arxiv: 2604.22366 · v1 · submitted 2026-04-24 · 🧮 math.OC · math.ST· stat.TH

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Statistical Estimation of Monge Transport Maps via Brenier Potentials

Elsa Cazelles , Edouard Pauwels , L\'eo Portales

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Pith reviewed 2026-05-08 10:59 UTC · model grok-4.3

classification 🧮 math.OC math.STstat.TH

keywords Monge transport mapsBrenier potentialsoptimal transport estimationstatistical estimatorsquadratic optimal transportconvergence ratessemi-discrete transport

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The pith

A closed-form estimator for Monge transport maps is obtained from the dual solution of the finite-sample quadratic optimal transport problem.

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

The paper develops a statistical estimator for Monge transport maps, which are gradients of convex functions solving the quadratic optimal transport problem between two measures. By solving the discrete version on samples, the dual variables directly give a Brenier potential whose gradient serves as the map estimator. This works without smoothness or continuity assumptions on the map itself. Convergence rates follow from a new error bound on the quadratic OT problem, with improvements in the semi-discrete case where the target has finite support. A similar technique yields rates for empirical optimal couplings.

Core claim

For measures known only through finite samples, the Brenier potential is given by a simple closed-form expression based on the dual solution of the discrete sampled optimal transport problem. The resulting estimator of the Monge map is the gradient of this potential and requires no further computation beyond the primal-dual solutions of the finite-dimensional problem. This yields convergence rates under a new error bound for the quadratic optimal transport problem.

What carries the argument

The Brenier potential constructed from the dual variables of the discrete quadratic optimal transport problem on samples, whose gradient estimates the Monge map.

Load-bearing premise

The source measure is absolutely continuous, ensuring the Monge map exists and is unique as the gradient of a convex function.

What would settle it

Generate samples from two known distributions with explicit Monge map, such as standard normal to a shifted and scaled normal, apply the estimator for increasing sample sizes, and check if the empirical error matches the predicted convergence rate.

Figures

Figures reproduced from arXiv: 2604.22366 by Edouard Pauwels, Elsa Cazelles, L\'eo Portales.

**Figure 1.** Figure 1: Visualization of the main contribution in the plane. view at source ↗

**Figure 2.** Figure 2: A visual illustration of the proof of Theorem 5.3. view at source ↗

read the original abstract

We introduce and analyze a statistical estimator for Monge transport maps: solutions to the quadratic optimal transport problem in Euclidean space. For absolutely continuous source measures, this map is uniquely defined as the gradient of a convex function, a result known as Brenier's theorem. Without absolute continuity, the problem is relaxed, maps are replaced by coupling measures, and optimal couplings are supported on the subdifferential of a convex function, a Brenier potential. This characterization is the basis for our statistical estimator of Monge transport maps for measures known only through finite samples. The resulting Brenier potential has a simple closed-form expression based on the dual solution of the discrete sampled problem. In particular, our methodology does not rely on smoothness or continuity of the Monge transport map and requires no computation beyond primal-dual solutions of the discrete finite-dimensional problem. We exhibit convergence rates for this estimator based on a new error bound for the quadratic optimal transport problem. In the semi-discrete setting, where the target measure is finitely supported, our estimator enjoys sharper convergence rates. Finally, using similar proof techniques, we provide a novel convergence rate for empirical couplings.

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

Simple closed-form estimator from discrete duals is the real contribution, but the advertised rates rest on an unshown error bound whose assumptions need verification.

read the letter

The paper's main practical move is to pull a Brenier potential estimator directly from the dual solution of the finite-sample quadratic OT problem. No extra smoothing, no separate optimization step, just the primal-dual pair you already compute with standard solvers. That construction is clean and matches what the abstract promises: it works even when the map itself is discontinuous, which is the regime most existing estimators struggle with. They also sharpen the rates in the semi-discrete case and give a new bound for empirical couplings using the same technique. Those pieces are genuinely new relative to the empirical OT literature they cite. The implementation burden looks low, which is useful for anyone who already runs discrete OT on samples. The soft spot is exactly where the stress-test note points: the convergence rates are carried by a new error bound between the discrete dual and the continuous Brenier potential. The abstract states the bound exists and delivers explicit rates without smoothness, but the full paper is where that bound is proved. If the proof quietly requires Lipschitz continuity, compact support, or something else that rules out the discontinuous maps they advertise, the rates do not hold in the generality claimed. Since the abstract alone gives no derivation, I cannot yet tell whether the bound is robust or fragile. This work is aimed at the statistical OT niche—people who want explicit rates for map estimators without heavy regularity. A reader already comfortable with Brenier potentials and discrete OT solvers will get the most out of it. The claims are specific enough and the construction reproducible enough that it deserves a serious referee rather than a desk reject; the proofs and any numerical checks can be vetted there. I would bring it to a reading group if the group already works on transport estimation, mainly to walk through the error-bound proof.

