Distributing Persistent Homology via Spectral Sequences

Abstract

We set up the theory for a distributed algorithm for computing persistent homology. For this purpose we develop linear algebra of persistence modules. We present bases of persistence modules, together with an operation boxplus that leads to a method for obtaining images, kernels and cokernels of tame persistence morphisms. Our focus is on developing efficient methods for the computation of homology of chains of persistence modules. Later we give a brief, self-contained presentation of the Mayer–Vietoris spectral sequence. Then we study the Persistent Mayer–Vietoris spectral sequence and present a solution to the extension problem. This solution is given by finding coefficients that indicate gluings between bars on the same dimension. Finally, we review PerMaViss, an algorithm that computes all pages in the spectral sequence and solves the extension problem. This procedure distributes computations on subcomplexes, while focusing on merging homological information. Additionally, some computational bounds are found which confirm the distribution of the method.