mlsanityData to 7 Oct 2026

Sampling via Gradient Flows in the Space of Probability Measures

arXiv 2310.03597 ↗First posted 5 Oct 2023stat.ML cs.LG cs.NAcited by 20

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Abstract (arXiv)

Sampling a target probability distribution with an unknown normalization constant is a fundamental challenge in computational science and engineering. Recent work shows that algorithms derived by considering gradient flows in the space of probability measures open up new avenues for algorithm development. This paper makes three contributions to this sampling approach by scrutinizing the design components of such gradient flows. Any instantiation of a gradient flow for sampling needs an energy functional and a metric to determine the flow, as well as numerical approximations of the flow to derive algorithms. Our first contribution is to show that the Kullback-Leibler divergence, as an energy functional, has the unique property (among all f-divergences) that gradient flows resulting from it do not depend on the normalization constant of the target distribution. Our second contribution is to study the choice of metric from the perspective of invariance. The Fisher-Rao metric is known as the unique choice (up to scaling) that is diffeomorphism invariant. As a computationally tractable alternative, we introduce a relaxed, affine invariance property for the metrics and gradient flows. In particular, we construct various affine invariant Wasserstein and Stein gradient flows. Affine invariant gradient flows are shown to behave more favorably than their non-affine-invariant counterparts when sampling highly anisotropic distributions, in theory and by using particle methods. Our third contribution is to study, and develop efficient algorithms based on Gaussian approximations of the gradient flows; this leads to an alternative to particle methods. We establish connections between various Gaussian approximate gradient flows, discuss their relation to gradient methods arising from parametric variational inference, and study their convergence properties both theoretically and numerically.

Key facts. Cited by 20 papers in the map (data through 7 Oct 2026). Builds on 51 of its 140 references that are in the map. First posted on arXiv 5 Oct 2023.

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51 of 140 references.

References · 51

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38 more
  1. 2022Variational inference via Wasserstein gradient flowscited by 84
  2. 2019On the geometry of Stein variational gradient descentcited by 82
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Citing papers per year: 2024: 8; 2025: 8; 2026: 4.

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  1. 2025Solving Inverse Problems via Diffusion-Based Priors: An Approximation-Free Ensemble Sampling Approachcited by 18
  2. 2024Fisher-Rao Gradient Flow: Geodesic Convexity and Functional Inequalitiescited by 13
  3. 2024Stein transport for Bayesian inferencecited by 11
  4. 2024Efficient, Multimodal, and Derivative-Free Bayesian Inference With Fisher-Rao Gradient Flowscited by 9
  5. 2025Fast Convergence for High-Order ODE Solvers in Diffusion Probabilistic Modelscited by 9
  6. 2024Sequential-in-time training of nonlinear parametrizations for solving time-dependent partial differential equationscited by 7
  7. 2024Importance Corrected Neural JKO Samplingcited by 7
  8. 2024Wasserstein Gradient Flows of MMD Functionals with Distance Kernel and Cauchy Problems on Quantile Functionscited by 7
  9. 2025Sequential Monte Carlo approximations of Wasserstein--Fisher--Rao gradient flowscited by 5
  10. 2024Ensemble-Based Annealed Importance Samplingcited by 4
  11. 2024Connections between sequential Bayesian inference and evolutionary dynamicscited by 4
  12. 2025A note on the unique properties of the Kullback--Leibler divergence for sampling via gradient flowscited by 4
8 more
  1. 2025New affine invariant ensemble samplers and their dimensional scalingcited by 3
  2. 2025Harnessing the Power of Reinforcement Learning for Adaptive MCMCcited by 2
  3. 2025Gradient Flow Sampler-based Distributionally Robust Optimizationcited by 2
  4. 2025An operator splitting analysis of Wasserstein--Fisher--Rao gradient flowscited by 2
  5. 2026Regularity of Solutions to Beckmann's Parametric Optimal Transportcited by 0
  6. 2026Properties and limitations of geometric tempering for gradient flow dynamicscited by 0
  7. 2026Nesterov acceleration in optimizing over probability measurescited by 0
  8. 2026Preservation of Log-Concavity and Convergence of Wasserstein-Fisher-Rao Gradient Flowscited by 0