Proximal Alternating Minimization for Analysis Dictionary Learning and Convergence Analysis

The sparse analysis model is an alternative approach to the sparse synthesis model that has emerged recently. Most analysis dictionary learning problems based on the sparse analysis model require solving a class of challenging nonsmooth and even nonconvex optimization problems. Despite the fact that many numerical methods have been developed for solving these problems, it remains an open problem to find a numerical method that is not only empirically fast, but also has mathematically guaranteed strong convergence. In this paper, to promote stronger sparsity in solutions than the ℓ
₁
-norm, we employ the nonsmooth and nonconvex ℓ
_1/2
-norm as a regularizer, and then we propose to use the proximal alternating minimization scheme for solving the nonconvex and nonsmooth problem, leading to an efficient and fast algorithm. More importantly, a rigorous convergence analysis shows that the proposed method satisfies the global convergence property: The whole sequence of iterates is convergent and converges to a critical point. The proposed algorithm has two stages: the analysis sparse-coding stage and the analysis dictionary-update stage. Besides the theoretical soundness, the practical benefit of the proposed method is validated by the synthetic and real-world data. Experiments show that the proposed method achieves better results with faster convergence compared to the state-of-the-art algorithms.