Schrödinger bridge based deep conditional generative learning

Conditional generative models represent a significant advancement in machine learning, enabling controlled data synthesis by incorporating additional information into the generation process. In this work, we introduce a novel Schrödinger bridge-based deep generative method for learning conditional distributions. Our approach begins with a unit-time diffusion process governed by a stochastic differential equation (SDE) that evolves a fixed point at time $t=0$ into a desired target conditional distribution at $t=1$. For effective implementation, we discretize the SDE using the Euler–Maruyama method, estimating the drift term nonparametrically with a deep neural network. We apply our method to both low-dimensional and high-dimensional conditional generation tasks. Numerical studies show that, although our method does not directly provide conditional density estimation, the samples generated exhibit higher quality than those from several existing methods. Furthermore, the generated samples can be effectively used to estimate the conditional density and related statistical quantities, such as the conditional mean and conditional standard deviation.