Overcoming Spurious Solutions in Semi-Dual Neural Optimal Transport: A Smoothing Approach for Learning the Optimal Transport Plan
Authors: Jaemoo Choi
Jaewoong Choi
Dohyun Kwon
Venue: ICML 2025 poster (2025)
Keywords: Optimal Transport
Spurious solution
Semi-dual formulation
Image-to-Image translation
Primary area: deep_learning->generative_models_and_autoencoders
TL;DR: We overcome fake solutions in semi-dual Neural Optimal Transport and propose a smoothing approach for learning the Optimal Transport Plan.

Abstract: We address the convergence problem in learning the Optimal Transport (OT) map, where the OT Map refers to a map from one distribution to another while minimizing the transport cost. Semi-dual Neural OT, a widely used approach for learning OT Maps with neural networks, often generates spurious solutions that fail to transfer one distribution to another accurately. We identify a sufficient condition under which the max-min solution of Semi-dual Neural OT recovers the true OT Map. Moreover, to address cases when this sufficient condition is not satisfied, we propose a novel method, OTP, which learns both the OT Map and the Optimal Transport Plan, representing the optimal coupling between two distributions. Under sharp assumptions on the distributions, we prove that our model eliminates the spurious solution issue and correctly solves the OT problem. Our experiments show that the OTP model recovers the optimal transport map where existing methods fail and outperforms current OT-based models in image-to-image translation tasks. Notably, the OTP model can learn stochastic transport maps when deterministic OT Maps do not exist, such as one-to-many tasks like colorization.

Overcoming Spurious Solutions in Semi-Dual Neural Optimal Transport: A Smoothing Approach for Learning the Optimal Transport Plan
Jaemoo Choi
Jaewoong Choi
Dohyun Kwon *3
Abstract
the optimal transport map directly using neural networks. OT has found extensive applications in various machine learning domains by appropriately defining source and tar- get distributions, such as generative modeling (Rout et al., 2022; Choi et al., 2023; 2024b; Choi & Choi, 2024; Lip- man et al., 2023),
We address the convergence problem in learn- ing the Optimal Transport (OT) map, where the OT Map refers map from one distribution to another while minimizing the transport cost. Semi-dual Neural OT, a widely used approach for learning OT Maps with neural networks, often generates spurious solutions that fail to transfer one distribution to another accurately. We identify a sufficient condition under which the max-mi min solution of Semi-dual Neural OT recovers the true OT Map. Moreover, to address cases when this sufficient condition is not satisfied, we propose novel method, ОTP, which learns both the OT Map and the Optimal Transport Plan, representing the optimal coupling between two distributions. Under sharp assumptions on the distributions, we prove that our model eliminates the spurious so- lution issue and correctly solves the OT problem. Our experiments show that the OTP model recov- ers the optimal transport map where existing meth- ods fail and outperforms current OT-based models in image-to-ing to-inage translation tasks. Notably, the OTP model can learn stochastic transport maps when deterministic OT Maps do not exist, such as one-to-man many tasks like colorization.
toegetto-mmage translation (Korotin et al., 2023b; Fan et al., 2022), point cloud completion (Lee et al., 2024), and domain adaptation (Flamary et al., 2016). The OT framework is particularly advantageous for unpaired dis- tribution transport tasks, as it relies solely on a predefined cost function to map one distribution to another, eliminating the need for paired data.
Among various approaches, the minimax algorithm, derived from the semi-dual formulation, has been widely investi- gated (Fan et al., 2022; Rout et al., 2022: Choi et al., 2023; 2024c). Formally, Fan et al. (2022); Rout et al. (2022) estab- lished the adversarial algorithm by leveraging the following max- min problem:
sup inf L(V,)) where L(V,T) V T:X-Y
J x c(xx clx, T(x)) -(=))())-((()))) (((())d(()))) /. V(w)du(y).
(I)
Here, the probability measures and represent the source and the target distribution, respectively. The function V y - IR and T approximates a Kantorovich potential (Kan- torovich, 1948), and an optimal transport map. respectively, Throughout this paper, we call these approaches the Semi- dual Neural OT (SNOT).
1. Introduction
When the optimal potential V* and the transport map T exist, it is well-known that
Optimal Transport (OT) theory (Villani et al., 2009; Peyré et al., 2019) addresses the problem of finding the cost- optimal transport map that transforms one probability dis- tribution (source distribution) into another (target distribu- tion). Recently, there has been growing interest in learning
Is arg mig L(V", .T).
as shown in Rout et al. (2022); Fan et al. (2022). Thus, the pair (V*,T*) is the solution to this max-min problem. However, a critical challenge arises: not all solutions of Eq.1 I correspond to the optimal potential and transport map pair. In other words, even the optimal solution in the SNOT framework may not recover the correct optimal transport map. We refer to this challenge as the spurious solution problem.
'Equal contribution 'Georgia Institute of Technology "Sungkyunkwan University 'University of Seoul 'Korea Insti- tute for Advanced Study. Correspondence to: Dohyun Kwon <dhhddhhynnkwe @ gmai,.com., Jacwoong Choi <jaewoong- choi @ skku.edu>.
Proceedings of the 42nd International Conference Machine Learning, Vancouver, Canada, PMLR 267, 2025. Copyright 2025 by the author(s).
In this paper, we analyze this fundamental issue of the spuri- ous solution problem in existing SNOT frameworks. Specif-

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Overcoming Spurious Solutions in Semi-Dual Neural Optimal Transport
ically, we identify a sufficient condition on the source distri- bution js that prevents the spurious solution problem. The key condition is that the source distribution should not place positive mass on measurable sets with Hausdorff dimension S d (see Thm 3. To the best of our knowledge, our work offers the first theoretical analysis of the sufficient condition under which the max-min solution of the SNOT framework can correctly learn the OT Map. Prior works were limited to the saddle point solution (Fan et al., 2023) or addressed a specific form of different OT problem (weak OT) (Korotin et al., 2023b;a) (see Appendix D for the re- lated works on spurious solution issues). Additionally, we comprehensively explore various failure cases when this condition not satisfied.
maps. In particular, we focus on approaches that leverage the semi-dual formulation (Rout et al., 2022; Fan et al., 2022).
Optimal Transport The Optimal Transport (OT) problem investigates transport maps that connect the source distri- bution μ and the target distribution (Villani et al., 2009; Santambrogio, 2015). The optimal transport map (OT Map or Monge Map) is defined as the minimizer of a given cost function among all transport maps between μ and v. For- mally, Monge (1781) introduced the OT problem with a deterministic transport map T as follows:
T(u, inf Ux cr,, T(o))d(o)) TzN=v
(3)
Note that the condition T# v/ indicates that the transport map T transforms fa to v, where TH denotes the push- forward distribution of μ under T'. However, the Monge OT problem is non-convex, and the existence of minimizer, i.e., the optimal transport map T is not always guaranteed depending on the assumption of μ. and v (Sec. 3.2.2).
Building on this condition, we develop a novel algorithm that ensures the learning of an optimal transport plan. We refer to our model as the Optimal Transport Plan model (OTP). Our method involves smoothing the source distribu- tion Me, so that the Neural OT models recover the correct optimal transport plan. Then, we gradually modify pe back to the original μ leveraging the convergence property. Our extensive experiments show that our OTP model accurately learns the optimal transport plan. Moreover, our model out- performs various (entropic) Neural OT models in diverse maage-to-image translation tasks. Our contributions can be summarized as follows:
To address this existence issue, Kantorovich (1948) pro- posed the following convex formulation of the OT problem:
C(u, v) = =EM(IMM)
inf Urxey c(x,
y) c(x,y)d u)
(4)
We refer to the joint probability distribution π E I((, as the transport plan between and v. Unlike the Monge OT problem, the optimal transport plan (OT Plan) x² is guaran- teed to exist under mild assumptions on (X, ()) and (V, v) and the cost function c (Villani et al., 2009). Intuitively, while the Monge OT (Eq. 3) covers only the deterministic transport map y T(x), the Kantorovich OT problem (Eq. 4) can represent stochastic transport map via the conditional distribution π (yyl=: for each x ~ J.. When the optimal trans- port map T* exists, the optimal transport plan also reduces to this deterministic transport map, i.e., x = (Id x T*) # p.
Our work is the first to identify sufficient condition under which the max-min solution of existing SNOT recovers the true OT Map.
We demonstrate diverse failure cases that occur when this sufficient condition is not satisfied.
We propose new algorithm that guarantees the learn- ing of the optimal transport plan.
Our experiments show that our model successfully re- covers the correct OT Plan in failure cases where exist- ing models fail.
Semi-dual Neural OT The goal of neural optimal trans- port (Neural OT) models is to learn the OT Map between I and v using neural networks. The semi-dual formulation of the OT problem is widely leveraged for learning OT Maps (Rout et al., 2022; Fan et al., 2022; Choi et al., 2023; Makkuva et al., 2020).
Notations and Assumptions Let (X,,s) ()) and (V.v) be Polish spaces where X' and y are closures of connected open sets in Rdd We regard and y as the source and target distributions. Unless otherwise described, we con- sider X = y = Rd and the quadratic transport cost e: x + R. c(x, = alI - y/12 for a given pos- itive constant a. For measurable map T, T fi represents the pushforward distribution of μ.. [[(,, v) denotes the set of joint probability distributions on X' x y whose marginals are μ and v, respectively. Moreover, we denote W2(+,-) as the 2- Wasserstein distance of two distributions.
R.
The semi-dual formulation of the OT problem is given as follows: For a general cost function c(:-)) that is lower semicontinuous and bounded below, the Kantorovich OT problem (Eq. 4) has the following semi-dual form (Villani et al., 2009, Thm. 5.10), (Santambrogio, 2015, Prop. 1.11):
2. Background
U/x (=(o)dulo)++ V V(y)de(w) (5)
v) =) VeS.
In this section, we present a brief overview of Optimal Trans- port theory (Villami et al., 2009; Santambrogio, 2015), and neural network approaches for learning optimal transport
where Se denotes the collection of c-concave functions e y - TR and Ve denotes the e-transform of V, i.e
2

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Overcoming Spurious Solutions in Semi-Dual Neural Optimal Transport
(=(x) inf (c(a, V(y)]- yEy
(6)
Theorem 3.1. Let μ E Pa(X), P2()) and c(a
2 || yll. Assume that ye does not give mass to the measur- able sets of Hausdorff dimension at most d -1 dimension.
The SNOT approaches utilize this semi-dual form (Eq. 5) for learning the OT Map T (Rout al., 2022; Fan et al., 2022; Makkuva et al., 2020). This formulation leads to max-min optimization problem, similar to GANs (Goodfellow et al., 2020). Specifically, these models parametrize the transport map Tg X + y and the potential V as follows:
(1) Then, there exists a unique OT Map T* in (Eq. 3) and a (possibly non-unique) Kantorovich potential V E Se in (Eq. 5).
(2) For the Kantorovich potential V* E Se a solution of the minimization problem,
To x + arg min [e(z, v) V& (v)]
yYY
D= == arg min [c(z, y) V (v)],
(10)
vz(2) = c (x)) - & (Tv()))
(8)
yEy
is uniquely determined u.a.s., i.e. D2 = [122} for p-a.s a E X. In particular, a map a + yv E D2 is a unique OT Map T* p-a.s..
Note that Ty- Ta-parametrization (Eq. 7) implies that the transform V can be expressed with the transport map To and the potential Vg. as shown in Eq. 8. From this, the SNOT models derive the following optimization problem Lv
Here, Dz corresponds to the Tp-parametrization in the SNOT framework. Therefore, the uniqueness of Dr for V implies that Ta-parametrizztion is fully characterized. Thm. 3.1 shows that the assumption on Jı, not on 2, enough to eliminate the ambiguity in mapping each x to (T( and this mapping corresponds to the OT Map. Further- more, the extension of Thm. 3.1 to general cost function is discussed in Thm. A.6 from Appendix A.1.
sup inf V.ES. To:K-y
TT) where C(V,T):=
(9)
- V J c -
(p(x) V(y)du(y).
