Near-Optimal Noisy Low-Tubal-Rank Tensor Completion via Singular Tube Thresholding

The problem of completing low-tubal-rank tensors from incomplete noisy observations is studied. To recover the underlying tensor, an iterative singular tube thresholding (ISTT) algorithm is proposed. To explore the statistical performance of the proposed algorithm, the estimation error in terms of the Frobenius norm is upper bounded non-asymptotically. The minimax optimal lower bound of the estimation error for tensors with limited tubal rank and bounded l∞-norm is also established. The lower bound indicates that our upper bound is near-optimal up to a logarithm factor. It is shown through numerical experiments that our upper bound can precisely predict the scaling behavior of the estimation error. Experiments on real-world video data demonstrates the effectiveness of the proposed algorithm.