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Distances have an ubiquitous role in persistent homology, from the directcomparison of homological representations of data to the definition andoptimization of invariants. In this article we introduce a family ofparametrized pseudometrics based on the algebraic Wasserstein distance definedin [arXiv:2006.16824] and phrase them in the formalism of noise systems. Thisis achieved by comparing $p$-norms of cokernels (resp. kernels) ofmonomorphisms (resp. epimorphisms) between persistence modules andcorresponding bar-to-bar morphisms. We use these pseudometrics to defineassociated stable invariants, called Wasserstein stable ranks, and compute themefficiently. Experimental results illustrate the use of Wasserstein stableranks on real and artificial data.

Algebraic Wasserstein distances and stable homological invariants of data