Computational Hardness and Fast Algorithm for Fixed-Support Wasserstein Barycenter

We study in this paper 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 answers an open question pertaining to the relationship between the FS-WBP and the minimum-cost flow (MCF) problem since it therefore proves that the FS-WBP in the standard LP form is not a 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} algorithm, with the complexity bound of $\widetilde{O}(mn^{7/3}\varepsilon^{-4/3})$ where $\varepsilon \in (0, 1)$ is the tolerance. This complexity bound is better than the best known complexity bound of $\widetilde{O}(mn^2\varepsilon^{-2})$ from the IBP algorithm in terms of $\varepsilon$, and that of $\widetilde{O}(mn^{5/2}\varepsilon^{-1})$ 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 \textsc{FastIBP} algorithm in practice.