The Score Kalman Filter

TL;DR

The paper introduces the Score Kalman Filter, a novel nonlinear Bayesian filtering method that avoids costly partition function calculations by combining score matching with Stein's identity, enabling efficient high-dimensional filtering.

Contribution

It presents a new filtering algorithm that simplifies the computation by eliminating the partition function, extending the classical Kalman filter to higher dimensions with improved accuracy.

Findings

SKF outperforms EKF, UKF, EnKF, and particle filters on synthetic benchmarks.

The method scales to 20 dimensions with lower RMSE.

It maintains the classical Kalman filter as a special case.

Abstract

A central obstacle in nonlinear Bayesian filtering is representing the belief distribution. Moment-based filters address this by propagating polynomial moments and reconstructing a density from them. Recent work completes the predict-update loop via the maximum-entropy (MaxEnt) principle, but each step requires the partition function and its gradient, both $n$-dimensional integrals whose cost scales exponentially, restricting the demonstrated MaxEnt moment filtering to $n \le 4$. We avoid the partition function entirely by combining score matching with Stein's identity. In our setting, score matching reduces the density fit to a single linear solve whose coefficients are assembled directly from the propagated moments. The same parameters then drive Stein's identity to close the moment hierarchy during prediction and to recover posterior moments after each Bayesian update, keeping the full predict-update loop free of partition function evaluation. The resulting Score Kalman Filter (SKF) reduces to the classical information-form Kalman filter as a special case and performs every step through linear algebra. On nonlinear coupled-oscillator networks, the SKF runs through $n=20$ and reports lower RMSE than the EKF, UKF, EnKF, and particle-filter baselines on the tested synthetic benchmarks.