the classical $d<n$ setting with the design matrix $X$ being rank $d$, the irrepresentable condition or the mutual incoherence condition can still be violated while the RIP type sparse-eigenvalue condition used in this paper holds trivially. In fact, this was pointed out in [@Zou06] as the main motivation of adaptive Lasso. Adaptive Lasso behaves similarly to the above mentioned post-processing, and thus suffers from the same bias problem.

The bias of Lasso is due to the looseness of convex relaxation for $L_0$ regularization. Therefore the remedy is to use a non-convex regularizer that is close to $L_0$ regularization. One drawback of using nonconvex optimization formulation is that we can only find a local optimal solution and different computational procedure may lead to a different local solution. Therefore the theoretical analysis has to be integrated with specific computational procedure to show that the local minimum obtained by the procedure has desirable properties (e.g., exact support recovery). Several nonconvex computational procedures have been analyzed in the literature, including an adaptive forward backward greedy procedure (referred to as FoBa) to approximately solve the regularization method (\[eq:L0\]) considered in [@Zhang08-foba], and the MC+ method in [@Zhang10-mc+] to solve a non-convex regularized problem using a 