Consequences of the Moosbauer-Poole Algorithms

TL;DR

Improved matrix multiplication schemes for various rectangular matrix formats are found using a flip graph search using a flip graph search.

Abstract

Moosbauer and Poole have recently shown that the multiplication of two $5\times 5$ matrices requires no more than 93 multiplications in the (possibly non-commutative) coefficient ring, and that the multiplication of two $6\times 6$ matrices requires no more than 153 multiplications. Taking these multiplication schemes as starting points, we found improved matrix multiplication schemes for various rectangular matrix formats using a flip graph search.

Figures (1)

• Figure 1: Left: Arai et al. obtained a good scheme for $(5,5,5)$ starting from a scheme of size $(2,2,2)$ by following the depicted path. Right: We performed flip graph searches for various formats, using starting points obtained along the indicated arrows from the schemes found by Moosbauer and Poole for $(5,5,5)$ and $(6,6,6)$. For some formats, we tried several paths.