A higher-order tensor allows several possible matricizations (reshapes into matrices). The simultaneous decay of singular values of such matricizations has crucial implications on the low-rank approximability of the tensor via higher-order singular value decomposition. It is therefore an interesting question which simultaneous properties the singular values of different tensor matricizations actually can have, but it has not received the deserved attention so far. In this paper, preliminary investigations in this direction are conducted. While it is clear that the singular values in different matricizations cannot be prescribed completely independent from each other, numerical experiments suggest that sufficiently small, but otherwise arbitrary perturbations preserve feasibility. An alternating projection heuristic is proposed for constructing tensors with prescribed singular values (assuming their feasibility). Regarding the related problem of characterising sets of tensors having the same singular values in specified matricizations, it is noted that orthogonal equivalence under multilinear matrix multiplication is a sufficient condition for two tensors to have the same singular values in all principal, Tucker-type matricizations, but, in contrast to the matrix case, not necessary. An explicit example of this phenomenon is given.

中文翻译：

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关于实张量高阶奇异值之间的相互联系

高阶张量允许几种可能的矩阵化（重塑为矩阵）。通过高阶奇异值分解，这种矩阵化的奇异值的同时衰减对张量的低秩近似性具有重要意义。因此，不同张量矩阵化的奇异值实际上可以同时具有的性质是一个有趣的问题，但到目前为止还没有受到应有的关注。本文在这方面进行了初步调查。虽然很明显，不同矩阵化中的奇异值不能完全独立地规定，但数值实验表明，足够小，否则任意扰动保持可行性。提出了一种交替投影启发式方法来构造具有指定奇异值的张量（假设它们的可行性）。关于在指定矩阵化中表征具有相同奇异值的张量集的相关问题，注意到多重线性矩阵乘法下的正交等价是两个张量在所有主要的 Tucker 型矩阵化中具有相同奇异值的充分条件，但是，与矩阵情况相反，没有必要。给出了这种现象的一个明确例子。值得注意的是，多重线性矩阵乘法下的正交等价是两个张量在所有主要的 Tucker 型矩阵化中具有相同奇异值的充分条件，但与矩阵情况相反，这不是必需的。给出了这种现象的一个明确例子。值得注意的是，多重线性矩阵乘法下的正交等价是两个张量在所有主要的 Tucker 型矩阵化中具有相同奇异值的充分条件，但与矩阵情况相反，这不是必需的。给出了这种现象的一个明确例子。