embed into distribution of l2

distribution.tex

@@ -27,7 +27,7 @@ SODA23 paper also embeds $(X,d)$ into distribution. We call this kind of embeddi
 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 $O(\log n)$.
+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.
@@ -39,6 +39,25 @@ We first ignore the outlier condition and see if stochastic embeddings break the
 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. 
+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:
+\begin{itemize}
+\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{itemize}
+
+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. 
+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 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.
 
 \end{document}