Exact weights, path metrics, and algebraic Wasserstein distances

Abstract

We use weights on objects in an abelian category to define what we call a path metric. We introduce three special classes of weight: those compatible with short exact sequences; those induced by their path metric; and those which bound their path metric. We prove that these conditions are in fact equivalent, and call such weights exact. As a special case of a path metric, we obtain a distance for generalized persistence modules whose indexing category is a measure space. We use this distance to define Wasserstein distances, which coincide with the previously defined Wasserstein distances for one-parameter persistence modules. For one-parameter persistence modules, we also describe maps to and from an interval module, and we give a matrix reduction for monomorphisms and epimorphisms.

accepted to Journal of Applied and Computational Topology, 33 pages, v4: made changes suggested by referees

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Cited by in corpus (2)

• Multi-Parameter Persistence Modules are Generically Indecomposable
• A stability theorem for bigraded persistence barcodes