×

Modified neural network operators and their convergence properties with summability methods. (English) Zbl 1440.41011

The authors study the approximation properties of Cardaliaguet-Euvrard type neural network operators. They first modify the operators in order to get the uniform convergence and use regular summability matrix methods in the approximation by means of these operators to get more general results than the classical ones. Also, some examples and graphical illustrations supporting their approximation results by neural networks operators are presented.

MSC:

41A25 Rate of convergence, degree of approximation
40F05 Absolute and strong summability
Full Text: DOI

References:

[1] Anastassiou, GA, Rate of convergence of some neural network operators to the unit-univariate case, J. Math. Anal. Appl., 212, 237-262 (1997) · Zbl 0899.68088
[2] Anastassiou, GA, Rate of convergence of some multivariate neural network operators to the unit, Comput. Math. Appl., 40, 1, 1-19 (2000) · Zbl 0958.68139
[3] Anastassiou, GA, Multivariate sigmoidal neural network approximation, Neural Netw., 24, 4, 378-386 (2011) · Zbl 1228.65018
[4] Anastassiou, GA, Multivariate hyperbolic tangent neural network approximation, Comput. Math. Appl., 61, 4, 809-821 (2011) · Zbl 1217.41035
[5] Anastassiou, GA, Rate of convergence of some neural network operators to the unit-univariate case, revisited, Mat. Vesnik, 65, 4, 511-518 (2013) · Zbl 1299.41025
[6] Anastassiou, GA, Rate of convergence of some multivariate neural network operators to the unit, revisited, J. Comput. Anal. Appl., 15, 7, 1300-1309 (2013) · Zbl 1290.68103
[7] Anastassiou, GA, Approximation by interpolating neural network operators, Neural Parallel Sci. Comput., 23, 1, 1-62 (2015)
[8] Aslan, I.; Duman, O., A summability process on Baskakov-type approximation, Period. Math. Hungar., 72, 2, 186-199 (2016) · Zbl 1389.40021
[9] Aslan, I.; Duman, O., Summability on Mellin-type nonlinear integral operators, Integral Transforms Spec. Funct., 30, 6, 492-511 (2019) · Zbl 1436.41013
[10] Aslan, I.; Duman, O., Approximation by nonlinear integral operators via summability process, Math. Nachr., 293, 3, 430-448 (2020) · Zbl 07198947
[11] Atlihan, OG; Orhan, C., Summation process of positive linear operators, Comput. Math. Appl., 56, 5, 1188-1195 (2008) · Zbl 1155.41308
[12] Boos, J., Classical and Modern Methods in Summability (2000), Oxford: Oxford University Press, Oxford · Zbl 0954.40001
[13] Cao, F.; Chen, Z., Scattered data approximation by neural networks operators, Neurocomputing, 190, 237-242 (2016)
[14] Cao, F.; Xie, T.; Xu, Z., The estimate for approximation error of neural networks: a constructive approach, Neurocomputing, 71, 4, 626-630 (2008)
[15] Cao, F.; Zhang, Y.; He, Z-R, Interpolation and rates of convergence for a class of neural networks, Appl. Math. Model., 33, 3, 1441-1456 (2009) · Zbl 1168.41302
[16] Cardaliaguet, P.; Euvrard, G., Approximation of a function and its derivative with a neural network, Neural Netw., 5, 2, 207-220 (1992)
[17] Cheang, GHL, Approximation with neural networks activated by ramp sigmoids, J. Approx. Theory, 162, 1450-1465 (2010) · Zbl 1203.82069
[18] Chen, Z.; Cao, F., The approximation operators with sigmoidal functions, Comput. Math. Appl., 58, 4, 758-765 (2009) · Zbl 1189.41014
[19] Chen, Z.; Cao, F., The construction and approximation of neural networks operators with Gaussian activation function, Math. Commun., 18, 1, 185-207 (2013) · Zbl 1311.41007
[20] Costarelli, D.; Spigler, R., Multivariate neural network operators with sigmoidal activation functions, Neural Netw., 48, 72-77 (2013) · Zbl 1297.41006
[21] Costarelli, D.; Spigler, R., Approximation results for neural network operators activated by sigmoidal functions, Neural Netw., 44, 101-106 (2013) · Zbl 1296.41017
