Concentration of kernel matrices with application to kernel spectral clustering
This work addresses the theoretical foundation for kernel methods in machine learning, particularly for clustering, by providing concentration bounds that handle non-identical distributions and non-zero means, which is incremental but important for practical applications.
The paper tackles the problem of random kernel matrix concentration by deriving nonasymptotic exponential concentration inequalities for Lipschitz kernels under independent data draws from multivariate distributions, including strongly log-concave ones, with a dimension-free and sharp bound. As an application, it bounds the misclassification rate of a kernel spectral clustering algorithm, showing high-dimensional consistency for a Gaussian kernel on noisy nested nonlinear manifolds.
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.