Structured variable selection in support vector machines

TL;DR

This paper introduces a structured variable selection framework for SVMs that enforces sparsity and heredity principles, improving interpretability and efficiency in high-dimensional classification tasks.

Contribution

It develops a linear programming approach to incorporate heredity constraints into SVMs and extends the framework to nonparametric settings.

Findings

The method respects heredity principles while maintaining sparsity.

Efficient linear programming formulation for structured SVMs.

Successful application to simulated and real datasets.

Abstract

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.