Sparse Scheduled Diffusion Guidance for Inverse Problems

Abduragim Shtanchaev; Albina Ilina; Yazid Janati; Arip Asadulaev; Martin Takac; Eric Moulines

cs.LGMar 9, 2026

Sparse Scheduled Diffusion Guidance for Inverse Problems

Authors: Abduragim Shtanchaev, Albina Ilina, Yazid Janati, Arip Asadulaev, Martin Takac, Eric Moulines

Organizations: MBZUAI, Abu Dhabi, United Arab Emirates · Institute of Foundation Models · EPITA, Paris, France

Abstract

Pretrained diffusion models are effective priors for Bayesian inverse problems, but posterior sampling with these priors is often costly because data-consistency guidance is applied throughout the full reverse trajectory. Existing methods have shown that vector-Jacobian products through the denoiser can sometimes be avoided, yet they typically still rely on dense guidance through the full trajectory or expensive inner solves. We introduce Sparse Scheduled Diffusion Guidance for Inverse Problems (Spin), a solver that avoids starting posterior sampling from pure noise. Spin first samples from a posterior time-marginal at an intermediate timestep $t_*$ , and then uses that state as a warm start for a guided reverse diffusion process. At guidance time, instead of enforcing the measurement constraint at every denoising step, Spin applies lightweight corrections only at scheduled timesteps where the denoiser can still clean up artifacts. The resulting procedure decouples prior refinement from data consistency: the prior supplies denoising, while lightweight pixel-space optimization enforces the measurement constraint without backpropagation through the denoiser or decoder. Across linear and nonlinear inverse problems on FFHQ and ImageNet, Spin achieves competitive reconstruction quality with a substantially better runtime--memory profile, running 2x faster on pixel-space models and up to 50x faster on latent diffusion models, with lower memory costs.

Figures & tables

Figure 1: Qualitative results on FFHQ and ImageNet. Comparison with baseline methods across Gaussian deblurring and center inpainting on FFHQ ( top and middle rows, respectively ), and motion deblurring on ImageNet ( bottom row ). Spin (ours) achieves more accurate and detailed restoration compared to existing approaches. See Appendix J / L for more samples.

Figure 2: Mean trajectories of posterior samplers on a 2D inverse problem, averaged over $300$ independent runs. Spin first builds a warm start near the data-consistent region, then uses its scheduler to take shorter guided steps toward the posterior mode; the baselines instead follow longer diffusion trajectories from Gaussian noise, leading to more variable paths.

Method	Inference Objective	Guidance Density	VJP $\hat{\mathbf{x}}^{\theta}_{0}(\cdot,\cdot)$ ?	VJP $\mathcal{E}$ / $\mathcal{D}$ ?	Assumptions on $\mathcal{A}$
DPS , PGDM	Score-based posterior approximation	Dense	Yes	Yes †	General / known $h^{\dagger}$
DAPS	Decoupled posterior annealing	Moderate + inner MCMC	No	Often †	General
RED-Diff , DDS	Variational / Krylov data consistency	Dense / iterative solve	No	No	General / linear
DiffPIR , PnP-DM	Plug-and-play / proximal posterior	Dense + inner loops	No	No	General
PSLD , ReSample , P2L , TReg , LATINO-PRO	Latent/text/consistency-model inverse solvers	Moderate–dense + inner solve	Varies	Varies †	Varies
FlowDPS , PnP-Flow , FLAIR	Flow / rectified-flow prior guidance	Dense / iterative solve	Varies	Often †	General / linear

Table 1: Comparison of training-free diffusion/consistency/flow inverse-problem solvers. VJP: backprop through denoiser

\hat{\mathbf{x}}_{0}(\cdot,\cdot)

and encoder/decoder

\mathcal{E}

\mathcal{D}

( † latent models only). DC Guidance Density: frequency of data-consistency or likelihood guidance application.

