具有一般运输成本的JKO计划
Cale Rankin1, Ting-Kam Leonard Wong2
1Department of Mathematics, Monash University, Victoria, Australia.
Calculus of variations and partial differential equations
|February 20, 2026
概括
我们修改了Wasserstein梯度流的JKO方案,通过在分流器上使用一般运输成本. 这种修改后的方案汇聚到里曼的福克-普朗克方程,提供计算优势.
科学领域:
- 数字分析 数字分析
- 不同几何学微分几何学
- 随机过程 随机过程
背景情况:
- JKO 方案是瓦瑟斯坦梯度流的时间分离.
- 瓦瑟斯坦距离在变形体上可能是计算密集的.
- 福克-普朗克方程模型扩散过程.
研究的目的:
- 在分流器上使用任意运输成本来概括JKO方案.
- 为了确定与里曼的福克-普朗克方程的收.
- 探索对里曼距离的计算替代方案.
主要方法:
- 修改JKO计划,将瓦瑟斯坦距离替换为一般运输成本.
- 在成本函数的Hessian条件下对收性质的分析.
- 在紧和完整的里曼的多元体上应用福克-普朗克方程.
主要成果:
- 修改后的JKO方案与里曼的福克-普朗克方程的趋同,当成本诱导里曼度量时.
- 在具有诺曼边界条件的紧子多元体和完整的里曼多元体上证明适用性.
- 成功地应用到使用布雷格曼分歧作为成本的赫西安分组.
结论:
- 概括的JKO方案提供了一个灵活的框架,用于对变频器的梯度流进行分离.
- 当里曼距离难以处理时,这种方法提供了计算优势.
- 该方法有效地连接了最佳运输,几何分析和数值方法.
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