PRISM：薛定谔桥模型的原则性参考识别

中文摘要 AI 辅助

薛定谔桥模型通过遵循参考过程的条件桥从降质观测中恢复干净信号，但该参考是启发式选择的，通常是带有手动调整调度的白噪声。我们开发了PRISM，这是一种桥参考设计的理论。我们刻画了那些在逐模式调度下仍完全可处理的时变高斯参考：恰好是那些其瞬时协方差可交换的参考。然后我们证明了不可见性原理：在精确漂移和无限求解器步数的情况下，每个可接受的参考都能恢复真实后验。因此，参考的选择仅在有限计算资源下才重要。对于固定步数预算，我们推导了有限步目标的闭式解，并证明每个最优噪声谱都与Pk成比例，其中Pk是传感器破坏的信息谱，具有与模式无关的常数x*(T) = (2 ln T)^-1/2 (1 + o(1))。分析表明，噪声颜色和时间调度是可互换的，正则化可证明地将最优参考向白噪声偏移。高斯设置中的实验证实了预测的排序和闭式损失下界。在FFHQ上，存在失真-感知权衡和谱定位转移，但白噪声优于匹配参考；一项改变训练机制的预先注册研究驳斥了脊白化作为解释。随后的2×2机制研究将反转追溯到真实图像的非高斯逐模式统计。PRISM将参考设计从超参数搜索转变为高斯 regime 中的计算，并精确定位了真实图像打破该理论的地方。

英文摘要

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.

发表机构

• Arizona State University（亚利桑那州立大学）

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