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Efficient Two-Parameter Persistence Computation via Cohomology

Authors Ulrich Bauer , Fabian Lenzen , Michael Lesnick



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Author Details

Ulrich Bauer
  • Department of Mathematics, TUM School of Computation, Information and Technology, and Munich Data Science Institute, Technical University of Munich, Germany
  • www.ulrich-bauer.org
Fabian Lenzen
  • Department of Mathematics, TUM School of Computation, Information and Technology, Technical University of Munich, Germany
Michael Lesnick
  • Department of Mathematics, SUNY Albany, NY, USA

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Ulrich Bauer, Fabian Lenzen, and Michael Lesnick. Efficient Two-Parameter Persistence Computation via Cohomology. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 15:1-15:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.SoCG.2023.15

Abstract

Clearing is a simple but effective optimization for the standard algorithm of persistent homology (ph), which dramatically improves the speed and scalability of ph computations for Vietoris-Rips filtrations. Due to the quick growth of the boundary matrices of a Vietoris-Rips filtration with increasing dimension, clearing is only effective when used in conjunction with a dual (cohomological) variant of the standard algorithm. This approach has not previously been applied successfully to the computation of two-parameter ph. We introduce a cohomological algorithm for computing minimal free resolutions of two-parameter ph that allows for clearing. To derive our algorithm, we extend the duality principles which underlie the one-parameter approach to the two-parameter setting. We provide an implementation and report experimental run times for function-Rips filtrations. Our method is faster than the current state-of-the-art by a factor of up to 20.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Algebraic topology
  • Theory of computation → Computational geometry
Keywords
  • Persistent homology
  • persistent cohomology
  • two-parameter persistence
  • clearing

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