Persistent and Zigzag Homology: A Matrix Factorization Viewpoint

TL;DR

This paper presents a matrix factorization approach to compute persistent and zigzag homology, enabling more flexible, parallelizable, and efficient algorithms for topological data analysis.

Contribution

It introduces a canonical matrix form framework for persistent and zigzag homology computations, extending capabilities beyond existing software.

Findings

Framework allows using arbitrary induced maps on homology

Demonstrates significant parallel speedups

Achieves improvements over existing software

Abstract

Over the past two decades, topological data analysis has emerged as a field of applied mathematics with new applications and algorithmic developments appearing rapidly. Two fundamental computations in this field are persistent homology and zigzag homology. In this paper, we show how these computations in the most general case reduce to finding a canonical form of a matrix associated with a type A quiver representation, which in turn can be computed using factorizations of associated matrices. We show how to use arbitrary induced maps on homology for computation, providing a framework that goes beyond the capabilities of existing software for topological data analysis. Furthermore, this framework offers multiple opportunities for parallelization which have not been previously exploited. We provide several examples of the utility of this framework, demonstrate parallel speedups, and report on significant improvements in comparison to existing software.