  When applying the support vector machine (SVM) to high-dimensional
classification problems, we often impose a sparse structure in the SVM to
eliminate the influences of the irrelevant predictors. The lasso and other
variable selection techniques have been successfully used in the SVM to perform
automatic variable selection. In some problems, there is a natural hierarchical
structure among the variables. Thus, in order to have an interpretable SVM
classifier, it is important to respect the heredity principle when enforcing
the sparsity in the SVM. Many variable selection methods, however, do not
respect the heredity principle. In this paper we enforce both sparsity and the
heredity principle in the SVM by using the so-called structured variable
selection (SVS) framework originally proposed in Yuan, Joseph and Zou (2007).
We minimize the empirical hinge loss under a set of linear inequality
constraints and a lasso-type penalty. The solution always obeys the desired
heredity principle and enjoys sparsity. The new SVM classifier can be
efficiently fitted, because the optimization problem is a linear program.
Another contribution of this work is to present a nonparametric extension of
the SVS framework, and we propose nonparametric heredity SVMs. Simulated and
real data are used to illustrate the merits of the proposed method.
