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.

中文翻译：

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齐变空间上等变 CNN 的一般理论

我们在欧几里得空间和球体等齐次空间上提出了群等变卷积神经网络 (G-CNN) 的一般理论。这些网络中的特征映射表示同构基础空间上的字段，层是字段空间之间的等变映射。该理论能够根据对称群、基空间和场类型对所有现有 G-CNN 进行系统分类。我们还考虑了一个基本问题：给定类型的特征空间（字段）之间最常见的等变线性映射是什么？在 Mackey 之后，我们展示了这种映射与使用等变内核的卷积一一对应，并表征了这种内核的空间。