Iterative estimation of constrained rank-one matrices in noise

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 simple iterative procedure that reduces the problem to a sequence of scalar estimation computations. The method is 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 estimates from the proposed iterative procedure is described by a simple scalar equivalent model, where the distribution of the estimates 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 method thus provides a computationally simple and general method for rank-one estimation problems with a precise analysis in certain high-dimensional settings.