On the geometrical properties of the coherent matching distance in 2D persistent homology

TL;DR

This paper introduces a new stable metric for bidimensional persistent homology based on coherent matchings, highlighting the importance of lines with slope 1 and providing a framework to handle monodromy phenomena.

Contribution

It develops a novel metric for 2D persistent homology, proves its stability, and links its computation to filtering functions along lines of slope 1, advancing theoretical understanding.

Findings

The new metric is stable under perturbations.

Computation relates to filtering functions on lines of slope 1.

Framework for managing monodromy in 2D persistence.

Abstract

In this paper we study a new metric for comparing Betti numbers functions in bidimensional persistent homology, based on coherent matchings, i.e. families of matchings that vary in a continuous way. We prove some new results about this metric, including its stability. In particular, we show that the computation of this distance is strongly related to suitable filtering functions associated with lines of slope 1, so underlining the key role of these lines in the study of bidimensional persistence. In order to prove these results, we introduce and study the concepts of extended Pareto grid for a normal filtering function as well as of transport of a matching. As a by-product, we obtain a theoretical framework for managing the phenomenon of monodromy in 2D persistent homology.