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

Yinshuang Xu, Jiahui Lei, Edgar Dobriban, Kostas Daniilidis
Proceedings of the 39th International Conference on Machine Learning, PMLR 162:24596-24614, 2022.

Abstract

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v162-xu22e, title = {Unified {F}ourier-based Kernel and Nonlinearity Design for Equivariant Networks on Homogeneous Spaces}, author = {Xu, Yinshuang and Lei, Jiahui and Dobriban, Edgar and Daniilidis, Kostas}, booktitle = {Proceedings of the 39th International Conference on Machine Learning}, pages = {24596--24614}, year = {2022}, editor = {Chaudhuri, Kamalika and Jegelka, Stefanie and Song, Le and Szepesvari, Csaba and Niu, Gang and Sabato, Sivan}, volume = {162}, series = {Proceedings of Machine Learning Research}, month = {17--23 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v162/xu22e/xu22e.pdf}, url = {https://proceedings.mlr.press/v162/xu22e.html}, abstract = {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.} }
Endnote
%0 Conference Paper %T Unified Fourier-based Kernel and Nonlinearity Design for Equivariant Networks on Homogeneous Spaces %A Yinshuang Xu %A Jiahui Lei %A Edgar Dobriban %A Kostas Daniilidis %B Proceedings of the 39th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2022 %E Kamalika Chaudhuri %E Stefanie Jegelka %E Le Song %E Csaba Szepesvari %E Gang Niu %E Sivan Sabato %F pmlr-v162-xu22e %I PMLR %P 24596--24614 %U https://proceedings.mlr.press/v162/xu22e.html %V 162 %X 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.
APA
Xu, Y., Lei, J., Dobriban, E. & Daniilidis, K.. (2022). Unified Fourier-based Kernel and Nonlinearity Design for Equivariant Networks on Homogeneous Spaces. Proceedings of the 39th International Conference on Machine Learning, in Proceedings of Machine Learning Research 162:24596-24614 Available from https://proceedings.mlr.press/v162/xu22e.html.

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