标题： 高维二次回归的正则化模型选择 显示英文标题

标题： Model Selection for High Dimensional Quadratic Regression via Regularization

摘要： 二次回归（QR）模型通过考虑协变量之间的交互效应自然地扩展了线性模型。 为了在QR中进行模型选择，保持主效应和交互效应之间的分层模型结构是很重要的。 现有的正则化方法通常通过解决复杂的优化问题来实现这一目标，但这些方法通常需要较高的计算成本，因此对于高维数据来说并不实用。 本文专注于高维QR中可扩展的正则化方法来进行模型选择。 我们首先考虑两阶段正则化方法，并建立了两阶段最小绝对收缩与选择算子（LASSO）的理论性质。 然后，提出了一种新的正则化方法，称为基于边缘原则的正则化算法（RAMP），以高效地计算保持层次结构的正则化解路径。 这两种方法还进一步被推广到求解广义QR模型。 数值结果也展示了这些方法的表现。 显示英文摘要

摘要： 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 paper 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.