Coresets for Kernel Clustering

TL;DR

This paper introduces a new coreset construction for kernel k-Means clustering that reduces computational complexity, enabling faster algorithms with maintained accuracy for non-linear clustering tasks.

Contribution

It presents a general, size-efficient coreset for kernel k-Means that improves upon previous results and can be constructed in near-linear time, facilitating faster clustering algorithms.

Findings

Coreset size is polynomial in k and 1/ε.

Algorithms achieve near-linear time complexity.

Coresets enable faster kernel k-Means++ and spectral clustering.

Abstract

We devise coresets for kernel $k$-Means with a general kernel, and use them to obtain new, more efficient, algorithms. Kernel $k$-Means has superior clustering capability compared to classical $k$-Means, particularly when clusters are non-linearly separable, but it also introduces significant computational challenges. We address this computational issue by constructing a coreset, which is a reduced dataset that accurately preserves the clustering costs. Our main result is a coreset for kernel $k$-Means that works for a general kernel and has size $\mathrm{poly}(k\epsilon^{-1})$. Our new coreset both generalizes and greatly improves all previous results; moreover, it can be constructed in time near-linear in $n$. This result immediately implies new algorithms for kernel $k$-Means, such as a $(1+\epsilon)$-approximation in time near-linear in $n$, and a streaming algorithm using space and update time $\mathrm{poly}(k \epsilon^{-1} \log n)$. We validate our coreset on various datasets with different kernels. Our coreset performs consistently well, achieving small errors while using very few points. We show that our coresets can speed up kernel $k$-Means++ (the kernelized version of the widely used $k$-Means++ algorithm), and we further use this faster kernel $k$-Means++ for spectral clustering. In both applications, we achieve significant speedup and a better asymptotic growth while the error is comparable to baselines that do not use coresets.