Stability and Oracle Inequalities for Optimal Transport Maps between General Distributions
Authors: Shubo Li
Yizhe Ding
Lingzhou Xue
Runze Li
Venue: NeurIPS 2025 poster ()
Keywords: optimal transport; sieve estimation
Primary area: other
TL;DR: We develop new theoretical guarantees for optimal transport map estimation under minimal regularity assumptions, providing the first provable error bounds for neural-OT estimators between general distributions.

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.

Stability and Oracle Inequalities for Optimal Transport Maps between General Distributions
Shubo Li Department of Statistics The Pennsylvania State University
Yizhe Ding Department of Statistics
The Pennsylvania State University
State College, PA 16803 sk16034Qpsu.
State College, PA 16803
yvd5212@psu.
Lingzhou Xue Department of Statistics
Runze Department of Statistics
The Pennsylvania State University 1zxue0psu. edu
The Pennsylvania State University rzli@psu. edu
Abstract
Optimal transport (OT) provides a powerful framework for comparing and trans- forming probability distributions, with wide applications in generative modeling. A14Science 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 pa- per, we introduce unified theoretical framework for semi-dual OT map estimation that relaxes both of these restrictions. Building sieved convex conjugate, our framework has two key contributions: (i) a new map stability bounds that holds without any second-order regularity assumptions the true Brenier potentials, and (ii) an oracle inequality that cleanly decomposes the estimation eπrоr into statistical eπror, sieved bias, and approximation error. Specifically, approximation error is measured in the - norm rather than Sobolev norm the existing results, aligning naturally with classical approximation theory. Leveraging these tools, we provide statistical error semi-dual estimators with mild and verifiable conditions the true OT map. Moreover, establish the first theoretical guarantee for deep neural network OT map estimator between general distributions, with Tanh network function class an example.
Introduction
Optimal Transport (OT) provides powerful framework for transforming one probability measure into another. Concretely, given two distributions and y on Rdd the OT problem seeks a map T : I - Rd that transports μ while minimizing cost function. Beyond its elegant mathematical foundation, OT has found broad applications in Ai4Science [39, 7, 29], computer vision [3, 23, 43, 12). and nonparametric statistical inference [10, 26, 24, 18), powering advances in scientific discovery and machine/deep learning. Despite its broad impact, rigorous theoretical guarantees for OT estimators under mild and realistic regularity conditions remain limited.
This paper aims to contribute to this direction by focusing on the dual-type OT map estimators, formally introduced in Section 2, which are based the characterization of the optimal transport map as the gradient established minimax optimality for mevelet-truncater dual-type OT estimator, but under restrictive conditions: y is supported on (0, 1]d with a bounded density, v is compactly supported, and the true
convex function (the so-called Brenier potential). The seminal work of [27]
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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OT map is both a-Hölder smooth and strongly convex. These assumptions exclude many practical OT applications.
In parallel, Gunsilius [25] introduced an alternative analysis using the Poincaré inequality, a funda- mental tool in PDEs and functional analysis. While still assuming compact and convex supports, this approach removed the need for Hölder smoothness or strong convexity of the Brenier potential. Building on this, [21] extended the theory to sub-exponential μs, under (a, convexiiy and (3, a)- smoothness conditions on the Brenier potential. More recently. [20] relaxed the Poincaré inequality assumption to accommodate heavy-tailed μ and v, and eliminated the (a, a) connexxtyy condition via a new sieved OT estimator, characterizing how tail thickness of and influences convergence.
We also note recent advances in plug-in in type OT estimators, which directly compute the OT map between estimated distributions [10, 17, 34, 37, 5]. In parallel with these theoretical advances, significant efforts have focused on improving the computational efficiency of OT, including entropic regularization [15, 1, 22, 11, 19] and sum-of-squares formulations [35, 41). Meanwhile, neural- network-based OT estimators have advanced rapidly, powering large-scale applications such image generation and translation [33, 30, 4, 31, 12].
Despite these advances, few core regularity assumptions the true Brenier potential PPu and distributions still limit the applicabilitt of dual-type estimators. To clarify these limitations, we decompose the overall estimation error into two components: statistical error and approximation error. The statistical error measures the discrepancy between the estimator and the best possible function in F. reflecting the randomness from finite samples. The approximation the true Brenier potential and its best approximation within F, since we do not assume that the true Brenier potential (o lies in F. Existing analyses of each error component often rely restrictive smoothness or convexity assumptions, which we discuss in more below.
captures the gap between
First, bounding the statistical error in existing works requires (3, b)-smohthsssss condition on (oo, i.e., polynomial bound on its Hessian. While this condition verifiable under strong log-concavity of μ and [8, 14, 13, 28], its validity beyond this class remains unclear. Moreover, Proposition 3.12 of [20] shows that this smoothness requirement constrains v/ to have tails no much heavier than those of μu, excluding many practical scenarios, such multivariate quantile [9, 10, 24], where transport from U(( 1/d) to distributions with unbounded support, or transporting from a Gaussian (with exponential tails) to a Student's is i-distribution (with polynomial tails).
Second, controlling the approximation error typically relies on (a, c)-covvexyty of Po. i.e. a gener- alization of strong convexity, which similarly imposes constraints on the relative tail behavior of and v [20]. For instance, OT map from an unbounded μ to a bounded v fails this condition. This is especially problematic in generative modeling, where the OT map is learned from Gaussian noise to image data of bounded values [40, 30, 38, 32). Moreover, existing approximation error bounds assume that both μ and y have bounded support, [27, 21].
These limitations raise a fundamental question in broadening the scope of optimal transport:
Can one bound the statistical and approximation errors of dual-type OT estimators for general distributions without such restrictive assumptions?
In this work, we provide a positive answer to the question posed above. Our two core innovations are a novel map stability inequality and refined oracle inequality. Building on these tools, we are the first to establish statistical and approximation error bounds for dual-type OT estimators under minimal regularity assumptions. Table I provides a side-by-side comparison of the regularity conditions required by our results versus those in the existing literature.
Table I: Regularity assumptions on the true Brenier potential Po and μ required establish statistical and approximation eтror bounds (Estat and Eapp) in prior work and in our results.
Assumption on 9o
Distributional Assumption
(B, b)- --oothh (for Extat) (o, convex (for Eoyp)
Tail thickness of μ
Poincaré incquality on μ
[27]
b ==0 0
==
Compact
Implicitly Used
[25]
Not Required
Not Required
Compact
Required
[21]
Required Required
Required
Sub-exponential
Required
[20]
Required
General
Poincaré-ty
-anpe inequality
Ours
Not Required
Not Required
General
Not Required

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Specifically, we summarize our main contributions as follows:
1. A new map stability inequality without (3, sssoothnnsss. Stability inequalities are key to linking OT estimation error to the semi-dual objective. Existing results rely on (8, b)ssmohhhssss of Po- In Proposition 3.3, we derive new inequality that removes this requirement, assuming only well-behaved OT map (Assumption 3.2), which holds unconditionally when y is bounded and under mild conditions for unbounded distributions.
A novel oracle inequality without (a, cc-oovexit Proposition 3.4 provides an oracle inequality that avoids the (a, a) convexityy assumption (Pu and decomposes the estimation error into statistical error, sieved bias, and approximation error. Unlike prior works [21, 20). which involve harder- to-analyze gradient terms like inf V Vepollza()): our bound relies on more tractable supremum error over compact region B: inf 1b (o)l making it more suitable for analysis using classical approximation theory.
3. First non-asymptotic bounds under minimal smoothness. With our new map stability inequality. Theorem 3.6 establishes the first non-asym -syyyptttoiic statistical error bound for dual-type OT estimators requiring only first-order condition on Po, without second-order smoothness assumptions.
4. First theoretical guarantee for neural OT estimators. convergence guarantee for neural OT estimators using smooth networks. For tanh neural networks (TNNs), Theorem 3.7 shows that if yoo E (RR)) for 2 2, then the neural OT estimator V between unbounded distributions satisfies:
With our new tools, we derive the first
E| V V Slogn n N..
(1)
Notably, this rate matches that of [25], but under significantly weaker assumptions: we do not require compact support for μ and v, nor a Poincaré inequality on μ..
Notations and Terminologies Throughout this paper, we use the following notations. For a E R. [a] (x: E Z :2 < a). For TR." we write ||u||2 for its lz norm and ||0]].c for its for norm. For matrix A E Rdixda, let ||All := max., |A If A is symmetric, ||Allop denotes its operator norm, Amin (A) denotes its smallest eigenvalue. For function ʃ: Rdd - IR and measure μí, we write ftf, F, (X), or f f du for its integral, and [|] == (f [1 dux for the 1,2 norm. For f: RR + R and set B C Rd we denote || 1.> (B) sup,en [=)))
We write a S b to mean a S Cb for some constant >> 0 independent of a and b, and a Stoge to omit polylogarithmic factors in n. We write = b if a X b and b S We use a V b and A b to denote the maximum and minimum of a and b, respectively. For sequences a, and b,, we write bn O(a,) if there exists constant >> O such that [6,| S ca., and write 6n Orog(O. (ax) if there exists constants C1, C2 > such that [bn] & cia, log(n))2 We use I to denote indicator function.
R. set (z) == 1 1 |z).. We define log, max(log-n 13. A function o E C2 is called (8, b)-mooth if p S 3(x)² for all æ, and (a, -coovex if Amin ( (V>)(z)) 2 a())".
For E RR, write (x)) 1 + ||=|||- When
A probability measure is (A, -e-sublWeibull if for X~ ns, P[ 2 1) S exp(-(1/K))) for all t 0. We refer such a distribution as X-sub-Weibull or simply nub-Weibull..
2 Background on Optimal Transport
Let μ and v/ be two probability measures on Rdd. A measurable map T : Rd - Rd is called a push-forward from μ to if for any measurable set A C (A) v(A). We denote this by = T μ.. Monge's problem, the optimal transport map is defined as the push-forward that minimizes the transport cost from j to v:
min TB R
|| ((x))12((da)
s.t. Typ= V;
However, Monge's problem may not admit a solution, for example, when l is discrete and is continuous. A natural relaxation is the Kantorovich formulation, which considers couplings r E I1(, v) with marginals y and v:
min /f || v1l2 (dxx, dy), s.t. T H(H,),)
(3)
3

