A Sparse Recovery Algorithm Using the Neurodynamic System With Predefined Time Convergence

Compressed sensing aims to compress sparse signals losslessly into significantly smaller samples and recover them if needed. One of the most critical approaches to recovering the sparse signal is minimizing $l_{1}$ -norm, the convex relaxation of $l_{0}$ -norm with the limit of linear measurement. However, sometimes such as the recovery of piecewise constant sparse signal, the need for controlling the convergence time has become an inevitable problem to solve. To solve this problem, the predefined time pseudo-inverse locally competitive algorithm (PPLCA) based on a nonlinear neurodynamic system is proposed in this brief, which is exceptionally effective if the signal is piecewise constant. Further, under the restricted isometry property (RIP), the predefined-time convergence rate of the proposed PPLCA is proven using Lyapunov's stability theory. Finally, the numerical simulations show that the PPLCA could converge to the stationary point within the predefined time.