Border Rank Nonadditivity for Higher Order Tensors
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Border Rank Nonadditivity for Higher Order Tensors. / Christandl, M.; Gesmundo, F.; Michałek, M.; Zuiddam, J.
In: SIAM Journal on Matrix Analysis and Applications, Vol. 42, No. 2, 2021, p. 503-527.Research output: Contribution to journal › Journal article › Research › peer-review
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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