On the interconnection between the higher-order singular values of real tensors

Wolfgang Hackbusch and André Uschmajew

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Submission date: 27. Sep. 2015 (revised version: April 2016)
Pages: 21
published in: Numerische Mathematik, 135 (2017) 3, p. 875-894 
DOI number (of the published article): 10.1007/s00211-016-0819-9
Bibtex
MSC-Numbers: 15A18, 15A21, 15A69
Keywords and phrases: tensors, spectrum, HOSVD
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Abstract:
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. It is shown that the singular values in different matricizations cannot be totally independent from each other, but numerical evidence is provided that this is at least locally the case. An alternating projection heuristic is proposed for constructing tensors with prescribed singular values (assuming their feasibility). Regarding the related problem of characterizing 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 phenomenom is given.