	$\downarrow$ LPIPS FFHQ
Task	Spin	DAPS	Red-Diff	PNP-DM	DPS	DDNM	PGDM	Diffpir
Gaussian Deblur	0.17 $\pm\,0.06$	0.19 $\pm\,0.06$	0.25 $\pm\,0.07$	0.20 $\pm\,0.07$	0.16 $\pm\,0.05$	0.84 $\pm\,0.04$	0.87 $\pm\,0.14$	–
Motion Deblur	0.13 $\pm\,0.04$	0.19 $\pm\,0.06$	0.20 $\pm\,0.06$	0.21 $\pm\,0.06$	0.21 $\pm\,0.06$	–	–	–
SR ( $\times$ 4)	0.20 $\pm\,0.07$	0.19 $\pm\,0.06$	0.44 $\pm\,0.08$	0.21 $\pm\,0.06$	0.22 $\pm\,0.07$	0.36 $\pm\,0.08$	0.30 $\pm\,0.07$	–
SR ( $\times$ 16)	0.35 $\pm\,0.08$	0.45 $\pm\,0.10$	0.59 $\pm\,0.08$	0.60 $\pm\,0.15$	0.36 $\pm\,0.08$	0.52 $\pm\,0.09$	0.42 $\pm\,0.06$	–
Box Inpainting	0.14 $\pm\,0.04$	0.12 $\pm\,0.04$	0.18 $\pm\,0.05$	0.55 $\pm\,0.07$	0.20 $\pm\,0.08$	0.18 $\pm\,0.05$	0.17 $\pm\,0.04$	0.14 $\pm\,0.04$

Table 2: Mean LPIPS for linear/nonlinear imaging tasks on the FFHQ datasets with

\sigma_{y}=0.05

. Lower metrics are better. HDR denotes High Dynamic Range. Blue factors are relative to Spin ; larger values mean greater time or memory savings.

	$\downarrow$ LPIPS ImageNet
Task	Spin	DAPS	Red-Diff	PNP-DM	DPS	DDNM	PGDM	Diffpir
Gaussian Deblur	0.34 $\pm\,0.10$	0.41 $\pm\,0.12$	0.44 $\pm\,0.11$	0.47 $\pm\,0.13$	0.35 $\pm\,0.14$	0.42 $\pm\,0.13$	1.06 $\pm\,0.10$	–
Motion Deblur	0.28 $\pm\,0.09$	0.40 $\pm\,0.11$	0.35 $\pm\,0.09$	0.50 $\pm\,0.13$	0.41 $\pm\,0.13$	–	–	–
SR ( $\times$ 4)	0.39 $\pm\,0.11$	0.41 $\pm\,0.12$	0.53 $\pm\,0.13$	0.49 $\pm\,0.14$	0.47 $\pm\,0.14$	0.27 $\pm\,0.09$	0.63 $\pm\,0.12$	–
SR ( $\times$ 16)	0.68 $\pm\,0.13$	0.80 $\pm\,0.13$	0.82 $\pm\,0.11$	0.88 $\pm\,0.10$	0.58 $\pm\,0.10$	0.68 $\pm\,0.17$	0.59 $\pm\,0.08$	–
Box Inpainting	0.32 $\pm\,0.05$	0.31 $\pm\,0.05$	0.36 $\pm\,0.06$	0.71 $\pm\,0.14$	0.46 $\pm\,0.13$	0.28 $\pm\,0.06$	0.28 $\pm\,0.05$	0.28 $\pm\,0.05$

Table 3: Mean LPIPS for linear/nonlinear imaging tasks on the ImageNet datasets with

\sigma_{y}=0.05

. Lower metrics are better. HDR denotes High Dynamic Range. Blue factors are relative to Spin ; larger values mean greater time or memory savings.

Figure 3: Impact of the initial timestep $t_{*}$ on Spin . The first row shows reconstruction quality for different $t_{*}$ values, while the second row shows the corresponding initial guesses. The plots report how quality metrics and runtime vary with $t_{*}$ .

Table 8

Appendix figures & tables35 assets

Supplementary material from the paper’s appendix.

