Miolane, N.
Statistics are mostly performed in vector spaces, flat structures where the computations are linear. When one wants to generalize this setting to the non-linear structure of Lie groups, it is necessary to redefine the core concepts. For instance, the linear definition of the mean as an integral can not be used anymore. In this thesis, we investigate three possible definitions depending of the Lie group geometric structure. First, we import on Lie groups the notion of Riemannian center of mass (CoM) which is used to define a mean on manifolds and investigate when it can define a mean which is compatible with the algebraic structure. It is the case only for a small class of Lie groups. Thus we extend the CoM’s definition with two others: the Riemannian exponential barycenter and the group exponential barycenter. This thesis investigates how they can define admissible means on Lie groups.
