text:
\section{Introduction}

Given a function $f$ defined on a smooth manifold $\mathcal{M}$, the cluster tree
of $f$ is a tree structure representing the creation and merging of connected components
of a level set $\{x: f(x)\geq f_0\}$ when we move down the level $f_0$ \citep{chen2016statistical}.
Because cluster trees keep track of the connected components of level sets,
the shape of a cluster tree contains topological information about the underlying function $f$.
Moreover, a cluster tree can be displayed on a two-dimensional plane regardless of the dimension of $\mathcal{M}$,
which makes it an attractive approach for visualizing $f$.
In this paper, we focus on the case where $f \equiv f_P$ is some function of the underlying distribution $P$.
In this context, the cluster tree of $f$ reveals information about $P$.

Most of cluster trees being studied in the literature
are the $\lambda$-tree of a distribution
\citep{balakrishnan2012,chaudhuri2010rates,chaudhuri2014consistent,chen2016statistical,kpotufe2011pruning,Stuezle03}.
The $\lambda$-tree of a distribution is the cluster tree of the density function $p$ of
that distribution.
In this case, the tree structure contains the topological information of $p$
and we can use the $\lambda$-tree to visualize a multivariate density function.
When we use an $\lambda$-tree for visualization purposes, it is also called a density tree
\citep{klemela2004visualization,klemela2006visualization,klemela2009smoothing}.

\cite{Kent2013} proposed a new type of cluster tree of a distribution--the
$\alpha$-tree. The $\alpha$-tree uses the function $\alpha(x) = P(\{y: p(y)\leq p(x)\})$ to construct a cluster tree.
When the distribution is nonsingular, the $\alpha$-tree and the $\lambda$-tree
are topologically equivalent (Lemma~\ref{lem::tree}),
so they both provide similar topological information for the underlying distribution.
To estimate an $\alpha$-tree, we use the cluster tree of the function estimator $\hat{\alpha}_n(x) = \hat{P}_n(\{y: \hat{p}_n(y)\leq \hat{p}_n(x)\})$
where $\hat{P}_n$ is the empirical measure and $\hat{p}_n$ is the kernel density estimator (KDE).
Namely,
we first use the KDE to estimate the density of each data point
and count the number of data points with a density below the density of that given point.

When a distribution is singular, the $\lambda$-tree is ill-defined because of the lack of a probability density function,
but the $\alpha$-tree is still well-defined under a mild modification.
For an illustrating example, see Figure~\ref{fig::Tree}.
These are
random samples from a distribution mixed with a point mass at $x=2$ with a probability of $0.3$
and a standard normal distribution with a probability of $0.7$.
Thus, these samples are from a singular distribution.
We generate $n=5\times 10^3$ (left), $5\times 10^5$ (middle), and $5\times 10^7$ (right) data points
and estimate the density using the KDE with the smoothing bandwidth selected by the default rule in R.
The estimated density and $\lambda$-trees (red trees) are displayed in the top row.
It can be seen that when the sample size increases, $\lambda$-trees become degenerated.
This is because there is no population $\lambda$-tree for this distribution.
However, the $\alpha$-trees are stable in all three panels (see the bottom row of Figure~\ref{fig::Tree}).

\begin{figure}
\includegraphics[width=1.6in]{figures/GT001_2-1}
\includegraphics[width=1.6in]{figures/GT001_4-1}
\includegraphics[width=1.6in]{figures/GT001_6-1}\\
\includegraphics[width=1.6in]{figures/GT002_2}
\includegraphics[width=1.6in]{figures/GT002_4}
\includegraphics[width=1.6in]{figures/GT002_6}
\caption{
Example of the estimated $\lambda$-tree and $\alpha$-tree of a singular distribution.
This is a random sample from a singular distribution where with a probability of $0.3$, we obtain a point mass at $x=2$,
and with a probability of $0.7$, we sample from a standard normal.
The top panel shows the density estimated by the KDE
and the red tree structure corresponds to the estimated $\lambda$-tree.
The bottom panel displays the estimated $\alpha$-tree.
From left to right, we increase the sample size from $5\times 10^3$, $5\times 10^5$, to $5\times 10^7$.
Because the distribution is singular, there is no population $\lambda$-tree so
when the smoothing bandwidth decreases (when the sample size increases), the estimated $\lambda$-tree
is getting degenerated.
On the other hand, the estimated $\alpha$-trees remain stable regardless of the smoothing bandwidth.
}
\label{fig::Tree}
\end{figure}

{\em Main Results.}
The main results of this paper are summarized as follows:
\begin{itemize}
\item When the distribution is nonsingular,
	\begin{itemize}
	\item[1.] We derive the convergence rate for the estimated level set
	when the level equals the density value of a critical point (Theorem~\ref{thm::critical}).
	\item[2.] We derived the convergence rate of $\hat{\alpha}_n$ (Theorem~\ref{thm::alpha_non}).
	\end{itemize}
\item When the distribution is singular,
	\begin{itemize}
	\item[3.] We propose a framework that generalizes $\alpha(x)$
	to define the $\alpha$-tree (Section~\ref{sec::singular})
	\item[4.] We show that after rescaling, the KDE is pointwisely, but
	not uniformly, consistent (Theorem~\ref{thm::KDE}).
	\item[5.] We prove that $\hat{\alpha}_n$ is inconsistent under the $L_\infty$ metric (Corollary~\ref{cor::alpha_infty_s})
	but consistent under the $L_1$ and $L_1(P)$ distance (Theorem~\ref{thm::alpha::Pro}).
	\item[6.] We identify a new type of critical points, the dimensional critical points (DCPs), which also contribute
	to the change in cluster tree topology. We analyze their properties in Lemma~\ref{lem::DCP::p}, \ref{lem::DCP}, and \ref{lem::DCP_level}.
	\item[7.] We demonstrate that the estimated $\alpha$-tree $T_{\hat{\alpha}_n}$
	is topologically equivalent to the population $\alpha$-tree with probability exponentially converging to $1$ (Theorem~\ref{thm::top::p}).
	\end{itemize}
\end{itemize}

{\em Related Work.}
There is extensive literature on theoretical aspects of the $\lambda$-tree.
Notions of consistency are analyzed in \cite{Hartigan81,chaudhuri2010rates,chaudhuri2014consistent,eldridge2015beyond}.
The convergence rate and
the minimax theory are studied in \cite{chaudhuri2010rates,balakrishnan2012,chaudhuri2014consistent}.
\cite{chen2016statistical} study how to perform statistical inference for a $\lambda$-tree.
The cluster tree is also related to the topological data analysis \citep{carlsson2009topology,edelsbrunner2012persistent}.
In particular, a cluster tree contains information about the zeroth order homology groups
\citep{cohen2007stability,fasy2014confidence,bobrowski2014topological,bubenik2015statistical}.
The theory of estimating a cluster tree is closely related to the theory of estimating a level set;
an incomplete list of literature is as follows:
\cite{polonik1995measuring,Tsybakov1997,Walther1997,mason2009asymptotic,singh2009adaptive,Rinaldo2010,steinwart2011adaptive}.

{\em Outline.}
We begin with an introduction of cluster trees and the geometric concepts used in this paper in Section~\ref{sec::background}.
In Section~\ref{sec::nonsingular}, we derive the convergence rate for the $\alpha$-tree estimator under nonsingular measures.
In Section~\ref{sec::singular}, we study the behavior of the KDE
and the stability of the estimated $\alpha$-tree under singular measures.
In Section~\ref{sec::topology}, we investigate critical points of singular measures and derive the topological consistency
of the estimated $\alpha$-tree.
We summarize this paper and discuss possible future directions in Section~\ref{sec::discussion}.

\section{Backgrounds}	\label{sec::background}

\subsection{Cluster Trees}		\label{sec::cluster_tree}
Here we recall the definition of cluster trees in \cite{chen2016statistical}.
Let $\K\subset \R^d$ and $f:\K\mapsto [0,\infty)$ be a function with support $\K$.
The cluster tree of function $f$ is defined as follows.

\begin{definition}[Definition 1 in \cite{chen2016statistical}]
For any $f: \K \mapsto [0,\infty)$
the {\em cluster tree} of $f$ is a function
$T_f: \mathbb{R} \mapsto 2^{\K}$, where
$2^{\K}$ denotes the set of all subsets of $\K$,
and $T_f(\lambda)$ is the set of connected components
of the upper level set $\{x \in \K: f(x) \geq \lambda\}$.
We define
the collection of connected components $\{T_f\}$,
as $\{T_f\}=\underset{\lambda}{\bigcup}~T_f(\lambda)$. Thus,  $\{T_f\}$
is a collection of subsets of $\K$ indexed by $\lambda$.
\label{def::tree}
\end{definition}

Clearly, the cluster tree $T_f$ has a tree structure,
because for each pair $C_{1},C_{2}\in T_f$, either
$C_{1}\subset C_{2}$, $C_{2}\subset C_{1}$, or $C_{1}\cap C_{2}=\phi$ holds.

\begin{figure}
\includegraphics[height=1.5in]{figures/Equiv_1}
\includegraphics[height=1.5in]{figures/Equiv_2_tree}
\caption{
Connected components, edges, and edge set of a cluster tree.
Left: We display connected components of level sets under five different levels
(indicated by the colors: magenta, yellow, sea green, sky blue, and gray).
The color of boundaries of each connected component denotes the edge they correspond to.
Right: The cluster tree. We color the three edges (vertical lines) red, blue, and black.
The edge set $E(T_f) = \{\mathbb{C}_{\sf red}, \mathbb{C}_{\sf blue}, \mathbb{C}_{\sf black}\}$
and we have the ordering $\mathbb{C}_{\sf red}\leq \mathbb{C}_{\sf black}$ and $\mathbb{C}_{\sf blue}\leq \mathbb{C}_{\sf black}$.
Note that the solid black horizontal line is not an edge set; it is a visual
representation of connecting the blue and red edges to the black edge.
The horizontal dashed lines indicate the five levels corresponding to the left panel.
In the left panel, the three connected components with red boundaries are elements of the edge $\mathbb{C}_{\sf red}.$
}
\label{fig::equiv}
\end{figure}

To get a geometric understanding of the cluster tree in Definition
\ref{def::tree}, we identify edges that constitute the cluster
tree. Intuitively, edges correspond to either leaves or internal
branches. An edge is roughly defined as a set of clusters whose
inclusion relation with respect to clusters outside an edge is
equivalent. So when the collection of connected components is
divided into edges, we observe the same inclusion relation between
representative clusters whenever any cluster is selected as
representative for each edge.

To formally define edges, we define an interval in the cluster
tree, and the equivalence relation in the cluster tree. For any two
clusters $A,B\in\{T_f\}$, the interval $[A,B]\subset\{T_f\}$ is
defined as a set clusters that contain $A$ and are contained in $B$,
i.e.
\[
[A,B]:=\left\{ C\in\{T_f\}:\,A\subset C\subset B\right\} ,
\]
The equivalence relation $\sim$ is defined as $A\sim B$ if and only if their inclusion relation with respect to clusters outside $[A,B]$ and $[B,A]$.
Namely,
$A\sim B$ if and only if
\begin{align*}
\forall C \in \{T_f\} \text{ such that }&C\notin[A,B]\cup[B,A],\\
&C\subset A \Leftrightarrow C\subset B, \quad A\subset C \Leftrightarrow B\subset C.
\end{align*}
It is easy to see that the relation $\sim$ is reflexive($A\sim
A$), symmetric$(A\sim B$ implies $B\sim A$), and transitive ($A\sim B$
and $B\sim C$ implies $A\sim C$). Hence the relation $\sim$ is indeed
an equivalence relation, and we can consider the set of equivalence
classes $\{T_f\}/_{\sim}$. We define the edge set (the collection of edges) $E(T_{f})$ as
$E(T_{f}):=\{T_f\}/_{\sim}$.
Each element in the edge set $\mathbb{C}\in E(T_f)$ is called an edge,
which contains many nested connected components of the cluster tree $\{T_f\}$ (i.e. if $C_1,C_2\in\mathbb{C}$, then
either $C_1\subset C_2$ or $C_2\subset C_1$).
Note that every element in an edge corresponds to a connected component of an upper level set of function $f$.

