Max-product neural network and quasi-interpolation operators activated by sigmoidal functions

Journal of Approximation Theory

Volumn 209, Issue , 2016, Pages 1-22

  Costarelli, Danilo ^a   Vinti, Gianluca ^a  

^a UNIVERSITY OF PERUGIA   (Italy)

Author keywords

Max product operators; Neural networks operators; Order of approximation; Sigmoidal functions; Uniform approximation

Indexed keywords

EID: 84971422843     PISSN: 00219045     EISSN: 10960430     Source Type: Journal    
DOI: 10.1016/j.jat.2016.05.001     Document Type: Article

References (56)

1. Anastassiou G.A. Rate of convergence of some neural network operators to the unit-univariate case. J. Math. Anal. Appl. 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., Coroianu L., Gal S.G. Approximation by a nonlinear Cardaliaguet-Euvrard neural network operator of max-product kind. J. Comput. Anal. Appl. 2010, 12(2):396-406.
4. φ-spaces in multidimensional setting. J. Math. Anal. Appl. 2009, 349(2):317-334.
5. Angeloni L., Vinti G. Approximation with respect to Goffman-Serrin variation by means of non-convolution type integral operators. Numer. Funct. Anal. Optim. 2010, 31(5):519-548.
6. Bardaro C., Butzer P.L., Stens R.L., Vinti G. Kantorovich-type generalized sampling series in the setting of Orlicz spaces. Sampl. Theory Signal Image Process. 2007, 6(1):29-52.
7. Bardaro C., Butzer P.L., Stens R.L., Vinti G. Prediction by samples from the past with error estimates covering discontinuous signals. IEEE Trans. Inform. Theory 2010, 56(1):614-633.
8. Bardaro C., Karsli H., Vinti G. On pointwise convergence of linear integral operators with homogeneous kernels. Integral Transforms Spec. Funct. 2008, 19(6):429-439.
9. C. Bardaro, J. Musielak, G. Vinti, Nonlinear Integral Operators and Applications, in: De Gruyter Series in Nonlinear Analysis and Applications, vol. 9, New York, Berlin, 2003.
10. Bardaro C., Vinti G. A general approach to the convergence theorems of generalized sampling series. Appl. Anal. 1997, 64:203-217.
11. Bardaro C., Vinti G. An abstract approach to sampling type operators inspired by the work of P.L. Butzer-Part I-linear operators. Sampl. Theory Signal Image Process. 2003, 2(3):271-296.
12. Barron A.R. Universal approximation bounds for superpositions of a sigmoidal function. IEEE Trans. Inform. Theory 1993, 39(3):930-945.
13. Butzer P.L., Nessel R.J. Fourier Analysis and Approximation I 1971, Academic Press, New York-London.
14. Butzer P.L., Ries S., Stens R.L. Approximation of continuous and discontinuous functions by generalized sampling series. J. Approx. Theory 1987, 50:25-39.
15. Cao F., Chen Z. The approximation operators with sigmoidal functions. Comput. Math. Appl. 2009, 58(4):758-765.
16. Cao F., Chen Z. The construction and approximation of a class of neural networks operators with ramp functions. J. Comput. Anal. Appl. 2012, 14(1):101-112.
17. Cardaliaguet P., Euvrard G. Approximation of a function and its derivative with a neural network. Neural Netw. 1992, 5(2):207-220.
18. Cheang G.H.L. Approximation with neural networks activated by ramp sigmoids. J. Approx. Theory 2010, 162:1450-1465.
19. Cluni F., Costarelli D., Minotti A.M., Vinti G. Enhancement of thermographic images as tool for structural analysis in earthquake engineering. NDT & E Int. 2015, 70:60-72.
20. Cluni F., Costarelli D., Minotti A.M., Vinti G. Applications of sampling Kantorovich operators to thermographic images for seismic engineering. J. Comput. Anal. Appl. 2015, 19(4):602-617.
21. Coroianu L., Gal S.G. Approximation by nonlinear generalized sampling operators of max-product kind. Sampl. Theory Signal Image Process. 2010, 9(1-3):59-75.
22. Coroianu L., Gal S.G. Approximation by max-product sampling operators based on sinc-type kernels. Sampl. Theory Signal Image Process. 2011, 10(3):211-230.
23. Coroianu L., Gal S.G. Saturation results for the truncated max-product sampling operators based on sinc and Fejér-type kernels. Sampl. Theory Signal Image Process. 2012, 11(1):113-132.
24. Coroianu L., Gal S.G. Saturation and inverse results for the Bernstein max-product operator. Period. Math. Hungar. 2014, 69:126-133.
25. Costarelli D. Sigmoidal functions approximation and applications 2014, (Ph.D. thesis), "Roma Tre" University, Rome, Italy.
