发表机构
ZJU-UIUC Institute, Zhejiang University; Zhejiang University（浙江大学ZJU-UIUC学院; 浙江大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究探究线性逆问题扩散模型的后验信息动力学，通过平滑似然力建立I-MMSE恒等式，经理论推导与实验验证条件作用对信息分配的影响及零空间轨迹特性。
AI 中文摘要
扩散模型被广泛用作线性逆问题的先验，但端点质量无法揭示测量信息何时进入反向去噪过程，也无法揭示其如何在信号方向间分配。我们通过平滑似然力（即每个噪声水平下精确后验得分与先验得分的差值）研究该过程。对于固定测量，其期望平方范数同时给出后验-先验相对熵耗散和反向路径相对熵增长；对所有测量取平均可得到信息-最小均方误差（I-MMSE）恒等式，将信息增益与去噪误差减少关联起来。在二阶矩有限的情况下，力能量及其与先验得分能量的比值，在高噪声下会随加噪核的信号系数呈二次衰减。可求解模型表明，条件作用会消除测量已解释的类别分离，将n个经验样本的均匀索引熵从log n降至H(I|r)，且即使奇异值相同，也会使信息同化依赖于算子-先验对齐。我们在具有可处理后验的模型中开展实验验证这些预测；在另一项使用冻结FFHQ模型的示例中，频谱相同的掩码会产生不同的先验归一化零空间轨迹统计量。
英文摘要
Diffusion models are widely used as priors for linear inverse problems, yet endpoint quality does not reveal when measurement information enters reverse denoising or how it is allocated across signal directions. We study this process through the smoothed likelihood force, the difference between exact posterior and prior scores at each noise level. For a fixed measurement, its expected squared norm gives both posterior--prior relative-entropy dissipation and reverse-path relative-entropy growth. Averaging over measurements yields an information--minimum mean-square error (I-MMSE) identity linking information gain to denoising-error reduction. Under finite second moments, the force energy and its ratio to prior-score energy decay quadratically in the noising kernel's signal coefficient at high noise. Solvable models show that conditioning removes class separation already explained by the measurement, reduces a uniform index entropy over \(n\) empirical samples from \(\log n\) to \(H(I\mid r)\), and makes assimilation depend on operator--prior alignment even for identical singular values. Experiments in models with tractable posteriors evaluate these predictions. In a separate illustration with a frozen FFHQ model, masks sharing the same spectrum yield different prior-normalized null-space trajectory statistics.
