Score-Based Manifold Projection for Diffusion-Based Inverse Problems
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Bibliographic record
Abstract
Inverse problems such as inpainting, deblurring, and super-resolution benefit significantly from generative diffusion models, which serve as powerful learned priors. However, incorporating measurement-consistency gradients naively can push the intermediate solutions off the high-likelihood manifold encoded by the diffusion model, leading to artifacts or suboptimal reconstructions. In this paper, we propose a scorebased manifold projection framework that leverages the internal score function of the diffusion model itself to preserve manifold fidelity. Specifically, we exploit the fact that the diffusion score <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\boldsymbol{s}_{\theta}(\boldsymbol{x}, t) \approx \nabla_{\boldsymbol{x}} \log p_{t}(\boldsymbol{x})$</tex> is ideally orthogonal to manifolds of constant log-likelihood at time <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$t$</tex>. By removing the component of the measurement gradient parallel to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\boldsymbol{s}_{\theta}$</tex>, our method constrains each update to remain (to first order) tangent to the learned data manifold. Theoretically, we prove that our approach preserves proximity to the manifold more effectively than an un-projected update, and empirically, we demonstrate improved robustness and quality on various inverse problems, including deblurring, inpainting, and super-resolution. Our results show that scorebased manifold projection not only reduces artifacts but also maintains the fidelity of the reconstructions, offering a simple yet effective enhancement to measurement-guided diffusion solvers.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it