  We propose a novel Riemannian manifold preconditioning approach for the
tensor completion problem with rank constraint. A novel Riemannian metric or
inner product is proposed that exploits the least-squares structure of the cost
function and takes into account the structured symmetry that exists in Tucker
decomposition. The specific metric allows to use the versatile framework of
Riemannian optimization on quotient manifolds to develop preconditioned
nonlinear conjugate gradient and stochastic gradient descent algorithms for
batch and online setups, respectively. Concrete matrix representations of
various optimization-related ingredients are listed. Numerical comparisons
suggest that our proposed algorithms robustly outperform state-of-the-art
algorithms across different synthetic and real-world datasets.
