Variational Gaussian approximation of the Kushner optimal filter
This work addresses state estimation in nonlinear systems, which is incremental as it builds on prior methods to extend linear filter results.
The authors tackled the problem of approximating the Kushner optimal filter for state estimation in nonlinear dynamical systems by proposing a variational Gaussian approximation method using proximal losses based on Wasserstein and Fisher metrics, resulting in a Gaussian flow that generalizes Kalman-Bucy and Riccati flows to nonlinear cases.
In estimation theory, the Kushner equation provides the evolution of the probability density of the state of a dynamical system given continuous-time observations. Building upon our recent work, we propose a new way to approximate the solution of the Kushner equation through tractable variational Gaussian approximations of two proximal losses associated with the propagation and Bayesian update of the probability density. The first is a proximal loss based on the Wasserstein metric and the second is a proximal loss based on the Fisher metric. The solution to this last proximal loss is given by implicit updates on the mean and covariance that we proposed earlier. These two variational updates can be fused and shown to satisfy a set of stochastic differential equations on the Gaussian's mean and covariance matrix. This Gaussian flow is consistent with the Kalman-Bucy and Riccati flows in the linear case and generalize them in the nonlinear one.