Composite Transportation Dissimilarity in Consistent Gaussian Mixture Reduction

Gaussian Mixtures (GMs) are a powerful tool for approximating probability distributions across a variety of fields. In some applications the number of GM components rapidly grows with time, so that reduction algorithms are necessary. Given a GM with a large number of components, the problem of Gaussian Mixture Reduction (GMR) consists in finding a GM with considerably less components that is not too dissimilar from the original one. There are many issues that make non trivial this problem. First of all, many dissimilarity measures exist for GMs, although most of them lack closed forms, and their numerical computation is a demanding task, especially for distributions in high dimensions. Moreover, some basic reduction actions can be simple or complex tasks depending on which dissimilarity measure is chosen. It follows that most reduction procedures proposed in the literature are made of steps that are aimed at maintaining low dissimilarity according to different measures, thus leading to a pipeline of actions that are not mutually consistent. In this paper Composite Transportation Dissimilarities are discussed and exploited to formulate a GMR framework that preserves consistency with a unique dissimilarity measure, and provides a generalization of the celebrated Runnalls’ GMR approach.