Projection Optimal Transport on Tree-Ordered Lines

We appreciate the reviewer’s feedback and have provided the following responses to address the concerns raised about our paper. Below, we summarize the weaknesses and questions highlighted by the reviewer and provide our answers accordingly.

W1. Some minor weaknesses are that there are no statistical analysis for the Tree Sliced-Wasserstein distance proposed, e.g. no sample complexity, nor topological analysis. One can wonder how it relates with other distances, it TSW-SL a lower-bound of the Sliced-Wasserstein or of the Wasserstein distance?

Answer. Given the paper's emphasis on the construction of tree systems and the corresponding Radon Transform, and considering the already extensive content, we have chosen to defer the analytical and statistical examination of TSW-SL to future work.

It is important to highlight that analyzing these aspects of TSW-SL presents unique challenges due to the inclusion of splitting maps. This feature is exclusive to Tree-Sliced Wasserstein variants and sets them apart from Sliced Wasserstein variants. We are actively investigating the properties of splitting maps, which appear to be a highly promising research direction.

Q1. When citing references about work which project on lower dimensional subspaces, some references are missing such as [1,2,3].

Answer. We appreciate the reviewer for highlighting those citations. We have incorporated them into our revised paper (Line 57).

Q2. I think in Equation (2), there is an abuse of notation as the Radon transform of a density is a density, and not a measure.

Answer. We acknowledge the reviewer's observation regarding the misuse of notation. We have carefully reviewed the paper and addressed similar errors in the revision.

Q3. The construction given in Algorithm 1 only samples a chain tree. Did you also test with sampling trees where nodes have different childs?

Answer. We evaluated the performance of models using alternative tree structures and found their performance to be comparable. Since the chain tree structure offers a straightforward approach for sampling trees and calculating Eq. (13), we opted to focus exclusively on the chain tree structure in the main body of the paper.

Considering the paper's emphasis on the construction of tree systems and the associated Radon Transform, and given its already comprehensive content, we have chosen to reserve further analysis on tree sampling and other aspects of TSW-SL for future work.

Q4. For the experiments on SW generators (Section 6.3), the number of projections (50 and 500) seems very low compared to the original paper, where I think they use something like 100K projections.

Answer. In the GAN experiment described in Section 6.3, we evaluated the performance of our TSW-SL method in comparison with SW using a total of 50 and 500 projection directions. In the existing literature, [1] also introduced SW with control variates for GANs and tested it using 10 and 1000 projection directions. Interestingly, we observed that increasing the number of projection directions from 500 to 1000 does not improve the FID score. For instance, [1] reported an FID score of 10.05 on CelebA when using SW with 1000 projection directions, which is higher than the FID score of 9.62 achieved with SW using 500 projection directions, as detailed in Section 6.3 of our paper. Therefore, we choose to conduct experiments to compare TSW-SL and SW with 50 and 500 projecting directions.

Reference.

[1] Khai Nguyen and Nhat Ho. Sliced wasserstein estimation with control variates. In The Twelfth International Conference on Learning Representations, 2024.

Once again, we sincerely thank the reviewer for their feedback. Please let us know if there are any additional concerns or questions from the reviewer regarding our paper.

We would like to thank the reviewer again for your thoughtful reviews and valuable feedback.

We would appreciate it if you could let us know if our responses have addressed your concerns and whether you still have any other questions about our rebuttal.

We would be happy to do any follow-up discussion or address any additional comments.

Q2. Can you give some stronger argument to the claim that tree systems allow avoiding a loss of topological information? Is this claim related with the number of lines that should be sampled ? What is the impact of this parameter onto the perfomances?

Answer. Given that the complexity of TSW-SL, as presented in Section 5.1 of our main text, is $\mathcal{O}(Lknlog⁡n+Lkdn)$ (L is the number of trees, k is the number of lines per tree, n is the number of supports for each distribution, and d is the number of dimensions in the original space), while the computational complexity of SW is $\mathcal{O}(Lnlog⁡n+Ldn)$ ($L$ is the total number of projection directions in SW), we conducted experiments such that the total number of projection directions in SW equals the product of the number of trees and lines in TSW-SL for a fair comparison. For example, when we set the total number of projection directions in SW to 50 in Section 6.3, we conducted experiments with TSW-SL using $3$ and $5$ lines per tree, selecting the number of lines accordingly to ensure that the total number of projection directions for both TSW-SL and SW was approximately the same. Additional experimental results on the impact of the number of lines per tree are provided in Table 5 in Appendix E.3, where we explore different configurations with varying numbers of lines per tree in TSW-SL. The results show that although the number of lines in each tree does have a different impact on performance, given the same number of total projection directions, our TSW-SL consistently yields better results compared to SW and Orthogonal SW.

Q3. Is it possible to derive additional properties of TSW-SL? For instance, does it metrizes the weak convergence? What are its statistical properties?

Answer: Given the paper's emphasis on the construction of tree systems and the corresponding Radon Transform, and considering the already extensive content, we have chosen to defer the analytical and statistical examination of TSW-SL to future work.

