Simultaneous codeword optimization (SimCO) for dictionary learning

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. In the literature, there are two benchmark mechanisms to update a dictionary. The first approach, for example the MOD algorithm, is characterized by searching for the optimal codewords while fixing the sparse coefficients. In the second approach, represented by the K-SVD method, one codeword and the related sparse coefficients are simultaneously updated while all other codewords and coefficients remain unchanged. We propose a novel framework that generalizes the aforementioned two methods. The unique feature of our approach is that one can update an arbitrary set of codewords and the corresponding sparse coefficients simultaneously: when sparse coefficients are fixed, the underlying optimization problem is the same as that in the MOD algorithm; when only one codeword is selected for update, it can be proved that the proposed algorithm is equivalent to the K-SVD method; and more importantly, our method allows to update all codewords and all sparse coefficients simultaneously, hence the term simultaneously codeword optimization (SimCO). Under the proposed framework, we design two algorithms, namely the primitive and regularized SimCO. Simulations demonstrate that our approach excels the benchmark K-SVD in terms of both learning performance and running speed.