To associate the edge set $E(T_{f})$ to a tree structure,
we define a partial order on the edge set as follows:
let $\mathbb{C}_1,\mathbb{C}_2\in E(T_f)$ be two edges, we write
$\mathbb{C}_1\leq \mathbb{C}_2$ if and only if $A\subset B$ for all $A\in \mathbb{C}_1$ and $B\in \mathbb{C}_2$.
Then the topology of the cluster tree (the shape of the cluster tree) is completely determined by the
edge set $E(T_f)$ and the partial order among the edge set.
Figure~\ref{fig::equiv} provides an example of the connected components, edges, and edge set of a cluster tree
along with a tree representation.

Based on the above definitions,
we define the topological equivalence between two cluster trees.
\begin{definition}
For two functions $f: \K \mapsto [0,\infty)$ and $g: \K \mapsto [0,\infty)$,
we say $T_f$ and $T_g$ are topological equivalent, denoted as $T_f\overset{T}{\approx} T_g$, if
there exists a bijective mapping $S: E(T_{f})\mapsto E(T_{g})$ such that for any $\mathbb{C}_1, \mathbb{C}_2\in E(T_f)$,
$$
\mathbb{C}_1 \leq \mathbb{C}_2 \Longleftrightarrow S(\mathbb{C}_1) \leq S(\mathbb{C}_2).
$$
\label{def::topology}
\end{definition}

For each $\mathbb{C}\in E(T_f)$, we define
$$
U(\mathbb{C}) = \sup\{\lambda: \exists C\in T_f(\lambda), C\in \mathbb{C}\}
$$
to be the maximal level of an edge $\mathbb{C}$.
We define the \emph{critical tree-levels} of function $f$ as
\begin{equation}
\mathcal{A}_f = \{U(\mathbb{C}): \mathbb{C}\in E(T_f)\}.
\label{eq::A_f}
\end{equation}
It is easy to see that $\mathcal{A}_f$ is the collection of levels of $f$ where
the creation of a new connected component or the merging of two connected components occurs.

In most of the cluster tree literature \citep{balakrishnan2012,chaudhuri2010rates,chaudhuri2014consistent,chen2016statistical,eldridge2015beyond},
the cluster tree is referred to as the $\lambda$-tree,
which uses the probability density function $p$ to build a cluster tree.
Namely, the $\lambda$-tree is $T_p$.

In this paper, we focus on the $\alpha$-tree \citep{Kent2013}
that uses the function
\begin{equation}
\alpha(x) = P(\{y:p(y)\leq p(x)\}) =  1- P(\{y: p(y)> p(x)\}) = 1- P(L_{p(x)})
\label{eq::alpha}
\end{equation}
to build the cluster tree $T_\alpha$ ($T_\alpha$ is called the \emph{$\alpha$-tree}).
The set $L_\lambda = \{x: p(x)\geq \lambda\}$ is the upper level set of $p$
(note that $P(\{y: p(y)> p(x)\}) = P(\{y: p(y)\geq p(x)\})$ when the density function $p$ is bounded).
A feature of the $\alpha$-tree is that the function $\alpha(x)$ depends only on the ordering of points within $\K$.
Namely, any function that assigns the same ordering of points within $\K$ as the density function $p$
can be used to construct the function $\alpha(x)$.
Specifically, let $\omega$ be an ordering such that for any $x_1,x_2\in\K$,
\begin{align*}
\omega(x_1)> \omega(x_2) &\Leftrightarrow p(x_1)>p(x_2),\\
\omega(x_1)< \omega(x_2) &\Leftrightarrow p(x_1)<p(x_2),\\
\omega(x_1)= \omega(x_2) &\Leftrightarrow p(x_1)=p(x_2).
\end{align*}
Then
\begin{equation}
\alpha(x) = 1- P(\{y: p(y)\geq p(x)\}) = 1- P(\{y: \omega(y)\geq \omega(x)\}).
\label{eq::alpha0}
\end{equation}
For instance, $\omega(x) = 2 p(x)$, or $\omega(x) = \log p(x)$ both yield the same $\alpha(x)$.
We will use this feature later to generalize equation \eqref{eq::alpha} to singular measures.

One feature of the $\alpha$-tree is that it is topological equivalent to the $\lambda$-tree.
\begin{lem}
Assume the distribution $P$ has density $p$.
Then the $\lambda$-tree and $\alpha$-tree are topological equivalent.
Namely,
$$
T_p \overset{T}{\approx} T_\alpha .
$$
\label{lem::tree}
\end{lem}
The proof of this lemma is simple so we ignore it. The main idea is that
by equation \eqref{eq::alpha}, $\alpha(x)$ is a monotonic transformation of the density $p$ so
the topology is preserved.

When we use the $\alpha$-tree, the induced upper level set
$$
\mathbb{A}_a = \left\{x: \alpha(x)\geq a\right\}
$$
is called an \emph{$\alpha$-level set}.

\begin{remark}[$\kappa$-tree]
\cite{Kent2013} also proposed another cluster tree--the $\kappa$-tree--which
uses the probability content within each edge set defined by an $\alpha$-tree (or a $\lambda$-tree)
to compute the function $\kappa(x)$.
Because it is just a rescaling from the $\alpha$-tree,
the theory of $\alpha$-tree also works for the $\kappa$-tree.
For simplicity, we only study the theory of the $\alpha$-tree in this paper.
\end{remark}

\subsection{Singular Measure}	\label{sec::singular}
When the probability measure is singular, the $\lambda$-tree is no longer well-defined
because there is no density function.
However,
the $\alpha$-tree
can still be defined.

A key feature for constructing the $\alpha$-tree is the ordering function $\omega(x)$.
Here we will use a generalized density function, the Hausdorff density \citep{preiss1987geometry,mattila1999geometry},
to define the $\alpha$-tree
under singular measures.
Given a probability measure $P$, the $s$-density ($s$ dimensional Hausdorff density) is
$$
\mathcal{H}_s (x) = \lim_{r\rightarrow 0}\frac{P(B(x,r))}{C_s\cdot r^s},
$$
where $C_s$ is the $s$-dimensional volume of an $s$-dimensional unit ball and $B(x,r) = \{y: \|y-x\|\leq r\}$.

For a given point $x$, we define the notion of generalized density using two quantities $\tau(x)$ and $\rho(x)$:
\begin{align*}
\tau(x) & = {\sf argmax}_{s\leq d} \mathcal{H}_s(x)<\infty\\
\rho(x) & = \mathcal{H}_{\tau(x)}(x).
\end{align*}
Namely, $\tau(x)$ is the `dimension' of the probability measure at $x$ and $\rho(x)$
is the corresponding Hausdorff density at that dimension.
Note that the function $\rho(x)$ is well-defined for every $x$.
For any two points $x_1,x_2\in\K$, we define an ordering $\omega$ by
$\omega(x_1)> \omega(x_2)$ if
$$
\tau(x_1)<\tau(x_2), \qquad \mbox{or}\quad \tau(x_1)=\tau(x_2), \quad \rho(x_1)>\rho(x_2).
$$
That is, for any pair of points,
we first compare their dimensions $\tau(x)$.
The point with the lower dimensional value $\tau$ will be ranked higher than the other point.
If two points have the same dimension, then we compare their
corresponding Hausdorff density.
When the distribution is nonsingular, $\tau(x)=d$ for every $x\in\K$
and $\rho(x) = p(x)$ is the usual density function.
Thus, the ordering function $\omega(x)$ can be chosen as the density function $p(x)$.

To define the $\alpha$-tree,
we use equation \eqref{eq::alpha0}:
\begin{equation}
\alpha(x) = P\left(\{y: \omega(y)\geq \omega(x)\}\right).
\label{eq::alpha1}
\end{equation}
Namely, $\alpha(x)$ is the probability content of regions where the ordering function $\omega$ is lower than or equal to $\omega(x)$.
As shown in equation \eqref{eq::alpha0}, equation \eqref{eq::alpha1} is the same as equation \eqref{eq::alpha} when $P$ is nonsingular.
Note that by equation \eqref{eq::alpha1},
the $\alpha$-level set $\mathbb{A}_a = \left\{x: \alpha(x)\geq a\right\}$ is
well-defined in a singular measure.

\subsection{Geometric Concepts} \label{sec::geometrics}

Based on the definition of $\tau(x)$, we decompose the support $\K$ into
\begin{equation}
\K = \K_d\bigcup \K_{d-1}\bigcup \cdots \bigcup \K_0,
\label{eq::support}
\end{equation}
where $\K_s = \{x: \tau(x)=s\}$.
Thus, $\{\K_0,\cdots,\K_d\}$ forms a partition of the entire support $\K$.
We call each $\K_s$ an $s$-dimensional support.
When we analyze the support $\K_s$,
any $\K_{s'}$ with $s'>s$ is called a higher dimensional support (with respect to $\K_s$) and
$s'<s$ will be called a lower dimensional support.

\begin{figure}
\center
\includegraphics[height=2in]{figures/ex02-1}
\caption{
The supports $\K_2,\K_1,$ and $\K_0$ for the bivariate random variable in Example \ref{ex::ex02}.
The yellow region is $\K_2 = [-1,1]^2 \backslash  R_{0.5}$, where $R_{0.5} = \{(x,y): x^2+y^2 = 0.5^2\}$.
The red ring is $\K_1 = R_{0.5}\backslash \{(0.5,0)\}$.
The blue dot is the location of $(0.5,0)$.
}
\label{fig::ex02}
\end{figure}

\begin{example}
Let $X$ be a bivariate random variable such that (i) with a probability of $0.7$, it is from a uniform distribution on $[-1,1]^2$,
(ii) with a probability of $0.25$, it is from a uniform distribution on the ring $R_{0.5} = \{(x,y): x^2+y^2 = 0.5^2\}$,
(iii) with a probability of $0.05$, it is equal to $(0.5,0)$.
Apparently, the distribution of $X$ is singular and the support $\K = [-1,1]^2$.
In this case, $\K_2 = [-1,1]^2 \backslash  R_{0.5}$, $\K_1 = R_{0.5}\backslash \{(0.5,0)\}$, and $\K_0 = \{(0.5,0)\}$.
Figure~\ref{fig::ex02} shows these supports under different colors.
The yellow rectangular region is $\K_2$, the red ring area is $\K_1$, and the blue dot denotes the location of $\K_0$.
\label{ex::ex02}
\end{example}

To regularize the behavior of $\rho(x)$ on each support $\K_s$,
we assume that the closure of the support $\overline{\K_s}$
is an $s$-dimensional smooth manifold
(properties of a smooth manifold can be found in
\citealt{lee2012introduction,tu2010introduction}).
For an $s$-dimensional smooth manifold $\mathcal{M}$,
the tangent space on each point of $\mathcal{M}$ changes smoothly \citep{tu2010introduction, lee2012introduction}.
Namely, for $x\in\mathcal{M}$,
we can find an orthonormal basis $\{v_1(x),\cdots, v_s(x): v_\ell(x)\in\R^d, \ell=1,\cdots, s\}$ such that
$\{v_1(x),\cdots, v_s(x)\}$ spans
the tangent space of $\mathcal{M}$ at $x$
and each $v_\ell(x)$ is a smooth (multivalued) function on $\mathcal{M}_s$.
For simplicity, for $x\in \K_s$, we denote $T_s(x)$ as the tangent space of $\K_s$ at $x$,
$N_s(x)$ as the normal space of $\K_s$ at $x$,
and $\nabla_{T_s(x)}$ to be taking the derivative in the tangent space.

For a function $f:\mathcal{M}\mapsto\R$ defined on a smooth manifold $\mathcal{M}$,
the function $f$ is a \emph{Morse function} \citep{milnor1963morse,morse1925relations,morse1930foundations}
if all its critical points are non-degenerate.
Namely, the eigenvalues of the Hessian matrix of $f$ at each critical point are away from zero.
When the function $f$ is a Morse function, its $\lambda$-tree is stable \citep{chazal2014robust,chen2016statistical}..

To link the concept of the Morse function to the Hausdorff density $\rho(x)$,
we introduce a generalized density
$$
\rho^\dagger_s:\overline{\K_s} \mapsto [0,\infty)
$$
such that $\rho^\dagger_s(x) = \lim_{x_n\in \K_s: x_n\rightarrow x} \rho(x_n)$.
It is easy to see that $\rho^\dagger_s(x) = \rho(x)$ when $x\in\K_s$
but now it is defined on a smooth manifold $\overline{\K_s}$.
We say $\rho(x)$ is a \emph{generalized Morse function} if
the corresponding $\rho^\dagger_s(x)$ is a Morse function for $s=1,\cdots, d$.
Later we will show that this generalization leads to a stable $\alpha$-tree for a singular measure.

