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Quadratic regression goes beyond the linear model by simultaneously includingmain effects and interactions between the covariates. The problem ofinteraction estimation in high dimensional quadratic regression has receivedextensive attention in the past decade. In this article we introduce a novelmethod which allows us to estimate the main effects and interactionsseparately. Unlike existing methods for ultrahigh dimensional quadraticregressions, our proposal does not require the widely used heredity assumption.In addition, our proposed estimates have explicit formulas and obey theinvariance principle at the population level. We estimate the interactions ofmatrix form under penalized convex loss function. The resulting estimates areshown to be consistent even when the covariate dimension is an exponentialorder of the sample size. We develop an efficient ADMM algorithm to implementthe penalized estimation. This ADMM algorithm fully explores the cheapcomputational cost of matrix multiplication and is much more efficient thanexisting penalized methods such as all pairs LASSO. We demonstrate thepromising performance of our proposal through extensive numerical studies.

Penalized Interaction Estimation for Ultrahigh Dimensional Quadratic Regression