text:
\section{Introduction}
\label{sec:intro}

Matrix completion is the problem of recovering a low-rank matrix from a sampling of its entries. In machine learning and signal processing, it has a wide range of applications including collaborative filtering \cite{rennie2005fast}, system identification \cite{liu2009interior} and dimension reduction \cite{candes2011robust}. In the era of big data, matrix completion has been proven to be an efficient and powerful framework to handle the enormous amount of information by exploiting low-rank structure of the data matrix.

Let $\bm M \in {\mathbb R}^{n \times m}$ be a rank $r$ matrix with \hlnew{$1 \leq r \leq \min(n,m)$}, and $\Omega = \{ (i,j) \mid M_{ij} \text{ is observed} \}$ be an index subset of cardinality $s$ such that $s \leq nm$. The goal is to recover the unknown entries of $\bm M$. Matrix completion can formulated as a linearly constrained rank minimization or a rank-constrained least squares problem \cite{candes2009exact}. Two popular approaches for solving the aforementioned matrix completion problem formulations are convex relaxation via nuclear norm and non-convex factorization. The former approach, motivated by the \hlnew{success} of compressed sensing, replaces the matrix rank by its convex surrogate (the nuclear norm). Extensive work on designing convex optimization algorithms with guarantees can be found in \cite{candes2009exact,ji2009accelerated,toh2010accelerated,cai2010singular,ma2011fixed}.
While on the theoretical side, the solutions of the relaxed problem and the original problem can be shown to coincide with high probability, on the practical side, computational complexity concerns \hlnew{diminish} the applicability of these algorithms. When the size of the matrix grows rapidly, storing and optimizing over a matrix variable become computationally expensive and even infeasible. In addition, it is evident this approach suffers from slow convergence \cite{koren2009bellkor,vu2019accelerating}.
In the second approach, the original rank-constrained optimization is studied. Interestingly, by reparametrizing the $n \times m$ matrix as the product of two smaller matrices $\bm M = \bm X \bm Y^\top \hspace{-0.05cm}$, for $\bm X \in {\mathbb R}^{n \times r}$ and $\bm Y \in {\mathbb R}^{m \times r}$, the resulting equivalent problem is unconstrained and more computationally efficient to solve \cite{burer2005local}. While this problem is non-convex, recent progress shows that for such problem
any local minimum is also a global \hlnew{minimum} \cite{sun2016guaranteed,ge2016matrix}. Thus, basic optimization algorithms such as gradient descent \cite{chen2015fast,sun2016guaranteed,ma2018implicit} and alternating minimization \cite{chen2012matrix,jain2013low,hardt2014understanding,hardt2014fast} can provably solve matrix completion under a specific sampling regime.
\hlnew{Alternatively, the original rank-constrained optimization problem can be solved without the aforementioned reparameterization via the truncated singular value decomposition.} \cite{jain2010guaranteed,goldfarb2011convergence,tanner2013normalized,jain2015fast,chunikhina2014performance,vu2019accelerating,vu2019local}.

Among the aforementioned algorithms, let us draw our attention to the gradient descent method due to its outstanding simplicity and scalability. The first global convergence guarantee is attributed to Sun and Luo \cite{sun2016guaranteed}. The authors proved that gradient descent with appropriate regularization can converge to the global optima of a factorization-based formulation at a linear rate.
Later on, Ma~\textit{et.~al.} \cite{ma2018implicit} proposed that even in the absence of explicit regularization, gradient descent recovers the underlying low-rank matrix by implicitly regularizing its iterates. \hlnew{The} aforementioned results, while \hlnew{establishing} powerful guarantees on the convergence behavior of gradient descent, impose several limitations.
For some methods, the linear convergence rate depends on \hlnew{constants that are not in closed-form and are} hard to verify in numerical \hlnew{experiments} even when the underlying matrix is known. Second, a solution-independent analysis of the convergence rate typically offers a loose bound when considered for a specific solution.
Third, the global convergence analysis requires certain assumptions on the underlying model which largely restrict the setting of the matrix completion problem in practice. Among such assumptions, one would consider the \hlnew{incoherence} of the target matrix, the randomness of the sampling set, and the fact that the rank $r$ and the condition number of $\bm M$ are small constants as $n,m \to \infty$.

To address these issues, we consider the local convergence analysis of gradient descent for factorization-based matrix completion. \hlnew{In the scope of this paper, we restrict our attention to the symmetric case. We identify the condition for linear convergence of gradient descent that depends only on the solution $\bm M$ and the sampling set $\Omega$.} In addition, we provide a closed-form expression for the asymptotic convergence rate that matches well with the convergence of the algorithm in practice. \hlnew{The proposed analysis does not require an asymptotic setting for matrix completion, e.g., large matrices of small rank}. We believe that our analysis can be useful in both theoretical and practical aspects of \hlnew{the} matrix completion problem.

\section{Gradient Descent for Matrix Completion}
\label{sec:prel}

\textit{\textbf{Notations.}} Throughout the paper, we use the notations $\norm{\cdot}_F$ and $\norm{\cdot}_2$ to denote the Frobenius norm and the spectral norm of a matrix, respectively. On the other hand, $\norm{\cdot}_2$ is used on a vector to denote the Euclidean norm. Boldfaced symbols are reserved for vectors and matrices. In addition, the $t \times t$ identity matrix is denoted by $\bm I_t$. $\otimes$ denotes the Kronecker product between two matrices, and $\vect(\cdot)$ denotes the vectorization of a matrix  by stacking its columns on top of one another. Let $\bm X$ be some matrix and $\bm F(\bm X)$ be a matrix-valued function of $\bm X$. Then, for some positive number $k$, we use $\bm F(\bm X) = \bm{\mathcal{O}}(\norm{\bm X}_F^k)$ to imply $\lim_{\delta \to 0} \sup_{\norm{\bm X}_F=\delta} \norm{\bm F(\bm X)}_F/\norm{\bm X}_F^k < \infty$.

We begin by introducing the low-rank matrix completion problem. For simplicity, we focus on the {\em symmetric} case where $\bm M$ is an $n \times n$ positive semi-definite (PSD) rank-$r$ matrix and the sampling set $\Omega$ is symmetric.\footnote{If the sampling set is not symmetric, one can symmetrize it by adding $(j,i)$, for any $(i,j) \in \Omega$, to $\Omega$ since $M_{ji} = M_{ij}$.} Let the rank-$r$ economy version of the eigendecomposition of $\bm M$ be given by
\begin{align*}
    \bm M = \bm U \bm \Lambda \bm U^\top \hspace{-0.05cm},
\end{align*}
where $\bm U \in {\mathbb R}^{n \times r}$ is a semi-orthogonal matrix and $\bm \Lambda \in {\mathbb R}^{r \times r}$ is a diagonal matrix containing $r$ non-zero eigenvalues of $\bm M$, i.e., $\lambda_1 \geq \ldots \geq \lambda_r > 0$.
Since $\bm M$ can be represented as
\begin{align*}
    \bm M = (\bm U \bm \Lambda^{1/2}) (\bm U \bm \Lambda^{1/2})^\top \hspace{-0.05cm} ,
\end{align*}
we can write $\bm M = \bm X^* {\bm X^*}^\top \hspace{-0.05cm}$, such that $\bm X^* = \bm U \bm \Lambda^{1/2} \in {\mathbb R}^{n \times r}$.
Therefore, the factorization-based \hlnew{formulation} for matrix completion can be \hlnew{described} using the following non-convex optimization:
\begin{align} \label{prob:P0}
    \min_{\bm X \in {\mathbb R}^{n \times r}} \frac{1}{4} \sum_{(i,j) \in \Omega} \bigl( [\bm X \bm X^\top \hspace{-0.05cm}]_{ij} - M_{ij} \bigr)^2 .
\end{align}
Denote ${\cal P}_\Omega : {\mathbb R}^{n \times n} \to {\mathbb R}^{n \times n}$ the projection onto the set of matrices supported in $\Omega$, i.e.,
\begin{align*}
    [{\cal P}_\Omega (\bm Z)]_{ij} = \begin{cases} Z_{ij} & \text{if } (i,j) \in \Omega, \\ 0 & \text{otherwise}. \end{cases}
\end{align*}
We can rewrite the objective function as $f(\bm X) = \frac{1}{4} \norm{{\cal P}_\Omega (\bm X \bm X^\top \hspace{-0.05cm} - \bm M)}_F^2$.
The gradient of $f(\bm X)$ is given by
\begin{align} \label{equ:df}
    \nabla f(\bm X) = {\cal P}_\Omega (\bm X \bm X^\top \hspace{-0.05cm} - \bm M) \bm X .
\end{align}
Starting from an initial $\bm X^0$ (usually through spectral initialization \cite{ma2018implicit}), the \hlnew{gradient descent} algorithm (see Algorithm~\ref{algo:GD}) simply updates the value of $\bm X$ by taking steps proportional to the negative of the gradient $\nabla f(\bm X)$.

\begin{algorithm}[t]
\caption{(Non-convex) \hlnew{Gradient Descent}}
\label{algo:GD}
\begin{algorithmic}[1]
\Require{$\bm X^0$, ${\cal P}_\Omega(\bm M)$, $\eta$}
\Ensure{$\{\bm X^k\}$}
\For{$k=0,1,2,\ldots$}
\State $\bm X^{k+1} = \bm X^k - \eta {\cal P}_\Omega \bigl(\bm X^k {\bm X^k}^\top \hspace{-0.05cm} - \bm M\bigr) \bm X^k$
\EndFor
\end{algorithmic}
\end{algorithm}

\section{Local Convergence Analysis}
\label{sec:result}

This section presents the local convergence result of Algorithm~\ref{algo:GD}. While recent work on the global guarantees of the algorithm has shown the linear behavior under certain statistical \hlnew{models}, we emphasize that no closed-form expression of the convergence rate was provided. Our result in this paper, on the other hand, does not make any assumption about the underlying model for $\bm M$ and $\Omega$, and \hlnew{provides} an exact expression of the asymptotic rate of convergence. Let us first introduce some critical concepts used in our derivation.

\begin{definition} \label{def:S}
Denote $\bar{\Omega} = \{ (i-1)n+j \mid (i,j) \in \Omega \}$.
The row selection matrix $\bm S$ is an ${s \times n^2}$ matrix obtained from a subset of rows corresponding to the elements in $\bar{\Omega}$ from the $n^2 \times n^2$ identity matrix $\bm I_{n^2}$.
\end{definition}

\begin{definition} \label{def:PU}
The orthogonal projection onto the null space of $\bm M$ is defined by $\bm P_{\bm U_\perp} = \bm I_n - \bm U \bm U^\top \hspace{-0.05cm}$.
\end{definition}

