Robust PCA as bilinear decomposition with outlier-sparsity regularization

Gonzalo Mateos, Georgios B. Giannakis

Research output: Contribution to journalArticlepeer-review

60 Scopus citations


Principal component analysis (PCA) is widely used for dimensionality reduction, with well-documented merits in various applications involving high-dimensional data, including computer vision, preference measurement, and bioinformatics. In this context, the fresh look advocated here permeates benefits from variable selection and compressive sampling, to robustify PCA against outliers. A least-trimmed squares estimator of a low-rank bilinear factor analysis model is shown closely related to that obtained from an ℓ 0 -(pseudo)norm-regularized criterion encouraging sparsity in a matrix explicitly modeling the outliers. This connection suggests robust PCA schemes based on convex relaxation, which lead naturally to a family of robust estimators encompassing Huber's optimal M-class as a special case. Outliers are identified by tuning a regularization parameter, which amounts to controlling sparsity of the outlier matrix along the whole robustification path of (group) least-absolute shrinkage and selection operator (Lasso) solutions. Beyond its ties to robust statistics, the developed outlier-aware PCA framework is versatile to accommodate novel and scalable algorithms to: i) track the low-rank signal subspace robustly, as new data are acquired in real time; and ii) determine principal components robustly in (possibly) infinite-dimensional feature spaces. Synthetic and real data tests corroborate the effectiveness of the proposed robust PCA schemes, when used to identify aberrant responses in personality assessment surveys, as well as unveil communities in social networks, and intruders from video surveillance data.

Original languageEnglish (US)
Article number6218787
Pages (from-to)5176-5190
Number of pages15
JournalIEEE Transactions on Signal Processing
Issue number10
StatePublished - 2012

Bibliographical note

Funding Information:
Manuscript received November 07, 2011; revised April 12, 2012; accepted June 08, 2012. Date of publication June 15, 2012; date of current version September 11, 2012. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Ignacio Santamaria. This work was supported by MURI (AFOSR FA9550-10-1-0567) grant. Part of the paper appeared in the Proceedings of the Fourty-Fourth Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, CA, November 7–10, 2010.


  • (Group) Lasso
  • outlier rejection
  • principal component analysis
  • robust statistics
  • sparsity

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