Virtual persistence diagrams, signed measures, Wasserstein distances, and Banach spaces
Peter Bubenik, Alex Elchesen
TL;DR
The paper develops a universal, functorial framework to extend Wasserstein-type distances from classical persistence diagrams to virtual and signed invariants on general metric pairs. It defines virtual persistence diagrams as Grothendieck group completions $K(X,A)$, extends $W_1$ (and $W_p$ under $p$-metric assumptions) to these objects and to Radon measures, and connects diagram distances to measure distances via canonical embeddings. It further constructs countable and complete diagram spaces $ar{D}_p(X,A)$ with universal properties, and builds universal Banach spaces (free Lipschitz spaces) capturing these objects, all while preserving isometric embeddings and functoriality. The work unifies persistence diagrams, signed measures, and optimal transport within a broad, universal categorical framework, enabling robust, algorithm-friendly extensions to generalized persistence. This provides a rigorous backbone for stable comparisons of generalized topological invariants across metric spaces and paves the way for practical tools that leverage Banach-space embeddings and Grothendieck completions in TDA and related fields.
Abstract
Persistence diagrams, an important summary in topological data analysis, consist of a set of ordered pairs, each with positive multiplicity. Persistence diagrams are obtained via Mobius inversion and may be compared using a one-parameter family of metrics called Wasserstein distances. In certain cases, Mobius inversion produces sets of ordered pairs which may have negative multiplicity. We call these virtual persistence diagrams. Divol and Lacombe recently showed that there is a Wasserstein distance for Radon measures on the half plane of ordered pairs that generalizes both the Wasserstein distance for persistence diagrams and the classical Wasserstein distance from optimal transport theory. Following this work, we define compatible Wasserstein distances for persistence diagrams and Radon measures on arbitrary metric spaces. We show that the 1-Wasserstein distance extends to virtual persistence diagrams and to signed measures. In addition, we characterize the Cauchy completion of persistence diagrams with respect to the Wasserstein distances. We also give a universal construction of a Banach space with a 1-Wasserstein norm. Persistence diagrams with the 1-Wasserstein distance isometrically embed into this Banach space.