确定性测度传输的平流Fisher-Rao几何
The Advective Fisher-Rao Geometry of Deterministic Measure Transport
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中文总结 AI 辅助
本文提出平流Fisher-Rao度量，从三视角推导其来源，经实验验证该度量可最优拟合概率密度，高斯-牛顿法可最优拟合速度场，为概率测度路径优化提供新几何工具。
中文摘要 AI 辅助
针对由连续性方程支配的概率测度路径上的优化任务，本文引入了一种新型平流Fisher-Rao度量，该度量可产生最优下降方向。研究表明，该度量可从三个不同视角自然导出：作为路径测度上Fisher-Rao度量的缩放零噪声极限，作为Freidlin-Wentzell大偏差速率泛函的二阶变分的期望值，以及作为动态最优传输中Benamou-Brenier作用泛函的Hessian。本文通过计算实验补充了这一几何构造，实证显示平流Fisher-Rao度量可实现概率密度的所需最优拟合，而高斯-牛顿法则可实现速度场的最优拟合。
英文摘要
A novel advective Fisher-Rao metric is introduced for optimization tasks on paths of probability measures governed by the continuity equation. This metric is shown to lead to optimal descent directions. It is then shown that this metric arises naturally from three different perspectives: As the rescaled zero-noise limit of the Fisher-Rao metric on path measures, as the expected value of the second variation of the Freidlin--Wentzell large deviation rate functional, and as the Hessian of the Benamou--Brenier action functional from dynamic optimal transport. We supplement this geometric construction with computational experiments. Here, we demonstrate empirically that the advective Fisher-Rao metric yields the desired optimal fitting of probability densities, whereas the Gauss--Newton method yields optimal fitting of velocity fields.
发表机构
- Institut für Mathematik, Technische Universität Berlin（柏林工业大学数学研究所）
- Max-Planck-Institut für Mathematik in den Naturwissenschaften（马克斯·普朗克数学研究所（自然科学领域））
机构由 AI 辅助整理，请以论文原文为准。
