Matrix coherency graph: A tool for improving sparse coding performance

Exact recovery of a sparse solution for an underdetermined system of linear equations implies full search among all possible subsets of the dictionary, which is computationally intractable, while ℓ
₁
minimization will do the job when a Restricted Isometry Property holds for the dictionary. Yet, practical sparse recovery algorithms may fail to recover the vector of coefficients even when the dictionary deviates from the RIP only slightly. To enjoy ℓ
₁
minimization guarantees in a wider sense, a method based on a combination of full-search and ℓ
₁
minimization is presented. The idea is based on partitioning the dictionary into atoms which are in some sense well-conditioned and those which are ill-conditioned. Inspired by that, a matrix coherency graph is introduced which is a tool extracted from the structure of the dictionary. This tool can be used for decreasing the greediness of sparse coding algorithms so that recovery will be more reliable. We have modified the IRLS algorithm by applying the proposed method on it and simulation results show that the modified version performs quite better than the original algorithm.