Optimal Reduction of Multivariate Dirac Mixture Densities

This paper is concerned with the optimal approximation of a given multivariate Dirac mixture, i.e., a density comprising weighted Dirac distributions on a continuous domain, by a Dirac mixture with a reduced number of components. The parameters of the approximating density are calculated by numerically minimizing a smooth distance measure, a generalization of the well-known Cramér–von Mises-Distance to the multivariate case. This generalization is achieved by defining an alternative to the classical cumulative distribution, the Localized Cumulative Distribution (LCD), as a smooth characterization of discrete random quantities (on continuous domains). The resulting approximation method provides the basis for various efficient nonlinear estimation and control methods.