Persistent (Co)Homology in Matrix Multiplication Time

TL;DR

This paper demonstrates that cycle representatives in persistent homology can be computed within matrix multiplication time, providing efficient algorithms for extracting these cycles in applications.

Contribution

It introduces faster algorithms for computing cycle representatives in persistent homology, matching matrix multiplication time bounds, and clarifies how to obtain standard and alternative representatives.

Findings

Cycle representatives can be computed in matrix multiplication time.

New algorithms simplify the reduction process for persistent homology.

Standard and alternative cycle representatives can be efficiently obtained.

Abstract

Most algorithms for computing persistent homology do so by tracking cycles that represent homology classes. There are many choices of such cycles, and specific choices have found different uses in applications. Although it is known that persistence diagrams can be computed in matrix multiplication time [8] for the more general case of zigzag persistent homology, it is not clear how to extract cycle representatives, especially if specific representatives are desired. In this paper, we provide the same matrix multiplication bound for computing representatives for the two choices common in applications in the case of ordinary persistent (co)homology. We first provide a fast version of the reduction algorithm, which is simpler than the algorithm in [8], but returns a different set of representatives than the standard algorithm [6] We then give a fast version of a different variant called the row algorithm [4], which returns the same representatives as the standard algorithm.