Foundations of Schrödinger Bridges for Generative Modeling
This work offers a foundational framework for generative modeling, potentially impacting researchers and practitioners in machine learning by unifying and extending existing methods, though it appears incremental in its theoretical development.
The paper tackles the problem of transforming a simple prior distribution into a complex target distribution in generative modeling by developing the mathematical foundations of Schrödinger bridges, which unify approaches like diffusion models and flow matching, and provides a toolkit for constructing these bridges to derive computational methods.
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.