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Entropic selection for optimal transport on the line with distance cost

Armand Ley

TL;DR

This work addresses the selection problem in entropic optimal transport on the real line with the distance cost. It introduces a rigorous one-dimensional decomposition of transport plans into independent marginal components and defines a natural limiting object, the Kellerer transport plan $\mathcal{K}(\mu,\nu)$, toward which entropic minimizers converge under suitable conditions. The authors prove that $\mathcal{K}(\mu,\nu)$ minimizes entropy among cyclically monotone transports and, when a finite-entropy OT plan exists, the entropic minimizers converge to it; cluster points are shown to exhibit weak multiplicativity, a structural property preserved under limits. They also extend the decomposition framework to general marginals (allowing atoms) and discuss higher-dimensional extensions via transport-ray decompositions, connecting to classical Sudakov-type decompositions. Collectively, the results provide a principled selection mechanism for entropic OT on the line and lay groundwork for higher-dimensional analysis.

Abstract

We study the small-regularisation limit of the entropic optimal transport problem on the line with distance cost. While convergence of entropic minimizers is well understood in the discrete setting and in the case where the cost is continuous and there is a unique optimal transport plan, the question of existence and characterization outside these settings remains largely open. We propose a natural candidate for the limiting object and establish its convergence under mutual singularity of the marginals. For arbitrary marginals, we moreover prove that every limit point of entropic minimizers obeys a structural condition known as weak multiplicativity. The construction of our candidate relies on a decomposition theorem for optimal transport plan that we believe is of independent interest. This article complements the previous work of Di Marino and Louet.

Entropic selection for optimal transport on the line with distance cost

TL;DR

This work addresses the selection problem in entropic optimal transport on the real line with the distance cost. It introduces a rigorous one-dimensional decomposition of transport plans into independent marginal components and defines a natural limiting object, the Kellerer transport plan , toward which entropic minimizers converge under suitable conditions. The authors prove that minimizes entropy among cyclically monotone transports and, when a finite-entropy OT plan exists, the entropic minimizers converge to it; cluster points are shown to exhibit weak multiplicativity, a structural property preserved under limits. They also extend the decomposition framework to general marginals (allowing atoms) and discuss higher-dimensional extensions via transport-ray decompositions, connecting to classical Sudakov-type decompositions. Collectively, the results provide a principled selection mechanism for entropic OT on the line and lay groundwork for higher-dimensional analysis.

Abstract

We study the small-regularisation limit of the entropic optimal transport problem on the line with distance cost. While convergence of entropic minimizers is well understood in the discrete setting and in the case where the cost is continuous and there is a unique optimal transport plan, the question of existence and characterization outside these settings remains largely open. We propose a natural candidate for the limiting object and establish its convergence under mutual singularity of the marginals. For arbitrary marginals, we moreover prove that every limit point of entropic minimizers obeys a structural condition known as weak multiplicativity. The construction of our candidate relies on a decomposition theorem for optimal transport plan that we believe is of independent interest. This article complements the previous work of Di Marino and Louet.

Paper Structure

This paper contains 25 sections, 38 theorems, 65 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. On the Monge-Kantorovitch problem and its entropic regularization
  3. A decomposition result for optimal transport plans
  4. Decomposition theorem for the set of martingale transport plans
  5. A decomposition result for cyclically monotone transport plan
  6. Back to the selection problem
  7. Convergence of minimizers for the $L^1$ entropic regularization on the real line
  8. Decomposition and regularized problem in higher dimension
  9. Organization of the paper
  10. General notation
  11. The decomposition result.
  12. The structure result of Di Marino--Louet
  13. Generalized decomposition result
  14. Applications of the decomposition result
  15. Definition of $\mathcal{K}(\mu,\nu)$ and characterization of $\mathcal{B}(\mu,\nu)$
  16. ...and 10 more sections

Key Result

Theorem A

Let $\mu$ and $\nu$ be two atomless measures on $\mathbb{R}$ such that $\min J < +\infty$, and define the marginal components $((\mu_k^+)_{k \in \mathcal{K}^+},(\mu_k^-)_{k \in \mathcal{K}^-},$$\mu^=)$ and $((\nu_k^+)_{k \in \mathcal{K}^+},(\nu_k^-)_{k \in \mathcal{K}^-},\nu^=)$ as in Equation def:c

Figures (5)

  • Figure 1: Representation of $F_\mu^+$ (in red) and $F_\nu^+$ (in green) and mass displacement of elements of $\mathcal{O}(\mu,\nu)$
  • Figure 2: Concentration region of optimal transport plans when $x \in \mathbb{R}$ is a barrier point.
  • Figure 3: Illustration of the concentration area of optimal transport plans.
  • Figure 4: Splitting of point 2 in Example \ref{['ex:guiding_example']}
  • Figure 5: Free crossings (in green) and non-free crossings (in red).

Theorems & Definitions (97)

  • Theorem A
  • Theorem B
  • Theorem C
  • Proposition 2.1: Structure result by Di Marino--Louet
  • Example 2.2
  • Definition 2.3
  • Remark 2.5
  • Theorem 2.6
  • Remark 2.7: Cyclicity and optimality of sub-measures
  • Definition 2.8: Barrier points
  • ...and 87 more