# Preservation of Log-Concavity and Convergence of Wasserstein-Fisher-Rao Gradient Flows

arXiv: https://arxiv.org/abs/2609.18118 · HTML: https://mlsanity.com/paper/2609.18118/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 32 references that are in the map.
- First posted on arXiv 16 Sep 2026.

## Builds on (most cited first, 19 of 19)

- [Optimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measures](https://mlsanity.com/paper/1508.07941/) (2015)
- [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)
- [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)
- [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)
- [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)
- [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)
- [An operator splitting analysis of Wasserstein--Fisher--Rao gradient flows](https://mlsanity.com/paper/2511.18060/) (2025)

## Abstract (arXiv)

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.