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

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.

Published in at http://dx.doi.org/10.1214/08-EJS177 the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (6)

• Lasso-type recovery of sparse representations for high-dimensional data
• High-dimensional generalized linear models and the lasso
• Sparsity oracle inequalities for the Lasso
• Aggregation for Gaussian regression
• The Dantzig selector and sparsity oracle inequalities
• Consistent selection via the Lasso for high dimensional approximating regression models

Cited by in corpus (1)

• Honest variable selection in linear and logistic regression models via ${\ell }_1$ and ${\ell }_1+{\ell }_2$ penalization