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

2025-11-22

AI Summary

This study systematically investigates how the ordering of Wasserstein (W) and Fisher–Rao (FR) operators in operator splitting schemes affects numerical convergence for Wasserstein–Fisher–Rao (WFR) gradient flows. Leveraging operator splitting, variational formulations, and gradient flow theory, we establish a quantitative convergence analysis framework for WFR splitting schemes. We derive, for the first time, sharp exponential decay bounds for WFR flows and prove preservation of log-concavity under the splitting dynamics. Our analysis reveals that, with appropriately chosen step sizes, the W–FR splitting outperforms FR–W; moreover, within specific parameter regimes, the splitting scheme converges faster than the full WFR flow itself. These results uncover the intrinsic coupling between operator ordering and the underlying geometric structure of the WFR metric space. The work provides novel theoretical foundations and practical design principles for efficient probability measure evolution and sampling algorithms.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

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 sequential 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.

Problem

• Analyzing operator splitting order impact on WFR gradient flows
• Comparing convergence rates between sequential splitting schemes
• Establishing log-concavity preservation under WFR gradient flows

Innovation

• Operator splitting for WFR gradient flows
• Sequential evaluation of W and FR operators
• Step size optimization for faster convergence