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.

该研究解决了在不增加状态维度的情况下，如何有效进行固定点平滑的问题。这项新方法提供了一个数值稳健的Cholesky形式，避免了传统方法的局限，实验表明其运行速度与现有最快的方法相当，同时保留了可靠性。此工作对动态系统中初始状态估计的应用具有重大影响。

无需状态扩展的数值稳健固定点平滑