Barriers for fast matrix multiplication from irreversibility

Determining the asymptotic algebraic complexity of matrix multiplication, succinctly represented by the matrix multiplication exponent \(\omega\), is a central problem in algebraic complexity theory. The best upper bounds on \(\omega\), leading to the state-of-the-art \(\omega \leq 2.37..\), have be...

Ausführliche Beschreibung

Gespeichert in:
Bibliographische Detailangaben
Veröffentlicht in:arXiv.org
Hauptverfasser: Christandl, Matthias, Vrana, Péter, Zuiddam, Jeroen
Format: Paper
Sprache:Englisch
Veröffentlicht: Ithaca Cornell University Library, arXiv.org 05.03.2022
Schlagworte:
Algebra
Barriers
Complexity theory
Lower bounds
Mathematical analysis
Multiplication
Multiplication & division
Tensors
Upper bounds
ISSN:2331-8422
Online-Zugang:Volltext
Tags: Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
Beschreibung
Zusammenfassung:Determining the asymptotic algebraic complexity of matrix multiplication, succinctly represented by the matrix multiplication exponent \(\omega\), is a central problem in algebraic complexity theory. The best upper bounds on \(\omega\), leading to the state-of-the-art \(\omega \leq 2.37..\), have been obtained via the laser method of Strassen and its generalization by Coppersmith and Winograd. Recent barrier results show limitations for these and related approaches to improve the upper bound on \(\omega\). We introduce a new and more general barrier, providing stronger limitations than in previous work. Concretely, we introduce the notion of "irreversibility" of a tensor and we prove (in some precise sense) that any approach that uses an irreversible tensor in an intermediate step (e.g., as a starting tensor in the laser method) cannot give \(\omega = 2\). In quantitative terms, we prove that the best upper bound achievable is lower bounded by two times the irreversibility of the intermediate tensor. The quantum functionals and Strassen support functionals give (so far, the best) lower bounds on irreversibility. We provide lower bounds on the irreversibility of key intermediate tensors, including the small and big Coppersmith--Winograd tensors, that improve limitations shown in previous work. Finally, we discuss barriers on the group-theoretic approach in terms of "monomial" irreversibility.
Bibliographie:SourceType-Working Papers-1
ObjectType-Working Paper/Pre-Print-1
content type line 50
ISSN:2331-8422
DOI:10.48550/arxiv.1812.06952