\begin{definition} \label{def:T}
Let $\bm T_{n^2}$ be an $n^2 \times n^2$ matrix where the $(i,j)$th block of $\bm T_{n^2}$ is the $n\times n$ matrix $\bm e_j \bm e_i^\top \hspace{-0.05cm}$ for $1 \le i,j \le n$. Then $\bm T_{n^2}$ can be used to represent the transpose operator as follows:
\begin{align*}
    \vect(\bm E^\top \hspace{-0.05cm}) = \bm T_{n^2} \vect(\bm E) \quad \text{ for any } \bm E \in {\mathbb R}^{n \times n} .
\end{align*}
\end{definition}
We are now in position to state our main result on the asymptotic linear convergence rate of Algorithm~\ref{algo:GD}.
\begin{theorem}
\label{theo:GD}
Denote $\bm P_1 = \bm I_{n^2} - \bm P_{\bm U_\perp} \otimes \bm P_{\bm U_\perp}$, $\bm P_2 = \frac{1}{2} \bigl(\bm I_{n^2} + \bm T_{n^2}\bigr)$, and $\bm P = \bm P_1 \bm P_2$. In addition, let
\begin{align*}
    \bm H = \bm P \Bigl( \bm I_{n^2} - \eta (\bm M \oplus \bm M) (\bm S^\top \hspace{-0.05cm} \bm S) \Bigr) \bm P , \numberthis \label{equ:H}
\end{align*}
where $\bm M \oplus \bm M = \bm M \otimes \bm I_n + \bm I_n \otimes \bm M$ is the Kronecker sum. Define the spectral radius of ${\bm H}$, $\rho(\bm H)$, as the largest absolute value of the eigenvalues of $\bm H$.
If $\rho(\bm H)<1$, then Algorithm~\ref{algo:GD} produces a sequence of matrices $\bm X^k {\bm X^k}^\top \hspace{-0.05cm}$ converging to $\bm M$ at an asymptotic linear rate $\rho(\bm H)$.
Formally, there exists a neighborhood ${\cal N}(\bm M)$ of $\bm M$ such that for any $\bm X^0 {\bm X^0}^\top \hspace{-0.05cm} \in {\cal N}(\bm M)$,
\begin{align*}
    \norm{\bm X^k {\bm X^k}^\top \hspace{-0.05cm} - \bm M}_F \leq C \norm{\bm X^0 {\bm X^0}^\top \hspace{-0.05cm} - \bm M}_F \rho(\bm H)^k, \numberthis \label{equ:XXM}
\end{align*}
for some numerical constant $C>0$.
\end{theorem}

\begin{remark}
Theorem~\ref{theo:GD} provides a closed-form expression of the asymptotic linear convergence rate of Algorithm~\ref{algo:GD}, which only depends on $\bm M$, $\Omega$ and the choice of step-size $\eta$.
We note that the condition for linear convergence, $\rho(\bm H)<1$, is fully determined given $\bm M$, $\Omega$, and $\eta$.
It would be interesting to establish a connection between this condition and the standard statistical model for matrix completion. For instance, how the incoherence of $\bm M$ and the randomness of $\Omega$ would affect the spectral radius of $\bm H$?
This exploration is left as future work.
\end{remark}

In our approach, the following lemma plays a pivotal role in the derivation of Theorem~\ref{theo:GD}, establishing the recursion on the error matrix $\bm X^k {\bm X^k}^\top \hspace{-0.05cm} - \bm M$:\footnote{We provide proofs of all the lemmas in the Appendix.}

\begin{lemma} \label{lem:EA}
Let $\bm E^k = \bm X^k {\bm X^k}^\top \hspace{-0.05cm} - \bm M$. Then
\begin{align*}
    \bm E^{k+1} = \bm E^k - \eta \bigl( {\cal P}_\Omega (\bm E^k) \bm M + \bm M {\cal P}_\Omega (\bm E^k) \bigr) + \bm{\mathcal{O}}(\norm{\bm E^k}_F^2) .
\end{align*}
Furthermore, denote $\bm A = \bm I_{n^2} - \eta (\bm M \oplus \bm M) (\bm S^\top \hspace{-0.05cm} \bm S)$ and $\bm e^k = \vect(\bm E^k)$, the matrix recursion can be rewritten compactly as
\begin{align*}
    \bm e^{k+1} & = \bm A \bm e^k + \bm{\mathcal{O}}(\norm{\bm e^k}_2^2) . \numberthis \label{equ:E4}
\end{align*}
\end{lemma}

\begin{remark}
Figure~\ref{fig1} illustrates the effectiveness of the proposed bound on the asymptotic rate of convergence given by Theorem~\ref{theo:GD}.
\hlnew{In} Fig.~\ref{fig1}, \hlnew{the low-rank solution matrix $\bm M$ is generated by taking the product of a $20 \times 3$ matrix $\bm X$ and its transpose, where $\bm X$ has $i.i.d.$ normally distributed entries. The sampling set $\Omega$ is obtained by randomly selecting the entries of $\bm M$ based on a Bernoulli model with probability $0.3$.
Next, we run the economy-SVD on $\bm M$ to compute $\bm X^* = \bm U \bm \Lambda^{1/2}$. The initialization $\bm X^0$ is obtained by adding $i.i.d.$ normally distributed noise with standard deviation $\sigma=10^{-2}$ to the entries of $\bm X^*$.}
Then we run Algorithm~\ref{algo:GD} \hlnew{with $\bm X^0$, ${\cal P}_\Omega(\bm M)$, and $\eta = 0.5/\norm{\bm M}_2$.}
It is noticeable from Fig.~\ref{fig1} that our theoretical bound $\norm{\bm e^0}_2 \rho(\bm H)^k$ \hlnew{given by the green line predicts successfully the rate of decrease in $\norm{\bm E^k}_F$, running parallel to the blue line as soon as $\norm{\bm E^k}_F < 10^{-2}$. As far as the approximations are concerned, we compare the changes in the error modeled by $\bm e^{k+1}=\bm A \bm e^k$ and the error modeled by $\bm e^{k+1}=\bm H \bm e^k$.
While the former (represented by $\norm{\bm A^k \bm e^0}_2$ in black) fails to approximate $\norm{\bm E^k}_F$ for $\norm{\bm E^k}_F < 10^{-2}$, the later (represented by $\norm{\bm H^k \bm e^0}_2$ in red) matches $\norm{\bm E^k}_F$ surprisingly well.}
\end{remark}

\begin{figure}[t]
  \centering
  \centerline{\includegraphics[width=9.1cm]{fig1_revised}}
  \caption{Linear convergence of gradient descent for matrix completion. \hlnew{The decrease in the norm of error matrix $\bm E^k$ through iterations is shown in the blue dashed line with triangle markers. The black solid line with square markers and the red dotted line with circle markers represent first-order approximations of the error using $\bm A$ and $\bm H$, respectively. Finally, the green dash-dot line is the theoretical bound (up to a constant) given by $\norm{\bm e^0}_2 \rho(\bm H)^k$. We use different markers, i.e., triangle versus circle, to better distinguish the blue line from the red line, respectively.}}
	\label{fig1}
\vspace{-9pt}
\end{figure}

In the rest of this section, we shall derive the proof of Theorem~\ref{theo:GD}. First, we present a major challenge met by the traditional approach that uses (\ref{equ:E4}) to characterize the convergence of the error towards zero. Next, we describe our proposed technique to overcome \hlnew{this difficulty. Finally, we show that our bounding technique recovers the exact rate of local convergence of Algorithm}~\ref{algo:GD}.

\vspace{-5pt}
\subsection{A challenge of establishing the error contraction}
\label{subsec:challenge}
The stability of the nonlinear difference equation (\ref{equ:E4}) is the key to analyze the convergence of Algorithm~\ref{algo:GD}. In essence, linear convergence rate is obtained by the following lemma:

\begin{lemma}
\label{lem:scalar}
(Rephrased from the supplemental material of \cite{vu2018adaptive}) Let \hlnew{$(a_n)_{n \in \mathbb{N}} \subset {\mathbb R}_+$} be the sequence defined by
\begin{align*}
a_{n+1} \leq \rho a_{n} + q a_{n}^2 \qquad \text{ for } n=0,1,2,\ldots ,
\end{align*}
where $0\le \rho<1$ and $q \ge 0$. Then $(a_n)$ converges to $0$ if and only if $a_0 < \frac{1-\rho}{q}$.
A simple linear convergence bound can be derived for $a_0 < \rho \frac{1-\rho}{q}$ in the form of
\begin{align*}
    a_n \le a_0 K \rho^n, \quad \text{for } K=\biggl(1 - \frac{a_0 q}{ \rho(1-\rho)}\biggr)^{-1} .
\end{align*}
\end{lemma}

\noindent In order to apply Lemma~\ref{lem:scalar} to (\ref{equ:E4}), one natural way is to perform the eigendecomposition $\bm A = \bm Q_{\bm A} \bm \Lambda_{\bm A} \bm Q_{\bm A}^{-1}$, where $\bm Q_{\bm A}$ is the square matrix whose columns are $n^2$ eigenvectors of $\bm A$, and $\bm \Lambda_{\bm A}$ is the diagonal matrix whose diagonal elements are the corresponding eigenvalues of $\bm A$. Then, left-multiplying both sides of (\ref{equ:E4}) by $\bm Q_{\bm A}^{-1}$ yields
\begin{align*}
    \bm Q_{\bm A}^{-1} \bm e^{k+1} & = \bm \Lambda_{\bm A} \bm Q_{\bm A}^{-1} \bm e^k + \bm{\mathcal{O}}(\norm{\bm e^k}_2^2) ,
\end{align*}
\hlnew{where $\bm Q_{\bm A}^{-1}$ does not affect the $\bm{\mathcal{O}}$ term since its norm is constant.}
Applying the triangle inequality\footnote{Given $a=b+c$, by triangle inequality, we have $\|a\| \le \|b\| + \|c\|$ and $\|a\| \ge \|b\| -\|c\|$ (since $b=a+(-c)$ and hence $\|b\| \le \|a\| + \|-c\| = \|a\|+\|c\|$ or $\|a\| \ge \|b\|-\|c\|$). Consequently, we can write $|\|a\| -\|b\|| \le \|c\|$ and hence $\|a\| = \|b\| + {\cal O}(\|c\|)$.} to the last equation leads to
\begin{align*}
    \norm{\bm Q_{\bm A}^{-1} \bm e^{k+1}}_2 & = \norm{\bm \Lambda_{\bm A} \bm Q_{\bm A}^{-1} \bm e^k}_2 + \mathcal{O}(\norm{\bm e^k}_2^2) . \numberthis \label{equ:eQA}
\end{align*}
With the definition of the spectral radius of $\bm A$ using the spectral norm of $\bm \Lambda_A$, we have
\begin{align*}
    \rho(\bm A) = \norm{\bm \Lambda_{\bm A}}_2 = \sup \biggl\{ \frac{\norm{\bm \Lambda_{\bm A} \tilde{\bm e}}_2}{\norm{\tilde{\bm e}}_2} : \tilde{\bm e} \in {\mathbb R}^{n^2}, \tilde{\bm e} \neq \bm 0 \biggr\} . \numberthis \label{equ:rhoA}
\end{align*}
Now, using (\ref{equ:rhoA}) and the fact that $\mathcal{O}(\norm{\bm e^k}_2^2) = \mathcal{O}(\norm{\bm Q_{\bm A}^{-1} \bm e^k}_2^2)$, (\ref{equ:eQA}) can be upper-bounded by
\begin{align*}
    \norm{\bm Q_{\bm A}^{-1} \bm e^{k+1}}_2 & \leq \rho(\bm A) \norm{\bm Q_{\bm A}^{-1} \bm e^k}_2 + \mathcal{O}(\norm{\bm Q_{\bm A}^{-1} \bm e^k}_2^2) . \numberthis \label{equ:boundA}
\end{align*}
If $\rho(\bm A) < 1$, then by Lemma~\ref{lem:scalar}, the sequence $\norm{\bm Q_{\bm A}^{-1} \bm e^k}_2$ converges to $0$ linearly at rate $\rho(\bm A)$. Unfortunately, one can verify that $\rho(\bm A) \geq 1$ by taking any vector $\bm v \in {\mathbb R}^{n^2}$ such that $v_i=0$ for all $i \in \bar{\Omega}$. Since $\bm A \bm v = \bm v$, $1$ must be an eigenvalue of $\bm A$.

