text:
\section{Proof of auxiliary lemmas: confidence intervals}

\label{sec:Analysis-of-CI}

\begin{comment}
@@@

\subsection{Tensor factors}

\label{subsec:Tensor-factors-CI}
\end{comment}

\subsection{Proof of Lemma \ref{lemma:u-entry-var-est-loss-neg}}

\label{subsec:u-entry-var-est-loss-neg}

Fix arbitrary $1\leq l\leq r$ and $1\leq k\leq d$. Before proceeding,
we pause to introduce some notation for simplicity of presentation.
Recalling the notation $\widetilde{\bm{U}}^{\star}:=\big[\bm{u}_{l}^{\star\otimes2}\big]_{1\leq l\leq r}\in\mathbb{R}^{d^{2}\times r}$
and $\widetilde{\bm{U}}:=\big[\bm{u}_{l}^{\otimes2}\big]_{1\leq l\leq r}\in\mathbb{R}^{d^{2}\times r}$,
we define two $d^{2}\times r$ matrices as follows
\begin{align}
\bm{V}^{\star} & :=\widetilde{\bm{U}}^{\star}(\widetilde{\bm{U}}^{\star\top}\widetilde{\bm{U}}^{\star})^{-1},\qquad\bm{V}:=\widetilde{\bm{U}}(\widetilde{\bm{U}}^{\top}\widetilde{\bm{U}})^{-1}.\label{def:V-true}
\end{align}
These allow us to express the covariance matrix as $\bm{\Sigma}_{k}^{\star}=\bm{V}^{\star}\bm{D}_{k}^{\star}\bm{V}^{\star}$
(resp.~$\bm{\Sigma}_{k}=\bm{V}\bm{D}_{k}\bm{V}$), where $\bm{D}_{k}^{\star}$
(resp.~$\bm{D}_{k}$) is defined in (\ref{eq:cov-matrix-m-diag})
(resp.~(\ref{eq:defn-Dk})). In addition, let us define
\[
s_{l,k}^{\star}:=\sqrt{(\bm{\Sigma}_{k}^{\star})_{l,l}}\qquad\text{and}\qquad s_{l,k}:=\sqrt{(\bm{\Sigma}_{k})_{l,l}}.
\]
Lemma~\ref{lemma:V-property} below collects several useful properties
regrading $\bm{V}_{:,l}$ and $\bm{V}_{:,l}^{\star}$; the proof is
deferred to the end of this section.

\begin{lemma}\label{lemma:V-property}Instate the assumptions and
notation of Lemma \ref{lemma:u-entry-var-est-loss-neg}. For each
$1\leq l\leq r$, one has
\begin{align}
 & \left\Vert \bm{V}_{:,l}^{\star}\right\Vert _{2}=\frac{1+o\left(1\right)}{\left\Vert \bm{u}_{l}^{\star}\right\Vert _{2}^{2}},\qquad\big\|\bm{V}_{:,l}^{\star}\big\|_{\infty}\lesssim\frac{\mu\sqrt{r}}{d}\frac{1}{\lambda_{\min}^{\star2/3}},\qquad\sum_{1\leq k\leq d}V_{(i,k),l}^{\star2}\lesssim\frac{\mu r}{d}\frac{1}{\lambda_{\min}^{\star4/3}};\label{eq:V-true-col-norm}\\
 & \left\Vert \bm{V}_{:,l}-\bm{V}_{:,l}^{\star}\right\Vert _{2}\lesssim\frac{\sigma_{\max}}{\lambda_{\min}^{\star}}\sqrt{\frac{d}{p}}\,\frac{1}{\lambda_{\min}^{\star2/3}},\qquad\left\Vert \bm{V}_{:,l}-\bm{V}_{:,l}^{\star}\right\Vert _{\infty}\lesssim\frac{\sigma_{\max}}{\lambda_{\min}^{\star}}\sqrt{\frac{\mu^{2}r\log d}{dp}}\,\frac{1}{\lambda_{\min}^{\star2/3}}.\label{eq:V-col-loss}
\end{align}
\end{lemma}

With these in place, we are ready to control $J_{l,k}$, which can
be expressed as
\[
J_{l,k}=\frac{u_{l,k}-u_{l,k}^{\star}}{s_{l,k}}-\frac{u_{l,k}-u_{l,k}^{\star}}{s_{l,k}^{\star}}=\left(u_{l,k}-u_{l,k}^{\star}\right)\frac{s_{l,k}^{\star2}-s_{l,k}^{2}}{s_{l,k}s_{l,k}^{\star}}\frac{1}{s_{l,k}^{\star}+s_{l,k}}.
\]
This suggest that we control both $u_{l,k}-u_{l,k}^{\star}$ and $s_{l,k}^{\star2}-s_{l,k}^{2}$.
\begin{itemize}
\item Regarding the estimation error of $u_{l,k}$, combining (\ref{eq:V-true-col-norm})
with the assumptions $\sigma_{\max}/\sigma_{\min}\asymp1$ and $\kappa\asymp1$
allows us to lower bound
\begin{equation}
s_{l,k}^{\star2}=\frac{1}{p}(\bm{V}_{:,l}^{\star})^{\top}\bm{D}_{k}^{\star}\bm{V}_{:,l}^{\star}\geq\frac{1}{p}\lambda_{\min}(\bm{D}_{k}^{\star})\left\Vert \bm{V}_{:,l}^{\star}\right\Vert _{2}^{2}\gtrsim\frac{\sigma_{\min}^{2}}{p\left\Vert \bm{u}_{l}^{\star}\right\Vert _{2}^{4}}.\label{eq:u-entry-var-LB}
\end{equation}
Hence, we know from (\ref{eq:U-loss-2inf}) and the conditions $\sigma_{\max}/\sigma_{\min}\asymp1$
and $\kappa\asymp1$ that
\[
\left|u_{l,k}-u_{l,k}^{\star}\right|\leq\big\|\bm{U}-\bm{U}^{\star}\big\|_{2,\infty}\lesssim\frac{\sigma_{\max}}{\lambda_{\min}^{\star}}\sqrt{\frac{\mu r\log d}{p}}\,\lambda_{\max}^{\star1/3}\lesssim s_{l,k}^{\star}\sqrt{\mu r\log d}.
\]
\item Next, we claim that
\begin{equation}
\left|s_{l,k}^{\star2}-s_{l,k}^{2}\right|\lesssim\Bigg\{\sqrt{\frac{\mu^{3}r^{2}\log^{2}d}{d^{2}p}}+\frac{\sigma_{\max}}{\lambda_{\min}^{\star}}\sqrt{\frac{\mu^{2}rd\log d}{p}}\Bigg\} s_{l,k}^{\star2}\ll s_{l,k}^{\star2};\label{claim:u-entry-var-loss}
\end{equation}
if this were true, then one would further obtain
\[
s_{l,k}\geq s_{l,k}^{\star}-|s_{l,k}-s_{l,k}^{\star}|\gtrsim s_{l,k}^{\star}.
\]
\item Putting the above bounds together reveals that
\[
\left|J_{l,k}\right|\lesssim\frac{\big|u_{l,k}-u_{l,k}^{\star}\big|\,\big|s_{l,k}^{\star2}-s_{l,k}^{2}\big|}{s_{l,k}^{\star3}}\lesssim\sqrt{\mu r\log d}\,\Bigg\{\sqrt{\frac{\mu^{3}r^{2}\log^{2}d}{d^{2}p}}+\frac{\sigma_{\max}}{\lambda_{\min}^{\star}p}\sqrt{\frac{\mu^{2}rd\log d}{p}}\Bigg\}
\]
as claimed.
\end{itemize}
Hence, the remainder of the proof boils down to establishing the claim
(\ref{claim:u-entry-var-loss}). Towards this, our starting point
is the following decomposition
\begin{align}
\frac{p}{2}\left(s_{l,k}^{2}-s_{l,k}^{\star2}\right) & =(\bm{V}_{:,l})^{\top}\bm{D}_{k}\bm{V}_{:,l}-(\bm{V}_{:,l}^{\star})^{\top}\bm{D}_{k}^{\star}\bm{V}_{:,l}^{\star}\nonumber \\
 & =\underbrace{(\bm{V}_{:,l})^{\top}(\bm{D}_{k}-\widehat{\bm{D}}_{k})\bm{V}_{:,l}}_{=:\,\beta_{1}}+\underbrace{(\bm{V}_{:,l})^{\top}\widehat{\bm{D}}_{k}\bm{V}_{:,l}-(\bm{V}_{:,l}^{\star})^{\top}\bm{D}_{k}^{\star}\bm{V}_{:,l}^{\star}}_{=:\,\beta_{2}},\label{eq:u-entry-var-est-loss-decomp}
\end{align}
where $\widehat{\bm{D}}_{k}\in\mathbb{R}^{d^{2}\times d^{2}}$ is
a diagonal matrix with entries given by
\begin{equation}
(\widehat{\bm{D}}_{k}){}_{(i,j),(i,j)}=p^{-1}E_{i,j,k}^{2}\chi_{i,j,k},\qquad1\leq i,j\leq d.\label{def:D-hat}
\end{equation}
In what follows, we shall control $\beta_{1}$ and $\beta_{2}$ separately.

\subsubsection{Bounding $\beta_{1}$}

To begin with, let us decompose
\begin{align*}
\beta_{1} & =\underbrace{(\bm{V}_{:,l}^{\star})^{\top}(\bm{D}_{k}-\widehat{\bm{D}}_{k})\bm{V}_{:,l}^{\star}}_{=:\,\gamma_{1}}+2\underbrace{(\bm{V}_{:,l}^{\star})^{\top}(\bm{D}_{k}-\widehat{\bm{D}}_{k})(\bm{V}_{:,l}-\bm{V}_{:,l}^{\star})}_{=:\,\gamma_{2}}\\
 & \quad+\underbrace{(\bm{V}_{:,l}-\bm{V}_{:,l}^{\star})^{\top}(\bm{D}_{k}-\widehat{\bm{D}}_{k})(\bm{V}_{:,l}-\bm{V}_{:,l}^{\star})}_{=:\,\gamma_{3}}.
\end{align*}

