Sparse Bayesian Methods for Low-Rank Matrix Estimation

• Published in: IEEE transactions on signal processing Vol. 60; no. 8; pp. 3964 - 3977
• Main Authors: Babacan, S. Derin, Luessi, Martin, Molina, Rafael, Katsaggelos, Aggelos K.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.08.2012; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Applied sciences; Bayesian analysis; Bayesian methods; Detection, estimation, filtering, equalization, prediction; Estimating; Estimation; Exact sciences and technology; Information, signal and communications theory; low-rankness; Mathematical analysis; Mathematical model; Matrices; matrix completion; Matrix decomposition; Matrix methods; outlier detection; Principal component analysis; Reconstruction; Recovery; robust principal component analysis; Robustness; Signal and communications theory; Signal, noise; sparse Bayesian learning; Sparse matrices; sparsity; Telecommunications and information theory; variational Bayesian inference; Bayes estimation; Performance evaluation; Rank statistic; High performance; State of the art; matrix completion; variational Bayesian inference; Theoretical study; low-rankness; Matrix factorization; robust principal component analysis; sparsity; Algorithm; outlier detection; Learning; Statistical method; Bayesian methods; Signal processing; Error detection; Bayes methods; sparse Bayesian learning; Principal component analysis
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2012.2197748

Recovery of low-rank matrices has recently seen significant activity in many areas of science and engineering, motivated by recent theoretical results for exact reconstruction guarantees and interesting practical applications. In this paper, we present novel recovery algorithms for estimating low-rank matrices in matrix completion and robust principal component analysis based on sparse Bayesian learning (SBL) principles. Starting from a matrix factorization formulation and enforcing the low-rank constraint in the estimates as a sparsity constraint, we develop an approach that is very effective in determining the correct rank while providing high recovery performance. We provide connections with existing methods in other similar problems and empirical results and comparisons with current state-of-the-art methods that illustrate the effectiveness of this approach.