Still, an analysis of joint contemporaneous LAT-HAWC data is desirable.
\subsubsection{VER J0521+211 \label{ver_j0521+211}}
\begin{figure}
\includegraphics[width=\hsize]{agnsurvey_hawc_f12.pdf}
\caption{Spectral fits of VER~J0521.7+2112 from {\em Fermi}-LAT (grey) and HAWC (blue) data. The best fit of the HAWC data is for an intrinsic power-law of index 2.0, which remains harder than the 3FHL log-parabola fit after considering EBL attenuation at the assumed redshift. The HAWC quasi-differential limit at (2.0-8.0)~TeV is for a $TS=9.1$.\label{verj0521-espectro}}
\end{figure}
VER~J0521+211 was discovered as a TeV $\gamma$-ray source by the VERITAS collaboration, during observations following-up its detection above 30~GeV by LAT~\citep{2009ATel.2260....1O}. 3FHL~J0521.7+2112 itself is associated to the radio-loud BL Lac object TXS~0518+211, and the X-ray {\it ROSAT} source RX~J0521.7+2112. The redshift survey of \citet{2013ApJ...764..135S} assigned $z=0.108$ to the optical counterpart. This is the value listed in the 3FHL catalog and assumed for this analysis. \citet{2013ApJ...776...69A} did not confirm this redshift, and neither did \citet{2017ApJ...837..144P}, who only set a lower limit $z>0.18$ using deep spectroscopy with the GTC 10.4~m. The VHE emission from this object has been measured at least up to 1~TeV, with no apparent decline in the spectrum~\citep{2015ICRC...34..864P}. This makes the high-energy characterization of VER~J0521+211 of particular interest.
The $\alpha=2.5$ search resulted in $TS=9.49$ for $K=(2.85\pm 0.93_{\rm stat} \pm 0.43_{\rm syst})\times 10^{-12}\,\rm TeV^{-1}cm^{-2}s^{-1}$. The best power-law fit for the intrinsic spectrum is,
\begin{equation}
{dN\over dE} = (2.39\pm 0.89_{\rm stat} \pm 0.36_{\rm syst})\times 10^{-12}\, \left(E \over 1\,{\rm TeV}\right)^{-2.01\pm 0.38_{\rm stat} \pm 0.10_{\rm syst} } {\rm TeV^{-1}cm^{-2}s^{-1}} \, ,
\end{equation}
with $TS=10.34$, representing a modest increase in the test-statistic $\Delta TS=0.85$ with respect to the default search. This is the hardest AGN spectrum for the $TS>9$ HAWC sample. The integrated observed photon flux is $N_{obs}(>0.5~{\rm TeV})=(1.5\pm 1.1)\times 10^{-12}\,\rm cm^{-2}s^{-1}$, attenuated by a factor of 2/3 due to the EBL. The energy flux and luminosity per solid angle cannot be accurately determined with the spectral index so close to $2.0$. If we take $E^{2}dN/dE$ at 1~TeV as indicative, we get an estimated $f_E\sim (3.8\pm 1.4)\times 10^{-12}\,\rm erg\,cm^{-2}s^{-1}$ and $d_L^{2}f_E\sim (8.4\pm 3.1)\times 10^{42}\,\rm erg\,s^{-1}sr^{-1}$. Notwithstanding the uncertain distance, this source appears particularly luminous in the GeV regime, with $L(10~{\rm GeV}-1~{\rm TeV})/\Delta\Omega \sim 1.4\times 10^{44}\,\rm erg\,s^{-1}sr^{-1}$, standing above the two Markarians in Figure~\ref{muestra-z-lumi}.
