Principal support vector machines for linear and nonlinear sufficient dimension reduction

Details Principal support vector machines for linear and nonlinear sufficient dimension reduction Principal support vector machines for linear and nonlinear sufficient dimension reduction

2012-03-13

arxiv.org/abs/1203.2790v1

Annals of Statistics 2011, Vol. 39, No. 6, 3182-3210

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.1214/11-AOS932 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of

Scientific paper

10.1214/11-AOS932

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, $\sqrt{n}$-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.

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