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arXiv · 2303.11193

Efficient two-parameter persistence computation via cohomology

Ulrich Bauer·Fabian Lenzen·Michael Lesnick
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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.

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Ulrich Bauer, Fabian Lenzen, Michael Lesnick. 2023-03-20. Efficient two-parameter persistence computation via cohomology. https://doi.org/10.4230/lipics.socg.2023.15

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