Abstract
In this work, we consider the Robust Principal Components Analysis, a popular method of dimensionality reduction. The corresponding optimization involves the minimization of \(l_0\)-norm which is known to be NP-hard. To deal with this problem, we replace the \(l_0\)-norm by a non-convex approximation, namely capped \(l_1\)-norm. The resulting optimization problem is non-convex for which we develop a reweighted \(l_1\) based algorithm. Numerical experiments on several synthetic datasets illustrate the efficiency of our algorithm and its superiority comparing to several state-of-the-art algorithms.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Candès, E.J., Li, X., Ma, Y., Wright, J.: Robust principal component analysis? J. ACM 58(3), 1–37 (2011)
Horn, R.A., Johnson, C.R.: Matrix Analysis, 2nd edn. Cambridge University Press, New York (2013)
Le Thi, H.A., Pham Dinh, T.: The DC (Difference of Convex Functions) programming and DCA revisited with DC models of real world nonconvex optimization problems. Ann. Oper. Res. 133(1–4), 23–46 (2005)
Le Thi, H.A., Pham Dinh, T., Le, H.M., Vo, X.T.: DC approximation approaches for sparse optimization. Eur. J. Oper. Res. 244(1), 26–46 (2015)
Lin, Z., Chen, M., Ma, Y.: The augmented lagrange multiplier method for exact recovery of corrupted low-rank matrices. arXiv:1009.5055 (2010)
Lin, Z., Ganesh, A., Wright, J., Wu, L., Chen, M., Ma, Y.: Fast convex optimization algorithms for exact recovery of a corrupted low-rank matrix. In: Intl. Workshop on Comp. Adv. in Multi-Sensor Adapt. Processing, Aruba, Dutch Antilles (2009)
Pham Dinh, T., Le Thi, H.A.: Convex analysis approach to dc programming: theory, algorithms and applications. Acta Math. Vietnamica 22(1), 289–355 (1997)
Pham Dinh, T., Le Thi, H.A.: A DC optimization algorithm for solving the trust-region subproblem. SIAM J. Optim. 8(2), 476–505 (1998)
Sun, Q., Xiang, S., Ye, J.: Robust principal component analysis via capped norms. In: Proceedings of the 19th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD 2013, pp. 311–319 (2013)
Wright, J., Ganesh, A., Rao, S., Peng, Y., Ma, Y.: Robust principal component analysis: exact recovery of corrupted low-rank matrices via convex optimization. Adv. Neural Inf. Process. Syst. 22, 2080–2088 (2009)
Zhao, Q., Meng, D., Xu, Z., Zuo, W., Zhang, L.: Robust principal component analysis with complex noise. In: Proceedings of the 31st International Conference on International Conference on Machine Learning - Volume 32, ICML 2014, pp. 55–63 (2014)
Zhou, Z., Li, X., Wright, J., Candes, E., Ma, Y.: Stable principal component pursuit. In: 2010 IEEE International Symposium on Information Theory, pp. 1518–1522 (2010)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Le, H.M., Xuanthanh, V. (2020). Reweighted \(\ell _1\) Algorithm for Robust Principal Component Analysis. In: Le Thi, H., Le, H., Pham Dinh, T., Nguyen, N. (eds) Advanced Computational Methods for Knowledge Engineering. ICCSAMA 2019. Advances in Intelligent Systems and Computing, vol 1121. Springer, Cham. https://doi.org/10.1007/978-3-030-38364-0_12
Download citation
DOI: https://doi.org/10.1007/978-3-030-38364-0_12
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-38363-3
Online ISBN: 978-3-030-38364-0
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)