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Topological and metric properties of spaces of generalized persistence diagrams

Peter Bubenik, Iryna Hartsock

TL;DR

The paper develops a comprehensive framework for generalized persistence diagrams on metric pairs $(X,d,A)$ equipped with the Wasserstein distances $W_p$, establishing detailed metric and topological properties. It defines finite and countable diagram spaces, characterizes optimal matchings via distance-minimizing subsets, and proves foundational results on completeness, separability, path-connectivity, and geodesic structure under broad geometric assumptions. It further derives compactness criteria, analyzes curvature and dimension (including CAT and Alexandrov-type results), and investigates embeddability into Hilbert spaces, highlighting limitations for infinite diagram collections. The work provides a unifying theory that informs both the theoretical understanding and practical use of persistence diagrams in topological data analysis and related domains, including implications for stochastic algorithms and kernel/embedding-based methods.

Abstract

Motivated by persistent homology and topological data analysis, we consider formal sums on a metric space with a distinguished subset. These formal sums, which we call persistence diagrams, have a canonical 1-parameter family of metrics called Wasserstein distances. We study the topological and metric properties of these spaces. Some of our results are new even in the case of persistence diagrams on the half-plane. Under mild conditions, no persistence diagram has a compact neighborhood. If the underlying metric space is $σ$-compact then so is the space of persistence diagrams. However, under mild conditions, the space of persistence diagrams is not hemicompact and the space of functions from this space to a topological space is not metrizable. Spaces of persistence diagrams inherit completeness and separability from the underlying metric space. Some spaces of persistence diagrams inherit being path connected, being a length space, and being a geodesic space, but others do not. We give criteria for a set of persistence diagrams to be totally bounded and relatively compact. We also study the curvature and dimension of spaces of persistence diagrams and their embeddability into a Hilbert space. As an important technical step, which is of independent interest, we give necessary and sufficient conditions for the existence of optimal matchings of persistence diagrams.

Topological and metric properties of spaces of generalized persistence diagrams

TL;DR

The paper develops a comprehensive framework for generalized persistence diagrams on metric pairs equipped with the Wasserstein distances , establishing detailed metric and topological properties. It defines finite and countable diagram spaces, characterizes optimal matchings via distance-minimizing subsets, and proves foundational results on completeness, separability, path-connectivity, and geodesic structure under broad geometric assumptions. It further derives compactness criteria, analyzes curvature and dimension (including CAT and Alexandrov-type results), and investigates embeddability into Hilbert spaces, highlighting limitations for infinite diagram collections. The work provides a unifying theory that informs both the theoretical understanding and practical use of persistence diagrams in topological data analysis and related domains, including implications for stochastic algorithms and kernel/embedding-based methods.

Abstract

Motivated by persistent homology and topological data analysis, we consider formal sums on a metric space with a distinguished subset. These formal sums, which we call persistence diagrams, have a canonical 1-parameter family of metrics called Wasserstein distances. We study the topological and metric properties of these spaces. Some of our results are new even in the case of persistence diagrams on the half-plane. Under mild conditions, no persistence diagram has a compact neighborhood. If the underlying metric space is -compact then so is the space of persistence diagrams. However, under mild conditions, the space of persistence diagrams is not hemicompact and the space of functions from this space to a topological space is not metrizable. Spaces of persistence diagrams inherit completeness and separability from the underlying metric space. Some spaces of persistence diagrams inherit being path connected, being a length space, and being a geodesic space, but others do not. We give criteria for a set of persistence diagrams to be totally bounded and relatively compact. We also study the curvature and dimension of spaces of persistence diagrams and their embeddability into a Hilbert space. As an important technical step, which is of independent interest, we give necessary and sufficient conditions for the existence of optimal matchings of persistence diagrams.
Paper Structure (31 sections, 120 theorems, 70 equations, 1 figure)

This paper contains 31 sections, 120 theorems, 70 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Persistence diagrams and Wasserstein distance
  3. Commutative monoids and persistence diagrams.
  4. Wasserstein distance
  5. Basic properties of spaces of persistence diagrams
  6. Persistence diagrams with essentially finite persistence
  7. Basic metric properties
  8. Basic topological properties
  9. Optimal matchings
  10. Paths in spaces of persistence diagrams
  11. Connectedness
  12. Path connectedness
  13. Rectifiable paths
  14. Length spaces
  15. Geodesic spaces
  16. ...and 16 more sections

Key Result

Theorem 1

Given a metric pair $(X,d,A)$, there exists an optimal matching between any two persistence diagrams if and only if $A$ is distance minimizing.

Figures (1)

  • Figure 1: A path connected pointed metric space $(X,d,x_0)$ whose space of persistence diagrams does not have a path from $\sum_{n=1}^{\infty} x_n$ to $0$. See Example \ref{['ex:circles']}.

Theorems & Definitions (232)

  • Theorem : \ref{['thm:optimal-matching']}
  • Theorem : \ref{['thm:path-connected', 'ex:circles']}
  • Theorem : \ref{['thm:length', 'prop:length']}
  • Theorem : \ref{['thm:geodesic', 'prop:X/A_geodesic', 'prop:unique_geodesic']}
  • Theorem : \ref{['prop:separability']}
  • Theorem : \ref{['thm:completeness_inf', 'thm:completeness_p', 'prop:completion', 'prop:complete']}
  • Proposition : \ref{['prop:D^n(X)_is_cmpt']}
  • Theorem : \ref{['thm:not_loc_compactness', 'thm:not_loc_compactness_D_p']}
  • Proposition : \ref{['prop:sigma-compact']}
  • Theorem : \ref{['thm:not_hemicompact']}
  • ...and 222 more