  The goal of tensor completion is to fill in missing entries of a partially
known tensor (possibly including some noise) under a low-rank constraint. This
may be formulated as a least-squares problem. The set of tensors of a given
multilinear rank is known to admit a Riemannian manifold structure, thus
methods of Riemannian optimization are applicable. In our work, we derive the
Riemannian Hessian of an objective function on the low-rank tensor manifolds
using the Weingarten map, a concept from differential geometry. We discuss the
convergence properties of Riemannian trust-region methods based on the exact
Hessian and standard approximations, both theoretically and numerically. We
compare our approach to Riemannian tensor completion methods from recent
literature, both in terms of convergence behaviour and computational
complexity. Our examples include the completion of randomly generated data with
and without noise and recovery of multilinear data from survey statistics.
