Shi Qiquan, Cheung Yiu-Ming, Lou Jian
IEEE Trans Cybern. 2022 Oct;52(10):10667-10682. doi: 10.1109/TCYB.2021.3067676. Epub 2022 Sep 19.
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.
张量奇异值分解（t-SVD）最近在部分和/或损坏观测下的张量恢复中越来越受欢迎。然而，现有的基于t-SVD的方法既没有利用秩先验，也没有提供准确的秩估计（RE），这会限制它们的恢复性能。从实际角度来看，张量RE问题是不平凡且难以解决的。因此，在本文中，我们旨在基于t-SVD从损坏观测中确定内在低秩张量的正确秩，并进一步利用估计的秩来改善恢复结果。具体来说，我们首先推导张量的张量核范数（TNN）与其f-对角张量的等价性。然后，我们同时最小化f-对角张量的重构误差和TNN，从而得到RE。随后，我们通过去除TNN正则化项来放宽我们的模型，以提高恢复性能。此外，我们通过提出具有RE的鲁棒张量主成分分析和鲁棒张量补全来考虑存在缺失数据和/或严重损坏的更一般情况。这些鲁棒方法可以通过用正确估计的秩来改进模型来实现成功恢复。实验结果表明，所提出的方法显著优于现有方法。
