Path representations in multiparameter persistent homology

TL;DR

This paper introduces a generalization of the matching distance in multiparameter persistent homology by replacing straight line filtrations with monotone path filtrations, supported by software implementations.

Contribution

It proposes a novel approach to measure distances in multiparameter persistent homology using monotone path filtrations, extending the traditional straight line method.

Findings

Generalization of matching distance to monotone paths

Development of software tools for the new method

Potential for capturing richer topological differences

Abstract

Multiparameter persistence module can capture more topological differences across data instances compared to using a single parameter, where the well-studied matching distance investigates the distance along a straight line in the multiparameter space that gives the biggest difference. We propose to generalize the straight line to a monotone path filtration and offer software implementations.