8,148 research outputs found
Reproducing Kernels of Generalized Sobolev Spaces via a Green Function Approach with Distributional Operators
In this paper we introduce a generalized Sobolev space by defining a
semi-inner product formulated in terms of a vector distributional operator
consisting of finitely or countably many distributional operators
, which are defined on the dual space of the Schwartz space. The types of
operators we consider include not only differential operators, but also more
general distributional operators such as pseudo-differential operators. We
deduce that a certain appropriate full-space Green function with respect to
now becomes a conditionally positive
definite function. In order to support this claim we ensure that the
distributional adjoint operator of is
well-defined in the distributional sense. Under sufficient conditions, the
native space (reproducing-kernel Hilbert space) associated with the Green
function can be isometrically embedded into or even be isometrically
equivalent to a generalized Sobolev space. As an application, we take linear
combinations of translates of the Green function with possibly added polynomial
terms and construct a multivariate minimum-norm interpolant to data
values sampled from an unknown generalized Sobolev function at data sites
located in some set . We provide several examples, such
as Mat\'ern kernels or Gaussian kernels, that illustrate how many
reproducing-kernel Hilbert spaces of well-known reproducing kernels are
isometrically equivalent to a generalized Sobolev space. These examples further
illustrate how we can rescale the Sobolev spaces by the vector distributional
operator . Introducing the notion of scale as part of the
definition of a generalized Sobolev space may help us to choose the "best"
kernel function for kernel-based approximation methods.Comment: Update version of the publish at Num. Math. closed to Qi Ye's Ph.D.
thesis (\url{http://mypages.iit.edu/~qye3/PhdThesis-2012-AMS-QiYe-IIT.pdf}
Statistical properties of the method of regularization with periodic Gaussian reproducing kernel
The method of regularization with the Gaussian reproducing kernel is popular
in the machine learning literature and successful in many practical
applications.
In this paper we consider the periodic version of the Gaussian kernel
regularization.
We show in the white noise model setting, that in function spaces of very
smooth functions, such as the infinite-order Sobolev space and the space of
analytic functions, the method under consideration is asymptotically minimax;
in finite-order Sobolev spaces, the method is rate optimal, and the efficiency
in terms of constant when compared with the minimax estimator is reasonably
high. The smoothing parameters in the periodic Gaussian regularization can be
chosen adaptively without loss of asymptotic efficiency. The results derived in
this paper give a partial explanation of the success of the
Gaussian reproducing kernel in practice. Simulations are carried out to study
the finite sample properties of the periodic Gaussian regularization.Comment: Published by the Institute of Mathematical Statistics
(http://www.imstat.org) in the Annals of Statistics
(http://www.imstat.org/aos/) at http://dx.doi.org/10.1214/00905360400000045
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