Theory on Score-Mismatched Diffusion Models and Zero-Shot Conditional Samplers
This addresses a theoretical gap for diffusion models in practical scenarios like zero-shot conditional sampling, offering foundational insights but is incremental as it builds on existing diffusion theory.
The paper tackles the problem of score-mismatched diffusion models, where target distributions differ from training distributions, by providing the first theoretical performance guarantees with explicit dimensional dependencies, showing an asymptotic bias proportional to the mismatch. It applies this to zero-shot conditional samplers and designs a bias-optimal sampler for linear models with convergence guarantees for distributions like Gaussian mixtures.
The denoising diffusion model has recently emerged as a powerful generative technique, capable of transforming noise into meaningful data. While theoretical convergence guarantees for diffusion models are well established when the target distribution aligns with the training distribution, practical scenarios often present mismatches. One common case is in the zero-shot conditional diffusion sampling, where the target conditional distribution is different from the (unconditional) training distribution. These score-mismatched diffusion models remain largely unexplored from a theoretical perspective. In this paper, we present the first performance guarantee with explicit dimensional dependencies for general score-mismatched diffusion samplers, focusing on target distributions with finite second moments. We show that score mismatches result in an asymptotic distributional bias between the target and sampling distributions, proportional to the accumulated mismatch between the target and training distributions. This result can be directly applied to zero-shot conditional samplers for any conditional model, irrespective of measurement noise. Interestingly, the derived convergence upper bound offers useful guidance for designing a novel bias-optimal zero-shot sampler in linear conditional models that minimizes the asymptotic bias. For such bias-optimal samplers, we further establish convergence guarantees with explicit dependencies on dimension and conditioning, applied to several interesting target distributions, including those with bounded support and Gaussian mixtures. Our findings are supported by numerical studies.