Neural network operators: Constructive interpolation of multivariate functions

Neural Networks

Volumn 67, Issue , 2015, Pages 28-36

  Costarelli, Danilo ^a  

^a UNIVERSITY OF PERUGIA   (Italy)

Author keywords

Irregular sampling scheme; Multivariate approximation; Multivariate interpolation; Neural networks operators; Order of approximation; Sigmoidal functions

Indexed keywords

NEURAL NETWORKS; THEOREM PROVING;

IRREGULAR SAMPLING; MULTIVARIATE APPROXIMATIONS; MULTIVARIATE INTERPOLATION; ORDER OF APPROXIMATION; SIGMOIDAL FUNCTIONS;

INTERPOLATION;

ARTICLE; ARTIFICIAL NEURAL NETWORK; DATA ANALYSIS; LINEAR SYSTEM; MATHEMATICAL COMPUTING; MATHEMATICAL MODEL; MATHEMATICAL PHENOMENA; MULTIVARIATE ANALYSIS; MULTIVARIATE APPROXIMATION; MULTIVARIATE INTERPOLATION; PRIORITY JOURNAL; SIGMOIDAL FUNCTION; SIGNAL PROCESSING; ALGORITHM; MACHINE LEARNING;

ALGORITHMS; MACHINE LEARNING; MULTIVARIATE ANALYSIS; NEURAL NETWORKS (COMPUTER);

EID: 84926293410     PISSN: 08936080     EISSN: 18792782     Source Type: Journal    
DOI: 10.1016/j.neunet.2015.02.002     Document Type: Article

References (59)

