\documentclass[12pt]{article}
\usepackage{chao}
\usepackage{algo}
\usepackage[normalem]{ulem}

\title{Outlier Embedding Notes}

\begin{document}

\section{Better Distortion with Distribution}
There is a well known lowerbound for the distortion of embedding a metric space $(X,d)$ into $\ell_1$.

\begin{theorem}
For any metric space $(X,d)$ on $n$ points, one has
\[(X,d) \lhook\joinrel\xrightarrow{\Omega(\log n)} \ell_1. \]
\end{theorem}

For $\ell_2$ the lowerbound is still $\Omega(\log n)$
\footnote{\url{https://web.stanford.edu/class/cs369m/cs369mlecture1.pdf}}.

Recall that we want to find a $(O(k),(1+\e)c)$-outlier embedding into $\ell_2$ for any metric space $(X,d)$ which admits a $(k,c)$-outlier embedding into $\ell_2$. If we can do this deterministically, we actually find an embedding of the outlier points into $\ell_2$ \sout{with distortion $O(k)$, which contradicts the lowerbound}. This is not true! The $\log k$ factor is required by SDP and only expansion bound is needed. We do not have to bound the contraction part. However, maybe we can do $O(k)$ via embedding into some distribution of $\ell_2$ metrics.

\begin{definition}[Expected distortion]
Let $(X,d)$ be the original metric space and let $\mathcal Y=\{ (Y_1,d_1),\ldots (Y_h,d_h) \}$ be a set of target spaces. Let $\pi$ be a distribution of embeddings into $\mathcal Y$. To be more precise, for each target space $(Y_i,d_i)$ we define an embedding $\alpha_i:X\to Y_i$ and define the probability of choosing this embedding to be $p_i$. The original metric space $(X,d)$ embeds into $\pi$ with distortion $D$ if there is an $r>0$ such that for all $x,y\in X$,
\[r\leq \frac{\E_{i\from \pi} [d_i(\alpha_i(x),\alpha_i(y))]}{d(x,y)}\leq Dr.\]
\end{definition}

Note that if we compute the minimum $D$ for all $x,y$ pair and take the average, the resulting value is called the average distortion.\footnote{\url{https://www.cs.huji.ac.il/w~ittaia/papers/ABN-STOC06.pdf}} There is an embedding into $\ell_p$ with constant average distortion for arbitrary metric spaces, while maintaining the same worst case bound provided by Bourgain's theorem.

The outlier paper (SODA23) also embeds $(X,d)$ into distribution. We call this kind of embeddings stochastic embedding.

\begin{lemma}
Let $\pi$ be a stochastic embedding into $\ell_p$ with expected expansion bound $\E_{i\from \pi}\|\alpha_i(x)-\alpha_i(y)\|_p\leq  c_{\E}d(x,y)$. Then there is a deterministic embedding into  $\ell_p$ with the same expansion bound.
\end{lemma}
\begin{proof}
We define a new averaged embedding $\alpha^*(x)=\sum_{i\from \pi} \alpha_i(x) p_i$. Consider the expansion bound for $\alpha^*$.
\begin{equation*}
\begin{aligned}
\| \alpha^*(x)- \alpha^*(y) \|_p & = \left\| \sum_{i\from \pi}  p_i ( \alpha_i(x) - \alpha_i(y) ) \right\|_p\\
    &\leq \sum_{i\from \pi} \| p_i ( \alpha_i(x) - \alpha_i(y) ) \|_p\\
    &= \sum_{i\from \pi}p_i  \| ( \alpha_i(x) - \alpha_i(y) ) \|_p\\
    &\leq c_{\E} d(x,y)
\end{aligned}
\end{equation*}
\end{proof}

Note that one cannot derive contraction bound for $\alpha^*$ from the stochastic embedding. So the distortion may not be the same.

\paragraph{Example: Random Trees}
Consider the problem of embedding some finite metric into a tree metric. We can get an $O(n)$ lowerbound via the unit edge length cycle $C_n$. However, if embedding into distortions is allowed, we can do $O(\log n)$.

\begin{theorem}[Bartal]
Let $(X,d)$ be a metric space on $n$ points, let $\mathcal D T$ be the set of tree metrics that dominate $d$, there is a distribution $\pi$ on $\mathcal D T$ such that $(X,d)$ embeds into $\pi$ with distortion $O(\log n)$.
\end{theorem}

Is there any other known result on expected distortion of embeddings besides Bartal's theorem?

% A kind of embedding problems which are closely related to outlier embeddings is Ramsey type embedding. Let $(X,d_X)$ be the original metric space and let $(Y,d_Y)$ be the target space. Given a fixed distortion $c$, Ramsey type embedding asks for the largest subset $Z$ of $X$ such that $(Z,d_X)$ embeds into $(Y,d_Y)$ with distortion at most $c$. This is the same as computing the smallest outlier set.

