Exact Posterior Score Estimation for Solving Linear Inverse Problems

2026-06-15

AI Summary

This work addresses the challenge of intractable posterior score computation in linear Gaussian inverse problems by deriving, for the first time, a closed-form expression that reformulates posterior sampling as a denoising task with anisotropic noise covariance and operator-dependent offset. The authors introduce a corresponding Expected Posterior Score (EPS) training objective that preserves standard denoising network architectures, enabling either training from scratch or fine-tuning of pretrained models without requiring likelihood gradients or projection operations during inference. Evaluated across five classes of linear inverse problems on FFHQ and ImageNet, the proposed method consistently outperforms existing approaches in fidelity, perceptual quality, and distributional alignment, while reducing the number of denoiser calls by nearly an order of magnitude.

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.

Problem

• linear inverse problems
• posterior score estimation
• diffusion models
• denoising prior
• Gaussian interpolants

Innovation

• posterior score estimation
• linear inverse problems
• diffusion models
• denoising prior
• Gaussian interpolants