Algebraic Wasserstein distances and stable homological invariants of data

TL;DR

This paper introduces algebraic Wasserstein distances for persistence modules, leading to stable invariants called Wasserstein stable ranks, with efficient computation methods and practical applications in data analysis.

Contribution

It develops a family of parametrized pseudometrics based on algebraic Wasserstein distances, bridging algebraic and combinatorial persistence modules, and introduces Wasserstein stable ranks as new stable invariants.

Findings

Wasserstein stable ranks are 1-Lipschitz stable under the introduced pseudometrics.

Low-rank approximation enables efficient computation of Wasserstein stable ranks.

Experimental results demonstrate the applicability of Wasserstein stable ranks in real data analysis.

Abstract

Distances have a ubiquitous role in persistent homology, from the direct comparison of homological representations of data to the definition and optimization of invariants. In this article we introduce a family of parametrized pseudometrics between persistence modules based on the algebraic Wasserstein distance defined by Skraba and Turner, and phrase them in the formalism of noise systems. This is achieved by comparing $p$-norms of cokernels (resp. kernels) of monomorphisms (resp. epimorphisms) between persistence modules and corresponding bar-to-bar morphisms, a novel notion that allows us to bridge between algebraic and combinatorial aspects of persistence modules. We use algebraic Wasserstein distances to define invariants, called Wasserstein stable ranks, which are 1-Lipschitz stable with respect to such pseudometrics. We prove a low-rank approximation result for persistence modules which allows us to efficiently compute Wasserstein stable ranks, and we propose an efficient algorithm to compute the interleaving distance between them. Importantly, Wasserstein stable ranks depend on interpretable parameters which can be learnt in a machine learning context. Experimental results illustrate the use of Wasserstein stable ranks on real and artificial data and highlight how such pseudometrics could be useful in data analysis tasks.