Abstract
Determining the exponent of matrix multiplication ω is one of the central open problems in algebraic complexity theory. All approaches to design fast matrix multiplication algorithms follow the following general pattern: We start with one "efficient" tensor T of fixed size and then we use a way to get a large matrix multiplication out of a large tensor power of T. In the recent years, several socalled barrier results have been established. A barrier result shows a lower bound on the best upper bound for the exponent of matrix multiplication that can be obtained by a certain restriction starting with a certain tensor.
We prove the following barrier over C: Starting with a tensor of minimal border rank satisfying a certain genericity condition, except for the diagonal tensor, it is impossible to prove ω = 2 using arbitrary restrictions. This is astonishing since the tensors of minimal border rank look like the most natural candidates for designing fast matrix multiplication algorithms. We prove this by showing that all of these tensors are irreversible, using a structural characterisation of these tensors. To obtain our result, we relate irreversibility to asymptotic slice rank and instability of tensors and prove that the instability of block tensors can often be decided by looking only on the sizes of nonzero blocks.
BibTeX  Entry
@InProceedings{blser_et_al:LIPIcs:2020:12686,
author = {Markus Bl{\"a}ser and Vladimir Lysikov},
title = {{Slice Rank of Block Tensors and Irreversibility of Structure Tensors of Algebras}},
booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
pages = {17:117:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771597},
ISSN = {18688969},
year = {2020},
volume = {170},
editor = {Javier Esparza and Daniel Kr{\'a}ľ},
publisher = {Schloss DagstuhlLeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/12686},
URN = {urn:nbn:de:0030drops126869},
doi = {10.4230/LIPIcs.MFCS.2020.17},
annote = {Keywords: Tensors, Slice rank, Barriers, Matrix multiplication, GIT stability}
}
Keywords: 

Tensors, Slice rank, Barriers, Matrix multiplication, GIT stability 
Collection: 

45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020) 
Issue Date: 

2020 
Date of publication: 

18.08.2020 