Score Kalman Filter beats state-of-the-art with pure linear algebra

Nonlinear Bayesian filtering has long been hindered by the computational cost of representing belief distributions. Moment-based filters propagate polynomial moments but must reconstruct a density at each step—traditionally via the maximum-entropy (MaxEnt) principle, which requires computing the partition function and its gradient via n-dimensional integrals. This exponential scaling restricted practical MaxEnt filtering to state dimensions n ≤ 4. Iwasaki, Bloch, Lee, and Ghaffari now circumvent this bottleneck entirely with the Score Kalman Filter (SKF). By employing score matching and Stein's identity, the SKF fits the density via a single linear solve from propagated moments, then uses the same parameters to close the moment hierarchy and recover posterior moments—all without ever evaluating the partition function. The filter reduces to the classical information-form Kalman filter as a special case.

In experiments on nonlinear coupled-oscillator networks, the SKF scales cleanly to n=20 and delivers lower root-mean-square error (RMSE) than the extended Kalman filter (EKF), unscented Kalman filter (UKF), ensemble Kalman filter (EnKF), and particle filters across synthetic benchmarks. Because the entire predict-update loop is performed through linear algebra, the method is computationally efficient and avoids the curse of dimensionality that plagues particle filters and grid-based methods. This breakthrough promises to make high-dimensional nonlinear filtering tractable for real-time control, robotics, navigation, and other domains where accurate state estimation is critical but computational budgets are tight.