It is important to highlight that analyzing these aspects of TSW-SL presents unique challenges due to the inclusion of splitting maps. This feature is exclusive to Tree-Sliced Wasserstein variants and sets them apart from Sliced Wasserstein variants. We are actively investigating the properties of splitting maps, which appear to be a highly promising research direction.

We appreciate the reviewer’s feedback and have provided the following responses to address the concerns raised about our paper. Below, we summarize the weaknesses and questions highlighted by the reviewer and provide our answers accordingly.

W2. There is also no discussion regarding the impact of the number of lines, which seems to be a critical aspect of the method—especially as using only one line recovers the standard SW. Additionally, the influence of the splitting maps is not addressed, despite being central to the new Radon transform.

Answer. Given that the complexity of TSW-SL, as presented in Section 5.1 of our main text, is $\mathcal{O}(Lknlog⁡n+Lkdn)$ ($L$ is the number of trees, k is the number of lines per tree, n is the number of supports for each distribution, and d is the number of dimensions in the original space), while the computational complexity of SW is $\mathcal{O}(Lnlog⁡n+Ldn)$ ($L$ is the total number of projection directions in SW), we conducted experiments such that the total number of projection directions in SW equals the product of the number of trees and lines in TSW-SL for a fair comparison.

For example, when we set the total number of projection directions in SW to $50$ in Section 6.3, we conducted experiments with TSW-SL using $3$ and $5$ lines per tree, selecting the number of lines accordingly to ensure that the total number of projection directions for both TSW-SL and SW was approximately the same. Additional experimental results on the impact of the number of lines per tree are provided in Table 5 in Appendix E.3, where we explore different configurations with varying numbers of lines per tree in TSW-SL.

The results show that although the number of lines in each tree does have a different impact on performance, given the same number of total projection directions, our TSW-SL consistently yields better results compared to SW and Orthogonal SW.

Q1 + Q2. What are the main arguments in favor of TSW-SL wrt other many variants of SW? Could you highlight the unique advantages of TSW-SL?"

Can you give some stronger argument to the claim that tree systems allow avoiding a loss of topological information?

Answer. The motivation for this paper arose from a simple yet intriguing idea: In the framework of Sliced Wasserstein (SW), a probability distribution on $\mathbb{R}^d$ is pushed forward onto a line. This raises the question: what does the resulting distribution reveal about the original one? It is evident that distinct distributions, when projected onto the same line, can become indistinguishable.

Now, let us compare a tree system composed of two lines, $a$ and $b$ in $\mathbb{R}^2$, along with a distribution $\mu$ on $\mathbb{R}^2$. For simplicity, assume that $\mu$ is a Dirac delta distribution.

• Many Dirac delta distributions in $\mathbb{R}^2$ become identical after being projected onto line $a$, and the same holds true for line $b$. However, in most cases, the projections of $\mu$ on both lines of the tree system can uniquely identify the original Dirac delta distribution. ("Most cases" excludes exceptional scenarios, such as when $a$ and $b$ are the same line.)
• The above reasoning might appear insufficient because it evaluates the tree system by considering only one of its lines at a time. What happens if we evaluate the tree system using both lines together? This is where the splitting map becomes essential. The splitting map enables a more versatile allocation of mass between the two lines, rather than concentrating all the mass onto one line.

In summary, with the same number of lines—and thus the same computational cost—tree systems in TSW-SL provide a significantly deeper understanding of probability distributions compared to individual lines in SW.

A natural question arises: if a better representation space is desired, why not replace one-dimensional lines with higher-dimensional subspaces of $\mathbb{R}^d$? The answer lies in computational feasibility. Optimal Transport in $\mathbb{R}^d$ for $d>1$ is computationally prohibitive due to the lack of a closed-form solution. In contrast, both SW and TSW-SL offer efficient closed-form expressions, making them more practical.

We believe this explanation adequately addresses the reviewer's concerns.

Comment 3. Line 467: The loss should not be denoted as $\mathcal{L}$ since this letter represents a line.

Answer. Thank you for your correction. We have made the adjustments in our manuscript.

Comment 4. Section 6.2: It is difficult to visually confirm that TSW-SL produces images that most closely resemble the target.

Answer. Figure 5 showcases both qualitative and quantitative results. Qualitatively, methods that yield images with colors more closely matching the target image are considered superior. The figure displays the generated images and the corresponding 2-Wasserstein distances at the final epoch, with a smaller 2-Wasserstein distance indicating better performance.

For a more detailed qualitative comparison of our methods with SW, MaxSW, and other SW variants, we provide an additional example of the color-transfer task in Appendix E.2. This example further demonstrates that our TSW-SL and MaxTSW-SL loss functions, when used to measure the distance between two color distributions, achieve superior results both qualitatively and quantitatively.

Comment 5. Appendix p.17, line 870: $\overline{L}$ should be $\mathcal{L}$.

Answer. Thank you for your correction. We have made the adjustments in our revision.

Comment 6. Example A.14 is unclear: $n_i$ should have length 6 (i.e., $i = 0$ to $5$). What do the arrows signify? Additionally, an arrow seems to be missing on line 1035. The progression through step $i$ is confusing.