Intuitively, To and V& serve similar roles to the generator and the discriminator in GANs. However, the OT Map T3 is addi- tionally trained to minimize the transport cost c (xx (z)), while GANs focus solely on learning the target distribution Typ- = y (Arjovsky et al., 2017; Gulrajani et al., 2017).
Note that Thm. 3.1 is sufficient for addressing the spurious solution problem. For the sake of completeness, we also present a sufficient condition where the SNOT framework admits unique max-min solution that corresponds to the correct OT Map (Cor. 3.3). In this case, the additional assumptions on v is also required. Here, we used the fact that the absolutely continuous measures with respect to the Lesbesgue measure satisfy the condition in Thm. 3.1.
3. A Analytical Results for Semi-dual Neural OT
A critical limitation of existing SNOT approaches is that the max-min solution Vv,T") of Eq. 9 may include not only the desired OT Map but also other spurious solutions (Rout et al., 2022). Formally, if the OT Map I exists, the optimal potential V and the OT Map T' become a max- min solution (Eq. 2). However, not all max-min solutions correspond to the true optimal potential and transport map, i..., [(V([( }) & ((V),T))) In particular, even 1"1 # does not hold in general (see Fig. 1), which means that Tt is not valid transport map from js to as in (Eq. 3). It is worth noting that any max-min solution satisfying T'## # = v is guaranteed to be the OT map (Fan et al., 2023, Thm 3).
Theorem 3.2. Suppose y C Bd is closure of a bounded open set. If v has a positive density almost everywhere with respect to the Lebesgue measure on y, then there exists unique Kantorovich potential V* See up to constant.
Cor. 3.3 is derived by combining Thm. 3.1 with the unique- ness of the optimal potential V in Thm.
Corollary 3.3. Suppose y Rd is a closure of a bounded open set. Suppose μ E P2(X) and v P2()) are abso- lutely continuous distributions that have positive density functions on their domain. Then, the solution (V*,T*) of equation (9) is unique. In other words, V E So is unique up to constant, and T is a deterministic OT Map.
We first investigate sufficient conditions to prevent spurious solution issues (Sec. 3.1), and present a comprehensive fail- ure case analysis of the SNOT approach (Sec. 3.2). Based on this, later in Sec. 4, we propose a method for learning an accurate Neural OT model that avoids such spurious solutions.
3.2. Failure Cases When Our Condition Is Not Met
In Thm 3.1, it is crucial to assume that μ does not give mass to the measurable sets of Hausdorff dimension at most d dimension. Without this assumption, SNOT may fail even when the deterministic OT Map T* uniquely exists. Specifically, the failure cases discussed in this section refer to scenarios where (Vi,Tt) is a max-min solution of Eq. but does not correspond to the OT Map T* (Eq. 3), i.c., (Id, Tt) #f4 fails to represent the OT Plan x (Eq. 4).
3.1. Sufficient Conditions for Ensuring Convergence of Semi-dual Neural OT
We provide sufficient conditions on the source distribution and the target distribution / to ensure a unique minimizer for the To- parametriizaoo (Eq. 7). This enables the SNOT objective to accurately recover the optimal transport plan.
3

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Overcoming Spurious Solutions in Semi-Dual Neural Optimal Transport
x~p
y-u
T(x)
(a) Perpendicular
(b) Parallel
(c) One-to-Many
(d) Grid
Figure 1. Visualization of failure cases by comparing the Optimal Transport map (1st row) and the max-min solution (2nd row) of Semi-dual Neural OT in the failure cases. The source data x ~ μ., target data y ~ p, and generated data T T(x) are represented in Blue, Orange, and Red. The max-min solution fails recover the correct OT Map.
3.2.1. DISCREPANCY BETWEEN MAX-MIN SOLUTION AND THE DETERMINISTIC OT MAP
target distribution.
V'(v)= = Elllll y
We first focus on the Monge OT problem (Eq. 3), where the deterministic OT Map 7 exists. Specifically, we investigate source and target distribution pairs where T exists, but
(12)
otherwise.
the max-min solution T of the SNOT objective (Eq. 9) fails to recover this optimal solution. Here, we provide two examples, depending on the uniqueness of T
By substituting V into V. the inner problem of SNOT (Eq.
9) be expressed as follows.
inf T - (2) || T(x) - du(x)
/
(13)
2 = 2 - [ du(x)
Example 1. [When T exists but is not unique] First, we introduce case where multiple optimal solutions 7 exist for the Monge OT problem. Assume that the source and target distributions are uniformly supported on A = [-1, 1] x {0} and B- = {0} x [,,, respectively (Fig. (a)).. this case, any transport map satisfying TH = v becomes an optimal transport map for the quadratic cost function. Formally, note that for any transport map T, the following holds:
inf Tst. supp(T x
inf Tst. supp(T x
||01|12 ((.TT(t/du(())
f 11111/
Note that T(x) E B for μ-a.s. in otherwise, the value of Eq. 13 becomes infinite. Therefore, (x, (zx)( = 0 and V*(T(x)) = [x))) Then, Eq. 9 becomes as follows
( dl, ((),(()))) 1 [1())d
f |1rl/ + S [.blloPaae
2
3 (14)
1 ff (eFPl)) du(x) + 2 [1y1)du(l))=
/
1
(11)
3.
Since V attains the same value of LV..T, T* in Eq. 11, V* is the optimal potential. Furthermore, by comparing Eq. 13 with T3 S--parmmetrization (Eq. 7). we can casily observe that any measurable map Ta - can be a max-mi solution of SNOT. In other words, there is no constraint ensuring that To v. For example, To(x) (0, 0) for Vz X is also a valid max-min solution. This means that the existing SNOT models cannot learn the optimal transport map between these two distributions (Fig. 3(a)).
The first equality follows from Ty p = v and (x, T(x)) O for all æ because A 1 B. Since every transport map achieves the same transport cost, any transport map becomes an optimal transport map T**
Then, prove that T does not correspond to . Specif- ically, we show that V (y) E Se' is the Kantorovich po- tential (Eq. 5) and that It is not guaranteed to generate the
Example 2. [When unique T* exists] Here, we present another failure case when there is a unique optimal transport map T Assume that the source and target distributions
Note that (2) is c-concave function since 4|lv| (v) convex and lower semicontinuous (Thm 1.21, (Santambrogio. 2015)).

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Overcoming Spurious Solutions in Semi-Dual Neural Optimal Transport
are uniformly distributed over A = =[-1, x {0) and B = [-1,1] (1), respectively (Fig. 1(b)). In this setup, the
Algorithm 1 Training algorithm of OTP
Require: Source distribution j and the target distribution
unique 7 is given by:
vi OT Map network T' and potential network Vg: Total number of iteration K; Number of inner-loop iterations KK
(x) = (21,1) for = (11, 0) X.
(15)
Kr; Decreasing sequence of noise levels {Ex)k=r
Thus, T(,, 2) = 2. Similar to Example 1, we show that ('(v) 2 -|1vall2 E Se with y = (y1, y2) is the optimal
1: for k = 0, 2.........4 K do
2: Sample a batch X ~ μ., y ~ V, 2 (O,II).
Kantorovich potential and analyze the max-min solution
+ x VCk2 or t / C2 + Vek2.
of Eq. 9. For this V*, the inner problem of SNOT can be
4: Update & to maximize Ls VV (To + Volu).
computed as follows:
5:
for j = 0, 1,...... KT do
6: Sample a batch x ~ μ, N(O,I).
inf / 11 -(=)) 1/, _ T x 2
|leell du(y)= (16) 2
7:
T レ x + V or I < v Exat + vEzz.
8:
Lg = c(t, To(o)) - V (To(E)) + Voly).
Because Eq. 16 achieves the same value as T(u, v), V is the optimal potential. For this V*, any transport map T((21, (21. a) for any X' becomes a max-min solution of the SNOT. In this case, the existing approach fails to even characterize the correct support of the target distribution V.
9: Update to minimize Lg.
10: end for
R for each (æ1, 222) E
11: end for
Stochastic Parametrization of OT Map In practice, a stochastic parametrization of Tg (2, 2) is often adopted to improve performance in the SNOT models (Korotin et al., 2023b; Choi et al., 2023). This stochastic parametriza- tion (1) introduces an additional noise variable = ~ N (O, I): To 2) E arg min(((z, v) - V* wEy
3.2.2. DISCREPANCY BETWEEN A MAX-MIN SOLUTION AND THE STOCHASTIIC OT MAP
The standard SNOT parametrizes the transport map with a deterministic function To (Eq. 7). When no deterministic OT Map T exists but only an OT Plan n² exists (Eq. 4), it is clear that the SNOT cannot accurately represent the stochastic OT Map (OT Plan).
(19)
(V(, a.s.. As a result, each is transported to multiple T(x, 2) values depending z. We point out that even a stochastic parametrization, such as To 2) with a noise variable 2 ~ N I), cannot address this limitation. For the formal statement, see Appendix B.
Example 3. [When only x" exists] Suppose the source and target distributions are uniform on A= = [0, 1] x {0} and B [0, 1] [1) 1] x {-1}, respectively (Fig. ((c)). In this case, it is clear that the OT Plan x² is given as follows:
Proposition 3.4 (Informal). Assume that the stochastic parametrization of To(o, is ideally trained as in equa- tion (19) for (W,NN)a.ss Dr in Eq. 10 may not uniquely determined and To [2( (2,2) may contain spurious solutions.
1 2 () + +2 ().))-1) where x ==(11, (17)
*" (1]2x)
The OT Plan n² (yylzr moves each x vertically either up or down with probability 1. without incurring additional cost from horizontal movement. Then, we show that (y) ||aall E S. with = (y1, 32) is the optimal potential. and The (V*)² and V* can be computed for μ and y as follows:
4. Method
In Sec. 3.2, we analyzed the sufficient condition to pre- vent failures in the existing SNOT framework. Building on this analysis, we propose a novel method for learning the OT Plan, called the Optimal Transport Plan (OTP) model, which is effective even when the conditions are not satisfied.
(V)))())=in(((()), wE>
V (y)) = inf ||- vnl -0 (18) y1 2
Vv(()= = } Illva 12 3 for Vy E y. By comparing the C for the optimal transport plan (Eq. 17) and the semi-dual form S for V*, we can easily verify that V* is the optimal Kantorovich potential.
4.1. Proposed Method
Our goal is to learn the OT Plan x= (Eq. 4) between the source distribution ş and the target distribution v. Note that the sufficient condition in Thm. 3.1 is inherent prop- erty of μ. When this condition is not satisfied, the existence of OT Map T is not guaranteed, and only x² exists. In this regard, our OTP model serves as a natural generalization of existing SNOT models.
Then, from Eq. 18, possible solutions for the 7 T-parametrization (Eq. 7) in the SNOT for V*, (x) == (I1, 1) and [2(x) := (21,-1). for x = (21, E X. These two candidates Ti, T2 only characterize a subset of the support of x ())1z)) Therefore, our deterministic To cannot learn the stochastic n (yylar).
that Tj and T2 are the two
Our method consists of two steps: First, we introduce a
5

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Overcoming Spurious Solutions in Semi-Dual Neural Optimal Transport
smoothed version of the source distribution pe-- 14 is con- structed to satisfy the sufficient conditions from Thm. 3.1. As result, the SNOT between μl and y recovers the correct OT Plan मे between them. Second, we gradually adjust pe back to the original source measure j.. This approach allows our method to learn the correct optimal transport plan, even in cases where the existing SNOT framework fails.
x (x)
2
V
2
Y2
Y3
Figure 2. Example of a stochastic transport map (OT Plan) task. e.g. -NNNST colorization.
ОTP Model As practical implementation of the high- level scheme described above, we propose a new method for learning the OT Plan πt" from ja to called Optimal Trans- port Plan (OTP) model. This method is based on Thm. 3.1 and Thm. 4.1, which require the following two conditions on the smoothed measure Pe::
mH. from our OTP model with either convolution above also weakly converges to n²,, along subsequence.
In this way, we can learn the optimal transport plan x² between μ and without falling into the spurious solu- tions of the max-min learning objective (Eq. 9). While the convergence theorem only guarantees convergence up to a subsequence (Thm. 4.1), our method exhibits decent conver- gence to ' in practice (Sec. 5). Specifically, our training algorithm progressively finetunes the transport network To and the potential network Va by adjusting the smoothing level. As result, the subsequence convergence does not pose any issues.
