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Nonconvex Optimization for Robust Tensor Completion from Grossly Sparse Observations

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Abstract

In this paper, we consider the robust tensor completion problem for recovering a low-rank tensor from limited samples and sparsely corrupted observations, especially by impulse noise. A convex relaxation of this problem is to minimize a weighted combination of tubal nuclear norm and the \(\ell _1\)-norm data fidelity term. However, the \(\ell _1\)-norm may yield biased estimators and fail to achieve the best estimation performance. To overcome this disadvantage, we propose and develop a nonconvex model, which minimizes a weighted combination of tubal nuclear norm, the \(\ell _1\)-norm data fidelity term, and a concave smooth correction term. Further, we present a Gauss–Seidel difference of convex functions algorithm (GS-DCA) to solve the resulting optimization model by using a linearization technique. We prove that the iteration sequence generated by GS-DCA converges to the critical point of the proposed model. Furthermore, we propose an extrapolation technique of GS-DCA to improve the performance of the GS-DCA. Numerical experiments for color images, hyperspectral images, magnetic resonance imaging images and videos demonstrate that the effectiveness of the proposed method.

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Notes

  1. http://sipi.usc.edu/database/.

  2. https://engineering.purdue.edu/~biehl/MultiSpec/hyperspectral.html.

  3. http://media.xiph.org/video/derf/.

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Acknowledgements

The authors would like to thank Professor Yuning Yang for sending us the code of the W-ST in [44].

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Authors and Affiliations

  1. School of Mathematics, Hunan University, Changsha, 410082, Hunan, People’s Republic of China

    Xueying Zhao & Minru Bai

  2. Department of Mathematics, The University of Hong Kong, Pok Fu Lam, Hong Kong

    Michael K. Ng

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  1. Xueying Zhao
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Correspondence to Minru Bai.

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Minru Bai: Research supported in part by the National Natural Science Foundation of China under Grant 11971159.

Michael K. Ng: Research supported in part by the HKRGC GRF 12306616, 12200317, 12300218, 12300519, 17201020, and HKU Grant 104005583.

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Zhao, X., Bai, M. & Ng, M.K. Nonconvex Optimization for Robust Tensor Completion from Grossly Sparse Observations. J Sci Comput 85, 46 (2020). https://doi.org/10.1007/s10915-020-01356-0

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