Change Action Derivatives in Persistent Homology

TL;DR

This paper generalizes the concept of the rank of the pair group in persistent homology for more complex filtrations, using categorical calculus to better understand the structure of persistence diagrams.

Contribution

It introduces a categorical framework for computing ranks in persistent homology for tame filtrations, extending classical methods to a broader class of data.

Findings

Generalization of the rank of the pair group to tame filtrations

Categorical calculus of finite differences applied to persistence

Enhanced understanding of persistence diagrams in complex filtrations

Abstract

Persistent homology is a popular technique in topological data analysis that tracks the lifespans of homological features in a nested sequence of spaces. This data is typically presented in a multi-set called a persistence diagram or a barcode. For single parameter filtrations with homology coefficient taken in a principal ideal domain, the persistence diagram/barcode can be computed using the presentation theorem for finitely generated modules over a PID. One way to reconstruct the persistence diagram/barcode is to consider the rank of the pair group at all intervals, as defined by Edelsbrunner and Harer, which counts the number of homology classes whose lifespans are precisely said intervals respectively. In this paper we generalize the rank of the pair group for suitably `tame' filtrations, described as functors from a partially ordered set to a category of chain complexes, and show how it can be captured by a categorical version of the calculus of finite-differences for abelian groups.