Appendix

Method	Target	Kernel / DC step	VJP den.?	VJP enc/dec?	Notes
DPS , PGDM
[ 3 , 33 ]	Posterior via DPS-style score approximation	Reverse diffusion + per-step likelihood guidance	Yes	Yes †	General $\mathcal{A}$ ; PGDM uses known/generalized $h^{\dagger}$ ; high memory/NFE
DAPS , RED-Diff
[ 40 , 22 ]	Decoupled / variational posterior	Annealed MCMC / variational update without denoiser VJPs	No	Often †	Inner sampling or optimization; DAPS latent variants may use encoder/decoder gradients
DDNM , DDRM
[ 38 , 17 ]	Linear posterior / projection	Denoise + pseudo-inverse / SVD projection (DC)	No	No	Needs linear $\mathcal{A}$ ; DDNM uses pseudo-inverse, DDRM uses SVD

Appendix

Table 7: Comparison of training-free diffusion/consistency-model inverse-problem solvers. “VJP” indicates backprop through the denoiser or encoder/decoder (latent diffusion models only).

Strategy	Weight Function $w_{i}$	Parameters
Uniform	$1$	—
Linear	$i+1$	—
Polynomial	$(i+1)^{p}$	$p>1$
Exponential	$\rho^{i}$	$\rho>1$
Gaussian	$\exp\!\left(\dfrac{(i-\mu\,M)^{2}}{2\sigma^{2}}\right)$	$\mu\in[0,1],\,\sigma>0$
Beta	$\mathrm{Beta}(x_{i};\,a,b)^{-1}$ , $x_{i}=\dfrac{i-\frac{1}{2}}{M}$	$a,b>0$

Appendix

Table 8: Weight functions for different scheduling strategies. Weights are computed for

i\in\{1,\ldots,M\}

and used to derive the timestep grid via the construction above.

Schedule	Deblur	Motion Deblur	SR4
Uniform	0.772 $\pm\,0.057$	0.768 $\pm\,0.057$	0.772 $\pm\,0.056$
Linear	0.692 $\pm\,0.068$	0.673 $\pm\,0.076$	0.691 $\pm\,0.073$
Polynomial	0.818 $\pm\,0.042$	0.754 $\pm\,0.041$	0.785 $\pm\,0.036$
Exponential	0.823 $\pm\,0.045$	0.808 $\pm\,0.040$	0.805 $\pm\,0.036$
Beta	0.759 $\pm\,0.034$	0.523 $\pm\,0.046$	0.717 $\pm\,0.032$
Gaussian	0.825 $\pm\,0.045$	0.824 $\pm\,0.041$	0.827 $\pm\,0.038$

Appendix

Table 9: SSIM scores for different noise schedules across tasks

Schedule	Deblur	Motion Deblur	SR4
Uniform	26.42 $\pm\,1.71$	26.24 $\pm\,1.72$	26.38 $\pm\,1.69$
Linear	23.63 $\pm\,1.58$	23.04 $\pm\,1.82$	23.48 $\pm\,1.68$
Polynomial	28.78 $\pm\,1.91$	27.93 $\pm\,1.64$	28.18 $\pm\,1.60$
Exponential	28.71 $\pm\,2.04$	28.52 $\pm\,1.85$	28.40 $\pm\,1.71$
Beta	27.84 $\pm\,1.58$	24.70 $\pm\,0.92$	27.19 $\pm\,1.31$
Gaussian	28.72 $\pm\,2.00$	28.57 $\pm\,2.00$	28.66 $\pm\,1.92$

Appendix

Table 10: PSNR scores for different noise schedules across tasks

Figure 4: Reproduction of Figure 2 for ease of reference. Mean trajectories of posterior samplers on the 2D inverse problem described in this section. Filled background: exact GMM prior density. Gray contours: relative posterior density. Dashed line: data-consistency subspace $\bm{A}\mathbf{x}=\mathbf{y}$ . Star: ground truth $\mathbf{x}^{\star}$ . Empty triangles ( $\triangle$ ) and filled circles ( $\bullet$ ) mark each sampler’s trajectory start and endpoint, respectively.

