Alternating Direction Method of Multipliers for Nonconvex Background/Foreground Extraction

In this paper, we study a general optimization model for extracting background and foreground from a surveillance video. This model covers a large class of existing models as special cases. In particular, it can be nuclear-norm-free, and can incorporate different possibly nonconvex sparsity inducing regularization functions for extracting the foreground, such as the $\ell_p$ quasi-norm for $0<p<1$. To solve the resulting possibly nonconvex optimization problem, we adapt the alternating direction method of multipliers (ADMM) with a general dual step-size to solve a reformulation that contains three blocks of variables, and analyze its convergence. We show that for any dual step-size less than the golden ratio, there exists a computable threshold such that if the penalty parameter is chosen above such a threshold and the sequence thus generated by our ADMM is bounded, then the cluster point of the sequence gives a stationary point of the nonconvex optimization problem. We achieve this via a potential function specifically constructed for our ADMM. Moreover, we establish the global convergence of the whole sequence generated if, in addition, this special potential function is a Kurdyka-{\L}ojasiewicz function. Furthermore, we present a simple strategy for initializing the algorithm to guarantee boundedness of the sequence generated. Finally, we perform numerical experiments comparing our ADMM with the proximal alternating linearized minimization (PALM) proposed in \cite{bst2014} on real data. The numerical results show that our ADMM with a dual step-size smaller than 1 performs better in the sense that it uses less time to solve all test problems.