Computing the Multicover Bifiltration

Corbet, René; Kerber, Michael; Lesnick, Michael; Osang, Georg

doi:10.4230/lipics.socg.2021.27

Cited by 1 publication

(1 citation statement)

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“…However, Vietoris-Rips ph is notoriously unstable to outliers. Besides other strategies [11,Section 1.7], a commonly proposed remedy for this is the introduction of a second filtration parameter controlling the local density of the point cloud, which leads to the notions of two-and multi-parameter ph [13,42,16,33,40,12]. A central computational problem of two-parameter ph, somewhat akin to the computation of a barcode, is the computation of a minimal free presentation or a minimal free resolution (mfr) of the ph module.…”

Section: Introductionmentioning

confidence: 99%

Efficient two-parameter persistence computation via cohomology

Bauer¹,

Fabian²,

Lesnick³

2023

Preprint

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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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