Entropic regularization of Monge's problem

Chenyang Zhong; Marcel Nutz

arxiv: 2604.21578 · v2 · submitted 2026-04-23 · 🧮 math.OC · math.AP· math.FA· math.PR

Entropic regularization of Monge's problem

Marcel Nutz , Chenyang Zhong This is my paper

Pith reviewed 2026-05-09 21:23 UTC · model grok-4.3

classification 🧮 math.OC math.APmath.FAmath.PR

keywords entropic optimal transportMonge problemregularization limittransport raysEuclidean distancevariational convergencesecond-order expansion

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The pith

For Euclidean distance, entropic optimal transport converges to a distinguished plan that solves a constrained problem on each transport ray.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies the limit of entropically regularized optimal transport as the regularization parameter ε approaches zero, using the Euclidean distance as the cost function in dimensions greater than one. It shows that the regularized minimizers converge to a specific optimal transport plan obtained by solving a separate constrained entropic problem along each transport ray. This explicit selection among possibly many optimal plans holds for any approximate minimizer whose value lies within o(ε) of the true minimum. The analysis also produces a second-order asymptotic expansion of the regularized transport cost that isolates the geometric contribution of the entropy term.

Core claim

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. This selection holds for all o(ε)-approximate minimizers, with sharp failure at the O(ε) scale. The framework additionally yields an explicit second-order expansion of the entropic transport cost whose leading correction term encodes the geometry of the regularization.

What carries the argument

The decomposition of the support into transport rays induced by the Euclidean distance, together with a variational convergence argument that reduces the global selection to independent constrained entropic problems on each ray.

If this is right

The limiting plan can be constructed ray by ray via independent constrained entropic problems.
The second-order expansion gives the precise asymptotic rate at which the regularized cost approaches the unregularized optimal value.
Any sequence of o(ε)-approximate minimizers still selects the same distinguished plan.
The selection property ceases to hold exactly when the approximation error reaches order ε.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The ray-wise construction may supply a practical method to pick one optimal plan when the unregularized problem is non-unique.
The second-order term could guide the choice of regularization strength in numerical schemes that trade off entropy against transport cost.
Similar ray-based selection might appear for other costs whose level sets permit an analogous decomposition, though the paper treats only the Euclidean case.

Load-bearing premise

The cost must be exactly the Euclidean distance in dimension greater than one so that the transport rays and their geometry can be used to localize the entropy effect.

What would settle it

An explicit pair of marginal measures possessing multiple optimal plans for which the limit of ε-minimizers fails to match the ray-constrained EOT solution, or a sequence of O(ε)-approximate minimizers whose limit differs from the distinguished plan.

Figures

Figures reproduced from arXiv: 2604.21578 by Chenyang Zhong, Marcel Nutz.

**Figure 2.** Figure 2: Construction for bounding u(x + ˜w) − u(x + w) in (1.13) The underlying geometric idea is illustrated in [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** Figure 3: Heuristic construction for the upper bound. The optimal transport kernel [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

**Figure 4.** Figure 4: Heuristic construction for the density p2. The “reverse” optimal transport kernel κ ′ 0,T′(·|y) sends y to a random point z ′ ∈ T ′ . We then diffuse orthogonally to T ′ by adding w ′ , drawn from a (d − 1)-dimensional Gaussian supported in OT′. The disintegration of p2 mimics a truncated version of this continuous kernel. Step 2: Balancing the marginals of the sub-coupling on each ray. The density p := mi… view at source ↗

**Figure 5.** Figure 5: Symmetry construction on a pair of transport rays. The symmetry points of [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗

**Figure 6.** Figure 6: Two-level anisotropic block approximation. Level 1 partitions [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives an explicit ray-by-ray characterization of the entropic limit for Euclidean cost and a second-order expansion of the regularized cost.

