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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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