众所周知，具有$m \le n$的$m \times n$矩阵的谱范数和Frobenius范数之间的最小可能比率是$1/\sqrt{m}$，并且（直到标量缩放）仅由具有成对正交行的矩阵获得。本文研究了$d$阶$n_1 \times \dots \times n_d$张量的谱模与Frobenius模之间的最小可能比，即最佳秩一逼近比.对于$n_1 \le \dots \le n_d$的大多数配置，确切值未知。使用实域上正交张量的自然定义（分别为，复域上的酉张量），它表明，当且仅当一个张量是正交的（分别，统一的）到缩放。正交张量或酉张量是否存在取决于维度$n_1,\dots,n_d$和场。三阶实正交张量的（不）存在性和复合代数上的经典Hurwitz问题之间的联系可以建立：大小为$\ell \times m \times n$的正交张量的存在性等价于三重$[\ell,m,n]$对Hurwitz问题的容许性。高阶张量的一些影响，然后给出。例如，$d \ge 3$阶的实正交$n \times \dots \times n$张量确实存在，但仅当$n = 1,2,4,8$。在复杂的情况下，情况更加激烈：只有当$\ell m \le n$时，才存在大小为$\ell \times m \times n$且具有$\ell \le m \le n$的酉张量。最后给出了谱范数计算的一些数值例子。
英文摘要：
As is well known, the smallest possible ratio between the spectral norm and the Frobenius norm of an $m \times n$ matrix with $m \le n$ is $1/\sqrt{m}$ and is (up to scalar scaling) attained only by matrices having pairwise orthonormal rows. In the present paper, the smallest possible ratio between spectral and Frobenius norms of $n_1 \times \dots \times n_d$ tensors of order $d$, also called the best rank-one approximation ratio in the literature, is investigated. The exact value is not known for most configurations of $n_1 \le \dots \le n_d$. Using a natural definition of orthogonal tensors over the real field (resp., unitary tensors over the complex field), it is shown that the obvious lower bound $1/\sqrt{n_1 \cdots n_{d-1}}$ is attained if and only if a tensor is orthogonal (resp., unitary) up to scaling. Whether or not orthogonal or unitary tensors exist depends on the dimensions $n_1,\dots,n_d$ and the field. A connection between the (non)existence of real orthogonal tensors of order three and the classical Hurwitz problem on composition algebras can be established: existence of orthogonal tensors of size $\ell \times m \times n$ is equivalent to the admissibility of the triple $[\ell,m,n]$ to the Hurwitz problem. Some implications for higher-order tensors are then given. For instance, real orthogonal $n \times \dots \times n$ tensors of order $d \ge 3$ do exist, but only when $n = 1,2,4,8$. In the complex case, the situation is more drastic: unitary tensors of size $\ell \times m \times n$ with $\ell \le m \le n$ exist only when $\ell m \le n$. Finally, some numerical illustrations for spectral norm computation are presented.
Jacobi-type algorithm for low rank orthogonal approximation of symmetric tensors and its convergence analysis
本文提出了一种求解对称张量低秩正交逼近问题的Jacobi型算法。该算法包括著名的Jacobi CoM 2算法的一个特殊情况下，近似正交对角化问题的对称张量。首先证明了该算法的弱收敛性，即\textit{i.e.}的任何聚点都是一个平稳点。然后研究了该算法在梯度序下的整体收敛：三阶对称张量的最佳秩2正交逼近，并证明了在一定条件下聚点是唯一的极限点.数值实验表明了该算法的有效性。
Laboratoire I3S, CNRS and the University of Nice, Sophia-Antipolis（I3S实验室，法国国家科学研究中心和尼斯大学）；Department of Computer Science and the Institute for Computational and Mathematical Engineering, Stanford University（计算机科学系和计算与数学工程研究所，斯坦福大学）；Projet GALAAD, INRIA（GALAAD项目，法国国家信息与自动化研究所）
Tensor Deflation for CANDECOMP/PARAFAC. Part 3: Rank Splitting
BSI-TOYOTA Collaboration Center, RIKEN（理化学研究所BSI-丰田合作中心）；System Research Institute, Warsaw（华沙系统研究所）；Institute of Information Theory and Automation, Prague（布拉格信息与自动化研究所）
GIPSA-Lab CNRS（GIPSA实验室 法国国家科学研究中心）；University of Chicago（芝加哥大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
Institute of Information Theory and Automation（信息与自动化研究所）；Brain Science Institute, RIKEN（理化学研究所脑科学研究所）；Systems Research Institute, Polish Academy of Sciences（波兰科学院系统研究所）