Sampler	Mean dist. to truth $\downarrow$	Mean data loss $\downarrow$
Spin (ours)	0.1431	2.79e-05
pgdm	0.3041	0.0184
daps	0.3108	0.1391
dps	0.3498	0.1260
ddnm	0.3819	0.0870

Appendix

Table 11: Mean endpoint metrics across

300

paired seeds for the 2D landscape problem of Figure 2 .

	SSIM FFHQ
Task	Spin	DAPS	Red-Diff	PNP-DM	DPS	DDNM	PGDM	Diffpir
Gaussian Deblur	0.83 $\pm\,0.04$	0.81 $\pm\,0.05$	0.81 $\pm\,0.04$	0.77 $\pm\,0.06$	0.72 $\pm\,0.07$	0.03 $\pm\,0.01$	0.14 $\pm\,0.09$	–
Motion Deblur	0.82 $\pm\,0.04$	0.79 $\pm\,0.05$	0.74 $\pm\,0.03$	0.75 $\pm\,0.06$	0.65 $\pm\,0.07$	–	–	–
SR ( $\times$ 4)	0.79 $\pm\,0.04$	0.80 $\pm\,0.05$	0.63 $\pm\,0.03$	0.75 $\pm\,0.05$	0.67 $\pm\,0.08$	0.69 $\pm\,0.03$	0.56 $\pm\,0.03$	–
SR ( $\times$ 16)	0.63 $\pm\,0.07$	0.55 $\pm\,0.08$	0.59 $\pm\,0.06$	0.50 $\pm\,0.07$	0.50 $\pm\,0.08$	0.62 $\pm\,0.07$	0.42 $\pm\,0.06$	–
Box Inpainting	0.76 $\pm\,0.04$	0.80 $\pm\,0.03$	0.70 $\pm\,0.03$	0.55 $\pm\,0.03$	0.72 $\pm\,0.06$	0.73 $\pm\,0.03$	0.70 $\pm\,0.03$	0.82 $\pm\,0.03$

Appendix

Table 12: Quantitative comparison of different methods across various inverse problems. Results are reported as SSIM

\pm

standard deviation. Higher is better. Blue factors are relative to Spin ; larger values mean greater time or memory savings.

	PSNR FFHQ
Task	Spin	DAPS	Red-Diff	PNP-DM	DPS	DDNM	PGDM	Diffpir
Gaussian Deblur	29.0 $\pm\,1.80$	28.2 $\pm\,2.08$	28.7 $\pm\,1.82$	25.2 $\pm\,2.76$	24.7 $\pm\,2.07$	7.8 $\pm\,0.10$	13.2 $\pm\,0.70$	–
Motion Deblur	28.1 $\pm\,1.76$	27.0 $\pm\,1.86$	27.8 $\pm\,1.22$	24.7 $\pm\,2.33$	22.2 $\pm\,1.83$	–	–	–
SR ( $\times$ 4)	28.2 $\pm\,1.62$	27.5 $\pm\,1.76$	26.1 $\pm\,0.92$	24.7 $\pm\,2.21$	22.8 $\pm\,2.05$	26.9 $\pm\,1.17$	24.6 $\pm\,1.22$	–
SR ( $\times$ 16)	21.6 $\pm\,1.69$	17.8 $\pm\,1.48$	21.2 $\pm\,1.44$	16.3 $\pm\,1.10$	17.9 $\pm\,1.65$	21.6 $\pm\,1.67$	18.4 $\pm\,1.23$	–
Box Inpainting	21.7 $\pm\,2.65$	22.4 $\pm\,2.90$	21.6 $\pm\,2.60$	12.5 $\pm\,0.74$	20.9 $\pm\,2.41$	22.5 $\pm\,2.78$	21.1 $\pm\,2.37$	22.6 $\pm\,3.42$

Appendix

Table 13: Quantitative comparison of different methods across various inverse problems. Results are reported as PSNR

\pm

standard deviation. Higher is better. Blue factors are relative to Spin ; larger values mean greater time or memory savings.

