Azimuthal Rotational Equivariance in Spherical CNNs

Carl Toft, Georg Bökman, Fredrik Kahl

Abstract: In this work, we analyze linear operators from $L^2(S^2) \rightarrow L^2(S^2)$ which are equivariant to azimuthal rotations, that is, rotations around the z-axis. Several high-performing neural networks defined on the sphere are indeed equivariant to azimuthal rotations, but not $SO(3)$. 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. Our theoretical findings are backed up with experimental results demonstrating that a state-of-the-art pipeline can be improved by making it fully equivariant to azimuthal rotations.

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