Numerically Robust Fixed-Point Smoothing Without State Augmentation
<|reference_start|>Numerically Robust Fixed-Point Smoothing Without State Augmentation: 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.<|reference_end|>
@article{krämer2024numerically,
    title={Numerically Robust Fixed-Point Smoothing Without State Augmentation},
    author={Nicholas Kr"amer},
    journal={arXiv preprint arXiv:2409.20004},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.20004},
    primaryClass={math.NA cs.LG cs.NA cs.SY eess.SY stat.CO stat.ML}
}