Natural Gradient Gaussian Approximation Filter with Positive Definiteness Guarantee

A team of researchers has published a critical fix for the NANO filter, an advanced Bayesian filtering algorithm. The original NANO filter, which avoids linearization errors common in Kalman filters for nonlinear systems, suffered from a fundamental instability: its natural gradient descent update could cause the posterior covariance matrix to lose its positive definiteness, leading the entire filter to diverge. The new paper, 'Natural Gradient Gaussian Approximation Filter with Positive Definiteness Guarantee,' identifies the root cause as the indefiniteness of the expected Hessian of the log-likelihood function.

To solve this, the authors propose two key mathematical remedies. First, they approximate the problematic Hessian using the Gauss-Newton method, representing it as a self-adjoint product of Jacobians, which is guaranteed to be positive semi-definite. Second, they reformulate the covariance update as an exponential-form update on the Cholesky factor, reconstructing the covariance via its Gram matrix to mathematically enforce positive definiteness. In experiments on three classical nonlinear systems, the revised NANO filter with these guarantees not only remained stable but also demonstrated superior performance compared to both the original, unstable NANO filter and popular members of the Kalman filter family like the Extended and Unscented Kalman Filters.