Algebraic Wasserstein distances and stable homological invariants of data

Jens Agerberg, Andrea Guidolin, Isaac Ren, Martina Scolamiero

Abstract: Distances have an ubiquitous role in persistent homology, from the direct comparison of homological representations of data to the definition and optimization of invariants. In this article we introduce a family of parametrized pseudometrics based on the algebraic Wasserstein distance and phrase them in the formalism of noise systems. This is achieved by comparing p-norms of cokernels (resp. kernels) of monomorphisms (resp. epimorphisms) between persistence modules and corresponding bar-to-bar morphisms. We use these pseudometrics to define associated stable invariants, called Wasserstein stable ranks, and compute them efficiently. Experimental results illustrate the use of Wasserstein stable ranks on real and artificial data.