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Border Rank Nonadditivity for Higher Order Tensors
SIAM Journal on Matrix Analysis and Applications ( IF 1.5 ) Pub Date : 2021-04-08 , DOI: 10.1137/20m1357366 M. Christandl , F. Gesmundo , M. Michałek , J. Zuiddam
SIAM Journal on Matrix Analysis and Applications ( IF 1.5 ) Pub Date : 2021-04-08 , DOI: 10.1137/20m1357366 M. Christandl , F. Gesmundo , M. Michałek , J. Zuiddam
SIAM Journal on Matrix Analysis and Applications, Volume 42, Issue 2, Page 503-527, January 2021.
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.
中文翻译:
高阶张量的边界秩非可加性
SIAM Journal on Matrix Analysis and Applications,第 42 卷,第 2 期,第 503-527 页,2021 年 1 月。
虽然矩阵秩在直接和下是可加的,但 1981 年 Schönhage 表明其对张量设置的概括之一,张量边界秩,可以对三阶张量是严格次可加的。对于高阶张量,边界等级是否可加仍然是开放的。在这项工作中,我们通过为四阶及更高阶张量提供 Schönhage 构造的类似物来解决这个问题。Schönhage 的工作源于对矩阵乘法计算复杂性的研究;我们讨论了我们的结果对矩阵乘法张量的高阶泛化的渐近秩的影响。
更新日期:2021-04-08
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.
中文翻译:
高阶张量的边界秩非可加性
SIAM Journal on Matrix Analysis and Applications,第 42 卷,第 2 期,第 503-527 页,2021 年 1 月。
虽然矩阵秩在直接和下是可加的,但 1981 年 Schönhage 表明其对张量设置的概括之一,张量边界秩,可以对三阶张量是严格次可加的。对于高阶张量,边界等级是否可加仍然是开放的。在这项工作中,我们通过为四阶及更高阶张量提供 Schönhage 构造的类似物来解决这个问题。Schönhage 的工作源于对矩阵乘法计算复杂性的研究;我们讨论了我们的结果对矩阵乘法张量的高阶泛化的渐近秩的影响。