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

TL;DR

This paper establishes minimax rates for high-dimensional linear regression with additive errors under -losses, demonstrating the optimality of the proposed estimator for weakly sparse parameters.

Contribution

It derives matching lower and upper bounds for -losses, showing the estimator's minimax optimality in high-dimensional errors-in-variables models.

Findings

Proposes an estimator that achieves minimax optimal rates.

Provides tight bounds for the -losses in the model.

Validates the estimator's optimality through theoretical analysis.

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.