  In statistical learning framework with regressions, interactions are the
contributions to the response variable from the products of the explanatory
variables. In high-dimensional problems, detecting interactions is challenging
due to combinatorial complexity and limited data information. We consider
detecting interactions by exploring their connections with the principal
Hessian matrix. Specifically, we propose a one-step synthetic approach for
estimating the principal Hessian matrix by a penalized M-estimator. An
alternating direction method of multipliers (ADMM) is proposed to efficiently
solve the encountered regularized optimization problem. Based on the sparse
estimator, we detect the interactions by identifying its nonzero components.
Our method directly targets at the interactions, and it requires no structural
assumption on the hierarchy of the interaction effects. We show that our
estimator is theoretically valid, computationally efficient, and practically
useful for detecting the interactions in a broad spectrum of scenarios.
