  The low-rank matrix completion problem can be solved by Riemannian
optimization on a fixed-rank manifold. However, a drawback of the known
approaches is that the rank parameter has to be fixed a priori. In this paper,
we consider the optimization problem on the set of bounded-rank matrices. We
propose a Riemannian rank-adaptive method, which consists of fixed-rank
optimization, rank increase step and rank reduction step. We explore its
performance applied to the low-rank matrix completion problem. Numerical
experiments on synthetic and real-world datasets illustrate that the proposed
rank-adaptive method compares favorably with state-of-the-art algorithms. In
addition, it shows that one can incorporate each aspect of this rank-adaptive
framework separately into existing algorithms for the purpose of improving
performance.
