Sinkhorn Barycenters with Free Support via Frank-Wolfe Algorithm
Giulia Luise, Saverio Salzo, Massimiliano Pontil, Carlo Ciliberto
Abstract
We present a novel algorithm to estimate the barycenter of arbitrary probability distributions with respect to the Sinkhorn divergence. Based on a Frank-Wolfe optimization strategy, our approach proceeds by populating the support of the barycenter incrementally, without requiring any pre-allocation. We consider discrete as well as continuous distributions, proving convergence rates of the proposed algorithm in both settings. Key elements of our analysis are a new result showing that the Sinkhorn divergence on compact domains has Lipschitz continuous gradient with respect to the Total Variation and a characterization of the sample complexity of Sinkhorn potentials. Experiments validate the effectiveness of our method in practice.
BibTeX
@inproceedings{NEURIPS2019_9f96f36b,
author = {Luise, Giulia and Salzo, Saverio and Pontil, Massimiliano and Ciliberto, Carlo},
booktitle = {Advances in Neural Information Processing Systems},
editor = {H. Wallach and H. Larochelle and A. Beygelzimer and F. d\textquotesingle Alch\'{e}-Buc and E. Fox and R. Garnett},
pages = {},
publisher = {Curran Associates, Inc.},
title = {Sinkhorn Barycenters with Free Support via Frank-Wolfe Algorithm},
url = {https://proceedings.neurips.cc/paper_files/paper/2019/file/9f96f36b7aae3b1ff847c26ac94c604e-Paper.pdf},
volume = {32},
year = {2019}
}