Skip to main content
SpringerLink
Log in

Approximation by radial bases and neural networks

  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

In this paper, we study approximation by radial basis functions including Gaussian, multiquadric, and thin plate spline functions, and derive order of approximation under certain conditions. Moreover, neural networks are also constructed by wavelet recovery formula and wavelet frames.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

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

    Google Scholar 

  2. G. Anastassiou, Rate of convergence of some multivariate neural network operators to the unit, Comput. Math. Appl., to appear.

  3. M.D. Buhmann, Multivariate cardinal interpolation with radial-basis functions, Construct. Approx. 6 (1990) 225–255.

    Google Scholar 

  4. M.D. Buhmann and C.A. Micchelli, Multiquadric interpolation improved, Comput. Math. Appl. 24 (1992) 21–25.

    Google Scholar 

  5. P. Cardaliaguet and G. Euvrard, Approximation of a function and its derivative with a neural network, Neural Networks 5 (1992) 207–220.

    Google Scholar 

  6. I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 61 (SIAM, Philadelphia, PA, 1992).

    Google Scholar 

  7. I.M. Gel'fand and G.E. Shilov, Generalized Functions, Vol. I (Academic Press, New York, 1964).

    Google Scholar 

  8. X. Li, Note on constructing near tight wavelet frames by neural networks, SPIE 2825 (1996) 152–162.

    Google Scholar 

  9. X. Li, On simultaneous approximations by radial basis function neural networks, Appl. Math. Comput. 95 (1998) 75–89.

    Google Scholar 

  10. C.A. Micchelli, Interpolatory subdivision schemes and wavelets, J. Approx. Theory 86 (1996) 41–71.

    Google Scholar 

  11. C.A. Micchelli, On a family of filters arising in wavelet construction, Appl. Comput. Harmonic Anal. 4 (1997) 38–50.

    Google Scholar 

  12. A. Pinkus, Approximation theory of the MLP model in neural networks, Acta Numerica (1999) 143–195.

  13. T. Poggio and F. Girosi, A Theory of Networks for Approximation and Learning, MIT AI Memo, No. 1140 (MIT Press, Cambridge, MA, 1989).

    Google Scholar 

  14. C. Rabut, Elementary m-harmonic cardinal B-splines, Numer. Algorithms 2 (1992) 39–62.

    Google Scholar 

  15. I.J. Schoenberg, Cardinal Spline Interpolation, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 12 (SIAM, Philadelphia, PA, 1973).

    Google Scholar 

  16. G.N. Watson, Theory of Bessel Functions, 2nd ed. (Cambridge Univ. Press, Cambridge, 1966).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, University of Nevada, Las Vegas, NV, 89154-4020, USA

    Xin Li

  2. Department of Mathematics and Statistics, State University of New York, The University at Albany, Albany, NY, 12222, USA

    Charles A. Micchelli

Authors
  1. Xin Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Charles A. Micchelli
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Li, X., Micchelli, C.A. Approximation by radial bases and neural networks. Numerical Algorithms 25, 241–262 (2000). https://doi.org/10.1023/A:1016685729545

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1016685729545

Search

Navigation