Minimization of ℓ1-2 for Compressed Sensing

We study minimization of the difference of l1 and l2 norms as a non-convex and Lipschitz continuous metric for solving constrained and unconstrained compressed sensing problems. We establish exact (stable) sparse recovery results under a restricted isometry property (RIP) condition for the constrained problem, and a full-rank theorem of the sensing matrix restricted to the support of the sparse solution. We present an iterative method for l1 − l2 minimization based on the difference of convex algorithm (DCA), and prove that it converges to a stationary point satisfying first order optimality condition. We propose a sparsity oriented simulated annealing (SA) procedure with non-Gaussian random perturbation and prove the almost sure convergence of the combined algorithm (DCASA) to a global minimum. Computation examples on success rates of sparse solution recovery show that if the sensing matrix is ill-conditioned (non RIP satisfying), then our method is better than existing non-convex compressed sensing solvers in the literature. Likewise in the magnetic resonance imaging (MRI) phantom image recovery problem, l1 − l2 succeeds with 8 projections. Irrespective of the conditioning of the sensing matrix, l1 − l2 is better than l1 in both the sparse signal and the MRI phantom image recovery problems.