the support set under the RIP type of conditions, which we are interested in here.

Although it is possible to achieve exact recovery using post-processing by thresholding the small coefficients of Lasso solution, this method is suboptimal under RIP in comparison to the $L_0$ regularization method (\[eq:L0\]) because it requires the smallest nonzero coefficients to be $\sqrt{\bar{k}}$ times larger than the noise level instead of only requiring the nonzero coefficients to be larger than the noise level with $L_0$ regularization in (\[eq:L0\]). This issue, referred to as the [*bias*]{} of Lasso for feature selection, was extensively discussed in [@Zhang10-mc+]. Detailed discussion can be found after Theorem \[thm:multi-stage-featsel\]. It is worth mentioning that under a stronger mutual coherence condition (similar to irrepresentable condition), this post-processing step does not give this bias factor $\sqrt{\bar{k}}$ as shown in [@Lounici08] (also see [@Zhang07-l1]). Therefore the advantage of bias removal for the multi-stage procedure discussed here is only applicable when RIP holds but when the irrepresentable condition and mutual incoherence conditions fail. A thorough discussion of various conditions is beyond the scope of the current paper, and we would like to refer the readers to [@GeerBuhlmann09-conditions]. Nevertheless, it is worth pointing out that even in 