Schur-Neural KF: Learned Schur-Consistent Corrections to the Extended Kalman Filter

2026-09-26

AI Summary

This study addresses the failure of probabilistic conditional interpretation and non-positive-definite covariance in the Extended Kalman Filter by proposing the SN-KF framework. This method introduces a novel Schur complement-based neural network correction mechanism that perturbs cross-covariance and noise factors to ensure the positive semi-definiteness of the joint predicted covariance, while integrating recurrent neural networks with amplitude-gated modulation for parameter optimization. Theoretical analysis proves that state uncertainty does not increase with additional measurements and that arbitrarily large corrections remain bounded. Experiments on dual-radar and unicycle platforms demonstrate that the proposed approach significantly expands the fault-free hyperparameter region, reduces small-sample RMSE, and achieves optimal detection accuracy alongside superior alarm rates.

Abstract

We present Schur-Neural KF (SN-KF), a learning-based correction to the extended Kalman filter (EKF) that preserves the probabilistic conditioning interpretation of the EKF. The method perturbs the predictive state-measurement cross-covariance and the Cholesky factor of the measurement noise covariance so that the resulting joint predictive covariance is always positive semidefinite. The positive semidefiniteness is ensured by a Schur complement-based parametrization. We instantiate the parametrization with a recurrent neural architecture whose matrix outputs are modulated by amplitude gates. We prove that incorporating a measurement does not increase the filter's state uncertainty, and show that no measurement can induce an arbitrarily large state correction relative to its statistical surprise. We also present a perturbative analysis suggesting SN-KF's structural strength in the data-scarce regime. We provide two numerical experiments to illustrate the practical benefits of SN-KF. In a two-radar experiment, enforcing Schur-consistency provides a much broader failure-free hyperparameter region and reduces RMSE for small training subsets, consistent with our theoretical analysis in the data-scarce regime. In the unicycle experiment, SN-KF achieves the best precision, recall, false alarm rate, and gated RMSE under innovation-based sensor-fault rejection.

Problem

• Extended Kalman Filter
• Schur complement
• positive semidefiniteness
• data-scarce regime
• sensor-fault rejection

Innovation

• Extended Kalman Filter
• Schur complement
• Recurrent neural network
• Positive semidefiniteness
• Data-scarce regime