论文标题
边境排名高阶张量
Border rank non-additivity for higher order tensors
论文作者
论文摘要
尽管矩阵等级在直接总和下是加性的，但在1981年，Schönhage表明，其对张量设置的概括之一，张量边界等级对于第三阶张量的张量可能是严格的亚粘附。边境等级是否为高阶张力量增加了。在这项工作中，我们通过为Schönhage的构造提供了四个及更高订单的张量的类似物来解决这个问题。 Schönhage的工作是由研究矩阵乘法的计算复杂性的动机。我们讨论结果对矩阵乘法张量的高阶概括的渐近等级的含义。
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 analogs 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.
