  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.
