  We develop latent variable models for Bayesian learning based low-rank matrix
completion and reconstruction from linear measurements. For under-determined
systems, the developed methods are shown to reconstruct low-rank matrices when
neither the rank nor the noise power is known a-priori. We derive relations
between the latent variable models and several low-rank promoting penalty
functions. The relations justify the use of Kronecker structured covariance
matrices in a Gaussian based prior. In the methods, we use evidence
approximation and expectation-maximization to learn the model parameters. The
performance of the methods is evaluated through extensive numerical
simulations.
