Posterior linearisation smoothing with robust iterations

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.