  We introduce the bilinear generalized vector approximate message passing
(BiG-VAMP) algorithm which jointly recovers two matrices U and V from their
noisy product through a probabilistic observation model. BiG-VAMP provides
computationally efficient approximate implementations of both max-sum and
sumproduct loopy belief propagation (BP). We show how the proposed BiG-VAMP
algorithm recovers different types of structured matrices and overcomes the
fundamental limitations of other state-of-the-art approaches to the bilinear
recovery problem, such as BiG-AMP, BAd-VAMP and LowRAMP. In essence, BiG-VAMP
applies to a broader class of practical applications which involve a general
form of structured matrices. For the sake of theoretical performance
prediction, we also conduct a state evolution (SE) analysis of the proposed
algorithm and show its consistency with the asymptotic empirical mean-squared
error (MSE). Numerical results on various applications such as matrix
factorization, dictionary learning, and matrix completion demonstrate
unambiguously the effectiveness of the proposed BiG-VAMP algorithm and its
superiority over stateof-the-art algorithms. Using the developed SE framework,
we also examine (as one example) the phase transition diagrams of the matrix
completion problem, thereby unveiling a low detectability region corresponding
to the low signal-to-noise ratio (SNR) regime.
