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

In this paper, we shall first derive the admissible control input of the multivariate feedback particle filter (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 \cite{YMM}, a special f-divergence, Kullback-Leibler (K-L) divergence, is used in the 1-dimensional nonlinear filtering problems. We show that the control input is invariant under the f-divergence class. That is, the control input satisfies exactly the same equations as those obtained by minimizing K-L divergence, no matter what f-divergence in use. In the latter half of this paper, we show the existence and uniqueness of the control input under suitable regular conditions. We confirm that the explicit expression of the control input given in \cite{YLMM} is the only admissible one.