Cholesky-KalmanNet: Model-Based Deep Learning With Positive Definite Error Covariance Structure

State estimation from noisy observations is crucial across various fields. Traditional methods such as Kalman, Extended Kalman, and Unscented Kalman Filter often struggle with nonlinearities, model inaccuracies, and high observation noise. This letter introduces Cholesky-KalmanNet (CKN), a model-based deep learning approach that considerably enhances state estimation by providing and enforcing transiently precise error covariance estimation. Specifically, the CKN embeds mathematical constraints associated with the positive definiteness of error covariance in a recurrent DNN architecture through the Cholesky decomposition. This architecture enhances statistical reliability and mitigates numerical instabilities. Furthermore, introducing a novel loss function that minimizes discrepancies between the estimated and empirical error covariance ensures a comprehensive minimization of estimation errors, accounting for interdependencies among state variables. Extensive evaluations on both synthetic and real-world datasets affirm CKN's superior performance vis-a-vis state estimation accuracy, robustness against system inaccuracies and observation noise, as well as stability across varying training data partitions, an essential feature for practical scenarios with suboptimal data conditions.