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# Optimal Coresets for Low-Dimensional Geometric Median

*Proceedings of the 41st International Conference on Machine Learning*, PMLR 235:262-270, 2024.

#### Abstract

We investigate coresets for approximating the cost with respect to median queries. In this problem, we are given a set of points $P\subset \mathbb{R}^d$ and median queries are $\sum_{p\in P} ||p-c||$ for any point $c\in \mathbb{R}^d$. Our goal is to compute a small weighted summary $S\subset P$ such that the cost of any median query is approximated within a multiplicative $(1\pm\varepsilon)$ factor. We provide matching upper and lower bounds on the number of points contained in $S$ of the order $\tilde{\Theta}\left(\varepsilon^{-d/(d+1)}\right)$.