The 3FHL data, shown together with the HAWC spectral fit in Figure~\ref{verj0521-espectro}, has strong detections up to the 150-500~GeV band, with a highest energy photon of 370~GeV. The 3FHL fit is a log-parabola, transiting from a hard spectrum at about 30~GeV to a very steep local spectral index of $3.7\pm 0.7$ at 1~TeV (4.3 when attenuating the 3FHL spectrum). The attenuated HAWC spectrum corresponds to an observed spectral index of $\sim 2.7$, mostly inconsistent with the LAT fit. The data may be reconciled through an intrinsic hardening at about 200 or 300~GeV. The HAWC quasi-differential bound in the 0.5-2.0~TeV band is a factor of 4.6 lower than the 3FHL limit in the same band. We note that the quasi-differential analysis gave $TS=9.1$ for the 2.0-8.0~TeV band, optimal in terms of the HAWC instrumental response. With the assumed redshift, $\tau$ is due to range from 1.6 to 3.2 in that energy interval, pointing to an intrinsically hard spectrum. The analysis gave $TS=0$ for 8.0-32.0~TeV, expected to be heavily attenuated for the assumed redshift. Additional HAWC data analysis should allow to further constrain the shape of the TeV spectrum VER~J0521+211, testing the redshift assumption. {We also note that VER~J0521+211 has a variability index $V_{\rm bayes}=4$ in the 3FHL catalog, indicating that a joint analysis of contemporaneous LAT and HAWC data would be relevant.}
{As mentioned in \S\ref{subsec:sample}, VER~J0521+211 is located $3.07^\circ$ from the Crab Nebula, the brightest source in the 2HWC and 3HWC catalogs. This angular distance corresponds to three times the 68\% containment angle $\psi_{68}$ for ${\cal B}=1$, and $> 6\psi_{68}$ for ${\cal B}>2$. We tested for potential contamination repeating the Maximum-Likelihood test with $\alpha=2.5$ at five locations equidistant from the Crab Nebula, forming together with VER~J0521+211 an hexagon around the Crab. These provided test-statistics $TS$ between $-4.46$ and $+1.54$, in contrast with $TS=+9.49$ at the location of VER~J0521+211.}
\subsubsection{1ES~1215+303}
\begin{figure}
\includegraphics[width=\hsize]{agnsurvey_hawc_f13.pdf}
\caption{High energy spectra of 1ES~1215+303 from the 3FHL catalog and HAWC observations. We note the intersection of both spectra at $\gtrsim 1\,\rm TeV$ and its decay, as it suffers EBL attenuation. This is the most distant source detected by HAWC so far.\label{1es1215-espectro}}
\end{figure}
1ES~1215+303 is one of six $\gamma$-ray emitting BL Lac objects located in the Northern part of the Coma Berenices constellation, five of them known to be VHE sources. The redshift of this HBL is now confirmed to be $z=0.130$, discarding the early measurement $z=0.237$ by~\citet{1993ApJS...84..109L}. The lower value was confirmed through optical spectroscopy at the GTC and, more recently, through the direct identification of a Ly$\,\alpha$ emission line~\citep{2017ApJ...837..144P,2019AJ....157...41F}. Catalogued as 3FHL~J1217.9+3006, this object is well detected up to 500~GeV, and modeled with a power-law of index $\alpha=2.3\pm 0.1$. It has a variability index $V_{\rm bayes}=2$.
The default HAWC search gave $TS=11.36$ for $K=(4.64\pm 1.38_{\rm stat} \pm 0.70_{\rm syst} )\times 10^{-12}\,\rm TeV^{-1}cm^{-2}s^{-1}$, as the intrinsic normalization. The optimized power-law fit for the intrinsic spectrum is rather softer,
\begin{equation}
{dN\over dE} = (3.78\pm 1.36_{\rm stat} \pm 0.57_{\rm syst} )\times 10^{-12}\, \left(E \over 1\,{\rm TeV}\right)^{-3.07\pm 0.37_{\rm stat} \pm 0.15_{\rm syst} } {\rm TeV^{-1}cm^{-2}s^{-1}} \, ,
\end{equation}
with $TS=12.80$, representing a test statistic increase $\Delta TS=1.44$. The attenuated photon flux is $N_{\rm obs}(>0.5\,{\rm TeV})=(2.53\pm 1.35)\times 10^{-12}\,\rm cm^{-2}s^{-1}$. The integrated energy flux results in a luminosity per solid angle of $L(>0.5~{\rm TeV})/\Delta\Omega = (3.9\pm 2.7)\times 10^{43}\,\rm erg\,s^{-1}sr^{-1}$, given the luminosity distance of 585~Mpc. This is about 30\% of the corresponding value between 10~GeV and 1~TeV. The 3FHL and HAWC spectra, compatible within the uncertainties, are shown in Figure~\ref{1es1215-espectro}.
