Abstract
We derive the l∞ 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.
Highlights
The Lasso is an l1 penalized least squares estimator in linear regression models proposed by Tibshirani [16]
Ritov and Tsybakov [1] have proved that the Dantzig selector of Candes and Tao [6] shares a lot of common properties with the Lasso
We show that under a sparsity scenario, it is possible to derive l∞ and sign consistency results even when the number of parameters is larger than the sample size
Summary
The Lasso is an l1 penalized least squares estimator in linear regression models proposed by Tibshirani [16]. The l2 consistency of Lasso with convergence rate has been proved in Bickel, Ritov and Tsybakov [1], Meinshausen and Yu [13], Zhang and Huang [20]. Ritov and Tsybakov [1] have proved that the Dantzig selector of Candes and Tao [6] shares a lot of common properties with the Lasso In particular they have shown simultaneous lp consistency with rates of the Lasso and Dantzig estimators for 1 p 2. We consider a high-dimensional linear regression model where the number of parameters can be much greater than the sample size. We show a sign concentration property of all the thresholded Lasso and Dantzig estimators simultaneously for a proper choice of the threshold if we assume that the non-zero components of the sparse target vector are large enough.
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