For $\overline{\K_s}$,
let $\mathcal{C}_s =\{x\in\overline{\K}: \nabla_{T_s(x)}\rho^\dagger_s(x)=0\}$ be the collection of its critical points.
Then the fact that $\rho^\dagger_s(x)$ is a Morse function
implies that the eigenvalues of the Hessian matrix $\nabla_{T_s(c)}\nabla_{T_s(c)}\rho^\dagger_s(c)$ are non-zero
for every $c\in\mathcal{C}_s$.
We call $g_s(x)=\nabla_{T_s(x)}\rho^\dagger_s(x)$ the generalized gradient and
$H_s(x)=\nabla_{T_s(x)}\nabla_{T_s(x)}\rho^\dagger_s(x)$ the generalized Hessian.
For the case $s=0$ (point mass), we define $ \mathcal{C}_0=\K_0 $.
The collection
$\mathcal{C} = \bigcup_{s=1,\cdots,d} \mathcal{C}_s$
is called the collection of\emph{ generalized critical points} of $\rho(x)$.
Each element $c\in\mathcal{C}$ is called a generalized critical point.

Finally, we introduce the concept of \emph{reach} \citep{Federer1959,chen2015density} for a smooth manifold $\mathcal{M}$.
The reach of $\mathcal{M}$ is defined as
$$
{\sf reach}(\mathcal{M}) = \sup\{r\geq0: \mbox{every point in }\mathcal{M}\oplus r\mbox{ has a unique projection onto $\mathcal{M}$}\},
$$
where $A\oplus r = \{x: d(x,A)\leq r\}$.
One can view reach as the radius of the largest ball that can roll freely outside $\mathcal{M}$.
More details about reach can be found in \cite{Federer1959,chen2015density}.
Reach plays a key role in the stability of a level set; see \cite{chen2015density} for more details.

\subsection{Estimating the $\alpha$ function and the $\alpha$-tree}
In this paper, we focus on estimating $\alpha$-trees via
the KDE:
$$
\hat{p}_n(x) = \frac{1}{nh^d}\sum_{i=1}^nK\left(\frac{\|x-X_i\|}{h}\right).
$$
Specifically, we first estimate the density by $\hat{p}_n$ and then
construct the estimator $\hat{\alpha}_n$:
\begin{equation}
\hat{\alpha}_n(x) = \hat{P}_n\left(\{y: \hat{p}_n(y)\leq \hat{p}_n(x)\}\right)
\label{eq::alpha_estimate}
\end{equation}
where $\hat{P}_n$ is the empirical measure and $\hat{L}_\lambda = \{x: \hat{p}_n(x)\geq \lambda\}$.
Note that when $x$ does not contain any point mass of $P$,
$\hat{\alpha}_n(x)= 1-\hat{P}_n\left(\hat{L}_{\hat{p}_n(x)}\right).$

To quantify the uncertainty in the estimator $\hat{\alpha}_n$, we consider three error measurements.
The first error measurement is
the \emph{$L_\infty$ error}, which is defined as
$$
\|\hat{\alpha}_n-\alpha\|_\infty=\sup_x|\hat{\alpha}_n(x)-\alpha(x)|.
$$
The $L_\infty$ error has been used in several cluster tree literature; see, e.g., \cite{eldridge2015beyond,chen2016statistical}.
An appealing feature of the $L_\infty$ error is that this quantity
is the same (up to some constant) as some other tree error metrics such as the merge distortion metric \citep{eldridge2015beyond}.
Convergence in the merge distortion metric implies the Hartigan consistency \citep{eldridge2015beyond},
a notion of consistency of a cluster tree estimator described in \cite{Hartigan81,chaudhuri2010rates,chaudhuri2014consistent}.
Thus, because of the equivalence between the $L_\infty$ error and the merge distortion metric, convergence in $L_\infty$ implies
the Hartigan consistency of an estimated cluster tree.

The other two errors are the integrated error and the probability error (probability-weighted integrated error).
Both are common error measurements for evaluating the quality of a function estimator \citep{wasserman2006all,scott2015multivariate}.
The \emph{integrated error} is
$$
\|\hat{\alpha}_n-\alpha\|_{\mu} = \int |\hat{\alpha}_n(x)-\alpha(x)| dx,
$$
which is also known as the integrated distance or $L_1$ distance.
The \emph{probability error (probability-weighted integrated error)} is
$$
\|\hat{\alpha}_n-\alpha\|_{P}=\int |\hat{\alpha}_n(x)-\alpha(x)| dP(x),
$$
which
is the integrated distance weighted by the probability measure, which is also known
as $L_1(P)$ distance.
The integrated error and the probability error
are more robust than the $L_\infty$ error--a large difference in a small region will not
affect on these errors much.

To quantify the uncertainty in the topology of $\alpha$-tree,
we introduce the notion of \emph{topological error}, which is defined as
$$
P\left(T_{\hat{\alpha}_n}\not\overset{T}{\approx} T_\alpha\right) = 1- P\left(T_{\hat{\alpha}_n}\overset{T}{\approx} T_\alpha\right).
$$
Namely, the topological error is the probability that the estimated $\alpha$-tree
is not topological equivalent to the population $\alpha$-tree.

Finally, we define the following notations.
For a smooth function $p$, we define $\|p\|_{\ell,\infty}$ as the supremum maximal norm of $\ell$-th derivative of $p$.
For instance,
$$
\|p\|_{0,\infty}=\sup_{x\in\K}p(x),\quad\|p\|_{1,\infty}=\sup_{x\in\K}\|g(x)\|_{\max},\quad\|p\|_{2,\infty}=\sup_{x\in\K}\|H(x)\|_{\max},
$$
where $g(x) = \nabla p(x)$ and $H(x) = \nabla \nabla p(x)$ are the gradient and Hessian matrix of $p(x)$, respectively.
A vector $\alpha =
(\alpha_1,\ldots,\alpha_d)$ of non-negative integers is called a
multi-index with $|\alpha| = \alpha_1 + \alpha_2 + \cdots + \alpha_d$
and the corresponding derivative operator is
\begin{equation}
D^\alpha = \frac{\partial^{\alpha_1}}{\partial x_1^{\alpha_1}} \cdots
\frac{\partial^{\alpha_d}}{\partial x_d^{\alpha_d}},
\label{eq::d1}
\end{equation}
where $D^\alpha f$ is often written as $f^{(\alpha)}$.
For a set $A$, define $\overline{A}$ to be the closure of $A$, $\partial A $ to be the boundary of set $A$,
and $A^C= \K\backslash A$ to be the complement of set $A$ restricted to the support $\K$.
We define $A\triangle B = (A\backslash B)\cup (B\backslash A)$ to be the symmetric difference for sets $A$ and $B$

\section{Theory for Nonsingular Measures}	\label{sec::nonsingular}

To study the theory for nonsingular measures, we
make the following assumptions.\\
{\bf Assumptions.}
\begin{itemize}
\item[(P1)] $p$ has a compact support $\K$ and is a Morse function with $\|p\|_{\ell,\infty}<\infty$ for $\ell=0,\cdots, 4.$
\item[(K1)] $K(x)$ has compact support and is non-increasing on $[0,1]$, and has at least fourth-order bounded derivative
and
$$
\int \|x\|^2 K^{(\alpha)}(\|x\|) dx <\infty, \qquad \int  \left(K^{(\alpha)}(\|x\|)\right)^2 dx <\infty
$$
for $|\alpha| \leq 2$.

\item[(K2)] Let
\begin{align*}
\mathcal{K}_r &= \left\{y\mapsto K^{(\alpha)}\left(\frac{\|x-y\|}{h}\right): x\in\mathbb{R}^d, |\alpha|=r\right\},
\end{align*}
where $K^{(\alpha)}$ is defined in equation \eqref{eq::d1}
and $\mathcal{K}^*_l = \bigcup_{r=0}^l \mathcal{K}_r$.
We assume that $\mathcal{K}^*_2$ is a VC-type class. i.e.
there exists constants $A,v$ and a constant envelope $b_0$ such that
\begin{equation}
\sup_{Q} N(\mathcal{K}^*_2, \cL^2(Q), b_0\epsilon)\leq \left(\frac{A}{\epsilon}\right)^v,
\label{eq::VC}
\end{equation}
where $N(T,d_T,\epsilon)$ is the $\epsilon$-covering number for a
semi-metric set $T$ with metric $d_T$ and $\cL^2(Q)$ is the $L_2$ norm
with respect to the probability measure $Q$.

\end{itemize}
Assumption (P1) is a common condition to guarantee the stability of critical points
\citep{chazal2014robust,chen2016statistical}.
We need the fourth-order derivative to ensure the estimated density Hessian matrix
converges to the population density Hessian matrix
(the bias in estimating the Hessian matrix depends on fourth-order derivatives).
Assumption (K1) is a standard condition on kernel function \citep{wasserman2006all,scott2015multivariate}.
Assumption (K2) regularizes the complexity of kernel functions so
we have uniform bounds on density, gradient, and Hessian estimation.
It was first proposed by \cite{Gine2002} and \cite{Einmahl2005}
and was later used in various studies such as \cite{genovese2009path,genovese2014nonparametric,chen2015asymptotic}.

We first study the error rates under nonsingular measures.
In the case of the $\lambda$-tree, error rates are well-studied, and
we summarize them in the following theorem.

\begin{thm}
Assume (P1, K1--2).
Then when $h\rightarrow 0,\frac{nh^{d+4}}{\log n} \rightarrow\infty$,
\begin{align*}
\|\hat{p}_n-p\|_\infty & =  O(h^2) + O_P\left(\sqrt{\frac{\log n}{nh^d}}\right)\\
\|\hat{p}_n-p\|_\mu & = O(h^2) + O_P\left(\sqrt{\frac{1}{nh^d}}\right)\\
\|\hat{p}_n-p\|_P & = O(h^2) + O_P\left(\sqrt{\frac{1}{nh^d}}\right)\\
P\left(T_{\hat{p}_n}\overset{T}{\approx} T_{p}\right)& \geq 1-\frac{c_0}{h^d}\cdot e^{-c_1\cdot nh^{d+4}},
\end{align*}
for some $c_0, c_1>0$.
\label{thm::lambda::non}
\end{thm}
The rate of consistency under the $L_\infty$ error can be found in \cite{chen2015density,Gine2002,Einmahl2005}.
The integrated error and probability error can be seen in \cite{scott2015multivariate}.
And the topological error bound follows Lemma 2 in \cite{chen2016statistical} and the concentration of $L_\infty$ metric
for the estimated Hessian matrix.

