Direction Identification and Minimax Estimation in High-Dimensional Sparse Regression via a Generalized Eigenvalue Approach

In high-dimensional (HD) sparse linear regression, parameter selection and estimation are addressed using a constraint l₀ on the direction of the parameter vector. We begin by establishing a general result that identifies this direction through the leading generalized eigenspace of specific measurable matrices. Using this result, we propose a novel approach to the selection of the best subsets by solving an empirical generalized eigenvalue problem to estimate the direction of the HD parameter. We then introduce a new estimator based on the RIFLE algorithm, providing a non-asymptotic bound for the estimation risk, minimax convergence, and a central limit theorem. Simulations demonstrate the superiority of our method over existing l₀ -constrained estimators.