Numerically robust Gaussian state estimation with singular observation noise
Nicholas Krämer, Filip Tronarp
TL;DR
This work addresses Gaussian state estimation under singular observation noise by transforming the problem into a lower-dimensional, nonsingular form using a sequence of QR-style decompositions combined with Bayes' rule. An offline model-reduction stage realigns the constrained and unconstrained state components, enabling a reduced, numerically stable online estimation algorithm that yields both filtering and smoothing capabilities along with marginal-likelihood computations. Key contributions include a numerically robust Gaussian conditioning framework, a linear-time estimation algorithm on the reduced model, and demonstrations of significant computational savings and improved numerical robustness in ill-conditioned scenarios, including time-varying systems. The method offers practical impact for scalable, stable inference in systems with singular observations, supported by simulations showing reduced runtime and enhanced robustness over traditional approaches.
Abstract
This article proposes numerically robust algorithms for Gaussian state estimation with singular observation noise. Our approach combines a series of basis changes with Bayes' rule, transforming the singular estimation problem into a nonsingular one with reduced state dimension. In addition to ensuring low runtime and numerical stability, our proposal facilitates marginal-likelihood computations and Gauss-Markov representations of the posterior process. We analyse the proposed method's computational savings and numerical robustness and validate our findings in a series of simulations.