We develop theory to understand an intriguing property of diffusion models
for image generation that we term critical windows. Empirically, it has been
observed that there are narrow time intervals in sampling during which
particular features of the final image emerge, e.g. the image class or
background color (Ho et al., 2020b; Georgiev et al., 2023; Raya & Ambrogioni,
2023; Sclocchi et al., 2024; Biroli et al., 2024). While this is advantageous
for interpretability as it implies one can localize properties of the
generation to a small segment of the trajectory, it seems at odds with the
continuous nature of the diffusion. We propose a formal framework for studying
these windows and show that for data coming from a mixture of strongly
log-concave densities, these windows can be provably bounded in terms of
certain measures of inter- and intra-group separation. We also instantiate
these bounds for concrete examples like well-conditioned Gaussian mixtures.
Finally, we use our bounds to give a rigorous interpretation of diffusion
models as hierarchical samplers that progressively "decide" output features
over a discrete sequence of times. We validate our bounds with synthetic
experiments. Additionally, preliminary experiments on Stable Diffusion suggest
critical windows may serve as a useful tool for diagnosing fairness and privacy
violations in real-world diffusion models.

通过发展理论来研究扩散模型的一个有趣特性 —— 关键时间窗口，文中提出了这些窗口的形式框架，并证明了一些数学方面的界限，从而对扩散模型进行了深入解释，并指出关键窗口可以用于诊断实际应用中的公平性和隐私问题。