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DOI: 10.4230/LIPIcs.SoCG.2026.82

A Persistent Version of Latschev’s Theorem

Authors: Steve Oudot and Lukas Waas

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

Abstract

Latschev’s theorem provides sufficient conditions on a metric space M and δ > 0 for the homotopy type of M to agree with that of the Vietoris-Rips complex ℛ^δ(𝕄) of any nearby space 𝕄 in the Gromov-Hausdorff distance. We prove a persistent version of this theorem, providing sufficient conditions on a pair (M, f:M → ℝ^N) and δ > 0 for the persistent homotopy type of the sublevel set filtration of (M, f) to be interleaved with that of the function-Rips complex ℛ^δ(𝕄^•) of any nearby pair (𝕄, 𝕗). In particular, our result answers a longstanding question on the related topic of estimating sublevel set persistent homology from finite point samples.

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Steve Oudot and Lukas Waas. A Persistent Version of Latschev’s Theorem. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 82:1-82:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{oudot_et_al:LIPIcs.SoCG.2026.82,
  author =	{Oudot, Steve and Waas, Lukas},
  title =	{{A Persistent Version of Latschev’s Theorem}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{82:1--82:17},
  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.2026.82},
  URN =		{urn:nbn:de:0030-drops-258891},
  doi =		{10.4230/LIPIcs.SoCG.2026.82},
  annote =	{Keywords: Topological data analysis (TDA), metric geometry, Vietoris-Rips complex, homotopy theory, multi-parameter persistent homology}
}

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A Persistent Version of Latschev’s Theorem

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