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GPARA: Graph-Posterior-Aligned Refinement and Active Acquisition for Grounding Diffusion Priors

Wangqian Chen, Hao Wang, Yumeng Zhang, Jiajia Guo, Junting Chen, Jun Zhang

arXiv:2609.34172v1cs.LG

TL;DR

Sparse measurements must both ground a frozen diffusion prior to the current target and be chosen for their effect on the unresolved reconstruction. GPARA reuses a learned graph-posterior response for refinement and acquisition, and experiments report improvements across radio-field reconstruction and depth completion.

  • Problem

    Active grounding must choose a limited set of measurements that improves the unresolved target, a response conventional inverse solvers do not generally expose explicitly.

    Source: §1 · pp. 1–2

  • Method

    GPARA learns a graph surrogate over prediction residuals, reusing its posterior response for refinement and weighted-risk acquisition with bounded residual calibration.

    Source: §1 · p. 2

  • Results

    Experiments on radio-field reconstruction and depth completion show improvements in both refinement and active sensing over evaluated baselines.

    Source: §5 · p. 9

  • Takeaways & Limitations

    A shared graph-posterior response supports measurement refinement and candidate evaluation, while ablations support step-wise refinement and weighted, analytically anchored acquisition.

    Source: §5 · p. 9

    The formulation uses equal-cost point observations under a limited sensing budget.

    Source: §2.2 · p. 3

Abstract

from arXiv · show

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.

1 INTRODUCTION

Sparse target measurements can ground a frozen diffusion prior to a particular realization, but choosing measurements requires understanding their effects across the unresolved target. GPARA uses a graph-based posterior response for both refinement and acquisition, with an analytic utility and bounded residual calibration.

1 INTRODUCTION

  • Conditional plausibility does not ensure agreement with the real target, because information absent from context can cause instance-specific prediction discrepancies.Direct target measurements provide complementary evidence about the current realization.Source: §1 · p. 1
  • Active grounding jointly selects a limited set of measurements and uses them to refine a frozen diffusion prior toward the current target realization.Acquisition must account for how a candidate affects the unresolved target, a response conventional inverse solvers do not generally expose explicitly.Source: §1 · pp. 1–2
  • Posterior-ensemble acquisition methods can estimate covariance-based utility, but AdaSense couples covariance estimation and acquisition cost to repeated ensemble regeneration as observations accumulate.Marginal uncertainty criteria also miss how cross-variable dependencies affect full-target reconstruction.Source: §1 · p. 2
  • GPARA learns a graph surrogate over prediction residuals, reusing its structured posterior response to propagate measurement innovations and evaluate candidate measurements with risk weights.Its analytic utility is retained as an anchor and augmented by bounded residual calibration.Source: §1 · p. 2
  • GPARA's contributions include graph-based residual refinement with a frozen diffusion backbone and acquisition whose analytic utility aligns with expected one-step reconstruction improvement.The framework also uses cached prior samples, stochastic probing, and sparse linear solves to avoid dense covariance construction and repeated posterior-ensemble generation at each acquisition round.Source: §1 · p. 2

2 BACKGROUND AND PROBLEM FORMULATION

The paper formulates active grounding as reconstruction from a frozen conditional diffusion prior combined with a limited budget of direct target measurements. It uses noisy point observations and seeks to jointly improve measurement selection and reconstruction.

2.1 CONDITIONAL DIFFUSION PRIOR

  • The conditional diffusion prior is pretrained and frozen at deployment, and the paper instantiates it with a DDPM while noting the framework is not restricted to that formulation.At inference, reverse diffusion starts from Gaussian noise; a conditional denoiser estimates the clean target at each reverse step before the standard reverse transition.Source: §2.1 · pp. 2–3
  • A DDPM progressively corrupts the clean target toward standard Gaussian noise over T forward steps, and its conditional denoising network predicts the injected noise.The conditioning context is denoted c and the clean target x0.Source: §2.1 · pp. 2–3

2.2 PROBLEM FORMULATION

  • Context-conditioned predictions can mismatch the actual target when environment dynamics, sensing imperfections, transient states, or fine-scale structures are not captured.Direct measurements are used to ground the offline prior to the current target.Source: §2.2 · p. 3
  • Because dense target measurements are costly or infeasible, the formulation limits acquisition to a budget Γ ≪ M and models selected observations as noisy measurements.The work considers equal-cost point observations.Source: §2.2 · p. 3
  • The population-level objective jointly chooses an acquisition policy under budget Γ and a reconstruction that minimizes reconstruction error.The policy may condition subsequent selections on information already obtained; Φ and ψ parameterize refinement and acquisition, respectively.Source: §2.2 · p. 3
  • The resulting problem couples posterior refinement, which determines how measurements modify the structured belief, with acquisition, which chooses where to measure for reconstruction benefit.The two decisions are made under the same sensing budget.Source: §2.2 · p. 3

3 METHOD

GPARA uses a graph posterior over prediction residuals as a shared response structure for measurement refinement and candidate evaluation. Its acquisition utility combines posterior-risk reduction with adaptive risk weights and bounded calibration, while the same conditioner supports refinement through reverse diffusion.

