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 by Coppersmith & Winograd (1990, §11)and explicitly by Bläser (2013, Problem 9.8) 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 3times 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}.
Highlights
The exponent ω of matrix multiplication is defined as ω := inf τ two n × n matrices may be multiplied using O(nτ ) arithmetic operations0123456789().: V,vol
We approach the problem via algebraic geometry and representation theory
Inspired by Conner et al (2019a), we introduce a new family of tensors, which are a skew-symmetric version of the small Coppersmith–Winograd tensors for every even q: (1.5)
Summary
The exponent ω of matrix multiplication is defined as ω := inf τ two n × n matrices may be multiplied using O(nτ ) arithmetic operations. The possibility to prove the upper bound ω < 2.3 using the second and third Kronecker power of Tcw,q for 3 ≤ q ≤ 10 was open, in the sense that the if the state-of-the-art lower bound on Tcwk,q were equal to an upper bound, Theorem 1.2 would have given an improvement. We show that this is not the case. There was no progress on upper bounds on the exponent until 2010 when, via a further refinement of the method, a series of improvements Stothers (2010), Williams (2012), Le Gall (2014) and Alman & Williams (2021) lowered the upper bound to the current state of the art ω < 2.373
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