Penalized interaction estimation for ultrahigh dimensional quadratic regression

Quadratic regressions extend linear models by simultaneously including the main effects and the interactions between the covariates. As such, estimating interactions in high-dimensional quadratic regressions has received extensive attention. Here, we introduce a novel method that allows us to estimate the main effects and the 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 formulae and obey the invariance principle at the population level. We estimate the interactions in matrix form under a 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 alternating direction method of multipliers algorithm to implement the penalized estimation. This algorithm fully exploits the cheap computational cost of the matrix multiplication and is much more efficient than existing penalized methods, such as the all-pairs LASSO. We demonstrate the promising performance of the proposed method using extensive numerical studies.