We thank the reviewer for their positive comments.
We have revised our manuscript to make the presentation more accessible to a machine-learning audience, and to make links between our result and other machine-learning results and/or applications clearer.
To expand on the latter point, we believe there are three main directions in which our complete representations for 3D point clouds can be employed outside of atomistic and molecular systems:
- **Classification tasks**: although we showed results for regression, which are the only relevant machine learning exercise in the prediction of microscopic properties, the Wigner kernel formalism could have a great impact in 3D object recognition [1], as 3D objects are most often represented as point clouds. This would be achievable through the use of a Gaussian process classifier [2] with a covariance function corresponding to the Wigner kernels.
- **Incorporation inside neural network architectures**: the Wigner iteration can be used as a layer in an equivariant neural network architecture, which would offer a more practical alternative to Gaussian process regression/classification. We believe this to be a very promising route to achieve richer and more complete representations in geometric machine learning.
- **Generalization to other groups and dimensions**: although our formalism is presented for the SO(3)/O(3)/E(3) groups, it is easily generalizable to other groups. In particular, the 2D implementation of the Wigner iteration is trivial, as the irreducible representations of the SO(2) group are all one-dimensional.
As a result, we believe that the proposed method could have a large impact on all areas of geometric machine learning.
Moving on to the questions:
- All the terms have now been defined in Eq. 2. We have also added a table of symbols in the appendices, which will be very helpful for the readers of our work who are not familiar with the SO(3) group formalism.
- Yes, they do. In the initial version, we decided to omit the chemical species indices in some key equations to make the presentation tidier. However, we realize that this might be confusing and we have now added them back. In particular, Eq. 5 shows that kernels between two individual atoms are symmetrized sums of integrals between densities of the same kind (that is, belonging to the same neighboring atom species). This amounts to considering each chemical element as its own dimension in the chemical species space, and it is the kernel equivalent of one-hot encoding of the chemical species of the neighbor atoms.
- We do not feel comfortable in providing training (or inference) timings for the models in Figs. 1 and 2 directly in our manuscript, even though they are typically larger than those for the Wigner kernel. This is because we implemented those models ourselves to ensure full consistency with the Wigner kernels, while more optimized (and often less accurate) packages exist. In many cases such implementations make approximations to enhance computational efficiency and usability, and they are therefore unsuitable for careful ablation and comparison tests. However, we can reassure the reviewer that, in the data regime we typically work on, training times are considered to be a significant strength of kernel models as, once the kernels are available, the fit consists of a single linear algebra operation. In our ablation studies, the training times of our method were comparable to those of other kernel and linear models, and much faster than those of neural networks trained on the same datasets. Of course, as we mention in the manuscript, increasing the train-set size requires the use of a projected-process approximation of the kernel, which we do not explore here as it would obfuscate the analysis of the converged-basis limit of body-ordered equivariant models by introducing the active space size as a further convergence parameter.
As a final note, the kernels in Eq. 5 are not completely novel, as they had been considered in Ref. [3]. Despite the fact that they had been known for a decade, nobody had computed these high-order kernels explicitly previously to our work due to their very high computational cost. The focus shifted instead into providing good approximations to the real-space density correlations that enter the kernels. Our Wigner iteration allows to compute these complete representations of geometric structures, showing that they indeed provide a very rich and complete representation of 3D objects.
[1] Qi, Charles R., et al. "Pointnet: Deep learning on point sets for 3d classification and segmentation." Proceedings of the IEEE conference on computer vision and pattern recognition. 2017.
[2] Williams, Christopher KI, and Carl Edward Rasmussen. Gaussian processes for machine learning. Vol. 2. No. 3. Cambridge, MA: MIT press, 2006.
[3] Bartók, Albert P., Risi Kondor, and Gábor Csányi. "On representing chemical environments." Physical Review B 87.18 (2013): 184115.