An operator splitting analysis of Wasserstein--Fisher--Rao gradient flows

TL;DR

This paper analyzes the numerical approximation of Wasserstein-Fisher-Rao gradient flows using operator splitting, revealing conditions under which splitting schemes can outperform the exact flow in convergence speed.

Contribution

It provides a quantitative analysis of operator splitting order effects on WFR flows and introduces variational formulae for their evolution.

Findings

Splitting schemes can converge faster than the exact WFR flow with proper step size and order.

The WFR flow preserves log-concavity and has a sharp decay bound.

Guidelines for choosing W-FR vs. FR-W splitting schemes are proposed.

Abstract

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 flow is evaluated over a given step size and then the FR flow (or vice versa). This works investigates the impact of the order in which the W and FR operator are evaluated and aims to provide a quantitative analysis. Somewhat surprisingly, we show that with a judicious choice of step size and operator ordering, the split scheme can converge to the target distribution faster than the exact WFR flow (in terms of model time). We obtain variational formulae describing the evolution over one time step of both splitting schemes and investigate in which settings the W-FR split should be preferred to the FR-W split. As a step towards this goal we show that the WFR gradient flow preserves log-concavity and obtain the first sharp decay bound for WFR flow.