论文标题
张量排名和复杂性
Tensor Rank and Complexity
论文作者
论文摘要
这些讲义旨在介绍张量等级的几个概念及其与矩阵乘法的渐近复杂性的联系。对后者进行了矩阵乘法的指数的研究，该指数将以张量（边界）等级，（边界）对称等级和某些张量的渐近等级表示。我们还介绍了张量的多线性等级，处理张量等效的概念，并通过castling变换来研究均匀媒介前矢量空间。此外，我们对可极性理论进行处理，并使用它来确定某些对称张量的对称等级（Waring等级）。
These lecture notes are intended as an introduction to several notions of tensor rank and their connections to the asymptotic complexity of matrix multiplication. The latter is studied with the exponent of matrix multiplication, which will be expressed in terms of tensor (border) rank, (border) symmetric rank and the asymptotic rank of certain tensors. We introduce the multilinear rank of a tensor as well, deal with the concept of tensor equivalence and study prehomogeneous vector spaces with the castling transform. Moreover, we treat Apolarity Theory and use it to determine the symmetric rank (Waring rank) of some symmetric tensors.
