Hellinger-Kantorovich Gradient Flows: Global Exponential Decay of Entropy Functionals
This provides a theoretical framework for gradient flows with applications in computational algorithms for statistical inference, optimization, and machine learning, but it is incremental as it builds on existing geometries and methods.
The paper tackles the analysis of gradient flows in Hellinger-Kantorovich geometry, characterizing global exponential decay of entropy functionals like KL and chi-squared, and develops a shape-mass decomposition to handle cases where standard methods fail.
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, $χ^2$) 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-)Łojasiewicz-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.