Wasserstein Barycenters Can Be Computed in Polynomial Time in Fixed Dimension
Abstract
Computing Wasserstein barycenters is a fundamental geometric problem with widespread applications in machine learning, statistics, and computer graphics. However, it is unknown whether Wasserstein barycenters can be computed in polynomial time, either exactly or to high precision (i.e., with $\textrm{polylog}(1/\varepsilon)$ runtime dependence). This paper answers these questions in the affirmative for any fixed dimension. Our approach is to solve an exponential-size linear programming formulation by efficiently implementing the corresponding separation oracle using techniques from computational geometry.
Cite
Text
Altschuler and Boix-Adsera. "Wasserstein Barycenters Can Be Computed in Polynomial Time in Fixed Dimension." Journal of Machine Learning Research, 2021.Markdown
[Altschuler and Boix-Adsera. "Wasserstein Barycenters Can Be Computed in Polynomial Time in Fixed Dimension." Journal of Machine Learning Research, 2021.](https://mlanthology.org/jmlr/2021/altschuler2021jmlr-wasserstein/)BibTeX
@article{altschuler2021jmlr-wasserstein,
title = {{Wasserstein Barycenters Can Be Computed in Polynomial Time in Fixed Dimension}},
author = {Altschuler, Jason M and Boix-Adsera, Enric},
journal = {Journal of Machine Learning Research},
year = {2021},
pages = {1-19},
volume = {22},
url = {https://mlanthology.org/jmlr/2021/altschuler2021jmlr-wasserstein/}
}