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Volumn 194, Issue 1, 2015, Pages 289-306

Approximation by series of sigmoidal functions with applications to neural networks

Author keywords

Multiresolution approximation; Neural networks approximation; Order of approximation; Sigmoidal functions; Truncation error; Wavelet scaling functions

Indexed keywords


EID: 84926311345     PISSN: 03733114     EISSN: 16181891     Source Type: Journal    
DOI: 10.1007/s10231-013-0378-y     Document Type: Article
Times cited : (39)

References (36)
  • 1
    • 78751604033 scopus 로고    scopus 로고
    • Univariate hyperbolic tangent neural network approximation
    • Anastassiou, G.A.: Univariate hyperbolic tangent neural network approximation. Math. Comput. Model. 53(5–6), 1111–1132 (2011)
    • (2011) Math. Comput. Model. , vol.53 , Issue.5-6 , pp. 1111-1132
    • Anastassiou, G.A.1
  • 2
    • 79651470747 scopus 로고    scopus 로고
    • Multivariate hyperbolic tangent neural network approximation
    • Anastassiou, G.A.: Multivariate hyperbolic tangent neural network approximation. Comput. Math. Appl. 61(4), 809–821 (2011)
    • (2011) Comput. Math. Appl. , vol.61 , Issue.4 , pp. 809-821
    • Anastassiou, G.A.1
  • 3
    • 0027599793 scopus 로고
    • Universal approximation bounds for superpositions of a sigmoidal function
    • Barron, A.R.: Universal approximation bounds for superpositions of a sigmoidal function. IEEE Trans. Inf. Theory 39(3), 930–945 (1993)
    • (1993) IEEE Trans. Inf. Theory , vol.39 , Issue.3 , pp. 930-945
    • Barron, A.R.1
  • 7
    • 84865627769 scopus 로고    scopus 로고
    • A neural network approach for solving Fredholm integral equations of the second kind
    • Buzhabadi, R., Effati, S.: A neural network approach for solving Fredholm integral equations of the second kind. Neural Comput. Appl. 21, 843–852 (2012)
    • (2012) Neural Comput. Appl. , vol.21 , pp. 843-852
    • Buzhabadi, R.1    Effati, S.2
  • 8
    • 67649743440 scopus 로고    scopus 로고
    • The approximation operators with sigmoidal functions
    • Cao, F., Chen, Z.: The approximation operators with sigmoidal functions. Comput. Math. Appl. 58(4), 758–765 (2009)
    • (2009) Comput. Math. Appl. , vol.58 , Issue.4 , pp. 758-765
    • Cao, F.1    Chen, Z.2
  • 9
    • 9644285073 scopus 로고
    • A constructive proof and an extension of Cybenko’s approximation theorem
    • Springer, New York
    • Chen, H., Chen, T., Liu, R.: A constructive proof and an extension of Cybenko’s approximation theorem. In: Computing Science and Statistics, pp. 163–168. Springer, New York (1992)
    • (1992) Computing Science and Statistics , pp. 163-168
    • Chen, H.1    Chen, T.2    Liu, R.3
  • 10
    • 51249165422 scopus 로고
    • Degree of approximation by superpositions of a sigmoidal function
    • Chen, D.: Degree of approximation by superpositions of a sigmoidal function. Approx. Theory Appl. 9(3), 17–28 (1993)
    • (1993) Approx. Theory Appl. , vol.9 , Issue.3 , pp. 17-28
    • Chen, D.1
  • 12
    • 84883037302 scopus 로고    scopus 로고
    • Solving Volterra integral equations of the second kind by sigmoidal functions approximations, to appear in
    • Costarelli, D., Spigler, R.: Solving Volterra integral equations of the second kind by sigmoidal functions approximations, to appear in J. Integral Eq. Appl. 25(2) (2013)
    • (2013) J. Integral Eq. Appl , vol.25 , Issue.2
    • Costarelli, D.1    Spigler, R.2
  • 13
    • 84882972393 scopus 로고    scopus 로고
    • Constructive approximation by superposition of sigmoidal functions
    • Costarelli, D., Spigler, R.: Constructive approximation by superposition of sigmoidal functions. Anal. Theory Appl. 29(2), 169–196 (2013)
    • (2013) Anal. Theory Appl. , vol.29 , Issue.2 , pp. 169-196
    • Costarelli, D.1    Spigler, R.2
  • 14
    • 84876320512 scopus 로고    scopus 로고