(cl) Me does not give mass to the measurable sets of Haus- dorff dimension at most --1 1 dimension (Thm. 3.1).
(c2) Mes weakly converges to j is - Do (Thm. 4.1).
For simplicity, we consider the absolute continuity condi- tion on Hl. as we did in Cor. 3.3. Motivated by diffusion models (Ho et al., 2020; Song et al., 2021), we consider two options for the smoothing distribution: (1) Gaussian con- volution Mec. = μ (0, e and (2) Variance-preseering convolution ue, = (v1 eyId) # u * (N ex./) with a predefined noise level < > 0. For noise-level scheduling, we follow Song et al. (2021). Note that both of these smooth- ing distributions satisfy conditions (cl) and (c2). Specifi- cally, for Gaussian convolution, for any μ E (2 (c1) Me is absolutely continuous with respect to the Lebesgue measure and has positive density on Rd. Moreover, (c2)
Importance of OT Plan in Neural OT Our OTP model is for learning the OT Plan, i.e., the stochastic transport map. In fact, OT Plans are not only a theoretical generalization of deterministic OT Maps, but are also inherently more suitable for various real-world machine learning applications. For instance, in image-to-image translation tasks, stochastic OT Plans can effectively model the diversity of plausible outputs. Similarly, in inverse problems such as colorization or image inpainting, stochastic OT Plans are also highly desirable because these tasks inherently involve multiple possible solutions. In See 5, our experiments show that our OTP model is effective in handling the stochastic transport map application in the MNIST-O-CMNIST image translation task (Fig. 2).
as 6 + O, Me - μ.. A similar argument works for the Variance- Variance-preserving convolution case. Then, we apply the SNOT framework to the smoothed measure Ma, and the target measure v. The learning objective is given as follows:
1/5 inf (x,,To())) -Vo (To (x)) (x)
CV
sup Vo
x To
+ / Voly)du(w) (20)
Algorithm We present our training algorithm for OTP (Algorithm 1). For each eg. we alternatively update the ad- versarial learning objective Lt, between the potential function Vo and the transport map To. similar to the GAN framework (Goodfellow et al., 2020). Note that the smooth source measure Mrs, corresponds the probability distribu- tion of the sum of the clean source measure μ (or the scaled source measure (vT- en. ld) # jd) and the Gaussian noise (0, T). Therefore, we can easily sample Fe, ~ Mec as follows (Line 3):
Then, we gradually decrease the noise level {Eu (E throughout training. The two conditions on Meg, i.e. (cl) and (c2), offer the following guarantees. First, for each noise level cx. the max-min solution of CT. recovers the op- timal transport map T and the Kantorovich potential Vi. Second, as k + oo, i.e. Ek '> O, the optimal transport plan #t = (Id, #e,, converges (up to subsequence) n².
Thm. 4.1 follows from combining Thm. 3.1 and Villani et al. (2009). See Appendix C for proof.
Tea = = VeNz ~ j + N((, or Tee v1 - ExI-
Theorem 4.1. Let (1ro) JveN be sequence of absolutely continuous probability measures, and Tī be the OT map from Pez to μ. If Pe, weakly converges to μ as k > oc, then पे = (Id, TT))e, weakly converges to the OT plan n between I and v, along a subsequence. Consequently,
(21)
where x ~ j and : ~~ No I). In practice, decreasing the noise level until a small positive constant Cmin > 0 0rovided better performance and training stability, compared to reduc- ing the noise level to exactly zero. For a fair comparison, we
6

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Overcoming Spurious Solutions in Semi-Dual Neural Optimal Transport
x-H
y-u
8~He
T(x)
(a) Perpendicular
(b) Parallel
(c) One-to-Many
(d) Grid
Figure 3. Qualitative comparison between OTM (1st row) and our model (2nd row) on failure cases in Sec 3.2. The noised source sample ë in Alg is denoted Green. While OTM falls into spurious solutions and fails to generate the target distribution correctly, our OTP model successfully learns the OT Plan.
compared the composition of the noising and transport map H Demin + (Temman) ), with the ground -truh optimal transport map x + T* (x) in the experiments (Sec. 5).
Table I. Quantitative comparison numerical accuracy on syn- thetic datascts. Each model is evaluated using two metries: trans- port cost error Denee (4) and target distribution error D Diarget (4).
Perpendicular
Oue-to-Many
Dimension
Model
Donat
Diargtt
Dosst
Diorget
5. Experiments
OTM
0.038
0.008
0.069
0.100
2
OTM-s
0.007
0.018
0.350
0.032
In this section, we evaluate our ОТР model from the fol- lowing perspectives. In Sec. 5.1, we evaluate whether OTP successfully learns the optimal transport plan. In Sec. 5.2, we demonstrate the scalability of OTP by assessing on the image-to-imet to-gge translation task. For implementation details of experiments, please refer to Appendix E.
Ours
0.019
0.007
0.002
0.110
OTM
0.043
0.039
0.100
0.090
OTM-s
0.033
0.065
0.038
0.089
0.009
0.033
0.094
4.97
73.23
d 16
OTM-s
0.061
4.85
97.49
99.57
Ours
0.058
0.59
0.06
0.65
OTM
2.13
19.37
21.92
32.94
5.1. OT Plan Evaluation on Failure Cases
d 64
OTM-s
2.74
18.79
0.20
12.21
Ours
0.97
10.09
0.14
9.98
First, assess whether model accurately learns the optimal transport plan n² between the source dis- tribution and the target distribution / in failure cases outlined in See 3.2. The evaluation is conducted in two settings: (I) Qualitative comparison in 2D cases and (2) Quantitative comparison high-dimensional cases. In each setting, our OTP model is compared against the existing SNOT framework (Eq. 9).
OTM
10.98
84,91
0.04
62.02
d= 256
OTM-s
16.45
81.68
0.25
61.87
Ours
5.05
63,36
0.33
61.27
transported to the target measure v. In Fig. 3, the vanilla OTM fails to learn the correct optimal transport plan in three cases except for the Parallel case. OTM fails to cover the target measure in the Perpendieular and Multi- perpendicular cases. In the One-to-Mane case, OTM does not learn the correct T*, i.c., the vertical transport.
Qualitative Comparison In Sec. 3.2, presented var- ious examples where the existing SNOT framework may fail to learn the OT Map (or Plan). Here, we demonstrate that the existing approaches indeed encounter these failures, while OTP successfully learns the correct OT Map. As a baseline, we compare our method against the standard OTM with deterministic transport map, i.c., (z)).
On the other hand, as we can see from a comparison with Fig. 1, our OTP successfully learns the optimal transport plan π². In particular, in the One-to-Many example, our model successfully recovers the correct stochastic transport map n (y|z) (z) by utilizing the initial noise guidance to either the upper or lower mode of the target distribution.
Fig. 3 presents qualitative results four datasets: Perpen- dicular (Ex. 1), Parallel (Ex.2), One-to-Many (Ex.3), and Grid. The first row shows the vanilla OTM results and the second row exhibit our OTP results. Note that our OTP de- creases the noise level until o = Emán > 0 (Sec. 4). Hence, the noised source samples Alg (Green in Fig 3) are
Quantitative Comparison to Ground-truth We evalu- ate the numerical accuracy of our ОTP, SNOT with deter- ministic generator (OTM (Rout et al., 2022)), and SNOT with stochastic generator (OTM-s, Eq. 19), by compar- ing them to the closed-form ground-trune solutions. Here, we measure two metrics: the transport cost error Deast

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Overcoming Spurious Solutions in Semi-Dual Neural Optimal Transport
8
8
8
8
8
8
8
Table 2. Image translation benchmark results compared to isting Neural (Entropic) OT models. indicates the results conducted by ourselves. DSBM scores are taken from (Gushchin al., 2024; De Bortoli et al., 2024a).
6
6
6
6
6
6
6
5
5
5
5
5
5
5
Data
Model
FID (()
1.PIPS (U)
(a) OTM-s (FID=
0336)
NOT (Koretin 2023b)
11.96
Male-to-Fomale (6-x644
OTM (Pan et et,,, 2022)
2022))
6.42
0.16
DIOTM' (Chui et al., 2024a)
4,48
0.20
9
9
9
9
9
9
9
OTP (Ours)
4.75
0.20
DSBM (Shi al., 2024)
20-
0.59
1
1
1
1
1
1
Wild-to-Cat (64x64)
OTMI (Fan et al., 2022)
12.42
0.47
DIOTM (Choi al., 2024a)
10.72
0.45
2
2
2
2
2
2.
2
OTP (Ours)
9.66
0.52
DSBM (Shi al., 2024)
37.8
0.25
ASBM (Gushchin et al., 2024)
16.08
(b) Ours (FID=3.18, LPIPS=0.32)
Holee-to-Female (128x128)
OTM! (Fan et al., 2022)
7.55
0.21
Figure 4. Experimental results on a stochastic transport map application, i.e., SEUIST-o-CMNISS translation.
DIOTM' (Chai el., 2024a)
7.40
0.25
OTP (Ours)
6,38
0.27
[W3 (H, [T))() - =1 dp())) and the target distribu- tion error Diorget = W (Tog (t, v). Deast assesses whether the model achieves the optimal transport cost, while Dtarget measures how accurately the model generates the target dis- tribution. Both models are tested on two synthetic datasets, Perpendicular and One-to-many (Fig. 3), with generalized dimensions of d E {2, 4, 16, 64, 256} (See Appendix E. for dataset details.).
consists of three colored variations (Red, Green, and Blue) for each grayscale image from the MNIST dataset (Fig. 2). Consequently, the desired OP plan should stochastically map each grayscale digit image to colored digit image of the same digit type (Fig. 2).
Fig. 4 illustrates the experimental results. Here, we intro- duced a stochastic generator to OTM (OTM-s) to provide the capacity to learn a stochastic transport map. However, OTM exhibited the mode collapse problem, transporting all grayscale images to blue-colored images. On the other hand, our OTP successfully learns the optimal transport plan '², achieving stochastic mapping to Red, Green, and Blue colors. This phenomenon is also observed in the quantitative metrics. Our model significantly outperforms OTM in FID score (4) (3.18 vs. 62.4) and archives better score in LPIPS (L) (0.32 vs. 0.36).
Tab. presents the quantitative results on the accuracy of the learned optimal transport plan Ty. Our OTP consistently achieves comparable or superior performance compared to both OTM and OTM-s across all metrics, particularly in sige-ddimnsional settings. Note that these experimental results confirm the challenges of the existing SNOT frame- work in accurately recovering the target distributions, as discussed in Sec. 3.2.1 and 3.2.2. Specifically, as shown in Dtarget, OTM and OTM-s models exhibit significantly larger target distribution errors in higher dimensions.
Image-to-Image Translation We assess model on three image-to-image translation benchmarks: Male-to- Female (Liu et al., 2015) (64 x 64, 128 x 128) and Wild- to-Cat (Choi et al., 2020) (64 x 64). For comparison, we include several OT models (NOT, OTM, and DIOTM) and Entropic OT models (DSBM and ASBM).
5.2. Neural OT Evaluation Unpaired Image-to-Inen Image Translation Tasks
In this section, we evaluate our model on the unpaired image-to-image translation task. The image-to-image translation is one of the most widely used machine learn- ing tasks in Neural OT models. The optimal transport map T* (or plan "T" can be understood as a generator of target distributions, mapping an input x to a similar counterpart y by minimizing the transport cost c(2, v). This mapping can be deterministic (y= T (z)) stochastic (y ~ nT" (-1z)). Therefore, the optimal transport map can naturally serve as a model for unpaired image-to-image translation.
