  We present a framework to simultaneously align and smooth data in the form of
multiple point clouds sampled from unknown densities with support in a
d-dimensional Euclidean space. This work is motivated by applications in
bioinformatics where researchers aim to automatically homogenize large datasets
to compare and analyze characteristics within a same cell population.
Inconveniently, the information acquired is most certainly noisy due to
mis-alignment caused by technical variations of the environment. To overcome
this problem, we propose to register multiple point clouds by using the notion
of regularized barycenters (or Fr\'{e}chet mean) of a set of probability
measures with respect to the Wasserstein metric. A first approach consists in
penalizing a Wasserstein barycenter with a convex functional as recently
proposed in Bigot and al. (2018). A second strategy is to transform the
Wasserstein metric itself into an entropy regularized transportation cost
between probability measures as introduced in Cuturi (2013). The main
contribution of this work is to propose data-driven choices for the
regularization parameters involved in each approach using the
Goldenshluger-Lepski's principle. Simulated data sampled from Gaussian mixtures
are used to illustrate each method, and an application to the analysis of flow
cytometry data is finally proposed. This way of choosing of the regularization
parameter for the Sinkhorn barycenter is also analyzed through the prism of an
oracle inequality that relates the error made by such data-driven estimators to
the one of an ideal estimator.
