Regularized 2-Wasserstein Barycenters on Discrete Metric Spaces

David Gentile, James M. Murphy

Session: General

Keywords: Optimal transport on graphs, regularized transport, signal processing on graphs

TL;DR: We use entropy regularized transport to synthesize Wasserstein barycenters on networks using connectivity based metrics

Abstract:

We apply entropically regularized Wasserstein geometry to the setting of a discrete metric space, where the cost functions are given by different choices of graph metric, which capture varying levels of connectivity information. We find that despite the degeneracy of the Monge-Kantorovich problem in the case of discrete metric spaces, regularized transport leads to geometric objects (e.g., geodesics and more generally barycenters) which convincingly resemble what one would expect from a Wasserstein-type geometry on the space of probability distributions defined for a network. We consider both synthesis (combining measures with respect to the regularized Wasserstein geometry) and analysis (decomposing signals with respect to fixed reference measures) on real geography data, demonstrating the utility of our approach. Our code is available on GitHub, https://github.com/dcgentile/fixed-support-barycenters

Submission Number: 12