Nonlinear Bayesian Filtering With Natural Gradient Gaussian Approximation.

Area of Science:

• * Bayesian filtering and state estimation
• * Nonlinear system analysis
• * Machine learning and optimization

Background:

• * Practical Bayes filters often use Gaussian assumptions for computational efficiency.
• * Traditional Gaussian filters like EKF and UKF struggle with nonlinear systems due to linearization errors.
• * Existing methods can introduce significant inaccuracies in state estimation for nonlinear models.

Purpose of the Study:

• * To develop a novel Gaussian filter that overcomes linearization errors in nonlinear systems.
• * To reconstruct prediction and update steps as optimization problems solvable via analytical conditions.
• * To introduce an iterative approach for minimizing update step objectives, avoiding linearization.

Main Methods:

• * Reconstruction of Gaussian filter prediction and update steps as optimization problems.
• * Utilizing Stein's lemma to find analytical solutions for optimal conditions.
• * Deriving a natural gradient on the Gaussian manifold using the Fisher information matrix for the update step.
• * Combining moment matching prediction with iterative natural gradient update (NANOfilter).

Main Results:

• * The proposed NANOfilter converges locally to the optimal Gaussian approximation at each time step.
• * Estimation error is proven to be exponentially bounded under specific conditions (nearly linear measurement, low noise).
• * Experimental results show NANOfilter reduces average root mean square error by approximately 45% compared to EKF, UKF, and others.
• * NANOfilter achieves these improvements with a comparable computational burden.

Conclusions:

• * NANOfilter offers a significant improvement in state estimation accuracy for nonlinear systems.
• * The method effectively avoids linearization errors inherent in traditional Gaussian filters.
• * NANOfilter presents a computationally efficient and accurate alternative for practical Bayesian filtering applications.