Skew-sparse matrix multiplication

05/13/2022

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by   Qiao-Long Huang, et al.

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Based on the observation that ℚ^(p-1) × (p-1) is isomorphic to a quotient skew polynomial ring, we propose a new method for (p-1)× (p-1) matrix multiplication over ℚ, where p is a prime number. The main feature of our method is the acceleration for matrix multiplication if the product is skew-sparse. Based on the new method, we design a deterministic algorithm with complexity O(T^ω-2 p^2), where T≤ p-1 is a parameter determined by the skew-sparsity of input matrices and ω is the asymptotic exponent of matrix multiplication. Moreover, by introducing randomness, we also propose a probabilistic algorithm with complexity O^(t^ω-2p^2+p^2log1/ν), where t≤ p-1 is the skew-sparsity of the product and ν is the probability parameter.