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Kantorovich formulation is always feasible [42]. As a linear program, its semi-dual form reads:
{ p du min pe (a)
/ 1p" du
min pu + vp"". Ge (p)
where op" is the convex conjugate (a.k.a. Legendre-Fencen Fenchel conjugate) of yo:
("(y) sup (1x, y) (xx)), for all y E R...
(5)
zER
The equivalence between Monge's 's problem (2), Kantorovich's 's formulation (3), and its semi-dual form (4) was established in the seminal work of Brenier [6]:
Theorem 2.1 (Brenier's Theorem). Let i and be probability measures on R., both with finite second moments, and assume that does not assign mass to small sets. Then there exists a convex function (o that uniquely (up constants) solves (4), Moreover, Vyp solves Monge's 's problem (2), and (Id x V4o) μ. solves the Kantorovich formulation (3).
Brenier's Theorem reveals that the OT map is the gradient of convex function ço. called the Brenier potential. Furthermore, since Po is the minimizer of (4), we can estimate using empirical measures Nn and Vm of μ and v, respectively. For chosen function class F., we define the Brenier potential estimator and the associated dual-type OT estimator as:
Pn,m
arg min P + Vm with OT map estimator PEF
P
However, the supremum over the entire space IR in the convex conjugate sp makes the optimization procedure unstable and sensitive. also introduces theoretical challenges when analyzing the convergence of dual-type OT estimator Ωn.m. To address this issue, [20] proposed the concept of sieved convex conjugate, which restricts the supremum to bounded region. Specifically, for some sieve radius R > 0, define:
(=)))== sup (x. (px)) for all E Rdd
Note that 40 (V<((x)) = (r. Vv(x)) - p(r).. By restricting x to B(0, Rt), we equivalently restrict the domain of /0 (2) to the compact set (ip(B(O, This compactness facilitates to analyze the estimation emror of sieved OT estimator.
Given sieve radius R, the corresponding sieved estimator is defined
pn,m == argmin/4m + Vm with sieved estimator V
(8)
PEP
Note that when R co, recover the original dual-type estimator in Equation (6). In the following. we may abbreviate it to Gn, when n = m.
3 Main Results
analytical tools: stability inequality for the sieved estimator (Proposition 3.3), and refined oracle inequality that cleanly decomposes the estimation error (Proposition 3.4). Building on these results, Section 3.3 establishes non-asymptotic statistical error rates for general function classes (Theorem 3.6). Finally, in Section 3.4, we show how the overall estimation error of neural OT estimators between general distributions can be controlled, using sanh-activated neural networks a concrete example (Theorem 3.7).
In Sections 3.1 and 3.2, develop
3.1
A New Map Stability Inequality
From statistical learning perspective, map stability inequalities essential for linking the semi- dual objective to estimation error. For the original dual-type estimator, denote its objective is (yp) pip + uyp". Existing results [27, 21, 20] typically show that for all y F,
Vipollia S (y.v(p)) Su,((po).
(9)

---

where Vyp is the true OT map to be estimated.
However, these results apply only the original estimator (6), not the sieved version in (8), which often performs better in practice. Moreover, while neither side of the inequality involves second-order derivatives, prior analyses still require (8, smoothhesss of yo, limiting their generality.
To accommodate sieved estimator and relax their assumptions, we consider the following conditions:
Assumption 3.1 (Envelope for Function Class). Let F (RR) be a function class. We assume there exists non-decreasing and pointwise finite function U2 : [0, 00) + [1, 00) such that:
sup sup PEF S=
VVp((z)lopp S (2)),,
sup |V>((0)] (2( PEF
Assumption 3.1 depends only on and is independent of po. l and v. is satisfied by many common function classes, for example,
1. Quadratic function class: When and are both Gaussian (or from the same elliptical family), the Brenier potential is quadratic and lies in the following quadratic family:
F Fquad= = fx +> x Be + (b,xx) x) B S Bloop I rı, ||b| S r2).
2. Smooth Neural Networks: Neural network function classes with all parameters bounded and activation functions that are -mooth, such as sigmoid, tanh, or softmax.
3. Reproducing Kernel Hilbert Spaces (RKHS): Let X' X R be a positive-definite kernel on its domain X' x X, and let HK be the corresponding RKHS with norm -| We assume F is C² and take F to be the unit ball of HK, i.c., F = { E HK ||ş||н S 1).
Assumption 3.1 and 3.2 together play a role similar to Assumption A.1 [21], but are strictly weaker. As result, the function classes discussed in Section 4 of [21], including parametric family, wavelet expansions, RKHS, and Barron Spaces, apply directly to our setting. We refer readers to Section 4 of [21] for detailed discussion.
Assumption 3.2 (Envelope for OT map). There is a function (p),,, such that for all E F,
[V+()> - V < u(x), for all x.
Define Ui(R) == sup aallallssR u(x). Additionally, let E (2( be an envelope function for FU(spo). i.e.
|p(x) < ((x), for all x E Rd and all EEE FUU
sufficient condition for Assumption 3.2 is that both Proposition A.2 and Assumption 3.1 hold. Specifically, Proposition A.2 ensures || [x)( ||2 to be bounded by some function Ln (r). Meanwhile, Assumption 3.1 provides envelope for v [12- denoted (xx) Then Assumption 3.2 follows via the triangle inequality (/)) - V-oo(=)l < (x) + L2(x). While Assumption 3.1 solely relies on F. typical distributions for Proposition A.2 hold include: μ can be normal or Student-t, and P can be any measure with certain tail decay. We refer to Appendix A for further details and an explicit example verifying Assumption 3.2.
We are now ready to present our new stability inequality for the sieved estimator:
Proposition 3.3 (Map Stability Inequality). Suppose Assumptions 3.1 and 3.2 hold. For any e >> define Re and sieve radius R2 such that
u(x) U2([|=112 + u(x))
x > Re) Se,
Re sup ln
{1ar112+
(10)
For any 1p E F, define the truncated excess risk:
T2(1)= = JA ((x) + P (Vio(z)) p(dx) - (o(()) (=)V (11)
Then the following inequality holds: for any yp F,
[ Vyolltt 520; + Un (Ra)) Ts re((p) + |1ull24()) _
(12)
5

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We first clarify the roles of R& and Re, which enable a more tractable approximation error in the next section. Specifically, the hyper-balls BO0 R₂) and V (B, R2)) serve as bounded pseudo- supports for y and v, respectively, over which the behavior of ip and its convex conjugate can be effectively controlled. Meanwhile, the sieve radius Re is introduced to handle any mismatch between (B(( R2)) and the pseudo-suppor of v, by better regulating the range of the sieved conjugate. In practice, if V closely approximates Vspo in _ norm (so that R 0), the mismatch is negligible and one can take Rs R Rg. In that case, it suffices to compute the convex conjugate over the pseudo-support of yu alone.
For simplicity, a sufficient choice for Rg Rs Rg + Ui(Re). In particular, if is supported on B0, Rt,, one may take Rg = R.
Equation (12) decomposes the estimation error into two components. The first arises from the truncated excess risk over the high- probabbityyyy region (0,, Rs), which can be well-controlled (see Proposition 3.4). Its coefficient (R, + U1 (RR)) originates from second-order Taylor expansion F on the ball (0, Rg + U1 (i(R)) (R.)). The second component accounts for the residual risk outside this region. As result, must be carefully selected balance the trade-off: a smaller reduces the residual risk but increases the truncated excess risk.
of y, capturing the "uniform curvature"
We defer the proof of Proposition 3.3 to Appendix B.
Comparison with existing map stability results. The most closely related results are Proposition 10 of [27], Proposition of [21]. and Lemma 3.15 of [20], all of which require (3, b)-smoothness assumptions on the true Brenier potential (Po and the candidate class F. In contrast, our Proposition 3.3 substantially relaxes these requirements by assuming only an upper bound on the true OT map V via function u(x)).
Another key distinction is the introduction the sieved convex conjugate PiR essential for providing a more accessible approximation error in the next section. This sieving remains necessary even when ju has bounded support, unless additional regularity conditions are imposed on fs and
Compared to map stability results for plug-in type estimators, our bound also holds under milder assumptions. For instance, Theorem 2.1 of [17] requires Vipo to be Lipschitz, while Theorem 3 of [5] assumes that the Brenier potential is strongly convex.
3.2 A New Oracle Inequality
Now, we present our second contribution of a new oracle inequality that helps cleanly decompose the estimation error into statistical error, sieved bias, and approximation error.
Proposition 3.4 (Oracle Inequality). Under Assumptions 3. I and 3.2, and let Re, Re be as defined in Equation (10). For the sieved OT estimator Pa. from Equation (8), its truncated excess risk Fn.m, re((n,m) defined in Equation (11), admits the decomposition:
Fn,m, == Te (P S Esttat + Esieve + Eupp.
(13)
where
Estat sup /
Sd(pm - 11) + sup gEg B(
gd(um - v),
(14a)
/EF / B()
Exieve =-218 -2Ex xmn . [x(X) 1[[|x||22 R.) 2 ~mm [G(Y) [VI(VV > Re),, (14b)
Eapp = inf 1|4 -ollI>(B(O...
(14c)
PEF
Here, F = {4p - 92 P1, 22 E F), g = (47 - Fgg 91,82 E F} and G(y) =
Re 1ollll2 + suP ((z).
Thus, with a careful choice of & and R. (which can be selected in a distribution- free manner, as discussed in Section 3.3), Propositions 3.3 and 3.4 together provide complete characterization of how the sample sizes, function class F, and the sieved convex conjugate collectively influence the estimation error of the sieved dual-type estimators.
Specifically, this result decomposes the truncated excess risk into three components. The first term, Estats reflects the statistical error from replacing the underlying distributions μ and with their

