Natural Gradient Gaussian Approximation Filter with Positive Definiteness Guarantee
Tianyi Zhang, Wenhan Cao, Shengbo Eben Li

TL;DR
This paper introduces a modified NANO filter that guarantees positive definiteness of the covariance matrix, improving stability and performance in nonlinear Bayesian filtering tasks.
Contribution
It proposes two novel methods to ensure positive definiteness in the NANO filter, addressing divergence issues caused by indefinite covariance updates.
Findings
The Gauss-Newton approximation guarantees positive semi-definite Hessian.
Exponential-form covariance update maintains positive definiteness.
The proposed filter outperforms traditional Kalman filters in experiments.
Abstract
Popular Bayes filters often apply linearization techniques, such as Taylor expansion or stochastic linear regression, to enable the use of the Kalman filter structure, but this can lead to large errors in strongly nonlinear systems. The recently proposed NANO filter addresses this issue by interpreting the prediction and update steps of Bayesian filtering as two distinct optimization problems and solving them through moment matching and natural gradient descent, thereby avoiding model linearization errors. However, the natural gradient update in NANO can occasionally diverge because the posterior covariance in its iteration may lose positive definiteness. Our analysis shows that the posterior covariance is the sum of the inverse prior covariance and the expected Hessian of the log-likelihood function, and that the indefiniteness of the latter term is the root cause of update failure. To…
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