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Solving Inverse Problems via Diffusion Optimal Control
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Frames diffusion-based inverse problem solving as discrete-time optimal control and derives an iLQR-inspired controller (diffusion optimal control) that works with any differentiable forward operator using pretrained unconditional diffusion models.
Problem
Diffusion-based inverse problem solvers rely on posterior sampling that needs an intractable conditional likelihood score, approximations of it, and accurate score networks, and degrade with fewer discretization steps.
Approach
Treat the discretized reverse diffusion process as a controlled dynamical system and solve for controls with an iLQR-style optimal controller using the terminal data-consistency cost -log p(y|x0); the conditional score emerges as the control, and posterior sampling is recovered as a special case. Adds randomized low-rank, matrix-free, and Adam-preconditioned updates to make high-dimensional iLQR t
Key results
- On FFHQ 256x256 4x super-resolution, the method (NFE=2500) reports FID 32.47 and LPIPS 0.171 versus DPS FID 39.35 and LPIPS 0.214 (NFE=1000).
- Ours (T=50) reaches PSNR 27.45 on 4x super-resolution and 31.84 on random inpainting, with the best PSNR among listed methods on most tasks except SSIM-leading DPS/PNP-ADMM rows.
- Performance is reported as stable across discretization steps T, with the method producing feasible solutions even for a randomly initialized diffusion model where DPS fails.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use diffusion optimal control rather than DPS | FFHQ inverse-problem reconstruction · FFHQ 256×256-1K, SR ×4 | FID: 39.35 -> 32.47 | Helped |
| Use diffusion optimal control rather than DPS | FFHQ inverse-problem reconstruction · FFHQ 256×256-1K, Random Inpainting | LPIPS: 0.168 -> 0.053 | Helped |
| Use diffusion optimal control rather than DPS | FFHQ inverse-problem reconstruction · FFHQ 256×256-1K, Box Inpainting | FID: 21.19 -> 20.22 | Helped |
| Use diffusion optimal control rather than DPS | FFHQ inverse-problem reconstruction · FFHQ 256×256-1K, Gaussian Deblurring | LPIPS: 0.257 -> 0.189 | Helped |
| Use diffusion optimal control rather than DPS | FFHQ inverse-problem reconstruction · FFHQ 256×256-1K, Motion Deblurring | FID: 39.92 -> 39.40 | Helped |
| Increase low-rank/matrix-free approximation rank from k=0 to k=1 (ablation) | Diffusion Optimal Control · FFHQ 256×256-1K, SR ×4 | LPIPS: 0.254 -> 0.171 | Helped |
| Increase low-rank/matrix-free approximation rank from k=0 to k=1 (ablation) | Diffusion Optimal Control · FFHQ 256×256-1K, Random Inpainting | LPIPS: 0.121 -> 0.053 | Helped |
| Increase rank from k=1 to k=16 (ablation) | Diffusion Optimal Control · FFHQ 256×256-1K, SR ×4 | LPIPS: 0.171 -> 0.170 | No clear effect |
| Increase rank from k=1 to k=16 (ablation) | Diffusion Optimal Control · FFHQ 256×256-1K, Random Inpainting | LPIPS: 0.053 -> 0.050 | No clear effect |
| Set Tikhonov regularization α=1e-7 instead of α=1e-4 (ablation) | Diffusion Optimal Control · FFHQ 256×256-1K, SR ×4 | LPIPS: 0.171 -> 0.173 | Hurt |
| Set Tikhonov regularization α=1 instead of α=1e-4 (ablation) | Diffusion Optimal Control · FFHQ 256×256-1K, SR ×4 | LPIPS: 0.171 -> 0.172 | No clear effect |
| Set α from Lemma instead of α=1e-4 (ablation) | Diffusion Optimal Control · FFHQ 256×256-1K, SR ×4 | LPIPS: 0.171 -> 0.170 | No clear effect |
Surprising findings
- The method produces reasonable solutions even with a randomly initialized diffusion model, where DPS cannot produce feasible solutions.
- Solution quality of the proposed method improves with more T in the ablation, with a large gain in FID/LPIPS at T=200 versus T=50 for super-resolution.
Limitations
- Higher runtime cost than probabilistic samplers, requiring equivalent-budget comparisons.
- Relies on approximations (low-rank, matrix-free, Adam preconditioning) for high-dimensional iLQR, which introduce deviation from exact iLQR.
- Theoretical recovery of posterior sampling holds only under assumptions (deterministic sampler, log p(y|x_t)=log p(y|x_0)).
Experimental setup
Datasets: FFHQ 256x256 MNIST
Benchmarks: FFHQ 256x256 1K held-out MNIST class-conditional inverse classification
Metrics: FID LPIPS PSNR SSIM MSE
Baselines: DPS DDRM MCG Score-SDE PSLD Flash-Diffusion DDNM PnP-ADMM ADMM-TV
Builds on: DPS (diffusion posterior sampling) iLQR Tweedie's formula DDRM randomized SVD
Closest papers by meaning
- Integrating Intermediate Layer Optimization and Projected Gradient Descent for Solving Inverse Problems with Diffusion Models (ICML 2025 · similarity 0.88)
- Self-diffusion for Solving Inverse Problems (NeurIPS 2025 · similarity 0.88)
- A Variational Perspective on Solving Inverse Problems with Diffusion Models (ICLR 2024 · similarity 0.88)
- InvFusion: Bridging Supervised and Zero-shot Diffusion for Inverse Problems (NeurIPS 2025 · similarity 0.88)
- Improving Diffusion Inverse Problem Solving with Decoupled Noise Annealing (CVPR 2025 · similarity 0.88)
- A Diffusion Model with State Estimation for Degradation-Blind Inverse Imaging (AAAI 2024 · similarity 0.88)
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