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Equivariance has emerged as a desirable property of representations ofobjects subject to identity-preserving transformations that constitute a group,such as translations and rotations. However, the expressivity of arepresentation constrained by group equivariance is still not fully understood.We address this gap by providing a generalization of Cover's Function CountingTheorem that quantifies the number of linearly separable and group-invariantbinary dichotomies that can be assigned to equivariant representations ofobjects. We find that the fraction of separable dichotomies is determined bythe dimension of the space that is fixed by the group action. We show how thisrelation extends to operations such as convolutions, element-wisenonlinearities, and global and local pooling. While other operations do notchange the fraction of separable dichotomies, local pooling decreases thefraction, despite being a highly nonlinear operation. Finally, we test ourtheory on intermediate representations of randomly initialized and fullytrained convolutional neural networks and find perfect agreement.

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