Concentration of kernel matrices with application to kernel spectral clustering

TL;DR

This paper establishes nonasymptotic concentration inequalities for random kernel matrices, including Lipschitz and Euclidean kernels, and applies these results to analyze the high-dimensional consistency of kernel spectral clustering.

Contribution

It derives sharp, dimension-free concentration bounds for kernel matrices under broad distributional assumptions and demonstrates their use in proving the consistency of kernel spectral clustering in high dimensions.

Findings

Dimension-free concentration inequalities for Lipschitz kernels.

High-dimensional consistency of kernel spectral clustering.

Applicability to nonparametric mixture models and noisy manifold data.

Abstract

We study the concentration of random kernel matrices around their mean. We derive nonasymptotic exponential concentration inequalities for Lipschitz kernels assuming that the data points are independent draws from a class of multivariate distributions on $\mathbb R^d$, including the strongly log-concave distributions under affine transformations. A feature of our result is that the data points need not have identical distributions or zero mean, which is key in certain applications such as clustering. Our bound for the Lipschitz kernels is dimension-free and sharp up to constants. For comparison, we also derive the companion result for the Euclidean (inner product) kernel for a class of sub-Gaussian distributions. A notable difference between the two cases is that, in contrast to the Euclidean kernel, in the Lipschitz case, the concentration inequality does not depend on the mean of the underlying vectors. As an application of these inequalities, we derive a bound on the misclassification rate of a kernel spectral clustering (KSC) algorithm, under a perturbed nonparametric mixture model. We show an example where this bound establishes the high-dimensional consistency (as $d \to \infty$) of the KSC, when applied with a Gaussian kernel, to a noisy model of nested nonlinear manifolds.