Universality of persistence diagrams and the bottleneck and Wasserstein distances

TL;DR

This paper establishes the universality of persistence diagrams under p-Wasserstein distances, demonstrating their fundamental algebraic structure and duality properties, applicable across various persistence representations.

Contribution

It proves that persistence diagrams with p-Wasserstein distances form the universal p-subadditive commutative monoid, extending to barcodes and multiparameter modules, and confirms the Kantorovich-Rubinstein duality for 1-Wasserstein.

Findings

Persistence diagrams form a universal p-subadditive monoid.

1-Wasserstein distance satisfies Kantorovich-Rubinstein duality.

Results apply to barcodes and multiparameter persistence modules.

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.