  We consider the problem of estimating a rank-one matrix in Gaussian noise
under a probabilistic model for the left and right factors of the matrix. The
probabilistic model can impose constraints on the factors including sparsity
and positivity that arise commonly in learning problems. We propose a family of
algorithms that reduce the problem to a sequence of scalar estimation
computations. These algorithms are similar to approximate message passing
techniques based on Gaussian approximations of loopy belief propagation that
have been used recently in compressed sensing. Leveraging analysis methods by
Bayati and Montanari, we show that the asymptotic behavior of the algorithm is
described by a simple scalar equivalent model, where the distribution of the
estimates at each iteration is identical to certain scalar estimates of the
variables in Gaussian noise. Moreover, the effective Gaussian noise level is
described by a set of state evolution equations. The proposed approach to
deriving algorithms thus provides a computationally simple and general method
for rank-one estimation problems with a precise analysis in certain
high-dimensional settings.
