Linear Time Small Coresets for k-Mean Clustering of Segments with Applications

David Denisov, Shlomi Dolev, Dan Feldman, Michael Segal

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 \(\varepsilon > 0\), an \(\varepsilon \)-coreset is a weighted subset \(C \subseteq \mathbb {R}^d\) that approximates \(D(\mathcal {S},X)\) within a factor of \(1 \pm \varepsilon \) 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 \(\varepsilon \), 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.

External IDs:dblp:conf/walcom/DenisovDFS26