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Function-Rips complexes in persistent homotopy theory: Local stability and Latschev theorems

Steve Oudot, Lukas Waas

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 $\mathcal{R}^δ(N)$ of any nearby space $N$ in the Gromov-Hausdorff distance. We prove a persistent version of this theorem, providing sufficient conditions on a pair $(M, f \colon M \to \mathbb{R}^N)$ 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 $\mathcal{R}^δ(N^{\bullet})$ of any nearby pair $(N,g)$. In particular, our result answers a longstanding question on the related topic of estimating sublevel set persistent homology from finite point samples.

Function-Rips complexes in persistent homotopy theory: Local stability and Latschev theorems

Abstract

Latschev's theorem provides sufficient conditions on a metric space and for the homotopy type of to agree with that of the Vietoris-Rips complex of any nearby space in the Gromov-Hausdorff distance. We prove a persistent version of this theorem, providing sufficient conditions on a pair and for the persistent homotopy type of the sublevel set filtration of to be interleaved with that of the function-Rips complex of any nearby pair . In particular, our result answers a longstanding question on the related topic of estimating sublevel set persistent homology from finite point samples.
Paper Structure (8 sections, 8 theorems, 11 equations, 2 figures)

This paper contains 8 sections, 8 theorems, 11 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Spaces of curvature bounded above
  4. Models for persistent homotopy types
  5. Persistent homotopy theory
  6. Interlude: Some remarks on persistent homotopy types
  7. Persistent simplicial sets
  8. Interleavings in the homotopy category

Key Result

Theorem 1

Let $M$ be a closed Riemannian manifold. Then there exists $\delta_0>0$ such that, for every $0<\delta\leq\delta_0$, there exists an $\varepsilon>0$ such that, for any metric space $\mathbb{M}$ with Gromov-Hausdorff distance to $M$ less than $\varepsilon$, $M$ is homotopy equivalent to the geometric

Figures (2)

  • Figure 1: Shrinking transformation arrows (colored) and the object's structure morphisms (black).
  • Figure 2: Illustration of equilateral triangles in curvatures $\kappa<0$, $\kappa=0$, and $\kappa>0$.

Theorems & Definitions (22)

  • Theorem : Latschev Latschev2001
  • Theorem 1.0.1: Persistent Latschev's theorem
  • Theorem : andre2025estimating
  • Theorem 1.0.2: Persistent Hausmann's Theorem
  • Theorem 1.0.3: Approximation of function-Rips persistent homotopy type
  • Remark 2.1.7
  • Definition 2.2.2
  • Example 2.2.5
  • Example 2.2.6: function-Rips complex
  • Definition 2.3.1
  • ...and 12 more