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arXiv · 1911.05401

Tropical Optimal Transport and Wasserstein Distances

Wonjun Lee·Wuchen Li·Bo Lin·Anthea Monod
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Abstract

We study the problem of optimal transport in tropical geometry and define the Wasserstein-$p$ distances in the continuous metric measure space setting of the tropical projective torus. We specify the tropical metric -- a combinatorial metric that has been used to study of the tropical geometric space of phylogenetic trees -- as the ground metric and study the cases of $p=1,2$ in detail. The case of $p=1$ gives an efficient computation of the infinitely-many geodesics on the tropical projective torus, while the case of $p=2$ gives a form for Fr\'{e}chet means and a general inner product structure. Our results also provide theoretical foundations for geometric insight a statistical framework in a tropical geometric setting. We construct explicit algorithms for the computation of the tropical Wasserstein-1 and 2 distances and prove their convergence. Our results provide the first study of the Wasserstein distances and optimal transport in tropical geometry. Several numerical examples are provided.

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BibTeXRIS

Wonjun Lee, Wuchen Li, Bo Lin, Anthea Monod. 2019-11-13. Tropical Optimal Transport and Wasserstein Distances. https://arxiv.org/abs/1911.05401

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