Venue: NeurIPS 2022 Conference

Reviews:

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: The paper provides a Donsker-typed theorem for Projected Wasserstein distance in $l^{\infty}$ norm over the projected directions ball $S^{d-1}$. From that, it derives Limit Theorems for some well-known Sliced Wasserstein distances. Some simulation studies are carried out to support the theoretical findings.
strengths_and_weaknesses: **Strength**: 

1. The results in this paper are important for Optimal Transport theory and application. The presentation of the paper is clean and easy to read. 

2. All the proofs are solid and written carefully. 

questions: 1. Throughout the paper, the distributions of interest $P$ and $Q$ are assumed to be compactly supported. A natural question is "can we extend the result to a more general setting where $P$ and $Q$ have arbitrary supports?". It seems like the compact condition of the supports is only needed to show the existence of the potential and to bound the entropy number. If we can extract those refined conditions and put them together into a more general result (maybe in Appendix), it would be more impactful to OT research. If it is not possible, it is helpful to have a paragraph to explain why.

2. It would be interesting to see some simulation studies about the limiting distribution of the empirical process of Max-Sliced Wasserstein distances, even in low-dimensional cases. It would be helpful to build some intuition about this non-Gaussian limit.
limitations: The paper is well written and has no major limitations.
review_title: null

final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: summary: Optimal transport distances (a.k.a. Wasserstein distance)  have recently drawn ample attention in statistics and machine learning communities as powerful discrepancy measures for probability distributions. One of the bottlenecks of Wasserstein distance consists of its expensive computational cost. An alternative to making this problem computationally tractable leads in the fact that it can be cast on averaging calculations of $1$-D Wasserstein distances. This latter approach is referred to as Sliced Wasserstein distance (SWD). In a nutshell, SWD relies on projecting the data sample on some $u$-direction from the unit sphere, ($u \in \mathbb{S}^{d-1}).$

This paper addresses a theoretical unified problem of proving a Central Limit Theorem for the empirical SWD distance under a compact supports condition satisfied by the distributions in question.  The authors show that  the asymptotic distribution is a tight-centered Gaussian process on the unit sphere $\mathbb{S}^{d-1}$.


strengths_and_weaknesses: The paper is easy to follow, and it presents a unified framework for proving central limit theorem for a family of SWD distance, including vanilla SWD, Max-Sliced, Trimmed SWD, etc. This result in itself is solid, especially for statistical applications involving SWD. 


questions: 
- For the discrete SWD:
    - can one get directly the central limit from the one verified by the Wasserstein distance?
    - does this limit process uniform with respect to $L$, the number of $u$-random direction projections. Namely, if there is $L \leq L'$ random projections, the limit is (approximately) the same?
- In Theorem 2.4., can one derive an order of the convergence rate to zero?
limitations: This is not applicable to the paper.
review_title: null

final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: summary: The authors present a unified method for providing distributional limits for processes constructed from the empirical Wasserstein distance between one dimensional projections of empirical distributions. To do so, they rely on the theory of empirical processes and on the Hadamard directional differentiability of various functions.
strengths_and_weaknesses: This article follows a natural set of results that have been proven for both classical and entropic optimal transport. The presentation is clear and the article is easy to read. I develop the strengths and weaknesses further in the following.

Strengths:
- The paper proposes a unified method to prove distributional limits for distances based on sliced distributions and 1D Wasserstein distance. The method is based on a previous work [20], that addressed the case of empirical optimal transport.

Weaknesses:
- The novelty of this article seems to be limited as most of the major results have been proven in [19,23].
- The assumption (CC) that the support of the projected measures is connected for all the vector in the unit sphere is very restrictive. I agree that the general case involves a complicated geometry of the distributions, however, the unified framework on which the paper is based concerns very few applied cases.
- The method for proving the results of the paper uses classical tools of weak convergence of empirical processes and does not particularly differ from the case of classical optimal transport.

Overall, the results presented here are restricted to probability measures under strict assumptions and are therefore rather limited: every one dimensional projection of probability distributions onto the unit sphere must have an interval as a support. Moreover, this paper seems to have been published in a hurry (see "Minor comments" below), there is notably no conclusion and no perspective of using such results for example.

Minor comments:
- The labels are visible in the margin.
- The reference [28,29] appears twice.
- l.94 : "uniquness"
- In the proof of Theorem 2.1 and in Corollary 2.3, the random variables $X_i$ and $Y_i$ are not defined.
- In section 3.2, it is question of MSW, $\tilde{W}$ and WPP (which is not defined). Standardisation of terminology is necessary.
- In Theorem 3.1, the dependence on $\delta$ of the trimmed SW disappeared.
- l. 190 : "given in section 4.1" should be removed/rewritten.
questions: - Can you specify where the assumption of compactness of the support of the probability measure in $\mathbb{R}^d$ is necessary?

