Hilbert-Scale Stability of Posterior Diffusion Priors for Ill-Posed Signal Recovery

Bahman Moraffah; Raha Moraffah

cs.LGOct 19, 2026

Hilbert-Scale Stability of Posterior Diffusion Priors for Ill-Posed Signal Recovery

Authors: Bahman Moraffah, Raha Moraffah

Organizations: Department of Computer Science, Data Science, and Artificial Intelligence, Worcester Polytechnic Institute

Abstract

Indirect and noisy observations are difficult to invert when the forward operator smooths out the unknown signal. In this case, the unstable modes are precisely the modes that the data reveals least, thus the success of any reconstruction method depends on how prior information interacts with the geometry of the operator across regularity scales. Diffusion priors have become ubiquitous for such problems; however, the current theory still does not determine when a learned score remains informative under ill-posed forward operators. In this paper, we study posterior diffusion for linear inverse problems on Hilbert scales and derive a stability theory under an explicit score-amplification condition. Under a smoothing condition on the forward map, a source condition on the truth, and weak-norm control of score error, we derive non-asymptotic recovery bounds in $\Hs^s$ that separate observation noise, score approximation, and reverse-time discretization. These bounds give a recoverable regularity threshold determined jointly by operator smoothing, truth regularity, score regularity, and the posterior correction. The threshold specifies the Hilbert-scale norms $\Hs^s$ in which weak-norm score error remains controlled after propagation through the posterior reverse dynamics. For larger regularity indices $s$ , the inverse geometry magnifies the same error, and the Hilbert-scale recovery estimate no longer gives uniform stability. We also prove that a posterior correction adapted to the forward operator improves stability relative to isotropic guidance in the regularity ranges where the smoothing action of the forward map matters most. The recovery estimates give a regularity-sensitive theory of posterior diffusion and show how learned score error, operator smoothing, and reconstruction norm jointly determine the stable range predicted by the analysis.

Figures & tables

Component	Setting
Dimensions	$d\in\{256,512,1024\}$
Forward smoothing	$r\in\{0.5,1,2,3\}$
Truth smoothness	$\beta\in\{1,2,3\}$
Noise levels	$\sigma\in\{10^{-3},10^{-2},3\times 10^{-2}\}$
Target norms	$s=0,0.05,\dots,2.50$
Trials per configuration	$120$

Table 1: Implementation details for Experiment 1.

Figure 1: Dense transition atlas. Each panel tracks normalized error across target regularity using a denser

s

grid and confidence ribbons.

Figure 2: Empirical recoverability heatmaps under two tolerance levels. The panels summarize how the critical regularity varies with $(r,\beta)$ .

$r$	$\beta=1$ post	$\beta=1$ learned	$\beta=1$ weak	$\beta=2$ post	$\beta=2$ learned	$\beta=2$ weak	$\beta=3$ post	$\beta=3$ learned	$\beta=3$ weak
$0.5$	0.30	0.30	0.30	0.55	0.55	0.55	0.81	0.81	0.81
$1.0$	0.34	0.34	0.34	0.58	0.58	0.58	0.85	0.85	0.85
$2.0$	0.40	0.40	0.40	0.62	0.62	0.62	0.89	0.89	0.89
$3.0$	0.46	0.46	0.46	0.70	0.70	0.70	0.93	0.93	0.93

Table 2: Empirical recoverability index

\widehat{s}_{\mathrm{crit}}

d=1024

\sigma=10^{-2}

, and tolerance

C_{\mathrm{tol}}=5

. Larger values indicate stable recovery deeper into stronger Hilbert norms.

$r$	Method	$s=0$	$s=1$	$s=2$
$0.5$	post	1.242e-03	2.657e-02	5.203e+00
$0.5$	oracle	1.291e-03	2.752e-02	5.362e+00
$0.5$	theta	1.243e-03	2.660e-02	5.476e+00
$0.5$	theta_weak	1.242e-03	2.657e-02	5.203e+00
$1.0$	post	2.600e-03	4.388e-02	5.452e+00
$1.0$	oracle	2.698e-03	4.550e-02	5.628e+00

Table 3: Mean reconstruction error at three target regularities for the representative case

d=1024

\beta=2

, and

\sigma=10^{-2}

Figure 3: Weak-score stress plot. Values above one indicate amplification of the strong-norm error relative to the exact posterior mean.

$d$	$\sigma$	$\mathcal{E}_{\theta,\mathrm{weak}}(2)/\mathcal{E}_{\mathrm{post}}(2)$
$256$	$1e-03$	0.998
$256$	$1e-02$	1.000
$256$	$3e-02$	1.000
$512$	$1e-03$	1.000
$512$	$1e-02$	0.999
$512$	$3e-02$	0.999

Table 4: Strong-norm stress ratio for the weakly perturbed score under the more ill-posed case

r=3

and

\beta=2

Component	Setting
Signal dimension	$d=256$
Noise level	$\sigma=0.04$
Train / validation / test	$1800/320/54$
Score features	$x$ , $x^{3}$ , $x^{5}$
Noise schedule	$10$ levels from $1.0$ to $0.03$
Reverse sampler	$4$ steps per level, $8$ chains

Table 5: Implementation details for Experiment 2.

