Optimal Transport for PDEs and Machine Learning

Optimal transport (OT) has recently gained significant interest in statistics and machine learning. It serves as a natural tool for comparing probability distributions in a geometrically faithful manner. However, OT faces challenges due to the curse of dimensionality, as it may require a sample size that grows exponentially with the dimension. This seminar will be divided into two parts:

• A tutorial on optimal transport, where I will review the Monge and Kantorovich formulations, and their connection to gradient flow PDEs via the minimizing movement scheme.
• A more advanced discussion on entropic regularization and the Schrödinger problem, which improves both the numerical complexity of computing OT and the ability to approximate OT with better sample complexity in high dimensions.

For more information and references, please visit the website of our book ‘Computational Optimal Transport’ at https://optimaltransport.github.io/.