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: This paper theoretically studies the operators on a manifold $\mathcal{M}$ that commute with the group of diffeomorphisms $\text{Diff}(\mathcal{M})$. Specifically, the author proves that in the case of a scalar fields, only point-wise non-linearities are the corresponding operators; and in the case of vector fields, those operators corresponds to the scalar multiplication. These result indicates that there is no universal class of non-linear operators (e.g., neural networks) that commute with any group of symmetries of $\mathcal{M}$
strengths_and_weaknesses: Strengths:
- This paper, especially the theory and its corresponding proofs, is well-written and easy to follow.
- The proved theory appears to be novel and solid, especially the result in the context of neural networks. I think this is a very nice contribution to the geometric deep learning theory.

Weaknesses and comments:
- Overall, I don't have any criticism on the theories, my only question is that since one of the result is that there is no universal class of non-linear operators that commute with 'any' group of symmetries, I wonder what can we say about the operators if we reduce the symmetries to some specific symmetry group? In this case, would that be possible to find some particular non-linear operators that commute with such symmetries? I think it would be great if there is a discussion on this.  

Minor comments:
- The operator $M$ (not the manifold $\mathcal{M}$) is mentioned in line 16 but it is defined later in line 64. 
questions: All the suggestions and questions are presented in the strength and weaknesses section.
limitations: Yes
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 studies operators mapping vector and scalar fields defined over a manifold. The author demonstrates that the non-linear operators that act on vector fields and commute with the group of diffeomorphisms are scalar multiplications, which implies that $Diff(\mathcal{M})$ is too rich to obtain non-trivial operators. In the case of vector fields $L^{p}_{w}(\mathcal{M}, T, \mathcal{M})$, the author demonstrates that these operators are the scalar multiplication.
strengths_and_weaknesses: As for the strengths of the manuscript, it is well-written and well-motivated. The statements in this paper are well-supported by proof. As for the weaknesses, I prefer to some experiments to verify the proposed the theorems, or provide some examples to show the usage of the proposed methods in the paper.
questions: Since I am not an expert in geometric deep learning, I have no more questions about this paper.
limitations: The limitations are not explicitly discussed in this 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: This paper contributes two main proofs that support the practice of using non linear activation functions in conjunction with linear operations in geometric deep learning. They also prove that Diff(M) is too rich and that there are no universal class of non-linear operators to motivate the design of Neural Networks over symmetries of M.
strengths_and_weaknesses: Strengths: The paper addresses important issues of symmetries in machine learning. It also proves that no universal class of nonlinear operators exist that can handle all possible symmetries in a manifold. This is important to know when designing neural networks. Thus the significance, quality, and originality of the paper are good.

Weakness: The paper is very dense mathematically and difficult to read for someone who is not embedded in the literature on this topic. 
questions: One thing I suggest to the authors to help with the clarity is to define more of their notation. Much of the notation is not defined in the main paper despite having an entire subsection dedicated to it. This could be improved as there is plenty of space in the paper.


limitations: The authors discussed some of the unsolved questions that remain.
review_title: null

Meta-review: metareview: Overall: studies operators mapping vector and scalar fields defined over a manifold M, and which commute with its group of diffeomorphisms Diff(M).

Reviews: The paper received three reviews, all of them on the positive side: Accept (less confident), Weak Accept (less confident), borderline accept (less confident). It seems that all reviewers are happy with the paper + the proposed changes by the authors during the rebuttal.. The reviewers found the paper is clear and has a clean presentation. The findings are interesting, and connect with the ML community. The authors have provided satisfactory answers to reviewers' comments, answering most of them successfully.


Confidence of reviews: Overall, the reviewers are fairly confident. We will put more weight to the reviews that got engaged in the rebuttal discussion period.
confidence: Less certain
Decision: Accept