Diffusion Posterior Sampling with Uncertain Forward Models

Wentao Shangguan; Vivek K Goyal

cs.CVOct 21, 2026

Diffusion Posterior Sampling with Uncertain Forward Models

Authors: Wentao Shangguan, Vivek K Goyal

Organizations: Department of Electrical and Computer Engineering, Boston University, Boston, MA 02215, USA

Abstract

Diffusion posterior samplers reconstruct signals by combining a pretrained diffusion prior with a measurement likelihood specified by a forward model and a noise model. Calibration and reference data can characterize uncertainty in the forward model's parameters, such as blur kernels or detector responses. Fixing these parameters at estimated values leaves their uncertainty out of the likelihood, and parameter errors are thus absorbed into the reconstruction. We introduce an uncertainty-aware likelihood (UAL), a Gaussian approximation to the marginal likelihood obtained by integrating over the uncertain parameters. For each candidate signal, UAL propagates parameter uncertainty into a structured measurement covariance, reducing the quadratic penalty along residual directions that parameter variation can explain. Existing samplers can use UAL in their data updates while retaining their pretrained diffusion priors and sampling schedules. Under a fixed Gaussian nuisance model, we bound likelihood error from linearization and average divergence between the exact marginal and UAL posteriors. We compare samplers with and without UAL across six inverse problems spanning Gaussian and motion deblurring, CT, MRI, ultrasound, and fluorescence microscopy, and find improvements in reconstruction and ensemble prediction. UAL reduces geometric mean MSE by 9% to 20% across samplers in the main Gaussian-deblurring setting and by 51% to 92% in motion deblurring when non-blind samplers use the population-mean kernel. In motion deblurring, DAPS and PnP-DM with UAL also give 6.2 to 6.4 dB higher ensemble-mean PSNR than published BlindDPS and Fast Diffusion EM. On the ablation set, UAL outperforms trace-matched isotropic inflation for PnP-DM in both deblurring tasks; in motion deblurring, it also exceeds the best-tested fixed scalar data weighting by 3.3 dB for PnP-DM and 4.2 dB for DAPS in average PSNR.

Figures & tables

Figure 1: Reconstruction examples and quantitative effects of UAL across six inverse problems. Reconstruction rows show selected individual trajectories; bars summarize each evaluation set on task-specific scales. Ultrasound reports CNR and gCNR computed within each reconstruction on the contrast phantom, with one SE across its 12 paired trajectories; the delay-and-sum reference (DAS75) is shown. Other tasks report PSNR and CRPS with paired-bootstrap whiskers.

Gaussian blur
Sampler / task	Version	PSNR T/M $\uparrow$	SSIM T/M $\uparrow$	CRPS $\downarrow$	Energy $\downarrow$	LVS $\downarrow$
	Published	29.49 / 30.60	.7416 / .8143	.02871	20.128	11.240
DAPS	UAL	29.88 / 30.94	.7468 / .8171	.02796	19.449	11.021
	Published	26.97 / 29.53	.5603 / .7616	.03209	20.874	15.286
PnP-DM	UAL	27.50 / 30.35	.5721 / .7776	.02956	18.803	14.252
	Published	29.13 / 30.28	.7687 / .8180	.02948	21.263	10.788

Table 1: Reconstruction and ensemble prediction. T/M: mean per-trajectory score / score of the ensemble-mean reconstruction. PSNR is in dB; CRPS, energy, and LVS are fair ensemble scores, with LVS scaled by

10^{3}

. Bold marks better UAL values.

Figure 2: Likelihood accuracy in Gaussian deblurring against numerical marginalization. (a,b) Final-state relative image-score error and absolute negative-log-likelihood error per measurement: means

\pm

one SE across the ablation set. (c) Score error at five saved states, using one trajectory per image. Errors are averaged over images within each uncertainty level. Lines and bands show the mean and range across

\tau=0.25,0.5,1,1.5

pixels. Generating and assumed width uncertainty vary together. Appendix D.1 gives the stored indices corresponding to panel (c)’s nominal iterations.

