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Measures determined by their values on balls and Gromov-Wasserstein convergence

Anne van Delft, Andrew J. Blumberg

Abstract

A classical question about a metric space is whether Borel measures on the space are determined by their values on balls. We show that for any given measure this property is stable under Gromov-Wasserstein convergence of metric measure spaces. We then use this result to show that suitable bounded subspaces of the space of persistence diagrams have the property that any Borel measure is determined by its values on balls. This justifies the use of empirical ball volumes for statistical testing in topological data analysis (TDA). Our intended application is to deploy the statistical foundations of van Delft and Blumberg (2023) for time series of random geometric objects in the context of TDA invariants, specifically persistent homology and zigzag persistence.

Measures determined by their values on balls and Gromov-Wasserstein convergence

Abstract

A classical question about a metric space is whether Borel measures on the space are determined by their values on balls. We show that for any given measure this property is stable under Gromov-Wasserstein convergence of metric measure spaces. We then use this result to show that suitable bounded subspaces of the space of persistence diagrams have the property that any Borel measure is determined by its values on balls. This justifies the use of empirical ball volumes for statistical testing in topological data analysis (TDA). Our intended application is to deploy the statistical foundations of van Delft and Blumberg (2023) for time series of random geometric objects in the context of TDA invariants, specifically persistent homology and zigzag persistence.
Paper Structure (4 sections, 13 theorems, 14 equations)

This paper contains 4 sections, 13 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Convergence of measures and values on balls
  3. Any measure on barcode space is determined by its value on balls
  4. Applications to persistent homology and to zigzag persistence

Key Result

Theorem 1.1

vDB23 Let $(X_t: t\in \mathbb{Z})$ be a Polish-valued stochastic process with law $\mu_{X}$ a locally finite Borel regular doubling measure. Then the process $\mu^X_{J}( B_\epsilon(\pi_{J} \circ X): \epsilon \ge 0)$, where $B_\epsilon(x)$ denotes the ball of radius $\epsilon$ with center $x$, chara

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.1
  • Corollary 1.2
  • Definition 2.1
  • Definition 2.2
  • Remark 2.1
  • Theorem 2.1
  • Lemma 2.1
  • ...and 9 more