Score-Based Manifold Projection for Diffusion-Based Inverse Problems

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$\boldsymbol{s}_{\theta}(\boldsymbol{x}, t) \approx \nabla_{\boldsymbol{x}} \log p_{t}(\boldsymbol{x})$is ideally orthogonal to manifolds of constant log-likelihood at time$t$. By removing the component of the measurement gradient parallel to$\boldsymbol{s}_{\theta}$, 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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