Now we turn to the consistency for $\alpha$-tree.
To derive the rate for the $\alpha$-tree,
we need to study the convergence rate of an estimated level set when the level is
the density value of a critical point (also known as a critical level).
The reason is that the quantity $\alpha(x) = 1-P(L_{p(x)})$
is the probability content of upper level set $L_{p(x)} = \{y: p(y)\geq p(x)\}$.
When $p(x)=p(c)$ for some critical point $c$ of $p$, we face the problem of analyzing the stability of
level sets at a critical level.

\begin{thm}[Level set error at a critical value]
Assume (P1) and (K1--2) and $d\geq 2$.
Let $\lambda$ be a density level corresponding to the density of a critical point.
Then
$$
\mu\left(\hat{L}_\lambda\triangle L_{\lambda}\right) = O_P\left(\|\hat{p}_n - p\|_{\mu}^{\frac{d}{d+1}}\right),
$$
where $\mu$ is the Lebesgue measure.
\label{thm::critical}
\end{thm}
The rate in Theorem~\ref{thm::critical} is slower than the usual density estimation rate.
This is because the boundary of $L_\lambda$ hits a critical point when $\lambda$ equals the density of a critical point.
The regions around a critical point have a very low gradient,
which leads to a slower convergence rate.
It is well-known \citep{wasserman2006all,scott2015multivariate} that under assumption (P) and (K1),
$$
\|\hat{p}_n - p\|_{\mu} = O(h^2) + O_P\left(\sqrt{\frac{1}{nh^d}}\right).
$$

In Theorem~\ref{thm::critical}, we see that when $d$ is large, the quantity $\frac{d}{d+1}\rightarrow 1$
so the error rate is similar to $\|\hat{p}_n - p\|_{\mu}$.
This is because the regions that slow down the error rate are areas around the critical points.
These areas occupy a small volume when $d$ is large, which decreases the difference in the rate.

\begin{remark}
The Hausdorff distance when $\lambda$ is the density of a saddle point is
${\sf Haus}\left(\hat{L}_\lambda, L_{\lambda}\right) = O_P\left(\|\hat{g}_n-g\|_{\infty}\right),$
where $$
{\sf Haus}(A,B) = \max\left\{\sup_{x\in A}d(x,B), \sup_{x\in B}d(x,A)\right\}$$ and $d(x,A) = \inf_{y\in A}\|x-y\|$.
Thus, the symmetric distance and the Hausdorff distance have different rates of convergence.
The reason is that the convergence rate of the Hausdorff distance is dominated by uncertainty around critical points, which
is at the gradient error rate.
When we consider the symmetric distance $\mu(A\triangle B)$, the uncertainty around critical points
is a minor contribution that can be ignored.
\end{remark}

\begin{remark}
Theorem~\ref{thm::critical} complements many existing level set estimation theories.
To our knowledge, no literature has worked on the situation where $\lambda$ equals the density of a critical point.
Level set theories mostly focus on one of the following three cases:
(i) the gradient on the level set boundary $\partial L_{\lambda} = \{x: p(x)=\lambda\}$
is bounded away from $0$ \citep{Molchanov1990,Tsybakov1997,Walther1997,Cadre2006,Laloe2012,mammen2013confidence,chen2015density},
(ii) a lower bound on the density changing rate around level $\lambda$ \citep{singh2009adaptive, rinaldo2012stability},
(iii) an $(\epsilon,\sigma)$ condition for density \citep{chaudhuri2010rates,chaudhuri2014consistent}.
When $\lambda$ equals a critical level, none of these assumptions hold.

\end{remark}

Based on Theorem~\ref{thm::critical}, we derive the convergence rate of $\hat{\alpha}_n$.
\begin{thm}
Assume (P1) and (K1--2) and $d\geq 2$.
Let $\mathcal{C} = \{x: \nabla p(x)=0\}$ be the collection of critical points
and let $a_n$ be a sequence of $n$ such that
$\|\hat{p}_n-p\|_\infty = o(a_n)$.
Then uniformly for all $x$,
$$
\hat{\alpha}_n(x) - \alpha(x) =
\begin{cases}
O_P\left(\|\hat{p}_n-p\|_\mu\right) \quad&\mbox{, if}\quad |p(x)-p(c)|>a_n\mbox{ for all $c\in\mathcal{C}$,}\\
O_P\left(\|\hat{p}_n-p\|_\mu^{\frac{d}{d+1}}\right) \quad&\mbox{, otherwise}.
\end{cases}
$$
\label{thm::alpha_non}
\end{thm}
Theorem~\ref{thm::alpha_non} shows uniform error rates for $\hat{\alpha}_n$.
When the density of a given point is away from critical levels,
the rate follows the usual density estimation rate.
When the given point has a density value close to some critical points,
the rate is slowed down by the low gradient areas around critical points.
Note that the sequence $a_n$ is to make the bound uniform for all $x$.
To obtain an integrated error rate (and the probability error rate) of $\hat{\alpha}_n$,
we can choose $a_n = \frac{1}{\log n}\left(O(h^2) + O_P\left(\sqrt{\frac{1}{nh^d}}\right)\right)$
which leads to the following result.

\begin{cor}
Assume (P1) and (K1--2) and $d\geq 2$.
Then when $h\rightarrow 0,\frac{nh^{d+4}}{\log n} \rightarrow\infty$,
\begin{align*}
\|\hat{\alpha}_n-\alpha\|_\infty & =  O\left(h^{\frac{2d}{d+1}}\right)+O_P\left(\left(\frac{\log n}{nh^d}\right)^{\frac{d}{2(d+1)}}\right)\\
\|\hat{\alpha}_n-\alpha\|_\mu & = O(h^2) + O_P\left(\sqrt{\frac{1}{nh^d}}\right)\\
\|\hat{\alpha}_n-\alpha\|_P & = O(h^2) + O_P\left(\sqrt{\frac{1}{nh^d}}\right)\\
P\left(T_{\hat{\alpha}_n}\overset{T}{\approx} T_\alpha\right)& \geq 1-\frac{c_0}{h^d}\cdot e^{-c_1\cdot nh^{d+4}},
\end{align*}
for some $c_0,c_1>0$.
\label{cor::alpha_non}
\end{cor}
Comparing Corollary \ref{cor::alpha_non} to Theorem~\ref{thm::lambda::non},
we see that
the only difference is the $L_\infty$ error rate.
This is because Theorem~\ref{thm::alpha_non} proves that, only at the level of a critical point, we will have
a slower convergence rate.
Thus, the $L_\infty$ error will be slowed down by these points.
However, the collection of points $\{x: p(x)=p(c)\mbox{ for some }c\in\mathcal{C}\}$ has Lebesgue measure zero
so the slow convergence rate does not translate to the integrated error and the probability error.
The topological error follows from Theorem~\ref{thm::lambda::non} and Lemma~\ref{lem::tree}:
$
T(\hat{p}_n)\overset{T}{\approx} T_{\hat{\alpha}_n},
T(p)\overset{T}{\approx} T_\alpha.
$

\section{Singular Measures: Error Rates}	\label{sec::singular}

Now we study error rates under singular measures.
When a measure is singular, the usual (Radon-Nikodym) density cannot be defined.
Thus, we cannot define the $\lambda$-tree.
However, as we discussed in Section~\ref{sec::singular},
we are still able to define the $\alpha$-tree.
Thus, in this section, we will focus on error rates for the $\alpha$-tree.

\subsection{Analysis of the KDE under Singular Measures}

To study the convergence rate, we first investigate the bias of smoothing in the singular measure.
Let $p_h(x) = \mathbb{E}(\hat{p}_n)$, which is also known as the smoothed density.

{\bf Assumption.}
\begin{itemize}
\item[(S)] For all $s<d$, $\overline{\K_s}$ is a smooth manifold with positive reach and $\K$ is a compact set.
\item[(P2)] $\rho(x)$ is a generalized Morse function and there exists some $\rho_{\min},\rho_{\max}>0$ such that
$0<\rho_{\min}\leq \rho(x)\leq\rho_{\max}<\infty$ for all $x$.
Moreover, for $s>0$, $\rho^\dagger_s$ has bounded continuous derivatives up to the fourth order.
\end{itemize}

Assumption (S) ensures that
the Hausdorff density $\rho(x)$ behaves well within each lower dimensional support $\K_s$.
Assumption (P2) is a generalization of (P1) to singular distributions.

\begin{lem}[Bias of the smoothed density]
Assume (S, P2).
Let $x\in\K_s$ and define
$m(x) = \min\{\ell> s: x\in\overline{\K_\ell}\}-s $.
Let $C^\dagger_\ell = (\int_{B_\ell}K(\|x\|)dx)^{-1}$, where $B_\ell = \{x: \|x\|\leq1,x_{\ell+1}=x_{\ell+2}=\cdots=x_d=0\}$
for $\ell=1,\cdots, d$ and $C^\dagger_0 = 1/K(0)$.
Then
for a fixed $x$, when $h\rightarrow 0$ and $m(x)>0$,
$$
C^\dagger_{\tau(x)}h^{d-\tau(x)}\cdot p_h(x) = \rho(x)+
\begin{cases}
O(h^2) + O(h^{m(x)}),\quad &{ if }\quad m(x)>0\\
O(h^2),\quad &{ if }\quad m(x)= 0\\
\end{cases}.
$$
Moreover, if $\overline{\K_\ell}\cap \K_s\neq \phi$, for some $s<\ell$,
then there exists $\epsilon>0$ such that
$$
\lim_{h\rightarrow 0}\sup_{x\in\K}|C_{\tau(x)}h^{d-\tau(x)}\cdot  p_h(x) - \rho(x)|>\epsilon>0.
$$
\label{lem::smooth::bias}
\end{lem}
Lemma~\ref{lem::smooth::bias} describes the bias of the KDE.
The scaling factor $C^\dagger_{\tau(x)}h^{d-\tau(x)}$ rescales the smoothed density
to make it comparable to the generalized density.
The first assertion is a pointwise convergence of smoothed density.
In the case of $m(x)>0$, the bias contains two components: $O(h^2)$, the usual smoothing bias,
and $O(h^{m(x)})$, the bias from a higher dimensional support.
This is because the KDE is isotropic, so the probability content outside $\K_s$ will also be
included, which causes additional bias.
The second assertion states that the smoothed density does not uniformly converge to the generalized density $\rho(x)$,
so together with the first assertion, we conclude that the smoothing bias converges pointwisely but not
uniformly.
Next, we provide an example showing the failure of uniform convergence of a singular measure.

\begin{example}[Failure of uniform convergence]
\label{ex::point}
We consider $X$ from the same distribution as in Figure~\ref{fig::Tree}:
with a probability of $0.3$, $X=2$, and with a probability of $0.7$, $X$ follows a standard normal.
For simplicity, we assume that the kernel function is the spherical kernel $K(x) = \frac{1}{2}I(0\leq x\leq 1)$
and consider the smoothing bandwidth $h\rightarrow 0$.
This choice of kernel yields $C^\dagger_1 = 1$.
Now consider a sequence of points $x_h = 2+\frac{h}{2}$.
Then the smoothed density at each $x_h$ is
\begin{align*}
p_h(x_h)
&= \frac{1}{h}P(x_h-h<X<x_h+h) \\
&= \frac{1}{h}P\left(2-\frac{h}{2}<X<2+\frac{3h}{2}\right) \\
&\geq \frac{1}{h}P\left(X=2\right)  = \frac{3}{10h},
\end{align*}
which diverges when $h\rightarrow 0$.
However,
it is easy to see that $\tau(x_h)=1$ and $\rho(x_h) =  \frac{7}{10}\phi(x_h)\rightarrow \frac{7}{10}\phi(1)$ which is a finite number.
Thus, $|\mathbb{E}(p_h(x_h)-\rho(x_h)|$ does not converge.
\end{example}

\begin{remark}
The scaling factor in Theorem \ref{lem::smooth::bias} $C^\dagger_{\tau(x)}h^{d-\tau(x)}$
depends on the support $\K_s$ where $x$ resides.
In practice, we do not know $\tau(x)$ so we cannot properly rescale $\hat{p}_n(x)$ to estimate $\rho(x)$.
However, we are still able to rank pairs of data points based on Theorem~\ref{lem::smooth::bias}.
To see this,
assume we want to recover the ordering of $x_1$ and $x_2$ using $\omega(x)$
(i.e., $\omega(x_1)>\omega(x_2)$ or $\omega(x_2)>\omega(x_1)$ or $\omega(x_1)=\omega(x_2)$).
When $x_1$ and $x_2$ are both in $\K_s$ for some $s$,
the scaling does not affect the ranking between them
so the sign of $\rho(x_1)-\rho(x_2) $ is the same as that of $p_h(x_1)-p_h(x_2)$.
When $x_1$ and $x_2$ are in different supports (i.e. $x_1\in\K_{s_1}, x_2\in\K_{s_2}$, where $s_1\neq s_2$),
$p_h(x_1)$ and $p_h(x_2)$ diverge at different rates, meaning that we can eventually distinguish them.
Thus, ordering for most points can still be recovered under singular measures.
This is an important property that leads to the consistency of $\hat{\alpha}_n$ under other error measurements.
\end{remark}

Due to the failure of uniform convergence in the bias, the $L_\infty$ error of $\hat{\alpha}_n$
does not converge under singular measures.
\begin{cor}[$L_\infty$ error for singular measures]
Assume (S, P2).
When $\overline{\K_d}\cap \overline{\K_s}\neq \phi$, for some $s<d$,
$\|\hat{\alpha}_n-\alpha\|_\infty$ does not converge to $0$.
Namely, there exists $\epsilon>0$ such that
$$
\liminf_{n,h} P(\|\hat{\alpha}_n-\alpha\|_\infty>\epsilon)>0.
$$
\label{cor::alpha_infty_s}
\end{cor}
The proof of Corollary~\ref{cor::alpha_infty_s} is a direct application of the failure of uniform convergence in
smoothing bias shown in Theorem~\ref{lem::smooth::bias}.
This corollary shows that for a singular measure,
the $L_\infty$ error of the estimator $\hat{\alpha}_n$ does not converge in general.
Thus, there is no guarantee for the Hartigan consistency of the estimated $\alpha$-tree.