3.1 GRAPH-STRUCTURED BAYESIAN RESIDUAL REFINEMENT

  • At each reverse step, GPARA represents the discrepancy from the clean-state prediction as a residual correction, with graph geometry conditioned on context and the known prior.The posterior mean combines observation consistency with graph propagation: innovations enter at observed locations and the response propagates them to graph-coupled unobserved variables.Source: §3.1 · pp. 3–4
  • The graph-Gaussian residual surrogate uses a learned symmetric nonnegative adjacency matrix, where stronger edges encourage measurement corrections to propagate between coupled target variables.Its graph precision is shared across reverse steps and learned jointly across timesteps.Source: §3.1 · p. 4
  • The graph-posterior correction balances correction magnitude, graph-structured residual consistency, and measurement fidelity, then replaces the clean-state prediction in the next reverse transition.Its correction strength is controlled by ρg, with element-wise clipping to the normalized range.Source: §3.1 · p. 4

3.2 POSTERIOR-ALIGNED ACTIVE ACQUISITION

  • Acquisition evaluates candidates using a weighted posterior risk: the graph posterior captures how uncertainty reduction propagates, while the diagonal risk matrix specifies where reduction is valuable.The utility combines local reducibility at the candidate with the reach of its posterior response across correlated target variables.Source: §3.2 · p. 4
  • The same posterior response that propagates a newly acquired innovation also determines its expected weighted acquisition value.For a candidate measurement, the posterior-mean change is the response vector multiplied by its innovation.Source: §3.2 · p. 5
  • For 0 < ρg < 2, maximizing Ui(Sk) maximizes its expected improvement; under the matched surrogate, the largest-utility candidate is expected to maximize one-step improvement in the same loss.This result motivates using such utility as the acquisition anchor.Source: §3.2 · p. 5
  • GPARA acquires measurements progressively using calibrated utility, with optional small batches introduced as a computational tradeoff to reduce update frequency.After the budget is reached, it fixes the observations and applies graph conditioning to predictions throughout reverse diffusion.Source: §3.2 · pp. 5–6
  • The deployment procedure caches a small prior ensemble once to update adaptive risk weights and evaluates covariance-dependent utility matrix-free without forming a dense posterior covariance.It uses stochastic probing and PCG, while posterior corrections use fixed-depth Jacobi iterations; no parameters are updated at deployment.Source: §3.2 · p. 6

4 EXPERIMENTS

Experiments evaluate GPARA on radio field reconstruction and RGB-guided depth completion using frozen diffusion priors. GPARA improves fixed-observation refinement and active acquisition over evaluated baselines, while ablations support contributions from graph refinement and calibrated, risk-weighted acquisition.

4.1 EXPERIMENTAL SETUP

  • RFR uses RadioMapSeer and DC uses NYU Depth V2; each task has a separately pretrained frozen DDPM, and active baselines include ADS and AdaSense.Fixed-mask baselines receive identical random measurements; metrics include RMSE and MAE for both tasks, with task-specific additional metrics.Source: §4 · p. 6, §4.1 · pp. 6–7

4.2 FIXED-OBSERVATION POSTERIOR REFINEMENT

  • As observation rates rise, GPARA’s RMSE gain over the strongest baseline grows to over 60% on RFR and nearly 80% on DC.Figure 3 compares RMSE across observed-location fractions using the same frozen prior and identical random observations at each rate.Source: Figure 3 · p. 7, §4.2 · p. 7
  • Under matched random observations, GPARA reduces RFR RMSE and MAE by 36.7% and 47.6% versus DiffPIR, and DC RMSE and AbsRel by 23.4% and 26.4%, while raising δ1 by 6.6 percentage points.Table 1 reports refinement under 491 observations, approximately 3% of RFR and 1% of DC target entries.Source: Table 1 · p. 7, §4.2 · p. 7

4.3 POSTERIOR-ALIGNED ACTIVE ACQUISITION

  • Against AdaSense+Graph using the same backend, GPARA achieves 22.2% and 11.9% lower RMSE, indicating that its response-aligned acquisition strategy adds benefit beyond the reconstruction backend itself.Source: §4.3 · p. 8

