65 lines
3.8 KiB
TeX
65 lines
3.8 KiB
TeX
\documentclass[12pt]{article}
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\usepackage{chao}
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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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Let $(X,d)$ be a finite metric space and let $\mathcal Y=\{ (Y_1,d_1),\ldots (Y_h,d_h) \}$ be a set of metric spaces. Let $\pi$ be a distribution of embeddings into $\mathcal Y$. 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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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 with diameter $\Delta$, 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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% 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 an algorithm that finds a desired embedding with probability at least $1/2$.\footnote{\url{https://home.ttic.edu/~harry/teaching/pdf/lecture3.pdf}} For the $\ell_2$ case, the embedding has the following bounds:
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\begin{itemize}
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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{itemize}
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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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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 any dimension $i$ the event that distance $|f_i(x)-f_i(y)|^2$ is much smaller than $d(x,y)^2$ happends with high probability.
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Now consider a subset $S_i$ by sampling each node in $X$ iid with probability $2^{-i}$.
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\end{document} |