统一扩散、分数匹配等生成模型的数学基础，揭示其内在原理。
Foundations of Schrödinger Bridges for Generative Modeling
- 从最优传输与路径优化出发，构建统一的随机桥框架。
- 通过最小熵偏差实现从简单分布到复杂分布的动态转换。
- 为生成模型提供可扩展的计算方法，适合研究者与工程师参考。
现代生成模型框架，包括扩散模型、分数基模型和流匹配，核心在于通过概率空间中的随机路径，将简单先验分布转化为复杂目标分布。薛定谔桥提供了一个统一原理，将该问题建模为在边际分布约束下，以最小熵偏离预定义参考过程的最优随机桥。本文系统阐述薛定谔桥问题的数学基础，结合最优传输、随机控制与路径空间优化，聚焦其动态形式，并与现代生成建模建立直接联系。我们从零构建薛定谔桥的完整工具包，展示其如何导出广义与任务特定的计算方法。
原文摘要 · Abstract (English)
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.
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