Group equivariant convolutional neural networks (G-CNNs) have recently emerged as a very effective model class for learning from signals in the context of known symmetries. A wide variety of equivariant layers has been proposed for signals on 2D and 3D Euclidean space, graphs, and the sphere, and it has become difficult to see how all of these methods are related, and how they may be generalized. In this paper, we present a fairly general theory of equivariant convolutional networks. Convolutional feature spaces are described as fields over a homogeneous base space, such as the plane $\mathbb{R}^2$, sphere $S^2$ or a graph $\mathcal{G}$. The theory enables a systematic classification of all existing G-CNNs in terms of their group of symmetry, base space, and field type (e.g. scalar, vector, or tensor field, etc.). In addition to this classification, we use Mackey theory to show that convolutions with equivariant kernels are the most general class of equivariant maps between such fields, thus establishing G-CNNs as a universal class of equivariant networks. The theory also explains how the space of equivariant kernels can be parameterized for learning, thereby simplifying the development of G-CNNs for new spaces and symmetries. Finally, the theory introduces a rich geometric semantics to learned feature spaces, thus improving interpretability of deep networks, and establishing a connection to central ideas in mathematics and physics.

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。

关于齐变卷积神经网络在同性空间上的一般理论