论文标题
距离矩阵的更快线性代数
Faster Linear Algebra for Distance Matrices
论文作者
论文摘要
相对于距离函数$ f $，数据集$ x $ $ n $点的距离矩阵表示$ f $引起的$ x $中点之间的所有成对距离。由于其广泛的适用性，矩阵的距离矩阵和相关家族一直是许多最近的算法作品的重点。我们继续进行这一研究线，并对距离矩阵的算法设计进行广泛的视图，目的是设计快速算法，该算法是针对距离矩阵定制的，用于基本线性代数基原始人。我们的结果包括用于计算各种距离矩阵的矩阵向量产品的有效算法，例如我们获得线性运行时的$ \ ell_1 $度量，以及$ω（n^2）$ lustermit boctionm $ \ ellgorithm的$ \ ellgorix $ \ el y ungrions $ \ efl ustrix $ \ elgoriThm的ungorions $ \ efl _ \ efl _ \ efl i _} $ \ ell _ {\ infty} $度量。我们的上限结果与矩阵矢量查询模型的最新作品结合使用，具有许多进一步的下游应用程序，包括用于计算$ \ ell_1 $和$ \ ell_2^2 $ ungortim ungorion的$ \ ell_1 $和$ \ ell_1 $和$ y y IMPARTIVE y IMPARTIVE y IMPARTIVE y IMPARTIVE y IMPARTIVE的距离的最快算法低级别近似值的相对差近似值。以及快速矩阵乘法的应用程序。我们还提供了用于构建距离矩阵的算法，并表明人们可以在时间上构造大约$ \ ell_2 $距离矩阵，速度比Johnson-Lindenstrauss Lemma所隐含的约束更快。
The distance matrix of a dataset $X$ of $n$ points with respect to a distance function $f$ represents all pairwise distances between points in $X$ induced by $f$. Due to their wide applicability, distance matrices and related families of matrices have been the focus of many recent algorithmic works. We continue this line of research and take a broad view of algorithm design for distance matrices with the goal of designing fast algorithms, which are specifically tailored for distance matrices, for fundamental linear algebraic primitives. Our results include efficient algorithms for computing matrix-vector products for a wide class of distance matrices, such as the $\ell_1$ metric for which we get a linear runtime, as well as an $Ω(n^2)$ lower bound for any algorithm which computes a matrix-vector product for the $\ell_{\infty}$ case, showing a separation between the $\ell_1$ and the $\ell_{\infty}$ metrics. Our upper bound results, in conjunction with recent works on the matrix-vector query model, have many further downstream applications, including the fastest algorithm for computing a relative error low-rank approximation for the distance matrix induced by $\ell_1$ and $\ell_2^2$ functions and the fastest algorithm for computing an additive error low-rank approximation for the $\ell_2$ metric, in addition to applications for fast matrix multiplication among others. We also give algorithms for constructing distance matrices and show that one can construct an approximate $\ell_2$ distance matrix in time faster than the bound implied by the Johnson-Lindenstrauss lemma.
