Simultaneous Codeword Optimization (SimCO) for Dictionary Update and   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, such as 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 similar to 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 us to update all codewords and all sparse coefficients simultaneously, hence the term simultaneous codeword optimization (SimCO). Under the proposed framework, we design two algorithms, namely, primitive and regularized SimCO. We implement these two algorithms based on a simple gradient descent mechanism. Simulations are provided to demonstrate the performance of the proposed algorithms, as compared with two baseline algorithms MOD and K-SVD. Results show that regularized SimCO is particularly appealing in terms of both learning performance and running speed. 
Wei Dai, Tao Xu, Wenwu Wang
cs.LG cs.IT math.IT