1ES~1215+303 was first detected as a VHE source by MAGIC in 2011~\citep{2011arXiv1110.6786L,2012A&A...544A.142A}. {It is usually observed together with PG~1218+304, located just 0.88 degrees away in the sky, for which we obtained a test statistic $TS=+2.24$. Given that both sources are known VHE emitters, the distance is not large enough to be certain that there is no overlap between both sources. With a redshift $z=0.184$, PG~1218+304 is more distant and prone to be heavily attenuated above 1~TeV. A dedicated study with improved HAWC analysis tools is now pending.}
Long-term monitoring of this source by VERITAS was presented by~\citet{2013ApJ...779...92A}, prior to the report of a single short and intense flare seen in 2014 with VERITAS and {\em Fermi}-LAT~\citep{2017ApJ...836..205A}. During this episode the VHE flux of this HBL reached 2.4 times that of the Crab Nebula, with a variability timescale $\lesssim 3.6\,{\rm hours}$. The MAGIC spectrum of~\citet{2012A&A...544A.142A}, ranging from 70~GeV to 1.8~TeV, had an intrinsic spectral index of $2.96$, de-attenuated with the model of~\citet{2011MNRAS.410.2556D}, used in here. The MAGIC and HAWC spectral indices are in good agreement. The long-term joint monitoring of this source by LAT and VERITAS, spanning more than ten years, confirms the spectral index and points to four strong and short flares that occurred during full HAWC operations by~\citet{2020ApJ...891..170V}. We checked the dates of these four episodes and did not find evidence for them in the HAWC data. An optimized time-dependent analysis of this source with HAWC, beyond the scope of this paper, will be subject of future work.
\subsection{Limits on sources of interest \label{sec:undetected}}
\begin{deluxetable*}{llcccccc}
\tablecaption{Upper limits on 3FHL AGN candidate targets. \label{undetected}}
\tablehead{ \colhead{3FHL entry} & \colhead{Source} & \colhead{z} & \colhead{$\pm\sqrt{TS}$} & \colhead{$K_{2\sigma}$} & \colhead{$N_{0.5}^{x}$} & \colhead{$N_{0.5}^{UL}$} }
\startdata
\hline
3FHL J0214.5+5145 & TXS~0210+515 & 0.049 & $+1.62$ & 5.43 & 2.40 & 6.50 \\
3FHL J0316.6+4120 & IC~310 & 0.019 & $+0.86$ & 1.01 & 4.37 & 1.60 \\
3FHL J1428.5+4240 & H~1426+428 & 0.129 & $+1.26$ & 8.18 & 1.21 & 4.67 \\
3FHL J1728.3+5013 & I~Zw~187 & 0.055 & $-0.22$ & 2.57 & 3.25 & 2.91 \\
3FHL J2250.0+3825 & B3~2247+381 & 0.119 & $-0.27$ & 2.54 & 2.37 & 1.59 \\
3FHL J2347.0+5142 & 1ES~2344+514 & 0.044 & $+1.45$ & 4.62 & 7.12 & 5.80 \\\hline
\enddata
\tablecomments{$K_{2\sigma}$, the spectrum normalization at 1~TeV, is in units of $10^{-12}\,\rm TeV^{-1}cm^{-2}s^{-1}$. $N_{0.5}$ are extrapolated photon fluxes above 0.5~TeV and the corresponding upper limit, in units of $10^{-12}\,\rm cm^{-2}s^{-1}$.}
\end{deluxetable*}
The extrapolation of 3FHL catalog spectra (Figure~\ref{3fhl-xtrapol}) and of IACT spectral fits allowed us to identify nine AGN targets for potential HAWC detection (\S\ref{extrapol}). We presented evidence of persistent TeV emission for three of those targets, Mkn~421, Mkn~501, and M87, and for two sources below the 30~mCrab reference limit (marked as a red line in Figure~\ref{3fhl-xtrapol}), VER J0521+211 and 1ES~1215+303. On the other hand, six of our target sources (IC~310, 1ES~2344+514, TXS~0210+515, 1ES~1727+502, B3~2247+381, H~1426+428) were not detected. As shown in Figure~\ref{ulims-fit}, the HAWC sensitivity is dependent on source declination. We note that while the sources with $\sqrt{TS}>9$ are all in the range $+12^\circ<\delta<+40^\circ$, five of the undetected candidates are either North of $\delta=+50^\circ$, or farther than $z=0.1$. Declination is of particular relevance in here, as the response of EAS arrays to low energy events is compromised at large zenith angles. Still, as shown in Table~\ref{undetected} and in~\S\ref{sec:extra-compa}, four of the corresponding upper limits are below the extrapolation of the corresponding 3FHL spectrum.
