提出一种无需状态扩展的数值稳定定点平滑方法
Numerically Robust Fixed-Point Smoothing Without State Augmentation
- 基于乔列斯基分解重构平滑算法，避免数值不稳
- 在保持常数内存的同时达到最快与最稳健的平衡
- 适合需要高精度初始状态估计的动态系统应用
过去60年中，高斯平滑算法的实际实现受到广泛关注。然而，几乎所有工作都聚焦于通过拉乌-通-斯特里贝尔平滑器估计完整时间序列（固定区间平滑，复杂度为 𝒪(K) 内存），很少关注估计初始状态（固定点平滑，复杂度为 𝒪(1) 内存）。由于固定点平滑是未知初始条件动态系统算法的关键组成部分，本文提出一种新的高斯固定点平滑器公式。与以往方法不同，本方法采用数值稳定的乔列斯基形式（无需降维操作），且不使用状态扩展，从而避免了状态空间模型的无谓膨胀，提升了固定点平滑代码的数值实用性。实验表明，使用JAX实现的该算法在运行时间上媲美最快速的方法，在数值鲁棒性上达到最稳健技术的水平，而现有实现始终需在二者间权衡。
原文摘要 · Abstract (English)
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.
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