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 $O(\frac k ε\log^2k)$ outliers and distortion at most $(32+ε)c$, for any $ε>0$. 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.

💡 Research Summary

This paper studies probabilistic embeddings of finite metric spaces into hierarchically separated trees (HSTs) while allowing a small set of outliers. For a metric ((X,d)) let (k) denote the size of the smallest subset whose removal enables an embedding of the remaining points into HSTs with expected distortion at most (c). The authors present an efficient algorithm that, given any target distortion (c) and any (\varepsilon>0), produces a probabilistic embedding with expected distortion at most ((32+\varepsilon)c) and an outlier set of size (O!\big(\frac{k}{\varepsilon}\log^{2}k\big)).

The technical contribution rests on two main ideas. First, the authors extend the notion of “nested embeddings”—originally developed for deterministic embeddings—into the probabilistic HST setting. A nested embedding combines a high‑quality embedding on a core subset (S\subseteq X) with embeddings on all small supersets of (S) (size at most (|X\setminus S|+1)). The challenge is to control both expansion and contraction simultaneously, because the linear program (LP) used for outlier‑aware embeddings contains outlier variables (\delta_i) that relax distance constraints when set to 1. By carefully designing the LP’s level‑wise labeling constraints, the authors guarantee that even fractional outliers must be assigned a representative within distance (2^i) at every level (i).

Second, the authors address the subtle issue that naïvely extending a random embedding of (S) to the whole set can increase expected distortion dramatically. They prove an “obliviousness” property of HSTs: when each partial embedding is sampled independently, the expected distance between any pair ((u,v)) in the final embedding equals the expectation of the distances in the individual samples, not the maximum over samples. This property allows the composition of partial embeddings without blowing up the distortion.

Algorithmically, the method proceeds as follows. An extended LP is solved to obtain (i) a fractional outlier set, (ii) a probabilistic embedding (D_S) on a non‑outlier core (S), and (iii) probabilistic embeddings (D_{K’}) for every subset (K’) of size at most (k+1). The set (X\setminus S) is randomly partitioned into groups (K_1,\dots,K_t). For each group a “close” representative (\gamma_i\in S) is selected, and the embedding on (K_i\cup{\gamma_i}) is merged with (D_S) using a carefully designed merge operation that preserves both expansion and contraction bounds. Repeating this over all groups yields a single probabilistic embedding (D) of the entire metric. The merge operation is proved to satisfy the nested‑embedding definition, guaranteeing that pairs inside (S) suffer at most a factor (4c_S) distortion, while arbitrary pairs incur distortion bounded by a constant times ((c_S\log k + \log^2 k)).

By plugging this nested‑embedding theorem (Theorem 2) into the LP framework of Munagala et al., the authors obtain the main bicriteria approximation (Theorem 1). The result holds for unweighted outlier sets as well as for weighted versions, where the cost of an outlier set is the sum of given vertex weights; in the weighted case the algorithm outputs an outlier set whose total weight is at most (O!\big(\frac{1}{\varepsilon}w^*\log^2 k\big)) while preserving the same distortion guarantee (Corollary 1).

The paper situates its contributions within a rich literature on metric embeddings, noting that classic results achieve (O(\log n)) uniform distortion for all metrics, whereas the present work achieves instance‑optimal distortion when a small number of outliers can be removed. It also contrasts with prior work on deterministic outlier embeddings and on embeddings with additive slack, emphasizing that multiplicative distortion is essential for algorithmic applications.

In summary, the authors provide a novel LP‑based algorithmic framework that (1) extends nested embeddings to the probabilistic HST setting, (2) controls both expansion and contraction for outlier‑aware embeddings, and (3) yields a practical bicriteria approximation: a small outlier set (polylogarithmic in its optimal size) and a constant‑factor increase over the target distortion. The techniques open avenues for tighter constants, dynamic settings, and extensions to other target spaces such as low‑dimensional Euclidean spaces.