Fréchet和Hausdorff距离下$k$ -中值聚类的核心集
Coresets for $k$-median clustering under Fréchet and Hausdorff distances
摘要：
当聚类中心被限制为低复杂度时，我们给出了计算多边形曲线（在离散和连续的Fréchet距离下）和点集（在Hausdorff距离下）的$(1+\varepsilon)$ -近似$k$ -中值聚类的核心集的算法。我们是第一个这样的结果，其中coreset的大小与要聚类的输入曲线/点集的数量无关（尽管它仍然取决于每个输入对象的最大复杂度）。具体来说，对于任何$δ> 0$，核心集的大小都是$Θ\left(\frac{k^3lm^δd}{\varepsilon^2}\log\left( \frac{kl}{\varepsilon}\right)\right)$，其中$d$是环境维度，$m$是输入曲线/点集中的最大点数，而$l$是聚类中心允许的最大点数。我们正式描述了一个一般条件的限制空间的集群中心-这有助于我们推广和应用的重要性采样框架，这是由Langberg和Schulman用于计算coresets的$k$ -中值聚类的$d$ -维点的赋范空间在$\mathbb{R}^d$，聚类曲线和点集使用的Fréchet和Hausdorff度量的问题。粗略地说，该条件对簇中心的有限空间可以击中的度量球的不同组合的数量设置了上限。我们还推导出下限的coreset的大小，给定的限制，coreset必须是一个子集的输入对象。
英文摘要：
We give algorithms for computing coresets for $(1+\varepsilon)$-approximate $k$-median clustering of polygonal curves (under the discrete and continuous Fréchet distance) and point sets (under the Hausdorff distance), when the cluster centers are restricted to be of low complexity. Ours is the first such result, where the size of the coreset is independent of the number of input curves/point sets to be clustered (although it still depends on the maximum complexity of each input object). Specifically, the size of the coreset is $Θ\left(\frac{k^3lm^δd}{\varepsilon^2}\log\left( \frac{kl}{\varepsilon}\right)\right)$ for any $δ> 0$, where $d$ is the ambient dimension, $m$ is the maximum number of points in an input curve/point set, and $l$ is the maximum number of points allowed in a cluster center. We formally characterize a general condition on the restricted space of cluster centers -- this helps us to generalize and apply the importance sampling framework, that was used by Langberg and Schulman for computing coresets for $k$-median clustering of $d$-dimensional points on normed spaces in $\mathbb{R}^d$, to the problem of clustering curves and point sets using the Fréchet and Hausdorff metrics. Roughly, the condition places an upper bound on the number of different combinations of metric balls that the restricted space of cluster centers can hit. We also derive lower bounds on the size of the coreset, given the restriction that the coreset must be a subset of the input objects.
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