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The Two Stage l1l_1 Approach to the Compressed Sensing Problem

Abstract

This paper gives new results on the recovery of sparse signals using l1l_1-norm minimization. We introduce a two-stage l1l_1 algorithm. The first step consists of the standard 1\ell_1 relaxation. The second step consists of optimizing the 1\ell_1 norm of a subvector whose components are indexed by the ρm\rho m largest components in the first stage. If ρ\rho is set to 14\frac14, an intuitive choice motivated by the fact that m4\frac{m}4 is an empirical breakdown point for the plain 1\ell_1 exact recovery probability curve, Monte Carlo simulations show that the two-stage 1\ell_1 method outperforms the plain 1\ell_1.

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