Hackbusch Wolfgang, Uschmajew André
Max-Planck-Institut Mathematik in den Naturwissenschaften, Inselstraße 22, 04103 Leipzig, Germany.
Hausdorff Center for Mathematics & Institute for Numerical Simulation, University of Bonn, 53115 Bonn, Germany.
Numer Math (Heidelb). 2017;135(3):875-894. doi: 10.1007/s00211-016-0819-9. Epub 2016 Aug 3.
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.
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同时衰减对于通过高阶奇异值分解的张量低秩逼近性具有关键意义。因此，一个有趣的问题是不同张量矩阵化的奇异值实际上可以具有哪些同时性属性，但到目前为止它尚未得到应有的关注。本文在这个方向上进行了初步研究。虽然很明显不同矩阵化中的奇异值不能完全相互独立地规定，但数值实验表明，足够小但其他方面任意的扰动保持可行性。提出了一种交替投影启发式方法来构造具有规定奇异值的张量（假设其可行性）。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，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，但与矩阵情况不同的是，它不是必要条件。给出了这种现象的一个明确例子。
