Diffusion Model's Generalization Can Be Characterized by Inductive Biases toward a Data-Dependent Ridge Manifold

When a diffusion model is not memorizing the training data set, how does it generalize exactly? A quantitative understanding of the distribution it generates would be beneficial to, for example, an assessment of the model's performance for downstream applications. We thus explicitly characterize what diffusion model generates, by proposing a log-density ridge manifold and quantifying how the generated data relate to this manifold as inference dynamics progresses. More precisely, inference undergoes a reach-align-slide process centered around the ridge manifold: trajectories first reach a neighborhood of the manifold, then align as being pushed toward or away from the manifold in normal directions, and finally slide along the manifold in tangent directions. Within the scope of this general behavior, different training errors will lead to different normal and tangent motions, which can be quantified, and these detailed motions characterize when inter-mode generations emerge. More detailed understanding of training dynamics will lead to more accurate quantification of the generation inductive bias, and an example of random feature model will be considered, for which we can explicitly illustrate how diffusion model's inductive biases originate as a composition of architectural bias and training accuracy, and how they evolve with the inference dynamics. Experiments on synthetic multimodal distributions and MNIST latent diffusion support the predicted directional effects, in both low- and high-dimensions.