Outlier-robust Kalman Filtering through Generalised Bayes

TL;DR

_outlier-robust Kalman Filtering through Generalised Bayes_ introduces WoLF, a robust online filtering framework that substitutes the standard log-likelihood with a weighted loss $\ell_t({\boldsymbol{\theta}}_t)$ within a generalised Bayes update. This yields closed-form Gaussian updates for the KF, EKF, and EnKF while remaining computationally competitive with traditional filters. The authors propose several weighting schemes, including IMQ, MD, and a thresholded MD, and prove bounded posterior influence under these weights, with empirical validation across 2D tracking, online MLP regression, and Lorenz96 benchmarks. The work demonstrates that WoLF can achieve superior or comparable robustness to existing methods at substantially lower computational cost, making it attractive for high-dimensional and nonlinear filtering in the presence of outliers.

Abstract

We derive a novel, provably robust, and closed-form Bayesian update rule for online filtering in state-space models in the presence of outliers and misspecified measurement models. Our method combines generalised Bayesian inference with filtering methods such as the extended and ensemble Kalman filter. We use the former to show robustness and the latter to ensure computational efficiency in the case of nonlinear models. Our method matches or outperforms other robust filtering methods (such as those based on variational Bayes) at a much lower computational cost. We show this empirically on a range of filtering problems with outlier measurements, such as object tracking, state estimation in high-dimensional chaotic systems, and online learning of neural networks.

Figures (20)

• Figure 1: First state component of the SSM \ref{['eq:noisy-2d-ssm']}. The grey dots are measurements sampled from \ref{['eq:noisy-2d-ssm']} and the red crosses are measurements sampled from an outlier measurement process. The dotted blue line shows the KF posterior mean estimate and the solid orange line shows our proposed WoLF posterior mean estimate. The regions around the posterior mean cover two standard deviations. For comparison, the dashed black line shows the true sampled state process.
• Figure 2: PIF for the 2d tracking problem of Section \ref{['experiment:2d-tracking']}. The last measurement ${\bm{y}}_t$ is replaced with ${\bm{y}}^{c}_t = {\bm{y}}_{t} + \boldsymbol{\epsilon}$, where $\boldsymbol{\epsilon} \in [-5, 5]\times[-5, 5]$. We observe that the PIF is asymetric for the weighted methods; this is because the weighting term is a function of the prior predictive and the measurement at time $t$. See Appendix \ref{['subsec:pif-plot-explain']} for a more detailed explanation.
• Figure 3: The left panel shows a sample path using the Student variant and the right panel shows a sample path using the mixture variant. The top left figure on each panel shows the true underlying state in black, and the measurements as grey dots.
• Figure 4: Distribution (across 500 2d tracking trials) of RMSE for first component of the state vector, $J_{T,0}$. Left panel: Student observation model. Right panel: Mixture observation model.
• Figure 5: RMedSE versus time per step (relative to the OGD minus $1$) across the corrupted UCI datasets.

Theorems & Definitions (13)

• Proposition 3.1
• Theorem 3.2
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4