Nested and outlier embeddings into trees

2026-01-21

AI Summary

This work addresses the problem of efficiently embedding metric spaces into hierarchical stochastic trees (HSTs) while allowing a small number of outliers to control overall distortion. We propose a probabilistic embedding framework with outliers that generalizes nested embeddings from deterministic to probabilistic settings and design an efficient algorithm achieving near-optimal trade-offs between the number of outliers and distortion. Our method guarantees low distortion on a core subset while bounding the distortion growth for the remaining points, yielding an embedding with at most $O(k/\varepsilon \log^2 k)$ outliers and distortion no more than $(32+\varepsilon)c$. This significantly improves instance-dependent approximation guarantees for problems such as buy-at-bulk network design and dial-a-ride.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Data Mining & Knowledge Management: Anomaly/Outlier Detection
• Search and Optimization: Distributed Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• outlier embeddings
• HSTs
• ultrametrics
• probabilistic embeddings
• metric embedding

Innovation

• nested embeddings
• probabilistic embeddings
• outlier embeddings
• HSTs
• metric embedding