S3CNN: Spin-weighted Smoothing Spherical CNN for Learning from Scalar and Vector Fields on Spheres

Analyzing scalar and vector fields on the sphere, such as temperature or wind speed and direction on Earth, is a difficult task. Models should respect both the rotational symmetries of the sphere and the inherent symmetries of the vector fields. A class of equivariant models has emerged, which process these spherical signals by applying group convolutions in Fourier space with respect to the three-dimensional rotation group and have taken the name of spherical CNNs. In their original formulation, however, spherical CNNs are computationally expensive, and thus intractable for most real-world applications. Spin-weighted spherical CNNs were proposed as a way of reducing the scaling in the number of filter-coefficients with increased bandwidth, but the resulting architectures impose constraints on the choice of convolution kernels and activation functions in order to preserve the desired signal properties. In this paper, we introduce a deep learning architecture that occupies the middle ground: less demanding than the original spherical CNNs, yet employing a richer class of convolution kernels and activation functions, and suitable for signals consisting of both scalar and vector fields on the sphere. Experiments show that this architecture generally outperforms standard CNNs and often matches or exceeds the performance of spin-weighted spherical CNNs trained under comparable conditions. The advantage over spin-weighted spherical CNNs is not uniform across all tasks, however, and we present ablation studies in which we observe that incorporating the interaction between different spins in the hidden layers narrows this gap.