Efficiency of deep networks for radially symmetric functions

We prove that radially symmetric functions in d dimensions can be approximated by a deep network with fewer neurons than the previously best known result. Our results are much more efficient in terms of the support radius of the radial function and the error of approximation. Our proofs are all constructive and we specify the network architecture and almost all of the weights. The method relies on space-folding transformations that allow us to approximate the norm of a high dimensional vector using relatively few neurons.