IC~310 is a target that is both nearby and culminates at an adequate zenith angle, but remained undetected in this HAWC analysis. The upper limit set in the photon flux is close to a third of the LAT extrapolation, and the HAWC upper limit in the 0.5-2.0~TeV band is 2.5 lower than the one in the 3FHL catalog. We note that while the 3FHL catalog reports a relatively low variability index, $V_{\rm bayes}=2$, this source is known to display extreme variability in the VHE range, on timescales as low as five~minutes, challenging models and severely constraining the emission region to scales smaller than its event horizon~\citep{2014Sci...346.1080A}.
In addition to the preselected targets, we point here to two additional sources of intrinsic interest: 3C~264 and NGC~1068. 3C~264 is a radio galaxy hosted by the elliptical galaxy NGC~3862, at a distance of about 90~Mpc. A compact radio source powers a relativistic jet imaged in radio and in the optical~\citep{2004A&A...415..905L,1993ApJ...402L..37C}. 3C~264 shows in the LAT data up to a highest energy photon of 97~GeV. It was listed as a TeV=N source in the 3FHL catalog, and considered as such in this analysis. However, it was later detected in the VHE range by the VERITAS Collaboration~\citep{2018ATel11436....1M}. The photon flux measured by VERITAS above 300~GeV indicates that this object should be too faint for HAWC. Even if the spectral index measured by LAT ($\alpha=1.65$) were to continue in the TeV range, its extrapolation would be $N(>0.5\,{\rm TeV})\lesssim 0.6\times 10^{-12}\,\rm cm^{-2}s^{-1}$. The HAWC data shows a $+1.9\sigma$ excess, statistically consistent with such a low flux and providing an upper limit $N(>0.5\,{\rm TeV}) < 1.42\times 10^{-12}\,\rm cm^{-2}s^{-1}$.
NGC~1068 has become a source of interest due to its coincidence with a hotspot in the IceCube all-sky map~\citep{2020PhRvL.124e1103A}. While two other starburst galaxies, NGC~253 and M82, have been detected in the VHE range as faint sources, with fluxes 1\% of the Crab Nebula~\citep{2016CRPhy..17..585O}, NGC~1068 remains undetected and an unlikely candidate for HAWC detection. It is a weak 5.3$\sigma$ detection in the 3FHL catalog, with practically no signal above 20~GeV and a rather steep spectral index \mbox{$\alpha=3.8\pm 1.0$}. The HAWC data has a $+1.2\sigma$ excess at the location of NGC~1068, for an upper limit normalization $K_{2\sigma}= 6.46\times 10^{-13}\,\rm TeV^{-1} cm^{-2}s^{-1}$ at 1~TeV. The quasi-differential HAWC limit in the common energy band, $N(0.5-2.0~{\rm TeV})\leq 2.32\times 10^{-12}\,\rm cm^{-2}s^{-1}$, is a factor of eight lower than the respective LAT limit.
