Bayesian Robust Principal Component Analysis

• Published in: IEEE transactions on image processing Vol. 20; no. 12; pp. 3419 - 3430
• Main Authors: Xinghao Ding, Lihan He, Carin, L.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.12.2011; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Applied sciences; Approximation; Bayesian analysis; Bayesian methods; Bayesian modeling; Computational modeling; Detection, estimation, filtering, equalization, prediction; Exact sciences and technology; Frames; Image processing; Information, signal and communications theory; low-rank matrix; Markov dependency; Markov processes; Matrix decomposition; Noise; Noise levels; Principal component analysis; Representations; Signal and communications theory; Signal processing; Signal, noise; Sparse matrices; sparsity; Statistics; Studies; Telecommunications and information theory; Bayes estimation; Performance evaluation; Markov process; State of the art; Noise; Noise reduction; Non stationary condition; sparsity; Markov dependency; Modeling; Implementation; Optimization; Noise level; Statistical method; Outlier; Bayesian modeling; Signal processing; Open market; Bayes methods; Hierarchical system; Principal component analysis; low-rank matrix
• ISSN: 1057-7149; 1941-0042
• DOI: 10.1109/TIP.2011.2156801

A hierarchical Bayesian model is considered for decomposing a matrix into low-rank and sparse components, assuming the observed matrix is a superposition of the two. The matrix is assumed noisy, with unknown and possibly non-stationary noise statistics. The Bayesian framework infers an approximate representation for the noise statistics while simultaneously inferring the low-rank and sparse-outlier contributions; the model is robust to a broad range of noise levels, without having to change model hyperparameter settings. In addition, the Bayesian framework allows exploitation of additional structure in the matrix. For example, in video applications each row (or column) corresponds to a video frame, and we introduce a Markov dependency between consecutive rows in the matrix (corresponding to consecutive frames in the video). The properties of this Markov process are also inferred based on the observed matrix, while simultaneously denoising and recovering the low-rank and sparse components. We compare the Bayesian model to a state-of-the-art optimization-based implementation of robust PCA; considering several examples, we demonstrate competitive performance of the proposed model.