An algebraic Wasserstein distance for generalized persistence modules

The Wasserstein distances are a family of $L^p$ distances, with $1 leq p leq infty$, for persistence diagrams. We define Wasserstein distances for persistence modules, the algebraic counterpart to persistence diagrams, and prove the following isometry theorem. The $p$-Wasserstein distance of a persistence module and its persistence diagram agree. Since our algebraic Wasserstein distances do not require computing a persistence diagram, they apply to persistence modules that are not interval decomposable and also to generalized persistence modules, such as multi-parameter persistence modules. We also prove structure theorems for maps from an interval module and maps to an interval module and show that for monomorphisms and epimorphisms of persistence modules there is an induced algebraic matching.