Distributed Continuous and Discrete Time Projection Neurodynamic Approaches for Sparse Recovery

Sparse representation acts as a fundamental data science methodology for solving a wide range of problems in machine learning and engineering. In this paper, we respectively propose novel distributed continuous-time and discrete-time projection neurodynamic approaches for sparse recovery by seeking the minimum <italic>l</italic><sub>1</sub>-norm solution with the undetermined linear measurement <inline-formula><tex-math notation="LaTeX">$Ax=b$</tex-math></inline-formula> in two cases over undirected networks. The proposed approaches only require the communication network to be undirected and connected, without the notion of central processing node, and no node visit the entire matrix <inline-formula><tex-math notation="LaTeX">$A$</tex-math></inline-formula>, so some privacy preserving efficiencies are guaranteed based on our approaches. First, for the one case that the rows of <inline-formula><tex-math notation="LaTeX">$A$</tex-math></inline-formula> are distributed, we propose a distributed continuous-time projection neurodynamic approach based on the projection operators and prove the global convergence and optimality of it. Then, a corresponding distributed discrete-time projection neurodynamic algorithm with a fixed step size is also presented, and the convergence and the scope of values of the step size of it are also analyzed. Second, for another case that the columns of <inline-formula><tex-math notation="LaTeX">$A$</tex-math></inline-formula> are distributed, a distributed continuous-time projection neurodynamic approach based on the projection operators and its derivative feedback is proposed and its global convergence is rigorously analyzed. Immediately following, we discuss its corresponding distributed discrete-time projection neurodynamic algorithm with a discrete version of the differential feedback term. Finally, the effectiveness and superiority of the proposed neurodynamic method are verified by the experiments of sparse signal and image reconstruction.