eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-07-17
35:1
35:21
10.4230/LIPIcs.CCC.2020.35
article
Geometric Rank of Tensors and Subrank of Matrix Multiplication
Kopparty, Swastik
1
Moshkovitz, Guy
2
3
Zuiddam, Jeroen
2
Rutgers University, Piscataway, NJ, USA
Institute for Advanced Study, Princeton, NJ, USA
DIMACS, Rutgers University, Piscataway, NJ, USA
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol169-ccc2020/LIPIcs.CCC.2020.35/LIPIcs.CCC.2020.35.pdf
Algebraic complexity theory
Extremal combinatorics
Tensors
Bias
Analytic rank
Algebraic geometry
Matrix multiplication