Fixed-Support Wasserstein Barycenters: Computational Hardness and Fast Algorithm
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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 Õ(mn^7/3ε^-4/3), where ε∈ (0, 1) is the tolerance. This complexity bound is better than the best known complexity bound of Õ(mn^2ε^-2) for the IBP algorithm in terms of ε, and that of Õ(mn^5/2ε^-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.
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