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 \textit{not totally unimodular} when m \geq 3 and n \geq 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 \geq 3 and n \geq 3 . We also develop a provably fast \textit{deterministic} variant of the celebrated iterative Bregman projection (IBP) algorithm, named \textsc{FastIBP}, with a complexity bound of \tilde{O}(mn {7/3}\varepsilon {-4/3}), where \varepsilon \in (0, 1) is the desired tolerance. This complexity bound is better than the best known complexity bound of \tilde{O}(mn 2\varepsilon {-2}) for the IBP algorithm in terms of \varepsilon, and that of \tilde{O}(mn {5/2}\varepsilon {-1}) from accelerated alternating minimization algorithm or accelerated primal-dual adaptive gradient algorithm in terms of n .