Approximation results for neural network operators activated by sigmoidal functions

Neural Networks

Volumn 44, Issue , 2013, Pages 101-106

  Costarelli, Danilo ^a   Spigler, Renato ^a  

^a ROMA TRE UNIVERSITY   (Italy)

Author keywords

Lipschitz classes; Neural networks operators; Order of approximation; Sigmoidal functions; Uniform approximation

Indexed keywords

APPROXIMATION RESULTS; HYPERBOLIC TANGENT FUNCTION; LIPSCHITZ; MULTIVARIATE FUNCTION; ORDER OF APPROXIMATION; SIGMOIDAL FUNCTIONS; UNIFORM APPROXIMATION; UNIFORM CONVERGENCE;

HYPERBOLIC FUNCTIONS;

NEURAL NETWORKS;

ANALYTICAL ERROR; ARTICLE; ARTIFICIAL NEURAL NETWORK; NEURAL NETWORK OPERATOR; PRIORITY JOURNAL; SIGMOIDAL FUNCTION; STATISTICAL MODEL;

NEURAL NETWORKS (COMPUTER);

EID: 84876320512     PISSN: 08936080     EISSN: 18792782     Source Type: Journal    
DOI: 10.1016/j.neunet.2013.03.015     Document Type: Article

References (36)

1. Anastassiou G.A. Rate of convergence of some neural network operators to the unit-univariate case. Journal of Mathematical Analysis and Applications 1997, 212:237-262.
2. Anastassiou G.A. Intelligent systems: approximation by artificial neural networks. Intelligent systems reference library 2011, vol. 19. Springer-Verlag, Berlin.
3. Anastassiou G.A. Multivariate hyperbolic tangent neural network approximation. Computers & Mathematics with Applications 2011, 61(4):809-821.
4. Anastassiou G.A. Multivariate sigmoidal neural network approximation. Neural Networks 2011, 24:378-386.
5. Anastassiou G.A. Univariate hyperbolic tangent neural network approximation. Mathematical and Computer Modelling 2011, 53(5-6):1111-1132.
6. Anastassiou G.A. Univariate sigmoidal neural network approximation. Journal of Computational Analysis and Applications 2012, 14(4):659-690.
7. Barron A.R. Universal approximation bounds for superpositions of a sigmoidal function. IEEE Transactions on Information Theory 1993, 39(3):930-945.
8. Butzer P.L., Nessel R.J. Fourier analysis and approximation 1971, Birkhäuser Verlag and Academic Press, New York, Bassel.
9. Cao F., Chen Z. The approximation operators with sigmoidal functions. Computers & Mathematics with Applications 2009, 58(4):758-765.
10. Cao F., Chen Z. The construction and approximation of a class of neural networks operators with ramp functions. Journal of Computational Analysis and Applications 2012, 14(1):101-112.
11. Cheang Gerald H.L. Approximation with neural networks activated by ramp sigmoids. Journal of Approximation Theory 2010, 162:1450-1465.
12. Chen D. Degree of approximation by superpositions of a sigmoidal function. Approximation Theory and its Applications 1993, 9(3):17-28.
13. Chui C.K., Li X. Approximation by ridge functions and neural networks with one hidden layer. Journal of Approximation Theory 1992, 70:131-141.
14. Costarelli, D., & Spigler, R. (2012). Approximation by series of sigmoidal functions with applications to neural networks (submitted for publication-a).
15. Costarelli, D., & Spigler, R. (2012). Constructive approximation by superposition of sigmoidal functions (submitted for publication-b).
16. Costarelli, D., & Spigler, R. (2013). A collocation method for solving nonlinear Volterra integro-differential equations of the neutral type by sigmoidal functions. Journal of Integral Equations and Applications (in press-a).
17. Costarelli, D., & Spigler, R. (2013). Solving Volterra integral equations of the 2nd kind by sigmoidal functions approximations. Journal of Integral Equations and Applications (in press-b).
18. Cybenko G. Approximation by superpositions of a sigmoidal function. Mathematics of Control, Signals, and Systems 1989, 2:303-314.
19. Gao B., Xu Y. Univariant approximation by superpositions of a sigmoidal function. Journal of Mathematical Analysis and Applications 1993, 178:221-226.
20. Girosi F., Anzellotti G. Rates of convergence for radial basis functions and neural networks. Artificial neural networks for speech and vision 1993, 97-113. Chapman & Hall, London. R.J. Mammone (Ed.).
21. Gnecco G. A comparison between fixed-basis and variable-basis schemes for function approximation and functional optimization. Journal of Applied Mathematics 2012, 17. ID 806945.
22. Gnecco G., Sanguineti M. On a variational norm tailored to variable-basis approximation schemes. IEEE Transactions on Information Theory 2011, 57:549-558.
23. Hahm N., Hong B. Approximation order to a function in C(R) by superposition of a sigmoidal function. Applied Mathematics Letters 2002, 15:591-597.
24. Jones L.K. Constructive approximations for neural networks by sigmoidal functions. Technical report series 7. Department of Mathematics, University of Lowell 1988.
25. Kainen P.C., Kurková V. An integral upper bound for neural network approximation. Neural Computation 2009, 21:2970-2989.
26. Kurková V. Complexity estimates based on integral transforms induced by computational units. Neural Networks 2012, 33:160-167.
27. Lenze B. Constructive multivariate approximation with sigmoidal functions and applications to neural networks. Numer. methods approx. theory 1992, 155-175. Birkhäuser Verlag, Basel, Boston, Berlin.
28. Lewicki G., Marino G. Approximation by superpositions of a sigmoidal function. Zeitschrift für Analysis und ihre Anwendungen Journal for Analysis and its Applications 2003, 22(2):463-470.
29. Lewicki G., Marino G. Approximation of functions of finite variation by superpositions of a sigmoidal function. Applied Mathematics Letters 2004, 17:1147-1152.
30. Li X. Simultaneous approximations of multivariate functions and their derivatives by neural networks with one hidden layer. Neurocomputing 1996, 12:327-343.
31. Li X., Micchelli C.A. Approximation by radial basis and neural networks. Numerical Algorithms 2000, 25:241-262.
32. Light W. Ridge functions, sigmoidal functions and neural networks. Approximation theory VII (Austin, TX, 1992) 1993, 163-206. Academic Press Boston, Boston, MA.
33. Makovoz Y. Uniform approximation by neural networks. Journal of Approximation Theory 1998, 95(2):215-228.
34. Mhaskar H.N., Micchelli C.A. Approximation by superposition of sigmoidal and radial basis functions. Advances in Applied Mathematics 1992, 13:350-373.
35. Mhaskar H.N., Micchelli C.A. Degree of approximation by neural and translation networks with a single hidden layer. Advances in Applied Mathematics 1995, 16:151-183.
36. Pinkus A. Approximation theory of the MLP model in neural networks. Acta Numerica 1999, 8:143-195.