Rank and border rank of Kronecker powers of tensors and 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 [Bläser, 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 ${\mathbb{C}}^3\otimes {\mathbb{C}}^3\otimes {\mathbb{C}}^3$.

Final Version

References in corpus (6)

• Barriers for Rank Methods in Arithmetic Complexity
• Barriers for fast matrix multiplication from irreversibility
• New lower bounds for matrix multiplication and the 3x3 determinant
• Tensors with maximal symmetries
• Limits on the Universal Method for Matrix Multiplication
• An improved upper bound for the Waring rank of the determinant

Cited by in corpus (2)

• The next gap in the subrank of 3-tensors
• Partial Degeneration of Tensors