Solving Inverse Problems via Diffusion-Based Priors: An Approximation-Free Ensemble Sampling Approach

Haoxuan Chen; Yinuo Ren; Martin Renqiang Min; Lexing Ying; Zachary Izzo

cs.LGJun 4, 2025

Solving Inverse Problems via Diffusion-Based Priors: An Approximation-Free Ensemble Sampling Approach

Authors: Haoxuan Chen, Yinuo Ren, Martin Renqiang Min, Lexing Ying, Zachary Izzo

Organizations: ICME, Stanford University, Stanford, CA 94305 · Machine Learning Department, NEC Labs America, Princeton, NJ 08540 · Department of Mathematics and ICME, Stanford University, Stanford, CA 94305

Abstract

Diffusion models (DMs) have proven to be effective in modeling high-dimensional distributions, leading to their widespread adoption for representing complex priors in Bayesian inverse problems (BIPs). However, current DM-based posterior sampling methods proposed for solving common BIPs rely on heuristic approximations to the generative process. To exploit the generative capability of DMs and avoid the usage of such approximations, we propose an ensemble-based algorithm that performs posterior sampling without the use of heuristic approximations. Our algorithm is motivated by existing work that combines DM-based methods with the sequential Monte Carlo (SMC) method. By examining how the prior evolves through the diffusion process encoded by the pre-trained score function, we derive a modified partial differential equation (PDE) governing the evolution of the corresponding posterior distribution. This PDE includes a modified diffusion term and a reweighting term, which can be simulated via stochastic weighted particle methods. Theoretically, we prove that the error between the true posterior and the empirical distribution of the generated samples can be bounded in terms of the training error of the pre-trained score function and the number of particles in the ensemble. Empirically, we validate our algorithm on several inverse problems in imaging to show that our method gives more accurate reconstructions compared to existing DM-based methods. Our code is available at the following Github repository~https://github.com/HaoxuanSteveC00/AFDPS-TMLR.

Figures & tables

Panel of Figure 1 — Figure 1: A roadmap for our posterior sampling method. I, II refers to the two stages of the proposed algorithm.

Panel of Figure 2 — Figure 2: Visualization of posterior samples by AFDPS on linear inverse problems. Upper: Original; Middle: Noisy (Measurement); Lower: Reconstructed.


Method	Gaussian Deblurring		Motion Deblurring		Super Resolution		Box Inpainting
PSNR ( $\uparrow$ )	LPIPS ( $\downarrow$ )	PSNR ( $\uparrow$ )	LPIPS ( $\downarrow$ )	PSNR ( $\uparrow$ )	LPIPS ( $\downarrow$ )	PSNR ( $\uparrow$ )	LPIPS ( $\downarrow$ )

DPS ( Chung et al., 2022 )	22.57	0.2976	21.00	0.3280	19.09	0.5627	21.57	0.3245
DCDP ( Li et al., 2024b )	24.77	0.2868	21.57	0.3487	21.23	0.5139	22.05	0.4525
SGS-EDM ( Wu et al., 2024c )	24.78	0.2776	23.45	0.3009	22.41	0.3225	23.69	0.2301

Table 1: Results on 4 linear inverse problems for 100 validation images from FFHQ-256.


Method	Gaussian Deblurring		Motion Deblurring		Super Resolution		Box Inpainting
PSNR ( $\uparrow$ )	LPIPS ( $\downarrow$ )	PSNR ( $\uparrow$ )	LPIPS ( $\downarrow$ )	PSNR ( $\uparrow$ )	LPIPS ( $\downarrow$ )	PSNR ( $\uparrow$ )	LPIPS ( $\downarrow$ )

DPS ( Chung et al., 2022 )	20.60	0.4351	20.46	0.5328	19.17	0.4940	22.70	0.3765
DCDP ( Li et al., 2024b )	22.34	0.4821	20.59	0.5338	20.26	0.5597	21.67	0.4344
SGS-EDM ( Wu et al., 2024c )	19.31	0.4807	20.54	0.4653	19.61	0.4986	21.42	0.4643

Table 2: Results on 4 linear inverse problems for 100 validation images from ImageNet-256.

Appendix figures & tables14 assets

Supplementary material from the paper’s appendix.