The failure of the aforementioned bounding technique is it overlooks the fact that $\bm E^k = \bm X^k {\bm X^k}^\top \hspace{-0.05cm} - \bm M$. By defining ${\cal E} = \{ \bm X \bm X^\top \hspace{-0.05cm} - \bm M \mid \bm X \in {\mathbb R}^{n \times r} \}$ and $\tilde{{\cal E}}_{\bm A} = \{ \bm Q^{-1}_{\bm A} \vect(\bm E) \mid \bm E \in {\cal E} \}$,
a tighter bound \hlnew{on $\norm{\bm \Lambda_{\bm A} \bm Q_{\bm A}^{-1} \bm e^k}_2 / \norm{\bm Q_{\bm A}^{-1} \bm e^k}_2$} can be obtained by
\begin{align} \label{equ:rho_constrained}
    \rho^{{\cal E}}(\bm A,\delta) = \sup \biggl\{ \frac{\norm{\bm \Lambda_{\bm A} \tilde{\bm e}}_2}{\norm{\tilde{\bm e}}_2} : \tilde{\bm e} \in \tilde{{\cal E}}_{\bm A}, \tilde{\bm e} \neq \bm 0, \norm{\tilde{\bm e}}_2 \leq \delta \biggr\} ,
\end{align}
for some constant $\delta > 0$.
Taking into account the structure of $\bm E^k$, one would expect $\rho^{\cal E} (\bm A) = \lim_{\delta \to 0} \rho^{{\cal E}}(\bm A,\delta)$ is a more reliable estimate of the asymptotic rate of convergence for (\ref{equ:E4}). Nonetheless, (\ref{equ:rho_constrained}) is a non-trivial optimization problem that has no closed-form solution to the best of our knowledge.

\subsection{Integrating structural constraints

\hlnew{To address the aforementioned issue, we propose to integrate the structural constraint on $\bm E^k$ into the recursion} (\ref{equ:E4}). \hlnew{As we shall show in the next subsection, this integration enables the application of Lemma}~\ref{lem:scalar} \hlnew{to the new recursion in order to obtain a tight bound on the convergence rate. First, let us characterize the feasible set of error matrices ${\cal E}$ as follows:}
\begin{lemma} \label{lem:E}
$\bm E \in {\cal E}$ if and only if the following conditions hold simultaneously:
\begin{enumerate}
    \item[(C1)] ${\cal P}_r(\bm M + \bm E) = \bm M + \bm E$, where ${\cal P}_r$ is the truncated singular value decomposition of order $r$ \cite{eckart1936approximation}.
    \item[(C2)] $\bm E^\top \hspace{-0.05cm} = \bm E$.
    \item[(C3)] $\bm v^\top \hspace{-0.05cm} (\bm M + \bm E) \bm v \geq 0$ for all $\bm v \in {\mathbb R}^n$.
\end{enumerate}
\end{lemma}
\noindent
Our strategy is to integrate three conditions in Lemma~\ref{lem:E} into the linear operator $\bm A$ so that the resulting recursion will implicitly enforce $\bm E^k$ to remain in ${\cal E}$. Specifically, for condition (C1), we linearize ${\cal P}_r$ using the first-order perturbation analysis of the truncated singular value decomposition \cite{vu2020perturbation}. For condition (C2), we leverage the linearity of the transpose operator. \hlnew{Finally, while handling condition (C3) is non-trivial, it turns out that this condition can be ignored.}
In the following lemma, we introduce the linear projection that ensures the updated error ${\bm E}^k$ remains near ${\cal E}$.
\begin{lemma} \label{lem:e}
Recall that $\bm P_1 = \bm I_{n^2} - \bm P_{\bm U_\perp} \otimes \bm P_{\bm U_\perp}$, $\bm P_2 = \frac{1}{2} \bigl(\bm I_{n^2} + \bm T_{n^2}\bigr)$. Then, the following statements hold:
\begin{enumerate}
    \item $\bm P_1$ corresponds to the orthogonal projection onto the tangent plane of the set of rank-$r$ matrices at $\bm M$.
    \item $\bm P_2$ corresponds to the orthogonal projection onto the space of symmetric matrices.
    \item $\bm P_1$ and $\bm P_2$ commute, and $\bm P = \bm P_1 \bm P_2 = \bm P_2 \bm P_1$ is also an orthogonal projection.
    \item For any $\bm E \in {\cal E}$, $\vect (\bm E) = \bm P \vect (\bm E) + \bm{\mathcal{O}}(\norm{\bm E}_F^2)$.
\end{enumerate}
\end{lemma}
\noindent By Lemma~\ref{lem:e}-4, we have $\bm e^{k} = \bm P \bm e^{k} + \bm{\mathcal{O}}(\norm{\bm e^{k}}_2^2)$ for all $k$. Using this result with $k+1$ instead of $k$ and replacing $\bm e^{k+1}$ from (\ref{equ:E4}) into the first term on the RHS, we have $$\bm e^{k+1} = \bm P \bigl(\bm A \bm e^k + \bm{\mathcal{O}}(\norm{\bm e^k}_2^2)\bigr) + \bm{\mathcal{O}}(\norm{\bm e^{k+1}}_2^2) .$$
Substituting $\bm e^{k} = \bm P \bm e^{k} + \bm{\mathcal{O}}(\norm{\bm e^{k}}_2^2)$ and using $\bm e^{k+1} = \bm{\mathcal{O}}(\norm{\bm e^k}_2)$, we obtain
\begin{align*}
    \bm e^{k+1} &= \bm P \bm A \bm P \bm e^k + \bm{\mathcal{O}}(\norm{\bm e^k}_2^2) . \numberthis \label{equ:E6}
\end{align*}
It can be seen from Lemma~\ref{lem:e}-1 and Lemma~\ref{lem:e}-2 that the projection $\bm P$ enforces the error vector $\bm e^k$ to lie in the space under conditions (C1) and (C2) in Lemma~\ref{lem:E}.
Now replacing the definition $\bm H = \bm P \bm A \bm P$, (\ref{equ:E6}) can be rewritten as
\begin{align}
    \bm e^{k+1} = \bm H \bm e^k + \bm{\mathcal{O}}(\norm{\bm e^k}_2^2) . \numberthis \label{equ:E7}
\end{align}
Similar to the derivation with $\bm A$, let $\bm H = \bm Q_{\bm H} \bm \Lambda_{\bm H} \bm Q_{\bm H}^{-1}$ be the eigendecomposition of $\bm H$ and define $\tilde{\bm e}^k = \bm Q^{-1}_{\bm H} \bm e^k$. Then, we have
\begin{align*}
    \norm{\tilde{\bm e}^{k+1}}_2 = \norm{\bm \Lambda_{\bm H} \tilde{\bm e}^k}_2 + \mathcal{O}(\norm{\tilde{\bm e}^k}_2^2) . \numberthis \label{equ:E8}
\end{align*}
In addition, denote $\tilde{{\cal E}}_{\bm H} = \{ \bm Q^{-1}_{\bm H} \vect(\bm E) \mid \bm E \in {\cal E} \}$, we can define
\begin{align*}
    \rho(\bm H) &= \sup \biggl\{ \frac{\norm{\bm \Lambda_{\bm H} \tilde{\bm e}}_2}{\norm{\tilde{\bm e}}_2} : \tilde{\bm e} \in {\mathbb R}^{n^2}, \tilde{\bm e} \neq \bm 0 \biggr\} \text{ and } \numberthis \label{equ:rhoH} \\
    \rho^{{\cal E}}(\bm H,\delta) &= \sup \biggl\{ \frac{\norm{\bm \Lambda_{\bm H} \tilde{\bm e}}_2}{\norm{\tilde{\bm e}}_2} : \tilde{\bm e} \in \tilde{{\cal E}}_{\bm H}, \tilde{\bm e} \neq \bm 0, \norm{\tilde{\bm e}}_2 \leq \delta \biggr\} . \numberthis \label{equ:rho_constrainedH}
\end{align*}
Since (\ref{equ:E4}) and (\ref{equ:E7}) are two different systems that describes the same dynamic for $\bm E^k \in {\cal E}$, one would expect they share the same asymptotic behavior. In particular, their linear rates of convergence should agree when the constraint $\bm E^k \in {\cal E}$ is considered.
\begin{lemma} \label{lem:rho}
Let $\rho^{\cal E} (\bm H) = \lim_{\delta \to 0} \rho^{{\cal E}}(\bm H,\delta)$. Then,
\begin{align*}
    \rho^{\cal E} (\bm H) = \rho^{\cal E} (\bm A).
\end{align*}
\end{lemma}
\noindent \hlnew{While using $\bm H$ instead of $\bm A$ preserves the system dynamic over ${\cal E}$, it provides updates of the error that ensure that it remains in ${\cal E}$. Consequently, we can ignore the constraints that are implicitly satisfied in our analysis when using $\bm H$.}

\subsection{Asymptotic bound on the linear convergence rate}
\hlnew{We have seen in Subsection}~\ref{subsec:challenge} that applying Lemma~\ref{lem:scalar} to (\ref{equ:E4}) \hlnew{fails to estimate the convergence rate due to the gap between $\rho^{\cal E}(\bm A)$ and $\rho(\bm A)$. In this subsection, we show that integrating the structural constraint helps eliminating the gap between $\rho^{\cal E}(\bm H)$ and $\rho(\bm H)$ (even when condition (C3) is omitted). Therefore, applying} Lemma~\ref{lem:scalar} to (\ref{equ:E8}) yields $\rho(\bm H)$ as a tight bound on the convergence rate. To that end, our goal is to prove the following lemma:
 \begin{lemma} \label{lem:tight}
As $\delta$ approaches $0$, we have $\rho(\bm H) - \rho^{{\cal E}}(\bm H,\delta) = \mathcal{O}(\delta)$. Consequently, it holds that $\rho(\bm H) = \rho^{{\cal E}}(\bm H)$.
\end{lemma}
\noindent \hlnew{Let us briefly present the key ideas and lemmas we use to prove Lemma}~\ref{lem:tight}.
\hlnew{Our proof relies on two critical considerations: (i) $\rho^{\cal E} (\bm H,\delta) \leq \rho(\bm H)$, (ii) there exists a maximizer $\tilde{\bm e}^\star$ of the supremum in} (\ref{equ:rhoH}) such that the distance from $\tilde{\bm e}^\star$ to $\tilde{{\cal E}}_{\bm H}$ is $\mathcal{O}(\delta^2)$. While (i) is trivial from (\ref{equ:rhoH}) and (\ref{equ:rho_constrainedH}), \hlnew{(ii) is proven by introducing ${\cal F}_\delta$ as a surrogate for the set ${\cal E}$ as follows:}
\begin{lemma} \label{lem:PSD}
\hlnew{Denote the eigenvector of $\bm H$ corresponding to the largest (in magnitude) eigenvalue by $\bm q_1$. Define $\bm G$ as the ${n \times n}$ matrix satisfying $\vect(\bm G) = \delta \bm q_1$.}
Let ${\cal F}_\delta$ be the set of $n \times n$ matrices satisfying the following conditions: (i) $\norm{\bm F}_F \leq 2\delta$; (ii) $\bm F^\top \hspace{-0.05cm} = \bm F$; (iii) $\norm{\bm P_{\bm U_\perp} \bm F \bm P_{\bm U_\perp}}_F \leq \frac{2}{\lambda_r} \delta^2$; and (iv) $\bm v^\top \hspace{-0.05cm} (\bm M + \bm F) \bm v \geq 0$ for all $\bm v \in {\mathbb R}^n$.
Then, there exists $\bm F \in {\cal F}_\delta$ satisfying
\begin{align*}
    \norm{\bm F - \bm G}_F = \mathcal{O}(\delta^2) .
\end{align*}
\end{lemma}
\begin{lemma} \label{lem:Pr}
For any $\bm F \in {\cal F}_\delta$, there exists $\bm E \in {\cal E}$ satisfying
\begin{align*}
    \norm{\bm E - \bm F}_F = \mathcal{O}(\delta^2) .
\end{align*}
\end{lemma}
\noindent \hlnew{From (i) and (ii), it follows that the difference between $\rho^{\cal E}(\bm H,\delta)$ and $\rho(\bm H)$ is $\mathcal{O}(\delta)$. Thus, $\rho(\bm H) = \rho^{\cal E}(\bm H)$ when taking the limit of $\rho^{\cal E}(\bm H,\delta)$ as $\delta \to 0$. Our derivation of Theorem}~\ref{theo:GD} is completed by directly applying Lemma~\ref{lem:scalar} to (\ref{equ:E8}).

