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The Persistent Topology of Optimal Transport Based Metric Thickenings

Henry Adams, Facundo Mémoli, Michael Moy, Qingsong Wang

Abstract

A metric thickening of a given metric space $X$ is any metric space admitting an isometric embedding of $X$. Thickenings have found use in applications of topology to data analysis, where one may approximate the shape of a dataset via the persistent homology of an increasing sequence of spaces. We introduce two new families of metric thickenings, the $p$-Vietoris-Rips and $p$-Čech metric thickenings for all $1\le p\le \infty$, which include all measures on $X$ whose $p$-diameter or $p$-radius is bounded from above, equipped with an optimal transport metric. The $p$-diameter (resp. $p$-radius) of a measure is a certain $\ell_p$ relaxation of the usual notion of diameter (resp. radius) of a subset of a metric space. These families recover the previously studied Vietoris-Rips and Čech metric thickenings when $p=\infty$. As our main contribution, we prove a stability theorem for the persistent homology of $p$-Vietoris-Rips and $p$-Čech metric thickenings, which is novel even in the case $p=\infty$. In the specific case $p=2$, we prove a Hausmann-type theorem for thickenings of manifolds, and we derive the complete list of homotopy types of the $2$-Vietoris-Rips thickenings of the $n$-sphere as the scale increases.

The Persistent Topology of Optimal Transport Based Metric Thickenings

Abstract

A metric thickening of a given metric space is any metric space admitting an isometric embedding of . Thickenings have found use in applications of topology to data analysis, where one may approximate the shape of a dataset via the persistent homology of an increasing sequence of spaces. We introduce two new families of metric thickenings, the -Vietoris-Rips and -Čech metric thickenings for all , which include all measures on whose -diameter or -radius is bounded from above, equipped with an optimal transport metric. The -diameter (resp. -radius) of a measure is a certain relaxation of the usual notion of diameter (resp. radius) of a subset of a metric space. These families recover the previously studied Vietoris-Rips and Čech metric thickenings when . As our main contribution, we prove a stability theorem for the persistent homology of -Vietoris-Rips and -Čech metric thickenings, which is novel even in the case . In the specific case , we prove a Hausmann-type theorem for thickenings of manifolds, and we derive the complete list of homotopy types of the -Vietoris-Rips thickenings of the -sphere as the scale increases.

Paper Structure

This paper contains 33 sections, 54 theorems, 151 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Metric spaces and the Gromov--Hausdorff distance.
  4. Simplicial complexes.
  5. Probability measures and Wasserstein distances.
  6. Fréchet means.
  7. Metric thickenings with $p=\infty$.
  8. Comparisons.
  9. Homology and Persistent homology.
  10. Interleavings
  11. The homotopy type distance
  12. The $p$-relaxation of metric thickenings
  13. The relaxed diameter and radius functionals
  14. The relaxed Vietoris--Rips and Čech metric thickenings
  15. A more general setting: controlled invariants
  16. ...and 18 more sections

Key Result

Proposition 2.4

Let $Z$ be a metric space (or more generally a first-countable space), let $X$ be a metric space, and let $f,g \colon Z \to \mathcal{P}_X$ be continuous. Then the linear homotopy $H \colon Z \times [0,1] \to \mathcal{P}_X$ given by $H(z,t) = (1-t)\,f(z) + t\,g(z)$ is continuous.

Figures (4)

  • Figure 1: The $p$-Vietoris--Rips bifiltration $\mathrm{VR}_{p}(X;r)$ for $X$ a metric space of three points in $\mathbb{R}^2$, with $\mathcal{P}_X$ visualized as the convex hull of $X$ in $\mathbb{R}^2$. Note $\mathrm{VR}_{p}(X;r)\subseteq\mathrm{VR}_{p'}(X;r')$ for $r\le r'$ and $p \ge p'$.
  • Figure 2: The $p$-C̆ech bifiltration $\mathrm{\check{C}}_{p}(X;r)$ for $X$ a metric space of three points in $\mathbb{R}^2$, with $\mathcal{P}_X$ visualized as the convex hull of $X$ in $\mathbb{R}^2$. Note $\mathrm{\check{C}}_{p}(X;r)\subseteq\mathrm{\check{C}}_{p'}(X;r')$ for $r\le r'$ and $p \ge p'$.
  • Figure 3: Homotopy types of both $\mathrm{VR}_{p}(Z_{n+1};\bullet)$ and $\mathrm{\check{C}}_{p}(Z_{n+1};\bullet)$.
  • Figure 4: The sequence of deformation retractions used in the proof of Theorem \ref{['thm_pCech_homotopy_equiv_simplicial_complexes']} collapses a subset of a simplex to a simplicial complex on its vertices. In this example, $n=2$ and the sequence of deformation retractions is $Y_2 \to Y_1 \to Y_0$.

Theorems & Definitions (134)

  • Definition 2.1: Uniform discrete metric space
  • Definition 2.2: $\varepsilon$-net
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Definition 2.7: frosini2017persistent
  • Proposition 2.8: frosini2017persistent
  • ...and 124 more