    • Approximation results for neural network operators activated by sigmoidal functions
    • Costarelli, D., Spigler, R.: Approximation results for neural network operators activated by sigmoidal functions. Neural Netw. 44, 101–106 (2013)
    • (2013) Neural Netw. , vol.44 , pp. 101-106
    • Costarelli, D.1    Spigler, R.2
  • 15
    • 84883048212 scopus 로고    scopus 로고
    • Multivariate neural network operators with sigmoidal activation functions
    • Costarelli, D., Spigler, R.: Multivariate neural network operators with sigmoidal activation functions. Neural Netw. 48, 72–77 (2013)
    • (2013) Neural Netw. , vol.48 , pp. 72-77
    • Costarelli, D.1    Spigler, R.2
  • 16
    • 0024861871 scopus 로고
    • Approximation by superpositions of a sigmoidal function
    • Cybenko, G.: Approximation by superpositions of a sigmoidal function. Math. Control Signals Syst. 2, 303–314 (1989)
    • (1989) Math. Control Signals Syst. , vol.2 , pp. 303-314
    • Cybenko, G.1
  • 17
    • 0003077036 scopus 로고
    • Ten Lectures on Wavelets
    • Society for Industrial and Applied Mathematics, SIAM, Philadelphia
    • Daubechies, I.: Ten Lectures on Wavelets, Regional Conference Series in Applied Mathematics 61. Society for Industrial and Applied Mathematics, SIAM, Philadelphia (1992)
    • (1992) Regional Conference Series in Applied Mathematics , vol.61
    • Daubechies, I.1
  • 18
    • 24944585477 scopus 로고    scopus 로고
    • A Practical Guide to Spline
    • Springer, New York
    • De Boor, C.: A Practical Guide to Spline, Applied Mathematical Sciences 27. Springer, New York (2001)
    • (2001) Applied Mathematical Sciences , vol.27
    • De Boor, C.1
  • 19
    • 0042892216 scopus 로고
    • Univariant approximation by superpositions of a sigmoidal function
    • Gao, B., Xu, Y.: Univariant approximation by superpositions of a sigmoidal function. J. Math. Anal. Appl. 178, 221–226 (1993)
    • (1993) J. Math. Anal. Appl. , vol.178 , pp. 221-226
    • Gao, B.1    Xu, Y.2
  • 20
    • 31244437685 scopus 로고    scopus 로고
    • Approximation order to a function in (Formula presented.) by superposition of a sigmoidal function
    • Hahm, N., Hong, B.: Approximation order to a function in (Formula presented.) by superposition of a sigmoidal function. Appl. Math. Lett. 15, 591–597 (2002)
    • (2002) Appl. Math. Lett. , vol.15 , pp. 591-597
    • Hahm, N.1    Hong, B.2
  • 22
    • 84882976061 scopus 로고
    • Constructive approximations for neural networks by sigmoidal functions
    • University of Lowell, Dep. of Mathematics
    • Jones, L.K.: Constructive approximations for neural networks by sigmoidal functions, Technical Report Series 7. University of Lowell, Dep. of Mathematics (1988)
    • (1988) Technical Report Series , vol.7
    • Jones, L.K.1
  • 23
    • 0011595675 scopus 로고
    • Constructive multivariate approximation with sigmoidal functions and applications to neural networks
    • Lenze, B.: Constructive multivariate approximation with sigmoidal functions and applications to neural networks. In: Numer. Methods Approx. Theory, Birkhauser Verlag, Basel-Boston-Berlin, pp. 155–175 (1992)
    • (1992) Numer. Methods Approx. Theory, Birkhauser Verlag, Basel-Boston-Berlin , pp. 155-175
    • Lenze, B.1
  • 24
    • 0041829446 scopus 로고    scopus 로고
    • Approximation by superpositions of a sigmoidal function
    • Lewicki, G., Marino, G.: Approximation by superpositions of a sigmoidal function. Z. Anal. Anwendungen J. Anal. Appl. 22(2), 463–470 (2003)
    • (2003) Z. Anal. Anwendungen J. Anal. Appl. , vol.22 , Issue.2 , pp. 463-470
    • Lewicki, G.1    Marino, G.2
  • 25
    • 10644262975 scopus 로고    scopus 로고
    • Approximation of functions of finite variation by superpositions of a sigmoidal function
    • Lewicki, G., Marino, G.: Approximation of functions of finite variation by superpositions of a sigmoidal function. Appl. Math. Lett. 17, 1147–1152 (2004)