\begin{itemize}
\item With respect to $\gamma_{1}$, the triangle inequality yields
\[
\left|\gamma_{1}\right|=\Big|\sum_{1\leq i,j\leq d}\big(\widehat{E}_{i,j,k}^{2}-E_{i,j,k}^{2}\big)p^{-1}\chi_{i,j,k}V_{(i,j),l}^{\star2}\Big|\leq\max_{(i,j,k)\in\Omega}\big|\widehat{E}_{i,j,k}^{2}-E_{i,j,k}^{2}\big|\sum_{1\leq i,j\leq d}p^{-1}\chi_{i,j,k}V_{(i,j),l}^{\star2}.
\]
From (\ref{eq:noise-est-inf-UB}), we know that
\[
\max_{(i,j,k)\in\Omega}\big|\widehat{E}_{ijk}-E_{ijk}\big|\lesssim\frac{\sigma_{\max}}{\lambda_{\min}^{\star}}\sqrt{\frac{\mu^{3}r^{2}\log d}{d^{2}p}}\,\lambda_{\max}^{\star}\ll\sigma_{\max}\sqrt{\log d},
\]
where the last inequality arises from the conditions $p\gg\mu^{3}r^{2}d^{-2}$
and $\kappa\asymp1$. By the standard results of the sub-Gaussian
random variables, one also has
\begin{align}
\left\Vert \bm{E}\right\Vert _{\infty} & \lesssim\sigma_{\max}\sqrt{\log d}\label{eq:noise-inf-UB}
\end{align}
with probability at least $1-O\left(d^{-20}\right)$. This reveals
that
\begin{equation}
\max_{(i,j,k)\in\Omega}\big|\widehat{E}_{i,j,k}^{2}-E_{i,j,k}^{2}\big|\leq\max_{(i,j,k)\in\Omega}\big|(\widehat{E}_{i,j,k}-E_{i,j,k})(\widehat{E}_{i,j,k}+E_{i,j,k})\big|\lesssim\sigma_{\max}^{2}\sqrt{\frac{\mu^{3}r^{2}\log^{2}d}{d^{2}p}}.\label{eq:noise-square-est-loss-UB}
\end{equation}
In addition, apply the Bernstein inequality to find that with probability
at least $1-O\left(d^{-20}\right)$,
\begin{align}
\sum_{1\leq i,j\leq d}p^{-1}\chi_{i,j,k}V_{(i,j),l}^{\star2} & \lesssim\big\|\bm{V}_{:,l}^{\star}\big\|_{2}^{2}+p^{-1}\log d\,\big\|\bm{V}_{:,l}^{\star}\big\|_{\infty}^{2}+\sqrt{p^{-1}\log d}\big\|\bm{V}_{:,l}^{\star}\big\|_{\infty}^{2}\big\|\bm{V}_{:,l}^{\star}\big\|_{2}^{2}\nonumber \\
 & \overset{(\mathrm{i})}{\asymp}\big\|\bm{V}_{:,l}^{\star}\big\|_{2}^{2}+p^{-1}\log d\,\big\|\bm{V}_{:,l}^{\star}\big\|_{\infty}^{2}\overset{(\mathrm{ii})}{\lesssim}\frac{1}{\lambda_{\min}^{\star4/3}}\left\{ 1+\frac{\mu^{2}r\log d}{d^{2}p}\right\} \overset{(\mathrm{iii})}{\asymp}\frac{1}{\lambda_{\min}^{\star4/3}},\label{eq:V-col-concentration}
\end{align}
where (i) follows from the AM-GM inequality, (ii) makes use of (\ref{eq:V-true-col-norm}),
and (iii) holds true as long as $p\gg\mu^{2}rd^{-2}\log d$. Combining
the above bounds, we arrive at
\begin{align*}
\left|\gamma_{1}\right| & \lesssim\frac{\sigma_{\max}^{2}}{\lambda_{\min}^{\star4/3}}\sqrt{\frac{\mu^{3}r^{2}\log^{2}d}{d^{2}p}}
\end{align*}
with probability at least $1-O\left(d^{-20}\right)$.
\item Turning to $\gamma_{2}$, one can use the triangle inequality and
Cauchy-Schwartz to obtain
\begin{align*}
\left|\gamma_{2}\right| & =\Big|\sum_{1\leq i,j\leq d}\big(\widehat{E}_{i,j,k}^{2}-E_{i,j,k}^{2}\big)p^{-1}\chi_{i,j,k}V_{(i,j),l}^{\star}\big(V_{(i,j),l}-V_{(i,j),l}^{\star}\big)\Big|\\
 & \leq\max_{(i,j,k)\in\Omega}\big|\widehat{E}_{i,j,k}^{2}-E_{i,j,k}^{2}\big|\left\Vert \bm{V}_{:,l}-\bm{V}_{:,l}^{\star}\right\Vert _{\infty}\sum_{1\leq i,j\leq d}\Big|\,p^{-1}\chi_{i,j,k}V_{(i,j),l}^{\star}\Big|\\
 & \leq\max_{(i,j,k)\in\Omega}\big|\widehat{E}_{i,j,k}^{2}-E_{i,j,k}^{2}\big|\left\Vert \bm{V}_{:,l}-\bm{V}_{:,l}^{\star}\right\Vert _{\infty}\sqrt{\sum_{1\leq i,j\leq d}p^{-1}\chi_{i,j,k}V_{(i,j),l}^{\star2}}\sqrt{\sum_{1\leq i,j\leq d}p^{-1}\chi_{i,j,k}}.
\end{align*}
It is straightforward to apply the Bernstein inequality to find that,
with probability exceeding $1-O\left(d^{-12}\right)$,
\begin{equation}
\sum_{1\leq i,j\leq d}p^{-1}\chi_{i,j,k}\lesssim d^{2}+p^{-1}\log d+\sqrt{d^{2}p^{-1}\log d}\asymp d^{2},\label{eq:obs-card-concentration}
\end{equation}
with the proviso that $p\gg d^{-2}\log d$. Taking this with (\ref{eq:V-col-loss}),
(\ref{eq:noise-square-est-loss-UB}) and (\ref{eq:V-col-concentration})
collectively, we conclude that
\begin{align*}
\left|\gamma_{2}\right| & \lesssim\sigma_{\max}^{2}\sqrt{\frac{\mu^{3}r^{2}\log^{2}d}{d^{2}p}}\cdot\frac{\sigma_{\max}}{\lambda_{\min}^{\star}}\sqrt{\frac{\mu^{2}r\log d}{dp}}\,\frac{1}{\lambda_{\min}^{\star2/3}}\cdot\frac{1}{\lambda_{\min}^{\star2/3}}\cdot d=\frac{\sigma_{\max}^{2}}{\lambda_{\min}^{\star4/3}}\frac{\sigma_{\max}}{\lambda_{\min}^{\star}}\sqrt{\frac{\mu^{5}r^{3}\log^{3}d}{dp^{2}}}.
\end{align*}
\item Regarding $\gamma_{3}$, we can develop an upper bound in an analogous
manner:
\begin{align*}
\left|\gamma_{3}\right| & =\Big|\sum_{1\leq i,j\leq d}\big(\widehat{E}_{i,j,k}^{2}-E_{i,j,k}^{2}\big)p^{-1}\chi_{i,j,k}\big(V_{(i,j),l}-V_{(i,j),l}^{\star}\big)^{2}\Big|\\
 & \leq\max_{(i,j,k)\in\Omega}\big|\widehat{E}_{i,j,k}^{2}-E_{i,j,k}^{2}\big|\left\Vert \bm{V}_{:,l}-\bm{V}_{:,l}^{\star}\right\Vert _{\infty}^{2}\sum_{1\leq i,j\leq d}p^{-1}\chi_{i,j,k}\\
 & \overset{(\mathrm{i})}{\lesssim}\sigma_{\max}^{2}\sqrt{\frac{\mu^{3}r^{2}\log^{2}d}{d^{2}p}}\left(\frac{\sigma_{\max}}{\lambda_{\min}^{\star}}\sqrt{\frac{\mu^{2}r\log d}{dp}}\,\frac{1}{\lambda_{\min}^{\star2/3}}\right)^{2}d^{2}\\
 & \overset{(\mathrm{ii})}{\ll}\frac{\sigma_{\max}^{2}}{\lambda_{\min}^{\star4/3}}\frac{\sigma_{\max}}{\lambda_{\min}^{\star}}\sqrt{\frac{\mu^{5}r^{3}\log^{3}d}{dp^{2}}}.
\end{align*}
Here, (i) uses (\ref{eq:V-col-loss}), (\ref{eq:noise-square-est-loss-UB})
and (\ref{eq:obs-card-concentration}), whereas (ii) holds as long
as $\sigma_{\max}/\lambda_{\min}^{\star}\ll\sqrt{p/(\mu^{2}rd\log d)}$.
\item Taking these bounds together, we demonstrate that with probability
at least $1-O\left(d^{-12}\right)$,
\begin{align*}
\left|\beta_{1}\right| & \lesssim\frac{\sigma_{\max}^{2}}{\lambda_{\min}^{\star4/3}}\left\{ \sqrt{\frac{\mu^{3}r^{2}\log^{2}d}{d^{2}p}}+\frac{\sigma_{\max}}{\lambda_{\min}^{\star}}\sqrt{\frac{\mu^{5}r^{3}\log^{3}d}{dp^{2}}}\right\} \asymp\frac{\sigma_{\max}^{2}}{\lambda_{\min}^{\star4/3}}\sqrt{\frac{\mu^{3}r^{2}\log^{2}d}{d^{2}p}},
\end{align*}
where the last step relies on the noise condition $\sigma_{\max}/\lambda_{\min}^{\star}\ll\sqrt{p/(\mu^{2}rd^{3/2}\log d)}$.
\end{itemize}

\subsubsection{Bounding $\beta_{2}$}

Next, we move on to the term $\beta_{2}$ defined in (\ref{eq:u-entry-var-est-loss-decomp}),
which admits the following decomposition
\begin{align*}
\beta_{2} & =(\bm{V}_{:,l})^{\top}\widehat{\bm{D}}_{k}\bm{V}_{:,l}-(\bm{V}_{:,l}^{\star})^{\top}\widehat{\bm{D}}_{k}\bm{V}_{:,l}^{\star}+(\bm{V}_{:,l}^{\star})^{\top}\widehat{\bm{D}}_{k}\bm{V}_{:,l}^{\star}-(\bm{V}_{:,l}^{\star})^{\top}\bm{D}_{k}^{\star}\bm{V}_{:,l}^{\star}\\
 & =2\underbrace{(\bm{V}_{:,l}^{\star})^{\top}\widehat{\bm{D}}_{k}(\bm{V}_{:,l}-\bm{V}_{:,l}^{\star})}_{=:\,\gamma_{4}}+\underbrace{(\bm{V}_{:,l}-\bm{V}_{:,l}^{\star})^{\top}\widehat{\bm{D}}_{k}(\bm{V}_{:,l}-\bm{V}_{:,l}^{\star})}_{=:\,\gamma_{5}}+\underbrace{(\bm{V}_{:,l}^{\star})^{\top}\big(\widehat{\bm{D}}_{k}-\bm{D}_{k}^{\star}\big)\bm{V}_{:,l}^{\star}}_{=:\,\gamma_{6}}.
\end{align*}
In the sequel, we shall upper bound each of these terms individually.
\begin{itemize}
\item To begin with, invoke the Cauchy-Schwartz inequality to bound
\begin{align*}
\left|\gamma_{4}\right| & =\Big|\sum_{1\leq i,j\leq d}E_{i,j,k}^{2}p^{-1}\chi_{i,j,k}V_{(i,j),l}^{\star}\big(V_{(i,j),l}-V_{(i,j),l}^{\star}\big)\Big|\\
 & \leq\left\Vert \bm{V}_{:,l}-\bm{V}_{:,l}^{\star}\right\Vert _{\infty}\sqrt{\sum_{1\leq i,j\leq d}E_{i,j,k}^{4}p^{-1}\chi_{i,j,k}V_{(i,j),l}^{\star2}}\sqrt{\sum_{1\leq i,j\leq d}p^{-1}\chi_{i,j,k}}.
\end{align*}
Applying the Bernstein inequality yields that, with probability at
least $1-O\left(d^{-13}\right)$,
\begin{align*}
\sum_{1\leq i,j\leq d}E_{i,j,k}^{4}p^{-1}\chi_{i,j,k}V_{(i,j),l}^{\star2} & \lesssim\sigma_{\max}^{4}\left\{ \big\|\bm{V}_{:,l}^{\star}\big\|_{2}^{2}+p^{-1}\log^{2}d\,\big\|\bm{V}_{:,l}^{\star}\big\|_{\infty}^{2}+\sqrt{p^{-1}\log d}\,\big\|\bm{V}_{:,l}^{\star}\big\|_{\infty}\big\|\bm{V}_{:,l}^{\star}\big\|_{2}\right\} \\
 & \overset{(\mathrm{i})}{\asymp}\sigma_{\max}^{4}\left\{ \big\|\bm{V}_{:,l}^{\star}\big\|_{2}^{2}+p^{-1}\log^{2}d\,\big\|\bm{V}_{:,l}^{\star}\big\|_{\infty}^{2}\right\} \\
 & \overset{(\mathrm{ii})}{\lesssim}\frac{\sigma_{\max}^{4}}{\lambda_{\min}^{\star4/3}}\left\{ 1+\frac{\mu^{2}r\log^{2}d}{d^{2}p}\right\} \overset{(\mathrm{iii})}{\asymp}\frac{\sigma_{\max}^{4}}{\lambda_{\min}^{\star4/3}},
\end{align*}
where (i) is due to the AM-GM inequality, (ii) uses (\ref{eq:V-true-col-norm}),
and (iii) holds as long as $p\gtrsim\mu^{2}rd^{-2}\log^{2}d$. This
combined with (\ref{eq:V-true-col-norm}) and (\ref{eq:obs-card-concentration})
further leads to
\[
\left|\gamma_{4}\right|\lesssim\frac{\sigma_{\max}}{\lambda_{\min}^{\star}}\sqrt{\frac{\mu^{2}r\log d}{dp}}\,\frac{1}{\lambda_{\min}^{\star2/3}}\cdot d\cdot\frac{\sigma_{\max}^{2}}{\lambda_{\min}^{\star2/3}}=\frac{\sigma_{\max}^{2}}{\lambda_{\min}^{\star4/3}}\frac{\sigma_{\max}}{\lambda_{\min}^{\star}}\sqrt{\frac{\mu^{2}rd\log d}{p}}.
\]
\item Next, we turn to the term $\gamma_{5}$. By (\ref{eq:V-col-loss}),
(\ref{eq:noise-inf-UB}) and (\ref{eq:obs-card-concentration}), the
following holds with probability at least $1-O\left(d^{-13}\right)$,
\begin{align*}
\gamma_{5} & =\sum_{1\leq i,j\leq d}E_{i,j,k}^{2}p^{-1}\chi_{i,j,k}\big(V_{(i,j),l}-V_{(i,j),l}^{\star}\big)^{2}\leq\left\Vert \bm{E}\right\Vert _{\infty}^{2}\left\Vert \bm{V}_{:,l}-\bm{V}_{:,l}^{\star}\right\Vert _{\infty}^{2}\sum_{1\leq i,j\leq d}p^{-1}\chi_{i,j,k}\\
 & \lesssim\sigma_{\max}^{2}\log d\cdot\left(\frac{\sigma_{\max}}{\lambda_{\min}^{\star}}\sqrt{\frac{\mu^{2}r\log d}{dp}}\,\frac{1}{\lambda_{\min}^{\star2/3}}\right)^{2}\cdot d^{2}\ll\frac{\sigma_{\max}^{2}}{\lambda_{\min}^{\star4/3}}\frac{\sigma_{\max}}{\lambda_{\min}^{\star}}\sqrt{\frac{\mu^{2}rd\log d}{p}},
\end{align*}
where the last step holds as long as $\sigma_{\max}/\lambda_{\min}^{\star}\ll\sqrt{p/(\mu^{2}rd\log^{3}d)}$.
\item As for $\gamma_{6}$, we can express $\gamma_{6}=\sum_{1\leq i,j\leq d}V_{(i,j),l}^{\star2}(p^{-1}E_{i,j,k}^{2}\chi_{i,j,k}-\sigma_{i,j,k}^{2})$
as a sum of independent zero-mean random variables. With the assistance
of (\ref{eq:V-true-col-norm}), one derives
\begin{align*}
B & :=\max_{1\leq i,j\leq d}\big\| V_{(i,j),l}^{\star2}(p^{-1}E_{i,j,k}^{2}\chi_{i,j,k}-\sigma_{i,j,k}^{2})\big\|_{\psi_{1}}\leq\frac{\sigma_{\max}^{2}}{p}\big\|\bm{V}_{:,l}^{\star}\big\|_{\infty}^{2}\lesssim\frac{\sigma_{\max}^{2}}{\lambda_{\min}^{\star4/3}}\frac{\mu^{2}r}{d^{2}p}.\\
V & :=\sum_{1\leq i,j\leq d}V_{(i,j),l}^{\star4}\mathbb{E}\big[(p^{-1}E_{i,j,k}^{2}\chi_{i,j,k}-\sigma_{i,j,k}^{2})^{2}\big]\lesssim\frac{\sigma_{\max}^{4}}{p}\big\|\bm{V}_{:,l}^{\star}\big\|_{\infty}^{2}\big\|\bm{V}_{:,l}^{\star}\big\|_{2}^{2}\lesssim\frac{\sigma_{\max}^{4}}{\lambda_{\min}^{\star8/3}}\frac{\mu^{2}r}{d^{2}p}.
\end{align*}
Applying the matrix Bernstein inequality, one has with probability
at least $1-O\left(d^{-20}\right)$,
\begin{align*}
\left|\gamma_{6}\right| & \lesssim B\log^{2}d+\sqrt{V\log d}\lesssim\frac{\sigma_{\max}^{2}}{\lambda_{\min}^{\star4/3}}\left\{ \frac{\mu^{2}r\log^{2}d}{d^{2}p}+\sqrt{\frac{\mu^{2}r\log d}{d^{2}p}}\right\} \asymp\frac{\sigma_{\max}^{2}}{\lambda_{\min}^{\star4/3}}\sqrt{\frac{\mu^{2}r\log d}{d^{2}p}},
\end{align*}
where the last step holds as long as $p\gtrsim\mu^{2}rd^{-2}\log^{3}d$.
\item Putting the bounds above together, we reach
\[
\left|\beta_{2}\right|\lesssim\frac{\sigma_{\max}^{2}}{\lambda_{\min}^{\star4/3}}\left\{ \sqrt{\frac{\mu^{2}r\log d}{d^{2}p}}+\frac{\sigma_{\max}}{\lambda_{\min}^{\star}}\sqrt{\frac{\mu^{2}rd\log d}{p}}\right\} .
\]
\end{itemize}

