Global and Local Views of the Hilbert Space Associated to Gaussian Kernel Huh, Myung-Hoe;
Consider a nonlinear transform of x in to Hilbert space H and assume that the dot product between and in H is given by < , >= 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 , which is defined by , > 0. Several numerical examples are given.
Data visualization;Hilbert space;Gaussian kernel;principal component analysis;
Gabriel, K. R. (1971). The biplot display of matrices with the application to principal component analysis, Biometrika, 58, 453-467.
Huh, M. H. (2013a). Arrow diagrams for kernel principal component analysis, Communications for Statistical Applications and Methods, 20, 175-184.
Huh, M. H. (2013b). SVM-guided biplot of observations and variables, Communications for Statistical Applications and Methods, 20, 491-498.
Huh, M. H. and Lee, Y. G. (2013). Biplots of multivariate data guided by linear and/or logistic regression, Communications for Statistical Applications and Methods, 20, 129-136.
Scholkopf, B., Smola, A. and Muller, K. R. (1998). Nonlinear component analysis as a kernel eigen-value problem, Neural Computation, 10, 1299-1319.