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Robust PCA Using Nonconvex Rank Approximation and Sparse Regularizer

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Abstract

We consider the robust principal component analysis (RPCA) problem where the observed data are decomposed to a low-rank component and a sparse component. Conventionally, the matrix rank in RPCA is often approximated using a nuclear norm. Recently, RPCA has been formulated using the nonconvex \(\ell _{\gamma }\)-norm, which provides a closer approximation to the matrix rank than the traditional nuclear norm. However, the low-rank component generally has sparse property, especially in the transform domain. In this paper, a sparsity-based regularization term modeled with \(\ell _1\)-norm is introduced to the formulation. An iterative optimization algorithm is developed to solve the obtained optimization problem. Experiments using synthetic and real data are utilized to validate the performance of the proposed method.

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Notes

  1. The codes of noncvxRPCA were downloaded from the Web site https://github.com/sckangz/noncvx-PRCA. As the codes of the LRSD-TNN algorithm are not available, we implemented this algorithm by ourselves. The codes of IALM were downloaded from http://perception.csl.illinois.edu/matrix-rank/sample_code.html.

  2. http://perception.i2r.astar.edu.sg/bk_model/bk_index.html.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (61906087), the Natural Science Foundation of Jiangsu Province of China (BK20180692), and the Natural Science Foundation of the Higher Education Institutions of Jiangsu Province of China (17KJB510025). The authors thank the associate editor and the anonymous reviewers for their contributions to improving the quality of the paper.

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Authors and Affiliations

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

    Jing Dong & Zhichao Xue

  2. Centre for Vision, Speech and Signal Processing, University of Surrey, Guildford, GU2 7XH, UK

    Wenwu Wang

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  1. Jing Dong
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  2. Zhichao Xue
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  3. Wenwu Wang
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Correspondence to Jing Dong.

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Dong, J., Xue, Z. & Wang, W. Robust PCA Using Nonconvex Rank Approximation and Sparse Regularizer. Circuits Syst Signal Process 39, 3086–3104 (2020). https://doi.org/10.1007/s00034-019-01310-y

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