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\title{Outlier Embedding Notes}

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\section{Better Distortion with Distribution}
There is a well known lowerbound for the distortion of embedding a metric space $(X,d)$ into $\ell_1$.

\begin{theorem}
For any metric space $(X,d)$ on $n$ points, one has
\[(X,d) \lhook\joinrel\xrightarrow{\Omega(\log n)} \ell_1. \]
\end{theorem}

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.

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. We call this kind of embeddings stochastic embedding.

\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 $O(\log n)$.
\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. 

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