Exact Posterior Score Estimation for Solving Linear Inverse Problems

Exact Posterior Score (EPS): Closed-form posterior sampling for linear inverse problems with diffusion models

A new arXiv preprint derives the exact posterior score in closed form for linear Gaussian inverse problems under general Gaussian interpolants, showing that posterior sampling reduces to a denoising problem at an operator-dependent shifted pivot under anisotropic noise covariance. The authors convert this identity into a training objective called Exact Posterior Score (EPS) that preserves the input/output structure of standard diffusion pretraining, enabling training from scratch or fine-tuning from a pretrained denoiser. EPS is evaluated on five linear inverse problems across FFHQ and ImageNet, outperforming both training-free and training-based baselines while requiring roughly an order of magnitude fewer denoiser evaluations than gradient-based posterior samplers.