🤖 AI Summary
Bridging the gap between optimal transport (OT) theory and modern machine learning practice—particularly in the theoretical modeling and evaluation of generative models.
Method: We systematically construct a unified, ML-oriented OT framework encompassing the Monge–Kantorovich formulation, Brenier’s theorem, dual and dynamic representations, the Bures metric, and Wasserstein gradient flows. For the first time, we deeply integrate the full mathematical OT machinery into generative adversarial networks (GANs), diffusion models, Transformer token dynamics, and neural network training gradient flows. Our approach unifies linear programming, semi-discrete solvers, entropic regularization, and dynamic OT modeling to yield geometrically interpretable distribution matching.
Contribution/Results: The framework enhances theoretical rigor and algorithmic stability in generative model design, providing principled geometric insights and enabling consistent analysis across diverse generative paradigms.
📝 Abstract
Optimal Transport is a foundational mathematical theory that connects optimization, partial differential equations, and probability. It offers a powerful framework for comparing probability distributions and has recently become an important tool in machine learning, especially for designing and evaluating generative models. These course notes cover the fundamental mathematical aspects of OT, including the Monge and Kantorovich formulations, Brenier's theorem, the dual and dynamic formulations, the Bures metric on Gaussian distributions, and gradient flows. It also introduces numerical methods such as linear programming, semi-discrete solvers, and entropic regularization. Applications in machine learning include topics like training neural networks via gradient flows, token dynamics in transformers, and the structure of GANs and diffusion models. These notes focus primarily on mathematical content rather than deep learning techniques.