We would like to first thanks you for your praises of our work such as finding our bilevel formulation "highly novel" and our numerical experiments "convincing evidence of the method’s effectiveness". We discuss below the main point that you addressed in your rebuttal regarding sample complexity.

Regarding sample complexity. We are not entirely sure we fully understand your question. We provide a tentative answer addressing two different points, regarding two different sample complexities, and we hope this will offer enough clarity. We remain available during the discussion period to better develop one of these two points.

• Regarding the sample complexity of the Monte Carlo approximation of the gradient. The Stein's approximation through DGSWP is done through a standard Monte Carlo approximation of the expectation. In this sense, the convergence rate of $\hat \nabla h_{\varepsilon, N}$ towards $\nabla h_\varepsilon$ has a rate of $O(1/\sqrt{N})$ where $N$ is the number of samples used (and crucially does not depends on the dimension of the features). Note that more refined use of Monte Carlo methods, for instance by using MCMC methods could lead to a better rate such as $O(1/N)$. We leave this issue to future works.

• Regarding the sample complexity of (sliced-)Wasserstein and DGSWP. The study of the statistical properties of (sliced-)Wasserstein has been the focus of considerable research in recent years. One significant result is that most sliced-based methods (including min-SWGG, see [1]) have a dimension-free sample complexity, which is not true for Wasserstein. This raises the question of the sample complexity of DGSWP. At this point, we do not have a definitive answer, as it depends on the function $\phi$ used to project the samples. If we consider a straight line, the dimension-free sample complexity of min-SWGG holds. Otherwise, it depends on the expressiveness of the neural network. For example, if the network can project samples onto a space of dimension $d \geq 2n$ where the samples are in general position, we would recover the sample complexity of Wasserstein. Then, we hypothesize that the resulting sample complexity will depend on the choice of the $\phi$ function. This is a very interesting question that we plan to explore in the near future.

[1] Mahey, G. (2024). Unbalanced and linear optimal transport for reliable estimation of the Wasserstein distance (Doctoral dissertation, INSA Rouen Normandie).

We are grateful for the reviewer's very positive feedback and are pleased that our work was well received. We would like to take this opportunity to address your comments and improve the manuscript based on your thoughtful suggestions.

Regarding a better exposition of our contributions. We will clarify our exact contribution more clearly. We want to emphasize that Definitions 1 and 2 are new objects that were never introduced in previous works. More precisely, [37] introduced SWGG that only tackles linear projection $\phi(x,\theta) = \langle x, \theta \rangle$, whereas [30] proposed generalized sliced Wasserstein that is a distinct metric from the one proposed in our submission. We will rephrase the first paragraph of Section 3 to better clarify this point.

Regarding comparison with other slicing methods. Except expected sliced Wasserstein (which we did not consider in the CFM experiment due to performance issues), sliced-based methods do not provide a transport plan. For CFM, getting such a plan is mandatory to sample trajectories. Then, they cannot be considered in this context. One major strength of DGSWP is its ability to provide good results in high dimensional problems, you are absolutely right. In the CFM experiments, the samples are set of images, one sample being one image seen as a vector of dimension 32 pixels $\times$ 32 pixels $\times$ 3 colors = 3072, which remains a challenging setting. Because we do not have access to extensive computation resources, we did not consider higher dimensional settings.

Regarding NFE values. We run additional experiments for a range of NFE values, which confirms the good performance of DGSWP. The following table gives the average values of FID over 3 runs +- std:

Regarding the ability of DGSWP to provide straight plans, it is indeed a desiderable behavior that is worth being investigating. Nevertheless, we believe that this question deserves a deeper investigation before raising complete conclusions, and we stick to the goal of this paper, which is to assess the ability of DGSWP to provide a meaningful plan.

Regarding evolutionary strategies. One of the points of our submission is to show that a typical gradient descent-type algorithm can be used to compute DGSWP. We fully agree that different methods could be considered, for instance evolutionary strategies or reinforcement learning methods. We want to point out that [37] used a 0-order method to compute minSWGG (that boils down to GSWP with a linear projection). We will add a comment that mention the work you indicated as a potential alternative to design an algorithm to solve the bilevel problem behind GSWP.

