mlsanityData to 7 Oct 2026

Fisher-Rao Gradient Flow: Geodesic Convexity and Functional Inequalities

arXiv 2407.15693 ↗First posted 22 Jul 2024math.AP cs.LG math.FAcited by 13

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Abstract (arXiv)

The dynamics of probability density functions have been extensively studied in computational science and engineering to understand physical phenomena and facilitate algorithmic design. Of particular interest are dynamics formulated as gradient flows of energy functionals under the Wasserstein metric. The development of functional inequalities, such as the log-Sobolev inequality, plays a pivotal role in analyzing the convergence of these dynamics. This paper aims to extend the success of functional inequality techniques to dynamics that are gradient flows under the Fisher-Rao metric, with various f-divergences serving as energy functionals. Such dynamics take the form of nonlocal differential equations, for which existing analyses critically rely on explicit solution formulas in special cases. We provide a comprehensive study of functional inequalities and the relevant geodesic convexity for Fisher-Rao gradient flows under minimal assumptions. A notable feature of our functional inequalities is their independence from the log-concavity or log-Sobolev constants of the target distribution. Consequently, the convergence rate of the dynamics (assuming well-posedness) remains uniform across general target distributions.

Key facts. Cited by 13 papers in the map (data through 7 Oct 2026). Builds on 26 of its 106 references that are in the map. First posted on arXiv 22 Jul 2024.

Builds on

26 of 106 references.

References · 26

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14 more
  1. 2022Mirror Descent with Relative Smoothness in Measure Spaces, with application to Sinkhorn and EMcited by 33
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Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 1; 2025: 6; 2026: 6.

13 papers.

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  1. 2025Hellinger-Kantorovich Gradient Flows: Global Exponential Decay of Entropy Functionalscited by 7
  2. 2025Sequential Monte Carlo approximations of Wasserstein--Fisher--Rao gradient flowscited by 5
  3. 2024Inclusive KL Gradient Flows: Otto-Wasserstein, Fisher-Rao-Gaussian, and Local-Estimator Dynamicscited by 4
  4. 2025A note on the unique properties of the Kullback--Leibler divergence for sampling via gradient flowscited by 4
  5. 2025Evolution of Gaussians in the Hellinger-Kantorovich-Boltzmann gradient flowcited by 3
  6. 2025Gradient Flow Sampler-based Distributionally Robust Optimizationcited by 2
  7. 2025An operator splitting analysis of Wasserstein--Fisher--Rao gradient flowscited by 2
  8. 2026To discretize continually: Mean shift interacting particle systems for Bayesian inferencecited by 1
  9. 2026On propagation of chaos for the Fisher-Rao gradient flow in entropic mean-field optimizationcited by 0
  10. 2026Regularity of Solutions to Beckmann's Parametric Optimal Transportcited by 0
  11. 2026Properties and limitations of geometric tempering for gradient flow dynamicscited by 0
  12. 2026Preservation of Log-Concavity and Convergence of Wasserstein-Fisher-Rao Gradient Flowscited by 0
1 more
  1. 2026Retraction-Based Gradient Projection Algorithms on Manifoldscited by 0