Answer. Thank you for your correction. We have revised the Example A.14 in our revision as follows:

Definition A.13. Let $T$ be a nonnegative integer, and $n_1, \ldots, n_T$ be $T$ positive integer. A sequence $s = \\{x_i\\}\_{i=0}^T$, where $x_i$ is a vector of $n_i$ nonnegative numbers, is called a tree representation if $x_0 = [1]$, and for all $1 \le i \le T$, $n_i$ is equal to the sum of all entries in vector $x_{i-1}$.

Example A.14. For $T = 5$ and $\{n_i\}_{i=1}^{5} = \{1,3,4,2,3\}$, the sequence $s ~ \colon ~ x_0 = [1] \rightarrow x_1 = [3] \rightarrow x_2 = [2,1,1] \rightarrow x_3 = [1,0,2,0] \rightarrow x_4 = [1,2] \rightarrow x_5 = [0,0,1].$ is a tree representation.

Comment 7. Page 25: Please provide more detail about the transition from eq. (46) to eq. (47).

Answer. We recall the transition from Eq. (46) to Eq. (47).

$\displaystyle \int_{\mathbb{T}} \biggl(\text{W}\_{d\_\mathcal{L},1}(\mathcal{R}^\alpha\_\mathcal{L} \mu\_1, \mathcal{R}^\alpha\_\mathcal{L} \mu\_2) + \text{W}\_{d\_\mathcal{L},1}(\mathcal{R}^\alpha\_\mathcal{L} \mu\_2, \mathcal{R}^\alpha\_\mathcal{L} \mu\_3) \biggr) ~d\sigma(\mathcal{L}) \ge \int_{\mathbb{T}} \text{W}\_{d\_\mathcal{L},1}(\mathcal{R}^\alpha\_\mathcal{L} \mu\_1, \mathcal{R}^\alpha\_\mathcal{L} \mu\_3) ~d\sigma(\mathcal{L})$

This transition comes directly from the inequality

$\text{W}\_{d\_\mathcal{L},1}(\mathcal{R}^\alpha\_\mathcal{L} \mu\_1, \mathcal{R}^\alpha\_\mathcal{L} \mu\_2) + \text{W}\_{d\_\mathcal{L},1}(\mathcal{R}^\alpha\_\mathcal{L} \mu\_2, \mathcal{R}^\alpha\_\mathcal{L} \mu\_3) \ge \text{W}\_{d\_\mathcal{L},1}(\mathcal{R}^\alpha\_\mathcal{L} \mu\_1, \mathcal{R}^\alpha\_\mathcal{L} \mu\_3),$

since the Wasserstein distance $\text{W}\_{d\_\mathcal{L},1}$ is a metric on the space of all measures on $\mathcal{L}$.

We sincerely thank the reviewer for the valuable feedback. If our responses adequately address all the concerns raised, we kindly hope the reviewer will consider raising the score of our paper.

Q5. It is stated in the beginning of section 6 that "the root is positioned near the mean of the target distribution": what is the impact of this setting?

Answer: Through empirical observations, we discovered that initializing the tree root near the mean of the target distribution results in more consistent performance for our models.

We believe a thorough investigation into the analytical and statistical properties of TSW-SL is necessary to provide a theoretical explanation for these findings. However, since the paper primarily concentrates on the construction of tree systems and the associated Radon Transform, and given its already extensive scope, we have decided to defer these aspects of TSW-SL to future research.

Comment 1. In the gradient flow experiment, differences in ground cost (e.g., $L_2$, tree metric) between methods make it challenging to compare results at a fixed number of iterations.

Answer. As described in Section 6.1 of the main text (Lines 408–413), the objective of this task is to update the source distribution to make it as close to the target distribution as possible. The empirical advantage of our TSW-SL over other SW-variants is demonstrated by measuring the 2-Wasserstein distance between the source and target distributions at steps 500, 1000, 1500, 2000, and 2500. In this task:

• TSW-SL, MaxTSW-SL, and other SW-variants are used as the loss function.
• The 2-Wasserstein distance serves as the evaluation metric.

It is crucial to note that the roles of the 2-Wasserstein distance (evaluation metric) and the other distances, such as TSW-SL, MaxTSW-SL, and SW-variants (loss functions), are separate. Hence, there is no difference when evaluating the distance between the updated source distribution and the target distribution.

Comment 2. Table 1: By Wasserstein distance, do you mean $W_2$? Additionally, could you clarify the timings reported in Table 2?

Answer. The Wasserstein mentioned in our paper is indeed $2-$Wasserstein distance $W_2$. In Table 2, the number reported is the average $2$-Wassersein distance over 10 runs. We have also added the average time to calculate the distance at one iteration over 10 runs in Table 2 of our manuscripts. The timings reported in Table 1 and Table 2 suggests that although the time per iteration for TSW-SL is slightly higher, the substantial reduction in Wasserstein distance underscores its efficiency across both datasets.

Table 2: Average Wasserstein distance between source and target distributions of 10 runs on high-dimensional datasets.