\subsubsection{Combining $\beta_{1}$ and $\beta_{2}$ to establish the claim (\ref{claim:u-entry-var-loss})}

Taking the bounds on $\beta_{1}$ and $\beta_{2}$ collectively yields
that, with probability exceeding $1-O\left(d^{-10}\right)$,
\begin{align*}
\left|s_{l,k}^{2}-s_{l,k}^{\star2}\right| & \lesssim\frac{\sigma_{\max}^{2}}{\lambda_{\min}^{\star4/3}p}\Bigg\{\sqrt{\frac{\mu^{3}r^{2}\log^{2}d}{d^{2}p}}+\frac{\sigma_{\max}}{\lambda_{\min}^{\star}}\sqrt{\frac{\mu^{2}rd\log d}{p}}\Bigg\}\lesssim\Bigg\{\sqrt{\frac{\mu^{3}r^{2}\log^{2}d}{d^{2}p}}+\frac{\sigma_{\max}}{\lambda_{\min}^{\star}}\sqrt{\frac{\mu^{2}rd\log d}{p}}\Bigg\} s_{l,k}^{\star2},
\end{align*}
where we have used the lower bound on $s_{l,k}^{\star2}$ (cf.~(\ref{eq:u-entry-var-LB}))
as well as the conditions $\sigma_{\max}/\sigma_{\min}\asymp1$ and
$\kappa\asymp1$ in the last step.

\subsubsection{Proof of Lemma \ref{lemma:V-property}}
\begin{enumerate}
\item We first consider the $\ell_{2}$ norm of $\bm{V}_{:,l}^{\star}$.
Let $\bm{\Lambda}^{\star}\in\mathbb{R}^{r\times r}$ be a diagonal
matrix with entries $\Lambda_{i,i}^{\star}=\left\Vert \bm{u}_{i}^{\star}\right\Vert _{2}^{3}$
for all $1\leq i\leq r$. We can then decompose
\begin{align*}
\bm{V}_{:,l}^{\star} & =\big(\widetilde{\bm{U}}^{\star}\bm{\Lambda}^{\star-4/3}\big)_{:,l}+\widetilde{\bm{U}}^{\star}\big((\widetilde{\bm{U}}^{\star\top}\widetilde{\bm{U}}^{\star})^{-1}-\bm{\Lambda}^{\star-4/3}\big)_{:,l}=\left\Vert \bm{u}_{l}^{\star}\right\Vert _{2}^{-4}\bm{u}_{l}^{\star\otimes2}+\widetilde{\bm{U}}^{\star}\big((\widetilde{\bm{U}}^{\star\top}\widetilde{\bm{U}}^{\star})^{-1}-\bm{\Lambda}^{\star-4/3}\big)_{:,l}.
\end{align*}
One can use the assumption (\ref{assumption:u-inf-norm}), as well
as the conditions (\ref{eq:U-true-tilde-spectrum}) and $\kappa\asymp1$,
to bound the second term
\begin{align}
\big\|\widetilde{\bm{U}}^{\star}\big((\widetilde{\bm{U}}^{\star\top}\widetilde{\bm{U}}^{\star})^{-1}-\bm{\Lambda}^{\star-4/3}\big)_{:,l}\big\|_{2} & \leq\big\|\widetilde{\bm{U}}^{\star}\big\|\big\|(\widetilde{\bm{U}}^{\star\top}\widetilde{\bm{U}}^{\star})^{-1}\big\|\big\|\widetilde{\bm{U}}^{\star\top}\widetilde{\bm{U}}^{\star}-\bm{\Lambda}^{\star4/3}\big\|\big\|\bm{\Lambda}^{\star-4/3}\big\|\nonumber \\
 & \lesssim\lambda_{\max}^{\star2/3}\cdot\frac{1}{\lambda_{\min}^{\star4/3}}\cdot r\max_{i\neq j}\left|\left\langle \bm{u}_{i}^{\star},\bm{u}_{j}^{\star}\right\rangle \right|^{2}\cdot\frac{1}{\lambda_{\min}^{\star4/3}}\lesssim\frac{1}{\lambda_{\min}^{\star2/3}}\frac{\mu r}{d}=\frac{o\left(1\right)}{\left\Vert \bm{u}_{l}^{\star}\right\Vert _{2}^{2}},\label{eq:U-true-tilde-Gram-Ind-diff}
\end{align}
where the last step arises from the condition $r=o\left(d/\mu\right)$.
Therefore, we obtain that $\big\|\bm{V}_{:,l}^{\star}\big\|_{2}=\left(1+o\left(1\right)\right)\left\Vert \bm{u}_{l}^{\star}\right\Vert _{2}^{-2}.$
\item Regarding the $\ell_{\infty}$ norm of $\bm{V}_{:,l}^{\star}$, we
can use (\ref{eq:U-true-tilde-norm}) and (\ref{eq:U-true-tilde-spectrum})
to upper bound
\[
\big\|\bm{V}_{:,l}^{\star}\big\|_{\infty}\leq\big\|\widetilde{\bm{U}}^{\star}\big\|_{2,\infty}\big\|(\widetilde{\bm{U}}^{\star\top}\widetilde{\bm{U}}^{\star})^{-1}\big\|\lesssim\frac{\mu\sqrt{r}\,\lambda_{\max}^{\star2/3}}{d}\frac{1}{\lambda_{\min}^{\star4/3}}\lesssim\frac{\mu\sqrt{r}}{d}\frac{1}{\lambda_{\min}^{\star2/3}}.
\]
\item Moreover, for any $1\leq i\leq d$, one can apply (\ref{eq:U-true-tilde-norm})
and (\ref{eq:U-true-tilde-spectrum}) again to demonstrate that
\begin{align*}
\sum_{1\leq k\leq d}V_{(i,k),l}^{\star2} & =\sum_{1\leq k\leq d}\big(\widetilde{\bm{U}}_{(i,k),:}^{\star}(\widetilde{\bm{U}}^{\star\top}\widetilde{\bm{U}}^{\star})_{:,l}^{-1}\big)^{2}\leq\sum_{1\leq k\leq d}\big\|\widetilde{\bm{U}}_{(i,k),:}^{\star}\big\|_{2}^{2}\big\|(\widetilde{\bm{U}}^{\star\top}\widetilde{\bm{U}}^{\star})^{-1}\big\|^{2}\\
 & \leq\frac{1}{\lambda_{\min}^{\star8/3}}\sum_{1\leq k\leq d}\sum_{1\leq s\leq r}\left(\bm{u}_{s}^{\star}\right)_{i}^{2}\left(\bm{u}_{s}^{\star}\right)_{k}^{2}\leq\frac{1}{\lambda_{\min}^{\star8/3}}\left\Vert \bm{U}^{\star}\right\Vert _{2,\infty}^{2}\max_{1\leq s\leq r}\left\Vert \bm{u}_{s}^{\star}\right\Vert _{2}^{2}\\
 & \lesssim\frac{1}{\lambda_{\min}^{\star8/3}}\cdot\lambda_{\max}^{\star2/3}\cdot\frac{\mu r}{d}\lambda_{\max}^{\star2/3}\lesssim\frac{\mu r}{d}\frac{1}{\lambda_{\min}^{\star4/3}},
\end{align*}
where the last line holds due to $\kappa\asymp1$.
\item Regarding the $\ell_{2}$ loss, invoke (\ref{eq:U-tilde-spectrum}),
(\ref{eq:U-tilde-op-loss}) and (\ref{eq:U-tilde-Gram-inv-op-loss})
to upper bound
\begin{align*}
\left\Vert (\bm{V}_{:,l}-\bm{V}_{:,l}^{\star})\right\Vert _{2} & \leq\big\|\widetilde{\bm{U}}(\widetilde{\bm{U}}^{\top}\widetilde{\bm{U}})^{-1}-\widetilde{\bm{U}}^{\star}(\widetilde{\bm{U}}^{\star\top}\widetilde{\bm{U}}^{\star})^{-1}\big\|\\
 & \leq\big\|\widetilde{\bm{U}}-\widetilde{\bm{U}}^{\star}\big\|\big\|(\widetilde{\bm{U}}^{\top}\widetilde{\bm{U}})^{-1}\big\|+\big\|\widetilde{\bm{U}}^{\star}\big\|\big\|(\widetilde{\bm{U}}^{\top}\widetilde{\bm{U}})^{-1}-(\widetilde{\bm{U}}^{\star\top}\widetilde{\bm{U}}^{\star})^{-1}\big\|\\
 & \lesssim\frac{\sigma_{\max}}{\lambda_{\min}^{\star}}\sqrt{\frac{d}{p}}\,\lambda_{\max}^{\star2/3}\cdot\frac{1}{\lambda_{\min}^{\star4/3}}+\lambda_{\max}^{\star2/3}\cdot\frac{\sigma_{\max}}{\lambda_{\min}^{\star3}}\sqrt{\frac{d}{p}}\,\lambda_{\max}^{\star2/3}\lesssim\frac{\sigma_{\max}}{\lambda_{\min}^{\star}}\sqrt{\frac{d}{p}}\,\frac{1}{\lambda_{\min}^{\star2/3}},
\end{align*}
which holds as long as $\kappa\asymp1$.
\item Finally, combining (\ref{eq:U-tilde-loss-2inf}), (\ref{eq:U-tilde-spectrum}),
(\ref{eq:U-true-tilde-norm}) and (\ref{eq:U-tilde-Gram-inv-op-loss})
allows us to upper bound the $\ell_{\infty}$ loss by
\begin{align*}
\left\Vert (\bm{V}_{:,l}-\bm{V}_{:,l}^{\star})\right\Vert _{\infty} & \leq\big\|\widetilde{\bm{U}}-\widetilde{\bm{U}}^{\star}\big\|_{2,\infty}\big\|(\widetilde{\bm{U}}^{\top}\widetilde{\bm{U}})^{-1}\big\|+\big\|\widetilde{\bm{U}}^{\star}\big\|_{2,\infty}\big\|(\widetilde{\bm{U}}^{\top}\widetilde{\bm{U}})^{-1}-(\widetilde{\bm{U}}^{\star\top}\widetilde{\bm{U}}^{\star})^{-1}\big\|\\
 & \lesssim\frac{\sigma_{\max}}{\lambda_{\min}^{\star}}\sqrt{\frac{\mu^{2}r\log d}{dp}}\,\lambda_{\max}^{\star2/3}\cdot\frac{1}{\lambda_{\min}^{\star4/3}}+\frac{\mu\sqrt{r}}{d}\lambda_{\max}^{\star2/3}\cdot\frac{\sigma_{\max}}{\lambda_{\min}^{\star3}}\sqrt{\frac{d}{p}}\,\lambda_{\max}^{\star2/3}\\
 & \asymp\frac{\sigma_{\max}}{\lambda_{\min}^{\star}}\sqrt{\frac{\mu^{2}r\log d}{dp}}\,\frac{1}{\lambda_{\min}^{\star2/3}},
\end{align*}
where the last step arises from $\kappa\asymp1$.
\end{enumerate}

\subsection{Proof of Lemma \ref{lemma:T-entry-var-est-loss-neg}}

\label{subsec:T-entry-var-est-loss-neg}

Fix any $1\leq i\leq j\leq k\leq d$. In order to control $K_{i,j,k}$,
we will apply an almost identical argument as in Appendix~\ref{subsec:u-entry-var-est-loss-neg}
for Lemma~\ref{lemma:u-entry-var-est-loss-neg}. We omit some details
of proof for the sake of conciseness.

