Coreset for Robust Geometric Median: Eliminating Size Dependency on Outliers

We study the robust geometric median problem in Euclidean space ℝd, 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 tildeO(varepsilon-2 · minvarepsilon-2, d) when n ≥ 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 tildeTheta(varepsilon-1/2 + fracmn 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.