Recursive Nonlinear Filtering via Gaussian Approximation with Minimized Kullback-Leibler Divergence

In order to solve various problems in a Bayesian framework efficiently, it is critical to approximate a posterior distribution. This work provides a Gaussian approximation of a general distribution via Kullback-Leibler divergence minimization by deterministic sampling. Two algorithms, feasible direction method and linearized alternating direction method of multipliers, each having its strengths, are proposed for the Gaussian approximation. Theoretical results of complexity, convergence, convergence rate, and guidelines for parameter selection of the proposed algorithms are also provided. Based on the Gaussian approximation, two recursive filters are developed for nonlinear dynamic systems. Examples are given to demonstrate the effectiveness and efficiency of the proposed Gaussian approximation and the related filters.