Keeping it sparse: Computing Persistent Homology revisited

TL;DR

This paper introduces new sparse matrix reduction algorithms for persistent homology computation, demonstrating improved efficiency and providing theoretical bounds that explain practical performance.

Contribution

It proposes two novel sparse reduction variants, swap and retrospective, with output-sensitive bounds and performance analysis in topological data analysis.

Findings

Sometimes significantly faster than existing methods

Retrospective variant has near-linear practical complexity

Certain data constructions favor specific variants

Abstract

In this work, we study several variants of matrix reduction via Gaussian elimination that try to keep the reduced matrix sparse. The motivation comes from the growing field of topological data analysis where matrix reduction is the major subroutine to compute barcodes, the main invariant therein. We propose two novel variants of the standard algorithm, called swap and retrospective reductions. We test them on a large collection of data against other known variants to compare their efficiency, and we find that sometimes they provide a considerable speed-up. We also present novel output-sensitive bounds for the retrospective variant which better explain the discrepancy between the cubic worst-case complexity bound and the almost linear practical behavior of matrix reduction. Finally, we provide several constructions on which one of the variants performs strictly better than the others.