Natural Gradient Gaussian Approximation Filter with Positive Definiteness Guarantee
For practitioners of nonlinear Bayesian filtering, this work provides a robust solution to a known instability issue in the NANO filter, improving reliability and accuracy.
The paper addresses the loss of positive definiteness in the posterior covariance during natural gradient updates in the NANO filter for nonlinear Bayesian filtering. Two remedies are proposed: a Gauss-Newton approximation of the log-likelihood Hessian and an exponential-form Cholesky factor update, both ensuring positive definiteness and outperforming existing Kalman filters and the original NANO filter on three nonlinear systems.
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.