  One of the key problems in tensor completion is the number of uniformly
random sample entries required for recovery guarantee. The main aim of this
paper is to study $n_1 \times n_2 \times n_3$ third-order tensor completion and
investigate into incoherence conditions of $n_3$ low-rank $n_1$-by-$n_2$ matrix
slices under the transformed tensor singular value decomposition where the
unitary transformation is applied along $n_3$-dimension. We show that such
low-rank tensors can be recovered exactly with high probability when the number
of randomly observed entries is of order $O( r\max \{n_1, n_2 \} \log ( \max \{
n_1, n_2 \} n_3))$, where $r$ is the sum of the ranks of these $n_3$ matrix
slices in the transformed tensor. By utilizing synthetic data and imaging data
sets, we demonstrate that the theoretical result can be obtained under valid
incoherence conditions, and the tensor completion performance of the proposed
method is also better than that of existing methods in terms of sample sizes
requirement.
