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Particle filters are a popular and flexible class of numerical algorithms tosolve a large class of nonlinear filtering problems. However, standard particlefilters with importance weights have been shown to require a sample size thatincreases exponentially with the dimension D of the state space in order toachieve a certain performance, which precludes their use in veryhigh-dimensional filtering problems. Here, we focus on the dynamic aspect ofthis curse of dimensionality (COD) in continuous time filtering, which iscaused by the degeneracy of importance weights over time. We show that thedegeneracy occurs on a time-scale that decreases with increasing D. In order tosoften the effects of weight degeneracy, most particle filters use particleresampling and improved proposal functions for the particle motion. We explainwhy neither of the two can prevent the COD in general. In order to address thisfundamental problem, we investigate an existing filtering algorithm based onoptimal feedback control that sidesteps the use of importance weights. We usenumerical experiments to show that this Feedback Particle Filter (FPF) by Yanget al. (2013) does not exhibit a COD.

How to avoid the curse of dimensionality: scalability of particle filters with and without importance weights