This note is an exposition of the recent preprints from OpenAI that establish the bound $\omega \leq 9/4$ on the matrix multiplication exponent. We argue that the heart of these preprints is a new technique, which we formulate as the untwisting lemma for semidirect sums of tensors. This gives a ... more >>>

The complexity of bilinear maps (equivalently, of $3$-mode tensors) has been studied extensively, most notably in the context of matrix multiplication. While circuit complexity and tensor rank coincide asymptotically for $3$-mode tensors, this correspondence breaks down for $d \geq 4$ modes. As a result, the complexity of $d$-mode tensors for ... more >>>