Title: Estimation of Stochastic Optimal Transport Maps
Venue: NeurIPS 2025
Accepted: yes
Keywords: optimal transport

map estimation

non-parametric statistics

robust statistics
Abstract: The optimal transport (OT) map is a geometry-driven transformation between high-dimensional probability distributions which underpins a wide range of tasks in statistics, applied probability, and machine learning. 
However, existing statistical theory for OT map estimation is quite restricted, hinging on Brenier's theorem (quadratic cost, absolutely continuous source) to guarantee existence and uniqueness of a deterministic OT map, on which various additional regularity assumptions are imposed to obtain quantitative error bounds. In many real‐world problems these conditions fail or cannot be certified, in which case optimal transportation is possible only via stochastic maps that can split mass.
To broaden the scope of map estimation theory to such settings, this work introduces a novel metric for evaluating the transportation quality of stochastic maps. Under this metric, we develop computationally efficient map estimators with near-optimal finite-sample risk bounds, subject to easy-to-verify minimal assumptions. Our analysis further accommodates common forms of adversarial sample contamination, yielding estimators with robust estimation guarantees. Empirical experiments are provided which validate our theory and demonstrate the utility of the proposed 
framework in settings where existing theory fails. These contributions constitute the first general-purpose theory for map estimation, compatible with a wide spectrum of real-world applications where optimal transport may be intrinsically stochastic.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces a new measure of error for transition kernels $\\kappa(y|x)$ used as stochastic transportation maps between $\\nu$ and $\\mu$, coined *Transportation Error* $\\mathcal{E}_{p}(\\kappa,\\mu,\\nu)$ which decomposes transportation cost into two terms:\n- the *optimality gap* induces by the mapping $\\kappa$ and the optimal $T^*$\n- the *feasibility gap* that quantifies the failure to meet the pushforward constraint $\\nu = T\\sharp \\mu$\n\nThis decomposition has several advantages:\n- better models of errors under TV or Wasserstein perturbations\n- \"weaker\" convergence than $L_p$ - which is strength for estimation from finite samples, like for Wasserstein cost\n- stability to perturbations wrt $\\nu$\n- stability to perturbations wrt $\\mu$\n- \"kernel decomposition\" property to link errors of $\\kappa_1$ and $\\kappa_2$ with $\\kappa_1\\circ\\kappa_2$ \n\nFinally, the authors propose an algorithm whose sample complexity approaches the optimal minimax rate. This algorithms relies on the convexity property of entropy-regularized OT, which yield a consistent estimator, well defined outside the support of the \"train set\".\n\nSTRENGTHS AND WEAKNESSES:\n## Strengths\n\nThe problem is well-exposed and well-motivated. To the best of my knowledge, this approach is new.   \n  \nThe flow is clear, and as pedagogical as one can be on these topics (e.g. Figure 2).  \n  \nFrom the paper: \"Adopting $\\mathcal{E}_{p}(\\kappa,\\mu,\\nu)$ as our objective enables kernel estimation without regularity or even existence of a Brenier map\" which may simplify assumptions or algorithms built upon it.  \n\nThe algorithm proposed starts with a textbook application of the properties of entropy-regularized OT, based on the work of Genevay et al, and then apply \"rounding\" trick to achieve better rates.  \n\nWhen $p\\geq 2$, using a Wasserstein distributionally robust optimization program, authors relaxed the frequent Lipschitz continuity assumption put on $T$ and $T^{-1}$. \n\nCode is given.  \n\n## Weaknesses\n\n### Missing related work\n\nThe proposed *Transportation Error* is closely related to the *Monge Gap* [Uscida & Cuturi] which performs a similar-looking decomposition between a *fitting term* $\\Delta(T\\sharp \\nu, \\mu)$ (playing the same role as the *feasibility gap*) and a *c-optimality* term which measures how different $T$ is from $T^*$ on the induced transportation cost. I acknowledge that their decomposition is not quite the same, nonetheless, the differences could be discussed and highlighted.  \n\n[Uscida & Cuturi] Uscidda, Théo, and Marco Cuturi. \"The monge gap: A regularizer to learn all transport maps.\" In International Conference on Machine Learning, pp. 34709-34733. PMLR, 2023.\n\n### Toy experiments\n\nPlease see the \"questions\" section.  \n\n## Recommendation\n\nOverall, the paper is clear and introduces a novel tool to measure the optimality of transition kernels $\\kappa$ used as stochastic transportation maps. These transportation maps are paramount to some modern generative models (e.g. GAN-like or flow matching). This paper introduces a new tool to evaluate their quality, with appealing theoretical properties, and a practical algorithm to evaluate it. I believe the NeurIPS community will find it useful.\n\nQUESTIONS:\n### Toy-ish experiments\n\nI'd be curious to see how your estimator behaves beyond the N=2,000 points limit, which is very small by today's standards. Can you plot results for $N\\geq 10^4$ points [you may reduce or remove bootstrap if computational budget is an issue]?  \n\nAlso, other \"challenging\" OT problems are documented in literature, with discontinuous OT maps. For example, the ones in Fig 3 of Makkuva et al. Also, the \"checkerboard\" dataset, used in e.g. Vesseron et al. It could be interesting to see how your estimator behaves on these datasets.   \n\n[Makkuva et al] Makkuva, Ashok, Amirhossein Taghvaei, Sewoong Oh, and Jason Lee. \"Optimal transport mapping via input convex neural networks.\" In International Conference on Machine Learning, pp. 6672-6681. PMLR, 2020.\n\nI'd be willing to increase my score if these experiments are done.  \n\n### Extension to deep learning\n\nNote that one of the paper's selling points is: \"*Adopting Ep as our objective enables kernel estimation without regularity or even existence of a Brenier map*\". Many folks in the deep learning community care about an objective which is:\n- statistically consistent, so it can be estimated from a finite sample in minibatch setup (as you did)  \n- is differentiable so it can be used as a loss function in a deep learning context\ne.g. with a kernel $\\kappa$ represented as a neural network.  \n  \nCan you discuss the use of $\\mathcal{E}_p$ in this context?\n\nLIMITATIONS:\nSome limitations are discussed, including the \"surprising\" efficiency of the NN estimator.  However, these experiments are toy, so while the transportation error $\\mathcal{E}_p$ is certainly a convincing theoretical tool, its practical relevance to \"large scale\" scenarios and realistic datasets still needs to be demonstrated. A lengthy discussion on this topic would be welcome.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors find optimal map $\\pi_{\\tau,n}$ in Entropic OT problem statement between data distributions and represent conditional kernel between data distributions through this map. Having armed with the kernel, they build a novel error metric $\\epsilon_{p}$  for validation of Optimal Transport (OT) solvers  that demonstrates better performance than $L_{p}$ with increasing of number of samples. Also, they create rounding estimator to increase the efficiency of developed methodology, providing comparison with nearest-neighbour estimator in two settings.\n\nSTRENGTHS AND WEAKNESSES:\n**strengths**\n\n1) The developed error does not require uniqueness or even existence of OT map.\n \n2) The developed error $\\epsilon_{1}$ demonstrates better performance than $L_{1}$\n\n**weaknesses**\n\n1) They solve Entropic OT to define conditional kernel in error metric. As far as known, Entropic OT has some drawbacks such as instability (exponents in  KL regularization),  sharpening of generated data (barycentric projection is averaging) and  tuning of hyper parameters.\n\n2)  Although the proposed methodology is applicable for any $p\\neq2$, the authors demonstrate experiments only for $p=1$. (see Fig.3)\n\n3)  Since 10 is the highest dimension for the experiments (see Sec.6, line 334), there is no confidence that this methodology is scalable. There is no Image experiments with MNIST or CIFAR-10 dataset.\n\n4) This methodology is applicable only for $\\sigma^{2}$-sub Gaussian data distribution.(line 120)\n\nQUESTIONS:\n1) Why do you use this error metric as validation function and not use as a training objective?\n\n2) Since barycentric projection is averaging of data, it seems that is poor representation for map. Is there opportunity to learn 2 neural networks, while the first approximates map the second learns conditional kernel?\n\n3) Is your error unbiased? Could you show the proof of this fact, please\n\n4) Why do you pay your attention on Entropic OT to get transport plan? Is there any ways to get transport plan to obtain map without barycentric projection?\n\n5) If I am not mistaken, the theory fails if we remove sub-Gaussian requirement, yes?\n\n6) I poorly understood sense of rounding-based estimators. Could you explain what is the main motivation of rounding-based estimator?\n\n\n7) Could you provide high-dimensional experiments with MNIST or CIFAR-10 dataset, please?\n\nLIMITATIONS:\nThe method has the same limitations as Entropic OT (see weakness 1).", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-01"}