4.4 MECHANISM AND ABLATION STUDIES

  • For posterior refinement, step-wise injection with a uniform graph lowers RMSE by 40.4% on RFR and 22.8% on DC versus pointwise conditioning; learned conductances add 2.1% and 1.9%.Moving from final-only to step-wise correction improves the learned-graph variant by 18.0% and 14.7%, respectively.Source: §4.4 · pp. 8–9, Table 4 · p. 9
  • For acquisition scoring, risk weighting adds 3.7% and 2.7% RMSE reduction, calibration adds 9.6% and 7.8%, and calibrated scoring beats direct learning by 18.4% and 11.9% on RFR and DC.The comparisons support retaining an analytic posterior-response anchor with learned residual corrections.Source: §4.4 · p. 9, Table 4 · p. 9

5 CONCLUSION

GPARA reuses a graph surrogate over prediction residuals to refine measurements and evaluate candidates for frozen diffusion priors. Experiments show improvements in refinement and active sensing, while the analysis remains limited to a graph-Gaussian surrogate and one-step acquisition.

5 CONCLUSION

  • GPARA improves refinement and active sensing on radio-field reconstruction and depth completion.Ablations also favor step-wise structured refinement over terminal correction and the weighted analytic acquisition criterion over a learned-only scorer.Source: §5 · p. 9
  • The graph surrogate over prediction residuals reuses a response structure for measurement refinement and candidate evaluation.Cached prior predictions, sparse conditioning, and bounded score calibration avoid repeated diffusion sample generation during acquisition.Source: §5 · p. 9
  • The analysis is limited to a graph-Gaussian surrogate and one-step acquisition; nonlinear observations and longer-horizon sensing remain future directions.The reproducibility statement says comparisons use the same frozen diffusion backbones and that code and configurations will be released publicly soon.Source: §5 · p. 9, REPRODUCIBILITY STATEMENT · p. 10

A ADDITIONAL RESULTS

The additional results examine reverse-step sensitivity and whether acquisition scores predict the benefit of a next measurement. Calibration improves deployed analytic one-step diagnostics on average, but these diagnostics do not directly predict full-budget closed-loop performance.

A ADDITIONAL RESULTS

  • GPARA reaches its best RFR performance within roughly 20 reverse steps, while DPS decreases from 0.031 RMSE at five steps to 0.018 at 500 steps.The comparison concerns full-region RMSE under 3% random observations over five paired seeds.Source: Table 5 · p. 13, A ADDITIONAL RESULTS · p. 14
  • The one-step diagnostic uses 60 matched prefix states across 20 test instances and evaluates 16 shared candidates per prefix.Each candidate is acquired independently and evaluated over four paired runs of Stage-II reconstruction; empirical benefit is the before-versus-after reconstruction MSE reduction.Source: A ADDITIONAL RESULTS · p. 14
  • Calibration raises deployed-score Spearman correlation from 0.35 to 0.40 on RFR and from 0.11 to 0.15 on DC, while reducing regret.The direct learned scorer has higher one-step rank on both tasks and lowest RFR regret; the calibrated score has lowest DC regret.Source: A ADDITIONAL RESULTS · p. 14
  • One-step diagnostics use shared random states, whereas full-budget rollouts evolve through policy-dependent states; the calibrated policy nevertheless achieves lower final RMSE than the direct scorer.Each acquisition changes the posterior and subsequent decisions, and the analytic anchor recomputes marginal value and redundancy from updated posterior geometry.Source: A ADDITIONAL RESULTS · p. 14

B.1 PROOF OF PROPOSITION 1

The proof establishes that the graph-conditioned residual posterior is proper and unique by showing its precision is positive definite. This holds even when the graph is disconnected.

B.1 PROOF OF PROPOSITION 1

  • For nonnegative symmetric graph conductances, the Laplacian is positive semidefinite, so τ > 0 gives graph precision Qg ⪰ τI ≻ 0.The posterior precision also includes positive-definite observation precision, yielding a positive-definite conditional precision.Source: §B.1 · p. 14
  • The posterior precision remains positive definite without requiring the graph to be connected.The proof concludes that AS ⪰ τI ≻ 0.Source: §B.1 · p. 14
  • Completing the square gives a proper Gaussian conditional residual posterior and establishes the posterior formulas and uniqueness.The derivation starts from the residual and observation model, expands the negative log-density, and completes the square.Source: §B.1 · p. 15

B.2 PROOF OF PROPOSITION 2

The proof connects a candidate measurement's innovation to a graph-posterior mean update and to weighted covariance-risk reduction. Under the surrogate, these describe the same one-step effect.

