Kolmogorov's spline network

IEEE Transactions on Neural Networks

Volumn 14, Issue 4, 2003, Pages 725-733

  Igelnik, Boris ^a   Parikh, Neel ^a  

^a PEGASUS TECHNICAL SERVICES INC   (United States)

Author keywords

Cubic splines; Ensemble of networks; Kolmogorov's superposition theorem (KST)

Indexed keywords

ALGORITHMS; APPROXIMATION THEORY; ERROR ANALYSIS; FUNCTIONS;

KOLMOGOROV SUPERPOSITION THEOREM (KST);

NEURAL NETWORKS;

EID: 0041524025     PISSN: 10459227     EISSN: None     Source Type: Journal    
DOI: 10.1109/TNN.2003.813830     Document Type: Article

References (36)

1. A. R. Barron, "Universal approximation bounds for superpositions of a sigmoidal function," IEEE Trans. Inform. Theory, vol. 39, pp. 930-945, Apr. 1993.
2. _, "Approximation and estimation bounds for artificial neural networks," Machine Learning, vol. 14, pp. 115-133, 1994.
3. A. N. Kolmogorov, "On the representation of continuous functions of many variables by superposition of continuous functions of one variable and addition," Dokl. Akad. Nauk SSSR, vol. 114, pp. 953-956, 1957.
4. D. A. Sprecher, "On the structure of continuous functions of several variables," Trans. Amer. Math. Soc., vol. 115, pp. 340-355, 1965.
5. G. G. Lorentz, M. von Golitschek, and Y. Makovoz, Constructive Approximation. Advanced Problems. New York: Springer-Verlag, 1996.
6. R. Hecht-Nielsen, "Counter propagation networks," Proc. IEEE Int. Conf. Neural Networks, vol. 2, pp. 19-32, 1987.
7. _, "Kolmogorov's mapping neural network existence theorem," Proc. IEEE Int. Conf. Neural Networks, vol. 2, pp. 11-13, 1987.
8. F. Girosi and T. Poggio, "Representation properties of networks: Kolmogorov's theorem is irrelevant," Neural Comput., vol. 1, pp. 465-469, 1989.
9. V. Kurková, "Kolmogorov's theorem is relevant," Neural Comput., vol. 3, pp. 617-622, 1991.
10. _, "Kolmogorov's theorem and multilayer neural networks," Neural Networks, vol. 5, pp. 501-506, 1992.
11. M. Nakamura, R. Mines, and V. Kreinovich, "Guaranteed intervals for Kolmogorov's theorem (and their possible relation to neural networks)," Interval Comput., vol. 3, pp. 183-199, 1993.
12. M. Nees, "Approximative versions of Kolmogorov's superposition theorem, proved constructively," J. Comput. Appl. Math., vol. 54, pp. 239-250, 1994.
13. D. A. Sprecher, "A numerical implementation of Kolmogorov's superpositions," Neural Neworks, vol. 9, pp. 765-772, 1996.
14. _, "A numerical implementation of Kolmogorov's superpositions II," Neural Networks, vol. 10, pp. 447-457, 1997.
15. S. Guarnieri, F. Piazza, and A. Uncini, "Multilayer neural networks with adaptive spline-based activation functions," in Proc. Int. Neural Network Soc. Annu. Meet., Washington, DC, 1995, pp. 1695-1699.
16. L. Vecci, F. Piazza, and A. Uncini, "Learning and approximation capabilities of adaptive spline activation function neural networks," Neural Networks, vol. 11, pp. 259-270, 1998.
17. A. Uncini, L. Vecci, P. Campolucci, and F. Piazza, "Complex-valued neural networks with adaptive spline activation function for digital radio links nonlinear equalization," IEEE Trans. Signal Processing, vol. 47, pp. 505-514, 1999.
18. S. Guarnieri, F. Piazza, and A. Uncini, "Multilayer feedforward networks with adaptive spline activation function," IEEE Trans. Neural Networks, vol. 10, pp. 672-683, May 1999.
19. B. Igelnik, "Some new adaptive architectures for learning, generalization, and visualization of multivariate data," in Quo Vadis Computational Intelligence? New Trends and Approaches in Computational Intelligence, P. Sincak and J. Vascak, Eds. Heidelberg, Germany: Physica-Verlag, 2000, pp. 63-78.
20. P. M. Prenter, Splines and Variational Methods. New York: Wiley, 1975.
21. A. B. Shidlovskii, Transcendental Numbers. Berlin, Germany: Walter de Gruyter, 1989.
22. B. Igelnik and N. Parikh, "Spline Neural Networks," U.S. Patent Bureau, Provisional Patent Application N 60/361 641, Mar. 5, 2002.
23. L. Breiman, "Hinging hyperplanes for regression, classification, and function approximation," IEEE Trans. Inform. Theory, vol. 39, pp. 999-1013, Apr. 1993.
24. L. K. Jones, "Good weights and hyperbolic kernels for neural networks, projection pursuit, and pattern classification: Fourier strategies for extracting information from high-dimensional data," IEEE Trans. Inform. Theory, vol. 40, pp. 439-454, Mar. 1994.
25. Y. Makovoz, "Random approximants and neural networks," J. Approx. Theory, vol. 85, pp. 98-109, 1996.
26. F. Scarcselli and A. C. Tsoi, "Universal approximation using feedforward neural networks: A survey of some existing methods and some new results," Neural Networks, vol. 11, pp. 15-37, 1998.
27. N. W. Townsend and L. Tarassenko, "Estimating of error bounds for neural-network function approximators," IEEE Trans. Neural Networks, vol. 10, pp. 217-230, Mar. 1999.
28. B. Igelnik, Y.-H. Pao, S. R. LeClair, and C. Y. Shen, "The ensemble approach to neural network learning and generalization," IEEE Trans. Neural Networks, vol. 10, pp. 19-30, Jan. 1999.
29. B. Igelnik, M. Tabib-Azar, and S. R. LeClair, "A net with complex coefficients," IEEE Trans. Neural Networks, vol. 12, pp. 236-249, Mar. 2001.
30. C. A. Hall, "On error bounds for spline interpolation," J. Approx. Theory, vol. 1, pp. 209-218, 1968.
31. J. H. Ahlberg, E. H. Nilson, and J. L. Walsh, The Theory of Splines and their Applications. New York: Academic, 1967.
32. D. E. Rumelhart, G. E. Hinton, and R. J. Williams, "Learning internal representations by error propagation," in Parallel Distributed Processing: Explorations in the Microstructure of Cognition, D. E. Rumelhart and J. L. McClelland, Eds. Cambridge, MA: MIT Press, 1988, vol. 1, pp. 318-362.
33. A. Albert, Regression and the Moore-Penrose Pseudoinverse. New York: Academic, 1972.
34. R. E. Shapire, "The strength of weak learnability," Machine Learning, vol. 5, pp. 197-227, 1990.
35. S. Ji and S. Ma, "Combinations of weak classifiers," IEEE Trans. Neural Networks, vol. 8, pp. 32-42, Jan. 1997.
36. L. Breiman, "Combining predictors," in Combining Artificial Neural Nets. Ensemble and Modular Neural Nets, A. J. C. Sharkey, Ed. London, U.K.: Springer-Verlag, 1999, pp. 31-48.