Title: Stability and Oracle Inequalities for Optimal Transport Maps between General Distributions
Venue: NeurIPS 2025
Accepted: yes
Keywords: optimal transport; sieve estimation
Abstract: Optimal transport (OT) provides a powerful framework for comparing and transforming probability distributions, with wide applications in generative modeling, AI4Science and statistical inference. However, existing estimation theory typically requires stringent smoothness conditions on the underlying Brenier potentials and assumes bounded distribution supports, limiting practical applicability. In this paper, we introduce a unified theoretical framework for semi-dual OT map estimation that relaxes both of these restrictions. Building on sieved convex conjugate, our framework has two key contributions: (i) a new map stability bounds that holds without any second-order regularity assumptions on the true Brenier potentials, and (ii) an oracle inequality that cleanly decomposes the estimation error into statistical error, sieved bias, and approximation error. Specifically, our approximation error is measured in the $L^\infty$ norm rather than Sobolev norm in the existing results, aligning more naturally with classical approximation theory. Leveraging these tools, we provide statistical error of semi-dual estimators with mild and verifiable conditions on the true OT map. Moreover, we establish the first theoretical guarantee for deep neural network OT map estimator between general distributions, with Tanh network function class as an example.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe goal of this paper is to obtain OT estimators under relaxed assumptions: namely, relaxing assumptions on the measures (boundedness and/or convex support) and/or on the potentials themselves (e.g. global smoothness, strong convexity ...) (see Table 1).\n\nTo do so, the paper relies on an estimator introduced in [20] that is based on a modified semi-dual formulation of (quadratic) OT: instead a taking a global Legendre transform, the infimum is performed over a some ball; note that implicitly, this sieved estimator was also already considered in Vacher et. al. 2024 (Proposition 3.3). This sieved estimator enjoys stability properties under weaker assumptions than previous works (Proposition 3.3). Its total error is controlled in Proposition 3.4 and is made more explicit in the case of a low metric entropy (Theorem 3.6). As an application, they show that the sieved estimator used with an appropriate neural network provides an error which scales well with the smoothness of the true OT potential yet without the need to assume compactness, smoothness and strong-convexity for instance (Theorem 3.7).\n\nSTRENGTHS AND WEAKNESSES:\nIn my opinion, the main strength of the paper is Theorem 3.7, which I even found quite surprising to be honest. \nThe weaknesses are:\n- it is not super clear that the sieved estimator is well-suited for practitioners: since the Legendre transform is restricted on a ball, the resulting sub-problem may be harder. However, I acknowledge that this is not the main point of the authors at all.\n- on a technical level, it is not so clear to me how this work is not a marginal improvement of [20].\n\nQUESTIONS:\n- Could Theorem 3.7 be obtained with [20]?\n- Is there a simpler class than the neural network described in Theorem 3.7 that could \"do the job\"?\n\nLIMITATIONS:\n- in the appendix, we can see that the network is actually quite large (exponential in the dimension). That would have been nice to say it explicitly somewhere.\n- in the same vein, computing the sieve estimator remains a hard problem. That would have been nice to say it explicitly somewhere.\nHowever, I fully understand that the contribution is not on the computational side hence it is a suggestion rather than a true limitation.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-16"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper aims to study the statistical estimation of Optimal Transport (OT) maps. \nPrevious results required some balance between requiring compact support, or light tails, for the involved measures, and between requiring some possibly restrictive smoothness properties.\nThe present paper aims to lessen these assumptions by studying the so-called sieved estimator, which solves the dual problem when restricted to a bounded domain.\nAs a result of their analysis, the authors are able to derive new estimation rates for approximating the OT map using neural networks with Tanh activation.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- Estimation of optimal transport is an interesting problem, and the results here should be interesting to the wider community.\n- The results seem to be potentially quite general, which is important for applications.\n- I particularly found the bound in Proposition 3.4 appealing. Expressing the error in terms of the Bernier potentials, rather than the OT map itself, seems like a useful thing.\n\nWeaknesses:\n- The paper is written in a very technical way, which sometimes makes it harder to synthesize the main ideas. For example, the way I see it, the use of the sieved estimator, rather than the standard dual estimator, should allow, in some sense, to reduce the problem into a compact support setting, where better results are known to hold. I think discussions of this sort and explanations of other crucial ideas can go a long way.