Efficient two-parameter persistence computation via cohomology

Clearing is a simple but effective optimization for the standard algorithm of persistent homology, which dramatically improves the speed and scalability of persistent homology 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 persistent homology. We introduce a cohomological algorithm for computing minimal free resolutions of two-parameter persistent homology 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.