Conditional generative models represent a significant advancement in the field of machine learning, allowing for the controlled synthesis of data by incorporating additional information into the generation process. In this work we introduce a novel Schr\"odinger bridge based deep generative method for learning conditional distributions. We start from a unit-time diffusion process governed by a stochastic differential equation (SDE) that transforms a fixed point at time $0$ into a desired target conditional distribution at time $1$. For effective implementation, we discretize the SDE with Euler-Maruyama method where we estimate the drift term nonparametrically using a deep neural network. We apply our method to both low-dimensional and high-dimensional conditional generation problems. The numerical studies demonstrate that though our method does not directly provide the conditional density estimation, the samples generated by this method exhibit higher quality compared to those obtained by several existing methods. Moreover, the generated samples can be effectively utilized to estimate the conditional density and related statistical quantities, such as conditional mean and conditional standard deviation.

翻译：条件生成模型代表了机器学习领域的一项重大进展，它通过在生成过程中融入额外信息，实现了数据的可控合成。本文提出了一种新颖的基于薛定谔桥的深度生成方法，用于学习条件分布。我们从一个单位时间扩散过程出发，该过程由随机微分方程（SDE）描述，将时间 $0$ 处的一个固定点转化为时间 $1$ 处期望的目标条件分布。为实现有效计算，我们采用欧拉-丸山方法对SDE进行离散化，并使用深度神经网络以非参数方式估计漂移项。我们将该方法应用于低维和高维条件生成问题。数值研究表明，尽管我们的方法不直接提供条件密度估计，但由此方法生成的样本质量优于多种现有方法所得样本。此外，生成的样本可有效用于估计条件密度及相关统计量，如条件均值和条件标准差。