Many kinds of real-world multi-way signal, like color images, videos, etc., are represented in tensor form and may often be corrupted by outliers. To recover an unknown signal tensor corrupted by outliers, tensor robust principal component analysis (TRPCA) serves as a robust tensorial modification of the fundamental PCA. Recently, a successful TRPCA model based on the tubal nuclear norm (TNN) (Lu et al. in IEEE Trans Pattern Anal Mach Intell 42:925–938, 2019) has attracted much attention thanks to its superiority in many applications. However, TNN is computationally expensive due to the requirement of full singular value decompositions, seriously limiting its scalability to large tensors. To address this issue, we propose a new TRPCA model which adopts a factorization strategy. Algorithmically, an algorithm based on the non-convex augmented Lagrangian method is developed with convergence guarantee. Theoretically, we rigorously establish the sub-optimality of the proposed algorithm. We also extend the proposed model to the robust tensor completion problem. Both the effectiveness and efficiency of the proposed algorithm is demonstrated through extensive experiments on both synthetic and real data sets.

多种现实世界中的多路信号（例如彩色图像，视频等）以张量形式表示，并且可能经常被异常值所破坏。为了恢复由异常值破坏的未知信号张量，张量鲁棒主成分分析（TRPCA）用作基本PCA的鲁棒张量修改。最近，基于输卵管核范数（TNN）的成功的TRPCA模型（Lu等人，在IEEE Trans Pattern Anal Mach Intell 42：925-938，2019）中因其在许多应用中的优越性而备受关注。但是，由于需要完全奇异值分解，因此TNN的计算量很大，从而严重限制了其对大张量的可伸缩性。为了解决这个问题，我们提出了一种新的TRPCA模型，该模型采用分解策略。从算法上讲 在收敛保证的基础上，提出了一种基于非凸增强拉格朗日方法的算法。从理论上讲，我们严格地建立了所提出算法的次优性。我们还将所提出的模型扩展到鲁棒的张量完成问题。通过对合成数据集和真实数据集进行大量实验，证明了该算法的有效性和效率。