Hellinger-Kantorovich Gradient Flows: Global Exponential Decay of Entropy Functionals
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| Format: | Preprint |
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2025
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| _version_ | 1866917904311648256 |
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| author | Mielke, Alexander Zhu, Jia-Jie |
| author_facet | Mielke, Alexander Zhu, Jia-Jie |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_17049 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hellinger-Kantorovich Gradient Flows: Global Exponential Decay of Entropy Functionals Mielke, Alexander Zhu, Jia-Jie Analysis of PDEs Machine Learning Optimization and Control 49Q22 (Primary) 35Q49 (Secondary) 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. |
| title | Hellinger-Kantorovich Gradient Flows: Global Exponential Decay of Entropy Functionals |
| topic | Analysis of PDEs Machine Learning Optimization and Control 49Q22 (Primary) 35Q49 (Secondary) |
| url | https://arxiv.org/abs/2501.17049 |