Title: How to avoid the curse of dimensionality: scalability of particle filters with and without importance weights,
Abstract: Particle filters are a popular and flexible class of numerical algorithms to
solve a large class of nonlinear filtering problems. However, standard particle
filters with importance weights have been shown to require a sample size that
increases exponentially with the dimension D of the state space in order to
achieve a certain performance, which precludes their use in very
high-dimensional filtering problems. Here, we focus on the dynamic aspect of
this curse of dimensionality (COD) in continuous time filtering, which is
caused by the degeneracy of importance weights over time. We show that the
degeneracy occurs on a time-scale that decreases with increasing D. In order to
soften the effects of weight degeneracy, most particle filters use particle
resampling and improved proposal functions for the particle motion. We explain
why neither of the two can prevent the COD in general. In order to address this
fundamental problem, we investigate an existing filtering algorithm based on
optimal feedback control that sidesteps the use of importance weights. We use
numerical experiments to show that this Feedback Particle Filter (FPF) by Yang
et al. (2013) does not exhibit a COD.