PRISM Improves Schrödinger Bridge Models for Image Restoration.

Schrödinger bridge models are powerful tools for restoring clean signals from noisy or degraded observations, often by following conditional bridges of a reference process. However, the selection of this reference, typically white noise with a manually tuned schedule, has largely been heuristic. Researchers have developed PRISM (Principled Reference Identification for Schrödinger Bridge Model), a new theoretical framework for designing these crucial bridge references. PRISM precisely characterizes the time-varying Gaussian references that remain exactly tractable with per-mode schedules, specifically those whose instantaneous covariances commute. The theory establishes an "invisibility principle," demonstrating that with perfect drift and unlimited computational steps, any admissible reference recovers the true posterior. This implies that the choice of reference becomes critical only when computational resources are limited. For a fixed computational budget, PRISM derives a closed-form objective and proves that the optimal noise spectrum is proportional to the spectrum of information destroyed by the sensor. Experiments confirm these theoretical predictions in Gaussian settings, though real-world images (like FFHQ) introduce non-Gaussian per-mode statistics that can alter the expected performance, highlighting the complexities of applying theoretical insights to practical data.