From 5502e7d20dd3c7b33e8bd3563649cbce1a7a0787 Mon Sep 17 00:00:00 2001
From: Yu Cong <sxlxcsxlxc@gmail.com>
Date: Tue, 22 Jul 2025 00:17:32 +0800
Subject: [PATCH] 18 log n lemma

---
 distribution.tex | 22 +++++++++++++++-------
 1 file changed, 15 insertions(+), 7 deletions(-)

diff --git a/distribution.tex b/distribution.tex
index bf4fbb2..1182e42 100644
--- a/distribution.tex
+++ b/distribution.tex
@@ -39,11 +39,11 @@ 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}
 
-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}
+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}} For the $\ell_2$ case, the embedding has the following bounds:
+\begin{enumerate}
+\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{enumerate}
 
 The contraction bound is almost tight. Let $K$ be the dimension of the target space. For the expansion bound, we have
 
@@ -58,8 +58,16 @@ The contraction bound is almost tight. Let $K$ be the dimension of the target sp
 
 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.
+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 $d(x,y)^2$ happens with high probability.
 
-Now consider a subset $S_i$ by sampling each node in $X$ iid with probability $2^{-i}$.
+Now consider a subset $S_j$ by sampling each node in $X$ iid with probability $2^{-j}$. We independently repeat this process $m=576\log n$ times and denote by $S_{ij}$ the sampled set for $i\in [m]$. A~free lemma is the following.
+
+\begin{lemma}
+For fixed $x,y\in X$ and $j$,
+\[
+\Pr[\text{for at least $18\log n$ values of $i$, $|f_{ij}(x)-f_{ij}(y)|\geq (\rho_j -\rho_{j-1})$}]\geq 1-\frac{1}{n^3},
+\]
+where $\rho_j$ is the smallest radius for which $|B(x,\rho_j)|\geq 2^j$ and $|B(y,\rho_j)|\geq 2^j$.
+\end{lemma}
 
 \end{document}
\ No newline at end of file