Kernel Representations and Kernel Density Estimation

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.