Tab. 2 presents the quantitative scores for the image-to- image translation tasks (See Appendix F.4 for qualitative examples). We adopted the FID (Heusel et al., 2017) and LPIPS (Zhang et al., 201 8) scores for quantitative evaluation. Note that these FID and LPIPS scores serve similar roles as D Denst and Dtarget in Sec 5. respectively. Our primary evaluation metric is the FID score because it measures how well the translated images align with the target semantics. As shown in Tab. 2, our model demonstrates state-of-un the- art FID scores and competitive I.PIS scores compared to existing (entropic) Neural OT models. Specifically, in the Male-to- Male-to-Feent -oeeaae (128x x 1 128) task, our OTP model achieves a FID score of 6.38, outperforming the SNOT model (OTM), the other Neural OT model (DIOTM), and entropic OT
MNISSS First, we demonstrate that our OTP model can learn stochastic transport mappings at the image scale. Specifically, we test our model on the MNIST-to- CMNIST translation task (See Appendix E.2 for dataset details). this task our Colored MNIST (CMNIST) dataset
8

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Overcoming Spurious Solutions in Semi-Dual Neural Optimal Transport
models (DSBM and ASBM). Although OTM achieves lower but comparable LPIPS score (0.21), its significantly worse FID score (7.55) suggests large target semantic emoor. Thus, we prioritize FID as our primary metric.
erative adversarial networks. In International conference machine learning, pp- 214-223. PMLR, 2017.
Choi, J. and Choi, J. Scalable simulation-free en- tropic unbalanced optimal transport. arXiv preprint arXiv:2410.02656, 2024.
6. Conclusion
Choi, J., Choi, J., and Kang. M. Generative modeling through the semi-dual formulation of unbalanced opti- mal transport. In Thirty- Conference on Neural Information Processing Systems, 2023.
In this paper. we provided the first theoretical analysis of the sufficient condition that prevents the spurious solution issue in Semi-dual Neural OT. Based on this analysis, we proposed our OTP model for learning both the OT Map and OT Plan, even when this sufficient condition is not satis- fied. Our experiments demonstrated that OTP successfully recovers the correct OT Plan when existing models fail and achieves state-of-the-ee performance in unpaired image-to- image translation. Our primary contribution is improving Neural OT frameworks by addressing their fundamental limitations, i.e., failing to recover the correct OT Map even with the ideal max-min min solution. One limitation of our work is that our convergence theorem holds up to a subsequence (Thm. 4.1). Nevertheless, in practice, our gradual training scheme (Alg 1) did not show any convergence issues. Also, our analysis provides sufficient condition, rather than necessary and sufficient one (Thm. 3.1), leaving room for further refinement in understanding the exact conditions under which spurious solutions occur.
Choi, J., Chen, Y., and Choi, J. Improving neural optimal transport via displacement interpolation. arXiv preprint arXiv:2410.03783, 2024a
Choi, J., Choi, J., and Kang, M. Scalable Wasserstein gra- dient flow for generative modeling through unbalanced optimal transport. In Proceedings of the 41st Interna- tional Conference on Machine Learning, volume 235 of Proceedings of Machine Learning Research. PMLR, 21-27 Jul 2024b.
Choi, J., Choi, J., and Kang, M. Analyzing and improv- ing optimal optimal-transpore ttt.--rrrnnttaaaee based adversarial networks. In The Twelfth International Conference on Learning Represen- tations, 2024c.
Acknowledgements
Choi, Y., Uh Y., Yoo, J., and Ha, J.-W. Stargan v2: Diverse image synthesis for multiple domains. In Proceedings of the IEFF/CVF conference on computer vision and pattern recognition, pp. 8188-8197, 2020.
JW was partially supported by the National Research Foun- dation of Korea(NRF) grant funded by the Korea govern- ment(MSIT) RKS-2024000349448 DK was partially sup- ported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. RS- 2023-00252516 and No. 25--2000000)00003,, the POSCO Science Fellowship of POSCO Park Foundation, and the Korea Institute for Advanced Study. JM is supported by National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) RCS-202400041066
De Bortoli, V., Korshunova, I., Mnih, A., and Doucet, A. Schr\ " odinger bridge flow for unpaired data translation. arXiv preprint S..iv:2449.094444 2024a.
De Bortoli, V., Korshunova, I., Mnih, A., and Doucet, A. Schrodinger bridge flow for unpaired data translation. Advances in Neural Information Processing Systems, 37: 103384-1000 103441, 2024b.
Impact Statement
Fan, J., Liu, S., Ma, S., Chen, Y., and Zhou, H.-M. Scal- able computation of monge maps with general costs. In ICLR Workshop on Deep Generative Models for Highly Structured Data, 2022.
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none of which we feel must be specifically highlighted here.
Fan, J., Liu, S., Ma, S., Zhou, H.-M., and Chen, Y. Neural monge map estimation and its applications. Transac- tions on Machine Learning Research, 2023. ISSN 2835- 8856. URL ht https:://// reopenreview.nee forum? id 1d=2m211@ssj3 Featured Certification.
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Overcoming Spurious Solutions in Semi-Dual Neural Optimal Transport
A. Proofs
A.I. A generalized version of Theorem 3.1
In this section, we introduce theorems Villani et al. (2009) and prove a generalized version Thm. 3.1 (See Thm. A.6). We believe that rediscovering and summarizing this generalized version in the machine learning literature will help point out other possible directions for developing algorithms to avoid spurious solutions.
To begin with, specify the assumptions the domain:
AO.
We consider X M,, where M is a smooth complete connected Riemannian manifold and X' is a closed subset of M, with dim(@x) z d - 1/ is arbitrary Polish space.
The following two lemmas, Lemma A.I and Lemma A.5, play pivotal role in the proof of Thm. 3.1. Furthermore, we introduce generalized version of the theorem.
We begin with Lemma 1. which establishes the existence of a Kantorovich potential and characterizes its optimality. Lemma A.1 (Theorem 5.10 in (Villani et al., 2009)). Let (X, and (V, v) be two Polish probability spaces and let e: X x y + R be a lower semi-contiuous cost function such that is lower bounded. Suppose the optimal cost V(H,v) == infxem(ue)) J edw is finite. Then,
VES. ( V + J ((Codduu))
C(u,v) = max
(22)
In other words, there exists e-concave function V that makes strong duality (Eq. 22) satisfy. Moreover, for c-cyclically monotone set C X x for any π E [T v) and c-concave function 1b
is optimal < xTT)= =1 * is optimal
r C Sas..
(23)
Assumptions on the Cost Function following assumptions:
Throughout the discussion, we assume that the cost function c(s)) satisfies the
A1. c: M x y - RR is a continuous cost function that is bounded below.
A2. e is superdifferentiable everywhere.
A3. V -2,(z,-) :) is injective where defined.
A4. e satisfies either the SC condition (see Definition A.2) or the Ho condition (see Definition A.3).
Note that Assumption A4 is required to ensure the existence of nonempty c-subdifferential, i.e., |0,b| > 1. We now provide the definitions of SC and H..
Definition A.2 (SC). We call the cost function e: X' x y IR is SC if c(x, y) is locally semi-concave as a funtion of x; uniformly in y, i.c.
3M E R s.t. c(z, v) M ||+|| is concave for all y E V.
(24)
Here, SC stands for semi-concavity.
Definition A.3 (H&)). We say the cost function c: X x y + IR satisfies the condition Ho if the cost satisfies the following two conditions:
H1. For any & and for any measurable set whose tangent space T.S is not contained in a half-space, there exists a finite collection of elements Æ1,..., F4 E and small ball containing x, such that for any y outside of a compact set,
inf c(w. v)
inf 15iSk
c(z,, v).
H2. For any and any neighborhood U of x, there exists a small ball B containing x such that
lim sup inf [o(2, y) c(w, v)] = -00. y weB =EU
12

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Overcoming Spurious Solutions in Semi-Dual Neural Optimal Transport
Remark A.4. Both HI and H2 deseribe the behavior of the cost function c(x; y) when y - co. Consequently, these two conditions automatically satisfied when y the support of μu is compact set.
Next, we state and prove generalized version of our main theorem, Theorem 3.1. We start with the following lemma: Lemma A.5 (Thm. 10.28 and Thm. 10.42 in (Villani al., 2009)). Assume A0-A4. Let μ E P(X) and P(Y) such that
A5. The optimal cost C(u, p) == inf =nf(a Jaxy cxx, v)dr( v) is finite;
A6. Any c-convex functionn is differentiable μ-almost surely (μ-aa.s)) on its domain of c-subdifferentiability.
Then, there exists a unique deterministic optimal coupling E MQ v) in law. Moreover, there exists c-concave function a such that
Vac(a,)) Vu((2)) 0, for a-llmost surely.
(25)
Ve((2) O, p-a.s. Moreover, ifT X + In other words, the Monge map T exists, and satisfies Vzc(x, (* (x)) satisfies
y
E V.d(x,)) Vib(x) 0}:
(26)
then T' is the unique Monge map (in law).
We now state and prove the generalized version of Thm. 3.1. Then, we discuss how this generalized theorem reduces to Thm. 3.1 under the standard quadratic cost setting.
Theorem A.6. Assume A0-A6.
(1) Then, there exists unique OT Map T in (Eq. 3) and (possibly non-unique) Kantorovich potential V E S. in (Eg.
5).
(2) For the Kantorovich potential V E Sc, solution of the minimization problem,
T(x) D, == arg inf (c(x,))~V'(g)) yEy
(27)
is uniquely determined p-a.s. i.e. Dr = }or p-a.s x X. In particular, a map T X + y given by x + yr Dr is unique OT Map T p.a.s..
Proof. From Lemma A.1 and A.5, there exists unique Monge map T (in law) and a Kantorovich potential V* Se. Let vb (V*)c, Because ((V*),, ye, V*) is pair of Kantorovich potentials, Lemma A. implies:
T (x) E
== ty E y (V*)==(y) = c(e, y) (V*)
(28)
Note that x = Vc is a c-concave, hence, by assumption A6, it is differentiable p-a.s.. For every E (z))). differentiating both sides of the equation (V*) = c(z, y) - (z)) with respect to x yields:
0 = Vac(z, v) v(V*)((z) vcc(,,)) -xb(()
(29)
Therefore, the condition in Lemma A.5 (i.c., Eq. 25) is satisfied. From the last statement of Lemma A.5, the e-subdifferential
(x)(x) is unique (i.e., (z)) = 1) for all x E X Z, where Z denotes the set of points where a is non non-differentiable. Because vb is differentiable p-almost surely, Z has zero measure with respect to Therefore, the mapping x + (z)))((
uniquely defined y-almost surely.
Now, consider:
yEY inf (c(z, () -
(V*) for yo E (x))
(30)
where the first equality follows from the definition c-transform and the second equality comes from Eq. 28. Since V assumed be c-concave, we have (V*jer V*. Therefore, the minimizer in
T(x) e arg min (c(æ, y) - V*(})} yey
(31)
unique jlllmost surely, due to the uniqueness of (v)))(() By Lemma A.5, this uniquely defined T a Monge map.
13

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Overcoming Spurious Solutions in Semi-Dual Neural Optimal Transport
A.2. Proofs of Thm. 3.1 and Thm. 3.2
In this section, we prove Thm. 3.1 by verifying that all the assumptions A0-A6 are satisfied. Moreover, Thm 3.1 is specific realization of Theorem A.6.
Proof of Thm. 3.1. Step 1. Check the basic assumptions: Since X' y are closures of connected opensets, trivially satisfies the domain assumption (A0). Assumption A6 is the assumption of the theorem. Since μ and have finite second moments, the optimal transport cost is finite (A5). Furthermore, since c is quadratic cost, it is continuous, bounded below (AI), superdifferentiable (A2), and V V,c(z, -) is injective for every E X' (A3). Trivially, e is semi-concave with respect to (A4).
Step 2. Realization of Assumption A6: Assumption A6 can be satisfied in various ways. For completeness, we refer to the theoretical results outlined in Remark 10.33 of Villani et al. (2009), which provide the following sufficient conditions for Assumption A6:
R1. e is Lipschitz on X' x y and y is absolutely continuous.
R2. is locally Lipschitz, μr, are compactly supported and μ is absolutely continuous.
R3. css locally semi-concan and satisfies the Ho condition, and does not give mass to to sets of Hausdorff dimension at most d 1.
