Venue: NeurIPS 2022 Conference

Reviews:

final_score: rating: 3: Reject: For instance, a paper with technical flaws, weak evaluation, inadequate reproducibility and incompletely addressed ethical considerations.
review_content: summary: This work attempted to extend the sliced Wasserstein distance to be the convolution sliced Wasserstein distance by replacing vectors with tensors and Radon transforms with convolution operators, respectively. Then, this work compared its proposed methods with different convolutions (usual convolution, convolution with stride, convolution with dilation) with the original sliced Wasserstein on image generation task with various datasets (CIFAR10, CelebA, STL10, CelebA-HQ), claiming to show favorable results. Its theorem showed that this new convolution sliced Wasserstein distance is a pseudo-metric, it is always less than or equal to the max sliced Wasserstein distance, and its expectation is upper bounded with a specific value.
strengths_and_weaknesses: - Even though this manuscript claimed that "We derive convolution sliced Wasserstein (CSW) and its variants" in the abstract, CSW was not derived, but was defined in this work (see Definition 5). While this work seems to assume that the distance between "images" are compared with the Wasserstein distance, there are a number of cases where "vectorized feature tensors" of two images are compared with the (sliced) Wasserstein distance (e.g., see [35]) and in that case, this work can be seen as a simplified version of these works. In that sense, it can be seen that there is no novelty in the definitions of this manuscript.
- While this defined CSW was shown to be a pseudo-metric in Theorem 1, it may be a serious weakness as a loss to be used in the optimization. Will the different image with the distance 0 be good to represent Unfortunately, there is no discussion on this issue. Non-linear extension of CSW was mentioned in lines 328-332, it does not seem to be meaningful as a metric considering that linear version of it is a pseudo-metric.
-  While its experiments were showing results on image generation task for various datasets, it was only compared with a conventional SW method, not a recent max SW or generalized sliced Wasserstein [23]. Moreover, there were many different convolutions defined in the main text and compared in the experiments, the results seems to suggest no clear winner among these convolutions. Thus, the experiments in its current form seem weak. Comparing with recent Wasserstein variants and investigating proper (or optimal) convolution operators could improve this manuscript.
questions: - Feel free to argue back to the weakness points above.
- This work argued the needs for using tensors instead of vectors in the sliced Wasserstein distance. While it might be making sense, it was not easy for me to be convinced about these arguments. Conventional image processing works have widely used vector representations in their models and tensors (i.e., reshapes) in their implementations. While this manuscript mentioned "a probability measure over images should be defined over the space of tensors instead of images" and "this extra step does not keep the spatial structures of the supports, which are crucial information of images", I need more explanation on why these sentences are justified. Even though images are represented as vectors, spatial structures can be considered in the operations on them. In that sense, these arguments seem relatively weak.
limitations: Even though the authors indicated that the limitations and potential negative societal impact of their work, I was not able to find them.
review_title: null

final_score: rating: 4: Borderline reject: Technically solid paper where reasons to reject, e.g., limited evaluation, outweigh reasons to accept, e.g., good evaluation. Please use sparingly.
review_content: summary: This paper proposed convolutional sliced Wasserstein distance and compared it with the conventional sliced Wasserstein distance. In addition, the authors also introduced a convolution base slicer, a convolution stride slicer, and its variant with dilation. The paper provided the details about how the convolutional sliced Wasserstein is calculated, and analyzed several properties. The experiment results on the multiple datasets demonstrate that for the generative tasks, the proposed CSW achieved better performance than SW while using similar computational resources.
strengths_and_weaknesses: The paper is well written and provides a lot of details about SW, which is very helpful for readers like me with little SW background.
The idea is well motivated and the math seems convincing to me.

I must apologize I don't have enough expertise to judge the merit of this paper. Below are some of my questions and concerns.
1. In L201, the authors did notice that convolution is a linear operation. Thus consecutive convolution operations can be replaced with a single convolution. Except for the non-linear convolution slicer, what is the purpose of using multiple linear convolution operations? Moreover, convolution is often implemented as matrix multiplication, which also flattens the image and converts the kernel into a Toeplitz matrix. So the proposed convolutional SW is similar to SW with a special R in Eq. 1?

2. I wonder how the kernel is determined? From L228-230, it seems the kernel is uniformly sampled from the set $K^{(l)}$. Do we sample once for all the images (that is, using the same set of kernels for all the images) or we repetitively sample kernels for each image?

3. Some training details are missing. What is the loss function? Based on the main script, it seems the network does not need a discriminator as CSW directly computes the distance between the generated images and real images and the network is trained to minimize the distance. However, in the supplementary, discriminators are still used (L732-739). If discriminators are used, I am a little confused as SW GAN [13] used discriminators to map images into 1D vector and compute SW, which is similar to nonlinear CSW introduced in this submission, except that kernels in [13] are learnable while the kernels in this submission is uniformly sampled.

