An inexact PAM method for computing Wasserstein barycenter with unknown supports

Wasserstein barycenter is the centroid of a collection of discrete probability distributions which minimizes the average of the $\ell_2$-Wasserstein distance. This paper focuses on the computation of Wasserstein barycenters under the case where the support points are free, which is known to be a severe bottleneck in the D2-clustering due to the large-scale and nonconvexity. We develop an inexact proximal alternating minimization (iPAM) method for computing an approximate Wasserstein barycenter, and provide its global convergence analysis. This method can achieve a good accuracy with a reduced computational cost when the unknown support points of the barycenter have low cardinality. Numerical comparisons with the 3-block B-ADMM in \cite{YeWWL17} and an alternating minimization method involving the LP subproblems on synthetic and real data show that the proposed iPAM can yield comparable even a little better objective values in less CPU time, and hence the computed barycenter will render a better role in the D2-clustering.