Weaknesses
W1. The equality in Eq. 3 seems not correct as it neglects permutation symmetry. If we consider positional features represented as matrices $\mathbf{X}\in\mathbb{R}^{n\times 3}$ (Line 77), the precise symmetry under consideration is a direct product of permutation group and Euclidean group $S_n \times SE(3)$ (Dym et al., 2020). I think it should be clarified that the equality in Eq. 3 is up to permutation of nodes.
W2. In Line 111, I think {-tree, -triangular, -subgraph} should be {tree-, triangular-, subgraph-}. Same applies to Line 603-604 and Line 612.
W3. In the definition of tree and triangular isometries (Line 114-118), is there any reason to use the term tree and triangular? I think the definitions are a straightforward extension of the subtree and overlap isomorphism proposed in Wijesinghe et al., 2022, respectively, and I see no reason to use different name. Especially, for the triangular isometry, I found the name misleading since the isometry condition is imposed on the entire set of triangles that share $e_{ui}$, rather than individual triangle.
W4. In the definition of triangular isometry (Line 116), the condition is specified for each edge in the intersection of the local subgraphs $e_{iu}\in\mathbf{S}\_{i-j}$. Please correct me if I am wrong, but to make sense, I think the condition should be for each edge in one of the local subgraphs, $e_{iu}\in\mathbf{S}\_{i}$.
W5. In Line 606, the term structure completeness is used without proper definition (although it is emphasized by boldface). Without clear definition, I was not able to properly understand and correctness of the argument in Line 608.
W6. The argument in Line 162-164 is ambiguous; do you mean a composition of Atoministc Cluster Expansion and invariant map constructed using scalarization by edge-wise equivariant frames is a universal approximator of SE(3) invariant functions, so it can serve as $\phi$ in Theorem 3.1? Is the proof of universality available in literature?
W7. For LSE, based on Appendix H, it seems that the hierarchy of local isomorphism proposed in this paper is a subcase of the geometric WL test proposed in Joshi et al., 2023. Since this paper provides 3-way classification while GWL provides a more sensitive separation of expressive power of geometric GNNs based on the orders of node tuple and feature tensors, one can argue that the local isomorphism hierarchy proposed in this paper is less of practical interest compared to GWL.
W8. For FTE, in Section 4, frame transition is introduced to solve the limitation of SE(3) invariant messages as outlined in Theorem 4.1. But as far as I know, a majority of practical equivariant geometric GNNs in literature are already using equivariant (tensor) messages (Thomas et al., 2018; Fuchs et al., 2020; Satorras et al., 2022; Geiger et al., 2022; Kohler et al., 2019), and the expressive power of some of them (TFN (Thomas et al., 2018) and SE(3)-Transformers (Fuchs et al., 2020)) are shown to be universal i.e., maximally expressive (Dym et al., 2020). Is the motivation of frame transition also relevant for these architectures? Also, is there specific reasons to favor the proposed frame based propagation over these already equivariant message passing algorithms, despite the added complexity of incorporating node- and edge-wise frames?
W9. In Line 214, does the frame here refer to frames in Riemannian manifold theory? Since there is no proper definition (Eq. 5 provides a description of key property but is not a definition), I was confused on what exactly are the frames mentioned here and was unable to verify whether Line 216-221 is correct. It seems the precise definition and descriptions of equivariant frame are completely deferred to Appendix B and F although it serves as a main component of the proposed architecture, which makes the paper hard to read and understand; I think key parts of the proposed algorithm should be described in the main text in a self-contained manner.
W10. Overall, I find that the writing of the paper has room for improvement in terms of organization and readability. There are many cross-references back and forth across the paper, so I couldn't serially read through each section without going through a large portion of the entire text multiple text. For example, Line 236-237 (Section 4) refers to Figure 3 (Section 5), Line 555 (Appendix A) refers to Eq. 21 (Appendix F), Line 592 (Appendix C) refers to Figure 3 (Section 5), Algorithm 1 (Appendix C) refers to Eq. 4 (Section 3), and so on.
W11. In Eq. 7, I don't see how the equation describes equivariance. I think it should be something like $\mathbf{m}(g\mathbf{x}\_u) = \sum\_{i=0}\^l\mathcal{M}\^i(g)\mathbf{m}\_i(\mathbf{x}\_u)$, is this a typo?
W12. Typo in Line 1 of Algorithm 1, gragh -> graph
W13. Specification of the output is missing in Algorithm 1, and the output in Algorithm 2 is not clear since normal and boldfaced h are mixed in the algorithm. Furthermore, In Line 7 of Algorithm 1, how do you scalarize the mutual 3D structure, which is a subgraph, based on edge frame $F_{ij}$? I don't see how this can be done in Eq. 8 which defines scalarization. Similarly, in Line 6 of Algorithm 2, how do you scalarize the mutual 3D structure without frame? Also, if the algorithm 2 is LSE only, why should one compute edge-wise frames in Line 5?
W14. In the proof of Theorem 5.1, I can see that the proposed parameterization is universal, but does it correctly guarantee equivariance as well (as specified in Theorem 5.1)? The current proof seems to clearly show universality, but I am not sure about equivariance.
Dym et al., On the universality of rotation equivariant point cloud networks (2020)
Wijesinghe et al., A new perspective on "how graph neural networks go beyond weisfeiler-lehman" (2022)
Joshi et al., On the Expressive Power of Geometric Graph Neural Networks (2023)
Thomas et al., Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds (2018)
Fuchs et al., SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks (2020)
Satorras et al., E(n) Equivariant Graph Neural Networks (2022)
Geiger et al., e3nn: Euclidean Neural Networks (2022)
Kohler et al., Equivariant Flows: sampling configurations for multi-body systems with symmetric energies (2019)