An evolutionary multiobjective method for low-rank and sparse matrix decomposition

Tao Wu; Yu Lei; Jiao Shi; Maoguo Gong

doi:10.3934/bdia.2017006

Article Contents

2017, Volume 2, Issue 1: 23-37. Doi: 10.3934/bdia.2017006

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An evolutionary multiobjective method for low-rank and sparse matrix decomposition

School of Electronics and Information, Northwestern Polytechnical University, 127 West Youyi Road, Xi'an Shaanxi, 710072, China

* Corresponding author: Tao Wu
* Corresponding author: Tao Wu

Published: January 2017

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Abstract

This paper addresses the problem of finding the low-rank and sparse components of a given matrix. The problem involves two conflicting objective functions, reducing the rank and sparsity of each part simultaneously. Previous methods combine two objectives into a single objective penalty function to solve with traditional numerical optimization approaches. The main contribution of this paper is to put forward a multiobjective method to decompose the given matrix into low-rank component and sparse part. We optimize two objective functions with an evolutionary multiobjective algorithm MOEA/D. Another contribution of this paper, a modified low-rank and sparse matrix model is proposed, which simplifying the variable of objective functions and improving the efficiency of multiobjective optimization. The proposed method obtains a set of solutions with different trade-off between low-rank and sparse objectives, and decision makers can choose one or more satisfied decomposed results according to different requirements directly. Experiments conducted on artificial datasets and nature images, show that the proposed method always obtains satisfied results, and the convergence, stability and robustness of the proposed method is acceptable.

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Mathematics Subject Classification: 90C29, 92D15.

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Figure 1. The distributions of nondominated solutions, dominated solutions and Pareto front in the objective space of the two objectives problem.

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Figure 2. Error of sparse component recovering by the ADM method for LRSMD with different choices of the parameter $\lambda$

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Figure 3. The Pareto front, two objective functions corresponding to the solutions and box-plot for $ErrLR$ and $ErrSP$ by running 30 times independent experiments. Three rows represent experiment on the data with size:$20\times 20$, rank: 5, sparsity: 0.2, size: $50\times 50$, rank: 5, sparsity: 0.2 and size: $100\times 100$, rank: 5, sparsity: 0.5 respectively.

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Figure 4. Box-plot about $ErrLR$ and $ErrSP$ for data with different noise. (a): number of noise points is 20, $\sigma=1$. (b): number of noise points is 20, $\sigma=5$. (c): number of noise points is 20, $\sigma=15$. (d): number of noise points is 50, $\sigma=5$. (e): number of noise points is 50, $\sigma=15$.

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Figure 5. The Pareto front and two objective functions corresponding to the solutions for noise data. Noise type: number of noise points is 50 and $\sigma =15$.

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Figure 6. The Pareto front and decomposed results with image lena. (a) is Pareto front of MOLRSMD and (b) is three different decomposed results selected form Pareto front, and they locate at the top, middle and bottom of Pareto front, respectively.

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Figure 7. The Pareto front and decomposed results with image face. (a) is Pareto front of MOLRSMD and (b) is three different decomposed results selected form Pareto front, and they locate at the top, middle and bottom of Pareto front, respectively.

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Figure 8. Results of the proposed MOLRSMD compared with ADM and ALM. The test dataset size is $50\times 50$, rank equals 5 and sparsity is 0.2.

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Table 1. The Parameters in the MOLRSMD used in the Experiments

Parameter	Meaning	Value
$N$	The number of subproblems	100
$T$	The number of neighbors	20
$t_{max}$	The maximum of generations	300
$pc$	The probability of crossover	0.8
$pd$	The differential multiplier	0.5
$pm$	The probability of mutation	0.2
$ps$	The probability of mutation selection	0.85

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Table 2. Errors of the proposed MOLRSMD compared with ADM and ALM.

Methods	$ErrLR$	$ErrSP$	$ErrT$
MOLRSMD	0.0100	0.1986	0
ADM	0.0299	0.1331	1.9135e-17
ALM	0.0148	0.0661	2.2412e-10

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