We present an algorithm for computing $ε$-coresets for $(k, \ell)$-median clustering of polygonal curves in $\mathbb{R}^d$ under the Fréchet distance. This type of clustering is an adaption of Euclidean $k$-median clustering: we are given a set of $n$ polygonal curves in $\mathbb{R}^d$, each of complexity (number of vertices) at most $m$, and want to compute $k$ median curves such that the sum of distances from the given curves to their closest median curve is minimal. Additionally, we restrict the complexity of the median curves to be at most $\ell$ each, to suppress overfitting, a problem specific for sequential data. Our algorithm has running time linear in $n$, sub-quartic in $m$ and quadratic in $ε^{-1}$. With high probability it returns $ε$-coresets of size quadratic in $ε^{-1}$ and logarithmic in $n$ and $m$. We achieve this result by applying the improved $ε$-coreset framework by Langberg and Feldman to a generalized $k$-median problem over an arbitrary metric space. Later we combine this result with the recent result by Driemel et al. on the VC dimension of metric balls under the Fréchet distance. Furthermore, our framework yields $ε$-coresets for any generalized $k$-median problem where the range space induced by the open metric balls of the underlying space has bounded VC dimension, which is of independent interest. Finally, we show that our $ε$-coresets can be used to improve the running time of an existing approximation algorithm for $(1,\ell)$-median clustering.
$k$ -median for lines问题的输入是$\mathbb{R}^d$中的$n$线的集合$L$，目标是计算$\mathbb{R}^d$中的一组$k$中心（点），以最小化$L$中每条线与其最近中心之间的平方距离之和。这是$k$ -中值问题的直接推广，其中输入是一组$n$点而不是线。我们建议第一个PTAS计算$(1+ε)$ -对于任何常数逼近误差$ε\in (0, 1)$和常数整数$k, d \geq 1$，在时间上对该问题进行逼近$O(n \log n)$。这是通过证明在$L$中总是存在$dk^{O(k)}\log (n)/ε^2$线的加权子集（称为coreset），其近似于从$L$到任何给定的$k$点集的平方距离之和。使用传统的合并-归约技术，该coreset意味着使用$n$中接近对数的存储器、更新时间和通信，以及删除任何行但使用线性空间，在一次通过（例如，云）中将行的流集合（可能是无限的）到$M$机器的结果。这些结果适用于其他距离函数，例如$k$ -中位数（距离总和）或忽略距离给定中心最远的$m$线以处理离群值。在Amazon EC2云上的10台机器上进行的实验结果表明，该算法在实际应用中表现良好。所有的算法和实验的开放源代码也提供。本文是以下已接受论文的扩展：“$k$ -Means Clustering of Lines for Big Data”，作者Yair M…
