Stable comparison of multidimensional persistent homology groups with torsion

TL;DR

This paper introduces a new pseudo-distance for comparing multidimensional persistent homology groups with torsion, providing a stable method that extends existing tools to more complex algebraic structures.

Contribution

It proposes a pseudo-distance d_T for persistent homology with torsion and proves its stability under changes in filtering functions, extending the scope of topological data analysis.

Findings

d_T is a pseudo-distance for groups with torsion

d_T is stable under perturbations of filtering functions

d_T relates to the 1D matching distance when coefficients are in a field

Abstract

The present lack of a stable method to compare persistent homology groups with torsion is a relevant problem in current research about Persistent Homology and its applications in Pattern Recognition. In this paper we introduce a pseudo-distance d_T that represents a possible solution to this problem. Indeed, d_T is a pseudo-distance between multidimensional persistent homology groups with coefficients in an Abelian group, hence possibly having torsion. Our main theorem proves the stability of the new pseudo-distance with respect to the change of the filtering function, expressed both with respect to the max-norm and to the natural pseudo-distance between topological spaces endowed with vector-valued filtering functions. Furthermore, we prove a result showing the relationship between d_T and the matching distance in the 1-dimensional case, when the homology coefficients are taken in a field and hence the comparison can be made.