The Kalman Evolve: Closing the Gap in Kalman Filtering via Interpretable Algorithm Discovery

2026-05-26

AI Summary

Traditional Kalman filtering is constrained by linear Gaussian assumptions, leading to suboptimal performance in nonlinear sensing scenarios such as Doppler radar and LiDAR, where mere parameter tuning cannot overcome inherent structural limitations. This work proposes the Kalman Evolve framework, which for the first time introduces algorithmic structure discovery into state estimation by jointly optimizing noise parameters and update structures. Leveraging large language models as structured priors over program space, the method employs program synthesis to generate interpretable, non-affine filtering algorithms that retain recursive form while adapting effectively to nonlinear dynamics. Experiments across diverse real-world and synthetic tracking tasks demonstrate up to a 12% reduction in root mean square error (RMSE), significantly outperforming strong existing baselines.

Abstract

State estimation is a fundamental problem in control and signal processing, for which the Kalman Filter provides an optimal solution under linear dynamics, Gaussian noise, and known noise covariances. However, these assumptions often fail in realistic sensing settings such as Doppler radar and LiDAR. In these cases, the optimal estimator is inherently nonlinear, which leads to systematic performance degradation. This creates a performance gap that cannot be eliminated by tuning the noise covariance parameters (i.e., the process and measurement noise in the Kalman Filter) alone. To address this limitation, we propose Kalman Evolve, a framework for discovering improved filtering algorithms by jointly optimizing both noise parameters and the update structure. Our approach leverages large language models (LLMs) as a structured prior over program space, enabling the generation of interpretable, non-affine modifications to the classical Kalman filter while preserving its recursive form. We provide analytical results establishing the suboptimality of affine estimators under common nonlinear sensing models, motivating the need for structure-aware updates. Across a range of synthetic and real-world tracking benchmarks, including Doppler radar, LiDAR-based localization, and pedestrian tracking, the discovered algorithms consistently improve over strong baselines such as the Optimized Kalman Filter, achieving up to 12\% reduction in RMSE. These results suggest that optimizing the structure of the Kalman filter, rather than only its parameters, provides a practical and interpretable way to improve state estimation.

Problem

• state estimation
• Kalman filter
• nonlinear sensing
• performance gap
• noise covariance

Innovation

• Kalman Evolve
• interpretable algorithm discovery
• nonlinear state estimation
• large language models
• structured program optimization