Tensor singular value decomposition (t-SVD) has recently become increasingly popular for tensor recovery under partial and/or corrupted observations. However, the existing t-SVD-based methods neither make use of a rank prior nor provide an accurate rank estimation (RE), which would limit their recovery performance. From the practical perspective, the tensor RE problem is nontrivial and difficult to solve. In this article, we, therefore, aim to determine the correct rank of an intrinsic low-rank tensor from corrupted observations based on t-SVD and further improve recovery results with the estimated rank. Specifically, we first induce the equivalence of the tensor nuclear norm (TNN) of a tensor and its f-diagonal tensor. We then simultaneously minimize the reconstruction error and TNN of the f-diagonal tensor, leading to RE. Subsequently, we relax our model by removing the TNN regularizer to improve the recovery performance. Furthermore, we consider more general cases in the presence of missing data and/or gross corruptions by proposing robust tensor principal component analysis and robust tensor completion with RE. The robust methods can achieve successful recovery by refining the models with correct estimated ranks. Experimental results show that the proposed methods outperform the state-of-the-art methods with significant improvements.

中文翻译：

---

稳健的Tensor SVD和具有秩估计的恢复。

张量奇异值分解（t-SVD）最近在部分和/或损坏的观察下恢复张量变得越来越流行。但是，现有的基于t-SVD的方法既未利用秩优先，也未提供准确的秩估计（RE），这将限制其恢复性能。从实践的角度来看，张量RE问题是不平凡的，而且难以解决。因此，在本文中，我们旨在基于t-SVD从损坏的观测值确定固有的低秩张量的正确秩，并进一步用估计秩提高恢复结果。具体来说，我们首先得出张量及其f对角张量的张量核范数（TNN）的等价关系。然后，我们同时最小化f对角张量的重建误差和TNN，从而导致RE。随后，我们通过删除TNN正则器来放松我们的模型，以提高恢复性能。此外，通过提出鲁棒的张量主成分分析和鲁棒的张量补全，我们考虑了在缺少数据和/或严重损坏的情况下的更一般情况。健壮的方法可以通过用正确的估计等级细化模型来实现成功的恢复。实验结果表明，所提出的方法比现有方法具有明显的改进。健壮的方法可以通过用正确的估计等级细化模型来实现成功的恢复。实验结果表明，所提出的方法比现有方法具有明显的改进。健壮的方法可以通过用正确的估计等级细化模型来实现成功的恢复。实验结果表明，所提出的方法比现有方法具有明显的改进。