	Gaussian blur			Motion blur
Metric	Published	Isotropic	UAL	Published	Isotropic	UAL
PSNR T (dB) $\uparrow$	26.944	27.020	27.475	17.872	24.084	26.641
CRPS ( $\times 10^{3}$ ) $\downarrow$	32.53	32.42	29.81	131.53	58.55	31.49

Table 2: Covariance structure at matched whitened trace on the ablation set. Appendix D.4 reports paired differences and standard errors.

Metric	Published	Fixed covariance	Derivatives stopped	UAL
PSNR T (dB) $\uparrow$	26.944	26.964	27.506	27.475
CRPS ( $\times 10^{3}$ ) $\downarrow$	32.53	34.84	29.76	29.81

Table 3: Gaussian PnP-DM covariance-update controls on the ablation set. The two recomputed treatments agree within two standard errors in per-trajectory PSNR.

Appendix figures & tables26 assets

Supplementary material from the paper’s appendix.

Appendix

Quantity	Point Gaussian likelihood	UAL
Mean	$\mathcal{A}(x,\psi_{0})$	$\bar{y}(x)=\mathcal{A}(x,\psi_{0})+J_{\psi}(x)(\mu-\psi_{0})$
Covariance	$\Sigma_{e}$	$S(x)=\Sigma_{e}+J_{\psi}(x)\Sigma_{\psi}J_{\psi}(x)^{\mathsf{T}}$
Negative log likelihood	$\tfrac{1}{2}d_{0}^{\mathsf{T}}\Sigma_{e}^{-1}d_{0}+\tfrac{1}{2}\log\det\Sigma_{e}$	$\tfrac{1}{2}d^{\mathsf{T}}S(x)^{-1}d+\tfrac{1}{2}\log\det S(x)$

Appendix

Table 4: Point likelihood and UAL at a fixed reference

\psi_{0}

. Here the residuals are

d_{0}=y-\mathcal{A}(x,\psi_{0})

and

d=y-\bar{y}(x)

. Both energies omit

m\log(2\pi)/2

;

\log\det\Sigma_{e}

is constant in

x

Task	Nuisance coordinates	Nuisance statistics	UAL construction
Gaussian blur	Kernel widths $(s_{x},s_{y})$	Width mean and covariance $\mu$ and ${\Sigma_{\psi}}=\tau^{2}I_{2}$	Width-sensitivity covariance (Sections C.1 – C.2 ).
Motion blur	Coefficients in the retained kernel basis	Mean and covariance from $131{,}072$ kernels	Gaussian moment matching in retained coordinates (Section C.1 ).
CT	Flat and dark reference fields	Mean and covariance across 20 flat and 20 dark frames	Linearized flat–dark calibration map (Section C.3 ).
MRI	Real coil-map coefficients	32 calibration replicates; calibrated maps for the mean prediction	Exact Gaussian marginal for a forward model affine in Gaussian coil-map coefficients (Section C.4 ).
Ultrasound	Sound speed $c_{s}$	Gaussian scale from CUBDL reference spread; truncated quadrature	Per-mode variances from nonlinear sound-speed perturbations (Section C.5 ).
Microscopy	Five pupil-phase coefficients (Noll modes 5, 6, 7, 8, 11)	Prescribed Gaussian phase law, also used to generate the synthetic observations	Derivatives of the normalized PSF; energy with candidate-dependent covariance (Section C.6 ).

Appendix

Table 5: Nuisance statistics and UAL constructions by task. The rows summarize the specifications below; the final column states how each task forms the UAL mean and covariance.

