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Border Rank Nonadditivity for Higher Order Tensors

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2021

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Christandl, Matthias
Gesmundo, Fulvio
Zuiddam, Jeroen

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SIAM Journal on Matrix Analysis and Applications. Society for Industrial and Applied Mathematics (SIAM). 2021, 42(2), pp. 503-527. ISSN 0895-4798. eISSN 1095-7162. Available under: doi: 10.1137/20M1357366

Zusammenfassung

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.

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510 Mathematik

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ISO 690CHRISTANDL, Matthias, Fulvio GESMUNDO, Mateusz MICHALEK, Jeroen ZUIDDAM, 2021. Border Rank Nonadditivity for Higher Order Tensors. In: SIAM Journal on Matrix Analysis and Applications. Society for Industrial and Applied Mathematics (SIAM). 2021, 42(2), pp. 503-527. ISSN 0895-4798. eISSN 1095-7162. Available under: doi: 10.1137/20M1357366
BibTex
@article{Christandl2021Borde-53609,
  year={2021},
  doi={10.1137/20M1357366},
  title={Border Rank Nonadditivity for Higher Order Tensors},
  number={2},
  volume={42},
  issn={0895-4798},
  journal={SIAM Journal on Matrix Analysis and Applications},
  pages={503--527},
  author={Christandl, Matthias and Gesmundo, Fulvio and Michalek, Mateusz and Zuiddam, Jeroen}
}
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