Nonconvex mixed matrix minimization

Meijuan Shang; Yanan Liu; Lingchen Kong; Xianchao Xiu; Ying Yang

doi:10.3934/mfc.2019009

Article Contents

2019, Volume 2, Issue 2: 107-126. Doi: 10.3934/mfc.2019009

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Nonconvex mixed matrix minimization

1.
Department of Mathematics, School of Science, Shijiazhuang University, Shijiazhuang 050035, China
2.
Department of Applied Mathematics, School of Science, Beijing Jiaotong University, Beijing 100044, China
3.
State Key Lab for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, College of Engineering, Peking University, Beijing 100871, China

^* Corresponding author: Xianchao Xiu
^* Corresponding author: Xianchao Xiu

Received: January 31, 2019

Published: July 2019

This work was supported in part by the National Natural Science Foundation of China (61633001, 11671029, 11601348).

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Abstract

Given the ultrahigh dimensionality and the complex structure, which contains matrices and vectors, the mixed matrix minimization becomes crucial for the analysis of those data. Recently, the nonconvex functions such as the smoothly clipped absolute deviation, the minimax concave penalty, the capped $ \ell_1 $-norm penalty and the $ \ell_p $ quasi-norm with $ 0<p<1 $ have been shown remarkable advantages in variable selection due to the fact that they can overcome the over-penalization. In this paper, we propose and study a novel nonconvex mixed matrix minimization, which combines the low-rank and sparse regularzations and nonconvex functions perfectly. The augmented Lagrangian method (ALM) is proposed to solve the noncovnex mixed matrix minimization problem. The resulting subproblems either have closed-form solutions or can be solved by fast solvers, which makes the ALM particularly efficient. In theory, we prove that the sequence generated by the ALM converges to a stationary point when the penalty parameter is above a computable threshold. Extensive numerical experiments illustrate that our proposed nonconvex mixed matrix minimization model outperforms the existing ones.

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Mathematics Subject Classification: Primary: 90C26, 90C46; Secondary: 90C90, 65K10.

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Table 1. Comparison Results for Synthetic Data

Cases		RMSE(B)			RMSE($ \gamma $)
$ r $	$ s $	LMM	LSMM	NonMM	LMM	LSMM	NonMM
5	0.01	0.0000	0.0000	0.0000	0.1806	0.1421	0.1101
	0.05	0.0456	0.0437	0.0274	0.6453	0.2742	0.2336
	0.1	0.0905	0.0688	0.0496	0.9162	0.3788	0.3176
	0.2	0.1314	0.1082	0.0911	1.0822	0.5063	0.4410
	0.5	0.2744	0.2152	0.1013	1.5868	0.7371	0.7092
10	0.01	0.0000	0.0000	0.0000	0.1927	0.1513	0.1217
	0.05	0.0781	0.05115	0.0382	0.7663	0.3312	0.2961
	0.1	0.1072	0.0749	0.0536	1.0452	0.3879	0.3401
	0.2	0.1339	0.1217	0.1161	1.2696	0.5305	0.4918
	0.5	0.2947	0.2440	0.1278	1.9561	0.7971	0.7330
15	0.01	0.0000	0.0000	0.0000	0.2197	0.1920	0.1405
	0.05	0.0982	0.0749	0.0654	0.8408	0.3223	0.2893
	0.1	0.1323	0.1130	0.1108	1.2501	0.3534	0.3293
	0.2	0.1846	0.1560	0.1272	1.6730	0.4988	0.4494
	0.5	0.3176	0.2812	0.2176	2.3608	0.7817	0.7502
20	0.01	0.0000	0.0000	0.0000	0.1936	0.1532	0.1509
	0.05	0.0987	0.0791	0.0732	0.9081	0.3168	0.2719
	0.1	0.1528	0.1454	0.1402	1.416	0.4583	0.3961
	0.2	0.1904	0.1821	0.1723	1.8304	0.4722	0.4148
	0.5	0.3876	0.2922	0.2134	2.4961	0.8049	0.7891
30	0.01	0.0000	0.0000	0.0000	0.1768	0.1620	0.1400
	0.05	0.1071	0.0932	0.0858	1.0089	0.2360	0.2232
	0.1	0.1608	0.1572	0.1469	1.5286	0.3241	0.3062
	0.2	0.2487	0.2371	0.2206	2.082	0.4920	0.4599
	0.5	0.4170	0.3943	0.3725	2.6203	0.7856	0.8041

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Table 2. Comparison Results for Real-world Data

Rate	LMM	LSMM	NonMM
5-fold	0.1321	0.1178	0.1025
10-fold	0.1972	0.1630	0.1589

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