  Diffusion models have demonstrated impressive results in both data generation
and downstream tasks such as inverse problems, text-based editing,
classification, and more. However, training such models usually requires large
amounts of clean signals which are often difficult or impossible to obtain. In
this work, we propose a novel training technique for generative diffusion
models based only on corrupted data. We introduce a loss function based on the
Generalized Stein's Unbiased Risk Estimator (GSURE), and prove that under some
conditions, it is equivalent to the training objective used in fully supervised
diffusion models. We demonstrate our technique on face images as well as
Magnetic Resonance Imaging (MRI), where the use of undersampled data
significantly alleviates data collection costs. Our approach achieves
generative performance comparable to its fully supervised counterpart without
training on any clean signals. In addition, we deploy the resulting diffusion
model in various downstream tasks beyond the degradation present in the
training set, showcasing promising results.
