In this paper, we consider the Tensor Robust Principal Component Analysis (TRPCA) problem, which aims to exactly recover the low-rank and sparse components from their sum. Our model is based on the recently proposed tensor-tensor product (or t-product) [14] . Induced by the t-product, we first rigorously deduce the tensor spectral norm, tensor nuclear norm, and tensor average rank, and show that the tensor nuclear norm is the convex envelope of the tensor average rank within the unit ball of the tensor spectral norm. These definitions, their relationships and properties are consistent with matrix cases. Equipped with the new tensor nuclear norm, we then solve the TRPCA problem by solving a convex program and provide the theoretical guarantee for the exact recovery. Our TRPCA model and recovery guarantee include matrix RPCA as a special case. Numerical experiments verify our results, and the applications to image recovery and background modeling problems demonstrate the effectiveness of our method.

中文翻译：

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新的Tensor核规范的Tensor鲁棒主成分分析

在本文中，我们考虑了张量鲁棒主成分分析（TRPCA）问题，该问题的目的是从它们的和中准确恢复低秩和稀疏成分。我们的模型基于最近提出的张量-张量积（或t积）[14] 。由t乘积推导，我们首先严格推导张量谱范数，张量核范数和张量平均秩，并证明张量核范数是张量谱范数的单位球内张量平均秩的凸包络。 。这些定义，它们之间的关系和性质与矩阵情况一致。配备了新的张量核规范，然后我们通过求解凸程序来解决TRPCA问题，并为精确恢复提供了理论保证。我们的TRPCA模型和恢复保证包括矩阵RPCA（作为特例）。数值实验验证了我们的结果，并将其应用于图像恢复和背景建模问题证明了该方法的有效性。