\subsection{Error Measurements}

Although Corollary~\ref{cor::alpha_infty_s} presents a negative result on estimating the $\alpha$-tree,
in this section, we show that the estimator $\hat{\alpha}_n$ is still consistent under other error measurements.
A key observation is that
there is a good region where the scaled KDE converges uniformly.

Let $\K_s(h) = \K_s \backslash(\bigcup_{\ell<s} \K_\ell\oplus h)$ be the set $\K_s(h)$
that is away from lower dimensional
supports.
We define $\K(h) = \bigcup_{s\leq d} \K_s(h)$, which is the union of each $\K_s(h)$.
Figure~\ref{fig::ex02-1} shows the good region of support $\K_s(h)$ and the original support $\K_s$ in Example~\ref{ex::ex02}.
Later we will show that the set $\K(h)$ is the good region.

\begin{figure}
\center
\includegraphics[height=1.5in]{figures/ex02-1}
\includegraphics[height=1.5in]{figures/ex02-2}
\caption{
Good regions $\K_s(h)$ for Example \ref{ex::ex02}.
Left: the original $\K_2, \K_1$, and $\K_0$.
Right: the corresponding $\K_2(h), \K_1(h)$, and $\K_0(h)$.
The yellow area is the set $\K_2(h)$, which are regions in $\K_2$ away from
lower-dimensional supports $\K_1$ and $\K_0$.
The red area is the set $\K_1(h)$, which are regions in $\K_1$ where regions close to $\K_0$ have been removed.
The blue dot is $\K_0(h)$, which is the same as $\K_0$ because there is no lower dimensional support.
Note that $\K(h)$ is the union of the three color regions in the right panel.
}
\label{fig::ex02-1}
\end{figure}

In Lemma~\ref{lem::smooth::bias}, the quantity
$$
m(x) = \min\{\ell\geq \tau(x): x\in\overline{\K_\ell}\}-\tau(x)
$$
plays a key role
in determining the rate of smoothing bias.
Only when $m(x)=1$, do we have a slower rate for the bias.
Thus, to obtain a uniform rate on the bias, we introduce the quantity
\begin{equation}
m_{\min} = \inf_{x\in \K, m(x)>0} m(x).
\label{eq::mx}
\end{equation}
If $m(x)=0$ for all $x\in \K$, we define $m_{\min}=2$.
We define the following quantities:
\begin{equation}
\begin{aligned}
\delta_{n,h,s} &= O(h^{2\bigwedge m_{\min}}) + O_P\left(\sqrt{\frac{\log n}{nh^s}}\right),\\
\delta^{(1)}_{n,h,s} &= O(h^{2\bigwedge m_{\min}}) + O_P\left(\sqrt{\frac{\log n}{nh^{s+2}}}\right),\\
\delta^{(2)}_{n,h,s} &= O(h^{2\bigwedge m_{\min}}) + O_P\left(\sqrt{\frac{\log n}{nh^{s+4}}}\right).
\end{aligned}
\label{eq::rate}
\end{equation}
We will show that these quantities act as the density estimation rate, the gradient estimation rate,
and the Hessian estimation rate.

\begin{thm}[Consistency of the KDE under singular measures]
Assume (S, P2, K1--2).
Let $C^\dagger_\ell$ be the constant in Lemma~\ref{lem::smooth::bias}.
Let $\delta_{n,h,s}, \delta^{(1)}_{n,h,s}, \delta^{(2)}_{n,h,s}$ be defined in equation \eqref{eq::rate}.
Then
when $h\rightarrow 0, \frac{nh^{d+4}}{\log n}\rightarrow \infty$,
\begin{align*}
\sup_{x\in\K_s(h)}\|C^{\dagger}_{s}h^{d-s}\hat{p}_n(x)-\rho(x)\| &= \delta_{n,h,s}\\
\sup_{x\in\K_s(h)}\|C^{\dagger}_{s}h^{d-s} \nabla_{T_s(x)}\hat{p}_n(x)-\nabla_{T_s(x)}\rho(x)\|_{\max} &= \delta^{(1)}_{n,h,s}\\
\sup_{x\in\K_s(h)}\|C^{\dagger}_{s}h^{d-s} \nabla_{T_s(x)}\nabla_{T_s(x)}\hat{p}_n(x)-\nabla_{T_s(x)}\nabla_{T_s(x)}\rho(x)\|_{\max} &= \delta^{(2)}_{n,h,s},
\end{align*}
where $\nabla_{T_s(x)}$ is taking gradient with respect to the tangent space of $\K_s$ at $x$.
\label{thm::KDE}
\end{thm}
Theorem~\ref{thm::KDE} shows that after rescaling, the KDE is uniformly consistent within the good region $\K_s(h)$
for density, gradient, and Hessian estimation.
A more interesting result is that, after rescaling, the error rate is the same as the usual $L_\infty$ error rate in the $s$-dimensional case
with a modified bias term
(bias is affected by a higher dimensional support).

\begin{remark}(Non-convergence of the integrated distance of the KDE)
\label{rm::IE_KDE}
One may wonder if
the scaled KDE ($C^\dagger_{\tau(x)}h^{d-\tau(x)}\cdot \hat{p}_n(x)$) converges to the generalized density $\rho(x)$
under the integrated distance.
In general, it does not:
$$
\int\|C^\dagger_{\tau(x)}h^{d-\tau(x)}\cdot \hat{p}_n(x)-\rho(x)\| dx = O_P(1).
$$
To see this, consider a point $x\in\K_s$ and let $\K_\ell$ be a higher order support ($\ell>s$)
with $x\in\overline{\K_\ell}$.
Then the region $B(x,h)\cap \K_\ell$ has $\ell$-dimensional volume at rate $O(h^{\ell-s})$.
For any point $y\in B(x,h)\cap \K_\ell$, $\tau(y)=\ell$ but
the KDE $\hat{p}_n(y)$ is at rate $O_P(h^{s-d})$.
Thus, the difference between the scaled KDE and the generalized density is
$$
C^\dagger_{\ell}h^{d-\ell}\cdot \hat{p}_n(y)-\rho(y) = O_P(h^{s-\ell}).
$$
Such $y$ has $\ell$-dimensional volume at rate $O(h^{\ell-s})$,
so the integrated error is at rate $O_P(h^{s-\ell})\times O(h^{\ell-s}) = O_P(1)$, which does not converge.
\end{remark}

Based on Theorem~\ref{thm::KDE}, we can derive a nearly uniform convergence rate of $\hat{\alpha}_n$.
\begin{thm}[Nearly uniform consistency of $\alpha$-trees]
Assume (S, P2, K1--2).
Let $\mathcal{C}_s$ be the collection of generalized critical points of $\K_s$.
Let $\delta_{n,h,s}$ be defined in equation \eqref{eq::rate}
and $r_{n,h,s}$ be a quantity such that $\frac{\delta_{n,h,s}}{r_{n,h,s}}=o_P(1)$.
Then when $h\rightarrow 0,\frac{nh^{d+2}}{\log n}\rightarrow \infty,$ uniformly for every $x\in\K_s(h)$,
$$
\hat{\alpha}_n(x)-\alpha(x) =
\begin{cases}
\delta_{n,h,s} \quad&\mbox{if}\quad \inf_{c\in\mathcal{C}_s}|\rho(x)-\rho(c)|>r_{n,h,s},\\
\left(\delta_{n,h,s}\right)^{\frac{s}{s+1}} \quad&\mbox{otherwise}.\\
\end{cases}
$$
\label{thm::alpha::point}
\end{thm}
The convergence rate in Theorem~\ref{thm::alpha::point} is similar to
that in Theorem~\ref{thm::KDE}: for a given point $x$
when $\alpha(x)$ is away from the $\alpha$ value of a generalized critical point (a critical $\alpha$ level),
we have the usual convergence rate.
When $\alpha(x)$ is close to a critical $\alpha$ level,
the convergence rate is slower.
The quantity $r_{n,h,s}$ behaves like the quantity $a_n$ in Theorem~\ref{thm::alpha_non},
which was introduced to guarantee the uniform convergence.
To derive the consistency of $\hat{\alpha}_n$ under the integrated error and the probability error,
we choose $r_{n,h,s} = \frac{\delta_{n,h,s}}{\log n}$,
which leads to the following theorem.

\begin{thm}[Consistency of $\alpha$-trees]
Assume (S, P2 , K1--2).

Then
\begin{align*}
\|\hat{\alpha}_n-\alpha\|_P &= \delta_{n,h,d},\\
\|\hat{\alpha}_n-\alpha\|_\mu &= \delta_{n,h,d}.
\end{align*}
\label{thm::alpha::Pro}
\end{thm}
Theorem~\ref{thm::alpha::point} shows that
the quantity $\alpha(x)$ can be consistently estimated by
$\hat{\alpha}_n(x)$ for the majority points.
This implies that
the ordering of points using $\hat{p}_n$ is consistent with the ordering from $\omega$ in most areas of $\K$.

\begin{remark}
In Theorem~\ref{thm::alpha::Pro}, $\hat{\alpha}_n$ converges under the integrated distance but
in Remark \ref{rm::IE_KDE}, the scaled KDE fails to converge.
Both the scaled KDE and $\hat{\alpha}_n$ rescale the original KDE
to adjust to the singular measure.
The rescaling in $\hat{\alpha}_n$ is with respect to the probability, which is
bounded by $1$, so the bad regions do not contribute
too much to the integrated error.
On the other hand, the scaled KDE is unbounded, meaning that the contribution from the bad regions
can be huge, which causes the failure of convergence in the integrated error.
\end{remark}

\section{Singular Measures: Critical Points and Topology}	\label{sec::topology}
Recall from Section~\ref{sec::cluster_tree} that
the topology of an $\alpha$-tree $T_\alpha$ is determined by its edge set $E(T_\alpha)$
and the relation among edges $\mathbb{C}\in E(T_\alpha)$.
The set $\mathcal{A}_\alpha$ (critical tree-levels) contains the levels where the upper level set
$\mathbb{A}_a=\{x: \alpha(x)\geq a\}$
changes its shape.
For nonsingular measures, $\mathcal{A}_\alpha$ corresponds
to the density value of some critical points.
For singular measures, this is not true even when $\rho(x)$ is a generalized
Morse function.

