Overcoming Spurious Solutions in Semi-Dual Neural Optimal Transport: A Smoothing Approach for Learning the Optimal Transport Plan

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.

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Experimental Designs Or Analyses:

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Q. One interesting fact explained in the paper is that, the noise scheduling (gradual reduction) is critical; constant noise leads to mode collapse.

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.

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Other Comments Or Suggestions:

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Q. Change "fake solution" to "spurious solution?"

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.

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.

We hope our responses to be helpful in addressing the reviewer's concerns.

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Other Strengths And Weaknesses:

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Q. It would be beneficial to include a comparison of training complexity and time.

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.

However, 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].

Reference

• [1] Choi, J., Choi, J., & Kang, M. Analyzing and Improving Optimal-Transport-based Adversarial Networks. ICLR 2024.

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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.

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.

Moreover, 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:

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:

• $c$ is $C^1$, strictly convex with bounded Hessian, and $\mu$ does not assign positive mass to sets of Hausdorff dimension $d-1$.
• $\mathcal{X}$ and $\mathcal{Y}$ are compact, $\mu$ is absolutely continuous, and $c$ is a $C^1$ strictly convex.
• $c(x,y) = h(x-y)$ 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 $d-1$.

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Questions For Authors:

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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 ?

A. Thank you for the thoughtful question. To avoid confusion, we revised the statement in Thm 3.1. as follows:

In particular, a map $x \mapsto y_{x} \in \cal{D}_{x}$ is a unique OT Map $T^{\star}$ $\mu$-a.s..

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.

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Q. For the second row of figure one, with what algorithm do you compute the max-min solution ? Are the results stochastic ?

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.

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Other Comments Or Suggestions:

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Q. Citations, Typos, and Presentations

A. We sincerely thank the reviewer for this detailed feedback. All comments regarding citations, typos, and presentations have been addressed as suggested.

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.

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Questions For Authors:

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Q. General theory in the appendix but quadratic cost in the main text?

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.

Nevertheless, 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:

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:

• 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 $d-1$;
• The sets $\mathcal{X}$ and $\mathcal{Y}$ are compact, $\mu$ is absolutely continuous, and $c$ is a $C^1$ strictly convex function;
• 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 $d-1$.

Then, the max-min solution of the SNOT problem recovers the correct OT map.

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Q. Relationship between the LPIPS metric and the transport cost.

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:

https://github.com/anonymousicml25/Overcome_Fake_Solution_OTP/tree/main

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Q. Additional experiments on the pathological cases (Table 1) with $d=256$ and on the benchmark introduced in [Korotin et al., 2022].

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:

https://github.com/anonymousicml25/Overcome_Fake_Solution_OTP/tree/main

Reference

• [1] Korotin et al., "Do Neural Optimal Transport Solvers Work? A Continuous Wasserstein-2 Benchmark.", NeurIPS, 2021.

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Q. Additional related work: Bridge matching and non-minimax approaches

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.

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Other Strength and Weakness:

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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?

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.

Regarding 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.

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Other Comments Or Suggestions:

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Q. Fake solutions may violate the marginal constraint or cost optimality.

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:

https://github.com/anonymousicml25/Overcome_Fake_Solution_OTP/tree/main