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

arXiv 2609.18118 ↗First posted 16 Sep 2026stat.ML cs.LG math.PRcited by 0

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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.

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References · 19

  1. 2015Optimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measurescited by 139
  2. 2018Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problemcited by 123
  3. 2019Accelerating Langevin Sampling with Birth-deathcited by 45
  4. 2022Birth-death dynamics for sampling: Global convergence, approximations and their asymptoticscited by 30
  5. 2024Sampling in Unit Time with Kernel Fisher-Rao Flowcited by 27
  6. 2023Sampling via Gradient Flows in the Space of Probability Measurescited by 20
  7. 2023An Explicit Expansion of the Kullback-Leibler Divergence along its Fisher-Rao Gradient Flowcited by 16
  8. 2023A connection between Tempering and Entropic Mirror Descentcited by 16
  9. 2024Fisher-Rao Gradient Flow: Geodesic Convexity and Functional Inequalitiescited by 13
  10. 2017Gradient Flows in Filtering and Fisher-Rao Geometrycited by 12
  11. 2024Measure transport with kernel mean embeddingscited by 11
  12. 2024Stein transport for Bayesian inferencecited by 11
7 more
  1. 2020McKean-Vlasov SDEs in nonlinear filteringcited by 7
  2. 2024Characterizing Dependence of Samples along the Langevin Dynamics and Algorithms via Contraction of Phi-Mutual Informationcited by 5
  3. 2025Sequential Monte Carlo approximations of Wasserstein--Fisher--Rao gradient flowscited by 5
  4. 2024Connections between sequential Bayesian inference and evolutionary dynamicscited by 4
  5. 2025A note on the unique properties of the Kullback--Leibler divergence for sampling via gradient flowscited by 4
  6. 2025Evolution of Gaussians in the Hellinger-Kantorovich-Boltzmann gradient flowcited by 3
  7. 2025An operator splitting analysis of Wasserstein--Fisher--Rao gradient flowscited by 2

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