From 64ab996e4983c18bad0c1780437e807f535616cd Mon Sep 17 00:00:00 2001 From: Yu Cong Date: Sun, 13 Jul 2025 13:10:24 +0800 Subject: [PATCH] expected distortion --- distribution.pdf | Bin 64921 -> 66751 bytes distribution.tex | 18 ++++++++++++++---- 2 files changed, 14 insertions(+), 4 deletions(-) diff --git a/distribution.pdf b/distribution.pdf index e8f9c932ea2dfc31a0aa1782036802d254886a45..8baf77e689a92a15c2e1f8112137ec26c1b0038c 100644 GIT binary patch delta 45785 zcmV)KK)Sz~`vbp;1dt;IGcz$TAd`^`Cx6XaZExea5&oWE;r2^qG}iI`<$xCG-R|}k zSoAhPd&nJdj;$m{96M(_m)(Eghax4BGG%%9QnX(zS`<0Mnc>VcLz2negiQW?K?a`R zTwQ$ijw&*ts+3IG$<^&dvxz36RMJeYmXqs(PJg`mTRUQMJ)?w>;+yJlsCLsCvwy63 zv)UiFyTfX`@&8{|hx@K)x4L<*`!J0orY4k@LTg8hl-6Kc(4S}ul4_z1E4AX2t4EN& zy03N>W>U6T?frA^A1~EmNBRJzrFR( z>hY#pE?1kogvetB<`=%bf0{3lk{XMDSs*6O@%6I zE78#U=96!Zo5I;NC!*MR#-Fy(>i*LI*={P|xS#KH z(^Jbz1DnAFrBTvr?titm!~{W%vvx{YO=oLdqG=~EO>iDKrmUmTeVME!vDa%jjk}KD zr~~{uSq{w>m}P`U%+6?OrGK8x5|#*=a!q}M8r3vp*Y%sBh4cI-6Qq=ykMA;3YRM+i 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zye(Jd*3BHiX`BAae8(}L@qc$7+?*Nb;9#TxmmwUpx~-oj2{#uUvo?u7D+wD3>%TcQ zXJ-!*ZgvtD5;ZtxStl!J5^j$F8Ay=m3y4bch_kS;i?eWZu!~BGiL-J_i1Knta`E!A zvha#>N(hns|CjvRMFIAIng|xo6tI5}p_r43a-m=-5jQV=WtQ1iLHBuj>FI%bD`1@> z9*In|b>xsCjvkh4HnU}h`cWh@sG{FO+aTI2tg-rhx!7Z~KA^BNgWe`gu%uDIDO@AJ zm7z!?%*%|=1+j@4~-v9sr diff --git a/distribution.tex b/distribution.tex index a3a5ac6..a32a9f7 100644 --- a/distribution.tex +++ b/distribution.tex @@ -18,17 +18,27 @@ For $\ell_2$ the lowerbound is still $\Omega(\log n)$ 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. -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 on $\mathcal Y$. The original metric space $(X,d)$ embeds into $\pi$ with distortion $D$ if there is an $r>0$ such that for all $x,y\in X$, +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 $x,y\in X$, \[r\leq \frac{\E_{i\from \pi} [d_i(\alpha_i(x),\alpha_i(y))]}{d(x,y)}\leq Dr.\] -SODA23 paper also embeds $(X,d)$ into distribution. +SODA23 paper also embeds $(X,d)$ into distribution. We call this kind of embeddings stochastic embedding. -\subsection{Example: Random Trees} +\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 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 $O(\log n\log \Delta)$. +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 $O(\log n)$. \end{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} + +We want to beat Bourgain's in terms of the expected distortion. \end{document} \ No newline at end of file