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: In this paper, the authors focus on the design and analysis of graph scattering networks with variable branching ratios and generic functional calculus filters. Spectrally-agnostic stability guarantees for node- and graph-level perturbations are established.


strengths_and_weaknesses: Strengths: 
This paper is well-organized. The theoretical results on stability guarantees and the experimental results on quantum chemical energies are interesting.

Weaknesses:
1. The authors present several upper bounds in Section 4 for stability guarantees. However, it is unknown the optimality of these bounds.

2. The section on higher order scattering needs more background knowledge to follow for readers.
questions: 1. The authors should add comments on the optimality of the upper bounds established in Section 4.

2. More background and details are needed for the section on higher order scattering.
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 paper proposes a generalization of graph scattering networks that goes beyond the graph wavelets setting. The authors provide stability guarantees for their generalized scattering transform, as well as layer-wise energy decay bounds. The authors propose a simple feature aggregation method to transform graphs into Euclidean space and briefly discussed taking into account higher-order scattering. The authors conducted experiments of their methods on a graph classification task and a graph regression task. 
strengths_and_weaknesses: - Originality: the idea, methods etc are original. The novelty/originality of the paper definitely meets the bar for publication at NeurIPS.
- Quality: the background and tools used are somewhat technical and motivated quite abstractly. The quality of the mathematics/derivations, even though I didn't check proofs line by line, are, to the best of my knowledge, sound.  
- Clarity: the organization and exposition of the article leaves much room for improvement. Details are given below. 
- Significance: Nowadays it is certainly interesting and significant to consider problems involving GCNs and graph networks. The topic of this paper is of sufficient significance to meet the bar for publication at NeurIPS. 

Overall, I think the originality, significance and mathematical soundness of the paper are fine, whereas the clarity of the paper leaves room for improvement, which I will detail in sections below. 
questions: Questions on the quality front: 

1. The lipschitz-type bounds in theorems 4.1 and 4.2 appear to be just iterative applications of a layer-wise lipschitz type condition. While this is certainly a valid bound, the constants in the upper-bound involves a product of N terms/terms raised to the Nth power (this exponential term also appears in theorem 4.5). I would argue that on a practical level, these exponential terms render the bounds rather impractical, unless matching lower bounds/some sharpness result can be shown. In particular, say N is 10. Then if I perturb say the input by some small constant, the output could have a order C^10 change to it, which can be astronomical. 

2. In the expressivity/energy section, the analysis was conducted in the limit of infinite network depth. The idea is quite neat, but it raises several important questions that the authors have to address:

-  the authors make the assumption that one can always choose an eigenvector of strictly positive entries. This of course follows from results in spectral graph theory. However, for the connected graph case, which I argue is probably the most important case, it is my understanding that the only eigenvector that satisfies this will be the eigenvector corresponding to the smallest eigenvalue, in which case the eigenvector has constant entries (they are all the same), and the corresponding eigenvalue is just 0, in which case the notation of defining m_n as the minimum and lambda_n etc becomes somewhat redundant. On the other hand, if the graph is disconnected, then the eigenvector that you pick will, to the best of my knowledge, have positive entries in one component and 0's in some other component, which would violate the strictly positive entry assumption. in either case, I think some change/clarification has to be made here. 

- While I think the energy bounds are interesting, it is unclear to me how useful/related this is to link to the expressivity of the network. The fact that the mapping only maps 0 to 0, when N goes to infinity, seems like a property that is only marginally related to expressivity in some bare-minimum way. One can probably come up with some kind of invertible linear-type transformation that also only maps 0 to 0. This property to me at first sight seems to just mean that the mapping is not contracting, but going from "no contraction" to "expressivity" seems a bit of a stretch claim to me. I think the authors could modify the wording of their conclusion here. 

In terms of the clarity of exposition, I think the general ML audience will find this paper difficult to read and apply, for the following reasons: 

1. There is too much material for a 9 page conference paper. Important aspects of the paper are delegated to the appendix, and there is not enough room for the authors to give the necessary treatment for background knowledge and definitions. As a result, only those that already have very substantial backgrounds in graph wavelets/graph networks and spectral graph theory will be able to understand it. 

2. The authors have the style of defining things in the broadest, most abstract and general version first, and then in the experimental section just make some very specific choices in their model that conform to their general theoretical results, but without justifying those experimental choices at all. I understand that this is a theoretical paper, but I think having one or two tables of pseudocodes on particular instantiations of your architecture, and providing more justification for why certain parameter or modeling choices are made (such as cross-validation etc) would help the users understand and adopt their method much more readily. In particular, the experimental results for the regression application is great, but for the classification is not very good. I wonder if the performance on the classification task could be improved with an alternative instantiation of their model, such as using other functions than sines and cosines, or changing the layer parameter, or using a different operator than the Laplacian etc. 


limitations: Main limitations are outlined in section above. 
No negative societal impact.
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: In this paper, the authors define a very general notion of scattering transform, with an application on graphs. They (re)define each building blocks of the scattering transform: spectral filters, output (low-pass) functions, non-linearities, etc., as well as proper projection operators when the domain changes between layers. Under appropriate assumption of Lipschitzness of each of these elements, they show stability of the resulting transform, as well as energy preservation, generalizing the classical Euclidean results. Some variants on graphs are presented: an aggregation strategy for graph classification and higher-order scattering. Experiments on real data show the effectiveness of the approach.
strengths_and_weaknesses: Strengths:

- a general approach, than can take into account many variants, domain changes, etc.
- all classical theoretical results on scattering hold
- a very complete supplementary material
- the experiments are convincing, especially for graph regression

Weaknesses:

- a bit paradoxically, the approach suffers from too much generality. The authors define very abstract operators and elements, and in fact nothing in particular is about graphs at all, until the discussions of sections 6 and 7 which are not the core of the approach. Furthermore, the actual choice of the filters, some combination of sin and cos, is quite hidden within the experiment section, and may seem a tad arbitrary. As a result, the reader is somewhat left wondering all along the paper what the actual architecture is, if this is just an abstract formulation of previous architecture or if there is something fundamentally new here. Examples of implementation on graphs along the abstract description could really help the understanding of the approach.
- many variants are described but, it seems, not tested in experiments (changing graphs, higher-order tensors...)
- the theorems are valid under many assumptions, but a minimal examples satisfying all of them is not given
questions: -The approach is interesting, but I would suggest either to reformulate the title and/or abstract to be less specialized on graphs, or on the contrary focusing far more the description of the architecture on graphs by giving minimal examples satifying all the formulated hypothesis along the way.

- if the higher-order architecture used in the regression experiment ? It is not clear.

- is the constant "$2$" in definition 4.3 arbitrary ?

- in theorem 4.2 (and related results), could the proximity of the normal operators be expressed in terms of spectral norm rather than Frobenius ? This is more often satisfied (eg by random graphs, or spectral graph coarsening, etc)
limitations: The authors are quite honest about the limitations in their first experiment, where they are often not state-of-the-art.
review_title: null

Meta-review: metareview: In the discussion, we reached a clear consensus that this paper is interesting for the NeurIPS community and should be accepted. The author's rebuttal and subsequent discussion were very useful and we are looking forward to the final version of the paper with the promised improvements implemented. 
confidence: Certain
Decision: Accept