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

Title & Authors
Global and Local Views of the Hilbert Space Associated to Gaussian Kernel
Huh, Myung-Hoe;

Abstract
Consider a nonlinear transform $\small{{\Phi}(x)}$ of x in $\small{\mathbb{R}^p}$ to Hilbert space H and assume that the dot product between $\small{{\Phi}(x)}$ and $\small{{\Phi}(x^{\prime})}$ in H is given by < $\small{{\Phi}(x)}$, $\small{{\Phi}(x^{\prime})}$ >= 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 $\small{K({\cdot},{\cdot})}$, which is defined by $\small{K(x,x^{\prime})={\exp}(-{\sigma}{\parallel}x-x^{\prime}{\parallel}^2)}$, $\small{{\sigma}}$ > 0. Several numerical examples are given.
Keywords
Data visualization;Hilbert space;Gaussian kernel;principal component analysis;
Language
English
Cited by
References
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