Gaussian blur: mean width 3 pixels, standard deviation 0.5 pixels
Sampler / task	PSNR gain 95% CI	CRPS reduction 95% CI $\times 10^{3}$
DAPS	$[0.20,0.61]$	$[-0.10,1.65]$
PnP-DM	$[.26,.84]$	$[1.79,3.49]$
DiffPIR	$[.59,1.46]$	$[2.35,5.31]$
BlindDPS	$[.80,.99]$	$[1.91,2.51]$
Motion blur

Appendix

Table 6: Pointwise 95% paired-bootstrap confidence intervals for the primary differences in Table 1 . PSNR gain is UAL minus Published for mean per-trajectory PSNR (dB); CRPS reduction is Published minus UAL, scaled by

10^{3}

$\tau$	Likelihood	iter. 0	iter. 25	iter. 50	iter. 74	iter. 99
0.25	Published	$36\pm 5$	$8.1\pm 0.9$	$6.1\pm 0.7$	$6.0\pm 0.6$	$6.2\pm 0.6$
	Isotropic	$33\pm 5$	$7.8\pm 0.7$	$6.9\pm 0.5$	$6.8\pm 0.5$	$6.9\pm 0.5$
	UAL	$17\pm 4$	$0.35\pm 0.06$	$0.14\pm 0.03$	$0.11\pm 0.01$	$0.12\pm 0.01$
0.5	Published	$37\pm 5$	$11\pm 1$	$8.2\pm 0.9$	$7.9\pm 0.8$	$8.3\pm 0.8$
	Isotropic	$34\pm 4$	$16\pm 1$	$17\pm 1$	$17\pm 1$	$17\pm 1$
	UAL	$18\pm 4$	$0.58\pm 0.08$	$0.24\pm 0.04$	$0.21\pm 0.03$	$0.22\pm 0.03$

Appendix

Table 7: Relative image-score error (percent) at five saved PnP-DM sampling states in Gaussian deblurring. Entries are mean

\pm

one SE across 16 images, with one trajectory per image at each

\tau

. Candidates are prior-step outputs from Published; iteration indices are zero-based. Figure 2 c labels the sampling states nominally as 1, 25, 50, 75, and 100. Numerical marginalization in Appendix D.1 provides the reference.

Sampler	Treatment	PSNR T/M $\uparrow$	SSIM T/M $\uparrow$	CRPS $\downarrow$	Energy $\downarrow$	LVS $\downarrow$
DAPS	Published	17.68 / 18.10	.3303 / .3779	.15043	96.013	57.076
	UAL	28.84 / 29.58	.7103 / .7726	.03502	23.388	12.774
	Oracle	30.65 / 32.50	.7698 / .8537	.02265	15.048	8.783
PnP-DM	Published	17.04 / 17.90	.2766 / .3897	.14485	91.587	61.032
	UAL	26.54 / 29.43	.5552 / .7628	.03236	20.943	14.959
	Oracle	27.65 / 30.68	.5752 / .7816	.02878	17.908	13.728

Appendix

Table 8: DAPS, PnP-DM, and DiffPIR motion deblurring on 50 images with 12 paired trajectories per image. Published uses the population-mean kernel; Oracle uses the true measurement kernel with the published sampler. UAL uses

P=256

covariance modes. T/M denotes mean per-trajectory score / score of the ensemble mean. PSNR is in dB; LVS is scaled by

