  This paper investigates the low-rank tensor completion problem, which is
about recovering a tensor from partially observed entries. We consider this
problem in the tensor train format and extend the preconditioned metric from
the matrix case to the tensor case. The first-order and second-order quotient
geometry of the manifold of fixed tensor train rank tensors under this metric
is studied in detail. Algorithms, including Riemannian gradient descent,
Riemannian conjugate gradient, and Riemannian Gauss-Newton, have been proposed
for the tensor completion problem based on the quotient geometry. It has also
been shown that the Riemannian Gauss-Newton method on the quotient geometry is
equivalent to the Riemannian Gauss-Newton method on the embedded geometry with
a specific retraction. Empirical evaluations on random instances as well as on
function-related tensors show that the proposed algorithms are competitive with
other existing algorithms in terms of recovery ability, convergence
performance, and reconstruction quality.