Regarding scaling. This is indeed an interesting question, thank you. It has been shown that, provided that i) we are able to sample the direction $\theta$ that minimizes SWGG ii) the samples are in general position (see [2]), min-SWGG is equal to Wasserstein when the dimension $d\geq 2n$. As DGSWP allows optimizing over $\theta$ (or a non-linear version of it), it is therefore natural to wonder under which conditions one could recover this result. However, it is hard to give a definite answer to it, for several reasons:

• as we face a non-convex problem, it is hard to guarantee that we will converge to the global optima
• to our knowledge, there is no network that guarantees that the embeddings will be in general position

Nevertheless, one can rely on this result to design a neural network architecture that is likely to provide good results, that is to say expressive enough to provide an embedding space in a large enough dimension. We propose to add a discussion about this on the paper.

[2] Cover, T. M. (1967). The number of linearly inducible orderings of points in d-space. SIAM Journal on Applied Mathematics, 15(2), 434-439.

We thank the reviewer for their comments. We remain available throughout the discussion period for any further clarifications, and we hope you will consider increasing your overall evaluation in the light of these clarifications.

Regarding the difficulty of our contribution. We thank the reviewer for highlighting the connection with GSW  [37] and minSWGG  [30].  While our approach builds on these ideas, and we obviously do not deny it, it is not a simple juxtaposition. The key contribution is to view this generalization of minSWGG as a bilevel optimization problem, and to carefully design a gradient-like method to solve it. It is well known that bilevel optimization in the context of non-strongly convex inner problem is difficult (see e.g., [1]) and we believe that our approximation results of GSWP are non-trivial. We agree that the proofs of Lemma 1 and Proposition 1 do not present technical difficulties, but we believe it is still important to formally state them, as they are necessary first steps before stating results on DGSPW. Despite these simple proofs, the approximation results that follow in Section 4 are less classical, and we believe that the proof of Proposition 2 contains interesting arguments (especially the fact that we obtain a gluing lemma "for free").

[1] Bolte, J., Lê, Q. T., Pauwels, E., & Vaiter, S. (2025). Geometric and computational hardness of bilevel programming. Mathematical Programming, 1-36.

Regarding the injectivity condition. We would like to recall that there is no injective map from $\mathbb{R}^d \to \mathbb{R}$ for $d > 1$, and we emphasize that we require only injectivity from the (potential) support to $\mathbb{R}$. This condition is satisfied almost surely in the linear case (all points in general configuration), and could be also satisfied with injective neural network [2]. Note that we require this property to prove that we obtain a metric, the object $d^\theta$ still make sense without this hypothesis. In particular, we did not observe performance collapse due to noninjective $\phi$ map, and we believe that it due that it is unlikely that the training procedure select a MLP that is degenerate with regard to this property. We do not know however to prove such result, since it involves structural properties of the training dynamics of neural networks. From a numerical point of view, we simply rely on the order given by the sort algorithm used by Python that can be interpreted as a selection map from the set of minimizers.

[2] Puthawala, Michael, et al. "Globally injective relu networks." Journal of Machine Learning Research 23.105 (2022): 1-55.

Regarding the choice of Conditional Flow Matching. The goal of the experimental section is to assess the performance of DGSWP in various contexts. The first experiment aims to provide a qualitative evaluation of the plan, while the second examines the convergence properties of DGSWP. Lastly, we seek to evaluate our method in a setting where a transport plan is required, and the number of samples is large. In this context, mini-batch OT is the main competitor, as, except for the expected sliced Wasserstein (see General comment), no sliced methods provide a transport plan. For these reasons, and to challenge DGSWP in a context where slicing methods have not been considered before, we focus on CFM, as [3] is well-positioned in the literature. Therefore, the primary motivation was not to deliver state-of-the-art performance on a generative modeling task but rather to investigate the performance of DGSWP across different settings. We propose to clarify this in the final version of the paper.

[3] Tong, A., Fatras, K., Malkin, N., Huguet, G., Zhang, Y., Rector-Brooks, J., ... & Bengio, Y. (2024). Improving and generalizing flow-based generative models with minibatch optimal transport. Transactions on Machine Learning Research, 1-34.

Regarding runtime comparison with OT solvers. You are perfectly right, we have not explicitly stated the complexity for our DGSWP method in the paper. We will fix that in the final version of the manuscript upon acceptance. Let $n$ be the number of samples in each distribution, $d$ be the dimensionality of the input space, $m$ be the number of Monte Carlo samples used for the gradient estimation in Eq. (8) and $p$ the number of gradient steps required to reach convergence, then the time complexity for the linear DGSWP is $O(p m n \log n + p m n d))$. Upon acceptance of the manuscript, we will add a visualization of the corresponding running times for both linear and NN variants of DGSWP.