Appendix


Term	AFDPS (Ours)	FK Corrector

Drift	$-F(t){\bm{x}}+V(t)^{2}{\bm{\phi}}_{\theta}({\bm{x}},t)$ $-\alpha_{t}V(t)^{2}\nabla_{{\bm{x}}}\mu_{{\bm{y}}}$	$-F(t){\bm{x}}+V(t)^{2}{\bm{\phi}}_{\theta}({\bm{x}},t)$
Reweighting	$\frac{1}{2}V(t)^{2}\left(\alpha^{2}_{t}\\|\nabla_{{\bm{x}}}\mu_{{\bm{y}}}\\|_{2}^{2}-\alpha_{t}\Delta_{{\bm{x}}}\mu_{{\bm{y}}}\right)$ $+\alpha_{t}\left(F(t){\bm{x}}-V(t)^{2}{\bm{\phi}}_{\theta}({\bm{x}},t)\right)^{\intercal}\nabla_{{\bm{x}}}\mu_{{\bm{y}}}-\alpha^{\prime}_{t}\mu_{{\bm{y}}}$	$-\frac{1}{2}V(t)^{2}\left(\alpha^{2}_{t}\\|\nabla_{{\bm{x}}}\mu_{{\bm{y}}}\\|_{2}^{2}-\alpha_{t}\Delta_{{\bm{x}}}\mu_{{\bm{y}}}\right)$ $+\alpha_{t}F(t){\bm{x}}^{\intercal}\nabla_{{\bm{x}}}\mu_{{\bm{y}}}-\alpha^{\prime}_{t}\mu_{{\bm{y}}}$

Appendix

Table 3: Drift and Reweighting Terms of AFDPS and FK Corrector

Panel of Figure 3 — Figure 3: Additional visual examples for the Gaussian deblurring problem on FFHQ and ImageNet.

Panel of Figure 4 — Figure 4: Additional visual examples for the motion deblurring problem on FFHQ and ImageNet.

Panel of Figure 5 — Figure 5: Additional visual examples for the super-resolution problem on FFHQ and ImageNet.

Panel of Figure 6 — Figure 6: Additional visual examples for the box inpainting problem on FFHQ and ImageNet.

Figure 7: Convergence of AFDPS towards the ground-truth posterior versus number of particles.

Panel of Figure 8 — Figure 8: Comparison between AFDPS and SGS-EDM ( Wu et al., 2024c ) on Fourier Phase Retrieval.


Method	Gaussian Deblurring		Motion Deblurring
PSNR ( $\uparrow$ )	LPIPS ( $\downarrow$ )	PSNR ( $\uparrow$ )	LPIPS ( $\downarrow$ )

DAPS (100 images)	22.10	0.4580	23.02	0.4661

AFDPS-SDE (100 images)	25.19	0.1291	24.07	0.1530

Appendix

Table 4: Results of the comparison between DAPS and AFDPS on two deblurring problems.

Panel of Figure 9 — Figure 9: Additional visual examples for the Gaussian deblurring problem on 1000 images from FFHQ

Panel of Figure 10 — Figure 10: Additional visual examples for the motion deblurring problem on 1000 images from FFHQ

Panel of Figure 11 — Figure 11: Additional visual examples for the super-resolution problem on 1000 images from FFHQ

Figure 12: Plot of ESS with respect to time on the Gaussian deblurring task

Figure 13: Tuning the number of particles in AFDPS-SDE on the Gaussian deblurring task

Figure 14: Tuning the stepsize of the corrector in AFDPS-ODE on the Gaussian deblurring task

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Provable diffusion-based posterior sampling for linear inverse problems via DDIM

Yuchen Jiao, Na Li, Changxiao Cai, Yuxin Chen+1

Department of Statistics and Data Science, Chinese University of Hong Kong, Hong Kong · Department of Industrial and Operations Engineering, University of Michigan, Ann Arbor, USA · Department of Statistics and Data Science, the Wharton School, University of Pennsylvania, PA, USA

Diffusion Models for Image Restoration Bayesian Inverse Problems Diffusion Sampling Diffusion-Based Inverse Problems Diffusion Posterior Sampling Inverse Problems

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Diffusion-based methods have achieved remarkable empirical success in solving inverse problems. However, many existing posterior samplers either lack rigorous theoretical guarantees or incur substantial computational overhead. We propose a simple and efficient algorithm, called \pddim, for solving linear inverse problems with diffusion priors via a DDIM-type sampler. Our method requires only lightweight, coordinate-wise modifications to the standard DDIM update, while explicitly incorporating the measurement model. The key idea is to perform posterior sampling separately along each singular direction of the measurement operator: for each direction, the sampler follows the learned diffusion prior when the observation signal-to-noise ratio (SNR) is below the corresponding diffusion SNR, and switches to a calibrated measurement-based predictor otherwise. We prove that the proposed sampler converges to the Bayesian posterior conditioned on the measurements. Empirical results show that the proposed sampler performs favorably against existing diffusion-based posterior samplers across a range of image restoration tasks, achieving the best performance on the majority of evaluation metrics considered. Overall, our results convert posterior sampling for noisy linear inverse problems to simple coordinate-wise DDIM updates, yielding an efficient, easy-to-implement algorithm with provable posterior consistency.

Abstract

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