Algorithms for Multiple Basis Pursuit Denoising
We address the problem of learning a joint sparse approximation of several signals over a dictionary. We pose the problem as a matrix approximation problem with a row-sparsity inducing penalization on the coefficient matrix. We propose a simple algorithm based on iterative shrinking for solving the problem. At the present time, such a problem is solved either by using a Second-Order Cone programming or by means of a M-Focuss algorithm. While the former algorithm is computationally expensive, the latter is efficient but present some pitfalls like presences of fixed points which are undesiderable when solving a convex problem. By analyzing the optimality conditions of the problem, we derive a simple algorithm. The algorithm we propose is efficient and is guaranteed to converge to the optimal solution, up to a given tolerance. Furthermore, by means of a reweighted scheme, we are able to improve the sparsity of the solution.