  In this paper a new Riemannian rank adaptive method (RRAM) is proposed for
the low-rank tensor completion problem (LRTCP) formulated as a least-squares
optimization problem on the algebraic variety of tensors of bounded
tensor-train (TT) rank. The method iteratively optimizes over fixed-rank smooth
manifolds using a Riemannian conjugate gradient algorithm from Steinlechner
(2016) and gradually increases the rank by computing a descent direction in the
tangent cone to the variety. Additionally, a numerical method to estimate the
amount of rank increase is proposed based on a theoretical result for the
stationary points of the low-rank tensor approximation problem and a definition
of an estimated TT-rank. Furthermore, when the iterate comes close to a
lower-rank set, the RRAM decreases the rank based on the TT-rounding algorithm
from Oseledets (2011) and a definition of a numerical rank. We prove that the
TT-rounding algorithm can be considered as an approximate projection onto the
lower-rank set which satisfies a certain angle condition to ensure that the
image is sufficiently close to that of an exact projection. Several numerical
experiments are given to illustrate the use of the RRAM and its subroutines in
{\Matlab}. Furthermore, in all experiments the proposed RRAM outperforms the
state-of-the-art RRAM for tensor completion in the TT format from Steinlechner
(2016) in terms of computation time.
