PRISM为扩散模型的参考噪声设计提供理论指导，揭示最优噪声与传感器破坏信息谱相关。
PRISM: Principled Reference Identification for Schrodinger Bridge Model
- 提出可计算的参考过程设计理论，基于模式间协方差可交换的高斯参考
- 证明有限步数下最优噪声谱正比于传感器破坏的信息谱，比例常数为(2 ln T)^-1/2
- 实验证明真实图像非高斯性导致白噪声优于匹配参考，揭示模型失效边界
Schrödinger桥模型通过遵循参考过程的条件路径从退化观测中恢复干净信号，但参考过程通常人为选择，如带调优时间表的白噪声。我们提出PRISM，一个桥参考设计的理论框架。我们刻画了在每模式独立调度下仍可精确求解的时间变高斯参考：即其瞬时协方差矩阵彼此可交换的情形。随后证明不可见性原理：在精确漂移和无限求解步数下，所有可接受的参考均能恢复真实后验。因此，参考选择仅在有限计算资源下才重要。对于固定步数预算，我们推导出闭式有限步目标函数，并证明所有最优噪声谱均与传感器破坏的信息谱Pk成正比，且模式无关常数x*(T) = (2 ln T)^-1/2 (1 + o(1))。分析表明噪声颜色与时间调度可互换，正则化会将最优参考推向白噪声。高斯设置实验验证了预测的排序及闭式损失下界。在FFHQ数据集上，失真-感知权衡与谱局域化迁移现象存在，但白噪声仍优于匹配参考；一项预注册研究排除了岭白化作为解释。2×2机制研究进一步将反演归因于真实图像的非高斯每模式统计特性。PRISM将参考设计从超参数搜索转变为高斯场景下的计算问题，并精确定位真实图像偏离该理论的临界点。
原文摘要 · Abstract (English)
Schrödinger bridge models restore a clean signal from a degraded observation by following the conditional bridges of a reference process, yet this reference is chosen heuristically, typically white noise with a hand-tuned schedule. We develop PRISM, a theory of bridge reference design. We characterize the time-varying Gaussian references that remain exactly tractable with per-mode schedules: precisely those whose instantaneous covariances commute. We then prove an invisibility principle: with the exact drift and unlimited solver steps, every admissible reference recovers the true posterior. The choice of reference therefore matters only under finite computational resources. For a fixed step budget, we derive the finite-step objective in closed form and prove that every optimal noise spectrum is proportional to Pk, the spectrum of information destroyed by the sensor, with a mode-independent constant x*(T) = (2 ln T)^-1/2 (1 + o(1)). The analysis shows that noise color and temporal scheduling are interchangeable, and regularization provably shifts the optimal reference toward white noise. Experiments in Gaussian settings confirm the predicted orderings and the closed-form loss floors. On FFHQ, the distortion-- perception trade-off and spectral localization transfer, but white noise outperforms the matched reference; a pre-registered study that changes the training regime refutes ridge whitening as the explanation. A 2x2 mechanism study then traces the inversion to the non-Gaussian per-mode statistics of real images. PRISM turns reference design from a hyperparameter sweep into a calculation in the Gaussian regime, and locates exactly where real images break it.
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