Measuring Effective Data Resolution in Guided Diffusion Posteriors

Ridham Patel; Defu Cao; Jiacheng Pang; Yan Liu

cs.LGOct 3, 2026

Measuring Effective Data Resolution in Guided Diffusion Posteriors

Authors: Ridham Patel, Defu Cao, Jiacheng Pang, Yan Liu

Organizations: Indian Institute of Technology Gandhinagar Gandhinagar, India · University of Southern California Los Angeles, USA

Abstract

Guided diffusion samplers are increasingly used to reconstruct physical fields from sparse observations, but standard diagnostics do not say how much of the reconstruction was actually determined by the data. We introduce effective data resolution for black-box generative posteriors: a comparison between the resolution warranted by the inverse problem, $\mathrm{dof}_{\mathrm{ref}}$ , and the resolution realised by the sampler, $\mathrm{dof}_{\mathrm{samp}}$ . A perturbation estimator measures $\mathrm{dof}_{\mathrm{samp}}$ and the spatial map $R(x,x)$ from sampler queries alone. We validate the estimator against exact references and use it to study guided diffusion. The resulting measurements show that guidance weight can strongly alter apparent information transfer, that mean, spread and resolution are not jointly corrected by one weight even with an exact prior and score, and that resolution fidelity does not follow reliably from the apparent principledness of a guidance rule.

Figures & tables

guidance	mean err	std err	var ratio	CRPS	cov@90%	$\rho_{\mathrm{res}}$
exact score (control)	0.040	0.046	0.97	0.309	0.812	0.985 $\pm$ .002
$\Pi$ GDM	0.041	0.047	0.96	0.307	0.812	1.022 $\pm$ .002
projection	0.051	0.049	0.94	0.308	0.812	1.019 $\pm$ .000
DPS $\zeta=0.3$	0.155	0.521	2.22	0.314	0.938	0.796 $\pm$ .038
DPS $\zeta=1$	0.074	2.712	13.8	0.356	1.000	1.219 $\pm$ .046
DPS $\zeta=3$	0.350	9.453	97.7	0.528	1.000	1.053 $\pm$ .129

Table 1: Guidance weight trades posterior accuracy, spread, and effective resolution in the controlled 1D backward-heat problem. “mean err” and “std err” are relative

L^{2}

errors in the posterior mean and pointwise standard deviation; “var ratio” is relative to the closed-form posterior, CRPS is the continuous ranked probability score, and “cov@90%” is empirical coverage of the nominal 90% interval. The fidelity ratio

\rho_{\mathrm{res}}=\mathrm{dof}_{\mathrm{samp}}/\mathrm{dof}_{\mathrm{ref}}

is measured at

\epsilon=\sigma_{y}

over

4

ensemble realisations;

\pm

denotes the across-realisation standard error.

Figure 1: Effective data resolution in the controlled 1D setting, at

\epsilon=\sigma_{y}

. (a) The spatial map

R(x,x)

, averaged over

16

ensemble realisations with

\pm 1

sd bands, showing where apparent data influence lands. The exact posterior spreads it almost uniformly (

0.107

0.138

) and the exact-score control recovers that to

0.002

with zero spread across realisations, which validates the apparatus. DPS at

\zeta=0.3

under-resolves roughly uniformly (averaged trace

6.64

against

7.85

), while DPS at

\zeta=3

concentrates influence in a band near

x=24

and starves the rest of the field (

0.019

0.464

); the latter’s map is not even stable across realisations, its per-realisation trace ranging from

-24.0

68.2

. (b) The scalar

\mathrm{dof}_{\mathrm{samp}}

against guidance weight (

4

realisations, error bars are across-realisation standard errors), with

\mathrm{dof}_{\mathrm{ref}}

as the horizontal line. The curve rises, crosses the reference, and falls back, so

\mathrm{dof}_{\mathrm{samp}}=\mathrm{dof}_{\mathrm{ref}}

has more than one root and matching resolution does not identify a guidance weight.

guidance	rel- $L^{2}$	std/ref	SW $\to$ ref	CRPS	cov@90%	$\rho_{\mathrm{res}}$
pCN (reference)	0.689	1.000	0.000	0.211	0.774	1.000
$\Pi$ GDM	diverged (implementation; see text)
projection	0.808	1.219	0.255	0.249	0.785	1.501
DPS $\zeta=0.3$	0.911	1.644	0.341	0.283	0.851	0.152
DPS $\zeta=1$	0.796	1.373	0.270	0.244	0.845	1.026
DPS $\zeta=3$	0.799	1.290	0.252	0.245	0.812	1.372

