Computing homology and persistent homology using iterated Morse decomposition

TL;DR

This paper introduces an innovative algorithm leveraging iterated Morse decomposition and discrete Morse theory to efficiently compute homology and persistent homology across all dimensions using simple graph operations.

Contribution

It presents a novel, correct algorithm for homology and persistent homology computation that simplifies traditional methods through iterated Morse decomposition.

Findings

Algorithm is provably correct in any dimension

Uses only simple graph theoretical operations

Improves computational efficiency for homology calculations

Abstract

In this paper we present a new approach to computing homology (with field coefficients) and persistent homology. We use concepts from discrete Morse theory, to provide an algorithm which can be expressed solely in terms of simple graph theoretical operations. We use iterated Morse decomposition, which allows us to sidetrack many problems related to the standard discrete Morse theory. In particular, this approach is provably correct in any dimension.