  The tensor train (TT) format enjoys appealing advantages in handling
structural high-order tensors. The recent decade has witnessed the wide
applications of TT-format tensors from diverse disciplines, among which tensor
completion has drawn considerable attention. Numerous fast algorithms,
including the Riemannian gradient descent (RGrad), have been proposed for the
TT-format tensor completion. However, the theoretical guarantees of these
algorithms are largely missing or sub-optimal, partly due to the complicated
and recursive algebraic operations in TT-format decomposition. Moreover,
existing results established for the tensors of other formats, for example,
Tucker and CP, are inapplicable because the algorithms treating TT-format
tensors are substantially different and more involved. In this paper, we
provide, to our best knowledge, the first theoretical guarantees of the
convergence of RGrad algorithm for TT-format tensor completion, under a nearly
optimal sample size condition. The RGrad algorithm converges linearly with a
constant contraction rate that is free of tensor condition number without the
necessity of re-conditioning. We also propose a novel approach, referred to as
the sequential second-order moment method, to attain a warm initialization
under a similar sample size requirement. As a byproduct, our result even
significantly refines the prior investigation of RGrad algorithm for matrix
completion. Lastly, statistically (near) optimal rate is derived for RGrad
algorithm if the observed entries consist of random sub-Gaussian noise.
Numerical experiments confirm our theoretical discovery and showcase the
computational speedup gained by the TT-format decomposition.
