Title: Kernel PCA for multivariate extremes
Authors: Richard Davis - Columbia University (United States)
Gennady Samorodnitsky - Cornell University (United States)
Marco Avella-Medina - Columbia University (United States) [presenting]
Abstract: Kernel PCA is proposed as a method for analyzing the dependence structure of multivariate extremes, and it is demonstrated that it can be a powerful tool for clustering and dimension reduction. We provide some theoretical insights into the preimages obtained by kernel PCA, demonstrating that under certain conditions, they can effectively identify clusters in the data. We build on these new insights to characterize rigorously the performance of kernel PCA based on an extremal sample, i.e., the angular part of random vectors for which the radius exceeds a large threshold. More specifically, we focus on the asymptotic dependence of multivariate extremes characterized by the angular or spectral measure in extreme value theory and provide a careful analysis in the case where the extremes a generated from a linear factor model. We give theoretical guarantees on the performance of kernel PCA preimages of such extremes by leveraging their asymptotic distribution and Davis-Kahan perturbation bounds. Our theoretical findings are complemented by numerical experiments illustrating the finite sample performance of our methods.