  We extend the generalized approximate message passing (G-AMP) approach,
originally proposed for high-dimensional generalized-linear regression in the
context of compressive sensing, to the generalized-bilinear case, which enables
its application to matrix completion, robust PCA, dictionary learning, and
related matrix-factorization problems. In the first part of the paper, we
derive our Bilinear G-AMP (BiG-AMP) algorithm as an approximation of the
sum-product belief propagation algorithm in the high-dimensional limit, where
central-limit theorem arguments and Taylor-series approximations apply, and
under the assumption of statistically independent matrix entries with known
priors. In addition, we propose an adaptive damping mechanism that aids
convergence under finite problem sizes, an expectation-maximization (EM)-based
method to automatically tune the parameters of the assumed priors, and two
rank-selection strategies. In the second part of the paper, we discuss the
specializations of EM-BiG-AMP to the problems of matrix completion, robust PCA,
and dictionary learning, and present the results of an extensive empirical
study comparing EM-BiG-AMP to state-of-the-art algorithms on each problem. Our
numerical results, using both synthetic and real-world datasets, demonstrate
that EM-BiG-AMP yields excellent reconstruction accuracy (often best in class)
while maintaining competitive runtimes and avoiding the need to tune
algorithmic parameters.
