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DOI: 10.4230/LIPIcs.SoCG.2026.62

Computing the Bottleneck Distance Between Persistent Homology Transforms

Authors: Michael Kerber and Elena Xinyi Wang

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract

The Persistent Homology Transform (PHT) summarizes a shape in ℝ^m by collecting persistence diagrams obtained from linear height filtrations in all directions on 𝕊^{m-1}. It enjoys strong theoretical guarantees, including continuity, stability, and injectivity. A natural way to compare two PHTs is to use the bottleneck distance between their diagrams as the direction varies. Prior work has either compared PHTs by sampling directions or, in 2D, computed the exact integral of bottleneck distance over all angles via a kinetic data structure. We improve the integral objective to Õ(n⁵) in place of the earlier Õ(n⁶) bound, where n denotes the number of simplices. For the max objective, we give an Õ(n³) expected-time algorithm in ℝ² and an Õ(n⁵) expected-time algorithm in ℝ³.

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Michael Kerber and Elena Xinyi Wang. Computing the Bottleneck Distance Between Persistent Homology Transforms. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 62:1-62:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{kerber_et_al:LIPIcs.SoCG.2026.62,
  author =	{Kerber, Michael and Wang, Elena Xinyi},
  title =	{{Computing the Bottleneck Distance Between Persistent Homology Transforms}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{62:1--62:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.62},
  URN =		{urn:nbn:de:0030-drops-258693},
  doi =		{10.4230/LIPIcs.SoCG.2026.62},
  annote =	{Keywords: Kinetic data structure, bottleneck distance, persistent homology transform, vineyards}
}

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Computing the Bottleneck Distance Between Persistent Homology Transforms

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