Batch Nonlinear Continuous-Time Trajectory Estimation as Exactly Sparse Gaussian Process Regression

In this paper, we revisit batch state estimation through the lens of Gaussian process (GP) regression. We consider continuous-discrete estimation problems wherein a trajectory is viewed as a one-dimensional GP, with time as the independent variable. Our continuous-time prior can be defined by any nonlinear, time-varying stochastic differential equation driven by white noise; this allows the possibility of smoothing our trajectory estimates using a variety of vehicle dynamics models (e.g., `constant-velocity'). We show that this class of prior results in an inverse kernel matrix (i.e., covariance matrix between all pairs of measurement times) that is exactly sparse (block-tridiagonal) and that this can be exploited to carry out GP regression (and interpolation) very efficiently. When the prior is based on a linear, time-varying stochastic differential equation and the measurement model is also linear, this GP approach is equivalent to classical, discrete-time smoothing (at the measurement times); when a nonlinearity is present, we iterate over the whole trajectory to maximize accuracy. We test the approach experimentally on a simultaneous trajectory estimation and mapping problem using a mobile robot dataset.Comment: Submitted to Autonomous Robots on 20 November 2014, manuscript # AURO-D-14-00185, 16 pages, 7 figure

Lifted Bayesian filtering in multi-entity systems

This thesis focuses on Bayesian filtering for systems that consist of multiple, interacting entites (e.g. agents or objects), which can naturally be described by Multiset Rewriting Systems (MRSs). The main insight is that the state space that is underling an MRS exhibits a certain symmetry, which can be exploited to increase inference efficiency. We provide an efficient, lifted filtering algorithm, which is able to achieve a factorial reduction in space and time complexity, compared to conventional, ground filtering.Diese Arbeit betrachtet Bayes'sche Filter in Systemen, die aus mehreren, interagierenden Entitäten (z.B. Agenten oder Objekten) bestehen. Die Systemdynamik solcher Systeme kann auf natürliche Art durch Multiset Rewriting Systems (MRS) spezifiziert werden. Die wesentliche Erkenntnis ist, dass der Zustandraum Symmetrien aufweist, die ausgenutzt werden können, um die Effizienz der Inferenz zu erhöhen. Wir führen einen effizienten, gelifteten Filter-Algorithmus ein, dessen Zeit- und Platzkomplexität gegenüber dem grundierten Algorithmus um einen faktoriellen Faktor reduziert ist