Enhanced Acceleration for Generalized Nonconvex Low-Rank Matrix Learning

Abstract

Matrix minimization techniques that employ the nuclear norm have gained recognition for their applicability in tasks like image inpainting, clustering, classification, and reconstruction. However, they come with inherent biases and computational burdens, especially when used to relax the rank function, making them less effective and efficient in real-world scenarios. To address these challenges, our research focuses on generalized nonconvex rank regularization problems in robust matrix completion, low-rank representation, and robust matrix regression. We introduce innovative approaches for effective and efficient low-rank matrix learning, grounded in generalized nonconvex rank relaxations inspired by various substitutes for the $\ell_{0}$-norm relaxed functions. These relaxations allow us to more accurately capture low-rank structures. Our optimization strategy employs a nonconvex and multi-variable alternating direction method of multipliers, backed by rigorous theoretical analysis for complexity and convergence. This algorithm iteratively updates blocks of variables, ensuring efficient convergence. Additionally, we incorporate the randomized singular value decomposition technique and/or other acceleration strategies to enhance the computational efficiency of our approach, particularly for large-scale constrained minimization problems. In conclusion, our experimental results across a variety of image vision-related application tasks unequivocally demonstrate the superiority of our proposed methodologies in terms of both efficacy and efficiency when compared to most other related learning methods.