\n- The necessary assumptions (Assumptions 3.1 and 3.2) seem a bit opaque to me, and there is no actual discussion surrounding their necessity or roles. The authors do give an example in Appendix A, but this is a very specialized case. It would be interesting to see whether those assumptions can hold for heavier polynomial tails, and in general, it is important to explain the exact extent to which these assumptions will hold. As I see it, compactly supported measures are perhaps less interesting in the overall setting of this paper, so it's better to have a larger class, or at least easy-to-verify properties.\n- As far as I understand, Theorem 3.7 is not algorithmic and requires searching over a very large function class. It's not a big disadvantage, since the paper mostly focuses on statistical perspectives, but I feel like some discussion about this point is in order. In particular, it would be good to explicitly state bounds on the overall complexity, depth and width.\n- Moreover, I found the smoothness assumption in Theorem 3.7 to be somewhat unnatural. I understand it is a technical necessity at the end, but since the paper aims to introduce an estimator which does not require a-priori smoothness assumptions, I would have expected to see an example which demonstrates this exact point.\n\nQUESTIONS:\nIn line with my comments above, I have the following questions:\n\n- How general are assumptions 3.1 and 3.2? Are there natural classes of measures, other than compactly supported, which satisfy these assumptions?\n- Is there a *natural* application of Theorem 3.6 for maps with no a-priori smoothness guarantees?\n- Theorem 3.6 will typically include dimensional factors. Can the authors explain the order of magnitude of the 'pre-factors' in Equation 18, under some proper normalization of the measures? The same thing would be useful for Equation 19.\n- I found proposition 3.4 to also be interesting in the case R = infity. Can the authors comment on that case, when compared to previous results in the literature?\n- Finally, it may be a bit naive on my side, but since Proposition 3.4 only depends on the Brenier potential, rather than the OT map itself, I hoped there could be a chance to consider less regular function classes, like ReLU networks. Can the authors comment on this point, and expand a bit on the discussion in Section 3.4?\n\nLIMITATIONS:\nNo societal impact.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-18"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper provides a new estimation theory with relaxed assumptions. Specifically, the paper provides a novel map stability inequality and a refined oracle inequality.\n\nSTRENGTHS AND WEAKNESSES:\nHonestly, I am not an expert in optimal transport theory, so I do not feel qualified to make a meaningful judgment on this paper. However, given its technical focus and theoretical depth, I believe it may be better suited for a probability-oriented journal, such as The Annals of Statistics, as the venue of [1].\n\n**Comments and Questions:**\n\n1. The paper should explain what dual-type OT map estimators are at the beginning of the introduction. Additionally, it would help to clarify what is meant by statistical error and approximation error, as these are central to the paper’s contribution.\n\n2. In the Introduction, the authors claim that the assumptions in [27] exclude many practical OT applications. It would be helpful to provide concrete examples of OT applications that are excluded.\n\n3. In Assumption 3.2, what is the envelope function? \n\n4. On line 150, the authors state that Assumptions 3.1 and 3.2 require only that the true OT map $\\nabla \\varphi_0$ be bounded. This is confusing, since Assumptions 3.1 and 3.2 are stated for all $\\varphi \\in \\mathcal{F}$.\n\n5. In Assumption 3.5, what does the notation $\\log_+$ mean? Please define it explicitly.\n\n6. What is the meaning of $\\mathcal{O}_{\\log}$ notation? It appears nonstandard and would benefit from a formal definition.\n\n7. In the training procedure, how do you enforce that the TNN functional class satisfies equation (23)? \n\n\n[1] Jan-Christian Hütter and Philippe Rigollet. Minimax estimation of smooth optimal transport\n400 maps. The Annals of Statistics, 2021.\n\nQUESTIONS:\nsee above\n\nLIMITATIONS:\nResults are shown only for tanh neural networks, which are less commonly used in modern deep learning compared to ReLU or more complex architectures.\n\nThe theoretical framework is mathematically dense and may be less accessible to non-specialists, which could limit adoption.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 1, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-19"}

comments: SUMMARY:
The paper presents a new error bound on an optimal transport map estimator based on sieved convex conjugate. In particular, they replace the standard conjugate in the semi-dual form of the Kantorovich problem with the sieved counterpart, showing that it helps relax the smoothness/convexity assumption of the true Brenier potential while still achieving stability. The new bound allows the authors to derive the first error bound for the class of tanh-activated deep neural network potentials while only assuming the smoothness of the true potential and the sub-Weibullness of the source distribution (which are mild and can be relaxed). Experiments on synthetic data with non-smooth/non-convex potentials align with the stability bound.