read the letter

The key advance here is a precise description of which optimal transport plan the entropic minimizers approach as the regularization parameter vanishes. For the Euclidean distance in dimension d greater than 1, the limit is the plan that solves a constrained entropic problem separately on each transport ray. This selection is stable for all approximations better than order epsilon and breaks exactly at order epsilon. They also derive an explicit second-order term in the expansion of the entropic cost that captures the asymptotic tradeoff between entropy and transport cost. Both results are new relative to earlier work on vanishing regularization limits. The variational framework they build to prove convergence looks solid because it exploits the straight-line geometry of the Euclidean cost to reduce the problem to independent one-dimensional subproblems along the rays. That reduction is the main technical step and it seems to work under standard assumptions on the marginals. The paper is careful about the rates and does not overclaim stability. The main limitation is that the whole argument is tied to the Euclidean cost; the ray decomposition does not carry over to general costs, so the selection result is specific to this setting. The second-order expansion is also stated for this cost, and it would be useful to see how the constants behave in low-dimensional examples. Overall the claims are focused and the approach is direct. This is for people working on regularized optimal transport, especially those who care about the vanishing regularization limit and its consequences for approximation. It is worth sending to a serious referee because the entropic selection question has been open and the paper supplies an explicit answer with matching rates.

Referee Report

1 major / 0 minor

Summary. The paper studies the vanishing-regularization limit of entropically regularized optimal transport (EOT) for the Euclidean distance cost in dimension d>1. It develops a variational convergence framework showing that EOT minimizers converge to a distinguished optimal transport plan characterized as the solution of a constrained EOT problem on each transport ray. This holds for o(ε)-approximate minimizers with sharp failure at O(ε), and provides an explicit second-order expansion of the entropic transport cost that encodes the geometry of the regularization.

Significance. If the results hold, this provides a resolution to the entropic selection problem in OT with precise rates and geometric insights from the ray decomposition. The explicit second-order expansion is a notable strength, offering an optimal asymptotic tradeoff between entropy and transport cost. The approach leverages the Euclidean geometry effectively under standard marginal regularity, contributing to the understanding of regularization effects in variational problems.

major comments (1)

The sharpness of the o(ε) vs O(ε) distinction for approximate minimizers is load-bearing for the selection claim; the manuscript should include a concrete example or construction showing failure at the O(ε) scale to substantiate the 'sharp failure' assertion.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading, positive assessment, and constructive suggestion. We respond to the major comment below.

read point-by-point responses

Referee: The sharpness of the o(ε) vs O(ε) distinction for approximate minimizers is load-bearing for the selection claim; the manuscript should include a concrete example or construction showing failure at the O(ε) scale to substantiate the 'sharp failure' assertion.

Authors: We appreciate the referee pointing out the value of an explicit illustration for the sharpness claim. Our proof establishes the o(ε) selection via a variational convergence argument showing that limit points of o(ε)-approximate plans must solve the ray-constrained EOT problem, while the O(ε) threshold arises from a quantitative error estimate that permits deviation at that scale. To make this distinction more concrete and accessible, we will add a simple explicit construction in dimension 2 (with uniform marginals supported on two parallel line segments) in the revised manuscript. This example will demonstrate an O(ε)-approximate plan whose limit fails to be the distinguished ray-constrained solution, thereby substantiating the sharpness. The addition will appear in the section on the variational convergence framework. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained variational analysis

full rationale

The paper develops a variational convergence framework for the ε→0 limit of entropically regularized OT with Euclidean cost. The central results—the explicit characterization of the limiting plan as a constrained EOT problem on each transport ray, the o(ε) selection property, and the second-order cost expansion—are obtained by decomposing the problem along rays using the geometry of the Euclidean distance and applying standard marginal regularity. No equation or claim reduces by construction to a fitted parameter, a self-referential definition, or a load-bearing self-citation whose content is itself unverified. The framework is externally falsifiable via the stated assumptions on marginals and cost, and the abstract and structure indicate independent mathematical derivation rather than renaming or smuggling of prior results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard background from optimal transport and measure theory; no free parameters, invented entities, or ad-hoc axioms are mentioned in the abstract.

axioms (1)

standard math Existence and basic properties of optimal transport plans for continuous cost functions on Euclidean space.
Implicitly used to set up the EOT problem and its limit.

pith-pipeline@v0.9.0 · 5443 in / 1181 out tokens · 30487 ms · 2026-05-09T21:23:12.928371+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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A., and Zimmer, J.From a large-deviations principle to the Wasserstein gradient flow: a new micro-macro passage.Comm

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A Wiley- Interscience Publication. [10]Caffarelli, L. A., Feldman, M., and McCann, R. J.Constructing optimal maps for Monge’s transport problem as a limit of strictly convex costs.J. Amer. Math. Soc. 15, 1 (2002), 1–26. [11]Carlier, G., Duval, V., Peyr ´e, G., and Schmitzer, B.Convergence of entropic schemes for optimal transport and gradient flows.SIAM J...

