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

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.