Quantitative Uniform Approximation by Activated Singular Operators

Abstract

In this article, we study the approximation properties of activated singular integral operators over the real line. We establish their convergence to the unit operator with rates. The kernels here derive from neural network activation functions and their corresponding density functions. The estimates are mostly sharp, and they are pointwise and uniform. The derived inequalities involve the higher order modulus of smoothness.

Keywords

• activation functions from neural networks
• best constant
• singular integral
• modulus of smoothness
• sharp inequality