Optimal Stable Coresets for Geometric Median via Uniform Sampling

2026-06-29

AI Summary

This work addresses the efficient construction of stable coresets for the constrained geometric median problem. To overcome limitations of existing methods whose sample complexity exhibits unfavorable dependence on dimensionality and constraint structure, the authors introduce a novel framework based on uniform sampling. They establish, for the first time, that an $(\varepsilon, O(\varepsilon))$-stable coreset can be constructed using only $O(\varepsilon^{-2} \log 1/\varepsilon)$ samples. By integrating the analysis framework of Carmel and Krauthgamer with iterative sample reduction and refined probabilistic arguments, the method achieves a balance between approximation accuracy and computational efficiency with high constant probability. The resulting sample complexity nearly matches the theoretical lower bound up to logarithmic factors, substantially narrowing the gap between stable and weak coresets, while naturally accommodating a broad class of structural constraints.

Abstract

The geometric median problem asks to find a point in $\mathbb{R}^d$ that minimizes the sum of Euclidean distances to an input set. It is a classical problem in computational geometry and appears as a subroutine in numerous optimization tasks, many of which require the solution to satisfy additional structural constraints. A common approach to reduce the input size is to construct a coreset, which is a small weighted subset that faithfully represents the input for a specific optimization problem. Strong coresets preserve the cost of every candidate solution but require linear time to construct; weak coresets admit sublinear construction, in fact by uniform sampling, but only preserve near-optimal solutions, which is insufficient when the solution is constrained. To address this, we focus instead on the recently introduced intermediate notion of a \emph{stable coreset}, which simultaneously handles all constrained variants. Currently, there is a large gap between the known sample sizes for stable and weak coresets. Our main result is that a uniform sample of size $O(ε^{-2} \log \tfrac{1}ε)$ is a stable $(ε, O(ε))$-coreset for the geometric median, with high constant probability, and this bound is tight up to the logarithmic factor. Our analysis adapts recent machinery of Carmel and Krauthgamer (ICLR 2026) for constructing stable coresets, which incurs an $O(\log d)$ factor. We show an iterative argument that progressively reduces the sample size, and eliminates this dependence on the dimension $d$. At a high level, this approach resembles the technique of iterative size reduction, which is applicable for strong coresets but not for weak coresets.

Problem

• geometric median
• stable coreset
• uniform sampling
• sample complexity
• computational geometry

Innovation

• stable coreset
• geometric median
• uniform sampling
• dimension-free
• iterative size reduction