Faster Algorithms for Structured Matrix Multiplication via Flip Graph Search
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2025
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| author | Khoruzhii, Kirill Gelß, Patrick Pokutta, Sebastian |
| author_facet | Khoruzhii, Kirill Gelß, Patrick Pokutta, Sebastian |
| contents | We give explicit low-rank bilinear non-commutative schemes for multiplying structured $n \times n$ matrices with $2 \leq n \leq 5$, which serve as building blocks for recursive algorithms with improved multiplicative factors in asymptotic complexity. Our schemes are discovered over $\mathbb{F}_2$ or $\mathbb{F}_3$ and lifted to $\mathbb{Z}$ or $\mathbb{Q}$. Using a flip graph search over tensor decompositions, we derive schemes for general, upper-triangular, lower-triangular, symmetric, and skew-symmetric inputs, as well as products of a structured matrix with its transpose. These schemes improve asymptotic constants for 13 of 15 structured formats. In particular, we obtain $4 \times 4$ rank-34 schemes for both multiplying a general matrix by its transpose and an upper-triangular matrix by a general matrix, improving the asymptotic factor from 8/13 (0.615) to 22/37 (0.595). Additionally, using $\mathbb{F}_3$ flip graphs, we discover schemes over $\mathbb{Q}$ that fundamentally require the inverse of 2, including a $2 \times 2$ symmetric-symmetric multiplication of rank 5 and a $3 \times 3$ skew-symmetric-general multiplication of rank 14 (improving upon AlphaTensor's 15). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_10786 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Faster Algorithms for Structured Matrix Multiplication via Flip Graph Search Khoruzhii, Kirill Gelß, Patrick Pokutta, Sebastian Symbolic Computation Data Structures and Algorithms 68Q25 F.2.1; F.1.3; G.1.3 We give explicit low-rank bilinear non-commutative schemes for multiplying structured $n \times n$ matrices with $2 \leq n \leq 5$, which serve as building blocks for recursive algorithms with improved multiplicative factors in asymptotic complexity. Our schemes are discovered over $\mathbb{F}_2$ or $\mathbb{F}_3$ and lifted to $\mathbb{Z}$ or $\mathbb{Q}$. Using a flip graph search over tensor decompositions, we derive schemes for general, upper-triangular, lower-triangular, symmetric, and skew-symmetric inputs, as well as products of a structured matrix with its transpose. These schemes improve asymptotic constants for 13 of 15 structured formats. In particular, we obtain $4 \times 4$ rank-34 schemes for both multiplying a general matrix by its transpose and an upper-triangular matrix by a general matrix, improving the asymptotic factor from 8/13 (0.615) to 22/37 (0.595). Additionally, using $\mathbb{F}_3$ flip graphs, we discover schemes over $\mathbb{Q}$ that fundamentally require the inverse of 2, including a $2 \times 2$ symmetric-symmetric multiplication of rank 5 and a $3 \times 3$ skew-symmetric-general multiplication of rank 14 (improving upon AlphaTensor's 15). |
| title | Faster Algorithms for Structured Matrix Multiplication via Flip Graph Search |
| topic | Symbolic Computation Data Structures and Algorithms 68Q25 F.2.1; F.1.3; G.1.3 |
| url | https://arxiv.org/abs/2511.10786 |