Wasserstein Fisher Rao Gradient Flows: Operating Splitting & Convergence Speed

Wasserstein-Fisher-Rao (WFR) gradient flows have been recently proposed as a powerful sampling tool that combines the advantages of pure Wasserstein (W) and pure Fisher-Rao (FR) gradient flows. Existing algorithmic developments implicitly make use of operator splitting techniques to numerically approximate the WFR partial differential equation, whereby the W part is evaluated over a given step size and then the FR part (or vice versa). This works investigates the impact of the order in which these operations are performed and aims to provide a quantitative analysis. It is shown that a judicious choice of order and step size can reduce time to equilibrium compared to the continuous time WFR flow. Several open questions are discussed.