    • (2004) Appl. Math. Lett. , vol.17 , pp. 1147-1152
    • Lewicki, G.1    Marino, G.2
  • 26
    • 0030221938 scopus 로고    scopus 로고
    • Simultaneous approximations of multivariate functions and their derivatives by neural networks with one hidden layer
    • Li, X.: Simultaneous approximations of multivariate functions and their derivatives by neural networks with one hidden layer. Neurocomputing 12, 327–343 (1996)
    • (1996) Neurocomputing , vol.12 , pp. 327-343
    • Li, X.1
  • 27
    • 0034561156 scopus 로고    scopus 로고
    • Approximation by radial bases and neural networks
    • Li, X., Micchelli, C.A.: Approximation by radial bases and neural networks. Numer. Algorithms 25, 241–262 (2000)
    • (2000) Numer. Algorithms , vol.25 , pp. 241-262
    • Li, X.1    Micchelli, C.A.2
  • 28
    • 33845415634 scopus 로고    scopus 로고
    • Numerical solution for high order differential equations using a hybrid neural network—optimization method
    • Malek, A., Shekari Beidokhti, R.: Numerical solution for high order differential equations using a hybrid neural network—optimization method. Appl. Math. Comput. 183, 260–271 (2006)
    • (2006) Appl. Math. Comput. , vol.183 , pp. 260-271
    • Malek, A.1    Shekari Beidokhti, R.2
  • 29
    • 84966210236 scopus 로고
    • Multiresolution approximations and wavelet orthonormal bases of (Formula presented.)
    • Mallat, S.G.: Multiresolution approximations and wavelet orthonormal bases of (Formula presented.). Trans. Am. Math. Soc. 315, 69–87 (1989)
    • (1989) Trans. Am. Math. Soc. , vol.315 , pp. 69-87
    • Mallat, S.G.1
  • 30
    • 0004168108 scopus 로고
    • Wavelets and Operators. Cambridge Studies in Advanced Mathematics 37
    • Meyer, Y.: Wavelets and Operators. Cambridge Studies in Advanced Mathematics 37, Cambridge (1992)
    • (1992) Cambridge
    • Meyer, Y.1
  • 31
    • 0000358945 scopus 로고
    • Approximation by superposition of sigmoidal and radial basis functions
    • Mhaskar, H.N., Micchelli, C.A.: Approximation by superposition of sigmoidal and radial basis functions. Adv. Appl. Math. 13, 350–373 (1992)
    • (1992) Adv. Appl. Math. , vol.13 , pp. 350-373
    • Mhaskar, H.N.1    Micchelli, C.A.2
  • 32
    • 0000194429 scopus 로고
    • Degree of approximation by neural and translation networks with a single hidden layer
    • Mhaskar, H.N., Micchelli, C.A.: Degree of approximation by neural and translation networks with a single hidden layer. Adv. Appl. Math. 16, 151–183 (1995)
    • (1995) Adv. Appl. Math. , vol.16 , pp. 151-183
    • Mhaskar, H.N.1    Micchelli, C.A.2
  • 33
    • 0000041417 scopus 로고    scopus 로고
    • Neural networks for optimal approximation of smooth and analytic functions
    • Mhaskar, H.N.: Neural networks for optimal approximation of smooth and analytic functions. Neural Comput. 8, 164–177 (1996)
    • (1996) Neural Comput. , vol.8 , pp. 164-177
    • Mhaskar, H.N.1
  • 34
    • 85011438572 scopus 로고    scopus 로고
    • Approximation theory of the MLP model in neural networks
    • Pinkus, A.: Approximation theory of the MLP model in neural networks. Acta Numer. 8, 143–195 (1999)
    • (1999) Acta Numer. , vol.8 , pp. 143-195
    • Pinkus, A.1
  • 35
    • 57649126218 scopus 로고    scopus 로고
    • Ten good reasons for using spline wavelets
    • Unser, M.: Ten good reasons for using spline wavelets. Wavelets Appl. Signal Image Process. 3169(5), 422–431 (1997)
    • (1997) Wavelets Appl. Signal Image Process. , vol.3169 , Issue.5 , pp. 422-431
    • Unser, M.1
  • 36
    • 84958754734 scopus 로고    scopus 로고
    • On the degree of approximation by wavelet expansions
    • Xiehua, S.: On the degree of approximation by wavelet expansions. Approx. Theory Appl. 14(1), 81–90 (1998)
    • (1998) Approx. Theory Appl. , vol.14 , Issue.1 , pp. 81-90
    • Xiehua, S.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.