$L^{p}$-convergence of Kantorovich-type Max-Min Neural Network Operators

TL;DR

This paper establishes the $L^{p}$-convergence of Kantorovich-type max-min neural network operators with sigmoidal kernels, providing approximation results for discontinuous functions and comparing their effectiveness to other neural network operators.

Contribution

It introduces and proves $L^{p}$-convergence for a new class of Kantorovich max-min neural network operators with sigmoidal kernels, including error estimates and practical applications.

Findings

Proven $L^{p}$-convergence for the operators.

Derived quantitative approximation error estimates.

Demonstrated advantages in noisy signal approximation, e.g., ECG signals.

Abstract

In this work, we study the Kantorovich variant of max-min neural network operators, in which the operator kernel is defined in terms of sigmoidal functions. Our main aim is to demonstrate the $L^{p}$-convergence of these nonlinear operators for $1\leq p<\infty$, which makes it possible to obtain approximation results for functions that are not necessarily continuous. In addition, we will derive quantitative estimates for the rate of approximation in the $L^{p}$-norm. We will provide some explicit examples, studying the approximation of discontinuous functions with the max-min operator, and varying additionally the underlying sigmoidal function of the kernel. Further, we numerically compare the $L^{p}$-approximation error with the respective error of the Kantorovich variants of other popular neural network operators. As a final application, we show that the Kantorovich variant has advantages compared to the sampling variant of the max-min operator and Kantorovich variant of the max-product operator when it comes to approximate noisy functions as for instance biomedical ECG signals.