Invariance properties of the multidimensional matching distance in Persistent Topology and Homology

TL;DR

This paper proves that the multidimensional matching distance in Persistent Topology remains invariant regardless of the choice of parameterization, ensuring robustness in comparing multidimensional homology features.

Contribution

The paper demonstrates the invariance of the multidimensional matching distance under different parameterizations, clarifying its robustness in multidimensional persistent homology analysis.

Findings

The multidimensional matching distance is invariant under different parameterizations.

This invariance ensures consistent comparison of multidimensional topological features.

The results generalize the stability properties of persistent homology.

Abstract

Persistent Topology studies topological features of shapes by analyzing the lower level sets of suitable functions, called filtering functions, and encoding the arising information in a parameterized version of the Betti numbers, i.e. the ranks of persistent homology groups. Initially introduced by considering real-valued filtering functions, Persistent Topology has been subsequently generalized to a multidimensional setting, i.e. to the case of $\R^n$-valued filtering functions, leading to studying the ranks of multidimensional homology groups. In particular, a multidimensional matching distance has been defined, in order to compare these ranks. The definition of the multidimensional matching distance is based on foliating the domain of the ranks of multidimensional homology groups by a collection of half-planes, and hence it formally depends on a subset of $\R^n\times\R^n$ inducing a parameterization of these half-planes. It happens that it is possible to choose this subset in an infinite number of different ways. In this paper we show that the multidimensional matching distance is actually invariant with respect to such a choice.