论文标题
张量和矩阵乘法的几何等级等级
Geometric rank of tensors and subrank of matrix multiplication
论文作者
论文摘要
由代数复杂性理论（例如矩阵乘法）和极端组合物（例如，盖集问题和向日葵问题）中的问题引起的，我们将几何等级介绍为张量和超读量研究的新工具。我们证明，几何等级是张量的子级别的上限和超图的独立性。我们证明，几何等级小于道的切片等级，并以渐近方式将几何等级与高沃尔人和狼的分析等级相关联。作为第一个应用程序，我们使用几何等级来证明矩阵乘法张量的（边框）子列的紧密上限，与Strassen的1987年众所周知的下限相匹配。
Motivated by problems in algebraic complexity theory (e.g., matrix multiplication) and extremal combinatorics (e.g., the cap set problem and the sunflower problem), we introduce the geometric rank as a new tool in the study of tensors and hypergraphs. We prove that the geometric rank is an upper bound on the subrank of tensors and the independence number of hypergraphs. We prove that the geometric rank is smaller than the slice rank of Tao, and relate geometric rank to the analytic rank of Gowers and Wolf in an asymptotic fashion. As a first application, we use geometric rank to prove a tight upper bound on the (border) subrank of the matrix multiplication tensors, matching Strassen's well-known lower bound from 1987.
