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Metric Geometry of Spaces of Persistence Diagrams

Mauricio Che, Fernando Galaz-García, Luis Guijarro, Ingrid Amaranta Membrillo Solis

Abstract

Persistence diagrams are objects that play a central role in topological data analysis. In the present article, we investigate the local and global geometric properties of spaces of persistence diagrams. In order to do this, we construct a family of functors $\mathcal{D}_p$, $1\leq p \leq\infty$, that assign, to each metric pair $(X,A)$, a pointed metric space $\mathcal{D}_p(X,A)$. Moreover, we show that $\mathcal{D}_{\infty}$ is sequentially continuous with respect to the Gromov-Hausdorff convergence of metric pairs, and we prove that $\mathcal{D}_p$ preserves several useful metric properties, such as completeness and separability, for $p \in [1,\infty)$, and geodesicity and non-negative curvature in the sense of Alexandrov, for $p=2$. For the latter case, we describe the metric of the space of directions at the empty diagram. We also show that the Fréchet mean set of a Borel probability measure on $\mathcal{D}_p(X,A)$, $1\leq p \leq\infty$, with finite second moment and compact support is non-empty. As an application of our geometric framework, we prove that the space of Euclidean persistence diagrams, $\mathcal{D}_{p}(\mathbb{R}^{2n},Δ_n)$, $1\leq n$ and $1\leq p<\infty$, has infinite covering, Hausdorff, asymptotic, Assouad, and Assouad-Nagata dimensions.

Metric Geometry of Spaces of Persistence Diagrams

Abstract

Persistence diagrams are objects that play a central role in topological data analysis. In the present article, we investigate the local and global geometric properties of spaces of persistence diagrams. In order to do this, we construct a family of functors , , that assign, to each metric pair , a pointed metric space . Moreover, we show that is sequentially continuous with respect to the Gromov-Hausdorff convergence of metric pairs, and we prove that preserves several useful metric properties, such as completeness and separability, for , and geodesicity and non-negative curvature in the sense of Alexandrov, for . For the latter case, we describe the metric of the space of directions at the empty diagram. We also show that the Fréchet mean set of a Borel probability measure on , , with finite second moment and compact support is non-empty. As an application of our geometric framework, we prove that the space of Euclidean persistence diagrams, , and , has infinite covering, Hausdorff, asymptotic, Assouad, and Assouad-Nagata dimensions.

Paper Structure

This paper contains 13 sections, 38 theorems, 128 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Metric pairs
  4. Commutative metric monoids and spaces of persistence diagrams
  5. Alexandrov spaces
  6. Gromov--Hausdorff convergence and sequential continuity
  7. Geodesicity
  8. Non-negative curvature
  9. Spaces of directions: the local geometry of noise
  10. Dimension of spaces of Euclidean persistence diagrams
  11. Completeness, separability, and Fréchet means
  12. Completeness and separability
  13. Fréchet means

Key Result

Theorem 1

The functor $\mathcal{D}_p$ , $1\leq p\leq \infty$, is sequentially continuous with respect to the Gromov--Hausdorff convergence of metric pairs if and only if $p=\infty$.

Figures (1)

  • Figure 1: The condition for a complete geodesic metric space $X$ to be an Alexandrov space with curvature $\geq \kappa$. Here, the curves $[pq]$, $[qr]$, $[rp]$, $[px]$, $[\widetilde{p}\widetilde{q}]$, $[\widetilde{q}\widetilde{r}]$, $[\widetilde{r}\widetilde{p}]$, $[\widetilde{p}\widetilde{x}]$ are geodesics, and the length of $[pq]$ (respectively, $[rp]$, $[qx]$, $[xr]$) is equal to the length of $[\widetilde{p}\widetilde{q}]$ (respectively, $[\widetilde{r}\widetilde{p}]$, $[\widetilde{q}\widetilde{x}]$, $[\widetilde{x}\widetilde{r}]$). Condition (T) then says that the length of $[\widetilde{p}\widetilde{x}]$ is not greater than the length of $[px]$.

Theorems & Definitions (93)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Corollary 6
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • ...and 83 more