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Fast and Accurate Matrix Completion via Truncated Nuclear Norm Regularization
Sept. 2013 (vol. 35 no. 9)
pp. 2117-2130
Yao Hu, State Key Lab. of CAD&CG, Zhejiang Univ., Hangzhou, China
Debing Zhang, State Key Lab. of CAD&CG, Zhejiang Univ., Hangzhou, China
Jieping Ye, Comput. Sci. & Eng. Dept., Arizona State Univ., Tempe, AZ, USA
Xuelong Li, State Key Lab. of Transicent Opt. & Photonics, Xi'an Inst. of Opt. & Precision Mech., Xi'an, China
Xiaofei He, State Key Lab. of CAD&CG, Zhejiang Univ., Hangzhou, China
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.
Index Terms:
minimisation,approximation theory,concave programming,convergence of numerical methods,convex programming,gradient methods,matrix algebra,accelerated proximal gradient line search method,truncated nuclear norm regularization,low-rank matrix approximation problem,rank operator,convex relaxation,nuclear norm minimization,rank of matrix,matrix completion algorithm,iterative procedures,TNNR-ADMM,TNNR-APGL,TNNR-ADMMAP,optimization problem,TNNR-AGPL,convergence,alternating direction method of multipliers,Optimization,Approximation methods,Minimization,Convergence,Acceleration,Matrix decomposition,Computer vision,accelerated proximal gradient method,Matrix completion,nuclear norm minimization,alternating direction method of multipliers
Citation:
Yao Hu, Debing Zhang, Jieping Ye, Xuelong Li, Xiaofei He, "Fast and Accurate Matrix Completion via Truncated Nuclear Norm Regularization," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 35, no. 9, pp. 2117-2130, Sept. 2013, doi:10.1109/TPAMI.2012.271
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