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 Ye et al. (IEEE Trans Signal Process 65:2317–2332, 2017) 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.