Diffusion Models Generate Images Like Painters: an Analytical Theory of Outline First, Details Later

How do diffusion generative models convert pure noise into meaningful images? In a variety of pretrained diffusion models (including conditional latent space models like Stable Diffusion), we observe that the reverse diffusion process that underlies image generation has the following properties: (i) individual trajectories tend to be low-dimensional and resemble 2D `rotations'; (ii) high-variance scene features like layout tend to emerge earlier, while low-variance details tend to emerge later; and (iii) early perturbations tend to have a greater impact on image content than later perturbations. To understand these phenomena, we derive and study a closed-form solution to the probability flow ODE for a Gaussian distribution, which shows that the reverse diffusion state rotates towards a gradually-specified target on the image manifold. It also shows that generation involves first committing to an outline, and then to finer and finer details. We find that this solution accurately describes the initial phase of image generation for pretrained models, and can in principle be used to make image generation more efficient by skipping reverse diffusion steps. Finally, we use our solution to characterize the image manifold in Stable Diffusion. Our viewpoint reveals an unexpected similarity between generation by GANs and diffusion and provides a conceptual link between diffusion and image retrieval.