- As proved in [Nadjahi], the sliced-Sinkhorn divergences do not depend on the entropy regularization parameter, unlike the entropy regularized optimal transport (or Sinkhorn divergence). Therefore, considering sliced-Sinkhorn divergences could also be of interest. As similar empicial process methods have been used for entropy regularized OT, would it be possible to extend you result to this particular framework?

- It could be made clear from the beginning that the proposed method is in any case limited to absolutely continuous probability measures, as their support must be connected.

- Since your results apply for various functions F, it might have been interesting to propose a new divergence to assert the interest of your generalisation. I however acknowledge that it is not an easy task to propose a new satisfactory distance.

- More could be said about the trimmed sliced Wasserstein distance, and the precise meaning of "robustification".

- In the appendix B.1, you consider the case $p=1$, which is not included in the Wasserstein distance setting that you develop (Section 2). In particular Theorem 2.1 is stated for the case $p>1$. Are the results of appendix B.1 can be obtain directly? If so, this could be clarified.

[Nadjahi] - "Statistical and Topological Properties of Sliced Probability Divergences", Nadjahi et al.
limitations: The authors have specified that their method is quite restrictive in the sense that it only applies to a rather small set of probability distributions.
review_title: null

final_score: rating: 5: Borderline accept: Technically solid paper where reasons to accept outweigh reasons to reject, e.g., limited evaluation. Please use sparingly.
review_content: summary: This paper defines the Sliced Wasserstein process, a stochastic process defined by the empirical Wasserstein distance between slices of empirical probability measures. A uniform distributional limit theorem is derived. Experimental results demonstrate the convergence towards the limiting Gaussian process.
strengths_and_weaknesses: Strengths: 
- This work build upon the mathematical techniques developed in [20] and adapt them well for the SW process. 
- The results seem correct (I just skimmed through the proofs).

Neutral:
- Numerical results are convincing but I would have expected more examples, maybe ones that show the theory also hold (or not?) in the unbounded case.

Weaknesses:
- The paper is not very clear: more space could have been given for defining mathematical objects; a sketch of proof of the main theorem could have been given knowing that there is a lot of space not used at the end of the paper.
- Contrary to what is said in the Checklist, the code is not given (it is specified it is given in a footnote shown page 6), and I would have liked to look at it.
- The authors limit themselves to compactly supported probability measures P and Q (I am not certain if it’s possible to go beyond compact, maybe sub-Gaussian?). At least the authors could have developed a bit on why they considered this assumption and where are the difficulty for the unbounded case.
- The authors did not verified the documents they sent: all the labels of equations and sections are visible…
- There are a lot of typos (for instance the title of the SM does not correspond to the paper's title).
- I don't see any real applications for these results: I know NeurIPS accepts very theoretical results, but it would be useful to know how this could improve or help us understand SW distances better given the already known results on the theory of SW.

questions: - A question I have regarding the SW process is the following: G_n defined in Equation 2 is said to enjoy a uniform central limit theorem. My naive question is: given the fact that $\mathbb{E}[W_p^p(P_{nu}, Q_{nu})]\neq W_p^p(P_u,Q_u)$ (due to the permutation/transport plan of the Wasserstein distance), then G_n is defined according to the "wrong expectation"? (see Corollary 2 of https://proceedings.neurips.cc/paper/2020/file/eefc9e10ebdc4a2333b42b2dbb8f27b6-Paper.pdf for instance where you need to compute a sample complexity). Maybe what I am saying is not true because the authors are considering a single slice. I would like to have their opinion on this.

- How hard would it be to go beyond the compact case? At least Sub-Gaussian.

- I suggest the authors should use the remaining space they have (1.5 pages!) to develop sketches of proof in the main paper, or do more experiments, maybe to show that their theory is valid for the unbounded case numerically?

I am willing to raise my score accordingly if the answers to my questions are satisfying and the weaknesses addressed.
limitations: No potential negative impact
review_title: null

Meta-review: metareview: After the rebuttal period, the reviewers have come to an agreement on the paper being novel, interesting, the contributions being significant. The rebuttal also addressed most of the concerns, though I agree with reviewer 84Jv on the comment that experiments on non-compact settings would be a plus to see the limits of the theory. Overall, I believe this is a nice continuation for the prior art and I recommend an acceptance for the paper. 
confidence: Certain
Decision: Accept