ECCC-Report TR20-029https://eccc.weizmann.ac.il/report/2020/029Comments and Revisions published for TR20-029en-usFri, 06 Mar 2020 17:03:24 +0200
Paper TR20-029
| Geometric Rank of Tensors and Subrank of Matrix Multiplication |
Guy Moshkovitz,
Swastik Kopparty,
Jeroen Zuiddam
https://eccc.weizmann.ac.il/report/2020/029Motivated 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.Fri, 06 Mar 2020 17:03:24 +0200https://eccc.weizmann.ac.il/report/2020/029