On the $L^p$-Convergence and Denoising Performance of Durrmeyer-Type Max-Min Neural Network Operators

TL;DR

This paper introduces Durrmeyer-type generalizations of max-min neural network operators and proves their convergence in $L^{p}$ for functions $f$ in $L^{p}([a,b],[0,1])$, providing pointwise, supremum, and $L^{p}$ convergence with explicit rate estimates. The nonlinear, non-homogeneous operators leverage a Durrmeyer-type weighting that enhances smoothing relative to Kantorovich variants, and the authors develop comprehensive quantitative bounds using modulus of continuity and $K$-functionals. Through numerical experiments and denoising tasks, the Durrmeyer-type max-min operators demonstrate smoother approximations and superior noise-filtering performance for both impulsive and Gaussian noise, including speech signals. These results highlight the operators’ potential for high-fidelity approximation and robust data processing, with multivariate extensions proposed for future work.

Abstract

In this paper, we investigate Durrmeyer-type generalizations of maximum-minimum neural network operators. The primary objective of this study is to establish the convergence of these operators in the $L^{p}$ norm for functions $f\in L^{p}([a,b],[0,1])$ with $1\leq p<\infty$. To this end, we analyze the properties of sigmoidal functions and maximum-minimum operations, subsequently establishing the convergence of the proposed operator in pointwise, supremum, and $L^{p}$ norms. Furthermore, we derive quantitative estimates for the rates of convergence. In the applications section, numerical and graphical examples demonstrate that the proposed Durrmeyer-type operators provide smoother approximations compared to Kantorovich-type and standard max-min operators. Finally, we highlight the superior filtering performance of these operators in signal analysis, validating their effectiveness in both approximation and data processing tasks.

Figures (4)

• Figure 1: Approximations by $D_{n}^{(m)}(f)$, $F_{n}^{(m)}(f)$ and $K_{n}^{(m)}(f)$.
• Figure 2: Comparison of the filtering capabilities of max-min type NN operators on signals corrupted by salt and pepper noise.
• Figure 3: Comparison of the filtering capabilities of max-min type NN operators on signals corrupted by Gaussian noise.
• Figure 4: Recorded speech and filtering result. Top: Original recorded waveform of the Turkish word “ merhaba”. Middle: The same signal after contamination with salt and pepper noise. Bottom: Reconstructed (filtered) signal obtained by applying the Durrmeyer-type maximum-minimum neural network operator.

Theorems & Definitions (20)

• Lemma 2.1
• Remark 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Lemma 2.7
• Lemma 3.1
• Theorem 3.2