On Tight Robust Coresets for $k$-Medians Clustering

2025-07-15

AI Summary

This paper studies the robust $k$-median clustering problem with $m$ outliers, aiming to construct compact coresets and characterize their optimal sizes across diverse metric spaces. We propose a novel data decomposition framework that, for the first time, jointly applies chaining arguments to multiple structural components, unifying the treatment of metric spaces with VC-dimension or doubling dimension $d$. Our theoretical analysis yields tight coreset size bounds: $O(m) + ilde{O}(kdvarepsilon^{-2})$ for VC-dimension or doubling-dimension-$d$ spaces, and $O(mvarepsilon^{-1}) + ilde{O}(min{k^{4/3}varepsilon^{-2},, kvarepsilon^{-3}})$ for Euclidean space—substantially improving upon prior state-of-the-art. Furthermore, our framework naturally extends to the $(k,z)$-clustering problem with outliers.

Technology Category

• Machine Learning: Clustering
• Data Mining & Knowledge Management: Anomaly/Outlier Detection
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

This paper considers coresets for the robust $k$-medians problem with $m$ outliers, and new constructions in various metric spaces are obtained. Specifically, for metric spaces with a bounded VC or doubling dimension $d$, the coreset size is $O(m) + ilde{O}(kdvarepsilon^{-2})$, which is optimal up to logarithmic factors. For Euclidean spaces, the coreset size is $O(mvarepsilon^{-1}) + ilde{O}(min{k^{4/3}varepsilon^{-2},kvarepsilon^{-3}})$, improving upon a recent result by Jiang and Lou (ICALP 2025). These results also extend to robust $(k,z)$-clustering, yielding, for VC and doubling dimension, a coreset size of $O(m) + ilde{O}(kdvarepsilon^{-2z})$ with the optimal linear dependence on $m$. This extended result improves upon the earlier work of Huang et al. (SODA 2025). The techniques introduce novel dataset decompositions, enabling chaining arguments to be applied jointly across multiple components.

Problem

• Constructs coresets for robust k-medians with outliers
• Optimizes coreset size in bounded VC or doubling dimensions
• Extends results to robust (k,z)-clustering with improved bounds

Innovation

• Optimal coreset size for robust k-medians
• Novel dataset decomposition techniques
• Improved coreset bounds in Euclidean spaces