Global and Local Views of the Hilbert Space Associated to Gaussian Kernel

Abstract

Consider a nonlinear transform Φ(x) of x in Rp to Hilbert space H and assume that the dot product betweenΦ(x) and Φ(x′) in H is given by < Φ(x);Φ(x′) >= K(x; x′). The aim of this paper is to propose a mathematicaltechnique to take screen shots of the multivariate dataset mapped to Hilbert space H, particularly suited to Gaussiankernel K(· ; ·), which is defined by K(x; x′) = exp(− ∥ x − x′∥2); > 0. Several numerical examples are given.