Finite Dimensional Estimation Algebra for Time-Varying Filtering System and Optimal Transport Particle Filter: A Tangent Flow Point of View

Ever since the technique of the Kalman filter was popularized, there has been a lot of research interest in finding more classes of finite-dimensional recursive filters. In past research, the estimation algebra method could only be used for time-invariant systems. In this article, we extend the estimation algebra method so that it applies to a general class of time-varying filtering systems. Then, the Wei-Norman method can be used to derive the explicit solution of the posterior distribution of state estimation. As a special control law, tangent flow is derived for the nonlinear filtering system based on the Monge-Amp$\grave{\text{e}}$re equation in optimal transport. As a result, we propose an optimal transportation filter by applying stochastic tangent flow to Yau filtering systems. The numerical experiments demonstrate the higher efficacy and accuracy of the proposed optimal transportation filter compared to common traditional algorithms such as the extended Kalman filter and feedback particle filter.