Quadratic regression goes beyond the linear model by simultaneously including main effects and interactions between the covariates. The problem of interaction estimation in high dimensional quadratic regression has received extensive attention in the past decade. In this article we introduce a novel method which allows us to estimate the main effects and interactions separately. Unlike existing methods for ultrahigh dimensional quadratic regressions, our proposal does not require the widely used heredity assumption. In addition, our proposed estimates have explicit formulas and obey the invariance principle at the population level. We estimate the interactions of matrix form under penalized convex loss function. The resulting estimates are shown to be consistent even when the covariate dimension is an exponential order of the sample size. We develop an efficient ADMM algorithm to implement the penalized estimation. This ADMM algorithm fully explores the cheap computational cost of matrix multiplication and is much more efficient than existing penalized methods such as all pairs LASSO. We demonstrate the promising performance of our proposal through extensive numerical studies.

中文翻译：

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超高维二次回归的惩罚交互估计

二次回归超越了线性模型，同时包含了协变量之间的主效应和相互作用。高维二次回归中的交互估计问题在过去十年中受到广泛关注。在本文中，我们介绍了一种新方法，该方法使我们能够分别估计主效应和交互作用。与现有的超高维二次回归方法不同，我们的提议不需要广泛使用的遗传假设。此外，我们提出的估计有明确的公式，并且在总体水平上遵循不变性原则。我们估计惩罚凸损失函数下矩阵形式的相互作用。即使协变量维度是样本大小的指数级，结果估计也显示为一致的。我们开发了一种有效的 ADMM 算法来实现惩罚估计。这种 ADMM 算法充分探索了矩阵乘法的廉价计算成本，并且比现有的惩罚方法（例如所有对 LASSO）更有效。我们通过广泛的数值研究证明了我们提案的有希望的性能。