Optimal partial transport for metric pairs

TL;DR

This paper extends the theory of optimal transport to metric pairs, establishing fundamental properties of measure spaces and their geometric features, and linking these to persistence diagram spaces with applications in geometric analysis.

Contribution

It generalizes classical optimal transport characterizations to metric pairs and demonstrates that measure spaces preserve key geometric properties, including embeddings of persistence diagrams.

Findings

Spaces of measures are complete, separable, and geodesic.

For p>1, measure spaces preserve non-branching properties.

For p=2, measure spaces preserve non-negative curvature.

Abstract

In this article we study Figalli and Gigli's formulation of optimal transport between non-negative Radon measures in the setting of metric pairs. We carry over classical characterisations of optimal plans to this setting and prove that the resulting spaces of measures, $\mathcal{M}_p(X,A)$, are complete, separable and geodesic whenever the underlying space, $X$, is so. We also prove that, for $p>1$, $\mathcal{M}_p(X,A)$ preserves the property of being non-branching, and for $p=2$ it preserves non-negative curvature in the Alexandrov sense. Finally, we prove isometric embeddings of generalised spaces of persistence diagrams $\mathcal{D}_p(X,A)$ into the corresponding spaces $\mathcal{M}_p(X,A)$, generalising a result by Divol and Lacombe. As an application of this framework, we show that several known geometric properties of spaces of persistence diagrams follow from those of $\mathcal{M}_p(X,A)$, including the fact that $\mathcal{D}_2(X,A)$ is an Alexandrov space of non-negative curvature whenever $X$ is a proper non-negatively curved Alexandrov space.