We consider a class of conditional forward-backward diffusion models for conditional generative modeling, that is, generating new data given a covariate (or control variable). To formally study the theoretical properties of these conditional generative models, we adopt a statistical framework of distribution regression to characterize the large sample properties of the conditional distribution estimators induced by these conditional forward-backward diffusion models. Here, the conditional distribution of data is assumed to smoothly change over the covariate. In particular, our derived convergence rate is minimax-optimal under the total variation metric within the regimes covered by the existing literature. Additionally, we extend our theory by allowing both the data and the covariate variable to potentially admit a low-dimensional manifold structure. In this scenario, we demonstrate that the conditional forward-backward diffusion model can adapt to both manifold structures, meaning that the derived estimation error bound (under the Wasserstein metric) depends only on the intrinsic dimensionalities of the data and the covariate.

翻译：本文研究一类用于条件生成建模的条件前向-后向扩散模型，即在给定协变量（或控制变量）条件下生成新数据的方法。为系统研究此类条件生成模型的理论性质，我们采用分布回归的统计框架，以刻画这些条件前向-后向扩散模型所导出的条件分布估计量的大样本性质。在此框架中，数据的条件分布被假定为随协变量平滑变化。特别地，在现有文献所涵盖的范围内，我们推导出的收敛速率在总变差度量下具有极小极大最优性。此外，我们通过允许数据和协变量变量可能具有低维流形结构来扩展理论。在此情形下，我们证明条件前向-后向扩散模型能够自适应于两种流形结构，这意味着所推导的估计误差界（在Wasserstein度量下）仅依赖于数据与协变量的本征维度。