Distributed Inertial Proximal Neurodynamic Approach for Sparse Recovery on Directed Networks

This article investigates a fully distributed inertial neurodynamic approach for sparse recovery. The approach is based on proximal operators and inertia items. It aims to solve the
$L_{1}$
-norm minimization problem with consensus and linear observation constraints over directed communication networks. The proposed neurodynamic approach has the advantages of only requiring the communication network to be directed and weight-balanced, does not involve a central processing node and global parameters, which means that no single node can access the entire network and observe it at any time, so it is fully distributed. To effectively deal with the nonsmooth objective function,
$L_{1}$
-norm, the proximal operator method is used here. For efficiently handling linear observation and consensus constraints, a primal-dual method is applied to the inertial dynamic system. With the aid of maximal monotone operator theory and Baillon-Haddad lemmas, it reveals that the trajectories of our approach can converge to consensus solution at the optimal solution, provided that the distributed parameters satisfy technical conditions. In addition, we aim to demonstrate the weak convergence of the trajectories in our proposed neurodynamic approach toward the zeros of the optimal operator in Hilbert space, using Opial’s lemma. Finally, comparative experiments on sparse signal and image recovery confirm the efficiency and effectiveness of our proposed neurodynamic approach.