Venue: ICLR 2023 Conference

Reviews:

final_score: rating: 6: marginally above the acceptance threshold
review_content: summary_of_the_paper: A novel variant of slice-based Wasserstein distance is proposed in the paper, namely the spherical sliced Wasserstein (SSW) distance. The SSW is particularly suited to evaluating the discrepancy between probability measures defined on spherical manifolds. By projecting measures defined on hyperspheres onto great circles with the proposed spherical Radon transform, the SSW is defined as the expectation of Wasserstein distances between obtained measures on great circles w.r.t to the uniform distribution over the Stiefel manifold. The proposed SSW has lower computational costs compared to standard Wasserstein distances on $R^d$, since Wasserstein distances on circles admit closed-form solutions. Experiment results show that the SSW indeed can better capture geometric information from data than the original SW when the data of interest lies on spherical manifolds.
strength_and_weaknesses: Strength:

The SSW is the first step towards defining slice-based Wasserstein distances for measures defined on manifolds. Closed-form solutions to projections onto great circles are derived in the paper and subsequently the spherical Radon transform is also defined analogously to the standard Radon transform.

Weakness:

There are a few things that can be improved in the paper. 
(1) The projection efficiency is not mentioned in the paper, but this is something important to slice-based Wasserstein distances. Can we quantify how many projections are needed to gain a good approximation of the distance? 
(2) What is the computational cost of SSW compared to the standard SW? Runtime comparison shown in Figure 2 does not include standard SW. From the experiments, the standard SW also produced competitive results.
(3) It is not explained clearly in the paper why the SSW has better performance than the standard SW. The overall argument is that the data manifold is taken into consideration in SSW, but detailed discussions and analysis to support this claim with visualisations, experiments, and theoretical analysis will make it a stronger paper.

clarity,_quality,_novelty_and_reproducibility: The paper is well-written, the proposed SSW is clearly explained, this work is the first step towards slice-based Wasserstein distances on spherical manifolds. The experiment results can be reproduced.
summary_of_the_review: This paper introduced an interesting variant of SW distances which accommodates well with measures defined on spherical manifolds. The method is novel, and experiments show its effectiveness on several tasks.
review_title: null

final_score: rating: 6: marginally above the acceptance threshold
review_content: summary_of_the_paper: The paper proposes a slice-based approach to efficiently compute the Wasserstein distance between two distributions $\nu$ and $\mu$ defined on the sphere. The method termed SSW (Spherical  Sliced Wasserstein) projects the distributions on great spheres and then, hinges on the efficient computation of Wasserstein distance on the circle to propose the sliced-variant of Wasserstein distance on the sphere. Along with SSW, the related spherical Radon Transform and its dual operator are formulated. Finally it is established that the proposed SSW defines a pseudo-distance. A numerical algorithm that summarizes the design of the random projection matrices, the computation of the samples' coordinate on the circle as well as the Wasserstein distance on the circle. Empirical evaluations on simulation dataset or on learning latent representations via auto-encoders highlight the potential of the proposed method.
strength_and_weaknesses: **Strengths**
- The rationale behind the proposed Spherical Sliced-Wasserstein (SSW) method is justified.
-  The approach aims to design an appropriate geodesic projection on great circles of distributions defined over sphere. That projection acts as the projection on straight line used in Sliced Wasserstein (SW). Overall the projection operators are formulated as matrices over Stiefel manifold. Using known results on Wasserstein distance on circle, a sliced version of Wasserstein distance on sphere is derived.
- As a contribution, the paper proposes a closed-form projection of the training samples from the sphere to the great circle using matrix on Stiefel manifold.
- Another interesting contribution is the proposal of the spherical Radon transform and related dual operator that allow to rephrase SSW as a function of the new Radon transform.  The pseudo-distance property of SSW is established in Proposition 4.
- Intensive empirical evaluations are conducted. They show that SSW exhibit a favorable runtime in a large scale setting. Used as a loss function, SSW allows for density estimation of simulated and real datasets with advantageous empirical results. Also SSW proves effective in generative modeling applied to classical Mnist, Fashion Mnist and Cifar datasets.

**Weaknesses**
- The plain metric property of SSW is not guaranteed.
- Topological properties of SSW compared to plain Wasserstein distance on sphere are not discussed. Can one theoretically or empirically relate SSW to its plain counterpart?
- For readability purpose, it would be interesting to formally introduce the notion of great circle in the paper. 

