Skip to main content
SpringerLink
Log in

Approximations by Multivariate Perturbed Neural Networks

  • Chapter
  • First Online:
Intelligent Systems II: Complete Approximation by Neural Network Operators

Part of the book series: Studies in Computational Intelligence ((SCI,volume 608))

  • 849 Accesses

Abstract

This chapter deals with the determination of the rate of convergence to the unit of each of three newly introduced here multivariate perturbed normalized neural network operators of one hidden layer.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. G.A. Anastassiou, Rate of convergence of some neural network operators to the unit-univariate case. J. Math. Anal. Appli. 212, 237–262 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  2. G.A. Anastassiou, Rate of convergence of some multivariate neural network operators to the unit. J. Comp. Math. Appl. 40, 1–19 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  3. G.A. Anastassiou, Quantitative Approximations (Chapman and Hall/CRC, Boca Raton, 2001)

    Google Scholar 

  4. G.A. Anastassiou, Rate of convergence of some neural network operators to the unit-univariate case, revisited. Vesnik 65(4), 511–518 (2013)

    MATH  MathSciNet  Google Scholar 

  5. G.A. Anastassiou, Rate of convergence of some multivariate neural network operators to the unit, revisited. J. Comput. Anal. Appl. 15(7), 1300–1309 (2013)

    MATH  MathSciNet  Google Scholar 

  6. G.A. Anastassiou, Approximation by Perturbed Neural Netwwork Operators. Submitted 2014

    Google Scholar 

  7. G.A. Anastassiou, Approximations by Multivariate Perturbed Neural Network Operators. Submitted 2014

    Google Scholar 

  8. A.R. Barron, Universal approximation bounds for superpositions of a sigmoidal function. IEEE Trans. Inf. Theory 39, 930–945 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  9. F.L. Cao, T.F. Xie, Z.B. Xu, The estimate for approximation error of neural networks: a constructive approach. Neurocomputing 71, 626–630 (2008)

    Article  Google Scholar 

  10. P. Cardaliaguet, G. Euvrard, Approximation of a function and its derivative with a Neural Network. Neural Netw. 5, 207–220 (1992)

    Article  Google Scholar 

  11. Z. Chen, F. Cao, The approximation operators with sigmoidal functions. Comput. Math. Appl. 58, 758–765 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  12. T.P. Chen, H. Chen, Universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its applications to a dynamic system. IEEE Trans. Neural Netw. 6, 911–917 (1995)

    Article  Google Scholar 

  13. C.K. Chui, X. Li, Approximation by ridge functions and neural networks with one hidden layer. J. Approx. Theory 70, 131–141 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  14. D. Costarelli, R. Spigler, Approximation results for neural network operators activated by sigmoidal functions. Neural Netw. 44, 101–106 (2013)

    Article  MATH  Google Scholar 

  15. D. Costarelli, R. Spigler, Multivariate neural network operators with sigmoidal activation functions. Neural Netw. 48, 72–77 (2013)

    Article  MATH  Google Scholar 

  16. G. Cybenko, Approximation by superpositions of sigmoidal function. Math. Control Signals Syst. 2, 303–314 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  17. S. Ferrari, R.F. Stengel, Smooth function approximation using neural networks. IEEE Trans. Neural Netw. 16, 24–38 (2005)

    Article  Google Scholar 

  18. K.I. Funahashi, On the approximate realization of continuous mappings by neural networks. Neural Netw. 2, 183–192 (1989)

    Article  Google Scholar 

  19. N. Hahm, B.I. Hong, An approximation by neural networks with a fixed weight. Comput. Math. Appl. 47, 1897–1903 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  20. S. Haykin, Neural Networks: A Comprehensive Foundation, 2nd edn. (Prentice Hall, New York, 1998)

    Google Scholar 

  21. K. Hornik, M. Stinchombe, H. White, Multilayer feedforward networks are universal approximations. Neural Netw. 2, 359–366 (1989)

    Article  Google Scholar 

  22. K. Hornik, M. Stinchombe, H. White, Universal approximation of an unknown mapping and its derivatives using multilayer feedforward networks. Neural Netw. 3, 551–560 (1990)

    Article  Google Scholar 

  23. M. Leshno, V.Y. Lin, A. Pinks, S. Schocken, Multilayer feedforward networks with a nonpolynomial activation function can approximate any function. Neural Netw. 6, 861–867 (1993)

    Article  Google Scholar 

  24. V. Maiorov, R.S. Meir, Approximation bounds for smooth functions in \(C\left( R^{d}\right) \) by neural and mixture networks. IEEE Trans. Neural Netw. 9, 969–978 (1998)

    Article  Google Scholar 

  25. Y. Makovoz, Uniform approximation by neural networks. J. Approx. Theory 95, 215–228 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  26. W. McCulloch, W. Pitts, A logical calculus of the ideas immanent in nervous activity. Bull. Math. Biophys. 7, 115–133 (1943)

    Article  MathSciNet  Google Scholar 

  27. H.N. Mhaskar, C.A. Micchelli, Approximation by superposition of a sigmoidal function. Adv. Appl. Math. 13, 350–373 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  28. H.N. Mhaskar, C.A. Micchelli, Degree of approximation by neural networks with a single hidden layer. Adv. Appl. Math. 16, 151–183 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  29. T.M. Mitchell, Machine Learning (WCB-McGraw-Hill, New York, 1997)

    MATH  Google Scholar 

  30. D.D. Stancu, On a generalization of the Bernstein polynomials. Studia Universitatis Babeş-Bolyai, Series Mathematica-Physica 14, 31–45 (1969)

    MATH  MathSciNet  Google Scholar 

  31. S. Suzuki, Constructive function approximation by three-layer artificial neural networks. Neural Netw. 11, 1049–1058 (1998)

    Article  Google Scholar 

  32. Z.B. Xu, F.L. Cao, The essential order of approximation for neural networks, Sci. China (Ser. F) 47, 97–112 (2004)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, University of Memphis, Memphis, TN, USA

    George A. Anastassiou

Authors
  1. George A. Anastassiou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to George A. Anastassiou .

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Anastassiou, G.A. (2016). Approximations by Multivariate Perturbed Neural Networks. In: Intelligent Systems II: Complete Approximation by Neural Network Operators. Studies in Computational Intelligence, vol 608. Springer, Cham. https://doi.org/10.1007/978-3-319-20505-2_24

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-20505-2_24

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-20504-5

  • Online ISBN: 978-3-319-20505-2

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation