Efficient computation of 2-parameter persistent cohomology
- Fabian Lenzen (TU Munich)
Persistent homology is one of the key concepts in topological data analysis and an active area of research in computational topology. It associates to a filtered simplicial complex the system of the homology vector spaces of each complex in the filtration. This collection can be viewed as a k[x]-module, and a common goal is to compute a free presentation of this module, which can be succinctly described by the so-called barcode. A common optimisation scheme in current software exploits the fact that the computation of persistent cohomology, albeit yielding equivalent results, can be carried out far more efficiently.
Analogously, the system of homology vector spaces of a two-parameter filtration can be viewed as a k[x, y]-module. Computing a free presentation of it is more involved, though, and the efficiency of existing implementations lags behind that of one-parameter persistent homology software. This is because optimisations using cohomology cannot be applied straightforwardly anymore, due to the fact that, unlike the one-parameter case, cochain modules are not free anymore.
I will show how cohomology can be used to develop efficient algorithms for two-parameter persistence nevertheless by considering free resolutions of cochain modules instead, using a result that links free resolutions of persistent homology and cohomology.