揭示流匹配与动作匹配在最优矢量场下的等价性
On the Equivalence of Optimal Transport Problem and Action Matching with Optimal Vector Fields
- 用最优矢量场重构生成模型中的流匹配方法
- 证明动作匹配可实现最优传输的精确解
- 为生成建模提供统一理论视角，适合研究者参考
生成建模中的流匹配（FM）方法通过构建任意概率分布间的插值路径，并学习定义该路径的微分方程矢量场来实现分布映射。最近研究表明，只要在损失最小化过程中仅考虑典型最优传输（OT）问题的最优矢量场，即可对任意初始插值实现最优分布映射。本文进一步表明，若仅考虑此类最优矢量场，动作匹配（AM）方法亦可实现最优传输。与需手动设计插值路径的流匹配不同，动作匹配直接学习给定分布序列所对应的微分方程矢量场，从而以另一种方式实现最优传输。
原文摘要 · Abstract (English)
Flow Matching (FM) method in generative modeling maps arbitrary probability distributions by constructing an interpolation between them and then learning the vector field that defines ODE for this interpolation. Recently, it was shown that FM can be modified to map distributions optimally in terms of the quadratic cost function for any initial interpolation. To achieve this, only specific optimal vector fields, which are typical for solutions of Optimal Transport (OT) problems, need to be considered during FM loss minimization. In this note, we show that considering only optimal vector fields can lead to OT in another approach: Action Matching (AM). Unlike FM, which learns a vector field for a manually chosen interpolation between given distributions, AM learns the vector field that defines ODE for an entire given sequence of distributions.
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