General multivariate arctangent function activated neural network approximations

George A. Anastassiou

doi:10.33993/jnaat511-1262

Authors

George A. Anastassiou University of Memphis, USA

DOI:

https://doi.org/10.33993/jnaat511-1262

Keywords:

arctangent function, multivariate neural network approximation, quasi-interpolation operator, Kantorovich type operator, quadrature type operator, multivariate modulus of continuity, abstract approximation, iterated approximation

Abstract views: 270

Abstract

Here we expose multivariate quantitative approximations 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 treat also the case of approximation by iterated operators of the last four types. These approximations are derived by establishing multidimensional Jackson type inequalities involving the multivariate modulus of continuity of the engaged function or its high order Frechet derivatives.

Our multivariate operators are defined by using a multidimensional density function induced by the arctangent function. The approximations are pointwise and uniform. The related feed-forward neural network is with one hidden layer.

Downloads

Download data is not yet available.

References

G.A. Anastassiou, Moments in Probability and Approximation Theory, Pitman Research Notes in Math., Vol. 287, Longman Sci. & Tech., Harlow, U.K., 1993.

G.A. Anastassiou, Rate of convergence of some neural network operators to the unit-univariate case, J. Math. Anal. Appli. 212 (1997), pp. 237-262, https://doi.org/10.1006/jmaa.1997.5494 DOI: https://doi.org/10.1006/jmaa.1997.5494

G.A. Anastassiou, Quantitative Approximations, Chapman&Hall/CRC, Boca Raton, New York, 2001.

G.A. Anastassiou, Inteligent Systems: Approximation by Artificial Neural Networks, Intelligent Systems Reference Library, Vol. 19, Springer, Heidelberg, 2011, https://doi.org/10.1007/978-3-642-21431-8 DOI: https://doi.org/10.1007/978-3-642-21431-8

G.A. Anastassiou, Univariate hyperbolic tangent neural network approximation, Mathematics and Computer Modelling, 53 (2011), pp. 1111-1132, https://doi.org/10.1016/j.mcm.2010.11.072 DOI: https://doi.org/10.1016/j.mcm.2010.11.072

G.A. Anastassiou, Multivariate hyperbolic tangent neural network approximation, Computers and Mathematics 61 (2011), pp. 809-821, https://doi.org/10.1016/j.camwa.2010.12.029 DOI: https://doi.org/10.1016/j.camwa.2010.12.029

G.A. Anastassiou, Multivariate sigmoidal neural network approximation, Neural Networks 24 (2011), pp.~378-386, https://doi.org/10.1016/j.neunet.2011.01.003 DOI: https://doi.org/10.1016/j.neunet.2011.01.003

G.A. Anastassiou, Univariate sigmoidal neural network approximation, J. of Computational Analysis and Applications, Vol. 14, No. 4, 2012, pp. 659-690.

G.A. Anastassiou, Approximation by neural networks iterates, Advances in Applied Mathematics and Approximation Theory, pp. 1-20, Springer Proceedings in Math. & Stat., Springer, New York, 2013, Eds. G. Anastassiou, O. Duman. DOI: https://doi.org/10.1007/978-1-4614-6393-1_1

G. Anastassiou, Intelligent Systems II: Complete Approximation by Neural Network Operators, Springer, Heidelberg, New York, 2016. DOI: https://doi.org/10.1007/978-3-319-20505-2

G. Anastassiou, Intelligent Computations: Abstract Fractional Calculus, Inequalities, Approximations, Springer, Heidelberg, New York, 2018. DOI: https://doi.org/10.1007/978-3-319-66936-6

H. Cartan, Differential Calculus, Hermann, Paris, 1971.

Z. Chen and F. Cao, The approximation operators with sigmoidal functions, Computers and Mathematics with Applications, 58 (2009), pp. 758-765, https://doi.org/10.1016/j.camwa.2009.05.001 DOI: https://doi.org/10.1016/j.camwa.2009.05.001

D. Costarelli, R. Spigler, Approximation results for neural network operators activated by sigmoidal functions, Neural Networks 44 (2013), pp.~101-106, https://doi.org/10.1016/j.neunet.2013.03.015 DOI: https://doi.org/10.1016/j.neunet.2013.03.015

D. Costarelli, R. Spigler, Multivariate neural network operators with sigmoidal activation functions, Neural Networks 48 (2013), pp. 72-77, https://doi.org/10.1016/j.neunet.2013.07.009 DOI: https://doi.org/10.1016/j.neunet.2013.07.009

S. Haykin, Neural Networks: A Comprehensive Foundation (2 ed.), Prentice Hall, New York, 1998.

W. McCulloch and W. Pitts, A logical calculus of the ideas immanent in nervous activity, Bulletin of Mathematical Biophysics, 7 (1943), pp. 115-133, https://doi.org/10.1007/bf02478259 DOI: https://doi.org/10.1007/BF02478259

T.M. Mitchell, Machine Learning, WCB-McGraw-Hill, New York, 1997.

L.B. Rall, Computational Solution of Nonlinear Operator Equations, John Wiley & Sons, New York, 1969