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Open AccessArticle

A Unified Proximity Algorithm with Adaptive Penalty for Nuclear Norm Minimization

1
College of Mathematics and Computer Science, Gannan Normal University, Ganzhou 341000, China
2
Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou 310018, China
*
Author to whom correspondence should be addressed.
Symmetry 2019, 11(10), 1277; https://doi.org/10.3390/sym11101277
Received: 9 September 2019 / Revised: 4 October 2019 / Accepted: 8 October 2019 / Published: 11 October 2019
(This article belongs to the Special Issue Fixed Point Theory and Computational Analysis with Applications)
The nuclear norm minimization (NNM) problem is to recover a matrix that minimizes the sum of its singular values and satisfies some linear constraints simultaneously. The alternating direction method (ADM) has been used to solve this problem recently. However, the subproblems in ADM are usually not easily solvable when the linear mappings in the constraints are not identities. In this paper, we propose a proximity algorithm with adaptive penalty (PA-AP). First, we formulate the nuclear norm minimization problems into a unified model. To solve this model, we improve the ADM by adding a proximal term to the subproblems that are difficult to solve. An adaptive tactic on the proximity parameters is also put forward for acceleration. By employing subdifferentials and proximity operators, an equivalent fixed-point equation system is constructed, and we use this system to further prove the convergence of the proposed algorithm under certain conditions, e.g., the precondition matrix is symmetric positive definite. Finally, experimental results and comparisons with state-of-the-art methods, e.g., ADM, IADM-CG and IADM-BB, show that the proposed algorithm is effective.
Keywords: nuclear norm minimization; matrix completion; alternating direction method; subdifferential; proximity operator nuclear norm minimization; matrix completion; alternating direction method; subdifferential; proximity operator
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Hu, W.; Zheng, W.; Yu, G. A Unified Proximity Algorithm with Adaptive Penalty for Nuclear Norm Minimization. Symmetry 2019, 11, 1277.

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