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

TL;DR

This paper introduces a geometry-aware Gaussian filtering method called NANO that uses natural gradient descent on the statistical manifold to improve state estimation in nonlinear systems, maintaining covariance positivity.

Contribution

It presents a novel natural gradient-based Gaussian filtering framework that respects the geometry of Gaussian distributions and connects to classical Kalman filtering.

Findings

NANO recovers the Kalman update exactly in linear-Gaussian cases.

The method improves nonlinear state estimation in satellite, SLAM, and robotic applications.

It preserves positive definiteness of the covariance matrix during updates.

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.