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Efficient Two-Parameter Persistence Computation via Cohomology

Authors: Ulrich Bauer, Fabian Lenzen, and Michael Lesnick

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

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.

Cite as

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)

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  author =	{Bauer, Ulrich and Lenzen, Fabian and Lesnick, Michael},
  title =	{{Efficient Two-Parameter Persistence Computation via Cohomology}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{15:1--15:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.15},
  URN =		{urn:nbn:de:0030-drops-178656},
  doi =		{10.4230/LIPIcs.SoCG.2023.15},
  annote =	{Keywords: Persistent homology, persistent cohomology, two-parameter persistence, clearing}
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