Machine Learning Techniques for High-Dimensional Optimal Transport
We present machine learning (ML) approaches for approximately solving optimal transport problems in the high-dimensional setting. Problems of this kind frequently arise in statistics, Bayesian inference, and generative modeling, yet progress has been limited due to the curse-of-dimensionality.
As our learning framework tackles the optimal control problem and its underlying Hamilton-Jacobi-Bellman equations directly, it does not require training data such as previous solutions. Our formulation uses Lagrangian PDE solvers to avoid space discretization, which is critical to prevent the curse-of-dimensionality. Our resulting scheme can be implemented in existing machine learning frameworks.
We illustrate our developments using several examples: First, the classical problem of finding an optimal transport map between two densities in high dimensions. Second, the problem of learning a high-dimensional probability distribution from samples, which is at the core of generative modeling. We will also present extensions of this approach to mean field games and multi-agent optimal control problems.
Lars Ruthotto is an Associate Professor of Mathematics and Computer Science at Emory University in Atlanta, GA. He received his diploma and his Ph.D. in mathematics from the University of Münster, Germany, in 2010 and 2012, respectively. His research interests include numerical analysis (particularly numerical methods for optimization, linear algebra, and partial differential equations) and scientific computing with applications in machine learning, and medical and geophysical imaging. He has authored and co-authored more than 35 peer-reviewed publications, is a recipient of several national grants including an NSF Early Career Award and an NSF REU/RET Site for Computational Mathematics and Data Science.