Exact minimal number of ring multiplications for general matrix multiplication formats

Determine, for arbitrary positive integers n, m, and p, the minimal number of multiplications in the coefficient ring required to compute the product of an n×m matrix and an m×p matrix—that is, the exact rank of the matrix multiplication tensor—for both asymptotically large matrices and for almost every specific finite format.

Background

The paper studies bilinear algorithms (matrix multiplication schemes) and their ranks, where the rank equals the number of ring multiplications needed to multiply matrices of given dimensions. While specific improvements are reported—for example, ranks 93 and 153 for square formats (5,5,5) and (6,6,6) and new records for many nearby rectangular formats—the general problem of determining the exact minimal number of multiplications for arbitrary formats is highlighted as unresolved.

This problem is equivalent to determining the tensor rank of the matrix multiplication tensor for dimensions (n, m, p). The authors emphasize that the uncertainty persists not only in the asymptotic regime but also for nearly all concrete small formats, underscoring the breadth of the open question.

References

More than half a century after the discovery of Strassen's algorithm, we still do not know how many multiplications in the ground ring are necessary for multiplying an $n\times m$ matrix with an $m\times p$ matrix. The question is not only open for asymptotically large matrices but for almost every specific matrix format.

— Consequences of the Moosbauer-Poole Algorithms (2505.05896 - Kauers et al., 9 May 2025) in Opening paragraph after the abstract (first page)