Gaussian Mixture Models (GMMs) are powerful tools for representing arbitrary distributions or data sets, especially in complex non-linear systems. They are often used as approximators due to their flexibility. However, in many cases, such as for dynamical systems, these must be propagated through non-linear functions. How we can do this is still an open problem.
In this work, we propose a scalable method for quantizing GMMs with formal bounds on the approximation error using the Wasserstein distance. This ensures the method is suitable for safety-critical applications such as autonomous vehicles and UAVs, where formal guarantees are essential. Our approach, called the multi-grids method, constructs local grids at locations of high density, which are identified using clustering techniques. Around each cluster, a hyperrectangular grid is formed, and the locations are placed ensuring minimal error in terms of the Wasserstein distance. This design allows the method to scale efficiently to GMMs of large sizes in terms of both the dimension and the number of components, a known limitation of existing methods.
We validate our method against the state of the art, across various settings, demonstrating significantly lower approximation errors in terms of the Wasserstein distance. Even at higher dimensions of 60 it can still find efficient quantizations with little computational costs. Additionally, we apply the multi-grids method to the uncertainty propagation problem in dynamical systems, including the benchmark Dubins Car, highlighting its practical effectiveness in real-time systems.