The coherent matching distance in 2D persistent homology

TL;DR

This paper introduces the coherent matching distance for 2D persistent Betti numbers, addressing limitations of the traditional multidimensional matching distance by preserving homological information and ensuring stability.

Contribution

It defines a new coherent matching distance that accounts for the continuity of filtrations and handles monodromy, improving the analysis of multidimensional persistence.

Findings

The coherent matching distance is well-defined.

It is stable under perturbations.

It better captures homological information.

Abstract

Comparison between multidimensional persistent Betti numbers is often based on the multidimensional matching distance. While this metric is rather simple to define and compute by considering a suitable family of filtering functions associated with lines having a positive slope, it has two main drawbacks. First, it forgets the natural link between the homological properties of filtrations associated with lines that are close to each other. As a consequence, part of the interesting homological information is lost. Second, its intrinsically discontinuous definition makes it difficult to study its properties. In this paper we introduce a new matching distance for 2D persistent Betti numbers, called coherent matching distance and based on matchings that change coherently with the filtrations we take into account. Its definition is not trivial, as it must face the presence of monodromy in multidimensional persistence, i.e. the fact that different paths in the space parameterizing the above filtrations can induce different matchings between the associated persistent diagrams. In our paper we prove that the coherent 2D matching distance is well-defined and stable.