Draisma, Jan; Ottaviani, Giorgio; Tocino, Alicia (2018). Best rankk approximations for tensors: generalizing EckartYoung. Research in mathematical sciences, 5(2) Springer 10.1007/s4068701801451

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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 rankatmostk tensors for f. When f is a matrix, the critical rankone matrices for f correspond to the singular pairs of f. The critical rankone tensors for f lie in a linear subspace Hf, the critical space of f. Our main result is that, for any k, the critical rankatmostk 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 rankone tensors. Since f itself belongs to Hf, we deduce that also f itself is a linear combination of its critical rankone 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: 
25220144 
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/s4068701801451 
BORIS DOI: 
10.7892/boris.125494 
URI: 
https://boris.unibe.ch/id/eprint/125494 