  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; Meng et al., 2022; Choi et al., 2022; Raya
& Ambrogioni, 2023; Georgiev et al., 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.
