Numerically Robust Fixed-Point Smoothing Without State Augmentation

2024-09-30

arXiv.org

AI Summary

Existing Gaussian fixed-point smoothing methods struggle to balance numerical stability and computational efficiency for dynamical systems with unknown initial states: faster variants suffer from instability, while stable ones incur high computational overhead. This paper proposes a novel Gaussian fixed-point smoothing framework that requires no state augmentation and achieves constant memory complexity—O(1). Its core innovation is the first derivation of numerically robust downward recursion via Cholesky decomposition, eliminating both ill-conditioned downward updates and state augmentation inherent in conventional approaches. The method integrates Gaussian recursive smoothing theory with JAX’s automatic differentiation and vectorized execution. Experiments demonstrate that our approach matches state-of-the-art numerical accuracy while attaining runtime performance comparable to the fastest existing algorithms—thereby breaking the long-standing trade-off between stability and speed.

Technology Category

• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

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.

Problem

• Gaussian Fixed-point Smoothing
• Dynamic Systems
• Stability and Efficiency

Innovation

• Gaussian fixed-point smoothing
• stability
• unknown initial state systems