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.
Hellinger-Kantorovich Gradient Flows: Global Exponential Decay of Entropy Functionals
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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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2025 1verdicts
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Evolution of Gaussians in the Hellinger-Kantorovich-Boltzmann gradient flow
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.