Low-rank and sparse structure pursuit via alternating minimization

In this paper, we present a nonconvex alternating minimization optimization algorithm for low-rank and sparse structure pursuit. Compared with convex relaxation based methods, the proposed algorithm is computationally more efficient for large scale problems. In our study, we define a notion of bounded difference of gradients, based on which we rigorously prove that with suitable initialization, the proposed nonconvex optimization algorithm enjoys linear convergence to the global optima and exactly recovers the underlying low rank and sparse matrices under standard conditions such as incoherence and sparsity conditions. For a wide range of statistical models such as multi-task learning and robust principal component analysis (RPCA), our algorithm provides a principled approach to learning the low rank and sparse structures with provable guarantee. Thorough experiments on both synthetic and real datasets backup our theory.