Venue: NeurIPS 2022 Conference

Reviews:

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: This paper proposes a method to express the stochastic representation over manifolds, using a novelly defined Riemannian Neural stochastic differential equation (RNSDE), the method is theoretically well-analyzed at each step, empirically RNSDE outperforms several baselines on some simple tasks. 
strengths_and_weaknesses: I’m not an expert on neural SDE, so some points maybe stupid :)
S1: the method is theoretically sound in most steps, except the learning step and some approximations;
S2: the extension from Euclidean neural SDE to Riemannian Neural SDE makes it easer to understand and follow;
S3: Empirically the proposed RNSDE outperforms some previous baselines on tasks including generative modeling, interpolation and reconstruction. 

W1: I think this paper is not that ‘readable’ for general NeurIPS audience, but only experts in this area, currently it’s heavy math, I would suggest add more explanations/backgrounds and intuitions to the text, for example the Eells-Elworthy-Malliavin interpretation of the diffusion process, and what each step in the Scheme is doing, this could significant help abstract more readers into this area. 

W2: RNSDE incurs more computations than the Euclidean case, including metric tensors, christoffee symbol and etc, which has to be computed for every local neighborhood as moving along the path, I would like to see some computation burden analysis or empirical results of RNSDE, plus compared to some baselines, particularly to one I mention in next question part. 

W3: The practical usages/applications of this method looks rare to me, what’s more,  the manifolds are also simple, though S^n is claimed to be one model, I only see n=2 case, and with a torus model, is it possible to test on high dimensional manifolds? 
questions: Q1: I understand that Euclidean Neural SDE gives results not on the manifold, but a very naive approach is to enforce a hard constraint (e.g. projection to the manifold) after each Euclidean neural SDE step, or something as a first order retraction? One can also move along local frameworks to get a trajectory on the manifold, do you include this baseline to compare with? Both performance and computations

Q2: I’m curious what’s the choice of the stopping time tau could affect in practice? How large it needs to be, how severe the gradient explosion is and how to choose this value? 
limitations: Mostly discussed above, the practical usage seems to be rare, and the computations are heavy, limited to a small set of manifolds and low dimensions.
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: This paper introduces a neural network method to learn and represent stochastic differential equations on manifolds. Applications are presented for density modeling on manifolds and learning latent paths on manifolds.
strengths_and_weaknesses: Strengths
-----------
+ The paper is presented very well. I appreciated the clear presentation, precise mathematical language, thorough writing, and informative figures.
+ The paper is very technically solid, as the core difficulty of extending Neural SDEs to manifolds is aptly handled with the correct Riemannian-geometric constructs.
+ The experiments are pretty well done, and the Vessel Route dataset is an interesting contribution to this space.

Weaknesses
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- There is work that slightly predates this one on extending SDEs to manifolds in the context of scored based generative models and diffusion (https://arxiv.org/abs/2202.02763). While I believe that the two works are different, the above work should be included in the discussion.
- For the density modeling experiments, some baselines are missing. In particular, I believe there should be a comparison with Moser Flow.

Verdict
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I generally lean on accepting the paper, as I think that it is a solid contribution to the space. However, I would want some more discussion with the baselines and previous methods mentioned above (as I think they sufficiently predate the work).
questions: My questions are covered in the weaknesses section. In particular, I would like to see a comparison with Moser Flow and a discussion with the Riemannian Score Based Modeling Paper.
limitations: Yes
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 submission proposes an approach for modeling stochastic differential equations on Riemannian manifolds using neural networks called Riemannian neural SDE (RNSDE). This approach allows for learning a probability distribution over a manifold and modeling stochastic processes happening on the manifold. RNSDE uses neural networks to model the terms in the  local representation of an SDE and allows for probability density estimation using the dual metrics formulation. Experiments show that RNSDE outperforms some of the related deep learning baselines in a set of illustrative toy tasks and in several real-world benchmarks.
strengths_and_weaknesses: ### Originality

According to the provided literature review, the submission proposes a novel approach to modeling SDEs on manifolds. Authors clearly state the difference between their method and related works, highlighting that RNSDE requires less trainable components to model evolution of the density and claiming that it provides potentially more flexible solution than some other restricted methods. However, the literature review does not cover diffusion models which seem to be very relevant to the proposed method, especially in its application to density estimation. Specifically, there’s a relevant work [1] studying diffusion models on Riemannian manifolds which would be worth to compare with or at least to mention as a competitive work and highlight the key differences.

### Quality

The submission is technically sound and complete with a solid theoretical basis and good enough experiments. Authors’s claims are mostly supported by experimental results and most of the baseline methods are appropriate. 

### Clarity

The presentation of the method is very decent, authors introduce all the important concepts and provide preliminaries required for understanding the method. They also support some of the involved concepts and derivations with intuitive explanations which are very helpful for a broader machine learning audience. Moreover, informative illustrations helps a lot to make sense of the method and experiments. However, a solid background in manifold theory is required to fully understand the method and its theoretical basis. 

### Significance

The proposed method addresses an important task of modeling SDEs on manifolds in an original way surpassing some of the existing methods. The literature review and experimental comparison suggests that the proposed method is fairly significant.

[1] De Bortoli, Valentin, et al. "Riemannian score-based generative modeling." arXiv preprint arXiv:2202.02763
 (2022).
questions: 1. It would be worth to consider diffusion models in related works, especially [1]
2. In the experiments with the Vessel route dataset, you choose Latent ODE and ODE RNN as the baselines. As far as I know the main feature of these models is that they can deal with irregular time series, i.e. where measurements are not equidistant wrt time. Does RNSDE capable to take this irregularity into account? If not, this comparison looks like comparing apples with oranges, where it is hard to make any conclusion why one model outperforms the others. I would highly recommend authors to add a comment on this in the text of the paper.
3. In the introduction section there’s a strong claim that conventional approaches which do not take into account geometry of the task are losing in their expressivity. However, there’s a little of validation for that in the experiments section. I would recommend to add this kind of conventional approaches as baselines to all the experiments to give a clear evidence of the importance of taking into account geometry of the data.
limitations: Authors clearly state  that they focus only on Euclidian manifolds while there’re other domains (non-compact Lie groups, product spheres) important for downstream applications to consider. There’re no discussion of the negative social impact provided.
review_title: null

Meta-review: metareview: There was a consensus towards weak acceptance among all the reviewers, and I agree with this consensus. This paper solves an important problem of applying SDEs to manifolds. It is clearly written, and all the reviewers agree that the claims are well-supported by strong experimental results. On the other hand, this clarity of writing perhaps relies overmuch on familiarity with the area, and some effort should be made to smooth out the presentation for a general NeurIPS audience. Beyond this, the weaknesses pointed out by the reviewers were well addressed by the author response, including an additional experimental result that provides a comparison to Moser Flow: there is no strong unaddressed weakness that would merit rejection. I think that the manifold learning community, as well as the Neural ODE community more broadly, at NeurIPS will find this work interesting and useful, as it expands the range of methods they can apply on manifolds. As such, I lean towards accepting this paper.
confidence: Less certain
Decision: Accept