Natural Gradient Bayesian Filtering: Geometry-Aware Filter for Dynamical Systems
For practitioners and researchers in state estimation for nonlinear dynamical systems, this work provides a principled geometric framework that improves upon standard Gaussian filters while maintaining computational tractability.
This tutorial introduces the Natural Gradient Gaussian Approximation (NANO) filter, a geometry-aware Gaussian filtering method that uses natural gradient descent on the statistical manifold of Gaussian distributions. It demonstrates that a single natural-gradient step exactly recovers the Kalman measurement update in linear-Gaussian cases and shows improved performance in nonlinear estimation problems such as satellite attitude estimation, SLAM, and robotic state estimation.
Bayesian filtering is a cornerstone of state estimation in complex systems such as aerospace systems, yet exact solutions are available only for linear Gaussian models. In practice,nonlinear systems are handled through tractable approximations,with Gaussian filters such as the extended and unscented Kalman filters being among the most widely used methods. This tutorial revisits Gaussian filtering from an information-geometric perspective, viewing the prediction and measurement update steps as inference procedures over state distributions. Within this framework, we introduce a geometry-aware Gaussian filtering approach that leverages natural gradient descent on the statistical manifold of Gaussian distributions. The resulting Natural Gradient Gaussian Approximation (NANO) filter iteratively refines the posterior mean and covariance while respecting the intrinsic geometry of the Gaussian family and preserving the positive definiteness of the covariance matrix. We further highlight fundamental connections to the classical Kalman filtering, showing that a single natural-gradient step exactly recovers the Kalman measurement update in the linear-Gaussian case. The practical implications of the proposed framework are illustrated through case studies in representative nonlinear estimation problems,including satellite attitude estimation, simultaneous localization and mapping, and state estimation for robotic systems including quadruped and humanoid robots.