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Efficient algorithms for robust and stable principal component pursuit problems

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  • Necdet Aybat
  • Donald Goldfarb
  • Shiqian Ma

Abstract

The problem of recovering a low-rank matrix from a set of observations corrupted with gross sparse error is known as the robust principal component analysis (RPCA) and has many applications in computer vision, image processing and web data ranking. It has been shown that under certain conditions, the solution to the NP-hard RPCA problem can be obtained by solving a convex optimization problem, namely the robust principal component pursuit (RPCP). Moreover, if the observed data matrix has also been corrupted by a dense noise matrix in addition to gross sparse error, then the stable principal component pursuit (SPCP) problem is solved to recover the low-rank matrix. In this paper, we develop efficient algorithms with provable iteration complexity bounds for solving RPCP and SPCP. Numerical results on problems with millions of variables and constraints such as foreground extraction from surveillance video, shadow and specularity removal from face images and video denoising from heavily corrupted data show that our algorithms are competitive to current state-of-the-art solvers for RPCP and SPCP in terms of accuracy and speed. Copyright Springer Science+Business Media New York 2014

Suggested Citation

  • Necdet Aybat & Donald Goldfarb & Shiqian Ma, 2014. "Efficient algorithms for robust and stable principal component pursuit problems," Computational Optimization and Applications, Springer, vol. 58(1), pages 1-29, May.
    • Handle: RePEc:spr:coopap:v:58:y:2014:i:1:p:1-29
    DOI: 10.1007/s10589-013-9613-0
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    References listed on IDEAS

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    1. NESTEROV, Yu., 2005. "Smooth minimization of non-smooth functions," LIDAM Reprints CORE 1819, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. K. Kiwiel & C.H. Rosa & A. Ruszczynski, 1995. "Decomposition via Alternating Linearization," Working Papers wp95051, International Institute for Applied Systems Analysis.
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    Cited by:

    1. Le Han & Shujun Bi & Shaohua Pan, 2016. "Two-stage convex relaxation approach to least squares loss constrained low-rank plus sparsity optimization problems," Computational Optimization and Applications, Springer, vol. 64(1), pages 119-148, May.
    2. N. Aybat & G. Iyengar, 2015. "An alternating direction method with increasing penalty for stable principal component pursuit," Computational Optimization and Applications, Springer, vol. 61(3), pages 635-668, July.
    3. Le Han & Shujun Bi, 2018. "Two-stage convex relaxation approach to low-rank and sparsity regularized least squares loss," Journal of Global Optimization, Springer, vol. 70(1), pages 71-97, January.

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