Multivariate Feedback Particle Filter and the Well-posedness of its Admissible Control Input

In this talk, we shall first derive the admissible control input of the multivariate feedback particle fifilter (FPF) by minimizing the f-divergence of the posterior conditional density function and the empirical conditional density of the controlled particles. On the contrast, in the original derivation, a special f-divergence, Kullback-Leibler (K-L) divergence, is used in the 1-dimensional nonlinear fifiltering problems. We show that the control input is invariant under the f-divergence class. That is, the control input satisfifies exactly the same equations as those obtained by minimizing K-L divergence, no matter what f divergence in use. In the later half of this talk, we show that if we restrict the control input to be the gradient of certain potential, then the existence and uniqueness of the control input is proven in some suitable functional space under certain regularity conditions. We confirm that the explicit expression of the control input given in [28] is indeed the unique one in some trivial situation.