  We consider two statistical problems at the intersection of functional and
non-Euclidean data analysis: the determination of a Fr\'echet mean in the
Wasserstein space of multivariate distributions; and the optimal registration
of deformed random measures and point processes. We elucidate how the two
problems are linked, each being in a sense dual to the other. We first study
the finite sample version of the problem in the continuum. Exploiting the
tangent bundle structure of Wasserstein space, we deduce the Fr\'echet mean via
gradient descent. We show that this is equivalent to a Procrustes analysis for
the registration maps, thus only requiring successive solutions to pairwise
optimal coupling problems. We then study the population version of the problem,
focussing on inference and stability: in practice, the data are i.i.d.
realisations from a law on Wasserstein space, and indeed their observation is
discrete, where one observes a proxy finite sample or point process. We
construct regularised nonparametric estimators, and prove their consistency for
the population mean, and uniform consistency for the population Procrustes
registration maps.
