Leveraging regularization, projections and elliptical distributions in optimal transport.

Summary The ability to manipulate and compare probability measures is essential for many applications in machine learning. Optimal Transport (OT) defines divergences between distributions based on the geometry of the underlying spaces: starting from a cost function defined on the space in which they are supported, OT consists in finding a coupling between the two measures that is optimal with respect to this cost. Because of its geometric anchoring, TO is particularly well adapted to machine learning, and is the subject of a rich mathematical theory. Despite these advantages, the use of TO for data science has long been limited by the mathematical and computational difficulties associated with the underlying optimization problem. To circumvent this problem, one approach is to focus on special cases that admit solutions in closed form, or can be solved efficiently. In particular, the TO between elliptic measures is one of the few cases for which the TO admits a closed form, defining the Bures-Wasserstein (BW) geometry. This thesis focuses on the BW geometry, with the aim of using it as a basic tool for applications in data science. To do so, we consider situations in which BW geometry is sometimes used as a tool for learning representations, extended from projections on subspaces, or regularized by an entropy term. In a first contribution, BW geometry is used to define plunges in the form of elliptic distributions, extending the classical representation in the form of R^d vectors. In a second contribution, we prove the existence of transports which extrapolate restricted applications to low dimensional projections, and show that these "optimal subspace" planes admit closed forms in the case of Gaussian measures. The third contribution of this thesis consists in obtaining closed forms for the entropy transport between non-normalized Gaussian measures, which constitute the first non-trivial expressions for entropy transport. Finally, in a last contribution we use entropy transport to impute missing data in a non-parametric way, while preserving the underlying distributions.