On $O( \max \{n_1, n_2 \}\log ( \max \{ n_1, n_2 \} n_3) )$ Sample Entries for $n_1 \times n_2 \times n_3$ Tensor Completion via Unitary Transformation

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.