In case, the cost function is quadratic, and therefore hence it is semi-concave. Morcover, discussed in Step 2, the quadratic cost satisfies the Hox condition. Finally, our assumption that μ does not assign positive mass to sets of Hausdorff' dimension at most d - 1 ensures that condition R3 is satisfied. Thus, all assumptions required by Thm. A.6 are met under the conditions of Thm. 3.1. Therefore, the set Dr a singleton j-almost surely.
Other Realization of the Cost Function Thm. A.6 is formulated in the general formulation with respect to the cost function. Thus, our theorem can be applied to various cost functions other than the standard quadratic cost. In this paragraph, we present several alternative realizations of cost functions that satisfy the required assumptions. We assume x,Y M gd.
R4. A1, A2 and A3 are satisfied when c is C² strictly convex function.
R5. On top of R4, e also has bounded Hessian and μ does not assign mass to sets of Hausdorff dimension at most d 1, then e SC, thus satisfies A4. Moreover, by Example 10.35 Villani et al. (2009), this setting also satisfies A6.
R6. Suppose v(x,v) = 1t( - y) where h Rd + R is convex, superlinear, strictly increasing with respect to |æ: y/. and radially symmetric function. Then, satisfies the Ho condition, hence satisfies A4 (See Example 10-19-10.21 in Villani et al. (2009)).
R7. R6 combined with R3 and R4, all the assumptions satisfied if: (i) c(x, v) h(x y) is C strictly convex and locally semi-coneave function where (ii) h is radially symmetric and strictly increasing function with respect to |r yl. (iii) does not assign mass to sets of Hausdorff dimension at most d 1.
R8. If the domains X' and y are both compact, then A4 is automatically satisfied by the definition of H Thus, combining R2 and R4, i.c. (i) X' and y are compact, (ii) y absolutely continuous, and (fii) is C* strictly convex function, then all the assumptions satisfied.
In summary, the following proposition outlines various conditions that ensure all assumptions are met: Proposition A.7. Let X,Y C M Rd satisfy Assumption AO. Suppose that any of the following conditions hold:
I. is C strictly convex with bounded Hessian. Moreover, i does not assign mass to sets of Hausdorff dimension at most d 1:
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Overcoming Spurious Solutions in Semi-Dual Neural Optimal Transport
2. X and y compact, p. is absolutely continuous, and is C+ strictly convex function;
3. c(x, y) = h(z - y) is a C' strictly convex and locally semi-concave function, where a radially symmetric and strictly increasing function with respect to
|| - yll, and p does not charge sets of dimension d.1.
Then, all assumptions (AI-A6) in Theorem A.6 are satisfied.
We emphasize that extending this analysis to Polish spaces (X, μ)) and ()/, v). with cost functions e: X x y + R satisfying the above conditions, presents a promising direction for future research and offers significant potential for theoretical advancement.
Proof of Theorem 3.2 Thm. 3.2 can be directly proved by leveraging Cor. 4 in Staudt et al. (2022):
Proof of Thm. 3.2. Let K C X be the compact set. Since y is also compact, there exists such that K2Y BR(O).
Then, for
[e(x, v1) - c(x, 12)ll S x (y1 - -22) + 1 -1v22 & Rlyy - -2ll + + 22 - y2ll = 2R - -22l. (32)
Thus, by using Thm. 5 of Staudt et (2022), our assumptions satisfy the conditions of Corollary Staudt et al. (2022). Therefore, there exists unique c-concave Kantorovich potential.
Note that Corollary 3.3 can be easily proved by simply combining Thm.s 3.1 and 3.2.
B. Non-convergence of Stochastic Parametrization in Semi-dual Neural OT
In this section, we further elaborate Prop. 3.4. Let xt (yl2z) denote the conditional distribution induced by To(x, :) (0) + where
Ta(x, z) E arg min(c( (v)),, (x,,2) ~ N(O, aaass.
(33)
yEy
Since the subdifferential of the c-transform of V is defined as
(V())) = (u E ev: (x) = c(x, v) V*)) = arg min (*))),
(34)
ye
Eq. 33 can be rewritten follows:
- (8.((V)))))) x) 1, p-a.s.
न P(X x satisfies तt (0.V")
(35)
Here, V* S. represents the optimal potential and 0.V* {x., 3) X' x (V*) V(y) c(x,1) 4) In summary, the stochastic map optimization problem in Eq. 33 is equivalent finding the joint distribution nt P(x x ))) that satisfies x (0.V** 1.
However, in general, this condition x =',VV*) = does not guarantee that π is the optimal transport plan for the Kan- torovich's problem (Eq. 4). As discussed in Lemma A.1, under the mild assumptions on the cost function x is optimal and only if (I') = for c-cyelic monotone 1', Since I C d.V*, we can say that one of the solution & is the optimal transport plan, however, the converse may not satisfy (See failure cases Sec. 3.2.2). Additionally, if a. (V))((zz) is unique J-almost surely, then there is unique (in law) deterministic optimal coupling of (p, v) (See Thm. 5.30 in Villani et al. (2009)). The discussion above can be summarized as follows:
Proposition B.1 (Formal). Suppose the assumptions of Lemma A. hold. Let V E Se be the Kantorovich potential. For n E P(x x Y), the condition m! (a.V*) = 1 does not imply that n/ is optimal transport plan.
C. Convergence of OТP
We present the convergence theorem for the optimal transport plan from Villani et al. (2009). Thm. 4.1 is a direct consequence of Thm. 3.1 and Thm. C.I.
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Overcoming Spurious Solutions in Semi-Dual Neural Optimal Transport
Theorem C.I (Villani et al. (2009), Thm. 5.20). Let e 2 0 be real-valued, continuous, and lower-bounded cost function. Consider a sequence of continuous cost functions {ck kee that uniformly converges Let { and (ux KeN be sequences of probability measures that weakly converge to μ and v, respectively. For each k. let Te be an optimal transport plan between Pk and Vk. If [ < oc, then, up to the extraction of a subsequence, Th converges weakly to some e-cyclically monotone transport plan x²' E [[, v). Moreover, if
S cadmi < 00,
lim inf KER
(36)
then the optimal transport cost between y and is finite and x² is an optimal transport plan.
Corollary C.2 (Corollary 4.1), Consider sequence absolutely continuous probability measures {Me, JREN such that {Mee} weakly converges to μ. Then, up to the extraction of subsequence, our OTP model, utilizing {He, TEN, weakly converges to the optimal transport plan n' between y and v.
Proof of Corollary 4.1. The absolutely continuous measure Heg does not assign positive mass to measurable sets of Haus- dorff dimension at most - 1. Therefore, by Thm. 3.1, our OTP model for noise level 6 (Eq. 20) correctly recovers the optimal transport plan के = (Id, T ( Then, by Thm. C.I, T weakly converges to the optimal transport plan x between y and
D. Related Works
D.1. Spurious Solution Problem
In this section, we overview previous attempts to address the spurious solution problem in the Semi-dual Neural OT approaches. The spurious solution problem refers to the problem where the solution of the max-min learning objective LV.,T, of Semi-dual Neural OT fails to fully characterize the correct optimal transport map. Rout et al. (2022); Fan et al. (2022) first identified this issue. They proved that while true optimal potential and transport map (V*)"*) are the solution to the max-min problem Cv,,r, (Eq. 9), the reverse does not hold. Furthermore, Fan et al. (2023) proved that, when μ is atomless, the saddle point solution (V Tind) (Eq. 37) of LV recovers the optimal transport map:
vt sad E arg max L(V,T) Tiad), Tt sad E arg min L(Vi T sads T)-
(37)
V
However, this does not hold for the max-min solution of LV,,T, In this paper, we establish the sufficient condition under which the max-min solution recovers the optimal transport map (Thm. 3.1). Furthermore, we suggest a method for learning the optimal transport plan n², which is applicable even when the optimal transport map T does not exist (Sec 4).
For the weak OT problem, Korotin al. (2023b;a) provided a theoretical analysis for spurious solutions in semi-dual approaches. Note that these analyses were conducted for the "--eak quadratic cost with y 0 and, therefore, do not cover the standard OT problem. Additionally, Korotin et al. (2023a) proposed alternative approach that modifies the cost function by introducing a positive definite symmetric kernel. While this approach addresses the spurious solution issue in the weak OT problem, it solves an inherently different problem due to the modified cost function. In contrast, our work is the first attempt to analyze the conditions under which spurious solutions occur in the standard OT problem (Eq. 4).
D.2. Neural Optimal Transport Models
For completeness, we provide an overview of various approaches for Neural OT solvers. First of all, several works proposed methods based on the semi-dual formulation of the OT problem (Rout et al., 2022; Fan et al., 2022; Choi et al., 2023; Makkuva et al., 2020; Fan et al., 2023). The spurious solution problem occurring in these formulations the main focus of this paper.
Moreover, there are dynamic Neural OT models, such as bridge matching (Shi al., 2024; De Bortoli et al., 2024b; Liu et al., 2022) and flow matching (Tong et al., 2024a;b; Lipman et al., 2023). These models learn continuous dynamics that transport the source distribution to the target distribution. Particularly, the bridge matching methods learn stochastic transport between two distributions based on the Entropic Optimal Transport (EOT) problem, i.e., the OT problem with an entropic regularizer. In contrast, our OTP model learns stochastic OT plans derived from the standard (non-entropic) OT problem, making the target formulation fundamentally different.
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Overcoming Spurious Solutions in Semi-Dual Neural Optimal Transport
Lastly, there recent Neural OT models based on non-minimax objective functions (Uscidda & Cuturi, 2023; Kassraie al., 2024). Uscidda & Cuturi (2023) introduces regularizer known as the Monge gap learn the OT map. Kassraie et al. (2024) proposes gradual learning scheme by solving sequence of EOT problems with decreasing entropic regularization. Although this gradual approach is conceptually related to our approach, Kassraie et al. (2024) solves the discrete OT problem, i.e., the OT problem between empirical distributions over finite samples. In contrast, our OTP model is designed for the continuous OT problem between two continuous distributions, where training data consists of i.id. samples drawn from these distributions.
E. Implementation Details
E.1. Synthetic Data Experiments
In this section, we explain the implementation details for the synthetic data experiments (Sec. 5.1), including dataset description, model architecture, and training hyperparameters.
Dataset Description Throughout this paragraph, let x, y E R and n = d/2. Then, let x = (x1, F22 and y = (y11 V2), where 31, T2, V/, 1/2 E RR. Moreover, let e1 = (1,0,..., E R".
Perpendicular: We generate & ~ as follows: 2 U ([-1, and zg 0. Similarly, we sample y ~ v by y1 and 1/2 U ([-1,1]").
Horizontal: We generate x μ as follows: T1 ([-11,1]') and 22 = 0. Similarly, we sample y v by y1 ~ ([-1, and y/2 = 61.
One-to-Many: We generate x ~ μ as follows: x1 ~ U y1 ~ U ([-1,1]") and y2 Cat((ei, -e1), (0.5, 0.5)).
and Z2 0. Similarly, we sample y ~ by
Multi-Perpendicular: Let P == Cat (( -1 2. 4. 1.111 (14)1 We generate p by sampling æ1 U ([-1, and 12 PP. Similarly, we sample y by y ~ and y2 ~ ([-1,11)"
For every experiments, we share the same network architecture and the same training hyperparameters. We employ one-hidden layer with ReLU activations for both potential function us and transport map Ta parametrization. For experiments of = 2 and d = 4, we use hidden dimension of 256. For higher dimensions, we use hidden dimension of 1024. We employ the batch size of 128, the number of iterations of 20K, the learning rate of 10-4, and Adam optimizer for (B1, 32) = (0, 0.9). To improve the optimization the inner loop the dual formulation of equation (9), we perform 20 updates T' for every single update Ug. i.e. Kr 20. For the experiment on 256, we use a 0.01 and X 1. Otherwise set
Training Details
and À 0.
For our experiment, we generate the perturbed data å as follows: a = + az where x μ1, (0, I). Here, we schedule the noise level a from Tmar = 0.2 to miin = 0.05. We update noise every 2K iterations, by the linear interpolation between initial noise to terminal noise. Specifically, in the k-th iteration, the noise level 0 is
σg (1 - t) + minn t = (P x [k/ 1)/K,
(38)
where P 2000, [:] the least integer function and K is the total iteration number.
