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.

从 Fourier 角度出发，我们在同质空间上引入了一种统一的群等变网络框架，其中考虑了张量值特征场及其卷积层前后的特征，通过利用提升特征场的 Fourier 系数的稀疏性，通过 Fourier 域统一推导卷积核并实现非线性激活，该方法在 spherical vector field regression、point cloud classification 和 molecular completion 等任务中达到了最先进的性能水平。

基于傅里叶变换的同变网络同构空间中的核与非线性设计

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

The big empirical success of group equivariant networks has led in recent
years to the sprouting of a great variety of equivariant network architectures.
A particular focus has thereby been on rotation and reflection equivariant CNNs
for planar images. Here we give a general description of $E(2)$-equivariant
convolutions in the framework of Steerable CNNs. The theory of Steerable CNNs
thereby yields constraints on the convolution kernels which depend on group
representations describing the transformation laws of feature spaces. We show
that these constraints for arbitrary group representations can be reduced to
constraints under irreducible representations. A general solution of the kernel
space constraint is given for arbitrary representations of the Euclidean group
$E(2)$ and its subgroups. We implement a wide range of previously proposed and
entirely new equivariant network architectures and extensively compare their
performances. $E(2)$-steerable convolutions are further shown to yield
remarkable gains on CIFAR-10, CIFAR-100 and STL-10 when used as a drop-in
replacement for non-equivariant convolutions.

该研究描述了在可旋转卷积神经网络框架中的 $E (2)$- 等变卷积，提出了转换特性表示描述特征空间变换法则的群表示。研究人员证明了这些约束可以通过使用不可约表示约简为任意群表示的约束，并通过实现一系列先前提出的和全新的等变网络架构进行了广泛比较，表明当用作非等变卷积的替代品时，在 CIFAR-10、CIFAR-100 和 STL-10 上使用 $E (2)$- 可旋转卷积可以取得显著的改进。