Linear time small coresets for k-mean clustering of segments with applications

Abstract

We study the $k$-means problem for a set $\mathcal{S}\subseteq {\mathbb{R}}^d$ of $n$ segments, aiming to find $k$ centers $X\subseteq {\mathbb{R}}^d$ that minimize $D(\mathcal{S},X):={\sum }_{S\in \mathcal{S}}{\min }_{x\in X}D(S,x)$, where $D(S,x):={\int }_{p\in S}|p-x|dp$ measures the total distance from each point along a segment to a center. Variants of this problem include handling outliers, employing alternative distance functions such as M-estimators, weighting distances to achieve balanced clustering, or enforcing unique cluster assignments. For any $\epsilon >0$, an $\epsilon$-coreset is a weighted subset $C\subseteq {\mathbb{R}}^d$ that approximates $D(\mathcal{S},X)$ within a factor of $1\pm \epsilon$ for any set of $k$ centers, enabling efficient streaming, distributed, or parallel computation. We propose the first coreset construction that provably handles arbitrary input segments. For constant $k$ and $\epsilon$, it produces a coreset of size $O({\log }^{2}n)$ computable in $O(nd)$ time. Experiments, including a real-time video tracking application, demonstrate substantial speedups with minimal loss in clustering accuracy, confirming both the practical efficiency and theoretical guarantees of our method.

First published in WALCOM 2026 by Springer Nature