Elements of Statistical Inference in 2-Wasserstein Space

This work addresses an issue of statistical inference for the datasets lacking underlying linear structure, which makes impossible the direct application of standard inference techniques and requires a development of a new tool-box taking into account properties of the underlying space.We present an approach based on optimal transportation theory that is a convenient instrument for the analysis of complex data sets. The theory originates from seminal works of a french mathematician Gaspard Monge published at the end of 18th century. This chapter recalls the basics on optimal transportations theory, explains the ideas behind statistical inference on non-linear manifolds, and as an illustrative example presents a novel approach of construction of non asymptotic confidence sets for so calledWasserstein barycenter, a generalized analogous of Euclidean mean to the case of non-linear space endowed with a particular distance belonging to a class of Earth-Mover distances that it is a main object of study in optimal transportation theory. The chapter is based on the paper [18].