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estimates An and V'n- Importantly, these estimates need not be empirical measures, allowing for smoothed or bootstrap-baser alternatives. The second term, Esieves accounts for the bias introduced by replacing the original convex conjugate (Equation (7)) with its sieved version. This bias vanishes when R2 = oc, where the sieved and original convex conjugates coincide.
The third term, Eupp. captures the approximation error of the function class F in representing the true potential (Po over the compact domain BO, Re). As usual, enlarging F reduces Eapp at the cost of increasing Estat- Notably, we express the approximation error in the 1 norm on a compact set, aligning naturally with classical approximation theory. In contrast, prior works [27, 21, 20] typically rely on gradient-based errors like pE V - V.Poll>?()) which remain challenging to analyze for rich function classes such as deep neural networks.
Proof ideas of Propositions 3.4 The sieve technique allows us treat B(0, Re) and Voo R₂)) as bounded pseudo-supports of μr and v, or equivalently, as bounded pseudo-domains for Po and P0- This reduction localizes the approximation problem to compact sets, where classical approximation theory applics. Another key benefit of this sieving approach is that eliminates the need for the (a, cc-onxxxitty assumption required to control the approximation error in prior works.
The sieve bias is then controlled via tail probability bounds, while the statistical error is handled using standard tools from empirical process theory. The full proof is deferred to Appendix C.
3.3 Non-Asymptotic Bounds of Statistical Error and Sieved Bias
In this section, derive bounds for the statistical error Estat and the sieved bias Esieve in Proposition 3.4, under minimal assumptions on the source distribution μ.. Notably, our results do not require smoothness of the true OT map Vipu, improving upon existing analyses.
We begin with the following assumption on the covering entropy of the function class F: Assumption 3.5 (Covering Entropy of F). For some 1) 0. 7 E (0, 2), o' 2 1 and Dy 1, the covering entropy of F satisfies: for any h 2 0,
log F, L ([- R, R < DF h h- l,,, (1/h)/ R".
(15)
We focus the Donsker regime (7 E (0, for theoretical simplicity, which already captures wide range of function classes, such as parametric families, wavelets, reproducing kernel Hilbert space (RKHS), and both shallow and deep neural networks. Here, D, represents the "effective dimension" of F, and D/ can be viewed as the effective sample size from the statistical learning theory perspective. Because most covering entropy results are stated for the L (,0, 1]d) norm, we introduce an additional R" factor to translate them to our unbounded setting in L ([-R, R. Rl Specifically, the exponent 1) measures how the envelope of F grows (c.g. 1) 2 if the envelope scales like x + z²). Theorem 3.6 (Statistical Error with Empirical Measures). Under assumptions of Proposition 3.4, and Assumption 3.5, the sieved estimator Pn,m from Equation (8), computed with empirical measures Nhn and Van satisfies:
Dr n
D3 P + Ma) log(m) m
Extat
I
R
+
((x) log(n) + HE R.)
sup
(16)
and
Egieve
$ 21(1 + [[G]] 2t.
(17)
where M := Rg supzem(.RR. [ + suPrEB(o.R.) ((x), and the suppressed constant de- pends only on T.
We defer the proof to Appendix D, where we also establish corresponding high-probabilityy guarantees.
In Theorem 3.6, we establish a standard convergence rate for estimation error, E[Estor] = Orog(VDD//n + /Dz/m), which is common in empirical process theory. Meanwhile, the mul- tiplicative prefactors in Extat, R + supzeB ((x)² à and R + M naturally capture the "diameter" of F on the pseudo-supports of μ and v.
Moreover, it tells that the sieve bias E is governed by the tail probability defined in Equation (10).
7

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Practical choice of RS and Rg. As shown in [20], RS can be selected in a distribution- manner. Here, we extend this idea to provide practical, joint selection strategy for both Rg and Rg:
RS
max 15
|X
R - Rg +
max ||x]ll2 15/5m
(18)
where C' 2 O is tuning parameter designed to account for the potential mismatch between the estimated and true OT maps, specifically the discrepancy @|V (Boo
The choice of Re can also be justified by our stability and oracle bounds (Propositions 3.3 and 3.4). In particular, n m, μ sub- Weibull, and both ((-). [|V grow at most polynomially, then the estimation error of the sieved estimator reads:
B Gn
VepolBBa Olog (nn/D)) Eapp + e
(19)
By exchangability of X, 's choosing Rg == maxisisi | yields x xnn so the sieved bias e is dominated by the statistical enror term (n/Dy)
While complete theoretical justification for our choice of RS remains open question, numerical results in Section 4 provide strong empirical support: our sieved estimator consistently outperforms the original dual-type estimator of [27, 25, 21] across range of settings.
3.4 Estimation Error of Neural OT Estimator
In this section, we derive the first a--mmmtttttic error bounds for dual-type OT map estimators parameterized by deep neural networks between general distributions.
ReL.U networks, despite their popularity, lack the smoothness needed recover the gradient of the estimated Brenier potential. Moreover, our analysis (Assumption 3.1) requires F to be C². Consequently, smooth activations are both practically and theoretically preferable.
While input convex neural networks (ICNNs) [2] exploit convexity of Brenier potential and have been applied at scale [33, 4, 31], their approximation theory remains underdeveloped. This gap limits rigorous control of approximation error in existing analyses of dual-type estimators [27, 25, 21, 20]. By contrast, our approach removes the convexity assumption F, allowing to employ standard fully connected networks.
Accordingly, we adopt the lanh-activated fully-connected neural network (TNN) as our function class F, due its 8000 smoothness and non-asymptotic universal approximation guarantees [16]:
f(x) L1 a LL-1 .... 0.0 (z(),,
(20)
where (x) g--e=
6"+e-= is applied entrywise, and Ce(x) = Aræ + be for l E [4]. where A E
R XPI bt E RP,, with p1 d, PL+1 1.
We define the TNN class F with bounded parameters as:
F(L, W, = { of form (20) ||Ae|l> V libelloo K, 5 W for all l E [4]},
(21)
where & > O is a truncation threshold used for technical convenience.
Theorem 3.7 (OT Estimation via Tanh Neural Network). Let = m, and suppose p is (A, K)- nub-Weibull, and the true Brenier potential (oo O (RRd) with 2 2 (3, b)-smooth. By setting Re, Re according to Equation (10), there exists a deep TNN function class = =FLL W,
where L. W. depend on n, d, a, B, b and are of order 3 S L = O(1), W = 0(n
Onn ddd-(lo (10142)2+
2(0420)
), such that the sieved estimator V&n with sieve radius
2a
Rs = CK... (logn) A, for some constant CK.da 2 4K
+
(22)
d 2a
satisfies
E V Vipollia(u) Slogn n ==20
(23)
Remark 3.8 (On mild regularity conditions). The sub- weihhll and (B, b)-smoothness assumptions used in our work primarily for analytical convenience, not because they are fundamentally

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required for the methodology. Notably, both conditions are widely used in the OT literature, see [21, 20, 27, 25, for (3. smmoohhssssss and [21, 17, 20] for sub-We Weibull condition.
The (B, bymmothhssssss of Go is needed solely to control approximation error through Lemma F.4, ensuring the error grows polynomially in Rg. Meanwhile, the jub-Weibull i guarantees that Re, Rs scale polynomially in log(1/e). Together, these two results ensure that the relevant Sobolev norm bounded by a polynomial in log(1/5) and the consequent approximation error has the order of
If (8.b)- smootthesss were lifted, one could still obtain approximation bounds, but they may degrade to O(e") for some a < 1. This complicates the statistical error, approximation error, and sieved estimation bias-variance trade-off off by introducing additional convergence exponent. Likewise, relaxing source distribution to be polynomially tails causes Re, R. scale polynomially in 1/E. While the p follow similar steps, the resulting convergence rate becomes much more complex (See Equation 3.10 in [20] for illustration of the added technical complications).
Remark 3,9 (On the choice of activation). While ReLU networks are widely used in practice our frumework currently focuses on the tanh activation due to theoretical considerations. Specifically, our estimator is defined via the gradient of potential function. Although PyTorch can handle ReLU networks computationally, the ReLU activation is not globally differentiable in mathematics, making this definition is ill posed. Moreover. even under weak derivative framework, ReLU networks lack second-order weak derivatives, whereas the existing optimal transport theory fundamentally relies on second-order regularity for statistical analysis.
The rate in Theorem 3.7 matches that of [25], but under substantially milder assumptions. Although slower than the minimax rate [27]. this expected: do not assume (a, c)-oovexiity of the true Brenier potential yoo compact support of and v/, or Poincaré inequality on μ. Moreover, the estimators in [27, 25, 21] are either computationally intensive or NP-hard. contrast, our deep TNN estimator is both implementable and scalable in practice.
We focus the TNN architecture due to the current lack of approximation theory for general smooth neural networks. The proof of Theorem 3.7 is provided in Appendix E. With minor modifications, the result also holds with high probability.
4 Numerical Examples
In this section, we present numerical simulations to evaluate the performance of our sieved-TNN estimator and compare against the original dual-type estimators from [27, 25, 21J. We consider four synthetic OT problems that pose challenges for existing theoretical frameworks:
* (on b)- smoooh maps:
1) + to. to +N, 1),
Uniform(O, 1) - N00, 1).
Non-(a, a) -cnvvex maps:
(O + Uniform (0, 1).
We evaluate performance in dimensions d == 5, 5,, 20, using i.i.d. samples X; ~ and Y, v with sample sizes n m (64, 128, 256, 512, 1024, 2048). Experiments were conducted a server with Intel Xeon Gold 6342 processors, requiring approximately 4,000 CPU core hours in total.
Implementation setup We implement our estimator in Py PyTcoc [36], following algorithms from [20] with objective function in Equation (8). The candidate class F consists of TNNs with two hidden layers of width 10 (for d 5). 20 (for = 10), and 30 (for d 20). Training proceeds in two phases: warm-up phase of max { 10000/nn], n/n 50) epochs at learning rate 5 x 10-3, followed by 300 epochs at 10-3 Mini-batch size is 64 for both μ and v, and the sieved convex conjugate subproblem is solved using 300 inner iterations (see Algorithm 2 in [20]). In each trial, 10% of the samples are held out for validation. We defer implementation details to Appendix G.
Sieve radius
As dicussed in Equation (18), we choose the sieve radius as
i
|| + C- max 1S
max
|Ygll2, 15150
C E 3,
Here, C corresponds to the setting in [20], while C = oo recovers the original dual-type estimator studied in [27, 25, 21], enabling direct comparison.

