Theory and Applications of the Geometry of Entropy Regularized Optimal Transport – THEATRE

Optimal transport, including its entropic regularization, plays an increasing role in our understanding of optimization schemes arising from machine learning problems. The good behavior of such optimization problems and the resulting algorithms are shrewdly related to the geometry induced by the metrics in play. If the geometry induced by classical optimal transport is well-known, the one of its regularized counterpart is much less understood, even though its efficiency is routinely showcased in applications.

The goal of this project is to study the geometry of entropy regularized optimal transport. From a theoretical viewpoint, the ultimate goal would be to establish conditions that would guarantee the convergence of gradient descents (flows) induced by the regularized optimal transport formulation. This requires first to study the different kinds of interpolations defined by regularized transport. This natural question is perfectly understood in the classical transport setting but seems much more challenging in the regularized case. From a practical standpoint, this will yield to the design of new optimization algorithms relying on regularized transport and will improve your grasp on existing algorithms that have showcase their usefulness in practice. This will lead to the production of open-source code to be integrated in libraries dedicated to optimal transport.