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We introduce a unified framework for group equivariant networks onhomogeneous spaces derived from a Fourier perspective. We considertensor-valued feature fields, before and after a convolutional layer. Wepresent a unified derivation of kernels via the Fourier domain by leveragingthe sparsity of Fourier coefficients of the lifted feature fields. The sparsityemerges when the stabilizer subgroup of the homogeneous space is a compact Liegroup. We further introduce a nonlinear activation, via an elementwisenonlinearity on the regular representation after lifting and projecting back tothe field through an equivariant convolution. We show that other methodstreating features as the Fourier coefficients in the stabilizer subgroup arespecial cases of our activation. Experiments on $SO(3)$ and $SE(3)$ showstate-of-the-art performance in spherical vector field regression, point cloudclassification, and molecular completion.

Unified Fourier-based Kernel and Nonlinearity Design for Equivariant Networks on Homogeneous Spaces