Abstract
We study the convergence of Wasserstein-Fisher-Rao (WFR) gradient flows for sampling from probability distributions known up to a normalisation constant. By combining Wasserstein transport with Fisher-Rao birth-death dynamics, WFR flows balance exploration and selection. These flows have been recognised as a promising mechanism to accelerate convergence beyond Langevin dynamics. We show that for a class of strongly log-concave target distributions satisfying additional curvature conditions, WFR flows preserve strong log-concavity, in contrast to Wasserstein flows which enjoy this property only in the Gaussian setting. Exploiting this result, we derive explicit non-asymptotic convergence rates for the symmetrised Kullback-Leibler divergence, without requiring a warm-start as required in current estimates. In particular, we show that the convergence rate decomposes additively into Wasserstein and Fisher-Rao contributions, thereby confirming a recent conjecture within this setting. These results provide refined convergence guarantees and further develop the theoretical foundations of WFR gradient flows for sampling and Bayesian inference.
Explore similar work
Nov 22, 2025stat.ML
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.
Francesca Romana Crucinio, Sahani Pathiraja
Apr 22, 2026stat.ML
We consider the problem of sampling from a probability distribution
π. It is well known that this can be written as an optimisation problem over the space of probability distributions in which we aim to minimise the Kullback--Leibler divergence from
π. We consider the effect of replacing
π with a sequence of moving targets
(πt)t≥0 defined via geometric tempering on the Wasserstein and Fisher--Rao gradient flows. We show that convergence occurs exponentially in continuous time, providing novel bounds in both cases. We also consider popular time discretisations and explore their convergence properties. We show that in the Fisher--Rao case, replacing the target distribution with a geometric mixture of initial and target distribution never leads to a convergence speed up both in continuous time and in discrete time. Finally, we explore the gradient flow structure of tempered dynamics and derive novel adaptive tempering schedules.
Francesca Romana Crucinio, Sahani Pathiraja
Oct 31, 2024stat.ML
Otto's Wasserstein gradient flow of the inclusive (forward) Kullback--Leibler (KL) divergence offers a principled framework for analyzing statistical inference algorithms, yet algorithms targeting the exclusive (reverse) KL divergence are rarely studied with such tools. We establish a unified gradient-flow and PDF framework for inclusive KL inference. We show that maximum mean discrepancy minimization can be viewed as inclusive KL inference with an approximate gradient estimator, and we develop the Fisher--Rao and Wasserstein--Fisher--Rao gradient flows that directly target the inclusive KL divergence. Restricting these flows to the manifold of Gaussian distributions yields explicit gradient-flow ODEs, providing a foundation for Gaussian variational inference. Building on this viewpoint, we further introduce a local-estimator Wasserstein gradient flow whose velocity is obtained by local nonparametric regression, free of density-ratio evaluation or kernel gradients, improving the algorithmic performance over the MMD-based particle method.
Jia-Jie Zhu