Abstract—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-(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. Out-liers 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 person-ality assessment surveys, as well as unveil communities in social networks, and intruders from video surveillance data. Index Terms—(Group) Lasso, outlier rejection, principal com-ponent analysis, robust statistics, sparsity. I.

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We derive a convex optimization problem for the task of segmenting sequential data, which explicitly treats pres-ence of outliers. We describe two algorithms for solving this problem, one exact and one a top-down novel ap-proach, and we derive a consistency results for the case of two segments and no outliers. Robustness to outliers is evaluated on two real-world tasks related to speech segmentation. Our algorithms outperform baseline seg-mentation algorithms. 1