## The Annals of Statistics

- Ann. Statist.
- Volume 39, Number 6 (2011), 2795-3443

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

Bing Li, Andreas Artemiou, and Lexin Li

#### 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.

#### Article information

**Source**

Ann. Statist. Volume 39, Number 6 (2011), 3182-3210.

**Dates**

First available: 5 March 2012

**Permanent link to this document**

http://projecteuclid.org/euclid.aos/1330958677

**Digital Object Identifier**

doi:10.1214/11-AOS932

**Zentralblatt MATH identifier**

06075594

**Mathematical Reviews number (MathSciNet)**

MR3012405

**Subjects**

Primary: 62-09: Graphical methods 62G08: Nonparametric regression 62H12: Estimation

**Keywords**

Contour regression invariant kernel inverse regression principal components reproducing kernel Hilbert space support vector machine

#### Citation

Li, Bing; Artemiou, Andreas; Li, Lexin. Principal support vector machines for linear and nonlinear sufficient dimension reduction. The Annals of Statistics 39 (2011), no. 6, 3182--3210. doi:10.1214/11-AOS932. http://projecteuclid.org/euclid.aos/1330958677.