By definition, one can express
\[
K_{i,j,k}=\frac{T_{i,j,k}-T_{i,j,k}^{\star}}{v_{i,j,k}^{1/2}}-\frac{T_{i,j,k}-T_{i,j,k}^{\star}}{v_{i,j,k}^{\star1/2}}=\left(T_{i,j,k}-T_{i,j,k}^{\star}\right)\frac{v_{i,j,k}^{\star}-v_{i,j,k}}{\sqrt{v_{i,j,k}v_{i,j,k}^{\star}}}\frac{1}{\sqrt{v_{i,j,k}^{\star}}+\sqrt{v_{i,j,k}}}.
\]
Recall the definitions $\bm{\Delta}:=\bm{U}\bm{\Pi}-\bm{U}^{\star}$
and $\widetilde{\bm{\Delta}}=\big[\bm{\Delta}_{l}^{\otimes2}\big]_{1\leq l\leq r}\in\mathbb{R}^{d^{2}\times r}$.
In view of the decomposition in (\ref{eq:T-entry-loss}), it is straightforward
to bound
\begin{align*}
\big|T_{i,j,k}-T_{i,j,k}^{\star}\big| & \lesssim\left\Vert \bm{\Delta}\right\Vert _{2,\infty}\Big(\big\|\widetilde{\bm{U}}_{(i,j),:}^{\star}\big\|_{2}+\big\|\widetilde{\bm{U}}_{(i,k),:}^{\star}\big\|_{2}+\big\|\widetilde{\bm{U}}_{(j,k),:}^{\star}\big\|_{2}\Big)\\
 & \quad+\Big|\big\langle\bm{U}_{i,:}^{\star},\widetilde{\bm{\Delta}}_{(j,k),:}\big\rangle+\big\langle\bm{U}_{j,:}^{\star},\widetilde{\bm{\Delta}}_{(i,k),:}\big\rangle+\big\langle\bm{U}_{k,:}^{\star},\widetilde{\bm{\Delta}}_{(i,k),:}\big\rangle+\big\langle\bm{\Delta}_{i,:},\widetilde{\bm{\Delta}}_{(j,k),:}\big\rangle\Big|\\
 & \overset{(\mathrm{i})}{\lesssim}\frac{\sigma_{\max}}{\lambda_{\min}^{\star}}\sqrt{\frac{\mu r\log d}{p}}\,\lambda_{\max}^{\star1/3}\Big(\big\|\widetilde{\bm{U}}_{(i,j),:}^{\star}\big\|_{2}+\big\|\widetilde{\bm{U}}_{(i,k),:}^{\star}\big\|_{2}+\big\|\widetilde{\bm{U}}_{(j,k),:}^{\star}\big\|_{2}\Big)\\
 & \quad+\left(\frac{\sigma_{\max}}{\lambda_{\min}^{\star}}\sqrt{\frac{\mu r\log d}{p}}\lambda_{\max}^{\star1/3}\right)^{2}\sqrt{\frac{\mu}{d}}\,\lambda_{\max}^{\star1/3}\\
 & \overset{(\mathrm{ii})}{\lesssim}\left\{ \sqrt{\mu r\log d}+\frac{\sigma_{\max}}{\lambda_{\min}^{\star1/3}}\frac{\mu^{3/2}r\log d}{\sqrt{dp}}\Big(\big\|\widetilde{\bm{U}}_{(n,l),:}^{\star}\big\|_{2}+\big\|\widetilde{\bm{U}}_{(m,l),:}^{\star}\big\|_{2}+\big\|\widetilde{\bm{U}}_{(m,n),:}^{\star}\big\|_{2}\Big)^{-1}\right\} \sqrt{v_{i,j,k}^{\star}}\\
 & \overset{(\mathrm{iii})}{\lesssim}\sqrt{\mu r\log d}\sqrt{v_{i,j,k}^{\star}},
\end{align*}
where (i) uses (\ref{eq:U-loss-2inf}), (\ref{eq:T-loss-dist-neg-part-part2})
and (\ref{eq:T-loss-dist-neg-part-part3}); (ii) follows from the
lower bound of $v_{i,j,k}^{\star}$ in (\ref{eq:T-entry-var-LB})
and conditions $\sigma_{\max}/\sigma_{\min},\kappa\asymp1$; and (iii)
arises from the assumption (\ref{eq:U-tilde-2norm-LB-rank-r}).

Then the claim (\ref{claim:T-entry-var-loss}) would immediately follow
as long as we could show that
\begin{align}
\frac{\big|v_{i,j,k}-v_{i,j,k}^{\star}\big|}{v_{i,j,k}^{\star}} & \lesssim\sqrt{\frac{\mu^{3}r^{2}\log d}{d^{2}p}}+\Big(\big\|\widetilde{\bm{U}}_{(i,j),:}^{\star}\big\|_{2}+\big\|\widetilde{\bm{U}}_{(i,k),:}^{\star}\big\|_{2}+\big\|\widetilde{\bm{U}}_{(j,k),:}^{\star}\big\|_{2}\Big)^{-1}\frac{\sigma_{\max}}{\lambda_{\min}^{\star1/3}}\sqrt{\frac{\mu^{4}r^{2}\log d}{dp}}=o\left(1\right).\label{claim:T-entry-var-loss}
\end{align}
Indeed, one can apply the triangle inequality to show that $v_{i,j,k}\asymp v_{i,j,k}^{\star}$,
and consequently obtain
\begin{align*}
\left|K_{i,j,k}\right| & \lesssim\frac{1}{v_{i,j,k}^{\star3/2}}\big|T_{i,j,k}-T_{i,j,k}^{\star}\big|\big|v_{i,j,k}-v_{i,j,k}^{\star}\big|\\
 & \lesssim\sqrt{\frac{\mu^{4}r^{3}\log^{2}d}{d^{2}p}}+\Big(\big\|\widetilde{\bm{U}}_{(i,j),:}^{\star}\big\|_{2}+\big\|\widetilde{\bm{U}}_{(i,k),:}^{\star}\big\|_{2}+\big\|\widetilde{\bm{U}}_{(j,k),:}^{\star}\big\|_{2}\Big)^{-1}\frac{\sigma_{\max}}{\lambda_{\min}^{\star1/3}}\sqrt{\frac{\mu^{5}r^{3}\log^{2}d}{dp}}
\end{align*}
as claimed.

Therefore, it remains to justify (\ref{claim:T-entry-var-loss}).
For notational convenience, we define the following $d^{2}\times d^{2}$
matrices:
\begin{align}
\bm{P} & :=\widetilde{\bm{U}}(\widetilde{\bm{U}}^{\top}\widetilde{\bm{U}})^{-1}\widetilde{\bm{U}}^{\top},\qquad\bm{P}^{\star}:=\widetilde{\bm{U}}^{\star}(\widetilde{\bm{U}}^{\star\top}\widetilde{\bm{U}}^{\star})^{-1}\widetilde{\bm{U}}^{\star\top}.\label{def:P}
\end{align}
We can then express
\begin{align}
v_{i,j,k}^{\star} & =\frac{2}{p}\,\Big(\bm{P}_{(i,j),:}^{\star}\bm{D}_{k}^{\star}\bm{P}_{:,(i,j)}^{\star}+\bm{P}_{(i,k),:}^{\star}\bm{D}_{j}^{\star}\bm{P}_{:,(i,k)}^{\star}+\bm{P}_{(j,k),:}^{\star}\bm{D}_{i}^{\star}\bm{P}_{:,(j,k)}^{\star}\Big),\label{eq:T-entry-var-expression}\\
v_{i,j,k} & =\frac{2}{p}\,\Big(\bm{P}_{(i,j),:}\bm{D}_{k}\bm{P}_{:,(i,j)}+\bm{P}_{(i,k),:}\bm{D}_{j}\bm{P}_{:,(i,k)}+\bm{P}_{(j,k),:}\bm{D}_{i}\bm{P}_{:,(j,k)}\Big),\label{eq:T-entry-var-est-expression}
\end{align}
where $\bm{D}_{k}^{\star}$ (resp.~$\bm{D}_{k}$) is defined in (\ref{eq:cov-matrix-m-diag})
(resp.~(\ref{eq:defn-Dk})) for each $1\leq k\leq d$. Lemma~\ref{lemma:P-property}
summarizes several bounds regarding $\bm{P}$ and $\bm{P}^{\star}$,
whose proof can be found at the end of the section.

\begin{lemma}\label{lemma:P-property}Instate the assumptions and
notations of Lemma \ref{lemma:T-entry-var-est-loss-neg}. For any
$1\leq i,j\leq d$, one has
\begin{align}
 & \big\|\bm{P}_{(i,j),:}^{\star}\big\|_{2}\lesssim\frac{1}{\lambda_{\min}^{\star2/3}}\big\|\widetilde{\bm{U}}_{(i,j),:}^{\star}\big\|_{2},\qquad\big\|\bm{P}_{(i,j),:}^{\star}\big\|_{\infty}\lesssim\frac{\mu\sqrt{r}}{d}\frac{1}{\lambda_{\min}^{\star2/3}}\big\|\widetilde{\bm{U}}_{(i,j),:}^{\star}\big\|_{2};\label{eq:P-true-norm}\\
 & \big\|(\bm{P}-\bm{P}^{\star})_{(i,j),:}\big\|_{2}\lesssim\frac{\sigma_{\max}}{\lambda_{\min}^{\star}}\sqrt{\frac{d\log d}{p}}\frac{\mu\sqrt{r}}{d},\qquad\big\|(\bm{P}-\bm{P}^{\star})_{(i,j),:}\big\|_{\infty}\lesssim\frac{\sigma_{\max}}{\lambda_{\min}^{\star}}\sqrt{\frac{d\log d}{p}}\frac{\mu^{2}r}{d^{2}}.\label{eq:P-loss}
\end{align}
\end{lemma}With these in mind, we are positioned to upper bound $v_{i,j,k}-v_{i,j,k}^{\star}$.
By (\ref{eq:T-entry-var-expression}), (\ref{eq:T-entry-var-est-expression})
and the triangle inequality, we will show below how to upper bound
$\bm{P}_{(i,j),:}\bm{D}_{k}\bm{P}_{:,(i,j)}-\bm{P}_{(i,j),:}^{\star}\bm{D}_{k}^{\star}\bm{P}_{:,(i,j)}^{\star}$.
The other two terms can be controlled analogously.

