Incoherent dictionary learning with log-regularizer based on proximal operators

In this study, we propose a novel dictionary learning algorithm with the log-regularizer and simultaneously with the coherence penalty based on proximal operators. Our proposed algorithm simply employs a decomposition scheme and alternating optimization, which transforms the overall problem into a set of single-vector variable subproblems, with either one dictionary atom or one coefficient vector. Although the subproblems are still nonsmooth and even nonconvex, remarkably they can be solved by proximal operators, and the closed-form solutions of the dictionary atoms and the coefficient vectors are obtained directly and explicitly. To the best of our knowledge, no previous studies of dictionary learning have applied proximal operators to sparse coding with the log-regularizer and simultaneously to dictionary updating with the coherence penalty. According to our analysis and simulation study, the main advantages of the proposed algorithm are its greater ability of recovering the dictionary and its faster convergence for reaching the values of the dictionary recovery ratios than state-of-the-art algorithms. In addition, for real-world applications, our proposed algorithm can obtain good performances on audio data and image classification.