The ridge function representation of polynomials and an application to neural networks

Acta Mathematica Sinica, English Series

Volumn 27, Issue 11, 2011, Pages 2169-2176

  Xie, Ting Fan ^a   Cao, Fei Long ^a  

^a CHINA JILIANG UNIVERSITY   (China)

Author keywords

approximation; neural network; polynomial; Ridge function

Indexed keywords

EID: 80053470591     PISSN: 14398516     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10114-011-9407-1     Document Type: Article

References (24)

1. Chui, C. K., Li, X.: Approximation by ridge functions and neural networks with one hidden layer. J. Approx. Theory, 70, 131-141 (1992).
2. Diaconis, P., Shahshahani, M.: On nonlinear functions of linear combinations. SIAM, J. Sci. Stat. Comput., 5, 175-191 (1984).
3. Lin, V. Y., Pinkus, A.: Fundamentality of ridge functions. J. Approx. Theory, 75(3), 295-311 (1993).
4. Maiorov, V. E.: On best approximation by ridge function. J. Approx. Theory, 99, 68-94 (1999).
5. Dahmen, W., Micchelli, C. A.: Some remarks on ridge functions. Approx. Theory Appl., 3, 139-143 (1987).
6. Kroo, A.: On approximation by ridge function. Constr. Approx., 13, 447-460 (1997).
7. Maiorov, V., Meir, R., Ratsaby, J.: On the approximation of functional classes equipped with a uniform measure using ridge functions. J. Approx. Theory, 99, 95-111 (1999).
8. Martin, D., Buhmann, W., Pinkus, A.: Identifying linear combinations of ridge functions. Adv. Applied Math., 22, 103-118 (1999).
9. Burger, M., Neubauer, A.: Error bounds for approximation with neural networks. J. Approx. Theory, 112, 235-314 (2001).
10. Ismailov, V. E.: On the representation by linear superpositions. J. Approx. Theory, 151, 113-125 (2008).
11. Konovalov, V. N., Leviatan, D., Maiorov, V. E.: Approximation by polynomials and ridge functions of classes of s-monotone radial functions. J. Approx. Theory, 152, 20-51 (2008).
12. Yin, H. X.: Error bounds of two smoothing approximations for semi-infinite minimax problems. Acta Mathematicae Applicatae Sinica, English Series, 25(4), 685-696 (2009).
13. Kelley, J. L.: General Topology, Graduate Texe in Math. 27, Springer-Verlag, New York, 1955.
14. Cybenko, G.: Approximation by superpositions of sigmoidal function. Math. Control Signals, System, 2, 303-314 (1989).
15. Funahashi, K. I.: On the approximate realization of continuous mappings by neural networks. Neural Networks, 2, 183-192 (1989).
16. Hornik, K., Stinchcombe, M., White, H.: Universal approximation of an unknown mapping and its derivatives using multilayer feedforward networks. Neural Networks, 3, 551-560 (1990).
17. Leshno, M., Lin, V. Y., Pinkus, A., et al: Multilayer feedforward networks with a nonpolynomial activation function can approximate any function. Neural Networks, 6, 861-867 (1993).
18. Chen, T. P., Chen, H.: Universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its application to dynamical system. IEEE Trans. Neural Networks, 6, 911-917 (1995).
19. Barron, A. R.: Universial approximation bounds for superpositions of a sigmodial function. IEEE Trans. Inform. Theory, 39, 930-945 (1993).
20. Mhaskar, H. N., Micchelli, C. A.: Approximation by superposition of sigmoidal and radial basis functions. Adv. Applied Math., 13, 350-373 (1992).
21. Mhaskar, H. N., Micchelli, C. A.: Degree of approximation by neural and translation networks with a single hidden layer. Adv. Applied Math., 16, 151-183 (1995).
22. Kåurková, V., Sanguineti, M.: Bounds on rates of variable-basis and neural-network approximation. IEEE Trans. Inform. Theory, 47, 2659-2665 (2001).
23. d) by neural and mixture networks. IEEE Trans. Neural Networks, 9, 969-978 (1998).
24. Cao, F. L., Xie, T. F., Xu, Z. B.: The estimate of approximation error for neural networks: a constructive approach. Neurocomputing, 71, 626-630 (2008).