Posterior linearisation smoothing with robust iterations

TL;DR

This work addresses iterative Bayesian smoothing in nonlinear state-space models with additive Gaussian noise by deriving a Gauss–Newton interpretation for the iterated posterior linearisation smoother (IPLS). It introduces Levenberg–Marquardt (LM) regularisation and Armijo–Wolfe line-search variants (LM–IPLS and LS–IPLS), showing that LM regularisation can be implemented via a pseudo-measurement within a modified state-space model and establishing a GN-based equivalence with IPLS. The authors provide explicit GN-cost formulations for IPLS, derive the gradient and LM connection, and present algorithms, including inner LM loops and line-search procedures. Simulations on coordinated-turn bearing-only problems demonstrate improved convergence and estimation accuracy under strong nonlinearity, at the expense of higher computational load. Overall, the paper advances robust, convergent smoothing methods for nonlinear systems by unifying IPLS with GN optimisation, LM regularisation, and line-search strategies.

Abstract

This paper considers the problem of iterative Bayesian smoothing in nonlinear state-space models with additive noise using Gaussian approximations. Iterative methods are known to improve smoothed estimates but are not guaranteed to converge, motivating the development of methods with better convergence properties. The aim of this article is to extend Levenberg-Marquardt (LM) and line-search versions of the classical iterated extended Kalman smoother (IEKS) to the iterated posterior linearisation smoother (IPLS). The IEKS has previously been shown to be equivalent to the Gauss-Newton (GN) method. We derive a similar GN interpretation for the IPLS and use this to develop extensions to the IPLS, with improved convergence properties. We show that an LM extension for the IPLS can be achieved with a simple modification of the smoothing iterations, enabling algorithms with efficient implementations. We also derive the Armijo--Wolfe step length conditions for the IPLS enabling an efficient inexact line-search method. Our numerical experiments show the benefits of these extensions in highly nonlinear scenarios.

Figures (5)

• Figure 1: Connection between GN optimisation and a general Gaussian smoother. Both the GN algorithm and the general Gaussian smoothers work by 1) linearising the problem and 2) solving the linearised problem analytically. The proofs of \ref{['thm:gn_ieks', 'thm:gn_ipls']} show that the linearised problems are identical.
• Figure 2: CT experiment with bearings only measurements. The two sensors are placed at $(-1.5, 0.5)^{\top}$ and $(1, 1)^{\top}$.
• Figure 3: Simulated CT model with bearings only measurements, see \ref{['fig:ct:realisation']} for setup. Curves show averaged RMSE and NEES across iterations averaged over 100 trials. Error bars correspond to the standard error, i.e. the estimated standard deviation scaled by $1 / \sqrt{100\xspace}$.
• Figure 4: Single realisation of the CT experiment with varying bearings-only measurements. At $k = 50, 100, \dots, 500$ only a single low noise measurement is observed. For this particular realisation, it is only the LM-regularised smoothers, LM--IEKS and LM--IPLS, which accurately estimate the general shape of the true trajectory.
• Figure 5: Simulated coordinated turn model with bearings only measurements. The plot shows averaged RMSE and NEES across iterations. Error bars correspond to the standard error, that is, the estimated standard deviation scaled by $1 / \sqrt{100\xspace}$.

Theorems & Definitions (8)

• Proposition 3.1
• Lemma 4.1
• Proposition 4.2
• proof
• Proposition 4.3
• proof
• Lemma 5.1
• proof