揭示扩散模型反向去噪中测量信息的进入时机与分配机制
Posterior Information Dynamics of Diffusion Models for Linear Inverse Problems
- 用平滑似然力分析测量信息如何随噪声水平进入反向过程
- 发现信息增益与去噪误差降低存在定量关系，且能量衰减速率为二次方
- 适用于研究逆问题中先验与观测对齐的理论分析，尤其适合图像重建场景
扩散模型广泛用作线性逆问题的先验，但其终点质量无法反映测量信息何时进入反向去噪过程，以及如何在信号方向间分配。本文通过平滑似然力（即各噪声水平下后验与先验得分之差）研究该过程。对于固定测量，其期望平方范数同时表征后验-先验相对熵耗散和反向路径相对熵增长。对测量取平均得信息-最小均方误差（I-MMSE）恒等式，将信息增益与去噪误差降低关联。在二阶矩有限条件下，似然力能量及其与先验得分能量之比在高噪声时随噪声核信号系数呈二次衰减。可解模型显示，条件化移除了测量已解释的类别分离，使n个经验样本的均匀索引熵从log n降至H(I|r)，且信息融合依赖于算子-先验对齐，即使奇异值相同亦然。具有可解析后验的模型实验验证了上述预测。另以冻结的FFHQ模型为例，具有相同谱的掩码产生不同的先验归一化零空间轨迹统计。
原文摘要 · Abstract (English)
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.
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