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APPROXIMATION BY SPHERICAL NEURAL NETWORKS WITH ZONAL FUNCTIONS

Part of: Approximations and expansions

Published online by Cambridge University Press:  26 April 2017

ZHIXIANG CHEN  and
FEILONG CAO
ZHIXIANG CHEN
Affiliation:
Department of Mathematics, Shaoxing University, Shaoxing 312000, Zhejiang Province, PR China email czx@usx.edu.cn
FEILONG CAO*
Affiliation:
Department of Applied Mathematics, China Jiliang University, Hangzhou 310018, Zhejiang Province, PR China email feilongcao@gmail.com
*
✉ feilongcao@gmail.com
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Abstract

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We address the construction and approximation for feed-forward neural networks (FNNs) with zonal functions on the unit sphere. The filtered de la Vallée-Poussin operator and the spherical quadrature formula are used to construct the spherical FNNs. In particular, the upper and lower bounds of approximation errors by the FNNs are estimated, where the best polynomial approximation of a spherical function is used as a measure of approximation error.

Keywords

approximationneural networkszonal functionerror

MSC classification

Primary: 41A25: Rate of convergence, degree of approximation

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 58 , Issue 3-4 , April 2017 , pp. 238 - 246
Copyright
© 2017 Australian Mathematical Society 

References

Cao, F. and Chen, Z., “Approximation by spherical neural networks with sigmoidal function”, J. Comput. Anal. Appl. 18 (2015) 390–396; https://scholar.google.com.au/scholar?cluster=16133814104893368681&hl=en&as_sdt=0,5&as_vis=1.Google Scholar
Cao, F. and Lin, S., “The capability of approximation for neural networks interpolant on the sphere”, Math. Methods Appl. Sci. 34 (2011) 469–478; doi:10.1002/mma.1373.Google Scholar
Dunkl, C. F. and Xu, Y., “Orthogonal polynomials of several variables”, in: Encyclopedia of Mathematics and its Applications (Cambridge University Press, Cambridge, 2001).Google Scholar
Filbir, F. and Themistoclakis, W., “Polynomial approximation on the sphere using scattered data”, Math. Nachr. 281 (2008) 650–668; doi:10.1002/mana.200710633.Google Scholar
Freeden, W., Gervens, T. and Schreiner, M., Constructive approximation on the sphere (Oxford University Press, New York, 1998).Google Scholar
Lin, S. and Cao, F., “Simultaneous approximation by spherical neural networks”, Neurocomputing 175 (2016) 348–354; doi:10.1016/j.neucom.2015.10.067.Google Scholar
Lin, S., Zeng, J. and Xu, Z., “Error estimate for spherical neural networks interpolation”, Neural Process. Lett. 42 (2015) 369–379; doi:10.1007/s11063-014-9361-x.Google Scholar
Maiorov, V. E., “On best approximation by ridge functions”, J. Approx. Theory 99 (1999) 68–94; doi:10.1006/jath.1998.3304.Google Scholar
Mhaskar, H. N., Narcowich, F. J. and Ward, J. D., “Approximation properties of zonal function networks using scattered data on the sphere”, Adv. Comput. Math. 11 (1999) 121–137;doi:10.1023/A:1018967708053.Google Scholar
Sloan, I. H., “Polynomial approximation on spheres – generalizing de la Vallée-Poussin”, Comput. Methods Appl. Math. 11 (2011) 540–552; doi:10.2478/cmam-2011-0029.Google Scholar
Stein, E. M. and Weiss, G., Introduction to Fourier analysis on Euclidean spaces (Princeton University Press, Princeton, NJ, 1971).Google Scholar