Blind denoising diffusion models and adaptive sampling algorithms

Denoising diffusion models (DDMs) are state-of-the-art methods to learn densities from data across numerous domains, yet many aspects of the training and sampling pipeline remain poorly understood. In particular, noise conditioning requires practitioners to incorporate contrived non-principled noise embeddings into neural network architectures and to use ad hoc noise schedules for sampling. To address these drawbacks, we provide a complete theory for blind denoising diffusion models (BDDMs): a variant of DDMs where the noise amplitude is not passed into the neural network during training nor sampling, obviating the need for the aforementioned design choices. We justify the correctness of BDDMs as a sampling algorithm under the main assumption of low intrinsic dimensionality of the underlying data distribution relative to the ambient dimension. This assumption arises through the introduction of the Bayesian problem of estimating noise levels from a single noisy sample, which might be of independent interest. We empirically compare the performance of BDDMs to standard DDMs, showcasing the benefits of an adaptive scheme which is rigorously justified by our analysis.