Group equivariant and steerable convolutional neural networks (regular and
steerable G-CNNs) have recently emerged as a very effective model class for
learning from signal data such as 2D and 3D images, video, and other data where
symmetries are present. In geometrical terms, regular G-CNNs represent data in
terms of scalar fields ("feature channels"), whereas the steerable G-CNN can
also use vector or tensor fields ("capsules") to represent data. In algebraic
terms, the feature spaces in regular G-CNNs transform according to a regular
representation of the group G, whereas the feature spaces in Steerable G-CNNs
transform according to the more general induced representations of G. In order
to make the network equivariant, each layer in a G-CNN is required to
intertwine between the induced representations associated with its input and
output space.
In this paper we present a general mathematical framework for G-CNNs on
homogeneous spaces like Euclidean space or the sphere. We show, using
elementary methods, that the layers of an equivariant network are convolutional
if and only if the input and output feature spaces transform according to an
induced representation. This result, which follows from G.W. Mackey's abstract
theory on induced representations, establishes G-CNNs as a universal class of
equivariant network architectures, and generalizes the important recent work of
Kondor & Trivedi on the intertwiners between regular representations.

本文提出了一个 G-CNNs 的数学框架，证明了如果输入和输出特征空间根据受激表示变换，则网络的层为卷积操作。这个结果确定了 G-CNNs 是一个通用的等变网络结构类，它一般化了最近对正则表示之间的交错操作的重要工作。