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Demystifying Latschev's Theorem: Manifold Reconstruction from Noisy Data

Sushovan Majhi

Abstract

For a closed Riemannian manifold $\mathcal{M}$ and a metric space $S$ with a small Gromov$\unicode{x2013}$Hausdorff distance to it, Latschev's theorem guarantees the existence of a sufficiently small scale $β>0$ at which the Vietoris$\unicode{x2013}$Rips complex of $S$ is homotopy equivalent to $\mathcal{M}$. 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 $\mathcal{M}$. 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 $\mathcal{M}$ as the sampling parameters. Our study also delves into the topological recovery of a closed Euclidean submanifold from the Vietoris$\unicode{x2013}$Rips complexes of a Hausdorff close Euclidean subset. As already known for Čech complexes, we show that Vietoris$\unicode{x2013}$Rips complexes also provide topologically faithful reconstruction guarantees for submanifolds.

Demystifying Latschev's Theorem: Manifold Reconstruction from Noisy Data

Abstract

For a closed Riemannian manifold and a metric space with a small GromovHausdorff distance to it, Latschev's theorem guarantees the existence of a sufficiently small scale at which the VietorisRips complex of is homotopy equivalent to . 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 in order to provide sampling conditions for to be homotopy equivalent to . In this paper, we prove a stronger and pragmatic version of Latschev's theorem, facilitating a simple description of using the sectional curvatures and convexity radius of as the sampling parameters. Our study also delves into the topological recovery of a closed Euclidean submanifold from the VietorisRips complexes of a Hausdorff close Euclidean subset. As already known for Čech complexes, we show that VietorisRips complexes also provide topologically faithful reconstruction guarantees for submanifolds.
Paper Structure (18 sections, 14 theorems, 63 equations)

This paper contains 18 sections, 14 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Related Work
  4. Our Contribution
  5. Preliminaries
  6. Metric Spaces
  7. Simplicial Complexes
  8. Barycentric Subdivision
  9. Riemannian Manifolds
  10. Abstract Manifold Reconstruction
  11. Jung's Theorem in Riemannian Manifolds
  12. Homotopy Equivalence
  13. Hausdorff Reconstruction of Euclidean Submanifolds
  14. Conclusion
  15. Acknowledgments
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $({\mathcal{M}},d_{\mathcal{M}})$ be a closed, connected Riemannian manifold. Let $(S,d_S)$ be a compact metric space and $\beta>0$ a number such that for some $0<\zeta\leq1/14$. Then, $\lvert \mathcal{R}_\beta(S) \rvert\simeq{\mathcal{M}}$.

Theorems & Definitions (28)

  • Theorem 1.1: Manifold Reconstruction under Gromov--Hausdorff Distance
  • Theorem 1.1: Submanifold Reconstruction under Hausdorff Distance
  • Definition 2.1: Diameter
  • Definition 2.2: Gromov--Hausdorff Distance
  • Definition 2.3: Convexity Radius
  • Theorem 2.4: Hausmann's Theorem hausmann_1995
  • Theorem 3.2: Extented Jung's Theorem Dekster1997
  • Proposition 3.3: Circumradius
  • Proposition 3.4: Circumcenters of Subsets
  • proof
  • ...and 18 more