Abstract
We consider the techniques behind the current best algorithms for matrix multiplication. Our results are threefold.
(1) We provide a unifying framework, showing that all known matrix multiplication running times since 1986 can be achieved from a single very natural tensor  the structural tensor T_q of addition modulo an integer q.
(2) We show that if one applies a generalization of the known techniques (arbitrary zeroing out of tensor powers to obtain independent matrix products in order to use the asymptotic sum inequality of SchÃ¶nhage) to an arbitrary monomial degeneration of T_q, then there is an explicit lower bound, depending on q, on the bound on the matrix multiplication exponent omega that one can achieve. We also show upper bounds on the value alpha that one can achieve, where alpha is such that n * n^alpha * n matrix multiplication can be computed in n^{2+o(1)} time.
(3) We show that our lower bound on omega approaches 2 as q goes to infinity. This suggests a promising approach to improving the bound on omega: for variable q, find a monomial degeneration of T_q which, using the known techniques, produces an upper bound on omega as a function of q. Then, take q to infinity. It is not ruled out, and hence possible, that one can obtain omega=2 in this way.
BibTeX  Entry
@InProceedings{alman_et_al:LIPIcs:2018:8360,
author = {Josh Alman and Virginia Vassilevska Williams},
title = {{Further Limitations of the Known Approaches for Matrix Multiplication}},
booktitle = {9th Innovations in Theoretical Computer Science Conference (ITCS 2018)},
pages = {25:125:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770606},
ISSN = {18688969},
year = {2018},
volume = {94},
editor = {Anna R. Karlin},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/8360},
URN = {urn:nbn:de:0030drops83609},
doi = {10.4230/LIPIcs.ITCS.2018.25},
annote = {Keywords: matrix multiplication, lower bound, monomial degeneration, structural tensor of addition mod p}
}
Keywords: 

matrix multiplication, lower bound, monomial degeneration, structural tensor of addition mod p 
Collection: 

9th Innovations in Theoretical Computer Science Conference (ITCS 2018) 
Issue Date: 

2018 
Date of publication: 

12.01.2018 