Coreset for Robust Geometric Median: Eliminating Size Dependency on Outliers
Ziyi Fang, Lingxiao Huang, Runkai Yang
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.