B.2 PROOF OF PROPOSITION 2

  • A prospective measurement's innovation updates the target posterior mean by the graph-posterior response scaled by the innovation.The response is expressed using the posterior covariance column for the candidate and the measurement-noise variance.Source: §B.2 · p. 15
  • An additional independent point measurement updates posterior precision through a rank-one change, enabling a covariance-risk reduction formula.The derivation applies the rank-one inverse identity while holding the current weighting matrix fixed across the hypothetical update.Source: §B.2 · p. 15
  • Under the surrogate, covariance reduction and expected squared posterior-mean change describe the same one-step effect.The prospective measurement is formed for the current observation set and candidate, with independent noise and residual as specified in the proof.Source: §B.2 · p. 15

B.3 PROOF OF COROLLARY 1

The proof shows that, under the unclipped matched comparison and a shared refinement strength in the stated range, expected one-step weighted-error improvement has the same maximizing candidates as the analytic utility.

B.3 PROOF OF COROLLARY 1

  • Adding candidate i changes the unclipped estimate by the graph-response term ρ_g u_iν_i.The proof then expands the weighted squared-error difference using this innovation-dependent change.Source: §B.3 · p. 16
  • When ρ_g ≠ 1, the scaled estimate is generally not the surrogate posterior mean, so the proof accounts for its residual error separately.It uses the zero conditional innovation mean and the stated cross-moment to simplify the expectation.Source: §B.3 · p. 16
  • For a common 0 < ρ_g < 2, expected weighted-error improvement and analytic utility have the same maximizing candidates, including ties.The candidate-independent factor ρ_g(2 − ρ_g) is positive; equality is of maximizing sets when ties occur.Source: §B.3 · p. 16

B.4 A BOUNDED-SCORE CONSEQUENCE

This appendix bounds how residual calibration can alter the analytic ranking, describes approximate matrix-free posterior calculations and adaptive risk weighting, and treats batching as an approximation to single-point acquisition.

B.4 A BOUNDED-SCORE CONSEQUENCE

  • If the analytic utility margin exceeds 2β_acq, bounded residual calibration cannot change the analytic maximizer.Calibration can refine close candidate rankings but cannot overcome a sufficiently strong analytic preference.Source: §B.4 · p. 16

B.5 MATRIX-FREE GRAPH SOLVES AND COVARIANCE PROBING

  • Posterior refinement and acquisition statistics use sparse linear solves; fixed-depth Jacobi approximates the mean, and the finite-probe stabilized utility ratio is generally biased.The raw moment estimators are unbiased under exact solves, but the stabilized ratio need not be unbiased at finite probe count.Source: §B.5 · p. 17

B.6 CONSTRUCTION OF THE OBSERVATION-SET-ADAPTIVE RISK MATRIX

  • The graph posterior describes how candidate measurements affect target variables, while risk weights emphasize locations with greater remaining disagreement; cached samples are reconditioned without regenerating diffusion samples.The small prior ensemble is generated once, and its conditioned variance adapts the risk emphasis to the evolving observation set.Source: §B.6 · pp. 17–18

B.7 BATCHED ACQUISITION BETWEEN POSTERIOR REFRESHES

  • Batches select up to eight candidates, penalizing response overlap after the first; measurements are not assimilated until the batch ends, so batching approximates rather than optimizes joint risk.The graph posterior, risk matrix, and scores are recomputed at the next refresh.Source: §B.7 · p. 18

C NETWORK ARCHITECTURES AND TRAINING

GPARA uses offline-trained graph and residual-scoring networks that remain fixed at deployment. The graph predictor is trained through differentiable conditioning, while the scorer calibrates analytic acquisition utility.

C NETWORK ARCHITECTURES AND TRAINING

  • The graph predictor and residual scorer are trained sequentially offline and remain fixed during deployment.The diffusion backbone is frozen while the graph predictor is learned, after which the graph predictor is frozen for scorer training.Source: C NETWORK ARCHITECTURES AND TRAINING · p. 19

C.1 NETWORK INTERFACES.

  • A lightweight UNet predicts four canonical edge maps for a sparse symmetric graph, while the scorer outputs a residual score for each target location.The scorer combines a convolutional encoder-decoder, graph-message-passing blocks, and budget conditioning.Source: §C.1 · p. 19

C.2 LEARNING THE GRAPH THROUGH POSTERIOR REFINEMENT.

  • The graph predictor is trained through differentiable conditioning, with losses for reconstruction, unobserved-region propagation, observation consistency, baseline preservation, and residual affinity.The learned conductances couple prediction-residual corrections rather than directly enforcing similarity between reconstructed target values.Source: §C.2 · pp. 19–20

C.3 LEARNING THE ACQUISITION RESIDUAL SCORER

  • The residual scorer trains on varied sampled observation sets against a one-step teacher, combining normalized analytic utility with bounded correction and distillation, selection, calibration, margin, and anchor-relative losses.The teacher uses a linear-response estimate of one-step improvement; the margin term separates an incorrect greedy winner, while the anchor-relative term prevents lower teacher-expected value than the analytic-anchor distribution.Source: §C.3 · pp. 20–21

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