Tractability of approximation by general shallow networks

Abstract

In this paper, we present a sharper version of the results in the paper [H. N. Mhaskar, Dimension independent bounds for general shallow networks, Neural Netw. 123 (2020) 142–152]. Let [Formula: see text] and [Formula: see text] be compact metric spaces. We consider approximation of functions of the form [Formula: see text], [Formula: see text], by [Formula: see text]-networks of the form [Formula: see text], [Formula: see text], [Formula: see text]. Defining the dimensions of [Formula: see text] and [Formula: see text] in terms of covering numbers, we obtain dimension independent bounds on the degree of approximation in terms of [Formula: see text], where also the constants involved are all dependent at most polynomially on the dimensions. Applications include approximation by power rectified linear unit networks, zonal function networks, certain radial basis function networks as well as the important problem of function extension to higher-dimensional spaces.