We consider the problem of constructing small coresets for $k$-Median in Euclidean spaces. Given a large set of data points $P\subset \mathbb{R}^d$, a coreset is a much smaller set $S\subset \mathbb{R}^d$, so that the $k$-Median costs of any $k$ centers w.r.t. $P$ and $S$ are close. Existing literature mainly focuses on the high-dimension case and there has been great success in obtaining dimension-independent bounds, whereas the case for small $d$ is largely unexplored. Considering many applications of Euclidean clustering algorithms are in small dimensions and the lack of systematic studies in the current literature, this paper investigates coresets for $k$-Median in small dimensions. For small $d$, a natural question is whether existing near-optimal dimension-independent bounds can be significantly improved. We provide affirmative answers to this question for a range of parameters. Moreover, new lower bound results are also proved, which are the highest for small $d$. In particular, we completely settle the coreset size bound for $1$-d $k$-Median (up to log factors). Interestingly, our results imply a strong separation between $1$-d $1$-Median and $1$-d $2$-Median. As far as we know, this is the first such separation between $k=1$ and $k=2$ in any dimension.

翻译：我们考虑在欧氏空间中构建$k$-中位问题的小型核心集问题。给定大规模数据点集$P\subset \mathbb{R}^d$，核心集是一个规模小得多的集合$S\subset \mathbb{R}^d$，使得任意$k$个中心相对于$P$和$S$的$k$-中位代价相近。现有文献主要关注高维情形，并在获得与维度无关的界方面取得重大成功，而小维情形仍待深入探索。鉴于欧氏聚类算法在低维场景中的广泛应用及当前研究缺乏系统性，本文研究了小维空间中$k$-中位问题的核心集。对于小维度$d$，自然产生的问题是：已有的近优维度无关界能否显著改进？我们针对一系列参数给出了肯定回答。同时，本文还证明了新的下界结果，这是目前小维情形下的最优下界。特别地，我们完全解决了$1$维$k$-中位问题的核心集大小界（忽略对数因子）。有趣的是，我们的结果表明$1$维$1$-中位与$1$维$2$-中位问题存在显著分离。据我们所知，这是任意维度下首次发现$k=1$与$k=2$之间的这种分离现象。