4. The authors might miss one reference about introducing convolution in Wasserstein to reduce the computational cost:
Justin Solomon, Fernando de Goes, Gabriel Peyré, Marco Cuturi, Adrian Butscher, Andy Nguyen, Tao Du, Leonidas Guibas, Convolutional Wasserstein Distances: Efficient Optimal Transportation on Geometric Domains, Proc. SIGGRAPH 2015.

Minor: L126 $R^{c\times \times d}$ should be $R^{c\times d\times d}$
questions: See the weakness section.

Overall this paper proposed an interesting idea to better compute the Wasserstein distance between two image sets. However I lack the related background knowledge and some experiment details are missing, I cannot judge the merit of this paper.
limitations: The authors did not provide the discussions or analysis about the limitations and potential negative societal impact (if I did not miss anything). Overall I think it is fine as the paper focuses more on the fundamental SW distance computation. See the weakness section for potential technical limitations, for example, the choice of kernels.
review_title: null

final_score: rating: 7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
review_content: summary: This paper presents a new methodology for comparing two probability measures over images.
The key idea of the paper is to apply convolution operators on probability measures over images.
The proposed method, named convolution sliced Wasserstein(CSW), makes use of the spatial structure of images and needs less slicing memory. 
The authors provide the metricity of CSW as well as its sample complexity, its computational complexity, and connection to sliced Wasserstein distance.
Beyond this theoretical contribution, the authors discuss numerical considerations and perform a thorough real data analysis, showing the favorable performance of their method.
strengths_and_weaknesses: The idea of applying the convolution operators on probability measures over images is a very natural idea. 
The proposed method is novel and simple to describe.
Also, the authors show the efficiency numerically. 
questions: **Major Comments**

* I am a little confused about the definition of the convolution-base slicer and its invariants.
In definition 3, when $d$ is even, the last kernel $K^{(N)}$ has dimension $1\times a \times a$.
However, $a=\frac{d}{2^{N-1}}$ and by definition $a$ is an integer.
Is there any guarantee about this? 
For example, what is the definition of these kernels when $d=18$?
In real data, the dimension $d=32,64,96,128$,
whose prime factor decomposition has at most one number which is not 2.
But $a=\frac{d}{2^{N-1}}$ won't be an integer when $d=18=2\times 3\times 3$.
This question also lies in the definition of other variants of convolution sliced-Wasserstein distances.

* The authors fix the stride size $s=2$ in definition of convolution-stride slicer.
There may be some trade-off between time consumption and estimation efficiency for different stride sizes $s$.
Some simple analysis may help us recognize it intuitively.
I am wondering how does the stride size affect the performance of the proposed method? Some discussion and empirical results are needed.

* This paper mainly considers probability measures over images. It would be convincing if the authors can provide other tasks in real data applications.


limitations:
review_title: null

final_score: rating: 7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
review_content: summary: While the Sliced-Wasserstein distance is widely used to compare distributions on images (which can be seen as tensors), it requires to perform a vectorization step using a reshape operation which is not suited to images and loses spatial structure, while being memory inefficient. In this work, authors propose to define a Convolutional Sliced-Wasserstein on the set of probabilities over tensors using convolution operations which are better suited to images. Some properties are derived such as the sample complexity, pseudo-distance. And many experiments are performed on classical image datasets.
strengths_and_weaknesses: This paper proposes to use convolutions to project images on the real line in order to use a SW distance between distributions over images. While the convolution idea is not novel since it has been widely used in neural networks for some time, the idea of using it in the context of optimal transport is new and very interesting. Moreover, the results seem quite convincing.

Strengths:
- Well written and very clear
- Using convolution to define a new SW distance.
- Application on different datasets which give good results
- Theoretical results

Weaknesses:
- Only a pseudo-distance?
questions: I have first a question on the theoretical results. In Theorem 1, it is stated that it is a pseudo-distance and that $CSW_p(\mu,\nu)=0 \nLeftrightarrow \mu=\nu$. I do not find the last statement very clear. I believe that it is meant that $CSW_p(\mu,\nu)=0$ does not imply that $\mu=\nu$. I did not find in the proof of the theorem any argument supporting this statement. Is there any counter example of $\mu\neq \nu$ such that $CSW_p(\mu,\nu)=0$?

A second question is on the experiments. While the results seem quite convincing, it seems like the experiments (to get e.g. the FID and the IS) were run only once. I believe it would be more robust and convincing to take a mean over several trainings for example.






Typos:
- Line 126: $\mu\mathbb{R}^{c\times \times d}$
limitations: Yes
review_title: null

Meta-review: metareview: The paper presents a new slicing methods for the Wasserstein distance between probability measures over images based on convolution operators. This way memory requirements can be reduced and locality can be better preserved. Experiments are conducted on generative modeling problems.
Reviewers noted that the idea of convolution operators on probability measures over images is natural and simple, yet novel and acknowledged theoretical and practical results. The rebuttals were in-depth and provided additional clarifications.
On the other hand reviewers note that CSW only defines a pseudo-metric.
Overall this paper is an interesting contribution to the NeurIPS community and should be accepted.
confidence: Less certain
Decision: Accept