Dirac Mixture Reduction Using Wasserstein Distances on Projected Cumulative Distributions

The reapproximation of discrete probability densities is a common task in sample-based filters such as the particle filter. It can be viewed as the approximation of a given Dirac mixture density with another one, typically with fewer samples. In this paper, the Wasserstein distance is established as a suitable measure to compare two Dirac mixtures. The resulting minimization problem is also known as location-allocation or facility location problem and cannot be solved in polynomial time. Therefore, the well-known sliced Wasserstein distance is introduced as a replacement and its ties to the projected cumulative distribution (PCD) are shown. An iterative algorithm is proposed to minimize the sliced Wasserstein distance between the given distribution and approximation.