A Wigner-Eckart Theorem for Group Equivariant Convolution Kernels

Group equivariant convolutional networks (GCNNs) endow classical convolutional networks with additional symmetry priors, which can lead to a considerably improved performance. Recent advances in the theoretical description of GCNNs revealed that such models can generally be understood as performing convolutions with $G$-steerable kernels, that is, kernels that satisfy an equivariance constraint themselves. While the $G$-steerability constraint has been derived, it has to date only been solved for specific use cases - a general characterization of $G$-steerable kernel spaces is still missing. This work provides such a characterization for the practically relevant case of $G$ being any compact group. Our investigation is motivated by a striking analogy between the constraints underlying steerable kernels on the one hand and spherical tensor operators from quantum mechanics on the other hand. By generalizing the famous Wigner-Eckart theorem for spherical tensor operators, we prove that steerable kernel spaces are fully understood and parameterized in terms of 1) generalized reduced matrix elements, 2) Clebsch-Gordan coefficients, and 3) harmonic basis functions on homogeneous spaces.