10^{3}

Pair	PSNR M (dB)	PSNR T (dB)	SSIM M	SSIM T	CRPS	Energy	LVS
DAPS with UAL vs BlindDPS	+6.37 [5.81, 6.95] 50	+6.72 [6.25, 7.19] 50	+0.1463 [0.1330, 0.1606] 50	+0.1544 [0.1385, 0.1713] 50	+28.90 [25.12, 32.81] 50	+23.24 [20.35, 26.31] 50	+5.52 [4.51, 6.58] 48
DAPS with UAL vs Fast Diffusion EM	+6.31 [5.69, 6.87] 48	+6.08 [5.53, 6.60] 49	+0.2201 [0.1960, 0.2435] 48	+0.1868 [0.1621, 0.2114] 48	+42.87 [37.43, 48.27] 48	+28.48 [24.98, 31.91] 48	+11.50 [10.15, 12.83] 49
PnP-DM with UAL vs BlindDPS	+6.22 [5.69, 6.75] 50	+4.42 [3.98, 4.85] 49	+0.1364 [0.1237, 0.1500] 50	$-$ 0.0007 [ $-$ 0.0199, 0.0190] 23	+31.55 [28.30, 34.94] 50	+25.69 [23.11, 28.33] 50	+3.33 [2.17, 4.45] 42
PnP-DM with UAL vs Fast Diffusion EM	+6.16 [5.58, 6.70] 50	+3.78 [3.25, 4.29] 47	+0.2103 [0.1859, 0.2338] 49	+0.0316 [0.0052, 0.0582] 31	+45.53 [40.55, 50.38] 50	+30.92 [27.58, 34.11] 50	+9.31 [7.73, 10.88] 47
Fast Diffusion EM with UAL vs BlindDPS	+7.52 [6.97, 8.05] 50	+6.98 [6.46, 7.51] 50	+0.1845 [0.1717, 0.1981] 50	+0.2076 [0.1941, 0.2215] 50	+36.84 [33.50, 40.45] 50	+28.30 [25.60, 31.16] 50	+8.66 [7.68, 9.74] 50
Fast Diffusion EM with UAL vs Fast Diffusion EM	+7.45 [6.83, 8.02] 50	+6.34 [5.71, 6.93] 48	+0.2583 [0.2329, 0.2832] 50	+0.2399 [0.2150, 0.2640] 49	+50.81 [45.32, 56.30] 50	+33.53 [30.09, 36.86] 50	+14.64 [13.21, 16.05] 50

Appendix

Table 9: Paired improvements over published blind samplers in motion deblurring, 50 images with 12 trajectories each: mean [95% paired bootstrap interval] and number of images improved. Values are UAL minus the blind baseline for PSNR and SSIM, and the blind baseline minus UAL for CRPS, energy, and LVS, so positive values denote improvement; CRPS and LVS are scaled by

10^{3}

. The first method in each pair uses UAL.

Figure 5: Selected Gaussian-deblurring reconstructions at $(\bar{s},\tau)=(3,0.5)$ pixels. Each row pair shows the observation and reference at left, followed by Published (upper) and UAL (lower) reconstructions from four samplers. Published and UAL use matched random seeds within each sampler. Numbers give PSNR (dB), SSIM, and CRPS, in that order. PSNR and SSIM describe the displayed trajectory; CRPS uses all 12 trajectories.

Figure 6: Selected motion-deblurring reconstructions for the same three images as Figure 5 , with $P=256$ covariance modes for UAL. Each row pair shows the observation and reference at left, followed by Published (upper) and UAL (lower) reconstructions. For each sampler, the displayed reconstructions are paired by random seed. Numbers give PSNR (dB), SSIM, and CRPS, in that order. PSNR and SSIM describe the displayed trajectory; CRPS uses all 12 trajectories.

Figure 7: Selected CT cases, with reconstruction draws paired by seed. Columns show cases. Each method panel combines the reconstruction in the upper-left half with its absolute difference from the reference in the lower-right half. Difference maps share one color scale. Numbers give PSNR (dB), SSIM, and CRPS, in that order. PSNR and SSIM describe the displayed draw; CRPS uses all 12 trajectories. DM4CT denotes Published.

Metric	Mean gain	95% interval
PSNR T (dB)	0.0127	$[0.0097,0.0160]$
PSNR M (dB)	0.0141	$[0.0104,0.0183]$
SSIM T	0.000313	$[0.000213,0.000420]$
SSIM M	0.000119	$[0.000026,0.000231]$
CRPS	0.0000331	$[0.0000239,0.0000435]$
Energy	0.0220	$[0.0160,0.0290]$

Appendix

Table 10: MRI gains and pointwise 95% paired-bootstrap confidence intervals across 50 scans, each with 12 paired trajectories. Gains are UAL minus Published for PSNR and SSIM, and Published minus UAL for the ensemble scores. T/M follows Table 1 .

