Conditional Diffusion Models are Minimax-Optimal and Manifold-Adaptive for Conditional Distribution Estimation
Year: 2024
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Authors: Tang, Rong, Lin, Lizhen, Yang, Yun
Language: English
Subfield: Computer Vision and Pattern Recognition
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4403815026