\(q\)-Deformed and \(\lambda\)-parametrized A-generalized logistic function induced Banach space valued multivariate multi layer neural network approximations
DOI:
https://doi.org/10.24193/subbmath.2024.3.08Abstract
Here we research the multivariate quantitative approximation of Banach space valued continuous multivariate functions on a box or \(\mathbb{R}^{N}\), \(N\in \mathbb{N},\) by the multivariate normalized, quasi-interpolation, Kantorovich type and quadrature type neural network operators. We investigate also the case of approximation by iterated multilayer neural network operators of the last four types. These approximations are achieved by establishing multidimensional Jackson type inequalities involving the multivariate modulus of continuity of the engaged function or its partial derivatives. Our multivariate operators are defined by using a multidimensional density function induced by a \(q\)-deformed and \(\lambda\)-parametrized \(A\)-generalized logistic function, which is a sigmoid function. The approximations are pointwise and uniform. The related feed-forward neural network are with one or multi hidden layers.
References
Anastassiou, G.A., Moments in Probability and Approximation Theory, Pitman Research Notes in Math., Vol. 287, Longman Sci. & Tech., Harlow, U.K., 1993.
Anastassiou, G.A., Rate of convergence of some neural network operators to the unit-univariate case, J. Math. Anal. Appl., 212(1997), 237-262.
Anastassiou, G.A., Quantitative Approximations, Chapman & Hall/CRC, Boca Raton, New York, 2001.
Anastassiou, G.A., Inteligent Systems: Approximation by Arti cial Neural Networks, Intelligent Systems Reference Library, Vol. 19, Springer, Heidelberg, 2011.
Anastassiou, G.A., Univariate hyperbolic tangent neural network approximation, Mathematics and Computer Modelling, 53(2011), 1111-1132.
Anastassiou, G.A., Multivariate hyperbolic tangent neural network approximation, Computers and Mathematics, 61(2011), 809-821.
Anastassiou, G.A., Multivariate sigmoidal neural network approximation, Neural Networks, 24(2011), 378-386.
Anastassiou, G.A., Univariate sigmoidal neural network approximation, J. of Computational Analysis and Applications, 14(2012), no. 4, 659-690.
Anastassiou, G.A., Approximation by neural networks iterates, Advances in Applied Mathematics and Approximation Theory, Springer Proceedings in Math. & Stat., Eds. G. Anastassiou, O. Duman, Springer, New York, 2013, 1-20.
Anastassiou, G.A., Intelligent Systems II: Complete Approximation by Neural Network Operators, Springer, Heidelberg, New York, 2016.
Anastassiou, G.A., Intelligent Computations: Abstract Fractional Calculus, Inequalities,
Approximations, Springer, Heidelberg, New York, 2018.
Anastassiou, G.A., General sigmoid based Banach space valued neural network approximation, J. of Computational Analysis and Applications, accepted, 2022.
Anastassiou, G.A., Banach Space Valued Neural Network, Springer, Heidelberg, New York, 2023.
Chen, Z., Cao, F., The approximation operators with sigmoidal functions, Computers and Mathematics with Applications, 58(2009), 758-765.
Costarelli, D., Spigler, R., Approximation results for neural network operators activated by sigmoidal functions, Neural Networks, 44(2013), 101-106.
Costarelli, D., Spigler, R., Multivariate neural network operators with sigmoidal activation functions, Neural Networks, 48(2013), 72-77.
Haykin, S., Neural Networks: A Comprehensive Foundation (2 ed.), Prentice Hall, New York, 1998.
McCulloch, W., Pitts, W., A logical calculus of the ideas immanent in nervous activity, Bulletin of Mathematical Biophysics, 7(1943), 115-133.
Mitchell, T.M., Machine Learning, WCB-McGraw-Hill, New York, 1997.
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