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A singular value p-shrinkage thresholding algorithm for low rank matrix recovery

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Abstract

In this paper, we propose an iterative singular value p-shrinkage thresholding algorithm for solving low rank matrix recovery problem, and also give its two accelerated versions using randomized singular value decomposition. The convergence result of the proposed singular value p-shrinkage thresholding algorithm is proved. Numerical results based on simulation data and real data show the effectiveness of all the three proposed algorithms compared to the existing state-of-the-art algorithms.

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Notes

  1. Please see Proposition 2.2 in Sect. 2.2 for details.

  2. When \(t=0\), we assign \(0^{0}\) the value 1.

  3. Here, the penalty function \(g_p^{\mu }\) can be constructed by using the Legendre–Fenchel transform of an antiderivative of the proximal operator \(s_p^{\mu }\).

  4. The definition of sampling ratio sr is given in Sect. 4.1.

  5. http://statweb.stanford.edu/~candes/svt/.

  6. http://www.math.nus.edu.sg/~mattohkc/NNLS.html.

  7. http://www4.comp.polyu.edu.hk/~cslzhang/code/WNNM_MC_code.zip.

  8. In the experiments, the iteration numbers of ADMc algorithm can not be reported, because there are a sequence of parameters {\(\lambda _{s} \)} created in the iteration, which uses a data driven regularization selection rules.

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Acknowledgements

The authors would like to thank Dr. Yangyang Xu and Zheng-Fen Jin for their kindly help to send us the code of t-IRucLq and ADMc algorithms, respectively. In addition, the authors deeply appreciate the anonymous referees for their contributions.

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Authors and Affiliations

  1. School of Mathematics (Zhuhai), Sun Yat-Sen University, Zhuhai, 519082, People’s Republic of China

    Yu-Fan Li

  2. College of Mathematics and Econometrics, Hunan University, Changsha, 410082, People’s Republic of China

    Kun Shang

  3. School of Mathematics, Tianjin University, Tianjin, 300072, People’s Republic of China

    Zheng-Hai Huang

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  1. Yu-Fan Li
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  2. Kun Shang
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  3. Zheng-Hai Huang
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Correspondence to Kun Shang.

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This work was supported in part by the National Natural Science Foundation of China under Grant Nos. 11431002, 11871051 and the Fundamental Research Funds for the Central Universities under Grant No. 18lgpy70.

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Li, YF., Shang, K. & Huang, ZH. A singular value p-shrinkage thresholding algorithm for low rank matrix recovery. Comput Optim Appl 73, 453–476 (2019). https://doi.org/10.1007/s10589-019-00084-y

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