\section{Conclusion and Future work}
\label{sec:conc}
We presented a framework for analyzing the convergence of the existing gradient descent approach for low-rank matrix completion. In our analysis, we restricted our focus to the symmetric matrix completion case. We proved that the algorithm converges linearly. Different to other approaches, we made no assumption on the rank of the matrix or fraction of available entries. Instead, we derived an expression for the linear convergence rate via the spectral norm of a closed-form matrix. As future work, using random matrix theory, the closed-form expression for the convergence rate can be further related to the rank, the number of available entries, and the matrix dimensions. Additionally, this work can be extended to the non-symmetric case.

\section{Appendix}

\subsection{Proof of Lemma~\ref{lem:EA}}
Recall the gradient descent update in Algorithm~\ref{algo:GD}:
\begin{align*}
    \bm X^{k+1} &= \bm X^k - \eta {\cal P}_\Omega \bigl(\bm X^k {\bm X^k}^\top \hspace{-0.05cm} - \bm M\bigr) \bm X^k \\
    &= (\bm I_n  - \eta {\cal P}_\Omega (\bm E^k)) \bm X^k . \numberthis \label{equ:GD}
\end{align*}
Substituting (\ref{equ:GD}) into the definition of $\bm E^{k+1}$, we have
\begin{align*}
    &\bm E^{k+1} = \bm X^{k+1} {\bm X^{k+1}}^\top \hspace{-0.05cm} - \bm M \\
    &\quad = \bigl( \bm I_n  - \eta {\cal P}_\Omega (\bm E^k)  \bigr) \bm X^k {\bm X^k}^\top \hspace{-0.05cm} \bigl( \bm I_n  - \eta {\cal P}_\Omega (\bm E^k)  \bigr)^\top \hspace{-0.05cm} - \bm M.
\end{align*}
From the fact that $\bm E^k$ is symmetric and $\Omega$ is a symmetric sampling, the last equation can be further expanded as
\begin{align*}
    &\bm E^{k+1} = \bm X^k {\bm X^k}^\top \hspace{-0.05cm} - \eta {\cal P}_\Omega (\bm E^k) \bm X^k {\bm X^k}^\top \hspace{-0.05cm} \\
    &- \eta \bm X^k {\bm X^k}^\top \hspace{-0.05cm} {\cal P}_\Omega (\bm E^k) + \eta^2 {\cal P}_\Omega (\bm E^k) \bm X^k {\bm X^k}^\top \hspace{-0.05cm} {\cal P}_\Omega (\bm E^k) - \bm M . \numberthis \label{equ:E1}
\end{align*}
Since $\bm X^k {\bm X^k}^\top \hspace{-0.05cm} = \bm M + \bm E^k$, (\ref{equ:E1}) is equivalent to
\begin{align*}
    \bm E^{k+1} &= \bm E^k - \eta \bigl( {\cal P}_\Omega (\bm E^k) \bm M + \bm M {\cal P}_\Omega (\bm E^k) \bigr) \\
    &\quad - \eta \bigl( {\cal P}_\Omega (\bm E^k) \bm E^k + \bm E^k {\cal P}_\Omega (\bm E^k) \bigr) \\
    &\quad + \eta^2 {\cal P}_\Omega (\bm E^k) \bm M {\cal P}_\Omega (\bm E^k) + \eta^2 {\cal P}_\Omega (\bm E^k) \bm E^k {\cal P}_\Omega (\bm E^k) . \numberthis \label{equ:E2}
\end{align*}
Note that $\norm{{\cal P}_\Omega (\bm E^k)}_F \leq \norm{\bm E^k}_F$. Hence, collecting terms that are of second order and higher, with respect to $\norm{\bm E^k}_F$, on the RHS of (\ref{equ:E2}) yields
\begin{align*}
    \bm E^{k+1} &= \bm E^k - \eta \bigl( {\cal P}_\Omega (\bm E^k) \bm M + \bm M {\cal P}_\Omega (\bm E^k) \bigr) + \bm{\mathcal{O}}(\norm{\bm E^k}_F^2) .
\end{align*}
Now by Definition~\ref{def:S}, it is easy to verify that
$$ \bm S \bm S^\top \hspace{-0.05cm} = \bm I_{n^2} \quad \text{and} \quad \vect\bigl({\cal P}_\Omega (\bm E^k)\bigr) = \bm S^\top \hspace{-0.05cm} \bm S \bm e^k .$$
Using the property
$\vect(\bm A \bm B \bm C) = (\bm C^\top \hspace{-0.05cm} \otimes \bm A) \vect(\bm B)$, (5) can be vectorized as follows:
\begin{align*}
    \bm e^{k+1} &= \bm e^k - \eta (\bm M \otimes \bm I_n) \vect\bigl({\cal P}_\Omega (\bm E^k)\bigr) \\
    &\quad - \eta (\bm I_n \otimes \bm M) \vect\bigl({\cal P}_\Omega (\bm E^k)\bigr) + \bm{\mathcal{O}}(\norm{\bm e^k}_2^2) .
\end{align*}
The last equation can be reorganized as
\begin{align*}
    \bm e^{k+1} = \Bigl( \bm I_{n^2} - \eta (\bm M \oplus \bm M) (\bm S^\top \hspace{-0.05cm} \bm S) \Bigr) \bm e^k + \bm{\mathcal{O}}(\norm{\bm e^k}_2^2) .
\end{align*}

\subsection{Proof of Lemma~\ref{lem:E}}
($\Rightarrow$) Suppose $\bm E \in {\cal E}$. Then for (C1), i.e., $\bm E^\top \hspace{-0.05cm} = \bm E$, $\bm E = \bm X \bm X^\top \hspace{-0.05cm} - \bm M$ is symmetric since both $\bm X \bm X^\top \hspace{-0.05cm}$ and $\bm M$ are symmetric.
For (C2), i.e., ${\cal P}_r (\bm M + \bm E) = \bm M + \bm E$, stems from the fact $\bm M + \bm E = \bm X \bm X^\top \hspace{-0.05cm}$ has rank no greater than $r$ for $\bm X \in {\mathbb R}^{n \times r}$. Finally, for any $\bm v \in {\mathbb R}^n$, we have
\begin{align*}
    \bm v^\top \hspace{-0.05cm} (\bm M + \bm E) \bm v = \bm v^\top \hspace{-0.05cm} (\bm X \bm X^\top \hspace{-0.05cm}) \bm v = \norm{\bm X^\top \hspace{-0.05cm} \bm v}_2^2 \geq 0 .
\end{align*}

\noindent ($\Leftarrow$) From conditions (C1) and (C3), $\bm M + \bm E$ is a PSD matrix. In addition, ${\cal P}_r(\bm M + \bm E) = \bm M + \bm E$ implies $\bm M + \bm E$ must have rank no greater $r$. Since any PSD matrix $\bm A$ with rank less than or equal to $r$ can be factorized as $\bm A = \bm Y \bm Y^\top \hspace{-0.05cm}$ for some $\bm Y \in {\mathbb R}^{n \times r}$, we conclude that $\bm E \in {\cal E}$.

\subsection{Proof of Lemma~\ref{lem:e}}
First, recall that any matrix $\bm \Pi \in {\mathbb R}^{n^2 \times n^2}$ is an orthogonal projection if and only if $\bm \Pi^2 = \bm \Pi$ and $\bm \Pi = \bm \Pi^\top \hspace{-0.05cm}$. Since $\bm P_{\bm U_\perp}^\top \hspace{-0.05cm} = \bm P_{\bm U_\perp}$, we have
\begin{align*}
    \bm P_1^\top \hspace{-0.05cm} &= \bigl(\bm I_{n^2} - \bm P_{\bm U_\perp} \otimes \bm P_{\bm U_\perp}\bigr)^\top \hspace{-0.05cm} \\
    &= \bm I_{n^2}^\top \hspace{-0.05cm} - \bm P_{\bm U_\perp}^\top \hspace{-0.05cm} \otimes \bm P_{\bm U_\perp}^\top \hspace{-0.05cm} \\
    &= \bm I_{n^2} - \bm P_{\bm U_\perp} \otimes \bm P_{\bm U_\perp} = \bm P_1 .
\end{align*}
In addition, since $\bm P_{\bm U_\perp}^2 = \bm P_{\bm U_\perp}$, we have
\begin{align*}
    \bm P_1 ^2 &= (\bm I_{n^2} - \bm P_{\bm U_\perp} \otimes \bm P_{\bm U_\perp}) (\bm I_{n^2} - \bm P_{\bm U_\perp} \otimes \bm P_{\bm U_\perp})^\top \hspace{-0.05cm} \\
    &= \bm I_{n^2}^2 - 2 \bm P_{\bm U_\perp} \otimes \bm P_{\bm U_\perp} + (\bm P_{\bm U_\perp} \otimes \bm P_{\bm U_\perp})^2 \\
    &= \bm I_{n^2} - 2 \bm P_{\bm U_\perp} \otimes \bm P_{\bm U_\perp} + (\bm P_{\bm U_\perp}^2 \otimes \bm P_{\bm U_\perp}^2) \\
    &= \bm I_{n^2} - 2 \bm P_{\bm U_\perp} \otimes \bm P_{\bm U_\perp} + \bm P_{\bm U_\perp} \otimes \bm P_{\bm U_\perp} \\
    &= \bm I_{n^2} - \bm P_{\bm U_\perp} \otimes \bm P_{\bm U_\perp} = \bm P_1.
\end{align*}
Second, using the fact that $\bm T_{n^2}^2 = \bm I_{n^2}$ and $\bm T_{n^2}$ is symmetric, we can derive similar result:
\begin{align*}
    \bm P_2^\top \hspace{-0.05cm} = \biggl( \frac{\bm I_{n^2} + \bm T_{n^2}}{2} \biggr)^\top \hspace{-0.05cm} = \frac{\bm I_{n^2} + \bm T_{n^2}}{2} = \bm P_2 ,
\end{align*}
and
\begin{align*}
    \bm P_2^2 &= \frac{(\bm I_{n^2} + \bm T_{n^2})^2}{4} \\
    &= \frac{\bm I_{n^2} + 2 \bm T_{n^2} + \bm T^2_{n^2}}{4} \\
    &= \frac{2\bm I_{n^2} + 2\bm T_{n^2}}{4} \\
    &= \frac{\bm I_{n^2} + \bm T_{n^2}}{2} = \bm P_2 .
\end{align*}

Third, we observe that $\bm P_1$ and $\bm P_2$ are the vectorized version of the linear operators
\begin{align*}
    \bm \Pi_1 (\bm E) = \bm E - \bm P_{\bm U_\perp} \bm E \bm P_{\bm U_\perp}
\end{align*}
and
\begin{align*}
    \bm \Pi_2 (\bm E) = \frac{1}{2} (\bm E + \bm E^\top \hspace{-0.05cm}) ,
\end{align*}
respectively, for any $\bm E \in {\mathbb R}^{n \times n}$. Hence, in order to prove that $\bm P_1$ and $\bm P_2$ commute, it is sufficient to show that operators $\bm \Pi_1$ and $\bm \Pi_2$ commute. Indeed, we have
\begin{align*}
    \bm \Pi_2 \bm \Pi_1 (\bm E) &= \frac{1}{2} \bigl( (\bm E - \bm P_{\bm U_\perp} \bm E \bm P_{\bm U_\perp}) + (\bm E - \bm P_{\bm U_\perp} \bm E \bm P_{\bm U_\perp})^\top \hspace{-0.05cm} \bigr) \\
    &= \frac{1}{2} (\bm E + \bm E^\top \hspace{-0.05cm}) - \bm P_{\bm U_\perp} \frac{1}{2} (\bm E + \bm E^\top \hspace{-0.05cm}) \bm P_{\bm U_\perp} \\
    &= \bm \Pi_1 \bm \Pi_2 (\bm E) .
\end{align*}
This implies $\bm \Pi_1$ and $\bm \Pi_2$ commute.
Since $\bm P$ is the product of two commuting orthogonal projections, it is also an orthogonal projection.

