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Universality of persistence diagrams and the bottleneck and Wasserstein distances

Peter Bubenik, Alex Elchesen

TL;DR

This work develops a category-theoretic foundation for persistence diagrams by treating them as free commutative monoids on metric pairs $(X,A)$ endowed with $p$-Wasserstein metrics $W_p$. The authors prove a universal property: persistence diagrams with $W_p$ are the universal $p$-subadditive commutative metric monoid generated by $(X,A)$, yielding a maximal, functorial, and stable distance framework that encompasses barcodes and multiparameter persistence. They establish a precise link between metric and monoid structures through $p$-subadditive monoids internal to symmetric monoidal categories, derive a left adjoint $D_p$ to the forgetful functor, and show that $W_p$ is the largest compatible metric extending the base metric. For $p=1$, Kantorovich-Rubinstein duality holds, connecting persistence diagrams to Lipschitz-function duality. Overall, the results provide a robust, universal, and computable approach to distances between persistence diagrams with broad applicability to various persistence models and stability analyses.

Abstract

We prove that persistence diagrams with the p-Wasserstein distance form the universal p-subadditive commutative monoid on an underlying metric space with a distinguished subset. This result applies to persistence diagrams, barcodes, and to multiparameter persistence modules. In addition, the 1-Wasserstein distance satisfies Kantorovich-Rubinstein duality.

Universality of persistence diagrams and the bottleneck and Wasserstein distances

TL;DR

This work develops a category-theoretic foundation for persistence diagrams by treating them as free commutative monoids on metric pairs endowed with -Wasserstein metrics . The authors prove a universal property: persistence diagrams with are the universal -subadditive commutative metric monoid generated by , yielding a maximal, functorial, and stable distance framework that encompasses barcodes and multiparameter persistence. They establish a precise link between metric and monoid structures through -subadditive monoids internal to symmetric monoidal categories, derive a left adjoint to the forgetful functor, and show that is the largest compatible metric extending the base metric. For , Kantorovich-Rubinstein duality holds, connecting persistence diagrams to Lipschitz-function duality. Overall, the results provide a robust, universal, and computable approach to distances between persistence diagrams with broad applicability to various persistence models and stability analyses.

Abstract

We prove that persistence diagrams with the p-Wasserstein distance form the universal p-subadditive commutative monoid on an underlying metric space with a distinguished subset. This result applies to persistence diagrams, barcodes, and to multiparameter persistence modules. In addition, the 1-Wasserstein distance satisfies Kantorovich-Rubinstein duality.

Paper Structure

This paper contains 20 sections, 34 theorems, 35 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related work
  3. Background
  4. Metric spaces
  5. Monoids
  6. $p$-norms
  7. Basic category theory
  8. Universal properties and adjunctions
  9. Symmetric monoidal categories and internal objects
  10. Metric pairs and pointed metric spaces
  11. Metric pairs, pointed metric spaces, and quotients
  12. The $p$-strengthened triangle inequality
  13. Commutative metric monoids and Wasserstein distance
  14. Commutative metric monoids
  15. Monoid objects in $\mathbf{Met_*^p}$
  16. ...and 5 more sections

Key Result

Proposition 1.1

Given a metric pair $(X,d,A)$, $(D(X,A),W_p,+,0)$ is a $p$-subadditive commutative metric monoid and the canonical inclusion $i:(X,d,A) \to (D(X,A),W_p,0)$ is $1$-Lipschitz. Furthermore, this construction is functorial.

Figures (1)

  • Figure 1: For $1\leq p\leq q\leq \infty$, the relationships between the categories $\mathbf{Met_*^p}$, $\mathbf{Met_*^q}$, $\mathbf{CMon(Met_*^p)}$, $\mathbf{CMon(Met_*^q)}$, and $\mathbf{Met_{pairs}}$ via the functors $U_r, D_r, \overline{U}_r, \overline{D}_r$, $S_r$ ($r \in \{p,q\}$), and $Q, S_{p,q}$. The solid triangles commute and the dashed triangles commute.

Theorems & Definitions (94)

  • Proposition 1.1: Functorial construction of persistence diagrams with Wasserstein distance
  • Theorem 1.2: Persistence diagrams with the Wasserstein distance as an adjoint
  • Theorem 1.3: Universality of persistence diagrams with the Wasserstein distance
  • Corollary 1.4: Wasserstein distance as largest subadditive distance
  • Theorem 1.5: Symmetric monoidal structures for pointed metric spaces
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • ...and 84 more