Venue: NeurIPS 2022 Conference

Reviews:

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: The paper proposes a general diffusion model for Riemannian manifolds. It introduces a variational framework based on Riemannian Feynman-Kac representation and Giransov’s theorem, with which a Rimannian continuous-time ELBO is derived. To compute the resulting ELBO, this paper further suggests a QR decomposition-based method for scalable unbiased estimation. Besides, it also provides a variance reduction technique to approximate the Remannian ELBO objective. 

strengths_and_weaknesses: Strength:

Based on SDE, this paper generalized the Euclidean diffusion model into a Riemannian one. The generalization seems to enjoy a solid mathematical background.

The experiments demonstrate the effectiveness of the proposed method on the data that are on spherical manifolds, product of spherical manifolds, hyperbolic spaces and orthogonal groups.
 
Weakness:

This paper requests a lot of effort to understand its background and the proposed idea.

The presentation of background (Sec.2) is also very challenging to understand. For example, regarding Euclidean diffusion models, it might be better to further clarify how the diffusion model can be formulated as a fixed Markov chain that is presented in Line 24-26. Besides, it should be necessary to indicate the meanings of \mu and \sigma after Eq. (1). The same suggestion could be applied to other equations like Eq.(2). 

As for the computation of the derived CT-ELBO, the suggested QR decomposition based method is an efficient approach. I cannot really understand why the paper further introduces a variance reduction technique to approximate the CT-ELBO. What is the major benefit of the used variance reduction over the QR decomposition based method? It seems that this paper does not study this benefit empirically.

questions: In Eq. (27) of Appendix, what’s the meaning of  \frac{\partial \varphi_{j}^{-1}}{\partial \tilde{x}_{i}}? If my understanding is right, \varphi_{j}^{-1} should be understood as (\iota \circ \varphi)_j^{-1}

Regarding the spherical dataset, it is not clear to me why the used data lie in a sphere? Maybe some explanation or references will help interested readers.

Small typos should be corrected. For example, Eq. (24) in Appendix, the v in LHS should be capitalized.

limitations: It seems this paper does not discuss its limitations. It will be highly appreciated if the authors can address my concerns/questions mentioned above.
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 extends the application of continuous-time diffusion models into Riemannian manifold. The paper proposes a new method of calculating the Riemannian divergence for computing the Riemannian lower variational bound objective function.  In particular, the authors also makes an insightful connection with Riemannian score matching.
strengths_and_weaknesses: Strengths:
- Overall, the paper was well organized and clearly written.
- The shadowed boxes on the salient equations help follow the derivations.
- The idea is a novel application/expansion of diffusion models as they have (to the extent of my knowledge) not have been explored on non-Euclidean spaces. 
- The connection between score-based models (in the Riemannian case) is a welcoming and insightful addition.

Weaknesses:
- The experiments show SOTA results and seem very promising, however, there seems to be a lack of analysis on the results. 
- Limited information on training/sampling complexity.
 
Additional Remarks:
- Pg. 4 Line 136 "showed that that..." typo
questions: - Why is there only one baseline model comparison in the tori results specifically not having the comparison with the Riemannian Score-Based Model?
- In some tasks, SGM have performed better in certain image generation than diffusion models (e.g. CIFAR-10, CelebA 256x256). In this task, it seems that this model performs better than its score-based counter part. Is there any reason to believe that diffusion models should theoretically perform better in the Riemann space?
- Projection can be a difficult task, especially projecting onto a high dimensional complex plane. Does this have any detrimental effects on the training/sampling time?
limitations: The authors address their limitations (i.e. algorithmic decisions).
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: The paper derives a variational framework for likelihood estimation of diffusion models on Riemannian manifolds. This is based on prior work on continuous time ELBOs and score matching. The authors generalize this to Riemannian manifolds and address computational issues including fast computation of the Riemannian divergence. The methods are experimentally demonstrated on several manifolds.
strengths_and_weaknesses: Strengths:
* well-written and technically sound paper
* addresses an important class of problems
* the presented techniques are mathematically interesting and non-trivial
* the paper couples stochastic process theory, variational inference and geometry

Weaknesses:
* the CT-ELBO was derived in earlier paper in the Euclidean setting. To some extend, the extension to the geometric context is not groundbreaking. However, it is in no way trivial either

In general, I found the paper a very interesting read.

questions: * line 172-173: I didn't get what is meant by "is evaluated only once as opposed to estimating the ELBO by integrating over the entire path of Y_s". What is evaluated only once? The entire sample path is still used?
limitations: yes
review_title: null

Meta-review: metareview: This paper presents a generalization of continuous-time diffusion models to Riemannian manifolds and derives a variational framework for likelihood estimation. The theoretical analysis is accompanied by numerical experiments.

Reviewers generally agree that the paper is novel, technically sound,  well-written, and would make a solid contribution to research on diffusion models.

Presentation: I would suggest that the presentation be improved to make it more accessible to the general ML community. Specific references should be given to concepts such as Riemannian divergence,  Divergence Theorem, etc , since these are not new.


confidence: Certain
Decision: Accept