最优确定性完全稀疏矩阵乘法
Optimal Deterministic Fully Sparse Matrix Multiplication
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中文总结 AI 辅助
该研究提出首个达最优运行时间指数的完全稀疏矩阵乘法确定性算法，将确定性适用的δ_out上限从0.642668提升至1.321334，运行时间接近二次。
中文摘要 AI 辅助
我们提出了首个达到最优运行时间指数的完全稀疏矩阵乘法确定性算法，该结果与此前已知的最优随机算法运行时间指数匹配。给定任意带单位元的结合环上的相容矩阵A和B，其中nnz(A)、nnz(B)=O(n^{δ_in})，nnz(AB)=O(n^{δ_out})，我们的算法可找到AB的支撑集并精确计算乘积，所需操作数为O(n^{β_R(δ_in,min{δ_out,2δ_in})+ε})，其中β_R(δ_in,δ)为所有满足a+b=δ的a,b∈[0,1]对应的δ_in与ω_{δ_in,R}(a,1,b)的最大值。对于交换环上的稠密输入，该界简化为O(n^{ω_R((δ_out-1)_+,1,1)+ε})。结合当前矩形矩阵乘法界，当δ_out≤1.321334时，该算法的运行时间接近二次，即O(n^{2+ε})，优于此前δ_out≤0.642668的确定性范围。为证明该结果，我们开发了一种通用确定性恢复技术，可在控制较密部分临时误差的同时，找到并修正未知矩阵的稀疏部分。
英文摘要
We give the first deterministic algorithm for fully sparse matrix multiplication that attains the optimal running-time exponent. This result matches the best previously known randomized algorithm running-time exponent. Given compatible matrices $A$ and $B$ over an arbitrary associative ring with identity, with $\operatorname{nnz}(A),\operatorname{nnz}(B)=O(n^{δ_{\mathrm{in}}})$ and $\operatorname{nnz}(AB)=O(n^{δ_{\mathrm{out}}})$, our algorithm finds the support of $AB$ and computes the product exactly in $$O\!\left(n^{β_R(δ_{\mathrm{in}},\min\{δ_{\mathrm{out}},2δ_{\mathrm{in}}\})+\varepsilon}\right)$$ operations, where $β_R(δ_{\mathrm{in}},δ)$ denotes the maximum of $δ_{\mathrm{in}}$ and $ω_{δ_{\mathrm{in}},R}(a,1,b)$ over all $a,b\in[0,1]$ satisfying $a+b=δ$. For dense inputs over a commutative ring, this bound simplifies to $O(n^{ω_R((δ_{\mathrm{out}}-1)_+,1,1)+\varepsilon})$. With the current rectangular matrix multiplication bounds, this is nearly quadratic, namely $O(n^{2+\varepsilon})$, for every $δ_\mathrm{out}\le1.321334$, improving the previous deterministic range of $δ_{\mathrm{out}}\le 0.642668$. To prove this result, we develop a general deterministic recovery technique that finds and fixes sparse parts of an unknown matrix while keeping temporary errors in denser parts under control.
