Measuring Effective Data Resolution in Guided Diffusion Posteriors

1 Introduction

• Accuracy and calibrated uncertainty do not reveal which parts of a reconstruction sensors determined, so plausible output may still rely mainly on the prior or contain unjustified detail.The paper frames data dependence as distinct from scoring the reconstruction and its marginals.Source: §1 · p. 1
• The problem-side reference, dofref, counts directions the data can constrain and is bounded by the sensor count, often substantially lower in ill-posed problems.It is computed from a Gauss–Newton Hessian without running a sampler.Source: §1 · pp. 1–2
• The sampler-side quantity, dofsamp, measures how strongly the reconstruction map responds to data and can differ from dofref, largely as a function of guidance weight.The paper notes this quantity is not currently reported and documents the guidance-weight dependence for an inspected DiffusionPDE configuration.Source: §1 · p. 2
• The resolution-fidelity ratio compares sampler-realised resolution with the inverse problem’s warranted resolution: near 1 matches, below 1 under-responds, and above 1 exceeds what noise justifies.The ratio is ρres = dofsamp/dofref.Source: §1 · p. 2
• A black-box perturbation estimator combines data perturbations with stochastic trace estimation, uses two reconstruction-map evaluations per probe regardless of field dimension, and yields a spatial influence map.The paper also reports validating the estimator against exact references.Source: §1 · p. 2
• In 1D, guidance moves effective resolution from 0.15 to 1.27 times warranted, with no weight jointly correcting mean, spread, and resolution; a comparable 2D span appears, but the controlled scheme ordering does not persist.The 1D setting uses an exact prior and score; the 2D setting has a non-Gaussian prior and trained score.Source: §1 · p. 2

2 Related work

• DiffusionPDE, score-based inverse-problem samplers, data assimilation, and function-space variants steer an unconditional prior using observations, but approximate the generally intractable exact guidance term.DPS uses a surrogate likelihood gradient at a denoised estimate; ΠGDM, DDNM, and DDRM motivate other approximations used in the experiments.Source: §2 · pp. 2–3
• Existing neural PDE surrogate and calibration work quantifies operator error or reported uncertainty, whereas this paper asks how much the sensors determined—a complementary property of data and sampler.The related work includes function-valued Gaussian-process uncertainty, calibration methods, pretrained PDE models, and neural operators.Source: §2 · p. 3
• Resolution operators and degrees of freedom for signal are established concepts, as are perturbation-based generalized degrees of freedom and trace estimators; the contribution is their adaptation to posteriors available only by sampling.The adaptation requires common random numbers, a noise-scaled perturbation window, and smooth coupling, and is certified against exact references.Source: §2 · p. 3

3 Effective data resolution

• The resolution operator records how a truth perturbation at one location propagates into the estimate at another; an ill-posed problem yields smoothing rather than perfect recovery.Its entries form an N × N object, and perfect recovery would give R = I.Source: §3 · p. 3
• The diagonal R(x, x) maps where data influence lands: values near 1 indicate the estimate tracks the truth locally, while values near 0 indicate stronger reliance on prior information.This is the spatial map of data influence.Source: §3 · p. 3
• Summing the resolution operator’s diagonal defines dofsamp, a property of the sampler at its guidance weight rather than of the inverse problem.Different samplers targeting the same posterior can return different dofsamp values.Source: §3 · p. 4
• The reference dofref is the inverse problem’s information budget: each direction contributes near 1 when well constrained and near 0 when the data do not see it.For nonlinear forward maps, the paper linearises at the posterior mean.Source: §3 · p. 4
• The ratio ρres compares sampler resolution with warranted resolution; values below or above 1 indicate less or more data response than warranted, respectively.In the reported settings, below-1 resolution coincides with weaker data response and above-1 resolution with variance ratios far above one and 100% coverage.Source: §3 · p. 4
• Resolution complements coverage, CRPS, and variance ratios because it measures reconstruction dependence on observations, which fixed-observation diagnostics do not directly access.Those conventional diagnostics score the distribution the sampler reports.Source: §3 · p. 4

