Equivariance has emerged as a desirable property of representations of objects subject to identity-preserving transformations that constitute a group, such as translations and rotations. However, the expressivity of a representation constrained by group equivariance is still not fully understood. We address this gap by providing a generalization of Cover's Function Counting Theorem that quantifies the number of linearly separable and group-invariant binary dichotomies that can be assigned to equivariant representations of objects. We find that the fraction of separable dichotomies is determined by the dimension of the space that is fixed by the group action. We show how this relation extends to operations such as convolutions, element-wise nonlinearities, and global and local pooling. While other operations do not change the fraction of separable dichotomies, local pooling decreases the fraction, despite being a highly nonlinear operation. Finally, we test our theory on intermediate representations of randomly initialized and fully trained convolutional neural networks and find perfect agreement.

本文通过对Cover函数计数定理的推广提出了一个方法，量化可以分配给物体的等变表现的线性可分和群不变的二元分裂的数量。作者发现可分割分割是由群作用修正的空间维度决定的。此外，作者将其理论应用于卷积神经网络的中间表示，并发现其与实际结果完全相符合。

Capacity of Group-invariant Linear Readouts from Equivariant
  Representations: How Many Objects can be Linearly Classified Under All
  Possible Views?

等变表示的群不变线性读出的容量：在所有可能的视角下，可以将多少个对象线性分类？