Part of Advances in Neural Information Processing Systems 33 (NeurIPS 2020)
Tianyi Lin, Nhat Ho, Xi Chen, Marco Cuturi, Michael I. Jordan
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≥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 \textit{deterministic} variant of the celebrated iterative Bregman projection (IBP) algorithm, named \textsc{FastIBP}, with a complexity bound of ˜O(mn7/3ε−4/3), where ε∈(0,1) is the desired 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 accelerated alternating minimization algorithm or accelerated primal-dual adaptive gradient algorithm in terms of n. Finally, we conduct extensive experiments with both synthetic data and real images and demonstrate the favorable performance of the \textsc{FastIBP} algorithm in practice.