Matroid Filtrations and Computational Persistent Homology

TL;DR

This paper presents a new algorithmic approach combining matroid theory, discrete Morse theory, and matrix factorizations to improve the efficiency of computing persistent homology in topological data analysis.

Contribution

It introduces a novel relationship between matroid theory and persistent homology computation, enhancing algorithm efficiency and implementation in the Eirene software package.

Findings

Demonstrates improved computational efficiency for persistent homology

Provides a new theoretical framework linking matroids and homological algebra

Benchmarks show performance gains over existing methods

Abstract

This technical report introduces a novel approach to efficient computation in homological algebra over fields, with particular emphasis on computing the persistent homology of a filtered topological cell complex. The algorithms here presented rely on a novel relationship between discrete Morse theory, matroid theory, and classical matrix factorizations. We provide background, detail the algorithms, and benchmark the software implementation in the Eirene package.