Non-parametric expectation-maximization for Gaussian mixtures

We propose a non-parametric EM algorithm, where nonparametric kernel density estimation is used instead of conventional parametric density estimation. Our proposal kernel function, the constructive elliptical basis function (CEBF), is an extension of the EBF and can effectively represent ill-scaled and non-separable distributions without a covariance matrix even in high dimensionality in a nonparametric manner. The overlapping CEBFs with a fixed smoothing parameter can be used as an approximation of Gaussian distribution in a statistical sense. Using CEBFs as kernel functions, we propose non-parametric expectation-maximization (NPEM) for the Gaussian mixture model (GMM). Then we show that NPEM obtains better estimation in terms of log likelihood than traditional EM algorithms when the given data set has high dimensionality or holds multiple components by numerical experiments.