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 ${\ell }_p(1\le p<\infty )$-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.