Summary
The paper proposes a class of G-equivariant neural networks for reductive lie groups that generalizes the previous ACE and MCAE models (which are designed to be equivariant with respect to the orthogonal group O(3)) to more general irreducible Lie groups. A software library (Lie-NN) has also been developed and released for implementation of the proposed method. The experimental results on jet tagging and 3d point cloud classification support the authors' claims about the equivariant model improving model performance.
Strengths
- The proposed approach seems novel; I am not aware of other works that design a G-equivariant network in this way.
- The paper is generally well-written and technically correct (although I have not performed an in-depth check for the proof of the universality theorem). Occasional variable definitions are missing but I had no trouble following the general flow of the paper.
- The experimental results appear to validate the claimed advantages of the proposed method (although more discussion on the computational aspects is needed to assess practicality; see below).
Weaknesses
- Since this work is a generalized version of the previous ACE and MACE methods, a brief review of ACE and MACE would be helpful to understand and evaluate the novelty of the method. The current version is presented in a way that contains elements of previous existing methods, without a clear delineation of the new and original aspects of the proposed method. Exactly what is the precise nature of the extensions to ACE and MACE should be made clearer.
- While the technical contents seem correct, some effort at providing intuitive explanation and justifications at key places would be helpful to understanding the paper. Descriptive figures, intuitive examples (e.g., basis including 1-particle basis or Clebsch-Gordan coefficients) come to mind as examples. In particular, I am curious about how to construct a 1-particle basis for specific Lie groups, e.g., SO(3).
- The formulation in the case of the product group in Section 4 seems to be missing, although there is a mention of the product group in the introduction and in Section 5. Is it trivial to design an equivariant model for the product group G_1 x G_2 with the formulation used in this method? My initial impression is that there may be some subtleties involved, such as the order of the group actions.
- A discussion of the computational aspects of this method is missing. When using this model, the Clebsch-Gordan coefficients must be calculated numerically (as mentioned in Section 5), and it would be helful to mention calculation times and errors. Does the calculation time vary depending on the number of basis components? I am also curious about how much time it takes for iteration on backpropagation compared to classic MLP models.
- Continuing with the above comment, since the Clebsch-Gordan coefficients are numerically calculated, some numerical calculation errors are inevitable. Is the equivariance of the model maintained in the presence of numerical errors? If not, some discussion, even qualitative, should be provided (e.g., to what extent the model remains equivariant) although quantitative results (e.g., experimental results, figures) would clearly be preferable.
Questions
These have been mentioned for the most part in the weaknesses section.
Rating
6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
Confidence
4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.
Limitations
No potential negative societal impact of this work as far as I can tell. As mentioned earlier, computational aspects of the method would be helpful. I don't want the authors to feel compelled to show that this method is immediately computationally practical, as that is not the only measure of the value and worth of any new idea, but it would be helpful to indicate to other researchers on what the computational limitations are, and to possibly spur interest in finding improvements.