Abstract: With wide-spread real-world applications, low-rank and sparse matrix recovery, where the concerned matrix with incomplete data is divided into a low-rank part and a sparse part, recently has attracted significant interest. To solve this structured non-convex optimization problem, we propose a non-monotone alternating Newton-like directional method which essentially updates two blocks of variables associated with the low-rank part using a single step of simple line-search along the Newton-like descent directions and another block of variables associated with the sparse part using a nonmonotone search. In particular, the non-monotone search technique helps our method find a better Newton-like descent direction in the next step. Moreover, we prove the global convergence of the proposed algorithm and discuss the iteration number in given precision under some mild conditions. Finally, computational results show the efficiency of the developed algorithm.

Key words: Low-rank and sparse, Matrix compressive recovery, Alternating Newton-like method, Non-monotone search