Existence and consistency of Wasserstein barycenters

By Thibaut Le Gouic, Jean-Michel Loubes. Probability theory and related fields, 2015

155 citations recorded by the source corpus.

Based on the Fréchet mean, we define a notion of barycenter corresponding to a usual notion of statistical mean. We prove the existence of Wasserstein barycenters of random probabilities defined on a geodesic space (E, d). We also prove the consistency of this barycenter in a general setting, that includes taking barycenters of empirical versions of the probability measures or of a growing set of probability measures.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/bdc4675873b229615d3e6ecf461500e869d2a8e3

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

A fixed-point approach to barycenters in Wasserstein space (2015, background): “This appealing concept shows a high potential for application, already considered in Artiﬁcial Intelligence or in Statistics (see, e.g., [16], [5], [10], [20], [7] or [3]).”
Statistical Aspects of Wasserstein Distances (2018, methodology): “The general consistency result of Le Gouic & Loubes (2017) is the important and necessary ﬁrst step in providing a sound statistical theory for random measures in Wasserstein space.”
Convergence rates for empirical barycenters in metric spaces: curvature, convexity and extendable geodesics (2018, background): “…of these algorithmic successes and the behavior of some statistical procedures in general (we refer to Bhattacharya and Patrangenaru, 2003; Kendall and Le, 2011; Le Gouic and Loubes, 2017; Altschuler et al., 2017; Dvurechensky et al., 2018; Bigot et al., 2018, for works in this direction).”
Statistical inference for Bures–Wasserstein barycenters (2019, methodology): “The original Bures metric appears in quantum mechanics in relation to fidelity measure between two quantum states and is used for measurement of quantum entanglement Marian and Marian (2008); Dajka et al.”
Bayesian Learning with Wasserstein Barycenters (2018, methodology): “Let us now recall the Wasserstein barycenter, introduced in [1] and further studied in [35, 25, 28], among others.”