Spherical CNNs are convolutional neural networks that can process signals on the sphere, such as global climate and weather patterns or omnidirectional images. Over the last few years, a number of spherical convolution methods have been proposed, based on generalized spherical FFTs, graph convolutions, and other ideas. However, none of these methods is simultaneously equivariant to 3D rotations, able to detect anisotropic patterns, computationally efficient, agnostic to the type of sample grid used, and able to deal with signals defined on only a part of the sphere. To address these limitations, we introduce the Gauge Equivariant Spherical CNN. Our method is based on the recently proposed theory of Gauge Equivariant CNNs, which is in principle applicable to signals on any manifold, and which can be computed on any set of local charts covering all of the manifold or only part of it. In this paper we show how this method can be implemented efficiently for the sphere, and show that the resulting method is fast, numerically accurate, and achieves good results on the widely used benchmark problems of climate pattern segmentation and omnidirectional semantic segmentation.
The paper proposes SO(3) equivariant layer, derived from the recently introduced Gauge equivariant CNN framework. The novel contributions are in taking the Gauge equivariance CNN and finding efficient ways to perform logarithmic mapping, parallel transport, and convolution by the equivariant kernel when applied to the sphere. An interpolation scheme for an improved approximation to global SO(3) symmetry is also discussed. Experimental results on spherical MNIST, climate pattern segmentation, and omnidirectional semantic segmentation demonstrate the usefulness of the proposed method for prediction on a sphere.
For the most part, the paper is very clearly written despite the challenging and technical nature of the topic. In particular, the first four pages provide a nice overview of related work and a clear explanation of the Gauge equivariance framework of Cohen et al’19. Two sections that can benefit from further clarification are the proposed "regular non-linearities" (where it would be nice to show in equation why we have equivariance), and equation (4), the logarithmic map (where I had a hard time mapping the discussion in words to the equation). 
However, the experiments, while satisfactory, are not impressive: one issue is that the results are mostly compared to the results of two relevant papers by Cohen and colleagues. In recent years, there have been other proposals for deep learning on the sphere, and I wonder why experiments do not try to compare with these works?  (see Kondor et al’18, Coors et al’18, and others cited in the paper.) Moreover, although in theory, the proposed framework improves the Icosahedral CNN of Cohen et al’19 (by directly operating on the sphere rather than an Icosahedral approximation), the practical improvements over the Icosahedral CNN seem to be often marginal (with one exception in spherical MNIST). Do you have any explanation for this? Is there any setup where you expect the proposed approach would give a substantial improvement?