Bilinear Factor Matrix Norm Minimization for Robust PCA: Algorithms and Applications

Fanhua Shang, James Cheng, Yuanyuan Liu, Zhi Quan Luo, Zhouchen Lin

Research output: Contribution to journalArticlepeer-review

46 Scopus citations


The heavy-tailed distributions of corrupted outliers and singular values of all channels in low-level vision have proven effective priors for many applications such as background modeling, photometric stereo and image alignment. And they can be well modeled by a hyper-Laplacian. However, the use of such distributions generally leads to challenging non-convex, non-smooth and non-Lipschitz problems, and makes existing algorithms very slow for large-scale applications. Together with the analytic solutions to \ell -{p} -norm minimization with two specific values of p , i.e., p=1/2 and p=2/3 , we propose two novel bilinear factor matrix norm minimization models for robust principal component analysis. We first define the double nuclear norm and Frobenius/nuclear hybrid norm penalties, and then prove that they are in essence the Schatten- 1/2 and 2/3 quasi-norms, respectively, which lead to much more tractable and scalable Lipschitz optimization problems. Our experimental analysis shows that both our methods yield more accurate solutions than original Schatten quasi-norm minimization, even when the number of observations is very limited. Finally, we apply our penalties to various low-level vision problems, e.g., text removal, moving object detection, image alignment and inpainting, and show that our methods usually outperform the state-of-the-art methods.

Original languageEnglish (US)
Article number8025394
Pages (from-to)2066-2080
Number of pages15
JournalIEEE Transactions on Pattern Analysis and Machine Intelligence
Issue number9
StatePublished - Sep 1 2018

Bibliographical note

Funding Information:
Fanhua Shang, James Cheng and Yuanyuan Liu were supported in part by Grants (CUHK 14206715 & 14222816) from the Hong Kong RGC, ITF 6904079, Grant 3132821 funded by the Research Committee of CUHK. Zhi-Quan Luo was supported in part by the National Science Foundation (NSF) under Grant CCF-1526434, the National Natural Science Foundation of China under Grants 61571384 and 61731018, and the Leading Talents of Guang Dong Province program, Grant No. 00201510. Zhouchen Lin was supported by National Basic Research Program of China (973 Program) (grant no. 2015CB352502), National Natural Science Foundation (NSF) of China (grant nos. 61625301, 61731018, and 61231002), and Qualcomm. Fanhua Shang and Yuanyuan Liu are the corresponding authors.


  • Frobenius/nuclear norm penalty
  • Robust principal component analysis
  • Schatten-p quasi-norm
  • alternating direction method of multipliers (ADMM)
  • double nuclear norm penalty
  • p-norm
  • rank minimization

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