We introduce an Ordinary Differential Equation (ODE) based deep generative
method for learning a conditional distribution, named the Conditional Follmer
Flow. Starting from a standard Gaussian distribution, the proposed flow could
efficiently transform it into the target conditional distribution at time 1.
For effective implementation, we discretize the flow with Euler's method where
we estimate the velocity field nonparametrically using a deep neural network.
Furthermore, we derive a non-asymptotic convergence rate in the Wasserstein
distance between the distribution of the learned samples and the target
distribution, providing the first comprehensive end-to-end error analysis for
conditional distribution learning via ODE flow. Our numerical experiments
showcase its effectiveness across a range of scenarios, from standard
nonparametric conditional density estimation problems to more intricate
challenges involving image data, illustrating its superiority over various
existing conditional density estimation methods.

我们介绍了一种基于常微分方程（ODE）的深度生成方法，称为条件 Follmer 流。该方法能够将标准高斯分布有效地转换为目标条件分布。在实现上，我们使用欧拉方法离散化流，并使用深度神经网络非参数地估计速度场。此外，我们推导出学习样本分布与目标分布之间的 Wasserstein 距离的非渐近收敛速率，为通过 ODE 流进行条件分布学习提供了首个全面的端到端误差分析。我们的数值实验展示了其在一系列场景中的有效性，从标准的非参数条件密度估计问题到涉及图像数据的更复杂挑战，证明了它在各种现有条件密度估计方法上的优势。