Rank and border rank of Kronecker powers of tensors and Strassen's laser method

TL;DR

This paper investigates the border rank of Kronecker powers of specific tensors, providing new bounds and insights that impact the understanding of matrix multiplication complexity and the potential of Strassen's laser method.

Contribution

It proves border rank multiplicativity for certain tensors, introduces a skew-symmetric tensor variant, and explores implications for the matrix multiplication exponent.

Findings

Border rank of Kronecker square of $T_{cw,q}$ is the square of its border rank for $q > 2$.

Border rank of Kronecker cube of $T_{cw,q}$ is the cube of its border rank for $q > 4$.

Introduces a skew-symmetric tensor $T_{skewcw,q}$ potentially useful for Strassen's laser method.

Abstract

We prove that the border rank of the Kronecker square of the little Coppersmith-Winograd tensor $T_{cw,q}$ is the square of its border rank for $q > 2$ and that the border rank of its Kronecker cube is the cube of its border rank for $q > 4$. This answers questions raised implicitly in [Coppersmith-Winograd, 1990] and explicitly in [Bl\"aser, 2013] and rules out the possibility of proving new upper bounds on the exponent of matrix multiplication using the square or cube of a little Coppersmith-Winograd tensor in this range. In the positive direction, we enlarge the list of explicit tensors potentially useful for Strassen's laser method, introducing a skew-symmetric version of the Coppersmith-Winograd tensor, $T_{skewcw,q}$. For $q = 2$, the Kronecker square of this tensor coincides with the $3\times 3$ determinant polynomial, $\det_3 \in \mathbb{C}^9\otimes \mathbb{C}^9\otimes \mathbb{C}^9$, regarded as a tensor. We show that this tensor could potentially be used to show that the exponent of matrix multiplication is two. We determine new upper bounds for the (Waring) rank and the (Waring) border rank of $\det_3$, exhibiting a strict submultiplicative behaviour for $T_{skewcw,2}$ which is promising for the laser method. We establish general results regarding border ranks of Kronecker powers of tensors, and make a detailed study of Kronecker squares of tensors in $\mathbb{C}^3\otimes \mathbb{C}^3\otimes \mathbb{C}^3$.