mlsanityData to 7 Oct 2026

Hellinger-Kantorovich Gradient Flows: Global Exponential Decay of Entropy Functionals

arXiv 2501.17049 ↗First posted 28 Jan 2025math.AP cs.LG math.OCcited by 7

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Abstract (arXiv)

We investigate a family of gradient flows of positive and probability measures, focusing on the Hellinger-Kantorovich (HK) geometry, which unifies transport mechanism of Otto-Wasserstein, and the birth-death mechanism of Hellinger (or Fisher-Rao). A central contribution is a complete characterization of global exponential decay behaviors of entropy functionals (e.g. KL, chi2) under Otto-Wasserstein and Hellinger-type gradient flows. In particular, for the more challenging analysis of HK gradient flows on positive measures -- where the typical log-Sobolev arguments fail -- we develop a specialized shape-mass decomposition that enables new analysis results. Our approach also leverages the (Polyak-)Lojasiewicz-type functional inequalities and a careful extension of classical dissipation estimates. These findings provide a unified and complete theoretical framework for gradient flows and underpin applications in computational algorithms for statistical inference, optimization, and machine learning.

Key facts. Cited by 7 papers in the map (data through 7 Oct 2026). Builds on 16 of its 56 references that are in the map. First posted on arXiv 28 Jan 2025.

Builds on

16 of 56 references.

References · 16

  1. 2016Variational Inference: A Review for Statisticianscited by 1,466
  2. 2016Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Conditioncited by 526
  3. 2015Optimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measurescited by 138
  4. 2016A Variational Perspective on Accelerated Methods in Optimizationcited by 108
  5. 2021The Bayesian Learning Rulecited by 95
  6. 2022Variational inference via Wasserstein gradient flowscited by 84
  7. 2020SVGD as a kernelized Wasserstein gradient flow of the chi-squared divergencecited by 64
  8. 2019Sparse Optimization on Measures with Over-parameterized Gradient Descentcited by 50
  9. 2018Fast yet Simple Natural-Gradient Descent for Variational Inference in Complex Modelscited by 37
  10. 2022Birth-death dynamics for sampling: Global convergence, approximations and their asymptoticscited by 30
  11. 2023Learning Gaussian Mixtures Using the Wasserstein-Fisher-Rao Gradient Flowcited by 25
  12. 2015Provable Bayesian Inference via Particle Mirror Descentcited by 23
4 more
  1. 2023An Explicit Expansion of the Kullback-Leibler Divergence along its Fisher-Rao Gradient Flowcited by 16
  2. 2024Fisher-Rao Gradient Flow: Geodesic Convexity and Functional Inequalitiescited by 13
  3. 2024Interaction-Force Transport Gradient Flowscited by 7
  4. 2024Inclusive KL Gradient Flows: Otto-Wasserstein, Fisher-Rao-Gaussian, and Local-Estimator Dynamicscited by 4

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2025: 4; 2026: 3.

7 papers.

  1. 2025Weighted quantization using MMD: From mean field to mean shift via gradient flowscited by 10
  2. 2026Verifier-Constrained Flow Expansion for Discovery Beyond the Datacited by 6
  3. 2025Evolution of Gaussians in the Hellinger-Kantorovich-Boltzmann gradient flowcited by 3
  4. 2025Gradient Flow Sampler-based Distributionally Robust Optimizationcited by 2
  5. 2025Wasserstein-Cram'er-Rao Theory of Unbiased Estimationcited by 2
  6. 2026A Unified Density Operator View of Flow Control and Mergingcited by 2
  7. 2026Neural Sampling with Reweighted Normalizing Flows via the Wasserstein--Fisher--Rao JKO Schemecited by 0