Principal Component Pursuit with Weighted Nuclear Norm

Abstract:

Principal Component Pursuit (PCP) recovers low-dimensional structures from a small set of linear measurements, such as low rank matrix and sparse matrix. Pervious works mainly focus on exact recovery without additional noise. However, in many applications the observed measurements are corrupted by an additional white Gaussian noise (AWGN). In this paper, we model the recovered matrix the sum a low-rank matrix, a sparse matrix and an AWGN. We propose a weighted PCP for the recovery matrix, which is solved by alternating direction method. Numerical results show that the reconstructions performance of weighted PCP outperforms the classical PCP in term of accuracy.

• Periodical: Applied Mechanics and Materials (Volumes 513-517)
• Pages: 1722-1726
• DOI: https://doi.org/10.4028/www.scientific.net/AMM.513-517.1722
• Online since: February 2014
• Authors: Qing Shan You*, Qun Wan
• Keywords: Alternating Direction Method, Low Rank Matrix, Nuclear Norm Minimization, Principal Component Pursuit, Sparsity

References

[1] E. J. Candes and T. Tao. The power of convex relaxation: Near-optimal matrix completion. IEEE Transactions on Information Theory, 56(5): 2053-2080, (2010).

DOI: 10.1109/tit.2010.2044061

[2] D. Gross. Recovering low-rank matrices from few coefficients in any basis. IEEE Transactions on Information Theory, 57(3): 1548-1566, (2011).

DOI: 10.1109/tit.2011.2104999

[3] J. Ellenberg. Fill in the blanks: Using math to turn lo-res datasets into hi-ressamples. Wired, March (2010).

DOI: 10.1515/9781400839544.75

[4] B. Zeng and J. Fu, Directional discrete cosine transforms: A new framework for image coding, IEEE Transactions on Circuits and Systems for Video Technology, 18(13), pp.305-313, (2011).

DOI: 10.1109/tcsvt.2008.918455

[5] H. Ji, C. Q. Liu, Z.W. Shen, and Y. H. Xu, Robust video denoising using low rank matrix completion, IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 1791-1798, (2010).

DOI: 10.1109/cvpr.2010.5539849

[6] X. Geng, K. S. Miles, Z. H. Zhou, and L. Wang, Face image modeling by multilinear subspace analysis with missing values, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 41(3): 881-892, (2011).

DOI: 10.1109/tsmcb.2010.2097588

[7] E. Candes, X. Li, Y. Ma, and J. Wright. Robust principal component analysis? Journal of the ACM, 58(1), 1-37. (2009).

[8] J.F. Cai, E. J. Candes, and Z. Shen. A singular value thresholding algorithm for matrix completion. SIAM Journal on Optimization, 20(4): 1956-1982, (2010).

DOI: 10.1137/080738970

[9] K. Toh and S. Yun. An accelerated proximal gradient algorithm for nuclear norm regularized least squares problems. Pacific Journal of Optimization, (2010).

[10] Q. S. You, Q. Wan, Y. P. Liu, a short note on strongly Convex Programming for Exact Matrix Completion and Robust Principal Component Analysis[J]. Inverse Problems and Imaging, 2013, Vol 7, No 1, pp: 305-306.

DOI: 10.3934/ipi.2013.7.305

[11] G. Stephane, L. Guillaume. Weighted algorithms for compressed sensing and matrix completion. http: /arxiv. org/abs/1107. 1638, (2011).

[12] M. Fazel, T. K. Pong, D. Sun, P. Tseng, Hankel matrix rank minimization with applications in system identification and realization, Available at http: /faculty. washington. edu/mfazel/, (2012).

DOI: 10.1137/110853996