4 Black-box estimation

• Cyclicity of the trace reduces the target to an n × n observation-space quantity, avoiding construction of the full N × N resolution operator.The dimension n is the sensor count, while N is the field dimension.Source: §4 · p. 4
• Averaging P independent Rademacher probes estimates the trace, with sensor count n and probe count P treated as independent quantities.The estimator uses the observation-space response rather than forming the field-space operator.Source: §4 · p. 4
• Each probe requires two reconstruction-map evaluations; with an ensemble of E trajectories per map estimate, the total is 2PE trajectories, while estimator evaluations 2P do not grow with field dimension.The field dimension still affects the cost of an individual sampler trajectory.Source: §4 · p. 4
• Using the n coordinate directions instead of random probes gives the spatial diagonal map in 2n evaluations without probe variance, suitable for sparse-sensor settings.This procedure recovers the columns of ∂m/∂y and hence R(x, x).Source: §4 · p. 5
• Stochastic-sampler perturbation runs require common random numbers and smooth coupling; the perturbation must also be scaled to observation noise and lie within a scheme-dependent valid window.Metropolis accept/reject discontinuities can split perturbed paths, while residual-normalised DPS has a lower-scale edge not seen in schemes without that normalisation.Source: §4 · p. 5

5 Validation

• At P = 128 probes, estimator bias is −0.009 ± 0.028 against the exact reference.The validation uses a backward-heat problem with a closed-form Gaussian posterior and dofref = 5.971.Source: §5 · p. 5
• Increasing P from 8 to 128 lowers per-run standard deviation from 0.897 to 0.179, a factor of 5.0 versus the predicted 4.0.The standard error follows the ordinary Monte-Carlo rate 1/P.Source: §5 · p. 5
• Preconditioned unadjusted Langevin gives 6.33 ± 0.59 against the exact 5.97, while matching the closed-form posterior to 0.3% in mean and 0.6% in standard deviation.Without low-rank posterior preconditioning, effective sample size was about 40 of 150k steps and the sampler was underdispersed by 2 to 8×; preconditioning raised ESS to 13,753.Source: §5 · p. 5
• Resolution measurements require a trustworthy posterior mean; guided samplers also require attention to perturbation scale because their estimates vary with ϵ.The exact-score control remained stable while high-weight DPS became realisation-sensitive at reduced probe budgets.Source: §5 · p. 5

6 Guidance controls apparent information content

• No DPS weight jointly matches mean, spread, and resolution: ζ = 1 has mean error 0.074 but 13.8× the correct variance, while ζ = 3 has ρres = 1.053 but 97.7× the correct variance and worst CRPS.The ζ = 0.3 weight with the best distributional match resolves only 0.80× the warranted degrees of freedom.Source: Table 1 · p. 6, §6 · pp. 6–7
• From 0.15 to 1.27 times dofref, DPS’s realised resolution varies across the guidance-weight sweep, which rises past the reference and then falls.The sweep uses ϵ = σy; Figure 1(b) shows multiple crossings of dofref.Source: Figure 1 · p. 6, §6 · p. 6
• At ζ = 3, DPS concentrates influence near x = 24 and spans 0.019 to 0.464, versus the exact posterior’s nearly uniform 0.107 to 0.138.Across 16 realisations, its trace ranges from −24.0 to 68.2; the exact-score control reproduces the closed-form map to 0.002 with zero spread.Source: Figure 1 · p. 6, §6 · p. 7
• The DPS data term is evaluated at the denoised estimate and scaled by residual norm, so changing ζ alters both data-pull strength and where corrections dominate diffusion.For this DPS implementation the local derivative is singular as ϵ → 0, making reported resolution a finite-scale sensitivity rather than an infinitesimal Jacobian.Source: §6 · p. 7
• Matching dofref is not a tuning rule: the non-monotone sweep has roots near ζ ≈ 0.6 and ζ ≈ 4, associated with very different samplers.The measurement reports a discrepancy but does not select a setting.Source: §6 · p. 7
• Exact guidance, ΠGDM, and projection land within 2% of dofref, while the controlled setup attributes deviations from the reference posterior to the guidance scheme.The 64-dimensional backward-heat field has 8 sensors, a Gaussian prior, a closed-form diffused score, and a known true posterior.Source: §6 · pp. 5–7
• Because the prior and score are exact in this experiment, whether these conclusions survive their mismatch remains open.The experiment establishes guidance-dependent resolution only within this controlled setting.Source: §6 · p. 7

