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

TL;DR

This paper studies the Kantorovich variant of max-min neural network operators with sigmoidal kernels, proving $L^{p}$-convergence for $1\le p<\infty$ and deriving rates of approximation for function classes, including discontinuous ones. It establishes convergence in sup norm for continuous inputs, provides Hölder- and $K$-functional-based error estimates, and analyzes nonlinearity and non-homogeneity inherent to the max-min framework. Through explicit examples and numerical experiments, the authors compare the Kantorovich max-min operator with max-product and linear variants, demonstrating comparable accuracy and superior denoising performance on noisy signals such as ECG data. The results support the use of Kantorovich max-min operators in signal processing and hint at extensions to higher-dimensional data such as images, where these nonlinear operators can offer robust approximation with noise suppression.

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.

Figures (4)

• Figure 1: Left: Approximation quality of Kantorovich type max-min operator $K_{n}^{(m)}(f)$ for increasing sampling rate $n$. Right: Approximation of $f$ by $K_{30}^{(m)}(f)$ using different sigmoidal functions.
• Figure 2: Left: errors for the approximation of the function $f$ in $L^{1}\left( \left[ 0,1\right] ,\left[ 0,1\right] \right)$ by $K_{n}^{(m)}(f),K_{n}^{(M)}(f)$ and $K_{n}(f)$. Right: Comparison of the computational times to calculate the functions $K_{n}^{(m)}(f)$, $K_{n}^{(M)}(f)$ and $K_{n}(f)$.
• Figure 3: Top row: original signal $f$ (left) and signal $f$ equipped with Gaussian noise (right). Bottom row: Kantorovich variant $K_{n}^{(m)}(f)$ (left) compared to the classical max-min variant $F_{n}^{(m)}(f)$ (right) of the max-min neural network operator applied to the noisy signal.
• Figure 4: Comparison of the Kantorovich forms of the max-min and max-product Neural Network operators

Theorems & Definitions (24)

• Lemma 2.1
• Remark 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Lemma 3.1
• Remark 3.1
• Theorem 3.2