Universal Equivariant Multilayer Perceptrons

Abstract

Group invariant and equivariant Multilayer Perceptrons (MLP), also known as Equivariant Networks, have achieved remarkable success in learning on a variety of data structures, such as sequences, images, sets, and graphs. Using tools from group theory, this paper proves the universality of a broad class of equivariant MLPs with a single hidden layer. In particular, it is shown that having a hidden layer on which the group acts regularly is sufficient for universal equivariance. Next, Burnside's table of marks is used to decompose product spaces. It is shown that the product of two G-sets always contains an orbit larger than the input orbits. Therefore high order hidden layers inevitably contain a regular orbit, leading to the universality of the corresponding MLP. It is shown that with an order larger than the logarithm of the size of the stabilizer group, a high-order equivariant MLP is equivariant universal.