Entropic regularization of Monge's problem

Abstract: We study the vanishing-regularization limit of entropically regularized optimal transport (EOT) for the Euclidean distance cost $c(x,y)=|x-y|$ in dimension $d>1$. We develop a comprehensive variational convergence framework that entails two main results. First, we resolve the longstanding entropic selection problem: the EOT minimizer converges to a distinguished optimal transport plan that is characterized explicitly as the solution of a constrained EOT problem on each transport ray. Denoting by $\varepsilon>0$ the regularization parameter, this selection holds for all $o(\varepsilon)$-approximate minimizers, with sharp failure at the $O(\varepsilon)$ scale. Second, we establish an explicit second-order expansion of the entropic transport cost. The second-order term encodes the geometry of the regularization and reveals the optimal asymptotic tradeoff between entropy and transport cost.

• The paper establishes a variational Γ-convergence framework proving that entropic regularized optimal transport converges to a uniquely selected entropic Monge plan as ε → 0+.
• The paper provides a detailed second-order expansion of the EOT cost, revealing a sharp correction term of −((d−1)/2) logε that depends on transport geometry.
• The paper highlights a delicate balance between minimal transport cost and diffusion effects, emphasizing the role of transport rays in ensuring solution stability.

Entropic Regularization of Monge's Problem: A Detailed Summary

Introduction and Problem Setting

The paper "Entropic regularization of Monge's problem" (2604.21578) rigorously analyzes the limit of entropic regularized optimal transport (EOT) for the Euclidean cost $c(x,y) = \|x-y\|$ in dimensions $d > 1$. The authors address the convergence of EOT plans and costs to Monge's problem as the regularization parameter $\varepsilon \to 0^+$, specifically targeting the 1-Wasserstein (Earthmover's) distance, and develop a fully formal variational (Γ-convergence) framework to characterize this limit.

Unlike the quadratic cost case, linear transport costs such as the Euclidean distance exhibit non-unique optimal plans in higher dimensions. The EOT cost, augmented with an entropy penalty ($H(\gamma | \mu \otimes \nu)$), always admits a unique minimizer for $\varepsilon > 0$. The main challenge is to describe which optimal plan is picked by the vanishing regularization, i.e., the entropic selection problem, and to characterize its fine properties. This fills an essential theoretical gap for both mathematical theory and computational practice, as EOT is widely used for scalable approximations via Sinkhorn-type algorithms.

Main Contributions

1. Variational Γ-Convergence and the Entropic Selection Problem

The authors prove $\Gamma$-convergence of the EOT functional towards Monge's linear cost problem with an explicit second-order expansion. Their framework involves disintegration along transport rays—maximal line segments along which the Kantorovich potential decreases at unit speed. Each transport plan can be described locally (on each ray), revealing the structure of the entropic limit.

Key Results:

• Entropic Monge Plan Construction: In the $d>1$ continuous setting with Lipschitz densities and disjoint supports, as $\varepsilon \to 0^+$, the EOT minimizer converges to a specific "entropic Monge plan." This plan is obtained, for almost every ray, as the solution to a novel constrained EOT problem with cost $-\frac{d-1}{2}\log\|x-y\|$ under a monotonicity constraint reflecting optimal raywise transport direction.
• Sharper Convergence Notions: The selection holds for all $o(\varepsilon)$-approximate minimizers, but convergence can fail at $d > 1$0; this threshold is sharp.

2. Second-Order Expansion of the EOT Cost

The paper establishes the asymptotics of $d > 1$1: $d > 1$2 with an explicit functional $d > 1$3 incorporating the geometry of the Monge problem and fine information about the limiting plan $d > 1$4. The first-order term only depends on the ambient dimension and the diffusion geometry (orthogonal to rays), in sharp contrast to quadratic or degenerate costs.

3. Fine Structure of the Selection and Tradeoff

Their analysis reveals that, contrary to discrete or $d > 1$5 cases, there is no separation into sequential minimization of cost and entropy. Instead, for the Euclidean cost in $d > 1$6, the limit reflects a delicate competition between small entropy (diffusion orthogonal to rays) and minimal transport cost. The selection functional $d > 1$7 encodes a "renormalized energy" akin to that arising in the Ginzburg–Landau theory.

Strong and Contradictory Claims

• Monotonicity Constraint Essential: The limiting plan is not simply the optimal coupling of minimal (finite or infinite) relative entropy; it is the unique solution to a monotone-constrained EOT problem on each transport ray.
• Second Order Term Depends on the Plan: In contrast to previous discrete or $d > 1$8 results, the second-order ($d > 1$9) term in the EOT expansion depends crucially on both marginals and the geometry of the Monge plan, not just the regularization parameter and cost.
• No Generic Two-Stage Selection Principle: The common two-stage selection principle (first minimize cost, then entropy) for limiting EOT fails in this non-uniqueness, $\varepsilon \to 0^+$0 case.

Numerical and Theoretical Strengths

The paper develops careful quantitative regularity estimates:

• The sharp rate in the entropy cost expansion ($\varepsilon \to 0^+$1).
• A unique local description of the limit in terms of raywise (monotone) constrained problems.
• Explicit sharp error regimes for approximate minimizers.

These results connect fine aspects of potential theory, geometric measure theory, and variational analysis, leveraging new regularity statements for the Kantorovich potential and the structure of transport rays. The multiscale block-approximation arguments for the upper bound and the second-order expansion mark a significant methodological advance.

Implications and Future Directions

Theoretical Implications

• Refined Understanding of EOT/OT Interface: The results close a theoretical gap regarding entropic selection in continuous, high-dimensional problems with non-unique Monge solutions.
• Raywise Localization as a General Tool: The local decomposition along transport rays provides a foundation for further generalization to other non-unique or degenerate settings and possibly other regularizations.
• Limits of Computational Approximation: The $\varepsilon \to 0^+$2 regime aligns with practical settings wherein Sinkhorn-type algorithms are terminated at limited precision.

Practical Implications

• Sensitivity of Selected Plans: Small changes in $\varepsilon \to 0^+$3 (or computation error) can select qualitatively different plans, emphasizing the need for understanding stability and error estimates when using EOT as a computational device.
• Guidance for Algorithmic Design: The multiscale analysis and explicit functional forms can be used to design sharper, more stable numerical solvers that mimic the entropic plan.

Prospects for Future Work

• Overlap and Degenerate Cases: The paper outlines (Remark 1.7) the substantial new technical challenges for marginals with overlapping supports, suggesting a refined multi-level block-approximation and new regularity theory near zero-length rays.
• General Cost Structures: Extending the local variational characterization and asymptotics to more general costs, possibly via local model analysis or PDE approximation, remains a challenging open problem.

Conclusion

This work delivers a precise variational limit for entropic regularization of Monge's linear optimal transport problem in dimensions $\varepsilon \to 0^+$4, solving a long-standing open problem and refining the theoretical foundation at the intersection of optimal transport and statistical regularization. The explicit construction of the entropic Monge plan, the sharpness of selection, and the second-order expansion have strong implications for both further theory and the stable computation of OT approximations in applied settings. The methods and results are expected to inform developments in high-dimensional statistics, computational optimal transport, and the analysis of interacting particle systems.