Active grounding of a frozen diffusion prior requires jointly determining where new measurements should be taken and how they should be used to refine the current reconstruction. Posterior-ensemble-based methods can estimate acquisition utility from generated samples, but require repeated ensemble generation as observations accumulate and capture posterior geometry only through empirical statistics. This paper proposes GPARA, which learns a context-dependent graph surrogate over diffusion prediction residuals, inducing an explicitly reusable posterior response operator that propagates measurement innovations to unobserved variables and evaluates candidate measurements through weighted posterior-risk reduction. Under the matched surrogate, we show that the same response operator also determines expected one-step acquisition benefit and yields an analytic ranking consistent with expected reconstruction improvement. A bounded learned residual calibrates the analytic utility to account for surrogate mismatch, while a small prior ensemble is generated once and reconditioned to update risk weights without repeated diffusion posterior sampling during acquisition. Experiments on two reconstruction tasks spanning physical field and computer vision show consistent improvements in refinement and active acquisition over the evaluated baselines. Ablations further support the complementary roles of step-wise graph refinement, adaptive risk weighting, and analytically anchored calibration.
Figures & tables
Figure 1: Active grounding of a frozen diffusion prior. A task-specific conditional prior is pretrained offline and frozen. At deployment, context alone may produce an instance-mismatched prediction, and sparse target measurements are thus acquired to ground the prior to the current realization.
Figure 2: Overview of GPARA. Stage I performs response-aligned active acquisition: the graph posterior specifies how information propagates, while a state-adaptive risk matrix specifies where uncertainty reduction is valuable. A bounded residual scorer calibrates the analytic utility. After reaching the sensing budget, Stage II fixes the acquired observations and reuses the same graph-posterior conditioner to refine prediction throughout the reverse diffusion process.
Figure 3: Posterior refinement across observation rates. RMSE as a function of the fraction of observed target locations for (a) RFR and (b) DC. All methods use the same frozen DDPM prior and identical randomly sampled observations at every observation rate.
Method
Radio-field reconstruction
Depth completion
RMSE ↓
MAE ↓
SSIM ↑
PSNR ↑
RMSE ↓
MAE ↓
AbsRel ↓
δ1↑
DDPM
0.035 ± 0.003
0.025
0.926
30.2
0.350 ± 0.005
0.273
0.113
0.878
DPS
0.031 ± 0.002
0.021
0.937
31.1
0.347 ± 0.005
0.271
0.112
0.881
DDNM
0.033 ± 0.003
0.023
0.919
30.6
0.344 ± 0.005
0.267
0.110
0.883
FPS
0.034 ± 0.003
0.024
0.921
30.4
0.345 ± 0.004
0.269
0.111
0.883
DiffPIR
0.030 ± 0.002
0.021
0.934
31.2
0.342 ± 0.004
0.266
0.110
0.885
Table 1: Posterior refinement under 491 matched random observations, approximately 3% of RFR and 1% of DC target entries. RMSE is reported as mean ± standard deviation across five independent seeds; other metrics show the mean values.
Figure 4: Qualitative comparison of active acquisition and reconstruction. The first two rows show RFR examples with 10% observations, and the last two rows are DC examples under 5% .
Method
Radio-field reconstruction
Depth completion
RMSE ↓
MAE ↓
SSIM ↑
PSNR ↑
RMSE ↓
MAE ↓
AbsRel ↓
δ1↑
ADS
0.026 ± 0.001
0.018
0.947
33.1
0.338 ± 0.002
0.267
0.111
0.882
AdaSense + DPS
0.026 ± 0.001
0.018
0.947
33.2
0.339 ± 0.002
0.268
0.111
0.881
AdaSense + DDNM
0.028 ± 0.002
0.020
0.937
32.7
0.336 ± 0.002
0.267
0.110
0.883
AdaSense + FPS
0.029 ± 0.002
0.021
0.935
32.5
0.336 ± 0.002
0.266
0.110
0.883
AdaSense + DiffPIR
0.026 ± 0.001
0.018
0.948
33.4
0.334 ± 0.002
0.264
0.110
0.885
Table 2: Active acquisition with a matched 491-point sensing budget across five independent seeds.
Figure 7
Model
Injection
RFR
DC
Point Bayes
Step-wise
0.0327
0.346
Uniform graph
Final-only
0.0240
0.309
Learn Graph
Final-only
0.0233
0.307
Uniform graph
Step-wise
0.0195
0.267
Learn Graph
Step-wise
0.0191
0.262
Table 4: Mechanism ablations of GPARA . (a) Compares posterior mechanisms under identical random observations. (b) Compares acquisition criteria while keeping others fixed.
Appendix figures & tables2 assets
Supplementary material from the paper’s appendix.