Figure 8: Magnitude reconstructions for selected MRI scans. Each triplet shows the reference, Published Ambient DPS, and Ambient DPS with UAL. Both methods use the same random seed for each scan. Numbers give PSNR (dB), SSIM, and CRPS, in that order. PSNR and SSIM describe the displayed draw; CRPS uses all 12 trajectories.

Figure 9: Selected fluorescence-microscopy sections. Reconstructions are paired by seed; difference maps share one scale. Numbers give PSNR (dB), SSIM, and CRPS. PSNR and SSIM use the displayed draw; CRPS uses all 12 trajectories.

Metric	Published	UAL
CNR (dB), higher	9.9292	12.3625
gCNR	0.9141	0.9743
Speckle SNR, target $1.91$	2.0616	2.0184
Axial FWHM (mm)	0.2506	0.2497
Lateral FWHM (mm)	0.5729	0.5722

Appendix

Table 11: Native ultrasound contrast and resolution metrics, averaged over 12 trajectories per phantom. CNR, gCNR, and speckle SNR use the contrast phantom; axial and lateral FWHM use the resolution phantom. FWHM is in mm and lower values are better. Speckle SNR is compared with the reference

1.91

. Bold marks better values.

Panel of Figure 10 — Figure 10: Pixelwise statistics for the microscopy section in Figure 9 , using all 12 paired random seeds. The left columns show the absolute difference between the ensemble mean and reference and the sample standard deviation (SD). The right columns show their ratio and a scatter plot of difference against SD. Red marks pixels above the 3-SD threshold, indicated by the dashed line; percentages give the pixel fractions on either side. UAL reduces the fraction above this threshold from 1.93% to 0.51%, so large reference differences relative to the ensemble spread are less frequent in this section.

$v$	1	3	6	12	24	48	96	192	384
PnP-DM	17.872	20.429	21.652	22.569	23.243	23.369	23.210	22.503	21.419
DAPS	18.441	21.098	22.431	23.448	24.113	24.413	24.360	23.965	23.245

Appendix

Table 12: Fixed data-term variance multipliers in motion deblurring: mean per-trajectory PSNR (dB) on the ablation set. For PnP-DM,

v

scales the observation-noise variance; for DAPS, it scales the data-fitting variance

\tau_{\rm D}^{2}

v=1

is the published sampler; UAL gives 26.641 dB for PnP-DM and 28.631 dB for DAPS.

Likelihood	PSNR T	PSNR M	CRPS	Energy	LVS
Published	26.944	29.388	32.53	21.327	15.22
Isotropic	27.020	29.472	32.42	21.078	15.12
UAL	27.475	30.259	29.81	19.052	14.18

Appendix

Table 13: Gaussian-deblurring covariance-structure control. T/M follows Table 1 ; PSNR is in dB, and CRPS and LVS are scaled by

10^{3}

. Lower ensemble scores are better.

Metric	Published	Quadratic only	UAL
PSNR T (dB) $\uparrow$	$17.8723\pm 0.7269$	$26.6157\pm 0.3040$	$26.6414\pm 0.3013$
PSNR M (dB) $\uparrow$	$18.7901\pm 0.7827$	$29.5099\pm 0.3551$	$29.5335\pm 0.3564$
SSIM T $\uparrow$	$0.30950\pm 0.02466$	$0.55399\pm 0.00805$	$0.55412\pm 0.00806$
SSIM M $\uparrow$	$0.43340\pm 0.03894$	$0.76189\pm 0.01441$	$0.76191\pm 0.01444$
CRPS ( $\times 10^{3}$ ) $\downarrow$	$131.533\pm 19.396$	$31.534\pm 1.491$	$31.489\pm 1.496$
Energy $\downarrow$	$83.874\pm 9.441$	$20.406\pm 0.882$	$20.351\pm 0.884$

Appendix

Table 14: Likelihood normalization in motion deblurring with PnP-DM at

P=256

. Values are means

\pm

one SE across the ablation set. Quadratic only omits the log determinant while retaining covariance recomputation and differentiation. T/M follows Table 1 .

