Nested and outlier embeddings into trees

In this paper, we consider outlier embeddings into HSTs. In particular, for metric $(X,d)$, let $k$ be the size of the smallest subset of $X$ such that all but that subset (the ``outlier set'') can be probabilistically embedded into the space of HSTs with expected distortion at most $c$. Our primary result is showing that there exists an efficient algorithm that takes in $(X,d)$ and a target distortion $c$ and samples from a probabilistic embedding with at most $O(\frac k \epsilon \log^2k)$ outliers and distortion at most $(32+\epsilon)c$, for any $\epsilon>0$. In order to facilitate our results, we show how to find good nested embeddings into HSTs and combine this with an approximation algorithm of Munagala et al. [MST23] to obtain our results.