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DOI: 10.4230/LIPIcs.SoCG.2026.6

Estimating the Persistent Homology of ℝⁿ-Valued Functions Using Function-Geometric Multifiltrations

Authors: Ethan André, Jingyi Li, David Loiseaux, and Steve Oudot

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

Abstract

Given an unknown ℝⁿ-valued function f on a metric space X, can we approximate the persistent homology of f from a finite sampling of X with known pairwise distances and function values? This question has been answered in the case n = 1, assuming f is Lipschitz continuous and X is a sufficiently regular geodesic metric space, and using filtered geometric complexes with fixed scale parameter for the approximation. In this paper we answer the question for arbitrary n, under similar assumptions and using function-geometric multifiltrations. Our analysis offers a different view on these multifiltrations by focusing on their approximation properties rather than on their stability properties. We also leverage the multiparameter setting to provide insight into the influence of the scale parameter, whose choice is central to this type of approach. From a practical standpoint, we show that our approximation results are robust to input noise, and that function-geometric multifiltrations have good statistical convergence properties. We also provide an algorithm to compute our estimators, and we use its implementation to conduct extensive experiments, on both synthetic and real biological data, in order to validate our theoretical results.

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Ethan André, Jingyi Li, David Loiseaux, and Steve Oudot. Estimating the Persistent Homology of ℝⁿ-Valued Functions Using Function-Geometric Multifiltrations. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{andre_et_al:LIPIcs.SoCG.2026.6,
  author =	{Andr\'{e}, Ethan and Li, Jingyi and Loiseaux, David and Oudot, Steve},
  title =	{{Estimating the Persistent Homology of \mathbb{R}ⁿ-Valued Functions Using Function-Geometric Multifiltrations}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{6:1--6:18},
  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.2026.6},
  URN =		{urn:nbn:de:0030-drops-258120},
  doi =		{10.4230/LIPIcs.SoCG.2026.6},
  annote =	{Keywords: Topological data analysis, multi-parameter persistent homology, function-Rips multifiltration}
}

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Estimating the Persistent Homology of ℝⁿ-Valued Functions Using Function-Geometric Multifiltrations

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