Optimal partial transport for metric pairs
Mauricio Che
TL;DR
The paper develops a comprehensive extension of optimal partial transport to metric pairs (X,A), proving fundamental geometric and measure-theoretic properties for the resulting spaces M_p(X,A) and Wb_p. It demonstrates that M_p(X,A) is complete, separable, and geodesic when X is, with non-branching for p>1 and Alexandrov nonnegative curvature for p=2, and it provides an isometric embedding of generalized persistence diagram spaces into M_p(X,A). The work further characterizes optimal partial transport via c-concave potentials and c-cyclical monotonicity, and it connects these spaces to persistence diagram geometry, enabling curvature results for D_2(X,A) in terms of X. These results extend and complement prior frameworks by handling metric pairs and offer new tools for topological data analysis and unbalanced OT. Applications include deriving curvature and geometric properties for spaces of persistence diagrams in settings where X is proper and nonnegatively curved.
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.