Supplementary Information for Autobahn: Automorphism-based Graph Neural Nets 1 Activations as functions on a group

In the Autobahn formalism, we make extensive use of the fact that the activations of a group-equivariant neural network can be treated as functions on the same group. Here we give a brief review for the unfamiliar reader. This formalism is also covered in detail in Sections 3 and 4 of Reference [6], although under slightly different conventions. Consider a space X acted on by a group G: at every point x in X, we can apply a group element g G, which maps x to another point in X . The action of the group on X induces an action on functions of X . For instance, for standard convolutional layers acting on images, each point on the space is a single pixel and the group of translation moves between pixels.