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Elements of statistical inference in 2-Wasserstein space. (English) Zbl 1445.62097

Hintermüller, Michael (ed.) et al., Topics in applied analysis and optimisation. Partial differential equations, stochastic and numerical analysis. Selected papers from the Joint CIM-WIAS workshop, TAAO’17, Lisbon, Portugal, December 6–8, 2017. Cham: Springer. CIM Ser. Math. Sci, 139-158 (2019).
Summary: 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 called Wasserstein 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 is a main object of study in optimal transportation theory.
The chapter is based on the paper [the authors, “Construction of non-asymptotic confidence sets in 2-Wasserstein space”, Preprint, arXiv:1703.03658].
For the entire collection see [Zbl 1433.35004].

MSC:

62G15 Nonparametric tolerance and confidence regions
49Q22 Optimal transportation
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