Notes on generative modeling: flow matching, diffusion, optimal transport and Schr{ö}dinger bridge

2026-06-29

AI Summary

This work aims to uncover the unifying mathematical principles underlying several prominent approaches in generative modeling. By constructing a high-level framework grounded in optimal transport theory, it systematically elucidates the intrinsic connections among diffusion models, flow matching, and Schrödinger bridges. The study demonstrates that these seemingly distinct methods can all be interpreted as special cases of optimal transport problems under varying constraints or approximations. This unified perspective not only clarifies the fundamental commonalities shared by diverse generative models but also establishes a theoretical foundation for their integration and further innovation.

Abstract

These notes recapitulate the high level mathematical principles behind different techniques for generative modeling. I show the connections between optimal transport and standard techniques such as Schr{ö}dinger bridge and flow matching.

Problem

• generative modeling
• optimal transport
• flow matching
• Schrödinger bridge
• diffusion

Innovation

• flow matching
• diffusion models
• optimal transport
• Schrödinger bridge
• generative modeling