  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 give a comprehensive convergence analysis on the
FOCUSS algorithm towards establishing a systematic convergence theory by
providing three primary contributions as follows. First, we give a rigorous
derivation for this algorithm exploiting the auxiliary function. Then, we prove
its convergence. Third, we systematically study its convergence rate with
respect to the sparsity parameter p and demonstrate its convergence rate by
numerical experiments.
