揭示了霍林格-坎托罗维奇梯度流中熵泛函的全局指数衰减规律。
Hellinger-Kantorovich Gradient Flows: Global Exponential Decay of Entropy Functionals
- 基于霍林格-坎托罗维奇几何，构建正测度梯度流的统一分析框架。
- 首次证明在正测度上霍林格型梯度流的熵泛函全局指数衰减。
- 提出形状-质量分解法，突破传统对数索博列夫不等式失效困境，适合算法设计者参考。
本文研究正测度与概率测度的一类梯度流，聚焦于统一奥托-瓦瑟斯坦传输机制与霍林格（或费舍尔-罗）生灭机制的霍林格-坎托罗维奇（HK）几何。核心贡献在于完整刻画了奥托-瓦瑟斯坦及霍林格型梯度流下熵泛函（如KL、χ²）的全局指数衰减行为。尤其针对正测度上更具挑战性的HK梯度流分析——传统对数索博列夫方法失效——我们提出了专用的形状-质量分解法，实现新的理论突破。研究还借助(Polyak-) Łojasiewicz型函数不等式和经典耗散估计的精细扩展，为统计推断、优化与机器学习中的计算算法提供了统一且完整的理论基础。
原文摘要 · Abstract (English)
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.
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