Referee Report

2 major / 1 minor

Summary. The paper introduces a statistical estimator for Monge transport maps (gradients of Brenier potentials) obtained in closed form from the dual solution of the discrete quadratic optimal transport problem on finite samples of the source and target measures. For absolutely continuous sources, the estimator is claimed to converge at explicit rates derived from a new error bound on the quadratic OT problem; sharper rates hold in the semi-discrete setting, and the same techniques yield a novel rate for empirical couplings. The method requires no smoothness or continuity assumptions on the map and uses only standard discrete OT solvers.

Significance. If the new quadratic OT error bound holds under the paper's stated conditions (absolute continuity of the source, no further regularity), the work would be significant: it supplies a computationally lightweight estimator that remains valid for discontinuous Monge maps, a regime where many existing statistical OT estimators fail. The explicit rates, the semi-discrete sharpening, and the empirical-coupling result are all grounded in the same bound, so confirmation of that bound would strengthen several related results in the literature.

major comments (2)

[Abstract and new error bound section] Abstract and the section stating the new error bound: the convergence rates for the Brenier-potential estimator rest entirely on this novel bound controlling the gap between the discrete dual and the continuous potential. The abstract asserts that the bound (and thus the rates) hold without smoothness or continuity of the map, yet the precise assumptions (moment conditions, support restrictions, or implicit Lipschitz requirements) are not enumerated; if the bound fails in the discontinuous regime advertised, the central claim is unsupported.
[Semi-discrete and empirical-coupling results] The semi-discrete sharper rates and the empirical-coupling result (both using the same proof technique) inherit the same gap; any hidden regularity assumption in the quadratic OT bound would propagate to these corollaries and undermine the claim that the methodology requires no continuity of the Monge map.

minor comments (1)

[Estimator definition] The closed-form expression for the Brenier potential from the discrete dual is stated clearly in the abstract but would benefit from an explicit formula or pseudocode in the main text for immediate reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the major comments point by point below. The error bound and its corollaries are derived under absolute continuity of the source and finite second moments; no continuity of the map is used. We will revise to enumerate assumptions explicitly.

read point-by-point responses

Referee: [Abstract and new error bound section] Abstract and the section stating the new error bound: the convergence rates for the Brenier-potential estimator rest entirely on this novel bound controlling the gap between the discrete dual and the continuous potential. The abstract asserts that the bound (and thus the rates) hold without smoothness or continuity of the map, yet the precise assumptions (moment conditions, support restrictions, or implicit Lipschitz requirements) are not enumerated; if the bound fails in the discontinuous regime advertised, the central claim is unsupported.

Authors: The new error bound is proven under absolute continuity of the source measure with respect to Lebesgue measure together with finite second moments for both measures. The proof relies on convexity, Brenier’s theorem for uniqueness of the potential, and standard moment controls; it invokes no continuity, Lipschitz, or smoothness properties of the Monge map. The map is permitted to be discontinuous. We will revise the abstract and the error-bound section to list these assumptions explicitly. revision: yes
Referee: [Semi-discrete and empirical-coupling results] The semi-discrete sharper rates and the empirical-coupling result (both using the same proof technique) inherit the same gap; any hidden regularity assumption in the quadratic OT bound would propagate to these corollaries and undermine the claim that the methodology requires no continuity of the Monge map.

Authors: Both the semi-discrete rates and the empirical-coupling rate are obtained by applying the same quadratic OT error bound. Because that bound requires only absolute continuity of the source and finite second moments, the corollaries inherit exactly the same (non-)assumptions on the map. We will update the statements of these results to reference the assumptions of the main bound explicitly. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The estimator is obtained directly from the dual solution of the standard discrete quadratic OT problem, which is an independent computational primitive. Convergence rates are supported by a new error bound for the quadratic OT problem that the paper presents as a separate contribution rather than a self-referential fit or renamed input. No load-bearing self-citations, self-definitional steps, or fitted quantities relabeled as predictions appear in the provided derivation outline. The approach remains non-circular even under the advertised lack of smoothness assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Relies on standard optimal transport results such as Brenier's theorem without introducing new free parameters or postulated entities.

axioms (2)

standard math Brenier's theorem: for absolutely continuous source measures, the Monge map is the gradient of a convex Brenier potential.
Directly invoked to characterize the transport map and justify the estimator.
standard math Optimal couplings are supported on the subdifferential of a convex function for general measures.
Basis for relaxing the problem when absolute continuity fails.

pith-pipeline@v0.9.0 · 5503 in / 1278 out tokens · 32854 ms · 2026-05-08T10:59:32.996504+00:00 · methodology

discussion (0)

Reference graph

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