Nested and outlier embeddings into trees

TL;DR

This paper introduces an efficient algorithm for embedding metric spaces into Hierarchically Separated Trees (HSTs) with outlier handling, achieving low distortion and few outliers, advancing metric embedding techniques.

Contribution

The paper presents a novel algorithm for probabilistic outlier embeddings into HSTs with controlled distortion and outlier count, combining nested embedding methods with existing approximation algorithms.

Findings

Efficient algorithm for outlier embeddings with low distortion.

Bound on outliers proportional to (k/psilon) log^2k.

Achieves probabilistic embeddings with at most O(k/psilon) outliers.

Abstract

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.