Consider the example in Figure~\ref{fig::DCP1}.
The solid box in the left panel indicates a new type of critical points, where merges between elements in different edge sets
occur (change in the topology of level sets occurs).
By the definition of $\mathcal{A}_\alpha$, this corresponds to an element in $\mathcal{A}_\alpha$, but it is clearly not a generalized critical point.
We call this type of critical points the \emph{dimensional critical points (DCPs)}.
In Figure~\ref{fig::DCP1}, the dimension $d=2$ and we have a $2D$ spherical distribution mixed with a $1D$ singular measure that
is distributed on the red curves $\K_1$.
The bluish contours are density contours of the $2D$ spherical distribution,
the crosses are locations of local modes,
and
the solid box is the location of a DCP.
To see how the solid box changes the topology of level sets,
we display two level sets in the middle and right panels.
In the middle panel, the level is high and there are two connected components (the gray area and the solid curve).
In the right panel, we lower the level, and now, the two connected components merge at the location of the solid box.
Although the location of the solid box does not belong to the collection of generalized critical points $\mathcal{C}$,
this point does correspond to the merging of connected components in the level sets.
Thus, this point corresponds to an element in $\mathcal{A}_a$.

\begin{figure}
\includegraphics[width=1.5in]{figures/S1_4}
\includegraphics[width=1.5in]{figures/S1_2}
\includegraphics[width=1.5in]{figures/S1_3}
\caption{Example of DCPs.
This is a $d=2$ case; there is a $2D$ spherical distribution mixed with
a $1D$ distribution on a line segment.
Left: the blue contours are density contours of the $2D$ spherical distribution
and the red line segment is $\K_1$, the support of the $1D$ singular distribution.
The two crosses are the density maxima at the $2D$ distribution and the $1D$ singular distribution.
The black square indicates a DCP.
To see how DCPs merge two connected components,
we consider the middle and the right panel, which are level sets of $\alpha(x)$
at two different levels.
Middle: the level set $\mathbb{A}_a$ where the level $a$ is high;
we can see that there are two connected components (left gray-black disk and the right line segment).
Right: we move down the level a little bit;
now the two connected components merge so there is only one connected component.
The merging point is the square point, which is defined as a DCP.
}
\label{fig::DCP1}
\end{figure}

Here is the formal definition of the DCP.
Recall that $\mathcal{C}$ is the collection of generalized critical points of $\rho(x)$
and from equation \eqref{eq::A_f}, $\mathcal{A}_\alpha$ is the collection of levels of function $\alpha(x)$
such that the creation of a new connected component or merging of connected components occurs.
For simplicity, we denote $\mathcal{A}=\mathcal{A}_\alpha$.
For $a\in\mathcal{A}$,
we define $\xi(a)$ to be the integer such that $\K_{s}\subset \mathbb{A}_a$ for all $s\leq \xi(a)$
and $\K_{\xi(a)+1}\not\subset  \mathbb{A}_a$.

\begin{definition}
For $a\in\mathcal{A}$, we say $x$ is a DCP if the following holds
\begin{itemize}
\item[(1)] $x\in\K_\ell$ for some $\ell\leq\xi(a)$.
\item[(2)] There exists an edge $\mathbb{C}\in E(T_\alpha)$ such that
	\begin{itemize}
	\item[(i)] $x\notin C$ for all $C\in\mathbb{C}$,
	\item[(ii)] $d(x,C_\epsilon)\rightarrow 0$ when $\epsilon\rightarrow 0$, where $C_\epsilon =\mathbb{C}\cap T_\alpha(a+\epsilon)$.
	\end{itemize}
\end{itemize}
Note that $x$ may not exist. In such a case, there is no DCPs for level $a$.
\end{definition}
The first requirement is to ensure that $x$ is on a lower dimensional support ($\K_\ell: \ell<\xi(a)$).
The second requirement is to show that the DCP $x$ is not in the same edge $\mathbb{C}$, but
its distance to the elements (connected components) of the edge is shrinking to $0$.
By the definition of $\alpha(x)$, the first requirement implies that $x$ is contained in
$\mathbb{A}_\alpha(a+\epsilon)$ for a sufficiently small $\epsilon$.
Therefore, we can find $\mathbb{C}'\in E(T_\alpha)$ such that every element $C\in\mathbb{C}'$ contains $x$.
Because $x$ is (i) in the elements of edge $\mathbb{C}'$, and (ii) not in any element of edge $\mathbb{C}$,
and because (iii) the distance from $x$ to the element of $\mathbb{C}$ converges to $0$
when the level decreases to the level $a$,
$x$ is a merging point of edges $\mathbb{C}'$ and $\mathbb{C}$
and the level $a$ is their merging level.

Let $\mathcal{C}^D$ be the collection of DCPs.
For a point $c\in\mathcal{C}^D$, we denote $\alpha^\dagger(c)$ as the level of $\alpha$ function
corresponding to the DCP at $c$.

\begin{remark}[Relation to the usual critical points]
The definition of DCPs is similar to that of saddle points or local minima, who contribute to the merging of level sets.
Saddle points (or local minima) that contribute to a merging of level sets can be defined as a point $x$ with the following properties:
\begin{itemize}
\item[(1)] $x\in\K_{\xi(a)+1}$.
\item[(2)] There exists two different edges $\mathbb{C}_1,\mathbb{C}_2\in E(T_\alpha)$ such that
	\begin{itemize}
	\item[(i)] $x\notin C_1,x\notin C_2$ for all $C_1\in\mathbb{C}_1$ and $C_2\in\mathbb{C}_2$,
	\item[(ii)] $d(x,C_{1,\epsilon})\rightarrow 0$ when $\epsilon\rightarrow 0$, where $C_{1,\epsilon} =\mathbb{C}_1\cap T_\alpha(a+\epsilon)$,
	\item[(iii)] $d(x,C_{2,\epsilon})\rightarrow 0$ when $\epsilon\rightarrow 0$, where $C_{2,\epsilon} =\mathbb{C}_2\cap T_\alpha(a+\epsilon)$.
	\end{itemize}
\end{itemize}
It is easy to see that for a Morse function, a point $x$ with the above properties must be a saddle point or a local minimum.
The main difference between this definition and that of DCPs is the support where $x$ lives--
if $x$ lives in a lower dimensional support $\K_\ell$, $\ell\leq \xi(a)$, then it is a DCP, and
if $x$ lives in the support $\K_{\xi(a)+1}$, then it is a saddle point or a local minimum.

\end{remark}

\begin{remark}	\label{rm::ex1}
Note that a DCP might be at the same position as a critical point.
Consider the example in Figure~\ref{fig::Tree} and Example \ref{ex::point}.
In this case, there is a DCP at $x=2$, which coincides with a local mode.
However, the DCP and the local mode correspond to different elements in $\mathcal{A}_\alpha = \{1,0.7,0.0319\}$.
The local mode corresponds to level $1$ and the DCP corresponds to level $0.0319$ because
the DCP represents the merging of the two connected components.
Note that the number $0.0319 = 0.7\times (\Phi_0(-2) + 1-\Phi_0(2))$, where $\Phi_0(x)$ is the cumulative distribution
of a standard normal.
\end{remark}

To analyze the properties of DCPs and their estimators,
we consider the following assumptions.

{\bf Assumptions.}
\begin{itemize}
\item[(A)] The elements in the collection $\mathcal{A}$ are distinct and each element corresponds
to one critical point or one DCP, but not both. And all DCPs are distinct.
\item[(B)] For every $x\in\partial \K_s$ ($s>0$) and $r>0$,
there is $y\in B(x,r)\cap K_s$ such that $\rho(y)>\rho(x)$.
\item[(C)] There exists $\eta_0>0$ such that
$$
\inf_{c\in\mathcal{C}_s}d(c, \K_\ell) \geq \eta_0
$$
for all $\ell<s$ and $s = 1,2,\cdots,d$.
\end{itemize}

Assumption (A) is to ensure that
no multiple topological changes will occur at the same level
so each level corresponds to only a merging or a creation.
Assumption (B) is to guarantee that no new connected component
at the boundary of a lower dimensional manifold will be created.
Thus, any creation of a new connected component of the level set $\mathbb{A}_a$
occurs only at a (generalized) local mode.
Assumption (C) is to regularize (generalized) critical points so that
they are away from lower dimensional supports.
This implies that when $h$ is sufficiently small, all critical points will be in the good region $\K(h)$.

\begin{lem}[Properties of DCPs]
Assume (S, P2, B).
The DCPs have the following properties
\begin{itemize}
\item If a new connected component of $\mathbb{A}_a$ is created at $a\in\mathcal{A}$, then there is a local mode $c$ of $\rho$ or an element in $\K_0$
such that $a=\alpha(c).$ Namely, DCPs only merge connected components.
\item For any value $a\in\mathcal{A}$, either $a= \alpha(c)$ for some $c\in\mathcal{C}$ or there is a DCP associated with $a$.
\end{itemize}
\label{lem::DCP::p}
\end{lem}

Lemma~\ref{lem::DCP::p} provides two basic properties of DCPs.
First, DCPs only merge connected components.
Moreover, when the topology of connected components of $\alpha$-level sets changes (when we decrease the level),
either a critical point or a DCP must be responsible for this.
Theorefore, as long as we control the stability of generalized critical points and DCPs, we
control the topology of an $\alpha$-tree.
Thus, in what follows, we will study the stability of generalized critical points and DCPs.

\begin{lem}[Stability of generalized critical points]
Assume (S, P2, K1--2, C).
Let $c\in\K_s$ be a generalized critical point with $n(s)$ negative eigenvalues of its generalized Hessian matrix.
Let $\hat{\mathcal{C}}$ be the collection of critical points of $\hat{p}_n$.
Let $m(c) = \min\{\ell\geq s: c\in\overline{\K_\ell}\}-s $.
Then when $h\rightarrow 0,\frac{nh^{d+4}}{\log n}\rightarrow \infty$ and $m(c)>1$,
there exists
a point $\hat{c}\in \hat{\mathcal{C}}$ such that
\begin{align*}
\|\hat{c} -c\|  &= O\left(h^{2\bigwedge m(c)}\right) + O_P\left(\sqrt{\frac{1}{nh^{s+2}}}\right)\\
\|\hat{\alpha}_n(\hat{c}) -\alpha(c)\|  &= (\delta_{n,h,s})^{\frac{s}{s+1}}
\end{align*}
and the estimated Hessian matrix at $\hat{c}$ has $n(c)+d-s$ negative eigenvalues.
The quantity $\delta_{n,h,s}$ is defined in equation \eqref{eq::rate}.

\label{lem::g_critical}
\end{lem}

Lemma~\ref{lem::g_critical} is a generalization of the stability theorem of
critical points given in Lemma 16 of \cite{chazal2014robust}.
The rate is similar to those in Theorem~\ref{thm::KDE} and \ref{thm::alpha::point}.
We have a modified bias due to the lower-dimensional support and the adaptive stochastic error rate of the dimension of the support
where $c$ resides in.
Because of the adaptive rate,
generalized critical points are more stable when they are on a lower dimensional support.

\begin{lem}[Properties of estimated critical points]
Assume (S, P2, K1--2, A, B).
Assume there are $k$ DCPs.
Let $\hat{\mathcal{C}}$ be the critical points of $\hat{p}_n$.
Define $\hat{\mathcal{G}}\subset \hat{\mathcal{C}}$ as the collection of estimated critical points
corresponding to the generalized critical points.
Let $\hat{\mathcal{D}} =\hat{\mathcal{C}}\backslash \hat{\mathcal{G}}$ be the remaining estimated critical points.
Then when $h\rightarrow 0, \frac{nh^{d+4}}{\log n}\rightarrow \infty$,
\begin{itemize}
\item $\hat{\mathcal{D}}\subset \K^C(h)$,
\item $\left|\hat{\mathcal{D}}\right|\geq k$, where $|A|$ for a set $A$ is the cardinality,
\item $\hat{\mathcal{D}}$ contains no local mode of $\hat{p}_n$.
\end{itemize}
\label{lem::DCP}
\end{lem}

Lemma~\ref{lem::DCP} provides several useful properties of the estimated critical points
(critical points of $\hat{p}_n$).
First, the estimated critical points are all in the bad region $\K^C(h)$, except for those converging to generalized critical points.
Second, the number of estimated critical points will (asymptotically) not be less than the total number of DCPs.
Third, all estimated local modes are estimators of generalized critical points.

\begin{lem}[Stability of critical tree-levels from DCPs]
Assume (S, P2, K1--2, A, B).
Let $c$ be a DCP and $\alpha^\dagger(c)\in\mathcal{A}$ be the associated level.
Let $\hat{\mathcal{D}}$ be defined as Lemma~\ref{lem::DCP}.
Then when $h\rightarrow 0,\frac{nh^{d+2}}{\log n}\rightarrow \infty$,
there exists
a point $\hat{c}\in \hat{\mathcal{D}}$ such that
$$
\|\hat{\alpha}_n(\hat{c}) -\alpha^\dagger(c)\|  = \delta_{n,h, \xi(\alpha_0(c))+1},
$$
where $\delta_{n,h,s}$ is defined in \eqref{eq::rate}.
Moreover, the $\hat{\mathbb{A}}_{\hat{\alpha}_n(\hat{c})+\epsilon}$ and $\hat{\mathbb{A}}_{\hat{\alpha}_n(\hat{c})}$
are not topological equivalent.
\label{lem::DCP_level}
\end{lem}

Lemma~\ref{lem::DCP_level} illustrates the stability of critical tree-levels from DCPs:
for every DCP, there will be an estimated critical point that corresponds to this DCP
and this estimated critical point also represents a merging of the estimated level sets.

