Disintegrated optimal transport for metric fiber bundles

TL;DR

This work defines disintegrated Monge–Kantorovich metrics $\\mathcal{M}^{\sigma}_{p,q}$ on probability measures supported on metric fiber bundles, generalizing classical OT concepts to dynamics along fibers. By disintegrating measures over the base and measuring costs fiberwise via MK distances, the authors establish a robust geometric framework: the metric space $(\\mathcal{P}^\sigma_{p,q}(E),\\mathcal{M}^{\sigma}_{p,q})$ is complete and separable (when $q<\infty$), geodesic under suitable fiber-base geometry, and possesses a dual Kantorovich formulation. They develop a comprehensive barycenter theory in this disintegrated setting, proving existence, duality, and, under additional regularity assumptions, uniqueness; this yields new, broad results for classical MK barycenters on complete, connected Riemannian manifolds with no curvature or injectivity radius constraints. The framework also connects to sliced and linearized OT via isometric embeddings and offers a foundation for variational and gradient-flow analyses of spatially inhomogeneous kinetic-type evolutions on manifolds and nonlinear fibers.

Abstract

We define a new two-parameter family of metrics on a subset of Borel probability measures on a general metric fiber bundle, called the $ \textit{disintegrated Monge--Kantorovich metrics}$. This family of metrics contains the classical Monge-Kantorovich metrics, linearized optimal transport distance, and generalizes the sliced and max-sliced Wasserstein metrics. We prove these metrics are complete, separable (except an endpoint case), geodesic spaces, with a dual representation. Additionally, we prove existence and duality for an associated barycenter problem, and provide conditions for uniqueness of the barycenter. These results on barycenter problems for the disintegrated Monge--Kantorovich metrics also yield the corresponding existence, duality, and uniqueness results for classical Monge--Kantorovich barycenters in a wide variety of spaces, including a uniqueness result on any connected, complete Riemannian manifold, with or without boundary; this is the first and only result with absolutely no restriction on the geometry of the manifold (such as on curvatures or injectivity radii). 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.

Theorems & Definitions (78)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Theorem 1.5
• Theorem 1.6
• Remark 1.7
• Corollary 1.8
• Theorem 2.1: Villani09*Theorem 6.9, Theorem 6.18
• Definition 2.2