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

Consider a nonlinear transform ${\Phi}(x)$ of x in $\mathbb{R}^p$ to Hilbert space H and assume that the dot product between ${\Phi}(x)$ and ${\Phi}(x^{\prime})$ in H is given by < ${\Phi}(x)$, ${\Phi}(x^{\prime})$ >= K(x, x'). The aim of this paper is to propose a mathematical technique to take screen shots of the multivariate dataset mapped to Hilbert space H, particularly suited to Gaussian kernel $K({\cdot},{\cdot})$, which is defined by $K(x,x^{\prime})={\exp}(-{\sigma}{\parallel}x-x^{\prime}{\parallel}^2)$, ${\sigma}$ > 0. Several numerical examples are given.