Optimal transport or OT has become a relevant tool in various machine learning applications. The rising popularity of OT is because it gives a principled approach in exploiting the underlying metric geometry to develop different notions of distances between probability distributions to be used in downstream applications. Consequently, there have been many works on developing computationally efficient tools to solve OT problems.

In this presentation, we look at the Riemannian approach to solving OT related optimization problems. The Riemannian approach has also got popular for solving structured nonlinear optimization problems. To that end, we discuss some of the recent works on exploiting Riemannian manifold structures to develop optimization-related ingredients for tackling OT formulations. We also look at various challenges and opportunities that lay ahead in making Riemannian tools a viable alternative for OT practitioners.