Additive Partial Matchings Induced by Persistence Maps

TL;DR

This paper introduces a new additive invariant for persistence maps in Topological Data Analysis, enabling more discriminative comparisons of persistence modules and efficient computation, especially in simplified complexes.

Contribution

It proposes a novel partial matching invariant for persistence maps that is additive, more discriminative than existing invariants, and computationally efficient.

Findings

Invariant is additive with respect to direct sum decomposition.

Invariant is more discriminative than the image invariant.

Implementation demonstrates efficiency in complex simplification scenarios.

Abstract

Persistent homology is a fundamental tool in Topological Data Analysis. The associated algebraic structure is the persistence module, a sequence of vector spaces connected by linear maps. Persistence modules admit a complete and fast-to-compute invariant known as the persistence diagram. However, this is no longer the case for maps between persistence modules (i.e. persistence maps). We propose a new invariant for persistence maps, consisting of a partial matching between the persistent diagrams of the domain and codomain modules. We show that this invariant is additive with respect to the direct sum decomposition of persistence maps, is more discriminative than the image invariant, and is computable in cubic time. Furthermore, we provide an implementation and demonstrate its efficiency by integrating it with edge collapse techniques for flag complexes (e.g., Vietoris-Rips complexes). As a key technical contribution, we describe how to induce a persistence map between two flag complexes that have been independently simplified via edge collapses, even when a direct simplicial map between them is no longer available.