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Unified Fourier-based Kernel and Nonlinearity Design for Equivariant Networkson Homogeneous Spaces
Yinshuang Xu · Jiahui Lei · Edgar Dobriban · Kostas Daniilidis
We introduce a unified framework for group equivariant networks on homogeneous spaces derived from a Fourier perspective. We address the case of feature fields before and after a convolutional layer being tensor valued, and present a unified derivation of kernels via the Fourier domain by taking advantage of 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 an activation method 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.

Author Information

Yinshuang Xu (University of Pennsylvania)
Jiahui Lei (University of Pennsylvania)
Edgar Dobriban (University of Pennsylvania)
Kostas Daniilidis (University of Pennsylvania)

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