Exact weights, path metrics, and algebraic Wasserstein distances

TL;DR

This paper introduces a new class of weights called exact weights in abelian categories, develops a path metric, and applies it to define Wasserstein distances for generalized persistence modules, unifying and extending previous metrics.

Contribution

It defines exact weights in abelian categories, establishes their equivalence with certain conditions, and applies these to develop Wasserstein distances for generalized persistence modules.

Findings

Exact weights are equivalent to three compatibility conditions.

A new distance for generalized persistence modules is introduced.

Wasserstein distances coincide with existing ones for one-parameter modules.

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.