2026-08-17T11:26:19Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:20018 2024-06-06T06:21:17Z ddc:004 open_access

Demystifying Latschev’s Theorem: Manifold Reconstruction from Noisy Data Majhi, Sushovan Vietoris-Rips complex submanifold reconstruction manifold reconstruction Latschev’s theorem homotopy Equivalence For a closed Riemannian manifold ℳ and a metric space S with a small Gromov-Hausdorff distance to it, Latschev’s theorem guarantees the existence of a sufficiently small scale β > 0 at which the Vietoris-Rips complex of S is homotopy equivalent to ℳ. Despite being regarded as a stepping stone to the topological reconstruction of Riemannian manifolds from a noisy data, the result is only a qualitative guarantee. Until now, it had been elusive how to quantitatively choose such a proximity scale β in order to provide sampling conditions for S to be homotopy equivalent to ℳ. In this paper, we prove a stronger and pragmatic version of Latschev’s theorem, facilitating a simple description of β using the sectional curvatures and convexity radius of ℳ as the sampling parameters. Our study also delves into the topological recovery of a closed Euclidean submanifold from the Vietoris-Rips complexes of a Hausdorff close Euclidean subset. As already known for Čech complexes, we show that Vietoris-Rips complexes also provide topologically faithful reconstruction guarantees for submanifolds. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Sushovan Majhi 2024 Is Part Of LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2024.73 urn:nbn:de:0030-drops-200188 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.73 eng https://creativecommons.org/licenses/by/4.0/legalcode