Numerically Robust Fixed-Point Smoothing Without State Augmentation
This work addresses a gap in practical algorithms for fixed-point smoothing, which is crucial for dynamical systems with unknown initial conditions, offering a solution that balances speed and numerical stability.
The paper tackles the lack of numerically robust fixed-point smoothing algorithms for dynamical systems with unknown initial conditions by introducing a new Gaussian fixed-point smoother formulation. The result is a method that matches the runtime of the fastest techniques and the robustness of the most robust ones, as demonstrated in experiments with a JAX implementation.
Practical implementations of Gaussian smoothing algorithms have received a great deal of attention in the last 60 years. However, almost all work focuses on estimating complete time series (''fixed-interval smoothing'', $\mathcal{O}(K)$ memory) through variations of the Rauch--Tung--Striebel smoother, rarely on estimating the initial states (''fixed-point smoothing'', $\mathcal{O}(1)$ memory). Since fixed-point smoothing is a crucial component of algorithms for dynamical systems with unknown initial conditions, we close this gap by introducing a new formulation of a Gaussian fixed-point smoother. In contrast to prior approaches, our perspective admits a numerically robust Cholesky-based form (without downdates) and avoids state augmentation, which would needlessly inflate the state-space model and reduce the numerical practicality of any fixed-point smoother code. The experiments demonstrate how a JAX implementation of our algorithm matches the runtime of the fastest methods and the robustness of the most robust techniques while existing implementations must always sacrifice one for the other.