Numerically Robust Fixed-Point Smoothing Without State Augmentation

TL;DR

This paper introduces a numerically robust Gaussian fixed-point smoothing algorithm that avoids state augmentation and downdates, improving stability and efficiency in estimating initial states in dynamical systems.

Contribution

It presents a new Cholesky-based formulation for fixed-point smoothing that enhances numerical robustness without increasing computational complexity or model size.

Findings

Matches runtime of fastest methods

Achieves high robustness in numerical computations

Does not require state augmentation or downdates

Abstract

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.