Tropical Optimal Transport and Wasserstein Distances

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'echet 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.