Natural Gradient Gaussian Approximation Filter with Positive Definiteness Guarantee

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 address this issue, we propose two remedies. The first approximates the log-likelihood Hessian using the Gauss-Newton method, representing it as the self-adjoint product of the Jacobian of the normalized measurement residual, which is guaranteed to be positive semi-definite. The second reformulates the covariance update as an exponential-form update of the Cholesky factor and reconstructs the covariance via its Gram matrix, which ensures positive definiteness. Experiments on three classical nonlinear systems demonstrate that the proposed NANO filter with guaranteed positive definiteness outperforms popular members of the Kalman filter family and original NANO filter.