Large-Scale Affine Matrix Rank Minimization With a Novel Nonconvex Regularizer

• Published in: IEEE transaction on neural networks and learning systems Vol. 33; no. 9; pp. 4661 - 4675
• Main Authors: Wang, Zhi, Liu, Yu, Luo, Xin, Wang, Jianjun, Gao, Chao, Peng, Dezhong, Chen, Wu
• Format: Journal Article
• Language: English
• Published: United States IEEE 01.09.2022; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Analytical models; Convergence; Data analysis; Data models; Data processing; Data recovery; Empirical analysis; Image processing; Inexact proximal step; low-rank minimization; Machine learning; Mathematical analysis; matrix completion; Minimization; novel nonconvex regularizer; Optimization; Optimization models; Principal components analysis; Recommender systems; robust principal component analysis (RPCA); Signal processing; Subtraction; tensor completion; Tensors
• ISSN: 2162-237X; 2162-2388; 2162-2388
• DOI: 10.1109/TNNLS.2021.3059711

Low-rank minimization aims to recover a matrix of minimum rank subject to linear system constraint. It can be found in various data analysis and machine learning areas, such as recommender systems, video denoising, and signal processing. Nuclear norm minimization is a dominating approach to handle it. However, such a method ignores the difference among singular values of target matrix. To address this issue, nonconvex low-rank regularizers have been widely used. Unfortunately, existing methods suffer from different drawbacks, such as inefficiency and inaccuracy. To alleviate such problems, this article proposes a flexible model with a novel nonconvex regularizer. Such a model not only promotes low rankness but also can be solved much faster and more accurate. With it, the original low-rank problem can be equivalently transformed into the resulting optimization problem under the rank restricted isometry property (rank-RIP) condition. Subsequently, Nesterov's rule and inexact proximal strategies are adopted to achieve a novel algorithm highly efficient in solving this problem at a convergence rate of <inline-formula> <tex-math notation="LaTeX">O(1/K) </tex-math></inline-formula>, with <inline-formula> <tex-math notation="LaTeX">K </tex-math></inline-formula> being the iterate count. Besides, the asymptotic convergence rate is also analyzed rigorously by adopting the Kurdyka- ojasiewicz (KL) inequality. Furthermore, we apply the proposed optimization model to typical low-rank problems, including matrix completion, robust principal component analysis (RPCA), and tensor completion. Exhaustively empirical studies regarding data analysis tasks, i.e., synthetic data analysis, image recovery, personalized recommendation, and background subtraction, indicate that the proposed model outperforms state-of-the-art models in both accuracy and efficiency.