STRENGTHS AND WEAKNESSES:
Strengths:
- The paper is well-written, theoretically sound and contains novel results.
- The bound for the class of neural OT estimators brings the proposed method closer to practicality.

Weaknesses
- The tanh activation is known to suffer from vanishing gradient and slow convergence. It might be helpful to investigate this empirically, e.g., increasing the number of layers in the experiments.

QUESTIONS:
- What is $t_6$ in the experiments?
- It might be helpful to empirically compare the proposed method with the original dual estimator on cases of smooth/convex potentials, as well as compare their computational speeds since they have two different optimization subproblems.
- The authors should highlight how the new stability bound with relaxed assumptions helps in Section 3.4.
- How does removing the smoothness assumption in Theorem 3.7 affect the error bound? Since the experiments also consider a non-smooth case, a comment on this might be necessary.

LIMITATIONS:
NA
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-06-26

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies the problem of estimators of optimal transport (OT) maps pushing a  measure $\\mu$ to a measure $\\nu$. Consider the semi-dual of the Kantorovich OT formulation,  $\\min_{\\phi \\in L^1(\\mu)} \\int \\phi \\, d\\mu + \\int \\phi^\\ast \\, d\\nu$, where $\\phi^\\ast$ is the convex conjugate of $\\phi$. The minimizer $\\phi_0$ of the above problem corresponds to the OT map via $T = \\nabla \\phi_0$. The authors propose a _sieved estimator_\n\\begin{equation*}\n    \\tilde \\phi_{n,m} := \\arg \\min_{\\phi \\in \\mathcal{F}} \\int \\phi \\, d \\mu_n + \\int \\phi^\\ast_R \\, d \\nu_m\n\\end{equation*}\nwhere $\\mu_n$ and $\\nu_m$ are the empirical measures of $\\mu$ and $\\nu$, with $n$ and $m$ samples, respectively, $\\mathcal{F}$ is some class of functions, and the sieved convex conjugate is defined as $\\phi^{\\ast}_{R}(y) := \\inf_{x : \\|x\\|_2 \\leq R} \\left \\{ < x, y >  - \\phi(x) \\right \\}$.\n\nUnder the existence of _envelopes_ for both the function class $\\mathcal{F}$ and the target OT map $T = \\nabla \\phi_0$, the authors prove four main results.\n\n**First**, a stability estimate, i.e., a bound on $\\| \\nabla \\phi - \\nabla \\phi_0 \\|_{L^2}^2$ in terms of the _truncated excess risk_, the classical excess risk associated to the above objective truncated at a radius $R > 0$. The bound also contains an error term, termed _residual risk_, in terms of the tail probability of $\\mu$ and the aforementioned envelopes. A trade-off can be seen here as larger $R$ minimize the residual risk but penalize for imperfect minimizers of the truncated excess risk.\n\n**Second**, the authors prove an oracle inequality, i.e., a bound on the truncated excess risk in terms of (a) the statistical error associated with the empirical measures $\\mu_n$ and $\\nu_m$, (b) the sieve error associated with a finite cutoff radius $0 < R < \\infty$, (c) the approximation error $\\min_{\\phi \\in \\mathcal{F}} \\| (\\phi - \\phi_0) \\, {1}{B_R(0)} \\|_{L^\\infty}$.\n\n**Third**, non-asymptotic statistical rates are established for the convergence of the statistical error using the oracle inequality above together with an assumption on the $L^\\infty$ covering number of the function class $\\mathcal{F}$. Typical rates of the order $n^{-1/2} + m^{-1/2}$ are obtained, up to $\\log n$ factors, with constants depending on the sieved radius $R$ and the envelope functions. The authors also show how to control the sieve error in terms of the sieve radius and the envelope of $\\mathcal{F} \\cup \\{\\phi_0\\}$. The derived rates are further used to motivate a choice for $R$ in terms of samples from $\\mu$ and $\\nu$. \n\n**Fourth**, the authors derive non-asymptotic statistical rates for the convergence of neural network OT map estimators. Specifically, to obtain such an estimator one solves the above sieved semi-dual objective with $\\mathcal{F}$ consisting of tanh-activated neural networks (TNNs). Here, the target potential is in $C^\\alpha(\\mathbb{R}^d)$ and the derived rate is $n^{-\\frac{\\alpha}{2\\alpha + d}}$.