A Fast Algorithm for Learning Overcomplete Dictionary for Sparse Representation Based on Proximal Operators

We present a fast, efficient algorithm for learning an overcomplete dictionary for sparse representation of signals. The whole problem is considered as a minimization of the approximation error function with a coherence penalty for the dictionary atoms and with the sparsity regularization of the coefficient matrix. Because the problem is nonconvex and nonsmooth, this minimization problem cannot be solved efficiently by an ordinary optimization method. We propose a decomposition scheme and an alternating optimization that can turn the problem into a set of minimizations of piecewise quadratic and univariate subproblems, each of which is a single variable vector problem, of either one dictionary atom or one coefficient vector. Although the subproblems are still nonsmooth, remarkably they become much simpler so that we can find a closed-form solution by introducing a proximal operator. This leads to an efficient algorithm for sparse representation. To our knowledge, applying the proximal operator to the problem with an incoherence term and obtaining the optimal dictionary atoms in closed form with a proximal operator technique have not previously been studied. The main advantages of the proposed algorithm are that, as suggested by our analysis and simulation study, it has lower computational complexity and a higher convergence rate than state-of-the-art algorithms. In addition, for real applications, it shows good performance and significant reductions in computational time.