  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.