\section{Stochastic Embedding into \texorpdfstring{$\ell_2$}{l2}}
We first ignore the outlier condition and see if stochastic embeddings break the $\Omega(\log n)$ lowerbound.

\begin{theorem}[Bourgain]
For any metric space $(X,d)$ and for any $p$, there is an embedding of $(X,d)$ into $\ell_p^{O(\log^2 n)}$ with distortion $O(\log n)$.
\end{theorem}

Bourgain develops a randomized algorithm that finds a desired embedding.\footnote{The expansion bound always holds. The contraction bound holds with probability at least $1/2$. See \url{https://home.ttic.edu/~harry/teaching/pdf/lecture3.pdf}}
Can we get better expected distortion by repeating the algorithm and uniformly selecting an embedding?
For the $\ell_2$ case, the embedding has the following bounds:
\begin{enumerate}
\item Expansion. $\|f(x)-f(y)\|_2\leq O(\log n) d(x,y)$
\item Contraction. $\|f(x)-f(y)\|_2 \geq \frac{d(x,y)}{O(1)}$
\end{enumerate}

The contraction bound is almost tight. Let $K$ be the dimension of the target space. For the expansion bound, we have

\begin{equation*}
\begin{aligned}
\|f(x)-f(y)\|_2 &= \left( \sum_{i=1}^{K} |f_i(x)-f_i(y)|^2\right)^{1/2}\\
    &\leq \left( \sum_{i=1}^{K} d(x,y)^2\right)^{1/2}\\
    &=\sqrt{K} d(x,y)\\
    &=O(\log n) d(x,y)
\end{aligned}
\end{equation*}

One thing we can try is to tighten the second line.

\begin{algo}
\underline{Bourgain's construction}:\\
$m=576\log n$\\
for $j=1$ to $\log n$:\\
\quad for $i=1$ to $m$:\\
\quad \quad choose set $S_{ij}$ by sampling each node in $X$ independently with probability $2^{-j}$\\
\quad \quad $f_{ij}(x)=\min_{s\in S_{ij}} d(x,s)$\\
$f(x)=\bigoplus_{j=1}^{\log n} \bigoplus_{i=1}^m f_{ij}(x)$ for all $x\in X$.
\end{algo}

% Recall that for each dimension $i$ a random subset $S_i\subset X$ is selected and the value of $f_i(x)$ is $\min_{s\in S_i} d(x,s)$.
We want to show that for any fixed $x,y\in X$ and $j$,
\[
\Pr[|f_{ij}(x)-f_{ij}(y)|\leq \frac{d(x,y)}{\polylog n}]\geq ???
\]

One can see that our desired event does not happen with high probability for any pair of $x,y$. Let the original metric space be a line metric with $n$ points. $x,y$ locate on two endpoints of an interval and the rest $n-2$ points locate on the middle of $xy$. Then our metric in the target space $|f_{ij}(x)-f_{ij}(y)|$ is a $\polylog n$ factor smaller than $d(x,y)$ if and only if both $x$ and $y$ are selected in $S_{ij}$, which happens with probability $4^{-j}$. This example shows that Bourgain's construction is tight up to a constant factor for some metric space.

\section{Grid}

Recall that we need an algorithm that outputs an embedding which extends a $(k,c)$-outlier embedding into $\ell_2$ and we want the extended embedding to have a good (expected) expansion bound.

\begin{conjecture}\label{conj:expansion}
Let $(X,d)$ be a metric space such that $|X|=n$ and $\alpha: X\setminus K \to \R^d$ be a $(k,c)$-outlier embedding of $(X,d)$ into $\ell_2^{d}$, where $K\subset X$ is the outlier set. Then there exist an embedding $\beta: X\to \R^d$ such that $\beta$ completes $\alpha$ and has expansion bound
\[
\max_{x,y\in X} \frac{\norm{\beta(x)-\beta(y)}_2}{d(x,y)}\leq O(c\sqrt{\log k}).
\]
\end{conjecture}

In their bi-criteria approximation the dimension $d$ is not important and therefore is considered as a fixed parameter. \autoref{conj:expansion} provides more tools than theorem 2.6, i.e. we know the coordinates of non-outlier points in the embedding $\beta$ and we can use coordinates in $\R^d$ instead of simply mapping non-outlier points to outliers.

A common and powerful method is to use grid. We divide $\R^d$ into identical hypercubes of some sidelength $s$ and working with grid cells instead of points. However, this method often involves the dimension $d$, which is not desirable...

\end{document}