Sufficient dimension reduction via principal L$q$ support vector machine

Abstract

Principal support vector machine was proposed recently by Li, Artemiou and Li (2011) to combine L$1$ support vector machine and sufficient dimension reduction. We introduce the principal L$q$ support vector machine as a unified framework for linear and nonlinear sufficient dimension reduction. By noticing that the solution of L$1$ support vector machine may not be unique, we set $q>1$ to ensure the uniqueness of the solution. The asymptotic distribution of the proposed estimators are derived for $q> 1$. We demonstrate through numerical studies that the proposed L$2$ support vector machine estimators improve existing methods in accuracy, and are less sensitive to the tuning parameter selection.