Unified Fourier-based Kernel and Nonlinearity Design for Equivariant Networks on Homogeneous Spaces
This work addresses the challenge of building efficient equivariant networks for applications in fields like computer vision and molecular modeling, representing an incremental improvement with a unified perspective.
The authors tackled the problem of designing group equivariant networks on homogeneous spaces by introducing a unified Fourier-based framework for kernel and nonlinearity design, achieving state-of-the-art performance in tasks like spherical vector field regression and point cloud classification.
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.