Disintegrated optimal transport for metric fiber bundles

TL;DR

This paper introduces a new family of metrics called disintegrated Monge--Kantorovich metrics on probability measures over metric fiber bundles, unifying and extending classical optimal transport distances with new geometric and duality properties.

Contribution

It defines a novel two-parameter metric family on fiber bundles that generalizes existing optimal transport metrics and establishes their fundamental geometric properties.

Findings

Metrics are complete, separable (except at an endpoint), and geodesic.

Includes isometric embeddings of sliced Wasserstein spaces.

Provides new duality and geodesic results in fibered Wasserstein spaces.

Abstract

We define a new two-parameter family of metrics on subsets of Borel probability measures on general metric fiber bundles, called the $ \textit{disintegrated Monge--Kantorovich metrics}$. This family contains the classical Monge-Kantorovich metrics, linearized optimal transport distance, and fibered Wasserstein distances, and certain cases admit isometric embeddings of the sliced and max-sliced Wasserstein spaces. We prove these metrics are complete, separable (except an endpoint case), and geodesic, with a dual representation. Our results cannot be obtained by applying the theory of $L^q$ maps valued in spaces of probability measures, in fact the $L^q$ map case can be recovered from our results by taking the underlying bundle as a trivial product bundle, and the geodesicness and duality results are new even in the fibered Wasserstein case.