In Lemma~\ref{lem::g_critical}, we derived the convergence rate of the estimated (generalized) critical points versus
the population critical points, but here we only derive the rate for
the critical tree-levels.
The reason is that critical points are solutions to a certain function (gradient equals to 0),
so we can perform a Taylor expansion to obtain the convergence rate.
However, for the DCPs, they are not solutions to some functions, so it is unclear how to derive the convergence rate
for their locations.

\begin{example}[A DCP and its estimator]	\label{ex::DCP}
Consider again the example in Figure~\ref{fig::Tree} and Example \ref{ex::point}.
We have a singular distribution mixed with a point mass at $x=2$ with a probability of $0.3$ and a standard normal with a probability of $0.7$.
In this case, as indicated, a DCP is located at $x=2$ with level $0.0319$ (see Remark \ref{rm::ex1}).
In every panel of the top row of Figure~\ref{fig::Tree}, there is a local minimum located in the region $x\in[1.5,2 ]$.
Moreover, when we increase the sample size (from left to right),
this local minimum is moving toward $x=2$.
This local minimum is an estimated critical point $\hat{c}\in\hat{\mathcal{D}}$, as described in Lemma~\ref{lem::DCP_level},
whose estimated $\alpha$-level is approaching the $\alpha$-level of the DCP at $x=2$.
\end{example}

\begin{thm}[Topological error of $\alpha$-trees]
Assume (S, P2, K1--2, A, B, C).
Then when $h\rightarrow 0, \frac{nh^{d+4}}{\log n}\rightarrow \infty$,
$$
P\left(T_{\hat{\alpha}_n}\overset{T}{\approx}T_\alpha\right) \geq 1- \frac{c_0}{h^d}\cdot e^{-c_1\cdot nh^{d+4}},
$$
for some $c_0,c_1>0$.
\label{thm::top::p}
\end{thm}
Theorem~\ref{thm::top::p} quantifies the topological error of the estimated $\alpha$-tree under singular measures.
The error rate is the same as that in the nonsingular measures (Theorem~\ref{cor::alpha_non}).
The topological error bound is similar to that in Corollary \ref{cor::alpha_non}.
Both have exponential concentration bounds
with a factor of $nh^{d+4}$, which is the Hessian estimation error rate.
The two concentration bounds are similar, because as shown in Theorem~\ref{thm::KDE}, the main
difference between singular and nonsingular measures lies in the bias part,
which will not contribute to the concentration inequality as long as $h\rightarrow0$.
The Hessian error rate is because we need to make sure the signs of eigenvalues of Hessian matrices around critical points
remain unchanged.

\begin{remark}
Theorem~\ref{thm::top::p} also implies that, under singular measures,
the cluster tree of the KDE $\hat{p}_n$ (estimated $\lambda$-tree) converges topologically
to a population cluster tree defined by the function $\alpha(x)$.
To see this, recall that by
Lemma~\ref{lem::tree}, $T_{\hat{p}_n}\overset{T}{\approx}T_{\hat{\alpha}_n}$. This, together with Theorem~\ref{thm::top::p},
implies
$$
P\left(T_{\hat{p}_n}\overset{T}{\approx}T_\alpha\right) \geq 1- \frac{c_0}{h^d}\cdot e^{-c_1\cdot nh^{d+4}} \rightarrow 1
$$
under a suitable choice of $h$.
This shows that even when the population distribution is singular,
the estimated $\lambda$-tree still converges topologically to the population $\alpha$-tree.
\end{remark}

\section{Discussion}	\label{sec::discussion}

In this paper, we study how the $\alpha$-tree behaves under singular and nonsingular measures.
In the nonsingular case,
the error rate under the $L_\infty$ metric is slower than other
metrics because of the slow rate of level set estimation around saddle points.
However, other error rates are the same as estimating the $\lambda$-tree.

When a distribution is singular,
we obtain many fruitful results for both the KDE and the estimated $\alpha$-tree.
In terms of the KDE, we prove that
\begin{itemize}
\item[1.] the KDE is a pointwisely consistent estimator after rescaling;
\item[2.] the KDE is a uniformly consistent estimator after rescaling for the majority of the support; and
\item[3.] the cluster tree from the KDE (estimated $\lambda$-tree) converges topologically to a population cluster tree defined by $\alpha$.
\end{itemize}
For the estimator $\hat{\alpha}_n(x)$ and the estimated $\alpha$-tree, we show that
\begin{itemize}
\item[1.] $\hat{\alpha}_n$ is a pointwisely consistent estimator of $\alpha$;
\item[2.] $\hat{\alpha}_n$ is a uniformly consistent estimator for the majority of the support;
\item[3.] $\hat{\alpha}_n$ is a consistent estimator of $\alpha$ under the integrated distance and probability distance; and
\item[4.] the estimated $\alpha$-tree converges topologically to the population $\alpha$-tree.
\end{itemize}
Moreover, we observe a new type of critical points--the DCPs--that also contribute to the merging of
level sets for singular measures.
We study the properties of DCPs and show that the estimated critical points from the KDE approximate these DCPs.

Finally, we point out some possible future directions.
\begin{itemize}
\item {\bf Persistence Homology.}
The cluster tree is closely related to the persistent homology of level sets \citep{fasy2014confidence,bobrowski2014topological}.
In the persistent homology, a common metric for evaluating the quality of an estimator is the bottleneck distance
\citep{cohen2007stability,edelsbrunner2012persistent}.
Because the bottleneck distance is bounded by the $L_\infty$ metric \citep{cohen2007stability, edelsbrunner2012persistent},
many bounds on the bottleneck distance are derived via bounding the $L_\infty$ metric
\citep{fasy2014confidence,bobrowski2014topological}.
However,
for $\alpha$-trees under singular measures,
the $L_\infty$ metric does not converge (Corollary~\ref{cor::alpha_infty_s}) but we do have topological consistency
(Theorem~\ref{thm::top::p}), which implies convergence in the bottleneck distance.
This provides an example where we have consistency under the bottleneck distance and inconsistency in the $L_\infty$ metric.
How this phenomenon affects the persistence homology is unclear and we leave that line of study
for future work.

\item {\bf Higher-order Homology Groups.}
Our definition of DCPs is for connected components, which are zeroth-order homology groups
\citep{cohen2007stability,fasy2014confidence,bubenik2015statistical}
and sufficient
for analyzing cluster trees.
However, critical points also contribute to the creation and elimination of higher order homology groups
such as loops and voids, which are not covered in this paper.
Thus, a future direction is to study whether the KDE is also consistent in recovering higher-order homology groups under singular measures.

\item {\bf Minimax theory.}
\cite{chaudhuri2010rates,chaudhuri2014consistent} derived the minimax
theory for estimating the $\lambda$-tree under nonsingular measures and
proved that the the k-nearest neighbor estimator is minimax.
When the distribution is nonsingular, the $\alpha$-tree and $\lambda$-tree are very similar so
we expect
the minimax theory to be the same.
However, for singular measures,
it is unclear how to derive the minimax theory so we plan to investigate this in the future.

\end{itemize}

redpajama_set_name: RedPajamaArXiv


Bangkok will not be a place you would choose to rest. It is such a busy city welcoming millions of visitors each year. Besides, I don't think you would travel to the other half of the world to rest.
This time, we just wanted to get away from the cold and snow and the easiest place to travel to is Bangkok. No visas required for us. So many choices in hotels. Many places to go to if you are restless and the choices are affordable.
But for us, this time, it was rest. We just finished a contract which lasted for 3 years and from there traveled to Spain and Portugal. Then, went home to Canada and from there, traveled to Florida. After that, we just wanted to rest while waiting for the next contract and knowing that the next assignment will be Asia, we just moved closer to where we will be next.
1. Cruise the Chao Phraya. There are many pier areas you can explore at your leisure. You can enjoy your sightseeing of Bangkok without much traffic. You can spend some time in the temples along the river or try out Thai food in the many restaurants in the pier areas. Here's more on Cruising the Chao Phraya.
2. Pray in the shrines around the city. There are always people praying in these shrines which are often in the busiest intersections. Some pray from on top the BTS walkway if you want to avoid the crowd.
3. Have lunch in the Embassy Central. Then, stay awhile and enjoy the peace and quiet and almost empty space. You can easily reach it by BTS.
4. Enjoy a massage. Try foot massage first so you have an idea of what you're getting.
Boat Cruise in the Chao Phraya. Copyright: aesta1.
Temples along the Chao Phraya. Copyright: aesta1.


As of September 1, Dr.
and Dr. Dolgin in their new roles.
year. In addition, our featured speaker, Dr.
head if you need assistance with car-pooling.
name will be checked off the visitor's list.
for car-pooling to expedite gate clearance.
gate check and traffic congestion at that hour.
Pearl Young Theater at NASA.
AFB . . . CARE . . . DEW . . . GEM . . .
JAG . . . LOOM . . . PA YE . . . SHAPE . . .
TQM, TQI, and VPTQI together.
through the use of empowered work groups.
Transcript T w T H o m a s N e l s o n •^COMMUNITY COLLEGE a biweekly newsletter for faculty and staff Vol. 9, No. 3 From the President As of September 1, Dr. Ann Dolgin will become Assistant to the President for Instructional Technology and Economic Development. Dr. J. P. "Jack" Hill will be the Interim Dean of Instruction for Thomas Nelson Community College. Both of these appointments are not expected to extend past the end of this fiscal year. These changes will allow Dr. Dolgin to pursue some special projects that she has been working on or planning but has not had the time to fully explore. In her role as assistant to the August 8,1995 president, she will be able to concentrate on these efforts. Dr. Dolgin has served as Dean of Instruction for the past six years. During that time, the college has experienced a 25 percent increase in professional development support of scholarly activities by faculty members, national accreditation for our Allied Health programs, establishment of special programs like Ford Asset, international education, and automation of the library. Dr. Hill, a long-time member of the campus community, will serve as Interim Dean of Instruction while we conduct a full Dr. Shirley R. Pippins search and screen for a new permanent Dean of Instruction. Dr. Hill, Professor Emeritus of the college and former head of the Public Administration and Human Services department, has been with the college since 1969. He retired from full-time teaching responsibilities in 1993, and has been managing the college's development programs. I ask all members of the college community to provide full support to Dr. Hill and Dr. Dolgin in their new roles. Fall Convocation I wish to take this opportunity to invite you to join me in the celebration of our college's achievements. For more than 25 years, Thomas Nelson Community College has been dedicated to providing the residents of the Virginia Peninsula a quality education that supports the economic demands of our community. This year the 1995 Fall Convocation will be a one-day event and will take place on Wednesday, August 16, at 8 a.m. in NASA's Pearl Young Theater. The morning's program will provide an overview of major college initiatives for the 1995-96 academic year. In addition, our featured speaker, Dr. Randi Levitz will present a keynote address on recruitment and retention. Further activities will take place on the college campus that afternoon with time designated for divisional and department meetings, Faculty Senate, and Classified Support Staff Association activities. We are pleased that NASA agreed to host Thomas Nelson Community College for this function. Because parking at NASA is limited, we are asking all faculty and staff to make car-pooling arrangements. Please contact your division chair or administrative by Dr. Ann B. Dolgin head if you need assistance with car-pooling. Due to security arrangements, you must enter NASA through Main Gate 4, where your name will be checked off the visitor's list. Should you already have a NASA/Langley permit, you may enter at another gate. These individuals are also encouraged to volunteer for car-pooling to expedite gate clearance. Please allow some extra travel time due to the gate check and traffic congestion at that hour. I look forward to greeting you on Wednesday, August 16,1995 at 8 a.m. in the Pearl Young Theater at NASA. TQM, TQI, VPTQI AFB . . . CARE . . . DEW . . . GEM . . . JAG . . . LOOM . . . PA YE . . . SHAPE . . . UCMJ . . . WYS . . . ZPG . . . . So many acronyms have infiltrated "our marvelous native tongue" that new ones, such as TQM, TQI, and VPTQI, are instinctively filed away in the back of our minds without hesitation; hence without consideration. As such, let's take a few minutes—I know you're busy, but it will be well worth your while—to decipher TQM, TQI, and VPTQI together. TQM, or Total Quality Management, is a management philosophy that has endured and prospered through the years, and has contributed to significant improvement in organizations, firms, and businesses that have by Melanie Parker embraced it. The key to competitiveness in the 1990s, TQM strategies, when consistently applied, reduce cost and increase profits by systematically finding and eliminating inherent waste, increase productivity by focusing on customer satisfaction and process improvement, and improve employee morale through the use of empowered work groups.


