On the interconnection between the higher-order singular values of real tensors.
On the interconnection between the higher-order singular values of real tensors.
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DOI：
10.1007/s00211-016-0819-9
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发表时间：
2017
影响因子：
2.1
通讯作者：
Uschmajew A
中科院分区：
文献类型：
--
作者：
Hackbusch W;Uschmajew A
高阶张量允许几种可能的矩阵化（重塑为矩阵）。这种矩阵化的奇异值的同时衰减对通过高阶奇异值分解的张量的低秩逼近性具有至关重要的影响。因此，不同张量矩阵化的奇异值究竟具有什么样的同时性质是一个有趣的问题，但至今尚未得到应有的重视。在本文中，在这个方向进行了初步的调查。虽然很明显，在不同的矩阵化的奇异值不能规定完全独立于彼此，数值实验表明，足够小，但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，用于构造具有指定奇异值的张量（假设其可行性）。关于表征在指定矩阵化中具有相同奇异值的张量集合的相关问题，需要注意的是，多线性矩阵乘法下的正交等价是两个张量在所有主要的塔克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，这不是必要的。给出了这种现象的一个明确的例子。
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.