	SSIM ImageNet
Task	Spin	DAPS	Red-Diff	PNP-DM	DPS	DDNM	PGDM	Diffpir
Gaussian Deblur	0.77 $\pm\,0.07$	0.68 $\pm\,0.13$	0.69 $\pm\,0.10$	0.63 $\pm\,0.14$	0.57 $\pm\,0.17$	0.60 $\pm\,0.15$	0.07 $\pm\,0.02$	–
Motion Deblur	0.72 $\pm\,0.05$	0.66 $\pm\,0.13$	0.65 $\pm\,0.05$	0.60 $\pm\,0.14$	0.49 $\pm\,0.17$	–	–	–
SR ( $\times$ 4)	0.71 $\pm\,0.05$	0.67 $\pm\,0.13$	0.59 $\pm\,0.05$	0.60 $\pm\,0.15$	0.51 $\pm\,0.17$	0.74 $\pm\,0.10$	0.27 $\pm\,0.05$	–
SR ( $\times$ 16)	0.58 $\pm\,0.14$	0.43 $\pm\,0.15$	0.45 $\pm\,0.13$	0.40 $\pm\,0.14$	0.34 $\pm\,0.16$	0.50 $\pm\,0.16$	0.21 $\pm\,0.09$	–
Box Inpainting	0.72 $\pm\,0.07$	0.73 $\pm\,0.05$	0.36 $\pm\,0.06$	0.51 $\pm\,0.04$	0.58 $\pm\,0.15$	0.76 $\pm\,0.05$	0.61 $\pm\,0.02$	0.78 $\pm\,0.04$

Appendix

Table 14: Quantitative comparison of different methods across various inverse problems. Results are reported as SSIM

\pm

standard deviation. Higher is better. Blue factors are relative to Spin ; larger values mean greater time or memory savings.

	PSNR ImageNet
Task	Spin	DAPS	Red-Diff	PNP-DM	DPS	DDNM	PGDM	Diffpir
Gaussian Deblur	27.3 $\pm\,2.57$	25.4 $\pm\,3.15$	25.8 $\pm\,2.99$	24.5 $\pm\,2.83$	22.6 $\pm\,3.06$	23.7 $\pm\,3.26$	9.7 $\pm\,0.64$	–
Motion Deblur	26.9 $\pm\,2.20$	24.7 $\pm\,2.97$	25.5 $\pm\,1.97$	23.4 $\pm\,2.68$	20.4 $\pm\,2.65$	–	–	–
SR ( $\times$ 4)	26.6 $\pm\,1.95$	25.1 $\pm\,2.91$	24.5 $\pm\,1.88$	23.8 $\pm\,2.68$	20.9 $\pm\,3.00$	26.4 $\pm\,3.33$	18.2 $\pm\,1.80$	–
SR ( $\times$ 16)	20.8 $\pm\,2.78$	16.9 $\pm\,1.90$	19.7 $\pm\,1.97$	15.1 $\pm\,1.37$	16.4 $\pm\,2.21$	20.4 $\pm\,2.38$	15.4 $\pm\,1.89$	–
Box Inpainting	18.3 $\pm\,1.99$	18.2 $\pm\,2.39$	18.0 $\pm\,2.82$	12.6 $\pm\,1.00$	17.1 $\pm\,2.18$	19.2 $\pm\,2.51$	16.4 $\pm\,1.85$	18.70 $\pm\,3.23$

Appendix

Table 15: Quantitative comparison of different methods across various inverse problems. Results are reported as PSNR

\pm

standard deviation. Higher is better. Blue factors are relative to Spin ; larger values mean greater time or memory savings.

Task group	$t_{*}$	$N$	$G_{\texttt{OPT}}$
Deblurring (Gaussian, Motion)	$[0.44,0.50]$	10	$[25,50]$
Super-resolution (SR4, SR16)	$[0.50,0.60]$	$[1,30]$	$[75,200]$
Inpainting (Box, Half)	$[0.70,0.80]$	$[5,30]$	50
JPEG / Phase Retrieval / HDR	$[0.44,0.70]$	$[5,20]$	$[25,200]$

Appendix

Table 16: Pixel-space hyperparameters for the ImageNet and FFHQ experiments, grouped by task family. Unless stated otherwise, we use

\eta_{\text{main}}=10^{-3}

\lambda=0

\eta_{\text{interleave}}=1.0

N_{\texttt{OPT}}^{\texttt{int}}=50

\eta_{\text{init}}=10^{-4}

M=30

, and a Gaussian guidance schedule with

\mu_{G}=0.4

and

\sigma_{G}=10.0

. Table ranges summarize task- and dataset-specific choices within each group.

