Turbo-Type Message Passing Algorithms for Compressed Robust Principal Component Analysis

Abstract

Compressed robust principal component analysis (RPCA), in which a low-rank matrix $\boldsymbol {L}$ and a sparse matrix $\boldsymbol {S}$ are recovered from an underdetermined amount of noisy linear measurements of their sum $\boldsymbol {L}+\boldsymbol {S}$ , arises in various applications such as face recognition and video foreground/background separation. This problem can be solved by Bayesian inference based iterative algorithms. However, most existing Bayesian algorithms factorize $\boldsymbol {L}$ into the product of two rank- $r$ matrices, and estimate the two rank- $r$ matrices (rather than $\boldsymbol {L}$ itself) in the iterative process, where $r$ is the rank of $\boldsymbol {L}$ . On one hand, this factorization is not essential to the original problem and so may cause a potential performance loss. On the other hand, the existing Bayesian algorithms assume a certain probability model for the low-rank matrix $\boldsymbol {L}$ and the sparse matrix $\boldsymbol {S}$ , whereas the probability model of $\boldsymbol {L}$ and $\boldsymbol {S}$ is usually difficult to acquire in real applications. In this paper, we develop a Bayesian message passing algorithm, termed turbo-type message passing (TMP), for the compressed RPCA problem. We show that the proposed TMP algorithm significantly outperforms the state-of-the-art compressed RPCA algorithms, and requires a much lower computational complexity. We also show that TMP does not assume any prior probability model for $\boldsymbol {L}$ and $\boldsymbol {S}$ ; TMP even does not require the knowledge of the environmental information, such as the rank of $\boldsymbol {L}$ and the sparsity level of $\boldsymbol {S}$ . Therefore, TMP gives a promising approach for real applications of the compressed RPCA problem.