86 lines
5.1 KiB
TeX
86 lines
5.1 KiB
TeX
\documentclass[12pt]{article}
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\usepackage{chao}
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\usepackage{algo}
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\title{Outlier Embedding Notes}
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\begin{document}
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\section{Better Distortion with Distribution}
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There is a well known lowerbound for the distortion of embedding a metric space $(X,d)$ into $\ell_1$.
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\begin{theorem}
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For any metric space $(X,d)$ on $n$ points, one has
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\[(X,d) \lhook\joinrel\xrightarrow{\Omega(\log n)} \ell_1. \]
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\end{theorem}
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For $\ell_2$ the lowerbound is still $\Omega(\log n)$
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\footnote{\url{https://web.stanford.edu/class/cs369m/cs369mlecture1.pdf}}.
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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 $\ell_2$ with distortion $O(k)$, which contradicts the lowerbound. However, maybe we can do $O(k)$ via embedding into some distribution of $\ell_2$ metrics.
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\begin{definition}[Expected distortion.]
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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 $x,y\in X$,
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\[r\leq \frac{\E_{i\from \pi} [d_i(\alpha_i(x),\alpha_i(y))]}{d(x,y)}\leq Dr.\]
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\end{definition}
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SODA23 paper also embeds $(X,d)$ into distribution. We call this kind of embeddings stochastic embedding.
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\paragraph{Example: Random Trees}
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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 $O(\log n)$.
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\begin{theorem}[Bartal]
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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 $O(\log n)$.
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\end{theorem}
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Is there any other known result on expected distortion of embeddings besides Bartal's theorem?
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% 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.
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\section{Stochastic Embedding into \texorpdfstring{$\ell_2$}{l2}}
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We first ignore the outlier condition and see if stochastic embeddings break the $\Omega(\log n)$ lowerbound.
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\begin{theorem}[Bourgain]
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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 $O(\log n)$.
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\end{theorem}
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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}}
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Can we get better expected distortion by repeating the algorithm and uniformly selecting an embedding?
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For the $\ell_2$ case, the embedding has the following bounds:
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\begin{enumerate}
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\item Expansion. $\|f(x)-f(y)\|_2\leq O(\log n) d(x,y)$
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\item Contraction. $\|f(x)-f(y)\|_2 \geq \frac{d(x,y)}{O(1)}$
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\end{enumerate}
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The contraction bound is almost tight. Let $K$ be the dimension of the target space. For the expansion bound, we have
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\begin{equation*}
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\begin{aligned}
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\|f(x)-f(y)\|_2 &= \left( \sum_{i=1}^{K} |f_i(x)-f_i(y)|^2\right)^{1/2}\\
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&\leq \left( \sum_{i=1}^{K} d(x,y)^2\right)^{1/2}\\
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&=\sqrt{K} d(x,y)\\
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&=O(\log n) d(x,y)
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\end{aligned}
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\end{equation*}
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One thing we can try is to tighten the second line.
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\begin{algo}
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\underline{Bourgain's construction}:\\
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$m=576\log n$\\
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for $j=1$ to $\log n$:\\
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\quad for $i=1$ to $m$:\\
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\quad \quad choose set $S_{ij}$ by sampling each node in $X$ independently with probability $2^{-j}$\\
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\quad \quad $f_{ij}(x)=\min_{s\in S_{ij}} d(x,s)$\\
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$f(x)=\bigoplus_{j=1}^{\log n} \bigoplus_{i=1}^m f_{ij}(x)$ for all $x\in X$.
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\end{algo}
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% Recall that for each dimension $i$ a random subset $S_i\subset X$ is selected and the value of $f_i(x)$ is $\min_{s\in S_i} d(x,s)$.
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We want to show that for any fixed $x,y\in X$ and $j$,
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\[
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\Pr[|f_{ij}(x)-f_{ij}(y)|\leq \frac{d(x,y)}{\polylog n}]\geq ???
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\]
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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.
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\end{document} |