Approximation results in Orlicz spaces for sequences of Kantorovich max-product neural network operators

TL;DR

This paper extends the theory of Kantorovich max-product neural network operators to Orlicz spaces, enabling approximation of a broader class of functions including non-continuous data, with quantitative error estimates and examples of activation functions.

Contribution

It generalizes existing approximation results to Orlicz spaces and introduces a K-functional for error estimation in this setting.

Findings

Extended approximation results to Orlicz spaces.

Introduced a K-functional for quantitative error bounds.

Analyzed sigmoidal activation functions in detail.

Abstract

In this paper we study the theory of the so-called Kantorovich max-product neural network operators in the setting of Orlicz spaces $L^{\varphi}$. The results here proved, extend those given by Costarelli and Vinti in Result Math., 2016, to a more general context. The main advantage in studying neural network type operators in Orlicz spaces relies in the possibility to approximate not necessarily continuous functions (data) belonging to different function spaces by a unique general approach. Further, in order to derive quantitative estimates in this context, we introduce a suitable K-functional in $L^{\varphi}$ and use it to provide an upper bound for the approximation error of the above operators. Finally, examples of sigmoidal activation functions have been considered and studied in details.