Finally, let us restrict $\bm E$ to belong to ${\cal E}$ and denote $\bm e = \vect (\bm E)$. Using Theorem~3 in \cite{vu2020perturbation}, we have
\begin{align*}
    {\cal P}_r (\bm M + \bm E) = \bm M + \bm E - \bm P_{\bm U_\perp} \bm E \bm P_{\bm U_\perp} + \bm{\mathcal{O}}(\norm{\bm E}_F^2) . \numberthis \label{equ:Pr}
\end{align*}
Since ${\cal P}_r (\bm M + \bm E) = \bm M + \bm E$, it follows from (\ref{equ:Pr}) that
\begin{align*}
    \bm P_{\bm U_\perp} \bm E \bm P_{\bm U_\perp} = \bm{\mathcal{O}}(\norm{\bm E}_F^2) .
\end{align*}
Vectorizing the last equation, we obtain
\begin{align} \label{equ:P1}
    (\bm P_{\bm U_\perp} \otimes \bm P_{\bm U_\perp}) \bm e = \bm{\mathcal{O}}(\norm{\bm E}_F^2) .
\end{align}
On the other hand, since $\bm E$ is symmetric,
\begin{align} \label{equ:P2}
    \bm e = \bm T_{n^2} \bm e = \Bigl( \frac{\bm I_{n^2} + \bm T_{n^2}}{2} \Bigr) \bm e .
\end{align}
From (\ref{equ:P1}) and (\ref{equ:P2}), we have
\begin{align*}
    \bm e &= (\bm I_{n^2} - \bm P_{\bm U_\perp} \otimes \bm P_{\bm U_\perp}) \bm e + \bm{\mathcal{O}}(\norm{\bm E}_F^2) \\
    &= (\bm I_{n^2} - \bm P_{\bm U_\perp} \otimes \bm P_{\bm U_\perp}) \Bigl( \frac{\bm I_{n^2} + \bm T_{n^2}}{2} \Bigr) \bm e + \bm{\mathcal{O}}(\norm{\bm E}_F^2) . \numberthis \label{equ:e_recur}
\end{align*}
Substituting
$$\bm P = \bm P_1 \bm P_2 = (\bm I_{n^2} - \bm P_{\bm U_\perp} \otimes \bm P_{\bm U_\perp}) \Bigl( \frac{\bm I_{n^2} + \bm T_{n^2}}{2} \Bigr) $$
into (\ref{equ:e_recur}) completes our proof of the lemma.

\subsection{Proof of Lemma~\ref{lem:rho}}
Let $\tilde{{\cal E}} = \{ \vect(\bm E) \mid \bm E \in {\cal E} \}$.
Recall that for any $\bm e \in \tilde{{\cal E}}$,
\begin{align*}
    \bm e = \bm P \bm e + \bm{\mathcal{O}}(\norm{\bm e}_2^2) .
\end{align*}
Therefore, by the triangle inequality, we obtain
\begin{align*}
    \norm{\bm A \bm e}_2 &= \norm{\bm A \bigl(\bm P \bm e + \bm{\mathcal{O}}(\norm{\bm e}_2^2) \bigr)} \\
    &\leq \norm{\bm A \bm P \bm e}_2 + \norm{\bm A \bm{\mathcal{O}}(\norm{\bm e}_2^2)}_2 .
\end{align*}
Since the second term on the RHS of the last inequality is $\mathcal{O}(\norm{\bm e}_2^2)$, it is also $\mathcal{O}(\delta^2)$ for any $\bm e \in \tilde{{\cal E}}$ such that $\norm{\bm e}_2 \leq \delta$. In other words,
\begin{align} \label{equ:uAe}
    \norm{\bm A \bm e}_2 \leq \norm{\bm A \bm P \bm e}_2 + \mathcal{O}(\delta^2) .
\end{align}
Similarly, we also have,
\begin{align*}
    \norm{\bm A \bm e}_2 &\geq \norm{\bm A \bm P \bm e}_2 - \norm{\bm A \bm{\mathcal{O}}(\norm{\bm e}_2^2)}_2 \\
    &= \norm{\bm A \bm P \bm e}_2 - \mathcal{O}(\delta^2) . \numberthis \label{equ:lAe}
\end{align*}
From (\ref{equ:uAe}) and (\ref{equ:lAe}), it follows that
\begin{align*}
    \frac{\norm{\bm A \bm e}_2}{\norm{\bm e}_2} &= \frac{\norm{\bm A \bm P \bm e}_2}{\norm{\bm e}_2} + \mathcal{O}(\delta) . \numberthis \label{equ:ulAe}
\end{align*}
Taking the limit of the supremum of (\ref{equ:ulAe}) as $\delta \to 0$ yields
\begin{align*}
    \rho^{\cal E}(\bm A) &= \lim_{\delta \to 0} \sup_{\substack{\bm e \in \tilde{{\cal E}} \\ \bm e \neq 0 \\ \norm{\bm e}_2 \leq \delta}} \frac{\norm{\bm A \bm e}_2}{\norm{\bm e}_2} \\
    &= \lim_{\delta \to 0} \sup_{\substack{\bm e \in \tilde{{\cal E}} \\ \bm e \neq 0 \\ \norm{\bm e}_2 \leq \delta}} \frac{\norm{\bm A \bm P \bm e}_2}{\norm{\bm e}_2} = \rho^{\cal E}(\bm A \bm P) . \numberthis \label{equ:A_AP}
\end{align*}
Now following similar argument in Lemma~\ref{lem:tight}, we have
\begin{align} \label{equ:PAP}
    \begin{cases} \rho^{\cal E}(\bm A \bm P) = \rho(\bm A \bm P) , \\
    \rho^{\cal E}(\bm P \bm A \bm P) = \rho(\bm P \bm A \bm P) .
    \end{cases}
\end{align}
Given (\ref{equ:A_AP}) and (\ref{equ:PAP}), it remains to show that $\rho(\bm A \bm P) = \rho(\bm P \bm A \bm P)$. Indeed, using Gelfand's formula \cite{gelfand1941normierte}, we have
\begin{align*}
    &\rho(\bm A \bm P) = \lim_{k \to \infty} \norm{(\bm A \bm P)^k}_2^{1/k} \\
    \text{ and } &\rho(\bm P\bm A \bm P) = \lim_{k \to \infty} \norm{(\bm P \bm A \bm P)^k}_2^{1/k} .
\end{align*}
By the property of operator norms,
\begin{align*}
    \norm{(\bm A \bm P)^k}_2 = \norm{\bm A (\bm P \bm A \bm P)^{k-1}}_2 \leq \norm{\bm A}_2 \norm{(\bm P \bm A \bm P)^{k-1}}_2 .
\end{align*}
Thus,
\begin{align*}
    \norm{(\bm A \bm P)^k}_2^{1/k} \leq \norm{\bm A}_2^{1/k} \Bigl(\norm{(\bm P \bm A \bm P)^{k-1}}_2^{1/(k-1)} \Bigr)^{(k-1)/k} .
\end{align*}
Taking the limit of both sides of the last inequality as $k \to \infty$ yields $\rho(\bm A \bm P) \leq \rho(\bm P \bm A \bm P)$.
Similarly, since
\begin{align*}
    \norm{(\bm P \bm A \bm P)^k}_2 = \norm{\bm P (\bm A \bm P)^k}_2 \leq \norm{(\bm A \bm P)^k}_2 ,
\end{align*}
we also obtain $\rho(\bm P \bm A \bm P) \leq \rho(\bm A \bm P)$. This concludes our proof of the lemma.

\subsection{Proof of Lemma~\ref{lem:tight}}

Without loss of generality, assume $\lambda_1$ is the eigenvalue with largest magnitude, i.e., $\abs{\lambda_1} = \rho(\bm H)$. By the definition of $\bm G$, we have $\norm{\bm G}_F = \delta$. Since $\bm H \vect(\bm G) = \lambda_1 \vect(\bm G)$ and $\bm H = \bm Q_{\bm H} \bm \Lambda_{\bm H} \bm Q^{-1}_{\bm H}$, it follows that
\begin{align} \label{equ:QHG}
    \bm Q_{\bm H} \bm \Lambda_{\bm H} \bm Q_{\bm H}^{-1} \vect(\bm G) = \lambda_1 \vect(\bm G).
\end{align}
Multiplying both sides of (\ref{equ:QHG}) by $\bm Q_{\bm H}^{-1}$, we obtain
\begin{align*}
    \bm \Lambda_{\bm H} \bm Q_{\bm H}^{-1} \vect(\bm G) = \lambda_1 \bm Q_{\bm H}^{-1}\vect(\bm G) .
\end{align*}
Taking the $L2$-norm and and reorganizing the equation yields
\begin{align*}
     \frac{\norm{\bm \Lambda_{\bm H} \bm Q_{\bm H}^{-1} \vect(\bm G) }_2}{\norm{\bm Q_{\bm H}^{-1} \vect(\bm G)}_2} = \abs{\lambda_1} = \rho(\bm H) . \numberthis \label{equ:Grho}
\end{align*}
Therefore, $\bm G$ leads to a solution of the supremum in (\ref{equ:rhoH}). We now prove that $\bm G$ is symmetric and $(\bm P_{\bm U_\perp} \otimes \bm P_{\bm U_\perp}) \vect(\bm G) = \bm 0$.
First, since $\bm P_1$, $\bm P_2$ and $\bm P = \bm P_1 \bm P_2$ are orthogonal projections, we have
\begin{align*}
    \bm P_2 \bm H &= \bm P_2 \bm P \bm A \bm P \\
    &= \bm P_2 \bm P_2 \bm P_1 \bm A \bm P \\
    &= \bm P_2 \bm P_1 \bm A \bm P \\
    &= \bm P_1 \bm P_2 \bm A \bm P \\
    &= \bm P \bm A \bm P = \bm H .
\end{align*}
Thus,
\begin{align*}
    \lambda_1 \vect(\bm G) &= \bm H \vect(\bm G) \\
    &= \bm P_2 \bm H \vect(\bm G) \\
    &= \lambda_1 \bm P_2 \vect(\bm G) . \numberthis \label{equ:Gt}
\end{align*}
Substituting $\bm P_2 = \frac{1}{2} \bigl(\bm I_{n^2} + \bm T_{n^2}\bigr)$ into (\ref{equ:Gt}) yields
\begin{align*}
    \vect(\bm G^\top \hspace{-0.05cm}) = \bm T_{n^2} \vect(\bm G) \quad \text{ or } \quad \bm G = \bm G^\top \hspace{-0.05cm} .
\end{align*}
Second, since $\bm P_1 \bm H = \bm H$, we obtain
\begin{align*}
    \lambda_1 \vect(\bm G) &= \bm H \vect(\bm G) \\
    &= \bm P_1 \bm H \vect(\bm G) \\
    &= \lambda_1 \bm P_1 \vect(\bm G) . \numberthis \label{equ:GP}
\end{align*}
Substituting $\bm P_1 = \bm I_{n^2} - \bm P_{\bm U_\perp} \otimes \bm P_{\bm U_\perp}$ into (\ref{equ:GP}) yields
\begin{align*}
    (\bm P_{\bm U_\perp} \otimes \bm P_{\bm U_\perp}) \vect(\bm G) = \bm 0 \quad \text{ or } \quad \bm P_{\bm U_\perp} \bm G \bm P_{\bm U_\perp} = \bm 0 .
\end{align*}
Since $\norm{\bm E - \bm G}_F \leq \norm{\bm E - \bm F}_F + \norm{\bm F - \bm G}_F$ (by the triangle inequality), Lemmas~\ref{lem:PSD} and \ref{lem:Pr} imply the existence of $\bm E \in {\cal E}$ such that $\norm{\bm E - \bm G}_F = \mathcal{O}(\delta^2)$.
Denote $\tilde{\bm e} = \bm Q^{-1}_{\bm H} \vect(\bm E) \in \tilde{{\cal E}}_{\bm H}$, we have
\begin{align*}
    \bm \Lambda_{\bm H} \tilde{\bm e} &= \lambda_1 \tilde{\bm e} - (\lambda_1 \bm I_{n^2} - \bm \Lambda_{\bm H}) \tilde{\bm e} \\
    &= \lambda_1 \tilde{\bm e} - (\lambda_1 \bm I_{n^2} - \bm \Lambda_{\bm H}) \bm Q^{-1}_{\bm H} \vect(\bm E) \\
    &= \lambda_1 \tilde{\bm e} - (\lambda_1 \bm I_{n^2} - \bm \Lambda_{\bm H}) \bm Q^{-1}_{\bm H} \vect(\bm E - \bm G) .
\end{align*}
where the last equality stems from the fact that $\lambda_1 \bm Q^{-1}_{\bm H} \vect(\bm G) = \bm \Lambda_{\bm H} \bm Q^{-1}_{\bm H} \vect(\bm G)$.
Next, using the triangle inequality, we obtain
\begin{align*}
    &\norm{\bm \Lambda_{\bm H} \tilde{\bm e}}_2 \geq \norm{\lambda_1 \tilde{\bm e}}_2 - \norm{(\lambda_1 \bm I_{n^2} - \bm \Lambda_{\bm H}) \bm Q^{-1}_{\bm H} \vect(\bm E - \bm G)}_2 \\
    &\qquad \geq \rho(\bm H) \norm{\tilde{\bm e}}_2 - \norm{\lambda_1 \bm I_{n^2} - \bm \Lambda_{\bm H}}_2 \norm{\bm Q^{-1}_{\bm H}}_2 \norm{\vect(\bm E - \bm G)}_2 .
\end{align*}
Dividing both sides by $\norm{\tilde{\bm e}}_2$ yields
\begin{align*}
    \frac{\norm{\bm \Lambda_{\bm H} \tilde{\bm e}}_2}{\norm{\tilde{\bm e}}_2} \geq \rho(\bm H) - \frac{\norm{\lambda_1 \bm I_{n^2} - \bm \Lambda_{\bm H}}_2 \norm{\bm Q^{-1}_{\bm H}}_2 \norm{\vect(\bm E - \bm G)}_2}{\norm{\tilde{\bm e}}_2} . \numberthis \label{equ:geqH}
\end{align*}
Since $\norm{\bm E - \bm G}_F = \mathcal{O}(\delta^2)$, (\ref{equ:geqH}) can be rewritten as
\begin{align*}
    \frac{\norm{\bm \Lambda_{\bm H} \tilde{\bm e}}_2}{\norm{\tilde{\bm e}}_2} \geq \rho(\bm H) - \mathcal{O}(\delta^2) . \numberthis \label{equ:geqH1}
\end{align*}
On the other hand, for any $\tilde{\bm e} \in \tilde{{\cal E}}_{\bm H}$, we also have
\begin{align*}
    \frac{\norm{\bm \Lambda_{\bm H} \tilde{\bm e}}_2}{\norm{\tilde{\bm e}}_2} \leq \rho^{{\cal E}}(\bm H,\delta) \leq \rho(\bm H) . \numberthis \label{equ:geqH2}
\end{align*}
Combining (\ref{equ:geqH1}) and (\ref{equ:geqH2}) yields $\rho(\bm H) - \rho^{{\cal E}}(\bm H,\delta) = \mathcal{O}(\delta)$.

