Border Rank Nonadditivity for Higher Order Tensors

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Standard

Border Rank Nonadditivity for Higher Order Tensors. / Christandl, M.; Gesmundo, F.; Michałek, M.; Zuiddam, J.

I: SIAM Journal on Matrix Analysis and Applications, Bind 42, Nr. 2, 2021, s. 503-527.

Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt

Harvard

Christandl, M, Gesmundo, F, Michałek, M & Zuiddam, J 2021, 'Border Rank Nonadditivity for Higher Order Tensors', SIAM Journal on Matrix Analysis and Applications, bind 42, nr. 2, s. 503-527. https://doi.org/10.1137/20M1357366

APA

Christandl, M., Gesmundo, F., Michałek, M., & Zuiddam, J. (2021). Border Rank Nonadditivity for Higher Order Tensors. SIAM Journal on Matrix Analysis and Applications, 42(2), 503-527. https://doi.org/10.1137/20M1357366

Vancouver

Christandl M, Gesmundo F, Michałek M, Zuiddam J. Border Rank Nonadditivity for Higher Order Tensors. SIAM Journal on Matrix Analysis and Applications. 2021;42(2):503-527. https://doi.org/10.1137/20M1357366

Author

Christandl, M. ; Gesmundo, F. ; Michałek, M. ; Zuiddam, J. / Border Rank Nonadditivity for Higher Order Tensors. I: SIAM Journal on Matrix Analysis and Applications. 2021 ; Bind 42, Nr. 2. s. 503-527.

Bibtex

@article{d9ad3b7a5fad40c5bf817016f708cc5c,
title = "Border Rank Nonadditivity for Higher Order Tensors",
abstract = "Whereas matrix rank is additive under direct sum, in 1981 Sch{\"o}nhage showed that one of its generalizations to the tensor setting, tensor border rank, can be strictly subadditive for tensors of order three. Whether border rank is additive for higher order tensors has remained open. In this work, we settle this problem by providing analogues of Sch{\"o}nhage's construction for tensors of order four and higher. Sch{\"o}nhage's work was motivated by the study of the computational complexity of matrix multiplication; we discuss implications of our results for the asymptotic rank of higher order generalizations of the matrix multiplication tensor.",
author = "M. Christandl and F. Gesmundo and M. Micha{\l}ek and J. Zuiddam",
year = "2021",
doi = "10.1137/20M1357366",
language = "English",
volume = "42",
pages = "503--527",
journal = "SIAM Journal on Matrix Analysis and Applications",
issn = "0895-4798",
publisher = "Society for Industrial and Applied Mathematics",
number = "2",

}

RIS

TY - JOUR

T1 - Border Rank Nonadditivity for Higher Order Tensors

AU - Christandl, M.

AU - Gesmundo, F.

AU - Michałek, M.

AU - Zuiddam, J.

PY - 2021

Y1 - 2021

N2 - Whereas matrix rank is additive under direct sum, in 1981 Schönhage showed that one of its generalizations to the tensor setting, tensor border rank, can be strictly subadditive for tensors of order three. Whether border rank is additive for higher order tensors has remained open. In this work, we settle this problem by providing analogues of Schönhage's construction for tensors of order four and higher. Schönhage's work was motivated by the study of the computational complexity of matrix multiplication; we discuss implications of our results for the asymptotic rank of higher order generalizations of the matrix multiplication tensor.

AB - Whereas matrix rank is additive under direct sum, in 1981 Schönhage showed that one of its generalizations to the tensor setting, tensor border rank, can be strictly subadditive for tensors of order three. Whether border rank is additive for higher order tensors has remained open. In this work, we settle this problem by providing analogues of Schönhage's construction for tensors of order four and higher. Schönhage's work was motivated by the study of the computational complexity of matrix multiplication; we discuss implications of our results for the asymptotic rank of higher order generalizations of the matrix multiplication tensor.

U2 - 10.1137/20M1357366

DO - 10.1137/20M1357366

M3 - Journal article

VL - 42

SP - 503

EP - 527

JO - SIAM Journal on Matrix Analysis and Applications

JF - SIAM Journal on Matrix Analysis and Applications

SN - 0895-4798

IS - 2

ER -

ID: 259831451