Q4 + W1. As noted in the conclusion, the paper "introduces a straightforward alternative to SW by replacing one-dimensional lines with tree systems," aiming to provide a more geometrically meaningful space. This objective aligns with several SW variants ... it remains unclear why and under what circumstances TSW-SL should be preferred over SW or its many variants.

It is stated in the conclusion that improved performances are expected by adapting recent advanced sampling schemes or techniques of SW to TSW-SL. It seems that the extension is not that straightforward: can you comment on that, and how improved performances can be expected?

Answer: Thanks for your nice question. We are eager to elaborate further on it. Roughly speaking, the tree-sliced framework in our paper is built on two key insights:

• Local Perspective: Each line in a tree system is treated similarly to a line in the Sliced Wasserstein (SW) framework. Splitting maps determine how the mass at each point is distributed across the lines, and then the projection of these mass portions onto the lines is processed in the same way as in SW. A significant challenge with this approach is verifying whether the injectivity of the corresponding Radon Transform is preserved, as this determines whether the proposed metric qualifies as a true metric or merely a pseudo-metric. However, we addressed this concern by providing a proof in the paper.
• Global Perspective: Tree structures and splitting maps establish connections between the lines, creating a cohesive system. This introduces a novel aspect compared to SW, enabling interaction and integration among the lines in a tree system. The Wasserstein distance can now be computed on this space with a closed-form expression, analogous to how lines are treated in SW.

From these insights, several follow-up directions for TSW-SL can be pursued, focusing on each perspective:

• Local Perspective: Consider, for example, the Generalized Sliced Wasserstein (GSW) distance [1]. In GSW, the SW framework is retained, but the projection mechanism is altered. Specifically, GSW generalizes the integration level set in the Radon Transform, replacing the level set defined by the inner product (representing orthogonal hyperplanes) with one defined by an arbitrary function. Similarly, in TSW-SL, which currently relies on the inner product, a framework could be developed that generalizes this by using an arbitrary function, offering new flexibility and applications.

• Global Perspective: More detailed analysis can aid in designing improved tree structures and splitting maps. For instance, splitting maps could be enhanced to account for both positional information from points and lines, rather than relying solely on lines as in the current implementation. This approach forms the core concept of one extension of the TSW-SL paper, which has also been submitted to this venue and can be accessed here.

It is important to note that while these ideas might seem straightforward, their development is non-trivial. For example, a critical property that must be preserved in TSW-SL variants is the injectivity of the corresponding Radon Transform. Additionally, from an empirical perspective, replacing lines with tree systems shows promising results. To illustrate this, let's consider the Denoising Diffusion Models task in our paper. We closely follow [2], which, to the best of our knowledge, is the most recent paper evaluating SW methods in diffusion models. Table 4 reports the following:

• The FID for the vanilla SW method is 2.90.
• The FIDs for four variants using random-path projecting directions from [2] are 2.70, 2.82, 2.87, and 2.88.

By merely replacing the foundation lines with tree systems, without introducing new techniques, our model using TSW-SL achieves an FID of 2.83. Models using TSW-SL with additional techniques (in the mentioned submission), achieve FIDs of 2.60 and 2.525.

We believe that our response regarding the both perspectives address the reviewer's question effectively. We hope this paper serves as a catalyst for a new research direction in the field, and we are actively working on advancing these ideas further.

References.

[1] Soheil Kolouri, Kimia Nadjahi, Umut Simsekli, Roland Badeau, and Gustavo Rohde. Generalized sliced wasserstein distances. Advances in neural information processing systems, 32, 2019.

[2] Khai Nguyen, Shujian Zhang, Tam Le, and Nhat Ho. Sliced wasserstein with random-path projecting directions. In Forty-first International Conference on Machine Learning, 2024.

We would like to thank the reviewer again for your thoughtful reviews and valuable feedback.

We would appreciate it if you could let us know if our responses have addressed your concerns and whether you still have any other questions about our rebuttal.

We would be happy to do any follow-up discussion or address any additional comments.

Q1. Could you explain how your training time (time per iteration, per epoch, overall time to convergence) compares with the one from DDGAN and the other baselines considered in the experiments.

Answer. Regarding the training time of our methods in DDGAN and other baselines in Table 4, we have already provided the training time per epoch of our methods compared with other baselines. We have further provided the training time per iteration as a new column in Table 4 of the revision of our paper. In terms of overall time to converge, it is impossible for us to provide in this discussion phase due to lack of time and resources since it requires reproducing the full training of all baselines mentioned in [1]. We will include the overall time to convergence of the remaining baseline in the final revision of our manuscript.

Q2. Could you please provide the convergence plots (FID as a function of epoch, e.g., once per several epochs) for your model vs. DDGAN vs. some other SW baseline? I would like to understand how stable is the overall training of your model compared with the baselines.

Answer. We have added the FID plots of our TSW-SL-DD compared to SW-DD in Figure 9 of Appendix E.4 of the revision of our paper. The results show that our TSW-SL-DD achieves a greater reduction in FID scores compared to SW-DD during the final 300 epochs. Due to the lack of time and resources during the discussion periods, we cannot provide FID score over epochs of other SW variants reported in the papers (since we follow the results reported in [10] to compare with our TSW-SL-DD results). Additionally, reproducing all baselines for DDGAN experiments would require time and resources far beyond what is feasible during the discussion period. We will include the FID over epochs of the remaining baselines in the final revision of our manuscript.

