A New Coreset Framework for Clustering

TL;DR

This paper introduces a new, simple coreset framework that improves approximation bounds for clustering problems across various metric spaces, including Euclidean and general metrics.

Contribution

The paper presents a novel coreset framework that enhances existing bounds for clustering in multiple metric spaces, simplifying the process.

Findings

Improved coreset bounds for Euclidean space.

Enhanced approximation guarantees for doubling and minor-free metrics.

Unified framework applicable to general metric spaces.

Abstract

Given a metric space, the $(k,z)$-clustering problem consists of finding $k$ centers such that the sum of the of distances raised to the power $z$ of every point to its closest center is minimized. This encapsulates the famous $k$-median ($z=1$) and $k$-means ($z=2$) clustering problems. Designing small-space sketches of the data that approximately preserves the cost of the solutions, also known as \emph{coresets}, has been an important research direction over the last 15 years. In this paper, we present a new, simple coreset framework that simultaneously improves upon the best known bounds for a large variety of settings, ranging from Euclidean space, doubling metric, minor-free metric, and the general metric cases.