$L^p$-convergence of Kantorovich-type Max-Min Neural Network Operators
<|reference_start|>$L^p$-convergence of Kantorovich-type Max-Min Neural Network Operators: 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.<|reference_end|>
@article{aslan2024$l^{p}$-convergence,
    title={$L^{p}$-convergence of Kantorovich-type Max-Min Neural Network Operators},
    author={.Ismail Aslan, Stefano De Marchi, Wolfgang Erb},
    journal={arXiv preprint arXiv:2407.03329},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.03329},
    primaryClass={math.NA cs.NA}
}