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We consider the data-driven dictionary learning problem. The goal is to seek an over-complete dictionary from which every training signal can be best approximated by a linear combination of only a few codewords. This task is often achieved by iteratively executing two operations: sparse coding and dictionary update. The focus of this paper is on the dictionary update step, where the dictionary is optimized with a given sparsity pattern. We propose a novel framework where an arbitrary set of codewords and the corresponding sparse coefficients are simultaneously updated, hence the term simultaneous codeword optimization (SimCO). The SimCO formulation not only generalizes benchmark mechanisms MOD and K-SVD, but also allows the discovery that singular points, rather than local minima, are the major bottleneck of dictionary update. To mitigate the problem caused by the singular points, regularized SimCO is proposed. First and second order optimization procedures are designed to solve regularized SimCO. Simulations show that regularization substantially improves the performance of dictionary learning.
Date of Publication: Dec. 2012