  Within the tensor singular value decomposition (T-SVD) framework, existing
robust low-rank tensor completion approaches have made great achievements in
various areas of science and engineering. Nevertheless, these methods involve
the T-SVD based low-rank approximation, which suffers from high computational
costs when dealing with large-scale tensor data. Moreover, most of them are
only applicable to third-order tensors. Against these issues, in this article,
two efficient low-rank tensor approximation approaches fusing randomized
techniques are first devised under the order-d (d >= 3) T-SVD framework. On
this basis, we then further investigate the robust high-order tensor completion
(RHTC) problem, in which a double nonconvex model along with its corresponding
fast optimization algorithms with convergence guarantees are developed. To the
best of our knowledge, this is the first study to incorporate the randomized
low-rank approximation into the RHTC problem. Empirical studies on large-scale
synthetic and real tensor data illustrate that the proposed method outperforms
other state-of-the-art approaches in terms of both computational efficiency and
estimated precision.