Appendix
Method
5 steps
10 steps
20 steps
50 steps
100 steps
500 steps
DDPM
0.035 ± 0.0034
0.030 ± 0.0006
0.030 ± 0.0014
0.030 ± 0.0010
0.030 ± 0.0009
0.031 ± 0.0009
DPS
0.031 ± 0.0023
0.028 ± 0.0006
0.028 ± 0.0013
0.027 ± 0.0009
0.024 ± 0.0003
0.018 ± 0.0002
DDNM
0.033 ± 0.0031
0.028 ± 0.0005
0.028 ± 0.0013
0.027 ± 0.0009
0.027 ± 0.0007
0.027 ± 0.0009
FPS
0.034 ± 0.0033
0.027 ± 0.0005
0.027 ± 0.0009
0.026 ± 0.0002
0.025 ± 0.0003
0.022 ± 0.0004
DiffPIR
0.030 ± 0.0021
0.027 ± 0.0006
0.026 ± 0.0008
0.026 ± 0.0003
0.025 ± 0.0003
0.021 ± 0.0004
GPARA
0.019 ± 0.0007
0.017 ± 0.0002
0.016 ± 0.0003
0.016 ± 0.0002
0.016 ± 0.0001
0.016 ± 0.0002
Appendix
Table 5: Sensitivity to the number of reverse diffusion steps on RFR under 3% random observations. Results are full-region RMSE (mean ± standard deviation) over five paired seeds.
Score
RFR
DC
Spearman ↑
Regret ( ×10−5 ) ↓
Spearman ↑
Regret ( ×10−4 ) ↓
High-accuracy analytic
0.42 ± 0.26
1.26 ± 1.45
0.19 ± 0.27
9.23 ± 15.60
Deployed analytic
0.35 ± 0.27
1.14 ± 1.35
0.11 ± 0.25
9.51 ± 15.52
Calibrated utility
0.40 ± 0.25
1.06 ± 1.25
0.15 ± 0.24
3.60 ± 4.97
Direct learned scorer
0.44 ± 0.29
0.94 ± 1.28
0.23 ± 0.27
6.08 ± 14.75
Appendix
Table 6: Alignment between acquisition scores and empirical one-step reconstruction benefit under shared random prefixes.
Diffusion posterior sampling conditions diffusion priors on measurements, but data-consistency updates are typically scaled by hand-tuned guidance weights and can destabilize sampling under stiff, operator-dependent curvature. We replace scalar guidance with a per-noise-level damped Gauss--Newton correction computed in diffusion-state coordinates. The correction pulls likelihood gradients back through the denoiser, uses a one-sided curvature model that avoids forward denoiser Jacobians, and applies diffusion-calibrated rank-one damping aligned with the denoiser residual. Each correction is solved with matrix-free GMRES using automatic differentiation, and sampling proceeds with a variance-preserving Langevin transition with a closed-form drift/noise split. On FFHQ and ImageNet across inverse problems, it achieves competitive PSNR/SSIM/LPIPS while running markedly faster than most of the compared baselines; on accelerated MRI reconstruction, it achieves the best PSNR/SSIM among the compared baselines.
Seunghyeok Shin, Minwoo Kim, Dabin Kim +1
Department of Electrical and Computer Engineering, Inha University, Incheon, 22212, South Korea.
From a Bayesian perspective, score-based diffusion solves inverse problems through joint inference, embedding the likelihood with the prior to guide the sampling process. However, this formulation fails to explain its practical behavior: the prior offers limited guidance, while reconstruction is largely driven by the measurement-consistency term, leading to an inference process that is effectively decoupled from the diffusion dynamics. We show that the diffusion prior in these solvers functions primarily as a warm initializer that places estimates near the data manifold, while reconstruction is driven almost entirely by measurement consistency. Based on this observation, we introduce \textbf{DAPS++}, which fully decouples diffusion-based initialization from likelihood-driven refinement, allowing the likelihood term to guide inference more directly while maintaining numerical stability and providing insight into why unified diffusion trajectories remain effective in practice. By requiring fewer function evaluations (NFEs) and measurement-optimization steps, \textbf{DAPS++} achieves high computational efficiency and robust reconstruction performance across diverse image restoration tasks.
Hao Chen, Renzheng Zhang, Scott S. Howard
University of Notre Dame, Notre Dame, IN 46556, USA
Training-free conditional diffusion provides a flexible alternative to task-specific conditional model training, but existing samplers often allocate computation inefficiently: independent guided trajectories can vary widely in quality, and additional function evaluations along a single trajectory may not recover from poor early decisions. We propose Tempered Guided Diffusion (TGD), an annealed sequential Monte Carlo framework for training-free conditional sampling with diffusion priors. TGD targets tempered posterior distributions over the clean signal, using noisy diffusion states only as auxiliary variables for proposing reconstructions and propagating particles. Particles are reweighted by incremental likelihood ratios, resampled, and propagated across noise levels, concentrating computation on trajectories plausible under both the prior and observation. Under idealized exact-reconstruction assumptions, full TGD yields a consistent particle approximation to the posterior as the number of particles grows. For expensive reconstruction tasks, Accelerated TGD (A-TGD) retains early particle exploration but prunes to a single high-likelihood trajectory partway through sampling. Experiments on a controlled two-dimensional inverse problem and image inverse problems show improved posterior approximation and favorable wall-clock speed-quality tradeoffs over independent multi-trajectory baselines.
Andreas Makris, Paul Fearnhead, Chris Nemeth
Department of Mathematics and Statistics Lancaster University, UK · Department of Mathematics and Statistics2026 Lancaster University, UK