\subsection{Proof of Lemma~\ref{lem:PSD}}
Denote $\bm P_{\bm U} = \bm U \bm U^\top \hspace{-0.05cm}$, for any $\bm v \in {\mathbb R}^n$, we can decompose $\bm v$ into two orthogonal component:
\begin{align*}
    \bm v = \bm v_{\bm U} + \bm v_{\perp},
\end{align*}
where $\bm v_{\bm U} = \bm P_{\bm U} \bm v$ and $\bm v_{\perp} = \bm P_{\bm U_\perp} \bm v$. Without loss of generality, assume that $\norm{\bm v}_2 = \norm{\bm v_{\bm U}}_2^2 + \norm{\bm v_{\perp}}_2^2 = 1$. Thus, we have
\begin{align*}
    \bm v^\top \hspace{-0.05cm} (\bm M + \bm G) \bm v &= (\bm v_{\bm U} + \bm v_{\perp})^\top \hspace{-0.05cm} (\bm M + \bm G) (\bm v_{\bm U} + \bm v_{\perp}) \\
    &= \bm v_{\bm U}^\top \hspace{-0.05cm} \bm M \bm v_{\bm U} + \bm v_{\bm U}^\top \hspace{-0.05cm} \bm G \bm v_{\bm U} + \bm v_{\bm U}^\top \hspace{-0.05cm} \bm G \bm v_{\perp} \\
    &\quad + \bm v_{\perp}^\top \hspace{-0.05cm} \bm G \bm v_{\bm U} + \bm v_{\perp}^\top \hspace{-0.05cm} \bm G \bm v_{\perp} , \numberthis \label{equ:vM1}
\end{align*}
where the last equation stems from the fact that $\bm M = {\cal P}_{\bm U} \bm M \bm P_{\bm U}$ and $\bm P_{\bm U} \bm P_{\bm U_\perp} = \bm 0$. Since $\bm P_{\bm U_\perp} \bm G \bm P_{\bm U_\perp} = \bm 0$, we have
\begin{align*}
    \bm v_{\perp}^\top \hspace{-0.05cm} \bm G \bm v_{\perp} = \bm v^\top \hspace{-0.05cm} \bm P_{\bm U_\perp} \bm G \bm P_{\bm U_\perp} \bm v = 0 .
\end{align*}
Thus, (\ref{equ:vM1}) is equivalent to
\begin{align} \label{equ:vM2}
    \bm v^\top \hspace{-0.05cm} (\bm M + \bm G) \bm v = \bm v_{\bm U}^\top \hspace{-0.05cm} \bm M \bm v_{\bm U} + \bm v_{\bm U}^\top \hspace{-0.05cm} \bm G \bm v_{\bm U} + 2 \bm v_{\bm U}^\top \hspace{-0.05cm} \bm G \bm v_{\perp} .
\end{align}
Now let us lower-bound each term on the RHS of (\ref{equ:vM2}) as follows. First, by the Rayleigh quotient, we have
\begin{align} \label{equ:vM21}
    \bm v_{\bm U}^\top \hspace{-0.05cm} \bm M \bm v_{\bm U} \geq \lambda_r \norm{\bm v_{\bm U}}_2^2 ,
\end{align}
and
\begin{align} \label{equ:vM22}
    \bm v_{\bm U}^\top \hspace{-0.05cm} \bm G \bm v_{\bm U} \geq \lambda_{\min}(\bm G) \norm{\bm v_{\bm U}}_2^2 \geq - \norm{\bm G}_F \norm{\bm v_{\bm U}}_2^2 .
\end{align}
Next, by Cauchy-Schwarz inequality,
\begin{align} \label{equ:vM23}
    \bm v_{\bm U}^\top \hspace{-0.05cm} \bm G \bm v_{\perp} \geq - \norm{\bm G}_2 \norm{\bm v_{\bm U}}_2 \norm{\bm v_{\perp}}_2 \geq - \norm{\bm G}_F \norm{\bm v_{\bm U}}_2 .
\end{align}
From (\ref{equ:vM21}), (\ref{equ:vM22}), and (\ref{equ:vM23}), we obtain
\begin{align*}
    \bm v^\top \hspace{-0.05cm} (\bm M + \bm G) \bm v \geq (\lambda_r - \norm{\bm G}_F) \norm{\bm v_{\bm U}}_2^2 - 2 \norm{\bm G}_F \norm{\bm v_{\bm U}}_2 . \numberthis \label{equ:vMG}
\end{align*}
Note that $\norm{\bm G}_F = \delta$ and the quadratic $g(t) = (\lambda_r - \delta) t^2 - 2 \delta t$ is minimized at
\begin{align*}
    t_* = \frac{\delta}{\lambda_r - \delta} , \quad g(t_*) = -\frac{\delta^2}{\lambda_r - \delta} .
\end{align*}
Combining this with (\ref{equ:vMG}) yields
\begin{align*}
    \bm v^\top \hspace{-0.05cm} (\bm M + \bm G) \bm v \geq -\frac{2}{\lambda_r} \delta^2 ,
\end{align*}
for sufficiently small $\delta$.
Let $\bm F = \bm G + \frac{2}{\lambda_r} \delta^2 \bm I_{n}$. Now we can easily verify that $\norm{\bm F - \bm G}_F = \mathcal{O}(\delta^2)$ and $\bm F \in {\cal F}$ .

\subsection{Proof of Lemma~\ref{lem:Pr}}
We shall show that the matrix $\bm E = {\cal P}_r(\bm M + \bm F) - \bm M$ belongs to ${\cal E}$ and satisfies
\begin{align} \label{equ:EF}
    \norm{\bm E - \bm F}_F = \mathcal{O}(\delta^2) .
\end{align}
First, since $\bm F \in {\cal F}_\delta$, $\bm M + \bm F$ must be PSD. Thus, ${\cal P}_r(\bm M + \bm F)$ is a PSD matrix of rank no greater than $r$ and it admits a rank-$r$ factorization ${\cal P}_r(\bm M + \bm F) = \bm Z \bm Z^\top \hspace{-0.05cm}$, for some $\bm Z \in {\mathbb R}^{n \times r}$. Therefore, by the definition of ${\cal E}$,
$$ \bm E = {\cal P}_r(\bm M + \bm F) - \bm M = \bm Z \bm Z^\top \hspace{-0.05cm} - \bm M \in {\cal E} .$$
Next, using (\ref{equ:Pr}), we have
\begin{align*}
    \bm E - \bm F &= {\cal P}_r(\bm M + \bm F) - \bm M - \bm F \\
    &= \bm P_{\bm U_\perp} \bm F \bm P_{\bm U_\perp} + \bm{\mathcal{O}}(\norm{\bm F}_F^2) .
\end{align*}
Since $\bm F \in {\cal F}_\delta$ implies $\bm P_{\bm U_\perp} \bm F \bm P_{\bm U_\perp} = \bm{\mathcal{O}}(\norm{\bm F}_F^2)$, we conclude that $\bm E - \bm F = \bm{\mathcal{O}}(\norm{\bm F}_F^2)$.

