Harmonic Persistent Homology

TL;DR

This paper introduces harmonic persistent homology spaces for filtrations of finite simplicial complexes, providing a stable, concrete way to associate cycle subspaces to barcode bars, and relating it to essential simplices.

Contribution

It develops harmonic persistent homology spaces, proving their stability and linking them to essential simplices, offering a new approach to analyze persistent homology.

Findings

Harmonic persistent homology spaces can be associated with each barcode bar.

These spaces are stable under small perturbations of the defining functions.

Harmonic representatives maximize the relative essential content among all cycle representatives.

Abstract

We introduce harmonic persistent homology spaces for filtrations of finite simplicial complexes. As a result we can associate concrete subspaces of cycles to each bar of the barcode of the filtration. We prove stability of the harmonic persistent homology subspaces under small perturbations of functions defining them. We relate the notion of "essential simplices" introduced in an earlier work to identify simplices which play a significant role in the birth of a bar, with that of harmonic persistent homology. We prove that the harmonic representatives of simple bars maximizes the "relative essential content" amongst all representatives of the bar, where the relative essential content is the weight a particular cycle puts on the set of essential simplices.