提出新型度量优化概率路径，可实现密度最优拟合。
The Advective Fisher-Rao Geometry of Deterministic Measure Transport
- 基于连续性方程构造新型流形度量，引导最优下降方向。
- 实验显示该度量能实现概率密度的最优拟合，优于高斯-牛顿法。
- 适用于动态最优传输、概率路径优化等场景的研究者。
针对由连续性方程控制的概率测度路径优化任务，提出一种新的对流型Fisher-Rao度量。该度量被证明可产生最优下降方向，并从三个不同视角自然导出：作为路径测度上Fisher-Rao度量的零噪声缩放极限，作为Freidlin--Wentzell大偏差率函数二阶变分的期望值，以及动态最优传输中Benamou--Brenier作用泛函的海森矩阵。通过计算实验验证，该度量在概率密度拟合上表现优异，而高斯-牛顿法仅能实现速度场的最优拟合。
原文摘要 · Abstract (English)
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.
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