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Demystifying Latschev’s Theorem: Manifold Reconstruction from Noisy Data

Author Sushovan Majhi

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Sushovan Majhi
  • George Washington University, Washington D.C., USA

Acknowledgements

The author would like to thank Henry Adams for providing constructive feedback on the manuscript, and for hosting the author at the University of Florida, Gainesville during the summer of 2023.

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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) https://doi.org/10.4230/LIPIcs.SoCG.2024.73

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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Algebraic topology
Keywords
  • Vietoris-Rips complex
  • submanifold reconstruction
  • manifold reconstruction
  • Latschev’s theorem
  • homotopy Equivalence

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