Summary
This paper proposes an extension to Sliced Wasserstein Distance (SW), an approach for measuring distances between distributions by computing the average of the energy of the 1-d Wasserstein distances between 1-d projections. The authors argue that moving towards non-uniform ways of sampling the projections is key, and subsequently show that the benchmark approach currently that does so (Distributional sliced Wasserstein) adds another optimization over a parametric distribution family, which may not yield stable results and is more computationally expensive. Subsequently, they propose a simple non-parametric extension to SW called Energy based SW (EBSW) that achieves the same objective of sampling non-uniformly, i.e. sampling projections with larger Wasserstein distance with a greater probability, by considering energy functions $f$ which are monotonically increasing w.r.t the sliced 1-d Wasserstein distance itself. Theoretical properties are highlighted, including the convergence properties of their proposed metric. Next, the authors propose a range of Monte Carlo estimation methods to approximate EBSW, including deriving the necessary gradient expressions to be used in some of their subsequent applications. Lastly, over multiple experiments ranging from point-cloud gradient flows to deep point-cloud reconstruction, the authors demonstrate that EBSW is faster (Importance Sampling approach), and yields distributions closer to the ground truth in most cases than other SW-based benchmarks.
Strengths
I really liked reading the paper and going through the various findings in it. The following are what I consider to be some of the main strengths of this work. (i) The paper is really well-written. All sections were intuitive to read and easy to understand for me, and this also applied to the Appendices, including the proofs of the Theorems and Propositions. (ii) The contributions of the paper are very clear and supported by the theoretical and empirical results. All proposed benefits of the approach have been verified by the authors in theory and practice. (iii) The experiments are well motivated and relatively exhaustive. The proposed approach is tested in a wide range of problems and the results convincingly showcase the benefits of EBSW. (iv) The background section is well organized and is an effective introduction to SW-based metrics in general.
Weaknesses
The paper is overall interesting and well-written. However, there are a few points noted below that can improve the work further.
1. I feel that the theoretical results can be given a bit more perspective w.r.t. the other SW-based metrics. Overall, I felt the theoretical results in Theorems 1, 2 and Propositions 1 and 2 are relatively intuitive to show. Some more specific insights on EBSW could be interesting (perhaps with constrained energy functions, i.e. Lipschitz constrained $f$).
2. I understand the overall direction for moving towards measures that look disproportionately at projections which have larger distances (lines 37-38). However, I feel that a few more words to elaborate on intuitively why one must move towards non-uniform distributions in the Sliced Wasserstein setting could be useful.
3. One of the main questions I have is the role of $p$, which is fixed for all experiments in this work. I feel that the authors can elaborate a bit on whether trying larger values of $p$ can achieve a similar goal to what EBSW-e achieves (More details in questions).
4. The result tables are convincing and informative w.r.t. the tested methods. However, in some of the visual depictions, I found it hard to see any significant differences between EBSW and some other variants (even in the Appendices). Mainly, the color transfer experiments were a bit hard to discern for me in terms of relative performance
Questions
In addition to the list of weaknesses above, below are some questions and general suggestions for the authors to address:
1. I noticed that the authors fix the value of $p$ to 2 for all experiments. I wonder if increasing $p$ can achieve a similar effect to using a monotonically increasing energy function (such as the exponential distribution in this work). Because intuitively, it seems to me that increasing $p$ eventually also gradually assigns more importance to the projections with values near to the maximum. I’m mainly asking this because increasing $p$ still falls under SW-p, and thus could perhaps have similar sample complexity and computational load. Could you please provide a theoretical explanation on how changing $p$ fundamentally differs from EBSW with an increasing energy function?
2. For Figure 2, the authors state that the gradient flows for EBSW are smoother than other approaches. Perhaps describing some visual cues to elaborate on that point can make it easier to follow.
3. Although four different ways of estimating the EBSW variants are proposed, I only see the IS-EBSW being reported in the main paper. Furthermore, having looked at the results in the appendix, it seems in almost all cases, the variants have roughly similar performance, and IS-EBSW seems to lead to better results in most cases. Furthermore, it also seems that the time complexity for all algorithms are similar, also seeing Table 3 in C.1, it seems that in most cases IS-EBSW is faster as well. Thus, my question is: in discussing these other approaches, are there benefits to the other proposed MCMC variants?
4. Are there any cases where the choice of an exponential energy function $f$ doesn't work? It would be intuitive to me that, considering a more general family of exponentials in $e^{(\lambda x)}$, the optimal "width" of the slicing distribution as set by $\lambda$ may be different for different cases. For instance, can there be any cases where the underlying distribution enforces a much larger or smaller $\lambda$ than one? Or is $\lambda=1$ in some sense universal?
5. As a follow-up to the previous question, I would assume that for the optimization results in Tables 1 and 2, as the generated distribution edges closer to the ground truth distribution, would it be profitable to increase $\lambda$ higher than one? As then the distances would all get very small, and potentially max-SW would start to look more appealing.
6. The theoretical results are interesting and important, however, could the authors put into perspective how the theoretical properties of EBSW compare to other SW based measures? It seems to me that both Theorems 1 and 2 may hold for SW based measures (at least SW, not sure about max-SW and DSW). Similarly, for Proposition 2 it seems from the proof that it would hold for both SW and max-SW (not sure about DSW). Additional theoretical results in this direction, or at the least a rough intuitive discussion on how these results would compare for the other SW counterparts, would be useful. This would put more into perspective how EBSW's theoretical properties compare to the other SW benchmarks.
Rating
6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
Confidence
4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.
Limitations
Some limitations have been discussed, but I feel there is scope to discuss more limitations of the proposed approach.