wrong lb

distribution.tex

@@ -1,6 +1,7 @@
 \documentclass[12pt]{article}
 \usepackage{chao}
 \usepackage{algo}
+\usepackage[normalem]{ulem}
 
 \title{Outlier Embedding Notes}
 
@@ -17,7 +18,7 @@ For any metric space $(X,d)$ on $n$ points, one has
 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 $\ell_2$ with distortion $O(k)$, which contradicts the lowerbound. However, maybe we can do $O(k)$ via embedding into some distribution of $\ell_2$ metrics.
+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 $x,y\in X$,
@@ -26,6 +27,23 @@ Let $(X,d)$ be the original metric space and let $\mathcal Y=\{ (Y_1,d_1),\ldots
 
 SODA23 paper 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 $O(\log n)$.