The Two Stage $l_1$ Approach to the Compressed Sensing Problem

This paper gives new results on the recovery of sparse signals using $l_1$-norm minimization. We introduce a two-stage $l_1$ algorithm. The first step consists of the standard $\ell_1$ relaxation. The second step consists of optimizing the $\ell_1$ norm of a subvector whose components are indexed by the $\rho m$ largest components in the first stage. If $\rho$ is set to $\frac14$, an intuitive choice motivated by the fact that $\frac{m}4$ is an empirical breakdown point for the plain $\ell_1$ exact recovery probability curve, Monte Carlo simulations show that the two-stage $\ell_1$ method outperforms the plain $\ell_1$.