Exploring the Meta Flip Graph for Matrix Multiplication

TL;DR

This work advances the search for low-rank matrix multiplication schemes by developing the meta flip graph, which combines flip graphs across formats via extension and projection edges. It employs randomized path traversals, starting from diverse schemes, and both extension and patching strategies to obtain rank reductions across numerous formats, including notable improvements for formats such as $(2,5,7)$, $(2,6,7)$, $(3,4,6)$, and $(3,6,8)$. A key contribution is showing how starting points beyond the standard $(2,2,2)$ and lifting results through $\mathbb{Z}_2$ to higher rings (and eventually to $\mathbb{Q}$) can yield practical, lower-rank schemes, with several formats requiring rational or modular denominators. The results, made publicly available, demonstrate that diverse search trajectories in the unified meta-graph yield complementary improvements and highlight both the power and limitations of lifting and projection in discovering tight rank bounds for small matrix-multiplication formats.

Abstract

Continuing recent investigations of bounding the tensor rank of matrix multiplication using flip graphs, we present here improved rank bounds for about thirty matrix formats.

Figures (4)

• Figure 1: Paths in the meta flip graph starting from the standard algorithm for format $(2,2,2)$ (black) that led to improvements for certain formats (boxed).
• Figure 2: Paths in the meta flip graph starting from a scheme found by moosbauer25 (black) that led to improvements for certain formats (boxed).
• Figure 3: Genealogy of the improvement found for $(3,6,8)$.
• Figure 4: Summary of results.