Bi-Lipschitz extensions and outlier embeddings into trees

We develop low distortion embeddings with outliers from arbitrary metrics into hierarchically separated trees (HSTs). In particular, we develop an efficient algorithm that for any $\epsilon>0$, given an input metric $(X,d)$, and a probabilistic embedding of all but $k$ points from $X$ into HSTs with distortion $c$, samples from a probabilistic embedding of all but $O(\frac{k}{\epsilon}\log k)$ points into HSTs that achieves distortion at most $(32+\epsilon)c$. Our results are based on two key technical components. First, we extend an algorithm of Munagala et al. [2023] for minimizing the distortion of embeddings without outliers into HSTs to the setting with outliers. We combine this with new results on bi-Lipschitz extensions into trees and $\ell_1$ space. In particular, we show that any probabilistic embedding into HSTs can be extended to $k$ additional points with only a factor $O(\log k)$ of additional distortion. This bi-Lipschitz extension result utilizes a new probabilistic partitioning scheme that we call onion partitioning.