A constrained optimization approach for complex sparse perturbed models

In this paper, we consider the problem of estimating a complex-valued signal having a sparse representation in an uncountable family of vectors. The available observations are corrupted with an additive noise and the elements of the dictionary are parameterized by a scalar real variable. By a linearization technique, the original model is recast as a constrained sparse perturbed model. An optimization approach is then proposed to estimate the parameters involved in this model. The cost function includes an arbitrary Lipschitz differentiable data fidelity term accounting for the noise statistics, and an l0 penalty. A forward-backward algorithm is employed to solve the resulting non-convex and non-smooth minimization problem. This algorithm can be viewed as a generalization of an iterative hard thresholding method and its local convergence can be established. Simulation results illustrate the good practical performance of the proposed approach when applied to spectrum estimation