Reducing complexes in multidimensional persistent homology theory

TL;DR

This paper extends discrete Morse theory algorithms to multidimensional filtrations in persistent homology, enabling complex reductions and demonstrating initial experimental results on triangular meshes.

Contribution

It introduces a framework and algorithm for Morse matchings in multidimensional filtrations, expanding the applicability of discrete Morse theory in persistent homology.

Findings

Algorithm correctness and complexity are established.

Reduction of complexes to smaller cellular complexes is achieved.

Initial experiments on triangular meshes demonstrate practical applicability.

Abstract

The Discrete Morse Theory of Forman appeared to be useful for providing filtration-preserving reductions of complexes in the study of persistent homology. So far, the algorithms computing discrete Morse matchings have only been used for one-dimensional filtrations. This paper is perhaps the first attempt in the direction of extending such algorithms to multidimensional filtrations. Initial framework related to Morse matchings for the multidimensional setting is proposed, and a matching algorithm given by King, Knudson, and Mramor is extended in this direction. The correctness of the algorithm is proved, and its complexity analyzed. The algorithm is used for establishing a reduction of a simplicial complex to a smaller but not necessarily optimal cellular complex. First experiments with filtrations of triangular meshes are presented.