Fixed-support Wasserstein barycenters: computational hardness and fast algorithm

We study the fixed-support Wasserstein barycenter problem (FS-WBP), which consists in computing the Wasserstein barycenter of m discrete probability measures supported on a finite metric space of size n. We show first that the constraint matrix arising from the standard linear programming (LP) representation of the FS-WBP is not totally unimodular when m≥ 3 and n≥ 3. This result resolves an open question pertaining to the relationship between the FS-WBP and the minimum-cost flow (MCF) problem since it proves that the FS-WBP in the standard LP form is not an MCF problem when m≥ 3 and n≥ 3. We also develop a provably fast deterministic variant of the celebrated iterative Bregman projection (IBP) algorithm, named FastIBP, with a complexity bound of O (mn7/3ε− 4/3), where ε∈(0, 1) is the tolerance. This complexity bound is better than the best known complexity bound of O (mn2ε− 2) for the IBP algorithm in terms of ε, and that of O (mn5/2ε− 1) from other accelerated algorithms in terms of n. Finally, we conduct extensive experiments with both synthetic and real data and demonstrate the favorable performance of the FastIBP algorithm in practice.