text:
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text:
Home News Entertainment 4 Underrated Directors Whose Movies You Need To Binge-Watch In 2023
4 Underrated Directors Whose Movies You Need To Binge-Watch In 2023
Mimi Clarkson
Source: bizjournals.com
The last two years have limited us a bit in going out, but with binge-watching movies, we felt at least a little easier. Still, even though we're used to watching blockbusters – we suggest you pay attention to some movies and directors that have been unjustifiably underestimated. So here are 4 underrated directors whose movies you need to binge-watch in 2023.
Discover New Directors And Fall In Love With Their Movies
We all love to watch movies, so in today's offer, it is not difficult for us to find something to enjoy while eating popcorn. However, when you get tired of the marathon of your favorite TV series or blockbusters – you can allow yourself to discover something new in the offer. Namely, there are so many movies and directors that have been completely unjustifiably underrated. Maybe a good idea would be to take a look at some of it and enjoy some new movie discoveries. Dealing with somewhat neglected directors, we discovered some of the films you could really enjoy. Some of these movies and directors certainly deserved far more attention than they've got. Therefore, here is our list of suggestions for binge-watching.
1. David Gordon Green
Source: ew.com
This middle-generation director showed his talent in various genres, and he also tried his hand as a co-author in writing screenplays. Known for dramas like George Washington, as well as the thriller Undertow – this director has shown his multiple talents. For all fans of the Halloween franchise, we will say that Green's film adaptation from 2018 was great, and there is also the latest part of the same franchise from 2023. The movie Halloween Kills is something you should not skip.
2. Scott Cooper
Source: superstarsbio.com
In genre terms, this is a director who is not afraid to go beyond his own borders. One of the best proofs is the film Hostiles – which Scott J Cooper himself considers to be one of his most serious works. Although his other movies like Out of the Furnace or Black Mass had a bigger commercial novelty – Hostiles is a real treat for all fans of historical drama and westerns. A powerful historical-political drama packaged in an era of conflict with the Indians – will win you over at first sight.
3. Emma Seligman
Source: thefullest.com
Young 26-year-old Canadian director Emma Seligman confirms that there are those among the new generations of directors who deserve attention. She drew attention with her debut work Shiva Baby. Despite the slightly confused and dark mood of the film, this is definitely a movie you must watch.
4. Rob Reiner
Source: tcm.com
Although already known and established as a director, Rob Reiner is also one of those whose work is not recognized to the extent that it is worth it. Although he made a success back in the '80s with films like Stand By Me and When Harry Met Sally – Reiner didn't get enough credit for his work. At the time, he focused on somewhat more serious dramas such as A Few Good Men and The Ghosts of Mississippi – but despite the praise, he remained insufficiently praised for his contribution to film culture. Still, we think he deserves more – so make sure to put some of his movies on your binge-watch list.
We hope that after our suggestions you will watch some of the movies of the mentioned directors. We think they deserve more credit than what has been shown in public – and you have the chance to confirm it by watching their films.
Binge-Watch
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T-Pain Says He Became Depressed After Usher Told Him He 'F**ked Up Music'
Usain Bolt & Kasi Bennett Share Photos Of Twin Newborn Sons


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Q: Make div all content below move down if div slides down I'm trying to make this notification div that slides down if it is active. It slides down using this script:
<script type="text/javascript">
    $(document).ready(function(){
        $('#notice').slideDown(1000);
         $('.exit').click(function(){
             $("#notice").slideUp(400);
         });
    });
</script>

And the style is:
#notice {
    background: #E0E0E0;
    color: #FFF;
    padding-left: 30px;
    padding-top: 20px;
    padding-bottom: 20px;
    padding-right: 30px;
    position: absolute;
    z-index: 999999;
    width: calc(100% - 60px);
    top: 0;
    font-weight: 200;
    letter-spacing: 1px;
    text-align: center;
    display: none;
}

But how can I make this move everything else down as it slides down, even though the div below is at state 'position: fixed'. And slide everything else back up when it is removed using the .exit link.

A: Here you are one option to do it
Example here http://jsfiddle.net/jvarj4gk/2/
$('#notice').slideDown(1000);
 var timer = setInterval(function () {
     $('#fixed').css({
         top: $('#notice').outerHeight()
     });
     if ($('#notice').outerHeight() == $('#fixed').css('top')) {
         clearInterval(timer);
     }

 }, 10);

 $('.exit').click(function () {
     $("#notice").slideUp(400, function(){
         $(this).remove();
     });
     $('#fixed').animate({
         top: 0
     }, 400);

 });

Update
To move the #body you can do it like this
Example http://jsfiddle.net/jvarj4gk/5/
$('#notice').slideDown(1000);
 var timer = setInterval(function () {
     $('#fixed, #body').css({
         top: $('#notice').outerHeight()
     });
     if ($('#notice').outerHeight() == $('#fixed').css('top')) {
         clearInterval(timer);
     }

 }, 10);

 $('.exit').click(function () {
     $("#notice").slideUp(400, function(){
         $(this).remove();
     });
     $('#fixed').animate({
         top: 0
     }, 400);

 });

A: You can animate() the div below
var h = '+=' + $('#notice').height();
$('#fixedDivId').animate({
    top: h;
},1000);

And almost the same stuff for slide up but h='-='+ $('#notice').height();
UPD
For your code it will look like this
 $(document).ready(function(){
    var h = $('#notice').height()+'px';
    $('#notice').slideDown({duration:1000, queue:false});
    $('#fixedDivId').animate({
       top: h
    },{duration:1000, queue:false});
    $('.exit').click(function(){
       $("#notice").slideUp(400);
    });
 });

Fiddle

redpajama_set_name: RedPajamaStackExchange


Thanks Lee! I'm really glad you're enjoying the site. I'm regularly adding new content on how to read Lenormand. For more hand-on help learning Lenormand, you might like my Learn Lenormand Forum.
I'm brazilian student and praticioner of lenormand. I'm starting in my website (www.cih.org.br) a section of lenormand cards. I would like to invite you to be a writer in this section. We can also translate (with credits, of course) your website content to help publicize the same, and put your link in our links section.
I wish I knew more Portuguese..! Thank you for the invite, Frater Goya. It is something to consider sometime in the future when I have more time available. I think the New Year should bring new opportunities and we can revisit then.
Best of luck on your website, I took a look around and you cover interesting subjects that do go hand in hand with what I plan to cover here.
Feliz Navidad y prospero año (hopefully my Spanish is better than my Portuguese).
I wanted to write you a little bit of note so as to say thank you for the breathtaking tactics you have shared on Lenormand. It was really generous of you to convey this information. Some people would have sold this as an ebook to make a profit on it. You could possibly have done it if you desired. The basics you give are a fantastic way to understand that other individuals have a similar passion for Lenormand just like my own. I know there are many more pleasurable occasions up for individuals that read your blog.
Thanks Dulcie! I'm glad you've enjoyed the information on this site. I do have a lot more information to share, like you mentioned. I hope you and others enjoy it just as much!
can you recommend an old book lenormand ? i;m interested about this tarot…i know only few knowledges about lenormand but i want impruve my " abilityes".
thankyou, and best regards from Freya.
Most books are in French or German. There aren't any I'd recommend in English yet, although andybc is supposed to be releasing one soon. I haven't read it though. I'm working on a book, but I don't know when it'll be ready for publication. Right now, the best way to learn Lenormand in English is from websites and forums.
So what is the 'stand alone' meaning for the rider?
rider+whip= mediation what do you mean by that?


text:
Requiescat in Pace: Father John J. Navone, SJ
Father John J. Navone, SJ, entered into God's peace December 25, 2016, at Providence Sacred Heart Medical Center in Spokane. He was 86.
A scholar and teacher in Catholic theology, prodigious writer and editor of several books and articles, and exquisite host; these descriptions only begin to tell the story of this deeply spiritual and gregarious Jesuit priest.
Born in Seattle on October 19, 1930, to James and Juliet (Micheli) Navone, John was the eldest of six siblings. He grew up in Seattle and graduated from Seattle Preparatory in 1948. He attended Seattle University for one year before entering the Jesuit Novitiate (Sheridan, Ore.) in August 1949. Fr. Navone followed the regular course of studies and teaching for Jesuits and was ordained a priest in Spokane on June 16, 1962.
Showing great interest in Catholic theology, Fr. Navone earned a doctorate in that field from the Gregorian University in Rome in 1966. He was accepted as a faculty member there a year later and remained there for 40 years. It was in Rome where he wrote most of his 23 books and numerous articles. His summers were regularly spent in Seattle or Spokane teaching summer school. When visitors came to Rome, Fr. Navone was a gracious host and tour guide. His fluency in the Italian language and knowledge of important places and people made these trips memorable.
Following his retirement in 2007, Fr. Navone came to Gonzaga University and was a scholar-in-residence for the Florence, Italy program for two years. He was always interested in discovering new ideas, meeting with people, and solving problems. Fr. Navone would regularly bring staff and faculty to lunch at the Jesuit residence to hear their stories and to share Jesuit history and philosophy with them.
Fr. Navone was preceded in death by his father and mother. He is survived by his sisters, Helen Gleason and Catherine Mullally, and his brothers, Joseph, James, and George Navone. There are many nephews, nieces, cousins, dear friends as well as his brother Jesuits who mourn his passing.
A funeral Mass for Fr. Navone will be celebrated at noon, January 21, at St. Aloysius Church, 330 E. Boone Ave., Spokane. Inurnment will follow at Mt. St. Michael Cemetery in Spokane. Donations in honor of Fr. Navone may be sent to the Senior Fund, Oregon Province, Society of Jesus, P.O. Box 86010, Portland, OR 97286. Gifts may also be made at www.jesuitswest.org/OREdonations on our secure donation site.
Loyola Institute for Spirituality
Loyola Institute for Spirituality (LIS), founded in 1997, is located in Orange, CA. LIS provides many programs and services for individuals, parishes, and dioceses throughout Southern California and beyond.


text:
Q: Convert hexadecimal to IEEE-754 single precision floating point binary scientific notation I am trying to convert these numbers to binary scientific notation, but I cannot figure out the process. Could someone please the process of going about solving this?

For IEEE 754 single precision floating point, what is the number, as written in binary scientific notation, whose hexadecimal representation is the following?
0061 0000

I can get this converted from hex to unsigned binary:

0000 0000 0110 0001 0000 0000 0000 0000

but I can't figure out how to properly represent this using binary scientific notation. Thanks in advance!

A: binary32 is broken into 3 sections: sign, exponent (biased) and significand (or fraction).
0000 0000 0110 0001 0000 0000 0000 0000
||        ||                          |
||        |\-- significand -----------/
| \ expo  /
\ sign

So in this case,
sign (negative) = 0, so number is positive
exponent (biased) = 0000 0000
significand = .1100001 0000 0000 0000 0000

If the exponent (power of 2) is at the highest value (1111 1111), that indicates the number is special: Infinity or Not-a-Number.
If the exponent is 0, the bias is -126, else the bias is -127 and an implied 1 should be added to the fraction.
sign = 0 (positive) or +1
exponent = 0 - 126
significand = 0.1100001 =  (binary) 1100001/10000000 = 97/128

+1 * pow(2, -126) * 97/128 = 8.9080431273251475213255815711373...e-39

Notes:
On-line converters available.  example
Endian: the order in which the bytes are to be interpreted can vary.  0061 0000 could be 00 00 61 00.  An assumption was made here with this example.


The Federation is a voluntary body that was formed in 1999 in order to exchange ideas, share information, and raise awareness of County Kildare's local history groups and their activities. More information on the federation can be found in the About KFLHG section.
On this website we provide a Local History Diary to share details of upcoming talks and events that our member groups have organised.
The News section contains events organised by non-member organisations and covers details of conferences, seminars, exhibitions, festivals, and lectures. In addition, notices of grants, competitions, and any other heritage/history/archaeology related news relevant to Co. Kildare, or surrounding counties, is posted in the news section.
The website also includes a section on each member group arranged by area. These sections have details of the individual groups, their upcoming events, publications, and in some cases a brief local history of the location is provided. Select an area from the links below or throughout the website from the side menu.