On the convergence of FOCUSS algorithm for sparse representation

FOCal Underdetermined System Solver (FOCUSS) is a powerful tool for sparse representation and underdetermined inverse problems, which is extremely easy to implement. In this paper, we provide a comprehensive convergence analysis on the FOCUSS algorithm towards establishing a systematic convergence theory for it. First, we give a rigorous derivation for this algorithm exploiting the auxiliary function. Then, we prove its convergence. In particular, we systematically analyze its convergence rate for different sparsity parameter p and demonstrate its convergence rate by numerical experiments.