Sparse Signal Recovery via Generalized Entropy Functions Minimization

Compressive sensing relies on the sparse prior imposed on the signal to solve the ill-posed recovery problem in an under-determined linear system. The objective function that enforces the sparse prior information should be both effective and easily optimizable. Motivated by the entropy concept from information theory, in this paper we propose the generalized Shannon entropy function and Rényi entropy function of the signal as the sparsity promoting objectives. Both entropy functions are nonconvex, and their local minimums only occur on the boundaries of the orthants in the Euclidean space. Compared to other popular objective functions such as the ‖x‖1, ‖x‖p, minimizing the proposed entropy functions not only promotes sparsity in the recovered signals, but also encourages the signal energy to be concentrated towards a few significant entries. The corresponding optimization problem can be converted into a series of reweighted l1 minimization problems and solved efficiently. Sparse signal recovery experiments on both the simulated and real data show the proposed entropy function minimization approaches are better than other popular approaches and achieve state-of-the-art performances.