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Evaluation
We assess the estimation error using unexplained variance proportion (UVP) [30]: ['.UVP(V.,)= = V / /Var, (IIY Lower values indicate better performance. For each experiment, we approximate the LV- using an independent set of 10 samples.
12:0MP vsn (sremnormal tytat)
124MP vs (srcrunfomm (gh-nooma))
[x]
D20A
124PP
20"44
30"
Number lamplexu
Number aamppii
(a) N Is
(b) Unif N
12:UVP (srent, nntmnor
124WP (srennormal tutrunttrm)
Col
20-14
04
120
226
512
3024
2043
64
120
226
5:2
3004
2048
Number samples..
Number samplas
(c) +
(d) + Unif
Figure 1: --UVVP when d 5. Each curve shows the mean LVP over 50 random trials with one standard deviation.
Due to space constraints, we report only the --- error versus sample size n for in Figure 1 here. The full set of numerical results is available in Appendix G.
Across all settings, sieved TNN estimator consistently converges and outperforms the original dual-type estimator (C' = oc), particularly in the small-sample regime. As shown in Figure 4a, sieve-based estimators yield lower I2-UVPP errors for small n. while the classical estimator only catches up as n grows large. This aligns with our theory: larger sieve radii may degrade convergence, and are only needed when the mismatching [ V 1,x is large.
These results also offer practical guidance: a modest sieve constant (c.g., C or 1) suffices, and performance of our sieved estimator remains stable across range of C.
5 Conclusion
In this paper, we develop new map stability and oracle inequalities for sieved dual-type OT estimators, without restrictive smoothness and convexity assumptions the true Brenier potential yoo in the literature. Under mild regularity conditions, we establish the first non-asymptotic error bounds for neural OT estimators, using tanh networks confirm the strong empirical performance of our sieved approach. Altogether, our unified framework advances the theoretical foundations of optimal transport and paves the way for future developments in both methodology and applications.
concrete example. Numerical experiments further
Acknowledgment: This research partially supported the U.S. National Science Foundation under the grants DMS 2514400 and DMS 2210775, and by the U.S. National Institutes of Health under the grants R01GM163244 and 1R01GM152812.
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Reviews:
[{"review_id": "czK4WeSq7F", "reviewer": "Reviewer_6m4y", "summary": "This 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 $\\mu$.", "questions": "Could 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.", "limitations": "Yes.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**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.", "was_revised": true, "final_justification": "I am quite satisfied with the authors' response to my earlier concerns, and I am now convinced that this paper provides useful contributions to the broader statistical theory of optimal transport map estimation. I am therefore raising my score.", "rebuttal": "We sincerely thank you for your thoughtful review and constructive comments. Below, we respond to your specific questions (Q1 to Q7):\n\n## Q1\nThanks for your comment. We will revise Assumption 3.1 to improve its clarity as follows:\n\n**Assumption 3.1 (Revised)** For function class $\\mathcal{F}\\subset C^2(\\mathbb{R}^d)$, suppose there exists a non-decreasing and pointwise finite function $U_2:[0,\\infty)\\to[1,\\infty)$ such that: $$\\sup_{\\varphi\\in\\mathcal{F}}\\sup_{\\\\|x\\\\|\\leq R}\\\\|\\nabla^2 \\varphi(x)\\\\|\\_{op}\\leq U_2(R),\\quad\\sup_{\\varphi\\in\\mathcal{F}}\\\\|\\nabla\\varphi(0)\\\\|\\_2\\leq U_2(R).$$\n\nWe explain the roles of the lower and upper bounds of $U_2$:\n\nThe upper bound of $U_2$ is critical for the stability inequality in Eq. 14. If $U_2(R)=\\infty$ for some $R>R_0$, the RHS of Eq. 14 is $\\infty$. So we require $U_2$ to be pointwise finite.\n\nThe lower bound $U_2\\geq 1$ is a technical assumption. To prove Proposition 3.3, we require a quadratic upper bound on $\\varphi(z)$ on L846. In degenerate cases where $\\sup_{\\varphi\\in\\mathcal{F}}\\sup_{\\\\|x\\\\|\\leq R}\\\\|\\nabla^2\\varphi(x)\\\\|\\_{op} =0$ (say $\\mathcal{F}$ consists of linear functions), any positive constant for $U_2$ suffices to ensure that the argument remains valid. Since such constants do not impact convergence rates, we impose $U_2(\\cdot)\\geq1$ to rule out vacuous bounds.\n\n## Q2 &  W1\nThanks for the constructive feedback. We now clarify how Assumptions 3.1 and 3.2 can be satisfied and provide concrete examples where they hold. Notably, they play a role similar to Assumption A.1 in [1], and are strictly weaker. As a result, the function classes discussed in Section 4 of [1] apply directly to our setting.\n\nAssumption 3.1 depends only on $\\mathcal{F}$ and is independent of $\\varphi_0$, $\\mu$ and $\\nu$. It is satisfied by many common function classes, including wavelets, RKHS, and neural networks with smooth activation functions and bounded parameters (see Section 4 of [1] for a detailed discussion).\n \nAlso, we clarify that a **sufficient condition for Assumption 3.2** is that both Proposition A.2 and Assumption 3.1 hold. Specifically, Proposition A.2 ensures $\\\\|\\nabla\\varphi_0(x) \\\\|\\_2$ to be bounded by some function $L_1(x)$. Meanwhile, Assumption 3.1 provides an envelope for $\\\\|\\nabla\\varphi(x)\\\\|\\_2$, denoted as $L_2(x)$. Then Assumption 3.2 follows via the triangle inequality $\\\\|\\nabla\\varphi(x)-\\nabla\\varphi_0(x)\\\\|\\_2\\leq L_1(x)+L_2(x)$. \n\nTypical cases where Proposition A.2 holds include: $\\mu$ can be normal or Student-t, and $\\nu$ can be any measure with certain tail decay. We refer to Appendix A for further details and an explicit example verifying Assumption 3.2.\n\nMoreover, there are several settings where $\\varphi_0\\in\\mathcal{F}$ and Assumption 3.2 holds automatically.\n\nA canonical example is the class of **quadratic functions**: $$\\mathcal{F}\\_{quad}=\\\\{x\\mapsto x^{\\top}Bx+\\langle b,x\\rangle:B\\in\\mathbb{S}_{+}^{d},\\\\|B\\\\|\\_{op}\\leq r_1, \\\\|b\\\\|\\_2\\leq r_2\\\\}.$$\nWhen $\\mu$ and $\\nu$ are Gaussian (or from the same elliptical family), the Brenier potential is quadratic. This case arises in applications such as distributionally robust estimation of covariance and precision matrices.\n\nAnother example $\\mathcal{F}$ is the class of **convex functions with uniformly bounded Lipschitz constants**, studied in [2]. For instance, if $\\nu$ is the uniform distribution on the unit ball, and the support of $\\mu$ is open and convex [3], then $\\varphi_0\\in\\mathcal{F}$ regardless of the tail behavior of $\\mu$. This setting arises in OT-rank estimation [4, 5].\n\nWe will incorporate these examples and clarifications.\n\n## Q3 & W2\nThanks for highlighting these important points. Notably, the $(\\beta,b)$-smoothness is widely used in the OT literature [1, 8-11], and the sub-Weibull assumption is also standard [1，7，8]. Both assumptions are used in our work primarily for analytical convenience, not because they are fundamentally required for the methodology.\n\nOur refined approximation result (Lemma F.4) relies on bounding the Sobolev norm $\\\\|\\varphi_0\\\\|\\_{W^{2,\\infty}([-\\tilde{R}\\_\\varepsilon,\\tilde{R}\\_\\varepsilon]^d)}$, and the $(\\beta,b)$\\-smoothness assumption ensures that this norm grows polynomially in $\\tilde{R}\\_\\varepsilon$. Meanwhile, the sub-Weibull $\\mu$ guarantees that $R\\_\\varepsilon,\\tilde{R}\\_\\varepsilon$ scale polynomially in $\\log(1/\\varepsilon)$. Together, these two results ensure that the relevant Sobolev norm is bounded by a polynomial in $\\log(1/\\varepsilon)$, and the consequent approximation error has the order of $\\varepsilon$. \n\nIf $(\\beta,b)$-smoothness were lifted, one could still obtain approximation bounds, but they may degrade to $O(\\varepsilon^a)$ for some $a<1$. This, in turn, complicates the bias-variance trade-off in the statistical analysis by introducing additional terms in the statistical error, approximation error, and sieved estimation bias trade-off. Thus, while not strictly necessary, the smoothness assumption simplifies the proof and keeps the convergence rate clean and interpretable.\n\nThe sub-Weibull assumption on source distribution relates to statistical error, approximation error, and sieved estimation error. Generally, a heavier tail in $\\mu$ makes these errors larger. The sub-Weibull tail ensures that any potential enlarging factors remain logarithmic in $n$. Importantly, this assumption is not limiting. Specifically, with polynomially tails, $R_\\varepsilon$ scales polynomially in $1/\\varepsilon$. While the proofs follow similar steps, the resulting convergence rate becomes much more complex. See Equation 3.10 in [8] for an illustration of the added technical complications.\n\nWe will incorporate this discussion into the revised manuscript to clarify the roles and implications of both assumptions.\n\n## Q4 & W3\nThanks for bringing up this important question.\n\nTo clarify, we present our observed convergence rates in experiments for $d=10$, $n\\geq 256$, by computing the slopes of logarithm of $L^2$-$UVP$ w.r.t to $\\log n$:\n\n|Source/Target|0|1|2|3|inf|\n|-|-|-|-|-|-|\n|normal/t|-0.38|-0.31|-0.31|-0.32|-0.38|\n|uniform/normal|-0.26|-0.22|-0.21|-0.20|-0.19|\n|t/normal|-0.51|-0.53|-0.52|-0.52|-0.52|\n|normal/uniform|-0.70|-0.69|-0.72|-0.71|-0.70|\n\nTwo conclusions can be made on this table:\n\n1. Across all settings, the observed rates exceed the rate $n^{-2/d} = n^{-0.2}$ for the plug-in estimator in [7], suggesting that our OT estimator effectively benefits from the smoothness of the TNN architecture.\n\n2. In the normal/uniform setting, the rates are much faster than $n^{-1/2}$, suggesting that optimization error dominates the overall convergence. As further evidence, when repeating experiments for $C=2$ and $C=\\infty$ with larger sample sizes ($n=1024, 2048, 3064, 4096, 5120$), the exponents become $-0.426$ and $-0.436$, respectively.\n\nSimilar phenomena have also been reported in Section 6.2 of [9]. \n\nWhile our experiments do not fully demonstrate the claimed theoretical rates, providing a complete explanation of OT estimation rates in practice is challenging and remains an open question. The numerical results are influenced by many factors, including neural network architectures, optimization methods, and other implementation choices.\n\n## Q5-Q7\nWe appreciate your attention to detail and will ensure all these corrections and clarifications are made in the revised manuscript.\n\n5.1 Please refer to our response to Q1 & W1.\n\n5.2 Eq. 42 provides the upper bound $\\nu\\_\\varepsilon(\\bar\\varphi\\_{\\tilde{R}\\_\\varepsilon}^*-\\varphi\\_0^\\*)\\leq\\\\|\\bar\\varphi-\\varphi\\_0 \\\\|\\_{L^\\infty(B(0,\\tilde{R}\\_\\varepsilon))}$\n\nEq. 43 gives the lower bound $\\nu\\_\\varepsilon(\\bar\\varphi\\_{\\tilde{R}\\_\\varepsilon}^*-\\varphi\\_0^\\*)\\geq-\\\\|\\bar\\varphi-\\varphi\\_0\\\\|\\_{L^\\infty(B(0,\\tilde{R}\\_\\varepsilon))}$. L872 follows by taking absolute values.\n\n5.3 The RHS in L930 is based on Chebyshev inequality and $Var(2\\Phi(X) \\mathbb{I}(\\\\|X\\\\|\\_2>R_\\varepsilon))\\leq E(4\\Phi^2(X)).$\n\n5.4 Introduced in [6], $N$ is the number of subdivisions per coordinate to localize the neural network approximation. It is fundamental to characterize the size of the TNN class.\n\n5.6 We define $\\langle R\\rangle=|R|+1$ for $R \\in \\mathbb{R}$. \n\n5.7 The sub-Weibull $\\mu$ ensures $R_\\varepsilon,\\tilde{R}\\_\\varepsilon$ are $O(\\log(1/\\varepsilon)^{1/\\lambda})$. By taking $\\varepsilon=O(n^{-\\frac{2\\alpha}{d+2\\alpha}})$, $R_\\varepsilon$ is logarithmic in $n$, so dominated by the polynomial terms in $n^{-1}$, and is suppressed. Please refer to our response to Q3 for a detailed discussion.\n\n5.8 L968 directly follows by Eq. 47. By plug-in $\\gamma=0$ and $\\gamma'=1$ as in L966, and combining Eq. 66, we then get Eq. 69. \n\n6.7 We define $\\log_+(x)=\\max\\\\{\\log x,1\\\\}$. \n\nAll the other typos and suggestions are addressed.\n\n## Reference\n[1] Divol et al. Optimal transport map estimation in general function spaces. Annals of Statistics (AoS) 2025\n\n[2] Kur et al. Convex regression in multidimensions: Suboptimality of least squares estimators. AoS 2024\n\n[3] Barrio et al. A note on the regularity of center-outward distribution and quantile functions. JMVA 2020\n\n[4] Chernozhukov et al. Monge–Kantorovich depth, quantiles, ranks and signs. AoS 2017\n\n[5] Ghosal and Sen. Multivariate ranks and quantiles using optimal transport: Consistency, rates and nonparametric testing. AoS 2022\n\n[6] Ryck et al. On the approximation of functions by tanh neural networks. Neural Networks 2021\n\n[7] Deb et al. Rates of estimation of optimal transport maps using plug-in estimators via barycentric projections. NeurIPS 2021\n\n[8] Ding et al. Statistical convergence rates of optimal transport map estimation between general distributions. ArXiv 2412.03722\n\n[9] Hutter and Rigollet. Minimax estimation of smooth optimal transport maps. AoS 2021\n\n[10] Gunsilius. On the convergence rate of potentials of Brenier maps. Econometric Theory 2022\n\n[11] Manole et al. Plugin estimation of smooth optimal transport maps. AoS 2024"}, {"review_id": "OMhmKQFHtW", "reviewer": "Reviewer_mmaX", "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.", "questions": "- What is $t_6$ in the experiments?