Tropical Optimal Transport and Wasserstein Distances

We study the problem of optimal transport in tropical geometry and define the Wasserstein-$p$ distances for probability measures 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 way to compute 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 on the ambient space of phylogenetic trees endowed with a tropical geometric structure. 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.