Beijing Institute of Mathematical Sciences and Applications

Shing Toung Yau

Wenhan Cao

Wednesday, October 23, 2024 8:00 PM - 8:30 PM

Online

Practical Bayes filters often assume the state distribution of each time step to be Gaussian for computational tractability, resulting in the so-called Gaussian filters. In nonlinear systems, Gaussian filters such as extended Kalman filter (EKF) or unscented Kalman filter (UKF) typically rely on certain linearization techniques, which can introduce large estimation errors. In this paper, we reconstruct the prediction and update steps of Gaussian filtering as solutions to two distinct optimization problems, whose optimal conditions are found to have analytical forms from Stein’s lemma. It is observed that the stationary point for the prediction step requires calculating the first two moments of the prior distribution, which is equivalent to that step in existing moment-matching filters. In the update step, instead of linearizing the model to approximate the stationary points, we propose an iterative approach to directly minimize the update step’s objective to avoid linearization errors. For the purpose of performing the steepest descent on the Gaussian manifold, we use the natural gradient descent that leverages Fisher information matrix to adjust the gradient direction, accounting for the curvature of the parameter space. Combining this update step with moment matching in the prediction step, we introduce a new filer called Natural Gradient Gaussian Approximation filter, or NANO filter for short. We prove that NANO filter can converge to the optimal Gaussian approximation at each time step, with errors bounded up to a second-order Taylor expansion. The estimation error is proven exponentially bounded for nearly linear measurement equation and low noise levels through constructing a supermartingale-like inequality across consecutive time steps. Real-world experiments demonstrate that, compared to popular Gaussian filters such as EKF, UKF, iterated EKF, and posterior linearization filter, NANO filter reduces the average root mean square error by approximately 45% while maintaining a comparable computational burden.