Per-Starting-Point Plausibility and Diversity Bounds for Score-Based Diffusion Models

Score-based diffusion is well-studied at the mixture level—averaged over the prior—but many practical pipelines (posterior sampling, SDEdit, OOD detection) fix the starting point and care about the per-start output. To our knowledge, this per-starting-point regime has no prior non-asymptotic analysis. We give two coupled bounds: a plausibility upper bound on how close the per-start output is to the data distribution, and a matching diversity lower bound on how distinguishable outputs from two different starts remain. The same score-contraction rate governs both—where plausibility is tight, diversity is vacuous, and vice versa—predicting a sharp horizon beyond which starting-point identity is erased. We illustrate the phenomenon qualitatively on 2D toy datasets, MNIST, and CIFAR-10.