\n- 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.\n- The authors should highlight how the new stability bound with relaxed assumptions helps in Section 3.4.\n- 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", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n- The paper is well-written, theoretically sound and contains novel results.\n- The bound for the class of neural OT estimators brings the proposed method closer to practicality.\n\nWeaknesses\n- 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.", "was_revised": true, "final_justification": "Overall, this is a solid paper with both theoretical interest and practical relevance. In the rebuttal, the authors provided additional experiments that address my earlier concerns, which would improve the paper’s quality. I therefore maintain my current score of 5, leaning toward acceptance.", "rebuttal": "We sincerely thank you for your thoughtful review, encouraging remarks, and constructive suggestions. We are grateful for your recognition of the theoretical contributions and practical relevance of our work. Below, we provide our point-by-point responses to your specific questions (Q1 to Q4):\n\n## Weakness\n\nThank you for raising this insightful point. Our additional experiments show that deeper TNN architectures can learn faster than 2-hidden-layer TNNs, and achieve slightly better ultimate L2-UVP when trained for sufficient epochs. Meanwhile, only minor gradient vanishing issues were observed in our setting.\n\nDue to the time limit, we only consider the setting of normal to uniform, $C=2$, $d = 5$. Five TNNs are compared in the experiment: Model 1 ($f_{2,10}$), Model 2 ($f_{8,10}$), Model 3 ($f_{8,4}$), Model 4 (Model 2 with residual connections), and Model 5 (Model 3 with residual connections), where $f_{D,W}$ denotes $D$ hidden layers and width $W$. The learning rate was set to $0.001$ and each model was trained for 300 epochs with $n = 256$. \n\nWe observed that the 8-layer TNN models (Models 2 and 4) demonstrated a faster decrease in loss compared to the 2-layer model. The addition of residual connections offered a slight improvement in learning efficiency, suggesting only a minor vanishing gradient issue in our setting. Throughout training, Models 2 and 4 consistently outperformed Model 1, while Models 3 and 5 performed notably worse.\n\n### Mean Loss over 50 Repetition\n| epoch | model1 | model2 | model3 | model4 | model5 |\n|-------|--------|--------|--------|--------|--------|\n| 1     | 29.03  | 28.99  | 29.00  | 28.86  | 28.85  |\n| 31    | 17.72  | 11.12  | 19.43  | 6.06   | 19.63  |\n| 61    | 5.80   | 4.86   | 10.33  | 4.09   | 10.17  |\n| 91    | 4.56   | 3.26   | 7.87   | 2.61   | 7.99   |\n| 121   | 3.38   | 2.56   | 6.57   | 2.07   | 6.53   |\n| 151   | 2.48   | 2.11   | 6.76   | 1.81   | 5.42   |\n| 181   | 2.01   | 1.89   | 5.06   | 1.69   | 4.89   |\n| 211   | 1.84   | 1.72   | 4.23   | 1.60   | 4.24   |\n| 241   | 1.75   | 1.63   | 4.15   | 1.53   | 3.92   |\n| 271   | 1.69   | 1.57   | 3.87   | 1.51   | 3.59   |\n| 300   | 1.64   | 1.55   | 3.40   | 1.49   | 3.35   |\n\n### L2-UVP\n| model1 | model2 | model3 | model4 | model5 |\n|--------|--------|--------|--------|--------|\n| 0.095  | 0.089  | 0.345  | 0.094  | 0.369  |\n\n## Q1\n\nThe notation $t_6$ refers to the Student’s $t$-distribution with 6 degrees of freedom, chosen in line with [DLX]. As a heavy-tailed distribution, it allows us to test the estimator beyond the compact support assumptions dominating the literature.\n\n## Q2\n\nThanks for this suggestion.\n\nTo compare the sieved estimator with original one on cases of smooth/convex potentials, we conducted additional experiments under settings as in [HR], where the source is a uniform distribution and the true OT map is either the identity or exponential mapping. The following table summarizes the median (standard deviation) of L2-UVP for source = uniform and OT = exp, d=10.\n\n### Median(std) of L2-UVP, uniform to exp(uniform), d = 10\n| C   | 128           | 256           | 512           | 1024          | 2048          |\n|-----|---------------|---------------|---------------|---------------|---------------|\n| 0   | 0.120 (0.017) | 0.071 (0.009) | 0.047 (0.005) | 0.031 (0.002) | 0.027 (0.004) |\n| 1   | 0.115 (0.020) | 0.065 (0.008) | 0.048 (0.006) | 0.033 (0.004) | 0.028 (0.005) |\n| 2   | 0.119 (0.023) | 0.065 (0.007) | 0.048 (0.006) | 0.033 (0.005) | 0.030 (0.005) |\n| 3   | 0.114 (0.023) | 0.065 (0.008) | 0.048 (0.006) | 0.034 (0.004) | 0.029 (0.005) |\n| inf | 0.122 (0.030) | 0.068 (0.011) | 0.049 (0.005) | 0.034 (0.005) | 0.031 (0.005) |\n\n\nIt demonstrates that the sieved estimator consistently outperforms the original one in the setting of compact support and smooth/convex OT maps.\n\nThe mean training time between the sieved estimator and the original estimator is summarized below:\n\n### Mean training time (second), normal to uniform, sieved estimator vs Original Estimator\n| C   | 64      | 128      | 256      | 512      | 1024      | 2048      |\n|-----|---------|----------|----------|----------|-----------|-----------|\n| 2   | 97.8096 | 159.7494 | 293.5467 | 587.9228 | 1107.5363 | 2159.0094 |\n| inf | 85.7766 | 140.1795 | 257.9078 | 513.5465 | 968.8996  | 1890.1076 |\n\nThe sieved estimator incurs an additional computational cost of approximately 15\\% due to the projection step. However, we believe this increase in computational time is moderate and acceptable, given the practical benefits of the sieved estimator.\n\n## Q3\n\nWe agree that the role of the new stability bound in relaxing assumptions should be emphasized more clearly. We will revise Section 3.4 accordingly.\n\nTo highlight the benefits of our new stability bound, our Theorem 3.7 improves the results in [DLX] in several ways:\n1. **Characterization of Approximation Error.** A crucial conceptual difference lies in how the approximation error is quantified. In [DLX], it is expressed in terms of the excess risk $\\inf_{\\varphi} (\\mu\\varphi + \\nu\\varphi^\\ast) - (\\mu\\varphi_0 + \\nu\\varphi_0^\\ast)$, which does not directly connect with classical approximation theory tools. \nIn contrast, we express approximation error in terms of the $L^\\infty$ norm $\\inf_{\\varphi}\\\\|\\varphi - \\varphi_0\\\\|\\_{L^\\infty(B(0,R\\_\\varepsilon))}$, which is easier to analyze.\n2. **Flexibility in Sieve Radius** Our result accommodates a wide range of sieve radii, allowing an adaptive selection based on the data and function class. In contrast, [DLX] fixes the sieve radius to the maximum norm of the source samples, $\\max_{1\\leq i\\leq n}\\\\|X_i\\\\|_2$, and their proof relies on this specific choice. \n3. **Assumptions on Sample Sizes.** While we assume $n = m$ in Theorem 3.7, it is only for theoretical analysis simplicity and can be relaxed (see Thm 3.6 for evidence). In contrast, [DLX] assumes $m \\lesssim n$ due to their choice of sieve radius. This assumption appears restrictive and somewhat counterintuitive, as it limits the use of larger sample sizes from $\\nu$, which should improve estimation rate. Upon closer inspection, this assumption in [DLX] is tightly coupled with their selection of sieve radius, and is challenging to remove.\n\n\n\n## Q4\n\nWe appreciate the reviewer’s interest in the smoothness assumptions underlying Theorem 3.7.\n\nWe impose that $\\varphi_0$ satisfies a $(\\beta,b)$-smoothness condition to facilitate the approximation error analysis. Removing this $(\\beta,b)$-smoothness may degrade the approximation error bound to $O(\\varepsilon^a)$ for $a<1$. In principle, one can repeat the proof to derive a convergence rate by balancing this additional exponent $a$ across the statistical error, approximation error, and sieve bias terms. Since the approximation error becomes larger, the overall convergence rates are expected to be slower. We agree with your suggestion and will add a discussion in the revised manuscript to acknowledge this limitation and empirical observation. \n\nIndeed, the $(\\beta,b)$-smoothness assumption is widely used in the OT estimation literature (e.g., [Gun]; [HR]; [Man+24]; [DNP]; [DLX]), as it provides a convenient and tractable framework for theoretical analysis. To our knowledge, there is currently no well-developed alternative for systematically characterizing the regularity or approximation properties of OT maps under weaker or more general conditions.\n\nWhile classical OT theory (e.g., [Caf92]; [Fig17]) offers sufficient conditions for $(\\beta,b)$-smoothness based on stringent assumptions on the source and target measures, these conditions are complex and difficult to verify in practice. Incorporating them directly into our framework would substantially complicate both the analysis and exposition, and may ultimately make our work less accessible.\n\nThus, while the $(\\beta,b)$-smoothness assumption introduces theoretical limitations, it remains a standard and necessary compromise in the current state of the literature. Exploring alternative regularity frameworks that can broaden applicability without sacrificing analytical tractability remains an important and challenging direction for future research.\n\n## Reference\n\n[HR] Jan-Christian Hütter, Philippe Rigollet \"Minimax estimation of smooth optimal transport maps,\" The Annals of Statistics 49.2 (2021): 1166-1194.\n\n[DLX] Ding, Yizhe, Runze Li, and Lingzhou Xue. \"Statistical convergence rates of optimal transport map estimation between general distributions.\" ArXiv 2412.08064.\n\n[DNP] Vincent Divol, Jonathan Niles-Weed, and Aram-Alexandre Pooladian. \"Optimal transport map estimation in general function spaces.\" The Annals of Statistics 53.3 (2025): 963-988.\n\n[Gun] Florian F Gunsilius. \"On the convergence rate of potentials of Brenier maps.\" Econometric Theory 38.2 (2022): 381-417.\n\n[Man+24] Manole, Tudor, et al. \"Plugin estimation of smooth optimal transport maps.\" The Annals of Statistics 52.3 (2024): 966-998.\n\n[Caf92] Luis A Caffarelli. \"The regularity of mappings with a convex potential.\" Journal of the American Mathematical Society 5.1 (1992): 99-104.\n\n[Fig17] Alessio Figalli. The Monge-Ampere equation and its applications. Zürich: European Mathematical Society, 2017.\n\n[Abd] Ahmed Abdeljawad. \"Uniform approximation with quadratic neural networks.\" Neural Networks (2025): 107742."}, {"review_id": "tyKmOLdVMV", "reviewer": "Reviewer_wHrA", "summary": "This paper provides a new estimation theory with relaxed assumptions. Specifically, the paper provides a novel map stability inequality and a refined oracle inequality.", "questions": "see above", "limitations": "Results 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.", "rating": 4, "confidence": 1, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Honestly, 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.", "was_revised": true, "final_justification": "This paper is a technical strong paper. However, its current format may need further polishing. Therefore, I maintain my scores.", "rebuttal": "We sincerely appreciate your thoughtful and encouraging comments on our work. \n\nOur main contributions lie at the intersection of statistics, machine learning, and optimization, making it highly relevant to the NeurIPS community. Our paper is motivated by practical learning problems and designed for algorithmic implementation, which we believe makes NeurIPS the ideal venue for this contribution.\n\nBelow, we provide our point-by-point responses to your specific questions (Q1–Q7) and mentioned limitation (L1).\n\n## Q1\n\nThank you for this helpful suggestion. We agree that clarifying these foundational concepts to enhance the accessibility and clarity of our manuscript.\n\nTo address your first point, we will revise the introduction to more clearly define dual-type OT map estimators. Specifically, we will include the following sentence:\n\n*This paper aims to contribute to this direction by focusing on the dual-type OT map estimators, formally introduced in Section 2, which are based on the characterization of the optimal transport map as the gradient of a convex function (the so-called Brenier potential).