A new nonconvex approach to low-rank matrix completion with application to image inpainting
- 410 Downloads
The problem of recovering a low-rank matrix from partial entries, known as low-rank matrix completion, has been extensively investigated in recent years. It can be viewed as a special case of the affine constrained rank minimization problem which is NP-hard in general and is computationally hard to solve in practice. One widely studied approach is to replace the matrix rank function by its nuclear-norm, which leads to the convex nuclear-norm minimization problem solved efficiently by many popular convex optimization algorithms. In this paper, we propose a new nonconvex approach to better approximate the rank function. The new approximation function is actually the Moreau envelope of the rank function (MER) which has an explicit expression. The new approximation problem of low-rank matrix completion based on MER can be converted to an optimization problem with two variables. We then adapt the proximal alternating minimization algorithm to solve it. The convergence (rate) of the proposed algorithm is proved and its accelerated version is also developed. Numerical experiments on completion of low-rank random matrices and standard image inpainting problems have shown that our algorithms have better performance than some state-of-art methods.
KeywordsLow-rank matrix completion Moreau envelope Proximal alternating minimization Image Inpainting
The authors would like to thank the editor-in chief and the anonymous reviewers for their insightful and constructive comments, which help to enrich the content and improve the presentation of this paper. This research was supported by the National Natural Science Foundation of China under the grants 11771347, 91730306, 41390454, and 11271297, and supported by the Nanhu Scholars Program for Young Scholars of XYNU.
- Bertsekas, D. (1999). Nonlinear optimisation, 2nd ed., Athena, Belmont, Massachusetts.Google Scholar
- Candès, E. J., & Wakin, M. (2008). An introduction to compressive sampling. IEEE Signal Processing Magazine, 25(2), 21–30.Google Scholar
- Cao, W. F., Sun, J., & Xu, Z. B. (2013). Fast image deconvolution using closed-form thresholding formulas of \(L_q\) \((q = 1/2, 2/3)\) regularization. Journal of Visual Communication and Image Representation, 24(1), 31–41.Google Scholar
- Chen, P., & Suter, D. (2004). Recovering the missing components in a large noisy low-rank matrix: Application to SFM source. IEEE Transactions on Pattern Analysis and Machine Intelligence, 26(8), 1051–1063.Google Scholar
- Fazel, M. (2002). Matrix rank minimization with applications. Ph.D. thesis, Stanford University.Google Scholar
- Goldberg, D., Nichols, D., Oki, B. M., & Terry, D. (1992). Using collaborative filtering to weave an information tapestry. Communications of the ACM, 35(12), 61–70.Google Scholar
- Hu, Y., Zhang, D. B., Ye, J. P., Li, X. L., & He, X. F. (2013). Fast and accurate matrix completion via truncated nuclear norm regularization. IEEE Transactions on Pattern Analysis and Machine Intelligence, 35(9), 2117–2130.Google Scholar
- Li, G., & Pong, T. (2016). Calculus of the exponent of Kurdyka–Łojasiewicz inequality and its applications to linear convergence of first-order methods. arXiv:1602.02915v1.
- Lu, Z. S., & Zhang, Y. (2015). Schatten-\(p\) quasi-norm regularized matrix optimization via iterative reweighted singular value minimization. http://www.optimization-online.org/DB_HTML/2015/11/5215.html.
- Lu, C. Y., Tang, J. H., Yan, S. C., & Lin, Z. C. (2015). Nonconvex nonsmooth low-rank minimization via iteratively reweighted nuclear norm. arXiv:1510.06895.
- Mohammadi, M. M., Zadeh, M. B., Amini, A., & Jutten, C. (2013). Recovery of low-rank matrices under affine constraints via a smoothed rank function. arXiv:1308.2293.
- Srebro, N. (2004). Learning with matrix factorizations. Ph.D. thesis, Massachusetts Institute of Technology.Google Scholar
- Tomasi, C., & Kanade, T. (1992). Shape and motion from image streams under orthography: A factorization method. International Journal of Computer Vision, 9(2), 137C154.Google Scholar
- Zhang, S., & Xin, J. (2014). Minimization of transformed \(\ell _1\) penalty: Closed form representation and iterative thresholding algorithms. arXiv preprint arXiv:1412.5240.
- Zhang, S., Yin, P. H., & Xin, J. (2015). Transformed Schatten-1 iterative thresholding algorithms for matrix rank minimization and applications. arXiv:1506.04444.