Entropic Regularization of Optimal Transport

Optimal transport (OT) is the general problem of moving one distribution of mass to another as efficiently as possible and is the continuum extension of the discrete problem of matching. A matching problem involves associating with each data point in one set (say patients), exactly one data point in another set (say hospitals). There is a cost incurred for this match to occur (say the distance the patient needs to travel to get to the hospital).

An optimal matching is one that minimizes the average cost. The mathematics of Monge Kantorovich OT has grown to be a unifying theme in many scientific disciplines, from purely mathematical areas such as analysis, geometry, and partial differential equations to revolutionary new methods in economics, statistics, machine learning and artificial intelligence.

For example, the impact of optimal transport in the study of geometric inequalities and Ricci curvature in Riemannian geometry was highlighted by two Fields medals (Villani 2010, Figalli 2018) in the last decade. On the other hand OT has established itself as a viable alternative to classical maximum likelihood based methods in analyzing high-dimensional data coming from a manifold.

Other notable applications include analysis of neural networks and adversarial networks in machine learning and artificial intelligence, new applications to stem cell biology, and economic applications such as generalizations of the Nobel prize winning Gale-Shapely algorithm, online auctions, and option pricing. Much of these are due to impressive leaps in computational methods in OT that hinge on entropy based regularizations.

This project focuses on new probability questions in this area of entropic regularization of optimal transport problems and their applications. The project also provides training opportunities for graduate students.

The PI proposes problems at the intersection of probability and Monge-Kantorovich OT that are broad and cut across several mathematical and applied disciplines. It brings new probabilistic tools and perspectives (such as exchangeability, Gaussian chaos expansions, Metropolis algorithm, mean-field particle interactions) to problems of mass transport that are of interest to statisticians, computer scientists, analysts and geometers.

The primary focus is on the asymptotic analysis of both discrete and continuous entropy-regularized OT, also called Schroedinger bridges, either as the temperature goes to zero or as the data size goes to infinity or both. One of the proposed problems involves Markov chain based computational method for OT, which, if resolved, will have immediate applications to various applied disciplines that depend on OT computations.

Other applications involve analysis of transaction costs in mathematical finance and robotics where the dynamics of a large number of interacting drones is described by a novel McKean-Vlasov type limit. A large mathematical audience will find the material close to their interests and the project is expected to spur further cross-disciplinary collaborations between probabilists and non-probabilists working in OT.

This is also facilitated by the PI-led PIMS Kantorovich Initiative that creates an infrastructure for interdisciplinary work based on OT located in the Pacific Northwest.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.