The paper presents the exact formula for the vector field that minimizes the
loss for the standard flow. This formula depends analytically on a given
distribution \rho_0 and an unknown one \rho_1. Based on the presented formula,
a new loss and algorithm for training a vector field model in the style of
Conditional Flow Matching are provided. Our loss, in comparison to the standard
Conditional Flow Matching approach, exhibits smaller variance when evaluated
through Monte Carlo sampling methods. Numerical experiments on synthetic models
and models on tabular data of large dimensions demonstrate better learning
results with the use of the presented algorithm.

给出了一种最小化标准流失函数的向量场精确公式，该公式在给定的分布 ρ₀和未知分布 ρ₁的基础上进行了分析，并提出了一种新的损失函数和算法。与标准条件流匹配方法相比，通过蒙特卡罗采样方法评估时，我们的损失函数表现出较小的方差。通过对合成模型和大维度表格数据模型的数值实验证明了使用该算法能够获得更好的学习结果。

智能流量匹配：流量匹配算法的理论及应用研究

Smart Flow Matching: On The Theory of Flow Matching Algorithms with  Applications

Tabular data is hard to acquire and is subject to missing values. This paper
proposes a novel approach to generate and impute mixed-type (continuous and
categorical) tabular data using score-based diffusion and conditional flow
matching. Contrary to previous work that relies on neural networks as function
approximators, we instead utilize XGBoost, a popular Gradient-Boosted Tree
(GBT) method. In addition to being elegant, we empirically show on various
datasets that our method i) generates highly realistic synthetic data when the
training dataset is either clean or tainted by missing data and ii) generates
diverse plausible data imputations. Our method often outperforms deep-learning
generation methods and can trained in parallel using CPUs without the need for
a GPU. To make it easily accessible, we release our code through a Python
library on PyPI and an R package on CRAN.

该论文提出了一种利用基于评分的扩散和条件流匹配生成和填补混合类型（连续和分类）表格数据的新方法。相对于依赖神经网络作为函数近似器的先前工作，我们使用了 XGBoost，一种流行的梯度提升树（GBT）方法。我们的方法在多个数据集上经验性地表明：当训练数据集是干净的或受到缺失数据影响时，它能够生成高度逼真的合成数据，并生成多样的合理数据填补。我们的方法经常优于深度学习生成方法，并且可以使用 CPU 并行训练，无需 GPU。为了方便使用，我们通过 PyPI 发布了我们的 Python 库和 CRAN 上的 R 包。