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Linearly Constrained Diffusion Implicit Models
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Proposes CDIM, a DDIM-based projection method for noisy linear inverse problems that adaptively sets the number and step size of measurement-consistency projections by matching the residual to its chi-squared forward-process distribution.
Problem
Diffusion-based linear inverse solvers (e.g., DPS) need many projection steps with network evaluations to enforce measurement consistency, which is slow under accelerated DDIM schedules and can diverge or fail to meet exact constraints in the noiseless case.
Approach
Align a computable proxy residual ||A x_t - y||^2 with its analytic chi-squared forward-process mean and variance to decide how many projection (gradient) steps to take and their size via line search, so projections stay in the plausible region without extra network evaluations.
Key results
- On FFHQ, CDIM with T'=50 reaches FID 26.09 on box inpainting and 29.68 on Gaussian deblur, with runtime of 6.4 s versus 70.42 s for DPS.
- CDIM with T'=25 runs in about 2.4 s on FFHQ and averages 27.2 total projection steps, versus 10-50x fewer projection steps than DPS-style methods.
- CDIM exactly satisfies noiseless measurement constraints even for out-of-distribution observations where DPS fails.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Increase denoising steps from T'=25 to T'=50 (ablation) | CDIM · FFHQ-1k val 4x super-resolution | FID: 33.87 -> 31.54 | Helped |
| Increase denoising steps from T'=25 to T'=50 (ablation) | CDIM · FFHQ-1k val box inpainting | FID: 27.51 -> 26.09 | Helped |
| Increase denoising steps from T'=25 to T'=50 (ablation) | CDIM · FFHQ-1k val Gaussian deblur | FID: 34.18 -> 29.68 | Helped |
| Increase denoising steps from T'=25 to T'=50 (ablation) | CDIM · FFHQ-1k val random inpainting | FID: 29.67 -> 28.52 | Helped |
| Use stopping threshold μR + 3σR instead of μR + σR (ablation) | CDIM T'=25 · FFHQ noiseless random inpainting | LPIPS: 0.171 -> 0.296 | Hurt |
| Use stopping threshold μR + 0.5σR instead of μR + σR (ablation) | CDIM T'=25 · FFHQ noiseless random inpainting | LPIPS: 0.171 -> 0.169 | Helped |
| Use stopping threshold μR + 3σR instead of μR + σR (ablation) | CDIM T'=25 · FFHQ noiseless random inpainting | total projection steps: 15 -> 24 | Mixed |
| Use stopping threshold μR + 0.5σR instead of μR + σR (ablation) | CDIM T'=25 · FFHQ noiseless random inpainting | total projection steps: 15 -> 25 | Mixed |
| Use CDIM rather than FPS-SMC | FFHQ 4x super-resolution · FFHQ-1k val 4x super-resolution | FID: 26.62 -> 31.54 | Hurt |
| Use CDIM rather than FPS-SMC | FFHQ box inpainting · FFHQ-1k val box inpainting | FID: 26.51 -> 26.09 | Hurt |
| Use CDIM rather than DPS | FFHQ random inpainting · FFHQ-1k val random inpainting | FID: 21.19 -> 28.52 | Hurt |
Surprising findings
- Using DPS with DDIM accelerated sampling yields blurry or divergent results, indicating DDIM acceleration alone does not speed up constrained sampling.
- Stopping criteria with more generous thresholds do not always reduce the number of projection steps.
Limitations
- Restricted to linear measurement operators; cannot handle non-linear constraints per the authors.
- Poisson noise handling relies on a Gaussian approximation of Pearson residuals that breaks down near zero intensities and at extreme noise levels.
- Proposition relies on assumptions (conditionally unbiased denoiser, Hanson-Wright concentration) and the noisy-case moment derivation uses substitutions that are approximate.
Experimental setup
Datasets: FFHQ 256x256 1k validation ImageNet 256x256 1k validation
Benchmarks: FFHQ 256 1k validation ImageNet 256 1k validation
Metrics: FID LPIPS PSNR MAE on observed pixels
Baselines: DPS MCG FPS-SMC DDRM PnP-ADMM Score-SDE ADMM-TV DSG DAPs Diff-PIR
Builds on: DDIM DDPM DPS Tweedie formula
Closest papers by meaning
- A Variational Perspective on Solving Inverse Problems with Diffusion Models (ICLR 2024 · similarity 0.90)
- Improving Diffusion-based Inverse Algorithms under Few-Step Constraint via Linear Extrapolation (NeurIPS 2025 · similarity 0.90)
- SITCOM: Step-wise Triple-Consistent Diffusion Sampling For Inverse Problems (ICML 2025 · similarity 0.89)
- Diffusion State-Guided Projected Gradient for Inverse Problems (ICLR 2025 · similarity 0.89)
- Unleashing the Denoising Capability of Diffusion Prior for Solving Inverse Problems (NeurIPS 2024 · similarity 0.89)
- Improving Diffusion Models for Inverse Problems Using Optimal Posterior Covariance (ICML 2024 · similarity 0.89)
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