Matrix Neurodynamic Approaches for Rank Minimization: Finite/Fixed-Time Convergence Technique

This article presents two innovative matrix neurodynamic approaches (MNAs) designed to tackle the rank minimization problem. First, by introducing the matrix norm-normalized sign function, two variants of MNA are developed: finite-time converging MNA (FINt-MNA) and fixed-time converging MNA (FIXt-MNA). Then, the proposed approaches are shown to guarantee the existence and uniqueness of solutions, and based on Lyapunov analysis, it is demonstrated that the proposed approaches converge to the optimal solution within FINt and FIXt. In addition, upper bounds on the settling time are determined using finite-time and fixed-time lemmas, with subsequent analysis examining the influence of tunable parameters on these bounds for the two approaches through the control variable method. Finally, numerical examples and an image completion experiment confirm the effectiveness and superiority of the proposed approaches compared with the existing MNA and two classical approaches.