Unified Fourier-based Kernel and Nonlinearity Design for Equivariant Networks on Homogeneous Spaces
Unified Fourier-based Kernel and Nonlinearity Design for Equivariant Networks on Homogeneous Spaces
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DOI：
10.48550/arxiv.2206.08362
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发表时间：
2022-06
期刊：
影响因子：
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通讯作者：
Yinshuang Xu;Jiahui Lei;Edgar Dobriban;Kostas Daniilidis
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文献类型：
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作者：
Yinshuang Xu;Jiahui Lei;Edgar Dobriban;Kostas Daniilidis
我们介绍了一个统一的框架，从傅立叶的角度来看，来自齐次空间上的群等变网络。我们考虑卷积层之前和之后的张量值特征场。我们提出了一个统一的推导内核通过傅立叶域利用稀疏的傅立叶系数的提升功能字段。当齐次空间的稳定子群是紧李群时，稀疏性出现。我们进一步引入了一个非线性激活，通过一个elementwise非线性的正规表示后，提升和投影回字段通过等变卷积。我们表明，其他方法处理功能的傅立叶系数的稳定子群是我们的激活的特殊情况。在$SO（3）$和$SE（3）$上的实验显示了球面向量场回归、点云分类和分子补全的最新性能。
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.