Research map / Probabilistic and scientific modeling / Variational and generative sampling
Flow Priors for Linear Inverse Problems via Iterative Corrupted Trajectory Matching
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Proposes ICTM, an iterative algorithm that approximates the MAP estimate under a pretrained flow-matching prior for linear inverse problems by decomposing the flow MAP objective into sequential local MAP objectives optimized with Tweedie-based gradients.
Problem
Computing flow-model likelihoods for MAP estimation requires backpropagating through an ODE solver, which is prohibitively slow for high-dimensional image inverse problems.
Approach
Approximate the flow-based MAP objective by a weighted sum of N local MAP objectives along the trajectory (with an auxiliary path between measurements and A(x0)), and optimize them sequentially using gradients of the log prior obtained via Tweedie's formula, avoiding full ODE backpropagation.
Key results
- On CelebA-HQ, ICTM reports PSNR 30.65 / SSIM 0.894 on random inpainting and 27.91 PSNR on 4x super-resolution, surpassing the listed baselines in most tasks.
- On HCP T2w compressed sensing at compression rate 1/4, ICTM reaches 27.03 PSNR / 0.733 SSIM versus 25.01 PSNR for DPS-ODE.
- ICTM takes about 2.72 h to recover 100 CelebA-HQ images versus 4.10 h for OT-ODE and 0.36 h for DPS-ODE.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use ICTM instead of ΠGDM | CelebA-HQ super-resolution · CelebA-HQ super-resolution | SSIM: 0.803 -> 0.805 | Helped |
| Use ICTM instead of ΠGDM | CelebA-HQ super-resolution · CelebA-HQ super-resolution | PSNR: 28.33 -> 27.91 | Hurt |
| Use ICTM instead of ΠGDM | CelebA-HQ random inpainting · CelebA-HQ random inpainting | PSNR: 29.98 -> 30.65 | Helped |
| Use ICTM instead of DPS-ODE | CelebA-HQ random inpainting · CelebA-HQ random inpainting | SSIM: 0.872 -> 0.894 | Helped |
| Use ICTM instead of RED-Diff | CelebA-HQ Gaussian deblurring · CelebA-HQ Gaussian deblurring | PSNR: 27.23 -> 26.54 | Hurt |
| Use ICTM instead of RED-Diff | CelebA-HQ Gaussian deblurring · CelebA-HQ Gaussian deblurring | SSIM: 0.765 -> 0.760 | Hurt |
| Use ICTM instead of ΠGDM | CelebA-HQ box inpainting · CelebA-HQ box inpainting | PSNR: 24.10 -> 24.34 | Helped |
| Use ICTM instead of ΠGDM | CelebA-HQ box inpainting · CelebA-HQ box inpainting | SSIM: 0.853 -> 0.866 | Helped |
| Use ICTM instead of DPS-ODE | HCP T2w compressed sensing · HCP T2w compressed sensing ν=1/4 | PSNR: 25.01 $\pm$ 1.87 -> 27.03 $\pm$ 1.77 | Helped |
| Use ICTM instead of DPS-ODE | HCP T2w compressed sensing · HCP T2w compressed sensing ν=1/4 | SSIM: 0.608 $\pm$ 0.08 -> 0.733 $\pm$ 0.04 | Helped |
| Use ICTM instead of DPS-ODE | HCP T2w compressed sensing · HCP T2w compressed sensing ν=1/10 | PSNR: 22.06 $\pm$ 1.66 -> 24.03 $\pm$ 1.23 | Helped |
| Use ICTM instead of DPS-ODE | HCP T2w compressed sensing · HCP T2w compressed sensing ν=1/10 | SSIM: 0.479 $\pm$ 0.09 -> 0.503 $\pm$ 0.04 | Helped |
Surprising findings
- The local prior term gives only modest gains; for super-resolution K=1 iteration is best, while compressed sensing needs K=10.
- Ours without prior underperforms DPS-ODE and several baselines on super-resolution (26.06 vs 27.85 PSNR), showing the prior term matters for some tasks.
Limitations
- Theory relies on optimal transport (straight) interpolation paths and exact trajectory compliance, which learned flows do not strictly satisfy.
- Only handles linear forward models; nonlinear inverse problems are not addressed.
- No uncertainty quantification; yields point estimates only.
Experimental setup
Datasets: CelebA-HQ HCP T2w FFHQ
Benchmarks: CelebA-HQ HCP T2w
Metrics: PSNR SSIM
Baselines: OT-ODE DPS-ODE RED-Diff PiGDM Wavelet prior TV prior D-Flow
Builds on: flow matching rectified flow diffusion posterior sampling OT-ODE
Closest papers by meaning
- Solving Inverse Problems with FLAIR (NeurIPS 2025 · similarity 0.91)
- Latent Refinement via Flow Matching for Training-free Linear Inverse Problem Solving (NeurIPS 2025 · similarity 0.91)
- Fast samplers for Inverse Problems in Iterative Refinement models (NeurIPS 2024 · similarity 0.89)
- PnP-Flow: Plug-and-Play Image Restoration with Flow Matching (ICLR 2025 · similarity 0.88)
- Beckmann Transport Models: From Autonomous Flows to One-Step Maps (NeurIPS 2026 · similarity 0.88)
- MAP Estimation with Denoisers: Convergence Rates and Guarantees (NeurIPS 2025 · similarity 0.87)
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