Sup-norm convergence rate and sign concentration property of Lasso and Dantzig estimators

TL;DR

This paper establishes the sup-norm convergence rates for Lasso and Dantzig estimators in high-dimensional linear regression, demonstrating their sign concentration properties under certain conditions on the design matrix and noise.

Contribution

It provides the first simultaneous derivation of $l_{}$ convergence rates and sign concentration properties for Lasso and Dantzig estimators under mutual coherence assumptions.

Findings

Derived $l_{}$ convergence rates for estimators.

Proved sign concentration property for thresholded estimators.

Validated results under Gaussian and finite variance noise assumptions.

Abstract

We derive the $l_{\infty}$ convergence rate simultaneously for Lasso and Dantzig estimators in a high-dimensional linear regression model under a mutual coherence assumption on the Gram matrix of the design and two different assumptions on the noise: Gaussian noise and general noise with finite variance. Then we prove that simultaneously the thresholded Lasso and Dantzig estimators with a proper choice of the threshold enjoy a sign concentration property provided that the non-zero components of the target vector are not too small.