## Principal support vector machines for linear and nonlinear sufficient dimension reduction.(English)Zbl 1246.62153

Summary: 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 machines 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, $$\surd$$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.

### MSC:

 62H25 Factor analysis and principal components; correspondence analysis 68T05 Learning and adaptive systems in artificial intelligence 46N30 Applications of functional analysis in probability theory and statistics 62A09 Graphical methods in statistics 62G08 Nonparametric regression and quantile regression 62H12 Estimation in multivariate analysis

Kernlab
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