Barriers for Fast Matrix Multiplication from Irreversibility

Determining the asymptotic algebraic complexity of matrix multiplication, succinctly represented by the matrix multiplication exponent ω, is a central problem in algebraic complexity theory. The best upper bounds on ω, leading to the state-of-the-art ω ≤ 2.37.., have been obtained via Strassen’s laser method and its generalization by Coppersmith and Winograd. Recent barrier results show limitations for these and related approaches to improve the upper bound on ω. 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 ω = 2. In quantitative terms, we prove that the best upper bound achievable is at least twice 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 grouptheoretic approach in terms of monomial irreversibility.