In the NOT (Korotin et al., 2023b) implementation, we introduce additional noise & ~ NN(0 1) into the network Te. set the dimensionality of the noise match the input data dimension. We simply concatenate the noise and the data. This augmented input is then passed through the transport network To.
E.2. Image Translation
MNIST-CCMM CMNIST In this paragraph, deseribe the implementation details of Fig. 4. We create red, green, and blue-colored MNIST datasets by isolating individual color channels. To achieve this, we assign the grayscale MNIST digit images to single color channel (red, green, or blue) while setting the other two channels to zero. We use the image size of 32, the number iterations of 50K, the batch size of 64, the learning rate of 10-4, Adam optimizer with (3,,8) = 0.9), = 10, Kr = 10, a = 0.1, dmax = 2, Jmin = 0.5, and P = 100. Note that we don't 't use any other techniques such as learning rate scheduling, exponential moving avarage, dropout, and clip.
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Overcoming Spurious Solutions in Semi-Dual Neural Optimal Transport
We adopt the network architectures of DCGAN (Radford et al., 2015), depicted follows: architecture. For the input embedding module, pass the input through the convolution layer, batch norm layer and activation layer. We employ four of downsample modules and four upsample modules, For every downsample module, we pass through convolution, activation, average plloing layer. For the upsample modules, we pass the inputs through upsample module, convolution, batch norm, activation layers. Note that as original UNet, we use the skip connections. For the last module, we pass through one convolution layer. For the activation function, we employ leaky ReLU with slope of 0.2. Every convolutional layers are x 3 convolutional layers. For average pooling, we use convolutional layer of 3 x 3 with stride of 2. For upsampling module, we simply use blockwise upsampling module.
For generator, we use UNet
For the potential network, use three downsample module. The settings of downsample module is same as the downsample module of the generator. After the downsample modules, we flatten it and pass it through linear layer. For every modules, note that the channel number (or the feature number) is fixed to 256.
Image-to- -oa-ttto--meeeeeee Translation In this paragraph, we describe the implementation details of Tab. 2. Most of the hyperpa- rameter except the network architecture and the number of iterations are shared. For every experiments, we use the batch size of 64, the learning rate of 10- Adam optimizer with (31,8,) = (0, 0.9). = 10, KT = 1, a = 0.001, Fmaz = 2. Mmin 0.2, and P 100. Note that we don't use any other techniques such as learning rate scheduling. exponential moving avarage, dropout, and clip. In the experiments for image size of 64, we follow the network architecture of the CIFAR-10 10 experiment in (Choi et al., 2023; 2024c). In the experiments for image size of 128, we follow the network architecture of the Ce experiment in (Choi et al., 2023). For Wild-Catt (64 x 64), Felae+eemale (64 x 64), and Male->Female (128 x 128), we take the number of iterations of 60K, 300K, and 500K, respectively.
Here, we use Variance- Prriance-Preserving noise scheduling, which is illustrated as follows: For x ~ ,,z ~ (O, I), we generate the perturbed data at k-th iterations as follows: & = VI Eg-T + VEEz where eg defined as follows:
Ej = 1 exp [-mmar
Tminn ,2 mminnt)
t= 1-1-(PX) [/
(39)
2
where P = 100, [:] is the least integer function and K is the total iteration number.
Evaluation metric We follow the evaluation metric of Choi et al. (2024a). Specifically, for the task, we transform Male in the test dataset, and use the transformed samples and the test samples of the Female data to calculate the evaluation metrics. For Cild->att, we generate 5000 samples from the test data of Wild dataset, and use the transformed samples and the training samples of the Cat data to calculate the evaluation metrics.
translation
F. Additional Results
F11 Input Convex Neural Networks
As discussed in Eq. 9, the search space for the potential V is constrained to S.. the set of c-concave potential functions. Strictly speaking, the potential Ve must be specifically parameterized to ensure it lies within S.. To address this, we parameterize Vs using an input convex neural network (ICNN) (Amos et al., 2017; Korotin et al., 2021a). The ICNN architecture fe ) - R is specifically designed with structural and weight constraints to enforce imput convexity, satisfying (x1) ) + should satisfy input convexity. To ensure this, we parametrize Vo as follows:
the property: (x1)
-1 (x))))))= 2 ʃ((1 ([(-)))a + tx2) for all i E j0, 1]. Given that c(x, v) = aIl - vl12, a | Vv)
Vo(y) - fo(y),
(40)
where fo is an ICNN. Using this parameterization, we conduct experiments on two-dimensional data d = 2 across several toy datasets. The results are presented in Fig. 5.
F.2. Ablation Studies: Constant Noise Scheduling
In practice, our OTP model decreases the noise level until it reaches a small constant Emiin > 0 (see Algorithm paragraph in Sec 4). Since the noise level is not reduced exactly to zero, an alternative approach is to train our OTP model directly at the minimum noise level Emin instead of gradually decreasing it. To investigate this alternative, we conduct an ablation study on noise scheduling, specifically Constant Noise Scheduling (CNS).
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Overcoming Spurious Solutions in Semi-Dual Neural Optimal Transport
(a) ICNN-OTM
(b) ICNN-OTP
Figure 5. Visualization failure cases with ICNN potential function fa(u) := alyy|l V (y). We compare the Optimal Transport map (a) and OTP model (b) in the failure cases. The source data μ, target data and generated data T(x) are represented in Blue, Orange, and Red. As illustrated in the figure, OTM fails to transport source to the target, i.e. Tz o v. On the other hand, our model successfully transports source μ larget
(a) Constant Noise Scheduling
(b) Decrcasing Noise Scheduling
Figure 6. We visualize the qualitative result between (a) constant noise scheduling, (amaz = 0.05) and (b) gradually decreasing noise scheduling (amaz = 0.2, Jmnn 0.05). We illustrate the results the data dimension d 64. We select the 1st and 33rd axis visualize the results effectively. The source data μa, target data and generated data T(x) are represented in Blue, Orange, and Red. The max-min solution fails to recover the correct OT map.
Fig and Tab 3 present the results. In Fig 6, our OTP model with CNS scheduling (OTP-CNS) shows the mode collapse problem. In the left figure of Fig 6(a), OTP-CNS generates only the bottom part the target distribution. Similarly, the right figure of Fig 6(a), OTP-CNS covers only the upper part of the target distribution. In contrast, Fig 6(b) demonstrates that original OTP model with decreasing noise scheduling successfully covers the entire target distribution without exhibiting the mode collapse problem. Tab 3 also shows this trend. Because of the mode collapse problem observed in Fig 6(a), OTP-CNS shows significantly larger target distribution error Dtarget- This ablation study shows that our noise-decreasing scheme is more effective choice for the OTP model.
F3. Benchmark Experiments
In this section, we present the experimental results on the Wasserstein-2 benchmark proposed in Korotin et al. (2021b). Specifically, we implemented our OTP model, based on the [MMv2:R] baseline in Korotin et al. (2021b), which solves the SNOT formulation (Eq. 20) using an ICNN architecture.
Note that this benchmark employs a Gaussian mixture as the source distribution, which is already absolutely continuous. Therefore, by our Thm 3.1, existing SNOT models can also recover the correct OT map. In this respect, this benchmark is OTP model, because additional input noise would be regarded error. Nevertheless,
slightly unfavorable shown in Table 4, our model shows comparable performance to [MMv2:R].
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Table 3. Quantitative comparison of numerical accuracy on synthetic datasets. The Const column stands for the constant noise scheduling, and Ours stands for our method which gradually decrease the noise level. Each model evaluated by target distribution emor (1b)
Dimension
Model
Perpendicular
One-to-Many
Constant
0.64
72.17
d == 16
Ours
0.59
0.65
Constant
12.36 10.09
20.72
64
Ours
9.98
Table 4. Quantitative comparison of numerical accuracy on benchmark dataset proposed in (Korotin al., 2021b).
Dimension
D=16
D=64
D=256
Metric
.VVP (4)
cos (1)
2VVVVP (L)
cos (1)
-VVVP (L)
cos (1)
41.6
0.73
63.9
0.75
67.4
0.77
QC
47.2
0.70
75.2
0.70
88.2
0.66
MM
2.2
0.99
3.2
0.99
4.1
0.99
MMv1
1.4
0.99
8.1
0.97
2.6
0.99
MMv2
3.1
0.99
10.1
0.96
2.7
0.99
MM-B
6.4
0.96
13.9
0.94
22.5
0.93
MMv2:R
7.7
0.96
6.8
0.97
2.8
0.99
OTPt
8.7
0.97
6.5
0.97
3.7
0.99
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Overcoming Spurious Solutions in Semi-Dual Neural Optimal Transport
F.4. Additional Qualitative Results
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(a) x p.
(b) Ta(x)
Pigure 7. Unpaired MNIST - CMNIST translation for 32 x 32 image.
(a) J. Pe
(b) (o(
Figure Unpaired Wild Cat translation for 64 x 64 image.
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Overcoming Spurious Solutions in Semi-Dual Neural Optimal Transport
(a) x ~ P4
(b) Tz
Figure 9. Unpaired Male Female translation for 64 64 image.
(a) T. ~ Pie
(b) To
Figure 10. Unpaired Male
Female translation for 128 x 128 image.
