Quadratically Regularized Optimal Transport on Graphs

Abstract

© 2018 Society for Industrial and Applied Mathematics. Optimal transportation provides a means of lifting distances between points on a geometric domain to distances between signals over the domain, expressed as probability distributions. On a graph, transportation problems can be used to express challenging tasks involving matching supply to demand with minimal shipment expense; in discrete language, these become minimum-cost network flow problems. Regularization typically is needed to ensure uniqueness for the linear ground distance case and to improve optimization convergence. In this paper, we characterize a quadratic regularizer for transport with linear ground distance over a graph. We theoretically analyze the behavior of quadratically regularized graph transport, characterizing how regularization affects the structure of flows in the regime of small but nonzero regularization. We further exploit elegant second-order structure in the dual of this problem to derive an easily implemented Newton-type optimization algorithm.