  We introduce a unified framework for group equivariant networks on
homogeneous spaces derived from a Fourier perspective. We consider
tensor-valued feature fields, before and after a convolutional layer. We
present a unified derivation of kernels via the Fourier domain by leveraging
the sparsity of Fourier coefficients of the lifted feature fields. The sparsity
emerges when the stabilizer subgroup of the homogeneous space is a compact Lie
group. We further introduce a nonlinear activation, via an elementwise
nonlinearity on the regular representation after lifting and projecting back to
the field through an equivariant convolution. We show that other methods
treating features as the Fourier coefficients in the stabilizer subgroup are
special cases of our activation. Experiments on $SO(3)$ and $SE(3)$ show
state-of-the-art performance in spherical vector field regression, point cloud
classification, and molecular completion.
