Exact Posterior Score Estimation for Solving Linear Inverse Problems
2026-06-15 • Machine Learning
Machine LearningComputer Vision and Pattern Recognition
AI summaryⓘ
The authors study how to improve solving linear inverse problems using diffusion and flow-based models, which usually learn to clean up noisy data. They found a way to exactly calculate the posterior information needed for these problems in a neat mathematical form, turning the solution into a special denoising task. Their approach, called Exact Posterior Score (EPS), keeps the standard structure of denoisers and can be trained from scratch or improved from existing models. When tested on image tasks, their method worked better than others and was much faster during use.
diffusion modelsflow-based modelslinear inverse problemsposterior scoreGaussian noisedenoisingscore matchingimage restorationsampling methodsFFHQ
Authors
Abbas Mammadov, Ozgur Kara, Kaan Oktay, Iskander Azangulov, Adil Kaan Akan, Hyungjin Chung, James Matthew Rehg, Yee Whye Teh
Abstract
Diffusion and flow-based models learn powerful data priors by training a denoiser to reverse Gaussian corruption. To use this prior to solve a linear inverse problem, one needs to sample from the posterior, but the score that the prior provides is the unconditional score, not the posterior score. Existing methods either steer a fixed pretrained denoiser with approximate measurement-matching corrections, or train a conditional restoration model that abandons the denoising structure of the prior. We derive the exact posterior score in closed form for linear Gaussian inverse problems under general Gaussian interpolants, and show that posterior sampling reduces to a denoising problem at an operator-dependent shifted pivot under an anisotropic noise covariance. We turn this identity into Exact Posterior Score (EPS), a denoising training objective that preserves the input/output structure of standard pretraining and can therefore be trained from scratch or fine-tuned from a pretrained denoiser. At inference, EPS uses the same sampler as the underlying backbone, with no likelihood gradients or projections. We evaluate EPS on five linear inverse problems across FFHQ and ImageNet, where it outperforms training-free and training-based baselines on fidelity, perceptual, and distributional metrics, while using roughly an order of magnitude fewer denoiser evaluations than gradient-based posterior samplers.