The gradient iteration for approximation in reproducing kernel Hilbert spaces |
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Authors: | Dodd, T. J. Harrison, R. F. |
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Affiliation: | Department of Automatic Control and Systems Engineering, The University of Sheffield, Sheffield S1 3JD, UK |
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Abstract: | Consider the bounded linear operator, L: F Z, where Z RN andF are Hilbert spaces defined on a common field X. L is madeup of a series of N bounded linear evaluation functionals, Li:F R. By the Riesz representation theorem, there exist functionsk(xi, ·) F : Lif = f, k(xi, ·)F. The functions,k(xi, ·), are known as reproducing kernels and F is areproducing kernel Hilbert space (RKHS). This is a natural frameworkfor approximating functions given a discrete set of observations.In this paper the computational aspects of characterizing suchapproximations are described and a gradient method presentedfor iterative solution. Such iterative solutions are desirablewhen N is large and the matrix computations involved in thebasic solution become infeasible. This is also exactly the casewhere the problem becomes ill-conditioned. An iterative approachto Tikhonov regularization is therefore also introduced. Unlikeiterative solutions for the more general Hilbert space setting,the proofs presented make use of the spectral representationof the kernel. |
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Keywords: | reproducing kernel Hilbert space gradient iteration function approximation. |
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