Recall the auxiliary matrix $\widehat{\bm{D}}_{k}$ (cf.~\ref{def:D-hat}).
One can then expand
\begin{align*}
\bm{P}_{(i,j),:}\bm{D}_{k}\bm{P}_{:,(i,j)}-\bm{P}_{(i,j),:}^{\star}\bm{D}_{k}^{\star}\bm{P}_{:,(i,j)}^{\star} & =\underbrace{\bm{P}_{(i,j),:}(\bm{D}_{k}-\widehat{\bm{D}}_{k})\bm{P}_{:,(i,j)}}_{=:\,\beta_{1}}+\underbrace{\bm{P}_{(i,j),:}\widehat{\bm{D}}_{k}\bm{P}_{:,(i,j)}-\bm{P}_{(i,j),:}^{\star}\bm{D}_{k}^{\star}\bm{P}_{:,(i,j)}^{\star}}_{=:\,\beta_{2}}.
\end{align*}
In what follows, we shall control $\beta_{1}$ and $\beta_{2}$ individually.
\begin{itemize}
\item For $\beta_{1}$, one decomposes it as follows
\begin{align*}
\beta_{1} & =\underbrace{\bm{P}_{(i,j),:}^{\star}(\bm{D}_{k}-\widehat{\bm{D}}_{k})\bm{P}_{:,(i,j)}^{\star}}_{=:\,\gamma_{1}}+2\underbrace{(\bm{P}-\bm{P}^{\star})_{:,(i,j)}(\bm{D}_{k}-\widehat{\bm{D}}_{k})\bm{P}_{:,(i,j)}^{\star}}_{=:\,\gamma_{2}}\\
 & \quad+\underbrace{(\bm{P}-\bm{P}^{\star})_{:,(i,j)}(\bm{D}_{k}-\widehat{\bm{D}}_{k})(\bm{P}-\bm{P}^{\star})_{:,(i,j)}}_{=:\,\gamma_{3}}.
\end{align*}
The term $\gamma_{1}$ can be bounded by
\begin{align*}
\left|\gamma_{1}\right| & \leq\max_{(s,l,k)\in\Omega}\big|\widehat{E}_{s,l,k}^{2}-E_{s,l,k}^{2}\big|\sum_{1\leq s,l\leq d}p^{-1}\chi_{s,l,k}P_{(i,j),(s,l)}^{\star2}.
\end{align*}
Using (\ref{eq:P-true-norm}), we know from the Bernstein inequality
and the AM-GM inequality that with probability at least $1-O\left(d^{-13}\right)$,
\begin{align}
\sum_{1\leq s,l\leq d}p^{-1}\chi_{s,l,k}P_{(i,j),(s,l)}^{\star2} & \lesssim\big\|\bm{P}_{(i,j),:}^{\star}\big\|_{2}^{2}+p^{-1}\log d\,\big\|\bm{P}_{(i,j),:}^{\star}\big\|_{\infty}^{2}+\sqrt{p^{-1}\log d}\,\big\|\bm{P}_{(i,j),:}^{\star}\big\|_{\infty}\big\|\bm{P}_{(i,j),:}^{\star}\big\|_{2}\nonumber \\
 & \asymp\big\|\bm{P}_{(i,j),:}^{\star}\big\|_{2}^{2}+p^{-1}\log d\,\big\|\bm{P}_{(i,j),:}^{\star}\big\|_{\infty}^{2}\lesssim\frac{1}{\lambda_{\min}^{\star4/3}}\left\{ 1+\frac{\mu^{2}r\log d}{d^{2}p}\right\} \big\|\widetilde{\bm{U}}_{(i,j),:}^{\star}\big\|_{2}^{2}\nonumber \\
 & \asymp\frac{1}{\lambda_{\min}^{\star4/3}}\big\|\widetilde{\bm{U}}_{(i,j),:}^{\star}\big\|_{2}^{2},\label{eq:P-concentration}
\end{align}
where the last step arises from the condition $p\gtrsim\mu^{2}rd^{-2}\log d$.
This combined with (\ref{eq:noise-square-est-loss-UB}) leads to
\[
\left|\gamma_{1}\right|\lesssim\frac{\sigma_{\max}^{2}}{\lambda_{\min}^{\star4/3}}\sqrt{\frac{\mu^{3}r^{2}\log^{2}d}{d^{2}p}}\,\big\|\widetilde{\bm{U}}_{(i,j),:}^{\star}\big\|_{2}^{2}.
\]
As for $\gamma_{2}$, invoking Cauchy-Schwartz and applying (\ref{eq:noise-square-est-loss-UB})
and (\ref{eq:P-concentration}) give
\begin{align*}
\left|\gamma_{2}\right| & \leq\max_{(s,l,k)\in\Omega}\big|\widehat{E}_{s,l,k}^{2}-E_{s,l,k}^{2}\big|\left\Vert (\bm{P}-\bm{P}^{\star})_{(i,j),:}\right\Vert _{\infty}\sqrt{\sum_{1\leq s,l\leq d}p^{-1}\chi_{s,l,k}P_{(i,j),(s,l)}^{\star2}}\sqrt{\sum_{1\leq s,l\leq d}p^{-1}\chi_{s,l,k}}\\
 & \lesssim\sigma_{\max}^{2}\sqrt{\frac{\mu^{3}r^{2}\log^{2}d}{d^{2}p}}\cdot\frac{\sigma_{\max}}{\lambda_{\min}^{\star}}\sqrt{\frac{d\log d}{p}}\frac{\mu^{2}r}{d^{2}}\cdot\frac{1}{\lambda_{\min}^{\star2/3}}\big\|\widetilde{\bm{U}}_{(i,j),:}^{\star}\big\|_{2}\cdot d\\
 & \lesssim\frac{\sigma_{\max}^{2}}{\lambda_{\min}^{\star2/3}}\frac{\sigma_{\max}}{\lambda_{\min}^{\star}}\sqrt{\frac{d\log d}{p}}\frac{\mu^{2}r}{d}\big\|\widetilde{\bm{U}}_{(i,j),:}^{\star}\big\|_{2},
\end{align*}
as long as $p\gtrsim\mu^{3}r^{2}d^{-2}\log^{2}d$. Regarding $\gamma_{3}$,
we can upper bound
\begin{align*}
\left|\gamma_{3}\right| & \leq\max_{(s,l,k)\in\Omega}\big|\widehat{E}_{s,l,k}^{2}-E_{s,l,k}^{2}\big|\left\Vert (\bm{P}-\bm{P}^{\star})_{(i,j),:}\right\Vert _{\infty}^{2}\sum_{1\leq s,l\leq d}p^{-1}\chi_{s,l,k}\\
 & \overset{(\mathrm{i})}{\lesssim}\sigma_{\max}^{2}\sqrt{\frac{\mu^{3}r^{2}\log^{2}d}{d^{2}p}}\left(\frac{\sigma_{\max}}{\lambda_{\min}^{\star}}\sqrt{\frac{d\log d}{p}}\frac{\mu^{2}r}{d^{2}}\right)^{2}d^{2}\\
 & \overset{(\mathrm{ii})}{\lesssim}\sigma_{\max}^{2}\left(\frac{\sigma_{\max}}{\lambda_{\min}^{\star}}\sqrt{\frac{d\log d}{p}}\frac{\mu^{2}r}{d}\right)^{2},
\end{align*}
where (i) uses (\ref{eq:noise-square-est-loss-UB}), (\ref{eq:obs-card-concentration})
and (\ref{eq:P-loss}); (ii) holds as long as $p\gtrsim\mu^{3}r^{2}d^{-2}\log^{2}d$.
Taking the above bounds for $\gamma_{1},\gamma_{2}$ and $\gamma_{3}$
together indicates that
\[
\left|\beta_{1}\right|\lesssim\frac{\sigma_{\max}^{2}}{\lambda_{\min}^{\star4/3}}\sqrt{\frac{\mu^{3}r^{2}\log^{2}d}{d^{2}p}}\,\big\|\widetilde{\bm{U}}_{(i,j),:}^{\star}\big\|_{2}^{2}+\frac{\sigma_{\max}^{2}}{\lambda_{\min}^{\star2/3}}\frac{\sigma_{\max}}{\lambda_{\min}^{\star}}\sqrt{\frac{d\log d}{p}}\frac{\mu^{2}r}{d}\big\|\widetilde{\bm{U}}_{(i,j),:}^{\star}\big\|_{2}+\sigma_{\max}^{2}\left(\frac{\sigma_{\max}}{\lambda_{\min}^{\star}}\sqrt{\frac{d\log d}{p}}\frac{\mu^{2}r}{d}\right)^{2}.
\]
\item Regarding $\beta_{2}$, we start by decomposing it as follows
\begin{align*}
\beta_{2} & =\bm{P}_{(i,j),:}\widehat{\bm{D}}_{k}\bm{P}_{:,(i,j)}-\bm{P}_{(i,j),:}^{\star}\widehat{\bm{D}}_{k}\bm{P}_{:,(i,j)}^{\star}+\bm{P}_{(i,j),:}^{\star}\widehat{\bm{D}}_{k}\bm{P}_{:,(i,j)}^{\star}-\bm{P}_{(i,j),:}^{\star}\bm{D}_{k}^{\star}\bm{P}_{:,(i,j)}^{\star}\\
 & =2\underbrace{\bm{P}_{(i,j),:}^{\star}\widehat{\bm{D}}_{k}(\bm{P}-\bm{P}^{\star})_{:,(i,j)}}_{=:\,\gamma_{4}}+\underbrace{(\bm{P}-\bm{P}^{\star})_{(i,j),:}(\widehat{\bm{D}}_{k}-\bm{D}_{k}^{\star})(\bm{P}-\bm{P}^{\star})_{:,(i,j)}}_{=:\,\gamma_{5}}+\underbrace{\bm{P}_{(i,j),:}^{\star}(\widehat{\bm{D}}_{k}-\bm{D}_{k}^{\star})\bm{P}_{:,(i,j)}^{\star}}_{=:\,\gamma_{6}}.
\end{align*}
To bound $\gamma_{4}$, we can combine (\ref{eq:P-true-norm}), (\ref{eq:P-loss})
and (\ref{eq:obs-card-concentration}) with the Cauchy-Schwartz inequality
and the Bernstein inequality, to obtain with probability at least
$1-O\left(d^{-13}\right)$,
\begin{align*}
\left|\gamma_{4}\right| & =\Big|\sum_{1\leq s,l\leq d}E_{s,l,k}^{2}p^{-1}\chi_{s,l,k}P_{(i,j),(s,l)}^{\star}\big(P_{(i,j),(s,l)}-P_{(i,j),(s,l)}^{\star}\big)\Big|\\
 & \leq\left\Vert (\bm{P}-\bm{P}^{\star})_{(i,j),:}\right\Vert _{\infty}\sqrt{\sum_{1\leq s,l\leq d}E_{s,l,k}^{4}p^{-1}\chi_{s,l,k}P_{(i,j),(s,l)}^{\star2}}\sqrt{\sum_{1\leq s,l\leq d}p^{-1}\chi_{s,l,k}}.\\
 & \leq\left\Vert (\bm{P}-\bm{P}^{\star})_{(i,j),:}\right\Vert _{\infty}\cdot\sigma_{\max}^{2}\left\{ \big\|\bm{P}_{(i,j),:}^{\star}\big\|_{2}+p^{-1/2}\log d\,\big\|\bm{P}_{(i,j),:}^{\star}\big\|_{\infty}\right\} \cdot d\\
 & \lesssim\frac{\sigma_{\max}^{2}}{\lambda_{\min}^{\star2/3}}\frac{\sigma_{\max}}{\lambda_{\min}^{\star}}\sqrt{\frac{d\log d}{p}}\frac{\mu^{2}r}{d}\big\|\widetilde{\bm{U}}_{(i,j),:}^{\star}\big\|_{2}
\end{align*}
as long as $p\gtrsim\mu^{2}rd^{-2}\log^{2}d$. As for $\gamma_{5}$,
combining (\ref{eq:noise-inf-UB}), (\ref{eq:obs-card-concentration})
and (\ref{eq:P-loss}) shows that with probability at least $1-O\left(d^{-13}\right)$,
\begin{align*}
\gamma_{5} & =\sum_{1\leq s,l\leq d}E_{s,l,k}^{2}p^{-1}\chi_{s,l,k}\big(P_{(i,j),(s,l)}-P_{(i,j),(s,l)}^{\star}\big)^{2}\\
 & \leq\left\Vert \bm{E}\right\Vert _{\infty}^{2}\left\Vert (\bm{P}-\bm{P}^{\star})_{(i,j),:}\right\Vert _{\infty}^{2}\sum_{1\leq s,l\leq d}p^{-1}\chi_{s,l,k}\lesssim\sigma_{\max}^{2}\left(\frac{\sigma_{\max}}{\lambda_{\min}^{\star}}\sqrt{\frac{d\log^{2}d}{p}}\frac{\mu^{2}r}{d}\right)^{2}.
\end{align*}
Finally, observe that $\gamma_{6}$ is a sum of independent random
variables. By (\ref{eq:P-true-norm}), invoking the Bernstein inequality
reveals that with probability at least $1-O\left(d^{-13}\right)$,
\begin{align*}
\left|\gamma_{6}\right| & \lesssim\sigma_{\max}^{2}\left\{ \frac{\log^{2}d}{p}\big\|\bm{P}_{(i,j),:}^{\star}\big\|_{\infty}^{2}+\sqrt{\frac{\log d}{p}}\big\|\bm{P}_{(i,j),:}^{\star}\big\|_{\infty}\big\|\bm{P}_{(i,j),:}^{\star}\big\|_{2}\right\} \\
 & \lesssim\frac{\sigma_{\max}^{2}}{\lambda_{\min}^{\star4/3}}\left\{ \frac{\mu^{2}r\log^{2}d}{d^{2}p}+\sqrt{\frac{\mu^{2}r\log d}{d^{2}p}}\right\} \big\|\widetilde{\bm{U}}_{(i,j),:}^{\star}\big\|_{2}^{2}\asymp\frac{\sigma_{\max}^{2}}{\lambda_{\min}^{\star4/3}}\sqrt{\frac{\mu^{2}r\log d}{d^{2}p}}\big\|\widetilde{\bm{U}}_{(i,j),:}^{\star}\big\|_{2}^{2},
\end{align*}
as long as $p\gtrsim\mu^{2}rd^{-2}\log^{3}d$. Therefore, we combine
bounds for $\gamma_{4},\gamma_{5}$ and $\gamma_{6}$ to conclude
that
\begin{align*}
\left|\beta_{2}\right| & \lesssim\frac{\sigma_{\max}^{2}}{\lambda_{\min}^{\star4/3}}\sqrt{\frac{\mu^{2}r\log d}{d^{2}p}}\big\|\widetilde{\bm{U}}_{(i,j),:}^{\star}\big\|_{2}^{2}+\frac{\sigma_{\max}^{2}}{\lambda_{\min}^{\star2/3}}\frac{\sigma_{\max}}{\lambda_{\min}^{\star}}\sqrt{\frac{d\log d}{p}}\frac{\mu^{2}r}{d}\big\|\widetilde{\bm{U}}_{(i,j),:}^{\star}\big\|_{2}+\sigma_{\max}^{2}\left(\frac{\sigma_{\max}}{\lambda_{\min}^{\star}}\sqrt{\frac{d\log^{2}d}{p}}\frac{\mu^{2}r}{d}\right)^{2}.
\end{align*}
\item Putting the above bounds for $\beta_{1}$ and $\beta_{2}$ together
reveals that
\begin{align*}
 & \big|\bm{P}_{(i,j),:}\bm{D}_{k}\bm{P}_{:,(i,j)}-\bm{P}_{(i,j),:}^{\star}\bm{D}_{k}^{\star}\bm{P}_{:,(i,j)}^{\star}\big|\\
 & \qquad\lesssim\frac{\sigma_{\max}^{2}}{\lambda_{\min}^{\star4/3}}\sqrt{\frac{\mu^{3}r^{2}\log d}{d^{2}p}}\,\big\|\widetilde{\bm{U}}_{(i,j),:}^{\star}\big\|_{2}^{2}+\frac{\sigma_{\max}^{2}}{\lambda_{\min}^{\star2/3}}\frac{\sigma_{\max}}{\lambda_{\min}^{\star}}\sqrt{\frac{d\log d}{p}}\frac{\mu^{2}r}{d}\big\|\widetilde{\bm{U}}_{(i,j),:}^{\star}\big\|_{2}+\sigma_{\max}^{2}\left(\frac{\sigma_{\max}}{\lambda_{\min}^{\star}}\sqrt{\frac{d\log^{2}d}{p}}\frac{\mu^{2}r}{d}\right)^{2}.
\end{align*}
\item Clearly, we can apply an analogous argument to bound $\bm{P}_{(i,k),:}\bm{D}_{j}\bm{P}_{:,(i,k)}-\bm{P}_{(i,k),:}^{\star}\bm{D}_{j}^{\star}\bm{P}_{:,(i,k)}^{\star}$
and $\bm{P}_{(j,k),:}\bm{D}_{i}\bm{P}_{:,(j,k)}-\bm{P}_{(j,k),:}^{\star}\bm{D}_{i}^{\star}\bm{P}_{:,(j,k)}^{\star}$.
Taken collectively with the lower bound of $v_{i,j,k}^{\star}$ (cf.~(\ref{eq:T-entry-var-LB}))
and the conditions $\sigma_{\max}/\sigma_{\min}\asymp1$ and $\kappa\asymp1$,
we obtain
\begin{align*}
\frac{\big|v_{i,j,k}-v_{i,j,k}^{\star}\big|}{v_{i,j,k}^{\star}} & \lesssim\sqrt{\frac{\mu^{3}r^{2}\log d}{d^{2}p}}+\Big(\big\|\widetilde{\bm{U}}_{(i,j),:}^{\star}\big\|_{2}+\big\|\widetilde{\bm{U}}_{(i,k),:}^{\star}\big\|_{2}+\big\|\widetilde{\bm{U}}_{(j,k),:}^{\star}\big\|_{2}\Big)^{-1}\frac{\sigma_{\max}}{\lambda_{\min}^{\star1/3}}\sqrt{\frac{d\log d}{p}}\frac{\mu^{2}r}{d}\\
 & \quad+\Big(\big\|\widetilde{\bm{U}}_{(i,j),:}^{\star}\big\|_{2}+\big\|\widetilde{\bm{U}}_{(i,k),:}^{\star}\big\|_{2}+\big\|\widetilde{\bm{U}}_{(j,k),:}^{\star}\big\|_{2}\Big)^{-2}\left(\frac{\sigma_{\max}}{\lambda_{\min}^{\star1/3}}\sqrt{\frac{d\log^{2}d}{p}}\frac{\mu^{2}r}{d}\right)^{2}.\\
 & \asymp\sqrt{\frac{\mu^{3}r^{2}\log d}{d^{2}p}}+\Big(\big\|\widetilde{\bm{U}}_{(i,j),:}^{\star}\big\|_{2}+\big\|\widetilde{\bm{U}}_{(i,k),:}^{\star}\big\|_{2}+\big\|\widetilde{\bm{U}}_{(j,k),:}^{\star}\big\|_{2}\Big)^{-1}\frac{\sigma_{\max}}{\lambda_{\min}^{\star1/3}}\sqrt{\frac{d\log d}{p}}\frac{\mu^{2}r}{d}=o\left(1\right),
\end{align*}
where the last line holds due to the assumptions (\ref{eq:U-tilde-2norm-LB-rank-r})
and $p\gg\mu^{3}r^{2}d^{-2}\log^{2}d$.
\end{itemize}