We sincerely thank the reviewer for the valuable feedback. If our responses adequately address all the concerns raised, we kindly hope the reviewer will consider raising the score of our paper.

We sincerely appreciate the reviewer’s thoughtful feedback and have provided the following responses to address the concerns raised about our paper. Below, we begin with a general reply and then proceed to address each of the reviewer’s questions individually.

It appears to us that the reviewer may not be fully familiar with the context of Optimal Transport, particularly the specific topic of our paper, which focuses on the sliced variants of the Wasserstein distance in Optimal Transport.

It seems to me that the main purpose of the proposed TSW-SL seems to be to compare the probability distributions, a feature which is primary needed for GAN training nowadays

The Wasserstein distance is known to have a supercubic computational complexity concerning the number of supports in the input measures. Specifically, for probability measures with at most $n$ supports, its computational complexity is $\mathcal{O}(n^3 \log n)$ [1]. This high computational cost has driven the development of sliced variants in Optimal Transport, which provide more computationally efficient alternatives to the original Wasserstein distance.

In the case of the original Sliced Wasserstein distance, the computational complexity is reduced to $\mathcal{O}(L n \log n + Ldn)$, where $d$ is the dimension of the supports and $L$ is the number of samples used in the Monte Carlo approximation. This represents a significant improvement in computational efficiency, reducing the complexity from $n^3 \log n$ to $n \log n$.

In our TSW-SL distance, the computational complexity is $\mathcal{O}(Lkn \log n + Lkdn)$, where $L$ is the number of samples in the Monte Carlo approximation, $k$ represents the number of tree levels, $n$ is the number of supports, and $d$ is the dimensionality of the supports. This complexity retains the same order with respect to the number of supports $n$, making it computationally efficient while incorporating additional structural information through the tree system.

This is also confirmed by the fact that most more practically-related papers (from CVPR, ECCV, etc.) still rely on vanilla/NS/WGAN loss with additional regularizations rather than other more complex losses (like SW-based) proposed by the community later. This suggests that (in 2024) the contribution of the current paper (TSW-SL as a GAN loss) may be relatively minor, so I am more on the negative side about the paper.

It is important to clarify that the primary contribution of our paper is not about introducing TSW-SL as a GAN loss as the reviewer stated. Instead, the paper focuses on introducing a novel integration domain, referred to as tree systems, as a replacement for the traditional lines used in the Sliced Wasserstein framework. The GAN experiments in our work primarily serve to highlight the advantages of this new integration domain when compared to lines.

As stated in our paper (Line 377): "It is worth noting that the paper presents a simple alternative by substituting lines in SW with tree systems, focusing mainly on comparing TSW-SL with the original SW, without expecting TSW-SL to outperform more recent SW variants."

Improving existing components in the SW framework ([1], [2], [3], [4], [5], [6], etc.) and extending one-dimensional lines to more complex domains such as one-dimensional manifolds or low-dimensional subspaces ([6], [7], [8], [9], etc.) are active and evolving areas of research in the Machine Learning community. Hence, we believe it is not entirely fair to evaluate a theoretical contribution like ours solely on one specific task. Instead, the focus should be on the generality and potential of the introduced approach.

References.

[1] Khai Nguyen, Nicola Bariletto, and Nhat Ho. Quasi-monte carlo for 3d sliced wasserstein. In The Twelfth International Conference on Learning Representations, 2024.

[2] Khai Nguyen, Nhat Ho, Tung Pham, and Hung Bui. Distributional sliced-wasserstein and applications to generative modeling. In International Conference on Learning Representations, 2021.

[3] Kimia Nadjahi, Alain Durmus, Pierre E Jacob, Roland Badeau, and Umut Simsekli. Fast approximation of the sliced-wasserstein distance using concentration of random projections. Advances in Neural Information Processing Systems, 34:12411–12424, 2021.

[4] Ishan Deshpande, Yuan-Ting Hu, Ruoyu Sun, Ayis Pyrros, Nasir Siddiqui, Sanmi Koyejo, Zhizhen Zhao, David Forsyth, and Alexander G Schwing. Max-sliced wasserstein distance and its use for gans. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 10648–10656, 2019.

[5] Soheil Kolouri, Kimia Nadjahi, Umut Simsekli, Roland Badeau, and Gustavo Rohde. Generalized sliced wasserstein distances. Advances in neural information processing systems, 32, 2019.

[6] Clément Bonet, Laetitia Chapel, Lucas Drumetz, and Nicolas Courty. Hyperbolic sliced-wasserstein via geodesic and horospherical projections. In Topological, Algebraic and Geometric Learning Workshops 2023, pages 334–370. PMLR, 2023

[7] David Alvarez-Melis, Tommi Jaakkola, and Stefanie Jegelka. Structured optimal transport. In International conference on artificial intelligence and statistics, pages 1771–1780. PMLR, 2018.

