Sharp Bounds on the Approximation Rates, Metric Entropy, and n-Widths of Shallow Neural Networks

In this article, we study approximation properties of the variation spaces corresponding to shallow neural networks with a variety activation functions. We introduce two main tools for estimating metric entropy, rates, and n-widths these spaces. First, notion smoothly parameterized dictionary give upper bounds on nonlinear their absolute convex hull. The depend upon order smoothness parameterization. This result is applied dictionaries ridge functions networks, they improve existing results in many cases. Next, provide method lower bounding entropy which contain certain classes gives sharp $$L^2$$ -approximation range important functions, including ReLU $$^k$$ sigmoidal bounded variation.