Metric	Published	Covariance weighting	Quadratic only	UAL
PSNR T (dB) $\uparrow$	22.76	23.80	28.49	29.10
PSNR M (dB) $\uparrow$	23.27	25.30	30.64	30.72
SSIM T $\uparrow$	.5236	.5663	.7537	.7635
SSIM M $\uparrow$	.5525	.6243	.8131	.8108
CRPS $\downarrow$	.07789	.05472	.02731	.02708
Energy $\downarrow$	51.865	37.077	18.360	18.332

Appendix

Table 15: Motion deblurring with Fast Diffusion EM on 50 images with 12 paired trajectories per image and

P=256

. All three adapted arms use the population-mean kernel and propagated covariance; covariance weighting holds the covariance fixed during differentiation and omits the log determinant, and the quadratic-only arm recomputes and differentiates the covariance but omits the log determinant. T/M follows Table 1 .

Task	Candidate	$c(x)$	$s_{1}^{2}/10^{4}$	$s_{2}^{2}/10^{4}$
Gaussian blur	Published	$0.189\pm 0.015$	$2.617\pm 0.266$	$1.098\pm 0.048$
	Isotropic	$0.185\pm 0.014$	$2.553\pm 0.248$	$1.088\pm 0.047$
	UAL	$0.197\pm 0.010$	$2.516\pm 0.134$	$1.357\pm 0.065$
Motion blur	Published	$20.085\pm 3.532$	$176.389\pm 44.334$	$57.841\pm 8.578$
	Isotropic	$3.244\pm 0.328$	$34.666\pm 4.084$	$12.922\pm 1.438$
	UAL	$4.626\pm 0.370$	$38.296\pm 3.533$	$19.090\pm 1.942$

Appendix

Table 16: Final-state covariance diagnostics, reported as mean

\pm

standard error across images. The leading squared singular values are scaled by

10^{-4}

Figure 12: PnP-DM sensitivity to nuisance moments in Gaussian deblurring. (a) Mean-width error with correct covariance. (b) Covariance-scale error with correct mean. Observations stay fixed; zero denotes the correct moment. Curves show mean per-trajectory PSNR

\pm

one SE across the ablation set. The dashed Published reference uses the correct mean in both panels. Mean PSNR changes little over the tested ranges.

Task	Covariance error	$\Delta$ PSNR T	$\Delta$ PSNR M	$\Delta$ CRPS (%)
Gaussian	$-20.1\%$	$+0.0002\pm 0.0002$	$+0.0001\pm 0.0002$	$-0.0003\pm 0.0008$
Gaussian	$-4.9\%$	$+0.0002\pm 0.0002$	$+0.0002\pm 0.0001$	$-0.0006\pm 0.0007$
Gaussian	$+5.1\%$	$+0.0002\pm 0.0001$	$+0.0002\pm 0.0001$	$-0.0010\pm 0.0006$
Gaussian	$+19.9\%$	$+0.0001\pm 0.0001$	$+0.0002\pm 0.0002$	$-0.0011\pm 0.0008$
Gaussian	$+99.9\%$	$+0.0001\pm 0.0001$	$+0.0002\pm 0.0002$	$-0.0009\pm 0.0010$
Motion	$-20.0\%$	$+0.0609\pm 0.0633$	$+0.0257\pm 0.0291$	$+0.0572\pm 0.0392$

Appendix

Table 17: Covariance-scale mismatch with fixed observations. Each task uses the ablation set. Signed changes (perturbed minus UAL at zero covariance error) are reported as mean

\pm

SE across images. PSNR is in dB. Relative CRPS changes divide the paired change and its SE by the reference mean CRPS.

