论文标题
三阶张量的三重分解和张量回收
Triple Decomposition and Tensor Recovery of Third Order Tensors
论文作者
论文摘要
在本文中，我们为三阶张量引入了一种新的张量分解，该分解以平衡的方式将三阶张量分解为三个三阶低级张量。我们称这种分解为三重分解，而相应的排名三重等级。对于三阶张量，其CP分解可以被视为其三重分解的特殊情况。三阶张量的三重等级不大于塔克等级的中间值，并且严格小于基本示例中塔克等级的中间值。这些表明，只要可以通过低等级CP或Tucker分解近似，就可以通过低等级的三重分解近似实用数据即可。该理论发现已被数值确认。数值测试表明，来自Internet流量和视频图像等实用应用程序的三阶张量数据是低三重等级的。提出了基于低等级三重分解的张量恢复方法。建立了其收敛性和收敛速度。数值实验证实了该方法的效率。
In this paper, we introduce a new tensor decomposition for third order tensors, which decomposes a third order tensor to three third order low rank tensors in a balanced way. We call such a decomposition the triple decomposition, and the corresponding rank the triple rank. For a third order tensor, its CP decomposition can be regarded as a special case of its triple decomposition. The triple rank of a third order tensor is not greater than the middle value of the Tucker rank, and is strictly less than the middle value of the Tucker rank for an essential class of examples. These indicate that practical data can be approximated by low rank triple decomposition as long as it can be approximated by low rank CP or Tucker decomposition. This theoretical discovery is confirmed numerically. Numerical tests show that third order tensor data from practical applications such as internet traffic and video 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.
