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