We study the robust geometric median problem in Euclidean space $\mathbb{R}^d$, with a focus on coreset construction.A coreset is a compact summary of a dataset $P$ of size $n$ that approximates the robust cost for all centers $c$ within a multiplicative error $\varepsilon$. Given an outlier count $m$, we construct a coreset of size $\tilde{O}(\varepsilon^{-2} \cdot \min\{\varepsilon^{-2}, d\})$ when $n \geq 4m$, eliminating the $O(m)$ dependency present in prior work [Huang et al., 2022 & 2023]. For the special case of $d = 1$, we achieve an optimal coreset size of $\tilde{\Theta}(\varepsilon^{-1/2} + \frac{m}{n} \varepsilon^{-1})$, revealing a clear separation from the vanilla case studied in [Huang et al., 2023; Afshani and Chris, 2024]. Our results further extend to robust $(k,z)$-clustering in various metric spaces, eliminating the $m$-dependence under mild data assumptions. The key technical contribution is a novel non-component-wise error analysis, enabling substantial reduction of outlier influence, unlike prior methods that retain them.Empirically, our algorithms consistently outperform existing baselines in terms of size-accuracy tradeoffs and runtime, even when data assumptions are violated across a wide range of datasets.

翻译：我们研究了欧几里得空间 $\mathbb{R}^d$ 中的鲁棒几何中位数问题，重点关注核心集的构建。核心集是大小为 $n$ 的数据集 $P$ 的紧凑摘要，能以乘法误差 $\varepsilon$ 近似所有中心 $c$ 的鲁棒代价。给定异常值数量 $m$，当 $n \geq 4m$ 时，我们构建了一个大小为 $\tilde{O}(\varepsilon^{-2} \cdot \min\{\varepsilon^{-2}, d\})$ 的核心集，消除了先前工作[Huang et al., 2022 & 2023]中存在的 $O(m)$ 依赖。对于 $d = 1$ 的特殊情况，我们实现了 $\tilde{\Theta}(\varepsilon^{-1/2} + \frac{m}{n} \varepsilon^{-1})$ 的最优核心集规模，这揭示了与[Huang et al., 2023; Afshani and Chris, 2024]中研究的普通情况存在明显分离。我们的结果进一步扩展到多种度量空间中的鲁棒 $(k,z)$-聚类问题，在温和的数据假设下消除了 $m$ 依赖性。关键的技术贡献是一种新颖的非分量误差分析方法，能够显著减少异常值的影响，这与保留异常值的先前方法不同。实证结果表明，即使在广泛的数据集上违反数据假设，我们的算法在规模-精度权衡和运行时间方面始终优于现有基线。