Border rank non-additivity for higher order tensors

TL;DR

This paper demonstrates that border rank, a tensor complexity measure, can be non-additive for tensors of order four and higher, extending previous results known for third-order tensors and impacting matrix multiplication complexity.

Contribution

The authors extend Sch"onhage's construction to higher order tensors, proving non-additivity of border rank beyond third order, and explore implications for matrix multiplication complexity.

Findings

Border rank is non-additive for tensors of order four and higher.

Extended Sch"onhage's construction to higher order tensors.

Implications for the asymptotic rank of matrix multiplication tensors.

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.