Gradient Flows in Uncertainty Propagation and Filtering of Linear Gaussian Systems
Provides a pedagogical exposition connecting variational schemes to gradient flows for researchers familiar with linear Gaussian systems.
This expository paper elucidates the JKO and LMMR variational schemes for uncertainty propagation and filtering, showing they can be understood as proximal operators realizing gradient flows. It recovers known results for linear Gaussian systems.
The purpose of this work is mostly expository and aims to elucidate the Jordan-Kinderlehrer-Otto (JKO) scheme for uncertainty propagation, and a variant, the Laugesen-Mehta-Meyn-Raginsky (LMMR) scheme for filtering. We point out that these variational schemes can be understood as proximal operators in the space of density functions, realizing gradient flows. These schemes hold the promise of leading to efficient ways for solving the Fokker-Planck equation as well as the equations of non-linear filtering. Our aim in this paper is to develop in detail the underlying ideas in the setting of linear stochastic systems with Gaussian noise and recover known results.