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Control-type particle filters have been receiving increasing attention overthe last decade as a means of obtaining sample based approximations to thesequential Bayesian filtering problem in the nonlinear setting. Here we analyseone such type, namely the feedback particle filter and a recently proposedapproximation of the associated gain function based on diffusion maps. The keypurpose is to provide analytic insights on the form of the approximate gain,which are of interest in their own right. These are then used to establish aroadmap to obtaining well-posedness and convergence of the finite $N$ system toits mean field limit. A number of possible future research directions are alsodiscussed.

Analysis of the feedback particle filter with diffusion map based approximation of the gain