26. Costarelli D. Interpolation by neural network operators activated by ramp functions. J. Math. Anal. Appl. 2014, 419:574-582.
27. Costarelli D. Neural network operators: constructive interpolation of multivariate functions. Neural Netw. 2015, 67:28-36.
28. Costarelli D., Spigler R. Constructive approximation by superposition of sigmoidal functions. Anal. Theory Appl. 2013, 29(2):169-196.
29. 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.
30. Costarelli D., Spigler R. Approximation results for neural network operators activated by sigmoidal functions. Neural Netw. 2013, 44:101-106.
31. Costarelli D., Spigler R. Multivariate neural network operators with sigmoidal activation functions. Neural Netw. 2013, 48:72-77.
32. 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.
33. Costarelli D., Spigler R. Convergence of a family of neural network operators of the Kantorovich type. J. Approx. Theory 2014, 185:80-90.
34. Costarelli D., Spigler R. Approximation by series of sigmoidal functions with applications to neural networks. Ann. Mat. Pura Appl. 2015, 194(1):289-306.
35. Costarelli D., Vinti G. Approximation by multivariate generalized sampling Kantorovich operators in the setting of Orlicz spaces. Boll. Unione Mat. Ital. (9) 2011, IV:445-468.
36. Costarelli D., Vinti G. Approximation by nonlinear multivariate sampling-Kantorovich type operators and applications to image processing. Numer. Funct. Anal. Optim. 2013, 34(8):819-844.
37. Costarelli D., Vinti G. Order of approximation for sampling Kantorovich operators. J. Integral Equations Appl. 2014, 26(3):345-368.
38. Costarelli D., Vinti G. Rate of approximation for multivariate sampling Kantorovich operators on some functions spaces. J. Integral Equations Appl. 2014, 26(4):455-481.
39. Costarelli D., Vinti G. Degree of approximation for nonlinear multivariate sampling Kantorovich operators on some functions spaces. Numer. Funct. Anal. Optim. 2015, 36(8):964-990.
40. Cybenko G. Approximation by superpositions of a sigmoidal function. Math. Control Signals Systems 1989, 2:303-314.
41. S. Dasgupta, Y. Shristava, Neural networks for exact matching of functions on a discrete domain, in: Proceedings of the 29th IEEE Conference on Decision and Control, Honolulu, 1990, pp. 1719-1724.
42. Gripenberg G. Approximation by neural network with a bounded number of nodes at each level. J. Approx. Theory 2003, 122(2):260-266.
43. Ismailov V.E. On the approximation by neural networks with bounded number of neurons in hidden layers. J. Math. Anal. Appl. 2014, 417(2):963-969.
44. Ito Y. Nonlinearity creates linear independence. Adv. Comput. Math. 1996, 5:189-203.
45. Ito Y. Independence of unscaled basis functions and finite mappings by neural networks. Math. Sci. 2001, 26:117-126.
46. Ito Y., Saito K. Superposition of linearly independent functions and finite mappings by neural networks. Math. Sci. 1996, 21:27-33.
47. Kainen P.C., Kurková V. An integral upper bound for neural network approximation. Neural Comput. 2009, 21:2970-2989.
48. Lewicki G., Marino G. Approximation by superpositions of a sigmoidal function. Z. Angew. Math. Phys. 2003, 22(2):463-470.
49. Llanas B., Sainz F.J. Constructive approximate interpolation by neural networks. J. Comput. Appl. Math. 2006, 188:283-308.
50. Maiorov V. Approximation by neural networks and learning theory. J. Complexity 2006, 22(1):102-117.
51. Makovoz Y. Uniform approximation by neural networks. J. Approx. Theory 1998, 95(2):215-228.
52. Sontag E.D. Feedforward nets for interpolation and classification. J. Comput. System Sci. 1992, 45:20-48.
53. Tamura S., Tateishi M. Capabilities of a four-layered feedforward neural network. IEEE Trans. Neural Netw. 1997, 8(2):251-255.
54. Vinti G., Zampogni L. Approximation by means of nonlinear Kantorovich sampling type operators in Orlicz spaces. J. Approx. Theory 2009, 161(2):511-528.
55. Vinti G., Zampogni L. A unifying approach to convergence of linear sampling type operators in Orlicz spaces. Adv. Differential Equations 2011, 16(5-6):573-600.
56. Vinti G., Zampogni L. Approximation results for a general class of Kantorovich type operators. Adv. Nonlinear Stud. 2014, 14:991-1011.