Regarding CIFAR-10 evaluation. Results were originally reported for one single run; we performed additional experiments to report results computed on 3 runs:

To achieve these results, we performed the computations on a cluster consisting of multiple GPUs (utilizing only 2 GPUs for each experiment) with different GPU types, using a job scheduler based on resource reservation and availability. As a result, it is challenging to provide a definitive answer to the question about the required runtime. However, we can assert that the CFM experiment with mini-batch OT and DGSWP has roughly the same runtime (as mentioned in the paper: "aggregate samples from 10 minibatches to compute the transport plan, and find that the additional computational cost remains negligible"). As for I-CFM, the running time is approximately 25% lower than that of OT-CFM.

We would like first to thanks the reviewer for their positive comments, acknowledging the quality of the writing and the novelty. We would like to address your main comments below.

Regarding statistical significance. We have provided gradient flows results with several repetitions. For CFM, the paper contains results only for one run. We performed additional runs to better assess the variability of the results. Below, we report the average +- std for 3 runs, which keeps the conclusion unchanged.

Regarding Expected Sliced Optimal Transport. When considering ESW as an additional competitor, we get the results from Tables S.1 and S.2 below: For the dataset in Fig. 1, varying the temperature parameter $\tau$ for Expected Sliced Wasserstein, we get the following results in terms of transportation cost: Table S.1

For the gradient flow experiment in Fig. 4, we get the following performance after 2,000 iterations for the Swiss Roll dataset (note that results are consistent across all 4 datasets).

Table S.2

In other words, in the experimental setup of Fig. 1 and Fig. 4, ESW is consistently outperformed by both min-SWGG and DGSWP in terms of transportation cost and convergence of the corresponding gradient flows (varying the hyper-parameter from $\tau=0$ to $\tau=10$). Based on these initial comparisons on toy datasets, we have decided not to include ESW in the comparison for larger-scale problems such as the CFM experiment.

Regarding control variate for Monte Carlo estimation of gradient. Thanks for this comment. We believe that the line of works that you mention is very interesting. Overall, we kept a discussion on the variance reduction aspect quite minimalistic, both from a theoretical and practical aspects. We believe that finding better, or even optimal, variance reduction method working alongside with Stein's approximation is an interesting topic of future works.

Regarding assessing the quality of a transportation plan. Assessing the quality of a transportation plan is hard. In this work, we do so by computing the cost associated to a plan, ie. $\langle \pi, C\rangle$ (cf. Fig 1). When doing so for the dataset in Fig. 1, varying the temperature parameter $\tau$ for Expected Sliced Wasserstein, we get the results obtained above in Table S.2: Notably, we observe that the cost for ESW is consistently larger than that for min-SWGG (which is expected, since ESW averages plans over drawn directions while min-SWGG retains the direction of minimal cost), and in practice larger than that of DGSWP (NN). Note that one could argue that assessing the similarity of a transport plan with respect to the ground truth one could be done by computing the matrix norm of the differences between plans, but we observe in practice that such a distance tends to favor uninformative uniform plans over more meaningful sharp ones, which is why we avoid such metrics in our work. If the reviewer has suggestions for an alternative way to assess the quality of an estimated transport plan, we would be happy to provide such metrics for the compared methods during the author-reviewer discussion period.

We would like to thanks the reviewer for these positive comments on our work, and the careful reading. We will improve the manuscrit based on all your minor comments, in particular regarding the presentation of the different objects introduced in our paper. We address in the following some specific comments that may merit discussions or clarifications.

Regarding injectivity: We recall that there is no injective map from $\mathbb{R}^d \to \mathbb{R}$ for $d > 1$, and we emphasize that require only injectivity from the (potential) support to $\mathbb{R}$. It is true that asking for injectivity of a fine grid would have numerical consequences, but here we merely need it on the real support of the transport problem. We would like to add that this condition is satisfied almost surely in the linear case, and could be also satisfied with injective neural network [1]. Note that we require this property to prove that we obtain a metric, the object $d^\theta$ still make sense without this hypothesis. In particular, we did not observe performance collapse due to noninjective $\phi$ map. From a numerical point of view, we simply rely on the order given by the sort algorithm used by Python that can be interpreted as a selection map from the set of minimizers.

[1] Puthawala, Michael, et al. "Globally injective relu networks." Journal of Machine Learning Research 23.105 (2022): 1-55.

Regarding the differentiability: Thanks for this comment. You're absolutely right. Formally, we should work in the context of Bolte & Pauwels [2] with conservative Jacobians. Since this kind of notions is far away from the interest of the OT community of NeurIPS, we believe that it is better to keep a differentiable hypothesis to simplify the exposition. We will add a comment in the revision in the direction of a formal proof with conservative Jacobians (that are exactly design to tackle for instance ReLU functions).