*Minor comments*

- In Figure 4 illustration, which dataset is shown?
- On page 8, the sentence "We see on this figure that the normalizing flows fitted are able to recover the principles modes of the data" is not clear. Where exactly on Figure 5 the recovery of the modes are highlighted?
- The paper should better introduce (at least in the supplementary material) some preliminary notions of Wasserstein distance on the circle. Especially, the parametrization by the interval [0, 1[ on the real line should be better explained.
- Some notations or abbreviations have to be clarified or formally defined within the main text. For instance, in equation (6), $C(\mathbb{R} \times S^{d-1})$ is not stated. Page 4, what $\mathcal{P}_{p, ac}$ stands for? Page 9, NF is for normalizing flows?
clarity,_quality,_novelty_and_reproducibility: Overall, the paper is well written and easy to follow up. It thoroughly  reviews the background and main works on Sliced Wasserstein and Wasserstein on circle to introduce optimal transport on sphere and its sliced version computation. The proposed contributions follow the classical procedure of deriving efficient sliced optimal transport computation. However the SSW departs from existing works by addressing the challenging Wasserstein distance between distribution defined on sphere, paving the way to the derivation of computationally efficient algorithm for OT on manifold. As such the paper is of interest and the contributions are somehow significative.
summary_of_the_review: The paper is well written and proposes a new sliced optimal transport on a sphere. It also proposes the related spherical Radon Transform. The new discrepancy measure of distributions over sphere is supported by its pseudo-metric property and empirical experiments showing its effectiveness. Topological properties of the proposed approach and some few details on the empirical  evaluations remain to be clarified.
review_title: null

final_score: rating: 8: accept, good paper
review_content: summary_of_the_paper: The paper introduces Spherical Sliced-Wasserstein (SSW) as a "pseudometric" to measure the dissimilarity between distributions defined on a hypersphere. At its core, the paper utilizes the prior work proposed by Rabin et al. 2011a that studies transport-based distances on circles and proposes efficient numerical schemes for calculating these distances. The authors then propose to slice distributions defined on a hypersphere by projecting them onto the great circles leading to spherical slices. Lastly, following the vanilla-sliced Wasserstein framework, the Spherical Sliced-Wasserstein is defined as the expected Wasserstein distance between the spherical slices of two distributions. According to Rabin et al. 2011a, the 1-Wasserstein distance between two distributions defined on a circle has a closed-form solution and can be obtained efficiently.  In addition, the authors show that the 2-Wasserstein distance between a distribution defined on a circle and the uniform distribution also has a closed-form and computationally efficient solution (Proposition 1). 

The paper provides extensive experiments (both in the main paper and in the appendix) showing that the proposed SSW "pseudometric" is effective in various applications, from density estimation to gradient flow and learning auto-encoders. The appendix also provides additional results on, for instance, self-supervised learning with spherical latent representations. The results indicate the robustness and effectiveness the proposed dissimilarity measure. 
strength_and_weaknesses: ### Strengths

* The paper addresses the exciting problem of comparing distributions defined on a hypersphere, which appears in many applications.
* The paper introduces a simple and computationally efficient pseudometric for comparing spherical distributions that is theoretically sound. 
* The paper is very well written, and it is easy to follow. 
* The authors show the application of the proposed SSW on a wide variety of applications. What I particularly appreciate is that while some of the experiments are not large-scale, they remain challenging to solve by existing methods. 

### Weaknesses

I don't see any significant weaknesses in this paper. Below are not necessarily weaknesses but points that could improve the paper. 

* It would have been great if the authors could show that the proposed Radon transform is invertible, and hence SSW is indeed a metric as opposed to a pseudometric. 
* I am surprised that SW distance works so well compared to SSW for spherical distributions. I was expecting a more significant performance gap. 

 
clarity,_quality,_novelty_and_reproducibility: * Clarity is one of the strengths of this paper. 

* While the transport-based distances on circles have been studied before, this paper introduces a novel spherical slicing that enables the definition of SSW. Hence, the proposed approach has sufficient novelty. 

* The algorithmic simplicity of the approach allows for a transparent implementation of the proposed pseudometric. In addition, the code was provided by the authors, and I found the results easily reproducible. 
summary_of_the_review: Overall, I think this is a fantastic paper. Here are my primary considerations in evaluating the work: 

* It is well-motivated, clear, and a pleasant read
* It is algorithmically simple while being theoretically sound
* It showcases the application of the proposed approach to a wide range of problems. 

