Minimax rates of $\ell_p$-losses for high-dimensional linear regression models with additive measurement errors over $\ell_q$-balls

Abstract

We study minimax rates for high-dimensional linear regression with additive errors under the $\ell_p\ (1\leq p<\infty)$-losses, where the regression parameter is of weak sparsity. Our lower and upper bounds agree up to constant factors, implying that the proposed estimator is minimax optimal.