OvaDiP: Controlling Structured Diffusion Paths, Separating Training and Sampling Contributions, and Tracking Their Dynamics

We introduce Operator-Valued Diffusion Paths (OvaDiP), a controlled analysis framework for structured diffusion paths whose final effects can vary across datasets. OvaDiP represents a structured path as an operator-valued deformation of a scalar reference and canonically separates its variation into common retiming and centered structural redistribution. Controlled factorial interventions along these two coordinates show that their effects depend on the sampling budget. We then use common-sampler cross-evaluation to separate the contributions associated with the trained model and the sampling path. Across AFHQv2 and FFHQ, small final Fréchet Inception Distance (FID) differences can conceal substantially larger opposing contributions from these two stages. Finally, we extend the same attribution across training and find that the factorial interaction can appear stable while the balance between training and sampling continues to evolve late in training. Together, these results show that a final generation metric alone can obscure how the observed effect of a structured diffusion path arises. OvaDiP provides a controlled framework for relating path interventions to training–sampling attribution and its evolution during training. Our implementation and evaluation code are available at https://anonymous.4open.science/r/ovadip.