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Low-rank and sparse matrix decomposition via the truncated nuclear norm and a sparse regularizer

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Abstract

Recovering the low-rank and sparse components from a given matrix is a challenging problem that has many real applications. This paper proposes a novel algorithm to address this problem by introducing a sparse prior on the low-rank component. Specifically, the low-rank component is assumed to be sparse in a transform domain and a sparse regularizer formulated as an \(\ell _1\)-norm term is employed to promote the sparsity. The truncated nuclear norm is used to model the low-rank prior, rather than the nuclear norm used in most existing methods, to achieve a better approximation to the rank of the considered matrix. Furthermore, an efficient solving method based on a two-stage iterative scheme is developed to address the raised optimization problem. The proposed algorithm is applied to deal with synthetic data and real applications including face image shadow removal and video background subtraction, and the experimental results validate the effectiveness and accuracy of the proposed approach as compared with other methods.

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Notes

  1. The code of our algorithm is available from https://github.com/xuezc/LRSD-TNNSR.

  2. As the code of LRSD-TNN is not available, we implemented it by ourselves. The codes of IALM and EALM [25] were downloaded from http://perception.csl.illinois.edu/matrix-rank/sample_code.html. The code of FALM [14] is available in the LRSLibrary [40] which can be downloaded from https://github.com/andrewssobral/lrslibrary. The experiments were all performed with MATLAB 2014a in Windows 7 running on an Intel i5-6500 CPU and 8G memory.

  3. The code of noncvxRPCA was downloaded from https://github.com/sckangz/noncvx-PRCA.

  4. The code of incPCP was downloaded from https://sites.google.com/a/istec.net/prodrig/Home/en/pubs/incpcp. The code of GRASTA was downloaded from https://github.com/andrewssobral/lrslibrary. The code of Prac-ReProCS was downloaded from http://www.ece.iastate.edu/~hanguo/PracReProCS.html. We thank the authors of [41] for sharing their code via email.

  5. http://bmc.iut-auvergne.com/

  6. The BMC Wizard can be downloaded from http://bmc.iut-auvergne.com/.

  7. The results of MOG-RPCA, PCA and DRMF are obtained from [3].

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Funding

This work was supported by the Natural Science Foundation of the Higher Education Institutions of Jiangsu Province of China (17KJB510025), the Natural Science Foundation of China (41676088) and the Major Basic Research Program for National Security of China (973 Program for National Defence, No. 613317).

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Authors and Affiliations

  1. College of Electrical Engineering and Control Science, Nanjing Tech University, Nanjing, Jiangsu, China

    Zhichao Xue, Jing Dong & Ryad Chellali

  2. College of Automation, Harbin Engineering University, Harbin, Heilongjiang, China

    Yuxin Zhao & Chang Liu

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  1. Zhichao Xue
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  2. Jing Dong
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  3. Yuxin Zhao
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  4. Chang Liu
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Correspondence to Jing Dong.

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Xue, Z., Dong, J., Zhao, Y. et al. Low-rank and sparse matrix decomposition via the truncated nuclear norm and a sparse regularizer. Vis Comput 35, 1549–1566 (2019). https://doi.org/10.1007/s00371-018-1555-1

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