Thresholded Lasso for high dimensional variable selection

Given $n$ noisy samples with $p$ dimensions, where $n \ll p$, we show that the multi-step thresholding procedure based on the Lasso -- we call it the {\it Thresholded Lasso}, can accurately estimate a sparse vector $\beta \in {\mathbb R}^p$ in a linear model $Y = X \beta + \epsilon$, where $X_{n \times p}$ is a design matrix normalized to have column $\ell_2$-norm $\sqrt{n}$, and $\epsilon \sim N(0, \sigma^2 I_n)$. We show that under the restricted eigenvalue (RE) condition, it is possible to achieve the $\ell_2$ loss within a logarithmic factor of the ideal mean square error one would achieve with an {\em oracle } while selecting a sufficiently sparse model -- hence achieving {\it sparse oracle inequalities}; the oracle would supply perfect information about which coordinates are non-zero and which are above the noise level. We also show for the Gauss-Dantzig selector (Cand\`{e}s-Tao 07), if $X$ obeys a uniform uncertainty principle, one will achieve the sparse oracle inequalities as above, while allowing at most $s_0$ irrelevant variables in the model in the worst case, where $s_0 \leq s$ is the smallest integer such that for $\lambda = \sqrt{2 \log p/n}$, $\sum_{i=1}^p \min(\beta_i^2, \lambda^2 \sigma^2) \leq s_0 \lambda^2 \sigma^2$. Our simulation results on the Thresholded Lasso match our theoretical analysis excellently.