A continuous-time neurodynamic approach in matrix form for rank minimization

Abstract

This article proposes a continuous-time neurodynamic approach for solving the rank minimization under affine constraints. As opposed to the traditional neurodynamic approach, the proposed neurodynamic approach extends the form of the variables from the vector form to the matrix form. First, a continuous-time neurodynamic approach with variables in matrix form is developed by combining the optimal rank r projection and the gradient. Then, the optimality of the proposed neurodynamic approach is rigorously analyzed by demonstrating that the objective function satisfies the functional property which is called as (2r,4r)-restricted strong convexity and smoothness ((2r,4r)-RSCS). Furthermore, the convergence and stability analysis of the proposed neurodynamic approach is rigorously conducted by establishing appropriate Lyapunov functions and considering the relevant restricted isometry property (RIP) condition associated with the affine transformation. Finally, through experiments involving low-rank matrix recovery under affine transformations and the completion of low-rank real image, the effectiveness of this approach has been demonstrated, along with its superiority compared to the vector-based approach.