Bayesian Nonparametric Diffusion Models

Score-based diffusion models have achieved strong generative performance across a wide range of applications, but their perturbation law, which determines the noisy states and score targets, is typically fixed to a standard Gaussian form. Recent work shows that alternative perturbation laws can improve modeling in specific data regimes. However, existing approaches either prescribe a perturbation family or learn restricted forward-process parameterizations, leaving open how to adapt the perturbation law itself to a particular learner and training regime. We address this problem with a Bayesian nonparametric (BNP) diffusion model that learns the perturbation distribution directly. We show that standard denoising score matching (DSM) contains a perturbation-dependent irreducible term and derive a corrected criterion for generalized Bayesian learning of the perturbation distribution. Theoretically, we show that the corrected criterion consistently targets finite-resource score risk, that the resulting generalized posterior concentrates around the corresponding optimal perturbation laws, and that the same score risk controls reverse-process error. Empirically, controlled experiments show that BNP adapts across Gaussian and non-Gaussian regimes and that the correction enables more accurate comparisons of score error across perturbation laws. On CIFAR-10 and HRRR, comparisons based on estimated corrected score risk show that the preferred perturbation varies across tasks and training budgets. Together, these findings support treating the perturbation law as a learnable component of diffusion models, adapted to the score learner and available training resources.