$L^{p}$-convergence of Kantorovich-type Max-Min Neural Network Operators
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| Format: | Preprint |
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2024
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| _version_ | 1866911943828176896 |
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| author | Aslan, İsmail De Marchi, Stefano Erb, Wolfgang |
| author_facet | Aslan, İsmail De Marchi, Stefano Erb, Wolfgang |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_03329 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $L^{p}$-convergence of Kantorovich-type Max-Min Neural Network Operators Aslan, İsmail De Marchi, Stefano Erb, Wolfgang Numerical Analysis 41A30, 41A25 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. |
| title | $L^{p}$-convergence of Kantorovich-type Max-Min Neural Network Operators |
| topic | Numerical Analysis 41A30, 41A25 |
| url | https://arxiv.org/abs/2407.03329 |