Designing optimal sparsifying dictionary using first order series expansion

Dictionary learning aims to represent a given signal database Y as a product of a dictionary D and a sparse coefficient matrix X, as follows, Y ≃ DX with a sparseness constraint on X. Some existing dictionary learning methods, like MOD, K-SVD, minimize the representation error, resulting a non-convex cost function, SBJ [8] applied a first order series expansion for the product DX resulting a convex cost function, but is a poor approximation to the original cost function. In this paper, we propose a new measure to make sure the approximation is more accurate, and a closed-form solution is derived to solve the resulting problem. Simulation shows that our proposed algorithm outperforms the exiting ones at a price of computational load and convergence rate.