Hellinger--Kantorovich gradient flows: Global exponential decay of entropy functionals
Authors
- Mielke, Alexander
ORCID: 0000-0002-4583-3888 - Zhu, Jia-Jie
2020 Mathematics Subject Classification
- 49Q22 35Q49 28A33 49J40 47J30
Keywords
- Optimal transport, gradient flow, Otto--Wasserstein, Hellinger, Fisher--Rao, unbalanced transport, optimization, calculus of variations, statistical inference, sampling
DOI
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 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.
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