The geometry of kernelized spectral clustering

TL;DR

This paper analyzes the geometric properties of kernelized spectral clustering, showing how it effectively recovers latent labels in mixture models by leveraging eigenspaces related to component densities.

Contribution

It provides a theoretical framework linking the geometry of spectral clustering to the separation of mixture components in nonparametric settings.

Findings

Principal eigenspace approximates component densities when overlap is small.

Embedded samples from different components are nearly orthogonal with large samples.

Spectral clustering can control mislabeling in finite mixture models.

Abstract

Clustering of data sets is a standard problem in many areas of science and engineering. The method of spectral clustering is based on embedding the data set using a kernel function, and using the top eigenvectors of the normalized Laplacian to recover the connected components. We study the performance of spectral clustering in recovering the latent labels of i.i.d. samples from a finite mixture of nonparametric distributions. The difficulty of this label recovery problem depends on the overlap between mixture components and how easily a mixture component is divided into two nonoverlapping components. When the overlap is small compared to the indivisibility of the mixture components, the principal eigenspace of the population-level normalized Laplacian operator is approximately spanned by the square-root kernelized component densities. In the finite sample setting, and under the same assumption, embedded samples from different components are approximately orthogonal with high probability when the sample size is large. As a corollary we control the fraction of samples mislabeled by spectral clustering under finite mixtures with nonparametric components.