Abstract
In the present paper, we develop the theory of max-product neural network operators in a Kantorovich-type version, which is suitable in order to study the case of L p-approximation for not necessarily continuous data. Moreover, also the case of the pointwise and uniform approximation of continuous functions is considered. Finally, several examples of sigmoidal functions for which the above theory can be applied are presented.
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References
Anastassiou, G.A.: Intelligent Systems: Approximation by Artificial Neural Networks, Intelligent Systems Reference Library 19. Berlin: Springer-Verlag (2011)
Anastassiou G.A., Coroianu L., Gal S.G.: Approximation by a nonlinear Cardaliaguet-Euvrard neural network operator of max-product kind. J. Comp. Anal. Appl. 12(2), 396–406 (2010)
Angeloni L., Vinti G.: Approximation in variation by homothetic operators in multidimensional setting. Differ. Integr. Equ. 26(5-6), 655–674 (2013)
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. 6(1), 29–52 (2007)
Bardaro C., Karsli H., Vinti G.: On pointwise convergence of linear integral operators with homogeneous kernels. Integral Transforms Spec. Funct. 19(6), 429–439 (2008)
Bardaro, C., Musielak, J., Vinti, G.: Nonlinear Integral Operators and Applications, New York, Berlin: De Gruyter Series in Nonlinear Analysis and Applications 9, (2003)
Barron A.R.: Universal approximation bounds for superpositions of a sigmoidal function. IEEE Trans. Inform. Theory 39(3), 930–945 (1993)
Cao F., Chen Z.: The approximation operators with sigmoidal functions. Comput. Math. Appl. 58(4), 758–765 (2009)
Cao F., Chen Z.: The construction and approximation of a class of neural networks operators with ramp functions. J. Comput. Anal. Appl. 14(1), 101–112 (2012)
Cao F., Liu B., Park D.S.: Image classification based on effective extreme learning machine. Neurocomputing 102, 90–97 (2013)
Cardaliaguet P., Euvrard G.: Approximation of a function and its derivative with a neural network. Neural Netw. 5(2), 207–220 (1992)
Cheang G.H.L.: Approximation with neural networks activated by ramp sigmoids. J. Approx. Theory 162, 1450–1465 (2010)
Cheney E.W., Light W.A., Xu Y.: Constructive methods of approximation by rigde functions and radial functions. Numer. Algorithms 4(2), 205–223 (1993)
Cluni F., Costarelli D., Minotti A.M., Vinti G.: Enhancement of thermographic images as tool for structural analysis in earthquake engineering. NDT & E Int. 70, 60–72 (2015)
Coroianu L., Gal S.G.: Approximation by nonlinear generalized sampling operators of max-product kind. Sampl. Th. Signal Image Process. 9(1-3), 59–75 (2010)
Coroianu L., Gal S.G.: Approximation by max-product sampling operators based on sinc-type kernels. Sampl. Th. Signal Image Process. 10(3), 211–230 (2011)
Coroianu L., Gal S.G.: Saturation results for the truncated max-product sampling operators based on sinc and Fejér-type kernels. Sampl. Th. Signal Image Process. 11(1), 113–132 (2012)
Coroianu L., Gal S.G.: Saturation and inverse results for the Bernstein max-product operator. Period. Math. Hungar 69, 126–133 (2014)
Costarelli D.: Neural network operators: constructive interpolation of multivariate functions. Neural Netw. 67, 28–36 (2015)
Costarelli D., Spigler R.: Multivariate neural network operators with sigmoidal activation functions. Neural Netw. 48, 72–77 (2013)
Costarelli D., Spigler R.: Convergence of a family of neural network operators of the Kantorovich type. J. Approx. Theory 185, 80–90 (2014)
Costarelli D., Spigler R.: Approximation by series of sigmoidal functions with applications to neural networks. Annali di Matematica Pura ed Applicata 194(1), 289–306 (2015)
Costarelli D., Vinti G.: Rate of approximation for multivariate sampling Kantorovich operators on some functions spaces. J. Int. Eq. Appl. 26(4), 455–481 (2014)
Costarelli D., Vinti G.: Degree of approximation for nonlinear multivariate sampling Kantorovich operators on some functions spaces. Numer. Funct. Anal. Optim. 36(8), 964–990 (2015)
Costarelli, D., Vinti, G.: Max-product neural network and quasi interpolation operators activated by sigmoidal functions, submitted (2015)
Cybenko G.: Approximation by superpositions of a sigmoidal function. Math. Control Signals Syst. 2, 303–314 (1989)
Gripenberg G.: Approximation by neural network with a bounded number of nodes at each level. J. Approx. Theory 122(2), 260–266 (2003)
Ismailov V.E.: On the approximation by neural networks with bounded number of neurons in hidden layers. J. Math. Anal. Appl. 417(2), 963–969 (2014)
Ito Y.: Independence of unscaled basis functions and finite mappings by neural networks. Math. Sci. 26, 117–126 (2001)
Kainen P.C., Kurková V.: An integral upper bound for neural network approximation. Neural Comput. 21, 2970–2989 (2009)
Llanas B., Sainz F.J.: Constructive approximate interpolation by neural networks. J. Comput. Appl. Math. 188, 283–308 (2006)
Maiorov V.: Approximation by neural networks and learning theory. J. Complexity 22(1), 102–117 (2006)
Makovoz Y.: Uniform approximation by neural networks. J. Approx. Theory 95(2), 215–228 (1998)
Pinkus A.: Approximation theory of the MLP model in neural networks. Acta Numer. 8, 143–195 (1999)
Sontag E.D.: Feedforward nets for interpolation and classification. J. Comp. Syst. Sci. 45, 20–48 (1992)
Vinti G., Zampogni L.: Approximation results for a general class of Kantorovich type operators. Adv. Nonlinear Stud. 14, 991–1011 (2014)
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Costarelli, D., Vinti, G. Approximation by Max-Product Neural Network Operators of Kantorovich Type. Results. Math. 69, 505–519 (2016). https://doi.org/10.1007/s00025-016-0546-7
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DOI: https://doi.org/10.1007/s00025-016-0546-7