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The Nonconvex Tensor Robust Principal Component Analysis Approximation Model via the Weighted \(\ell _p\)-Norm Regularization

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Abstract

Tensor robust principal component analysis (TRPCA), which aims to recover the underlying low-rank multidimensional datasets from observations corrupted by noise and/or outliers, has been widely applied to various fields. The typical convex relaxation of TRPCA in literature is to minimize a weighted combination of the tensor nuclear norm (TNN) and the \(\ell _1\)-norm. However, owing to the gap between the tensor rank function and its lower convex envelop (i.e., TNN), the tensor rank approximation by using the TNN appears to be insufficient. Also, the \(\ell _1\)-norm generally is too relaxing as an estimator for the \(\ell _0\)-norm to obtain desirable results in terms of sparsity. Different from current approaches in literature, in this paper, we develop a new non-convex TRPCA model, which minimizes a weighted combination of non-convex tensor rank approximation function and the weighted \(\ell _p\)-norm to attain a tighter approximation. The resultant non-convex optimization model can be solved efficiently by the alternating direction method of multipliers (ADMM). We prove that the constructed iterative sequence generated by the proposed algorithm converges to a critical point of the proposed model. Numerical experiments for both image recovery and surveillance video background modeling demonstrate the effectiveness of the proposed method.

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The authors confirm that the data supporting the findings of this study are available within the article and its supplementary materials.

Notes

  1. https://brainweb.bic.mni.mcgill.ca/.

  2. http://trace.eas.asu.edu/yuv/.

  3. http://jacarini.dinf.usherbrooke.ca/dataset2012/.

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Acknowledgements

The authors would like to thank three anonymous referees for their very helpful comments and suggestions.

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Authors and Affiliations

  1. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, People’s Republic of China

    Minghui Li, Wen Li & Yannan Chen

  2. Department of Mathematics, Southern Illinois University-Carbondale, Carbondale, IL, 62901, USA

    Mingqing Xiao

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  1. Minghui Li
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Correspondence to Wen Li.

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This research was supported by the National Natural Science Foundations of China (12071159, 11671158, U1811464) and the NSF-DMS 1854638 of the United States.

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Li, M., Li, W., Chen, Y. et al. The Nonconvex Tensor Robust Principal Component Analysis Approximation Model via the Weighted \(\ell _p\)-Norm Regularization. J Sci Comput 89, 67 (2021). https://doi.org/10.1007/s10915-021-01679-6

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  • DOI: https://doi.org/10.1007/s10915-021-01679-6

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