Simplification of complexes for persistent homology computations

TL;DR

This paper introduces preprocessing techniques like elementary collapses, coreductions, and acyclic subspace methods to simplify complexes, making persistent homology computations more efficient by reducing complex size before boundary matrix generation.

Contribution

It adapts existing homology reduction techniques for persistent homology, providing a framework to preprocess complexes and improve computational efficiency.

Findings

Reduction methods decrease complex size significantly.

Preprocessing speeds up persistent homology calculations.

Techniques are applicable to various types of complexes.

Abstract

In this paper we focus on preprocessing for persistent homology computations. We adapt some techniques which were successfully used for standard homology computations. The main idea is to reduce the complex prior to generating its boundary matrix, which is costly to store and process. We discuss the following reduction methods: elementary collapses, coreductions (as defined by Mrozek and Batko) and acyclic subspace method (introduced by Mrozek, Pilarczyk and \.Zelazna).