Computing Sparse Representation in a Highly Coherent Dictionary Based on Difference of L1 and L2

We study analytical and numerical properties of the L1−L2 minimization problem for sparse representation of a signal over a highly coherent dictionary. Though the L1 −L2 metric is non-convex, it is Lipschitz continuous. The difference of convex algorithm (DCA) is readily applicable for computing the sparse representation coefficients. The L1 minimization appears as an initialization step of DCA. We further integrate DCA with a non-standard simulated annealing (SA) methodology to approximate globally sparse solutions. Non-Gaussian random perturbations are more effective than standard Gaussian perturbations for improving sparsity of solutions. In numerical experiments, we conduct an extensive comparison among sparse penalties such as L0, L1, Lp for p ∈ (0, 1) based on data from three specific applications (over-sampled discreet cosine basis, differential absorption optical spectroscopy, and image denoising) where highly coherent dictionaries arise. We find numerically that the L1 − L2 minimization persistently produces better results than L1 minimization, especially when the sensing matrix is ill-conditioned. In addition, the DCA method outperforms many existing algorithms for other nonconvex metrics. ∗Department of Mathematics, UC Irvine, Irvine, CA 92697, USA. The work was partially supported by NSF grants DMS0928427 and DMS-1222507. Manuscript revised on July 19, 2014.