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

Area of Science:

• * Control Theory
• * Matrix Analysis
• * Machine Learning

Background:

• * Rank minimization is a critical problem in various fields, including signal processing and machine learning.
• * Existing matrix neurodynamic approaches (MNAs) often lack guaranteed convergence times or uniqueness of solutions.

Purpose of the Study:

• * To develop novel matrix neurodynamic approaches (MNAs) for efficient rank minimization.
• * To introduce finite-time converging MNA (FINt-MNA) and fixed-time converging MNA (FIXt-MNA) variants.
• * To analyze the convergence properties and settling time bounds of the proposed MNAs.

Main Methods:

• * Introduction of a matrix norm-normalized sign function to create FINt-MNA and FIXt-MNA.
• * Lyapunov stability analysis to prove the existence, uniqueness, and convergence to optimal solutions.
• * Application of finite-time and fixed-time lemmas to determine settling time bounds.
• * Control variable method to analyze parameter influence on settling times.

Main Results:

• * The proposed FINt-MNA and FIXt-MNA guarantee the existence and uniqueness of solutions.
• * Lyapunov analysis confirms convergence to the optimal solution within finite and fixed times, respectively.
• * Settling time bounds were derived and analyzed concerning tunable parameters.
• * Numerical simulations and an image completion task demonstrated superior performance over existing methods.

Conclusions:

• * The developed FINt-MNA and FIXt-MNA are effective and superior for rank minimization.
• * These novel approaches offer guaranteed convergence times and enhanced performance.
• * The findings have significant implications for applications requiring efficient rank minimization.