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A Persistent Version of Latschev’s Theorem

Authors Steve Oudot , Lukas Waas

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LIPIcs.SoCG.2026.82.pdf
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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Algebraic topology
Keywords
  • Topological data analysis (TDA)
  • metric geometry
  • Vietoris-Rips complex
  • homotopy theory
  • multi-parameter persistent homology

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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.

Cite As Get BibTex

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) https://doi.org/10.4230/LIPIcs.SoCG.2026.82

Author Details

Steve Oudot
  • Inria Saclay, and École Polytechnique, Palaiseau, France
Lukas Waas
  • University of Oxford, UK

Funding

  • Waas, Lukas: Lukas Waas acknowledges support from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2181/1 - 390900948 (the Heidelberg STRUCTURES Excellence Cluster). Furthermore, Lukas Waas is a member of the Oxford/Max Planck collaboration and this research was funded in in part by EPSRC international centre to centre collaboration grant EP/Z531224/1.

Acknowledgements

This work started when the two authors attended the TDA week at Kyoto University in June 2025. It then continued as the two authors participated in the thematic program Topological Data Analysis, Persistence And Representation Theory Intertwined (TP25TD) at the Okinawa Institute of Science and Technology, from June to August 2025.

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