Neurodynamic Approaches with Fixed-Step Optimality for Rank Minimization: A Data-Driven Framework

Rank minimization is a fundamental yet NP-hard problem with applications in recommendation systems, signal processing, and video denoising. Neurodynamic approaches provide a principled framework by modeling optimization as a continuous-time dynamical system, but they are often sensitive to discretization parameters and require a large number of iterations in practice. In this paper, we propose a unified data-driven framework that integrates continuous-time Matrix Neurodynamic Approaches (MNA) with deep unfolding networks, thereby combining the interpretability and convergence guarantees of neurodynamic approaches with the adaptability of learning-based models. Firstly, we discretize the MNA via the forward Euler scheme, deriving two discrete algorithms: the Fixed-Step Discrete Matrix Neurodynamic Approach (FDMNA) and the Variable-Step Discrete Matrix Neurodynamic Approach (VDMNA). For both approaches, we establish explicit optimality conditions and analytically characterize the interplay among iteration number, error, and step-size parameters. Secondly, to enable end-to-end optimization, we unfold FDMNA and VDMNA into trainable deep architectures in which all algorithmic parameters are learned directly from data. Remarkably, this data-driven training allows both networks to achieve fixed-step optimality while preserving theoretical interpretability. To ensure differentiable learning, we further introduce the Logistic-Threshold Function (LT-Func), which is a smooth surrogate for the non-smooth rank-$r$ projection operator, thus enabling adaptive singular value selection within the network. Finally, extensive experiments on low-rank signal recovery and high-resolution video compression and reconstruction verify both the effectiveness and scalability of the proposed framework, demonstrating superior performance over conventional approaches.