International Journal of Computer Vision

, Volume 111, Issue 3, pp 315–344 | Cite as

Practical Matrix Completion and Corruption Recovery Using Proximal Alternating Robust Subspace Minimization

  • Yu-Xiang WangEmail author
  • Choon Meng Lee
  • Loong-Fah Cheong
  • Kim-Chuan Toh


Low-rank matrix completion is a problem of immense practical importance. Recent works on the subject often use nuclear norm as a convex surrogate of the rank function. Despite its solid theoretical foundation, the convex version of the problem often fails to work satisfactorily in real-life applications. Real data often suffer from very few observations, with support not meeting the randomness requirements, ubiquitous presence of noise and potentially gross corruptions, sometimes with these simultaneously occurring. This paper proposes a Proximal Alternating Robust Subspace Minimization method to tackle the three problems. The proximal alternating scheme explicitly exploits the rank constraint on the completed matrix and uses the \(\ell _0\) pseudo-norm directly in the corruption recovery step. We show that the proposed method for the non-convex and non-smooth model converges to a stationary point. Although it is not guaranteed to find the global optimal solution, in practice we find that our algorithm can typically arrive at a good local minimizer when it is supplied with a reasonably good starting point based on convex optimization. Extensive experiments with challenging synthetic and real data demonstrate that our algorithm succeeds in a much larger range of practical problems where convex optimization fails, and it also outperforms various state-of-the-art algorithms.


Low rank matrix completion Robust matrix factorization Non-convex optimization SfM Photometric stereo 



This work was supported by the Singapore PSF grant 1321202075.


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Yu-Xiang Wang
    • 1
    • 2
    Email author
  • Choon Meng Lee
    • 2
  • Loong-Fah Cheong
    • 2
  • Kim-Chuan Toh
    • 2
  1. 1.Carnegie Mellon UniversityPittsburghUSA
  2. 2.National University of SingaporeSingaporeSingapore

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