Kernel Representations and Kernel Density Estimation

author: Peter J. Bickel, Department of Statistics, UC Berkeley
published: Dec. 18, 2008,   recorded: December 2008,   views: 6905


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There has been a great deal of attention in recent times particularly in machine learning to representation of multivariate data points x by K(x, ·) where K is positive and symmetric and thus induces a reproducing kernel Hilbert space.The idea is then to use the matrix

K(Xi , Xj )as a substitute for the empirical covariance matrix of a sample X1 , . . . , Xn for PCA

and other inference.(Jordan and Fukumizu(2006) for instance. Nadler et. al(2006) connected this approach to one based on random walks and diffusion limits and indicated a connection to kernel density estimation.By making at least a formal connection to a multiplication operator on a function space we make further connection and show how clustering results of Beylkin ,Shih and Yu (2008) which apparently differ from Nadler et al. can be explained.

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