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Rank of the 3×3 matrix multiplication tensor

Determine the exact tensor rank of the trilinear matrix multiplication tensor for multiplying 3×3 matrices.

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Background

To design fast algorithms via Schönhage’s asymptotic sum inequality, one ideally bounds the ranks of matrix multiplication tensors. Direct rank bounds, however, are difficult. As a canonical example, the exact rank of the 3×3 matrix multiplication tensor remains unresolved, which is why modern approaches rely on indirect methods such as the laser method applied to structured tensors like CW_q.

This open problem underscores a core obstacle behind rank-based algorithm design and motivates the indirect strategies developed and extended by the authors.

References

However, directly bounding the rank of a matrix multiplication tensor is quite difficult (for example, even determining the rank of the tensor for multiplying 3 × 3 matrices is still an open problem today), and so algorithms since the work of Coppersmith and Winograd [cw90] have used an indirect approach:

— More Asymmetry Yields Faster Matrix Multiplication (2404.16349 - Alman et al., 25 Apr 2024) in Section 2.1 (Technical Overview: The Laser Method and Asymptotic Sum Inequality)