Abstract

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.

Keywords

  1. tensor rank
  2. border rank
  3. matrix multiplication

MSC codes

  1. 14N07
  2. 15A69

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Information & Authors

Information

Published In

cover image SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Matrix Analysis and Applications
Pages: 503 - 527
ISSN (online): 1095-7162

History

Submitted: 4 August 2020
Accepted: 23 November 2020
Published online: 8 April 2021

Keywords

  1. tensor rank
  2. border rank
  3. matrix multiplication

MSC codes

  1. 14N07
  2. 15A69

Authors

Affiliations

Funding Information

H2020 European Research Council https://doi.org/10.13039/100010663 : 818761
Division of Computing and Communication Foundations https://doi.org/10.13039/100000143 : CCF-1900460
National Science Foundation https://doi.org/10.13039/100000001 : DMS-1638352
Villum Fonden https://doi.org/10.13039/100008398 : 10059

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