Optimal State Estimation in the Presence of Non-Gaussian Uncertainty via Wasserstein Distance Minimization

Abstract: This paper presents a novel distribution-agnostic Wasserstein distance-based estimation framework. The goal is to determine an optimal map combining prior estimate with measurement likelihood such that posterior estimation error optimally reaches the Dirac delta distribution with minimal effort. The Wasserstein metric is used to quantify the effort of transporting from one distribution to another. We hypothesize that minimizing the Wasserstein distance between the posterior error and the Dirac delta distribution results in optimal information fusion and posterior state uncertainty. Framework validation is demonstrated by the successful recovery of the classical Kalman filter for linear systems with Gaussian uncertainties. Notably, the proposed Wasserstein filter does not rely on particle representation of uncertainty. Furthermore, the classical result for the Gaussian Sum Filter (GSF) is retrieved from the Wasserstein framework. This approach analytically exhibits the suboptimality of GSF and enables the use of nonlinear optimization techniques to enhance the accuracy of the Gaussian sum estimator.