Vector valued optimal transport: from dynamic to static formulations

TL;DR

This work develops a cohesive theory of vector-valued optimal transport on product spaces $\Omega\times\mathcal{G}$, where $\mathcal{G}$ is a finite weighted graph encoding label interactions. It introduces three vvOT distances—a dynamic Benamou–Brenier-type distance and two static lift-based distances—establishing sharp inequalities $W_{\Omega\times\mathcal{G}}\le D_{\mathbb{R}^d\times\mathcal{G}}\le W_{2,\mathcal{W}}$ and proving bi-Hölder equivalence on bounded domains, with all metrics dominating the bounded-Lipschitz distance. The framework leverages a lifting to the simplex $\Delta^{n-1}$ to connect graph OT with spatial transport via a projection $\mathfrak{P}$ and canonical lifts, enabling a static Kantorovich-type view and a dynamic PDE-based view to be unified. The paper also develops boundary-avoidant approximations of lifted geodesics and demonstrates how lifted solutions project to vector-valued continuity equations with strictly lower action, providing a path toward gradient-flow PDEs for multispecies problems and practical linearization (LOT) for scalable data analysis. Together, these contributions unify prior vvOT approaches, illuminate the role of graph geometry in transport distances, and enable gradient-flow interpretations and computational acceleration in multispecies and labeled-data contexts.

Abstract

Motivated by applications in classification of vector valued measures and multispecies PDE, we develop a theory that unifies existing notions of vector valued optimal transport, from dynamic formulations (à la Benamou-Brenier) to static formulations (à la Kantorovich). In our framework, vector valued measures are modeled as probability measures on a product space $\mathbb{R}^d \times G$, where $G$ is a weighted graph over a finite set of nodes and the graph geometry strongly influences the associated dynamic and static distances. We obtain sharp inequalities relating four notions of vector valued optimal transport and prove that the distances are mutually bi-Hölder equivalent. We discuss the theoretical and practical advantages of each metric and indicate potential applications in multispecies PDE and data analysis. In particular, one of the static formulations discussed in the paper is amenable to linearization, a technique that has been explored in recent years to accelerate the computation of pairwise optimal transport distances.

Figures (2)

• Figure 1: An illustration of the isometry $\mathbf{p}$ between the simplex $\Delta^2 \subseteq {\mathord{\mathbb R}}^2$ and probability measures on a three node weighted graph $\mathcal{G}$. Left: Two examples of probability measures, $\sum_{i=1}^3 p_i \delta_i \in {\mathcal{P}}(\mathcal{G})$. Right: The corresponding two points in the simplex. When the simplex is endowed with the distance coming from the Wasserstein metric on ${\mathcal{P}}(\mathcal{G})$, Gangbo, Li, and Mou gangbo2019geodesics showed that there exist geodesics between points in the interior that touch the boundary.
• Figure 2: An illustration of the way in which probability measures on ${\mathord{\mathbb R}} \times \Delta^{2}$ project down to vector valued measures ${\mathcal{P}}({\mathord{\mathbb R}} \times \mathcal{G})$, in the case of a three node graph. Bottom: a vector valued measure, chosen so that the distribution of yellow mass coincides in physical space with the distribution of the first bump of red mass. Top left: the canonical lifting, where the color of the mass determines the corner of the simplex it is lifted to; shading is used to indicate the distribution of the mass, which is peaked at the center of each bump. Top right: an alternative lifting, where the yellow bump and first red bump are lifted to a distribution along a line halfway between the edge corresponding to pure yellow mass and the edge corresponding to pure red mass.

Theorems & Definitions (63)

• Theorem 1.2
• Proposition 1.3
• Theorem 1.4
• Corollary 1.5: Relations between (semi)-metrics on ${\mathcal{P}}_2(\Omega \times \mathcal{G})$
• Proposition 1.6: Examples of sharpness and inequality
• Proposition 1.8
• Remark 1.9
• Definition 2.1: Solution of graph continuity equation
• Remark 2.2: Comparison with arithmetic mean
• Remark 2.3: Vanishing at boundary