High-dimensional Interactions Detection with Sparse Principal Hessian Matrix
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In statistical methods, interactions are the contributions from the products of the predictor variables to the response variable. In high-dimensional problems, detecting interactions is challenging due to the combinatorial complexity in conjunction with limited data information. We consider detecting interactions by exploring their connections with the principal Hessian matrix. To handle the challenging high dimensionality, a one-step synthetic approach is proposed to estimate the principal Hessian matrix by a penalized M-estimator. An alternating direction method of multipliers (ADMM) is proposed to efficiently solve the encountered regularized optimizations. Based on the sparse estimator, we propose a new method to detect the interactions by identifying the nonzero components. Our method directly targets at the interactions, and it requires no structural assumption on the hierarchy between the interactions effects and those from the original predictor variables. We show that our estimator is theoretically valid, computationally efficient, and practically useful for detecting the interactions in a broad spectrum of scenarios.
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