While the experiments might be far from being large-scale, they remain relevant and challenging.
review_title: null

final_score: rating: 6: marginally above the acceptance threshold
review_content: summary_of_the_paper: Following the well-established Sliced-Wasserstein (SW) used as a Wasserstein-based metric to compare measures supported on the Euclidean space $\mathbb{R}^d$, this paper introduces a natural Spherical-Sliced-Wasserstein (SSW) distance to compare measures supported on an hypersphere. 

The construction technique follows the SW-one, namely : 
- First, observe that comparing measures supported on the circle $S^1$ can be done fairly efficiently (akin to the comparison of measures supported on $\mathbb{R}$). 
- In dimension $d$, intersect the hypersphere $S^{d-1}$ with 2d-planes (parametrized by the Stiefel manifold $V_{d,2}$) to get a (great) circle, project measures of interest $\mu,\nu$ on these circles, compute (efficiently) the Wasserstein distance between those measures. Eventually, integrate over all the circles. 

The authors show that this construction defines a pseudo-metric on $P(S^{d-1})$ (probability measures supported on the hypersphere) ; provide implementation details and showcase their approach in a variety of experiments.
strength_and_weaknesses: # Strengths : 
- The paper is overall very well written, with appropriate level of mathematical rigor (though this may be a matter of taste). 
- The construction is very natural yet (perhaps surprisingly) new to the best of my knowledge. 
- While I did not proofread the appendices in full details, proofs seems well-written and I did not identify a major flaw at first glance. 
- [minor but still] The figures are of high-quality. 

# Weaknesses : 
- (1) To some extend, one may argue that the contributions of the work are somewhat limited, in sense that the construction follows the exact step of the usual SW distance and, up to few technical modification brought to the Radon transform, there is no specific "breakthrough" in this paper. 
- (2) The setting is restricted to the spherical case. While this yield elegant formulas and possible implementation overall, it is also somewhat disappointing in sense that the proposed method may be generalized quite well (typically intersecting [insert assumptions here] hypersurfaces with $V_{d,2}$). From a theoretical standpoint, that may lead to a more instructive construction. 
- (3) A reference and discussion to the recent preprint _Intrinsic sliced wasserstein distances for comparing collections of probability distributions on manifolds and graphs_, Rustamov et al., may be relevant. In this work as well, the authors build a SW-like distance on manifolds. They follow a fairly different approach though (relying on the eigenfunctions of the Laplace-Beltrami operator on the manifold). It is not clear to me how the two methods compare. While the approach proposed in the current paper seems (to me) more natural and easier to implement, the one of Rustamov et al., has the benefits of being more general and quite original. 
clarity,_quality,_novelty_and_reproducibility: # Clarity 

As said above, the paper is very clear and well-written. 

# Quality

I think this is a competent paper, the proposed approach is robust and motivated, and experiments are convincing overall. 

# Novelty

As said above, the proposed approach is novel to the best of my knowledge, though it is naturally adapted from the usual SW distance. I overall think that the novelty is somewhat limited. See **Weaknesses** above. 

# Reproducibility 

The clarity of the paper makes the work reasonably easy to reproduce as far as I can tell. 
Some experimental details in the **Runtime Comparison** paragraph are lacking though. Saying that SSW is faster (or not) than Sinkhorn is not very inisghtful given that these algorithms compute different things. Sinkhorn approximates the true Wasserstein distance while "Monte-Carlo SSW" approximate the true SSW which, as far as I can tell, is not directly related to the true Wasserstein distance between the two measures. 
summary_of_the_review: I think this is a well-written paper that present a natural, well-motivated and easy-to-understand approach.

On the downsides, I would say that the paper does not bring any brand new idea from the theoretical side that would shed a new light on "numerical OT on manifolds" ; and while the experiments are interesting, I cannot assess the potential impact for numerical OT practitioners (though it clearly is non-zero).
review_title: null

Meta-review: metareview:_summary,_strengths_and_weaknesses: In this paper, the authors propose the Spherical Sliced-Wasserstein (SSW), which is a  "pseudometric" to measure the dissimilarity between distributions defined on a hypersphere. The idea of the sliced Wasserstein distance on a sphere is a new and novel. Moverover, the reviewers overall agree to accept the paper. So, I also vote for acceptance.

In the final version, we highly encourage authors update the paper based on the reviewer's comments. 
justification_for_why_not_higher_score: The sliced Wasserstein is a popular topic in Wasserstein distance. The work presented in this paper is its variant. The idea is overall interesting, however, the number of potential researchers who would be interested in this work is slightly limited. Thus, I recommend it with Accept (poster).
Decision: Accept: poster