	MSE $\downarrow$			PSNR $\uparrow$			CRPS $\downarrow$
$\tau$ (pixels)	Published	Isotropic	UAL	Published	Isotropic	UAL	Published	Isotropic	UAL
.25	.007301	.007312	.007202	27.508	27.502	27.551	29.93	29.97	29.23
.50	.008459	.008263	.007348	26.944	27.020	27.475	32.53	32.42	29.81
1.00	.014066	.010826	.008800	25.368	26.018	26.818	40.95	38.24	33.26
1.50	.020463	.012810	.012655	24.043	25.318	25.789	49.58	43.24	39.90

Appendix

Table 18: Matched generating and UAL width uncertainty, on the ablation set. MSE and PSNR (dB) are averaged per trajectory; CRPS is scaled by

10^{3}

. Isotropic is the trace-matched control of ( 77 ), run at each level.

		DAPS		PnP-DM		DiffPIR		BlindDPS
$(\bar{s},\tau)$	Version	PSNR $\uparrow$	CRPS $\downarrow$	PSNR $\uparrow$	CRPS $\downarrow$	PSNR $\uparrow$	CRPS $\downarrow$	PSNR $\uparrow$	CRPS $\downarrow$
	Published	29.49	28.71	26.97	32.09	29.13	29.48	28.31	26.98
$(3,0.5)$	UAL	29.88	27.96	27.50	29.56	30.12	25.83	29.21	24.76
	Published	28.16	32.89	26.35	35.58	27.52	32.67	26.96	32.38
$(5,0.5)$	UAL	28.65	32.22	26.84	33.93	28.60	28.64	27.37	29.79
	Published	28.05	35.42	25.80	38.56	26.70	43.55	28.43	27.01

Appendix

Table 19: Gaussian-width effects on reconstruction and ensemble prediction on 50 images, with 12 paired trajectories per image. Entries are per-trajectory PSNR (dB) and CRPS (

\times 10^{3}

) for the published sampler and UAL;

(\bar{s},\tau)

is in pixels. Bold marks better UAL values at the displayed precision.

$(5,0.5)$ pixels
Sampler	Version	PSNR T/M $\uparrow$	SSIM T/M $\uparrow$	CRPS $\downarrow$	Energy $\downarrow$	LVS $\downarrow$
DAPS	Published	28.16 / 29.32	.6888 / .7685	.03289	22.999	12.581
	UAL	28.65 / 29.67	.6949 / .7702	.03222	22.512	12.631
PnP-DM	Published	26.35 / 28.55	.5325 / .7290	.03558	23.459	16.258
	UAL	26.84 / 29.12	.5366 / .7338	.03393	22.008	15.759
DiffPIR	Published	27.52 / 29.11	.7028 / .7712	.03267	22.846	11.400

Appendix

Table 20: Full Gaussian-width results for

(5,0.5)

and

(3,1)

pixels on 50 images, with 12 paired trajectories per image. T/M follows Table 1 ; PSNR is in dB and LVS is scaled by

10^{3}

. Bold marks better UAL values at the displayed precision.

Sampler	Treatment	PSNR T/M $\uparrow$	SSIM T/M $\uparrow$	CRPS $\downarrow$	Energy $\downarrow$	LVS $\downarrow$
DAPS	Published	18.44 / 18.89	.3677 / .4190	0.13913	88.578	52.203
	UAL, $P=256$	28.63 / 29.34	.7030 / .7660	0.03572	24.046	13.079
	UAL, $P=2048$	28.56 / 29.29	.7002 / .7659	0.03568	23.962	12.990
PnP-DM	Published	17.87 / 18.79	.3095 / .4334	0.13153	83.874	54.395
	UAL, $P=256$	26.64 / 29.53	.5541 / .7619	0.03149	20.351	14.766
	UAL, $P=2048$	26.25 / 29.31	.5420 / .7517	0.03209	20.788	14.997

Appendix

Table 21: Motion deblurring with the published samplers and UAL. All treatments use the ablation set. T/M follows Table 1 ; PSNR is in dB. CRPS, energy, and LVS are fair ensemble scores, with LVS scaled by

10^{3}

Gaussian deblurring
Sampler / task	Published (s)	UAL (s)
PnP-DM	77.3	76.9
DAPS	39.4	116.6
DiffPIR	6.0	6.1
BlindDPS	129.6	371.8
Motion deblurring

Appendix

Table 22: Mean wall time per trajectory on NVIDIA L40S GPUs with one reconstruction job per GPU.