\subsubsection{Proof of Lemma \ref{lemma:P-property}}

Fix any $1\leq i,j\leq d$.
\begin{itemize}
\item We start with the norms of the rows of $\bm{P}^{\star}$ (cf.~\ref{def:P}).
By (\ref{eq:U-true-tilde-spectrum}), it is straightforward to deduce
that
\end{itemize}
\begin{align*}
\big\|\bm{P}_{(i,j),:}^{\star}\big\|_{2} & \leq\big\|\widetilde{\bm{U}}_{(i,j),:}^{\star}\big\|_{2}\big\|(\widetilde{\bm{U}}^{\star\top}\widetilde{\bm{U}}^{\star})^{-1}\big\|\big\|\widetilde{\bm{U}}^{\star}\big\|\lesssim\frac{1}{\lambda_{\min}^{\star2/3}}\big\|\widetilde{\bm{U}}_{(i,j),:}^{\star}\big\|_{2};\\
\big\|\bm{P}_{(i,j),:}^{\star}\big\|_{\infty} & \leq\big\|\widetilde{\bm{U}}_{(i,j),:}^{\star}\big\|_{2}\big\|(\widetilde{\bm{U}}^{\star\top}\widetilde{\bm{U}}^{\star})^{-1}\big\|\big\|\widetilde{\bm{U}}^{\star}\big\|_{2,\infty}\lesssim\frac{\mu\sqrt{r}}{d}\frac{1}{\lambda_{\min}^{\star2/3}}\big\|\widetilde{\bm{U}}_{(i,j),:}^{\star}\big\|_{2}.
\end{align*}

\begin{itemize}
\item Next, we move on to the $\ell_{2}$ norm of $(\bm{P}-\bm{P}^{\star})_{(i,j),:}$,
which can be decomposed as
\begin{align}
(\bm{P}-\bm{P}^{\star})_{(i,j),:} & =\widetilde{\bm{U}}_{(i,j),:}(\widetilde{\bm{U}}^{\top}\widetilde{\bm{U}})^{-1}\widetilde{\bm{U}}^{\top}-\widetilde{\bm{U}}_{(i,j),:}^{\star}(\widetilde{\bm{U}}^{\star\top}\widetilde{\bm{U}}^{\star})^{-1}\widetilde{\bm{U}}^{\star\top}\nonumber \\
 & =(\widetilde{\bm{U}}-\widetilde{\bm{U}}^{\star})_{(i,j),:}(\widetilde{\bm{U}}^{\top}\widetilde{\bm{U}})^{-1}\widetilde{\bm{U}}^{\top}+\widetilde{\bm{U}}_{(i,j),:}^{\star}\big((\widetilde{\bm{U}}^{\top}\widetilde{\bm{U}})^{-1}-(\widetilde{\bm{U}}^{\star\top}\widetilde{\bm{U}}^{\star})^{-1}\big)\widetilde{\bm{U}}^{\top}\nonumber \\
 & \quad+\widetilde{\bm{U}}_{(i,j),:}^{\star}(\widetilde{\bm{U}}^{\star\top}\widetilde{\bm{U}}^{\star})^{-1}(\widetilde{\bm{U}}-\widetilde{\bm{U}}^{\star})^{\top}.\label{eq:P-P-true-decomp}
\end{align}
By the triangle inequality, we shall control these three terms separately.
From (\ref{eq:U-tilde-loss-2inf}), one has
\begin{align*}
\big\|(\widetilde{\bm{U}}-\widetilde{\bm{U}}^{\star})_{(i,j),:}(\widetilde{\bm{U}}^{\top}\widetilde{\bm{U}})^{-1}\widetilde{\bm{U}}^{\top}\big\|_{2} & \leq\big\|(\widetilde{\bm{U}}-\widetilde{\bm{U}}^{\star})_{(i,j),:}\big\|_{2}\big\|(\widetilde{\bm{U}}^{\top}\widetilde{\bm{U}})^{-1}\big\|\big\|\widetilde{\bm{U}}\big\|\\
 & \lesssim\frac{\sigma_{\max}}{\lambda_{\min}^{\star}}\sqrt{\frac{\mu^{2}r\log d}{dp}}\,\lambda_{\max}^{\star2/3}\cdot\frac{1}{\lambda_{\min}^{\star4/3}}\cdot\lambda_{\max}^{\star2/3}.
\end{align*}
As for the remaining two terms, from (\ref{eq:U-true-tilde-norm}),
(\ref{eq:U-tilde-spectrum}) and (\ref{eq:U-tilde-Gram-inv-op-loss}),
one has
\begin{align*}
\big\|\widetilde{\bm{U}}_{(i,j),:}^{\star}\big\|_{2}\big\|(\widetilde{\bm{U}}^{\top}\widetilde{\bm{U}})^{-1}-(\widetilde{\bm{U}}^{\star\top}\widetilde{\bm{U}}^{\star})^{-1}\big\|\big\|\widetilde{\bm{U}}\big\| & \lesssim\frac{\sigma_{\max}}{\lambda_{\min}^{\star3}}\sqrt{\frac{d}{p}}\,\lambda_{\max}^{\star2/3}\cdot\lambda_{\max}^{\star2/3}\big\|\widetilde{\bm{U}}_{(i,j),:}^{\star}\big\|_{2}\\
 & \lesssim\frac{\sigma_{\max}}{\lambda_{\min}^{\star5/3}}\sqrt{\frac{d}{p}}\,\big\|\widetilde{\bm{U}}_{(i,j),:}^{\star}\big\|_{2}.
\end{align*}
Combining (\ref{eq:U-true-tilde-norm}), (\ref{eq:U-true-tilde-spectrum})
and (\ref{eq:U-tilde-op-loss}) yields
\begin{align*}
\big\|\widetilde{\bm{U}}_{(i,j),:}^{\star}\big\|\big\|(\widetilde{\bm{U}}^{\star\top}\widetilde{\bm{U}}^{\star})^{-1}\big\|\big\|\widetilde{\bm{U}}-\widetilde{\bm{U}}^{\star}\big\| & \lesssim\frac{1}{\lambda_{\min}^{\star4/3}}\cdot\frac{\sigma_{\max}}{\lambda_{\min}^{\star}}\sqrt{\frac{d}{p}}\,\lambda_{\max}^{\star2/3}\big\|\widetilde{\bm{U}}_{(i,j),:}^{\star}\big\|_{2}\\
 & \lesssim\frac{\sigma_{\max}}{\lambda_{\min}^{\star5/3}}\sqrt{\frac{d}{p}}\,\big\|\widetilde{\bm{U}}_{(i,j),:}^{\star}\big\|_{2}.
\end{align*}
The above bounds taken collectively allow us to obtain
\[
\big\|(\bm{P}-\bm{P}^{\star})_{(i,j),:}\big\|_{2}\lesssim\frac{\sigma_{\max}}{\lambda_{\min}^{\star5/3}}\sqrt{\frac{d}{p}}\left(\frac{\mu\sqrt{r\log d}}{d}\lambda_{\max}^{\star2/3}+\big\|\widetilde{\bm{U}}_{(i,j),:}^{\star}\big\|_{2}\right)\asymp\frac{\sigma_{\max}}{\lambda_{\min}^{\star}}\sqrt{\frac{d\log d}{p}}\frac{\mu\sqrt{r}}{d},
\]
where the last step arises from the incoherence condition that $\|\widetilde{\bm{U}}^{\star}\|_{2,\infty}\lesssim\mu\sqrt{r}\lambda_{\max}^{\star2/3}/d$
and $\kappa\asymp1$.
\item Finally, let us look at the $\ell_{\infty}$ norm of $(\bm{P}-\bm{P}^{\star})_{(i,j),:}$.
Armed with the decomposition in (\ref{eq:P-P-true-decomp}), we can
bound
\begin{align*}
 & \big\|(\bm{P}-\bm{P}^{\star})_{(i,j),:}\big\|_{\infty}\leq\big\|(\widetilde{\bm{U}}-\widetilde{\bm{U}}^{\star})_{(i,j),:}\big\|_{2}\big\|(\widetilde{\bm{U}}^{\top}\widetilde{\bm{U}})^{-1}\big\|\big\|\widetilde{\bm{U}}\big\|_{2,\infty}\\
 & \qquad\qquad+\big\|\widetilde{\bm{U}}_{(i,j),:}^{\star}\big\|_{2}\big\|(\widetilde{\bm{U}}^{\top}\widetilde{\bm{U}})^{-1}-(\widetilde{\bm{U}}^{\star\top}\widetilde{\bm{U}}^{\star})^{-1}\big\|\big\|\widetilde{\bm{U}}\big\|_{2,\infty}+\big\|\widetilde{\bm{U}}_{(i,j),:}^{\star}\big\|_{2}\big\|(\widetilde{\bm{U}}^{\star\top}\widetilde{\bm{U}}^{\star})^{-1}\big\|\big\|\widetilde{\bm{U}}-\widetilde{\bm{U}}^{\star}\big\|_{2,\infty}\\
 & \qquad\overset{(\mathrm{i})}{\lesssim}\frac{\sigma_{\max}}{\lambda_{\min}^{\star}}\sqrt{\frac{\mu^{2}r\log d}{dp}}\,\lambda_{\max}^{\star2/3}\cdot\frac{1}{\lambda_{\min}^{\star4/3}}\cdot\frac{\mu\sqrt{r}}{d}\lambda_{\max}^{\star2/3}+\frac{\sigma_{\max}}{\lambda_{\min}^{\star3}}\sqrt{\frac{d}{p}}\,\lambda_{\max}^{\star2/3}\cdot\frac{\mu\sqrt{r}}{d}\lambda_{\max}^{\star2/3}\big\|\widetilde{\bm{U}}_{(i,j),:}^{\star}\big\|_{2}\\
 & \qquad\quad+\frac{1}{\lambda_{\min}^{\star4/3}}\cdot\frac{\sigma_{\max}}{\lambda_{\min}^{\star}}\sqrt{\frac{\mu^{2}r\log d}{dp}}\,\lambda_{\max}^{\star2/3}\big\|\widetilde{\bm{U}}_{(i,j),:}^{\star}\big\|_{2}\\
 & \qquad\overset{(\mathrm{ii})}{\lesssim}\frac{\sigma_{\max}}{\lambda_{\min}^{\star5/3}}\sqrt{\frac{\mu^{2}r\log d}{dp}}\left(\frac{\mu\sqrt{r}}{d}\lambda_{\max}^{\star2/3}+\big\|\widetilde{\bm{U}}_{(i,j),:}^{\star}\big\|_{2}\right)\overset{(\mathrm{iii})}{\lesssim}\frac{\sigma_{\max}}{\lambda_{\min}^{\star}}\sqrt{\frac{\mu^{2}r\log d}{dp}}\frac{\mu\sqrt{r}}{d}.
\end{align*}
Here, (i) relies on (\ref{eq:U-tilde-property}) and (\ref{eq:U-tilde-Gram-inv-op-loss}),
(ii) is due to the condition $\kappa\asymp1$, whereas (iii) arises
from (\ref{eq:U-true-tilde-norm}) and $\kappa\asymp1$.
\end{itemize}


TITLE: Decomposition of a group into left and right cosets
QUESTION [2 upvotes]: $G$ is a group with subgroup $H$. For $a_i \in G$ there is a decomposition of the group into left cosets $G = a_1H \bigcup a_2H ... \bigcup a_sH$
Show that $Ha_1^{-1} \bigcup Ha_2^{-1} ... \bigcup Ha_s^{-1}$ is a decomposition of the group into right cosets.
 I know that there are the same number of left and right cosets but I don't know how to proceed.