22


Reviews:
[{"review_id": "jraMax3VVJ", "reviewer": "Reviewer_YbFr", "summary": "This paper investigates convergence issues that arise when learning Optimal Transport (OT) maps using the Semi-Dual formulation. Specifically, the authors:\n- Identify a sufficient condition under which the standard Semi-dual Neural OT approach converges to the correct OT map, resolving issues with inaccurate (\"fake\") solutions.\n- Propose a method, OTP, which jointly learns both the OT map and/or the optimal transport plan, overcoming \"fake\" solutions.\n- Provide theoretical guarantees showing that, under precise assumptions, OTP correctly solves the OT problem.\n- Demonstrate experimentally that OTP successfully recovers OT maps where previous methods fail, on synthetic and image-to-image translation tasks.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank the reviewer for carefully reading our manuscript and providing valuable feedback. We greatly appreciate the reviewer's acknowledgment that \"this paper presents a nice contribution that would be relevant for the conference\". We hope our responses to be helpful in addressing the reviewer's concerns.\n\n$ $\n\n---\n### **Questions For Authors:**\n\n$ $\n\n>**Q.** General theory in the appendix but quadratic cost in the main text?\n\n**A.** We appreciate the reviewer for providing helpful suggestions to improve our paper. In the original manuscript, we chose to focus on the quadratic cost function in the main text to **maintain consistency with our failure case analysis and experimental results**, which are all based on the squared Euclidean cost. Accordingly, in line 96, we defined $c$ as the quadratic function. Meanwhile, a more general setting is presented in Theorem A.4 in the appendix.\n\nNevertheless, we agree with the reviewer that presenting a broader statement than Thm 3.1 would strengthen the paper. In this regard, instead of specifying a particular assumption in Thm 3.1, **we added the following proposition in the appendix that outlines additional settings under which Thm 3.1 holds** as follows:\n> **Proposition A.7** Let $\\mathcal{X}, \\mathcal{Y} \\subset M = \\mathbb{R}^d$ satisfy Assumption A0. Suppose that any one of the following conditions holds: \n>- The cost function $c$ is $C^1$, strictly convex with bounded Hessian, and the source distribution $\\mu$ does not assign positive mass to sets of Hausdorff dimension $d-1$;\n>- The sets $\\mathcal{X}$ and $\\mathcal{Y}$ are compact, $\\mu$ is absolutely continuous, and $c$ is a $C^1$ strictly convex function;\n>- The cost function $c(x,y) = h(x-y)$ is a $C^1$ strictly convex and locally semi-concave function, where $h$ is a radially symmetric and strictly increasing function with respect to $\\Vert x-y\\Vert$, and $\\mu$ does not assign positive mass to sets of Hausdorff dimension $d-1$.\n>\n> Then, the max-min solution of the SNOT problem recovers the correct OT map.\n\n$ $\n\n---\n>**Q.** Relationship between the LPIPS metric and the transport cost.\n\n**A.** Thank you for the insightful comment. We interpret the **LPIPS metric as a proxy for the transport cost in the feature space**. Due to the rebuttal length constraint, a more detailed explanation is provided at the following anonymous link:\n\nhttps://github.com/anonymousicml25/Overcome_Fake_Solution_OTP/tree/main\n\n$ $\n\n---\n> **Q.** Additional experiments on the pathological cases (Table 1) with $d=256$ and on the benchmark introduced in [Korotin et al., 2022].\n\n**A.** We appreciate the reviewer for providing an insightful comment. Following the reviewer's suggestion, we conducted two additional experiments: (1) benchmark experiments from [1] and (2) 256-dim version of the failure case experiments in Table 1. Due to the rebuttal length constraint, we provide the full results at the following anonymous link:\n\nhttps://github.com/anonymousicml25/Overcome_Fake_Solution_OTP/tree/main\n\nReference\n- [1] Korotin et al., \"Do Neural Optimal Transport Solvers Work? A Continuous Wasserstein-2 Benchmark.\", NeurIPS, 2021.\n\n$ $\n\n---\n> **Q.** Additional related work: Bridge matching and non-minimax approaches\n\n**A.** We appreciate the reviewer for suggesting these relevant works. We incorporated additional related works, such as bridge matching approaches and non-minimax approaches in Appendix D.\n\n$ $\n\n---\n### **Other Strength and Weakness:**\n\n$ $\n\n> **Q.** Thm 3.2 and Cor 3.3 seem essentially to be straightforward reformulations of Brenier's theorem. Additionally, why the assumption that the target measure has a density is required here? \n\n**A.** We agree with the reviewer that Thm 3.2 and Cor 3.3 can be derived from classical results such as Brenier’s theorem. As noted in Lines 125–129, we included these results for the sake of **completeness within the context of SNOT problems**. In this regard, we would like to emphasize that, to the best of our knowledge, Cor 3.3 is the first result to explicitly state a sufficient condition for the **uniqueness of the max-min solution in the SNOT objective**.\n\nRegarding the assumption that the target measure has a positive density on its domain $\\mathcal{Y}$, **this condition ensures that the Kantorovich potential is unique (up to a constant) within $\\mathcal{Y}$**. If the target distribution has disconnected support, the potential can differ by locally constant functions across its connected components. \n\n$ $\n\n---\n### **Other Comments Or Suggestions:**\n\n$ $\n\n>**Q.** Fake solutions may violate the marginal constraint or cost optimality.\n\n**A.** Thank you for the great question. Interestingly, **any max-min solution satisfying the marginal constraint is guaranteed to be the optimal transport map**. Due to the rebuttal length constraint, a more detailed explanation is provided at the following anonymous link:\n\nhttps://github.com/anonymousicml25/Overcome_Fake_Solution_OTP/tree/main", "claims_and_evidence": "All experimental claims are backed up with experiments on synthetic and real data. See more details in the Experimental Designs Or Analyses section.", "methods_and_evaluation_criteria": "See the Experimental Designs Or Analyses section.", "theoretical_claims": "All theoretical claims are clearly supported by rigorous and, to my knowledge, correct proofs. Moreover, any informal assumptions or limitations of the theory are explicitly acknowledged, including:\n\n> Proposition 3.4 (L263-266)\n> *While the convergence theorem only guarantees convergence up to a subsequence (Thm. 4.1), our method exhibits decent convergence to $\\pi^\\star$ in practice (Sec. 5).* (L321-324)", "experimental_designs_or_analyses": "- The experiments on synthetic data clearly illustrate and motivate the problem. However, they currently (i) rely on rather pathological cases and (ii) do not explore dimensions beyond $d=64$. I would suggest evaluating the proposed method on a benchmark for which the OT map is known in closed form, such as the one proposed in [Korotin+2022], where the OT map is given explicitly as the gradient of an ICNN. Additionally, this benchmark covers dimensions up to $d=256$. Given that the authors advocate for improved OT solutions, this additional experiment would clearly assess this claim, which in my opinion is not yet fully supported.\n\n- To measure performance on image-to-image translation, the authors report FID and LPIPS. While I understand that FID (to some extent) measures the fitting of the marginal constraint $T\\sharp\\mu=\\nu$, it is unclear to me why LPIPS can be interpreted as a measure of optimal transport cost. I suggest explicitly adding the following quantity as a metric to Table 2:\n$$\n\\mathcal{C}(T) = \\int_{\\mathcal{X}} c(x, T(x)) d\\mu(x)\n$$\nto properly investigate whether the proposed method achieves better transport-cost optimality.", "relation_to_broader_scientific_literature": "The author did a thorough background section on the literature about learning OT maps through the Semi-Dual OT formulation in Section 2, and notably adding more details in Section D of the Appendix. Yet, in my opinion, they lack relationship to other neural OT solvers, that are especially non-min-max optimization based.\n\nFirst, while they include DSBM [Shi+2024] as a baseline in the experiments, they do not link their approach to bridge matching based neural OT solvers, such as DSBM [Shi+2024] itself, or [De Bortoli+2024], [Liu+2023], and flow matching based neural solvers, such as [Pooladian+2024], [Tong+2024:a,b], [Klein+2024]. In my opinion, this is a problem as I think that the the experimental gains observed are mainly due to the iterative refinement procedure, where we learn a map between a sequence of smoothed distributions. Especially, as the author advocate for learning stochastic OT maps, i.e. couplings, the bridge matching methods are particularly relevant.\n\nSecond, the author do not mention that some recent neural OT solvers [Uscidda+2023],  [Kassraie+2024] based relying on non-minimax objective functions that can approximate the OT maps as well and showed strong performances on benchmarks [Korotin+2022] where the OT map is known in closed form as the gradient of an ICNN. [Kassraie+2024] is of particular relevance to this work, as it learns a transport map between two distributions by solving a sequence of OT problems with decreasing entropic regularization.\n\n[Shi+2024] Shi, Y., De Bortoli, V., Campbell, A., and Doucet, A. Diffusion schrodinger bridge matching. Advances in Neural Information Processing Systems, 36, 2024.\n\n[Uscidda+2023] Théo Uscidda and Marco Cuturi. The monge gap: A regularizer to learn all transport maps, 2023.\n\n[Tong+2024;a]  Alexander Tong, Nikolay Malkin, Guillaume Huguet, Yanlei Zhang, Jarrid Rector-Brooks, Kilian Fatras, Guy Wolf, and Yoshua Bengio. Conditional flow matching: Simulation-free dynamic optimal transport, 2024.\n\n[Tong+2024;b] Alexander Tong, Nikolay Malkin, Kilian Fatras, Lazar Atanackovic, Yanlei Zhang, Guillaume\nHuguet, Guy Wolf, and Yoshua Bengio. Simulation-free schrodinger bridges via score and flow matching.\n\n[Kassraie+2024] P Kassraie, AA Pooladian, M Klein, J Thornton, J Niles-Weed, M Cuturi. Progressive entropic optimal transport solvers.", "essential_references_not_discussed": "N/A.", "other_strengths_and_weaknesses": "**Strengths:**\n\nThe paper proposes a very interesting idea of learning a sequence of transport map between sequences of noise-perturbed source and target distribution, with decreasing noise levels, akin to a diffusion models. They propose two smoothing scheme, and relate them VP and VE diffusion processes, making the method interpretable as a kind of diffusion model. Additionally, as the method eventually learn a \"one-shot\" single map between the source and the target distribution, the method could be interpreted as learning a learning a diffusion model and a \"one-step\" model distillation of it simultaneously during training, which is very interesting. For these reasons, I truly that this paper present a nice contribution that would be relevant for the conference. Yet, I have some concerns that I think should be fully in favor of acceptance.  \n\n**Weaknesses:**\n\n***Min-Max optimization.*** \n\nA major weakness of the proposed methodology is its reliance on adversarial training, which is known to be challenging. This difficulty has notably motivated the recent preference for alternative approaches, such as diffusion models or flow/bridge matching frameworks.\n\n***Theoretical Framework*** \n\nI have mixed feelings regarding the theoretical framework presented in this paper. In my view, the statements in the main text appear somewhat oversold. However, upon examining the appendix, it becomes clear that the authors have significantly simplified their results for presentation in the main text, and the motivation behind this simplification is unclear to me.\n\nFor example, in the main text, all the theoretical results are stated explicitly for the squared Euclidean cost $c(x, y) = \\alpha \\|x - y\\|^2 $ with measures $\\mu$ and $\\nu$ supported on $\\mathbb{R}^d$. More concretely, Theorem3.2 (L135--138) and Corollary~3.3 (L142--147) seem essentially to be straightforward reformulations of Brenier's theorem restricted to a compact subset of $\\mathbb{R}^d$. Additionally, could the authors clarify why the assumption that the target measure has a density is required here? I may be misunderstanding this point, so some explanations would be appreciated.\n\nOn the other hand, in the appendix, the authors introduce a more general theoretical framework: measures are supported on a closed subset of a smooth, complete, connected Riemannian manifold, and the cost function is considered within a broader family of costs, including the squared geodesic distance. Given this broader framework, I am also curious why the authors did not explicitly state their results for more general cost functions of the form $c(x, y) = h(x - y)$ with $h$ strictly convex. Indeed, as mentioned in AppendixA.2 (L635--650), the Gangbo--McCann theorem guarantees the existence and uniqueness of the Monge map under this more general class of costs, provided that the source measure $\\mu$ has a density. It thus appears that the main text results could naturally be generalized to this broader family of cost functions. Alternatively, if the authors specifically prefer the squared Euclidean cost $c(x, y) = \\alpha \\|x - y\\|^2$, it might have been clearer to explicitly replace all generic cost references directly with this squared Euclidean form throughout the main text. Could the authors comment on this choice?\n\nFinally, while it is indeed interesting that the authors introduce a weaker assumption---requiring only that the source distribution does not give mass to measurable sets of Hausdorff dimension at most $d - 1$---it seems that, in practice, their approach ultimately relies on the stronger condition that the source distribution has a density. This stronger condition is precisely what is used in the experiments, where the source measure is smoothed by introducing decreasing noise levels (whether in the VP or VE SDE framework), ensuring that the perturbed measure has a density. To fully appreciate the generality of their theoretical framework, it would have been interesting if the authors had also proposed perturbation schemes of the source measure satisfying their weaker condition, without necessarily imposing the stricter requirement of generating a density. Could the authors provide some insights into this decision as well?\n\n***Evaluation.***  See the Experimental Designs Or Analyses section.\n\nI would be more than willing to increase my score if the authors address these concerns. See the questions section.", "other_comments_or_suggestions": "-  I would suggest explicitly clarifying this point in the abstract. Indeed, stating that the method \"often generates fake solutions that fail to transfer one distribution to another accurately\" (L18–19) might imply that the issue arises solely from violating the marginal constraint $T\\sharp\\mu=\\nu$. However, as the authors correctly point out, fake solutions can potentially both violate the marginal constraint and fail to be cost-optimal, which is an important distinction to highlight.", "questions_for_authors": "- Could the authors clarify why their choice of deriving general theory is presented in the appendix, yet restricted specifically to the squared Euclidean cost and measure on $\\mathbb{R}^d$ for the main text?