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 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.

翻译：去噪扩散模型最近已成为一种强大的生成技术，能够将噪声转化为有意义的数据。尽管当目标分布与训练分布一致时，扩散模型的理论收敛性保证已得到充分确立，但实际场景中常出现失配情况。一个常见案例是零样本条件扩散采样，其中目标条件分布不同于（无条件）训练分布。这类分数失配扩散模型在理论视角下仍很大程度上未被探索。本文首次针对一般分数失配扩散采样器提出了具有显式维度依赖性的性能保证，重点关注具有有限二阶矩的目标分布。我们证明分数失配会导致目标分布与采样分布之间存在渐近分布偏差，其大小与目标分布和训练分布之间累积的失配程度成正比。该结果可直接应用于任意条件模型的零样本条件采样器，且与测量噪声无关。有趣的是，推导出的收敛上界为设计线性条件模型中新型偏差最优零样本采样器提供了实用指导，该采样器能最小化渐近偏差。对于此类偏差最优采样器，我们进一步建立了具有显式维度与条件依赖关系的收敛性保证，并将其应用于若干有趣的目标分布，包括有界支撑集分布与高斯混合分布。我们的研究结果得到了数值实验的支持。