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

arXiv: https://arxiv.org/abs/2511.18060 · HTML: https://mlsanity.com/paper/2511.18060/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 60 references that are in the map.
- First posted on arXiv 22 Nov 2025.

## Builds on (most cited first, 26 of 26)

- [Optimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measures](https://mlsanity.com/paper/1508.07941/) (2015)
- [Analysis of Langevin Monte Carlo via convex optimization](https://mlsanity.com/paper/1802.09188/) (2018)
- [Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problem](https://mlsanity.com/paper/1802.08089/) (2018)
- [Accelerating Langevin Sampling with Birth-death](https://mlsanity.com/paper/1905.09863/) (2019)
- [High-Order Langevin Diffusion Yields an Accelerated MCMC Algorithm](https://mlsanity.com/paper/1908.10859/) (2019)
- [Birth-death dynamics for sampling: Global convergence, approximations and their asymptotics](https://mlsanity.com/paper/2211.00450/) (2022)
- [Sampling in Unit Time with Kernel Fisher-Rao Flow](https://mlsanity.com/paper/2401.03892/) (2024)
- [Sampling via Gradient Flows in the Space of Probability Measures](https://mlsanity.com/paper/2310.03597/) (2023)
- [Stochastic Proximal Langevin Algorithm: Potential Splitting and Nonasymptotic Rates](https://mlsanity.com/paper/1905.11768/) (2019)
- [An Explicit Expansion of the Kullback-Leibler Divergence along its Fisher-Rao Gradient Flow](https://mlsanity.com/paper/2302.12229/) (2023)
- [A connection between Tempering and Entropic Mirror Descent](https://mlsanity.com/paper/2310.11914/) (2023)
- [Adaptive Importance Sampling meets Mirror Descent: a Bias-variance tradeoff](https://mlsanity.com/paper/2110.15590/) (2021)
- [Fisher-Rao Gradient Flow: Geodesic Convexity and Functional Inequalities](https://mlsanity.com/paper/2407.15693/) (2024)
- [Gradient Flows in Filtering and Fisher-Rao Geometry](https://mlsanity.com/paper/1710.00064/) (2017)
- [Measure transport with kernel mean embeddings](https://mlsanity.com/paper/2401.12967/) (2024)
- [Stein transport for Bayesian inference](https://mlsanity.com/paper/2409.01464/) (2024)
- [Unbiased Kinetic Langevin Monte Carlo with Inexact Gradients](https://mlsanity.com/paper/2311.05025/) (2023)
- [Efficient, Multimodal, and Derivative-Free Bayesian Inference With Fisher-Rao Gradient Flows](https://mlsanity.com/paper/2406.17263/) (2024)
- [Nonreversible Langevin Samplers: Splitting Schemes, Analysis and Implementation](https://mlsanity.com/paper/1701.04247/) (2017)
- [McKean-Vlasov SDEs in nonlinear filtering](https://mlsanity.com/paper/2007.12658/) (2020)
- [Characterizing Dependence of Samples along the Langevin Dynamics and Algorithms via Contraction of Phi-Mutual Information](https://mlsanity.com/paper/2402.17067/) (2024)
- [Sequential Monte Carlo approximations of Wasserstein--Fisher--Rao gradient flows](https://mlsanity.com/paper/2506.05905/) (2025)
- [Connections between sequential Bayesian inference and evolutionary dynamics](https://mlsanity.com/paper/2411.16366/) (2024)
- [A note on the unique properties of the Kullback--Leibler divergence for sampling via gradient flows](https://mlsanity.com/paper/2507.04330/) (2025)
- [Evolution of Gaussians in the Hellinger-Kantorovich-Boltzmann gradient flow](https://mlsanity.com/paper/2504.20400/) (2025)
- [Tuning-Free Sampling via Optimization on the Space of Probability Measures](https://mlsanity.com/paper/2510.25315/) (2025)

## Built on by (most cited first, 2 of 2)

- [Properties and limitations of geometric tempering for gradient flow dynamics](https://mlsanity.com/paper/2604.20301/) (2026)
- [Preservation of Log-Concavity and Convergence of Wasserstein-Fisher-Rao Gradient Flows](https://mlsanity.com/paper/2609.18118/) (2026)

## Abstract (arXiv)

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.