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Jun 10, 2026

A Stabilized Path-Space Approach to Diffusion-Based Posterior Sampling

Diffusion models provide expressive data-driven priors for Bayesian inverse problems, but many diffusion posterior samplers rely on heuristic guidance approximations that can fail for nonlinear operators and multimodal posteriors. In this work, we develop a stabilized path-space framework for diffusion-based posterior sampling. Starting from a base diffusion process whose terminal marginal represents the prior, we define a likelihood-weighted target measure on trajectories and cast posterior sampling as learning a controlled stochastic process whose path measure matches this target. This formulation connects diffusion posterior sampling to stochastic optimal control while preserving the Bayesian structure needed for uncertainty quantification. We introduce a time reparameterization that makes the path-space control problem well posed by removing the bias induced by the unknown initial value function, without auxiliary training. We then learn the control via a trust-region path-space optimization method with log-variance objectives. The path-space perspective also unifies our learned control approach with existing guidance-based samplers, quantifies the sampling error induced by approximate controls, and yields importance sampling corrections for asymptotically exact posterior expectations. We evaluate the proposed framework on a suite of benchmark inverse problems with analytically characterized or high-quality reference posteriors, enabling principled assessment of sampling accuracy and uncertainty quantification. These experiments provide insight into the behavior of diffusion-based posterior samplers and demonstrate improved accuracy and robustness over leading approaches.

Evan Scope Crafts, Umberto Villa, Saviz Mowlavi +3

Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX · Department of Biomedical Engineering, The University of Texas at Austin, Austin, TX · Mitsubishi Electric Research Laboratories, Cambridge, MA

Bayesian Inverse Problems Diffusion-Based Inverse Problems Diffusion Posterior Sampling

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Jul 21, 2026

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

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.

Yuchen Jiao, Na Li, Changxiao Cai +2

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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May 12, 2026

Improving Diffusion Posterior Samplers with Lagged Temporal Corrections for Image Restoration

Diffusion-based posterior sampling (PS) is a leading framework for imaging inverse problems, combining learned priors with measurement constraints. Yet, its standard formulations rely on instantaneous data-consistent estimates, which induce temporal variability in the reverse dynamics. We reinterpret PS from a dynamical perspective, showing that the standard PS update corresponds to a first-order discretization of the diffusion dynamics plus a residual correction capturing the mismatch between the denoised prediction and the data-consistent estimate. A second-order discretization, however, naturally introduces a temporal correction based on the variation of consecutive estimates. Building on this, we propose LAMP, combining the second-order update with the residual correction characterizing a PS technique. LAMP thus inherits a lagged temporal correction, and it can be implemented as a modular plug-in over the PS backbone. We show that LAMP preserves the structure of a posterior sampler, and we perform a one-step risk analysis to characterize when LAMP improves the reverse transition via a bias-variance trade-off. Experiments across multiple imaging tasks demonstrate consistent improvements over strong baselines such as DiffPIR and DDRM, without increasing the number of denoising evaluations.

Davide Evangelista, Elena Morotti, Francesco Pivi +1

Dept. of Computer Science and Engineering University of Bologna · Dept. of Political and Social Sciences University of Bologna

Diffusion Sampling Image Restoration Diffusion-Based Inverse Problems Diffusion Posterior Sampling Image Inverse Problems

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Abstract

Diffusion-Based Inverse Problems

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A Stabilized Path-Space Approach to Diffusion-Based Posterior Sampling

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

Improving Diffusion Posterior Samplers with Lagged Temporal Corrections for Image Restoration