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December 2011 Principal support vector machines for linear and nonlinear sufficient dimension reduction
Bing Li, Andreas Artemiou, Lexin Li
Ann. Statist. 39(6): 3182-3210 (December 2011). DOI: 10.1214/11-AOS932

Abstract

We introduce a principal support vector machine (PSVM) approach that can be used for both linear and nonlinear sufficient dimension reduction. The basic idea is to divide the response variables into slices and use a modified form of support vector machine to find the optimal hyperplanes that separate them. These optimal hyperplanes are then aligned by the principal components of their normal vectors. It is proved that the aligned normal vectors provide an unbiased, √n-consistent, and asymptotically normal estimator of the sufficient dimension reduction space. The method is then generalized to nonlinear sufficient dimension reduction using the reproducing kernel Hilbert space. In that context, the aligned normal vectors become functions and it is proved that they are unbiased in the sense that they are functions of the true nonlinear sufficient predictors. We compare PSVM with other sufficient dimension reduction methods by simulation and in real data analysis, and through both comparisons firmly establish its practical advantages.

Citation

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Bing Li. Andreas Artemiou. Lexin Li. "Principal support vector machines for linear and nonlinear sufficient dimension reduction." Ann. Statist. 39 (6) 3182 - 3210, December 2011. https://doi.org/10.1214/11-AOS932

Information

Published: December 2011
First available in Project Euclid: 5 March 2012

zbMATH: 1246.60061
MathSciNet: MR3012405
Digital Object Identifier: 10.1214/11-AOS932

Subjects:
Primary: 62-09 , 62G08 , 62H12

Keywords: Contour regression , invariant kernel , inverse regression , principal components , ‎reproducing kernel Hilbert ‎space , Support Vector Machine

Rights: Copyright © 2011 Institute of Mathematical Statistics

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Vol.39 • No. 6 • December 2011
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