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Bi-Lipschitz Extensions and Outlier Embeddings into Trees

Authors Shuchi Chawla , Arnold Filtser , Kristin Sheridan , Yonatan Trachtenberg

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ACM Subject Classification
  • Theory of computation → Rounding techniques
Keywords
  • metric embeddings
  • hierarchically separated trees
  • outliers

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Abstract

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 ε > 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((k/ε)log k) points into HSTs that achieves distortion at most (32+ε)c. 
Our results are based on two key technical components. First, we extend an algorithm of Munagala et al. [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 𝓁₁ 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.

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Shuchi Chawla, Arnold Filtser, Kristin Sheridan, and Yonatan Trachtenberg. Bi-Lipschitz Extensions and Outlier Embeddings into Trees. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 17:1-17:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2026.17

Author Details

Shuchi Chawla
  • Department of Computer Science, University of Texas at Austin, TX, USA
Arnold Filtser
  • Department of Computer Science, Bar-Ilan University, Ramat Gan, Israel
Kristin Sheridan
  • Department of Computer Science, University of Texas at Austin, TX, USA
Yonatan Trachtenberg
  • Department of Computer Science, Bar-Ilan University, Ramat Gan, Israel

Funding

  • Chawla, Shuchi: This research was funded in part by NSF awards CCF-2217069 and CCF-2225259.
  • Filtser, Arnold: This research was supported by the ISRAEL SCIENCE FOUNDATION (grant No. 1042/22).
  • Sheridan, Kristin: This research was funded in part by NSF awards CCF-2217069 and CCF-2225259 and by JPMorgan Chase & Co. Any views or opinions expressed herein are solely those of the authors listed, and may differ from the views and opinions expressed by JPMorgan Chase & Co. or its affiliates. This material is not a product of the Research Department of J.P. Morgan Securities LLC. This material should not be construed as an individual recommendation for any particular client and is not intended as a recommendation of particular securities, financial instruments or strategies for a particular client. This material does not constitute a solicitation or offer in any jurisdiction.

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