A family of the subgradient algorithm with several cosparsity inducing functions to the cosparse recovery problem

• The hot cosparse analysis model is discussed.
• A new alternative way replacing the l0-norm based on the cosparsity inducing function is presented.
• A new constrained optimal model and a family of subgradient algorithm for the cosparse recovery problem is given.
• The convergence analysis of the proposed algorithm is given.
• Simulations on the recovering of the unknown signal indicate its better performance.

In the past decade, there has been a great interest in the sparse synthesis model for signal. The researchers have obtained a series of achievements about the sparse representation. The cosparse analysis model as the corresponding version of the sparse synthesis model has drawn much attention in recent years. Many approaches have been proposed to solve this model. In some conventional general, these methods usually relaxed l0-norm to l1-norm or l2-norm to represent the cospasity of signal, from which some reasonable algorithms have been developed. Furthermore, this work will present a new alternative way to replace the l0-norm based on the cosparsity inducing function, which is closer to l0-norm than l1-norm and l2-norm. Based on this function, we firstly construct the objective function and give a constrained optimal model of the cosparse recovery problem. Then we propose a subgradient algorithm – cosparsity inducing function (CIF) algorithm, which belongs to a two-layer optimization algorithm. Specifically, through converting the constrained optimal problem into the unconstrained case, we firstly obtain a temporary optimal variable, in which the cosparsity inducing function is approximated using its local linear approximation in order to avoid its nonconvex property. Secondly, a new cosupport is given by projecting the temporary optimal variable into the cosparse subspace and then keeping the l smallest elements. Besides, the desired signal is estimated using a conjugate gradient algorithm on the new cosupport. Moreover, we study the relative theoretical analysis about CIF algorithm. Simulations on the recovering of the unknown signal in the cosparse analysis model indicate its better performance at last.