mlsanityData to 14 Oct 2026

A note on the unique properties of the Kullback--Leibler divergence for sampling via gradient flows

arXiv 2507.04330 ↗First posted 6 Jul 2025stat.ME cs.LG math.STcited by 4

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Abstract (arXiv)

We consider the problem of sampling from a probability distribution π which admits a density w.r.t. a dominating measure. 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 a divergence from π. The optimisation problem is normally solved through gradient flows in the space of probability distributions with an appropriate metric. We show that the Kullback--Leibler divergence is the only divergence in the family of Bregman divergences whose gradient flow w.r.t. many popular metrics does not require knowledge of the normalising constant of π.

Key facts. Cited by 4 papers in the map (data through 14 Oct 2026). Builds on 27 of its 39 references that are in the map. First posted on arXiv 6 Jul 2025.

Builds on

27 of 39 references.

References · 27

  1. 2016Variational Inference: A Review for Statisticianscited by 1,472
  2. 2017Stein Variational Gradient Descent as Gradient Flowcited by 152
  3. 2018Analysis of Langevin Monte Carlo via convex optimizationcited by 127
  4. 2019Maximum Mean Discrepancy Gradient Flowcited by 127
  5. 2017Measuring Sample Quality with Kernelscited by 119
  6. 2022Variational inference via Wasserstein gradient flowscited by 85
  7. 2019On the geometry of Stein variational gradient descentcited by 82
  8. 2020SVGD as a kernelized Wasserstein gradient flow of the chi-squared divergencecited by 64
  9. 2021Kernel Stein Discrepancy Descentcited by 57
  10. 2019Accelerating Langevin Sampling with Birth-deathcited by 45
  11. 2022Birth-death dynamics for sampling: Global convergence, approximations and their asymptoticscited by 30
  12. 2023Learning Gaussian Mixtures Using the Wasserstein-Fisher-Rao Gradient Flowcited by 27
15 more
  1. 2024Sampling in Unit Time with Kernel Fisher-Rao Flowcited by 27
  2. 2022Discrete Langevin Sampler via Wasserstein Gradient Flowcited by 24
  3. 2015Provable Bayesian Inference via Particle Mirror Descentcited by 23
  4. 2023Sampling via Gradient Flows in the Space of Probability Measurescited by 20
  5. 2022Sampling with Mollified Interaction Energy Descentcited by 17
  6. 2023An Explicit Expansion of the Kullback-Leibler Divergence along its Fisher-Rao Gradient Flowcited by 16
  7. 2023A connection between Tempering and Entropic Mirror Descentcited by 16
  8. 2021Adaptive Importance Sampling meets Mirror Descent: a Bias-variance tradeoffcited by 13
  9. 2024Fisher-Rao Gradient Flow: Geodesic Convexity and Functional Inequalitiescited by 13
  10. 2018Alpha-Beta Divergence For Variational Inferencecited by 12
  11. 2025Weighted quantization using MMD: From mean field to mean shift via gradient flowscited by 12
  12. 2024Stein transport for Bayesian inferencecited by 11
  13. 2024Provable Convergence and Limitations of Geometric Tempering for Langevin Dynamicscited by 10
  14. 2025Sequential Monte Carlo approximations of Wasserstein--Fisher--Rao gradient flowscited by 5
  15. 2024Inclusive KL Gradient Flows: Otto-Wasserstein, Fisher-Rao-Gaussian, and Local-Estimator Dynamicscited by 4

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2025: 2; 2026: 2.

4 papers.

  1. 2025Towards understanding Accelerated Stein Variational Gradient Flow -- Analysis of Generalized Bilinear Kernels for Gaussian target distributionscited by 2
  2. 2025An operator splitting analysis of Wasserstein--Fisher--Rao gradient flowscited by 2
  3. 2026Properties and limitations of geometric tempering for gradient flow dynamicscited by 0
  4. 2026Preservation of Log-Concavity and Convergence of Wasserstein-Fisher-Rao Gradient Flowscited by 0