Fast and Accurate Matrix Completion via Truncated Nuclear Norm Regularization

• Published in: IEEE transactions on pattern analysis and machine intelligence 2013-09, Vol.35 (9), p.2117-2130
• Main Authors: Hu, Yao, Zhang, Debing, Ye, Jieping, Li, Xuelong, He, Xiaofei
• Format: Article
• Language: English
• Subjects: accelerated proximal gradient method; Acceleration; Algebra; Algorithms; alternating direction method of multipliers; Applied sciences; Approximation; Approximation methods; Artificial intelligence; Computer science; control theory; systems; Computer systems and distributed systems. User interface; Computer vision; Convergence; Exact sciences and technology; Intelligence; Linear and multilinear algebra, matrix theory; Mathematical analysis; Mathematics; Matrix completion; Matrix decomposition; Minimization; Norms; nuclear norm minimization; Operations research; Optimization; Pattern analysis; Pattern recognition. Digital image processing. Computational geometry; Sciences and techniques of general use; Search methods; Software; Studies
• ISSN: 0162-8828; 1939-3539; 2160-9292
• DOI: 10.1109/TPAMI.2012.271

Recovering a large matrix from a small subset of its entries is a challenging problem arising in many real applications, such as image inpainting and recommender systems. Many existing approaches formulate this problem as a general low-rank matrix approximation problem. Since the rank operator is nonconvex and discontinuous, most of the recent theoretical studies use the nuclear norm as a convex relaxation. One major limitation of the existing approaches based on nuclear norm minimization is that all the singular values are simultaneously minimized, and thus the rank may not be well approximated in practice. In this paper, we propose to achieve a better approximation to the rank of matrix by truncated nuclear norm, which is given by the nuclear norm subtracted by the sum of the largest few singular values. In addition, we develop a novel matrix completion algorithm by minimizing the Truncated Nuclear Norm. We further develop three efficient iterative procedures, TNNR-ADMM, TNNR-APGL, and TNNR-ADMMAP, to solve the optimization problem. TNNR-ADMM utilizes the alternating direction method of multipliers (ADMM), while TNNR-AGPL applies the accelerated proximal gradient line search method (APGL) for the final optimization. For TNNR-ADMMAP, we make use of an adaptive penalty according to a novel update rule for ADMM to achieve a faster convergence rate. Our empirical study shows encouraging results of the proposed algorithms in comparison to the state-of-the-art matrix completion algorithms on both synthetic and real visual datasets.