矩阵乘法边界秩的新下界
计算复杂性
2013-06-04 v3 代数几何
摘要
n×n矩阵的矩阵乘法算子的边界秩是其复杂度的标准度量。利用代数几何和表示论的技术，我们证明边界秩至少为2n²-n。对于所有n>2，我们的界优于先前(Lickteig于1985年)的下界3/2 n² + n/2 -1。这些界是通过寻找小边界秩双线性映射必须满足的新方程获得的，即三重Segre乘积的割线簇的新方程，而矩阵乘法不满足这些方程。
引用
@article{arxiv.1112.6007,
title = {New lower bounds for the border rank of matrix multiplication},
author = {J. M. Landsberg and Giorgio Ottaviani},
journal= {arXiv preprint arXiv:1112.6007},
year = {2013}
}
备注
9 pages. Version 1 contained an error in the proof of its main theorem and in the course of fixing it we proved a stronger statement. v3: proof of main theorem moved up