Bounds on complexity of matrix multiplication away from Coppersmith–Winograd tensors

Abstract

We present three families of minimal border rank tensors: they come from highest weight vectors, smoothable algebras, and monomial algebras. We analyze them using Strassen's laser method and obtain an upper bound 2.431 on ω. 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.