title: Universal approximations of permutation invariant/equivariant functions by deep neural networks
abstract: In this paper, we develop a theory about the relationship between $G$-invariant/equivariant functions and deep neural networks for finite group $G$. Especially, for a given $G$-invariant/equivariant function, we construct its universal approximator by deep neural network whose layers equip $G$-actions and each affine transformations are $G$-equivariant/invariant. Due to representation theory, we can show that this approximator has exponentially fewer free parameters than usual models. 
authors: Akiyoshi Sannai Yuuki Takai Matthieu Cordonnier
hIndex: Akiyoshi Sannai 10