Foundations of Schrödinger Bridges for Generative Modeling

• URL: http://arxiv.org/abs/2603.18992v1
• Date: Thu, 19 Mar 2026 14:59:56 GMT
• Title: Foundations of Schrödinger Bridges for Generative Modeling
• Abstract summary: Schrdinger bridges transform a simple prior distribution into a complex target distribution through paths in probability space.<n>This guide develops the mathematical foundations of the Schrdinger bridge problem.<n>It focuses on its dynamic formulation with direct connections to modern generative modeling.
• Score: 0.7953056533753116
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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