Feedback Particle Filter Design Using a Differential-Loss Reproducing Kernel Hilbert Space

The feedback particle filter was created to avoid the complex and problem-dependent “resampling” that arises in other particle filter formulations. Practical implementation of this approach to nonlinear filtering requires an approximation of the innovations gain. It is known that this gain has a representation as the gradient to the solution to a version of Poisson's equation. This paper introduces a new approach to approximating the gradient with minimum mean-square error within a Reproducing Kernel Hilbert Space (RKHS) setting. The input to the new algorithm is the particles generated by the filter, and its output is a direct approximation to the gain; that is, the approximation of the gradient of the solution of Poisson's equation is obtained directly, without differentiation. The approach is based on two key ideas: reformulating the mean-square loss so that its dependence on the true solution to Poisson's equation is removed, and applying a recent variation of RKHS theory that allows for derivatives in the loss function. The methodology is illustrated with numerical experiments.