Vector valued optimal transport: from dynamic to static formulations

2025-05-06

AI Summary

Unifying dynamic (Benamou–Brenier-type) and static (Kantorovich-type) formulations of vector-valued optimal transport to support multi-species PDE modeling and vector-valued metric classification. Method: Vector-valued measures are modeled as probability measures on ℝᵈ × G, where G is a weighted finite graph; the framework integrates geometric measure theory, weighted graph embeddings, and dynamic–static coupling analysis. Contribution/Results: First proof that four vector-valued transport distances are mutually bi-Hölder equivalent—revealing the decisive role of graph structure in distance geometry. A linearizable static formulation is introduced, overcoming classical computational bottlenecks. A sharp inequality system rigorously quantifies the equivalence between dynamic and static distances, yielding a scalable, computationally tractable metric framework for multi-species PDEs and high-dimensional vector-valued data classification.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Security and Privacy: Large-scale security measurements

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 (`a la Benamou-Brenier) to static formulations (`a la Kantorovich). In our framework, vector valued measures are modeled as probability measures on a product space $mathbb{R}^d imes G$, 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-H""older 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.

Problem

• Unify dynamic and static vector valued optimal transport theories
• Model vector measures as probability measures on product spaces
• Explore applications in multispecies PDE and data analysis

Innovation

• Unifies dynamic and static vector valued optimal transport
• Models measures on product space with weighted graph
• Proves bi-Hölder equivalence between transport distances