论文标题
热-SVD：基于张量张量产品的张量的高阶T线值分解
Hot-SVD: Higher-Order t-Singular Value Decomposition for Tensors based on Tensor-Tensor Product
论文作者
论文摘要
本文考虑了一种将三阶张量（被视为管矩阵）的T-SVD推广到任意阶n的张量的方法（可以类似地将其视为订单（n-1）的输卵管张量）。 \ color {black}这样的概括与t-svd不同，对于大于三个的量张量[Martin，Shafer，Larue，Siam J. Sci。 Comput。，35（2013），A474 - A490]。该分解称为HOT-SVD，因为它可以被识别为HOSVD的张量张量产品。证明了热智能的存在。为此，引入了针对三阶张量的新转置。这种转齿对于验证热智能至关重要，因为它是输卵管张量与其展开之间的桥梁。我们建立了类似于HOSVD的热智能的一些特性，并在此过程中强调了输管张量的视角。然后引入了截断和依次截断的热-SVD，对于$（n+1）$ - th订单张量，其错误范围为$ \ sqrt {n} $。我们提供数值示例，以验证热-SVD，截短的热vd和依次截短的热-SVD。
This paper considers a way of generalizing the t-SVD of third-order tensors (regarded as tubal matrices) to tensors of arbitrary order N (which can be similarly regarded as tubal tensors of order (N-1)). \color{black}Such a generalization is different from the t-SVD for tensors of order greater than three [Martin, Shafer, Larue, SIAM J. Sci. Comput., 35 (2013), A474--A490]. The decomposition is called Hot-SVD since it can be recognized as a tensor-tensor product version of HOSVD. The existence of Hot-SVD is proved. To this end, a new transpose for third-order tensors is introduced. This transpose is crucial in the verification of Hot-SVD, since it serves as a bridge between tubal tensors and their unfoldings. We establish some properties of Hot-SVD, analogous to those of HOSVD, and in doing so we emphasize the perspective of tubal tensors. The truncated and sequentially truncated Hot-SVD are then introduced, whose error bounds are $\sqrt{N}$ for an $(N+1)$-th order tensor. We provide numerical examples to validate Hot-SVD, truncated Hot-SVD, and sequentially truncated Hot-SVD.
