Iterative ℓ1 minimization for non-convex compressed sensing

An algorithmic framework, based on the difference of convex functions algorithm (DCA), is proposed for minimizing a class of concave sparse metrics for compressed sensing problems. The resulting algorithm iterates a sequence of ℓ1 minimization problems. An exact sparse recovery theory is established to show that the proposed framework always improves on the basis pursuit (ℓ1 minimization) and inherits robustness from it. Numerical examples on success rates of sparse solution recovery illustrate further that, unlike most existing non-convex compressed sensing solvers in the literature, our method always out-performs basis pursuit, no matter how ill-conditioned the measurement matrix is. Moreover, the iterative ℓ1 (IL1) algorithm lead by a wide margin the state-of-the-art algorithms on ℓ1/2 and logarithimic minimizations in the strongly coherent (highly ill-conditioned) regime, despite the same objective functions. Last but not least, in the application of magnetic resonance imaging (MRI), IL1 algorithm easily recovers the phantom image with just 7 line projections.