用数学视角统一解释生成模型中的流匹配方法
Flow Matching: Markov Kernels, Stochastic Processes and Transport Plans
- 通过耦合、马尔可夫核和随机过程三种方式学习分布间的速度场
- 揭示流匹配与贝叶斯反问题中条件瓦瑟斯坦距离的关联
- 适合对生成模型数学基础感兴趣的科研人员
在生成神经模型中，流匹配因其简单适用性和良好扩展性脱颖而出。该方法学习连接简单先验分布与目标分布的连续曲线的速度场，随后可通过常微分方程从先验样本采样得到目标分布样本。本文从数学角度综述了在瓦瑟斯坦几何中学习绝对连续曲线速度场的不同技术。我们展示了速度场可通过：i) 先验与目标分布之间的传输计划（耦合），ii) 马尔可夫核，以及 iii) 随机过程来表征与学习，其中后两者包含耦合方法但通常更广泛。此外，本文还展示了流匹配如何用于求解贝叶斯反问题，其中条件瓦瑟斯坦距离起核心作用。最后，简要讨论了连续归一化流与得分匹配技术，它们从不同方向逼近速度场的学习。
原文摘要 · Abstract (English)
Among generative neural models, flow matching techniques stand out for their simple applicability and good scaling properties. Here, velocity fields of curves connecting a simple latent and a target distribution are learned. Then the corresponding ordinary differential equation can be used to sample from a target distribution, starting in samples from the latent one. This paper reviews from a mathematical point of view different techniques to learn the velocity fields of absolutely continuous curves in the Wasserstein geometry. We show how the velocity fields can be characterized and learned via i) transport plans (couplings) between latent and target distributions, ii) Markov kernels and iii) stochastic processes, where the latter two include the coupling approach, but are in general broader. Besides this main goal, we show how flow matching can be used for solving Bayesian inverse problems, where the definition of conditional Wasserstein distances plays a central role. Finally, we briefly address continuous normalizing flows and score matching techniques, which approach the learning of velocity fields of curves from other directions.
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