Regularized Non-Parametric Multivariate Density and Conditional Density Estimation

In this paper, a distance-based method for both multivariate non-parametric density and conditional density estimation is proposed. The contributions are the formulation of both density estimation problems as weight optimization problems for Gaussian mixtures centered about samples with identical parameters. Furthermore, the minimization is based on the modified Cram\'{e}r-von Mises distance of the Localized Cumulative Distributions, removing the ambiguity of the defi- nition of the multivariate cumulative distribution function. The minimization problem is amended with a regularization term penalizing the densities' roughness to avoid overfitting. The resulting estimation problems for both densities and conditional densities are shown to be phrasable in the form of readily implementable quadratic programs. Experimental comparison against EM, SVR, and GPR based on the log-likelihood and performance in benchmark recursive filtering applications show high quality of the densities and good performance at less computational cost, i.e., the density representations are sparser.