  In this paper we consider the dictionary learning problem for sparse
representation. We first show that this problem is NP-hard by polynomial time
reduction of the densest cut problem. Then, using successive convex
approximation strategies, we propose efficient dictionary learning schemes to
solve several practical formulations of this problem to stationary points.
Unlike many existing algorithms in the literature, such as K-SVD, our proposed
dictionary learning scheme is theoretically guaranteed to converge to the set
of stationary points under certain mild assumptions. For the image denoising
application, the performance and the efficiency of the proposed dictionary
learning scheme are comparable to that of K-SVD algorithm in simulation.
