Penalized Interaction Estimation for Ultrahigh Dimensional Quadratic Regression

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