\n\nThe authors conduct experiments using the above framework to learn OT maps that are either non-smooth or non-convex. The TNN estimator disused above is used. An \"$L^2$ unexplained variance\" objective is used to evaluate the performance of the estimator; $L^2$-UVP is plotted against the number of samples $n$.\n\nIt is notable that the the first three results above, i.e. the stability estimate, the oracle inequality and the convergence of the statistical error, only depend on the existence of envelope functions for $\\mathcal{F}$ and $T$. In contrast to previous work, no explicit assumptions on the tail behavior or the smoothness or convexity of the OT map are made. However, the convergence results on the TNN estimator require a $(\\beta, b)$-smooth target potential and a sub-Weibull source measure $\\mu$.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n- This is an interesting paper with mostly well-structured mathematical arguments.\n- The goal is clear and valuable to the statistical OT community.\n- The tools used are fitting and the results are interesting and intuitive---for example, the trade-offs between the truncated excess risk and the residual risk or the scaling of the sieve radius with the sub-Weibull parameter $\\lambda$ of the source measure, in Theorem $3.6$.\n- Moreover, the paper gives what might be the (?) first non-asymptotic statistical rates for a neural OT map estimator, a significant contribution to the literature.\n\n**Weaknesses**\n\nThe paper has two major and one minor weaknesses. \n- First, Assumptions $3.1$ and $3.2$, i.e. the existence of the envelope functions, are quite opaque. It is a priori not clear what function classes admit envelope functions or for what measures $\\mu, \\nu$ the OT map $T$ admits an envelope. Now if the assumptions are satisfied, the transport map $|\\nabla \\phi_0$ is bounded by the envelope functions.  In Appendix A, the authors demonstrate that such a bound holds for a large class of OT maps, namely maps between $\\mu \\propto e^{-\\frac{\\|x\\|^{2p}}{2}}$ and $\\nu$ with polynomial tails.\nHowever, this reviewer maintains that this is a _necessary_ and not a _sufficient_ condition for the assumptions to hold. Here, we would like to see examples of $\\mu$, $\\nu$, _and_ $\\mathcal{F}$ such that the assumptions hold.\n\n- Second, the TNN estimator result does, in fact, assume a smooth target potential and a sub-Weibull source measure. The authors argue that the smoothness assumption is for \"convenience\" since it is used to \"only bound the approximation error and not the statistical rate.\"\nWe would like this to be discussed further.\n\n- Third, the experiments merely show that the sieved TNN estimator outperforms the un-sieved TNN estimator, but do not demonstrate the claimed rates.\n\nQUESTIONS:\nCould the authors please:\n\n1. Address a potential typo in Assumption 3.1: the function $U_2$ is not assumed to belong to any function class making the assumption, as stated, vacuous. Is the assumption that $U_2$ is point-wise finite?