A Closed-Form Framework for Schr\"odinger Bridges Between Arbitrary Densities

TL;DR

This paper introduces a unified closed-form framework for Schr"odinger Bridges that simplifies their computation, enabling efficient modeling of complex distribution transformations in fields like genomics and image processing.

Contribution

The authors develop a novel closed-form formulation for Schr"odinger Bridges, unifying previous solutions and enabling simulation-free inference directly from data samples.

Findings

Framework encompasses known solutions like Schr"odinger F"ollmer process and Gaussian SB.

Developed a simulation-free algorithm for SB dynamics inference.

Applied to single-cell genomics and image restoration tasks.

Abstract

Score-based generative models have recently attracted significant attention for their ability to generate high-fidelity data by learning maps from simple Gaussian priors to complex data distributions. A natural generalization of this idea to transformations between arbitrary probability distributions leads to the Schr\"odinger Bridge (SB) problem. However, SB solutions rarely admit closed-form expressios and are commonly obtained through iterative stochastic simulation procedures, which are computationally intensive and can be unstable. In this work, we introduce a unified closed-form framework for representing the stochastic dynamics of SB systems. Our formulation subsumes previously known analytical solutions including the Schr\"odinger F\"ollmer process and the Gaussian SB as specific instances. Notably, the classical Gaussian SB solution, previously derived using substantially more sophisticated tools such as Riemannian geometry and generator theory, follows directly from our formulation as an immediate corollary. Leveraging this framework, we develop a simulation-free algorithm that infers SB dynamics directly from samples of the source and target distributions. We demonstrate the versatility of our approach in two settings: (i) modeling developmental trajectories in single-cell genomics and (ii) solving image restoration tasks such as inpainting and deblurring. This work opens a new direction for efficient and scalable nonlinear diffusion modeling across scientific and machine learning applications.