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Constructive Approximation manuscript No. (will be inserted by the editor) Power Series Kernels

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Titel: Constructive Approximation manuscript No. (will be inserted by the editor) Power Series Kernels
Autoren: Barbara Zwicknagl
Weitere Verfasser: The Pennsylvania State University CiteSeerX Archives
Quelle: http://www.mis.mpg.de/preprints/2006/preprint2006_155.pdf.
Publikationsjahr: 2006
Bestand: CiteSeerX
Schlagwörter: multivariate polynomial approximation, Bernstein theorem, dot prod- uct kernels, reproducing kernel Hilbert spaces, error bounds, convergence orders
Beschreibung: We introduce a class of analytic positive definite multivariate kernels which includes infinite dot product kernels as sometimes used in machine learning, certain new nonlinearly factorizable kernels and a kernel which is closely related to the Gaussian. Each such kernel reproduces in a certain ‘native ‘ Hilbert space of multivariate analytic functions. If functions from this space are interpolated in scattered locations by translates of the kernel, we prove spectral convergence rates of the interpolants and all derivatives. By truncation of the power series of the kernel-based interpolants, we constructively generalize the classical Bernstein theorem concerning polynomial approximation of analytic functions to the multi-variate case. An application to machine learning algorithms is presented.
Publikationsart: text
Dateibeschreibung: application/pdf
Sprache: English
Relation: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.86.5528; http://www.mis.mpg.de/preprints/2006/preprint2006_155.pdf
Verfügbarkeit: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.86.5528
http://www.mis.mpg.de/preprints/2006/preprint2006_155.pdf
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  Data: Constructive Approximation manuscript No. (will be inserted by the editor) Power Series Kernels
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  Data: <searchLink fieldCode="AR" term="%22Barbara+Zwicknagl%22">Barbara Zwicknagl</searchLink>
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  Data: The Pennsylvania State University CiteSeerX Archives
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  Data: <i>http://www.mis.mpg.de/preprints/2006/preprint2006_155.pdf</i>.
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  Data: <searchLink fieldCode="DE" term="%22multivariate+polynomial+approximation%22">multivariate polynomial approximation</searchLink><br /><searchLink fieldCode="DE" term="%22Bernstein+theorem%22">Bernstein theorem</searchLink><br /><searchLink fieldCode="DE" term="%22dot+prod-+uct+kernels%22">dot prod- uct kernels</searchLink><br /><searchLink fieldCode="DE" term="%22reproducing+kernel+Hilbert+spaces%22">reproducing kernel Hilbert spaces</searchLink><br /><searchLink fieldCode="DE" term="%22error+bounds%22">error bounds</searchLink><br /><searchLink fieldCode="DE" term="%22convergence+orders%22">convergence orders</searchLink>
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  Data: We introduce a class of analytic positive definite multivariate kernels which includes infinite dot product kernels as sometimes used in machine learning, certain new nonlinearly factorizable kernels and a kernel which is closely related to the Gaussian. Each such kernel reproduces in a certain ‘native ‘ Hilbert space of multivariate analytic functions. If functions from this space are interpolated in scattered locations by translates of the kernel, we prove spectral convergence rates of the interpolants and all derivatives. By truncation of the power series of the kernel-based interpolants, we constructively generalize the classical Bernstein theorem concerning polynomial approximation of analytic functions to the multi-variate case. An application to machine learning algorithms is presented.
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      – Text: English
    Subjects:
      – SubjectFull: multivariate polynomial approximation
        Type: general
      – SubjectFull: Bernstein theorem
        Type: general
      – SubjectFull: dot prod- uct kernels
        Type: general
      – SubjectFull: reproducing kernel Hilbert spaces
        Type: general
      – SubjectFull: error bounds
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      – SubjectFull: convergence orders
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      – TitleFull: Constructive Approximation manuscript No. (will be inserted by the editor) Power Series Kernels
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