Bounds on complexity of matrix multiplication away from CW tensors

TL;DR

This paper introduces new families of tensors derived from algebraic structures, analyzes their border ranks using Strassen's laser method, and provides an improved upper bound of 2.431 on the matrix multiplication exponent, exploring alternatives to CW tensors.

Contribution

It presents three novel families of minimal border rank tensors and demonstrates their analysis with the laser method, offering new directions for tensor-based matrix multiplication bounds.

Findings

Upper bound of 2.431 on matrix multiplication exponent

New tensor families from algebraic structures analyzed

Degeneration techniques can outperform direct laser method applications

Abstract

We present three families of minimal border rank tensors: they come from highest weight vectors, smoothable algebras, or monomial algebras. We analyse them using Strassen's laser method and obtain an upper bound $2.431$ on $\omega$. We also explain how in certain monomial cases using the laser method directly is less profitable than first degenerating. Our results form possible paths in the search for valuable tensors for the laser method away from Coppersmith-Winograd tensors.