Venue: NeurIPS 2022 Conference

Reviews:

final_score: rating: 9: Very Strong Accept: Technically flawless paper with groundbreaking impact on at least one area of AI/ML and excellent impact on multiple areas of AI/ML, with flawless evaluation, resources, and reproducibility, and no unaddressed ethical considerations.
review_content: summary: The paper proposes a score-based generative model (SGM) on compact Riemannian manifolds. The paper first points out three major components that contribute to the success of SGMs in Euclidean spaces.
Linear SDEs in Euclidean spaces (more precisely, OU processes) have stationary distributions and are Normal distributions.
For the SDEs, the reverse diffusions exist, whose initial distributions are the stationary distributions above, and the form of the reverse SDEs is known, yielding efficient parameterizations.
The solutions of OU processes are Normal distributions; thus, it is easy to sample and compute loss in denoising score matching (DSM).
The parametric reverse diffusions can be trained via minimizing various score matching losses, including DSM.
Thus, the paper proposes those counterparts for SGMs on compact Riemannian manifolds, specifically by identifying and solving the non-trivialities of the counterparts.

Then, the paper shows that linear diffusions on compact Riemannian manifolds have reverse diffusions, and the resulting SDEs are the same form as diffusions in Euclidean spaces.
However, the paper identifies that for the linear diffusions on compact Riemannian manifolds, its solution for a given data at time 0 doesn't have analytical forms, unlike OU processes. This makes two problems: the difficulty of simulation x_t given x_0 and the difficulty of computing denoising score matching loss.

To address this, the paper proposes using Geodesic random walks (GRW) for simulating both forward noising and reverse denoising processes; note that it is approximate sampling, unlike efficient sampling of the forward noising processes in SGMs on Euclidean spaces.

Fourth, the paper proposes two approximate methods to approximate the conditional log density gradients (or score) $\nabla_{x_t} p(x_t|x_0)$ of the forward noising process. When we know the eigenvalues and eigenvectors of the manifold of interest, one can approximate the conditional scores using the first N eigenvectors, i.e., the Truncation method. One can employ Varadhan's asymptotic, trading-off approximation errors when we don't have such information.

Next, the papers introduce DSM and implicit score matching (ISM) losses, whose computation will involve the simulation and approximation techniques above.

Finally, the paper introduces the neural network-based score function parameterizations suitable for compact Riemannian manifolds. Briefly speaking, when d-dimensional Euclidean spaces can describe the manifold, the suitable parameterization requires n > d sets of vector components.

For the experiments, the paper demonstrates the effectiveness of the proposed method, especially by comparing it with other manifold-valued generative models.
strengths_and_weaknesses: **Strength**

In my understanding, the paper's contributions are clear, and I also consider that the results are essential for several reasons:
  - The paper provides novel manifold-valued generative models based on SGMs.
 - The paper motivates each proposed method well so that readers can understand how each method addresses the challenges of designing manifold-valued SGMs.
 - I found that the paper has a well-organized structure that makes it clear to understand the proposed methods.
 - The paper provides sufficient information about their theoretical claims, helping readers understand the proposed methods' characteristics and limitations.

**Weakness**
The followings can be improved.
 - Clarifying the terminologies will help readers to understand the paper more clearly. For instance, one can mention that "heat kernel" means "(fundamental) solutions of (linear) SDEs". I understand such terms are widely accepted in SDE literature but possibly confuse the ML communities.
 - The papers can introduce notations before introducing the main theoretical claims. As the paper proposes technically demanding methods, the clarifications of the notations seem necessary beforehand—for instance, exponential mapping notation is introduced in line 203 but appears in Eq 7.
questions: For Varadhan's asymptotic, What can we do when $t$ is large? Does Eq. 7 mean that using the approximation results in biased learning signals when we use the asymptotic to estimate score matching losses?

limitations: Please find the comments on the Weakness section above. In general, the paper's contributions wrt the novelty are clear, and the proposed methods are well-defined. In addition, I found that the paper has a well-organized structure that makes it clear to understand the proposed methods. Thus, I'm inclined to accept the paper.
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 explore continuous-time score-based generative models in non-Euclidean space specifically on the Riemannian manifold. The paper defines the forward process of the SGM model directly on the Riemannian manifold and integrates Geodesic Random Walks (GRW) to project the Brownian motion onto a manifold for sampling on the reverse diffusion process. The authors validate their proposal by testing on multiple datasets (e.g. Earth and climate science dataset, Synthetic data on tori, and orthogonal group).
strengths_and_weaknesses: Strengths:
- The overall paper was clear and organized well. The paper did a good job of translating and introducing the SGM forward noising process for the Euclidean space to the Riemannian forward process. 
- The author provides a nice comprehensive overview of related literature works with useful insight on each past work.
- The paper is highly motivated and novel as there have not been any past papers that have applied score-based/diffusion models in a non-Euclidean space.
- Derivations are concise and eloquent, but it would be nice to see the elaborated derivations that are available in the Appendix.

Weaknesses:
- Many of the experiments do not have many benchmark models to compare with the proposed model. 
- There is not an explanation on the stereographic Score-Based method. A quick explanation on this model would be helpful.

