A global optimization approach for rational sparsity promoting criteria

We consider the problem of recovering an unknown signal observed through a nonlinear model and corrupted with additive noise. More precisely, the nonlinear degradation consists of a convolution followed by a nonlinear rational transform. As a prior information, the original signal is assumed to be sparse. We tackle the problem by minimizing a least-squares fit criterion penalized by a Geman-McClure like potential. In order to find a globally optimal solution to this rational minimization problem, we transform it in a generalized moment problem, for which a hierarchy of semidefinite programming relaxations can be used. To overcome computational limitations on the number of involved variables, the structure of the problem is carefully addressed, yielding a sparse relaxation able to deal with up to several hundreds of optimized variables. Our experiments show the good performance of the proposed approach.