Flow matching (FM) is a family of training algorithms for fitting continuous normalizing flows (CNFs). A standard approach to FM, called conditional flow matching (CFM), exploits the fact that the marginal vector field of a CNF can be learned by fitting least-square regression to the so-called conditional vector field specified given one or both ends of the flow path. We show that viewing CFM training from a Bayesian decision theoretic perspective on parameter estimation opens the door to generalizations of CFM algorithms. We propose one such extension by introducing a CFM algorithm based on defining conditional probability paths given what we refer to as ``streams'', instances of latent stochastic paths that connect pairs of noise and observed data. Further, we advocates the modeling of these latent streams using Gaussian processes (GPs). The unique distributional properties of GPs, and in particular the fact that the velocities of a GP is still a GP, allows drawing samples from the resulting stream-augmented conditional probability path without simulating the actual streams, and hence the ``simulation-free" nature of CFM training is preserved. We show that this generalization of the CFM can substantially reduce the variance in the estimated marginal vector field at a moderate computational cost, thereby improving the quality of the generated samples under common metrics. Additionally, we show that adopting the GP on the streams allows for flexibly linking multiple related training data points (e.g., time series) and incorporating additional prior information. We empirically validate our claim through both simulations and applications to two hand-written image datasets.

本研究解决了流匹配（FM）在训练连续归一化流（CNF）中的不足。通过引入基于“流”的条件概率路径的CFM扩展，利用高斯过程（GP）模拟潜在流，将流匹配训练的“无模拟”特性保持，同时显著降低了估计的边际向量场的方差，提升生成样本的质量。研究结果在手写图片数据集上得到实证验证，展现了该方法的潜在影响。

基于贝叶斯决策理论的流级匹配