A fast algorithm for general matrix factorization

Matrix factorization algorithms are emerging as popular tools in many applications, especially dictionary learning method for recovering biomedical image data from noisy and ill-conditioned measurements. We introduce a novel dictionary learning algorithm based on augmented Lagrangian (AL) approach to learn dictionaries from exemplar data and it can be extended to general matrix factorization problems due to different constraints. Specically, we use the alternating minimization strategy to decouple the dictionary learning scheme into three main subproblems, which can be solved efficiently. The proposed algorithm can accommodate arbitrary priors on the dictionary, which enables us to inject prior information into the learning process. We validate the algorithm using simulated data and demonstrate its utility in the context of denoising. Comparisons with existing methods show a considerable speedup over other methods. More importantly, we observe that the proposed algorithm is able to recover the dictionaries correctly, even at high sparsity levels and is relatively insensitive to initialization.