Neurodynamic approaches with derivative feedback for sparse signal reconstruction

Abstract

This paper addresses the reconstruction issue of sparse signal by developing centralized and distributed neurodynamic approaches. An $l_{1}$-minimization problem can be employed for reconstructing sparse signal, and for solving this problem, a centralized neurodynamic approach with derivative feedback is designed. Considering the fact that the distributed approaches decompose the large-scale problem into small-scale one without central processing node in centralized ones, it effectively reduces the computational complexity of a single node. According to the distributed consensus and graph theory, the original $l_{1}$-minimization problem can be equivalently transformed as a distributed optimization problem. Then, based on our proposed centralized approach, a distributed neurodynamic approach with derivative feedback is further proposed. Through the convex optimization theory and Lyapunov method, we indicate that the optimal solution of $l_{1}$-minimization problem is equivalent to the equilibrium point of centralized or distributed approach, and that each neurodynamic approach globally converges to its equilibrium point. Furthermore, by comparing with several state-of-the-art neurodynamic approaches, our proposed approaches demonstrate their effectiveness and superiority by simulation results in reconstructing sparse signals and images.