Nonconvex Robust High-Order Tensor Completion Using Randomized Low-Rank Approximation

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.

Guaranteed Tensor Recovery Fused Low-rankness and Smoothness

The tensor data recovery task has thus attracted much research attention in recent years. Solving such an ill-posed problem generally requires to explore intrinsic prior structures underlying tensor data, and formulate them as certain forms of regularization terms for guiding a sound estimate of the restored tensor. Recent research have made significant progress by adopting two insightful tensor priors, i.e., global low-rankness (L) and local smoothness (S) across different tensor modes, which are always encoded as a sum of two separate regularization terms into the recovery models. However, unlike the primary theoretical developments on low-rank tensor recovery, these joint L+S models have no theoretical exact-recovery guarantees yet, making the methods lack reliability in real practice. To this crucial issue, in this work, we build a unique regularization term, which essentially encodes both L and S priors of a tensor simultaneously. Especially, by equipping this single regularizer into the recovery models, we can rigorously prove the exact recovery guarantees for two typical tensor recovery tasks, i.e., tensor completion (TC) and tensor robust principal component analysis (TRPCA). To the best of our knowledge, this should be the first exact-recovery results among all related L+S methods for tensor recovery. Significant recovery accuracy improvements over many other SOTA methods in several TC and TRPCA tasks with various kinds of visual tensor data are observed in extensive experiments. Typically, our method achieves a workable performance when the missing rate is extremely large, e.g., 99.5%, for the color image inpainting task, while all its peers totally fail in such challenging case.