Optimal transportation through saddlepoints and some robustness issues

Abstract
We showcase some unexplored connections between saddlepoint approximations, measure transportation, and some key topics in information theory by reviewing selectively some fundamental results available in the literature. We start with the link between Esscher's tilting (which is a result rooted in information theory and that lies at the heart of saddlepoint approximations) and the solution of the dual Kantorovich problem (which lies at the heart of measure transportation theory) via the Legendre transform of the cumulant generating function. We then investigate these links in the framework of M-estimators and quantile regression. The unveiled connections offer the possibility to view saddlepoint approximations from different angles, putting under the spotlight the links to e.g. convex analysis (via the notion of duality) or differential geometry (via the notion of geodesic). Finally, we discuss some robustness issues of optimal transportation and their connections to penalization methods. Joint work with Davide La Vecchia and Andrej Ilievski.