Best rank-k approximations for tensors: generalizing Eckart-Young

Draisma, Jan; Ottaviani, Giorgio; Tocino, Alicia (2018). Best rank-k approximations for tensors: generalizing Eckart-Young. Research in mathematical sciences, 5(2) Springer 10.1007/s40687-018-0145-1

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Given a tensor f in a Euclidean tensor space, we are interested in the critical points of the distance function from f to the set of tensors of rank at most k, which we call the critical rank-at-most-k tensors for f. When f is a matrix, the critical rank-one matrices for f correspond to the singular pairs of f. The critical rank-one tensors for f lie in a linear subspace Hf, the critical space of f. Our main result is that, for any k, the critical rank-at-most-k tensors for a sufficiently general f also lie in the critical space Hf. This is the part of Eckart–Young Theorem that generalizes from matrices to tensors. Moreover, we show that when the tensor format satisfies the triangle inequalities, the critical space Hf is spanned by the complex critical rank-one tensors. Since f itself belongs to Hf, we deduce that also f itself is a linear combination of its critical rank-one tensors.

Item Type:

Journal Article (Original Article)

Division/Institute:

08 Faculty of Science > Department of Mathematics and Statistics > Institute of Mathematics

UniBE Contributor:

Draisma, Jan

Subjects:

500 Science > 510 Mathematics

ISSN:

2522-0144

Publisher:

Springer

Language:

English

Submitter:

Olivier Bernard Mila

Date Deposited:

08 May 2019 11:00

Last Modified:

24 Oct 2019 03:57

Publisher DOI:

10.1007/s40687-018-0145-1

BORIS DOI:

10.7892/boris.125494

URI:

https://boris.unibe.ch/id/eprint/125494

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