The adaptive and the thresholded Lasso for potentially misspecified models

Abstract

We revisit the adaptive Lasso as well as the thresholded Lasso with refitting, in a high-dimensional linear model, and study prediction error, ${\ell }_q$-error ($q\in \{1,2\}$), and number of false positive selections. Our theoretical results for the two methods are, at a rather fine scale, comparable. The differences only show up in terms of the (minimal) restricted and sparse eigenvalues, favoring thresholding over the adaptive Lasso. As regards prediction and estimation, the difference is virtually negligible, but our bound for the number of false positives is larger for the adaptive Lasso than for thresholding. Moreover, both these two-stage methods add value to the one-stage Lasso in the sense that, under appropriate restricted and sparse eigenvalue conditions, they have similar prediction and estimation error as the one-stage Lasso, but substantially less false positives.

arXiv comment45 pages

References in corpus (5)

• Nearly unbiased variable selection under minimax concave penalty
• Simultaneous analysis of Lasso and Dantzig selector
• On the conditions used to prove oracle results for the Lasso
• The Dantzig selector and sparsity oracle inequalities
• Thresholded Lasso for high dimensional variable selection and statistical estimation

Cited by in corpus (4)

• High-dimensional covariance estimation based on Gaussian graphical models
• Thresholded Lasso for high dimensional variable selection and statistical estimation
• Sharp Support Recovery from Noisy Random Measurements by L1 minimization
• Support estimation in high-dimensional heteroscedastic mean regression