  The sparsity of natural signals and images in a transform domain or
dictionary has been extensively exploited in several applications such as
compression, denoising and inverse problems. More recently, data-driven
adaptation of synthesis dictionaries has shown promise in many applications
compared to fixed or analytical dictionary models. However, dictionary learning
problems are typically non-convex and NP-hard, and the usual alternating
minimization approaches for these problems are often computationally expensive,
with the computations dominated by the NP-hard synthesis sparse coding step. In
this work, we investigate an efficient method for $\ell_{0}$ "norm"-based
dictionary learning by first approximating the training data set with a sum of
sparse rank-one matrices and then using a block coordinate descent approach to
estimate the unknowns. The proposed block coordinate descent algorithm involves
efficient closed-form solutions. In particular, the sparse coding step involves
a simple form of thresholding. We provide a convergence analysis for the
proposed block coordinate descent approach. Our numerical experiments show the
promising performance and significant speed-ups provided by our method over the
classical K-SVD scheme in sparse signal representation and image denoising.
