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: The paper proposes an architecture to model the dynamics of multi-body systems.
The main characteristics are equivariance with respect to Euclidean transformations $g \in \textbf{E}(n)$ of the inputs and ability to model the hierarchy present in such systems. This is achieved through EMMP and E-Pool/E-UnPool layers.
The main novelty claimed is that this is the first architecture to be _both_ equivariant and hierarchical.
strengths_and_weaknesses: ### Strengths
- The paper is well written and describes in detail the architecture.
- The authors empirically show that this architecture is capable of modeling complex hierarchical systems as their original goal.
- Clustering of the nodes might help investigate and interpret the results of the architecture.

### Weaknesses
- The EMMP layer is only a slight variation of EGNN to directional matrices. Additionally EGNNs are still used in the proposed architecture as the external EMMPs. It's unclear whether using the new EMMP layer is beneficial at all.
- There are many parameters such as number of clusters $k$ and adjacency matrix construction which have to be manually provided. The authors describe how $k$ might affect the performances of the model in the ablation studies but provide no guidance regarding their choice.
questions: - What is the performance of the architecture if the external EMMPs are not relaxed as EGNNs?
- Can you provide a better evaluation and discussion on the impact of the choice of number of clusters? "Ablation Studies/Q1" briefly describes the degradation with $K = 3$. Perhaps it's possible to show a plot for multiple $K$s?

limitations: N/A
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 introduces a hierarchical message passing neural network that is equivariant to Euclidean transformations. The building blocks of these networks are i) an equivariant message passing layer that generalises EGNN, ii) a pooling layer that can roughly be seen as an E(3)-equivariant adaptation of Diffpool, and iii) an unpooling layer that uses the clustering coefficients learned by the pooling layers. 
The resulting architecture (called EGHN) is benchmarked on future-state prediction tasks in several settings. 

strengths_and_weaknesses: **Summary**: this is a overall a good work, but the ablation study is not properly performed. I am willing to raise my rating if the authors successfully demonstrate the need for the hierarchical architecture.


**Strenghts**:
  - The paper is well written and easy to understand.  
  - To my knowledge, it is the first work that introduces E(3) equivariant hierarchical message passing networks. 
  - The work is well motivated, and looks particularly relevant to learning tasks on proteins.
  - The presented experimental results are good, and the presented method seems to consistently outperform previous work. However, it is not clear that this is due to the hierarchical structure, and not to the fact that equation 3) and 4) provide a better representation power than EGNN. 

**Weaknesses**: 

  - The ablation study is not satisfactory. Since the pooling and unpooling layers contain learnable parameters and update the representation of each node, the hierarchy ablation study should not remove these layers, but rather set the cluster assignments to the identity  ($s_i = \mathbb 1_i$).
Furthermore, since the hierarchical aspect is key to your contribution, this ablation should be performed for all datasets, and not only for the motion capture. At first sight, it is for example not so clear that a hierarchical structure will be useful for the n-body system. 

The ablation is all the more important as hierarchical graph neural networks are known to often have disappointing performance (cf. “Rethinking pooling in graph neural networks”, Mesquita et al. 2020). You therefore need to demonstrate that it is key to good performance, at least for some of the tasks. 
  - While most relevant works are cited in the related work section, credit is not really paid where it should be inside Section 3. For example, there is a strong similarity between Diffpool and the equivariant pooling in 3.3., and the two should be compared in details (cf. also the question on Equations 3 and 4 below).

**Some advice**:
  - It should be clear from the abstract that you are referring to E(3) equivariance. There are other works such as “Multiresolution equivariant graph variational autoencoder” from Hy and Kondor (2021) that deal with permutation equivariance, which can introduce some confusion.
  - “Hierarchical inter message-passing for learning on molecular graphs” (Fey et al. 2020) is a relevant related work.



questions: The originality of the work is not easy to assess. The pooling and unpooling layers (as well as the reconstruction loss) are in my opinion moderately original and have a strong similarity to Diffpool and Graph-U nets. I however find equations 3) and 4) very interesting, as they generalize EGNNs without requiring to compute spherical harmonics. Are these equations novel or are they borrowed from existing work?

  - If they are novel, this should be clearly stated, as they are in my opinion one of the biggest contributions of the paper. 
     - If possible, you should also give intuition on i) what can be computed with these equations that cannot be computed easily with a EGNN ii) why they are compatible with equivariance. 
    - In the ablation study, it would be useful to assess by how much these equations improve upon EGNN. 
    - These equations should be connected to “Frame averaging for invariant and equivariant network design” (Puny et al. 2022) -- they also have some similarity to “Sign and basis invariant networks for spectral graph representation learning” (Lim, Robinson et al. 2022).

  - If these equations are already used in existing work, this should be clearly stated as well. 


Could you also provide a comparison of the training time for the different architectures? It seems that your method is more costly than flat MPNNs, but it is not clear by how much.
limitations: There is no clear potential negative societal impact to this work.
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 authors propose a architecture for E(n) equivariant graph neural networks with two improvements:
1) Compared to the EGNN, the method includes as node features - besides a set of scalars - not a single E(n)-equivariant vector as feature, but a matrix of M such vectors. The message passing done with an unconstrained MLP that takes as input the invariant 2Mx2M matrix created by taking the pairwise norms of those vectors of both source and target nodes.
2) The authors propose to use attention to pool to and from finer and coarser hierarchies and carefully normalize the attention weights to ensure equivariance.

The authors perform a thorough empirical evaluation on datasets with synthethic, motion capture and molecular data and demonstrate state-of-the-art performance.
strengths_and_weaknesses: # Originality
The method is an incremental improvement over prior work, so has limited originality.

# Quality
The paper is of very high quality. The proposed method is sound and thoroughly evaluated on a diverse set of tasks.
Two points for improvement:
- I would have liked to see a comparison to [1], as they also generalize over the EGNN and claim they improve its performance.
- I'd think that the hierarchical architecture constitutes a U-Net with skip-connections. Could the authors add this to the text?

[1] Brandstetter, Johannes, et al. "Geometric and physical quantities improve e (3) equivariant message passing." (2021).

# Clarity
The paper is very clearly written. Some minor potential points of improvement:
- The experimental parameter $J$ "different systems with varying combinations of ..." is not clearly explained.
- I'd like a more extensive explanation of the second loss term in (16)

# Significance
As the method is easy to implement and improves performance compared to prior work on a domain - equivariant pointcloud prediction - that is widely applicable, this method can have significant impact in the community.

# Conclusion
The proposed method is well-motivated, simple and achieves significant performance benefits over prior work. Therefore, I recommend acceptance of this paper.
questions: no questions
limitations: Contrary to what's stated in the checklist, the authors do not give any limitations of their work. This should happen in the main paper, not the appendix.
review_title: null

Meta-review: metareview: All reviewers agree that this work makes progress on the front of E3-equivariant message-passing neural networks. Though not surprising in light of previous work (e.g., on graph coarsening and pooling for GNNs), the introduced modifications appear to be novel in the context of EGNN. The proposed ideas are well motivated and explained while also yielding numerical benefits.
confidence: Certain
Decision: Accept