REPLY [2 votes]: The subsets $H a_j^{-1}$ are certainly right cosets. The only thing to show is that they are disjoint since, as you observed, the number of left and right cosets are the same. If $H a_i^{-1} \cap H a_j^{-1} \neq \emptyset$, for $i \neq j$, then for some $h_1, h_2 \in H$ we must have $h_1 a_i^{-1} = h_2 a_j^{-1}$ which implies $a_i^{-1} = h_1^{-1} h_2 a_j^{-1}$. Then $a_i = a_j h_2^{-1} h_1$ and therefore $a_i \in a_j H$. But by assumption, $a_i H \cap a_j H = \emptyset$ which contradicts the conclusion that $a_i \in a_j H$. This completes the proof that the right cosets $H a_i^{-1}$ are pairwise disjoint.


text:
\section{Property of Group Automorphism which Fixes Identity Only/Corollary 1}
Tags: Property of Group Automorphism which Fixes Identity Only

\begin{theorem}
:$\forall x \in G: \exists g \in G: x = g^{-1} \, \map \phi g$
\end{theorem}

\begin{proof}
Let $\psi: G \to G$ be the [[Definition:Mapping|mapping]]:
:$\forall x \in G: \map \psi x = x^{-1} \, \map \phi x$
From [[Property of Group Automorphism which Fixes Identity Only]]:
:$\forall x, y \in G: x^{-1} \, \map \phi x = y^{-1} \, \map \phi y \implies x = y$
That is, $\psi$ is an [[Definition:Injection|injection]].
From [[Injection from Finite Set to Itself is Surjection]], it follows that $\psi$ is a [[Definition:Surjection|surjection]].
That is:
:$\forall x \in G: \exists g \in G: x = \map \psi g$
which is a restatement of the hypothesis.
{{qed}}
\end{proof}

config: wiki


TITLE: Given three points on a plane and two points on a line that intersects the plane. I need the angle between the plane and the line.
QUESTION [2 upvotes]: A real world problem where the angular orienation of the axis of a feature to a reference plane is needed but is absent from a blueprint. Thank you.

REPLY [1 votes]: Similar to augurar's answer, but using vector products:
Let $p_1, p_2, p_3$ be the points defining the plane, and let $l_1, l_2$ be the points defining the line.

Let $\vec{a_1} = p_1-p_3$ and $\vec{a_2} = p_2-p_3$.
Let $\vec{c} = \vec{a_1} \times \vec{a_2}$ so $\vec{c}$ is orthogonal to the plane.
Let $\vec{u} = l_1 - l_2$.
Angle is $\cos^{-1}\left(\left|\dfrac{\vec{c} \cdot \vec{u}}{|c||u|}\right|\right)$


TITLE: Vectors on Different Coordinate Systems?
QUESTION [0 upvotes]: Consider the product of vectors coordinated relative to a given coordinate frame, defined by
   $$\vec{a}\square\vec{b}=((a_{1},b_{1})\square(a_{2},b_{2})):=(a_{1}b_{1},a_{2}b_{2})$$
  Explain why that in the context of vectors this is no good.

My work:
I think that it is no good because if you change the coordinate system, you may change the vector. (Like a vector in Cartesian coordinates would be a different vector in polar coordinates [Not a 100% sure about that, but seems to make sense]).
If anyone could expand as to why this is true, it would be greatly appreciated. Also please note, that I am fairly new to vectors, so if the advanced physics jargon could be kept to a minimum it would be appreciated as well.

REPLY [1 votes]: You can define a product in any way you want, so your definition is perfectly fine.  The question is whether that product is useful.  And one of the things we usually require is that a product be invariant under rotations, for example.  What this means is that if you rotate each of the vectors that goes into the product, you should get the same result, but rotated.
So for example, suppose we write the rotated version of $\vec{a}$ and $\vec{b}$ as $R(\vec{a})$ and $R(\vec{b})$.  Then we might reasonably demand that your product is invariant: $R(\vec{a}) \square R(\vec{b}) = R(\vec{a} \square \vec{b})$.  Now we have to check whether this is true.
If we rotate by an angle $\theta$, we can write $R(\vec{a}) = (a_1\, \cos\theta - a_2\, \sin\theta, a_1\, \sin\theta + a_2\, \cos\theta)$, and similarly for $R(\vec{b})$.  With your definition, we have
\begin{equation}
\begin{split}
R(\vec{a}) \square R(\vec{b})
 = \big( (a_1\, \cos\theta - a_2\, \sin\theta)\, (b_1\, \cos\theta - b_2\, \sin\theta), \\ (a_1\, \sin\theta + a_2\, \cos\theta)\, (b_1\, \sin\theta + b_2\, \cos\theta) \big)~.
\end{split}
\end{equation}
But we also have
\begin{equation}
R(\vec{a} \square \vec{b})= \big( (a_1\, b_1\, \cos\theta - a_2\, b_2\, \sin\theta), (a_1\, b_1\, \sin\theta + a_2\, b_2\, \cos\theta) \big)~,
\end{equation}
which is not the same.  (Your job is to show why these things are true, and show that they are different.)
So your product is not invariant under rotations.  And different coordinate systems might just express these vectors as rotations of one another.  So the meaning of your product depends on the particular coordinate system you are using.  That's usually not a good idea.  You can check that $R(\vec{a}) \cdot R(\vec{b}) = R(\vec{a} \cdot \vec{b}) = \vec{a} \cdot \vec{b}$ and $R(\vec{a}) \times R(\vec{b}) = R(\vec{a} \times \vec{b})$ for the usual dot and cross products (the latter of which requires three dimensions).  That's why they are so useful.


TITLE: perfect groups with non-trivial group homology over rational coefficients
QUESTION [3 upvotes]: A group $G$ is perfect if $G=[G,G]$. For perfect groups, we know that the first group homology $H_1(G, \mathbb{Z})=G/[G,G]=0$. A group $G$ is called acyclic if its group homology $H_i(G, \mathbb{Z})=0$ for $i \geq 1$ and $H_0(G, \mathbb{Z})=\mathbb{Z}$. It is well-known that the group $A_5$ is a perfect group which is not acyclic.
However, I am interested in computing group homology with rational coefficients. I would like to know an example of a perfect group which has non-trivial rational group homology. It seems to me that such a group cannot be a finite perfect group, because one can use the universal coefficient theorem
$$0 \rightarrow H_i(G,\mathbb{Z})\otimes \mathbb{Q}\rightarrow H_i(G, \mathbb{Q})→Tor(H_{i-1}(G,\mathbb{Z}),\mathbb{Q})→0$$
and deduce the rational group homology vanish for $i \geq 1$ inductively.
So I am wondering if a perfect group is actually rationally acyclic or a counterexample to this exisit?

REPLY [0 votes]: You can find an example where second homology group of a perfect group is not zero in this paper of Milnor. p. 79.
http://www.maths.ed.ac.uk/~aar/papers/milnor7.pdf
Milnor consider the cohomology of the discrete group $PSL(2,R)$ (in his terminology $PSL(2,R)$ made discrete) and it shows that there is a class in $H^2$ which is not trivial. The  duality implies that you have an homology class $\geq 1$ which is not trivial.


TITLE: Distribution of the index of the variable which achieves the minimum of exponential random variables
QUESTION [6 upvotes]: I am reading Exponential distribution from Wiki, and it is said that the index of the variable which achieves the minimum is distributed according to the law
$$P(k|X_k=min\{X_1,X_2,...,X_n\})=\frac{\lambda_k}{\lambda_1+...+\lambda_n}$$
I don't know how to prove this property. I try the case $n=2$ in different ways.
First, I find $P(X_1\le X_2)=\frac{\lambda_1}{\lambda_1+\lambda_2}$, but I can't change this to the conditional probability formally.
Second, I try to prove this through pdf. Let $Y=min\{X_1,X_2\}$, I want to calculate $f_{X_1|Y}(x_1,y)$. However I find that there should be infinite value of $f_{X_1|Y}(x_1,y)$ at $x_1=y$ , since the conditional probability is actually a discrete distribution. I don't know how to obtain the discrete distribution from continuous pdf.
Please tell me how can I continue my proof or give another formal proof, Thanks!!

REPLY [0 votes]: The probability density of an exponential random variable with parameter $\lambda$ is $\lambda e^{-\lambda x}$ which is equal to $\lambda$ at $0$. It means that the probability that it will be less than $\delta$ is equal to $\lambda \delta$ for $\delta \rightarrow 0$. So if we have $n$ independent exponential random variables with parameters $\lambda_i$ the probability that any of them will be less than $\delta$ is equal to $\lambda_i\delta$ for $\delta \rightarrow 0$. Now, the probability that two of them are less than $\delta$ is significantly smaller than any of the $\lambda_i\delta$ for $\delta \rightarrow 0$, thus the probability that the $i$-th variable will be smaller than $\delta$ conditioned on the event that one of them is approaches:
$$\frac{\lambda_i\delta}{\lambda_1\delta+\lambda_2\delta+\dots+\lambda_n\delta}=\frac{\lambda_i}{\lambda_1+\lambda_2+\dots+\lambda_n}$$
as $\delta \rightarrow 0$. Therefore, by memoryless property, it is also the probability that the $i$-th variable will be the smallest one.


text:
\section{Reconstruction of FV-solution}\label{sec:econs}

This section focuses on the computation of some function $\phi_i(\cdot,t ; \mu)$  for a target value of the parameter $\mu \in \Pc$  from given precomputed snapshot $u(\cdot, t;\mu_i), \mu_i \neq \mu$.  To this end, let $\varphi(\mu,\mu_i)$ be a parameter dependent transformation from $\Omega \times I$ to $\Omega$ given by
\begin{equation}
\label{eq:spaceshift}
\phi(\cdot,t ; \mu,\mu_i) = u(\cdot, t,\mu_i) \circ \varphi(\mu,\mu_i)(\cdot,t), \qquad t \in I.
\end{equation}
In Section \ref{sec:RTA}, we transpose this idea to a FV framework with the {\it Recontruct-Transform-Average} method. This approach is a first step toward  ROM for discretized hyperbolic conservation laws as discussed in Section \ref{sec:torom}.

\subsection{Recontruct-Transform-Average algorithm}\label{sec:RTA}

Given a precomputed FV approximation $\{\Vu^k(\mu_i)\}_{k=0}^K$, satisfying Equation \eqref{eq:conservativeForm} for one instance $\mu_i \in \Pc$ of the parameter, we reconstruct the sequence of vectors $\{\Vphi^k_i(\mu)\}_{k=0}^K \subset \RR^N$ from this snapshot by means of the transformation $\varphi(\mu,\mu_i)$.  To mimic
 the continuous transformation  $\varphi(\mu,\mu_i)$ in the discrete setting, the Reconstruct-Transform-Average (RTA) method is introduced here. In the lines of the REA method. At given time $t^k$, it consists of three steps. First, the global approximation $u_N^k(\mu_i)$ in $V_N$ at time $t^k$ is deduced from $\Vu^k(\mu_i)$. Second, the resulting function is transformed with $\varphi(\mu, \mu_i)(\cdot, t^k)$. Third,  the components  $\Vphi^k_{i,j}(\mu)$  of the vector $ \Vphi^k_i(\mu)$ are obtained  by averaging  the transformed snapshot in each grid cell  $C_j$. The overall RTA procedure is summarized in Algorithm \ref{alg:RTA}.

\begin{algorithm}[RTA algorithm] \label{alg:RTA}~\\[0.2cm]
Let be given $\mu, \mu_i \in \Pc$,  $\Vu^k(\mu_i)$ the snapshot associated with $\mu_i$ at time $t^k$, and the space transformation $\varphi(\mu,\mu_i)$. Compute $\Vphi_i^k (\mu)$ from the snapshot $\Vu^k(\mu_i)$ as follows.
\begin{enumerate}
\item \emph{Reconstruct} $u^k_N(\mu_i)\in V_N$ such that
\begin{equation}
u^k_N(\mu_i)(x) =  u_j^k(\mu_i), \text{ for } x \in C_j.
 \label{eq:reconstruct}
\end{equation}
\item \emph{Transform} the snapshot $u^k_N(\mu_i)$
\begin{equation}
\tilde  \phi^k_N(\mu,\mu_i)(x) =u^k_N(\mu_i) \circ \varphi(\mu,\mu_i)(x,t^k), \text{ for }  x \in \overline{\Omega}.
 \label{eq:transform}
\end{equation}
\item \emph{Average} $\tilde \phi^k_N(\mu)$ over each grid cell $C_j$ to obtain the components $\phi^k_{i,j}(\mu)$ of $\Vphi^k_i(\mu) \in \RR^N$ given by
\begin{equation}
 \phi^k_{i,j}(\mu) = \dfrac{1}{\Delta x} \int_{C_j} \tilde \phi^k_N(\mu,\mu_i)(x) dx, \quad j = 1, \dots, N.
 \label{eq:average}
 \end{equation}
\end{enumerate}
\end{algorithm}

\begin{remark}
For a transport problem, when $\varphi(\mu,\mu_i)$ is chosen to be a space shift and the snapshots are computed with the upwind scheme given by Equation \eqref{eq:upwind_matrix_form},  the RTA method is proven to be Total Variation Bounded (TVB), see Section \ref{sec:stab}. The main avantage of the RTA algorithm in this case is that the approximation $\Vphi_i^k(\mu)$ reconstructed from the snapshot $\Vu^k_N(\mu)$  will not exhibit any spurious numerical oscillations.
\end{remark}

\begin{remark}
The extension of the RTA procedure to approximations of higher order is straighforward on cartesian grids. Moreover, it is not limited to any particular boundary condition or transformation form. Such extensions are the object of future works.
\end{remark}

\subsection{Toward ROM} \label{sec:torom}

As motivated in the introduction, one possible application of the RTA algorithm is nonlinear ROM for FV approximation of parameter dependent conservation laws. The objective is to properly approximate the solution manifold at each instant by means of a suitable parameter (and time) dependent reduced space.  The RTA algorithm can be seen as a first step for the construction of such adapted and parameter-dependent reduced basis, required in the approximation \eqref{eq:approxt}. At this point, it remains to define the transformation maps $\varphi(\mu,\mu_i)$. The particular case of space shift transformation in the line of \cite{Welper2017,Rim2018,Rim2018-2,Rim2018Feb,Reiss2018,Black2019,Black2020Nov,Sarna2020Mar} is discussed.  \\

The scalar transport equation with periodic boundary conditions considered in Section \ref{FVtransport}, admits the following exact solution
$$
u(x,t; \mu)=u_0 (x-a(\mu)t),
$$
which is a translation of the initial condition $u^0$ along the characteristic $x = x_0+a(\mu)t$ starting from $x_0$.