comments: SUMMARY:
This paper introduces a refined $p$-Wasserstein-type error functional that incorporates an additional transportation kernel. Various forms of stability for marginal distributions with respect to the $p$-Wasserstein distance are established. Furthermore, the author studies the error when the reference transportation is close to the optimal one, under certain regularity assumptions. A synthetic experiment with a discontinuous optimal transport map is performed to test the accuracy of this error functional.

STRENGTHS AND WEAKNESSES:
Strengths:
This article is mathematically sophisticated and sound, with a well-organized presentation. The refined error functional sheds new light on the errors in approximating optimal transport, providing insights that were not captured in previous results.

Weakness:
Even though the new error functional $\mathcal{E}_p(\kappa; \mu, \nu)$ can be used to analyze the error near optimal transport maps (for instance, for the minimizer in the entropic estimator in Section 3), it does not appear to provide an algorithm for obtaining better approximations or a more scalable method for large data sets.

QUESTIONS:
In Section 4, line 243, the authors assume Hölder continuity of the optimal kernel for the $W_p$ problem. This seems quite limiting, as it may not hold in practice. Could the authors provide justification for this assumption?

LIMITATIONS:
Yes
TITLE: Official Review
CLARITY: 3
IMPACT: 2
ORIGINALITY: 3
DATE: 2025-07-02

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper develops a new theoretical framework for estimating _stochastic_ optimal transport (OT) maps. It broadens the classical deterministic Monge setting where OT maps existence is due to Brenier’s theorem, and it encompass scenarios where no unique deterministic map exists. Their framework relies on breaking down the estimation error into an _optimality gap_ (the excess transport cost over the true OT value) and a _feasibility gap_ (the Wasserstein distance between the map’s push-forward and the true target). \n\nThe analysis consists of breaking the overall error into two parts:\n\n- The optimality gap: This is characterized by studying the **entropic kernel estimator**, which is used to study the optimality gap, leveraging entropic OT with a bias–variance trade-off and its sample complexity has been studied by Genevay et al. 2019 and Peyre et al. 2013.\n- The feasibility gap: This is characterized by the **rounding estimator**, which discretizes the source, trims the target, solves a near-optimal discrete OT problem, then “rounds” back to the continuous domain. The sample complexity of this is analyzed Hölder continuous kernels.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n\n- The paper is very well written.\n- Studies an important problem of characterizing OT maps when Brenier theorem does not hold, which is a common occurrence in real-world, finite-sample settings.\n- The proposed error functional is general, and it applies when no deterministic OT map exists, covering fully stochastic real-world settings common in domain adaptation and single-cell genomics.\n- Finite-sample upper bounds match minimax lower bounds under the assumption of sub-Gaussianity (or finite moments) without requiring density smoothness.\n- The stability lemmas for the error functional enable decouple the local (Wasserstein) and global (TV) robustness, yielding optimal rates under strong corruption models.\n- The rates are corroborated empirically. Both entropic and rounding estimators are implemented via existing entropic OT solvers or discrete OT routines.\n\n\n**Weaknesses**\n- None that I can point to. \n- Given my limited expertise in theory of OT, I cannot judge the novelty of the theoretical claims made in this work. I hope that the other reviewers and the AC could evaluate these aspects.\n\nQUESTIONS:\n**Questions to the Authors**\n\nI think the work tackles an important problem of studying finite-sample bounds for stochastic OT problems. I have the following question:\n- As an OT practitioner, I have the following question, could the finite-sample rates inform how OT maps are parametrized and fitted?   Could you characterize the error functional (i.e. which error term dominates when) at different sample ranges? \n- Does this sample complexity bounds hold for conditional OT maps where there is a family of OT maps is fitted instead of a single one? If not, could the authors provide intuition on how it may generalize to conditional OT problems?\n\nLIMITATIONS:\nNot relevant", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-08"}