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arXiv · 2007.05458

Border rank non-additivity for higher order tensors

Matthias Christandl·Fulvio Gesmundo·Mateusz Michałek·Jeroen Zuiddam
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Abstract

Whereas matrix rank is additive under direct sum, in 1981 Sch\"onhage 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 analogs of Sch\"onhage's construction for tensors of order four and higher. Sch\"onhage'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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BibTeXRIS

Matthias Christandl, Fulvio Gesmundo, Mateusz Michałek, Jeroen Zuiddam. 2020-07-10. Border rank non-additivity for higher order tensors. https://doi.org/10.1137/20m1357366

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