Abstract
In this paper an approximation of multivariable functions by Hermite basis is presented and discussed. Considered here basis is constructed as a product of one-variable Hermite functions with adjustable scaling parameters. The approximation is calculated via hybrid method, the expansion coefficients by using an explicit, non-search formulae, and scaling parameters are determined via a search algorithm. A set of excessive number of Hermite functions is initially calculated. To constitute the approximation basis only those functions are taken which ensure the fastest error decrease down to a desired level. Working examples are presented, demonstrating a very good generalization property of this method.
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References
Beliczynski, B.: Properties of the Hermite activation functions in a neural approximation scheme. In: Beliczynski, B., Dzielinski, A., Iwanowski, M., Ribeiro, B. (eds.) ICANNGA 2007, Part II. LNCS, vol. 4432, pp. 46–54. Springer, Heidelberg (2007)
Hlawatsch, F.: Time-Frequency Analysis and Synthesis of Linear Signal Spaces. Kluwer Academic Publishers, Dordrecht (1998)
Ma, L., Khorasani, K.: Constructive feedforward neural networks using Hermite polynomial activation functions. IEEE Transactions on Neural Networks 16, 821–833 (2005)
Kwok, T., Yeung, D.: Constructive algorithms for structure learning in feedforward neural networks for regression problems. IEEE Trans. Neural Netw. 8(3), 630–645 (1997)
Kwok, T., Yeung, D.: Objective functions for training new hidden units in constructive neural networks. IEEE Trans. Neural Networks 8(5), 1131–1148 (1997)
Kreyszig, E.: Introductory functional analysis with applications. J. Wiley, Chichester (1978)
Beliczynski, B., Ribeiro, B.: Some enhanencement to approximation of one-variable functions by orthonormal basis. Neural Network World 19, 401–412 (2009)
Lorentz, R.: Multivariate hermite interpolation by algebraic polynomials: A survey. Journal of Computational and Applied Mathematics 122, 167–201 (2000)
Reed, R.: Pruning algorithms - a survey. IEEE Trans. on Neural Networks 4(5), 740–747 (1993)
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Beliczynski, B. (2011). Approximation of Functions by Multivariable Hermite Basis: A Hybrid Method. In: Dobnikar, A., Lotrič, U., Šter, B. (eds) Adaptive and Natural Computing Algorithms. ICANNGA 2011. Lecture Notes in Computer Science, vol 6593. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20282-7_14
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DOI: https://doi.org/10.1007/978-3-642-20282-7_14
Publisher Name: Springer, Berlin, Heidelberg
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