Numerically Robust Fixed-Point Smoothing Without State Augmentation

TL;DR

This work targets fixed-point smoothing for Gaussian state-space models with unknown initial conditions, highlighting memory and numerical robustness gaps in existing methods. It proposes a novel recursion for p(x0|xK,y1:K) and implements it in a Cholesky-based form that avoids state augmentation, achieving $O(1)$ memory with $O(K D^3)$ runtime while maintaining flexibility across Gaussian parametrisations. The two concrete implementations (covariance-based and Cholesky-based) demonstrate comparable speed to the fastest fixed-point approaches and superior numerical robustness in challenging settings, with the Cholesky version excelling in probabilistic numerics and boundary-value problem tests. Three experiments—including efficiency, robustness against ill-conditioning, and a tracking parameter-estimation case study—validate memory savings, runtime competitiveness, and stability, suggesting practical impact for differential-equation solvers and dynamical-system parameter inference.

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.

Figures (4)

• Figure 1: Fixed-point (left) versus fixed-interval smoothing problem (right) as factor graphs. The shaded variables are observed. "QoI": "Quantity of interest".
• Figure 2: Outline of the memory-, runtime-, and robustness-related demonstrations.
• Figure 3: 15th Boundary value problem mazzia2015fortran.
• Figure 4: Initial distributions $p(\theta_1, \theta_2 \mid y_{1:K})$ of the car tracking model after running the fixed-point smoother (recall $x\coloneqq(\theta_1, \theta_2, \dot \theta_1, \dot \theta_2)$). Left to right: After three iterations, the combination of expectation maximisation with the fixed-point smoother finds the correct initial mean $\theta = (\theta_1, \theta_2)$ of the state-space model. Top: First coordinate $p(\theta_1 \mid y_{1:K})$. Bottom: second coordinate $p(\theta_2 \mid y_{1:K})$. "PDF": "Probability density function".

Theorems & Definitions (2)

• Proposition 1
• proof