Natural Gradient Bayesian Filtering: Geometry-Aware Filter for Dynamical Systems

2026-05-04

AI Summary

This work addresses the absence of analytical solutions for Bayesian filtering in nonlinear dynamical systems, where existing Gaussian approximation methods often neglect the underlying geometric structure of probability distributions. From an information-geometric perspective, the authors model prediction and measurement update steps as inference processes on the manifold of Gaussian distributions and propose the geometry-aware NANO filter. This approach iteratively refines the posterior mean and covariance via a single-step natural gradient descent, preserving covariance positive definiteness while exactly recovering the Kalman update in the linear Gaussian case. The method demonstrates superior accuracy and robustness across diverse applications, including satellite attitude estimation, SLAM, and state estimation for quadrupedal and humanoid robots.

Abstract

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.

Problem

• Bayesian filtering
• nonlinear dynamical systems
• Gaussian approximation
• information geometry
• state estimation

Innovation

• Natural Gradient
• Information Geometry
• Gaussian Filtering
• Bayesian Inference
• State Estimation