Ambient Diffusion Posterior Sampling: Solving Inverse Problems with
  Diffusion Models trained on Corrupted Data
Asad Aali Giannis Daras Brett Levac Sidharth Kumar Alexandros G. Dimakis Jonathan I. Tamir
We provide a framework for solving inverse problems with diffusion models
learned from linearly corrupted data. Our method, Ambient Diffusion Posterior
Sampling (A-DPS), leverages a generative model pre-trained on one type of
corruption (e.g. image inpainting) to perform posterior sampling conditioned on
measurements from a potentially different forward process (e.g. image
blurring). We test the efficacy of our approach on standard natural image
datasets (CelebA, FFHQ, and AFHQ) and we show that A-DPS can sometimes
outperform models trained on clean data for several image restoration tasks in
both speed and performance. We further extend the Ambient Diffusion framework
to train MRI models with access only to Fourier subsampled multi-coil MRI
measurements at various acceleration factors (R=2, 4, 6, 8). We again observe
that models trained on highly subsampled data are better priors for solving
inverse problems in the high acceleration regime than models trained on fully
sampled data. We open-source our code and the trained Ambient Diffusion MRI
models: https://github.com/utcsilab/ambient-diffusion-mri .