Saturation Classes for Max-Product Neural Network Operators Activated by Sigmoidal Functions

In the present paper, we obtain a saturation result for the neural network (NN) operators of the max-product type. In particular, we show that any non-constant, continuous function on the interval [0, 1] cannot be approximated by the above operators Fn(M)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F^{(M)}_n$$\end{document}, n∈N+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \in \mathbb {N}^+$$\end{document}, by a rate of convergence higher than 1 / n. Moreover, since we know that any Lipschitz function f can be approximated by the NN operators with an order of approximation of 1 / n, here we are able to prove a local inverse result, in order to provide a characterization of the saturation (Favard) classes.