OpenAI research collection · 3 manuscripts · Source checked 2026-10-19
Matrices are grids of numbers used throughout computing. These papers claim improved upper bounds on the number of arithmetic operations needed to multiply large matrices, including an exponent of at most 2.25 over the complex numbers.
Matrix multiplication is central to AI, but an asymptotically faster algorithm does not automatically accelerate today's GPUs. Constants, memory use and implementation still matter.
Proves ω\le9/4 over ℂ, giving O_ε(n^{9/4+ε}) arithmetic operations for square matrix multiplication. In characteristic zero, some inner dimension n^(a) with a > 0.465 permits n^{2+o(1)} rectangular multiplication. Further square bounds give ω < 2.258 outside finitely many positive characteristics and ω < 2.371054886006746 over every fixed field.
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We prove that the exponent of matrix multiplication over the complex numbers is at most 9/4.
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, secondary writeup Over every field of characteristic zero, we prove that the square matrix-multiplication exponent satisfies ω < 2.258, the dual exponent satisfies α > 0.465, and ω(1,0.709,1)< 2.092. The strict square and k = 0.709 rectangular bounds also hold over every field except possibly in one finite set of positive…
, secondary writeup Over every field of characteristic zero, we prove that the square matrix-multiplication exponent satisfies ω < 2.258, the dual exponent satisfies α > 0.465, and ω(1,0.709,1)< 2.092. The strict square and k = 0.709 rectangular bounds also hold over every field except possibly in one finite set of positive characteristics, in the arithmetic-operation model.
This PDF is listed in OpenAI's formalization catalogue. See the scope and dependencies before interpreting coverage.
We prove that the arithmetic exponent of square matrix multiplication over every fixed field satisfies ω < 2.371054886006746. This includes every positive characteristic.
This PDF is listed in OpenAI's formalization catalogue. See the scope and dependencies before interpreting coverage.
New mathematical reasoning could help AI tackle harder scientific and computational problems. How reliably these skills transfer beyond mathematics will determine how much they contribute to general intelligence.
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