Direction Identification and Minimax Estimation by Generalized Eigenvalue Problem in High Dimensional Sparse Regression

Sauvenier, Mathieu;Van Bellegem, Sébastien
(2023) , 53 pages

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In high-dimensional sparse linear regression, the selection and the estimation of the parameters are studied based on an L0−constraint on the direction of the vector of parameters. We first establish a general result for the direction of the vector of parameters, which is identified through the leading generalized eigenspace of measurable matrices. Based on this result, we suggest addressing the best subset selection problem from a new perspective by solving an empirical generalized eigenvalue problem to estimate the direction of the high-dimensional vector of parameters. We then study a new estimator based on the RIFLE algorithm and demonstrate a nonasymptotic bound of the L2 risk, the minimax convergence of the estimator and a central limit theorem. Simulations show the superiority of the proposed inference over some known l0 constrained estimators.
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Sauvenier, M., & Van Bellegem, S. (2023). Direction Identification and Minimax Estimation by Generalized Eigenvalue Problem in High Dimensional Sparse Regression (LIDAM Discussion Paper CORE 2023/05). https://hdl.handle.net/2078.5/102052