A Two-Layer Distributed Algorithm Using Neurodynamic System for Solving L 1-Minimization

Abstract

This brief considers a distributed algorithm for solving <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${L_{1}}$ </tex-math></inline-formula> -minimization problem based on nonlinear neurodynamic system. Compared with centralized algorithms, distributed algorithms have great potential in data privacy protection, distributed storage and processing of data. In this brief, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${L_{1}}$ </tex-math></inline-formula> -minimization problem is transformed into a distributed problem by using multiagent consensus theory. For the distributed optimization problem, a two-layer distributed algorithm is designed by utilizing neurodynamic system, projection matrix and derivative feedback technique. Compared with the existing distributed neurodynamic algorithm, the proposed algorithm has a simpler structure and has fewer neurons on the premise that the calculation error does not increase. Besides, the proposed algorithm converges to a minimal point of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${L_{1}}$ </tex-math></inline-formula> -minimization problem and is Lyapunov stable. Finally, the comparative examples of sparse signal reconstruction show that the proposed distributed algorithm is effective and superior.