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Matrix multiplication with exponent at most $9/4$
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GAME #107
Matrix multiplication with exponent at most $9/4$
Multiply two giant matrices faster than anyone ever! The exponent is now at most 2.25.
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| Matrix multiplication with exponent at most 9/4. Proves $\omega\le9/4$ over ℂ, giving $O_\varepsilon(n^{9/4+\varepsilon})$ arithmetic operations for square matrix multiplication. In characteristic zero, some inner dimension na 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 $\omega(1,0.709,1)\lt 2.092$. 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.
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We prove that the arithmetic exponent of square matrix multiplication over every fixed field satisfies ω < 2.371054886006746. This includes every positive characteristic.
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