\n\n2. Give examples of function classes $\\mathcal{F}$ and measures $\\mu, \\nu$ such that Assumptions 3.1 and 3.2 are satisfied. It is not necessary, but recommended, to also give examples where the $\\mathcal{F}$ contains the target potential $\\phi_0$.\n\n3. Further discuss how to lift the smoothness assumption on the target potential in the TNN estimator results. Also, discuss significance of the sub-Weibull assumption on the source measure and why that is not limiting.\n\n4. Discuss why the experiments do not demonstrate the claimed rates.\n\n5. Address these questions on the proofs:\n- In the proof of Proposition 3.3 why is it that $D_x \\geq 1$? How do you avoid degenerate cases where $U_2 \\equiv 0$, for example.\n- In the conclusion of line 872 how is equation (42) used? It seems that this follows directly from (43) and the fact that $\\mu$ is a probability measure.\n- In line 930 you claim to be using Chebyshev but the bound below below line 929 is on the mean not the variance. Please clarify.\n- What is capital $N$ appearing in the statement of Theorem 3.7 in the appendix? It does not appear on the original formulation of the paper.\n- To obtain the relations below line 947 you are using the face that $\\phi_0$ is $(\\beta, b)$ not that it is in $C^\\alpha$.\n- The term $\\langle R_\\epsilon \\rangle$ appears quite a lot but it is an abuse of notation: you defined $\\langle x \\rangle = \\|x\\| + 1$ but $R_\\epsilon$ is a scalar, not a vector. Mention this.\n- What is the role of the expression $R_\\epsilon = K (\\log 2/\\epsilon)^{1/\\lambda}$ in the proof of Theorem 3.7? I am assuming this relationship is important because once $\\epsilon$ is chosen to be polynomial in $n$ then $R_\\epsilon$ is logarithmic in $n$ and so it can be suppressed. If so, then say this. This is important as it is the manifestation of the sub-Weibull assumption.\n- What is $\\gamma'$ in the display below line 968? I thought that by combining the display below line 966 with assumption 3.5 we have $\\gamma = 0, \\gamma' = 1$.\n- In the statement of Lemma F.2 the right hand side should be $\\mathcal{F}$ not $\\bar{\\mathcal{F}}$. The same mistake appears in the proof. Also in line 960 it should be $h/6F$ not $h/6B$.\n\n6. Other typos, just FYI:\n-  Display on line 855 two commas in $B(0, R_\\epsilon)$.\n- Line 863 Brenier should be capitalized.\n-  Line 843 ``the'' is missing.\n-  In equations (26-28) I am assuming the $\\nu \\, f$ is meant to mean $\\int f \\, d\\nu$? Look below as to why this is bad notation.\n- Line 881 instead of \"the the\" it should be \"the same.\"\n- Your final result under line 882 involves $\\sup_{g \\in \\bar{\\mathcal{G}}} \\int_{\\nabla \\phi_0 (B(0, R_\\epsilon))} \\ldots$ where the integral is over the image of the ball $B(0, R_\\epsilon)$ under the OT map $\\nabla \\phi_0$. However, the stated result in the main text has the same integral but over the ball $B(0, R_\\epsilon)$. Please reconcile this.\n- If $\\log_+ = \\max\\{0, \\log\\}$ then equation (52) has a mistake: the upper bound in the integral should be $1$ not $e$.\n    \n\n7. Other suggestions:\n- For a measure $\\mu$ avoid the shorthand $\\mu \\, f$ for the integral $\\int f \\, d\\mu$. This looks like a scalar multiplying a function. Instead, opt for $\\langle \\mu, f \\rangle $  which is way more standard in analysis, specifically distribution theory. \n- Be more explicit when using non-trivial results. For example, in equation (31) say you are using the Fenchel-Young inequality.\n- Line 926: ``since $\\gamma' \\geq 1$ _and_ $D_{\\mathcal{F}} > 1$, re-organizing \\dots''\n\n\nI am ready to raise my overall rating if Questions 2--4 above are addressed, especially Question 2.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}