[8] Franc¸ois-Pierre Paty and Marco Cuturi. Subspace robust wasserstein distances. In International conference on machine learning, pages 5072–5081. PMLR, 2019.

[9] Jonathan Niles-Weed and Philippe Rigollet. Estimation of wasserstein distances in the spiked transport model. Bernoulli, 28(4):2663–2688, 2022.

[10] Khai Nguyen, Shujian Zhang, Tam Le, and Nhat Ho. Sliced wasserstein with random-path projecting directions. In Forty-first International Conference on Machine Learning, 2024.

[11] K. Nguyen and N. Ho. Sliced wasserstein estimation with control variates. In The Twelfth International Conference on Learning Representations, 2024.

We would like to thank the reviewer again for your thoughtful reviews and valuable feedback.

We would appreciate it if you could let us know if our responses have addressed your concerns and whether you still have any other questions about our rebuttal.

We would be happy to do any follow-up discussion or address any additional comments.

• GAN Experiment Results and Their Significance: We acknowledge your concern regarding the modest numerical gains in FID scores and the possible variability of these metrics. Nevertheless, we believe that the improvements—22.43% over DDGAN and 2.4% over SW—are significant, especially considering the already low baseline FID scores. Furthermore, when placing our results within the broader context of image generation benchmarks, such as those on CIFAR-10, our approach ranks among the top 49 methods. This highlights the effectiveness of our method in advancing diffusion models.

• Log scale of our newly added figure for FID plots over epochs: As noted in Appendix E4 of our manuscripts, we utilize a logarithmic scale in FID plots due to the wide range of values (from over 400 in initial epochs to less than 3.0 in final epochs). We think that this scale offers clearer visualization, and we will include the FID plot in the original scale in the final revision of our manuscript.

Thank you again for your thoughtful feedback, and we are happy to address any further concerns regarding our work.

Best regards,
The Authors

References

[1] Soheil Kolouri, Kimia Nadjahi, Umut Simsekli, Roland Badeau, and Gustavo Rohde. Generalized sliced wasserstein distances. Advances in neural information processing systems, 32, 2019.

[2] Nadjahi, K., Durmus, A., Jacob, P. E., Badeau, R., & Simsekli, U. (2021). Fast approximation of the sliced-Wasserstein distance using concentration of random projections. Advances in Neural Information Processing Systems, 34, 12411-12424.

[3] Khai Nguyen and Nhat Ho. Sliced wasserstein estimation with control variates. In The Twelfth International Conference on Learning Representations, 2024.

[4] Khai Nguyen, Nhat Ho, Tung Pham, and Hung Bui. Distributional sliced-wasserstein and applications to generative modeling. In International Conference on Learning Representations, 2021.

[5] Khai Nguyen, Nicola Bariletto, and Nhat Ho. Quasi-monte carlo for 3d sliced wasserstein. In The Twelfth International Conference on Learning Representations, 2024a.

[6] Khai Nguyen, Shujian Zhang, Tam Le, and Nhat Ho. Sliced wasserstein with random-path projecting directions. In Forty-first International Conference on Machine Learning, 2024.

[7] Izquierdo, S., & Civera, J. (2024). Optimal transport aggregation for visual place recognition. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (pp. 17658-17668).

[8] Wu, X., Dong, X., Nguyen, T. T., & Luu, A. T. (2023, July). Effective neural topic modeling with embedding clustering regularization. In International Conference on Machine Learning (pp. 37335-37357). PMLR.

[9] He Zhao, Dinh Phung, Viet Huynh, Trung Le, and Wray Buntine. Neural topic model via optimal transport. arXiv preprint arXiv:2008.13537, 2020.

Dear Reviewer MtTT,

Thank you for your detailed feedback and thoughtful analysis. We appreciate the time you’ve taken to evaluate our work and provide constructive insights.

We would like to further address your concerns regarding the significance of our proposed framework below.

• Application to ML-related problems: While we understand the importance of demonstrating the relevance of our framework to real-world ML-related problems, we emphasize that the primary contribution of our paper is the introduction of tree systems as a novel integration domain within the Sliced Wasserstein (SW) framework. As noted in the revised paper (Line 377), our goal is to present this as an alternative to traditional lines in SW, with the focus being on comparing TSW-SL with the original SW, rather than developing a loss that can achieve state-of-the-art results in a specific domain like generative models.

• Soundness of Our Proposed Distance in the Optimal Transport Field: To the best of our knowledge, our work introduces the first method to formally construct a tree system that preserves topological properties, accommodates dynamic supports, and achieves computational efficiency comparable to the Sliced Wasserstein (SW) distance. TSW-SL serves as a bridge between SW and TSW, leveraging the increased flexibility of TSW while retaining the dynamic adaptability characteristic of SW. We believe our method is non-trivial for several reasons. Firstly, it opens up new avenues for efficient computation in Optimal Transport, overcoming the limitations of both SW and TSW. As discussed in our global aspect of our General Response, the framework we present lays the groundwork for future variants of TSW-SL, which could retain its advantages while incorporating more sophisticated improvements. For example, future enhancements could involve refining the splitting maps to incorporate positional information from both points and lines, rather than relying solely on lines as in the current implementation. This idea is a central part of an extension to the TSW-SL framework, referenced by Reviewer 7kaA, and is also being submitted to this venue, with access available here.

