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A Unified Kantorovich Duality for Multimarginal Optimal Transport

Yehya Cheryala, Mokhtar Z. Alaya, Salim Bouzebda

TL;DR

This work develops a complete Kantorovich duality theory for multimarginal OT (MOT) on general Polish product spaces with bounded continuous costs. It proves dual attainment and that optimal potentials can be chosen within the $c$-conjugate class, using a convex-analytic reformulation and, in the non-compact case, a truncation-tightness strategy based on $c$-splitting sets. The results unify the compact and non-compact theories, with a canonical representation of dual potentials that underpins stability and asymptotic analyses under marginal perturbations. This framework supports robust statistical and computational treatments of MOT, including differentiability and empirical process considerations.

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.

A Unified Kantorovich Duality for Multimarginal Optimal Transport

TL;DR

This work develops a complete Kantorovich duality theory for multimarginal OT (MOT) on general Polish product spaces with bounded continuous costs. It proves dual attainment and that optimal potentials can be chosen within the -conjugate class, using a convex-analytic reformulation and, in the non-compact case, a truncation-tightness strategy based on -splitting sets. The results unify the compact and non-compact theories, with a canonical representation of dual potentials that underpins stability and asymptotic analyses under marginal perturbations. This framework supports robust statistical and computational treatments of MOT, including differentiability and empirical process considerations.

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 -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 -conjugate potentials. The argument relies on a refined use of -splitting sets and their equivalence with multimarginal -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 -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.
Paper Structure (25 sections, 19 theorems, 153 equations)

This paper contains 25 sections, 19 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Organization of the paper.
  4. Preliminaries and weak duality
  5. Duality: compact case
  6. Proof of Theorem \ref{['KANTO']}
  7. Legendre-Fenchel transform of $\Theta$.
  8. Legendre-Fenchel transform of $\Xi$.
  9. Duality: non-compact case
  10. Proof sketch.
  11. Proof of Theorem \ref{['KANTO2']}
  12. Optimal dual potentials via -conjugacy
  13. Compact case
  14. Proof sketch.
  15. Proof of Theorem \ref{['KKK']}
  16. ...and 10 more sections

Key Result

Lemma 1

Under the above assumptions on the measurable spaces $(X_k)_{k=1}^K$, the marginals $(\mu_k)_{k=1}^K$, and the bounded measurable cost function $c:X_1\times\cdots\times X_K\to\mathbb R$, the following weak duality relation holds:

Theorems & Definitions (28)

  • Lemma 1
  • Theorem 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Theorem 2
  • Remark 1
  • Remark 2
  • Definition 1: $c$-conjugate, inspired by pass2022
  • ...and 18 more