Task group	$t_{*}$	$N_{\texttt{OPT}}^{\texttt{int}}$	$G_{\texttt{OPT}}$	$p$
Deblurring (Gaussian, Motion)	$[0.50,0.70]$	$[50,300]$	3000	$[2.0,2.5]$
Super-resolution (SR4, SR16)	$[0.30,0.40]$	300	$[1000,2000]$	$[2.5,4.0]$
Inpainting (Box, Half)	0.44	$[800,1000]$	800	$[2.5,3.0]$
JPEG / Phase Retrieval / HDR	$[0.40,0.70]$	$[50,500]$	$[500,3000]$	3.0

Appendix

Table 17: Latent-space hyperparameters for the FFHQ experiments, grouped by task family. Unless stated otherwise, we use

\lambda=0

\eta_{\text{init}}=10^{-4}

\eta_{\text{main}}=10^{-4}

\eta_{\text{interleave}}=1.0

N=30

M=10

, and a polynomial guidance schedule

\textsc{Poly}(p)

. Table ranges summarize task-specific choices within each group.

Figure 5: Reconstructions for half mask inpainting on FFHQ dataset.

Figure 6: JPEG dequantization with QF = 2 on FFHQ dataset.

Figure 7: Reconstructions for Gaussian deblurring on ImageNet dataset.

Figure 8: Gaussian Deblurring on FFHQ dataset.

Figure 9: Motion Deblurring on FFHQ dataset.

Figure 10: Super-resolution ( $\times$ 4) on FFHQ dataset.

Figure 11: Super-resolution ( $\times$ 16) on FFHQ dataset.

Figure 12: Box Inpainting on FFHQ dataset.

Figure 13: Half Inpainting on FFHQ dataset.

Figure 14: JPEG Compression (QF=2) on FFHQ dataset.

Figure 15: Phase Retrieval on FFHQ dataset.

Figure 16: Gaussian Deblurring on FFHQ dataset with LDM prior.

Figure 17: Motion Deblurring on FFHQ dataset with LDM prior.

Figure 18: Super-Resolution ( $\times$ 4) on FFHQ dataset with LDM prior.

Figure 19: Box Inpainting on FFHQ dataset with LDM prior.

Figure 20: Half Inpainting on FFHQ dataset with LDM prior.

Figure 21: Deblurring on ImageNet dataset.

Figure 22: Deblurring on ImageNet dataset.

Figure 23: Super-resolution ( $\times$ 4) on ImageNet dataset.

Figure 24: Center inpainting on ImageNet dataset.

Figure 25: JPEG compression (QF=2) on ImageNet dataset.

Figure 26: High Dynamic Range on ImageNet dataset.

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Diffusion models (DMs) have recently shown remarkable performance on inverse problems (IPs). Optimization-based methods can fast solve IPs using DMs as powerful regularizers, but they are susceptible to local minima and noise overfitting. Although DMs can provide strong priors for Bayesian approaches, enforcing measurement consistency during the denoising process leads to manifold infeasibility issues. We propose Noise-space Hamiltonian Monte Carlo (N-HMC), a posterior sampling method that treats reverse diffusion as a deterministic mapping from initial noise to clean images. N-HMC enables comprehensive exploration of the solution space, avoiding local optima. By moving inference entirely into the initial-noise space, N-HMC keeps proposals on the learned data manifold. We provide a comprehensive theoretical analysis of our approach and extend the framework to a noise-adaptive variant (NA-NHMC) that effectively handles IPs with unknown noise type and level. Extensive experiments across four linear and three nonlinear inverse problems demonstrate that NA-NHMC achieves superior reconstruction quality with robust performance across different hyperparameters and initializations, significantly outperforming recent state-of-the-art methods. The code is available at https://github.com/NA-HMC/NA-HMC.

Abstract

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