SIAM Journal on Matrix Analysis and Applications, Volume 42, Issue 1, Page 299-329, January 2021.
Motivated by the Tucker decomposition, in this paper we introduce a new tensor decomposition for third order tensors, which decomposes a third order tensor to three third order factor tensors. Each factor tensor has two low dimensions. We call such a decomposition the triple decomposition, and the corresponding rank the triple rank. The triple rank of a third order tensor is not greater than the middle value of the Tucker rank. The number of parameters in the bilevel form of standard triple decomposition is less than the number of parameters of Tucker decomposition in substantial cases. The theoretical discovery is confirmed numerically. Numerical tests show that third order tensor data from practical applications such as internet traffic and image are of low triple ranks. A tensor recovery method based on low rank triple decomposition is proposed. Its convergence and convergence rate are established. Numerical experiments confirm the efficiency of this method.

中文翻译：

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三阶张量的三重分解和张量恢复

SIAM 矩阵分析与应用杂志，第 42 卷，第 1 期，第 299-329 页，2021 年 1 月。
受 Tucker 分解的启发，本文引入了一种新的三阶张量张量分解，将三阶张量分解为三个三阶因子张量。每个因子张量都有两个低维度。我们称这种分解为三重分解，相应的秩为三重秩。三阶张量的三阶不大于 Tucker 秩的中间值。在实际情况下，标准三重分解的双层形式的参数数量少于 Tucker 分解的参数数量。理论发现得到了数值证实。数值测试表明，来自互联网流量和图像等实际应用的三阶张量数据属于低三阶。提出了一种基于低秩三元分解的张量恢复方法。其收敛性和收敛速度成立。数值实验证实了该方法的有效性。