• Broader impacts of our works in real-world applications: Rather than generative models for images, the need to efficiently compare probability distributions in applications such as Visual Place Recognition [7] and natural language processing tasks [8, 9, 10], where slow algorithms like Sinkhorn are often used despite SW’s promise since its computation relies on the projection of the supports in the original space into 1-dimensional space, which causes the loss of topological information. We believe that the introduction of our TSW-SL and other extensions of TSW-SL in the future will be a better option in assisting to advance these fields.

• The choice of experimental tasks: We choose to report the superiority of our methods compared with other SW-variants, varying from the synthetic task (gradient flow) and real tasks (color transfer and generative models including GAN and DDGAN) since each experiment aims to provide better aspects to highlight the empirical advantage of our methods compared with SW. From synthetic tasks to real-world problems such as generative models, we conduct experiments across various datasets, ranging from simpler datasets like CIFAR-10 (32×32) to more complex datasets like STL-10 (96×96) and CelebA (64×64). Most results show that our methods consistently outperform SW. Recognizing that experiments with SN-GAN are sufficient for simpler models, we extend our experiments to DDGAN due to its superior image generation quality. The results also indicate that our models yield better results than SW. We do not report the FID score in mean and standard deviation in DDGAN as in SN-GAN since the experiments with DDGAN take a long time to train. Most of the baselines we compare with are from [6], and this work does not provide the standard deviation for the DDGAN experiment due to long training time. With a resource constraint, we cannot repeat the training multiple times to report mean and standard deviation results.

Of all the broader impacts mentioned above, we choose to conduct our experiments on gradient flow, color transfer, and generative models since in the SW literature, there are existing works that validate their empirical advantages through these experimental settings (gradient flow [1, 2], Generative Adversarial Networks [3, 4], color transfer [5], Denoising Diffusion Model [6]). We think that comparing our methods with the same setting as existing works can help better for a fair comparison and alleviate the need to re-implement all previous methods in new tasks, thus completing the broad picture of our evaluation.

W1. The poor evaluation is the main weakness of the paper. The authors provided several experiments, but only Tables 1 and 2 offer an expressive justification of the proposed method.

Answer. The paper proposes a straightforward alternative to Sliced Wasserstein (SW) by substituting lines with tree systems, focusing primarily on comparing Tree-Sliced Wasserstein (TSW-SL) with the original SW. A significant advantage of tree systems is that they serve as a simple replacement for lines while preserving all the beneficial properties of lines in SW.

Enhancements to TSW-SL could be achieved by incorporating advanced techniques designed for SW, as tree systems can be locally considered as lines. Techniques applied to lines, such as the Generalized Sliced Wasserstein [2], could similarly be adapted for TSW-SL. However, while these extensions may appear straightforward, their implementation involves significant challenges. For instance, it is crucial to ensure that any TSW-SL variant retains the injectivity of the corresponding Radon Transform.

Given the extensive research and development of improved SW variants over the years, we do not anticipate TSW-SL to surpass the performance of more recent SW variants. Our focus remains on establishing the foundational aspects of TSW-SL, leaving further developments for future research.

Q4. Table 4 does not demonstrate that the proposed method actually provides better results in FID.

Regarding Table 4, which relates to Denoising Diffusion Models, we closely follow [3], the most recent paper evaluating SW methods in diffusion models, to the best of our knowledge. Table 4 reports:

• The FID for the vanilla SW method is 2.90.
• The FIDs for four variants using random-path projecting directions from [3] are 2.70, 2.82, 2.87, and 2.88.

By merely replacing the foundation lines with tree systems, without introducing new techniques, our model using TSW-SL achieves an FID of 2.83. As noted by reviewer 7kaA, extensions of TSW-SL are also under consideration at this venue. One such submission can be found at this link, where models using TSW-SL with additional techniques achieve FIDs of 2.60 and 2.525.

For these reasons, we believe the paper should be evaluated by the development of tree systems and the corresponding Radon Transform aspects (i.e., compare apple-to-apple), rather than focusing on numerical comparisons with state-of-the-art methods.

We sincerely thank the reviewer for the valuable feedback. If our responses adequately address all the concerns raised, we kindly hope the reviewer will consider raising the score of our paper.

We appreciate the reviewer’s feedback and have provided the following responses to address the concerns raised about our paper. Below, we summarize the weaknesses and questions highlighted by the reviewer and provide our answers accordingly.

Q1. What should I observe in Figure 5? The images appear identical to me. More expressive examples of the color transfer are necessary.

Answer. The primary objective of the color-transfer task is to navigate along the curve connecting the color distributions of the source and target images. In details:

• We employ our TSW-SL and MaxTSW-SL loss functions to minimize the distance between the color distributions of the generated and target images.

• The 2-Wasserstein distance serves as the evaluation metric for the similarity between the generated and target images, with a smaller 2-Wasserstein distance indicating better performance.

