Model Selection for High-Dimensional Quadratic Regression via Regularization

ABSTRACT

Quadratic regression (QR) models naturally extend linear models by considering interaction effects between the covariates. To conduct model selection in QR, it is important to maintain the hierarchical model structure between main effects and interaction effects. Existing regularization methods generally achieve this goal by solving complex optimization problems, which usually demands high computational cost and hence are not feasible for high-dimensional data. This article focuses on scalable regularization methods for model selection in high-dimensional QR. We first consider two-stage regularization methods and establish theoretical properties of the two-stage LASSO. Then, a new regularization method, called regularization algorithm under marginality principle (RAMP), is proposed to compute a hierarchy-preserving regularization solution path efficiently. Both methods are further extended to solve generalized QR models. Numerical results are also shown to demonstrate performance of the methods. Supplementary materials for this article are available online.

KEYWORDS:

• Generalized quadratic regression
• Interaction selection
• LASSO
• Marginality principle
• Variable selection

Acknowledgments

Ning Hao and Yang Feng contribute equally to this work. The authors are grateful to the editor, AE, and referees for their helpful suggestions.

Additional information

Funding

The authors are partially supported by NSF grants DMS-1309507 (Hao and Zhang), DMS-1308566 and DMS-1554804 (Feng), DMS-1418172 and NSFC 11571009 (Zhang).