Nearly Optimal Clustering Risk Bounds for Kernel K-Means

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"om kernel k-means, and prove that it achieves the same statistical accuracy as the exact kernel k-means considering only Omega(√nk) Nystr"om 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.