Nearly Optimal Clustering Risk Bounds for Kernel K-Means

TL;DR

This paper establishes nearly optimal statistical bounds for kernel k-means clustering, including analysis of approximation effects, significantly advancing theoretical understanding in clustering risk analysis.

Contribution

It provides the first sharp excess clustering risk bounds for kernel and approximate kernel k-means, improving prior theoretical results.

Findings

Achieves nearly optimal excess clustering risk bounds.

Shows Nyström kernel k-means with  ext{(} ext{)}} landmark points matches exact kernel k-means accuracy.

Provides theoretical analysis of computational approximation effects.

Abstract

In this paper, we study the statistical properties of kernel $k$-means and obtain a nearly optimal excess clustering risk bound, substantially improving the state-of-art bounds in the existing clustering risk analyses. We further analyze the statistical effect of computational approximations of the Nystr\"{o}m kernel $k$-means, and prove that it achieves the same statistical accuracy as the exact kernel $k$-means considering only $\Omega(\sqrt{nk})$ Nystr\"{o}m landmark points. To the best of our knowledge, such sharp excess clustering risk bounds for kernel (or approximate kernel) $k$-means have never been proposed before.