Analysis of Deep Neural Networks with Quasi-optimal polynomial approximation rates

We show the existence of a deep neural network capable of approximating a wide class of high-dimensional approximations. The construction of the proposed neural network is based on a quasi-optimal polynomial approximation. We show that this network achieves an error rate that is sub-exponential in the number of polynomial functions, $M$, used in the polynomial approximation. The complexity of the network which achieves this sub-exponential rate is shown to be algebraic in $M$.