  The goal of tensor completion is to fill in missing entries of a partially
known tensor under a low-rank constraint. In this paper, we mainly study low
rank third-order tensor completion problems by using Riemannian optimization
methods on the smooth manifold. Here the tensor rank is defined to be a set of
matrix ranks where the matrices are the slices of the transformed tensor
obtained by applying the Fourier-related transformation onto the tubes of the
original tensor. We show that with suitable incoherence conditions on the
underlying low rank tensor, the proposed Riemannian optimization method is
guaranteed to converge and find such low rank tensor with a high probability.
In addition, numbers of sample entries required for solving low rank tensor
completion problem under different initialized methods are studied and derived.
Numerical examples for both synthetic and image data sets are reported to
demonstrate the proposed method is able to recover low rank tensors.
