Robust PCA as Bilinear Decomposition with Outlier-Sparsity Regularization

TL;DR

This paper introduces a robust PCA framework using bilinear decomposition and outlier-sparsity regularization, enhancing outlier detection and subspace tracking in high-dimensional data, with applications in surveys, social networks, and surveillance.

Contribution

It develops a novel outlier-aware PCA method based on convex relaxation and regularization, enabling scalable, real-time, and kernelized robust subspace estimation.

Findings

Effective outlier detection in synthetic and real data

Robust community detection in social networks

Intruder identification in surveillance videos

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 $\ell_0$-(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 neat 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.