\section{Summary \label{sec:summary}}
The HAWC Gamma-Ray Observatory has performed an extensive follow-up survey of known GeV $\gamma$-ray emitting active galaxies at TeV energies. We investigated all AGN in the 3FHL catalog with a redshift lower than 0.3 and transiting within $40^\circ$ of latitude $19^\circ\,\rm N$, the HAWC zenith, searching for TeV $\gamma$-ray emission averaged over a 4.5~year period. The HAWC data show clear signals from Mkn~421 and Mkn~501, from which we quantified their long-term averaged spectra. In addition, we obtained evidence for TeV emission from the radiogalaxy M87 and the BL Lac objects VER~J0521+211 and 1ES~1215+303. The fluxes estimated for these sources are compatible with values previously reported for mid or low activity states. When excluding Mkn~421 and Mkn~501, we find collective evidence for long-term averaged TeV emission from radiogalaxies and BL Lac objects with a p-value $\sim 1\%$, and for known VHE emitters (TeV=P) with p-value $\sim 10^{-5}$. No evidence was found for other source classes or for LAT sources not previously reported in the VHE range.
Upper limits were set for the whole sample assuming intrinsic power-law spectra of index 2.5 attenuated by the interaction of $\gamma$ rays with extragalactic background radiation. These limits are dependent on the declination and redshift of the sources, confirming a redshift attenuation of exponential scale $z_h \simeq 0.1$ for HAWC. Bounds on observed photon fluxes in three energy intervals (0.5-2.0)~TeV, (2.0-8.0)~TeV and (8.0-32.0)~TeV are also given. HAWC measurements were compared with the mostly non-contemporaneous 3FHL catalog long-term data and with specific IACT observations.
As the exposure of HAWC continues to deepen, the increased sensitivity will allow to perform deeper searches of extragalactic sources. Long-term variability is an area of opportunity for ground based EAS $\gamma$-ray observatories not explored in this paper. Analyses of multi-year AGN light curves are underway in order to expand the investigation presented here to the time regime. The continuous and improved operation of HAWC is leading to a better understanding of EAS arrays, and in particular of water Cherenkov detectors. New analysis tools to improve the sub-TeV sensitivity of HAWC have been developed, and are now been implemented. These will provide improved energy response and the reduction of noise at low energies, two requirements for a deeper access to the extragalactic sky. These upgrades will allow HAWC to build on the analysis presented here, laying the path to future TeV survey instruments, in both the Northern and Southern Hemispheres.
\acknowledgments
We acknowledge the support from: the US National Science Foundation (NSF); the US Department of Energy Office of High-Energy Physics; the Laboratory Directed Research and Development (LDRD) program of Los Alamos National Laboratory; Consejo Nacional de Ciencia y Tecnolog\'ia (CONACyT), M\'exico, grants 271051, 232656, 260378, 179588, 254964, 258865, 243290, 132197, A1-S-46288, A1-S-22784, C\'atedras 873, 1563, 341, 323, Red HAWC, M\'exico; DGAPA-UNAM grants IG100317, IG101320, IN111315, IN111716-3, IN111419, IA102019, IN112218; VIEP-BUAP; PIFI 2012, 2013, PROFOCIE 2014, 2015; the University of Wisconsin Alumni Research Foundation; the Institute of Geophysics, Planetary Physics, and Signatures at Los Alamos National Laboratory; Polish Science Centre grant, DEC-2017/27/B/ST9/02272; Coordinaci\'on de la Investigaci\'on Cient\'ifica de la Universidad Michoacana; the Royal Society - Newton Advanced Fellowship 180385; la Generalitat Valenciana, grant CIDEGENT/2018/034. Chulalongkorn University's CUniverse (CUAASC) grant. Thanks to Scott Delay, Luciano D\'iaz and Eduardo Murrieta for technical support.
This work used data from the Fermi Science Support Center and the TeVCat online source catalog (http://tevcat.uchicago.edu). It also made use of the SIMBAD, NED (NASA/IPAC Extragalactic Database), ADS, and SDSS databases.
\vspace{5mm}
\facilities{The High Altitude Water Cherenkov (HAWC) Gamma-Ray Observatory}
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2024-02-18T23:40:24.874Z
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2020-09-22T02:02:15.000Z
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{"paloma_paragraphs":[]}
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{"arxiv_id":"2009.09039","language":"en","timestamp":1600740135000,"url":"https:\/\/arxiv.org\/abs\/2009.09039","yymm":"2009"}
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proofpile-arXiv_000-10105
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{"provenance":"002.jsonl.gz:10106"}
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