*\n\nWe will also clarify **statistical error** and **approximation error** explicitly in our updated manuscript:\n1. **Approximation error** refers to the inherent gap between the true Brenier potential and the best possible approximation within the candidate function class $\\mathcal{F}$. This arises because we do not assume that the true Brenier potential $\\varphi_0$ lies in $\\mathcal{F}$.\n2. **Statistical error** captures the discrepancy between the estimator and the best possible approximator in $\\mathcal{F}$. It reflects the order of magnitude of randomness introduced by the use of finite samples, i.e., the difference between the empirical and population loss functionals.\n    \n## Q2\n\nWe appreciate your suggestions. For example, in image generation, generative models often sample from Gaussian white noise as the source distribution $\\mu$, which is unbounded and violates compact support assumptions.\n\nAs another example, consider $\\mu=N(0,1)$ and $\\nu=U[0,1]$. The OT map from $\\mu$ to $\\nu$ is the CDF of $\\mu$, which is not $(\\alpha,a)$-convex for any $a\\geq0$. This is because the corresponding Hessian is not bounded below by 0. Conversely, the OT map from $\\nu$ to $\\mu$, as the quantile function of $\\mu$, is not $(\\beta,b)$-smooth for any $b\\geq0$, as it diverges at the boundaries of $[0,1]$.\n\nThese examples show that existing assumptions on distributions and OT maps, such as compact supports and $(\\alpha,a)$-convexity or $(\\beta,b)$-smoothness, exclude canonical OT applications in deep generative models and multivariate statistical inference (a.k.a. OT rank and OT quantile). We can incorporate these examples into the revised introduction.\n\nWe also refer you to our response to Question 2 from Reviewer qHh5, where we further emphasize how our results provide theoretical backbones for such OT applications, with a focus on OT rank estimation as an example, and theoretical guarantees not covered by existing works.\n\n## Q3\n\nThank you for this helpful question.\n\nIn Assumption 3.2, the envelope function refers to a real-valued, non-negative function $\\Phi \\in L^2(\\mu)$ that uniformly bounds the absolute values of all functions in the class $\\mathcal{F}\\cup\\{\\varphi_0\\}$. Specifically, it satisfies $$|\\varphi(x)|\\leq\\Phi(x),\\quad\\text{for all $x$ and all $\\varphi\\in\\mathcal{F}\\cup\\\\{\\varphi_0\\\\}$}.$$\n\nThis function $\\Phi$ provides a uniform control over the magnitude of the entire function class $\\mathcal{F}\\cup\\\\{\\varphi_0\\\\}$ and plays a key role in applying the empirical process theory. We will clarify this definition explicitly in the revised version.\n\n## Q4\n\nWe appreciate the reviewer’s comment that our original explanation of Assumptions 3.1 and 3.2 lacked clarity. To address this, we have revised the statements of Assumptions 3.1 and 3.2 in the updated manuscript to better reflect their intended meaning.\n\nAssumption 3.1 depends only on $\\mathcal{F}$ and is independent of $\\varphi_0$, $\\mu$ and $\\nu$. It is satisfied by many common function classes, including wavelets, RKHS, and neural networks with smooth activation functions and bounded parameters (see Section 4 of [DNP] for a detailed discussion).\n \nAlso, we clarify that a **sufficient condition for Assumption 3.2** is that $\\\\|\\nabla\\varphi_0(x)\\\\|_2$ is bounded by some function $L_1(x)$ and Assumption 3.1 holds. Assumption 3.1 provides an envelope for $\\\\|\\nabla\\varphi(x)\\\\|_2$, denoted as $L_2(x)$. Then, Assumption 3.2 follows from the triangle inequality: $\\\\|\\nabla\\varphi(x) - \\nabla\\varphi_0(x)\\\\|_2 \\leq L_1(x) + L_2(x)$. In our appendix, Proposition A.2 shows that a bound for $\\\\|\\nabla\\varphi_0(x)\\\\|_2$ can be obtained under mild distributional assumptions on $\\mu$ and $\\nu$. We refer to Appendix A for further details and for an explicit example verifying Assumption 3.2.\n\n## Q5\n\nThank you for catching this. We define $\\log_+(x):=\\max\\\\{\\log x,1\\\\}$, ensuring the quantity remains non-negative and bounded below by 1. We will include this definition in the updated manuscript.\n\n## Q6\n\nThank you for raising this point. We use $\\mathcal{O}\\_{\\log}$ as shorthand for “up to logarithmic factors.” For example, $$\\mathcal{O}\\_{\\log}\\bigl((n/D\\_{\\mathcal{F}})^{-1/2}\\bigr)$$\nmeans there exist constants $c_1,c_2>0$ such that the expression is bounded by $$c_1\\cdot (n/D\\_{\\mathcal{F}})^{-1/2}\\log(n/D\\_{\\mathcal{F}})^{c_2}.$$\nWe will define this notation explicitly in the revised manuscript to ensure clarity.\n\n## Q7\n\nThank you for your interest in the implementation details of our TNN-based estimator. It is important to explain the dependence of the network architecture parameters on the sample size $n$, dimension $d$, and the smoothness $\\alpha$ of the target potential $\\varphi_0$ in the manuscript.\n\nIn all our experimental settings, no explicit truncation of the TNN parameters was required in practice. According to our mathematical analysis (see lines 942–943 and 969 in the supplement), the hyperparameter $\\kappa$, which controls the range of network parameters in the TNN class, should be chosen as $$\\kappa = O\\left(n^{\\frac{d(d+(\\lfloor\\alpha\\rfloor+2)^2+4)+2}{2d+4\\alpha}}\\right).$$\nIn our experiments, all underlying potentials $\\varphi_0$ are infinitely smooth. Taking the limit $\\alpha\\to\\infty$ yields $\\kappa\\to\\infty$.\n\nTo further investigate the effect of truncation, we conducted supplementary experiments in which all network parameters were clamped by $\\pm 10$. Interestingly, we observed that truncation has a  negligible impact on TNN performance. Due to space constraints, we report here the L2-UVP and the mean of the maximum parameter magnitude in the $\\mathcal{N}(0,1)$ to $t_6$ setting with $d=10$ across 50 repetitions. The results suggest that parameter values remain small and increase slowly with sample size, further supporting that truncation is primarily a theoretical device, and is unnecessary in practice.\n\n### Normal to t (d=10), No truncation vs Truncation at 10.\n|C|64|128|256|512|1024|2048|\n|-|-|-|-|-|-|-|\n|2 (No)|0.41|0.26|0.14|0.09|0.06|0.06|\n|2 (10)|0.43|0.26|0.14|0.09|0.07|0.06|\n|inf (No)|0.58|0.42|0.23|0.10|0.06|0.06|\n|inf (10)|0.58|0.44|0.25|0.10|0.07|0.06|\n\n### Normal to t (d=10), Mean of Max Parameter over 50 repetition\n|C|64|128|256|512|1024|2048|\n|-|-|-|-|-|-|-|\n|0|1.19|1.32|1.74|2.39|3.10|4.02|\n|1|1.31|1.55|2.04|2.81|3.62|4.53|\n|2|1.33|1.58|2.06|2.86|3.71|4.55|\n|3|1.34|1.59|2.09|2.86|3.73|4.65|\n|inf|1.357|1.62|2.11|2.85|3.68|4.56|\n\n## L1\n\nWe acknowledge that tanh networks are less commonly used in practice, and bridging theoretical insights with practical architectures remains one of our long-term goals in optimal transport.\n\nHowever, as discussed in our response to Q5 from Reviewer qHh5, the mathematical formulation of OT maps as gradients of potential functions is incompatible with ReLU networks, which lack second-order (weak) derivatives. Indeed, the second-order regularity of the candidate function class is foundational in current OT theory for conducting statistical analysis.\n\nBeyond such reasons from theoretical analysis, we have also run one setting in our numerical study with larger ReLU networks. Shockingly, they are significantly worse than TNNs: \n\n### Median L2-UVP, 3-Hidden Layer and 20-neurons ReLU, Uniform to Normal, d = 10.\n|C|64|128|256|512|1024|2048|\n|-|-|-|-|-|-|-|\n|0|0.62|0.43|0.32|0.28|0.23|0.18|\n|1|0.61|0.43|0.34|0.29|0.24|0.19|\n|2|0.59|0.42|0.34|0.29|0.23|0.19|\n|3|0.59|0.41|0.33|0.29|0.24|0.19|\n|inf|0.55|0.41|0.33|0.29|0.24|0.20|\n\nPractically, ReLU is often preferred for its computational efficiency. However, recent findings suggest that smooth activations may improve approximation capabilities. For instance, [Abd] shows that ReQU networks can achieve the same approximation accuracy as ReLU with a square root number of neurons. Combining the theoretical hardness and our shocking empirical evidence, OT estimation may benefit from smooth tanh activation. This observation may motivate further exploration, especially given the more powerful GPUs available today compared to the early DL era (say AlexNet in 2012).\n\n## L2\n\nThanks for this comment. We will include more explanations of our crucial ideas (e.g., the pseudo-compact support brought by sieved estimator, linking OT estimation error to Brenier potential estimation error) in the revised manuscript. \n\n## Reference\n\n[DNP] Vincent Divol, Jonathan Niles-Weed, and Aram-Alexandre Pooladian. \"Optimal transport map estimation in general function spaces.\" The Annals of Statistics 53.3 (2025)\n\n[Abd] Abdeljawad. Uniform approximation with quadratic neural networks. Neural Networks 2025"}, {"review_id": "z92z37k9Ot", "reviewer": "Reviewer_qHh5", "summary": "The 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.", "questions": "In 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?", "limitations": "No societal impact.", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\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.", "was_revised": true, "final_justification": "I maintain my position that this is an interesting paper with potentially useful results.", "rebuttal": "We sincerely appreciate your detailed and constructive feedback. Below, we address your identified weaknesses (W1–W4) and questions (Q1–Q5).\n\n## W1\n\nWe will revise the discussion following Eq. 7, where the sieved estimator is first introduced, to clarify its main idea as following:\n\nNote that $\\varphi^*(\\nabla\\varphi(x))=\\langle x,\\nabla\\varphi(x)\\rangle-\\varphi(x)$. By restricting $x$ to $B(0,\\tilde{R})$, we equivalently restrict the domain of $\\varphi\\_{\\tilde{R}}^\\*(y)$ to the compact set $\\nabla\\varphi(B(0,\\tilde{R}))$. This compactness facilitates us to analyze the estimation error of sieved OT estimator.\n\nAlso, we will include more explanations of our crucial ideas (e.g., linking OT estimation error to Brenier potential estimation error) in the revised manuscript. \n\n## W2 & Q1\n\nWe have followed your helpful comments to revise the statements and added more explanation and examples. \n\nAssumption 3.1 depends only on the function class $\\mathcal{F}$ and is independent of the true OT map $\\nabla \\varphi_0$, as well as the measures $\\mu$ and $\\nu$. It primarily characterizes the regularity of the function class and is essential for controlling the second-order Taylor expansion, which is a key tool in linking the OT estimation error to the Brenier potential estimation error. Additionally, Assumption 3.1 guarantees an envelope function for the candidate function class, which is important for empirical process analysis.\n\nThis assumption is not restrictive and is satisfied by many common function classes, including wavelets, RKHS, and neural networks (possibly with engineering modifications such as parameter truncation). For detailed discussion and explicit examples, we refer to Section 4 of [6], whose constructions apply directly to our setting. \n\nAssumption 3.2 quantifies the maximum discrepancy between the estimated OT map $\\nabla\\varphi$ and the true OT map $\\nabla\\varphi_0$. This assumption requires that the candidate function class does not include functions that are too distant from $\\varphi_0$.\n\nA sufficient condition for Assumption 3.2 is that both Proposition A.2 in Appendix and Assumption 3.1 hold. We clarify that Proposition A.2 allows both $\\mu$ and $\\nu$ to be heavy-tailed, e.g., Student-t, Pareto distributions. Specifically, Proposition A.2 does not force the density of $\\mu$ to be exponentially decay ($V$ can be a logarithmic function), and Definition A.1 accommodates polynomial-tailed $\\nu$. \n\n## W3\n\nOur current analysis, based on empirical process, assumes access to the global minimizer and does not account for optimization error. We will discuss this point more explicitly in the revised manuscript. In addition, we will include explicit bounds on the network complexity, which are originally stated in the Appendix, Theorem E.1.\n\n## W4\n\nThe current state of OT theory relies on the smoothness assumption for approximation error (but not for statistical error and sieve bias). Due to space limitations, we kindly refer you to our response to Q4 from Reviewer mmaX, where we outline the theoretical challenges without such assumptions.\n\nThat said, we fully agree it is important to demonstrate how our framework can operate without a-priori smoothness assumptions. As in our response to your Q2, we provide an example of OT rank estimation. Specifically, when the target $\\nu$ is the uniform measure on the unit hyper-ball, and the source $\\mu$ satisfies some mild conditions, the smoothness of $\\varphi_0$ follows automatically from [1], eliminating the need for a-priori assumptions presented in Theorem 3.7. We also refer you to Q4 from Reviewer mmaX on this point.\n\nIn conclusion, while smoothness assumptions remain crucial for establishing approximation error, they are unnecessary in some key OT applications. We will revise the manuscript to clarify these distinctions and highlight such examples.