Designing a convolution for a spherical neural network requires a delicate tradeoff between efficiency and rotation equivariance. DeepSphere, a method based on a graph representation of the sampled sphere, strikes a controllable balance between these two desiderata. This contribution is twofold. First, we study both theoretically and empirically how equivariance is affected by the underlying graph with respect to the number of vertices and neighbors. Second, we evaluate DeepSphere on relevant problems. Experiments show state-of-the-art performance and demonstrates the efficiency and flexibility of this formulation. Perhaps surprisingly, comparison with previous work suggests that anisotropic filters might be an unnecessary price to pay.
The paper presents DeepSphere, a method for learning over spherical data via a graphical representation and graph-convolutions. The primary goal is to develop a method that encodes equivariance to rotations, cheaply. The graph is formed by sampling the surface of the sphere and connecting neighbors according to a distance-based similarity measure. The equivariance of the representation is demonstrated empirically and theoretical background on its convergence properties are shown. DeepSphere is then demonstrated on several problems as well as shown how it applies to non-uniform data.
The paper is interesting and clear. The projection of structured data to graphical representations is both efficient in utilizing existing algorithmic techniques for graph convolutions and useful for approaching the spherical structure of the data. The theoretical analysis and discussion of sampling is interesting, though should be more clearly stated throughout and potentially visualized in figures.
The experiments performed are thorough and interesting. The approach both outperforms baselines in inference time and accuracy. However, one wonders the performance on the well-researched tasks such as the performance on 3D imagery, e.g., Su & Grauman, 2017; Coors et al., 2018. 
The unevenly sampled data is a nice extension showing the generality of the approach. How does the approach work for data connected within a radius rather than a k-nearest approach?
Minor:
- A figure detailing the parameters and setup for theorem 3.1 and figure 2 would be useful.
- The statement on the dispersion of the sampling sequence states “the smallest ball in \R^3 containing \sigma_i”, but I believe it should be “containing only \sigma_i”.