DROPS

Document

DOI: 10.4230/LIPIcs.SoCG.2024.73

Demystifying Latschev’s Theorem: Manifold Reconstruction from Noisy Data

Authors: Sushovan Majhi

Published in: LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

Abstract

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.

Cite as

Sushovan Majhi. Demystifying Latschev’s Theorem: Manifold Reconstruction from Noisy Data. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 73:1-73:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{majhi:LIPIcs.SoCG.2024.73,
  author =	{Majhi, Sushovan},
  title =	{{Demystifying Latschev’s Theorem: Manifold Reconstruction from Noisy Data}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{73:1--73:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-316-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{293},
  editor =	{Mulzer, Wolfgang and Phillips, Jeff M.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.73},
  URN =		{urn:nbn:de:0030-drops-200188},
  doi =		{10.4230/LIPIcs.SoCG.2024.73},
  annote =	{Keywords: Vietoris-Rips complex, submanifold reconstruction, manifold reconstruction, Latschev’s theorem, homotopy Equivalence}
}

Search Results

Documents authored by Majhi, Sushovan

Demystifying Latschev’s Theorem: Manifold Reconstruction from Noisy Data

Abstract

Cite as

Thanks for your feedback!

Could not send message