Convex geometry of the Coding problem for error constrained Dictionary Learning

In this article we expose the convex geometry of the class of coding problems that includes the likes of Basis Pursuit Denoising. We propose a novel reformulation of the coding problem as a convex-concave min-max problem. This particular reformulation not only provides a nontrivial method to update the dictionary in order to obtain better sparse representations with hard error constraints, but also gives further insights into the underlying geometry of the coding problem. Our results shed provide pointers to new ascent-descent type algorithms that could be used to solve the coding problem.