Given some snapshot of the solution for an instance of the parameter $\mu_i \in \Pc$ at time $t$, the target solution for the value $\mu \in\Pc$ reads
\begin{equation}
\label{eq:reecriture}
u(x,t; \mu)=u(\varphi(\mu,\mu_i)(x,t), t; \mu_i),
\end{equation}
with the transformation map $\varphi(\mu,\mu_i)= x -  (a(\mu)-a(\mu_i)) t$,  corresponding to a shift in the space domain. For such a choice, the approximation provided by Equation \eqref{eq:approxt} is exact with only a rank $r$ equal to $1$.  Transposing this observation to the FV discrete setting, the RTA algorithm can be expected to provide, up to some discretization error depending on $\Delta x$, an approximation  $\Vphi_i^k(\mu)$ of the true numerical solution $\Vu^k(\mu)$  reconstructed from the snapshot  $\Vu^k(\mu_i)$ \cite[Example 3.6]{Rim2019}. The corresponding procedure with online-offline implementation is discussed in the forthcoming section. In practice, it provides a very simple strategy, and allows to efficiently compute  approximations of FV solutions of parameter dependent linear conservation laws from pre-computed FV solutions.

For nonlinear conservation laws, the characteristics are generally complex to compute, or even non explicitly known. In that case, efficient ROM strategies computing at the same time both reduced approximation of the solution and suitable approximation of the transformation should be preferred \cite{Cagniart2017,Rim2019}.

config: arxiv


TITLE: Can one quantify the total luminosity of the cosmological horizon?
QUESTION [4 upvotes]: If I integrate all the power (radiation, matter particles, neutrinos) radiated from the cosmological horizon into the universe, what number do I get?
Is it true that the integral power/luminosity is of the order of $c^5/G = 3.6 \cdot 10^{52}$ Watt?
Has the integral horizon power been constant over the past evolution of the universe?

REPLY [0 votes]: First I will answer your first question, calculating a value for the radiated power.
I was unable to find any discussion of "power" in
https://en.wikipedia.org/wiki/Cosmological_horizon .
I think your concept has a flaw. When you calculate black body radiation power ($P$) in watts from an object, the object will have a surface area ($A$) and a temperature ($T$). The watts per unit area depends on temperature.
http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/stefan.html
$$P = \sigma A T^4$$
$\sigma$ is the Stephan-Boltzmann constant
https://en.wikipedia.org/wiki/Stefan%E2%80%93Boltzmann_constant
$$σ = 5.670374419 \times 10^8 \, \mathrm{ W/(m^2 K^4)}$$
I suggest you choose a value for the radius ($R_{\mathrm{ou}}$) of the observable universe and a value for the time of radiation, $t>0$. Then calculate a value for the scale factor value $a(t)$.
https://en.wikipedia.org/wiki/Observable_universe
$R_{\mathrm{ou}}\mathrm{(now)} = 46.5\, \mathrm{ Gly}$
$R_{\mathrm{ou}}\mathrm{(now)}$ is the current radius of the spherical surface that did the radiation at time $t$.  You then need to calculate the radius $R_{\mathrm{ou}}(t)$.
$$R_{\mathrm{ou}}(t) = R_{\mathrm{ou}}\mathrm{(now)} / a(t).$$
The time t chosen should be well into the era when radiation mass equivalent density dominates matter (and also of course dark energy).  Therefore,
$$T(t) = T(\mathrm{now}) / a(t)$$
Now calculate the power radiated.
$$R = \sigma  4 \pi R^2_{\mathrm{ou}}(t)  T^4_{\mathrm{ou}}(t)$$
Now for the second question.  I think this is just wrong. The radiation is watts per area, and there is no area in your calculation, presumably because $t=0$.
Now for the third question. I am not sure what you are intending to ask. There are two possible interpretations.

Is the radiation from time $t$ still radiating? The answer is no. It stopped radiating immediately after time $t$. However, a little later, say at $t+dt$, there will be radiation observed at the same observation place (at the center of the observable universe sphere)  at little later than "now".

Is the radiation from a sphere at radius $R_{\mathrm{ou}}(t)$ detectable at the center of the sphere at all times since t, and with the same intensity? The answer is yes, it continues to be observable at all times since $t$, but the observed intensity changes with the time "now". For different times "now" the observed intensity depends upon how far the radiation travels to the observer, $R_{\mathrm{ou}}\mathrm{(now)}$. The observed intensity of radiation will be inversely proportion to the square of the distance: $1/R^2_{\mathrm{ou}}\mathrm{(now)}$.


text:
\section{Introduction}

\subsection{Motivations}
Consider the problem of estimating a binary vector $\sgl\in\left\{ -1,1\right\} ^{p}$ from noisy linear measurements in the form of
\begin{equation}
\vy=\dmtx\sgl+\noise.\label{eq:linear_model}
\end{equation}
Here, $\dmtx\in\R^{n\times p}$ is a known sensing matrix
and $\noise \sim\mathcal{N}(\boldsymbol{0}, \sigma_p^2 \mI_n)$ denotes an unknown noise vector. This is a prototypical model for multi-user detections in MIMO communication systems \cite{viterbi1995cdma, rusek2012scaling}. It also arises in other applications such as compressed sensing \cite{das2013finite}, source separation \cite{aissa2015sparsity}, and image processing \cite{ahn2016compressive}.

Various algorithms have been proposed to solve (\ref{eq:linear_model}). Examples include sphere decoding \cite{pohst1981computation}, zero-forcing \cite{grotschel2012geometric}, approximate message passing \cite{jeon2015optimality}, Markov chain Monte Carlo methods \cite{hassibi2014optimized}, and semidefinite programming \cite{luo2010semidefinite }. Among them, a convex-optimization based method, known as the box-relaxation decoder \cite{tan2000box,thrampoulidis2016ber,thrampoulidis2018symbol}, is popular in practice due to its simplicity and efficiency.
The method consists of merely two
steps: (1) solve a box-constrained least squares problem
\begin{equation}
\optx=\argmin{\vx\in[-1,1]^{p}}\frac{1}{2}\|\vy-\mA\vx\|^{2},\label{eq:box_LASSO}
\end{equation}
and (2) obtain an estimate of $\sgl$ by taking the sign of $\optx$, \emph{i.e.}, $\sol=\sgn(\optx).$

The performance of this algorithm can be measured by the bit error rate (BER):
\begin{equation}\label{eq:BER}
\text{BER} = \frac{1}{p}\sum_{i=1}^{p}\I_{\{\soli_{i}\neq\beta_{i}\}},
\end{equation}
where $\I_{\{\cdot\}}$ denotes the indicator function. The achievable BER depends on two key parameters: the noise variance $\sigma_p^2$, and the sampling ratio $\delta_p \bydef n/p$.

Under the assumption that the sensing matrix $\mA$ has i.i.d. normal entries, the authors of  \cite{thrampoulidis2016ber,thrampoulidis2018symbol} analyzed the asymptotic BER achieved by the box-relaxation decoder. They show that, as $n,p\to\infty$ with $\delta_p \to \delta \in (\frac{1}{2}, \infty)$ and $\nsdp^2 \equiv \sigma^2>0$, the BER converges in probability to a deterministic limit, \emph{i.e.},
\begin{equation}\label{eq:asymp_BER}
\text{BER} \cip \mathcal{E}(\delta, \sigma^2) \in\left(0, \tfrac{1}{2}\right).
\end{equation}
This means that for any $\sigma^2>0$ and $\delta>\frac{1}{2}$, the algorithm can asymptotically achieve a \emph{weak recovery} of $\sgl$: it is better than random guess, but $\sol$ always contains a nonzero fraction of errors. Moreover, one can show that
\begin{equation}\label{eq:asymp_E}
\lim_{\delta \to \infty} \mathcal{E}(\delta, \sigma^2)  = \lim_{\sigma^2 \to 0} \mathcal{E}(\delta, \sigma^2) = 0.
\end{equation}

The expressions in \eref{asymp_E}, together with \eref{asymp_BER}, suggest that the asymptotic BER can be made arbitrarily small if we increase the number of measurements or reduce the noise variance. This then raises a tantalizing question: is there a regime of $(\delta_p, \sigma_p^2)$ such that the box-relaxation decoder can \emph{perfectly} recover the target signal? Existing results in \cite{thrampoulidis2016ber,thrampoulidis2018symbol} cannot answer this question, for two reasons. First, $\text{BER} \overset{p \to \infty}{\longrightarrow} 0$ only guarantees that the \emph{number of error bits}
\begin{equation}
\neb\bydef\sum_{i=1}^{p}\I_{\{\soli_{i}\neq\beta_{i}\}},\label{eq:NEB}
\end{equation}
is sublinear in $p$, but it contains no information about the actual distribution of $N_e$, including whether $N_e = 0$. The second issue is subtle but important. It has to do with the specific order with which the limits are taken in \eref{asymp_BER} and \eref{asymp_E}. There, we first send the dimension $p \to \infty$ \emph{before} letting $\delta_p \to \infty$ or $\sigma_p^2 \to 0$. In practice, $p$ is large but always finite, and thus the speed with which $\delta_p \to \infty$ and $\sigma_p^2 \to 0$ [\emph{e.g.}, $\sigma_p^2 = \mathcal{O}(1/p)$ vs. $\sigma_p^2 = \mathcal{O}(1/\log p)$] makes all the difference.

The goal of this paper is to present a precise asymptotic characterization of the probability distribution of $N_e$. We show that, in certain scaling regimes of $(\delta_p, \sigma_p^2)$, the distribution of $N_e$ converges to a Poisson law. Moreover, we derive conditions under which the exact recovery of $\sgl$ is possible and provide an asymptotic formula for $\P(\neb=0)$ in the form of a Gumbel distribution.


TITLE: Is there any name for the vector space does not suppose zero vector?
QUESTION [2 upvotes]: I know the definition of the vector space includes the existence of the zero vector. Is there any theory on the space satisfying all the definitions of the vector space but the existence of the zero vector? Thank you.

REPLY [3 votes]: As discussed in the comments, affine space is essentially a vector space where we forget about the origin. Addition is no longer completely possible, but there is an associated vector space that acts on the affine space by translation, and we can still take linear combinations if the sum of the coefficients is $1$.


TITLE: Conditions of validity of basic fluid equations
QUESTION [0 upvotes]: I easily get confused about the conditions of validity of basic fluid equations. I list the conditions for these equations to hold true, as I could understand them.

$\rho v S=\mathrm{constant}$: steady flow
$ v S=\mathrm{constant}$: steady flow, $\rho=\mathrm{constant}$ (incompressible fluid)
$ \frac{1}{2}\rho v^2+\rho  g h+ p=\mathrm{constant}$: steady flow, $\rho=\mathrm{constant}$ (incompressible fluid), no viscosity $(\eta=0)$
$Q=\frac{\pi R^4 \Delta p}{8 \eta L}$: steady flow, laminar motion

Firstly I'm not sure if 2. really holds without the condition of $\eta=0$. That means that anytime 4. is valid (for a fluid with $\eta \neq 0$) then 2. is valid too (and so is 1.). Is this true?
Secondly, is there a condition on $\rho$ in 4.? Must $\rho$ be constant or not?
If I forgot any condition please suggest it, as I can understand better when I can use one or another equation.

REPLY [1 votes]: Equation 2 is the equation of continuity which does not depend on the viscosity of the fluid.
Equation 4 is the Poiseuille equation which assumes that the fluid is incompressible (density = constant) and the flow is laminar.

REPLY [1 votes]: Steady flow in a channel of constant cross sectional area.  Flat velocity profile, unless v is the average axial velocity.
Same as 1, except incompressible.
Also valid along streamline of steady viscous flow
Steady incompressible laminar flow of viscous fluid in a straight circular cylindrical horizontal pipe.  Constant temperature.