Optimal Transport as a Reduction Technique for Deterministic Nonlinear Filtering

The solution to the state estimation problem is given by the Bayesian recursive relations (BRRs). Recently, ensemble Gaussian mixture filters have shown to be an accurate and consistent solution to the state estimation problem. In this type of filters, the BRRs are solved by approximating the state probability density function (PDF) via Gaussian mixtures (GMs) and point masses (PMs). Throughout the propagation and measurement update steps, the approximated state PDF is constantly switching between GMs and PMs. Therefore, a key step for this solution involves optimally sampling PMs from GMs. For onboard applications, verifiable and computationally inexpensive sampling techniques are crucial. In previous work, a deterministic sampling technique was developed by minimizing a distance metric known as the modified Cramér-von Mises distance (MCVMD), yielding a verifiable solution. However, the computationally feasibility of this solution for onboard use was not considered. This work introduces a new sampling strategy that is both deterministic and computationally inexpensive compared to MCVMD approach. By solving the approximate optimal transport problem via an iterative Sinkhorn-Knopp algorithm, this new technique is able to sub-optimally sample from a GM, providing a computationally inexpensive filter.