High-Quality Assumed Gaussian Filtering Based on Wasserstein Barycentric Interpolation

In this paper, we introduce a novel Gaussian Assumed Density Filter (GADF) for high-quality state estimation in discrete-time stochastic nonlinear dynamic systems, with a primary focus on the measurement update. Rooted in optimal transport theory, the Wasserstein distance is employed as a powerful metric for comparing probability distributions. Building on this foundation, we utilize the unique, explicit Wasserstein barycentric interpolation between Gaussian distributions to parameterize an initial Gaussian Process (GP) in the joint measurement/prior state space. Deterministic samples drawn from the true joint measurement/state density are then used with likelihood-based parameter estimation techniques to optimize the parameters of this Gaussian Process. As a result, the derived Gaussian Process provides a local non-Gaussian approximation to the true joint density. This approach eliminates the need for a second Gaussian assumption on the joint density and avoids an explicit likelihood function, making it a higher-quality plug-in replacement for the commonly used Linear Regression Kalman Filter (LRKF).