Exponential Separations in Symmetric Neural Networks

In this work we demonstrate a novel separation between symmetric neural network architectures. Specifically, we consider the Relational Network \parencite{santoro2017simple} architecture as a natural generalization of the DeepSets \parencite{zaheer2017deep} architecture, and study their representational gap. Under the restriction to analytic activation functions, we construct a symmetric function acting on sets of size N with elements in dimension D, which can be efficiently approximated by the former architecture, but provably requires width exponential in N and D for the latter.