Homotopy method for matrix rank minimization based on the matrix hard thresholding method

Zhengshan Dong; Jianli Chen; Wenxing Zhu

doi:10.3934/naco.2019015

Article Contents

2019, Volume 9, Issue 2: 211-224. Doi: 10.3934/naco.2019015

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Homotopy method for matrix rank minimization based on the matrix hard thresholding method

1.
Ruijie Networks Co., Ltd, Fuzhou 350108, China
2.
Center for Discrete Mathematics and Theoretical Computer Science, Fuzhou University, Fuzhou 350108, China

^* Corresponding author: wxzhu@fzu.edu.cn
^* Corresponding author: wxzhu@fzu.edu.cn

Received: May 2018

Revised: August 2018

Published: January 2019

This research is supported by the National Natural Science Foundation of China under Grant 61672005

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Abstract

Based on the matrix hard thresholding method, a homotopy method is proposed for solving the matrix rank minimization problem. This method iteratively solves a series of regularization subproblems, whose solutions are given in closed form by the matrix hard thresholding operator. Under some mild assumptions, convergence of the proposed method is proved. The proposed method does not depend on a prior knowledge of exact rank value. Numerical experiments demonstrate that the proposed homotopy method weakens the affection of the choice of the regularization parameter, and is more efficient and effective than the existing sate-of-the-art methods.

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Mathematics Subject Classification: Primary: 15A29, 65F10, 90C27.

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Table 4. Numerical results on random uniform matrices with size $ m = n = 500 $ and sampling ratio $ \tau = 0.5. $

	r=5				r=10
Alg.	NS	time	rank	rel.err.	NS	time	rank	rel.err.
FPCA	50	6.71	5	9.22E-06	50	13.61	10	2.66E-05
PD	50	24.08	5	7.70E-05	50	33.65	10	8.96E-05
HIHT	50	19.37	5	5.33E-05	50	16.12	10	4.78E-05
	r=30				r=50
Alg.	NS	time	rank	rel.err.	NS	time	rank	rel.err.
FPCA	0	43.74	16.6	3.80E-02	-	-	-	-
PD	50	39.55	30	7.42E-05	50	53.83	50	1.00E-04
HIHT	50	36.34	30	3.52E-05	50	47.36	50	9.93E-06
	r=70				r=90
Alg.	NS	time	rank	rel.err.	NS	time	rank	rel.err.
FPCA	-	-	-	-	-	-	-	-
PD	50	87.17	70	1.51E-04	50	294.05	90	2.38E-04
HIHT	50	70.3	70	6.31E-05	50	82.79	90	2.02E-04

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Table 1. Influence of the regularization parameter $ \lambda $

$ \eta $	IHT				HIHT
$ \eta $	NS	time	rank	rel.err.	NS	time	rank	rel.err.
50	4	122.43	38.56	1.29E-01	50	35.33	40	1.17E-05
20	50	18.83	40.00	5.82E-05	50	19.86	40	1.17E-05
10	47	23.69	40.06	2.52E-03	50	23.25	40	1.38E-05
5	0	52.92	227.10	6.33E-01	0	45.41	227.72	6.34E-01

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Table 2. Influence of maximum number of inner iterations on Algorithm 2

	HIHT
$ step $	NS	time	rank	rel.err.
5	50	16.72	40	4.35E-05
10	50	18.13	40	2.09E-05
20	50	18.97	40	1.35E-05
30	50	19.29	40	1.16E-05
40	50	19.27	40	1.16E-05
50	50	19.24	40	1.17E-05
60	50	19.72	40	1.01E-05
70	50	19.71	40	1.02E-05

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Table 3. Numerical results on random Gaussian matrices with size $ m = n = 500 $ and sampling ratio $ \tau = 0.5 $

	r=5				r=10
Alg.	NS	time	rank	rel.err.	NS	time	rank	rel.err.
FPCA	50	4.69	5	1.24E-06	50	5.19	10	2.90E-06
PD	50	95.92	5	5.92E-05	50	102.83	10	8.54E-05
HIHT	50	10.91	5	7.49E-05	50	12.54	10	7.02E-05
	r=30				r=50
Alg.	NS	time	rank	rel.err.	NS	time	rank	rel.err.
FPCA	50	5.78	30	1.44E-05	50	255.32	50	1.61E-04
PD	50	102.38	30	9.40E-05	50	106.19	50	1.50E-04
HIHT	50	18.57	30	6.63E-05	50	24.13	50	2.67E-05
	r=70				r=90
Alg.	NS	time	rank	rel.err.	NS	time	rank	rel.err.
FPCA	50	279.55	70	2.23E-04	50	332.90	90	2.71E-04
PD	50	164.06	70	1.78E-04	50	471.48	90	2.51E-04
HIHT	50	38.23	70	5.01E-05	50	84.32	90	2.08E-04

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Table 5. Computational results on random Gaussian (or uniform) matrices with size $ m = n = 4000 $, $ r = 200 $, and sampling ratio $ \tau = 0.3. $

	Gaussian matrix				uniform matrix
Alg.	NS	time	rank	rel.err.	NS	time	rank	rel.err.
FPCA	50	284.36	200	8.28E-05	0	1887.44	1.06	2.35E-02
PD	50	13232.85	200	1.02E-04	50	5976.50	200	9.89E-05
HIHT	50	4427.63	200	1.63E-05	50	7637.74	200	1.70E-05

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