Abbas Mammadov, Ozgur Kara, Kaan Oktay, Iskander Azangulov, Adil Kaan Akan, Hyungjin Chung, James Matthew Rehg, Yee Whye Teh
For linear Gaussian inverse problems under general Gaussian interpolants, we derive the exact posterior score in closed form and propose the EPS training objective, which achieves superior performance with fewer computations than existing methods.
Diffusion or flow-based models learn data priors by training a denoiser to reverse Gaussian corruption. To use this prior for linear inverse problems (e.g., image super-resolution, inpainting), one needs to sample from the posterior, but the prior provides the unconditional score, not the posterior score. Existing methods either use approximate measurement-matching corrections or require training a separate conditional restoration model.
We derive the exact posterior score in closed form for linear Gaussian inverse problems under general Gaussian interpolants (e.g., diffusion processes). This reduces to a denoising problem at an operator-dependent shifted pivot under an anisotropic noise covariance. Based on this identity, we propose the Exact Posterior Score (EPS) denoising training objective, which preserves the input/output structure of standard pretraining and can be trained from scratch or fine-tuned from a pretrained denoiser. At inference, EPS uses the same sampler as the backbone, with no likelihood gradients or projections.
Evaluated on five linear inverse problems across FFHQ and ImageNet, EPS outperforms training-free and training-based baselines on fidelity, perceptual, and distributional metrics. It also achieves comparable performance with roughly an order of magnitude fewer denoiser evaluations than gradient-based posterior samplers.