Exact Posterior Score Estimation for Solving Linear Inverse Problems
English summary
This paper derives the exact posterior score in closed form for linear Gaussian inverse problems under general Gaussian interpolants, reducing posterior sampling to a denoising problem at an operator-dependent shifted pivot with anisotropic noise covariance. The method, Exact Posterior Score (EPS), defines a denoising training objective that mirrors standard pretraining, enabling training from scratch or fine-tuning a pretrained denoiser. At inference, EPS uses the identical sampler as the base model, eliminating the need for likelihood gradients or projections. Evaluated on five linear inverse tasks across FFHQ and ImageNet, EPS surpasses both training-free and training-based baselines in fidelity, perceptual, and distributional metrics while requiring roughly an order of magnitude fewer denoiser evaluations than gradient-based posterior samplers.
Chinese summary
该论文推导了线性高斯逆问题在一般高斯插值下的闭式精确后验评分，将后验采样简化为在算子相关偏移锚点及各向异性噪声协方差下的去噪问题。精确后验评分（EPS）方法定义了一个与标准预训练结构一致的去噪训练目标，支持从头训练或微调预训练去噪器。推理时，EPS直接使用骨干采样器，无需似然梯度或投影。在FFHQ和ImageNet上的五种线性逆任务评估中，EPS在保真度、感知和分布指标上均优于无训练和基于训练的基线，同时去噪器评估次数比基于梯度的后验采样器减少约一个数量级。
Key points
Derives the exact closed-form posterior score for linear Gaussian inverse problems, removing the need for approximations.
推导出线性高斯逆问题的闭式精确后验评分，消除近似需求。
Reformulates posterior sampling as a denoising task at a shifted pivot with anisotropic noise, enabling reuse of standard denoiser training.
将后验采样重构成偏移锚点下的各向异性噪声去噪任务，可复用标准去噪器训练。
EPS training objective matches standard pretraining, allowing training from scratch or fine-tuning and inference without extra components.
EPS训练目标与标准预训练一致，支持从头训练或微调，推理时无需额外组件。
On FFHQ and ImageNet across five inverse problems, EPS outperforms baselines with an order of magnitude fewer denoiser evaluations.
在FFHQ和ImageNet的五种逆问题上，EPS以少于一个数量级的去噪器评估次数，性能超越基线。