An Efficient Algorithm for Low Rank Matrix Restoration Problem with Unknown Noise Level

JIN Zheng-fen, WANG Duo, SHANG You-lin, LV Jin-man

doi:10.13371/j.cnki.chin.q.j.m.2021.04.002

Chinese Quarterly Journal of Mathematics >

2021 , Vol. 36 >Issue 4: 356 - 368

DOI: https://doi.org/10.13371/j.cnki.chin.q.j.m.2021.04.002

An Efficient Algorithm for Low Rank Matrix Restoration Problem with Unknown Noise Level

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1. School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471023, China; 2. LMIB of the Ministry of Education, School of Mathematical Sciences, Beihang University, Beijing 100191, China; 3. State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing 100044, China; 4. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

JIN Zheng-fen (1987-), female, native of Luoyang, Henan, lecturer of Henan University of Science and Technology, postdoctor of Beihang University, engages in nonlinear programming; WANG Duo (1995-), female, native of Nanyang, Henan, PHD of Beijing Jiaotong University, engages in linear mixed integer programming; SHANG You-lin (1963-), male, native of Luoyang, Henan, professor of Henan University of Science and Technology, doctoral supervisor, engages in operations research, cybernetics, systems science and engineering; LV Jin-man (1991-), female, native of Shangqiu, Henan, PHD of Wuhan University, engages in bi-level programming.

Received date: 2021-07-01

Online published: 2021-12-30

Supported by

Supported by the National Natural Science Foundation of China (Grant No. 11971149, 12101195, 12071112, 11871383);

Natural Science Foundation of Henan Province for Youth (Grant No. 202300410146).

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Abstract

Recovering an unknown high dimensional low rank matrix from a small set of entries is widely spread in the fields of machine learning, system identification and image restoration, etc. In many practical applications, the few observations are always corrupted by noise and the noise level is also unknown. A novel model with nuclear norm and square root type estimator has been proposed, which does not rely on the knowledge or on an estimation of the standard deviation of the noise. In this paper, we firstly reformulate the problem to an equivalent variable separated form by introducing an auxiliary variable. Then we propose an efficient alternating direction method of multipliers(ADMM) for solving it. Both of resulting subproblems admit an explicit solution, which makes our algorithm have a cheap computing. Finally, the numerical results show the benefits of the model and the efficiency of the proposed method.

Key words： Matrix restoration; Alternating direction method of multipliers; Square root least squares; Matrix completion

Cite this article

JIN Zheng-fen, WANG Duo, SHANG You-lin, LV Jin-man . An Efficient Algorithm for Low Rank Matrix Restoration Problem with Unknown Noise Level[J]. Chinese Quarterly Journal of Mathematics, 2021 , 36(4) : 356 -368 . DOI: 10.13371/j.cnki.chin.q.j.m.2021.04.002

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