Kernel k-means

TLDR

Kernel k‑means and spectral clustering are used to find non‑linearly separable clusters, yet they have remained loosely related despite extensive research. The study establishes an explicit theoretical link between kernel k‑means and spectral clustering. The authors generalize the weighted kernel k‑means objective and show that the normalized‑cut spectral clustering objective is a special case. The derived connection yields a weighted kernel k‑means algorithm that monotonically decreases normalized cut, obviating the need for eigenvector computations and enabling faster, higher‑quality clustering, as demonstrated on gene‑expression and handwriting data.

Abstract

Kernel k-means and spectral clustering have both been used to identify clusters that are non-linearly separable in input space. Despite significant research, these methods have remained only loosely related. In this paper, we give an explicit theoretical connection between them. We show the generality of the weighted kernel k-means objective function, and derive the spectral clustering objective of normalized cut as a special case. Given a positive definite similarity matrix, our results lead to a novel weighted kernel k-means algorithm that monotonically decreases the normalized cut. This has important implications: a) eigenvector-based algorithms, which can be computationally prohibitive, are not essential for minimizing normalized cuts, b) various techniques, such as local search and acceleration schemes, may be used to improve the quality as well as speed of kernel k-means. Finally, we present results on several interesting data sets, including diametrical clustering of large gene-expression matrices and a handwriting recognition data set.