FixedSupport Wasserstein Barycenters: Computational Hardness and Fast Algorithm
Abstract
We study the fixedsupport Wasserstein barycenter problem (FSWBP), 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 FSWBP 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 FSWBP and the minimumcost flow (MCF) problem since it proves that the FSWBP 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 primaldual 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.
 Publication:

arXiv eprints
 Pub Date:
 February 2020
 arXiv:
 arXiv:2002.04783
 Bibcode:
 2020arXiv200204783L
 Keywords:

 Computer Science  Computational Complexity;
 Computer Science  Data Structures and Algorithms;
 Statistics  Machine Learning
 EPrint:
 Accepted by NeurIPS 2020