On the computation of Wasserstein barycenters
18 Pages Posted: 28 Nov 2018 Last revised: 1 Dec 2018
Date Written: November 29, 2018
In recent years, the Wasserstein barycenter has become an important notion in the analysis of high dimensional data with a broad range of applications in applied probability, economics, statistics and in particular to clustering and image processing. We give a new criterion for the explicit construction of barycenters generalizing the Gaussian case. Based on the n-coupling problem and an iterative version of the swapping algorithm, we introduce a new and simple algorithm to compute Wasserstein barycenters. We show in some examples that our approach is able to provide an accurate and fast visualization of barycenters even for a large number of marginals relevant for applications. The algorithm also provides an approximate solution for more complex optimization problems like the k-barycenter problem.
Keywords: Wasserstein barycenter, swapping algorithm, optimal transportations, k-means clustering, image processing
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