7 Does resolution behaviour transfer? A nonlinear test

• The test uses 576 field unknowns, 64 sensors, a nonlinear Darcy map, a non-Gaussian prior, and a trained score, with latent pCN providing an exact posterior reference.The reference resolution is dofref = 43.15, or 7.5% of the field unknowns.Source: §7 · p. 7
• Across DPS weights, ρres ranges from 0.15 to 1.37; projection over-resolves by 50%, while DPS at ζ = 1 is closest to warranted resolution at ρres = 1.026.The range is comparable to the controlled setting’s 0.15 to 1.27; ΠGDM diverged in this implementation and is excluded.Source: §7 · p. 7, Table 2 · p. 8
• Unlike the controlled case, projection does not retain its resolution advantage; refreshing JJ⊤ every 25, 5, or 1 steps gives ρres = 1.501, 1.482, or 1.479.The authors report that the projection result is not caused by the stale-Jacobian approximation.Source: §7 · pp. 7–8

8 Use and scope

• For residual-normalised DPS, the small-scale derivative is singular, so the study reports perturbations from 0.1 to 1 times observation-noise scale and makes no claim about other scalings.In 2D Darcy, using the 1D absolute perturbation ϵ = 10^-2 perturbs data by 30× the noise and gives dofsamp = 10.0 against 43.1.Source: §8 · p. 8
• The estimator requires a reproducible, smoothly coupled sampler under common random numbers; pCN accept/reject discontinuities can split perturbed paths, so the study uses its Gauss–Newton reference instead.Coordinate-wise estimation costs 2n evaluations, while the calibration suggests about 12 probes for 10% accuracy or 289 for 2%.Source: §8 · p. 8
• Claims are limited to the implemented guidance schemes and require a trustworthy dofref; nonlinear resolution is interpreted as a posterior-mean linearisation, with severe over-dispersion limiting measurability.The comparison uses analytic resolution in 1D and latent pCN in 2D; other applications require a defensible reference resolution.Source: §8 · p. 8

9 Conclusion

• Guided diffusion can transfer the wrong amount of data information: guidance weight shifts apparent resolution, no single weight jointly corrects mean, spread, and resolution even with an exact score and prior, and controlled method ordering need not persist in nonlinear learned settings.The paper estimates ρres and the spatial map R(x, x) from black-box sampler queries by comparing dofsamp with dofref.Source: §9 · p. 8

A The perturbation scale: window and singular behaviour

• The operating scale balances secant bias against across-realisation variance: ϵ = σy is used in 1D, while 2D uses ϵ = 0.5 σy and retains some probe-scale sensitivity at larger ζ.For residual-normalised DPS, the derivative is singular as ϵ approaches zero; the authors caution that other guidance scalings were not tested.Source: A The perturbation scale: window and… · pp. 10–11

B Three formulations that do not work

• Three alternatives fail in distinct ways: zero-prior variance diverges, variance removal duplicates the variance map for stationary priors, and elasticity downweights well-informed directions.The elasticity instead measures where the prior still dominates, the complement of the target question.Source: B Three formulations that do not work · p. 11

C Certification details

• On the stochastic sampler, preconditioned unadjusted Langevin returns 6.33 ± 0.59 versus the exact 5.97; over a 16× probe increase from P = 8 to 128, per-run standard deviation falls 0.897 to 0.179.This is a factor of 5.0, against the 4.0 predicted by 1/√P.Source: C Certification details · p. 11

D Full guidance-weight sweep

• Across the guidance-weight sweep, effective resolution is non-monotone and spans 0.15 to 1.27 times dofref; the ζ = 10 estimate is especially noisy, with standard error larger than its value.The tested weights are practical configuration choices, including hand-tuned DiffusionPDE weights spanning three orders of magnitude.Source: D Full guidance-weight sweep · p. 12

• The random compressed-sensing check uses an exact Gaussian prior and posterior reference, with warranted resolution dofref = 15.78.The random measurements are informative under this prior.Source: E Additional breadth and stability… · p. 12
• The compressed-sensing check is not a third application benchmark and does not address released checkpoints, independent codebases, or image-domain solvers.Source: E Additional breadth and stability… · p. 12
• ρres provides information not contained in CRPS, coverage, or posterior variance.Source: E Additional breadth and stability… · p. 12
• Projection has nearly correct ρres but about 2.1× posterior variance, while DPS can have near-reference ρres with much larger spread.The exact-score control again recovers dofref.Source: Table 4 · p. 13
• At P = 12 across six realisations, the exact control stays stable, low-weight DPS remains below warranted resolution, and larger weights are noisy.This stress test uses half the main-table probe budget, so its estimates are not headline estimates.Source: E Additional breadth and stability… · p. 12, Table 5 · p. 13
• 97.7× posterior variance coincides with DPS realisation values from −9.94 to 17.21; across-realisation variation dominates probe standard error once over-dispersed.Source: E Additional breadth and stability… · p. 13