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. Univariate hyperbolic tangent neural network approximation. Mathematical and Computer Modelling 2011, 53(5-6):1111-1132.
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. Intelligent systems: approximation by artificial neural networks. Intelligent systems reference library 2011, vol. 19. Springer-Verlag, Berlin.
6. Anastassiou G.A. Univariate sigmoidal neural network approximation. Journal of Computational Analysis and Applications 2012, 14(4):659-690.
7. Antsaklis P.J., Sartori M.A. A simple method to derive bounds on the size and to train multilayer neural networks. IEEE Transactions on Neural Networks 1991, 2(4):467-471.
8. Babri H.A., Huang G.B. Feedforward neural networks with arbitrary bounded nonlinear activation functions. IEEE Transactions on Neural Networks 1998, 9(1):224-229.
9. Bardaro C., Butzer P.L., Stens R.L., Vinti G. Kantorovich-type generalized sampling series in the setting of Orlicz spaces. Sampling Theory in Signal and Image Processing 2007, 6(1):29-52.
10. Bardaro C., Musielak J., Vinti G. Nonlinear integral operators and applications. Nonlinear analysis and applications 2003, Vol. 9. De Gruyter, New York, Berlin.
11. Bardaro C., Vinti G. A general approach to the convergence theorems of generalized sampling series. Applicable Analysis 1997, 64:203-217.
12. Bardaro C., Vinti G. An abstract approach to sampling type operators inspired by the work of P.L. Butzer - part I - linear operators. Sampling Theory in Signal and Image Processing 2003, 2(3):271-296.
13. Barron A.R. Universal approximation bounds for superpositions of a sigmoidal function. IEEE Transaction on Information Theory 1993, 39(3):930-945.
14. Butzer P.L., Nessel R.J. Fourier analysis and approximation. Pure and applied mathematics 1971, vol. 40. Academic Press, New York-London.
15. Butzer P.L., Ries S., Stens R.L. Approximation of continuous and discontinuous functions by generalized sampling series. Journal of Approximation Theory 1987, 50:25-39.
16. Cao F., Chen Z. The approximation operators with sigmoidal functions. Computers & Mathematics with Applications 2009, 58(4):758-765.
17. 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.
18. Cardaliaguet P., Euvrard G. Approximation of a function and its derivative with a neural network. Neural Networks 1992, 5(2):207-220.
19. Cheang G.H.L. Approximation with neural networks activated by ramp sigmoids. Journal of Approximation Theory 2010, 162:1450-1465.
20. Cheney E.W., Light W.A., Xu Y. On kernels and approximation orders. Approximation theory 1992, 227-242. Marcel Dekker, New York. G. Anastassiou (Ed.).
21. Cheney E.W., Light W.A., Xu Y. Constructive methods of approximation by rigde functions and radial functions. Numerical Algorithms 1993, 4(2):205-223.
22. Cluni F., Costarelli D., Minotti A.M., Vinti G. Enhancement of thermographic images as tool for structural analysis in earthquake engineering. NDT & E International 2015, 70:60-72.
23. Cluni F., Costarelli D., Minotti A.M., Vinti G. Applications of sampling Kantorovich operators to thermographic images for seismic engineering. Journal of Computational Analysis and Applications 2015, 19(4):602-617.
24. Costarelli D. Sigmoidal functions approximation and applications 2014, (Ph.D. thesis), "Roma Tre" University, Rome, Italy.
25. Costarelli D. Interpolation by neural network operators activated by ramp functions. Journal of Mathematical Analysis and Application 2014, 419:574-582.
26. Costarelli D., Spigler R. Constructive approximation by superposition of sigmoidal functions. Analysis in Theory and Applications 2013, 29(2):169-196.
27. Costarelli D., Spigler R. Solving Volterra integral equations of the second kind by sigmoidal functions approximations. J. Integral Equations Appl. 2013, 25(2):193-222.
28. Costarelli D., Spigler R. Approximation results for neural network operators activated by sigmoidal functions. Neural Networks 2013, 44:101-106.
29. Costarelli D., Spigler R. Multivariate neural network operators with sigmoidal activation functions. Neural Networks 2013, 48:72-77.
30. Costarelli D., Spigler R. A collocation method for solving nonlinear Volterra integro-differential equations of the neutral type by sigmoidal functions. J. Integral Equations Appl. 2014, 26(1):15-52.
31. Costarelli D., Spigler R. Convergence of a family of neural network operators of the Kantorovich type. Journal of Approximation Theory 2014, 185:80-90.
32. Costarelli D., Spigler R. Approximation by series of sigmoidal functions with applications to neural networks. Annali di Matematica Pura ed Applicata 2015, 194(1):289-306. 10.1007/s10231-013-0378-y.
33. Costarelli D., Vinti G. Approximation by multivariate generalized sampling Kantorovich operators in the setting of Orlicz spaces. Bollettino della Unione Matematica Italiana. Serie 9 2011, IV:445-468.
34. Costarelli D., Vinti G. Approximation by nonlinear multivariate sampling-Kantorovich type operators and applications to image processing. Numerical Functional Analysis and Optimization 2013, 34(8):819-844.
35. Costarelli D., Vinti G. Order of approximation for sampling Kantorovich operators. Journal of Integral Equations and Applications 2014, 26(3):345-368.
36. Costarelli D., Vinti G. Rate of approximation for multivariate sampling Kantorovich operators on some functions spaces. Journal of Integral Equations and Applications 2014, 26(4):455-481.
37. D. Costarelli, G. Vinti, (2014c). Sampling Kantorovich operators and their applications to approximation problems and to digital image processing. In Proceedings of 8th international conference on applied mathematics, simulation, modelling (ASM'14), Florence, Italy November 22-24, 2014. Recent advances in applied mathematics, modelling and simulation (pp. 256-260).
38. Cybenko G. Approximation by superpositions of a sigmoidal function. Mathematics of Control, Signals, and Systems 1989, 2:303-314.
39. S. Dasgupta, Y. Shristava, (1990). Neural networks for exact matching of functions on a discrete domain. In Proceedings of the 29th IEEE conference on decision and control, Honolulu (pp. 1719-1724).
40. Gripenberg G. Approximation by neural network with a bounded number of nodes at each level. Journal of Approximation Theory 2003, 122(2):260-266.
41. 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.
42. Ismailov V.E. On the approximation by neural networks with bounded number of neurons in hidden layers. Journal of Mathematical Analysis and Applications 2014, 417(2):963-969.
43. Ito Y. Nonlinearity creates linear independence. Advances in Computational Mathematics 1996, 5:189-203.
44. Ito Y. Independence of unscaled basis functions and finite mappings by neural networks. The Mathematical Scientist 2001, 26:117-126.
45. Ito Y., Saito K. Superposition of linearly independent functions and finite mappings by neural networks. The Mathematical Scientist 1996, 21:27-33.
46. Kainen P.C., Kurková V. An integral upper bound for neural network approximation. Neural Computation 2009, 21:2970-2989.
47. Kurková V. Complexity estimates based on integral transforms induced by computational units. Neural Networks 2012, 33:160-167.
48. Lenze B. Constructive multivariate approximation with sigmoidal functions and applications to neural networks. Numerical methods of approximation theory 1992, 155-175. Birkhauser Verlag, Basel, Boston, Berlin.
49. Lewicki G., Marino G. Approximation by superpositions of a sigmoidal function. Zeitschrift fur Analysis und ihre Anwendungen 2003, 22(2):463-470.
50. Llanas B., Sainz F.J. Constructive approximate interpolation by neural networks. Journal of Computational and Applied Mathematics 2006, 188:283-308.
51. Maiorov V. Approximation by neural networks and learning theory. Journal of Complexity 2006, 22(1):102-117.
52. Makovoz Y. Random approximants and neural networks. Journal of Approximation Theory 1996, 85:98-109.
53. Makovoz Y. Uniform approximation by neural networks. Journal of Approximation Theory 1998, 95(2):215-228.
54. Pinkus A. Approximation theory of the MLP model in neural networks. Acta Numerica 1999, 8:143-195.
55. S. Ries, R.L. Stens, (1984). Approximation by generalized sampling series. In Constructive theory of functions '84, Sofia (pp. 746-756).
56. Sontag E.D. Feedforward nets for interpolation and classification. Journal of Computer and System Sciences 1992, 45:20-48.
57. Tamura S., Tateishi M. Capabilities of a four-layered feedforward neural network. IEEE Transactions on Neural Networks 1997, 8(2):251-255.
58. Vinti G. A general approximation result for nonlinear integral operators and applications to signal processing. Applicable Analysis 2001, 79:217-238.
59. Vinti G., Zampogni L. A unifying approach to convergence of linear sampling type operators in Orlicz spaces. Advances in Differential Equations 2011, 16(5-6):573-600.