Convolutional Neural Networks Approximation in Quasi-Orlicz Spaces on Sphers
DOI:
https://doi.org/10.31642/JoKMC/2018/110101%20Keywords:
Approximation, Quasi-Orlicz, Modulus of Smoothness, Convolution Neural NetworkAbstract
It is necessary to study the theoretical bases of an approximation deep convolutional neural networks, because of its interesting developments in vital domains. The approximation abilities of deep-convolution neural networks produced by downsampling operators in quasi- Orlicz spaces have been studied, since this space is wider and more important than other spaces. In this paper, we define quasi-Orlicz norm on spherical spaces. In addition, modulus of smoothness is also studied in terms of quasi-Orlicz norm. Finally, Function approximation theorems are studied by using convolution neural networks with
Downloads
References
W. Orlicz, "Über eine gewisse Klasse von Räumen vom Typus B," Bull. Int. Acad. Pol. Ser. A, vol. 8, no. 9, pp. 207-220, 1932.
H. Nakano, Topology and linear topological spaces. Maruzen Company, 1951.
W. A. J. Luxemburg, "Banach function spaces," 1955.
W. Luxemburg and A. Zaanen, "Conjugate spaces of Orlicz spaces," in Indagationes Mathematicae (Proceedings), 1956, vol. 59: Elsevier, pp. 217-228.
H. Hudzik and L. Maligranda, "Amemiya norm equals Orlicz norm in general," Indagationes Mathematicae, vol. 11, no. 4, pp. 573-585, 2000.
A. M. Aljanabi, Almurieb, Hawraa Abbas, "Orlicz Approximation by Convolutional Neural Networks," Evolutionary Intelligence ,Springer Nature, February 2023.
F. Dai and Y. Xu, "Moduli of smoothness and approximation on the unit sphere and the unit ball," Advances in Mathematics, vol. 224, no. 4, pp. 1233-1310, 2010.
Z. Ditzian and V. Totik, "Springer Series in Computational Mathematics," vol. 9, ed: Springer-Verlag New York, 1987.
D.-X. Zhou, "Theory of deep convolutional neural networks: Downsampling," Neural Networks, vol. 124, pp. 319-327, 2020.
Z. Fang, H. Feng, S. Huang, and D.-X. Zhou, "Theory of deep convolutional neural networks II: Spherical analysis," Neural Networks, vol. 131, pp. 154-162, 2020.
U. o. I. a. U.-C. C. f. S. Research, Development, and G. Cybenko, Continuous valued neural networks with two hidden layers are sufficient. 1988.
D.-X. Zhou, "Universality of deep convolutional neural networks," Applied and computational harmonic analysis, vol. 48, no. 2, pp. 787-794, 2020.
A. M. Aljanabi, Almurieb, Hawraa Abbas, "The Degree of Best Downsampled Convolutional Neural Network Approximation in terms of Orlicz Modulus of Smoothness," journal of Mathematical Sciences, 26May -2023.
Downloads
Published
How to Cite
Issue
Section
Categories
License
Copyright (c) 2024 Amna Manaf AL-Janabi, Hawraa Abbas Almurieb
This work is licensed under a Creative Commons Attribution 4.0 International License.
which allows users to copy, create extracts, abstracts, and new works from the Article, alter and revise the Article, and make commercial use of the Article (including reuse and/or resale of the Article by commercial entities), provided the user gives appropriate credit (with a link to the formal publication through the relevant DOI), provides a link to the license, indicates if changes were made and the licensor is not represented as endorsing the use made of the work.