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.

翻译：扩散模型与流模型通过训练去噪器逆转高斯损坏来学习强大的数据先验。若要利用该先验求解线性逆问题，需要从后验分布中采样，但先验提供的是无条件分数而非后验分数。现有方法要么通过近似测量匹配校正来引导固定的预训练去噪器，要么训练抛弃先验去噪结构的条件恢复模型。我们在一般高斯插值条件下推导了线性高斯逆问题的精确后验分数闭式解，并证明后验采样可归结为各向异性噪声协方差下、在算子依赖的偏移枢轴点处的去噪问题。我们将该恒等式转化为精确后验分数——一种保留标准预训练输入/输出结构的去噪训练目标，因此可从零训练或由预训练去噪器微调获得。在推理时，EPS使用与基础骨干网络相同的采样器，无需似然梯度或投影。我们在FFHQ和ImageNet上的五个线性逆问题中评估EPS，其在保真度、感知质量和分布度量上均优于免训练和基于训练的基线方法，同时梯度后验采样器所需的去噪器评估次数约少一个数量级。