Additional Remarks:
- Section 5.2: I believe that "torii" should be spelled "tori" as it is the plural form of "torus".
questions: - How are stereographic score-based models projected and could a similar method (albeit a different projection) be possible for comparison on the tori experiment?
- On the torus experiment, why were the Moser Flows the only benchmark comparison?
limitations: The authors discusses and addresses the current limitations on their model (e.g. incompatible manifolds).
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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 work extends score-based generative modeling with time-reversed diffusion processes from Euclidean geometry to compact Riemannian manifolds. Several modifications have been made to adapt the method to compact Riemannian manifolds, including using Brownian motion in forward pass, uniform base distribution, and geodetic random walks in forward/backward sampling.  Experiments on real and synthetic datasets show the effectiveness of proposed methods in modeling the symmetry in data. 
strengths_and_weaknesses: Strength: the originality and quality of the theoretical part seem good, and the paper is clearly written.  The extension to the Riemannian manifold is a reasonable idea to consider. The experiments show that this algorithm can work in some cases.

Weakness: The main weakness of the paper is its significance. First, very few datasets really lie on some compact Riemannian manifold without boundary (very very restrictive in terms of math, in fact. ). This setting is more like a mathematical toy to me. This is why the experiments are mainly about synthetic datasets. Second, there are some questions about its feasibility. For example, why uniform distribution on a compact manifold is easy to sample? It highly depends on how the manifold is represented in the memory. So this paper is more about math toys not about solving real machine learning problems. 

questions: 
#### Strengths
- This work proposes novel methods for extending score-based generative modeling techniques beyond Euclidean geometry.  
- This paper is clearly written, covering both the big picture of the considered research topic and the technical details of the proposed method.
- Both theoretical convergence analysis and empirical experimental results are included in the paper to show the validity of the proposed approach.
  
#### Weaknesses
- It is not quite clear to me whether the assumption of compactness of the Riemannian manifold can be too limited for the consideration of generative modeling for real-world applications. It would be more helpful to provide mode detailed analysis from this perspective.
- For the consideration of scalability, stochastic approximations are needed to obtain a tractable training objective. 
- Most of the experimental results are on synthetic datasets, and the choices of real-world datasets satisfying the assumptions of this work seem to be limited.


limitations: - Have the authors tried different parametrizations for the score function, i.e. using the clean data or noise as the prediction target similar to DDPM?
- How general are the proposed approach in terms of the applicability of various samplers or ODE solvers? like DDIM and other advanced solvers.
- How large the gap is to extend this work to Riemannian manifolds which are non-compact or with boundaries or implicitly defined?
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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: In this paper, the authors propose an extension of score based generative modeling to data supported on a compact Riemannian manifold. To that end, concepts the need to be extended from the classical Euclidean setting include an easy to sample reference distribution (uniform on the manifold), that can be further seen as the limit of a noising process (brownian motion on the manifold), as well a an SDE reverting the noising process, in order to be able to generate samples by integrating this SDE thanks to geodesic random walks. The authors also present how to obtain score estimation on the manifold by using the heat kernel, as well as the parametization of vector fields as neural networks, and finally a way to obtain exact likelihood computations, as in the Euclidean case. They obtain a convergence result showing that when the score estimation cost function gets smaller, their model obtains a density that gets arbitrarily close to the true one in terms of Wasserstein-1 distance. They show that their model compares favorably in terms of computional complexity to riemannian continuous normalizing flows or Moser flows. Several experiments real goescience data (on the sphere) and synthetic data on the torus and the special orthogonal group support those claims.
strengths_and_weaknesses: Strengths :

- The authors are able to extend score based generative models that have now become very popular to compact Riemannian manifolds by generalizing or adapting all the necessary ingredients to this setting.

- The theoretical construction and guarantees are extensive and show that the propose method converges and has a better computational complexity than existing alternatives for manifold valued data. In particular one appealing feature is the scalability to higher dimensional manifolds that is verified in the experiments (up to 200 on the torus data and 64 in the special orthogonal data in the synthetic experiments), while competing approaches cannot deal efficiently (or in the extreme case cannot even run) with such high dimensional non euclidean data


Weaknesses :

- The approach is only limited to compact manifolds, which encompass a number of interesting cases already but leave out many more (SPD matrices, hyperbolic spaces…) . The main difficulties are the convergence of the Browian motion to the uniform distribution that only holds in the compact case, and the manipulation of the transition kernel of the Brownian motion in that case. 

- The only real case study with geoscience data does not show clear improvement over other methods, most likely because the problem is too simple and the dimension of the manifold (2 here) is not high enough for the computational benefits of the proposed approach to be visible. Maybe it would be possible to find a real application in higher dimensions where the method can be shown to clearly outperform others? 
questions: - Where does the denoising score matching (equation below equation 2) identity come from ? I couldn’t find the same equation in [Song et al. 2021b], where the score matching objective is obtained from an integration by parts?

- Here it makes sense to consider a uniform distribution as the simple distribution to start from, thanks to the convergence result of Brownian motion on compact manifolds to that distribution. Could other possibilities be envisioned? Has this case already been treated in the Euclidean case, where typically one chooses a normal distribution for the base distribution (and if so, how does it compare to the classical score based with a gaussian base distribution)?
limitations: The main limitation of the method is the restriction to compact manifolds, which is clearly stated. Other manifolds of practical interest are not covered. Nevertheless, the construction for compact manifolds is already sound and covers a certain number of practical cases of interest. Especially, this construction translates into what seems to be efficient algorithms, which is often a drawback of geometric methods operating on manifold valued data.
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Meta-review: metareview: The manuscript generalizes score-based generative models to compact Riemannian manifolds. All referees agree that the method is novel, technically sound, and gives nice results. I recommend the paper to be accepted to the conference. 
confidence: Certain
Decision: Accept