Foundations of Schrödinger Bridges for Generative Modeling

2026-03-19

AI Summary

This work investigates how to efficiently transform a simple prior distribution into a complex target distribution that satisfies boundary constraints via stochastic trajectories in probability space, while ensuring path optimality. Building upon Schrödinger bridge theory, we develop a first-principles generative modeling framework that achieves distributional transformation by minimizing entropy deviation. Our approach establishes a unified mathematical foundation linking Schrödinger bridges with modern generative models—including diffusion models, score matching, and flow matching—and introduces a generalizable, task-oriented dynamic construction method. By integrating optimal transport, stochastic control, and path-space optimization, we devise an efficient computational toolkit for dynamic Schrödinger bridges, offering both theoretical grounding and practical improvement pathways for existing generative models.

Technology Category

• Machine Learning: Deep Generative Models & Autoencoders
• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• Web Mining and Content Analysis: Models for Web evolution

Abstract

At the core of modern generative modeling frameworks, including diffusion models, score-based models, and flow matching, is the task of transforming a simple prior distribution into a complex target distribution through stochastic paths in probability space. Schrödinger bridges provide a unifying principle underlying these approaches, framing the problem as determining an optimal stochastic bridge between marginal distribution constraints with minimal-entropy deviations from a pre-defined reference process. This guide develops the mathematical foundations of the Schrödinger bridge problem, drawing on optimal transport, stochastic control, and path-space optimization, and focuses on its dynamic formulation with direct connections to modern generative modeling. We build a comprehensive toolkit for constructing Schrödinger bridges from first principles, and show how these constructions give rise to generalized and task-specific computational methods.

Problem

• Schrödinger bridges
• generative modeling
• optimal transport
• stochastic control
• probability space

Innovation

• Schrödinger Bridges
• generative modeling
• optimal transport
• stochastic control
• path-space optimization