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arXiv · 1704.08200

Quadratically-Regularized Optimal Transport on Graphs

Montacer Essid·Justin Solomon
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Abstract

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; state-of-the-art techniques employ entropic regularization on the transportation matrix. In this paper, we explore a quadratic alternative to entropic regularization for transport 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.

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Montacer Essid, Justin Solomon. 2017-04-26. Quadratically-Regularized Optimal Transport on Graphs. https://arxiv.org/abs/1704.08200

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