One-Dimensional Reduction of Multidimensional Persistent Homology

TL;DR

This paper extends size functions to higher homology modules, reducing multidimensional persistent homology to a one-dimensional case, enabling stable distance computation and providing insights on homological critical values.

Contribution

It introduces a method to reduce multidimensional persistent homology to a one-dimensional form, facilitating stability analysis and critical value insights.

Findings

Reduction of multidimensional to 1D persistent homology

Introduction of a stable distance for multidimensional persistence

Insights on i-essentiality of homological critical values

Abstract

A recent result on size functions is extended to higher homology modules: the persistent homology based on a multidimensional measuring function is reduced to a 1-dimensional one. This leads to a stable distance for multidimensional persistent homology. Some reflections on i-essentiality of homological critical values conclude the paper.