Figure 4: Top: representative reconstructions for the missing-frequency operator. Middle: spectral energy of the truth, reconstructions, and reconstruction error. Bottom: high-frequency band error across forward operators.

Operator	Method	$H^{0}$	$H^{1}$	$H^{2}$	$E_{\mathrm{high}}$
A1	Tik	8.010e+01	4.802e+05	1.170e+10	1.408e+01
A1	Sob	1.849e+02	8.573e+04	9.019e+07	1.802e-02
A1	DPS	2.553e+02	9.791e+04	2.497e+08	1.556e-01
A1	GeoPD	2.676e+02	9.814e+04	2.395e+08	1.215e-01
A2	Tik	1.597e+02	8.466e+04	9.028e+07	1.838e-02
A2	Sob	2.158e+02	8.640e+04	9.021e+07	1.802e-02

Table 6: Reconstruction metrics for non-Gaussian one-dimensional signals. The geometry-aware sampler is most distinct from isotropic guidance in stronger norms and in the high-frequency band.

Component	Setting
Spectral domain	$d=512$ , $\beta=2$ , $\sigma=10^{-2}$ , $r\in\{0.5,1,2,3\}$
Learned-signal domain	$d=256$ , operators $\mathcal{A}_{2}$ and $\mathcal{A}_{3}$
Guidance sweep	$\gamma\in\{0.25,0.40,0.55,0.70,0.85,1.00,1.20,1.45,1.70,2.00,2.30\}$
Sweep norms	$s\in\{0,0.5,1,1.5,2\}$
Reported matched norms	$H^{0}_{d}$ , $H^{1}_{d}$ , $H^{2}_{d}$
Reverse sampler	$9$ levels, $4$ steps per level, $10$ chains

Table 7: Implementation details for Experiment 3.

Domain	Scenario	$\gamma_{\mathrm{iso}}$	$\gamma_{\mathrm{geo}}$	Residual gap
spectral	r=0.5	1.70	2.30	3.396e-04
spectral	r=1.0	1.70	2.30	3.969e-03
spectral	r=2.0	1.00	2.00	7.609e-04
spectral	r=3.0	0.40	0.70	2.247e-03
learned_signal	A2	0.85	0.85	1.558e-05
learned_signal	A3	1.00	0.40	6.957e-04

Table 8: Guidance strengths selected for the matched-residual comparison

Figure 5: Residual-error tradeoff curves for representative severe cases in Experiment 3. Each curve traces the effect of guidance strength before the matched-residual point is selected.

Domain	Scenario	Method	Residual	$H^{0}$	$H^{1}$	$H^{2}$
spectral	r=0.5	isotropic	5.297e-01	7.801e-01	4.592e+04	7.389e+09
spectral	r=0.5	geometry_aware	5.447e-01	8.454e-01	4.572e+04	7.317e+09
spectral	r=1.0	isotropic	4.774e-01	7.793e-01	4.662e+04	7.528e+09
spectral	r=1.0	geometry_aware	4.449e-01	7.328e-01	4.676e+04	7.550e+09
spectral	r=2.0	isotropic	5.381e-01	8.918e-01	4.218e+04	6.670e+09
spectral	r=2.0	geometry_aware	4.721e-01	7.765e-01	4.224e+04	6.664e+09

Table 9: Experiment 3 matched-residual comparison of isotropic and geometry-aware posterior guidance. Residuals are tuned to be comparable within each scenario before comparing reconstruction stability.

Domain	Scenario	$s$	Gain ratio
spectral	r=0.5	$0$	0.923
spectral	r=0.5	$1$	1.004
spectral	r=0.5	$2$	1.010
spectral	r=1.0	$0$	1.064
spectral	r=1.0	$1$	0.997
spectral	r=1.0	$2$	0.997

Table 10: Relative gain ratio

\|\hat{x}_{\mathrm{iso}}-x^{\dagger}\|_{H^{s}}^{2}/\|\hat{x}_{\mathrm{geo}}-x^{\dagger}\|_{H^{s}}^{2}

at matched residual in Experiment 3.

Figure 6: Matched-residual gain heatmaps for Experiment 3. Values above one favor geometry-aware guidance.

Figure 7: Mode-wise reconstruction error and posterior variability for representative severe cases in Experiment 3.

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Diffusion and flow-based models learn powerful data priors by training a denoiser to reverse Gaussian corruption. To use this prior to solve a linear inverse problem, one needs to sample from the posterior, but the score that the prior provides is the unconditional score, not the posterior score. Existing methods either steer a fixed pretrained denoiser with approximate measurement-matching corrections, or train a conditional restoration model that abandons the denoising structure of the prior. We derive the exact posterior score in closed form for linear Gaussian inverse problems under general Gaussian interpolants, and show that posterior sampling reduces to a denoising problem at an operator-dependent shifted pivot under an anisotropic noise covariance. We turn this identity into Exact Posterior Score (EPS), a denoising training objective that preserves the input/output structure of standard pretraining and can therefore be trained from scratch or fine-tuned from a pretrained denoiser. At inference, EPS uses the same sampler as the underlying backbone, with no likelihood gradients or projections. We evaluate EPS on five linear inverse problems across FFHQ and ImageNet, where it outperforms training-free and training-based baselines on fidelity, perceptual, and distributional metrics, while using roughly an order of magnitude fewer denoiser evaluations than gradient-based posterior samplers.

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