Azimuthal Rotational Equivariance in Spherical Convolutional Neural Networks

Paper in proceeding, 2022

In this work, we analyze linear operators on the space of square integrable functions on the sphere. Specifically, we characterize the operators which are equivariant to azimuthal rotations, that is, rotations around the z-axis. Several high-performing neural networks defined on the sphere are equivariant to azimuthal rotations, but not to full SO(3) rotations. Our main result is to show that a linear operator acting on band-limited functions on the sphere is equivariant to azimuthal rotations if and only if it can be realized as a block-diagonal matrix acting on the spherical harmonic expansion coefficients of its input. Further, we show that such an operation can be interpreted as a convolution, or equivalently, a correlation in the spatial domain. Our theoretical findings are backed up with experimental results demonstrating that a state-of-the-art pipeline can be improved by making it equivariant to azimuthal rotations.