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Ambient Diffusion Posterior Sampling: Solving Inverse Problems with Diffusion Models Trained on Corrupted Data
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Extends Ambient Diffusion to train MRI diffusion models from Fourier-subsampled multi-coil k-space and proposes Ambient DPS, a posterior sampler that uses such corrupted-data priors for inverse problems with different forward operators.
Problem
Solving inverse problems with diffusion priors when only linearly corrupted (e.g., Fourier subsampled MRI) training data is available, and whether priors trained on one corruption type can solve inverse problems with a different forward operator.
Approach
Replace the clean denoiser in Diffusion Posterior Sampling with an Ambient denoiser trained on corrupted measurements (with extra Fourier-domain line deletion), and apply it to arbitrary linear inverse problems; the paper also argues that priors trained on higher corruption can outperform clean-data priors at high acceleration.
Key results
- At R=8 inference acceleration, A-DPS (trained at R=2) achieves conditional FID 0.71 versus 4.16 for FS-DPS (trained on fully sampled data).
- At low acceleration (R=2,4), FS-DPS outperforms A-DPS; at high acceleration (R=6,8) A-DPS outperforms FS-DPS on LPIPS/DISTS, e.g., R=8 LPIPS 0.204 for A-DPS vs 0.231 for FS-DPS (R=8 row of LPIPS table).
- On natural images, Ambient models trained with erasure probability p=0.6 outperform clean-data models for compressed sensing at very low measurement counts, though clean-data models win with more measurements.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Train the MRI diffusion prior on subsampled data and use A-DPS instead of clean-data FS-DPS | MRI posterior sampling · Inference acceleration R=8 | Conditional FID: 4.16 -> 0.71 | Helped |
| Train the prior on subsampled MRI data at R=6 or R=8 instead of L1-Wavelet reconstructions | MRI unconditional generation · R=6 and R=8 | FID: 51.43 -> 31.50; 102.98 -> 48.15 | Helped |
| Train the prior on subsampled MRI data instead of clean data | MRI posterior reconstruction · High inference acceleration, R=8 | LPIPS: 0.231 -> 0.150 | Helped |
| Train the prior on subsampled MRI data instead of clean data | MRI posterior reconstruction · Low inference acceleration, R=2 and R=4 | NRMSE, SSIM, and PSNR | Hurt |
| Increase the training-data erasure probability (ablation) | Ambient models on natural images · Compressed sensing at low measurement counts | MSE and LPIPS | Helped |
| Increase the training-data erasure probability (ablation) | Ambient models on natural images · Compressed sensing at high measurement counts | MSE and LPIPS | Hurt |
| Train with corrupted inpainted data rather than clean data (ablation) | Natural-image restoration models · Super-resolution at 2x | MSE and LPIPS | Hurt |
| Train with corrupted inpainted data rather than clean data (ablation) | Natural-image restoration models · More severe downsampling | MSE and LPIPS | Helped |
| Train with higher corruption rather than clean(er) data (ablation) | Natural-image restoration models · Low NFEs, compressed sensing and downsampling | MSE and LPIPS | Helped |
| Train with higher corruption rather than clean(er) data (ablation) | Natural-image restoration models · Higher NFEs, compressed sensing and downsampling | MSE and LPIPS | Hurt |
| Use Ambient Diffusion rather than clean-data EDM for unconditional generation | MRI diffusion models · R=2 and R=4 | FID: 10.41 -> 30.34; 10.41 -> 32.31 | Hurt |
Surprising findings
- Models trained on highly corrupted (subsampled) MRI data can be better priors for solving inverse problems at high acceleration than models trained on fully sampled clean data.
- Ambient models trained on inpainted natural images outperform clean-data models in the very low measurement regime for compressed sensing, with no known theoretical explanation.
Limitations
- No theoretical explanation for why high-corruption-trained priors outperform clean priors on other corruption types.
- Only tested with DPS-style posterior sampling, not other inverse solvers.
- Depends on availability of Ambient Diffusion models, which are fewer than clean-data models.
Experimental setup
Datasets: FastMRI T2-weighted brain multi-coil k-space CelebA FFHQ AFHQ abdominal MRI (undersampled, no ground truth)
Benchmarks: FastMRI brain reconstruction AFHQ CelebA FFHQ
Metrics: NRMSE SSIM PSNR LPIPS DISTS FID conditional FID MSE
Baselines: FS-DPS FS-ALD L1-Wavelet compressed sensing MoDL-Sup L1-DPS SSDU 1D-Partitioned SSDU A-ALD A-OS
Builds on: Ambient Diffusion Diffusion Posterior Sampling EDM Noisier2Noise SSDU
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