矩阵乘法与二进制空间划分树的探讨
Matrix Multiplication and Binary Space Partitioning Trees : An Exploration
摘要：
在这里，我们探讨了一个双树算法的矩阵乘法的$A\in \mathbb{R}^{M\times D}$和$B\in\mathbb{R}^{D\times N}$，非常狭窄的有效性，如果规范化的行的$A$和列的$B$，作为$\mathbb{R}^{D}$中的向量，落入集群的顺序成比例的$Ω(D^τ)$与半径小于$\arcsin(ε/\sqrt{2})$的表面上的单位$D$ -球。该算法利用必要的修剪规则，以保证$ε$精度与结果矩阵中的向量幅度乘积成比例。\textit{ Unfortunately, if the rows and columns are uniformly distributed on the surface of the unit $D$-ball, then the expected points per required cluster approaches zero exponentially fast in $D$; thus, the approach requires a great deal of work to pass muster.}
英文摘要：
Herein we explore a dual tree algorithm for matrix multiplication of $A\in \mathbb{R}^{M\times D}$ and $B\in\mathbb{R}^{D\times N}$, very narrowly effective if the normalized rows of $A$ and columns of $B$, treated as vectors in $\mathbb{R}^{D}$, fall into clusters of order proportionate to $Ω(D^τ)$ with radii less than $\arcsin(ε/\sqrt{2})$ on the surface of the unit $D$-ball. The algorithm leverages a pruning rule necessary to guarantee $ε$ precision proportionate to vector magnitude products in the resultant matrix. \textit{ Unfortunately, if the rows and columns are uniformly distributed on the surface of the unit $D$-ball, then the expected points per required cluster approaches zero exponentially fast in $D$; thus, the approach requires a great deal of work to pass muster.}
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