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On spectral distribution of kernel matrices related to radial basis functions

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Abstract

This paper focuses on spectral distribution of kernel matrices related to radial basis functions. By relating a contemporary finite-dimensional linear algebra problem to a classical problem on infinite-dimensional linear integral operator, the paper shows how the spectral distribution of a kernel matrix relates to the smoothness of the underlying kernel function. The asymptotic behaviour of the eigenvalues of a infinite-dimensional kernel operator are studied from a perspective of low rank approximation—approximating an integral operator in terms of Fourier series or Chebyshev series truncations. Further, we study how the spectral distribution of interpolation matrices of an infinite smooth kernel with flat limit depends on the geometric property of the underlying interpolation points. In particularly, the paper discusses the analytic eigenvalue distribution of Gaussian kernels, which has important application on stably computing of Gaussian radial basis functions.

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Authors and Affiliations

  1. Numerical Analysis Group, The University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, England

    Andrew J. Wathen

  2. Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, P.O.Box 8009, Beijing, 100088, China

    Shengxin Zhu

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  1. Andrew J. Wathen
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  2. Shengxin Zhu
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Correspondence to Shengxin Zhu.

Additional information

This research is partially supported by Nature Science Foundation of China (No. 61170309, No.61472462, and No.91430218) and the Laboratory of Computational Physics.

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Wathen, A.J., Zhu, S. On spectral distribution of kernel matrices related to radial basis functions. Numer Algor 70, 709–726 (2015). https://doi.org/10.1007/s11075-015-9970-0

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