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Barycenters in Disintegrated optimal transport

Jun Kitagawa, Asuka Takatsu

TL;DR

This work introduces a fiber-bundle perspective on optimal transport by formulating disintegrated Monge–Kantorovich metrics $\mathcal{M}^{\sigma}_{p,q}$ on probability measures supported on a metric fiber bundle. It proves existence and duality for barycenters in this setting, and, notably, establishes a unique barycenter on any complete, connected Riemannian manifold (with or without boundary) under mild absolute continuity assumptions, without geometric constraints such as curvature bounds or injectivity radius. The results extend classical Agueh–Carlier duality to broad metric spaces and provide sharp, geometry-free uniqueness in the Riemannian context, with the $p=q$ case achieving dual-attainment by a simple dual choice. The methodology hinges on a robust duality framework for disintegrated metrics, tightness via Heine–Borel properties, and fibrewise optimal transport maps that yield uniqueness through single-valued transport on almost every fibre.

Abstract

We prove existence and duality on a wide class of metric spaces, and uniqueness results on any connected, complete Riemannian manifold, with or without boundary, for classical Monge--Kantorovich barycenters. In particular, this is the first and only uniqueness result with no restriction on the geometry of the manifold aside from connectedness and completeness. We obtain these via the corresponding results for barycenter problems associated to a new two-parameter family of metrics on probability measures on a general metric fiber bundle, called the $\textit{disintegrated Monge--Kantorovich metrics}$ (previously introduced by the authors).

Barycenters in Disintegrated optimal transport

TL;DR

This work introduces a fiber-bundle perspective on optimal transport by formulating disintegrated Monge–Kantorovich metrics on probability measures supported on a metric fiber bundle. It proves existence and duality for barycenters in this setting, and, notably, establishes a unique barycenter on any complete, connected Riemannian manifold (with or without boundary) under mild absolute continuity assumptions, without geometric constraints such as curvature bounds or injectivity radius. The results extend classical Agueh–Carlier duality to broad metric spaces and provide sharp, geometry-free uniqueness in the Riemannian context, with the case achieving dual-attainment by a simple dual choice. The methodology hinges on a robust duality framework for disintegrated metrics, tightness via Heine–Borel properties, and fibrewise optimal transport maps that yield uniqueness through single-valued transport on almost every fibre.

Abstract

We prove existence and duality on a wide class of metric spaces, and uniqueness results on any connected, complete Riemannian manifold, with or without boundary, for classical Monge--Kantorovich barycenters. In particular, this is the first and only uniqueness result with no restriction on the geometry of the manifold aside from connectedness and completeness. We obtain these via the corresponding results for barycenter problems associated to a new two-parameter family of metrics on probability measures on a general metric fiber bundle, called the (previously introduced by the authors).
Paper Structure (7 sections, 9 theorems, 187 equations)

This paper contains 7 sections, 9 theorems, 187 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminary definitions
  3. Discussion of techniques and literature
  4. Disintegrated barycenters
  5. Existence of disintegrated barycenters
  6. Duality for disintegrated barycenters
  7. Uniqueness of disintegrated barycenters

Key Result

Theorem 1.1

Fix $K\in \mathbb{N}$, $K\geq 2$, $(\lambda_k)_{k=1}^K\in \Lambda_K$, $1\leq p<\infty$. Let $(Y,\mathop{\mathrm{\mathrm{d}}}\nolimits_Y)$ be a complete, separable metric space and fix some subset $\{\mu_k\}_{k=1}^K\subset\mathcal{P}_p(Y)$.

Theorems & Definitions (26)

  • Theorem 1.1
  • Definition 1.2
  • Definition 1.3: KitagawaTakatsu25a*Definition 1.2
  • Theorem 1.4
  • Remark 1.5
  • proof : Proof of Theorem \ref{['thm: barycenters']} \ref{['thm: disint bary exist']}
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • ...and 16 more