Sample set design for nonlinear Kalman filters viewed as a moment problem

For designing sample sets for nonlinear Kalman filters, i.e., Linear Regression Kalman Filters (LRKFs), a new method is introduced for approximating Gaussian densities by discrete densities, so called Dirac Mixtures (DMs). This approximating DM should maintain the mean and some higherorder moments and should homogeneously cover the support of the original density. Homogeneous approximations require redundancy, which means there are more Dirac components than necessary for fulfilling the moment constraints. Hence, some sort of regularization is required as the solution is no longer unique. Two types of regularizers are possible: The first type ensures smooth approximations, e.g., in a maximum entropy sense. The second type we pursue here ensures closeness of the approximating density to the given Gaussian. As standard distance measures are typically not well defined for discrete densities on continuous domains, we focus on shifting the mass distribution of the approximating density as close to the true density as possible. Instead of globally comparing the masses as in a previous paper, the key idea is to characterize individual Dirac components by kernel functions representing the spread of probability mass that is appropriate at a given location. A distance measure is then obtained by comparing the deviation between the true density and the induced kernel density. As a result, the approximation problem is converted to an optimization problem as we now minimize the distance under the desired moment constraints.