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。

深度条件生成学习：模型和误差分析