  In this paper, we aim at the problem of tensor data completion. Tensor-train
decomposition is adopted because of its powerful representation ability and
linear scalability to tensor order. We propose an algorithm named Sparse
Tensor-train Optimization (STTO) which considers incomplete data as sparse
tensor and uses first-order optimization method to find the factors of
tensor-train decomposition. Our algorithm is shown to perform well in
simulation experiments at both low-order cases and high-order cases. We also
employ a tensorization method to transform data to a higher-order form to
enhance the performance of our algorithm. The results of image recovery
experiments in various cases manifest that our method outperforms other
completion algorithms. Especially when the missing rate is very high, e.g.,
90\% to 99\%, our method is significantly better than the state-of-the-art
methods.
