Fisher-Rao梯度流：测地凸性与函数不等式
Fisher-Rao Gradient Flow: Geodesic Convexity and Functional Inequalities
AI总结：
本文研究Fisher-Rao度量下的梯度流动力学，探讨测地凸性与函数不等式，提出不依赖目标分布log-concavity或log-Sobolev常数的分析方法，确保动力学收敛速率在一般分布下保持一致。
摘要：
概率密度函数的动力学在计算科学和工程中被广泛研究，以理解物理现象并促进算法设计。特别关注的是在Wasserstein度量下以能量泛函为形式的梯度流动力学。发展如log-Sobolev不等式等函数不等式在分析这些动力学收敛性中起关键作用。本文旨在将函数不等式技术的成功扩展到Fisher-Rao度量下的梯度流动力学，其中各种f-分歧度作为能量泛函。此类动力学以非局部微分方程形式出现，现有分析在特殊情况下依赖于显式解公式。本文在最小假设下对Fisher-Rao梯度流的函数不等式和相关测地凸性进行了全面研究。本文函数不等式的一个显著特点是不依赖目标分布的log-concavity或log-Sobolev常数。因此，假设动力学良好定义的情况下，收敛速率在一般目标分布下保持一致。
英文摘要：
The dynamics of probability density functions have been extensively studied in computational science and engineering to understand physical phenomena and facilitate algorithmic design. Of particular interest are dynamics formulated as gradient flows of energy functionals under the Wasserstein metric. The development of functional inequalities, such as the log-Sobolev inequality, plays a pivotal role in analyzing the convergence of these dynamics. This paper aims to extend the success of functional inequality techniques to dynamics that are gradient flows under the Fisher-Rao metric, with various $f$-divergences serving as energy functionals. Such dynamics take the form of nonlocal differential equations, for which existing analyses critically rely on explicit solution formulas in special cases. We provide a comprehensive study of functional inequalities and the relevant geodesic convexity for Fisher-Rao gradient flows under minimal assumptions. A notable feature of our functional inequalities is their independence from the log-concavity or log-Sobolev constants of the target distribution. Consequently, the convergence rate of the dynamics (assuming well-posedness) remains uniform across general target distributions.
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