  Control-type particle filters have been receiving increasing attention over
the last decade as a means of obtaining sample based approximations to the
sequential Bayesian filtering problem in the nonlinear setting. Here we analyse
one such type, namely the feedback particle filter and a recently proposed
approximation of the associated gain function based on diffusion maps. The key
purpose 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 a
roadmap to obtaining well-posedness and convergence of the finite $N$ system to
its mean field limit. A number of possible future research directions are also
discussed.
