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

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arxiv 2501.17049 v1 pith:5A2HRVK7 submitted 2025-01-28 math.AP cs.LGmath.OCstat.ML

Alexander Mielke , Jia-Jie Zhu This is my paper · ORCID
classification math.APcs.LGmath.OCstat.ML
keywords flowsgradientanalysiscompletedecayentropyexponentialfunctionals
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abstract

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, $\chi^2$) 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-)\L{}ojasiewicz-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.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Evolution of Gaussians in the Hellinger-Kantorovich-Boltzmann gradient flow

    math.AP 2025-04 conditional novelty 7.0 of 10

    The HK-Boltzmann gradient flow preserves Gaussianity, and the reduced equations for mean, covariance, and mass admit exponential convergence rates with explicit dependence on the geometry parameters.

  2. Gradient Flow Sampler-based Distributionally Robust Optimization

    math.OC 2025-10 conditional novelty 5.0 of 10

    Entropy-regularized Wasserstein DRO can be solved by sampling from a Gibbs worst-case distribution with gradient-flow samplers, giving new WFR/SVGD algorithms and a principled recovery of WRM.

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