Open Access
August 2009 Kernel dimension reduction in regression
Kenji Fukumizu, Francis R. Bach, Michael I. Jordan
Ann. Statist. 37(4): 1871-1905 (August 2009). DOI: 10.1214/08-AOS637


We present a new methodology for sufficient dimension reduction (SDR). Our methodology derives directly from the formulation of SDR in terms of the conditional independence of the covariate X from the response Y, given the projection of X on the central subspace [cf. J. Amer. Statist. Assoc. 86 (1991) 316–342 and Regression Graphics (1998) Wiley]. We show that this conditional independence assertion can be characterized in terms of conditional covariance operators on reproducing kernel Hilbert spaces and we show how this characterization leads to an M-estimator for the central subspace. The resulting estimator is shown to be consistent under weak conditions; in particular, we do not have to impose linearity or ellipticity conditions of the kinds that are generally invoked for SDR methods. We also present empirical results showing that the new methodology is competitive in practice.


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Kenji Fukumizu. Francis R. Bach. Michael I. Jordan. "Kernel dimension reduction in regression." Ann. Statist. 37 (4) 1871 - 1905, August 2009.


Published: August 2009
First available in Project Euclid: 18 June 2009

zbMATH: 1168.62049
MathSciNet: MR2533474
Digital Object Identifier: 10.1214/08-AOS637

Primary: 62H99
Secondary: 62J02

Keywords: consistency , Dimension reduction , Positive definite kernel , regression , reproducing kernel

Rights: Copyright © 2009 Institute of Mathematical Statistics

Vol.37 • No. 4 • August 2009
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