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.

翻译：在张量奇异值分解（T-SVD）框架下，现有的鲁棒低秩张量补全方法已在科学与工程的多个领域取得了重大进展。然而，这些方法均涉及基于T-SVD的低秩逼近，在处理大规模张量数据时计算成本高昂。此外，其中绝大多数方法仅适用于三阶张量。针对上述问题，本文首先在d阶（d≥3）T-SVD框架下设计了两种融合随机化技术的高效低秩张量逼近方法。在此基础上，我们进一步研究了鲁棒高阶张量补全（RHTC）问题，开发了一种双非凸模型及其相应的具有收敛性保证的快速优化算法。据我们所知，这是首次将随机低秩逼近融入RHTC问题的研究。在大规模合成张量和真实张量数据上的实验表明，本文所提方法在计算效率和估计精度两方面均优于其他当前最优方法。