We present a general theory of Group equivariant Convolutional Neural
Networks (G-CNNs) on homogeneous spaces such as Euclidean space and the sphere.
Feature maps in these networks represent fields on a homogeneous base space,
and layers are equivariant maps between spaces of fields. The theory enables a
systematic classification of all existing G-CNNs in terms of their symmetry
group, base space, and field type. We also consider a fundamental question:
what is the most general kind of equivariant linear map between feature spaces
(fields) of given types? Following Mackey, we show that such maps correspond
one-to-one with convolutions using equivariant kernels, and characterize the
space of such kernels.

本研究提出了关于群等变卷积神经网络（G-CNNs）在同种空间如欧几里德空间和球面上的总体理论。这些网络中的特征映射表示同种基本空间上的场，层是场空间之间的等变映射。该理论使得所有现有的 G-CNNs 都能按照它们的对称群、基础空间和场类型进行系统分类。我们还考虑了一个根本性问题：什么是给定类型的特征空间（场）之间等变线性映射的最普遍类型？我们证明这样的映射与使用等变核进行卷积一一对应，并且表征了这些核的空间。