Data-driven regularization of Wasserstein barycenters with an application to multivariate density registration
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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 bio-informatics where researchers aim to automatically normalize large datasets to compare and analyze characteristics within a same cell population. Inconveniently, the information acquired is 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 barycenter (or Fréchet mean) of a set of probability measures with respect to the Wasserstein metric which allows to smooth such data and to remove mis-alignment effect in the sample acquisition process. 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 modify the Wasserstein metric itself by using an entropically regularized transportation cost between probability measures as introduced in Cuturi (2013). The main contribution of this work is to propound 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.
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