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Computing the Bottleneck Distance Between Persistent Homology Transforms

Authors Michael Kerber , Elena Xinyi Wang

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LIPIcs.SoCG.2026.62.pdf
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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Algebraic topology
  • Theory of computation → Computational geometry
Keywords
  • Kinetic data structure
  • bottleneck distance
  • persistent homology transform
  • vineyards

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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) https://doi.org/10.4230/LIPIcs.SoCG.2026.62

Author Details

Michael Kerber
  • Institute of Geometry, Graz University of Technology, Austria
Elena Xinyi Wang
  • Institute of Geometry, Graz University of Technology, Austria
  • University of Fribourg, Switzerland

Funding

  • Kerber, Michael: Supported by the Austrian Science Fund (FWF) 10.55776/P33765.
  • Wang, Elena Xinyi: Supported by the Austrian Science Fund (FWF) 10.55776/P33765.

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