\n\n## Q2\n\nLet us elaborate using the OT-rank problem [2,3] as an example, which provides an important generalization for the notion of ranks to multivariate settings. However, to estimate the OT-rank from $\\mu$ to target $\\nu$, existing works assume both distributions have compact and convex supports. These assumptions exclude canonical cases, such as unbounded univariate distributions (e.g., Gaussian or Student’s $t$) or the banana-shaped bivariate distributions often used to motivate the OT-rank concept.\n\nWhen $\\nu$ is compactly supported (common for OT-rank to mimic the role of the uniform measure in univariate ranks), the true OT map $\\nabla\\varphi_0$ must be uniformly bounded regardless of $\\mu$. In such cases, any $\\mathcal{F}$ with uniformly bounded gradients and Hessians ensures that Assumptions 3.1 and 3.2 hold, with constants $U_1$, $U_2$. Assumption 3.5 is independent of the smoothness. Consequently, Theorem 3.6 holds without a-priori smoothness assumptions.\n \nThus, with a proper selection of sieve radius $\\tilde{R}$ and $\\mathcal{F}$, Theorem 3.6 highlights the promise of sieved estimators in improving existing OT-rank estimators.\n\n## Q3\n\nWe first emphasize that Eq. 19, quantifying the sieve bias, has no \"pre-factors\". It arises solely from the choice of the sieve radius and is independent of $d$.\n\nRegarding Eq. 18, we clarify that $d$ does not appear explicitly, despite implicitly upon $D_\\mathcal{F}$ and $M$ (to be explained). It is common for statistical error not to directly depend on $d$ in statistical learning theory. For example, [9] analyzed robust ReLU network estimators using Huber loss and also found that both the statistical error and the bias are dimension-free. The suppressed constant in Eq. 18 depends only on $\\gamma$, $\\gamma^\\prime$, coming from empirical process theory, and is often neglected. \n\nNevertheless, $D_\\mathcal{F}$ and $M$ may depend on $d$. For instance, if $\\mathcal{F}$ is a neural network class with Lipschitz activation, $D_\\mathcal{F}$ typically scales linearly with the number of neurons. As $d$ increases, achieving a good approximation of $\\varphi_0$ may require increasing the network size, causing $D_\\mathcal{F}$ to depend on $d$. Similarly, $M$ may scale with $d$ through the term $\\sup\\\\|\\nabla\\varphi_0(x)\\\\|\\_2$, as shown in Proposition A.2.\n\nIn summary, while the statistical error in Eq. 18 is nominally dimension-free, some dimension dependence still arises indirectly via the complexity of $\\mathcal{F}$ and regularity properties of $\\varphi_0$. \n\n## Q4\n\nWhen sieve radii are $\\infty$, our sieve estimator reduces to the dual-type estimator studied in prior works [4-7]. In this regime, the sieve bias term $\\mathcal{E}_{sieve}$ vanishes, and Proposition 3.4 is the same as in these works.\n\nBut our formulation offers a slightly different bound for approximation error, where we express it as $2\\inf_{\\varphi\\in\\mathcal{F}}\\\\|\\varphi-\\varphi_0\\\\|\\_{L^\\infty},$ which connects more directly to classical function approximation theory. In contrast, prior works either represent it with the excess risk $\\inf_{\\varphi}(\\mu\\varphi+\\nu\\varphi^\\ast)-(\\mu\\varphi_0+\\nu\\varphi_0^\\ast);$ or based on the gradient $\\inf_{\\varphi\\in\\mathcal{F}}\\\\|\\nabla\\varphi-\\nabla\\varphi_0\\\\|\\_{L^2(\\mu)},$ which are harder to analyze.\n\n## Q5\n\nWe will discuss in Section 3.4 to clarify why our framework currently focuses on tanh activation. Indeed, incorporating ReLU networks into the current framework poses two challenges:\n1. **Gradient-based definition of the OT:** Our estimator is defined via the gradient of a potential function. While PyTorch can handle ReLU networks computationally, the ReLU activation is not globally differentiable in mathematics. So, this definition is ill-posed for ReLU networks.\n2. **Second-order regularity:** Even if we use a weak derivative to define OT, ReLU networks do not admit second-order weak derivatives. However, existing OT theory fundamentally relies on second-order regularity to perform statistical analysis.\n\nThat said, we conducted additional experiments with larger ReLU networks and found that our smaller TNNs with smooth activations still outperformed them (see our response to Reviewer wHrA, comment L1). This aligns with recent findings (e.g., [8] shows that ReQU networks can achieve the same approximation accuracy as ReLU using fewer neurons). These results suggest that smooth activations like tanh may offer advantages in OT estimation, both theoretically and empirically, and may warrant deeper investigation in future work.\n\n## Typo\n\nLast but not least, thanks to your careful reading, we have also corrected the typo in Proposition A.2. \n\n## Reference\n\n[1] Del Barrio et al. A note on the regularity of center-outward distribution and quantile functions. JMVA 2020\n\n[2] Chernozhukov et al. Monge–Kantorovich depth, quantiles, ranks and signs. AoS 2017\n\n[3] Ghosal and Sen. Multivariate ranks and quantiles using optimal transport: Consistency, rates and nonparametric testing. AoS 2022\n\n[4] Ding et al. Statistical convergence rates of optimal transport map estimation between general distributions. ArXiv 2412.08064.\n\n[5] Hütter and Rigollet. Minimax estimation of smooth optimal transport maps. AoS 2021\n\n[6] Divol et al. Optimal transport map estimation in general function spaces. AoS 2025\n\n[7] Gunsilius. On the convergence rate of potentials of Brenier maps. ET 2022\n\n[8] Abdeljawad. Uniform approximation with quadratic neural networks. Neural Networks 2025\n\n[9] Fan et al. How do noise tails impact on deep relu networks? AoS 2024"}, {"review_id": "85lCgFwDhh", "reviewer": "Reviewer_oQWg", "summary": "The 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).", "questions": "- 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\"?", "limitations": "- 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.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "In 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].", "was_revised": true, "final_justification": "I maintain my score: I think the framework is nice and that Theorem 3.7 is novel.", "rebuttal": "We sincerely appreciate your thoughtful evaluation and constructive suggestions. Below, we respond to the identified weaknesses (W1-W2), limitations (L1-L2), and questions (Q1-Q2), and offer clarifications based on your valuable feedback.\n\n## W1 & L2\n\nWe first clarify the practical implementation of the sieved estimator, which we believe addresses your concerns regarding computational feasibility.\n\nThe sieve estimator involves solving the inner optimization problem: for some $R\\in(0,\\infty]$ and function class $\\mathcal{T}$, $$\\varphi\\_R^\\ast(y)=\\sup_{x\\in\\mathbb{R}^d:\\|x\\|_2\\leq R}\\langle x,y \\rangle-\\varphi(x).$$\nWhen $\\varphi$ is convex (as is the case when using input convex neural networks, a common and practical choice in neural OT estimation), this is a convex optimization problem and can be efficiently solved using gradient descent. In fact, when $R = \\infty$, corresponding to the classical Legendre transform, gradient-based methods have been effectively applied in the deep learning literature for years (e.g., [Hua+]).\n\nThe sieved version simply introduces an additional projection step onto the $\\ell_2$-ball of radius $R$ at each iteration of gradient descent. In another word, the computation of $\\varphi_R^\\ast$ can be achieved with projected gradient descent. As presented in our response to Q2 of mmaX, supplementary experiments show that the sieved estimator incurs only a modest computational overhead of about 15\\% due to this projection. We emphasize that this moderate cost is justified by the consistent improvements in estimation accuracy and stability observed across all scenarios. \n\nFurthermore, we note that related works in neural OT estimation reformulate it as a min-max problem, e.g., let $\\text{id}$ be the identity mapping, $$\\hat\\varphi_{n,m}=\\min_{\\varphi\\in\\mathcal{F}}\\sup_{T\\in\\mathcal{T}}\\mu_n \\varphi+\\nu_m(\\langle \\text{id}, T\\rangle-\\varphi\\circ T), $$\nas seen in [Mak+, Tar+]. If the function class $\\mathcal{T}$ is sufficiently rich and its outputs are uniformly bounded in $\\ell_2$-norm by $R$, achievable via simple tricks like output truncation, then this formulation can be interpreted as another practical implementation of the sieve conjugate.\n\nIn summary, while the sieved estimator offers theoretical advantages, it also remains computationally practical, incurring negligible additional burden for practitioners. Now, we move on to explaining your questions.\n\n## W2 and Q1\n\nThank you for this insightful question regarding the distinction between our results, especially Theorem 3.7, and related results in [DLX]. While our results may appear similar to [DLX], the two are fundamentally different in several key aspects:\n\n1. **Characterization of Approximation Error.** A crucial conceptual difference lies in how the approximation error is quantified. In [DLX], it is expressed in terms of the excess risk functional $\\inf_{\\varphi} (\\mu\\varphi + \\nu\\varphi^\\ast) - (\\mu\\varphi_0 + \\nu\\varphi_0^\\ast)$, which does not directly connect with classical approximation theory tools. In contrast, we express approximation error in terms of the $L^\\infty$ norm $\\inf_{\\varphi}\\\\|\\varphi - \\varphi_0\\\\|\\_{L^\\infty}$, making it easier to analyze using known approximation rates of TNNs. \n\n2. **Flexibility in Sieve Radius Selection.** Our result accommodates a wide range of sieve radii $\\tilde{R}\\_\\varepsilon$, allowing the choice of radius to be adapted based on the data and function class. In contrast, [DLX] fixes the sieve radius to the maximum empirical norm of the source sample, i.e., $\\max_{1\\leq i\\leq n}\\\\|X_i\\\\|\\_2$, and their proof techniques rely on this specific construction. It remains unclear whether this choice yields the same performance in our context, with TNNs as an example as the focus in our Theorem 3.7.\n\n3. **Assumptions on Sample Sizes.** While we assume $n = m$ (equal sample sizes from $\\mu$ and $\\nu$) in Theorem 3.7 for theoretical analysis simplicity, this condition can be relaxed in our analysis (see Theorem 3.6). In contrast, Theorem 3.17 in [DLX] assumes $m \\lesssim n$ due to their choice of sieve radius. This assumption appears restrictive and somewhat counterintuitive, as it limits the use of larger sample sizes from the target measure $\\nu$, which should theoretically improve estimation performance. Upon closer inspection, we found this condition to be tightly coupled with the selection of sieve radius in [DLX], making it challenging to remove.\n\n\nWhile [DLX] provides valuable insights into the sieved estimator and presents appealing results on statistical error, their work does not address the analysis of approximation error. Without our novel techniques for selecting the sieve radius and for characterizing approximation error, it would not be possible to derive results such as Theorem 3.7. Our contributions fill this important gap, provide a unified framework to  capture both statistical error and approximation error in the estimation of OT maps.\n\n## Q2\n\nThank you for this thoughtful question regarding the function class used in Theorem 3.7. We focus on TNNs with smooth activation in Theorem 3.7 because this is an important setting in the literature where the approximation error is explicitly characterized in terms of the network’s complexity under smooth activation constraints. If one replaces TNNs with another nonparametric function class that is sufficiently expressive and satisfies the regularity assumptions in Assumptions 3.1 and 3.2, then analogous results could likely be established. Examples may include classical function spaces such as RKHS or wavelets, as studied in [DNP].\n\n\n\n## L1\n\nThanks for point this out. In our revised manuscript, we have explicitly listed the network architectures in Theorem 3.7, which is originally stated in Appendix E.\n\n\n## Reference\n\n[DNP] Vincent Divol, Jonathan Niles-Weed, and Aram-Alexandre Pooladian. \"Optimal transport map estimation in general function spaces.\" The Annals of Statistics 53.3 (2025): 963-988.\n\n[DLX] Ding, Li, and Xue. Statistical convergence rates of optimal transport map estimation between general distributions. ArXiv 2412.08064\n\n[Hua+] Huang et al. Convex potential flows: Universal probability distributions with optimal transport and convex optimization. ICLR 2021\n\n[Mak+] Makkuva et al. Optimal transport mapping via input convex neural networks. ICML 2020 \n\n[Tar+] Tarasov et al. A statistical learning perspective on semi-dual adversarial neural optimal transport solvers. ArXiv 2502.01310"}]

