论文标题
统一的基于傅立叶的内核和非线性网络的同质空间上的非线性设计
Unified Fourier-based Kernel and Nonlinearity Design for Equivariant Networks on Homogeneous Spaces
论文作者
论文摘要
我们在从傅立叶角度得出的同质空间上引入了一个统一的框架。我们考虑在卷积层之前和之后考虑张量值的特征场。我们通过利用提升特征场的傅立叶系数的稀疏性来提出通过傅立叶域的统一推导。当同质空间的稳定子亚组是一个紧凑的谎言组时，稀疏性就会出现。我们进一步引入了非线性激活，通过元素悬浮在定期表示并通过均等卷积向场上投射回现场的非线性非线性。我们表明，其他将特征视为稳定器亚组中傅立叶系数的方法是我们激活的特殊情况。 $ SO（3）$和$ SE（3）$进行的实验显示了球形矢量场回归，点云分类和分子完成中的最新性能。
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.