Table 2: Resolution fidelity under a nonlinear map with a trained score, against an exact pCN reference. “rel-

L^{2}

” is the relative

L^{2}

error of the posterior mean, “std/ref” is ensemble spread relative to the pCN reference, “SW

\to

ref” is the sliced-Wasserstein distance to that reference, CRPS is the continuous ranked probability score, and “cov@90%” is empirical coverage of the nominal 90% posterior interval. The span of

\rho_{\mathrm{res}}

is comparable to the controlled setting, but projection over-resolves by

50\%

while DPS at

\zeta=1

is the closest listed method.

\Pi

GDM diverged in our implementation, so we exclude it rather than claim a result about the method.

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

$\epsilon/\sigma_{y}$	$10^{-6}$	$10^{-4}$	$10^{-2}$	$0.1$	$1$	$10$
$\mathrm{dof}_{\mathrm{samp}}$	$-5.25\times 10^{6}$	$4.27\times 10^{4}$	$462.2$	$10.94$	$10.91$	$8.35$

Appendix

Table 4

$P$ (probes)	solves	bias	sd of mean	per-run sd
8	17	$-0.094$	0.142	0.897
32	65	$-0.011$	0.070	0.442
128	257	$-0.009$	0.028	0.179

Appendix

Table 3: Certification of the black-box estimator. Backward heat,

N=128

n=12

, exact posterior,

\mathrm{dof}_{\mathrm{ref}}=5.971

;

40

independent probe sets per row. From

P=8

P=128

the per-run standard deviation falls by a factor

5.0

against the

4.0

predicted by

1/\sqrt{P}

$\zeta$	$\mathrm{dof}_{\mathrm{samp}}$	stderr	$\rho_{\mathrm{res}}$
$0.1$	1.755	0.103	0.224
$0.2$	4.028	0.370	0.513
$0.3$	6.247	0.300	0.796
$0.5$	7.231	0.575	0.921
$0.7$	8.405	0.628	1.071
$1$	9.565	0.363	1.219

Appendix

Table 6

guidance	mean err	var ratio	CRPS	cov@90%	$\rho_{\mathrm{res}}$
exact score	0.102	0.96	0.271	0.812	0.991 $\pm$ .001
$\Pi$ GDM	0.246	1.17	0.283	0.828	0.899 $\pm$ .004
projection	0.603	2.11	0.484	0.797	1.014 $\pm$ .000
DPS $\zeta=0.3$	0.697	21.4	0.521	0.922	1.023 $\pm$ .064
DPS $\zeta=1$	0.409	17.0	0.402	1.000	1.079 $\pm$ .091
DPS $\zeta=3$	0.126	33.4	0.393	1.000	0.839 $\pm$ .066

Appendix

Table 4: Non-PDE breadth check: random compressed sensing with an exact prior, exact score and exact posterior reference. Resolution is estimated with

\epsilon=\sigma_{y}

P=24

probes, ensembles of

512

, and

4

independent realisations. The exact-score control again recovers

\mathrm{dof}_{\mathrm{ref}}

, while distributional spread and resolution fidelity separate: projection has nearly correct

\rho_{\mathrm{res}}

but about

2.1\times

posterior variance, and DPS can have near-reference

\rho_{\mathrm{res}}

with much larger spread.

guidance	$\mathrm{dof}_{\mathrm{samp}}$	$\rho_{\mathrm{res}}$	sd $(\rho_{\mathrm{res}})$	median probe se
exact score	$7.77\pm 0.03$	0.990	0.009	0.039
DPS $\zeta=0.3$	$5.50\pm 0.39$	0.701	0.123	1.128
DPS $\zeta=1$	$8.92\pm 1.94$	1.137	0.606	3.743
DPS $\zeta=3$	$6.23\pm 4.32$	0.793	1.347	10.282

Appendix

Table 5: Higher-replicate 1D stability stress test, with

\mathrm{dof}_{\mathrm{ref}}=7.85

\epsilon=\sigma_{y}

P=12

probes, ensembles of

512

, and

6

realisations. The exact control remains stable. Low-weight DPS remains below the warranted resolution, while larger weights are noisy at this reduced probe budget, consistent with the unstable high-

\zeta

maps in Figure 1 .

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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

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