设施选址和单链接聚类的随机近似约简
Randomized Dimensionality Reduction for Facility Location and Single-Linkage Clustering
摘要：
随机降维是一种多功能的工具，用于加速高维问题的算法。我们研究了它的应用到两个聚类问题：设施定位问题，和单链接层次聚类问题，这是相当于计算最小生成树。我们表明，如果我们将输入点集$X$投影到一个随机的$d = O(d_X)$维子空间（其中$d_X$是双倍维度的$X$），那么在投影空间中的最佳设施选址成本近似于原始成本的常数因子。我们显示了一个类似的最小生成树的声明，但与尺寸$d$有一个额外的$\log \log n$长期和逼近因子任意接近$1$。此外，我们将这些结果扩展到近似的解决方案，而不仅仅是他们的成本。最后，我们提供实验结果来验证解决方案的质量和由于降维的加速比。与以前的几篇论文研究这种方法的背景下，$k$ -手段和$k$ -中位数，我们的维数界不依赖于集群的数量，但只对内在的维数$X$。
英文摘要：
Random dimensionality reduction is a versatile tool for speeding up algorithms for high-dimensional problems. We study its application to two clustering problems: the facility location problem, and the single-linkage hierarchical clustering problem, which is equivalent to computing the minimum spanning tree. We show that if we project the input pointset $X$ onto a random $d = O(d_X)$-dimensional subspace (where $d_X$ is the doubling dimension of $X$), then the optimum facility location cost in the projected space approximates the original cost up to a constant factor. We show an analogous statement for minimum spanning tree, but with the dimension $d$ having an extra $\log \log n$ term and the approximation factor being arbitrarily close to $1$. Furthermore, we extend these results to approximating solutions instead of just their costs. Lastly, we provide experimental results to validate the quality of solutions and the speedup due to the dimensionality reduction. Unlike several previous papers studying this approach in the context of $k$-means and $k$-medians, our dimension bound does not depend on the number of clusters but only on the intrinsic dimensionality of $X$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