Author rebuttal:

Meta-review: This paper develops new theoretical guarantees for optimal transport map estimation under minimal regularity assumptions. All reviewers agree that it is a solid and valuable contribution, and that the paper should be accepted.

The work has many merits. It addresses a timely and important problem, and its theoretical contributions are both strong and original. In particular, it establishes novel stability results and a clean oracle inequality that aligns well with classical approximation theory. The mathematical framework is well-structured, the proofs are both fitting and intuitive, and the results are likely to be useful not only to the statistical OT community but also to applied work where neural estimators are employed. The rebuttal was detailed and constructive, addressing earlier concerns and including additional experiments that further reinforce the contribution.

Some weaknesses were noted. The writing is highly technical, which may make the paper less accessible to non-specialists, and certain assumptions could be presented more transparently. The empirical validation, while helpful, remains limited, focusing mainly on improvements of the sieved TNN estimator over the unsieved version without broader benchmarking. In addition, questions remain about the practical suitability of the sieved estimator, and further discussion would strengthen the paper. Most of these concerns were clarified during the rebuttal, and the authors have committed to improving the exposition and clarifying assumptions in the final version, which should broaden the paper’s impact.

In sum, this is a theoretically strong and timely contribution that advances the understanding of optimal transport map estimation. I recommend acceptance.
Decision: Accept (poster)