Algorithm Design and Comparative Test of Natural Gradient Gaussian Approximation Filter

Abstract: Popular Bayes filters typically rely on linearization techniques such as Taylor series expansion and stochastic linear regression to use the structure of standard Kalman filter. These techniques may introduce large estimation errors in nonlinear and non-Gaussian systems. This paper overviews a recent breakthrough in filtering algorithm design called \textit{N}atural Gr\textit{a}dient Gaussia\textit{n} Appr\textit{o}ximation (NANO) filter and compare its performance over a large class of nonlinear filters. The NANO filter interprets Bayesian filtering as solutions to two distinct optimization problems, which allows to define optimal Gaussian approximation and derive its corresponding extremum conditions. The algorithm design still follows the two-step structure of Bayes filters. In the prediction step, NANO filter calculates the first two moments of the prior distribution, and this process is equivalent to a moment-matching filter. In the update step, natural gradient descent is employed to directly minimize the objective of the update step, thereby avoiding errors caused by model linearization. Comparative tests are conducted on four classic systems, including the damped linear oscillator, sequence forecasting, modified growth model, and robot localization, under Gaussian, Laplace, and Beta noise to evaluate the NANO filter's capability in handling nonlinearity. Additionally, we validate the NANO filter's robustness to data outliers using a satellite attitude estimation example. It is observed that the NANO filter outperforms popular Kalman filters family such as extended Kalman filter (EKF), unscented Kalman filter (UKF), iterated extended Kalman filter (IEKF) and posterior linearization filter (PLF), while having similar computational burden.