In this paper, we derive an optimal transportation particle filter for linear time-varying systems with correlated noises. This method can be regarded as the extension of the feedback particle filter with optimal transportation structure. However, the particles in our method are evolved in a deterministic way while we need to generate random particles in feedback particle filter. Consequently, we only need a very few particles to obtain the satisfying results and this property is especially significant for high dimensional problems. The error analysis of our method and feedback particle filter have been carried out when the system is time-invariant. Compared with feedback particle filter and ensemble Kalman filter, our method shows great efficiency in numerical experiments including both scalar case and high dimensional case.

In this article, we propose the novel neural stochastic differential equations (SDEs) driven by noisy sequential observations called neural projection filter (NPF) under the continuous state-space models (SSMs) framework. The contributions of this work are both theoretical and algorithmic. On the one hand, we investigate the approximation capacity of the NPF, i.e., the universal approximation theorem for NPF. More explicitly, under some natural assumptions, we prove that the solution of the SDE driven by the semimartingale can be well approximated by the solution of the NPF. In particular, the explicit estimation bound is given. On the other hand, as an important application of this result, we develop a novel data-driven filter based on NPF. Also, under certain condition, we prove the algorithm convergence; i.e., the dynamics of NPF converges to the target dynamics. At last, we systematically compare the NPF with the existing filters. We verify the convergence theorem in linear case and experimentally demonstrate that the NPF outperforms existing filters in nonlinear case with robustness and efficiency. Furthermore, NPF could handle high-dimensional systems in real-time manner, even for the $100$ -D cubic sensor, while the state-of-the-art (SOTA) filter fails to do it.

This note introduces the Modified Extended Kalman Filter (MEKF), reformulating the EKF update step within a nonlinear regression framework. We propose a novel outlier-robust scheme, MCIC-MEKF, utilizing the minimum correntropy-induced cost criterion. We provide a theoretical analysis of its outlier robustness through stochastic stability, proving exponentially bounded mean square posterior estimation error under natural conditions. Additionally, we present a technical approximation for the adaptive Kalman gain, enhancing efficiency without compromising performance. Simulation results confirm MCIC-MEKF's robustness against various non-Gaussian noises with large outliers, outperforming several filtering benchmarks.

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 can only be used for time-invariant systems. In this paper, we extend the estimation algebra method so that it applies to a general class of time-varying filtering systems. Then 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 EKF and FPF.