\bibliographystyle{IEEEbib}

redpajama_set_name: RedPajamaArXiv


text:
Top ten fundraising tips
Howard Lake offers advice for newcomers on how to raise money for charity
Juliet Shedden
Thu 9 Nov 2000 14.19 EST First published on Thu 9 Nov 2000 14.19 EST
1. The first rule of fundraising is don't be afraid to approach the "right people" for money, ie those who are sympathetic to your cause. You need to carefully consider how much to ask them for. Some will be able to donate money; some their time; others may leave a legacy or bequest.
2. This means researching people before approaching them. Ask yourself: "Does this person or organisation have an interest in my charity?"
3. Ask people who have already given, but think about when they last donated. Do not ask them for money a week afterwards.
4. To encourage people to donate regularly, suggest that they set up a monthly direct debit.
5. Ask people to give tax efficiently with a "gift aid" form. This form has been in use since 1989 but the minimum donation was recently reduced to £1.
6. Ask other people to help you. Challenge members of your charity's board of trustees to "give, get, or get off".
7. Communicate regularly with your supporters. You may use direct mail, but do not mass produce letters. Write to your supporters personally, and tell them how their money is being spent. This will encourage them to continue donating.
8. Think about how you might address different audiences and seek donations from different sources, eg grant-making trusts, individuals, the national lottery, the UK government, Europe and fundraising events.
9. Do not forget to learn your charity law. There are a number of requirements, responsibilities and duties you need to know about before you begin to raise funds. For information contact the Charity Commission which provides up to date guides.
10. Finally, do not forget to thank donors. This is just as important as asking for money. It is polite, ethical and good business. It is also an effective way of making donors feel good about supporting your charity.
• Howard Lake heads Fundraising UK which operates the UK Fundraising website.
Public services policy

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text:
Signposts and Junctions
MCC_Vets 1972-1975
The Manchester Community College Veterans Club
Manchester, Connecticut
MCC Vets Club Softball Team - 1974
Left-right - rear, Tim Ahern, Don Davis, Mark Lawrence, Ray Jenack, Mike Chartier, Sam Silver. Front, Bob Kenton, Doug Cianci, Chuck Scribner, Wayne Vojeck, Bob Buccino, Terry Nolan.
I recently came across this old picture of the 1974 MCC Vets Club softball team. The school had an active Veterans Club, of which I was a member, and this club sported a broad membership of other Vietnam Veterans from all branches of the Armed Services. I had joined the MCC Vets Club soon after beginning classes in September of 1972. Many Vietnam Era Veterans were showing up on campus in the early 1970s as those years marked the highpoint of the American troop presence in Southeast Asia. We studied and partied hard, enjoying the camaraderie and empathy that all of us found in the company of each other. We enjoyed many activities as a group, including outings at the shore, canoeing, camping, and hiking trips, and we fielded a softball team that competed in the local slow-pitch softball leagues.
In the summer and fall of 1972, the softball team went 1 and 13. The following year we were 7 and 7, and in 1974 we went 13 and 1 to win the Dusty 'Sofball' League Championship. In the town playoffs we lost in an early round to Center Billiards (or a team made up of some of their players). We held our own but lost to a better squad, the way it should be. You can't tell from the picture that Tim Ahern was severely injured by a land mine in Vietnam, but Don Davis and Mark Lawrence made sure he attended every game and team event with the rest of us, so for three years he suffered along with our defeats and celebrated our victories as a member of the squad.
That was the beginning of the end for friends were going off to four-year colleges, moving to new areas of the country, starting families, and searching out careers. I graduated in December of 1975, and over the years the Diaspora continued and I found myself in contact with fewer and fewer friends from the old days at school, yet I remain grateful for the support and friendship they provided me when I was leaving the memories of the Vietnam Era behind and finding a new direction for myself after three years in the Army.
We were soldiers once, as well as students, teammates, and friends. And yes, we were also young.
Laudizen King
Copyright 2009 Laudizen King all rights reserved


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Home » Relationships » 9 Lies Most Girls Tell On First Dates
9 Lies Most Girls Tell On First Dates
Admin 9th August 2017 9 Lies Most Girls Tell On First Dates2017-08-09T06:50:04-07:00 Relationships
Notice we say "most," not "all." If this doesn't apply to you, rock on.
There is a lot of pressure for girls to give off a good impression during the first date. As they say, you only have one chance at making a good first impression and so you can't squander your opportunity. That is the reason a lot of women will be on their absolute best behavior during the first date. They will want to project the best versions of themselves in an effort to make a really good first impression. However, some women tend to take it to a whole new level and they will exaggerate who they really are in an effort to overcompensate for what they lack. In other words, they will use some seemingly innocent white lies to win over the affections of the guy.
A lot of these lies tend to be very harmless, but it all really depends on the situation. You just have to be the judge of whether you can overlook her minor dishonesty on your very first meeting. Some men will find these efforts to lie quite cute and endearing. To others, dishonesty can be a blatant deal breaker. Different tastes for different folks. But it never hurts to be in the know. Be wary. Be careful. These are some examples of common lies that women tend to use on their first dates.
1. "I love the outdoors!"
Chances are, she doesn't. She's just trying to portray an image of an active and adventurous girl who likes to go out and explore the world. A lot of girls aren't really going to be so fond of that active life. In fact, a lot of girls practically abhor outdoor activities, but they know that this is something that a lot of men find attractive. And so they will try to oversell their passion for the great outdoors.
2. "I have a really small appetite."
Of course, if a girl has a hearty appetite, then she isn't going to want to let you know about it right on the first date. A lot of girls are made to believe that men like them to be dainty, demure, conservative, and reserved especially when it comes to their table manners. That's why a lot of girls will tend to understate their love for food.
3. "I have never gotten drunk in my life."
No one wants to date the girl who can't handle her alcohol well at parties. But chances are, she's probably had her fair share of nights wherein alcohol got the best of her. Of course, she's not going to be willing to admit that to you on the first date. She also won't want to give you any ideas about getting her drunk on your first date so you can get lucky.
4. "I don't really pay much attention to my diet."
Of course she has. Every girl has paid attention to what they eat at least once in their lives. She doesn't want to come off as someone who is insecure or self-conscious about her figure, and so she's going to be a little nonchalant about it. But she's probably lying. Of course she cares about how she looks. She cares a whole lot more than she is letting on.
5. "I get ready fairly quickly in the mornings."
She doesn't. No girl does. A lot of girls are always going to lie about how fast it takes them to get ready before going out. They know that men don't like to be kept waiting for them to get ready and so in an effort to impress you, she is severely going to downplay her slow preparations.
6. "I'm not really active in the dating world."
Then why would she be dating you if she wasn't being active in the dating world? This is just a common lie that women use to make the man that they're with feel special. Also, she wouldn't want you to feel threatened at the thought of her still seeing other people. She would want you to feel that she's exclusively dating you even though she isn't.
7. "I love that too!"
That band that you really love? Yeah, she doesn't love them even though she claims that she does. Rock climbing? CrossFit? She doesn't love doing those things. She only tells you that she does because she knows that that's what you want to hear. Don't fret though. It means she's really desperate for your affections when she lies to you like this.
8. "I really love to cook!"
A lot of girls will lie about their cooking skills because they know that a way to a man's heart is through his stomach. The way that she talks about her cooking prowess – she's going to make it seem like she's Gordon Ramsey's daughter or something.
9. "I'm an avid bookworm."
In an effort to come off as cultured and intelligent, she is going to tell you that she reads a lot – or perhaps that she's fond of renaissance art or some other deep cultural activities. She doesn't want you to think that she's just some basic girl even though she truly is.
Via RelRules
http://the-conscious-mind.com/9-lies-most-girls-tell-on-first-dates/http://the-conscious-mind.com/wp-content/uploads/2017/08/lies-e1502286549674.jpghttp://the-conscious-mind.com/wp-content/uploads/2017/08/lies-150x150.jpg 2017-08-09T06:50:04-07:00 AdminRelationshipsfirst dates,girls,lies,relationships
Notice we say "most," not "all." If this doesn't apply to you, rock on. There is a lot of pressure for girls to give off a good impression during the first date. As they say, you only have one chance at making a good first impression and so you can't...
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Home / Business / N Korea's Trade with China Contracts in 2015
N Korea's Trade with China Contracts in 2015
Posted on February 1, 2016 by Korea Bizwire in Business, Top News with 1 Comment
The North Korea-China trade volume was valued at US$4.9 billion in the January-November period, down 14.8 percent from $5.76 billion a year earlier, marking the first two-digit on-year drop since 2000. (Image : Yonhap)
SEJONG, Feb.1 (Korea Bizwire) - North Korea's trade with China dipped nearly 15 percent last year apparently due to a chilly bilateral relationship between the two neighboring countries, a report showed Sunday.
The North-China trade volume reached US$4.9 billion in the January-November period, down 14.8 percent from $5.76 billion a year earlier, marking the first double-digit on-year drop since 2000, according to a report by state-run think tank Korea Development Institute (KDI).
Pyongyang's shipments to its neighbor sank 12.3 percent to $2.28 billion over the cited period, while imports from China plunged 16.8 percent to $2.63 billion.
The trade between the allies has risen an average of 22.4 percent between 2000 and 2014. Only in 2009 and 2014 did it shrink on-year.
The KDI report attributed the sharp decline to sluggish raw material exports, as shipments of anthracite coal and iron ore fell 6.3 percent and 68.5 percent, respectively.
"The chilly relationship between Pyongyang and Beijing and a slowdown in the Chinese economy seemed to affect North Korea's sluggish trade with China," said the report. "North Korean leader Kim Jong-un's New Year message, which called for using home-made products and rejecting foreign-made ones, also had some influence on the downbeat trend."
The alliance between Pyongyang and Beijing had been described as being "forged in blood," since China fought alongside North Korea in the 1950-53 Korean War. China is the only country that provides crude oil to the reclusive North.
But their political relations have become strained since 2013, partly because of the North's defiant pursuit of nuclear weapons and a series of purges of pro-Chinese officials in North Korea.
For 2016, the KDI report noted that there is a higher possibility that bilateral trade will contract further following Pyongyang's nuclear tests on Jan. 6, as the global community including the United Nations is set to impose sanctions against the reclusive regime.
"North Korean trade will be dragged down by international economic sanctions sparked by the North's latest nuclear test in the first half of this year," the KDI said. " North Korea-China trade has shrank to some extent, following sanctions by the U.N."
(Yonhap)
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One thought on "N Korea's Trade with China Contracts in 2015"
t--t.info February 1, 2016 at 1:36 pm
Please Dear God Help Me


Don't forget that this coming Thursday, December 1st, is our last guild meeting of the year! Be ready for some games and prizes. Hope you can join us! Don't forget to renew your membership before the end of December to be included in the new directory for 2012.


