Compression for 2-parameter persistent homology

Ulderico Fugacci, Michael Kerber, Alexander Rolle

Research output: Contribution to journalArticlepeer-review

Abstract

Compression aims to reduce the size of an input, while maintaining its relevant properties. For multi-parameter persistent homology, compression is a necessary step in any computational pipeline, since standard constructions lead to large inputs, and computational tasks in this area tend to be expensive. We propose two compression methods for chain complexes of free 2-parameter persistence modules. The first method extends the multi-chunk algorithm for one-parameter persistent homology, returning the smallest chain complex among all the ones quasi-isomorphic to the input. The second method produces minimal presentations of the homology of the input; it is based on an algorithm of Lesnick and Wright, but incorporates several improvements that lead to substantial performance gains. The two methods are complementary, and can be combined to compute minimal presentations for complexes with millions of generators in a few seconds. The methods have been implemented, and the software is publicly available. We report on experimental evaluations, which demonstrate substantial improvements in performance compared to previously available compression strategies.

Original languageEnglish
Article number101940
JournalComputational Geometry: Theory and Applications
Volume109
DOIs
Publication statusPublished - Feb 2023

Keywords

  • Matrix reduction
  • Minimal presentations
  • Multi-parameter persistent homology

ASJC Scopus subject areas

  • Computational Mathematics
  • Control and Optimization
  • Geometry and Topology
  • Computer Science Applications
  • Computational Theory and Mathematics

Fields of Expertise

  • Information, Communication & Computing

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