  We study the problem of matrix completion in this paper. A spectral scaled
Student prior is exploited to favour the underlying low-rank structure of the
data matrix. Importantly, we provide a thorough theoretical investigation for
our approach, while such an analysis is hard to obtain and limited in
theoretical understanding of Bayesian matrix completion. More precisely, we
show that our Bayesian approach enjoys a minimax-optimal oracle inequality
which guarantees that our method works well under model misspecification and
under general sampling distribution. Interestingly, we also provide efficient
gradient-based sampling implementations for our approach by using Langevin
Monte Carlo which is novel in Bayesian matrix completion. More specifically, we
show that our algorithms are significantly faster than Gibbs sampler in this
problem. To illustrate the attractive features of our inference strategy, some
numerical simulations are conducted and an application to image inpainting is
demonstrated.
