On the spectral distribution of kernel matrices related to radial basis functions
This paper focuses on the 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 kerne...
| Main Authors: | , |
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| Format: | Report |
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Oxford preprint
2013
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| _version_ | 1797059987050070016 |
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| author | Wathen, A Zhu, S |
| author_facet | Wathen, A Zhu, S |
| author_sort | Wathen, A |
| collection | OXFORD |
| description | This paper focuses on the 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 have important application on stable computing of Gaussian radial basis functions. |
| first_indexed | 2024-03-06T20:11:24Z |
| format | Report |
| id | oxford-uuid:2aac74e7-ea73-4fca-b3cc-a8eea861dcce |
| institution | University of Oxford |
| last_indexed | 2024-03-06T20:11:24Z |
| publishDate | 2013 |
| publisher | Oxford preprint |
| record_format | dspace |
| spelling | oxford-uuid:2aac74e7-ea73-4fca-b3cc-a8eea861dcce2022-03-26T12:26:26ZOn the spectral distribution of kernel matrices related to radial basis functionsReporthttp://purl.org/coar/resource_type/c_93fcuuid:2aac74e7-ea73-4fca-b3cc-a8eea861dcceMathematical Institute - ePrintsOxford preprint2013Wathen, AZhu, SThis paper focuses on the 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 have important application on stable computing of Gaussian radial basis functions. |
| spellingShingle | Wathen, A Zhu, S On the spectral distribution of kernel matrices related to radial basis functions |
| title | On the spectral distribution of kernel matrices related to radial basis functions |
| title_full | On the spectral distribution of kernel matrices related to radial basis functions |
| title_fullStr | On the spectral distribution of kernel matrices related to radial basis functions |
| title_full_unstemmed | On the spectral distribution of kernel matrices related to radial basis functions |
| title_short | On the spectral distribution of kernel matrices related to radial basis functions |
| title_sort | on the spectral distribution of kernel matrices related to radial basis functions |
| work_keys_str_mv | AT wathena onthespectraldistributionofkernelmatricesrelatedtoradialbasisfunctions AT zhus onthespectraldistributionofkernelmatricesrelatedtoradialbasisfunctions |