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Biographical Article (10)
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1-13 of 13 Results for:
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Bromfield, Louis (1896-1956)
Maker: Carl Van Vechten
Bromfield, Louis (1896-1956), novelist, experimental farmer, and newspaper columnist
Louis Bromfield Photograph by Carl Van Vechten, 1933. Courtesy of the Library of Congress (LC-USZ62-103721).
David D. Anderson
Bromfield, Louis (27 December 1896–18 March 1956), novelist, experimental farmer, and newspaper columnist, was born in Mansfield, Ohio, the son of Charles Bromfield, a banker and local Democratic office holder, and Annette Marie Coulter. His father was from an old New England family, and his mother was the daughter of a pioneer family of Richland County, Ohio; both ancestries would influence his later fiction. Bromfield attended Mansfield public schools, spending summers on his mother's family's farm. In 1914–1915 he studied agriculture at Cornell University and then briefly attended Ohio Northern University in Ada, Ohio. He then studied journalism at Columbia University until his enlistment in the U.S. Army Ambulance Service in June 1917. He served with Section 577, attached to the French army, from December 1917 to February 1919. He participated in seven major battles during World War I and was awarded the Croix de Guerre. He was discharged in June 1919 while still in France....
Chamberlain, William Isaac (1837-1920), agriculturalist and editor
Raymond M. Hyser
Chamberlain, William Isaac (11 February 1837–30 June 1920), agriculturalist and editor, was born in Sharon, Connecticut, the son of Jacob Chamberlain and Anna Nutting, farmers. When Chamberlain was only fifteen months old, his parents moved from Connecticut to Hudson, Ohio, where they purchased and maintained a 147-acre farm. He received an A.B. from Western Reserve College (located in Hudson) in 1859 and an A.M. from the same school two years later. For three years he taught Greek and Latin at Shaw Academy in East Cleveland, Ohio, then became a member of the Western Reserve College faculty teaching the same languages. In 1863 he became superintendent of the Cuyahoga Falls, Ohio, schools, a position he held for two years, but with a decline in his health and his elderly parents needing more care Chamberlain resigned that position to teach languages at Western Reserve College and to maintain the family farm, which he had purchased in 1863. Also in 1863 he married Lucy Jones Marshall, daughter of publisher David Marshall, in Pittsburgh, Pennsylvania. They had five children....
Cobb, Cully Alton (1884-1975), agricultural educator, editor, and publisher
Sandra S. Vance
Cobb, Cully Alton (25 February 1884–07 May 1975), agricultural educator, editor, and publisher, was born in a log cabin on the farm of his paternal grandfather near Prospect, Tennessee, the son of Napoleon Bonaparte Cobb, a farmer and rural minister, and Mary Agnes Woodward. Cobb attended public school in Giles County, Tennessee, and Decatur, Alabama. He entered Mississippi Agricultural and Mechanical College (now Mississippi State University) in 1904 and graduated in 1908 with a bachelor of science degree in agriculture. From 1908 to 1910 he served as principal of Chickasaw County Agricultural High School at Buena Vista, Mississippi. The first of fifty such institutions established in the state between 1908 and 1920, the school afforded rural youths a college-preparatory education as well as practical training in farming. In 1910 he married Ora May "Byrdie" Ball, with whom he had two children....
Colman, Norman Jay (1827-1911), agricultural journalist and first secretary of agriculture
Bruce E. Seely
Colman, Norman Jay (16 May 1827–03 November 1911), agricultural journalist and first secretary of agriculture, was born near Richfield Springs, New York, the son of Hamilton Colman and Nancy Sprague, farmers. He attended local academies and soon was teaching school himself. In 1847 Colman left home for Kentucky, where plans to open a school in Owensboro were ended by an illness. He recovered and directed a public school in Brandenburg before enrolling at the University of Louisville Law School, earning a degree in 1849. He opened a successful law office in New Albany, Indiana, married Clara Porter in 1851, and in 1852 was elected district attorney. A year later they moved to St. Louis, Missouri, and purchased a farm, although Colman continued to practice law and was elected an alderman within two years. The couple had two children before Clara Colman died in 1863....
Dymond, John (1836-1922), planter, publisher, and politician
Edward L. Lach, Jr.
Dymond, John (03 May 1836–05 March 1922), planter, publisher, and politician, was born in Canada (exact location unrecorded), the son of Richard Dymond, a Methodist minister, and Anne Hawkens. During his early childhood Dymond's family moved to Zanesville, Ohio, where he was educated in the local public schools and the Zanesville Academy before entering Bartlett's College, a business school in Cincinnati. Following his graduation from Bartlett's in 1857 he took a job as a clerk with his father, who had by that time established himself in the mercantile trade. He then toyed with the cotton manufacturing business in partnership with Homer White (trading under the name White & Dymond) before moving to New York City, where, on the eve of the Civil War, he took a job as a traveling salesman with a firm whose name has not survived. He returned to Zanesville on 3 June 1862 to marry Nancy Elizabeth Cassidy; they had six children....
London, Jack (1876-1916)
Maker: Arnold Genthe
London, Jack (1876-1916), writer, war correspondent, and agronomist
Jack London. Photograph by Arnold Genthe. Courtesy of the Library of Congress (LC-G4085- 0411 P&P).
Earle Labor
London, Jack (12 January 1876–22 November 1916), writer, war correspondent, and agronomist, was born in San Francisco, California, the son of Flora Wellman and, allegedly, William Henry Chaney, a reformer and professor of astrology. Chaney separated from his common-law wife when he learned of her pregnancy, angrily denying his paternity and later insisting (in two 1897 letters written in response to London's inquiries) that he had been impotent at the time of the child's conception. Wellman nevertheless named her son "John Griffith Chaney" on his birth certificate....
Mapes, James Jay (1806-1866)
Mapes, James Jay (1806-1866), chemist and writer
James J. Mapes. Daguerreotype from the studio of Mathew B. Brady. Courtesy of the Library of Congress (LC-USZ62-109830).
Mark R. Finlay
Mapes, James Jay (29 May 1806–10 January 1866), chemist and writer, was born in Maspeth, New York, the son of Jonas Mapes, a merchant and importer, and Elizabeth Tylee. While at a boarding school on Long Island, Mapes lived for a time with the English reformer William Cobbett. As a scientist, however, he was largely self-taught....
Robinson, Solon (1803-1880), author, agricultural journalist, and Indiana pioneer
Donald B. Marti
Robinson, Solon (21 October 1803–03 November 1880), author, agricultural journalist, and Indiana pioneer, was born in Tolland, Connecticut, the son of Jacob Robinson, a farmer and cooper, and Salinda Ladd. His father died when Solon Robinson was about six, and then his mother married James Robinson, one of her deceased husband's cousins. After his mother died and her second husband refused further responsibility for his stepchildren, Solon Robinson was in the care of William Bottom. He worked on his guardian's farm, got a little education in a country school near Lisbon, Connecticut, and briefly worked as a carpenter's apprentice, which was harder labor than his health could stand. In 1818, for unknown reasons, Solon successfully petitioned that Vine Robinson, an uncle in Brooklyn, Connecticut, be his guardian. Solon's later devotion to temperance may have been learned from his uncle, but little more is known about the next few years of his life....
Rodale, J. I. (1898-1971), health food publisher
Rodale, J. I. (16 August 1898–07 June 1971), health food publisher, was born Jerome Irving Cohen in New York City, the son of Michael Cohen, a capmaker and grocer, and Bertha Rouda. Both parents were Polish immigrants. Rodale studied at New York and Columbia Universities but did not earn any degrees. At age twenty he became an auditor for the Internal Revenue Service, and at twenty-one he moved to Pittsburgh, where he worked in a private accounting firm for three years. He wanted, however, for some vague reason that he never explained, to be a farmer and publisher. In 1920 he traveled to Kentucky on business and became enchanted with the Bluegrass State. "Being among farmers and in farm country I was more and more imbued with the ambition of some day having my own farm and riding to town with my children on a buck-board drawn by two trusty horses."...
Stockbridge, Horace Edward (1857-1930), agricultural chemist and editor
Stockbridge, Horace Edward (19 May 1857–30 October 1930), agricultural chemist and editor, was born in Hadley, Massachusetts, the son of Levi Stockbridge and Joanna Smith. He grew up on the college farm of the Massachusetts Agricultural College (now part of the University of Massachusetts at Amherst), where his father was president and a prominent leader in American agricultural education. In 1878 Horace earned a B.S. degree from Massachusetts Agricultural College in a joint program with Boston University. In 1882 he traveled to Göttingen, Germany, where he earned his Ph.D. in agricultural chemistry under Wilhelm Henneberg in 1884. Upon his return to the United States, Stockbridge was married in 1885 to Arabella "Belle" Lamar, member of a prominent political family from Sumter County, Georgia. The couple eventually had eight children, only four of whom survived to adulthood....


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Home > Georgia > Marietta > Browning, Thomas J.
Browning, Thomas J.
Updated: October 18th, 2019 Marietta Lawyer List B, Browning & Smith, Family Law 0 review
Address: 31 Atlanta St. SE. Ste. 201, Marietta, GA 30060
Lawyer Firm: Browning & Smith
Website: http://www.browningsmith.com
Areas of Practice Family Law
Mr. Browning has gained over thirty years of experience in domestic relations. He is well respected among Family Law lawyers. He has represented many notable clients while handling numerous cases in the Court of Appeals and Supreme Court of Georgia.
Since 2011 Mr. Browning has been recognized as a Georgia Super Lawyer. He is also an AV rated lawyer by Martindale Hubbell.
Mr. Browning received his Bachelor's Degree in Liberal Arts from Oglethorpe University in 1967. In 1973, he graduated Magna Cum Laude and received his Juris Doctorate from Woodrow Wilson College of Law, where he served as editor of the Woodrow Wilson Yearbook, Verdict, that same year. Mr. Browning was then admitted to the State Bar of Georgia. He worked as a Law Clerk to Judge Luther C. Hames of the Cobb Superior Court for two years and learned the invaluable ins and outs of the justice system.
Following his clerkship with Judge Hames, he entered private practice with former Governor Roy E. Barnes where they worked together for twenty-three years before Barnes became Governor. For seventeen years he was associated with former Supreme Court Justice, George T. Smith and in 2002 Browning & Smith was formed. In 2010 Justice Smith passed away.
Mr. Browning is actively involved in several civic organizations in the City of Marietta. Mr. Browning has held leadership positions including; trustee at Joseph T. Walker School, chairman of Downtown Marietta Development Authority and a member of the board for Atlanta Lyric Theatre. Mr. Browning is a member of the Marietta First United Methodist Church and served as director of Georgia State Bank. He has fifteen years experience as Chairman of the Institute of Continuing Legal Education Successful Trial Practice Seminar. Mr. Browning is known for his ability and is highly regarded, both personally and professionally. As founder of Browning & Smith, LLC, it is imperative that his commitment to excellence is shared by everyone associated with the firm.
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Det norske secures Norway's approval to develop Jette field
admin 20 February 2012
Det norske oljeselskap has received approval from the Norwegian authorities to develop the Jette field located in blocks 25/7 and 25/8, in the mid-part of the North Sea.
The field, located in a water depth of 127m will be produced with two horizontal wells, which will be tied back to the Jotun floating production, storage and offloading vessel.
Det norske has secured finance for the project which is its first field development as an operator.
ExxonMobil will oversee the day-to-day operations at the field which is scheduled to commence production during first quarter of 2013.
Det norske CEO Erik Haugane said: "This approval represents a significant step in the development of a Det norske toward a fully-fledged exploration and development company operating on the Norwegian Shelf."
Jette contains about 14m barrels of oil equivalents, based on a 30% recovery rate and daily production will constitute approximately 14,000 barrels in the first year.
The development of Jette is estimated to entail an investment of approximately NOK2.5bn ($440.21m) and the field may generate gross revenues of NOK8bn ($1.4bn).
Det norske holds 63.3% stake in the venture, while its partner Dana Petroleum, Bridge Energy and Petoro hold 19.2%, 6% and 11.5% respectively.
Earlier in October 2011, Det norske has signed supplier agreements with three contractors which include Cameron, Subsea 7 and Aker Solutions.
Cameron will supply the subsea production system which includes fabricating the Christmas trees and protective structures.
Subsea 7 will undertake the engineering, procurement, fabrication and installation of a 6km flexible flow line and jumpers, installation of a 6km integrated services umbilical, Christmas trees and protective structures and supply diver assisted tie-ins.
Aker Solutions will fabricate the 6km integrated services umbilical.
Image: The Jette field development and operation will be led by Det norske's headquarter in Trondheim. Photo: Courtesy of Det norske oljeselskap.
Ward Operations
Ward Operations is a provider of high-voltage cable jointing solutions...
Ward Operations is a provider of high-voltage cable jointing solutions for application in offshore and other heavy industry environments.
Telemar (UK)
Telemar Scotland, based in Aberdeen and Grangemouth, is a business...
Telemar Scotland, based in Aberdeen and Grangemouth, is a business unit of Telemar (UK) and part of the global group of Telemar companies, which specialises in marine communications and navigation equipment and systems.
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Travelling with Thirsty Swagman is a once-in-a-lifetime experience. It's meant to be fun for every traveller, and therefore we have a "code of conduct" that all travellers must adhere to, and always respect. It means that everyone can enjoy the tour, and nobody spoils it for anyone else. Above all, it encourages you to have fun, and be safe!
Be on time for departures, and ready for flights, bus tours etc.
Thirsty Swagman will have complete discretion to decide whether you are guilty of the above behaviours, irrespective of whether you are charged or convicted by the local law authorities.
You may also be reported to the local law authorities for any of the above.
Learn about the place you're visiting, and respect their culture. The world is an amazing place, and many places that you may visit are very different to where you're from. Enjoy that difference, and leave on a positive note. You'll probably want to come back one day!
Tip the locals – it's become increasingly accepted practice in all areas of the service industry, but particularly in restaurants, bars and taxis worldwide.