[22] Costarelli, D.; Spigler, R., Convergence of a family of neural network operators of the Kantorovich type, J. Approx. Theory, 185, 80-90 (2014) · Zbl 1297.41003
[23] Costarelli, D.; Vinti, G., Max-product neural network and quasi-interpolation operators activated by sigmoidal functions, J. Approx. Theory, 209, 1-22 (2016) · Zbl 1350.41001
[24] Costarelli, D.; Vinti, G., Convergence for a family of neural network operators in Orlicz spaces, Math. Nachr., 290, 2-3, 226-235 (2017) · Zbl 1373.47010
[25] Costarelli, D.; Vinti, G., Estimates for the neural network operators of the max-product type with continuous and \(p\)-integrable functions, Results Math., 73, 1, 10 (2018) · Zbl 1390.41020
[26] Costarelli, D.; Vinti, G., Saturation classes for max-product neural network operators activated by sigmoidal functions, Results Math., 72, 3, 1555-1569 (2017) · Zbl 1376.41014
[27] Costarelli, D.; Vinti, G., Convergence results for a family of kantorovich max-product neural network operators in a multivariate setting, Math. Slovaca, 67, 6, 1469-1480 (2017) · Zbl 1505.41005
[28] Cybenko, G., Approximations by superpositions of sigmoidal functions, Math. Control Signals Syst., 2, 4, 303-314 (1989) · Zbl 0679.94019
[29] Duman, O., Summability process by Mastroianni operators and their generalizations, Mediterr. J. Math., 12, 1, 21-35 (2015) · Zbl 1321.40003
[30] Gal, SG, Approximation by max-product type nonlinear operators, Stud. Univ. Babeş-Bolyai Math., 56, 2, 341-352 (2011) · Zbl 1249.41039
[31] Gokcer, T. Y., Duman, O.: Approximation by max-min operators: a general theory and its applications. Fuzzy Sets Syst. (2019). 10.1016/j.fss.2019.11.007 · Zbl 1452.41012
[32] Gripenberg, G., Approximation by neural network with a bounded number of nodes at each level, J. Approx. Theory, 122, 2, 260-266 (2003) · Zbl 1019.41018
[33] Hardy, GH, Divergent Series (1949), Oxford: Clarendon Press, Oxford · Zbl 0032.05801
[34] Keagy, TA; Ford, WF, Acceleration by subsequence transformations, Pacific J. Math., 132, 2, 357-362 (1988) · Zbl 0669.40001
[35] Lorentz, GG; Zeller, K., Strong and ordinary summability, Tohoku Math. J. Second Ser., 15, 4, 315-321 (1963) · Zbl 0185.13301
[36] Mhaskar, H.N.: Approximation Theory and Neural Networks, Wavelets and Allied Topics, 247-289. Narosa, New Delhi (2001) · Zbl 1037.41011
[37] Mhaskar, HN; Micchelli, CA, Degree of approximation by neural and translation networks with a single hidden layer, Adv. Appl. Math., 16, 151-183 (1995) · Zbl 0885.42012
[38] Mohapatra, RN, Quantitative results on almost convergence of a sequence of positive linear operators, J. Approx. Theory, 20, 239-250 (1977) · Zbl 0351.41010
[39] Makovoz, Y., Uniform approximation by neural networks, J. Approx. Theory, 95, 2, 21-228 (1998) · Zbl 0932.41016
[40] Sakaoğlu, I.; Orhan, C., Cihan Strong summation process in \(L_p\)-spaces, Nonlinear Anal., 86, 89-94 (2013) · Zbl 1283.41017
[41] Silverman, LL, On the definition of the sum of a divergent series, University of Missouri Studies, Math. Ser., I, 1-96 (1913) · JFM 44.1116.09
[42] Smith, DA; Ford, WF, Acceleration of linear and logarithmic convergence, SIAM J. Numer. Anal., 16, 2, 223-240 (1979) · Zbl 0407.65002
[43] Swetits, JJ, On summability and positive linear operators, J. Approx. Theory, 25, 186-188 (1979) · Zbl 0422.41019
[44] Toeplitz, O., Über die lineare Mittelbildungen, Prace mat.-fiz., 22, 113-118 (1911) · JFM 44.0281.02
[45] Vanderbei, RJ, Uniform Continuity is Almost Lipschitz Continuity, Technical Report SOR-91 11. Statistics and Operations Research Series (1991), Princeton: Princeton University, Princeton
[46] Wimp, J., Sequence Transformations and Their Applications (1981), New York: Academic Press, New York · Zbl 0566.47018
[47] Yu, DS, Approximation by neural networks with sigmoidal functions, Acta Math. Sin. (Engl. Ser.), 29, 10, 2013-2026 (2013) · Zbl 1311.41015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.