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.

翻译：我们提出了一种基于常微分方程的深度生成方法用于学习条件分布，命名为条件福尔默流（Conditional Follmer Flow）。从标准高斯分布出发，该流能够在时刻1高效地将其转化为目标条件分布。为有效实现，我们采用欧拉方法对该流进行离散化，并通过深度神经网络非参数地估计速度场。进一步地，我们推导出了学习样本分布与目标分布之间Wasserstein距离的非渐近收敛速率，首次为基于常微分方程流的条件分布学习提供了全面的端到端误差分析。我们的数值实验展示了该方法在多种场景下的有效性——从标准非参数条件密度估计问题到涉及图像数据的更复杂挑战，均优于现有的多种条件密度估计方法。