A Unified Kantorovich Duality for Multimarginal Optimal Transport

TL;DR

This paper develops a comprehensive Kantorovich duality theory for multimarginal optimal transport on general Polish spaces, extending classical duality principles and providing foundational results for statistical and machine learning applications.

Contribution

It introduces a unified duality framework for MOT on general spaces, including dual attainment and regularization of potentials, extending classical two-marginal results to the multimarginal setting.

Findings

Established duality on Polish spaces with bounded cost functions.

Proved dual attainment and primal-dual equality for general MOT.

Provided a structural foundation for stability and differentiability analysis.

Abstract

Multimarginal optimal transport (MOT) has gained increasing attention in recent years, notably due to its relevance in machine learning and statistics, where one seeks to jointly compare and align multiple probability distributions. This paper presents a unified and complete Kantorovich duality theory for MOT problem on general Polish product spaces with bounded continuous cost function. For marginal compact spaces, the duality identity is derived through a convex-analytic reformulation, that identifies the dual problem as a Fenchel-Rockafellar conjugate. We obtain dual attainment and show that optimal potentials may always be chosen in the class of $c$-conjugate families, thereby extending classical two-marginal conjugacy principle into a genuinely multimarginal setting. In non-compact setting, where direct compactness arguments are unavailable, we recover duality via a truncation-tightness procedure based on weak compactness of multimarginal transference plans and boundedness of the cost. We prove that the dual value is preserved under restriction to compact subsets and that admissible dual families can be regularized into uniformly bounded $c$-conjugate potentials. The argument relies on a refined use of $c$-splitting sets and their equivalence with multimarginal $c$-cyclical monotonicity. We then obtain dual attainment and exact primal-dual equality for MOT on arbitrary Polish spaces, together with a canonical representation of optimal dual potentials by $c$-conjugacy. These results provide a structural foundation for further developments in probabilistic and statistical analysis of MOT, including stability, differentiability, and asymptotic theory under marginal perturbations.