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August 2004 Statistical properties of the method of regularization with periodic Gaussian reproducing kernel
Yi Lin, Lawrence D. Brown
Ann. Statist. 32(4): 1723-1743 (August 2004). DOI: 10.1214/009053604000000454

Abstract

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.

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Yi Lin. Lawrence D. Brown. "Statistical properties of the method of regularization with periodic Gaussian reproducing kernel." Ann. Statist. 32 (4) 1723 - 1743, August 2004. https://doi.org/10.1214/009053604000000454

Information

Published: August 2004
First available in Project Euclid: 4 August 2004

zbMATH: 1045.62026
MathSciNet: MR2089140
Digital Object Identifier: 10.1214/009053604000000454

Keywords: asymptotic minimax risk , Gaussian reproducing kernel , nonparametric estimation , rate of convergence , Sobolev Spaces , White noise model

Rights: Copyright © 2004 Institute of Mathematical Statistics

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Vol.32 • No. 4 • August 2004
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