\n\n- Could the authors expand upon the relationship between the LPIPS metric and the transport cost? It's unclear how LPIPS can be naturally interpreted as a transportation cost.\n\n- It would be beneficial if the authors reported the transport cost for the image-to-image experiments, in addition to the LPIPS metric. This would allow for clearly distinguishing the source of experimental gains, determining whether improvements come from better satisfaction of the marginal constraint $T\\sharp\\mu=\\nu$ (which is measured by FID), or from achieving better optimality through minimizing the transport cost itself. \n\n- Could the authors apply their method to the benchmark introduced in [Korotin et al., 2022]? While the experiments in Table 1 provide convincing motivational examples, the constructed measures correspond to rather pathological cases and may not adequately demonstrate learning capabilities. Moreover, the current experiments only explore dimensions up to $d=64$, whereas this benchmark extends up to $d=256$, offering a more thorough evaluation.\n\n- Lastly, could the authors elaborate on the connection between their approach and methods based on bridge matching?\n\nI would be more than willing to increase my score if the authors address these concerns.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Vr77XWoVc0", "reviewer": "Reviewer_C1Wd", "summary": "The manuscript presents a minimax algorithm used to find solutions to the semi-dual optimal transport plan problem. A primary drawback of this method is that not all solutions of the semi-dual formulation yield optimal transport maps and optimal potential functions; in other words, the formulation can lead to “fake” solutions. The authors identify a sufficient condition on the source distribution to avoid such fake solutions. Based on this condition, they propose a new algorithm with theoretical guarantees for approximating the optimal transport plan.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank the reviewer for carefully reading our manuscript and providing valuable feedback. We greatly appreciate the reviewer's acknowledgment that our method is **theoretically founded**. Moreover, we’re pleased to hear that the reviewer **appreciated reading the examples, as they provide great insight while remaining straightforward.** \n\nWe hope our responses to be helpful in addressing the reviewer's concerns.\n\n$ $\n\n---\n### **Other Strengths And Weaknesses:**\n\n$ $\n\n> **Q.** It would be beneficial to include a comparison of training complexity and time.\n\n**A.** We appreciate the reviewer for the insightful comment. **Due to the use of the gradual training scheme with noise scheduling, our OTP model requires more training iterations** compared to existing SNOT models. For example, as described in Appendix F, in our Male-to-Female(64x64) experiment, our model uses 300K training iterations, while the OTM uses 60K iterations. Since both models adopt the same backbone network, the training time ratio is the same as the training iteration ratio. \n\nHowever, we would like to emphasize that **OTM achieves its best performance when trained for 60K iterations**. In practice, the OTM model is known to suffer from training instability under long training [1]. Hence, in our experiments, increasing the number of iterations beyond 60K led to training failure with loss explosion, as observed in [1].\n\nReference\n- [1] Choi, J., Choi, J., & Kang, M. Analyzing and Improving Optimal-Transport-based Adversarial Networks. ICLR 2024.\n\n$  $\n\n---\n> **Q.** The novelty of Theorem 3.1 is somewhat limited, as it follows from Lemma A.3. The proof verifies that the conditions of Lemma A.3 are satisfied; perhaps it should be regarded as a corollary instead.\n\n**A.** Thank you for the thoughtful comment. We would like to clarify that Thm A.4 is an auxiliary result for proving Thm 3.1. Hence, Thm A.4 also contributes to the novelty of Thm 3.1. We chose to present Theorem 3.1 as a theorem to emphasize its **importance in the context of the Neural Optimal Transport**, where the quadratic cost function is widely adopted and the transport map is parametrized as follows (Eq 7 or 23):$$T_{\\theta}: x \\mapsto \\arg \\min_{y \\in \\cal{Y}} [c(x,y) - V_{\\Phi}(y)]$$Thm A.4 shows that the **conclusion of Lemma A.3 (Eq. 22) implies that this transport map parametrization in SNOT (Eq 23) recovers the correct OT Map**. In this regard, Thm 3.1 plays a key role in identifying the sufficient condition under which the SNOT model avoids fake solutions.\n\nMoreover, while responding to the Reviewer YbFr, we were able to characterize additional cases where the SNOT model avoids fake solutions. **As an addtional contribution, we incorporated these additional results into the appendix as the following proposition**:\n\n> **Prop A.7** Suppose $\\mathcal{X},\\mathcal{Y}\\subset M =\\mathbb{R}^d$ satisfy Assumption A0. Then, the max-min solution of the SNOT problem recovers the correct OT map if any of the following conditions hold:\n>- $c$ is $C^1$, strictly convex with bounded Hessian, and $\\mu$ does not assign positive mass to sets of Hausdorff dimension $d-1$.\n>- $\\mathcal{X}$ and $\\mathcal{Y}$ are compact, $\\mu$ is absolutely continuous, and $c$ is a $C^1$ strictly convex.\n>- $c(x,y) = h(x-y)$ is a $C^1$ strictly convex and locally semi-concave function, where $h$ is a radially symmetric and strictly increasing function w.r.t $\\Vert x-y\\Vert$, and $\\mu$ does not assign positive mass to sets of Hausdorff dimension $d-1$.\n\n$ $\n\n---\n### **Questions For Authors:**\n\n$ $\n\n> **Q.** Could you comment on the fact that the uniqueness in Thm 3.1. is in law, is this a limitation ? Especially, is it the same definition of uniqueness that is used in example 1, 2 ?\n\n**A.** Thank you for the thoughtful question. To avoid confusion, we revised the statement in Thm 3.1. as follows: \n\n> In particular, a map $x \\mapsto y_{x} \\in \\cal{D}_{x}$ is a unique OT Map $T^{\\star}$ $\\mu$-a.s..\n\n**This is not a limitation, but rather a natural property of the OT Map.** The uniqueness of the OT Map $T^{\\star}$ is always defined up to $\\mu$-a.s., because modifying $T^{\\star}$ on a set of measure zero does not affect the transport cost (Eq. 3). Specifically, this is the same definition of uniqueness that is used in example 1, 2.\n\n$ $\n\n---\n> **Q.** For the second row of figure one, with what algorithm do you compute the max-min solution ? Are the results stochastic ?\n\n**A.** **The second row of Fig 1 presents manually visualized examples**, based on the analytical form of the max-min solution in Sec 3.2, rather than results by a numerical algorithm. In contrast, the first row of Fig 3 shows empirical results obtained by training the OTM model.\n\n$ $\n\n---\n### **Other Comments Or Suggestions:**\n\n$ $\n\n> **Q.** Citations, Typos, and Presentations\n\n**A.** We sincerely thank the reviewer for this detailed feedback. All comments regarding citations, typos, and presentations have been addressed as suggested.", "claims_and_evidence": "The claims are supported by theoretical results, experiments, and references to relevant literature.", "methods_and_evaluation_criteria": "The method is theoretically founded, and the evaluation criteria include various metrics across different datasets. It would be beneficial to report standard deviations where possible and additional analysis on training time or complexity.", "theoretical_claims": "I verified the proofs of Theorems 3.1 and 3.2 but did not verify the references cited in those proofs.", "experimental_designs_or_analyses": "The experiments and analyses are reasonable, including various comparisons and datasets.", "relation_to_broader_scientific_literature": "This work proposes a new method for learning an optimal transport (OT) map based on the semi-dual formulation. OT is utilized in various fields, including generative models and biology; thus, improving the quality of an estimator can have a significant impact.", "essential_references_not_discussed": "I cannot think of any essential references that are missing.", "other_strengths_and_weaknesses": "- The manuscript is overall well written, and the structure is very clear. I appreciated reading the examples, as they provide great insight while remaining straightforward.\n- The experiments are interesting. While the datasets seem relatively simple, they include various settings and dimensions. Overall, the method appears to improve on existing approaches.\n- The novelty of Theorem 3.1 is somewhat limited, as it follows from Lemma A.3. The proof verifies that the conditions of Lemma A.3 are satisfied; perhaps it should be regarded as a corollary instead.\n- It would be beneficial to include a comparison of training complexity and time.", "other_comments_or_suggestions": "- Line 11. It has recently also been used with generative model like flow matching.\n- Line 67, You could mention that T_\\# is a push-forward operator.\n- The citation formatting on line 100 seems odd.\n- Potentially a missing word on line 141 \"if OT Map T...\"\n- The phrasing of Thm 3.1 could be reviewed to improve clarity. \n- Line 311 missing word \"*be a sequence absolutely*\"\n- Overall the manuscript would benefit from another read and revision. \n- I believe it is generally preferred to avoid starting a sentence with a mathematic symbol.  \n- It would be best to uniformize the table presentation, for example, in tab1 the number of decimals are not always the same.", "questions_for_authors": "- Could you comment on the fact that the uniqueness in Thm 3.1. is in law, is this a limitation ? Especially, is it the same definition of uniqueness that is used in example 1, 2 ?\n\n- For the second row of figure one, with what algorithm do you compute the max-min solution ? Are the results stochastic ?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ezmtpjnn8I", "reviewer": "Reviewer_v2np", "summary": "Semidual Neural OT (SNOT) is one of the NN-based formulation for computing transport maps for general OT problems. The paper seeks to address the issue of spurious solutions for SNOT formulation, a problem that has been known before. For example, when the source distribution places mass on low-dimensional sets, or when stochastic transport is needed (e.g., one-to-many mappings like colorization).. The paper also proposes a smoothing algorithm  to first smear the source distribution in order to satisfy the sufficient conditions they identified. Proofs and illustrative examples, and image-to-image translation examples are used to illustrate the effectiveness of the proposed method.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank the reviewer for carefully reading our manuscript and providing valuable feedback. We are especially grateful for the reviewer's recognition that our work is the **first to identify a reasonable sufficient condition for the avoidance of spurious solutions** for the minimax formulation used in SNOT. In addition, we greatly appreciate the reviewer' acknowledgment that our work has done **a good job illustrating the effectiveness** of the proposed smoothing algorithm.\n\n$ $ \n\n---\n### **Experimental Designs Or Analyses:**\n\n$ $\n\n> **Q.** One interesting fact explained in the paper is that, the noise scheduling (gradual reduction) is critical; constant noise leads to mode collapse.\n\n**A.** Thank you for highlighting this point. We also found this phenomenon to be quite interesting. Our interpretation is that adding a certain level of noise to the input serves as a form of regularization, which helps stabilize the training of the OT map. Since our goal is to learn the OT Map that transports each source sample $x \\sim \\mu$ to a nearby target sample $y \\sim \\nu$, **the initial noise can guide the model to explore all modes of the target distribution**. Gradually reducing the noise then allows the model to refine the transport map without experiencing mode collapse.\n\n$ $\n\n---\n### **Other Comments Or Suggestions:**\n\n$ $\n\n> **Q.** Change \"fake solution\" to \"spurious solution?\" \n\n**A.** Thank you for suggesting the appropriate terms. We agree that \"spurious solution\" is a more appropriate term than \"fake solution.\" We have **revised the title and updated the terminology** throughout the manuscript accordingly.", "claims_and_evidence": "SNOT has been in existence for a while, but the paper is the first to identify a reasonable sufficient condition for the avoidance of spurious solutions for the minimax formulation.  The paper identifies that spurious solutions are avoided if the source distribution does not concentrate on sets with Hausdorff dimension <= d-1. This result is interesting but not surprising. The proposed smoothing algorithm is quite natural and the paper has done a good job illustrating its effectiveness. The paper also proves that the smoothed distributions weakly converge to the original, ensuring the learned transport plan approaches the true OT plan.", "methods_and_evaluation_criteria": "NA", "theoretical_claims": "The identified sufficient condition is use for guiding the proposal of the smoothing algorithms.", "experimental_designs_or_analyses": "One interesting fact explained in the paper is that, the noise scheduling (gradual reduction) is critical; constant noise leads to mode collapse.", "relation_to_broader_scientific_literature": "NA", "essential_references_not_discussed": "NA", "other_strengths_and_weaknesses": "NA", "other_comments_or_suggestions": "change \"fake solution\" to \"spurious solution?\"", "questions_for_authors": "NA", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: The paper presents a thoughtful and well-motivated approach to overcoming known issues in semi-dual neural optimal transport, specifically addressing the occurrence of spurious solutions. The proposed method, OTP, introduces a smoothing-based strategy with solid theoretical backing, including a sufficient condition for convergence to the correct transport map. The paper also delivers practical insights through a sequence of carefully designed experiments, demonstrating improved stability and performance compared to standard baselines in both synthetic and real-world tasks. The reviewers highlighted several strengths: clear identification of a key limitation in existing semi-dual OT methods, meaningful theoretical contributions (e.g., the condition on Hausdorff dimension), and the design of a simple yet effective regularization scheme via noise scheduling. The authors’ responses during the rebuttal were thorough and constructive. They expanded the theoretical scope of their results, clarified design choices, conducted additional experiments on established benchmarks, and addressed reviewer suggestions regarding terminology and related work. Overall, this is a technically sound, well-presented, and timely contribution that advances our understanding and practical handling of neural OT models. I recommend acceptance.
Decision: Accept (poster)