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This research provides a rigorous mathematical analysis of operator splitting techniques used to approximate Wasserstein-Fisher-Rao gradient flows for sampling tasks. The authors demonstrate that specific ordering and step-size choices can accelerate convergence beyond the exact flow, offering new variational formulae and decay bounds for practitioners in probabilistic machine learning.
An operator splitting analysis of Wasserstein--Fisher--Rao gradient flows
arXiv:2511.18060v3 Announce Type: replace-cross 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.
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.
Machine Learning (stat.ML); Machine Learning (cs.LG); Methodology (stat.ME)
arXiv:2511.18060 [stat.ML]
(or arXiv:2511.18060v3 [stat.ML] for this version)
From: Sahani Pathiraja Dr. [view email] [v1] Sat, 22 Nov 2025 13:37:53 UTC (125 KB) [v2] Fri, 27 Feb 2026 01:28:42 UTC (250 KB) [v3] Wed, 16 Sep 2026 04:36:22 UTC (152 KB)
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