---
title: "NANO Filter 原始存档"
created: 2026-06-22
type: raw
arxiv: "2410.15832"
source: "https://arxiv.org/abs/2410.15832"
---

# Nonlinear Bayesian Filtering with Natural Gradient Gaussian Approximation

- **作者**: Wenhan Cao, Tianyi Zhang, Zeju Sun, Chang Liu, Stephen S.-T. Yau, Shengbo Eben Li
- **机构**: 清华大学（车辆与运载学院、数学科学系）、北京大学（工学院）、BIMSA
- **arXiv**: 2410.15832 [eess.SY]
- **提交**: 2024-10-21 | 最新版本 v4: 2026-03-15
- **DOI**: https://doi.org/10.48550/arXiv.2410.15832

## 摘要

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. When facing 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. To address this issue, this paper reconstructs 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 derive its natural gradient 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 iterative filter for nonlinear systems called **N**atural Gr**a**dient Gaussia**n** Appr**o**ximation filter, or NANO filter for short. We prove that NANO filter locally converges to the optimal Gaussian approximation at each time step. Furthermore, the estimation error is proven exponentially bounded for nearly linear measurement equation and low noise levels through constructing a supermartingale-like property across consecutive time steps.

## 关键概念

- Natural gradient descent on Gaussian manifold
- Fisher information matrix
- Moment matching (prediction step)
- Stein's lemma for optimality conditions
- Gibbs posterior for robustness
- Pseudo-Huber loss for outlier handling
- Convergence proof & exponential error bound
