Particle Filtering on Riemannian Manifolds

Particle filtering is an approximate Monte Carlo method implementing the Bayesian Sequential Estimation. It consists in online estimation of the a posteriori distribution of the system state given a flow of observed data. The popularity of the particle filter method stems from its simplicity and flexibility to deal with non linear/non Gaussian dynamical models. However, this method suffers from the curse of dimensionality. In general, the system state lies in a constrained subspace which dimension is much lower than the whole space dimension. In this contribution, we propose an implementation of the particle filter with the constraint that the system state lies in a low dimensional Riemannian manifold. The sequential Bayesian updating consists in drawing state samples while moving on the manifold geodesics. An Affine Generalized Hyperbolic regression process is proposed to model the transition dynamics on the manifold. It is a parametric family able to cover a wide range of tail behaviors of real signal distributions.