Figure 5 showcases both qualitative and quantitative results. Qualitatively, methods that yield images with colors more closely matching the target image are considered superior. The figure displays the generated images and the corresponding 2-Wasserstein distances at the final epoch.

For a more detailed qualitative comparison of our methods with SW, MaxSW, and other SW variants, we provide an additional example of the color-transfer task in Appendix E.2. This example further demonstrates that our TSW-SL and MaxTSW-SL loss functions, when used to measure the distance between two color distributions, achieve superior results both qualitatively and quantitatively.

Q2. Why did you consider your distance as the regularization for the GAN model?

Answer. In the experiment with the GAN model discussed in Section 6.3, we use TSW-SL as the loss function to compute the distance between the distributions of the target image and the generated image. Specifically, we employ it as the generator loss in the GAN model, not as a regularization term. This approach is closely based on the methodology of the Sliced Wasserstein generator [1], with details provided in the Appendix E.3.

Q3. In my understanding, gradient flow methods from section 6.1 in higher dimensions would provide better evidence. The paper demonstrates how the proposed distance can help decrease the distance to the target datasets. However, there is no evaluation showing that the computed distance is actually closer to the real Wasserstein distance provided.

Answer. It appears that the reviewer may have misunderstood the role of TSW-SL, MaxTSW-SL, and other SW-variants in our experiments. To clarify, these are utilized as loss functions.

As described in Section 6.1 of the main text (Lines 408–413), the objective of this task is to update the source distribution to make it as close to the target distribution as possible. The empirical advantage of our TSW-SL over other SW-variants is demonstrated by measuring the 2-Wasserstein distance between the source and target distributions at steps 500, 1000, 1500, 2000, and 2500. In this task:

• TSW-SL, MaxTSW-SL, and other SW-variants are used as the loss function.
• The 2-Wasserstein distance serves as the evaluation metric.

It is important to emphasize that the roles of the 2-Wasserstein distance (evaluation metric) and the other distances, including TSW-SL, MaxTSW-SL, and SW-variants (loss functions), are distinct. Therefore, it is not the objective of this task to identify a distance that more closely approximates the true Wasserstein distance.

References.

[1] Deshpande, I., Zhang, Z., & Schwing, A. G. (2018). Generative modeling using the sliced wasserstein distance. In Proceedings of the IEEE conference on computer vision and pattern recognition (pp. 3483-3491).

[2] Soheil Kolouri, Kimia Nadjahi, Umut Simsekli, Roland Badeau, and Gustavo Rohde. Generalized sliced wasserstein distances. Advances in neural information processing systems, 32, 2019.

[3] Khai Nguyen, Shujian Zhang, Tam Le, and Nhat Ho. Sliced wasserstein with random-path projecting directions. In Forty-first International Conference on Machine Learning, 2024.

We would like to thank the reviewer again for your thoughtful reviews and valuable feedback.

We would appreciate it if you could let us know if our responses have addressed your concerns and whether you still have any other questions about our rebuttal.

We would be happy to do any follow-up discussion or address any additional comments.

Thank you for addressing my concerns. I'm raising my score.

Q2. Evaluation of TSW-SL.

The paper proposes a straightforward alternative to Sliced Wasserstein (SW) by substituting lines with tree systems, focusing primarily on comparing Tree-Sliced Wasserstein (TSW-SL) with the original SW. A significant advantage of tree systems is that they serve as a simple replacement for lines while preserving all the beneficial properties of lines in SW.

Enhancements to TSW-SL could be achieved by incorporating advanced techniques designed for SW, as tree systems can be locally considered as lines. Techniques applied to lines, such as the Generalized Sliced Wasserstein [1], could similarly be adapted for TSW-SL. However, while these extensions may appear straightforward, their implementation involves significant challenges. For instance, it is crucial to ensure that any TSW-SL variant retains the injectivity of the corresponding Radon Transform.

From an empirical perspective, replacing lines with tree systems shows promising results. To illustrate this, let's consider the Denoising Diffusion Models task in our paper. We closely follow [2], which, to the best of our knowledge, is the most recent paper evaluating SW methods in diffusion models. Table 4 reports the following:

• The FID for the vanilla SW method is 2.90.
• The FIDs for four variants using random-path projecting directions from [2] are 2.70, 2.82, 2.87, and 2.88.

By merely replacing the foundation lines with tree systems, without introducing new techniques, our model using TSW-SL achieves an FID of 2.83. Models using TSW-SL with additional techniques (in the mentioned submission), achieve FIDs of 2.60 and 2.525.

Given the extensive research and development of improved SW variants over the years, we do not anticipate TSW-SL to surpass the performance of more recent SW variants. Our focus remains on establishing the foundational aspects of TSW-SL, leaving further developments for future research.

Reference.

[1] Soheil Kolouri, Kimia Nadjahi, Umut Simsekli, Roland Badeau, and Gustavo Rohde. Generalized sliced wasserstein distances. Advances in neural information processing systems, 32, 2019.

[2] Khai Nguyen, Shujian Zhang, Tam Le, and Nhat Ho. Sliced wasserstein with random-path projecting directions. In Forty-first International Conference on Machine Learning, 2024.

We are glad to answer any further questions you have on our submission.