Theory of Outcome Boundary Prediction in a Determinstic Diffusion ODE

Are there patterns in the unstructured noise from which a diffusion samples? Surprisingly, deterministic ODE samplers impose structure in the noise space. When the data distribution consists of separable gaussian clusters, a deterministic ODE-based sampler partitions the sampling noise based on the class the noise belongs to. The boundaries that separate these clusters can be mathematically characterized to represent decision thresholds in the noise space where the score field transitions between distinct modes, causing nearby seeds to produce ambiguous or hallucinated outputs. We formalize this correspondence and introduce a method that traces this mapping, which we call pullback. Using pullback, we can predict hallucinations, or classes in any mixture-of-Gaussian setting.