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T., and Pavon, M.On the relation between optimal transport and Schr¨ odinger bridges: a stochastic control viewpoint.J

[13]Chen, Y., Georgiou, T. T., and Pavon, M.On the relation between optimal transport and Schr¨ odinger bridges: a stochastic control viewpoint.J. Optim. Theory Appl. 169, 2 (2016), 671–691. 193 [14]Chewi, S., Niles-Weed, J., and Rigollet, P.Statistical optimal transport, vol. 2364 of Lecture Notes in Mathematics. Springer, Cham,

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Partial Differential Equations 48, 6 (2023), 895–943

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[22]Eckstein, S., and Nutz, M.Quantitative stability of regularized optimal transport and convergence of Sinkhorn’s algorithm.SIAM J. Math. Anal. 54, 6 (2022), 5922–5948. [23]Eckstein, S., and Nutz, M.Convergence rates for regularized optimal transport via quan- tization.Math. Oper. Res. 49, 2 (2024), 1223–1240. [24]Erbar, M., Maas, J., and Renger, D. R. ...

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[28]F ¨ollmer, H.Random fields and diffusion processes. In ´Ecole d’ ´Et´ e de Probabilit´ es de Saint- Flour XV–XVII, 1985–87, vol. 1362 ofLecture Notes in Math.Springer, Berlin, 1988, pp. 101–

work page 1985

[8] [8]

[29]F ¨ollmer, H., and Gantert, N.Entropy minimization and Schr¨ odinger processes in infinite dimensions.Ann. Probab. 25, 2 (1997), 901–926. [30]Fournier, N., and Guillin, A.On the rate of convergence in Wasserstein distance of the empirical measure.Probab. Theory Related Fields 162, 3-4 (2015), 707–738. 194 [31]Ghosal, P., Nutz, M., and Bernton, E.Stabi...

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[37]L ´eonard, C.From the Schr¨ odinger problem to the Monge-Kantorovich problem.J

[36]Lacker, D.Mean field games via controlled martingale problems: existence of Markovian equilibria.Stochastic Processes and their Applications 125, 7 (2015), 2856–2894. [37]L ´eonard, C.From the Schr¨ odinger problem to the Monge-Kantorovich problem.J. Funct. Anal. 262, 4 (2012), 1879–1920. [38]L ´eonard, C.A survey of the Schr¨ odinger problem and some...

work page arXiv 2015

[10] [10]

Entropic e stimation of optimal transport maps

Recorded presentation,https://www.imsi.institute/videos/ entropic-selection-in-optimal-transport. 195 [48]Nutz, M., and Wiesel, J.Entropic optimal transport: convergence of potentials.Probab. Theory Related Fields 184, 1-2 (2022), 401–424. [49]Pal, S.On the difference between entropic cost and the optimal transport cost.Ann. Appl. Probab. 34, 1B (2024), 1...

work page arXiv 2022

[11] [11]

[53]Sandier, E., and Serfaty, S.From the Ginzburg-Landau model to vortex lattice problems. Comm. Math. Phys. 313, 3 (2012), 635–743. [54]Santambrogio, F.Optimal transport for applied mathematicians, vol. 87 ofProgress in Nonlinear Differential Equations and their Applications. Birkh¨ auser/Springer, Cham,

work page 2012

[12] [12]

¨Uber die Umkehrung der Naturgesetze

[55]Schr ¨odinger, E. ¨Uber die Umkehrung der Naturgesetze. Sitzungsberichte Preuss. Akad. Wiss.Akad. Wiss., Berlin. Phys. Math. 144(1931), 144–153. [56]Sudakov, V. N.Geometric problems in the theory of infinite-dimensional probability distri- butions.Proc. Steklov Inst. Math., 2 (1979), i–v, 1–178. [57]Trudinger, N. S., and Wang, X.-J.On the Monge mass t...

work page 1931