[2] Bolte, Jérôme, and Edouard Pauwels. "Conservative set valued fields, automatic differentiation, stochastic gradient methods and deep learning." Mathematical Programming 188.1 (2021): 19-51.

Regarding practical benefits over SWGG. You are right, we erroneously introduce ASWD as AWD in Sec. 2.2 which impairs understanding. This will be fixed in the next iteration of the paper. Regarding comparison between min-SWGG (optim) and DGSWP (linear) at equal timings (sorry, figures are not allowed), we get (note that the iteration for DGSWP (linear) is chosen here so as to lead to a timing lower or equal to that of the corresponding min-SWGG run):

We will add such numbers in the next iteration of the paper and discuss the fact that min-SWGG can be more efficient early in the process and DGSWP (linear) becomes more accurate as running time grows.

Regarding the comparaison with Expected Sliced Wasserstein [35]. When considering ESW as an additional competitor, we get the results from Tables S.1 and S.2 below: For the dataset in Fig. 1, varying the temperature parameter $\tau$ for Expected Sliced Wasserstein, we get the following results in terms of transportation cost:

Table S.1

For the gradient flow experiment in Fig. 4, we get the following performance after 2,000 iterations for the Swiss Roll dataset (note that results are consistent across all 4 datasets).

Table S.2

In other words, in the experimental setup of Fig. 1 and Fig. 4, ESW is consistently outperformed by both min-SWGG and DGSWP in terms of transportation cost and convergence of the corresponding gradient flows (varying the hyper-parameter from $\tau=0$ to $\tau=10$). Based on these initial comparisons on toy datasets, we have decided not to include ESW in the comparison for larger-scale problems such as the CFM experiment.

Regarding function evaluations. In the CFM context, at generation time, given a model for $v(x, t)$, different solvers can be used to integrate the path from $x_0$ to $x_1$. It is common in the CFM litterature to evaluate the complexity of the generation steps as the number of calls to $v$ required for the integration [13, 34, 54]. Note that in all our experiments, $v$ is implemented using a UNet architecture, which is the same across methods. In practice, between I-CFM, OT-CFM and DGSWP-CFM, the only difference lies in the training procedure that is used to train $v$. As a consequence, NFE, which is the number of times the model $v$ is called is the adequate measure of complexity for the generation step.

Regarding local minima w.r.t $\theta$. You are absolutely right, DGSWP is non-convex and it converges towards a local minima. Even with simple scenario with 4 samples can lead to non-convex objective function. To assess the impact of the initialization, we perform several runs for each experiments. We notice that, in practice, results are similar across runs, even for the CFM experiment with is in dimension $d> 3000$.

Regarding positive curvature. It is indeed a very interesting question. DGSWP necessitates to define a $\phi$ function that projects the sample $\in \mathbb{R}^d \to \mathbb{R}$, and uses a sliced mechanism to obtain the permutations (i.e. it suffices to sort the samples according to their projection). It is not straightforward to adapt DGSWP to the context of spherical slicing: for instance, [4] projects on a circle (and not on a line), which necessitates to employ the SW distance on the circle; [5] uses stereographic projections that could be used but is not naturally rotationaly invariant. We believe that some adaptations of DGSWP should allows it to be adapted in this context but we leave it for future work. Regarding positive curvature spaces, and again relying the formulation of [6] to define manifold feature maps $\mathcal{M} \to \mathbb{R}$, one could naturally use DGSWP to compare SPD matrices (see [7] for an example with sliced-Wasserstein).

[4] Bonet, C., Berg, P., Courty, N., Septier, F., Drumetz, L., & Pham, M. T. Spherical Sliced-Wasserstein. ICLR.

[5] Tran, H., Bai, Y., Kothapalli, A., Shahbazi, A., Liu, X., Diaz_Martin, R. P., & Kolouri, S. (2024). Stereographic Spherical Sliced Wasserstein Distances. ICML.

[6] Katsman, I., Chen, E., Holalkere, S., Asch, A., Lou, A., Lim, S. N., & De Sa, C. M. (2023). Riemannian residual neural networks. NeurIPS.

[7] Bonet, C., Malézieux, B., Rakotomamonjy, A., Drumetz, L., Moreau, T., Kowalski, M., & Courty, N. (2023). Sliced-Wasserstein on symmetric positive definite matrices for M/EEG signals. ICML