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Title: Penalized interaction estimation for ultrahigh dimensional quadratic regression Authors:  Binyan Jiang - The Hong Kong Polytechnic University (Hong Kong) [presenting]
Abstract: Quadratic regression goes beyond 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. We introduce a novel method which allows us to estimate the main effects and interactions separately. Unlike existing methods for ultrahigh dimensional quadratic regressions, the proposal does not require the widely used heredity assumption. In addition, the 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 hence is much more efficient than existing penalized methods under heredity constraints. We demonstrate the promising performance of our proposal through extensive numerical studies.