Keeping it sparse: Computing Persistent Homology revised

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. We propose two novel variants of the standard algorithm, called swap and retrospective reductions, which improve upon state-of-the-art techniques on several examples in practice. 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.