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 santoro2017simple architecture as a natural generalization of the DeepSets 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.