Gradient Flows in Filtering and Fisher-Rao Geometry
For researchers in filtering and information geometry, this work offers a geometric reinterpretation of existing filtering equations, but it is incremental as it extends previous work on Wasserstein metrics to the Fisher-Rao metric.
This paper derives filtering evolution equations as proximal operators under the Fisher-Rao metric, providing new geometric interpretations for known filtering equations. The linear Gaussian case is developed in detail, showing that a two-step optimization procedure still applies.
Uncertainty propagation and filtering can be interpreted as gradient flows with respect to suitable metrics in the infinite dimensional manifold of probability density functions. Such a viewpoint has been put forth in recent literature, and a systematic way to formulate and solve the same for linear Gaussian systems has appeared in our previous work where the gradient flows were realized via proximal operators with respect to Wasserstein metric arising in optimal mass transport. In this paper, we derive the evolution equations as proximal operators with respect to Fisher-Rao metric arising in information geometry. We develop the linear Gaussian case in detail and show that a template two step optimization procedure proposed earlier by the authors still applies. Our objective is to provide new geometric interpretations of known equations in filtering, and to clarify the implication of different choices of metric.