Sparsity control for robust principal component analysis

Principal component analysis (PCA) is widely used for high-dimensional data analysis, with well-documented applications in 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 component analysis model is shown closely related to that obtained from an ℓ₀-(pseudo)norm-regularized criterion encouraging sparsity in a matrix explicitly modeling the outliers. This connection suggests efficient (approximate) solvers based on convex relaxation, which lead naturally to a family of robust estimators subsuming Huber's optimal M-class. Outliers are identified by tuning a regularization parameter, which amounts to controlling the sparsity of the outlier matrix along the whole robustification path of (group)-Lasso solutions. Novel algorithms are developed to: i) estimate the low-rank data model both robustly and adaptively; and ii) determine principal components robustly in (possibly) infinite-dimensional feature spaces. Numerical tests corroborate the effectiveness of the proposed robust PCA scheme for a video surveillance task.