Extrapolation of Wasserstein geodesics

Extending geodesics in the Wasserstein space, namely the space of probability measures endowed with the quadratic Wasserstein distance, can be useful for designing high order accurate time discretizations of gradient flows or acceleration techniques à la Nesterov. However, this is not a well-defined operation in general. In this talk we will discuss different alternatives and focus in particular on a recent variational model we proposed. This is obtained by allowing for negative coefficients in the classical variational characterization of Wasserstein barycenters. We show that this problem, despite being non-convex along linear interpolations, admits two equivalent convex formulations. In particular, it can be recast as an instance of weak optimal transport. We take advantage of this formulation to devise an efficient approach to approximate the extrapolation on point clouds, via entropic regularization and a variant of the Sinkhorn algorithm. This is a joint work with Thomas Gallouët and Andrea Natale.