License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CCC.2020.35
URN: urn:nbn:de:0030-drops-125874
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12587/
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Kopparty, Swastik ; Moshkovitz, Guy ; Zuiddam, Jeroen

Geometric Rank of Tensors and Subrank of Matrix Multiplication

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Abstract

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.

BibTeX - Entry

@InProceedings{kopparty_et_al:LIPIcs:2020:12587,
  author =	{Swastik Kopparty and Guy Moshkovitz and Jeroen Zuiddam},
  title =	{{Geometric Rank of Tensors and Subrank of Matrix Multiplication}},
  booktitle =	{35th Computational Complexity Conference (CCC 2020)},
  pages =	{35:1--35:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-156-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{169},
  editor =	{Shubhangi Saraf},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12587},
  URN =		{urn:nbn:de:0030-drops-125874},
  doi =		{10.4230/LIPIcs.CCC.2020.35},
  annote =	{Keywords: Algebraic complexity theory, Extremal combinatorics, Tensors, Bias, Analytic rank, Algebraic geometry, Matrix multiplication}
}

Keywords: Algebraic complexity theory, Extremal combinatorics, Tensors, Bias, Analytic rank, Algebraic geometry, Matrix multiplication
Collection: 35th Computational Complexity Conference (CCC 2020)
Issue Date: 2020
Date of publication: 17.07.2020

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